Did you know . . .?

Long before the calculator, logarithms were great mathematical labor-saving devices!

Although there is evidence that logarithms were known in 8th century India, their invention as an aid to calculation is attributed to a Scottish nobleman named John Napier (1550-1617) in his Mirifici logarithmorum canonis descriptio (1614) and Mirifici logarithmorum canonis constructio (published posthumously in 1619). In collaboration with Oxford professor Henry Briggs, Napier refined his logarithms by constructing tables for logarithms in base 10. Napier is also credited with creating one of the earliest calculating machines ("Napier's bones") and with the first systematic use of the decimal point.

Not a bad mathematical pedigree for a man who never finished university and who considered his most important work to be his Plaine Discovery of the Whole Revelation of St. John (1593)!

John Napier.

Napier lived during an age of great innovation in the world of astronomy. Copernicus had published his theory of the solar system in 1543, and many astronomers were eagerly involved in calculating and re-calculating planetary positions based in the wake of Copernicus's ideas. Their calculations took up pages and pages and hours and hours of work. Johannes Kepler (1571-1630) still had to fill nearly 1000 large pages with dense arithmetical computations to obtain his famous laws of planetary motions! Napier's logarithms helped ease that burden.

Because they are exponents, logarithms allow tedious calculations (like multiplying and dividing very large numbers) to be replaced by the simpler process of adding and subtracting the corresponding logarithms.

Not that mathematicians simply put down their pens after Napier. Many objected to using logarithms because no one knew understood they worked (an objection similar to one made to the use of computers in the 1960s)!

"Napier's bones" multiplication device.

In his own words:

On logarithms:

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three. --John Napier, Mirifici logarithmorum canonis descriptio (1614) (from the preface of the first English translation in 1616).

On the decimal point:

In computing tables, these large numbers may again be made still larger by placing a period after the number and adding ciphers. ... In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many ciphers after it as there are figures after the period. --John Napier, Mirifici logarithmorum canonis constructio (1619)

An explanation of logarithms together with an automatic logarithm calculator: http://www.math.utah.edu/~alfeld/math/log.html

Another demonstration of how to use Napier's bones to multiply: http://www.cee.hw.ac.uk/~greg/calculators/napier/about.html

And to use Napier's bones in any base: http://www.cut-the-knot.com/blue/Napier.html

http://www-ppg.dcs.st-and.ac.uk/Languages/Napier88/JohnNapier/