mathematics

You will be glad to hear that I enjoy Mathematics now.

7639With this was included mathematics, astronomy and astrology, and even the magic arts.

4928The former was professor of mathematics at Bologna, and published, among other works, a treatise on the infinitesimal calculus.

4030Of more general interest, however, are his labours in pure mathematics, which appear for the most part in Crelle's Journal from 1828 to 1858.

2915On his return to the capital Peter, in order to see what progress his son had made in mechanics and mathematics, asked him to draw something of a technical nature for his inspection.

2220His knowledge of the higher mathematics was acquired by his own unaided efforts after he had left the college.

2114He divides geography into The Spherical Part, or that for the study of which mathematics alone is required, and The Topical Part, or the description of the physical relations of parts of the earth's surface, preferring this division to that favoured by the ancient geographers - into general and special.

2012At Leipzig, GÃ¶ttingen and Halle he studied for four years, ultimately devoting himself to mathematics and astronomy.

1712"The hours are the same, and the lathe, and also the mathematics and my geometry lessons," said Princess Mary gleefully, as if her lessons in geometry were among the greatest delights of her life.

1512After taking his degree he wavered between classics and mathematics, but finally chose the latter.

1411Each problem was something unique; the elements of transition from one to another were wanting; and the next step which mathematics had to make was to find some method of reducing, for instance, all curves to a common notation.

116As I have said before, I had no aptitude for mathematics; the different points were not explained to me as fully as I wished.

119Scarcely any member of the Arabian circle of the sciences, including theology, philology, mathematics, astronomy, physics and music, was left untouched by the treatises of Avicenna, many of which probably varied little, except in being commissioned by a different patron and having a different form or extent.

1111Higher education is represented by the provincial university, which teaches science and mathematics, holds examinations, distributes scholarships, and grants degrees in all subjects.

105During a long and active life, he played many parts: professor of mathematics at the Elphinstone college (1854) founder of the Rast Goftar newspaper; partner in a Parsi business firm in London (1855); prime minister of Baroda (1874); member of the Bombay legislative council (1885); M.P. for Central Finsbury (1892-1895), being the first Indian to be elected to the House of Commons; three times president of the Indian National Congress.

106He was then appointed to the ordinary chair of mathematics successively at Basel (1863), Tubingen (1865) and Leipzig (1868).

109It is to her that the Principles of Philosophy were dedicated; and in her alone, according to Descartes, were united those generally separated talents for metaphysics and for mathematics which are so characteristically co-operative in the Cartesian system.

1010A refugee Pole, Zamosz, taught him mathematics, and a young Jewish physician was his tutor in Latin.

1010He was himself always occupied: writing his memoirs, solving problems in higher mathematics, turning snuffboxes on a lathe, working in the garden, or superintending the building that was always going on at his estate.

1010A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble.

1010Not until the age of seventeen did he attack the higher mathematics, and his progress was much retarded by the want of efficient help. When about sixteen years of age he became assistant-master in a private school at Doncaster, and he maintained himself to the end of his life in one grade or other of the scholastic profession.

95"This won't do, Princess; it won't do," said he, when Princess Mary, having taken and closed the exercise book with the next day's lesson, was about to leave: "Mathematics are most important, madam!

99They write shorthand, but speak no English; they have a smattering of higher mathematics, yet are ignorant of book-keeping.

88Mathematics and natural science - 1,364 3,500

810If Miss Keller is fond of language and not interested especially in mathematics, it is not surprising to find Miss Sullivan's interests very similar.

812While at Oxford Wren distinguished himself in geometry and applied mathematics, and Newton, in his Principia, p. 19 (ed.

75Four out of his five papers on applied mathematics were sent up absolutely blank.

751832), became in 1858 Privatdozent, and in 1863 extraordinary professor of mathematics at Halle.

77From his sixth to his ninth year he was given over to the care of learned foreigners, who taught him history, geography, mathematics and French.

78Princess Mary had ceased taking lessons in mathematics from her father, and when the old prince was at home went to his study with the wet nurse and little Prince Nicholas (as his grandfather called him).

69In preparation for these he spent the winter of 1877-1878 in reading up original treatises like those of Laplace and Lagrange on mathematics and mechanics, and in attending courses on practical physics under P. G.

53After working under Leopold Gmelin at Heidelberg, and Liebig at Giessen, he spent three years in Paris studying the higher mathematics under Comte.

53Gregory wrote Hints for the Use of Teachers of Elementary Mathematics (1840, new edition 1853), and Mathematics for Practical Men (1825), which was revised and enlarged by Henry Law in 1848, and again by J.

53Yet he contrived to write his great commentary on the Pentateuch and other books of the Bible, treatises on philosophy (as the Yesodh mora), astronomy, mathematics, grammar (translation of Ilayyu j), besides a Diwan.

54He now employed himself in making optical glasses, and in engraving on metal, devoting his spare time to the perusal of works on mathematics and optics.

54For history, applied mathematics - for anything, in fact, outside the exact sciences - he felt something approaching to contempt.

54He was educated at Winchester and University College, Oxford, where he took a first class in classics and a second in mathematics, besides taking a leading part in the Union debates.

55He at first taught mathematics at Yale; but in 1895 was made assistant professor of political economy, and in 1898 professor.

55Two regions become prominent in the working out of intuitionalism, if still more prominent in the widely differing philosophy of Kant - the regions of mathematics and of morals.

55The English translation renders the definition thus: " Geography is that part of mixed mathematics which explains the state of the earth and of its parts, depending on quantity, viz.

55At the end of 1709 he went to Dresden for twelve months for finishing lessons in French and German, mathematics and fortification, and, his education completed, he was married, greatly against his will, to the princess Charlotte of BrunswickWolfenbiittel, whose sister espoused, almost simultaneously, the heir to the Austrian throne, the archduke Charles.

56In 1873 he was called to Rome to organize the college of engineering, and was also appointed professor of higher mathematics at the university.

58On the 1st of October 1824 he was elected fellow of Trinity, and in December 1826 was appointed Lucasian professor of mathematics in succession to Thomas Turton.

58For more detailed bibliographical information see Apercu des travaux zoo-ge'ographiques, published at St Petersburg in connexion with the Exhibition of 1878; and the index Ukazatel Russkoi Literatury for natural science, mathematics and medicine, published since 1872 by the Society of the Kiev University.

58In the same year he went to Paris, where he was appointed to the chair of philosophy in the Gervais College in 1631, and two years later to the chair of mathematics in the Royal College of France.

43He was professor of mathematics at Gratz (1864-1867), of physics at Prague (1867-1895), and of physics at Vienna (1895-1901).

43The comprehensive scheme of study included mathematics also, in which he advanced as far as the conic sections in the treatise of L'Hopital.

43By this time he had ceased to devote himself to pure mathematics, and in company with his friends Mersenne and Mydorge was deeply interested in the theory of the refraction of light, and in the practical work of grinding glasses of the best shape suitable for optical instruments.

47As results of Roberval's labours outside the department of pure mathematics may be noted a work on the system of the universe, in which he supports the Copernican system and attributes a mutual attraction to all particles of matter; and also the invention of a special kind of balance which goes by his name.

47In 1721 he entered Merton College, Oxford, as a gentleman commoner, and studied philosophy, mathematics, French, Italian and music. He afterwards studied law at the Inner Temple, but was never called to the bar.

48He graduated in 1840 from Lafayette College, where he was tutor in mathematics (1840-1842) and adjunct professor (1843-1844).

48Perhaps I shall take up these studies later; but I've said goodbye to Mathematics forever, and I assure you, I was delighted to see the last of those horrid goblins!

49Both Mr. Gilman and Mr. Keith, the teachers who prepared her for college, were struck by her power of constructive reasoning; and she was excellent in pure mathematics, though she seems never to have enjoyed it much.

411Cajori, History of Mathematics (London, 1894); M.

35Great as is the difference when we pass from mathematics to morality, yet there are striking similarities, and here again intuitionalism claims to find much support.

