The Calculus of Wikipedia

Yesterday, I was recounting my early childhood experiences with calculus for a book I'm writing about learning. (Punchline: I fell in love with calculus, even though I can barely balance a checkbook; it turns out lots of people who fall in love with calculus have a hard time adding 2 + 2). I digress. In any case, I googled to find out some quick-and-basic information about calculus and, of course, went to Wikipedia and read an entirely new history of the development of the fundamental ideas of calculus. Surprise! That history isn't just a scandal among the Founding Fathers, the great debate about whether Leibniz stole ideas (and maybe even a manuscript) from Newton. Back in 1800 BC ancient Egyptians were using calculating methods for the volume of a pyramidal frustrum that include principles we now associate with modern calculus. I'm reblogging the historical part of the Wikipedia calculus entry below. But, guess what, after spending now close to something like an hour on the internet, I've not found a more interesting account elsewhere. Yes, I know. Wikipedia has a lot of problems. Yes, I know, without Wikiscanner we wouldn't know how the FBI, CIA, and Corporate America are all fudging and editing entries (but, as I said, I wish we had a Wikiscanner-detector for ABC, NBC, New York Times, etc). Still, Wikipedia has done more to make knowledge in the West less Eurocentric than anyone could have anticipated. I have an email in to a professional reference librarian for other online sources, but, for the present, I'm smiling to know that Zu Chongzhi was calculating the volume of a sphere in the 5th century AD and that Indian mathematician Araybhata was working on infinitesimals in 499 AD. The "science of fluxions" was not invented solely by the great Western wigged ones. (And go to the "Comment" section of the blog for the history of calculus from the online Encyclopedia Britannica. Only Newton and Leibniz all over again. . .)

Development

Main article: History of calculus

The history of calculus falls into several distinct periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian worked out the volume of a pyramidal frustrum.[1] [2] Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes. Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2]

In the medieval period, the Indian mathematician Aryabhata used the notion of infinitesimals in 499 AD and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around 1000 AD, the Iraqi mathematician Ibn al-Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus.[6] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[8] which are treated in the text Yuktibhasa.[9][10][11]

In the modern period, independent discoveries in calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method of exhaustion. In Europe, the second half of the 17th century was a time of major innovation. Calculus provided a new opportunity in mathematical physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668.

Gottfried Wilhelm Leibniz was originally accused of plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus.

Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The basic insight that both Newton and Leibniz had was the fundamental theorem of calculus.

When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus the "the science of fluxions".

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue further generalized the notion of the integral.

Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development.[12]

[edit] Significance

While some of the ideas of calculus were developed earlier, in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce the basic principles of calculus. This work had a strong impact on the development of physics.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.

[edit] Foundations

In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.

There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.