iv. If s is any class and zero is a member of it, also if when x is a

cardinal number and a member of s, also x+1 is a member of s, then the whole class of cardinal numbers is contained in s.

v. If a and b are cardinal numbers, and a + 1 = b + 1, then a = b.

vi. If a is a cardinal number, then a + 1 [/=] 0.

It may be noticed that (iv) is the familar principle of mathematical induction. Peano in an historical note refers its first explicit employment, although without a general enunciation, to Maurolycus in his work, _Arithmeticorum libri duo_ (Venice, 1575).

But now the difficulty of confining mathematics to being the science of number and quantity is immediately apparent. For there is no self-contained science of cardinal numbers. The proof of the six premisses requires an elaborate investigation into the general properties of classes and relations which can be deduced by the strictest reasoning from our ultimate logical principles. Also it is purely arbitrary to erect the consequences of these six principles into a separate science. They are excellent principles of the highest value, but they are in no sense the necessary premisses which must be proved before any other propositions of cardinal numbers can be established. On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science. It is no more than an interesting subdivision in a general theory.

_Ordinal Numbers._--We must first understand what is meant by "order," that is, by "serial arrangement." An order of a set of things is to be sought in that relation holding between members of the set which constitutes that order. The set viewed as a class has many orders. Thus the telegraph posts along a certain road have a space-order very obvious to our senses; but they have also a time-order according to dates of erection, perhaps more important to the postal authorities who replace them after fixed intervals. A set of cardinal numbers have an order of magnitude, often called _the_ order of the set because of its insistent obviousness to us; but, if they are the numbers drawn in a lottery, their time-order of occurrence in that drawing also ranges them in an order of some importance. Thus the order is defined by the "serial" relation. A relation (R) is serial[4] when (1) it implies diversity, so that, if x has the relation R to y, x is diverse from y; (2) it is transitive, so that if x has the relation R to y, and y to z, then x has the relation R to z; (3) it has the property of connexity, so that if x and y are things to which any things bear the relation R, or which bear the relation R to any things, then _either_ x is identical with y, _or_ x has the relation R to y, _or_ y has the relation R to x. These conditions are necessary and sufficient to secure that our ordinary ideas of "preceding" and "succeeding" hold in respect to the relation R. The "field" of the relation R is the class of things ranged in order by it. Two relations R and R´ are said to be ordinally similar, if a one-one relation holds between the members of the two fields of R and R´, such that if x and y are any two members of the field of R, such that x has the relation R to y, and if x´ and y´ are the correlates in the field of R´ of x and y, then in all such cases x´ has the relation R´ to y´, and conversely, interchanging the dashes on the letters, i.e. R and R´, x and x´, &c. It is evident that the ordinal similarity of two relations implies the cardinal similarity of their fields, but not conversely. Also, two relations need not be serial in order to be ordinally similar; but if one is serial, so is the other. The relation-number of a relation is the class whose members are all those relations which are ordinally similar to it. This class will include the original relation itself. The relation-number of a relation should be compared with the cardinal number of a class. When a relation is serial its relation-number is often called its serial type. The addition and multiplication of two relation-numbers is defined by taking two relations R and S, such that (1) their fields have no terms in common; (2) their relation-numbers are the two relation-numbers in question, and then by defining by reference to R and S two other suitable relations whose relation-numbers are defined to be respectively the sum and product of the relation-numbers in question. We need not consider the details of this process. Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number. This relation-number is the ordinal number corresponding to n; let it be symbolized by n. Thus, corresponding to the cardinal numbers 2, 3, 4 ... there are the ordinal numbers 2, 3, 4.... The definition of the ordinal number 1 requires some little ingenuity owing to the fact that no serial relation can have a field whose cardinal number is 1; but we must omit here the explanation of the process. The ordinal number 0 is the class whose sole member is the null relation--that is, the relation which never holds between any pair of entities. The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals. Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations. Thus the illusory nature of the traditional definition of mathematics is again illustrated.

_Cantor's Infinite Numbers._--Owing to the correspondence between the finite cardinals and the finite ordinals, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point. But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by [aleph]0, where [aleph] is the Hebrew letter aleph. Similarly, a class of serial relations, called _well-ordered_ serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite. These relation-numbers are the infinite ordinal numbers. The arithmetic of the infinite cardinals does not correspond to that of the infinite ordinals. The theory of these extensions of the ideas of number is dealt with in the article NUMBER. It will suffice to mention here that Peano's fourth premiss of arithmetic does not hold for infinite cardinals or for infinite ordinals. Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.

_The Data of Analysts._--Rational numbers and real numbers in general can now be defined according to the same general method, If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n × x = m × y. Thus the rational number one, which we will denote by 1_r, is not the cardinal number 1; for 1_r is the relation 1/1 as defined above, and is thus a relation holding between certain pairs of cardinals. Similarly, the other rational integers must be distinguished from the corresponding cardinals. The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication. But the desire to obtain general enunciations of theorems without exceptional cases has led mathematicians to employ entities of ever-ascending types of elaboration. These entities are not created by mathematicians, they are employed by them, and their definitions should point out the construction of the new entities in terms of those already on hand. The real numbers, which include irrational numbers, have now to be defined. Consider the serial arrangement of the rationals in their order of magnitude. A real number is a class ([alpha], say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of [alpha]. Thus, consider the class of rationals less than 2_r; any member of this class precedes some other members of the class--thus 1/2 precedes 4/3, 3/2 and so on; also the class of predecessors of predecessors of 2_r is itself the class of predecessors of 2_r. Accordingly this class is a real number; it will be called the real number 2_R. Note that the class of rationals less than or equal to 2_r is not a real number. For 2_r is not a predecessor of some member of the class. In the above example 2_R is an integral real number, which is distinct from a rational integer, and from a cardinal number. Similarly, any rational real number is distinct from the corresponding rational number. But now the irrational real numbers have all made their appearance. For example, the class of rationals whose squares are less than 2_r satisfies the definition of a real number; it is the real number [root]2. The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multiplication. Except for the immediate purposes of an explanation, such as the above, it is unnecessary for mathematicians to have separate symbols, such as 2, 2_r and 2_R, or 2/3 and (2/3)_R. Real numbers with signs (+ or -) are now defined. If a is a real number, +a is defined to be the relation which any real number of the form x + a bears to the real number x, and -a is the relation which any real number x bears to the real number x + a. The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows. Finally, we reach a complex number of the nth order. Such a number is a "one-many" relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers 1, 2 ... n respectively. If such a complex number is written (as usual) in the form x1e1 + x2e2 + ... + x_n e_n, then this particular complex number relates x1 to 1, x2 to 2, ... x_n to n. Also the "unit" e1 (or e2) considered as a number of the system is merely a shortened form for the complex number (+1) e1 + 0e2 + ... + 0e_n. This last number exemplifies the fact that one signed real number, such as 0, may be correlated to many of the n cardinals, such as 2 ... n in the example, but that each cardinal is only correlated with one signed number. Hence the relation has been called above "one-many." The sum of two complex numbers x1e1 + x2e2 + ... + x_n e_n and y1e1 + y2e2 + ... + y_n e_n is always defined to be the complex number (x1 + y1)e1 + (x2 + y2)e2 + ... + (x_n + y_n)e_n. But an indefinite number of definitions of the product of two complex numbers yield interesting results. Each definition gives rise to a corresponding algebra of higher complex numbers. We will confine ourselves here to algebraic complex numbers--that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra. The product of two complex numbers of the second order--namely, x1e1 + x2e2 and y1e1 + y2e2, is in this case defined to mean the complex (x1y1 - x2y2)e1 + (x1y2 + x2y1)e2. Thus e1 × e1 = e, e2 × e2 = -e1, e1 × e2 = e2 × e1 = e2. With this definition it is usual to omit the first symbol e1, and to write i or [root]-1 instead of e2. Accordingly, the typical form for such a complex number is x + yi, and then with this notation the above-mentioned definition of multiplication is invariably adopted. The importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers. This is exactly the same reason as that which has led mathematicians to work with signed real numbers in preference to real numbers, and with real numbers in preference to rational numbers. The evolution of mathematical thought in the invention of the data of analysis has thus been completely traced in outline.

_Definition of Mathematics._--It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that "mathematics" is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ "mathematics" in the general sense[5] of the "science concerned with the logical deduction of consequences from the general premisses of all reasoning."

_Geometry._--The typical mathematical proposition is: "If x, y, z ... satisfy such and such conditions, then such and such other conditions hold with respect to them." By taking fixed conditions for the hypothesis of such a proposition a definite department of mathematics is marked out. For example, geometry is such a department. The "axioms" of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions. The special nature of the "axioms" which constitute geometry is considered in the article GEOMETRY (_Axioms_). It is sufficient to observe here that they are concerned with special types of classes of classes and of classes of relations, and that the connexion of geometry with number and magnitude is in no way an essential part of the foundation of the science. In fact, the whole theory of measurement in geometry arises at a comparatively late stage as the result of a variety of complicated considerations.

_Classes and Relations._--The foregoing account of the nature of mathematics necessitates a strict deduction of the general properties of classes and relations from the ultimate logical premisses. In the course of this process, undertaken for the first time with the rigour of mathematicians, some contradictions have become apparent. That first discovered is known as Burali-Forti's contradiction,[6] and consists in the proof that there both is and is not a greatest infinite ordinal number. But these contradictions do not depend upon any theory of number, for Russell's contradiction[7] does not involve number in any form. This contradiction arises from considering the class possessing as members all classes which are not members of themselves. Call this class w; then to say that x is a w is equivalent to saying that x is not an x. Accordingly, to say that w is a w is equivalent to saying that w is not a w. An analogous contradiction can be found for relations. It follows that a careful scrutiny of the very idea of classes and relations is required. Note that classes are here required in extension, so that the class of human beings and the class of rational featherless bipeds are identical; similarly for relations, which are to be determined by the entities related. Now a class in respect to its components is many. In what sense then can it be one? This problem of "the one and the many" has been discussed continuously by the philosophers.[8] All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class--or relation--existence or being in any sense in which the entities composing it--or related by it--exist. Thus, to say that a pen is an entity and the class of pens is an entity is merely a play upon the word "entity"; the second sense of "entity" (if any) is indeed derived from the first, but has a more complex signification. Consider an incomplete proposition, incomplete in the sense that some entity which ought to be involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional function; and, if [phi]x be a propositional function, the undetermined variable x is the argument. Two propositional functions [phi]x and [psi]x are "extensionally identical" if any determination of x in [phi]x which converts [phi]x into a true proposition also converts [psi]x into a true proposition, and conversely for [psi] and [phi]. Now consider a propositional function F_[chi] in which the variable argument [chi] is itself a propositional function. If F_[chi] is true when, and only when, [chi] is determined to be either [phi] or some other propositional function extensionally equivalent to [phi], then the proposition F_[phi] is of the form which is ordinarily recognized as being about the class determined by [phi]x taken in extension--that is, the class of entities for which [phi]x is a true proposition when x is determined to be any one of them. A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments. It is then possible to define by a parallel elaboration what is meant by classes of classes, classes of relations, relations between classes, and so on. Accordingly, the number of a class of relations can be defined, or of a class of classes, and so on. This theory[9] is in effect a theory of the _use_ of classes and relations, and does not decide the philosophic question as to the sense (if any) in which a class in extension is one entity. It does indeed deny that it is an entity in the sense in which one of its members is an entity. Accordingly, it is a fallacy for any determination of x to consider "x is an x" or "x is not an x" as having the meaning of propositions. Note that for any determination of x, "x is an x" and "x is not an x," are neither of them fallacies but are both meaningless, according to this theory. Thus Russell's contradiction vanishes, and an examination of the other contradictions shows that they vanish also.

