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mathematical

Space-time in simple terms is a mathematical model that combines space and time into a single continuum.

3122All these rest upon the facts of mathematical geography, and the three are so closely inter-related that they cannot be rigidly separated in any discussion.

139In England, Robert Recorde had indeed published his mathematical treatises, but they were of trifling importance and without influence on the history of science.

1210It has been stated that Napier's mathematical pursuits led him to dissipate his means.

103The angle which the earth's axis makes with the plane in which the planet revolves round the sun determines the varying seasonal distribution of solar radiation over the surface and the mathematical zones of climate.

105The fundamental geographical conceptions are mathematical, the relations of space and form.

75It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life.

67With this machine movable type shuttles can be used, and one can have several shuttles, each with a different set of characters--Greek, French, or mathematical, according to the kind of writing one wishes to do on the typewriter.

54On his return to his native city he devoted himself to mathematical research.

46This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.

46Papers by him have appeared in the mathematical journals of Italy, France, Germany and England, and he has published several important works, many of which have been translated into other languages.

46He was president of the mathematical and physical section of the British Association at Bradford in 1873 and of the London Mathematical Society in 18 741876.

32Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected.

33Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected.

33The science of geodesy is part of mathematical geography, of which the arts of surveying and cartography are applications.

34His first appointment was as elementary mathematical master at the gymnasium and lyceum of Cremona, and he afterwards obtained a similar post at Milan.

35We may now investigate the mathematical expression for the disturbance propagated in any direction from a small particle upon which a beam of light strikes.

21In 1852 he graduated at Harvard, and became computer to the American Ephemeris and Nautical Almanac. He made his name by contributions on mathematical and physical subjects in the Mathematical Monthly.

22He also made important contributions to the mathematical theory of electrodynamics, and in papers published in 1845 and 1847 established mathematically the laws of the induction of electric currents.

22The disturbing conditions of will, life and organic forces are eliminated from the problem; he starts with the clear and distinct idea of extension, figured and moved, and thence by mathematical laws he gives a hypothetical explanation of all things.

22In 1852 he graduated at Harvard, and became computer to the American Ephemeris and Nautical Almanac. He made his name by contributions on mathematical and physical subjects in the Mathematical Monthly.

22Before applying the solution to a mathematical investigation of the present question, it may be well to consider the matter for a few moments from a more general point of view.

23Peacock's original contributions to mathematical science were concerned chiefly with the philosophy of its first principles.

23Thus early commenced the separation between what were long called mathematical and political geography, the one subject appealing mainly to mathematicians, the other to historians.

23Dalton, who was a mathematical physicist even more than a chemist, had given much thought to the study of gases.

24From 1863 to 1870 he was secretary and recorder to the American Academy of Arts and Sciences, and in the last year of his life he lectured on mathematical physics at Harvard.

24His life was devoted to the study of higher geometry and reforming the more advanced mathematical teaching of Italy.

24Napier's three mathematical works are reprinted by N.

24He gained the senior mathematical scholarship in 1851.

24His early mathematical and physical papers, written under the name of J.

12He won the Ireland scholarship in 1848 and obtained a first class in both the classical and the mathematical schools in 1849.

12To the public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications which have become standard works.

12To these lists should be added a paper on the mathematical basis of logic, published in the Mechanic's Magazine for 1848.

12Only two systematic treatises on mathematical subjects were completed by Boole during his lifetime.

12His writings include: Mathematical Investigations in the Theory of Value and Prices (1892); Elements of Geometry (with A.

12The above statements, though correct as far as they go, are an imperfect account of the nature of the radiation from a coupled antenna, but a mathematical treatment is required for a fuller explanation.

121 The mathematical theory of this antenna was given by J.

1224) followed Eratosthenes rather than Aristotle, but with sympathies which went out more to the human interests than the mathematical basis of geography.

12Ptolemy used the word geography to signify the description of the whole oekumene on mathematical principles, while chorography signified the fuller description of a particular region, and topography the very detailed description of a smaller locality.

12Sebastian Munster, on the other hand, in his Cosmographia universalis of 1544, paid no regard to the mathematical basis of Munster.

