A Short Account of the History of Mathematics · W. W. Rouse Ball
Among the more important of his other researches are those on the legitimate use of imaginary quantities; the determination of the number of real and imaginary roots of any algebraic equation within a given contour; his method of calculating these roots approximately; his theory of the symmetric functions of the coefficients of equations of any degree; his a priori valuation of a quantity less than the least difference between the roots of an equation; his papers on determinants in 1841, which assisted in bringing them into general use; and his investigations on the theory of numbers. Cauchy also did something to reduce the art of determining definite integrals to a science; the rule for finding the principal values of integrals was enunciated by him. The calculus of residues was his invention. His proof of Taylor's theorem seems to have originated from a discussion of the double periodicity of elliptic functions. The means of showing a connection between different branches of a subject by giving complex values to independent variables is largely due to him.
He also gave a direct analytical method for determining planetary inequalities of long period. To physics he contributed memoirs on waves and on the quantity of light reflected from the surfaces of metals, as well as other papers on optics.
Argand. I may mention here the name of Jean Robert Argand, who was born at Geneva on July 18, 1768, and died at Paris on August 13, 1822. In his Essai, issued in 1806, he gave a geometrical representation of a complex number, and applied it to show that every algebraic equation has a root. This was prior to the memoirs of Gauss and Cauchy on the same subject, but the essay did not attract much attention when it was first published. An even earlier demonstration that sqrt(-1) may be interpreted to indicate perpendicularity in two-dimensional space, and even the extension of the idea to three-dimensional space by a method foreshadowing the use of quaternions, had been given in a memoir by C. Wessel, presented to the Copenhagen Academy of Sciences in March 1797; other memoirs on the same subject had been published in the Philosophical Transactions for 1806, and by H. Kühn in the Transactions for 1750 of the Petrograd Academy.
I have already said that the idea of a simple complex number like a + bi where i^2 = -1 was extended by Kummer. The general theory has been discussed by K. Weierstrass, H. A. Schwarz of Berlin, J. W. R. Dedekind, H. Poincaré, and other writers.
Hamilton. In the opinion of some writers the theory of quaternions will be ultimately esteemed one of the great discoveries of the nineteenth century in pure mathematics. That discovery is due to Sir William Rowan Hamilton, who was born in Dublin on August 4, 1805, and died there on September 2, 1865. His education, which was carried on at home, seems to have been singularly discursive. Under the influence of an uncle who was a good linguist, he first devoted himself to linguistic studies; by the time he was seven he could read Latin, Greek, French, and German with facility; and when thirteen he was able to boast that he was familiar with as many languages as he had lived years. It was about this time that he came across a copy of Newton's Universal Arithmetic. This was his introduction to modern analysis, and he soon mastered the elements of analytical geometry and the calculus. He next read the Principia and the four volumes then published of Laplace's Mécanique céleste. In the latter he detected a mistake, and his paper on the subject, written in 1823, attracted considerable attention. In the following year he entered at Trinity College, Dublin. His university career is unique, for the chair of Astronomy becoming vacant in 1827, while he was yet an undergraduate, he was asked by the electors to stand for it, and was elected unanimously, it being understood that he should be left free to pursue his own line of study.
His earliest paper on optics, begun in 1823, was published in 1828 under the title of a Theory of Systems of Rays, to which two supplements were afterwards added; in the latter of these the phenomenon of conical refraction is predicted. This was followed by a paper in 1827 on the principle of Varying Action, and in 1834 and 1835 by memoirs on a General Method in Dynamics--the subject of theoretical dynamics being properly treated as a branch of pure mathematics. His lectures on Quaternions were published in 1852. Some of his results on this subject would seem to have been previously discovered by Gauss, but these were unknown and unpublished until long after Hamilton's death. Amongst his other papers, I may specially mention one on the form of the solution of the general algebraic equation of the fifth degree, which confirmed Abel's conclusion that it cannot be expressed by a finite number of purely algebraical expressions; one on fluctuating functions; one on the hodograph; and, lastly, one on the numerical solution of differential equations. His Elements of Quaternions was issued in 1866: of this a competent authority says that the methods of analysis there given show as great an advance over those of analytical geometry, as the latter showed over those of Euclidean geometry. In more recent times the subject has been further developed by P. G. Tait (1831--1901) of Edinburgh, by A. Macfarlane of America, and by C. J. Joly in his Manual of Quaternions, London, 1905.
Hamilton was painfully fastidious on what he published, and he left a large collection of manuscripts which are now in the library of Trinity College, Dublin, some of which it is to be hoped will be ultimately printed.
Grassmann. The idea of non-commutative algebras and of quaternions seems to have occurred to Grassmann and Boole at about the same time as to Hamilton. Hermann Günther Grassmann was born in Stettin on April 15, 1809, and died there in 1877. He was professor at the gymnasium at Stettin. His researches on non-commutative algebras are contained in his Ausdehnungslehre, first published in 1844 and enlarged in 1862. This work has had great influence, especially on the continent, where Grassmann's methods have generally been followed in preference to Hamilton's. Grassmann's researches have been continued and extended, notably by S. F. V. Schlegel and G. Peano.
The scientific treatment of the fundamental principles of algebra initiated by Hamilton and Grassmann was continued by De Morgan and Boole in England, and was further developed by H. Hankel (1839--1873) in Germany in his work on complexes, 1867, and, on somewhat different lines, by G. Cantor in his memoirs on the theory of irrationals, 1871; the discussion is, however, so technical that I am unable to do more than allude to it. Of Boole and De Morgan I say a word or two in passing.
