NICOLAUS I (1687–1759)

Although in the years following he was concentrating on his course to qualify in law, he was at the same time diligently exploring the Theory of Probability. In 1709 he qualified Licencié on the strength of a dissertation in which he applied the Theory of Probability to legal issues; the work was given immediate publication under the title De usu artis conjectandi in Jure, Basel 1709. It was under his supervision that Jacob's great work in Probability Ars Conjectandi was published in Basel in 1713. He was awarded the degree of Doctor of Jurisprudence in 1717.

In 1712 Nicolaus travelled through the Netherlands, England and France. In Paris he met Montmort, with whom there developed both friendship and cooperation. In the booklet Essai d'analyse sur les jeux de hazard published by Montmort there is included the correspondence from Nicolaus in which is presented for the first time the problem in Probability later known as the St. Petersburg Paradox.

In 1716 he succeeded Jacob Hermann as Professor of Mathematics in Padova, but staying only a few years he returned to Basel in 1719 to take the post of Professor of Logic. However, he retained the strong bonds he had formed with Italian scientists, through a sustained extensive correspondence. Twelve years later, in 1731, he made a sideways move when he exchanged the Chair of Logic for the Chair of Law—still in the University of Basel. Meantime during these years at Basel he served five terms as Rector of the University.

Clearly he shouldered more than his share of responsibilities during his career—responsibilities which narrowed his scope for doing mathematical research. We should not be surprised, therefore, that we do not have from his hand a large volume of published work— particularly when compared with that of his two uncles. But he gave a lot of his energy to getting the work of Jacob into print. Besides the work of Jacob on Probability which he edited and made ready for the printer, he was an important support in seeing that the entire work of Jacob was published: it was under the editorship of G. Cramer that it appeared (Opera, Geneva, 1744) and it included an appendix, compiled from Jacob's diary. This diary, entitled Meditationes, Annotationes, Animadversiones ... was Jacob's personal scientific journal. Before his death, Jacob had been selecting from his journal certain items which he clearly intended for publication. Of these, three were transcribed by Jacob himself into a second manuscript: possibly under Jacob's direction, or soon after his death, nine additional items were transcribed by Jacob Hermann into this latter script. Many years later, this was augmented by a further twenty items, selected and transcribed by Nicolaus. Thus was formed the final collection (of 32 items) to which Nicolaus gave the title Varia Posthuma in the Appendix to the Opera.

It is possible that it is in his correspondence—particularly in that significant part to both Leibniz (5 items) and Euler (8 items)—that he shows most clearly his prowess as a mathematician. He would merit his well-won fame even if it were only to acknowledge his pre-eminence when it came to precise analysis. In particular he recognised that in dealing with infinite series it is necessary to address the question of convergence. He is found repeatedly stressing this point—and with good reason—to both giants, Leibniz and Euler.

Of particular interest is his contribution to what Spiess has designated a "spezifisch baslerisches Problem", namely, the summation of the series of the quadratic reciprocals, , the solution of which had evaded both of his famous uncles. By the mid-thirties Euler had arrived at the correct result , which he communicated in a letter to Daniel Bernoulli in 1736. Euler's result appeared in CP VII, 1734/5 (1740), later followed by the derivation in CP IX, 1737 (1744). But it was Nicolaus who provided the elementary and elegant derivation published in CP X, 1738 (1747). The announcement of the result by Euler led to much discussion and correspondence among the mathematicians of Basel, and Nicolaus' elegant derivation places him in the front ranks of that distinguished company. In his published work there is a series of dissertations in which he made an intense study of the differential equation named for Jacopo Riccati, and another series devoted to questions in Mechanics, directed particularly at Orthogonal Trajectories. In this published work it is clear that he would not yield to any of his contemporaries in his mastery of Mechanics, though the extent of his contributions does not compare with Euler, Daniel Bernoulli or Clairaut.

In the controversy between Leibniz and Newton, Nicolaus along with his uncle Johann had unequivocally taken their stance on the side of Leibniz. In their critical scrutiny of Newton's Principia, they made an independent derivation (for the particular case of the semi-circle) of the solution for motion under gravity in a resisting medium (Example 1 to Proposition X in the second book of the 1687 Principia). For the crucial ratio measuring resistance against gravity they arrived at a coefficient 3/2 where Newton had the factor 1. On the occasion of Nicolaus' visit to London in 1712 this result was communicated to Newton who quickly recognized its validity though it took him some time to detect the source of his error. While the Bernoulli derivation gave them priority in furnishing the first correct solution to this problem, their diagnosis of the source of the error in Newton's work was invalid. They mistakenly attributed the error to a mishandling of coefficients in the series expansion— and by implication of the higher derivatives: Johann was to maintain this plausible but erroneous perception long after the appearance of the second edition of the Principia (1713) where the derivation of Proposition X is rectified to yield the correct result. The whole issue is dealt with in detail by D. T. Whiteside in Vol. VIII (pp. 312–424) of his 1980 edition of The Mathematical Papers of Isaac Newton.

In spite of the small volume of published work, it nevertheless leaves no doubt as to his position as a mathematician of the first rank who was far in advance of his contemporaries in recognising precision as an inflexible requirement in mathematical analysis.

Ó Mathúna, 1999