206 



NA TURE 



{Dec. 27, i88j 



crystal there is an ideal or root form proper to one or other of 

 the five kinds of internal symmetry which have been presented, 

 from w hich root form the actual form can be derived by a proper 

 proportionate increase nf dimension in one or more directions. 



It is evident thit, while our path must become more and more 

 intricate as we endeavour to establish in the cases of more 

 complex compounds relatione similar to those abive traced, the 

 reference of who'e classes of analogous forms, differing only in 

 their angles, to one root form, removes a very important difficulty, 

 and the wide api'licability which it confers on the five kinds of 

 internal symmetiy with u hich we started appears in the fact that 

 there is no crystal form which cannot be thus referred to an 

 approjiriate root form in harmony with one or other of these five 

 kinds of internal symmetry.' 



One more ca^e maybe mentioned in which a probable internal 

 symmetry can be assigned to a compound in harmony with its 

 actual cry^tal form ; it is a mire diflicult one. 



The molecule of Ictland or calc-ipar is usually believed to 

 consist of one atom of calcium, one atom of carbon, and three of 

 oxygen. We shall, however, take a liberty, and suppose that the 

 atoms of calcium or the atoms of carbon have but half the mass 

 attributed to them ; that in the formula of this compound we 

 should write either two .atoms of calcium or two atoms of carbon 

 in place of one.- 



Making this supposition, we observe that if the calcium and 

 carbon atoms were alike we should have six atoms, three of one 

 kind, three of another ; in other words, we should have equal 

 proportions of two kinds of atoms, from which, since the form 

 of Iceland spar is but little removed from a cube, we naturally 

 argue that just before crystalli^ation its atoms were arr.mged 

 according to the fir.-t or second kind of internal symmetry ; these 

 two kinds being, it will be remembered, those in harmony with 

 the cubic form which admit of very symmetiical arrangement of 

 particles of two kinds present in equal numbers. 



Since Iceland spar is auniaxal crystal, the arrangement of the 

 three kinds of atoms, whatever it is, must be symmetrical about 

 one axis only ; and we shall now endeavour to show that the 

 atoms can be thus arranged in either the first or second kind of 

 symmetry. 



We will show first that they can be -bus arranged in the 

 second kind. 



Where there are but two kinds of particles present in equal 

 numbers, symmetry require; that the alternate layers of this 

 kind of symmetry (see Fig. 3) shall consist entirely of similar 

 kinds ^nd therefore in the case before us, one set of alternate 

 layers will represent oxygen atoms ; the other, atoms of c:ilcium 

 and carbon. Now particles present, as we suppose the calcium 

 and carbon atoms to be, in the proportion i : 2 can be ((uite 

 .symmetrically arranged in these layers (plan/), as the sphere 

 centres were in the layers depicting the fourth kind of symmetry 

 (plan <;), and therefore the only question remaining is the relative 

 disposition of the layers of calcium and carbon atoms with 

 respect to one another. 



Now the spheres in alternate layers of the second kind of 

 symmetry considered alone have the relative arrangement of the 

 third kind of symmetry (Fig. 4), and in determining the rela- 

 tive disposition of the calcium and carbon atoms, we may 

 therefore neglect the oxygen atoms, and treat the case as belong- 

 ing to the third kind of symmetry. The two spiral arrange- 

 ments in this kind of symmetry, in which the less numerous 

 sph-res in the fourth layer are vertically over those in the first 

 (see anlc), have the necessary symmetry about a single axis, 

 and if the calcium and, carbon atoms have one of these arrange- 

 ments, the requirements of the case are entirely met. 

 ^..We will now show that the three kinds of atoms can also be 

 arranged symmetrically about a single axis in the first kind of 

 symmetry. 



One half the spheres depicting this kind of symmetry will in 

 this case reiiresent the oxygen atoms, and the remaining half the 

 atoms of calcium and carbon (see Fig. 2), and, as previously 

 noticed, the arrangement of either half will be that of the second 

 kind of symmetry. It follows that the question of the relative 

 disposition of the atoms of calcium and carbon is simply the 

 question of the symmetrical arrangement about a single axis of 

 atoms of two kinds present in the proportion 2 : I in the second 



* The very symmetrical form the pentagonal dodecahedron is not in 

 harmony with either of the five kinds of symmetry, nor is it found in 

 crystals. 



