**Giovanni Saccheri**'s father, Giovanni Felice Saccheri, was a lawyer. As a child Saccheri 'was notably precocious'. He entered the Jesuit Order at Genoa on 24 March 1685 so entering an Order highly involved in higher education. In 1554 the Company of Jesus of Genoa had begun taking an interest in higher education and had founded its own college. The building in which Saccheri studied was designed by Bartolomeo Bianco and built around 1640; today it is called the Palazzo Balbi and forms the main building of the University of Genoa. For the first two years he was totally taken up with his studies, but from 1687 onwards he began teaching at the College as well as studying philosophy and theology. In 1690 the Superiors of the Company of Jesus sent him to the Collegio di Brera in Milan. This College was founded by the Jesuits in 1571, designed by Martino Bassi and continued by Francesco Maria Ricchini. The Jesuits made the College into a school of higher education able to award doctorates. It received a papal bull in 1572 from Pope Gregory XIII. The topics taught there were Holy Scripture, Scholastic theology, mathematics, philosophy, Greek, Hebrew, the humanities, rhetoric and grammar.

In this Jesuit College Saccheri taught grammar and studied philosophy and theology. It was while he was studying in Brera College that he was encouraged to take up mathematics by one of his teachers Tommaso Ceva who suggested that he read Christopher Clavius's edition of Euclid's *Elements*. There were two Ceva brothers who had both been educated at the Brera College in Milan. Tommaso had remained at the College and taught there for forty years, while his brother Giovanni Ceva had been appointed as mathematician to the Duchy of Mantua in 1684. Both exerted a major influence on Saccheri, Tommaso through personal contacts at Brera College and Giovanni through correspondence. Mathematics played an important part in the lives of both Ceva brothers, and they soon imparted this enthusiasm to Saccheri. Perhaps Giovanni Ceva had the greatest mathematical influence for his passion for geometry, seen in his book *De lineis rectis* (1678), encouraged Saccheri to work in this area. It was through the influence of Giovanni Ceva that Saccheri published his first mathematical work *Quaesita geometrica* (1693), although the book was also written with considerable advice and help from Tommaso Ceva. In this book, which Saccheri dedicated to Guzman who was the governor of Milan, he solved many problems in elementary geometry. It was not a particularly significant work but showed that Saccheri was becoming deeply involved in thinking about Euclidean geometry. We should also mention at this point that Tommaso Ceva also encouraged Saccheri to correspond with the mathematician Vincenzo Viviani who worked in Florence.

Saccheri was ordained a priest in March 1694 at Como and then later in the year he was sent by the Superiors of the Jesuit Order to teach philosophy at the Jesuit College in Turin. This College, built by Guarini in 1679, operated until 1759 when it was handed over to the Academy of Sciences. While at the College, Saccheri made the acquaintance of Victor Amadeus II, the Duke of Savoy who played a major role in diplomacy that eventually led to the creation of the Italian State. The Duke often called on him when needing someone to undertake hard mathematical calculations. Saccheri taught at the College in Turin from 1694 to 1697, three important years for they led to the publication of *Logica demonstrativa* (1697). The work was dedicated to Count Filippo Archintio, who was on the Senate of Milan. Alberto Pascal writes [13]:-

Fruit of these three years of philosophic teaching was a little book which well merits to be better known: perhaps its extreme rarity has contributed to this oblivion, even since Giovanni Vailati, in1903, brought to light its superlative merit.

In fact the first edition of the work does not seem to be written by Saccheri since it appears under the name of Count Gravere who was Saccheri's student. When Count Gravere's theses were examined and published, Saccheri took the opportunity to publish the course on logic that he had been delivering at the College in Turin. However he seemed happy to let it appear as if was, like the theses, written by Count Gravere. Only one copy of this work has survived on which it is written "Author Father Hyeronymo Saccherio Societatis Jesu." George Halsted writes [11]:-

Saccheri, astute and prudent, had his reasons for issuing this three-year child of his genius under the count's cloak. Then as professor he changed subjects and residence, and only four years afterward did the book appear with his name. The first issue Saccheri never mentions. The second edition, so called, he refers to repeatedly and insistently. It differs from the first by some suppressions, especially in the preface, but no thought has been added during these four years of waiting.

The aim of this work is to study definitions. Saccheri distinguishes between two different types of definitions: the first he calls 'definitiones quid nominis' or 'nomindes' which are only intended to give the meaning of the term being defined; the second he calls 'definitiones quid rei' or 'reales' which in addition to giving the meaning of the term also claims that the concept being defined actually exists. For example we could define the mid-point of a line segment as the point which divides the segment into two equal segments. This would be a definition of the second type, namely one where we also know that the mid-point so defined could be constructed. Of course if this definition were made before a method of construction were known, then it would be a 'nomindes' which would become a 'reales' after the construction was given. Saccheri make it clear that Euclid understood this distinction for in Book I of *The Elements* he defines a square, but he does not assume its existence until after he has given a proof. However, Saccheri warns that other authors have not fully appreciated the distinction and have been led to false proofs by giving a definition which they assume to exist when in actual fact it is impossible. Vailati, who rediscovered this work in 1903, writes:-

This gives him[Saccheri]the right to an eminent place in the history of modern logic.

How high the merit of having been far the first to envisage this difficult matter and to have proffered an analysis of the various forms of fallacy to which its non-recognition may give rise!