_Applied Mathematics._--The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications. For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting. It is only recently that the _succession_ of processes which is involved in any act of counting has been seen to be irrelevant to the idea of number. Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. A legend of the Council of Nicea[10] illustrates this point: "When the Bishops took their places on their thrones, they were 318; when they rose up to be called over, it appeared that they were 319; so that they never could make the number come right, and whenever they approached the last of the series, he immediately turned into the likeness of his next neighbour." Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analogous considerations. In the first place, some physical process of addition is presupposed, involving some inductively inferred law of permanence during that process. Thus in the theory of masses we must know that two pounds of lead when put together will counterbalance in the scales two pounds of sugar, or a pound of lead and a pound of sugar. Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of real numbers. Indeed, mathematicians now reserve "continuity" as the term for the latter kind of continuity; the mere property of having an infinite number of terms between any two terms is called "compactness." The compactness of the series of rational numbers is consistent with quasi-gaps in it--that is, with the possible absence of limits to classes in it. Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. In calculations the latter hypothesis is made because of its mathematical simplicity. But, the assumption has certainly no a priori grounds in its favour, and it is not very easy to see how to base it upon experience. For example, if it should turn out that the mass of a body is to be estimated by counting the number of corpuscles (whatever they may be) which go to form it, then a body with an irrational measure of mass is intrinsically impossible. Similarly, the continuity of space apparently rests upon sheer assumption unsupported by any a priori or experimental grounds. Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity.

In one sense there is no science of applied mathematics. When once the fixed conditions which any hypothetical group of entities are to satisfy have been precisely formulated, the deduction of the further propositions, which also will hold respecting them, can proceed in complete independence of the question as to whether or no any such group of entities can be found in the world of phenomena. Thus rational mechanics, based on the Newtonian Laws, viewed as mathematics is independent of its supposed application, and hydrodynamics remains a coherent and respected science though it is extremely improbable that any perfect fluid exists in the physical world. But this unbendingly logical point of view cannot be the last word upon the matter. For no one can doubt the essential difference between characteristic treatises upon "pure" and "applied" mathematics. The difference is a difference in method. In pure mathematics the hypotheses which a set of entities are to satisfy are given, and a group of interesting deductions are sought. In "applied mathematics" the "deductions" are given in the shape of the experimental evidence of natural science, and the hypotheses from which the "deductions" can be deduced are sought. Accordingly, every treatise on applied mathematics, properly so-called, is directed to the criticism of the "laws" from which the reasoning starts, or to a suggestion of results which experiment may hope to find. Thus if it calculates the result of some experiment, it is not the experimentalist's well-attested results which are on their trial, but the basis of the calculation. Newton's _Hypotheses non fingo_ was a proud boast, but it rests upon an entire misconception of the capacities of the mind of man in dealing with external nature.

_Synopsis of Existing Developments of Pure Mathematics._--A complete classification of mathematical sciences, as they at present exist, is to be found in the _International Catalogue of Scientific Literature_ promoted by the Royal Society. The classification in question was drawn up by an international committee of eminent mathematicians, and thus has the highest authority. It would be unfair to criticize it from an exacting philosophical point of view. The practical object of the enterprise required that the proportionate quantity of yearly output in the various branches, and that the liability of various topics as a matter of fact to occur in connexion with each other, should modify the classification.

Section A deals with pure mathematics. Under the general heading "_Fundamental Notions_" occur the subheadings "_Foundations of Arithmetic_," with the topics rational, irrational and transcendental numbers, and aggregates; "_Universal Algebra_," with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and "_Theory of Groups_," with the topics finite and continuous groups. For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS. Under the general heading "_Algebra and Theory of Numbers_" occur the subheadings "_Elements of Algebra_," with the topics rational polynomials, permutations, &c., partitions, probabilities; "_Linear Substitutions_," with the topics determinants, &c., linear substitutions, general theory of quantics; "_Theory of Algebraic Equations_," with the topics existence of roots, separation of and approximation to, theory of Galois, &c.; "_Theory of Numbers_," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY. Under the general heading "_Analysis_" occur the subheadings "_Foundations of Analysis_," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "_Theory of Functions of Complex Variables_," with the topics functions of one variable and of several variables; "_Algebraic Functions and their Integrals_," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "_Other Special Functions_," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "_Differential Equations_," with the topics existence theorems, methods of solution, general theory; "_Differential Forms and Differential Invariants_," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "_Analytical Methods connected with Physical Subjects_," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "_Difference Equations and Functional Equations_," with the topics recurring series, solution of equations of finite differences and functional equations. For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF. Under the general heading "_Geometry_" occur the subheadings "_Foundations_," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "_Elementary Geometry_," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "_Geometry of Conics and Quadrics_," with the implied topics; "_Algebraic Curves and Surfaces of Degree higher than the Second_," with the implied topics; "_Transformations and General Methods for Algebraic Configurations_," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "_Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry_," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "_Differential Geometry: applications of Differential Equations to Geometry_," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, _AXIOMS OF_; GEOMETRY, _EUCLIDEAN_; GEOMETRY, _PROJECTIVE_; GEOMETRY, _ANALYTICAL_; GEOMETRY, _LINE_; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; SURFACE; TRIGONOMETRY.

This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject. Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations. A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connexion with "number" and "quantity," while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily "apply" theorems of number. But the complexity of the idea of number is practically illustrated by the fact that it is best studied as a department of a science wider than itself.

_Synopsis of Existing Developments of Applied Mathematics._--Section B of the _International Catalogue_ deals with mechanics. The heading "_Measurement of Dynamical Quantities_" includes the topics units, measurements, and the constant of gravitation. The topics of the other headings do not require express mention. These headings are: "_Geometry and Kinematics of Particles and Solid Bodies_"; "_Principles of Rational Mechanics_"; "_Statics of Particles, Rigid Bodies, &c._"; "_Kinetics of Particles, Rigid Bodies, &c._"; "_General Analytical Mechanics_"; "_Statics and Dynamics of Fluids_"; "_Hydraulics and Fluid Resistances_"; "_Elasticity_." For the subjects of this general heading see the articles MECHANICS; DYNAMICS, ANALYTICAL; GYROSCOPE; HARMONIC ANALYSIS; WAVE; HYDROMECHANICS; ELASTICITY; MOTION, LAWS OF; ENERGY; ENERGETICS; ASTRONOMY (_Celestial Mechanics_); TIDE. Mechanics (including dynamical astronomy) is that subject among those traditionally classed as "applied" which has been most completely transfused by mathematics--that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.

Every branch of physics gives rise to an application of mathematics. A prophecy may be hazarded that in the future these applications will unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena. This reflection is suggested by the following articles: AETHER; MOLECULE; CAPILLARY ACTION; DIFFUSION; RADIATION, THEORY OF; and others.

The applications of mathematics to statistics (see STATISTICS and PROBABILITY) should not be lost sight of; the leading fields for these applications are insurance, sociology, variation in zoology and economics.

_The History of Mathematics._--The history of mathematics is in the main the history of its various branches. A short account of the history of each branch will be found in connexion with the article which deals with it. Viewing the subject as a whole, and apart from remote developments which have not in fact seriously influenced the great structure of the mathematics of the European races, it may be said to have had its origin with the Greeks, working on pre-existing fragmentary lines of thought derived from the Egyptians and Phoenicians. The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities. The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them--namely, ratio, irrationality, continuity, the point, the straight line, the plane. This period lasted[11] from the time of Thales, c. 600 B.C., to the capture of Alexandria by the Mahommedans, A.D. 641. The medieval Arabians invented our system of numeration and developed algebra. The next period of advance stretches from the Renaissance to Newton and Leibnitz at the end of the 17th century. During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation. The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace. This period may be conceived as continuing throughout the first quarter of the 19th century. It was remarkable both for the brilliance of its achievements and for the large number of French mathematicians of the first rank who flourished during it. The next period was inaugurated in analysis by K. F. Gauss, N. H. Abel and A. L. Cauchy. Between them the general theory of the complex variable, and of the various "infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces. The applied mathematical sciences of light, electricity and electromagnetism, and of heat, were now largely developed. This school of mathematical thought lasted beyond the middle of the century, after which a change and further development can be traced. In the next and last period the progress of pure mathematics has been dominated by the critical spirit introduced by the German mathematicians under the guidance of Weierstrass, though foreshadowed by earlier analysts, such as Abel. Also such ideas as those of invariants, groups and of form, have modified the entire science. But the progress in all directions has been too rapid to admit of any one adequate characterization. During the same period a brilliant group of mathematical physicists, notably Lord Kelvin (W. Thomson), H. V. Helmholtz, J. C. Maxwell, H. Hertz, have transformed applied mathematics by systematically basing their deductions upon the Law of the conservation of energy, and the hypothesis of an ether pervading space.

