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Practical mathematicians and mathematical practice in later seventeenth-century London
- PHILIP BEELEY
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- Journal:
- The British Journal for the History of Science / Volume 52 / Issue 2 / June 2019
- Published online by Cambridge University Press:
- 14 June 2019, pp. 225-248
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- June 2019
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Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of mathematical disciplines, ranging from algebra to arithmetic, from merchants’ accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of interdependency even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of ‘the London mathematicians’ in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching practices and relations to the London book trade, which supplied inexpensive practical mathematical books to a seemingly insatiable public.
Carcavi, Pierre de (ca.1600–1684)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
- Published online:
- 05 January 2016
- Print publication:
- 01 January 2015, pp 84-85
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Carcavi was born in Lyon, the son of Jean de Carcavi, receiver-general of the districts of Languedoc, Guyenne, and Lyonnais. There are no records of university education, but evidently he studied law and early on acquired a strong interest in mathematics. In 1632, through the help of his father, he was made conseiller to the Parlement of Toulouse, where he first met Pierre de Fermat. They remained close friends throughout their lives, and Fermat later sent him copies of his mathematical texts. Carcavi was encouraged by Fermat to write to other mathematicians such as Descartes, Roberval, and Torricelli, and he later developed an extensive correspondence network. In 1636, with money from his father, he bought the office of conseiller to the Grand Conseil and moved to Paris.
Paris provided Carcavi with a fertile intellectual environment. He developed close ties to Roberval, as well as to Mersenne and the young Blaise Pascal, and later became a member of the Académie de Montmor. In 1648 his career suffered a temporary setback, when financial difficulties of his father forced him to sell his position at the Grand Conseil. For a while he traded books before entering the service of Duc de Liancourt. Through a stroke of luck, another of Roger du Plessis's protégés, the Abbé Amable de Bourzéis, introduced him to Colbert who was seeking someone to catalog Cardinal Mazarin's library. After thus demonstrating his aptitude, Carcavi was appointed custodian of the Royal Library by Colbert in 1663. He held this post for twenty years and played a decisive role in enriching the library's holdings.
Carcavi's relations to Descartes were seldom straightforward. After the death of Mersenne in 1648, he offered his extensive correspondence to Descartes. The following year, Carcavi informed Descartes of Roberval's objections to his Geometry. Descartes replied with a refutation of his claims. When Carcavi publicly defended Roberval, Descartes severed all ties with him. He died in Paris April 1684.
See also Geometry; Mersenne, Marin; Roberval, Gilles Personne de
Debeaune (de Beaune), Florimond (1601–1652)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
- Published online:
- 05 January 2016
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- 01 January 2015, pp 182-183
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Debeaune was born in Blois to Florimond de Beaune senior, Seigneur de Goulioux. He was educated in Paris, where he studied law, before entering military service. His second marriage to Marguerite du Lot in 1623, following the death of his first wife, brought him considerable wealth. On the basis of his legal studies, he bought himself the office of conseiller to the court of justice in Blois. An amateur mathematician and landowner, he built up an extensive library and constructed an observatory on his estate near Blois. Through both the medium of correspondence and regular visits to Paris, he participated actively in contemporary scientific discourse. He exchanged letters with men such as Mydorge, Billy, and Mersenne, while Descartes and Erasmus Bartholin visited him in Blois to discuss mathematical topics.
Little of his mathematical work survives, apart from what is published in the Schooten's Latin translation of the Geometry. Descartes believed that no one understood his work better than his wealthy friend. Debeaune's Notes brèves, translated by Schooten in 1649, served to clarify numerous difficult passages and played an important part in the reception of the work. Shortly before Debeaune died, he met with Bartholin, leading to the publication of two short papers on algebra in Schooten's second edition. Of Debeaune's other work, a treatise of mechanics is mentioned by Mersenne, and a treatise on dioptrics is mentioned by Schooten; neither has survived.
