# Origin of delta

Why does delta mean change? What is the origin of delta? I understand that upper-cased delta is used in this way and that delta is the fourth letter of the Greek alphabet. I also read that delta is the first letter of a Greek word that means "difference." The question remains: why does delta now symbolize change?

• If it's the first letter of a word that means "difference", then that would seem to explain it. $\qquad$ – Michael Hardy Feb 19 '16 at 4:18
• @MichaelHardy As "difference" (Eng., Fr.) and "διαφορά" (Greek) are words that mean difference, and all have the first letter 'd', err $\delta$, ... :) @ OP, what more needs to be said? The change in something is the difference between two of its states. Why uppercase $\Delta$ rather than $\delta$? There's no accounting for taste, and perhaps It's easier to write if you don't write Greek. – BrianO Feb 19 '16 at 4:22
• Perhaps one should add that although that would seem to explain it, that doesn't prove that it's historically what happened. Sometimes the actual history is a colorful story that doesn't seem like the immediately plausible explanations. $\qquad$ – Michael Hardy Feb 19 '16 at 4:26
• That is true. While this is a plausible story, some evidence would be nice, to elevate it from folk etymology. It can be hard to track such things down. In this case, I'd probably start with accounts of Newton's &/or Leibniz's work. – BrianO Feb 19 '16 at 4:32
• Cauchy used $\delta$ for différence, or "difference". – Henricus V. Feb 19 '16 at 4:56

The origin of the symbol $dx$ for the differential is with Leibniz.

See:

• Gottfried Wilhelm Leibniz, The Early Mathematical Manuscripts, (J.M. Child editor, 1920, Dover ed), Elementa calculi novi pro differentiis et summis, tangentibus et quadraturis, Ms. no date (ca.1680) page 137:

Let $CC$ be a line, of which the axis is $AB$, and let $BC$ be ordinates perpendicular to this axis, these being called $y$, and let $AB$ be the abscissae cut off along the axis, these being called $x$. [see figure]

Then $CD$, the differences [latin: differentia] of the abscissae, will be called $dx$.

The first Leibniz's publication using it is Nova methodus pro maximis et minimis (1684).

Compare with a "mature" explanation (1710):

Hic $dx$ significat elementum, id est incrementum vel decrementum (momentaneum) ipsius quantitatis $x$ (continue) crescentis. Vocatur et differentia [...].

According to the historian Florian Cajori, History of Mathematical Notations (1928–1929 - Dover ed.), II vol., page 222:

§593. Partial derivatives appear in the writings of Newton, Leibniz, and the Bernoullis, but as a rule without any special symbolism. To be sure, in 1694, Leibniz, in a letter to De l'Hospital, wrote "$\delta m$" for the partial derivative [...].

in 1776 Euler" uses $\dfrac {\delta^{\lambda}} p V$ to indicate the $\lambda$th derivative, partial with respect to the variable $p$.

And page 225:

The use of the rounded letter $\delta$ in the notation for partial differentiation occurs again in 1786 in an article by A.M. Legendre.

For Cauchy:

§610. More fully developed is [Cauchy] notation of 1844. Taking $s$ as a function of many variables, $x, y, z, \ldots, u, v, w$, Cauchy designates:

Partial increments $\Delta_1 s, \Delta_{11}s,\ldots$.

Différentielles partielles $d_1 s, d_{11}s, \ldots$.

For finite difference, see:

§640. In place of Leibniz' small Latin letter $d$, [Euler] used the corresponding Greek capital letter $\Delta$ which Johann Bernoulli had used previously for differential coefficient