Boole. George Boole, born at Lincoln on November 2, 1815, and died at Cork on December 8, 1864, independently invented a system of non-commutative algebra, and was one of the creators of symbolic or mathematical logic. From his memoirs on linear transformations part of the theory of invariants has developed. His Finite Differences remains a standard work on that subject.
De Morgan. Augustus de Morgan, born in Madura (Madras) in June 1806, and died in London on March 18, 1871, was educated at Trinity College, Cambridge. In 1828 he became professor at the then newly-established University of London (University College). There, through his works and pupils, he exercised a wide influence on English mathematicians. He was deeply read in the philosophy and history of mathematics, but the results are given in scattered articles; of these I have made considerable use in this book. His memoirs on the foundation of algebra; his treatise on the differential calculus published in 1842, a work of great ability, and noticeable for his treatment of infinite series; and his articles on the calculus of functions and on the theory of probabilities, are worthy of special note. The article on the calculus of functions contains an investigation of the principles of symbolic reasoning, but the applications deal with the solution of functional equations rather than with the general theory of functions.
Galois. A new development of algebra--the theory of groups of substitutions--was suggested by Evariste Galois, who promised to be one of the most original mathematicians of the nineteenth century, born at Paris on October 26, 1811, and killed in a duel on May 30, 1832, at the early age of 20.
The theory of groups, and of invariant subgroups, has profoundly modified the treatment of the theory of equations. An immense literature has grown up on the subject. The modern theory of groups originated with the treatment by Galois, Cauchy, and J. A. Serret (1819--1885), professor at Paris; their work is mainly concerned with finite discontinuous substitution groups. This line of investigation has been pursued by M. E. C. Jordan (1838--1922) of Paris and E. Netto of Strassburg. The problem of operations with discontinuous groups, with applications to the theory of functions, has been further taken up by (among others) F. G. Frobenius of Berlin, F. C. Klein of Göttingen, and W. Burnside formerly of Cambridge and now of Greenwich.
Cayley. Another Englishman whom we may reckon among the great mathematicians of this prolific century was Arthur Cayley. Cayley was born in Surrey, on Aug. 16, 1821, and after education at Trinity College, Cambridge, was called to the bar. But his interests centred on mathematics; in 1863 he was elected Sadlerian Professor at Cambridge, and he spent there the rest of his life. He died on Jan. 26, 1895.
Cayley's writings deal with considerable parts of modern pure mathematics. I have already mentioned his writings on the partition of numbers and on elliptic functions treated from Jacobi's point of view; his later writings on elliptic functions dealt mainly with the theory of transformation and the modular equation. It is, however, by his investigations in analytical geometry and on higher algebra that he will be best remembered.
In analytical geometry the conception of what is called (perhaps, not very happily) the absolute is due to Cayley. As stated by himself, the "theory, in effect, is that the metrical properties of a figure are not the properties of the figure considered per se but its properties when considered in connection with another figure, namely, the conic termed the absolute"; hence metric properties can be subjected to descriptive treatment. He contributed largely to the general theory of curves and surfaces, his work resting on the assumption of the necessarily close connection between algebraical and geometrical operations.
In higher algebra the theory of invariants is due to Cayley; his ten classical memoirs on binary and ternary forms, and his researches on matrices and non-commutative algebras, mark an epoch in the development of the subject.
Sylvester. Another teacher of the same time was James Joseph Sylvester, born in London on Sept. 3, 1814, and died on March 15, 1897. He too was educated at Cambridge. Like Cayley, with whom he was on intimate terms of friendship, he was called to the bar, and yet preserved all his interests in mathematics. He held professorships successively at Woolwich, Baltimore, and Oxford. He had a strong personality and was a stimulating teacher, but it is difficult to describe his writings, for they are numerous, disconnected, and discursive.
On the theory of numbers Sylvester wrote valuable papers on the distribution of primes and on the partition of numbers. On analysis he wrote on the calculus and on differential equations. But perhaps his favourite study was higher algebra, and from his numerous memoirs on this subject I may in particular single out those on canonical forms, on the theory of contravariants, on reciprocants or differential invariants, and on the theory of equations, notably on Newton's rule. I may also add that he created the language and notation of considerable parts of those subjects on which he wrote.
The writings of Cayley and Sylvester stand in marked contrast: Cayley's are methodical, precise, formal, and complete; Sylvester's are impetuous, unfinished, but none the less vigorous and stimulating. Both mathematicians found the greatest attraction in higher algebra, and to both that subject in its modern form is deeply indebted.
Lie. Among the great analysts of the nineteenth century to whom I must allude here, is Marius Sophus Lie, born on Dec. 12, 1842, and died on Feb. 18, 1899. Lie was educated at Christiania, whence he obtained a travelling scholarship, and in the course of his journeys made the acquaintance of Klein, Darboux, and Jordan, to whose influence his subsequent career is largely due.
In 1870 he discovered the transformation by which a sphere can be made to correspond to a straight line, and, by the use of which theorems on aggregates of lines can be translated into theorems on aggregates of spheres. This was followed by a thesis on the theory of tangential transformations for space.