36published in 1524, and subsequently edited and added to by Gemma Frisius under the title of Cosmographia, based the whole science on mathematics and measurement.

36For pure mathematics he had a special gift - almost a passion.

36I still found more difficulty in mastering problems in mathematics than I did in any other of my studies.

36Mr. Gilman had agreed that that year I should study mathematics principally.

37Abandoning the conception of cause, mathematics seeks law, that is, the property common to all unknown, infinitely small, elements.

39He taught mathematics at Columbia, and in 1845 was admitted to the bar, but, owing to defective eyesight, never practised.

24Mathematics has influenced the form and the terminology of the science, and has sometimes been useful in analysis; but mathematical methods of reasoning, in their application to economics, while possessing a certain fascination, are of very doubtful utility.

101344), called Ralbag, the great commentator on the Bible and Talmud, in philosophy a follower of Aristotle and Averroes, known to Christians as Leo Hebraeus, wrote also many works on halakhah, mathematics and astronomy.

14He was educated at Pembroke College, Oxford, of which college (after taking a first class in mathematics in 1840 and gaining the university mathematical scholarship in 1842) he becalm fellow in 1844 and tutor and mathematical lecturer in 1845.

14He then returned to Pavia, where he pursued his studies at the university under Francesco Brioschi, and determined to seek a career as teacher of mathematics.

00The majority of them are addressed to Mersenne, and deal with problems of physics, musical theory (in which he took a special interest), and mathematics.

00JOSEF BEM (1795-1850), Polish soldier, was born at Tarnow in Galicia, and was educated at the military school at Warsaw, where he especially distinguished himself in mathematics.

00Then Bem escaped to Paris, where he supported himself by teaching mathematics.

00He next devoted himself to medicine, but his natural inclination proved too strong for him, and within a year he resolved to give his whole time to mathematics.

00He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an arithmetical, and the quantities of acids saturating a given base a geometrical, progression.

00Mathematics were cultivated by the Chinese, Indians and Arabs, but nearly all the sciences based on the observation of nature, including medicine, have remained in a very backward condition.

00In later life he was accustomed to say that he knew as much about mathematics when he was eighteen as ever he knew; but his reading embraced nearly the whole round of knowledge - history, travels, poetry, philosophy and the natural sciences.

00From about 1796 Ampere gave private lessons at Lyons in mathematics, chemistry and languages; and in 1801 he removed to Bourg, as professor of physics and chemistry, leaving his ailing wife and infant son at Lyons.

00In the same year he was appointed professor of mathematics at the lycee of Lyons.

00Delambre, whose recommendation obtained for him the Lyons appointment, and afterwards (1804) a subordinate position in the polytechnic school at Paris, where he was elected professor of mathematics in 1809.

00Of this school, which had Lagrange for its professor of mathematics, we have an amusing account in the life of Gilbert Elliot, 1st earl of Minto, who with his brother Hugh, afterwards British minister at Berlin, there made the acquaintance of Mirabeau.

00His father, Nathaniel, though a barber, was a man of some education, for Jeremy was "solely grounded in grammar and mathematics" by him.

00Cantor's histories of mathematics, and more elaborate analyses are those of Nesselmann (Die Algebra der Griechen, Berlin, 1842) and G.

00For the mathematics of Kepler's problem see E.

00As a fellow and lecturer of his college he remained in Cambridge for two years longer, and then left to take up the professorship of mathematics at Queen's College, Belfast.

00Here you find articles in the encyclopedia on topics related to mathematics.

00Apart from decided signs of proficiency in mathematics, he showed no special ability.

00His natural parts were excellent; and a strong bias in the direction of abstract thought, and mathematics in particular, was noticeable at an early date.

00are: - (1) The extensive work on the fundamental notions of physics, called Communia Naturalium, which is found in the Mazarin library at Paris, in the British Museum, and in the Bodleian and University College libraries at Oxford; (2) on the fundamental notions of mathematics, De Cornmunibus Mathematicae, part of which is in the Sloane collection, part in the Bodleian; (3) Baconis Physica, contained among the additional MSS.

00Charles, however, has given good grounds for supposing that it is merely a preface, and that the work went on to discuss grammar, logic (which Bacon thought of little service, as reasoning was innate), mathematics, general physics, metaphysics and moral philosophy.

00After that, apparently, logic was to be treated; then, possibly, mathematics and physics; then speculative alchemy and experimental science.

00In his youth he went to the continent and taught mathematics at Paris, where he published or edited, between the years 1612 and 1619, various geometrical and algebraical tracts, which are conspicuous for their ingenuity and elegance.

00Mathematics.

00In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

00Allman, Greek Geometry from Thales to Euclid (1889); Florian Cajori, History of Mathematics (New York, 1894); M.

00For Pythagorean mathematics see further Pythagoras.

00In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the theorems afterwards published in his Geometria organica.

00In 1717 he was elected professor of mathematics in Marischal College, Aberdeen, as the result of a competitive examination.

00The following year he was elected professor of mathematics in the university of Edinburgh on the urgent recommendation of Newton.

00After serving for a short time in the artillery, he was appointed in 1797 professor of mathematics at Beauvais, and in 1800 he became professor of physics at the College de France, through the influence of Laplace, from whom he had sought and obtained the favour of reading the proof sheets of the Mecanique celeste.

00Save for the barest rudiments of reading and writing, he tells us that he had no master; yet we find him at Verona in 1521 an esteemed teacher of mathematics.

00ATHANASY LAVRENTEVICH (?-1680) ORDUIN - NASHCHOKIN, Russian statesman, was the son of a poor official at Pskov, who saw to it that his son was taught Latin, German and mathematics.

00But he seems to have been well cared for, and he was at the age of fourteen sufficiently advanced "in algebra, geometry, astronomy, and even the higher mathematics," to calculate a solar eclipse within four seconds of accuracy.

00in 1833, as first class in classics and second class in mathematics.

00He was appointed professor of mathematics at Messina in 1649 and at Pisa in 1656.

00She was well versed in mathematics, which she studied at the university of Moscow, and in general literature her favourite authors were Bayle, Montesquieu, Boileau, Voltaire and Helvetius.

00ROBERT EMMET (1778-1803), Irish rebel, youngest son of Robert Emmet, physician to the lord-lieutenant of Ireland, was born in Dublin in 1778, and entered Trinity College in October 1793, where he had a distinguished academic career, showing special aptitude for mathematics and chemistry, and acquiring a reputation as an orator.

00In 1812 he entered the university of Edinburgh, where he distinguished himself specially in mathematics.

00It has been usual to define mathematics as "the science of discrete and continuous magnitude."

00Even Leibnitz,' who initiated a more modern point of view, follows the tradition in thus confining the scope of mathematics properly so called, while apparently conceiving it as a department of a yet wider science of reasoning.

00Thus every subdivision of mathematical science would appear to deal with quantity, and the definition of mathematics as "the science of quantity" would appear to be justified.

00The Principles of Mathematics, by Bertrand Russell (Cambridge, 1903).

00But now the difficulty of confining mathematics to being the science of number and quantity is immediately apparent.

00Thus the illusory nature of the traditional definition of mathematics is again illustrated.

00It 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.

00But 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' of the "science concerned with the logical deduction of consequences from the general premisses of all reasoning."

00By taking fixed conditions for the hypothesis of such a proposition a definite department of mathematics is marked out.

00The foregoing account of the nature of mathematics necessitates a strict deduction of the general properties ' The first unqualified explicit statement of part of this definition seems to be by B.

00Peirce, "Mathematics is the science which draws necessary conclusions" (Linear Associative Algebra, § i.

00The full expression of the idea and its development into a philosophy of mathematics is due to Russell, loc. cit.

003 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.

00It is more fully explained by him, with later simplifications, in Principia mathematics (Cambridge).

00Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity.

00In one sense there is no science of applied mathematics.

00Thus 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.

00For no one can doubt the essential difference between characteristic treatises upon "pure" and "applied" mathematics.

