## Introduction a l'Analyse des Lignes Courbes Algébriques.

Geneve: Freres Cramer & Cl. Philbert, 1750. First edition of “the most complete exposition of algebraic curves existing at that time” (Struik), It contains ‘Cramer’s paradox’ that, for example, two cubic curves generally intersect in 9 points, but that, in general, giving 9 points on a cubic curve determines it uniquely. It also contains ‘Cramer’s rule’ for solving systems of linear equations.

First edition of this major treatise on analytic geometry, containing Cramer’s rule and paradox, “the most complete exposition of algebraic curves existing at that time” (Struik), and “a worthy successor to Newton’s Enumeratio.” (Boyer). According to Cantor, this together with Euler’s Introductio, forms the first actual text-book on algebraic curves, and it “contains the earliest demonstration that a curve of the n^{th} degree is in general determined if ½n(n+3) points on it be given.” (Ball).

Honeyman 775; Sotheran’s Catalogue 770, 1917 (‘very rare’).

(DSB:) “Cramer’s major publication *Introduction à l’analyse lignes courbes algébriques*, was published in 1750 … The first chapter of the Introduction defines regular, irregular, transcendental, mechanical, and irrational curves and discusses some techniques of graphing, including our present convention for the positive directions on coordinate axes. The second chapter deals with transformations of curves, especially those which simplify their equations, and the third chapter develops a classification of algebraic curves by order or 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 solution of equations, diameters, branch points and singular points, tangents, points of inflection, maxima, minima, and curvature. Cramer claims that he gives no example without a reason, and no rule without an example.

“The third chapter of Cramer’s Introduction uses a triangular arrangement of the terms of complete equations of successively higher degree as the basis for deriving the formula *v ^{2}/2 + 3v/2* for the number of arbitrary constants in the general equation of the

*v*th degree. This is the sum of

*v*terms of the arithmetic progression 2 + 3 + 4 +… derived from the rows of the triangle by regarding one coefficient, say

*a*, as reduced to unity by division. From this he concludes that a curve of order

*v*can be made to pass through

*v*points, a statement that he says needs only an example for a demonstration. In his example Cramer writes five linear equations in five unknowns by substituting the coordinates of five points into the general second-degree equation. He then states that he has found a general and convenient rule for the solution of a set of

^{2}/2 + 3v/2*v*linear equations in

*v*unknowns; but since this is algebra, he has put it into appendix 1. … 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. …“Cramer’s paradox was the outgrowth of combining the formula

*v*with the theorem, which Cramer attributes to Maclaurin, that

^{2}/2 + 3v/2*m*th- and

*n*th- order curves intersect in

*mn*points. The formula says, for example, that a cubic curve is uniquely determined by nine points; the theorem says that two different cubic curves would intersect in nine points.”

DSB III, p.460; Struik, A Source Book in Mathematics 1200-1800, pp.180-81; Boyer, History of Analytic Geometry, pp.194-96; Ball, A Short Account of the History of Mathematics, pp. 371-72; Coolidge, History of Geometrical Methods, pp.132-33; Cajori, A History of Mathematics, p.241.

4to: 234 x 190 mm. Contemporary calf, richly gilt spine, lower capitil a little chipped. Pp. XIII, (1:blank), 680, XI, (1:errata) and 33 engraved folding plates. A fine copy.

Item #2618

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Price:
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