Aerarium Philosophiae Mathematicae, In Quo Elementa Philosophiae Geometricae de Planis, Curvis, & Solidis figuris Applicata, Et Ornata Usibus eximiis in omni Scientiarum, & Artium genere, novis Praxibus, Paradoxis, locis Aristotelicis, & aliorum Philosophorum, & Scriptorum, Corollariis, Scholiis, Eruditionibus, Moralitatibus, Demonstrationibus novis, facillimis, & universalissimis confirmata, Methodo Iucundiore, ac breviore in Tres Tomos distributa sunt. Intercessere Ingeniosae inventionis Exodia Horaria 3 In Quo Reliqui quatuor libri elementares de planis applicati, &c. Epilogus Planimetricus, Breviarium speculativum, & practicum de curvis, & solidis cum facillimis, ac novis demonstrationibus, & Materiae plurium Tomorum indicatae: Cum Indicibus viginti communibus Secundo, et Tertio huic Tomo.

Bologna: G.B. Ferroni, 1648.

First edition, the superb Macclesfield copy printed on thick paper, of this rare compendious scholastic mathematical work by the Jesuit mathematician Mario Bettini (1582-1657), encompassing all the major fields of mathematics, such as the flight of projectiles, the construction of fortifications, the art of navigation, horology, and astronomical theories, among many others, all of which illustrated the ways in which geometric principles pervaded the world. Following Paul Guldin’s death in 1643, Bettini had taken the lead in the Jesuits’ opposition to the method of indivisibles, and the Aerarium contains an important attack on Cavalieri’s theory. The work had gone through the Collegio Romano’s rigorous censorship under the eye of Christoph Grienberger (d. 1636), who as Clavius’ successor was responsible for enforcing adherence to Aristotle in matters of natural philosophy, and was a highly accomplished mathematician himself. “In Bettini’s view, the philosophus mathematicus relies on results from pure mathematics (and especially from geometry) to carry out the study of the natural world. The advantage of this procedure resides in its geometrical abstraction. Unlike the philosopher, who considers sensible matter in given spatial and temporal circumstances, the philosopher-geometer abstracts from sensible matter. He is therefore able to examine a large number of working theories, either possible or impossible (e.g, quadrationis circuli) and with no temporal or spatial constraints (e.g., representing the earth in the center of the universe as a point at the center of a circle is not based on concrete observation) [vol. 1, pp. 18-23]. From this point of view, Bettini states in a somewhat baroque style, ‘when philosophizing on the subject of physics, the mathematician-philosopher-geometers have this numerous, illustrious and prolific [geometric] abstraction as an invulnerable protection through which they securely support, avoid and ridicule the arms of sophisms that the ignorance of so illustrious and remarkable philosophy [i.e., mathematical philosophy] in vain dares to throw against them” [vol. 1, p. 22]” (Carolino, p. 198). At the time, the most important threat to the primacy of geometry was the method of indivisibles, and Bettini devoted Book 5 of vol. III to an attack on Cavalieri’s theory. This was the Jesuits’ response to Cavalieri’s rebuttal, in his Exercitationes geometricae sex (1647), of Guldin’s attack on indivisibles by Bettini’s in his Centrobaryca (1641). Only one other copy of the Aerarium has sold at auction since the Riccardi/Kenney copy in 1971 (which lacked the famous ‘tree of life’ plate, present in this copy). OCLC records North American copies at Berkeley, Wesleyan University, Harvard, Michigan, the Adler, and Linda Hall. “This work is very rare” (UCL Catalogue).

Provenance: The Earls of Macclesfield, Shirburn Castle (blind stamp on first three leaves, bookplates on front paste-downs). Sotheby’s, June 10, 2004, lot 347, £11,400.

European scientific thought during the period 1620-1660 was dominated by the revolutionary investigations of Galileo and others into the nature of the solar system. The Jesuits, as the chief proponents of traditional Aristotelian teachings, were hailed by many as the intellectual champions of the Catholic Church, the philosophical mathematical and Scriptural arguments of Jesuit writers against the Copernican theory of a sun-centred solar system the main driving forces of seventeenth century inquiry.
At the beginning of the seventeenth century to understand meant to grasp regularities, and this knowledge was based largely on the accepted authority of ancient texts. However, as the worldview expanded the certainty of such views began to diminish. For the philosophically well-educated mathematics appeared as one of the few refuges of eternal truths untainted by the possibility of such dissent. Jesuit colleges, such as that at Parma where Bettini taught, were among the most important and prestigious of all educational institutions in the sixteenth and seventeenth centuries. The establishment of colleges was part of the Jesuits’ Counter-Reformatory mission, which aimed to display to protestants an intimidating mixture of intellectual and cultural sophistication.

