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iMacrura)-e.g. Crayfish (q.v.), Lobster; short. Friars. They came to England in the 13th century, tailed (Brachyura), Crabs.

and had monasteries in London (which still gives Distribution in Space and Time.-(a) Deep- name to a street), Oxford, and Reigate. sea forms are very abundant, and often remarkable Cruveilhier, JEAN, physician, born at 'for their colossal size, their bizarre forms, and Limoges in 1791, became professor of Pathology brilliant red colouring.' Blind species are known at Montpellier in 1824, and of Pathological Ana. to occur in the depths, and others are brilliantly tomy in Paris in 1836. Besides his great work, phosphorescent. (6) Pelagic surface Crustaceans Anatomie Pathologique du Corps Humain (2 vols. (especially Schizopods and Entomostraca) are very 1828-42), he published several other works on abundant, and often form a large part of the food

anatomy, which were for many years the most of fishes. They are often beautifully transpar valuable French contributions to their subject, ent, and hardly to be seen in the water. Less and also a Life of Dupuytren (1840). He died 6th frequently they are brilliantly coloured (as in March 1874. Sapphirina) or phosphorescent. Some of them are

Crwth (pronounce the w as French u), an old remarkable for their large eyes. One Amphipod

Welsh stringed instrument. Four of its six strings Crustacean presents a curious mimicry of a Medu

were played with a bow, the other two being soid form. “Decapods are most abundant in the

twitched by the thumb. warmer waters. (c) Crustaceans form an important part of the relatively sparse and uniform fauna of

Cryolite, a mineral which exists in great abund. lakes. They occur both on the surface and at the

ance on the coast of Greenland. It consists mainly bottom, the latter being generally more sluggish.

of a fluoride of aluminium in combination with The surface forms, at anyrate, are usually per

Fluoride of Sodium, 6NaF, Al,F. The metal fectly transparent, with the exception of the eye.

| Aluminium (q.v.) was formerly largely obtained

| from it, but it is now most important as a source (d) The catalogue of terrestrial Crustaceans, which includes species of Amphipods, Isopods, and Deca

of alum and of soda bicarbonate ; much of it, also, pods, is relatively a very short one.

is melted and made into a kind of glass. The Crustacea date back to Canıbrian times, but Cryophorus (Gr. kryos, cold,' and phero, ‘I the highest forms (Decapods ) were not firmly

carry') is an instrument consisting of a glass tube established till the Tertiary period. Some 800

with a bulb at both ends, used for showing the fossil species, as against over 5000 living forms, are

diminution of temperature in water by evapora. known. Some of the genera-e.g. Estheria—from

tion. In constructing it the whole of the air is the Devonian, are marvellously persistent, and

extracted, leaving practically a vacuum inside. A survive from ancient epochs as still very successful

little water is present in one of the bulbs, and and widely distributed forms. The Trilobites

when the second bulb, containing only water(q.v.) are not now regarded as true Crustaceans.

vapour, is placed in a freezing mixture, the vapour (For distribution, see Heilprin, Inter. Sc. Series,

condenses, which causes more vapour to rise from 1887.)

the water in the first bulb. The result of this Pedigree. It is usually believed that Crustacea

vaporation from the first bulb is the abstraction of are descended from a primitive Phyllopod-like

much heat, and ultimately the remaining water ancestor, and this from a segmented worm-type. | passes into a frozen state. The very constant occurrence of a Nauplius larva Crypt (Gr. krypto, 'I hide'). In the early days has led zoologists to regard it as representing a of persecution the Christians were accustomed for remote ancestor. The lines of differentiation chiefly security to worship in the catacombs or crypts where consisted in the development and manifold modifi. | they buried their dead (see CATACOMBS). When cation of the fundamentally similar appendages, persecution ceased, this custom led to the erection and in the perfecting of the exoskeleton as a base of churches over the graves of martyrs and saints; for muscular attachment. (See Herdman's Classi. but at a later date the bodies of the saints were fication of Animals, Lond. 1885.)

