This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as ‘scholarship standard’. I hope that it may be useful to other classes of readers, but it is this class whose wants I have considered first. It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.
A considerable space is occupied with the discussion and application of the fundamental ideas of the Infinitesimal Calculus, Differential and Integral. But the general range of the book is a good deal wider than is usual in English treatises on the Calculus. There is at present hardly room for a new Calculus of an orthodox pattern. It is indeed not many years since there was urgent need of such a book, but the want has been met by the excellent treatises of Professors Gibson, Lamb, and Osgood, to all of which, I need hardly say, I am greatly indebted. And so I have included in this volume a good deal of matter that would find a place in any Traité d’Analyse, though in English books it is usually separated from the Calculus and classed as ‘Higher Algebra’ or ‘Trigonometry’.
In the first chapter I have discussed in some detail the various classes of numbers included in the arithmetical continuum. I have not attempted to include any account of any purely arithmetical theory of irrational number, since I believe all such theories to be entirely unsuitable for elementary teaching. My aim in this chapter is a more modest one: I take the ‘linear continuum’ for granted and assume the existence of a definite number corresponding to each of its points; and all that I attempt to do is to analyse and distinguish the various classes of numbers whose existence these assumptions involve.
Chapter 1. Real variables.
Chapter 2. Functions of real variables.
Chapter 3. Complex numbers.
Chapter 4. Limits of functions of a positive integral variable.
Chapter 5. Limits of functions of a continuous variable: Continuous and discontinuous functions.
Chapter 6. Derivatives and integrals.
Chapter 7. Additional theorems in the differential and integral calculus.
Chapter 8. The convergence of infinite series and infinite integrals.
Chapter 9. The logarithmic and exponential functions of a real variable.
Chapter 10. The general theory of the logarithmic, exponential, and circular functions.
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