## Exercitationes quaedam mathematicae. Venice: Domenico Louvisa, 1724. [Bound with:] SUZZI, Giuseppe. Disquisitiones mathematicae. Venice: Domenico Louvisa, 1725. [Bound with:] DA RIPA [or RIVA], Lodovico. Miscellanea. (1. De Meteoro ignito, quod in Agro Tarvisino apparuit. 2. De vi Vaporum in Hygrometris. 3. Demonstrationes Theorematum ad quadraturas spectantium. A summao geometra Joah: Bernoullio. 4. Prolusio habita in Gymnasio Patavino). Venice: Domenico Louvisa, 1725. [Bound with:] MICHELOTTI, Pietro Antonio. Apologia in qua summum geometram Jo: Bernoullium motricis fibrae in musculorum motu inflatae curvaturam rectissimè supputasse defenditur, & Ric. Mead Georgii II. Magnae Britanniae Regis Archiatri longe eruditissimi objectionibus respondetur, ... Accedit rari ex utero morbi historia una cum necessariis medicis animadversionibus ab eodem Michelotto perscripta. Venice: Gabriel Hertz & Giovanni Manfrè, 1727.

Venice: Domenico Louvisa; Gabriel Hertz & Giovanni Manfrè, 1724-27.

Sammelband of four very rare works, all first editions, representing the work of the great Swiss mathematician Daniel Bernoulli, and of those of his friends and adversaries, during the period 1723-25 he spent in Italy. Bernoulli’s *Exercitationes* is his first mathematical publication, the success of which resulted in his appointment to the Academy of Sciences in St Petersburg. The work is in four parts, and is notably polemical. In the first part Bernoulli develops the theory of recurrent series and applies it to problems of probability involving the card game Faro; here Bernoulli disputes the position of Giovanni Rizzetti, famed as a critic of Newton’s theory of light. This was “evidence of his early interest in the work on the theory of probability done by his predecessors Montmort and De Moivre, which had been nourished by discussions with his cousin Nikolaus I” (DSB). In the second part, Bernoulli addresses the hydrodynamic problem of the outflow of water through a hole at the bottom of a vessel. Bernoulli supports the opinions of Pietro Antonio Michelotti; it was to study medicine under Michelotti that Bernoulli travelled to Venice in 1723, and Bernoulli’s interest in this problem was probably motivated by his study of blood flow and blood pressure. Opposing Jacopo Riccati, Bernoulli supports the (erroneous) theory put forth by Newton in the first edition of the *Principia*. In the second edition, Newton revised his opinion and accepted Riccati’s theory as correct, and Bernoulli himself revised his position in his *Hydrodynamica *(1738). The third part of the *Exercitationes* is a response to, and solution of, Riccati’s work on the differential equations named after him: Bernoulli identifies the equations for which separation of variables occurs, in which case finding the solution is reduced to an integration. While in Venice, Bernoulli had proposed to Riccati’s pupil Guiseppe Suzzi the problem of the quadrature of ‘lunulae’ (figures bounded by two circular arcs). In the fourth part of the *Exercitationes*, Bernoulli takes issue with Suzzi, giving his own treatment of the problem; Suzzi published his own results the next year in *Disquisitiones mathematicae*, “the first Italian treatise where some currency was given to ‘evanescent quantities’ and ‘fluent and fluxions’” (Guicciardini, p. 291). Daniel Bernoulli’s father Johann, and uncle Jakob, were notorious for their academic disputes, but the *Exercitationes* was Daniel’s only polemical writing. Later he developed “an aversion to all controversies” (*Werke* I, p. 238). Ripa’s *Miscellanea* is a collection of four works, the most important of which is a determination of the ‘ballistic curve’, the path followed by a projectile moving under gravity in a resistive medium. Michelotti’s *Apologia* is a defence of Johann Bernoulli’s doctoral thesis *De motu musculorum* against the attacks of Newton’s friend and personal physician Richard Mead. ABPC/RBH list only the Macclesfield copy of the Bernoulli (Sotheby’s 2004, £5,400), and no copies of any of the other three works.

