THE SQUARING OF THE CIRCLE. 



121 



employed in the solution of the five problems mentioned. Geom- 

 etry simply limits itself to the presupposition that these problems 

 are solvable, and regards a complicated problem as solved if, upon 

 a specification of the constructions of which the solution consists, 

 no other requirements are demanded than the five above mentioned. 

 Since, accordingly, geometry does not itself furnish the solution of 

 these five problems, but rather exacts them, they are termed postu- 

 lates* All problems of plane geometry are not reducible to these 

 five problems alone. There are problems that can be solved only 

 by assuming other problems as solvable which are not included in 

 the five given ; for example, the construction of an ellipse, having 

 given its centre and its major and minor axes. Many problems, 

 however, possess the property of being solvable with the assistance 

 of the above-formulated five postulates alone, and where this is the 

 case they are said to be "constructive with straight edge and com- 

 passes," or " elementarily " constructible. 



After these general remarks upon the solvability of problems 

 of geometrical construction, which an understanding of the history 

 of the squaring of the circle makes indispensable, the significance 

 of the question whether the quadrature of the circle is. or is not 

 solvable, that is elementarily solvable, will become intelligible. 

 But the conception of elementary solvability only gradually took 

 clear form, and we therefore find among the Greeks as well as 

 among the Arabs endeavors, successful in some respects, that aimed 

 at solving the quadrature of the circle with other expedients than 

 the five postulates. We have also to take these endeavors into 

 consideration, and especially so as they, no less than the unsuccess- 

 ful efforts at elementary solution, have upon the whole advanced 

 the science of geometry, and contributed much to the clarification 

 of geometrical ideas. 



* Usually geometers mention only two postulates (Nos. i and 2). But since to 

 geometry proper it is indifferent whether only the eye, or additional special mechan- 

 ical instruments are necessary, the author has regarded it more correct in point of 

 method to assume- five postulates. 