transferred to chambers, constructed to resemble Economic Importance.-Crabs, lobsters, crayfish, the catacombs, under the sanctuary or altar of the shrimps, prawns, &c. form part of our food-supply. new churches, in order to add to their sanctity. Others are indirectly useful as important parts of These crypts and their sacred shrines were visited the food of herrings and other fishes. Many are by numerous pilgrims, and were frequently condoubtless useful in purifying the water from structed for the accommodation of the devotees, of organic debris, while others are the hosts of im sufficient size to admit a number at a time, who portant parasites-e.g. the Cyclops species, which descended by one stair, and ascended by another. contains Dracunculus medinenis.

In other cases the crypts were so placed that the See ACORN-SHELLS, BARNACLE, BRINE-SHRIMP, CIRRI.

shrine of the saint could be seen from the aisles of PEDIA, COPEPODA, CRAB, CRAYFISH, CYPRIS, CYCLOPS,

the choir, the floor of which was necessarily raised LOBSTER, PRAWN, SHRIMP, WATER-FLEA, &c. ; also Con. considerably above the level of the nave. Crypts MENSALISM, PARASITISM, PARTHENOGENESIS, &c., and of these kinds were usual in the early centuries, references under above articles. For further details, and many examples of them have been preserved in consult general text-books of Brooks (Boston, 1882),

Italy and France, even where the churches over Claus, Gegenbaur, Huxley, Rolleston and Hatchett them have been rebuilt. The crypt of the Circular Jackson; also Baird, British Entomostraca (Ray Soc.

church of St Bénigne at Dijon was one of the 1850); Balfour's Embryology; Bell, British Stalk-eyed Crustacea (Lond. 1856); Challenger Reports (several);

largest. There a great circular aperture in the Claus, Genealogy of Crustacea (1876); Dana, Crustacea

centre of the floor of the upper church enabled a of U.S. Exploring Expedition (Phila. 1852); Gerstäcker very large congregation of pilgrims, including those in Bronn's Thierreich; Huxley's Crayfish (1881); Milne. in the crypt, to see the shrine of the saint, and Edwards, Histoire Naturelle des Crustacés (Paris, 183+ | witness any ceremony taking place there. 40); the Monographs of the Naples Station ( several); | After the 13th century crypts were not so much F. Müller, Facts for Darwin (1869); Sars, Fresh-water | in use. The great cathedrals were regarded as

1 of Norway (Christiania, 1867); Spence Bate much in the light of civil as of ecclesiastical edifices, and Westwood, British Sessile-cycd Crustacea (1863–68) and the floor of the choir was brought down to the Stebbing, A History of Crustacea (1893).

level of the rest of the building. It sometimes Crutched Friars, an order of friars, carry. happened that owing to the slope of the site con. ing in their hand a staff, on the top of which siderable underbuilding was required under the was a cross, received the name of Croisiers (Fr.choir, in which case an under church was concroix, 'cross '), corrupted into Crouched or Crutched I structed, which was called by the old name of the 598



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crypt, and was generally used for sepulchral pur- key to decipher letters so written, to what purposes. The crypt of Canterbury is one of the finest pose would they be intercepted by such a deed! of this kind. The crypt of Glasgow Cathedral is In these modern times, however, there has been so

great an improvement in the morals of governments that the custom of killing foreign-office messengers for the sake of their despatch-bags is entirely obsolete in diplomacy, and statesmen have ceased to pillage post-offices or rifle portmanteaus for cryptographic messages.

