102 THE SQUARING OF THE CIRCLE. 



compasses, tbeii we would produce lo the amount of such a part a 

 straight line exactly three times larger thau the diameter, and should 

 thus obtain a straight line exactly equal to the circumference of the 

 circle. But as a matter of fact, and as has actually been demonstrated, 

 there do not exist two whole numbers, be they ever so great, that exactly 

 represent by their proportion to one another the number t:. Conse- 

 quently, a rectification of the kind just described does not attain the 

 object desired. 



It might be asked here, whether from the demonstrated fact that tbe 

 number -n is not equal to the ratio of two whole numbers however great, 

 it does not immediately follow that it is impossible to construct a 

 straight line exactly equal in length to the circumference of a circle ; 

 thus demonstrating at once the impossibility of solving the problem. 

 This question is to be answered in the negative. For there are in 

 geometry many sets of two lines of which the one can be easily con- 

 structed from the other, notwithstanding tlie fact that no two whole 

 numbers can be found to represent the ratio of the two lines. The side 

 and the diagonal of a square, for instance, are so constituted. It is 

 true the ratio of the latter two magnitudes is nearly that of 5 to 7. 

 But this proportion is not exact, and there are in fact no two numbers 

 that represent the ratio exactly. Nevertheless, either of these two 

 lines can be easily constructed from the other by the sole employment of 

 ruler and compasses. This might be the case, too, with the rectifica 

 tiou of the circle; and consequently from the impossibility of represent- 

 ing TT by the ratio between two whole numbers the impossibility of the 

 problem of rectification is not inferable. 



The quadrature of the circle stands and falls with the problem of 

 rectification. This is based upon the truth above mentioned, that a cir- 

 cle is equal in area to a right-angle triangle, in which one side is equal 

 to the radius of the circle and the other to the circumference. Sup- 

 posing, accordingly, that the circumference of the circle were rectified, 

 then we could construct this triangle. But every triangle, as is taught 

 in the elements of planimetry, can, with the help of ruler and com- 

 passes, be converted into a square exactly equal to it in area. So that, 

 therefore, supposing the rectification of the circumference of a circle 

 were successfully performed, a square could be constructed that would 

 be exactly equal in area to the circle. 



The dependence upon one another of the three problems of the com- 

 putation of the number ;r, of the quadrature of the circle, and its recti 

 fication, thus obliges us, in dealing with the history of the quadrature, 

 to regard investigations with respect to the value of -. and attemi)ts to 

 rectify the circle as of equal importance, and to consider them accord- 

 ingly. 



Conditions of the geometrical solution. — We have used repeatedly in 

 the course of the discussion the expression " to construct with ruler and 



