**Giovanni Saccheri** was an Italian Jesuit who did important early work on non-euclidean geometry.

- 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.
- In this Jesuit College Saccheri taught grammar and studied philosophy and theology.
- Both exerted a major influence on Saccheri, Tommaso through personal contacts at Brera College and Giovanni through correspondence.
- Perhaps Giovanni Ceva had the greatest mathematical influence for his passion for geometry, seen in his book "De lineis rectis" Ⓣ(Straight lines) (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" Ⓣ(The search for geometry) (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.
- Saccheri taught at the College in Turin from 1694 to 1697, three important years for they led to the publication of Logica demonstrativa Ⓣ(Logic demonstrated) (1697).
- 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.
- The first issue Saccheri never mentions.
- 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.
- 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.
- In recognition of his appreciation of the appointment by the Senate of Milan, Saccheri dedicated his next work Neo-statica Ⓣ(Neo-statics), published in 1708, to them.
- 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.
- It matters little whether he was aware of Omar Khayyam's insights, for Saccheri's work is certainly a masterpiece.
- Saccheri took a line segment, say ABABAB, with two equal segments ACACAC and BDBDBD each perpendicular to ABABAB.
- Saccheri knew that if he could prove that the angles at CCC and DDD were right angles without using the Parallel Postulate, then he could deduce the Parallel Postulate from the other axioms.
- Saccheri then adopted his famous approach: he assumed that the angles at CCC and DDD were not 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.
- Saccheri well understood these extra assumptions of Euclid, and he too made these assumptions as well as the first four of Euclid's axioms.
- Next Saccheri assumed that the two equal angles in his quadrilateral were each less than a right angle.
- Saccheri proved many theorems in these seventy pages.
- 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.
- As we wrote above, after 1697 Saccheri worked in Pavia for the rest of his life.
- Saccheri chose not to return.
- 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.
- 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.

Born 5 September 1667, San Remo, Genoa (now Italy). Died 25 October 1733, Milan (now Italy).

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Geometry, Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive