## Solutionem tergemini problematis arithmetici, geometrici et astronomici, una cum adnexis ex universam mathesi corollariis.

Basel: Mechel, [1687].

First edition, exceptionally rare, of this dissertation which Bernoulli (1655-1705) submitted in order to secure the chair of mathematics at the University of Basel. This was the beginning of a remarkable career in mathematics, in which he “greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability [and] was one of the most significant promoters of the formal methods of higher analysis” (DSB). The dissertation, ‘The solution of a triplet of problems, arithmetical, geometrical, and astronomical, together with corollaries from general mathematics,’ treats three elementary problems in number theory, one arising from arithmetic, one from geometry, and one from astronomy/navigation. “Jacob Bernoulli’s research on elementary mathematics, taken as a whole, constitutes a work of no mean importance, very diverse in content, lacking organic unity to be sure, but also of exceptional historic interest. Indeed, only from very few of the mathematicians who have left a lasting mark do we have the documentation which allows us to examine carefully the process of their scientific education … However, in contrast, from Jacob Bernoulli we actually possess some ‘exercises’ that he wrote, beginning during the early years of his education. This is how we might characterize some of his *Meditationes* which he worked on with great diligence and singular ability. The interest of these exercises lies not only in their relationship to the general state of mathematics of the time, but also (and perhaps more) in the way in which they represent an almost complete psychological picture of the formation of a great mathematician – which he certainly was – toward the end of the XVIIth century” (*Werke* 2, p. 15). These *Meditationes* remained unpublished until the twentieth century, except for the three published, with much additional detail, in the present pamphlet. On the last page of the dissertation is a list of ‘Corollaries’, one of which concerns the values of expectations in a lottery, and is of particular interest in view of Bernoulli’s posthumously-published *Ars conjectandi* (1713), the founding work of mathematical probability. “The academic dissertation *Solutionem tergemini problematis arithmetici, geometrici et astronomici*, which was presented at the University of Basel (4.2.1687), secured him the desired teaching post. Therefore this paper has particular biographical interest” (*ibid*., p. 18). OCLC lists only 5 copies worldwide (Yale only in US); not on COPAC.

“Let us recall briefly what we know of Jacob Bernoulli's education before he obtained, at the age of 34, the chair in mathematics in his hometown Basel. Jacob was born in this city on December 27th, 1654 (according to the old calendar) in a protestant family of spice traders who had fled the Spanish low lands after the fall of the Duke of Alba. Complying with the wish of his father Nicolas Bernoulli, a state adviser and magistrate, Jacob studied philosophy and then theology until 1676. As was common at the time, he chose a motto. His came from Phaeton who drew the solar carriage *Invito patre sidera verso*, which may be translated by ‘Despite my father, I am among the stars’. Rather than exaggerated modesty, this motto was a proud affirmation of superiority.

“The young Jacob fully benefited from what Daniel Roche calls ‘culture de la mobilité’, promoted in the second half of the XVII century by new institutions which facilitated the movement of individuals and the spread of knowledge. Starting in August 1676, he traveled by horse to Geneva where he remained for twenty months preaching, instructing a blind young girl, Elisabeth von Waldkirch, and serving as an opponent during the theological disputations. He relates his experience teaching mathematics to the blind in an article published in the *Journal des Savants* in 1685. This article is probably a reaction to an account by Spon published in the same journal in 1680, in which the author attributes to the father of the blind girl the writing system that was in fact developed by Jacob. It is here that Jacob meets Nicolas Fatio de Duillier, a life long friend who recalled, in a letter dating from July 22nd, 1700, that he had seen Jacob play court tennis in Geneva, a game on which Jacob later wrote a

famous letter …

“In June 1678, Jacob continues his extensive traveling in France, residing in the

Limousin (in Nède with the marquis de Lostanges, where he constructs two sundials in the castle courtyard), then in Bordeaux and a few weeks in Paris. During this journey, he begins, in 1677, to write his mathematical journal, *Meditationes, annotationes, animadversiones theologicae et** philosophicae*, which contains 236 articles … The journal is a precious testimony from this early phase of Jacob's scientific training which only really began when he encountered the Cartesian

environment, initially in France, later mainly in the Netherlands (Amsterdam and Leiden) and in England during a second journey (April 1681- October 1682). In August 1682, Jacob attended a meeting of the Royal Society in London. Jacob started out by acquainting himself with the Cartesian philosophy of nature after which he turned to geometry …

“After his return to Basel in 1682, Jacob gave up the idea of a career in the clergy and decided to devote himself to mathematics. At the University of Basel he gave courses in experimental physics, as can be gathered by a pamphlet printed in Basel in 1686.

