Quaesita geometrica a comite Rugerio de Vigintimillis omnibus proposita.
Milan: Milan: Marc’Antonio Pandolfo Malatesta, 1693. First edition, exceptionally rare, of Saccheri’s first published work on geometry, a precursor to his famous treatise on non-Euclidean geometry, Euclides ab omni naevo vindicatus (1733). “Saccheri proceeded in his youthful work with a ‘severa methodo’, aiming towards the construction of a geometry solidly based on its axioms and certain in its conclusions … for him the problem is that of deepening the meaning and the rigour of the logical structure of Euclidean geometry and of the ‘demonstrative logic’ which allows mathematical truths to be linked with absolute rigour to the postulates.The frequent and systematic use of the ‘reductio ad impossibile’ in the demonstrative procedures used in the Quaesita geometrica shows that the Jesuit mathematician, already in 1693, was above all interested in putting to the test those logical instruments which he would use with such skill later” (Brigaglia & Nastasi, p. 37, our translation). The Quaesita elegantly solves, using synthetic Euclidean methods, some geometric problems that had been proposed by the brilliant, but now practically unknown, Sicilian mathematician Count Ruggero Ventimiglia, which he had proposed in a work published in 1692. The problems themselves are less important than the methods Saccheri uses to solve them, methods which show Saccheri to be a significant figure in the history of projective geometry. “The solutions of the first two problems gave Saccheri, assisted by his friend and colleague Giovanni Ceva, the opportunity to rediscover, by a systematic employment of the properties of harmonic and anharmonic ratio, many important theorems of Desargues, La Hire, and others … The recent edition, moreover, of Newton’s manuscript work on ‘Geometria’, and Whiteside’s careful notes upon it [Papers, Vol. VII, Ch. 2] have given us a new perspective on the ‘refound’ interest in classical geometry during the later seventeenth century … We have now verified to our satisfaction not only that the development of Ceva’s and Saccheri’s mathematical thinking derived in an organic way from the study of the pole-polar properties of conics in the Italian tradition, but that they were also greatly influenced (in their search for unifying theoretical principles) by the ideas of René Descartes, even while refusing to follow his preference for algebraic technicalities” (ibid., p. 7). Saccheri was certainly familiar with Cartesian methods. His solutions of the 4th and 6th problems strongly suggest that he first treated these problems analytically and then converted the solutions into synthetic form. Again, in the 3rd problem, Saccheri gives the sought for hyperbola without explanation, and only verifies a posteriori that it satisfies the required properties. “But this is even more evident from his correspondence, in which Saccheri, for example, studies in a completely analytical way a non-trivial sextic … in many Jesuit colleges, particularly Spanish ones (and Pavia was then Spanish), geometry was taught according to analytical methods” (ibid., p. 31). A second edition was published in the following year under the title Sphinx Geometra, seu quaesita geometrica proposita, et solute, to which Ventimiglia’s original questions were added. OCLC lists 6 copies worldwide (Bayerische Staatsbibliothek (2 copies), BnF, Biblioteca Nazionale Centrale, Rome (2 copies, from the Jesuits’ Collegio Romano and Casa Professa), University of Turin (Giuseppe Peano’s copy)) – no copy in the US. No other copy on RBH. “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. “One of Saccheri’s teachers at the Brera was Tomasso Ceva, best known as a poet but also well versed in mathematics and mechanics. Through him Saccheri met his brother Giovanni, a mathematician living at the Gonzaga court in Mantua. This Ceva is known for his theorem in the geometry of triangles (1678). Under Ceva’s influence Saccheri published his first book, Quaesita geometrica (1693)” (DSB). After the dedication to the Spanish viceroy of Milan, in which the author declares to have solved the Quesiti in the space of two days (although this was an exaggeration), the Ad lectorem follows. This “is an irreplaceable document attesting to the cultural atmosphere in which the Milanese group worked at the end of the 17th century (the brothers Giovanni and Tommaso Ceva, Saccheri himself and Pietro Paolo Caravaggio) and above all to Saccheri’s programme of research, which