Isaac Newton’s great work, Philosophiae Naturalis Principia
Mathematica (“Mathematical Principles of Natural Philosophy”), published in 1687. It contains laws of motion and universal gravitation, basically asserting that the same laws apply both to small objects on the surface of the earth and to all bodies in space including the earth. This perspective was widely accepted as the absolute and final truth until the Michelson-Morley Experiment 200 years later deconstructed that synthesis. Einstein’s work early in the following century re-ordered our understanding of large-scale physics, but in the everyday space-time environment, Newton’s formulations endure.
Newton’s life and works were diverse and multifaceted. In the Principia Mathematica, a hefty three-volume work, the author stated his laws of motion and from Johannes Kepler’s empirical Laws of Planetary Motion, Newton derived his law of universal gravitation. This theoretical edifice is part of the picture, but from our present standpoint, an equally important contribution was Newton’s invention of calculus.
For years, an intense controversy obscured the matter. It was widely believed that Gottfried Leibniz (1646-1716) should receive full credit for this great innovation, and the controversy degenerated into interminable discussions of who said or wrote what in which year. At various times each of these profound thinkers accused the other of stealing his work. Currently it is generally held that the two great theoreticians developed their different but closely related systems independently.
Prior to the Leibniz-Newton inventions, the word “calculus” referred generically to any type of mathematics. Following their innovation, it came to mean the study of infinitesimal differentiation and integration.
Newton developed his version of calculus in response to issues he perceived pertaining to his physics. Calculus for Newton was descriptive, a tool for understanding phenomena. Leibniz, in contrast, investigated the inner nature of tangents. His motivation was to find a metaphysical explanation for change.
Their values and goals were different, one continental and the other quintessentially British. In the end both thinkers succeeded in formalizing the inverse relation between the integral and differential of a function, to the eternal at least temporary bafflement of beginning math majors.
Newton called his discipline the science of fluxions. That was his terminology. He determined the area under a curve based on the moment-to-moment rate of change. Leibniz conceived of the tangent as a ratio (like Newton), seeing it as the ratio between ordinates and abscissas. The integral, accordingly, resolved into the sum of an infinite series of rectangles.
Leibniz developed a far better (from our point of view) system of notation. Newton’s contribution was to bring calculus into the concrete universe of attraction and motion, and that is why he is seen as the greatest seventeenth-century thinker.
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