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Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri
	The frontispiece of "Euclides ab omni nævo vindicatus" (1733).
Born	September 5, 1667; San Remo, Republic of Genoa
Died	25 October 1733 (aged 66); Milan, Duchy of Milan
Nationality	Italian
Alma mater	Jesuit College of Brera, Milan
Known for	Non-Euclidean geometry; Saccheri quadrilateral; Saccheri–Legendre theorem
Parent(s)	Giovanni Felice Saccheri; Maria Saccheri
	Scientific career
Fields	Mathematics; Geometry; Logic
Institutions	University of Turin; University of Pavia
Notable students	Luigi Guido Grandi

Giovanni Girolamo Saccheri (Italian pronunciation: [dʒoˈvanni dʒiˈrɔːlamo sakˈkɛːri]; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He is considered the forerunner of non-Euclidean geometry.^[2]^[3]

Biography

The son of a lawyer, Saccheri was born in Sanremo, Genoa (now Italy) on September 5, 1667.^[4] From his youth he showed extreme precociousness and a spirit of inquiry.^[2] He entered the Jesuit novitiate in 1685. He studied philosophy and theology at the Jesuit College of Brera in Milan.^[5]

His mathematics teacher at the Brera college was Tommaso Ceva, who introduced him to his brother Giovanni.^[4] Ceva convinced Saccheri to devote himself to mathematical research and became the young man's mentor. Saccheri was in close scientific communion with both brothers. He used Ceva's ingenious methods in his first published work, 1693, solutions of six geometric problems proposed by the Sicilian mathematician Ruggero Ventimiglia (1670-1698).^[6]

Saccheri was ordained as a priest in March 1694. He taught philosophy at the University of Turin from 1694 to 1697 and philosophy, theology and mathematics at the University of Pavia from 1697 until his death.^[3] He published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708). Saccheri died in Milan on 25 October 1733.^[4]

The Logica demonstrativa, reissued in Turin in 1701 and in Cologne in 1735, gives Saccheri the right to an eminent place in the history of modern logic.^[7] According to Thomas Heath “Mill’s account of the true distinction between real and nominal definitions was fully anticipated by Saccheri.”^[8]

Geometrical work

Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century.^[9]

The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a reductio ad absurdum proof of any alternative to Euclid's parallel postulate. To do so, he assumed that the parallel postulate was false and attempted to derive a contradiction.^[3]

Since Euclid's postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°.

The first led to the conclusion that straight lines are finite, contradicting Euclid's second postulate. So Saccheri correctly rejected it. However, the principle is now accepted as the basis of elliptic geometry, where both the second and fifth postulates are rejected.

The second possibility turned out to be harder to refute. In fact he was unable to derive a logical contradiction and instead derived many non-intuitive results; for example that triangles have a maximum finite area and that there is an absolute unit of length. He finally concluded that: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Today, his results are theorems of hyperbolic geometry.^[10]

There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.

One tool that Saccheri developed in his work (now called a Saccheri quadrilateral) has a precedent in the 11th-century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid (Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis). Khayyam, however, made no significant use of the quadrilateral, whereas Saccheri explored its consequences deeply.^[11]

Works

Quaesita geometrica (in Latin). Milano: Marc'Antonio Pandolfo Malatesta. 1693.
Logica demonstrativa (in Latin). Pavia: eredi Carlo Francesco Magri. 1701.
Neostatica (in Latin). Milano: Giuseppe Pandolfo Malatesta. 1708.
Euclides ab omni naevo vindicatus (in Latin). Milano: Paolo Antonio Montani. 1733.
Euclide liberato da ogni macchia. Testo latino a fronte; a cura di Pierangelo Frigerio, introduzione di Imre Tóth ed. Elisabetta Cattanei, Milano, Bompiani, 2001.
Logica dimostrativa. Testo latino a fronte; a cura di Paolo Pagli e Corrado Mangione, Milano, Bompiani, 2011.

References

Bibliography

Saccheri, Gerolamo (2014). Euclid Vindicated from Every Blemish: Edited and Annotated by Vincenzo De Risi. Translated by G.B. Halsted and L. Allegri. Cham: Springer International Publishing. doi:10.1007/978-3-319-05966-2. ISBN 978-3-319-05965-5.

Halsted, George Bruce (1900). "Non-Euclidean Geometry". The American Mathematical Monthly. 7 (5): 123–133. doi:10.2307/2970500. JSTOR 2970500.
Segre, Corrado (1903). "Congetture intorno all'influenza di Girolamo Saccheri sulla formazione della geometria non-euclidea". Atti della R. Accademia delle Scienze. XXXVIII: 535–547.
Vailati, Giovanni (1903). "Di un'opera dimenticata del P. Girolamo Saccheri («Logica Demonstrativa» 1697)". Rivista Filosofica. 4 (VI): 528–540.
Roberto Bonola (1912) Non-Euclidean Geometry, Open Court, Chicago. English translation by H. S. Carslaw.
Emch, Arnold F. (1935). "The Logica Demonstrativa of Girolamo Saccheri". Scripta Mathematica. 3: 51–60, 143–152, 221–233.
Fitzpatrick, Mary of Mercy (1964). "Saccheri, forerunner of non-Euclidean geometry". The Mathematics Teacher. 57 (5): 323–332. doi:10.5951/MT.57.5.0323. JSTOR 27957056.
Brigaglia, Aldo; Nastasi, Pietro (1984). "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 (1): 7–44. doi:10.1007/BF00348433. JSTOR 41133716. S2CID 120682965.
Martin Gardner, Non-Euclidean Geometry, Chapter 14 of The Colossal Book of Mathematics, W. W.Norton & Company, 2001, ISBN 0-393-02023-1
M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 1st ed. 1974, 2nd ed. 1980, 3rd ed. 1993, 4th edition, W. H. Freeman, 2008.
Girolamo Saccheri, Euclides Vindicatus (1733), edited and translated by G. B. Halsted, 1st ed. (1920);^[12] 2nd ed. (1986), review by John Corcoran: Mathematical Reviews 88j:01013, 1988.
Angelelli, Ignacio (1995). "Saccheri's "Postulate"". Vivarium. 33 (1): 98–111. doi:10.1163/1568534952579812. JSTOR 42569929.
De Risi, Vincenzo (2013). "Saccheri, Giovanni Girolamo". Il Contributo italiano alla storia del Pensiero: Scienze. Istituto dell'Enciclopedia Italiana. Retrieved 12 August 2023.
Roero, Clara Silvia (2017). "SACCHERI, Giovanni Girolamo". Dizionario Biografico degli Italiani, Volume 89: Rovereto–Salvemini (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.