104 THE SQUARING OF THE CIRCLE. 



problems of planimetry are not reducible to these five })roblem8 alond. 

 There are problems that cau be solved only by assuming other prob- 

 lems as solvable which are not included in the five given : for example, 

 the construction of an ellipse, having given its center and its major 

 and minor axes. Many problems, however, possess the property of be- 

 ing solvable with the assistance solely of the five postulates above for- 

 mulated, and where this is the case they are said to be " constructible 

 with ruler and compasses," or "elementarily" constructible. 



After these general remarks upon the solvability of problems of geo- 

 metrical construction, which an understanding of the history of the 

 squaring of the circle makes indispensably necessary, the significance 

 of the question whether the quadrature of the circle is or is not solva- 

 ble, that is, elementarily solvable, will become intelligible. But the 

 conception just discussed 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 solv- 

 ing 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 unsuccessful efforts at ele- 

 mentary solution, have upon the whole advanced the science of geome- 

 try and contributed much to the clarification of geometrical ideas. 



III.-HISTORICAL ATTEMPTS. 



The Egyptian Quadrature. — In the oldest mathematical work that we 

 possess we find a rul>^ that tells us how to make a square which is equal 

 in area to a given circle. This celebrated book, the Papyrus Rhind of 

 the British Museum, translated and explained by Eisenlohr (Leipsic, 

 1887), was Avritten, as it is stated in the work, in the thirty-third year of 

 the reign of King Ra-a-us, by a scribe of that monarch, named Ahmes. 

 The composition of the work falls accordingly into the period of the two 

 Hiksos dynasties, that is, in the period between 2000 and 1700 b. c. But 

 there is another important circumstance attached to this, Ahmes men- 

 tions in his introduction that he composed his work after the model 

 of old treatises, written in the time of King Raenmat ; whence it appears 

 that the originals of the mathematical expositions of Ahmes, are half 

 a thousand years older yet than the Papyrus Rhind. 



The rule given in this papyrus for obtaining a square equal to a cir- 

 cle, specifies that the diameter of the circle shall be shortened one- 

 ninth of its length and upon the shortened line thus obtained a square 

 erected. Of course, the area of a square of this construction is only ap- 

 proximately equal to the area of the circle. An idea may be obtained 

 of the degree of exactness of this original, primitive quadrature by our 

 remarking that if the diameter of the circle in question is one metre 

 in length, the square that is supposed to be equal to the circle is a lit- 

 tle less than half a square decimetre larger; an approximation not so 

 accurate as that computed by Archimedes, yet much more correct than 