In 1872 he became professor at Christiania. His earliest researches here were on the relations between differential equations and infinitesimal transformations. This naturally led him to the general theory of finite continuous groups of substitutions; the results of his investigations on this subject are embodied in his Theorie der Transformationsgruppen, Leipzig, three volumes, 1888--1893. He proceeded next to consider the theory of infinite continuous groups, and his conclusions, edited by G. Scheffers, were published in 1893. About 1879 Lie turned his attention to differential geometry; a systematic exposition of this is in course of issue in his Geometrie der Berührungstransformationen.
Lie seems to have been disappointed and soured by the absence of any general recognition of the value of his results. Reputation came, but it came slowly. In 1886 he moved to Leipzig, and in 1898 back to Christiania, where a post had been created for him. He brooded, however, over what he deemed was the undue neglect of the past, and the happiness of the last decade of his life was much affected by it.
Hermite. Another great algebraist of the century was Charles Hermite, born in Lorraine on December 24, 1822, and died at Paris, January 14, 1901. From 1869 he was professor at the Sorbonne, and through his pupils exercised a profound influence on the mathematicians of to-day.
While yet a student he wrote to Jacobi on Abelian functions, and the latter embodied the results in his works. Hermite's earlier papers were largely on the transformation of these functions, a problem which he finally effected by the use of modular functions. He applied elliptic functions to find solutions of the quintic equation and of Lamé's differential equation.
Later he took up the subject of algebraic continued fractions, and this led to his celebrated proof, given in 1873, that e cannot be the root of an algebraic equation, from which it follows that e is a transcendental number. F. Lindemann showed in a similar way in 1882 that pi is transcendental. The proofs have been subsequently improved and simplified by K. Weierstrass, D. Hilbert, and F. C. Klein.
To the end of his life Hermite maintained his creative interest in the subjects of the integral calculus and the theory of functions. He also discussed the theory of associated covariants in binary quantics and the theory of ternary quantics.
So many other writers have treated the subject of Higher Algebra (including therein the theory of forms and the theory of equations) that it is difficult to summarise their conclusions.
The convergency of series has been discussed by J. L. Raabe (1801--1859) of Zürich, J. L. F. Bertrand, the secretary of the French Academy; E. E. Kummer of Berlin; U. Dini of Pisa; A. Pringsheim of Munich; and Sir George Gabriel Stokes (1819--1903) of Cambridge, to whom the well-known theorem on the critical values of the sums of periodic series is due. The last-named writer introduced the important conception of non-uniform convergence; a subject subsequently treated by P. L. Seidel.
Perhaps here, too, I may allude in passing to the work of G. F. B. Riemann, G. G. Stokes, H. Hankel, and G. Darboux on asymptotic expansions; of H. Poincaré on the application of such expansions to differential equations; and of E. Borel and E. Cesàro on divergent series.
On the theory of groups of substitutions I have already mentioned the work, on the one hand, of Galois, Cauchy, Serret, Jordan, and Netto, and, on the other hand, of Frobenius, Klein, and Burnside in connection with discontinuous groups, and that of Lie in connection with continuous groups.
I may also mention the following writers: C. W. Borchardt (1817--1880) of Berlin, who in particular discussed generating functions in the theory of equations, and arithmetic-geometric means. C. Hermite, to whose work I have alluded above. Enrico Betti of Pisa and F. Brioschi of Milan, both of whom discussed binary quantics; the latter applied hyperelliptic functions to give a general solution of a sextic equation. S. H. Aronhold (1819--1884) of Berlin, who developed symbolic methods in connection with the invariant theory of quantics. P. A. Gordan of Erlangen, who has written on the theory of equations, the theories of groups and forms, and shown that there are only a finite number of concomitants of quantics. R. F. A. Clebsch (1833--1872) of Göttingen, who independently investigated the theory of binary forms in some papers collected and published in 1871; he also wrote on Abelian functions. P. A. MacMahon, formerly an officer in the British army, who has written on the connection of symmetric functions, invariants and covariants, the concomitants of binary forms, and combinatory analysis. F. C. Klein of Göttingen, who, in addition to his researches, already mentioned, on functions and on finite discontinuous groups, has written on differential equations. A. R. Forsyth of Cambridge, who has developed the theory of invariants and the general theory of differential equations, ternariants, and quaternariants. P. Painlevé of Paris, who has written on the theory of differential equations. And, lastly, D. Hilbert of Göttingen, who has treated the theory of homogeneous forms.
No account of contemporary writings on higher algebra would be complete without a reference to the admirable Higher Algebra by G. Salmon (1819--1904), provost of Trinity College, Dublin, and the Cours d'algèbre supérieure by J. A. Serret, in which the chief discoveries of their respective authors are embodied. An admirable historical summary of the theory of the complex variable is given in the Vorlesungen über die complexen Zahlen, Leipzig, 1867, by H. Hankel, of Tübingen.
Analytical Geometry. It will be convenient next to call attention to another division of pure mathematics--analytical geometry--which has been greatly developed in recent years. It has been studied by a host of modern writers, but I do not propose to describe their investigations, and I shall content myself by merely mentioning the names of the following mathematicians.
