Logica demonstrativa. Quam una cum Thesibus ex tota Philosophia Decerptis. Beato Aloysio Gonzagae Societatis Jesu.
Turin: Domenici Paulini, 1699. First edition, probable third issue, of “Saccheri’s wonderful Logica demonstrativa” (Halsted) which “grants Saccheri the right to an eminent place in the history of modern logic” (Vailati). All three issues of the first edition are of the greatest rarity: no copy of any issue of the first edition has appeared at auction for at least a century, and none is listed on OCLC; in his census, Paolo Pagli located only two other complete copies of this issue, both in Italian institutional collections. Saccheri’s most famous book, the Euclides ab omni naevo vondicatus (1733), can be seen as an application of the methods developed in the Logica. The principal innovation in the earlier work was the consequentia mirabilis, according to which the truth of a proposition can be established by showing that its denial leads logically to the assertion of its truth; in the Euclides Saccheri applied this principle in an attempt to prove the Parallel Postulate of Euclidean geometry, in the process establishing many of the basic results of non-Euclidean geometry. “Three jewels have been contributed to logical theory by Saccheri in the Logica demonstrativa: his postulate stipulating the existence of various types of predicates; his proposed ‘nobler’ method for achieving, without the postulate, the same results as obtained with the postulate’ and, as a bonus, a brilliant use of the consequentia mirabilis in at least some of the applications of the nobler method … The other themes have been neglected, perhaps overshadowed by the ‘admirable consequence,’ although they alone would suffice to secure for our author an outstanding place in logic” (Angelelli, p. 98). “Saccheri’s two most important books, the Logica and the Euclides, were virtually forgotten until they were rescued from oblivion – the Euclides by E. Beltrami in 1889 and the Logica by G. Vailati in 1903. They show Euclid’s fifth postulate (equivalent to the parallel axiom) intrigued Saccheri throughout his life. In the Logica it led him to investigate the nature of definitions and in the Euclides to an attempt to apply his logic to prove the correctness of the fifth postulate. Although the fallacy in this attempt is now apparent, much of Saccheri’s logical and mathematical reasoning has become part of mathematical logic and non-Euclidean geometry. The Logica demonstrativa … is an attempt, probably the first in print, to explain the principles of logic more geometrico. Stress is placed on the distinction between definitiones quid nominis (nominal definitions), which simply define a concept, and definitiones quid rei (real definitions), which are nominal definitions to which a postulate of existence is attached. But when we are concerned with existence, the question arises whether one part of the definition is compatible with another part. This may be the case in what Saccheri called complex definitions. In these discussions he was deeply influenced by Euclid’s Elements, notably by the definition of parallelism of two lines. He warned against the definition, given by G. A. Borelli (Euclides restitutus, 1658) of parallels as equidistant straight lines. Thus, Saccheri was one of the first to draw explicit attention to the question of consistency and compatibility of axioms. To test whether a valid proposition is included in a definition, Saccheri proposed reasoning seemingly analogous to the classical reductio ad absurdum … his demonstration resulted from the fact that, reasoning from the negation, we obtain exactly the proposition to be proved, so that this proposition appears as the consequence of its own negation” (DSB). The study of the consistency of axiom systems initiated by Saccheri is, of course, of great contemporary interest. One only need only remember the work of Kurt Gödel and Paul Cohen on the Axiom of Choice and the Continuum Hypothesis, for example. Provenance: Contemporary Italian ownership inscription at foot of title, “del Sig.r Medico Caleri,” indicating early ownership by an otherwise unidentified northern Italian physician Caleri, probably active in or near Turin around 1700. Later history untraced; in the later 20th or early 21st century in the library of a modern scholar whose research in Italian collections on the printing history of Saccheri’s Logica demonstrativa is cited (anonymously) in the present bookseller’s description. “In 1685 Saccheri (1667-1733) entered the Jesuit novitiate in Genoa and after two years taught at the Jesuit college in that city until 1690. Sent to Milan, he studied philosophy and theology at the Jesuit College of the Brera, and in March 1694 he was ordained a