Part 2 — CHAPTER LII MATHEMATICS

There is no single book in the English language, save the Britannica, in which the whole body of mathematical knowledge is examined and classified with special reference to the inter-relation of its various parts and to the results obtained in the neighboring domains of physics, chemistry, and engineering. Text-books necessarily have a somewhat narrow purpose, namely to teach the student how to solve problems in a single given field; wide views over the surrounding country can, therefore, seldom be afforded. The Britannica, however, does for English readers, what the Encyclopädie der Mathematischen Wissenchaften does for German, and more, in that in the Britannica the shadowy borderlands are illuminated and the roads cleared which connect the mathematical and the experimental sciences. In fact if anyone possessed every mathematical text-book that had ever been published, he would still find 317the articles full of suggestion to him, for in them the whole subject has been presented, in all its complex bearings, logically and as a whole.

History

It is nearly 4,000 years since a mathematician was last deified in the person of Amenophis, and as far as can be ascertained only one other of his calling ever received this honour, and he also was an Egyptian who had entered into his godship a full thousand years earlier (Vol. 9, p. 46). To the ancient Egyptians mathematics owes the first fragmentary ideas of arithmetic and mensuration, but little else, for despite their amazing mechanical achievements very little record of purely mathematical knowledge has come down from them. It was the Greeks, starting with Thales (600 B.C.), who really created the sciences of geometry and numbers. To them we owe the great abstract ideas which dominate the science. The Greek period lasted till the capture of Alexandria by the Mohammedans, A.D. 640, at which time the Arabian school took shape, and to it we owe the development of algebra (al-jebr-wa’l-muqubala, which means the transposition and removal [of terms of an equation]). With the Renaissance the centre of scientific research shifted to Western Europe and from then on the boundaries of mathematical knowledge were rapidly extended, till to-day the subject is the common ground on which all the physical sciences meet. The student is referred to the article Mathematics (Vol. 17, p. 878), by A. N. Whitehead, fellow and senior lecturer in mathematics, Trinity College, Cambridge, for a brilliant exposition of the foundations of the subject.

The professed mathematician will, of course, not need any set guide to his reading, but it may be well to point out one or two articles which he will find especially worthy of his attention.

Leading Articles

The article Probability, (Vol. 22, p. 376), by Professor Edgeworth, author of Mathematical Psychics, and numerous papers on the calculus of probabilities, gives, to the best of our belief, the only statement of the whole problem in the English language. That on Algebraic Forms (Vol. 1, p. 620), by Major Macmahon, former president of the London Mathematical Society, includes a number of results not previously published. The article Elasticity (Vol. 9, p. 141), by A. E. H. Love, professor of natural philosophy in the University of Oxford, embodies the experience of a distinguished mathematician who has made this subject the object of his special study for years. Sir George Darwin (son of Charles Darwin) in the article Tide (Vol. 26, p. 938) summed up the results of his life’s work. The new electrical theory of the properties of Matter (Vol. 17, p. 891) is discussed by Sir J. J. Thomson, professor of physics, Cambridge, who has done more than anyone else to develop it. There are many other valuable articles, e.g., Geometry, Axioms (Vol. 11, p. 730), and Geometry, Non-Euclidean (Vol. 11, p. 724), by A. N. Whitehead; Units, Dimensions of (Vol. 27, p. 736), by Professor J. A. Fleming; Energy and Energetics (Vol. 9, p. 398 and p. 390), by Sir Joseph Larmor; Groups, by Prof. Burnside, author of Theory of Groups of Finite Order. Articles which will be found highly useful to the engineer are Mensuration (Vol. 18, p. 134); Earth, Figure of (Vol. 8, p. 801); Geodesy (Vol. 11, p. 607); Strength of Materials (Vol. 25, p. 1007).

Leading Contributors

The mathematician will at once recognize the peculiar fitness of the contributors to deal with the subjects allotted to them, and this fitness is the more noticeable in the following list, arranged in alphabetical order, which names and briefly describes the distinguished 318mathematicians who have collaborated in the Britannica, and indicates the principal articles written by each.

