34 Prof. Tait on Listing's Topologie. 



about one axis, so that each of the other two is inverted. Such 

 an operation does not change their relative situation. 



But to invert one only, or all three, of the axes requires that 

 the body should (as it were) be pulled through itself, a process 

 perfectly conceivable from the kinematical, but not from the 

 physical point of view. By this process the relative situation 

 of the axes is changed. 



When we think of the rotation about the axis of x which 

 shall bring Oy where Oz was, we see that it must be of oppo- 

 site character in these two cases. And it is a mere matter of 

 convention which of the two systems we shall choose as our 

 normal or positive one. 



That which seems of late to have become the more usual is 

 that in which a quadrantal rotation about x (which may be 

 any one of the three) shall change Oy into the former Oz (i. e. 

 in the cyclical order x, y, z), when it is applied in the sense in 

 which the earth turns about the northern end of its polar axis. 

 Thus we may represent the three axes, in cyclical order, by a 

 northward, an upward, and an eastward line. So that we 

 change any one into its cyclical successor by seizing the 

 positive end of the third, and, as it were, unscrewing through 

 a quadrant*. 



The hands of a w r atch, looked at from the side on which 

 the face is situated, thus move round in the negative direc- 

 tion; but if we could see through the watch, they would appear 

 to move round in the positive direction. This universally 

 employed construction arises probably from watch-dials having 

 been originally made to behave as much as possible like sun- 

 dials, on which the hours follow the apparent daily course of 

 the sun, i. e. the opposite direction to that of the earth's rota- 

 tion about its axis. 



(4) This leads us into another very important elementary 

 branch of our subject, one in which Listing (it is to be 

 feared) introduced complication rather than simplification, by 

 his endeavours to extricate the botanists from the frightful 

 chaos in which they had involved themselves by their irrecon- 

 cilable descriptions of vegetable spirals. [He devotes a good 

 many pages to showing how great was this confusion.] 



When Ave compare the tendrils of a hop with those of a 

 vine, we see that while they both grow upw r ards, as in coiling 

 themselves round a vertical pole, the end of the hop tendril goes 



* These relations, and many which follow, were illustrated by models, 

 not by diagrams ; and the reader who wishes fully to comprehend them 

 will find no r< aeon to grudge the little trouble involved in constructing 

 such models for himself. 



