Diversarum speculationum mathematicarum, & physicarum liber [Theoremata arithmetica; De rationibus operationum perspectivae; De mechanicis; Disputationes de quibusdam placitis Aristotelis; In quintum Euclidis librum; Physica & mathematica responsa per epistolas].

Turin: heirs of Niccolò Bevilaqua, 1585.

First edition, very rare, of Benedetti’s major work, his Book of Various Mathematical and Physical Ideas, which include theories of motion and experiments on falling bodies predating Galileo’s studies on the subject, as well as studies of perspective presented in three-dimensional terms, fifty years before Desargues articulated his theories of projective geometry. “Benedetti’s final work, containing the most important Italian contribution to physical thought prior to Galileo, was the Diversarum speculationum (1585)” (Stillman Drake in DSB).

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First edition, very rare, of Benedetti’s major work, his Book of Various Mathematical and Physical Ideas, which include theories of motion and experiments on falling bodies predating Galileo’s studies on the subject, as well as studies of perspective presented in three-dimensional terms, fifty years before Desargues articulated his theories of projective geometry. “Benedetti is of special significance in the history of science as the most important immediate forerunner of Galileo … Benedetti’s final work, containing the most important Italian contribution to physical thought prior to Galileo, was the Diversarum speculationum (1585)” (Stillman Drake in DSB). “Giovanni Battista Benedetti is counted as one of the most brilliant mathematical and philosophical minds of the late Italian Renaissance. However, the theoretical and historical relevance of his work is still obscure in many respects. This is due to several factors, principal among which is the relative rarity of his major work, Diversarum speculationum mathematicarum et physicarum liber (Turin, 1585). This work was a major contribution to Renaissance science, especially due to its insights on mechanics, the mathematical approach to natural investigation, and the connection of celestial and terrestrial dynamics from a post-Copernican perspective” (Omodeo & Renn). Following a section on arithmetic, Benedetti treats some standard perspective problems, “but the manner in which they are treated is entirely new. Benedetti jumps straight into three dimensions … his 3-dimensional diagrams explicitly introduce the center of projection and therefore clearly bring us closer to seeing perspective as an example of projective geometry (that is, to the work of Desargues)” (Field (1997), pp. 162 & 171). In his treatment of mechanics, Benedetti “assert[ed] clearly and for the first time that the impetus of a body freed from rapid circular motion is rectilinear and tangential in character, a conception of fundamental importance” (DSB). Benedetti then turns to a discussion of falling bodies. “For equality of speed of different weights falling in vacuo, Benedetti proposed a thought experiment that is often said to be identical with Galileo’s, although the difference is considerable. Benedetti supposes two bodies of the same weight connected by a line and falling in vacuo at the same speed as a single body having their combined weight; he appeals to intuition to show that whether connected or not, the two smaller bodies will continue to fall at the same speed … Benedetti correctly holds that natural rectilinear motion continually increases in speed because of the continual impression of downward impetus, whereas Galileo wrongly believed that acceleration was an accidental and temporary effect at the beginning of fall only, an error which vitiated much of the reasoning in De motu and was corrected only in his later works” (ibid.). After a section on Euclid’s first five books, the Speculationum concludes with a series of letters on various topics. “Benedetti published no separate work on astronomy, but his letters in the Speculationum show that he was an admirer of Copernicus and that he was much concerned with accuracy of tables and precise observation” (ibid.). In one of the letters (pp. 270-271), he describes the use of concave mirrors and convex lenses in conjunction with the camera obscura to correct the inverted image. RBH lists only this copy and the Macclesfield copy, in an 18th century binding (Sotheby's, June 10, 2004, lot 328, £7200 = $13,162).

Born in 1530, Benedetti received most of his education from his father, described by Luca Guarico as a Spaniard, philosopher, and physicus, probably in the sense of “student of nature” but possibly meaning “doctor of medicine.” Benedetti “had no formal education beyond the age of seven, except that he studied the first four books of Euclid’s Elements under Niccolò Tartaglia, probably about 1546-1548. Their relations appear to have been poor, for Tartaglia nowhere mentions Benedetti as a pupil; Benedetti named Tartaglia in 1553 only ‘to give him his due’ and severely criticized his writings in later years” (DSB). From 1558 he served as court mathematician to Duke Ottavio Farnese, but in 1567 he was invited to Turin by Duke Emanule Filiberto of Savoy and remained there until his death. “Benedetti died on 20 January 1590, two years earlier than he had predicted from his own horoscope. It is interesting that on his deathbed he made a recomputation and concluded that there must have been an error of four minutes in the original data – presumably the exact hour of his birth” (Drake & Drabkin, p. 35).

