THE SQUARING OF THE CIRCLE. 



I2 7 



to obtain by calculation the perimeter of a regular dodecagon, and 

 then the perimeter of a figure having double the number of sides of 

 that, and so on. Treating, then, the circumscribed polygons in a 

 similar manner, and proceeding with both series of polygons up to 

 a regular 96-sided polygon, he discovered on the one hand that the 

 ratio of the perimeter of the inscribed g6-sided polygon to the 

 diameter was greater than 6336 : 2017^, and on the other hand, that 

 the corresponding ratio with respect to the circumscribed g6-sided 

 polygon was smaller than 14688:4673^. He inferred from this, 

 that the number n, the ratio of the circumference to the diameter, 

 was greater than the fraction ^-^ and smaller than . Reducing 

 the two limits thus found for the value of ft, Archimedes then 

 showed that the first fraction was greater than 3^, and that the 

 second fraction was smaller than 3^, whence it followed with cer- 

 tainty that ,the value sought for n lay between 3} and -$\\. The 

 larger of these two approximate values is the only one usually 

 learned and employed. That which fills us with most astonish- 

 ment in the case of Archimedes's computation of n, is, first, the 

 great acumen and accuracy displayed by him in all the details of 

 the computation, and secondly the unwearied perseverance which 

 he exercised in calculating the limits of n without the help of the 

 Arabian system of numerals and the decimal notation. For it must 

 be considered that at many stages of the computation what we call 

 the extraction of roots was necessary, and that Archimedes could 

 only by extremely tedious calculations obtain ratios that expressed 

 approximately the roots of given numbers and fractions.* 



With regard to the mathematicians of Greece that follow Archi- 

 medes, all refer to and employ the approximate value of 3^ for it, 

 without, however, contributing anything essentially new to the 

 problems of quadrature and of cyclometry. Thus Hero of Alex- 

 andria, the father of surveying, who flourished about the year 

 100 B. C., employs for purposes of practical measurement some- 



* For Archimedes's actual researches, see Rudio, Archimedes, Huygens, Lam- 

 bert, Legendre, vier Abhand. iiber die Kreismessung (Leipsic, 1892), where 

 translations of the works of these four authors on cyclometry will be found.- -TV. 