priest at Como. In the same year he was sent to teach philosophy first at Turin and, in 1697, at the Jesuit College of Pavia. In 1699 he began teaching philosophy at the university, where until his death he occupied the chair of mathematics” (DSB). The Logica was based on Saccheri’s teaching at Turin and, as we shall see, it continued to be used there even after he moved to Pavia. “Following the alleged example of Aristotle, Saccheri divided the Logica into (1) Prior Analytics, (2) Posterior Analytics, (3) Topics, and (4) Sophistics, including under these, respectively, the rules of correct argumentation, the method to be followed in scientific, thinking, the grounds of probable knowledge, and the fallacies which are ordinarily committed in argumentation. These four sections were in turn subdivided into 27 chapters as follows: the Prior Analytics into 14 chapters, on terms and their properties, propositions, rules of supposition and equipollence, etc., principles of opposition, conversion, inference, etc., and a new and ‘more excellent’ method of demonstration; the Posterior Analytics into eight chapters, on the principles of knowledge, analytics, definition, axiom, postulate, hypothesis, division, etc.; the Topics into two chapters on the nature and kinds of dialectical argument; and finally the Sophistics into three chapters on the various kinds of fallacies committed in argument. “Thus intent on finding out the true principles of logic, Saccheri began with the Prior Analytics, which he maintained is ‘the foundation of the rest’, and in a manner analogous to the method of geometers, developed at first a system of logic in the form of a series of demonstrations based upon definitions and postulates ‘proper to it’ and axioms ‘common to all the sciences,’ and later, in a manner more ingenious, supplanted the system with another based on a ‘more excellent’ method ‘without the assistance of any postulate.’ ‘When I speak of demonstrative logic,’ Saccheri explains, ‘I wish you to think of geometry – that rigorous method of demonstration which grudgingly admits first principles, and allows nothing that is not clear, not evident, not indubitable. “Galen it was who, some 1500 years before the advent of the Logica, first suggested a logica ordine geometrica demonstrata, but Saccheri seems to have been the first to actually formulate such a system. The Cartesians, of course, and especially Pascal, had found in geometry the ideal of demonstrative science, and Leibniz, a contemporary of Saccheri, had already published his Dissertatio de arte combinatoria, but no one had yet undertaken the Euclidean line of proof to demonstrate the very principles of logic. In this respect, the Logica was unique. Here logic was not only an ‘instrument’ of knowledge whose validity was assumed, a propadeutic ‘serving the interests of all the sciences,’ but itself a science presumably comparable to geometry. This was a radical departure from Aristotle” (Emch, pp. 57-59). Saccheri begins by clarifying the role of definitions in his system of logic. “In his Logica demonstrativa, Saccheri lays down the clear distinction between what he calls definitio quid nominis and definitio quid rei, namely, that the former are only intended to explain the meaning that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it. The definitio quid nominis becomes a definitio quid rei ‘by means of a postulate, or when we come to the question whether the thing exists and it is answered affirmatively’” (Halsted, p. xviii). Saccheri gives the example of a square, which can be defined as a plane figure with four equal sides and each interior angle a right-angle. This nominal definition becomes a real definition only when the existence of a square is established, as is done in Euclid I, 46. But in non-Euclidean geometry, no such figure exists. “Confusion between the nominal and the real definition as thus described, i.e., the use of the former in demonstration before it has been turned into latter by the necessary proof that the thing defined exists, is, according to the Saccheri, one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the ‘complexity’ of the definition, i.e., the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e., the simultaneous presence of which in a given figure can be proved, by means of other postulates etc., forming part of the basis of the science, to be impossible. “This signal anticipation of Mill’s famous distinction would alone justify … Vailati in saying of Saccheri: ‘This gives him the right to an eminent place in the history of modern logic.’ “But in additional elaboration Saccheri broadens the matter, clearly recognising the more general question relative to the necessity of excluding the possible existence of incompatibility among the fundamental postulates made the basis of a demonstrative science; and not merely their directly contradicting one another, but whether the falsity at one of them could be proved by means of the others, a thing not directly recognisable. “These questions, far from having grown old, are acquiring an ever greater importance with the accentuation of the modern tendency to regard as the function of mathematics, the development, logically coherent, of the consequences flowing from a given set of premises, whether or not these be susceptible of direct interpretation or experimental verification. “Since actually, in this case, the postulates assume the character of simple hypotheses subject to the condition of being mutually compatible, that is, of neither directly nor indirectly contradicting one another, the question relative to the means of ascertaining whether such compatibility really exists, ceases to be, in Vailati’s phrase, a pure question de luxe, upon its solution having come to depend the legitimacy and even the possibility of assuming a given system of hypotheses as basis of a demonstrative science. “How high the merit of having been far the first to envisage this difficult matter and to have proffered analysis of the various forms of fallacy to which its non-recognition may give rise! “Precisely to such subject is dedicated the final chapter of the Logica demonstrativa. And so ultra-modern and yet unfinished is this whole question here raised and entered upon first, that it beckons with rising interest to mathematicians and philosophers toward this little book, so near to vanishing unrecognised from the Earth … “The fallacy of ‘complex definitions,’ such as attribute to the thing defined the simultaneous possession of diverse properties, as for example Borelli’s of ‘parallel’ the property of being a straight line, and that of being also the locus of points of a plane equidistant from another given straight line, consists in supposing that such definitions can be adopted unchecked in the demonstrations, without the compatibility of the properties themselves having first been ascertained. “Obviously, in case such compatibility is lacking, in case the existence of an object possessing simultaneously the properties in question can be proved impossible (by means of the other hypotheses anteriorly postulated as a basis for the discussion), any argumentation among whose premises figure such definitions combined with the aforesaid hypotheses ceases to have value, being based on contradictory premises. “The forms of illusory reasoning examined and probed by Saccheri under the name of fallaciae duplicis hypothesis are precisely those which consist in believing that consequences worth considering can be deduced from systems of hypotheses incompatible with one another (i.e., such that among them are some whose negation can be deduced from the others); and of such he passes in review various types, beginning with the simplest, namely that of a syllogism whose premises directly contradict one another, and going on to the more complicated cases in which the contradiction can be revealed only by the successive development of the consequences of the system of hypotheses, or postulates, assumed as basis of the entire matter. “In the investigation of the independence of postulates the method consists in finding a case or a particular interpretation in which the proposition we wish to prove not deducible from others given, ceases to be true while all the others remain true. If such is found, we conclude that the proposition cannot be deduced from the others, else it would be true whenever they were. “This use and construction of examples to show the independence of a certain proposition from others given has of late assumed the importance of an ordinary and indispensable procedure in the strictly rigorous elaboration of any deductive theory. But Saccheri was the first to use this procedure and constructs his example without leaving the field of formal logic. If his treatment of his ‘hypothesis of acute angle’ is another case, it is the most marvellous in the world” (Halsted, pp. xix-xxi). “The question as to the consistency of a set of propositions whose members are to be employed as promises of a deductive system could hardly have occurred to Saccheri had he regarded every premise of the system as certain and evident, for he stated explicitly that any theorem which is derived from a set of premises whose members are each certain an evident is in turn certain and evident … But Saccheri the geometrician was well aware that this certainty and evidence was not always present, as was amply shown by the attitude of geometricians towards the fifth postulate of Euclid. It is well known that to the successors of Euclid ‘this axiom had signally failed to appear self-evident.’ Even with Euclid himself there is a reason to believe that he did not regard this postulate as being quite so fundamental, or quite so self-evident, as his other postulates. For, although he states it with the other postulates, he avoids using it until the twenty-ninth theorem of the first book. It was precisely on account of the controversy over the certainty and evidence of this postulate that Saccheri, 36 years after the publication of the Logica, endeavoured to show by an imposing series of demonstrations in the Euclides ‘that not without cause was that famous axiom assumed by Euclid as known per se’” (Emch, p. 148). According to Angelelli, the most important single innovation in the Logica, which Saccheri was the first to envisage, is the consequentia mirabilis, the ’more excellent method’ of deduction by which he was be able to proceed ‘without the assistance of any postulate’. “The idea seems to have been suggested to him in certain particulars by his study of Clavius. In a scholion to the 12th proposition of the ninth book of Euclid, Clavius objects to Cardanus’s claim to originality in employing a method which derives a proposition by assuming the contradictory of the proposition to be proved. The specific words of Cardanus are: ‘this has never been done by anyone; indeed, it seems clearly impossible, and it is the most wonderful thing that has been devised since the creation of the world, viz., to prove something from its opposite, the demonstration not leading to an impossibility’” (Emch, pp. 151-152). “In its most general terms, this method involves an operation by which the truth of a proposition is ostensibly proved by assuming precisely its opposite. Vailati has correctly observed that even though this method is closely analogous to that designated reductio ad absurdum, in so far at least as one begins in both with the supposition that the proposition to be proved is not true, the two methods should nevertheless be distinguished. Proof by the ordinary method of reductio ad absurdum consists in showing that the contradictory of the proposition to be proved leads to a conclusion which is contradictory to some other principle previously proved or assumed, whereas proof by the method of Saccheri consists in showing that the contradictory of the proposition to be proved leads precisely to its reassertion. Thus, contrary to an opinion which seems to have been held by some mathematicians, Saccheri’s method, at least in its original form, is not strictly reductio ad absurdum, but rather a procedure more correctly referred to as reaffirmation through denial … “Saccheri was convinced that his rather recondite use of reaffirmation through denial, was one of the most important ameliorations introduced by him into the field of logic. In this respect, he was not entirely mistaken, for … [he] introduced in the attempt a technique, which, in its subsequent adaptation in the Euclides, has become tremendously important in its consequences for both methodology and geometry … Although Saccheri did not have a clear perception of this principle, nevertheless he employed it so that by making an assumption contrary to Euclid’s parallel postulate he derived many non-Euclidean theorems which were subsequently obtained by Lobachevski and Bolyai” (Emch, pp. 227-231). The history of the publication of the Logica has hitherto been something of a mystery. Until recently, it was believed that there were three editions before the twentieth century, the first published in 1697, the second in 1701, and the third in 1735, two years after Saccheri’s death. In 2009, however, Paolo Pagli reported the discovery of ‘Two unnoticed editions of Girolamo Saccheri’s Logica demonstrativa’, one undated, the other being the 1699 work offered here. As we believe the following discussion demonstrates, these three ‘editions’ are, in fact, issues of the same (first) edition, and that the order in which they were published was: 1697, undated, 1699. This conclusion differs from Pagli’s conjectured order of publication, but we are in possession of information that apparently was not available to him. As the book is so rarely seen, it may be useful to include here some information about all three issues. Some of this information, which is based on extensive research in Italian libraries, was provided to us by the previous owner of our copy. In all three issues the main text is paginated 1-264, and both the text and the paper used are identical in all three issues; only the preliminary leaves and the philosophical ‘Theses’ at the end of the book vary from one issue to another. This clearly demonstrates that we are dealing with