H. F. Baker, Fellow and Lecturer of St. John’s College, Cambridge. Cayley Lecturer in Mathematics in the University. Author of Abel’s Theory and the Allied Theory, etc.:

Differential Equation; Function, Functions of Complex Variables.

Ludwig Boltzmann, formerly Professor of Theoretical Physics in the Universities of Munich, Vienna, and Leipzig. Author of Lectures on the Theory of Gas; Lectures on Maxwell’s Theory of Electricity and Light:

Model.

W. Burnside, Professor of Mathematics, Royal Naval College, Greenwich. Hon. Fellow of Pembroke College, Cambridge. Author of the Theory of Groups of Finite Order, etc.:

Groups, Theory of

Arthur Cayley, formerly Professor of Pure Mathematics in the University of Cambridge. See the biographical article (Vol. 5, p. 589):

Curve (in part); Determinant; Equation; Numbers, Partition of; Surface (in part); Gauss, K. F.; Monge, G.

George Chrystal, Professor of Mathematics and Dean of the Faculty of Arts, Edinburgh University, Hon. Fellow and formerly Fellow and Lecturer, Corpus Christi College, Cambridge:

Perpetual Motion; Pascal (in part); Riemann, Georg.

Col. A. R. Clarke, Royal Medal of Royal Society 1887; in charge of trigonometrical operations of the Ordnance Survey 1854–1881:

Earth, Figure of the (in part); Geodesy (in part); Map, Projections (in part).

Agnes Mary Clerke, Author of History of Astronomy in the 19th Century; The System of the Stars; Problems in Astrophysics; and many other astronomical books. See the biographical article (Vol. 6, p. 497):

Astronomy, History: Zodiac; Brahe, Tycho; Copernicus; Flamsteed; Halley; Huygens; Kepler, etc.

Lt. Col. C. F. Close, head of the Geographical Section, British General Staff, formerly British Representative on the Nyasa-Tanganyika Boundary Commission. Author of Text-Book of Topographical Surveying, etc.:

Maps, Projections (in part).

W. E. Dalby, Professor of Civil and Mechanical Engineering at the City and Guilds of London Institute, Central Technical College, South Kensington. Author of The Balancing of Engines, etc.:

Mechanics, Applied (in part); and several engineering subjects.

Sir George H. Darwin, late Fellow of Trinity College, Cambridge, and Plumian Professor of Astronomy and Experimental Philosophy in the University. President of the British Association, 1905. Author of The Tides and Kindred Phenomena in the Solar System, etc.:

Tide.

F. Y. Edgeworth, Professor of Political Economy in the University of Oxford, etc. Author of Mathematical Psychics, and numerous papers on the Calculus of Probabilities in the Philosophical Magazine, etc.:

Probability.

E. B. Elliott, Waynflete Professor of Pure Mathematics, and Fellow of Magdalen College, Oxford. Formerly Fellow of Queen’s College, Oxford. President of the London Mathematical Society, 1896–1898. Author of Algebra of Quantics, etc.:

Curve, (in part); Geometry, IV Analytical Geometry.

C. Everitt, Magdalen College, Oxford:

Algebra, History: Density; Light, Introduction, History, etc.

319J. A. Ewing, Director of (British) Naval Education. Hon. Fellow of King’s College, Cambridge. Formerly Professor of Mechanism and Applied Mechanics in the University of Cambridge. Author of the Strength of Materials, etc.:

Strength of Materials, and several engineering subjects.

J. A. Fleming, Pender Professor of Electrical Engineering in the University of London. Fellow of University College, London. Formerly Fellow of St. John’s College, Cambridge, and Lecturer on Applied Mechanics in the University. Author of Magnets and Electric Currents, etc.:

Units, Physical; and many articles on Electrical Science.

Rev. A. H. Frost:

Magic Square.

W. Garnett, Educational Adviser to the London County Council; formerly Fellow and Lecturer of St. John’s College, Cambridge. Principal and Professor of Mathematics, Durham College of Science. Author of Elementary Dynamics, etc.:

Energy (in part); Hydrometer; Kelvin, Lord.

