International Routier, May 29, 2010

Having discussed the introduction of the decimal point, and then cleared up a small matter of me taking the absolute values of negative powers, let’s have a look at their first popular application.

Logarithms were developed at the turn of the seventeenth century independantly by a number of people as a way of simplifying long multiplication or division by turning it into a simple series of additions or subtractions. Like the decimal point, John Napier is credited with popularising the use of Logarithms, helped in no small way be Johannes Keppler publishing a booklet promoting and explaining Napier’s work.

Put simply, as my old TAFE Mathematics 3 and 4 teacher would have it:

The logarithm is the power to which the base should be raised to equal the number.

The textbooks explained it as

Log (A x B) = log A + log B, and

Log (A / B) = log A – log B.

Assuming a base of 10, 100 is the same as 10^{2} so the logarithm_{base 10} of 100 is 2. Suppose we want to carry out the difficult computation of 100 x 1000. We could do it longhand but if it is exceptionally difficult, we could simplify it using logarithms, so the equation becomes

2+3=5

and then look up the antilogarithm of 5 in the tables, finding that it is 100,000.

Napier derived what are now called Natural Logarithms. Although not calculated that way, Napier’s logarithms ended up with the inverse of the mathematical constant *e* (ie, 1 on about 2.71828) as the base. His first set of tables (not published until 1619) were to seven decimal places but due to rounding in his computations were only accurate to about five. His later tables, published in 1614 were correct to 14 places. A friend, Edward Wright translated Napier’s work into English and Napier was so pleased with the translation, he wrote in the foreword

I… finde it to bee most exact and precisely conformable to my minde and the originall.

One of the difficulties with his tables was the use of the obscure base. Rather than the comfortable log_{10} 1 = 0, in Napier’s logarithms, log_{1/e} 10^{7}=0. Napier delegated the rescaling of logarithms using 10 as the base to another friend, Henry Briggs. Briggs then published logarithms for the numbers 1–20,000 and 90,000–100,000 in 1624.

Michael Stifel had earlier published *Arithmetica Integra* in Nuremberg in 1544 which contains his discovery of logarithms but doesn’t seem to have gone anywhere with it. Like Napier, Joost Bürgi independently discovered logarithms (Bürgi in about 1588); however, he did not publish his discovery until four years after Napier in 1620. In his book, Bürgi used Napier’s decimal point convention so had seen Napier’s work at some stage. Brigg’s set of tables was republished by Adriaan Vlacq in London in 1628, Vlacq calculating the missing values himself, filling in the gap between 20,000 and 89,999. Vlacq fled back to Holland at the outbreak of the civil wars.

Napier’s original work, *Mirifici logarithmorum canonis constructio: et eorum ad naturales ipsorum* is at http://books.google.com/books?id=VukHAQAAIAAJ with a good modern English translation at 17 Century Maths.

Edward Wright’s translation, *A Description of the Admirable Table of Logarithmes*, (1616) is at http://johnnapier.com/table_of_logarithms_001.htm

*Arithmetica Logarithmica*, (1624), by Henry Briggs is at http://www.17centurymaths.com/contents/albriggs.html

## Further reading

Wikipedia – Logarithms accessed 10 May 2010

Wikipedia – Adriaan Vlacq accessed 10 May 2010

Wikipedia – Edward Wright accessed 4 May 2010