MATHEMATICS

in the great principia mathe- matica, published in 1911, mathematics js rigorously deduced from a few primitive propositions of logic. since then, not to speak of improvements of detail effected by sheffer and wicod, the principia has been subjected to fundamental criticisms which have induced many workers to abandon altogether its line of approach. i. mathematical logic theory of types.—in attempting to provide a firm logical basis for the theories of infinite aggregates and irrational number de- veloped during the 19th century by cantor and dedekind, mathematics the chief theoretica] difficulty is to avoid the famous paradoxes of the theory of aggregates. these are direct contradictions which arise as a result of pursuing apparently ordinary and legiti- mate mathematical arguments. such a state of things is intoler- able, and shows that there must be something radically wrong with our fundamental ideas. whitehead and russell discovered that behind each contradiction lay a “ vicious-circle fallacy,” due to neglecting the fundamental principle that what involves the whole of a given totality cannot itsclf be a member of the totality. in order to avoid such fallacies they put forward their theory of types, which consisted of two rather different parts. the first part asserted that things were of different logical types, and that for any predicate there was only one type of subject, of which the predicate could significantly be predicated; ap- plying it toa subject of any other type would give not falsehood but literally nonsense. according to this principle many of the paradoxes consist of sentences, which, though grammatically correct, are nevertheless strictly meaningless, and the contra- dictions are thus completely removed. but not all the para- doxes could be dealt with thus, and the others required the second part of the theory of types, which distinguished further different orders of predicates applicable to the same type of subject. for instance, the quality of having all qualities of a certain sort cannot itself be a quality of that sort, or it would be involved in its own definition, but must be a quality of a higher order. so, in order to avoid vicious circles, we must never speak of all qualities generally, but only of all qualities of a certain order. if all statements of this kind are made precise as to the order of qualities involved, all the paradoxes which escape the first part of the theory of types in their turn disappear. axiom of reducibility—the theory of types is thus com- pletely successful in avoiding the contradictions, but this is only half the problem. the arguments leading to contradiction must be invalidated, but this must be done without also invalidating the arguments used in accepted mathematics; and in this white- head and russell failed, for the restrictions imposed by the sec- ond part of the theory of types upset an enormous part of ordi- nary analysis, and this catastrophe was only averted in principia by the introduction of the axiom of reducibility. this axiom asserts that for any predicate of higher order there is a predicate of the lowest order having the same extension (i.e., applying to the same subjects) as the one of higher order. this implies that any class defined as the extension of a higher order predicate will also be the extension of a lowest order predicate; so that thus the totality of classes, whose members are of a given type, can be ob- tained from the totality of lowest order predicates applicable to that type and will bea legitimate totality. in this way the theory of irrational numbers and the other parts of mathematics threat- ened by the theory of types can be saved, if we are prepared to accept the axiom of reducibility. but this axiom cannot possibly be regarded as self-evident, and there is really no reason what- ever for supposing it to be true, so that anything which can only be proved by assuming it cannot be regarded as proved at all. intuitionist views——this point was especially emphasised by weyl in das kontinuum (leipzig, 1918) who went so far as to reject not merely as unproved, but as evidently unprovable, much of the general theory of real numbers, bounds, limits and continuous functions, to say nothing of more recent advances. he proposed to confine mathematics to the narrower body of truth, which could be proved not merely without the axiom of reducibility but without using higher order predicates at all. since then weyl has moved still further from the standpoint of principia mathematica, and in his latest work (e.g., in mwathe- matische zeitschrift, 1921) he appears as a follower of brouwer, who is regarded as the leader of the “ intuitionist ”’ school. the excluded middle —brouwer’s principal thesis is his denial of the law of excluded middle, which may be illustrated by a case of the following kind. suppose we wish to establish the existence of a number having a certain property; then we may argue thus: a certain condition, say », must either be fulfilled or not; if is true, we can make a construction which will give a number of the required kind; if # is false, we can also make a construction for mathematics such a number, though a different construction from the previous one. so we shall conclude that in either case there exists a num- ber having the required property. such an argument would, according to brouwer, be illegitimate, because if, as often hap- pens, we could not tell whether # were true or false, we should not know how to set about actually constructing the number in question. it would be wrong, he thinks, to argue by the law of excluded middle that, because it could not be true in either case that all numbers have not got this property, there must be one number at least which has it. this criticism, which is supported by several eminent authori- tics, would annihilate considerable portions of modern mathe- matics. there are, however, many thinkers to whom brouwer’s criticism seems to have no force whatsoever, and who are deter- mined to find a theory of the foundations of mathematics which will not involve the rejection of any generally accepted parts of the subjects; and of their views we must now give some account. axiom systems.