00In pure mathematics the hypotheses which a set of entities are to satisfy are given, and a group of interesting deductions are sought.

00In "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.

00Accordingly, 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.

00Section A deals with pure mathematics.

00Under 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.

00This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject.

00Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics.

00Mechanics (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.

00Every branch of physics gives rise to an application of mathematics.

00The history of mathematics is in the main the history of its various branches.

00Viewing 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.

00A Short History of Mathematics, by W.

00In 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.

00- References to the works containing expositions of the various branches of mathematics are given in the appropriate articles.

00Bertrand Russell, The Principles of Mathematics (Cambridge, 1903), and his article on "Mathematical Logic" in Amer.

00Peano (with various collaborators of the Italian school), Formulaire de mathematiques (Turin, various editions, 1894-1908; the earlier editions are the more interesting philosophically); Felix Klein, Lectures on Mathematics (New York, 1894); W.

00For the history of mathematics the one modern and complete source of information is M.

00Ball, A Short History of Mathematics (London 1st ed., 1888, three subsequent editions, enlarged and revised, and translations into French and Italian).

00In 1756 he was appointed by Leopold, grand-duke of Tuscany, to the professorship of mathematics in the university of Pisa, a post which he held for eight years.

00In 1764 he was created professor of mathematics in the palatine schools at Milan, and obtained from Pope Pius VI.

00PIERRE JULES CESAR JANSSEN (1824-1907), French astronomer, was born in Paris on the 22nd of February 1824, and studied mathematics and physics at the faculty of sciences.

00But the latter contented hilnself with an annual stipend which would enable him to devote all his time to his favorite studies of mathematics and astronomy.

00The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.

00The effect of this was to co-ordinate many branches of mathematics and greatly to increase the number of workers.

00References For The Theory Of Determinants.-T.Muir's "List of Writings on Determinants," Quarterly Journal of Mathematics.

00Journal of Mathematics, Baltimore, Md.

00RICHARD HILDRETH (1807-1865), American journalist and author, was born at Deerfield, Massachusetts, on the 28th of June 1807, the son of Hosea Hildreth (1782-1835), a teacher of mathematics and later a Congregational minister.

00Besides the ordinary studies of the monastic scholar, he devoted himself to mathematics, astronomy and music, and constructed watches and instruments of various kinds.

00This double cultivation of his scientific powers had the happiest effect on his subsequent work; for the greatest achievements of Riemann were effected by the application in pure mathematics generally of a method (theory of potential) which had up to this time been used solely in the solution of certain problems that arise in mathematical physics.

00where he took the first prize in mathematics and physics; at the Ecole Polytechnique, where he stood first at the exit examination in 1819; and at the Ecole des Mines (1819-1822), where he began to show a decided preference for the science with which his name is associated.

00His emoluments as treasurer at war, together with his wife's fortune, provided him with ample means, which he lost by rash speculations, a circumstance regarded by his son as the prelude to his own good fortune; for had he been rich, he used to say, he might never have known mathematics.

00The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions.

00In the advancement of almost every branch of pure mathematics Lagrange took a conspicuous part.

00His publications show him to have been a man of original and active mind with a singular facility in applying mathematics to practical questions.

00naturalis et matheseos, 1472-1875 (Budapest, 1878), where the number of Magyar works bearing on the natural sciences and mathematics printed from the earliest date to the end of 1875 is stated to be 3811, of which 106 are referred to periodicals.

00He accordingly obtained for him an appointment as professor of mathematics in the Ecole Militaire of Paris, and continued zealously to forward his interests.

00LORENZO MASCHERONI (1750-1800), Italian geometer, was professor of mathematics at the university of Pavia, and published a variety of mathematical works, the best known of which is his Geometria del compasso (Pavia, 1797), a collection of geometrical constructions in which the use of the circle alone is postulated.

00al-jebr wa'l-mugabala, transposition and removal (of terms of an equation), the name of a treatise by Mahommed ben Musa al-Khwarizmi), a branch of mathematics which may be defined as the generalization and extension of arithmetic.

00This appears to have been due in the first instance to Albert Girard (1595-1632), who extended Vieta's results in various branches of mathematics.

00These attempts at the unification of algebra, and its separation from other branches of mathematics, have usually been accompanied by an attempt to base it, as a deductive science, on certain fundamental laws or general rules; and this has tended to increase its difficulty.

00Neither mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science.

00As to the teaching of algebra, see references under Arithmetic to works on the teaching of elementary mathematics.

00The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded.

00The fame of this astronomer and mathematician rests on his work, the Aryabhattiyam, the third chapter of which is devoted to mathematics.

00It is of great interest to the historical student, for it exhibits the influence of Greek science upon Indian mathematics at a period prior to Aryabhatta.

00After an interval of about a century, during which mathematics attained its highest level, there flourished Brahmagupta (b.

00598), whose work entitled Brahma-sphuta-siddhanta (" The revised system of Brahma ") contains several chapters devoted to mathematics.

00Moritz Cantor has suggested that at one time there existed two schools, one in sympathy with the Greeks, the other with the Hindus; and that, although the writings of the latter were first studied, they were rapidly discarded for the more perspicuous Grecian methods, so that, among the later Arabian writers, the Indian methods were practically forgotten and their mathematics became essentially Greek in character.

00Mathematics was more or less ousted from the academic curricula by the philosophical inquiries of the schoolmen, and it was only after an interval of nearly three centuries that a worthy successor to Leonardo appeared.

00In it he mentions many earlier writers from whom he had learnt the science, and although it contains very little that cannot be found in Leonardo's work, yet it is especially noteworthy for the systematic employment of symbols, and the manner in which it reflects the state of mathematics in Europe during this period.

00These works are the earliest printed books on mathematics.

00The renaissance of mathematics was thus effected in Italy, and it is to that country that the leading developments of the following century were due.

00The 17th century is a famous epoch in the progress of science, and the mathematics in no way lagged behind.

00Notable service was also rendered by Augustus de Morgan, who applied logical analysis to the laws of mathematics.

00- The history of algebra is treated in all historical works on mathematics in general (see MATHEMATICS: References).

00He devoted his youth to the study of history, chronology, mathematics, astronomy, philosophy and medicine.

00Other works of his, chiefly on mathematics and astronomy, are still in manuscript only.

00Born on the 15th of February 1514, he studied at Tiguri with Oswald Mycone, and afterwards went to Wittenberg where he was appointed professor of mathematics in 1537.

00He projected numerous other works, as is shown by a letter to Peter Ramus in 1568, which Adrian Romanus inserted in the preface to his Idea of Mathematics.

00Borrell entrusted him to the care of a Bishop Hatto, under whose instruction Gerbert made great progress in mathematics.

00WILLIAM STUBBS (1825-1901), English historian and bishop of Oxford, son of William Morley Stubbs, solicitor, of Knaresborough, Yorkshire, was born on the 21st of June 1825, and was educated at the Ripon grammar school and Christ Church, Oxford, where he graduated in 1848, obtaining a first-class in classics and a third in mathematics.

00He soon distinguished himself as a student and made rapid progress, especially in mathematics.

00professor of mathematics in the school in which he had been a pupil.

00As a politician Fourier achieved uncommon success, but his fame chiefly rests on his strikingly original contributions to science and mathematics.

00Here Jacques Davy received his education, being taught Latin and mathematics by his father, and learning Greek and Hebrew and the philosophy then in vogue.

00As a foundation his education must be thorough in the natural and physical sciences and mathematics.

00These miners' schools (Bergschule, ecoles des mineurs) give elementary instruction in chemistry, physics, mechanics, mineralogy, geology and mathematics and drawing, as well as in such details of the art of mining as will best supplement the practical information already acquired in underground work.

00The invention of the mechanical air-pump is generally attributed to Otto von Guericke, consul of Magdeburg, who exhibited his instrument in 1654; it was first described in 1657 by Gaspar Schott, professor of mathematics at Wurttemberg, in his NI echanica hydraulico-pneumatica, and afterwards (in 1672) by Guericke in his Experimenta nova Magdeburgica de vacus spatia.

