## History of Nabla and Other Math Symbols

From NANET 26 Jan 1998

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From: Arnold Neumaier <neum@cma.univie.ac.at>
Date: Mon, 19 Jan 1998 21:38:43 +0100
Subject: History of Nabla

Last week I posted the question:

Does anyone know the history of using the symbol $\nabla$ for the gradient, and the meaning of the symbol outside of mathematics?

to na-net, and got a number of interesting answers. Thanks to all who replied. A summary appears below; the full text of the replies I got is available at

Corrections to the information given below are welcome.

Arnold Neumaier
Homepage

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The most definite information came from Avinoam Mann (MANN@VMS.HUJI.AC.IL) who had posted a contribution to a nabla discussion on the academia mailing list (ACADEMIA@techunix.technion.ac.il), and it was communicated to me by Dani Censor (CENSOR@bguee.ee.bgu.ac.il), the maintainer of the list. Mann refers to two web sites by Jeff Miller,

where one can find the following:

These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. Written sources are listed on a separate page. The most important written source is the definitive A History of Mathematical Notations by Florian Cajori.

The Hamiltonian operator. The symbol , which is also called a "del," "nabla," or "atled" (delta spelled backwards), was introduced by William Rowan Hamilton (1805-1865) in 1853 in Lectures on Quaternions, according to Cajori vol. 2, page 135.

David Wilkins has found the symbol used earlier by Hamilton in the Proceedings of the Royal Irish Academy of the meeting held on July 20, 1846. The volume appeared in 1847. However the symbol is rotated 90 degrees.

The word NABLA (for the "del" or Hamiltonian operator) was suggested humorously by James Clerk Maxwell, according to one source. According to a post in sci.math by Noam D. Elkies, the term was coined by Tullio Levi-Civita (1873-1941). A nabla is the name of an Egyptian harp. Cajori (vol. 2, page 135) says Heaviside called the symbol a nabla.

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Garry Tee mentioned that the standard biographies of Kelvin by S. P. Thompson (1910) and Andrew Gray, and by Crosbie Smith in "Energy and Empire" (CUP 1989) say something to the effect that the symbol $\nabla$ was invented (c1870) by William Thomson (later Baron Kelvin), as a modification of the symbol $\delta$ which he used for the Laplacian operator. The symbol suggests the shape of a harp, and so Thomson gave it the Greek name.

But these references are many years later than Hamilton.

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As regards language, nabla is the Greek word for some sort of harp. David Schaps (dschaps@mail.biu.ac.il) points out that the greek word does not derive from the related Hebrew word nevel=nebel for harp since it can be found already in the work of Sophocles. But probably the common origin of both words is aramaic. Indeed, S I Ben-Abraham (benabr@BGUMAIL.BGU.AC.IL) writes:

I venture to add that was borrowed to Greek via its Aramaic definite form (analogous to .

And Alexandre Chorin (chorin@math.berkeley.edu) writes:

I just had a conversation with Vivian Roumani (a librarian at UC Berkeley) and Morton Denn (Chair, Chem. Eng. at UC Berkeley), who told me that the symbol Nabla was invented by Hamilton; it is supposed to a drawing of an ancient hebrew harp (Nevel in Hebrew, Nabla in Aramaic).

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