Anyone who studies physics and/or mathematics has often encountered the following conundrum:

How do you distinguish 18th-century French mathematicians with surnames beginning with an “L”? (I call these E.C.F.M.W.S.B.W.A.L.’s)

For example, you might recall that an E.C.F.M.W.S.B.W.A.L. invented the calculus of variations, some time around 1760. Was it Legendre? Lagrange? Laplace? Or maybe you remember that an E.C.F.M.W.S.B.W.A.L. was the father of probability theory, and worked on the Buffon needle problem. Was that Laplace? Legendre? Lagrange?

So as a public service, I’ve sorted this out for you. I henceforth talk about these three great mathematicians, and hope to distinguish them in your mind.

**Lagrange: perhaps the best mathematician of the 1700’s.**

Lagrange is the oldest of the E.C.F.M.W.S.B.W.A.L.’s, born in 1736. Some call him the greatest mathematician of the century, although I might give that title to Euler. In any case, he’s responsible for a host of discoveries: he pretty much invented an entire branch of mathematics, the calculus of variations; he used this tool to reformulate classical mechanics (think L = T – V) making it suitable for non-Cartesian coordinates, such as polar; he invented Lagrange multipliers, an elegant way to deal with constraints in differential equations; and he introduced the f(x),f'(x),f”(x)…notation for derivatives.

His greatest work was* Mécanique analytique; *all of the above achievements are found in this book. Hamilton described the work as a “a scientific poem,” for its elegance is astounding.

Lagrange was rigorous and abstract: he bragged that the *Mécanique analytique* did not have a single diagram. To Lagrange, math was an art; the aesthetics of a theory took precedence over utility.

**Laplace: the “applied” mathematician**

Laplace was seven years younger than Lagrange, born in 1749. He also is associated with classical mechanics, but unlike Lagrange, he did not reformulate the field *per se*. Rather, he took Newtonian mechanics to its “apex” with his work *Mécanique céleste. *This work is brilliant, but it’s also clunky and difficult. It analyzes the orbits of all known bodies in the solar system, and concludes that there is no need of God to keep the whole mess going. In fact, Napoleon supposedly asked why Laplace didn’t mention God in the *Mécanique céleste.* He reportedly said “I have no need for that hypothe*sis.”*

Laplace didn’t place as much emphasis on “beauty” in mathematics. To him, math was just a tool. Not surprisingly, he contributed to the “applied” field of probability theory; in fact, he’s arguably the founder of probability theory as we know it today.

**Legendre: the elliptic integral guy**

Although highly regarded in his day, Legendre (b. 1752) is really a tier below the first two guys. Basically, he worked out how to do some elliptic integrals, and he introduced the Legendre transformation, which is used in many branches of physics. For example, you can go back and forth between the Hamilton and Lagrange approaches of classical mechanics by means of Legendre transformation. Also, such transformations are ubiquitous in thermodynamics (think U → H → A → G).

Legendre is also know for the portrait debacle. Only a single known image of Legendre exists, and that image is not flattering:

Every other supposed portrait of Legendre is actually the picture of some obscure politician, because of a mistake which has propagated forward for 200 years.

In summary:

**Lagrange**: the beauty of math; reformulated mechanics in the *Mécanique analytique*

**Laplace**: math as a tool; Newtonian mechanics reaches its zenith in *Mécanique céleste*; probability theory

**Legendre**: the creepy looking elliptic integral guy

Note: I have not mentioned Lavoisier (b. 1743) because he was a chemist. But if you really need him:

**Lavoisier**: a chemist who was guillotined in the French Revolution.

[Note added Dec. 4, 2014] I could have included L’Hopital (French, died 1704) but all he did was write a textbook. Laguerre was French, but he was born in 1834; Lebesgue was French, but he was born in 1875.

on November 24, 2014 at 4:42 PM |drakodocWhy “E.C.F.M.W.S.B.W.A.L.’s?” I might be missing an obvious inside joke?

on November 24, 2014 at 5:50 PMMatthew Ravelook at the previous sentence… It stands for “Eighteenth century French mathematicians whose surnames begin with an L”

on November 24, 2014 at 7:57 PM |drakodocDuh…

on November 25, 2014 at 6:17 AM |john zandeI want the backstory to that sketch of Legendre. One surely couldn’t be happy with such a representation…

on November 25, 2014 at 2:17 PMMatthew RaveThere’s a discussion on Wikipedia, which is how I found out about it. Someone 200 years ago found a picture of the politician “Louis Legendre” and put it in a book, saying it was the mathematician. No one noticed until 2005, at which point, no other images of him remained…

on November 25, 2014 at 2:24 PMjohn zandeOh, I read that part in the link. What i want to know is why the artist drew the

realLegendre like some sadistic ghoul 🙂on November 25, 2014 at 3:23 PMMatthew RaveI don’t know; I wish I did!

on November 26, 2014 at 7:28 AM |Laplace, Lagrange und Legendre: Was ist der Unterschied? – Astrodicticum Simplex[…] möchte, wie man zwischen den FMDAJDNMLBs unterscheidet, für den habe ich einen Lesetipp: Im Blog Many Worlds Theory werden die Leistungen der Mathematiker und vor allem die Unterschiede zwischen ihnen kurz und knapp […]

on November 30, 2014 at 6:22 AM |AnonymousActually, Lagrange was “Italian”: http://en.wikipedia.org/wiki/Joseph-Louis_Lagrange

But this would ruin the nice acronym… oh, nevermind.

on November 30, 2014 at 9:47 AMMatthew RaveWell he did gain French citizenship, so he was French too.

on November 30, 2014 at 2:32 PM |Paolook, he became French because Napoleone Bonaparte conquered half Europe 😀 I just meant to point out this almost unknown (and unimportant) detail.

Anyway, thanks for the short and practical summary!

on February 20, 2015 at 7:23 PM |Rich BeveridgeI would include the work Lagrange and Legendre did on quadratic residues which Gauss drew from (extensively) for Disquisitiones Arithmeticae

on March 29, 2019 at 11:05 AM |SAlso, for people who care about the relative dates —

Lagrange: 1736 – 1813

Laplace: 1749 – 1827

Legendre: 1752 – 1833

— a convenient mnemonic I discovered is that both birth and death order follow the alphabetical order.

I don’t know about comparing Legendre to Lagrange or Laplace, but he did a lot of good work besides elliptic integrals: the first published account of the least squares method, quadratic residues and (conjectured) the quadratic reciprocity law (thus the Legendre symbol (r|p) for whether r is a quadratic residue mod p), the conjectured prime number theorem, Legendre polynomials, and a book Éléments de géométrie that replaced Euclid as the leading geometry textbook for nearly a century.