00catena, a chain), in mathematics, the curve assumed by a uniform chain or string hanging freely between two supports.

00Trained for the scholastic profession, he was appointed assistant professor at the Academy of Paris in 1831, professor of mathematics at Lyons in 1834, rector of the Academy of Grenoble in 1835, inspector-general of studies in 1838, rector of the Academy of Dijon and honorary inspectorgeneral in 1854, retiring in 1862.

00Cournot was the first who, with a competent knowledge of both subjects, endeavoured to apply mathematics to the treatment of economic questions.

00Bacon, with bibliography of mathematics of economics by Irving Fisher, 1897) was published in 1838.

00Notwithstanding Cournot's just reputation as a writer on mathematics, the Recherches made little impression.

00PETER BARLOW (1776-1862), English writer on pure and applied mathematics, was born at Norwich in 1776 and died on the 1st of March 1862.

00The development of astronomy implies considerable progress in mathematics; it is not surprising, therefore, that the Babylonians should have invented an extremely simple method of ciphering or have discovered the convenience of the duodecimal system.

00The first forty-two years of his life are obscure; we learn from incidental remarks of his that he was a Sunnite, probably according to the IIanifite rite, well versed in all the branches of natural science, in medicine, mathematics, astronomy and astrology, in.

00In natural science, geography, natural history, mathematics and astronomy he took a genuine interest.

00quadrator, squarer), in mathematics, a curve having ordinates which are a measure of the area (or quadrature) of another curve.

00In1824-1828he was professor of mathematics and natural philosophy at Brown University, acting as president in 1826-1827; in1828-1831was president of Transylvania University, Lexington, Kentucky; and in1831-1837was president of the University of Alabama at Tuscaloosa, where he organized the Alabama Female Athenaeum.

00The study of mathematics learned from Greece and India was developed by Arabian writers, who in turn became the teachers of Europe in the 16th century.

00The university, founded in 1338, has faculties of law, medicine, mathematics and philosophy and literature, and is to this day one of the most famous in Italy.

00All these have faculties of letters and law, and San Marcos has in addition faculties of theology, medicine, mathematics and science, philosophy and administrative and political economy.

00Godin, a member of the French commission for measuring an arc of the meridian near Quito, became professor of mathematics at San Marcos in 1750; and the botanical expeditions sent out from Spain gave further zest to scientific research.

00In 1900 she entered Radcliffe College, and successfully passed the examinations in mathematics, &c. for her degree of A.

00is given up to mathematics, under which head are included music, geometry, astronomy, astrology, weights and measures, and metaphysics.

00Thence he journeyed to Bagdad, where he learned Arabic and gave himself to the study of mathematics, medicine and philosophy, especially the works of Aristotle.

00Having studied mathematics under John Machin and John Keill, he obtained in 5708 a remarkable solution of the problem of the " centre of oscillation," which, however, remaining unpublished until May 5754 (Phil.

00Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the " calculus of finite differences."

00Grant was the best horseman of his class, and took a respectable place in mathematics, but at his graduation in 1843 he only ranked twenty-first in a class of thirty-nine.

00To the great dissatisfaction of his parents, he resolved to return to Paris (1816), and to earn his living there by giving lessons in mathematics.

00This and two other engagements as a teacher of mathematics secured him an income of some £400 a year.

00Comte's series or hierarchy is arranged as follows: (i) Mathematics (that is, number, geometry, and mechanics), (2) Astronomy, (3) Physics, (4) Chemistry, (5) Biology, (6) Sociology.

00They are thus the double key of The double Comte's systematization of the philosophy of all the key of sciences from mathematics to physiology, and his positive analysis of social evolution, which is the base of philo= sociology.

00The main principles of the Comtian system are derived from the Positive Polity and from two other works, - the Positivist Catechism: a Summary Exposition of the Universal Religion in Twelve Dialo ues between a Woman and a The g, g Elvis f Priest of Humanity; and, second, The Subjective Synthesis (1856), which is the first and only volume of a work upon mathematics announced at the end of the Positive Philosophy.

00He worked hard at his classical lessons, and supplemented the ordinary business of the school by studying mathematics in the holidays.

00He holds a high place in the history of humanism by the foundation of the College de France; he did not found an actual college, but after much hesitation instituted in 1530, at the instance of Guillaume Bude (Budaeus), Lecteurs royaux, who in spite of the opposition of the Sorbonne were granted full liberty to teach Hebrew, Greek, Latin, mathematics, &c. The humanists Bude, Jacques Colin and Pierre Duchatel were the king's intimates, and Clement Marot was his favourite poet.

00In 1724 he was offered the chair of mathematics in the university of Upsala, which he declined, on the ground that it was a mistake for mathematicians to be limited to theory.

00Among the most representative are: the Popular Science Monthly, New York; the monthly Boston Journal of Education; the quarterly American Journal of Mathematics, Baltimore; the monthly Cassier's Magazine (1891), New York; the monthly American Engineer (1893), New York; the monthly House and Garden, Philadelphia; the monthly Astrophysical Journal, commenced as Sidereal Messenger (1882), Chicago; the monthly American Chemical Journal, Baltimore; the monthly American Naturalist, Boston; the monthly American Journal of the Medical Sciences, Philadelphia; the monthly Outing, New York; the weekly American Agriculturist, New York; the quarterly Metaphysical Magazine (1895) New York; the bi-monthly American Journal of Sociology, Chicago; the bi-monthly American Law Review, St Louis; the monthly Banker's Magazine, New York; the quarterly American Journal of Philology (1880), Baltimore; the monthly Library Journal (1876), New York; the monthly Public Libraries, Chicago; Harper's.

00After acting for a short time as assistant in Harvard College Observatory, he was appointed assistant professor of mathematics in the U.S. Naval Academy in 1866, and in the following year became director of the Allegheny Observatory at Pittsburg, a position which he held until his selection in 1887 as secretary of the Smithsonian.

00bi-, bis, twice, and nomen, a name or term), in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a+b, a-b.

00At Oxford, as at Eton, he read literature from natural liking, and he paid some attention to mathematics.

00While he taught during the day at Stote's Hall, he studied mathematics in the evening at a school in Newcastle.

00In 1773 he was appointed professor of mathematics at the Royal Military Academy, Woolwich, and in the following year he was elected F.R.S.

00After his Tables of the Products and Powers of Numbers, 1781, and his Mathematical Tables, 1785, he issued, for the use of the Royal Military Academy, in 1787 Elements of Conic Sections, and in 1798 his Course of Mathematics.

00His Mathematical and Philosophical Dictionary, a valuable contribution to scientific biography, was published in 1795 (2nd ed., 1815), and the four volumes of Recreations in Mathematics and Natural Philosophy, mostly a translation from the French, in 1803.

00CHRISTIAN KARL AUGUST LUDWIG VON MASSENBACH (1758-1827), Prussian soldier, was born at Schmalkalden on the 16th of April 1758, and educated at Heilbronn and Stuttgart, devoting himself chiefly to mathematics.

00Here he heard Luther preach, but was more attracted by Melanchthon, who interested him in mathematics and astrology.

00This practice continued to prevail till the 17th century, when, at the instance of the Jesuit Schall, president of the tribunal of mathematics, they adopted the European method of dividing the day into twenty-four hours, each hour into sixty minutes, and each minute into sixty seconds.

00In science and theology, mathematics and poetry, metaphysics and law, he is a competent and always a fair if not a profound critic. The bent of his own mind is manifest in his treatment of pure literature and of political speculation - which seems to be inspired with stronger personal interest and a higher sense of power than other parts of his work display.

00In the course of a century eight of its members successfully cultivated various branches of mathematics, and contributed powerfully to the advance of science.

00While at Geneva he taught a blind girl several branches of science, and also how to write; and this led him to publish A Method of Teaching Mathematics to the Blind.