James Booth (1806--1878) and James MacCullagh (1809--1846), both of Dublin, were two of the earliest British writers in this century to take up the subject of analytical geometry, but they worked mainly on lines already studied by others. Fresh developments were introduced by Julius Plücker (1801--1868) of Bonn, who devoted himself especially to the study of algebraic curves, of a geometry in which the line is the element in space, and to the theory of congruences and complexes; his equations connecting the singularities of curves are well known; in 1847 he exchanged his chair for one of physics, and subsequently gave up most of his time to researches on spectra and magnetism.
The majority of the memoirs on analytical geometry by A. Cayley and by Henry Smith deal with the theory of curves and surfaces; the most remarkable of those of L. O. Hesse (1811--1874) of Munich are on the plane geometry of curves; of those of J. G. Darboux of Paris are on the geometry of surfaces; of those of G. H. Halphen (1844--1889) of Paris are on the singularities of surfaces and on tortuous curves; and of those of P. O. Bonnet are on ruled surfaces, curvature, and torsion. The singularities of curves and surfaces have also been considered by H. G. Zeuthen of Copenhagen, and by H. C. H. Schubert of Hamburg. The theory of tortuous curves has been discussed by M. Nöther of Erlangen; and R. F. A. Clebsch of Göttingen has applied Abel's theorem to geometry.
Among more recent text-books on analytical geometry are J. G. Darboux's Théorie générale des surfaces, and Les Systèmes orthogonaux et les coordonnées curvilignes; R. F. A. Clebsch's Vorlesungen über Geometrie, edited by F. Lindemann; and G. Salmon's Conic Sections, Geometry of Three Dimensions, and Higher Plane Curves; in which the chief discoveries of these writers are embodied.
Plücker suggested in 1846 that the straight line should be taken as the element of space. This formed the subject of investigations by G. Battaglini (1826--1892) of Rome, F. C. Klein, and S. Lie. Recent works on it are R. Sturm's Die Gebilde ersten und zweiten Grades der Liniengeometrie, 3 volumes, Leipzig, 1892, 1893, 1896, and C. M. Jessop's Treatise on the Line Complex, Cambridge, 1903.
Finally, I may allude to the extension of the subject-matter of analytical geometry in the writings of A. Cayley in 1844, H. G. Grassmann in 1844 and 1862, G. F. B. Riemann in 1854, whose work was continued by G. Veronese of Padua, H. C. H. Schubert of Hamburg, C. Segre of Turin, G. Castelnuovo of Rome, and others, by the introduction of the idea of space of n dimensions.
Analysis. Among those who have extended the range of analysis (including the calculus and differential equations) or whom it is difficult to place in any of the preceding categories are the following, whom I mention in alphabetical order. P. E. Appell ADPPBiographies of Appell, Darboux, Picard, and Poincaré, with bibliographies, by E. Lebon, were issued in Paris in 1909, 1910. of Paris; J. L. F. Bertrand of Paris; G. Boole of Cork; A. L. Cauchy of Paris; J. G. Darboux ADPP of Paris; A. R. Forsyth of Cambridge; F. G. Frobenius of Berlin; J. Lazarus Fuchs (1833--1902) of Berlin; G. H. Halphen of Paris; C. G. J. Jacobi of Berlin; M. E. C. Jordan (1838--1922) of Paris; L. Königsberger of Heidelberg; Sophie Kowalevski (1850--1891) of Stockholm; M. S. Lie of Leipzig; E. Picard ADPP of Paris; H. Poincaré ADPP of Paris; G. F. B. Riemann of Göttingen; H. A. Schwarz of Berlin; J. J. Sylvester; and K. Weierstrass of Berlin, who developed the calculus of variations.
The subject of differential equations should perhaps have been separated and treated by itself. But it is so vast that it is difficult--indeed impossible--to describe recent researches in a single paragraph. It will perhaps suffice to refer to the admirable series of treatises, seven volumes, on the subject by A. R. Forsyth, which give a full presentation of the subjects treated.
A recent development on integral equations, or the inversion of a definite integral, has attracted considerable attention. It originated in a single instance given by Abel, and has been treated by V. Volterra of Rome, J. Fredholm of Stockholm, D. Hilbert of Göttingen, and numerous other recent writers.
Synthetic Geometry. The writers I have mentioned above mostly concerned themselves with analysis. I will next describe some of the more important works produced in this century on synthetic geometry.
Modern synthetic geometry may be said to have had its origin in the works of Monge in 1800, Carnot in 1803, and Poncelet in 1822, but these only foreshadowed the great extension it was to receive in Germany, of which Steiner and von Staudt are perhaps the best known exponents.
Steiner. Jacob Steiner, "the greatest geometrician since the time of Apollonius," was born at Utzensdorf on March 18, 1796, and died at Bern on April 1, 1863. His father was a peasant, and the boy had no opportunity to learn reading and writing till the age of fourteen. He subsequently went to Heidelberg and thence to Berlin, supporting himself by giving lessons. His Systematische Entwickelungen was published in 1832, and at once made his reputation: it contains a full discussion of the principle of duality, and of the projective and homographic relations of rows, pencils, &c., based on metrical properties. By the influence of Crelle, Jacobi, and the von Humboldts, who were impressed by the power of this work, a chair of geometry was created for Steiner at Berlin, and he continued to occupy it till his death. The most important of his other researches are contained in papers which appeared in Crelle's Journal: these relate chiefly to properties of algebraic curves and surfaces, pedals and roulettes, and maxima and minima: the discussion is purely geometrical. Steiner's works may be considered as the classical authority on recent synthetic geometry.