BIBLIOGRAPHY.--References to the works containing expositions of the various branches of mathematics are given in the appropriate articles. It must suffice here to refer to sources in which the subject is considered as one whole. Most philosophers refer in their works to mathematics more or less cursorily, either in the treatment of the ideas of number and magnitude, or in their consideration of the alleged a priori and necessary truths. A bibliography of such references would be in effect a bibliography of metaphysics, or rather of epistemology. The founder of the modern point of view, explained in this article, was Leibnitz, who, however, was so far in advance of contemporary thought that his ideas remained neglected and undeveloped until recently; cf. _Opuscules et fragments inédits de Leibnitz. Extraits des manuscrits de la bibliothèque royale de Hanovre_, by Louis Couturat (Paris, 1903), especially pp. 356-399, "Generales inquisitiones de analysi notionum et veritatum" (written in 1686); also cf. _La Logique de Leibnitz_, already referred to. For the modern authors who nave rediscovered and improved upon the position of Leibnitz, cf. _Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet von Dr G. Frege, a.o. Professor an der Univ. Jena_ (Bd. i., 1893; Bd. ii., 1903, Jena); also cf. Frege's earlier works, _Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens_ (Halle, 1879), and _Die Grundlagen der Arithmetik_ (Breslau, 1884); also cf. Bertrand Russell, _The Principles of Mathematics_ (Cambridge, 1903), and his article on "Mathematical Logic" in _Amer. Quart. Journ. of Math._ (vol. xxx., 1908). Also the following works are of importance, though not all expressly expounding the Leibnitzian point of view: cf. G. Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre," _Math. Annal._, vol. xxi. (1883) and subsequent articles in vols. xlvi. and xlix.; also R. Dedekind, _Stetigkeit und irrationales Zahlen_ (1st ed., 1872), and _Was sind und was sollen die Zahlen?_ (1st ed., 1887), both tracts translated into English under the title _Essays on the Theory of Numbers_ (Chicago, 1901). These works of G. Cantor and Dedekind were of the greatest importance in the progress of the subject. Also cf. G. Peano (with various collaborators of the Italian school), _Formulaire de mathématiques_ (Turin, various editions, 1894-1908; the earlier editions are the more interesting philosophically); Felix Klein, _Lectures on Mathematics_ (New York, 1894); W. K. Clifford, _The Common Sense of the exact Sciences_ (London, 1885); H. Poincaré, _La Science el l'hypothèse_ (Paris, 1st ed., 1902), English translation under the title, _Science and Hypothesis_ (London, 1905); L. Couturat, _Les Principes des mathématiques_ (Paris, 1905); E. Mach, _Die Mechanik in ihrer Entwickelung_ (Prague, 1883), English translation under the title, _The Science of Mechanics_ (London, 1893); K. Pearson, _The Grammar of Science_ (London, 1st ed., 1892; 2nd ed., 1900, enlarged); A. Cayley, _Presidential Address_ (Brit. Assoc., 1883); B. Russell and A. N. Whitehead, _Principia Mathematica_ (Cambridge, 1911). For the history of mathematics the one modern and complete source of information is M. Cantor's _Vorlesungen über Geschichte der Mathematik_ (Leipzig, 1st Bd., 1880; 2nd Bd., 1892; 3rd Bd., 1898; 4th Bd., 1908; 1st Bd., _von den ältesten Zeiten bis zum Jahre 1200, n. Chr._; 2nd Bd., _von 1200-1668_; 3rd Bd., _von 1668-1758_; 4th Bd., _von 1795 bis 1790_); W. W. R. Ball, _A Short History of Mathematics_ (London 1st ed., 1888, three subsequent editions, enlarged and revised, and translations into French and Italian). (A. N. W.)

FOOTNOTES:

[1] Cf. _La Logique de Leibnitz_, ch. vii., by L. Couturat (Paris, 1901).

[2] Cf. _The Principles of Mathematics_, by Bertrand Russell (Cambridge, 1903).

[3] Cf. _Formulaire mathématique_ (Turin, ed. of 1903); earlier formulations of the bases of arithmetic are given by him in the editions of 1898 and of 1901. The variations are only trivial.

[4] Cf. Russell, _loc. cit._, pp. 199-256.

[5] The first unqualified explicit statement of _part_ of this definition seems to be by B. Peirce, "Mathematics is the science which draws necessary conclusions" (_Linear Associative Algebra_, § i. (1870), republished in the _Amer. Journ. of Math._, vol. iv. (1881)). But it will be noticed that the second half of the definition in the text--"from the general premisses of all reasoning"--is left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell, _loc. cit._

[6] "Una questione sui numeri transfiniti," _Rend. del circolo mat. di Palermo_, vol. xi. (1897); and Russell, _loc. cit._, ch. xxxviii.

[7] Cf. Russell, _loc. cit._, ch. x.

[8] Cf. _Pragmatism: a New Name for some Old Ways of Thinking_ (1907).

[9] Due to Bertrand Russell, cf. "Mathematical Logic as based on the Theory of Types," _Amer. Journ. of Math._ vol. xxx. (1908). It is more fully explained by him, with later simplifications, in _Principia mathematica_ (Cambridge).

[10] Cf. Stanley's _Eastern Church_, Lecture v.

[11] Cf. _A Short History of Mathematics_, by W. W. R. Ball.

MATHER, COTTON (1663-1728), American Congregational clergyman and author, was born in Boston, Massachusetts, on the 12th of February 1663. He was the grandson of Richard Mather, and the eldest child of Increase Mather (q.v.), and Maria, daughter of John Cotton. After studying under the famous Ezekiel Cheever (1614-1708), he entered Harvard College at twelve, and graduated in 1678. While teaching (1678-1685), he began the study of theology, but soon, on account of an impediment in his speech, discontinued it and took up medicine. Later, however, he conquered the difficulty and finished his preparation for the ministry. He was elected assistant pastor in his father's church, the North, or Second, Church of Boston, in 1681 and was ordained as his father's colleague in 1685. In 1688, when his father went to England as agent for the colony, he was left at twenty-five in charge of the largest congregation in New England, and he ministered to it for the rest of his life. He soon became one of the most influential men in the colonies. He had much to do with the witchcraft persecution of his day; in 1692 when the magistrates appealed to the Boston clergy for advice in regard to the witchcraft cases in Salem he drafted their reply, upon which the prosecutions were based; in 1689 he had written _Memorable Providences Relating to Witchcraft and Possessions_, and even his earlier diaries have many entries showing his belief in diabolical possession and his fear and hatred of it. Thinking as he did that the New World had been the undisturbed realm of Satan before the settlements were made in Massachusetts, he considered it natural that the Devil should make a peculiar effort to bring moral destruction on these godly invaders. He used prayer and fasting to deliver himself from evil enchantment; and when he saw ecstatic and mystical visions promising him the Lord's help and great usefulness in the Lord's work, he feared that these revelations might be of diabolic origin. He used his great influence to bring the suspected persons to trial and punishment. He attended the trials, investigated many of the cases himself, and wrote sermons on witchcraft, the _Memorable Providences_ and _The Wonders of the Invisible World_ (1693), which increased the excitement of the people. Accordingly, when the persecutions ceased and the reaction set in, much of the blame was laid upon him; the influence of Judge Samuel Sewall, after he had come to think his part in the Salem delusion a great mistake, was turned against the Mathers; and the liberal leaders of Congregationalism in Boston, notably the Brattles, found this a vulnerable point in Cotton Mather's armour and used their knowledge to much effect, notably by assisting Robert Calef (d. c. 1723) in the preparation of _More Wonders of the Invisible World_ (1700) a powerful criticism of Cotton Mather's part in the delusion at Salem.

Mather took some part as adviser in the Revolution of 1689 in Massachusetts. In 1690 he became a member o£ the Corporation (probably the youngest ever chosen as Fellow) of Harvard College, and in 1707 he was greatly disappointed at his failure to be chosen president of that institution. He received the degree of D.D. from the University of Glasgow in 1710, and in 1713 was made a Fellow of the Royal Society. Like his father he was deeply grieved by the liberal theology and Church polity of the new Brattle Street Congregation, and conscientiously opposed its pastor Benjamin Colman, who had been irregularly ordained in England and by a Presbyterian body; but with his father he took part in 1700 in services in Colman's church. Harvard College was now controlled by the Liberals of the Brattle Street Church, and as it grew farther and farther away from Calvinism, Mather looked with increasing favour upon the college in Connecticut; before September 1701 he had drawn up a "scheme for a college," the oldest document now in the Yale archives; and finally (Jan. 1718) he wrote to a London merchant, Elihu Yale, and persuaded him to make a liberal gift to the college, which was named in his honour. During the small-pox epidemic of 1721 he attempted in vain to have treatment by inoculation employed, for the first time in America; and for this he was bitterly attacked on all sides, and his life was at one time in danger; but, nevertheless, he used the treatment on his son, who recovered, and he wrote _An Account of the Method and further Success of Inoculating for the Small Pox in London_ (1721). In addition he advocated temperance, missions, Bible societies, and the education of the negro; favoured the establishing of libraries for working men and of religious organizations for young people, and organized societies for other branches of philanthropic work. His later years were clouded with many sorrows and disappointments; his relations with Governor Joseph Dudley were unfriendly; he lost much of his former prestige in the Church--his own congregation dwindled--and in the college; his uncle John Cotton was expelled from his charge in the Plymouth Church; his son Increase turned out a ne'er-do-well; four of his children and his second wife died in November 1713; his wife's brothers and the husbands of his sisters were ungodly and violent men; his favourite daughter Katherine, who "understood Latin and read Hebrew fluently," died in 1716; his third wife went mad in 1719; his personal enemies circulated incredible scandals about him; and in 1724-1725 he saw a Liberal once more preferred to him as a new president of Harvard. He died in Boston on the 13th of February 1728 and is buried in the Copps Hill burial-ground, Boston. He was thrice married--to Abigail Phillips (d. 1702) in 1686, to Mrs Elizabeth Hubbard (d. 1713) in 1703, and in 1715 to Mrs Lydia George (d. 1734). Of his fifteen children only two survived him.

Though self-conscious and vain, Cotton Mather had on the whole a noble character. He believed strongly in the power of prayer and repeatedly had assurances that his prayers were heard; and when he was disappointed by non-fulfilment his grief and depression were terrible. His spiritual nature was high-strung and delicate; and this condition was aggravated by his constant study, his long fasts and his frequent vigils--in one year, according to his diary, he kept sixty fasts and twenty vigils. In his later years his diaries have less and less of personal detail, and repeated entries prefaced by the letters "G.D." meaning Good Device, embodying precepts of kindliness and practical Christianity. He was remarkable for his godliness, his enthusiasm for knowledge, and his prodigious memory. He became a skilled linguist, a widely read scholar--though much of his learning was more curious than useful--a powerful preacher, a valued citizen, and a voluminous writer, and did a vast deal for the intellectual and spiritual quickening of New England. He worked with might and main for the continuation of the old theocracy, but before he died it had given way before an increasing Liberalism--even Yale was infected with the Episcopalianism that he hated.