—there are first zermelo, frinkel and others, who are working at the construction of axiom systems for the theory of aggregates; they propose to cease using the word class or aggregate in its ordinary sense, whatever that may be, but to mean by it anything satisfying certain axioms; these axioms will be constructed so as to permit the deduction of all the ordi- nary theory of aggregates, but avoid all the contradictions. but from a philosophical point of view this shirks the real problem, which is to clear from contradiction the notion of class and other allied notions which occur in mathematics, when, for instance,we speak of the class of prime numbers, as well as outside it. formalists—the most prominent opponents of the intuition- ists are, however, the formalists, of whom hilbert (see especially mathematische ainnalen, 1922) is the leader. this school regards mathematics as consisting of meaningless symbols manipulated according to formal rules, thus reducing it to something of the nature of a game of chess. such an account of mathematics is wholly inadequate; since if “ two ” and ‘‘ three ”’ were meaning- less marks on paper there could be no difference in meaning between ‘‘ two shillings ” and “ three shillings.”? but the formal- ist attitude, although hopeless philosophically, may be useful in dealing with particular questions, and the essential novelty of hilbert’s work consists not in his formalist theory, but in the propositions he has proved or hopes to prove about mathemati- cal formulae. the chief of these propositions is that it is impossi- ble to deduce a contradiction by permissible operations from the - axioms of mathematics, including the law of excluded middle. this demonstration is only required to convince the sceptics or intuitionists, who will not be satisfied with the simple argu- ment that the axioms are true and that therefore no contradiction can be logically deduced from them. hence it is essential not to use in the demonstration any principle which the sceptics doubt. this appears to be possible, since, owing to the finite nature of any piece of mathematical reasoning, in proving that it does not lead to contradiction we shall only have to employ the law of excluded middle for finite ranges, for which everyone admits its validity. the science on which hilbert is now (1926) engaged, which takes for its subject matter the meaningless for- mulae of mathematics, he calls metamathematics, and believes it to be capable of establishing many important results relating to the multiplicative axiom or axiom of selections and to the continuum problem. interesting, however, as his work is, it does not offer any satisfactory account of the foundations of mathematics as a science, but only deals with it as a game, and contributes no more to the solution of the main problem than does the annihilat- ing scepticism of the intuitionists. mathematical inductton.—so that, in view of the bankruptcy of all other proposals, it would seem that the truc account of the matter must be on the lines of that of whitehead and russell, although this cannot be correct in detail owing to the unsatis- factory nature of the axiom of reducibility. of great interest from this point of view is the new work in the second edition of principia mathematica (vol. 1, 1925) and in particular the new theory of mathematical induction, which by most ingenious arguments is established without using the axiom of reduci- 831 bility. unfortunately the authors do not see their way to con- structing a similar account of dedekindian series, for which and for all that part of analysis which depends upon it we are left without any justification. in this new edition of principia alathematica some use is made of the theories of tractatus logico-philosophicus by ludwig witt- genstein, which, although mainly concerned with more purely philosophical problems, also contains contributions of funda- mental importance to our subject. of one of the principal prob- lems in particular he has given an original and possibly final solution; but in order to understand this it is necessary to go back a little. mathematics and logic-—one of the chief objects of the work of frege and russell was to reduce mathematics to logic, to show that all mathematical concepts could be defined in terms of “ logical constants,’ and that all mathematical propositions could be deduced from those of formal logic; so that mathe- matics and logic became one and the same subject. granting the validity of this reduction, doubt still remains as to what is the peculiar characteristic of logical or mathematical proposi- tions, and how they differ from all others. this question russell answered in 1903 by saying that logic or mathematics consisted of all true propositions, which were hypothetical in form and contained no constants except logical constants; this amounts practically to saying that any completely general true proposi- tion is a proposition of logic. it subsequently became clear that this definition was much too wide, and that it was necessary to distinguish the propositions which could be stated in purely logical terms and were in fact true, from those which logic alone could assert to be true, which form the science of logic and mathe- matics. and the clistinguishing characteristic of this smaller group of propositions which logic can assert to be true had yet to be found. wittgenstein’s work.—this problem wittgenstein has probably solved; his answer is complicated by the fact that he does not accept the reduction of mathematics to logic, and gives different accounts of the propositions of logic and those of mathematics. of these his theory of logical propositions is the more important; and if, as is quite possible, his criticisms of the reduction of mathematics to logic are unsound, it would entirely supersede his theory of mathematics, which is given only in barest outline. wittgenstein’s account of the propositions of logic is that they are tautologies, in a sense of which a precise definition is given in terms of his theory of propositions in general. any proposition, he maintains, ts the expression of agreement and disagreement with certain ultimate possibilities, the possibilities of existence and non-existence of atomic facts. a proposition which agrees with every possibility is a tautology; one which disagrees with every possibility is a contradiction. an ordinary proposition asserts something to be the case and is true if it is the case; a tautology on the other hand asserts nothing but agrees with every possibility and is therefore true unconditionally. this account can be made to cover not only what are called in prin- cipia mathematica “ elementary propositions,” but also general propositions which involve apparent variables. symbolism.