Synopsis of Existing Developments of Pure Mathematics.-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. The classification in question
was drawn up by an international committee of eminent mathematicians,
and thus has the highest authority. It would be unfair
to criticize it 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 connexion
with each other, should modify the classification.
Section A deals with pure mathematics. 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, quaternions, 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, UNIVERSAL; GROUPS,
THEORY Or; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS;
VECTOR ANALYSIS. Under the general heading “Algebra and
Theory of Numbers ” occur the subheadings “ Elements of Algebra, ”
with the topics rational polynomials, permutations, &c., partitions,
probabilities; “ Linear Substitutions, " with the topics determinants,
&c., linear substitutions, general theory, of quantics; “ Theory
of Algebraic Equations, " with the topics existence of roots, Separation
Of and approximation to, theory of Galois, &c. “ 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; DETERMINANTS;
EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS;
MAGIC SQUARE; PROBABILITY. 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 o Complex Variables, ” with the topics
functions of one variabe and of several variables; “Algebraic
Functions and their Integrals, " with the topics algebraic functions
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;
“ Diferential Equations, ” with the topics existence theorems,
methods of solution, general theory; “ Diferential Forms and
Diferential I niiariants, " with the topics differential forms, including
Pfaftians, transformation of differential forms, including tangential
(or contact) transformations, differential invariants; “Analytical
Methods connected with Physical Sub 'ects, ” with the topics harmonic
analysis, Fourier'S series, the differential equations of applied
mathematics, Dirichlet's roblem; “ Dilference 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 DIFFERENTIAL EQUA-TIONS;
FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION;
FUNCTION or REAL VARIABLES; FUNCTION COMPLEX; GROUPS,
THEORY or; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA;
SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS,
CALCULUS OF. 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 Conjigurations,
" 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 configurations
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; “ Diferential Geometry:
applications of Dijferential Equations to Geometry, " with the topics
curves on surfaces, minimal surfaces, surfaces determined by differential
properties, conformal and other representation of surfaces
on Others, deformation of surfaces, Orthogonal and isothermic
surfaces. For the subjects under this heading see the articles
CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY
GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective;
GEOMETRY, Analytical; GEOMETRY, Line; KNOTS,

MATHEMATICAL THEORY Or; MENSURATION; MODELS; PRO-JECTION; SURFACE; TRIGONOMETRY.

This survey of the existing developments of pure mathematics confirms the conclusions arrived 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 1

a special property. Thus the modern ideas, which have so power full extended and unified the subject, have loosened its Connexion with, “number” and “quantity, ” while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily “ apply ” theorems of number; 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.

Synopsis of Existing Developments of Applied Mathematics.-Section B of the International Catalogue deals with mechanics. The heading “ Measurement of Dynamical Quantities " includes the topics units, measurements, and the constant of gravitation. The topics of the other headings do not require express mention. These headings are: “Geometry and Kinematics of Particles and Solid Bodies "; “ Principles of Rational Mechanics "; “ Statics of Particles, Ri id Bodies, é1'c."; “Kinetics of Particles, Rigid Bodies, &rc."; “ General Analytical Mechanics "; “Statics and Dynamics of Fluids”; “Hydraulics and Fluid Resistances "; “ Elasticity." For the subjects of this general heading see the articles MECHANICS; DYNAMICS, ANALYTICAL; GYROSCOPE; HARMONIC ANALYSIS; WAVE; HYDROMECHANICS; ELASTICITY; MOTION, LAWS OF: ENERGY; ENERGETICS; ASTRONOMY (Celestial Mechanics);TIDE. Mechanics (including dynamical astronomy) is that subject among those traditionally classed as “ applied “ which has been most completely transfused by mathematics-that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician. Every branch of physics gives rise to an application of mathematics. A pro hecy may be hazarded that in the future these applications wili unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena. This reflection is suggested by the following articles: AETHER; MOLECULE; CAPILLARY ACTION; DIFFUSION; RADIATION, THEORY OF; and others.

The applications of mathematics to statistics (see STATISTICS and PROBABILITY) should not be lost sight of; the leading fields for these applications are insurance, sociology, variation in zoology and economics.

The History of Mathematics.-The history of mathematics is in the main the history of its various branches. A short account of the history of each branch will be found in Connexion with the article which deals with it. Viewing the Subject as a whole, and apart from remote developments which have not in fact Seriously influenced the great structure of the mathematics of the European races, it may be Said to have had its origin with the Greeks, working on pre-existing fragmentary lines of thought derived from the Egyptians and Phcenicians. The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities. The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them-namely, ratio, irrationality, continuity, the point, the straight line, the plane. This period lastedl from the time of Thales, c. 600 B.c., to the capture of Alexandria by the Mahommedans, A.D. 641. The medieval Arabians invented our system of numeration and developed algebra. The next period of advance stretches from the Renaissance to Newton and Leibnitz at the end of the 17th century. During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation. The 18th century witnessed a rapid development of analy-Sis, and the period culminated with the genius of Lagrange and Laplace. This period may be conceived as continuing throughout the first quarter of the 19th century. It was remarkable both for the brilliance of its achievements and for the large number of French mathematicians of the first rank who flourished during it., The next period was inaugurated in analysis by K. F. Gauss, N. H. Abel and A. L. Cauchy. Between them the general theory of the complex variable, and of the various “ infinite” processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of Surfaces. The applied mathematical sciences of light, electricity and electromagnetism, 1 Cf. A Short History of Mathematics, by W. W. R. Ball.