would conclude with the well-known Euclides ab omni naevo vindicatus (Milan, 1733) … the Jesuit mathematician expressed very clearly his understanding of the fact that through the study, for example, of the harmonic ratio, a uniform and fairly general doctrine could be created” (Brigaglia & Nastasi, p. 8). Saccheri writes (our translation): ‘You have here, my Reader, Geometer, the solution of the problems proposed to everyone by Count Ruggero Ventimiglia, in the Journal of Parma, which I received last April. The talent of this noble young man and his understanding of geometrical matters are sufficiently indicated by the solution of the twelve questions that he published formerly while covering himself with the veil of anonymity. The problems, or, to say more truly, the enigmas that he formerly proposed are drawn from the arcana of the highest geometry. One will judge their novelty and their depth, by the solutions we give them … I have changed the order of the propositions.There you will find two theorems due to my great friend, Girolamo Ceva, demonstrated and constructed by a new method that he invented.He has printed [De lineis rectis, Milan, 1678], which I do not hesitate to rank among the beautiful discoveries of our time.’ The ’harmonic ratio’ is a concept from projective geometry. Based on the idea of projecting a figure from one plane to another, projective geometry was initially the concern of artists. The fundamental quantities of classical geometry, such as length and angle, are not preserved by projection, so they have no meaning in projective geometry. Projective geometry can discuss only things that are preserved by projection, such as points and lines, and also conics (parabolas, hyperbolas, and ellipses). A fundamental concept in projective geometry is a “ratio of ratios” of lengths called the harmonic-ratio; it is preserved by projection, and many other concepts in projective geometry can be derived from it. If A, B, C, D are four points on a straight line, their harmonic ratio is AC x BD / AD x BC, where AC, etc. are signed distances. If this harmonic ratio is equal to –1, then D is said to be the harmonic conjugate of C with respect to A and B. A conic is a curve in the plane such that, if P is a point not on C, and if a variable line through P intersects C at points A and B, then the variable harmonic conjugate of P with respect to A and B traces out a line. The point P is called the pole of that line of harmonic conjugates, and this line is called the polar line of P with respect to the conic. Some of these concepts will be mentioned below. Although one mathematical result about projections was discovered by the Greek mathematician Pappus around 300CE, the most important advances were made by the French mathematician Girard Desargues about 1640. His ideas were used by Pascal in his study of conics, but they were ahead of their time and it was not until Philippe de la Hire published his account of them in Nouvelle Méthode en Géométrie (1676) that they were assimilated into mainstream mathematics. The proposer of the problems solved by Saccheri in the Quaesita, Ruggero Ventimiglia, (1670-98), came from a prominent Sicilian family that had connections with the famous mathematicians Francesco Maurolico (1494-1575) and Giovanni Borelli (1608-79). In 1690, he published Enodationes duodecim problematum a Geometra post tabulam latente propositorum, the solutions to 12 problems posed by an anonymous Dutch geometer around 1675. Ventimiglia, then twenty years old, solved these problems by means of a systematic use of the harmonic ratio and of the Sectio determinate of Apollonius.Two years later, he published six new problems of his own in a book that now appears to be lost, Geometram quaero, and it is these problems which Saccheri solves in the present work. Saccheri recognized that Ventimiglia’s problems were not chosen at random, but “problems articulated for a precise purpose, which imply a precise programme, a programme on which Saccheri himself was studying intensively in those years” (ibid., p. 9). The six problems treated in the Quaesita are divided into three groups. The first two problems refer to the polar properties of conics: Prob. 1: let A, B, C, D be points on a conic, let the tangents to the conic at A and B intersect at K, let AD and BC intersect at H, and let AC and BD intersect at S; we have to show that K, S, and H lie on a straight line. Prob. 2: let A, B, C, D be points on a conic (not a hyperbola), let AC and BD intersect at