J. W. L. Glaisher, Fellow of Trinity College, Cambridge. Formerly President of the Cambridge Philosophical Society and the Royal Astronomical Society, Editor of Messenger of Mathematics and the Quarterly Journal of Pure and Applied Mathematics:

Logarithm; Table, Mathematical; Legendre, A. M.; Napier, John.

J. H. Grace, Lecturer in Mathematics at Peterhouse and Pembroke College, Cambridge. Fellow of Peterhouse:

Geometry, Line Geometry.

Sir A. G. Greenhill, formerly Professor of Mathematics in the Ordnance College, Woolwich. Author of Differential and Integral Calculus with Applications; Hydrostatics; Notes on Dynamics, etc.:

Ballistics; Gyroscope and Gyrostat; Hydromechanics.

Sir Thomas Little Heath, Assistant-Secretary to the Treasury, London. Fellow of Trinity College, Cambridge. Author of Apollonius of Perga; Treatise on Conic Sections; The Thirteen Books of Euclid’s Elements, etc.:

Anthemius; Apollonius of Perga; Archimedes; Hero of Alexandria; Pappus of Alexandria; Porism, etc.

F. R. Helmert, Professor of Geodesy in the University of Berlin:

Earth, Figure of the (in part); Geodesy (in part).

O. M. F. Henrici, Professor of Mechanics and Mathematics in the Central Technical College of the City and Guilds of London Institute. Author of Vectors and Rotors; Congruent Figures, etc.:

Calculating Machines; Geometry, I. Euclidean; II. Projective; III. Descriptive; Perspective; Projection.

E. W. Hobson, Fellow and Tutor in Mathematics, Christ’s College, Cambridge. Stokes Lecturer in Mathematics in the University:

Fourier’s Series; Spherical Harmonics; Trigonometry.

A. E. Jolliffe, Fellow, Tutor and Mathematical Lecturer, Corpus Christi College, Oxford. Senior Mathematical Scholar, 1892:

Continued Fractions; Maxima and Minima; Series.

H. Lamb, Professor of Mathematics, University of Manchester, formerly Fellow and Assistant Tutor of Trinity College, Cambridge; Member of Council of Royal Society, 1894–1896. Royal Medallist, 1902. President of London Mathematical Society 1902–1904. Author of Hydrodynamics, etc.:

Dynamics; Harmonic Analysis; Mechanics, I. Theoretical; Vector Analysis; Wave.

320A. E. H. Love, Sedleian Professor of Natural Philosophy in the University of Oxford. Hon. Fellow of Queen’s College; formerly Fellow of St. John’s College, Cambridge; Secretary to the London Mathematical Society:

Elasticity; Variations, Calculus of; Function, Functions of Real Variables; Infinitesimal Calculus.

W. H. Macaulay, Fellow and Tutor of King’s College, Cambridge:

Motion, Laws of.

Major P. A. Macmahon, Deputy Warden of the Standards, Board of Trade. Joint General Secretary, British Association. Formerly Professor of Physics, Ordnance College. President of London Mathematical Society, 1894–1896:

Algebraic Forms; Combinatorial Analysis; Cayley, Arthur.

G. B. Mathews, formerly Professor of Mathematics, University College of N. Wales, sometime Fellow of St. John’s College, Cambridge:

Algebra, Special Kinds of Algebra; Number.

J. Clerk Maxwell, former Professor of Experimental Physics in the University of Cambridge. See biographical article (Vol. 17, p. 929):

Capillary Action (in part); Diagram.

Simon Newcomb, former Professor of Mathematics and Astronomy, Johns Hopkins University, etc. See the biographical article (Vol. 19, p. 474):

Astronomy, Descriptive; and many other astronomical subjects.

J. H. Poynting, Professor of Physics and Dean of the Faculty of Science in the University of Birmingham. Formerly Fellow of Trinity College, Cambridge. Joint-author of Text-Book of Physics:

Acoustics; Gravitation (in part); Sound.