Daniel Bernoulli (1700-82) was the most distinguished member of the second generation of the Bernoulli family, the son of Johann I. Daniel gained his baccalaureate in 1715 and master's degree in 1716 at Basle University, but, while studying philosophy at Basle, he began learning about the calculus from his father and his older brother Nikolas. He studied medicine at Heidelberg in 1718, Strasbourg in 1719, and then returned to Basle in 1720 to complete his doctorate. After completing his medical studies in 1721, he applied for a chair at Basel, but like his father before him, he lost out in a lottery. Having failed to obtain an academic post, Daniel went to Venice to study practical medicine under Michelotti (1673-1740). While in Venice he continued his mathematical studies, and made the acquaintance of Christian Goldbach (1690-1764), who assisted him with the publication of his *Exercitationes quaedam mathematicae* in July 1724.

The first part of the *Exercitationes* is evidence of Bernoulli’s early interest in probability – his best known contribution in this field is his formulation of the ‘St. Petersburg paradox’ in 1738. In the present work Bernoulli studies the card game ‘Faro’ (or ‘Pharaoh’), a late 17th-century French gambling game using cards which some regard as a precursor of poker. It is descended from Basset, and belongs to the Lansquenet and Monte Bank family of games due to the use of a banker and several players. Winning or losing occurs when cards turned up by the banker match those already exposed.

Faro had been analysed by Montmort in his *Essay d'Analyse sur les Jeux de Hazard* (1708, 1713). Bernoulli here gives his own analysis of the game, and includes several letters exchanged with Rizzetti (1675-1751) disputing his analysis of the same problem. Modern historians have traced the origin of the dispute between the two men as a difference between them of the concept of randomness; in the following year Rizzetti published his results in his *Ludorum scientia*, which some have seen as anticipating the strong law of large numbers. Bernoulli’s analysis in the *Exercitationes* is based on the use of ‘recurrent series’ – these are series in which, starting after a given term, the coefficient of any power of the variable is expressed as a linear function of a fixed number of the coefficients of lesser powers. The general term of such a series can be determined by using power series and partial fractions. Bernoulli later made use of these series for the approximate calculation roots of algebraic equations.

The second part of the *Exercitationes* prefigures Bernoulli’s most important contribution to science, his *Hydrodynamica* (1738). It involves a dispute between Daniel Bernoulli and Michelotti on one side, and Jacopo Riccati (1676-1754) on the other, regarding Newton’s treatment in the *Principia* of the flow of fluid from an orifice in a vessel. Born to a noble family who lived near Venice, Riccati began his pursuit of mathematics at the University of Padua, although he never held a formal university post. Michelotti studied at Padua under Jakob Hermann (1678-1733), who had been a disciple of Daniel’s uncle Jakob Bernoulli (1655-1705) at Basel; it was Hermann who introduced Michelotti to differential and integral calculus.

Evangelista Torricelli (1608-47) had found by experiment that the velocity with which the fluid exits the orifice is that which a body would acquire in falling a distance equal to the height of the water column above the orifice. Isaac Newton (1643-1727) found by his own experiments that the velocity of the fluid was only that due to half the depth of the fluid, explaining, in the first edition of *Principia* (1687), Torricelli’s observations as due to the contraction of the jet of fluid after it exits the orifice. In the second edition (1713), Newton changed his mind and accepted Torricelli’s observations.

“The hydraulic section of the *Exercitationes *comprises 42 of the altogether 90 pages of the pamphlet (pp. 30-71) with the following structure: an introduction, containing a description of the circumstances of the origin of the discussion between Daniel Bernoulli and Michelotti on the one hand and Riccati on the other hand, and a survey of the essence of the controversy, including summaries of the contents of the first two letters from the correspondence between Bernoulli and Riccati; a letter by Riccati to Bernoulli which dates, apparently, from the end of February 1724, and contains a denial of arguments advanced by Bernoulli; five particular objections of Bernoulli to this letter; Bernoulli’s reply to this letter, written in early March 1724; Riccati’s next letter of 20 March 1724; and nine objections of Bernoulli’s to this last letter …

