The Calculus Gallery: Masterpieces from Newton to Lebesgue Book

Newton was more interested in integration than differentiation and did most of his work using infinite series. (Infinite series had been discovered independently in India at least 150 years before, but I have yet to read that the Indians knew anything about calculus. China and the Islamic world were no longer major players in mathematics after the Mongol invasions.) Leibniz created most of the modern notation, including dy/dx for differentiation, the long s for integration and the Leibniz rule d(u v) = u dv + v du. The Fundamental Theorem of Calculus was in the air, but it didn't really say anything yet because there was no proper definition of the integral.


Calculus was originally treated as a branch of geometry, as dealing with the properties of curves. It was Cauchy who pointed out that differentiation, integration and infinite series are all examples of limits. He is therefore the father of real analysis as we now know it, although many of the questions he raised were not resolved until later.


The first truly analytical definition of integration was due to Riemann, and he also determined what kind of functions could be integrated. Others investigated what kind of functions could be differentiated, and showed that even when the derivative exists, it can't necessarily be integrated in the Riemann fashion.


Finally, Lebesgue defined a more general definition of integration and filled the last gaps in the Fundamental Theorem of Calculus by showing that where the derivative of a function exists, it can always be Lebesgue-integrated and always restores the original function.


So this book contains all the important ideas, but for that very reason it risks being a first year real analysis textbook.