Von Staudt. A system of pure geometry, quite distinct from that expounded by Steiner, was proposed by Karl Georg Christian von Staudt, born at Rothenburg on Jan. 24, 1798, and died in 1867, who held the chair of mathematics at Erlangen. In his Geometrie der Lage, published in 1847, he constructed a system of geometry built up without any reference to number or magnitude, but, in spite of its abstract form, he succeeded by means of it alone in establishing the non-metrical projective properties of figures, discussed imaginary points, lines, and planes, and even obtained a geometrical definition of a number: these views were further elaborated in his Beiträge zur Geometrie der Lage, 1856--1860. This geometry is curious and brilliant, and has been used by Culmann as the basis of his graphical statics.
As usual text-books on synthetic geometry I may mention M. Chasles's Traité de géométrie supérieure, 1852; J. Steiner's Vorlesungen über synthetische Geometrie, 1867; L. Cremona's Éléments de géométrie projective, English translation by C. Leudesdorf, Oxford, second edition, 1893; and Th. Reye's Geometrie der Lage, Hanover, 1866--1868, English translation by T. F. Holgate, New York, part i, 1898. A good presentation of the modern treatment of pure geometry is contained in the Introduzione ad una teoria geometrica delle curve piane, 1862, and its continuation Preliminari di una teoria geometrica delle superficie by Luigi Cremona (1830--1903): his collected works, in three volumes, may be also consulted.
The differences in ideas and methods formerly observed in analytic and synthetic geometries tend to disappear with their further development.
Non-Euclidean Geometry. Here I may fitly add a few words on recent investigations on the foundations of geometry.
The question of the truth of the assumptions usually made in our geometry had been considered by J. Saccheri as long ago as 1733; and in more recent times had been discussed by N. I. Lobatschewsky (1793--1856) of Kasan, in 1826 and again in 1840; by Gauss, perhaps as early as 1792, certainly in 1831 and in 1846; and by J. Bolyai (1802--1860) in 1832 in the appendix to the first volume of his father's Tentamen, but Riemann's memoir of 1854 attracted general attention to the subject of non-Euclidean geometry, and the theory has been since extended and simplified by various writers, notably by A. Cayley of Cambridge, E. Beltrami (1835--1900) of Pavia, by H. L. F. von Helmholtz (1821--1894) of Berlin, by S. P. Tannery (1843--1904) of Paris, by F. C. Klein of Göttingen, and by A. N. Whitehead of Cambridge in his Universal Algebra. The subject is so technical that I confine myself to a bare sketch of the argument from which the idea is derived.
The Euclidean system of geometry, with which alone most people are acquainted, rests on a number of independent axioms and postulates. Those which are necessary for Euclid's geometry have, within recent years, been investigated and scheduled. They include not only those explicitly given by him, but some others which he unconsciously used. If these are varied, or other axioms are assumed, we get a different series of propositions, and any consistent body of such propositions constitutes a system of geometry. Hence there is no limit to the number of possible Non-Euclidean geometries that can be constructed.
Among Euclid's axioms and postulates is one on parallel lines, which is usually stated in the form that if a straight line meets two straight lines, so as to make the sum of the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles. Expressed in this form the axiom is far from obvious, and from early times numerous attempts have been made to prove it. All such attempts failed, and it is now known that the axiom cannot be deduced from the other axioms assumed by Euclid.
The earliest conception of a body of Non-Euclidean geometry was due to the discovery, made independently by Saccheri, Lobatschewsky, and John Bolyai, that a consistent system of geometry of two dimensions can be produced on the assumption that the axiom on parallels is not true, and that through a point a number of straight (that is, geodetic) lines can be drawn parallel to a given straight line. The resulting geometry is called hyperbolic.
Riemann later distinguished between boundlessness of space and its infinity, and showed that another consistent system of geometry of two dimensions can be constructed in which all straight lines are of a finite length, so that a particle moving along a straight line will return to its original position. This leads to a geometry of two dimensions, called elliptic geometry, analogous to the hyperbolic geometry, but characterised by the fact that through a point no straight line can be drawn which, if produced far enough, will not meet any other given straight line. This can be compared with the geometry of figures drawn on the surface of a sphere.
Thus according as no straight line, or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line, we have three systems of geometry of two dimensions known respectively as elliptic, parabolic or homaloidal or Euclidean, and hyperbolic.
In the parabolic and hyperbolic systems straight lines are infinitely long. In the elliptic they are finite. In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle. In the elliptic system all straight lines are of the same finite length; any two lines intersect; and the sum of the angles of a triangle is greater than two right angles.
In spite of these and other peculiarities of hyperbolic and elliptical geometries, it is impossible to prove by observation that one of them is not true of the space in which we live. For in measurements in each of these geometries we must have a unit of distance; and if we live in a space whose properties are those of either of these geometries, and such that the greatest distances with which we are acquainted (ex. gr. the distances of the fixed stars) are immensely smaller than any unit, natural to the system, then it may be impossible for us by our observations to detect the discrepancies between the three geometries. It might indeed be possible by observations of the parallaxes of stars to prove that the parabolic system and either the hyperbolic or elliptic system were false, but never can it be proved by measurements that Euclidean geometry is true. Similar difficulties might arise in connection with excessively minute quantities. In short, though the results of Euclidean geometry are more exact than present experiments can verify for finite things, such as those with which we have to deal, yet for much larger things or much smaller things or for parts of space at present inaccessible to us they may not be true.
