# Who first defined the "equal-delta" or "delta over equal" ($\triangleq$) symbol?

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:

• 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. Sep 5, 2016 at 1:45
• Mathsym jeff560.tripod.com/mathsym.html has only $=_{\mathrm{Def}}$ (from 1894) but not the Delta version. Sep 5, 2016 at 13:44
• @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". Mar 19, 2018 at 19:42
• @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!) Jun 28, 2018 at 23:43
• The symmetric way understand it is to read it as "this equality holds by definition". From that perspective there is no "definiens" or "definiendum". Jan 7, 2021 at 16:20

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 $$\overset=}{=}$$ 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.
• After I posted this, I noticed Gerald Edgar's comment which points to an 1894 use of $=_{\rm Def}$ to mean equal by definition. Aug 29, 2021 at 20:07