Nicholas Oresme (1323 - 1382)

We know little of the early life of Oresme. We do know that he was Bishop of the Norman city of Lissieux but his other achievements are many and wide: he was interested in taxation, and understood that the Earth revolved around the Sun (and not vice versa) over 100 years before Copernicus.

Little known now, he was a leading thinker not afraid of the consequences of his revolutionary work.

Oresme moved from Normandy to Paris and, after gaining an arts degree from the University of Paris, began a steady ascent through the ranks of academia and the clergy.

He became friendly with Charles the fifth, King of France and went on to become his advisor. In this role he is renowned as a great medieval economist fighting a battle against currency debasement and interference by the King.

He was a learned man and his translation of many classical works from Latin to French means that he created the words for many mathematical ideas. Uniquely in the medieval world, he questioned the teachings of Aristotle.

He made pre-Galilean attempts to derive laws of motion and also invented a type of coordinate geometry before Descartes - tabulating and graphing values. His work went on to influence many of the advances in European mathematics.

Oresme’s mathematics

Oresme introduced the ‘+’ sign for addition and extended the index notation for powers, 3^n, to fractional and negative powers, so that for example,

3^{1 \over 2} is simply \sqrt{3} since 3^{1 \over 2}.3^{1 \over 2}=3^1=\sqrt{3}.\sqrt{3}

and

3^{-\frac{1}{2}}={1 \over 3^{1 \over 2}}={1\over{\sqrt{3}}}

He is credited with the discovery that the so-called Harmonic sequence is divergent. The sequence is the sum of the reciprocals of the whole numbers,

H_r=1+{1\over2}+{1\over3}+\cdots{1\over{r}}

What, wondered Oresme, happens as r increases without limit? The numerical evidence suggests that the sum remains finite:

H_{10}=2.928\cdots;\ H_{100}=5.187\cdots;\ H_{1000}=7.486\cdots

it simply does not look like a divergent sequence, and yet it is. This is Oresme’s proof, which relies on some ingenious re-grouping followed by a simple inequality

H_\infty=1+{1\over2}+\left({1\over3}+{1\over4}\right)+\left({1\over5}+{1\over6}+{1\over7}+{1\over8}\right)+\left({1\over9}\cdots+{1\over16}\right)+\cdots

\quad\ >1+{1\over2}+\left({1\over4}+{1\over4}\right)+\left({1\over8}+{1\over8}+{1\over8}+{1\over8}\right)+\left({1\over16}\cdots+{1\over16}\right)+\cdots

\quad\ >1+{1\over2}+{1\over2}+{1\over2}+{1\over2}+\cdots;

this grows slowly but inexorably larger and so the sequence must be divergent.

There are many, many different proofs of this. A proof of a very different kind - for those familiar with the integral calculus - is given by an integral:

\int\limits_{-\infty}^0{e^z\over{1-e^z}}dz=\int\limits_{-\infty}^0e^z\sum\limits_{k\ge0}e^{kz}dz=\int\limits_{-\infty}^0\sum\limits_{k\ge1}e^{kz}dz=\sum\limits_{k\ge1}\int\limits_{-\infty}^0e^{kz}dz

=\sum\limits_{k\ge1}\left[{e^{kz}\over{k}}\right]_{-\infty}^0=H_{\infty} ;

but the integral is easy to evaluate otherwise,

\int\limits_{-n}^0{e^z\over{1-e^z}}dz=\left[-\log\left|1-e^z\right|\right]_{-n}^0 \to \infty \mbox{ as } n \to \infty

Either way, the Harmonic sequence diverges. This proof also shows that the sum is linked to the logarithm; in fact,

H_r\approx\log r-\gamma

where \gamma is a constant (\gamma=0.57721\cdots) called Euler’s constant.