In this book, the author intends to explain everything with the help of pictures, metaphors, and anecdotes. We also get exposed to some of the finest equations and proofs in human history. One of the first things introduced is the “infinity principle,” where things are broken down into infinite simpler parts, analyzed, and then added back together to produce the whole. Calculus can be thought of as a methodological theme consisting of a mystery of curves, the mystery of motion, and the mystery of change.<br><br>We start with the work of Archimedes from about the 3rd century BCE. We see here the beginnings of integral calculus, where triangles and parabolic regions are apparently and mysteriously equivalent. Eighteen hundred years passed until a new Archimedes appeared, whom we know as Galileo Galilei. It was interesting to learn about the law of odd numbers rule, which led Galileo to conclude that the total distance fallen is proportional to the square of the time elapsed. What Galileo did for the motion of objects, Johannes Kepler did for the motion of the planets. Both channeled the spirit of Archimedes, “carving solid objects in their minds into imaginary thin wafers, like so many slices of salami.”<br><br>We see the arrival of algebra in Europe around 1200 from Asia and the Middle East. Hindu mathematicians invented the concept of zero and the decimal place system. Algebraic techniques for solving equations came from Egypt, Iraq, Persia, and China. But the study of algebra as a symbolic system began to flourish in Renaissance Europe around the 1500s. Analytic geometry makes its appearance with Pierre de Fermat, and Rene Descartes. Fermat actually invented the ideas that led to the concept of derivatives.<br><br>From here we delve into functions – power and exponential, for example. There are some interesting basics of the relationship of logarithms to exponents. And then there is the natural logarithm, which grows as a rate precisely equal to the function itself. The author notes that “exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful.” This leads into a more detailed discussion of the derivative. By the time we get to Newton, we see the concept of the fundamental theorem. Newton’s brainstorm was to invite time and motion into the picture and let the area flow. And now we are into integral calculus. The author notes that the reason integration is so much harder than differentiation relates to the distinction between local and global, which he clearly demonstrates in the book. I think the author has done a great job of showing us just how these concepts arose and how to make sense of them. You won’t get this is in your typical calculus book.<br><br>After this, we delve into differential equations – ordinary and partial. The author gives a clear explanation of what these beasts are and some real-world examples to help us understand. In talking about the future of calculus, the author discusses some applications, such as nonlinearity (biology, sociology) and chaos, where you have an inherent sensitivity to initial conditions. He concludes by taking us to the “Twilight Zone” for three examples of the eerie effectiveness of calculus.