By Alan George

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Yet his work is essential, not only because it solved Cramer's paradox, but also because it put forward simple intuitive facts with important implications. 2. Cramer and the Birth of the Theory of Determinants In the same year, Gabriel Cramer published a treatise entitled Introduction à l'Analyse des Courbes Algébriques. Except for a letter by Leibniz, from 1693, but only published for the first time in 1850, this text was the first one in which a specific notation was given in order to write a system of linear equations with nonspecified coefficients9.

In fact, the maximal order r of non-evanescent minors in a system of p linear equations in n unknowns gives the number n-r of arbitrary unknowns to be chosen to describe the set of solutions of a consistent system and represents the maximal number of independent equations in the system. Such ideas had become 12 well known by the middle of the 19th century. The classical method consisted in isolating the part of the equations corresponding to a nonzero minor of maximal order and then in using Cramer's rule, with the other unknowns as parameters appearing in the second members.

N). BB'+ ... NN' = 0 25 ist. C + ... N = (a + b + c + ... S. 26 16 JEAN-LUC DORIER Möbius did not intend to build an algebraic theory; he aimed at providing mathematicians and physicists with a tool in order to improve the solving of geometry or mechanics problems. This is why he was not interested in investigating specific cases which seem to have only a theoretical relevance. From this perspective, his work was quite a success, but it also induced a theoretical breakthrough. In particular, it inspired Karl von Staudt, in his discovery of projective coordinates, which were essential in order to separate projective geometry from metric considerations and, therefore, to better understand the nature of this very important approach to geometry.