Most of the odd knacks, contrivances, decoys, blinds, now employed by cryptographers were to some extent known to and employed by the ancients. Substituting points for vowels; arranging threads, knots, or ink-spots at determinate distances ; substituting one letter for another; inventing new arbitrary characters for whole words or even sentences—now made use of extensively in telegraph codes; abbreviating words in their prefixes and affixes; writing a long sentence of nonsense, with a clue to find the words which gave the proper sense-all were brought into requisition. Perhaps the most amusing of all cryptographs was the one mentioned by Herodotus. Histiæus, a Greek at the Persian court, being desirous of sending a secret

message to Aristagoras at Miletus, selected a slave Crypt, Canterbury Cathedral

who was afflicted with bad eyes, and shaved his head,

pretending that it was necessary for his recovery. also a very beautiful example, the vaulting over

In performing this, Histiæus inprinted his secret the shrine of St Mungo being pointed out by Sir

intention in legible characters on the man's head, Gilbert Scott in his lectures as one of the best

and kept him in close confinement till his hair grew specimens of its class.

again, when he sent him to Aristagoras for a perfect Cryptogamia. This term was introduced by

cure. Aristagoras repeated the shaving, read the Linnæus as the twenty-fourth and last class of his

writing underneath, and thus obtained the desired system of classification, and broadly with its

information by means of the unconscious mes. present contents. The name, however (Gr. kryp

senger. tos, concealed,' and gamos, marriage'), in opposi. One of the simplest methods of cryptography is tion to flowering plants (Phanerogamia, q.v.), to use instead of each letter of the alphabet a records its donor's well-founded expectation that certain other letter at a regular interval in advance. sexual reproduction would one day be discovered. Such was a mode of secret writing adopted by Jussieu proposed to distinguish them as Acotyle- Julius Caesar. As a variety of this plau, the alphadones from monocotyledons and dicotyledons; but | bet is used invertedly, z for a, y for b, c for c, the term has necessarily lapsed for many reasons. and so on ; or, while the first seven letters are De Candolle distinguished them into two great

represented by the second seven, the next six may groups, Cellulares and Vasculares ; while End.

be represented by the last six. And many other licher's separation of the vegetable kingdom into

variations may be adopted. But the deciplierment Thallophytes and Cormophytes still further re

of such messages is naturally not difficult, and with cognised their vast morphological range. Armed

a little consideration of the peculiarities of the with the microscope more recent investigators English language, all the ups and downs of many have determined the life-history and mode of

an interesting love story related in cipher in the reproduction of all the leading types, so not only

columns of the Times can be followed from start to amply confirming the hypothesis of Linnæus, or finish with comparative ease. It is known that e is even still further increasing their morphological the most frequent letter; that the is the commonest importance as compared with Phanerogams, but word ; that ea and ou are the double vowels which entirely revolutionising our interpretation of the most frequently occur ; that the consonants most flowering plants themselves, since leading us to common at the ends of words are r, s, and t, &c. view them as more profoundly cryptogamic than We also know that a word of a single letter must the cryptogams. The separate groups of crypto be either the pronoun I, or the vowels a or 0 ; that gamic plants are outlined in the articles ALGÆ, an, at, on, to, of, and in are the most common words SEAWEEDS, BACTERIA, FUNGI, LICHENS, MOSSES, of two letters; that the and and are the most freFERNS, RHIZOCARPS, HORSE-TAIL, LYCOPODIUM,

quent words in three letters; that the most usual SELAGINELLA ; while their relation to higher plants doubled letters are ee, oo, ll, ss, ff; that double is explained under PHANEROGAMIA, FLOWER, vowels are mostly followed by l, m, n, r; that the GYMNOSPERMS.

letter a begins three two-letter words in very exCryptography, the art of secret writing, also tensive use-an, as, at; that the letter o begins or called Writing in Cipher, Hieroglyphic Writing, ends eight two-letter words in very common useSecret Writing, Steganography, Polygraphy, has do, go, no, so, to, of, on, or ; that more words in a been in use from an early date in correspondence sentence of average English begin with t than with between diplomatists and others engaged in import any other letter; that in about three-fourths of ant affairs requiring secrecy. Every government all the words in a sentence, either the first or the used to employ its staff of decipherers, who availed second letter is a vowel; that among consonants, themselves of extraordinary means for interpreting d and h are most largely used, after which come despatches which (fairly or unfairly) came into their n, s, r, t; that the letter q is always followed by possession. The cipherers and the decipherers u ; that no English word of two letters or more waged a constant struggle to outwit each other; ends with i. All these considerations will guide us the one by constructing new difficulties; the other to the solution of any simple cipher, enabling a by conquering the difficulties as soon as constructed. skilful decipherer to read almost any ordinary How often we hear of a courier being murdered piece of cryptographic writing in a very short and his despatches carried off! And without the time.





