Johann Faulhaber
Encyclopedia
Johann Faulhaber 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 wheel
s in his home town and geometrical instruments for the military. Faulhaber supervised the first publication of Henry Briggs's
logarithm
s 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 .
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 :
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 [of cracking the code] 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.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
.
Born in Ulm
Ulm
Ulm is a city in the federal German state of Baden-Württemberg, situated on the River Danube. The city, whose population is estimated at 120,000 , forms an urban district of its own and is the administrative seat of the Alb-Donau district. Ulm, founded around 850, is rich in history and...
, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
and Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
. Besides his work on the fortifications of cities (notably Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...
and Frankfurt
Frankfurt
Frankfurt am Main , commonly known simply as Frankfurt, is the largest city in the German state of Hesse and the fifth-largest city in Germany, with a 2010 population of 688,249. The urban area had an estimated population of 2,300,000 in 2010...
), Faulhaber built water wheel
Water wheel
A water wheel is a machine for converting the energy of free-flowing or falling water into useful forms of power. A water wheel consists of a large wooden or metal wheel, with a number of blades or buckets arranged on the outside rim forming the driving surface...
s in his home town and geometrical instruments for the military. Faulhaber supervised the first publication of Henry Briggs's
Henry Briggs
Henry Briggs may refer to:*Henry Briggs *Henry Briggs , *Henry Shaw Briggs , Union general in the American Civil War...
logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s 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
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
. He was a Rosicrucian
Rosicrucian
Rosicrucianism is a philosophical secret society, said to have been founded in late medieval Germany by Christian Rosenkreuz. It holds a doctrine or theology "built on esoteric truths of the ancient past", which, "concealed from the average man, provide insight into nature, the physical universe...
, 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
Michael Stifel
Michael Stifel or Styfel was a German monk and mathematician. He was an Augustinian who became an early supporter of Martin Luther. Stifel was later appointed professor of mathematics at Jena University...
, Bürgi
Joost Bürgi
Jost Bürgi, or Joost, or Jobst Bürgi , active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomical instruments and a mathematician....
and Napier
John Napier
John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...
. He made the first German publication of Briggs'
Henry Briggs (mathematician)
Henry Briggs was an English mathematician notable for changing the original logarithms invented by John Napier into common logarithms, which are sometimes known as Briggsian logarithms in his honour....
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
Ars Conjectandi
Ars Conjectandi is a combinatorial mathematical paper written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, most notably among other combinatorial topics, probability theory: indeed, it is widely regarded as...
(published in Basel in 1713), where the Bernoulli numbers (so named by De Moivre) appear .
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 :
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
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
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 formulae up to k = 23, but his formulae for k = 24 and k = 25 appear to be wrong.