( 409 ) 



The foII()\viiii>' ('onsidorjilion with I'elcrciice to llgure 1 {)roves ihat 



Fig. 1. 



this is true, (hi, Oh, (k tbrni the tirst fuiulamental system of axes; 

 ABC and A' /}'(■ represent the two planes for which the principle 

 of the rational indices holds, i.e. their intercepts oji the axes fnllil 

 equation (1). For a simple |)i-oof it is essential that we let the two 

 planes cut the (>r-axis in the same point, so that /n := ir' and e(piatioJi 

 (1) assumes the form 



1 



(2) 



As second system of axes we take the lines BO, BA and EC, 

 and as second pair of surfaces, which likewise cut the BC-axis in 

 one point, the surfaces A' 11' C and A<J(J. If the prijiciple is to lead 

 to no contradiction, from (2) must follow 



,11 



t V 

 t n 



(3) 



in which the notation to the left is explained by the iigure and 



$1, C^ and ?., are likewise whole numbers. 



If we imderstand by /■ and o I'atioual fractions we can expect 

 that 



u' v' f v" 



— : — = ?• may follow from — : — = ^, 



11 V * t V 



while at the same time 



— ^ r' follows from'' 



V V 



The latter is apparent ; for by the llgure v' -\- if = v and hence 

 r' -j-q' = '1, follows; consequently if /' is rational, then q' is also 

 rational. 



