================ History of nabla ================ ------------------------------------------------------------------------ On Jan. 19, 1998, 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 appended afterwards. Corrections to the information given below are welcome. Arnold Neumaier http://solon.cma.univie.ac.at/~neum ------------------------------------------------------------------------ (added January 7, 2004:) Earliest Known Uses of Some of the Words of Mathematics (N)

now contains the following extensive story: Tait used the symbol for "the very singular operator devised by Hamilton" in An Elementary Treatise on Quaternions (1867, p. 221). Tait made very effective use of the operator in a series of papers, including "On Green's and other allied theorems" (1870) Scientific Papers I, p. 136. The story of its naming is related in Cargill Gilston Knott\x{2019}s Life and Scientific Work of Peter Guthrie Tait (1911): From the resemblance of this inverted delta to an Assyrian harp Robertson Smith suggested the name Nabla. The name was used in playful intercourse between Tait and Clerk Maxwell, who in a letter of uncertain date finished a brief sketch of a particular problem in orthogonal surfaces by the remark "It is neater and perhaps wiser to compose a nablody on this theme which is well suited for this species of composition." [...] It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his homorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla," that is, Tait. In a letter from Maxwell to Tait on Nov. 7, 1870, Maxwell wrote, "What do you call this? Atled?" In a letter from Maxwell to Tait on Jan. 23, 1871, Maxwell began with, "Still harping on that Nabla?" Maxwell and Tait were school friends and the background to the fun was Maxwell\x{2019}s search for a suitable term to use in his own work. In the event Maxwell called the operator the slope: see A Treatise on Electricity and Magnetism (1873, pp.15-16.) In "On the importance of quaternions in physics" (1890) Scientific Papers II, pp. 303-4 Tait wrote "we must have a name attached to [this remarkable operator] and I shall speak of it as Nabla." Heaviside also used "nabla" but without enthusiasm, "The fictitious vector ... is very important. Physical mathematics is very largely the mathematics of . The name Nabla seems, therefore, ludicrously inefficient." ("On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field," Philosophical Transactions of the Royal Society of London. A, 183, (1892), p. 431.) This supersedes most of what is below. ------------------------------------------------------------------------ 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, Earliest Known Uses of Some of the Words of Mathematics

Earliest Uses of Various Mathematical Symbols

where one can find the following: http://members.aol.com/jeff570/mathsym.html 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. [Cajori, Florian. A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929.] http://members.aol.com/jeff570/calculus.html [from mathsym.html] 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. http://members.aol.com/jeff570/m-r.html [from mathword.html] 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. ----- But Noam D. Elkies (elkies@math.harvard.edu) says he ``cannot reconstruct the source of the claim about Levi-Civita and nabla'', and in view of the discussion here, ``while it still makes sense for L-C to be the coiner, the case is not as strong.'' Two alternative contenders: Garry Tee mentions 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. Michele Benzi refers to p. 143 of Cargil Gilston Knott, "Life and Scientific Work of Peter Guthrie Tait", Cambridge, England, 1911. that nabla was the name suggested to P. G. Tait by Robertson Smith because of the similarity of the symbol to an Assyrian harp. ----- 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). John Clark (http://www.finerandd.com/) writes: If the term is of biblical origin, they have got their Hebrew mildly wrong. A lyre or some such instrument is variously ''navvel'' or ''nevvel'' (or in the Sephardi Hebrew dialect favoured by the King James translators of the Bile into English, nabbel or nebbel). ''Nabla'' is roughly the Sephardi rendition of the Hebrew word for carcass, which would be ''navla''\x in the more common Israeli and Ashkenazi dialects of Hebrew. That clearly is not what the mathematicians had in mind! ----- Now I wonder whether the harp is Assyrian, Egyptian, Greek, Hebrew, or Phoenician, all that being suggested, with hebrew apparently ruled out. ======================================================================== ======================================================================== ======================================================================== To my query From: Arnold Neumaier Date: Tue, 13 Jan 