TREATISE

ON

MINERALOGY.

EDINBURGH:
PRINTED AT THE CALEDONIAN MERCURY PRESS-

TREATISE

MINERALOGY,

NATURAL HISTORY OF THE MINERAL KINGDOM.

BY

FREDERICK MOHS,

PROFESSOR IN THE MINING ACADEMY OF FREIBERG.

Translated from the German, with considerable Additions,

BY

WILLIAM HAIDINGER, F.R.S.E.

VOL. i.

EDINBURGH :

PRINTED FOR ARCHIBALD CONSTABLE AND CO. EDINBURGH
AND HURST, ROBINSON, AND CO. LONDON.
1825,

•
: .;

PREFACE.

THE spirit and zeal with which Mineralogy and
the kindred sciences are cultivated in Britain and
America, and the numerous opportunities afforded
to the inhabitants of these countries of visiting the
most remote regions of the globe, have made the
author of the present work anxious to contribute
his share to the more general1 diffusion of Mineralo-
gicai Science, by publishing in the English lan-
guage the elements of a method which places Mi-
neralogy within the reach of those who wish to be-
come acquainted with minerals, without the assist-
ance of lectures or extensive collections. With
the view therefore of fulfilling the promise of Mr
MOHS, given in the Introduction to his Character-
istic, p. vi., a translation of his Grundriss der
Mineralogie is now laid before the English pub-
lic. The original Work appeared in two volumes^
the first in 1 8%%, and the second in the autumn of
1824. A considerable portion of it was translated
from the manuscript under the eye of the author,
and the remainder from the printed sheets which

II PREFACE.

were sent over during the progress of publication.
In consequence of a continued correspondence with
Professor MOHS, and the present rapid progress of
the science itself, the translator found it necessary
to make many alterations, improvements, and addi-
tions ; so that this Treatise on Mineralogy may be
considered in many respects as a second edition,
rather than as a mere translation of the original
work.

The principles according to which Mineralogy is
here treated, are so different from those generally
received, that, in order to prepare the public for
the reception of his method, the author found it ne-
cessary to give a full developement of his ideas in
a Preface of considerable length ; and this was the
more indispensable, as the second volume was not
published along with the first. In conformity with
the views of Mr MOHS, the translator has endea-
voured to attain the same object, by publishing
in the Transactions of the Royal Society of Edin-
burgh, and in the scientific journals of that city,
several papers drawn up in strict accordance with
these principles, and shewing their application in
particular cases. These papers were designed to con-
vey a just idea of the leading principles of the pre-
sent work, from which even the substance of some
of them is extracted. From these considerations, it
would be superfluous to give here the translation
of that part of the German Preface which regards

PREFACE. Ill

the exposition of the principles, notwithstanding its
high importance. It will only be necessary to ad-
vert to those passages in it, which refer more par-
ticularly to the arrangement of some of the de-
partments of the work itself.

In the systematic nomenclature, introduced by
Professor MOHS, and employed in the present
work, the compound names and denominations
express the degree of connexion in which the spe*
cies stand to each other, and faithfully represent
their resemblance. In the trivial nomenclature,
the name applied to the species does not express
any thing of that connexion, and it must be a
single word, if it shall be convenient for use, in cases
where we do not intend to apply it in Natural History
to any scientific purpose ; consequently those are se*
lected which, according to the rules of §. 241., may
be considered as unexceptionable, and are added to
the specific characters in the Characteristic, referring
at the same time to the page of the second and
third volume, where the species is more amply de-
scribed, and other synonymes added. Where no
good trivial names existed, the names or denomina-
tions used by Professor JAMESON in the third edi-
tion of his system, or those adopted by other Mine-
ralogical authors, or by Chemists, have been intro
duced in their place.

The actual employment of the Characteristic to

IV PREFACE.

the purpose for which it is intended, the determina-
tion of minerals occurring in nature, cannot be too
strongly recommended to the beginner. This alone
will make him accustomed to observe with his own
eyes the characters upon which depend the identity
or difference between several species. The present
work is the only one hitherto published, which ena-
bles those who have studied the Terminology, to
determine every mineral by a philosophical process,
although they should never have seen it before.

The synonymes quoted in the General Descrip-
tions of the species are confined to a very few
works. Among those in the English language,
the works of Professor JAMESON are no doubt the
principal ones. The synonymes selected for the use
of the present publication are contained in the third
edition of his valuable System of Mineralogy, and
Manual, in which he has adopted the system
of the method of Natural History. To these syn-
onymes are added the names in the third edition of
Mr PHILLIPS'" Elementary Introduction to the know-
ledge of Minerals, which appeared too late to be
attended to in the German edition. The German
works noticed, are the System of WERNER, as con-
tained in the Handbuch der Mineralogie by HOFF-
MANN, continued by Mr BREITHAUPT, and the Sys-
tem of Professor HAUSMANN, these works being
framed according to the most original views. The
former, in particular, has met with a very general

reception for a long series of years. The System
of Mr VON LEONHARD is of a later date, and is
recommended by its comprehensive references to
mineralogical works, and other interesting notices.
Among the French works on Mineralogy, those
of the Abbe HAUY have been selected, as being
most generally received and understood. The
nomenclature used in both editions of his Traite de
Mineralogie, and in the Tableau Comparatif, may
of itself be considered as an abstract of the history of
his system.

The works and memoirs of Messrs BROOKE,
LEVY, and PHILLIPS, have been consulted in regard
to numerous and highly valuable crystallographic
observations on several substances, which had not
before been examined with sufficient exactness, and
which were unknown to Professor MOHS, at the
publication or the composition of the German ori-
ginal.

It was not till the publication of the first vo-
lume of this work, in 1822, that the axis of any
mineral was ascertained, by actual measurement,
to be in an inclined position towards the base:
and although that fact, which was first intro-
duced into crystallography by Mr MOHS, is
there indicated in the characters of some of the
species, it had not yet been so generally ascer-
tained, nor could it be so fully developed in

VI PIIEFACK.

the crystallographic department of the work, as it
requires. The formulas which in the German are
given in the Preface, are inserted in this translation
in their proper place, but without changing the num-
bers and distribution of the paragraphs. The names
and denominations also remain ; in the characters
alone the necessary changes are made, the expression
Prismatic in a more general signification being
employed as including the prismatic, hemi-prisma-
tic, and tetarto-prismatic forms. In regard to such
hemi-prismatic forms as have their axis inclined
to the base, it should be observed, that the angles
of horizontal prisms indicated, are those which the
face of the prism includes with the inclined axis,
like BAM and B'AM, Vol. I. Fig. 41.

Those simple forms which have been observed in
nature, are noted with an asterisk. For these, and
also for each of the combinations, some locality is
mentioned in which they have been discovered. In
a few instances another variety is substituted in the
translation for one in the original, when a certain lo-
cality could be obtained for an uncertain one, by a
comparison with the specimens in the cabinet of Mr
ALLAN.

The letters inclosed and printed in italics refer
to the figures, or to the works of HAUY, sometimes
also to particular papers, and in this case the title
of the latter has been mentioned. Some of the

PREFACE. VH

figures represent the combinations, where they are
quoted themselves ; others are only similar to them,
in so far as they have the same general appearance,
but different angles.

To distinguish the compound varieties from the
simple ones, is a matter of the highest importance ;
and they must therefore be kept perfectly separate.
This is the point where Mineralogy ends and Geo-
logy begins, two sciences which have nothing com-
mon with each other. Geology presupposes Mine-
ralogy ; but it considers the productions of the Mine-
ral Kingdom in quite another point of view, and
according to different principles ; without which
it would not be a distinct science.

With the enumeration of the compound varieties,
the General Description of the species is completed.
But there exists besides, a great variety of informa-
tion in regard to the productions of the Mineral
Kingdom, belonging in part to Natural History,
but partly also foreign to this science ; the latter
nevertheless is generally deemed an essential part
of a work on Mineralogy. Something of this in-
formation is contained in the Observations added
to every species, and which may require here a few
explanations.

The first of them properly belongs to the pro-
vince of Natural History. It comprehends some re-

VIII PREFACE.

marks on crystallographic subjects, or on the his-
tory of the species. Here the species are also com-
pared with the determinations and divisions into
sub-species and kinds, as contained in the Werne-
rian system, which will enable the reader not only
to understand the principles of these divisions, but
also to form an idea of their contents in reference to
the varieties occurring in nature. These distinctions
are not susceptible of strictness and precision; the
only purpose, therefore, in treating of them, is to
convey the ideas with brevity and distinctness.

Then follow some of the chemical properties of
the species, as exhibited before the blowpipe, or
when acted upon by acids, &c., and one or more
analyses, instituted by the most celebrated chemists,
in many cases accompanied by the formulae and cor-
responding proportions among the ingredients, as
proposed by BERZELIUS. To Professor MITSCHER-
LICH the translator is indebted for several interesting
facts regarding the circumstances under which cer-
tain species still continue to be formed or may be
produced at will, in laboratories and furnaces.

In the third place, something of the geological
position of the species is mentioned ; it does not
contain every thing known in this respect, but only
so much as will suffice for giving a general idea of
the modes of occurrence in nature, peculiar to the
species.

I'ilEFACE. IX

The geographic distribution of mineral species
is of far less importance than the distribution of
plants or animals, in which so much depends on the
geographic position, climate, soil, the particular
place of growth or residence, and other accidental
circumstances. It is the subject of the fourth class
of observations, which are confined to the state-
ment of a few localities only, since it cannot fall
within the scope of an elementary work to enumerate
all the localities of the different varieties of the
species.

Under the fifth head, some of the applications of
the species are mentioned, and sometimes a sixth
number is added, containing notices of species,
nearly allied to the one just treated of, but which
have not yet been received into the system. Some-
times one or several of these classes of observations
are wanting, or joined in a single number.

The first Appendix, which follows immediately
after the system, contains such minerals as will
probably, when farther examined and compared,
be received into the system as particular species.
They are arranged in alphabetical order ; and in
some of them the order, or even the genus, is
mentioned, to which they will probably be found
to belong. Their great number cannot surprise
those who are aware how imperfectly many minerals
which were long ago known, have been hitherto ex-

X PKEFACK.

amined and described ; consequently it is less advis-
able to receive at once newly discovered minerals
into the system, when we see that even those deter-
minations, which were usually considered most firmly
established, have frequently been found, on more
accurate examination, to be erroneous. The species
contained in this appendix must be viewed exactly
in the same light as the plantce mcertce sedis in the
natural system in Botany (not in artificial systems,
which cannot admit of any appendix), which are
not included in any of the systematic unities, not-
withstanding the advantage that the examination of
one, or of two individuals at the most, should here
be sufficient for knowing the species to its full ex-
tent.

The properties of the minerals contained in the
second Appendix are such, that we cannot expect
that they will ever form particular species, since
they are not susceptible of a natural-historical deter-
mination. Some minerals of this description have
been enumerated in the Observations annexed to
those species, to the decomposition of which they
owe their existence, as, for instance, Porcelain-Earth,
which is noticed under the head of Prismatic Feld-
spar.

The copper-plates, which have been extremely well
executed by Mr MILLER, are intended not merely to
represent the figures quoted in the general descrip-

PREFACE. XI

tions of the species, but also for producing a gene-
ral survey of the combinations most commonly oc-
curring among the simple forms found in nature.
They are disposed in the order of the systems of
crystallisation, and provided on the opposite page
with the explanations given in crystallographic signs.
The figures in the last five plates are not ar-
ranged in this order, since they were added only
in the course of printing the work. They refer
either to some remarkable varieties of species de-
scribed in the system, or they have been rendered
necessary by the reception of several species into the
first Appendix, which had been described by vari-
ous authors, the greater part of them since the pub-
lication of the German original.

In comparing many of these with nature, the
cabinet of Mr ALLAN, as will appear from the
frequent references made to it, has afforded the
translator the most important assistance; and he
trusts it will not be found out of place, if he embraces
the present opportunity of expressing to that gentle-
man, the deep sense he must ever entertain of the
many marks of kindness he has received from him.
To him he has not only been indebted for a home
in a foreign land, but also for much assistance, and
many valuable observations, in the progress of this
work. To Dr BREWSTER he is under the greatest
obligations, both for many interesting additions,
concerning the optical and other properties of mi-

XII PREFACE.

nerals, and for the perseverance and patience with
which he has aided him in the correction of the
press. Dr TURNER also has favoured him occa-
sionally with his valuable assistance.

The translator feels it an agreeable duty to ac-
knowledge, in the present place, that the addi-
tions to this work, and likewise the papers, which
he composed previous to its publication, owe the
greater share of any merit they may possess to
Professor MOHS, whose constant tuition in Mine-
ralogy, since the year 1812, he has had the good
fortune to enjoy, and of whose continued friendship
he has every reason to be proud.

The following words of Professor MOHS, at the
end of his Preface to the first and to the second
volumes of the original, will form the best conclu-
sion of these prefatory observations. " The present
Treatise on Mineralogy is founded on principles so
** different from those generally received in treating
«' of minerals, being in part in direct opposition to
" them, that it is not without hesitation that I have
" determined to lay it before the public. I have
" endeavoured to unite accuracy, correctness, and
u perspicuity, with as much of precision as I could
ec command ; yet I am perfectly aware that I have
« not everywhere succeeded, and that this Trea-
" tise is in many respects imperfect. The task I
« had to perform was nothing less than to apply

1'IIKFACE. XIII

" witli consistency to a whole science, a method
" which, though not new in itself, had yet been dis-
" regarded in Mineralogy, and to remove all the
" difficulties arising from deficiencies I had to
" supply, and from errors I had to correct. This
" problem, however, requires so much time and
" labour, that the person who undertakes to re-
" solve it, must leave many parts to subsequent
" investigation, while those who judge of the merits
" of a first attempt of this kind, will be disposed
" to relax in the severity of their criticism. Yet
" I wish that this work may be subjected even
44 to the strictest examination, provided it be can-
" did, well grounded, and does not omit to con-
" sider, that at the present moment the disposi-
" tion~of the whole must be of greater importance
" than the minuter details of the various depart-
" meats of the work. I know none of the im-
" perfections still to be met with, which could
" not be removed by future labours, and which
" will not soon disappear, if I have been for-
" tunate enough to call the attention of natu-
" ralists towards the exact knowledge of the phy-
" sical qualities of minerals, and to induce them
" to investigate these more closely and accurately
" than has hitherto been customary. Like every
" other department of Natural History, Minera-
" logy is a charming science. But its charms are
" grounded only upon its exactness ; and nothing
" has a more baneful influence on the science itself,

XIV PREFACE.

" than a superficial view of it. My only intention
" is to forward the scientific progress of Mineralogy,
" which is chiefly dependent on the purity of
" method necessary in every science ; on the cor-
" rectness of principles already demonstrated in the
" other departments of Natural History ; and on
" the consistency of the different parts of the science
" among themselves, objects which I have en-
" deavoured to attain. In this condition, Minera-
" logy answers every purpose that can be reason-
" ably expected from any part of Natural History.
" I trust that the results already obtained, however
" insufficient they may be, will induce naturalists
" to take advantage of the first step towards the
" construction of that edifice, of which I have laid
" down the plan in the present Treatise."

Charlotte Square, EDINBURGH,
25th March, 1825.

CONTENTS OF VOL. I.

INTRODUCTION.

§. 1. Nature * . . \

§. 2. Natural History . . .1

§. 3. Natural-Historical Properties . 2

§. 4. Natural Productions . . .3

§. - 5. Design of Natural History . . 3

§. 6. Individuals . . .4

§. 7. Organic and Inorganic Natural Productions 5

§. 8. Animals and Plants . . 6

§. 9. Minerals . . . .6

§. 10. Natural Kingdoms . . 7

§. 11. Division of Natural History . . 7

§. 12. Principal Heads of Natural History . 8

§. 13. Terminology . . .9

§. 14. Theory of the System . . 10

§. 15. Nomenclature . . .11

§. 16. Characteristic . . . 12

§. 17. Physiography . . .13

§. 18. Idea of Natural History . . 14
§. 19. Method of studying the Natural History of the Mi-

neral Kingdom . . .15

PART I.
TERMINOLOGY.

General Consideration of Minerals. Their distinction into Simple,
Compound, and Mixed. Division of their Natural-Historical
Properties.

§. 20. Power of Crystallisation, and its Products 19

§.21. Minerals decomposed and imperfectly formed 20

XVI CQXTKNTS.

§. 22. Simple Mineral . . .22

§. 23. Compound Mineral . , ' 22

§. 24. Mixed Mineral . . . 2.3

§. 25. Division of the Natural-Historical Properties 24

SECTION I.

The Natural-Historical Properties of Simple Minerals.

CHAPTER I.

.

Of the Regular Forms of Minerals.

1. &EXERAL. CONSIDERATION OF THE REGULAR FORMS.

§. 26. Crystal . . . .27

§. 27- Object of Crystallography . . 28

§. 28. Forms and Faces . . .28

§. 29. Edges ... 29

§. 30. Solid Angles . . .29

§. 31. Homologous Faces -. . 29

§. 32. Homologous or Equal Edges . .30

§. 33. Denomination of the Solid Angles . 30

§. 34. Simple and Compound Forms . .31

§. 35. The Compound Forms consist of the Simple 31

§. 36. Tangent Planes . . 32

§. 37. Sections . . .32

§. 38. Homologous Sections . . 33

§. 39. Axes . . . .35

§. 40. Homologous Axes . . 35

§. 41. Principal and subordinate Axes . . 3G

§. 42. Upright Position . . 37

§. 43. Parallel Position , . .37

§. 44. Horizontal Projection . 37

§. 45. Regularity . . .38

§. 46. Degrees of Regularity • . 38

§. 47. Determination of the Degrees of Regularity 39

§. 48. Classification of Simple Forms . 40

§. 49. Nomenclature of Simple Forms . . 40

CONTENTS. XVII
II. OF SIMPLE FORMS Itf PARTICULAR.

Consideration of Simple Forms, and some of their Geometrical
Relations.

§. 50. The Rhombohedron . - . .41

§.51. Pyramids in general . . 43

§. 52. Isosceles Four-sided Pyramids . . 44

§. 53. Scalene Four-sided Pyramids . 45

§. 54. Isosceles Six-sided Pyramids . . 46

§. 55. Scalene Six-sided Pyramids . . 47

§. 56. Scalene Eight-sided Pyramids , . 48

§. 57. The Tetrahedron . . 49

§. 58. The Hexahedron . . * 50

§. 59. The Octahedron . . 50

§. 60. Dodecahedrons in general . . 51

§.61. Trigonal-dodecahedrons . . 51

§. 62. Tetragonal-dodecahedrons . . 52

§. 63. The Monogrammic Tetragonal-dodecahedron- 52

§. 64. Digrammic Tetragonal-dodecahedrons . 53

§. 65. Pentagonal-dodecahedrons . . 54

§. 66. Hexahedral Pentagonal-dodecahedrons . 54

§. 67. Tetrahedral Pentagonal-dodecahedrons . 55

§. 68. Icositetrahedrons in general * . 56

§. 69. Trigonal-icositetrahedrons . . 57

§. 70. Tetrahedral Trigonal-icositetrahedrons . 57

§.71. Hexahedrai Trigonal-icositetrahedrons . 58

§. 72. Octahedral Trigonal-icositetrahedrons .- 59

§. 73. Tetragonal-icositetrahedrons . 59

§. 74. Digrammic Tetragonal-icositetrahedrons . 60

§. 75. Trigrammic Tetragonal-icositetrahedrona 61

§,76- Pentagonal-icositetrahedrons . . 62

§. 77. Tetraconta- octahedrons . . 63

Consideration of the Connexion among Simple Formr, and of the Re-
lations upon which it depends*

§. 78. Observations . . .64

§. 79. Derivation ... 66

§. 80. First Process of Derivation , . 67

XVIII < ONTKNT*.

§.81. Second Process . . .67

§. 82. Third Process 68

§. 83. Fourth Process . . .68

§. 84. Position of the Derived Forms . 69

§. 85. Series . . . .69

§. 86. Limits . . . 70

§. 87. Fundamental Form . . .71

1» Derivations from the Scalene Four-aided Pyramid.

§. 88. Derivation of more acute and more obtuse Pyramids,

of similar bases with the Fundamental Form 72

§. 89. Ratio between the Derived and the Fundamental

Form ... 73

§. 90. Series of Scalene Four-sided Pyramids, whose bases

are similar to the base of the Fundamental Form 7-1
§. 91. Limits of the Series of Scalene Four-sided Pyramids 77
§. 92. Derivation of Scalene Four-sided Pyramids of

dissimilar transverse Sections . 78

§. 93. The Ratio of the Diagonals of the Bases is de-
pendent on m . . 81
§. 94. Ratio of the Derived and the Fundamental Form 82
§. 95. Series of Derived Pyramids of a dissimilar trans-
verse Section with that of P. Other method
of deriving Pyramids of the same kind . 84
§. 96. Subordinate Series . . 89
§. 97. Horizontal Prisms . . .91
§. 98. Series of Horizontal Prisms, and their limits. In-
clination of the Axis . .93

2. Derivations front the Isosceles Four-sided Pyramid.

§. 99. Derivation of Homogeneous Forms . 97

§. 100. Ratio between the Derived and the Fundamen-
tal Form . . .98
§.101. Series of Isosceles Four-sided Pyramids 99
§. 102. Limits of the Series . .101
§. 103. Derivation of Scalene Eight-sided Pyramids 102
.$ 104. The bases of the Scalene Four-sided Pyramids

depend upon m . • . 104

(JOXTENTS. XIX

§. 105. Series of Scalene Eight-sided Pyramids 105
§. 10'J. Limits of the Series of Scalene Eight-sided Pyra-
mids . . . 10&
§. 107- Subordinate Series . . 107

3. Derivations from the Rhombohedron.

§.103. Derivation of Homogeneous Forms . 108

§.100. Ratio of the Derived Ithombohedrons . 109

§.110. Series of Ithombohedrons . . Ill

§. 111. Limits of the Series of Rhombohedrons 112

§.112. Derivation of Scalene Six-sided Pyramids . 114

§. 113. The transverse Sections of the Scalene Six-sided

Pyramids depend on m . . 116

§.114. Series of Scalene Six-sided Pyramids . 116

§.115. Limits of the Series of Scalene Six-sided Pyramids 118
§.116. Subordinate Series . . .121

§. 117- Derivation of Isosceles Six-sided Pyramids 123

§. 118. Series of Isosceles Six-sided Pyramids . 124

4. Derivations from the Hexdliedron.

§. 119. Different Positions of a moveable Plane . 125
§. 120. Production of the Forms of several Axes 128

§. 121. The Octahedron . . .129

§. 122. The Dodecahedron . . 129

§. 123. The Octahedral Trigonal-icositetrahedron . 130

§. 124. The Hexahedron . . 131

§. 125. The Digrammic Tetragonal-icositetrahedron 131

§. 126. The Hexahedral Trigonal-icositetrahedron 132

§. 127. The Tetracontaoctahedron. Designation of the

Tessular Forms . . .133

§. 128. Resolution of Forms belonging to the first degree

of regularity . . 135

§. 129. The Tetrahedron . . .138

§. 130. The Hexahedral Pentagonal-dodecahedron 138

§. 131. The Digrammic Tetragonal-dodecahedron 139

f . 132. The Trigonal-dodecahedron . 140

XX CONTENTS.

§. 133. The Tetrahedral Trigonal-icositetrahedron, the
Trigrammic Tetragonal-icositetrahedron, and
the Pentagonal-icositetrahedron , 140

§. 134. The Tetrahedral Pentagonal-dodecahedrons 142

General Ideas of Simple Forms.

§. 135. System of Crystallisation . . 145

§. 136. Series of Crystallisation . . 146

§. 137. The System of Crystallisation determined from a

single Form . . .147

III. OF COMBINATIONS.
Of Combinations in general.

§. 138. Definition . . . 148

§. 139. First Law of Combination . .150

§. 140. Second Law of Combination .' 151

§. 141. Symmetry of Combinations . . 152

§. 142. Edges of Combination . . 153

§. 143. Developement of Combinations . 153

§. 144. Magnitude of the Edges of Combination 158

Of the Combinations of the different Systems of Crystallisation.

§. 145. llhombohedral Combinations • 160

§. 146. Di-rhombohedral Combinations . '173
§. 147. Hemi-rhombohedral and Hemi-di-rhombohedral

Combinations . . 174
§. 148. Developement of llhombohedral and Di-rhombo-
hedral Combinations . .179
§. 149. Pyramidal Combinations . . 183
§. 150. Hemi-pyramidal Combinations . 191
§. 151. Developement of Pyramidal Combinations 193
§, 152. Prismatic Combinations . . 195
§, 153. Hemi-prismatic Combinations . .199
§. 154. Tetarto-prismatic Combinations . 201
§. 155. Developement of Prismatic Combinations 203
§. 156. Tessular Combinations . 205
§.157- Semi-tessular Combinations . 207

CONTENTS. XXI

IV. OF THE IMPERFECTIONS OF CRYSTALS IN RESPECT TO
THEIR FOHM.

§, 158. Two kinds of this Imperfection . 209

§. 159. Deviations from Regularity, depending upon the

Formation of the Individuals themselves 210

§. 160. Deviations from Regularity, depending upon the

Contact with other Individuals . 216

CHAPTER II.

Of the Structure of Minerals.

§. 161. Explanation of Structure . . 219

§. 162. Cleavage . . .221

§. 163. Faces of Cleavage . . 221

§. 164. Direction of Cleavage . . 222

§. 165. Character of Cleavage . . 223

§. 166. Faces of Cleavage parallel to Faces of Crystalli-
sation . . . 224
§. 167. Form of Cleavage . . 225
§. 168. Forms of Cleavage, distinguished according to

the quality of their Faces . . 227

§. 169. Forms of Cleavage Members of the Series of

Crystallisation . . 229

§. 170. Designation and Nomenclature of the Forms of

Cleavage . . .231

§.171. Fracture ... 232

§. 172. Faces of Fracture . . ,233

§. 173. Character of Fracture . . 234

k CHAPTER III.

Of Surface.

§. 174. Surface in general . . .235

§. 175. Striated Faces of Crystallisation . 236

§. 176. Diverse qualities of the Faces of Crystallisation 238
§. 177. Faces of Composition . , 240

XX If COXTKNTS.

{SECTION II.

The Natural- Historical Properties of Compound Minerah.

§. 178. Regular and Irregular Composition . 242

§. 179- Regular Composition. Twin-crystals . 243
§. 180. Irregular Composition. Groupe and Geode of

Crystals . . . 252

§. 181. Imitative Shapes . . 253
§. 182. Imitative Shapes, originating in the Groupes of

Crystals . . . 253
§. 183. Imitative Shapes arising out of the Geodes of

Crystals ... 254

§. 184. Amorphous Compositions . . 257

§. 185. Accidental Imitative Shapes . 258
§. 186. Regular accidental Imitative Shapes. Pseudo-

morphoses . . . 259

§. 187. Irregular accidental Imitative Shapes . 263

§.188. Particles of Composition . . 265

§. 189. Single and Multiple Composition . 268

§. 190. Characteristic Marks of Composition . 269

§. 191. Structure of Compound Minerals . 272

§. 192. Composition is of little value in Natural History 273

SECTION III.

The Natural-Historical Properties common to loth Simple and Com-
pound Minerals.

§. 193. Division . . .276

CHAPTER I.

Of the Optical Properties of Mineral*.

§. 194. Explanation . . .276

§. 195. Lustre, Colour, Transparency . . 277

§. 196. Kind and Intensity of Lustre . 278

§. 197. Series in the Differences of Lustre . 281

§. 198. Colour, properly so called, and Streak . 282

§. 199. Division of Colours f . 282

COXTKNTi. TXIII

§. 200. Metallic Colours . . 283

§.201. Non-metallic colqurs . . 284

§. 202. Series of Colours . . 292
§. 203. Several other peculiarities in the occurrence of

Colours . . . 293

§. 204. The Streak . ... 296

§. 205. Degrees of Transparency . . 297

CHAPTER II.

Oftlie Physical Properties of Miner ah.

§. 206. Explanation . . .298

§. 207. State of Aggregation . . 299

§. 208. Hardness . . .300

§. 209. Specific Gravity . . 307

§. 210. Magnetism . . .310

§. 211. Electricity ... 310

§. 212. Taste . . . .311

§. 213. Odour . . 312

PART II.
THEORY OF THE SYSTEM.

§. 214. Identity , . . .314

§.215. Difference . . . 314

§. 216. Degrees of Difference . . 315
§. 217. Mutual relations of the Natural-Historical Pro-

perties in certain Individuals . 317
§. 218. Individuals brought under the idea of Identity 318

§. 219. Connexion of several Series of Individuals . 320

§. 220. Species ... 324

§.221. Transitions . . .325

§. 222. Homogeneity from the Transitions . 326

§. 223. Principle of Classification . . 329

§. 224. Degrees of Natural-Historical Resemblance 330

§. 225. Genus . . .331

§. 226. Mineral Kingdom . . 334

§.227. Order . . * . . 337

XXIY CONTENTS.

§. 228. Class . . , 338

§. 229. Mineral System . . . 340

PART III.
NOMENCLATURE.

§. 230. Definition ... 346

§. 231. Object of the Systematic Denomination . 347

§. 232. Properties of the Systematic Denomination 349

§. 233. Object of the Names . . 351

§. 234. Name of the Order . . 352

§. 235. Selection of the Names of the Orders . 354

§. 236. Signification of the Names of the Orders 355

§. 237. Name of the Genus . . 358

§. 238. Denomination of the Species . 360
§. 239. Representation of the Species through its Denomi-
nation . . .361

§. 240. Systematic Nomenclature how to be judged of 363

§.241. Trivial Nomenclature . . 365

PART IV.
CHARACTERISTIC.

§. 242. Definition . . .368

§. 243. Natural and Artificial Characters . 369

§. 244. Properties of the Characters . . 370

§. 245. Absolute and conditioned Characteristic Marks 373

§. 246. Arrangement of the Characters of the Species 374

§. 247- No Characteristic before the System . 379

§. 248. Base of a perfect Characteristic . 381

§. 249. Use of the Characteristic . . , 383
§. 250. Determination of Individuals, by means of the

Characteristic. Example . . 384

§. 251. Immediate and mediate Determination. Example 388

§. 252. Base of the mediate Determination . 390

Characters of the Classes, Orders, Genera, and Species 391

INTRODtT&PJON.

§. 1. NATURE.

In the sciences, the word Nature is used under
three different significations. In the first, it de-
notes the general idea of the natural bodies alto-
gether, or the compass of natural existence ; in the
second, the assemblage of the properties of a single
body, or the constitution and appearances of things ;
in the third, it is used for expressing the power or
cause which produces them.

These significations are contained in the following ex-
amples. " There are bodies in Nature very much resem-
bling, and yet different from, each other." " It Is in the
nature of gold to be ductile, heavy, &c." " Nature pro-
duces different species of animal, vegetable, and mineral
bodies."

§. & NATURAL HISTORY.

The object of Natural History is Nature consi-
dered as the assemblage of all material bodies.

The name of Natural History, does not express the es-
sential properties of the science to which it is applied, and
has therefore been used in a very improper sense. Na-
tural History is by no means a historical science ; it has
no business with accidents or facts, but refers to objects, of
which it is indifferent whether they exist contemporane-
ously or consecutively ; and it considers these objects either
singly, or in such relations as they are brought into, by

VOL. i. A

2 INTRODUCTION. §. £

the application of the science itself. Of this kind is the
connexion produced amongst the natural objects by the
Theory of the System, or the second principal head of the
present work. Natural History by no means considers
those connexions among different bodies, in which some of
them produce alterations in the others, or contain the causes
of certain events.

The peculiar character of History, consists in being a nar-
rative or a relation of facts, arranged according to the suc-
cession of time. Natural History has nothing to relate,
and takes no notice of the succession of events.

The impropriety of the words Natural History, has ex-
ercised a prejudicial influence upon the developement
and the progress of the science itself, and has given rise
to many misconceptions. All these misconceptions disap-
pear as soon as the notion of Natural History is circum-
scribed within proper limits. Supposing the existence of a
definition of this kind, the name of Natural History can
be retained, particularly since it has not, till now, been
superseded by another more appropriate expression.

§. 3. NATURAL-HISTORICAL PROPERTIES.

The properties of a body, in as far as they are
considered and made use of in Natural History,
are called Natural-Historical Properties.

The Natural-Historical properties are those with which
, Nature has endowed the bodies which it produces, pro-
vided these properties, as well as the bodies themselves,
remain unaltered during their examination. A body is
said to be in its natural state, while it continues to shew
these properties. The natural state of a body is either
constant, or it is variable during a certain period of time.
In the first case, Natural History at once selects such of
the invariable properties, as may serve its purpose agree-
ably to the principles of the science. In the second, it de-
termines before hand the state of the highest perfection,
or of the full developement of these bodies, and then makes

§. 4. 5. INTRODUCTION. 3

the same choice. Properties thus selected are the natural-
historical properties of a body. Hence every natural-his-
torical property is one of those appertaining to a body in
its natural state ; but every one of the properties to be
met with in a body is not on that sole account also a natu-
ral-historical property. The assemblage of all the natural-
historical properties of a natural production, is its Natural,
or Natural-Historical quality.

Properties not subservient to the use of Natural His-
tory, are considered in other sciences, which, in respect to
their fundamental principles, entirely differ from that of
Natural History.

§. 4. NATURAL PRODUCTIONS.

Material Bodies, in consequence of their being
produced by Nature, are called Natural Produc-
tions.

It is Nature alone which produces bodies. Art only
modifies certain properties of the bodies produced by Na-
ture. Natural productions, modified or altered by the as-
sistance of art, are called artificial productions. A tree is a
natural production ; a table, into the form of which the
wood of the tree has been fashioned, is an artificial pro-
duction. If a gem undergoes a chemical analysis, it ceases
by that process to be a natural production. If it is cut
and polished, abstraction being made of its artificial form,
it still must be considered as a natural production ; whilst,
in respect to the form itself, it becomes an artificial one.

§. 5. DESIGN OF NATURAL HISTORY.

Natural History considers the natural produc-
tions as they are, and not how they have been
formed.

Natural History does not inquire into the mode of for-
mation of natural productions, but only into their natu-

4» INTRODUCTION. §. 6-

ral-historical properties, because herein consists the only
object of its consideration. Yet this is not on account of
the difficulties, which attend the explanation of the mode
in which natural productions have been formed, but be-
cause it acknowledges principles, which entirely exclude all
explanations of this kind. Thus the principles of Natural
History fix the extent and the limits of that science ;
limits which it cannot transgress without inconvenience.
Yet it is not thereby too much confined, since whatever
may thus be excluded, does not belong to Natural His-
tory itself, but to other sciences. Every addition from
these would only serve to contaminate the fundamental
principles of Natural History. It is a matter of the highest
importance to keep the sciences perfectly distinct from each
other, and strictly within their respective limits, in order to
become acquainted with their stronger and their weaker
parts, and to assist wherever it should be necessary ; but
the philosopher must not possess them separately. The
sciences might be compared to working tools set in dif-
ferent handles, and subservient to different purposes. The
intelligent naturalist is like an able artist, who knows how
to employ them conformably to his design.

§. 6. INDIVIDUALS.

A natural production, in as far as it is a single
body, and, as such, by itself fit to be an object of
natural-historical consideration, is termed a Natu-
ral+Historical Individual*

Natural productions, which are not individuals, or whose
individuals are no more recognisable, may, nevertheless,
be objects of examination, according to the principles of
Natural History. The idea of individuality implies unity
of form ; and by this an individual becomes an independ-
ent whole, whose natural-historical consideration does not
presuppose the existence of, or at least not the con-
nexion with, another individual. In Natural History, »

§. 7. INTRODUCTION. 5

tree is an individual; not the trunk, nor a branch, nor
the fruit of that tree. For the first of these is, by itself,
an object of natural-historical consideration, the others
only in as far as they are parts of the tree.

In water and other fluid bodies, individuals are at least
not observable. Water and other fluid bodies produced
by Nature, though whole masses of them (which may,
nevertheless, consist of individuals) are without individu-
ality, on all accounts remain natural productions, and, as
such, objects of Natural History.

§. 7. ORGANIC AND INORGANIC NATURAL PRO-
DUCTIONS.

The most conspicuous difference which presents
itself in Natural History, is that which exists be-
tween bodies either organic or organised) and inor-
ganic.

An organised body is composed of organs ; that is to say,
of vessels and instruments suitable to the subsistence, in-
crement, and reproduction of themselves and of the whole.
During a certain variable period of time, called Life,
the organised body is beyond the reach of those powers
which affect inanimate matter, if removed from that condi-
tion. Matter, in as far as it forms a part, or is the pro-
duct, of an organised body, is called organic matter ; and
a body consisting of it, an organic body. An inorganic
body consists of inanimate (not organic) matter. Here the
powers actuating it, have finished their effect, and are
therefore in equilibrium ; it exists in itself in an invariable
state, and can be altered only by external force. Certain
products of organised bodies, as resins, &c. are not orga-
nised themselves ; that is to say, do not consist of organs ;
yet they do not cease to be organic bodies, because tljey
ponsist of organic matter. However, they are not by
themselves objects of a natural-historical consideration
(.§• <*•)•

6 INTRODUCTION. §. 8. 9.

§. 8. ANIMALS AND PLANTS.

A farther difference takes place among the or-
ganised natural productions, depending upon their
mode of generation, subsistence, growth, propaga-
tion, and upon the quality and utility of their or-
gans. One part of them are termed Animals, the
other Plants.

§. 9. MINERALS.

There is no such difference among the inorganic
bodies. The inorganic productions of Nature are
altogether comprehended under the name of Mi-,
nerals.

Some naturalists have attempted to introduce a distinc-
tion among the inorganic productions of Nature, similar to
that mentioned above in respect to the organised bodies ;
yet the characters upon which this distinction was founded,
do not refer to those bodies themselves, or to their natural-
historical properties ; but arise merely from their con-
nexion with each other, from local relations, &c. ; and
hence the distinction itself is foreign to Natural History.

Those inorganic productions of Nature which have been
separated from the minerals, and provided with a particular
name, are the AtmospJicrUia^ or those bodies which constitute
the atmosphere, in the same way in which the others form
the solid parts of the globe. Agreeably to the preceding
considerations, this difference, the only one between the
two classes of natural bodies, is quite inadmissible in Natu-
ral History ; for Natural History does not consider the na-
tural productions in so far as they constitute the solid mass
of the globe, or the fluid mass of the atmosphere, but in
so far as, taken separately, they possess certain natural-his-
torical properties. Hence the atmospherilia cannot be se-
parated from the minerals. In a subsequent paragraph it

§. 10. 11. INTRODUCTION. 7

will appear, that a distinction of this kind would be con-
trary to the very idea of a mineral.

The Wernerian school has applied the term Fossil to
those minerals, which constitute the solid part of our globe.
Commonly this name is given to the remains of organic
bodies, which are dug out from the earth, as " fossil wood,
fossil bones, &c." and this is indeed the right use of the
expression. The name of a fossil becomes entirely inap-
plicable, if, agreeably to the principles of Natural History,
the atmospherilia are united with the minerals. Moreover,
the meteoric masses of iron, being the only varieties we
know, of the species of octahedral Iron, cannot be calkd
fossil bodies.

§. 10. NATUEAL KINGDOMS.

Natural History considers the differences men-
tioned in §. §. 8. and 9., as the foundation of divi-
ding the natural productions. Each member of this
division is called a Kingdom. That division which
comprehends the animals is termed the Animal;
that which contains the plants, the Vegetable ; and
that which comprises the minerals, the Mineral
Kingdom.

§. 11. DIVISION OF NATUEAL HISTOEY.

The distinction among the natural productions,
in §. 10., has occasioned a division of Natural His-
tory, according to these three Kingdoms. That
part of Natural History which considers the Ani-
mal Kingdom, is called Zoology ; that which con-
siders the Vegetable Kingdom, Botany ; and that
whose object is the Mineral Kingdom, Mineralogy 9
or the Natural History of the Mineral Kingdom.

8 INTRODUCTION. §. 12.

This division of Natural History, is founded upon the
difference of the objects, to which the single parts of that
science refer. It has no influence upon the principles and
upon the method ; or, properly speaking, it is not a conse-
quence from these, which are identical for all the three
parts of Natural History.

There is, however, another division required in Natural
History, which does not depend upon differences among
the objects considered, but is founded upon the being of
the science, and is therefore equally applicable in Zoology,
in Botany, and in Mineralogy. This is the division in
Determinative and Descriptive Natural History, which will
be explained more fully hereafter. It appears from the pre-
ceding observations, that respecting the mineral kingdom,
Anorgauography does not signify the same as Mineralogy,
but applies merely to the descriptive part of it. Oryctog-
nosy, however, means the doctrine of what is dug out of
the earth, that is to say, according to the mode of expres-
sion mentioned in §. 9., of the fossils, and cannot therefore
be applied with more propriety than the other, to the Na-
tural History of the Mineral Kingdom.

Mineralogy, or the Natural History of the Mineral
Kingdom, does not allow of any other subdivision than
that which has just been considered. Jience Geology is
not a part of Mineralogy, but of Physical Astronomy;
Mineralogical Chemistry is not a part of Mineralogy, but
of Chemistry ; Economical Mineralogy is not a part of
Mineralogy, but of Economy ; nor is Mineralogical Geo-
graphy a part of Mineralogy, but of Physical Geography,
which belongs to Physical Astronomy.

§. 12. PRINCIPAL HEADS OF NATURAL HISTORY.

The method of Natural History in general, and
each of its departments in particular, is develop-
ed under the following heads : 1. Terminology,
2. Theory of the System, 3. Nomenclature, 4. Cha-
racteristic, 5, Physiography.

§. 13. INTRODUCTION. 9

If this be effected in a general way for all the three king-
doms, it produces the method of Natural History in gene-
ral ; if applied to each of them, it gives the method of the
Natural History of the kingdom concerned.

As yet, the method of Natural History in general, has
not been treated of separately, nor is it an object which
requires to be more circumstantially developed in the pre-
sent place. This method would be for the whole compass
of the productions of Nature, what the Philosophia Botanica
of Linnaeus is for the vegetable kingdom.

The method of the Natural History of any particular
kingdom, is contained in that of Natural History in gene-
ral, and differs from the method of the other kingdom,
only by its being applied to different bodies. This will be
perfectly evident, if we reflect that the different parts of
Natural History could not be parts of one and the same
science, should their methods be different. Indeed, the
method according to which the aggregate of various infor-
mation, commonly called Mineralogy, has hitherto been
treated, is different from the method of Natural History
in general. Mineralogy, however, treated in this manner,
is not the Natural History of the Mineral Kingdom, but
is a compound not contained within a single science, and
which altogether cannot be traced to constant principles,
by any regular process of reasoning.

§. 13. TERMINOLOGY.

Terminology is the explanation of the natural-
historical properties, in as far as they are employed
in recognising, distinguishing, and describing the
productions of Nature, and in developing those
general ideas, which the method requires.

Terminology teaches the language adapted to the pecu-
liar use of the science, and explains the meaning of what
has been called the Technical Terms.

In this scientific language, fixed expressions are connect-
ed with accurately determined ideas, and, vke versa, accu-

|0 INTRODUCTION. §. 14?.

rately determined ideas with fixed expVessions. It is as
necessary in Natural History, as it is in Geometry ; and it
may be said to be in respect to the former, what the DefiV
nitions are in respect to the latter. In Natural History,
however, the Terminology has to surmount many more
difficulties than in Geometry, since it refers principally
to empiric notions. Hence, the more geometrical ideas can
be introduced into the mineralogical Terminology, the
greater advantage will be obtained ; because, by this means,
its explanations will approach the nearer to the character
of geometrical definitions. None of the other parts of
Natural History allow of the introduction of geometrical
ideas to so great an extent as the Natural History of the
Mineral Kingdom.

§. 14. THEORY OF THE SYSTEM.

The Theory of the System determines the idea
of the Species in Natural History. It fixes the
principle of classification ; and upon the idea of
the species, it founds, according to this principle,
the ideas of the Genus, the Order, the Class, and
the Kingdom, in both the natural and the artificial
systems ,- the difference of these it likewise indi-
cates and explains. Lastly, by applying all these
ideas to Nature, the outline of the system thus con-
structed, is furnished with its contents, in confor-
mity to our knowledge of the productions of Nature,
as obtained from immediate observation.

The Theory of the System contains the reasoning, or
philosophical part of the science, and consists in the pro.
*"" (duction of ideas of a greater extent, than those derived
immediately from observation. These are the ideas men-
tioned above. The fundamental proposition, in this part
pf the science, is the following: All things arc identical

§. 15. INTRODUCTION. 11

which, in their natural state (§. 3.), do not differ from each other
in any of their properties ; and this may be considered as an
Axiom in Natural History. This mode of reasoning is com-
mon to all the three parts of the science. There occur,
however, differences in respect to the application of these
ideas to Nature. They arise out of the different qualities
of the natural productions contained in each of the King,
doms.

The possibility of introducing mathematical ideas into
the Terminology of the Mineral Kingdom, is particularly
beneficial to the establishment of these systematic ideas,
in as much as their precision, in some measure, extends to
the latter ; and imparts to the most important of them
all, to the Idea of the Species, a degree of evidence, which
seems to be wanting in the other kingdoms, both vegetable
and animal, and which it is scarcely possible to supersede
by any other considerations. In this part of Natural His.
tory, the Theory of the System takes the place of the
Axioms and Theorems of Geometry.

The name of Classification has been sometimes applied
to the systematic reasonings in Natural History. Yet,
properly speaking, classification is only that part of the
Theory of the System which refers to the idea of genera,
orders, &c. under which the species shall be finally ar-
ranged, and in their application to Nature.

§. 15. NOMENCLATURE.

Nomenclature, is the assemblage of rules, ac-
cording to which Names and Denominations are
applied to natural productions. By these names
and denominations, the ideas of the system are
fixed ; or the one can be substituted as representa^
tives instead of the other.

The scientific nomenclature in Natural History is syste-
matical. Any nomenclature, not systematical, is termed a
trivial nomenclature, and does not belong to the science.

12 INTRODUCTION. §. 16.

The necessity of a systematic nomenclature in Natural
History needs no demonstration. Fundament um Botanices
duplex cst: Dispositio ct Denominatio. LINN. Phil. Bot.
151. The systematical nomenclature is the base upon
which the existence and the progress of the science is
founded, which, without it, must fall into confusion. This
is more obvious, indeed, in Zoology and Botany, than in
Mineralogy, yet by no means legs true in this part of
Natural History, as is sufficiently proved by long conti-
nued experience.

No systematical nomenclature has hitherto, existed in
Mineralogy ; and even the fragments of it* to be met with
here and there, dp not deserve oui" attention, because they
refer to systems foreign to Natural History.

Trivial names* are not fit for any scientific use, but they
are very convenient for common usage, particularly if they
are well chosen.

§. 16. CHARACTERISTIC.

The Characteristic furnishes us with the peculiar
terms or marks, by which we are able to distinguish
objects from each other, in so far as they are com-
prehended in the ideas established by the Theory
of the System. The Characteristic is peculiar to
the Determinative part %f Natural History (§. 11.).

The Characteristic presupposes the general notions or
ideas of Natural History to have been developed and appli-
ed to the data of observation, and therefore is not the
source of these ideas, nor of any other. The natural-his-
torical properties, or those assemblages of them, by which
we can distinguish the different species of one genus, the
different genera of one order, the different orders of one

* What Linnceus calls trivial names, will be explained in
its proper place.

§. 17. INTRODUCTION. 13

class, the different classes of one kingdom, are termed
Characters ; while the single properties made use of, or con-
tained in them, are called their Characteristic terms or marks.
The Characteristic is intimately connected with, and indeed
presupposes the existence of the system. A character re-
ferring to a natural system is called a natural Character, and
one which .refers to an artificial system, an artificial Cha-
racter.

Hitherto there has never existed a Characteristic in Mi-
neralogy, nor was it even possible to be successful in the
attempt of its construction ; because there never has been
a system, to which a Characteristic oould have been applied.

§. 17. PHYSIOGRAPHY.

Physiography is the description of natural pro-
ductions, and consists in the enumeration of all
their natural-historical properties. Physiography is
peculiar to the Descriptive part of Natural His-
toy (§. 11.)-

Dcscriptio est totius plantcc character naluralls, qui dcscr'ibat
omncs cjusdem paries externas. LINN. Phil. Bot. 326.

Physiography is intended to produce, by its descriptions,
a distinct image -of the natural productions. If considered
as a mere description, the object of Physiography is the
individual (§. 6.) ; and these descriptions do not require
any thing but Terminology, and the correct idea of Natu-
ral History itself. No systematical ideas are wanted, and
any names may be employed, if they be only fit to be kept
in a constant, though in itself arbitrary, reference to the
object described. Very little advantage, however, is deriv-
ed from such descriptions, for the Natural History of the
Mineral Kingdom. In order to answer the purposes of
Physiography, their object must be the Species ; and the
result obtained by that means, is the Collective or General
Description of the species, which unites in itself the descrip-
tion of all its individuals or varieties. Under this supposi-
thm, it requires also a correct notion of the natural-his-

14 INTEODUCTION. §. IS.

torical species. But since Physiography is entirely inde-
pendent of the system, and consequently also of the
systematic nomenclature ; the general descriptions will he
applicable to any system, provided the terminology employ-
ed be sufficiently accurate. Any nomenclature can be
made use of in this part of the science, because the arbitrary
names in every instance can easily be exchanged for the
systematic denominations. The Descriptive part of Mi-
neralogy has been hitherto the only one, towards which the
labours of naturalists have been directed ; and it is solely
to them that we are indebted for the progress of our infor-
mation respecting the products of inorganic Nature.

To the Descriptive part of Mineralogy must be referred
all those representations of the objects, as drawings, models,
&c. which are executed with the view of giving a more com-
prehensive idea of these objects themselves ; but they belong
to Terminology, if they are intended to elucidate certain
particular properties of the minerals, as the drawings and
models in Crystallography, which are employed for the sake
of developing the whole theory of Crystallisation.

§. 18. IDEA OF NATUKAIXHISTORY.

Natural History is the science, which enables
us to find the Systematic Denomination of any
natural production (§. 3.), if its Natural-Histori-
cal Properties be given or known ; and, vice versa,
from the Denomination being given, to find the
Natural Quality of a body. Mineralogy being one
of the three departments of Natural History, is
the same to the Mineral Kingdom (§. 10.), as
Natural History in general, is to the whole material
Nature (§. &).

Lcge artis mutuo noscatur planta ex nomine ct nomcn ex
planta ; utrumque ex proprto characlere ; in illo scripto, in
hac delineato ; tcrtiits non admittatur. LINN, Phil, Bot. 261.

§. 19. INTRODUCTION, 15

Hence the application of Natural History, to tlie objects
of observation, essentially consists in the process of con-
necting the natural-historical properties of the natural pro-
ductions with their systematic denominations ; or, on the
contrary, in that of joining the denominations with the
individual or collective descriptions (§. 170- The first re-
quires the assistance of the System and of the Charac-
teristic ; the other can be immediately effected, and does
not require the application of a particular proceeding.

From the manner in which Mineralogy has hitherto been
treated, it was impossible to obtain any other but an em-
pirical knowledge of Minerals, which consists in the re-
collection of having already met with a similar object, to
which a certain arbitrary name had been given.

It is very difficult to attain a correct knowledge of the
productions of the Mineral Kingdom, if we confine our-
selves to empiricism. Besides, it is a waste of time, and
the information thus acquired, is at the best uncertain.
The bad consequences of having chosen an unscientific
mode of proceeding of this kind, increases with the actual
enlargement of our information, in respect to the produc-
tions of inorganic Nature.

§. 19. METHOD OF STUDYING THE NATURAL
HISTORY OF THE MINERAL KINGDOM.

The only scientific way of studying Mineralogy
is, to proceed according to the principles, and con-
formably to the method of the science itself. This
requires some practice in several observations, re-
lative to certain properties of Minerals ; it presup-
poses some acquaintance with the mathematics ; and
a little tuition will greatly facilitate its application.

Every person who intends to acquire solid information
in Mineralogy, must endeavour to become conversant

16 INTRODUCTION. §. 19.

with the principles of Natural History; without which, it
is but too easy to miss the way to its attainment. An-
other very material object is the correct application of
these principles to the Mineral Kingdom ; for the best
and most perfect instrument is of no utility to those who
are not acquainted with its employment.

Lastly, a certain degree ef skill is required in recog-
nising and finding out the connexion of those forms which,
in Mineralogy, are called regular. In order to facilitate
the acquisition of this, some mathematical knowledge is ne-
cessary.

After having become sufficiently acquainted with Ter-
minology, the surest and shortest way for the beginner
lo proceed, is to apply at once to Nature itself. This may
be effected by means of the Characteristic, which, accor-
ding to the rules laid down, under the fourth Head, must
be employed in order to acquire a certain degree of prac-
tice, in the determination of individuals occurring in Na-
ture. This leads to an intimate acquaintance with the
minutest details, and thus becomes the basis of informa-
tion of greater extent.

If the student has an opportunity of examining well
arranged collections, he will be enabled to acquire general
ideas, and form general views, in a much shorter period
of time than would be possible by the comparatively slow,
yet detailed and sure processes of the Determinative part
of Mineralogy. In collections of this kind, the determina-
tion of the species must be correct, and their arrangement
conformable to the general principles of Natural History.
Collections otherwise arranged, can be of little use to the
beginner ; on the contrary, they may be prejudicial to his
future progress, in as much as they confound his ideas ;
and indeed they may be said to be useful only to those, who
wish to enlarge their information, by observation and in-
quiry.

There exist but few Mineralogical works, which can pro-
perly be recommended to a beginner. The following enu-
meration contains those most useful for this purpose.

§. 19. INTRODUCTION. 17

FOR THE STUDY OF TERMINOLOGY.

Von den aeusserlichen Kennzeichen der Fossilien, von A. G.
Werner. Leipzig. 1774. This work has been translated
into several languages. It has been translated into Eng-
lish, under the title of—

A Treatise on the External Characters of Fossils, of Abraham
Gottlob Werner, by Thomas Weaver. Dublin. 1805.

A Treatise on the External, Chemical, and Physical Cha-
racters of Minerals, by Robert Jameson. Second Edi-
tion. Edinburgh. 1816.

Cristallographie, ou Description des Formes Propres a tous
les Corps du Regne Mineral. Avec Figures et Tableaux
Synoptiques de tous les Cristaux connus. Par M. de
Rome' de 1'Isle. 2de Edition. Paris. 1783.

Traite de Mineralogie, par le Cen. Haiiy, &c. En cinq
volumes, dont un contient 86 planches. Paris. 1801.

De la Cristallisation conside're'e geome'triquement et phy-
siquement ; ou Traite abrege7 de Cristallographie, &c.
Par A. J. M. Brochant de Villiers. Strasbourg. 1819.

Versuch einesABC Buchs der Krystallkunde von Karl
von Raumer. Berlin. 1820.

Nachtrage zu dem ABC Buche der Krystallkunde von
Karl von Raumer. Berlin. 1821.

Various Memoirs in the Journal and the Annales des
Mines, &c. by Messrs Haiiy, Monteiro, &c. Also in the
Memoirs of the Academy and the Society of Berlin, in
the Journal f iir Chemie und Physik of Schweigger, &c.
by Messrs Weiss, Bernhardi, &c.

Since the original publication of this work in German, there
has appeared —

A Familiar Introduction to Crystallography; by Henry
James Brooke. London. 1823.

FOR THE STUDY OF THE THEORY OF THE SYSTEM.

Caroli Linnaei Philosophia Botanica, &c. Holmise. 1751.

Des Caracteres Exterieurs des Mine'raux, ou existe-t-il
dans les Substances du Regne Mineral des Caracteres
qu'on puisse regarder comme spe'cifiques, &c. Par M.
de Rome* de 1'Isle. Paris. 1784,

VOL. I. B

18 INTRODUCTION. §. 19.

Most of the works relating to the subject of the Mineral
System, and Classification in general, require the utmost
attention on the part of the beginner, who intends to per-
use them.

As to the principles of Nomenclature and the Characte-
ristic, the study of the works of Linnaeus is particularly to
be recommended, and, above all, of his Philosophia Bota-
nica and Critica Botanica.

In the second volume of this Treatise, those works are
quoted, which regard the Descriptive part of Mineralogy,
and which partly also have been referred to in the descrip-
tions of the single species.

PART I.
TERMINOLOGY.

GENERAL CONSIDERATION OF MINERALS. THEIR DISTINC-
TION INTO SIMPLE, COMPOUND, AND MIXED. DIVISION OF
THEIR NATURAL-HISTORICAL PROPERTIES.

§. 20. POWER OF CRYSTALLISATION, AND ITS PRO-
DUCTS.

That power which produces the individual (§. 6.)
in the Mineral Kingdom, is termed the Power of
Crystallisation.

This name has been applied, because the most eminent
and perfect productions of that power are Crystals (§. 26.).
The power of Crystallisation is included in the general
idea of the Individualising power, which tends to produce
individuals in all the three kingdoms of nature ; and which
refers equally well to regular crystals, and to such indivi-
duals of the mineral kingdom, as are produced by the power
of crystallisation, although they are not crystals them-
selves, as will be shewn in another part of this work.

Individuality does not require regularity, but it implies
unity of form (§. 6.). An individual, whatever may be its
form, fills the space occupied by that form, with a certain
matter (§. 23.), and thus it represents a whole, being co-
herent in itself, and limited towards the outside. For this
reason, the individual is a single body, and, as such, by it*
self a fit object for the consideration of Natural History.

When minerals pass into the state of individuality, they
at the same time are endowed with the rest of those natu-
ral-historical properties which are peculiar to them in that
state of distinct existence ; and hence these properties
must likewise be considered as produced by the power of

20 TERMINOLOGY. §. 21.

crystallisation. The assemblage of those properties, is the
mineral, or the natural production itself; at least, in as
far as it is an object of Natural History.

Minerals, upon which the power of crystallisation has
not exercised its action, are without individuality, and
therefore do not possess any of the properties connected
with this state of existence. They want unity of space ;
they are not single bodies, and, as such, by themselves fit
objects of Natural History. As mere shapeless masses,
with certain inherent properties, they can be considered as
objects of Natural History, only because they are natural
productions (§. 4.).

Temperature has a great influence over the power of
crystallisation. Several minerals, as water, fluid mercury,
&c. pass into the state of individuality, and become solid,
if the degree of temperature be sufficiently reduced ; on
the contrary, by an increase of temperature, hexahedral
silver, octahedral bismuth, &c. leave this state, and become
liquid, and others elastic. * For that reason, in treating of
Natural History, it is necessary to fix the degree of tem-
perature in which the productions of the mineral kingdom
are considered; and this is the ordinary temperature, at
which water is fluid, and the most fusible crystals are solid. -j-

§. 21. MINERALS DECOMPOSED AND IMPERFECTLY
FORMED.

The productions of the power of crystallisation
continue to be objects of natural-historical conside-
ration, so long as they retain the properties pecu-
liar to them, which they have derived from the ac-
tion of this power. By the loss of some or of seve-

* If the change produced on a mineral by the application
of heat, affects more than the mere state of aggregation,
the consideration of this change makes part of another sci-
ence, and has no reference to Natural History.

f This is the reason why water, and not ice, has received
a place in the system.

§. 21. PRELIMINARY OBSERVATIONS. 21

ral of these properties, they cease to be suitable ob-
jects for the consideration of Natural History.

A mineral possessed of the properties it received from the
power of crystallisation, is in its natural or original state
(§. 3.). A mineral which has lost these properties more or
less, is decomposed, and ceases to be an object of natural-
historical consideration.

Minerals thoroughly decomposed commonly appear in
the form of powder, or as shapeless masses, without present-
ing any regular structure, or lustre, or determined and con-
stant degrees of hardness or specific gravity ; and the co-
hesion of their particles is destroyed. They form part of
the friable or earthy minerals. An example of an earthy
mineral we have in Porcelain-earth, a substance produced by
the decomposition of prismatic Feld-spar.* The decompo-
sition of minerals, however, does not in all cases proceed so
far. Some minerals retain their form, whilst colour, lustre,
hardness, &c. are changed ; as in several varieties of hexa-
hedral and prismatic Iron-pyrites. All, even the slightest,
of these alterations, exercise an influence upon the natural-
historical consideration of those bodies. It is in direct oppo-
sition to the principles of Natural History, to consider de-
composed varieties of one species, as varieties of another ;
but this, nevertheless, has been but too often the case in
Mineralogy. Thus, decomposed varieties of hexahedral and
prismatic Iron-pyrites, and of brachytypous Parachrose-
baryte, have been taken for varieties of prismatic Iron-ore.
In most cases it is possible to determine what the decom-
posed minerals have been in their natural or original state,
though indeed, for that purpose, we have often to recur to
considerations foreign to Natural History.

It seems that the substance of several minerals has, in
the period of their formation, not arrived at that state of
perfection which distinguishes the finished productions of

* Another class of friable minerals consists of very small
fragments of crystals, and grains of fresh or not decomposed
minerals. Such are fine sand, &c.

22 TERMINOLOGY. §. 22. 23.

crystallisation. In respect to Natural History, they must
be classed with those which are decomposed. Minerals im-
perfectly formed, may be compared to animals or plants
mutilated, defective, monstrous ; while those that are decom-
posed, having ceased to retain their original state, may be
compared to the animal or the plant which has ceased to
live. They may elucidate facts, both in Zoology and Bo-
tany, though in that state they are not in themselves ob-
jects of inquiry in Natural History. 1^ is therefore per-
fectly evident, that the distinction introduced by some
naturalists among minerals, into crystallised, crystalline,
and amorphous, depends upon accidental circumstances in
the formation of these bodies ; and, therefore, is not essen-
tial.

§. 22. SIMPLE MINERAL.

A mineral consisting of one single individual, or
forming a part thereof, is termed a Simple Mineral.

This is the idea of a simple mineral in Natural History.
The simple mineral must be distinguished from what is
called simple in Chemistry ; and, likewise, from what Mi-
neralogists commonly call simple. The last frequently
consists of several individuals, and is therefore not simple
in the sense of Natural History. Examples of simple mi-
nerals are crystals and grains of dodecahedral Garnet, or
of octahedral Diamond. The particles of which granular
Limestone is composed, are each simple minerals belong-
ing to the species of rhombohedral Lime-haloide ; while
those of Coccolite are also simple minerals, belonging to
the species of paratomous Augite-spar, &c.

§. 23. COMPOUND MINERAL.

A mineral consisting of more than one indivi-
dual of the same quality, * is termed a Compound
Mineral.

* The term Homogeneous individuals would be more

§. 24. PRELIMINARY OBSERVATIONS. 23

The compound mineral consists of simple ones. It is
produced when several individuals of the same quality are
formed in a common space, either at the same time, or one
after the other ; one being the support of, or at least con-
tiguous to the other. It is, therefore, not one simple mi-
neral, but a composition of several. If many of these sim-
ple minerals come into contact, they prevent each other mu-
tually from assuming their regular form. Compound mi-
nerals, therefore, which consist of many simple ones, do
not possess regularity.

Examples of compound minerals are frequently met with,
as in the above mentioned varieties of rhombohedral Lime-
haloide, and paratomous Augite-spar ; also the globular
masses of hexahedral and prismatic Iron-pyrites, and the
stalactitic masses of rhombohedral Quartz, called Calcedony,
&c. may serve as examples of compound minerals.

§. 24. MIXED MINERAL.

A mineral, consisting of several individuals of
different qualities, is termed a Mixed Mineral.

The mixed mineral consists of simple minerals, like the
compound. The mixed mineral, as such, is not an object
of Natural History, because its constituent parts, the sim-
ple minerals, have already been considered by themselves,
and received their appropriate places in the system of Na-
tural History. For the same reason it becomes necessary,
from the principles of Natural History, to exclude even com-

exact, if, in the present place, we could avail ourselves of
that expression. In order to understand what is meant
here, it will be sufficient to consider individuals of the
same quality, to be such as are contained in the examples
quoted in the preceding paragraph of rhombohedral Lime-
haloide, and paratomous Augite-spar. Individuals of dif.
ferent quality, are such as exhibit notable differences in their
natural-historical properties ; as, for instance, Granite,
where the component individuals of rhombohedral Quartz,
prismatic Feld-spar, and rhombohedral TaJc-mica, widely
differ in appearance and character.

24 TERMINOLOGY. §. 25.

pound minerals from these considerations. It is necessary,
however, to distinguish correctly between the simple and
the compound minerals; and since this cannot be done
otherwise, than by knowing all the details respecting these
bodies themselves, their consideration must not entirely be
neglected.

The union among the simple minerals in the mixed mi-
neral, is sometimes so close, and the particles of the mixture
so diminutive, that it becomes impossible to ascertain their
reality by simple ocular inspection. Many Mineralogists
in this case consider mixed minerals as simple, and clasa
them as such in their systems. But this is not the only
error of the kind, occurring in such arrangements. Both
mixed and decomposed minerals are by themselves no
longer objects of the method ; yet there are even mixtures
of decomposed minerals, which have been introduced into
the systems, and to which particular places have been
assigned.

Examples of mixed minerals we have in many varieties
of rocks ; in granite, gneiss, porphyry, &c. ; also in many of
those masses which constitute beds and veins. Examples
of close or impalpable mixtures, are found in Iron-flint
and Heliotrope, both varieties of rhombohedral Quartz ;
the first of which is mixed with oxide of iron, the other
with Green Earth, a variety of prismatic Talc-mica. Mix-
tures of decomposed minerals we have in Clay, Yellow-
Earth, Tripoli, &c.

§. 25. DIVISION OF THE NATURAL-HISTORICAL
PROPERTIES.

The natural-historical properties of minerals are
divided into ; 1. such as refer to simple ; 2. such
as refer to compound minerals ; 3. such as are
common to both.

The natural-historical properties of minerals compre-
hend their colour, the different degrees of hardness, the

§. 25. PRELIMINARY OBSERVATIONS. 25

different kinds of lustre, the regukr forms, the circum.
stances and relations, under which the particles of the in-
dividuals can be separated from each other, &c. ; because
these are the properties of minerals exhibited in their na-
tural state, and may be considered without producing any
change or alteration on the mineral.

Properties which can only be observed during, or after
a change, cannot be employed agreeably to the principles
of Natural History, and must therefore be excluded from
Mineralogy ; because, in observing them, we transfer the
object itself from its natural state, into another, in which it
ceases altogether to be an object of Natural History, Pro-
perties of this kind are, the fusibility of minerals examined
before the blow-pipe, or by the assistance of some other
apparatus, and the concomitant phenomena ; — their solu-
bility in acids ; — phosphorescence produced by heat, if, after
the first experiment, it cannot be observed any longer ; —
chemical analysis instituted to ascertain the quality or re-
lative quantity of the component parts, and the results
of that process : — every thing, in short, must be excluded,
which alters the natural state of a mineral. There are
properties to be met with in minerals in their natural state,
which, although not altered by examination, yet are of no
utility in Natural History ; such as the size of crystals ;
the irregular enlargement, and figure of some of the faces
depending upon it ; the accidental forms minerals assume
by being broken, rubbed down, water-worn, decomposed, &c.
Such properties are accidental, because the identity (§. 14.)
of the individual is not destroyed by their occurrence.

The natural-historical properties include the greater part
of the characters commonly called external, and some of
those called physical.

As to the distribution of those properties among the dif-
ferent heads mentioned above, the first will include those
which can be observed only in an individual itself, or in a
fragment of an individual. These are the geometrical pro-
perties, or such as refer to Space ; the relations of Struc-
ture, those of Surface, and the phenomena of Refraction,
in so far as they depend upon the regular form of minerals.

26 TERMINOLOGY. §. 2.5.

To the second belong the rektions of Composition, the Forms
of compound minerals, the mode of Junction of the indivi-
duals in these compositions, &c. these properties being
such as are only to be met with in compound minerals.
The third comprehends those in which the simple or com-
pound state of the mineral has no influence upon the con-
sideration of the properties ; as Colour, Lustre, Transparen-
cy in general, Hardness, Specific Gravity^ the Stale of Aggre-
gation, Taste, $c.

Terminology includes, therefore, three Sections, within
which each of the above mentioned properties is considered
in a separate Chapter.

SECTION I.

THE NATUBAL HISTORICAL PROPERTIES OP SIMPLE
MINERALS.

CHAPTER I.

OF THE REGULAR FORMS OF MINERALS.

I. GENERAL CONSIDERATION OF THE REGULAR FORMS.

§. 26. CRYSTAL.

In Mineralogy, the term Crystal is applied to a
body, which consists of continuous and homoge-
neous matter, and occupies, from its origin, a re-
gularly limited space.

Crystals assume a regularly limited space in their origin,
that is to say, in the very act of their formation. A mine-
ral which appears in a regularly limited space only after
a part of its homogeneous matter has been detached from
it, is not contained under the preceding definition, and
therefore no Crystal.

The matter contained within the regularly limited space,
is termed homogeneous, if it be everywhere of the same qua-
lity ; and it occupies or fills the space with Continuity, if in
its interior it allows no particles to be distinguished from
one another, of which the whole mass might be said to be
composed. There are minerals occupying a regularly li-
mited space, with homogeneous matter, but without conti-
nuity ; because in their interior, particles can be observed,
which are evidently distinct from each other (§. 186.). Mi-
nerals which are found thus to consist of homogeneous mat-

28 TERMINOLOGY. §. 27. 28.

ter within a regularly limited space, yet want continuity,
and cannot therefore be called crystals in the signification
of that term, as defined above.

§. 27. OBJECT OF CRYSTALLOGRAPHY.

The object of the science of Crystallography, is
to ascertain the regularly limited space, that is to
say, the Form of the Crystals, not the matter,
which occupies that space.

Since the object of Crystallography is nothing but figur-
ed Space, and in this nothing is to be considered besides
geometrical quantities, and their relations to each other ;
it appears that Crystallography is a pure geometrical
science.

§. 28. FORMS AND FACES.

The regularly limited space occupied by a crys-
tal, is termed a Form of Crystallisation, and the
limits or planes. Faces of Crystallisation.

In Crystallography, the faces of crystallisation are con-
sidered as perfect Planes, although this is not always the
case in nature.

They are termed Faces of Crystallisation, in order to
distinguish them from certain other faces of minerals,
which, though they exhibit regular shapes, yet are no crys-
tals (§. 26.).

The faces of crystallisation receive particular names, ac-
cording to the forms which they limit, as, for instance,
Faces of the Rhombohedron, of the Octahedron, &c. They are
called Faces, without any nearer determination, if the
form to which they belong is understood to be a form of
crystallisation.

§.29.30. 31. OF FOKMS IN GENERAL. 29
§. 29. EDGES.

The limits of the faces, or their intersections
with each other, are termed Edges.

The Edges of forms are always supposed to be straight
lines, although in nature they are not always straight.

The Edges are denominated not only according to the
forms to which they belong, but also according to their par-
ticular situation in respect to these forms. If a form of
crystallisation contain edges of only one kind, these bear
simply the name of the form ; as, for instance, Edges of the
Hexahedron. If it contain several kinds of edges, they are
distinguished from each other by their name, for in-
stance, Terminal and Lateral Edges of the Rhombohe-
dron, &c.

§. 30. SOLID ANGLES.

The limits or terminal points of the edges are
Solid Angles.

The solid angles are named according to the forms in
which they are found, and receive a nearer determination,
by some epithet expressing their particular situation and
quality. Thus we say, Solid Angles of the Hexahedron ;
also rhombohedral, pyramidal, prismatic Solid Angles, &c.

§. 31. HOMOLOGOUS FACES.

Faces, equal and similar to each other, and simi-
larly situated, are termed homologous ; such as are
not equal and similar, or assume a different situa-
tion in the forms, are not liomologous.

In nature, the homologous faces are not always equal
and similar to each other, yet they are always similarly
situated. Sometimes a single face of a crystalline form k

30 TERMINOLOGY. §. 3£. 33.

enlarged, and assumes a figure different from what it
should be, and dependent upon this enlargement. Crys.
tallography takes no notice of these irregularities, in as
much as they are accidental, and because this science is in-
tended to promote the study of the forms in their peculiar
Regularity and Perfection, in order to enable us to develope
their relations to each other, and to facilitate the applica-
tion of both to the phenomena of Nature*

§. 32. HOMOLOGOUS OR EQUAL EDGES.

Edges are said to be of equal magnitude, or of
equal value, if the faces meeting in them are equally
inclined to each other, or produce an equal angle of
incidence ; they are said to be of equal length, if
they are formed by equal sides of the faces ; and if
they are both of equal magnitude and equal length,
and at the same time similarly situated, they are
termed equal or homologous.

The inclination at the edges is invariable in nature ; and
upon this constancy of the angles of incidence, is founded
the application of crystallography to nature. The length
of the edges is subject to variation, as well as the figure of
the faces. In the crystals themselves, all the edges of
equal quantity which are similarly situated, although per-
haps not of equal absolute length, are considered as homo-
logous.

§. 33. DENOMINATION OF THE SOLID ANGLES.

Solid angles are denominated according to the
number of faces contiguous to them, or according
to the quality of the edges produced by the inter-
section of these faces. Solid angles formed by homo-
logous faces, are said to be Homologous themselves

§. 34'. 35. OF FORMS IN GENERAL. 31

A solid angle formed by the intersection, or consisting
of three, four, five, &c. faces, is said to be a solid angle of
three faces, a solid angle of four faces, &c. A solid angle
is equiangular ', if the plane angles contiguous to it are equal ;
it is unequiangitlar, if they are different from each other. A
solid angle, resulting from the junction of two, three, &c.
different kinds of edges, is said to be digrammic, trigrammic,
&c. ; and a solid angle having all its edges equal, or which
possesses only one kind of edge, is, in opposition to the
latter, termed a monogrammic solid angle.*

§. 34. SIMPLE AND COMPOUND FOEMS.

A form contained under homologous faces
(§. 31.) is termed a Simple Form ; one that is con-
tained under faces which are not homologous, a
Compound Form.

We have examples of the former, in the Hexahedron, the
Octahedron, as Geometry considers those solids, Fig. 1. 2.,
and in several others besides. Of the latter in the same,
if their angles or edges, or both, are replaced by faces not
belonging to their own form, Figs. 3. 4., or in general, if the
form is limited by more and other faces, than is required
for a simple form.

§. 35, THE COMPOUND FORMS CONSIST OF THE
SIMPLE.

A compound form consists of two or more simple
ones. Those faces of the compound, which are

* MonogrammtCi single-edged or one-edged, from
single, and y^PP** a line ; digramtnic, double-edged or two-
edged, from $/? , double, and <ygct/u.{A$i ; trigrammic, triple-
edged or three-edged, from r^j, triple, and y^a^rj ; referring
to the number of different kinds in those lines or edgesy
which terminate in the solid angle.

32 TERMINOLOGY. §. 36. 37-

homologous to each other, belong to one and the
same simple form.

The hexahedron, whose angles are replaced by equilateral
triangles, or by equiangular hexagons, Figs. 3. 4., is a com-
pound form. The faces of a four-sided or eight-sided figure,
homologous to each other, are faces of the hexahedron,
which is one of the simple forms ; the triangles or hexa-
gons, again homologous to each other, are faces of the se-
cond simple form, which is the octahedron, and the com-
pound form is said to consist of both.

It is possible, that a compound form may assume the as-
pect of a simple one, in so far as it may be contained under
faces, which, according to the given definition, are homo-
logous. The particular circumstances, under which this
happens, and the reasons, why a form of that kind, never-
theless is considered as compound, will be given afterwards.

§. 36. TANGENT PLANES.

A plane, which touches a simple form in one of
its edges, is called a Tangent Plane.

The edge of the simple form lies in the tangent plane ;
and the latter is always supposed to be equally inclined to
both the faces meeting in the edge of the simple form, un-
less it be expressly mentioned otherwise.

§. 37. SECTIONS.

A plane, which intersects a simple form, is term-
ed a Section. A. Principal Section divides the form
into two equal halves, without dissecting an edge ;
a Transverse Section is perpendicular to a certain
line within the solid.

The knowledge of Sections is very useful, in a more de-
tailed examination of the forms themselves ; and the Prin-

§. 38. OF FORMS IN GENERAL. 83

cipal sections, in particular, allow of many interesting ap-
plications, both in Crystallography and in Optics. In Op-
tics, however, the term principal section is applied only to
those planes, which pass through the principal axis.

In the hexahedron, Fig. 1., the principal section ACEG
passes through the parallel diagonals AC and EG of two
opposite faces, and through the edges joining them AE and
CG, forming an oblong or rectangle. In the rhombohe-
dron, Fig. 7«> one principal section ABXC passes through
those diagonals of two parallel faces AB and XC, which join
different solid angles with each other, and through the in-
termediate edges AC and BX, forming a rhomboid. An-
other principal section C'C'^'B", passes through the dia-
gonals of parallel faces C'C" and B'B", joining equal solid
angles with each other, and through the intermediate edges
C'B" and C"B', forming a rectangle. Two or more prin-
cipal sections, of equal and similar figure, and similarly
situated, are accounted as one. Some forms, as the rhom-
bohedron, have more than one ; others, as the tetrahedron,
no principal section at all. The consideration of these sec-
tions is not of equal importance in all forms.

It is not necessary to carry the distinction of these sec-
tions any farther, than to such as yield regular, or at least
equiangular or equilateral figures. If, therefore, sections
in general are mentioned, only sections of that description
are to be understood,

§. 38. HOMOLOGOUS SECTIONS.

Sections, which either possess similar figures, or
which assume them, if reduced to the same distance
from the centre of the solid, or in which the junc-
tion of certain points, by straight lines, produces si-
milar figures, are termed sections of the same
kind, or Homologous Sections.

There are two sets of homologous sections in the hexa-
hedron, containing on one side all the squares, as A'B'C'D',,
VOL. i. c

34 TERMINOLOGY. §. 38.

Fig. 1. ; on the other all the equilateral triangles, as RST,
Fig. 5. But the equiangular hexagons, as R/'R/S"S"F/T',
produced by sections parallel to the triangles, are homolo-
gous with them ; for triangles of this kind can be inscribed
in them, or they are transformed into such triangles, if
brought to the same distance from the centre of the form.

The sections of the hexahedron, which appear as oblongs
or rectangles, as HSU'S' Fig. 6., are likewise homologous
to each other ; for they are similar, if made equidistant from
the centre. Let the edge of the hexahedron AD be = 1,
and AS that part of it through the end of which the sec-
tion passes, = — j— ; the figure of this section will be a

y 2

square. A section of that kind, however, if made equidis-
tant from the centre with a rectangle, is likewise transform-
ed into a rectangle, and therefore homologous with these
figures, and not with the squares above mentioned.

In every oblong, a rhomb can be inscribed, if we join
the centres of its sides by straight lines. Hence sections
of a rhombic figure are homologous with sections of an ob-
long or rectangular figure.

Besides the sections described in the hexahedron, there
are none to be met with in any other solid whatever ; or
those which may be met with in other solids, can always be
traced to one of these. The different kinds of sections are,
therefore :

1. Such as are either equilateral Triangles themselves, or
in which equilateral triangles may be inscribed ; as regu-
lar hexagons, or equiangular hexagons, whose alternate
sides, or equilateral hexagons, whose alternate angles, are
equal ; dodecagons of the same description, &c.

2. Such as are either Squares themselves, or into which
squares may be inscribed, as regular octagons, or equiangu-
lar octagons, whose alternate sides are equal, or equilateral
ones, whose alternate angles are equal, &c.

3. Such as are Rectangles or Rhombs, or in which rectangles
or rhombs may be inscribed. It must be remarked here,
that if among the rectangular sections, there is only one, or
two squares, as in the tetrahedron and in the hexahedron,

§. 39. 40. OF FORMS IN GENERAL. 35

the rectangular, as the greater number, determine the kind
of the sections.

The different kinds of sections will be furnished with
appropriate and expressive names in the following §§. 50,
62, 53.

§. 39. AXES.

The straight line passing through the centres of
two parallel sections, if it be perpendicular to their
planes, is termed an Axis.

Suppose a hexahedron, Fig. 1., to be intersected by a
plane A' B' C' D' parallel to one of its faces ; the section
will be a square. The straight line PQ through M and P
the centres of this, and of a parallel square, will be an
axis. Take from a solid angle of the hexahedron, Fig. 5.
equal parts AH, AS, AT upon the edges terminating in
this angle, and lay a section through the points thus de-
termined. The straight line AG through the centres M , M'
of this and of a parallel section R' S" S' T" T' R", even
though the figure of the latter should be no triangle, is
likewise an axis. Take equal parts AR, AS ; ER', ES'
of the parallel edges of a hexahedron, Fig. 6., beginning
from two adjacent solid angles A and E, and lay a plane
through the points thus determined. Its figure will be a
rectangle, or, at a certain distance from the centre of the
hexahedron, it will be a square (§. 38.) ; and the straight
line NO through M and M' the centres of this and of a
parallel section, is again an axis.

Every axis passes through the centre of the solid.

In the centre of the solid, all axes, which are perpendicu-
lar to homologous sections (§. 38.), intersect each other at
equal angles.

§. 40. HOMOLOGOUS AXES.

An axis belongs to that section, in the centre of

36 TERMINOLOGY. §. 41.

which it is perpendicular to its plane. Axes be-
longing to homologous sections, are said to be
themselves Tiomologous*

The axes belonging to the equilateral triangles, &c. may
for the present be called axes of the first ; those belonging
to the squares, &c., of the second ; and those belonging to
the rhombs or rectangles, &e., axes of the third kind.

Some forms contain only one, others two, and others
three kinds of axes in different number. The number in
which the axes of the first kind appear, is one or four ; that
in which those of the second kind are found, one or three ;
and that in which those of the third kind are contained in
the solids, one, three, four, or sir.

The hexahedron contains four axes of the first, three of
the second, and six of the third kind ; the tetrahedron four
of the first, none of the second, three of the third ; the
rhombohedron contains only one axis of the first kind.

§. 41. PRINCIPAL AND SUBORDINATE AXES.

Principal Axes are those whose sections are re-
gular, or such figures as allow regular figures to be
inscribed into them ; Subordinate Axes such whose
sections are no regular figures themselves, and in
which no regular figures can be inscribed. If a
form contains no principal axis properly so called,
one of the subordinate axes is considered as the
principal axis.

The axes of the third kind, in whatever number they may
appear, are always subordinate axes, if they occur at the
same time with others. But if they occur alone in a form,
their number is in no case greater than three ; and then
two of them are subordinate ; the third is the principal axis.

§. 4$. 43. 44. OF FORMS IN GENERAL. 37

§. 42. UPRIGHT POSITION.

A form is said to be in its Uprigfit Position
when one of its principal axes is vertical.

Forms that have only one principal axis are upright but
in a single one ; such as have more than one are upright
in several positions. If a form contains only axes of the
third kind, it is upright, when that axis is in the vertical
position, which is considered as its principal axis.

In the subsequent inquiries, all forms, simple and com-
pound, are supposed to have been previously brought into
an upright position.

§. 43. PARALLEL POSITION.

Two or several forms are in Parallel Position,
if the axes of the one are parallel to the homolo-
gous axes of the other.

Two or several forms are in parallel position, if of the
axes of the one, only two are parallel to two homologous
axes of the other. For all the homologous axes intersect
each other in the centre of the form, at equal angles (§. 39.)

The parallel position cannot in general be perfectly de-
termined in forms which possess only one axis. Several
forms, moreover, may be considered in different positions.
It will be pointed out hereafter by what means, in these
cases, the parallel position must be determined. The diffe-
rent positions of forms are of great importance in all crys-
tallographic researches, if the object of these be more than
the consideration of one form at a time.

Similar forms in parallel position, have their faces pa-
rallel.

§. 44. HORIZONTAL PROJECTION.

Place any given form in its upright position.

88 TERMINOLOGY. §. 45. 46.

Draw from the angles of this form, perpendicular
lines to a horizontal plane, and join all the points
thus determined by straight lines. The greatest
plane figure obtained by this proceeding is the Ho-
rizontal Projection of the form.

The horizontal projection belongs to the vertical axis,
and is homologous with the sections to which the axis be-
longs (§. 40.), and to which the projection is parallel.

A form possesses as many different horizontal projections
As it has kinds of principal axes.

The Side of the horizontal projection is the unity of most
of the subsequent calculations referring to the dimensions
of crystalline forms.

§. 45. REGULARITY.

The Regularity of simple forms is their greater
or lesser agreement with the regular solids of geo-
metry.

Regularity refers only to simple forms. The regularity
of the simultaneous existence of these in the compound, is
termed the Symmetry of combinations, which will be con-
sidered more at large in §. 141.

The irregularities so frequently occurring in crystals,
must be abstracted, in our consideration of them, and the
forms reduced to their peculiar regularity.

§, 46. DEGREES OP REGULARITY.

The regularity of simple forms allows of being
arranged in Several Degrees.

Geometry considers solids whose angles are not altogether
situated in the surface of one sphere, to be less regular than
those whose angles are all touched by the surface of a single

§. 4)7. OF FORMS IN GENERAL. 39

sphere ; and thus, it likewise acknowledges different degrees
of regularity. The degrees of regularity in Crystallography
are not the same with these. By the peculiar method of
treating its object, Crystallography is forced to ascribe the
same degree of regularity to forms, the angles of one of
which may lie in one, of another in two, of a third in three
different spheres, as to the hexahedron, to the monogram-
mic Tetragonal-dodecahedron (§. 63.), and to the Tetracon-
ta-octahedron (§. 77-) ; and it ascribes to others, as to the
Tetrahedron, a lesser degree of regularity, although its
angles altogether should be situated in the face of one and
the same sphere.

§. 47. DETERMINATION OF THE DEGREES OF
REGULARITY.

The degrees of regularity of simple forms, are
determined according to the Kind and the Number
of their Axes.

There are four degrees of regularity to be distinguished
in simple forms.

Forms of the first degree of regularity contain four axes
of the first kind, three of the second, and six of the third ;
of the second degree of regularity, four of the first kind, some
of them at the same time three of the second, some three of
the third, some none besides those of the first ; of the third
degree of regularity, only one axis of either the first or the
second kind, and an undetermined number of axes of the
third kind ; of the fourth degree, three axes of the third
kind.

Thus, first 436

(430

second 44 0 3

(400

third jj J undetermined,
fourth 003

40 TERMINOLOGY. §. 48. 49.

With these degrees of regularity, the rest of the quality
of simple forms is in the closest agreement, as will be seen
hereafter. The first degree of regularity contains two of
those forms, and the second one of those forms, which are
geometrically regular.

§. 48. CLASSIFICATION OF SIMPLE FOEMS.

Simple forms are divided, according to the num-
ber of their principal axes, into such as have only
One Principal Axis, and such as have Several.

The forms with one axis are of the third and fourth,
those with several axes are of the first and second degree
of regularity.

§. 49. NOMENCLATIVE OF SIMPLE FOBMS.

The forms of one axis receive their names ac-
cording to the figure of their faces, or according to
some general property ; those of several axes, ac-
cording to the number of their faces ; and when
a more accurate determination is necessary, accord-
ing to certain peculiarities of these forms them-
selves.

Systematic nomenclature remedies the want of confer-
mity and precision, which has hitherto prevailed in the
method of denominating crystalline forms ; and at the same
time produces a distinct idea of the forms themselves, since
it is in fact their abridged description. This shews the use-
fulness of the systematic nomenclature, and justifies its in-
troduction.

Forms of a single axis, whose faces are rhombs, are term-
ed Rhombohedrons ; others, whose faces are triangles, are
called Pyramids.

A form of several axes, which is contained under four faces,
is a Tetrahedron, or the Tetrahedron, because there exists

§. 50. OF SIMPLE FORMS IN PARTICULAR. 41

only one, or because all tetrahedrons are similar ; a form
contained under six faces, is the Hexahedron ; a form con-
tained under eight faces, is the Octahedron ; a form con-
tained under twelve faces, is a Dodecahedron^ because there
are several.varieties, or because not all dodecahedrons are
similar ; a form contained under twenty-four faces is an
Icositetrahedron ; and a form contained under forty-eight
faces, a Tetracontaoctahcdron.

The denominations which denote the different kinds and
varieties of simple forms, according to their peculiar pro»
perties, are formed from these names by composition, or by
the addition of adjectives.

II. OF SIMPLE FORMS IN PARTICULAR.

CONSIDERATION OF SIMPLE FORMS, AND SOME OF THEItt
GEOMETRICAL RELATIONS.

§. 50. THE RHOMBOHEDRON.

The rhombohedron. Fig. 7., is a form contained
under six equal and similar rhombic faces ; or the
rhombohedron is contained under six equal and si-
milar rhombs.

1. Any six rhombs, which are equal and similar to each
other, limit one, and if the obtuse angle of their figure is
less than 120°, two rhombohedrons.

2. All rhombohedrons belong to the same kind of forms.

3. The solid angles A, X, produced by equal plane angles
and equal edges (§. 33.) of the rhombohedron, are termed
its Apices.

4. The straight line AX through the apices, is the Axis of
the rhombohedron. The Rhombohedron has only one axis,
and this is of the first kind (§. 40.). Since all forms con-
nected with the rhombohedron possess axes of this kind,
these in future will be designated by the denomination of
rhombohcdral Axes.

4:2 TERMINOLOGY. §. 50.

5. A solid angle, through which a rhonibohedral axis
passes, is termed a rhornbohedral solid angle. This applies
equally to forms which are not rhombohedrons themselves.

6. The edges CA, C'A, C"A, BX, B'X, B"X, contigu-
ous to the terminal points of the axis, are Terminal Edges ;
while CB', B'C", C"B, &c. or those which do not intersect
or meet with the axis, are Lateral Edges.

7- The diagonals of the faces of a rhombohedron, are
commonly said to be the diagonals of the rhombohedron
itself. Those which are horizontal, like CC", C'C", &c.
when the rhombohedron is in its upright position (§. 42.),
are termed the Horizontal Diagonals ,- those which, on the
same supposition, assume a direction inclined to the axis, like
AB, AB', &c. are called the Inclined Diagonal* of that form.

8. The rhombohedron has two principal sections. The
first and most useful is a rhomboid, bounded by two parallel
terminal edges, and the inclined diagonals contained be-
tween them, as ABXC ; the second is a rectangle, as
C'C"B'B". The other sections are of the first kind
(§. 38.), and termed Rhonibohcdral Sections. That through
the centre of the form, or the transverse section, is a regu-
lar Hexagon.

9. The horizontal projection of the rhombohedron is a
Regular Hexagon, equal to that circumscribed about the
transverse section.

10. Of two rhombohedrons, that with a greater plane
angle at the apex, C'AC", is termed the more obtuse ; that
with a lesser, the more acute of these forms. The same dis-
tinction applies also to pyramids.

11. The sections CC'C" and BB'B", through contiguous
horizontal diagonals, are perpendicular to the axis, and di-
vide it in three equal parts, AP, PQ, and Q,X.

12. If, as it is supposed in all calculations concerning
the rhombohedron, the side of the horizontal projection is
= 1 ; the horizontal diagonal is = A/ 3.

13. Let the axis AX be = a ; the angle of inclination at
the terminal edge = x ; we obtain :

cos.x = 2-±!^f.
4 a2+ 0

§. 51. OF SIMPLE FORMS IN PARTICULAR.

14. Let a. be the plane angle at the apex ; we have
2 a2— 9
2(aa+ 9>

cos. a.

§. 51. PYRAMIDS IN GENERAL.

A pyramid is contained under equal and simi-
lar triangles.

1. The whole number of these triangles, as well as its
half, is an even number.

2. The triangles are either isosceles or scalene. A pyra-
mid contained under isosceles triangles, is termed an Isosce-
les pyramid; one contained under scalene triangles, a scalene.
pyramid.

3. The angles at the vertex of these triangles, are the
Apices of the pyramid.

4. The straight line through the apices is the Principal
Axis,

5. The edges contiguous to the terminal points of the
axis, are called Terminal Edges ; they are equal in isosceles,
and unequal in scalene pyramids. The remaining edges of
the pyramid either lie in a plane perpendicular to the prin-
cipal axis, or they are situated like the lateral edges of a
rhombohedron (§. 50. 6.). They are called Lateral Edges,
the first of them sometimes Edges at the Base.

