“Because we tend only to see the final product of an intellectual endeavor, it is easy to forget that even titans such as Archimedes cannot know the final answer before they get their hands dirty with the problem at hand.”

very general summary of *Infinite Powers*, by Steven Strogatz, might describe the book as an exposition of the history and application of calculus, one of the most fundamental branches of mathematics ever discovered by mankind. But such a clinical synopsis would miss the vibrant, personal story that Strogatz tells, reminding us that scientific, philosophical, and mathematical progress is necessarily a human endeavor. Although we stand on the shoulders of giants, those giants were as prone to ego, insecurity, fervor, and mysticism as any of us mere mortals. The stories in *Infinite Powers* demonstrate that the intellectual rigor by which discoveries are made—and the drama that infuses any tale of people—cannot be separated. The growth of knowledge and humanity are, and have been, forever intertwined. Strogatz reminds us that even the seemingly infallible heroes of our intellectual lineage are flesh and blood, one with the rest of us.

**Archimedes (288-212 BC)**

It is quite clear in Strogatz’s telling of Archimedes’ mathematical accomplishments that he admires the ancient thinker—and with good reason. The ancient Greek discovered both the principle of buoyancy and the law of the lever (the reader should consult the book for details), and he was the first to exploit what Strogatz dubs *the Infinity Principle*. This is the strategy whereby, in order to “shed light on any continuous shape, object, motion process, or phenomenon…reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.” Strogatz details Archimedes’ masterful exploitation of the principle to solve problems. Successfully playing with infinity literally thousands of years before the invention of full-fledged calculus reveals the mathematician’s creative willingness to forego any previously established rules and to follow his intuitions wherever they took him. In this ancient thinker, freedom of mind reigned supreme.

###### Intuitions may be the stimulus of an eventual solution—or they might lead to a dead-end. The real work is everything in between.

But Strogatz does not only focus on his mathematical accomplishments. The author describes a letter that Archimedes had written to his friend, Eratosthenes. The crux of the letter, as Strogatz describes it, is that Archimedes admits to following his intuitions when trying to solve problems. Because we tend only to see the final product of an intellectual endeavor, it is easy to forget that even titans such as Archimedes cannot know the final answer before they get their hands dirty with the problem at hand. Intuitions may be the stimulus of an eventual solution—or they might lead to a dead-end. The real work is everything in between.

Archimedes gave civilization a mature science of static systems—those that remain unchanged over time. Significant progress in the study of dynamical, changing systems would have to wait well over a millennium.

**Galileo Galilei (1564-1642 AD) and Johannes Kepler (1571-1630 AD)**

Strogatz then launches the reader some eighteen hundred years into the future, landing squarely in Renaissance Italy. Prior to this period, it was rare for thinkers to employ the tools of mathematics to study Nature. Furthermore, humanity’s understanding of the physical world was limited to regularities that were fixed in space and time—those problems that Archimedes had dealt with.

Enter two heroes of civilization, Galileo Galilei and Johannes Kepler. While the Italian Galileo studied movements of terrestrial objects such as pendulums and balls rolling down ramps, the German Kepler observed the wanderings of planets. Each sought to describe systems that were moving across time and space in terms of equations.

The imaginative leap required to mathematize our understanding of Nature was no small feat, obvious though it may be to our 21st century minds. The two scientists especially respected geometry, the mathematics studied by their Greek forefathers. Galileo famously regarded the universe as, “this grand book…written in the language of mathematics…” Kepler, the more religious of the pair, felt that geometry “supplied God with patterns for the creation of the world.”

After reviewing Galileo’s numerous scientific accomplishments, Strogatz goes on to emphasize the revolutionary nature of Galileo’s experimentation. In setting up his apparatus for rolling balls down incline planes, the Italian carefully ensured that the friction between the ball and the plane was minimized. In other words, he abstracted away any noise that might interfere with his interrogation of Nature. Galileo took the nascent scientific method and demonstrated how it works—first ask a good question, and then conduct a precise experiment that tests potential answers (hypotheses) to the question, while simultaneously abstracting away unnecessary complications.

Meanwhile, Kepler was studying the motions of planets and struggling mightily to explain the data of their movements. Good fortune struck at the dawn of the seventeenth century, when he inherited more accurate astronomical data from his assistant, Tycho Brahe. Even then, the pious scientist continued to struggle. Finally, by 1609, Kepler discovered the first two of his laws of planetary motion. The third and final law took another decade and “came to him after months of furiously renewed calculations and more than twenty years after his agonizing near miss with the Platonic solids.” His religious fervor, which fostered his faith in an orderly universe, had paid off at last.

While each scientist took humanity’s understanding of the universe farther than it had ever been—Galileo in the earthly realm, Kepler in the night sky—their contrasting dispositions—Galileo mechanical, Kepler mystical—reveal a peculiar feature of our universe, namely, that it does not matter how one chooses to initially approach a problem. What matters, ultimately, is whether one solves it.

**René Descartes (1596-1650 AD) and Pierre de Fermat (1607-1665 AD)**

Strogatz later takes the reader to seventeenth century France, where two intellectual powerhouses had begun to feud.

Although he was born into wealth, René Descartes was mostly bedridden as a child, leading to a taste for reading and thinking that never abated into adulthood. He grew into a man of ambition with a healthy disrespect of authority on the one hand, and ego and a paranoid sensitivity on the other.

