**Gregory of Saint-Vincent** was a Flemish Jesuit who wrote a book covering many aspects of contemporary mathematics.

- We know nothing of his parents and his early life.
- He began his studies at the Jesuit College of Bruges in 1595.
- Saint-Vincent became a Jesuit novice in 1605 in Rome and entered the Jesuit Order in 1607.
- He was a student of Christopher Clavius at the Collegio Romano in Rome and his talents were quickly spotted by Clavius who persuaded him to remain at the College to study mathematics, philosophy and further advanced topics in theology.
- This was a period when science and theology struggled to come to terms with arguments that the sun rather than the earth was at the centre of the universe.
- Galileo's results obtained by turning the newly invented telescope on the moon and planets had not contributed to the heliocentric argument except to show that, if the earth was the centre of the universe, then not all heavenly bodies revolved round the centre since he had seen Jupiter's moons orbiting Jupiter.
- Leaders of the Jesuit Order asked Jesuit mathematicians on the faculty of the Collegio Romano, for their opinion on Galileo's discoveries.
- Saint-Vincent and others in the Collegio Romano were fascinated by Galileo's work and expressed views supporting a heliocentric universe which certainly did not please the leader of the Jesuit Order who insisted that they support Aristotle's world-view.
- At the beginning of February 1612 Saint-Vincent's teacher Clavius died and later that year he went to Louvain to complete his theology degree.
- Then he continued to teach Greek in a number of Jesuit Colleges - in Bois-le-Duc (now 's-Hertogenbosch in the Netherlands) in 1614, and Coutrai (now Kortrijk in Belgium) in 1615.
- The next year he was appointed chaplain to the Spanish troops stationed in Belgium which must have been a difficult job since this was a period of Dutch revolt against Spain.
- Saint-Vincent next spent three years teaching mathematics at the Jesuit College in Antwerp, first as François de Aguilon's assistant becoming his successor after his death in 1617.
- During this time Saint-Vincent published Theses cometis Ⓣ(Theses on comets) (1619) and Theses mechanicae Ⓣ(Mechanical theses) (1620).
- In 1621 the College in Antwerp moved to Louvain where Saint-Vincent spent four years teaching mathematics.
- During his years in Louvain, Saint-Vincent worked on mathematics and developed methods which were important in setting the scene for the invention of the differential and integral calculus.
- elaborated the theory of conic sections on the basis of Commandino's editions of Archimedes (1558), Apollonius (1566), and Pappus (1588).
- He also developed a fruitful method of infinitesimals.
- His students Gualterus van Aelst and Johann Ciermans defended his 'Theoremata mathematica scientiae staticae' (Louvain, 1624); and two other students, Guillaume Boelmans and Ignaz Derkennis, aided him in preparing the 'Problema Austriacum', a quadrature of the circle, which Gregorius regarded as his most important result.
- He requested permission from Rome to print his manuscript, but the general of the order, Mutio Vitelleschi, hesitated to grant it.
- Vitelleschi's doubts were strengthened by the opinion that Christoph Grienberger (Clavius's successor) rendered on the basis of preliminary material sent from Louvain.
- Among Saint-Vincent's students mentioned in this quote Johann Ciermans (1602-1648) and Guillaume Boelmans (born 7 October 1603 in Maastricht, died 20 October 1638 at Louvain) are perhaps the most important.
- Saint-Vincent made a request to Mutio Vitelleschi, the Sixth Superior General of the Society of Jesus, to publish his manuscript.
- This led Vitelleschi to ask Saint-Vincent to prepare a submission for Christoph Grienberger, professor of mathematics at the Collegio Romano and censor of all mathematical works written by Jesuit authors, asking him to give an opinion on the value of Saint-Vincent's new methods.
- The Bibliotheque Royale de Belgique still contains the manuscript prepared by Guillaume Boelmans under Saint-Vincent's directions which was sent to Grienberger.
- It is interesting to note that this document contains the first recorded use of polar coordinates.
- As he did for all such submissions, Grienberger returned corrections and changes which he recommended that Saint-Vincent incorporate in his work before it could be published.
