Origin of delta

Question

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?

Answer 1

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 [...].

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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