Among his four hundred or more published works, many of which are sermons, tracts and letters, the most notable is his _Magnalia Christi Americana: or the Ecclesiastical History of New England, from Its First Planting in the Year 1620 unto the Year of Our Lord, 1698_. Begun in 1693 and finished in 1697, this work was published in London, in 1702, in one volume, and was republished in Hartford in 1820 and in 1853-1855, in two volumes. It is in seven books and concerns itself mainly with the settlement and religious history of New England. It is often inaccurate, and it abounds in far-fetched conceits and odd and pedantic features. Its style, though in the main rather unnatural and declamatory, is at its best spontaneous, dignified and rhythmical; the book is valuable for occasional facts and for its picture of the times, and it did much to make Mather the most eminent American writer of his day. His other writings include _A Poem Dedicated to the Memory of the Reverend and Excellent Mr Urian Oakes_ (1682); _The Present State of New England_ (1690); _The Life of the Renowned John Eliot_ (1691), later included in Book III. of the _Magnalia; The Short History of New England_ (1694); _Bonifacius_, usually known as _Essays To Do Good_ (Boston, 1710; Glasgow, 1825; Boston, 1845), one of his principal books and one which had a shaping influence on the life of Benjamin Franklin; _Psalterium Americanum_ (1718), a blank verse translation of the Psalms from the original Hebrew; _The Christian Philosopher: A Collection of the Best Discoveries in Nature, with Religious Improvements_ (1721); _Parentator_ (1724), a memoir of his father; _Ratio Disciplinae_ (1726), an account of the discipline in New England churches; _Manuductio ad Ministerium: Directions for a Candidate of the Ministry_ (1726), one of the most readable of his books. He also left a number of works in manuscript, including diaries, a medical treatise and a huge commentary on the Bible, entitled "Biblia Americana."

See _The Life of Cotton Mather_ (Boston, 1729), by his son, Samuel Mather; William B. O. Peabody, _The Life of Cotton Mather_ (1836) (in Jared Sparks's "Library of American Biography," vol. vi.); Enoch Pond, _The Mather Family_ (Boston, 1844); John L. Sibley, _Biographical Sketches of Graduates of Harvard University_, vol. iii. (Cambridge, 1885); Barrett Wendell, _Cotton Mather, the Puritan Priest_ (New York, 1891), a remarkably sympathetic study and particularly valuable for its insight into (and its defence of) Mather's attitude toward witchcraft; Abijah P. Marvin, _The Life and Times of Cotton Mather_ (Boston, 1892); M. C. Tyler, _A History of American Literature during the Colonial Period_, vol. ii. (New York, 1878); and Barrett Wendell, _A Literary History of America_ (New York, 1900).

Cotton Mather's son, SAMUEL MATHER (1706-1785), also a clergyman, graduated at Harvard in 1723, was pastor of the North Church, Boston, from 1732 to 1742, when, owing to a dispute among his congregation over revivals, he resigned to take charge of a church established for him in North Bennett Street.

Among his works are _The Life of Cotton Mather_ (1729); _An Apology for the Liberties of the Churches in New England_ (1738), and _America Known to the Ancients_ (1773). (W. L. C.*)

MATHER, INCREASE (1639-1723), American Congregational minister, was born in Dorchester, Massachusetts, on the 21st of June 1639, the youngest son of Richard Mather.[1] He entered Harvard in 1651, and graduated in 1656. In 1657, on his eighteenth birthday, he preached his first sermon; in the same year he went to visit his eldest brother in Dublin, and studied there at Trinity College, where he graduated M.A. in 1658. He was chaplain to the English garrison at Guernsey in April-December 1659 and again in 1661; and in the latter year, refusing valuable livings in England offered on condition of conformity, he returned to America. In the winter of 1661-1662 he began to preach to the Second (or North) Church of Boston, and was ordained there on the 27th of May 1664. As a delegate from Dorchester, his father's church, to the Synod of 1662, he opposed the Half-Way Covenant adopted by the Synod and defended by Richard Mather and by Jonathan Mitchell (1624-1668) of Cambridge; but soon afterwards he "surrendered a glad captive" to "the truth so victoriously cleared by Mr Mitchell," and like his father and his son became one of the chief exponents of the Half-Way Covenant. He was bitterly opposed, however, to the liberal practices that followed the Half-Way Covenant and (after 1677) in particular to "Stoddardeanism," the doctrine of Solomon Stoddard (1643-1729) that all "such Persons as have a good Conversation and a Competent Knowledge may come to the Lord's Supper," only those of openly immoral life being excluded. In May 1679 Mather was a petitioner to the General Court for the call of a Synod to consider the reformation in New England of "the Evils that have Provoked the Lord to bring his Judgments,"[2] and when the "Reforming Synod" met in September it appointed him one of a committee to draft a creed; this committee reported in May 1680, at the Synod's second session, of which Mather was moderator, the Savoy Declaration (slightly modified, notably in ch. xxiv., "Of the Civil Magistrate"), which was approved but was not made mandatory on the churches by the General Court, and in 1708 was reaffirmed at Saybrook, Connecticut. With the Cambridge Platform of 1646, drafted by his father, the Confession of 1680, for which Increase Mather was largely responsible, was printed as a book of doctrine and government for the churches of Massachusetts.

After the threat of a _Quo Warranto_ writ in 1683 for the surrender of the Massachusetts charter, Mather used all his tremendous influence to persuade the colonists not to give up the charter; and the Boston freemen unanimously voted against submission. The royal agents immediately afterwards sent to London a treasonable letter, falsely attributed to Mather; but its spuriousness seems to have been suspected in England and Mather was not "fetch'd over and made a Sacrifice." He became a leader in the opposition to Sir Edmund Andros, to his secretary Edward Randolph, and to Governor Joseph Dudley. He was chosen by the General Court to represent the colony's interests in England, eluded officers sent to arrest him,[3] and in disguise boarded a ship on which he reached Weymouth on the 6th of May 1688. In London he acted with Sir Henry Ashurst, the resident agent, and had two or three fruitless audiences with James II. His first audience with William III. was on the 9th of January 1689; he was active in influencing the Commons to vote (1689) that the New England charters should be restored; and he published _A Narrative of the Miseries of New-England, By Reason of an Arbitrary Government Erected there Under Sir Edmund Andros_ (1688), _A Brief Relation for the Confirmation of Charter Privileges_ (1691), and other pamphlets. In 1690 he was joined by Elisha Cooke (1638-1715) and Thomas Oakes (1644-1719), additional agents, who were uncompromisingly for the renewal of the old charter. Mather, however, was instrumental in securing a new charter (signed on Oct. 7, 1691), and prevented the annexation of the Plymouth Colony to New York. The nomination of officers left to the Crown was reserved to the agents. Mather had expressed strong dissatisfaction with the clause giving the governor the right of veto, and regretted the less theocratic tone of the charter which made all freemen (and not merely church members) electors. With Sir William Phips, the new governor, a member of Mather's church, he arrived in Boston on the 14th of May 1692. The value of his services to the colony at this time is not easily over-estimated. In England he won the friendship of divines like Baxter, Tillotson and Burnet, and effectively promoted the union in 1691 of English Presbyterians and Congregationalists. He was at heavy expense throughout his stay, and even greater than his financial loss was his loss of authority and control in the church and in Harvard College because of his absence.

Mather had been acting president of Harvard College in 1681-1682, and in June 1685 he again became acting president (or rector), but still preached every Sunday in Boston and would not comply with an order of the General Court that he should reside in Cambridge. In 1701 after a short residence there he returned to Boston and wrote to the General Court to "think of another President for the Colledge." The opposition to him had been increasing in strength, his resignation was accepted, and Samuel Willard took charge of the college as vice-president, although he also refused to reside in Cambridge. That Mather's administration of the college was excellent is admitted even by his harsh critic, Josiah Quincy, in his _History of Harvard University_.[4] The Liberal party, which now came into control in the college repeatedly disappointed the hopes of Cotton Mather (q.v.) that he might be chosen president, and by its ecclesiastical laxness and its broader views of Church polity forced the Mathers to turn from Harvard to Yale as a truer school of the prophets.

The Liberal leaders, John Leverett (1662-1724), William Brattle (1662-1713)--who graduated with Leverett in 1680, and with him as tutor controlled the college during Increase Mather's absence in England--William Brattle's eldest brother, Thomas Brattle (1658-1713), and Ebenezer Pemberton (1671-1717), pastor of the Old South Church, desired an "enrichment of the service," and greater liberality in the matter of baptism. In 1697 the Second Boston Church, in which Cotton Mather had been his father's colleague since 1685, upbraided the Charlestown Church "for betraying the liberties of the churches in their late putting into the hands of the whole inhabitants the choice of a minister." In 1699 Increase Mather published _The Order of the Gospel_, which severely (although indirectly), criticized the methods of the "Liberals" in establishing the Brattle Street Church and especially the ordination of their minister Benjamin Colman by a Presbyterian body in London; the Liberals replied with _The Gospel Order Revived_, which was printed in New York to lend colour to the (partly true) charge of its authors that the printers of Massachusetts would print nothing hostile to Increase Mather.[5] The autocracy of the Mathers in church, college, colony and press, had slipped from them. The later years of Mather's life were spent almost entirely in the work of the ministry, now beginning to be a less varied career than when he entered on it. He died on the 23rd of August 1723. He married in 1662 Maria, daughter of Sarah and John Cotton. His first wife died in 1714; and in 1715 he married Ann Lake, widow of John Cotton, of Hampton, N.H., a grandson of John Cotton of Boston.

Increase Mather was a great preacher with a simple style and a splendid voice, which had a "Tonitruous Cogency," to quote his son's phrase. His style was much simpler and more vernacular than his son's. He was an assiduous student, commonly spending sixteen hours a day among his books; but his learning (to quote Justin Winsor's contrast between Increase and Cotton Mather) "usually left his natural ability and his education free from entanglements." He was not so much self-seeking and personally ambitious as eager to advance the cause of the church in which he so implicitly believed. That it is a mistake to consider him a narrow churchman is shown by his assisting in 1718 at the ordination of Elisha Callender in the First Baptist Church of Boston. Like the most learned men of his time he was superstitious and a firm believer in "praesagious impressions"; his _Essay for the Recording of Illustrious Providences: Wherein an Account is Given of many Remarkable and very Memorable Events which have Hapned in this Last Age, Especially in New England_ (1684) shows that he believed only less thoroughly than his son in witchcraft, though in his _Cases of Conscience Concerning Evil Spirits_ (1693) he considered some current proofs of witchcraft inadequate. The revulsion of feeling after the witchcraft delusion undermined his authority greatly, and Robert's Calef's _More Wonders of the Spiritual World_ (1700) was a personal blow to him as well as to his son. With Jonathan Edwards, than whom he was much more of a man of affairs, and with Benjamin Franklin, whose mission in England somewhat resembled Mather's, he may be ranked among the greatest Americans of the period before the War of Independence.