—but the importance of wittgenstein’s work does not only consist in his having defined the propositions of logic, and hence possibly those of mathematics, but also in his theory of symbolism, which seems capable of far-reaching applications to the theory of types. the theory of symbolism may indeed be expected to be relevant for the following reason. there is an obvious division of the paradoxes into two groups, which is en- tirely overlooked in principia mathematica. the first group of contradictions, such as those about the class of classes not con- taining themselves and the greatest ordinal, can be stated in purely logical or mathematical terms. they would actually occur in a mathematical treatise if it investigated the problems in question. the remaining contradictions, such as richard’s paradox, involve, besides mathematical terms, terms drawn from psychology or epistemology, such as knowing, asserting, naming or meaning. these contradictions are not therefore purely me- chanical, and the responsibility for them need not, as must that 832 for the first group, lie with faulty mathematics or logic, but may lie in faulty ideas of knowledge and symbolism. this was the view taken of them by peano; but it cannot be regarded as satis- factory until an adequate explanation is given of where exactly the fault lies. the probability that it hes in the epistemology and not in the mathematics, is strengthened by the fact that this grouping of the contradictions into purely mathematical and mixed coincides with their grouping according to whether their solution in principia requires the first or second part of the the- ory of types. so that the first part of the theory, which is un- doubtedly sound, will suffice for the purely mathematical con- tradictions, and if the mixed contradictions can be shown to be due to the epistemological element, there may no longer be any need for the second part of the theory of types, which requires the axiom of reducibility and is so responsible for the whole trouble. by using the work of wittgenstein a solution has been con- structed along these lines, 01 which the theory of types is so modified that all need for an axiom of reducibility disappears, and mathematics consists entirely of tautologies in wittgen- stein’s sense. bibl1ography.—mathematische zettschrift (1921); mathematische annalen (1922); l. wittgenstein, tractatus logico-philosophicus, ger. and eng. (1922); principia mathematica (1925). (f. p. r.) il. theory of numbers in the article number (19.847) an excellent summary is given of the classical theory. modern mathematics has seen the rise of a new theory, the ‘‘ analytic ” theory, which has de- veloped with great rapidity, and has almost monopolised the attention of arithmeticians. (a) theory of primes —the modern developments of the theory of numbers depend in the main on the application to the theory of the ideas of the theory of functions of a complex variable (see 11.301). it was in the theory of the distribution of primes that these ideas first bore fruit. it is usual to write (x) for the number of primes less than x. it has been known since euclid that the number of primes is infinite, that is to say that w(x) tends to infinity with x. the central problem of the theory has been the determination of the order of magnitude of «(x) when x is large, and its solution is embodied in the przm- zuhlsaiz, or ‘‘ prime number theorem,” expressed by the formula 5 w(x) nige ‘ where the symbol“ expresses the fact that the ratio of the two functions tends to unity. this theorem, conjectured by a. ml. legendre (1798) and c. f. gauss (about 1792), was first proved by j. hadamard and ch. j. de la vallee poussin in 1896. the real founder of the modern theory, however, was b. riemann, who, in a famous memoir published in 1859, first indicated the road along which subsequent research has progressed. riemann did not prove the prime number theorem; strangely enough, he did not mention it, his object being to obtain, not an asymptotic formula for (x) but an exact expression in the form of an infinite series. nor did riemann attain the goal at which he aimed, his analysis, profound and beautiful as it is, being altogether incomplete and inconclusive. but it was riemann who first recognised where the key to the solu- tion lay, viz., in the study of the ‘ riemann zeta-function,”’ rs) =e +i) =2e-r= 11.) (where n=i, 2, 3, ..., and » runs through the series of primes), considered as a function of the complex variable s. riemann estab- lished some, and conjectured others, of the properties of {(s); one famous conjecture, that all the complex zeros of ¢(s) lie on the line ¢ = 3, remains unsettled to this day. riemann’s memoir bore no fruit for over 30 years, when the way was cleared by the researches of hladamard in the theory of analytic functions (see function, 11.301 seg.). these researches led hada- mard himself, de la vallee poussin, and other writers, to a proof not only of the prime number theorem but of very much more. thus de dt : 2 log £ represents w(x) with an error of lower order than x(log x), where k is any number however large. he also investigated the distribu- tion of primes of a linear form am-+6 or a quadratic form am?