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=i rt. 

 Z, ACB= rt. Z, and BC=*. 



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 



