DK. LEESON ON ISOMORPHISM &C. XV.-On Isomorphism +. and on a simple Zaw governing all crystalline forms by H. B.LEESON, M.D. MY former papers were more especially intended to conduce to a correct reading of crystalline forms and to show the relationship which all crystals possess to certain lines of direction termed gubernatorial axes such axes not being arbitrarily chosen but coinciding with the directions of the aggregating forces magnetic and electric and evidenced by the state of tension existing in the interior of the crystals as exhibited when such crystals are examined by polarized light. I am the more anxious to impress upon all those advancing doctrines having reference to crystalline form the necessity of a correct understanding of the crystals on which their observations are based because I observe in numerous papers on Isomorphism Dimorphism &c.that forms have been considered primary although onIy secondary modifications ; and that substances have been ranked as dimorphous although in fact only crystallized in different varieties of the same system or class. If Isomorphism as the name would imply had reference simply to the same external configuration then all substances would be polymorphous ; for although only one form may have hitherto been observed in any particular substance still the instances are so numerous in which the same substance (as for instance fluor spar sulphate of barytes or carbonate of lime,) does crystallize in a variety of forms that judging from analogy we have reason to believe every substance may be similarly varied; or to convey our meaning in other words; that each substance may crystallize in uniaxial biaxial and triaxial forms such forms being still further varied by imperfect or defective development elongation and composition for the understanding of which I must refer to my former papers as printed in the Memoirs of the Chemical Society.The term isomorphous is now then generally employed with reference only to the system in which a substance is supposed to crystal- lize; and such system or class must be determined by the position and length of the gubernatorial axes. Perhaps I may be excused for suggesting that the term omo-axed or sirnil-axed would better convey the idea intended and prevent that misconception which evidently exists in the minds of some observers.Since I first noticed the similarity of the series of forms occurring in each class I have been enabled very considerably to enlarge my collection by specimens exhibiting a regular gradation of such forms. This has enabled me to apply the goniometer formerly described DR. LEESON ON ISOMORPHISM &C. to the messurement of the inclination of edges-as well as of planes in a very extensive series of crystals. Thus it is that I have been led to observe a very simple relation indicating the law by which so far as my own observations extend every variety of crystalline form seems to be engendered.* Premising that the law itself is masked (as explained in my former papers) by the unequal development of particular planes; that such unequal development sometimes occasions the defect of certain planes and that by elongation or combination of separate forms the external aspect of a crystal may be still further complicated I proceed to enunciate the principle according to which as I believe all perfect forms are produced.Of course in examining a particular crystal those planes must be selected which belong to the same simple form then such form must be considered as it would exist if all the planes were equally developed and any defective planes supplied all this will be the more readily accomplished when the method of Nature’s proceeding and the series of forms is fully understood. To enunciate then the principle.The perfect simple forms constitute a series c0mmencin.y ; First with the uniaxial form and subsequently composed of six pyramids of four or eight sides placed oiie at each extremity of the three gubernatorial axes. Such pyramids succeeding each other by a similar and regular gradation WHATEVER BE THE SYSTEM or disposition and normal length of the gubernatorial axes. The series then may be considered as composed FIRST,of the uniaxial form. A six-sided parallelopiped described in my former papers as produced by a plane placed on each extremity of the gubernatorial axes so as to be parallel to the other two axes. The planes of this form may be considered as the lower limit or zero in height of the biaxialf- pyramids. SECONDLY, of alternate triaxial and biaxial forms consisting each * In my former papers I have already shewn that uniformity in the measurement of the inclination of planes is far from constant and that the mathematical accuracy supposed to be attainable is not to be expected.My own goniometer and the microscope have still further convinced me that planes apparently perfect and brilliant to the naked eye are full of inequalities ; and I believe that in fluor want of attention thereto and of microscopic examination has caused inclinations to be taken for planes which really consisted of step-like diminutions whilst in that substance as in bismnth and other metals forms have been considered rhombic resulting merely from the unequal deposition or subsequent removal (by solution or otherwise) of lamina on one or more edges of the crystal.t By referring to my preceding papers it will be understood that the term ‘‘biaxial” means that the planes cut two of the gubernatorial axes and are parallel to the third axis whilst ‘‘ triaxial” planes cut all three axes. DR. LEESON ON ISOMORPHISM &C. of six four-sided pyramids replacing or surmounting as it were