The first authority for the life of Increase Mather is the work of his son Cotton Mather, _Parentator: Memoirs of Remarkables in the Life and Death of the Ever Memorable Dr Increase Mather_ (Boston, 1724); there are also a memoir and constant references in Cotton Mather's _Magnalia_ (London, 1702) especially vol. iv.; there is an excellent sketch in the first volume of J. L. Sibley's _Biographical Sketches of Graduates of Harvard University_ (Cambridge, 1873), with an exhaustive list of Mather's works (about 150 titles); there is much valuable matter in Williston Walker's _Ten New England Leaders_ (New York, 1901) and in his _Creeds and Platforms of Congregationalism_ (New York, 1893); for literary criticism of the Mathers see ch. xii. of M. C. Tyler's _History of American Literature, 1607-1676_ (New York, 1878), and Barrett Wendell's _Cotton Mather_ (New York, 1891). Mather's worth has been under-estimated by Josiah Quincy, Justin Winsor and other historians out of sympathy with his ecclesiastical spirit, who represent him as only an ambitious narrow-minded schemer. (R. We.)

FOOTNOTES:

[1] He was so christened "because of the never-to-be-forgotten increase, of every sort, wherewith God favoured the country about the time of his nativity." He often latinized his name, spelling it _Crescentius Matherus_.

[2] That is, King Philip's War, the Boston fires of 1676, when Mather's church and home were burned, and 1679, the threatened introduction of Episcopacy, and the general spiritual decay of the country.

[3] He had previously been arrested and acquitted on a charge of having attributed the forged letter to Randolph.

[4] Mather led the resistance to the royal demand instigated by Edward Randolph in 1683, for the annulment of the college charter, and after its vacation in 1684 strove for the grant of a new charter; King James promised him a confirmation of the former charter; the new provincial charter granted by William and Mary confirmed all gifts and grants to colleges; in 1692 Mather drafted an act incorporating the college, which was signed by Phips but was disallowed in England; and in 1696, 1697, 1699, and 1700, Mather repeated his efforts for a college charter.

[5] Mather was made a licenser of the Press in 1674 when the General Court abolished the monopoly of the Cambridge Press.

MATHER, RICHARD (1596-1669), American Congregational clergyman, was born in Lowton, in the parish of Winwick, near Liverpool, England, of a family which was in reduced circumstances but entitled to bear a coat-of-arms. He studied at Winwick grammar school, of which he was appointed a master in his fifteenth year, and left it in 1612 to become master of a newly established school at Toxteth Park, Liverpool. After a few months at Brasenose College, Oxford, he began in November 1618 to preach at Toxteth, and was ordained there, possibly only as deacon, early in 1619. In August-November 1633 he was suspended for nonconformity in matters of ceremony; and in 1634 was again suspended by the visitors of Richard Neile, archbishop of York, who, hearing that he had never worn a surplice during the fifteen years of his ministry, refused to reinstate him and said that "it had been better for him that he had gotten Seven Bastards." He had a great reputation as a preacher in and about Liverpool; but, advised by letters of John Cotton and Thomas Hooker, and persuaded by his own elaborate formal "Arguments tending to prove the Removing from Old-England to New ... to be not only lawful, but also necessary for them that are not otherwise tyed, but free," he left England and on the 17th of August 1635, and landed in Boston after an "extraordinary and miraculous deliverance" from a terrible storm. As a famous preacher "he was desired at Plimouth, Dorchester, and Roxbury." He went to Dorchester, where the Church had been greatly depleted by migrations to Windsor, Connecticut; and where, after a delay of several months, in August 1636 there was constituted by the consent of magistrates and clergy a church of which he was "teacher" until his death in Dorchester on the 22nd of April 1669.

He was an able preacher, "aiming," said his biographer, "to shoot his arrows not over his people's heads, but into their Hearts and Consciences"; and he was a leader of New England Congregationalism, whose policy he defended and described in the tract _Church Government and Church Covenant Discussed, in an Answer of the Elders of the Severall Churches of New England to Two and Thirty Questions_ (written 1639; printed 1643), and in his _Reply to Mr Rutherford_ (1647), a polemic against the Presbyterianism to which the English Congregationalists were then tending. He drafted the Cambridge Platform, an ecclesiastical constitution in seventeen chapters, adopted (with the omission of Mather's paragraph favouring the "Half-way Covenant," of which he strongly approved) by the general synod in August 1646. In 1657 he drafted the declaration of the Ministerial Convention on the meaning and force of the Half-way Covenant; this was published in 1659 under the title: _A Disputation concerning Church Members and their Children in Answer to XXI. Questions_. With Thomas Welde and John Eliot he wrote the "Bay Psalm Book," or, more accurately, _The Whole Booke of Psalmes Faithfully Translated into English Metre_ (1640), probably the first book printed in the English colonies.

He married in 1624 Katherine Hoult or Holt (d. 1655), and secondly in 1656 Sarah Hankredge (d. 1676), the widow of John Cotton. Of six sons, all by his first wife, four were ministers: SAMUEL (1626-1671), the first fellow of Harvard College who was a graduate, chaplain of Magdalen College, Oxford, in 1650-1653, and pastor (1656-1671, excepting suspension in 1660-1662) of St Nicholas's in Dublin; NATHANIEL (1630-1697), who graduated at Harvard in 1647, was vicar of Barnstaple, Devon, in 1656-1662, pastor of the English Church in Rotterdam, his brother's successor in Dublin in 1671-1688, and then until his death pastor of a church in London; ELEAZAR (1637-1669), who graduated at Harvard in 1656 and after preaching in Northampton, Massachusetts, for three years, became in 1661 pastor of the church there; and INCREASE MATHER (q.v.). Horace E. Mather, in his _Lineage of Richard Mather_ (Hartford, Connecticut, 1890), gives a list of 80 clergymen descended from Richard Mather, of whom 29 bore the name Mather and 51 other names, the more famous being Storrs and Schauffler.

See _The Life and Death of That Reverend Man of God, Mr Richard Mather_ (Cambridge, 1670; reprinted 1850, with his _Journal_ for 1635, by the Dorchester Antiquarian and Historical Society), with an introduction by Increase Mather, who may have been the author; W. B. Sprague's _Annals of the American Pulpit_, vol. i. (New York, 1857); Cotton Mather's _Magnalia_ (London, 1702); an essay on Richard Mather in Williston Walker's _Ten New England Leaders_ (New York, 1901); and the works referred to in the article on Increase Mather. (R. We.)

MATHERAN, a hill sanatorium in India, in the Kolaba district of Bombay, 2460 ft. above the sea, and about 30 m. E. of Bombay city. Pop. (1901), 3060. It consists of several thickly wooded ridges, on a spur of the Western Ghats, with a magnificent outlook over the plain below and the distant sea. First explored in 1850, it has since become the favourite resort of the middle classes of Bombay (especially the Parsis) during the spring and autumn months. It has recently been connected by a 2 ft. gauge mountain line with Neral station on the Great Indian Peninsula railway, 54 m. from Bombay.

MATHESON, GEORGE (1842-1906), Scottish theologian and preacher, was born in Glasgow in 1842, the son of George Matheson, a merchant. He was educated at the university of Glasgow, where he graduated first in classics, logic and philosophy. In his twentieth year he became totally blind, but he held to his resolve to enter the ministry, and gave himself to theological and historical study. His first ministry began in 1868 at Innellan, on the Argyllshire coast between Dunoon and Toward. His books on _Aids to the Study of German Theology, Can the Old Faith live with the New? The Growth of the Spirit of Christianity from the First Century to the Dawn of the Lutheran Era_, established his reputation as a liberal and spiritually minded theologian; and Queen Victoria invited him to preach at Balmoral. In 1886 he removed to Edinburgh, where he became minister of St Bernard's Parish Church. Here his chief work as a preacher was done. In 1879 the university of Edinburgh conferred upon him the honorary degree of D.D., and the same year he declined an invitation to the pastorate of Crown Court, London, in succession to Dr John Cumming (1807-1881). In 1881 he was chosen as Baird lecturer, and took for his subject "Natural Elements of Revealed Theology," and in 1882 he was the St Giles lecturer, his subject being "Confucianism." In 1890 he was elected a fellow of the Royal Society of Edinburgh, Aberdeen gave him its honorary LL.D., and in 1899 he was appointed Gifford lecturer by that university, but declined on grounds of health. In the same year he severed his active connexion with St Bernard's. One of his hymns, "O love that will not let me go," has passed into the popular hymnology of the Christian Church. He died suddenly of apoplexy on the 28th of August 1906. His exegesis owes its interest to his subjective resources rather than to breadth of learning; his power lay in spiritual vision rather than balanced judgment, and in the vivid apprehension of the factors which make the Christian personality, rather than in constructive doctrinal statement.

MATHEW, THEOBALD (1790-1856), Irish temperance reformer, popularly known as Father Mathew, was descended from a branch of the Llandaff family, and was born at Thomastown, Tipperary, on the 10th of October 1790. He received his school education at Kilkenny, whence he passed for a short time to Maynooth; from 1808 to 1814 he studied at Dublin, where in the latter year he was ordained to the priesthood. Having entered the Capuchin order, he, after a brief time of service at Kilkenny, joined the mission in Cork, which was the scene of his religious and benevolent labours for many years. The movement with which his name is most intimately associated began in 1838 with the establishment of a total abstinence association, which in less than nine months, thanks to his moral influence and eloquence, enrolled no fewer than 150,000 names. It rapidly spread to Limerick and elsewhere, and some idea of its popularity may be formed from the fact that at Nenagh 20,000 persons are said to have taken the pledge in one day, 100,000 at Galway in two days, and 70,000 in Dublin in five days. In 1844 he visited Liverpool, Manchester and London with almost equal success. Meanwhile the expenses of his enterprise had involved him in heavy liabilities, and led on one occasion to his arrest for debt; from this embarrassment he was only partially relieved by a pension of £300 granted by Queen Victoria in 1847. In 1849 he paid a visit to the United States, returning in 1851. He died at Queenstown on the 8th of December, 1856.

See _Father Mathew, a Biography_, by J. F. Maguire, M.P. (1863).