Benedetti enjoyed considerable fame during his lifetime as a prominent mathematician and mathematical philosopher. “Among his admirers, the astronomer Johannes Kepler regarded him as one of the few Italians to significantly contribute to the advancement of mathematics in his time: ‘The Italians are asleep with the sole exceptions of Commandino and Giovanni Battista Benedetti.’ For his part, the mathematician of the Collegio Romano, Christopher Clavius, extolled Benedetti’s scientific merits in the 1589 edition of his reputed commentary of Euclid. In the dedicatory letter to Carlo Emanuele I of Savoy, he praised ‘his court mathematician’ Benedetti as ‘very expert in mathematics’” (Omodeo & Renn, p. 7).

“The dedicatory epistle of the Diversae speculationes begins with an acknowledgment of the generosity of Emanuele Filiberto of Savoy, with whom Benedetti had particularly good relations. As one reads, they often talked about mathematical issues pertaining to arithmetic, geometry, optics, music, and astrology. Since the Diversae speculationes appeared after Emanuele Filiberto’s death, it was dedicated to his successor, Carlo Emanuele I. Benedetti reports that both dukes encouraged his inquiries and their questions motivated his investigation of specific questions. This is the reason why Benedetti’s Speculationes have an occasional character and are not ordered in a systematic manner” (ibid., p. 63).

The most important contribution of the Speculationes to mathematics is Book 2, De rationibus operationum perspectivae, “an elegant, rigorous, and original piece of mathematics …the most important aspect of this originality is that unlike all his predecessors Benedetti treats the problem of perspective construction as three-dimensional” (Field (1985), p. 95). “Benedetti is indeed a remarkable figure in the history of perspective. He came to the conclusion that theoretically, a perspective construction is a question of determining the image of an arbitrary point, and he incorporated more than the theory of similar triangles from the body of geometrical knowledge in his proofs. In particular he drew upon results from three-dimensional geometry … I have the distinct impression, although I cannot prove it, that Guidobaldo [del Monte] and Stevin read Benedetti’s work and found it inspirational” (Andersen, p. 152). Benedetti’s work on perspective clearly foreshadows that of Girard Desargues (1591-1661), famous for his study of conics by means of projection, which enabled him to state theorems which apply to all types of conics simultaneously, whereas Apollonius in his Conics had to treat each type separately. “Before he published his work on conics, Desargues had worked on linear perspective, and it is clear that his projective treatment of conics arose out of his work on perspective. For Desargues considers the actual three-dimensional configuration, relating both object and picture to the eye” (Field (1985), p. 74), whereas the manuals for painters used a two-dimensional representation of the physical world.

Benedetti is best known today for his studies on mechanics, in which he is the most important precursor of Galileo. These studies are contained in the third and fourth books of the Speculationes. The third book, ‘On mechanics’ (De mechanicis) “commences with an ‘explanation’ of the law of the lever based on a concept of the degree to which a weight pushes or pulls upon the center (Chs. 1-5). In the discussion of the bent lever (Ch. 3), the basic notion of statical moment is developed. The rest of the section is taken up largely with a running commentary and critique (Chs. 10-25) of the Aristotelian Questions of Mechanics. There are also some strictures (Chs. 7-8) on Tartaglia’s edition of the Liber Jordani de ratione ponderis, which had appeared in 1565. The subject of mechanics figures prominently in other sections as well, e.g., in the anticipation of Pascal’s hydrostatic principle (p. 287)” (Drake & Drabkin, p. 41).

“Chapter 14 [of Book 3] provides a long discussion of problem 8 of the Aristotelian Mechanical Problems … At the end of the chapter, Benedetti discussed the question of why a potter’s wheel set into motion by an external force will continue to rotate for a time but not forever. In his response he took into account the friction with the support of the wheel and with the surrounding air. But he also discussed reasons that are more deeply concerned with the nature of such motion. He claimed, in particular, that the rotational motion is not a natural motion of the wheel, evidently making reference to the Aristotelian distinction between natural and violent motions. He also claimed that a body moving by itself because an impetus has been impressed upon it by an external force has a natural tendency to move along a rectilinear path. This statement seems to comes close to the principle of inertia of classical physics” (Omodeo & Renn, p. 105).

