Further reading

  • Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. The reference work on the origin of each symbol; almost every date in this chapter can be checked in it.
  • Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton: Princeton University Press, 2014. An accessible history on the same theme as this book.
  • Ifrah, Georges. The Universal History of Numbers. London: Harvill Press, 1998. On number systems and zero.
  • Klein, Jacob. Greek Mathematical Thought and the Origin of Algebra. Cambridge, MA: MIT Press, 1968. On the passage from Diophantus to Viète.
  • Boyer, Carl B.; Merzbach, Uta C. A History of Mathematics. 3rd ed. Hoboken: Wiley, 2011.
  • Crowe, Michael J. A History of Vector Analysis. Notre Dame: University of Notre Dame Press, 1967. On the dispute between quaternions and vectors.
  • Maor, Eli. Trigonometric Delights. Princeton: Princeton University Press, 1998. On the history of the sine and the names of trigonometry.
  • Grabiner, Judith V. The Origins of Cauchy’s Rigorous Calculus. Cambridge, MA: MIT Press, 1981. On Berkeley, Cauchy and the limit.
  • Miller, Jeff. Earliest Uses of Various Mathematical Symbols. A page maintained since the 1990s, with the earliest known occurrences of each sign.
  • MacTutor History of Mathematics, University of St Andrews: biographies of al-Khwarizmi, Fibonacci, Recorde, Viète, Harriot, Oughtred, Cayley, Boole, Frege and Peano.

The old texts on this site about geometry, trigonometry, the square root, quadratic equations, logarithms, derivatives, the chain rule, integrals, limits, functions, differential equations, groups, matrices, vectors and quaternions were the basis for rewriting this chapter. From them come the story of radix quadrati, the completing-the-square figure, Pythagoras’ theorem in Latin, the etymology of cathetus, hypotenuse, minute and second, the passage from $\Sigma$ to $\int$, the proof that $1=2$, Hooke’s law, the “canon of heat” and the propagation of errors, the abbreviation of logarithmus to $\log$, the etymology of derivative and function, reading a function as a sentence with a hole, the group table of the triangle’s moves, the dispute between quaternions and vectors, and the ideas of active and sterile notation.

Image sources

Image Source Licence
Page of the Liber abbaci (Biblioteca Nazionale Centrale di Firenze) Wikimedia Commons public domain
Opening of al-Khwarizmi’s book Wikimedia Commons public domain
Page of La Géométrie, René Descartes (1637) Wikimedia Commons public domain
Portrait of François Viète Wikimedia Commons public domain
Portrait of René Descartes, Frans Hals Wikimedia Commons public domain
The equals sign in The Whetstone of Witte, Robert Recorde (1557) Wikimedia Commons public domain
The first equation with an equals sign, Robert Recorde (1557) Wikimedia Commons public domain
Portrait of Isaac Newton, Godfrey Kneller (1689) Wikimedia Commons public domain
Portrait of Leibniz, Christoph Bernhard Francke Wikimedia Commons public domain
Nova methodus pro maximis et minimis, Leibniz (1684) Wikimedia Commons public domain
Portrait of Euler, Jakob Emanuel Handmann (1753) Euler’s Eyes collection; see Euler image credits public domain

The diagrams (place value, area of the square, median, completing the square, the triangle’s moves and quantifiers) and the interactive timeline were made for this book. The notations shown in the timeline are teaching reconstructions, not facsimiles of the manuscripts.

Chapter index · ← Boole, Frege and Peano: writing down reasoning · Chapter 2: the mathematical sign →