Synopsis of Existing Developments of Pure Mathematics

SYNOPSIS OF EXISTING DEVELOPMENTS OF PURE MATHEMATICS The International Catalogue.—A complete classification of mathematical sciences, as they at present exist, is to be found in the International Catalogue of Scientific Literature promoted by the Royal Society, and was drawn up by an international com mittee of eminent mathematicians, and has the highest authority. It must not be criticized from an exacting philosophical point of view. The practical object of the enterprise required that the proportionate quantity of yearly output in the various branches, and that the liability of various topics as a matter of fact to occur in connection with each other, should modify the classification.

Fundamental Notions.—Section A deals with pure mathe matics. Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal Algebra," with the topics complex numbers, quater nions, Ausdehnungslehre, vector analysis, matrices and algebra of logic ; and "Theory of Groups," with the topics finite and continuous groups. For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC NUMBERS ; GROUPS; CALCULUS; NUMBER; NUMBERS, THEORY OF; QUATERNIONS ; VECTOR ANALY SIS.

Algebra.—Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, etc., partitions, prob abilities; "Linear Substitutions," with the topics determinants, etc., linear substitutions, general theory of quantics ; "Theory of Algebraic Equations," with the topics existence of roots, separa tion of and approximation to, theory of Galois, etc. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANT; EQUATIONS, THEORY OF; FRACTION ; CONTINUED FRACTIONS; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROB ABILITY AND ERROR.

Analysis.

Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic f unctions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Dif ferential Equations," with the topics existence theorems, meth ods of solution, general theory ; "Differential Forms and Differen tial Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tan gential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equa tions of applied mathematics, Dirichlet's problem ; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations. For the subjects of this heading see the articles DIF

FERENTIAL EQUATIONS ; FOURIER SERIES ; CONTINUED FRAC TIONS; FUNCTIONS; GROUPS; CALCULUS; MAXIMA AND MINIMA; SERIES; NUMBER SEQUENCES; SPHERICAL HARMONICS; TRIGO NOMETRY; CALCULUS OF VARIATIONS; DIFFERENTIAL FORMS.

Geometry.—Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry ; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Con figurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic con figurations in hyperspace ; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geo metry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces deter mined by differential properties, conformal and other representa tion of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading sce the articles CONIC SECTIONS; CIRCLE; CURVE; CURVES, SPECIAL; GEOMETRY : Axioms; GEOMETRY : Non-Euclidean; PROJECTIVE GEOMETRY; ANALYTIC GEOMETRY; LINE GEOMETRY; KNOTS; MENSURATION; MATHEMATICAL MODELS; PROJECTION; SURFACE; TRIGONOMETRY.

Conclusion of the Survey.—This survey of the existing developments of pure mathematics confirms the conclusions ar rived at from the previous survey of the theoretical principles of the subject. Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations. A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connection with "number" and "quantity," while bringing ideas of form and structure into in creasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. But the complexity of the idea of number is practically illustrated by the fact that it is best studied as a department of a science wider than itself.