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| Newtonian fluxions | |
|---|---|
| Name | Newtonian fluxions |
| Inventor | Isaac Newton |
| Introduced | 17th century |
| Field | Mathematics |
| Related | Calculus, Differential calculus, Integral calculus, Fluxion method |
Newtonian fluxions Newtonian fluxions denote the early form of differential calculus devised by Isaac Newton in the late 17th century, developed contemporaneously with work by Gottfried Wilhelm Leibniz and others. The method framed rates of change as "fluxions" of "fluents", situating the theory within problems treated by contemporaries such as John Wallis, Isaac Barrow, and students of the Royal Society. Newton's fluxional notation and approach influenced debates involving figures like Edmond Halley, Robert Hooke, William Jones, and institutions including the Royal Society and the Royal Society of London.
Newton first articulated fluxions during the 1660s and 1670s while corresponding with Isaac Barrow and Henry Pemberton and communicating results to Edmond Halley and John Collins. Early manuscripts such as the De methodis and the Method of Fluxions circulated among members of the Royal Society and contemporaries like Christopher Wren and Robert Boyle. The priority dispute with Gottfried Wilhelm Leibniz involved publications by Leibniz in the Acta Eruditorum and later interventions by the Royal Society under figures such as Samuel Pepys and John Conduitt. Debates engaged mathematicians across Europe including Jakob Bernoulli, Johann Bernoulli, Brook Taylor, Jean le Rond d'Alembert, and Leonhard Euler, and were entangled with print culture from publishers like John Smith and William Innys.
Newton defined a fluent as a varying quantity and its fluxion as the instantaneous rate of change, introduced alongside symbols for second and higher fluxions. His notation contrasted with that of Gottfried Wilhelm Leibniz and later commentators such as Joseph-Louis Lagrange and Pierre-Simon Laplace. Manuscripts circulated to correspondents such as Humphry Ditton and John Wallis show conventions for representing velocities of geometric points and the fluxions of polynomial fluents. Contemporary expositors including Colin Maclaurin and Brook Taylor translated Newtonian terms into algebraic formulations used in works by Adrien-Marie Legendre and Augustin-Louis Cauchy.
Fluxional analysis provided rules for differentiation of sums, products, quotients, and powers, and procedures for higher-order fluxions analogous to modern derivatives used by Leonhard Euler and Joseph Fourier. Techniques for tangents, maxima, minima, and rectification appeared in Newton's applications to problems studied by Christiaan Huygens, Johannes Kepler, and Blaise Pascal. Newton applied fluxions to series expansions, infinite series work paralleled by James Gregory and John Wallis, and to the binomial theorem as extended by Abraham de Moivre and Brook Taylor. Methods for fluxional integration corresponded to antidifferentiation approaches later formalized by Augustin-Louis Cauchy and Karl Weierstrass.
The fluxional method stood in contrast to the differential notation introduced by Gottfried Wilhelm Leibniz, and the ensuing priority conflict involved agents such as Richard Bentley, John Locke, and officials of the Royal Society. European schools led by Johann Bernoulli and Gottfried Leibniz adopted the dx, dy notation, while British mathematicians often retained fluxional symbols in textbooks by John Colson, Colin Maclaurin, and Thomas Simpson. Later syntheses by Joseph-Louis Lagrange and critiques by Jean le Rond d'Alembert and Pierre-Simon Laplace reframed both traditions, with further rigorous foundations developed by Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass.
Newton applied fluxions to classical mechanics problems connected to the work of Galileo Galilei, Christiaan Huygens, and Johannes Kepler, treating motion, planetary orbits, and gravitational attraction in the context of the Principia Mathematica. Fluxional methods solved curvature and tangent problems of interest to René Descartes and Blaise Pascal, and they underpinned work on series and approximations pursued by Leonhard Euler, James Stirling, and Brook Taylor. Applied problems addressed by later adopters included fluid resistance and elasticity studied by Daniel Bernoulli and Leonhard Euler and optimization issues appearing in the work of Joseph-Louis Lagrange and Adrien-Marie Legendre.
Although the dx/dy notation of Gottfried Wilhelm Leibniz became dominant in continental Europe, Newtonian fluxions shaped the pedagogy of British mathematics through the 18th century via authors like Colin Maclaurin, John Colson, and Thomas Simpson. Historical controversies involving the Royal Society and pamphlets by George Peacock, Augustus De Morgan, and John Herschel contributed to historiography later surveyed by Carl Boyer and E. T. Bell. The conceptual move from fluxions to rigorous limits was developed by Augustin-Louis Cauchy, formalized in the 19th century by Karl Weierstrass and expanded in analysis by Bernhard Riemann and Émile Borel, while applications proliferated in fields cultivated by Joseph Fourier, Sofia Kovalevskaya, and David Hilbert.