With regard to the mathematicians of Greece that follow Archimedes, all refer to and employ the approximate value of 3-1/7 for π, without however, contributing anything essentially new or additional to the problems of quadrature and of cyclometry. Thus Heron of Alexandria, the father of surveying, who flourished about the year 100 B. C., employs for purposes of practical measurement sometimes the value 3-1/7 for π and sometimes even the rougher approximation π = 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 universally recognised as correct down to the time of Copernicus, was the only one who furnished a more exact value; this he designated, in the sexigesimal system of fractional notation which he employed, by 3, 8, 30,—that is 3 and 8/60 and 30/3600, or as we now say 3 degrees, 8 minutes (partes minutae primae), and 30 seconds (partes minutae secundae). As a matter of fact, the expression 3 + 8/60 + 30/3600 = 3-17/120 represents the number π more exactly than 3-1/7; but on the other hand, is, by reason of the magnitude of the numbers 17 and 120 as compared with the numbers 1 and 7, more cumbersome. IV. #Among the Romans.# In the mathematical sciences, more than in any other, the Romans stood upon the shoulders of the Greeks. Indeed, with respect to cyclometry, they not only did not add anything to the Grecian discoveries, but often evinced even that they either did not know of the beautiful result obtained by Archimedes, or at least did not know how to appreciate it. For instance, Vitruvius, who lived during the time of Augustus, computed that a wheel 4 feet in diameter must measure 12-1/2 feet in circumference; in other words, he made π equal to 3-1/8. And, similarly, a treatise on surveying, preserved to us in the Gudian manuscript of the library at Wolfenbüttel, contains the following instructions to square the circle: Divide the circumference of a circle into four parts and make one part the side of a square; this square will be equal in area to the circle. Aside from the fact that the rectification of the arc of a circle is requisite to the construction of a square of this kind, the Roman quadrature, viewed as a calculation, is more inexact even than any other computation; for its result is that π = 4. #Among the Hindus.#