6. The pyramids are divided according to the whole,
and denominated according to half the number of their faces,
as follows :

g

Number of
Faces.

Denomination.

Figure of the
Triangles.

1

Eight,

Four-sided,

( isosceles.
( scalene.

N
PI

Twelve,

Six-sided,

f isosceles.
( scalene.

Sixteen,

Eight-sided,

scalene.

44 TIUIMINOLOGY. §. 52.

7- The term Pyramid has not exactly the same signifi-
cation in Crystallography, as in Geometry. In Geometry,
it means a solid, bounded by any number of triangular
planes meeting in one point, and terminating at one poly-
gonal plane, as its base. In Crystallography, it must be
restricted to simple forms ; and, therefore, cannot be ap-
plied to any other but those, which have hitherto been call-
ed double pyramids. There are no simple pyramids, as
simple forms, to be considered in Crystallography. The te-
trahedron, which has been called a simple three-sided pyra-
mid, is no pyramid at all, but is a form of several axes, and
in the closest connexion with other forms of that kind,
particularly with the octahedron. The epithet double, there-
fore, is superfluous, since the crystallographer has on no oc-
casion to distinguish between simple and double pyramids,
as two different classes of simple forms.

§. 52. ISOSCELES FOUR-SIDED PYRAMIDS.

The isosceles four-sided pyramids, Fig. 8., are
contained under eight isosceles triangles.

1. The isosceles four-sided pyramids, have two principal
flections, one of which, BCB'C', is a square, the other,
ACXC, or AB'XB, a rhomb.

2. The remaining sections, belonging to the principal
axis, are also squares ; and the axis is therefore of the se-
cond kind (§. 40.). This axis itself, the solid angles through
which it passes, and the sections belonging to it, are term-
ed Pyramidal^ because all forms in connexion with the isos-
celes four-sided pyramid, contain axes, solid angles, and sec-
tions of the same kind. The same denomination applies to
every form possessing similar axes, solid angles, and sec-
tions, although this form be not an isosceles four-sided py-
ramid.

3. Besides these, the isosceles four-sided pyramids con-
tain four axes of the third kind, two of which, BB' and
CC', are the diagonals, the two others HH'" and H'H"

§. 53. OF SIMPLE FORMS IN PARTICULAR. 45

parallel to the sides, of the base. These axes intersect
each other at angles of 45°.

4. The horizontal projection is a square, equal to the
base, or to the square principal section.

5. Let the side of the horizontal projection be = 1 ; the
axis = a, the terminal edge = x ; the lateral edge, or that
at the base = z ; we obtain :

1 I n J

cos. x = _ . cos. z =

1 + a2 1 + a2

§. 53. SCALENE FOUR-BIDED PYRAMIDS.

The scalene four- sided pyramids, Fig. 9., are con-
tained under eight scalene triangles.

1. The scalene four-sided pyramids have three principal
sections, ABXB', ACCSIC and BC'B'C, all of which are
rhombs.

2. The remaining sections, all parallel to the principal
ones, are likewise rhombs : the axes AX, BB', CC' there-
fore are of the third kind. These axes, the solid an-
gles through which they pass, and the sections belonging
to them, are called Prismatic, on account of the great
number and variety of oblique angular four-sided prisms
existing among the forms in connexion with the scalene
four-sided pyramid, all of which possess axes, solid angles,
and sections of this kind. Those denominations of axes,
solid angles, and sections, are likewise made use of in such
forms as are not connected with the scalene four-sided
pyramid.

3. Any axis of the scalene four-sided pyramid can be
assumed as the principal one, or any solid angle can be
considered as the apex. After the principal axis has been
obtained, the subordinate axes, apices, solid angles, the base
and its diagonals, are thereby ascertained, and remain inva-
riable in all considerations of a determined form of this kind.

46 TERMINOLOGY. §. 54.

4. Among the terminal edges, the greater is said to be
the obtuse edge, the lesser the acute edge ; which is likewise
the case in the scalene six-sided pyramids (§. 55.), and in
the eight-sided pyramids (§. 5G.).

5. The horizontal projection is equal and similar to the
base, or to that principal section, which is perpendicular to
the principal axis.

5. Let the axis AX of a scalene four-sided pyramid be
=. a ; BB' one of its diagonals = b ; CC' the other dia-
gonal = c ; the terminal edge AB contiguous to b = y ;
the terminal edge AC contiguous to c = x ; the edge BC
at the base = z : then

a2 b2 + (a2+b2)c2
cos. x = a*c2-(a2+c2)b2 .
a2 c2 + (a2+c2)b2

b2 c2 + (b2 +C2)a2
and

cos. y + cos. x + cos. z «= — 1.
Hence

cos. y SB — (1 + cos. x + cos. z) ;
cos. x = — (1 + cos. y + cos. z) ;
cos. z = — (1 + cos. y + cos. x),

7« Suppose cos. y = « ; cos. x = ft ; cos. z = y : the fol-
lowing ratio among the diagonals will be obtained :
a:b:c= ^ [ (1 + «) 0 + 0) 1 ' V [ 0 + «) (1 + 7)]
: V [ (1 + A) (1 + y) ]•

§. 54}. ISOSCELES SIX-SIDED PYRAMIDS.

The isosceles six-sided pyramids, Fig. 10., are
contained under twelve isosceles triangles.

1. The isosceles six-sided pyramids have two principal
sections : one of them AHXZ, &c. is a rhomb, the other
HORZNT a regular hexagon. The latter is at the same
time the base of the pyramid.

55. OF SIMPLE FORMS IN PARTICULAR. 47

2. The remaining sections are rhombohedral (§. 50.) ami
prismatic (§. 53.), as also the axes. Of the prismatic axes,
three HZ, ON and RT pass through the lateral solid angles,
and three IS, KU and LV through the centres of the la-
teral edges.

3. The horizontal projection is equal and similar to the
base, or to the rhombohedral principal section.

4. The side of the horizontal projection being = 1
(§. 50. 12.), let the axis be = m.a (the axis of a rhombo-
hedron being designated by a, and a certain constant co-
efficient by m) ; the terminal edge = x ; the lateral edge
=2 z : we have

t2 + 6
i2 + 6

m2. a2 — 3

cos. z

(m2. a2 — 3 \
l
m2. a2 + 3 /

§. 55. SCALENE SIX-SIDED PYRAMIDS.

The scalene six-sided pyramids, Fig. 11., are
contained under twelve scalene triangles.

1. The principal section A'BX'C, &c. is a rhomboid.

2. The remaining sections are rhombohedral; those
which pass only through terminal edges are equilateral
hexagons of alternately equal angles ; that through the
centre, or the transverse section, is an equilateral dodeca-
gon, likewise of alternately equal angles.

3. The lateral edges of this form are disposed like the
lateral edges of a rhombohedron.

4. The horizontal projection is a regular hexagon.

5. The side of the horizontal projection being = 1 ; let
the axis A'X' be = m.a (where a signifies the axis AX of
a rhombohedron, whose lateral edges coincide with the la-
teral edges CB, BC, &c. of the pyramid, and m a variable
co-efficient) : the obtuse terminal edge = y ; the acute ter-
minal edge = x ; the lateral edge = z : the following for-
mulae will be derived.

48 TERMINOLOGY. §. 56.

cos. y » - (_<

2 [ (3 m* + 1) a" + 9]

cos. x , __

2 [ (3 m2

^I)»- + "»\ .

+ 1) a2 + 9] /

_

(3 m2 + 1) a2 + 9
cos. y = 1 + cos. x + cos. z — 2. ^ [(1 + cos x) (1 — cos. z)] ;
cos.x= 1 + cos.y+cos.z — 2. A/[(! + cos.y)(l — cos.z)];
cos. z = — (1 + cos. y + cos. x + 2. ,J [ (1 +cos.y)(l + cos.x)]).

§. 56. SCALENE EIGHT-SIDED PYRAMIDS.

The scalene eight-sided pyramids, Fig. 12., are
contained under sixteen scalene triangles.

1. The scalene eight-sided pyramids have three different
principal sections ; the first of these B'SC'S'BS^CS" is an
equilateral octagon, of alternately equal angles ; the other
two A'CXX'C and A'B'X/B on one side, and A'SX"S'" and
A'S'X'S" on the other are rhombs.

2. The remaining sections are pyramidal and prismatic ;
so are likewise the axes. Of the prismatic axes, every two
B'B, C'C, and SS'", S'S", pass through equal solid angles.

3. Those edges which are not terminal, are edges at the
base.

4. The horizontal projection is equal and similar to the
base or the pyramidal principal section.

5. Let the axis A'X' of the scalene eight-sided pyramid
be = m. a (where a is = AX, the axis of an isosceles four-
sided pyramid, the side of the horizontal projection of
which, SSy is = 1, and m a variable co-efficient, greater
than 1 + *J 2 (§ 103.)) ; the acute terminal edge = y ;
the obtuse = x ; the edge at the base = z : we obtain

(m» + l)a9 + 2
/(m* + l)a»-2y
\m« + 1) a2 + 2/

§. 57. OF SIMPLE FORMS IX PARTICULAR. 49

And

cos. y = — 4 (1 + cos. z

+ 2. J [_ (cos. x + cos. z) (1 + cos. x)}) ;
cos. x = — ^ 0 4- cos. z

-f 2. ^ [— (cos. y + cos. z) (1 + cos. y)]) ;
cos. z as — (3 + 2. cos. y + 2. cos. x

-f 2. V [2 (1 + cos. y) (1 + cos. x)]).

§. 57. THE TETRAHEDRON.

The Tetrahedron, Figs. 13. 14-., is contained un-
der four equilateral triangles.

1. The plane angles a,a,a of the tetrahedron are = 60° ;
the angles of incidence, at the edges A, A, &c. (their magni-
tude) = 70° 31' 44".

2. The sections of the tetrahedron are rhombohedral and
prismatic; one of the latter, through the centre, is a
square.

3. The principal axes are rhombohedral ; they join the
solid angles with the centres of the opposite faces ; their
number is four, and they intersect each other at angles of
109° 28' 1C", and 70° 31' 44". These angles of intersection
are general for the rhombohedral axes^ •whenever more than one
occur in the same form. The subordinate axes are prisma-
tic ; they join the centres of opposite edges ; their number
is three, and they are perpendicular to each other.

4. The tetrahedron is a regular solid of geometry. *

5. This form occurs, either by itself or in combina-
tions, in tetrahedral Copper-glance, in dodecahedral Gar-
net-blende, &c.

* The principal sections and horizontal projections of
the forms of several axes being of comparatively little use,
and besides very easily ascertained, I have thought it
superfluous to enter here into a greater detail.

VOL. I. D

50 TERMINOLOGY. §. 58. 59.

§. 58. THE HEXAHEDRON.

The Hexahedron, Fig. 1., is contained under six
squares.

1. All the angles of the hexahedron, those of the faces
as well as those of the edges, are = 90°.

2. The sections are rhombohedral, pyramidal, and pris-
matic : so are the axes.

3. The rhombohedral axes pass through the solid angles ;
the pyramidal axes, whose number is three, through the
centres of parallel faces, and these are perpendicular to each
Other ; and this again is general to the pyramidal axes, whenever
more than one occur in the same form. The prismatic axes,
whose number is six, pass through the centres of parallel
edges ; those which belong to parallel edges, intersect each
other at right angles ; those which belong to edges that are
not parallel, at angles of 60° and 120° ; and tJwse are again
general angles for the prismatic axes.

4. The hexahedron or cube is a regular solid of geometry.

5. This form is frequently met with in nature, as in
octahedral Fluor-haloide, hexahedral Iron-pyrites, &c.

§. 59. THE OCTAHEDRON.

The Octahedron, Fig. 2., is contained under
eight equilateral triangles.

1. The plane angles of the octahedron are = 60° ; the
edges or angles of incidence = 109° 28' 16". The angles
of incidence of the octahedron and of the tetrahedron are
supplemental to each other (to 180°). These angles are
the same as those at which the rhombohedral axes inter-
sect each other (§. 57. 3.>

2. The sections and axes are the same as in the hexahe-
dron ; only the rhombohedral axes pass through the centres
of parallel faces, and the pyramidal axes through the solid
angles.

§. 60. 61. OF SIMPLE FORMS IN PARTICULAR. 51

3. The octahedron is a regular solid of geometry.

4. This form occurs very frequently in different species,
as in octahedral Corundum, octahedral Iron-ore, &c.

§. 60. DODECAHEDRONS IN GENERAL.

The Dodecahedrons are contained under twelve
equal and similar faces, the figure of which deter--
mines the kind of the dodecahedrons. A dodeca-
hedron whose faces are triangles, is termed a
Trigonal-dodecahedron ; one whose faces are tetra-*
gons, a Tetragonal-dodecahedron ; and one whose
faces are pentagons, a Pentagonal-dodecahedron.

1. None of these dodecahedrons are regular in the geo-
metrical sense of the word ; for their faces are not regular
polygons ; besides, they have at least two different kinds of
angles, and, one of the dodecahedrons only excepted, they
have also at least two kinds of edges.

§. 61. TRIGONAL-DODECAHEDRONS.

The Trigonal-dodecahedrons, Figs. 15. 16., are
contained under equal and similar isosceles triangles.

1. The trigonal-dodecahedrons possess the general aspect
of the tetrahedron, and their sections and axes are of the
same kind, and in the same situation.

2. They contain four solid angles of three, and four of
six faces ; both of them being equiangukr. The first are
monogrammic, and correspond to the centres of the faces ?
the others are digrammic, and correspond to the solid angles
of the tetrahedron.

3. Of the two kinds of edges of the trigonal-dodecahe-
drons, the first, or those joining the angles of six faces,
have the situation of the edges of the tetrahedron ; the
others meet in the solid angles of three faces, upon the
centre of the faces of that form.

52 TERMINOLOGY. §. 62. 63.

4. There are two known varieties of these dodecahe-
drons, whose dimensions are as follows :

a.» b. A. B.

1. 117° 2' 8". 31° 28' f>6". 109° 28' 16". 146° 26' 33".

2. 112° 53' 7". 33° 33' 261". 129° 31' 16". 129° 31' 16".

5. Of the first variety of this form we have examples in
tetrahedral Copper-glance ; of the second variety, in dode-
eahedral Garnet-blende.

§. 62. TETRAGONAL-DODECAHEDRONS.

The Tetragonal-dodecahedrons are contained
under equal and similar tetragons.

1. There are two kinds of these forms.

2. Of the faces of the one, two are always parallel to
each other, and they contain two pairs of equal angles. Of
the faces of the other, no two faces are parallel, and they
contain only one pair of equal angles, the remaining two
being also different betwixt themselves. All the edges of
the former are equal, while the latter possess two kinds of
different edges.

3. From this last mentioned difference, the denomina-
tions of the two kinds are derived ; the first containing the
monogrammic, the second the digrammic Tetragonal-dodeca-
hedrons.

§. 63. THE MONOGRAMMIC TETRAGONAL-DODECA-
HEDRON.

The faces of the monogrammic Tetragonal-do-
decahedron, or of the Dodecahedron, Fig. 3L, are
rhombs.

* The small letters a, b, signify the plane angles of the
faces, and the large ones A, B, the angles of the incidence
at the edges, as referring to the figures.

§. 64. OF SIMPLE FORMS IN PARTICULAR. 53

1. The plane angles of these rhombs are = 109° 28' 16"
and 70° 31' 44", equal to the edges of the octahedron (§. 59.
1.) and of the tetrahedron. They are equal also to the
angles of intersection of the rhombohedral axes (§. 57. 1.
3.). The edges are all = 120°.

2. The raonogrammic Tetragonal-dodecahedron has eight
solid angles formed by three, and six formed by four faces ;
both of them are equiangular. The first are situated like
the solid angles of the hexahedron, the second like those of
the octahedron.

3. The sections and axes are as in these. The rhombohe-
dral axes pass through the solid angles of three, the pyrami.
dal axes through the solid angles of four faces, and the pris-
matic axes through the centres of parallel faces of the solid.

4. There is only one variety of this form, which is com-
monly expressed by the name of the Dodecahedron.

5. The dodecahedron is not a rare form ; it is found in
dodecahedral Garnet, hexahedral Gold, &c.

§. 64-. DIGRAMMIC TETRAGONAL-DODECAHEDRONS.

The faces of the digrammic Tetragonal-dodeca-
hedrons, Figs. 17. 18., possess the outlines of those
inscribed in a Trapezium.

1. The digrammic Tetragonal-dodecahedrons have the
general aspect of the tetrahedron.

2. They contain two kinds of solid angles formed by three
faces, four of each. Both kinds are equiangular. The
more acute correspond to the solid angles, the more obtuse
to the centres of the faces of the tetrahedron. They possess
moreover six solid angles of four faces, which are equian-
gular, but digrammic, and situated above the centres of the
edges of the tetrahedron.

3. Of the two kinds of edges of these forms, the more ob-
tuse join in the obtuse, the more acute in the acute solid
angles formed by three faces, and both in the solid angles
consisting of four faces.

54 TERMINOLOGY. §. 65. 66.

4. The sections and axes are as in the tetrahedron. The
prismatic axes pass through the solid angles of four faces.

5. There is only one variety known of this form, whose
dimensions are the following :

a. b. e. A. B.

90°. 118° 4' 10". 75° 67' 55". 90°. 152° 44' 2".

6. It has been observed in dodecahedral Garnet-blende.

§. 65. PENTAGONAL-DODECAHEDRONS.

The Pentagonal-dodecahedrons are contained un-
der equal and similar pentagons.

1. There are two kinds of Pentagonal-dodecahedrons.

2. In the one, every face has the opposite one parallel to
it, a property which is not to be met with in the other.

3. The first have the general aspect of the hexahedron,
and are therefore termed Iwxahedral ; the other that of the
tetrahedron ; and accordingly they bear the denomination
of tetraJiedral Pentagonal-dodecahedrons.

§.66. HEXAHEDRAL PENTAGONAL-DODECAHEDRONS.

The faces of the hexahedral Pentagonal-dodeca-
hedrons, Figs. 19. 20., have two pairs of equal angles,
and four equal sides. The single angle is opposite
to the single side.

1. All the solid angles of these Pentagonal-dodecahe-
drons, are bounded by three faces ; eight of them are equi-
angular and monogrammic, and correspond to the solid
angles of the hexahedron. The other twelve are formed
by two equal angles, and the single angle ; they are di-
grammic, and pairs of them may be conceived to be situated
upon the faces of the hexahedron, in the direction of
planes, which pass through two pyramidal axes of that form.

2. Of the two kinds of edges, those opposite to the single
angle are the Characteristic Edges of this form ; because

§. 07. OF SIMPLE FORMS IN PARTICULAR. 55

the examination of these yields the best means to distill-
guish the different varieties of the hexahedral Pentagonal-
dodecahedrons. The other edges meet in the monogram-
mic solid angles.

3. The sections and axes are as in the tetrahedron. The
prismatic axes pass through the centres of the characteris-
tic edges.

4. There are three varieties of this form, whose dimen-
sions are the following :

a. b. c.

1 . 102° 35' 40". 108° 24' 30". 110° IT 40".

2. 121° 35' 18". 106° 36' 2". 102° 36' 19".

3. 141° 16' 50". 103° 20' 33". 96° 1' 2".

A. B.

1. 112° 37' 12". 117°29/11//.

2. 126° 52' 12". 113° 34' 41".

3. 143° 7' 48". 107° 27' 27".

5. The first and second variety are found in hexahedral
Iron-pyrites, the angles of the third depend upon the third
variety of the icositetrahedrons, §. 71- 130.

§. 67. TETRAHEDRAL PENTAGONAL-DODECAHE-
DRONS.

The faces of the tetrahedral Pentagonal-dodeca-
hedrons, Figs. 21. 22. 23. 21, have no equal angles ;
but they possess two pairs of equal sides.

1. The tetrahedral Pentagonal-dodecahedrons have three
kinds of solid angles, all of which are formed by three faces.
The first are equiangular, monogrammic, four in number,
and they correspond to the solid angles of the tetrahedron ;
the second, of the same description, but more obtuse, cor.
respond to the centres of the faces of the same form. The
third are not equiangular ; they are trigrammic, twelve in
number, and pairs of them are situated between the more
acute equiangular solid angles.

50 TERMINOLOGY. §. 68.

2. This form contains three kinds of edges ; the first meet
in the more acute, the second in the more obtuse equian-
gular solid angles, and the third join those which are form-
ed by three different plane angles.

3. Its sections are rhombohedral ; and it does not pos-
sess any other but rhombohedral axes, in conformity with
the sections.

4. These solids are remarkable, on account of their being
as it were twisted, some to the Right, others to the Left.
They are equal and similar to each other ; but every part of
the one, has exactly the reverse situation 0f the other.

5. The dimensions of the three varieties pf this form, are
as follows :

a. b. c. d.

1. 116° 6' 13". 111° 50' 44". 93° 49' 21". 143° 11' 29".

2. 113° 21' 46". 113° 43' 28". 99° 35' 38". 130° 12' 11".
& 113° 34' 41". 128° 20' 44". 97° 59' 19". 136° 39' 57".

e. A. B. C.

1. 75° 2' 13;;. 141° 47' 12". 94° 5' 45". 106° 36' 2".

2. 83° 6' 57". 131° 4' 57". 78° 2?' 46". 115° 22' 37",

3. 66° 25' 19". 131° 48' 37". 95° 27' 54". 121° 35' 18".

6. This form has not yet been found in nature ; and the
angles of the mentioned varieties depend upon those of the
tetraconta-octahedrons, §. 77' 134.

§. 68. ICOSITETRAHEDRONS IN GENERAL.

The Icositetrahedrons are contained under twen-
ty-four equal and similar faces, the figure of which
determines the kinds of icositetrahedrons. An
icositetrahedron, whose faces are triangles, is
termed a TVigwiaZ-icositetrahedron ; one whose
faces are tetragons, a TWrag-owaZ-icositetrahedron ;
and one whose faces are pentagons, a Pentagonal-
icositetrahedron.

§. 69. 70. 01- ' SIMl'LE FORMS IX PARTICULAR. 57

1. None of these kosi tetrahedrons are geometrically re-
gular.

§. 69. TRIGONALrlCOSITETRAHEDRONS.

The Trigonal-icositetrahedrons are contained un^
der equal and similar, isosceles or scalene triangles.

1. This species of icositetrahedrons comprises three
kinds, different from each other by their general aspect,
and the situation of their faces.

2. The varieties of the first kind have no parallel faces ;
they exhibit the general aspect of the tetrahedron, and are
therefore said to be tetrahedral ; the varieties of the second
have parallel faces, and the general aspect of the hexahe-
dron ; these are termed hcxahcdral ; the varieties of the
third possess also parallel faces, but the general aspect of
the octahedron, and these are termed octahedral Trigoual-
icositetrahedrons.

§. 70. TETRAHEDRAL TRIGONAL-ICOSITETRAHE^
DRONS.

The faces of the tetrahedral Trigonal-icositetra-
hedrons, Figs. 25. 26., are scalene triangles.

1. These forms have four solid angles, and six faces, all
of which are equiangular and digram mic. Those contained
by four faces, six in number, are situated above the
centres of the edges ; the more obtuse solid angles of six
faces, four in number, above the centres of the faces of the
tetrahedron, and the more acute solid angles formed by
the same number of faces, also four in number, correspond
to the solid angles of this form.

2. There are three different kinds of edges in this form.
The longest join those solid angles of six faces which are
not similar to each other, the intermediate ones the more
acute, and the shortest the more obtuse of these solid angles
with those which are bounded by four faces.

58 TERMINOLOGY. §. 71.

3. The sections and axes are as in the tetrahedron. The
prismatic axes pass through the solid angles of four faces.

4. There are three varieties, of the following dimensions:

a.
1. 5G° 15' 4".
2. 53° 46' 42".
3. 64° 21' 34".
A.
1. 110° 55' 29".
2. 122° 52' 42".
3. 124° 51' 0".

b.
82° 23' 19".
82° IT 58".
85° 19' 19".
B.
158° 12' 48".
152° 20' 22".
144° 2' 58".

c.
41° 21' 37".
43° 55' 20".
40° 19' 7".
C.
158° 12' 48".
152° 20' 22".
162° 14' 50".

5. The third variety of this form has been observed in
hexahedral Boracite ; the other two depend upon the first
and second variety of the tetraconta-octahedron, §. 77. 133.

§. 71. HEXAHEDRAL TRIGONAL-ICOSITETRAHE-
DRONS.

The faces of the hexahedral Trigonal-icositetrahe-
drons, Fig. 32., are isosceles triangles.

1. The solid angles consist either of four or of six faces,
and are all equiangular. The first, six in number, are mo-
nogrammic, and situated above the centres of the faces ;
the second, eight in number, and digrammic, are situated
like the solid angles of the hexahedron.

2. Those edges of the form which correspond to the
edges of the hexahedron, join the angles of six faces with
each other; the others join the solid angles of six faces
with those of four faces.

3. The sections and axes are as in the hexahedron.

4. There are three varieties of these forms, whose dimen-
sions are the following ;

a. b. A. B.

1-79° 31' 28". 50° 14' 16". 157° 22' 48" 133° 48' 47".

2. 83° 37' 14". 48° 11' 23". 143° 7' 48" 143° 7' 48".

3- 86° 58' 59". 46° 30' 30|". 1<26° 52' 12". 154° 9' 29".

§. 72. 73. OF SIMPLE FORMS IN PARTICULAR. 59

. 5. The second variety occurs in dodecahedral Garnet,
the third in octahedral Fluor-haloide ; the angles of the
first depend upon those of the first variety of the dodeca-
hedrons, §. 66. 130.

§. 72. OCTAHEPRAL TRIGONAL-ICOSITETRAHEDRONS.

The faces of the octahedral Trigonal-icositetra-
hedrons, Fig. 33., are isosceles triangles.

1. Their solid angles consist of either three or eight
faces, and are equiangular. The first, eight in number,
are monogrammic, and correspond to the centres of the
faces ; the second, six in number, are digrammic, and cor-
respond to the angles of the octahedron.

2. Those edges, which have the situation of those of the
octahedron, join the solid angles of eight faces with each
other ; the other edges unite two dissimilar solid angles.

3. The sections and axes are as in the octahedron.

4. There is only one variety known, of the following
dimensions :

a. b. A. B.

118° 4' 10". 30° 57' 55". 141° 3' 28". 152° 44' 2".
4. Examples of this form are found in octahedral Fluor-
haloide, hexahedral Lead^glance, &c.

§. 73. TETRAGONAL-ICOSITETRAHEDRONS.

The Tetragonal-icositetrahedrons, are contained
under equal and similar tetragonal faces.

1. This species of icositetrahedrons comprises two kinds,
the varieties of which are distinguished from each other by
the figures of their faces, and by several properties depend-
ing upon them, chiefly by the diversity of their edges, ac-
cording to which, they also receive their denominations.

2. The varieties of the first kind, contain only two dif-
ferent edges, and are termed digrammic ; whilst the second,

60 TERMINOLOGY. §. 74?.

or trigrammic Tetragonal-icositetrahedrons, possess three dif-
ferent kinds of edges.

§. 74. DIGRAMMIC TETRAGONAL-ICOSITETRAHE-
DRONS.

The digrammic Tetragonal-icositetrahedrons,
Fig. 34., are contained under tetragonal faces,
which can be divided by one of their diagonals,
in two isosceles triangles.

1. These icositetrahedrons possess three different kinds
of solid angles, one of which is formed by three, the others
by four faces : all of them are equiangular. The first are
monogrammic, eight in number, and correspond to the solid
angles of the hexahedron. Of the second, six are mono-
grammic, and correspond to the solid angles of the octahe-
dron ; the other twelve are digrammic, and correspond to
the centres of the faces of the dodecahedron, (§. 63.).

2. These forms possess two kinds of edges, the one
terminating in the solid angles of three faces, the other in
those which are produced by four equal edges.

3. The sections and axes are the same as in the hexahe-
dron, the octahedron, &c. The rhombohedral axes pass
through the solid angles of three faces, the pyramidal axes
through the monogrammic, and the prismatic axes through
the digrammic solid angles consisting of four faces.

4. There are two varieties known in nature, of the follow-
ing dimensions :

a. b. c. A. B.

1. 78° 27' 46". 82° 15' 3". 117° 2/8//. 131° 48'36". 146°26'33".

2. 84°15'39". 8l°25'37". 112°53'7". 144°54'11". 129° 31' 16".

5. Examples of the first variety of these forms we have
in hexahedral Kouphone-spar and dodecahedral Garnet;
of the second, in octahedral Fluor-haloide and dodecahedral
Corundum ; of both, in hexahedral Iron-pyrites.

§. 75. OF SIMPLE FORMS IN PARTICULAR. Gl

§. 75. TRIGRAMMIC TETRAGONAL-ICOSITETRAHE-
DRONS.

The trigrammic Tetragon al-icosi tetrahedrons,
Figs. 27. 28., are contained under tetragonal faces,
which cannot be divided in two isosceles triangles by
any of their diagonals.

1. The angles of these forms consist of either three or
four faces. The first are monogrammic, equiangular, eight
in number, and they are situated like the solid angles of
the hexahedron. Of the solid angles of four faces, six are
equiangular and digrammic, and they are distributed like
the solid angles of the octahedron ; the other twelve are un-
equiangular and trigrammic, and they have the situation of
the digrammic solid angles in the hexahedral pentagonal-
dodecahedron (§. 66. 1.).

2. Of the three different kinds of edges, the first termi-
nate in the solid angles consisting of three faces ; the first
and second in the digrammic, and the first, second, and
third, in the trigrammic solid angles, bounded by four faces.

3. The mutual inclination NOP of the longest or greatest
edges, in the digrammic solid angle, is the Characteristic Angle
D of the trigrammic tetragonal-icositetrahedron.

4. The sections and axes are the same as those of the
hexahedral pentagonal-dodecahedrons; the rhombohedral
axes pass through the solid angles of three faces, the pris-
matic axes through the digrammic solid angles of four faces.

5. There are three varieties of these forms, whose dimen-
sions are as follows :

a. b. c. d.

1- 106° 59' 7". 79° 53' 50". 116° 6' 13". 57° 0' 50".

2. 104° 38' 25". 84° 12' 32". 113° 21' 46". 57° 47' 17".

3. 96° 13' 37". 83° 46' 23". 113° 34' 41". 66° 25' 19".

A. B. C. D.

1. 148° 59' 50". 115° 22' 37". 141° 47' 12". 112° 37' 12".

2. 160° 32' 13". 118° 59' 9". 131° 4' 57". 118° 4' 10".

3. 154° 47' 28". 128° 14' 48". 131° 48' 37". 126° 52' 12".

6:2 TERMINOLOGY. §. 76.

6. All these varieties are met with in hexahedral Iron-
pyrites ; the first also in hexahedral Cobalt-pyrites.

§. 76, PENTAGONAT>ICOSITETRAHEDRONS.

The Pentagonal-icositetrahedrons, Figs. 29. 30.,
are contained under irregular pentagonal faces, all
the angles of which are different, but which possess
two pairs of equal sides.

1. These forms contain three kinds of solid angles, two
of which consist of three, and one of four faces. Eight of
those formed by three faces are equiangular and mono-
grammic ; these are situated like the solid angles of the
hexahedron : the other twenty-four are unequiangular and
trigrammie ; the situation of these is similar to that of the
trigrammic solid angles in the tetrahedral pentagonal-do-
decahedron (§, 67. !.)• The six so^ angles of four faces
are equiangular, monogrammie, and correspond to the solid
angles of the octahedron.

2. There are three different kinds of edges ; the first
terminate in those solid angles which are produced by the
concurrence of three equal edges ; the second terminate in
the solid angles of four faces ; and the third join those solid
angles with each other, which do not consist of equal plane
angles.

3. The sections are rhombohedral and pyramidal, as
also their corresponding axes. The pyramidal axes pass
through those solid angles which contain four, the rhom-
bohedral axes through those which contain three equal
plane angles. These forms possess no prismatic axes at all,
and agree with the tetrahedral pentagonal-dodecahedrons,
In this particular as well as in the occurring difference be-
tween right and left.

4. The dimensions of the three varieties of these forms
are as follows ;

§. 77. OF SIMPLE FORMS IN PARTICULAR. C3

a. b. c. d.

1. 77° 26' 40". 126° 18' 53". 116° 6' 13". 93° 49' 21".
& 80° 24' 22". 132° 54' 46". 113° 21' 46". 113° 43' 28".
3« 82° 14' 1". 115° 18' 18". 113° 34' 41". 123° 12' 38*.

e. A. B. C.

1. 126° 18' 53". 130° W 19". 141° 47' 12". 141° 47' 12".

2. 99° 35' 38". 135° 35' 43". 131° 4' 57". 145° 57' 8",

3. 105° 40' 22". 149° 37' 57". 131° 48' 37". 135° 35' 43".

5. This form has not yet been found in nature ; the
angles of the three varieties depend upon those of the te-
traconta-octahedrons, §. 77- 130.

§. 77. TETRACONTA-OCTAHEDR03JS.

The Tetraconta-octahedrons, Fig. 35., are con-
tained under forty-eight scalene triangles.

1. The solid angles of these forms are bounded by four,
six, or eight faces ; they are equiangular and digrammic.
Twelve consist of four faces, and are situated above the
centres of the faces of the dodecahedron ; eight consist of
six faces, and correspond to the solid angles ef the hexahe-
dron ; and the remaining six, which consist of eight faces,
are distributed like the solid angles of the octahedron.

2. Of the three different kinds of edges of these forms,
the first, being the longest, join the solid angles of six faces
with those of eight faces ; the second or intermediate join
the solid angles of eight faces with those of four faces ; and
the third, which are the shortest, unite the solid angles of
six faces with those of four faces.

3. The sections and axes are as in the hexahedron, In the
octahedron, &c. The rhombohedral axes pass through the
solid angles of six faces, the pyramidal axes through the
solid angles of eight faces, and the prismatic axes through
the solid angles of four faces.

4. There are three varieties of these forms, of the fol-
lowing dimensions ;

64 TERMINOLOGY. §. 78.

a. b. c.

1. 86° 5G' 25". 56° 15' 4". 3G3 48' 31".

2. 87° 34' 49". 53° 46' 42". 38° 38' 29".

3. 85° 50' 23". 54° 21' 34". 39° 48' 3".

A. B. C.

1. 158° 12' 48". 148° 59' 50". 158° 12' 48".

2. J 52° 20' 22". 1GO°32'13". 152° 20' 22".

3. 162° 14' 50". 154° 47' 28". 144° 2' 58".

5. The first of these varieties occurs in dodecahedral
Garnet; the third in octahedral Fluor-haloide ; the se-
cond depends upon the second variety of the icositetrahe-
drons, §. 7o. 133."

CONSIDERATION OF THE CONNEXION AMONG SIMPLE FORMS,
AND OF THE RELATIONS, UPON WHICH IT DEPENDS.

§. 78. OBSERVATIONS.

There exists a very remarkable connexion among
several simple forms, which depends not only upon
the kind, but also upon the relative dimensions of
these simple forms.

It is a matter of fact, sufficiently demonstrated by nu-
merous observations, that certain crystalline forms are pe-

* The preceding enumeration of the varieties of tessular
, forms, as occurring in nature, is by no means complete.
Several varieties of the digrammic tetragonal-icositetra-
hedron (§. 74.), of the tetraconta-octahedroii (§. 77-)» and of
other forms, have already been observed ; for instance, in
octahedral Fluor-haloide, in dodecahedral Garnet, in hexa-
hedral Iron-pyrites, &c. but not with a sufficient degree of
accuracy. It is to be expected, that our knowledge of these
forms will be considerably enlarged by a more accurate ex-
amination of nature.

§. 78. OF THE CONNEXION OF FORMS. 65

culiar to certain mineral species, whilst others are never
found in the same substances. Thus hexahedral Gold is
found in hexahedrons, but never in rhombohedrons ; rhom-
bohedral Lime-haloide in rhombohedrons, never in hexa-
hedrons.

Experience proves quite as generally that varieties of one
and the same accurately determined mineral species, may as-
sume several different forms of crystallization ; hexahedral
Gold, beside the form of the hexahedron, assumes also that
of the octahedron, of the dodecahedron, of the digrammic
tetragonal -icositetrahedron, &c. ; rhombohedral Lime-ha-
loide, besides rhombohedrons, exhibits also several isosceles
and scalene six-sided pyramids, and regular six-sided prisms;
and we may frequently observe, that even in one and the
same individual of such species, several of those simple
forms appear at the same time, or in connexion with each
other : thus, in hexahedral Gold, the hexahedron occurs in
one individual with the octahedron; in rhombohedral Lime-
haloide, rhombohedrons are found with pyramids, with
prisms, &c.

It is likewise demonstrated by experience, that two or
more simple forms, if they appear at the same time, in a
species or an individual, do really possess certain dimen-
sions or relations towards each other, and that other forms,
though of the same kind with the preceding, are exclud-
ed from such species, merely on account of their dimen-
sions. Thus the species of rhombohedral Lime-haloide does
not present indiscriminately the forms of any rhombohe-
dron, or of any six-sided pyramid whatever ; but we find
only such as possess certain dimensions, upon which the
symmetry of their combinations depends.

Natural History does not lead us to inquire into the fi-
nal cause of that remarkable fact, why the crystals of hexa-
hedral Gold should be hexahedrons, octahedrons, &c. ; and
why those of rhombohedral Lime-haloide should be rhom-
bohedrons and six-sided pyramids of certain dimensions.
Such questions, supposing even that they were capable of
being answered, are not within the province of Natural

VOL. I. E

66 TERMINOLOGY. §. 79-

History (§. 5.). But Natural History endeavours to de-
termine the relations under which crystalline forms of cer-
tain dimensions appear in the individuals of the same spe-
cies, or come into connexion with each other. These re-
searches not only form part of the peculiar object of Na-
tural History ; but this science derives the greatest advan-
tage from them in its farther developement.

§. 79. DERIVATION.

The method employed in Natural History for
determining the kind and the relations of crystal-
line forms, which occur in the individuals of the
same species, or come into connexion with each
other, is called the Derivation.

To derive one simple form from another, is to shew-
how, according to a certain general rule, it arises, or is
produced from it. The processes of derivation consist in
geometrical constructions, which are not gratuitously ima-
gined, but deduced from observation ; and their correct-
ness and applicability, though evident from their very ori-
gin, is thoroughly confirmed by the exactness with which
the phenomena in nature can be explained.

There are several of these rules or methods of proceed-
ing by which the derivation can be effected. Of these
different methods, those must be selected which will ap-
ply to the quality of the form from which the derivation
is to start, and which is termed the given form. The pro-
duct of derivation is called the derived form. The derived
and given forms are either of the same, or of different kinds.
The derived form is a simple form, like the given one ; or,
should this not be the case, it must be resolved into two or
more simple forms. The derived form having thus been
developed, the relations existing between this and the
given one are to be determined.

§. 80. 81, OF THE CONNEXION OF FORMS. 67

§. 80. FIRST PROCESS OF DERIVATION.

The first process of derivation requires tangent
planes (§. 36.) to be placed on certain edges of the
given form, and enlarged till they limit the space
either entirely, or at least as far as the number and
situation of the faces will allow.

If the edges to which the tangent planes are applied, be
equal or homologous, as, for instance, the terminal edges
of the rhonibohedrons, and of the isosceles four-sided pyra-
mids, or the acute and obtuse terminal edges of the scalene
six or eight-sided pyramids ; this process will yield a simple
form at once, which is the derived form itself.

If, on the contrary, the edges in which the tangent planes
are to be laid, be not homologous, as is the case in the
acute and the obtuse terminal edges of the scalene four-
sided pyramids ; this process will not give a simple form,
but a compound one, which is contained under faces
not homologous with each other. Compound forms of
this kind are not the derived forms themselves (§. 79-)*
though they either contain them, or at least may be em-
ployed for their ulterior derivation. They are considered
as Auxiliary or Intermediate Forms.

All intermediate forms belong to that given form, from
which they result by the above mentioned process.

§. 81. SECOND PROCESS.

The second process requires the axis of a form
contained under tetragonal faces, to be produced
on both sides, to an undetermined but equal length ;
straight lines to be drawn from the lateral angles
of the tetragonal faces towards the terminal points
of the lengthened axis, and planes to be laid on
every contiguous pair of them. The derived form
is contained under these planes,

68 TERMINOLOGY. §. 82. 83.'

This process is not limited to forms which, like the rhom-
bohedron, are originally contained under tetragonal faces ;
but it can be extended to such as are originally contained
under triangles, and is therefore applicable to pyramids of
every description. In this case, however, the given form
requires a certain preparation, the nature of which will be
explained in its proper place.

Forms produced in this way, if simple, are the derived
forms themselves; if compound, they are, like those in
§. 80., considered as intermediate or auxiliary forms, and
made use of accordingly.

Of intermediate forms in general, it may here be re-
marked, that, by enlarging their homologous faces, till the
rest disappear, they may be resolved, and by that means the
simple forms which they contain, may be extracted.

§. 82. THIRD PROCESS.

The third process requires planes to be laid on
the terminal edges of the given form, which may
likewise be an intermediate one (§. SO. 81.) ; their
number and inclination being such, that the inter-
sections of the faces from both apices produce a
plane figure, similar and parallel to the horizontal
projection of the given form. The derived form is
contained under these planes.

The number of faces contiguous to every terminal edge,
as employed in this process, is either one or two ; more
than two faces can never be applied to one terminal edge.
This process in some cases affords a determined solution of
a problem, which it would be impossible to obtain from a
process analogous to that of §. 80.

§. 83. FOURTH PROCESS.

The fourth process requires the consideration of
those differences which take place in the situation

§. 84. 85. OF THE CONNEXION OF FORMS. 69

of a Moveable Plane, tangent to the uppermost
point of a vertical rhombohedral axis in a form of
several axes.

The last process refers only to those forms which possess
several axes ; while the three methods of derivation de-
scribed above, are more particularly intended for such as
have only one axis. The fourth process produces only
simple forms.

§. 84. POSITION OF THE DERIVED FORMS.

By the application of these processes of deriva-
tion, the derived forms are obtained in such posi-
tions in respect to the given one, as will enable
them to produce symmetrical combinations, both
with the given form, and among each other.

In forms of several axes, this is the parallel position
(§. 43.). In those of only one axis, it must be determined
in particular, according to the quality of the forms con-
cerned. It is sufficiently demonstrated by all compound
forms occurring in the individuals of the mineral kingdom,
that the simple forms of which they consist, in every in-
stance are found in such positions as are assigned to them
by the derivation and by the connexion which it produces
between forms of a certain quality.

§. 85. SERIES.

If one of the processes described above yields
a derived form of the same kind as the given form,
the same process may be applied also to this
derived form ; and not only to this, but also to the
new product of the derivation, and so on. The as-
semblage of forms thus produced, and following
each other, is termed a Series.

70 TERMINOLOGY. §. 86.

Series may also be produced, although the derived form
be not of the same kind as the given form ; yet this does
not take place so immediately as under the circumstances
noticed.

These series form a peculiar feature, and are of the
greatest importance in the Method of Crystallography de-
veloped in this work.

A constant ratio exists between every two subsequent
members of those series. The general expression of this
ratio is the Law of the Series.

Upon the series themselves is founded the method of
Crystallographic Designation (§. 90.),

§. 86. LIMITS.

The limits of the series of those forms which pos-
sess one axis, are Prisms of infinite axes.

There is no reason why a series produced by derivation
(§. 85.), should stop at a member, as long as another ulte-
rior one can still be derived from it. This is always pos-
sible, as long as those dimensions, which are altered by
the derivation, remain finite. All members in which this
is the case, are termed finite members. If a member re-
ceives infinite dimensions, the derivation can no longer be
continued. The limits of derivation, and consequently
the limits of the series arising from it, are therefore at-
tained, if the dimensions of these forms become infinite.

The dimensions of forms which most conveniently may
be supposed to grow infinite, or infinitely small, are the
axes ; if these be infinite, the form becomes a prism ; on
the contrary, if they be infinitely small, it becomes a plane.
Prisms of infinite axes, and planes of infinite extent, are,
therefore, limits of all the series of those forms which
contain one axis.

Forms of infinite dimensions can never appear by them-
selves. Those which occur in nature, and consist of termi-
nal and lateral faces, are only segments, or parts of those
prisms of infinite axes. The lateral faces of the prism
represent the limit of the series on one side ; the termi-

§. 87. OF THE CONNEXION OF FORMS. 71

nal faces or the base, the limit of the series on the other.
Hence they are not simple, but compound forms ; and this
is the reason why they have not been enumerated among
the simple forms. The whole series is comprised within
the two limits. The particular mode in which the limits
of the series of different forms result, depends upon the
quality of those forms themselves, and will be explained in
particular in every series.

Forms of several axes cannot have limits of this descrip-
tion. However, if we suppose the different varieties of ho-
mogeneous simple forms of variable dimensions (§. 70. 71« 73.
77«) to constitute series ; the limits of these series will be
represented by those forms of many axes, whose dimen-
sions are constant (§. 58. 59. 63.).

§. 87. FUNDAMENTAL FORM.

That form, which serves as the base of the de-
rivation (§. 81.), is termed the Fundamental Form.

The idea of the Fundamental Form in the present me-
thod, is well defined, and perfectly determined. Hence
mere simple forms, or such as have hitherto been common-
ly called fundamental, primitive or primary forms, must
not be confounded with this idea.

Fundamental forms must possess the following proper-
ties. They must be,

1. Simple forms;

2. Forms not derivable from another fundamental form ;

3. Forms which do not possess infinite axes; and

4. Forms contained under the least possible number of

faces, provided the form itself be not objectionable
from other considerations.

According to these characters, the fundamental forms of
the mineral kingdom will be,

1. The Scalene Four-sided Pyramid,

2. The Isosceles Four-sided Pyramid,

3. The Rhombohedron, and

4. The Hexahedron.

TERMINOLOGY. §. 88.

1. DERIVATIONS FROM THE SCALENE FOUR-SIDED PYRAMID.

§. 88. DERIVATION OF MORE ACUTE AND MORE OB-
TUSE PYRAMIDS, OF SIMILAR BASES WITH THE
FUNDAMENTAL FOKM.

From every scalene four-sided pyramid may be
derived a more obtuse pyramid of the same kind,
possessing a similar base with the fundamental form.

First of all, one of the prismatic axes of the given py-
ramid is fixed upon as its principal axis (§. 41.) ; according
to this the pyramid itself is brought into its upright po-
sition (§. 42.). Apply, after the first process (§. 80.), tan-
gent planes to the terminal edges AB, AC, &c. of this form
ABCB'CX, Figs. 37., and enlarge them, till they intersect
each other from all sides. There will arise a form AFGIHX,
contained under eight isosceles triangles, which, by four
and four, are equal and similar to each other. If the form
be considered as a four-sided pyramid, its base FGIH has
the form of an oblong or rectangle. This is the intermediate
form (§. 80.). It will be proper to observe here, that these
intermediate forms are not mere geometrical conceptions,
but that they are very frequently found in nature, and
will be farther explained in §. 97«

Place now, after the third process (§. 82.), one plane
on each of the terminal edges AF, AG, &c. of this inter-
mediate form, the inclination of this plane to the faces of
the form, and to each other, being such as to enable them
to intersect each other, after the necessary enlargement, in
the plane of the base, and thus to produce a plane figure
QSra'C', similar and parallel to BCB'C', the base of the
fundamental form. These planes will contain the required
form, namely, a scalene four-sided pyramid.

This process may also be applied inversely, that is to
say, for any given scalene four-sided pyramid, we may find
the one from which it is derivable, according to the method

§. 89- OF THE CONNEXION OF FORMS. 73

described above. For this purpose inscribe that intermediate
form which belongs to the pyramid sought, in that which
is given. This is effected by bisecting each of the lateral
edges, and joining the points thus determined by straight
lines ; after which, lines must be drawn from the angles of
the oblong figure, which has thus been inscribed in the
base of the pyramid, to the terminal points of the axis of"
the fundamental form. A rhomb is now inscribed in the
rectangular base of the intermediate form ; the angles of
the rhomb coinciding with the centres of the sides of the
oblong. This rhomb will be similar and parallel to the
base of the fundamental form. But it likewise represents
the base of the derived form, which will be completed, if we
draw straight lines from the angles of this rhombic figure,
towards the terminal points of the axis of the fundamental
form, and lay planes on every two of those lines, which
are adjacent or contiguous to each other.

§. 89. BATIO BETWEEN THK DERIVED AND THE
FUNDAMENTAL FO11M.

The axes of two scalene four-sided pyramids, of
which the one is derived from the other, according
to §. 88., are towards each other in the ratio of
J: 1, if the derived pyramid is more obtuse; in
the ratio of 2 : 1, if the derived pyramid is more
acute than the given one. The horizontal projec-
tions of all these forms are supposed equal ; and
the axis of the fundamental pyramid =: I.

Let BCB'C', Fig. 37., represent the base of the fundamen-
tal form, which bisects the axis AX in M, the centre of
the pyramid ; GAF will be a plane tangent to the termi-
nal edge AC, HAF another plane tangent to the terminal
edge AB, and consequently AFGIHX the intermediate
form, whose base is the rectangle FGIH.

Circumscribe about this rectangle, a rhomb 10C1B'C'? sirai-

74 TERMINOLOGY. §. 90.

lar to the base of the fundamental form, and draw the
lines BA, CA, &c. These lines BA, CA, &c. will be
terminal "edges ; the planes BAC, CAB', &c. faces, and
BCB'C' the base of the derived pyramid, which is repre-
sented by ABCB'C'X.

The triangle BMC is equal to the triangle BFC = A
FCC = A BBF. Hence A BMC = 4. A BMC, and
BCB'C'" = 4. BCB'C'. Therefore BC = 2. BC, and BM
= 2. BM.

In the plane BAM, draw the line BA' parallel to ISA ;
the triangle BA'M will be similar to the triangle 30AM, and

BM:BM=MA:MA',
hence

MA' = £ MA.

If ABCB'C'X be now supposed the fundamental pyra-
mid, AFGIHX will be the inscribed auxiliary form, and
ABCB'C'X the more acute derived pyramid, to which the
auxiliary form belongs (§. 80.).

In the bases of both these pyramids, the triangle BMC
is = J A BMC; therefore BC = £ BC, and BM = ^ BM.

Lengthen now the axis MA to the point 3, and draw the
line B2t in the plane B9LM. This is the terminal edge of
the more acute derived pyramid, if its horizontal projection
be supposed equal to that of the fundamental form.

But on account of the similarity of the triangles BAM
and B&M, the following proportion takes place :

BM : BM = MA : M3,
and therefore,

Ma = 2. MA.

§. 90. SERIES OF SCALENE FOUR-SIDED PYRAMIDS,
WHOSE BASES ARE SIMILAR TO THE BASE OF THE
FUNDAMENTAL FORM.

If the derivation (§. 88.) be continued, a series
of scalene four-sided pyramids of similar bases will
arise, whose axes increase and decrease, like the

§. 90. OF THE CONNEXION OF FORMS. 75

powers of the number 2 ; the horizontal projections
always being supposed equal.

There exist certain constant ratios in the homologous di-
mensions of any two scalene four-sided pyramids, thus de-
rived from each other ; for the sake of an easier comparison,
they can be expressed by constant ratios between their axes,
if referred to one and the same horizontal projection.

Supposing the horizontal projection to be equal in all
the pyramids considered, designate the fundamental form
by A ; and by B, C, D ... the derived pyramids whose axes
are decreasing, by B', C',[D' ... those whose axes are increas-
ing : a fragment of the series, containing such members as
are nearest to the fundamental form, will be represented by

...D, C, B, A, B', C', D'...
Let the axis of A be = a, the axis of B will be = |. a,
that of C = |. A. a = \. a, &c., that of B' = 2. a, that of
C' = 2. 2. a = 4. a, &c. ; hence the fragment of the series
given above, as expressed by the axes of its members :

... |. a, £. a, |. a, a, 2. a, 4. a, 8. a ...
and their ratio to each other =

... | : i : £ : 1 : 2 : 4 : 8 ...
that is to say =

... 2— 3 :2— 2 : 2— l : 2° : 21 : 22 : 23 ...
The general member of this series, or the expression of
the axis of an indeterminate nth member, will be = 2«. a,
where a is the axis of the fundamental form, and 2« the
Law of Progression. The number 2 is the Fundamental
Number of the series.

Upon laws of this kind is founded the method of Crys-
tallograpJiic Designation^ which is comprised under the fol-
lowing rules. The fundamental form is expressed by any
arbitrary letter. The same letter serves also for denoting
such derived forms as are of the same quality as the funda-
mental one, as is the case in the present instance, where the
derived and the fundamental scalene four-sided pyramids
possess similar bases ; but if the derived forms are of an-
other kind, the letter is transferred to these derived mem*

76 TERMINOLOGY. §. 90.

bers, along with the modification or alterations thereby ren-
dered necessary. Under both circumstances, the place of
the member in the series to which it belongs, is expressed
by the appropriate exponents of the fundamental form, to-
gether with + or — , their positive or negative signs. In the
fundamental form, the exponent is = 0, and therefore not
expressly indicated.

Let P designate the fundamental form of the above-
mentioned series ; the fragment of that series will be
... P — 3, P — 2, P _ 1, P, P + 1, P + 2, P + 3 ...

An indeterminate nth member receives the designa-
tion P + n, the (n + l)th, P + n + 1, where the number n
may be either positive or negative.

Since the ratio of the diagonals of the base b : c (§. 53.)
is known from the dimensions of P, and remains the same
in all the members ; the dimensions of any required mem-
ber can be found, if the ratio of its axis to that of the fun-
damental form be known, and this ratio is indicated by
the designation. One of the advantages of this designation
consists, therefore, in affording a distinct idea of the forms
themselves, speaking as it were to the eye ; at the same
time, it expresses the connexion existing among them, in
as far as they are derived from each other ; and, moreover,
it contains every thing required for calculating the dimen-
sions of any member, if those of the fundamental form, or
of any other member, be known.

The expressions of the cosines of the edges, as given for
the scalene four-sided pyramid (§. 53.), refer to those of the
pyramid P. If, instead of a2, 22n a2 is substituted in these
formulae, they are changed into the following expressions,
which refer to the pyramid P + n,

22»a2 b2 — (22na2 + b2) c2
cos. y = ^ '- — ;

22.1 a2 1,2 + (2*1 a2 + fc2) C2

cos x = 22na2 c2 — (22" a2 + c2) b2 .
2*n a2 C2 + (2a» a2 + c2) b2 '
cos z = b2 c2 — (b2 4- c2) 2«»_a8
""• b2 c2 + (b2 + c2) 22n~a2 *
In order to find the cosine of any of these angles for a

§. 91. OF THE CONNEXION OF FORMS. 77

certain determined member of the series, of which P is the
fundamental form, there is nothing required but to substi-
tute for n that number which denotes the peculiar place of
the member in the series. These general formulae are very
useful in all crystallographic calculations.

§. 91. LIMITS OF THE SERIES OF SCALENE FOUR-
SIDED PYRAMIDS.

The limits of the series, §. 90., are on one side an
oblique-angular four- sided prism, whose transverse
section is equal and similar to the base of the fun-
damental form, and its axis infinite ; on the other
side a plane, perpendicular to that axis.

The series (§. 90.) may be continued on both sides, as
long as the members obtained, or as long as their axes, are
finite quantities ; and there can be no reason why we should
consider o»e of these pyramids as the last, because the de-
rivation always will produce new members of the series.
But when the axis becomes infinite, no more new members
are produced ; and in this case the series breaks off, or ar-
rives at its limits (§. 86.). These limits are therefore sca-
lene four-sided pyramids of known bases and infinite axes.

But suppose now the axis to decrease, the terminal edges
will approach to the parallelism with the diagonals of the
base, the inclination of the faces in these lines, or the ter-
minal edges, to 180°, and the magnitude of the lateral edges
to 0. The final term of these approximations is obtained,
when the axis becomes infinitely small ; in this case the
pyramid is transformed into a plane figure, similar to the
base.

The prism of an infinite axis is infinitely distant from P in
the series of pyramids; or, in other words, there is an infinite
number of members of the series between the fundamental
form and that prism. The number n in that form, is there-
fore = co . In the same manner, n is = — eo for the prism

78 TERMINOLOGY. §. 92.

of an infinitely small axis. The crystallographic signs for
these limits are P 4- co and P — oo , and the series itself is
thus represented between its limits :

P— OS ... P + 11 ... P + CO.

The algebraic expressions in the preceding §. yield the
plane angles of the base of P, which is equal to the trans-
verse section of the prism of infinite axis, if n be supposed
SB + oo, as follows :

b2 c2

cos. y = " S. .

b2 + c2'

— b2

.

c2 + b2

The value of cos. z = — 1, indicates that one face of the
pyramid contiguous to the upper apex, and one contiguous
to the lower apex, coincide in a single plane parallel to the
axis, by which the lateral edge z becomes = 1 80°.

Several members of this series, together with their in-
termediate forms and limits, have been observed in nature.
Thus, prismatic Topaz, presents three consecutive mem-
bers, the intermediate form belonging to the most acute
of them, and both the limits of the series. Two consecu-
tive members, with their intermediate forms and limits,
are known in several species, as in prismatic Lime-haloide,
prismatic Lead-baryte, diprismatic Copper-glance, and
others.

§. 92. DERIVATION OF SCALENE FOUR-SIDED PYRA-
MIDS OF DISSIMILAR TRANSVERSE SECTIONS.

The members of the series of §. 90. serve as a
foundation to several other derivations. From every
one of them, several Pairs of scalene four-sided Py-
ramids may be derived, the bases of which are dissi-
milar to that of the fundamental form, and partly
also amongst themselves.

The derivation is effected by the second process (§. 81.);

§. 92. OF THE CONNEXION OF FORMS. 79

but this form being contained under triangular faces, must
undergo a preliminary operation, before the process can
be applied.

Let AX, Fig. 40., be the axis, BCB'C' the base of the
fundamental form, of which the faces BAG', C'AB', &c. are
contiguous to the upper, and BXC', CXB', &c. to the lower
apex of the pyramid. Enlarge now the planes of these
faces upwards and downwards, beyond the edges BC', C'B',
&c. ; and in these enlargements describe the triangles B A'C',
BX'C', &c. and C'A"B', COfB', &c. equal and similar to
the faces of the fundamental form. This process deter-
mines the situation of the points A', A", &c. X', X", &c.
which, being joined by straight lines, will produce rectan-
gular figures, similar and parallel to the base of the inter-
mediate form (§. 90.). These rectangular figures are per-
pendicular to the axis of the fundamental form, which they
intersect in the points A and X. This mode of transform-
ing triangular planes in such as are rhomboidal, is the pre-
paration of forms mentioned above (§. 81.).

After this preparation, let the axis of the fundamental
form be produced on both sides to an indefinite but equal
length, so as to have A& = X£ or Ma = M£ ; and
draw straight lines from the points A, A", &c. of the lower
rectangle towards 21, which is the upper point, from the
points X', X", &c. of the upper rectangle towards £, which
is the lower terminal point of the lengthened axis, and
from the angles B, C, B', C', of the base of the fundamen-
tal form, towards both these extremities. If planes be now
laid on every contiguous pair of these lines, those faces
which are inclined towards the upper apex, will intersect
those which are inclined towards the lower apex, in the
lines BS, SC', C'S', &c., and thus produce a form con-
tained under sixteen scalene triangles.

The triangles BM9L and C'Ma are rectangular in M,
and the line Ma is common to both. But BM is either
greater or less than C'M ; therefore, also, B& will be
greater or less than C'JJ. Hence the two faces B2S
and C'3S of the derived form, contiguous to the edge 3S,

80 TERMINOLOGY. §. 92.

are not homologous with each other ; and the form conse-
quently is not a simple one.

It is the intermediate or auxiliary form, mentioned in
§.81. This compound form can be resolved, or the simple
forms, contained in it, can be extracted, as follows : — Produce
first the lines C'S, and CS", to their intersection in OB ;
C'S' and CS'" to their intersection in 113', and draw OB&
and OB'a : MC', C'BOB', &c. will be faces, BCB'C' the
base of the pyramid aBCB'C% which is one of those
sought for. On the other hand, produce the lines BS" and
B'S"' to their intersection in C ; BS and B'S' to their intersec-
tion in C', and draw CSJ and C'SL : BEC', C'SB', &c. will
be faces, BC'B'C the base of aBC'B'C£, &c. which is the
other pyramid contained in the compound form. The
transverse sections of this pair of derived pyramids are
dissimilar, or differ from each other, as well as from that of
the fundamental form.

The former of these two pyramids has the same short
diagonal CC', the latter the same long diagonal BB', as the
fundamental form. The latter is therefore said to apper-
tain or to refer to the long diagonal, while the other is said
to appertain or to refer to the short diagonal of the funda-
mental form.

The axis 2l£, common to these pyramids, may be con-
sidered as being = m. AX, a product of the axis AX of
the fundamental form, and a certain number m, which is
called the Number of Derivation. This number must be
positive, and greater than 1, either whole or fractionary.
The values of this number most commonly, though not
exclusively, occurring in nature, are 3, 4, and 5. The crys-
tallographic signs of the pyramids thus obtained, are com-
posed of the sign of that fundamental or derived member
of the series (§. 90.), upon which they depend, which is in-
cluded in a parenthesis, and of the number of derivation m,
added to it in the form of an exponent. The signs o and -,
placed above the letter referring to the fundamental form,
denote the diagonal to which the derived pyramids belong.
The first indicates the long, the second the short diagonal
of the fundamental form. Thus,

§. 93. OF THE CONNEXION OF FORMS. 81

(P + n)«> and (P + n)™

are the crystallographic signs for the derived pair of pyra-
mids.

The axis common to both these forms is = 2n. m. a; 2". a
being the axis of P + n, and a the axis of P.

§. 93. THE RATIO OF THE DIAGONALS OF THE
BASES IS DEPENDENT ON m. . ^

If m be supposed equal, the bases of all (? + n)m,
and, on the other hand, the bases of all (P -f n)m,
are equal and similar to each other.

It is evident from the preceding paragraph, that the
figures of the bases, or their dimensions, are determined
by the situation of the points S, S', S", S'", or, which is the
same, by the length of the lines MS, MS', &c. Draw the
line AA' which bisects BC', the lateral edge of the fun-
damental form in H, and the lines HM and A'X, perpen-
dicular to the axis, it will follow that

A'X = 2. HM = V (b* + c2).

From the similarity of the triangles 2CSM, and aA'X,
we obtain

aX : XA' = 3M : MS ; or
(m + 1) a : V (b2 + c2) = m. a : MS.
Therefore

MS=_?1_ VC>2 + c2>
m 4- 1

From this expression it appears, that for a given ratio
of b and c in the base of P, the quantity of the axis a en-
ters for nothing in the determination of the bases of the
derived forms, and consequently, that the angles of these
depend only upon the number m.

If, according to this process, pyramids of dissimilar bases
are derived from several scalene four-sided pyramids, ac-
cording to a determined m ; the bases of all these derived
pyramids will be equal and similar to each other, in as

vor.. r. F

82 TERMINOLOGY. §. 94?.

far as they belong to one and the same diagonal ; because
the axes of the fundamental pyramids have no influence
upon the dimensions of the bases. It will be exactly the
same, if the axes are in the ratio of the powers of the
number 2, that is to say, if the fundamental pyramids are
members of one series; as, for instance, P, P + 1,
P + 2, &c.

§. 94. EATIO OF THE DERIVED AND THE FUNDA-
MENTAL FORM.

If in the pyramid P, the ratio of the axis, the
longer, and the shorter diagonal is expressed by

a : b : c,
or in P -f n, by

2n. a : b : c;

the ratio of the analogous lines in the same succes-
sion will be,

for (?)m = m. a : b : m. c, or
for(f + n)m=2n.m.a : b : m. c;
for (P)m = m. a : m. b : c, or
for(P-f n)m = 2n.m.a :m.b : c.
Since, Fig. 40., in the ratio of

3M : MB : MC'

SIM and MC' are known quantities, being expressed by a,
c and m (for 3M is = m. a, or = 2n. m. a, and MC' = c) ;
the only thing still to be effected, is to express MB in the
same manner, or only by b and m.

Draw the line SN parallel to BM ; the triangles SC^N,
3BC'B', will be similar to each other, and
C'NcSN^C'B'rBB',

= C'B' : MB -f MB'.
But

C'N = C'B' — NB' = CB' — MS

m -f 1

§. 94). OF THE CONNEXION OF FORMS.

c=; and

m 4- 1
SN = MB' = b.
Therefore

— ^W(b2 + c2):b = V(b2 +c'):(m+ 1) b ;
m-H 1

and

(m + 1) b = MB 4- MB' = MB + b,
from which follows

MB = m. b.

For (P -j- n) m, therefore, will follow that
aM : MB : MC' = 2". m. a : m. b : c, from which, by
the mere permutation of the diagonals, the ratio of the
analogous lines of (P 4- n)m is found to be
= 2n. m. a : b : m. c.

Since

AM : BM = aM : BM ;
AM : CM = £M ; C'M ;

the triangle AMB is similar to aMB, and AMC' to aM£',
and aB parallel to AB, aC' parallel to AC'. If therefore
from any member of the series §. 90., according to whatever
m, a pyramid of dissimilar base with the fundamental form
be derived ; the terminal edges contiguous to similarly situ-
ated, although unequal diagonals, b and m. b or c and m. c.
of the two pyramids, will always be parallel to each other.

The number m may be so great, that m. c becomes great-
er than b. Nevertheless m. c remains the line corres-
ponding to the diagonal c, which is here supposed to be the
short one. The correspondence between two diagonals
must not therefore be judged of according to their absolute
length, but according to their situation. This will require
some attention, in order to avoid being confounded by the
apparently different position of such pyramids.

If the ratios obtained just now between the axis and the
diagonals of (? + n)m, and (P + n)m be substituted in the
general formulae for the edges of the scalene four-sided
pyramid (§. 51.); the result will be other formulae, similar
to those in §. 90., and as general ; and they will refer to sca«

B4 TERMINOLOGY. §. 95.

lene four-sided pyramids, derived according to the above
mentioned process.

§. 95. SERIES OF DERIVED PYRAMIDS OF A DIS-
SIMILAR TRANSVERSE SECTION, WITH THAT OF P.
OTHER METHOD OF DERIVING PYRAMIDS OF THE
SAME KIND.

The pairs of scalene four-sided pyramids, deriv-
ed after one and the same m, from the members of
the series §. 90. form two series, which proceed ac-
cording to the law of the series §. 90., and are
similarly limited.

The same method which from P produces (P)m and (P)'",
if applied to P + n, yields (P + n)m and (P 4- n)m.
The axis of (P)m is therefore to that of (P 4- n)m in the ra-
tio of the axis of P to the axis of P + n, or in that of 1 : 2".
Hence 2 is the fundamental number, 2^ the law of pro-
gression of the series.

If the positive and negative value of n becomes infinite,
(P + n)m and (£ + n)m are changed into (P + cc)m,
(P — ^c)m and (P + cc)in, (P — co )™. According to §. 91.
these farms are oblique-angular four-sided prisms, whose
transverse sections are equal and similar to the bases of
(P + n)m and (P + n)ra. Their plane angles are obtained by
the algebraic expressions in the preceding paragraph, if n is
supposed = 03 . The signs (P — ec )ra and (P — co )m re-
fer to the face perpendicular to the axis, already expressed ;
which face, however, more generally is designated by
P —CD, the S'gn obtained in §. 91. The complete designa-
tion of the two series between their limits, is therefore
P _ co ... (P + n)m ... (P + 03 )m ;
P — co ... (P + n)m ... (P + co )m.

There exists, however, still another method of deriving
pyramids of dissimilar bases from the fundamental form ;
and although this method does not produce any new forms,

§. 95. OF THE CONNEXION OF Foil MS. 85

yet it is very well calculated to shew the agreement be-
tween the forms derivable from the scalene, and those de-
rivable from the isosceles four-sided pyramid. This me-
thod also tends to preserve the sameness of the value of m
in both, at least in respect to those numbers of derivation,
which are most commonly met with in nature.

This method of derivation consists in applying the process
§. 92., not to the pyramid P itself, but to the interme-
diate form which belongs to that pyramid. It is exactly
the same as that by which the scalene four-sided pyramids
of dissimilar bases are obtained from the fundamental py-
ramids themselves, and therefore requires no particular
description.

The first result is a compound form, as obtained above,
which, by a further resolution, yields a pair of scalene
four-sided pyramids, one of which refers to the long, the
other to the short diagonal of the fundamental form ? al-
though in the derivation, none of these diagonals remain
unchanged. The correspondence of these pyramids to the
diagonals of the fundamental form, is determined as in §. 92.
Their crystallographic designation, in as far as they are
obtained by the application of the last mentioned process,
is (Pr + n)m and (Pr + n) m.

The ratio of the diagonals of the bases is entirely de-
pendent upon m. For, considering one and the same m,
aH the bases of (Pr + n)ra on one side, and all the bases of
(Pr + n)m on the other, are equal and similar to each other,
as may easily be deduced from §. 93. The two lines MS,
MS", Fig. 39., are in the same ratio as the diagonals c and b.
From the consideration of the figure it appears, that

TVTCJ 2 m

Mo = . c.

m+ 1

MS"- 2m .b.
m + 1

The co-efficients of the three perpendicular lines in this
derivation are different from those obtained in §. 94. If the
axis, the longer, and the shorter diagonal of P + n be in the
ratio of

86 TERMINOLOGY. §. 95.

2". a : b : c,

the ratio of the analogous lines in the same succession
will be

for (Pr + n)m = m + 1. 2n. a : b : m_±l. c :

2 m — 1

for (Pr + n> = m"t_L 2". a :^±J. b : c.

2 m— 1

Let S'B'S, Fig- 39., be half of the base of the derived
pyramid (Pr + n)m ; in the ratio of the three lines
3M : MB' : MS

9LM and MS are already Imown, that is to say, expressed
by a, c, and m ; and the only expression still wanting, is
that of MB' by means of a function of m and b.

The triangle CIS is similar to the triangle MB'S;
therefore

CS : C'I = MS : MB'.
But we hare

C'S = MS — MC = — ?_m_. c — c = m_TL_1. c.

m + 1 m+1

Therefore

2m ».

: b = -. e

m -f 1 m + 1 m — 1

and

Since this is the required expression, it follows that
: M1B' : MS = 2" m. a : 2 m .b : 2 m -

m — 1 m+1

or if the co-efficient of that diagonal to which the pyramid
belongs, is supposed = 1 , the same ratio will be expressed

by

= ?±1. 2-a: ^Ltib : c.
2 m— 1

By exchanging the diagonals b and c, the ratio of the
three lines of (Pr 4- n)m will be obtained

= m+1- 2- a : b : m + *- c.
2 m — 1

The co-efficients of the axis and the two diagonals, as de-
veloped here, if substituted in the expressions for the co-

§, 95. OF THE CONNEXION OF FORMS. 87

sines of the edges of a scalene four-sided pyramid (§. 53.),
will produce similar expressions for the edges of the de-
rived pyramids, in as far as the process of derivation has
been applied to the intermediate form.

The pairs of scalene four-sided pyramids thus produced,
according to an equal m, from all the intermediate forms,
which belong to the members of the series §. 90., will them-
selves likewise form two series, which proceed according
to the general law of the former, and are limited by planes
perpendicular and parallel to the axis. The complete de-
signation of the two series between their limits, is

P — co ... (Pr + n)m ... (Pr + co)™;

P — co ... (Pr + n)ra ... (Pr + co)<".

It has already been stated, that by these two different
modes of derivation, we obtain exactly the same forms.
This may be demonstrated, by proving how one form, whose
derivation from P is expressed by the sign (P + n)m, may
likewise be derived from the intermediate form Pr -f n',
under the sign (Pr + n')m/, or that (P + n)m may be
= (Pr + ri')m/.

The ratio of the axis and the diagonals, that is to say, of
the three lines perpendicular to each other, is in

(P 4- n)m = m. 2n. a : m. b : c ; in

!^±i. 2-.a: "L+2. b : c.
2 m' — 1

If we suppose the co-efficients of b in the two pyramids

to be equal, we obtain m' = ^"t-l , which being sub.
m — 1

stituted in the ratios for (Pr + n')m/ gives

= m . 2»'. a : m. b : c.
m—1

The equality of the three lines supposes therefore also

-J!L_. 2«' to be = m.2n; and consequently 2*':= (m—1) 2".
m — l

Any scalene four-sided pyramid, which belongs to P -f n,
under the sign of (P 4- n)m, may therefore likewise be con.
sidered as deriving from the intermediate form of P + n,

88 TERMINOLOGY. §. 95.

viz. from Pr + n under the sign (in — 1) (Pr 4- n) m — '.
It. depends here upon the value of m, whether the deriva-
tion proceeds from an intermediate form belonging to the
principal series, or to a subordinate one (§. 96.).

Let m be = 2, and the secondary pyramid therefore
= (P 4- n)2 ; we have

(m — 1) (Pr 4- n)^T == (Pr 4- n)3 ;
the ratio of the perpendicular lines = 2. 2* . a : 2. b : c,
exactly as in (P + n)2. Let m be = |, it will follow, that

(m — 1) (Pr 4- n) m— * = 4 (Pr 4- n)5 = (Pr 4- n — I)5,
and the ratio of the lines given above = 3 . 2n — *. a : f . b
: c = f . 2n. a : |. b : c, as in (P 4- n)*.
In both these examples, Pr 4- n belongs to the principal

scries. But suppose m = 4 ; (m — 1) (Pr 4- n) ™ — *

will be = 3 (Pr 4- n)3, and the ratio of the three lines will be
= |. 2n. a : 4. b : c, of which the first member still must
be multiplied by 3 to make it equal to that of (P 4- n)4, as
the crystallographic sign requires. Here Pr 4- n belongs
to that subordinate series, whose co-efficient is the num-
ber 3.

In the two first of these examples, it appears that one
pyramid may be transformed into another, whose number
of derivation is greater ; in order to obtain forms in every
respect more analogous to the rest of those contained within
the compass of forms derived from the scalene four-sided
pyramid. The last example shews how a pyramid de-
rived from a member of a subordinate series, may be trans-
formed into another which belongs to a member of the
principal one. These two kinds of permutation sufficiently
account for the utility in the employment of the double
mode of deriving and of designating forms, which, for their
absolute dimensions, might be considered as being identical.

The values of m most commonly found in nature are 3,
4 and 5, particularly the first of them ; and there is scarcely
a species to be found presenting forms in connexion with

§. DC. OF THE CONNEXION OF FORMS. 89

the scalene four-sided pyramid, which does not afford ex-
amples of their occurrence. Prismatic and prismatoidal
Hal-baryte, prismatic Topaz, prismatic Chrysolite, may be
quoted as examples. Besides these, (P + n)G and (P + n)8
occur in prismatic Hal-baryte, (Pr -f n)7 in prismatoidal
Antimony-glance, (P + n)'J in prismatoidal Manganese-

§. 96. SUBORDINATE SERIES.

There exist several series of forms, homogeneous
and of similar bases with that of §. 90. and belonging
to it, in reference to which the latter is termed the
Principal or Fundamental Series, while the others
are said to be Subordinate.

A Subordinate Series is a succession of homogeneous
forms, whose bases are equal and similar to those of the
members of the principal series, but possessing axes,
which, on account of their relative magnitude, are ex-
cluded from the principal series : these members, however,
may form another series among themselves, which follows
the law of progression of the principal one.

The members of the subordinate series may be derived
from those of the principal series, either directly, or by the
interposition of certain other forms, which are produced
from that series. In the present case, the first of these
methods being more simple, will find its application.

Let AX, Fig. 38., be the axis, BCB'C' the base of the
fundamental form, and the points A', A", &c., X7, X", &c.,
be determined, as has been shewn in the preceding deriva-
tions.

Through those points lay the two rhombs FGIH and
F'GTH', similar and parallel to the base C'B'CB : produce
the axis, so as to have a£ = m. AX; draw the lines
F3, G21, &c. F% G% &c. and lay planes into these in
such situations that they include a space by themselves.
This space will have the form of a scalene four-sided pyra.

90 TERMINOLOGY. §. 96.

mid, the base of which is similar to that of P. The same
result is obtained by applying at once the third method of
derivation (§. 82.) to the intermediate form in §. 92. This
would require to lay planes on the edges formed by the
intersection of the faces of (£ + n)m with those of (P + n) m,
the inclination of these planes being such, as to make their
common intersection a rhoHib, similar and parallel to the
base of the fundamental form.

It is now wanted to determine the ratio of the axis of
the derived pyramid to that of P, the horizontal projections
of the two pyramids being supposed equal.

For this purpose draw the lines : AA', bisecting the edge
BC' in K ; MK in the plane of the base, and XA' in the
plane of the lower rhomb, parallel to MK. From the si-
milarity of the triangles AMK, AX A' follows

XA' = 2. MK.

Draw now the line A'a, and another K&' parallel to it ;
and M9L' will be half the axis of the derived pyramid, its
horizontal projection being reduced to BCB'C'. But from
the similarity of the triangles EA'X and 2TKM follows

XA' : MK = Xa : Ma'
or, since

aX = &M + MX = (m + 1) a :

2 : 1 = (m + 1) a :
and consequently,

a'M = "^±1 a.
2

The number — __ is termed the co-efficient of the sub-
2

ordinate series, and prefixed in the crystallographic sign of
one of its members, to the sign of the member of the
principal series, from which the derivation started, so that

m~*~ P 4- n is the designation of an indeterminate mem-

ber of the subordinate series.

If m 4- 1 becomes a power of the number 2, the deriva-
tion yields a member of the principal series itself; and if

§. 97. OF THE CONNEXION OF FO11MS. 91

m+ is a power of the number 2 greater than 1, n is in-

creased for the exponent of this power. Any other m pro-
duces one member of a subordinate series from every mem-
ber of the principal one. In this case is divided by

that power of the number 2, which, in the series of these
powers, differs least from the mentioned number ; n is in-
creased for the exponent of that power, or diminished if
that power be negative, and the quotient thus obtained is
now considered as the co-efficient of the member of the
subordinate series.

The expressions for the cosine of the edges referring to
those members of the subordinate series, are developed, as
has been pointed out before in several similar occasions.

The limits of the subordinate series evidently coincide
with those of the principal series.

From the value of m = 3, = 4 and = 5, the co-efficients
of the subordinate series are found = f and = f . These
and their inverse | and * have already been found in na-
ture ; |, for instance, in prismatic Hal-baryte, £ in pris-
matic Lime-haloide, f and £ in prismatic Sulphur.

§. 97. HORIZONTAL PRISMS.

To every scalene four- sided pyramid, derived
from P, as well as to P kself, belong two Horizon-
tal Prisms, one of which refers to the long, the
other to the short diagonal of the base of the fun-
damental form.

In any scalene four-sided pyramid, we may suppose one
of the diagonals of the base to be increased continually,
while the other remains unchanged. The value of the ter-
minal edges changes with the increase of the diagonal. The
edge which is contiguous to the unchanged diagonal ap-
proaches to 180°. That contiguous to the increasing one
approaches to equality with the angle of the principal

92 TERMINOLOGY. §. 97-

section through the axis and the unchanged diagonal, and
these limits are attained when the increasing diagonal be-
comes infinite. The pyramid is thus transformed into a
prism, the axis of which is the infinite diagonal : its situa-
tion is horizontal, and on this account the whole form is
termed a Horizontal Prism. Since each of the diagonals
may thus be supposed to increase till it becomes infinite,
there will be two horizontal prisms belonging to every sca-
lene four-sided pyramid, and each of these prisms is refer-
red to that diagonal, which remains unchanged, while the
other increases to infinity.

This mode of considering the matter will suffice for
giving a general idea of horizontal prisms. But it has no
connexion with the relations of forms developed in the
preceding paragraphs, where there exists nowhere an ab-
solute increment of a diagonal, this being always a conse-
quence of a simultaneous increment of the axis (§. 93. 95.).
In the principal series, whose members also possess their
appropriate horizontal prisms, the diagonals do not change
at all, while the axes may be increased to infinity.

There are, however, two methods of obtaining horizon-
tal prisms, in connexion with other forms : either the inter-
mediate form (§. 90.) is resolved by enlarging its homolo-
gous faces, or tangent planes are laid, not on all, but only
on the homologous, terminal edges of the given scalene
four-sided pyramid. The result is the same in both pro-
cesses.

The designation of horizontal prisms is in general
Pr + n ; it is Pr + n, if they belong to the longer, it is
Pr -f- n, if they belong to the shorter diagonal of P. If
the faces of Pr + n and those of Pr + n appear in combina-
tion with each other, and produce the intermediate form,
their relative breadth is in the ratio of those diagonals
to which their axes are parallel. This intermediate form,
in as far as it is a compound form, receives the com-
pound sign Pr + n. Pr + n; but in as far as it is em-
ployed in the derivation of other forms, as in §. 95., the
sign Pr -f n is applied to it, because the reference to
the diagonals is only taken into consideration afterwards.

§. 98. OF THE CONNEXION OF FORMS. 93

§. 98. SERIES OF HORIZONTAL PRISMS, AND THEIR
LIMITS. INCLINATION OF THE AXIS.

Every series of scalene four-sided pyramids has
two concomitant series of horizontal prisms. The
limits of these series are planes, which are perpen-
dicular to those diagonals to which they belong,
if + n becomes infinite, and perpendicular to the
axis of the fundamental form, if — n becomes in-
finite.

The designation of an indeterminate member in each of

these series is m_±_! pr + n an(j m + 1 pr _j_ n> wnich, if

2 2

tn = 1, signifies a member of the principal series. From
every other value of m, provided m + 1 be not a power of
the number 2, a member of a subordinate series is obtain-
ed. If m + 1 be a power of the number 2 greater than 1,
yet the co-efficient nevertheless will remain = 1, as in
every other member of the principal series ; but the n in
its crystallographic sign is augmented for the exponent of
that power, just as has been mentioned above, in respect
to members of subordinate series.

There is no particular designation required for such hori-
zontal prisms as belong to pyramids of dissimilar bases,
§. 92. and §. 95., because their principal sections coincide
with those of the pyramids, already expressed by the above
mentioned signs. This may be proved, for some of them,
by considering in general the ratio between the axis and the
two diagonals of pyramids, derived according to an undeter-
mined m ; and for others, by taking the determined value
of m, as obtained from observation.

A horizontal prism is determined by the cosine of that
terminal edge, which is contiguous or parallel to the infinite
diagonal, or by that angle of the principal section of the
pyramid, which lies in the terminal point of the axis.

The values of these cosines are obtained, if in the gene-

94 TERMINOLOGY. §. 98.

ral expressions of these quantities for ™ P 4- n, one

of the diagonals after the other is supposed infinite. In the
horizontal prisms belonging to P, the values of the cosines
are as follows :

« a2_b2

for Pr, cos. y =

for Pr,

As to the limits of the horizontal prisms, it is evident,
that in the same proportion in which the axis of the pyra-
mid Si-. P + n increases, the angle of the horizontal

prism at the axis must diminish ; and that it must entirely
disappear, when the axis becomes infinite. The supple-
ment of this evanescent angle is = 180°, and the hori-
zontal prism therefore is transformed in two unlimited pa-
rallel planes, perpendicular to those diagonals to which they
belong. If on the other side the axis decreases, the same
angle becomes greater and greater, and at last = 180°, if
the axis is infinitely small. The supplement of this angle
is = 0 ; the faces of the horizontal prism contiguous to the
opposite ends of the axis, coincide with the plane of the
basis ; and they appear as faces perpendicular to the axis.

This is the result of m +1 Pr + co and m+1 Pr + co,

and of m.±! Pr - co and m-±I Pr - oo.
2 2

The series of horizontal prisms between their limits, are
expressed by

P _ co ... ELL! Pr + n ... P r + n,

P — co ... m_±i Pr + n ... Pr + eo.

The face perpendicular to the axis has received its sign
as the limit of the principal series ; and in the limits for
n = + co, it is unnecessary to attend to the co-efficients of
the different series.

§. 98. OF THE CONNEXION OF FORMS. 95

The method of derivation, applied to the scalene four-
sided pyramid, is so general, that it will remain unaltered,
and yield similar forms and relations, even though the axis
of the fundamental form should be inclined at an angle to
the plane of the base.

This Inclination of the Axis may take place, either in the
plane of only one of the diagonals of the rhombic base,
as in Fig. 41., or in the planes of both diagonals, as in
Fig. 42. In the first two principal sections, ACA'C' and
BCB'C' are rhombs, and one ABA'B' is a rhomboid;
in the second, only BCB'C' is a rhomb; and both the
others, ACA'C' and ABA'B' are rhomboids. A third case
is still possible, where all the three principal sections would
yield rhomboidal figures, upon which supposition, how-
ever, the diagonals CC' and BB' themselves intersect each
other at oblique angles in the point M. From the want
of sufficiently accurate observations, it is at present impos-
sible to decide which of the two last cases, or whether
perhaps both of them take place in nature, while the
fact of an inclination of the axis in the plane of one of
the diagonals has already been established by numerous ob-
servations. According to the principles laid down in §. 87.
as respects the determination of fundamental forms, it will
be impossible to limit the number of these to four, because
forms whose axis is inclined, cannot be derived from others
whose axis is perpendicular to the base by any of the given
processes of derivation. Without entering here into the
full developement of the theory of these forms, and without
drawing all the necessary consequences from this import-
ant fact, it may be useful to mention some of the alge-
braic formulae, dependent upon the inclination of the axis
in the plane of one of the diagonals.

Let a : b : c : d denote the ratio between the four
lines AP, BM, CM, and MP, in Fig. 41., the following
formulae will be obtained :

cos. y =

a2 (b2 + c2) + c2 (b + d)

cos. y' a2(b2 — c2)-c2(b —

a2 (b2 + c2) + c2(b_d)-

96 TERMINOLOGY. §. 98.

a2 (c2 — b2) — c2 (b2 — d2)

cos.x =

cos.z= cM^-d'O-a2 (b*+c2)

-_ d)2)]'

The angles of the principal sections are found by means
of the following formulae :

a2 + d2 + c2

cos. CBC' = — ~c2 ;
b2 + c2

cos. BAB' = ^1

In most cases, it will be convenient to have the angles
BAP and B'AP separately, as given by the formulae :
tang BAP =A±J? ;

tang B'AP = b. "A
a

For finding the Angle of Inclination, or that at which the
axis AA' is inclined to the line AP, perpendicular upon
the base, we have

tang MAP = —

ct

The terminal edge of the horizontal prism belonging to
the diagonal c, is expressed in the formula :

a2 c2

cos. y == _ ;

a2 + c2

that lateral edge of P -f 03 which is contiguous to the dia-
gonal b, in the formula :

cos. y = ^-^-±-^-1
a2 b2 + c2 (a2 + d2)

In such forms as Fig. 42., where the axis is inclined in a
plane which, if it intersects the base at right angles, passes
through neither of its diagonals, the formulae become more
complicated by the introduction of a new variable quantity
e = PR, yet the general processes of derivation are still
applicable to the same extent in this apparently irregular
figure, as they are to the scalene four-sided pyramid, whose
axes are perpendicular to each other.

§. 99. OF THE CONNEXION OF FORMS. 97

2. DERIVATIONS FROM THE ISOSCELES FOUR-SIDED
PYRAMID.

§. 99. DERIVATION OF HOMOGENEOUS FORMS.

From every isosceles four-sided pyramid another
form of the same kind may be derived, which is
more obtuse, and in a diagonal position to the fun-
damental one.

The derivation is effected by laying tangent planes on
the terminal edges AB, AC, &c. of the given pyramid
ABCB'C'X, Fig. 45., and enlarging them till they wholly
include the space AFGF'G'X (§. 80.). The result of this
operation is at once the derived form itself, because the
terminal edges of the fundamental pyramid are equal to
each other.

In this case, also, the process may be inverted. Draw for
this purpose the perpendicular lines from the apices to the
lateral edges upon each of the faces of the given isosceles
four-sided pyramid, and by planes, laid through every two
adjacent ones of these lines, detach those parts of the form
which are situated towards their outside. The remaining
form is the same isosceles four-sided pyramid, from which,
after the process described above, the given more obtuse
pyramid has been derived.

The terminal edge of the given pyramid coincides with
the perpendicular line, drawn upon the face of the derived
form, from the apices towards the lateral edges, as is evi-
dent from the circumstance that these planes touch the
given pjTamid in its edges. The base of the derived
pyramid is the square circumscribed about the base of the
fundamental form ; it is inscribed, if the method has been
applied inversely. The two pyramids, and thus every
two which are in the same relation, assume such a posi-
tion towards each other, that the diagonals of the base of
the more acute pyramid are parallel to the sides of the base

98 TERMINOLOGY. §. 100.

of the more obtuse one, and vice versa. This Is termed
the diagonal position. If from the more obtuse pyramid ano-
ther still more obtuse is derived, and from the more acute one
another, still more acute ; these new pyramids are diagonal-
ly situated to that one from which they are derived ; but to
the fundamental form they are in such positions as to have
their sides and diagonals parallel to the analogous lines of
the other, and this is termed the parallel position.

§. 100. EATIO BETWEEN THE DERIVED AND THE
FUNDAMENTAL FOEM.

The axis of an isosceles four-sided pyramid,
whose faces touch the edges of another, is to the
axis of this latter pyramid, in the ratio of */i : 1 ;
the axis of that pyramid, whose edges are touched
by the planes of another, is to the axis of the latter,
in the ratio of *J2 : I. In both cases, the sides of
the horizontal projection are supposed equal.

Let AM, Fig. 45., be half of the axis, BCB'C' the base
of the fundamental form ; FAG will be a plane, laid on
the terminal edge AB, GAF' another, kid on the termi-
nal edge AC, &c. : therefore FGF'G' will be the base, and
FA, GA, &c. the terminal edges of the derived pyramid.
The axis AM is common to both.

The square FGF'G' = 2. BCB'C',
hence FG = GF' = BCV2,
and FG : BC' = MG : MB = V 2 : 1-

Describe from M with the distance MB, the arc BB"; it
will follow, that

MB" = MB = —.

V2

Draw the line B"A' parallel to GA ; MA' will be half of
the axis of the derived pyramid, its horizontal projection
being equal to BCB'C'. In the similar triangles AGM
and A'B"M, is

§. 101. OF THE CONNEXION OF FORMS. 99

GM : MA = B"M : MA',

or

GM : MA = MG. : MA',

V 2

therefore

MA' == .MfL

In order to find the ratio of the axis of the more acute
pyramid, from which the more obtuse one may be derived,
upon the supposition of their horizontal projections being
equal, let FGF'G' be the base of the latter ; BCB'C' will
be the base of the former, if the axes of the two pyramids
are supposed equal.

But we have BCB'C' = | FGF'G'.

Therefore BC' = F-^,

and BC' : FG = MB : MG = 1 : 1.

Produce the line MB, and with the distance MG, de-
scribe from the point M the arc GG", MG" will be = MG
= MB. V 2.

Now produce the axis AM, and draw G"A" parallel to
B A ; MA" will be half the axis of the more acute pyramid,
its horizontal projection being equal to FGF'G'. In the
similar triangles BAM, G"A"M, the following proportion
takes place :

BM : MA = G"M : MA",
or

BM : MA = MB. J 2 : MA",
therefore

MA" = MA. V 2.

§. 101. SERIES OF ISOSCELES FOUR^SIDED PYRAMIDS.

Every derived pyramid may again be considered
as a fundamental form, and the derivation maybe con-
tinued. This will produce a series of icosceles four-
sided pyramids, whose axes increase and decrease

100 TERMINOLOGY. §. 101.

like the powers of the square root of 2 ; the hori-
zontal projections of these forms being always sup-
posed equal.

