Alex's Adventures in Numberland - Alex Bellos [79]
The convenience that logarithms brought made them the most significant mathematical invention of Napier’s time. Science, commerce and industry benefited massively. The German astronomer Johannes Kepler, for example, used logs almost immediately to calculate the orbit of Mars. It has recently been suggested that he might not have discovered his three laws of celestial mechanics without the ease of calculation offered by Napier’s new numbers.
In his 1614 book A Description of the Admirable Table of Logarithmes, Napier used a slightly different version of logarithms than those used in modern mathematics. Logarithms can be expressed as a power of any number, which is called the base. Napier’s system used an unnecessarily complicated base of 1 – 10–7 (which he then multiplied by 107). Henry Briggs, England’s top mathematician in Napier’s day, visited Edinburgh to congratulate the Scot on his discoveries. Briggs went on to simplify the system by introducing base-ten logarithms – which are also known as Briggsian logarithms, or common logarithms, because ten has been the most popular base ever since. In 1617 Briggs published a table of the logs of all numbers from 1 to 1000 to eight decimal places. By 1628 Briggs and the Dutch mathematician Adriaan Vlacq had extended the log table to 100,000, to ten decimal places. Their calculations involved laborious number-crunching – although, once the sums were done correctly, they never needed to be done again.
Page of Briggs’s log tables from 1624.
That is, until 1792, when the young French republic decided to commission ambitious new tables – the log of every number to 100,000 to 19 decimal places,nd rom 100,000 to 200,000 to 24 decimal places. Gaspard de Prony, the man who headed the project, claimed that he could ‘manufacture logarithms as easily as one manufactures pins’. He had a staff of nearly 90 human calculators, many of whom were former servants or wig dressers whose pre-revolutionary skills had become redundant (if not treasonous) in the new regime. Most of the calculations were finished by 1796, but by then the government had lost interest, and de Prony’s gigantic manuscript was never published. Today it is housed in the Paris Observatory.
Briggs’s and Vlacq’s tables remained the basis for all log tables for 300 years, until the Englishman Alexander J. Thompson in 1924 began work manually on a new set accurate to 20 places. Yet instead of giving an old concept a modern sheen, Thompson’s work was already outdated when he finished it, in 1949. By then computers could generate the tables easily.
When you plot the digits 1 to 10 on a ruler positioned to their log values, you get the following pattern:
We can carry on like this, say, up to 100.
This is what is known as a logarithmic scale. In the scale, numbers get progressively closer together the higher they are.
Some scales of measurement are logarithmic, which means that for every unit you go up on the scale, it represents a tenfold change in what it is measuring. (In the second scale above, the distance between 1 and 10 is equal to the distance between 10 and 100.) The Richter scale, for example, which measures the amplitude of waves recorded by seismographs, is the most commonly used logarithmic scale. An earthquake that registers 7 on the Richter scale triggers an amplitude that is ten times more than an earthquake that registers a 6.
In 1620 the English mathematician Edmund Gunter was the first person to mark the logarithmic scale on a ruler. He noticed that he could multiply by adding lengths of this ruler. If a compass was placed with the left spike at 1, and the right at a, then when the left spike was moved to b, the right spike pointed to a × b. The diagram below shows the compass set to 2 and then positioned with the left spike at 3, putting the right spike at 2×3 = 6.
Gunter multiplication.
Not long afterwards William Oughtred, an Anglican minister, improved on Gunter’s idea. He dispensed with the compasses, instead