Other systems of Non-Euclidean geometry might be constructed by changing other axioms and assumptions made by Euclid. Some of these are interesting, but those mentioned above have a special importance from the somewhat sensational fact that they lead to no results inconsistent with the properties of the space in which we live.
We might also approach the subject by remarking that in order that a space of two dimensions should have the geometrical properties with which we are familiar, it is necessary that it should be possible at any place to construct a figure congruent to a given figure; and this is so only if the product of the principal radii of curvature at every point of the space or surface be constant. This product is constant in the case (i) of spherical surfaces, where it is positive; (ii) of plane surfaces (which lead to Euclidean geometry), where it is zero; and (iii) of pseudo-spherical surfaces, where it is negative. A tractroid is an instance of a pseudo-spherical surface; it is saddle-shaped at every point. Hence on spheres, planes, and tractroids we can construct normal systems of geometry. These systems are respectively examples of elliptic, Euclidean, and hyperbolic geometries. Moreover, if any surface be bent without dilation or contraction, the measure of curvature remains unaltered. Thus these three species of surfaces are types of three kinds on which congruent figures can be constructed. For instance a plane can be rolled into a cone, and the system of geometry on a conical surface is similar to that on a plane.
In the preceding sketch of the foundations of Non-Euclidean geometry I have assumed tacitly that the measure of a distance remains the same everywhere.
The above refers only to hyper-space of two dimensions. Naturally there arises the question whether there are different kinds of hyper-space of three or more dimensions. Riemann showed that there are three kinds of hyper-space of three dimensions having properties analogous to the three kinds of hyper-space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surfaces, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface; a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface.
Foundations of Mathematics. Assumptions made in the Subject. The discussion on the Non-Euclidean geometry brought into prominence the logical foundations of the subject. The questions of the principles of and underlying assumptions made in mathematics have been discussed of late by J. W. R. Dedekind of Brunswick, G. Cantor of Halle, G. Frege of Jena, G. Peano of Turin, the Hon. B. A. W. Russell and A. N. Whitehead, both of Cambridge.
Kinematics. The theory of kinematics, that is, the investigation of the properties of motion, displacement, and deformation, considered independently of force, mass, and other physical conceptions, has been treated by various writers. It is a branch of pure mathematics, and forms a fitting introduction to the study of natural philosophy. Here I do no more than allude to it.
I shall conclude the chapter with a few notes--more or less discursive--on branches of mathematics of a less abstract character and concerned with problems that occur in nature. I commence by mentioning the subject of Mechanics. The subject may be treated graphically or analytically.
Graphics. In the science of graphics rules are laid down for solving various problems by the aid of the drawing-board: the modes of calculation which are permissible are considered in modern projective geometry, and the subject is closely connected with that of modern geometry. This method of attacking questions has been hitherto applied chiefly to problems in mechanics, elasticity, and electricity; it is especially useful in engineering, and in that subject an average draughtsman ought to be able to obtain approximate solutions of most of the equations, differential or otherwise, with which he is likely to be concerned, which will not involve errors greater than would have to be allowed for in any case in consequence of our imperfect knowledge of the structure of the materials employed.
The theory may be said to have originated with Poncelet's work, but I believe that it is only within the last twenty years that systematic expositions of it have been published. Among the best known of such works I may mention the Graphische Statik, by C. Culmann, Zürich, 1875, recently edited by W. Ritter; the Lezioni di statica grafica, by A. Favaro, Padua, 1877 (French translation annotated by P. Terrier in 2 volumes, 1879--85); the Calcolo grafico, by L. Cremona, Milan, 1879 (English translation by T. H. Beare, Oxford, 1889), which is largely founded on Möbius's work; La statique graphique, by M. Levy, Paris, 4 volumes, 1886--88; and La statica grafica, by C. Sairotti, Milan, 1888.
The general character of these books will be sufficiently illustrated by the following note on the contents of Culmann's work. Culmann commences with a description of the geometrical representation of the four fundamental processes of addition, subtraction, multiplication, and division; and proceeds to evolution and involution, the latter being effected by the use of the equiangular spiral. He next shows how the quantities considered--such as volumes, moments, and moments of inertia--may be represented by straight lines; thence deduces the laws for combining forces, couples, &c.; and then explains the construction and use of the ellipse and ellipsoid of inertia, the neutral axis, and the kern; the remaining and larger part of the book is devoted to showing how geometrical drawings, made on these principles, give the solutions of many practical problems connected with arches, bridges, frameworks, earth pressure on walls and tunnels, &c.
The subject has been treated during the last twenty years by numerous writers, especially in Italy and Germany, and applied to a large number of problems. But as I stated at the beginning of this chapter that I should as far as possible avoid discussion of the works of living authors I content myself with a bare mention of the subject.
Analytical Mechanics. I next turn to the question of mechanics treated analytically. The knowledge of mathematical mechanics of solids attained by the great mathematicians of the last century may be said to be summed up in the admirable Mécanique analytique by Lagrange and Traité de mécanique by Poisson, and the application of the results to astronomy forms the subject of Laplace's Mécanique céleste. These works have been already described. The mechanics of fluids is more difficult than that of solids and the theory is less advanced.
Theoretical Statics, especially the theory of the potential and attractions, has received considerable attention from the mathematicians of this century.
