Tobias Dantzig on Jacob Bernoulli

“The tract in which Leibnitz deals with series appeared late in the seventeenth century and was among the first on the subject. ...the question of their convergence or divergence ...was in those days more or less ignored. ...It was not until the publication of Jacques Bernoulli's work on infinite series in 1713 that a clearer insight into the problem was gained. ...Bernoulli's work directed attention towards the necessity of establishing criteria of convergence. The evanescence of the general term, i.e., of the generating sequence, is certainly a necessary condition, but this is generally insufficient. Sufficient conditions have been established by d'Alembert and Maclauren, Cauchy, Abel, and many others. ...to recognized whether a series converges or diverges is even today rather difficult in some cases.”

“The great invention... Descartes gave to the world, the analytical diagram, ...gives at a glance a graphical picture of the law governing a phenomenon, or of the correlation which exists between dependent events, or of the changes which a situation undergoes in the course of time. ...the invention of Descartes not only created the important discipline of analytic geometry, but it gave Newton, Leibnitz, Euler, and the Bernoullis that weapon for the lack of which Archimedes and later Fermat had to leave inarticulate their profound and far-reaching thoughts.”