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CRAMER



Gabriel Cramer (1704 – 1752) was a Swiss mathematician, the author of the famous Cramer's rule for decision of system of liner equation.
Cramer was born in Geneva, he was a son of physician Jean Cramer. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. He published his best-known work in his forties. This was his treatise on algebraic curves «Introduction in the analysis of algebraic curves», published in 1750. It contains the earliest demonstration that a curve of the n-th degree is determined by n(n + 3)/2 points on it, in general position. He edited the works of the two elder Bernoulli’s; and wrote on the physical cause of the spheroidal shape of the planets and the motion of their apsides (1730), and on Newton's treatment of cubic curves (1746). He was professor at Geneva, and died at Bagnols-sur-Ceze. He did extensive travel throughout Europe in the late 1730s which greatly influenced his works in mathematics.
At 20-years age Cramer has proposed the candidature on a vacant position of the teacher on faculty of philosophy of the Geneva University. Nominees was three, all was made with good impression, and city council adopted the "Solomon’s decision": to found separate faculty of mathematics and to direct there (on one rate) two "extra", including Cramer, with the right to travel by turns at own expense.
1727: Cramer has taken advantage of this right and 2 years travelled across the Europe, at the same time adopting experience at leading mathematicians – Johann Bernoulli and Euler in Basel, Halley and de Moivre in London, and others. After return it enters with them the correspondence, its short life proceeded all.
1728: Cramer discovers a solution so-called St.-Petersburg paradox (the paradox illustrating a divergence of a population mean of a prize with a "common" assessment by people), close to the decision which 10 years later is published by Daniel Bernoulli.
1729: Cramer comes back to Geneva and renews teaching work. It participates in the competition declared by the Parisian Academy, the task in which: whether there is a communication between ellipsoid the form of the majority of planets and their displacement of aphelion? Cramer's Work takes the second seat (the first prize has received Johann Bernoulli). In free from teaching time Cramer writes numerous articles on the most different subject matters: geometry, history of mathematics, philosophy, appendices of probability theory. Cramer also publishes work on the heavenly mechanics (1730) and the comment to Newtonian classifications of curves of the third order (1746). About 1740 Johann Bernoulli charges to Cramer efforts under the edition of the collection of assembly of the works. In 1742 Cramer publishes the collection in 4 volumes, and soon (1744) lets out the similar (posthumous) collection of works of Jacob Bernoulli and the two-volume book of correspondence Leibniz with Johann Bernoulli. All these editions had a huge resonance in the scientific world.
Most known of Cramer's works — the treatise published shortly before his death «Introduction in the analysis of algebraic curves» (1750). In it for the first time it is proved, that the algebraic curve of n-th order generally is completely certain, if are set it n (n + 3)/2 points. For the proof Cramer builds system of the linear equations and solves it by means of the algorithm named after its name: Cramer's method.
Cramer has considered system of any quantity of the linear equations with a square matrix. It has presented the decision of system in the form of a column of fractions with the general denominator – a determinant of a matrix. The term "determinant" then still did not exist (it has entered Gauss per 1801), but Cramer has given precise algorithm of its calculation: the algebraic sum of every possible works of elements of a matrix, on one of every line and each column. The sign composed in this sum, on Cramer, depends on number of inversions of appropriating substitution of indexes: plus, if number of inversions even. As to numerators in a column of decisions they are counted up in a similar way: n-th numerator is a determinant of the matrix received by replacement of n-th column of an initial matrix on a column of free members.
Cramer's methods at once have received the further progress in works Bezout, Vandermonde and Cayley which have finished creation of bases of linear algebra. The theory of determinants has quickly found set of appendices in astronomy and the mechanic (the century equation), at the decision of algebraic systems, research of forms, etc.
Cramer has lead classification of algebraic curves up to the fifth order inclusive. It is curious, that in all things the substantial research of curves Cramer anywhere does not use the mathematical analysis though he undoubtedly owned these methods.
In 1751 Cramer has received a serious trauma after road accident with the carriage. The doctor recommended it to have a rest on the French resort, but there its condition has worsened, and on January, 4th, 1752 Gabriel Cramer has died.


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Last updated: August 5, 2012