in British history, cryptography has at no time disguise of music, the notes, rests, expressionbeen in greater requisition than during the Civil | marks, &c., standing for letters. War. Charles I.'s celebrated letter to the Earl of All the methods, however, of cryptography may Glamorgan (afterwards Marquis of Worcester), in thus be summarised : (1) By invisible ink; (2) which he made some compromising concessions to the by superfluous words ; (3) by misplaced words ; Catholics of Ireland, was composed in an alphabet (4) by vertical and diagonal reading ; (5) by arti(sometimes supposed to be Charles's own, but more ficial word-grouping ; (6) by stencil-plates cut out probably Worcester's invention) of twenty-four so as to show certain words beneath; (7) by using short strokes variously situated upon a line (see two letters (Lord Bacon's cipher); (8) by transOGAM). Other letters by the same monarch are posing the letters ; (9) by substitution of letters ; to appearance a mere series of numbers of two or (10) by counterpart tabulations; (11) by mixed three figures divided by semicolons. In such cases symbols ; (12) by a printed key and code-book, it was necessary that the two parties engaging in used chiefly in telegrams; (13) by the employment the correspondence should have previously con- of numerals. certed what words each number was to represent. The present century has seen the decline of

In the reign of William III. the Jacobites in cryptography for all practical purposes, and the art vented many curious ciphers to enable them to is now only regarded as a curious study, closely communicate with their exiled king. All the connected with the history of all nations. Jacobite clubs had distinct methods of their own

Cryptomeria, or JAPANESE CEDAR. This their great aim being to write in such a manner lofty and beautiful hardy coniferous tree (C. that the very ciphers themselves should pass japonica) is widely distributed in mountain dis. through their enemies' hands without suspicion. i tricts of Japan and China, as well as cultivated This they accoinplished by means of sympathetic in many varieties. It was introduced by Robert inks. A favourite Jacobite cipher was the use of Fortune in 1842, and has since passed into cultivaparables, conveying, by means of ordinary lan- tion. Although originally confused with the guage, a double meaning, which only the person cypresses, it is nearly allied to Sequoia and Taxo. acquainted with the writer's views would think of. dium. See CONIFERÆ. The use of cryptography for purposes of state in

Cryptoprocta, a fierce carnivorous animal of England ended, it may be said, with the Peace of

Madagascar, forming a genus and species by itself. 1815. During the Peninsular war the government

Semi-plantigrade, and with beautiful fur, it reattached a cryptographer to the office of the Minister

sembles a large polecat, three feet long, and for Foreign Affairs to read and write the ciphers !

attacks the largest animals with great ferocity. received and despatched. It is said that on more than one occasion the minister was unable to com.

Crystalline Rocks, a name given to all rocks prehend his own cipher.

having a crystalline structure. The crystalline The earliest elaborate treatise on writing in

texture may either be original or superinduced. cipher is the Steganographia (Frankf. 1606) of the

Thus some crystalline rocks, such as certain cal. abbot John Trithemius, à MS. copy of which was

careous masses, owe their origin to chemical prebought for a thousand crowns at Antwerp by Dr Dee

cipitation from water, while others again, such as in 1563. Lord Bacon, who esteemed cryptography

lavas, have consolidated from a state of igneous one of the most useful arts of his time, frained

fusion. There is another large class of crystalline what he believed a not easily penetrable cipher

rocks, the crystalline granules of which present in which he employed only a and b, arranging each

a remarkable foliated character-that is, they are of these letters in groups of five, in such collocations

arranged in more or less parallel layers (see as to represent all the twenty-four letters. Thus

SCHISTS). This peculiar schistose structure appears uabab, ababa, babba conveyed the word fly. In

to have been superinduced-the original rocks his De Augmentis he styled this an omnia per omnia

having been either fragmental or crystalline or cipher, believing that in this case preconcertment

both-and the result of great heat and pressure.