12The above statements, though correct as far as they go, are an imperfect account of the nature of the radiation from a coupled antenna, but a mathematical treatment is required for a fuller explanation.

12This confusion Lotze, who had been trained in the school of mathematical reasoning, tried to dispel.

13Archimedes died at the capture of Syracuse by Marcellus, 212 B.C. In the general massacre which followed the fall of the city, Archimedes, while engaged in drawing a mathematical figure on the sand, was run through the body by a Roman soldier.

13In the year following his marriage Mendelssohn won the prize offered by the Berlin Academy for an essay on the application of mathematical proofs metaphysics, although among the competitors were Abbt and Kant.

13Being especially interested in mathematical science, the father gave his son his first lessons; but the extraordinary mathematical powers of George Boole did not manifest themselves in early life.

13(1) Mathematical geography, which deals with the form, size and movements of the earth and its place in the solar system; (2) Moral geography, or an account of the different customs and characters of mankind according to the region they inhabit; (3) Political geography, the divisions according to their organized governments; (4) Mercantile geography, dealing with the trade in the surplus products of countries; (5) Theological geography, or the distribution of religions.

14P. Mahaffy, Descartes (1902), with an appendix on Descartes's mathematical work by Frederick Purser; Victor de Swarte, Descartes directeur spirituel (Paris, 1904), correspondence with the Princess Palatine; C. J.

00He also wrote the introduction to the collected edition of Clifford's Mathematical.

00They claim probability - moral certainty - mathematical certainty.

00And, as the sympathizers with Hegel try to force mechanical necessity into the garb of absolute or ideal necessity, so they seek to show that moral necessity is only an inferior form of absolute or ideal or, we might say, mathematical necessity.

00From the underlying abstract mathematical considerations all through the superimposed physical, biological, anthropo.

00Its author, with a considerable mathematical and mechanical bias, reckoned entirely with the quantity, not with the quality of his units, and relied almost implicitly upon his formulae.

00Some idea of his activity as a writer on mathematical and physical subjects during these early years may be gathered from the fact that previous to this appointment he had contributed no less than three important memoirs to the Philosophical Transactions of the Royal Society, and eight to the Cambridge Philosophical Society.

00Airy's writings during this time are divided between mathematical FIG.

00Amongst the most important of his works not already mentioned may be named the following: - Mathematical Tracts (1826) on the Lunar Theory, Figure of the Earth, Precession and Nutation, and Calculus of Variations, to which, in the second edition of 1828, were added tracts on the Planetary Theory and the Undulatory Theory of Light; Experiments on Iron-built Ships, instituted for the purpose of discovering a correction for the deviation of the Compass produced by the Iron of the Ships (1839); On the Theoretical Explanation of an apparent new Polarity in Light (1840); Tides and Waves (1842).

00In 1746 and 1748 he published in the Memoirs of the Academy of Berlin "Recherches sur le calcul integral," a branch of mathematical science which is greatly indebted to him.

00He also wrote several literary articles for the first two volumes of the Encyclopaedia, and to the remaining volumes he contributed mathematical articles chiefly.

00It is to be observed, moreover, that as Alembert confined himself chiefly to mathematical articles, his work laid him less open to charges of heresy and infidelity than that of some of his associates.

00He 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.

00He at once took a leading position in the mathematical teaching of the university, and published treatises on the Di f ferential calculus (in 1848) and the Infinitesimal calculus (4 vols., 1852-1860), which for long were the recognized textbooks there.

00Educated at Trinity College, Cambridge, where he took a first-class both in the mathematical tripos and in the 2nd part of the moral sciences tripos, he remained at Cambridge as a lecturer, and became well known as a student of mathematical philosophy and a leading exponent of the views of the newer school of Realists.

00A condition of tenure attached to this chair was that the holder should propose mathematical questions for solution, and should resign in favour of any person who solved them better than himself; but, notwithstanding this, Roberval was able to keep the chair till his death, which occurred at Paris on the 27th of October 1675.

