## Jakob I Bernoulli

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## Bernoulli, Jakob (Jacques) I

# Bernoulli, Jakob (Jacques) I

(*b*. Basel, Switzerland, 27 December 1654; *d*. Basel, 16 August 1705)

*mathematics, mechanics, astronomy*.

Bernoulli came from a line of merchants. His grandfather, Jakob Bernoulli, was a druggist from Amsterdam who became a citizen of Basel in 1622 through marriage. His father, Nikolaus Bernoulli, took over the thriving drug business and became a member of the town council and a magistrate; his mother, Margaretha Schönauer, was the daughter of a banker and town councillor. Jakob was married in 1684 to Judith Stupanus, the daughter of a wealthy pharmacist; their son Nikolaus became a town councillor and master of the artists’ guild.

Bernoulli received his master of arts in philosophy in 1671, and a licentiate in theology in 1676; mean-while, he studied mathematics and astronomy against the will of his father. In 1676 he went as a tutor to Geneva, where in 1677 he began his informative scientific diary, *Meditationes*; he then spent two years in France, familiarizing himself with the methodological and scientific opinions of Descartes and his followers, among whom Was Nicolas Malebranche. Bernoulli’s second educational journey, in 1681–1682, took him to the Netherlands, where he met mathematicians and scientists, especially Jan Hudde, and to England, where he met Robert Boyle and Robert Hooke. The scientific result of these journeys was his inadequate theory of comets (1682) and a theory of gravity that was highly regarded by his contemporaries (1683).

After returning to Basel, Bernoulli conducted experimental lectures, concerning the mechanics of solid and liquid bodies, from 1683 on. He sent reports on scientific problems of the day to the *Journal des sçavans* and the *Acta eruditorum*, and worked his way through the principal mathematical work of those days, *Geometria*, the Latin edition of Descartes’s *Géométrie*, which had been edited and provided with notes and supplements by Frans van Schooten (2nd ed., Amesterdam, 1659–1661). As a result of this work, Bernoulli contributed articles on algebraic subjects to the *Acta eruditorum*. His outstanding achievement was the division of a triangle into four equal parts by means of two straight lines perpendicular to each other (1687). After these contributions had been extended and supplemented, they were published as an appendix to the *Geometria* (4th ed., 1695).

In four disputations published from 1684to 1686, Bernoulli presented formal logical studies that tended toward the sophistical. His first publication on probability theory dates from 1685. By working with the pertinent writings of John Wallis (those of 1656, 1659, and 1670–1671) and Isaac Barrow (1669–1670), concerning mathematical, optical, and mechanical subjects, Bernoulli was led to problems in infinitesimal geometry.

In the meantime his younger brother Johann began attending the University of Basel after an unsuccessful apprenticeship as a salesman. As respondent to one of Jakob’s scholarly logic debates, Johann earned his master of arts degree in 1685 and by order of his father, studied medicine. Simultaneously, however, he secretly studied mathematics under his brother, becoming well versed in the fundamentals of the field. In 1687 Jakob became professor of mathematics at Basel, and with his brother he studied the publications of Leibniz and of Ehrenfried Walther von Tschirnhaus in *Acta eruditorum* (1682–1686), which had in essence been limited to examples and intimations of infinitesimal mathematics and its application to mechanics and dynamics. After much effort, Bernoulli was able to make himself master of those new methods, which he erroneously believed to be merely a computational formalism for Barrow’s geometrical treatment of infinitesimals, His mathematical studies reached a first peak about 1689 with the beginnings of a theory of series, the law of large numbers in probability theory, and the specials stress on complete induction.

Bernoulli showed his mastery of the Leibnizian calculus with his analysis (in May 1690) of the solutions given by Huygens in 1681 and by Leibniz in 1689 to the problem of the curve of constant descent in a gravitational field. (It was in that analysis that the term “integral” was first used in its present mathematical sense.) The determination of the curve of constant descent had been posed as a problem by Leibniz in 1687. As a counter problem Bernoulli raised the determination of the shape of the catenary, to which he had, perhaps, been directed by Albert Girard’s notes to the *Oeuvres* of Simon Stevin (1634); Girard claimed that the catenary is a parabola. Leibniz promptly referred to the significance of this counterproblem, which he had spontaneously solved (1690) and which was later treated by Johann Bernoulli, Huygeus and himself in the *Acta eruditorem* (1691). Jakob, who found himself at that time in difficulties at the university because of his open criticism of university affairs and saw himself being overshadowed by his brother, did not take part directly, but proposed generalizations of the problem, allowing the links of the chain to be elastic or of unequal weight. He also announced a treatise on the *elastica*, the form of a bent elastic beam, which, under certain conditions, satisfies the differential equation *dy/dx* = . Later he investigated this thoroughly, supposing arbitrary functions of elasticity (1694). In two notable contributions to differential calculus (1691), he examined the parabolic spiral (in polar coordinates: , the elliptical integral for the curve length with its charateristic feature of symmetry) and the logarithmic spiral.

