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IN this book we have sought to give an account of a department of mathematics which is now generally regarded as fundamental. A list of the men to whom the successive advances of the subject are due, includes, with few exceptions, the names of the greatest French and German mathematicians of the century, from Cauchy and Gauss onward. And in line with these advances lie the chief fields of mathematical activity at the present day. The most legitimate extensions of elementary analysis lead so directly into the Theory of Functions, that recent writers on Algebra, Trigonometry, the Calculus, etc., give theories which are indispensable parts of our subject.