-+bmn +cn? where a, b, c are integers without common factor, showing, for example, that the primes are, on the average, equally distributed between the various arithmetical progressions am+1, am-+2, «««., as had been conjectured long before by p. g. lejeune la vallee poussin proved that the logarithm integral lix = mathematics dirichlet. there is a corresponding theory for the “ prime ideals ”’ of the “ corpus ” associated with any algebraic number, the ana- logue of riemann’s zeta-function was discovered by r. dedekind, but it is only recently that, in the hands of e. hecke and e. landau, the development of the theory has been pushed to a point corre- sponding with that of the ordinary theory. the outstanding unsolved problem of the theory is that of the determination of the order of the difference r(x) —lix. this problem is bound up essentially with riemann’s unproved hypcthcsis con- cerning the zeros of ¢(s). if riemann’s hypothesis is true, the max- imum order of the difference differs from that of ¥« by iicgarithmic factors only. in any case the difference assumes values ef etiher sien which tend to infinity with x. this theorem, proved by j. e. littlewood in 1914, disposes of the old conjecture of gauss and b. goldschmidt that w(x) is always less than li(x). apart from applications to the theory of primes, there is a large literature connected with the pure theory of €(s). it has been shown by h. bohr, e. landau and f. carlson that (to put it roughly) nearly all the zeros lie very near the critical line; by g. h. hardy and j. e. littlewood that (equally roughly) a considerable propor- tion lie actually on it. but the hypothesis itself remains unproved. (b) additive theory.—the “ additive ” theory of numbers in- cludes combinatory analysis (see 6.752), partitions (sce 19.865), the theory of the representation of numbers by sums of squares, cubes, or higher powers, and so forth. _ the central problem is that of determining (exactly or approx- imately) the number of representations of an arbitrary pcsitive integer # in the form aj;ta:+ ... -+a,, where the a's are num- bers of some special type (e.g. squares), and s may be fixed or un- restricted, according to the particular problem envisaged. there isa fundamental difference between the “ additive " theory and what may be called the “‘ multiplicative’ theory, in which the central idea is that of the resolution of a number into prime factors, analyti- cally, this difference expresses itself as follows: the multiplicative theory depends on the theory of ‘ dirichlet’s series ” of the type senn-*, the additive theory on that of power series la.x", a great deal of the additive theory is purely algebraic, and is intimately bound up with the theory of elliptic functions. this side of the theory (founded by l, euler) has been developed to a high pitch by english mathematicians, notably a. cayley, j. j. sylvester, and p. a. mac- mahon, while more recently the methods of complex functicn theory have been applied to the theory and an ‘‘ analytic additive " theory has been founded. among many curious results we may menticn the theorem of s. ramanujan, that the numbers cf the unrestricted partitions of numbers of the forms 5%+4, 7m4+5 and t1m-+6 are divisible by 5, 7 and 11 respectively. one of the most remarkable problems of the additive theory is ‘“‘waring’s problem.” it was asserted by f-. waring (1782) that an number x is the sum of at most 4 squares, g positive cubes, 19 fourt powers, and, generally, g(#) powers, where g(&) is a number de- pending on & alone and noton xn. this problem—in so far as it simply asserts the existence of some such number g(k)— was solved by d. hilbert in aes j. l. lagrange (1774) proved that g(2)=4 (any © number is the sum of 4 squares, and some numbers not of less), and e. wieferich (1909) that g(3)=9 and g(4)$37. only a finite number of numbers (probably only 23 and 239) require more cubes than 8 (e. landau, 1908), while an infinite number require 4 at least; and only a finite number of numbers require more than 19 fourth powers (g. h. hardy and j. e. littlewood, 1924), while an infinite number require 16 at least; and asymptotic formulae for the number of representations have been found. but our knowledge of this field is still extremely incomplete. the ‘empirical theorem” of chr. goldbach, that every even number is the sum of two primes, has also received a considerable amount of attention, but is still unproved. among other unsolved problems of the same character may be mentioned that of proving the existence of an infinity of primes of the form m?+1 or (more generally) am?+bm+c. this problem is not to be confused with the problem of primes am?-+-bmn-+cn?, solved by de la vailee poussin. (c) miscellaneous investigations —the work of dirichlet and l. kronecker on the approximation of irrational numbers by rationals has led to extensive investigations lying on the border line between arithmetic and analysis, developed above all by h. minkowski under the titles of diophantische approximation and geometrie der zahlen. the central idea in this theory is that of the lattice (gitter). a lattice point (gitterpunkf) in space of any number of dimensions is a point with integral co-ordinates, and most difficult and tec matnle problems arise when we consider the number of lattice points whic lie within a volume of specified form in 2-dimensional space. thus minkowski proved that any convex figure in space of two dimensions with symmetry about a centre, its centre at a lattice point, and of area 4, includes other lattice points besides its centre; with a whole series of corresponding theorems concerning more general configura- tions. another class of lattice-point problems is exemplified by the mathematics “circle ’’ problem of gauss and w. sierpinski, that of determining approximately the number of lattice points inside the circle x? + y? = when n is large. a first approximation is naturally given by wn, the area of the circle, but the estimation of the error is a problem of exceptional difficulty. this problem and the analogous problem for the hyperbola xy=n (dirichlet’s divisor problem) were connected with the theory of ¢(s) [see (a) supra] by ee these problems also are susceptible of manifold generalisation. and in al! these problems, we observe the dominating and irresistible tendency of modern higher arithmetic, the tendency to abandon its ancient tradi- tion of