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Presidential Address on the Theory of Proportion

Ideas Leading up to Euclid’s Fifth and Seventh Definitions

Published online by Cambridge University Press:  03 November 2016

M. J. M. Hill
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Extract

In the second edition of my Contents of the Fifth and Sixth Books of Euclid I have taken the ideas stated in the preceding article as my starting point, and built up the conditions involved in the 5th and 7th definitions in the following way:

  1. 1. The ratio of a magnitude A to another magnitude B of the same kind is a real number, rational or irrational, determined in the manner explained in what follows. It is denoted by the symbol A : B.

  2. 2. If the magnitudes A and B have a common measure G, so that A = aG, B = bG, where a, b are positive whole numbers, then A : B is defined to be the rational number .

Type
Research Article
Information
The Mathematical Gazette , Volume 6 , Issue 100 , October 1912 , pp. 360 - 368
Copyright
Copyright © Mathematical Association 1912 

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References

page 362 note * I am indebted to Mr. Kose-Innes for the introduction of this article at this stage of the argument.

page 363 note * It may be noticed that Euclid does not in his Fifth Book assume, as has been done here, the divisibility of any magnitude into any number of equal parts, an assumption which is required for the proofs of the propositions in Art. 16 (6) (see my Euclid V. and VI., 2nd Edition, Art. 44). He does, however, make this assumption in the 6th Proposition of the Tenth Book (Heath, l.c. Vol. III. p. 26).

page 364 note * Dedekind, Essays on Number, translated by Beman, p. 40.

page 365 note * See also my Euclid V. and VI. 2nd edition, p. 113, where a proof substantially the same as Simson’s is given.