Stacy Langton, PhD

My research is in the history of mathematics, and particularly the work of the 18th-century Swiss mathematician Leonhard Euler. I also study the work of other 18th-century mathematicians. I recently translated a Latin article published in 1728 by Euler’s colleague and friend Daniel Bernoulli (another Swiss mathematician). The title of the article is “On recurrent series”. In addition to the transla tion, I wrote a commentary, or “reader’s guide” to the article.

Bernoulli’s “recurrent series” are what we now call linear recurrence relations. You may have encountered these series in Combinatorics. Bernoulli’s article contains some very nice results, in particular what is, as far as I know, the first publication of the now-famous formula for the nth Fibonacci number.

The result in Bernoulli’s article of which I think he was most proud was a method for solving polynomial equations numerically. Today this method is known as “Bernoulli’s method”. Unlike Newton’s method for solving equations, which you may have seen in Numerical Analysis, Bernoulli’s method does not require an accurate initial guess for the root being found. On the other hand, Bernoulli’s method does not converge as fast as Newton’s. 

Another article that I have worked on recently is a 1720 article by the Prussian mathematician Christian Goldbach. Goldbach was a correspondent of both Euler and Daniel Bernoulli (and of Nicolas Bernoulli, too). You may have heard of “Goldbach’s conjecture”, which he suggested in a letter to Euler: that every even number greater than 2 is the sum of two primes. Despite much effort, this conjecture has never been proved or disproved.

A longer-term project concerns an article published in 1755 by the Hungarian mathematician Johann Andreas Segner, “On the theory of tops” (in Latin, “Specimen theoriae turbinum”). A correspondent asked me a few years ago whether I knew of an English translation of Segner’s article, and I had to respond that I did not know of one. So I thought that perhaps I should translate the article myself.

Segner’s article gives the first proof of an important theorem of Linear Algebra, which is known as the Spectral Theorem, or the Principal Axis Theorem. It says that every symmetric matrix can be diagonalized, and in fact can be diagonalized by an orthogonal matrix. This means that an n × n symmetric matrix has n linearly independent eigenvectors, which form an orthonormal system.

However, this is not the way that Segner formulated his theorem. In 1755, mathematicians did not know about matrices, or eigenvalues or eigenvectors. What Segner actually showed was that any rigid body has three mutually orthogonal axes of free rotation. The symmetric matrix that Segner worked with is now called the inertia tensor of the rigid body. (This matrix is always symmetric, regardless of the shape of the body.) The axes of free rotation are the lines determined by the eigenvectors of that matrix.

In 1750, Euler had published an article in which he gave the first explicit statement of what physicists call “Newton’s second law”, in the form F = ma. As an application, Euler gave the first statement and proof of the equations of motion for a rigid body. But Euler did not know how to diagonalize a matrix, and consequently his statement of the equations of a rigid body was quite complicated. But after Segner’s result had been published, Euler was able to go back and give a much simpler and more direct treatment of the motion of a rigid body.

He teaches courses in mathematics, computer science and statistics.  His method in teaching is to stress the fundamentals of each subject.