Variable quantities are those that continually increase or decrease; and, on the contrary, constant quantities are those that remain the same while the others change.

— Marquis de l'Hôpital, Analysis of the Infinitely Small (1696), definition I

The unknown is a letter that stands still: it hides a value, and the value does not change. The variable is a letter that moves. In $y=x^2-4x+3$, nobody asks “what is $x$?”. The question is a different one: “when $x$ changes, what happens to $y$?”.

From equation to curve

The passage from one idea to the other happened in the seventeenth century. When Descartes and Fermat, around 1637, linked equations to curves, an equation with two letters, such as $y=x^2$, stopped having one answer and came to have infinitely many: all the pairs $(x,y)$ that make up the parabola. The $x$ was no longer sought; it was travelled along. For each position, the equation says where $y$ is.

Quantities that flow

Newton gave this idea a physical image. For him a variable was a quantity that flows with time, like the position of a moving body, and he called it a fluent; the speed at which it flows was the fluxion. Leibniz spoke of quantitas variabilis, and the first calculus textbook, l’Hôpital’s of 1696, opens with the definition quoted at the top of this page: variables increase or decrease, constants stay still.

The word carries an image of motion. That is why we say “as $x$ tends to $0$”, “$y$ grows”, “the function goes up”, as if the letters walked.

One letter depends on the other

With two variables comes the idea of dependence. In 1755, in the preface to the Foundations of Differential Calculus, Euler wrote that when some quantities depend on others in such a way that, when the latter change, the former change too, the former are called functions of the latter. In $y=x^2-4x+3$:

  • $x$ is the independent variable: we choose its value freely;
  • $y$ is the dependent variable: its value is imposed by the rule.

The choice is not written in the letters. In $A=\pi r^2$ we usually think of the radius as independent, but if the question is “what radius gives an area of $10$ m²?”, we swap the roles: $r=\sqrt{A/\pi}$.

Weierstrass’s variable that stands still

The image of motion is useful, but it brought problems. What does “$h$ tends to zero” mean if $h$ is just a letter on paper? In the nineteenth century Karl Weierstrass and others removed time from the definition. A variable became simply a letter that can be replaced by any element of a set, its domain. Nothing moves: what exists is the collection of all possible replacements. This is the reading behind the limit definition of Cauchy and Weierstrass, with its $\varepsilon$ and $\delta$.

Today the two images live side by side. The teacher who says “as $x$ keeps growing” is using Newton’s variable; the definition in the book uses Weierstrass’s. The notation is the same, and the reader needs to know how to pass from one to the other.

Exercise: find the rule from the table

A table records: $x=0\to y=1$, $x=1\to y=3$, $x=2\to y=5$, $x=3\to y=7$. When $x$ increases by $1$, $y$ increases by $2$, and at $x=0$ the value of $y$ is $1$. The rule is $y=2x+1$. Note that a finite table never proves the rule: another rule could agree at these four points and differ at $x=4$. The table suggests; the formula asserts.

Exercise: who depends on whom?

A car travels at $80$ km/h. The distance covered is $d=80t$. If the question is “how far does it go in $2.5$ h?”, $t$ is the independent variable and $d=200$ km. If the question is “how long does it take to cover $300$ km?”, the independent variable becomes $d$, and $t=\frac{d}{80}=3.75$ h. There is one formula; the dependence is chosen by the question.

Chapter index · ← Letters that generalise: from 3 + 5 = 5 + 3 to a + b = b + a · Parameters: letters that stay still while others move →