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The Methods Used by Archimedes for Approximating to Square-Roots
Published online by Cambridge University Press: 03 November 2016
Extract
It is hardly necessary for me to express here my indebtedness to the patience and learning displayed by Sir Thomas Heath in his edition of the Works of Archimedes (Cambridge, 1897); but for his statements and suggestions I should never have even known that the greatest master of Greek geometry had left so fascinating a problem for the speculations of later mathematicians. In what follows I can only claim to have tried to frame my conjectures on the hypothesis that so consummate a geometer was at least on the same intellectual plane as such men as Newton or Gauss, so far as a humble admirer can compare those who have towered as giants in the subject.
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- Copyright © Mathematical Association 1928
References
page note 253 * Of course the decimal notation becomes indispensable (or certain modern methods for long computations, which depend on logarithms and calculating machines.
page note 254 * The results are used in his estimates for 
; see Works (pp. 94 and 96), and the summary in § 4 below.
page note 255 * See pp. lxxx-xcix of Sir Thomas Heath’s Introduction to the Works.
page note 255 † It is usually enough to take c0 − a as the integer next below b2/2a; thus in the two examples below:
‡
Works, p. 94: these numbers arise from the fact that 
.
§ The limit 
is admissible: but it does not improve the final value, while the subsequent stages would be made slightly heavier by the change from 
(See § 4 below.)
page note 256 * Works, p. 97.
page note 256 † The choice of 
instead of the slightly nearer value 
is discussed in § 4 below.
page note 257 * After what has been said, the reader will have no difficulty in finding simple reasons for the choice of the two fractions and; fuller details are given in pp. lxxxvi-lxxxix of Sir Thomas Heath’s Introduction to the Works.