Faulhaber

Johann Faulhaber

Born: 5 May 1580 in Ulm, Germany
Died: 1635 in Ulm, Germany

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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 was a Rosicrucian, 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 and influenced his thinking.

Faulhaber's most major contribution, however, was in studying sums of powers of integers. Let N = n(n+1)/2. Define Sigman^k to be the sum Sigmai^k where the sum is from 1 to n. Then N = Sigman^1. In 1631 Faulhaber published Academia Algebra in Augsburg. It was a German text despite the Latin title.

In Academia Algebra Faulhaber gives Sigman^k 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 Jacobi was influenced by Faulhaber's work, 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, eight years after Jacob Bernoulli died, where the Bernoulli numbers (so named by De Moivre) appear.

Academia Algebra contains a generalisation of sums of powers. Faulhaber gave formulas for m-fold sums of powers defined as follows. Define Sigma^0n^k = n^k and

Sigma^m^+^1n^k = Sigma^m1^k + Sigma^m2^k + ... + Sigma^mn^k.
Faulhaber gives formulas for many of these m-fold sums including giving a polynomial for Sigma^1^1n^6. Knuth, in [3] remarks:-
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 Sigman^k as far as k = 25. He gives the formulas in the form of a secret code, which was common practice at the time. Knuth, in [3], suggests he is the first to crack the code (the task [of cracking the code] is relatively easy with modern computers ) and shows that Faulhaber had the correct formulas up to k = 23, but his formulas for k = 24 and k = 25 appear to be wrong.

Other Web sites:

Rice University, USA

References:

  1. Dictionary of Scientific Biography
  2. H Keefer, Johannes Faulhaber, der bedeutendste Ulmer Mathematiker und Festungsbaumeister, Württembergische Schulwarte 4 (1928), 1-12.
  3. D E Knuth, Johann Faulhaber and Sums of Powers, Mathematics of Computation 61 (1993), 277-294.
  4. A F Beardon, Sums of powers of integers, Amer. Math. Monthly 103 (1996), 201-213.
  5. A W F Edwards, Sums of powers of integers : a little history, Mathematical Gazette 66 (1982), 22-29.
  6. A W F Edwards, A quick route to sums of powers, Amer. Math. Monthly 93 (1986), 451-455. 
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