I have previously mentioned that the introduction of the idea of the potential is due to Lagrange, and it occurs in a memoir of a date as early as 1773. The idea was at once grasped by Laplace, who, in his memoir of 1784, used it freely and to whom the credit of the invention was formerly, somewhat unjustly, attributed. In the same memoir Laplace also extended the idea of zonal harmonic analysis which had been introduced by Legendre in 1783. Of Gauss's work on attractions I have already spoken. The theory of level surfaces and lines of force is largely due to Chasles, who also determined the attraction of an ellipsoid at any external point. I may also here mention the Barycentrisches Calcul, published in 1826 by A. F. Möbius (1790--1868), who was one of the best known of Gauss's pupils. Attention must also be called to the important memoir, published in 1828, on the potential and its properties, by G. Green (1793--1841) of Cambridge. Similar results were independently established, in 1839, by Gauss, to whom their general dissemination was due.
Theoretical Dynamics, which was cast into its modern form by Jacobi, has been studied by most of the writers above mentioned. I may also here repeat that the principle of "Varying Action" was elaborated by Sir William Hamilton in 1827, and the "Hamiltonian equations" were given in 1835; and I may further call attention to the dynamical investigations of J. E. E. Bour (1832--1866), of Liouville, and of J. L. F. Bertrand, all of Paris. The use of generalised co-ordinates, introduced by Lagrange, has now become the customary means of attacking dynamical (as well as many physical) problems.
As usual text-books I may mention those on particle and rigid dynamics by E. J. Routh, Cambridge; Leçons sur l'intégration des équations différentielles de la mécanique by P. Painlevé, Paris, 1895; Intégration des équations de la mécanique by J. Graindorge, Brussels, 1889; and C. E. Appell's Traité de mécanique rationnelle, Paris, 2 vols., 1892, 1896. Allusion to the treatise on Natural Philosophy by Sir William Thomson (later known as Lord Kelvin) of Glasgow, and P. G. Tait of Edinburgh, may be also here made.
On the mechanics of fluids, liquids, and gases, apart from the physical theories on which they rest, I propose to say nothing, except to refer to the memoirs of Green, Sir George Stokes, Lord Kelvin, and von Helmholtz. The fascinating but difficult theory of vortex rings is due to the two writers last mentioned. One problem in it has been also considered by Sir J. J. Thomson, of Cambridge, but it is a subject which is as yet beyond our powers of analysis. The subject of sound may be treated in connection with hydrodynamics, but on this I would refer the reader who wishes for further information to the work first published at Cambridge in 1877 by Lord Rayleigh.
Theoretical Astronomy is included in, or at any rate closely connected with, theoretical dynamics. Among those who in this century have devoted themselves to the study of theoretical astronomy the name of Gauss is one of the most prominent; to his work I have already alluded.
Bessel. The best known of Gauss's contemporaries was Friedrich Wilhelm Bessel, who was born at Minden on July 22, 1784, and died at Königsberg on March 17, 1846. Bessel commenced his life as a clerk on board ship, but in 1806 he became an assistant in the observatory at Lilienthal, and was thence in 1810 promoted to be director of the new Prussian Observatory at Königsberg, where he continued to live during the remainder of his life. Bessel introduced into pure mathematics those functions which are now called by his name (this was in 1824, though their use is indicated in a memoir seven years earlier); but his most notable achievements were the reduction (given in his Fundamenta Astronomiae, Königsberg, 1818) of the Greenwich observations by Bradley of 3222 stars, and his determination of the annual parallax of 61 Cygni. Bradley's observations have been recently reduced again by A. Auwers of Berlin.
Leverrier. Among the astronomical events of this century the discovery of the planet Neptune by Leverrier and Adams is one of the most striking. Urbain Jean Joseph Leverrier, the son of a petty Government employé in Normandy, was born at St. Lô on March 11, 1811, and died at Paris on September 23, 1877. He was educated at the Polytechnic school, and in 1837 was appointed as lecturer on astronomy there. His earliest researches in astronomy were communicated to the Academy in 1839: in these he calculated, within much narrower limits than Laplace had done, the extent within which the inclinations and eccentricities of the planetary orbits vary. The independent discovery in 1846 by Leverrier and Adams of the planet Neptune by means of the disturbance it produced on the orbit of Uranus attracted general attention to physical astronomy, and strengthened the opinion as to the universality of gravity. In 1855 Leverrier succeeded Arago as director of the Paris observatory, and reorganised it in accordance with the requirements of modern astronomy. Leverrier now set himself the task of discussing the theoretical investigations of the planetary motions and of revising all tables which involved them. He lived just long enough to sign the last proof-sheet of this work.
Adams. The co-discoverer of Neptune was John Couch Adams, who was born in Cornwall on June 5, 1819, educated at St. John's College, Cambridge, subsequently appointed Lowndean professor in the University, and director of the Observatory, and who died at Cambridge on January 21, 1892.
There are three important problems which are specially associated with the name of Adams. The first of these is his discovery of the planet Neptune from the perturbations it produced on the orbit of Uranus: in point of time this was slightly earlier than Leverrier's investigation.
The second is his memoir of 1855 on the secular acceleration of the moon's mean motion. Laplace had calculated this on the hypothesis that it was caused by the eccentricity of the earth's orbit, and had obtained a result which agreed substantially with the value deduced from a comparison of the records of ancient and modern eclipses. Adams shewed that certain terms in an expression had been neglected, and that if they were taken into account the result was only about one-half that found by Laplace. The results agreed with those obtained later by Delaunay in France and Cayley in England, but their correctness has been questioned by Plana, Pontécoulant, and other continental astronomers. The point is not yet definitely settled.
The third investigation connected with the name of Adams, is his determination in 1867 of the orbit of the Leonids or shooting stars which were especially conspicuous in November, 1866, and whose period is about thirty-three years. H. A. Newton (1830--1896) of Yale, had shewn that there were only five possible orbits. Adams calculated the disturbance which would be produced by the planets on the motion of the node of the orbit of a swarm of meteors in each of these cases, and found that this disturbance agreed with observation for one of the possible orbits, but for none of the others. Hence the orbit was known.
Other well-known astronomers of this century are G. A. A. Plana (1781--1864), whose work on the motion of the moon was published in 1832; Count P. G. D. Pontécoulant (1795--1871); C. E. Delaunay (1816--1872), whose work on the lunar theory indicates the best method yet suggested for the analytical investigations of the whole problem, and whose (incomplete) lunar tables are among the astronomical achievements of this century; P. A. Hansen (1795--1874), head of the observatory at Gotha, who compiled the lunar tables published in London in 1857 which are still used in the preparation of the Nautical Almanack, and elaborated the methods employed for the determination of lunar and planetary perturbations; F. F. Tisserand (1845--1896) of Paris, whose Mécanique céleste is now a standard authority on dynamical astronomy; and Simon Newcomb (1835--1909), superintendent of the American Ephemeris, who re-examined the Greenwich observations from the earliest times, applied the results to the lunar theory, and revised Hansen's tables.
Other notable work is associated with the names of Hill, Darwin, and Poincaré. G. W. Hill, until recently on the staff of the American Ephemeris, determined the inequalities of the moon's motion due to the non-spherical figure of the earth--an investigation which completed Delaunay's lunar theory. Hill also dealt with the secular motion of the moon's perigee and the motion of a planet's perigee under certain conditions; and wrote on the analytical theory of the motion of Jupiter and Saturn, with a view to the preparation of tables of their positions at any given time. Sir G. H. Darwin (1845--1912), of Cambridge, wrote on the effect of tides on viscous spheroids, the development of planetary systems by means of tidal friction, the mechanics of meteoric swarms, and the possibility of pear-shaped planetary figures. H. Poincaré (1854--1912), of Paris, discussed the difficult problem of three bodies, and the form assumed by a mass of fluid under its own attraction, and is the author of an admirable treatise, the Mécanique céleste, three volumes. The treatise on the lunar theory by E. W. Brown, Cambridge, 1896; his memoir on Inequalities in the Motion of the Moon due to Planetary Action, Cambridge, 1908; and a report (printed in the Report of the British Association, London, 1899, vol. lxix, pp. 121--159) by E. T. Whittaker on researches connected with the solution of the problem of three bodies, contain valuable accounts of recent progress in the lunar and planetary theories.
Within the last half century the results of spectrum analysis have been applied to determine the constitution of the heavenly bodies, and their directions of motions to and from the earth. The early history of spectrum analysis will be always associated with the names of G. R. Kirchhoff (1824--1887) of Berlin, of A. J. Ångström (1814--1874) of Upsala, and of George G. Stokes of Cambridge, but it pertains to optics rather than to astronomy. How unexpected was the application to astronomy is illustrated by the fact that A. Comte in 1842, when discussing the study of nature, regretted the waste of time due to some astronomers paying attention to the fixed stars, since, he said, nothing could possibly be learnt about them; and indeed a century ago it would have seemed incredible that we could investigate the chemical constitution of worlds in distant space.
During the last few years the range of astronomy has been still further extended by the art of photography. To what new results this may lead it is as yet impossible to say. In particular we have been thus enabled to trace the forms of gigantic spiral nebulae which seem to be the early stages of vast systems now in process of development.
The constitution of the universe, in which the solar system is but an insignificant atom, has long attracted the attention of thoughtful astronomers, and noticeably was studied by William Herschel. Recently J. C. Kapteyn of Groningen has been able to shew that all the stars whose proper motions can be detected belong to one or other of two streams moving in different directions, one with a velocity about three times as great as the other. The solar system is in the slower stream. These results have been confirmed by A. S. Eddington and F. W. Dyson. It would appear likely that we are on the threshold of wide-reaching discoveries about the constitution of the visible universe.
Mathematical Physics. An account of the history of mathematics and allied sciences in the last century would be misleading if there were no reference to the application of mathematics to numerous problems in heat, elasticity, light, electricity, and other physical subjects. The history of mathematical physics is, however, so extensive that I could not pretend to do it justice, even were its consideration properly included in a history of mathematics. At any rate I consider it outside the limits I have laid down for myself in this chapter. I abandon its discussion with regret because the Cambridge school has played a prominent part in its development, as witness (to mention only three or four of those concerned) the names of Sir George G. Stokes, professor from 1849 to 1903, Lord Kelvin, J. Clerk Maxwell (1831--1879), professor from 1871 to 1879, Lord Rayleigh, professor from 1879 to 1884, Sir J. J. Thomson, professor from 1884, Sir Joseph Larmor, professor from 1903, and Sir Ernest Rutherford, professor from 1919.
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