In Johann Bernoulli’s study concerning the focal line of incident parallel rays of light on a semicircular mirror (1692), there is reference to Jakob’s general procedure for determination of evolutes. This procedure is based on the generation of an algebraic curve as the envelope of its circles of curvature, and this procedure is worked out fully in the case of the parabola. Here Bernoulli corrected a mistake made by Leibniz (1686)—the statement that the circle of curvature meets the curve at four coinciding points—but he himself made a mistake in his assertion that the radius of curvature becomes infinite at every point of inflection. This error, corrected in 1693 by G. F. A. de L’Hospital, was the occasion for Bernoulli’s removing of the singularity *a*^{2}*x*^{3} = *y*^{5} origin (1697). Almost simultaneously, and independently of each other, the brothers recognized that the form of a sail inflated by the wind is described by (*dx*/*ds*)^{3} = *ad*^{2}*y*/*dx*^{2}. Jakob made a preliminary report in 1692 and a thorough one in 1695.

Further investigations concerned evolutes and caustics, first of the logarithmic spiral (*spira mirabilis*) and the parabola (1692), and later of epicycloids (1692) and diacaustic surfaces (1693), this in connection with Johann’s similar studies. These last were included in his private instruction to L’Hospital (1691–1692). Here, for the first time, public reference was made to the *theorema aureum*, which had been developed in the spring of 1692. The theorem, which gives the radius of curvature as (*ds*/*dx*)^{3}: (*d*^{2}*y*/*dx*^{2}), was published in 1694. Bernoulli’s solution of the differential equation proposed by Johann Bernoulli (1693), *xdy* – *ydx/yds* = *a/b*, was completed by Huygens (1693). In treating the paracentric isochrone, a problem proposed by Leibniz (1689) that leads to the differential equation

Bernoulli separated the variables by substituting

x^{2}+y^{2}=r^{2},ay=rt

and was able to relate the solution to the rectification of the *elastica;* later he found the reduction to the rectification of the lemniscate,

These and other studies—among which the kinetic geometrical chord construction for the solution (1696) of *dy/dx* = *t*(*x*)/*a* and the solution (1696) of the so called Bernoullian differential equation *y′* = *p*(*x*)*y* + *q*(*x*)*y*^{n} (1696) merit special attention—are proof of Bernoulli’s careful and critical work on older as well as on contemporary contributions to infinitesimal mathematics and of his perseverance and analytical ability in dealing with special pertinent problems, even those of a mechanical-dynamic nature.

Sensitivity, irritability, a mutual passion for criticism, and an exaggerated need for recognition alienated the brothers, of whom Jakob had the slower but deeper intellect. Johann was more gifted in working with mathematical formulations and was blessed above all with a greater intuitive power and descriptive ability. Johann was appointed a professor at the University of Groningen in 1695, and in 1696 he proposed the problem to determine the curve of quickest descent between two given points, the brachistochrone. In connection with this he replied to the previous gibes of his brother with derisive insinuations. Jakob gave a solution (1697) that was closely related to that given by Leibniz (1697). It is based on the sufficient but not necessary condition that the extreme-value property of the curve in question (a common cycloid) is valid not only for the entire curve but also for all its parts. As a counter problem Jakob set forth the so-called isoperimetric problem, the determination of that curve of given length between the points *A*(– *c*; 0), *B*(+ *c*; 0) for which takes a maximum value. Johann, in the *Histoire desouvrages de savants* (1697), through a misunderstanding of the difficulty of the problem and of its nature (calculus of variations), gave a solution based on a differential equation of the second degree. A differential equation of the third degree is necessary however, After showing that a third-degree equation is required (1701), Jakob was able also to furnish the proof which Johann and Leibniz had been seeking in vain, that the inexpansible and homogeneous catenary is the curve of deepest center of gravity between the points of suspension.

Johann Bernoulli may have comprehended the justification for his brother’s argument soon after publication of the dissertation of 1701 (*Analysis magni problematis isoperimetrici*), but he remained silent. Only after Brook Taylor had adopted Jakob’s procedure (1715) was he induced to accept Jakob’s point of view. In the 1718 series of the *Mémoires* of the Paris Académic des Sciences, of which the brothers had been corresponding members since 1699, Johann gave a presentation, based on Jakob’s basic ideas but improved in style and organization. It was not superseded until Leonhard Euler’s treatment of the problems of variations (1744).

