Calculus
Relying on intuition and logic alone, it is not possible to achieve a deep understanding of the universe. A tool is needed that is capable of revealing simple patterns in complex and subtle natural phenomena. That mathematical tool is calculus. The development of calculus as a branch of mathematics is at the root of all scientific and technological advances in modern times. Without calculus, it is impossible to understand the behavior of gravity, the motion of the planets, the nature of light, electricity, or magnetism.
The development of calculus can trace its beginnings back to the time of the ancient Greeks. The mathematician Archimedes discovered formulas for determining the area and volume of geometric shapes such as cylinders and spheres. The methods he used were similar to the methods of today's integral calculus.
By the sixteenth and seventeenth centuries, the scientific revolution was in full swing and new phenomena were being observed and discovered at a rapid pace. Mathematicians were tackling increasingly complex algebraic problems, without much success, involving changing quantities of motion, mass and energy. This mathematical log-jam was finally broken in the second half of the seventeenth century. Almost simultaneously, Isaac Newton in England, Gottfried Leibniz in Germany, and Seki Kowa in Japan introduced a new mathematical way of analyzing quantities that change over time: calculus.
The development of calculus was not a smooth process, but basically the technique evolved into two main branches, differential calculus and integral calculus. Differential calculus provides methods for calculating the rate of change in one quantity due to a change in another quantity. For example, velocity is the rate of change in distance with respect to the change in time, while acceleration is the rate of change in velocity, again with respect to the change in time. One method for studying change is to examine the slope of a graph of a function, or the slope of the tangent line at a particular point on the graph. Differential calculus has applications to problems of optimization as well as to problems of motion.
Integral calculus, on the other hand, solves the opposite problem of determining an unknown quantity for which only the rate of change is known. It is essentially the calculation of sums of infinitesimally small quantities, and is useful for computing quantities like distance, area, and volume. Both the differential calculus and the integral calculus involve using only a finite number of points on the graph of a function to approximate the behavior at a particular point or over an interval of points. Analyzing more and more points yields a more accurate description of the curve and the event it represents. The final answer is obtained by using the method called convergence to a limit.
The powerful concepts of calculus enabled Newton to prove, mathematically, the statements of Johannes Kepler 's three laws of planetary motion and to develop a theory of gravity that has stood for over two centuries. Such a comprehensive theory would have been impossible without calculus.
The debate over who really invented calculus, Newton or Leibniz, raged for a hundred years and swept up many mathematicians. Jean Bernoulli and Jacques Bernoulli were among the first to achieve a full understanding of differential calculus and promoted the Leibnizian methods in Europe. Other mathematicians, mostly British, championed the Newtonian version. Centuries of use have favored the Leibnizian notation system over Newton's, but historians of science give equal credit to both men for the invention of calculus.
The techniques of calculus are so wide-ranging, and their applications to science so vast, that constant refinements and improvements to the system have been made over the decades, and continue to this day. France has been the center of much of this work. The mathematician Jean Le Rond d'Alembert devoted most of his career to investigating the techniques of integral calculus. He also developed the theory of partial differential equations, which was later elaborated upon by the Russian mathematician Sonya Kovalevsky. Augustin-Louis Cauchy clarified the use of limits and continuity. And more recently, Henri Lebesque refined an earlier integral theory of Georg Riemann, in which Lebesque included a measure-theoretic viewpoint of integration which became useful in several branches of mathematics, such as curve rectification and the theory of trigonometric series. Calculus was the tool scientists needed to expose the world of forces and energy for study and analysis.
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