The Calculus Diaries - Jennifer Ouellette [11]
So the area under a curve corresponds to the integral, while the slope of the tangent line to a point on that curve corresponds to the derivative. With the merging of algebra and geometry, the stage was set for calculus to make its grand entrance. Ultimately, the credit for inventing calculus is given jointly to Isaac Newton and Gottfried Wilhelm Leibniz, who independently made their revolutionary discoveries in the 1660s and 1670s, giving rise to an epic intellectual battle for the title of Inventor of Calculus.
CLASH OF THE TITANS
Isaac Newton hardly needs an introduction. He is almost universally recognized as the father of modern physics via his masterpiece, the Principia, as well as his work on the nature of light published in Opticks toward the end of his illustrious career. The Principia is inarguably one of the most influential scientific books ever written—eighteenth-century mathematician Joseph-Louis Lagrange declared it “the greatest production of a human mind”—yet it is one of the least read. Three volumes of mathematical theory on the nature of gravity and the laws of motion, rendered in excruciatingly pedantic seventeenth-century Latin prose and chock-full of equations, are hardly summer beach reading. Apparently Newton made it deliberately difficult “to avoid being baited by little smatterers in mathematics.” The Great Newton despised dilettantes.
The son of a yeoman farmer in Lincolnshire, England, who could neither read nor write, Newton was born in 1642, two months after the death of his father, and so premature and small that hardly anyone expected him to survive. His mother, Hannah, married a clergyman named Barnabas Smith when Isaac was only three years old and promptly moved away with her new husband to start a new family, leaving young Isaac behind with his grandparents.
Hannah wanted him to become a farmer, and when the boy was seventeen, he was expected to take over the family farm. But he proved disastrous at minding the sheep or cows, feeding the chickens, or taking produce to market. Invariably he would be found sprawled under a shady tree with a book, jotting his thoughts down in a notebook, or jumping from one spot to another in the field, trying to determine the length of those jumps. He invented methods for producing chalk and gold ink, and a technique “to make birds drunk,” as well as a phonetic alphabet; he “contrived water wheels and dams” and dabbled in magic tricks. In short, he did anything but the various chores a competent farmer must master.
Hannah relented and packed Newton off to Cambridge University to pursue the life of the mind, where he earned his undergraduate degree in science and math in 1665. His graduate studies were interrupted by the outbreak of the plague in Cambridge. Students and professors alike fled the city, and Newton returned home for the ensuing year, until the panic (and danger) had passed. He later described this period as “the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since.” He wasn’t exaggerating. Not only did he work out his three laws of motion and a universal theory of gravity; he also invented the mathematical tool he needed to achieve those insights: calculus.
“Rather than thinking of a curve as a simple geometrical shape or construction on paper, Newton began to think of curves in real life—not as static structures like buildings or windmills, but as dynamic motions with variable quantities,” Jason Bardi writes in The Calculus Wars. Take that famous (and possibly apocryphal) anecdote about Newton observing an apple falling from a tree and coming up with his critical insights into gravity. The position and speed of the apple are changing at every moment8: The apple is still on the tree at what physicists call time zero. (That’s shorthand for “the value of the variable t for time is 0.”) A fraction of a second later, it has started its fall, and another fraction of a second finds it midway from branch to ground, and so forth. The apple’s descent progresses in tiny increments