Such highly altered rocks occur in the neighbour. would be necessary; but in reality any clever modern decipherer could have read any letter com

hood of masses of granite, and cover wide regions, posed in such a manner if it were of any length.

where there is abundant evidence to show that the Mr Donnelly, in his work The Great Cryptogram,

strata have been subjected to enormous compresendeavonrs to prove that Bacon inserted a cipher

sion, crushing, and crumpling-having been folded in the Shakespearian plays - which he claims is

and fractured and pushed violently over each other the work of the great philosopher-but the cipher

for distances of sometimes 15 miles and more. It is of so elaborate a kind that nobody but Mr

is therefore believed that pressure and the heat Donnelly has been able to follow its intricacies.

engendered by great earth-movements, and the inThe unfortunate Earl of Argyll used a mode of trusion of plutonic igneous matter, are among the secret writing which consisted in setting down the

most potent agencies in the production of schistose words at certain intervals, which he afterwards

structure. filled up with other words, making of the whole Crystallites, minute non-polarising bodies (the something intelligible, but of no use to any one result of incipient crystallisation ) occurring in the else reading the message. The Marquis of 'Wor. vitreous portions of igneous rocks. See IGNEOUS cester invented a cipher composed of dots and lines

Rocks. variously ordered within a geometrical figure; while Crystallography (from the Greek krustallos, Dr Blair made one of three dots, placed over, under, ice,' an idea among the ancients being that or on the line, by which he could represent no rock-crystal, which may be taken as a type of fewer than eighty-one letters, figures, or words. crystalline minerals, resulted from the subjection The Doctor, in his able article in Rees's Cyclopædia, of water to intense cold). Minerals, salts, and in. declares this cipher to be as nearly as possible organic bodies generally (examples, rock-crystal, undecipherable by strangers; but two years after fluor-spar, alum, and sugar) exist in the crystalline wards, Mr Gaye, of Norwich, published a pamphlet state ; and when we examine all crystals, whether on purpose to solve Dr Blair's riddle. As he occurring naturally or obtained artificially, certain devoted fourteen closely printed octavo pages to laws have been discovered, and phenomena observed, the explanation, any description of the cipher is and these laws and phenomena coustitute the beyond the limits of this article. Mr Thicknesse, science of crystallography. The following are the & well-known expert of the 18th century, also more important laws and principles of the science : devised a plan of conveying information in the (1) Luw of Constancy of Angles.-Crystals of the

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same substance may differ much in general appear. Cubic System.-Nine planes of symmetry-fluorance, but when the angles between their faces are spar, galena, and alum. measured these angles are found constant. Thus *(3) Law of Rationality of Indices. The various the crystals A and B (fig. 1), when cut through in planes of crystals, as explained below, are indicated

in the Millerian system by three numbers, which together form the symbol of the plane. Thus we have planes represented by 1 2 3, by 11 l, by 1 1 0, &c. Now the law of rationality asserts that the symbol of a plane must be represented by numbers which are rational-i.e. numbers which can be expressed exactly, not those like 12, 14, &c., which can only be obtained approximately. Thus by the law of rationality, no plane of a crystal can have such a symbol as 1 V3 5, 1 V2 0, &c.

Crystallographic Notation.-Several methods of representing planes of crystals by symbols are in use. Two of these only need be mentioned-viz. Miller's notation and Naumann's notation. In both systems the planes are referred to three axes corresponding in direction to three edges of the crystal.