As in §. 90. the fundamental pyramid is designated by
P, the more obtuse members in their succession by P — 1,
P — 2, P — 3, &c. the more acute members by P + 1,
P + 2, P + 3, &c. This designation not only rests upon
the same principles as that in §. 90., but is in fact exactly
the same. Since it is necessary to know before hand
whether P is an isosceles or a scalene four-sided pyramid,
if the forms derived from it are to be taken into consider-
ation ; this identity of the designation can neither in this
nor in any other case, admit of, or give rise to, any ambigui-
ty. Suppose the axis of P = a ; the series of pyramids, and
that of their axes, will appear as follows :
... P — 3, P — 2, P — 1, P, P + 1, P + 2, P 4- 3 ...

V2-a, 2. a, 2 J 2. a...

2^2 2 V 2

The ratio of the axes is

1

2 V2 2 V 2

that is to say

The axis of an undetermined nth member, or of P + n, is

n

= A/ 2n. a = 25. a ; and this expression is the Law of Pro-
gression of the series, whose fundamental number is

V 2 = 2».

It is evident, that subsequent members of the series are
in a diagonal position, alternating members in a parallel
position ; and since the position of P may be taken for
normal, all members of an even exponent will be in a
parallel position, those of an odd exponent in a diagonal
position.

The algebraic expressions in §. 52. refer to the edges of
P. Those of P + n are obtained by substituting 2". a2 in
the place of a2.

§. 102. OF THE CONNEXION OF FORMS. 101

§. 102. LIMITS OF THE SERIES.

The limits of the series §. 101. are, on one side
a plane perpendicular to the axis, on the other two
rectangular four- sided prisms, one of which is in
a parallel, the other in a diagonal position with
the fundamental form.

The origin of these prisms is sufficiently evident from
§. 91. If the diagonals of the base of the pyramid be sup-
posed equal, a rectangular four-sided prism is obtained,
instead of an oblique-angular one. It appears likewise, from
the calculations in §. 101., that an isosceles four-sided pyra-
mid of an infinite axis is transformed into a rectangular
four-sided prism.

The series of scalene four-sided pyramids is limited on one
side by a plane perpendicular to the axis, on the other by
a single prism, because there exists no difference in the
position of its members. But there is a difference of that
kind, in the series of isosceles four-sided pyramids, in which
the position of two subsequent members changes from
the parallel to the diagonal, and from the diagonal again to
the parallel position ; and since the last member, or the
limit of the series, may be considered in the one, as wel|
as in the other of these positions, it becomes necessary to
assume two rectangular four-sided prisms of infinite axes,
as limits of this series, one for the parallel, the other for
the diagonal position of two prisms limiting the series.
This supposition is exactly conformable to experience.

The limits on the opposite side are squares equal to the
horizontal projection, because the isosceles four-sided py-
ramid, if its axis becomes infinitely small, is transformed
into a square plane figure. Here the difference of the
position can no longer be considered, because this pyramid
being nothing but a plane figure, cannot appear by itself in
nature, and receives its boundaries from the intersection
with the planes of other pyramids and prisms.

The designation of the limits is, agreeably to what has

102 TERMINOLOGY. §. 103

already been stated, P — o? and P + cc. Since in the latter
case it is impossible to argue from the exponent upon the
position of the form, it is necessary to express this difference
of the two four-sided prisms, by some other contrivance in the
designation. The prism in the parallel position receives the
sign as it has been employed above, the sign of that in the
diagonal position is inclosed in crotchets, and the designa-
tion of the series of isosceles four-sided pyramids between
its limits is therefore as follows :

P_»...P + n...{^^

Five members of this series have been observed in pyra-
midal Garnet, four of them being consecutive, in the same
species also the limits on both sides of the series. Four
consecutive members are known in pyramidal Copper-
pyrites, three in pyramidal Lead-baryte, pyramidal Tin-
ore, pyramidal Titanium-ore. The limits of the series
occur still more frequently, even in those species, which
exhibit fewer members of finite dimensions.

§. 103. DERIVATION OF SCALENE EIGHT-SIDED
PYRAMIDS.

From every member of the series §. 101., seve-
ral scalene eight-sided pyramids may be derived.

This derivation is effected according to the second method
(§. 81.), and it is applied after a preparation of the form, as
described in §. 92.

The faces of the isosceles four-sided pyramids being iso-
sceles triangles, the figures BAC'A', C'AB'A", &c., Fig. 40.,
will be rhombs, and the points A', A", &c., X', X", &c. will
be situated in the angles of two squares, whose planes, like
those of the rectangular figures, §. 92., are perpendicular to
the axis in the terminal points A and X. These squares
A'A"A'"A"" and X'X"X'"X"" are equal and parallel to
the square circumscribed about the base BCB'C'.

Draw now from the points A', A", &c. towards the upper,
from the points X', X", &c. towards the lower terminal

§. 103. OF THE CONNEXION OF FORMS. 103

point of the produced axis, the straight lines A'a, A"3,
&c., X% X'% &c. These lines, and accordingly their
intersections S, S', &c. will be situated in planes, which are
perpendicular to the faces of the isosceles four-sided pyra-
mids ; and the lines BS, C'S, &c. are therefore equal to
each other. Thus likewise the triangles B3S, SaC', &c.
are equal and similar to each other, and the form obtained
by the derivation is a simple one, namely, the scalene
eight-sided pyramid a'BSC'S'B'S'"CS'%

The designation of the scalene eight-sided pyramids is
(P -f n)ra, in agreement with §. 92. It comprehends as it
were at once the two pyramids (P + n)m and (P + n)m of the
mentioned paragraph, which forms, in the present case, would
be equal and similar. The axis of the scalene eight-sided

pyramid is = 25. m. a ; where 25. a is the axis of P + n.

The relative length of the axis of the eight-sided and the
four-sided pyramids, is expressed by the number m, as in
§. 92. The only values of this number yet ascertained by
observation relative to the pyramids, are 3, 4, and 5 ; and
although it cannot be determined in general, yet it must
always be rational, positive, and greater than 1 + *J 2
(§. 56. 5.). This supposition is necessary for making it possible
to determine the position of scalene eight-sided pyramids
among themselves, and towards isosceles four-sided pyra-
mids. If m is equal to 1 + >J 2, the eight-sided pyramid
is isosceles ; if it is less than 1 + ^/2, the acute terminal
edges are transformed into the pbtuse ones, and vice versa.
And since every scalene four-sided pyramid, derivable from
P + n, according to a certain m less than 1 + ^/2, can like-
wise be derived according to another m greater than l+y^/2,
from another more obtuse isosceles four-sided pyramid P -|- n'
connected with P ; this supposition, by excluding the above
mentioned values of m, produces at once simplicity and clear*
ness in the consideration of these forms. By the supposition
of m being greater than 1 + fj 2, it is also possible to avoid
a double designation of the same kind as that mentioned in
§. 95. These considerations, however, yield a formula
for changing any pyramid (P + n)m, in which m is less

104} TERMINOLOGY. §. 104.

than 1 + >J 2 into another (P -f n')m/, in which m' is great-
er than 1 + ^/2. It is

(P + n')m/ = (m — 1) (P + n — 1) «-'.
Thus, for instance, 2 being less than 1 -f- *J 2, the sign
of a pyramid (P + I)2 may be transformed into another

2 + 1

(2 — 1) (P + 1 — 1)2-1 _ (P)3j where the exponent 3
is greater than 1 + ^2.

The position in which the scalene eight-sided pyramid is
obtained from the isosceles four-sided pyramid, supposing
m to be greater than 1 + ^ 2, is the parallel position, that
which differs from it for 45°, the diagonal position. In the pa-
rallel position, a plane through the axis and the acute termi-
nal edge of the eight-sided pyramid, passes at the same time
through the acute terminal edge of another eight-sided py-
ramid, or through the perpendicular line from the apex,
upon the face of an isosceles four-sided one, whilst in the
diagonal position, the same plane passes through the obtuse
terminal edge of the other eight-sided, or through the ter-
minal edge of the four-sided pyramid.

Pyramids expressed by the sign (P + n)3 are frequently
met with in nature, as in pyramidal Garnet, in pyramidal
Zircon, £c. ; those dependent upon other values of m oc-
cur more sparingly, as (P + n)4 in pyramidal Garnet, and
(P + n)6 in pyramidal Tin-ore.

§. 104. THE BASES OF THE SCALENE FOUR-SIDED
PYRAMIDS DEPEND UPON m.

For one and the same m, the bases of all forms
contained under the sign of (P -f- n)ni3 are equal
and similar to each other.

The demonstration of this proposition follows from §. 93.
The lines denoted in that paragraph by b and c, obtain
here the determined value = *J 2 ; and hence we have

,ro 2. m
MS = .

m+ 1

§. 105. 106. OF THE CONNEXION OF FORMS. 105

The magnitude of the plane angles of the base, and there-
fore of all sections perpendicular to the principal axis of the
scalene eight-sided pyramid, depend consequently solely
upon m, and these bases are equal, whenever m is the same.

§. 105. SERIES OF SCALENE EIGHT-SIDED PYRA-
MIDS.

Every member of the series §. 101., gives for
every determined m likewise a determined scalene
eight-sided pyramid. The forms of this kind, de-
rived from consecutive members of the series, ac-
cording to one and the same m, produce a parti-
cular series among themselves, the axes of which

n

increase and decrease, as the powers of V 2, or as 2J.

These series arise like those in §. 95. The axes of their
members are products of m, the number of derivation,
with the axes of the members of the series of isosceles four-
sided pyramids, = 22. m. a ; and since m. a is a factor
common to them all, they will be among each other in

the ratio of 9*.

n

If in the algebraic expressions §. 56., 25.a be substituted
for a, the result will be expressions of the same kind for
the cosines of the edges of (P+ n) m.

§. 106. LIMITS OF THE SERIES OF SCALENE EIGHT-
SIDED PYRAMIDS.

The limits of the series of scalene eight-sided py-
ramids are, on one side a plane perpendicular to the
axis, on the other unequiangular eight-sided prisms,
whose transverse sections are equal and similar to
those of the members of the series, and their axes
infinite. The latter must be considered in two dif-
ferent positions.

106

TERMINOLOGY.

§. 106.

That these limits are prisms of the kind above mentioned,
follows from §. 95. But, as has been shewn in §. 102. re-
garding the rectangular four-sided prisms, the eight-sided
prisms must, for the same reasons, be considered in
either position, the parallel as well as the diagonal one ;
and, therefore, two limits of infinitely long axes must be
assumed for the series of scalene eight-sided pyramids, one
in a parallel, the other in a diagonal position to the funda-
mental form. The positions of these prisms are determined,
like the positions of scalene eight-sided pyramids.

If n becomes = co, (P 4- n)m is transformed into (P + co) «,
(P — n)m into (P — co)m ; the latter, not being different
from P — oa, is denoted by that sign. In (P 4- co) m the
position, which cannot follow from n = co, must be indicat-
ed by the designation, and this is effected as in four-sided
prisms, by giving to the parallel prism the sign (P + co)m,
to the diagonal prism the sign [(P + co)m]. The designa-
tion of the whole series between its limits, is therefore :

The above-mentioned algebraic expressions give for
n = + <», the cosines of the angles in the transverse sec-
tions of the unequiangular eight-sided prisms. Thus,

CO,

-_

m2 + 1

COS.* --

m2 4- 1

The following result is obtained, for the above-mention-
ed determined values of m (§. 103.).

PRISMS.

cos. y.

COS. X.

y-

X.

(P+CO)3

— 1

— 1

126° 52' 12"

143° 7' 48"

(P+CO)4

-T8T

-If

118° 4' 10"

151° 55' 50"

(P4-»)«

-TS*

— «

11 2° 37' 12"

157° 22' 48"

§. 107. OF THE CONNEXION OF FORMS. 107

Examples of (P 4- os)3 we have in pyramidal Garnet, in
both positions ; also in pyramidal Copper-pyrites, pyra-
midal Tin-ore, &c. ; of (P + co)5 in pyramidal Lead-baryte,
and pyramidal Tin-ore.

§. 107. SUBORDINATE SERIES.

There are several Subordinate Series of isosceles
four-sided pyramids, belonging to that in §. 103.,
which, in reference to these, is termed the Princi-
pal Series.

The derivation of the members of these subordinate se-
ries is exactly the same as that employed for the scalene
four-sided pyramids, §. 96. ; only being here applied to
members of the series of isosceles four-sided pyramids, the
result will be the required subordinate series of isosceles
pyramids. The co-efficient thus obtained is likewise

= m ; and the subordinate series themselves proceed

according to the law of the principal one, and are bounded
by the same limits.

The same members of the subordinate series may also be
obtained by laying tangent planes on the homologous termi-
nal edges of the scalene eight-sided pyramids, &c. The latter
process would be the same as that employed in §. 116., for
the derivation of subordinate series of rhombohedrons from
the principal one. The results of this and of the preceding
process are identical. For if the tangent plane be laid on
the acute edges of the scalene eight-sided pyramid, the co-
rn 4- 1
efficient obtained will be — «j — ; if it be laid on its obtuse

edges, the co-efficient will be m . By substituting seve-

v 2

ral values instead of m, for instance, those which are most
commonly found in nature, we obtain members belonging to
the same subordinate series. It is therefore sufficient to
assume one of these terms as the general algebraic expres-

108 TERMINOLOGY. §. 108.

sion of the co-efficient. If the co-efficients become powers
of the number 2, the members belong to the principal
series. The members of the subordinate series in particu-
lar are designated by m P + n, agreeably to the rules

given in §. 96.

The position of the members of the subordinate series in
respect to those of the principal series, follows easily from
their derivation ; and the expressions of the cosines of their

edges are found by substituting m+ . a instead of a in

the formulae §. 56.

The values of the co-efficients hitherto ascertained by
observation are, ~-^ f , -^-2, and f . Members of the series
— j-^P + n occur in pyramidal Zircon and pyramidal Cop-
per-pyrites ; of the series | P + n in pyramidal Tin-ore ;
of the series ^pP + n in pyramidal Lead-baryte, pyra-
midal Kouphone-spar, of the series f P + n in pyramidal
Kouphone-spar and pyramidal Titanium-ore. It may be
remarked here, that the two series, ^/^P + n and
£ P + n are obtained together with the principal series,
if, according to the first method of derivation, tangent
planes are applied to the terminal edges of those eight-
sided pyramids, which depend upon m = 3, m = 4, and
m = 5.

3. DERIVATIONS FROM THE RHOMBOHEDRON.
§. 108. DERIVATION OF HOMOGENEOUS FORMS.

From every rhombohedron, another form of the
same kind, but more obtuse, may be derived. The
derived rhombohedron is in a transverse position to-
wards the fundamental form.

The first method, §. 81., is applied here without any
further preparation ; and it is evident that the form thus

§. 109. OF THE CONNEXION OF FORMS. 109

obtained will be a rhombohedron, which is more obtuse
than the given one.

The inclined diagonals of this more obtuse rhombohedron,
assume the situation of the terminal edges of the other ;
while the horizontal projections of both are parallel. One
of these forms is said to be in a transverse position towards the
other, because this position may be obtained, by turning
a rhombohedron from its original position, round the prin-
cipal axis, for an angle of 60° or 180°. If a rhombohedron is
in the transverse position towards another, it may be brought
into the parallel position, only by turning it again round
the axis for the same quantity.

In the parallel position, a plane through the axis, and
the inclined diagonal, or the terminal edge of one rhom-
bohedron, passes at the same time through the inclined dia-
gonal or the terminal edge of the other ; in the transverse
position, on the contrary, the plane through the axis and
the inclined diagonal of the one at the same time passes
through the terminal edge of the other rhombohedron.

In order to invert the process of derivation given above,
draw the inclined diagonals upon the faces of the rhombo-
hedron, and take away, by planes passing through every two
adjacent diagonals, those parts of the form which lie on
their outside. The remainder is the more acute rhombo-
hedron, from which the given one in its due position may be
derived according to the first method of derivation, as em-
ployed above.

§. 109- RATIO OF THE DERIVED RHOMBOHEDRONS.

The axis of the rhombohedron, whose faces
touch the terminal edges of another, is to the
axis of this rhombohedron == J : 1 ; and the axis
of the rhombohedron whose terminal edges are
touched by the planes of another, is to the axis of
this in the ratio of 2 : 1 ; the horizontal projections
always being supposed equal.

110 TERMINOLOGY. §. 109.

Let AX, Fig. 45., be the axis, and ACXB the principal
section (§. 37.) of the given rhombohedron ; AC will be
one of its terminal edges.

According to the method of derivation applied, the ter-
minal point B' of the inclined diagonal of the derived rhom-
bohedron will be situated in the prolongation of the termi-
nal edge AC of the fundamental form. But this point lies
also in the prolongation of Q.S', a line perpendicular to the
axis in Q, AQ, being = § AX (§. 50. 11.). Therefore the
terminal point B' of the inclined diagonal of the derived
rhombohedron coincides with the intersection of the two
produced lines AC and QS', and AB' is this diagonal
itself.

If in the same way, the lines XB and PS are pro-
duced till they intersect each other in C' ; XC' is the other
inclined diagonal of the derived rhombohedron parallel to
the former ; and the lines joining the point X with B', and
A with C', represent the terminal edges, AB'XC' accord-
ingly the principal section of the derived rhombohedron.

The line AX, or the axis, is common to both principal
sections ACXB, and AB'XC' ; but the side of the horizon-
tal projection of the given rhombohedron is PC, while
that of the derived rhombohedron is QB'.

In the similar triangles APC, AQB' we have
PC : QB7 = AP : AQ = ^ AX : § AX = 1 : 2 (§. 50. 11.).

Hence the side of the horizontal projection of the deriv-
ed rhombohedron is double the same line of the given
rhombohedron, their axes being equal ; and the inclined
diagonal AB' = XC' of the derived rhombohedron, is
double the terminal edge AC = XB of the given rhombo-
hedron.

Draw now the line BA' parallel to C'A, and B"A' paral-
lel to B'A ; BA'B"X will be the principal section, and
A'X the axis of the derived rhombohedron ; provided the
two rhombohedrons have the same horizontal projection,
whose side is = BQ..

But in the similar triangles AC'X and A'BX the follow-
ing proportion takes place :

§. 110. OF THE CONNEXION OF FORMS. Hi

AX : A'X = C'X : BX = 2 : 1.

Hence the axis of the derived rhombohedron is equal to
half the axis of the fundamental one, if the horizontal pro-
jections are equal.

In order to find the ratio between the axis of the more
acute rhombohedron and that of the fundamental form, let
AB'XC' be the principal section of the latter, the side of
the horizontal projection being C'P. In this case ACXB
will be the principal section of the derived rhombohedron,
the side of its horizontal projection = BQ, and its axis equal
to that of the fundamental rhombohedron.

From the point C' draw the line C'B'" parallel to the
axis, produce AB to B'", and complete the rhomboid
AB'X'B"'. This is the principal section of the derived
rhombohedron, the side of the horizontal projection being
B'"Q' = C'P.

The similar triangles ACX and AB'X' give
AX : AX' = AC : AB' =1:2.

The axis of the more acute rhombohedron is conse-
quently double the axis of the fundamental form, their ho-
rizontal projections being equal.

§. 1 10. SERIES OF RHOMBOHEDRONS.

By continuing the derivation, a series of rhom-
bohedrons is obtained, whose axes increase and de-
crease as the powers of the number 2 ; their hori-
zontal projections being always supposed to be
equal.

This series corresponds exactly to the series of scalene
four-sided pyramids (§. 90.), in respect to the ratio be-
tween the axes of its members, in as much as it depends
upon the same fundamental number.

Designate the fundamental form by R ; R 4- n will be
the general expression of an indeterminate member of the
series.

Fig. 44. will assist us in giving a clearer idea of this se-

TERMINOLOGY. §. 111.

ries. Let ACXB be the principal section of R. Produce
the axis AX, and from the points B and C draw lines paral-
lel to it ; every line drawn perpendicularly from these upon
the axis, will be equal to the side of the horizontal projec-
tion of R.

The inclined diagonal of II becomes the terminal edge
of R + 1. The inclined diagonal of R + 1 passes through
the point S, the centre of the inclined diagonal XC of R.
Hence draw the line AB' through S, and produce this line
till it arrives at the parallel from C ; AB' will represent
the inclined diagonal, BAB'X' the principal section, and
AX' = 2. AX consequently the axis of R + 1.

The inclined diagonal of R + 1 becomes the terminal
edge of R 4- 2. The inclined diagonal of R + 2 passes
through S', &c. Draw the line AB" till it arrives at the
parallel from B ; AB" will be the inclined diagonal of
R + 2, AB'X"B" its principal section, and AX" = 2. AX'
== 4. AX its axis.

Thus, considering AB" as the terminal edge of R + 3, it
will be found, that AB'" is its inclined diagonal, AB"X'"B"'
the principal section, and AX"' = 2. AX" = 4. AX' =
8. AX the axis of R + 3 ; and in this manner we may con-
tinue the series, as long as we please.

The axis of R being = a, that of R + 3 is = 23.a, that
of R + n = 2n. a, that of R + n + 1 == 2" + ». a. These
values substituted in the expressions §. 50., give the cosines
of the terminal edges for any required member of the series
of rhombohedrons.

§.111. LIMITS OF THE SERIES OF RHOMBOHEDRONS.

The limits of the series §. 110. are, on one side
a plane perpendicular to the axis, on the other a
regular six-sided prism of an infinite axis. The
transverse section of that prism is equal and paral-
lel to the transverse section of the fundamental
form ; the figure of the plane perpendicular to the

§. 111. OF THE CONNEXION OF FORMS. 113

axis is a regular hexagon, equal and similar to its
horizontal projection.

Lay planes of intersection through the horizontal dia-
gonals of a rhombohedron, whose axis is longer than the
side of its horizontal projection. These planes will detach
those parts of the rhombohedron which are contiguous to
the terminal edges of this form. The remainder, contigu-
ous to the lateral edges, is contained under two equilateral
triangles in the direction of the axis, and under six isosce-
les triangles, being halves of the faces of the rhombohedrons.
The equal sides of these triangles are the lateral edges of
that form. This solid is the Central Part of the rhombo-
hedron.

In the central part of a more acute rhombohedron, the
angles at the bases of the isosceles triangles are greater, but
the angles at the vertex are less ; and the horizontal projec-
tion always being constant, the sum of the first approaches to
two right ones, the latter to nothing, the more the axis
of the rhombohedron is elongated. The equal sides in this
case approach nearer and nearer to the parallelism with
each other and with the axis, and to the equality with
one-third of it, which is contained in the central part of the
rhombohedron.

The limits to which these approximations lead, cannot
be obtained, while the axis remains a finite quantity. But
when the axis becomes infinitely long, these limits are ob-
tained ; the triangles are transformed into unlimited paral-
lelograms, and contain a regular six-sided prism, which is
still unlimited in the direction of its axis.

As to the transverse section of the prism, we may ima-
gine, that in the proportion in which the axis of the rhom-
bohedron increases, its faces turn round certain immoveable
lines. These lines are the sides of the transverse section of
the rhombohedron ; and therefore they are likewise the
sides of the transverse section of the prism.

Let HORZ, Fig. 46., be part of the horizontal projec-
tion, and the vertical lines C'D, EB', &c. through the points

vot.. i. H

TERMINOLOGY. §.

H, O, R, Z, the edges of that regular six-sided prism, in
whose planes are situated the lateral edges CB, C'B' of two
rhombohedrons. These lateral edges, intersecting each
other at the points M, N, turn as it were round these, and
consequently the faces of the rhombohedron turn round the
line MN, in a ratio, dependent upon the length of the axis.
If the axis becomes infinitely long, the lateral edges of the
rhombohedron assume the situation C"B" ; the rhombohe-
dron is transformed into a regular six-sided prism, upon the
faces of which are drawn the horizontal lines MN, ML,
&c. ; and these lines, the sides of the transverse section of
the rhombohedron, are therefore likewise the sides of the
transverse section of the prism, whose position is thus de-
termined in respect to the rhombohedrons of the series, and
to their horizontal projection.

As to the opposite limit, it is evident, that if the axis
becomes infinitely small, all the faces of the rhombohedron
coincide in a single plane, and that this form is therefore
changed into a regular hexagon, equal and similar to the
horizontal projection of the fundamental form.

The crystallographic signs of the limits are R + co and
R — 03, those of the series between its limits,
R — - co ... R -f n ... R + os.

Many examples are found in nature, illustrative of this
series. Thus, rhombohedral Lime-haloide presents five
consecutive members, and both the limits ; in rhombohe-
dral Tourmaline and rhombohedral Ruby-blende, four con-
secutive members and both limits have been observed ;
two or three consecutive members occur in many species ;
and in most of those, affecting forms which are in connexion
with the rhombohedron, we likewise frequently meet with
the limits on either side of the series.

§. 112. DERIVATION OF SCALENE SIX-SIDED PYRA-
MIDS.

From the members of the series in the preced-
ing paragraph, several scalene six-sided pyramids

§, 112. OF THE CONNEXION OF FORMS. 115

may be derived, the lateral edges of which agree
in their situation with the lateral edges of the rhom-
bohedron.

We employ here the second method (§. 81.), without any
farther preparation. Produce the axis of the rhombohe-
dron on both sides, to an indefinite but equal length ; or,
what is the same thing, multiply the axis of the rhom-
bohedron by the number of derivation m, which must be
rational, positive, and greater than 1 ; draw from the angles
of the rhombohedron, straight lines towards the terminal
points of the axis produced, and lay planes on every ad-
jacent pair of these lines. The result will be a scalene six-
sided pyramid, whose lateral edges coincide with those of
the rhombohedron.

Every determined prolongation of the axis of the rhom-
bohedron, or every determined m, determines a scalene
six-sided pyramid. A rhombohedron, and all the scalene
six-sided pyramids derived from it, which therefore agree
in the situation of their lateral edges, and also the pyramids
among each other, are said to be forms belonging together or
co-ordinate.

The position in which the scalene six-sided pyramid is
placed by the derivation towards the rhombohedron, is
termed the parallel position. The pyramid is in a transverse
position towards rhombohedrons, which immediately pre-
cede or follow that from which it is derived, because the
rhombohedrons themselves are in a transverse position to-
wards each other. In general the pyramids partake of the
position of the rhombohedron from which they are derived,
and pyramids belonging together are in a parallel position.

In general, two scalene six-sided pyramids, or one pyra-
mid and a rhombohedron, are said to be in a parallel position,
when a plane through the obtuse terminal edge and the
axis of the pyramid, intersects the face of the rhombohe-
dron in its inclined diagonal, or in the other pyramid like-
wise passes through its obtuse terminal edge, and in both
forms at the same time also through the axis. The same

116 TERMINOLOGY. §.113.

forms are in a transverse position when this plane passes
through the terminal edge of the rhombohedron, or through
the acute terminal edge of the other pyramid.

The crystallographic sign of a scalene six-sided pyramid
derived from P + n according to m is (P + n)m ; its axis
is = 2n. m. a ; where 2n. a is the axis of II + n, and a the
axis of R.

The value of m is frequently found in nature to be = 2,
= 3, or = 5, all of which occur in rhombohedral Lime-ha-
loide. (P + n)3 is also found in rhombohedral Tourma-
line, in rhombohedral Ruby-blende, (P -f n)5 in rhombo-
hedral Iron-ore. Besides these, we have m = 7 in rhom-
bohedral Lime-haloide, m = § in rhombohedral Fluor-
haloide, rhombohedral Quartz, rhombohedral Tourmaline,
m = § in the two first of these species, m = V and = f
in rhombohedral Quartz.

§. 113. THE TRANSVERSE SECTIONS OF THE SCA-
LENE SIX-SIDED PYRAMIDS DEPEND ON m.

For one and the same m, the transverse sections
of all forms contained under the sign of (P + n)m
are similar to each other.

Let ABXC, Fig. 47., represent the principal section of
the rhombohedron from which the scalene six-sided pyra-
mid has been derived ; A3, X£ the prolongations of the
axis, and consequently 9LB, %C the obtuse terminal edges,
20, £B the acute terminal edges of this pyramid, and
aB£C its principal section.

Draw from M, the centre of the axis, in the plane of the
transverse section, the line MG parallel to QB. This line
will be intersected in F by the obtuse terminal edge 3LB ;
and F will therefore be that point in the transverse sec-
tion, or MF that line situated in its plane, upon which de-
pends the magnitude of the angle of the transverse section
at the place of the obtuse terminal edge of the pyramid.

§. 114. OF THE CONNEXION OF FORMS. 117

Draw now the line BG, parallel to the axis ; we have
MG = QB = 1 ; and FG = 1 — MF.

In the similar triangles FBG and F3M, we have
GB : GF = M3L : MF,

: 1 _ MF » 1 : MF,
6 2

and consequently

3 m + 1

from which it appears that the angles of the transverse
section are dependent solely upon m, and that consequent-
ly they are the same in all pyramids derived according to
the same m, whatever may be the fundamental rhombohe-
dron.

If the section does not intersect the lateral edges of the
pyramid, its figure is an unequiangular, but equilateral
hexagon. The angle at the obtuse edge is as above ; but
the angle at the acute edge is likewise dependent upon m.
For let CPF' be in the plane of the section ; the lines CP
and PF' will determine its figure. But CP is = 1 ; and

PF = 3 m — \ as it follows from the similarity of the
3 m + 1

triangles Q3B and P£F'. Now

CP : PF = 3 m + 1 : 3 m — 1,

a ratio solely dependent upon m. Hence all scalene six-
sided pyramids, derived according to the same m, have
their sections through the terminal edges similar to each
other.

§. 114. SERIES OF SCALENE SIX-SIDED PYRAMIDS.

From every member of the series §. 110., several
scalene six-sided pyramids may be derived. The
axes of those which are derived according to one
and the same m, and consequently the pyramids
themselves, produce a series which proceeds agree-

118 TERMINOLOGY. §. 115.

ably to the law of the series of rhombohedrons
(§. 110.).

The axes of the members of this series may be consider-
ed as products of the axes of II + n by m, that is to say,
as being equal to 2". m. a ; and since m and a are common
factors to them all, the axes are to each other in the ratio
of2»(§. 110.).

If these values, namely 2n. a instead of a, are substitut-
ed in the expressions §. 55., the cosines of the edges are
obtained for any particular member of the series.

§. 115. LIMITS OF THE SERIES OF SCALENE SIX-
SIDED PYRAMIDS.

The limits of the series of scalene six-sided pyra-
mids are, on the one side unequiangular twelve-sided
prisms of infinite axes, whose transverse sections
are determined by m, and on the other side plane
figures equal and similar to the horizontal projec-
tion.

The axis of a scalene six-sided pyramid, which belongs
to a rhombohedron, whose axis is in a finite ratio to the
side of its horizontal projection, can never become infinite,
unless m itself should be infinite, a supposition, however,
which is excluded from the relations to be taken here into
consideration, in as much as such an m would give no
series, or rather a series, all the members of which are
equal to each other. Therefore the limits of the series of
these pyramids must arise from rhombohedrons, whose
axes are themselves infinite, or from the regular six-sided
prism (§. 111.).

Lay planes of intersection through the terminal points
of the lateral edges of a scalene six-sided pyramid derived
from a rhombohedron, whose axis is longer than the side of
its horizontal projection ; and thus detach the apices of

§. 115. OF THE CONNEXION OF FORMS. 119

the pyramids, so that only the central part analogous to
that in §. 111. remains, to which the lateral edges are ad-
jacent.

This part contains one-third of the axis of the rhom-
bohedron, from which the pyramid is derived. It is limit-
ed perpendicularly to the axis by equilateral hexagons,
whose alternating angles are equal (§. 1 13.), and from the
sides by twelve scalene triangles, whose bases are the sides
of those hexagons, the longer sides being the entire lateral
edges, and the shorter sides parts of the obtuse terminal
edges of the scalene six-sided pyramid.

The central part of a pyramid derived according to the
same m from a more acute rhombohedron, although con-
tained under faces of the same kind, will yet differ from
the preceding by the sum of the angles at the base of the
triangles being greater, the angle at the vertex smaller,
and the sides approaching nearer to parallelism and equa-
lity with each other, and to one-third of the axis of the
rhombohedron.

The limits to which the pyramids approach, when thus
derived according to the same m, from more and more acute
rhombohedrons, are : the sum of the angles at the base of
the triangles must be = 180°, the angle at the vertex = 0 ;
and the sides equal and parallel among themselves, and
to one-third of the axis of the rhombohedron. These li-
mits are obtained when the central part of the pyramid,
containing one-third of the axis of the rhombohedron,
from which the pyramid is derived, and therefore the axis
of the rhombohedron itself becomes infinite, or when the
rhombohedron is changed into the regular six-sided prism.
If therefore (P 4- n)m is changed into a prism, the expres.
sion of its axis 2n. m. a, will be changed into 2* . m. a, because
n and not m is = oo>

While the scalene triangles, the lateral faces of the cen-
tral part, by the continual increase of the axis, are chang-
ed into unlimited parallelograms, the central part itself
is changed into a twelve-sided prism, unlimited in the direo
tion of the axis. Through all these changes, however, m

120 TERMINOLOGY. §. 115.

remains unaltered ; and since the angles of the transverse
section are solely dependent upon m, the unequiangular
twelve-sided prism will have the same transverse section as
any other member of that series, whose limit it represents.
As to the opposite limit, it is evident, that if the height
of the central part, = i of the axis of the rhombohedron,
disappears, the axis of this rhombohedron itself must also
be = 0 ; and consequently that the rhombohedron must be
a plane figure, equal and similar to the horizontal projec-
tion. Now the axis of the pyramid sought is m. 0 = 0 ;
and the pyramid therefore likewise a plane figure, equal
and similar to the horizontal projection.

The designation of the series between its limits is

R _ co ... (P + n)« ... (P + cc )™.
If an unequiangular twelve-sided prism, from its original
position, is transferred into another different from it for
60° or 180°, its faces and edges resume exactly the situa-
tion they had before. Hence there is only one position
existing for these forms : in which property it agrees with
the regular six-sided prisms, the limit of the series of rhom-
bohedrons, from which it derives its origin.*

If in the algebraic expressions mentioned in the last
paragraph, n is made = co ; the angles are obtained for the
transverse section of the unequiangular twelve-sided prism,
being the limit of the respective series of the scalene six-
sided pyramid. Thus we find

m2 + 6 m—

* Another regular six-sided prism, which in every respect,
but the position, agrees with the former, will be considered
in §. 1 18. This form, however, is in no immediate connexion
with the scalene six-sided pyramids ; and consequently no
unequiangular twelve-sided prism can be considered in, or
referred to, a position analogous to that regular six-sided
one, although the angles of their transverse sections should
seem to indicate a similar position.

§.116.

OF THE CONNEXION OF FORMS.

The values of the cosines, and ihe angles of the trans-
verse sections of the limits, relative to the above-mentioned
series occurring in nature, are the following :

PRISMS.

cos. y.

cos. z.

y-

z.

(P + C3)1

(P + co)*

-II

~H

— u
— «

162° G' 12"
158° 12' 48"

137° 53' 48"
141° 47' 12"

(p+=)2

(P 4- »)*

— H

-11

— n
-ft

152° 12' 15"
147° 47' 45"

147° 47' 45"
152° 12' 15"

(P + CO)3

(P+eo)1'

-H
— H

-i!

-— §1

J419 47' 12"
137° 53' 48"

158° 12' 48"
162° 6' 12"

(P + ®)5

— ii

— §i

133° 10' 25"

106° 49' 35"

(P-f CD)'

— n

riH

129° 25' 48"

170° 34' 12"

Few ,of these prisms have as ye£ been observed in na-
ture. Those which have been observed are (P + co)B and
(P + co)3 in rhombohedral Fluor-haloide, (P -f cs)3 in
rhombohedral Lime-haloide and in rhombohedral Tour-
maline, (P + co)5 in rhombohedral Quartz, (P -f co)5in
rhombohedral Corundum. The angles of other prisms have
not been exactly ascertained.

The angles of the first and sixth of the preceding prisms,
those of the second and fifth, and those of the third and
fourth, are inversely equal to each other. In general,
the number of derivation which produces the one, may be
found from that which produces the other, by the formula :

3 (m' — 1)
§. 116. SUBORDINATE SERIES.

To the series of rhombohedrons, §. 110., belong
several subordinate series, in reference to winch
tl>e former is termed the principal series.

TERMINOLOGY. §. 116.

For the derivation of these subordinate series, the first
process is applied to the scalene six-sided pyramids.

Let ABXC, Fig. 47., represent the principal section of a
rhombohedron, 21B£C that of a scalene six-sided pyramid
derived from it according to a certain m.

If tangent planes are laid on the terminal edges 3B, &c.
of this pyramid ; these edges become the inclined diagonals
of the resulting rhombohedron. Suppose its axis = a' ; that
part of it which corresponds to the inclined diagonal 2tB
will be

and

ao, = |. a' = Ma + MQ=3 mtJL

6

,_, 3m+ 1 3m+ 1
a_2. -_.a .a,

the side of the horizontal projection BQ being = I.

If on the other side tangent planes be laid on the acute
terminal edges aC, &c. ; these terminal edges again will
become the inclined diagonals of the derived rhombohe-
dron. The axis of this rhombohedron being = a" ; that
part of it which corresponds to the inclined diagonal aC
willb

ap = §. a" = Ma — MP = L!Lrzi a

and

a" = L?_rJ. a

4
for the same horizontal projection.

Hence 3 m +J- and 3 m "ll are the co- efficients, with

4 4

which a, the axis of R, or 2?. a, the axis of R + n must be
multiplied for obtaining members of the subordinate series.
When these co-efficients become powers of the number 2,
the rhombohedron produced is a member of the principal
series ; when they are not powers of the number 2, mem,
bers are produced belonging to a subordinate series, which
is determined by m.

Designate the subordinate series by — — R 4- »•

The quantity 3 m "Li. 2". a substituted for 2". a in the

§. 117. OF THE CONNEXION OF FORMS.

above mentioned algebraic expressions, gives the cosines of
the edges for the members of the subordinate series.

The position of the members of the subordinate series
towards each other, and towards those of the principal se-
ries, follows from their derivation. The limits are com-
mon to the principal, and to all subordinate series.

§. 117. DERIVATION OF THE ISOSCELES SIX-SIDED
PYRAMIDS.

From every rhombohedron an isosceles six-sided
pyramid may be derived, whose axis is to the axis
of the rhombohedron in the ratio of 2 : 3, the ho-
rizontal projections of the two forms being sup-
posed equal.'

The third method of derivation (§. 82.) is applicable to
the present case.

Let ABXC, Fig. 48., represent the principal section of
the given rhombohedron, and suppose a horizontal plane
to pass through M, the centre of its axis. In this plane
is situated the base of the six-sided pyramid, which is to
be derived.

The terminal edge AC of the rhombohedron, being
produced to Z, will be changed in AZ the terminal
edge of the pyramid. In the same way XB is changed
into XH, so that, if we draw AH and XZ, AZXH will
be the principal section of the derived isosceles six-sided
pyramid, its axis being equal to that of the rhombohedron,
the side of its horizontal projection MZ = MH.

Draw the lines BG and CG' perpendicularly to HZ ;
MG' will be = MG = PC, equal to the side of the hori.
zontal projection of the given rhombohedron ; and if now
the lines GA', G'A', GX', G'X', be drawn parallel to the
sides of the principal section of the pyramid, A'X' will re-
present the axis of this form for the side of its horizontal
projection MG' = PC, that is to say, equal to the horizon-
tal proje«tion of the given rhombohedrpn.,

124 TERMINOLOGY. §. 118.

The two triangles APC, A'MG' are equal and similar to
each other. Therefore A'M = AP ; which if the axis of
the pyramid be called a', may be expressed thus :

k- a' = *- a,
and consequently

a' = |. a.

The above-mentioned constant co-efficient (§. 54. 4.) ac-
cordingly is = f .

§. 118. SERIES OF ISOSCELES SIX-SIDED PYRAMIDS.

There is a series of isosceles six-sided pyramids
in connexion with the principal series of rhombo-
hedrons, with which it proceeds after the same law,
and is limited by infinite six-sided prisms.

The axes of the members producing this series, are equal
to the axes of the rhombohedrons, multiplied by §, the ho-
rizontal projections being always supposed equal ; so that,
if P + n denotes an undetermined nth member of the se-
ries, its axis will be = § 2n. a. In the expression for the
axis of any member, the common factor §. a is contained ;
and by dividing with it, we find that the axes of the series
increase and decrease like the powers of the number 2 ;
and 2n consequently expresses also in this series the law of
progression.

The limits of this series are isosceles six-sided pyramids
belonging to rhombohedrons, whose axes are on one side
infinitely long, on the other infinitely short. It is evident
that an isosceles six-sided pyramid of an infinitely long axis
can be nothing else but a regular six-sided prism, whose
transverse section is equal and similar to the horizontal
projection of the pyramid ; and that an isosceles six-sided
pyramid of an infinitely short axis can be nothing else but
a plane figure perpendicular to the axis, and equal and si-
milar to the same horizontal projection.

The regular six-sided prism, which limits the series of
isosceles six-sided pyramids, can be distinguished by its

§. 119. OF THE CONNEXION OF FORMS. 125

position from the other regular six-sided prism, which li-
mits the series of rhombohedrons. The faces of the two
prisms differ in their situation for 30° and 150° : and the
prisms therefore themselves are two different well defined
forms, which must not be confounded with each other.

If, in the algebraic expressions in §. 54., m is made = |,
and 22n. a2 is substituted for a2 ; the expressions produced
refer to the cosines of the edges for P + n. They are :

cos.x = _(?l!^±L7\;
V4.22«.a2 + 27'

cos.z = _(ll?±^ZL?7Y
V4.22«.a2 + 27'

The series of isosceles six-sided pyramids between its
limits, receives the following designation :
R_ co ... P + n ... P+ co.

Several members of this series, together with its limits,
occur in rhombohedral Fluor-haloide, rhombohedral Quartz,
rhombohedral Corundum, &c. There exist also series
appertaining to rhombohedrons of certain subordinate se-
ries, and which, on that account, receive a co-efficient in
their representative sign, like the rhombohedrons from
which they are derived. Examples of the series | P + n,
3 P + n, and § P + n, have been found in rhombohedral
Corundum ; of the two first also in rhombohedral Quartz,
of the first in rhombohedral Iron-ore, &c.

4. DERIVATIONS FROM THE HEXAHEDRON.

§. 119. DIFFERENT POSITIONS OF A MOVE ABLE
PLANE.

A plane, moveable round the terminal point of
a rhombohedral axis of the hexahedron, is liable
to assume four different classes of positions. One
of these is exactly determined, and admits of only

12C TERMINOLOGY. §. 119.

one case ; the others allow of a farther distinction in

two cases.

Let ACBCOrC'B'X, Fig. 3G., represent a hexahedron,
which is brought into an upright position by supposing one
of its rhombohedral axes AX vertical. AC, AC', &c. are
therefore the terminal edges, AB, &c. the inclined diagonals
of this hexahedron, if considered as a rhorabohedron of 90°.

Direct now the planes MNOO', PQRH' and TUVV
through the axis AX, so as to make them pass through
the inclined diagonals AB, AB', and through the terminal
edges AC", AC' which are opposite to these edges. The
planes will intersect each other at angles of 60° and 120°.

The part MNSS' of the plane MNOO' turned towards
the observer, may be termed the Section of the Face, the
part PQSS of the plane PQRR', similarly situated, the
Section of the Edge, in so far as they refer to the upper apex
A ; because the former passes through the inclined diagonal
AB, and bisects the face, while the latter passes through
the terminal edge AC of the hexahedron, and bisects the
angle at which the faces meet.

The sections of the face divide every face of the hexa-
hedron into two equal and similar triangles, as ABC,
ABC', &c. and thus the solid angle A may be conceived to
consist of six faces, which, for the sake of derivation, are
considered moveable, and their situation is ascertained in
respect to both the sections, to that of the face, and that
of the edge. Whatever results are found for one of these
faces, likewise applies to the other, because the hexahe-
dron is a solid of several axes, and it will therefore be suf-
ficient to consider the situation of one of these six faces,
because the rest must assume an analogous position. This
refers evidently not only to those contiguous to A, but
also to those belonging to the other solid angles B, C, &c.
of the hexahedron.

The moveable plane may assume the following situations :

1. Perpendicular to loth sections.

Upon this supposition, ABC will be perpendicular to

§. 119. OF THE CONNEXION OF FORMS. 127

the intersection of MNSS' and PQSS', which is the line
SS', or the rhoinbohedral axis of the hexahedron. All the
six planes contiguous to that point, will necessarily coin-
cide in that plane.

2. Perpendicular to tlie section of the edge ; but inclined to
the section of tlie face.

Here every two faces situated like ABC and AB'C, &c.
coincide in a single plane, which, though always perpendi-
cular upon FQSS', yet may be differently inclined to AX.

3. Perpendicular to the section of the face, inclined to the section
of the edge.

In this case again, pairs effaces like ABC and ABC' coin-
cide in a single plane perpendicular to MNSS7, inclined to
AX.

4. Perpendicular to none of the sections.

No two planes contiguous to the same solid angle of the
hexahedron coincide, but every two meeting in the same
section, as ABC and AB'C in PQ.SS', or ABC and ABC'
in MNSS', are inclined to that section at the same angle.

In the first of the above mentioned cases, the situation
of the moveable plane is perfectly determined.

In the second case, the plane must either

a) touch AC, the edge of the hexahedron, or

b) the line of its intersection with PQ.SS' must include
an angle with AX, which is greater then CAX.*

Supposing the first to take place, two faces of the solid
angle C, coincide in one single plane, with two faces of
the solid angle A, for instance CC'A with ABC, and
CC"A with AB'C. This does not take place upon the
latter supposition.
In the third case, the moveable plane may either

a) pass through the diagonal AB, and consequently coin-
cide with the face of the hexahedron itself, or

* Should this angle be less than CAX, it would be neces-
sary to refer the whole derivation, from the solid angle A,
to the solid angle C, where the case is confined within the
one above mentioned.

128 TERMINOLOGY. §. 120.

b) its intersection with MNSS' may include an angle
with AX, greater than BAX.

In the first case, two planes of the solid angle B coincide
with two planes of the solid angle A, as for instance BAG
with ABC, and BAC' with ABC', and consequently two
from C likewise with two from C', being altogether eight
planes coinciding in a single one, which is not the case in
the second.

The differences which may occur in the fourth case are
the following :

a) the intersection of every two faces, as ABC and AB'C,
with the plane PQ.SS', may coincide with the edge of the
hexahedron, or

b) it includes an angle with AX greater than CAX,
Suppose the former to take place ; two faces from the

solid angle A will coincide with two faces from the solid
angle C, viz : ABC with CC'A, and ACB' with CC"A ;
which is not the case upon the latter supposition.

The different situations, which a plane moveable round
the point A may assume, are therefore the following :

1. Perpendicular to both the sections ;

2. Perpendicular to the section of the edge, touching the

edge of the hexahedron ;

3. Perpendicular to the section of the edge, intersecting

the edge of the hexahedron ;

4. Perpendicular to the section of the face, in the face of

the hexahedron ;
o. Perpendicular to the section of the face, not in the face

of the hexahedron ;
C. Inclined to both the sections, touching the edge of the

hexahedron ;
7» Inclined to both the sections, intersecting the edge of

the hexahedron.

§. 120. PRODUCTION OF THE FORMS OF SEVERAL
AXES.

Whatever situation of those mentioned in the
preceding paragraph the moveable plane may as-

§. 121, 122. OF THE CONNEXION OF FORMS. 129

sume, it corresponds to the face of a form of seve-
ral axes.

We obtain or derive the forms of several axes from the
hexahedron, by considering the space limited by all those
faces which are homologous to the one whose situation has
been ascertained.

Hence there will exist as many different kinds of forms
of several axes, as there are possible situations of the
moveable plane, and no more ; and we obtain, therefore,
the complete number of these forms, whilst at the same time
every form is excluded which does not belong to this as-
semblage.

In the preceding paragraphs 57 — 77> we have met with
more than seven forms of several axes. Those which are
not immediately produced according to the present consi-
deration, are nevertheless contained in its results, the
mode of which will be explained in the paragraphs 128 —
134.

§. 121. THE OCTAHEDRON.

In the first situation the moveable plane is the
face of the Octahedron (§. 59.).

Of the forty-eight faces which are moveable round the
eight solid angles of the hexahedron, every six conti-
guous to one of these solid angles coincide in one and the
same plane, perpendicular to a rhombohedral axis of the
form (§. 59. 2.).

§. 122. THE DODECAHEDRON.

In the second situation the moveable plane is
the face of the Dodecahedron (§. 63.).

A pair of faces from every solid angle of the hexahedron
coincides with another pair of faces contiguous to an adja-

130 TERMINOLOGY. §. 123.

cent solid angle in a plane which touches the edge of the
hexahedron, and joins these two solid angles. Hence, of the
forty-eight faces, four and four will coincide, and the solid
obtained will be limited by twelve faces. The prismatic axes
pass through the centres of the edges of the hexahedron,
and consequently also through the centres of the faces «ow
obtained. Thus the faces of the derived form become
perpendicular to the prismatic axes, and are themselves
the faces of the monogrammic Tetragonal-dodecahedron
(§. G3. 3.).

§. 1&3. THE OCTAHEDRAL TRIGONAL-ICOSITETRA-
HEDRON.

In the third position the moveable plane is the
face of an octahedral Trigo?ial-icositetrahedron
(§. 72.)-

In this case there are no pairs of faces from one solid
angle, coinciding with pairs from another ; but of the six
faces contiguous to one and the same solid angle, two and
two faces will coincide. Hence the number of faces of this
form is twenty -four. Each of these faces is intersected by
the two other faces contiguous to the same, and by one
contiguous to the adjacent solid angle ; with the last of
these faces it produces an edge in the direction of the
greater diagonal of the dodecahedron, or in the direction of
the edge of the octahedron. Its faces therefore assume the
figure of isosceles triangles ; the rhombohedral solid angles
of the fonn consist of three faces, and they are monogram,
mic ; the pyramidal solid angles are formed by eight faces,
and they are digrammic ; the form itself is an octahedral
Trigonal-icositetrahedron (§. 72. 1. 2.).

The different varieties of octahedral trigonal-icositetra-
hedrons may be considered as forms intermediate between
the dodecahedron and the octahedron. If the angle
measuring the inclination of the moveable plane to the axis
AX, Fig, 36., becomes greater than CAX, the face of the

§. 124. 125. OF THE CONNEXION OF FORMS. 131

monogrammic tetragonal-dodecahedron is divided into two
isosceles triangles, whose common base is the longer diagonal
of the rhomb. The triangles retain their isosceles figure,
though the angles may vary, till the moveable plane inter,
sects the axis of the form at an angle of 90°. In this case,
all the faces contiguous to the same solid angle coincide in
a single plane, which is the face of the octahedron. All
possible varieties of octahedral Trigonal-icositetrahedrons
are therefore contained between the two forms just men-
tioned, and the dimensions of their varieties depend upon
the magnitude of the above mentioned angle.

§. 124. THE HEXAHEDRON.

In the fourth situation, the moveable plane is
the face of tlie Hexahedron (§. 58.).

In this situation pairs of faces from all the four solid
angles A, B, C, C' coincide in a plane perpendicular to the
pyramidal axis (§. 58. 3.).

§. 125. THE DIGRAMMIC TETRAGONAL-ICOSITE-
TRAHEDRON.

In the fifth situation, the moveable plane is the
face of a digrammic Tctragonal-icositetrahedron
(§. 74.).

The pairs of faces from the angles A and B, and those
from the angles C and C', do not coincide, but they inter,
sect each other at equal angles in a determined point of the
lengthened pyramidal axis of the hexahedron. A solid
angle of three faces is produced at the point*A. The
edges which produce these two kinds of solid angles unite
with each other in the prismatic axes prolonged, and thus
produce solid angles, which contain likewise four faces, but
two different kinds of edges.

Each face is intersected by four other faces, two of

TERMINOLOGY, §. 12G.

which are contiguous to the same solid angle, the other two
to the adjacent ones. The faces are four-sided ; and on ac-
count of the two kinds of edges which the form contains,
it is a digrammic Tetragonal-icositetrahedron (§. 74. 2.).

The digrammic Tetragonal-icositetrahedrons may be con-
sidered as forms intermediate between the hexahedron and
the octahedron. For if the angle measuring the inclination
of the moveable plane to the axis AX becomes greater
than BAX, a digrammic Tetragonal-icositetrahedron will be
produced, and the varieties of this form will succeed each
other, till the angle just mentioned becomes = 90°, when the
derived form is changed into the octahedron. The dimen-
sions of the different varieties, are dependent upon the
value of that angle.

§. 126. THE HEXAHEDRAL TRIGONAL-ICOS1TETRA-
HEDRON.

In the sixth situation the moveable plane is the
face of a tiexaliedral Trigonal-icositetrdhedron
(§. 71.).

This icositetrahedron is produced by the coincidence of
two faces contiguous to adjacent solid angles. From every
edge of the hexahedron faces rise towards the prolongation
of the pyramidal axes, at which they form a solid angle of
four faces, intersecting each other at equal angles, while
the general aspect of the hexahedron is retained in the de-
rived form. The rhombohedral solid angles are equiangu-
lar, but they consist of two kinds of edges.

Each of the faces of this form is intersected by three
other faces, of which one is contiguous to the same, and one
to an adjacent solid angle of the hexahedron, the third face
being common to both these solid angles. The faces of
this form are consequently triangular, and intersect each
other at equal angles in its pyramidal solid angles. The
form, therefore, will be a hexahedral Trigonal-icositetrahe-
dron.

§. 127. OF THE CONNEXION OF FORMS. 133

The hexahedral Trigonal-icositetrahedrons are forms in-
termediate between the hexahedron and the dodecahe-
dron. The moveable plane passes through the edge of the
hexahedron ; the limits to which its inclination approaches
will therefore be on one side a plane perpendicular to the
section of the edge, on the other a plane perpendicular to
the section of the face, of which the first produces the
dodecahedron (§. 122.) ; and the other the hexahedron
(§• 124.).

The varieties are determined according to the mutual
inclination of the faces at the place of the edge of the hex-
ahedron. Every different inclination, greater than 90°
and less than 180°, yields a particular hexahedral Trigonal-
icositetrahedron .

§. 127. THE TETRACONTAOCTAHEDRON. DESIGNA-
TION OF THE TESSULAR FORMS.

In the seventh situation, the moveable plane is
the face of a Tetracontaoctahedron (§. 77.).

No two faces coincide in a single plane. The form there-
fore is contained under forty-eight faces, which are scalene
triangles, on account of their intersection with two faces
from the same, and with one face from the adjacent solid
angle of the hexahedron. The rhombohedral solid angles
are formed by six faces, the pyramidal solid angles by
eight, and the prismatic solid angles by four. All of them
are equiangular and digrammic. Hence the form will be a
Tetracontaoctahedron.

It may here be observed, that the seven forms thus con-
nected with the hexahedron, the hexahedron itself being
one of the number, perfectly agree with each other in re-
spect to the kind, number, and situation of their axes,
which is an immediate consequence from the method of
derivation employed. The same property does not extend
to the rest of the forms of several axes; and thus the num-
ber of the different kinds of axes in their peculiar situation,

184 TERMINOLOGY. §.

becomes the character by which it is possible to ascertain
whether or not a form of several axes belongs to the com-
pass of those which may be derived immediately from the
hexahedron. All these fonns are contained under the first
degree of regularity (§. 47.).

If, instead of a rhombohedron, we substitute the hexa-
hedron, or, instead of an isosceles four-sided pyramid the oc-
tahedron, and apply to them the modes of derivation describ-
ed above (§. 80. 81. 82.) : certain combinations between the
derived forms will likewise represent the whole compass of
those obtained by the process of the moveable plane ; some
of them even with their determined dimensions, which may
be considered as an advantage of this method. It requires,
however, some knowledge of compound forms (§. 34.),
upon the resolution or developement of which it depends.

The designation of the forms of several axes has been
framed upon a principle different from that followed in the
designation of the rest of simple forms. The great num-
ber of distinct kinds of forms, and the few varieties of each,
known or occurring in the compass of derived forms, have
been the reason why it was impossible to trace in the crys-
tallographic signs, all those relations of the forms to each
other which distinguish the designation of the fonns
derived from the four-sided pyramids and from the rhom-
bohedrons. The following method, although it does not
possess the advantage of expressing these relations, yet
is recommended by its brevity and distinctness.

Designate the hexahedron by the letter H, the octahe-
dron by O, the dodecahedron by D, the tetracontaoctahe-
dron by T ; these being the initial letters of their respec-
tive names. The initial letters of the icositetrahedrona
cannot be employed in the same way, because there are
three different kinds of such forms among those immediate-
ly derived from the hexahedron ; — namely, the hexahe-
dral and the octahedral trigonal-icositetrahedrons, and the
digrammic tetragonal-icositetrahedron. Designate the first
of these by A, the second by B, the third by C ; and add
. to them, as Avell as to the sign of the tetracontaoctahedrons ,

§. 128. OF THE CONNEXION OF FORMS. 135

the number which denotes the variety of these forms, as
contained in the preceding paragraphs 71-77*

§. 128. RESOLUTION OF FORMS BELONGING TO THE
FIRST DEGREE OF REGULARITY.

To resolve a form of several axes, means to pro-
duce from it two or more equal and similar forms
of several axes, the faces of which agree in num-
ber and situation with one-half or one-fourth of the
faces of the original form. These forms reproduce
the original one, if combined in the required po*
sition.

A form of several axes, produced by the resolution of an-
other, if it contains half the number of its faces, is termed
a Half; if it contains only one-fourth of the faces, it is
termed a Fourth of the resolved or original form.

Those halves must not be taken for half forms, or such
as might be obtained by cutting in two, one of the original
forms, as would be a simple pyramid. Nor are the fourths
real quarters of original forms, because they have not been
obtained by cutting in two, one of the preceding halves.

The method by which the resolution is effected, is the
following :

Place the given form in an upright position, so as to
make one of its rhombohedral axes vertical.

Call the upper terminal point of this axis a Principal
Point, the lower one a Subordinate Point, and transfer those
names to all the terminal points of rhombohedral axes, dis-
tant from the vertical one, for 109° 28' 16". In the hexa-
hedron, as represented Fig. 3G., the principal points are
A, B, B', B", and the subordinate points, X, C, C', C".

Enlarge now,

1. all the faces contiguous to the principal points, till those con-

tiguous to the subordinate points disappear ; or,

2. the alternating faces from the principal points, and those from

TERMINOLOGY. §.

the subordinate ones which arc parallel to them, again, till
the rest disappear : or,

3. the alternating faces from the principal points, and those
from the subordinate ones, which arc not parallel to them,
till the rest disappear.

The enlarged faces, if they can geometrically include a
space by themselves, will produce a form of many axes,
which is a Half, if only one of the three methods has been
applied ; a Fourth, if two at the same time, or subsequent-
ly, have been employed in resolving the original form.

If, in the first mode of resolution, instead of enlarging
the faces contiguous to the principal points, we enlarge
those from the subordinate ones, the result from the same
original form will be another half, equal and similar to the
first, but in a different situation from the other form.
The two halves can be brought into a parallel position, by
inverting the perpendicular axis of one of them. The posi-
tion now mentioned is called the inverse, in reference to the
other or normal position ; and one half of this kind is said
to be the Inverse of the other, which has been obtained in
the normal position.

A similar result takes place, if, in applying the second
mode of resolution, those faces are made to disappear,
which produce the half in the normal position, while the
others are enlarged. Both these kinds of halves are re-
markable for the parallelism of their faces, which, however,
is a consequence of the method of resolution applied.

The third method of resolution, if treated in the same
manner, enlarging those faces which had been made to dis-
appear before, and vice -versa, does not yield exactly the
same result. In respect to position, there is no difference
among the two halves ; but there is a difference according
to Right and Left, as mentioned in §. §. 67. and 76. The
same relation exists in the Fourths, which, like the halves
of the first and third method of resolution, have no paral-
lel faces.

Some of the forms derived from the hexahedron, do not
allow of any resolution at all ; either because half the num-

§. 128. OF THE CONNEXION OF FORMS. 137

her of their faces would not be sufficient to include a space
from all sides ; or because none of the methods mentioned
is applicable to them. The first is the reason why the
hexahedron, the other why the dodecahedron, have no
halves. Besides, it is not every form that can be resolved
by every one of the above-mentioned methods ; but cer-
tain properties of the form are required to render one of
these methods applicable.

Thejirst method supposes the faces of the form which is
to be resolved not to touch the terminal points of two rhom-
bohedral axes ; or, which is the same, not to touch a prin-
cipal point and a subordinate one at the same time. For
as it is required to effect the reverse on one of those points,
from what has been done on the other, it would thus be
requisite, that one and the same face should at the same
time be made to enlarge itself and disappear. For this
reason, the hexahedral trigonal-icositetrahedron cannot be
resolved according to the first method.

The second and third method supposes the number of
faces at the rhombohedral solid angles to be such as will
render it possible to enlarge the alternating ones. This
cannot take place, if the solid angles are formed of three
faces. In this case, the resolution too is impossible ; and
therefore, the two methods require the rhombohedral solid
angles to consist of six faces. The third method requires
moreover the condition of the first method to take place,
else it would be necessary to enlarge all the faces ; and con-
sequently no resolution at all could take place. By this
last condition, the hexahedral trigonal-icositetrahedron is
excluded, and the method becomes applicable only to the
tetracontaoctahedron, which, however, can be resolved ac-
cording to both the other methods.

The axes undergo very remarkable changes by the reso-
lution. The rhombohedral ones remain unaltered ; the
prismatic axes disappear entirely in all the halves ; the
changes in the pyramidal axes are various. If the third
method has been applied, they remain constant like the
rhombohedral axes ; they are changed into prismatic axes

138 TERMINOLOGY. §. 129. 130.

in the first and second processes ; and then there are no
more than three axes of this kind to be found in the solid.
In the resolution of halves in fourths, or in the application
of two of the methods at once, they disappear entirely.
The peculiar character of the halves is, therefore, that they
have six axes less ; of the fourths, that they have nine axes
less than the original forms.

The halves arising from the first method of resolution,
and the fourths, into the formation of which this method like-
wise enters, assume the general aspect of the tetrahedron.

The crystallographic signs of these forms are obtained
by indicating a division by the numbers 2 and 4, the re-
ferences as to position by the signs + and — , and those to
Bight and Left, by the letters r and 1 prefixed to the crys-
tallographie sign of the original form.

§. 129. THE TETRAHEDRON.

The half of the octahedron is the Tetrahedron
(§. 57.)-

The octahedron allows of the application of the first pro-
cess. The number of faces of its half is four, and these
faces are perpendicular to the rhombohedral axes.

The crystallographic signs of the two tetrahedrons, of
which one is in the normal, the other in the inverse posi-
tion, Figs. 13. 14., as halves of the octahedron, are — - (o)
and — 5L ((/).

§. 130. THE HEXAHEDRAL PENTAGONAL-DODECA-
HEDRON.

The half of the hexahedral trigonal-icositetrahe-
dron is the hexahedral Pentagonal-dodecahedron
(§. 66.).

Here the second process must be applied. If the al-

§. 131. OF THE CONNEXION OF FORMS. 139

ternating faces contiguous to the principal points disap-
pear, and at the same time those which are parallel to the
former at the subordinate points; every one of the re-
maining enlarged faces is intersected by five others, and
thus assumes a pentagonal figure. The number of faces ig
evidently twelve ; the form produced will therefore be a
pentagonal-dodecahedron, which is a hexahedral one because
the second mode of resolution does not change the general
aspect of the form. The latter property also might be
derived from the equality and similarity of the eight solid
angles of three faces, which correspond to those of the
hexahedral trigonal-icositetrahedron, which are formed by
six faces (§. 66. !.)•

The crystallographic signs of the hexahedral pentagonal-
dodecahedrons, one of them being in the normal, the other

in the inverse position, Figs. 19. 20., are — — (a) and

— — — (a'), where n denotes the variety which is to be
expressed.

§. 131. THE DIGBAMMIC TETRAGONAL-DODECAHE-
DRON.

The half of the octahedral trigonal-icositetrahe-
dron is the digrammic Tetragonal-dodecahedron
(§. 64.).

The resolution is effected after the first method.

Each of the enlarged faces is intersecte'd by four others,
of which two belong to the same, and two to other princi-
pal points. Th us they become four-sided, and their number
is twelve. Hence the form is a tetragonal-dodecahedron ;
and since it assumes the general aspect of a tetrahedron,
the first mode of resolution having been applied, it will be
that described in §. 64. 1., or the digrammic tetragonal-
dodecahedron.

The crystallographic signs of these forms in the normal

140 TERMINOLOGY. §. 132. 133.

and in the inverse position, Figs. 17« 18., are — (&) and

§. 132. THE TRIGONAL-DODECAHEDRON.

The half of the digrammic tetragonal-icositetra-
hedron is the Trigonal-dodecahedron (§. 61.)

This resolution is likewise effected after the first me-
thod.

Each of the enlarged faces is intersected by two faces
of the same, and one of another principal point. These
faces, isosceles triangles, are twelve in number ; and the
half therefore is a trigonal-dodecahedron.

The crystallographic signs of these forms in the normal

and in the inverse position, Figs. 15. 16., are — (c)

§. 133. THE TETRAHEDRAL TRIGONAL-ICOSITETRA-
HEDRON, THE TRIGRAMMIC TETRAGONAL-ICOSI-
TETRAHEDRON, AND THE PENTAGONAL-ICOSITE-
TRAHEDRON.

The halves of the tetracontaoctahedron are,

1. The tetrahedral Trigonal-icositetrahedron
(§. 70.);

2. The trigrammic Tetragonal-icositetrahedron
(§. 75.) ; and

3. The Pentagonal-icositetrahedron (§. 76.).

Any one of the three methods of resolution may be ap-
plied to the tetracontaoctahedron; and this form conse-
quently has three kinds of halves, which at first sight seem
to agree with each other only in their being icositetrahe-
drons.

§. 133. OF THE CONNEXION OF FORMS. 1-il

The first mode yields the tetrahedral trigonal-icositetra-
hedron. The solid angles of six faces of the original form
are likewise to be found in the half, because according to
this method all the faces contiguous to the principal points
are enlarged. The faces are intersected by two other faces
of the same principal point, and by one face contiguous
to another, exactly as in the preceding case ; the faces re-
main triangles, which are likewise scalene, but not similar
to those of the original form. The half therefore is a tri-
gonal-icositetrahedron, which owes its tetrahedral aspect to
the application of the first process.

In the designation of these forms, it is necessary to indicate
the mode of resolution upon which they depend. The sign of
the tetrahedral trigonal-icositetrahedrons in both, the normal

rp

and the inverse positions, Figs. 25. 2G., will therefore be — _

T1*!

and r . The trigrammic tetragonal-icositetrahedron is

the result of the second mode of resolution. The intersec-
tion of the enlarged faces takes place with two faces of the
same principal points, and with two faces contiguous to adja-
cent subordinate points. In comparing the pyramidal solid
angle of the tetracontaoctahedron with the corresponding
solid angle in its half, we find, that of the eight faces which
constitute the first, alternating Pairs of faces are enlarged,
and not the alternating single faces. From this it becomes
evident that the angle formed by four faces above the face
of the hexahedron cannot consist of equal edges, and that
consequently the faces of the tetragonal-icositetrahedron
thus formed cannot by a straight line be divided in two isosce-
les triangles. Besides the two kinds of edges meeting in the
solid angle of four faces, the form contains still another kind
of edges, which meet in the rhombohedral solid angle form-
ed by three planes ; it is therefore a trigrammic tetragonal-
icositetrahedron (§. 75. 2.).

The crystallographic signs of the trigrammic tetragonal
icositetrahedrons in both, the normal and the inverse posi-
tions, Figs. 27. 28., are — and — —
2 ii 2 ii

TEHMIXOLOGY. §. 134.

If we apply the third method, the result will be a pen-
tagonal-icositetrahedron. Here all the alternating faces
of the original form are enlarged, while the rest disappear,
so that each face is intersected by five others, of which two
belong to the same principal point, the rest to the adjoin-
ing subordinate points, and the faces assume therefore a
pentagonal figure. Of the pyramidal solid angle of the te-
tracontaoctahedron likewise the alternating faces are en-
larged ; this solid angle therefore remains a pyramidal one,
and the axis which passes through it, a pyramidal axis.
These properties will suffice for determining the form to be
a pentagonal icositetrahedron.

The crystallographic signs of the pentagonal-icositetra-
hedrons, in reference to their being as it were twisted to

the right or to the left, Figs. 29. 30., are r IL^ and 1 ~~.

2 2

It would be superfluous to add here the number iii for in-
dicating the mode of resolution, except in comparing these
forms with other halves of the tetracontaoctahedron.

§. 134-. THE TETRAHEDRAL PENTAGONAL-DODE-
CAHEDRONS.

The three icositetrahedrons of the preceding pa-
ragraph, which are halves of the tetracontaoctahe-
dron, allow of a farther resolution, and then yield
the fourths of that form, The fourths of the te-
tracontaoctahedron, are the tetrahedral Pentagonal-
dodecahedrons (§. 67.).

The resolution of the tetrahedral trigonal-icositetrahe-
dron, is effected by enlarging its alternating faces, till they
limit the space by themselves. Each of the enlarged faces
is intersected by five others ; and the resultant fourth is
therefore a pentagonal-dodecahedron, whose general aspect
is that of the tetrahedron, on account of the application of
the first process. The fourth itself is the tetrahedral
pentagonal-dodecahedron.

§. 134". OF THE CONNEXION OF FORMS.

This method of resolution produces the same result as if
the first and the second, or the first and the third, had been
applied to the tetracontaoctahedron. The first requires
the faces of the subordinate points to disappear ; the others
require in this case only the enlargement of the alternating
faces of the remainder.

If in the icositetrahedron considered above, we enlarge
those faces which have disappeared, and vice versa, let those
disappear which have been enlarged before ; the result
in respect to that obtained first, will be a Left tetrahedral
pentagonal-dodecahedron. But the icositetrahedron may be
resolved both in the normal and in the inverse position.
Hence both the differences, as to Right and Left, and as
to Normal and Inverse, come into consideration in the te-
trahedral pentagonal-dodecahedron.

The trigrammic tetragonal-icositetrahedron may be re-
solved after the first process, by enlarging all the faces
contiguous to its principal points, &c. Each of these faces;
is intersected by five others, two of which belong to the same,
the other three to adjacent principal points. For the rest,
every thing is as above ; and the trigrammic tetragonal-
icositetrahedron yields exactly the same fourths.

The pentagonal-icositetrahedron is resolved according to
the first method, by enlarging all the faces contiguous to
the principal points, &c. Each of these faces again is in-
tersected by five others, and the result of the resolution is
likewise a tetrahedral pentagonal-dodecahedron.

Thess four pentagonal-dodecahedrons, different on one
side as to right and left, on the other as to their normal
or inverse position, reproduce in binary combinations the
icositetrahedron s, and in a quadruple combination the tetra-
contaoctahedron itself, from the resolution of whicli they
have been obtained.

The first of these differences is expressed by the letters
r and 1, the second by the signs -4- and — , prefixed to

— the general notation of one-fourth of the tetraooiHa-
4

octahedron. The four dodecahedrons will therefore be ;

144 TERMINOLOGY. §. 184.

+ rlE (0 Fig. 21., -r^(0 Fig. 22.,
4 4

+ 1 IE (*'') Fig. 23., — 1 IE (*'") Fig. 24.
4 4

These four solids yield six binary aggregates :

1. + r IE. — r IE, which is = 1 IE Fig. 30. ;

4 4 2iii

2. + r IE. + 1 IE, which is = + — Fig. 25. ;

4 4 2i

3. + r IE. _ 1 IE, which is = — IE Fig. 28. ;

44 2 ii

4. — rlE. +1 IE, which is = + — Fig. 27. ;

44 2 ii

5. -rlE. _ 1 IE, which is = -IE Fig. 26. ;

4 4 2i

6. + 1 IE. — 1 IE, which is = r IE Fig. 29.

44' 2iii

Of these, 1 and 6 are pentagonal-icositetrahedrons, 1 is
the left, and 6 the right one ; 2 and 5 are tetrahedral
trigonal-icositetrahedrons, 2 is in the normal, and 5 in the
inverse position ; and 3 and 4 are trigrammic tetragonal-
icositetrahedrons, of which 4 is in the normal, and 3 in
the inverse position. Every two homogeneous forms of
these six reproduce by combination the tetracontaoctahe-
dron itself.

The halves and fourths belong to the second degree of
regularity.

The preceding methods of resolving the original forms
of several axes yield all those forms which have been de-
scribed above (§. 57. — 77«)> and which could not be ob-
tained by immediate derivation. Thus, resolution com-
pletes what by derivation would have remained imperfect ;
and we are entitled to consider as complete the number
of simple forms of several axes.

The method of resolving simple forms, is not confined
to those which possess several axes, in as much as it may
also be applied to pyramids of every description, and even

§. 135. GENERAL IDEAS OF FORMS. 145

to rhombohedrons. This requires, however, certain re-
strictions, which will be mentioned along with the results
of that process, in some of the paragraphs referring to the
Character of Combinations.

GENEHAL IDEAS OP SIMPLE FORMS.
§. 135. SYSTEM OF CRYSTALLISATION.

The assemblage of simple forms derivable from
one fundamental form (§. 87.), independent of all
consideration of its dimensions, is termed a System
of Crystallisation, and denominated after the fun-
damental form, from which it is derived.

The term System of Crystallisation has often been made
use of in a sense different from that of the present defini-
tion.

A System of Crystallisation is not a mere aggregation of
forms, according to their different kinds, or according to
certain properties peculiar to them ; but it is the Assemblage
of those Relations which take place among certain simple
forms, in as far as they are derived from one fundamental
form.

From the above mentioned four fundamental forms, there
arise four different Systems of Crystallisation ; and no more
systems are possible, if there exist only four forms of this
kind. We have no reason to assume a new fundamental
form in Crystallography, unless we have discovered or ob-
served a form, which cannot be derived from any one of
those which are known. As this is the case with the sca-
lene four-sided pyramids with an inclined axis (§. 90.), the
number of systems of crystallisation will be increased to
six, or perhaps to seven. In none of these systems can
there be any objection against considering all those forms

VOL. i. K

14)6 TERMINOLOGY. §. 13G.

which geometrically must enter into it, as really belonging
to them, although nature should not as yet have produced
them as simple forms.

The System of Crystallisation derived from the rhombo-
hedron is termed the Rhombohcdral System. The system
whose fundamental form is the isosceles four-sided pyramid,
is called the quadrato-pyramidal, or, in shorter terms, the
Pyramidal System ; that from the scalene four-sided pyra-
mid is the rhombeo-pyramidal system, which, on account of
the great variety of prisms which it contains (§. 91. 95. 98.),
receives the denomination of the Prismatic System ; and
that from the hexahedron is called the Tessular System, not
the hexahedral one, in order to intimate, that experience
has not as yet given any reason for assuming another sys-
tem of tessulur forms, although geometrically we may con-
ceive a system of forms of several axes, which stands in the
same relation to the regular dodecahedron of Geometry, in
which the tessular system is to the hexahedron. The other
systems, comprising the Hemiprismatic (§. 153.) and the
Tetartoprismatic (§. 154.) forms, have not as yet been pro-
vided with particular denominations.

§. 136. SERIES OF CRYSTALLISATION.

The fundamental form being supposed to possess
determined dimensions, the assemblage of derived
forms becomes a Series of Crystallisation.

The system of crystallisation is an idea of very great ex-
tent, but it is liable to certain restrictions in particular de-
terminate cases. These restrictions consist in ascertaining
or fixing the dimensions of the fundamental forms in the
different systems : for the relations of the derived forms
among each other and to the fundamental one are general,
and must remain unchanged in all series of crystallisation,
which belong to the same system. If, therefore, the funda-
mental form is supposed to possess determined dimensions,
the derivation will yield a System of Crystallisation, which.

§. 137. GENERAL IDEAS OF FORMS. 147

is determined for a particular case; or, which is the same
thing, it will produce a Series of Crystallisation.

From these considerations it follows that the tessular sys-
tem, being the only one in which the simple forms possess
invariable dimensions, will comprehend only one Series of
Crystallisation ; while all the other systems, which possess
variable dimensions, may comprehend an unlimited num-
ber of such series, that is to say, as many as there may be
differences in the dimensions of their fundamental forms.

Since these series represent the systems of crystallisa-
tion themselves, though determined for particular dimen-
sions ; it is plain that they must not be confounded with
the series of homogeneous simple forms (§. 85.) considered
above, as, for instance, with the series of rhombohedrons,
or of the different pyramids. It is also evident that, if
different members of the same series, for instance R,
R 4- 3, II — 1, &c., be considered as fundamental forms,
they will not yield different series of crystallisation, because
upon this supposition, the results obtained by derivation
will be identical.

§. 137. THE SYSTEM OF CRYSTALLISATION DETER-
MINED FROM A SINGLE FORM.

From the observation of any single form, except
the right rectangular prism, the System of Crystal-
lisation to which this form belongs may be inferred.
This extends to the Series of Crystallisation, if the
dimensions of the form be given or known.

The facility of ascertaining the System of Crystallisation1,
by observing one single form, is obvious, and is founded
upon the difference among those simple forms, which consti-
tute the different systems. Thus the rhombohedral system
is composed of rhombohedrons, of six-sided pyramids, and of
six-sided and twelve-sided prisms ; the pyramidal system, of
isosceles four-sided pyramids, and of scalene eight-sided py-

US TERMINOLOGY. §. 138.

ranikls, besides rectangular four-sided prisms, and eight-
sided prisms of alternately equal angles, &c. It is therefore
not very difficult from one form being known, to trace or
find out the system to which it belongs. The only excep-
tion occurring here, is the right rectangular four-sided
prism, which may belong, as a simple form, or as a com-
pound one, to three, or at least to two different systems,
if we abstract from the plane perpendicular to the axis.
As a simple form, it is the hexahedron, and belongs to the
tessular system ; as a compound form, which consists of two
simple ones (P — os and P -j- »), it is a right rectangular
prism, and belongs to the pyramidal system ; and as a com-
pound form, consisting of three simple ones (P — eo.
Pr -f oo. Pr 4- GO.), it is likewise a right rectangular prism,
but belongs to the prismatic system. The abstract geome-
trical consideration of these forms, yields no characters by
which they could be distinguished from one another, though
the means will be afterwards (§. 159.) pointed out, by which
this uncertainty may be removed, and which principally
depend upon the connexion of certain forms with each
other, and upon several peculiarities occurring along with
them in natural bodies.

If the given form possesses finite dimensions, these are
either known, or may be found by immediate measure-
ment ; in both cases, therefore, it is possible to obtain those
of the fundamental form, and consequently also of the se-
ries to which it belongs.

III. OF COMBINATIONS.

OF COMBINATIONS IN GENERAL.

§. 138. DEFINITION.

A compound form is termed a Combination
(§. 34.).

§. 138. OF COMBINATIONS. 149

A mineral occurring in a compound form is nevertheless
a simple mineral (§. 21.), and may be an individual; for
that composition, which is the subject of our present consi-
deration, regards only the external form of the mineral.
Yet an individual, affecting a compound form, or more than
one simple form at a time, may be imagined to represent two,
three, or more individuals, if we suppose that all the na-
tural historical properties of the individual, with the ex-
ception of the geometrical ones, are connected with every
one of the simple forms contained in the combination.

A combination may, in some instances, assume the as-
pect of a simple form, being contained under faces which
are equal and similar to each other (§. 35.). This takes
place when two equal and similar forms combine in differ-
ent positions, which positions, nevertheless, are always pe-
culiar to the system in which these forms are found. Com-
binations of this kind may be ascertained to be such, and dis-
tinguished from really simple forms, either by the number
of their faces, which is greater than that produced by deri-
vation, or by their relations and the position, which exclude
these apparently simple forms from the series to which, as
simple forms, they would necessarily belong.

The form of a combination is the space contained at the
same time within all the simple forms constituting it.
Hence, none of the angles of incidence of a combination
can be greater than 180°, or re-entering angles. Such angles
are produced, though not always, if two or more indivi-
duals, of the same or of different forms, are connected in
different positions; these compositions will be properly
considered in §. 178. &c.

The number of forms entering into a combination, is un-
determined. There may be only two, but there also may
be a great number of them. A combination containing two
simple forms is also termed a Unary combination ; one con-
taining three simple forms, a triple combination, &c. The
exact knowledge of binary combinations is the most in-
teresting department of Crystallography, in as far as it
refers to compound forms. The knowledge of binary com-

150 TKEMIXQLOGY. §. 139.

binations may be considered as the elements of the know-
ledge of multiple combinations ; and a triple combination
may be resolved into two, a quadruple combination into

six, one consisting of n simple forms, into n . .~Z — '
binary combinations.

§. 139. FIRST LAW OF COMBINATION.

The first Law of Combination is : That the

combined simple Jbrms must belong not only to the

same System (§. 135.), but also to the same Series of

Crystallisation (§. 136.);'*Aiy must be derived

from one and the same Fundamental Form.

If one of the simple forms contained in a combination,
belong to a certain System of Crystallisation, the rest of
them also must belong to the same system. Let, for in-
stance, this form be a regular six-sided prism ; the rest of
the forms combined with it will belong to the rhombohe-
dral system ; and thus one single form recognised in a
combination, though it be a limiting one, will be sufficient
to determine the System of Crystallisation (§. 137-). But
the form recognised may be a finite one, and moreover it
may be, according to its dimensions, a member of the Se-
ries of Crystallisation peculiar to rhombohedral Lime-ha-
loide ; the rest of the forms will belong to the same series
(though from this only it does not necessarily follow that
the individual is rhombohedral Lime-haloide) ; and their
dimensions may be calculated, as soon as we know in what
relation they stand to that which has been determined.
Hence also, in this case, the observation of one single form
suffices for determining the whole Series of Crystallisation
to which all the simple forms of the combination belong
(§. 137.). Nature confirms this law in all combinations,
without any exception.

The only combinations to be considered in Crystallogra-

§. 140. OF COMBINATIONS. 151

phy are therefore such as are produced by forms belonging
to the same Series of Crystallisation.

§. 140. SECOND LAW OF COMBINATION.

The second Law of Combination is : That the
simple forms contained in the combination, must be
in such Positions towards each other as are peculiar
to them in the systems to which they belong.

According to the preceding derivations, the simple forms
of every system are obtained in certain determined posi-
tions. In these positions, and only in these, they join in
combinations. Thus, in the rhombohedral system we have
the parallel position and the transverse position ; in the py-
ramidal system the parallel position and the diagonal posi-
tion, &c. In most cases, the combined forms assume those
positions, in which they have been derived. Thus, in the
rhombohedral system, the subsequent rhombohedrons R and
11 4- 1 are in a transverse position, the alternating rhombohe-
drons 11 and 11 -f 2 in a parallel position, in regard to each
other. Yet there are some exceptions in this respect. The
position of It, the fundamental form of the rhombohedral
system, is considered as the normal one, to which the position
of all other forms is referred ; yet this rhombohedron some-
times appears in a transverse position, whilst other forms,
though according to the derivation obtained in the trans-
verse position, nevertheless affect the parallel one in the
combinations. If thus one and the same simple form appears
in both positions at once in a combination, a remarkable re-
sult will be obtained, after the necessary enlargement of
their faces, all other faces having been made to disappear.
They produce a form contained under equal and similar
faces, which assumes the aspect of a simple one (§. 138.),
though it is really compound, as results from the process
by which it has been obtained.

Nature confirms this second law relative to the position

TERMINOLOGY. §. 141.

of the forms as the preceding one (§. 139.)} in every combi-
nation.

§. 141. SYMMETRY OF COMBINATIONS.

The Symmetry of combinations is founded upon
the two laws in §. 139. and §. 140.

The Symmetry of combinations consists in the same-
ness of disposition of the faces, edges, and angles of each of
those simple forms which they contain, in respect to the
homologous parts of the other ; or it consists in the same-
ness of situation of the different edges and angles produced
by the combination of these simple forms. Symmetry re-
fers only to combinations, Regularity only to simple forms
(§• 45.).

All combinations produced by nature are symmetrical,
and experience thus confirms the truth of the two above-
mentioned laws ; since the symmetry of combinations de-
pends upon the relative dimensions and the position of
simple forms. These Laws of Combination, and not the
Symmetry of the latter, are fundamental laws in Crystallo-
graphy, because the latter is a mere and necessary conse-
quence of the former.

Sometimes there occur in nature apparent exceptions
to this Symmetry. Yet they are merely accidental, and
arise from an unequal and disproportionate enlargement of
certain faces of Crystallisation ; and this sometimes goes so
far, as to cause some of these faces entirely to disappear.
Sometimes, however, certain faces are enlarged, and others
diminished, according to constant laws, and then the sym-
metry is not destroyed altogether, though it assumes a pe-
culiar character, different from what it has been before.
The differences thus produced, are called the Character of
Combinations, and will be considered afterwards in greater
detail.

In considering the combinations, we must abstract from
all casual deviations from symmetry, as in like manner has

§. 142. 143. OF COMBINATIONS. 153

been done with certain irregularities of simple forms (§. 45.),
and reduce the combinations themselves to their peculiar
symmetry.

§. 142. EDGES OF COMBINATION.

The edges, in which the faces of two different
forms contained in a combination, meet or intersect
each other, are termed Edges of Combination.

Compound forms contain sometimes a great many edges
of combination : in this case they are determined and dis-
tinguished from each other by attending to the simple
forms, between the faces of which they are situated, or
whose intersections they represent. What has been said
in §. 29. 32. of edges in general, applies likewise to edges
of combination.

It sometimes happens that the edges of the simple forms
disappear entirely in a compound form, so that every edge
to be met with is an edge of combination.

§. 143. DEVELOFEMENT OF COMBINATIONS.

The developement of a combination consists in
determining, 1 . the kind of all the simple forms con-
tained in it ; 2. their peculiar position ; and 3. their
relations to each other.

It is not difficult to recognise the kind of simple forms
contained in a combination. For this purpose, enlarge one
set of homologous faces which it contains after the other,
till the rest disappear ; and the kind of the form will be-
come evident, from their number and disposition. Each
of these forms is obtained at the same time in their peculiar
position.

The determination of the mutual relations of the forms
is in many cases somewhat more circumstantial. They
might be derived immediately from their dimensions,

154 TERMINOLOGY. §. 143,

comparing them with each other, as obtained from the ob-
servation of the angles of incidence at the edges of these
simple forms themselves, or at the edges of combination :
this process, however, being deficient in geometrical preci-
sion, cannot lead to any generality, and is therefore not an
appropriate foundation for a scientific method of Crystallo-
graphy. Besides, it would suppose a great number of mea-
surements, which, in establishing general laws, we must
avoid as much as possible, because the crystals themselves
are very seldom found in such perfection, and under such
circumstances, as to allow of any observations of this kind,
upon the correctness of which we might rely.

The method of Crystallography intended for the use of
the Natural History of the Mineral Kingdom, and em-
ployed in the present work, is founded solely upon the si-
tuation of those edges, in which the faces of several simple
forms intersect each other ; that is to say, of the edges of
combination ; and this method is therefore independent of
all measurement. The situation of those edges is a conse-
quence of the relations among the simple forms, and it is
changed as soon as any change takes place in these. It
will be possible, therefore, to determine the simple forms
contained in the combination, provided the situation of
the edges affords sufficient data. Only if these be wanting,
it will be necessary to resort to immediate measurement.

The number of data required for this purpose, de-
pends entirely upon the quality of the form itself. Thus
a rhombohedron depends upon a single datum ; for in or-
der to determine it exactly, nothing is required but to
know the relative length of its axis, or what place it occu-
pies in one of the series developed above. At the same
time, it appears whether it be a member of the principal
series or of a subordinate one. For a scalene six-sided
pyramid, two data are required, in order to determine, first,
the rhombohedron from which it is derived, and then with
what number the axis of this rhombohedron must be mul-
tiplied in the derivation. In general, since the situation
of any plane is perfectly determined by three points given

§. 143. OF COMBINATIONS. 155

in it, there is no possible case, in which more than two data
are required, each of these data consisting in the situation
of a straight line upon one of the faces of that form.

This method of determining the relations of simple
forms contained in a combination, is evidently founded
upon the knowledge of the series produced by these forms,
which have been explained above : by these it acquires a
perfect generality, because in every stage the same rela-
tions exist among the members.

The developement itself may be effected either analyti-
cally or synthetically. The synthetical method speaks more
plainly to the eye, and is therefore particularly recom-
mended to beginners. The analytical method is more easy,
elegant, and general. Several examples of the synthetical
method are contained in the course of this work ; and since
it would far exceed its limits to treat of these methods at
large, I shall only subjoin a short sketch of the process of
the analytical method.

Let ABC, A'B'C', Fig. 49., represent the faces of two
forms of the rhombohedral system, for instance, of two sca-
lene six-sided pyramids, whose horizontal projection is the
same, and which are placed in a parallel position. . The
lines CB, C'B' will intersect each other in the point G, and
six points situated like G will be common to both the forms.
The points G, G &c. are situated in a horizontal plane, per.
pendicular to the axis AX in M, the centre of the form ;
they are constant in all forms of the rhombohedral sys-
tem ; for though the situation of the points C, C' and
B, B' may vary, yet this never can have any influence
upon their intersection in G. In the other systems of
crystallisation, the situation of the points G, G &c. is
not invariable ; but it may easily be shewn, that this si-
tuation depends upon the diagonals of the bases of the
forms combined.

The acute terminal edges AC, A'C', intersect each other
in the points G', G', &c., which points, therefore, are likewise
common to both the forms. The situation of these points
is variable, and depends upon the relations of the axes, be-

156 TERMINOLOGY. §. 143.

longing to the combined forms. If A, or A', or both at the
same time change their places, G', G', &c. necessarily must
change theirs likewise.

The straight line GG', joining the points G' and G, is
the intersection of the faces of the two forms, and repre-
sents, therefore, their edge of combination (§. 142.) ; it lies,
on that account, both in the plane ABGG', or in the face
ABC of the one, and in the plane A'B'GG', or in the face
A'B'C' of the other pyramid. Hence it appears that the
situation of the line GG' depends upon the relations of the
bases, or of the horizontal projections, and upon those of the
axes ; or in gen'eral, upon the dimensions of the combined
forms themselves.

If we now produce the obtuse terminal edges AB and
A'B' of the combined pyramids, till they intersect each
other in F ; F will again be a point common to both the
forms. Hence it follows, that F, G and G' must be situ-
ated in one straight line ; and that if the one of the va-
riable points F and G' moves, the other likewise must be
affected by this alteration. This demonstrates the imme-
diate dependence of the situation of F upon the dimensions
of the combined forms.

The horizontal plane HZ intersects the obtuse terminal
edges of the pyramids in E and E'. The situation of these
points, or their distance from the centre M, is likewise va-
riable ; but it depends upon the dimensions of the combined
forms, exactly as the rest of the variable points, and be-
comes determined for determined forms.

Thus the length of the lines EF or E'F will be perfectly
determined, being a function of the above-mentioned re-
lations.

The line EF or EF' is termed the Line of Combination.
Its length can be measured by comparing it with the ter-
minal edge, or with the diagonal, of which it is a part, or
in which it lies, if produced to a sufficient length. A single
equation is sufficient for expressing this line in the rhom-
bohedral system. Two expressions are required in the
pyramidal system, on account of the differences arising

§• 14-3. OF COMBINATIONS. 157

from the parallel and the diagonal position of the forms
themselves, and of their horizontal projection, which dif-
ferences cannot be comprehended in a single formula. A
single equation again is sufficient in the prismatic system,
where there exists no such difference.

These equations contain every possible case in respect to
the kind and the position of forms within one and the same
system, and to certain differences in the edges of combina-
tion, in as much as these may be produced by faces conti-
guous either to the same, or to different apices ; and which
again belong either to that side which may be conceived to
be turned towards the observer, or to the opposite side
of the forms under consideration. These differences are
expressed in the equations by the addition of the signs 4-
and — .* The possibility of thus comprehending every
binary combination of a system in one, or, at the utmost,
in two expressions, is at the same time the most convincing
proof of the simplicity and generality of the method.