Debeaune's legacy consists primarily in his work on two problems. He was the first to point out that the properties of a curve can be deduced from the properties of its tangent, and he achieved important results on the determination of the tangent to an analytically defined curve. Furthermore, he considered for the first time the upper and lower limits of the roots of numerical equations.
He had the reputation of being the finest instrument maker of his day. In 1639 Descartes asked Debeaune to build the machine for grinding hyperbolic lenses, which he had described in Dioptrics (AT II 511–12). Having embarked on this project, at the beginning of 1640, Debeaune cut his hand badly on glass. Poor health set in. Around 1648 he retired his post of conseiller. He died following the amputation of a foot.
See also Dioptrics; Geometry; Mathematics; Mersenne, Marin
Roberval, Gilles Personne de (1602–1675)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- The Cambridge Descartes Lexicon
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- 05 January 2016
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- 01 January 2015, pp 655-657
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Roberval was born as Gilles Personne in the small village of Roberval near Senlis, France. His father was a poor farmer or farmworker, and his mother is said to have given birth to him in a field. At the age of fourteen, his intelligence recognized, Gilles Personne was given instruction in mathematics and languages by a local priest. He later earned his living as an itinerant teacher of mathematics. After witnessing the siege of La Rochelle, he supplemented his mathematical knowledge through studies in fortification and ballistics. Arriving in Paris shortly afterward, in 1628, Roberval soon became a member of the circle around Mersenne and later of the Académie de Montmor. Around 1630 he was given permission to append “de Roberval” to his surname. In 1632 Roberval was appointed professor of philosophy at Collège de Maître Gervais in Paris, a small institution attached to the university, and he lived in austere rooms there until his death. He never married. Roberval competed successfully for the Ramus chair in mathematics at the Collège Royal and was appointed in 1634. Required by the statutes to submit to open competition every three years, he nonetheless kept this post until the end of his life.
Descartes and Roberval disliked each other intensely. Unimpressed by the Geometry, Roberval sent his critical comments to its author through their mutual friend Mersenne. Their animosity increased further when Roberval defended Fermat's method of tangents against criticism from Descartes (AT II 104–14). From this point onward, the two men engaged in dispute on many questions from methods for determining centers of gravity to the nature of space and body. When Roberval published what was purportedly the Latin version of a newly found Arabic script of a lost work of Aristarchus, but was in fact his own affirmation of the Copernican hypothesis, Descartes poured ridicule on the book. In particular, he noted that the supposed properties of matter were inconsistent with a spherical universe (AT IV 402).
Fermat, Pierre de (1607–1665)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
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- 05 January 2016
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- 01 January 2015, pp 289-291
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Fermat was born in Beaumont-de-Lomagne, son of the wealthy leather merchant Dominique de Fermat and his second wife Claire, née de Long. No records exist of his early education or of his path into mathematics. He studied law at the universities of Toulouse and Orléans, where he was awarded the degree of bachelor of civil of law in 1631. In the same year, he married his fourth cousin removed, Louise de Long.
During a sojourn in Bordeaux, at the end of the 1620s, he actively participated in the scientific circle there around Étienne d'Espagnet. Early versions of mathematical texts, including parts of his restoration of Apollonius's Plane loci, were circulated among members such as Pierre Prades and Jean Beaugrand. After his return to Toulouse, Fermat briefly studied Galileo's Dialogo and produced work on the topic of free fall.
Fermat's career was in law and civic administration. By the time of his marriage he had purchased the offices of conseiller to the Parliament of Toulouse and of commissioner of requests to the palace. In the course of his civic career he reached the highest councils, eventually dividing his time between Toulouse and the court of justice in Castres.
In Toulouse he met Pierre de Carcavi, a fellow conseiller, who shared his interest in mathematical science. When Carcavi went to Paris to take up the post of royal librarian, he provided a glowing account of Fermat to Mersenne. Thus persuaded to initiate a correspondence with the Toulouse mathematician, Mersenne asked Fermat to share his findings with him and other members of his circle.
