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	Index using Historical Classification Systems

Here are the top-level areas of mathematics and related fields classified according to two systems used early in the twentieth century.

The first is the system used in the "Catalogue of Scientific Papers 1800-1900" published by the Royal Society of London in 1908; they continued using this classification system for another decade or so as Schedule A (Pure Mathematics) of the International Catalogue of Scientific Literature. The system often subdivides these roughly 200 headings into unnumbered subheadings, not shown.

The second is the set of about a hundred headings used by the "Jahrbuch über die Fortschritte der Mathematik" near the time it slowed publication about 1931. (Publication began about 1880, and ended around 1945.)

These systems are included primarily for historical interest; many of the most active fields of research during the twentieth century correspond to no field in these schemes. However, traces of these systems can be seen in the Mathematics Subject Classification (MSC) scheme developed several decades later and, with modifications, still used to classify current publications. The MSC is the basis for the organization of materials at this site. (In addition, the Library of Congress scheme includes several passages nearly identical to portions of the Royal Society schedules.)

Not shown in the Royal Society Scheme are the classifications of topics in applied mathematics, such as mechanics. (But no classification existed for Mathematical Logic, Ring Theory, Algebraic Topology, Numerical Analysis, Quantum Theory, and all the other fields of today's mathematics which were at the time at best minor or isolated topics!)

The headings in the Royal Society system are

• ARITHMETIC AND ALGEBRA
  • Foundations of Arithmetic
  • Universal Algebra
  • Theory of Groups
• ALGEBRA AND THEORY OF NUMBERS
  • Elements of Algebra
  • Linear Substitutions
  • Theory of Equations
  • Theory of Numbers
• ANALYSIS
  • Foundations of Analysis
  • Theory of Functions of Complex Variables
  • Algebraic Functions and their Integrals
  • Other Special Functions
  • Differential Equations
  • Differential Forms and Differential Invariants
  • Analytical Methods connected with Physical Problems
  • Difference Equations and Functional Equations
• GEOMETRY
  • Foundations
  • Elementary Geometry
  • Geometry of Conics and Quadrics
  • Algebraic Curves and Surfaces of degree higher than the second
  • Transformations and General Methods for Algebraic Configurations
  • Infinitesimal Geometry; applications of Differential and Integral Calculus to Geometry
  • Differential Geometry; applications of Differential Equations to Geometry

Finer divisions within the headings above were enumerated; these are shown at the end of this document. The system used by the Jahrbuch used the following headings:

