Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/27

CHAPTER II

The First Period

Earliest traces of the problem

The earliest traces of a determination of $\pi$ are to be found in the Papyrus Rhind which is preserved in the British Museum and was translated and explained^[1] by Eisenlohr. It was copied by a clerk, named Ahmes, of the king Raaus, probably about 1700 B.C., and contains an account of older Egyptian writings on Mathematics. It is there stated that the area of a circle is equal to that of a square whose side is the diameter diminished by one ninth; thus $\textstyle A=({\frac {8}{9}})^{2}d^{2}$, or comparing with the formula $\textstyle A={\frac {1}{4}}\pi d^{2}$ this would give

$\textstyle \pi ={\frac {256}{81}}=3{\cdot }1604\ldots .$

No account is given of the means by which this, the earliest determination of $\pi$, was obtained; but it was probably found empirically.

The approximation $\pi =3$, less accurate than the Egyptian one, was known to the Babylonians, and was probably connected with their discovery that a regular hexagon inscribed in a circle has its side equal to the radius, and with the division of the circumference into $6\times 60\equiv 360$ equal parts.

This assumption ($\pi =3$) was current for many centuries; it is implied in the Old Testament, 1 Kings vii. 23, and in 2 Chronicles iv. 2, where the following statement occurs:

"Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about."

The same assumption is to be found in the Talmud, where the statement is made "that which in circumference is three hands broad is one hand broad."

1. Eisenlohr, Ein mathematisches Handbuch der alten Ägypter (Leipzig, 1877).