I shall now shortly explain the use of the Line of
Combination, in the developement of compound forms,
which is here reduced to the determination of the relations
among the simple forms contained in the compound one,
that are already known as to their kind and position.

Let ABC, A'B'C', Fig. 50., represent the faces of the
same forms as in Fig. 49. ; and the points G, G', F be
identical with those in the same figure which are marked
by the same letters. Combine now a third form with these,
whose face is A'^'C", and whose dimensions are such as
to have the points G and G' common to all these three
forms. The points G must always keep the same place, if
we suppose the horizontal projection to be equal. The
edge of combination thus produced between the new form,
and any one of the others, will evidently coincide with that

* Vide Gilbert's Annalen der Physik. 1821. 8., where
these formulae have been published, along with examples
of their application.

158 TERMINOLOGY. §. 144.

in which tLe faces of the two first forms meet. Hence the
algebraic expression of the Line of Combination must be
exactly the same for the third form and the first, or the
third and the second, which is found when calculating the
Line of Combination for the first and the second form
among themselves.

After these preparations, the length of the Line of Com-
bination EF or EF' must be calculated from the known or
given dimensions of the two first forms. An expression
will be obtained for it, finite or infinite, positive or nega-
tive, in which every single quantity is determined.

The same line EF or EF' must also be calculated
from the dimensions, either of the first and the third, or of
the second and the third form. An expression similar to
that mentioned above will be found, in which every thing
is known, except what refers to, or is dependent upon,
the dimensions of the third form, to ascertain whose re-
lations is the object of the inquiry. The two values of
EF. or EF' being equal to each other, they maybe joined
in an equation to be resolved for one of the unknown quan-
tities.

If this equation contains only one unknown quantity,
the resolution yields its value at once, and by this the
perfect determination of the third form is obtained. But
if there are two unknown quantities, the process must be
repeated, by constructing another equation, for a case in
which the unknown form which is to be determined com-
bines in the same way with other known forms ; the second
equation is procured in the same way as the first.

In the application of this process, we may take advantage
of many circumstances, which may very often render the
second equation superfluous ; this, however, is so obvious
in every particular case, that it will not be necessary here
to enter upon this subject more at large.

§. 144. MAGNITUDE OF THE EDGES OF COMBINATION.

To a complete knowledge of a combination is also

§. 144. OF COMBINATIONS. 159

required the knowledge of the angles at which the
faces of different forms intersect each other, or of
the edges of combination.

In every particular case, the magnitude of the edges of
combination may easily be calculated from the dimensions
of the simple forms. There is, however, also a general so-
lution of this problem, effected in every system by the same
number and kind of equations as those for the Line of
Combination.* Those parts by which the simple forms
are determined, enter into these equations as variable quan-
tities, whose real value is found by the developement
(§. 143.). These values being substituted instead of the
variable quantities, we obtain trigonometrical functions for
the Edge of Combination.

The application of these equations pre-supposes the di-
mensions of one of the forms to be known. These dimen-
sions must be found by immediate measurement, whenever
the form belongs to one of the systems whose dimensions are
variable. If in a species one of the forms, for instance, the
fundamental one, is known in respect to its dimensions, no
new measurement is required for the combinations of this
species, provided the situation of the edges contain suffi-
cient data for their developement. Hence it appears that
we must endeavour to ascertain the dimensions of the fun-
damental form, with the utmost accuracy, but at the same
time also that the measurement of different forms of the
same series may be useful in correcting each other. This
subject, however, will be treated of more at large in the
Elementary Treatise on Crystallography.

The designation of compound forms must be founded
upon the relations of the simple forms among each other.
It will therefore be sufficient to indicate, by their peculiar
signs, the simple forms, in order to express the combinations

* Gilbert' s Annalcn, 1. c.

160 TERMINOLOGY. §. 14)5.

which they produce. The signs of the simple forms follow
each other in the designation of the compound one in a cer-
tain order, so as to let those precede which refer to forms
whose faces are more inclined to the axis, and those follow
whose faces are less inclined or parallel to the same line.
The signs of two different forms are separated by a full
stop. The sign of a combination of R and R + 1 will
therefore be

R. R + 1.

An example of the designation of a compound form con.
taining a greater number of simple ones will be found in
§. 148.

From the preceding observations it is evident, that the
designation contains every thing required for calculating
the combinations.

OF THE COMBINATIONS OF THE DIFFERENT SYSTEMS OF
CRYSTALLISATION.

§. 145. RHOMBOHEDRAL COMBINATIONS.

A combination of the rhombohedral system is
more particularly said to possess a Rhombohedral
Character, if each of the simple forms contained in
it appears only in one position, but with the whole
number of their faces.

Hence a combination of rhombohedrons, six-sided pyra-
mids, and prisms, exhibits a rhombohedral character, if these
forms are found in only one position, and if those which
cannot assume any other position, as the isosceles six-sided
pyramid, and the six-sided and the twelve-sided prisms,
appear with the number of faces peculiar to them. In all
other cases the combinations receive a particular denomina-
tion, provided the changes which take place in respect to
the enlargement of some of their faces, are not owing to

§. 145. OF COMBINATIONS. 161

an accidental irregularity (§. 31.). In general, and suppos-
ing, for the sake of brevity, only one series of rhombo-
hedrons, the binary combinations of this system are
i. 11 + n. R + n',

ii. R + n. (P 4- n')m/,

iii. R + n. P 4- n',

iv. (P 4- n)<". (P + n')m/,

v. (P 4- n)m. P + n',

vi. P + n. P 4- n'.

Some of the most common and remarkable of these com-
binations may be shortly noticed.

i. R + n. R + n'.

1. Let n' be = n + 1. The two forms are consecutive
members of the series §. 110., and as such in a transverse
position to each other. The edges of combination which they
produce are parallel among themselves, and at the same
time also to the terminal edges of the more acute rhombohe-
dron, and to the inclined diagonals of the more obtuse one.
Example, R — 1 (n) and R (P), or R (P) and R 4- 1 (r)
in rhombohedral Kouphone-spar. Vol. II. Fig. 120.
We may also argue inversely : when two rhombohe-
drons join in a transverse position, and produce edges
of combination of the above mentioned kind, that is to
say, parallel to each other, parallel to the terminal edges of
the more acute rhombohedron, and parallel also to the in-
clined diagonals of the more obtuse rhombohedron, the re-
lations between the two forms must be the same as those
between two consecutive members of the series §. 110.
This is a direct consequence from the derivation of those
forms (§. 108.).

2. Let n' be = n + 2 r, where r may be any whole
number. In this case an odd number of members of the
series §. 110. is wanting between the two combined forms,
the forms therefore are in a parallel position, and the edges
of combination produced are horizontal. Example, R (P)
and R 4- 2 (m) in rhombohedral Lime-haloide. Vol. II.
Fig. 115. A similar result is obtained from rhombohedrons

VOL. i. L

162 • TERMINOLOGY. §. 145.

belonging to different series (§. 1 1G), provided they are in a
parallel position. Any two rhombohedrons in a parallel
position intersect each other in horizontal edges, and their
dimensions therefore cannot be determined solely from a
combination of this kind.

The reason why the edges of combination here are ho-
rizontal is evident, because the horizontal projections being
supposed equal, those faces of two rhombohedrons in a
parallel position, which are contiguous to the same apex,*
must intersect each other in the line GG (§. 143.); and
this line itself is horizontally situated in the upright com-
bination.

3. Let n' be = — w, R + n' therefore = R — cc. This
form appears as a face perpendicular to the axis of the
rhombohedron R + n (§. 111.), and the resulting edges of
combination must evidently be horizontal, whatever value
n may affect. Example, R — co (o) and R (P) in rhom-
bohedral Alum-haloide. Vol. II. Fig. 111.

4. Let n' be = + co. Upon this supposition, R + n'
is = 11 + co, or a regular six-sided prism, the limit of the se-
ries of rhombohedrons (§. 111.). The edges of combination
produced by its alternating faces with those of the upper
apex of R -4- n are horizontal ; so are the similarly situat-
ed ones between the rest of its faces and the faces of the
rhombohedron contiguous to the lower apex ; and their
situation is independent of the dimensions of R + n (2.).
There are also inclined edges of combination to be found
here, which depend upon R + n. If we consider the re-
gular six-sided prism as a rhombohedron of an infinite axis
(§. 111.), they are produced by the intersection of those of
its faces which are contiguous to the lower apex, and
those faces of R+ n which are contiguous to the upper apex

* It would lead us too far to consider more at large the
intersections of faces contiguous to opposite apices, and
the edges of combination which they produce. A few ex-
amples contained in the subsequent part of the work, will
shew their application.

§. 145. OF COMBINATIONS. 163

of the combination, and vice versa. Ex. R (P) and II -f oo (c)
in rkomboliedral Lime-haloide. Vol. II. Fig. 114. The co-
sine of the angle at the horizontal edge, is equal to double
the cosine of the angle at the inclined edge of combination.
From the horizontal edges of combination between a rhom-
bohedron and a regular six-sided prism, we may infer that
the latter is R + oo and not P + OD (§. 118.). The demon-
stration of this depends upon §. 111.

ii. R + n. (P + n')m'.

1. Let n' be = n. Upon this supposition the forms be-
come co-ordinate (§. 112.). The edges of combination
which they produce are parallel to the edges of the rhom-
bohedron, and to the lateral edges of the pyramid, whatever
be the value of m'. The figure of the faces of the rhom-
bohedron remains a rhomb, and they appear contiguous to, or
in the place of the apices of tbje pyramid. Ex. R (P) and
(P)3 (r), or It (P) and (P)5 (y) in rhombohedral Lime-ha-
loide. Vol. II. Fig. 116. From the rhombic figure of the
faces, which is a consequence of the situation of the edges
of combination, follow the relations of the combined forms,
as is immediately evident from the derivation (§. 112.).

2. Let n or n' be = — 03. One of the forms becomes
= R — 03, and the edges of combination are horizontal
(i. 3.).

3. Let nbe = +oo;R + n therefore = R + eo. The
figure which the faces of this prism assume in the combi-
nation with a pyramid, is that of an irregular tetragon,
which may be divided by a horizontal line into two isosceles
triangles. The relative heights of these triangles are to
each other in the ratio of m' — 1 : m' -f 1. The more ob-
tuse triangle is produced by the intersection of the faces of
R + 03 with the upper faces of (P 4- n')m', while the more
acute one results from the intersection of the same faces of
R -f oo with the lower ones of (P + n^'. Ex. R + 03 (c)
and (P)5 (y) in rhombohedral Lime-haloide. Vol. II.
Fig. 116. The figure of the faces of the regular six-sided
prism c at once indicates it to be R -I- oo, and not P + so.

164 TERMINOLOGY. §. 145.

4. Let n' be = n — 1, and m' = 3. The combination
is = R + n. (P 4- n — I)3. Under these circumstances,
the forms are in a transverse position, because R + n — 1,
the rhombohedron from which the pyramid is derived, is
itself in a transverse position towards R 4- n. The faces
of R + n have the situation of the more acute terminal
edges of the pyramid ; and the edges of combination are pa-
rallel to each other, to the above mentioned acute terminal
edges, and to the inclined diagonals of the rhombohedron
Ex. R _ 1 (z) and (P _ 2)3 (i) in rhombohedral Ruby-
blende. Vol. II. Fig. 126. Inversely from the situation
of the edges, in which the faces of the two forms meet in
the given position, we may infer the above mentioned re-
lation to exist between the two combined forms.

In order to demonstrate this, let ABXC, Fig. 47., be
the principal section of the rhombohedron, from which the
pyramid is derived, AX its axis, and M3L half the axis of
the pyramid : 3LC becomes its acute terminal edge, and at
the same time the inclined diagonal of the rhombohe-
dron, whose plane touches the pyramid in this termi-
nal edge, if the horizontal projections of the two forms are
equal. Let now a = AX, be the axis of that rhombohe-
dron, from which the pyramid is derived ; and a' the axis
of the rhombohedron sought ; it follows in respect to the
pyramid, that

ar = 3m'--J a,

6

in respect to the rhombohedron, that

aP=§.a';
and on account of the equality of both expressions, that

3m' — 1 , ,

-y-

and a = . a'.

3m'— 1
If now we suppose m' = 3,

a becomes = £. a', or a' = 2. a.
and n' = 11 — 1.

But, let m' be = 2 ; the result, obtained in the same
way, will be

§. 145. OF COMBINATIONS. 165

a = f . a', or a' = |. a = f . 2°. a ;

and R + n = f R + n', or that member of the first subor-
dinate series, which belongs to R + n'.

Let now m' be = 5 ; we find that

a = f . a', or a' = |. a = |. 21. a,

and R + n = | R + n' + 1, or that member of the second
subordinate series, which belongs to R 4- n' + 1.

Each of the two co-efficients, f and that immediately fol-
lowing |, determines a particular subordinate series, which
may be distinguished by the name of the first and the second.
In itself it is quite arbitrary which of their members are
considered to be in the nearest relation to members of the
principal series. But it is very useful to fix upon a cer-
tain member, and this has been done here by supposing,
that, when the axis of R + n of the principal series is = 2n. a,
that of R + n of the first subordinate series is f . 2". a,
and that of R + n of the second subordinate series |. 2n. a.
Members of these dimensions are said to belong together,
or to be co-ordinate (§. 116.).

5. Let n' be = n — 2 and m' = 5. The sign of the
combination will be R + n. (P + n — 2)s. Under these
circumstances, the forms are in a parallel position. The
faces of the rhombohedron appear in the place of the more
obtuse terminal edges of the pyramid. The edges of combi-
nation are parallel with each other, with the above mentioned
terminal edges of the pyramid, and with the inclined dia-
gonals of the rhombohedron. Ex. R + 2 (m) and (P)5 (y)
in rhombohedral Lime-haloide. Vol. II. Fig. 116. We
may infer inversely from this situation of the edges, that
the above mentioned relations really take place.

For, making again use of Fig. 47., &B will represent the
obtuse terminal edge of the pyramid, but at the same
time also the inclined diagonal of that rhombohedron,
whose faces touch the more obtuse terminal edges of the
pyramid, their horizontal projections always being suppos-
ed equal. Hence, for the pyramid, we have

ad = ^LtJ. a,

166 TERMINOLOGY". §. 145.

for the rhombohedron

aQ=5- a';
therefore

6 3 m' + 1

If now we suppose m' = 5,

a becomes = £. a', or a' = 4. a,
and n' = n — 2.

If here we suppose m' = 3, we find

a = f. a'; a' = f. a = |. 2>. a,

and R + n = f R + n' + 1, the meml)er of the first sub-
ordinate series belonging to II 4- n' + 1.

For m' = 2, follows

a = f . a' or a' = |. a = |. 2°. a,

and R + n as £ R + n', that member of the second sub-
ordinate series, which belongs to 11 + n'.

6. Let n' be = n — 2, m' = 3. The combination is
R 4- n. (P + n — 2)3. The forms are in a parallel position ;
the more acute terminal edges of the pyramid coincide with
the terminal edges of the rhombohedron, and the edges
of combination are parallel as well among themselves as
also with both the mentioned terminal edges. Ex. R (P)
and (P — 2)3 (t) in rhombohedral Ruby-blende. Vol. II.
Fig. 126. From this situation and position we may in-
versely conclude, that the given relations really take place
among the forms.

For let 2CC be the more acute terminal edge of the pyra-
mid, in which the terminal edge of the rhombohedron is
situated. We shall have for the pyramid

for the rhombohedron

a? = ^. a',

and therefore 3 m' "" 1. a = £. a'.

6
Now m' being = 3, we obtain

a == |. a', or a' = 4. a = 22. a,
and 11 + n = R + n' + 2.
If m' is supposed = 2, we find

§. 145. OF COMBINATION;?. 167

a = f. a', or a' = f . a = .f. 21. a
and R + n = f & 4- n' + 1.

The number 5 substituted for m', gives

a = i. a', or a' = 7- a = |. 22. a,
and R 4- n = 1 11 + n' + 2.

7. Let n' be = n — 3, m' = 5. The combination is
R 4- n. (P 4- n — 3)5. The forms are in a transverse posi-
tion. The more obtuse terminal edges of the pyramid are
parallel to the terminal edges of the rhombohedron, and at
the same time also to the edges of combination arising from
the intersection of their faces.

Suppose the horizontal projections of the forms to be the
same ; the terminal edge of the rhombohedron will lie in
the more obtuse terminal edge of the pyramid. Hence
for the pyramid we have

~~6~~~

for the rhombohedron

&Q, = £. a',

and consequently IJ^-U. a = £. a'. •

. 6

For m' ;== 5 this expression gives

a = $. a', or a' = 8. a = 23. a,
and R 4- n = R 4- n' 4- 3 ;
for m' = 3,

a = \. a', or a' = 5. a = £. 22. a,
and R 4- n = f R 4- n' 4- 2 ;
for m' = 2,

a = f. a', or a' = £. a = |. 21. a,
and R 4- n = I R 4- n' 4- 1.

iii. R 4- n. P 4- n;.

1. For n' = n, the combination is R + n. P 4- n. The
forms are co-ordinate, and the faces of the pyramid appear
in pairs in the place of the terminal edges of the rhombo-
hedron. The edges of combination are parallel to these,
to the alternating terminal edges of the pyramid, and
among each other. In a combination, which, besides these

168 TERMINOLOGY. §. 145.

two forms, contains P + co (4.), the faces of the rhombohe-
dron retain their rhombic figure ; and this may be con-
sidered as a proof of the existence of the above mentioned
relations. This, however, follows immediately from the
derivation (§. 117.). Ex. R (s) and P (P, z) in rhombohe-
dral Quartz. Vol. II. Fig. 145.

2. Let n' be = n + 1. The combination is R + n.
P + n + 1. The faces of the rhombohedron appear instead
of the alternating terminal edges of the pyramid, and the
edges of combination are parallel to each other, to the al-
ternating terminal edges of the pyramid, and to the inclined
diagonals of the rhomhohedron. Ex. R (P) and P + I (•/•)
in rhombohedral Corundum. Vol. II. Fig. 121. From
this parallelism again follows the equal inclination upon
the axis of the alternating terminal edges in the pyramid,
and of the diagonals in the rhombohedron, and from this
the relations existing among the axes of the two forms.

3. n or n' = — co produces horizontal edges of combina-
tion (i. 3.).

4. Let n' be == + co , the combination R + n. P + ce.
The regular six-sided prism P + co is the limit of the
series of isosceles six-sided pyramids (§. 118.), and its faces
appear with parallel edges of combination in the place of the
lateral edges of R, + n, or the faces of the rhombohedron
terminate the regular six-sided prism, their figure being that
of a rhomb. Ex. R + 1 (r) and P + co (*) in rhombohe-
dral Emerald-malachite. Vol. II. Fig. 118. This rhombic
figure of the faces, or the situation of the edges in general,
affords the means of distinguishing a combination of R + n
and P + co from a combination of R + n and It + co
(i. 4.). The difference depends upon the position of P + 03
in respect to R +• n, and P + n (§. 118.).

5. Let n be = + co . The combination is R + n. P + n'.
The faces of R + co appear as rhombs in place of the late-
ral angles of P -f n'. This figure by itself is a sufficient
character for distinguishing the prism R + co in this com-
bination from P + co .

Q. If n and n' at the same time are = + ce, the com-

§. 145. OF COMBINATIONS. 16*9

bination therefore II + es. P + os ; the result is an equian-
gular, and if the faces are of equal extent, it is a regular
twelve-sided prism ; because the faces of one of the six-
sided prisms appear under equal inclinations in the place of
the edges of the other prism, their edges of combination being
parallel to each other and to the axis. Ex. B, + os (e) and
P + os (M) in rhombohedral Fluor-haloide. Vol. II. Fig. 149.
Hence the twelve-sided prism is a compound form, and not
a simple one. Also this combination depends upon the dif-
ferent situations of the two prisms.

iv. (P 4- n)m. (P 4- n')ra'.

1. Let n' be = n. The combination is (P 4- n)m.
(P 4- n)m/ ; the forms will be co-ordinate scalene six-sided
pyramids. The faces of the more acute pyramid are situ-
ated in the place of the lateral edges of the more obtuse
one. The edges of combination are parallel to each other and
to the lateral edges of both the pyramids. Ex. (P)3 (r) and
(P)5 (y) in rhombohedral Lime-haloide. Vol. II. Fig. 11G.
If, on the contrary, the edges assume the mentioned po-
sition, we may infer that n is = n', which is exactly what
follows from their derivation.

2. Let m' be = m, the combination will be (P + n)m .
(P + n')m. Suppose at the same time the forms to be in a
parallel position, or n' = n + 2 r (i. 2.). Under these cir-
cumstances, the edges of combination which they produce
become horizontal. For the transverse section of one of
these pyramids is similar to the transverse section of the
other (§. 113.), and therefore a plane passing through
those edges in which the faces of the two forms meet, will
be perpendicular to their axis. Ex. (P — 2)3 (t) and (P)3 (r)
in rhombohedral Lime-haloide. Vol. II. Fig. 129. This
situation of the edges cannot take place, if the scalene six-
sided pyramids are in a transverse position. From such ho-
rizontal edges, therefore, we infer not only the parallel
position of the two forms, but also that they are derived
according to the same m, or that m' is = m.

3. The horizontal situation of those edges is not altered,

170 TERMINOLOGY. §.

if n.or n' should become infinite and negative, whatever
may be the value of m and m'.

4. But let m' be = m, and n' or n = -f co : the combi-
nation therefore (P + co)™. (P + n')m or (P + n)m.
( P + co ) n . The edges of combination between, those faces
of the two forms, which are contiguous to the same apex,
must here likewise be horizontal, because the unequiangu-
lar twelve-sided prism is the limit of the series of pyramids,
and as such contains an identical transverse section (§. 115.).
Ex. (P)* («) and (P + os)* (c) in rhombohedral Fluor-ha-
loide. Vol. II. Fig. 148. The intersections of the faces from
two different apices, however, assume an inclined situation,
dependent upon the dimensions of the simple forms (i. 4.).
The inferences drawn in 2. extend also to the present case.

5. In the series of scalene six-sided pyramids,

... (P + n)5, (P + n + 1)», (P + n + 2)», (P + n + 1)*...
the law of progression is evident. Instead of n any whole
number, positive or negative, may be substituted, and the
series arbitrarily continued on either side. If now from
the above mentioned series we select a combination of any
two subsequent members, as (P + n)s and (P + n + I)3,
or (P + n + I)3 and (P + n + 2)2, &c. ; the' obtuse ter-
minal edges of every more obtuse pyramid appear in the
place of the acute terminal edges of the more acute mem-
ber, the edges of combination being parallel to each
other, and to the mentioned terminal edges of the two py-
ramids. Examples occur in rhombohedral Lime-haloide,
between (P)3 (r) and (P + I)2 (#), &c.

The situation of the edges is a consequence of the trans-
verse position of every two subsequent members of the
above mentioned series, and of its peculiar property, that
the more obtuse terminal edge of every lower member is
inclined to the axis at the same angle as the more acute
one of the higher member. We have in Fig. 47.
sin. 3BQ = __ (Sjnj^-.a __
VI(3ra + iy. 2««. a' + 36]'

rin. 3CP =

,/ ((3 ra' — I)2. 2"'. a» + 36]

§. 14)5. OF COMBINATIONS. 171

If in the hypothesis of (P + n)5. (P + n + I)3, we sub-
stitute in sin. &BQ, the number 5 instead of m, and in
sin. SCP the expression n + 1 for n', and 3 for m' ; we
obtain

4.22".a2 +36)'
two values which are equal.

If the above mentioned relations take place among the
consecutive members of the series, the parallelism of the
edges of combination always must follow. But these re-
lations are not a necessary consequence of that parallelism,
and many pyramids really exist and produce in their regular
positions, parallel edges of this kind ; and yet they are not
derived from rhombohedrons of the same series at all, or ac-
cording to other values of m, than those of 2, 3, and 5. This is
the case in the combinations of (P — I)3 (a) and (f P — I)3 (6)
in rhombohedral Ruby-blende. Vol. II. Fig. 126. In the
above mentioned series, one datum has been assumed,
upon which the rest is dependent. If this be changed, the
consequences also must be altered, without, however, in
the least affecting the parallelism of the edges of combi-
nation. Supposing the pyramids to be derived from the
members of the principal series, we may argue with per-
fect security, from the parallelism of the edges to the va-
lues of m, and determine the one whenever we know the
other. Restrictions of this kind frequently occur in general
solutions of crystallographic problems.

v. (P + n)m. P -f n'.

1. Let m be = 5, n' = n + 3. We have the combina-
tion (P -|- n)5. P + n + 3. The faces of the scalene pyra-
mid meeting in its obtuse terminal edges, appear in the
place of the alternating terminal edges of the isosceles one.
The edges of combination are parallel to each other, and to
the above mentioned terminal edges of the two forms. For
on these suppositions the inclination of the terminal edges of

172 TERMINOLOGY. §. 145.

the isosceles pyramid to the axis, is equal to the inclination
of the more obtuse terminal edges of the scalene pyramid
to that line. For the latter we have (iv. 5.)

sin. 2BQ = (3j*±Jl2l.t__ .

V [(3m+ I)2. 22». a2 + 36]'
and for the former

sin.2lBQ,= m/-2n/'a — .

^/(m'2. 22n. a2 + 4)'

two values which become equal, if in that for the scalene
pyramid we substitute 5 instead of m, and in that for the
isosceles pyramid the expressions, | instead of m, and
n 4- 3 instead of n'. Examples occur in (P — 2) s and
P 4- 1 (n) of rhombohedral Iron-ore.

2. Let m be = 3, n' = n 4- 2 ; the combination will be
(P 4- n)3. P + n 4- 3. In this case, similar to the pre-
ceding, the faces of the isosceles pyramid appear in the
place of the more acute terminal edges of the scalene one ;
the edges of combination being parallel among each other,
and to the above mentioned terminal edges, from the same
reason as in 1., because the inclinations of the terminal
edges to the axis are equal in these two pyramids. This
may be shewn by effecting the necessary substitutions in
the values of sin. <3CP, which here takes the place of
sin. 3BQ.

3. For n or n' = — co all the edges of combination be-
come horizontal (i. 3.).

4. Let n' be = 4- cs. The combination is(P 4- n)m. P + os.
The edges of combination are parallel to the lateral edges
of the pyramids. Ex. (P)3 (h) and P 4- co («) in rhombo-
hedral Ruby-blende. Vol. II. Fig. 126. The situation of
the edges distinguishes P 4- oo from R + co in ii. 3.

vi. P + n. P 4- n'.

1. The edges of combination are always horizontal, what-
ever may be the value of n or n', even though this be 4- 03
or — os. Ex. P + 1 (r) and P 4- 2 (ft) in rhombohedral
Corundum. Vol. II. Fig. 122.

§. 146. OF COMBINATIONS. 173

§. 146. DI-RHOMBOHEDRAL COMBINATIONS.

A combination of the rhombohedral system is
said to possess a Di-rhombohedral Character, if one
or more of the simple forms contained in it appear
at once in both their positions (§. 108.).

The rhombohedrons and the scalene six-sided pyramids
are the only forms of this system which may assume two
different positions ; for if the position of the other forms is
supposed to change, their faces resume the situation they
had before, or, properly speaking, another face exactly
takes the place of that which just has been removed from it.

1. Suppose, therefore, in §. 145. i., n' to be = n, or the
combination, R + n. — R -f n. The sign — indicates, that
the two rhombohedrons are in a transverse position to each
other. The combination assumes the aspect of an isosceles
six-sided pyramid (§. 35.), which is a simple form, and the
edges of combination are parallel to those lines, which, in
the faces of the single rhombohedrons, join the apices with
the centres of the lateral edges. This form is now desig-
nated by the name of a Di-rhombohedron, and its crystallo-
graphic sign is either as above 11 + n. — II + n, or it is
2 (R + n). The difference of this form from the isosceles
six-sided pyramids consists in the position, and in the rela-
tions existing between their axes and horizontal projec-
tions. From the di-rhombohedrons the combinations are
denominated, in which these forms occur.

Let the terminal edge of the rhombohedron be = x, that
of the di-rhombohedron = C, being the edge of combina-
tion ; we obtain the formulae

cos. x = 3 cos. C -J- 2,

cos. C = COS'X- 2;
3

which are useful for finding the dimensions of any di-rhom-
bohedron from those of its rhombohedron, and vice versa.
In §. 145. iii. 1., the faces of the isosceles six-sided py-

TERMINOLOGY. §. 147.

ramids are transformed into rhombs, if, instead of R -f n,
2 (II + n) enters into the combination.

2. If in §. 145. iv., n' is made = n, and m' = m, the
combination is (P + n)™. — (P -f n)m ; the forms are consi-
dered here in two different positions. The combination as-
sumes the aspect of a scalene twelve-sided Pyramid, Fig. 51. ;
the edges of combination are parallel to lines which, in the
simple pyramids, join the apices with the centres of the la-
teral edges. A form of this kind receives the name of a
Di-pyramid, and is designated by (P + n)m. — (P 4- n)m or
by 2 ((P + n);"). The di-pyramids form part of the di-
rhombohedral forms, and combinations, which contain them,
are likewise considered as di-rhombohedral.

Di-rhombohedral combinations occur in rhombohedral
Emerald. Vol. II. Fig. 150.

§. 147. HEMI-RHOMBOHEDRAL AND HEMI-DI-RHOM-
BOHEDRAL COMBINATIONS.

A combination of the rhombohedral system is said
to be hemi-rhombohedrat, if only half the number
of the faces appear of some of the simple forms
which it contains. The combination is termed he-
mi-di-r7iombo7iedral9 if one or several of the di-rhom-
bohedral forms constituting it, enter with only half
the number of their faces into the combination.

It has already been observed (§. 141.), that such combi-
nations are perfectly symmetrical : hence appearances of
this kind are by no means in opposition to the symmetry of
the combinations.

The rhombohedron itself cannot assume a hemi-rhombo-
hedral appearance in the same way as other forms of this
system, because three faces cannot be distributed symme-
trically on two different apices.

If in a scalene six-sided pyramid we enlarge the alternat-
ing faces contiguous to one of the apices, the symmetry

§. 147. OF COMBINATIONS. 175

will require us also to enlarge three alternating faces on the
opposite side ; but the enlarged faces may be either those
which are parallel to the former, or those which are not
parallel to them.

The first process produces a form exactly similar to a
rhombohedron, if considered as a geometrical solid, which
nevertheless cannot be considered as a rhombohedron in
Crystallography, because in a compound form its faces can
never assume the position of the faces of a rhombohedron.
A combination in which one or several such forms appear,
is more particularly designated by the expression of a hemi-
rhombohedral form of parallel faces. In a combination of
this kind, particular attention must be given to the situa-
tion of the faces, in as much as two such forms similar to
a rhombohedron arise from the resolution of a single pyra-
mid, whose faces are situated either to the right or to the
left of a face of the fundamental form, or of any other
complete form contained in the combination.

The second process yields two forms, contained under
irregular trapezoidal faces, which on that account are called
three-sided Trapezohedrons. They are equal and similar, but
distinguished from each other by the character of being
twisted as it were, to the Right or to the Left (§. 67. 4.).
Fig. 53. represents a Right Trapezohedron, which is pro-
duced by the enlargement of the faces, a, a, &c. Fig. 11, while
Fig. 54. shews a Left one, which is contained under the
faces &, &, &c. of the same pyramid. A combination partly
or entirely consisting of such forms, is termed a hemi-
rhombohedral one of inclined' faces. The contorted aspect of
the trapezohedrons extends likewise to these combinations.

The same process applied to the isosceles six-sided pyra-
mid, gives in the first case, or by means of the enlargement
of parallel faces, forms likewise similar to a rhombohedron,
which yet, for the reasons mentioned above, cannot be con-
sidered as rhombohedrons. If, however, we enlarge those
faces, which are not parallel, the result is an isosceles three-
sided Pyramid^ Fig. 52. The faces of those rhombohedron-
like forms, as well as those of the three-sided pyramids,

176 TERMINOLOGY. §. 147.

keep the situation of those of the isosceles six-sided pyra-
mid, from which they are derived, and appear as such in
the combinations ; and this is the character by which they
may be recognised. The combinations themselves receive
the same denominations as above.

The regular six-sided prism P + co may assume a hemi-
rhombohedral aspect, like a finite isosceles six- sided pyra-
mid. In this case its faces must be symmetrically distri-
buted in the combination, like those of an equilateral three-
sided prism in a position which is characteristic and pecu-
liar to it.

If the unequiangular twelve-sided prism enters a hemi-
rhombohedral combination, its faces likewise must be sym-
metrically distributed, which comprises two cases. By en-
larging the alternating faces, it will appear either as a regular
six-sided prism, different from R + co and P + co by its
position ; or by enlarging the alternating pairs of faces, it as-
sumes the aspect of a six-sided prism, whose alternating
angles only are equal.

The rhombohedron itself may produce a hemi-di-rhombo-
hedral combination, but only one of inclined faces, because
if the parallel faces of the di-rhombohedron are enlarged,
the rhombohedron will be reproduced, and the combina-
tion itself will be simply rhombohedral (§. 145.). The re-
suit is an isosceles three-sided pyramid, differing both in
position and dimensions, from that which may be obtained
from the isosceles six-sided pyramid. In the same way
the di-pyramid (§. 146.) also enters into a hemi-di-rhom-
bohedral combination, although its halves and fourths may
appear both as forms of inclined faces, and as forms of pa-
rallel faces. For if we enlarge the alternate faces conti-
guous to the upper apex of the di-pyramid, and those con-
tiguous to the lower apex, which are parallel to the former,
two forms will arise, which, though exactly similar to iso-
sceles six-sided pyramids, are yet different from these forms
in respect to the situation of their faces. Combinations of
this kind are said to be hemi-di-rhombohedral of parallel faces.
The characteristic form of the hemi-di-rhombohedral combi-

§. 117. OF COMBINATIONS. 177

nations of inclined faces is a six-sided Trapezohedron^ a solid
contained under twelve equal and similar trapezoidal
faces. Pairs of them are obtained by enlarging the alter-
nating faces contiguous to one of the apices, and those of
the other which are not parallel to them ; they shew, like
some of the forms considered above, the differences of
Right and Left. A Right Trapezohedron of this kind is
represented in Fig.SJ., a Left one in Fig. 58. The first is
obtained by enlarging the faces, a, a ; and a', a', &c. ; the
second, by enlarging the faces marked &, b ; and &', &', &c.

The three-sided trapezohedrons, Figs. 53. and 54., are
obtained by enlarging all those faces of the six-sided ones
which are marked a, a, &c. or &, J, &c. in Fig. 51. ; but if
we enlarge the faces «, «, &c. or b7 &, &c. contiguous to the
upper apex, and the faces a', a', &c* or V, &', &c. conti-
guous to the lower apex, forms will result, like Fig. 55.
and Fig. 5C., and which are right and left trapezohedrons
like the former, but which contain faces of the scalene
six-sided pyramids in both positions. The situation of
these faces determines their existence in the combinations.

In order to re-obtain the simple forms from a di-pyramid,
it is necessary to enlarge the alternating pairs of faces,
a and 6, or those which meet in the obtuse terminal edges
of the six-sided pyramid from the upper apex, and the al-
ternating ones, a and 6 from the lower apex. If, on the
contrary, we enlarge those on the lower apex which pro-
duce with the former horizontal edges of combination
like a' and &', the result will be scalene six-sided py-
ramids, whose bases are hexagons of alternately equal
angles, similar to the figure produced by a section perpen-
dicular to the axis which does not intersect the lateral edges.
Simple minerals may assume this form, although they have
as yet not been found in nature ; but they are very com-
mon in compound minerals, and as such they will be the
subject of farther investigations.

Hemi-rhombohedral and hemi-di-rhombohedral combina-
tions occur in rhombohedral Fluor-haloide, and in rhombo-
hedral Quartz.

VOL. I. M

178 TERMINOLOGY. §. 147.

The mode in which the rhombohcdrons themselves, and
the limits of their series, produce hemi-rhombohedral com-
binations, is particularly remarkable, and very distinct from
what we have seen till now. Here all the faces belonging
to one apex enter into the combination, whilst all those con-
tiguous to the opposite apex disappear. The results are
combinations of a dissimilar configuration in their opposite
terminations. The crystallisations of rhombohedral Tourma-
line give the most generally known examples of this pecu-
liarity, which, however, is likewise frequently met with
in rhombohedral Ruby-blende. It is evident that the six-
sided prism R + CD, if subject to this modification, will
only shew half the number of its faces ; and that the
three-sided prism occurring in these two species, can only be
explained upon the supposition, that it is a rhombohedron
of an infinite axis, three faces of which belong to one of
the 'apices of the combination, and are enlarged, while those
belonging to the other apex disappear.

The designation of hemi-rhombohedral and hemi-di-rhom-
bohedral forms, depends upon the same principle as that

of the halves in the tessular system (§. 129 133.). The

number 2 is added to the sign of the entire, simple form,
as a divisor ; and by the signs + and — , or r and 1, is in-
dicated the parallel and the transverse position, or the dif-
ference between right and left.

The following signs refer to the figures 51 — 58, com-
prehending the different forms of a di-rhombohedral, hemi-
rhombohedral, or hemi-di-rhombohedral character :
n)">)Fig. 51.;

* P + n w

1 P + n ftr + R + n

Fig. 52. ;
Fig. 54.;
Fig. 56. ;
- Fig. 58.

r 2 '

12 2
FI>- M. ; ! (P + n)m

r 2

+ r (P + n) m

' 1 2
Fig.,, . +l(P+n)»

— r 2

r 2((P+n

'-1 2
)m)FV.,7. 1 2((P + n)»

r 2

0 ' r 2

§. 148. OF COMBINATIONS. 179

§. 148. DEVELOPEMENT OF RHOMBOHEDRAL AND
DI-RHOMBOHED11AL COMBINATIONS.

The developement of the combinations containing
three or more simple forms, is founded upon the
knowledge of binary combinations.

A few examples will be the best means of instructing
the beginner how to proceed in similar cases. The example
chosen for rhombohedral combinations, shews at once the
sufficiency of the few particular cases mentioned in §. 145.
— 147- of binary combinations for the developement of such
as consist of a greater number of simple forms.

The 59th figure represents a rhombohedral combination
(§. 145.), consisting of four rhombohedrons, two scalene six-
sided pyramids, and a regular six-sided prism. Its inde-
terminate designation is

11 + n. R + nT. R + mi. R -I- nTH. (P + n™)<*.
a c e d b

(P + nV)>»'. R + co.
/ g

The only form immediately determined in this combina-
tion, is R + cc . The edges of combination between the
faces of this form and those of the rhombohedrons R + nr«,
R -|- n11 and R 4- n are horizontal (§. 145. i. 4.), whereas
P + 03, if it were contained in the combination, would
produce edges of combination parallel to the lateral or
terminal edges of these rhombohedrons (§. 145. iii. 4.).

Among the rhombohedrons, one must be selected and
fixed upon as the fundamental form, and the letter n in
its sign therefore, must be made = 0. The figure repre-
sents a crystal of rhombohedral Lime-haloide, which mine-
ral is cleavable (§. 162.), parallel to the faces of the rhom-
bohedron here designated by R -f- n1 = 105° 5'. According
to this we determine the rhombohedron R + nT to be the

180 TERMINOLOGY. §. 148.

fundamental form, and consequently n' to be = 0, and
R 4- n' = R. The position of R is considered as the nor-
mal position.

The rhombohedron R 4- n is in a transverse position to-
wards R ; and since the edges of combination between R,
(P 4- niv)m and R + n are parallel to the terminal edges of
R and to the inclined diagonals of R 4- n, those between
R and R 4- n will assume the same situation, if the faces
of (P 4- niv)m disappear. Hence the two forms are in
the relation of R 4- n : R + n — 1 (§. 145. i. 1.) ; or, for
n = 0, in that of R : R — 1. Consequently n is = — 1
and R 4- n = R — 1.

Suppose the faces of R and those of R + niv to be en-
larged till they intersect each other.* R 4- niv is in the
same position towards R as R — 1 ; the two forms will
produce edges of combination parallel to the inclined dia-
gonals of R, and consequently to the terminal edges of
R 4- niv. The forms R + niv and R are again in the
ratio of R 4- n and R 4- n — 1 (§. 145. i. 1.). And since
for R, the expression n — 1 is = 0, niv for R 4- nlv will
be = 1, and R 4- niv = R 4- 1.

The three rhombohedrons R — 1, R and R 4- 1 are
consecutive members of one and the same series.

The edges of combination between (P 4- nv) n/ and R are
parallel to the terminal or to the lateral edges of R, and to
the lateral edges of (P 4- nv)ra' ; the scalene six-sided pyramid
belongs to the rhombohedron (§. 145. ii. 1.). In (P 4- nv)m'
therefore nv is = 0, and the pyramid itself == (P)m'.

The rhombohedron R 4- 1 is in a transverse position to-
wards this pyramid, which is itself parallel to R, and the
faces of R 4- 1 take away the more acute terminal edges of
(P)m' with parallel edges of combination. The relation of
the forms is therefore as in §. 145. ii. 4. ; and we have

3 m/ - l a • 2 a
a = ,. 2. a,

* The faces of R may easily be enlarged by cleavage,
(§. 162.).

148. OF COMBINATIONS. 181

and hence m' = 3. The pyramid accordingly will be per-
fectly designated by the crystallographic sign (P)3.

The more obtuse pyramid produces with (P)3 horizontal
edges of combination, m is therefore = in7 = 3 (§. 145.
iv. 2.); and (P 4- niv)m = (P 4- n")3.

But the faces of R — 1 appear with parallel edges of
combination, in the place of the more acute terminal edges
of this pyramid. If, therefore, a' be the axis of the rhom-
bohedron to which the pyramid belongs, whilst a is the
axis of R, we have from §. 145. ii. 4.,

3.3 — 1 a, _ _2_ a .
~~6"~ 3 ~2~'

from which follows

a'= i. a = 2-2. a;
niv is therefore = — 2, and (P + 11^)3 _ (p __ 2)3.

The edges of combination between the pyramid just now
determined and the rhombohedron R -f- n" are parallel to
the more obtuse terminal edges of the former, and to the
terminal or lateral edges of the latter. The two forms
rank under the head of §. 145. ii. 7-

Let a' be the a^sis of the rhombohedron R -f n". We
have

and accordingly,

n11 is therefore = 0 ; but the rhombohedron belongs to the
first subordinate series, and R + n11 is = £ R.

If now we write down the signs of the forms as we have
found them in the developement in their regular order ;
we obtain the crystallographic designation of the combina-
tion, disposed according to the different angles which per-
pendicular lines drawn upon the faces produce with the axis,

R — 1. (P — 2)3. R. | R. R + l. (P)3. R + 03.
a I c d e f g

The 60th figure represents a di-rhombohedral combiua-

182 TERMINOLOGY. §. 14:8.

tion of rhombohedral Emerald, designated indeterminately
thus :

R — os. P + n. P + n1. 2 (II + n»). P + cc.
aid c e

Here R — 03 (a) and P 4- co (e) the limits of the series of
isosceles six-sided pyramids are immediately determined.
The only rhombohedron which it contains is R + n11 (c).
It is present in both positions ; the combination therefore
assumes a di-rhombohedral character (§. 146.). If this
rhombohedron be considered as the fundamental form, the
value of n11 wiU be = 0, and 2 (R + n" ) therefore = 2 (R).

The faces of P + n (&) if duly enlarged, appear as
rhombs in the place of the apices of the di-rhombohedron ;
the faces of the latter likewise would be rhombs, were they
not intersected by the faces of other forms. Hence n is
also = 0, or the pyramid P + n, and the di-rhombohedron
2 (R) are co-ordinate forms (§. 146. 1. ; and §. 145. iii. 1.).
P + n therefore is = P.

The faces of the di-rhombohedron appear with parallel
edges of combination in the place of the terminal edges of
P + n1 (d). The relations of the pyramid and the rhom-
bohedron will therefore be those considered in §. 145. iii. 2.
From these it follows that n1 is = n11 + 1 = 0 4- 1 = 1.
The pyramid will be = P + 1.

The determined designation of the developed di-rhombo-
hedral combination is therefore :

R — oo. P. 2 (R> P + 1. P + os.

a be d e

These developements, as represented by the signs, con-
tain every thing required for calculations referring to the
compound forms ; since the designation contains all those
determined definite values of m and n, which must be sub-
stituted in the general equations referred to in §. 144.

The developement of the combinations is peculiarly ap-

§. 149. OF COMBINATIONS. 183

plicable, if it is not limited to the forms of a single in-
dividual, but if it refers to all the crystalline varieties
known in a natural-historical species. It will be treat-
ed of more at large than the proposed limits of this work
Avould allow, in a particular work on Crystallography.
In general, those simple forms which are known from one
developed combination, become the foundation of every
farther developement ; and this method of proceeding is
also used in compound forms of a single individual, if some
of its forms can be identified with others, which have been
developed in other crystalline varieties of the same species.

§, 149. PYRAMIDAL COMBINATIONS.

A combination of the pyramidal system is more
particularly said to possess a Pyramidal character,
if the simple forms contained in it appear with the
full number of their faces in their peculiar position.

The binary combinations of this system, under the same
restrictions as to subordinate series as in §. 145., are in
general

i. P + n. P 4- n',
ii. P + n. (P 4- n')m,
iii. (P + n)m. (P + n')m/.

i. P + n. P + n'.

1. Let n be = n + 1. Under these circumstances, the
forms will be consecutive members of the series §. 101.
As such, they are in a diagonal position, and the edges of
combination which they produce, must be parallel among
each other, but at the same time they must be parallel also
to the terminal edges of the more acute pyramid, and to
those lines in the faces of the more obtuse one, which may
be drawn perpendicularly from the apices to the lateral
edges. Ex. P — 1 (f) and P (P) in pyramidal Zircon.
Vol. II. Fig. 99. Inversely from the described situation
of the edges follow the above mentioned relations of the two

184 TERMINOLOGY. §. 149.

forms among each other; which is immediately evident
from the derivation.

2. JLet n' be = n ±_ 2 r. The forms are in a parallel posi-
tion. The edges of combination therefore will be horizon-
tal. Ex. P (c) and P + 2 (ft) in pyramidal Garnet. Vol. II.
Fig. 96. They retain this situation, even though the com-
bined forms belong to different series, and their relative di-
mensions therefore in such a combination remain undeter-
mined. The reason why the faces of all parallel pyramids in-
tersect each other in horizontal edges, consists in the pa-
rallel situation of their respective lateral edges, or of the
sides of their horizontal projections.

3. Let n or n' be = — os. One of the forms under
these circumstances appears as a face perpendicular to the
axis of the other ; and the edges of combination, inde-
pendently of the dimensions of the forms, must be horizon-
tal, as is evident from 2. Ex. P — oo («) and -l^JL P _ 3 (6)
in pyramidal Lead-baryte. Vol. II. Fig. 92.

4. Let n or n' be = + co ; one of the forms becomes a re-
gular four-sided prism. In a parallel position, the edges in
which the faces of the two forms meet, must be parallel to
the lateral edges of the finite form, and therefore horizon-
tal (2). Ex. P (P) and P + oo (J), or P + 1 (*) and [P + co] (g)
in pyramidal Tin-ore. Vol. II. Fig. 102. In a diago-
nal position they are parallel to the rhombic principal sec-
tion (§. 53. 2.) of the finite member, and their situation
depends on its dimens'ons. Ex. P + 1 (*) and P + oo (Z)
in pyramidal Tin-ore. Vol. II. Fig. 102. If the faces of
the prism do not intersect eac|i other, they appear as
rhombs. The combinations P — cc. P + oo, and P — co.
[P + oo ], are right rectangular four-sided prisms.

5. If both n and n' are == + or, and the forms in a dia-
gonal position, the faces of one of the prisms appear in the
place of the edges of the other ; their combination produces
an equiangular, and if the faces are of equal extent, a re-
gular eight-sided prism. Ex. P + co (Z) and [P -f ooj (g)
in pyramidal Tin-ore. Vol. II. Fig. 102.

§. 149. OF COMBINATIONS. 185

ii. P + n. (P + n')"'.

1. Let n' be = n. The forms in this case are co-ordinate,
and as such, in a parallel position. The faces of the four-
sided pyramid appear in the place of the apices of the eight-
sided one as rhombs, their edges of combination being
parallel to the terminal edges of the four-sided pyramid.
Ex. P (c) and (P)3 (*), or P (c) and (P)4 (x) in pyramidal
Garnet. Vol. II. Fig. 96. The situation of the edges
produced between the two forms, is independent of m', but
in the parallel position already mentioned, it equally re-
quires the above relation of the forms, whatever may be
the value of m.

If the forms are in a diagonal position, and therefore not
co-ordinate ones, there exists for every sea ne eight-sided
pyramid a particular four-sided pyramid, whose faces
appear as rhombs in the place of the apices of the former.
Ex. P — 1 (o) and (P — 2)3 (a), or P (P) and (P— I)3 (2)
in pyramidal Garnet. Vol. II. Fig. 96. In these combina-
tions the rhombic figure of the faces of the isosceles four-
sided pyramids depends on a certain relation of m', n'
and n, which is expressed in the equation :

n — n'4- 1

m' = 2 ~ + 1.

This equation is very useful, from two of these quantities
being known, to find the value of the third.

2. Let n or n' be = — co . One of the forms becomes
= P — co, and as a face perpendicular to the axis, it pro-
duces horizontal edges of combination (i. 3.).

3. Let n be = + cs. The pyramid P -f n in this case
appears as a rectangular four-sided prism. In either posi-
tion its faces assume a rhombic figure, by their intersec-
tion with those faces which form the lateral solid angles of
the pyramid. In the parallel position the angles of those
rhombs, and therefore the situation of the edges of combi-
nation are altogether dependent on m', and different in
different pyramids. In the diagonal position, however,
these angles and edges of combination do not depend on
m', since the angles are equal to the angles of the rhombic

186 TERMINOLOGY. §. 149.

principal section, and the edges of combination consequent-
ly parallel to the alternating terminal edges of that isosce-
les four-sided pyramid, from which the scalene eight-sided
pyramid is derived, or to which it belongs. Examples of
both cases are contained in pyramidal Zircon, Vol. II.
Fig. 99. The scalene eight-sided pyramids (P)3 Or), (P)4 (?/)
and (P)5 (~) are in a parallel position with P + cc (/), but
they are in a diagonal position with [P 4- oo] (A).

If, therefore, the edges of combination produced by a
scalene eight-sided pyramid and an isosceles four-sided one,
in a parallel position, are found to be parallel to those
edges which are produced by the same eight-sided pyramid,
and a rectangular four-sided prism in a diagonal position ;
it follows that the isosceles four-sided pyramid and the
eight-sided pyramid must be co-ordinate forms.

4. Let n' be = n — 2, and m' = 3. The combination
will be P + n. (P -f n — 2)3, and the forms in a pa-
rallel position. The faces of the four-sided pyramid ap-
pear in the place of the more acute terminal edges of the
eight-sided pyramid ; the edges of combination being pa-
rallel among themselves, to the above-mentioned terminal
edges of the eight-sided pyramid, and to those lines in the
four-sided pyramid, which, from its apices, can be drawn
perpendicularly to its lateral edges. Examples occur in
pyramidal Zircon, of (P)3 and P 4- 2.

Suppose A'M, Fig. 69., to be half the axis, A'C the
more acute terminal edge of the eight-sided pyramid. If
CM be made equal to half the side of the horizontal pro-
jection, MA7 becomes half the axis, A'C the above-mention-
ed peiy.endicular Hue upon the face of the four-sided pyra-
mid ; and consequently, supposing MD, half the side of
the horizontal projection = |, we have

n / . i n'

MA = 2'. a = m+- 2*. a.

2

If, according to the supposition, m' be = 3, the values
of n and n' will follow thus :

n = n' + 2, and n' = n — 2.
But if we substitute 4, instead of m', we obtain

§. 149. OF COMBINATIONS. 187

22 = |. 22 =|. 2>. 22 = f. 2 »;• .,

and P 4- n = | P 4- n' 4- 2, or that member of the second
subordinate series (§. 107-), which belongs to P 4- n' 4- 2.
For m' — 5, ths result is

n' * s

= 3. 25 = _J_. 2*.

and P + n = -^j- P 4- n' + 3, that member of the first

subordinate seiics (§. 107-), which belongs to P + n' + 3.

5. Let n' be = n — 3, m' == 4. The combination is
P + n. (P + n — 3)4 ; under these circumstances, the
forms will be in a diagonal position. The faces of the
four-sided pyramids appear in the more obtuse terminal
edges of the eight-sided ones. The edges of combination
are parallel to these, to the perpendicular lines upon the
faces of the four-sided pyramid, and among each other.

Suppose, Fig. 70., MA' to be half the axis of the eight-
sided pyramid, and A'B its more obtuse terminal edge. If
now BM is half the side of the horizontal projection, we
have in MA' half the axis, and in A'B the perpendicular line
upon the face of the four-sided pyramid ; and consequent-
ly, if half the side of the horizontal projection MD is sup-
posed = £, MA will be

n *'-*

= 2 2. a = m'. 2 * . a.
But we have m' = 4 ; therefore

n = n' 4- 3, and n' = n — 3.
If m' be = 3, it will follow that

n'— 1 n'—l n' + 2

2* = 3. 2~ir = _JL_. 2l. 2-ir _ _JL_. 2— ;

and thus P 4- n becomes z /a P 4- n' 4- 2, or that mem-
ber of the first subordinate series, which belongs to
P 4- n' 4- 2.

m' = 5 makes

„ n'-T n'-l n'4-3

25 = 5. 2 2 = |. 22. 2 2 = |. 2 2 .
P 4- n therefore becomes = | P 4- n' 4- 3, which is that

188 TERMINOLOGY. §. 149.

member of the second subordinate series which belongs to
P + n' + 3.

6. Let n' be = n — 3, m' = 3, the combination there-
fore P -f n. (P + n — 3)3. The forms again are in a dia-
gonal position, and the more acute terminal edges of the
eight-sided pyramid, therefore, fall into the same vertical
plane, which passes through the terminal edges of the four-
sided pyramid. The edges of combination, arising between
the faces of the two forms, become parallel among them-
selves, and to the above-mentioned terminal edges of the
pyramids. Ex. P + 2 (5) and (P — I)3 (2) or P + 4 (r)
and (P + I)3 (e) in pyramidal Garnet. Vol. II. Fig. 96.

For, the rest being as in the other example, let A'C,
Fig. 69., represent the terminal edge of the four-sided py-
ramid : it will follow that

MA = 2^. a = m^-1- 2~^~. a.

If now, according to the supposition, m' be = 3; we
have

n = n' + 3, and n' = n — 3.

But if m' is = 4 ; P + n becomes = | P -f n7 + 3, or
that member of the second subordinate series, which be-
longs to P + n' + 3 ; if m' is = 5, the pyramid becomes
— -=— ^ + n' + 4, or that member of the first subordinate
series, which belongs to P -f n' + 4.

7- Let n' be = n — 4, m' = 4, or the combination
P + n. (P + n — 4)4. The forms are in a parallel posi-
tion ; the more obtuse terminal edges of the eight-sided
pyramid coincide with the terminal edges of the four-sided
pyramid. In this situation of the faces, the edges of com-
bination between the two forms are parallel to each other,
and to both the mentioned terminal edges. Ex. P + 4 (r)
and (P)4 (x) in pyramidal Garnet. Vol. II. Fig. 96.

For we have

%
MA = 2?. a = m'. 2'. a,

from which, 4 being substituted for m', we obtain

§. 149- OF COMBINATIONS. 189

n = n' 4- 4, or n' = n — 4 ;
m' = 3 gives P + n =— L_P + n'-f 3, the member of

2 V'2

the first subordinate series belonging to P + n' + 3 ;
m = 5 gives P + n = \ P + n' + 4, or that member of
the second subordinate series which belongs to P + n' + 4.

iii. (P + n)". (P + n')m/.

1. Let n' be = n. The combination is (P + n)m.
(P + n)m'; the forms are co-ordinate pyramids, and as
such in a parallel position. The situation of the edges in
which the faces of the two pyramids intersect each other,
do not depend upon m or m' (ii. 1.). The situation of the
edges produced between any one of those pyramids, and the
four-sided one from which they are derived, is such that
the faces of the latter become rhombs, or the edges of
combination are parallel to their terminal edges. The
edges of combination between two such co-ordinate scalene
eight-sided pyramids, will therefore likewise be parallel to
the edges of that isosceles four-sided pyramid to which
they belong ; and if more than two appear in one and the
same compound form, the above mentioned parallelism will
be observable in the edges of combination between them all.
Ex. The pyramids a1, ?/, and 2, already mentioned in pyra-
midal Zircon. Vol. II. Fig. 99. From the observed pa-
rallelism, we may decide inversely whether the pyramids
are co-ordinate forms or not, and whether or not they are
derived from that isosceles four-sided pyramid, with which
they enter into combination.

2. Let m be = m'. The combination will be (P + n)m.
(P 4- n')m. If, moreover, the forms are in a parallel position,
the edges produced by the intersection of their faces be-
come horizontal. Ex. (P — I)3 (z) and (P 4- I)3 (e) in
pyramidal Garnet. Vol. II. Fig. 96. The transverse sec-
tions of the two forms are similar to each other, since these
forms are members of one and the same series (§. 105.).
The observations in respect to scalene six-sided pyramids
in §. 145. iv. 2. extend likewise to this case.

3. If one of the combined forms, by n becoming ~ +_ co

190 TERMINOLOGY. §. 149-

is changed either into an unequiangultir eight-sided prism,
or into a plane perpendicular to the axis, the situation of
the edges in 2. remains nevertheless unchanged, provided
in the case of the prism, the position still remains the
parallel one and m = m'. Ex. of the latter, (P + I)3 (e)
and [(P 4- co)3] (/) in pyramidal Garnet. Vol.11. Fig. 96.

4. There exists in the pyramidal system, a series of sca-
lene eight-sided pyramids analogous to that of the scalene six-
sided pyramids in the rhombohedral system (§. 145. iv. 4.).
Its members succeed each other in the following order:
... (P + n)s, (P 4- n + I)4, (P 4- n +2)3, (P + n + I)5 ...
in which the consecutive members assume a diagonal posi-
tion towards each other. Ex. (P)4 (.r) and (P + I)3 (e) in
pyramidal Garnet. Vol. II. Fig. 96. The edges of combi-
nation between the faces of every two subsequent pyramids
are parallel to each other, to the more obtuse terminal edges
of the lower, and to the more acute terminal edges of the
higher member in the series. The demonstration of this
property depends upon the same suppositions as in the six-
sided pyramids.

Let A'B, Fig. 70., represent the more obtuse terminal
edges of (P 4- n)m ; we find the algebraic expression of

n

. i/T>,r in. 25. a
sin A'BM =

a2 + 2)

If in the same way we suppose A'C, Fig. 69., to be the
more acute terminal edge of (P 4- n')!tl/ ; a similar algebraic
expression will give

(m' 4- 1) 2'. a

sinA'CM= ililJL.^

l(m' + I)2. 2< a2 -f 4]
These two expressions become equal, if in the first we
substitute 5 for m, and in the second n 4- 1 for n', and 4
for m'. They again become equal if in the first we suppose
m = 4 ; in the second n' = n + 1 and m' = 3 : and again
for m = 3 in the first, and n' = n — 1, and m' = 5 in the
second expression. For the rest, the remarks of §. 145. iv. 5.
find here equally their full application.

§. 150. OF COMBINATIONS. 191

§. 150. HEMI-PYRAMIDAL COMBINATIONS.

A combination of the pyramidal system possesses
a Hemi-pyramidal Character, if one or more of the
simple forms contained in it, appear with only half
the number of their faces.

The isosceles four-sided pyramid may be resolved into
halves, like the octahedron (§. 129.). The result is a pair
of forms contained under four equal and similar isosceles tri-
angles, Figs. 61. 62., the first being produced from the isosce-
les four-sided pyramid, Fig. 8., by the enlargement of the faces
a and «, the second by the enlargement of & and b. None
of their faces are parallel to each other. This form is ana-
logous to the tetrahedron. A combination, containing one
or several forms of this kind, is termed a hemi-pyramidal
combination of inclined faces. If we enlarge parallel faces, no
finite forms can be obtained ; and this seems to be the reason
why hemi-pyramidal forms thus possessing parallel faces
have not yet been found in nature.

The scalene eight-sided pyramid, if resolved by enlarg-
ing its alternate faces, gives forms contained under eight
irregular trapezoidal faces, Figs. 63. 64., which on that ac-
count receive the denomination of four-sided Trapczohcdro™.
These forms agree exactly with the six-sided trapezohedrons
obtained by the same mode of resolution from the di-pyra-
mid, §. 146., except in the number of their faces. The two
forms thus produced from the eight-sided pyramid, are also
distinguished from each other by the difference of Right and
Left, as the forms of the rhombohedral system already
mentioned. Fig. 63. represents a right four-sided Trape-
zohedron, produced from the eight-sided pyramid, Fig. 12.,
by the enlargement of the faces a, a, while Fig. 64. is the
left one which arises from the enlargement of 6 and ft.

By enlarging parallel faces we obtain two forms absolute-
ly similar to isosceles four-sided pyramids, except in their
position, since the faces of such hemi-pyramidal forms are
always situated like the faces of that eight-sided pyramid,

TERMINOLOGY. §. 150.

whose halves they are. Combinations, into which forms of
this kind enter, are termed hemi-pyramidal combinations
of parallel faces.

But a scalene eight-sided pyramid may also be resolved
according to the process, applied in §. 146. to the di-pyramid,
for reproducing the simple forms which it contains. This
is effected by enlarging the alternate pairs of faces which
meet in the acute terminal edges of the one, and those of the
other apex which are not contiguous to them. The result-
ing forms are contained under eight equal and similar sca-
lene triangles, all their faces being inclined to one another,
Figs. Co. 66. ; hence they likewise change those pyramidal
combinations in which they are contained, into hemi-pyra-
midal ones of inclined faces. These forms are in respect to
the eight-sided pyramids, what the forms analogous to the
tetrahedron considered above are to the four-sided pyra-
mids, and occur along with them in the same combinations.

Hemi-pyramidal combinations of parallel faces occur in
pyramidal Scheelium-baryte, and hemi-pyramidal combina-
tions of inclined faces in pyramidal Copper-pyrites. No ex-
ample has yet been found in nature of di-pyramidal combi-
nations, or such as would contain one and the same form of
the pyramidal system in two different positions ; nor have
we any reason to suspect their existence, because no ra-
tional number of derivation can produce from any pyrami-
dal form, another diagonally situated, and equal and similar
to that pyramidal form.

Hemi-pyramidal forms, like the hemi-rhombohedral ones,
are designated by adding the divisor 2 to the crystallogra-
phic signs of the entire forms. The situation of those faces
which occur in the combination, are moreover indicated by
the signs 4- and — , or r and 1.

The following signs refer to the figures 61 — 66 :

+ L±J? Fig. 61. ; - Lt^ Fig. 62. ;
_L

Fig.C3.; Fig. 64.;

_ Fig. 65. ; - _- Fi>. 66.

§. 151. OF COMBINATIONS. 193

§. 151. DEVELOPEMENT OF PYRAMIDAL COMBI-
NATIONS.

The following developements will shew how the
knowledge of binary combinations relative to the
pyramidal system in §. 149-, is to be applied in
particular cases.

Fig. 07- represents a pyramidal combination, whose un-
determined designation is

P + n. P + n1. (P + n»)«". P + w. [P + oo].
a b c d e

First of all, the more obtuse of the two four-sided pyra-
mids, or P + n (#), is supposed to be the fundamental form.
The value of n will therefore be = 0, and P -f n = P.
This determination must precede that of the vertical prisms,
which, however, is now very easily effected ; the prism d,
or that whose intersections with P are horizontal, being
P + co, whilst <?, the other prism, is [P -f- co], and produces
intersections with the faces of P, which are parallel to the
terminal edges of this form (§. 149. i. 4.).

The edges of combination between P and P 4- n1 (6), are
parallel among themselves, but at the same time also to
the perpendicular lines drawn upon the faces of the former,
and to the terminal edges of the latter pyramid. The
forms are in a diagonal position to each other ; and they
are therefore in the relation of P + n and P + n + 1 ; and
since n = 0, P + n1 will be = P + 1, as follows immediate-
ly from the derivation.

The scalene eight-sided pyramid c belongs to P ; for it is
in a parallel position with it, and the faces of the four-sided
pyramid appear as rhombs in the place of the apices of the
eight-sided pyramid (§. 149. ii. 1.). For n11 = 0, (P + nn)m
becomes = (P)m.

P + 1 is in a diagonal position with (P)m ; its faces, how-

VOL. I. N

TERMINOLOGY. §. 151.

ever, likewise appear as rhombs in the combination, for the
edges between the two forms are parallel to the terminal
edges of P + 1- From the equation given in §. 149., we

have

i — o + i

m = 2 + 1 = 2 + 1 = 3, and (P)™ = (P)3.

The perfectly determined designation of the developed
compound form, is therefore

P. P + 1. (P)3. P + co. [P + «].

a b c d c

The indeterminate designation of the compound form, re-
presented in Fig. C8., is

P + n. P + n1. (P + n")m. P + oo. [P + cc].
a f c d e

If we compare the present combination -with the pre-
ceding one, we find a perfect identity between the forms
P + n and P, and between the forms (P -I- n11)*1 and (P)3,
the rectangular prisms likewise being common to both ;
and accordingly we may consider these forms as already
known, so that in

P. P + n1. (P)3. P + oo. [P -f w].

a f e d e

the only form still to be determined is the four-sided pyra-
mid P + n1 (/).

The faces of the scalene eight-sided pyramid (P)3, meet-
ing in their more obtuse terminal edges, are found in the
combination to appear in the place of the terminal edges of
P + n1. The two forms are therefore in a parallel position,
and the relation existing among them, if compared with
those considered above, is comprised under the case §. 149.
ii.7-

If the number 3 be substituted for m, the axis of the

§. 152. OF COMBINATIONS. 195

n

isosceles four-sided pyramid will be = 3. 2.5. a, or = 3. a if
n = 0. Hence the pyramid is that member of the first
subordinate series which belongs to P + 3 (§. 149. ii. 4.),
that is to say, it is —-£—f + 3.

The complete designation of the compound form follows
according to this developement, thus :

P. -^ P 4- 3. (P)3. P + co. [P + ao ].
a fed e

§. 152. PRISMATIC COMBINATIONS.

A combination of the prismatic system is said more
particularly to possess a Prismatic Character, if
the forms contained in it appear with the whole
number of their faces.

The number of binary combinations in the prismatic
system is so great, on account of the great variety of differ-
ent relations among its forms, that it becomes impossible
in the present place to consider them all, even though this
should be done in the most general manner. We can
therefore notice only as many as will be sufficient for ex-
plaining the greater part of the cases commonly occurring
in nature, and which, at the same time, shew how to pro-
ceed in the application of the methods of developement
mentioned above (§. 143.). These binary combinations are :

i.

P + n.

P + n',

ii.

P + n.

(P + n')m',

iii.

P + n.

(P + n')m',

iv.

P + n.

(Pr + n')«",

v.

P+n.

(Pr + nOm/,

vi.

P + n.

Pr + n',

vii.

P + n.

Pr + n',

viii.

Pr + n.

Pr + n'.

196 TERMINOLOGY. §. 152.

i. P + n. P 4- n'.

The bases of the forms contained in this combination
are similar to each other, because these forms are members
of one and the same series, and no different position can
have any influence upon them, whatever members of the
series may be combined. The edges of combination, there-
fore, become horizontal for every value of n and n', even
though this be = 4- co or = — co. The same reasoning
applies to all such combinations as are produced by simple
forms of similar bases, and inversely, horizontal edges of
combination may be considered as a certain character of
this property of forms, as is evident from the derivation of
the series itself, and of its limits. Ex. P — co (/»),
|P — 1 (s\ P (o) and P 4- co (M ) in prismatic Topaz.
Vol. II. Fig. 34.

ii. P + n. (£ 4- n')<

Let n' be = n ; and the forms accordingly co-ordinate
ones. The faces of P + n, meeting in their more obtuse
terminal edges, appear in the place of those terminal edges
of (P 4- n)m/, which are situated contiguous to the prolonged
diagonal, and the edges of combination will be parallel
among themselves, and to the above mentioned edges of the
two pyramids. This parallelism follows from the simulta-
neous increment of the axis and of the variable diagonal of
(? + n)-' (§. 94.).

iii. P + n. (P + n')m'.

Let n' be = n. Every thing is as in ii. ; except that the
faces of P 4- n meeting in the more acute terminal edges,
appear in the place of the similarly situated terminal edges
of (P 4- n)m/. This likewise follows from §. 94. Ex. P (P)
and (P)3 (a) in prismatic Melane-glance. Vol. II. Fig. 34.

iv. P 4- n. (Pr 4- n')<

Let n' be = n ; m == 3. The combination will be
P + n. (Pr 4- n)3. The faces of P 4- n meeting in its
more obtuse terminal edges, are situated in the place

§. 152. OF COMBINATIONS. 197

of the corresponding terminal edges of (Pr 4- n)3. The
edges of combination are parallel to each other, and to the
above mentioned edges of the pyramids. For, if in the
general co-efficients of (Pr + n)ra (§. 95.), m is supposed

= 3, the ratio m + l 2». a : m_±L c becomes equal
2 m — 1

to 2n. a : c, which is identical with the ratio of the ana-
logous lines in P + n. Ex. P (P) and (Pr)3 («) in Serpen-
tine. Vol. II. Fig. 33.

If m' be = 5, and n' = n — 1, the same situation of the
edges takes place. A similar result is obtained by substi-
tuting 4 instead of m'. In this case, however, the pyra-
mid, from which (P -f- n'):n/ is derived, would not be one be-
longing to the same series as P 4- n, but it would be a py-
ramid, belonging to that subordinate series of which f is
the co-efficient. This becomes evident from a comparison
of the general co-efficients. The observed parallelism of
the edges in one direction alone is therefore insufficient for
the determination of the form, if there is not another da-
tum supplying this want from another side.

v. P + n. (Pr -f n')m'.

Let n' be = n ; m' = 3, or the combination P -f n. (Pr 4- n)3.
As the preceding case (iv.) refers to the more obtuse ter-
minal edges, so the present one applies to the more acute
ones ; which is evident from the comparison of the general
co-efficients of the forms concerned. The situation of the
edges being as described here, if m be supposed to assume
such values as do not make m' 4- 1 equal to a power of the
number 2 ; the pyramids P 4- n and P 4- n' will not be-
long to one and the same series.

vi. P 4- n. Pr 4- n'.

The pyramids and horizontal prisms considered here are
supposed to belong to one and the same series.

Let n' be = n. The faces of the horizontal prism ap-
pear in the place of the acute terminal edges of the pyra-
mid, and the edges of combination are parallel among each

198 TERMINOLOGY. §.

other, and to the above mentioned terminal edges of the
pyramid, which is evident from the derivation. The same
takes place for (Pr + n)m. Pr + n', and if m be = 3, also
for (Pr + n)m. Pr -f n'; and it is therefore a very useful
and evident datum for the determination of forms. Ex.
P (P) and £ r (o) in diprismatic Olive-malachite. Vol. II.
Fig. 5.

vii. P -f n. Pr -f n'.

1. What has been said in vi. of the acute terminal edges
of P 4- n, applies here on the same supposition to the ob-
tuse ones, in the combination P + n. Pr + n, as well as in
those of (P + n) m. Pr + n, and, in the case of m = 3, also
to (Pr + n)m. Pr + n. Ex. P (o) and Pr (P) in diprisma-
tic Iron-ore. Vol. II. Fig. 4.

2. Suppose now a triple combination, whose crystallo-
graphic sign is

P -f n. Pr + n'. Pr + n",

and in this n' = n, n" = n — 1 ; the faces of Pr + n" will
assume a rhombic figure in the combination.

Let AM, Fig. 71-5 represent part of the axis, MB one of
the diagonals, and BG part of that terminal edge of the pyra-
mid P + n, which is contiguous to BG ; FGHI, FGH'F
will be the faces of the horizontal prism Pr 4- n.

The rhomb AQB'P is a face of the horizontal prism
Pr + n" ; AN is = NB', the triangle NB'N' therefore simi-
lar to the triangle AB'M', and equal and similar to the
triangle NGA. Hence N'N = NG = | N'G.

The line N'B' is at the same time the diagonal of the
pyramid P + n, and of the pyramid P + n", to which the
horizontal prism Pr + n" belongs. N'G therefore repre-
sents the axis of the first, N'N that of the second pyramid,
and from the ratio of these = 2 : 1, we infer that n" is
«n — 1.

3. If instead of P -f n, the triple combination contains
the vertical prism P + co, and n' is = n", the faces of all
the three forms become rhombs. The rhombic figure of the
faces of a horizontal prism, if produced by the intersection

§. 153. OF COMBINATIONS. 199

with such forms as are known, will always suffice for its
perfect determination.

viii. Pr + n. Pr + n'.

1. Let n' be = n. The combination Pr -f- n. Pr + n will
be the intermediate form belonging to P + n (§. 97-)«

2. If the two forms do not belong to one and the same
series, due attention must be given to the co-efficients of
the different series. If n or n' become infinite, the edges
of combination are parallel to the terminal edges of that
scalene four-sided pyramid, to which the finite prism be-
longs, whatever may be its co-efficient.

3. If both n and n' are = + co, the combination P — oo.
Pr + co. Pr 4- oo is transformed into a right rectangular
prism, whose transverse section is an oblong. This prism
must not be confounded with the right rectangular four-
sided prism of the pyramidal system, whose transverse sec-
tion is a square, and whose crystallographic sign may be
either P — co. P + po or P — . co, [P + oo].

§. 153. HEMI-PRISMATIC COMBINATIONS.

A combination possesses a Hemi-prismatic Cha-
racter, if one or several of the forms contained in
it, and bearing to each other the general relations
of those in the prismatic system, appear only with
half the number of their faces, or in which these
faces shew differences in their angles referring to an
axis which is inclined in a plane perpendicular upon
the base, and passing through one of its diagonals.

The hemi-prismatic combinations depend upon the fun-
damental form, §. 98. Fig. 41., whose axis is inclined in a
plane perpendicular to the base, and passing through one of
its diagonals.

The hemi-prismatic forms are designated like the pris-
matic ones, with that difference only, that the signs of

200 TERMINOLOGY. §. 1£3.

those forms, of which only half the number of faces appear,
receive the additional divisor 2 ; and that those faces which
are turned towards the observer, contiguous to the upper
apex, are provided with the sign -f, while those contigu-
ous to the same apex, but on the opposite side, are distin-
guished by the sign — .

Only a few observations shall be made in the present
place on these combinations, in order to explain their ge-
tferal appearance.

If the inclination of the axis be supposed = 0, the com-
bination of P — 03 with P + OD, or with (P + cc)m,
(Pr + so)3, &c. will be a right oblique-angular four-sided
prism, which is a compound form, bearing altogether the
character of prismatic combinations, considered above.

But if the axis be inclined in the plane of the greater
diagonal, or ,if the inclined face of the horizontal prism

_? terminate the prism P -)- co ; then the combination
2

will assume the appearance of an oblique-angular four-sided
prism, the basis of which is inclined to its acute lateral
edges. A similar prism is produced by a combination of

P — co or of r+n with an oblique-angularfour-sided prism,
only that the oblique terminal face is inclined towards the
obtuse lateral edges of the prism. If in ?^±!? or ^r + n

n becomes = 4- co ; the horizontal prism is transformed
into a pair of planes parallel to the axis of the four-sid-
ed prism, and these faces appear with parallel edges of

combination, in the case of — — ? instead of its acute la-
teral edges, in the case of — — l- instead of its obtuse ones,

the prism itself remaining unlimited in the direction of its
axis. Combinations of this kind cannot be distinguished
from the prismatic ones P -f co. Pr + co, and P 4- cc.
Pr + co, unless some other faces be present, which, by their

§. 154. OF COMBINATIONS. 201

inclination upon the axis, shew the hemi-prismatic character
of the combination.

Supposing again the inclination of the axis to be = 0,
the triple combination P — as. Pr + co. Pr -f cs will be a
right rectangular four-sided prism, its base an oblong rect-
angular face, and the combination itself a prismatic one, as it
appears from what has been stated above, §. 152. The same
triple combination P — co. Pr + oo. Pr -^ cs , upon the sup-
position of the axis being inclined in the plane of the
longer, or the shorter diagonal, or the triple combinations

?L±f. Pr + c=. Pr + co or £L±_n. Pr + co. Pr + co,
2 2

appear as oblique rectangular four-sided prisms, two faces
of which are perpendicular to the rectangular base, while
the two others produce with it horizontal edges of combi-
nation, supplemental to each other.

The axis is not always inclined to the base at the same
angle, but varies according to the different species in
which it occurs.

Prismatoidal Gypsum-haloide, prismatic Azure-mala-
chite, paratomous, hemi-prismatic and prismatoidal Augite-
spar, may be quoted as examples of hemi-prismatic combi-
nations.

§. 154. TETAIITO-PEISMATIC COMBINATIONS.

The Tetar to-prismatic Character of a combina-
tion requires, that of the forms which constitute it,
the scalene four-sided pyramids shew only one-
fourth, and the prisms, both horizontal and ver-
tical, only one-half the number of their faces, or
that these faces are distinguished from each other
by their angles, which refer to an axis inclined in a
plane perpendicular upon the base, and passing
through neither of its diagonals.

TERMINOLOGY. §. 15 k

Tetarto-prismatic combinations consist, like the prisma-
tic and the hemi-prismatic combinations, of forms, Avhose
general relations to each other are those developed for the
prismatic system.

The tetarto-prismatic combinations depend upon the
fundamental form, §. 98., Fig. 42., whose axis is inclined
in a plane which is perpendicular to the base, and passes
through neither of its diagonals.

The designation of tetarto-prismatic forms must dis-
tinguish all the faces of the pyramid contiguous to one and

the same apex. Thus the face BAC being turned towards

p

the observer on his right hand, is noted r — , BAG' to his

4
p

left is noted 1 _ ; on the opposite side the face B'AC turn-

p

ed to his right hand is distinguished by — r — , the face

4

B'AC' turned to the left by — 1 _ . In the same manner

also the front and back faces of the horizontal prisms be-
longing to the diagonal BB', and the right and left faces
of the horizontal prisms belonging to the diagonal CC' are

indicated by ± J-L+JL, and r_gr + n the latter of
2 2

which extends also to those prisms whose axis is parallel
to the principal axis AA' of P.

The tetarto-prismatic combination — iLB. P + <x> gives

4

an oblique oblique-angular four-sided prism, in which the
alternating edges mutually are supplemental of each other.
Two of them assume a horizontal position, if the inclina-
tion of the axis is = 0. But if the angle of inclination is
an appreciable magnitude, or if these forms do not belong
to one and the same series, none of the edges of combina-
tion can become horizontal.

The tetarto-prismatic combination . Pr + «c .

Pr -f os is an oblique four-sided prism, in which none of the

§. 155. OF COMBINATIONS. 203

edges of combination are horizontal. This prism is rect-
angular if the base of the fundamental form is a rhomb,
but its transverse section will differ more or less from a
rectangular figure, if this base be a rhomboid.

Examples of tetarto-prismatic combinations occur in te-
tarto-prismatic Vitriol-salt, in several species of the genus
Feld-spar, in prismatic Axinite, and other species.

§. 155. DEVELOPEMENT OF PRISMATIC COMBINA-
TIONS.

The following developement will fully explain
the application of what has been stated above in
respect to binary combinations.

Fig. 72. represents a prismatic combination, indetermi-
nately designated by

Among the simple forms, constituting this compound
one, two are immediately determined, P + co (/) and
Pr -f co (/*).

If we suppose n1 = 0, P + n1 (e) becomes = P, that is
to say, the fundamental form. On account of the parallel
edges of combination between P and Pr + nm (c), n111 is

• In the indeterminate designation both the signs
(§. 192. 195.) are made use of before the forms have
been determined, till it appears from the developement ac-
cording to the reasons given above, which of the two is to
be retained.

204; TERMINOLOGY. §. 155.

also = n1 = 0 (§. 152. vi.) ; and consequently £ r + nm
= Pr.

The faces of the horizontal prism Pr + nlv («) appear as
rhombs if combined only with P and Pr. This horizontal
prism therefore is = Pr — 1 (§. 152. vii.).

The same horizontal prism produces parallel edges of
combination with P + n (6), in the place of its more ob-
tuse terminal edges. Hence Pr — 1 and P + n are co-
ordinate forms, n is = — 1, and P + n = P — 1.

The horizontal prism Pr belongs to P. But at the same
time it also belongs to, or produces parallel edges of com-
bination with d, or that scalene four-sided pyramid of a
dissimilar section with P, whose double representative sign
has been expressed, either by (P + nn)'n or by (Pr 4- nn)m.

If we suppose the corresponding finite diagonals of the
horizontal prism, and the mentioned pyramid to be equal ;
the axes of the two forms must necessarily be also equal,
and since Pr belongs to P, the same applies to this funda-
mental pyramid ; and hence we infer that the ratio of the
said diagonal in the secondary pyramid to its axis is the
same, which takes place in the analogous lines of the fun-
damental form itself.

Suppose in the pyramid, which is to be determined, the
ratio of the three perpendicular lines, equal to a' : b' : c'.
(§. 53. 6.) ; we have

a' : b' == a : b.

The horizontal prism Pr — 1 belongs to P — 1, but, on
account of the parallel edges of combination, also to the
pyramid d. If we proceed in comparing the axis and the
diagonals as above, we find the ratio of
a' : cf = | a : c = a : 2. c ;
and therefore the ratio of all the three lines
a' : b' : c' = a : b : 2. c.

If now we compare the co-efficients of this ratio with
the general co-efficients for (P + n)m (§. 94.) ; that is to
say

1:1:2,
with 2". m : 1 : m ;

§. 156. OF COMBINATIONS. 205

we find m = 2, n = — 1 ; and therefore

(P + n11)™ = (P — 1)2.

By comparing them in the same way with those for
(Pr + n)m (§. 95.), or

1 : 1 : 2

with m + '. qn . i . ELtl;

2 m—1

we have m = 3, n = — 1, and

(Pr + nn)m = (Pr — I)3.

In respect to the dimensions of the forms, it is quite in-
different which of the two designations we employ, for
, they both express exactly the same thing. Yet on account of
the number of derivation 3, for the analogy with the pyra-
midal system, we rather prefer the latter. The vertical
prisms become evident from the consideration of their be-
longing to the pyramids. One of them, /, is P 4- 03 on
account of its horizontal edges at the intersection with P ;
while the other, g, is (Pr + o>)3, because the edges of com-
bination of this prism with (Pr — I)3 are horizontal.

The definite designation of this compound form will
therefore be, according to the preceding developement,

Pr — 1. P — 1. Pr. (Pr — I)3. P.
a led e t

P + 03. (Pr + C3)3. Pr + 03.
/ S h

§. 156. TESSULAR COMBINATIONS.

A combination of the tessular system is more
particularly said to possess a Tessular Character,
if it contains the faces of the original forms pecu-
liar to this system (§.121 — 127.), without any
halves or fourths (§. 128.).

It would be superfluous to enter here into a minuter
detail of the binary combinations, comprised under this

TERMINOLOGY. §. 156.

head. Every thing necessary to know of them follows im-
mediately from their derivation from the hexahedron, if
\ve only attend to the different positions, which the faces
of the forms combined assume in respect to the different
axes. Hence even the angles of incidence at the edges of
combination may immediately be deduced for those forms
whose dimensions are invariable, their faces being perpen-
dicular to one of the three kinds of axes (§. 40.). For
these angles of intersection between the faces of such
forms, and the angles at the centre produced by those
axes, which are perpendicular to these faces, must be sup-
plemental to each other. The algebraic formulae given for
the different systems of variable dimensions, may also be
employed for obtaining the angles both of simple forms
and of combinations of the tessular system. For this pur-
pose, the simple forms peculiar to the tessular system, or
rather parts of them contained under faces similarly situ-
ated in respect to a single axis considered as the principal
one, may be considered as forms belonging to one of the
preceding systems. In this case every thing applies to
them, that has been above stated in respect to binary com-
binations. If, for instance, the hexahedron be considered
as a rhombohedron = It ; the horizontal faces of the octa-
hedron will represent R — co, the inclined ones R + 1 ;
and the combination of the hexahedron and the octahedron
supposed to be a rhombohedral combination, will be ex-
pressed by II — co. R. R + 1. As a tessular combina-
tion, its crystallographic sign is H.O (§. 121. 124.). If we
consider the octahedron as an isosceles four-sided pyramid
of the pyramidal system, and designate it accordingly by
P, the horizontal faces of the hexahedron will assume the
situation of P — co, while the vertical ones assume that of
[P + co] ; thus, for the sake of applying the calculations,
P — eo. P. [P 4- co] will express the same combination
of the hexahedron and the octahedron.

This process also extends to combinations produced by
more than two simple forms. Suppose, for instance, a com-
bination of the hexahedron, the octahedron, the dodecahe-

§. 157. OF COMBINATIONS. 207

dron, and the first variety of the digrammic tetragonal-ieo-
sitetrahedrons, to be considered as a rhombohedral combina-
tion, of which the hexahedron is the fundamental form
R = 90°. The octahedron is consequently =11 — co.
R + 1 ; the dodecahedron = 11 — 1. P + co ; and the di-
grammic tetragonal-icositetrahedron = 11 — 2. (P — I)3.
R + co. The entire tessular combination expressed as
a compound form of the rhombohedral system, is therefore
= R — co. R _ 2. R — 1. R. (P — I)3. R + 1. R + co.
P -f co. It may here be observed, that besides other forms,
this combination contains four consecutive members of one
series of rhombohedrons, one of which is R = 90°. The
designation of this compound form, as belonging to the tes-
sular system, is : H. O. D. Ai.

It is evident that this may likewise yield a method of
finding the dimensions of the different varieties of such
forms, as possess faces not perpendicular to any axis, of
icositetrahedrons, of tetracontaoctahedrons, &c.

§. 157. SEMI-TESSULAR COMBINATIONS.

A combination of the tessular system assumes a
Semi-tessular Character, if it contains one or more
Halves. The semi-tessular combinations must far-
ther be distinguished into semi-tessular combina-
tions of parallel faces, and of 'inclined faces ', accord-
ing to the kind of halves which they contain (§. 128.).

Among those binary semi-tessular combinations, which
contain only one Half, there are two in particular deserving
of notice. The first of them is the combination of the octahe-
dron with one of the hexahedral pentagonal-dodecahedrons.
The faces of the octahedron appear as equilateral triangles
in the place of the rhombohedral solid angles of the penta-
gonal-dodecahedron. If all the faces of the combination
become triangles, the form produced is contained under

208 TERMINOLOGY. §. 157.

eight equilateral triangles, originating from the octahedron,
and twelve isosceles triangles originating from the hexahe-
dral pentagonal-dodecahedron. This form has been called
the Icosahedron of Mineralogy. The icosahedron, however,
is not a simple form ; and, on this account, it receives no
particular name in the systematic nomenclature of forms

(§• «•>

The other is the combination of the hexahedron and the

trigrammic tetragonal-icositetrahedron. The faces of the
hexahedron appear in the figure of rhombs in the place of
the prismatic solid angles of the icositetrahedron ; and if
they are enlarged till all the faces limiting the combination
become tetragons, the result is that form which has been
called the Trlacontahedron of Mineralogy. This form is con-
tained under six rhombs and twenty-four trapezoidal faces,
the one and the other equal and similar among themselves.
It is not a simple form ; and therefore as little entitled to
a peculiar systematic name as the icosahedron.

If two forms occur at the same time in a combination,
we must attend to their position, whether they are both in
the normal position, or whether one of them is in the nor-
mal while the other is in the inverse position ; for the ge-
neral appearance of the combination is very much influ-
enced by this difference.

The two semi-tessular combinations of parallel faces,
Figs. 75. 76., contain the same halves ; with this difference
only, that one of them in Fig. 76. is the inverse of the

same in Fig. 7^., the former being the combination '- (c).

2 ii

— (f), and the latter the combination — — (c'\ JL1 (/) :
2 ii 2 ii 2 ii

and although the general appearance of the two forms is
very different, yet an accurate comparison of the faces
will easily shew their identity. Both of them occur in
hexahe&lral Iron-pyrites.

The semi-tessular combinations of inclined faces, Figs.
77- 78., likewise contain similar halves, but Fig. 77- will be

§. 159. OF THE IMPERFECTIONS OF CRYSTALS. &09

\

designated by -— (P). — (J)> while Fig. 78- is expressed
2 2i

O C*

by — (P) -- _ . (r), since the trigonal-dodecahedron r in

the latter combination is the inverse of the trigonal-dode-
cahedron I in the former, the tetrahedron in the normal
position being common to both. These combinations oc-
cur in tetrahedral Copper-glance.

IV. OF THE IMPERFECTIONS OF CRYSTALS IN RESPECT TO
THEIR FORM.

§. 158. TWO KINDS OF THIS IMPERFECTION.

The imperfections of crystals in respect to their
form, originate either in the very formation of the
crystals themselves, or they are the consequence of
the contact of these with other minerals.

The imperfections of the crystalline forms are deviations
from that regularity which has been supposed to take place
in the preceding considerations of forms. This regularity
requires the faces of crystals to be planes of a certain
figure and extent, and the edges in which they intersect
each other to be straight lines. It is very seldom met with
in nature, perhaps never, if we examine the natural pro-
ductions with the utmost accuracy ; the deviations which it
presents are founded in some cases upon the formation of
the crystals themselves, if we find ourselves entitled to sup-
pose that nothing external has had any influence upon the
quality of the form ; in other cases they depend upon the
contact into which one crystal has come with another ; for
this is the means by which one of them could influence
the shape of the other. The most interesting of these ob-
jects is the consideration of the first kind of these imper-
fections. The other will be considered more at large when
VOL. i. o

210 TERMINOLOGY. §. 159.

treating of compound minerals ; at present it is only ne-
cessary to examine in what shape an individual will ap-
pear, which is prevented from assuming its regular form
by some external obstacle.

§• 159. DEVIATIONS FROM REGULARITY, DEPEND-
ING UPON THE FORMATION OF THE INDIVIDUALS
THEMSELVES.

Those deviations from the regularity of crystal-
line forms, which arise from the formation of the
individuals themselves, refer both to simple forms
and to combinations. They appear either in the size
and figure, or in the physical quality of their faces.

The deviations from regularity take place in two differ-
ent ways :

1, By the disproportionate and irregular enlargement or
decrease of some of the faces, or

2, By their curvature, or in general by the property of
not being mathematical planes.

With respect to the first, it must here be observed, that the
regular enlargement of certain faces, considered above in the
combinations called hemi-rhombohedral, hemi-pyramidal,
&c., does not enter within the limits of our present exami-
nation ; on the contrary, simple forms and combinations of
that kind may, as well as any others, be subject to those
deformities of which we are now treating.

We find many examples of such irregularities in simple
forms. The faces of the hexahedron, for instance, very
often are not squares, but oblong or rectangular figures ;
sometimes only two of them are squares. Of the faces of
the octahedron, four become irregular tetragons, and two
equiangular hexagons. The whole form of the dodecahe-
dron is sometimes elongated or shortened in the direction
of one of its axes : if this be a rhombohedral one, the
form will assume the aspect of a combination of the rhom-

§. 159. OF THE IMPERFECTIONS OF CRYSTALS.

bohedral system ; if it be a pyramidal axis, the dodecahe-
dron will assume the appearance of a combination of the
pyramidal system ; and if it be a prismatic axis, it will have
the aspect of a combination of the prismatic system. The
same changes are sometimes met with in the digrammic
tetragonal-ieositetrahedron. It is also a case not unfre-
quently occurring, that single faces are enlarged in the
manner just described, of which the isosceles six-sided pyra-
mids of rhombohedral Quartz may be quoted as a remark-
able instance.

