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First Edition of 'Cramer's Rule.' [xxiv], 680 pp, xi[i=errata]; text figs.; 33 folding plates engraved in red and black; folding table. Contemporary full leather, 4to. Very Good. 'Cramer's major publication . That he made little use of Euler's work is supported by the rather surprising fact that throughout his book Cramer makes essentially no use of the infinitesimal calculus in either Leibniz' or Newton's form, although he deals with such topics as tangents, maxima and minima, and curvature, and cites Maclaurin and Taylor in footnotes. One conjectures that he never accepted or mastered the calculus. . . . The first chapter . . . defines regular, irregular, transcendental, mechanical and irrational curves and discusses some techniques of graphing, including our present convention for the positive directions of coordinate axes. The second chapter deals with curves, especially those which simplify their equations, and the third chapter develops a classification of algebraic curves by order of degree, abandoning Descartes's classification by genera. Both Cramer's rule and Cramer's paradox develop out of this chapter. The remaining ten chapters include discussions of the graphical solutions of equations, diameters, branch points and singular points, tangents, points of inflection, maxima, minima, and curvature. . . . He . . . states that he has found a general and convenient rule for the solution of a set of v linear equations in v unknowns; but since this is algebra, he has put it in to appendix 1. Figure 2 [in the D.S.B. article] shows the first page of this appendix. The use of raised numerals as indices, not exponents, applied to coefficients represented by capital letters enabled Cramer to state his rule in general terms and to define the signs of the products in terms of the number of inversions of these indices when the factors are arranged in alphabetical order. Although Leibniz had suggested a method for solving systems of linear equations in a letter to L'Hospital in 1693, and centuries earlier the Chinese had used similar patterns in solving them, Cramer has been given priority in the publication of this rule. However, Boyer has shown . . . that an equivalent rule was published in Maclaurin's Treatise of Algebra in 1748. He thinks that Cramer's superior notation explains why Maclaurin's statement of this rule was ignored even though his book was popular. Another reason may be that Euler's popular algebra text gave Cramer credit for this 'très belle règle' ' (D.S.B. III: 459-62; this article is illustrated by two figures printed in Cramer's book). '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. . . . 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' (MacTutor History of Mathematics Web site). '[A] large volume containing the most complete exposition of algebraic curves existing at that time, going far beyond Newton's Enumeratio' (Struik, Source Book in Mathematics, 180-83). Boyer, History of Analytic Geometry, 194-7. Smith, History of Mathematics, p. 520. A. A. Kosinski, 'Cramer's Rule Is Due to Cramer' (Mathematics Magazine, Vol. 74, No. 4, Oct., 2001, 310-12). Sotheran 892 ('Rare'). One specialist catalogued a copy at 1200 pounds, then just five catalogues later, offered another that was significantly better, but also now 4250 pounds!
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