Euclides ab omni nævo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universæ geometriæ principia.
Milan: Pauli Antonii Montani, 1733. First edition, notoriously rare, of this “masterpiece of eighteenth-century geometry” (De Risi, p. 40), “the first treatise on non-Euclidean geometry … this rare and inestimable treasure” (George Bruce Halstead). “Today it is generally recognized by mathematicians as one of the important documents in the history of geometry” (Emch, p. 53). In it, Saccheri attempted to give a demonstration of the famous Parallel Postulate of Euclidean geometry. Proofs had been attempted since the time of the ancient Greeks, but “Saccheri tried a wholly new way, and thus his book marks an epoch” (ibid.). The new way was to use the method of ‘consequentia mirabilis’ which Saccheri had studied in detail in his Logica demonstrativa (1697): assume the Parallel Postulate is false, make deductions from this assumption, and conclude that the Parallel Postulate must, in fact, be true (this method is closely related to, but distinct from, ‘reductio ad absurdam’). Saccheri believed, incorrectly, that his Proposition XXXII led to such a contradiction. However, as the great Italian geometer Corrado Segre wrote: “the first seventy pages, up to Proposition 32 inclusive, constitute an ensemble of logic and of geometric acumen which may be called perfect.” These pages, indeed, constitute essentially a textbook of non-Euclidean geometry, written a century before Bolyai and Lobachevsky. “Saccheri did not find the supposed contradiction, as it was nowhere to be found, but he was unable to convince himself that the new geometry he had erected might in fact be a reasonable alternative to Euclid’s Elements rather than a green-eyed monster: consequently, he pointed to a contradiction of his own making … This effort notwithstanding, the sacrilege, so to speak, had already been committed, and Saccheri’s outstanding achievements towards the construction of hyperbolic geometry, while disowned by their author and relegated to a book printed in quite few copies, sneaked into European mathematical culture and poisoned the minds of certain more acute, unprejudiced, or simply more modern geometers. One century after the Jesuit’s death, these scholars eagerly welcomed Saccheri’s ‘monster’ in their writings, thus celebrating the triumph of non-Euclidean geometry. Following this widespread story, Saccheri unwittingly (yet brilliantly) anticipated one of the most momentous conceptual revolutions in the genesis of contemporary mathematics” (De Risi, p. 4). The philosophical importance of non-Euclidean geometry was that it greatly clarified the relationship between mathematics, science and observation. Our present-day understanding of models of axioms, relative consistency and so on can all be traced back to the advent of non-Euclidean geometry. Before hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true. Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment. Non-Euclidean geometry paved the way for Riemannian geometry, which in turn paved the way for Einstein's General Theory of Relativity. Saccheri’s work is divided into two Books. The first, Propositions I–XXXIX, is devoted to the Parallel Postulate. The second, now less well known, is a defence of the profound treatment of ratio and proportion in Book V of the Elements. Of this second part, Halsted writes: “It shows again Saccheri’s wisdom, penetration and modernity.” In 1621 the Oxford scholar and mathematician Sir Henry Savile described the Parallel Postulate and the theory of proportion as “two blemishes, two moles” spoiling Euclid’s “beautiful body”. Saccheri referred to Savile’s comments in the very title pf his book, Euclid vindicated from every blemish. OCLC lists 6 copies in the US (UC Berkeley, Brown, Columbia, Huntington, Indiana, Michigan). Library Hub lists BL and UCL. No copies listed on RBH, although the Stanhope-Halsted copy sold at Christie’s New York, 25 February 2003, $26,290. In about 300 BC Euclid wrote the Elements, a book which “has exercised an influence upon the human mind greater than that of any other work except the Bible” (DSB). Euclid stated five postulates from which he deduced all the theorems and propositions in his work: Euclid was dissatisfied with the fifth postulate, also known as the Parallel Postulate, and he tried to avoid its use as long as possible. In fact, the first 28 propositions of The Elements are proved without using it, including results about congruence and the isosceles triangle theorem, but neither of the most famous results of Euclidean geometry, that the angle sum of a triangle is π (180°), and Pythagoras’s theorem. Because the parallel postulate is not transparently true, attempts were made to deduce it from the first four axioms. There were many such attempts. There