21 



secting in the fixed point of the figure, this last must necessarily 

 have a third characteristic rotation round another symmetry-axis 

 passing through 0. The last rota- 

 tion is equivalent to the former 

 two, executed in succession. Thus 

 from the existence of two axes of 

 symmetry, the existence of a third 

 necessarily follows. 



But if the figure considered has 

 two axes of the second order, in- 

 tersecting in 0, the resulting mani- 

 pulation must be an ordinary rota- 

 tion around an axis of symmetry 

 of the first order, passing through 

 0. For by the first operation of 

 the second order the figure F is 

 transformed into its mirror-image 

 F', and this is in its turn changed 

 by the operations corresponding to 

 the second mirror-axis into a figure 

 congruent with the initial one. 

 Executed successively they are 

 therefore together equivalent to 

 an ordinary rotation; and this is 

 evidently the same thing as the 

 assertion already made (page 16) 

 that a figure cannot possess more than one symmetry-property 

 of the second order alone, without the simultaneous possession of 

 those of the first order. 



From both these examples it will be clearly understood that there are 

 definite mathematical relations connecting the different symmetry- 

 properties of one and the same figure simultaneously present. Thus the 

 simultaneous existence of two such properties always requires the 

 existence of a third one equivalent to the other two combined. It is 

 precisely the existence of such relations, that makes the whole doctrine 

 of symmetry a possibility and permits us to deduce the general 

 classes of symmetrical types by mere mathematical reasoning. 



8. However in this connection the question might rise: what 

 will happen, if those simultaneous axes of symmetry do not intersect, 

 but for instance, if they cross in space? 



Fig. 12. 



