108 THE SQUARING OF THE CIRCLE. 



212 B. c. wlieii lie fell by the band of a Roman soldier wbom be bad for- 

 bidden to destroy tbe figures be bad drawn in tbe sand. To tbe greatest 

 performances of Arcbimedes tbe successful computation of tbe number r. 

 unquestionably belong. Like Bryson be started witb regular inscribed 

 and circumscribed polygons. He sbovved bow it was possible, begin- 

 ning witb the perimeter of an inscribed bexagon, wbicb is equal to six 

 radii, to obtain by way of calculation tbe perimeter of a regular dodec- 

 agon, and tben tbe i)erimeter of a figure baving double the number of 

 sides of the preceding one. Treating, tben, tbe circumscribed polygons 

 in a similar manner, and proceeding witb both series of polygons up to 

 a regular 96 sided polygon, be perceived on tbe one baud that tbe ratio 

 of the perimeter of the inscribed 96-sided polygon to the diameter was 

 greater than 6336: 20171, and on tbe other band, that tbe correspond- 

 ing ratio with respect to tbe circumscribed 96-sided polygon was 

 smaller than 14688 : 4673.}. He inferred from this, that the number n, 

 the ratio of tbe circumference to tbe diameter, was greater than the 

 fraction aVrVV '^"*^ smaller than i||f|. Reducing the two limits thus 

 found for the value of tt, Archimedes then showed that the first frac- 

 tion was greater than ^\\^ and that the second fraction was smaller 

 than 3i, whence it followed witb certainty that tbe value sought for n 

 lay between 3i and 3i5. The larger of these two approximate values 

 is the only one usually learned and employed. That which fills us 

 most with astonishment in the Arcbimedean computation of ::, is, first, 

 the great acumen and accuracy displayed in all the details of tbe com- 

 putation, and then the unwearied perserverance that be must have 

 exercised in calculating the limits of -n: without the advantages of the 

 Arabian system of numerals and of 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 approxi- 

 mately the roots of given numbers and fractions. 



The later mathematicians of Greece. — With regard to the mathemati- 

 cians of Greece that follow Arcbimedes, all refer to and employ the 

 approximate value of 3^ for tt, without however contributing any- 

 thing essentially new or additional to the problems of quadrature 

 and of cyclometry. Thus Heron of Alexandria, tbe father of sur- 

 veying, who flourished about the year 100 u. c, employs for pur- 

 poses of practical measurement sometimes the value 3i for tt and 

 sometimes even the rougher approximation r=3. The astronomer 

 Ptolemy, who lived in Alexandria about the year 150 A. D., and 

 who was famous as being tbe author of the planetary system univer- 

 sally recognized as correct down to the time of ('opernicus, was the 

 only one who furuisbed a more exact value ; this be designated, in the 

 sexigesimal system of fractional notation wbicb be employed, by 3, 8, 

 30— that is 3 and -.^ and s%%u^ or as we now say 3 degrees 8 minutes 

 {partes minutce prlmce) and 30 seconds {partes minutce secundw). As 



