what precedes it is clear that the ordinary process of clearing of fractions leads either to an equivalent or to a redundant equation. VI. A FEW REFERENCES 30. Text-books. The algebraic equation occupies a prominent place in algebra and some of its elementary properties are developed in the text-books on algebra for the secondary schools. More extensive developments of these properties may be found in the advanced text-books on this subject, such as (1) Chrystal, Algebra, an elementary text-book, 2 vols., 2d edition, 1900. (2) Capelli, Istituzioni di analisi algebrica, 4th edition, 1909. (3) Weber, Lehrbuch der Algebra, 2 vols., 2d edition, 1898–99. (4) Serret, Cours d'algèbre supérieure, 2 vols., 6th edition, 1910. The last two of these works include a treatment of the Galois theory of equations while the first two omit this theory, but they give an elementary introduction to the theory of substitutions. In the first this introduction is very brief and incomplete. A large number of special treatises on the general theory of the algebraic equation have appeared, beginning with the works of Vieta in the early part of the seventeenth century. Among the modern works in the English language Burnside and Panton's Theory of Equations is probably the most generally known. The first three editions of this work appeared in one volume and excluded the Galois theory, while the fourth and fifth appeared in two volumes and include an introduction to substitution groups and the Galois theory of equations. Among the other treatises on this subject we may mention (1) Dickson, Introduction to the Theory of Algebraic Equations, 1903. (2) Cajori, An Introduction to the Modern Theory of Equations, 1904. (3) Mathews, Algebraic Equations, 1907. (4) Netto-Cole, Theory of Substitutions and its Applications to Algebra, 1892. (5) Barton, An Elementary Treatise on the Theory of Equations, 2d edition, 1903. (6) Bianchi, Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois, 1900. (7) Vogt, Leçons sur la résolution algébrique des equations, 1895. (8) Matthiessen, Grundzüge der antiken und modernen Algebra der litteralen Gleichungen, 1878. The last of these works contains an account of many of the ancient methods which were used to solve equations and is rich in historical material. As a result of the rapid growth of historical knowledge during recent years some of this material has been found not entirely reliable. Certain phases of the theory of equations are presented in a very instructive manner in Klein's Elementarmathematik vom höheren Standpunkte aus, Autogr., 1908-09; and also in Bôcher's Introduction to Higher Algebra, 1907. An extensive list of treatises on this and other mathematical subjects may be found in the Mathematischer Bücherschatz by Ernst Wölffing. This extensive work is supposed to give a systematic list of the principal books and monographs appearing during the nineteenth century. It appeared in the Abhandlungen zur Geschichte der Mathematischen Wissenschaften, 1903. 31. Articles. (1) Pierpont, Galois "Theory of algebraic equations," Annals of Mathematics, Vols. I and II, 1900, pp. 113 and 22. (2) Bôcher, Gauss's "Third proof of the fundamental theorem of algebra," Bulletin of the American Mathematical Society, Vol. I, 1895, p. 205. (3) Sylvester, "On an elementary proof and generalization of Sir Isaac Newton's hitherto undemonstrated rule for the discovery of imaginary roots," Proceedings of the London Mathematical Society, Vol. I, 1865, p. 1. (4) Van Vleck, "A sufficient condition for the maximum number of imaginary roots of an equation of the nth degree," Annals of Mathematics, Vol. IV, 1903, p. 191. (5) Baker, “A balance for the solution of algebraic equations," American Mathematical Monthly, Vol. II, 1904, p. 224. (6) Emch, "Hydraulic solution of an algebraic equation of the nth degree," ibid., Vol. VIII, 1901, p. 58. (7) Moritz, "On certain proofs of the fundamental theorem of algebra," ibid., Vol. X, 1903, p. 159. (8) McClintock, "A method for calculating simultaneously all the roots of an equation," American Journal of Mathematics, Vol. XVII, 1895, p. 89. (9) Tanner, "A graphical representation of the theorems of Sturm and Fourier," Messenger of Mathematics, Vol. XVIII, 1889, p. 95. (10) Kellogg, "A necessary condition that all the roots of an algebraic equation are real," Annals of Mathematics, Vol. XI, 1908, p. 97. (11) Lambert, "On the solution of algebraic equations in infinite series," Bulletin of the American Mathematical Society, Vol. XIV, 1908, p. 467. (12) Allardice, "On a limit of the roots of an equation that is independent of all but two of the coefficients," ibid., Vol., XIII, 1907, p. 443. (13) Dickson, "On the theory of equations in a modular field," ibid., Vol. XIII, 1906, p. 8. (14) Bauer, "Ueber die verschiedenen Wurzeln einer algebraischen Gleichung," Mathematische Annalen, Vol. LII, 1899, p. 113. (15) Dedekind, "Ueber Gleichungen mit rationalen Coefficienten," Jahresbericht der deutschen Mathematiker-Vereinigung, Vol. I, 1892, p. 33. (16) Lucas, "Résolution electromagnétique des equations," Comptes rendus de l'Académie des Sciences, Paris, Vol. CXI, 1890, p. 965. A very extensive list of additional references to articles may be found in the Royal Society of London Subject Index Catalogue of Scientific Papers, Vol. I, 1908, pp. 156-87. A selected list of treatises and articles is contained in Felix Müller's Führer durch die mathematische Literatur, 1909, pp. 55-62. 1-3, The need of a unifying conception in elementary math- ematics. 4-18 29-32 Functions with discontinuous graphs (10); 11-17, Classification of functions; The location of the elementary functions (12–14); Applications in collegiate teaching (15-17); III. THE FUNDAMENTAL NOTIONS OF THE CALCULUS 19-21, The derivative and its interpretations; 31-32, Relations between functions and their graphs. 262 |