- It has already been remarked that the crystal form of fluor-spar favours 

 the supposition that calcium has half the atomic weight usually attributed 



kind of symmetry (Fig. 3). And since the layers of spheres 

 depicting this kind of symmetry have a triangular arrangement 

 (plan b). It is evident that this can be accomplished here ju.-t as 

 in the former case. 



In tither of the two arrangements just described we have only 

 to suppose that when the sym-jietrically placed atoms change 

 volume at the time of crystallisation the dimensions transversely 

 to the axis of symmetry are increa-ed relatively to those in the 

 direction of this axis, and we have an obtuse rhombohedron 

 where formerly we hal a cube. And the significant fact that 

 the angle of a rhombohedron of calc-spar diminishes when the 

 crystal is heated supports this theory of its production. Per- 

 haps the arrangement of the atoms according to the first kind of 

 internal symmetry is the more probable of the two, as this would 

 give the cleavage directions coincident with the directions of 

 layers of similar atoms (oxygen). 



An important fact supporting our conclusions is that certain 

 definite relations as to their proportions which are found sub- 

 sisting between the allied forms taken by crystals of the same 

 substance are found inherent in one or other of the five kinds of 

 internal symmetry. 



Thus it is well known that if a particular substance is found 

 crystallised in hexa,;onal pyramids of various kinds — that is, 

 whose sides have various different degrees of inclination to the 

 b.ase — the number of kinds is strictly limited, and they are strictly 

 related to each other. If x be the side of the hexagonal base of 

 the pyramid and y the height for the same substance, while x 

 remains constant, y has not more than fourteen different values, 

 seven related thus : c,\ c,\c, \ c, \ c, -j-V c, jV <^ I ^'"-l ''^^ other 

 seven sin lilarly related thus : d, \d, hd, \ d, 5 d, y'n d, y'5 d ; 

 and c bearing to d the ratio 2 : ^^'3. 



Now, if we turn to the fourth kind of internal symmetry 

 (Fig. 5) to ascertain the possible varieties of inclination of the 

 sides of hexagonal pyramids which c.in be depicted, we find that 

 the greatest possible height to which we can build a hexagonal 

 pyramid of equal spheres is exactly double the height of a tetra- 

 hedron with the same side as the hexagonal base of the pyramid. 

 Thus, if twenty-five spheres form each side of the hexagonal 

 base, giving twenty-four equal distances betvieen the sphere 

 centres in any one side, we find that the highest possible pyramid 

 has forty-nine layers of spheres giving forty-eight equal spaces 

 between consecutive layers. 



If we call this height f,.it is evident that pyramids corresponding 

 with the first of the above series of actually observed forms will 

 have respectively — 



49 layers of balls, giving 48 spaces between consecutive layers. 

 37 ., ,, 36 



25 .. ,, 24 



13 .. ,, 12 



7 „ ,. 6 



5 .. » 4 



4 ., ,, 3 



We find, moreover, that such a series can be readily depicted, 

 and that, upon examination, no additional terms appear ad- 

 missible. 



Again, a further inspection of the stack of spheres shows us 

 that w ith the same heights — that is, with the respective numbers 

 of layers just enumerated — we may, in place of the base layer 

 which forms a hexagon whose sides have twenty-five spheres 

 each, have a base derived from this in which each of the six 

 spheres at the angles becomes the centre of a side, the outline of 

 the base layer being now a larger hexagon described about the 

 hexagon which bounded the former base layer. The sides of 

 this new base thus bear to the sides of the old the ratio subsisting 

 between the side and the perpendicular of an equilateral triangle, 

 i.e. the ratio 2 : 1/3. And finally, since the distance between 

 the planes containing the centres in successive layers bears to the 

 distance between centres in the same layer the same ratio which 

 the perpendicular from the angle of a tetrahedron upon its 

 opposite face bears to its edge, that is the ratio \'2 : v'3> it 

 follows — 



That the two allied series of possible altitudes of hexagonal 

 pyramids thus formed, if we take t/ie same length of side a for 

 both, will be — 



First Series 



?dla-^-^a- ^a- "-il a; ^ a ■ ^ a ; ^ a. 



^i\ ' 2sl2 ' V3 2^/3 4-^^3 6\/3 8^3 



Second Series 



\'2.a ; I \'2 a ; J •/2 . a ; i\/2 a ; IV2 a ; i\, ■y/z a ; ^\ V2 . a. 