In Book 4 of the Speculationes, entitled Disputationes de quibusdam placitis Aristotelis (Disputations on Some Opinions Held by Aristotle), “Benedetti does not limit himself to criticism but rather seeks to provide a new approach to and foundation of physics and cosmology, beginning with the theory of motion … The first twelve chapters are a lengthy discussion of Aristotle’s Physics IV 8. This section deals with the ratio of velocities of bodies moving through different media or the void [see below]. Secondly, from chapter 13 to chapter 18, Benedetti challenges Physics VII 5 on further problems linked with the theory of motion. The third subdivision (chapters 19 to 22) deals with basic philosophical matters (the void, infinity, place, and time) … Another subdivision (chapters 23-26) deals with local motion and the shortcomings of the Aristotelian theory of natural places. The fifth and last subdivision presents cosmological ideas. It deals with the ‘sphere’ as a geometrical-cosmological figure, as well as with the (apparent) motion of the sun, with stars, meteorological aspects linked to the sun, the propagation of light in the cosmos, and other issues connected with astronomy in a broad sense. The Copernican system is discussed in the second part of this last section (chapters 35 to 39), along with other innovative theses such as the plurality of inhabited worlds akin to the earth and the reciprocity of the observational points in the universe” (ibid., pp. 141-142).

The first twelve chapters of the Disputations constitute the definitive presentation of Benedetti’s ‘buoyancy theory of fall,’ first put forward in his Resolutio omnium Euclidis problematum (1553). According to this theory, the distance traveled in a given time by a body falling freely in a medium is proportional to the difference between the specific weight of the body and that of the medium. It follows that bodies of the same shape and material, but of different size, would fall with the same speed. This result ignores the resistance of the medium – in his Demonstratioproportionum motuum localium (1554) Benedetti asserted that this resistance is proportional, not to the weight of the body, but to its surface area. “Benedetti’s ultimate expansion of his discussion of falling bodies in the Speculationum is of particular interest because it includes an explanation of their acceleration in terms of increments of impetus successively impressed ad infinitum. That conception is found later in the writings of Beeckman and Gassendi, but it appears never to have occurred to Galileo” (DSB).

In chap. 33, Pythagoreorum opinionem de sonitu corporum coelestium non fuisse ab Aristotele sublatam, Benedetti denied that the ‘sound of celestial bodies’ is the material production of any sounds. Unlike Kepler, he thought there was no harmonic proportion among planetary motions, as there was no perfect astronomical geometry. Both Benedetti and Kepler were convinced of the geometrical-rational structure of the universe first taught by Pythagoras; but the former believed that Divine Providence must manifest itself through infinite space, whereas, according to the latter, God’s creation must be finite, harmonic, and proportional. Furthermore, Benedetti did not suppose that nature can fully realize mathematical perfection, whereas Kepler held the opposite view.

“Chapter 35, Motum rectum curvo posse comparari (Straight and curvilinear motions are comparable), is a crucial chapter, since it is here that Benedetti, almost at the end of his Disputations, introduces the Copernican theory … ‘[Aristotle] is wrong when he says that straight motion cannot be compared to the curvilinear (Physics VII 4), where he mistakenly also says that one cannot find any lines equal to the circumference of a circle’ (p. 194). It is directed against Aristotle’s denial that a straight and a circular motion could be compared, thus hinting at the qualitative difference between celestial circular motions and the vertical tendency of the elements in the sublunary sphere … Benedetti first appeals to Archimedes’s De quadratura circuli to argue that the circle and the straight line are comparable: ‘If, then, this quadrature can exist, there can also exist, for the reason already given, a straight line equal to the circumference of that circle’ (p. 194) … These considerations offer Benedetti the occasion to expand on the velocity of celestial motions. According to the commonly held opinion [i.e., Aristotelian cosmology], the heavens would have to cover an immense distance within the 24 hours of the daily rotation … ‘But this difficulty does not occur in the most beautiful system of Aristarchus of Samos that has been so divinely expounded by Nicolaus Copernicus’ (p. 195)” (Omodeo & Renn, pp. 165-166).