The antagonism between the brothers Soon led tougly critical remarks. In 1695 Jakob failed to appreciate the significance of Johann’s extraordinarily effective series expansion (1694), which is based on iterated integration by parts and leads to a remainder in integral form. On the other hand, Johann, who, in 1697 had challenged the criterion for geodetic lines on convex surfaces, complained in the following year that his brother knew how to solve the problem “only” on rotation surfaces. Other items of disagreement were the determination of elementary quadrable segments of the common cycloid and related questions. The brothers argued over this in print in 1699 and 1700. The formulas for the multisection of angles are connected with these problems. In his ingenious use of Wallis’ incomplete induction (1656), Jakob presented 2 · cos *nα* and 2 · sin *nα* as functions of 2 · sin α. This is related to his notes from the winter of 1690–1691, in which, furthermore, the exponential series was derived from the binomial series in a bold but formally unsatisfactory manner.

Jakob Bernoulli’s decisive scientific achievement lay not in the formulation of extensive theories but in the clever and preeminently analytical treatment of individual problems. Behind his particular accomplishments there were, of course, notions of which Bernoulli was deeply convinced, primarily concerning continuity of all processes, of nature (*natura non facit saltum*). Although Bernoulli assigned great significance to experimental research, he limited himself—for example, in investigations of mechanics—to a few basic facts to which he tried to cling and an which he sought to base full theories. For this reason his final results were intellectually interesting, and as points of departure they were significant for further investigation by his contemporaries and subsequent generations. Naturally enough, they usually do not conform to more modern conclusions, which rest on far wider foundations. It is to be regretted, however, that Bernoulli’s contributions to mechanics are hardly ever mentioned in the standard works.

The theory that seeks to explain natural phenomena by assuming collisions between particles of the ether, developed in the *Dissertatio de gravitate aetheris* (1683), of course does not mean much to a later generation. There are extensive discussions about the center point of oscillation, which had been determined correctly for the first time by Huygens in his *Horologium oscillatorium* (1673), but this was strongly debated by some of the members of the Cartesian school. On this subject Bernoulli expressed his opinions first in 1684, and then in more detail in 1686 and 1691; finally he succeeded in developing a proof from the properties of the lever (1703–1704). Important also is his last work, on the resistance of elastic bodies (1705). Supplementary material from his scientific diary is contained in the appendix to his *Opera*. Additional, but unpublished, material deals with the center of gravity of two uniformly moved bodies, the shape of a cord under the influence of several stretching forces, centrally accelerated motion (in connection with the statements of Newton in his *Principia*[1687]), and the line of action and the collective impulse of infinitely many shocks exerted on a rigid arc in the plane.

In the field of engineering belongs the 1695 treatment of the drawbridge problem (the curve of a sliding weight hanging on a cable that always holds the drawbridge in balance), stemming from Joseph Sauveur and investigated in the same year by L’Hospitaland Johann Bernoulli. Leibniz was also interested in the problem. In Bernoulli’s published remains, the contour is determined upon which a watch spring is to be developed so that the tension always remains the same for the movements of the watch.

The five dissertations in the *Theory of Series* (1682–1704) contain sixty consecutively numbered propositions. These dissertations show how Bernoulli (at first in close cooperation with his brother) had thoroughly familiarized himself with the appropriate formulations of questions to which he had been led by the conclusions of Leibniz in 1682 (series for *π*/4 and log 2) and 1683 (questions dealing with compound interest). Out of this there also came the treatise in which Bernoulli took into account short-term compound interest and was thus led to the exponential series. He thought that there had been nothing printed concerning the theory of series up until that time, but he was mistaken: most conclusions of the first two dissertations (1689, 1692) were already to be found in Pietro Mengoli (1650), as were the divergence of the harmonic series (Prop. 16) and the sum of the reciprocals of infinitely many figurate numbers (Props. 17–20).

The so-called Bernoullian inequality (Prop. 4), (1 + *x*)^{n} > 1 + *nx*, is intended for *x* > 0, *n* as a whole number > 1. It is taken from Barrow’s seventh lecture in the *Lectiones geometricae* (1670). Bernoulli would have been able to find algebraic iteration processes for the solution of equations (Props. 27–35) in James Gregory’s *Vera… quadratura* (1667). The procedure of proof is still partially incomplete because of inadmissible use of divergent series. At the end of the first dissertation Bernoulli acknowledged that he could not yet sum up in closed form (Euler succeeded in doing so first in 1737); but he did know about the majorant , which can be summed in elementary terms. In Proposition 24 it is written that equals (2^{m} – 1)/1 (*m* integer > 1), and that diverges more rapidly than . Informative theses, based on Bernoulli’s earlier studies, were added to the dissertations: and theses 2 and 3 of the second dissertation are based on the still incomplete classification of curves of the third degree according to their shapes into thirty-three different types.

