The Theory of Integration

THE integral calculus was founded two thousand years ago by the Greeks, who applied it with much success to the determination of areas and volumes. Its first appeal to the mathematical world was that of a new and powerful instrument of calculation. When interest was re-aroused in science after the Middle Ages, the infinitesmal calculus developed rapidly under the stimulus of new symbolism, progress being mainly in manipulation until the end of the eighteenth century. Up to then there was little advance on the rigour of the ancients, which, though possibly misplaced, was very severe. There followed a period of criticism, initiated by Cauchy's theory of limits, when the logical ideas at the foundation of the subject were examined. It was found that many of the results obtained by the methods of the calculus would not stand the scrutiny of the new analysis. Such ideas as the treatment of infinite series by the processes of finite algebra were found to need revision.The Theory of Integration.By L. C. Young. (Cambridge Tracts in Mathematics and Mathematical Physics, No. 21.) Pp. viii + 53. (Cambridge: At the University Press, 1927.) 5s. net.