## John Napier (1550 - 1617)

Napier was born into a privileged Scots family. He is one of few mathematicians to have worked from his own castle. Educated initially at St Andrews he left for the continent to complete his education as did so many rich Scots of the time.

He was probably fortunate to have escaped, for in the years he was away Scotland went through a turbulent time with the dethronement of Mary Queen of Scots and the accession of James.

Typical for the time, religious strife lay at the heart of the troubles. Napier himself was a fervent Protestant. More than anything else he saw himself as an advocate for the Protestant church and contributed books on the subject as well as helping to devise weapons to ensure that Scotland remained Protestant.

In his spare time however, he was a keen mathematician. He was one of the first to use the decimal place system, and certainly used the most systematic method then known. He also devised a set of rods that aided multiplication and was a precursor to the slide rule.

Typically, for a man with so little spare time, his great idea was the invention of logarithms - which considerably speeded up calculations. This work, probably completed as early as 1594 developed a system whereby roots, products and quotients could be quickly calculated using a set of tables with a fixed number used as a base.

Napier was clearly an inventive man, but to his contemporaries he seemed just plain strange. Despite his great religious work, persistant rumours circulated that he had mysterious supernatural powers, was a warlock and was never seen without his black rooster.

## Napier’s mathematics

In essence logarithms work like this. Consider the graph of the function,

f(x)=2^x

If we take the values of x = 0,1,2 and 3 then this function takes the values,

f(0)=1,\;f(1)=2,\;f(2)=4\;\mbox{and}\;f(3)=8

From these values, we can sketch its graph:

Suppose, by way of a very simple example, that we want to multiply 1.5 by 2.5. We make the two lower constructions shown on the graph, and read off that:

So now we have,

(1.5).(2.5)\approx{2^{0.58}}.2^{1.32}=2^{0.58+1.32}=2^{1.9}

and the remaining construction now gives the answer, 3.7.

We have converted a multiplication into an addition, which is always easier to carry out. Powers can be done by repeated addition; division by subtraction. Napier used rods with markings on them - much like the axes in the graph above - as the basis for his system. Later would come sets of tables - developed to considerable accuracy - to carry out the same procedure. These tables were called logarithm tables after the inverse function:

f(x)=2^x\ \mbox{has the inverse function}\ f^{-1}(x)=\log{x}.

Logarithm tables were in continuous use right up to the 1970s when they were replaced by calculators.

The method simplified and made possible the large calculations which were necessary for astronomical research - the inspiration for much mathematics and the mathematicians engaged in it.