the uniaxial planes; and placed on each extremity of the gubernatorial axes in such a manner as that a line joining the apices of the opposite pyramids corresponds to the gubernatorial axis ;whilst the four sides of each pyramid are parallel in the biaxial forms to the four edges of the uniaxial plane they respectively surmount ;and in the triaxial forms to the diagonals of such plane.These forms succeed each other in a series produced by a continual replacement of the four lateral edges of the pyramid by planes; so that each succeeding or preceding pyramid differs or revolves as it were 45O on theuniaxial plane. And THIRDLY, of a series of triaxial forms composed of six eight- sided pyramids produced as it were by a duplicature or repetition of the pyramids of the triaxial forms in the preceding or second series. Each eight-sided pyramid consisting of planes joining the four lateral edges of a pyramid of one of the triaxial forms in the preceding series with the four lateral edges of another equal and similar pyramid placed in the reverse or biaxial direction so that the bases of the two pyramids thus joined together differ from each other 45O.The primary triaxial form may be considered as the first pyramid of the series or point of departure; from which by the continual replacement of edges by planes the most usual and descending series of pyramids of lower height is produced; whilst by replacement of planes by edges an ascending series of more acute pyramids may be obtained. The ascending series will necessarily consist of only two pyramids joined base to base otherwise re-entering angles as subsequently explained would be produced which is inconsistent with a perfeet form. Ae an illustration of the principle I will describe the series as occurring in the regular or as I term it the rectangular equiaxial system as that in which the succession may be most easily comprehended; but I hope it will be fully understood that the same series exists in each of the other systems and that I possess numerous specimens to substantiate such position particularly in the oolique and right oblique classes; figures of which I am preparing for publication in which the planes belonging to each member of the series will be designated by a particular colour so as to exhibit at one glance the various perfect forms entering into the composition of any particular crystal.To describe then the series as occurring in the regular system. The figures referred to being those given with my former papers. FiasT.-Primary uniaxial form Cube Fig. 7,Pi. VII. SECOND.-P~~WUW~ triaxial form Octahedron Fig.7 consisting DH. LEESON ON ISOhlORPHISM &C. of six four-sided pyramids placed diagonally on the faces of the cube so as to bisect the four edges of each face. The height of the pyramid being such that the planes of the three pyramids,/ur- rounding each corner of the cube coincide in one plane; thus each plane of the octahedron may be considered as composed of the planes of three separate pyramids and corresponds to the lower limit or zero of the three-sided pyramid to be observed in each SUC-ceeding triaxial form the solid angles of the cube being the upper limit or zero of height of the three-sided pyramid. Third.-Primary biaxial form Rhomboidal dodecahedron Fig. 8. By replacing the twelve edges of the octahedron by planes the rhom- boidal dodecahedron is produced and it will be readily observed in Fig 8 that it consists of six four-sided pyramids placed with their sides parallel to the edges of the cube the height of the pyramid being such that the planes of the two pyramids adjacent to the same edge coincide and thus form one plane.If the pyramid were of lower height as in the succeeding biaxial forms the planes would as it were double over the edge and thus give rise to twenty-four instead of twelve planes. If the pyramid were of greater height the angle across the edge would be a re-entering angle. Hence this hrm is the limit of the height of the pyramid in the biaxial forms. POURTH.-&COTZ~ triaxial form Trapezohedron Fig. 21 P1. X. This form results from a replacement of the edges of the rhomboidal dodecahedron by planes and may be easily observed to consist of six four-sided pyramids the sides of which are parallel to the diagonals of the sides of the cube; but in consequence of the manner in which the planes necessarily intersect each other at their bases the planes themselves lose their triangular outline becoming in fact trapezoidal and thus obscuring the pyramidal nature of the form.The height of the pyramid being less than in the primary triaxial form originates a three-sided pyramid corresponding to each face of the octahedron and the form might thus be considered as composed of eight three-sided pyramids. The solid angles of the cube or uniaxial form being the upper limit or zero in height of these three- sided pyramids.A line joining the opposite pyramids corresponds of course to a diagonal of the uniaxial form. FIFTH.-&?cOnd biaxiul form Pyramidal hexahedron Fig. 7 P1. X. produced by planes replacing the edges of the last form and evidently consisting of six four-sided pyramids of lower height than those of the primary biaxial form and therefore doubling over the edges of the uniaxial form as already explained. This doubling originate8 a six-sided pyramid corresponding to each face of the octahedron or 152 DR. LEESON ON ISOMORPHISM &C. primary triaxial form. Hence the diagonals of the uniaxial form join the apices of the opposite six-sided pyramids. SIXTH.-T~~~~ Triaxialform a more obtuse trapezohedron. SEvE"rH.-!ird biaxial form a more obtuse pyramidal hexa- hedron.EZGHTH.