“At the beginning of the introduction, Daniel Bernoulli notes the absence of a consensus among scientists with regard to the force generating the outflow of jets from vessels. He refers, in particular, to the fact that Newton changed (he assumes under the influence of Huygens) during the preparation of the second edition of the *Principia*, his initial point of view regarding the acting force. Bernoulli launches here the traditional attack on the English, asserting that Michelotti disproved their opinion in *De separation fluidorum* (1721) and demonstrated there the correctness of Newton’s initial point of view, namely that the force generating the outflow of water through an orifice was equal to the weight of the column of water lying above the orifice and having a section equal to the area of the orifice. Next, Bernoulli introduces his main opponent – Riccati. In Bernoulli’s words, Riccati initially doubted the correctness of Newton’s point of view as expressed in the second edition of the *Principia*. Later on, Riccati he sent Michelotti a letter in which he took Newton’s side and defended the modified point of view regarding the force generating the outflow of fluid from vessels. A copy of this letter was forwarded to Bernoulli, after he had arrived in Venice, with a request to state his opinion on this matter. To Daniel’s politely formulated retort, Riccati replied, ostensibly, with severe criticism which provided an immediate reason to publish the ensuing discussion. Bernoulli repeats Newton’s reasoning from the first edition of the *Principia*, in which the latter had given a purely hydrostatic exclamation of the fact that the force causing outflow of water equals the weight of the column the floor is located above the orifice. If one plugs the orifice, then the head of the water on the bottom of the vessel supposedly does not change, and the pressure in the stagnant fluid will, of course, be distributed uniformly over the bottom. Bernoulli expresses already here his categorical dissent with the objections expressed by Riccati, that ‘the internal motion of the water is oblique and increases the pressure’ and that it is necessary ‘to distinguish between the states of rest and motion.’ In confirmation of his point of view, Daniel refers to the argument contributed by his father Johann against Jurin and included in Michelotti’s treatise …

“Daniel Bernoulli cites Riccati’s accurate words from the latter’s letter to Michelotti explaining Newton’s amended point of view … Daniel Bernoulli confronts this explanation with two objections – apparently convincing to him – which he assumed to be indisputable and on which he was to base all following statements. Firstly, he may not have understood why Newton chose in his reasoning the time interval equal to the time of free fall of a body from the height of the water surface in the vessel above the orifice … Secondly, Daniel Bernoulli did not agree with the fact that the rate of flow of the issuing fluid equals the product of the velocity of outflow and the area of the orifice. At this point, he begins a discussion about the role of jet contraction … [He] presents a long excerpt from his letter of 16 January 1723 to a friend he does not name [Michelotti] … According to Bernoulli, … the cross-section of the jet should be contracted during its ejection, while its length is maintained. During this process – as Bernoulli suggests – the rate of flow of the ejected fluid will be exactly half the product of the velocity of outflow and the area of the orifice (this assumption he refers to as his theorem)” (*Werke* I, pp. 240-241).

The third part of the *Exercitationes* deals with the Riccati equation. As one of the major areas of research arising from the new calculus, differential equations were an area of great interest for much of the mathematical community in the early eighteenth century. Until around 1730, no systematic methods of solution were known other than separation of variables, and so nearly all studies on differential equations until that time focused on the search for transformations to make equations separable – a separable differential equation is one of the form *dy/dx =A*(*x*)/*B*(*y*), so the solution is reduced to integration, **∫***A*(*x*)*dx* = **∫***B*(*y*)*dy*.

Riccati’s most important work was on differential equations, and he is most famous for his work on the particular class of differential equations named after him,

*dy/dx = A*(*x*)y^{2} + *B*(*x*)*y + C*(*x*),

where *A*, *B*, and *C* are known functions of the variable *x*. He first proposed this equation in a letter to his friend Rizzetti on New Year’s Eve 1720, but it first appeared in print in 1724 in his ‘Animadversiones in aequationes differentiales secundi gradus’ (‘Investigations on differential equations of the second order’), published in in the *Supplementa *to the* Acta Eruditorum*. This paper was immediately followed, in the same volume of the Acta, by Daniel Bernoulli’s first work on the Riccati equation, ‘Notata in praecedens schediasma Ill. Co. Jacobi Riccati’ (‘Note on the preceding sketch by the illustrious Count Jacopo Riccati’). “Bernoulli took up the challenge issued by Jacopo Riccati in the paper printed immediately before his own: to find infinitely many *n* for which the equation *ax ^{n} dx + y*