- In an attempt to make his manuscript acceptable for publication, Saint-Vincent went to Rome in 1625 but two years later, having failed to get his material into a form Grienberger deemed to be satisfactory for publication, he returned to Louvain.
- These were years of great difficulty since his health was poor but there was also severe tensions between the fervent Catholicism of Ferdinand and the Protestant nobles.
- Not long after taking up the post, Saint-Vincent suffered a stroke but slowly recovered and his request to have his former student Theodor Moret appointed as his assistant was granted.
- After spending time as Saint-Vincent's assistant, he taught at the Academy in Olomouc.
- By this time Saint-Vincent's reputation was high and the Madrid Academy made him a tempting offer of a position in 1630.
- Sadly his health was still not robust enough to allow him to accept such an offer and he was forced to decline.
- Gustav took Munich in May 1632 and his ally, the Protestant elector of Saxony, attacked Prague.
- As the Protestant forces entered Prague, Saint-Vincent fled to Vienna leaving in such haste that he left behind many of his important mathematical papers.
- He moved to the Jesuit College in Ghent where he taught from 1632 for the rest of his life.
- Ten years after he abandoned his papers in Prague they were returned to him by Father de Amagia and they were published as Opus geometricum quadraturae circuli sectionum coni Ⓣ(Geometric work on the quadrature of the circle of of conic sections) in Antwerp in 1647.
- Despite his poor health, Saint-Vincent turned to another of the classical problems of mathematics, namely the duplication of the cube.
- One of the problems which arose was the trisection of an angle.
- In contrast with classical Greek mathematics, St Vincent thus accepts, for the first time in the history of mathematics, the existence of a limit.
- But what a book! It contains more than 1200 pages (in folio), and thousands of figures.
- There was uneasiness in the learned world because no one in that world still believed that under the specific Greek rules the quadrature of a circle could possibly be effected, and few relished the thought of trying to locate an error, or errors, in 1200 pages of text.
- Four years later, in 1651, Christiaan Huygens found a serious defect in the last book of 'Opus geometricum', namely in Proposition 39 of Book X, on page 1121.
- There are many topics covered in the book including a study of circles, triangles, geometric series, ellipses, parabolas and hyperbolas.
- His book also contains his quadrature method which is related to that of Cavalieri but which he discovered independently.
- He gives a method of squaring the circle which we can now see is essentially integration.
- Let us give a few more details of this remarkable work.
- He applies his results to a number of interesting problems such as the trisection of an angle which he achieves through an infinite series of bisections.
- He also applies his summation of series to the classical Greek problem of Zeno, namely Achilles and the tortoise.
- This connection was in fact made by Saint-Vincent's pupil Alphonse Antonio de Sarasa (1618-1667).
- In Book IX he extends his methods to give volumes of cylindrical bodies.
- Of course, as far as Saint-Vincent was concerned, his aim was to square the circle and he reaches that in Book X.
- In 1647, ten years after the publication of Descartes' 'La Géométrie', algebraic methods were rapidly gaining ground and the form and manner of presentation of Grégoire's work was not such as to make it easy reading.
- many who read it, however, became fascinated by the geometric integration methods and went on to make a deeper study of the entire work.
- Amongst those who gained much from the 'Opus geometricum' should be counted Blaise Pascal whose 'Traité des trilignes rectangles et leurs onglets' is based essentially on the 'ungula' of Grégoire.
- Huygens recommended the section on geometric series to Leibniz who later came to make a thorough study of the entire work.
- Tschirnhaus, friend and associate of Leibniz during his Paris years, found in the 'ductus in planum' a valuable foundation for the development of his own algebraic integration methods.

Born 8 September 1584, Bruges, Spanish Netherlands (now Belgium). Died 27 January 1667, Ghent, Spanish Netherlands (now Belgium).

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Ancient Greek, Geometry, Origin Belgium, Number Theory, Puzzles And Problems, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive