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The Logical Eye and the Mathematical Eye: Their Outlook on Euclid’s Theory of Proportion: Presidential Address to the Mathematical Association, 1928
Published online by Cambridge University Press: 03 November 2016
Extract
My address last year was devoted to the consideration of a few rules or statements made in teaching, with, as it seemed to me, insufficient explanation, in the hope that I might suggest some line of action, in a few particular cases, which would be helpful to members of our Association engaged in elementary teaching.
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- Copyright © Mathematical Association 1928
References
page 36 note * Cajori, History of Mathematics, Second Edition, p. 331.
page 36 note † De Morgan, Connexion of Number and Magnitude, p. 55. (To be referred to later as De M. I.)
page 37 note * De Morgan, I, pp. 55-56.
page 37 note † Penny Cyclopaedia, vol. xix (1841). (To be referred to later as De M. II.)
page 37 note ‡ Vol. ii, pp. 121-124.
page 37 note § See Article 21.
page 37 note ∥ Penny Cyclopaedia (1841), vol. xix. p. 52, col. 2.
page 37 note ¶ Vorlesungen über allgemeine Arithmetik (1885), part i, p. 87.
page 37 note ** See my Contents of the Fifth and Sixth Books of Euclid (Second Edition), p. 29; my Theoty of Proportion, p. 37; and the Cambridge Philosophical Transactions, vol. xxii, No. x. p. 187 (1917).
page 38 note * It will be supposed in what follows that the logical errors in Propositions 10 and 18 pointed out by Simson have been removed.
page 38 note † De Morgan, I. p. 62.
page 39 note * De Morgan, I. pages 25-27.
page 39 note † Nos. 8 (the latter part), 10 as corrected by Simson, 13, 14, 20 and 21.
page 39 note ‡ Nos. 4, 7,11, 12, 15, 17, and 18 as corrected by Simson.
page 40 note * A “number” here means a “whole number.”
page 40 note † De Morgan, I. p. 1.
page 41 note * Dedekind, The Nature and Meaning of Numbers, p. 39 of the English Translation.
page 42 note * A “number” here means a “whole number.”
page 44 note * It can be deduced from the consideration that, if two ratios are unequal, then some rational fraction falls between them, or is equal to one of them but not equal to the other.
page 45 note * It will be seen that a portion of the earlier part of this proposition, which part, however, does not contain any reference to ratio, will be required.
page 46 note * It is not necessary to include Proposition 11 in this list, since ratio has been denned as a number, and this proposition expresses nothing more than that numbers which are equal to the same number are equal to one another.
page 48 note * The argument up to this point is the same as that required for the proofs of Propositions 22, 23. It will be noted that it is at this stage that the earlier part of Eue. V. 8 has been used.
page 48 note † It will be seen that the rest of the argument in this case is obtainable from the corresponding portion of the argument in the previous case by reversing the sign of inequality.
page 50 note * Camb. Phil. Trans., vol. xxii. No. xxiv, p. 460; Camb. Phil. Soc. Proc, vol. xxi, pt. v, p. 474; Camb. Phil. Soc. Proc., vol. xxiii, pt. vii, p. 779.
page 51 note * In each of these Propositions, having given certain data, the object is to prove the equality of two ratios.
page 51 note † In these two Propositions, it is given that two ratios are equal, and certain properties of the magnitudes appearing therein have to he proved.
page 52 note * For proofs, see my Contents of Euclid, Books V. and VI, and my Theory of Proportion.
page 52 note † Euclid V. 11 is not required when ratio has been defined as a number.
page 52 note ‡ We have assumed this as a fundamental idea of ratio. We may also deduce it from the test for equal ratios.
page 53 note * Hilbert’s foundations of feometry, pp. 41-45.
page 53 note † Forder’s foundations of Euclidean Geometry, pp. 41-45.
page 53 note * Hilbert’s foundations of feometry, pp. 20.
page 54 note * See Appendix III.