-FOU& triaxial form a trapezohedron still more obtuse. NINTH.-F$~ biaxial form a pyramidal hexahedron with still flatter pyramids. Unless in fluor what I consider as step like strk be considered planes I have no specimens carrying this second series beyond the fifth biaxial form. THIRD SERIES.-F~~S~, a duplicature of the primary triaxial form or octahedron. The Triaxisoctahedron Fig. 24 P1. VIII. formed of planes joining the edges of the primary triaxial form or octahedron with those of another equal and similar octahedron resolved 45O thus forming a figure compounded of six eight-sided pyramids placed on the extremities of the gubernatorial axes or of eight three-sided pyramids placed on the faces of the octahedron.The line joining thc apices of the opposite three-sided pyramids corresponds to the diagonals of the uniaxial form. These three-sided pyramids are it will be observed in reverse position to those belonging to the trape- zohedron. Compare Fig. 24 P1. VIII. with Fig. 7 P1. VII Second form a duplicature of the second triaxial form or trape-zohedron producing the Tetraconta Octahedron Fig. 21 P1. VIIL composed of 48 planes joining the four lateral edges of the six pyramids of the second triaxial form with the four lateral edges of six other equal and similar pyramids placed in the reverse or biaxial direction each plane of the cube or primary uniaxial form is thus surmounted or replaced by an eight-sided pyramid and each solid angle of the cube or face of the octahedron by a six-sided pyramid.Third form a Tetracontahedron derived from a duplicature of the third triaxial form and hence more obtuse. Fourth a still more obtuse Tetracontahedron derived from the fourth triaxial form. Beyond this point I do not possess nor have I met with any specimen. Thus it is evident that if one number of the series can be fairly made out all the others may be deduced from it; and I have greatly assisted my labours by preparing and keeping by me a table of the successive angles of inclination assuming 78O 20' as the most frequently occurring acute angle of the oblique uniaxial forms. In conclusion I would refer to a few circumstances which have tended as I believe to erroneous conclusions.DR. LEESON OM ISOMORPHISM &C. First too much reliance on the perfection of planes which on examination by the microscope would have been found imperfect. Next assigning a primary position to planes of a secondary character. I have already sufficiently indicated in my former paper many instances of this description. 1 have in my collection perfect rhomboids of fluor spar pro- duced by the defect of two p€anes of the octahedron as indicated in Fig. 23 plate XII.; and yet no one has hitherto ventured to pronounce fluor dimorphous. Again I have cubes of fluor apparently rhombic from the unequal deposition of laminz before referred to and yet similar discrepancies in bismuth and antimony have induced some to consider such crystals as not belonging to the regular system.But again since by defect of planes consequent on the undue development of the planes of any two opposite four-sided pyramids every form may become octahedral. It is evident that the apparent proportional length of the axes in different supposed octahedra of the same substance may be greatly varied and thus a substance actually belonging to the regular system may be referred to the pyramidal merely in consequence of the octahedron examined not being the primary triaxial variety but a portion of another form. I have crystals both of copper and of silver illus- trating this position; and which if they had occurred to other observers would I have no doubt have induced them to class these substances as like tin similarly circumstanced dimorphous.Cleavage in such cases would evidently afford no assistance ; and here again I may observe too much reliance must not be placed on the coincidence of clcavage planes with those of the uniaxial form. In fluor it is well known the cleavage planes correspond with the primary triaxial planes; and I believe that in the right oblique system they frequently coincide with the primary biaxial planes. Lastly I would caution persons not to consider crystals as pseudomorphous merely because they are of unusual occurrence amongst the specimens of the substance. If there is no reason to suppose the crystals have been actually moulded in the matrix of some other crystal the term is not appropriate.I have primary uniaxial and triaxial specimens of quartz which others would term pseudomorphous but which I believe have not been formed in any matrix and are only SO termed because persons have not been accustomed to believe in the existence of such forms. MR. CHAPMAN ON March 19 1849. The President in the Chair. The following presents were announced “Memoirs upon Natural History,” collected by W. Haidinger and published by subscription and ‘‘Reports upon the Communications of the Friends of Natural Science in Vienna,” from W. Haidinger. Messrs. C. F. Birnaud G. Simpson E. Packard and R. Prosser were elected members of the Society. The following papers were read Analysisof Berlin Porcelain by MR.WILLIAM WILSON,Student in the Royal College of Chemistry.-The specimen analysed was taken from an evaporating dish and the quantities of the different ingredients were estimated by the usual analytical processes.The specimen as shown by the composition detailed below was richer in silica and protoxide of iron than is usually the case while there is a deficiency in alumina and potash. Silicic acid . . 71.34 Alumina . . 23.763 Protoxide of iron . .. 1.743 Lime ..... . 0.5686 Magnesia ... . 0.1923 Potash .... 2.001