Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian
Jesuit
, image = Ihs-logo.svg
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, caption = ChristogramOfficial seal of the Jesuits
, abbreviation = SJ
, nickname = Jesuits
, formation =
, founders ...
priest,
scholastic philosopher
Scholasticism was a medieval school of philosophy that employed a Organon, critical organic method of philosophical analysis predicated upon the Aristotelianism, Aristotelian categories (Aristotle), 10 Categories. Christian scholasticism eme ...
, and
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
.
Saccheri was born in
Sanremo
Sanremo (; lij, Sanrémmo(ro) or , ) or San Remo is a city and comune on the Mediterranean coast of Liguria, in northwestern Italy. Founded in Roman times, it has a population of 55,000, and is known as a tourist destination on the Italian Rivie ...
. He entered the Jesuit order in 1685 and was ordained as a priest in 1694. He taught philosophy at the
University of Turin
The University of Turin (Italian: ''Università degli Studi di Torino'', UNITO) is a public research university in the city of Turin, in the Piedmont region of Italy. It is one of the oldest universities in Europe and continues to play an impo ...
from 1694 to 1697 and philosophy, theology and mathematics at the
University of Pavia
The University of Pavia ( it, Università degli Studi di Pavia, UNIPV or ''Università di Pavia''; la, Alma Ticinensis Universitas) is a university located in Pavia, Lombardy, Italy. There was evidence of teaching as early as 1361, making it one ...
from 1697 until his death. He was a protégé of the mathematician
Tommaso Ceva
Tommaso Ceva (December 20, 1648 – February 3, 1737) was an Italian Jesuit mathematician from Milan. He was the brother of Giovanni Ceva.
Biography
From a wealthy Milanese family, Ceva entered the Society of Jesus in 1663. He taught math ...
and published several works including ''Quaesita geometrica'' (1693), ''Logica demonstrativa'' (1697), and ''Neo-statica'' (1708).
Geometrical work
He is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, ''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.
The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a ''
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'' proof of any alternative to
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's
parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...
. To do so, he assumed that the parallel postulate was false and attempted to derive a contradiction.
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
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
.
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
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
'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.
See also
*
Saccheri–Legendre theorem
In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of th ...
*
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
*
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...
*
List of Jesuit scientists
This is a list of Catholic clergy throughout history who have made contributions to science. These churchmen-scientists include Nicolaus Copernicus, Gregor Mendel, Georges Lemaître, Albertus Magnus, Roger Bacon, Pierre Gassendi, Roger Joseph ...
*
List of Roman Catholic cleric–scientists
This is a list of Catholic clergy throughout history who have made contributions to science. These churchmen-scientists include Nicolaus Copernicus, Gregor Mendel, Georges Lemaître, Albertus Magnus, Roger Bacon, Pierre Gassendi, Roger Joseph Bo ...
References
*
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
, ''Non-Euclidean Geometry'', Chapter 14 of ''The Colossal Book of Mathematics'', W. W.Norton & Company, 2001,
* 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);
2nd ed. (1986), review by
John Corcoran: ''Mathematical Reviews'' 88j:01013, 1988.
Works
*
*
External links
*
*
{{DEFAULTSORT:Saccheri, Giovanni Gerolamo
1667 births
1733 deaths
People from Sanremo
17th-century Italian mathematicians
18th-century Italian mathematicians
Geometers
17th-century Italian Jesuits
18th-century Italian Jesuits
Italian philosophers
Jesuit scientists