Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician.
Born in Ulm, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Besides his work on the fortifications of cities (notably Basel and Frankfurt), Faulhaber built water wheels in his home town and geometrical instruments for the military. Faulhaber supervised the first publication of Henry Briggs's logarithms in Germany. He died in Ulm.
Faulhaber's major contribution involved calculating the sums of powers of integers. Jacob Bernoulli makes references to Faulhaber in his Ars Conjectandi.
Faulhaber collaborated with Kepler and van Ceulen. He was a Rosicrucian, a member of a brotherhood combining elements of mystical beliefs with an optimism about the ability of science to improve the human condition. He made a major impression on Descartes with both his scientific and Rosicrucian beliefs, and influenced his thinking.
Faulhaber was a "Cossist", an early algebraist. He is important for his work explaining logarithms associated with Stifel, Bürgi and Napier. He made the first German publication of Briggs' logarithms.
Faulhaber's most major contribution, however, was in studying sums of powers of integers. Let N = n(n+1)/2. Define Σ nk to be the sum Σ ik where the sum is from 1 to n. Then N = Σ n1.
In 1631 Faulhaber published Academia Algebra, which was in German despite the Latin title. This book gives Σ nk as a polynomial in N, for k = 1, 3, 5, ... ,17. He also gives the corresponding polynomials in n. Faulhaber states that such polynomials in N exist for all k, but gave no proof. This was first proved by Jacobi in 1834. It is not known how much Faulhaber's work influenced Jacobi, but we do know that Jacobi owned Academia Algebra, since his copy of it is now in the University of Cambridge.
Faulhaber did not discover the Bernoulli numbers, but Jacob Bernoulli refers to Faulhaber in Ars Conjectandi (published in Basel in 1713), where the Bernoulli numbers (so named by De Moivre) appear (Smith 1959).
Academia Algebra contains a generalisation of sums of powers. Faulhaber gave formulae for m-fold sums of powers defined as follows.
Define Σ0 nk = nk and Σm+1 nk = Σm 1k + Σm 2k + ... + Σm nk.
Faulhaber gives formulae for many of these m-fold sums including giving a polynomial for Σ11 n6. Knuth remarks (Knuth 1993, p. 14):
His polynomial ... turns out to be absolutely correct, according to calculations with a modern computer. ... One cannot help thinking that nobody has ever checked these numbers since Faulhaber himself wrote them down, until today.
At the end of Academia Algebra Faulhaber states that he has calculated polynomials for Σ nk as far as k = 25. He gives the formulae in the form of a secret code, which was common practice at the time. Donald Knuth suggests he is the first to crack the code: (the task is relatively easy with modern computers) and shows that Faulhaber had the correct formulae up to k = 23, but his formulae for k = 24 and k = 25 appear to be wrong.