The third dissertation was defended by Jakob Hermann, who wrote Bernoulli’s obituary notice in *Acta eruditorum* (1706). In the introduction L’Hospital’s *Analyse* is praised. After some introductory propositions, there appear the logarithmic series for the hyperbola quadrature (Prop. 42), the exponential series as the inverse of the logarithmic series (Prop. 43), the geometrical interpretation of

(Prop. 44), and the series for the arc of the circle and the sector of conic sections (Props. 45, 46). All of these are carefully and completely presented with reference to the pertinent results of Leibniz (1682; 1691). In 1698 previous work was supplemented by Bernoulli’s reflections on the catenary (Prop. 49) and related problems, on the rectification of the parabola (Prop. 41), and on the rectification of the logarithmic curve (Prop. 52).

The last dissertation (1704) was defended by Bernoulli’s nephew, Nikolaus I, who helped in the publication of the *Ars conjectandi* (1713) and the reprint of the dissertation on series (1713) and became a prominent authority in the theory of series. In the dissertation Bernoulli first (Prop. 53) praises Wallis’ interpolation through incomplete induction. In Proposition 54 the binomial theorem is presented, with examples of fractional exponents, as an already generally known theorem. Probably for this reason there is no reference to Newton’s presentation in his letters to Leibniz of 23 June and 3 November 1676, which were made accessible to Bernoulli when they were published in Wallis’ *Opera* (Vol. III, 1699). In proposition 55 the method of indeterminate coefficients appears, without reference to Leibniz (1699). Propositions 56–58 and 60 deal with questions related to the *elastica*.

In Proposition 59 it is stated that the series

for log 2 should be replaced by

which converges more rapidly. From the letter to Leibniz of 2 August 1704, we know that in Proposition 59 Bernoulli used an idea of Jean-Christophe Fatio-de-Duillier (1656–1720), an engineer from Geneva, for the improvement of convergence. The procedure was expanded by Euler in the *Institutiones calculi differential is* (1755) to his so-called series transfonnation. In the dissertations on series Bernoulli apparently wished to reproduce everything he knew about the subject. In this he was primarily concerned with the careful rendering of the results and not so much with originality.

The *Ars conjectandi* is Bernoulli’s most original work, but unfortunately it is incomplete. The first part is basically a first-rate commentary on Huygens’ *De ratiociniis in aleae ludo*, which was published as an appendix to van Schooten’s *Exercitationes mahematicae* (1657). In the second part Bernoulli deals with the theory of combinations, based on the pertinent contributions of van Schooten (1657), Leibniz (1666), Wallis (1685), and Jean Prestet’s *Élémens de mathématiques* (1675; 2nd ed., 1689). The chief result here is the rigid derivation of the exponential series through complete induction by means of the so-called Bernoullian numbers. In the third part Bernoulli gives twentyfour examples, some simple, some very complicated, on the expectation of profit in various games.

The fourth part contains the philosophical thoughts on probability that are especially characteristic of Bernoulli: probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori and a posterioti probability; expectation of winning when the players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers, and reference to the *Art de penser (Logique de Port Royal*, Antoine Arnauld and Pierre Nicole, eds., 1662). The last section contains a penetrating discussion of *jeu de paume*, a complicated predecessor of tennis that was very popular. This part is Bernoulli’s answer to the anonymous gibes occasioned by his debate of 1686 on scholarly logic.

Bernoulli’s ideas on the theory of probability have contributed decisively to the further development of the field. They were incorporated in the second edition of Rémond de Montmort’s *Essai* (1713) and were considered by Abraham de Moivre in his *Doctrine of Chances* (1718).

Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability. He was self-willed, obstinate, aggressive, vindictive, beset by feelings of inferiority, and yet firmly convinced of his own abilities. With these characteristics, he necessarily had to collide with his similarly disposed brother. He nevertheless exerted the most lasting influence on the latter.

Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity. The following lines taken from the *Arsconjectandi* (published posthumously in 1713) are not without a certain grace, however, and represent a nearly statement, made with wit and clarity, of the boundaries of an infinite series.

Ut non-finitam Seriem tinita cöercet,

Summula, & in nullo limite limes adest:

Sic modico immensi vestigia Numinis haerent

Corpore, & angusto limite limes abest.

Cernere in immenso parvum, dic, quanta voluptas!

In parvo immensum cernere, quanta, Deum!

Even as the finite encloses an infinite series

And in the unlimited limits appear,

So the soul of immensity dwells in minutia

And in narrowest limits no limits inhere.

What joy to discern the minute in infinity!

The vast to perceive in the small, what divinity!

## BIBLIOGRAPHY

I. Original Works. Bernoulli’s most famous single writing is *Ars conjectandi* (Basel, 1713; Brussels, 1968). His *Opera*, G. Cramer, ed, (Geneva. 1744; Brussels. 1968), contains all his scientific writings except the *Neuerfundene**Anleitung, wie man den Lauff der Comet-oder Schwantzsternen in gewisse grundmässige Gesätze einrichten und ihre Erscheinung vorhersagen Könne (Basel, 1681), as well as a Prognosticon*. Its contents were incorporated in the *Conamen novi systematis cometarum*… (Amsterdam, 1682), which is reproduced in the *Opera* as part 1.

