Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/35

the multiples of 100,

100, 200,…900 by ρ′, σ′, τ′, υ′, φ′, χ′, ψ′, ω′, ϡ′.

The intermediate numbers were expressed by juxtaposition, representing here addition, the largest number being placed on the left, the next largest following, and so on in order. There was no sign for zero. Thousands were represented by the same letters as the first nine integers but with a small dash in front and below the line; thus for example ͵δ′ was 4000, and 1913 was expressed by ͵αιγϡ′ or ͵αιγϡ. 10000 and higher numbers were expressed by using the ordinary numerals with Μ or Μν as an abbreviation for the word μυριάς; the number of myriads, or the multiple of 10000, was generally written over the abbreviation, thus 349450 was λδΜ͵θων′. A variety of devices were employed for the representation of fractions^[1].

The determinations of square roots such as ${\sqrt {3}}$ by Archimedes were much closer than those of earlier Greek writers. There has been much speculation as to the method he must have employed in their determination. There is reason to believe that he was acquainted with the method of approximation that we should denote by

$a\pm {\frac {b}{2a}}>{\sqrt {a^{2}\pm b}}>a\pm {\frac {b}{2a\pm 1}}$.

Various alternative explanations have been suggested; some of these suggest that a method equivalent to the use of approximation by continued fractions was employed.

A full discussion of this matter will be found in Sir T. L. Heath's work on Archimedes.

The treatise of Archimedes on the measurement of the circle must be regarded as the one really great step made by the Greeks towards the solution of the problem; in fact no essentially new mode of attack was made until the invention of the Calculus provided Mathematicians with new weapons. In a later writing which has been lost, but which is mentioned by Hero, Archimedes found a still closer approximation to $\pi$.

The essential points of the method of Archimedes, when generalized and expressed in modern notation, consist of the following theorems:

(1) The inequalities $\sin {\theta }<\theta <\tan {\theta }$.

1. For an interesting account of the Arithmetic of Archimedes, see Heath's Works of Archimedes, Chapter iv.