This book was first published in 1908 by Cambridge University Press and was intended “for the first-year calculus students, whose abilities reach or approach the scholarship standard”. The book came as a third text in a series of Cambridge books that defined a revolution in the teaching of calculus (or rather mathematical analysis), after Whittaker’s “A Course of Modern Analysis” in 1902 and Hobson’s “The theory of functions of a real variable” in 1907 (all three books are still in print today). Ever since Leibniz and Newton started playing with fluxes and other derivative-like things, mathematicians have been trying to put calculus, including the basic concept of the definition of real numbers, on solid grounds. This immensely difficult task was essentially finished, thanks to great minds such as Bolzano, Weierstrass, Cauchy, Abel, Dirichlet, Riemann, Cantor and Peano, by the end of the 19th century. Naturally, by the outbreak of the 20th century, it was time to establish, for the first time ever, a rigorous university course in mathematical analysis. Such a course was defined by Hardy’s book and had an immense affect on British, and later continental, teaching and understanding of the principles of analysis. At the beginning, the audience that could follow the book must have been, naturally, quite small. However, the book became a primer source of analysis and great inspiration to entire generations of mathematicians and it is very much so even today.
Hardy, a giant after whom fundamental mathematical notions are called - such as the very important inequality for sums and integrals, the celebrated maximal operator (a true blessing in real analysis) and function spaces (indispensable in harmonic analysis) - writes in a vigorous and enthusiastic and yet still precise style, with a lot of comments on how the stuff, brand new at the time, should be viewed by the reader. Often, there is a variety of points of view from which certain topics could be regarded (see, for instance, Section 16 on the definition of real numbers). The reader feels safe and well-led. Moreover, Hardy’s missionary style is an amusing read. If only contemporary authors had time and courage to write analysis textbooks in a similar style. If I may quote the wonderful foreword to the new edition by T. W. Körner (from which I have been borrowing heavily anyway), the fact that the Cambridge University Press published in 2008 a centenary edition of this book, is not an act of piety. The reason is that, in a hundred years, the book has lost none of its power. It is still a great reading and a unique inspiration. May the generations of young mathematicians for which Hardy’s book will be the gate to analysis continue forever.