Notation

In 1928-29, Florian Cajori published his two-volume A History of Mathematical Notations: Notations in Elementary Mathematics (vol. I) and Notations Mainly in Higher Mathematics (vol. II). These were republished by Dover in 1993, in a single volume and this remains the standard authority on the matter of notation in mathematics. On the matter of Hindu-Arabic numerals, Cajori quotes Pierre Simon Laplace (1749-1827) as follows:

It is from the Indians that there has come to us the ingenious method of expressing all numbers, in ten characters, by giving them, at the same time, an absolute and a place value; an idea fine and important, which appears indeed so simple, that for this very reason we do not sufficiently recognize its merit. But this very simplicity, and the extreme facility which this method imparts to all calculation, places our system of arithmetic in the first rank of the useful inventions. How difficult it was to invent such a method one can infer from the fact that it escaped the genius of Archimedes and of Apollonius of Perga, two of the greatest men of antiquity.

Florian Cajori, A History of Mathematical Notations, p. 70

Laplace’s remark underscores the central role that good notation plays in mathematics. Notation can obscure or illuminate the concepts that underpin mathematics. In Tunstall’s De arte supputandi libri quatuor (Paris, 1538), the earliest mathematical work in the Edward Worth Library, we can see both familiar and unfamiliar manners of displaying common arithmetical procedures. Today, with the use of electronic calculators, there is little need to labour over the detail of ‘extracting’ square roots, and so such computation has already vanished from the repertoire of the public at large. Long division is still widely taught, but not using the same schema as in the sixteenth century. Even in the following 130 years, the reader can appreciate that the ways of carrying calculations in arithmetic had changed by the time Leybourn wrote The art of numbring by speaking-rods: vulgarly termed Nepeirs bones (London, 1667).

Robert Steell, A treatise of conic sections (Dublin, 1723), preface.

This page from the preface of Steell’s A treatise of conic sections (Dublin, 1723) shows the inclusion of a list of notation in a mathematical textbook in the early eighteenth century.

If developments of notation in arithmetic can be easily discerned, the changes in algebraic and geometrical notation are even more evident. In the webpage on Kersey, attention is drawn to the wordiness of the explanation of algebra, relative to what is considered the norm today. Today, it is not uncommon for students to think of the Pythagorean theorem as a² = b² + c², where the letters here are readily explained in terms of a right-angled triangle; however, the sense here is quite different from the statement of Proposition 47 of Book I of the Elements, illustrated on the Euclid page.

It is no exaggeration to say that a revolution in mathematical notation took place in the seventeenth century. Evidence of this is clear from works in the Edward Worth Library, in particular those of Viète, Harriot and Descartes, as discussed on the pages devoted to these mathematicians (with some further detail on Harriot’s notation elaborated on the Algebra page). With the publication of La Géométrie in 1637, René Descartes (1596-1650), made the connection between algebra and geometry more explicit than ever before by assigning coordinates to points in the plane and applying the rules of algebra to the quantities that arise. This was indeed a powerful innovation, but one that requires a good notation to facilitate it. Descartes had this too. His work on notation transformed mathematics forever, and opened the way for Newton’s and Leibniz’s invention of the calculus later in the 1660s and 1670s.

Sources

Cajori, Florian, A History of Mathematical Notations, vols I-II (New York: Dover, 1993).

Stedhall, Jacqueline, Mathematics Emerging: A Sourcebook 1540-1900 (Oxford University Press, 2008).

Text: Maurice OReilly.