FIG. 42. 



SCIENCE ABSOLUTE OF SPACE. 47 



structed, manifestly may be compared like 

 straights, and in this respect considered equiv 

 alent to straights). 



Moreover, a quadrilateral, ex. gr. regular 

 = n is constructed as follows: 



Take ABC=rt.Z, BAC= rt. 

 Z, ACB=i rt. Z, and ~BC=x. 



By mere square roots, X (from 

 31, II) can be expressed and (by 

 37) constructed; and having X 

 (by 38 or also 29 and 35), x itself can be 

 determined. And octuple A ABC is manifestly 

 = n , and by this a plane circle of radius s is 

 geometrically squared by means of a recti 

 linear figure and uniform lines of the same 

 species (equivalent to straights as to compari 

 son inter se) ; but an F form circle is plani- 

 fied in the same manner: and we have either 

 the Axiom XI of Euclid true or the geomet 

 ric quadrature of the circle, although thus 

 far it has remained undecided, which of these 

 two has place in reality. 



Whenever tan 2 ^ is either a whole number, 

 or a rational fraction, whose denominator (re 

 duced to the simplest form) is either a prime 

 number of the form 2 m +l (of which is also 

 2=2+l), or a product of however many prime 

 numbers of this form, of which each (with the 



