Grégoire de Saint-Vincent () - in Latin : Gregorius a Sancto Vincentio, in Dutch : Gregorius van St-Vincent - (8 September 1584
Bruges
Bruges ( , ; ; ) is the capital and largest city of the province of West Flanders, in the Flemish Region of Belgium. It is in the northwest of the country, and is the sixth most populous city in the country.
The area of the whole city amoun ...
– 5 June 1667
Ghent
Ghent ( ; ; historically known as ''Gaunt'' in English) is a City status in Belgium, city and a Municipalities of Belgium, municipality in the Flemish Region of Belgium. It is the capital and largest city of the Provinces of Belgium, province ...
) was a Flemish
Jesuit
The Society of Jesus (; abbreviation: S.J. or SJ), also known as the Jesuit Order or the Jesuits ( ; ), is a religious order (Catholic), religious order of clerics regular of pontifical right for men in the Catholic Church headquartered in Rom ...
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, mathematical structure, structure, space, Mathematica ...
. He is remembered for his work on
quadrature of the hyperbola. He is also known as Gregorio a San Vincente.
[
Grégoire gave the "clearest early account of the summation of ]geometric series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
."[ Margaret E. Baron (1969) ''The Origins of the Infinitesimal Calculus'', Pergamon Press, republished 2014 by ]Elsevier
Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, ...
Google Books preview
/ref> He also resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus had a finite sum.[
]
Life
Grégoire was born in Bruges
Bruges ( , ; ; ) is the capital and largest city of the province of West Flanders, in the Flemish Region of Belgium. It is in the northwest of the country, and is the sixth most populous city in the country.
The area of the whole city amoun ...
8 September 1584. After reading philosophy in Douai, he entered the Society of Jesus
The Society of Jesus (; abbreviation: S.J. or SJ), also known as the Jesuit Order or the Jesuits ( ; ), is a religious order of clerics regular of pontifical right for men in the Catholic Church headquartered in Rome. It was founded in 1540 ...
21 October 1605. His talent was recognized by Christopher Clavius in Rome. Grégoire was sent to Louvain in 1612, and was ordained a priest 23 March 1613. Grégoire began teaching in association with François d'Aguilon in Antwerp
Antwerp (; ; ) is a City status in Belgium, city and a Municipalities of Belgium, municipality in the Flemish Region of Belgium. It is the capital and largest city of Antwerp Province, and the third-largest city in Belgium by area at , after ...
from 1617 to 20. Moving to Louvain in 1621, he taught mathematics there until 1625. That year he became obsessed with squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
and requested permission from Mutio Vitelleschi to publish his method. But Vitelleschi deferred to Christoph Grienberger, the mathematician in Rome.
On 9 September 1625, Grégoire set out for Rome to confer with Grienberger, but without avail. He returned to the Netherlands in 1627, and the following year was sent to Prague
Prague ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...
to serve in the house of Emperor Ferdinand II. After an attack of apoplexy, he was assisted there by Theodorus Moretus. When the Saxons raided Prague in 1631, Grégoire left and some of his manuscripts were lost in the mayhem. Others were returned to him in 1641 through Rodericus de Arriaga.
From 1632 Grégoire resided with The Society in Ghent
Ghent ( ; ; historically known as ''Gaunt'' in English) is a City status in Belgium, city and a Municipalities of Belgium, municipality in the Flemish Region of Belgium. It is the capital and largest city of the Provinces of Belgium, province ...
and served as a mathematics teacher.[Herman van Looy (1984) "A Chronology and Historical Analysis of the mathematical Manuscripts of Gregorius a Sancto Vincentio (1584–1667)", Historia Mathematica 11: 57–75]
:The mathematical thinking of Sancto Vincentio underwent a clear evolution during his stay in Antwerp. Starting from the problem of trisection of the angle and the determination of the two mean proportional, he made use of infinite series, the logarithmic property of the hyperbola, limits, and the related method of exhaustion. Sancto Vincentio later applied this last method, in particular to his theory ''ducere planum in planum'', which he developed in Louvain in the years 1621 to 24.[
]
Ductus plani in planum
The contribution of ''Opus Geometricum'' was in
:making extensive use of spatial imagery to create a multitude of solid
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
s, the volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
s of which reduce to a single construction depending on the ''ductus'' of a rectilinear figure, in the absence of lgebraic notation and integral calculussystematic geometric transformation fulfilled an essential role.[
For example, the " ungula is formed by cutting a right circular cylinder by means of an oblique plane through a diameter of the circular base." And also the "’''double ungula'' formed from cylinders with axes at right angles."][ Ungula was changed to "onglet" in French by ]Blaise Pascal
Blaise Pascal (19June 162319August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest ...
when he wrote ''Traité des trilignes rectangles et leurs onglets''.[
Grégoire wrote his manuscript in the 1620s but it waited until 1647 before publication. Then it "attracted a great deal of attention...because of the systematic approach to volumetric integration developed under the name ''ductus plani in planum''."][ "The construction of solids by means of two plane surfaces standing in the same ground line" is the method ''ductus in planum'' and is developed in Book VII of ''Opus Geometricum''][
"The study of mean proportionals ed San Vincenteto one of his greatest discoveries, the logarithmic properties of a hyperbola."][Meskens, Ad J. (2021) ''Between Tradition and Innovation: Gregorio a San Vincente and the Flemish Jesuit mathematics school'', page 98, ]Brill Publishers
Brill Academic Publishers () is a Dutch international academic publisher of books, academic journals, and Bibliographic database, databases founded in 1683, making it one of the oldest publishing houses in the Netherlands. Founded in the South ...
In the matter of quadrature of the hyperbola, "Grégoire does everything save give explicit recognition to the relation between the area of the hyperbolic segment and the logarithm."[
The manuscript also claimed to solve the ancient problem of ]squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
, for which it was criticized by others, including Vincent Léotaud in his 1654 work ''Examen circuli quadraturae''.
Quadrature of the hyperbola
Saint-Vincent found that the area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
under a rectangular hyperbola (i.e. a curve given by ) is the same over