1998 10:08:45 +0100 Subject: Nabla Does anyone know the history of using the symbol $\nabla$ for the gradient, and the meaning of the symbol outside of mathematics? Arnold Neumaier neum@cma.univie.ac.at I got the following replies: ======================================================================== ======================================================================== ======================================================================== From: "Dr. Thomas Seidman" Date: Sat, 17 Jan 1998 18:49:25 -0500 (EST) I saw your question in NANews. I cannot tell you anything about the history of the use of this symbol, but note that the term `nabla' is a (biblical) Hebrew word for a variety of hand-held harp whose appearance is approximately like the symbol. -- _/_______/ ___ V | / \ . | / ____ _ _ \___ _ _ | _ _ _ _ / / \ | \/ \ \ |_/ | / \| | \/ \ / | | \ \___/__/\__/ | | |_/ \___/ _/\__/\/\_/\/| | |_/\/\/| |_/ Prof. Thomas I. Seidman UMBC --- Dept. Math/Stat http://math.umbc.edu/~seidman/index.html Baltimore, MD 21250 (1-410)-455-2438 [FAX: -1066] ----- From: chorin@math.berkeley.edu (Alexandre Chorin) Date: Sat, 17 Jan 1998 16:05:06 -0800 (PST) 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). ----- From: Garry Tee Date: Sun, 18 Jan 1998 16:08:01 +1300 Dear Dr Neumaier, 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 (probably of Semitic origin) for a type of harp. Fuller details are given in the standard biographies of Kelvin by S. P. Thompson (1910) and Andrew Gray, and by Crosbie Smith in "Energy and Empire" (CUP 1989). Yours sincerely, Garry J. Tee, Department of Mathematics, University of Auckland, New Zealand. ----- From: Allan Pinkus Date: Sun, 18 Jan 1998 08:56:09 +0200 (IST) I was always told that "nabla" had its etymological roots in the Hebrew word "nevel" which mean's harp. (Put some strings in and it does look like a harp. ----- From: "Raymond E. Rogers" Date: Sun, 18 Jan 1998 16:51:04 -0500 Does anyone know the history of using the symbol $\nabla$ for the gradient, and the meaning of the symbol outside of mathematics? Could you pass along any good answers. I think it is a good question:) Enjoy Ray ----- From: "Dr K.M. Briggs" Date: Mon, 19 Jan 1998 11:27:29 +0000 (GMT) I thought it was Hebrew for `harp', from the shape of the letter. Keith -- Keith Briggs, Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EA, England. Email: kmb28@cam.ac.uk My personal webpage: http://www-epidem.plantsci.cam.ac.uk/~kbriggs ----- Date: Tue, 20 Jan 1998 14:54:03 -0700 (MST) From: Michele Benzi Dear Dr. Neumaier, The Nabla was an ancient musical instrument, similar to a lyre. If I am not mistaken, some information about the origin of the name is provided in the book "A history of vector analysis", by M. Crowe, published by Dover. I have the book at home, so I can send you more information tomorrow. I don't think the symbol is used outside of Mathematics. Best regards, --Michele --------------------------------------------------------------- Michele Benzi Scientific Computing Group (CIC-19) phone : (505) 665-4778 MS B256 fax : (505) 667-1126 Los Alamos National Laboratory e-mail: benzi@lanl.gov Los Alamos, NM 87545 -- USA http://www.c3.lanl.gov/~benzi/benzi.html --------------------------------------------------------------- ----- From: CENSOR@bguee.ee.bgu.ac.il Date: Mon, 19 Jan 1998 16:22 +0200 dear Arnold Neumaier i am running the "academia" list in israel, on which recently the subject of "nabla" was extensively discussed. i hope this will be useful. your message has been forwarded to me by my brother yair censor, a mathematics prof in haifa univ. dani censor ----- dear colleagues. i need help for a friend in finland who investigates the historical roots of mathematics. for the benefit of curious readers, i will give here some background: the symbol in question is an inverted upper case delta, like this: ____ \ / \/ which we call NABLA. i believe a bible researcher or a jewish history specialist with strong background in classical greek could help me. does a word NABLA or thereabouts exist in the classical greek, and what does it mean? my contention is that NABLA is related to the biblical hebrew word NEVEL (i.e., hebrew for harp, lyre). for those more interested, please read on. ======================================================= Subj: NEBEL, NABLA AND KINNOR dear ismo, the advantage i have over you, regarding the "nabla" business, is that i can read hebrew and english side by side. the original bible is written in an archaic style and wording, but my hebrew (being almost my mother tongue, i say almost, because my mother did not speak hebrew, she spoke only german, the language of the country from which she had to escape) is good to the extent that i can read the bible freely (almost). in the english, i refer to the KJV (king james version). like martin