The Development of Calculus



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Юрий Светлозаров Донев фак. № 43551

ФМИ, Информатика, I курс, 7 група

Курсова работа по английски език 2003/2004

The Development of Calculus

Calculus is a branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a certain point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Works on calculus go back as far as the Antiquity when first the Egyptians discovered the rule for the volume of a pyramid as well as an approximation of the area of a circle and later in ancient Greece Archimedes proved that if ‘c’ is the circumference and ‘d’ the diameter of a circle, then 31/7d1/11d.

The beginning of the 17th century saw some very important discoveries in the area. First, the French mathematician René Descartes’ La Géométrie (‘The Geometry’) appeared. In this important work, Descartes showed how to use algebra to describe curves and obtain an algebraic analysis of geometric problems. A codiscoverer of this analytic geometry was the French mathematician Pierre de Fermat, who also discovered a method of finding the greatest or least value of some algebraic expressions—a method close to those now used in differential calculus. About 20 years later, the English mathematician John Wallis published The Arithmetic of Infinites. His colleague at the University of Cambridge was Newton’s teacher, the English mathematician Isaac Barrow, who published a book that displayed geometrically the inverse relationship between problems of finding tangents and areas, a relationship known today as the fundamental theorem of calculus.

Although many other mathematicians of the time came close to discovering calculus, the real founders were Newton and Leibniz. Newton’s discovery combined infinite sums, the binomial theorem for exponents, and the algebraic expression of the inverse relation between tangents and areas into methods we know today as calculus. Newton, however, was not eager to publish, so Leibniz became recognized as a codiscoverer because he published his discovery of differential calculus in 1684 and of integral calculus in 1686. It was Leibniz, also, who replaced Newton’s symbols with those familiar today.

In the following years, one problem that lead to new results and concepts was that of describing mathematically the motion of a vibrating string. Leibniz’s students, the Bernoulli family of Swiss mathematicians, used calculus to solve this and other problems, such as finding the curve of quickest descent connecting two given points in a vertical plane. Despite these advances in technique, calculus remained without logical foundations. Only in 1821 did the French mathematician Cauchy succeed in giving a secure foundation to the subject by his theory of limits, a purely arithmetic theory that did not depend on geometric intuition. Cauchy then showed how this could be used to give a logical account of the ideas of derivatives and integrals. In the next decade, the Russian mathematician Lobachevsky and German mathematician Dirichlet both gave the definition of a function as a correspondence between two sets of real numbers, and the logical foundations of calculus were completed by the German mathematician Dedekind in his theory of real numbers, in 1872.
Microsoft® Encarta® Reference Library 2003. © 1993-2002 Microsoft Corporation. All rights reserved
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Юрий Светлозаров Донев фак. № 43551

Glossary

variable – something that does not have fixed numerical value

derivative - the limit approached in the ratio of a function and its variable

curve - a line whose points are defined by an equation and whose coordinates are functions of an independent variable

slope of a curve- the first derivative of a curve at a point

computation – calculation

approximationa figure that is not exact, but is only slightly higher or lower than a given amount

circumference – the distance around the edge of a circle

tangent - a line that touches another curve or surface but does not cross or intersect it

infinite – very great in size, number, degree, or extent

binomial - mathematical expression made up of two terms and a plus or minus sign

plane – two-dimensional surface or space

limitsnumerical value approached by a mathematical function as the independent variable of the function approaches infinity or some specified value

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Юрий Светлозаров Донев фак. № 43551


Exercise

Fill in the gaps with the appropriate word from the list below. Some of the sentences refer to information from the article above:





subtraction dimension computation circumferences

approximations division curves plane

coordinate system area



  1. Archimedes was one of the first men to calculate ________ of circles. However, his results were not precise, they were ________.

  2. The ________ of complex mathematical expressions, which took scientists days to accomplish in the past, is now performed by computers in just a few seconds.

  3. The ________ of a triangle can be calculated by many different formulae, but maybe the most interesting of them all is the one discovered by Hero of Alexandria since it involves only the lengths of the sides.

  4. Spatial figures always seem a little awkward when drawn on a black board, because it is a sort of a ________ and therefore lacks a third ________.

  5. The Cartesian ________ is the most suitable one for drawing ________ of functions since its axes are perpendicular to each other.

  6. Although ________ and ________ are considered to be fundamental mathematical operations they are actually defined as the inverse processes of multiplication and addition.


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