Algebra

Lagrange's legacy

Lagrange is the lofty pyramid of the mathematical sciences. — Napoleon Bonaparte, according to his contemporaries Lagrange left no anecdotes of precocious genius like Pascal, nor a body of work the size of Euler’s. He left something else: a way of doing mathematics. Where others drew, he calculated; where others found a trick for each problem, he looked for the rule that explained all the tricks. Almost everything he did is still in the textbooks under his name: the Euler–Lagrange equation, the Lagrangian, the Lagrange points, Lagrange’s theorem, the Lagrange remainder, Lagrange multipliers.

October 14, 2026 Read

Image credits for Lagrange

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Joseph-Louis Lagrange, engraving by an unknown artist: source page · public domain. Medallion with Lagrange’s profile, bronze by Pierre-Jean David d’Angers, Musées d’Angers: source page · CC0. Title page of the Méchanique analitique, Paris, 1788: source page · public domain. Statue of Lagrange in Turin, photo by Twice25, 2009: source page · CC BY-SA 3.0. Blog figures The cover, the diagram of the five Lagrange points and the graph of curves of equal area with the constraint line are original illustrations, with versions in each of the site’s languages. The diagram of the points is not to scale: L1 and L2 are in reality a hundred times closer to the Earth than the Sun is.

October 14, 2026 Read

References and credits

Further reading Usiskin, Zalman. Conceptions of school algebra and uses of variables. In: Coxford, A. F.; Shulte, A. P. (eds.). The Ideas of Algebra, K–12. Reston: NCTM, 1988, pp. 8–19. Küchemann, Dietmar. Algebra. In: Hart, K. M. (ed.). Children’s Understanding of Mathematics: 11–16. London: John Murray, 1981, pp. 102–119. Clement, John; Lochhead, Jack; Monk, George S. Translation difficulties in learning mathematics. The American Mathematical Monthly, 88(4), 1981, pp. 286–290. MacGregor, Mollie; Stacey, Kaye. Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33, 1997, pp. 1–19. Philipp, Randolph A. The many uses of algebraic variables. The Mathematics Teacher, 85(7), 1992, pp. 557–561. Kieran, Carolyn. The learning and teaching of school algebra. In: Grouws, D. A. (ed.). Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan, 1992, pp. 390–419. Kleiner, Israel. Evolution of the function concept: a brief survey. The College Mathematics Journal, 20(4), 1989, pp. 282–300. Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. On the cossic signs, the letters of Viète and Descartes, $f(x)$ and Herschel’s $\sin^{-1}$. Heath, Thomas L. Diophantus of Alexandria: A Study in the History of Greek Algebra. Cambridge: Cambridge University Press, 1910. Viète, François. In artem analyticem isagoge. Tours: Jamet Mettayer, 1591. l’Hôpital, Guillaume de. Analyse des infiniment petits, pour l’intelligence des lignes courbes. Paris: Imprimerie Royale, 1696. The older texts on functions on this site supplied the idea of the function as a sentence with a hole in it, used in chapter 1 and taken up again here.

October 15, 2026 Read

How to read a letter: a guide and a glossary

This chapter showed that a letter can do at least five different jobs and that the writing hardly ever says which. The guide below gathers the questions that help to find out. Five questions for a letter Does it hide a value or stand for any value? If it hides one, it is an unknown, and the task is to find it. If it stands for any, the sentence is a general law or an identity. Does it move during the problem? If so, it is a variable, and it is worth asking what it depends on and what depends on it. Does it stay still while the others move? Then it is a parameter or a constant. It is worth asking: fixed for ever, like $\pi$, or only in this problem, like the $a$ in $ax^2$? Is it tied to an operator? If it appears in a $\Sigma$, an $\int$, a $\forall$ or a $\mapsto$, it is a bound letter, and it can be swapped for another without changing anything. Which word in the text announces the role? “Find”, “for every”, “let”, “when”, “given”. If none appears, the conventions of the subject decide, and it is good to make them explicit. A full reading of y = ax² + bx + c $y$ and $x$: variables. $x$ is the independent one, $y$ the dependent one. Together they run through the points of a parabola. $a$, $b$ and $c$: parameters. Fixed during the problem, they choose which parabola. The $a$ must be non-zero, or the curve becomes a straight line. Change the question and the roles change. In “where does the parabola cross the $x$-axis?”, we set $y=0$, and $x$ becomes an unknown. In “which parabola passes through $(1,2)$, $(2,3)$ and $(3,6)$?”, the parameters become the unknowns, and $x$ and $y$ become data: $a+b+c=2$, $4a+2b+c=3$, $9a+3b+c=6$, which give $a=1$, $b=-2$, $c=3$. No letter is a dummy. They all matter: swapping $a$ for $k$ requires swapping it throughout the text. One short line, three different roles, and each question rearranges everything.