00Chemistry, as well as mathematics, seems to have been the object of his early attention; and in the year 1690 he published a dissertation on effervescence and fermentation.

00His independent discoveries in mathematics are numerous and important.

00Although he had declined a professorship in Germany, he now accepted an invitation to the chair of mathematics at Groningen (Commercium Philosophicum, epist.

00His inaugural discourse was on the "new analysis," which he so successfully applied in investigating various problems both in pure and applied mathematics.

00It is, however, his works in pure mathematics that are the permanent monuments of his fame.

00His writings were collected under his own eye by Gabriel Cramer, professor of mathematics at Geneva, and published under the title of Johannis Bernoulli Operi Omnia (Lausan.

00Meanwhile the study of mathematics was not neglected, as appears not only from his giving instruction in geometry to his younger brother Daniel, but from his writings on the differential, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands.

00Both were appointed at the same time professors of mathematics in the academy of St Petersburg; but this office Nicolas enjoyed for little more then eight months.

00After his'return,though only twenty-four years of age, he was invited to become president of an academy then projected at Genoa; but, declining this honour, he was, in the following year, appointed professor of mathematics at St Petersburg.

00He studied law and mathematics, and, of ter travelling in France,was for five years professor of eloquence in the university of his native city.

00On the death of his father he succeeded him as professor of mathematics.

00In Italy he formed a friendship with Lorgna, professor of mathematics at Verona, and one of the founders of the Societe' Italiana for the encouragement of the sciences.

00His fame now rests, however, entirely upon his achievements in mathematics.

00He then devoted himself with astonishing ardour to mathematics, chemistry, natural history, technology and even political economy.

00In 1842 he took a "double-first" and was elected fellow of B alliol, and lecturer in mathematics and logic. Four years later he took orders, and with the aim of helping forward the education of the very poor, he accepted the headship of Kneller Hall, a college which the government formed for the training of masters of workhouse and penal schools.

00in 1841, first in classics and philosophy and bracketed first in mathematics.

00FIGURATE NUMBERS, in mathematics.

00He graduated at Harvard in 1817, was tutor in mathematics there in 1820-1821, was admitted to practice in the court of common pleas in December 1821, and began the practice of law in Newburyport, Mass., in 1824.

00In this way the principle of continuity, which is the basis of the method of Fluxions and the whole of modern mathematics, may be applied to the analysis of problems connected with material bodies by assuming them, for the purpose of this analysis, to be homogeneous.

00Considering only those states of the system which have a given value of E2, it can be proved, as a theorem in pure mathematics,' that when s, s', ...

00COUNT MIKHAIL MIKHAILOVICH SPERANSKI (1772-1839), Russian statesman, the son of a village priest, spent his early days at the ecclesiastical seminary in St Petersburg, where he rose to be professor of mathematics and physics.

00Of his sons, Thomas (1616-1680) was born at Copenhagen, where, after a long course of study in various universities of Europe, he was appointed successively professor of mathematics (1647) and anatomy (1648).

00Another son, Erasmus (1625-1698), born at Roskilde, spent ten years in visiting England, Holland, Germany and Italy, and filled the chairs of mathematics and medicine at Copenhagen.

00The sciences of mathematics, astronomy and medicine were also cultivated with assiduity and success at Alexandria, but they can scarcely be said to have their origin there, or in any strict sense to form a part of the peculiarly Alexandrian literature.

00Alexandria continued to be celebrated as a school of mathematics and science long after the Christian era.

00In 1827 he became extraordinary and in 1829 ordinary professor of mathematics at Konigsberg, and this chair he filled till 1842, when he visited Italy for a few months to recruit his health.

00It was in analytical development that Jacobi's peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle's Journal and elsewhere from 1826 onwards will sufficiently indicate.

00In 1836 he entered Marischal College, and came under the influence of John Cruickshank, professor of mathematics, Thomas Clark, professor of chemistry, and William Knight, professor of natural philosophy.

00His college career was distinguished, especially in mental philosophy, mathematics and physics.

00In 1845 he was appointed professor of mathematics and natural philosophy in the Andersonian University of Glasgow.

00This classification, though it is of high value in the clearing up of our conceptions of the essential contrasted with the accidental, the relation of genus, differentia and definition and so forth, is of more significance in connexion with abstract sciences, especially mathematics, than for the physical sciences.

00This being so, not only were physics and mathematics impossible as sciences of necessary objective truth, but our apparent consciousness of a permanent self and object alike must be delusive.

00The students number some 750, and there are five faculties of theology, law, medicine, mathematics and science, and letters.

00His great reputation and the influence of Sir William Boswell, the English resident, with the states-general procured his election in 1643 to the chair of mathematics in Amsterdam, whence he removed in 1646, on the invitation of the prince of Orange, to Breda, where he remained till 1652.

00His chief works are: Astronomical History of Observations of Heavenly Motions and Appearances (1634); Ecliptica prognostica (1634); Controversy with Longomontanus concerning the Quadrature of the Circle (1646?); An Idea of the Mathematics, 12m0 (1650); A Table of Ten Thousand Square Numbers (fol.; 1672).

00It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

00Vitality can only be retained by close association with more abstract branches of mathematics.

00Castle, Manual of Practical Mathematics (1903); F.

00C. Clarke, Practical Mathematics (1907); C. T.

00He was educated at the university of Moscow, and in 1859 became professor of mathematics in the university of St Petersburg, a position from which he retired in 1880.

00Having studied literature, he afterwards devoted himself entirely to mathematics and natural philosophy.

00He became professor of philosophy, mathematics, and Oriental languages at Wurzburg, whence he was driven (1631) by the troubles of the Thirty Years' War to Avignon.

00Through the influence of Cardinal Barberini he next (1635) settled in Rome, where for eight years he taught mathematics in the Collegio Romano, but ultimately resigned this appointment to study hieroglyphics and other archaeological subjects.

00His voluminous writings in philology, natural history, physics and mathematics often accordingly have a good deal of the historical interest which attaches to pioneering work, however imperfectly performed; otherwise they now take rank as curiosities of literature merely.

00Next follow chapters on the literary renaissance of the nation, its progress in art, mathematics, chemistry and natural science; the magnificent development of agriculture, modern industry, commerce and finance; and in particular its flourishing selfgovernment, " which will be exercised in the fullest freedom," and in which " the communal organization embodies in the highest degree the conception of self-government " (p. 234), and " the independent sphere of activity unlimited in its fundamental principle " (p. 235) in that " State control is exercised seldom and discreetly " (p. 236).

00Varenius studied at the gymnasium of Hamburg (1640-42), and at Konigsberg (1643-45) and Leiden (1645-49) universities, where he devoted himself to mathematics and medicine, taking his medical degree at Leiden in 1649.

00In 181 9 he was appointed professor of mathematics at the athenaeum of Brussels; in 1828 he became lecturer at the newly created museum of science and literature, and he continued to hold that post until the museum was absorbed in the free university in 1834.

00de Valder in the chair of philosophy and mathematics at Leiden.

00In 1788 he entered the corps of noble cadets in the artillery and engineering department, where his ability, especially in mathematics, soon attracted attention.

00Despite extreme penury, he then continued to study indefatigably ancient and modern languages, history and literature, finally turning his attention to mathematics and astronomy.

00In 1613 he succeeded his father Rudolph Snell (1546-1613) as professor of mathematics in the university of Leiden.

00Soon afterwards he was appointed professor of mathematics in the Ecole Militaire at Paris, and he was afterwards professor in the Ecole Normale.

00It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics.

00He studied theology, and was for some years a dissenting minister at Tonbridge, but on the death of his father he devoted himself to the congenial study of mathematics.

00Meanwhile, at Oxford a proposal practically making Greek optional with all undergraduates was rejected, in November 1902, by 189 votes to 166; a preliminary proposal permitting students of mathematics or natural science to offer one or more modern languages in lieu of Greek was passed by 164 to 162 in February 1904, but on the 29th of November the draft of a statute to this effect was thrown out by 200 to 164.

