BiBuoGRAplly.—The history of such special topics as algebra, arith metic, geometry, trigonometry, calculus, functions, and number will be found treated as extensively as the space permits under these several heads. In addition to these articles the reader will find it desirable to consult the following: M. Cantor, V orlesungen fiber Geschichte der Mathematik (Leipzig, Teubner, 1880-19o8), with various re visions. (The standard treatise for scholars. In consulting the work, see the numerous suggestions for change or correction in each number of the Bibliotheca Mathematica, 3rd series.) Sir T. L. Heath, History of Greek Mathematics (Cambridge, 1921). (The leading work on the subject.) J. C. Poggendorff, Handworterbuch zur Geschichte der exacten Wissenschaften (Leipzig, 1858-1926). (The standard work on mathematical biography.) J. Tropfke, Geschichte der Elementar Mathematik (2nd ed.). (The leading collection of facts on the subject.) F. Cajon, History of Mathematics, 2nd. ed. (New York, Macmillan, 1919). (A general survey of the subject.) D. E. Smith, History of Mathematics (Boston, U.S.A., Ginn, 1923, 1925). (Chiefly elementary mathematics. Many illustrations, including facsimiles.) S. Gunther and H. Wieleitner, Geschichte der Mathematik (Leipzig, 1908, 1921). (A general summary for students.) W. W. R. Ball, History of Mathematics, various editions (London, Macmillan). (A well-written introduction to the subject). For an extensive bibliog raphy consult Smith's history cited above, and the current volumes of the Jahrbuch iiber die Fortschritte der Mathematik (Berlin) ; Isis (Ghent and Brussels), and the Revue Semestrielle (Amsterdam). For the earlier bibliography of historical journals and source material see Smith, op. cit. See also MATHEMATICAL SOCIETIES AND PERIODICALS.
(D. E. S.) MATHEMATICS, NATURE OF, the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. (The word mathematics is derived from the Greek pahmuarvi sc. art.) It has been usual to define mathematics as "the science of discrete and con tinuous magnitude." Even Leibnitz,' who initiated a more modern point of view, follows the tradition in thus confining the scope of mathematics properly so called, while apparently conceiving it as a department of a yet wider science of reasoning. A short con sideration of some leading topics of the science will exemplify both the plausibility and inadequacy of the above definition. Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensu rable numbers. It would seem that "the general theory of discrete and continuous quantity" is the exact description of the topics of 'Cf. La Logique de Leibnitz, ch. vii., by L. Couturat (Paris, 19o1).
these sciences. Furthermore, can we not complete the circle of the mathematical sciences by adding geometry? Now geometry deals with points, lines, planes and cubic contents. Of these all except points are quantities. Also, as the Cartesian geometry shows, all the relations between points are expressible in terms of geo metric quantities. Accordingly, at first sight it seems reasonable to define geometry in some such way as "the science of dimensional quantity." Thus every subdivision of mathematical science would appear to deal with quantity, and the definition of mathematics as "the science of quantity" would appear to be justified. We have now to see why the definition is inadequate.
Types Relating to Numbers.—What are numbers? We can talk of five apples and ten pears. But what are "five" and "ten"
apart from the apples and pears? Also in addition to the cardinal numbers there are the ordinal numbers : the fifth apple and the tenth pear claim thought. What is the relation of "the fifth" and "the tenth" to "five" and "ten"? "The first rose of summer" and "the last rose of summer" are parallel phrases, yet one explicitly introduces an ordinal number and the other does not. Again, "half a foot" and "half a pound" are easily defined. But in what sense is there "a half," which is the same for "half a foot" as "half a pound"? Furthermore, incommensurable numbers are defined as the limits arrived at as the result of certain procedures with rational numbers. But how do we know that there is any thing to reach? We must know that J 2 exists before we can prove that any procedure will reach it.
Types Relating to Geometry.—Also in geometry, what is a point? The straightness of a straight line and the planeness of a plane require consideration. Furthermore, "congruence" is a diffi culty. For when a triangle "moves," the points do not move with it. So what is it that keeps unaltered in the moving triangle? Thus the whole method of measurement in geometry as described in the elementary textbooks and the older treatises is obscure to the last degree. Lastly, what are "dimensions"? All these topics require thorough discussion before we can rest content with the definition of mathematics as the general science of magnitude; and by the time they are discussed the definition has evaporated. An outline of the modern answers to questions such as the above will now be given. A critical defence of them would require a volume.' Nature of Cardinal Numbers.—A one-one relation between the members of two classes a and 0 is any method of correlating all the members of a to all the members of 0, so that any mem ber of a has one and only one correlate in 3, and any member of 0 has one and only one correlate in a. Two classes between which a one-one relation exists have the same cardinal number and are called cardinally similar; and the cardinal number of the class a is a certain class whose members are themselves classes—namely, it is the class composed of all those classes for which a one-one correlation with a exists. Thus the cardinal number of a is itself a class, and furthermore a is a member of it. For a one-one rela tion can be established between the members of a and a by the simple process of correlating each member of a with itself. Thus the cardinal number one is the class of unit classes, the cardinal number two is the class of doublets, and so on. Also a unit class is any class with the property that it possesses a member x such that, if y is any member of the class, then x and y are identical. A doublet is any class which possesses a member x such that the modified class formed by all the other members except x is a unit class. And so on for all the finite cardinals, which are thus defined successively. The cardinal number zero is the class of classes with no members; but there is only one such class, namely—the null class. Thus this cardinal number has only one member. The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 0, satisfying the conditions (I) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and /3 two other suitable classes whose cardinal numbers are defined to be respectively the 'Cf. The Principles of Mathematics, by Bertrand Russell (Cam bridge, 5903).