CHANGING IDENTITIES 



ard, must contain such a number of standard 

 squares as is contained in the series of the increase of 

 squares. 



The following is the beginning of the series : 



I 2 = 1 = 1 It is easy to see that 



2*- 4 (i_|_ 3) the additional identical 

 3 2 = 9 = ( 4 + 5) squares required in the 

 4 2 = 16 = ( 9 + 7) increase of squares form 

 5 2 = 25 =(16+ 9) an arithmetical progres- 

 6 2 = 36 = (25 + 11) sion whose common dif- 

 T 49 = (36 + 13) ference is two. 

 8 2 = 64 (49 + 15) When we divide a pie, 

 9 2 = 81 =(64 + 17) we multiply the pieces of 

 10 2 =100= (81 + 19) which it consists, but if 



it is a square pie to be 



cut into square pieces the number of pieces must 

 again fall into that fixed series of squares, or they 

 cannot be both square and equal in area. Thus a 

 given square area may not be divided into six 

 equal square areas any more than a square area 

 may be built up of six identical square areas. 

 Therefore, the increase of squares is not, etc. 



PKOPOSITION III. 



The increase of cubes as designated ~by numbers 

 is not arbitrary, but a fixed series. 



One is an identical cube, but it takes an addi- 

 tion of seven identical cubes to make the next 

 larger identical cube. 



39 