• 1. HISTORY OF MATHEMATICS, PHILOSOPHY, EDUCATION
  • 1.1 History
    • 1.1.1 General History
    • 1.1.2 Antiquity
    • 1.1.3 East Asia
    • 1.1.4 Maya
    • 1.1.5 Orient
    • 1.1.6 Greek
    • 1.1.7 Medieval
    • 1.1.8 Contemporary
  • 1.2 Philosophy of mathematics and of natural sciences
  • 1.3 Education
  • 1.4 Biographies
• 2. LOGIC, ARITHMETIC, ALGEBRA, NUMBER THEORY
  • 2.1 Logic
  • 2.2 Elementary logic and algebra
  • 2.3 Linear algebra, invariant theory
    • 2.3.1 Determinants
    • 2.3.2 Matrices
    • 2.3.3 Forms and invariants
  • 2.4 Polynomials and algebraic equations
  • 2.5 Group theory
  • 2.6 Abstract algebra
    • 2.6.1 Lattices
    • 2.6.2 Rings
    • 2.6.3 Ideals
    • 2.6.4 Algebras
    • 2.6.4 Fields
    • 2.6.6 Universal algebra and vector analysis
  • 2.7 Number fields and function fields
  • 2.8 Number theory
    • 2.8.1 Elementary number theory
    • 2.8.2 Diophantine equations
    • 2.8.3 Forms
    • 2.8.4 Additive number theory
    • 2.8.5 Analytic number theory
    • 2.8.6 Density
    • 2.8.7 Number-theoretic functions
    • 2.8.8 Lattice point problems
    • 2.8.9 Algebraic number
    • 2.8.10 Distribution of primes
    • 2.8.11 Geometry of numbers
    • 2.8.12 Diophantine approximation
    • 2.8.13 Magic squares
• 3. GENERAL AND COMBINATORICS
  • 3.1 Combinatorics
  • 3.2 Games
  • 3.3 Recreational mathematics
• 4. PROBABILITY AND STATISTICS
  • 4.1 Probability
  • 4.2 Statistics
  • 4.3 Applications to economics, finances and assurances
• 5. ANALYSIS
  • 5.1 Set theory
  • 5.2 General theory of real functions
  • 5.3 Differentiation und integration of real functions
  • 5.4 Series
  • 5.5 Fourier analysis
  • 5.6 Almost periodic functions
  • 5.7 Convex functions
  • 5.8 Special functions
  • 5.9 Complex analysis
  • 5.10 Difference equations
  • 5.11 Integral equations
  • 5.12 Integral transformations
  • 5.13 Ordinary differential equations
  • 5.14 Partial differential equations
    • 5.14.1 Differential equations of higher order
    • 5.14.2 Differential equations of first order
    • 5.14.3 Elliptic differential equations
    • 5.14.4 Potential theory
    • 5.14.5 Parabolic differential equations
    • 5.14.6 Hyperbolic differential equations
    • 5.14.7 Boundary value problems
  • 5.15 Calculus of variations
  • 5.16 Numerical analysis and scientific computing
  • 5.17 Functional analysis
• 6. GEOMETRY
  • 6.1 Foundations of Geometry, Non-Euclidean Geometry
  • 6.2 Elementary Geometry
  • 6.3 Analytic and projective geometry
  • 6.4 Algebraic geometry
  • 6.5 Vector and tensor calculus
  • 6.6 Differential geometry
  • 6.7 Riemannian manifolds
  • 6.8 Applied geometry
  • Topology
• 7. MATHEMATICAL PHYSICS AND OTHER APPLICATIONS
  • 7.1 Mechanics of Rigid Bodies
  • 7.2 Elasticity
  • 7.3 Acoustics
  • 7.4 Optics
  • 7.5 Thermodynamics
  • 7.6 Hydro- and aerodynamics
  • 7.7 Astronomy
  • 7.8 Electrodynamics
  • 7.9 Relativity and Gravitation
  • 7.10 Quantum theory
  • 7.11 Statistical mechanics
  • 7.12 Geodesy
  • 7.13 Geophysics

Here are the numbered subdivisions of the Royal Society system:

• 0000 Philosophy
• 0010 History. Biography
• 0020 Periodicals. Reports of Institutions, Societies, Congresses, etc.
• 0030 General Treatises, Text Books, Dictionaries, Tables, Collected Works
• 0032 Bibliographies
• 0035 Tables of Mathematical Functions
• 0040 Addresses, Lectures, etc. of a general character
• 0050 Pedagogy
• 0060 Institutions
• 0070 Nomenclature
• 0080 Instruments, including Calculating Machines. Models.
• 0090 Aids to Calculation, Graphical Processes
• ARITHMETIC AND ALGEBRA
• Foundations of Arithmetic
• 0400 General
• 0410 Rational numbers; arithmetical operations
• 0420 Existence of irrational and transcendental numbers; infinite processes adapted to rational numbers
• 0430 Aggregates
• Universal Algebra
• 0800 General
• 0810 Calculus of Operations
• 0820 General theory of complex numbers (See also 2870)
• 0830 Quaternions
• 0840 Ausdehnungslehre; vector analysis
• 0850 Matrices
• 0860 Other special sorts of complex numbers
• 0870 Algebra of Logic
• Theory of Groups
• 1200 General
• 1210 Discrete groups of finite and infinite order (including groups of permutations) (See also 2450, 4440)
• 1230 Continuous groups of finite and infinite order
• ALGEBRA AND THEORY OF NUMBERS
• 1590 General
• Elements of Algebra
• 1600 General
• 1610 Rational polynomials; divisibility; reducibility
• 1615 Algebraic inequalities
• 1620 Permutations, combinations, partitions, distributions. Binomial and multinomial coefficients
• 1625 Finite summation. Recurring series
• 1630 Probabilities (including combination of observations)
• 1635 Theory of statistics. Actuarial mathematics.
• 1640 Calculus of differences; interpolation
• Linear Substitutions
• 2000 General
• 3010 Determinants
• 2020 Discriminants and resultants
• 2030 Characteristic properties of linear substitutions; types of linear substitutions (See also 1210)
• 2040 General theory of quantics
• 2050 Binary forms
• 2060 Ternary forms
• 2070 Special developments associated with forms in more than three variables
• Theory of Equations
• 2400 General
• 2410 Elements of the theory of algebraic equations; existence of roots, symmetric functions; rational fractions; partial fractions
• 2420 Reality, multiplicity, separation, of roots
• 2430 Equations of the second, third, and fourth orders; other particular equations
• 2440 Numerical solution of equations (including graphical methods)
• 2450 General resolution of equations; theory of Galois (See also 1210)
• 2460 Simultaneous equations; including linear equations
• 2470 Transcendental equations
• Theory of Numbers
• 2800 General
• 2810 Divisibility; linear congruences
• 2815 Continued fractions and indeterminate equations
• 2820 Quadratic residues
• 2830 Quadratic binary forms
• 2840 Quadratic forms of three of more variables; bilinear forms (See also 2060, 2070)
• 2850 Congruences other than linear; cubic and higher residues
• 2860 Forms of higher degree which cannot be considered as products of linear factors
• 2870 Forms of higher degree which can be considered as products of linear factors; algebraic numbers; ideals (See also 0820)
• 2880 Application of trigonometrical functions to arithmetic; cyclotomy
• 2890 Application of other transcendental functions to arithmetic
• 2900 Distribution of prime numbers
• 2910 Special numbers and numerical functions
• 2920 Irrationality and transcendence of particular numbers, such as e and \pi
• ANALYSIS
• 3190 General
• Foundations of Analysis
• 3200 General
• 3210 Theory of functions of real variables
• 3220 Infinite series; infinite products and other infinite processes (See also 5610, 5620)
• 3230 Principles and elements of the differential calculus
• 3240 Taylor's series. Maxima and minima; other analytical applications of the differential calculus
• 3250 Principles and elements of the integral calculus. Approximate integration. Mechanical quadrature
• 3260 Definite integrals (simple)
• 3270 Multiple integrals
• 3280 Calculus of variations
• Theory of Functions of Complex Variables
• 3600 General
• 3610 Uniform functions of one variable
• 3620 Multiform functions of one variable; Riemann surfaces (See also 4060, 4620, 8075)
• 3630 Expansions in series of functions, other than powers of the variable
• 3640 Functions of several variables.
• Algebraic Functions and their Integrals
• 4000 General
• 4010 Algebraic functions of one variable
• 4020 Algebraic functions of several variables
• 4030 Logarithmic, circular, exponential functions
• 4040 General properties of elliptic functions and single theta functions; addition-theorem (See also 8050, 8060)
• 4050 Multiplication, division, transformation of elliptic functions; modular functions
• 4060 Abelian integrals (See also 8050, 8060)
• 4070 Periodic functions of several variables; general theta functions
• Other Special Functions
• 4400 General
• 4410 Eulerian functions
• 4420 Legendre's functions; Bessel's functions; hypergeometric functions
• 4430 Polymorphic functions. Other functions which may be defined by definite integrals (See also 4860)
• 4440 Automorphic functions (See also 1210, 4050)
• 4450 Other functions which may be defined by linear differential equations. Lamé's functions
• 4460 Functions which may be defined by functional equations (See also 6030)
• 4470 Integral functions
• Differential Equations