isojation and assimilate itself so far as possible to the theory of functions, in order to utilise the immensely powerful weapons which the latter theory alone can provide. there is one famous problem in which no such reduction of arithmetic to analysis has been effected. ‘‘ fermat’s last theorem ”’ asserts that there is no integral solution of x°-++-y" =2" (other than the trivia) solution x=z, y=o) for any value of # greater than 2. it was the attempt to prove this theorem that led to the whole development of the theory of algebraic numbers; but, in spite of the wide-spread attention which it has excited, and the extreme impor- tance of the general theories of which it has been the starting point, the theorem itself remains unproved, though important additions have been made recently to our knowledge by a. wieferich, d. miri- manov, l. e. dickson, and h. s. vandiver. thus wieferich proved that the theorem holds for odd prime values of 2, and values of x, y, 2, not divisible by », unless 2" "'—1 is a multiple of n?. one old conjecture has been definitely disposed of. mersenne asserted that 2"—1, where # is a prime not exceeding 257, is prime when, and only when, n=i, 2, 3, 5, 7, 13, 17, 19, 3t, 67, 127, 257. this statement contains at least four errors, relating to the values 61, 67, 89, 107; and it need no longer be taken seriously. bibliography.—an indispensable work for the serious student of higher arithmetic (on any of its sides) is l. e. dickson, history of the theory of numbers, 1920. this work is not, however, specially concerned with the analytic theory. for general accounts of the theory of primes see the article ‘‘ die neuere entwicklung der analytischen zahlentheorie ’’ by h. bohr and h. cramer in the encyklopddie der mathematischen wissen- schafien (2c8); e. landau, handbuch der lehre von der verteilung der primzahien (1909), and einfiihrung in die elementare und analytische theorie der algebratschen zahlen (1918). for the additive theory see p. a. macmahon, combinatory analysis (1915-6), and an introduction to combinatory analysts (1921); p. bachmann, niedere zahlentheorie: 2. (additive zahlen- theorie) (1910); g. h. hardy, some famous problems of the theory ef numbers (1920). for fermat's last problem see p. bachmann, das fermatproblem (1917); l. j. mordell, four lectures on fermat's last problem (1921). comparatively little of recent work is accessible in a connected form, and the study of the original memoirs is indispensable. iti. theory of series the most striking modern developments in the theory of series (see 24.668; 10.753; 12.956) have also been suggested by the development of the theory of functions. the theory of functions of a real variable has been revolution- ised by the ideas of e. borel and h. lebesgue, and this has in- spired a corresponding revolution in the theory of fourier’s series and ‘ series of orthogonal functions ”’ generally. a system of functions ¢n(x)(m=1, 2, 3, ...) is said to be 0 bn (x)dulax)dx =o(m xn (1) the simplest examples are obtained by taking ¢:(x) to be cos mx or sin mtx and the interval (a, 6) to be (0, 27); or en(x) to be legen- dre’s polynomial pm(x) and (a, 5) to be (—1, 1). there is then a simple procedure by which we may endeavour to expand an ar- bitrary function f(x) in the form of a series sam¢m(x), viz., by mul- tiplying this series by ¢m(x) and integrating over the interval (a, 5): the formula thus suggested is f(x) =zandm(t), am=(f 2fle)bm (x) dx) '( {pom (sax) (2) ; a more accurate analysis of this procedure raises a multitude of profoundly interesting and difficult questions. on the one hand we may start from a series with arbitrary coefficients @m, and inquire whether there exists a function which stands to it in the relation expressed by the equation (2). in particular, given a trigonometrical series zamncosmx or lbasinmx or, more generally z(a,,cosmx +bmsinmx), with arbitrary coefficients, we may ask whether it is a fourier's series, that is to say, whether there is a function /(x) such that adm and bm are given by fourier’s integral formulae. on the other hand, we may start not from an arbitrary series but from an arbitrary function f(x), form the coefficients (am or bm) by yourier’s formulae or the more general formulae (2), and then inquire whether the forma! development thus obtained is convergent, and whether, if convergent, it represents the function f(x) and so forth. the problems thus raised are among the most difficult of modern mathematics; and a very cursory examination of them is enough to orthogonal if 833 show that the methods of the older analysis are not sufficiently powerful for their solution. it is essential that we should enlarge our conceptions, on the one hand, by taking account of the modern gener- alisations of the notion of an integral, and, on the other, by adopting a broader view as to what is meant by the “‘ sum ”’ of an infinite series. the modern theory of functions of a complex variable (see 11.301) points to the same conclusion. a function f(z) of the complex variable z, regular for g=29, is defined throughout a certain circle whose centre is 2 by a power-series zan(z—29)"; but the region of existence of the function is very generally more extensive than the circle of convergence of the series; and this fact has led, during the last generation, to a mass of work on the problem of “ analytic continuation.” this problem is that of discovering analytic repre- sentations of the function, whether by integrals, or by continued fractions, or by series of a different form, which are valid throughout a wider region than that in which it is represented by the original power series, here also we are confronted by the need for a scientific theory of divergent scries. there are passages in the older analysts (e.g., in l. euler) which suggest a half-conscious anticipation of modern ideas. but it is roughly true to say that they did not concern themselves with the precise meaning of the infinite series of which they made such effec- tive use. a. l. cauchy and n. h. abel were the first to give a precise definition of the ‘‘sum” of a series a@o+ai+a2+ ... or lan, viz., asthe limit of s,a=aotait ... +a, when n tends to infinity (7-+e). such a series as i—i+1— ... has then mo sum, for sn is alternately 1 and 0; and it was the tendency, for many years after cauchy and abel, to banish such series from analysts entirely. a school of mathematicians survived, among whom one may cite a. de morgan, who viewed this tendency with obvious discontent, but there was no escape from the conclusion that the followers of cauchy and abel were right. it is impossible to say ‘the sum of ya, is so-and-so ”’ except after framing an accurate definition of ‘‘ sum ’’; the definition of cauchy and abel was the only definition: and, until some new and wider definition was offered, that was the end of the matter. we may define the meaning of a mathematical word or symbol as we please, provided only that the definition is free from contradiction, given a sequence of numbers a, @2,... we may associate with the sequence a number s in any manner that we please, and we may say, if we like, that s is the ‘sum ” of the series. we might say, for in- stance, that the ‘‘ sum ” of every infinite series is, by definition, zero, this definition would be perfectly legitimate but futile, because it would reduce all equations involving infinite series to the trivial form 0=0; and confusing, because it would conflict with cauchy's definition. cauchy's definition is only one among many, but it is admittedly the most important, and a new definition is only likely to be of value if it is consistent with the standard definition. it must satisfy what is called the condition of consistency; it must apply to all convergent series, and give a ‘sum " equa? to their sum in the ordinary sense. its value for analysis will then be measured by the extent and importance of the class of non-convergent series to which it attributes a “ sum.” the simplest and most important of the definitions which have been given is that of the “ first arithmetic mean.” suppose that sn=qotait+ ... +@n and on=(sotsi+ ... +5n)/(n+1), the arithmetic mean of the first »-+1 values of s,. if sna tends to a limit, on tends to a limit also, and the two limits are the same; but o, may tend to a limit when s, docs not. for example, if da=(—1)*, son = 1 and seng1 =o, and ss, does not tend to a limit; but on tends to the limit 3. if now we agrce to call the limit of on, whenever it exists, the ‘‘ sum ” of the serics lan, our new definition is in perfect accord with cauchy’s definition, but is applicable to an extensive class of series for which cauchy’s definition fails. it therefore fulfils the conditions required for a theory of divergent series. the most striking illustration of the importance of these ideas is to be found in the theory of fourier’s series (see 10.753). the fourier’s series of a continuous function f(x) is not necessarily con- vergent; further conditions on f(x), of a much more artificial charac- ter, are required to ensure convergence. it was, however, shown by l. fejer that the fourier series of any continuous function is “summable " by the procedure indicated above; that is to say, that the arithmetic mean on tends to a limit equal to the value of the function; and this fundamental result has been the starting point of a inass of modern research. another important definition attributes to the series as “sum” the value of the limit of the power series danx" when x tends to i through positive values lessthan 1. a third (of particular importance in complex function theory) was advanced by borel; and all of these definitions have entailed a host of still more general definitions, bibliography.---for the general theory of divergent series sce e. borel, lecons sur les series divergentes (1901); t. j. 1’a. bromwich, introduction to the theory of infinite series, ch. x. (1908); g. h. hardy and m, riesz, the general theory of dirichlet’s series (1915). for the theory of fourier’s series, h. lebesgue, lecgons sur les series trigonometriques (1912); ch. j. de la vallee poussin, cours d'analyse infinitesimale, 2nd ed., vol. 2 (1916); e. w. hobson, the theory of functions of a real variable (1907, 2nd ed., 1925). the general theory of series of orthogonal functions is, for the most part, still only to be read in the original memoirs, or in works on the theory of integral equations. a very important generalisation of the con- cept of a fourier’s series has been developed recently by h. bohr in a series of memoirs in the acta mathematica. iv. theory of functions the theory of functions (see 11.301, 14.53) has two great branches, the real and the complex theories. recent advances in the complex theory, important as they are, have been of too technical a character for rapid summary. the real theory, on the other hand, has been remodelled from its foundations. the older form of the theory was cumbrous and unattractive. the modern theory has the aesthetic character required of a first- rate mathematical science, and its development has been per- haps the most striking achievement of modern analysis. 1. sets of poitnts.—the theory of functions of a real variable is based upon the theory of aggregates (see 19.847-850) and in particu- lar the theory of “‘ sets of points.’’ a set of points s is an aggregate of real numbers x, such as the aggregate of rational numbers, or of irrational numbers, in the interval (0, 1). a number & ts said to bea ‘limit point ” (heufungstelle) of s if every ‘‘ neighbourhood ” of &, that is to say every interval (e—e, +) including £, contains points of s other than e itself. a limit point of s may or may not belong itself to s. thus every number of (0, 1), rational or irrational, is a limit point of the set s of rationals of (0, 1). if every limit point of s$ belongs to s, s is closed. if every point of s is a limit point, s is compact or dense. a set which is both closed and compact is perfect. the continuum, the aggregate of all real numbers, is perfect. an idea of dominating importance in the theory of functions is that of the content or measure of a set of points. suppose, for sim- plicity, that the set s in question is contained in (0, 1). then cantor defined the content of sas follows:—divide (0, 1) in any manner into a