F. Purser, formerly Fellow of Trinity College, Dublin; Professor of Natural Philosophy in the University of Dublin; Member of the Royal Irish Academy:

Surface (in part).

J. Purser, formerly Professor of Mathematics in Queen’s College, Belfast. Member of the Royal Irish Academy:

Surface (in part).

W. J. M. Rankine, former Professor of Civil Engineering at Glasgow University. See the biographical article (Vol. 22, p. 894):

Mechanics, Applied (in part).

Hon. B. A. W. Russell, formerly Fellow of Trinity College, Cambridge. Author of Foundations of Geometry; Principles of Mathematics, etc.:

Geometry, VI. Non-Euclidean (in part).

W. F. Sheppard, Senior Examiner in the Board of Education; formerly Fellow of Trinity College, Cambridge; Senior Wrangler, 1884:

Algebra, Principles of Ordinary Algebra; Arithmetic; Differences, Calculus of; Interpolation; Mensuration.

P. G. Tait, late professor of Natural Philosophy, Edinburgh University. Author of Elementary Treatise on Quaternions. Joint author with Lord Kelvin of Treatise on Natural Philosophy:

Knot; Quaternions (in part); Hamilton, Sir William; Maxwell, James Clerk.

Rev. Charles Taylor, formerly Master of St. John’s College, Cambridge. Vice-Chancellor, Cambridge University, 1887–1888. Author of Geometrical Conics, etc.:

Geometrical Continuity.

H. M. Taylor, Fellow of Trinity College, Cambridge; formerly Tutor and Lecturer. Smith’s Prizeman, 1865. Editor of the Pitt Press Euclid:

Newton, Sir Isaac.

Sir J. J. Thomson, Cavendish Professor of Experimental Physics and Fellow of Trinity College, Cambridge. President 321of the British Association, 1909–1910. Author of A Treatise on the Motion of Vortex Rings; Application of Dynamics to Physics and Chemistry:

Matter; and several articles on Electrical Science.

J. Walker, Christ Church, Oxford. Demonstrator in the Clarendon laboratory. Formerly Vice-President of the Physical Society. Author of The Analytical Theory of Light, etc.:

Polarization of Light; Refraction, Double Refraction.

A. N. Whitehead, Fellow and Lecturer in Mathematics, Trinity College, Cambridge. Author of A Treatise on Universal Algebra, etc.:

Geometry VI. Non-Euclidean Geometry (in part); Geometry VII. Axioms on Geometry; Mathematics.

These are the men who are responsible for the mathematical sections of the Britannica. A fuller list of articles on mathematical subjects is given below.

Abel, Niels Henrik

Abscissa

Acceleration

Agnesi, Maria Gaetana

Aguillon, F. D.

Algebra

Algebraic Forms

Aliquot

Allen, or Alleyn T.

Amicable Numbers

Anderson, Alexander

Angle

Anthemius

Apollonius of Perga

Archimedes

Argument

Arithmetic

Autolycus of Pitane

Axis

Babbage, Charles

Baldi, Bernardino

Ballistics

Barlow, Peter

Barrow, Isaac

Bernoulli (family)

Bessel Function

Binomial

Biquadratic

Bisectrix

Boole, George

Borda, Jean Charles

Boscovich, Roger J.

Bouguer, Pierre

Bowditch, Nathaniel

Brachistochrone

Briggs, Henry

Buxton, Jedediah

Calculating Machines

Camus, Charles E. L.

Cardan, Girolamo

Cardioid

Castel, Louis Bertrand

Catenary

Cauchy, A. L., baron

Cayley, Arthur

Charles, J. A. C.

Chebichev, P. L.

Circle

Cissoid

Clairault, A. C.

Clifford, William K.

Cocker, Edward

Colburn, Z.