Pierre de Fermat, meanwhile, lived the life of a family man, going to work and coming home to his wife and children each day. He’d spend a few hours at night working on math problems for his own pleasure. Compared with Descartes’ monumental aspirations, those of Fermat were almost adorably tepid. He was satisfied with simply doing math during the hours of the moon, having no desire to publish his work.

The opposing personalities and ambitions of the two men were such that, when Descartes heard that some amateur intellectual had apparently invented analytic geometry years before he did, trouble was bound to follow. Given Descartes’ ruthless ego and Fermat’s naiveté, one can predict this story ends. Descartes went on to sully Fermat’s name with utter disregard. Slander, as it happens, was as effective in the seventeenth century as it is now.

To the victor go the spoils, and to this day the coordinates of analytic geometry are described as “Cartesian,” even though Fermat had discovered them first. Let it not be said that intellectual progress is devoid of human-shaped scars.

What is this analytic geometry that has earned both Descartes and Fermat a portrait in the intellectual hall of fame? The subject is both an enormous step forward in our intellectual history as well as a unification of two previously disparate fields: algebra and geometry. While we take the fruitfulness of plotting one variable against another for granted now, the idea was not always so obvious. As Strogatz notes, the very axes of the Cartesian plane are *number lines*—two concepts that the ancient Greeks had kept firmly distinct. In another revolutionary insight, both Descartes and Fermat noticed that one could take an *algebraic *equation and convert it to a *curve *on this new Cartesian plane. Again, this is an idea that any modern student of mathematics finds quite simple, but it required extraordinary radicalism initially to conceive of it.

**Isaac Newton (1643-1727 AD) and Gottfried Leibniz (1646-1716 AD)**

After much incremental progress, calculus at last grew wings of its own with the work of Newton and Leibniz in the latter part of the seventeenth century. Once again, with brilliant insights came heated rivalry.

Isaac Newton is another example of a future ground-breaking scientist with unfortunate beginnings. Strogatz tells the reader that he was left to be raised by his grandmother when his mother remarried. With few childhood friends and feeling abandoned by his mother, Newton, in this solitude, “pursued scholarly investigations on his own, building sundials in the farmhouse, measuring the play of light and shadows on the wall.” As a teenager working at his family farm, he “would often lie in the fields and read by himself.”

Despite—or, perhaps, because of—these troubled beginnings, Newton went on to make some of the greatest contributions to science and mathematics of anyone in history. As Strogatz writes, “…inventing modern calculus wasn’t enough to keep his mind occupied. He also discovered the inverse-square law of gravity…and showed experimentally that white light is composed of all the colors of the rainbow.” In creating some of the mathematical tools of calculus, Newton “unified, synthesized, and generalized ideas from his great predecessors,” including the Infinity Principle of Archimedes and the Cartesian plane of Descartes (and Fermat). Every intellectual monument rests atop the beautiful temples that are its precursors.

But despite these achievements, Newton remained a relatively unknown figure, much by his own design. He was antisocial, and, like Descartes, paranoid and sensitive to criticism.

Leibniz heard about some of Newton’s mathematical results via word of mouth and requested from Newton a proof of his work. Although Leibniz’s request was one of admiration, Newton responded obnoxiously by providing intentionally long and impenetrable proofs.

Without some of Newton’s most cherished secrets, Leibniz went on to formulate calculus on his own. Unlike poor Fermat, Leibniz has received plenty of credit for the invention of calculus, alongside the uncooperative Newton. In fact, if not for Newton’s self-imposed isolation, Leibniz may never have become the co-father of the subject in the first place. He ended up publishing a comprehensible work on calculus, and he attracted followers that would go on to disseminate knowledge of the subject to more and more people. Neither writing for the lay audience nor gathering followers was in Newton’s repertoire. This time, the pricklier member of the pair did not completely swallow the gentler one in the annals of history.

**Conclusion**

In *Infinite Powers*, Steven Strogatz writes a very human story of a particular strand of our knowledge, namely, calculus. Along the way, readers will learn about the ancient Greeks’ fascination with geometry, how astronomy led the Scientific Revolution, and how spats between geniuses are just as gossip-worthy as any contemporary celebrity feud. What a beautiful history our civilization’s pursuit of knowledge has. Like any great play, it is full of flawed characters, mistakes, tragedies, serendipitous encounters, and triumphs. Strogatz recounts some episodes in this unfinished story that anyone interested in our species’ intellectual genealogy will take delight in reading*.*

*Logan Chipkin is a freelance writer and PhD candidate studying evolutionary theory. His writings can be found at Areo, Arc Digital, and Medium, and he can be followed on Twitter @ChipkinLogan.*

Well done Logan. Infinite Powers is a tremendous literary achievement for all stated reasons and, in my view, most importantly because of the conceptual framework in which Strogatz so expertly describes the ‘reasons’ for the various operations of calculus.

I, as someone who, similar to Fermat (though with far less mathematical intuition that he had) studies mathematics for the pleasure of it, found that, despite all of the previous reading, videos, and tutorials I have processed on the matter, I had yet to encounter such clear and concise conceptual descriptions of the divisions of calculus.

As we know from critical rationalism, all begins with theory and no observation/experiment has context without it.

Thus, my view is that every calculus course, or perhaps mathematics course broadly speaking, should begin with a reading of Infinite Powers. As the conceptual understanding the reader will develop, from Strogatz’s brilliant mode of explanation, will dramatically enrich the practical process of assembling/deriving the formulas and doing the calculations.