CRYSTALLOGRAPHY. 



face, let another plate be applied fimilar to the firft, but 

 lefs than it by a rov/ of integrant molecules ; fo that each 

 fide contains only thirteen fquares, and the whole only 1 69 

 fquares. Let fiK other plates be applied in fucceflion to 

 each of the faces, decreafing by a row of little cubes all 

 round ; fo that the fides confift of eleven, nine, feven, five, 

 thsree, and one fquares refpeftively. It is obvious, that by 

 this procefs we have raifed upon each of fix faces of the 

 cube a four-fided pyramid, the faces of which, inftead of 

 being fmooth, will, by their conftant diminution of bulk, 

 reprefent the fteps of flairs. If, however, we conceive 

 the molecules to be extremely fmall, and the number of de- 

 creafing lamina; to be increaf.d, the fteps of the ftairs may 

 be fo fmall as to be imperceptible to the eye, in which cafe 

 the furfaces of the pyramids will appear fmooth. 



Thefe oyramids having each four faces conftitute twenty- 

 four triangular faces, fo that the cube is converted isi.to a 

 new cryftal. Inftead, however, of having twenty -four faces, 

 the decrements having been equal on each edge, the triangular 

 faces in each adjacent pyramid will be in the fame plane, 

 and form together a rhomb, which w ill be evident from the in- 

 fpeftion oijigs. 28 and 29 ; the cube will therefore be converted 

 into a rhomboidal dodecahedron. The cubic nucleus I' I', 

 O O', E E', Jig. 29. is reprel'cnted with the pyramids raifed 

 on three of its faces. When complete, it will have the form 

 reprefented in Plate II. Jig. 27. If the decrement had taken 

 place by two ranges on each of the lamina:, when applied to 

 the cube the pyramids would have been lower ; and their 

 adjacent faces being no longer in the fame plane, the 

 fecondary cryftal would have terminated in twenty-four dif- 

 tin(^ triangles. 



In the example given {Jg. 29. ) it wiU be feen, that as each 

 of the lamina: decreafes by one row on each of its edges, 

 I'iz,. one on I O, and another on the inferior row I' O', and 

 the fame on the other edges, it is obvious that the pyramid 

 decreafes by two rows in breadth for every row in height ; 

 therefore the height wiU be equal to half the breadth at the 

 bafe. 



The terms decrement in breadth, and decrement in height, 

 are thus explained by Haiiy. Decrements in breadth are 

 thofe in which the thicknefs or height of each plate or 

 lamina is only equal to one integrant molecule ; and the 

 refult of the decrement is by one, two, three, or more 

 ranges in the direction of the breadth. 



Decrement in height implies a decrement of one row in 

 breadth on each of the fuccefGve plates ; but each of thefe 

 rows may have the thicknefs or height of two, three, or more 

 molecules. In the latter cafe, the decrement is faid to take 

 place by two, three, or more ranges in height. 



Thefe two kinds of decrement are often combined toge- 

 ther, of which we have an example in iron pyrites with 

 twelve pentagonal faces. ( Plate lll.Jg. 30. ) This variety 

 has a cube for the nucleus, as reprefented Jig. 31 ; and may 

 be conceived to be formed, as reprefented Jig. 32, by decre- 

 ments of two ranges in breadth in one direftion, and by 

 decrements of two ranges in height in the other. The 

 decrements in breadth by two ranges tend to produce a 

 more inchned face than the decrements by two ranges in 

 height ; the confequence refulting is, that the cryftal will 

 not terminate in pyTamidal points, but in wedges, as is feen 

 at qp. Jig. 32. The ftrufture of this cryftal is more par- 

 ticularly delcribed under the article Crystal ; but for 

 Plate!. Nos. 14, 15, 16. r. Plate 11. Jig. ^o, 31, 32. 

 Cr^allography. 



