10^ TIIK SQUARING OF THE CIRCLE. 



212 IJ. c. when lie fell by tlie band of a lioman soldier wboin be bad for- 

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

 performances of Archimedes tbe successful computation of tbe number - 

 uncjuestionably belong. Like Bryson be started with regular inscribed 

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

 ning with the perimeter of an inscribed hexagon, which is equal to six 

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

 agon, and then the perimeter of a tigure having double tbe number of 

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

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

 a regular 9G sided polygon, be perceived on the one band that the ratio 

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

 greater than G33G:2017j, and on the other band, that the correspond- 

 ing ratio with respect to the circumscribed OG-sided polygon was 

 smaller than 14G88 : 4673i. He inferred from this, that the number tt, 

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

 fraction -.[\f{{'^ and smaller than |^m|. Reducing tbe two limits thus 

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

 tion was greater than 3\f, and that the second fraction was smaller 

 than 31, whence it followed with certainty that the value sought for ;: 

 lay l)etween 3i and 315. Tbe larger of these two approximate values 

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

 most with astonishment in the Archimedean computation of -, is, first, 

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

 l)utatiou, and then the unwearied perserverance that he must have 

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

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

 be considered that at many stages of tbe 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 tbe mathemati- 

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

 approximate value of 31 for tt, without however contributing any- 

 thing essentially new or additional to the problems of quadrature 

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

 veying, who nourished about tbe year 100 u. c, employs for pur- 

 l)Oses of practical measurement sometimes the value 3\ 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 the author of the planetary system univer- 

 sally recognized as correct down to tbe time of C-'opernicus, was the 

 only one who furnished a more exact value ; this he designated, in the 

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

 30 — that is 3 and -,^„ and Trlsn, f>i" as we now say 3 degrees 8 minutes 

 [partes minutx prima) and 30 seconds {partes minuta; secunda). As 



