28 Mr. C. M. Willich on the Partition of the Cube. 



being cut into two portions by a diagonal line will produce a 

 body which I have assumed as a unit (A) for the construction 

 of many geometrical and crystalline bodies. The models laid 

 before the Association show some of the forms produced. The 

 rhomboidal cube (J), and the rhomboidal dodecahedron (L) with 

 pyramidal faces (containing in mass one half of the cube from 

 which it is derived) , may be considered interesting; but the 

 various crystalline figures which may be formed by a combina- 

 tion of my unit (A) I cannot even estimate — though probably all 

 geometrical solids and even many, if not all, crystalline bodies 

 may be included, if we use sections of bodies produced by a par- 

 tition of the cube. 



It may be observed that the pyramid (B), or one-sixth of the 

 cube, which contains two units, may itself be divided into four 

 bodies by sections parallel to the sides, each of which is one-third 

 of a cube containing one-eighth of the mass of the cube from 

 which it was derived ; so that in fact we may go on dividing and 

 reproducing bodies of a similar shape, and still retaining the 

 same angles as in the portion from the original cube. How far 

 this subdivision may be carried in nature, or how much further 

 than our powers of vision will reach, I will not venture an opinion. 

 We can imagine that the commencing atoms may be infinitely 

 small when we remember the wonders revealed by the microscope. 



I entertain a sanguine hope that, should the attention of phi- 

 lophers be drawn to this subject, the further development may 

 perhaps be the means of throwing some unexpected light as to 

 the shape of an atom. I incline, however, to think that atoms 

 may differ in shape in the three kingdoms of nature — mineral, 

 vegetable, and animal. 



As to the practical use in education, I am of opinion that the 

 study of geometry would be simplified by the use of models 

 showing the relative value as to the solidity of geometrical bodies, 

 and thus convey knowledge to the youthful mind by means of 

 the eye more readily than by any description, as when convinced 

 by the sight the mind would understand with greater facility. 



LIST OF MODELS ACCOMPANYING THE ABOVE PAPER. 



A. The unit or one- twelfth part of the cube having an edge of one inch. 



B. 1st union of two units, forming a low four-sided pyramid of which six 



make up a cube. 



C. 2nd union of two units, forming a high four-sided pyramid of which 



six also equal cube. 



D. 3rd union of two units, forming right-handed solid, being one-sixth of 



cube. 



E. 4th union of two units, forming left-handed solid, being also one- sixth 



of cube. 



F. 5th union of two units, forming part of body G, which is the fourth part 



of cube. 



