Johann Müller Regiomontanus (1436 - 1476)

Regiomontanus was born Johann Müller of Königsberg and it is from this town that he gets his name (in Latin Königsberg, meaning King’s mountain, is Regio Monte).

Regiomontanus changed the way astronomy worked in Europe and in so doing helped to contribute to many mathematical advances. That was the first of his achievements: the second was his skills in tracking down and making available classic works in mathematics and astronomy.

He was an astronomer in Vienna and went on to work for the King of Hungary. He kept detailed records of a comet we now know as Halley’s comet, helped to manufacture astronomical instruments and also cast horoscopes for the king to predict the future.

His work was built on an understanding of Greek mathematics and he was particularly interested in triangles and perfect numbers, just as Ptolemy had been many centuries before.

His mathematical and astronomical knowledge helped him to translate the work of Ptolemy from Arabic to Latin.

This laid the foundations for the explosion of astronomical research which then occurred. He also wrote an insightful introduction to Ptolemy which tried to make his model more three dimensional which would give better mathematical predictions of planetary movement. This work was to inspire Copernicus and Galileo.

The Epitome of the Almagest, written by Regiomontanus was one of the most important Renaissance sources on ancient astronomy and mathematics.

Regiomontanus was interested in scholarship of the highest standard and he went to great lengths and trouble to ensure that he provided accurate information. His dislike of sloppily translated Greek work inspired him to set up an early printing press. This published classical and modern mathematical works, helping to systemise the mathematics known and also pioneered the appearance of astronomical diagrams and mathematical tables in print.

One of the old books which Regiomontanus had come across in 1462 while he was in Venice, was an incomplete copy of Diophantus’s Arithmetica.

Diophantus’s/Regiomontanus’ mathematics

Diophantus’s work in mathematics was concerned with problems that resulted in equations with whole numbers as answer. Let us take a modern example based on an old theme - are there any triangular numbers that are also square?

We start by inspection: the triangular numbers are,

1, 3, 6, 10, 15, 21, 28, 36 … yes 36 is square. What about the next one ?

Now the nth triangular number is,

n(n+1)\over{2}

and the mth square is just m^2 so we require the solutions of the equation,

m^2=n(n+1)\over{2}

for whole numbers, n and m. In the example we found earlier, m=6 and n=8. Such equations are called Diophantine equations after Diophantus. It turns out that the solution of this equation is given by the amazing expression,

{1\over{32}}\Bigg[\;{(17+12\sqrt{2})^n}+{(17-12\sqrt{2})^n}-2 \Bigg]\;

When n=2 this gives the solution we found above - what does it give when n=1? Why?

Regiomontanus never translated Diophantus’s Arithmetica and he never found a complete version. Indeed nobody has ever discovered a complete version, but this important discovery by Regiomontanus marks the beginning of the Arithmetica becoming known in Europe. It is certainly the way that Fermat had learnt about such problems, and the motivation for his own work and legacy.