Another example of decrements on the edges is deferving 

 particular attention : it is afforded by that peculiar kind of 

 cryftal of calcareous fpar, commonly called dog-tooth fpar, 

 7 



or which Haiiy denominates the metaftatic cryftal. (Plalell. 

 Jig. 22.) In this cryftal, the edges E O, O I, I K, where 

 the two oppofite pyramids join, coincide with the edges of 

 the primitive nucleus, as may be feen in_^^. 23. The decre- 

 ments fet out from thefe edges, and do not take place on the 

 other fix edges of the nucleus. Now it is eafy to conceive, 

 that the edges of the plates, laid upon the primitive nucleus, 

 form as many triangles, E j O, I j' O, E j' O, &c. refting 

 upon the edges from which they fet out ; and as there are 

 fix in number, there will be twelve triangles, fix above and 

 fix below; and thefe will all be fcalene, in confequence of 

 the obliquity of the edges from which they fet out. 



With refpeft to the other edges of the plates of fuper- 

 pofition, fo far from experiencing any decrement they will 

 increafe ; becaufe they muft always remain contiguous to 

 the axis of the cryftal, which is an imaginary hne drawn 

 from s to s. It is from calculation combined with obferv- 

 ation, that we muft; determine the law of decrement on 

 which this dodecahedral form depends. If we fuppofe a 

 decrement by one range, it may be deraonftrated, that the 

 two faces produced on each fide of the edge from which the 

 decrement takes place will be in the fame plane, and parallel 

 to the axis of the primitive cryftal, conditions which do not 

 apply to the prefent form. The moft fimple hypothefis is 

 that which fuppofes a decrement by two ranges in breadth. 

 This will be more clear from infpefting Plate III. Jig. 33 : 

 it reprefents the upper pyramid of this cryftal, placed on the 

 upper planes of the primitive nucleus, which being partly 

 vifible, admits us to perceive more clearly the progreffive 

 effefts of the decrement by two ranges. 



Each edge of the nucleus, as K I, I O, O E, is divided 

 into ten ; from whence it follows, that each face is an aflem- 

 blage of one hundi-ed fmall rhom.bs, which are the exterior 

 planes of fo many molecules. This conftruftion requires 

 only eight plates of fuperpofition for each face of the 

 nucleus ; and thefe plates being joined to each other, three 

 and three on the edges, which correfpond with the upper 

 edges of the nucleus, form decreafing envelopes, covering 

 each other in fucceflion ; the laft of which is compofed of 

 eight little rhomboids. If we confider the pofition of the 

 line E s, which reprefents an edge of this pyramid, as it 

 appears to the eye, and E j', fuch as it really exifts, we may 

 obferve that the geometrical fummit of the pyramid s is 

 placed a little above the true fummit j' ; but this difference 

 is imperceptible, on account of the extreme minutenefs of 

 the molecules : and for the fame reafon, the channels or 

 fteps upon the pyramid are not vifible. There are cafes, 

 however, in which the channels may be perceived by the 

 naked eye. 



For determining the form of fecondary cryftals by cal- 

 culation, it is only necefTary to take the decrements at their 

 commencement, in order to have as many planes, which, if 

 we conceive them to be extended until they meet, would 

 form a complete polyhedral cryftal ; and in this manner 

 we only conlider the initial effefts of decrements mathema- 

 tically, a method more fimple and expeditious than that of 

 reafoning. It is ufeful, however, to explain in detail the 

 ftrufture of a cryftal, in fuch a manner as may enable us 

 to arrange a number of fmaU foh'ds fimilar to primitive 

 molecules to form a nucleus, in an order conformable to 

 that of nature, and thus to imitate the procefs of cryf- 

 tallizaticn. We fhall give another example from that variety 

 of calcareous fpar, called by Haiiy equiaxe. 



This variety, the fecondary cryftal, is a rhomboid, much 

 more obtufe than the necleus, the greater angle being 

 114° 18' 56". It is reprefented {Plate 111. fg. t^^^.) fur. 

 rounding the nucleus. To extraA the latter at once, it is 