MATHEWS, CHARLES (1776-1835), English actor, was born in London on the 28th of June 1776. His father was "a serious bookseller," who also officiated as minister in one of Lady Huntingdon's chapels. Mathews was educated at Merchant Taylors' School. His love for the stage was formed in his boyhood, when he was apprentice to his father, and the latter in 1794 unwillingly permitted him to enter on a theatrical engagement in Dublin. For several years Mathews had not only to content himself with thankless parts at a low salary, but in May 1803 he made his first London appearance at the Haymarket as Jabel in Cumberland's _The Jew_ and as Lingo in _The Agreeable Surprise_. From this time his professional career was an uninterrupted triumph. He had a wonderful gift of mimicry, and could completely disguise his personality without the smallest change of dress. The versatility and originality of his powers were admirably displayed in his "At Homes," begun in the Lyceum theatre in 1818, which, according to Leigh Hunt, "for the richness and variety of his humour, were as good as half a dozen plays distilled." Off the stage his simple and kind-hearted disposition won him affection and esteem. In 1822 Mathews visited America, his observation on his experiences there forming for the reader a most entertaining portion of his biography. From infancy his health had been uncertain, and the toils of his profession gradually undermined it. In 1834 he paid a second visit to America. His last appearance in New York was on the 11th of February 1835, when he played Samuel Coddle in _Married Life_ and Andrew Steward in _The Lone House_. He died at Plymouth on the 28th of June 1835. In 1797 he had married Eliza Kirkham Strong (d. 1802), and in 1803 Anne Jackson, an actress, the author of the popular and diverting _Memoirs, by Mrs Mathews_ (4 vols., 1838-1839).

His son CHARLES JAMES MATHEWS (1803-1878), who was born at Liverpool on the 26th of December 1803, became even better known as an actor. After attending Merchant Taylors' School he was articled as pupil to an architect, and continued for some years nominally to follow this profession. His first public appearance on the stage was made on the 7th of December 1835, at the Olympic, London, as George Rattleton in his own play _The Humpbacked Lover_, and as Tim Topple the Tiger in Leman Rode's _Old and Young Stager_. In 1838 he married Madame Vestris, then lessee of the Olympic, but neither his management of this theatre, nor subsequently of Covent Garden, nor of the Lyceum, resulted in pecuniary success, although the introduction of scenery more realistic and careful in detail than had hitherto been employed was due to his enterprise. In the year of his marriage he visited America, but without receiving a very cordial welcome. As an actor he held in England an unrivalled place in his peculiar vein of light eccentric comedy. The easy grace of his manner, and the imperturbable solemnity with which he perpetrated his absurdities, never failed to charm and amuse; his humour was never broad, but always measured and restrained. It was as the leading character in such plays as the _Game of Speculation_, _My Awful Dad_, _Cool as a Cucumber_, _Patter versus Clatter_, and _Little Toddlekins_, that he specially excelled. In 1856 Mme Vestris died, and in the following year Mathews again visited the United States, where in 1858 he married Mrs A. H. Davenport. In 1861 they gave a series of "At Homes" at the Haymarket theatre, which were almost as popular as had been those of the elder Mathews. Charles James Mathews was one of the few English actors who played in French successfully,--his appearance in Paris in 1863 in a French version of _Cool as a Cucumber_, written by himself, being received with great approbation. He also played there again in 1865 as Sir Charles Coldcream in the original play _L'Homme blasé_ (English version by Boucicault, _Used up_). After reaching his sixty-sixth year, Mathews set out on a tour round the world, in which was included a third visit to America, and on his return in 1872 he continued to act without interruption till within a few weeks of his death on the 24th of June 1878. He made his last appearance in New York at Wallack's theatre on the 7th of June 1872, in H. J. Byron's _Not such a Fool as he Looks_. His last appearance in London was at the Opéra Comique on the 2nd of June 1877, in _The Liar_ and _The Cosy Couple_. At Stalybridge he gave his last performance on the 8th of June 1878, when he played Adonis Evergreen in his own comedy _My Awful Dad_.

See the _Life of Charles James Mathews_, edited by Charles Dickens (2 vols., 1879); H. G. Paine in _Actors and Actresses of Great Britain and the United States_ (New York, 1886).

MATHEWS, THOMAS (1676-1751), British admiral, son of Colonel Edward Mathews (d. 1700), and grandson on his mother's side of Sir Thomas Armstrong (1624-1684), who was executed for the Rye House Plot, was born at Llandaff Court, Llandaff. He entered the navy and became lieutenant in 1699, being promoted captain in 1703. During the short war with Spain (1718-20) he commanded the "Kent" in the fleet of Sir George Byng (Lord Torrington), and from 1722 to 1724 he had the command of a small squadron sent to the East Indies to repress the pirates of the coast of Malabar. He saw no further service till March 1741, when he was appointed to the command in the Mediterranean, and plenipotentiary to the king of Sardinia and the other courts of Italy. It is impossible to understand upon what grounds he was selected. As an admiral he was not distinguished; he was quite destitute of the experience and the tact required for his diplomatic duties; and he was on the worst possible terms with his second in command, Richard Lestock (1679?-1746). Yet the purpose for which he was sent out in his double capacity was not altogether ill performed. In 1742 Mathews sent a small squadron to Naples to compel King Charles III., afterwards king of Spain, to remain neutral. It was commanded by commodore, afterwards admiral, William Martin (1696?-1756), who refused to enter into negotiations, and gave the king half an hour in which to return an answer. In June of the same year a squadron of Spanish galleys, which had taken refuge in the Bay of Saint Tropez, was burnt by the fireships of Mathews' fleet. In the meantime a Spanish squadron of line-of-battleships had taken refuge in Toulon, and was watched by the British fleet from its anchorage at Hyères. In February 1744 the Spaniards put to sea in company with a French force. Mathews, who had now returned to his flagship, followed, and an engagement took place on the 11th of February. The battle was highly discreditable to the British fleet, and not very honourable to their opponents, but it is of the highest historical importance in the history of the navy. It marked the lowest pitch reached in discipline and fighting and efficiency by the fleet in the 18th century, and it had a very bad effect in confirming the pedantic system of tactics set up by the old Fighting Instructions. The British fleet followed the enemy in light winds on the 10th of February, and became scattered. Mathews hoisted the signal to form the line, and then when night fell, to lie to. At that moment Lestock, who commanded in the rear, was at a considerable distance from the body of the fleet, and he ought undoubtedly to have joined his admiral before lying to, but he obeyed the second order, with the result, which it is impossible not to feel that he foresaw and desired, that when morning came he was a long way off the flag of Mathews. The enemy were within striking distance of the van and centre of the British fleet, and Mathews attacked their rear. The battle was ill fought, as it had been ill prepared. Lestock never came into action at all. One Spanish line-of-battleship, the "Poder" (74), was taken, but afterwards burnt. Several of the British captains behaved very badly, and Mathews in a heat of confused anger bore down on the enemy out of his line, while the signal to keep the line was still flying at his mast head. The French and Spaniards got away, and were not pursued by Mathews, though they were of inferior strength.

Deep indignation was aroused at home by this naval miscarriage, and the battle led to more than twenty courts-martial and a parliamentary inquiry. The evils which had overrun the navy were clearly displayed, and in so far some good was done. It was shown for instance that one of the captains whose ship behaved worst was a man of extreme age who was nearly blind and deaf. One of the captains was so frightened at the prospect of a trial that he deserted on his way home and disappeared into Spain. Mathews resigned and returned home after the battle. In consequence of the parliamentary motion for inquiry, Lestock was brought to trial, and acquitted on the ground that he had obeyed orders. Then Mathews was tried in 1746, and was condemned to be dismissed the service on the ground that he had not only failed to pursue the enemy but had taken his fleet into action in a confused manner. He had in fact not waited till he had his fleet in a line with the enemy before bearing down on them, and he had disordered his own line. To the country at large it appeared strange that the admiral who had actually fought should be condemned, while the admiral who had kept at a distance was acquitted. Mathews looked upon his condemnation as the result of mere party spirit. Sheer pedantry on the part of the officers forming the court-martial affords a more satisfactory explanation. They judged that a naval officer was bound not to go beyond the Fighting Instructions as Mathews had undoubtedly done, and therefore condemned him. Their decision had a serious effect in fixing the rule that all battles, at any rate against enemies of equal or nearly equal numbers, were to be fought on one pattern. Mathews died on the 2nd of October 1751 in London. There is a portrait of him in the Painted Hall at Greenwich.

In Beatson's _Naval and Military Memoirs_, vol. i., will be found a fair account of the battle of February 1744. It is fully dealt with by Montagu Burrows in his _Life of Hawke_. The French account may be found in Tronde's _Batailles Navales de la France_. The Spanish view is in the _Vida de Don Josef Navarro_ by Don Josef de Vargas. The battle led to a violent pamphlet controversy. The charges and findings at the courts-martial on both Lestock and Mathews were published at the time. The minor trials arising out of the action are collected in a folio under the title "Copies of all the Minutes and Proceedings taken at and upon the several Tryals of Captain George Burrish" (1746). A "Narrative" was published by, or for, Lestock in 1744, and answered by, or on behalf of, Mathews under the title "Ad----l M----w's Conduct in the late Engagement Vindicated" in 1745. (D. H.)

MATHY, KARL (1807-1868), Badenese statesman, was born at Mannheim on the 17th of March 1807. He studied law and politics at Heidelberg, and entered the Baden government department of finance in 1829. His sympathy with the revolutionary ideas of 1830, expressed in his paper the _Zeitgeist_, cost him his appointment in 1834, and he made his way to Switzerland, where he contributed to the _Jeune Suisse_ directed by Mazzini. On his return to Baden in 1840 he edited the _Landtagszeitung_ at Carlsruhe, and in 1842 he entered the estates for the town of Constance. He became one of the opposition leaders and in 1847 helped to found the _Deutsche Zeitung_, a paper which eventually did much to further the cause of German unity. He took part in the preliminary parliament and in the assembly of Frankfort in 1848-1849, where he supported the policy of H. W. A. von Gagern, and after the refusal of Frederick William IV. to accept the imperial crown he still worked for the cause of unity. He was made finance minister in Baden in May 1849, but was dismissed after a few days of office. He then applied his financial knowledge to banking business in Cologne, Berlin, Gotha and Leipzig. He was recalled to Baden in 1862, and in 1864 became president of the new ministry of commerce. He sought to bring Baden institutions into line with those of northern Germany with a view to ultimate union, and when in 1866 Baden took sides with Austria against Prussia he sent in his resignation. After the war he became president of a new cabinet, but he did not live to see the realization of the policy for which he had striven. He died at Carlsruhe on the 3rd of February 1868.

His letters during the years 1846-1848 were edited by Ludwig Mathy (Leipzig, 1899), and his life was written by G. Freytag (Leipzig, 2nd ed., 1872).

