Legendre who?

In Chapter 10 of my book, Euler’s Gem, I give Adrien-Marie Legendre‘s beautiful proof of Euler’s polyhedron formula: for any (convex) polyhedron with V vertices, E edges, and F faces, V-E+F=2. His use of spherical geometry to prove the theorem is extremely elegant. On page 88 I include the portrait of Legendre shown at right….

Computing integer sums using l’Hôpital’s rule

Now that the busy semester is over, I’ve been able to catch up on some reading. Yesterday I read William Dunham’s article “When Euler Met l’Hôpital,” in the February 2009 issue of Mathematics Magazine. The aim of the article is to showcase some of Euler’s applications of l’Hôpital’s rule in his Institutiones calculi differentialis (1755)….

The sum of kth powers

Everyone loves the “baby Gauss story” in which Gauss amazes his teacher by quickly summing the first 100 positive integers in a flash of brilliance—he adds the first to the 100th, the second to the 99th, and so on to get the sum of fifty 101s to obtain 5050. (Brian Hayes has a great article…

Königsberg today

In 1736 Leonhard Euler solved the now-famous bridges of Königsberg problem.  It is often hailed as the birth of topology and graph theory. Although no graph appears in Euler’s paper, his argument is topological in spirit. (He proved another famous topological theorem a decade and a half later.) The problem is stated as follows. If a resident of Königsberg…