**Gregory of Saint-Vincent** was known as Grégoire and also by the Latin version of his name, Gregorius a Sancto Vincentio. We know nothing of his parents and his early life. He began his studies at the Jesuit College of Bruges in 1595. After six years, he went to Douai in northern France in 1601, studying mathematics and philosophy there. 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. He was ordained a priest in 1613.

From 1613 he began his career as a teacher, first at Brussels where he taught Greek. 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. The revolt was, however, a long-term struggle which was on hold at this time, the war with Spain only being actively resumed in 1621. 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. Ad Meskens writes [6]:-

De Aguilon(1567-1617)had published 'Opticorum Libri Sex'(1613), an influential book on optics and binocular vision. St Vincent's first mathematical investigations deal with problems related to refraction and reflection.

During this time Saint-Vincent published *Theses cometis* (1619) and *Theses mechanicae* (1620). His [1]:-

'Theses de cometis'(Louvain,1619)and 'Theses mechanicae'(Antwerp,1620)were defended by his student Jean Charles de la Faille, who later made them the basis of his highly regarded 'Theoremata de centro gravitatis'(1632).

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. He [1]:-

... 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. He then spent six years in Prague as chaplain to the Holy Roman Emperor Ferdinand II from 1626 until 1632. 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. Moret (1602-1667), from Antwerp, was a fine mathematician and had joined the Jesuit Order when he was twenty years old. 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.

In September 1631 Gustav II of Sweden led his Protestant army to attack the Catholic forces in Saxony. Victories saw German Protestant princes join forces with him as he moved south. 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* 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. He suffered a second stroke in 1659 and his book was still incomplete when he died from a third stroke in 1667. The book *Opus geometricum ad mesolabium* was published in 1668, over a year after his death.

Let us turn to Saint Vincent's mathematical contributions. Ad Meskens writes [6]:-

St Vincent's first investigations had to do with reflection and refraction. One of the problems which arose was the trisection of an angle. In searching for ways to obtain a trisection, St Vincent came across the series

^{1}/_{1}-^{1}/_{2}+^{1}/_{4}-^{1}/_{8}+....

This series according to St Vincent equals^{2}/_{3}, which he called the terminus. In contrast with classical Greek mathematics, St Vincent thus accepts, for the first time in the history of mathematics, the existence of a limit. While Euclid writes that "one will obtain at last something smaller than the smallest quantity", St Vincent goes further and boldly writes: "the quantity will be exhausted"

Saint-Vincent's main work, the *Opus geometricum quadraturae circuli sectionum coni* (1647), is a book over 1250 pages long. The title indicates that Saint-Vincent's main aim was to square the circle and this, as Jean Dhombres writes in [5], proved a major stumbling block to its important methods being influential:-

[

Saint-Vincent]is essentially a man of one book: 'Opus geometricum'. But what a book! It contains more than1200pages(in folio), and thousands of figures. It was printed in Antwerp in1647, but was never republished. One thing about the book immediately stirred some uneasiness: the addition to the title, namely: 'quadraturae circuli'. The engraved frontispiece shows sunrays inscribed in a square frame being arranged by graceful angels to produce a circle on the ground: 'mutat quadrata rotundis'. 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, in1200pages of text. Four years later, in1651, Christiaan Huygens found a serious defect in the last book of 'Opus geometricum', namely in Proposition39of Book X, on page1121. This gave the book a bad reputation.

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. Book I is, naturally, an introductory one in which Saint-Vincent sets the scene with material on circles and triangles. Also in this Book is an introduction to transformations treated in a geometrical way. Book II looks at geometric series which Saint-Vincent is able to sum using the transformations he introduced in the first Book. 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.

In Books III, IV, V and VI Saint-Vincent treats conic sections, the circle, ellipse, parabola, and hyperbola, using classical Greek geometric methods. He treats the hyperbola summing the area under the curve using a sequence of ordinates in geometric progression [2]:-

Basically these procedures are closely related to the development of the integral calculus and the numerical methods which were subsequently developed for the calculation of logarithms form a fascinating study.

Basically, Saint-Vincent showed that the area under a rectangular hyperbola *xy* = *k* over the interval [*a*, *b*] is the same as that over the interval [*c*, *d*] if ^{a}/_{b} = ^{c}/_{d}. He therefore integrated *x* ^{-1} in a geometric form that, although he does not make the connection, is easily recognised as the logarithmic function. This connection was in fact made by Saint-Vincent's pupil Alphonse Antonio de Sarasa (1618-1667).

In Book VII Saint-Vincent gives full details of his geometric method of integration, giving a large collection of interesting examples. He considers solid figures generated by two plane parallel surfaces on a common base where the solid is bounded by equidistant parallels. 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. It was here that he made the error which led him to believe that he had squared the circle using ruler and compass methods. We have already noted in an above quote that the error was detected by Huygens in 1651. Margaret Baron writes [2]:-

Unfortunately the delayed publication of the 'Opus geometricum' prevented it from receiving the attention it would certainly have merited had it appeared twenty years earlier. In1647, 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. Those who obtained the book did so mainly to study the faulty circle quadrature. 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.

**Article by:** *J J O'Connor* and *E F Robertson*

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