36 ELEMENTS OF SCIENCE 



Then the triangle ACB will be the triangle required, 

 i.e., it will be an equilateral triangle drawn upon a given 

 straight line of a certain definite length namely, from 

 AtoB. 



This is and must be so, for the following reasons : 



Since A is the centre of the circle BCDEF, it follows, 

 from premiss 3, that the two lines AB and AC must 

 be equal, since they are both lines which pass from the 

 centre to the circumference of the same circle, i.e., the 

 circle BCDEF. 



Similarly, because B is the centre of the circle ACGHI, 

 the lines AB and BC must be equal, because they both 

 pass from the centre to the circumference of a circle, 

 i.e., of the circle ACGHI. 



But we have already seen that the line AC is equal 

 to the line AB, therefore (by the 4th premiss) the line 

 A C must be equal to the line BC since both AC and 

 BC are* each equal to AB. It follows then that the three 

 lines AB, AC, BC, are equal to one another. Therefore 

 the triangle they form is an equilateral one described 

 upon a given straight line of a definite length namely, 

 upon the line AB. 



Proofs analogous to the above, support all the proposi- 

 tions of Euclid, and the results are absolutely certain and 

 true. In nature, the properties of bodies as regards 

 their occupation of space or, as it is called, their "ex- 

 tension" correspond as accurately with the laws of 

 geometry as their material conditions render possible. 

 Obviously the lines and surfaces which can be made in 

 some substances are less definite and exact than those 

 which can be formed in others, and in no substance can 

 lines and surfaces of ideally perfect straightness, &c., be 

 produced. 



But such deviations from ideal perfection, in no way 