It will not be amiss to mention here the following pre-
cautions, in order to avoid the errors into which such irregu-
larities might bad. First of all, those angles must be care-
fully examined, in which the faces of the forms intersect
eacli other. Suppose, for instance, a form, exhibiting the
aspect of a vertical oblique-angular four-sided prism, com-
bined with a horizontal one, which belongs to the long dia-
gonal of the former, but whose edges are all = 109° 28' 16"
and 70° 31' 44"; this form will be the octahedron. If in a
form representing a rhombohedral combination of R + n and
P + os, or in a pyramidal one of P + n and [P + co], all
the edges are = 120°, this form will be the dodecahedron.
A form, which seems to be hemi-prismatic, and composed
of an oblique-angular four-sided prism, and half the num-
ber of the faces of a horizontal one, if the edges of combi-
nation prove to be equal to those of the prism, is not what
it appears, but it is a rhombohedron. If, on the contrary,
in a solid contained under six rhombic faces, those edges
which represent the terminal edges of the rhombohedron,
be not equal, the solid itself will not be a rhombohedron,
but a hemi-prismatic form of the description given above.

In the second place, it is necessary to attend to those
forms which enter into combinations with the one, respect-
ing the determination of which there exists some uncer-
tainty.

If in a right rectangular four-sided prism, instead of one
or more of its solid angles, we observe equilateral triangles,
the form will be the hexahedron ; if these triangles be iso-

TERMINOLOGY. §. 159.

sceles, the form will be the right rectangular four-sided
prism of the pyramidal system ; and should they be scalene,
we have reason to suppose (§. 150.), that the form in ques-
tion will be the right rectangular four-sided prism of the
prismatic system.

In similar cases, the forms of cleavage (§. 167-) allow
very often the same application as crystalline forms. The
octahedral Fluor-haloide may serve as an example; the
angles of a right rectangular four-sided prism of this spe-
cies may be taken away, or broken off, by equilateral tri-
angles, which are faces of cleavage. The figure of these
triangles proves the form to be the hexahedron, although
perhaps not one of its faces is a square. The following
chapter will contain farther observations on cleavage.

In the third place, the quality of the faces (Chap. III.)
must be considered. Nothing is more easy than to decide
whether all the faces under which a form is contained, are
of the same quality, or whether they differ from each other
in this respect. If this quality is the same in all the faces,
the form may be a simple one ; if it is different, the form
must be a combination of at least as many simple forms, or
symmetrical halves and fourths, as there are differences ex-
isting in the qualities of the faces. A right rectangular four-
sided prism, contained under faces of three different quali-
ties, must belong to the prismatic system. If the faces pre-
sent only two different qualities, the form may belong to the
pyramidal system, though by this description of the faces it
is not yet excluded from the prismatic system ; and it may
belong to the tessular system, if all its forms are exactly
of the same quality. In this last case, however, the form
is not excluded from any one of the other two systems.

The same is evidently applicable to combinations. It
sometimes happens that some of the faces belonging to the
simple forms which the combination contains, are irregu-
larly increased, whilst other faces belonging to the same
forms, diminish till even they entirely disappear. This
will naturally produce differences in the figure of the faces.
The method to be followed in occurrences of this kind con-

§. 159. OF THE IMPERFECTIONS OF CRYSTALS.

sists in reducing the homologous faces to their regular size
and figure, which is effected by giving them an equal dis-
tance from the centre of the form, and by adding, if neces-
sary, those faces which do not appear at all in the combina-
tion ; provided the observation of one or more of the same
simple form entitles us to suppose this form to be one be-
longing to the combination. In thus completing irregu-
larly limited compound forms, it will always be necessary
to reflect on the possibility of the forms possessing a hemi-
rhombohedral, a hemi-pyramidal, a semi-tessular, &c. cha-
racter. If, for instance, in a hexahedron four of the solid
angles are replaced by equilateral triangles corresponding
to the faces of the tetrahedron, we are not entitled to sup-
pose that the four remaining solid angles too should be
truncated, because in this case it is the tetrahedron, and
not the octahedron, which is contained in the combination ;
but if only one more of the remaining solid angles be re-
placed by a triangle of the same description, then we are
fully entitled to add the rest of the faces required for the
production of the octahedron, or perhaps of two tetrahedrons?
since either the octahedron or two tetrahedrons are really
contained in the combination.

Sometimes the combinations, like the simple forms, are
elongated or depressed in the direction of one of their axes ;
and they assume the aspect of such forms as belong to the
system to which the lengthened or shortened axes refer.
Examples of this kind occur very frequently in the combi-
nations of the hexahedron and the octahedron of hexahedral
Lead-glance. The crystals are sometimes elongated in the
direction of a prismatic axis, or depressed in the direction
of a rhombohedral one. The latter assume the appearance
of a rhombohedral combination, while the former possess the
aspect of a combination of the prismatic system. The most
common form of rhombohedral Quartz is a combination of an
isosceles six-sided pyramid P with the regular six-sided
prism P + os (Vol. II. Fig. 145., abstraction being made of
the faces s and s"). Very often this combination appears flat-
tened in the direction of a prismatic axis, and then it takes

TERMINOLOGY. §. 159*

the aspect of a combination of Pr. P. (Pr + oc)3. Pr + co of
the prismatic system, much resembling Vol. II. Fig. 16.
Also in this case the means quoted above will secure us from
errors which a little practice very soon teaches to avoid in
imperfectly formed varieties of crystallisations.

The curvature of faces, if occurring in simple forms,
in general affects all the faces at once. Thus it is in the
hexahedrons of octahedral Fluor-haloide ; in the dodeca-
hedrons, icositetrahedrons, and tetracontaoctahedrons of
octahedral Diamond ; in the rhombohedrons of the two
species of Parachrose-baryte, &c. Similar imperfections
produce the lenticular forms, particularly the saddle-shaped
lens of the above mentioned species ; which is more correctly
represented in Fig. 79- than in any of the drawings and
models hitherto published, in which, for the greater part, it
is given with four corners, instead of six.

In combinations, the curvature always takes place upon
homologous faces, while the rest are not affected by this
deformity. Examples may be found in the species of pris-
matoidal Gypsum-haloide, of paratomous Augite-spar, of
octahedral Diamond, &c.

Curved edges are produced by the intersection of curved
faces ; rounded edges, too, arise evidently from the curva-
ture of the adjoining faces.

From the preceding considerations of the irregularities
of crystals, it is plain that it is necessary to observe the
greatest precautions in ascertaining the measures of their
angles, if we wish to obtain useful and correct results.
The inaccuracies of so many of these measurements, are not
always errors arising from the imperfection of the instru-
ment, or from the operation ; but very often they are the
consequences of the imperfection of the crystals them-
selves. Small crystals are commonly less subject to these
irregularities than large ones ; and to this, in particular,
the great advantage must be ascribed, which the lleflective
Goniometer possesses over the common one, because, in
applying the former, we may make use of crystals which
are smaller, and therefore in general more perfectly formed,

§. 159- OF THE IMPERFECTIONS OF CRYSTALS. 215

while the common goniometer requires crystals of a larger
size, if it shall be applicable at all, and these are but rarely
found in the necessary perfection.

The reflective goniometer must necessarily be employed
in those accurate and desultory observations, upon which
are founded the determinations of the angles of the funda-
mental form, and consequently also the dimensions of a
natural-historical species ; the common goniometer, how-
ever, will always be found sufficiently accurate, if our ob-
ject be only to discriminate individuals, or to find out the
place of a given one in the system, by the assistance of the
Characteristic.

Notwithstanding all the variability in the size and in
the figure of the faces depending upon it, both of which
may be referred to the imperfect formation of the crystals ;
yet the situation of those faces towards each other will be
always found constant. The faces of forms, both simple
and compound, constantly intersect each other at the same
angles which they would produce if the form had arrived
at the highest possible degree of perfection, which depends
upon the exact equality and similarity of the homologous
faces. The magnitude of the angles is constant. This re-
markable fact has first been ascertained and demonstrated
by the celebrated ROME DE L'!SLE ; it is the basis upon
which is founded the possibility of applying crystallography
to the mineral kingdom. Doubts have been raised against
the correctness of that law, derived from an apparent transi-
tion of certain crystalline forms into others, by continual
changes in the magnitude of the angles: these doubts,
however, immediately disappear, if we consider the crystals
in their greatest geometrical perfection, and not affected
by any of those irregularities to which they are subject.
Hence we may infer, that the crystallisation of minerals,
from the simplicity and constancy of its laws, under the
appearance of the greatest variability, deserves on that ac-
count to be called the most remarkable of those phenomena
which inorganic nature presents to the observer.

216 TERMINOLOGY. §. 160.

§. 160. DEVIATIONS FROM REGULARITY, DEPEND-
ING UPON THE CONTACT WITH OTHER INDI-
VIDUALS.

Crystals may either touch on all sides, those mi-
nerals to which they are adjoining, or they may ad-
here to them only by some of their parts.

Crystals, surrounded and inclosed by the solid mass in
which they are found, or in which they have been formed,
are in contact with this mass on all sides. This mass
may either be homogeneous (§. 23.) to that of the crystals,
or may not be homogeneous with it. In the first case,
the regularity of the form is almost without any exception
so much disfigured, that not even a trace of it will re-
main. One of the individuals prevents the other individual,
by their contact, from assuming that regular form which is
peculiar to it ; and, in fact, we have very often occasion to
observe, that the individual really assumes this regular
form, whenever a part of it emerges from the contact with
other individuals.

Examples of this are frequently found in the compound
varieties of rhombohedral Lime-haloide, and of other spe-
cies. The individuals in these compositions are real crys-
tals, which only have been prevented by their mutual contact,
from assuming their peculiar regular forms. Very often we
find cavities or empty spaces in the interior of sucli com-
pound varieties ; the individuals lining these cavities pre-
sent regular forms, wherever they do not touch the rest of
the compound mass.

In these compositions the crystals sometimes lose only
the regularity of their form, while they still continue to
present its general aspect. Thus, for instance, the particles
of such species whose forms belong to the tessular system,
often Jhave their three dimensions nearly equal, while many
of those of other systems have two dimensions greater or
less than the third. The System of Crystallisation, however,
cannot be inferred from this observation. If one or two of

§. 160. Or THE IMPERFECTIONS OF CRYSTALS. £17

the dimensions dimmish much in size, the individuals be-
come thin laminae or fibres, of which the latter very often
are much thinner than a human hair.

Nay, they may even withdraw themselves entirely from
observation, if the third dimension too nearly disappears.
Yet the compound mineral can never on this account be
transformed into a simple one. This subject will be ex-
plained more at large in Section II.

If the mass which surrounds a crystal, and this crystal
itself, are not homogeneous, the regularity of the latter is
not always impaired. A crystal which, under such cir-
cumstances, has retained its regular form, is said to be
formed imbedded, and if separated from its support, it is
termed a loose Crystal.

Crystals of this kind may be taken out of the mass which
surrounds them, and if they do not cohere with any
particles of the mass, a smooth print of their form will re-
main. Loose crystals, if not perhaps imperfect on some
other account, may be considered as the most perfect pro-
ductions of inorganic nature. But we rarely find such crys-
tals. Commonly they are imperfectly formed of them-
selves, or part of their perfection has been lost in the con-
tact with the surrounding mass. Those individuals, whose
dimensions are nearly equal, appear in this case as round-
ish masses, more or less spheroidal, or as angular masses,
and bear the names of Grains or Angular Pieces. Both the
grains and the angular pieces, therefore, are nothing else
but crystals imperfectly formed.

Besides these minerals, which indeed are nothing but
imperfectly formed crystals, there exist a great many
others, which likewise assume more or less a spheroidal
shape, or that of grains and angular pieces. These, how-
ever, must be carefully distinguished from real grains and
angular pieces, because they are not simple, but compound
minerals.

Crystals which are formed in an empty space, and ad-
here only with some of their parts to the support, which
is, in most cases, different from the mass of crystals, are

218 TERMINOLOGY. §. 160.

termed implanted crystals. Implanted crystals are always
incomplete ; because those parts are wanting in which the
crystals are attached to the supporting mass. They can-
not be removed from it, so as to leave behind a print of
their form; they only may be broken off from the support
with which they cohere more or less firmly.

Implanted crystals must be duly completed, for the
purpose of a crystallographic consideration ; so must also
those crystals, which, by some accident, or on purpose, have
been broken or rendered incomplete. The only rules we
must attend to in this process are those of symmetry, by
which a perfect equality and similarity is established as to
the number and situation of f?_es between those parts of
the crystal which are wanting, and those which may be ob-
served. The most common crystallisation of rhombohedral
Quartz, consists of an isosceles six-sided pyramid, which is
combined in a parallel position with a regular six-sided
prism. As these crystals very often occur implanted, the
observation of one end of the pyramid only is possible ;
evidently the opposite termination of the crystal must be
completed, by supposing it equal and similar to that which
has been observed. Simple pyramids of rhombohedral
Quartz, (and in similar cases also the forms of other mine-
rals), if they present only one of their apices to the ob-
Lsrver, must likewise be completed according to the rules
of symmetry ; and we can never be entitled to assume or
consider such things as simple pyramids, because those do
not exist among the productions of nature, nor are they
obtained from the different processes of derivation (§. 80 —
83.). Similar examples occur in pyramidal Garnet, in octa-
hedral Fluor-haloide, in prismatic Hal-baryte, &c. ; which
must be completed according to the method explained
above.

There are cases, however, in which it becomes necessary
to allow of an exception of that rule. These comprehend
the crystals, in which two opposite solid angles possess a dif-
ferent configuration (§. 147-)- Evidently this difference
always must remain within the range of the series of crys-

§. 161. OF THE STRUCTURE OF MINERALS.

tallisation. It has been observed, that certain crystals,
part of which is differently formed from another one, which
is similarly situated, likewise present differences in their
electric action, on being exposed to an elevated temperature.
Prismatic Zinc-baryte, prismatic Topaz, rhombohedral
Tourmaline, and tetrahedral Boracite, may be quoted as
examples of this peculiarity. Could this observation be
established as a general law, it might prove useful in com-
pleting crystals thus imperfectly formed, though it would
not indicate what faces are to be added on that termination
which is opposite to the observed one.

Another case, in which the two opposite terminations of
crystals are differently formed, does not refer to the pre-
sent place, in as much as it is found only in compound mi-
nerals. It will be treated of more at large in §. 179.

The preceding ones are the most simple modes of the
occurrences of minerals in nature.

CHAPTER II.

OF THE STRUCTURE OF MINERALS.
§. 161. EXPLANATION OF STRUCTURE.

Structure represents the mechanical connexion
among the particles of a simple mineral. It may
be observed, if we destroy this connexion, or sepa-
rate the particles from each other.

We have to distinguish here between the regular and the
irregular structure.

If we break a crystal of hexahedral Lead-glance, or of
rhombohedral Lime-haloide, we observe particles detach-
ed which are contained under even, smooth, and shining
faces. The property of allowing these particles to be se«

TERMINOLOGY. §. 161.

parated, but not the particles themselves, exists in the mi-
nerals, previous to its having been rendered visible by
mechanical force. The particles of a mineral therefore
necessarily must stand in some regular mechanical con-
nexion, because the faces in which they separate are parallel
to faces of regular forms. It is by means of the division
that we acquire a knowledge of this connexion, or of what
is understood by the regular structure of individuals.

If we divide the particles of an individual in other direc-
tions than in those of regular structure, the division takes
place not in even faces, but in uneven faces of different
descriptions, and by this the regular structure is not ren-
dered observable, although it does, or at least may take
place in the same individual. These particles no longer
possess the property of being regularly divisible parallel to
faces obtained by this kind of division ; and the quality of
the faces therefore demonstrates, that no regular mechani-
cal connexion of the particles can take place in these di-
rections. The connexion between the particles in these
directions, is also termed the irregular structure.

It is sometimes attended with considerable difficulty to as-
certain the regular structure, and very often it is no less dif-
ficult to observe the irregular structure. The particles of
certain minerals, as of hexahedral Lead-glance, of rhombohe-
dral Lime-haloide, &c., so very readily separate in the direc-
tion of their regular structure, that it becomes almost impos-
sible to produce any divisions in another direction ; al-
though they may allow of such a division. On the con-
trary, others separate with much greater facility in every
direction but that of the regular structure, so that it
likewise becomes difficult to observe even traces of it,
though by analogy we are led to suppose the existence of
the regular structure.

The regular structure of minerals is observed in their
Cleavage, the irregular structure appears in their Fracture ;
both, fracture and cleavage are comprehended under the
more general idea of Structure.

§. 162. 163. OF THE STRUCTURE OF MINERALS.
§. 162. CLEAVAGE.

An individual is said to be cleavable, or to admit
of Cleavage, if by a mechanical separation of its
particles, the regular structure can be rendered vi-
sible.

Certain individuals may be cleaved with great facility ;
and with only the blow of a hammer, they will divide
into fragments contained under even faces, like those men-
tioned in the preceding examples. This is not the case
with others, in which the mere percussion yields only irre-
gular faces. In these, however, we are not yet forced to
assume the non-existence of cleavage, but by the help of
delicate chisels, or of other appropriate instruments, and by
a careful examination of the resulting faces, we have to de-
termine whether or not cleavage occurs in the individual.
It is very useful to expose such faces to an intense light,
the reflection of which will very soon decide which of these
is the case. A certain degree of skill is required in cleav-
ing minerals, which, however, a little practice will teach
more accurately than could be done by many words ; and
therefore we may omit here, as superfluous, all farther
particulars.

§. 163. FACES OF CLEAVAGE.

The faces obtained in cleaving a mineral, are
termed its Faces of Cleavage.

The faces of cleavage are distinguished from each other
in respect to their properties or their relative aspect.

These properties depend upon the perfection of the
faces, in as far as they may be compared to mathematical
planes, and upon the degree of lustre which they possess.
It is very easy to tell, from the mere ocular inspection of
faces, whether their quality be the same, or whether they
differ more or less from each other. These differences,

TERMINOLOGY. §. 164.

however, particularly in respect to lustre, will be consider-
ed more accurately on another occasion.

Faces of cleavage, which are of the same quality, and ap-
pear in one and the same individual, or in one and the
same species, are said to be homologous ; faces of cleavage of
different qualities, if they appear under the same circum-
stances, are considered as being faces of cleavage not homo-
logous with each other.

Sometimes faces of cleavage appear to be curved. Irre-
gularities of this kind require to be considered, like those
mentioned above in respect to the faces of crystallisation.
But. very often curved faces of cleavage result from the
composition of several individuals in a position little differ-
ent from the parallel one. Fig. 80. shews a remarkable in-
stance in rhombohedral Lime-haloide, where the axes of
the individuals diverge very little from a common centre,
by which the compound product of cleavage assumes the
appearance of a rhombohedron, of which three faces are
convex, and the opposite ones concave.

§. 164. DIRECTION OF CLEAVAGE.

The direction in which the individuals of a spe-
cies allow themselves to be cleaved, is the Direction
of Cleavage.

The direction of the faces of cleavage is constant ; its
situation in respect to the fundamental form, or any other
derived crystalline form of the species, is determined, and
not subject to any alteration.

The directions of cleavage in every individual are not
found in the same number. The individuals of most of
those species which constitute the order Mica, can be
cleaved only in one direction, and therefore possess only
one direction of cleavage. Many species of the order Spar
contain two ; rhombohedral Lime-haloide and hexahedral
Lead-glance contain three ; octahedral Fluor-haloide con-
tains four ; dodecahedral Garnet-blende six ; and in many

§. 165. OF THE STRUCTURE OF MINERALS.

minerals cleavage may be effected in still more directions,
differing in number, and often in quality. The direc-
tion of cleavage more particularly regards such as produce
the most apparent faces of cleavage. If two directions
of cleavage exist at the same time in a single individual,
wherever the faces corresponding to them are obtained,
they always intersect each other at the same constant
angles. This is a necessary consequence of the parallelism
of all those faces of cleavage, which lie in one and the same
direction.

§. 165. CHARACTER OF CLEAVAGE.

The Character- of cleavage consists in the con-
stancy of its direction (§. 164.), and in the possibility
of separating the particles of individuals in this di-
rection, as long as the acuteness of our senses, and
the delicacy of our instruments will allow.

There are minerals whose particles may be separated
from each other in faces which are regularly situated in
respect to the crystalline forms, but which exist previous
to the actual division. Those masses, however, which are
contained between two such faces, allow of no farther cleav-
age. This property of certain minerals will be considered
more particularly in §. 1 79. It is, however, very distinct
from real cleavage, the character of which consists in the
possibility of continuing it, as long as our senses may per-
ceive it, or our instruments and contrivances may answer
the purpose.

Experience shews that cleavage indeed does possess this
property, since it may be effected, to whatever point of the
cleavable individual the instrument is applied in the re-
quired position. If, therefore, an individual is cleavable
in the direction of a certain plane, it must also be cleavable
in any other plane parallel to the former, the distance of
these planes being less than any given straight line.

224) TERMINOLOGY. §. 166.

Several minerals may be cleaved into exceedingly delicate
laminae, others do not admit of cleavage to such an extent.
Among the first, several species of the genus Talc-mica are
particularly remarkable. Prismatoidal Gypsum-haloide may
also be cleaved into uncommonly thin laminae ; and we
might succeed in attenuating them still more, if instru-
ments could be found of sufficient delicacy. Cleavage may
be continued so far in these cases, because, except that single
cleavage, there are no other directions in which the minerals
cleave with the same facility, or, what is the same thing, be-
cause it is very difficult to separate their particles at all in
other directions. The other class of cleavable minerals com-
prehends the individuals of such species as present more than
one direction of cleavage, or whose particles may be more
easily separated in uneven irregular faces. The facility with
which the particles may be separated from each other in
more than one direction of cleavage, or in irregular faces,
prevents the cleavage from being more apparent, and ob-
tained with greater facility in one of the directions. Hexa-
hedral Lead-glance, rhombohedral Lime-haloide, &c. may
be quoted as examples of minerals, which cleave with equal
facility in more than one direction.

§. 166. FACES OF CLEAVAGE PARALLEL TO FACES
OF CRYSTALLISATION.

Every direction of cleavage (§. 164.) is parallel
to the face of a form of the Series of* Crystallisa-
tion of that species, to which the cleavable indivi-
dual belongs.

In the species of octahedral Fluor-haloide, the solid
angles of the hexahedron may be broken off by cleavage
with the greatest facility ; in the place of every one of
those solid angles, there will appear an equilateral triangle,
or an equiangular hexagon, which is the face of cleavage. On
account of its figure, it is situated perpendicularly to a rhom-
bohedral axis, that is to say, like a face of the octahedron, a

§. 167. OF THE STRUCTURE OF MINERALS. 225

form which in fact belongs to the series of crystallisation
of octahedral Fluor-haloide. The solid angles of the oc-
tahedrons cannot in this species be broken off by faces of
cleavage, but this form may be cleaved parallel to its own
faces in four directions, intersecting each other at angles of
109° 28' 16" and 70° 31' 44".

The directions of the faces of cleavage in which the pyra-
mid (P)3 of rhombohedral Lime-haloide may be cleaved,
are parallel to those of faces passing through every two con-
secutive lateral edges of the pyramid. But this is the di-
rection peculiar to the faces of the rhombohedron 11. The
faces of cleavage are therefore parallel to the faces of the
rhombohedron R, which is the fundamental form in the
series of crystallisation of the species. In the rhombohe-
dron, the cleavage may be continued parallel to its own
faces, in three directions, intersecting each other at angles
of 105° 5' and 74° 55'.

llhombohedrai Talc-mica, pyramidal Kouphone-spar,
prismatic Topaz, &c. admit of a cleavage perpendicular to
their axes. The faces of cleavage will be parallel either to
It — os or to P — oo, both of them being limits of their
respective series of crystallisation.

It appears, from the given examples, that not every
crystalline form is cleavable parallel to some one or the
other of its faces. Several of them, however, shew this
property. Simple forms of finite dimensions, if cleavable
parallel to their own faces, have on that account by prefer-
ence been chosen for fundamental forms, even though such
forms should not as yet have been produced by nature
among the crystalline forms of the species.

§. 167. FORM OF CLEAVAGE.

A form contained only under faces of cleavage, is
termed a Form of Cleavage.

The forms of cleavage may be either simple forms or

VOL. I. p

226 TERMINOLOGY. §. 167.

combinations. Both may be either perfect (finite) forms,
or imperfect (infinite) forms.

If we continue cleaving an individual of octahedral Fluor-
haloide, till it is comprehended on all sides within faces of
cleavage, the result will be the perfect form of cleavage
belonging to the species ; it will be a simple form, because
the faces of cleavage homologous to each other, are parallel
only to the faces of the octahedron, and not to those of any
other simple form. It is exactly the same with the forms
of cleavage of hexahedral Lead-glance, of hexahedral
Rock-salt, of octahedral Corundum, of dodecahedral Gar-
net-blende, and many other species.

Peritomous Ruby-blende cleaves in the direction of the
regular six-sided prism R + co. The form of cleavage is
therefore a simple one; but it is incomplete or infinite,
and terminated in the direction of the axis, either by faces
of separation, which are not faces of cleavage, or by faces of
crystallisation.

Prismatic Topaz cleaves parallel to the faces of P ~ co.

The form of cleavage is a simple one, but imperfect ; and

• is contained in the directions parallel to the axis under

faces of crystallisation, or under such faces of separation,

as do not owe their existence to cleavage.

Rhombohedral Fluor-haloide cleaves parallel to the faces
of P + co ; but at the same time, also to those of R — co.
The form of cleavage, R — co. P + co, is therefore a com-
bination. But it is a perfect form, since by these faces the
space is limited on all sides.

Pyramidal Feld-spar and pyramidal Garnet are cleav-
able in the direction of two rectangular four-sided prisms ;
and at the same time perpendicularly to their axis. Their
form of cleavage therefore, P — co. P + cc. [P + cc], is a
combination and at the same time a perfect form of cleavage.

Paratomous Augite-spar, and in most cases also hemi-
prismatic Augite-spar, cleave in the direction of vertical
oblique-angular four-sided prisms ; and at the same time in
the direction of planes, passing through the axes and the
diagonals of the prisms. Their forms of cleavage are ex-

§. 1G9. OF THE STRUCTURE OF MINERALS.

pressed by the combination (Pr 4- oo)3. Pr + 09. Pr + 09
which is imperfect, that is to say, unlimited in the direc-
tion of its axis, except by faces of crystallisation, or by
faces of cleavage different from those contained in the repre-
sentative sign of the compound form, or at last by faces
not produced by cleavage at all.

§, 168. FORMS OF CLEAVAGE, DISTINGUISHED AC-
CORDING TO THE QUALITY OF THEIR FACES.

Faces of cleavage, which belong to one and the
same simple form, are homologous (§. 163.). Faces
of cleavage which are not homologous, belong to
different simple forms of cleavage.

If we find an opportunity ol comparing the form of
cleavage of an individual with its form of crystallisation, it
will not be difficult to decide which of the faces of cleavage
belong to one, and which to different simple forms ; and
indeed the certainty and generality of the present propo-
sition depends upon observations of this kind.

In many instances, however, we only can examine forms
of cleavage, belonging to minerals which are not crystal-
lised, or such individuals as present no regular external
forms, and in which therefore the forms of cleavage are of
still higher value for the natural-historical determination.
Here the above mentioned proposition becomes of the great-
est utility, since it allows the forms of cleavage to be con-
sidered in the right point of view, even though only a few
of their faces should present themselves to the observer.

As to the first, we find that, all the faces of cleavage in
the individuals of rhombohedral Lime-haloide, of octahe-
dral Diamond, of dodecahedral Garnet-blende, of prismatic
Topaz, and of many others, are of exactly the same qua-
lity, so that they can by no means whatever be distinguish-
ed from each other ; these faces being found to possess the
same properties throughout the whole individual. A very
remarkable difference, however, is found in several species,

228 TERMINOLOGY. §. 168.

relative to the quality of the faces of cleavage in different
varieties. One of the most striking examples of this oc-
curs in hexahedral Iron-pyrites. Several varieties of this
species cleave very readily in faces which are pretty even,
shining, &c. ; others only with the greatest difficulty in
faces very much interrupted by asperities. In both cases,
however, the resultant form of cleavage is the hexahe-
dron ; and the faces under which it is contained are of the
same perfection and quality, at least in each individual ta-
ken separately, as it must be on account of their belonging
to one and the same simple form.

-The circumstance that two or more faces of cleavage
possess exactly the same properties, cannot be consider-
ed as a proof of their belonging to one and the same
simple form. An example of this may be taken from pris-
matic Gypsum-haloide. The form of cleavage of this
species is represented by the combination P — OD. Pr + oc.
Pr + os : the two latter, although faces of two different
forms, shew almost exactly the same quality ; and yet the
same two forms in prismatoidal Gypsum-haloide, where
they likewise appear as forms of cleavage, differ very much
in their aspect.

On the contrary, in respect to the second part of the
proposition, the difference of the simple forms to which they
belong, follows in every instance from the different quali-
ty of the faces of cleavage, or from their not being homo-
logous. In the above mentioned compound form of cleav-
age of pyramidal Feld-spar, the face perpendicular to the
axis appears very different from those parallel to it, being
less even, &c. : in a similar way in rhombohedral Emerald,
the face perpendicular to the axis is more perfect and even
than those parallel to it. As a very remarkable example,
the form of cleavage of prismatoidal Gypsum-haloide de-
serves our particular attention. The form considered by
itself appears as a right oblique-angular four-sided prism.
It is more natural, however, to consider it in reference
to its ci'ystalline forms, as the hemi-prismatic oblique-rect-
angular four-sided prism (§. 153.), similar to Vol. II.

§. 169. OF THE STRUCTURE OF MINERALS. 229

Fig. 4G., of the dimensions given in the Characteristic.
From this point of view it becomes at once evident,
that only those pairs of parallel faces which are of
the same quality, can belong to the same simple forms,
which could not agree with the hypothesis of a right
oblique-angular four-sided prism. The crystalline forms
being always supposed to have been previously brought
into an upright position, the most even, smooth, and shin-
ing face of cleavage will correspond to the form Pr + oo.
The other two forms, likewise very different from each
other, though less so than from Pr 4- cc, correspond to

Pr + co and to il Many individuals, containing only

one very apparent face of cleavage, as, for instance, the one
perpendicular to the axis in pyramidal Euchlore-mica, in
prismatic Topaz, &c. besides shew traces of other forms
of cleavage, which here likewise may serve as examples.

But the most interesting of all the consequences to
be drawn from these and similar examples, is the full
confirmation of the theory of crystalline forms, as it has
been given in the preceding chapter. According to that
theory, there exist forms which, although they appear in
a single face, or in two or more, in such directions that
they cannot include the space from all sides, yet must be
considered as peculiar simple forms. The physical quality
of the faces of cleavage in the individuals incontestibly
proves the correctness of that method of considering forms,
Avhich nevertheless has been merely the consequence of geo-
metrical inquiries.

§. 169. FORMS OF CLEAVAGE MEMBERS OF THE
SERIES OF CRYSTALLISATION.

The forms of cleavage represent members of the
series of crystallisation of those species, from the
individuals of which they have been extracted.

The demonstration of this proposition follows immedi-

230 -* TEEMINOLOGY. §. 169.

ately from §. 166. For, if the faces containing a form of
cleavage are parallel to the faces of a crystalline form in
the same species, the form obtained by cleavage must itself
necessarily be similar to the form of crystallisation ; it must
possess the same dimensions and relations, and therefore
be capable of being substituted for the member in the series.
This is applicable to both simple and compound forms of
cleavage ; because nature produces combinations of such
simple forms only as are members of the same series (§. 139.).
Cleavage therefore extends the application of crystallo-
graphy to the productions of the mineral kingdom, and
enables us not only to determine the system, but also
very often even the series of crystallisation of such indivi-
duals or species, in which crystals are either not known at all,
or at least are not the immediate object of our observation.
Hence the study of cleavage is particularly recommended
to those who intend to apply the Characteristic to nature,
and to acquire that degree of skill which is required for
determining .with facility and certainty the productions of
the mineral kingdom by the assistance of the Characteristic.

An accurate knowledge of the peculiarities of cleavage is
moreover very useful for recognizing and completing such
crystalline forms as occur indistinct, imperfect, cohering
with others, &c. Some single faces, fissures in the interior,
strise arising from the superposition of laminae and other
observations of that kind, are very often sufficient for as-
signing the true position to a crystalline form, and by this
means to acquire a correct knowledge of its nature.

The preceding observations indisputably shew, that cleav-
age in itself is a highly remarkable phenomenon of in-
organic nature. Yet this will appear still more strikingly,
if we consider it in connexion with the forms of crystallisa-
tion, in as far as they both refer to the natural-historical
species. If we attend also to the less apparent directions
of cleavage, or those faces which are less distinct, though
subject to the same laws, we are led to the conclusion
that cleavage represents the phenomenon of crystallisa-
tion to its full extent. The Jattcr is therefore not a

§. 170. OF THE STllUCTURE OF MINERALS. 231

mere accident of the exterior form, but it is a property in-
timately related to the existence of the species itself; and
every one of the forms which it is capable of assuming, is
deeply founded in its interior, or in the regular structure
of the mineral ; so that a face of crystallisation is parallel
to every direction of cleavage, and a direction of cleav-
age, more or less distinct, parallel to every face of crystal-
lisation of the species. The first of these results has al-
ready been perfectly confirmed by observation. More ac-
curate information with respect to the second will be ob-
tained by future investigations of this interesting subject.

§. 170. DESIGNATION AND NOMENCLATURE OF THE
FOE MS OF CLEAVAGE.

The forms of cleavage are designated like those
of crystallisation ; several particular cases of its oc-
currence have been provided with appropriate verb-
al expressions, for the purpose of the Systematical
Nomenclature and of the Characteristic.

Since the crystallographic designation of forms of crys-
tallisation and of forms of cleavage is exactly identical ; it
becomes necessary to indicate whether the sign refers to
cleavage or to crystallisation.

In the systems of variable dimension, the cleavage is said
to be A.rotomous*, if it consists of a single face perpendicular
to the axis, or parallel to the base of the fundamental form.
The same expression may be employed, although, beside
this single cleavage, others should appear parallel to the
axis, or including an angle with it, yet it is always required,
that such faces should be less distinct, and thus form a con-
trast with the single one* This observation extends also
to the following expressions relative to some other pecu*
liarities of cleavage.

* From, £g«y, the axis, and ri/ttw, I cut or cleave.

TERMINOLOGY. §. 171.

Cleavage is termed Prismatoldal, if it takes place in a
single direction parallel to the axis, whatever may be the
simple form corresponding to this direction.

It is said in general to be Monotonous*, if it consists of
a single face, which may be perpendicular, or inclined, or
parallel to the axis.

The term Paratomons-\- refers to faces of cleavage of an
indeterminate number, parallel to the faces of a finite
form, and which therefore are not vertical, nor perpendicular
to the axis of the fundamental form. . This expression may
also be employed if the directions of cleavage correspond
only to half the number of faces of a simple form.

Cleavage is termed Peritomous $, if it takes place in more
than one direction parallel to the axis, and if the faces are
all of the same quality and perfection. The result of this
cleavage is a vertical prism.

If the directions of cleavage are parallel to the faces of a
vertical oblique-angular four-sided prism, and at the same
time to those of a horizontal one, the cleavage is said to be
Di-prismatl-c. This di-prismatic cleavage and the prismatoi-
dal cleavage is confined to prismatic, hemi-prismatic, and
tetarto-prismatic forms. The rest may occur in any of the
systems of variable dimensions, which, if necessary, is ex-
pressed by Cleavage, rhombohcdral and paratomous, &c. In
the peritomous cleavage, the form obtained may be a com-
bination ; it is a combination in- the di-prismatic cleavage.

The expressions employed for the designation of the par-
ticulars of cleavage in the tessular system, are evident from
themselves, and therefore need no farther explanation.

§. 171. FRACTURE.

To break a mineral, in order to obtain its Frac-

* From ftova;, single, and ripvu, I cleave.
•f From vrxou, about, and rituvu, I cleave.
£ From <r*^x, round, and TJ^W, I cleave.

§. 172. OF THE STRUCTURE OF MINERALS. 233

turc, is to make its irregular structure appear, by
a mechanical separation of its particles.

Every individual, cleavable or not, may be broken. Ir-
regular structure, that is to say, fracture, may be much
more generally observed than cleavage. Fracture, how-
ever, on account of the want of regularity essential to it,
can only be of very limited use in the Natural History of
the Mineral Kingdom. It may here be useful, from this ob-
servation, to derive the important consequence, that it is
not a generality or a variety, but only a regularity and con-
stancy in the differences occurring in a natural-historical proper-
ty, tcJtich renders it applicable, and determines its value as a dis-
tinctive character in Natural History.

Fracture is considered here as a property of individuals,
or of simple minerals in general, agreeably to the principles
of Natural History. The greater part of the varieties of
fracture, quoted in books on Mineralogy, on that account
must here be excluded, because they refer to compound
minerals, which in fact also may be broken in pieces.
These, however, will be considered in another more conve-
nient place.

§. 172. FACES OF FRACTURE.

The faces in which the particles of the individuals
separate when broken, are termed Faces of Frac-
ture.

The kinds of fracture are determined according to the
quality of its faces. The irregularities of these faces are
either round or angular. The first sometimes represent the
aspect of the inside of a shell. That kind of fracture which
is formed by such faces, is termed the Conchoidal fracture,
and provided with peculiar adjectives referring to the size,
concavity, lustre, &c. ; all this, however, without any useful
consequences. The angular irregularities cannot be com-
pared to any thing at all. The kind of fracture formed by

23!< TERMINOLOGY. §. 173.

these asperities, has received the denomination of the Vn.
even fracture, and has been divided again according to dif-
ferences in the size of its grain, but this too is of very
little value. There exists an immediate transition be-
tween the two kinds of fracture, in which only arbitrary
limits can be fixed. The other kinds of fracture, the even
fracture, the fibrous fracture, the splintery fracture, either
do not refer to simple minerals, or they do not belong to
structure at all. The hackly fracture is not produced by
breaking, but by tearing a mineral in pieces. Foliated
fracture is cleavage ; the passage of the folia means the same as
the directions of cleavage ; regular fragments are the forms
of cleavage ; and the varieties of radiated fracture are the
same, only referring to compound minerals.

§. 173. CHARACTER OF FRACTURE.

The faces of fracture preserve no constant direc-
tion. In this particular, fracture is essentially dif-
ferent from cleavage.

Individuals without cleavage, or with a very indistinct
cleavage, or such individuals whose particles possess only a
small degree of coherence, may be broken in any direction,
that is to say, no direction could be indicated beforehand,
and determined in reference to the situation of any line or
plane, in or parallel to which the mineral would break.
Fracture, therefore, has no constant or determined direction ;
neither can it be continued according to parallel planes, al-
though, by a continual diminution in size, we may carry on
the separation of the particles to an indefinite extent.

The same applies also to those individuals which allow of
cleavage, if they be broken, that is to say, if their particles
be separated in any other directions than those of cleavage.

The irregular structure is very often observable, along
with the regular structure in the same individuals.

§. 174. OF SURFACE. 235

CHAPTER III.

OF SURFACE.
§. 174. SURFACE IN GENERAL.

The Faces of Crystallisation (§. 28.) arc the
most interesting and useful among all those faces
which limit the forms occurring in the productions
of the mineral kingdom,

Beside the faces of crystallisation, Terminology con-
siders the Faces of Cleavage ', of Fracture^ and of Composition.
An explanation of the three first has been given in §. 28.
163. 172. Faces of Composition are those in \ihich several
individuals touch one another ; they belong to tli£ indi-
viduals (§. 158.), and therefore require to be considered in
this part of the work. Besides, it is necessary to attend to
the difference existing between these faces of composition,
and those of crystallisation, of cleavage, and of fracture,
in order to distinguish forms contained only under faces of
this kind, from such as are contained under faces of crys-
tallisation, of cleavage, or of fracture. This will in parti-
cular be necessary if we have to consider individuals,
limited merely by faces of composition, by themselves or
singly, and in their original connexion with others.

Of all sorts of faces, the most interesting ones are thof e
which are even, since, in the mineral kingdom, the uneven
faces are not subject to any constant law, not being curved
like the surface of the geometrical solids, the sphere, the
cone, or the cylinder. The only even faces are the faces
of crystallisation and the faces of cleavage. The latter
exhibit very little remarkable differences in their quality ;
while the former shew a peculiarity, the more deserving of
consideration, because it is closely connected with the
phenomenon of crystallisation itself. The faces of crystal*

TERMINOLOGY. §. 175.

lisation, therefore, will form the most interesting subject of
our present inquiries.

The different qualities of even faces consist in their
being either smooth, without elevations or depressions ; or in
their being provided with certain elevations and depressions,
which, however, are so faint, that the general appearance
of evenness and continuity of the faces is not affected by
their occurrence. Smooth faces also are called perfect,
particularly if they refer to cleavage ; and cleavage is said
to be the more perfect, the more its faces possess this
property. This is at the same time the reason why smooth
faces of cleavage are commonly much more easily obtain-
ed, than such as shew opposite properties.

The quality of irregular faces likewise depends upon the
greater or less degree of smoothness of its inequalities ; and
according to this measure in particular, the conchoidal frac-
ture is said to be more or less perfect. The insensible dimi-
nution of this perfect smoothness produces the passage from
the conchoidal fracture into the uneven fracture (§. 172.)-

The intensity or the degree of lustre of the faces, is
proportional to the degree of their perfection.

Those faces which are not smooth, may be striated, or
rough, or drusy. The most remarkable of these are the
striated faces.

§. 175. STRIATED FACES OF CRYSTALLISATION.

The Striae upon the faces of crystallisation are
produced by the alternating re-appearance of the
faces of those simple forms, which are contained in
a compound one, and they are always parallel to
edges of combination.

One of the forms most commonly occurring in rhombo-
liedral Quartz, is the combination of the regular six-sided
prism, with an isosceles six-sided pyramid, P. P 4- ». The
faces of the prism are streaked horizontally, if the forms
be brought into the upright position.

§. 175. OF SURFACE. 237

These striae are produced in the following way. Instead
of the faces of the pyramid, P, 2, Fig. 73., which in
the perfect combination would continue Avithout being
interrupted from the edges of combination to the apices,
the faces of the prism r, r' re-appear. These faces, how-
ever, do not reach very far, but are again exchanged
for the faces of the pyramid, which in their turn must yield
to the faces of the prism, and this alternately, till at last
the faces of the pyramid meet in the apices, as it is repre-
sented in Fig. 73- If now we suppose the faces altogether,
and more particularly those of the pyramid, to become very
narrow, we may form an idea of those delicate striae, which
are so often met with in nature. It is, however, not very
rare, to be able to observe immediately the formation of
the striae, on the large scale now described, which per-
fectly confirms the above explanation.

It is evident, that the striae thus produced must be pa-
rallel to the edges of combination between the faces of the
pyramid arid those of the prism ; because the faces of those
forms alternate in their edges of combination.

The situation of the faces (viz. those of the prism), is
indeed altered by these striae ; yet this is inconsiderable if
the striae become very delicate, and the more so, if those
faces which belong to the opposite apex of the pyramid,
likewise alternate with the former, and exercise their in-
fluence upon the streaking, as may be seen in Fig. 74. It
will require some attention if we have to apply measuring
instruments to crystalline faces disfigured by these striae.

Another very remarkable example of striated faces oc-
curs in hexahedral Iron-pyrites, in the combination of the
hexahedron, and of the hexahedral pentagonal-dodecahe-
dron. Vol. II. Fig. 1G5. In this example, the striae are
parallel to the edges of combination ; hence they are parallel
to each other upon parallel faces, and perpendicular to each
other upon such faces as are not parallel. If, instead of the
hexahedron, the form combined with the dodecahedron is a
trigrammic tetragonal-icositetrahedron, whose characteristic-
angle is equal to the characteristic edge of the dodecahe-

38 TERMINOLOGY. §. 170.

itron, the striae produced upon the faces of the latter, are
parallel to perpendicular lines, which may be drawn from
the single plane angle of the faces towards the opposite
edge ; upon the faces of the icositetrahedron, the striae are
parallel to the'longer one of those edges, which join two
different solid angles formed by four faces. It would be
superfluous to mention here any more examples ; for every
case referring to this subject may be explained with as
great facility as the preceding combinations of rhombohe-
dral Quartz, and of hexahedral Iron-pyrites.

The curvature of the faces (§. 1,59.) sometimes is pro-
duced by streaking. Thus, in rhombohedral Tourmaline,
the three-sided prisms, with convex faces, are produced by

numerous striae between the faces of . * °?(E)andP-f so (-?).

Arol. II. Fig. 138.

The streaking of the faces may be very useful in rinding
out those which are homologous, since homologous faces al-
ways shew similar occurrences of this phenomenon ; a fact
proved by numerous examples in rhombohedral Lime-ha-
loide, in dodecahedral Garnet, &c.

§. 170. DIVERSE QUALITIES OF THE FACES OF
CRYSTALLISATION.

The property of the surface of crystalline forms,
designated by the terms rough and drusy> arises
from elevations projecting from the faces of crys-
tals. They differ from each other only in the size
of the elevated particles.

In the species of octahedral Fluor-haloide, octahedral
crystals, sometimes of considerable size, seem entirely to
consist of minute hexahedrons. The faces of such octahe-
drons cannot be planes ; but they consist of the faces of
the hexahedrons, which are perpendicular upon each other ;
in such situations, that a plane passing through their solid

§. 176. OF SURFACE. 239

angles is parallel to the face of the octahedron. The more
the size of these hexahedrons diminishes, and consequently
the more their number increases, the more the faces them-
* selves will assume the appearance of exact planes. They
are said to he drusy, if the asperities upon the faces are still
easily distinguishable ; they are termed rough, if they may
only be perceived with difficulty, or if the existence of such
asperities merely can be inferred from the want of lustre of
the resultant faces.

The faces of the hexahedron exhibit in the same species
a phenomenon connected with the former, which in many
respects is very remarkable. They are someti mes covered as
it were with small very flat four-sided pyramids, whose late-
ral edges are parallel to the edges of the hexahedron ; and
of which only the upper part is visible. The faces of these
pyramids are the faces of the hexahedral trigonal-icosite-
trahedron, a form not uncommon in this species. If they
become very minute, they may render the faces drusy or
rough, or they may produce striae parallel to the edges of
the hexahedron.

Another very remarkable fact, which in particular is
very often met with in rhombohedral Quartz, must be
classed along with the preceding peculiarities. Many crys-
tals of the same form, P. P + co, appear as if grouped pa-
rallel to each other, or round a larger crystal of the same
form. Exact parallelism is here understood, as it is in
every occurrence of this kind. Sometimes, however, the
aggregation of crystals of rhombohedral Quartz depends
upon regular composition (§. 179.). The preceding obser-
vations, in respect to striated surfaces^ as to the quality of
homologous faces, applies likewise to such as are rough or
drusy.

Those particles which project from the faces of the
crystals, must not be considered as single individuals ; and
crystals with drusy faces therefore are not compound mi-
nerals. They indicate rather the gradual progress of the
formation of crystals, from the interruption in which they
arise. If we suppose in the octahedrons of octahedral Fluor-

240 TERMINOLOGY. §. 177.

haloide, homogeneous matter of the crystal to fill up the
interstices between the faces of those minute hexahedrons,
so that the formation is terminated ; no character remains
by which a crystal thus formed could be distinguished from
a simple mineral. We may imagine that, in the progress
of the formation, a simple mineral may give rise to a com-
pound one ; but it is utterly impossible that a simple one
could be formed out of a compound mineral. Crystals
with drusy faces may consequently be simple minerals.

§. 177. FACES OF COMPOSITION.

The quality of the faces of composition is acci-
dental.

The faces of composition sometimes are even, yet this is
very rare. Even faces of composition may easily be dis-
tinguished from faces of cleavage, because those particles
which are contained between two faces of composition, can
no more be cleaved in the same direction ; provided they
do not possess besides a cleavage of that kind ; in this case,
however, the quality of the two kinds effaces would suffice
for their distinction.

They are rarely smooth ; and if this happens, we find it
only in single, not continuous parts of the faces. More
commonly they are streaked ; but the striae are irregular,
without any determined or constant direction. Very often
we meet with rough faces of composition, their lustre
being of a very low degree, or even sometimes entirely
wanting ; this may be used as a distinctive character be-
tween the faces of cleavage and those of composition, if, in
a mineral, these two kinds of faces should happen to be pa-
rallel. Lastly, they often are uneven, or contain more or
less considerable elevations and depressions. Faces of this
kind must not be confounded with uneven faces of fracture ;
this, however, may be very easily avoided by comparing
them with real faces of fracture in the same individual.

The character by which the faces of composition essen^

§. 177. OF SURFACE.

tially differ from those of crystallisation and of cleavage,
consists in the circumstance, that generally they preserve no
determined direction, and do not produce any regular forms.
Here we must except those faces, in which parallel indi-
viduals touch one another, or those which depend upon re-
gular composition, and which will afterwards be considered
more at large (§. 179.)- These faces, indeed, preserve a con-
stant and determined direction, and are occasionally of a
peculiar degree of evenness, although, in other respects,
they possess all the properties peculiar to faces of composi-
tion.

Very often the individuals cohere so very strongly in
their faces of composition, that they will rather separate
in faces of cleavage or of fracture, than in those of compo-
sition. If the individuals, on account of their minuteness,
withdraw themselves from observation, or become im-
palpable, the faces of composition likewise disappear. It is
evident, from the preceding observations, that by this pro-
cess a compound mineral can never be transformed into a
simple one. At all events, we must carefully distinguish
between the faces of composition and those of crystallisa-
tion and cleavage, the first of which are present in every
compound mineral.

VOL. I.

TERMINOLOGY. §. 178.

SECTION II.

THE NATURAL-HISTORICAL PROPERTIES OF COM-
POUND MINERALS.

§. 178. REGULAR AND IRREGULAR COMPOSITION.

The mode of composition in which the indivi-
duals of the mineral kingdom appear, is said to be
regular , if the form produced by their connexion
is a regular one, and if this regularity is a neces-
sary consequence of the composition ; if the con-
trary takes place, the composition is said to be
irregular.

If two or more homogeneous individuals join in a com-
pound form, regularly and symmetrically, at least if duly
completed ; the composition is in every respect perfectly
determined. For we may indicate, with the greatest accu-
racy, in which faces of the simple forms, or in which plane
the individuals cohere, even though this plane should not
be parallel to a face of any simple form of that species
to which the individuals belong. In general we may ob-
tain the situation of the individuals required, or necessary
in order to produce the compound form. A composition
of this kind is said to be regular.

The composition is irregular, if the forms are not con-
nected in the manner now described, and if, therefore,
they do not produce any regular or symmetrical forms.
Two or more individuals joined in this way, are said to be
merely aggregated, an expression which intimates, that there
is no regularity in their composition.

There are compound minerals, which affect regular ex-
ternal forms, although their composition is in fact irre-
gular. The regularity of the form in such cases evidently
does not follow from the composition, but it must originate

§. 179. OF COMPOUND MINERALS. 243

from something which is foreign to the mineral. Compo-
sitions of this kind cannot be called regular in the sense
of the word now explained.

§. 179. REGULAR COMPOSITION. TWIN-CRYSTALS.

The regular composition of two homogeneous
individuals, joined in one crystalline form, has
been designated by the name of a Twin-Crystal.

It is unnecessary to consider by themselves regular com-
positions of three, four, five, or more individuals, because,
although compositions of that kind, according to several
laws, should take place in nature, yet they always can be
reduced to, and explained by the regular composition of two
individuals.

The property peculiar to the twin-crystals consists in
the close and exact connexion of the Face of Compo-
sition (§. 177')? dr that in which the individuals join, with
the series of crystallisation of the species. The face of
composition is either parallel to the face of a form' belong,
ing to this series, or it is perpendicular to a certain edge.
The situation of the two individuals themselves is obtain-
ed, if we first suppose both to be in the parallel position,
and then turn one of them round a certain line, likewise of
a determined direction, under an angle of ISO0, while the
other remains unmoved. This line is termed the Axis of
Revolution. It is either perpendicular to the face of com-
position, or it coincides with this face, while it is parallel
to a crystallographical axis of the individual. The angle
of 180° is the Angle of Revolution.

These properties are necessary to, and characteristic
of, the twin-crystals, which are very easily distinguished
from any other compositions occurring in nature, in which
the junction of two or more individuals takes place in other
faces and in other directions, the aggregation of the indi-
viduals being accidental.

It^is moreover necessary that the individuals be homoge-

244 TERMINOLOGY. §. 179.

neous. Two individuals not homogeneous, even though
they were supposed to possess the same form, and to join
according to some rule, can never produce a twin-crystal ;
they cannot be considered as compound, but must be classed
among the mixed (§. 14.) minerals.

The forms of the single individuals of a twin-crystal
must therefore contain members of one and the same series
of crystallisation : They may be simple forms or combina-
tions. Commonly they either are or both contain the same
members, so that in most cases the forms of the two indi-
viduals may be considered merely as parts of one and the
same crystalline form, part of which only has assumed an
extraordinary yet determined situation.

This has been the foundation of another method employ-
ed for explaining the form of regularly composed individuals.
A plane is imagined to bisect in a determined situation the
form of the simple mineral, and one of the halves to be
turned in this plane through a certain number of degrees,
while the other remains in its former situation. The num-
ber of degrees is equal to half the circumference, or = 180°,
hence twin-crystals considered in this point of view, have
been called Hemitrope Crystals.

The change in the situation, which is produced in the
parts of the combined individuals, occasions, under certain
circumstances, the production of angles greater than 180°,
which are not to be met with in the forms of simple indivi-
duals as considered above. These angles are said to be re-
entering or re-entrant ; and they are commonly taken for a
character of a twin-crystal, or of a hemitrope one.

Angles of this kind, however, greater than 180°, may
arise if two individuals of a species of the same or of dif-
ferent forms, are joined in a parallel position ; as, for in-
stance, in rhombohedral Quartz, where it sometimes happens
that two crystals of the common form have all their faces pa-
rallel, and the axis of the one in the prolongation of the axis
of the other. The product, nevertheless, must not be con-
sidered as a twin-crystal, or as a hemitrope one. The two
apparent individuals in a composition of this kind, form parts

§. 179. OF COMPOUND MINERALS.

of a single one ; and the whole is nothing else but a parti-
cular case of those mentioned in the explanation of the striae
(§. 175.), that is to say, it is an alternating repetition of
the faces of the simple forms contained in the combination.

On the other hand, we find real hemitrope or twin-
crystals, which yet do not contain any re-entering angles,
because the existence of these depends upon the situation
of the faces of the simple forms. It is not the re-entering
angles, therefore, that form the essential character of the
twin-crystals, but the situation of their different parts,
which cannot be explained without assuming that they are
formed from the composition of two individuals.

The law according to which twin-crystals are formed, may
also be expressed by crystallographic signs ; since it is re-
quired only to mention the situation of the axis of revolu-
tion and the plane of composition in reference to the crys-
talline forms occurring in a species. For this purpose
the crystallographic sign of the face, parallel to which the
regular composition takes place, is included in braces ;
the direction of the axis of revolution is added to it, and
separated by the sign : , if it should not be perpendicular
to the face of composition.

After this general consideration of twin-crystals, a few
examples will be sufficient for illustration, in so far as is
required for our purpose, and for exemplifying the em-
ployment of the crystallographic signs.

Suppose the face of composition to be parallel to a face of
crystallisation, and the axis of revolution, perpendicular to it, to
beat the same time an axis of crystallisation.

If we join two octahedrons in a parallel position in such
situations, that they come into contact with their own faces,
the face of composition will be parallel to one of the faces
of the octahedrons, and one of the rhombohedral axes will
be common to both these forms. Although the faces of
one of them produce re-entering angles with some of the
faces of the other, yet the assemblage of the two octahe-
drons cannot be considered as a twin-crystal, because the
faces of the two forms are exactly parallel to each other.

246 TERMINOLOGY. §. 179.

This must be referred to the case explained above in the
examples of octahedral Fluor-haloide, and rhombohedral
Quartz.

Turn now one of these octahedrons round the common
rhombohedral axis through an angle of 180°, this axis being
considered as the axis of revolution, while the other re-
mains unmoved ; the face of composition being perpendi-
cular to the axis of revolution. A twin-crystal will now
be formed, because the individuals can no more be consi-
dered the one as the continuation of the other, since their
respective homologous parts have assumed a different, yet
determined situation towards each other, which is the pe-
culiar character of a twin-crystal. The faces of the one
produce re-entering angles with those of the other, equal
to double the edge of the octahedron, or = 218° 56' 32"
= 360° — 141° 3' 28".

The same result is obtained, if we bisect an octahedron
by a plane through its centre, parallel to two of its faces,
or perpendicular to one of its rhombohedral axes, and allow
one of the halves to make a revolution of 180° round that
rhombohedral axis, upon which the section is perpendicular,
while the other half remains unmoved. The plane of the
section is the face of composition itself.

In the preceding case, the term twin-crystal is more ap-
propriate to the first, that of a hemitrope crystal, more to
the last mode of explanation. The first supposes that every
twin-crystal consists of two different individuals, which re-
quire to be joined in a certain determined situation, in
order to produce the compound crystal ; the other sup-
poses only one individual, in which the situation of some
of its parts has undergone a regular change by an opera-
tion which can never have been employed by nature. This
is the reason why the first, and the name twin-crystal re-
ferring to it, has been preferred in the present work, to that
of a hemitrope crystal, particularly since it is applicable
to many cases, where the latter term would produce an
erroneous idea. The expression regular composition being
more general than either of them, is very often useful in its
application.

§. 179. OF COMPOUND MINERALS. 247

The above mentioned twin-crystals occur in octahedral
Iron-ore, in octahedral and dodecahedral Corundum, &c.
Vol. II. Fig. 156. Its crystallographic sign is O,{§}.

Instead of the octahedrons, we may substitute two hexa-
hedrons, without changing any thing in the law of compo-
sition ; the resulting twin-crystal will have its face of
composition parallel to a face of the octahedron, and that
rhombohedral axis, which is common to the individuals,
for its axis of revolution. The re-entering angles are
= 250° 31' 44" = 360°— 109° 28' 16".

Regular compositions of this kind are found in hexahe-
dral Lead-glance, also such twin-crystals as result from com-
binations of the hexahedron and the octahedron, or other
forms of the species. The crystallographic signs of the
first are H, {£} ; of the second H.O, {?}.

The dodecahedron, if substituted instead of the octahe-
dron, produces similar twin-crvstals, which are found in
octahedral Diamond, dodecahedral Garnet-blende, &c.
Vol. II. Fig. 163. Their designation is D,{°}. ,

If we join two rhombohedrons in a transverse position,
the principal axis being the axis of revolution, the twin-
crystal produced will be of the same kind as the preceding
ones, which becomes evident, if we imagine a rhombohe-
dron to take the place of the above mentioned hexahedron.
The face of composition, which is perpendicular to the
rhombohedral axis, and, as such, analogous to that of the
octahedron, is parallel to H — 03. We may now substi-
tute any finite rhombohedral form, for instance (P)3, or
any combination of rhombohedrons, pyramids, and prisms,
instead of the rhombohedron ; and the result will yield
forms of the same kind, of which many examples are
found in rhombohedral Lime-haloide. Vol. II. Fig. 129.
The crystallographic sign of this regular composition is

(P)«,{R_«.}.

Let every thing be as before, only the axis of revolution
not parallel to an axis of crystallisation.

Twin-crystals of this kind are produced, if two rhombo-

TERMINOLOGY. §. 179-

hedrons of the form R + n join in their own faces, so
that their axes include an angle. They are brought into
this situation by supposing the axis of revolution to be
perpendicular, the face of composition parallel, to one of
the faces of R + ri. Of this composition, the sign is
R, {!}• The law of composition remaining the same,
we may substitute in the place of R + n, any one of the
forms R + n + 1, R -f n — 1, R + co, &c. ; in short, every
member of the series of crystallisation derived from R + n,
and every combination which they possibly may produce.
On the other hand, we may likewise alter the situation of
the face in which the individuals join ; and, provided it
does not coincide with R — co, the resultant twin-crystals
still belong to the present section. Nature produces
many examples of this law ; for instance, in rhombohedral
Lime-haloide, R — 1 composed in the face of R, of which
the crystallographic sign is R — 1,|||; the combina-
tion R — cc. R + CD, composed in the face of R, Vol. II.
Fig. 132., or in the face of R — 1, Vol. II. Fig. 133., the crys-
tallographic sign of the former being R — co. R + co,{!J} ;
that of the latter, R — co. 11 + co,{?iZll}; ft, jn the

face of R— 1, designated by R { — - }. Vol. II. Fig. 130.;
(P)3, in the face of R + 1, which is designated by

(P)S{1±±}; &C.

The present law is not confined to the rhombohedral
system. The pyramid P in pyramidal Copper-pyrites, if
composed in one of the faces of P, produces a twin -crystal,
very much resembling that of the octahedron in octahedral
Iron-ore, but differing from it, in as much as the axis
of revolution in pyramidal Copper-pyrites, is not at the
same time an axis of crystallisation. Its sign is P, { ? } . In
the common twin-crystals of pyramidal Tin-ore two in-
dividuals of the crystalline form P + 1, or P + 1. [P + co],
are joined in a face of P ; the axis of revolution is
perpendicular to this face, and is not an axis of crystallisa-

§. 179. OF COMPOUND MINERALS. £49

tion. The crystallographic sign becomes P + 1 { 5 } , or

p + ].[]?+ »],{;}.

Sometimes the faces of infinite forms are faces of com-
position. Thus, in paratomous Augite-spar the crystalline

-p w
form — (Pr + co)3. Pr + co is composed in the face of

Pr + 05, the axis of revolution being perpendicular to this

face. The re-entering angles are produced by the hemi-

p

prismatic form — The crystallographic sign of this re-
gular composition is (Pr + co)3. Pr + co, {Pr + cs}.

Hemi-prismatic Augite-spar gives a similar example. The
form is - . _£. (Pr + co)3. Pr + co, the face of composi-
tion is parallel to Pr + co, and the axis of revolution per-
pendicular to it. This composition, however, by a curious

anomaly, commonly does not present any re-entering angles,

p
although they should occur, both on account of _ and

of-*.
2

To this class likewise belong the well known cruciform
twin-cr}Tstals of paratomous Kouphone-spar and prismatoi-
dal Garnet. The crystalline form of the first is P. Pr -f co.
Pr + co ; the face of composition is P 4- co, and the axis of
revolution is perpendicular to it, Vol. II. Fig. 40., without
the faces t. The form of the other is P — co. P + co.
Pr + co ; the face of composition is one of the faces of f Pr,
and the axis of revolution is perpendicular to it.

We may observe, in respect to the latter two twin-crys-
tallisations, that they represent in some measure -the double
of what we have seen in the preceding cases, so that a more
symmetrical assemblage of the two individuals is produced,
which is not the case with the greater number of the other
twin-crystals. This peculiarity is owing to the circum-
stance that the individuals do not terminate at the face of

250 TERMINOLOGY. §. 179.

composition, but that they are continued beyond it, which
causes their cruciform aspect. In the crystallographic sign
of similar compositions, the number 2 is prefixed to the
sign of the face of composition. Thus the twin-crystal of
paratomous Kouphone-spar is expressed by P. Pr + cc..
Pr -f 05, 2{P-|^]; that of prismatoidal Garnet by P— -co.

P + cc. fr + co, 2 -j ? — > . Other systems present similar

occurrences, as, for instance, one of the most remarkable in
the hexahedral pentagonal-dodecahedrons of hexahedral
Iron-pyrites.
The crystallographic sign of the cruciform pentagonal.

dodecahedron is — 1, 2{s}» the face of composition being

one of the faces of the dodecahedron (§. 63.).

Another kind of twin-crystals results, if the face of com-
position is perpendicular to an edge of the crystalline form, and
the axis of revolution parallel to this edge^ or, which is the same
thing, perpendicular to the face of composition. Rhombo-
hedral Ruby-blende affords examples of this law ; the crys-
talline form R — 1. P + co is composed in a face perpen-
dicular to one of the terminal edges of R — 1, or to the
inclined diagonal of R — 2, the axis of revolution being
paralleltothisline.Vol.il. Fig. 139. The crystallographic
sign is R — 1. P + co,!*""'^*"1}; where R — 2. R — 1

indicates those edges, which at the saml time lie in the
planes of R — 1, and of R — 2.

According to this law may be explained the twin-crystals
of di-prismatic Lead-baryte, prismatic Lime-haloide, pris-
matic Melane-glance, &c., whenever they assume a cruci-
form aspect. Under these circumstances, the present law
is as it were complementary to the preceding one, and
either of them may be applied, although in many cases the
preceding one will be found more simple. The law of the
composition of di-prismatic Lead-baryte, Vol. II. Fig. 38.,
may therefore be expressed either by 2{Pr}, or by

§. 179. OF COMPOUND MINERALS. 251

2 1 P. Pr}; of which the former indicates the face of com-
position to be parallel to both the faces of the horizontal prism
Pr, wjiile the latter refer to faces of composition perpendi-
cular to those edges of P, in which the pyramid is touched
by the faces of Pr. The' continuation of the individuals
beyond the faces of composition produces the identity of
the results of the two laws.

Prismatic Feld-spar gives an interesting example of an-
other law of regular composition, in which the face of compo-
sition is parallel to a face of crystallisation^ but the axis of revo-
lution lies in the same face^ and coincides with an axis of t/ie
crystalline form. Its crystalline form is the combination

1?L±? (y> _ *L (P). (Pr + 0=) 3 (r, 0- Pr + oo (M).

Vol. II. Fig. 61. The face of composition is Pr + co ; the
axis of revolution is situated in it, and parallel to the prin-
cipal axis of P. After having placed^ two crystals of this
form in an upright and parallel position, we must turn the
one round its vertical axis, while the other remains un-
moved. In the position thus produced, the junction of the
two individuals will produce a twin-crystal. But we may
join them either with those faces of Pr 4- co (M\ which are
situated to the right, or with those which are situated to the
left of the face y, which is conceived to be turned towards
the observer, and contiguous to the upper apex of the
form. The product in the first case will be the twin-
crystal represented, Vol. II. Fig. 80., and its crys-

tallographic sign IJ-1±_?. — ££.. (Pr 4- os)3, Pr + OB,

{ r, Pr -f 03 : Pr + oo. Pr + os } ; the product of the second is
the twin-crystal, Vol. II. Fig. 81., and its sign * + 2.

.— — (Pr+co)3.Pr+03,{l,Pr + c» : Pr + oo. Pr + w}.

The first part of the expression within the braces, indicates :
the right or the left face of Pr + oa to be the face of com-
position, while the second expresses the direction of £he

252 TERMINOLOGY. §. 180.

axis of revolution by the edge produced between Pr+ oo
and Pr + co, which is parallel to the axis of P.

Cleavage confirms to the full extent of its application,
every thing that has been said here on the subject of twin-
crystals. For it is possible to extract from the twin- crystals
compound forms of cleavage, in which those parts which
belong to one individual represent the real forms of cleavage
of the species.

§. 180. IRREGULAR COMPOSITION. GROUPE AND
GEODE OF CRYSTALS.

If several loose or imbedded crystals are merely
aggregated (§. 178.), so that the one becomes the
support of the other, while there exists no general
support ; the assemblage is termed a Groupe of
Crystals ; if, however, several crystals of that kind
are fixed to a common basis, so as to produce a ge-
neral support for them all, the assemblage is said
to be a Geode of Crystals.

The difference between these two sorts of assemblages
is the same as that existing between an imbedded and an
implanted crystal.

There is sometimes a certain order observable in these
groupes of crystals, although this is never geometrical re-
gularity (§. 178.); and no regular form is produced in the
assemblage. Both the groupes and the geodes refer only
to compound minerals, never to such as are mixed.

These compositions consist of individuals of considerable
size, which, therefore may be very easily recognized ; they
assume always their regular form, as soon as they are dis-
engaged, or cease to touch other individuals. Upon these
assemblages is founded the explanation of several forms of
compound minerals found in nature.

§.181.182. OF COMPOUND MINERALS.
§. 181. IMITATIVE SHAPES.

The shape of a compound mineral is called an
imitative or particular external Shape, if it bears
some resemblance to the shape of another natural
or artificial body. Some of these forms are pro-
duced in a space not incumbered with matter, and
depend upon the properties peculiar to the minerals
themselves, without being influenced by any con-
tiguous matter ; others owe their shape to that
extraneous or foreign matter, with which they are
surrounded. The latter of these have been called
extraneous imitative Shapes.

The groupes and geodes are the simplest modes in which
the irregularly compound minerals appear in nature. If
the individuals thus connected are diminished in size, and
if their number at the same time increases, imitative
forms are produced from the groupes of crystals ; which,
although they are founded in the nature of the individuals
themselves, yet. cannot be employed to any useful purpose
in Natural History. The extraneous imitative forms can-
not be brought into any connection with these groupes
at all ; they do not depend upon the natural forms of
the individuals ; on the contrary, in most cases where they
are observable, we find them quite contradictory to the na-
ture of the individuals which they contain. For in these
the form depends entirely upon the shape of the space pre-
viously existing, and is accordingly entirely accidental.

§. 182. IMITATIVE SHAPES ORIGINATING IN THE
GROUPES OF CRYSTALS.

The imitative shapes which originate from the
groupes of crystals, are loose or imbedded, and
more or less regular, globular or spheroidal masses.

TERMINOLOGY. §. 186.

If the individuals connected with each other become very
small, but join in a great number into a groupe of crystals,
globular forms result, which are sometimes perfect, some-
times very imperfect. Their surface is drusy, or covered
with asperities, where it has not been disfigured in its
formation, or by subsequent accidental circumstances. In
their interior we may still discover the direction of the
constituent individuals, which, in most cases, corresponds
to the direction of the radii of a sphere ; they begin in the
centre, and terminate at the surface. Imbedded globular
shapes, like imbedded crystals, are complete on all sides,
and leave an impression of their form in the mass from
which they have been detached.

Several globular masses of this kind, if attached to one
another, may produce reniform and botryoidal shapes,
which, however, require to be distinguished from those de-
scribed in §. 183.

The loose or imbedded globular shapes differ from grains
and angular masses (§. 160.), in as much as they are not
simple minerals. Examples of imbedded globular shapes oc-
cur in prismatic Iron-pyrites, in prismatic Azure-malachite,
and other species ; the same Malachite presents also reni-
form and botryoidal shapes formed from imbedded crystals.

§. 183. IMITATIVE SHAPES ARISING OUT OF THE
GEODES OF CRYSTALS.

There are three different kinds of imitative
shapes resulting from geodes of crystals : 1. Those
in which the individuals spring from, or are attached
to a common point of support ; 2. Those in which
the individuals form one the support of the other ;
and 3. Those in which the support is cylindrical,
sometimes a simple line, sometimes a tube.

Among those of the first division we find the implanted
globular shapes. They arise, if very thin, capillary crystals, or

§. 183. OF COMPOUND MINERALS. 255

in general, such as have one of their dimensions considerably
surpassing the others, are fixed with one of their ends to a
common point of support, from which they diverge in every
direction. The mode of the formation of such globular
shapes is more apparent, if the number of the individuals
is not so great that they touch each other on all sides.
The implanted globules must necessarily be incomplete,
because the implanted crystals of which they consist, are
themselves incomplete, and therefore they leave no im-
pression when detached from their support. Globular
shapes of this kind occur very frequently in prismatic
Kouphone-spar, in macrotypous Lime-haloide, in prismatic
Hal-baryte, &c.

If, during the formation of several globules, they come
into contact with each other, there will arise reniform and
botryoidal shapes, which therefore are nothing else than
several implanted globules joined together. The single
globules are separated from each other by faces of composi-
tion. Rhombohedral Iron-ore very often affects shapes of
this description, in which species they are known under
the name of Haematites. They occur also in the varieties
of rhombdhedral Quartz, called Calcedony. In these very
often the individuals are so delicate, that they withdraw
themselves from observation.

Into the present class belong also the fruticose shapes,
which possess some resemblance with parts of certain
plants, and most of those commonly called dendritic, the
latter of which may penetrate throughout the whole mass,
or only be superficial.

The second division contains, among others, the denti-
form, the filiform, and the capillary shapes. These arise, if
one implanted crystal is the support of another, this of a
third, and so on ; so that rows of such crystals are pro-
duced, as we may observe them very often differently bent
in hexahedral Silver, in octahedral Copper, and also in
octahedral Iron ; in the last of these, however, they have
not yet been found disengaged. If the crystals join so very
intimately, that it is no longer possible to distinguish them

253 TERMINOLOGY. §. 183.

from each other, those imitative shapes result, which are
not unfrequently met with in the above mentioned species.

Sometimes several rows of individuals thus composed
join within one and the same plane in certain constant di-
rections, so that the individuals of the one of these series
do not join with those of the other, but remain separate.
Thus the dendritic shapes are produced, of which most dis-
tinct varieties occur in hexahedral Silver and hexahedral
Gold. The same minerals evidently shew the formation
of the dendritic, as also that of the dentiform, filiform,
and capillary shapes, in those varieties where the indivi-
duals constituting them still may be distinguished in the
compositions. To this division belong also some of the
superficial dendritic shapes formed in fissures.

If the rows of individuals, thus arranged, approach so
near each other that they at last meet, so as to form a con-
tinuous mass, they are said to occur in the external shape
of leaves or membranes, which are among The most common
shapes found in hexahedral Gold, where they exhibit va-
rious modifications. In some of them we may still discern
the individuals ; in others there are striae in certain direc-
tions, indicating their composition, so that their mode of
formation does not remain doubtful. From external shapes
of this description, we may infer that those likewise whose
smooth surfaces no longer present any traces of composi-
tion, are yet owing to the aggregation of several indi-
viduals.

Compound minerals, like those now described, may
again join in a new composition, in which consequently the
individuals are arranged in the direction of different planes ;
in most cases at right angles to each other. Thus the re-
ticulated shapes arise, of which the most distinct specimens
occur in octahedral Cobalt-pyrites. In these very often
the composition itself is still observable. Some of the re-
ticulated forms, however, allow of a different explanation,
if, instead of rows of individuals, they consist of capillary
crystals, like those of peritomous Titanium-ore.

The third division comprehends the stalactit'tc and coralloidal

§.184. OF COMPOUND MINERALS. 257

shapes. The first of these consist of individuals which are
perpendicular to every point of a straight cylindrical or linear
support in its whole circumference, as appears from many
examples in prismatic Iron-ore, in prismatic Iron-pyrites,
where the composition commonly is still observable, and in
the varieties of rhombohedral Quartz, called Calcedony,
where the individuals no longer can be distinguished. On
a very large scale they are not uncommon in limestone
caves, and consist of varieties of rhombohedral Lime-ha-
loide. The coralloidal shapes consist of individuals inclined
at an angle to their support, which, although linear, is not
straight ; they are fixed upon this support in every part of
the circumference, exactly as is the case in the stalactitic
shapes. This vsort of imitative shapes occurs not unfre-
quently in prismatic Lime-haloide, particularly in those
varieties which have been called Flos-ferri.

There occur many more imitative shapes in nature than
those contained in the preceding examples, and many more
have been distinguished and described by mineralogists.
But these few examples will be sufficient to shew the me-
thod of explaining all forms similarly composed.

§. 184. AMORPHOUS COMPOSITIONS.

If the mass, formed by the junction of several
individuals, is not only of an irregular shape, but if
even in this we cannot trace any resemblance with
the shape of another body, the mineral is said to
be massive.

Massive minerals are amorphous irregular compositions of
individuals of the same species, which are in contact with
each other on all sides. The difference between massive
minerals, and those forms resulting from the groupes of
crystals, which deviate more or less from the spheroidal
shape, consists merely in the strong adhesion of the fyr-
mer to the surrounding masses of other species. It is
formed, however, and assumes a shape corresponding to

258 TERMINOLOGY. §. 185.

its own inherent powers, and does not depend upon' its
support, in as much as we are led to suppose both of them
to he of contemporaneous origin.

Massive minerals of a smaller size are also called dissemi-
nated minerals, which have again been subdivided according
to the size of the particles. Very large masses of amor-
phous minerals sometimes enter into the composition of
rocks, as rhombohedral Lime-haloide and prismatoidal Gyp-
sum-haloide, several varieties of Iron-ores, &c. Under
these circumstances they assume the shape of beds, &c.,
the consideration of which is no longer an object of Natu-
ral History.

§. 185. ACCIDENTAL IMITATIVE SHAPES.

The accidental imitative shapes presuppose an
empty space, which has been filled up by the indi-
viduals of compound minerals, to which is trans-
ferred the form of the pre-existing space.

In this case, the shape which the mineral assumes is not
a consequence of the properties inherent in the mineral,
or peculiar to its nature, but it merely belongs to that space,
in which the formation takes place. The sides of this space
serve as support for the individuals. Thus, at first a coat-
ing is formed, which consists of small, but in many cases
very perceptible crystals, whose apices are turned towards
the inside of the empty space. This accounts for the hol-
lowness of many imitative forms of this kind, of which the
cavities are still lined with crystals. Sometimes also we find
in the interior of such specimens, implanted globular, reni-
form or botryoidal shapes, &c., in short, imitative forms de-
pending upon the crystallisation of the mineral itself. Yet
the external shape of the whole, or of the compound mi-
neral, must always be considered as an accidental imitative
shape.

If the whole of the space is entirely filled up, there re-
mains nothing else but the mode of composition of the in-

§. 186. OF COMPOUND MINERALS. 259

dividuals, from which we may judge, without giving any
attention to the surrounding mass, whether the imitative
shape be the result of crystallisation, or whether it has been
influenced by other circumstances.

The space in which the accidental imitative shapes are
formed, may be either regular or irregular. A regular
space cannot be produced except by crystallisation ; and
this may be either in the interior of a real crystal, or it is
the cast of a crystal in the surrounding mass (§. 186.). The
first is not uncommon, particularly in large crystals of
rhombohedral Quartz, where part of the space of the crys-
tals has remained empty, and is regularly limited by the
surrounding crystalline mass.

The irregular spaces sometimes consist of accidental fis-
sures, cracks, and other similar openings : sometimes they
depend upon the? structure of the surrounding mass, which
in many instances belongs to the class of rocks ; others at
last -arise from moulds of various minerals, and also of
organic bodies.

The different description of the space in which compound
minerals are formed, produces a distinction of their forms
into regular and irregular accidental imitative forms.

§. 186. REGULAR ACCIDENTAL IMITATIVE SHAPES.
PSEUDOMORPHOSES.

The regular imitative shapes of the preceding
paragraph have been called Pseudomorphoses, or
Supposititious Crystals. The latter denomination
seems to be rather improper, since they share so
very little in the properties of real crystals.

No pseudomorphoses are formed in such impressions as
originate from imbedded crystals, which can be separated
from their surrounding mass (§. 160.) ; at least experience
has not as yet furnished us with any well authenticated in-
stances which could not be explained, but on that suppo-
sition.

260 TERMINOLOGY. §. 18G.

But if an implanted crystal (§. 160.) is covered over by
the mass of another mineral, which has been formed after
the production of the first, the deposite of new individuals
will at first constitute a Coating, which consists of minute
crystals, and through which the form of the implanted crys-
tal still continues to be perceptible ; the mineral may yet
proceed in its formation, and become massive, or it may as-
sume any other imitative shape, in which the form of the
original implanted crystal entirely disappears. The crys-
tal is moulded in this mass ; and, if it be taken away, or
decomposed, it will leave an Impression of its form. Rhom-
bohedral Quartz, and many other minerals, present in-
stances of similar impressions. From the form of the im-
pression, we may very often infer by what mineral they
have been occasioned. What has been called the ramose
shape of the octahedral Iron from Siberia, is nothing else
but the result of impressions produced by crystals and
grains of prismatic Chrysolite.

The crystals sometimes are again decomposed in the
place of their formation, and thus leave the impressions of
their form on other minerals, which have escaped this de-
struction. If in these empty spaces a new compound mine-
ral is formed, it must necessarily assume the shape of the
space already existing, since the sides of this become the
support of the individuals newly formed. Thus pseudomor-
phoses are formed, which appear in the shape of implanted
crystals, if the mass containing the impressions, by what-
ever circumstances, happens to disappear.

All the peculiarities of the pseudomorphoses can easily
be explained from the mode of their formation now de-
scribed.

The form of the pseudomorphoses has no relation at all
to the nature of the mineral in which it occurs. For it
is entirely accidental, from what mineral the impression is
derived in which the new individuals have been deposited.
Hence it is not derived from the composition of the indi-
viduals contained in the pseudomorphosis (§. 17&). Thus
in rhombohedral Quartz we meet with forms originating

186.

OF COMPOUND MINERALS.

from rhomboheclral Lime-haloide, from octahedral Fluor-
haloide, from prismatoidal Gypsum-haloide, &c. ; which is
sufficient to prove, that the forms of the pseudomorphoses
cannot by any means be members in the series of crystal-
lisation of those species (§. 136.), to which they belong.

The quality of the surface of the pseudomorphosis depends
only upon its form, and not upon its substance or its mode
of composition. For the elevations and depressions of the
mould are likewise expressed in the cast, which in this case
is the pseudomorphosis. The quality of this surface tends
very often to indicate the mineral, from which the form is
derived, particularly if these forms belong to the tessular
system, which occurs in several species. Thus rhombo-
hedral Quartz presents not unfrequently pseudomorphic
hexahedrons, which as such may originate from various
minerals. But on a closer inspection, we observe upon the
faces of some of them, the obtuse apices of isosceles tetra-
gonal pyramids, which, as it is mentioned in §. 176., belong
to the hexahedrai tetragonal-icositetrahedron, a form not
uncommon in octahedral Fluor-haloide. "We may hence
infer that the form of the pseudomorphosis of rhombohe-
dral Quartz now under consideration, is owing to the spe-
cies of octahedral Fluor-haloide.

The surface of the pseudomorphosis is never drusy in
the sense of §. 176., but only in the way described above.
But sometimes it bears a new coating of very minute crys-
tals, of the species of which the pseudomorphosis consists.
This is not at all uncommon in many of those pseudomor-
phoses of rhombohedral Quartz, which affect the form of
rhombohedral Lime-haloide. In general the surface of the
pseudomorphoses is less smooth and shining than that of
real crystals of the species. This, however, is merely ac-
cidental, and does not deserve to be classed among the pe-
culiar and constant characters of pseudomorphoses.

The pseudomorphoses are very often hollow in their in-
terior ; the cavities are lined with crystals, or with reni-
form and other imitative shapes of that species, which con-
stitutes the pseudomorphoses. There are crystals which

262 TERMINOLOGY. §. 186.

contain cavities, either empty or filled with water and other
fluids. These are always in close relation to the external
form of the crystals themselves, which is not the case in
the pseudomorphoses. Another class of opening", in the in-
terior, occasioned by other minerals being included, must
be referred to the impressions.

The pseudomorphoses are compound minerals, even
though, on account of the minuteness of the individuals, the
composition should no longer be perceptible. But they
are also very often mixed, since several species may be de-
posited in an impression at the same time, in the same
way in which several species may enter into the composi-
tion of a geode.

The pseudomorphoses cohere immediately with the adja-
cent mass, and therefore seem only to be implanted.

This also is the case in certain real crystals ; but here the
crystals form only those parts of the individuals constituting
the support, which have reached the free space, and which
for that reason have assumed a regular form.

Mere coatings of crystals must not be enumerated along
with the pseudomorphoses, since the latter are produced by
the process of subsequent formation in a mould, as it has
been explained before. Nor can it be allowed to consider
decomposed or otherwise destroyed varieties of one species,
as pseudomorphoses of another (§. 21.). Thus the decom-
posed varieties of hexahedral Iron-pyrites can never be-
come pseudomorphoses of prismatic Iron-ore, nor those of
paratomous Augite-spar pseudomorphoses of Green-earth,
the latter being a variety of prismatic Talc-mica.

The origin of another remarkable appearance, is so very^
nearly related to that of the pseudomorphoses, that there
is no place more adapted than this for its explanation.

Sometimes it happens that also the regular structure of a
simple mineral is impressed into the mass of another, which
enters into fissures parallel or dependent upon this struc-
ture. If now the simple mineral, by whatever accident, is
decomposed, the remaining compound one will represent
a shape which entirely depends on the structure of the

§. 187- OF COMPOUND MINERALS. 263

decomposed individuals. The same takes place if the indi-
viduals of compound minerals do not cohere from all sides,
so that they allow of the interposition of foreign matter.
Thus the cellular shapes arise, of which again the former
have been called regular, and the latter irregular cellular
shapes. The sides of the alveolae again are sometimes
lined with minute .crystals of a third mineral, and this
among others is the case in what has been called the cellu-
lar P}'rites. In that mineral the sides of the alveolae are
perpendicular to each other, because they express the struc-
ture of hexahedral Lead-glance ; they consist of rhombohe-
dral Quartz, and are lined with crystals sometimes of
hexahedral, sometimes of prismatic Iron-pyrites. On this
account it is necessary to refer some varieties of cellular
pyrites to the one, some to the other species of the genus
Iron-pyrites.

The crystals of Steatite are considered as real crystals by
some mineralogists, by others as pseudomorphoses : nothing
decisive, however, has been brought forward in respect to
this point ; and they require therefore a very accurate exa-
mination, to prevent us from forming an erroneous opinion
of their nature.

§. 187- IRRKGULAll ACCIDENTAL IMITATIVE
SHAPES.

According to the quality of the space, in which
these imitative shapes have been formed, they may
be distinguished into : 1, such whose form is entire-
ly accidental ; 2, such whose form depends upon
particular openings in other minerals, which are not
simple ones ; and 3, such whose form depends upon
bodies, not belonging to the mineral kingdom.

In the mass of rocks, and in that of beds and veins, we
very often meet with cracks or fissures, which seem to
have once been open, or which still continue so. Com-

TERMINOLOGY. §. 187.

monly this appearance is explained by supposing them to
be real fissures, or that the coherence of the particles in
the rocky mass has, in their place, by whatever means,
been resolved. If a mineral is formed in a fissure of that
kind, it must necessarily assume its form ; and the mineral
appearing in this shape, is said to occur in Plates. Mixed
minerals likewise may affect this shape ; and the veins
themselves might be quoted as examples, if their consi-
deration did not belong to another science. These fissures
sometimes are so very narrow, that a fluid can scarcely
enter between their sides ; a mineral formed in such a
space is said to occur superficial, which in fact is nothing
else, but a very thin plate. Examples occur in hexahe-
dral Silver, octahedral Copper, &c. both of plates, and of
superficial varieties.

There are instances where the sides of these fissures
are nearly even, and possess a certain degree of polish.
Fissures of that* description very seldom seem to have
been filled up with other minerals : on the contrary, the
sides are commonly in immediate contact with each other.
Minerals are said to occur specular, if specimens of them
shew part of such polished sides of fissures. The specular
faces sometimes shew a particular sort of striae, which
would deserve very well to be noticed by geologists. As
examples, we may quote specular hexahedral Iron-py-
rites, hexahedral Lead-glance, brachytypous Parachrose-
baryte, &c.

Several rocks contain vesicular cavities. In these ca-
vities minerals are formed, which consequently must as-
sume their shape, and appear as more or less spheroidal
masses. Sucii globules very often consist of the varieties of
more than one species, and are sometimes hollow inside.
They must' be accurately distinguished from the grains
(§. 160.), and from the globules described above (§. 183.).
Among the present we must class also the Agate-balls, and
the balls of other varieties of rhombohedral Quartz, as of
flint, of Egyptian jasper, &c.

If this kind of globular concretions is not hollow inside,

g. 188. OF COMPOUND MINERALS. 265

and at the same time very irregular, so as to exhibit some
resemblance with the roots of certain plants, the forms
arising are called tuberose ; of which flint is one of the
most common examples.

To this class also we must refer the irregular cellular
shapes (§. 188.). The present subject is of little value in
scientific Mineralogy, and does not require, therefore, very
nice divisions.

Those shapes which depend upon forms foreign to the
mineral kingdom, are the ^!nf actions. There is no differ-
ence between the formation of the greater part of petri-
factions, and of the pseudomorphoses or the accidental
imitative forms, and it does not require any particular ex-
planation. Mineralized organic substances cannot be class-
ed among real petrifactions. These are not formed like
pseudomorphoses, in which the space left empty by the
decomposition of one body is filled up by another, but the
organic mass is metamorphosed or changed into that of
the mineral. Mineralized organic bodies, besides their
original shape, also may retain their original structure, as „
numerous varieties, particularly of bituminous Mineral-coal.
Several minerals, even after their formation, assume
other forms, which, however, are quite accidental. Thus
Pebbles are formed when fragments of minerals are carried
along by water, till, by attrition, they acquire a more or
less roundish or globular shape. Simple, compound, and
mixed minerals, are found in the shape of pebbles.

§. 188. PARTICLES O¥ COMPOSITION.

The individuals of which a compound mineral
consists are called its Particles of Composition.

The particles of composition are true crystals, which, by
their contact, have prevented each other from assuming
their regular form (§. 160.).

The particles of composition have also been called .Dw-
tinct Concretions. The other expression, however, is by far

266 TEIIMINOLOGT. §. 188.

preferable, in as much as by itself it intimates, that they
can only appear in compound minerals. It conveys like-
wise more readily the idea of individuals, since distinct
concretions might also refer to simple minerals. The idea
of the individuality of minerals is one of those which have
for a long time remained unsettled, and yet the possibility
of a scientific mineralogical method greatly depends upon
this idea.

The particles of composition are distinguished, according
to their length, breadth, and thickness, into granular, colum-
nar, and lamellar particles of composition. The granular
particles have all their dimensions nearly equal, or at least
not very different. We may omit here all those distinc-
tions which mineralogists have introduced, in respect to
the particular shape of these granular particles, because
this being not a regular one, it signifies but little in the mi-
neral kingdom. Granular limestone (rhombohedral Lime-
haloide), Coccolite (paratomous Augite-spar), dodecahedral
Garnet, &c. contain many examples of granular particles
of composition.

In the columnar particles the length is greater than both
breadth and thickness. Sometimes they are rather thicker
on one end ; sometimes also thev are broad. This, how-
ever, for the above mentioned reasons, does not occasion
any farther difference. As to their direction, they are
either parallel or diverging. Examples of columnar par-
ticles of composition we find in rhombohedral Lime-haloide,
in a variety of prismatic Topaz called Picnite, in rhombo-
hedral and prismatic Iron-ore, &c.

In the lamellar particles, the length and breadth surpass
the thickness. These likewise are sometimes thicker on
one end, and thus approach to the columnar particles of
composition ; in general, these three kinds of particles of
composition are not contained within precise limits, but
they pass insensibly into each other. There are straight
and curved lamellar particles of composition. The latter
are not individuals, but of themselves they are already
composed, which distinguishes them from the former, even

§. 188. OF COMPOUND MINERALS. 2G7

in case they should themselves be straight, while the
others are curved or bent. Straight lamellar particles, or
such as in themselves are simple, are found in prismatic Kal-
baryte, axotomoiis Kou phone-spar, rhombohedrai Lime-ha-
loide (the varieties called Slate-spar), &c.

The size of the particles of composition varies consider-
ably. Sometimes they are so minute that they entirely
withdraw themselves from observation. Yet a compound
mineral, consisting of such impalpable individuals, does not
for that reason cease to be a compound one. This subject
requires our particular attention. A specimen of hexahe-
dral Lead-glance being given, which consists of consider-
ably large particles of composition, we may very easily
find another, in which these particles are smaller, and a
third, in which the size of the particles is still more dimi-
nished. These specimens differ only in the size of their
constituent individuals. We may continue, and discover a
fourth variety, a fifth, &c., every one of them being in a
similar relation to that which immediately precedes it ;
very soon we arrive at such varieties, whose individuals are
scarcely perceptible to the naked eye. Yet the immediate
connexion with the other varieties, and a magnifying glass,
demonstrate that they are all the same mineral, viz. va-
rieties of hexahedral Lead-glance. No reason can be as-
signed why these varieties should be the limit of the
series ; the members immediately following will be the va-
rieties of compact Lead-glance ; and in these very often the
individuals are so miitute, that they withdraw themselves
from observation even through a magnifying glass. Com-
pact Lead-glance, therefore, is not a simple mineral, but a
compound one. From the same point of view, we must
consider the compact varieties of rhombohedrai Lime-ha-
loide (compact Limestone), of octahedral Fluor-haloide
(compact Fluor), of prismatic Hal-baryte (compact Heavy-
spar), of rhombohedrai Quartz (Flint, Hornstone, Chryso-
prase, &c.), and of other species. The composition which
really takes place, cannot be observed only on account of
the minuteness of the particles of composition.

