Ideas about mathematical analysis took a long hiatus after the Greeks. It didn’t begin to grow again until the 16th century, when the need to examine mechanics problems became important. For example, German astronomer and mathematician Johannes Kepler (1571–1630) needed to calculate the area of sectors in an ellipse in order to understand planetary motion. (Interestingly, Kepler thought of areas as the sums of lines—a kind of crude form of integration; even though he made two errors in his work, they canceled each other out and he was still able to determine the correct numbers.)

By the 17th century, many mathematicians had begun to contribute to the field of mathematical analysis. For example, French mathematician Pierre de Fermat (1601–1665) made contributions that eventually led to differential calculus. Bonaventura Cavalieri presented his method of indivisibles, one he developed after examining Kepler’s integration work. English mathematician Isaac Barrow (1630–1677) worked on tangents that formed the foundation for Newton’s work on calculus. Italian mathematician Evangelista Torricelli (1608–1647) added to differential calculus and many other facets of mathematical analysis. (In fact, collections of paradoxes that arose through the inappropriate use of the new calculus were found in his manuscripts. Unfortunately for mathematics, Torricelli died young of typhoid.) And, of course, the one name most associated with calculus— Isaac Newton—developed some of his most brilliant work during the 17th century.