1733 de Lagny Analyse Generale, 1stEd Mathematical Analysis, Academie Royale

Description:

Paris. la Compagnie des Librairies, 1733, Tome XI, First edition.
In French.
Period binding of full leather, mottled brown calf, ribbed spine with gilt ornate decorations and titles, marbled endpapers. Quarto (26 x 20.5 cm). xv, 428, [10] pages.

Note: This is the large quarto edition. A smaller format edition of this work was also published by the same publisher and also dated 1733. That edition has the same contents but on more pages.

"There are certain obscurities in our knowledge of Lagny's life, talented calculator though he was... His collaboration with L'Hospital and his first publications concerning the approximate calculation of irrationals show that he was a good mathematician... Lagny's work belonged to a type of computational mathematics at once outmoded and unappreciated. While the idea of the function was gaining dominance, Lagny continued to approach mathematical problems ? both ancient problems such as the solution of equations and new ones such as integration ? with the aid of numerical tables. Employing with great skill the property possessed by algebraic forms of corresponding to tables in which the differences of a determined order are constant, he recognized the existence of transcendental numbers in the calculation of series. He made pertinent observations on convergence, in connection with the series that he utilized to calculate the first 120 decimal places in the value of pi. He attempted to establish trigonometric tables through the use of transcription into binary arithmetic, which he termed "natural logarithm" and the properties of which he discovered independently of Leibniz... His works retain a certain didactic value". (DSB VII pp. 558/559)

---- Méthode nouvelle pour résoudre les équations de tous les degrez à l'infini par des tables - Avantages de ces tables - Méthode nouvelle pour résoudre par des tables les problèmes déterminez ou les équations de tous les degrez à l'infini et même dans le cas irréductible - De la nature et du nombre des racines des équations de tous les degrez, de leur genres et de leurs espèces - Usage des tables pour résoudre les équations de tous les degrez à l'infini - Résolution du cas irréductible du troisième degré - tables pour la résolution des équations du second, du troisième et du cinquième degré - L'analyse générale ou les règles générales de l'analyse - Des problèmes déterminez de tous les degrez ou des équations composées de tous les degrez à l'infini - Résolution des équations en général et en particulier - Formation et résolution des équations du 3e degré - Méthode générale et nouvelle pour résoudre les équations de tous les degrez, à l'infini par le terme dominant - Méthode générale pour résoudre les équations de tous les degrez à l'infini par les progressions arithmétiques appliquée aux équations du 2d et du 4e degré - Méthode générale d'approximation pour trouver les racines irrationelles des puissances imparfaites et des équations irrationelles de tous les degrez par des formules rationelles ? etc.

Thomas Fantet de Lagny's father was Pierre Fantet, a royal official in Grenoble, while his mother was Jeanne d'Azy, the daughter of a physician from Montpellier. He was taught first by his paternal uncle, then he entered a Jesuit College in Lyon. He came top of every class he attended at the College. While at the College he composed Greek verse, and also studied mathematics texts such as Euclid's Elements and an algebra text by Jacques Pelletier which, Fontenelle writes, that he bought one day by chance.

He had nobody to help him in his study of mathematics, no mathematics teacher, nor anyone with whom he could discuss mathematical ideas, yet while at the College he was able to teach himself a broad range of mathematical techniques. However, he never seems to have become familiar with the latest mathematical developments which were taking place at this time and he always adopted a fairly classical approach to problems he tackled. After his time at the Jesuit College in Lyon, he studied law in Toulouse for three years, then went to Paris. He published Dissertation sur l'or de Toulouse in the Annales de la ville de Toulouse in 1687.

In 1686 De Lagny became a mathematics tutor to the Noailles family in Paris, a position he held for about 10 years. By this stage he had acquired the title of de Lagny which came from a property which he had acquired. He collaborated with de L'Hôpital while in Paris and it was during this time that he began to publish mathematics papers. He returned to Lyons and was there when, on 11 December 1695, he became a member of the Academy of Sciences. Then two years later he became professor of hydrography at Rochefort, a town to the south of La Rochelle. The town is situated on the right bank of the Charente River, 16 km from the sea, but an important military port and arsenal had been created there by Jean-Baptiste Colbert, minister to Louis XIV, in the 17th century. Lagny worked at the Institute of Hydrography which was associated with the military port.

He also held positions as librarian at Bibliothéque du Roi for a time and spent two years from 1716 to 1718 as deputy director of the Banque Générale. It was the Marechal Duc de Noailles, president of the Conseil des Finances, who set up this post for Lagny. He had been taught by Lagny during his time as tutor to the Noailles family in Paris. When the Banque Générale became the Banque Royale in 1718, Lagny resigned his position at the Bank. In the following year he was awarded a pension by the Academy of Sciences so could undertake research in mathematics without having to earn his living. In 1723 he became a pensionnaire géomètre at the Academy, replacing Varignon who had died in December 1722.

De Lagny is well known for his contributions to computational mathematics, calculating ? to 120 places and also making useful comments on the convergence of the series he was using. In about 1690 he developed a method of giving approximate solutions of algebraic equations and, in 1694, Halley published a twelve page paper in the Philosophical Transactions of the Royal Society giving his method of solving polynomial equations by successive approximation which is essentially the same as that given by Lagny a few years earlier. One should note that although methods based on the differential calculus were being developed at this time, neither Lagny not Halley used these new ideas. Lagny's publications on this topic are Méthodes nouvelle infiniment générale et infiniment abrégée pour l'extraction des racines quarrées, cubique (1691) and Méthodes nouvelles et abrégée pour l'extraction et l'approximation des racines (1692).

Lagny constructed trigonometric tables and used binary arithmetic in his text Trigonométrie française ou reformée published in Rochefort in 1703. In 1733 he examined the continued fraction expansion of the quotient of two integers and, as an example, considered adjacent Fibonacci numbers as the worst case expansion for the Euclidean algorithm in his paper Analyse générale ou Méthodes nouvelles pour résoudre les problèmes de tous les genres et de tous les degrés à l'infini.
Article by: J J O'Connor and E F Robertson