﻿Elements of Geometry. 29 



than with scratches ; and if any of the propositions were 

 doubted and put to the test of experiment, it would certainly 

 be by means of edges ; the extension to scratches would be 

 by means of the contiguity of edges with them. Euclid's 

 methods here undoubtedly indicate that he is leaving, 

 perhaps consciously, the realities of experimental science 

 tor the pure ideas of mathematics. But he has made 

 so little progress towards the new peak that, if he is to be 

 restored to safety, it is far easier to drag him back to that 

 which he lias never left completely than to guide him 

 through the bog in which the two sciences are confused 

 to the very distant goal. 



13. Only a few disconnected remarks will be offered 

 here on the process of deducing the Euclidean propositions 

 from the fundamental laws that have been sketched. Of 

 course, we should employ the " application " (or contiguity) 

 method of Prop. I. 4 wherever possible, instead of trying 

 to avoid it ; for it is based directly on the fundamental 

 notions. Again, we should not commit Euclid's error of 

 supposing that strictly similar triangles can be brought 

 into contiguity; we should apply the mirror image first 

 to one triangle and then to the other. There would be 

 no need to introduce area to prove Prop. I. 47. A Greek 

 writer was forced to do so, because, not being familiar with 

 the multiplication table, he could describe in no other w r ay 

 the relation between a number and its product by 'itself. 

 We should proceed from Prop. I. 34 to Book VI. and prove 

 Prop. I. 47 by drawing the perpendicular from the right 

 angle to the hypoteneuse and using the relations of similar 

 triangles, treated by algebra. For nowadays, since w r e 

 admit no incommensurable magnitudes, we can dispense 

 altogether with Euclid's very beautiful and ingenious 

 subtle! ies about ratios. A ratio in experimental science 

 is nothing but a value taken from the multiplication table, 

 which is established by the measurement of number, i. e. by 

 counting. The laws of the measurement of number are 

 involved in those of the measurement of every "continuous" 

 magnitude. 



April 22, 1922. 