Two collections of letters are printed: those to Leibniz in Leibniz’ *Mathematische Schriften*, Vol. III, C. I. Gerhardt, ed. (Halle, 1855: Hildesheim, 1962); those to his brother Johann, in *Der Briefwechsel Von Johann Bernoulli*, Vol. I, O. Spiess, ed. (Basel, 1955).

His MSS at the library of the University of Basel are *Reisebüchlein* (1676–1683); *Meditationes, annotationes, animadversiones* (1677–1705); *Stammbuch* (1678–1684); *Tabulae gnomicae. Typus locorum hypersolidorum*, which concerns classification of the curves of the third degree into thirty-three types; *Memorial über die Missbräuche an der Universität* (1691); and *De arte combinatoria* (1692) (all unpublished manuscripts); and “De historia cycloidis” (1701), in *Archiv für Geschichte der Mathematik und Naturwissenschaften*, **10** (1927–1928), 345 ff.

Bernoulli’s unpublished manuscripts at the library of the University of Geneva are lectures on the mechanics of solid and liquid bodies, *Acta collegii experimentalis* (1683–1690), parts of which have been transcribed.

The collected works are in preparation. Included are Bernoulli’s correspondence with Nicolas Fatio-de-Duillier (1700–1701) and with Otto Mencke (1686, 1689). The most important correspondence with L’Hospital and Pierre Varignon seems to have been lost.

Translations from the *Opera* are in the series Ostwald’s Klassiker der Exacten Wissenschaften: *Unendliche Reihen* (1689–1704), translated into German by G. Kowalewski, no. 171 (Leipzig, 1909); and *Abhandlungen über das Gleichgewicht und die Schwingungen der ebenen elastischen Kurven von Jakob Bernoulli (1691, 1694, 1695) und Leonh. Euler (1744)*, translated into German by H. Linsenbarth, no. 175 (Leipzig, 1910). Other translations are Abhandlungen über Variations-Rechnung. Erster Theil: *Abhandlungen von Joh. Bernoulli (1696), Jac. Bernoulli (1697) und Leonhard Euler*, translated into German by P. Stäckel, no. 46 (Leipzig, 1894; 2nd ed., 1914); and *Jakob Bernoulli: Wahrscheinlichkeitsrechnung*, translated into German by R. Haussner, nos. 107 and 108 (Leipzig, 1899). The latter was translated into English by Fr. Masères in his *Doctrine of Permutations and Combinations* (London, 1795) and in *Scriptores logarithmici*, Vol. III (London, 1796).

II. Secondary Literature. Works concerning Bernoulli and his contributions to mathematics are P. Dietz, “The Origins of the Calculus of Variations in the Works of Jakob Bernoulli, in *Verhandlungen der Naturforschenden Gesellschaft in Basel*, **70** (1959), 81–146, a dissertation presented at the University of Mainz in 1958; J. O. Fleckenstein. *Johann und Jakob Bernoulli* (Basel, 1949), which is supp. 6 to the journal *Elemente der Mathematik*; J. E. Hofmann, *Uber Jakob Bernoullis Beiträge zur Infinitesimalmathematik* (Geneva, 1956), no. 3 in the series *Monographies de I’Enseignement Mathématique;* and O. Spiess, “Bernoulli,” in *Neue deutsche Biographie*, II (Berlin, 1955), 128–129, and “Jakob Bernoulli,” *ibid*., pp. 130–131, which include supplementary bibliographical material.

The poem “On Infinite Series,” Helen M. Walker, trans., appears in D. E. Smith. *A Source Book in Mathematics*; repr. by permission Harvard University Press.

J. E. Hofmann

## Bernoulli, Jakob (Jacob, Jacques, James) I

# BERNOULLI, JAKOB (JACOB, JACQUES, JAMES) I

Basel, Switzerland, 27 December 1654; *d*. Basel, 16 August 1705)

*mathematics, mechanics, astronomy*. For the original article on Bernoulli see *DSB*, vol. 2.

When the original *DSB* article was written, the collected works of Jakob Bernoulli were still in preparation, although volume 1 of *Die Werke von Jakob Bernoulli*, on astronomy and natural philosophy, actually appeared before volume 2 of the *DSB*. Work on the papers of the Bernoulli family had been underway for many years and the results were known to some scholars, but after 1970 many scholars, including especially the editors of *Die gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli*, published on the Bernoullis using a wealth of newly or more easily available evidence. The volumes of Jakob Bernoulli’s *Werke* include not only his published books and journal articles but also passages from his meticulous research notebook and the hard-to-find pamphlets he had published in connection with lectures and disputations at the University of Basel. The introductions to and commentaries on the texts, as well as the work in editing previously unpublished materials, are of very high quality. Needless to say, materials that appeared in the *Werke* of Johann and Daniel Bernoulli are also important for understanding Jakob Bernoulli. Because J. E. Hofmann’s article in the original *DSB* describes the main points of Jakob Bernoulli’s mathematical and scientific work, this postscript will emphasize historiographic, social, and institutional factors.

