On the Discovery of the Logarithmic Series and Its Development in England up to Cotes

Summary

To the expert of today the logarithmic series appears to be a very non-essential detail. In its time it was a very notable discovery as regards itself alone, as well as in the framework of the general theory of series. It was discovered circa 1667 by Newton and independently by Mercator. Huygens and Gregory were close to the same discovery but they were anticipated by the other two. Newton was then 24 years old, Mercator 47. For Newton the logarithmic series was a beginning, for Mercator the climax.

Nicolas Mercator

(1620-1687)

Mercator's life work is almost forgotten today, certainly unjustly. Mercator was a distinguished mathematician, physicist and astronomer. Shortly after his arrival in London the much-traveled man was received into the Royal Society. Products of that period are his new astronomical theory [10], the edition of Euclid [11], the navigation problems [12] and the calculation of logarithms [13]. We shall be concerned here with the latter.

The Logarithmo-technia is divided into three very unequal parts. The first two sections, which had already been published separately in 1667, are devoted entirely to the calculation of a system of common logarithms. In the presentation of logarithms Mercator proceeds very intuitively and clearly according to the then generally customary usage. He divides the number domain between 1 and 10 by insertion of geometric means (he calls them ratiunculae) into 10 million parts. Thus the logarithm of a number between 1 and 10 is determined from the number of ratiunculae between 1 and this number.