finite number of intervals 6, and these intervals 6 into two classes 6, and &, according as they do or do not include points of s; and let c(5) be the sum of the lengths of the intervals 6,. then the content of s is the limit of c(8) when the intervals 6 tend to zero, if this limit should exist. there is a striking defect in this definition, the full implications of which were first perceived by e. borel. the content of the sum of two sets is not generaily the sum of their contents. thus the rationals of (o, 1) have content i (since every 4 is obviously a 5), and ltkewise the irrationals. the sum of the contents is 2, whereas the content of the sum is 1. the rationals of (0, 1) cannot be included in a finite set of intervals whose aggregate length is less than 1. if we abandon the restriction that the set of intervals must be finite, the situation is completely changed. thus borel observed that we may include the ae ++ , and that the sum of all rational p/g in the interval = ae these intervals may be made as small as we please by choice of ¢; and this simple remark has revolutionised the theory of functions. the first step was to frame a satisfactory definition of measure, and this concept, which has entirely superseded cantor's “ content,” is now defined as follows. we consider sets s included in (0, 1). let s be enclosed, 4” any manner whatsoever, in a system o of intervals 5; let m(o) be the sum of the intervals of «; and ict m. be the lower bound (or “inferior limit ’’}) of the aggregate of values of m(c). then m. is the extertor measure of s. the interior measure mj; 1s 1—m’'., where m’. is the exterior measure of s;, the set complemen- tary to s, z.e., the set of points of (0,1) which do not belong to s. if me=mi, the set s is measurable, and its measure is m, the common value of me and m;. this definition (due to h. lebesgue) is of ex- treme generality, and no example of a non-measurable set is known. measure, thus defined, has the properties which measure ought to have, but which cantor’s content lacked. in particular the sum of two mutually exclusive and measurable sets is measurable, and its measure is the sum of the measures of the component scts. the measure of any enumerable set, and in particular of the rationals, is zero. the definition may be extended to sets in space of any num- ber of dimensions. 2. integration.—the new theory of measure has led to new theories of integration, in the light of which the older theories are of historical or didactic interest only. the most important of these theories are due to h. lebesgue and w. h. young. (a2) lebesgue’s definition of an integral is as follows. a function f(x), defined in an interval (4,5), is measurable if the set of points s(a) for which f > a is measurable for every a. all known functions are measurable. we now suppose that f is bounded, so that (say) h<f<h, and we divide up the interval (4, h) into a finite number of intervals (¢:, i41) or 6:. [t is this subdivision of the range of variation of f(x), instead of (as in the older theory) that of x, that is charac- teristic of lebesgue’s procedure. the set of points for which (/i:s f<tis) is measurable. if we denote its measure by mi, write j = yiimti and suppose that the intervals 6; tend to zero. then j tends to a limit 1, and we write j =f" f(x)dx mathews the integral so defined 1s a bonea-fide generalisation of the integral of riemann, for it exists whenever riemann’s integral exists and agrees with it in value. but it is far more general: thus the function fc) which is unity when x is a rational of (0,1), and zero otherwise, has no riemann integral, but has a lebesgue integral equal to zero. the definition is capable of many-sided generalisation, to unbounded functions, and functions of many variables; it throws entirely new light on the relations between integration and differentiation; and it has proved itself adapted for a mass of analytical applications of the most far-reaching importance, tn particular in the theory of fourier’s scries and the theory of integral equations. (b) a different definition was proposed by w.h. young. he ad- heres to a subdivision of the range of variation (a,b) of the independ- ent variable; but, instead of dividing it into a finite number of inter- vals, divides it into a finite or infinite number of measurable sets. this procedure leads to results roughly equivalent to those of lebesgue’s theory; but it is somewhat more general and is certainly a more natural development of the older theory of measure. 3. geometrical applications.—those new theories have led in- evitably to a searching re-examination of the concepts of ‘‘ curve,” “surface,” “ length,” *‘ area,’’ and so forth, which were generally accepted without question by the older analysts on the supposed evidence of geometrical intuition. this unreflective attitude has now been abandoned, and it is recognised that analysis is in no sense dependent upon geometry. the notion of a curve was first made precise by c. jordan. a curve is a set of points (x,y), that is an aggregate of pairs of real numbers x, y, where x and y are functions of a single variable ¢, subject to appropriate restrictions. a simple closed continuous curve is a curve for which (1) x=x(¢) and y= y(#) are continuous for sts t: (2) x(t.) =x(ti) and y(t) =y(h), and (3) it is false that x(t’) =x(?"’) and y(t’) =y(t’’) for any pair of values ?’, t” other than 4, & a fundamental theorem, due in substance to jor- dan, asserts that such a curve c divides the plane into two “ regions "’ d and d’ separated by the curve. two points which lie in the same region can be connected by a continuous curve which has no point in common with c: but powe which lie in different regions cannot be thus connected. we thus define the tnside and outside of a closed curve in strictly analytical terms. a similar account has been given of the concepts of area and length. in particular the simple closed continuous curve c has both an area and a length if x(t) and y(e) are functions of ‘bounded variation.” 4. integral equations.