MATILDA (1102-1164), queen of England and empress, daughter of Henry I. of England, by Matilda, his first wife, was born in 1102. In 1109 she was betrothed to the emperor-elect, Henry V., and was sent to Germany, but the marriage was delayed till 1114. Her husband died after eleven years of wedlock, leaving her childless; and, since both her brothers were now dead, she was recalled to her father's court in order that she might be recognized as his successor in England and Normandy. The Great Council of England did homage to her under considerable pressure. Their reluctance to acknowledge a female sovereign was increased when Henry gave her in marriage to Geoffrey Plantagenet, the heir of Anjou and Maine (1129); nor was it removed by the birth of the future Henry II. in 1133. On the old king's death both England and Normandy accepted his nephew, Stephen, of Mortain and Boulogne. Matilda and her husband were in Anjou at the time. They wasted the next few years in the attempt to win Normandy; but Earl Robert of Gloucester, the half-brother of the empress, at length induced her to visit England and raise her standard in the western shires, where his influence was supreme. Though on her first landing Matilda only escaped capture through the misplaced chivalry of her opponent, she soon turned the tables upon him with the help of the Church and the barons of the west. Stephen was defeated and captured at Lincoln (1141); the empress was acclaimed lady or queen of England (she used both titles indifferently) and crowned at London. But the arrogance which she displayed in her prosperity alienated the Londoners and the papal legate, Bishop Henry of Winchester. Routed at the siege of Winchester, she was compelled to release Stephen in exchange for Earl Robert, and thenceforward her cause steadily declined in England. In 1148, having lost by the earl's death her principal supporter, she retired to Normandy, of which her husband had in the meantime gained possession. Henceforward she remained in the background, leaving her eldest son Henry to pursue the struggle with Stephen. She outlived Henry's coronation by ten years; her husband had died in 1151. As queen-mother she played the part of a mediator between her sons and political parties. Age mellowed her temper, and she turned more and more from secular ambitions to charity and religious works. She died on the 30th of January 1164.

See O. Rössler, _Kaiserin Mathilde_ (Berlin, 1897); J. H. Round, _Geoffrey de Mandeville_ (London, 1892). (H. W. C. D.)

MATILDA (1046-1115), countess or margravine of Tuscany, popularly known as the Great Countess, was descended from a noble Lombard family. Her great-grandfather, Athone of Canossa, had been made count of Modena and Reggio by the emperor Otto I., and her grandfather had, in addition, acquired Mantua, Ferrara and Brescia. Her own father, Boniface II., the Pious, secured Tuscany, the duchy of Spoleto, the county of Parma, and probably that of Cremona; and was loyal to the emperor until Henry plotted against him. Through the murder of Count Boniface in 1052 and the death of her older brother and sister three years later, Matilda was left, at the age of nine, sole heiress to the richest estate in Italy. She received an excellent education under the care of her mother, Beatrice of Bar, the daughter of Frederick of Lorraine and aunt of Henry III., who, after a brief detention in Germany by the emperor, married Godfrey IV. of Lorraine, brother of Pope Stephen IX. (1057-1058). Thenceforth Matilda's lot was cast against the emperor in the great struggle over investiture, and for over thirty years she maintained the cause of the successive pontiffs, Gregory VII., Victor III., Urban II., Paschal II., with varying fortune, but with undaunted resolution. She aided the pope against the Normans in 1074, and in 1075 attended the synod at which Guibert was condemned and deprived of the archbishopric of Ravenna. Her hereditary fief of Canossa was the scene (Jan. 28, 1077) of the celebrated penance of Henry IV. before Gregory VII. She provided an asylum for Henry's second wife, Praxides, and urged his son Conrad to revolt against his father. In the course of the protracted struggle her villages were plundered, her fortresses demolished, and Pisa and Lucca temporarily lost, but she remained steadfast in her allegiance, and, before her death, had, by means of a league of Lombard cities which she formed, recovered all her possessions. The donation of her estates to the Holy See, originally made in 1077 and renewed on the 17th of November 1102, though never fully consummated on account of imperial opposition, constituted the greater part of the temporal dominion of the papacy. Matilda was twice married, first to Godfrey V. of Lorraine, surnamed the Humpbacked, who was the son of her step-father and was murdered on the 26th of February 1076; and secondly to the 17-year-old Welf V. of Bavaria, from whom she finally separated in 1095--both marriages of policy, which counted for little in her life. Matilda was an eager student: she spoke Italian, French and German fluently, and wrote many Latin letters; she collected a considerable library; she supervised an edition of the Pandects of Justinian; and Anselm of Canterbury sent her his _Meditations_. She combined her devotion to the papacy and her learning with very deep personal piety. She died after a long illness at Bodeno, near Modena, on the 24th of July 1115, and was buried in the Benedictine church at Polirone, whence her remains were taken to Rome by order of Urban VIII. in 1635 and interred in St Peter's.

The contemporary record of Matilda's life in rude Latin verse, by her chaplain Domnizone (Donizo or Domenico), is preserved in the Vatican Library. The best edition is that of Bethmann in the _Monumenta germ. hist. scriptores_, xii. 348-409. The text, with an Italian translation, was published by F. Davoli under the title _Vita della granda contessa Matilda di Canossa_ (Reggio nell' Emilia 1888 seq.).

See A. Overmann, _Gräfin Mathilde von Tuscien; ihre Besitzungen ... u. ihre Regesten_ (Innsbruck, 1895); A. Colombo, _Una Nuova vita delta contessa Matilda in R. accad. d. sci. Atti_, vol. 39 (Turin, 1904); L. Tosti, _La Contessa Matilda ed i romani pontefici_ (Florence, 1859); A. Pannenborg, _Studien zur Geschichte der Herzogin Matilde von Canossa_ (Göttingen, 1872); F. M. Fiorentini, _Memorie della Matilda_ (Lucca, 1756); and Nora Duff, _Matilda of Tuscany_ (1910). (C. H. Ha.)

MATINS (Fr. _matines_, med. Lat. _matutinae_, sc. possibly vigiliae, morning watches; from _matutinus_, "belonging to the morning"), a word now only used in an ecclesiastical sense for one of the canonical hours in the Roman Breviary, originally intended to be said at midnight, but sometimes said at dawn, after which "lauds" were recited or sung. In the modern Roman Catholic Church, outside monastic services, the office is usually said on the preceding afternoon or evening. The word is also used in the Roman Catholic Church for the public service held on Sunday mornings before the mass (see BREVIARY; and HOURS, CANONICAL). In the Church of England since the Reformation matins is used for the order of public morning prayer.

MATLOCK, a market town in the western parliamentary division of Derbyshire, England, on the river Derwent, 17 m. N. by W. of Derby on the Midland railway. Pop. (1901), of urban district of Matlock, 5979; of Matlock Bath and Scarthin Nick, 1819. The entire township includes the old village of Matlock, the commercial and manufacturing district of Matlock Bridge, and the fashionable health resorts of Matlock Bath and Matlock Bank. The town possesses cotton, corn and paper mills, while in the vicinity there are stone-quarries and lead mines. A peculiar local industry is the manufacture of so-called "petrified" birds' nests, plants, and other objects. These are steeped in water from the mineral springs until they become encrusted with a calcareous deposit which gives them the appearance of fossils. Ornaments fashioned out of spar and stalactites have also a considerable sale.

MATLOCK BATH, one and a half miles south of Matlock, having a separate railway station, overlooks the narrow and precipitous gorge of the Derwent, and stands in the midst of woods and cliffs, deriving its name from three medicinal springs, which first became celebrated towards the close of the 17th century. They were not known to the Romans, although lead-mining was carried on extensively in the district in the 1st and 2nd centuries A.D. The mean temperature of the springs is 68° F. Extensive grounds have been laid out for public use; and in the neighbourhood there are several fine stalactite caverns.

Sheltered under the high moorlands of Darley, MATLOCK BANK has grown up about a mile north-east of the old village, and has become celebrated for the number and excellence of its hydropathic establishments. A tramway, worked by a single cable, over a gradient said to be the steepest in the world, affords easy communication with Matlock Bridge.

MATOS FRAGOSO, JUAN DE (1614?-1689), Spanish dramatist, of Portuguese descent, was born about 1614 at Alsito (Alemtejo). After taking his degree in law at the university of Evora, he proceeded to Madrid, where he made acquaintance with Perez de Montalbán, and thus obtained an introduction to the stage. He quickly displayed great cleverness in hitting the public taste, and many contemporaries of superior talent eagerly sought his aid as a collaborator. The earliest of his printed plays is _La Defensa de la fé y principe prodigioso_ (1651), and twelve more pieces were published in 1658. But though his popularity continued long after his death (January 4, 1689), Matos Fragoso's dramas do not stand the test of reading. His emphatic preciosity and sophistical insistence on the "point of honour" are tedious and unconvincing; in _La Venganza en el despeño_, in _Á lo que obliga un agravio_, and in other plays, he merely recasts, very adroitly, works by Lope de Vega.

MATRASS (mod. Lat. _matracium_), a glass vessel with a round or oval body and a long narrow neck, used in chemistry, &c., as a digester or distiller. The Florence flask of commerce is frequently used for this purpose. The word is possibly identical with an old name "matrass" (Fr. _materas_, _matelas_) for the bolt or quarrel of a cross-bow. If so, some identity of shape is the reason for the application of the word; "bolt-head" is also used as a name for the vessel. Another connexion is suggested with the Arabic _matra_, a leather bottle.

MATRIARCHATE ("rule of the mother"), a term used to express a supposed earliest and lowest form of family life, typical of primitive societies, in which the promiscuous relations of the sexes result in the child's father being unknown (see FAMILY). In such communities the mother took precedence of the father in certain important respects, especially in line of descent and inheritance. Matriarchate is assumed on this theory to have been universal in prehistoric times. The prominent position then naturally assigned women did not, however, imply any personal power, since they were in the position of mere chattels: it simply constituted them the sole relatives of their children and the only centre of any such family life as existed. The custom of tracing descent through the female is still observed among certain savage tribes. In Fiji father and son are not regarded as relatives. Among the Bechuanas the chieftainship passes to a brother, not to a son. In Senegal, Loango, Congo and Guinea, relationship is traced through the female. Among the Tuareg Berbers a child takes rank, freeman's or slave's, from its mother.

BIBLIOGRAPHY.--J. F. McLennan, _Patriarchal Theory_ (London, 1885); T. T. Bachofen, _Das Mutterrecht_ (Stuttgart, 1861); E. Westermarck, _History of Human Marriage_ (1894); A. Giraud-Teulon, _La Mère chez certains peuples de l'antiquité_ (Paris, 1867); _Les Origines du mariage et de la famille_ (Geneva and Paris, 1884); C. S. Wake, _The Development of Marriage and Kinship_ (London, 1889); Ch. Letourneau, _L'Évolution du mariage et de la famille_ (Paris, 1888); L. H. Morgan, _Systems of Consanguinity and Affinity of Human Family_, "Smithsonian Contributions to Knowledge," vol. xvii. (Washington, 1871); C. N. Starcke, _The Primitive Family_ (London, 1889).

