John Napier (1550–1617)

Napier's "Bones", a set of wooden rods used for muliplication

If you have studied logarithms, you have John Napier to thank for their discovery. He died 400 years ago today.

John Napier was a Scottish landowner, mathematician, physicist, and astronomer, but is probably best known as the discoverer of logarithms.

Before calculators became widespread in the second half of the 20th century, logarithms were widely used to perform multiplication of numbers, particularly large numbers. Logarithms were looked up in "log tables".

Logarithms have many uses and come into many areas of modern maths, and if you have studied beyond GCSE level, you will be very familiar with them. The natural logarithm (log base e) is sometimes called the Napierian logarithm, although Napier didn't explicitly work with base e.

Napier also invented "Napier's bones", which were a set of wooden rods inscribed with digits, and were another tool for performing multiplication. Napier was also one of the earlier mathematicians to make frequent use of the decimal point in his work and helped to popularise this idea.

Napier's birthplace, Merchiston Tower in Edinburgh, is now a part of Edinburgh Napier University, which is named after him.

Napier died in 1617 at his home at Merchiston Castle from the effects of gout, but his revolutionary work on what became a key part of modern mathematics will ensure he is remembered for a long time to come.

e

Leonhard Euler is widely considered one of the greatest mathematicians. (Photo credit: Wikipedia)

In Our Time on Radio 4 introduces us to e, the irrational number that underpins so much of our modern mathematics.

e is an infinite decimal, like pi, and is approximately 2.718. It is irrational, transcendental and a part of what is often described as the most beautiful equation ever written.

Pi has been known about since the times of the ancient Greeks, but e was not discovered until the 17th century, because the mathematics required simply did not exist. The ancient Greeks' fear of the infinite was a part of the reason they did not stumble across it.

Jakob Bernoulli was the first mathematician to discover e cropping up in his studies of compound interest. A little later, John Napier invented logarithms, using a base approximately equal to 1/e. These huge tables of numbers took Napier 20 years of his life to devise and were designed to make multiplication easier and more accurate. Logarithms were later refined by Henry Briggs.

e also appears a lot in modern calculus. The function e to the power x has a gradient which always has the same value as the function itself. When Leibniz and Newton discovered calculus they used infinite sums to form the derivative of the exponential function.

Later the natural logarithm was found to be the area under the curve y=1/x.

e was named by Euler, the great mathematician of the 18th century, possibly the greatest mathematician of all time and probably the most prolific in terms of publications. But he did not, we are told, name e after himself. Instead, he thought e to be the first letter of the alphabet not widely in use in mathematics. Euler showed e was the sum of infinite series and introduced his famous Euler identity, which many mathematicians consider the most beautiful equation ever written.

To give just a few applications, e crops up in radioactive decay, in the normal distribution in statistics, and in the prime number theorem, which tells us roughly how many primes there are below any given integer.