Prof. Tait on Listing's Topologie. 33 



(2) But there is a very simple method of turning a man's 

 right hand into his left, and vice versa, and of shifting his heart 

 to the right-hand side, without waiting for the (problematical) 

 results of untold ages of development or evolution. We have 

 only to look at him with the assistance of a plane mirror 

 or looking-glass, and these extraordinary transformations are 

 instantly effected. Behind the looking-glass the world and 

 every object in it are perverted (yerkehrt, as Listing calls it). 

 Seen through an astronomical telescope, every thing is 

 inverted merely (umgekehri). Particular cases of this distinc- 

 tion, which is of very considerable importance, were of course 

 known to the old geometers. For two halves of a circle are 

 congruent ; one semicircle has only to be made to rotate 

 through two right angles in its own plane to be superpo- 

 sable on the other. But how about the halves of an isosceles 

 triangle formed by the bisector of the angle between the equal 

 sides? They are equal in every respect except congruency; 

 one has to be turned over before it can be exactly superposed on 

 the other. 



Listing gives many examples of this distinction, of which 

 the following is the simplest: — 



Inversion: — (English) V and (Greek) A. 



Perversion: — (English) R and (Russian) K. 



Inversion and perversion: — (English) L and (Greek) T. 



He also gives an elaborate discussion of the different rela- 

 tive situations of two dice whose edges are parallel, taking 

 account of the points on the various sides. 



When weflype a glove (as in taking it off when very wet, 

 or as we skin a hare), we perform an operation which (not de- 

 scribable in English by any shorter phrase than " turning out- 

 side m") changes its character from a right-hand glove to a 

 left. A pair of trousers or a so-called reversible waterproof 

 coat is, after this operation has been performed, still a pair of 

 trousers or a coat, but the legs or arms are interchanged; 

 unless the garments, like those of " Paddius a Corko," are 

 buttoned behind*. 



(3) The germ of the whole of this part of the subject lies in 

 the two ways in which we can choose the three rectangular 

 axes of x, y, z\ and is intimately connected with the kinema- 

 tical theory of rotation of a solid. 



Thus we can make the body rotate through two right angles 



* When a Treatise comes to be written (in English) on this science, 

 great care will have to be taken in exactly denning the senses in which 

 such words as inversion, reversion, perversion, &c. are to be employed. 

 There is much danger of confusion unless authoritative definitions be 

 given once for all, and not too late. 



Phil. Mag. S. 5. Vol. 17. No. 103. Jan. 1884. D 



