![\"Portraits](\"\/\/media.nature.com\/w800\/magazine-assets\/d41586-019-01038-4\/d41586-019-01038-4_16578934.jpg\")
<\/div>\n<\/div>Isaac Newton (left) and Gottfried Wilhelm Leibniz each independently invented calculus.<\/span>Credit: Left, DeAgostini\/Getty; Right, Lombard\/ullstein bild via Getty<\/p>\n <\/p>\n <\/p>\n<\/figcaption><\/figure>\n Infinite Powers: How Calculus Reveals the Secrets of the Universe<\/b>Steven Strogatz<\/i>\u00a0Houghton Mifflin Harcourt (2019)<\/p>\n <\/p>\n Midway through\u00a0Infinite Powers<\/i>, Steven Strogatz writes that Isaac Newton and Gottfried Wilhelm Leibniz both \u201cdied in excruciating pain while suffering from calculi \u2014 a bladder stone for Newton, a kidney stone for Leibniz\u201d. It was a cruelly ironic end for the scientists who independently invented calculus: the word comes from the Latin for \u2018small stone\u2019, in reference to pebbles once used for counting.<\/p>\n Such fascinating anecdotes abound in\u00a0Infinite Powers<\/i>. Strogatz, a mathematician working in nonlinear dynamics and complex systems, has written a romp through the history of calculus \u2014 the study of how things change. Starting with the ancient Greeks, the book ends with connections between the field and artificial intelligence and machine learning. Calculus was key to working with Newton\u2019s laws of motion, which stimulated the Industrial Revolution. It is also central to quantum mechanics, which underpins the modern revolution in computers and communications. The book is a roll call of luminaries, including Galileo Galilei, Johannes Kepler, Ren\u00e9 Descartes and Pierre de Fermat.<\/p>\n Strogatz\u2019s telling is very much that of an applied mathematician: he acknowledges that a historian or pure mathematician might disagree with it. He anchors the story in how the evolution of calculus was intertwined with attempts to understand nature. That is, the geometry of curved lines and surfaces; the study of the planets, accelerating and decelerating in their orbits; and the laws of change (concerning, for example, the flow of fluids).<\/p>\n <\/p>\n Sofia Kovalevskaya, the first woman to win a mathematics PhD, explored the limits of calculus.<\/span>Credit: Alamy<\/p>\n<\/figcaption><\/figure>\n <\/p>\n <\/p>\n The book\u2019s pace varies, slowing down to focus on early groundwork and speeding up to discuss modern applications. It devotes a chapter to classical mathematician Archimedes, including his \u201cheroic\u201d calculations of\u00a0\u03c0<\/i>, the ratio of a circle\u2019s circumference to its diameter. There\u2019s an exquisite discussion of his method for expressing the area of a segment of a parabola in terms of the area of a simpler shape made up of straight lines. He broke the problem into smaller and smaller parts \u2014 flirting with infinity \u2014 and added it all back up to arrive at an answer. This presaged integral calculus. Strogatz then zips through the modern use of such principles in facial surgery and computer animation in films such as\u00a0Toy Story<\/i>\u00a0(1995), where surfaces are modelled as composed of smaller and smaller triangles, and volumes as composed of tiny tetrahedrons.<\/p>\n The early-seventeenth-century rivalry between Descartes and Fermat is illuminating. We see them vying to combine geometry and algebra to give us analytical geometry \u2014 essential for present-day calculus. We now take for granted graphs that have one variable on the\u00a0x<\/i>-axis and the other on the\u00a0y<\/i>; back then, it wasn\u2019t obvious that you could plot equations in this way. (Descartes, for all his contributions, emerges as mean-spirited and intent on demeaning Fermat.)<\/p>\n The invention of modern calculus by Newton and Leibniz some three decades later makes for snappy reading. Newton comes across as an astonishing genius, but secretive and paranoid. He found a way to calculate areas under curves while still a student at the University of Cambridge, yet was wary of sharing his insights. A few years later, Leibniz, \u201cthe most versatile genius in a century of geniuses that included Descartes, Galileo, Newton and Bach\u201d, came up with his own methods.<\/p>\n <\/p>\n Katherine Johnson used calculus in her work for NASA from the 1950s onwards.<\/span>Credit: NASA\/Donaldson Collection\/Getty<\/p>\n<\/figcaption><\/figure>\n <\/p>\n <\/p>\n Female scholars, too-often unheralded, are here given their due. African American mathematician Katherine Johnson (featured in Margot Lee Shetterly\u2019s 2016 book\u00a0Hidden Figures<\/i>, and the film of the same name) used calculus to plot the trajectory of US astronaut John Glenn\u2019s 1962 Earth orbit (see go.nature.com\/2dvorxu). At the turn of the nineteenth century, French mathematician Sophie Germain used a male pseudonym to obtain lecture notes, but was the first woman to win a prize from the Paris Academy of Sciences. She solved the puzzle of Chladni patterns \u2014 standing-wave formations on vibrating rigid surfaces. Russian mathematician Sofia Kovalevskaya \u2014 the first woman ever to gain a maths PhD (in 1874, at the University of G\u00f6ttingen in Germany) \u2014 showed that calculus has limits, by proving that one can\u2019t predict the behaviour of certain physical systems, even if they follow deterministic laws.<\/p>\n Strogatz uses the right amount of technical detail to convey complex concepts with clarity. If you can grapple with simple graphs and algebraic equations, you can grasp the details of the origins, genesis and meaning of calculus and the role of infinity. Anyone who can\u2019t might find the going tough.<\/p>\n When it comes to applications, the book occasionally feels forced. Take the invention of the computed tomography scanner. It\u2019s true that calculus is essential for inferring and reconstructing the structure of tissue in the path of X-rays, but the physics, materials science and engineering are equally important and exciting.<\/p>\n That said, Strogatz excels when he explains the role of calculus in understanding how DNA coils and twists and how enzymes act on it, and why calculus was essential to modelling how the number of T cells in the immune system infected with HIV changes after patients are given a protease inhibitor. It was calculus that showed the virus could be fought more effectively by three drugs at once, leading to the now-standard triple-drug therapy. If calculus is the \u201clanguage God talks\u201d, as physicist Richard Feynman put it, nowhere is this more obvious than when it meets biology.<\/p>\n Barring a few odd notes,\u00a0Infinite Powers<\/i>\u00a0evocatively conveys how calculus illuminates the patterns of the Universe, large and small.<\/p>\n <\/p>\n <\/p>\n<\/div>\n Nature<\/span>\u00a0568<\/strong>, 32 (2019)<\/p>\n <\/p>\n <\/p>\n <\/p>\n (\uc6d0\ubb38: \uc5ec\uae30<\/a>\ub97c \ud074\ub9ad\ud558\uc138\uc694~)<\/p>\n <\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":"![\"Black](\"\/\/media.nature.com\/w800\/magazine-assets\/d41586-019-01038-4\/d41586-019-01038-4_16584330.jpg\")
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