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Long before Napier, the link between a geometric and arithmetic series was known. Let us look at the simplest of the series which can be connected - the natural or counting numbers, and the powers of the number 2.

The bottom row will be the powers of 2, and the top will be the indices:

 0 1 2 3 4 5 6 7 8 9 10 1 2 4 8 16 32 64 128 256 512 1024

Now, the bottom row also represents a geometric series, while the top an arithmetic one. If you take any two numbers in the arithmetic series, and add them up, this would indicate the term in the corresponding geometric series that is the product of the two corresponding terms.

Let's see how this works:

Take 4 and 6, add them up: 4+6=10.

Now look at the corresponding members of the geometric series: for

10 ... 1024

4 ... 16

6 ... 64

If you multiply the corresponding members of the geometric series, you get 16x64=1024

John Napier published for a first time a book Mirifici Logarithmorum Canonis Descriptio, in which he described the method of connecting arithmetic with a geometric series.

Click on the picture to get a full-size copy

The meaning of the word logarithm is derived from two Greek words - logos - meaning ratio, and arithmos - meaning number.

Later on in the same year when Napier's book was published, a London mathematician Henry Briggs (who was a professor of geometry at Gresham College at the time) started working on the same topic. He went to Scotland to work with Napier, and when Napier died in 1617, took over the work on logarithms. He popularised the concept of logarithms and published a modified book of tables in 1624 under the title Arithmetica Logarithmica. Within no more than few decades from this publication, the use of logarithms was spread around the world. They became as popular as calculators in our time, because they offered an easy way of calculation.

Click on the picture to get a full-size copy

The most common bases of logarithms are 2, e and 10. Briggs called common logarithms those who are based on powers of 10 - we still use the same title. The logarithms with the base e are written as ln and are called natural logarithms.

You can see the copy of the original fronticepiece of Napier's work on Logarithms, or the work of his successor here.

Some other interesting mathematical concepts and their invention are described here.

Or just climb the mathematical three and see what fruit of knowledge you can pick up!

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