Bettini was, with André Tacquet and Paul Guldin, one of the leading Jesuit mathematicians of his time. He was cited and praised by Grienberger, by Athanasius Kircher inArs magna lucis; by Gaspar Schott inMechanica hydraulico-pneumatica, where he describes a hydraulic clock machine invented by Bettini (this is also quoted by Schott in his Magia Universalis); by Marin Mersenne inCogitata physico-mathematica; andby Giovanni Battista Riccioli inGeographicae crucis fabrica et usus (Severino, p. 2). Like Guldin and Tacquet, Bettini was a fierce critic of indivisibles. By the time Aerarium was published, Cavalieri and Guldin were both dead. “But the death of the two chief protagonists, along with the passing of Torricelli in 1647, did nothing to tamp down the debate. Mathematicians could come and go but the Society’s determination to extinguish the infinitely small remained the same, and the role of the chief critic of indivisibles was simply handed on to another Jesuit mathematician, Mario Bettini … The theory of indivisibles was not a natural fit in [this] practically oriented and eclectic collection, but it was nevertheless the focus of book 5 of volume 3 of the Aerarium. This was, after all, one year after the publication of Cavalieri’s rebuttal of Guldin, and it was important that the Society respond and keep up the pressure on the champions of the infinitely small.

“It is very likely that Bettini and Cavalieri knew each other personally, and there is much to suggest that their relationship was far from friendly. In 1626, Cavalieri was appointed prior of the Jesuit house in Parma, where Bettini was professor at the university, and it is hard to imagine that the two mathematicians did not cross paths in this modest-sized city. Cavalieri entertained hopes of being appointed to a mathematical chair at the Univserity of Parma, but, as he complained to Guldin on August 7 of that year, it all came to naught: ‘As for the lectureship in mathematics,’ he wrote, ‘were the Jesuit fathers not here I would have great hope, because of the great inclination of Monsignor Cardinal Aldobrandini to favor me … but as [the univesrity] is under the rule of the Jesuit fathers, I cannot hope any longer.’ There can be little doubt that among the Jesuit fathers who scuttled Cavalieri’s appointment was their own leading mathematician, Mario Bettini …

“What Bettini lacked in mathematical sophistication he made up for in fervency. Guldin, and later Tacquet, kept the debate largely within the bounds of technical mathematics, but Bettini did not hesitate to use blunt language and warn darkly of dire consequences if his admonitions were not heeded. It is possible that the bitterness of his personal history with Cavalieri led him to go beyond his mandate of a sedate mathematical critique of indivisbles, but whatever his personal motivation, Bettini’s attitude was probably more in line with the true tenor of the Jesuit campaign against the infinitely small. He merely gave it voice.

“[Bettini] hammered away at one point alone: that ‘infinity to infinity has no proportion,’ and it therefore made no sense to compare the infinite lines of one figure with the infinite lines of another. Since this procedure is at the heart of the method of indivisibles, Bettini insisted that it was imperative that students and novices be warned against this tempting but false approach. ‘In order to set forth the elements of geometry,’ he writes, ‘I point out [these] hallucinations, so that novices will learn to distinguish (as in the proverb) ‘what separates the false coin from the true’ in geometrical philosophy.’ Indivisibles, according to Bettini, were a dangerous fantasy that was best ignored if at all possible. Under the circumstances, however, ‘being pressed, I respond to the counterfeit philosophizing about geometrical figures by indivisibles. Far, far be it from me to wish to make my geometrical theorems useless, lacking demonstrations of truth. Which would be to compose … figures and philosophize about them by indivisibles.’ To avoid undermining all demonstrations and subverting geometry itself, one must steer clear of the dangerous hallucination – the method of indivisibles” (Alexander, pp. 157-159).