T, let AB and CD intersect at P, and let PT intersect the conic at H and K; we have to show that PT is the harmonic mean between PK and PH, i.e., PK : PH = KT : TH. The second group (Prob. 3 and 4) refers to the construction of conics (‘solid loci’) by projective methods.Aside from some technical complications, the problem is to determine: Prob. 3: given a circle, a point A on it, a perpendicular to the diameter through A, as the point G on the line varies, find the locus of the fourth harmonic F between A, the point C of intersection between A and the circle, and G itself – i.e., AF : AC = GF : GC (the result is a hyperbola);Prob. 4: given two lines r and s, an angle a and a point D outside them, find the locus of the third vertex M of a triangle that has two vertices (A, N) that slide on s and r respectively, with N being the point of intersection between AC and the line r, angle CAN = a and CA2 : CN x AM is a given ratio (the required locus is generally a parabola).Finally, problems 5 and 6 are both ‘insertion’ problems: Prob. 5: given two circles, draw a straight line cutting the first at A and B, say, and tangent to the second at C such that AB : BC is a given ratio;Prob. 6: given two straight lines and a point P, find the circle passing through P, tangent to the first straight line and cutting off a chord on the other line of a given length. “Having examined the problems, it is not difficult to understand the proposer’s goal: it is a question of finding the most general properties (of projective type) relating to the harmonic ratio and to apply them to the systematic study of conics … This systematic character was immediately noticed by Saccheri who comments as follows [Ad lectorem]: ‘It often happens that the same method lets one discover many different things very worthy of being known.This is what has happened, I believe, in the present solutions … As to harmonic ratios, which Girolamo Ceva especially has treated in a superior manner, we have discovered here much more than what the ancients and the moderns had found until now.’ “We believe that it is especially worth noting the enthusiastic emphasis on the use of the harmonic ratio and the conviction that in this work one finds many more of its properties than what has been done so far.Saccheri probably does not know the works of Desargues and Pascal, but the enthusiastic tone (so rare in a man so shy and reluctant to publish) does not allow for doubts: he is convinced of having found something of very general importance and not only of having solved some interesting problems.And it is for this reason that he publishes with great haste a work that is far from perfect (two out of six problems are unfinished).In fact, he believes that the first three problems (which are treated in a much better way than the others and take up 19 pages out of 34) would be enough to give universal meaning to his solutions” (ibid., pp. 9-10). “In fact, the [Quaesita] can certainly be considered as the most mature sequel to the famous work of Ceva [De lineis rectis, 1678] … What is worth noting here is the absolute continuity between the work of 1678 and that of 1693. In the Quaesita geometrica, Ceva and Saccheri clearly demonstrate that the work of 1678 was not an isolated episode: the more recent work proves that a large number of results regarding the study of conics can be obtained through the use of the projective techniques so profitably devised 15 years earlier.In fact, one could say that the Quaesita constitute the logical culmination of the De lineis.Exactly as in every modern text of projective geometry, the segmental properties of lines and triangles constitute the foundation on which to build the edifice of the systematic study of the polar properties of conics.In fact, Ceva and Saccheri not only resolve the very general questions posed by Ventimiglia, but also produce, during the demonstrations, a series of fundamental and now very well-known results on those properties. Even a cursory examination of the solutions to the 1st and 2nd problems of the Geometram quaero immediately demonstrates how Ceva and Saccheri had gone well ahead and what was still in its infancy in 1678 had found its completion [in the Quaesita] …” (ibid., p. 15). “If we move from the first two problems of Ventimiglia to the second pair, we see that it deals with another group of questions, those known as ‘solid loci’.Examining this second group of problems we realize that they correspond to beautiful constructions of the hyperbola and of the