The invention of the calculus. When 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 important 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 somewhat 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.