10 



[Jan. 18, 



three pairs having edges of lengths respectively equal to QB and QC, 

 QC and QA, QA and QB. It is the third of these pairs that is shown 

 in fig. 7. Remark now that the sum of the six angles of the six tetra- 

 hedrons at the edge equal to any one of the lengths QP, QD, QE, 

 QF is four right angles. Remark also that the sum of the four 

 angles at the edge of length QA in the two pairs of tetrahedrons in 

 which the length QA is found is four right angles, and the same 

 with reference to QB and QC. Remark lastly that the two tetra- 

 hedrons of each pair are equal and dichirally* similar, or enantio- 

 morphs as such figures have been called by German writers. 



10. Now, suppose any one pair of the tetrahedrons to be taken 

 away from their positions in the primitive parallelepiped, and, by 

 purely translational motion, to be brought into position with their 

 edges of length QD coincident, and the same to be done for each of 

 the other two pairs. The sum of the six angles at the coincident 

 edges being two right angles, the plane faces at the common edge 

 will fit together, and the condition of parallelism in the motion of 

 each pair fixes the order in which the three pairs come together in 

 the new position, and shows us that in this position the three pairs 

 form a parallelepiped essentially different from the primitive paral- 

 lelepiped, provided that, for simplicity in our present considerations, 

 we suppose each tetrahedron to be wholly scalene, that is to say, the 

 seven lengths found amongst the edges to be all unequal. Next 

 shift the tetrahedrons to bring the edges QE into coincidence, and 

 next again to bring the edges QF into coincidence. Thus, including 

 the primitive parallelepiped, we can make four different parallele- 

 pipeds in each of which six of the tetrahedrons have a common edge. 



11. Now take the two pairs of tetrahedrons having edges of 

 length equal to QA, and put them together with these edges coinci- 

 dent. Thus we have a scalene octahedron. The remaining pair of 



A pair of gloves are dichirally similar, or enantiomorphs. Equal and similar 

 right-handed gloves are chirally similar. 