00Daunou (October 1795), which divided the pupils of the " central schools " into three groups, according to age, with corresponding subjects of study: (r) twelve to fourteen, = drawing, natural history, Greek and Latin, and a choice of modern languages; (2) fourteen to sixteen, - mathematics, physics, chemistry; (3) over sixteen, - general grammar, literature, history and constitutional law.

00The law of the 1st of May 1802 brought the lycees into existence, the subjects being, in Napoleon's own phrase, " mainly Latin and mathematics."

00In these schools the subjects of study included mathematics and natural sciences, geography and history, and modern languages (especially French), with riding, fencing and dancing; Latin assumed a subordinate place, and classical composition in prose or verse was not considered a sufficiently courtly accomplishment.

00Siivern recognized four principal co-ordinated branches of learning: Latin, Greek, German, mathematics.

00The new gymnasium aimed at a wider education, in which literature was represented by Latin, Greek and German, by the side of mathematics and natural science, history and religion.

00In 1875 Wiese was succeeded by Bonitz, the eminent Aristotelian scholar, who in 1849 had introduced mathematics and natural science into the schools of Austria, and had substituted the wide reading of classical authors for the prevalent practice of speaking and writing Latin.

00In 1788 Pfaff became professor of mathematics in Helmstedt, and so continued until that university was abolished in 1810.

00From that time till his death on the 21st of April 1825 he held the chair of mathematics at Halle.

00His brother, Johann Wilhelm Andreas Pfaff (1774-1835), was professor of pure and applied mathematics successively at Dorpat, Nuremberg, Wurzburg and Erlangen.

00At Harrow he obtained in 1842 a Lyon scholarship, and at Oxford in 1845 a first-class in mathematics, in 1846 the junior and in 1847 the senior university mathematical scholarship. In 1846 he left Oxford to take his father's place in the business, in which he was engaged until his death.

00His papers, numbering over 100, were published principally in the Philosophical Transactions, Proceedings of the Royal Society, Quarterly Journal of Mathematics, Proceedings of the London Mathematical Society and Crelle, and one or two in the Comptes rendus of the Paris Academy; a list of them, arranged according to the several journals in which they originally appeared, with short notes upon the less familiar memoirs, is given in Nature, xxvii.

00His Son, Johann Ernst Immanuel (1725-1778), studied Semitic languages at Jena, and also natural science and mathematics.

00He graduated in 1865 at the Lawrence Scientific School of Harvard, where for the next two years he was a teacher of mathematics.

00At Glasgow his favourite studies had been mathematics and natural philosophy; but at Oxford he appears to have devoted himself almost entirely to moral and political science and to ancient and modern languages.

00At the age of eleven he was entered as a student at St Andrews, where he devoted himself almost exclusively to mathematics.

00In May 1803, after attending further courses of lectures in Edinburgh, and acting as assistant to the professor of mathematics at St Andrews, he was ordained as minister of Kilmany in Fifeshire, about 9 m.

00In 1805 he became a candidate for the vacant professorship of mathematics at Edinburgh, but was unsuccessful.

00X yos, word, ratio, and &pc0,u6s, number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the equation a^x = m, then a is called the base, and x is said to be the logarithm of m to the base a.

00Between John Craig and John Napier a friendship sprang up which may have been due to their common taste for mathematics.

00Now Mark Napier found in the library of the university of Edinburgh a mathematical work bearing a sentence in Latin which he translates, " To Doctor John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted with various and excellent learning, professor of medicine, and exceedingly skilled in the mathematics, Tycho Brahe bath sent this gift, and with his own hand written this at Uraniburg, 2d November 1588."

00The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject.

00In the Messenger of Mathematics, vol.

00Mathematics >>

00His early mastery of classical literature led him to the study of classic monuments in classic lands, while his equally conspicuous talent for mathematics gave him the laws of form and proportion in architectural design.

00A vacancy among the fellows is filled up by the provost and a select number of the fellows, after examination comprised in five principal courses, mathematics, experimental science, classics, mental and moral science and Hebrew.

00The scholars on the foundation (or "of the House") are chosen from among the undergraduates, for merit in classics, mathematics or experimental science.

00From 1813 to 1820 he was extraordinary professor of astronomy and mathematics at the new university and observer at the observatory, becoming in 1820 ordinary professor and director.

001854) studied mathematics at Dorpat, and became in 1883 assistant, and in 1890, on his father's retirement, astronomer at the observatory at Pulkowa.

00John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon's Conic Sections.

00formed an epoch in the history of mathematics generally, and had, of course, a marked influence on after investigations regarding circle-quadrature.

00A much less wise class than the 7r-computers of modern times are the pseudo-circle-squarers, or circle-squarers technically so called, that is to say, persons who, having obtained by illegitimate means a Euclidean construction for the quadrature or a finitely expressible value for 7r, insist on using faulty reasoning and defective mathematics to establish their assertions.

00Such persons have flourished at all times in the history of mathematics; but the interest attaching to them is more psychological than mathematical.2 It is of recent years that the most important advances in the theory of circle-quadrature have been made.

00He had seen Cyrene from the sea, probably on his voyage from Puteoli to Alexandria, where he remained a long time, probably amassing materials, and studying astronomy and mathematics.

00mathematics, with the conclusion that good is one, it appeared to.

00He gradually became a logician out of his previous studies: out of metaphysics, for with him being is always the basis of thinking, and common principles, such as that of contradiction, are axioms of things before axioms of thought, while categories are primarily things signified by names; out of the mathematics of the Pythagoreans and the Platonists, which taught him the nature of demonstration; out of the physics, of which he imbibed the first draughts from his father, which taught him induction from sense and the modification of strict demonstration to suit facts; out of the dialectic between man and man which provided him with beautiful examples of inference in the Socratic dialogues of Xenophon and Plato; out of the rhetoric addressed to large audiences, which with dialectic called his attention to probable inferences; out of the grammar taught with rhetoric and poetics which led him to the logic of the proposition.

00But at any rate the process was gradual; and Aristotle was advanced in metaphysics, mathematics, physics, dialectics, rhetoric and poetics, before he became the founder of logic.

00The commentators themselves were doubtful about the order: Boethus proposed to begin with Physics, and some of the Platonists with Ethics or Mathematics; while Andronicus preferred to put Logic first as Organon (Scholia, 25 b 34 seq.).

003), and in the classification would stop at mathematics, which we still call exact science: in the wide sense, on the other hand, it extends to the whole of the necessary and to the usual contingent, but excludes the accidental (Met.

00E 2), and would in the classification include not only metaphysics and mathematics, but also physics, ethics, economics, politics, necessary and fine art; or in short.

00Aristotle, who made this great discovery, must have had great difficulty in developing the new investigation of reasoning processes out of dialectic, rhetoric, poetics, grammar, metaphysics, mathematics, physics and ethics; and in disengaging it from other kinds of learning.

004, 1 359 b To; cf, 1 35 6 b 9, 1 357 a 30, b 25); and in the Metaphysics he evidently refers to it as " the science which considers demonstration and science," which he distinguishes from the three speculative sciences, mathematics, physics and primary philosophy (Met.

00The mind, especially in mathematics, abstracts numbers, motions, relations, causes, essences, ends, kinds; and it over-abstracts things mentally separate into things really separate.

00Here he introduced many improvements in map-making, and gained a scientific reputation which led (in 1751) to his election to the chair of economy and mathematics in the university of Gottingen.

00He afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to his going to Cambridge.

00After studying at Bologna, he became professor of mathematics at Modena, and in 1831 was appointed inspector-general of studies in the duchy.

00The same year, in his Analyst, he attacked the higher mathematics as leading to freethinking; this involved him in a hot controversy.

00Having come to London by the advice of Dr Henry Pemberton (1694-1771), who had recognized his talents, he for a time maintained himself by teaching mathematics, but soon devoted himself to engineering and the study of fortification.

