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The word calculus is originally Latin for a small stone, referring to those used to perform calculations on an abacus or similar tool. Today in mathematics it has come to mean any system of rules for symbolic manipulation, something vastly important in a subject where the use of symbolic notation is crucial (see symbolism). The two major divisions in logic are known as the propositional calculus and the predicate calculus, and together these comprise the logical calculus. Calculi are also important in the study of semantics in the theory of computing.
Usually, the term is used to refer to the infinitesimal calculus unless qualified. Although this is often thought of as the invention of Newton (1642 - 1727) and Leibniz (1646 - 1716) (and to which of them it should be ascribed has been the subject of much debate), it was really the product of the work of many 17th-century mathematicians attempting to solve two major problems in applied mathematics. Newton and Leibniz\'s fundamental contribution was to realize that the two problems were intimately connected.
The first problem is that of the differential calculus, to determine the tangent lines to a curve. For a particular point on a curve, there may be a straight line through the point which just touches the curve without crossing it. Such a line is called the tangent. (For some curves there are no such straight lines at some points.) The problem is, given a curve and a point on it, to discover whether the tangent exists and, if it does, to find its slope, which is known as the derivative of the curve at the point. This is important physically because the velocity of a moving body is always along the tangent to the curve described by its path. A knowledge of differential calculus is therefore essential to the study of mechanics.
The second problem, known as the problem of quadrature, is that of the integral calculus. The problem is to determine the area within a curve. In a Cartesian co-ordinate system in two dimensions (that is, with x and y co-ordinates), a curve is usually written as an equation, as y = f(x), where f is a function. Then the integral of the curve (from a to b) is taken to be the area of the figure given by the curve itself between (a,f(a)) and (b,f(b)), and the straight lines joining (a,f(a)) to (a,0), (b,f(b)) to (b,0) and (a,0) to (b,0) along the x-axis. This is written as ~a ^b f(x)dx.
The intimate connection between the two problems, discovered independently by Newton and Leibniz, is known as the fundamental theorem of the (infinitesimal) calculus. A function F(x) is defined to be the integral of the curve f(x) between a and x (for some fixed number a), so that F(x) = {img src=show_image.php?name=22A5.gif }âx f(x)dx. The fundamental theorem of the calculus is simply that F\'(x) = f(x), where F\'(x) is the derivative of F at the point (x,F(x)). In other words, the process of integration takes us from the function f(x) to the function F(x), and the process of differentiation takes us back from F(x) to f(x). There is in fact no difference between the two; there is only one calculus.
Newton and Leibniz used infinitesimal numbers (ones smaller in size than any other, but not actually zero); Leibniz connected the theorem with mystical ideas about infinitesimals that led to the calculus being regarded as not really a true part of mathematics with the same status as, say, algebra. This problem was eventually circumvented by Bolzano, Cauchy and Weierstrass in the early 19th century, and the mathematical theory of analysis was born. SMcL |
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