Let abc (fig. 3) represent parts or parameters

cut off from three axes xyz, then in Miller's Fig. 1. Drawings of two crystals differing much in appearance, but

with angles at a shown to be constant when similar sections are made.



the direction xy at right angles to the prism, give the sections shown at A', B'; and in each section the angles a will be found the same-viz. 120°; or again, if the angles between the faces ab, bc, or ac, be measured, they will be found identical in both crystals.

(2) Law of Symmetry.-Suppose we cut a crystal in two, and then place the two parts with their cut surfaces on a mirror. The mirror will reflect each part, and may or may not produce the appearance

Fig. 3.

Fig. 4. of the original crystal. If the mirror will produce

The plane 111 in Miller's The plane 1 2 3 in Miller's notation,

notation, the appearance of the original crystal, we have severed the crystal in a plane of symmetry. Thus with a cube, if we cut it in either of the planes system the plane 11 l represents a plane which

abc, def, ghk, Imn, opg, cuts the x axis at one-oneth of a, the y axis at
rhm, nhg, lkn, gmk, and one-oneth of b, and the
place in each case the z axis 'at one-oneth of c.
two severed parts on a Such a plane is indicated
mirror in the way de- / by pgr. The plane 1 2 3
scribed the reflection means a plane which cuts
together with the object the x axis at one-oneth of
will reproduce a cúbe.a, the y axis at one-half of
There are then in the 6, and the 2 axis at one-
cube nine planes of sym- | third of c. Such a plane
metry. The octahedron is represented by stu, fig.
and dodecahedron simi. 4. The plane 1110 means

larly have nine planes of a plane which cuts the x
Fig. 2.

symmetry. With such axis at one-oneth of a, the Each of the planes represented by dotted lines is a plane of * a form as a common y axis at one-oneth of b,

Fig. 5. symmetry.

brick there are three and the z axis at one. The plane 1 1 0 in Mile's planes of symmetry, noughth of c-i.e. does not

notation. while with other forms varying numbers of planes cut c at all, or is parallel of symmetry may be found, until with a sphere to it. Such a plane is represented by uvus in fig. 5. there are an infinite number of planes of symmetry, In Naumann's system some form is selected as for it is obvious that if a sphere be cut anywhere the fundamental pyraby a plane passing through its centre, and the mid of the crystal, and half thus obtained be laid upon a mirror, the his pyramid, which corappearance of a complete sphere will be produced. responds to Miller's Now examining all (holohedral) crystals, it is form, 111, is represented found that they fall into one of the following six by the letter P in all categories or systems : (1) Anorthic System.--No systems but the cubic (in plane of symmetry-examples, copper sulphate this system it is called and anorthite. (2) Oblique System. One plane of 0) and the rhombohedsymmetry-gypsum and washing-soda. (3) Pris. ral (in this system it is matic System. — Three planes of symmetry at right called R). Thus the

Fig. 6. angles to each other-barytes, salt petre, and native planes marked P (fig. 6) A crystal with the faces marked sulphur. (4) Rhombohedral System. - Three planes form the fundamental of symmetry at 120° to each other-calcite, quartz, pyramid, the planes P and ice. (5) Pyramidal System.-Five planes of are those of a pyramid one-half the height, while symmetry-cassiterite, zircon, and idocrase. (6) | the basal plane is represented by oP or a pyramid

in Naumann's notation.

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of no height, while the planes o P represent a the sphere. These great circles correspond to the pyramid of infinite height.

planes of symmetry of the cube (fig. 2) and other **Drawing and Mapping of Crystals.- Various forms of the cubic system. These stereographic modes of representing crystals have been adopted. maps, as will be seen by reference to treatises on Perspective drawings are made by projecting the axes according to the rules of Projection (9.v.), then the various planes are indicated, and from these their intersections are known, and these intersections form the drawing of the crystal. Fig. 7 represents one octant of the form 2 1 1

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the subject, convey a good deal of information respecting the crystals they portray.