Combinatorial Analysis

Conchoid

Cone

Conic Section

Conoid

Continued Fractions

Cotes, Roger

Cremona, Luigi

Cube

Curve

Cycloid

Cylinder

Demoivre, Abraham

De Morgan, Augustus

Determinant

Diagonal

Diagram

Diameter

Differences, Calculus of

Differential Equation

Dimension

Diophantus of Alexandria

Ditton, Humphry

Dodecahedron

Dynamics

Earth, Figure of the

Elasticity

Ellipse

Ellipsoid

Emerson, William

Energetics

Energy

Epicycloid

Equation

Euclid

Euler, Leonhard

Fermat, Pierre de

Figurate Numbers

Focus

Folium

Fourier, J. B. J.

Fourier’s Series

Frisi, Paolo

Frustum

Function

Galloway, Thomas

Galois, Evariste

Gauss, Karl Friedrich

Geodesy

Geometrical Continuity

Geometry

Gnomon

Graphical Methods

Gravitation

Greaves, John

Gregory (family)

Gregory, Olinthus G.

Groups, Theory of

Gunter, Edmund

Gyroscope and Gyrostat

Hachette, J. N. P.

Hamilton, Sir W. R.

Harmonic

Harmonic Analysis

Harriot, T.

Hero of Alexandria

Hodograph

Hutton, Charles

Huygens, Christiaan

Hydrodynamics

Hydromechanics

Hydrostatics

Hyperbola

Icosahedron

Inaudi, Jacques

Infinite

Infinitesimal Calculus

Interpolation

Inversion

Involution

Ivory, Sir James

Jacobi, Karl G. J.

Kelvin, William Thomson, 1st baron

Kinematics

Kinetics

Kircher, Athanasius

Knot

Kovalevsky, Sophie

Lagrange, Joseph L.

Landen, John

Laplace, P. S., de

Lardner, Dionysius

Legendre, Adrien Marie

Lemniscate

Leonardo of Pisa

Leslie, Sir John

Lever

Lie, Marius Sophus

Limaçon

Line

Lobachevskiy, N. I.

Locus

Logarithm

Logocyclic Curve, Strophoid or Foliate

MacCullagh, James

Maclaurin, Colin

Magic Square

Map

Mascheroni, Lorenzo

Mathematics

Matter

Maupertuis, Pierre de

Maxima and Minima

Maxwell, J. Clerk

Mechanics

Mensuration

Mersenne, Marin

Monge, Gaspard

Montucia, Jean-Étienne

Motion, Laws of

Murphy, Robert

Napier, John

Newton, Sir Isaac

Nicomachus of Gerasa

Number

Numbers, Partition of

Numeral

Octahedron

Ordinate

Oughtred, William

Oval

Pantograph

Pappus of Alexandria

322Parabola

Peacock, George

Peirce, Benjamin

Pell, John

Perpetual Motion

Perspective

Pfaff, J. F.

Playfair, John

Plücker, Julius

Poinsot, Louis

Poisson, Siméon Denis

Polygon

Polygonal Numbers

Polyhedral Numbers

Polyhedron

Poncelet, Jean Victor

Porism

Price, Bartholomew

Prism

Probability

Projection

Quadratrix

Quaternions

Recorde, Robert

Riccati, J. F., count

Riemann, G. F. B.

Roberval, G. P. de

Robins, Benjamin

Roulette

Routh, Edward John

Russell, John Scott

Salmon, George

Saunderson, N.

Serenus of Antissa

Series

Serpentine

Simpson, Thomas

Simson, Robert

Smith, H. J. S.

Smith, Robert

Snell, Willebrord

Sphere

Spherical Harmonics

Spheroid

Spiral

Spottiswoode, W.

Statics

Steiner, Jakob

Stevinus, Simon

Stirling, James

Stokes, Sir George G.

Strength of Materials

Sturm, J. C. F.

Surface

Sylvester, J. J.

Table, Mathematical

Tait, Peter G.

Tartaglia, Niccolo

Taylor, Brook

Tetrahedron

Theodosius of Tripolis

Thompson, T. P.

Tide

Todhunter, Isaac

Triangle

Trigonometry

Trisectrix

Units, Dimensions of

Units, Physical

Variations, Calculus of

Vector Analysis

Vernier, Pierre

Vieta (or Viète), F.

Wallace, William

Wallis, John

Wave

Witch of Agnesi

Zero