Relations with Descartes were less equanimous. When Beaugrand, in May 1637, sent galley proofs of the Dioptrics to Fermat, the recipient soon reported that the work contained two considerable errors. In particular, the derivation of the sine law of refraction assumed more than Descartes acknowledged. Over the following months arguments were traded, both men being supported by their respective camps: Fermat principally by Beaugrand, Étienne Pascal, and Roberval; Descartes principally by Mydorge. In November 1637, Roberval and Mersenne solicited Fermat to send his method for determining maxima and minima and tangents to curved lines to them in Paris.
Desargues, Girard (1591–1661)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
- Published online:
- 05 January 2016
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- 01 January 2015, pp 189-190
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Desargues was born in Lyon. His father was a royal notary, and the family was clearly wealthy, with various properties in and around Lyon. Little is known about Desargues’ early life and education. By 1626 he was in Paris and moving in mathematical circles. Later, he is recorded as having regularly attended Mersenne's meetings with men such as Étienne Pascal, Mydorge, and Hardy, and probably much of his early mathematical work was produced for distribution among such contemporaries.
In 1636 he published Une methode aisée pour apprendre et enseigner à lire et escrire la musique, which was later included in Mersenne's Harmonie universelle. The same year also saw publication of Méthode universelle de metre en perspective les objets donnés réellement, in which Desargues formulated the mathematical rules of perspective that had been developed by painters and architects during the Renaissance. Desargues’ most important work, in which he set out the foundations of projective geometry, is the Brouillon project d'une atteinte aux événemens des rencontres du cone avec un plan. This “Rough Draft” is short and somewhat impenetrable; beginning with topics such as the range of points on a line, Desargues proceeds to show that conics can be discussed by means of properties that are invariant under projection. Remarkable thereby is the rigorous treatment of cases involving infinite distances. Not many copies were printed, and few apart from Blaise Pascal recognized its significance. For Descartes, Desargues’ failure to employ algebra limited the scope of his approach.
Descartes corresponded with Desargues indirectly through Mersenne and praised his articulation of the principles of gnomonics or dialing. For his part, Desargues, in 1638, joined forces with Mydorge and Hardy in defending Descartes against attacks from Roberval and Étienne Pascal during their dispute with him over Fermat's method of determining maxima and minima.
In 1640 Desargues published under the Brouillon project an essay on stone cutting and on gnomonics, showing how his graphical method was to be used as a means to simplifying the construction of sundials. This was an area of mathematical practice traditionally governed by the laws of trade guilds and is indicative of his concern for applications of mathematics.
A dispute over the publication, in 1642, of Dubreuil's La perspective pratique, in which Desargues found his method copied and distorted, led him to entrust the engraver Abraham Bosse with spreading his methods and defending his work.
Geometry
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
- Published online:
- 05 January 2016
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- 01 January 2015, pp 330-331
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The only mathematical work written by Descartes, La géométrie, was published by the Leiden printer Jan Maire in 1637 as the last of three tracts appended to the Discourse on Method. While the Dioptrics and the Meteors were said to use and exemplify Descartes’ philosophical method, the Geometry was actually held to demonstrate it (AT I 478, 621; CSMK 77–78).
Before 1636, there is no mention in Descartes’ writings of a publication corresponding to the Geometry. Nonetheless, the published work does take up ideas found already in the Rules and contemporary correspondence such as the numerical expression of powers and the construction of mean proportionals by the intersection of a circle and a parabola. Anticipating claims he would later make for the Geometry itself, he already announces to Isaac Beeckman in March 1619 his plan to formulate an “entirely new science” by which all problems that can be posed concerning all kinds of quantity, continuous or discrete, can be generally solved (AT X 156). In effect, his aim then, as later in the Geometry, was to show that after him nothing would remain to be discovered in mathematics (AT I 340, CSMK 51; AT II 361–62; AT X 157).
