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 the 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. [Sidenote: 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. [Sidenote: 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). [Sidenote: 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 mathematicians 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. J. 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. A. 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 of 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 Parabola 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