268 TERMINOLOGY, §. 189.

The columnar and lamellar particles are exactly in the
same case. The columnar particles may still very easily be
traced in the stalactitic and reniform shapes, called brown
Hematite. But in compact brown Iron-stone, they have
entirely disappeared. Of this vanishing arid impalpable
composition, we have a very remarkable example in those
varieties of rhombohedral Quartz, which have been called
Calcedony, and occur in reniform and stalactitic shapes.
Commonly there is not a trace of their composition left
in the interior ; but in some of its varieties this com-
position is still observable. Among these, the fibrous
Carnelian is one of the most well known instances.

It has been mentioned above (§§. 171. 172.)? that the co-
lumnar composition has sometimes been confounded with cer-
tain relations of structure. Fibrous fracture is always colum-
nar composition ; and the difference between what has been
called the foliated and the radiated fracture, consists in no-
thing else but that the first refers to simple minerals or gra-
nular compositions, while the second is confined to columnar
compositions.

§. 189. SINGLE AND MULTIPLE COMPOSITION.

The single composition takes place, if a com-
pound mineral consists of individuals ; but if the
particles of composition are again composed, then
the composition is multiple.

The compositions treated of in the preceding paragraphs,
are single compositions.

But there exist granular particles of composition, which
are again composed of granular particles ; and these only
are real individuals. They join into those masses, which
again on a larger scale produce a. granular composition.
Macrotypous Lime-haloide exhibits examples of this com-
position. Sometimes the granular particles consist of co-
lumnar particles, diverging from the centre, or from one of

§. 190. OF COMPOUND MINERALS. 269

the corners of the former. Among the first are several va-
rieties of rhombohedral Lime-haloide, called Oolite, of he-
mi-prismatic Augite-spar, called Actinolite, &c. ; among the
second the varieties of several species of Kouphone-spar,
of Wood-tin, a variety of pyramidal Tin-ore, &c. In
other cases, the granular particles of composition again con-
sist of lamellar particles, as in prismatic Hal-baryte and
axotomous Kouphone-spar.

Columnar particles of composition sometimes consist again
of columnar ones, as in several varieties of prismatic Iron-
ore. The preceding observations will suffice for explain-
ing many occurrences of this kind, among which the curved
lamellar particles of composition are the most remarkable.
The curvature of their surface corresponds to that of reni-
form or globular shapes, so that the quantity of its deviation
from a mathematical plane depends upon the radius of cur-
vature. Commonly they consist of columnar, sometimes of
lamellar and granular particles. They occur in octahedral
Fluor-haloide ; in prismatic Iron-ore ; in prismatic Hal-
baryte, called curved lamellar Heavy-spar ; in rhombohe-
dral Quartz, called Calcedony ; in rhombohedral Lime-ha-
loide, in rhombohedral Antimony, &c.

The composition is very often still more complicated ;
we may dispense, however, with entering into a greater
detail, since the given examples are perfectly sufficient
for explaining every other case.

§. 190. CHARACTERISTIC MARKS OF COMPOSITION.

Imitative shapes, and the want of cleavage, are
the chief characters, from the presence of v ich
composition may be inferred, if this should not be
observable at first sight.

An individual formed under such circumstances as to
be beyond the reach of any foreign influence, will always
assume a regular form.

If, therefore, we meet with minerals which evidently

270 TERMINOLOGY. §. 1QO.

have not been acted upon by any such circumstance, and
which nevertheless do not present any regular form, we
may infer with perfect security that the mineral is not a
simple one, but that it is a compound of several indivi-
duals. This proposition is demonstrated on one side by
all crystals which are simple, on the other by all the imi-
tative shapes dependent upon the nature of the mineral
itself (§. 182. 183.), which are compound.

With regard to the accidental imitative shapes, it is evi-
dent, that not even those which are regular, can be the forms
of simple minerals, because they are altogether accidental,
whereas the forms of simple minerals are founded in the na-
ture of the individuals themselves. Hence, the imitative
shapes, of whatever kind they may be, are, in every instance,
infallible characters from which the composition of the mine-
rals may be inferred. But we could suppose that a com-
pound mineral might consist of particles in a perfectly
parallel position, but so small, that on account of their
minuteness, the composition can no longer be observed, so
that the directions of cleavage of the single particles or
supposed individuals in one of them are the continuation
of those in the other. In this case, the whole mass will be
cleavable, and the whole will therefore be a single indi-
vidual, and not a composition, agreeably to the definitions
m §. 176. Hence cleavable minerals are simple ; and the
want of cleavage in varieties of such species as commonly
allow of cleavage, is a mark of their composition ; because
here one individual assumes a situation different from that
of another, so that their respective faces of cleavage can
have no continuity among one another. From the same
reason, compact Limestone, compact Fluor, compact Heavy-
spar, compact Lead-glance, are not cleavable, although the
simple varieties of the same species may be cleaved with
the greatest facility.

The same applies to the pseudomorphoses.

Among the other characters of composition, we may
mention, that compound minerals, in which the composi-
tion can no longer be observed, are most intimately con-

§. 190. OF COMPOUND MINERALS. £71

nected in all their properties with those in which it is still
visible, and that commonly they possess lower degrees of*
transparency and lustre, than simple varieties of the same
species. Examples occur in hexahedral Lead-glance,
rhombohedral Lime-haloide, rhombohedral Quartz, &c.

The following observations will furnish characters in
most cases perfectly sufficient for distinguishing mixed mi-
nerals and compound minerals, in both of which the par.
tides disappear on account of their minuteness.

The different ingredients of the mixture are sometimes
found separated from the rest in more or less pure masses,
by which the mixture ceases to be uniform. If we find
an opportunity for observing mixed masses of this kind on
a larger scale, we may very often find those particles en-
tirely disengaged, or separated from each other, as is the
case with rhombohedral Iron-ore, and rhombohedral Quartz,
in the original repositories of Iron-flint, which is an inti-
mate mixture of these two species. Thus we infer Basalt
to consist of several species of the genus Feld-spar, and
hemi-prismatic or paratomous Augite-spar, because Green-
stone and the Syenitic rocks in which the particles of mix-
ture have only more extension, really do consist of indi-
viduals of the above mentioned species, and differ from
Basalt merely by their coarser grain.

Moreover the mixed minerals partly possess the properties
of the one, partly also those of the other of the simple mi-
nerals of which they consist, without entirely agreeing
with any of them, as, for instance, Iron-flint, which pos-
sesses some of the properties of rhombohedral Quartz, &c.
or they assume such properties as never occur in simple
minerals, as, for instance, the columnar shapes of Basalt, of
Porphyry, the globular concretions of Green-stone, of Sye-
n'te, &c., which by themselves prove those minerals to be
compound, even though the component individuals should
no longer be perceptible.

272 TERMINOLOGY. g. 191.

§. 191. STRUCTURE OF COMPOUND MINERALS.

Thtit kind of structure which has been consider-
ed above in the simple minerals, does not occur in
the compound ones. If we break them, however,
we produce what has been called their Fracture ;
and the particles of the mineral separate in the
Faces of Fracture.

If the particles are still distinguishable as individuals,
they must be considered according to their respective regu-
lar or irregular structure, to their faces of composition, and
to every other character which they present to the observer ;
in short, they must be considered as simple minerals. In
the present place, therefore, only those compound minerals
shall be treated of, in which, on account of their minute-
ness, the individuals are no longer distinguishable. In
these the following kinds of fracture have been distin-
guished.

1, The Conchoidal Fracture, together with its various
modifications, which depend upon size, perfection, rela-
tive depression (§. 172.)-

2, The Uneven Fracture, which has been subdivided ac-
cording to the size of the asperities, into coarse-grained,
small-grained, and fine-grained uneven fracture.

3, The Even Fracture, which arises if the elevations and
depressions upon the face of separation nearly approach to
evenness. These even parts of the fracture must not be
confounded with faces of cleavage, because they do not
keep a constant direction, and are only observable in com-
pound minerals. This variety of fracture is so very rare,
that it is difficult to quote good examples of it. Sometimes
it occurs in compact Lead-glance.

4, The Splintery Fracture, which is produced if upon the
face of separation, detached scaly particles remain, joined to
the mass by their thicker end. These particles are render-
ed visible by that portion of light which passes through

§. 192. OF COMPOUND MINERALS. 273

them, and the splintery fracture therefore does not occur
in perfectly opaque minerals. It may occur at the same
time with the conchoidal, or another kind of fracture.
This has been expressed by the phrases : conchoidal in the
grca^ splintery in the small, &c.

5. The Hackly Fracture, which has been sufficiently ex-
plained above (§. 172.).

G. The Slaty Fracture, which resembles imperfect faces
of cleavage (§. 163.), and partly arises from it. It is met
with in the different kinds of Slate, which, for the greater
part, are compound minerals, or even mixed, although they
appear to be simple. The slaty fracture keeps a constant
direction, and is in this respect analogous to cleavage.

7. The Earthy Fracture, which is the same as the uneven
fracture, except that it occurs in decomposed minerals.

The particles into which compound minerals may be
broken, are termed Fragment!!, and their shape is irregular.
According to the quality of their edges, they have been di-
vided into sharp-edged and Hunt-edged fragments. Slaty frac-
ture produces tabular fragments ; thin columnar composi-
tion produces splintery fragments.

§. 19&. COMPOSITION IS OF LITTLE VALUE IN NA-
TURAL HISTORY.

It is impossible to derive characteristic terms for
the determination of the natural-historical species,
from the occurrences of composition. Hence they
are of use in the Natural History of the Mineral
Kingdom, only in so far as their knowledge is ne-
cessary for ascertaining the existence of the simple
mineral in the compound.

Among the various impediments that have retarded the
progress of the science, and, in particular, the correct de-
termination of the natural-historical species, one of the most
conspicuous was the occurrence of individuals in compound

VOL. i. s

274 TERMINOLOGY. §.

varieties, as long as this composition had not been explain-
ed, nor the individuals themselves traced in it, whose di-
minutive size very often considerably adds to the difficulty.
If we consider the varieties of rhombohedral Iron-ore, or
of many other species, and if we do riot possess an exact
knowledge of the individual in the mineral kingdom, that
is to say, if we do not sufficiently distinguish between
simple and compound minerals ; we find some of them so
very different, that it would seem to be in opposition to all
the principles of Natural History, if nevertheless we would
unite them into one species, since they differ in almost
every one of their properties. This has indeed been the
reason why Specular Iron-ore, lied Iron-ore, and many
others, have really been considered and distinguished as
particular species. At that time the transitions (§. 221.),
which might have led to the knowledge of errors commit-
ted in these determinations, had also not yet been dig-
tinctly developed ; and it must be avowed that those mi-
neralogists who have escaped similar errors, awe this
to inquiries and considerations of minerals, very different
from such as are carried on according to the principles of
Natural History.

A few examples, once given of the dismemberment of a
natural-historical species, could not but produce many si-
milar errors in subsequent instances. The consequences
of this mode of proceeding, have been numberless species,
or aggregates erroneously so called, being connected Avith
each other by transitions. Incapable of being character-
ised, or distinguished from one another, they have only
served to incumber the nomenclature, and to degrade the
systems into mere registers of words. If one single erro-
neous idea is capable of producing such confusion, we must
bestow all possible attention on the establishment of cor-
rect ideas, in a science in which they are or might be so
pure and simple, as in the Natural History of the Mineral
Kingdom.

Zoology and Botany have not been subject to similar er-
rors. They are almost impossible in the first, and in the

§. 192. OF COMPOUND MINERALS.

second they would be obvious at first sight. It would be
an error of this kind, if a person should consider a corn-
field or a bank of trees for an individual, and establish ac-
cordingly these corn-fields and banks of trees into particu-
lar species. A corn-field or a bank of trees is exactly the
same as Red Hematite (the fibrous Red Iron-ore, a com-
pound variety of rhombohedral Iron-ore), if compared with
real individuals of their respective species.

From the preceding observations we may infer, that
from the composition of minerals, the Natural History of
the Mineral Kingdom cannot derive any characteristic pro«
perties for the determination of the species. This would
be an error sufficiently powerful to shake it to the very
foundations, and to degrade it from the rank it assumes in
the sciences as a part of Natural History.

SECTION III.

THE NATURAL-HISTORICAL PROPERTIES, COMMON
TO BOTH, SIMPLE AND COMPOUND MINERALS.

§. 193. DIVISION.

Those natural-historical properties, which are
common to both, the simple and the compound mi-
nerals, may be divided into the Optical Properties,
and into the Physical Properties of minerals, or such
as refer more particularly to their Mass or Substance.

CHAPTER I.

OF THE OPTICAL PROPERTIES OF MINERALS.
§. 194. EXPLANATION.

Optical properties are such as depend upon light,
and are not observable except in its presence.

The consideration of the natural-historical properties in
general, presupposes the presence of light. Yet all of them
do not depend upon its presence. This is the case, however,
with the properties to be considered in this chapter. "We
cannot maintain that a mineral possesses in the dark that
same colour, lustre, or transparency, which it exhibits
when observed under the influence of light. This charac-
ter, therefore, is sufficient for distinguishing the optical
properties from all other properties of the minerals.

§. 195. OPTICAL PROPERTIES OF MINERALS. 277
§. 195. LUSTKE, COLOUR, TRANSPARENCY.

The phenomena observable in minerals, with
respect to reflected and transmitted light, are com-
prehended under the heads of Lustre, Colour, and
Transparency.

These subjects are treated of in Natural History, only
in so far as they allow some application to its own peculiar
purposes, of discriminating and describing minerals. Their
explanation belongs to Natural Philosophy.

In order to employ lustre, colour, and transparency,
agreeably to the purposes of the Natural History of the
Mineral Kingdom, it is necessary to determine and to pro-
vide with certain denominations, those differences which
may be distinguished in these properties, in respect to both,
their kind and their degrees of intensity. This will require
us to fix a certain impression upon our mind, and always to
designate this impression with the same name, or to recal it
to our memory, whenever we read this name, or hear it ut-
tered. It is necessary therefore to have experienced these
impressions upon our own mind, and explanations or de-
scriptions cannot be successfully substituted in their place.

An acquaintance with the colours occurring in the Mi-
neral Kingdom, with the different kinds of lustre, &c., may
be acquired from the consideration of bodies, which are not
minerals. It may indeed be effected in more than one
way ; but it seems to answer the purpose best, if we em-
ploy these bodies themselves, this being not only a more
sure, but also an easier way of fulfilling our intention.
To those who do not intend to go any farther than to become
capable of making use of the Characteristic for discriminat-
ing varieties occurring in nature, a very small number of
specimens will suffice, the choice of which is not attended
with any difficulties.

The relative value we have to attach to the employment
of these properties, depends upon their generality and con-
stancy, and consequently upon the possibility of their

278 TERMINOLOGY. §. 196.

application. The properties most important, in Deter-
minative Mineralogy (§. 11.) are the different kinds of
lustre, the metallic colours, and particularly the phe-
nomena of simple and double refraction, as observed in
common and polarised light. As yet we cannot dispense
in the characters with the first and the second, although
they are of comparatively less value. The third are
of great consequence : in many cases they are sufficient
for determining the system of crystallisation, in such
varieties as do not present any regular form, or any
traces of cleavage ; they very often yield neat and de-
cisive characters, where the minerals are much resem*
bling each other in the rest of their properties, and
they are most useful in ascertaining whether a mineral is
simple or compound. For the Descriptive Mineralogy
(§. 11.) all the optical properties of the minerals are of
equal consequence, and at the same time not inferior to
any other properties. These descriptions have not the pur-
pose of distinguishing objects, but that of producing an
image of them ; and to this the colours confer no less than
the forms, the knowledge of lustre and transparency no
less than the degrees of hardness or of specific gravity.

The optical properties must therefore by no means be ne-
glected, although many of them are of less influence than
those deriving from the forms to the scientific progress of
Mineralogy. Very often by their assistance we may dis-
pense with the use of the Characteristic, because they are
very well calculated for recalling to our mind such va-
rieties of the same species as we have already determin-
ed. They are obvious at first sight, and therefore easily
observed and determined under whatever circumstances
they may be found, they are in particular recommended to
those who wish to acquire an extensive general knowledge
of the productions of the mineral kingdom.

§. 196. KIND. AND INTENSITY OF LUSTBE.

The lustre of minerals is considered in respect
to its Kmdy and in respect to its Intensity.

§. 196. OPTICAL PKOPEKTIES OF MINERALS. 279

The kinds of lustre are :

1, Metallic lustre,

2, Adamantine lustre,

3, Resinous lustre,

4, Vitreous lustre,

5, Pearly lustre.

Metallic lustre is subdivided into perfect, and wiper-
feet metallic lustre. The first occurs in all the species
of the orders Metal, Pyrites, and Glance, and in some of
those of the order Ore. The same lustre occurs in wrought
metals and metallic alloys, as silver, brass, copper, &c.
The second is found in several ores, as Tantalum-ore,
Uranium-ore, &c.

Adamantine lustre is subdivided into metallic adamantine
lustre, and common adamantine lustre. Examples of the
first may be seen in the dark-coloured varieties of several
species of the order Blende, and in some, particularly the
grey varieties of di-prismatic Lead-baryte ; the common
adamantine lustre is peculiar to octahedral Diamond, to the
pale-coloured varieties of Ruby -blende and Garnet-blende,
and to some varieties of di-prismatic Lead-baryte.

Resinous lustre is that which a body presents, if besmeared
with oil or fat. It occurs in dodecahedral Garnet, still more
distinctly in pyramidal Garnet ; also in the varieties of
empyrodox Quartz, called Pitchstone.

Vitreous lustre is that of common glass, and may be ob-
served in rhombohedral Quartz, rhombohedral Emerald,
prismatic Chrysolite, and in several other species of the
order Gem.

Pearly lustre is divided into common, and metallic pearly
lustre. The first occurs in prismatoidal and hemi-pris-
matic Kouphone-spar, in prismatic Disthene-spar, in rhom-
bohedral Talc-mica, and in other species of the order
Mica ; the second is found in several species of Schiller-
spar, and in several varieties of rhombohedral Talc-mica.

As to the intensity of lustre, we have to distinguish the
following degrees :

280 TERMINOLOGY. §. 196.

1, Splendent,

2, SJtlnlng,

3, Glistening,

4, Glimmering,

5, Dull.

Splendent faces, or those which possess the highest degree
of lustre in the whole mineral kingdom, produce distinct
and well defined images of the objects, provided they pos-
sess the required extension and evenness. Such faces are
contained in many varieties of dodecahedral Garnet-blende,
of rhombohedral Iron-ore, of rhombohedral Quartz, and
other species.

Shining is a less degree of lustre ; it is still lively, but
does not produce a distinct image. This degree is very
common in several species of the orders Spar, Haloide,
Baryte, &c.

Glistening reflects light still more disorderly ; but al-
though it does not any longer produce an image, yet it
reflects it in pretty well defined patches. This degree of
lustre is found in most of those compound minerals, in
which the particles of composition are still observable, or
at least in which they have not yet entirely disappeared.
Examples are pyramidal Copper-pyrites, tetrahedral Cop-
per-glance, &c.

Glimmering does not reflect defined patches" of light, but
a mass of undefined light seems spread over the glimmer-
ing surface. This degree of lustre is peculiar to the very
thin columnar composition (commonly called fibrous frac-
ture), and to several other compound minerals, in which
the composition disappears, as in the varieties of rhombo-
hedral Quartz, called flint, calcedony, hornstone, in com-
pact hexahedral Lead-glance, and other minerals. Com-
monly this degree is a sign of a compound mineral, the
individuals of which are so very small, as nearly to disap-
pear. It is produced by the reflection of light from every
one of the impalpable component particles.

Dull possesses no lustre at all. This perfect absence
of lustre is almost entirely confined to decomposed mine-

§. 197- OPTICAL PROPERTIES OF MINERALS. 281

rals, as Porcelain-earth, which is a decomposed Feld-spar,
and to some compound minerals, as Chalk, where it de-
pends upon some particular circumstances in the formation.

§. 197- SERIES IN THE DIFFERENCES OF LUSTRE.

The gradations in the kinds and in the degrees
of lustre in the varieties of one and the same natu-
ral-historical species, produce continuous series.

In general, neither the kinds nor the degrees of lustre
admit of rigorous limits. It is necessary to determine
them in some particularly distinct examples, and to com-
pare with them such as are less distinct.

If there occur several kinds or degrees of lustre in the
varieties of a species, these will be in an uninterrupted
connexion, and they will pass insensibly into one another,,
so that in no place we are capable of observing any inter-
ruption or want of continuity.

From the succession in these gradations, the above men-
tioned series arise, llhombohedral Uuby-blende presents a
striking example of a similar series. In some of its varieties
the lustre is nearly metallic, in others decidedly adaman-
tine. Between these there are a great number of grada-
tions of metallic adamantine lustre, by which the two
kinds of lustre are so .closely connected, that it is impos-
sible to say where the one begins, and where the other ter-
minates.

The series in the gradations of lustre allow of the same
application as the series in the varieties of colour mention-
ed in §, 202. The results of crystallography in respect to
simple forms, are exactly confirmed by the occurrences of
lustre in single individuals, as to the existence of forms,
which, although they appear with a number of faces, not
sufficient to limit the space from all sides, yet must be
considered as simple forms (§. 1G8.). For if we abstract
from what is merely accidental, homologous faces entirely
.agree as to the kind and to the degrees of intensity of

282 TERMINOLOGY. §. 198. 199.

their lustre ; and, vice versa, such faces which do not agree
with each other in the same respects, are not homologous,
or they do not belong to one and the same simple form.
This is equally applicable to faces of crystallisation, and
to faces of cleavage, as is shewn by numerous examples in
prismatoidal Gypsum-haloide, in several species of the
order Mica, in several species of the genus Kouphone-spar,
&c. Pearly lustre is the most remarkable among the dif-
ferent kinds, since, in a high state of perfection, it appears
in simple minerals only upon single faces of crystallisation,
as well as of cleavage. Such faces, therefore, are parallel
either to the axis, or to the base of the fundamental form
of the species. A single face which possesses a distinc t
pearly lustre, if it is a face of cleavage, is also termed
Eminent. The pearly lustre of compound minerals very
often is a consequence of this composition.

§. 198. COLOUR, PROPERLY SO CALLED, AND STREAK.

We have to distinguish between the colour of
the entire mineral and that of its powder. The
first is the Colour of the mineral, properly so call-
ed, while the second has been designated by the
name of the Streak.

§. 199. DIVISION OF COLOURS.

The colours have been divided into metallic and
into non-metallic colours.

This division is not rigorously correct, because the dif-
ference does not so much lie in the colours themselves, as
in the kinds of lustre joined to them. It is, however, very
useful, since it separates what is as yet indispensable, from
what is merely useful in the process of discriminating mi-
nerals.

For the sake of a better distinction of the colours, eight
-principal colours have been assumed by the celebrated

§.200. OPTICAL PROPERTIES OF MINERALS. 283

WERNER, to whose labours this part of Terminology is
particularly indebted. These colours are, White, Grey,
Black, Blue, Green, Yellow, Red, and Brown. Each of them
comprehends several varieties, of which again the one con-
sidered the purest, is called the cliaractcrlstlc colour. The
names or denomination of the varieties are either derived
from such bodies in which they are found often or by pre-
ference, or they are formed by composition. Examples of
the first are, rose-red, apple-green, gold-yellow ; of the latter,
reddish-brown, yellowish-brown, greyhJi-wlntc, &c.

The Wernerian method in the determination of colours
is as generally introduced and received as it deserves. It
is not advisable to change or alter any thing without the
most urgent necessity, even though, from other reasons,
these alterations should be improvements ; since there is
nothing required, but to recal a certain impression upon
our mind : and the best plan therefore will be to keep to
such expressions to which we have been accustomed.

§. 200. METALLIC COLOURS.

The metallic colours are: 1. Copper-red; 2.
Bronze-yellow; 3. Brass-yellow, and 4. Gold-yel-
low ; 5. Silver-white, and 6. Tin-white ; 7. Lead-
grey, and 8. Steel-grey, and 9. Iron-Hack.

1. Copper-red, the colour of metallic copper. Examples,
Octahedral Copper ; less distinct prismatic Nickel-pyrites.

2. Bronze-yellow, the colour of several metallic, alloys
called Bronze and Speise, in particular the alloy of copper
and tin. Very distinct in hexahedral and prismatic Iron-
pyrites.

3. Brass-yellow, the colour of brass. Ex. Pyramidal Cop-
per-pyrites. This colour is never found in hexahedral Gold.

4. Gold-yellow, the colour of pure gold. Distinct, but
exclusively in hexahedral Gold. The gold-yellow colour
sometimes becomes pale, and then approaches to silver-
white.

28 * TERMINOLOGY. §. 201.

5. Silver-white, the colour of pure silver. Distinct in
hexahedral Silver ; less distinct in prismatic Arsenical-py-
rites ; inclining to red in hexahedral Cobalt-pyrites.

C. Tin-white, the colour of pure tin, particularly not
mixed with lead. Ex. Fluid Mercury ; rhombohedral
Antimony, and also native Arsenic, in this, however, inclin-
ing a little to lead-grey.

7. Lead-grey, the colour of metallic lead. Of this colour
three different shades have been distinguished.

a, Whitish lead-grey ; b, pure lead-grey ; and c, blackish
lead-grey. Whitish lead-grey is found in the compact
varieties of hexahedral Lead-glance ; pure lead-grey in
the common varieties of the same species, which consist
of larger individuals than the former, also in rhombohe-
dral Molybdena-glance, &c. ; blackish lead-grey in hexa-
hedral Silver-glance, in prismatic Copper-glance, &c.

8. Steel-grey, nearly the colour of fine grained steel
upon a recent fracture. Ex. Native Platina and prismatic
Antimony-glance.

9. Iron Had; nearly the colour of highly carboniferous
east iron. Ex. Octahedral Iron-ore ; less distinct rhombo-
hedral Iron -ore, and tetrahedral Copper-glance.

§. 201. NON-METALLIC COLOURS.

The non-metallic colours are considered in the
consecutive order of the principal kinds (§. 199.),
which represent the general series of colours.

The following are the non-metallic colours :

A. White.

1. Snow-white. The purest white colour. Nearly the
colour of newly fallen snow. Ex. llhombohedral Lime-
haloide (Carrara marble), prismatic Lime-haloide (Flosferri).

2. Reddish-white. White (though not always of the purest
tint) inclining somewhat to red. Ex. Several varieties of
rhombohedral and macrotypous Lime-haloide, of rhombo-
hedral Quartz, &c.

§. 201. OPTICAL PROPERTIES- OF MINERALS. 285

3. Yellowish-white. White (though not always of the
purest) inclining to yellow. Ex. Several varieties of rhom-
bohedral Lime-haloide and of uncleavable Quartz.

4. Greyish-white. White, inclining to grey. Common
in rhombohedral Lime-haloide, particularly in the com-
pound varieties called granular Limestone, and in rhombo-
hedral Quartz, particularly in Common Quartz.

5. Greenish-white. White, somewhat inclining to green.
Very distinct in several varieties of hemi-prismatic Augite-
spar, particularly in Amiantus, and in the varieties of pris-
matic Talc-mica, called Common Talc.

6. Milk-white. White somewhat inclining to blue, the
colour of skimmed milk. Ex. Several varieties of uncleav-
able Quartz, called Common Opal.

B. Grey.

1. Blucish-grcy. Grey inclining to a dirty blue colour.
Seldom distinct. Sometimes in the varieties of rhombohe-
dral Quartz, called splintery Hornstone, and in several
compound varieties of rhombohedral Lime-haloide.

2. Pearl-grey. Grey, mixed with red and blue. In the
pearls this colour is very pale. Sometimes it is very dis-
tinct in hexahedral Pearl-kerate ; less distinct in several
varieties of rhombohedral Quartz, and of prismatic Hal-
baryte.

3. Smoke-grey. Grey mixed with brown ; the colour of
thick smoke. This colour occurs particularly in the dark
varieties of Flint, which belong to the species of rhombo-
hedral Quartz.

4. Greenish-grey. Grey mixed with green. Ex. Seve-
ral varieties of rhombohedral Quartz, particularly Cats-eye ;
several varieties of rhombohedral Talc-mica, &c.

5. Yellowish-grey. Grey mixed with yellow. This co-
lour is not uncommon in several compound varieties of
rhombohedral Lime-haloide (compact Limestone) and of
rhombohedral Quartz (Flint).

6. Ash-grey. The purest grey colour, a mixture of
white and black. It is seldom distinct. E.v. Prismatoidal

TERMINOLOGY. §. 201.

Augite-spar, called Zoisite, and trapezoidal Kouphone-
spar.

C. Black.

1. Greyish-black. Black mixed with grey (without any
green, brown, or blue tints). Ex. Basalt ; Lydian-stone,
Avhich is an impure variety of rhombohedral Quartz ; An-
thrakolite, an impure variety of rhombohedral Lime-haloide.

2. Velvet-black. The purest black colour. It is the co-
lour of black velvet. Ex. Empyrodox Quartz, called Ob-
sidian ; rhombohedral Tourmaline, called Schorl.

3. Greenish-black. Black mixed with green. A very
common colour in several species of the genus Augite-spar.

4. Brownish.black. Black mixed with brown. Ex. Se-
veral varieties of rhombohedral Talc-mica; bituminous
Mineral-coal.

5. Blueish-black. Black mixed with blue. It is a rare
colour, and scarce ever found except in the botryoidal and
reniform varieties of Black Cobalt from Saalfeld in Thu-
ringen.

D. Blue.

\.Blackish-blue. Blue mixed with black. Ex. The dark
coloured varieties of prismatic Azure-malachite.

2. Azure-blue. A very bright blue colour, mixed with a
little red. Ex. The pale varieties of prismatic Azure-ma-
lachite, and the bright varieties of Lapis lazuli.

3. Violet-blue. Blue mixed with red. Ex. .Rhombohe-
dral Quartz (Amethyste), and octahedral Fluor-haloide.

4. Lavender-blue. Blue with a little red, and a great deal
of grey. Ex. JLithomarge, and some varieties of Porcelain
Jasper.

5. Plum-blue. A colour inclining somewhat to brown,
and very difficult to describe. It is something like the co-
lour of certain varieties of plums. It is very rare, and oc-
curs only in a few varieties of dodecahedral Corundum, and
of octahedral Fluor-haloide.

6. Prussian-blue, or Berlin-blue. The purest blue colour.

§. 201. OPTICAL PROPERTIES OF MINERALS. 287

Ex. llhonibohedral Corundum (the bright coloured varieties
of Sapphire); prismatic Disthene-spar ; and hexahedral
Rock-salt.

7. Smalt-blue. The colour of a pale coloured sort of
smalt, called Eschcl. Ex. Several varieties of prismatoidal
Gypsum-haloide.

8. Indigo-blue. Blue mixed with black and gi'een. The
colour of several coarser sorts of indigo. Ex. Prismatic
Iron-mica, particularly the decomposed or imperfectly form-
ed varieties called Blue Iron-earth.

9. Duck-blue. Blue, with a great deal of green, and a
little black. Ex. Several varieties of dodecahedral Corun-
dum, called Ceylanite ; also several varieties of prismatic
Talc-mica, under the denomination of common Talc.

10. Sky-blue. A pale blue colour, with a little green.
It is called Mountain blue by painters ; it is the colour of
the clear sky. Ex. Prismatic Larocone-malachite, some-
times also octahedral Fluor-haloide.

E. Green.

1. Verdigris-green. A green colour, very much inclining
to blwe. It is the colour of verdigris (Acetite of Copper),
Ex. Amazon-stone, which is a variety of prismatic Feld-
spar, and prismatic Lirocone-malachite.

2. Celandine-green. A green colour, mixed with blue
and grey. Ex. Prismatic Talc-mica, called Green-earth ;
several varieties of rhombohedral Emerald.

3. Mountain-green. Green, with a great proportion of
blue. Ex. Rhombohedral Emerald ; prismatic Topaz, the
oriental Aqua-marine.

4. Leek-green. Green, with a little brown ; the colour
of the leaves of garlick. Very distinct in rhombohedral
Quartz, called Prasem.

5. Emerald-green. The purest green colour. Very dis-
tinct in rhombohedral Emerald ; less characteristic in some
varieties of hemi-prismatic Habroneme-malachite.

6. Apple-green. A light green colour, with a trace of yellow.
Very distinct in rhombohedral Quartz, called Chrysoprase.

288 TERMINOLOGY. §.'201.

7. Grass-green. Green, mixed with a little more of yel-
low ; the fresh colour of grass. Very distinct in Green
Diallage, and other varieties of paratomous and hemi-pris-
matic Augite-spar ; sometimes in pyramidal Euchlore-mica
and in hemi-prismatic Habroneme-malachite.

8. Pistachio-green. Green, with yellow and a little brown.
Ex. Prismatic Chrysolite ; sometimes prismatoidal Augite-
spar.

9. Asparagus-green. Pale green, with a great proportion
of yellow. Ex. Prismatic Corundum and the varieties
of rhombohedral Fluor-haloide from Spain and Salzburg,
called Asparagus-stone.

10. Blackish-green. Green, with black. Ex. Paratomous
Augite-spar ; sometimes also Serpentine.

^11. Olive-green. Pale green, with a great deal of brown
and yellow. Ex. Prismatic Chrysolite, the varieties called
Olivine ; several varieties of dodecahedral Garnet ; hexa-
hedral Lirocone-malachite, and Pitchstone, a variety of
empyrodox Quartz.

12. Oil-green. A green colour, still lighter, with more
of yellow, and less of brown. The colour of olive oil.
Ex. Dodecahedral Garnet-blende ; rhombohedral Emerald ;
empyrodox Quartz, called Pitchstone.

13. Siskin-green. A light, green colour, very much in-
clining to yellow. Very distinct in pyramidal Euchlore-
mica ; also in some varieties of rhombohedral Lead-
baryte.

F. Yellow.

1. SiilpJtur-yellow. The colour of pure sulphur. Ex.
Prismatic Sulphur.

2. Straw-yellow. A rare colour; light yellow, with a
little grey. Nearly the colour of straw. Ex. Some varieties
of prismatic Topaz, called Pycnite.

3. Wax-yellow. Yellow, with grey and a little brown.
The colour of pure yellow wax. Ex. Pyramidal Lead-
baryte ; several varieties of uncleavable Quartz, called
common Opal.

§.201. OPTICAL PROPERTIES OF MINERALS. 289

4. Honey -yellow. Yellow, with a little red and brown ;
the dark colour of honey. Ex. llhombohedral Lime-
haloide ; octahedral Fluor-haloide ; pyramidal Melichrone-
resin.

5. Lemon-yellow. The purest yellow colour. Ex. Pris-
matoidal Sulphur, and the decomposed varieties of uncleav-
able Uranium-ore, called Urane-ochre.

C. Ochre-yellow. Yellow, with brown. Ex. The varieties
of rhombohedral Quartz and of uncleavable Quartz, if
mixed with oxide of iron, from which this colour is derived.

7- Wine-yellow. A pale yellow colour, with a little red
and grey. The colour of several sorts of white wine.
Ex. Prismatic Topaz from Saxony and from Asia Minor ;
octahedral Fluor-haloide.

8. Cream-yellow. A pale yellow colour, with a little red
and very little brown. Rare. Sometimes in Lithomarge,
and Bolus from Strigau in Silesia.

9. Orange-yellow. Yellow, very much inclining to red.
The colour of ripe oranges. Ex. Several varieties of pyra-
midal Lead-baryte from Hungary and Carinthia.

G. Red.

1. Aurora-red. Red with a great deal of yellow. Very
distinct in several varieties of hemi-prismatic Sulphur.

2. Hyacinth-red. Red with yellow and a little brown.
Ex. Pyramidal Zircon, called Hyacinth ; dodecahedral
Garnet.

3. Brick-red. Red with yellow, brown, and grey. The
colour of newly baked bricks. Ex. Hemi-prismatic and
prismatoidal Kouphone-spar ; also Porcelain-jasper, and
other varieties of burnt clay.

4. Scarlet-red. The brightest red colour, but not with-
out a tint of yellow. It is the colour of Cinnabar, or
of the streak of peritomous Ruby -blende.

5. Blood-red. Red with a little of yellow and black. The
colour of blood. Ex. Dodecahedral Garnet, called Pyrope.

6. Flesh-red. A pale red colour, with grey and a little
vellow. E.v. Prismatic Hal-barvte.

TERMINOLOGY. * §. 201.

7- Carminc-rcd. The purest red colour, that of carmine.
Hare. Ex. Dodecahedral Corundum. This colour occurs less
distinct in the capillary varieties of octahedral Copper-ore.

8. Cochineal-red. Red with a little blue and grey. Ex.
Rhombohedral Ruby-blende ; dodecabedral Garnet.

9. Rose-red. A pale red colour, mixed with white and a
little grey, the colour of the flowers of rosa centifoUa. Ex.
Rhombohedral Quartz, called Hose-quartz, macrotypous
Parachr ose-bary te..

10. Crimson-red. Red with a little blue, a particularly
fine colour. Ex. Rhombohedral Corundum (the bright co-
loured varieties of Ruby) ; prismatic Cobalt-mica.

11. Peacliblossom-rcd. Red with white and more of grey
than rose-red. The colour of peachblossom. Ex. Prismatic
Cobalt-mica ; also Lepidolite.

12. ColumUnc-red. Red with a little blue, and a great
deal of black. Distinct in dodecahedral Garnet.

13. Cherry-red. A dark red colour, mixed with a great
deal of blue, and a little brown and black. Ex. Prismatic
Purple-blende.

14. Brownish-red. Red with a great deal of brown. The
colour of reddle, a well known substance for drawing. Ex.
Iron-flint, a mixture of rhombohedral Quartz and oxide
of iron. This colour occurs besides almost exclusively in
undeterminable varieties of rhombohedral Iron-ore.

H. Brown.

1. Reddish-lrown. Brown mixed with a great deal of
red. Ex. Dodecahedral Garnet-blende ; pyramidal Zircon.

2. Clove-brown. Brown with red and a little blue. Very
distinct in prismatic Axinite j also in several varieties of
rhombohedral Quartz.

3. Hair-brown. Brown with a little yellow and grey.
Ex. Prismatic Iron-ore ; several varieties of uncleavable
Quartz, called Wood-opal.

4. Broccoli-brown. A brown colour mixed with blue,
red, and grey, hardly to be defined, and scarcely ever to be
met with, except in pyramidal Zircoiir

§. 201. OPTICAL PROPERTIES OF MINERALS. 291

5. Chcsnut-brown. The purest brown colour. Ex. Egyp-
tian Jasper, a variety of rhombohedral Quartz, mixed
with oxide of iron.

6. Yellowish-brown. Brown with a great deal of yellow.
Ex. Iron-flint and common Jasper, both varieties of rhom-
bohedral Quartz, mixed with oxide of iron.

7- Pinchbeck-brown. Yellowish-brown, with a metallic
lustre, or with a metallic-pearly lustre. Ex. Several varie-
ties of rhombohedral Talc-mica. In this mineral, at least,
pinchbeck-brown does not deserve the name of a metallic
colour, since it is only superficial, and is changed in the
streak into white or grey.

8. Wood-brown. Brown with yellow and grey. The
colour of old, nearly rotten wood. Very distinct in several
varieties of hemi-prismatic Augite-spar, called Mountain
Wood ; sometimes also in Bituminous Wood.

9. Liver -brown. Brown with grey and a little green.
Ex. Common Jasper, a variety of rhombohedral Quartz,
mixed with oxide of iron and clay ; it occurs also in brown
Earthy Cobalt, which is oxide of cobalt mixed with clay.

10. Blackish-brown. Brown with a great deal of black.
Ex. Several varieties of black Mineral-resin, and of bitu-
minous Mineral-coal, called Brown-coal.

The mentioned varieties of colours represent as many
fixed points, between which there exist in nature a great
number of shades or varieties. Such colours are expressed
by the indications of those two, with which they agree near-
est. It is not necessary, however, that these two colours
should be consecutive ones in the above series ; they may
even be varieties of different principal colours. If the oc-
curring colour differs but little from one of these fixed
points, it is said to represent, or to be that colour, only
inclining, or passing, into another.

Colours may be different in their intensity, though be-
longing to one and the same variety. Differences of this
kind are indicated by the expressions, pale, light, deep, dark,
which expressions do not require any further explanation.

292 TERMINOLOGY. §. 202.

§. 202. SERIES OF COLOURS.

The varieties of colours occurring in the indivi-
duals of one and the same species, form an unin-
terrupted series, which is called the Series of Co-
lours of that Species.

If we consider the colours occurring in a species, which is
pretty complete in this respect, we find that they insen-
sibly pass into each other, or that every one of them is in-
termediate between two others. Thus they represent an
uninterrupted succession of the shades of colours, and this
it is what is meant by the series of colours.

The occurrence of the series of colours is the most im-
portant fact in respect to the present subject for the use of
Natural History. In order to obtain these series, by ab-
stracting from the rest of the natural-historical properties
of the individuals in any complete species, it is necessary to
exclude all those colours, which are derived from a mixture
with heterogeneous minerals ; as, for instance, the lemon-
yellow, and the blood-red colours of prismatic Hal-baryte,
the existence of which is owing to an admixture of certain
species of the order Sulphur ; or the yellowish-brown and
reddish-brown colours of rhombohedral and uncleavable
Quartz, arising from oxides of iron, &c.

The series of colours cannot be described ; they must
necessarily be studied from nature ; but the little trouble
which this requires will be amply rewarded. In the
species of the order Gem, the most striking examples
occur. Octahedral Diamond, rhombohedral Corundum,
prismatic Topaz, rhombohedral Emerald, dodecahedral
Garnet, and rhombohedral Tourmaline, may serve as ex-
amples for the illustration of these series. The series
of colours of octahedral Fluor-haloide is one of the most
common, and very easily completed, at least to a certain
extent. This series comprehends a great variety of co-
lours, and resembles, in some respect, the series of colours
of octahedral Diamond and of rhombohedral Fluor-haloide.

§.203. OPTICAL PROPERTIES OF MINERALS. £93

It is deserving of notice, that very often all the species of
one genus possess nearly the same series of colours. Thus
the several species of the genus {Jarnet entirely agree with
each other ; the three most common species of the genus
Augite-spar, and more particularly the hemi-prismatic and
the paratomous one, almost entirely coincide in their series
of colours.

The metallic colours do not form any series at ail, or at
least, the series which they form are very limited. This is
the reason why they are more applicable in the Character-
istic than the non-metallic colour, the employment of which
is almost entirely confined to the Descriptive part of Mine-
ralogy. There are series containing both metallic and non-
metallic colours, as, for instance, those of rhombohedral
lluby-blende, of rhombohedral Iron-ore, and others.

The series of colours differ very essentially from the
series of homogeneous forms (§. 85.) or from the series of
.crystallisation themselves (§. 136.). The latter are deriv-
able from a single form, which is given or has been observ-
ed, and can be obtained in their whole extent between their
limits; whereas the former arise by the interpolation of
new members between known ones ; and these consequent-
ly cannot be produced with security beyond that extent,
which is given by immediate observation.

§. 203, SEVERAL OTHER PECULIARITIES IN THE
OCCURRENCE OF COLOURS.

The Play of Colours, the Change of Colours,
the Opalescence, the Iridescence, the Tarnish, and
the Delineations of Colours, must be considered as
properties very remarkable in themselves, though
of comparatively little use in Natural History.

The only use that can be made of these properties, is in
the Descriptive part of Natural History, and even here
it is very limited.

TERMINOLOGY. §. 203.

The Play of Colours is produced, if the mineral in certain
directions reflects as it were coloured points of great inten-
sity, which change with the position of the mineral, or with
the direction of the rays of light. Of this property, octa-
hedral Diamond, if cut, and precious Opal, a variety of un-
cleavable Quartz, both cut and in its natural state, are
quoted as examples. The play of colours in octahedral
Diamond depends upon the reflexion of refracted light,
occasioned by the artificial facets ; in precious Opal it is
more analogous to the change of colours and the opalescence.

The Change of Colours consists in the reflection of bright
hues of colour, in certain directions depending upon the
structure of the mineral. The mineral which presents the
change of colour in the most remarkable degree, is La-
bradore felspar, a species of the genus Feld-spar.

The Opalescence consists in a kind of milky light, which
certain minerals reflect, either if cut en cabocfion, or upon
plane faces both natural and artificial. It is, like the preced-
ing property, analogous to the play of colours in uncleavable
Quartz. In the varieties of rhombohedral Quartz, called
Cats eye, it depends upon composition ; in prismatic Co-
rundum, and in the transparent varieties of prismatic Feld-
spar, called Moonstone, it depends upon the crystalline
structure. Upon this structure it likewise depends in
rhombohedral Corundum, and in dodecahedral Garnet :
this appears in particular in the six-sided and four-sided
stars of light, from which the varieties of the former have
received the name of Asteria.

The Iridescence shews the colours of the rainbow, simi-
lar to those produced by the refraction of light, through a
prisjn of glass. It presupposes fissures or separations in
the interior of the minerals, which may depend on struc-
ture or on composition, or which may even be entirely ac-
cidental. The included space, not filled up by the mineral,
shews the phenomenon of the coloured rings, sometimes
very bright as in rock-crystal, a variety of rhombohedral
Quartz, where it is occasioned by accidental fissures in the
interior. Another variety of the same species, called the

§. 208. OPTICAL PROPERTIES OF MINERALS. 295

l?ainbow Calcedony, shews similar colours but more faint,
and here they depend upon composition.

Another remarkable property of certain minerals is, that
they shew different colours, if examined by transmitted light
in different determined directions, which demonstrates that
it is intimately connected with their forms and structure.
This property of minerals has been called their Dichroism.
Rhombohedral Tourmaline, prismatic Quartz, rhombohe-
dral and prismatic Talc-mica, are among the most distinct
examples. Several varieties of the first are nearly opaque
in the direction of the axis, while they shew different de-
grees of transparency, and different green, brown, and blue
colours, in the directions perpendicular to it. Prismatic
Quartz is blue in the direction of the axis, yellowish-grey
perpendicular to it. Prismatic Talc-mica is sometimes
green in the direction of the axis, and brown perpendicular
to this line, &c. The application of this property is greatly
extended by examining minerals in polarised b'ght, where
many minerals shew dichroism, which exhibit in common
light the same colour in every direction.

The Tarnish consists in the alteration of the colour of a
mineral upon its surface. It is necessary to be acquainted
with this peculiarity of certain minerals, in order to avoid
confounding it with their real colours. Minerals with a
perfect metallic lustre are almost the only ones subject to
become tarnished ; in these it produces many shades of
bright colours, the further distinction of which, however,
is of very little use in Natural History. Several minerals
become tarnished in a very short time, if a new fracture
has been effected. Among these we observe native Ar-
senic.

Simple minerals very seldom present more than one colour
at a time. There are, however, examples of the occurrence
of two colours, as in rhombohedral Corundum, prismatic To-
paz, rhombohedral Tourmaline, prismatic Disthene-spar, and
a few others. Compound minerals, on the contrary, are very
often variegated ; and the Delineation of Colours comprises the
figures which the different colours produce. It is super-

296 TERMINOLOGY. §. 204.

fluous to enter, in this respect, into a minute detail. With
regard to the dendritic delineations, however, it is worth
noticing that they are real imitative forms (§. 183.), and
that therefore they do not refer to the mineral upon which
they are found : they may be only superficial, or be distri-
buted throughout the whole mass of the specimen.

The delineations of the Florentine ruin marble, a com-
pound variety of rhombohedral Lime-haloide, represent on
a small SCA!C a very interesting phenomenon, which occurs
very often in nature on a larger scale ; this, however, is
a subject foreign to the Natural History of the Mineral
Kingdom.

§. 204. THE STKEAK.

If we scratch a mineral with a sharp instrument,
either a powder will be produced, or the scratched
place assumes a higher degree of lustre. Both
these phenomena are comprehended -under the ge-
neral expression of the Streak.

The lustre is increased by the streak in malleable me-
tals, in several species of the order Glance, and in several
varieties of black Mineral-resin. This is likewise the case
with clay, and with several other decomposed minerals.

The best method for observing the colour of the powder,
is to rub the mineral upon a plate of porcelain biscuit, or
upon a file, till the powder appears. In those minerals
which are too hard for a process of this kind, the streak itself
is of no great consequence.

Some minerals retain their colour in the streak ; others
change it. Among the former are most of those belonging
to the orders Glance, Haloide, Spar, and all those of a white
colour ; among the latter, several of the orders Ore, fy-
rites, Blende, &c. The former are said to be unchanged in
the streak ; of the latter, the alteration of the colour in
the streak is indicated. A white or grey streak of mine-
rals is said to be uncolourcti.

§. 205. OPTICAL PROPERTIES OF MINERALS.
§. 205. DEGREES OF TRANSPARENCY".

With regard to the transparency of minerals, we
have to observe the relative quantity of light which
is transmitted through their substance. The use of
the degrees of transparency is confined to the De-
scriptive part of Natural History.

These degrees are,

1, Transparent, if the light is transmitted in a sufficient
quantity to enable us to distinguish small objects placed
behind the mineral.

2, Semi-transparent, if it is possible to see an object behind
the mineral, without, however, being able to distinguish
more of it, than its general outline.

8, Translucent, if a small quantity of light only falls into
the mineral, but without allowing an object behind it to
be seen, except in so far as it in general may prevent the
Kght from falling upon the mineral.

4, Translucent on the edges, if only the most acute edges
of a mineral receive some light, while the interior remains
perfectly dark. This degree of transparency has moreover
been distinguished into strongly and feebly translucent
on the edges ; and it is upon distinctions of this kind that
are founded the differences between the varieties of rhom-
bohedral Quartz, called flint, hornstone, jasper, &c.

5, Opaque, if a mineral transmits no light at all.

The species of the orders Metal, Glance, and Pyrites,
consequently most of those which possess a perfect metal-
lic lustre, are entirely opaque. This is, however, not quite
general for all the minerals of a metallic appearance, as
for instance the lamellar varieties of rhombohedral Iron-ore,
which transmit sometimes in the sun a very bright red
colour.

Minerals of a non-metallic appearance, are not entirely
opaque, a few species of the order Ore, perhaps except-
ed. Yet accidental impurities influence so much their

298 TERMINOLOGY. §. 206.

transparency, that this property becomes almost entirely
useless in the Determinative part of Natural History.
The best employment to be made of it seems to be, in the
distinction of compound varieties from simple ones, if the
minuteness of the particles of composition prevents them
from being observed immediately. Commonly in the same
species (§. 190.) the compound varieties possess a less
degree of transparency than the simple ones. A very dis-
tinct example of this we have in the varieties of rhombo-
hedral Quartz. Almost all its single individuals, provided
they are not impure, shew higher degrees of transparency
than flint, hornstone, calcedony, and other compound va-
rieties.

CHAPTER II.

OF THE PHYSICAL PROPERTIES OF MINERALS.
§. 206. EXPLANATION.

The properties of the substance of minerals, or
those which have by preference been termed their
physical properties, comprehend all those which
neither depend upon their form and the space
which they fill up, nor upon the presence or ab-
sence of light.

Among these are the State of Aggregation, Hardness,
fic Gravity, Magnetism, Electricity, Taste, and Odour.

It is almost unnecessary to observe, that the expressions
mass, or substance, must not be conceived in the chemical
sense of the word, and that these properties are not meant to
be more essential to the minerals than any of those which
have been considered above, which perhaps might be in»

§.207. PHYSICAL PROPERTIES Olr MINERALS. 299

ferred from the consideration, that these properties refer
to the minerals themselves, even though we should ab-
stract from their geometrical and optical properties.

§. 207. STATE OF AGGREGATION.

In respect to the state of aggregation, we distin-
guish solid and fluid minerals. The former are
either brittle, or sectile, or malleable, or flexible, or
elastic ; the latter are either liquid or expansible.

A solid mineral is said to be — .

1, Brittle, if in the experiment of detaching small par-
tides of it with a knife or a file, these particles lose their
coherence, and separate with a grating noise, while they fly
about in the state of powder. The particles therefore cannot
alter their respective sitviations without separating en-
tirely. Ex. All the species cf the orders Gem, Spar, Py-
rites, several of those of Ore, Haloide, &c.

2, Malleable, if the particles detached by the knife, do
not lose their connexion. From a malleable mineral, we
may detach slices as we do from metallic lead. Ex. Many
metals, hexahedral Pearl-kerate, hexahedral Silver-glance,
and several varieties of black Mineral-resin.

3, Sectile, if in the above experiment the particles lose
their connexion, and do not allow the separation of any
slices ; but if at the same time they do not fly about with
a noise, but quietly remain upon the instrument we have
applied. The sectile minerals form an intermediate stage
between the malleable and the brittle ones. This state of
aggregation is commonly not distinguished by natural philo-
sophers, though it is very useful in the characters of several
species. Examples of sectile minerals we have in most of
the species of the orders Mica and Glance, in some of the
orders Haloide, Baryte, &c.

4, Ductile, if it can be wrought into Sheets or wire; so
that by the application of a greater or lesser force, the par*

300 TERMINOLOGY, §. 208.

tides of the mineral may change their relative situation,
.without absolutely losing their connexion. Ex. Several
metals, as hexahedral Gold and hexahedral Silver.

5, Flexible, if the particles, whose relative situation has
been changed, do not resume their former situation. There
are flexible minerals, which are neither ductile nor malleable.
Ex. Several metals, hexahedral Silver-glance, and several
varieties of prismatic Talc-mica.

6, Elastic, if the particles, whose relative situation has
been changed, resume their former situation. Ex. Several
varieties of rhombohedral Talc-mica, and of black Mineral-
resin.

A fluid mineral is more particularly said to be —

1, Liquid, if in pouring it out from a vessel, perfect
round drops are formed. Ex. Water, several Acids, fluid
Mercury, and several varieties of black Mineral -resin.

2, Viscid, if the drops are not round, but ropy. Ex. Se-
veral varieties of black Mineral-resin.

Expansible minerals do not shew any further differences
in this respect. They comprehend the Gases and some
Acids.

It is evident that all these properties are subject to small
variations, and that they pass into each other by insensible
gradations.

§. 208. HARDNESS.

Hardness in general may be defined to be the
resistance of solid minerals to the displacement of
their particles. The magnitude of this resistance is
their Degree of Hardness.

Hardness is a very useful property in the Natural His-
tory of the Mineral Kingdom, particularly so in its deter-
minative part.

Nothing is attended with greater difficulties, than the
establishment of an accurate scale for the degrees of bard-

§. 208. PHYSICAL PROPERTIES OF MINERALS. 301

ness. It is necessary therefore to endeavour, even without
an accurate scale of that kind, to become capable of ascer-
taining and indicating these differences, at least with a de-
gree of accuracy and certainty sufficient for the wants of
the Natural History of the Mineral Kingdom.

The existence of differences in the degrees of hardness
among the minerals, is very easily discovered, by the
simple experiment of scratching one of them by the other.
A sharp corner of rhombohedral Quartz will produce a deep
cut in the mass of rhombohedral Lime-haloide ; whilst a
sharp corner of the latter species does not injure the sur-
face of the former. Hence we infer, that rhombohedral
Quartz possesses a higher degree of hardness than rhom-
bohedral Lime-haloide ; and in general, that of two mine-
rals, the harder one scratches the other, but cannot inversely
be scratched by it. Some precautions, however, are neces-
sary in drawing general inferences from these observations.

If we proceed upon this principle, we may obtain a Scale
for the degrees of hardness, answering in every respect the
purposes of Mineralogy. This is effected by choosing a
certain number of suitable minerals, of which every pre-
ceding one is scratched by that which follows it, while the
latter does not scratch the former ; taking care always that
the intervals between every two members of the scale be
not so disproportionate, as either to render its employment
more difficult, or to hinder it altogether.

The following scale possesses these properties :

1, Prismatic Talc-mica, the common Talc of mineralogists,
of a whitish or greenish colour.

2, Prismatoidal Gypsum-haloidc, a variety imperfectly
cleavable, of an inferior degree of transparency, and not
crystallised; crystals and perfectly transparent varieties
being rather too soft. This degree of hardness is exactly
that of hcxalicdral Rock-salt, which mineral therefore may
be very useful, either in being immediately employed in
the determination of hardness, or at least in finding out
such varieties of the above-mentioned species, as exactly
possess the required degree of hardness.

302 TECHNOLOGY. §. SOS.

3, Rhomlohcdral Limc-haloide. Any cleavable variety.
The minerals called Brown-spar (macro typous Lime-ha-
loide), or llhomb-spar (macrotypous and brachytypous
Lime-haloide), cannot be employed in its place, the hard-
ness of these being considerably higher.

4, Octahedral Fluor-haloide. Any cleavable variety.

5, Rhomboliedral Fluor-haloide. The variety from Salzburg
called Asparagus-stone, possessing a conchoidal fracture.
The Apatite from Saxony or Bohemia, will seldom be
found to answer the purpose, though of exactly the same
degree of hardness.

6, Prismatic Feld-spar. A perfectly cleavable variety of
Adularia.

7, Rhombohedral Quartz. Limpid and transparent.

8, Prismatic Topaz. Any simple variety.

9, Rhombohedral Corundum. The easily cleavable variety
from Bengal, called Corundum-stone.

10, Octahedral Diamond.

The minerals, which represent the units of this scale,
have been chosen among those species, which may be most
easily acquired with the necessary qualifications, excepting
perhaps only rhombohedral Fluor-haloide. Yet it has been
impossible to find out another which might be as useful in
its place.

The intervals between the members of the scale are not
everywhere of the same magnitude. Octahedral Diamond
is evidently much harder, if compared with rhombohedral
Corundum, than octahedral Fluor-haloide, if compared with
rhombohedral Lime-haloide. This, however, is of no con-
sequence in the case above mentioned ; for there exists no
mineral of a hardness intermediate between the degrees
represented by the two first of these species. But the
interval between rhombohedral Fluor-haloide and prismatic
Feld-spar is likewise greater than it should be. In this case
it would be very desirable to have another mineral which
might allow of being employed instead of rhombohedral
Fluor-haloide. But in general it is very difficult to ascer-
tain the perfect equality of the intervals between the differ-

§.808. PHYSICAL PROPERTIES OF MINERALS. 803

ent degrees of hardness, and on that account also it is
very difficult to be obtained. Yet all these imperfections
are by no means prejudicial to the useful employment of
the scale.

The degrees of hardness are expressed by means of those
numbers which in the above enumeration are prefixed to
them. Thus the hardness of rhombohedral Lime-haloide is
= 3, that of rhombohedral Corundum = 9.

The intervals between each two subsequent members may
be divided into ten equal parts ; and these tenths determin-
ed by estimate. It will very seldom be required to value
the hardness to more or less than 0. 5 ; but it will al-
ways be possible to proceed so far as we find it necessary to
answer our purpose.

The state of liquidity may be considered as the zero of
the scale.

If, in employing the scale, we endeavour to find the de-
gree of hardness of a given mineral, by trying which
member of the series is scratched by it, and which of
them injures the surface of the given one, it will appear
that the specimens employed should possess certain proper-
ties, in many cases difficult to be found. They should all
have faces perfectly smooth and even, and solid angles or
corners of the same form, and be equally durable.

As to the faces, those produced by cleavage seem the
most eligible, if they possess a pretty high degree of per-
fection. Faces of crystallisation are commonly uneven or
streaked ; cut and polished faces, however, in many in-
stances shew a less degree of hardness than the mineral
really possesses.

It is still more difficult to obtain the corners with the
constant quality which is required. Even in a determined
form these are sometimes liable to be so much influenced
by structure, that they give very uncertain results. In
this respect, the solid angles of the tetrahedron, and those
of the octahedron of octahedral Fluor-haloide, shew quite
different phenomena. The corners of compound varieties,
in which the individuals become impalpable or disappear,

304 TERMINOLOGY. §. 208.

such as Calcedony, Flint, and others, are commonly found
very powerful, much more so than similarly formed corners
of simple varieties. But if the composition is still observ-
able, the particles very often separate in the experiment of
scratching another mineral, and the corner of a compound
mineral cannot produce the effect of that of the simple
mineral. The application of the edges is subject to similar
difficulties.

Numerous experiments of determining the degree of
hardness, by the mere scratching of one substance with
the other, have completely established, that this process
alone is not sufficient, if we intend to make a more sure
and extensive application of the characters that may be
taken from hardness, than that which has hitherto been
common in Mineralogy.

But if we take several specimens of one and the same
mineral, and pass them over a fine file, we shall find that
an equal force will everywhere produce an equal effect, pro-
vided that the parts of the mineral in contact with the file
be of a similar size, so that the one does not present to
the file a very sharp corner, while the other is applied to
it by a broad face. It is necessary also that the force applied
in this experiment, be always the least possible.

Every person, however little accustomed, will experience
a very marked difference, if comparatively trying in this
way any two subsequent members of the above scale, and
thus the difference in their hardness will be easily perceiv-
ed. A short practice is sufficient for rendering these per-
ceptions more delicate and perfect, so that in a short time
it is possible to determine differences in the hardness very
much less than those between two subsequent members of
the scale.

Upon these observations is founded the application of
the scale, the general principle of which consists in this,
that the degree of hardness of the given mineral is com-
pared with the degrees of hardness of the members of the
scale, not immediately, by their mutual scratching, but
mediately, through the File, and determined accordingly.

The process of this determination is as follows :

§. 208. PHYSICAL PROPERTIES OF MINERALS. 305

First we try, with a corner of the given mineral, to
scratch the members of the scale, beginning from above,
in order that we may not waste unnecessarily the spe-
cimens representing lower members. After having thus
arrived at the first, which is distinctly scratched by the
given mineral, we have recourse to the file, and compare
upon it the hardness of this degree, that of the next higher
degree, and of the given mineral. Care must be taken
to employ specimens of each of them nearly agreeing in
form and size, and also as much as possible in the quality
of their angles. From the resistance these bodies oppose
to the file, and from the noise occasioned by their passing
over it, we argue with perfect security upon their mutual
relations in respect to hardness. The experiment is repeat-
ed with all the alterations thought necessary, till we may
consider ourselves arrived at a fair estimate, which is at
last expressed by the number of that degree with which it
has been found to agree nearest, the decimals being like-
wise added, if required.

The files answering best for the purpose are fine and
very hard ones. Their absolute hardness is of no conse-
quence ; hence every file will be applicable, whose hard-
ness is in the necessary relation with that of the mineral.
For it is not the hardness of the file with which we have
to compare that of the minerals, but the hardness of ano-
ther mineral, by the medium of the file. From this ob-
servation it appears, that the application of the file widely
differs from the methods of determining the hardness of
minerals which have hitherto been in use ; as scratching
glass, striking fire with steel, cutting with a knife, scratch-
ing with the nail, &c.

Besides an appropriate form, there is another necessary
property of the minerals to be determined, consisting in
their state of purity. Neither the degree of hardness, nor
that of specific gravity, can be correctly ascertained, if we
employ impure substances. For the same reason it would
be wrong to make use of minerals which have undergone a
total or even partial decomposition ; and in general every
VOL. i. u

306 TERMINOLOGY. §. 208.

circumstance which might influence the hardness, must be
duly attended to, if we intend to arrive at a useful and
correct result.

Minerals that cleave with particular facility in only one
direction, very often shew a less degree of hardness upon
the perfect face of cleavage, than in other directions. Pris-
matic Disthene-spar sometimes is scratched by octahedral
Fluor-haloide upon the eminent face of cleavage, whilst an
angle of the very same individual scratches not only rhom-
bohedral Fluor-haloide, but even sometimes prismatic
Feld-spar. If we intend to determine a mineral of this de-
scription by the help of the Characteristic, it will be the
best plan to take a mean term between the two degrees
measured, or rather to keep nearer to the higher one. It
would be wrong to receive them into a scale of hardness,
like the preceding one, since this would betray a want
of acquaintance with the scale itself, and with its employ,
ment.

Supposing all the precautions necessary in determining
the degrees of hardness to have been taken, and the cir-
cumstances well attended to, which might have exercised
some influence ; we find that those individuals which be-
long to one and the same species, admirably agree with
each other in respect to this property ; and that deviations
from an exact coincidence, if they happen to occur, do not
take place per saltum, but that they are joined with each
other by intermediate members. These members produce
a series, in most cases between very narrow limits. This
observation seems to be contradicted by the authority of
several mineralogical works. But there are indeed few
subjects with regard to the properties of minerals, which
have been treated with more indifference or even careless-
ness than their hardness, and on this account little or no
credit is due to what most of the mineralogical works con-
tain of its indications.

KIRWAN, DE LA METHERIE, and HOME DE L'IsLE,
have each endeavoured to construct scales of hardness. A
comparative table of the hardness of different substances.

§. 209. PHYSICAL PROPERTIES OF MINERALS. 307

is contained in the works of HAUY. A glance at them will
suffice for enabling us to form an idea of their applicability.

§. 209. SPECIFIC GRAVITY.

If we suppose the absolute weight of one of two
bodies, possessing the same volume, to be = 1 ;
the ratio of the absolute weight of the other to
this unit, is termed its Specific Gravity.

The determination of the specific gravity depends upon
the comparison between absolute weights and volumes.
They cannot be instituted at all, or at least not to a suf-
ficient degree of accuracy, merely by sight or estimate.
We must avail ourselves of the assistance of appropriate
instruments, if we wish the determination to be of use.

The instruments intended for ascertaining the specific
gravity of solid bodies, are the Hydrostatic Balance and
Nicholson's Araeometer. That of a liquid is determined
by weighing in it a solid body, whose specific gravity we
know, and which is not soluble in the liquid. Instruments
have likewise been constructed for this purpose. The de-
termination of the specific gravity of expansible fluids re-
quires very delicate operations, and instruments that are
not within the reach of every body.

The arrangement of the two above mentioned instru-
ments, their use, and the whole process of taking the specific
gravity of bodies, are very generally known, or at least they
may be found described at large in every treatise on Natural
Philosophy. Each of them possesses particular advantages.

The hydrostatic balance allows of a high degree of ac-
curacy, and is very convenient in its use. The instru-
ment being correct in itselfj and delicate as a common
balance, its delicacy as a hydrostatic one will depend upon
the thinness of the thread by which the vessel is suspended,
which bears the body immersed in the water. A human
hair is sufficiently strong for supporting a weight of three
hundred grains, and therefore very useful in taking the

308 TERMINOLOGY. §. 209-

specific gravity of such bodies as do not possess any great
absolute weight.

The pin, which supports the uppermost cup of the araeo-
meter, destined for the reception of the weights and of
the body to be weighed, must possess a certain diameter,
since it acts not only as a supporter, but also in the
capacity of a real weight, according to the depth to which
it is immersed in the water. This diameter, however,
must remain within certain limits, if the instrument shall
not lose its niceness.

The hydrostatic balance will therefore be more eligible for
more accurate inquiries, either for obtaining a greater num-
ber of decimal figures, or for determining the gravity of a very
small specimen : hence it must always be employed, if our ob-
ject is to fix the limits of the range in the specific gravities
of a natural-historical species, for the sake of the determina-
tion of its varieties ; and this has been done in the species
contained in the subsequent Characteristic. For the com-
mon use of determining the specific gravity of minerals, in
order to find out their denominations by the assistance of
this Characteristic, the araeometer will be found both suf-
ficient and preferable, because in this case we may ac-
quiesce in most cases in the first decimal figure of the spe-
cific gravity. The instrument is besides recommendable,
on account of its being cheap and portable. The size of
the specimens, the specific gravity of which may be taken
by help of the araeometer, cannot exceed certain limits,
determined on one side by the absolute weight it will bear,
till it be immersed to the sign marked upon the pin, on the
other by the niceness of the instrument itself.

In taking the specific gravity, we must likewise observe
the degree of temperature. The changes of temperature
render it necessary to determine the normal weight, or that
which is required for depressing the araeometer to a cer-
tain point, at every experiment, in the same way as it is
necessary in the hydrostatic balance, which likewise, previ-
ous to every experiment, must be brought into equilibrium.

The minerals, of which we intend to take the specific

§. 209. PHYSICAL PROPERTIES OF MINERALS. 809

gravity, must be perfectly pure. The greatest care there-
fore must be taken in removing as much as possible what-
ever heterogeneous substances may adhere to them, or at
least, if this should be impossible, not to neglect consi-
dering the influence of such an admixture upon the coi>
rectness of the results. Moreover, all the vacuities or
empty spaces within the specimens, must carefully be
opened. In order to get rid of these, the minerals ought
to be broken down, till, even by the assistance of a mi-
croscope, we can no longer detect a want of continuity in
the fragments. Compound varieties are more subject to
contain similar vacuities than simple minerals; for this
reason the composition must be removed, at least in so far
that it cannot have any more influence upon the accuracy
of our results. Yet the minerals must not be too much
reduced in size, since this might lead into an opposite er-
ror, in supposing those minerals lighter than water, which
swim upon it, when reduced to an impalpable powder.

These precautions have been very often neglected in
taking many of those specific gravities quoted in mineralo-
gical works, and thus numberless erroneous and inaccurate
statements have been introduced, which render their em-
ployment at least uncertain, and on that account use-
less for Mineralogy. Another source of error, for which
many examples might be quoted, consists in the incorrect
determination of the natural-historical species to which
the specific gravities have been referred, and which have
passed from one work into another.

A certain degree of attention is required, both in select-
ing the specimens and in the operation of ascertaining the
specific gravity. But from this it will appear that the re-
sults of the single experiments made upon specimens of
homogeneous minerals, coincide in a remarkable degree ;
and thus we may argue upon the great importance of the
application which this property will allow in the Natural
History of the Mineral Kingdom.

310 TERMINOLOGY. §. 210. 811.

§. 210. MAGNETISM.

Some minerals act upon the magnetic needle, if
they are brought within the sphere of its attraction.
Others become magnets themselves. These pheno-
mena are made use of as characters, under the
name of Magnetism.

The only minerals hitherto known, which exercise a con-
eiderable action upon the magnetic needle, are the octahe-
dral Iron, and the octahedral Iron-ore. Rhombohedral
Iron-ore, rhombohedral Iron-pyrites, and several others,
likewise act upon it, but with less energy.

Instead of a needle, the magnetic bars may be applied
in examining minerals, which in this case must be convert-
ed into a fine powder, in order to extract from them such
particles as possess magnetic properties.

§. 211. ELECTRICITY,

Several minerals produce electric phenomena;
some of them by friction, others by pressure, others
by communication, and others by heat. Some are
idio-electric ; others are conductors of electricity.
These phenomena may be usefully applied as
characters of minerals.

Vitreous electricity is produced by friction in most mi-
nerals of the orders Gem, Spar, Mica, Bary te, &c. in seve-
ral Haloides, and even in Salts. In the same way those of
the orders Sulphur, Resin, and Coal, shew the phenomena
of resinous electricity. As conductors of electricity, we
may notice the minerals of the orders Metal, Pyrites, and
Glance. Those of the orders Blende, Ore, and several
others, do not appear quite uniform in this respect.

Heat produces electric phenomena in prismatic Topaz,
in rhombohedral Tourmaline, in prismatic Kouphone-spar,

PHYSICAL PROPERTIES OF MINERALS. 311

ill axotomous Triphane-spar, in prismatic Zinc-baryte, &c.
The opposite extremities of the crystals assume in these
species opposite kinds of electricity, and they possess there-
fore electric axes. Tetrahedral Boracite shews four electric
axes, coinciding with the rhombohedral axes of the hexa-
hedron. This difference in the electric phenomena is very
often accompanied by a different configuration of the oppo-
site terminations of crystals (§. ICO.).

The processes employed in producing and observing the
electric phenomena, and the small apparatus required, may
be found in many works, both described and illustrated by
figures. Perhaps these phenomena will prove in future
more useful for the purposes of Natural History, than has
hitherto been the case, since, in respect to minerals, they
have been too generally considered as mere physical cu-
riosities.

§. 212. TASTE.

Several minerals, solid as well as fluid, produce
a sensible taste. Most of the solid ones are taste-
less. This difference yields very useful general
characters.

All the Acids and Salts produce some taste. The salts
found in nature, commonly not shewing any of the charac-
ters required for their exact determination, their taste is
almost the only one left to which we possibly may recur ;
and for this reason the differences in the kinds of taste have
been provided with particular denominations. The follow.
ing expressions have been employed :

1, Astringent for the taste of vitriol;

2, Sweetish for the taste of alum ;

3, Saline for the taste of common salt ;

4, Alkaline for the taste of soda ;

5, Cooling for the taste of saltpetre ;

6, Sitter for the taste of epsom salt ;

312 TERMINOLOGY. §.

7, Urinoiu for the taste of sal ammoniac ;

8, Sour for the taste of sulphuric acid, or of carbonic acid.
Besides, the intensity or other peculiarities of several

kinds of taste may be indicated, which is sufficiently plain
by the manner in which it is effected in this work.

Pure artificial salts are most eligible as examples for the
different kinds of taste. Some caution is required in ascer-
taining this character in unknown minerals. Since in most
cases it is quite sufficient to know, whether or not a mine-
ral excites some taste, we may also dissolve them in water,
because all sapid minerals are soluble in a small quantity
of this fluid.

§. £13. ODOUE.

There are minerals which, either spontaneously
or when rubbed, emit some odour, which likewise
in particular cases, may afford useful characters.

Several varieties of black Mineral-resin possess a bitu-
minous odour. The species of the genus Iron-pyrites emit
a sulphureous odour, when strongly rubbed, as takes place
in striking fire. The Arsenical-pyrites under the same cir-
cumstances yield an arsenical or garlick smell. Several
varieties of rhombohedral Lime-haloide, of prismatic Hal-
baryte, of prismatoidal Gypsum-haloide, &c., if rubbed with
hard substances, emit an empyreumatic odour ; pebbles of
rhombohedral Quartz, and other hard bodies, if rubbed
against each other. Several Resins produce a peculiar
odour, if rubbed against soft substances.

Certain species of Gas, of expansible Acids, possess a
peculiar kind of odour ; that of rotten eggs, of rotten fish,
of burning sulphur, &c.

Besides the characters treated of till now, there are still
some more phenomena which have been employed as such,
by mineralogists. Among these, the Adhesion to the tongue
is almost exclusively met with in decomposed minerals;

§. 213. PHYSICAL PROPERTIES OF MINERALS. 813

the Unctuons and Meagre toiich are used for distinguishing
certain friable minerals; and the Phosphorescence produced by
heat, is also employed in those minerals in which the natural-
historical properties properly so called are not observable.
It would be superfluous to dwell any longer upon these
subjects ; the more so since every Treatise on Mineralogy
may be consulted for all their particulars.

PART II.
THEORY OF THE SYSTEM.

§. 214. IDENTITY.

Natural productions, which do not differ from
each other in any of their natural-historical proper-
ties, are identical (§. 14.).

This proposition is self-evident ; and it is the foundation
of the whole Theory of the System in Natural History.

By considering in this science two bodies as identical, it
is meant that every one of them may be substituted in the
place of the other in every natural-historical respect ; so
that if the one belongs to a certain class, to a certain order,
genus, or species, the other likewise must necessarily belong
to the same class, to the same order, genus, and species.

In considering the identity of two bodies, we must ab-
stract all accidental differences (§. 25.). Such are, besides
the size of crystals, also the disproportionate enlargement
of some of their faces (§. 159.), their junction with other
individuals, their being implanted or imbedded, &c. Indi-
viduals, which differ only in properties of this kind, must
be taken for identical ones, as well as those which agree
also in respect to these accidental circumstances.

§. 215. DIFFERENCE.

Individuals, which do not agree in all their na-
tural-historical properties, are not identical.

This proposition is an immediate consequence of the
preceding one.

If two individuals agree in every one of their properties,
except in their crystalline form, or in colour, or iu hardness,

§. 216. TIIEOEY OF THE SYSTEM. 315

or in specific gravity, &c., so as to differ only in one of these
properties, nevertheless they will not be identical. For
the above mentioned properties are natural-historical ones,
and upon these depends their identity or their difference
(§. 214.). Hence the difference among individuals may be
produced by a difference, however small, in any one of their
natural-historical properties; and in this case, from the
one nothing can be argued in respect to the other. It is
almost superfluous to observe, that accidental differences
also in the present place cannot have any influence upon
the difference or identity of bodies.

The Natural History of the Mineral Kingdom does not
require any foreign assistance for determining what is acci-
dental or not, and agrees in this respect with the Natural
History of the Vegetable and of the Animal Kingdoms.

§. 216. DEGREES OF DIFFERENCE.

The difference among those individuals which
are not identical (§. 215.), does not everywhere
take place in the same degree.

Suppose two crystals of dodecahedral Garnet, to agree
in all their natural-historical properties except in their
crystalline forms ; the form of the one being the dodecahe-
dron, while that of the other is a digrammic tetragonal-
icositetrahedron. These individuals are evidently different.
Now, suppose one of those crystals again, to be compared with
a crystal of hexahedral Gold. There is also a difference be-
tween these two individuals ; and nobody will hesitate in
pronouncing the degree of difference in the latter case to
be higher than that in the preceding one ; even though
the form of the crystal of hexahedral Gold should be exact-
ly the same as that of the crystal of dodecahedral Garnet.
Many examples of this kind might be quoted, which indubit-
ably demonstrate the degrees of difference not, to be the same
in every two different individuals. We may very easily
perceive what influence this must have upon tfce further

810 THEORY OF THE SYSTEM. §. 216.

consideration of mineral productions, and what would be
the consequence if, on the contrary, we should meet every-
where with exactly the same degrees of difference.

These degrees of difference must not be valued accord-
ing to the number or the kind of properties in which those
individuals differ which are not identical. They depend
rather upon certain relations of these properties with each
other, which will be explained afterwards. A crystal of
hexahedral Iron-pyrites is much more different from a
crystal of prismatic Iron-pyrites, though they should ex-
actly agree in every property except in the form, and what
depends upon it, than one crystal of rhombohedral Corun-
dum, of the variety called Sapphire, from another of the
same species called Adamantine spar. And yet the form
of the first is an isosceles six-sided pyramid, that of the
other a regular six-sided prism. The first presents almost
no trace of cleavage, while the other cleaves very easily
parallel to the faces of a rhombohedron : they differ more-
over in colour, in transparency, and in many other charac-
ters. Such examples are common ; and whoever therefore
would determine the degrees of the natural-historical affi-
nity according to the number, or even to the kind of pro-
perties not agreeing, considering the one as more essential
than the other, would act contrary to the principles of
Natural History.

If it were possible to invent a scale for measuring with
accuracy the degrees of difference among the non-identical
individuals, this would afford most useful assistance in clas-
sifying the productions of inorganic nature. But there
exists no such scale. We must contrive, therefore, to col-
lect several of the non-identical individuals, adapted to this
purpose in respect to their properties, and to bring iliem
under tlic idea of identity (§. 214.). This will enable us to
extend the inferences which may be drawn from identi-
cal individuals, to such as by themselves do not exactly
agree in all their properties. Without this process of ex-
tending the idea of identity, it would be impossible to
derive from it a sufficiently useful employment in Natural

§. 817. THEORY OF THE SYSTEM. 817

History. In order to effect this, however, some prepara-
tions will be necessary.

§. 217. MUTUAL RELATIONS OF THE NATURAL-
HISTORICAL PROPERTIES IN CERTAIN INDIVIDUALS.

Individuals, which are different from each other
in their natural-historical properties, so that their
differences constitute members of one and the same
series, may thoroughly agree with each other in
the rest of their properties.

Among the different cases comprised under this head,
there is one more remarkable than the rest, if the differ,
ences of the individuals consist in the forms, and if these
forms are members of a series. It deserves to be considered
more in particular, since all the others may very easily be
explained upon a similar principle.

Experience confirms, in numerous examples, that forms,
which are different, members of the same series, may, in
other respects, possess properties entirely equal and similar.
A demonstration, however, may be given of this proposition
with more generality, and which therefore will receive a
greater degree of evidence than that which it could acquire
by any number or description of examples quoted.

It has been observed above (§. 139.), that such forms,
as are members of one series, may enter into combina-
tion with each other, and inversely, that all the combina-
tions produced by nature contain only such simple forms
as belong to, or represent members of, one and the same
series.

An individual appearing in a compound form, appears at
the same time in as many simple forms as the combina-
tion contains ; and, in respect to these, it may be consi-
dered as representing as many individuals at a time (§. 138.).
But with every one of these simple forms we find connected
the rest of the natural-historical properties of the individual
under consideration. These assemblages of properties repre-

318 THEORY OF THE SYSTEM. §. 218.

sent individuals, which, in so far as their forms are members
of the same series, differ only in these, and in none of their
other properties. Every combination occurring in nature
confirms, therefore, the above proposition ; and if we are
capable of deriving useful arguments from it, these will
be perfectly general, on account of the perfect generality
of the laws according to which combinations are formed
(§. 130. 140.).

The preceding observations are not limited to the series
of crystallisation ; but they extend to every natural-histo-
rical property, by the differences or gradations of which
series are produced. It applies, however, equally to any na-
tural-historical property whatever: their gradations may
produce series or not ; because those which give no series
at all, or at least not in every instance, may yet be consi-
dered as series of equal members. In the present inquiry
the series of forms have been chosen by preference, because
they allow of a mathematical mode of treatment, and there-
fore impart a full evidence to the arguments derived from
them. Together with the arguments, also, this evidence
is transferred to other series of properties treated in the
same manner.

§. 218. INDIVIDUALS BROUGHT UNDER THE IDEA
OF IDENTITY*

Individuals, whose forms are members of a se-
ries, their remaining natural-historical properties
being entirely coincident, may be brought under
the idea of identity.

It cannot be liable to any objection, that every indivi-
dual, not excepting those which appear in compound forms,
is identical with itself. But if in this combination we al-
low all the simple forms to disappear, except one, and con-
tinue this process with every simple form contained in the
combination ; we develope a scries of individuals, each of
which is exactly in the same relation to the idea of iden-.

§. 218. THEORY OF THE SYSTEM. 319

tity as the fundamental individual, and which therefore
exactly agree in respect to this idea, although they are
not absolutely identical. It is indifferent which of the
simple forms we may ascribe to the one or to the other ;
and thus we may arbitrarily exchange these forms with each
other^ without in the least altering any thing in respect to that
relation. Two or more of these individuals become abso-
lutely identical (§. 214.), if we suppose them to possess one
and the same form. Herein consists the process by which,
under the supposed circumstances, individuals, though not
absolutely identical, may yet be brought under the idea
of identity.

If, on the contrary, a number of different individuals is
given, agreeing in every natural-historical property except
the forms of crystallisation, and if these forms are mem-
bers of the same series, we are entitled to consider all
those individuals as a single one, whose form is a combina-
tion of the different simple forms of the single individuals,
with which the rest of the properties exactly agree. The
individual in the compound form is identical with itself, and
the single individuals contained in it will consequently be
in an exactly similar relation to the idea of identity.
This proves that by the above mentioned process, that is to
say, by substituting one form instead of the other, they
may be collected under the idea of identity.

In order to explain this by an example, let us suppose a
crystal of octahedral Fluor-haloide to possess the form of
a hexahedron. If we substitute the octahedron in its stead,
the relation of the individual to the idea of identity is
not altered, because the two forms, the hexahedron and
the octahedron, are members of the same series of crystal-
lisation. Notwithstanding this and other similar substitu-
tions, the individual does not cease to be octahedral Fluor-
haloide. But if, instead of the hexahedron, we suppose a
rhombohedron, or any other form not belonging to the
tessular system, to take the place of the hexahedron, the
relation of the individual to the idea of identity is indeed
altered, and it can no longer be maintained that the indivi-

320 THEORY OF THE SYSTEM. §.

dual still remains octahedral Fluor-haloide. For if we sup-
pose a number of individuals, to agree with each other in
all their properties, excepting their forms, these however
not being members of the same series ; we are not entitled
to apply to them the above process ; because the differences
existing among them cannot be removed or made to dis-
appear by the idea of a series, and accordingly the indi-
viduals themselves cannot be brought under the idea of
identity. The degree of difference (§. 216.) between such
individuals is therefore much higher than it would be, if
under the same circumstances the forms were members of
the same series.

As an example of the latter case, we may quote the hexa-
hedral and the prismatic Iron-pyrites. There exist indivi-
duals in these two species exactly agreeing with each other in
every one of their natural-historical properties, except their
crystalline forms. But as these forms belong to different
systems, and are therefore incompatible with each other,
the difference between the individuals appears greater than
it would be, if the forms should belong to the same series.

The process explained in the members of the series of
crystalline forms, naturally applies likewise to every pro-
perty, from the gradual differences of which series arise.
Thus we are provided with the means of discovering such
individuals, as, though not absolutely identical, may yet
be brought under the idea of identity, and of separating
them from all the rest. Such individuals might be col-
lected under particular ideas ; but such ideas would be
of little use, on account of their very limited application to
experience. Nevertheless they lead the way to that idea,
which it is the particular object of the present inquiry, to
develope according to the principles of Natural History.

§. 219. CONNEXION OF SEVERAL SERIES OF INDIVI-
DUALS.

An individual which, on account of its form and
the rest of its natural-historical properties, is a

§ 219. THEORY OF THE SYSTEM.

member of a series of individuals (§. 218.), differ-
ing in nothing but their forms, may at the same
time be a member of another series of individuals,
differing only in the gradations of their colours,
See., the rest of the natural-historical properties
being supposed exactly to agree.

Let A, B, C, D, . . .

represent a Series of Individuals •, in which the forms and the
colours are not yet determined, so that every one of those
letters signifies the aggregate of the remaining properties,
which are exactly the same in all of them. Hence in this
respect they differ from each other only by their succes-
sion, that is to say, by their not being one and the same
thing. Suppose, now, every individual to have the same
colour a, but different forms, without the latter of which
they would not be different individuals. According to our
supposition, these forms must be members of the same se-
ries, and may therefore be expressed by

X, X + 1, X + 2, X + 3,

where X may denote any fundamental form whatever.
The designation of the series of individuals, as above, only
including their forms and colours, will therefore be
I. A.a.X; B.a.(X+l); C.a.(X + 2); D.a.(X+3); ...
A fragment of another series of individuals may, under
the same restrictions as those mentioned above, be desig-
nated by

. . . P, Q, R, S, . . .

P, Q, R, &c. being similar aggregates of properties, as
A, B, C, &c. in the preceding series. Suppose the differ-
ence among the individuals to consist only in their colours,
which, according to the supposition, are members of one
series of colours. The colours of the individuals

. . . b, c, d, e, . . .

which represent members of a series of colours, may be
joined to a form X + n, common to them all, and to the
above mentioned aggregates, so that the entire designation
VOL. i. x

THEORY OF THE SYSTEM. §. 219-

of the series of individuals, every property being taken
into consideration, becomes

II. ... P. b. (X + n) ; Q. c. (X + n); R. d. (X + n) ;
S. e. (X + n) ; . . .

The individual connecting the two series, or that which
at the same time is a member of the series I. and a mem-
ber of the series II. ; must necessarily possess the form
(X + n) and the colour a, the rest of its properties coincid-
ing exactly with those of the two series. If, for the indi-
vidual above mentioned, we designate that aggregate by N,
the individual itself will be :

N. a. (X + n).

For under these circumstances its properties, excepting
the form, agree entirely with those of series I. ; and this
form is a member of the series X, X 4- 1, &c. ; whilst in
the same manner, excepting its colour, it agrees exactly
with series II. ; its colour being a member of the series
b, c, . * .

We are led by experience to assume such relations as
those mentioned above. Suppose, for instance, A, B, C, ...
P, Q, R, ... in the above signification of the letters, to be
varieties of octahedral Fluor-haloide. Let the members
of the series of forms in I. be * the dodecahedron (D), the
octahedron (O), a digrammic tetragonal-icositetrahedron
(I), a tetraeonta-octahedron (T) ... and the colour grass-
green (gg) ; the series of individuals will be

A.gg.O; B.gg.D; C.gg.I; D.gg.T . . .

In the series II. the series of colours may be apple-
green (ag), mountain-green (mg), verdigris-green (vg), sky-
blue (sb)... and their form the hexahedron (H) ; the series
of individuals therefore

. . . P. ag. H ; Q. mg. H ; R. vg. H ; S. sb. H . . .

The above mentioned forms and colours have not only

* We may choose whatever forms of the series of crys-
tallisation, and whatever varieties of the series of colours
of octahedral Fluor-haloide ; we shall always derive the
same results.

§. 219. THEORY OF THE SYSTEM. 323

been really observed in octahedral Fluor-haloide, and ac-
knowledged as members of their respective series ; but we
know also, from §. 218., that without in the least affecting
the relation to the idea of identity, they may arbitrarily be
exchanged with each other, and that we were even entitled to
produce or suppose the members of the two series I. and II.
if we had not had any occasion of observing them in nature.
The individual, whose remaining quality is expressed by N,
becomes thus = N. gg. H ; and this likewise is either an
object of our immediate observation, or it may be produ-
ced by connecting single observations.* Thus experience
confirms to its full extent, that several series of individuals
may be connected with each other in the manner described.

An individual N. y. (X + n), can therefore be at the
same time a member of two different series, only under the
following conditions. Those of its properties which have not
been mentioned by name, and which are here expressed by
N, must agree with the properties analogous to them in the
two series ; and those which have been named (in the pre-
ceding case, forms, and colours), must be members of the
respective series, produced by the properties in the two
series of individuals. Under these circumstances, N. y.
(X + n) may be brought under the notion of identity
(§. 218.) with the members of the first, but at the same
time also with the members of the second series. From
this we draw the inference, that all the members of om of
these series may be brought under the idea of identity^ with all
the members of the other.

If we continue this process, and extend it upon all those
properties which form series by their gradations, we may in-
clude the assemblage of all those individuals, which, notwith-
standing their differences, may yet be brought under the idea
of identity. At the same time those individuals which do
not allow the process to be applied to them, are excluded

* It is evident, that if it be found necessary thus to ob-
tain a determined object, the individuals of every well de-
termined natural-historical species may serve as examples.

324< THEORY OF THE SYSTEM. §. 220.

with perfect distinctness and accuracy. An assemblage of
individuals formed in this way does not contain any thing
foreign, nor does it want any thing capable of being united
with it on account of its natural-historical properties.

§. 220. SPECIES.

An assemblage of individuals, brought under the
idea of identity by the process of §. 219., is term-
ed a Species ; and the individuals belonging to it
are homogeneous individuals.

This is the pure natural-historical and invariable idea
of the species in the Mineral Kingdom. The series of
characters do not every one of them allow of a mathemati-
cal treatment. This, however, has no influence upon their
application for producing the idea of the natural-historical
species ; and nothing is lost of the peculiar evidence of this
idea, which immediately flows from the mode of its forma-
tion (§. 218. 219.). Under these circumstances, the idea of
the species is capable of becoming a certain foundation to
the whole scientific Mineralogy ; it must likewise be the
fixed point, from which every inquiry has to start, whose
object it is to procure some knowledge of the productions
of the Mineral Kingdom, of whatever kind this knowledge
may be, if we wish to preserve a certain unity in the
acquirement of our information.

We must not pass over unnoticed any of the series, in
short none of the natural-historical properties, in pro-
ducing the idea of the natural-historical species ; be-
cause this would render the idea itself incomplete ; the
variety of nature could not be explained sufficiently and
to its full extent, nor could it be demonstrated, by a ge-
neral developement, that we are really entitled to consider
certain lodies under the required circumstances, as belonging to
one and tlie same species, although they differ in their natural-
historical properties ; and this nevertheless is the very pro-
position which was to be proved by the preceding consi-

§. 221. THEORY OF THE SYSTEM. 325

derations. It would be contrary to the principles of Na-
tural History, to determine the idea of the species ac-
cording to single properties, of whatever kind these may
be. An idea thus determined is not scientific, and cannot
be but incomplete. It will never be found sufficient in its
application, and thus open the way to the introduction of
other considerations, foreign to Natural History, in pro-
ducing the idea of the species. This has been the source
of the contamination which the science has suffered through
the introduction of heterogeneous principles, the disagree-
able consequences of which have long ago been sufficiently
conspicuous. Moreover, the determination, according to
single characters, will unavoidably introduce a distinction
among essential and accidental properties, which cannot be
allowed to take place, either in developing the idea of
the species, or in considering the identity of individuals
(§.215.).