parabola. The problems all determine one-to-one correspondences between a straight line and the conic and can therefore be seen as analogues of the organic constructions of Newton (Lemmas XX and XXI of the Principia). Saccheri’s solutions are not entirely satisfactory.For problem 3 he determines the circle and then, given the hyperbola, which he probably determined algebraically, he demonstrates that it is the sought locus” (ibid., p. 25). For problem 4, Saccheri confesses his failure [p. 34]: ‘I have pursued this beast by his tracks, and now I end the search.It is true that the author whom you seek to speak of is an ingenious author, this is the innermost part of his secret’. Bosmans (p. 16, note 1) gives a modern statement of problem 4, which caused Saccheri so much difficulty. “Given two lines AB, AC, and a fixed point D. The vertex S of a constant angle moves along AB.One of the sides of the angle passes through the fixed point D and meets AC at G. On the second side of the angle we take a point M such that SM2 = SD x DG.We ask for the locus of point M.” The problem was later taken up by the Marquis de L'Hospital using Cartesian methods [Traité analytiques des sections coniques (1720), pp. 254-259]. Problems 5 and 6 are both ‘insertion problems’. A simple example of such a problem is as follows. Suppose we have two parallel straight lines and a point O between them. We draw another straight line passing through O. Unless this is parallel to the given lines it will intersect them, say in points A and B. We want to choose this third line so that AB is equal to a given distance. In other words, we have to ‘insert’ a line segment of given length between the two given parallel lines. It turns out that this is not a problem that can be solved by ruler and compasses: it is a ‘solid’ problem. Nevertheless, Pappus allowed the existence of the solution to such problems, and the great French mathematician François Viète stated in his Supplementum geometriae (1593) that any solid problem could be solved using this additional postulate. Newton’s manuscript ‘Geometria’ was largely an account of what could be achieved by using it. Problem 5 turns out to be a ‘planar’ problem and was quickly solved by Saccheri. As to problem 6, Saccheri failed to find a solution, writing [pp. 25-26]: ‘I do not see how this problem can be universally solved by a single rule and circle … I have often analysed this case using analytical calculus, and I have always come across a solid equation. Therefore, I wait for a solution from the proposer himself.’ Saccheri thus recognized that problem 6 is a ‘solid’ problem, in analytical terms one that is equivalent to solving an equation of the third or fourth degree. His response shows the Quaesita to be part of the classical tradition (despite his evident familiarity with Descartes): for Saccheri, a problem can be ‘solved’ only if a solution can be found using ruler and compasses only, i.e., only if it is planar (equivalent to the solution of a quadratic equation). For Newton and Viète, the acceptance of the insertion postulate was a question of broadening the number of Euclidean postulates of constructibility (and therefore of existence) of geometric entities, in order to create a system capable of adapting to the new needs of mathematics.Saccheri expresses a different point of view: for him the problem is that of deepening the meaning and the rigor of the logical structure of Euclidean geometry and of the ‘demonstrative logic’ which allows mathematical truths to be linked with absolute rigor to the postulates.While this might seem to be looking to the past, it was precisely this attitude which enabled Saccheri 40 years later to analyse in his Euclides vindicatus the consequences of omitting the parallel postulate from the usual Euclidean axioms, and thus to anticipate the discovery of non-Euclidean geometry a century later. Bosmans, ‘Le géomètre Jerome Saccheri S. J. (1667-1733),’ Révue des Questions Scientifiques 4 (1925), pp. 401-430. Brigaglia & Nastasi, ‘Le soluzioni di Girolamo Saccheri e Giovanni Ceva al 'Geometram quaero' di Ruggero Ventimiglia: Geometria proiettiva italiana nel tardo seicento,’ Archive for History of Exact Sciences 30 (1984), pp. 7-44.
4to (230 x 166 mm), pp. [12], 37, [3] (first and last leaves blank), with 8 folding plates, woodcut device on title, previous owner’s signature on title (one plate with pale discolouration). Contemporary calf, spine gilt in compartments (rubbed). A very good copy.
Item #6350
Price: $18,000.00
