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


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”).


Algebraic Thinking Binomial Theorem Biographical Note Indian Mathematician Parabolic Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baron, M. E. (1969). The Origins of the Infinitesimal Calculus. Oxford: Pergamon Press.Google Scholar
  2. Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development. New York: Dover Publications Inc.Google Scholar
  3. Davenport, J. H. (1981). On the Integration of Algebraic Functions. Berlin: Springer-Verlag.Google Scholar
  4. Edwards, Jr., C. H. (1979). The Historical Development of the Calculus. New York: Springer-Verlag.Google Scholar
  5. Hofmann, J. E. (1974). Leibniz in Paris, 1672–1676. London: Cambridge University Press. His growth to mathematical maturity, Revised and translated from the German with the assistance of A. Prag and D. T. Whiteside.Google Scholar
  6. Kahn, D. (1967). The Codebreakers. London: Weidenfeld and Nicholson.Google Scholar
  7. Robinson, A. (1966). Non-standard Analysis. Amsterdam: North-Holland Publishing Co.Google Scholar
  8. Westfall, R. S. (1980). Never at Rest. Cambridge: Cambridge University Press. A biography of Isaac Newton.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

Personalised recommendations