Abstract
THE technical use of the term ‘limit’ in mathematics retains much of the original meaning of the Latin limes, a boundary line, from which it is derived. The notion of limit is also a boundary in another sense, for it marks the frontier between elementary and higher mathematics, where the adjectives must not be taken as necessarily synonymous with easy and difficult. The student who arrives at this frontier has before him an important choice of routes. One route leads to an introduction of limits by an appeal to geometrical intuition by means of which, if the illustrations are judiciously chosen and if the language is sufficiently vague, most of the theorems of the calculus can be deduced with a generality which is almost embarrassing. Advance by this route may seem to be rapid, but the cost is a mistrust of mathematical processes which appear to lead to paradoxical results. The other route proceeds to a somewhat longer but rigorous building of foundations on a secure arithmetical basis. The start may be fairly arduous but the reward is a feeling of security founded on understanding. That there is an increasing tendency to realise the value of the more rigorous treatment of foundations is evidenced by the above three books, for (1) is in its fifth edition, while (2) and (3) are in their second.
(1) A Course of Pure Mathematics.
By Prof. G. H. Hardy. Fifth edition. Pp. xii + 455. (Cambridge: At the University Press, 1928.)12s. 6d. net.
(2) The Elementary Differential Geometry of Plane Curves.
By R. H. Fowler. (Cambridge Tracts in Mathematics and Mathematical Physics, No. 20.) Second edition. Pp. ix + 105. (Cambridge: At the University Press, 1929.) 6s. net.
(3) An Elementary Treatise on Differential Equations and their Applications.
By Prof. H. T. H. Piaggio.(Bell's Mathematical Series: Advanced Section.) Pp. xviii + 256 + xxvii. (London: G. Bell and Sons, Ltd., 1928.) 12s. net.
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MILNE-THOMSON, L. Pure Mathematics. Nature 124, 683 (1929). https://doi.org/10.1038/124683a0
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DOI: https://doi.org/10.1038/124683a0