Il8 THE SQUARING OF THE CIRCLE. 



equal to the area of a triangle produced by drawing in the circle a 

 radius, erecting at the extremity of the same a tangent, that is, in 

 this case, a perpendicular, cutting off upon the latter the length 

 of the circumference, measuring from the extremity, and joining 

 the point thus obtained with the centre of the circle. It follows 

 from this that the area of a circle is as many times larger than the 

 square upon its radius as the number n amounts to. 



The numerical rectification and numerical quadrature of the 

 circle based upon the computation of the number n are to be clearly 

 distinguished from problems that require a straight line equal in 

 length to the circumference of a circle, or a square equal in area to 

 a circle, to be constructively produced from its radius or its diameter; 

 problems which might properly be called "constructive rectifica- 

 tion" or "constructive quadrature." Approximately, of course, by 

 employing an approximate value for TT, these problems are easily 

 solvable. But to solve a problem of construction in geometry, 

 means to solve it with mathematical exactitude. If the value n 

 were exactly equal to the ratio of two whole numbers to each 

 other, the constructive rectification would present no difficulties. 

 For example, suppose the circumference of a circle were exactly 

 3^ times greater than its diameter ; then the diameter could be di- 

 vided into seven equal parts, which could easily be done by the 

 principles of planimetry with straight edge and compasses ; then 

 by prolonging to the amount of such a part a straight line exactly 

 three times as long as the diameter, we should obtain a straight 

 line exactly equal to the circumference of the circle. But as a mat- 

 ter of fact, and this has actually been demonstrated, there do 

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

 represent by their proportion to each other the number n. Con- 

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

 the object desired. 



It might be asked here, whether from the demonstrated fact 

 that the 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 circum- 

 ference of a circle ; thus demonstrating at once the impossibility of 



