Section 1.1 The invention of the calculus
ΒΆWhen the student of mathematics pauses to look back upon the achievements of mathematicians of the past he must be impressed with the fact that the seventeenth century was a most impor tant epoch in the development of modern mathematical analysis, since to the mathematicians of that period we owe the invention of the differential and integral calculus. At first the calculus theory, if indeed at that time it could be called such, consisted of isolated and some what crude methods of solving special problems. In the domain of what we now call the integral calculus, for example, an Italian mathematician named Cavalieri (1598-1647) devised early in the seventeenth century a summation process, called the method of indivisibles, by means of which he was able to calculate correctly many areas and volumes. His justification of his device was so incomplete logically, however, that even in those relatively uncritical times his contemporaries were doubtful and dissatisfied. Somewhat later two French mathematicians, Roberval (1602-75) and Pascal (1623-62), and the Englishman Wallis (1616-1703), improved the method and made it more like the summation processes of the integral calculus of today. In the case of the differential calculus we find that before the final quarter of the seventeenth century Descartes (1596-1650), Roberval, and Fermat (1601-65) in France, and Barrow (1630-77) in England, all had methods of constructing tangents to curves which were pointing the way toward the solution of the fundamental problem of the differential calculus as we formulate it today, namely, that of determining the slope of the tangent at a point of a curve.
At this important stage there appeared upon the scene two men of extraordinary mathematical insight, Newton (1642-1727) in England, and Leibniz (1646-1716) in Germany, who from two somewhat different standpoints carried forward the theory and applications of the calculus with great strides. It is a mistake, though we often find it an easy convenience, to regard these two great thinkers as having invented the calculus out of a clear sky. Newton was in fact a close student of the work of Wallis, and a pupil of Barrow whom he succeeded as professor of mathematics at Cambridge, while we know that Leibniz visited Paris and London early in his career and that he there became acquainted with the most advanced mathematics of his day. No one could successfully contest the fact, however, that these two men were the most able spokesmen and investigators of the seventeenth-century school of mathematicians to which we owe the gradual evolution of the calculus.
In spite of the great abilities of Newton and Leibniz the underlying principles of the calculus as exposed by them seem to us from our modern viewpoint, as indeed to their contemporaries and immediate successors, some what vague and confusing. The difficulty lies in the lack of clearness at that early time, and for more than a century thereafter, in the conceptions of infinitesimals and limits upon which the calculus rests, a difficulty which has been overcome only by the systematic study of the theory of limits inaugurated by Cauchy (1789-1857) and continued by Weierstrass (1815-97), Riemann (1826-66), and many others.