MATHEMATICS. 184 MATHER. ics, including subjects discussed in the articles on SUBSTITITIOXS; QlATEKXlO.N'S; SURFACES; Curve; Complex Number: Determinants; Functions; and the more general articles on Algebra, Geometry, Trigonometry, Number, and Calculus. Classification. No entirely satisfactory classification of uiathematics is possible. The various braiubes are so interrelated that exact lines of separation cannot be drawn, a fact of api)arent and great advantage to the science. The most recent attempt at classification is that made in the Enc-yklopadie der mathciiiatischen ^Vissc>l.1ch(lf)c)l. The following scheme covers the principal subjects discussed: I. pure mathematics. A. Ai-ithmeiic and Algebra. (a) Arithmetic (q.v.). (1) Fundamental operations with pure numbers. See NuMBKR : Arithmeth*. (2) The conil)inati>r.v thei)iy. IrolnilinK combl- n.Ttioiis, permutations, determin.int8. See Per- ]lIlfl-.rioN8 AND ('kMBINATIONS. (3) Irrationals ami the eDUverjiency question. See Number; Irrational Number. (4) romidex numbers (<!. v.). (5) Menttenlehre, literally the "multitude theory'; OS ol a multitude (unlimited number) ot points. (6) Finite discrete groups. See Sudstitutios. (b) Algebra (q.v.). (1) Fundamental concepts, including rational func- tions. See Function. (2) Theory of invariants. See Forms. (3) Theory of equations. See Equation. (c) Theory of numbers. See Number. (d) Theory of probabilities. See Probability. B. Analysis. (a) Analysis of real quantities. (1) nitfi'H'ntial and iiiteKral calculus. SeeCALCCLUS. (■2) Diff'M-i'iiIial equations. See EtiUATION. (.')) Cdiitiiiuous transformation groups. See Substi- tution. (4) Infinite series. (6) Calculus of variations. (b) Analysis of complex quantities. (1) Oeneral theory of functions. See Function. (2) S|iei.ial kinds of functions, elliptic, Abellan, auto- morphii*. etc. (3) Funetional equations and operations. C. Geometry (q.v.). (a) Pure geometry. See Geometry. (1) (ienerril iirlnclples and elementary geometry. (2) Positional geometry. (3) rrojcctive geometry. See Geometry; Projec- tion. (4) Descriptive geometry. See Oeomktrv. (b) Algebra and analysis as applied to geom- etry. See Analytic Geometry. (11 Codrdlnateey stems. See Coordinates. (2) Conies. (3) Algebraic curves and surfaces. (4) Space of n dimensions. .See Geometry. (c) DilTeiential geometry, including tran- sc-endent curves and surfaces. II. APPLIED mathematics. A. ilcchnnics (q.v.). including kinematics, ki- netics, statics, the vector analysis (see Qua- ternions), hydrodynamics, and the theory of ehisticity. li. Physirs (q.v.), including thernuxlynamics, molecular physics, electricity, optics. C. Geodesy and Gcophy.iirs. including naviga- tion, geodetic mensuration, cartograpliy, magnetism. £). Astronomy. Bibliography. Select special bibliograpliies may be found in most of the articles on mallie- matical topics. Following are some of the best- known general works on the history of mathe- matics: Cantor, ^'ortesungcn iiber Gcschiehle der Malhematik (Leipzig, 1880-02) : Fink, His- tory of Mtithcmatics (Chicago, 1000), form- ing a brief compendium of Cantor's work; Ball, A Short Account of the History of Mathe- matics (London, 1001); Smith, "History of Modern Mathematics," in Merriman and Wood- ward's Higher Mathematics (New York, 1890); Suter, Geschichte der mathenialischcn Wis- senschaftoi (Zurich, 1873-75) ; Hankel, Ziir Geschichte der Mathematik im Altertum vnd Mittelaltcr (Leipzig, 1874) ; Zeuthen, Die Lchre ron den Kegrlschnitten im Altrrtuni (Co- penhagen, 1880) : Zeuthen, Vorlesuniicn. iiber die Geschiclite der Matheiiialilc (Copenhagen, Ger. trans, in 1895) ; Giinther. Geschichte dcs mathe- matischen Unterrichts im deutschen Mittclalter bis zum Jahre 132o (Berlin. 1887) ; Cajori, A History of ^ Mathematics (New York. 1894) ; Cajori, A History of Elementary Mathematics (New York, 180G) ; Ahhandlungen zur Ge- schichte der Mathematik (Leipzig. 1877 et seq.). The Bihliotheea Mathematica (Leipzig), edited by Enestriim, is devoted to the liistory of the subject. For the general bibliograpliy of the i science, consult the elaborate Encyhloimdie der maihematischen ^yissen^cllaften, the publication of which was begun at Leipzig in 1808. Com- plete records of the recent publications touch- ing niathematical subjects may be found in the Jahrhiich iiber die Fortschritte der Mathematik (Berlin, since 1871 ) . MATHER, mfiTii'er. Cotton (1663-1728). A. colonial divine and author, eldest son of Increase Mather (q.v.) and Jlaria. daugliter of John Cotton (q.v.). He was bom in Boston. Febni- ary 12. Ul(i3. He was very [ireciHious anil w.is unfortunately overestimated and praised, with the result that he became morbidly self-con- scious. An omnivorous reader from the first, he entered Harvard at eleven, and graduated in 1678 at fifteen. At sixteen he studied medicine, despairing of being al)le to enter the ministry on account of a propensity to stammering. This he conquered by methods of deliberate speedi. and at seventeen prcnchcd his first seriiuni and be- came an assistant to liis father, lie dsik his master's degree in 1681. refused a call to New Haven, and became associate pastor ^yith liis father in the North Church of Boston, 'in 1686 he marrieil: two years later his father's mis- sion to Enghrnd left liim at the age of twenty- five in sole charge of the North Church, and probably the most important man in liostim. lie was widely celebrated as a scholar and was the olivious leader of the conservative element among the Puritans of the day. He had also liegun to lake a great interest in the subject of witchcraft, his Memonililc I'rorideners /{elating In Witchcraft and Possessions appearing in 168!). During the witchcraft epidemic at Sah'iii in 1602 he became an infatuated investigator of suspected cases, a constant adviser of the mag- istrates, and wrote his 11"o)irfrr.i of the Inrisitile World (1603) to confute all doubters. In 1603 Mather planned his great ecclesiastical history of New Enghind. the Mnynalin. which was finished in 1607, and finally appeared in 1702.