Planes of crystals form a zone when the inter. sections of the planes (i.e. the edges) are parallel to each other. Thus, in fig. 6 the faces oP, 4P, P, and COP form a zone. Now in Miller's notation these forms have the indices 001, 112, 111, 110, and it will be noticed that all these symbols have a common ratio-thus, the first and second

index are equal to each other. It may be shown that Fig. 7.

this is universally true; hence, knowing the indices Mode of drawing a crystal from projection of axes. of a plane, we can say whether it is on a particular

zone, or knowing that å plane lies in two zones, we drawn by this method. Some writers represent can determine its indices. Thus, the planes 1 2 3, crystal forms by orthographic projections—that is, 1 2 4, 1 2 5, &c., are all in one zone, as the represent them in plan and front elevation. Of symbols have the common ratio 1:2, and the plane all methods, however, of representing crystals 3 4 5 cannot be on this zone, because its symbol from measurements made with the goniometer, does not contain the ratio 1:2. the most elegant and convenient is that of spher Holohedrism and Hemihedrism.-Crystals which ical projections. Two kinds of spherical projec. have all faces present as required by the law of tion are in use-viz. the gnomic and the stereo. symmetry are termed holohedral. Where, as is graphic. Imagine a glass sphere placed within a often the case, only one-half of these faces are crystal, as in fig. 8, and suppose the faces of the present, the crystal is said to be hemihedral; while

crystal to move parallel if only one-fourth of the full number of faces are
to their original posi. present, the crystal is said to be tetartohedral.
tions until they touch the Physical Crystallography.The physical pro-
sphere, and where the perties of crystals have some interesting relations
faces touch let dots be to the symmetry and form of the crystal, and
marked on the sphere. | these properties are included generally with crystal.
Thus the face a will pro. | lography. Thus, if in the regular system a face
duce the dot a', the face is striated or has any peculiarity, this striation or
o the dot o', and so on. peculiarity will be found on each face which is
When the sphere is thus present by the law of symmetry. Again, most
marked with dots corre. crystals cleave (i.e. break easily ) in certain direc-
sponding to the several tions, and the cleavage planes follow the law of

faces, the next thing is to symmetry. Again, when examined by polarised Fig. 8.

make a map of the dots | light, other properties of crystals in relation to Sphere within a crystal. in their proper position. symmetry are brought out. Thus, crystals of the When planes are moved If the map is to be made regular system (except in a few certain cases) do they touch the sphere where on the gnomic projection, not doubly refract light, no matter in what direcdots are marked.

the sphere is supposed tion the light is incident. With crystals of the

to be placed on the paper rhombohedral system and the pyramidal system on which the map is to be made, and the eye light is not doubly refracted when it falls parallel is then placed at the centre of the sphere. The to the vertical axis, but in other directions it is various dots when projected on to the paper as doubly refracted ; while in the remaining systems seen by the eye placed at the centre of the sphere two directions can be found in which the crystals produce the map. If the map is to be made on of these systems do not doubly refract light, though the stereographic projection, suppose a piece of they do so in all other directions. Again, heat glass to pass through the centre of the sphere as is conducted differently in different systems of in fig. 9, and let the eye be placed touching the crystals. Suppose crystals turned in a lathe into sphere at E, then the dots as they appear on the spheres, and that the centre is made suddenly hot, glass to the eye at E form the map. Such a map then in the regular system the heat spreads equally, of the crystal of fig. 8 is given in fig. 10. In the and after a time the surface of the sphere is stereographic projection all great circles on a uniformly raised in temperature; with other sys. sphere are represented on the map by either straight tems the effect is different; with the pyramidal lines or arcs of circles, whereas in the gnomic pro- | and rhombohedral systems a similar experiment jection they are represented by straight lines. The would result in the surface of the sphere being map (fig. 10) shows not only the position of the heated uniformly over belts corresponding to an dots or poles, but also great circles passing through I equator and parallels of latitude, but the tempera.

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