SQUARING THE CIRCLE 13 



vided the number is one of which it is possible to extract 

 the square root. Thus, if we have a circle which contains 

 100 square feet, a square with sides of 10 feet would be 

 exactly equal to it. But the ascertaining of the area of the 

 circle is the very point where the difficulty comes in ; the 

 dimensions of circles are usually stated in the lengths of 

 the diameters, and when this is the case, the problem re- 

 solves itself into another, which is : To find the area of a 

 circle when the diameter is given. 



Now Archimedes proved that the area of any circle is 

 equal to that of a triangle whose base has the same 

 length as the circumference and whose altitude or height 

 is equal to the radius. Therefore if we can find the length 

 of the circumference when the diameter is given, we are in 

 possession of all the points needed to enable us to " square 

 the circle." 



In this form the problem is known to mathematician's 

 that of the rectification of the curve. 



In a practical form this problem must have presented 

 itself to intelligent workmen at a very early stage in the 

 progress of operative mechanics. Architects, builders, 

 blacksmiths, and the makers of chariot wheels and vessels 

 of various kinds must have had occasion to compare the 

 diameters and circumferences of round articles. Thus 

 in I Kings, vii, 23, it is said of Hiram of Tyre that "he 

 made a molten sea, ten cubits from the one brim to the 

 other; it was round all about * * * and a line of 

 thirty cubits did compass it round about," from which it 

 has been inferred that among the Jews, at that time, the 

 accepted ratio was 3 to i, and perhaps, with the crude 

 measuring instruments of that age, this was as near as could 

 be expected. And this ratio seems to have been accepted 