Bettini’s attack did not go unanswered, despite the death of Cavalieri and Torricelli. In 1658, Stefano degli Angeli, a Jesuit himself, attacked Bettini in an ‘Appendix pro indivisibilibus’ attached to his book Problemata geometrica sexaginta. He compared Bettini’s warnings against ‘conterfeit philosophizing’ to a hysterical exorcist trying to fight off demons with furious incantations.

Bettini’s attack on indivisibles is, of course, only a small part of the 1100 pages of this encyclopaedic work. We cannot discuss the multitude of beautiful, and sometimes strange, inventions and phenomena he describes. The important contribution of the work to gnomonics has been analysed by Severino, and gives some idea of the wealth of material in Bettini’s work. Severino lists the main contributions of Bettini in this field as follows (p. 53):

1) A new geometric way of finding the points of the astronomical hour lines on the equinoctial line with a single compass operation;

2) A new geometric method for describing diurnal declination curves on a polar clock; 3) A new gnomonic instrument based on a polar clock to describe sundials;

4) Bettini writes about asymptotes two decades before the official dates reported in the encyclopedias;

5) A universal tool for describing sundials;

6) The geometric method for finding the points of the Italic hour lines on the horizontal line, described perhaps for the first time by Bettini;

7) The way of obtaining the time from the height sundials drawn flat, by means of a ruler;

8) The Mira Horaria;

9) Sacrobosco's definition of planetary hours and the distinction from temporal hours; 10) The universal gnomonic sandal;

11) The ‘Horary zither’, as a gnomonic tool to build wall sundials;

12) The gnomon bow, an unknown gnomon instrument by Grienberger;

13) The hour cylinder with the ecliptic Planetary hours.

Of these contributions to gnomonics, perhaps the most extraordinary is a kind of portable sundial called the ‘Sandalium exodium horarium’ (10), described and illustrated in a tract with separate title and pagination at the end of vol. II. A model of this ‘gnomonic sandal’, with a sundial inscribed on the sole of the sandal, and made in the twentieth century by a certain R. Walden, is kept in the Museum of the History of Science in Oxford. Its use is explained by Severino (pp. 32-36).

Bettini included in the Aerarium a ‘Scholion Parergicon’ eulogising Grienberger, one of the major mathematicians of the period, although little was published under his own name. Bettini compares him to Archimedes, adding that “Grienberger has no greater enemy than his own modesty, by which it has come to pass that his ingenious inventions have been neglected, and he will be consigned to oblivion.” Bettini added that “It was a remarkable characteristic of [Grienberger] that, following the example of Archimedes, he combined most acute theories with extraordinary practices”, and his claims for Grienberger’s achievements in designing instruments and machines are closely echoed by other contemporary mathematical authors. How much of the present work can be accredited to Grienberger is difficult to say, but Bettini acknowledges the debt and the astronomical typinium and many other mechanical and measuring devices illustrating the work are probably from his hand.

BL 17th Century Italian 103 (vol. II only); Kenney 1471; Riccardi I, 125; Sommervogel I, 1428. Alexander, Infinitesimal, 2014. Carolino, ‘Cristoforo Borri and the epistemological status of mathematics in seventeenth-century Portugal,’ Historia Mathematica 34 (2007), pp. 187-205. Severino, La gnomonica di Mario Bettini, 2009.



7 parts in 3 volumes, bound in 2, 4to (230 x 165mm). Vol. I: pp. [lvi], 701, [1, blank], [16]; Vol. II: [iv], 474; [iv], 60; Vol. III: [xxvi], 225, [3, blank], 227-354, [2, blank]; 70, [2, blank]; [viii], 115, [2, blank]; [xii], 42, [2, blank], 43-54, 33, [1, colophon], [1, blank], with identical engraved additional pictorial titles by Francesco Curti of Bologna to vols. I and II only [that, also identical, to vol. III discarded in this copy because vols. II and III are bound together], 2 engraved portraits of bishops Madruzzi & Zeccadori, engraved plate with a genealogical tree, 3 folding engraved plates attached to edges of ff. in vol. III, profusely illustrated with woodcut and engraved diagrams, including 10 fine full-page and some half-page engraved illustrations of scientific instruments. Contemporary Dutch vellum, blind-stamped centre-pieces, flat spines, lacking ties.

Item #2435

Price: $17,500.00