**The Multiple Bernoullis.** Jakob Bernoulli was the first famous Bernoulli scientist or mathematician, but he was followed into science and mathematics first by his brother Johann and then by many later Bernoullis of the next generations, starting with their nephew Nikolaus and then Johann Bernoulli’s sons. Because the family chose to repeat the same given names from generation to generation, and because these given names appear in many languages (so that Jakob in German may appear as Jacob in Latin, Jacques in French, James in English, and so forth), confusion between different Bernoullis is not infrequent. In order to create some order, it is standard to number the most famous Bernoullis, starting with Jakob I, the subject of this postscript, his brother Johann I, their nephew Nikolaus I, and so forth. A frequently repeated error is to say that the nephew, Nikolaus I Bernoulli (1687–1759), helped in the posthumous publication of *Ars Conjectandi* in 1713. When Jakob I Bernoulli died in 1705, there had been so much struggle between Jakob I and Johann I over their respective contributions to mathematics that Jakob’s wife and son resolved to keep his unpublished papers out of the hands of other family members. After many years, Jakob’s *son* Nikolaus (1687–1769, an artist and not a mathematician and therefore without a number, but sometimes labeled “Nikolaus the younger” to distinguish him from his uncle “Nikolaus the elder,” 1662–1716, also a painter and the father of Nikolaus I) finally sent the manuscript of *Ars Conjectandi* to be printed. Only when the printing was nearly complete did Jakob’s *nephew* Nikolaus I return from a long period of travel, compile one page of errata, and write a two-page introduction to the reader for the book.

Because Nikolaus I had been active in continuing his uncle’s work in mathematical probability in the years between Jakob’s death in 1705 and the publication of *Ars Conjectandi* in 1713, it is not unimportant to know that

the book was published with no emendation by Nikolaus I beyond the one page of errata. Although one might wonder at Nikolaus I’s extensive use of his uncle’s unpublished work, first in his law dissertation *De Usu Artis Conjectandi in Jure*(1709) and then in extensive letters to Pierre Rémond de Montmort, published in the second edition of Montmort’s *Essay d’analyse sur les jeux de hazard*(1713; just after *Ars Conjectandi*), many historians credit Nikolaus I with helping to establish and disseminate the new discipline of mathematical probability, particularly in its applications to law. Given that Johann I in correspondence repeatedly downplayed the importance of publishing *Ars Conjectandi*, Nikolaus I’s promotion of that work, even while borrowing from it, was an important contribution. Nikolaus I helped spread Jakob’s probability work not only to Montmort but also to Abraham De Moivre, who took the next giant steps in developing the mathematics of probability.

Although the very useful articles on the Bernoullis available from the St. Andrews MacTutor History of Mathematics Archive acknowledge their significant indebtedness to *DSB* articles, repeated copying of the MacTutor articles on other Web sites may lead to a cancerous spread of errors. As an example, the misinformation has proliferated that when Johann I left his professorship of mathematics in Groningen to return to Basel in 1705, he was unaware that Jacob I had “died of tuberculosis.” Tellingly, the idea that tuberculosis was the cause of Jakob I’s death appears only in Web biographies of Johann I and not in those of Jakob I himself.

**Conflict between Jakob and Johann.** More seriously, the present popular image of Jakob Bernoulli is much distorted by being seen through the eyes of Johann I, who survived Jakob I and served as the professor of mathematics at Basel for more than forty years after Jakob’s early death at age fifty in 1705. It is beyond doubt that the brothers spoke sharply about each other in print concerning their respective contributions to the development of the calculus of variations (see the *Streitschriften* listed below). Otto Mencke, editor of the *Acta Eruditorum*, at a certain point made a firm decision that the journal would publish no more articles in which the brothers attacked each other. The Académie Royale des Sciences at Paris admitted Jakob and Johann at the same time in 1699 *on the condition* that they make peace and end their mutual public vituperations, which were thought to give mathematicians a bad name. Gottfried Wilhelm Leibniz, Pierre Varignon, and others wrote to the brothers pleading with them to act in a more fraternal manner. Nevertheless, Johann I was probably more responsible for the falling out between the brothers than Jakob I, thirteen years his senior and Johann’s most important mathematics teacher.

