Introduction a l'Analyse des Lignes Courbes Algébriques.

Geneve: Freres Cramer & Cl. Philbert, 1750.

First edition of this major treatise on analytic geometry, containing Cramer’s rule and paradox. It is “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 nth degree is in general determined if n(n+3)/2 points on it be given” (Ball). “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 … 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 v2/2 + 3v/2 for the number of arbitrary constants in the general equation of the vth degree … From this he concludes that a curve of order v can be made to pass through v2/2 + 3v/2 points, a statement that he says needs only an example for a demonstration … Cramer’s paradox was the outgrowth of combining the formula v2/2 + 3v/2 with the theorem, which Cramer attributes to Maclaurin, that mth- and nth-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 … He then states that he has found a general and convenient rule [‘Cramer’s rule’] for the solution of a set of 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” (DSB).

“Cramer’s volume resembles strongly in aim and content the work of de Gua and Euler. The author, however, says that his book had been almost completed before the appearance of the Usage d’analyse in 1740, and that the Introductio in 1748 was published too late for him to make use of it. The first half of this statement is belied to some extent by the effective use Cramer made of the analytical triangle, for the introduction of which he complements de Gua; but the latter half is borne out by his failure to employ the analytical trigonometry of Euler. Apparently Cramer was influenced primarily by Newton’s Enumeratio and Stirling’s Commentary. He did not apply the calculus, and so his book rivals that of Euler as a resume of the application of analytical geometry to the study of curves. It opens with a general theory and classification of algebraic curves. Transcendental curves are illustrated by the curves y = bax and y√2+ y = x, but these are not systematically considered. A whole chapter is devoted to the transformation of axes, but trigonometric symbolism is not here used. Cramer was among the first to make formal use of two axes and to define the two coordinates simultaneously and symmetrically in terms of these. The coordinates are designated by the terms coupée and appliquée, as well as by the more modern names abscissa and ordinate.

“One finds an intruding chapter on the odd subject of the construction of equalities, with a table indicating the curves of minimum degree necessary for the graphical solution of equations in the traditional manner. The table runs from quadratic equations to those of degree 100, for the last of which two 10th degree curves are indicated as necessary. Cramer even quotes the rule given in l’Hospital’s Conics to cover the favourite Cartesian topic of the simplest possible curves … He suggests as an alternative the Newtonian idea that it is less the algebraic simplicity than the geometric facility of description which one should seek. This being so, he doesn’t see why one should reject the type of graphical solution of polynomial equations which is found in L’Hospital’s Conics and which he ascribes to Jacques Bernoulli. This method makes use of the line x = a and the curve x = by + cy2 + dy3 + … to solve the polynomial equation a = by + cy2 + dy3 + … A score of cubic and quartic polynomial equations of this form are plotted graphically. Cramer pointed out that ‘this construction is simple and of an easy practicality,’ but he himself used it, following L’Hospital and Stirling, largely to illustrate the character of the roots of equations as determined by its coefficients. No examples are given in which the coefficients are simple numbers. Cramer did, however, express a well-warranted surprise that such graphical solutions should be rejected (presumably by the algebraists of his day).

“Like Euler, Cramer devoted much space to the properties of plane curves, both elementary and higher – their diameters, singular points, and infinite branches. He made extensive use of the Newton-de Gua triangle and of series developments for singularities, both at the origin and at infinity. Specific cases of equations with numerical coefficients are given frequently, an exceptional practice in that day. The general equation of the circle, rarely given by writers of the time, appears in the form (y – a)2 + (b – x)2 = rr; but neither of the curves of elementary geometry – the line and the circle – was studied systematically. The straight line appears in the general form a = ± by ± cx, and various cases (cutting across each of the four quadrants, respectively) are geometrically represented. The equations of the axes, x = 0 and y = 0 are specifically mentioned.

“In determining a curve of order n through n(n + 3)/2 points, Cramer suggested the method of undetermined coefficients. Thus to find the conic [n = 2] through 5 points, the coordinates of the point are substituted into the equation A + By + Cx + Dyy +Exy + xx = 0; and by means of the five linear equations thus obtained one calculates the five unknown coefficients. Admitting that the calculation is fairly long, Cramer adds the comment, ‘I believe I have found for this a rule which is quite convenient and general when one has any given number of equations and unknowns none of which exceeds the first degree.’ Using exponents as distinguishing indices instead of powers, Cramer wrote the equations as

A1= Z1z + Y1y + X1x + V1v + ….

A2= Z2z + Y2y + X2x + V2v + ….

A3= Z3z + Y3y + X3x + V3v + ….

‘Then to find the values of the unknowns one forms fractions as follows: in the denominator one writes all possible products of a Z, a Y, an X, a V, etc., always written with letters in order and with signs determined according as the number of inversions in superscripts in odd and even. In the numerator, one substitutes the A’s for the coefficients of the unknowns desired.’ Through this device, now known as Cramer’s rule, the author is entitled to credit as a rediscoverer of the value of the determinant notation of Leibniz. Leibniz in a letter to L’Hospital had shown how to take advantage of a harmonious notation in eliminating two unknowns from three linear equations, boasting that this ‘little pattern’ showed that Viète and Descartes were not aware of all the mysteries. Determinant patterns were occasionally used in the latter half of the eighteenth century, but it was almost a hundred years after the time of Cramer before mathematicians generally realised the role that determinants can play in analytic geometry.

“Among the properties of curves cited in the Introduction is one which has become known as ‘Cramer’s paradox,’ although it had been referred to in a memoir by Euler two years before, and was mentioned also by Maclaurin. This dilemma pointed out that in general two algebraic curves of degree n intersect in n2 points, but that the number n2 is usually greater than n(n + 3)/2, the number of points by which each of the curves presumably should be uniquely determined. The only exceptions are the conics (for which n2 is 4 and n(n + 3)/2 is 5) and the cubics (for which both numbers are 9). Cramer seems to have realized, in a vague sort of way, that the question of independence of points was involved here; but it remained for the nineteenth century to give a clear explanation of this paradoxical situation” (Boyer, pp. 194-6).

Both Cramer’s rule and Cramer’s paradox were, in fact, given earlier by the great Scottish mathematician Colin Maclaurin: the former in his Treatise ofAlgebra (1748) (although Maclaurin only gives the rule in some special cases, and does not use any special notation like that introduced by Cramer); and the latter in his Geometria organica (1720) – Cramer in fact refers to Maclaurin for the paradox in the Introduction, but it is Cramer’s name that is attached to the phenomenon.

Honeyman 775; Sotheran’s Catalogue 770, 1917 (‘very rare’). DSB III, p.460; Ball, A Short Account of the History of Mathematics, pp. 371-72; Boyer, History of Analytic Geometry, pp. 194-96; Coolidge, History of Geometrical Methods, pp. 132-33; Cajori, A History of Mathematics, p. 241; Struik, A Source Book in Mathematics 1200-1800, pp. 180-81.



4to (234 x 190 mm), pp. xiii, [1, blank], 680, xi, [1, errata], with 33 folding engraved plates. Contemporary calf, richly gilt spine (lower capital a little chipped). A fine copy.

Item #2618

Price: $3,250.00