Giovanni Girolamo Saccheri
Encyclopedia
Giovanni Girolamo Saccheri (September 5, 1667, Sanremo
– October 25, 1733) was an Italian
Jesuit priest, scholastic philosopher, and mathematician
.
Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin
from 1694 to 1697, and philosophy, theology, and mathematics at Pavia
from 1697 until his death. He was a protege of the mathematician Tommaso Ceva
and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).
He is primarily known today for his last publication, in 1733 shortly before his death. Now considered the second work in 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.
Many of Saccheri's ideas have precedent in the 11th Century Persian polymath Omar Khayyam
's Discussion of Difficulties in Euclid (Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis), a fact ignored in most Western sources until recently.
It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. The Saccheri quadrilateral
is now sometimes referred to as the Khayyam-Saccheri quadrilateral.
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 this 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, today this principle is 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
.
There is some minor argument on whether Saccheri really meant this, as he published his work in the final year of his life, came extremely close to discovering non-Eucliean geometry and was a logician. Some believe Saccheri only concluded in such way in an intent to avoid criticism that might come from seemingly illogical aspects of hyperbolic geometry
.
Sanremo
Sanremo or San Remo is a city with about 57,000 inhabitants on the Mediterranean coast of western Liguria in north-western Italy. Founded in Roman times, the city is best known as a tourist destination on the Italian Riviera. It hosts numerous cultural events, such as the Sanremo Music Festival...
– October 25, 1733) was an Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
Jesuit priest, scholastic philosopher, and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
.
Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin
Turin
Turin is a city and major business and cultural centre in northern Italy, capital of the Piedmont region, located mainly on the left bank of the Po River and surrounded by the Alpine arch. The population of the city proper is 909,193 while the population of the urban area is estimated by Eurostat...
from 1694 to 1697, and philosophy, theology, and mathematics at Pavia
Pavia
Pavia , the ancient Ticinum, is a town and comune of south-western Lombardy, northern Italy, 35 km south of Milan on the lower Ticino river near its confluence with the Po. It is the capital of the province of Pavia. It has a population of c. 71,000...
from 1697 until his death. He was a protege of the mathematician Tommaso Ceva
Tommaso Ceva
Tommaso Ceva was an Italian Jesuit mathematician from Milan. He was the brother of Giovanni Ceva....
and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).
He is primarily known today for his last publication, in 1733 shortly before his death. Now considered the second work in non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami
Eugenio Beltrami
Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...
in the mid-19th Century.
Many of Saccheri's ideas have precedent in the 11th Century Persian polymath Omar Khayyam
Omar Khayyám
Omar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....
's Discussion of Difficulties in Euclid (Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis), a fact ignored in most Western sources until recently.
It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. The Saccheri quadrilateral
Saccheri quadrilateral
A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclid vindicatus , an attempt to prove the parallel postulate using the method Reductio ad absurdum...
is now sometimes referred to as the Khayyam-Saccheri quadrilateral.
The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a reductio ad absurdum
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
proof of any alternative to Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's parallel postulate
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...
. To do this 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, today this principle is accepted as the basis of elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
, 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
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
There is some minor argument on whether Saccheri really meant this, as he published his work in the final year of his life, came extremely close to discovering non-Eucliean geometry and was a logician. Some believe Saccheri only concluded in such way in an intent to avoid criticism that might come from seemingly illogical aspects of hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
See also
- Saccheri–Legendre theoremSaccheri–Legendre theoremIn absolute geometry, the Saccheri–Legendre theorem asserts 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 the axiom that is equivalent to the parallel postulate...
- Hyperbolic geometryHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
- Parallel postulateParallel postulateIn 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...
- List of Jesuit scientists
- List of Roman Catholic scientist-clerics