ORDINARY MEETING, JUNE 3, 1867.

CAPTAIN E. G. FISHBOURNE, R.N., C.B., IN THE CHAIR.

The Minutes of the last Meeting were read and confirmed, after which the following paper was read by the author:

ON THE GEOMETRICAL ISOMORPHISM OF CRYSTALS AND THE DERIVATION OF ALL OTHER FORMS FROM THOSE OF THE CUBICAL SYSTEM. By REV. WALTER MITCHELL, M.A.

1. WHEN elementary substances, or their chemical combinations, pass from a state of vapour; or from a fluid condition into that of a solid; or if they are deposited by evaporation from a fluid holding them in solution, there is a tendency of their particles to arrange themselves according to certain laws of symmetry.

2. Thus solids more or less symmetrical, and with few exceptions bounded by smooth, plane, or flat surfaces, are produced. Such solids are called crystals, and their plane surfaces are termed faces.

3. Some crystals are remarkable for perfect symmetry of form. Among these may be found solids formed with mathematical accuracy, whose geometrical properties had fascinated the ancient geometers ages before they were known to exist in the productions of nature. Others are exceedingly complex, being formed by the combination of faces parallel to those belonging to several simpler forms; the relative positions of these simpler forms to each other being regulated by certain mathematical laws.

4. The more complex forms being reduced to the combination of the simplest from which they can be derived, it is found that all the simpler forms can be grouped together in six distinct classes or systems.

5. The crystals of any one substance may generally be reduced to forms belonging to one system; but there seems to be no limit to the number of combinations of different species of these forms which may take place in any individual crystal.

6. To the rule that all the crystals of a particular substance should have their faces parallel to those of the forms of one system, there are numerous exceptions,

7. The following are the six systems :

1st. The Cubical; called also the tesseral, tessular, octahedral, regular, isometric, and monometric. 2nd. The Pyramidal; called also the tetragonal, square prismatic, quadratic, monodimetric, dimetric, fourmembered, viergliedrig, and the two-and-one axial. 3rd. The Rhombohedral; called also the hexagonal, monotrimetrical, sechsgliedrig, and the three-andone axial.

4th. The Prismatic; called also the rhombic, trimetric, binary, unisometric, orthotype, orthorhombic, zweigliedrig, and one-and-one axial.

5th. The Oblique; called also the monoclinohedric, hemiprismatic, hemiorthotype, clinorhombic, hemihedric-rhombic, augitic, zwei-und-eingliedrig, and the two-and-one-membered.

6th. The Anorthic; called also the doubly oblique, triclinic, triclinohedric, anorthotype, clinorhomboidal, tetarto-prismatic, tetarto-rhombic, eingliedrig, and the one-and-one-membered.

CUBICAL SYSTEM.

8. The forms of the cubical system possess the highest possible degree of symmetry when compared with those of the other systems. They are divided into two groups,―the holohedral, or perfectly symmetrical, and the hemihedral, or half-symmetrical; the latter being derived from the former by being parallel to, or possessing only half their faces, grouped together after certain laws.

9. The holohedral, or perfectly symmetrical forms, are seven in number, and are shown on Plate I. Of these, three-the cube (fig. 1), the octahedron (fig. 7), and the rhombic dodecahedron (fig. 8), are invariable forms, each having but one species, and each the same invariable angles, either of their faces or inclination of their faces.

The remaining four forms are not invariable, and there are an infinite variety of species, each differing from the other in the angles of their faces and their inclinations to each other. The half-symmetrical, or hemihedral forms, are represented in figs. 15, 17, 19, 21, 23, and 25, Plate III.

Holohedral forms, cubical system.

10. The CUBE (fig. 1, Plate I.) is bounded by six equal faces, each face, such as 0105004, being a perfect square;

it has therefore eight solid angles, 01, 0, &c., O., each angle being formed by the union of three planes; and twelve equal edges, such as 0,02, 0203, &c. The inclination of any face to another is measured by the angle contained between two perpendiculars drawn from any point in the edge made by the intersection of the two faces, each on one of the adjacent faces. In the cube this inclination of two adjacent faces is 90°. The facial angles, or the angles between two edges of a face, such as 0,0,0, are always 90°.

11. The OCTAHEDRON (fig. 7, Plate I.) is bounded by eight equal faces, each face, such as CCC, shown on a plane surface (fig. 33, Plate IV.), being an equilateral triangle. It has six solid angles, C1, C2, &c., Ce, each formed by the union of four planes, and twelve equal edges; the inclination of adjacent faces is an angle of 109° 28', and the facial angle, such as C12, is 60°.

12. The RHOMBIC DODECAHEDRON (fig. 5, Plate I.) is bounded by twelve equal faces; each face, such as 0C0C, (fig. 30, Plate IV.), is a geometrical rhomb bounded by four equal lines, o,C being parallel to oCs, and o,Cs to oC. The greater angles of the rhomb C20,C, and C0C, being 109° 28', and the lesser, oС0 and 0,С05, 70° 32'. It has twenty-four equal edges, such as C101, C102, &c., eight solid angles, 01, 02, &c., o, formed by the union of three planes, and six solid angles, C1, C2, &c., C, formed by the union of four planes. The inclination of adjacent faces is 120°. This form is called by some German writers the granatoëdron, as being a characteristic form of the garnet.

13. These three forms, the cube, octahedron, and rhombic dodecahedron, are called invariable forms, as, though differing in size, they always have similar faces and angles; that of the cube being a square, that of the octahedron an equilateral triangle, and that of the rhombic dodecahedron a rhomb whose larger angle is 109° 28'.

14. The four other forms (figs. 2, 3, 4, and 6, Plate I.) are called variable, each presenting an infinite variety of species, differing from each other in their angles of inclination and those of their faces.

15. The THREE-FACED OCTAHEDRON (fig. 6, Plate I.) is bounded by 24 equal faces, each being an isosceles triangle, o1CC (fig. 32, Plate IV.). These faces are so grouped together as to form a solid having eight solid angles, formed by the union of three planes, 01, 02, 03, &c., 0, (fig. 6); the plane angles being the largest of the isosceles triangles; and six solid angles, C1, C2, &c., C, each formed by the union of eight of the equal angles of the isosceles triangles.

There are 12 longer edges, such as CC, CC, &c., and 24 shorter, such as o,C1, o,C, &c. The 12 longer edges are the edges of an octahedron. It may be formed by placing on every face of the octahedron a three-faced pyramid on a equilateral triangular base. The angles of these isosceles triangles differ in different species of the three-faced octahedron, within certain limits to be described hereafter.

The synonyms for this form are the pyramidal octahedron, triakisoctahedron, trioctahedron, and galenoid.

16. The FOUR-FACED CUBE (fig. 2, Plate I.) is bounded like the last by 24 equal faces, each being an isosceles triangle, such as Coo, (fig. 34, Plate IV.), but grouped so together as to form a solid having six solid angles, C1, C., &c., C (fig. 2), each formed by the union of four of the largest angles of the isoscles triangles, and eight solid angles, 0, 0, &c., o, (fig. 2), formed by the union of six of the equal angles of the isosceles triangles. This form has 24 shorter edges, such as Co, C10, &c., and 12 longer ones, such as 0,0,, 0,05, &c. The 12 longer edges are those of a cube.

It may be formed by placing on every face of the cube a four-faced pyramid on a square base.

The angles of the isosceles triangles differ for each particular species of the four-faced cube.

Synonyms. Pyramidal cube, hexatetrahedron, tetrakishexahedron, and fluoride.

17. The TWENTY-FOUR-FACED TRAPEZOHEDRON (fig. 4, Plate I.) is bounded by 24 equal faces, each face being a deltoid or trapezium, Cido,d, (fig. S9, Plate IV.); that is, a four-faced figure having two longer equal sides, Cd, and C1d2, and two shorter equal sides, o,d, od. These 24 equal trapeziums are so grouped together as to form a solid having six solid angles, C1, C2, &c., C, formed by the union of the plane angles of four trapeziums, equal to dC,d; eight solid angles, 0, 0, &c., 08, formed by the union of the plane angles of three trapeziums, equal to do,d; and 12 solid angles, di, do, &c., d12, formed by the union of the plane angles of four trapeziums, equal to Cdo. This form has 24 equal longer edges, such as Cd, Cd, and 24 shorter edges, such as o,d, od, &c. The angles of the deltoids or trapeziums differ for each particular species of the twenty-four-faced trapezium.

Synonyms.-Icositessarahedron, icositetrahedron, trapezohedron, and leucitoid.

18. The SIX-FACED OCTAHEDRON (fig. 3, Plate I.) is bounded by 48 equal faces, each face being a scalene triangle, Cod (fig. 36, Plate IV.). These 24 triangular faces are so grouped together as to form a solid having six solid angles, C1, C, &c.,

C, each formed by the union of eight equal plane angles at the points C1, C2, &c.; eight solid angles, formed by the union of six equal plane angles at the points 0, 0, &c., o,; and 12 solid angles, formed by the union of four plane angles at the points d1, da, &c., d12.

This form has 24 edges, each equal to the edge Cid, 24 each equal to the edge Co, and 24 each equal to od.

The angles of the triangular faces of this form differ for each particular species of the six-faced octahedron.

Synonyms.-Hexakis-octahedron, hexoctahedron, tetrakontaoktaedron, pyramidal granatohedron, triagonal polyhedron, and adamantoid.

19. These seven forms, grouped together on Plate I., have this relation in nature, that any substance forming crystals of any one of these forms may, and does sometimes, form crystals of any one of the other forms, or parallel to their faces. But when these forms are combined on any one crystal, as in fig. 29*, Plate IV.*, the forms to which the faces are parallel, except in the case of what are called twin crystals, always have a certain fixed position with regard to each other. These forms have not only this natural relationship to each other, but they have also certain geometrical relations, which we shall proceed to describe.

20. Looking at Plate I., the forms present no relationship to each other. Plate II. shows them connected together by beautiful geometrical laws.

21. In Plate II. we see that each of the six other forms can every one of them be inscribed, as geometers term it, in the cube.

Fig. 8, Plate II., shows the cube having each of its faces divided into eight equal triangles, by joining the opposite angles of each square by two diagonals, such as 0,08, 0,05, meeting in C2, the centre of the face, and by two other lines, such as DD, DD, also meeting in C1, and joining the centres D1, D, of the edges 010, 050, and D., D, the centres of the edges 0,0, and 0.0.

Fig. 9, Plate II., shows the Four-faced cube inscribed in the cube, and we see that the six solid angles of the twenty-four faced cube, C1, Ca, &c., Ca touch the six centres of the six faces. of the circumscribing cube.

Fig. 10. The Six-faced octahedron inscribed in the cube, six of its solid angles, C1, C2, &c., Ca, touching the centres of the six faces of the circumscribing cube.

Fig. 11. The Twenty-four-faced trapezohedron inscribed in the cube, six of its solid angles, C1, C2, &c., Ce, touching the centres of the six faces of the circumscribing cube.

Fig. 12. The Rhombic dodecahedron inscribed in the cube,