THE SQUARING OF THE CIRCLE. 101 



II.— NATURE OF THE PROBLEM. 



Numerical reef ijicat ion. — If we have a circle before us, it is easy for us 

 to determine tbe leugtb of its radius or of its diameter, wbicb must be 

 double that of the radius; and the question next arises to find the num- 

 ber that represents how many times larger its circumference, that is the 

 length of the circular line, is than its radius or its diameter. From the 

 fact that all circles have the same shape it follows that this proportion 

 will always be the same for both large and small circles. Now, since 

 the time of Archimedes, all civilized nations that have cultivated math- 

 ematics have called the number that denotes how many times larger 

 than the diameter the circumference of a circle is, tt — the Greek initial 

 letter of the word periphery. To compute ::, therefore, means to calcu- 

 late how many times larger the circumference of a circle is than its 

 diameter. This calculation is called "the numerical rectificatlou of the 

 circle." 



The numerical quadrature. — Next to the calculation of the circumfer- 

 ence, the calculation of the superficial contents of a circle by means of 

 its radius or diameter is perhaps most important; that is, the computa- 

 tion of how much area that part of a plane which lies within a circle 

 measures. This calculation is called the " numerical quadrature." It 

 depends, however, upon the problem of numerical rectification; that is, 

 upon the calculation of the magnitude of n. For it is demonstrated in 

 elementary geometry that the area of a circle is equal to the area of a 

 triangle produced by drawing in the circle a radius, erecting at the ex- 

 tremity 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 center 

 of the circle. But it follows from this that the area of a circle is as 

 many times larger than the square upon its radius as the number t: 

 amounts to. 



Constructive rectification and quadrature. — The numerical rectification 

 and numerical quadrature of the circle based upon the computation of 

 the number r are to be clearly distinguished from problems that require 

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

 equal in area to a circle, to be constructively produced out of its radius 

 or its diameter; problems which might properly be called "constructive 

 rectification" 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 tt were exactly 

 equal to the ratio of two whole numbers to one another, the constructive 

 rectification would present no difficulties. For example, suppose the 

 circumference of a circle were exactly 3i times greater than its diam- 

 eter; then the diameter could be divided into seven equal parts, which 

 could be easily done by the principles of planimetry with ruler and 



