Differential equations: the law in an anagram, ∂ and ∇

In 1676 the Englishman Robert Hooke published, at the end of a book on helioscopes, instruments for observing the Sun, a list of discoveries he did not yet want to reveal. One of them read:

It was an anagram: the letters of a Latin sentence, shuffled and put in alphabetical order. It was a way of recording priority for a discovery without handing it to rivals. Two years later, in 1678, Hooke revealed the sentence: ut tensio, sic vis, “as the extension, so the force”. The more a spring is stretched, the harder it pulls back. Today we write it in four signs:

The anagram is the exact opposite of a notation: a script made to hide. The formula is made to show. The same law went from one extreme to the other in little more than a century.

A question whose answer is a function

Attach a body of mass $m$ to the spring, pull and let go. Where will the body be at each instant $t$? Combining Hooke’s law with Newton’s law, $F=m\cdot a$, and writing acceleration as the second derivative of position, we get

This is a differential equation. It looks strange at first, because the unknown is not a number but a function: the rule $x(t)$ that gives the position at each instant. The old texts on this site posed this kind of question as a riddle: “there is a function $h$ whose second derivative, added to the function itself, gives zero; what function is it?”. The answer is the sine, or the cosine, because differentiating either twice gives it back with its sign changed:

For the spring, the solution is $x(t)=A\cos(\omega t)+B\sin(\omega t)$, with $\omega=\sqrt{k/m}$. Leonhard Euler, in the 1740s, showed how to solve the whole family of these equations at once, swapping the question about functions for an ordinary algebraic equation. It was only possible because Euler’s $f(x)$ and Leibniz’s $\frac{d}{dx}$ already gave names to functions and their derivatives: an equation between functions needs a script in which functions are objects.

Units are signs too

How do we know that $\omega=\sqrt{k/m}$ is right, and not $\sqrt{m/k}$? By the units. The constant $k$ measures force per length, $\frac{\mathrm N}{\mathrm m}=\frac{\mathrm{kg}}{\mathrm s^2}$. Divided by a mass, it gives $\frac1{\mathrm s^2}$, and the root gives $\frac1{\mathrm s}$: exactly the unit of something that is multiplied by time inside a cosine, since the argument of a cosine must be a pure number. With $\sqrt{m/k}$ we would get seconds, and $\cos(\text{seconds}\cdot\text{seconds})$ makes no sense.

This check, dimensional analysis, was systematised by Joseph Fourier in 1822, and James Clerk Maxwell popularised the bracket script: $[M]$ for mass, $[L]$ for length, $[T]$ for time. Force becomes $[F]=[M][L][T]^{-2}$. The site’s old texts liked to point out that many physics entrance-exam questions can be solved by this route alone. From a semiotic point of view, units are a second layer of signs stuck to the numbers, a layer that is not for calculating but for checking: a formula whose units do not match is wrong before any calculation.

∂ and the harp

When a quantity depends on several others at once, like the volume of a gas, which depends on temperature $T$ and pressure $P$, Leibniz’s derivative needs a warning: “derivative with respect to $T$, holding $P$ fixed”. In 1786 the Frenchman Adrien-Marie Legendre proposed a rounded $d$, $\partial$, for these partial derivatives, and soon gave it up. In 1841 the German Carl Jacobi revived it, and this time it stuck. With it, the total change in volume is written

a formula the old texts on this site called the “canon of heat”. It says, in its very shape, that the total change is the sum of the changes in each direction.

The most curious sign in this family is $\nabla$, an upside-down $\Delta$, which Hamilton used in his work on quaternions. It gathers the partial derivatives in every direction into a single sign, and with it we write Laplace’s equation, $\nabla^2\Psi=0$, and the four Maxwell equations we saw in the previous text. The name nabla was suggested to Peter Guthrie Tait by a friend who studied the Bible: it is the Greek name of an ancient triangular harp. Maxwell, joking, called the operator atled, “delta” backwards.

Sources: Florian Cajori, A History of Mathematical Notations · Robert Hooke, MacTutor · Joseph Fourier, MacTutor · Earliest Uses of Symbols of Calculus, Jeff Miller.

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