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This is the first volume of a series of four volumes covering all stages of development of the Calculus, from the last year at school to degree standard. The books are written for students of science and engineering as well as for specialist mathematicians, and are designed to bridge the gap between the works used in schools and more advanced studies, with their emphasis on rigour. This volume is concerned with the basic ideas and applications of differentiation and integration in relation to algebraic and trigonometric functions, but excluding logarithmic and exponential functions. Integration begins on the 'Riemann integral' basis, and the treatment of curves combines accuracy with simplicity, without shirking the awkward problems of sign. Each section has examples; at the end of each chapter there are problems from school-leaving and open scholarship examinations.