Although the evidence is incomplete, it seems likely that a huge source of tension between the brothers was that when Johann I traveled to France in the early 1690s, he signed a very lucrative contract with the Marquis de l’Hôpital stipulating that he would teach l’Hôpital Leibniz’s new methods of calculus and provide l’Hôpital with new mathematical results, while keeping secret the extent of his contributions. Johann I wrote to l’Hôpital that Jakob I was jealous of the payment he was receiving from l’Hôpital, but it is clear that Jakob believed that some of the methods and results Johann I was providing to l’Hôpital secretly were Jakob’s own intellectual property. No wonder that, whereas at first the brothers published each other’s results, inserting such phrases as “when he saw my work, my brother suggested …,” Jakob began to try much more carefully to distinguish his own work from that of his brother. In an autobiography he wrote many years after Jakob’s death, Johann claimed to be responsible for the introduction of the word “integral” for the operation opposite to differentiation, for issuing challenge problems that were in fact first published by Jakob, and much more. Thus began a confusion of credit between the brothers, which helped lead to the creation of, in Jeanne Peiffer’s terms, the “chimera” of “the Bernoulli brothers.” Writing of “the Bernoulli brothers” may be thought to finesse the problem of distributing credit between them and may help explain why inventions of or quotations from Jakob are often attributed to Johann and vice versa.

**Relationship to their Father.** It is commonly said that both Jakob I and Johann I studied mathematics secretly and against their father’s will, while, following their father’s dictates, Jakob I obtained a degree in theology and Johann in medicine. It is true that Jakob Bernoulli chose as his motto *Invito patre sidera verso* or “Against the will of my father, I turn [or disturb] the heavens.” This should be understood, however, in connection with Bernoulli’s emblem, which showed the mythical phaeton disastrously attempting to drive the chariot of his father, the sun-god Apollo. Jakob’s contemporaries, moreover, would almost certainly have paired his phrase *invito patre* with the common phrase *invita Minerva*, meaning “against one’s natural bent.” The real point is that both Jakob and Johann became academic mathematicians because that was their natural bent and they strongly believed that one should follow one’s talents, natural or God-given, in choosing a vocation, Jakob being the model for Johann in this regard. Their father was not necessarily opposed to mathematics—at one point he was named *Rechenherr*, something like an auditor, in the town government of Basel—so much as he wanted his sons to pursue educations leading to jobs (so-called *Brotstudia*) by which they could support their families. The position of professor of mathematics at the university of Basel in the seventeenth and eighteenth centuries was not well paid (and there was only one such position), and both Jakob and Johann Bernoulli had to take on additional jobs or rent rooms in their houses to supplement their incomes. They were lucky to get the Basel chair of mathematics in turn, while most of the later Bernoullis had to find other jobs (Johann II Bernoulli succeeded his father in the chair of mathematics).

Thanks to the publications on Jakob Bernoulli after 1970, much more could be added to this postscript, but scholars had only just begun to profit from the publication of *Die gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli*, which as of 2007 continued, so that researchers could expect continued growth in their knowledge of the contributions of the Bernoullis in the future.

## SUPPLEMENTARY BIBLIOGRAPHY

### WORKS BY BERNOULLI

*Collected Works and Correspondence Die Werke von Jakob Bernoulli*. Vol. 1, *Astronomie: Philosophia naturalis*, edited by Joachim Otto Fleckenstein (1969); vol. 2, *Elementarmathematik*, edited by Clara Silvia Roero and Tulio Viola (1989); vol. 3, *Wahrscheinlichkeitsrechnung*, edited by Bartel Leendert van der Waerden (1975); vol. 4, *Reihentheorie*, edited by André Weil, (1993); vol. 5, *Differentialgeometrie*, edited by André Weil and Martin Mattmüller (1999). Basel, Switzerland: Birkhäuser. The last volume, which covers mechanics, had not appeared at this writing.

*Die Streitschriften von Jacob und Johann Bernoulli*. Edited by Herman H. Goldstine. Basel: Birkhäuser, 1991.

*Der Briefwechsel von Jacob Bernoulli*. Edited by André Weil, with contributions from Clifford Truesdell and Fritz Nagel. Basel: Birkhäuser, 1993. The previously unpublished correspondence of Jakob Bernoulli with his brother Johann (there are only four extant letters, all of which are from Johann to Jakob, the last written in the autumn of 1691 from Geneva) were included in *Der Briefwechsel von Johann Bernoulli*, vol. 1 (Basel: Birkhäuser, 1955). Because the correspondence of Jakob with l’Hôpital and with Varignon has been lost, it is important, for understanding the relationship of the brothers, to read the correspondence between Johann and l’Hôpital in this volume, as well as the correspondence between Johann and Varignon: part 1, 1692–1702, in vol. 2 (Basel: Birkhäuser, 1988) and part 2, 1702–1714, in vol. 3 (Basel: Birkhäuser, 1992). The correspondence with Varignon reveals the way in which Jakob and Johann were admitted at the same time to the Académie Royale des Sciences in Paris on the condition that they end their public quarrel. In the later correspondence, Varignon’s efforts to supervise Jakob’s son Nikolaus when he was in Paris studying painting are a repeated topic.