is a lost work of Archimedes that was titled ‘On Parallel Lines’ It is conjectured by some that in it Archimedes replaced Euclid’s Parallel Postulate with one stating that two lines are parallel if and only if the distance between them is constant (this is equivalent to the postulate stated above in the presence of the other Euclidean axioms). The Syrian mathematician Posidonius attempted a proof of the Parallel Postulate based on such an assumption. Ptolemy also tried to prove the Postulate. He did not use the equidistant assumption, but he assumed that if two parallel lines are cut by a transversal, the interior angles on the same side of the transversal sum to two right-angles. Another attempt to prove the Parallel Postulate was made by the Greek scholar Proclus (410-485). Along with the Mathematical Collection of Pappus, the Commentary on the First Book of Euclid’s Elements by Proclus serves as a crucial historic source for ancient Greek geometry in addition to providing valuable mathematical notes on Euclid’s work. In his commentary on the Elements, Proclus (410-485) noted several incorrect attempts to deduce the fifth postulate from the other four, including one by Ptolemy, and gave a false proof of his own. Several Islamic writers in the Middle-Ages, perhaps most notably Umar Khayyam and Nasir al-Din al-Tusi, also attempted to prove the Parallel Postulate. Both of these writers considered what later became known as a ‘Saccheri quadrilateral’, as it played an important role in Saccheri’s work. This is a quadrilateral ABCD in which AB and CD are perpendicular to the line BD. Al-Tusi considers three hypotheses, according to whether the equal angles BAC and DCA are acute, obtuse, or right-angles, and attempts to refute the first two hypotheses, leaving the third which holds in Euclidean geometry. For seventeenth century mathematicians, the Parallel Postulate and the definitions of ‘ratio’ and ‘proportion’ were the two parts of elementary mathematics most in need of clarification and foundation. In his Praelectiones tresdecim in principium Elementorum Euclidis (Oxford, 1621), Sir Henry Savile describes them as “in pulcherrimo Geometriea corpore duo sunt naevi,” “two blemishes, two moles in the most beautiful body of geometry.” Savile even went so far as to establish a mathematics chair in Oxford designed to produce the research needed to wash the ugly blemishes away. Half a century later, John Wallis (1616-1703), the Savilian Professor of Geometry, in the second of his two lectures given on the evening of 11 July 1663, tried to prove the parallel postulate as originally stated by Euclid. Wallis concluded that he had accomplished the task assigned to him by his remote benefactor: in his words, he had finally “vindicated” Euclid. Wallis’s proof was correct, but he had made an unspoken assumption, namely that one can draw arbitrary similar triangles. This additional axiom turns out to be equivalent to the Parallel Postulate, in the presence of the other Euclidean axioms. Wallis’s ‘proof’ was included in the second volume of his Opera mathematica (1693), a book known to Saccheri. But the most important attempt to prove the Parallel Postulate was undoubtedly Saccheri’s. In the Euclides vindicatus, Saccheri applied the principles he had set out in his work Logica demonstrativa (1697) to the axioms of Euclid’s Elements. The Logica was important because it treated questions related to the compatibility of definitions. “To test whether a valid proposition is included in a definition, Saccheri proposed reasoning seemingly analogous to the classical reductio ad absurdum, using for his example Elements IX, 12 … [Saccheri’s] 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). An example of precisely this type of reasoning in the Euclides vindicatus will be seen below. Saccheri started with the Euclidean axioms except the Parallel Postulate and considered the three hypotheses studied by al-Tusi: the hypothesis of the acute angle (HAA), the hypothesis of the right angle (HRA), and the hypothesis of the obtuse angle (HOA). These are equivalent, in the presence of the other Euclidean axioms, to the statement that a triangle has angle sum <, =, or > π. Of these, HRA holds in Euclid’s Elements. Saccheri refutes the HOA by showing that, under this hypothesis, as well as under the HRA, the parallel postulate holds.: ‘The hypothesis of obtuse angle is completely false, because it destroys itself’ (Prop. XIV). Actually, it is now recognised that Saccheri’s argument here makes the assumption that straight lines of arbitrary length exist. Indeed, the HOA is valid in ‘elliptic geometry’, essentially the geometry on the surface of a sphere, on which the maximum length of a ‘straight’ line is the circumference of the sphere. That left him with HAA. If he could show that it too led to a contradiction then he would have shown that Euclid’s was the only consistent geometry. Saccheri’s attempt on the HAA contains many true, proved and unexpected results. For example: ‘If the angle-sum in one triangle be equal to, greater than, or less than two right angles, so will it be in every triangle.’ Saccheri also considered the location of points that are equidistant from a straight line and thus arrived at Lobachevsky’s concept of ‘horocycles’ (under the HAA, these equidistant curves are not straight lines, as they are in Euclidean geometry). Most importantly for his attempted proof of the Parallel Postulate, Saccheri introduces the concept of the ‘asymptotic parallel’. “To explain this, observe that the HAA geometry is one in which given a line l and a point P not on the line, there are infinitely many lines through P that never meet l. All these lines may be called parallels to l through P. Saccheri showed [in Prop. XXXII] that each one of them has a common perpendicular with the base line l, except for one in each direction, which separates the parallels from the lines that meet l. These lines he called the asymptotic parallels through the point P to the line l” (Gray, p. 82). Saccheri’s book climaxes in the very next proposition: ‘The hypothesis of acute angle is absolutely false, because repugnant to the nature of the straight line.’ His argument is that asymptotic parallels have a common perpendicular ‘at infinity’, contrary to the nature of straight lines. However, Saccheri appears to recognise that this proof is not completely convincing. He therefore continues in Part I by proving a series of five lemmas to produce ‘the most exact demonstration’. Saccheri drew all this together and deduced the existence of a unique straight line perpendicular to a given straight line at a given point and observed that it cannot split into two. This enabled him to spell out exactly why the HAA was repugnant to the nature of the straight line. But even now he is not satisfied. At the end of Part I he writes: ‘And here I might safely stop. But I do not wish to leave any stone unturned, that I may show the hostile hypothesis of acute angle, torn out by the very roots, contradictory to itself’ [Scholion, p. 86, Halsted’s translation]. He therefore goes on in Part II to give another proof that HAA is untenable. Unfortunately, this second proof contains an error. "Saccheri considers the locus of points in the plane equidistant from a straight line. Unlike his predecessors, he is aware that under the acute-angle hypothesis this line is neither a straight line nor a circle. In computing the length of an arc of this curve by means of infinitesimals Saccheri makes a mistake and concludes that the required length is equal to the distance between the feet of the perpendiculars dropped from the ends of the arc to the straight line. On the other hand, Saccheri has shown that the perpendiculars move apart, so that the distance between the ends of the arc, to say nothing of the arc length, is greater than the distance between the feet of the perpendiculars. Having found this contradiction Saccheri again declares [Proposition XXXVIII] that: ‘The hypothesis of acute angle is absolutely false, because it destroys itself’” (Rosenfeld, p. 99). Saccheri’s concluding remark shows that he is still not entirely satisfied: ‘It is well to consider here a notable difference between the foregoing refutations of the two hypotheses. For in regard to the hypothesis of obtuse angle the thing is clearer than midday light; since from it assumed as true is demonstrated the absolute universal truth of the controverted Euclidean postulate, from which afterward is demonstrated the absolute falsity of this hypothesis; as is established from P. XIII. and P. XIV. But on the contrary I do not attain to proving the falsity of the other hypothesis, that of acute angle, without previously proving; that the line, all of whose points are equidistant from an assumed straight line lying in the same plane with it, is equal to this straight, which itself finally I do not appear to demonstrate from the viscera of the very hypothesis, as must be done for a perfect refutation’ [Scholion, p. 98]. “As regards the theory of proportions, the subject of Book Two, we know that Saccheri was engaged in the topic since his youth, to the point where he refers in Euclid Vindicated to some results already obtained in the Demonstrative Logic from 1697. And if it is true that those references are quite unspecific, and that the Logic does not contain any relevant mathematical proof in the theory of ratios, it is equally true that the strengths of Book Two of Euclid Vindicated are not those few and poorly executed demonstrations, but precisely its logical structure and its epistemological considerations. In the end, it