Descartes’ catalyst for writing the Geometry was the Pappus problem that Jacob Golius sent to him in 1631. The ancient solution of this problem was unknown to the seventeenth century, and Descartes recognized the importance of the interplay of algebraic equations and geometry in solving it. The solution that he discovered in 1632 was accorded a prominent place in the Geometry and served to illustrate the power of his geometrical calculus (AT I 323, 244, 478; CSMK 78).
Mydorge, Claude (1585–1647)
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- By Philip Beeley, Oxford University
- Edited by Lawrence Nolan, California State University, Long Beach
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- Book:
- The Cambridge Descartes Lexicon
- Published online:
- 05 January 2016
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- 01 January 2015, pp 531-533
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Mydorge was born in Paris to one of the wealthiest families in France. He was educated at the Jesuit College of La Flèche and subsequently trained as a lawyer, before embarking on a legal and administrative career. After serving as conseiller to the court of the Grand Châtelet, he became treasurer of the généralité of Amiens, the collector general being a direct agent of the king. Mydorge's chosen employment allowed him sufficient time to combine public office with the life of a savant. Residing in what remained of the ancient Palais des Tournelles, he first met Descartes around 1625, becoming one of his most faithful friends and helping to establish his reputation in Paris. The mathematician Claude Hardy, a leading figure in the scientific circles around Mersenne, Roberval, and Étienne Pascal, lodged with him while he was producing his edition of Euclid's Elements.
Mydorge shared with Descartes a strong interest in optics and the nature of vision. It is well known that in order to promote his friend's investigations on these topics, he commissioned the production of innumerable parabolic, hyperbolic, oval, and elliptic lenses, reputedly spending in excess of 100,000 écus on optical instruments over the years. Both men were interested particularly in refraction, and when Descartes, independently of Snell, discovered the law of refraction, he persuaded Mydorge to have a hyperbolic glass made in order to test his discovery. It is possible that Mydorge contributed to the results that Descartes achieved, although his approach to refraction differed fundamentally from the law published in Dioptrics, which employs the ratio of the sine of the angle of incidence to the sine of the angle of refraction.
Mydorge's first major work was the Examen du livre des Récréations mathématiques, published in 1630. As the title suggest, it was a work on recreational mathematics and was effectively a critique of Laurechon's book on the theme. However, it was through his work on conic sections that Mydorge made the greatest scientific impact. His motivation for these studies came from his investigations on catoptrics, the optics of mirrors. While employing ancient techniques in dealing with conics, he achieved considerable success in simplifying the proofs of Apollonius.
3 - A Philosophical Apprenticeship: Leibniz's Correspondence with the Secretary of the Royal Society, Henry Oldenburg
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- By Philip Beeley, Lecturer in History of Science University of Hamburg
- Edited by Paul Lodge, Mansfield College, Oxford
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- Book:
- Leibniz and his Correspondents
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- 02 September 2009
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- 26 April 2004, pp 47-73
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Still scarred by the ravages of war and politically disunited, Germany at the time of the young Leibniz was in danger of losing touch with philosophical and scientific developments elsewhere in Europe. In a letter written in August 1670 to the diplomat Johann Christian von Boineburg (1622–72) who since his reconciliation with Johann Philipp von Schönborn (1605–73) some two and a half years earlier was once more in the service of the Elector of Mainz, Henry Oldenburg (1618?–77) expresses his concerns over what he sees as the deplorable state of philosophy in the country of his birth:
Would that those who excel in litigation and in the sciences in Germany make their contributions towards the restoration and perfection of philosophy with a better will than they have shown hitherto, and would eagerly imitate in this the example of England, France, and Italy herself in turning to experiments. What we are about is no task for one nation or another singly. It is needed that the resources, labors, and zeal of all regions, princes, and philosophers be united, so that this task of comprehending nature may be pressed forward by their care and industry.
(OCH VII, 107–8/108–9)Oldenburg's concerns were not without justification. The new, mechanistic world picture had, after all, found only few adherents, most notably Erhard Weigel (1625–99) in Jena, and the creation of a focal point of scientific activity such as the Royal Society in London or the Académie Royale des Sciences in Paris was still a long way off.