October 15, 2026 Read

When the letter misleads: students, professors and apples

In 1981 three researchers at the University of Massachusetts, John Clement, Jack Lochhead and George Monk, set first-year engineering students a simple problem: Write an equation using the variables $S$ and $P$ to represent the following statement: “There are six times as many students as professors at this university.” Use $S$ for the number of students and $P$ for the number of professors. About 37% got it wrong. And almost all of those who got it wrong wrote the same thing:

October 15, 2026 Read

Dummy letters: the k of the sum and the t of the integral

These two expressions have the same value: $$\sum_{k=1}^{4}k^2=1+4+9+16=30,\qquad\sum_{j=1}^{4}j^2=1+4+9+16=30.$$Swapping $k$ for $j$ changed nothing. The letter is there only to mark where the calculation passes; once the sum is done, it disappears. Letters like these are called dummy, or bound. They are the opposite of unknowns: an unknown keeps a value that matters; a dummy letter keeps no value at all once the operation is over. Free and bound Now compare:

October 15, 2026 Read

f, f(x) and x ↦ x²: three ways to name a function

Chapter 1 told how Euler, in 1734, invented $f(x)$ and how the function became a sentence with a hole in it (Euler and functions). This text looks at the ways of naming a function, and at the confusions each one brings. The function and its value Compare two sentences: “the function $f$ is increasing”; “$f(x)$ is positive for $x>0$”. In the first, $f$ is the whole function, the rule, the object that can be differentiated, composed, drawn. In the second, $f(x)$ is a number: the value the function gives when the input is $x$. The difference is the same as between “the calculator” and “the number on its display”.

October 15, 2026 Read

Parameters: letters that stay still while others move

In the equation $$y=ax^2+bx+c$$there are five letters, and they are not on the same level. The $x$ and the $y$ vary: they run through the points of a parabola. The $a$, the $b$ and the $c$ stay still while this happens, but they can be chosen at will beforehand. Each choice gives a different parabola. Letters like these are called parameters, from the Greek pará, beside, and métron, measure: what stands beside the measure, regulating it.

October 15, 2026 Read

The variable: the letter that moves

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 $y$?”.

October 15, 2026 Read

Letters that generalise: from 3 + 5 = 5 + 3 to a + b = b + a

Add $3+5$ and $5+3$: you get $8$ both times. Try $12+7$ and $7+12$, $100+1$ and $1+100$. It always works. But how many examples do you need to be sure? A thousand? A million? No number of examples is enough, because there are infinitely many pairs of numbers. The sentence $$a+b=b+a$$solves the problem in another way. It speaks of no particular pair: it speaks of all of them at once. The letters $a$ and $b$ here do not hide a value, as the unknown does; they stand for any value. This is what Usiskin called generalised arithmetic.

October 15, 2026 Read

The unknown: a place for what is being sought

An unknown is, literally, something in-cognita, not known. But it is not just anything: it is a definite number that we do not yet know. The equation $x+3=7$ does not speak of all numbers; it speaks of one, and the task is to find out which. Names for what is not known For a long time the unknown had the name of a word, not of a letter. Around the year 250, Diophantus of Alexandria, in the Arithmetica, called the unknown arithmós, “the number”, and abbreviated it with a sign like a final sigma, $\varsigma$. He also had abbreviations for the square of the unknown, $\Delta^{\Upsilon}$ (dýnamis, power), and for its cube, $K^{\Upsilon}$ (kýbos). In Baghdad, around 820, al-Khwarizmi called it shay’, thing, and its square māl, wealth. He wrote everything out in words, without any symbol. In the fifteenth and sixteenth centuries the Italians translated shay’ as cosa, and the German algebraists who copied the Italians were nicknamed cossists. In 1544 Michael Stifel still used different signs for the thing, its square and its cube: The “cossic” signs of Michael Stifel’s Complete Arithmetic (1544): each power of the unknown had its own symbol, with no visible relation between them.Michael Stifel · Wikimedia Commons · public domain In these signs, the square and the cube of the thing do not look like the thing. It is a sterile notation, in the sense of this book: to know that the cube times the square gives the fifth power, one had to memorise a table. When Descartes wrote $x$, $x^2$, $x^3$, the rule $x^3\cdot x^2=x^5$ became visible in the exponents. Chapter 1 tells this passage in more detail, in the text from rhetoric to symbol.

October 15, 2026 Read

One letter, five roles

Look at these five lines: $$x+3=7,\qquad A=b\cdot h,\qquad a+b=b+a,\qquad y=kx,\qquad \sin^2x+\cos^2x=1.$$All of them use letters, and all of them look like algebra. But in none of them do the letters do the same thing. In $x+3=7$, the $x$ is a fixed, unknown number. Only $4$ will do. It is an unknown. In $A=b\cdot h$, the letters hide nothing: they say how three quantities are related. It is a formula, and the letters are quantities that vary together. In $a+b=b+a$, the letters are arbitrary numbers. The sentence holds for every pair. The letter generalises. In $y=kx$, $x$ and $y$ move, and $k$ stays still. The $k$ is a parameter: fixed in each problem, but arbitrary. In $\sin^2x+\cos^2x=1$, the $x$ is also arbitrary, but the sentence is not an equation to solve: it is an identity, true for every $x$. Usiskin’s conceptions In 1988 the American educator Zalman Usiskin, in an article that became a classic, proposed that school algebra mixes at least four conceptions, each giving the letter a different role:

October 15, 2026 Read