—among the remaining developments of modern analysis, perhaps the most remarkable are in the theory of integral equations. the typical integral equation is b f(x) = f ke otat (1) where f(x) and k(«,t) are given and the unknown function ¢(¢) is to be determined. ‘this equation is called an integral equation of the first kind; but it has been found that equations of the form b o(x) =f(x) +a if : k(x) (dt (2) known as equations of the second kind, are better adapted for the foundation of a general theory, it was shown by i. fredholm that, if fand k satisfy certain conditions, there is in general one and only one continuous solution ¢{t); the exceptions arise when } is a zero of a certain transcendental function d(a). when d has one of these exceptional values, the equation o(x) = af k (x,t)o(e)de a has a continuous solution other than the obvious solution o(f) =0, otherwise this is the only solution. the theory has been widely developed by fredholm, d. hilbert, v. volterra and other writers. bibliography.—h. lebesgue, legons sur vintegration (1904) and lecons sur les series trigonometriques (1906); e. borel, legons sur la theorie des fonctions (2nd ed., rg14) and legons sur les fonctions de variables reelles (1905); ch. j. de la vallee poussin, cours d’analyse infinitestmale (1909, 1912) and integrales de lebesgue etc. (1916); m. becher, an introduction to the study of integral equations (1909); a. kneser, die integralgleichungen und thre anwendungen in der mathematischen physik (1911); t. lalesco, introduction a la theorte des equations integrales (1912); h. b. heywood and m. frechet, l' equation de fredholm et ses applications a la physique mathematiq ue (1912); d. hilbert, grundztige einer allgemeinen theorie der linearen integralgleichungen (1912); h. hahn, theorze der reelen funktionen (1921); e. w. hobson, the theory of functions of a real variable (2nd ed. vol. i, 1921); e. goursat, cours d’ analyse (ard ed., vol. ini., 1923). (g. h. h. mathews, shailer (1863-— ), american educationist and theologian, was born at portland, me., may 26 1863. he gradu- ated from colby college (a.b., 1884; a.m., 1887) and continued his studies at newton theological institution and the university of berlin. he was associate professor of rhetoric (1887-9) and matisse—matter professor of history ‘and political economy (1889-94) at colby college. in 1894 he went to the university of chicago as associate professor, and in 1897 became professor of new testament his- tory and interpretation. in 1908 he was made dean of the divinity school. in 1912 he was president of the federal council of the churches of christ in america, and he visited japan in 1915 as representative of that body. from 1903 to 1911 he was editor of the world today; from 1913 to 1920, of the biblical world. he has written a number of books including the french revolution (1901); itistory of new testament times in palestine (1908); social teachings of jesus (1910); spiritual interpretation of itstory (1916); and fazth of modernism (1924). (see baptists.) matisse, henri (1860- ), french painter, was born at cateau (nord) dec. 31 1869. he studied at the ecole des beaux-arts, and under gustave moreau. he soon showed revo- lutionary tendencies, and was recognised as the boldest member of the group known as “ les fauves.”? though a painter of light, he did not treat it as the impressionists had done, by means of the juxtaposition of minute touches of colour, but by employing pure tones on a large scale. in this way he produced the effect of modelling, and, by the contrast of values, was able to give the illusion of space. he was particularly successful in his use of expressive distortion. matisse spent two years in morocco, but most of his work was carried on in southern france, at collioure and nice. later, he reduced the size of his canvases, painting still life and landscapes, as well as small feminine figures and brilliantly illuminated interiors. he is entitled to be considered as the most eminent master of the contemporary french school. in his drawing he exhibits sometimes a nervous restrained man- ner, and by his use of shadow succeeds in reproducing the actual bulk of bodies without having to insist on their contour. ma- tisse’s lithographic work is also of considerable importance. bibliography.—e, faure, henri matisse (1920); marcel sembat, henrt matisse: trente reproductions precedees d'une etude critique (1920); a. basler, henrt matisse (1924). matsudaira, tsuneo (1877- ), japanese diplomat, was born in tokyd. he joined the imperial university of t6kys6 as a student of economics, and in 1902 he entered the foreign office. his first experience abroad was gained in peking where he was a secretary of legation. his determination, however, coupled with his wide theoretical knowledge, pushed him for- ward until he became first secretary in london and later in paris; but his criticism of the extreme nationalism of many of his countrymen stood in the way of his receiving imperial honours. from 1918-9 he was japanese high commissioner in siberia and in 1920 was appointed director of the bureau for european and american affairs at the foreign office in toekys. in 1921-2 matsudaira was chief secretary of the japanese dele- gation to the washington conference. in 1924, after the dip- lomatic blunder committed by the japanese ambassador to the united states, mr. hanihara, it was considered that some means must be found to improve the situation. as a result, in 1925 matsudaira was appointed ambassador and he undertook his duties with the avowed intention of bettering american- japanese relations. matsukata, masayoshi (1835-10924), japanese statesman (see 17.890), was from 1917-22 keeper of the privy seal, and on resigning from this post was created a prince. he died in toky6 july 2 1924. mattei, tito (1841-1914), italian musician and composer, was born at campobasso, near naples, may 24 1841. he became at an early age a professor at the santa cecilia academy of music at rome, and subsequently had several successful european tours as a pianist. in 1863 he finally settled in london, where he remained for the rest of his life. he composed several hundred songs and pianoforte pieces, many of which became very popular. he died in london march 30 rorq.