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Calculus

  • John StillwellEmail author
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

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The shift towards algebraic thinking was not only a revolution in geometry. It was decisive in the second and greatest mathematical revolution of the 17th century: the invention of calculus. It is true that some results we now obtain by calculus were known to the ancients; for example, the area of the parabolic segment was found by Archimedes. But the systematic computation of areas, volumes, and tangents became possible only when symbolic computation—that is, algebra—became available. The dependence of calculus on algebra is particularly clear in the work of Newton, whose calculus is essentially the algebra of infinite polynomials (power series). Moreover, Newton’s starting point was a basic theorem about the polynomial (1 + x)n, the binomial theorem, which he extended to fractional values of n. The calculus of Leibniz was likewise based on algebra—in his case the algebra of infinitesimals. Despite doubts about the meaning and existence of infinitesimals, Leibniz and his followers obtained correct results by computing with them. Results that we now obtain through a combination of algebra and limit processes were obtained by Leibniz through the algebra of infinitesimals. Our derivative dy/dx was, for Leibniz, literally the quotient of the infinitesimal dx by the infinitesimal dy. And our integralf (x) dx was, for Leibniz, literally the sum of the infinitesimals f (x) dx (hence the symbol ∫, which is an elongated S for “sum”).

Keywords

Algebraic Thinking Binomial Theorem Biographical Note Indian Mathematician Parabolic Segment 
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References

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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