In chap. 36, Benedetti deals with the question of the plurality of worlds. “After a section on the motion of light through the cosmic void (chap. 37) and one on the geometry of the elements (chap. 38), the concluding section of the Disputations (chap. 39) attacks a Peripatetic dogma: the unalterability of the heavens. In De caelo I 22 Aristotle remarked that no change in the heavens was ever observed. This is, according to Benedetti, not a valid argument. One should rather assume a principle of relativity of the point of observation. In fact, the earth would be invisible from the eighth heaven (that of the fixed stars), even though, by supposition, it was endowed with a light equal to that of the sun. The distance hinders us from perceiving changes that occur on other worlds (p. 197). With this rejection of the distinction between a sublunary and a heavenly realm, Benedetti’s criticism of Aristotelian physics is complete” (ibid., p. 167).

“The last book of the Diversae speculationes is a large collection of letters … The epistles are not organized chronologically (actually the dates are almost always omitted) but rather according to the importance of the addressees, some of whom were already dead at the time of publication. The first letter was directed to the duke Emanuele Filiberto, the second to his son Carlo Emanuele I, and the following four letters to the powerful nobleman Andrea Provana de Leyní. The topics are linked to Benedetti’s role as court mathematician and mathematical advisor … Michela Cecchini and Clara Silvia Roero, in their accurate reconstruction, came to the following assessment: ‘The variety of themes that were discussed and of the professions of the participants in the debates shows that Giovanni Battista Benedetti was a man of culture and practice. He was ready to engage in a fruitful debate with exponents of the scientific world in the broadest sense (such as mathematicians, physicians, jurists, and philosophers) and with politicians, diplomats, and ambassadors, as well as with experts of military art and religion. Moreover, he did not dislike architects, artisans, constructors of instruments and fortifications, surveyors, and astrologers’” (ibid., pp. 76-77).

Benedetti made an important contribution to the theory of sound in “two letters on music written to Cipriano da Rore and preserved in the Speculationum. Those letters, in the opinion of [the Flemish madrigalist] Claude Palisca, entitle Benedetti to be considered the true pioneer in the investigation of the mechanics of the production of musical consonances … Departing from the prevailing numerical theories of harmony, Benedetti inquired into the relation of pitch, consonance, and rates of vibration. He attributed the generation of musical consonances to the concurrence or cotermination of waves of air. Such waves, resulting from the striking of air by vibrating strings, should either agree with or break in upon one another. Proceeding thus, and asserting that the frequencies of vibration of two strings under equal tension vary inversely with the string lengths, Benedetti’s empirical approach to musical theory, as applied to the tuning of instruments, anticipated the later method of equal temperament and contrasted sharply with the rational numerical rules offered by Gioseffo Zarlino” (ibid.).

“Pierre Duhem has observed that Giovanni Battista Benedetti came close to appreciating the principle underlying the hydraulic press when reflecting on his practical experience with fountains, a printed version of his thoughts appearing in 1585 [in a letter to Giovanni Paolo Capra, a gentleman of Novara, p. 287]. Duhem argued that Pascal may well have known of Benedetti’s reflections either by reading him directly or through Mersenne” (Chalmers, p. 94). Benedetti considers two connected cylinders of different cross-sectional areas. He replaces the column of water by a piston of the same weight, first in the narrow cylinder, then in the wider one. The only difference in Pascal’s account is that he made this substitution in both cylinders at the same time.

Benedetti also made significant contributions to meteorology in the Speculationes. “Benedetti also attributed winds to changes in the density of air, caused by alterations of heat. In opposition to the view that clouds are held in suspension by the sun, he applied the Archimedean principle and stated that clouds seek air of density equal to their own; he also observed that bodies are heated by the sun in relation to their degree of opacity” (DSB)