luther's german translation of the bible, which was one of the fundaments of "high german" (hochdeutsch), also the KJV bible was one of the pillars on which the "king's english" was founded as a normative and grammatical language. let us look at the following verse in book samuel one: 1 Samuel 10:5 After that thou shalt come to the hill of God, where is the garrison of the Philistines: and it shall come to pass, when thou art come thither to the city, that thou shalt meet a company of prophets coming down from the high place with a psaltery, and a tabret, and a pipe, and a harp, before them; and they shall prophesy: what the KJV bible tranlates as harp, appears in the hebrew bible as KINNOR. what the KJV translates as psaltery, appears in the hebrew as NEVEL (which in your fig. 399 is transliterated as NEBEL, in hebrew the v is a soft b, so this is the same word). at the time the book samuel one was written, there was no known way of the greek language entering the bible, so the NEVEL and KINNOR are of hebrew origin. this is not true for others parts of the bible. in the book of daniel (which by the way is originally written partly in hebrew, and partly in arameic, which is close to hebrew, but a different language), we find: Daniel 3:5 That at what time ye hear the sound of the cornet, flute, harp, sackbut, psaltery, dulcimer, and all kinds of musick, ye fall down and worship the golden image that Nebuchadnezzar the king hath set up: the corresponding arameic for harp is KITROS (the archaic of the modern guitar), the psaltery is called PSANTERIN, the similarity in the sound of the two words is obvious, and both stem from the greek, the dulcimer is in the arameic referred to as SUMPONIA (the obvious derivative of the greek synphonia). so at the time of the book of daniel, greek influence was obvious. what all this leads to is the following: KINNOR and NEVEL (NEBEL) are of early biblical hebrew origin. the NABLA is the greek derivative of the hebrew NEVEL. the name of the "nabla" we use in mathematics is therefore about 3000-4000 years old. i hope this will add some color to your lectures on fields... dani ----- Date: Fri, 12 Sep 1997 13:25:00 0200 From: Dani Censor Subject: NABLA/NEVEL To: ACADEMIA@techunix.technion.ac.il david schaps asked me to post for him his reply to my querry. dani ... From: "dschaps@mail.biu.ac.il" 12-SEP-1997 11:39:30.93 To: CENSOR@BGUEE.EE.BGU.AC.IL <= dschaps@mail.biu.ac.il (* SMTP.MAIL ASCII) CC: Subj: Re: NABLA=NEVEL? Dear Dani, Yes, "nabla" is the Greek equivalent of "nebel". It appears both in the Septuagint as a translation of "nebel" and in classical Greek sources (first in Sophocles) -- so it was not from the Bible that the Greeks first learned the word. Many Semitic words can be found in ancient Greek, mostly from Phoenician (which was so close to Hebrew that they may have been mutually intelligible) but some from Aramaic or other languages. -- David M. Schaps Department of Classical Studies Bar Ilan University Ramat Gan, Israel FAX: -972-3-534-7601 ----- Date: Sun, 14 Sep 1997 16:58:38 +0200 From: S I Ben-Abraham To: ACADEMIA@techunix.technion.ac.il Hi, OF COURSE! This happens to be a trivial case. The root is NBL and the symbol is called that because if its shape. I, for one, take it quite badly when people around me (Hebrew speakers) call it "del" to demonstrate that they have been americanized, Who and when invented the term I don't know at the moment (Sir william Hamilton????) but I promise to try to find out [I happen to be good at this sort of thing]. Tioll then cheers SIBA ----- Date: Sun, 14 Sep 1997 17:05:25 +0200 From: S I Ben-Abraham To: ACADEMIA@techunix.technion.ac.il I venture to add that was borrowed to Greek via its Aramaic definite form (analogous to . SIBA ----- Date: Mon, 15 Sep 1997 00:48:18 +0300 From: Eddy Zemach To: ACADEMIA@techunix.technion.ac.il You may like to know how Nabla is used in logic: "Nabla(p)" means "'p' has no truthvalue". "Delta(p)" means "'p' is true or else false". The symbols were intorduced by Gareth Evans. ----- Date: Mon, 15 Sep 1997 02:35:00 +0300 From: Avinoam Mann To: ACADEMIA@techunix.technion.ac.il There is a web page for history of mathematical terms and symbols, at http://members.aol.com/jeff570/mathword.html, maintained by Jeff Miller. (for symbols, change "word" to "sym"). According to it, the symbol was introduced by William Rowan Hamilton in 1853, in his book _Lectures on Quaternions_. About the word nabla he says: Nabla wa suggested humourously by James Clerk Maxwell, according to one source.According to a post in sci.math by Noam D. Elkies, the term was coined by Tulio Levi-Civita. A nabla is the name of an Egyptian harp. Cajori (vol.2, p.135) says Heavyside called the symbol a nabla. End of quotation. The reference Cajori is to the book by F.Cajori, _History of Mathematical