From 1682 on, he also submitted short articles to the *Journal des Savants* – reactions to the works of others that he presented or criticized – initially in the area of natural philosophy (machines for breathing under water, to elevate water, to weigh air, oscillation center), then from 1685, in mathematics …

“[Jacob] slowly acquired a knowledge of mathematics, at first through his readings of the second Latin edition of Descartes' *Géométrie* (1659-61), later that of Arnauld and his *Logique,* Malebranche and Prestet … Jacob is confronted with precise problems, often stemming from the area of applied mathematics. Solving these leads him to discover general methods. He begins by a thorough study of previous works, which will serve him as a springboard to make further headway and produce new results. On several occasions, Jacob voices the opinion that it is necessary to base one’s own progress on the knowledge of what has been done in the past. Accordingly, in the memoir entitled *Solutionem tergemini problematis arithmetici, geometrici et astronomici* [offered here], presented on February 4th, 1687, in order to obtain the mathematics chair in Basel, he describes his own way of proceeding in the following way: ‘In reality, he who embraces a career as a mathematician is not the one who copies the inventions of others, remembers them and recites them on occasion, but the one who is truly innovative and is able to invent by using the divine algebra and thus to revolutionize what has been studied by others’” (Peiffer).

“Like other graduates of the University of Basel in the seventeenth century, Bernoulli had a broad, if not deep, knowledge of all the disciplines of the liberal arts – an education he would draw on in writing *Ars conjectandi* (1713). The university at this time had a small faculty of philosophy, including only nine chairs, in logic, rhetoric, eloquence, Greek, mathematics, physics, history, ethics, and Hebrew. So, although he had only an elementary education in mathematics as an undergraduate, Bernoulli had a knowledge of Greek, logic, rhetoric, and other subjects, and was able to write polished Latin.

“It was not uncommon at this time for a man to take a university post in a less desirable discipline if the chair he would have preferred was not available. He would then move, if the opportunity arose, to the preferred discipline, without having to do everything required of a new applicant for a university position, such as paying to print lists of theses to be defended in a disputation … The lists of theses that Jacob Bernoulli proposed to defend publicly in the years after his return to Basel in October 1682 reflect this list of open positions. Thus, in January 1684, of the 100 theses that Bernoulli proposed to defend, 34 were logical and 18 oratorical … Again, in September 1685 and in February 1686, Bernoulli’s theses were heavily logical. Finally, after the death of the professor of mathematics in 1686, Bernoulli applied for, disputed, and won the chair” (Sylla, pp. 6-7).

The first of the three problems treated by Bernoulli in his *Solutionem tergemini problematis* is: ‘to find, without the aid of algebra, with the help of numerical arithmetic alone, a number such that, if we divide it into the numbers 12 and 36, and then add 8 to each of these numbers, the resulting sums are in the ratio of 3 to 5.’ The problem is generalized to arbitrary sets of given numbers.

The second problem is more challenging. It “calls for the construction of a quadrilateral inscribed in a semi-circle, with sides and diagonals of rational lengths (in other words, commensurable to the radius)” (*Werke *2, p. 18).

The third problem is one which could arise in navigation: ‘It is observed somewhere at the sixth hour of the sun, that it has an altitude of 12 degrees above the horizon, and that one hour and 12 minutes after the moment of this observation the sun sets. The question is, at what latitude (and at what time of the year) was the observation made?’

On the last page of text, under ‘Corollaria’, Bernoulli lists some of his other *Meditationes* – problems dealing with logic, physics, meteorology, geometry (referring to Propositions 55-57 of Euclid, Book X), stereometry, mechanics (perpetual motion machines), dioptrics, perspective (referring to Bosse and Desargues), gnomonics (referring to Münster and Sturm), ballistics, ‘ars conjecturandi’ (probability), and figurate numbers (giving the very large example 1,580,972).

The problem on probability “concerned the slowness with which the values of expectations in a lottery increase as more and more blank slips are drawn out of the urn. In an urn containing 16,000 slips, if it initially cost 7½ to participate, 1000 blank slips in a row would have to be drawn out before the person could sell his expectation to someone else for 8” (Sylla, p. 27).

Peiffer, ‘Jacob Bernoulli, teacher and rival of his brother Johann,’ *Electronic Journal for History of Probability and Statistics* 2, 2006. Sylla (tr.), *Jacob Bernoulli. The Art of Conjecturing*, 2006.

Small 4to (198 x 156 mm), pp. [16], including a full page of geometrical diagrams printed on title verso. Contemporary marbled wrappers.

Item #5601

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Price:
$4,500.00
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