The species itself is the proper object of classification, or
the thing which is to be classified. The idea of the spe-
cies, therefore, cannot be produced by the classification,
as some naturalists seem to believe, who begin and termi-
nate their classification, without previously having pro-
duced the idea of the species. This idea is constant every-
where, in all sciences, concerning the productions of the
Mineral Kingdom ; and it must be the foundation of every
system, whatever may be the principles followed in its
construction. The correct determination of the natural-
historical species thus appears to be of the greatest mo-
ment in the Mineral Kingdom.

§. 221. TRANSITIONS.

The progress of the gradations in the properties
of homogeneous individuals is termed a Transition
or Passage ; and we say of individuals, in which
such a progress may be demonstrated, that they pass
into each other.

326 THEOltY OF THE SYSTEM. §. 222.

The transitions arise from the series of characters. Simple
transitions take place only in one character, compound transi-
tions in more than one character at the same time. The
simple transitions are very evident, but comparatively rare.
The compound ones are more common, but they must be fol-
lowed up in every simple transition of which they consist,
if we wish to draw consequences upon which we may rely.
This is effected by supposing the differences in all the pro-
perties to disappear, except in the single one, in which the
transition is to be considered. If these differences consti-
tute members of one and the same series, there exists a
transition in this property ; if they cannot be joined in one
and the same series, we are not entitled to assume a transition.
After having thus followed up and demonstrated the simple
transitions in every one of those properties which present
differences in a number of individuals, we may consider
the compound transition, with the greatest security, as
really existing, and the individuals themselves as passing
into each other.

§. 222. HOMOGENEITY FROM THE TRANSITIONS.

Individuals, connected by transitions, are homo-
geneous, or belong to one and the same species.

A transition in a single character, for instance in the
forms of crystallisation, arises, if these forms of the indi-
viduals are members of the same series of crystallisation ^
all the remaining properties being equal. Under these cir-
cumstances the individuals are homogeneous.

The colours form a transition, if, in several individuals,
exactly agreeing in the rest of their characters, they repre-
sent members of the same series of colours. But in this
case again the individuals are homogeneous.

Hence the individuals also are homogeneous, if they are
joined by compound transitions.

It is not necessary that the members of the series of
crystallisation representing the transition, be such as im-

§, 222. THEORY OF T11E SYSTEM. 327

mediately follow each other, or produce among themselves
a coherent fragment of the series. Thus, not only R,
R + 1, R -|- 2 ... but also R ... R + n ... (P + n)m ...
R + eo will answer the idea of what is meant by a transi-
tion. It is the same with the transitions in any other se-
ries, for instance, in the shades of colour, &c. ; though in
these it must be applied with the necessary degree of cir-
cumspection. In the forms of crystallisation, this like-
wise becomes necessary in respect to limiting forms, since
these are common to some of the series, in some cases even
to all the series of the same system of crystallisation.

Transitions exist only within the species, as it evidently
follows from the preceding considerations ; hence there can
lc no transition from one species into another.

Many examples of this kind of transitions may be found,
not in nature, but in several mineralogical books. Of
these it may be maintained, that wherever the transition is
correct, the determination of the species is erroneous, and
•vice versa,) that the transition is falsely stated, if the de-
termination of the species be correct.

From the continuity of the transitions, or of the series
of characters from which they depend, we may infer, that
there is a remarkable connexion within the natural-histori-
cal species, by which all the differences occurring in its in-
dividuals may be joined into a whole. Thus we become
capable of comprehending the variety of inorganic nature.
For the very same reason also, it is contrary to the real
interest of Mineralogy to divide or subdivide the species,
and to distinguish sub-species and kinds. The purpose of
such divisions is to facilitate the general survey of the
species ; but this indeed would rather be assisted by establish-
ing the connexion between its individuals, if this should
happen to be still wanting, than by such divisions, which
render it less evident.

With those divisions into sub-species and kinds, which
have hitherto been in use, it has very often been the case,
that an individual was really found to belong to the species,
and yet to none of its sub-species. This has been the con-

328 THEORY OF THE SYSTEM. §. 222.

sequence of the divisions not having been effected in a
single series of characters, but in several series at once.

Let the forms of several individuals in a species be re-
presented by R, R + 1 ... ; their colours by a, a' ... ; the
rest of those characters which produce series by p, p' ... ;
the individuals themselves will be represented by the fol-
lowing aggregates :

R; R + 1; R + 2; R + 3 ...

a a' a" a'" ...

p p' p" p"' ...

If we divide here only in the series of the forms, so as
to ascribe the members R and R + 1 to the first division
or sub-species, R + 2 and R + 3 to the second, the in-
dividuals uniting the rest of the characters will fall either
in the first or in the second of these divisions. But
if the division at the same time extends to the co-
lours, and determines, that besides the mentioned forms,
the first sub-species should be of the colours a and a' ;
the second, besides its peculiar forms, of the colours a"
and a'" ; all the compositions like (R + 1). a", (R + 2). a'
will not belong to any one of those divisions, although
they occur as generally in the species as any one of
those contained in the two sub-species. Many divisions
are found in mineralogical works, of the description
above mentioned. These divisions within the species,
in whatever character they may be effected, must always
remain entirely arbitrary, and on this account they never
will be unanimously received. ' It is much more useful to
suppress them altogether, which is the plan adopted in the
course of this work.*

* The application of Mineralogy to the objects of every
day's occurrence, may require to give particular attention
to certain varieties, which have been employed in the arts,
and provided with particular names. This will be proper-
ly attended to in the second volume of this work ; but not
being of any scientific value, it is foreign to our present
consideration.

Ur

§. 223. THEORY OF THE SYSTEM. 329

§. 223. PRINCIPLE OF CLASSIFICATION.

The principle of classification in Natural History
is the Natural-Historical Resemblance.

Several bodies are similar, or resemble each other, which
approximate more or less in their properties ; and this re-
semblance is the greater, the higher we find the degree of
approximation.

In Geometry, similarity consists in the equality of the
relations among homogeneous quantities, and allows of
no variation. The idea of similarity in Natural History
is not so simple ; it cannot be expressed by a single ratio,
because here a great many properties must be taken into
consideration. It receives a certain latitude, in which
there may occur some variation. This, however, has no
prejudicial influence, either upon its evidence, or upon
its applicability. On the contrary, if we apply this idea
of similarity to nature, we find that only owing to the greater
extent ascribed to it in a natural-historical consideration,
it is capable of being the principle of classification in Natural
History,

It is not difficult to decide the question, whether or not
the natural-historical resemblance should be fixed upon as
the principle of classification in Natural History. In every
science the classification must rest upon such relations as are
objects of the science, and therefore it must represent nature
according to the image expressed by these relations. Natu-
ral History refers to none but the natural-historical proper-
ties ; hence the approximation of heterogeneous bodies in
these properties, or the natural-historical resemblance, is the
only relation expressed among or by means of the produc-
tions of nature. For this reason Natural History is forced,
not only to apply this resemblance as its principle of classifi-
cation, but also carefully to explain and illustrate it, in order
to render the classification a true and sufficient representa-
tion of nature. In this respect a classification may be called
natural. The organic kingdoms of nature have always had

330 THEORY OF THE SYSTEM. §. 22

the advantage of similar classifications, and those depart-
ments of Natural History which refer to them, have long
ago made a scientific progress. In the Mineral Kingdom,
unfortunately, a different way has been proceeded upon ;
and as a science, the Natural History of the Mineral King-
dom has not been promoted.

In the Natural History of the Mineral Kingdom, the
classifiable objects are not the individuals of this kingdom ;
but according to the preceding inquiries, the natural-histo-
rical species (§. 220.). The natural-historical resemblance
has therefore nothing to do with individuals. These, in-
deed, notwithstanding their homogeneity, bear in many
instances so little resemblance to each other, that accord-
ing to the principle of similarity, they rather should be
divided than joined. It becomes necessary indeed to demon-
strate their connexion by means of the transitions, that
is to say, by considering their series of characters, in order
to convince ourselves of their homogeneity. Hence the
two ideas of Resemblance and of Homogeneity are essentially
different, and the former is not a higher degree, or a nearer
restriction of the latter. We must carefully avoid con-
founding them with each other. In another science, refer-
ring to the Mineral Kingdom, another principle of classi-
fication may replace the natural-historical resemblance ;
but there is not another idea which can be substituted for
that of homogeneity.

§. 224. DEGREES OF NATURAL-HISTORICAL RESEM-
BLANCE.

The degrees of natural-historical resemblance
in different species, are not everywhere the same.

If we consider the species as unities to be classified, and
compare them with each other in respect to their natural-
historical properties ; we perceive that some of them are
more, some of them less allied to each other in resemblance.
Thus hexahedral Iron-pyrites is more similar to prismatic

§. 225. THEORY OF THE SYSTEM. 331

Iron-pyrites, than to rhombohedral Lime-haloide. The
latter species again is more similar to the rest of the Lime-
haloides, than to prismatic Feld-spar, or to rhombohedral
Corundum, &c.

This relation of similarity is not unavoidably necessary.
One of the species, for instance, might present the same de-
gree of resemblance to every one of the others, and vice
versa ,• so that among these species there could not be per-
ceived even the consecutive order of series. As there really
exist several species, among which this is the case, the same
might occur among all. This would not limit the variety
of nature, although that variety would then appear under
a different form.

On the other hand, the relation of similarity might be
found different in every particular case; so that if we
would suppose a certain species to bear a certain degree of
resemblance to another, a second one could not be found,
among which and the first, the same degree of resem-
blance would prevail. The consequence of this would be
the impossibility of producing any other arrangement among
the species than that of a series, in which they would fol-
low each other according to their different degrees of re-
semblance.

A single glance at the species of the Mineral Kingdom,
and at the above examples, will suffice to shew, that nei-
ther of these two suppositions takes place in nature. There
exist different degrees of resemblance, by which a series
may be produced, but not a series of the single species.

The different degrees of resemblance'lay the foundation
for the higher ideas of the Theory of the System ; that
is to say, for the ideas of classification.

§. 225. GENUS.

An assemblage of species, connected by the high-
est degree of natural-historical resemblance, is
termed a Genus.

m THEORY OF THE SYSTEM. §. 225.

The genus is the resemblance of different species of bo-
dies. In Botany, this idea is commonly limited to the
similar formation of the organs of fructification. In Mi-
neralogy, such a restriction is impossible, because the pro-
ductions of the Mineral Kingdom do not present any parts,
different in the same manner from the rest, as is the case
in plants. But even supposing their existence, Minera-
logy will not admit of any such restriction, on account of
the necessity to preserve the idea of the species in its
original generality, which admits of no exceptions, devia-
tions, or ambiguities, necessarily connected with a restric-
tion to single characters. For the rest, the idea of the
species in Mineralogy is identical with that in Botany ;
and there is nothing more required, but to shew that it is
equally applicable. Thus we shall find ourselves enabled to
employ it with the same security in the Mineral Kingdom,
as in the Vegetable one.

The species of hexahedral Iron-pyrites agrees so very
closely with that of prismatic Iron-pyrites in every charac-
ter, except the forms-, that but for this difference in their
systems of crystallisation, they would join into one and the
same species. They possess that degree of resemblance
which requires their union into the same genus ; a de-
gree of similarity expressed in the present instance by the
perfect agreement of all the natural-historical properties,
except the crystalline forms. The same degree of resem-
blance prevails among the two species of Emerald ; but
here, beside the difference in the systems of crystallisation,
there is also a difference in the specific gravity. In other
genera, as, for instance, in those of Garnet, of Kouphone-
spar, and others, we observe differences in many characters
at once ; and yet the resemblance is here as great as in the
above mentioned examples of Iron-pyrites and Emerald ;
a resemblance which becomes evident upon ocular in-
spection, the only method of ascertaining its existence.
The examples quoted prove, that there may exist differ-
ences sometimes only in a few, sometimes in many charac-
ters at a time, without having any influence upon the dc-

§. 225. THEORY OF THE SYSTEM. 333

gree of resemblance itself. On this account it becomes im-
possible to express this resemblance by the agreement in
one or a certain number of characters. This does not,
however, prevent the application of the idea of the ge-
nus to the natural productions altogether ; for this appli-
cation does not pre-suppose the idea to be limited to
single characters; but it allows, and even requires, to
preserve it in its full generality. The genera being thus
founded upon the resemblance of the species, as is already
demonstrated in Zoology and Botany, render all the ser.
vices which Natural History possibly may expect ; and the
same will be likewise confirmed by their introduction in fu-
ture in Mineralogy.

The natural-historical idea of the genus is peculiar to
Natural History, and is solely intended for the purpose of
promoting that science. Hence the natural-historical ge-
nera must not be compared with ideas of this kind other-
wise determined ; not even with those of the same deno-
mination, Avhich have been hitherto applied in Minera-
logy. For, the establishment of these genera is founded in
part upon determinative reasons, foreign to Natural His-
tory ; so that the inferences drawn from them must be in-
consistent with the principles of that science, even though
it were demonstrated that in every case they correspond
to the natural-historical resemblance. Other sciences must
consider the natural-historical genus in the same point of
view. In a chemical system of minerals, the genus must
have a chemical foundation. It is not necessary that it
should agree with the genus in Natural History ; though
the species, determined according to chemical ideas in the
one, and according to natural-historical ideas in the other
science, must be identical in both (§. 220.). The method
of connecting several points of view into a single one, does
not promote the sciences, and gives full scope to all sorts
of hypotheses. Different sciences, which refer to the same
subject, must follow without deviation the course deter-
mined by their peculiar principles, or they will cease to be
different sciences. Their way will finally converge in one

THEOltY OF THE SYSTEM. §.

great aim, which is the discovery of truth, perhaps hidden
for ever, if we endeavour to arrive at it according to any
other method.

§. 226. MINERAL KINGDOM.

The Mineral Kingdom is represented by a Se-
ries of Natural Historical Genera.

The object of mineralogy in producing its higher degrees
of classification, is first, to acquire a clear idea of the Mineral
Kingdom, which consists in a general survey of all its pro-
ductions ; and secondly, to become capable of collecting
every one of these productions with facility and security,
under the above mentioned ideas.

If we examine the systems of Natural History, we find
that for the most part they are founded upon the idea of
a series. Yet it is difficult to decide, whether in the mi-
neral systems this is a series of genera or of species : for-
in these systems both genus and species are founded upon
reasons so uncertain, that it becomes utterly impossible to
derive any clear idea from the existence of their determi-
nations. A series of species is in direct opposition with that
of a series of genera, in fact to the idea of the genus itself, in
as much as it supposes that there exist no equal degrees of
resemblance among different species. In order to convince
ourselves in this respect, the best contrivance will be, to
produce a series of species in nature, in which those placed
nearest must resemble each other most, and where we may
begin at any chosen member. Evidently for this process a
single variety cannot represent the species, but we must
employ the whole species as completely known as pos-
sible. In endeavouring to produce a series of this kind,
very soon we shall meet with species which render it
doubtful whether the one or the other, or even a third,
a fourth, &c. should follow the preceding species ; and
at last we must either entirely abandon the experiment
or we must suppose that two, three, or more spe-

§. 226. THEORY OF THE SYSTEM. 335

eies occupy the same place in the series. The Croupes
of species thus formed are the natural-historical genera
(§. 225.). In the real existence of genera in the Mineral
Kingdom, we discover the reason why there is no series of
single species. A more evident demonstration of the pre-
ceding observations may be acquired by considering a few
examples, for instance, the genera Schiller-spar, Disthene-
spar, Triphane-spar, Dystome-spar, Kouphone-spar, Peta-
line-spar, Feld-spar, Augite-spar, &c., or the genera of the
order Bary te, or of any other somewhat more comprehen-
sive order of the natural-historical system of Mineralogy.
The series thus representing the Mineral Kingdom is a se-
ries of genera, exactly as in the Animal and Vegetable
Kingdoms ; and like these it does not contain a series of
single species.

The Mineral Kingdom is constituted by a series of natu-
ral-historical genera, each of which contains similar species
(if it contain more than one); every one of these, again,
being the assemblage of homogeneous individuals. Thus
the idea of the Mineral Kingdom receives its fullest evi-
dence, and requires in this respect no other notions inter-
mediate between that of the genus and that of the Mineral
Kingdom.

The idea of a series requires a beginning and a termi-
nal point. There can be no objection to the received or-
der of the three kingdoms, in which the Animal King-
dom is followed by the Vegetable one, and this again by
the Mineral Kingdom ; and it deserves, in fact, that gene-
ral reception which it has always found. Those produc-
tions of the Mineral Kingdom, which resemble most some
of the Vegetable Kingdom, immediately will follow these,
and form one of the terminal points in the series of genera
which constitutes the Mineral Kingdom. Thus, always
according to the principle of similarity, we obtain the whole
series of genera, as it is contained in the natural-historical
system of the present work.

It will be useful to add here a few remarks on the recep-
tion of what has been called the Atmosphcrilia into the Mine-

33G THEORY OF THE SYSTEM. §. 226.

ral Kingdom. This evidently depends upon the very idea or
definition of a mineral. If we examine this idea as contain-
ed in most of the mineralogical works, we find several cha-
racters, which are not natural-historical ones, and which
therefore by no means can be an object of any natural-histo-
rical inquiry. If we omit all these as being foreign to the
science, the only character still remaining is that of an Inor-
ganic Natural Production ; so that the mineral kingdom con-
tains the inorganic natural productions altogether, and
among these consequently also the atmospherilia.

Hence we cannot exclude Water, the different kinds
of Gas, of Acid, and of other productions of inorganic na-
ture, from the mineral system, because upon this sup-
position it would be impossible to define the idea of a mi-
neral, in conformity with the requisites of Natural History,
and containing therefore solely Natural-Historical Characters.
Moreover, it is impossible to say upon what should be
founded the difference between the Mineral Kingdom and
that of the Atmospherilia ; nor would the idea of Natu-
ral History itself allow to follow the example of most of
the naturalists, to pass with silence over the latter.

With the assistance of all the ideas developed till now,
it cannot be difficult to decide, whether or not a natural
production belongs to the Mineral Kingdom. It will be
attended with as little difficulty to determine the species of
a mineral, if we know its genus. It is more difficult, from
the idea of the Mineral Kingdom to reach that of the
genus, that is to say, to determine the Genus of a Mineral ,•
and therefore some preparations are wanted, consisting in
several intermediate ideas, by which it becomes more
easy to descend, as it were, from the highest to the low-
est of these ideas. On account of their employment,
they have been called the Degrees of Classification. These
intermediate ideas must be founded upon the general
principle of classification in Natural History, exactly like
the idea of the genus itself. Their purpose is not to
illustrate, but only to render more easy the application of
the general ideas of the genus and of the species in the

§. &27. THEORY OF THE SYSTEM. 337

Mineral Kingdom. These are the ideas of the Order
and of the Class.

§. 227. ORDER.

The Order is an assemblage of similar genera.

The order is in respect to the genera, what the genus is
in respect to the species. The idea of the order is there-
fore perfectly evident from the preceding inquiries ; and the
only object that requires some consideration in the present
place, is to shew its application to the Mineral Kingdom.

The genus Iron-pyrites, in the peculiar place it occupies
in the general series of genera, is surrounded by several
other genera, which exhibit so high a degree of resemblance
to each other, that they seem to have been formed after a
common type or original. These are, the genera Nickel-
pyrites, Cobalt-pyrites, Arsenical-pyrites, and Copper-py-
rites. There is not another genus to be found in the whole
Mineral Kingdom, as hitherto known, which could be
enumerated along with them, without destroying the
idea produced by the assemblage of the above-mentioned
genera. In a similar manner the genus Iron-ore is con-
nected on one side with the genus Manganese-ore, on the
other side with the genera Chrome-ore, Cerium-ore, Urani-
um-ore, Tantalum-ore, Copper-ore, Scheelium-ore, Tin-ore,
Zinc-ore, and Titanium-ore. Thus likewise round the ge-
nus Feld-spar are assembled the other genera of Spars un-
der similar circumstances. Every groupe of this kind
which is an assemblage of genera similar to each other, is
an order.

These orders are as distinct from one another, and as re-
markable as the genera in the Mineral Kingdom, but they
require also, like these, to be observed in nature. They re-
present in the Mineral Kingdom the natural families of the
Vegetable Kingdom ; and their reception afld determina-
tion in the one and the other, depends upon the same prin-
ciple.

The orders, like the genera, must not be compared with

VOL. I. Y

338 THEORY OF THE SYSTEM. §. 228.

what formerly has been designated by this name in Mine-
ralogy. The chemical principles of classification, received
in these systems, more particularly exercise their influence
in their higher divisions. In general, Mineralogy must
not be compared with any science except Zoology and Bo-
tany, and it has nothing to fear, if it is found to admit of
this comparison. For this will demonstrate that it has ap-
plied the general principles of Natural History in conformity
•with itself to its object, which is its peculiar, but at the same
time also its only business.

§. 228. CLASS.
The Class is an assemblage of similar orders.

What the genus is to the species, or the order to the
genera, the class is in respect to the orders. Genenim ge-
nus est Ordo, ordinum autem genus Classis est. LINN. Phil.
Bot. 204. The idea of the class is so comprehensive, that
it becomes difficult to judge of its applicability, without
the direct inspection of the objects themselves. This in-
spection proves, that every one of the three classes of
the natural-historical system in Mineralogy does contain
orders which are connected by a greater degree of simi-
larity, with each other, than with those of other classes.

The idea of the class likewise depends solely upon natu-
ral-historical considerations, and does not admit of any fo-
reign principle. It is analogous to the classes in the orga-
nic kingdoms, in as far as these are not artificial, or pro-
duced by a mere division. These ideas, however, do not
allow of any comparison with the classes hitherto used in
Mineralogy; which, not being framed according to the
principles of Natural History, are partly founded upon
hypotheses, and partly on principles which the present
state of Chemistry no longer admits.

The systematical ideas of the species, of the genus, the
order, and the class, are all that is necessary for producing
what has been called a System of Nature ; and it appears
that they possess the same requisites in all the three king-

THEORY OF THE SYSTEM. 339

doms of nature. To form the classes into Kingdoms, and
to comprehend these within the still higher idea of Nature,
is a problem belonging to Natural History in general, from
which the Natural History of every particular kingdom
must borrow these higher ideas.

In order to give a general view of the subject, it will be
useful shortly to repeat the whole process of developing
these ideas, and the systematical unities themselves.

All material bodies are distinguished, according to their
most general differences, into organic and inorganic natu-
ral productions (§. 70-

Organic nature comprehends two kingdoms, the Animal
and the Vegetable (§. 8.) : inorganic nature comprehends
only one, the Mineral Kingdom (§. 9.).

The Mineral Kingdom is a series of natural-historical
genera, the succession of which is determined according to
their greater or less agreement or similarity (§. 226.). It
contains three classes.

Every class comprehends part of the series of genera col-
lected into several orders. The classes are not of the same
extent ; and the orders which they contain are joined by
an equal degree of similarity (§. 228.).

Every order is an assemblage of several genera in their
regular succession ; hence it is likewise a portion of the
general series of genera. The genera comprised within
an order, present equal degrees of similarity (§. 227.).

Every genus is an assemblage of similar species ; it is a
unity in the series of genera. The species within the ge-
nera are connected by equal degrees of similarity (§. 225.).

Every species is an assemblage of homogeneous indi-
viduals ; the individuals of a species are connected by the
series of characters, that is to say, by real natural-historical
transitions (§. 221. 222.).

The individual is the simple mineral, produced by na-
ture, either singly (§. 160.), or in various compositions
(§. 178... 189.). It is the only systematic idea which imme-
diately refers to nature, or to which an object of observation

340 THEORY OF THE SYSTEM. §. 229.

corresponds. In respect to the whole of the species, the in-
dividual is called a variety.

It must here be observed, that none of these ideas have
been obtained, or deduced from the others by means of
a division. For, in order to arrive at them, we have not
begun with the highest, but with the lowest one, which
is that of the individual, and then we have first deter-
mined the idea of the species according to the idea of ho-
mogeneity, those of the genus, the order, &c. according to
different degrees of natural-historical resemblance ; and
the whole of them by aggregation or assemblage. Be-
sides the idea of the species, a division would have pre-
supposed also that of the Mineral Kingdom, and it would
have required a principle, according to which it might
have been effected with consistency. In another place
it will appear, that these conditions in fact may be ful-
filled; yet, by such a division, it would not have been
possible to obtain the same classes, orders, and genera,
which have been obtained by the other process ; and the
degrees of classification thus obtained, would not have
been suitable to the purpose of giving a general view of
inorganic nature, in agreement with the similarity which
exists among its productions ; and this nevertheless is the
last and highest aim of Natural History. Mcthodus Na-
turatts ultimus finis Botanices est et erit. LINX. Phil. Bot. 206.

§. 229. MINERAL SYSTEM.

The Mineral System is the collection of the na-
tural-historical ideas, conformably to the degree of
their generality, and applied to the productions of
the Mineral Kingdom.

The mineral system is a representation of the Mineral
Kingdom by means of general ideas. These ideas must be
clear, precise, complete, and correctly subordinate to one
another, so as to become consistent with each other, and
applicable to experience.

§. 229. THEORY OF THE SYSTEM. 341

The nature of these ideas evidently depends upon the
preceding observations. If, therefore, in general, and
from reasons of Natural History, no objection can be made
to the ideas of the species, the genus, the order, or the
class ; the applicability of the mineral system, arising
from the connexion of these ideas, will depend solely
upon their consistent application to the present state of
knowledge.

To apply the ideas of the species, the genus, &c. to
experience, is to collect the individuals occurring in na-
ture, agreeably to their homogeneity, into species ; to join
these species into genera, according to the highest de-
gree of natural-historical resemblance which occurs among
them, &c. ; or, in other words, accurately to determine
the more particular contents of the system. This sup-
poses an accurate examination and a careful comparison
of the individuals among each other ; and in that respect
the only mode in which the mineral system may arrive
at its perfection, is by a thorough knowledge of the ob-
jects themselves. This knowledge, however, being an em-
pirical one, must always remain incomplete, and is really
so at present in a great measure : a perfect mineral sys-
tem, therefore, is an object, which, though we may ap-
proach, we never can reach.

We must proceed with some caution in thus determin-
ing the contents of the mineral system : a few rules in this
respect will not be superfluous. The first of these rules
requires, that we attach the highest importance to the
correct determination of the species, because this is the
foundation of all the other ideas, and therefore of the sys-
tem itself. According to another of these rules, we must
endeavour to unite newly discovered individuals, along
with other species already determined, till we may con-
vince ourselves by an accurate examination, that this is no
longer possible ; because it is a general rule in Natural
History not to multiply the number of the species, without
an evident necessity. In fact, one of the greatest errors
of many mineral systems consists in their containing too

342 THEORY OF THE SYSTEM. §. 229.

many species, which on that account cannot be well de-
termined, and which, besides other bad consequences, only
serve to render the nomenclature difficult and inapplicable.
A third rule recommends us not to enter newly determin-
ed species into the system with too much precipitation, but
to follow the example of other cautious naturalists, and to
expect some more extended knowledge from future obser-
vations, in order not to hazard precipitous determinations.
Indeed it seems better that something, though it were
known for some time, should be wanting in the system ;
than to allow an ill determined species to injure the con-
nexion of the whole. For the rest, mere hypotheses or the
results of other sciences, should never be relied on in these
determinations, because it is below the dignity of a science,
which admits of the application of mathematics, to build
its frame upon hypotheses. Natural History too posses-
ses so many means of assistance peculiar to itself, that if
well applied (which has not always been the case in
Mineralogy) it may rest solely upon its own determi-
nations.

A system thus produced is what has been termed the
Natural System, because it expresses the different degrees
of natural-historical resemblance with which nature itself
has stamped its productions. This, however, is the only
part nature takes in a natural system, and it is therefore
not the System of Nature. This idea is merely an imagi-
nary one, and no object corresponds to it. Nature pro-
duces only different bodies, but no abstract ideas ; and the
system of nature, mentioned by several natural philoso-
phers, are only words without ideas, or ideas without ob-
ject. Opposed to the natural system, there are also Arti-
ficial Systems. The natural system produces its general
ideas by the process of aggregation, while in the artificial
ones, the assemblages depend upon general ideas, in as
much as they are obtained by the process of division.

The expressions of natural and artiftcutl systems, though
generally received, do not convey the exact idea of what
they have to express. They have on that account been the

§. 229. THEOKY OF THE SYSTEM. 343

source of many differences of opinion, according to the cor-
rect or erroneous ideas attached to them by naturalists.
For more than one reason, it would therefore be advisable
to substitute in their stead, the expressive denominations
of syntJictical and analytical systems.

Doubts have been raised against the possibility of arti-
ficial systems in Mineralogy. The experiments indeed
which had formerly been instituted, seemed only to de-
monstrate that these doubts were well founded. The sys-
tems of a mixed or double principle do not even belong to
these, for they are neither analytical nor synthetical, be-
cause they do not possess the unity of principle required
in every system.

The production of any system requires the previous de-
termination of the species. The species therefore must
have already been established, if we intend to build also an
artificial or analytic system. Upon this supposition, any
analytic system in the Mineral Kingdom has to resolve one
single problem of importance, which is to effect the first
division, without impairing the unity, or destroying the con-
nexion within the species. There is no great difficulty in
this respect in effecting the subsequent lower divisions.

If therefore we fix upon the systems of crystallisation
as a principle of the first division ; we obtain the base of
an analytic system of Mineralogy, in which the species
remains entire, and in which the farther subdivisions may
be effected without difficulty.

The natural-historical similarity is entirely lost in ana-
lytic systems. Although systems of this kind, if well
managed, greatly facilitate the determination of individu-
als occurring in nature, yet they possess so very little of
the other requisites of a system, that they do not at pre-
sent deserve to be considered in greater detail. Analytic
systems may be compared to registers, in which the objects
follow each other according to certain single properties, like
words in a dictionary, without regard to signification;
whereas in the synthetic system, the succession of objects is
determined by their natural-historical resemblance, no at-
tention being given to single properties.

344 THEORY OF THE SYSTEM. §.

I shall not at present venture to examine, whether a na-
tural-historical system, partly analytical and partly synthe-
tical, like the Linnean system in Botany, might not unite
the advantages of both, whilst it avoids their inconveni-
ences. A system of this kind would be less objectionable
indeed, and more useful than those founded at the same
time upon two different principles ; but it could not unite
those properties which the synthetical system presents, if
it only approaches perfection to a certain degree.

The double purpose for which the systematic ideas
(§. 22C.) have been produced, likewise will require our at-
tention in constructing a mineral system. The first de-
mands a general view of the variety of nature, collected
within different unities ; the other requires a method of
recognising individuals occurring in nature, or of assigning
to them their peculiar place in the system, and of provid-
ing them with the names and denominations connected
with these places.

It seems, that attention has been paid only to the
first of these purposes in several of the systems hitherto
constructed. The natural-historical system of the present
work perfectly answers this requisite. It represents na-
ture according to the different degrees of resemblance
which, notwithstanding all their variety, exist among its
productions. Several mineral systems do not represent the
relations among the bodies or natural productions them-
selves ; but among the results of their chemical analyses,
which are not objects of Natural History.

A mineral system may be found to suffice as to the first
requisite, without answering the second ; it may facilitate
the general survey of the productions of the Mineral King-
dom, without assisting in the determination of individuals.
On the contrary, it may answer to the second, without ful-
filling the first ; it may assist in the determination, with-
out producing a general view ; and this is the case in ana-
lytic systems.

If we examine the systems hitherto published in respect
to the second point, we find ourselves completely dissatis-
fied. An unknown plant may be determined by the help

§. 229. THEORY OF THE SYSTEM. 845

of the Linnean system; but there exists no system,
by the assistance of which we might determine an un-
known mineral. In order to find the class, we must ana-
lyse the mineral ; and by this analysis, the mineral is de-
stroyed : nothing remains to determine the order, the ge-
nus, and the species. The difficulty is not removed by as-
serting that only a small portion of the whole is to be taken
for the examination ; because here, our object is not to de-
termine a mineral in a particular case, but to fix the gene-
ral and scientific method of determination. Thus, in the
systems noticed, we must immediately proceed to the
species, for there is no methodical way to arrive at it
through the intermediate degrees of the class, the order,
and the genus. In short, we must content ourselves with
acquiring an empirical knowledge of minerals, how small
soever the scientific value of this knowledge may be, be-
cause indeed it is better to know the natural productions in
this way, than not to know them at all. It is evident that
a description can be of no use for this purpose ; and thus it
appears, that systems of the kind mentioned above do not
possess either of the two necessary requisites. How far
the determination of occurring individuals is facilitated by
the natural-historical system, will be examined in another
part of the present Treatise.

PART III.
NOMENCLATURE.

§. 230. DEFINITION.

The Systematic Nomenclature is the assemblage
of those denominations which Natural History ap-
plies to natural productions, and which refer to a
natural-historical system.

There is only one mode in which Natural History may
provide the productions of nature with denominations ; but
this mode is in the closest connexion with the whole being
of the science. The whole object of Natural History in
this respect, is to resolve that single problem : to com-
prehend the unities of observation within certain ideas,
which may be formed either by assemblage, or by division
(§. 229.). We may say that we know a natural production,
if we are capable to tell to which of these assemblages or
divisions it belongs ; but we may say the same thing, if
we know its denomination. This- denomination must there-
fore be intimately connected with the above-mentioned
ideas ; it must express the relation, in which that natural
production, to which it is applied, stands to others, with
whom it agrees more or less in respect to their natural-his-
torical resemblance.

According to the preceding observations, the natural or
synthetical systems represent this connexion. That only
mode of nomenclature, therefore, which expresses this con-
nexion, will deservedly be called the systematic nomen-
clature.

The systematic nomenclature alone is capable of fulfilling
those conditions which Natural History requires from no-
menclature in general. For it provides every natural pro-

§. 231. NOMENCLATURE. 347

duction with a denomination, and represents, by these de-
nominations, the natural-historical resemblance by which
these bodies are connected in the system.

In every part of Natural History, nomenclature is the
mirror, which reflects an image of the whole science. The
image hitherto produced by Mineralogy, has not been an
agreeable one, or calculated to invite the zoologist or bota-
nist, or any enlightened naturalist, to bestow particular at-
tention on Mineralogy. A mass of names and denominations,
formed arbitrarily or accidentally, and subject to perpetual
change, retard the solid progress of the science, and are
a great impediment to the acquisition of knowledge in its
purity. The want of a well constructed systematic nomen-
clature, is therefore an essential defect in the Natural His-
tory of the Mineral Kingdom ; and the present attempt to
remove it, how imperfect soever it may be, is founded upon
the very idea of Natural History (§. 18.), which cannot
exist without it.

§. 231. OBJECT OF THE SYSTEMATIC DENOMINA-
TION.

The object to which the systematic denomination
must refer, is to express the correctly determined
natural-historical species.

The natural-historical species is the foundation^ the syste-
matic nomenclature the verbal expression of the system.
The species, therefore, is the object to which the systema-
tic denomination refers.

Nomenclature requires that the species be previously
correctly determined, according to the principles of Natu-
ral History. For the necessary connexion among several
of these unities, which is to be represented by a verbal ex-
pression, cannot take place upon any other supposition ;
and the systematic nomenclature is degraded into a mere
jumble of words, to which no object corresponds. In this
state, it cannot any longer be useful to Natural History ;

348 NOMENCLATURE. §. 231.

but it really becomes an impediment, to remove which, the
trivial nomenclature (§. 241.), has been resorted to.

In most of the mineral systems, the species has as yet
not been correctly determined. In these systems, therefore,
neither a name nor a denomination appertains to a well determin-
ed species.

Such systems contain names and denominations of single
varieties of a species ; as, for instance, Rock-crystal,
Flint, Chrysoprase, &c. ; Common Quartz, Conchoidal
Hornstone, &c. ; but neither a name nor a denomination
exists for the whole species of rhombohedral Quartz, to
which all these varieties belong.

If it be necessary to denominate a newly discovered spe-
cies, it is still more necessary to provide a species with a
new denomination, which has been corrected, because none
of the old ones will express it, and because it is impos-
sible to apply all of them at once. The same mode of rea-
soning must be applied, if a species has not been cor-
rectly determined, and contains varieties of several natural-
historical species.

In a scientific treatment of Mineralogy, this becomes
again a new and urging reason for altering the nomencla-
ture ; and it could not be forgiven, if, under these circum-
stances, we should not at the same time endeavour to give
the new nomenclature a systematical arrangement.

The object of nomenclature in general, and more particu-
larly that of the systematic nomenclature, is to express ly
words, or to denominate those things or bodies (the species),
of which other sciences, Natural Philosophy, Chemistry,
Geology, &c., afford more detailed and particular informa-
tion. This is effected by substituting names and denomi-
nations, instead of the characters and the general descrip-
tions of the species. These names and denominations, there-
fore, must possess such properties as will enable us to find
them out, or to recognise them, whenever the characters
or natural-historical properties of a natural production are
given. This is effected by means of the Characteristic, to
be explained hereafter. If they be meant to excite, or to

§. 232. NOMENCLATURE. 34)9

produce, an image of the natural-historical quality of the
objects to which they refer, or to remind us of those which
are more or less similar to them, they must indicate the
place which the species occupy in the general assemblage
of the natural productions belonging to the kingdom.

This is the point of view from which we must consider
and develope the arrangement of the systematic nomen-
clature in general, and the properties of the systematic de-
nominations in particular.

§. 232. PROPERTIES OF THE SYSTEMATIC DENOMI-
NATION.

The systematic denomination must be composed
of several words, the order of which expresses the
connexion between the denominated object and
several others, to which it is more x>r less similar.

In order to recognise a given individual, or to determine
the place which it occupies in the system, it is necessary
to proceed with it through all the general ideas of this sys-
tem, from the highest degree to the lowest one. For this
is the means by which we learn the connexion in which it
is with others.

If we have to express this connexion by words, we must
construct the denomination in such a manner that it may
tell the unities of all the above-mentioned ideas, in as far
as it is necessary (§. 234.) ; it must therefore consist of se-
veral words ; and since the ideas, according to their con-
tents, are subordinate to each other (§. 228.), these words
must follow each other according to the same order. That
word which expresses the highest idea must precede, and
that which expresses the lowest idea must follow, in the or-
der of these expressions. This distribution of words should
agree with the general spirit of the language ; and the Latin
language, therefore, would be preferable in the mineralo-
gical nomenclature, as well as in the zoological, or in the
botanical one. This language, however, has been so very

350 XOMFA'CLATURE. §. 232.

little employed in Mineralogy, particularly of late years,
when the science itself has been more particularly cultivat-
ed and enriched, that it would be very difficult to produce
a Latin systematic nomenclature, without introducing al-
most endless innovations. This has been the reason why,
in the present first attempt at constructing a systematic no-
menclature, the German language has been made use of, to
the spirit of which the English language in this respect
exactly corresponds. In these two languages, however,
the successive order of the words is exactly contrary to
that required in the Latin language. In the English no-
menclature, therefore, the highest idea will be expressed
by the last word, while the lowest idea is indicated in the
first word.

That word, with which we designate a single object, or a
single species, without regard to its genus, or a single genus
independently from its connexion with others in one and
the same order, &c., is termed its Name. If a name be re-
stricted by means of an adjective, it is transformed into a
Denomination. A name consisting of a single word, is a
simple name ; one consisting of two words, is a compound
name. Simple names can never express the connexion of
those bodies to which they are given ; this may, however,
be effected by compound names, or by denominations.
Hence the mere simple names are of no use in a systema-
tic nomenclature ; but it will require compound names or
denominations, or even both of them. The simple name
designates the highest idea occurring in the nomenclature,
and inversely, this idea must always be expressed by a
simple name. The compound name designates a lower de-
gree, the denomination the lowest degree of the ideas ex-
pressed by the systematic nomenclature. Agreeably to the
observation in §. 234., it may be considered as a rule of
the systematic nomenclature in general, that compound
names, in the event of their being applied, should never
contain more than two words, and that a denomination
should never admit of more than one adjective.

§, 233. NOMENCLATURE. 351

§. £33. OBJECT OF THE NAMES.

The ideas expressed by the names are the higher
unities of classification, immediately preceding that
of the species.

In order to enable us to add the proper restrictions in
the denomination of the species, we must apply the name
to the genus, or to the order, in general to one of the high-
er unities of classification. The name, therefore, is not
fixed upon a single natural production, or upon an indivi-
dual, not even upon a single species ; but it is applied to
an assemblage of a greater extent, and is allowed to be
transferred upon the species or upon the individual, only
in so far as the one and the other of these belong to the
above-mentioned more extensive assemblage, in virtue of
their natural-historical properties. This is a peculiar pro-
perty of every systematic nomenclature, and it is a charac-
ter by which in particular it differs from the trivial nomen-
clature. In the latter, the species, or in general the low-
est systematic idea bears the name, without any regard
to the genus, or to the order in which it is contained.
The trivial nomenclature therefore bestows its names upon
the objects themselves, without indicating the connexion
Avhich exists between them and other bodies, in one and
the same system. This definition of the trivial nomencla-
ture explains its farther arrangement, as shewn in §.241.

The trivial nomenclature allows of an unlimited arbi-
trariness in providing the species with names ; which is
limited in a systematic nomenclature. This alone would
suffice to shew the usefulness of its general reception, even
though it were not recommendable on account of its other
qualities. A newly discovered mineral which does not be-
long to any known species, may perhaps belong to a known
genus, and assume its name ; or it may belong at least to
a known order, and as such receive the name of that or-
der. And even if neither of these be the case, the syste-

352 NOMENCLATURE, §. 234.

matic nomenclature contains the rules, after which it may
be provided with a correct denomination.

§. 234. NAME OF THE ORDER.

In the Natural History of the Mineral King-
dom, the Order is the highest idea expressed in the
systematic nomenclature. The order consequently
will bear the simple name.

It depends entirely upon the kind of the system and of
the classified objects themselves, whether the genus or the
order should be that higher idea above the species, which
bears the simple name. If the order be produced by the
analytical process of division, as in the Linnean system in
Botany, or if the system itself, to which the nomenclature
refers, be an analytical system (§. 229.), and if moreover
the orders contain a vast number of genera, and these ge-
nera again a vast number of species : then it will be more
advisable to fix the name upon the genus, and to be con-
tented with the advantages thus arising to the mode of
denomination. By placing the name upon the order, the
nomenclature would be rendered more difficult, without
becoming more useful. For in an analytical or artificial
system, there exists no similarity among the classified ob-
jects (§. 229.) ; and a nomenclature expressing the order
would not serve to simplify or illustrate the general sur-
vey of the objects ; it would rather serve to produce confu-
sion. On the other hand, it could not but be useful, even in
Botany, to fix the name upon the order, supposing the sys-
tem to which the nomenclature refers to be the natural sys-
tem. Si classes naturalcs essent inventce omnes, maximc arridc-
ret terminatio nominum conformis in ajpnibits, iii nimia mnfat'/o
prohibcret. LINN. Crit. Bot. In the Natural History of
the Mineral Kingdom, however, the system itself, and con-
sequently also the order, is a natural or synthetical one ;
hence there exists a connexion among the classified objects,
which may, and even must be expressed by means of the

§. 234. NOMENCLATURE. 353

systematic nomenclature. Thus it becomes necessary to
fix the name upon the order. Only a trifling advan-
tage, indeed, would be gained, if in Mineralogy the genus
should be fixed upon as the unity of classification which
has to bear the name, since most of the genera contain
comparatively only very few species. The systematic
nomenclature would thus not find room to shew its useful.
ness, and to yield the services required by Natural His.
tory.

The higher the degree of classification is, upon which
we may fix the name, the more we shall be enabled to ex-
press, in the denomination of the species, the extent of the
connexion among the classified objects. This, however,
must have its limits, and should not be carried on so far that
it ceases to be either useful or convenient. It is very useful
to express the order in the Natural History of the Mineral
Kingdom ; and this may be effected without any inconve-
nience. But if we would endeavour also to express the
class, this would be both, of little utility, in as much as
there are only three classes in the natural-historical system
of Mineralogy ; and very inconvenient, because,, in order
to denominate a species, this would require four different
words. Moreover, the last of these words being in most
cases the same, the constant repetition would produce at
least a very disagreeable monotony.

It has been considered as a rule in Natural History, that
the genus should bear the name. This rule, as we have
seen above, is not a general one. We may consider it,
however, as a general rule, that the systematic nomencla-
ture should unite as much as possible all the advantages,
and render all those services which Natural History rea-
sonably can expect. The name, therefore, should be so
placed, as best to fulfil these demands. From the rea-
sons developed, and under the circumstances above men-
tioned, this really takes place, if in Botany it is the genus,
if in Mineralogy it is the order which bears the name. For
the sake of a greater conformity, however, with the method
received in Botany, a compound name is fixed in Minera-

voi>. i. 7.

354 NOMENCLATURE. §. 235.

logy upon the genus, or the name of the genus receives such
an arrangement, that it expresses at the same time the or-
der ; a contrivance which cannot be considered as objection-
able, according to the principles of Zoology or Botany.

§. 235. SELECTION OF THE NAMES OF THE ORDERS.

The simple names are the foundation of the
whole nomenclature. In their selection, therefore,
we must proceed with the necessary precaution and
attention.

I have endeavoured to introduce as few new names as pos-
sible, and I have derived therefore as many from ancient Mi-
neralogy, as I found to answer the purpose. Nomina veternm
plantis imposita lando, ad conspectum vcro recentiorum plurhtm
horreo. Hcec cnim maximam partcm stint nihil nisi Chaos
confusionis, cujus Mater barlaries, Pater anthoritas, Nutrix
praejudicium. LiNNi Crit. Bot. Several of these ancient
names have been of late severely censured, others have
been entirely rejected : yet I have thought it right to select
some of them which possessed the necessary qualities, and
to introduce them into the systematic nomenclature of the
present work. These names are, Gas, Water, Acid, Salt,
Baryte, Malachite, Mica, Spar, Gem, Ore, Metal, Pyrites,
Glance, Blende, Sulphur, Resin, and Coal. These names are
sufficient for all the orders hitherto known, except two ;
and it seems that a spirit of innovation cannot be with pro-
priety reproached to a new nomenclature, if it does not
contain more than two new names, particularly if we re-
collect that it refers to a new system, or rather to a sys-
tem in which a new application of principles has been made,
which, though not new in themselves, have yet been ne-
glected in Mineralogy. These new names are Haloide and
Kerate. They would not have been made use of, had any
others of a similar signification occurred in the older mine-
ralogy.

§. 23G. NOMENCLATURE. 355

§. 236. SIGNIFICATION OF THE NAMES OF THE
ORDERS.

The simple names receive their signification in
agreement with the ideas of the orders. They
must always be used in these, never in any other
significations.

It may be useful in this place to add a few remarks on
the meaning attached to several of these names as applied
to the orders. The name of Ore has been hitherto applied
to a great many minerals, and it is one of those, whose sig-
nification has been particularly vague and uncertain. Even
its technical signification had disappeared ; for though in
this respect we understand very well what is meant by
Iron-ore, Copper-ore, Manganese- ore, and Chrome-ore ;
yet it will be very difficult to say what we should understand
by Liver-ore, Tinder-ore, Horn-ore, Tile ore, Oliven-ore,
Pea-ore, Pitch-ore, &c. Names of that kind have indeed
no signification at all, as long as there does not exist an
order Ore, to the genera of which they may be referred.

If, on the contrary, we attempt to comprise in one single
expression all the minerals which have hitherto been de-
signated by the name of Ore, and add such as are naturally
allied to them in their properties and resemblance, in order
to produce the natural-historical idea of the name of Ore ;
the idea conveyed by this term would necessarily become
so complex and varied as to be utterly devoid of clearness
and consistency with itself. It would moreover-include so
many species, that the greater part of the minerals would
become ores, and thus the whole Mineral Kingdom would
not admit of another idea of that kind.

If therefore the name of Ore shall not lose both signifi-
cation and applicability in the Natural History of the Mi-
neral Kingdom ; it must be confined to a few only of those
minerals which hitherto have been called ores. Among
these are Red Copper-ore (octahedral Copper-ore), Tin-ore
(pyramidal Tin-ore), the Iron-ores (rhombohedral, octahe-

356 NOMENCLATURE. §. 236.

dral and prismatic Iron-ore), Grey Manganese- ore (prisma-
toidal Manganese-ore), &c. The same name of Ore must
likewise be applied to other species, which are connected
with the preceding ones by equal degrees of natural-histo-
rical similarity. Such are Rutile (peritomous Titanium-
ore), Anatase (pyramidal Titanium-ore), Wolfram (prisma-
tic Scheelium-ore), Uranite (uncleavable Uranium-ore), &c.
All these species form together one natural "order, which
bears the name. The name acquires, therefore, its signi-
fication and application, from the nature of the genera
and species which the order contains ; and is transferred
upon them in so far as they belong to that order. Hence
every thing must also bear the name of Ore, which belongs
to the order Ore, or, in other words, it will be said to be.
an Ore ; and a mineral is called an Ore, only because it be-
longs to this order. Thus the idea of the name of Ore
becomes exactly determined in conformity to the idea of
the order. A newly discovered mineral, which, on account
of its natural-historical properties, belongs to the order Ore,
assumes this name, or comes as it were with this name to
the notice of the world, and is thus secured from a burden
which arbitrariness very soon would load upon it, as long
as there does not exist a systematical nemenclature.

The signification of the name Pyrites has been better main-
tained in its purity. Most of the minerals hitherto called
Pyrites, really belong to that natural order, upon which the
name of Pyrites has been fixed. Such are Copper-pyrites
(pyramidal Copper-pyrites), Iron-pyrites (hexahedral and
prismatic Iron-pyrites), Arsenical pyrites (axotomous and
prismatic Arsenical-pyrites), &c. Other species, that ne-
vertheless belong to the same order, have hitherto not been
called pyrites ; as Cobalt-glance (hexahedral Cobalt-py-
rites), white Cobalt-ore (octahedral Cobalt-pyrites), Kup-
fernickel (prismatic Nickel-pyrites). Cobalt and Nickel are
metals, and minerals which bear these names must belong
to the order Metal. But if Iron-pyrites be a Pyrites, that
is to say, if it belong to a certain natural order, all the rest
likewise must belong to that order, and consequently as-

§. 23G. NOMENCLATURE. 357

sume its name. They are therefore compelled as it were, by
their natural-historical properties, to part with their former
names.

No other name has been subject to more ambiguities, and
to abuses, than the name of Spar. The word Spar is ge-
nerally said to signify a certain structure which has been
called the spathose one. We should never derive a name —
a word, which designates a very general idea, — from this
property ; for no property is less constant in a determined
signification, as its constancy does not extend beyond the
species, if we except the tessular system. In a more com-
prehensive or indeterminate signification, again, it is so ge-
neral, that it occurs not only in all the systems of crystal-
lisation, but also in many different orders.

The name of Felspar or Fcld-spar is almost generally re-
ceived in many languages, and may be useful on that ac-
count in deriving the name of an order. Hence only those
minerals are said to be Spars, -which belong with Feld-spar to
one and the same natural order, and not those species only
which shew the so called spathose structure.

Mica signifies a mineral, which may be cleaved with fa-
cility into thin shining laminae. Common Mica (rhombo-
hedral Talc-mica), Uran-mica (pyramidal Euchlore-mica),
Copper-mica (rhombohedral Euchlore-mica), are of this kind.
Although this relation of structure alone is not sufficient
for determining a natural order ; yet the order Mica con-
tains only such species as present it in a high degree of
perfection, and it bears therefore deservedly this name.

A mineral which may with propriety bear the name of a
Metal, must really be a metal, or it must present the pro-
perties peculiar to metals. Hence this name must not
be given to species belonging to the orders Pyrites,
Glance, Blende or Ore, because they do not possess the
properties of metals, although metals actually may be
extracted from them. The classification of metallic sub-
stances in the greater part of the systems hitherto pub-
lished, regard much more the rcguli obtained by chemical
aid, than the productions of nature ; and thus they betray

358 NOMENCLATURE. §. 237.

that they are not meant to be systems of Natural History,
but something which indeed cannot always be defined or
reduced to clear ideas.

The rest of the names need no further remarks. Some
of them belong entirely to the English idiom ; others have
long ago been introduced by mineralogical authors. The use
of these names is similar to that of the words explained be-
fore ; the ideas which they express being dependent upon
the contents of the orders to which they have been applied.

The only two names newly introduced are Halolde and
Kerate. The first of these has been used by several che-
mists for certain compositions of Muriatic acid with other
bodies. The name, however, does not indicate this acid to
be contained in the mixture. It means only a substance
resembling salt, like octahedral Fluor-haloide, prismatoidal
Gypsum-haloide, rhombohedral Lime-haloide, &c.

The above mentioned examples shew that there exist
mineral productions in nature, so very similar to the salts,
that is to say, to the minerals contained in the fourth order of
the first class, in respect to their natural-historical proper-
ties, that without a nearer examination, they might be
easily mistaken for one another ; and which, therefore, in-
deed possess that resemblance to salt in a remarkably high
degree. These substances form the first order of the se-
cond class, and have received the name of Haloides.

Kerate is a translation of Horn-ore (to which Quicksilver
Horn-ore likewise belongs), suppressing only the word ore,
a name to which these substances are not entitled. In
Mineralogy, we find very often comparisons with horn,
sometimes not in a proper place, as in hornstone, horn-
blende, hornslate, &c. It may be tolerated in the Ke-
rates, though its application is not the consequence of its
particular merits.

§. 237, NAME OF THE GENUS.

In the genus, the name of the order is more re-
stricted by connecting another word with the name

§. 237. XOMEXCLATUUE. 359

of the order ; and thus the product is a compound
name, which is the Name of the Genus, or the Ge-
neric Name.

The generic name should refer to the natural-historical
properties of the genus ; and it should therefore express, as
much as possible, some striking feature of its resemblance
with other bodies. Such is the name Garnet-blende. The
genus, designated by this name, belongs to the order Blende ;
the individuals which it contains very often look like garnet.
This garnet-like appearance, however, is not a similarity of
that kind, upon which the classification of the genus might
depend ; besides, this classification must precede both no-
menclature (§. 240.) and characteristic (§. 247-). There
can be no doubt, that for the nomenclature of a synthetical
system, no better mode of constructing the generic names
could be invented. Nomlna generlca^ quce cliaracterem essen-
tialcm vcl habit urn (plantce) cxhibcnt, optima sunt. Habitus in-
dlcat slmllltudinem, qua excltatur idea, et ex idea nomen. LINN.
Phil. Bot. 240. This mode of nomenclature, however, could
not be applied to every genus, without introducing a great
many new words, which is attended with all kinds of disad-
vantages, and must therefore be carefully avoided. On this
account, two different methods have been resorted to. In
many instances common trivial names have been employed
in forming generic names ; in others these names have been
derived from a property which is not a natural-historical
one. According to the first process were obtained the names
of Feld-spar, Augite-spar, Triphane-spar, Disthene-spar,
Azure-spar ; according to the second process, Iron-pyrites,
Cobalt-pyrites, Lead-glance, M olybdena-glance, Scheelium-
bary te, Lead-bary te, and others. The latter of these names
really have a chemical sound : but they have in the present
place no chemical meaning; and it is only the meaning
which must here be taken into consideration. The genera
Iron-pyrites, Nickel-pyrites, Cobalt-pyrites, the genera
Uranium-ore, Chrome-ore, Manganese-ore, &c. are not
assembled because they contain the metals of which they

3()0 NOMENCLATURE. .§. 238.

bear the names ; but because the first are Pyrites, the
second Ores. Many minerals indeed contain the said me-
tals, and yet are not joined to these species within the
same orders. Thus the above mentioned names not only
become innocuous, but they are even useful, in as much as
they convey the idea, though not a natural-historical one, of
the quality of the contents to which these bodies owe their
application in the arts. Yet they would not have been
employed, but for the above mentioned rule in the construc-
tion of a new nomenclature, not to introduce too many
new words.

In the names of two of the orders, Gem and Metal,
an exception seems to have been made from the general
process of applying compound names to the genera, which
at the same time should express the orders. The name of
the order is, however, only suppressed in these, because
to a certain extent it is understood from itself. Every
body knows Gold, Silver, Bismuth, Tellurium, &c. to be
metals, and the names Gold-metal, Silver-metal, &c. would
certainly not meet with general approbation. It is the
same with Diamond, Topaz, JZircon, &c. Nobody would
approve of the names Diamond-gem, Topaz-gem, Zircon-
gem, &c. The name of a metal, and the name of a gem,
therefore, signify by themselves the order to which the
one and the other belongs. The only genus of the order
Sulphur has as yet no generic name.

§. 238. DENOMINATION OF THE SPECIES.

The nearer restriction of the generic name to
the species is effected by an adjective. Thus the
Denomination of the Species is produced.

The adjective, with which the species is designated with-
in its genus, must be taken from its natural-historical pro-
perties, and if possible so selected, that it refers to one
of those properties of the species which are most useful in
distinguishing it from other species of the same genus. To

§. 239. NOMENCLATURE. 361

this end, the most desirable are the systems of crystallisa-
tion and the relations of cleavage. Examples are, hexahe-
dral, prismatic, rhombohedral Iron-pyrites ; rhombohedral,
octahedral, dodecahedral, prismatic Corundum ; rhombohe-
dral, octahedral, prismatic Iron-ore, and many others. If
two or three species of a genus belong to one system of
crystallisation, one of them retains the adjective express-
ing the system ; the others receive a denomination taken
from a general property of crystallisation, or of cleav-
age, as in peritomous and pyramidal Titanium-ore, in
prismatic, prismatoidal, hemi-prismatic, and paratomous
Augite-spar, in rhombohedral and peritomous Ruby -blende,
in rhombohedral, macrotypous, brachytypous and parato-
mous Lime-haloide, and several others. In the same man-
ner have also been employed the adjectives axotomous,
diatomous, diprismatic, and prismatoidal, agreeably to the
explanations given above. The adjectives nndcavable and
native should be retained, only till the cleavage or the form
of those species to which they are now applied, will be as-
certained. Only a few, for want of better, have been taken
from colours, none from localities or persons, nor has, in
any instance, the adjective common been employed, which
is indeed the worst of alL The natural quality of the spe-
cies contained in the three first orders of the first class,
and in the second order of the third class, has produced the
necessity of deriving adjectives from other relations, which
would else not have been applied.

§. 239. REPRESENTATION OF THE SPECIES THROUGH
ITS DENOMINATION.

The systematic denomination produces a repre-
sentation of the species. This nevertheless cannot
be substituted in the place of the character in the
Characteristic, nor of the general description in the
Physiography.

It is not a small advantage of that systematic nomenda-

362 NOMENCLATURE. §. 239-

ture, which refers to a natural or synthetical system, to create
an image of the object to which it refers, hy means of its de-
nominations, which never can be erroneous, though it may
be insufficient. The trivial nomenclature can never render
this service, how perfect soever it may be in respect to its
peculiar properties. If we hear the name Jlutlle, and do not
know the species itself, to which it belongs, we never can
imagine any thing like a representation of the object,
though for the rest, our knowledge of Mineralogy may be
very extensive. If, on the contrary, we hear peritomous
Titanium-ore, and have only an idea of the order Ore, this
at once will produce a general image of the species, which
will be still more restricted, if we have some idea of the
genus Titanium-ore. But we may indeed suppose that every
person possesses an image of the orders and genera in his mind,
who is in some degree acquainted with the Natural History of
the Mineral Kingdom. If, besides, we attend to the cleav-
age, which is peritomous, this will suffice, by the mere de-
nomination, for distinguishing the varieties of peritomous
Titanium-ore from those of the prismatic or the pyrami-
dal Titanium-ore. The denominations may be still more
useful, if they express the cleavage more minutely, or if
they refer to the system of crystallisation. By the deno-
mination hexahedral Iron-pyrites we learn, that in the
order Pyrites, the species belongs to the genus Iron-pyrites,
and the adjective hexahedral signifies that its forms belong
to the tessular system, and that cleavage takes place in the
direction of the faces of the hexahedron. An image, thus
produced of the species by the mere denomination, is in-
deed very useful. Yet it cannot serve as a substitute to
the character in respect to the process of determining an
individual, by the assistance of the Characteristic, nor can
it be employed instead of the General Description, for re-
presenting the varieties of the species. For commonly it
contains only one character, in most cases a very general
one, and refers besides to images or representations, which
can only be acquired by ocular inspection, and never can
receive the precision of a character, nor that exact idea of

§. 240. NOMENCLATURE. 363

the species, which is produced by the general description.
The characters contain every thing required for a correct •
distinction within their sphere : the general descriptions
contain every thing required to a perfect natural-historical
knowledge of the species itself.

It will not remain unnoticed by those who consider an
empirical knowledge of the productions of the Mineral
Kingdom as a valuable acquirement, that the methodical
way is the only one which leads to this end with the great-
est facility ; and, what is still more important, with the
greatest exactness and security, while every other attempt
must remain fruitless.

§. 240. SYSTEMATIC NOMENCLATURE HOW TO BE
JUDGED OF.

The systematic nomenclature presupposes a sys-
tem, to which it refers ; and upon the due conside-
ration of this system must necessarily depend a
complete judgment of the nomenclature. The sys-
tem requires a systematic nomenclature, in order
to be applicable to the objects of experience.

Although the first part of this proposition immediately
follows from the very idea of a systematic nomenclature
(§, 230.), yet it seems necessary to add here a few remarks.

The reason why as yet there has not existed a systema-
tic nomenclature in Mineralogy, was the want of a system,
capable of serving as basis to a systematic nomenclature, a
system which for that purpose would have required to
contain correctly determined species, arranged according
to the general principles of Natural History. Several
attempts have been made to construct a Latin syste-
matic nomenclature ; they have not succeeded, because
the systems to which they referred, did not possess the
necessary properties. The celebrated ABBE HAUY ac-
knowledges indeed the great value of a systematic no.

364? NOMENCLATURE. §. 240.

menclature, yet only a part of that nomenclature, which
he applies himself, is systematic, the rest is trivial no-
menclature. This mixed nomenclature is a consequence
of a mixed system ; and proves that the nomenclature can-
not acquire uniformity, unless the system rest upon simple
principles. Mineralogy, as hitherto treated, has amply
demonstrated this observation. Zoology and Botany prove
the reverse. These parts of Natural History have always
proceeded according to one and the same principle, and on
this account they have long ago possessed the advantage of
a systematic nomenclature.

It appears evidently from the preceding observations,
that the systematic nomenclature must be judged of ac-
cording to the system to which it refers. If this be found-
ed upon relations appertaining to the science, and if its
different parts be consistent with each other ; that is to
say, if it fulfil the above mentioned demands of Natural
History ; the whole business of nomenclature will be to
express this system by words, so that it becomes possible
from the denomination of a species to infer the connexion in
which it stands with others. The nomenclature, moreover,
should not be contrary to the spirit of the language ; its
expressions should be concise and intelligible, and the de-
nominations, if possible, should be expressive of the objects
themselves. These are the chief properties of nomencla-
ture, upon which its applicability depends. It will thus
allow of all those improvements and refinements, or, in
general, of all those changes, which are rendered necessary
by the continual advancement of our knowledge regarding
the productions of the Mineral Kingdom.

The object of the systematic nomenclature is to promote
and facilitate the application of the system to nature, or to
the data of observation. For while we attribute certain
names or denominations to the natural productions, we ar-
range them at the same time under the general ideas of
the system ; the system, however, is constructed for the
purpose of collecting the variety of nature within its ge-
neral ideas, and thus reducing it to a unity, in order to

§. 241. NOMENCLATURE. 365

enable us to survey and to understand it, and to acquire
some more exact knowledge of it, than that which consists in
a mere work of memory. The system would lose its applica-
tion if there was no nomenclature, and both, System and
Nomenclature, appear therefore equally important in re-
spect to the idea of Natural History, and they are connect-
ed with each other by means of the Characteristic.

§. 241. TRIVIAL NOMENCLATURE.

In the trivial nomenclature the name is fixed
upon the species.

The trivial nomenclature does not express the connexion
among those bodies, which it provides with names. Any
name, not destined to expi'ess this connexion, is termed a
Trivial Name, which rests accordingly upon the lowest
idea of the system, that is to say, upon the species. The
meaning attached here to a trivial name is somewhat differ-
ent from the trivial name as defined by LINN^F.US. The
latter consists of a mere adjective, substituted for the cha-
racter of the species : it is not properly a name (§. 232.),
and can therefore never be used by itself, but only in
connexion with a name, as a denomination.

It is a custom generally received, and in no way objec-
tionable, to provide the natural productions, and, above all,
those which are used in the arts of life, with particular
names, which, on account of their conciseness and simplici-
ty, are more convenient for use than the long and compound
systematic denominations. Moreover, it is supposed here,
that we are already acquainted with the object thus named,
or the name at least is not meant to express some farther
information, and so they are destined as it were, for a less
strict or scientific employment. But it would be blame-
able indeed, if, for the trivial nomenclature, we should ne-
glect the systematic one, and thus betray an indifference
towards the science itself, which could not but produce
evil consequences.

306 NOMENCLATURE. §. 24 1 .

Together with the knowledge of the bodies themselves,
the trivial nomenclature likewise supposes that of the con-
nexion in which they are with others. Hence it supposes
the systematical nomenclature ; and it is evident, there-
fore, that although it may exist beside it, yet it never can,
instead of it, fulfil the demands of Natural History.

The properties requisite in trivial names, may be very
easily inferred from the preceding observations. Their
chief recommendation consists in their simplicity ; they
must be simple names (§. 232.). For a compound name ex-
presses a connexion or a relation with other objects, with
which trivial nomenclature has nothing to do, and refers
to a system, which does not exist, at least not in respect to
the trivial nomenclature. The names, Spinel, for dodeca-
hedral Corundum ; Eudas, for prismatic Emerald ; Rutik,
for peritomous Titanium-ore, are excellent trivial names.
The name Hornblende, if applied to a species, supposes a
genus Blende, which does not exist in any of the systems
in which that name has been used ; if it be supposed to re-
fer to one or to several varieties, it will suppose the exist-
ence of a species Blende ; to which, however, in these sys-
tems, hornblende does not belong. Examples of this kind,
of which a great many more might be quoted, are calculat-
ed to shew, that there exist rules even in respect to the tri-
vial nomenclature, which it is indispensable to observe, if
we intend not to confound those ideas, which it is the pur-
pose of the system to explain, and the purpose of nomen-
clature to preserve in their purity. Compound trivial
names, moreover, without either the necessity or the advan-
tage, produce all the difficulties of a systematic nomencla-
ture. And yet, the only motive of introducing a trivial
nomenclature is to avoid these difficulties ; hence it appears
that compound trivial names are entirely to be rejected.

It requires' but a limited knowledge of the object to un-
derstand, that the difficulties, connected with the construc-
tion of a good trivial nomenclature, possessing the required
properties, by far surpass those which attend the construc-
tion of a systematic nomenclature. Many names contain.

§. 241. NOMENCLATURE. 367

ed in several of the systems of Mineralogy hitherto pub-
lished, might he of use in the construction of a good trivial
nomenclature ; they should be collected and properly com-
pleted. The difficulty of introducing a nomenclature of
this kind, even though it were acknowledged as useful,
will be probably found greater, than if it had been syste-
matic. On account of the great difficulties in establishing
certain and general rules in the selection of trivial names,
there will always remain some arbitrariness, which is un-
avoidable, and will form the principal impediment of a
general agreement in this respect.

The natural-historical determination of natural produc-
tions, does not go beyond the species (§. 222.). The syste-
matic nomenclature, therefore, must stop at the denomina-
tion, the trivial nomenclature at the name of the species.
The disadvantage arising to the systematic nomenclature,
from a want of attention to this rule, consists in the cir-
cumstance, that the denominations become composed of
too many words, because they require at least two adjec-
tives. If the trivial nomenclature applies names to parti-
cular varieties, as Amethyst, Prase, Adularia, Amiantus,
Anhydrite, &c. the idea of the species becomes too much
dismembered. If it produces denominations by adding
adjectives to the names of the species, it assumes the ap-
pearance of a systematic nomenclature, neither of which pro-
perties would serve to its recommendation.

PART IV.
CHAR ACTERISTIC.

§. 242. DEFINITION.

The Characteristic is the assemblage of certain
natural- historical properties, arranged according to
a certain system, for the purpose of distinguish-
ing the unities contained in this system.

Any single natural-historical property, or a collection of
several of them, if it be subservient to the distinction
of several species of a genus, or of several genera of an
jorder, or of several orders of a class, &c. is termed a Cha-
racter, and the single properties it contains are its Charac-
teristic Terms or Marks. If a character contains only one
characteristic mark, this mark itself represents the character.

Without a system, there cannot exist a character ; and
therefore likewise no Characteristic, because the distinction
of several bodies, by means of characters, takes place only
within the unities of the system. Thus it becomes possible
that a character may contain, or be limited to a single one, or
a small number of characteristic marks. A character calcu-
lated for distinguishing one single species from all other
species, to whatever genera or orders they might belong ;
that is to say, a general character, would require the enu-
meration of all the natural-historical properties of the spe-
cies, as its characteristic marks. But this enumeration of
all the properties of a natural production, is its Description
(§. 27.), which is beyond the limits of the Characteristic,
and enters those of Physiography.

The term natural character is applied by LIXNJEITS to
the description. Among its properties, he mentions that
it contains all the characteristic marks, of the genus in Bo-

§. 243. CHARACTERISTIC. 869

tany, to which every thing LINNJEUS says of characters,
more particularly refers; Naturalis character notas omnes
gcncricas possibiks allegat ; adeoque Essentialcm et Factitium
inchtdit. LINN. Phil. Bot. 189. He likewise says, that it
is invariable and not dependent upon the system ; and that
it may serve for every system ; inservit omni systematl ;
Basin stcrnit novis systematibus, immutatus persistit, licet in*
finita genera nova detegcrentur. LINN. Phil. Bot. ibid. These
properties do not belong to a Character properly so called ;
but they are essential to the General Description. The
preceding observations contain the reasons why I have
thought necessary to abandon the denominations used by

LlNNJEUS.

§. 243. NATURAL AND ARTIFICIAL CHARACTERS.

If the system to which the character refers is
the natural or synthetical system, also the cha-
racter is said to be a natural one ; if the system is
an artificial or analytical one, it likewise contains
artificial characters.

This is the correct idea of a natural and of an artificial
character, which gives no occasion to ambiguities. Essen-
tialis character nnica idea distinguit Genus a congeneribus suis
sub eodem ordine naturali. LINN. Phil. Bot. 187- Factitius
character Genus db aliis, ejusdem tantum ordinis artificialis, dis-
tinguit. LINN. Phil. Bot. 188. This natural character,
therefore, must not be considered as something produced
by nature, for nature does not institute comparisons be-
tween its productions, from which the natural character
might be derived. It seems not to be exactly in harmony
with the idea of this character, to call it an essential one,
since it depends upon the properties of the objects (orders,
genera, species) compared, and is a result of their compa-
rison ; so that by the discovery of new genera in an order,
or of new species in a genus, it may be subject to altera-
VOL. i. 2 A

370 CHARACTERISTIC. §. 244.

tions, which will never cease, and therefore always hinder the
character from being infallible, till every thing is known
which nature has produced within that kingdom. Nidlus
character infallibilis est, antequam secundum omnes suas species
directus est. LINK. Phil. Bot. 193. It would be possible,
on the contrary, to call the artificial character an essential
one, at least in respect to a certain system, since, as it will
appear afterwards, it is the foundation of the divisions in
that artificial system. The division is effected according
to general properties, and every individual necessarily be-
longs to one or to the other of these divisions, in as far
as it contains the characteristic property. Thus the artifi-
cial character is not dependent upon the enlargement of
our knowledge by experience. Yet even here it is better
to avoid this expression, because commonly it gives rise to
accessory considerations, which may lead to erroneous
ideas.

The denomination of the characters corresponds with
their object, so that we have Characters of the Orders, Generic
Characters, &c.

§. 244. PROPERTIES OF THE CHARACTERS.

The characters must be sufficient to a precise
distinction within their respective sphere, and as
short as the necessary degree of evidence in the de-
termination of the species will allow.

The first requisite is an immediate consequence of the
very idea of a character. Characters are entirely useless,
if they serve for the distinction only of some of the species
contained within their genus, or of some of the genera con-
tained within their order. If a single characteristic mark
suffice for a general distinction, this mark will represent
the character itself : if not, several of them must be ap-
plied in connexion, and thus form the character. A com-
pound character of this kind can only belong to a natural

§. 24)4. CHARACTERISTIC. * 371

system, because only in this it is possible, that one single
characteristic term should not be found sufficient for a ge-
neral distinction. This natural character it is, which re-
quires the greatest possible Conciseness and Uniformity.
Character essentialis, quo brevior, eo etiam prcestantior est.
LINN. Phil. Bot. 187. The shorter the character is, the
more facility and certainty it will afford to the distinction ;
and this facility and certainty it is, which we demand in a
character. The more uniformly we arrange the charac-
ters of the same kind, the less likely we are to omit any
characteristic marks. Hence the characters should not
contain any thing, but what is unavoidably required for
the distinction and the evidence in the determination of the
species ; and every superfluous word, every word of an am-
biguous signification, is reprehensible ; so is every restric-
tion in regard to time, or other relations, and, above all,
every verbal exception, quite contrary to the idea of a cha-
racter. Even now, at least in respect to the Natural His-
tory of the Mineral Kingdom, it is not superfluous to add :
Oratorio stylo in charactcre, nil magis abominabile. LINN.
Phil. Bot. 199.

The higher the degree of classification, within the sphere
of which the distinction is to take place, the more it is ne-
cessary to bestow all possible attention upon the above
mentioned properties of the characters. For if the first
distinction be not evident and correct, the subsequent ones
will be still less worthy our consideration. It is obvious
that the characters will acquire the properties of concise-
ness and uniformity to a comparatively higher degree, the
more the system corresponds to nature, in proportion like-
wise to the correctness and consistency within itself, with
which it expresses the relations of the natural-historical
resemblance. The characters of the classes and of the ge-
nera possess these properties to a considerable extent. Only
the characters of some of the orders in the second class are
longer, they contain more characteristic marks, than it is
desirable they should contain. Though in this degree of
classification the variety in the connexion among the

374- CHARACTERISTIC. §, 246.

" Solid : taste ;" where solidity is the condition under which
the property of exciting some taste must necessarily take
place. The characters, and every single mark which they
contain, must be taken literally ; and they admit of no expla-
nation or other accessory significations, but that which is
really expressed by the words. In the instance just men-
tioned, it would be erroneous to infer, that if a mineral
which shall belong to the first class is not solid, it must be
insipid. The character does not express this ; and it is
therefore quite indifferent whether, if not solid, the mine-
ral has any taste or not. Sometimes the conditioning part,
sometimes the conditioned part of the characteristic mark,
at other times both of them, are compound. Yet the em-
ployment of the conditioned characteristic mark is not dif-
ferent from that of the absolute ones, as explained above.

§. 246. ARRANGEMENT OF THE CHARACTERS OF
THE SPECIES.

The arrangement of the characters of the species
must be such, that, by their assistance, the deter-
mination of the individuals receives the greatest evi-
dence which the science can possibly produce.

The only thing we may reasonably demand from the
characters of the classes, the orders and genera, is, that they
should exclude every individual which does not belong to
them, and that they should not exclude those which these uni-
ties comprehend. It is quite indifferent by what properties,
and in what manner this is effected, provided the properties
be sufficient for a general distinction within their sphere, and
the method agreeable to the principles of Natural History.
The character of the species requires something more.
For here the object of our inquiry is not only to know that
a given individual is not excluded from a certain species,
but we wish to find out, and to convince ourselves, that it
really does belong to that species. For this reason the

§. 246. CHARACTERISTIC. 375

character of the species should contain marks, which, if not
.always, at least very often may be superfluous in respect to
the mere process of distinction. The genus Emerald con-
tains two species, prismatic and rhombohedral Emerald.
For the sake of mere distinction, the character of each of
these species need not to contain any thing besides the
system ef crystallisation, or the limits of specific gravity,
for these likewise would suffice for distinguishing the two
species. A third species, however, might exist, besides the
two species above mentioned ; a species which, on account
of its natural-historical properties, did belong to the genus
Emerald, and that species might agree with one or the
other of these in the above mentioned characteristic marks.
In order to assure ourselves that an individual belonging to
the genus Emerald enters either within the species of the
prismatic or of the rhombohedral Emerald, their characters
are made to contain a greater number of marks, whose pro-
perties leave no doubt, upon the supposition of the indi-
vidual being an Emerald, whether and to which of the two
species the individual belongs. This arrangement, more-
over, produces an uniformity in the specific characters,
which, according to what has been stated above, is one of
their principal qualities.

The specific characters, therefore, consist chiefly of three
marks of this kind, which, wherever the quality of the
species would allow, have been given in all instances.
These are the crystalline forms (including cleavage), the
degrees of hardness, and the specific gravity. The first
characteristic mark in the specific character is the system
of crystallisation. Then follows, together with its angles
(if these be known), the fundamental form, from which all
the other simple and compound forms of the species may
be derived. Of rhombohedrons the terminal edge is given ;
for instance, in rhombohedral Lime-haloide, R = 105° 5' ;
of an isosceles four-sided pyramid, first the terminal edge,
then the lateral edge ; as in pyramidal Zircon, P = 123°
19', 84° 20' ; of a scalene four-sided pyramid the obtuse
terminal edge, the acute terminal edge, and the lateral

376 CHARACTERISTIC. §. 846.

edge in succession; thus, in prismatic Topaz, P = 141°
7', 101° 52', 90° 55', &c. This is observed in every spe-
cies, where the fundamental form commonly occurs in na-
ture, and therefore may be often observed ; or where the
combinations do not present any peculiar character, as the
hemi- and tetarto-prismatic, or the di-rhombohedral ones,
&c. If the fundamental form is seldom observable, its angles
are not indicated ; in their stead, however, the characters
contain the angles of such derived forms, as commonly are
to be met with in nature, as in prismatic Hal-baryte
fr = 105° 6'; (Pr+co)3=77° 27'. In every instance re-
garding horizontal prisms, that angle is given which is con-
tiguous to the axis of the fundamental form, in vertical
prisms the angle corresponding to the obtuse terminal edge
of the fundamental form. If the combinations of a species
possess a particular character, the Characteristic indicates
only those dimensions of the fundamental form, which cor-
respond to that character, and therefore are immediately

p
observable, as in paratomous Augite-spar, _ = 120° 0' ; in

a like manner in rhombohedral Fluor-haloide, 2 (11) =
131° 14', 111° 20', &c.

In those forms which assume a hemi-prismatic character,
it is very often the case, that the axis of the fundamental
form is not perpendicular upon the base (§. 98.). The in-
clination of the axis takes place in a plane through the
axis, and one of the diagonals of the base. This plane bi-
sects the angle produced by those faces of the simple forms,
which appear in the combination, and which therefore is in-
dicated in the designation of the form. Together with
the magnitude of this terminal edge, the angle of incli-
nation itself is likewise contained in the character. Thus

p
in prismatic Azure-malachite " — =117° 37', Inclination

(of the axis), = 23 21'" signifies, that those faces of the
fundamental form, which necessarily appear together in
the combinations, are contiguous to the short diagonal, and
meet under angles of 117° 37'; and that the axis of P

§. 84{>. CHARACTERISTIC. 377

includes an angle of 2° 21' in the plane of the short dia-
gonal with the perpendicular line erected in the centre of
the base. " Inclination = 0" means, either that there exists
no inclination at all, or that this inclination has not been
as yet attended to in the present determination of the crys-
talline forms.

The angle indicated in horizontal prisms that assume
a hemi-prismatic character, is the one contained between
the occurring face and the axis, together with the situation
of the face by means of the signs + and — , agreeably to

Pr

§. 1 53. For instance, in paratomous Augite-spar, — =

73° 54', where the sign + is understood. For designating
vertical prisms in hemi-prismatic combinations, the lateral
angle is given, which corresponds to that terminal edge of
the fundamental form, which occurs in the combinations.

With respect to cleavage, the expression " Cleavage, R,"
for instance, in rhombohedral Lime-haloide, means, that
this mineral has its cleavage parallel to the faces of a rhomr
bohedron, similar to the fundamental form of the species ;
" Cleavage, P — oo. P 4- oo. [P + w]" in pyramidal Gar-
net, means, that this mineral has its cleavage parallel to
the faces of two rectangular prisms, and at the same time
perpendicular to their axis ; " Cleavage, ?r + os" in pris-
matic Chrysolite, indicates, that the cleavage of this mineral
is parallel to a plane passing at the same time through the
axis and the short diagonal of the prism P + oo ; " Clear*-
" age, (Pr + c;)3 =87° 42'. Pr -f oo. Pr + co" in parato-
mous Augite-spar, expresses that the individuals of this
species may be cleaved, first, parallel to the faces of an ob-
lique-angular four-sided prism, of the given angles ; and
secondly, parallel to planes, which pass through the axis
and both diagonals of the prism P + cc ; or, what comes to
the same, parallel to the faces of a rectangular prism.

Cleavage is sometimes found to be hemi-prismatic or
tctartO'prismatic. Paratomous Augite-spar shews an ex-
ample of the first. The hemi-prismatic faces of P, express-

378 CHARACTERISTIC. §. £46.

ed above by — = 120° 0', under which angle they meet,

appear in several varieties as faces of cleavage. This rela-
tion, and all others of the same kind, are expressed by signs
analogous to those used in the forms of crystallisation. In
general, every face of cleavage is expressed by the sign of
the face of crystallisation to which it corresponds. The
degree of perfection of the faces of cleavage has likewise
been indicated, and needs no farther explanation. The
student, however, who intends to employ the Characteris-
tic for the determination of occurring individuals, has par-
ticularly to attend to those faces of cleavage, which are
most apparent.

The degrees of hardness, which are in general expressed by
H., and those of specific gravity, expressed by G., are given
in the Characteristic with their limits, or those points be-
tween which the hardness and the specific gravity of the
varieties are found to be contained ; observation will very
seldom yield these limits themselves; and only in such
cases, where it cannot have any prejudicial influence upon
the determinative process itself. Evidently this must apply
in a still higher degree to the characters of the orders and the
genera. Those who make use of the Characteristic, must
not therefore compare one character with another* but they
have to compare occurring individuals of the species con-
tained in it with these characters.

Besides these three characteristic terms, the characters
of some species contain also the indication of several occur-
rences of colour, more particularly the streak ; also the
lustre, or the general aspect ; sometimes also the state of
aggregation, taste, &c. In most cases they would have
been superfluous, had a more accurate knowledge of the
forms existed. In this particular, we have reason to expect
a great deal from future and accurate observations, which
will enable us to keep the characters free from all such
marks, as do not allow of a perfectly strict definition.

The specific characters of Jluid minerals require another

§. &17. CHARACTERISTIC. 879

process, because in these bodies two of the most valuable
marks in the characters of solid minerals, form and hard-
ness, are wanting. They have not as yet been brought to
any degree of perfection ; and indeed there is little to
be done in this respect, our knowledge of their natural-
historical properties being still very defective.

The specific characters should not contain either condi-
tioned or exclusive characteristic marks, though this may be,
and indeed is, the case with the characters of the orders
and the genera. In the latter characters there occur some-
times terms mutually excluding each other, as in the genus
Corundum, " Tessular, rhombohedral, prismatic^ or in the ge-
nus Iron-ore, " Streak red, brown. Mack." This evidently
means, that the forms of an individual belonging to the ge-
nus Corundum, must be either tessular, or rhombohedral,
or prismatic ; and that an individual of the genus Iron-ore
must yield a streak either red, or brown, or black ; because
in one and the same individual, two different kinds of the
same characteristic property are impossible. In another
place, an example will be given to shew, that characters
thus arranged do really convey all possible security ; which
example will at the same time serve to illustrate the use
of the Characteristic in this respect.

§. 247. NO CHARACTERISTIC BEFORE THE SYSTEM.

The Characteristic presupposes in its full extent
the system, to which it refers.

In the natural or synthetic system, or that whose foun-
dation is the natural-historical similarity, those objects are
placed nearest, which are connected by the highest degrees
of resemblance, or which are most similar to each other.
In arranging them, no attention is paid to single properties,
and perhaps least of all to such as might be useful in the
distinctive characters. Indeed, the conformity of the differ-
ent parts of the system would be very soon lost, should we
allow such accessory views to be introduced. First of all.

380 CHARACTERISTIC, §. 247.

the system, as fur as experience allows, must have been
completed ; then only it becomes possible to compare the
different homologous unities which it contains, with each
other, classes with classes, orders with orders, genera with
genera, species with species, in order to discover the charac-
teristic marks in which they differ, and from which their
characters must be formed.

Hence our mode of considering the Characteristic in the
natural system will only then be correct, if we keep in
mind that the order, the genus, &c. is not produced and
determined by the character, but that the character de-
pends upon the order, the genus, <&c. Scias characterem non
constitucre Genus, sed Genus characterem. Characterem non esse
ut Genus fiat, sed ut Genus noscatur. LINN. Phil. Bot. 169.
We must not, therefore, look for the reasons for which the
unities of the system have been adopted and determined,
in these characters, from which they never can be deduced,
because they consist solely in the relations of natural-his-
torical similarity, by which the objects either approach to,
or recede from each other, a matter brought to full evi-
dence in the preceding paragraphs. The only object of the
Characteristic is to collect with facility the individuals oc-
curring in nature, under the ideas of the system. This is
effected without regard to any thing, except the distinctive
characters. The idea of the species, or of any higher uni-
ty, does not come into consideration, since in general the
Characteristic has nothing to do with the developement or
establishment of general ideas, which belongs to the
Theory of the System. Here we do not ask, which pro-
perties are peculiar to the bodies, but only what are the
properties in which they differ. The characters of a spe-
cies, or of any other of the systematic unities, must not be
considered as defective or erroneous, if they should not con-
tain so many characteristic marks as are necessary for ex-
citing the idea of the species ; for this is not its object, and
belongs to the General Description. Every character is
perfect, which affords a general distinction within its
sphere, and thus attains its object. It would be an error to

§. 248. CHARACTERISTIC. 881

collect in it superfluous marks, which are of no use in the
process of distinguishing individuals.

Thus it appears that the Characteristic presupposes the
existence of the system to its full extent, its only object
being to distinguish minerals occurring in nature, while
that of the Theory of the System is to produce the syste-
matic ideas agreeably to the principle of natural-histo-
rical resemblance with consistency. Both of them must
keep strictly in their peculiar course, and will then become
the more useful as departments of the science ; by these
properties, the Characteristic will become the link between
the systematic ideas and the systematic names and deno-
minations, while both the Characteristic and Theory of
the System will produce the connexion between the na-
tural-historical properties and the same systematic names
and denominations.

§. 248. BASE OF A PERFECT CHARACTERISTIC.

The perfection of the Characteristic depends
upon the perfection and accuracy of our natural-
historical knowledge of natural productions.

The truth of this proposition, in regard to the Charac-
teristic of both natural and artificial systems, is so very
evident, that it would be superfluous to add any explana-
tory remarks. But as a consequence of this truth, we may
mention, that the most useful, or rather the only means of
.bringing the Characteristic nearer perfection, consists in
the continued study*, and in the accurate investigation of
nature. The more we inquire into the nature of bodies,
and the more our knowledge becomes accurate and exten-
sive, the more the ideas of the system will advance towards
purity and correctness, and afford in proportion a higher
degree of facility and certainty to the process of distin-
guishing them from one another by means of the Charac-
teristic. The want of an accurate knowledge is still per-

382 CHARACTERISTIC. §. 248.

ceptible in many of the indications of forms, which in many
species are either not determined at all, or at least not to a
sufficient degree of accuracy. From such imperfect infor-
mation, the greater part of the difficulties derive, which we
have to encounter in the construction and application of
the Characteristic. This has no doubt been one of the
reasons which has deterred naturalists from following that
path in Mineralogy which has been found the right one in
Botany and Zoology, and they have considered according-
ly as impracticable, every attempt towards the construction
of a Characteristic. It is too soon as yet to expect it to
be perfect ; yet it is not too soon to make the first step,
and the science itself requires that it should be done, in
order to obtain in its regular scientific form the third,
and not least important part of Natural History, that of
the Mineral Kingdom. The imperfections of the Cha-
racteristic appear more strikingly in the first class than
in the second, or even in the third; but in that class
we know so very little of the natural-historical properties
of the bodies which it contains, that it has been introdu-
ced, almost entirely for the sake of exhibiting with some de-
gree of completeness, what the Characteristic should con-
tain. Thus also, the systematic nomenclature, which is
always in proportion to our knowledge of the objects them-
selves, is here more imperfect than in any other part of the
system. It is to be expected, however, from the progress
already made, that these difficulties will entirely disappear ;
and this will take place the sooner, the more we convince
ourselves that in order to remove those which are still re-
maining, we are not compelled to recur to foreign assist-
ance, a process by which the purity of the principles
of Natural History would be entirely sacrificed ; and yet
this purity has been the only source from which flows
even that little, which has as yet been effected.

§. 249. CHARACTERISTIC. 383

§. 84$. USE OF THE CHARACTERISTIC.

The use of the Characteristic is the same in Mi-
neralogy as in Zoology and Botany.

It will be useful to give a short explanation of the pro-
cess used in the determination of minerals.

If a mineral is to be determined, first its Form, if this
be regular, must be ascertained, at least as far as to know
the system to which it belongs. Then Hardness arid Speci-
fic Gravity must be tried with proper accuracy, and express-
ed in numbers. It is sufficient, however, to know the lat-
ter to one or two decimals. The specific character requires
these data ; they are also of use in the characters of the
classes, orders, and genera. After this examination, the
Characteristic may be applied, and it will at the same time
point out what other characters are still wanting ; so that
a mere inspection of the mineral, or a very easy experi-
ment, as, for instance, to try the streak upon a file, or still
better, upon a plate of porcelain biscuit, will very often be
sufficient. The given individual is now carried through
the subordinate characters of the classes, orders, genera,
and species, one after the other, comparing its properties
with the characteristic marks contained in the characters of
these systematic unities. From their agreement with
some, and their difference from other characters, we in-
fer, that the individual belongs to one of the classes, to
one of the orders, to one of the genera, and to one of the
species. Having advanced in this manner to the character
of the species, it will in some instances be necessary, and
in all cases advisable, for the sake of certainty (§. 246.), to
have recourse to the dimensions of the forms. This is
particularly necessary, if the genus, to which the mineral
belongs, contain several species having forms of the same
system, as is the case in the genus Augite-spar. The com-
mon goniometer in most cases will suffice for determining
the dimensions of the forms, the differences in the angles

384? CHARACTERISTIC. §. 250.

being in general so great, that they cannot easily be miss-
ed, even by the application of this instrument. If the
differences be small, and their distinction require on that
account a higher degree of accuracy, it will be necessary to
recur to the reflective goniometer.

It will seldom be necessary to read over the whole of any
character of a class, order, genus, or species, excepting
those which comprise the individual ; one term that does
not agree sufficing for its exclusion. Thus even the cha-
racters of the orders, though the longest, will not be found
troublesome.

The application of the Characteristic has been facilitated
in a great measure by separating the absolute characteristic
marks from the conditioned ones. It becomes still more
easy and expedititious, by taking particular notice of some
characters, which might be termed prominent. Such are a
metallic appearance ; a high degree of specific gravity,
particularly if the appearance be not metallic ; and a high
degree of hardness. The observation of these will imme-
diately decide whether an individual can belong to any
particular class, order, genus, or species. It is understood,
that if it be not thereby excluded, the other characters
must next be examined, till either an excluding one be
found, or if not, the individual may be considered as be-
longing to that class, order, &c. with which it has been
compared and found to agree.