seems as though Saccheri could very well have held the contents of Book Two in his hands from the beginning of his mathematical career, although the actual composition must have taken place much later” (De Risi, p. 32). The Euclides vindicatus made little impact at the time of its publication. Undoubtedly, this was partly because eighteenth-century mathematical research in analysis, algebra, and mechanics was rapidly moving in new and promising directions, and scholars were loath to interrupt its progress to scrutinize the foundations of elementary geometry. However, in 1763 Abraham Kästner, then regarded as the best teacher of mathematics in Germany, “encouraged his pupil Simon Klügel to write a dissertation reviewing all the (failed) attempts to prove the axiom. Klügel’s Recensio from 1763 represented a turning point for Euclid Vindicated, which was the work most extensively discussed therein. Klügel provides a summary of the general structure and an abridgement of several theorems from Saccheri’s work, then proceeds to level all sorts of objections against it. The mathematical objections are correct, and succeed in uncovering the fallacies in Saccheri’s arguments … Most importantly, Klügel laments that Saccheri devotes so much time in proving theorems in the acute angle hypothesis – that is, in hyperbolic geometry – which are not strictly necessary for the Jesuit’s conclusions: a staunch Euclidean, Klügel thus overlooked what we (along with Saccheri, perhaps) view as the most beautiful fruits of Euclid Vindicated. In any case, it was Klügel’s dissertation that precipitated the general discussion of the parallel axiom in Germany, a discussion that would yield legions of essays and inconclusive demonstrations over the next fifty years, and that would eventually culminate in Gauss’ work … “In addition, interpreters have debated the question of whether Gauss, Lobachevsky and Bolyai were acquainted with Euclid Vindicated. We know that a copy of the book existed in Göttingen, and it is therefore possible – even likely – that Gauss read it … Lobachevsky and Bolyai, on the one hand, do not discuss elliptic geometry at all: this aligns them with Legendre, whom they certainly knew, rather than with Clavius and Saccheri, who distinguished three hypotheses instead of two (Euclidean versus non-Euclidean) … In any case, it does not seem necessary to conjecture that these thinkers drew directly on a work as obscure as Saccheri’s. We should also note that these new developments in hyperbolic geometry, which in some sense mark its official origin, take as their starting point an absolute definition of parallel lines (valid for both Euclidean and hyperbolic space), which hinges on the concept of an asymptotic limit straight line. Such a line is also the subject of the last theorem proved by Saccheri under the acute angle hypothesis. Hence, Saccheri’s last word represents the basis for and beginning of Gauss’s, Lobachevsky’s and Bolyai’s researches. Euclid Vindicated, thereby, fulfilled in some sense the function of a ladder that one must throw away after he has climbed it: a new era of mathematics had begun” (De Risi, pp. 52-54). “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). Publication of Euclides vindicatus could only take place after approval by the Inquisition and it received this on 13 July 1733. The work then passed to the Provincial Company of Jesus on 16 August 1733 for their approval. Saccheri died in Milan two months later. We do not know if he ever saw the printed volume. De Risi (ed.), Girolamo Saccheri. Euclid Vindicated from Every Blemish, 2014. Emch, ‘The Logica demonstrativa of Girolamo Saccheri,’ Scripta Mathematica 3 (1935), pp. 51-60, 143-152 & 221-233. Gray, Worlds out of Nothing. A Course in the History of Geometry in the 19th Century, 2007. Halsted (ed.), Girolamo Saccheri’s Euclides Vindicatus, 1920. Rosenfeld, A History of Non-Euclidean Geometry, 1988. Segre, ‘Congetture intorno all'influenza di Girolamo Saccheri sulla formazione della geometria non-euclidea,’ Atti della Reale Accademia delle Scienze 38 (1903), pp. 535–547.
‘According as an angle inscribed in a semicircle is right, obtuse or acute, the hypothesis of right, obtuse or acute angle is true.’
‘With the hypothesis of the right angle, two distinct straight lines intersect, except in the one case in which a transversal cuts them under equal corresponding angles. With the hypothesis of the obtuse angle, two straight lines always intersect. With the hypothesis of the acute angle there are infinitely many straight lines through a given point not on the given straight line, which do not meet the given straight line.’
4to (237 x 184 mm), pp. xvi, 142, with 6 folding plates (light browning). Contemporary calf, spine gilt, marbled endpapers (slightly worn).
Item #6374
Price: $60,000.00