“Benedetti’s astronomical considerations are not systematic. They are scattered throughout the volume in different sections. In spite of the difficulty of ordering them and obtaining an overview, they were very much appreciated among his contemporaries. Apart from Kepler’s eulogy of Benedetti’s ingenuity, the broad European success of the astronomical parts of this work is documented in other references. A few years after the publication of the Diversae speculationes, Brahe must have had a copy of it in Denmark, as he quoted it extensively and accurately on two occasions. In his correspondence with Landgrave William IV and the Hesse-Kassel court mathematician Christopher Rothmann, he referred to Benedetti’s observation of the light of Venus reflected on the part of the lunar disc not presently enlightened by the sun [letter to Baron Emanuele Filiberto Pingone, pp. 256-257] … A second long direct quotation of Benedetti can be found in Brahe’s book on the nova of 1572, which was part of the Astronomiae Instauratae Progymnasmata (1602). The Danish astronomer … entirely reproduced Benedetti’s letter and diagrams on the star in Cassiopeia (pp. 371-374). This letter was directed against Annibale Raimondo – an author whom Brahe also criticized – and demonstrated that the nova appeared above the sublunary sphere … Another reader of the Diversae speculationes was the English scholar of magnetism William Gilbert. In De mundo nostro sublunari philosophia nova (1651), he discussed Benedetti’s views on the spots on the surface of the moon, in a chapter trying to determine which parts of it were seas and continents” (Omodeo & Renn, pp. 125-126).

“The heliocentric system is not the main issue at stake in the Disputations, although that theory becomes part of a general program of reform for natural philosophy. Far from being a mere ‘Copernican enterprise,’ Benedetti’s visionary project is much more complex. It is an ambitious attempt to build a new physics, in the wide Renaissance meaning of the term, out of a criticism of Aristotelian physics” (ibid., pp. 167-169).

“Benedetti’s scientific originality and versatility leave little doubt that his work afforded a basis for the overthrow of Aristotelian physics. The extent of its actual influence on others, however, presents very difficult questions … The case of Galileo is the most perplexing. It is widely held that he was directly indebted to Benedetti for the ideas underlying De motu, but the resemblances of those ideas are easily accounted for by the Archimedean principle and the medieval impetus theory, easily accessible to both men independently, while the differences, particularly with respect to acceleration and the accumulation of impressed motion, are hard to explain if the young Galileo had the work of Benedetti before him. The absence of Benedetti’s name in Galileo’s books and notes, where other kindred spirits such as Gilbert and Guido Ubaldo are praised, is suggestive: much more so is the fact that Benedetti is not mentioned to or by Galileo in the vast surviving correspondence of his time. Jacopo Mazzoni has been proposed as a positive link – he was a colleague and friend of Galileo’s at Pisa about 1590, and certainly knew Benedetti’s work by 1597; but since Galileo left Pisa for Padua in 1592, the connection is uncertain. Benedetti appears to have remained unknown to Galileo’s teacher at Pisa, Francesco Buonamico, who in 1591 published a treatise, Demotu, of over a thousand pages. On the whole, it appears that Benedetti’s Speculationum was not widely read by his contemporaries, despite its outstanding achievements in extending the horizons of mathematics, physics, and astronomy beyond the Peripatetic boundaries” (DSB).

The Speculationum was issued again under slightly different titles at Venice in 1586 (Speculationum mathematicarum et physicarum tractatus) and, posthumously, in 1599 (Speculationum liber). The 1580 edition, mentioned by Riccardi and Smith, is a ghost.

Censimento 16 CNCE 5170; STC Italian, p. 82; Adams 655; Riccardi I, 111 (“raro ed assai pregiato”); Smith, Rara arithmetica, p. 364. Andersen, The Geometry of an Art, 2007. Chalmers, One Hundred Years of Pressure: Hydrostatics from Stevin to Newton, 2017. Drake & Drabkin, Mechanics in Sixteenth-Century Italy, 1969. Field, ‘Giovanni Battista Benedetti on the mathematics of linear perspective,’ Journal of the Warburg and Courtauld Institutes, Vol. 48 (1985), pp. 71-99. Field, The Invention of Infinity, 1997. Omodeo & Renn, Giovan Battista Benedetti’s Diversarum speculationum mathematicarum et physicarum liber (Turin, 1585), 2019.



Folio (300 x 210 mm), pp. [viii], 426, [2, errata]. Contemporary vellum, manuscript title on spine, device on title, numerous woodcut diagrams in text, manuscript inscription of a Jesuit college on title (first few leaves browned).

Item #4919

Price: $48,000.00