Logarithms Invented

In 1614, the Scottish mathematician John Napier, Laird of Merchiston, published Mirifici Logarithmorum Canonis Descriptio ("Description of the Wonderful Canon of Logarithms"). The book introduced logarithms to the world and included tables that let users replace laborious multiplication and division with addition and subtraction. For astronomers, navigators, and surveyors, who spent enormous effort on long calculations, it was one of the most practical mathematical advances of the early modern era.

The need was real. By the late sixteenth century, astronomy and navigation relied on trigonometry, and computing with sines and large numbers often meant many digits and many opportunities for error. Mathematicians had already used tricks such as prosthaphaeresis, which turned products of trigonometric values into sums using angle identities. Napier, a landowner and theologian who pursued mathematics as a private passion, reportedly worked on the problem for about two decades. He was aware of the pattern linking arithmetic and geometric progressions, which underlies the idea.

Napier's approach was geometric and kinematic rather than based on the exponents and bases familiar today. He imagined two points moving along lines, one at a steady speed and the other at a speed proportional to its remaining distance, and defined the logarithm through their relationship. His logarithms differ from modern natural logarithms, though they are closely related, and the concept of a "base" had not yet been formulated. The 1614 work explained the idea briefly and presented tables of logarithms of sines. A later work on the construction of the tables was published after his death in 1617.

The reception was swift and enthusiastic. The English mathematician Henry Briggs traveled to Scotland to meet Napier, and the two reportedly discussed adjusting the system so that the logarithm of 1 would be zero and the logarithm of 10 would be a power-of-ten benchmark. Briggs went on to publish tables of base-10 logarithms, which became the standard for practical computation. The Swiss instrument maker Jost Bürgi had independently developed a similar idea, apparently earlier, but he published his tables in 1620, and Napier's work had the greater influence.

Logarithms spread quickly across Europe, and Johannes Kepler used them in his astronomical tables. Edmund Gunter and William Oughtred built on the principle to create the slide rule, which remained a staple of science and engineering for more than three centuries until electronic calculators replaced it in the 1970s. Later mathematicians connected logarithms to exponential functions and calculus, giving them a central place in pure mathematics as well. The idea still underlies scales used to measure earthquakes, sound, and acidity, a lasting legacy of one Scottish nobleman's patient labor.