**Gabriel Cramer** worked on analysis and determinants. He is best known for his formula for solving simultaneous equations.

- Gabriel certainly moved rapidly through his education in Geneva, and in 1722 while he was still only eighteen years old he was awarded a doctorate having submitted a thesis on the theory of sound.
- The competition for the chair was between three men; the eldest was Amédée de la Rive while the other two were both young men, Giovanni Ludovico Calandrini who was twenty-one years old and Cramer who was one year younger.
- Clearly they were looking to the future and seeing in Cramer and Calandrini two men who would make important future contributions to the Academy.
- De la Rive was offered the philosophy chair, which after all was what he had applied for in the first place, while Cramer and Calandrini were offered the mathematics chair on the understanding that they shared the duties and shared the salary.
- The magistrates put another condition on the appointment too, namely that Cramer and Calandrini each spend two or three years travelling and while one was away the other would take on the full list of duties and the full salary.
- It was a good plan for not only did it successfully attract all three men to the Academy, but it also gave Cramer the opportunity to travel and meet mathematicians around Europe and he was to take full advantage of this which both benefited him and the Academy.
- Cramer and Calandrini divided up the mathematics courses each would teach.
- Cramer taught geometry and mechanics while Calandrini taught algebra and astronomy.
- We must not give the impression that Cramer just fitted into an existing pattern of teaching.
- Appointed in 1724, Cramer followed the conditions of his appointment and set out for two years of travelling in 1727.
- Cramer then visited England where he met Halley, de Moivre, Stirling, and other mathematicians.
- His discussions with these mathematicians and the continuing correspondence with them after he returned to Geneva had a big influence on Cramer's work.
- From England Cramer made his way to Leiden where he met 'sGravesande, then he moved on to Paris where he had discussions with Fontenelle, Maupertuis, Buffon, Clairaut, and others.
- These two years of travelling were to set the tone for Cramer's career for he was highly regarded by all the mathematicians he met, he corresponded with them throughout his life, and he was to perform many extremely valuable major tasks as an editor of their works.
- Back in Geneva in 1729, Cramer was at work on an entry for the prize set by the Paris Academy for 1730, which was "Quelle est la cause de la figure elliptique des planètes et de la mobilité de leurs aphélies?" Ⓣ(What is the cause of the elliptic figure of the planets and the movement of their aphelion?) Cramer's entry was judged as the second best of those received by the Academy, the prize being won by Johann Bernoulli.
- In 1734 the "twins" split up when Calandrini was appointed to the chair of philosophy and Cramer became the sole holder of the Chair of Mathematics.
- Cramer lived a busy life, for in addition to his teaching and correspondence with many mathematicians, he produced articles of considerable interest although these are not of the importance of the articles written by most of the top mathematicians with whom he corresponded.
- There are two areas of Cramer's mathematical work which we should highlight.
- Johann Bernoulli died in 1748, only three or so years before Cramer, but he arranged for Cramer to publish his Complete Works before his death.
- It shows how much respect Bernoulli had for Cramer that he insisted that no other edition of his works be published by any editor other than Cramer.
- Johann Bernoulli's Complete Works was published by Cramer in four volumes in 1742.
- Not only did Johann Bernoulli arrange for Cramer to publish his Complete Works but he also requested that he edit Jacob Bernoulli's works.
- Jacob Bernoulli had died 1705 and Cramer published his Works in two volumes in 1744.
- In 1745, jointly with Johann Castillon, Cramer published the correspondence between Johann Bernoulli and Leibniz.
- Cramer also edited the five volume work by Christian Wolff, first published between 1732 and 1741 with a new edition appearing between 1743 and 1752.
- Finally we should describe Cramer's most famous book "Introduction à l'analyse des lignes courbes algébraique" Ⓣ(Introduction to the analysis of algebraic curves).
- It is a work which Cramer modelled on Newton's memoir on cubic curves and he praises highly a commentary on Newton's memoir written by Stirling.
- Of course Euler's book was only published in 1748 at which time much of Cramer's book might well have been written.
- The suggestion that Cramer never mastered the calculus must be considered doubtful, particularly given the high regard that he was held in by Johann Bernoulli.
- After an introductory chapter in which types of curves are defined and techniques for drawing their graphs are discussed, Cramer goes on to a second chapter in which transformations to simplify curves are studied.
- The third chapter looks at a classification of curves and it is in this chapter that the now famous "Cramer's rule" is given.
- This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution.
- We should remark, of course, that Cramer was certainly not the first to give this rule.
- The other "well known" part of Cramer's work is his description of Cramer's paradox.
- This, says Cramer, is a paradox, but his attempt to explain the paradox is incorrect.
- Cramer's name has sometimes been attached to another problem, namely the Castillon-Cramer problem.
- This problem, proposed by Cramer to Castillon, asked how to inscribe a triangle in a circle so that it passed through three given points.
- Castillon solved the problem 25 years after Cramer's death, and the problem went on to various generalisations about inscribed polygons in a conic section.
- Cramer had worked extremely hard over a long period with writing his "Introduction à l'analyse" Ⓣ(Introduction to analysis) and undertaking the large amount of editorial work in addition to all his normal duties.

Born 31 July 1704, Geneva, Switzerland. Died 4 January 1752, Bagnols-sur-Cèze, France.

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Algebra, Origin Switzerland

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