^{2}

*dx = b*

*dy*is separable [where now

*a*and

*b*are constants]. He succeeded, but in the ‘Notata’ he gave the solution only in the form of an obscure anagram, saying he wanted to stake a priority claim for the first solution while still leaving the problem open for others. It was not until he wrote

*Exercitationes quaedam mathematicae*, his first major mathematical book, later in the same year, that he revealed his method …

“In the chapter on the Riccati equation, titled ‘Auctoris explanatio notationum suarum . . . una cum ejusdem solutione problematis Riccatiani’ (‘Explanation by the author of his ‘Notata’ … together with a solution of the problem of Riccati’), Bernoulli gave a way of finding infinitely many *n* for which the equation

*ax ^{n} dx + y*

^{2}

*dx = b*

*dy*

is separable. This method relies upon two lemmas” (Cretney, pp. 19-20). In *Lemma primum* and *lemma secundum* [p. 78], he showed by means of clever substitutions that if the original equation can be solved by separation of variables, so can the equation with the exponent *n *replaced by –*n/n* + 1, and also with *n *replaced by – *n* – 4. Starting with the trivial case *n* = 0 and using these two lemmas, Bernoulli showed that the equation is separable when *n* = – 4*m*/2*m* + 1, where *m* is an integer.

Cretney shows that Euler’s reading of this work of Bernoulli on the Riccati equation probably led to his discovery of the famous continued fraction expansion of *e*, the base of natural logarithms, given in his ‘De fractionibus continuis dissertatio’ (1737).

The lunulae, or lunes, discussed in the fourth and final part of Bernoulli’s work were first studied by Hippocrates of Chios (*ca*. 470-410 BCE): they are regions contained between two arcs of circles. Hippocrates wanted to know when such lunes are ‘quadrable’, i.e., when their area can be determined exactly by using ruler and compasses only. This was obviously related to the problem of the quadrature of the circle, one of the great unsolved problems of antiquity. Whether a lune is quadrable depends on the angular ratio α : β, where α and β are the angles subtended by the two arcs at the centre of their respective circles. Hippocrates showed that the lune is quadrable when the ratio is 2 : 1, 3 : 1, or 3 : 2. The first case, for example, concerns the lune the arcs of which are those of a half-circle and a quarter-circle and the corresponding radii are in the ratio 1 : √2.

While Bernoulli was in Italy, he made the acquaintance of the young mathematician Giuseppe Suzzi (1701-64). Suzzi, together with Lodovico Riva (1698-1746), had been taught infinitesimal calculus by Riccati at Padua in 1722-23; Riccati’s lectures were published later in his *Opere* (vol. 1, 1761). Suzzi asked Bernoulli to propose a good mathematical problem. Bernoulli suggested that of finding lunes, other than those found by Hippocrates, whose area could be determined exactly. He then proceeded to publish his own results on the question in the fourth part of the *Exercitationes*. Bernoulli showed that the lunes with ratios 5 : 1 and 5 : 3 are also quadrable. In 1771 Euler studied the problem and conjectured that no further quadrable lunes exist. This was shown to be the case in papers by the Russian mathematicians N. G. Chebotarev and A. V. Dorodnov published in 1933 and 1947.

Suzzi published his own work on lunes in his *Disquisitiones mathematicae, *including correspondence between himself and Bernoulli on the matter; the work also includes numerous letters exchanged with Riccati on various topics in the infinitesimal calculus. “Suzzi’s little pamphlet was composed as a collection of letters between Riccati and his pupil. These dealt with some of Riccati’s previous research and with the controversies (on second-order fluxions, central forces, or motion in resisting media) which had divided Leibnizians from Newtonians, and which had separated the position of Riccati from that of Johann Bernoulli in the early 1710s. The approach of Suzzi and Riccati was clearly anti-Bernoullian: ‘it is not Newton who makes mistakes, … but Bernoulli’ [p. 38]. They rejected the criticisms formulated against Newton by Johann Bernoulli very much according to the manner that was canonical in philo-Newtonian writings, even though they employed more civilized terms than those adopted by John Keill” (Guicciardini, pp. 291-292).

