Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Biography

Cramer was born on 31 July 1704 in Geneva, Republic of Geneva to Jean-Isaac Cramer, a physician, and Anne Mallet. The progenitor of the Cramer family in Geneva was Jean-Ulrich Cramer, Gabriel's great-grandfather, who immigrated from

Strasbourg Strasbourg ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen ...

in 1634. Cramer's mother, a member of the Mallet family, was of

Huguenot The Huguenots ( , ; ) are a Religious denomination, religious group of French people, French Protestants who held to the Reformed (Calvinist) tradition of Protestantism. The term, which may be derived from the name of a Swiss political leader, ...

origin. Cramer showed promise in mathematics from an early age. In 1722, aged 18, he received his doctorate from the Academy of Geneva, and at 20 he was made co-chair (along with Jean-Louis Calandrini) of mathematics at the Academy. He became the sole professor of mathematics in 1734 and was appointed professor of philosophy at the Academy in 1750. Cramer was also involved in the politics of the Republic of Geneva, entering first the Council of Two Hundred in 1734 then the Council of Sixty in 1750. He was a member of the science academies of

Bologna Bologna ( , , ; ; ) is the capital and largest city of the Emilia-Romagna region in northern Italy. It is the List of cities in Italy, seventh most populous city in Italy, with about 400,000 inhabitants and 150 different nationalities. Its M ...

Lyon Lyon (Franco-Provençal: ''Liyon'') is a city in France. It is located at the confluence of the rivers Rhône and Saône, to the northwest of the French Alps, southeast of Paris, north of Marseille, southwest of Geneva, Switzerland, north ...

, and

Montpellier Montpellier (; ) is a city in southern France near the Mediterranean Sea. One of the largest urban centres in the region of Occitania (administrative region), Occitania, Montpellier is the prefecture of the Departments of France, department of ...

, as well as a foreign member of the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

of London and the Royal Academy of Sciences of Berlin. Cramer died on 4 January 1752 at

Bagnols-sur-Cèze Bagnols-sur-Cèze (, "Bagnols-on-Cèze"; ) is a Communes of France, commune in the Gard Departments of France, department in the Occitania (administrative region), Occitania Regions of France, region in Southern France. History A small regiona ...

while traveling in southern France to restore his health.

Contributions to mathematics

In 1728, Cramer 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 Daniel Bernoulli ( ; ; – 27 March 1782) was a Swiss people, Swiss-France, French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applicati ...

. He did extensive travel throughout Europe in the late 1730s, which greatly influenced his works in mathematics. Cramer published his best-known work in his forties. This included his treatise on

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s (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 (see Cramer's theorem (algebraic curves)). This led to the misconception that is Cramer's paradox, concerning the number of intersections of two curves compared to the number of points that determine a curve. Cramer edited the works of the two elder Bernoullis, and wrote on the physical cause of the spheroidal shape of the

planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...

s and the motion of their apsides (1730), and on Newton's treatment of

cubic curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...

s (1746). In 1750 he published

Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...

, giving a general formula for the solution for any unknown in a linear equation system having a unique solution, in terms of determinants implied by the system. This rule is still standard.

Selected works

* ''Quelle est la cause de la figure elliptique des planètes et de la mobilité de leur aphélies?'', Geneva, 1730 * . Geneva: Frères Cramer & Cl. Philibert, 1750

Notes

References

* "Gabriel Cramer", i
''Rousseau et les savants genevois''
p. 29 * W. W. Rouse Ball, ''A Short Account of the History of Mathematics'', (4th Edition, 1908) * Isaac Benguigui, ''Gabriel Cramer : illustre mathématicien, 1704–1752'', Genève, Cramer & Cie, 1998 * * Johann Christoph Strodtmann, �
Geschichte des Herrn Gabriel Cramer
», in ''Das neue gelehrte Europa ��', 4th part, Meissner, 1754 Also digitized by e-rara.ch

External links

* * {{DEFAULTSORT:Cramer, Gabriel 1704 births 1752 deaths 18th-century mathematicians from the Republic of Geneva 18th-century politicians from the Republic of Geneva University of Geneva alumni Academic staff of the University of Geneva Foreign members of the Royal Society Linear algebraists