00In mathematics he was twenty-fourth wrangler, Isaac Todhunter being senior.

00Though he never became either a scholar or a mathematician, he did enough accurate work to be placed in the honorary fourth class both in classics and in mathematics.

00In mathematics, the term "mean," in its most general sense, is given to some function of two or more quantities which (1) becomes equal to each of the quantities when they themselves are made equal, and (2) is unaffected in value when the quantities suffer any transpositions.

00This view involves the denial of force as a cause, and the assertion that all we know about force is that the acceleration of one mass depends on that of another, as in mathematics a function depends on a variable; and that even Newton's third law of motion is merely a description of the fact that two material points determine in one another, without reciprocally causing, opposite accelerations.

00This being so, he finds in mathematics two kinds of transcendence - real, where the transcendent, though not actual in experience, can become partly so, e.g.

00He began life as a clerk, but, obtaining an appointment to a cadetship at West Point in 1825, he graduated there in 1829, and acted as assistant professor of mathematics 1829-1832.

00He was then called to the bar, but in 1836 became professor of mathematics and natural philosophy at Cincinnati College.

00His taste for mathematics early developed itself; and he acquired Latin that he might study Newton's Principia.

00He was offered, but declined, the professorship of mathematics and astronomy at Harvard.

00Having studied law at Leipzig, Helmstadt and Jena, and mathematics, especially geometry and mechanics, at Leiden, he visited France and England, and in 1636 became engineer-in-chief at Erfurt.

00It is based on Descartes' fundamental principle that knowledge must be clear, and seeks to give to philosophy the certainty and demonstrative character of mathematics, from the a priori principle of which all its claims are derived.

00As a boy he showed great aptitude for study, and at first devoted himself to theology, but under the influence of Wolff's writings he took up mathematics and philosophy on the lines of Wolff and Leibnitz.

00In 1721, after two years' study under Wolff, he became professor of philosophy at Halle, and in 1724 professor of mathematics.

00Like most of the great metaphysicians of the 17th century, Malebranche interested himself also in questions of mathematics and natural philosophy, and in 1699 was admitted an honorary member of the Academy of Sciences.

00Roger Bacon, his pupil, speaks highly of his attainments in theology and mathematics.

00At the age of thirteen he entered King's College, Aberdeen, where the first prize in mathematics and physical and moral sciences fell to him.

00He was intended for the church, but the bent of his mind was towards mathematics, and, when a prospect opened of his succeeding to the mathematical chair at the university of Glasgow, he proceeded to London for further study.

00After a year in London he returned to Glasgow, and in 1711 was appointed by the university to the professorship of mathematics, an office which he retained until 1761.

00He studied at Reims under Gerbert, afterwards Pope Silvester II., who taught him mathematics, history, letters and eloquence.

00But he had received some instruction in mathematics from a distant relative, Elihu Robinson, and in 1781 he left his native village to become assistant to his cousin George Bewley who kept a school at Kendal.

00Mainly through John Gough (1757-1825), a blind philosopher to whose aid he owed much of his scientific knowledge, he was appointed teacher of mathematics and natural philosophy at the New College in Moseley Street (in 1889 transferred to Manchester College, Oxford), and that position he retained until the removal of the college to York in 1799, when he became a "public and private teacher of mathematics and chemistry."

00He was educated in the Jesuit college at Parma, and showed at first a great aptitude for mathematics.

00FOLIUM, in mathematics, a curve invented and discussed by Rene Descartes.

00MARIUS SOPHUS LIE (1842-1899), Norwegian mathematician, was born at Nordfjordeif, near Bergen, on the 17th of December 1842, and was educated at the university of Christiania, where he took his doctor's degree in 1868 and became extraordinary professor of mathematics (a chair created specially for him) four years later.

00In 1768 Monge became professor of mathematics, and in 1771 professor of physics, at Mezieres; in 1778 he married Mme Horbon, a young widow whom he had previously defended in a very spirited manner from an unfounded charge; in 1780 he was appointed to a chair of hydraulics at the Lyceum in Paris (held by him together with his appointments at Mezieres), and was received as a member of the Academie; his intimate friendship with C. L.

00Although pressed by the minister to prepare for them a complete course of mathematics, he declined to do so, on the ground that it would deprive Mme Bezout of her only income, from the sale of the works of her late husband; he wrote, however (1786), his Traite elementaire de la statique.

00Kastner, professor of mathematics and also an epigrammatist of repute.

00In 1669 he resigned his mathematical chair to his pupil, Isaac Newton, having now determined to renounce the study of mathematics for that of divinity.

00He was undoubtedly a clear-sighted and able mathematician, who handled admirably the severe geometrical method, and who in his Method of Tangents approximated to the course of reasoning by which Newton was afterwards led to the doctrine of ultimate ratios; but his substantial contributions to the science are of no great importance, and his lectures upon elementary principles do not throw much light on the difficulties surrounding the border-land between mathematics and philosophy.

00Having graduated at Harvard College in 1844, he studied mathematics and astronomy under C. F.

00Having completed his education at the university of Edinburgh, where he was distinguished in mathematics, Robert was induced to enter a banking-house in order to acquire a practical knowledge of business, but his ambition was really academic. In 1769 he gave up business pursuits and accepted the rectorship of Perth academy.

00For many years, however, by private arrangement with his colleague Professor Copland, Hamilton taught the class o £ mathematics.

00In each of these universities there are five faculties, namely, law, theology, medicine, science and mathematics, and literature and philosophy, the courses for which are respectively four, five, eight, and six or seven years for the two last named.

00Among the literary and scientific associations of Copenhagen may be mentioned the Danish Royal Society, founded in 1742, for the advancement of the sciences of mathematics, astronomy, natural philosophy, &c., by the publication of papers and essays; the Royal Antiquarian Society, founded in 1825, for diffusing a knowledge of Northern and Icelandic archaeology; the Society for the Promotion of Danish Literature, for the publication of works chiefly connected with the history of Danish literature; the Natural Philosophy Society; the Royal Agricultural Society; the Danish Church History Society; the Industrial Association, founded in 1838; the Royal Geographical Society, established in 1876; and several musical and other societies.

00John was educated at Leiden, and early displayed remarkable talents, more especially in mathematics and jurisprudence.

00The propositions of mathematics seem to be independent of this or that special fact of experience, and to remain unchanged even when the concrete matter of experience varies.

00Hume's theory of mathematics - the only one, perhaps, which is compatible with his fundamental principle of psychology - is a practical condemnation of his empirical theory of perception.

00Mathematics and classics are taught in them and the masters are allowed to take boarders.

00He studied at Geneva, Leyden and Paris, before becoming (1700) professor of philosophy and mathematics at the academy of Lausanne, of which he was four times rector before 1724, when the theological disputes connected with the Consensus led him to accept a chair of philosophy and mathematics at Groningen.

00His scheme was first to work out, in a separate treatise De corpore, a systematic doctrine of Body, showing how physical phenomena were universally explicable in terms of motion, as motion or mechanical action was then (through Galileo and others) understood - the theory of motion being applied in the light of mathematical science, after quantity, the subject-matter of mathematics, had been duly considered in its place among the fundamental conceptions of philosophy, and a clear indication had been given, at first starting, of the logical ground and method of all philosophical inquiry.

00Taking up mathematics when not only his mind was already formed but his thoughts were crystallizing into a philosophical system, Hobbes had, in fact, never put himself to school and sought to work up gradually to the best knowledge of the time, but had been more anxious from the first to become himself an innovator with whatever insufficient means.

00With the translation,' in the spring of 1656, he had ready Six Lessons to the Professors of Mathematics, one of Geometry, the other of Astronomy, in the University of Oxford (E.W.

00All these controversial writings on mathematics and physics represent but one half of his activity after the age of p y g Years.

00Upon every subject that came within the sweep of his system, except mathematics and physics, his thoughts have been productive of thought.

00In yet another branch of pure mathematics Pascal ranks as a founder.

