Grégoire De Saint-Vincent
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Grégoire de Saint-Vincent () - in Latin : Gregorius a Sancto Vincentio, in Dutch : Gregorius van St-Vincent - (8 September 1584
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– 5 June 1667
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) was a Flemish
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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
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/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
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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,
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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 xy = k) is the same over ,b/math> as over ,d/math> when :\frac = \frac. This observation led to the hyperbolic logarithm. The stated property allows one to define a function A(x) which is the area under said curve from 1 to x, which has the property that A(xy) = A(x)+A(y). This functional property characterizes logarithms, and it was mathematical fashion to call such a function A(x) a
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
. In particular when we choose the rectangular hyperbola xy = 1, one recovers the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
. A student and co-worker of Saint-Vincent, A. A. de Sarasa noted that this area property of the hyperbola represented a logarithm, a means of reducing multiplication to addition. The property may be observed as invariance of area of
hyperbolic sector A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbol ...
s under
squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping. For a fixed p ...
. In 1651
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
published his ''Theoremata de Quadratura Hyperboles, Ellipsis, et Circuli'' which referred to the work of Saint-Vincent. The quadrature of the hyperbola was also addressed by James Gregory in 1668 in ''True Quadrature of Circles and Hyperbolas'' While Gregory acknowledged Saint-Vincent's quadrature, he devised a convergent sequence of inscribed and circumscribed areas of a general
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
for his quadrature. The term ''natural logarithm'' was introduced that year by Nicholas Mercator in his ''Logarithmo-technia''. Saint-Vincent was lauded as ''Magnan'' and "Learned" in 1688: “It was the great Work of the Learned ''Vincent'' or ''Magnan'', to prove that distances reckoned in the Asymptote of an Hyperbola, in a Geometrical Progression, and the Spaces that the Perpendiculars, thereon erected, made in the Hyperbola, were equal one to the other.” A historian of the calculus noted the assimilation of natural logarithm as an area function at that time: :As a consequence of the work of Gregory St. Vincent and de Sarasa, it seems to have been generally known in the 1660s that the area of a segment under the hyperbola y = \frac is proportional to the logarithm of the ratio of the ordinates at the ends of the segment.C.H. Edwards, Jr. (1979) ''The Historical Development of the Calculus'', page 164, Springer-Verlag,


Works

* *


See also

* Gabriel's horn (1643) * History of logarithms


References

* Karl Bopp (1907) "Die Kegelschnitte der Gregorius a St. Vincentio", ''Abhandlungen zum Geschichte der mathematische Wissenschaft'', XX Heft. *
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
(1651
Examen de la Cyclométrie du três savant Grégoire de Saint-Vincent
''Oeuvres Complètes'', Tome XI, link from
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. * David Eugene Smith (1923) ''History of Mathematics'', Ginn & Co., v.1, p. 425. * Hans Wussing (2008) ''6000 Jahre Mathematik: eine kulturgeschichtliche Zeitreise'', S. 433, Springer, .


External links


Gregory Saint Vincent, and his polar coordinates
from ''Jesuit History, Tradition and Spirituality'' by Joseph F. MacDonnell. * {{DEFAULTSORT:Saint-Vincent, Gregoire de 1584 births 1667 deaths Flemish Jesuits Mathematicians from the Spanish Netherlands French geometers Logarithms University of Douai alumni Jesuit scientists