The *Miscellanea *of Lodovico Ripa consists of four parts, the first of which gives an account of a meteor seen in the skies over Treviso; Ripa cites Riccati and Rizzetti in support of his observations. In the second part, Ripa considers a simple hygrometer consisting of a weight suspended at the middle of a chord fixed at both ends. A change in the humidity of the air causes the weight to rise or fall in a straight line as the chord shortens or lengthens, but the amount of rise or fall is not proportional to the change in humidity. To correct this, Ripa considers suspending the weight from a different point on the chord; the weight then traces out a curved line as the humidity changes, a curve which Ripa studies by geometrical and analytical methods. In the third part Ripa gives his own proof of the shape of the ‘ballistic curve’ – the path of a projectile moving under gravity in a resistive medium. This had been treated by Newton in the *Principia* for the case in which the resistance is proportional to velocity, and he gave approximate solutions when the resistance is proportional to the square of the velocity. Johann Bernoulli in his 1719 paper in the *Acta Eruditorum*, ‘Responsio ad nonneminis provocationem, ejusque solutio quaestionis ipsi ab eodem propositae de invenienda linea curva quam describit projectile in medio resistente’ (pp. 216-226) treated the case in which the resistance is proportional to any power of the velocity. Ripa gives his own solution for this case, and agrees with Bernoulli that this problem demonstrates the greater applicability and superiority of Leibniz’s calculus over that of Newton. The last part is an oration delivered by Ripa at the University of Padua, on the occasion of his appointment to the professorship of astronomy and meteorology.

In 1694 Johann Bernoulli presented at Basel his inaugural dissertation, *De motu musculorum*. Heavily influenced by Giovanni Alfonso Borelli (1608-79), it “employed differential calculus to explain muscle function and described the contraction as follows: ‘When the mind wishes that a limb of the body moves, some agitation of animal spirits occurs in the brain so that, by twitching the origin of some nerve, they shake the spirituous juice contained inside over its whole length and, because of the irritation of the origin of the nerve, the last droplet of nervous juice is driven out by the slight vibration of the other orifice’” (Tipton, p. 12).

Through Hermann, Michelotti was able to establish epistolary ties with Johann Bernoulli in 1714; this correspondence continued until the end of 1725 and was valuable to him for the compilation of his *Animadversiones *which he annexed to the second edition (Venice, 1721) of Bernoulli’s thesis. The *Animadversiones* were a frontal attack on the objections of James Keill (1673-1719) to Bernoulli’s twitch doctrine; in 1727 Michelotti sent Hermann, then stationed in St. Petersburg, his *Apologia* in which he responded to other criticisms made by Richard Mead (1673-1754).

[Bernoulli:] Macclesfield 335. Cretney, ‘The origins of Euler’s early work on continued fractions,’ *Historia Mathematica*, 41 (2014), pp. 139–156. [Ripa:] Houzeau & Lancaster 8847; Riccardi I, 381. [Suzzi:] Riccardi I, 479/480. Guicciardini, ‘The reception of Newton’s method of series and fluxions in eighteenth-century Europe,’ Ch. 11 in *The Reception of Isaac Newton in Europe*, Pulte & Mandelbrote (eds.), 2019. [Michelotti:] Riccardi I, 158. Tipton, *History of Exercise Physiology*, 2014.

4to (258 x 193 mm). [Bernoulli:] pp. 96, with engraved vignette on title, four engraved initials and four engraved headpieces, and one folding engraved plate. [Michelotti:] pp. [1-2], 3-45, [1, errata], [2, blank], 47-59, with one folding engraved plate. [Ripa:] pp. [viii], 79 with one folding engraved plate. [Suzzi:] pp. 80, with two folding engraved plates. contemporary vellum with gilt lettering to spine, marbled edges (some soiling to boards, corners very slightly rubbed). A fine copy.

Item #5535

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Price:
$9,500.00
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