Mercator and Euler: Logarithm Function

"The Logarithmotechnia of Nicolas Mercator (1620-1687)was published in 1668. The first two parts of this book were devoted entirely to the calculation of a table of common logarithms.(...)

It is the very different third part of the Logarithmotechnia that is now of principal interest. Here Mercator finds his famous series (apparently used previously by Newton)

for the area under the hyperbola

over the interval from 0 to x."(Edwards, pag. 162)

It was known in the 1660s, as a consequence of the work of Gregory St. Vincent and de Sarasa, that there is a relation between the area under the hyperbola and the logarithm.

Mercator's series for the Logarithm aproximates the function only between 0 and 2. More than that, its rate of convergence is very slow and it is not practical to use this serie to calculate logarithms.

Euler replaced x with -x in Mercator's series and then subtract logarithms to obtain

Euler used a rational function

His series converges everywhere.

We can see in the mathlet how fast this series converges.

REFERENCES

C. H. Edwards - The Historical Development of the Calculus - Springer-Verlag