Logarithms: the trick that turns multiplication into addition

Today, when we have a large multiplication to do, we take out our mobile phones. Four hundred years ago, astronomers, navigators and land surveyors had to multiply and divide seven-digit numbers by hand. A single calculation could take hours, and the smallest error spoiled the whole work. In 1614, a landowner and mathematician from Scotland, John Napier, presented a method that halved this labour. His method is called the logarithm.

The mathematician of Merchiston

According to the MacTutor archive of the University of St Andrews, Napier was born in 1550 at Merchiston Castle in Edinburgh and died in Edinburgh on 4 April 1617. He was not a professional teacher. He managed his estate, and mathematics was his hobby. MacTutor writes that most of his work on logarithms was done while he lived at Gartness after 1574.

In 1614 he published a book in Latin called “Mirifici logarithmorum canonis descriptio”, meaning “Description of the Wonderful Table of Logarithms”. In 1616, Edward Wright translated it into English.

What is the method?

The basic idea of the logarithm is simple. Multiply 10 by 10 and you get 100, that is, 10 taken twice. In 1,000, 10 is multiplied three times. Now, to multiply 100 by 1,000, we only need to add “two” and “three”: five. So the answer has 10 multiplied five times, and that is one lakh (100,000). These “two” and “three” are the logarithms of those numbers.

The same principle applies to every number, even one that is not a straight power of 10. If someone has a table giving the logarithm of every number, he looks up the logarithms of two numbers, adds them, finds the sum in the table again and reads off the answer. Multiplication becomes addition, and division becomes subtraction. To find a root, the logarithm is halved.

According to MacTutor, Napier drew this idea not from algebra but from an example of motion: two points move along two lines in different ways, and the relation between their distances gives the logarithm. He set the length of the base line at 10 million (10 to the power of seven), because the best trigonometric tables of the time were accurate to seven places. In his system, the logarithm of 1 was not zero, and, unlike today, no “base” was fixed.

Henry Briggs and the base of ten

When Napier’s book reached London, Henry Briggs, Professor of Geometry at Gresham College, was captivated by it. MacTutor quotes his remark that he had never seen a book which pleased and amazed him more. In the summer of 1615 he travelled from London to Edinburgh and stayed with Napier for about a month. He went again in 1616.

In these meetings the two agreed that new tables should use the base of ten and that the logarithm of 1 should be zero. These are now called “common logarithms”. Napier died in 1617, so Briggs carried the work forward. In that same year he published the first short table, covering a thousand numbers. In 1624, in “Arithmetica logarithmica”, he gave logarithms to 14 decimal places for the numbers from 1 to 20,000 and from 90,000 to 100,000. According to MacTutor, the gap in between was filled in 1628 by Adriaan Vlacq of Holland.

Another inventor

Jost Bürgi, a Swiss clockmaker and instrument maker, also developed a similar system independently. According to MacTutor, his method differed from Napier’s. Most historians believe he invented it around 1588, but his work was printed in 1620, six years after Napier’s book.

From astronomy to the slide rule

Astronomy benefited most from logarithms. According to MacTutor, the German astronomer Johannes Kepler was able to work out his observations with the help of logarithms, and this work later became the basis of Newton’s theory of gravity. The French mathematician Laplace is famous for saying that logarithms, by reducing labour, doubled the life of the astronomer.

A device also came out of logarithms. According to MacTutor, the English mathematician Edmund Gunter made a logarithmic scale on a strip in 1620. William Oughtred devised a way of placing two such strips side by side and sliding them over each other, and in 1630 he made the circular slide rule. Oughtred also popularised the “×” sign for multiplication. Sliding the two scales of a slide rule adds logarithms, and the answer to the multiplication appears. According to the Victoria and Albert Museum in London, slide rules remained in common use until the 1970s, when electronic calculators replaced them.

Logarithms today

Today, printed logarithm tables and slide rules are kept in museums. The logarithm itself remains in mathematics and science books, and the “log” button on a calculator still recalls it. Napier’s idea was simple: turn a great labour into small steps. After four centuries, science still benefits from that idea.