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 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.

.—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 have 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 et 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 1799); 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).

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 (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 of 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 smallpox 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