Notations_. Avinoam Mann ------------------------------------------------------------------------ Date: Wed, 21 Jan 1998 14:43:26 -0700 (MST) From: Michele Benzi Dear Dr. Neumaier, I have read the replies to your query on Nabla, and I noticed that the information I have is somewhat different from that provided by others. For your convenience, I am going to reproduce here part of the content of note 27 to chapter four of "A history of vector analysis" by Michael J. Crowe (originally published by University of Notre Dame Press in 1967, and republished by Dover in 1985). The note, on page 146 of the Dover edition, reads: Nabla was the name suggested to [P. G.] Tait by Robertson Smith because of the similarity of the symbol to an Assyrian harp. See [1; 143]. Maxwell used the word only once in his published writings, and that was in a poem, "To the Chief Musician upon Nabla, A Tyndallic Ode." The "Chief Musician upon Nabla" was Tait. The poem was published in Nature and is given in [1; 171--174]. (...) (the note continues with some information about the origin of the terms "del" and "atled"). Reference [1] above is Cargil Gilston Knott, "Life and Scientific Work of Peter Guthrie Tait", Cambridge, England, 1911. As you see, the information offered by Crowe (based on Knott) is somewhat at variance with the one provided by other people who have answered your query. Do you think you can get this straight?? Best regards, --Michele Benzi --------------------------------------------------------------- Michele Benzi Scientific Computing Group (CIC-19) phone : (505) 665-4778 MS B256 fax : (505) 667-1126 Los Alamos National Laboratory e-mail: benzi@lanl.gov Los Alamos, NM 87545 -- USA http://www.c3.lanl.gov/~benzi/benzi.html --------------------------------------------------------------- -------- From: "Sergey P. Shary" Date: Thu, 22 Jan 1998 19:34:21 +0600 Arnold, Nabla is a harp in ancient Greek. Sergey -------- Date: Thu, 22 Jan 1998 20:58:01 -0500 From: elkies@MATH.HARVARD.EDU (Noam Elkies) Dear A.Neumaier, I cannot reconstruct the source of the claim about Levi-Civita and nabla. Since I knew the Hebrew cognate NEVEL of "nabla", and knew that Levi-Civita was Jewish, it made sense when I first ran across it. But I see in the correspondence quoted at > http://solon.cma.univie.ac.at/~neum/contrib/nabla.txt that "nabla" was also borrowed from Aramaic into ancient Greek -- so, while it still makes sense for L-C to be the coiner, the case is not as strong. The circumstances of the use of the word by Tait suggest that the word was already familiar in the mathematical community by 1911, but give no clue as to the originator's identity. The only thing I can add to the discussion at this point is the tangential remark that PSANTERIN (the Aramaic rendition of "psalterion" [or whatever the correct Greek termination is] in the Book of Daniel) gave rise to the modern Hebrew PSANTER for "piano". NB a piano has the same harp shape when viewed from above. Sincerely, --Noam D. Elkies -------- From: "Hans Oser" Date: Tue, 27 Jan 1998 10:09:27 -0500 Lieber Herr Dr. Neumaier, Meine erste Begegnung mit dem Nabla Operator geht zurueck in die 50er Jahre zu den ersten Vorlesungen ueber Theoretische Physik. In der neunten Auflage des Lehrbuches der Theoretischen Physik von Georg Joos (Akademische Verlagsgesellschaft Geest & Portig, Leipzig 1956, Seite 26) im Paragraphen 10 ueber Vektorrechnung erwaehnt er den symbolischen Operator als nach dem ihm aehnlichen phoenizischen Saiteninstrument "NABLA". Nicht viel mehr zu sagen, aber vieleicht hilft's in Ihren weiteren Forschungen ueber den Ursprung dieses Symbols, das sich in den angelsaechsischen Laendern als Del-Operator eingebuergt hat. Dr. Hans J. Oser Potomac, Maryland/USA -------- Date: Wed, 11 Feb 1998 10:54:36 -0700 (MST) From: Jon Rokne Dear Dr. Neumaier, I noticed your question on nabla. Although you might already have many answers here is another contribution. At the location http://forum.swarthmore.edu/epigone/sci.math/twendzhermkhil/5thpbf$5ug@bell. maths.tcd.ie they have this explanation. Subject: Re: How does one read \nabla Author: Michael Carley <_m_j_c_a_r_l_e_y_@maths.tcd.ie> Organization: Dept. of Maths, Trinity College, Dublin, Ireland. Date: 21 Aug 1997 17:11:27 +0100 dorfsmay@cuug.ab.ca writes: >And out of interest, where does this symbol come from ? The symbol \nabla comes from Syriac and was introduced by William Rowan Hamilton, mathematician, polymath, bridge defacer and all round good egg. - -- ``Permitt not your schollars to ramble abroad, especially lett them not soe much as peepe into a tavern or tipleing house'' (Provost Loftus). Michael Carley, Mech. Eng., TCD, IRELAND. m.carley@leoleo.mme.tcd.ie Home page Best greetings Jon Rokne -------- my home page (http://solon.cma.univie.ac.at/~neum)

Arnold Neumaier (neum@cma.univie.ac.at)