In 1697, the year that *Logica demonstrativa* was first published, Saccheri was sent to the Jesuit College of Pavia to teach philosophy and theology. He taught at the College in Pavia until his death, but he also held the chair of mathematics at the University of Pavia from 1699 until his death. We note that Pavia was known as Ticinum in Roman times and the university, founded in 1361, was also called the Università Ticinese. The residencies named *Collegio Borromeo* and *Collegio Ghislieri* were built in the 16^{th} century. This university appointment was by the Senate of Milan so perhaps dedicating *Logica demonstrativa* to Count Filippo Archintio on the Senate had been a good move. In recognition of his appreciation of the appointment by the Senate of Milan, Saccheri dedicated his next work *Neo-statica*, published in 1708, to them. It is a work on statics of relatively little importance.

However, the main reason that Saccheri is remembered today is because of another work which was only rediscovered by Eugenio Beltrami 150 years after its first publication. In *Euclides ab Omni Naevo Vindicatus* (Euclid cleared of every defect), published in 1733, he did important early work on non-euclidean geometry, although he did not see it as such, rather an attempt to prove the parallel postulate of Euclid. This work, also dedicated to the Senate of Milan, can in many ways be seen as following the logical methods which he had introduced in *Logica demonstrativa*. It is unclear whether Saccheri was aware of Omar Khayyam's *Discussion of Difficulties in Euclid *or whether his approach was totally innovative. He was certainly aware of the work of John Wallis and Nasir al-Din al-Tusi on the Parallel Postulate since he criticises both in his book. It matters little whether he was aware of Omar Khayyam's insights, for Saccheri's work is certainly a masterpiece. Let us look at how he approached the question of the Parallel Postulate.

Saccheri took a line segment, say *AB*, with two equal segments *AC* and *BD* each perpendicular to *AB*. Join *CD* to obtain a quadrilateral. Saccheri knew that if he could prove that the angles at *C* and *D* were right angles without using the Parallel Postulate, then he could deduce the Parallel Postulate from the other axioms. He was easily able to prove that the angles at *C* and *D* were equal but proving that they were right angles was not at all easy. Saccheri then adopted his famous approach: he assumed that the angles at *C* and *D* were not right angles. His aim now was, working with all of Euclid's axioms except the Parallel Postulate, to obtain a contradiction. He had then two cases to consider: either the two equal angles at *C* and *D* were less than right angles or, alternatively, they were greater than right angles.

The second of these alternatives Saccheri was able to dispose of fairly quickly although it took him 13 propositions. Readers who are familiar with the basics of non-Euclidean geometry may be rather puzzled by this statement for they will know of geometries in which the angles in a triangle add to more than two right angles, making it possible for the two angles in Saccheri's quadrilateral each to be greater than a right angle. However, one has to realise that Euclid made more assumptions than the five axioms that he wrote down. Euclid had also assumed the Archimedean property, namely that if one extends a line segment by a given length sufficiently often it will exceed any given length, he had also assumed that every straight line is unbounded, is infinite, and divides the plane into two parts. Saccheri well understood these extra assumptions of Euclid, and he too made these assumptions as well as the first four of Euclid's axioms. It was these extra assumptions which allowed him to dispose of the obtuse angle hypothesis, for with this assumption he was able to show that straight lines were of finite length.

Next Saccheri assumed that the two equal angles in his quadrilateral were each less than a right angle. In this case, after 20 more propositions he was unable to obtain any contradiction and he developed many theorems of non-Euclidean geometry. As Corrado Segre wrote:-

Nevertheless the first seventy pages(apart from a few isolated phrases), up to Proposition32inclusive, constitute an ensemble of logic and of geometric acumen which may be called perfect.

For the constructive part of Saccheri's work, the first seventy pages, through Proposition32, all connoisseurs have enthusiastically expressed their admiration, their delight in its elegance, its exquisite artistic finish.

Saccheri proved many theorems in these seventy pages. For example, let us quote three:

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.

*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.

Now eventually Saccheri convinced himself that he had the contradiction that he was looking for to rule out the case that in a triangle the sum of the angles are less than two right angles. He wrote, rather weakly, that:-

... the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines.

However, he was not satisfied with this second case as is evident by what he then wrote:-

It is well to consider here a notable difference between the foregoing refutations of the two hypotheses. For in regard to the hypothesis of the obtuse angle the thing is clearer than midday light. ... But on the contrary, I do not attain to proving the falsity of the other hypothesis, that of the 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 line.

In fact *Euclides ab Omni Naevo Vindicatus * is in two parts, the first part consisting of 39 propositions, contains the material we have just described. The second part of the work defends Euclid's treatment of proportion in Book V of *The Elements*. Of this second part, Halsted writes [11]:-

It shows again Saccheri's wisdom, penetration and modernity.

As we wrote above, after 1697 Saccheri worked in Pavia for the rest of his life. However, there were attempts to tempt him to chairs at other universities. In 1713 Victor Amadeus II, the Duke of Savoy, tried to bring him back to Turin by offering him the chair of mathematics there. Saccheri chose not to return. There is another story, which seems unlikely to be true, that Victor Amadeus II offered Saccheri a bishopric if he would return to Turin. This seems untrue but the story may well have arisen because Victor Amadeus II was so keen to again have Saccheri's assistance. What is certainly true is that Saccheri was offered the chair of mathematics at the University of Padua (the university of the Republic of Venice) that had been filled by Galileo almost a century earlier. This was a prestigious chair, but again Saccheri chose to remain in Pavia. However, he was a member of the Academia Claelia Vigilantium in Milan and went to Milan in university vacations to spend time at the Colleggio di Nobili.

Publication of *Euclides ab Omni Naevo 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 and only 170 years later was the significance of the work realised. It is fair to say that the discovery of non-Euclidean geometry by Nikolai Lobachevsky and János Bolyai was not due to this masterpiece by Saccheri. Neither seems to have ever heard of him.

**Article by:** *J J O'Connor* and *E F Robertson*

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