issues rather than editions. All three issues were printed by Domenicus Paulinus at Turin, using the same sheets for the main text. In the 1697 issue the same paper stock is used throughout the book; in the undated issue, the same paper is used for the main text and ‘Theses’, but a different stock is used for the preliminaries; and in the 1699 issue the paper used for both the preliminaries and the ‘Theses’ is different from that of the main text, and different from the preliminaries of the undated issue. The ‘Theses’ in the 1697 and undated issues are identical, but those in the 1699 issue are different, both textually and in terms of pagination. In our view, this clearly establishes the order of publication. The presence of the ‘Theses’ indicates that the book was to be used by students in their public defence, a requirement for being awarded a degree. Garibaldi has pointed out that the chance of a book being published was much higher when it was used as a public defence book than without this feature, because the examinee had to pay for the printing of the book, or at least part of it. The name of such a student, Joannes Franciscus Caselette, ‘Graveriarum Comes’ (Count of Gravere), appears in the title of the 1697 issue, which does not mention the name of the author. The name of a different student, Marcus Antonius Grondana, appears in the title of the undated edition, and again the author’s name does not appear. In the 1699 issue, Saccheri’s name appears for the first time, albeit in the form of the pseudonym Carolus Iosephus Saccarellus. Pagli notes (p. 334) that “apart from the consonance with ‘Saccherius’, Saccarellus is one of the three highest mountains in Liguria, the one near the city of San Remo, Saccheri’s birth-place. So this rather clear pseudonym is the first (very timid!) public and official indication of the true author”. Why Saccheri chose not to have his name stated (explicitly) on the title pages of any of the three issues is still unclear, although Pagli ascribes it to Saccheri’s timidity, which he was able to partially overcome only after his appointment to the chair of mathematics at the University of Pavia in 1699. The 1697 and undated issues were both dedicated to Prince Victor Amadeus I, Duke of Savoy, and the full dedications (a little shorter in the undated issue) are in the same bombastic style, with an emphasis on the virtues of the Prince in war, peace, and culture. After the dedication, both issues contain the foreword to the Logica, ‘Lectori Benevolo.’ The 1699 issue was dedicated to the Blessed Aloysius Gonzaga, S. J. (1568-91). His beatification had taken place in 1605, and he was to be canonised in 1726, while Saccheri was still alive. This was the first mention of the Jesuits in any of the issues. The 1699 issue does not mention the name of any student on the title page, but it is clear that it had the same function as the other two, since at the end of the ‘Theses’ there is space to fill in the date of the public discussion into the following printed text: Disputabuntur publicè in Collegio Taurinensi Societatis Iesu Anno [...] Mense [...] Die [...] Hora [...]. This shows that the book was still used in Turin after Saccheri had moved to Pavia. The 1701 edition published at Turin, and the posthumous 1735 edition published at Cologne, are both printed in 8vo format, rather than the 12mo format used for the first edition, and both lack the philosophical theses. Pagli located 3 copies of the 1697 issue, 7 of the undated (one of which lacks a leaf of text), and three of the 1699 (one of which lacks the first and last quires) – all of these are in Italian institutional collections. Angelelli, ‘Saccheri’s Postulate,’ Vivarium 33 (1995), pp. 98-111. Emch, ‘The Logica Demonstrativa of Girolamo Saccheri,’ Scripta Mathematica 3 (1935), pp. 51-60, 143-152, 221-233. Garibaldi, 'Il contributo dei Gesuiti alla didattica e alla critica dei principi della natematica da Prof. Dipartimento Clavio a Saccheri?,’ pp. 194-207 in: Atti del Convegno 'Il pensiero matematico nella ricerca storica italiana, Ancona 2-28 marzo 1992 (Frosali & Ottaviani eds.), 1993. Halsted, Girolamo Saccheri’s Euclides Vindicatus, 1920. Pagli, ‘Two unnoticed editions of Girolamo Saccheri’s Logica demonstrativa’, History and Philosophy of Logic 30 (2009), pp. 331-340. Vailati, ‘Di un’opera dimenticata del P. Girolamo Saccheri (‘Logica Demonstrativa’ 1697), Rivista Filosofica 6 (1903), pp. 212-219.
12mo (140 x 79 mm), pp. [1-9], 10-12, 264, 12 (browning, mostly light but heavier towards the end of the main text, two leaves with unimportant stains). Contemporary calf, spine gilt with orange lettering-piece (slightly worn). A very good copy of an incredibly rare book.
Item #6348
Price: $38,500.00
