Jennifer Lehrman

Footnote 18

MAT 267 10:40

Newton and Leibniz: The Founders of Calculus

Sir Isaac Newton and Gottfried Leibniz, probably two of the greatest mathematicians of the 17^th century, have been credited with the invention of calculus by mathematicians for centuries.  Yet earlier ideas of integration had already been explored by ancient Greeks such as Archimedes and Eudoxus, and basic methods for finding tangents had first been developed by Pierre Fermat, Isaac Barrow (Newton’s professor at Cambridge), and others.  But it was not until Newton and Leibniz that an application for derivation and integration was employed and thus fore providing the basis for what calculus is today.

            Sir Isaac Newton, an analyst and mathematical physicist, was born in Woolsthorpe, England on Christmas day in the year 1642.  At eighteen years old, he attended Trinity College, Cambridge in 1661.  It was here that Newton was the student under Isaac Barrow, who had already made advances in mathematics in methods for finding tangents.  In 1665, the university was closed due to the bubonic plague.  This seemingly unfortunate event forced Newton to continue his research and studies back home, where, in fact, his development of calculus began.  These first discoveries made in early 1665 were the result of his ability to express functions in terms of infinite series and rate of change (fluxion or differential calculus) of continuously varying quantities (fluents).  These two discoveries of infinite series and rate of change, Newton linked together and called “my method.”  Another significant discovery of Newton’s in 1664 or 1665, the Binomial Theorem, was an indirect approach to infinite analysis.  This finding led Newton to determine that infinite series could be operated in the same way as finite polynomial expressions.  Because of this new infinite analysis allowing him to exploit the relationship between slope and area, Newton became the effective inventor of calculus.  Even though mathematicians from Torricelli to Barrow sensed this relationship, Newton was the first to give a general applicable procedure for determining instantaneous rate of change (derivatives) and to invert this in the case of problems involving summations (integrals).  However, it was not until 1711 that his book, De Analysi per Aequationes Numero Terminorum Infinitas, which explained his calculus, was published.  But what Newton is perhaps most famous for in the mathematical world is his method of fluxions, written in 1671 and published in 1736.  With this method, maxima and minima, tangents to curves, curvature of curves, points of inflection, and convexity and concavity of curves could be determined.  It also provided a method for approximating values of real roots for either algebraic or transcendental numerical equations.

            Gottfried Wilhelm Leibniz was born in 1646 at Leipzig, Germany.  He entered the University of Leipzig at only fifteen years old, and had earned his bachelor’s degree by age seventeen.  But when he was refused to pursue a degree of doctor of laws, due to his young age, at Leipzig, he moved to Nuremberg.  Most of Leibniz’s early work consisted largely of infinite series.  On October 29, 1675, he first used the modern integral sign ∫ derived from the Latin word summa (sum) to indicate the sum of Cavalieri’s indivisibles.  In 1684, Leibniz published the first account of differential calculus, beating Newton by nearly 27 years.  Two years later, the Acta Eruditorum was published, which explained integral calculus in which quadratures were shown to be special cases of inverse methods of tangents.  Less than a decade later, in 1693, Leibniz is said to have discovered a theory of determinants, despite a similar consideration ten years prior by Seki Kōwa in Japan.   This theory carried over into an analysis idea of infinitely small quantities of different orders, based upon the principle of homogeneity.

            With Newton’s discoveries in determining fluxion (derivative) and Leibniz’s discoveries in differentials, calculus had taken form and provided an actual application of its concepts.  Although Newton and Leibniz had somewhat different approaches to solve problems relating to calculus, both provided invaluable discoveries to the subject.  Leibniz applied his use of definite integrals in a philosophical perspective such as monads (ultimate particles of matter) while Newton applied most of his calculus to physics and a basis in the notion of velocity and used his free use of series to represent functions (indefinite integrals).  But thanks to the contributions of both mathematicians, calculus was now built on algebraic concepts which reduced four main problems (rates, tangents, maxima and minima, summation) to differentiation and anti-differentiation and no longer an extension of Greek geometry, but an independent science capable of handling a large spectrum of problems. 

References

Boyer, Carl, and Uta Merzbach. A History of Mathematics. New York: John Wiley, 1987.

Boyer, Carl. The History of the Calculus and Its Conceptual Development. New York : Dover, 1959.

Eves, Howard. An Introduction to the History of Mathematics. 4th ed.. New York: Saunders, 1990.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972.

MAT266, Calculus II, Spring 2007 - Kathryn Jordan, "Newton and Leibnez:  A Background and Contrast"