American Mathematical Society Colloquium Lectures Elementary Theory of Equations

Abstract

THESE works are very different in scope, but may be noticed together as examples of the way in which mathematics is studied in the United, States. Prof. Dicksons treatise on the theory of equations is suited for the average university student, and it is interesting to see how an accurate and distinguished mathematician like the author chooses and discusses his topics. He begins with graphics; and here he does what so many fail to do gives a warning example to show the risk of drawing wrong conclusions from free and easy plotting. Graphical methods are very useful in this theory in connection with such things as Fouriers theorem, Newtons approximation rule, etc.; they are not trustworthy substitutes for calculation when the real roots of a given equation have to be found. Other things dealt with are the elementary theory of cubics and quartics, symmetric functions, separation and calculation of real roots, determinants, systems of linear equations, resultants. The proof of the fundamental theorem for symmetric functions is the proper one: namely, by establishing the one one correspondence of the highest part of any given symmetric function to that of a definite product of coefficients not using the theorem about the sums of powers of the roots. The proof that every equation has a root is substantially that of Gauss; this is a rather noteworthy fact. The omission of the Galois theory is quite natural; but we are inclined to regret that the author did not comment on the solutions of the cubic and quartic so as to show their common features, and prepare the way for group-theory.

American Mathematical Society. Colloquium Lectures.

Vol. iv., The Madison Colloquium, 1913. (i.) On Invariants and the Theory of Numbers. By L. E. Dickson. (ii.) Topics in the Theory of Functions of Several Complex Variables. By W. F. Osgood. Pp. iii + iii + 110 + iii + 111230. (New York: American Mathematical Society, 1914.)

Elementary Theory of Equations.

By Prof. L. E. Dickson. Pp. v + 184. (New York: J. Wiley and Sons, Inc.; London: Chapman and Hall, Ltd., 1914.) Price 7s. 6d. net.