• 4800 General
• 4810 Existence theorems for ordinary and partial differential equations
• 4820 Methods of solution and reduction of ordinary differential equations
• 4830 Methods of solution and reduction of partial differential equations of first order, including the differential equations of theoretical dynamics
• 4840 Methods of solution and reduction of partial differential equations of the second and higher orders
• 4859 General theory of ordinary linear equations (See also 4450)
• 4860 Integration of ordinary linear equations by definite integrals (See also 4430)
• 4870 General theory of ordinary equations, not linear, of the first order
• 4880 General theory of ordinary equations, not linear, of order higher than first
• Differential Forms and Differential Invariants
• 5200 General
• 5210 Linear differential forms; Pfaffians
• 5220 Differential forms of the second and higher orders (See also 8450)
• 5230 Transformation of differential forms, including tangential (or contact) transformations
• 5240 Differential invariants (See also 1230)
• Analytical Methods connected with Physical Problems
• 5600 General
• 5610 Harmonic Analysis; Fourier's series (See also 3220)
• 5620 Harmonic Analysis; series other than Fourier's. Spherical and ellipsoidal harmonics (See also 3220)
• 5630 Generalities on the differential equations of mathematical physics
• 5640 Integration of the differential equations of mathematical physics by series
• 5650 Integration of the differential equations of mathematical physics by definite integrals
• 5650 Dirichlet's problem and analogous problems affected by boundary conditions
• Difference Equations and Functional Equations
• 6000 General
• 6020 Solution of equations of finite differences
• 6030 Solution of functional equations (See also 4460)
• GEOMETRY
• 6390 General
• Foundations
• 6400 General
• 6410 Principles of geometry; non-Euclidean geometries; hyperspace
• 6420 Topology of space and hyperspace (See also 3620, 8075)
• 6430 Methods of analytical geometry (See also 0840)
• Elementary Geometry
• 6800 General
• 6810 Planimetry; straight lines, and circles
• 6820 Stereometry; straight lines, planes, and spheres; polyhedra
• 6830 Trigonometry, plane and spherical
• 6840 Descriptive geometry; perspective
• Geometry of Conics and Quadrics
• 7200 General
• 7210 Metrical and projective properties of conics
• 7230 Systems of conics (See also 8070 and 8090)
• 7240 Metrical and projective properties of quadric surfaces
• 7260 Systems of quadric surfaces (See also 8070, 8090)
• Algebraic Curves and Surfaces of degree higher than the second
• 7600 General
• 7610 Metrical and projective properties of algebraic plane curves of degree higher than the second
• 7630 Special plane algebraic curves
• 7640 Algebraic surfaces of degree higher than the second
• 7650 Special algebraic surfaces
• 7660 Skew algebraic curves
• Transformations and General Methods for Algebraic Configurations
• 8000 General
• 8010 Collineation; duality
• 8020 Other algebraic transformations (See also 6810, 7650)
• 8030 Groups of points on an algebraic curve; genus of curves; principle of correspondence
• 8040 Groups of curves and points on an algebraic surface; genus of surfaces (See also 7640)
• 8050 Applications of transcendental functions to algebraic curves (See also 4040, 4060, 7660)
• 8060 Application of transcendental functions to algebraic surfaces (See also 4040, 4060, 7650)
• 8070 Enumerative geometry (See also 7230, 7260, 7600-7660)
• 8075 Special configurations of points, lines, planes, or other elements. Space partitioning. (See also 3620 6820, 7210, 7610, 7640)
• 8080 Line geometry. Connexes, complexes, congruences; higher elements of space (See also 7650)
• 8090 Systems (linear, and not linear) of curves and surfaces
• 8100 Algebraic configurations in hyperspace
• Infinitesimal Geometry; applications of Differential and Integral Calculus to Geometry
• 8400 General
• 8410 Principles of infinitesimal geometry
• 8420 Kinematic geometry (See also Mechanics 0420)
• 8430 Curvature of plane curves; other applications of the differential calculus to plane curves (See also 7210)
• 8440 Curvature of skew curves; other applications of the differential calculus to skew curves (See also 7240, 8420)
• 8450 Curvature of surfaces; curvilinear coordinates, and other applications of the differential calculus to surfaces (See also 5220, 8810, 8840, 8850)
• 8455 Differential geometry of congruences and other applications of the differential calculus to elements of space
• 8460 Rectification and quadrature of curves; areas and volumes of surfaces. Other applications of the integral calculus to geometry
• 8470 Special transcendental curves
• 8480 Special transcendental surfaces
• 8490 Hypergeometric configurations and higher elements of hyperspace
• Differential Geometry; applications of Differential Equations to Geometry
• 8800 General
• 8810 Determination of curves on surfaces
• 8820 Minimal surfaces
• 8830 Surfaces determined by relations of curvature and by other differential properties
• 8840 Conformal and other representations of surfaces on others (See also 3600, 6840 and Geography 7095)
• 8850 Deformation of surfaces
• 8860 Orthogonal and isothermic surfaces
• 8870 Hypergeometric configurations and higher elements of hyperspace.