Giovanni Girolamo Saccheri

Giovanni Girolamo Saccheri was born in Sept, 1667, in Genoa (Italy). He is known for making the logical deductions that lead to non-euclidean geometry.

Saccheri entered the Jesuit order in 1685, and two years later started teaching at the Jesuit college until 1690. From there he went to Milan, and learned was taught philosophy and theology at the Jesuit college of Brera. One of Saccheri’s teachers was Tommaso Ceva, best known as a poet, but also a mathematician.(2) Through Tommaso, Saccheri met Tommaso’s brother Giovanni, a mathematician who is known for his theorem in the geometry of triangles (1678). The Ceva brothers imparted their enthusiasm in mathematics to Saccheri.(3) Through influence from Giovanni, and with assistance in writing from Tommaso, Saccheri wrote his first mathematical work Quaesita geometrica (1693), in which he solved problems from elementary, and coordinate geometry. Ceva sent this book to Vincenzo Viviani, one of the last surviving pupils of Galileo, who in 1692 had challenged the learned world with a problem in analysis known as the Window of Viviani.(2) Although it had been solven by others, Viviani published his own solution, and sent one to Saccheri in exchange for the Quaesita. Two letters from Saccheri to Viviani have been preserved, one of which shows Saccheri’s solution. In 1694 he was ordained a priest at Como, he was then sent to teach Philosophy in Turin. Here Saccheri wrote Logica Demonstrativa (1679), which was on definitions, Saccheri distinguishes between two definitions the first ‘definitiones quid nominis’ or ‘nominis’ which are supposed to give the meaning of the term defined, and the second ‘definitiones quid rei’ or ‘reales’ which gives the meaning of the term, and claims that the concept exists. In the same year, he was sent to the Jesuit College of Pavia. In 1699, he started teaching philosophy at the university (again), at which he occupied the chair of mathematics until his death. At Pavia Saccheri wrote two books Neo-statica (1708), and Euclides ab omni naevo vindicatus (1733), the second of which contains the classic text that made Saccheri the precursor to non-euclidean geometry. Saccheri’s two books the Logica and the Euclides showed his interest in Eulclid’s fifth postulate, with the Logica investigating the nature of definitions, and the Euclides attempting to prove the fifth postulate(the parallel postulate).(2)

Euclid combined all the known information on mathematics in 13 books.(1) But, before he could arrange this information into theorems he had to articulate the unprovable premises that everyone took for granted like points and lines.(1) To do this he made 10 premises the first 5 “postulates” which dealt with geometry, and the second 5 “axioms” which were common in geometry and mathematics.(1) Nine of the premises were simple and convincing, but the 5th postulate was long and convoluted, compared to the rest and looked like a theorem.(1) An example is the first postulate which states “Two points determine one unique straight line.”, while the parallel postulate states “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced infinitely, meet on the side on which the angles are together less than two right angles”, this long postulate sounds like a theorem crying out for truth, and many geometers tried to provide one.(1)

Saccheri was one of these geometers, but he used a different method. Most of the geometers considered it more of an aesthetic problem than a logical one, but Saccheri was the first geometer to impose rigorous rules of logic in his attempt to get rid of Euclid’s “flaw”.(1) First Saccheri makes a quadrilateral “Given a line segment AB, construct segments AC and BD on the same side of AB such that AC = BD and both AC and BD are perpendicular to AB (Figure 9. 9). Then join C and D forming what is known today as a Saccheri Quadrilatera”(4) (4)
Without using the Parallel Postulate Saccheri was able to prove that the angles ACD and BDC were congruent, and he called these summit angles. He observed that only one of these following statements is true:
1) Right Angle Hypothesis: The summit angles are right angles.
2) Obtuse Angle Hypothesis: The summit angles are obtuse angles.
3) Acute Angle Hypothesis: The summit angles are acute angles. (4)
Saccheri was able to prove that the Parallel Postulate followed the Right Angle Hypothesis, and he planned to prove it right by showing that the two other hypotheses were untrue. (4) He was able to show that the Obtuse Angle Hypothesis was false because it contradicted the infinite length of a line, but he was never able to reach a contradiction. (4) The closest he got was “... the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines.” and later on he stated “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.”

Even though Saccheri was never able to prove the Parallel Postulate, it is important to note that his reasoning on this subject have become part of mathematical logic (even though the mathematicians who discovered non-euclidean geometry had never heard of him ).(3;4)

Works Cited

Saccheri's Flaw while eliminating Euclid's "Flaw" The Evolution of Non-Euclidean Geometry http://www.faculty.fairfield.edu/jmac/sj/sacflaw/sacflaw.htm
Saccheri, (Giovanni) Girolamo https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/saccheri-giovanni-girolamo
Giovanni Girolamo Saccheri http://www-groups.dcs.st-and.ac.uk/history/Biographies/Saccheri.html
Giovanni Girolamo Saccheri http://www.robertnowlan.com/pdfs/Saccheri,%20Giovanni%20Girolamo.pdf