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The Approach to the Calculus
Published online by Cambridge University Press: 03 November 2016
Extract
Now that the Infinitesimal Calculus is becoming part of regular school courses in Mathematics, the question of how the subject should be introduced takes on correspondingly increased importance. And the question is really of fundamental importance; for it is only when the average person, as distinguished from those of special aptitude for a subject, has to be considered, that we are driven back upon the necessity for a sound common-sense basis of theory. Recent articles in the Gazette are evidence of the growing interest in this question—discussion of the perennial issue of e being only a part of this wider inquiry.
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- Copyright © Mathematical Association 1927
References
page note 407 * The writer has published a treatment of this subject in a little book, The Number-System of Arithmetic and Algebra (Melbourne Univ. Press; London, Macmillan & Co.), of which there is a copy in the M.A. Library : referred to below as “N.S.”
page note 407 † See Gazette, vol. x. No. 144, p. 10 (January, 1920).
page note 408 * Essays on the Theory of Numbers (trans. Open Court Co.), especially Essay I. on “Continuity and Irrational Numbers,” Sections III. and IV.
page note 409 * Or, either a given number, the other being a function of x.
page note 409 † This important fundamental proposition is very closely related to the definitions of a a and og a b in the general real case. A proof can be given which is comparatively simple. See “N.S.” ch. v. and app. iv.
page note 410 * The general proofs of the Involution and Logarithmation identities are not elementary. But the forms are fundamental and the general theorems are always assumed in the most ordinary use of Logarithm Tables [see “N.S.” §§ 8, 9, 23, 24].
page note 410 † “N.S.,” appendix ill., on “Nought and Infinity,” p. 70.
page note 411 * The third case of this type is y > Y if | x − c | <: h, for y → ∞ when x→ c.
page note 411 † See Gazette, vol. xl. No. 106, p. 387 (October, 1923), for the writer’s comments on this question.
page note 411 ‡ Tor elementary theory, we have to exclude the possibility that, in § 7, 2, ´y and, in § 7, 3, Swmay vanish in the immediate neighbourhood of ´x = 0. See Carslaw, Gazette, vol.xii. No. 170 (May, 1924), p. 92. This consideration does not affect § 8,1, 3.
page note 413 * See last footnote on p. 411.
page note 413 † See footnote to § 3, 7, 2, 2 and § 3, 7, 2, 3.
page note 413 ‡ § 3, 7, 2, 3.
page note 414 * Cp. § 8, 2, 3.
page note 414 ‡ I remember Mr. J. P. Gabbatt, then professor of Mathematics in Christchurch, N.Z., showing me a purely analytical proof of this inequality—of which the detail was strikingly close to that of the above elementary geometrical proof.