page_title: History of mathematical notation
section_title: Euler and prime notations
hierarchical_section_title: History of mathematical notation / Symbolic stage / High division operators and functions / Euler and prime notations
caption_attribution_description: Leonhard Euler's signature (Swiss mathematician)
context_page_description: The history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.
  The development of mathematical notation can be divided in stages. The "rhetorical" stage is where calculations are performed by words and no symbols are used. The "syncopated" stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the post-classical age, bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light.
context_section_description: Leonhard Euler was one of the most prolific mathematicians in history, and also a prolific inventor of canonical notation. His contributions include his use of e to represent the base of natural logarithms.  It is not known exactly why  was chosen, but it was probably because the four letters of the alphabet were already commonly used to represent variables and other constants.  Euler used  to represent pi consistently.  The use of  was suggested by William Jones, who used it as shorthand for perimeter.  Euler used  to represent the square root of negative one, although he earlier used it as an infinite number.  For summation, Euler used sigma, Σ. For functions, Euler used the notation  to represent a function of . In 1730, Euler wrote the gamma function. In 1736, Euler produces his paper on the Seven Bridges of Königsberg initiating the study of graph theory.
  The mathematician William Emerson would develop the proportionality sign. Much later in the abstract expressions of the value of various proportional phenomena, the parts-per notation would become useful as a set of pseudo units to describe small values of miscellaneous dimensionless quantities. Marquis de Condorcet, in 1768, advanced the partial differential sign.  In 1771, Alexandre-Théophile Vandermonde deduced the importance of topological features when discussing the properties of knots related to the geometry of position. Between 1772 and 1788, Joseph-Louis Lagrange re-formulated the formulas and calculations of Classical "Newtonian" mechanics, called Lagrangian mechanics. The prime symbol for derivatives was also made by Lagrange.
caption: Leonhard Euler's signature
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