§. 250. DETERMINATION OF INDIVIDUALS, BY MEANS
OF THE CHARACTERISTIC. EXAMPLE.

An individual, which has been carried through
the characters of the classes, orders, genera and spe-
cies, and whose systematic denomination has thus
been found, is said to have been determined. The
determination is complete, if the individual has been
traced to a species ; it is incomplete, if it has only
been brought under a certain order or genus.

§. 250. CHARACTERISTIC. 385

It is not difficult to arrive in this way at the determina-
tion of an individual, provided those properties can be as-
certained, which the complete determination requires. The
determination will be defective only in consequence of the
impossibility of observing at all, or at least with sufficient
accuracy, one or more of the characteristic marks in the
mineral.

In illustration of this, let us take the following example.
Let the form of the mineral which is to be determined, be a
combination of a scalene eight-sided pyramid, of an isosceles
four-sided pyramid, and of a rectangular four-sided prism ;
the cleavage parallel to the faces of two rectangular four-
sided prisms, in diagonal position to each other ; form and
cleavage therefore pyramidal, or belonging to the pyramidal
system. Let Hardness be = 6-5 ; Specific Gravity = 6*9.

In this case, both hardness and specific gravity are pro-
minent characters, and exclude the individual at once from
the first and third, but not from the second class : with
the characters of this class, its other properties also perfectly
agree. Hence the individual belongs to the second class.

Comparing the properties of the individual with the cha-
racters of the orders in the second class ; hardness and spe-
cific gravity will be found too great for the order Haloide ;
hardness too great for the orders Baryte and Kerate ; both
of them too great for the orders Malachite and Mica;
and specific gravity too great for the orders Spar and Gem.
But in the character of the order Ore, both hardness and
specific gravity fall between the fixed limits, and cannot
exclude the individual from this order. The other parts of
this character are now to be taken into consideration. If the
appearance of the individual be metallic, its colour must
be black, otherwise it cannot belong to the order Ore. But
the appearance is not metallic ; therefore the colour of the
individual is quite indifferent ; that is, this conditional cha-
racteristic mark does not affect the individual, and conse-
quently cannot decide. Since the appearance is not me-
tallic, the individual must exhibit adamantine or imperfect
metallic lustre. The first will be found, particularly in the

VOL. I. 2 B

386 CHAEACTEEISTIC. §. 250.

fracture. The following characteristic marks refer to mi-
nerals of a red, yellow, brown, or black streak ; and as the
individual gives none of these, its streak being uncoloured,
these characteristic marks do not come into consideration.
The next mark requires, that if hardness be =± 4*5 and less,
the streak should be yellow, red or black ; but hardness
is = 6-5, therefore the colour of the streak indifferent. If
hardness be = 6*5 and more, and streak uncoloured ; then
specific gravity must be = 6-5 and more. New this condi-
tion takes place ; hardness is = 6-5 ; streak is uncoloured.
But also the conditioned character takes place, specific
gravity being = 6-9, which is greater than 6-5.

In regard to the individual, which is to be determined,
all the characteristic marks constituting the Character of
the order Ore, may be divided into two parts. The first
part contains those which refer to the individual ; the se-
cond those which do not ; the last evidently cannot be de-
cisive. But with the first, all the properties of the indi-
vidual concur. These properties agree consequently with
the whole character of the order, as far as it is applicable
to the individual, and determine it to belong to the order
Ore, or, in shorter terms, to be an Ore.

It will be advisable to beginners, who do not yet possess a
sufficient practice in the use of the Characteristic, also to
compare the characters of the remaining orders, which will
enable them to find out any error they might have com-
mitted in the comparison of the individual with the charac-
ters of the preceding orders. In the present case, the
non-metallic appearance excludes the individual from the
orders Metal, Pyrites and Glance; hardness from the order
Blende; and both hardness and specific gravity from the
order Sulphur. This fully confirms the above determina-
tion, and we must now return to the order Ore for com-
paring the properties of the individual with the generic
characters which the order contains.

Considering again hardness and specific gravity as pro-
minent, the individual will be immediately excluded from
the genera Titanium-ore, Zinc-ore, and Copper-ore, but

§. 250. CHARACTERISTIC. 387

not from the genus Tin-ore. The form of the pyramidal
system, and the uncoloured streak, shew that it belongs to
this genus. If we compare the individual with the remain-
ing generic characters, we find that it is excluded from the
genus Scheelium-ore by its too great hardness, and too
little specific gravity ; from the genera Tantalum-ore, Ura-
nium-ore, Cerium-ore, Chrome-ore, Iron-ore, and Man-
ganese-ore, by hardness and specific gravity, both of
them being too great; as also by its uncoloured streak,
which only agrees with that genus from which the indivi-
dual differs most by its hardness and specific gravity. From
all this we infer that the individual cannot belong to any
other than to the fourth genus, and that we are therefore
entitled to give it the name of Tin-ore.

This genus contains but one species. The conclusion that
the individual must belong to this species, might never-
theless be erroneous. There could exist a second species of
this genus. Hence we must accurately consider the di-
mensions of the forms. If these coincide with the angles
given in the character, the highest degree of certainty, that
the individual belongs to or is pyramidal Tin-ore, will be
obtained.

The perfect determination of an individual depends, as the
above example has shewn, upon the possibility of correct-
ly ascertaining those three properties : viz. form, includ-
ing cleavage ; hardness ; and specific gravity. If one or
the other of these characteristic marks be wanting, the de-
termination will remain incomplete. It does not, however,
become prejudicial to the method, that minerals of this
kind cannot thoroughly be determined by its assistance.
It is exactly the same in the other parts of Natural His-
tory, in Zoology and Botany. The characteristic proper-
ties must be completely observable, otherwise a complete
determination will be impossible. In Mineralogy the
Characteristic affords sometimes more : it leads to a cor-
rect determination, even if the knowledge of the forms re-
mains imperfect, or if it is entirely wanting. But such a
determination wants evidence (§. 246.) j and for this reason

CHARACTERISTIC. §.

it will be a useful rule for beginners to occupy themselves
at first with the determination of such individuals as pre-
sent properties, which may be easily and fully investigated.
The rest will come of itself, when their knowledge of the
Mineral Kingdom, and particularly of the properties of
minerals, increases, and when they have, by experience,
acquired the skill to judge properly of form and cleavage,
at least so far as is necessary for the determination of the
system of crystallisation, even in those cases where form
and cleavage are somewhat difficult to be observed. This
exercise is particularly recommendable to every person
who intends to acquire a satisfactory knowledge of mine-
rals, with the help of the Characteristic.

§. 251. IMMEDIATE AND MEDIATE DETERMINA-
TION. EXAMPLE.

If a mineral can be determined without the help
or intervention of one or of several other minerals,
the determination is said to be an immediate one.
If, on the contrary, we must employ one or several
other minerals for this purpose, we only obtain a
mediate determination.

The immediate determination has been explained and
illustrated by an example in §. 250. An example will
likewise be useful in the mediate determination.

The variety of hemi-prismatic Augite-spar, which has
received the name of Amiantus, occurs in such very delicate
crystals, that even supposing they should be regular, their
form could not be observed, even through the most power-
ful magnifying instruments : it is the same with cleavage.
The crystals are flexible, like fibres of flax, their hardness
accordingly not to be ascertained. Their surface is so large
compared to their bulk, that wherever they may be placed in
water, or in another liquid, they neither sink nor rise :
although their specific gravity is not inconsiderable ; but
we halve no means to ascertain it. However, there are va-

§. 251. CHARACTERISTIC. 389

rieties, for the rest exactly agreeing with Amiantus, in
which the crystalline filaments are somewhat coarser. They
are no longer flexible, but still too weak to stand the ex-
periment of determining their hardness. Others are still
thicker : we may discern traces of their regular structure ;
yet on account of their minuteness, we cannot apply the
goniometer for taking their dimensions. They sink in
water, scratch prismatoidal Gypsum-haloide, but they
lose their coherence, if we try to pass them over a face of
rhombohedral Lime-haloide. At last we meet with va-
rieties, whose form and cleavage are more apparent
and observable, whose specific gravity is about three times
that of water, and the hardness between 5-0 and 6'0.
These allow of an immediate determination, and will be
placed by that process within the species of hemi-prismatic
Augite-spar. The mode of reasoning applied here will be
the following. 1 he variety preceding the last is the same
as that which has been determined ; those immediately pre-
ceding are again the same as the one immediately preceding
the last ; and thus we finally arrive at the Amiantus itself.
The determination of this mineral is effected by the assist-
ance of a greater or less number of varieties, interposed be-
tween one that is immediately determined, and another
which cannot be determined immediately ; the method em-
ployed is therefore that of the Mediae Determination. The
more general our knowledge of the productions of the Mi-
neral Kingdom, the greater facility we shall experience in
the mediate determination. Through this means, a great
number of minerals may be determined, and reduced to their
respective species, which could never have been ascertained
by immediate determination. The mediate determination
has indeed been hitherto very often applied, though it was not
clearly reduced or brought in connexion with the immediate
determination, upon which nevertheless both the correctness
and certainty of the mediate determination depends. The
mediate determination is peculiar to, and intimately con-
nected with, the natural-historical method of Mineralogy ;
hence we may infer, that nothing can escape this, which may
be determined by any other method.

CHARACTERISTIC. §.

it will be a useful rule for beginners to occupy themselves
at first with the determination of such individuals as pre-
sent properties, which may be easily and fully investigated.
The rest will come of itself, when their knowledge of the
Mineral Kingdom, and particularly of the properties of
minerals, increases, and when they have, by experience,
acquired the skill to judge properly of form and cleavage,
at least so far as is necessary for the determination of the
system of crystallisation, even in those cases where form
and cleavage are somewhat difficult to be observed. This
exercise is particularly recommendable to every person
who intends to acquire a satisfactory knowledge of mine-
rals, with the help of the Characteristic.

§. 251. IMMEDIATE AND MEDIATE DETERMINA-
TION. EXAMPLE.

If a mineral can be determined without the help
or intervention of one or of several other minerals,
the determination is said to be an immediate one.
If, on the contrary, we must employ one or several
other minerals for this purpose, we only obtain a
mediate determination.

The immediate determination has been explained and
illustrated by an example in §. 250. An example will
likewise be useful in the mediate determination.

The variety of hemi-prismatic Augite-spar, which has
received the name of Amiantus, occurs in such very delicate
crystals, that even supposing they should be regular, their
form could not be observed, even through the most power-
ful magnifying instruments : it is the same with cleavage.
The crystals are flexible, like fibres of flax, their hardness
accordingly not to be ascertained. Their surface is so large
compared to their bulk, that wherever they may be placed in
water, or in another liquid, they neither sink nor rise :
although their specific gravity is not inconsiderable ; but
we have no means to ascertain it. However, there are va-

§. 251. CHARACTERISTIC. 389

rieties, for the rest exactly agreeing with Amiantus, in
which the crystalline filaments are somewhat coarser. They
are no longer flexible, but still too weak to stand the ex-
periment of determining their hardness. Others are still
thicker : we may discern traces of their regular structure ;
yet on account of their minuteness, we cannot apply the
goniometer for taking their dimensions. They sink in
water, scratch prismatoidal Gypsum-haloide, but they
lose their coherence, if we try to pass them over a face of
rhombohedral Lime-haloide. At last we meet with va-
rieties, whose form and cleavage are more apparent
and observable, whose specific gravity is about three times
that of water, and the hardness between 5-0 and 6'0.
These allow of an immediate determination, and will be
placed by that process within the species of hemi-prismatic
Augite-spar. The mode of reasoning applied here will be
the following. 1 he variety preceding the last is the same
as that which has been determined ; those immediately pre-
ceding are again the same as the one immediately preceding
the last ; and thus we finally arrive at the Amiantus itself.
The determination of this mineral is effected by the assist-
ance of a greater or less number of varieties, interposed be-
tween one that is immediately determined, and another
which cannot be determined immediately ; the method em-
ployed is therefore that of the Mediae Determination. The
more general our knowledge of the productions of the Mi-
neral Kingdom, the greater facility we shall experience in
the mediate determination. Through this means, a great
number of minerals may be determined, and reduced to their
respective species, which could never have been ascertained
by immediate determination. The mediate determination
has indeed been hitherto very often applied, though it was not
clearly reduced or brought in connexion with the immediate
determination, upon which nevertheless both the correctness
and certainty of the mediate determination depends. The
mediate determination is peculiar to, and intimately con-
nected with, the natural-historical method of Mineralogy ;
hence we may infer, that nothing can escape this, which may
be determined by any other method.

390 CHARACTERISTIC. §. 252.

§. 252. BASE OF THE MEDIATE DETERMINATION.

The mediate determination entirely depends upon
the transitions in the series of characters (§.221.),

The mediate determination is effected by a series or con-
catenation of varieties, whose terminal member on one side
is immediately determinable. This series of varieties is
produced by the gradation in the differences of their pro-
parties, which likewise represent members of connected se-
ries, as it has been amply demonstrated above. But in
these series we observe the transitions ; and thus they
appear as the base upon which the mediate determination
is founded.

The transitions must always be employed with the neces-
sary precautions, as mentioned in §.221. But upon this sup-
position, the mediate determination is effected with a secu-
rity by no means inferior to that of the immediate deter-
mination, with which it is in the closest connexion. Mi-
neralogy is not the only, part of Natural History which
makes use of the mediate determination. It is necessary
also in Zoology and Botany, in both of which it is em-
ployed; yet it does not occur so frequently in these
sciences, because the individuals of the organic kingdoms
do not constitute compound masses ; the only case excepted
if the individual to be determined has not yet arrived at
the state of greatest perfection. In that case the botanist
compares a plant, which is not in flower, with another
which presents the perfect flower, and with other in-
dividuals, representing intermediate stages of efflores-
cence, between the perfect immediately determinable plant,
and that which he wants to determine, in perfect agree-
ment with the rules developed above ; and he knows by
experience how far he may extend this comparison, in or-
der to obtain results, upon the accuracy of which he may
rely.

CHARACTERS

OF THE

CLASSES, ORDERS, GENERA,
AND SPECIES.

CHARACTERS OF THE CLASSES.

CLASS I.

G. under 3-8.
No bituminous odour.
Solid : taste.

CLASS II.

G. above 1-8.
Tasteless.

CLASS III.

G. under 1-8.

Fluid : bituminous odour.
Solid : no taste.

392 CHARACTERISTIC. CLASS I,

CHARACTERS OF THE ORDERS.

CHARACTERS OF THE ORDERS OF CLASS I.

I. ORDER. GAS.

a = O'Oooi ... o-ooi4

Expansible.
Not acid.

II. ORDER. WATER.

G. = 1-0.

Liquid.

Without odour or taste.

III. ORDER. ACID.

G. = 0-0015 ... 3-7.
Acid.

IV. ORDER. SALT.

G. = 1-2 ... 2-9.

Solid.

Not acid. ,

CLASS II. OfcDERS. 395

CHARACTERS OF THE ORDERS OF CLASS II.

I. OEDER. HALOIDE.

Non-metallic.
Streak uncoloured.
H. = 1-5 ... 5-0.
G. = 2-2 ... 3-3.

Pyramidal or prismatic : H. == 4-0 and less,
cleavage imperfect, in oblique directions.
Tessular : H. = 4-0.
Cleavage monotomous, eminent : G. = 2-4

and less.

H. under 25 : G. = £-4 and less.
G. = 2-4 and less : H. under 2-5, no re-
sinous lustre.

CHAKACTEBISTIC, CLASS II.

II. ORDER. BARYTE.

Non-metallic.

Streak uncoloured or orange-yellow.

H. = 2-5... 5-0.

G. = 3-3 ... 7-3.

Cleavage monotomous : G. = 4-0 and less ;

or = 5*0 and more.
Lustre adamantine or imperfect metallic :

G. = 5-0 and more.

Streak orange-yellow : G. = 6*0 and more.
H. = 5-0 : G. under 4r5.
G. under 4-0 ; and H. = 5-0 : cleavage di-
prismatic.

III. ORDER. KERATE.

Non-metallic.

Streak uncoloured.

Cleavage not monotomous, not perfect peritomous.

H. = 1-0... 20.

G. above 5*5.

CLASS II.

ORDERS. 395

IV. ORDEB. MALACHITE.

Non-metallic.

Colour blue, green, brown.

Cleavage not monotomous.

H. = 2-0 ...5-0.

G. = 2-0 ... 4-6.

Colour or streak brown : H. = 3-0 and less,
G. above £*5.

Streak blue : H. = 4-0 and less.

Streak uncoloured : G. == 2-2 and less,
H. under 8-0.

V. OBDEB. MICA.

Cleavage monotomous, eminent.
H. = 1-0... 4-5.
G. = 1-8 ...3-2.

Metallic : G. under S-2.

Non-metallic : G. above

H. — 3*0 and more : rhombohedral.

G. under 2-5 : metallic.

396

CHARACTERISTIC. CLASS II.

VI. ORDER. SPAR.

Non-metallic.

Streak uncoloured ... brown, blue.

H. = 3-5 ... 7-0.

G. rr 2-0 ... 3-7.

Tessular : G. = 3-0 and less.
Rhombohedral : G. — 2-2 and less ; or H.

= 6-0.
H. = 4-0 and less : cleavage monotomous,

eminent.
H. above 6-0 : pearly lustre ; G. under 2-5

or above 2-8.

G. above 3-3 : forms hemi- or tetarto-pris-
matic; or H. = b'-O; no adamantine
lustre.

G. = 2-4 and less : not without traces of
form and cleavage.

CLASS IT. OllDERS. 397

VII. ORDER, GEM.

Non-metallic.

No metallic adamantine lustre.

Streak uncoloured.

H. = 5-5 ... 10-0.

G. = 1-9 ... 4-7.

H. = 6-0 and less : tessular, G. = 3-1
and more ; or G. = 2'4 and less, and no
traces of form and cleavage.
G. under 3-8 : no pearly lustre upon faces
of cleavage.

VIII. ORDER. ORE.

No green streak.
H. —2-5 ... 7-0.
G. =3-4 ... 7-4-.

Metallic : colour black.
Non-metallic : lustre adamantine or imper-
fect metallic.
Streak yellow or red : H = 3-5 and more,

G. = 4-8 and more.
Streak brown or black : H. = 5-0 and more ;

or cleavage monotomous.
H. = 4-5 and less : streak yellow, red or

black.

H. — 6-5 and more, and streak uncoloured :
G. = 6-5 and more.

398 CHARACTERISTIC. CLASS II.

IX. ORDER. METAL.

Metallic.
Colour not black.
H. = 0-0... 5-0.
G. = 57 ...20-0.

Colour grey : malleable, G. = 74 and
more.

H. above 4*0 : malleable.

X. ORDER. PYRITES.

Metallic.

H. = 3-0 ... 6-5.

G. =4-1 ,..7-7.

H. = 4-5 and less : G. under 5*3.

G. = 5'3 and less : colour yellow or red.

XI. ORDER. GLANCE.

Metallic.

Colour grey, black.
H. = 1-0 ... 40.
G. = 4-2 ... 7-6.

Cleavage monotomous ; G. being under 5-0
colour lead-grey.

G. above 74 : colour lead-grey.

CLASS II. ORDERS. 399

XII. ORDER. BLENDE.

Streak green, red, brown, uncoloured.
H. = 1-0 ... 4-0.
G. = 3-9 ... 8-2.

Metallic : colour black.

Non-metallic : lustre adamantine.

Streak green : colour black.

Streak brown ... uncoloured : G. between
4-0 and 4-2, form tessular.

Streak red : H. = 2-5 and less.

G. = 4-3 and more : streak red.

XIII. ORDER. SULPHUR.

Non-metallic.

Colour yellow, red, brown.

Prismatic.

H. = 1-0 ... 2-5.

G. = 1-9 ... 3-6.

Cleavage monotomous : G. = 3*4 and more,
G. above £•! : streak yellow or red

400 CHARACTERISTIC. CLASS III.

CHARACTERS OF THE ORDERS OF CLASS III.

I. ORDER. RESIN.

H. = 0-0 ... 2-5.
G. = 0-7 ... 1-6.

G. = !•& and more : streak uncoloured.

II. ORDER. COAL.

Streak brown, black,

H. = 1-Q ... 2-5.
G. = 1«2 ... 1-5.

ORDER I. GENERA AND SPECIES. 401

CHARACTERS

OF THE

GENERA AND SPECIES

OF THE

ORDERS OF CLASS I.
I. ORDER. GAS.

I. HYDROGEN-GAS. OdoUr.

G. = 0-0001 ... 0-0014.

1. PURE. Odour of hydrogen.

G. = 0-00012.

Pure Hydrogen-Gas. JAMESON. Vol. ii. p. 17.

2. EMPYREUMATIC. Empyreumatic odour.

G. = 0-0008.

Empyreumatic Hydrogen-Gas. J. ii. 18.

3. SULPHUROUS. Odour of putrid eggs.

G. = 0-00135.

Sulphuretted Hydrogen-Gas. J. ii. 19.

4. PHOSPHOROUS. Odour of putrid fish. ,

G. unknown.

Phosphuretted Hydrogen-Gas. J. ii. 10.

II. ATMOSPHERIC-GAS. Without odour or sapidity.

G. =0-001 ... 0-0015.

VOL. I. 2 C

402 CHARACTERISTIC. CLASS I.

1. PURE. As above.

Pure Atmospheric- Air. J. ii. 20.

II. ORDER. WATER.

I. ATMOSPHERIC-WATER. Without odour or
sapidity.

1. PURE. As above.

Water. ii. 21.

III. ORDER. ACID.

I. CARBONIC-ACID. Taste slightly acid.

G. = 0-0018.

1. GASEOUS. Expansible.
Taste acidulous, pungent.

Aeriform Carbonic Acid. J. ii. 22.

II. MURIATIC- ACID. Odour pungent.

Taste strongly acid.
G. = 0-0023.

1. GASEOUS. Expansible.
Odour pungent.

Aeriform Muriatic Acid. J. ii. 23.

III. SULPHURIC-ACID.

G. = 0-0025 ...1-9.

ORDER IV. GENERA AND SPECIES. 403

Expansible : odour sulphurous.
Liquid : taste strongly acid.

1. GASEOUS. Expansible.

G. = 0-0028.

Aeriform Sulphuric Acid. J. ii. 23.

2. LIQUID. Liquid.

G. = 1-8 ... 1-9.

Liquid Sulphuric Acid. J. ii. 24.

IV. BoRAcic-AciD. Solid.

G. — 1-4 ... 1-5.

1. PRISMATIC. Prismatic.

Taste acidulous, afterwards bitter and cooling, lastly

sweetish.
G. = 1-4 ... 1-5.

Sassolinc, ii, 25.

V. ARSENIC-ACID. Solid.

G. above 3-0.

1. OCTAHEDRAL. Tessular.
Cleavage, octahedron.
Taste sweetish astringent.
H. = 1-5.
G. = 3-6 ... 3-7.

Octahedral Arsenic-Acid. J. ii. 26.

IV. ORDER. SALT

I. NATRON-SALT. Prismatic.
Taste pungent, alealine.
H. = 1-0 ... 1-5.
G. = 1-4 .., 1-6.

404 CHARACTERISTIC. CLASS I.

1. HEMI.PRISMATIC. Henri-prismatic. — =79° 41'. In-

clination == 3° 0'.
Cleavage, — = 58° 52'. Less distinct, Pr + CD and

(&r +03)3 =76° 28'.
H. = 1*0 ... 1-5.
G. = 1-4 ... 1-5.

Prismatic Natron. J. ii. 27-

2. PRISMATIC. Prismatic. £ r = 83° 50', (£r + os)s =

107° 50'.

Cleavage, Pr + <», very indistinct.
H. = 1-5.
G. = 1-5 ... 1-6.

Prismatic Natron. J. ii. 29.

II. GLAUBER-SALT. Prismatic.

Taste cool, then saline and bitter, weak.
H. = 1-5 ... 2-0.
G. = 1-4 ... 1-5.

1. PRISMATIC. Hemi-prismatic. — = 93° 12'. Incli-
nation = 14° 4K
Cleavage, f r 4- co, perfect. Traces of — — = 72° 15'.

and of Pr -f 03.

Prismatic Glauber-Salt. J. ii. 31.

III. NITRE-SALT. Prismatic.
Taste saline, cool.
H. = 2-0.
G. = 1-9 ... 2-0.

1. PRISMATIC. Prismatic. P =* 132° 22', 91° 15', 107° 43'.

ORDER IV. GENERA AND SPECIES. 405

Cleavage, P + w = 120°. Somewhat more distinct
Pr + co.

Nitre. ii. 34.

IV. ROCK-SALT. Tessular.

Taste saline.

H. = 2-0.

G. = 2-2 ... 2-3.

1. HEXAHEDHAL. Tessular.
Cleavage, hexahedron.

Hexahedral Rock-Salt. J. ii. 36.

V. AMMONIAC-SALT. Tessular.

Taste saline, pungent.
H. = 1-5 ... 2-0.
G. = 1-5 ... 1-6.

1. OCTAHEDRAL. Tessular.
Cleavage, octahedron.

Octahedral Sal Ammoniac. J. ii. 39.

VI. VITRIOL-SALT. Prismatic.

Taste astringent.
H. = 2-0 ... 2-5.
G. = 1-8 ... 2-3.

M

1. HE MI-PRISMATIC. Hemi-prismatic. — = 101° 35'.

Inclination = 14° 20'.
Cleavage, P — oo. Somewhat less distinct, P + co =

82° 21'. Inclination of P — co on P + oo = 99° 23'.
Colour green.
H. = 2-0.
G. = 1-8 ... 1-9.

Green Vitriol. J. ii. 41

406 CHARACTEHISTIC. GLASS I.

2. TETARTO-PRISMATIC. Tetarto-prismatic.

Cleavage, two faces, one of them more apparent ; both

indistinct. Inclination = 124° 2'.
Colour blue.
H. = 2-5.
G. = 2-2 ... 2-3.

Blue Vitriol. J. ii. 44.

3. PRISMATIC. Prismatic. P + 03 =90° 42'.

Cleavage, Pr + co, perfect.
Colour white.
H. = 2-0 ... 2-5.
G. = 2-0 ... 2-1.

White Vitriol. J. ii. 46.

VII. EPSOM-SALT. Prismatic.

Taste saline, bitter.
H. = 2-0 ... 2-5.
G. = 1-7 ... 1-8.

1. PRISMATIC. Prismatic. P + co = 90° 38'.
Cleavage, Pr + co, perfect.

Prismatic Epsom Salt. J. ii. 48.

VIII. ALUM-SALT. Tessular.
Taste sweetish, astringent.
H. = 2-0 ... 2-5.

G. = 1-7 ... 1-8.

1. OCTAHEDRAL. Tessular..
Cleavage, octahedron.

ii. 5Q.

IX. BORAX-SALT. Prismatic.

Taste sweetish, feebly alcaline.
H, = 2-0 ... 2-5.
G. = 1-7 ... 1-8.

OHDER IV. GENEEA AND SPECIES. 407

1. PRISMATIC. Hemi-prismatic. '.- = 120° 23'. Incli-
nation = 0.

Cleavage, (Pr + oo)3 = 80° 9'. Somewhat more dis-
tinct, Pr + co.

Borax. ii. 52.

X. BRITHYNE-SALT. Prismatic.

Taste saline, feebly astringent.
H. = 2-5 ... 3-0.
G. = 2-75 ... 2-85.

fe

1. PRISMATIC. Hemi-prismatic. — = 120° 22'. Incli-
nation = 22° 49'.

Cleavage, P — . oo, perfect. Traces of P + oo = 80° 6'.
Inclination of P — oo on P + oo = 104° 28'.

Glaulerite. ii. p. 54.

Bloedite. iii. 79.

Mascagninc. iii. 125.

Nitrate of Soda, of Chemists. iii. 132.

Polyhalite. iii. 141.

Sulphate of Cobalt. Chem. iii. 145.

Sulphate of Potash. Chem. iii. 159.

Trona. iii. 164.

408 CHARACTERISTIC. CLASS II,

CHARACTERS

OF THE

GENERA AND SPECIES

OF THE

ORDERS OF CLASS II.

I. ORDER. HALOIDE.

I. GYPSUM-HALOIDE. Prismatic.
H. =. 1-5 ... 3-5.
G. = 2-2 ... 3-0.

G. above 2-5 : cleavage in three direc-
tions, perpendicular to each other, one
of them being less distinct.

•p

1. PRISMATOIDAL. Hemi-prismatic. __=143°52'. In-
clination = 9° IT.

Cleavage, £r + co, perfect and eminent. — — = 66*

2
52'. Pr + co.

H. = 1-5 ... 2-0.
G. = 2-2 ... 2-4.

Gypsum. ii. 57.

ORDEK I. GENERA AND SPECIES. 409

2. PRISMATIC. Prismatic. P = 121° 32', 108° 35', 99° 7'.
Cleavage, Pr + os. Pr + oo. Less distinct, P _ os.

Traces of P + » = 100° 8'.
H. = 3-0 ... 3-5.
G. = 2-7 ... 3-0.

Anhydrite. ii. 62.

II. CRYONE-HALOIDE. Prismatic.

Cleavage in three directions, perpendicular
to each other, one of them being more
distinct.

H. = 2-5 ... 3-0.

G. = £-9 ... 3-0.

1. PRISMATIC. Prismatic.

Cleavage, P — co. Less distinct, Pr -f os. Pr + os
Traces of P.

Cryolite. ii. 66.

III. ALUM-HALOIDE. Rhombohedral.

H. = 5-0.

G. = 2-5 ... 2-8.

1. RHOMBOHEDRAL. Rhombohedral. R = 92° 50'.
Cleavage, R — oo. Less distinct, R.

Rhoniboidal Alumstone. J. ii. 67.

IV. FLUOE-HALOIDE. Tessular, rhombohedral.

H. = 4-0 ... 5-0.
G. = 3-0 ... 3-3.

Rhombohedral : cleavage peritomous.

1. OCTAHEDRAL. Tessular.
Cleavage, octahedron.
H. = 4-0.
G. = 3-0 ... 3-3.

Fluor.

410 CHARACTERISTIC. CLASS II.

2. .RHOMBOHEDRAL. Di-rhombohedral. P = 142° 2<y,
80° 25'. (P + n)m hemi-di-rhombohedral with pa-
rallel faces.

Cleavage, Jl •— ». P + oo.
H. = 5-0.
G. = 3-0 ... 3-3.

Apatite. ii. 73.

V. LIME-HALOJDE. Rhombohedral, prismatic.

Cleavage, rhombohedral and paratomous, or

prismatoidal.
H.= 3-0 ... 4-5.
G. = 2-5 ... 3-2.

H. above 4-0 : G. = 2-8 and more.

G. = 2-9 and more : H. == 3-5 and more.

1. PRISMATIC. Prismatic. P = 112° 39', 93° 33X, 123° 34'.

Cleavage, Pr _ 1 - 108° 8'. (Pr + os)3 = 63° 44'.

More distinct Pr + co.
H. = 3-5 ... 4-0.
G. = 2-6 ... 3-0.

Arragonite. ii. 79.

2. RHOMBOHEDRAL. Rhombohedral. R = 105° 5'.

Cleavage, R.

H. = 3-0.

G. = 2-5 ... 2-3.

Rhomboidal Limestone. J. ii. 83.

3. MACRO TYPO us. Rhombohedral. R = 106° 15'.

Cleavage, R.
H. = 3-5 ... 4-0.
G. = 2-8 ... 2-95.

Dolomite. ii. 93.

4. BRACHYTYPOUS. Rhombohedral. R = 107° 22;.

Cleavage, R.

ORDER II. GENERA AND SPECIES. 411

H. = 4-0 ... 4-5.
G. = 3-0 ... 3-2.

Breunnerite. ii. Of?.

4. PARATOMOUS. Rhombohedral. II = 106' 12'.
Cleavage, R.
H. = 3-5 ... 4-0.
G. = 2-95 ... 3-1.

Ankerite. ii. 100.

Childrenite ? iii. 85.

Flucllite. iii. 101.

Hopette. iii. 109.

Magnesite. iii. 121.

Pharmacolite? iii. 135.

Rosclite. iii. 147.

Wavellitc. iii. 169.

II. ORDER. BARYTE.

I. PARACHROSE-BARYTE. Rhombohedral.
Cleavage paratomous.
H. = 3-5 ...4-5.
G. =3-3 ... 3-9.

1. BRACK YTYPOUS. Rhombohedral. R = 107° 0'.

Cleavage, R.
H. = 3-5 ... 4-5.
G. = 3-6 ... 3-9.

Rhomboidal Sparry Iron. J. ii. 101.

2. MACROTYPOUS. Rhombohedral. R = 106° 51'.

Cleavage, R.

H. = 3-5.

G. = 3-3 ... 3-6.

Rhomboidal Red Manganese, J. ii. 10G.

CHARACTERISTIC. CLASS II.

II. ZINC-BARYTE. Rhombohedral, prismatic.

H. = 5-0.

G. =3-3 ...4-5.

Rhombohedral : G. above 4-0.

1. PRISMATIC. Prismatic. Pr _ 1 = 128° 27'.

Cleavage, Pr = 116° 40'. Somewhat more distinct,
(£r + os) 3 = 76° 7'-

H. =: 5-0.

G. = 3-3 ... 3-6.

Prismatic Calamme. J. ii. 108,

2. RHOMBOHEDRAL. Rhombohedral. R = 107° 4(X.

Cleavage, R.

H. = 5*0.

G. = 4-2 ... 4-5.

Rhomboidal Calamme. J. ii. 111.

III. SCHEELIUM-BARYTE. Pyramidal.

H. = 4-0 ... 4-5.
G. = 6-0.... 6-1.

1. PYRAMIDAL. Pyramidal. P = 107° 27', 113° 35'.

Combinations hemi-pyramidal with parallel faces.
Cleavage, P + 1 = 100° 8', 130° 20'. P. J» _- w.
Pyramidal Tungsten. J. ii. 113.

IV. HAL-BARTTK. Prismatic.

H. =3-0 ... 3-5.
G. = 3-6 ... 4-7.

1. PERITOMOUS. Prismatic.

Cleavage, P +00 5= 117° 19'. Less distinct, £r.

Traces of Pr + 05.
H. = 3-5.
G. = 3-6 ... 3-8.

Stronticmite. ii. 116.

ORDER II. GENERA AND SPECIES. 413

2. Di-PniSMATic. Prismatic.

Cleavage, P -f co = 118° 30'. Pr + 03. Pr.
H. = 3-0 ... 3-5,
G. = 4-2 ... 4-4.

Witherite. ii. 119.

3. PRISMATIC. Prismatic. Pr = 105° 6' ; (Pr + oo)3

= 77° 27'.
Cleavage, Pr = 78° 18'. Somewhat easier, Pr + co.

Traces of P — oo.
H. = 3-0 ... 3-5.
G. = 4-1 ... 4-7-

Heavy. Spar. J. ii, 121.

4. PRISMATOIDAL. Prismatic. Pr=103°58'. (Pr + oo)s

= 78° 35'.
Cleavage, Pr = 76° 2'. More^pparent, Pr + w. Less

distinct, P — 03.
H. = 3-0 ... 3-5.
G. = 3-6 ... 4-0.

Celestine. ii. 126.

V. LEAD-BARYTE. Rhombohedral, pyramidal,

prismatic.
H. = 2-5 ... 4-0.
G. = 6-0 ... 7-3.

H. above 3-5 : G. = 6-5 and more.

1. DI-PRISMATIC.. Prismatic. P = 130° (X, 108° 28,

92° 19'.

Cleavage, Pr = 117° 13'. (Pr + OD)» = 69° 207.
H. = 3-0 ... 3-5.
G. = 6-3 ... 6-6.

Di-Prismatic Lead-Spar. J. ii, 130.

2. RHOMBOHEDRAL. Di-rhombohedral. P = 142° 12',

80° 44'.
Cleavage, P. P -f co. Both very indistinct.

414 CHARACTERISTIC. CLASS II.

H. = 3-5 ... 4-0.
G. = 6-9 ... 7'3.

Rliomloidal Lead-Spar. J. ii. 133.

p

3. HEMI-PRISMATIC. Hemi-prismatic. = 119° 0'.

Inclination = 12° 307.

Cleavage, P + o> = 93° 40'. Pr + os. Pr 4- *.
Streak orange yellow.
H. = 2-5.
G. = 6-0 ... 6-1.

Red Lead-Spar. J. ii. 137-

4. PYRAMIDAL. Pyramidal. P = 99° 40', 131° 35'.

Cleavage, P. Less distinct P — oo.

H. = 3-0.

G. = 6-5 ... 6-9.

Yellow Lead-Spar. J. ii. 140.

5. PRISMATIC. Prismatic. Pr = 104° 55'; (Pr -f cc)3 =

78° 45'.

Cleavage, Pr = 76° 11'. Pr + oo.
H. = 3-0.
G. = 6-2 ... 6-3.

Sulphate of Lead. J. ii. 142.

p

6. AXOTOMOUS. Hemi-prismatic. - = 72° 3C'. In-

clination = 0° 29'.

Cleavage, P — oo, perfect and eminent.
H. = 25.
G. = 6-2 ... 6-4.

Sulpliato-tri-Carlonate of Lead. BROOKE. ii. 144.

VI. ANTIMONY-BARYTK. Prismatic.
H. = 2-5 ... 3-0.
G. = 5-5 ... 5-6.

1. PRISMATIC. Prismatic. Pr = 70° 32'.

Cleavage, (Pr + oa)3 = 136° 58', highly perfect. Pr 4- oo.

ORDER III. GENERA AND SPECIES. 415

Prismatic Wliitc Antimony-. J. ii. 151.

Corneous Lead. J. ii. 150.

Cupreous Sulphate of Lead. BROOKE. ii. 149.
Cupreous Sulpliato- Carbonate ofLead.'BnoOK'E. ii. 1 49.

Fluate of Cerium 9 Chem. iii. 100.

Hemi-prismatic Hal-barytc. iii. 76.

Peritomous Lead-laryte. ii. 151.

Plombgomme 9 iii. 140.

Stromnite 9 iii. 159.

Sulphato-Carlonate of Lead. BKOOKE. ii. 148.

Tungstate of Lead. Chem. iii. 165.

Yttro-Ccrite9J. iii. 172.

III. ORDER. . KERATE.

I. PEARL-KERATE. Tessular, pyramidal.
H. = 1-0 ... 2-0.
G, = 5-5 ... 6-5.

1. HEXAHEDRAL. Tessular.

Cleavage, none.
Malleable.
H. = 1-0 ... 1-5.
G. = 5-5 ... 5-6.

Hexahedral Corneous Silver. J. ii. 154.

2. PYRAMIDAL. Pyramidal. P = 126° 31;, 79* 3'.

Cleavage, P 4-'o5, imperfect.

Sectile.

H. = 1-0 ... 2-0.

G. = 6-4 ... 6-5.

Calomel. ii. 15G.

416 CHARACTERISTIC. CLASS II,

IV. ORDER. MALACHITE.

I. STAPHYLINE-MALACHITE. Amorphous.
H. = 2-0 ... 3-0.
G. = 2-0 ...

Cleavage none.
Fracture conchoidal.

Chrysocolla. ii. 158.

II. LIROCONE-MALACHITE. Tessular, prismatic.

H. = 2-0 ... 2-5.
G. = 2-8 ... 3-0.

1. PRISMATIC. Prismatic.

Cleavage, Pr = 71° 59'. P + co = 1 19° 45' imperfect.
Streak pale verdigris-green ... sky-blue.
H. = 2-0 ... 2-5.
G. = 2-8 ... 3-0.

Prismatic Liricomte. J. ii. 160.

2. HEXAHEDRAL. Semi-tessular with inclined faces,

Cleavage, hexahedron, imperfect.
Streak pale olive-green ... brown.
H. = 2-5.
G. = 2-9 ... 3-0.

Hexdhedral Liriconite. J. ii. 162.

III. OLIVE-MALACHITE. Prismatic.

Colour or streak neither blue nor bright

green.

H. = 3-0 ... 4-0.
G. = 3-6... 4-6.

1. PRISMATIC. Prismatic.

Cleavage, Pr = 110° 50'. P + oo =* 92° 30'. Both

very indistinct.
Streak olive-green ... brown.

ORDER IV. GENERA AND SPECIES. 417

H. = 3-0.

G. = 4-2 ... 4-6.

Prismatic or Acicular Olivenite. J. ii. 164.

2. DI-PRISMATIC. Prismatic, fr = 111° &&. P+w

= 95° 2'.

Cleavage, ?r + CD. Pr + eo. Both very indistinct.
Streak olive-green.
H. = 4-0.
G. »= 3-6 ... 3-8.

Di-prismatic Olivenite. J. ii. 166.

IV. AZURE-MALACHITE. Prismatic.

Colour blue.
H. = 3-5 ... 40.
G. = 3-7 ... 3-9.

1. PRISMATIC. Henri-prismatic. _ = 117° 37'. Incli-
nation = 2° 21'.
Cleavage, (£r + os)3 = 59° 14'. Less distinct, P — , oo.

Traces of £ r = 99° 32'.
Streak blue.

Prismatic Blue Malachite. J. ii. 167.

V. EMERALD-MALACHITE. Rhombohedral.

H. = 5-0.

G. = 3-2 ...3-4.

1. RHOMBOHEDHAL. Rhombohedral. R + 1 = 95° 48'.
Cleavage, R = 126° 17'.
Streak green.

Dioptase. ii, 171,

VI. HABRONEME-MALACHITE. Prismatic.

Colour or streak bright green.

VOL. I. 2 D

418 CHARACTERISTIC. CLASS II.

H. = 3-5 ... 5-0.
G. = 3-6 ... 4-3.

1. PRISMATIC. Hemi-prismatic. (Pr + os)3 = 38° 56'.

pr j

Cleavage, traces of . and Fr + «.

Streak emerald-green.

H. = 5-0.

G. = 4-0 ... 4-3.

Prismatic Green Malachite. J. ii. 173.

2. HEMI-PRISMATIC. Hemi-prismatic. _ = 139° 17'«

Inclination = 0. P + co = 103° 42'.
Cleavage, — 5 = 61° 49', and Pr + oo highly perfect.

Streak grass-green, apple-green.
H. = 3-5 ... 4-0.
G. = 3-6 ... 4-05.

Common Malachite. J. ii. 175-

Atacamite. . iii- 74.

Brochantit€. iii. 81-

Euchroite. iii. 94.

Green Iron-Earth ? WERNER. iii. 106.

Radiated Acicular Olivenite. J. iii. 144.

Scorodite? iii. 149.

Vauquelinite? iii. 167-

Velvet-Slue Copper ? J. iii. 168,

V. ORDER. MICA.

I, EucHLORK-MicA. Rhombohedral, pyramidal,
prismatic.

ORDEB V. GENEBA AND SPECIES. 419

Streak green ... yellow.
H. = 1-0 ... 2-5.
G. = 2-5 ... 3-2.

Streak green : G. = 2-6 and less; or
= 3-0 and more.

1. RHOMBOHEDRAL. Rhombohedral. R = 68° 45'.

Cleavage, R — co.

Streak emerald-green, apple-green.

H. = 2-0.

G. =2-5 ...2-8.

Hemi-prismatic Copper-Mica. J. ii. 178.

2. PRISMATIC. Prismatic.

Cleavage, P — co.
Laminae flexible.
Streak pale, apple-greeto.
H. = 1-0 ... 1-5.
G. = 3-0 ... 3-2.

Kupferschaum. WERNER. ii. 18(T.

3. PYRAMIDAL. Pyramidal. P = 95° 46', 143° 2'.

Cleavage, P — co.
Laminae not flexible.
Streak green ... yellow.
H. = 2-0 ... 2-5.
G. = 3-0 ... 3-2.

Uranite. ii. 182.

II. COBALT-MICA. Hemi-prismatic.

Cleavage parallel to the plane of inclination.

H. = 2-5.

G. = 2-9 ... 3-1.

1. PRISMATIC. Hemi-prismatic. — == 118° 23'. Inclina-
tion = 9° 47V- ?E = 55° 9f.
2-

420 CHARACTERISTIC. CLASS II.

Cleavage, Pr -}- os.
Streak red ... green.

Prismatic Red Cobalt. J. ii. 184.

III. IRON-MICA. Prismatic.

Streak uncoloured ... blue.
H. = 2-0.
G. = 2-6 ... 2-7.

1. PRISMATIC. Hemi-prismatic — = 119° 4'. Inclina-
tion = 10° 53'. — = 54° 13'.

2
Cleavage, Pr + «.

Vivianite. ii. 188.

IV. GRAPHITE-MICA. Rhombohedral.

H. = 1-0 ... 2-0.
G. = 1-8 ... 2-1.

1. RHOMBOHEDRAL. Di-rhombohedral.
Cleavage, R — 03.
Metallic.
Streak black.

Plumbago. ii. 191.

V. TALC-MICA. Rhombohedral, prismatic.

Streak uncoloured ... green.
H. = 1-0 ... 2-5.
G. = 2-7 ... 3-0.

Streak green : G. = 2-8 and less.
Hemi-prismatic: cleavage perpendicular
to the plane of inclination.

1. PRISMATIC. Prismatic. P + os = 120° (nearly).
Cleavage, P — os.

0111>E11 VI. GENEllA AND SPECIES. 421

Laminae flexible.

Streak uncoloured ... green.

H. = 1-0 ... 1-5.

G. = 2-7 ... 2-8.

Prismatic Talc-Mica. J. ii. 193.

2. RHOMBOHEDRAL. Di-rhombohedraL
Cleavage, R — w.
Laminae elastic.
Streak uncoloured.
H. = 2 0 ... 2-5.
G. = 2-8 ... 3-0.

Rhomboidal Talc-Mica. J. ii. 198.

VI, PEARL-MICA. Rhombohedral.
H. = 3-5 ... 4-5.
G. = 3-0 ... 8-1. .

1. RHOMBOHEDRAL. Di-rhombohedral.
Cleavage, R — co.
Streak uncoloured.

Margarite. ii. 204.

Cronstedtite ? iii. 90.

Hydrate of Magnesia. Chem. iii. . 1 1 2.

Pyrosmalite. iii. 143.

VI. ORDER. SPAR.

I. SCHILLER-SPAB. Prismatic.

Cleavage monotomous, eminent.
H. = 3-5 ... 6-0.
G. = 2-6 ... 3-4.

CHARACTERISTIC. CLASS II.

1. DIATOMOUS. Prismatic.

Cleavage prismatoidal.
Lustre metallic-pearly.
H. = 3-5 ... 4-0.
G. = 2-6 ... 2-8.

Diatomous Schiller-Spar. J. ii. 206.

•p

2. HEMI-PRJSMATIC. Hemi-prismatic. — .

Cleavage, Pr + oo. Less distinct, _ = 72° and P + co

= 94°. Traces of Pr + vs.
Lustre metallic-pearly.
H. = 4-0 ... 5-0.
G. == 3-0 ... 3-3.

Bronztte. ii. 207-

3. PRISMATOIDAL. Prismatic.

Cleavage, Pr + co. Less distinct, P + co = 93° (near-
ly). Pr + co.
Lustre metallic-pearly.
H. = 6-0.
G. = 3-3 ... 3-4.

Hypcrsthene. ii. 210.

4. PRISMATIC. Prismatic.

Cleavage, Pr + co. Somewhat less distinct, P + co =

124° 30'. fr-fos.
Lustre almost metallic-pearly.
H. = 5-0 ... 5-5.
G. = 3-0 ... 3-3.

Anthophyllite. ii. 211.

II. DiSTHENE-SpAR. Prismatic.
H. = 5-0 ... 7-0.
G. = 6-0 ... 3-7.

1. PRISMATIC. Tetarto-prismatic.

Cleavage, two faces, one of them more distinct, per-
fect and eminent. Inclination = 100° 507.

Kyanite. ii. 213.

ORDER VI. GENEEA AND SPECIES.

III. TRIPHANE-SPAR. Prismatic.

Cleavage, somewhat more distinct in one

direction.
Colour not blue.
H. - 6-0 ... 7-0.
G. = 2-8 ... 3-1.

1. PRISMATIC. Prismatic.

Cleavage, P + co = 93°. Somewhat more distinct,

Pr + co.

H. = 6-5 ... 7'0.
G. = 3-0 ... 3-1.

Spodumene. ii. 216.

2. AXOTOMOUS. Prismatic.

Cleavage, P + os = 99° 30'. More distinct, P — ee.
H. = 6-0 ... 7-0.
G. = 2-8 ... 3-0.

Prebnite. ii. 217.

IV. DYSTOME-SPAR. Prismatic.

Cleavage difficult ; lustre of the fracture re-
sinous.

Colour not blue.
H. = 50 ... 5-5.

G. = 2-9 ... 30.

*

I. PRISMATIC. Hemi-prismatic. _ =122°0/. Inclina-
tion = 1° 41' 30".

Cleavage, P + wj= 71° 307, very indistinct ; a little
more distinct, Pr + co.

Datolite. ii. 220.

V. KOUPHONE-SPAR. Tessular, rhombohedral,

pyramidal, prismatic.

CHARACTERISTIC. CLASS II.

H. = 3-5 ... 6-0.
G. = 2-0 ... 2-5.

Pyramidal : cleavage axotomous, eminent.

H. = 6-0 : tessular.

1. TRAPEZOIDAL. Tessular.

Cleavage, hexahedron, dodecahedron, imperfect.
H. = 5-5 ... 6-0.
G. = 2-4 ... 2-5.

Leucite. ii. 224.

2. DODECAHEDRAL. Tessular.

Cleavage, dodecahedron distinct.
H. = 5-5 ... 6-0.
G. = 2-25 ... 2-35.

Sodalite. ii. 225.

3. HEXAHEDRAL. Tessular.

Cleavage, hexahedron, imperfect.

H. = 5-5.

G. = 2-0 ... 2-2.

Analcime. ii. 227-

4. PARATOMOUS. Prismatic.

Cleavage, P. Pr + co. Somewhat easier Pr + ep.

Imperfect.
H. = 4-5.
G. = 2-3 ... 2-4.

Harmotome. ii. 229.

5. RHOMBOHEDRAL. RhombohedraL R = 94° 46'.

Cleavage, R.
H. = 4-0 ... 4-5.
G. = 2-0 ... 2-1.

Chabasite. ii. 232.

6. DIATOMOUS. Hemi-prismatic. P + co = 86° 15'.

Cleavage, Pr -f oo. Traces of Pr + o».
H. unknown.
G. = 2-3 ... 2-4.

Laumonite. ii. 234.

OltDEll VI. GENERA AND SPECIKS. 425

7- PRISMATIC. Prismatic. P = 143° 20', 142° 40', 53° 20'.
Cleavage, P + co = 91° 0'.
H. = 5-0 ... 5-5.
G. = 2-2 ... 2-3.

Mesotype. ii. 236.

8. PRISMATOIDAL. Prismatic. P = 119° 15', 114° 0', 96° 0'.

Cleavage, Pr + co, eminent.
H. = 3-5 ... 4-0.
G. = 2-0 ... 2-2.

Stmtoe. ii. 239.

9. HE MI-PRISMATIC. Hemi-prismatic. Irregular six-

sided prism of 129° 40', 116° 20', and 114° 0'.
Cleavage, Pr + co, very eminent.
H. = 3-5 ... 4-0.
G. = 2-0 ... 2-2.

Heulandite. ii. 242.

10. PYRAMIDAL. Pyramidal. P = 104° 2', 121° 0'.

Cleavage, P — co, eminent. [P + co J imperfect.
H. = 4-5 ... 5-0.
G. = 2-2 ... 2-5.

Apophyllite. ii. 244.

1J. AXOTOMOUS. Prismatic. P=106°52',101037',120°34'.

Cleavage, P — co, eminent. Less distinct, ?r + co.

Pr + co.

H. = 4-5 ... 5-0.
G. = 2-2 ... 2-5.

Apophyllite. ii. 246.

Brewsterite. iii. 0.

Comptonite. iii. 89.

Gmelinite. iii. 104.

Levyne. iii. 120.

Mesolc. iii. 126.

Mesolinc. iii. 126.

Sarcolite. iii. 147.

Thomsonite. iii. \Q2.

426 CHARACTERISTIC. CLASS II.

VI. PETALINE-SPAR. Prismatic.

H. = 6-0 ... 6-5.
G. = 2-4 ... 2-5.

1. PRISMATIC. Prismatic.

Cleavage, a prism of 95° (nearly). More distinct,
Pr + 05.

Petalite. ii. 248.

VII. FELD-SPAR. Rhombohedral, pyramidal,

prismatic.
H, = 5-0 ... 6-0.
G. = 2-5 ... 2-8.

H. = 5-5 and less : form pyramidal,
cleavage not axotomous.

1. RHOMBOHEDRAL. Di-rhombohedral. P = 139° 19*,

88° 6'.

Cleavage, R — . oo. R + oo.
H. = 6-0.
G. = 2-5 ... 2-6.

Nepheline. ii. 250.

«

2. PRISMATIC. Hemi-prismatic, = 126° 12'. Inclina-

tion = 0.

Pr

Cleavage, — _ = 64° 34', perfect. Pr + os perfect,

but often interrupted. (Pr + co)3 = 118° 52', im-
perfect.

H. = 6-0.

G. = 2-5 ... 2-6.

Prismatic Felspar. J. ii. 251.

3. PX-RAMIDAL. Pvramidal. P = 136° 7;, 63° 48'.

Cleavage, P + oo. [P + oo]. Traces of P — oo.
H. = 5-0 ... 5-5.
G. = 2-5 ... 2-8.

ORDER VI. GENERA AND SPECIES.

Wernerlte. ii. 264.

Albite. ii. 255.

Anorthite. iii. 71.

Elaolite. ii. 93.

Felspar from Baveno. Vulg. ii. 258.

Felspar from the Saualpe. Vulg. ii. 257.

Labradorite. ii. 257.

LatroUte. iii. 118.

Nuttallite. iii. 133.

VIII. AUGITE-SPAR. Prismatic.
Lustre not metallic-pearly.
H. = 4-5 ... 7-0.
G. = 2-7 ... 3-5.

H. above 6-0 : G. = 3-2 and more.
G. under 3 2 : cleavage oblique-angular
peritomous, perfect.

w w

1. PARATOMOUS. Hemi-prismatic. — = 120° (X. •_£ =

2 2

73° 54'. Inclination = 0.
Cleavage, (Pr + os)^ = 87° 5'. Pr +00. Pr + oo.

P

Sometimes _.
2

H. *= 5-0 ... 6-0.
G. = 3-2 ... 3-5.

Pyroxene. ii» 268.

£

2. HEMI-PRISMATIC. Hemi-prismatic. — . = 148° 39'.

& = 75° 2'. Inclination = 0.
2

Cleavage, (?r + oo) 3 = 1 24° 34'. Less distinct, Pr + w .
Pr + oe.

4-28 CHARACTERISTIC. CLASS II.

H. = 5-0 ... 6-0.
G. = 2-8 ... 3-2.

Amphibole. ii. 274.

* «r

3. PRISMATOIDAL. Hemi-prismatic. _ = 70° 33'- _

2 2

= 63° 43'. Inclination = 0° 33'.

Cleavage, — _ = 64° 36'. More distinct, £ r -f 05.

H. = 6-0 ... 7-0.
G. = 3-2 ... 3-5.

Epidote. ii. 282*

4. PRISMATIC. Prismatic.

Cleavage, perfect in two directions, one of them being

more easily obtained. Inclination s= 95° 25'.
H. = 4-5 ... 5-0.
G. = 2-7 — 2-9.

Wollastonite. ii. 286.

Acmite. iii. 67.

Arfvedsonlte. iii. 73.

Babingtonite* iii. 75.

Indianite. iii. 113.

Jeffersonite. iii. 115.

Mangane&e-Spar ? J. iii. 122.

Withamite. iii, 170.

IV. AZUEE-SPAE. Tessular, prismatic.
Colour blue.
H. = 5O ... 6-0.
G. = 2-9 ... 8-1.

1. DODECAHEDRAL.

Cleavage, impejfect.
Colour bright.
Streak blue.

OHDKR VI. GENERA AND SPECIES. 429

H. = 5-5 ... 6-0.
G. = 2-9 ... 3-0.

Azurestone or Lapis Lazuli. J. ii. 288.

2. PRISMATIC. Prismatic.

Cleavage, P + oo = 91° 30*, imperfect
Colour bright.
Streak uncoloured.
H. = 5-0 ... 5-5.
G. = 3-0 ... 3-1.

LazuKte. ii. 290.

3. PRISMATOIDAL. Prismatic.

Cleavage, prismatoidal, imperfect.
Colour pale.
Streak uncoloured.
H. = 5-5 ... 6-0.
G. = 3-0 ... 3-1.

Prismatoidal Azure-Spar or Blue-Spar. J. ii. 292.

Amblygonite. iii. 70.

Bergmannite. iii. 77.

Bucklandite $ iii. 83.

Calaite. iii. 83.

Chiastolite. iii. 84.

Diaspore. iii. 92.

Eudlalyte. iii. gg.

Gehlenite. iii. 102.

Hauyne. . iiL 107.

Karpholite? iii. H6.

Nephrite. iii. 131.

Saussurite. iii. 143.

Sillimanite. iii. 153.

Somervillite. Hi. 154.

iii. 162.

430 CHARACTERISTIC. CLASS II.

VII. ORDER. GEM.

I. ANDALUSITE. Prismatic.

Cleavage not pr is mate-id al.

H. = 7-5.

G. = 3-0 ... 3-2.

1. PRISMATIC. Prismatic.

Cleavage, P + oo = 91° 33'. £ r +00. Pr + oa.

Andalusite. ii. 293.

II. CORUNDUM. Tessular, rhombohedral, pris-

matic.

H. = 8-0 ... 9-0.
G. = 3-5 ... 4-3.

Prismatic : G. =. 3-65 and more; H. = 8-5.

Colour red or brown ; G. = 3-7 and more :
H. = 9-0.

1. DODECAHEDRAL. Tessular.

Cleavage, octahedron, difficult.

H. = 8-0.

G. = 3-5 ... 3-8.

Spindle. ii. 295.

2. OCTAHEDRAL. Tessular.

Cleavage, octahedron, perfect.

H. = 8-0.

G. = 4-1 ... 4-3.

Gahnite. ii. 298.

3. RHOMBOHEDRAL. Rhombohedral. R = 86° 6'.

Cleavage, R. Sometimes, R — w.

H. = 9-0.

G. = 3-9 ... 4-05.

Corundum. "• 299.

ORDER VII. GENERA AND SPECIES. 431

4. PRISMATIC. Prismatic. Pr = 119° 48'. (p" + os)3

= 70° 41'.

Cleavage, ?r + os. Less distinct, Pr -f 93,
H. = 8-5.
G. =3-65 ...3-8.

Clirysdberyl. ii. 304.

IT I. DIAMOND. Tessular. .
H. = 10-0.
G. = 3-4 ... 3-6.

1. OCTAHEDRAL. Tessular.

Cleavage, octahedron, perfect.

Diamond. ii. 306.

IV. TOPAZ. Prismatic.

Cleavage, axotomous.

H. = 8-0.

G. = 3-4 ... 3-6.

1. PRISMATIC. Prismatic. P = 141°7', 101° 52', 90° 55'.

P + os = 124° 19'.
Combinations sometimes different in the opposite ends

of the crystals.
Cleavage, P — os, highly perfect.

Topaz. ii. 308,

V. EMERALD. Rhombohedral, prismatic.

Cleavage, rhombohedral axotomous and
peritomous, or prismatoidal of a high
degree of perfection.

H. = 7-5 ... 8-0.

G. = 2-6 ...3-2.

CHARACTERISTIC. CLASS II.

1. PRISMATIC. Hemi-prismatic.

Cleavage, £r + co, of a high degree of perfection.

?£ = 40° 52'.

2

H. = 7'5.
G. = 2-9 ... 3-2.

Euclase. ii. 313.

2. RHOMBOHEDRAL. Di-rhombohedral. P = 151° 9', 59° 47'-

Cleavage, R — eo. Less distinct, P + co.
H. = 7'5 ... 8-0.
G. = 2-6 ... 2-8.

Emerald. ii. 316.

VI. QCARTZ. Rhombohedral, prismatic.
Cleavage, not axotomous.
H. = 5-5 ... 7-5.
G. = 19 ...'2-7.

1. PRISMATIC. Prismatic.

Cleavage, P + co = 120° (nearly). £r 4- co.
Dichroism, parallel and perpendicular to the axis.
H. = 7-0 ... 7-5.
G. = 2-5 ... 2-6.

Cordierite. ii. 319.

2. RHOMBOHEDRAL. Rhombohedral. R = 75° 55'.

Combinations, hemi-rhombohedral and hemi-di-rhom-
bohedral ; R + n and (P + n')m with inclined faces,

P + n" with parallel faces. — = 94° 15'.

Cleavage, P = 133° 44', 103° 35'. Commonly —

somewhat less distinct. P + co.
H. = 7*0.
G. = 2-5 ... 2-7-

Quartz. ii. 321.

3. UNCLE AV ABLE. Reniform ... massive.

Cleavage, none.

ORDER VIJ. GENERA AND SPECIES. 433

H. = 5'5 ... 6-5.
G. = 1-9 ... 2-2.

Opal ii. 332.

4. EMPYRODOX. Grains ... massive.
Cleavage, none.
H. = 6-0 ... 7-0.
G. = 2-2 ... 2-4.

Fusible Quarts. J. ii. 337.

VII. AXINITE. Prismatic.

Lustre pure vitreous.
H. = 6-5 ... 7-0.
G. = 3-0 ... 3-3.

1. PRISMATIC. Tetarto-prismatic.

Cleavage, two faces, one of them more distinct. In-
clination = 101° SCK.

Axinite. ii. 341.

VIII. CHRYSOLITE. Prismatic.
Lustre pure vitreous.
H. = 6-5 ... 7-0.

G. = 3-3 ... 3-5.

1. .PRISMATIC. Prismatic. Pr = 80° 53'. (Pr + co)3 =
130° 2'.

Cleavage, Pr + 05. Traces of Pr + 09.

Chrysolite. ii. 345.

IX. BORACITE. Tessular.

H. ^ 70.

G. = 2-8 ... 3-0.

1. TETRAHEDRAL. SemUessular, with inclined faces.
Cleavage, octahedron, imperfect.

Boradte. H. 347.

VOL. I. 2 E

434 CHARACTERISTIC. CLASS It.

X. TOURMALINE. Rhombohedral.

H. = 7-0 ... 7-5.
G. = 3-0 ... 3-2.

1. RHOMBOHEDRAL. Rhombohedral. R = 133° 26'.

Combinations, the opposite ends of the crystals con-
taining different faces.
Cleavage, R. P + os. Imperfect

Tourmaline. ii. 349.

XI. GARNET. Tessular, pyramidal, prismatic.

Lustre, not pure vitreous.
H. = 6-0 ... 7-5.
G. = 3-1 ... 4-3.

Colour black : G. = 3 9 and less.

H. = 7*5 : colour red or brown.

G. under 3-3 : form tessular.

1. PYRAMIDAL. Pyramidal. P = 129° 29', 74° 14'.

Cleavage, P — . os. P + 03. [P + oa].

H. = 6-5.

G. = 3-3 ... 3-4.

Idocrase. ii. 354.

2. TETRAHEDRAL. Semi-tessular, with inclined faces.

Cleavage, octahedron, imperfect.
H. == 6-0 ... 6-5.
G. = 3-1 ... 3-3.^

Helvine. . ii. 357.

3. DODECAHEDRAL. Tessular.

Cleavage, dodecahedron, imperfect.
H. = 6-5 ... 7'5.
G. = 3-5 ... 4-3.

Garnet. ii. 359.

., Prismatic. Pr = 70° 32'. (I5r + «),*

129° 3K

OBDER VII. GENEKA AND SPECIES. 435

Cleavage, Pr «= os, perfect.
H. = 7-0 ... 7-5.
G. *= 3-3 ... 3-9.

Staurolitc. ii. 366.

XII. ZIRCON. Pyramidal.

H. = 7-5.

G. =. 4-5 ... 4-7.

1. PYRAMIDAL. Pyramidal. £ =* 123° 19*, 84° 2<K
Cleavage, P. P + 03.

Zircon. ii. 368.

XIII. GADOLINITE. Prismatic.
Colour black.

H. = 6-5 ... 7-0.
G. = 4-0 ... 43.

1. PRISMATIC. Hemi-prismatic.-
Cleavage almost none.
Fracture conchoidal.

Gadolinite. ii. 371.

Aflome. Ii. 364.

Chondroditej iii. 87.

Essonite. ji. 364.

Fibrolite ? iL 99.

Forsterite. iii. 102.

Jfyalosiderite ? iii. 111.

KnebeliteZ iii. 118.

Ligurite? iii. 121.

Mellilite? iii. 125.

SphoeruliteZ iii. 155.

Spmellanef iii. 156.

Zeagonite. iiu 174.

466 CHARACTERISTIC. CLASS II.

VIII. ORDER. ORE.

I. TITANIUM-ORE. Pyramidal, prismatic.

Streak uncoloured ... very pale brown.
H. = 5-0 ... 6-5.
G. =3-4 ... 4-4.

G. under 4-2 : streak uncoloured.

1. PRISMATIC. Hemi-prismatic. — == 113° 20'. Inclina-

tion = 8° 18'. (Pr -1- os)3 =136° 8'.
Cleavage, —. ?£ = 28° 7', difficult.

Streak uncoloured.
H. = 5-0 ... 5-5.
G. = 3-4 ... 3-6.

Sphene. ii. 373.

2. PERITOMOUS. Pyramidal. P= 117° 2', 95° 13'.

Cleavage, P 4- os. [P + »].
Streak pale brown.
H. = 6-0 ... 6-5.
G. = 4-2 ... 4-4.

Rutile. ii. 376.

3. PYRAMIDAL. Pyramidal. P = 97° 56', 130° 22'.

Cleavage, P — oo. P.
Streak uncoloured.
H. = 5-5 ... 6-0.
G. = 3-8 ... 3-9.

Anatase. ii. 379.

II. ZINC-ORE. Prismatic.

Streak orange-yellow.
H. = 4-0 ... 4-5.
G. - 54... 5-5.

QKDKIl VIII GEXEUA A\JD SPECIES. 437

1. PRISMATIC. Prismatic.

Cleavage, P + co = 125° (nearly). Traces of Pr + oo.
Prismatic Zinc-Ore. J. ii. 380.

III. COPPER-ORE. Tessular.

Streak brownish-red.
H. = 2-5 ... 4-0.
G. = 5-6 ... 6-0.

1. OCTAHEDRAL. Tessular.
Cleavage, octahedron.

Octahedral Red Copper-Ore. J. ii. 381.

IV. TIN-ORE. Pyramidal.

Streak not black.
H. = 6-0 ... 7-0.
G. = 6-3 .,. 7-1.

1. PYRAMIDAL. Pyramidal. P = 133° 26', 67° 59'.
Cleavage, P + oo. [P + oo].
Streak uncoloured ... pale brown.

Pyramidal Tin-Ore. J. ii. 384.

V. SCHEELIUM-ORE. Prismatic.

Streak reddish -brown, dark.
H. =50 ... 5-5.
G. = 7-1 ... 74.

1. PRISMATIC. Hemi-prismatic. £LZT_1 = 62° 40'. In-
clination = 0. P + oo = 101° 5'.

Cleavage, P r + co, perfect.

Prismatic Wolfram. J. ii. 387.

<-

VI. TANTALUM-ORE. Prismatic.

Streak brownish-black.

438 CHARACTEllISTIC. CLASS II.

H. = 6-0.

G. =6-0... 6-3.

1. PRISMATIC. Prismatic.
Cleavage, prismatoidal.

Tantalite. ii. 390.

VII. URANIUM-ORE. Form not determinable.

Streak black.

H. = 5-5.

G. = 64 ... 6-6.

1. UNCLEAVABLE. Renfform, massive.
Cleavage, none.

Uncleavdble Uranium-Ore. J. ii. 393.

VIII. CERIUM-ORE. Form not determinable.
Streak uncoloured.

H. sp 5-5.

G. =4-9 ... 5-0.

1. UNCLEAVABLE. Massive.
Cleavage, none.

Cerite. ii. 394.

IX. CHROME-ORE. Tessular.

Streak brown.

H. = 55.

G. = 4-4 ... 4-5.

1. OCTAHEDRAL. Tessular.
Cleavage, octahedron.

Prismatic Chrome-Ore or Chromate of Iron. J. ii. 396.

X. IRON-ORE. Tessular, rhombohedral, prismatic.

Streak red, brown, black.

OllDER VIII. GENERA AND SPECIES. 439

H. = 50... 6-5.
G. =3-8 ... 5-3.

Streak brown: G. = 4-8 and less, or

4*8 and more.

G. under 4-3 ; the colour being black :
streak without lustre.

1. AXOTOMOUS. Rhombohedral. R = 85° 59'.

Combinations hemi-rhombohedral, with parallel faces.

P + l = 91° 20'.
2

Cleavage, 11 — oo, perfect. Traces of R.

Streak black.

Weak action upon the magnetic needle.

H. = 5-0 ... 5'5.

G. = 4-4 ... 4-8.

Titanitic Iron. Vulg. ij. 397.

2. OCTAHEDRAL. Tessular.

Cleavage, octahedron.

Streak black.

Strong action upon the magnetic needle.

H. = 5-5 ... 6-5.

G. = 4-8 ... 5-2.

Octahedral Iron-Ore. J. ii. 399.

3. DC-DECAHEDRAL. Tessular.

Cleavage, octahedron, very indistinct.

Streak brown.

Weak action upon the magnetic needle.

H. = 6-0 ... 6-5.

G. = 5-0 ... 5-1.

FranWnite. ii. 403.

4. RHOMBOHEDRAL. Rhombohedral. R =s 85° 58'.

Cleavage, R. Sometimes, R-.cc.

Streak red ... reddish-brown.

Sometimes a weak action upon the magnetic needle.

440 CHARACTERISTIC. CLASS II.

H. = 5-5 ... 6-5.
G. = 4-8 ... 5-3.

RJiomboidal Iron-Ore. J. ii. 404.

5. PRISMATIC. Prismatic.

Cleavage, P + 03.

Streak yellowish-brown.

No action upon the magnetic needle.

H. = 5-0 ... 5-5.

G. = 3-8 ... 4-2.

Prismatic Iron-Ore. J. ii. 410.

6. DJ. PRISMATIC. Prismatic. P = 139°37', 117° 38, 77° 16'.

Cleavage, Pr = 113° 2'. P + oo = 112° 37'. Some-
what more distinct, P — oo. £r + oo. Altogether
imperfect.

Streak black, sometimes greenish or brownish.

No action upon the magnetic needle.

H. = 5-5 ... 6-0.

G. = 3-8 ... 4-1.

Lievrite. ii. 414.

XI. MANGANESE-()BE. Pyramidal, prismatic.
Streak dark brown, black.
No action upon the magnetic needle.
H. = 2-5 ... 6-0.
G. = 4-0 ... 4-8.

Streak brown : G. = 4-7 and more,

H. = 4'0 and more.

H. above 4-0 ; the streak being black :
lustre in the streak.

PYRAMIDAL. Pyramidal. P = 105° 25', 117° 54'.
Cleavage, P — so. Traces of P — 1 = 114° 51', 99°

11', and of P.
Streak brown.

ORDER JX. GENERA AND SPECIES. 441

H. = 5-0 ... 5-5.
G. = 4-7 ... 4-8.

Black Manganese. J. ii- 416.

UNCLEAVABLE. Reniform, botryoidal, massive.
Cleavage, none.
Streak brownish-black, shining.
H. = 5-0 ... 6-0.
G. = 4-0 ... 4-1.

Black Manganese. J. "• 418.

3. PIUSMATOIDAL. Prismatic.

Cleavage, Pr + oo perfect, less distinct P + w = 99° 40'.
Streak black.
H. = 2-5 ... 3-5.
G. = 4-4 ... 4-8.

Grey Manganese. J. ii. 419.

Allamte. iii. 68.

Brookite. iii. 82.

Fergusonite. iii. 98.

Orthite. iii. 133.

Phosphate of Manganese. Chem. iii. 136.

Stilpnosidcrite. iii. 158.

YUro-TantaUtc. J. iii. 173.

IX. ORDER. METAL.

/

I. ARSENIC, Form unknown.
Colour tin-white.
H. = 3-5.
G. = 5-7 ... 5-8.

1 . - N AT i VE. Reniform, massive.

Arsenic. "• 423.

442 CHARACTERISTIC. CLASS II.

II. TELLURIUM. Form unknown.

Colour tin-white.
H. = 2-0 ... 2-5.
G. = 6-1 ... 6-2.

1. NATIVE. Massive.

Tellurium. ii. 424.

III. ANTIMONY. Rhombohedral, prismatic,

Not malleable.

Colour white, not inclining to red.

H. = 3-0 ... 3-5.

G. == 6-5 ... 10-0.

1. RHOMBOHEDRAL. Rhombohedral. R = 117° 15'.

Cleavage, R — w perfect. R. Traces of R — 2 and

P + co.

H. = 3-0 ... 3-5,
G. = 6-5 ... 6-8.

Antimony. ii. 426.

2. PRISMATIC. Prismatic.

Cleavage, P — co. Pr. Less distinct, P 4- oe.

H. = 3-5.

G. = 8-9 ... 10-0.

Prismatic Antimony or Antimonial Silver. J. ii. 427-

IV. BISMUTH. Tessular.

Colour silver-white, inclining to red.
H. = 2-0 ... 2-5.
G. = 9-6 ... 9'8.

1. OCTAHEDRAL. Semi-tessular with inclined faces.
Cleavage, octahedron, perfect.

Bismuth. iif 430,

ORDER IX. GENERA AND SPECIES. 443

V. MERCURY. Tessular, fluid.
Not malleable.
Colour white.
H. = 0-0 ... 3-0.
G. = 10-5 ... 15-0.

1. DODECAHEDRAL. Tessular.

Cleavage, none.
Colour silver-white.
H. = 1-0 ... 3-0.
. G. = 10-5 ... 12-5.

Dodecahedral Mercury or Native Amalgam. J. ii. 431.

2. FLUID. Fluid.

Colour tin-white.

H. = 0-0.

G. = 12-0 ... 15-0.

Mercury. ii. 432.

VI. SILVER. Tessular.

Ductile.

Colour silver-white.
H. = 2-5 ... 3-0,
G. = 10-0 ... 10*5.

1. HEXAHEDRAL. Tessular.
Cleavage, none.

Silver. ii. 433.

VII. GOLD. Tessular.

Colour gold-yellow.
H. = 2-5 ... 3-0,
G. = 12-0 ... 200.

1. HEXAHEDRAL. Tessular.
Cleavage, none.

Gold. ii. 436,

CHARACTERISTIC. CLASS II.

VIII. PLATINA. Form unknown.
Colour steel-grey.

H. - 4-0 ... 4*5.
G. = 16-0 ... 20-0.

1. NATIVE. Massive.
Cleavage, none.

Platina. ii. 441.

IX. IRON. Tessular.

Colour pale steel-grey.

H. = 4-5.

G. = 7-4 ... 7-8.

1. OCTAHEDRAL. Tessular.
Cleavage, none.

Iron. ii. 442.

X. COPPER. Tessular.

Colour copper-red.
H. = 2-5 ... 3-0.
G. = 8-4 ... 8-9.

1. OCTAHEDKAL.

Cleavage, none.

Copper. ii. 444.

Iridium. iii. 114.

Lead. iii. 129.

Palladium. iii. 134.

ORDER X. GENERA AND SPECIES. 445

X. ORDER. PYRITES.

I. NICKEL-PYRITES. Prismatic.

Colour copper- red.
H. = 5-0 ... 5-5.
G. = 7-5 ... 7-7.

1. PRISMATIC. Prismatic.
Cleavage, indistinct.

Prismatic Nickel Pyrites. J. ii. 446.

II. ABSENICAL-PYRITES. Prismatic.

Colour not inclining to red.
H. = 5 0 ... 6-0.
G. =5-7... 7-4.

Colour white or grey : G. under 6*3 or
above 7*0.

1. AXOTOMOUS. Prismatic. Pr = 51° 20'. P + co =

122° 26'.

Cleavage, P — co. Less distinct, Pr = 86° 10'. Tra-
ces of P + oo.

H. = 5-0 ... 5-5.

G. = 7'1 ... 7'4.

Axotomous Arsenical Pyrites. J. ii. 448.

2. PRISMATIC. Prismatic.

Cleavage, P — cc. (Pr + co)3 =5 111° 53'.
H. = 5-5 ... 6-0.
G. = 5-7 ... 6-2.

Prismatic Arsenical Pyrites. J. ii. 449.

III. COBALT-PYRITES. Tessular.

Colour white, inclining to steel-grey or red.

H. = 5-5.

G. = 6-1 ... 6-6.

446 CHARACTERISTIC. CLASS II.

1. OCTAHEDRAL. Tessular.

Cleavage, hexahedron, octahedron, dodecahedron, in-

distinct.

Colour white, inclining to grey.
H. = 5-5.
G. = 6-4 ... 6-6.

Tin-White Cobalt. J. iii. 452.

2. HEXAHEDRAL. Semi-tessular with parallel faces.

Colour white, inclining to red.

H. = 5-5.

G. =6-1 ... 6-35.

Silver-White Cobalt. J. ii. 455.

IV. IRON-PYRITES. Tessular, rhombohedral, pris-
matic.
Colour yellow, sometimes inclining to cop-

jter-red.

H. = 3-5... 6-5.
G. = 4-4 ... 5-05.

1. HEXAHEDRAL. Semi-tessular with parallel faces.

Cleavage, hexahedron, octahedron.
Colour bronze-yellow.
H. =s 6-0 ... 6-5.
G. = 4-9 ... 5-05.

Hexaliedral Iron*Pyrites. J. ii. 45?.

2. PRISMATIC. Prismatic. £r = 114° 19'.

Cleavage, Pr = 106° 36', distinct. Traces of P + w

*= 98° 13'.

Colour bronze-yellow.
H. = 6-0 ... 6-5.
G. = 4-65 ... 4-9.

Prismatic Iron-Pyrites. J. ii. 461.

3. RHOMBOHEDRAL. Di-rhombohedral.

Cleavage, R — c». Less distinct, P + co.
Colour bronze-yellow, inclining to copper-red.

OllDEB XI. GENERA AND SPECIES.

H. = 3-5 ... 4-5.
G. = 4-4 ... 4-7-

Rhomboidallron- Pyrites or Magnetic Pyrites. J. ii. 465.

V. COPPER-PYRITES. Tessular, pyramidal.
Colour brass-yellow, copper-red.
H. = 3-0 ... 4-0.
G. = 4-1 ... 5-1.

1. OCTAHEDRAL. Tessular.

Cleavage, octahedron, very indistinct.

Colour copper-red.

H. = 3-0.

G. = 4-9 ... 5-1.

Variegated Copper. J. ii. 467.

2. PYRAMIDAL. Hemi-pyramidal with inclined faces. P

= 109° 53', 108° 40'.
Cleavage, P + 1 = 101° 49', 126° 11'.
Colour brass-yellow.
H. = 3-5 ... 4-0.
G. ^4-1 ...4-3.

Pyramidal Copper-Pyrites. J. ii. 469.

Cobalt Kies 9 J. iii. 88.

Nickeliferous Grey Antimony. J. iii. 131.

XI. ORDER. GLANCE.

I. COPPER-GLANCE. Tessular, prismatic.

Colour blackish lead-grey, steel-grey, black.
Cleavage, indistinct, not axotomous.
H. = 2-5 ... 4-0.
G. = 44 ... 5-8.

448 CHAllACTEllISTIC. CLASS II,

1. TETRAHEDRAL. Semi-tessular with inclined faces.

Cleavage, octahedron.
Colour steel-grey ... iron -black.
H. = 3-0 ... 4-0.
G. = 4-4 ... 5'2.

Tetrahedral Copper-Glance. J. iii. 1.

2. PRISMATOIDAL. Prismatic.

Cleavage, ?r + GO.

Colour blackish lead-grey.

Brittle.

H. = 3-0.

G. = 5-7 ... 5-8.

Prismatoidal Copper-Glance. J. iii. 4.

3. DI-PRISMATIC. Prismatic. Pr— 1 =87° 8'; (£r + oo)3

= 96° 31'.

Cleavage, Pr + oo. Pr + oo. The former rather

more distinct.

Colour steel-grey, inclining to lead-grey or iron-black.
Brittle.

H. = 2-5 ... 3-0.
G. = 5-7 ... 5-8.

Bournonite. iii. 5.

4. PRISMATIC. Prismatic. £r = 119° 35'; (Pr + oo)3 =

63° 48'.

Cleavage, Pr, very imperfect.
Colour blackish lead-grey.
Very sectile.
H. = 2-5 ... 3-0.
G. = 5-5 ... 5-8.

Prismatic Copper-Glance or Vitreous Copper. J. iii. 8.

II. SILVER-GLANCE. Tessular.
Colour blackish lead-grey.
H. = 2-0 ... 2-5.
G. =6-9 ... 7-2.

ORDER XI. GENERA AND SPECIES.

1. HEXAHEDRAL. Tessular.

Cleavage, traces of the dodecahedron.
Malleable.

Hexahcdral Silver -Glance. J. iii. 11.

III. LEAD-GLANCE. Tessular.

Colour pure lead-grey.

H. = 2-5.

G. = 7-4 ... 7'6.

1. HEXAHEDRAL. Tessular.

Cleavage, hexahedron, perfect.

Hexahedral Gakna or Lead-Glance. J. iii. 13.

IV. TELLURIUM-GLANCE. Prismatic.

Colour blackish lead-grey.
Cleavage monotomous, perfect.
H. = 1-0 ... 1-5.
G. = 7-0 ... 7-1.

1. PRIS3IATIC. Prismatic.

Cleavage, axotomous or prismatoidal.

Prismatic Tellurium-Glance. J. iii. 16.

V. MOLYBDENA-GLANCE. Rhombohedral.

Colour pure lead-grey.
Thin laminae very flexible.
H. = 1-0 ... 1-5.
G. = 4-4 ... 4-6.

1. RHOMBOHEDRAL. Di-rhombohedral.
Cleavage, R — co, perfect.

Rhomboidal Molybdcna. J. iii. 18.

VI. BISMUTH-GLANCE. Prismatic.

Colour pure lead- grey.

450 CHARACTERISTIC. CLASS II.

H, = 2-0 ... 2-5.
G. = 6-1 ... 6-4.

1. PRISMATIC. Prismatic.

Cleavage, P -f co = 90°, nearly; also Pr + co and
Pr + co, one of them highly perfect.

Prismatic Bismuth-Glance. J. iii. 19.

VII. ANTIMONY-GLANCE. Prismatic.

Colour lead-grey, not blackish, steel-grey.
Cleavage, perfect.
H. = 1-5 ... 2-5.
G. = 4-2 ... 5-8.

G. under 5-3 : H. = 2-0 : thin laminae

not very flexible.
G. above 5-3 : colour steel-grey.

1. PRISMATIC. Prismatic.

Cleavage, Pr + co, perfect in a high degree. Less ap-
parent, Pr + co.
Colour pure steel-grey.
H. = 1-5 ... 2-0.
G. = 5-7 ... 5-8.

Prismatic Antimony Glance. J. iii. 21.

2. PRISMATOIDAL. Prismatic. P = 109° 16', 108° 10',

110° 59'.
Cleavage, ?r + «, highly perfect. Less apparent,

P — co. P + co as 90° 45'. Pr + co.
Colour lead-grey.
H. = 2-0.
G. = 4-2 ... 47.

Grey Antimony. J. iii. 23.

3. AXOTOMOUS. Prismatic.

Cleavage, P — co, perfect. P + co = 101° 207; Pr -I- so.
Colour steel-grey.

OltOEll XII. GENERA AND SPECIES. 451

H. = 2-0 ... 2-5.
G. = 5-5 ... 5-8.

Jamesonite. iii. 26.

VIII. MELANE-GLANCE. Prismatic.
Colour iron-black.
H. = 2-0 ... 2-5.
G. = 5-9 ... OH.

1. PRISMATIC. Prismatic. Pr = 115° 39'.

Cleavage, (Pr + os)3 = 72° 13'. Pr + oo. Indistinct.
Prismatic Melane-Glance. J. iii. 27«

Argentiferous Copper-Glance. J. iii. 73.

Bismuthic Silver $ J. iii. 78.

Cdbaltic Galena. J. iii. 88.

Cupreous Bismuth* 3, iii. 91.

Eucairite. iii. 94.

Flexible Sulphurct of Silver. PHILL. iiL 30.

MolyMena-Silver. J. iii. 127.

Native Nickel. J, iiL 129.

Needle-Ore. J. iii. 130.

Seleniuret of Copper. PHILL. iii. 150.
Sulphuret of Silver and Antimony. PHILL. iii. 30.

Tennantite. iii. 161.

Tin-Pyrites. J. iii. 163.

Yellow Tellurium? J. iii. 171.

XII. ORDEB. BLENDE.

I. GLANCE-BLENDE. Tessular.
Streak green.
H. = 3-5 ... 4-0.
G. =3-9 ... 4-05.

452 CHARACTERISTIC. CLASS IT.

1. HEXAHEDRAL. Tessular.

Cleavage, hexahedron, perfect.

Prismatic Manganese-Blende. J. iii. 3 1 .

II. GARNET-BLENDE. Tessular.

Streak uncoloured ... reddish-brown.
H. = 35 ... 4-0.
G. — 4-0 ... 4-2.

1. DODECAHEDRAL. Semi-tessular with inclined faces.
Cleavage, dodecahedron, highly perfect.

Dodecahedral Zinc-Blende. J. iii. 32.

III. PURPLE-BLENDE. Prismatic.

Streak cherry-red.
H. =r 1-0 ... 1-5.
G. =4-5 ... 4-6.

1. PRISMATIC. Hemi-prismatic.
Cleavage prismatoidal.

Red Antimony. J. iii. 36.

IV. RUBY-BLENDE. Rhombohedral, prismatic.

Streak red.

H. = 2-0 ... 2-5.

G. = 5-2 ... 8-2.

1. RHOMBOHEDRAL. Rhombohedral. R = 108° 18'.
Combinations sometimes different on the opposite ends

of the crystals.
Cleavage, R.
Streak, cochineal-red.
H. = 2-5.
G. = 5-4 ... 5-9.

Rhomboidal Ruty-Bknde or Red Silver. J. iii. 38.

OKDER XIII. GENERA AND SPECIES. 453

2. HEMI-PRISMATIC. Hemi-prismatic. P + co = 8C° 4'.

Inclination of P — co on the acute edge = 101° 6'.

Cleavage, 1 — 1— !"_ and Pr + co, imperfect.

Streak dark cherry-red.
H. = 2-0 ... 2-5.
G. = 5-2 ... 5-4.

Var. of Dark Red Silver. Vulg. Hi. 42.

3. PER'ITOMOUS. Rhombohedral. 11 = 71° 47'-

Cleavage, 11 + co, highly perfect.
Streak scarlet-red.
H. = 2-0 ... 2-5.
G. = 6-7 ... 8-2.

Cinnabar. iii. 44.

XIII. ORDER. SULFUR,

I, SULPHUR. Prismatic.
H. = 1-5 ... 2-5.
G. = 1-9 ... 3-6.

1. PRISMATOTDAL. Prismatic. Pr = 83° 37'. P + co

= 117° 49'.

Cleavage, Pr -f co, eminent-
Streak lemon-yellow.
H. = 1-5 ... 2-0.
G. = 3-4 ... 3-6.

Orpiment. iii. 47.

p

2. HEMI-PRISMATIC. Hemi-prismatic. — = 130° 0.

Inclination = 4° I'.
leavage, — ?!L = 6U
P -f co = 74° 30'. Pr -f co. Imperfect.

Cleavage, — — = 60° 44'. Pr + co. Less distinct,

454

CHARACTERISTIC.

CLASS II.

Streak orange-yellow ... aurora-red.
H. = 1-5 ... 2-0.
G. =: 3-5 ... 3-6.

Realgar. iii. 49.

3. PRISMATIC. Prismatic. P = 106° 38', 84° 58', 143° 17'.
Cleavage, P. P + co = 101° 59'. Imperfect.
Streak uncoloured .., sulphur-yellow.
H. = 1-5 ... 2-5.
G. = 1-9 ... 2-1.

Sulohur. iii. 52.

OllDKR I. GENKRA AND SPECIES. 455

CHARACTERS

OF THE

GENERA AND SPECIES

OF THE

ORDERS OF CLASS III.

I. ORDER. RESIN.

I. MELICHRONE-RESIN. Pyramidal.

H. = 2-0 ... 2-5.
G. = 1-4 ... 1-6.

1. PYRAMIDAL. Pyramidal. P = 118° 4', 93° 22'.
Cleavage, P, imperfect.

Mellite. iii. 56.

II. MINERAL-RESIN. Amorphous.

H. = 0-0 ... 2-5.
G. =0-8... !•«.

1. YELLOW. Solid.

Colour yellow ... white.
Streak uncoloured.

450 CHARACTERISTIC. CLASS III.

H. = 2-0 ... 2-5.
G. = 1-0 ... 1-1.

Amber. iii. 57-

2. BLACK. Solid ... fluid.

Colour black, brown, red, grey. v
Streak black, brown, yellow, grey.
H. = 0-0 ... 2-6.
G. = 0-8 ... 1-2.

Black Mineral-Resin. J. iii. 59.

Retinite. iii. 146.

II. ORDER. COAL.

I. MINERAL-COAL. Amorphous.
H. = 1-0 ... 2-5.
G. =1-2 ... 1-5.

1. BITUMINOUS. Colour brown, black.

Lustre resinous.
Odour bituminous.
H. = 1-0 ... 2-5.
G. = 1-2 ... 1-5.

Bituminous Mineral Coal. J. iii. 61.

2. NON-BITUMINOUS. Colour black.

Lustre imperfect metallic,
Odour not bituminous.
H. = 2-0 ... 2-5.
G. = 1/3 ... 1-5.

Anthracite. iii. C4.

Among the remaining minerals contained in the Appen-
dix,' Professor MOHS proposes to form two new Orders,
which will comprehend nearly the following species :

AFPENP. GENERA AND SPECIES. 457

i. Order, to be Inserted between Kerate and Malachite.

Black Cobalt-Ochre. J. iii. 78.

Kupferindig. BREITHAUPT. iii. 118.

Cupreous Manganese. J. iii. 92.

Black Wad. J. ii. 421.

Plumbago *. ii. 191.

Hisingerite. iii. 108.

Pyrorthite. iii. 142.

Chloropal. iii. 85.

Chlorophccitc. iii. 86.

Iron-sinter. J. iii. 115.

Chrysocolla*. ii. 158.

Allophane. iii. 69.

ii. Order, to le inserted Icttvcen Mica and Spar.

Steatite. iii. 157.

Agalmatolitc. iii. 100.

Serpentine. iii. 151.

Fahlunitc. iii. 97.

Gieseckite. iii. 104.

Finite. iii. 139.

Killinite. iii. 117.

Gttbsite? iii. 103.

Marmolite? iii. 124.

Picrolite. iii. 136.

Ficrosmine. iii. 137.

Pyrallolite. iii. 141.

Many of these substances have been too imperfectly de-
scribed to enable us to receive them as yet in the system.
This is still more the case with the following minerals,
some of which, moreover, possess properties apparently so
discrepant from any of the orders comprised in the system,

* Hitherto contained in V. and IV. Orders.
VOL. i. 2 &

458 CHARACTERISTIC. APPEND.

that no place can yet be assigned to them with any degree
of probability.

Aluminitc. iii. 70.

Ajjhritc. iii. 72.

Arsenical Bismuth. WERN. iii. 74.

BreislaTcite. iii. 80.

Hatchctinc. iii. J06.

Huniboldtlne. iii. 110.

Leelite. iii. 119.

Sordawatitc. iii. 155.

Torrelite. iii. 164.

Turnerite. iii. 166.

Wagnerite. iii. 169.

Zurlite. iii, 176.

END OF THE F-IRST VOtTTME.

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