MATRIMONY (Lat. _matrimonium_, marriage, which is the ordinary English sense), a game at cards played with a full whist pack upon a table divided into three compartments labelled "Matrimony," "Intrigue" and "Confederacy," and two smaller spaces, "Pair" and "Best." These names indicate combinations of two cards, any king and queen being "Matrimony," any queen and knave "Intrigue," any king and knave "Confederacy"; while any two cards of the same denomination form a "Pair" and the diamond ace is "Best." The dealer distributes a number of counters, to which an agreed value has been given, upon the compartments, and the other players do likewise. The dealer then gives one card to each player, face down, and a second, face up. If any turned-up card is the diamond ace, the player holding it takes everything on the space and the deal passes. If not turned, the diamond ace has only the value of the other three aces. If it is not turned, the players, beginning with the eldest hand, expose their second cards, and the resulting combinations, if among the five successful ones, win the counters of the corresponding spaces. If the counters on a space are not won, they remain until the next deal.

MATRIX, a word of somewhat wide application, chiefly used in the sense of a bed or enclosing mass in which something is shaped or formed (Late Lat. _matrix_, womb; in classical Latin _matrix_ was only applied to an animal kept for breeding). Matrix is thus used of a mould of metal or other substance in which a design or pattern is made in intaglio, and from which an impression in relief is taken. In die-sinking and coining, the matrix is the hardened steel mould from which the die-punches are taken. The term "seal" should strictly he applied to the impression only on wax of the design of the matrix, but is often used both of the matrix and of the impression (see SEALS). In mineralogy, the matrix is the mass in which a crystal mineral or fossil is embedded. In mathematics, the name "matrix" is used of an arrangement of numbers or symbols in a rectangular or square figure. (See ALGEBRAIC FORMS.)

In med. Latin _matrix_ and the diminutive _matricula_ had the meaning of a roll or register, particularly one containing the names of the members of an institution, as of the clergy belonging to a cathedral, collegiate or other church, or of the members of a university. From this use is derived "matriculation," the admission to membership of a university, also the name of the examination for such admission. _Matricula_ was also the name of the contributions in men and money made by the various states of the Holy Roman Empire, and in the modern German Empire the contributions made by the federal states to the imperial finances are called _Matrikularbeiträge_, matricular contributions. (See GERMANY: _Finance_.)

MATROSS, the name (now obsolete) for a soldier of artillery, who ranked next below a gunner. The duty of a matross was to assist the gunners in loading, firing and sponging the guns. They were provided with firelocks, and marched with the store-wagons, acting as guards. In the American army a matross ranked as a private of artillery. The word is probably derived from Fr. _matelot_, a sailor.

MATSUKATA, MARQUIS (1835- ), Japanese statesman, was born at Kagoshima in 1835, being a son of a _samurai_ of the Satsuma clan. On the completion of the feudal revolution of 1868 he was appointed governor of the province of Tosa, and having served six years in this office, was transferred to Tokyo as assistant minister of finance. As representative of Japan at the Paris Exhibition of 1878, he took the opportunity afforded by his mission to study the financial systems of the great European powers. On his return home, he held for a short time in 1880 the portfolio of home affairs, and was in 1881 appointed minister of finance. The condition of the currency of Japan was at that time deplorable, and national bankruptcy threatened. The coinage had not only been seriously debased during the closing years of the Tokugawa régime, but large quantities of paper currency had been issued and circulated, both by many of the feudal lords, and by the central government itself, as a temporary expedient for filling an impoverished exchequer. In 1878 depreciation had set in, and the inconvertible paper had by the close of 1881 grown to such an extent that it was then at a discount of 80% as compared with silver. Matsukata showed the government the danger of the situation, and urged that the issue of further paper currency should be stopped at once, the expenses of administration curtailed, and the resulting surplus of revenue used in the redemption of the paper currency and in the creation of a specie reserve. These proposals were acted upon: the Bank of Japan was established, and the right of issuing convertible notes given to it; and within three years of the initiation of these financial reforms, the paper currency, largely reduced in quantity, was restored to its full par value with silver, and the currency as a whole placed on a solvent basis. From this time forward Japan's commercial and military advancement continued to make uninterrupted progress. But _pari passu_ with the extraordinary impetus given to its trade by the successful conclusion of the war with China, the national expenditure enormously increased, rising within a few years from 80 to 250 million yen. The task of providing for this expenditure fell entirely on Matsukata, who had to face strong opposition on the part of the diet. But he distributed the increased taxation so equally, and chose its subjects so wisely, that the ordinary administrative expenditure and the interest on the national debt were fully provided for, while the extraordinary expenditure for military purposes was met from the Chinese indemnity. As far back as 1878 Matsukata perceived the advantages of a gold standard, but it was not until 1897 that his scheme could be realized. In this year the bill authorizing it was under his auspices submitted to the diet and passed; and with this financial achievement Matsukata saw the fulfilment of his ideas of financial reform, which were conceived during his first visit to Europe. Matsukata, who in 1884 was created Count, twice held the office of prime minister (1891-1892, 1896-1898), and during both his administrations he combined the portfolio of finance with the premiership; from October 1898 to October 1900 he was minister of finance only. His name in Japanese history is indissolubly connected with the financial progress of his country at the end of the 19th century. In 1902 he visited England and America, and he was created G.C.M.G., and given the Oxford degree of D.C.L. In September 1907 he was advanced to the rank of Marquis.

MATSYS (MASSYS or METZYS), QUINTIN (1466-1530), Flemish artist, was born at Louvain, where he first learned a mechanical art. During the greater part of the 15th century the centres in which the painters of the Low Countries most congregated were Bruges, Ghent and Brussels. Towards the close of the same period Louvain took a prominent part in giving employment to workmen of every craft. It was not till the opening of the 16th century that Antwerp usurped the lead which it afterwards maintained against Bruges and Ghent, Brussels, Mechlin and Louvain. Quintin Matsys was one of the first men of any note who gave repute to the gild of Antwerp. A legend relates how the smith of Louvain was induced by affection for the daughter of an artist to change his trade and acquire proficiency in painting. A less poetic but perhaps more real version of the story tells that Quintin had a brother with whom he was brought up by his father Josse Matsys, a smith, who held the lucrative offices of clockmaker and architect to the municipality of Louvain. It came to be a question which of the sons should follow the paternal business, and which carve out a new profession for himself. Josse the son elected to succeed his father, and Quintin then gave himself to the study of painting. We are not told expressly by whom Quintin was taught, but his style seems necessarily derived from the lessons of Dierick Bouts, who took to Louvain the mixed art of Memlinc and Van der Weyden. When he settled at Antwerp, at the age of twenty-five, he probably had a style with an impress of its own, which certainly contributed most importantly to the revival of Flemish art on the lines of Van Eyck and Van der Weyden. What particularly characterizes Quintin Matsys is the strong religious feeling which he inherited from earlier schools. But that again was permeated by realism which frequently degenerated into the grotesque. Nor would it be too much to say that the facial peculiarities of the boors of Van Steen or Ostade have their counterparts in the pictures of Matsys, who was not, however, trained to use them in the same homely way. From Van der Weyden's example we may trace the dryness of outline and shadeless modelling and the pitiless finish even of trivial detail, from the Van Eycks and Memlinc through Dierick Bouts the superior glow and richness of transparent pigments, which mark the pictures of Matsys. The date of his retirement from Louvain is 1491, when he became a master in the gild of painters at Antwerp. His most celebrated picture is that which he executed in 1508 for the joiners' company in the cathedral of his adopted city. Next in importance to that is the Marys of Scripture round the Virgin and Child, which was ordered for a chapel in the cathedral of Louvain. Both altar-pieces are now in public museums, one at Antwerp, the other at Brussels. They display great earnestness in expression, great minuteness of finish, and a general absence of effect by light or shade. As in early Flemish pictures, so in those of Matsys, superfluous care is lavished on jewelry, edgings and ornament. To the great defect of want of atmosphere such faults may be added as affectation, the result of excessive straining after tenderness in women, or common gesture and grimace suggested by a wish to render pictorially the brutality of gaolers and executioners. Yet in every instance an effort is manifest to develop and express individual character. This tendency in Matsys is chiefly illustrated in his pictures of male and female market bankers (Louvre and Windsor), in which an attempt is made to display concentrated cupidity and avarice. The other tendency to excessive emphasis of tenderness may be seen in two replicas of the "Virgin and Child" at Berlin and Amsterdam, where the ecstatic kiss of the mother is quite unreal. But in these examples there is a remarkable glow of colour which makes up for many defects. Expression of despair is strongly exaggerated in a Lucretia at the museum of Vienna. On the whole the best pictures of Matsys are the quietest; his "Virgin and Christ" or "Ecce Homo" and "Mater Dolorosa" (London and Antwerp) display as much serenity and dignity as seems consistent with the master's art. He had considerable skill as a portrait painter. Egidius at Longford, which drew from Sir Thomas More a eulogy in Latin verse, is but one of a numerous class, to which we may add the portrait of Maximilian of Austria in the gallery of Amsterdam. Matsys in this branch of practice was much under the influence of his contemporaries Lucas of Leiden and Mabuse. His tendency to polish and smoothness excluded to some extent the subtlety of modulation remarkable in Holbein and Dürer. There is reason to think that he was well acquainted with both these German masters. He probably met Holbein more than once on his way to England. He saw Dürer at Antwerp in 1520. Quintin died at Antwerp in 1530. The puritan feeling which slumbered in him was fatal to some of his relatives. His sister Catherine and her husband suffered at Louvain in 1543 for the then capital offence of reading the Bible, he being decapitated, she buried alive in the square fronting the cathedral.

Quintin's son, Jan Matsys, inherited the art but not the skill of his parent. The earliest of his works, a "St Jerome," dated 1537, in the gallery of Vienna, the latest, a "Healing of Tobias," of 1564, in the museum of Antwerp, are sufficient evidence of his tendency to substitute imitation for original thought.

MATTEAWAN, a village of Fishkill township, Dutchess county, New York, U.S.A., on the eastern bank of the Hudson river, opposite Newburgh and 15 m. S. of Poughkeepsie. Pop. (1890), 4278; (1900), 5807 (1044 foreign-born); (1905, state census), 5584; (1910), 6727. The village is served by the Central New England railway, and is the seat of the Matteawan state hospital for the criminal insane, the Highland hospital, and the Sargeant industrial school. The Teller House dates back to the beginning of the 18th century. Near Matteawan is Beacon Hill, the highest of the highlands, which has an electric railway to its summit. There are manufactures of hats, rubber goods, machinery (notably "fuel-economizers"), &c., water-power being furnished by Fishkill Creek. The village owns its waterworks, the supply for which is derived from Beacon Hill. Matteawan was incorporated as a village in 1886.