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The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \triangleq m$ (generally) means: "$t$ is defined to be $m$" or "$t$ is equal by definition to $m$" (often under certain conditions).

In a similar sense, some use $:=$ or $=:$ (see for instance Symbols based on equality). Yet, this Delta variant is more important to me.

The SE. Maths post What is meant by the delta equivalent sign? proposes a slight distinction (not crystal-clear to me) between the above similar senses:

Sometimes it is used with the slightly different meaning of "equal by definition", to underline the difference w.r.t. "$:=$ " which is the definition itself. i.e.

$$ a:=3;\\ 5+a \triangleq 5 + 3 = 8 $$

I always took for granted that the $\Delta$ stood for letter "D", i.e. for the initial of "definition". Indeed, one sometimes finds $\overset{\mathrm{def}}{=}$ too. In German apparently, one also uses $≙$ (Entspricht-Zeichen, with Unicode U+2259).

Based on these prior hints, my questions are as follows:

  • Who introduced this dual symbol first in science, and where (which source)?
  • What motivated the Greek $\Delta$ notation? The abbreviation of some word, a symbol (why not a latin notation)?
  • Why not merge the lower bar of the Delta with the upper bar of the equal sign, to save some ink, and create a lighter symbol?

References: the symbol itself was already discussed in StackExchange:

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    $\begingroup$ This notation seems very strange, as it suggests symmetry between the definiens and the definiendum. I am more used to := or =:, with the colon on the side of the definiendum. $\endgroup$
    – Margaret Friedland
    Sep 5, 2016 at 1:45
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    $\begingroup$ Mathsym jeff560.tripod.com/mathsym.html has only $=_{\mathrm{Def}}$ (from 1894) but not the Delta version. $\endgroup$
    – Gerald Edgar
    Sep 5, 2016 at 13:44
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    $\begingroup$ @MargaretFriedland The asymmetry is given by what is before the symbol and what is after the symbol. Thus $A \triangleq B$ always means "A is defined to be B". $\endgroup$
    – juanrga
    Mar 19, 2018 at 19:42
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    $\begingroup$ @MargaretFriedland, I am sympathetic to your objection, and, in fact, would go so far as to object somewhat to the $:=$ usage, since it lends itself to typos so easily. Rather, I'd strongly prefer that the context clarify any asymmetry (such as a definition or notational abbreviation) in a mathematical equality. (Mathematicians seem unlikely to take these asymmetries as seriously as programmers may, so I don't necessarily trust them!) $\endgroup$
    – paul garrett
    Jun 28, 2018 at 23:43
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    $\begingroup$ The symmetric way understand it is to read it as "this equality holds by definition". From that perspective there is no "definiens" or "definiendum". $\endgroup$
    – Michael Bächtold
    Jan 7, 2021 at 16:20

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This is really a comment rather than a full answer.

Florian Cajori's encyclopedic tome A History of Mathematical Notations does not seem to discuss this symbol specifically; the closest thing is the following remark in Section 269:

L. Gustave du Pasquier (Comptes Rendus du Congrès International des Mathématicians (Strasbourg, 22–30 Septembre 1920), p. 164) in discussing general complex numbers employs the sign of double equality $\overset{\displaystyle =}{=}$ to signify “equal by definition.”

I checked MathSciNet, and the review of the 1953 paper "Zwei Klassen von Flächen, deren Bestimmung von einem Integral der Telegraphengleichung abhängt" by Hans Jonas (MR0058259) uses the notation $=_{\text{def}}$ (but note that Jonas's paper itself does not use that notation). This seems to be the earliest occurrence of $=_{\text{def}}$ or $\overset{\text{def}}=$ in MathSciNet. But I do not have access to a print version to verify that the notation was actually used back in the 1953 edition of Mathematical Reviews.

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  • $\begingroup$ I have been waiting for a long time, let me validate yours, until... $\endgroup$
    – Laurent Duval
    Aug 29, 2021 at 16:47
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    $\begingroup$ After I posted this, I noticed Gerald Edgar's comment which points to an 1894 use of $=_{\rm Def}$ to mean equal by definition. $\endgroup$
    – Timothy Chow
    Aug 29, 2021 at 20:07

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