00Pascal's work as a natural philosopher was not less remarkable than his discoveries in pure mathematics.

00He was educated at the university of Turin, where he qualified as an engineer and became a doctor of mathematics.

00He held many college offices, becoming successively lecturer in Greek (1651), mathematics (1653),andhumanity('655), praelector (1657), junior dean (1657), and college steward (1659 and 1660); and according to the habit of the time, he was accustomed to preach in his college chapel and also at Great St Mary's before the university, long before he took holy orders.

00grammar, music, painting, sculpture, medicine, geometry, mathematics and optics; c. 2 is on the general principles of architectural design; c. 3 on the considerations which determine a design, such as strength, utility, beauty; c. 4 on the nature of different sorts of ground for sites; c. 5 on walls of fortification; c. 6 on aspects towards the north, south and other points; c. 7 on the proper situations of temples dedicated to the various deities.

00Accordingly we find that Arabian philosophy, mathematics, geography, medicine and philology are all based professedly upon Greek works (Brockelmann, Gesch.

00Besides the subjects taught at the Azhar university, instruction is given in literature, mathematics and physical science.

00Apart from a few calculations and accounts, practically all the materials for our knowledge of Egyptian mathematics before the Hellenistic period date from the Middle Kingdom.

00In later life, he gave up speculative thought and turned to scientific research, especially in mathematics, physics and astronomy.

00MARIA GAETANA AGNESI (1718-1799), Italian mathematician, linguist and philosopher, was born at Milan on the 16th of May 1718, her father being professor of mathematics in the university of Bologna.

00Though the wish was not gratified, she lived from that time in a retirement almost conventual, avoiding all society and devoting herself entirely to the study of mathematics.

00d'Antelmy, with additions by Charles Bossut (1730-1814), appeared at Paris in 1775; and an English translation of the whole work by John Colson (1680-1760), the Lucasian professor of mathematics at Cambridge, was published in 1801 at the expense of Baron Maseres.

00to the chair of mathematics and natural philosophy at Bologna.

00The legislature, thanks to the efforts of Joseph Carrington Cabell, a close personal friend of Jefferson, adopted the plan in 1818 and 1819, and seven independent schools - ancient languages, modern languages, mathematics, natural philosophy, moral philosophy, chemistry and medicine - were opened to students in March 1825; a school of law was opened in 1826.

00In 1804 he was appointed professor of mathematics at the Lycee, in 1809 professor of analysis and mechanics, and in 1816 examiner at the Ecole Polytechnique.

00In addition to publishing a number of works on geometrical and mechanical subjects, Poinsot also contributed a number of papers on pure and applied mathematics to Lionville's Journal and other scientific periodicals.

00He cared little for any of the professors, except Sir John Leslie, from whom he learned some mathematics.

00He did not seek re-election in 1893, but devoted himself thenceforward to mathematics, helping to make known in France the theories of Giusto Bellavitis.

00PEDRO NUNEZ (PETRUS Nomus) (1492-1577), Portuguese mathematician and geographer, was born at Alcacer do Sal, and died at Coimbra, where he was professor of mathematics.

00He landed at Macao in 1610, and while waiting a favourable opportunity to penetrate into China busied himself for three years in teaching mathematics.

00His knowledge of mathematics caused him to be employed on the coast survey in 1834.

00He followed his father's trade, but found time to acquire a knowledge of Latin, Greek, mathematics, physics, anatomy and other subjects.

00His father, an avocat au parlement, gave him an excellent education at the college Mazarin, and encouraged his taste for natural science; and he studied mathematics and astronomy with N.

00His duties were light, and he employed his leisure in the study of philology, mathematics, philosophy, history, political economy, natural science and natural history, for which he made large collections.

00These names have in the mathematics tripos survived the procedure.

00Rouse Ball in his History of the Study of Mathematics at Cambridge (1889), p. 193, states that he can find no record of any European examinations by means of written papers earlier than those introduced by R.

00A further distinction is important, especially in such subjects as mathematics or foreign languages, in which it is legitimate to ask what precise power on the part of a candidate the passing of an examination shall signify.

00Of recent years the Thesls thesis has been introduced into lower examinations; it is required for the master's degree at London in the case of internal students, in subjects other than mathematics (1910); both at Oxford and London, the B.Sc. degree, and at Cambridge the B.A.

00Having studied theology in the academy of the Moravian brethren at Niesky, and philosophy at Leipzig and Jena, he travelled for some time, and in 1806 became professor of philosophy and elementary mathematics at Heidelberg.

00In 1816 he was invited to Jena to fill the chair of theoretical philosophy (including mathematics and physics, and philosophy proper), and entered upon a crusade against the prevailing Romanticism.

00The grand-duke, however, continued to pay him his stipend, and in 1824 he was recalled to Jena as professor of mathematics and physics, receiving permission also to lecture on philosophy in his own rooms to a select number of students.

00The word was reintroduced in modern philosophy probably by Rene Descartes (or by his followers) who, in the search for a definite self-evident principle as the basis of a new philosophy, naturally turned to the familiar science of mathematics.

00This collection, alphabetically arranged, comprised annotations on classical authors, passages from newspapers, treatises on morals and mathematics from the standard works of the period.

00Besides philosophy, he once at least lectured on mathematics.

00John is said to have owed his education in philosophy, mathematics and theology to an Italian monk named Cosmas, whom Sergius had redeemed from a band of captive slaves.

00De Morgan was one of his colleagues, but he resigned in 1840 in order to become professor of mathematics in the university of Virginia.

00But he failed to obtain either of two posts - the professorships of mathematics at the Royal Military Academy and of geometry in Gresham College - for which he applied in 1854, though he was elected to the former in the following year on the death of his successful competitor.

00Three years later he was appointed professor of mathematics in the Johns Hopkins University, Baltimore, stipulating for an annual salary of $5000, to be paid in gold.

00At Baltimore he gave an enormous impetus to the study of the higher mathematics in America, and during the time he was there he contributed to the American Journal of Mathematics, of which he was the first editor, no less than thirty papers, some of great length, dealing mainly with modern algebra, the theory of numbers, theory of partitions and universal algebra.

00His early life was occupied in mastering the curriculum of theology, jurisprudence, mathematics, medicine and philosophy, under the approved teachers of the time.

00He studied mathematics and physics in his native town, Groningen, where in 1879 he took his doctor's degree on presenting a dissertation entitled New Proofs of the Earth's Rotation.

00That which is object of thought cannot be outside consciousness; just as in mathematics -V - is an unreal quantity, so "things-in-themselves" are ex hypothesi outside consciousness, i.e.

00Complete or perfect knowledge is confined to the domain of pure thought, to logic and mathematics.

00Jefferson carried with him from the college of William and Mary at Williamsburg, in his twentieth year, a good knowledge of Latin, Greek and French (to which he soon added Spanish, Italian and Anglo-Saxon), and a familiarity with the higher mathematics and natural sciences only possessed, at his age, by men who have a rare natural taste and ability for those studies.

00Notwithstanding his many official duties, he found time to publish more than three hundred works, several of them extensive treatises, and many of them memoirs dealing with the most abstruse branches of pure and applied mathematics.

00There are few branches of mathematics to which he did not contribute something, but it was in the application of mathematics to physical subjects that his greatest services to science were performed.

00In pure mathematics, his most important works were his series of memoirs on definite integrals, and his discussion of Fourier's series, which paved the way for the classical researches of L.

00Born at Edinburgh in 1710 and originally educated for the church, Short attracted the attention of Maclaurin, professor of mathematics at the university, who permitted him about 1732 to make use of his rooms in the college buildings for experiments in the construction of telescopes.

00At an early age he manifested a preference for the study of mathematics, but this was gradually superseded by an interest in natural science, which led him ultimately to the study of medicine.

00The university, in Calle Uruguay, has faculties of law, medicine, letters, mathematics, engineering, and some minor groups of studies, including agriculture and veterinary science.

00