*Translations of* Ars Conjectandi *Jacques Bernoulli et l’Ars Conjectandi*. Translated by Norbert Meusnier. Rouen, IREM, 1987. Includes a French translation of part 4.

*Christian Huygens et Jacques Bernoulli: La première partie de l’ArsConjectandi (1657–1713)*. Translated by Norbert Meusnier. Paris, CAMS, 1992. Includes a French translation of part 1, which consists of Christiaan Huygens, *De ratiociniis in ludo aleae*, together with Bernoulli’s notes.

*The Art of Conjecturing, together with Letter to a Friend on Sets inCourt Tennis*. English translation with an introduction and notes by Edith Dudley Sylla. Baltimore, MD: Johns Hopkins University Press, 2006.

### OTHER SOURCES

Hess, Heinz-Jürgen, and Fritz Nagel. *Der Ausbau des Calculus durch Leibniz und die Brüder Bernoulli*. Symposion der Leibniz-Gesellschaft und der Bernoulli-Edition der Naturforschenden Gesellschaft in Basel, 15–17 June 1987. Studia Leibnitiana, Sonderheft 17. Stuttgart: Franz Steiner Verlag Wiesbaden GMBH, 1989. See especially Clara Silvia Roero, “The Passage from Descartes’ Algebraic Geometry to Leibniz’s Infinitesimal Calculus in the Writings of Jacob Bernoulli” and Heinz-Jürgen Hess, “Die Briefwechsel von Jacob Bernoulli und Leibniz: Eine Kurz-charakteristik.”

Meusnier, Norbert. “Nicolas, neveu exemplaire.” *JournalElectronique d’Histoire des Probabilités et de la Statistique*2, no. 1 (June 2006). Available from http://www.jehps.net See also other articles in this volume and in vol. 2, no. 2 (November 2006), resulting from a conference commemorating the three hundredth anniversary of Jacob Bernoulli’s death.

Nagel, Fritz. “Jacob Bernoullis Vorschläge zur Universitätsreform während der Basler Unruhen von 1691.” In *Der Briefwechsel von Jacob Bernoulli*, edited by André Weil. Basel: Birkhäuser, 1993.

O’Connor, J. J., and E. F. Robertson. “Jacob (Jacques) Bernoulli.” MacTutor History of Mathematics Archive. Available from http://turnbull.mcs.st-and.ac.uk/history/ A convenient starting point but should be used with caution.

Peiffer, Jeanne. “Jacob Bernoulli, Teacher and Rival of His Brother Johann.” *Journal Electronique d’Histoire des Probabilités et de la Statistique* 2, no. 2 (November 2006). Available from http://www.jehps.net

Roero, Carla Silvia. “Jacob Bernoulli attento studioso delle opere di Archimede: Le note marginali all’edizione de Barrow del 1675.” *Bollettino di Storia delle Scienze Matematiche* 3, no. 1 (1983): 77–125. Reprinted in *Die Werke von Jakob Bernoulli*, vol. 2, *Elementarmathematik*, edited by Clara Silvia Roero and Tulio Viola. Basel: Birkhäuser, 1989.

Sylla, Edith Dudley. Introduction to *The Art of Conjecturing together with Letter to a Friend on Sets in Court Tennis* by Jacob Bernoulli. Baltimore: Johns Hopkins University Press, 2006. Contains much more detail about Jakob’s work in probability mathematics and its context.

Truesdell, Clifford. *The Rational Mechanics of Flexible or ElasticBodies, 1638–1788. Leonhardi Euleri Opera Omnia, Series 2*. Vol. 11, part 2. Zürich: Orell Füssli, 1960. Contains the detailed examination of texts supporting Truesdell’s praise of Jakob Bernoulli in the following article

. ———. “A Program toward Rediscovering the Rational Mechanics of the Age of Reason.” *Archive for History of Exact Sciences* 1 (1975): 1–36. Truesdell states (p. 14) that “the creation of rational mechanics is due as much to James Bernoulli as to Newton, though James Bernoulli’s work nowadays is little known.” He goes on to say that Jakob’s paper of 1703 (sent in parts to the Académie Royale des Sciences) was “second only to the *Principia* itself in its influence on the later growth of the discipline.”

———. “Mechanics, Especially Elasticity, in the Correspondence of Jacob Bernoulli and Leibniz.” In *Der Briefwechsel von Jacob Bernoulli*, edited by André Weil. Basel: Birkhäuser, 1993.

Yushkevich, Adolf P. “Nicholas Bernoulli and the Publication of James Bernoulli’s *Ars Conjectandi*.” *Theory of Probability and Its Applications* 31, no. 2 (1987): 286–303.

*Edith Dudley Sylla*

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