004 For a very complete exposition of the operation of valves in the horn, and of the mathematical proportions to be observed in construction, see Victor Mahillon's "Le Cor," also the article by Gottfried Weber in Caecilia (1835), to which reference was made above.

00Having settled at Cambridge in 1796, Gregory first acted as sub-editor on the Cambridge Intelligencer, and then opened a bookseller's shop. In 1802 he obtained an appointment as mathematical master at Woolwich through the influence of Charles Hutton, to whose notice he had been brought by a manuscript on the "Use of the Sliding Rule"; and when Hutton resigned in 1807 Gregory succeeded him in the professorship. Failing health obliged him to retire in 1838, and he died at Woolwich on the 2nd of February 1841.

00It is emphatically a speculation; it cannot be demonstrated by observation or established by mathematical calculation.

00His examination for mathematical honours exhibited some of the peculiarities of his character and mental powers.

00This is one of the few purely mathematical papers he published, and it exhibited at once to experts the full genius of its author.

00Thomson) had, in 1846, shown that a totally different assumption, based upon other analogies, led (by its own special mathematical methods) to precisely the same results.

00Reid graduated at Aberdeen in 1726, and remained there as librarian to the university for ten years, a period which he devoted largely to mathematical reading.

00The "Essay on Quantity, occasioned by reading a Treatise in which Simple and Compound Ratios are applied to Virtue and Merit," denies the possibility of a mathematical treatment of moral subjects.

00He reverted in his old age to the mathematical pursuits of his earlier years, and his ardour for knowledge of every kind remained fresh to the last.

00His father began to teach him Latin, but ceased on discovering the boy's greater inclination and aptitude for mathematical studies.

00lettres," and shortly afterwards he received an appointment as assistant mathematical master in the college.

00In estimating Brewster's place among scientific discoverers the chief thing to be borne in mind is that the bent of his genius was not characteristically mathematical.

00His earliest work dealt mainly with mathematical subjects, and especially with quaternions (q.v.), of which he may be regarded as the leading exponent after their originator, Hamilton.

00In addition, quaternions was one of the themes of his address as president of the mathematical section of the British Association in 1871.

00But he also produced original work in mathematical and experimental physics.

00Copious extracts from a diary kept by him at this time are given by Bain; they show how methodically he read and wrote, studied chemistry and botany, tackled advanced mathematical problems, made notes on the scenery and the people and customs of the country.

00Mathematics 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.

00His originality lies in trying to discover an exact mathematical relation between them.

00Sensations, he argued, thus being representable by numbers, psychology may become an "exact" science, susceptible of mathematical treatment.

00JOSIAH WILLARD GIBBS (1839-1903), American mathematical physicist, the fourth child and only son of Josiah Willard Gibbs (1790-1861), who was professor of sacred literature in Yale Divinity School from 1824 till his death, was born at New Haven on the 11th of February 1839.

00Returning to New Haven in 1869, he was appointed professor of mathematical physics in Yale College in 1871, and held that position till his death, which occurred at New Haven on the 28th of April 1903.

00His first contributions to mathematical physics were two papers published in 1873 in the Transactions of the Connecticut Academy on "Graphical Methods in the Thermodynamics of Fluids," and "Method of Geometrical Representation of the Thermodynamic Properties of Substances by means of Surfaces."

00The name of Willard Gibbs, who was the most distinguished American mathematical physicist of his day, is especially associated with the "Phase Rule," of which some account will be found in the article Energetics.

00He corresponded with some of the most eminent scholars of his time on mathematical subjects; and his house was generally full of pupils from all quarters.

00Among the special collections are the George Ticknor library of Spanish and Portuguese books (6 393 vols.), very full sets of United States and British public documents, the Bowditch mathematical library (7090 vols.), the Galatea collection on the history of women (2193 vols.), the Barton library, including one of the finest existing collections of Shakespeariana (3309 vols., beside many in the general library), the A.

00He was wanting in mathematical ability, and never displayed in any remarkable degree the still more important power of scientific generalization, which, whether accompanied by mathematical skill or not, never fails to mark the highest genius in physical science.

00The mathematical fragments are collected by Fr.

00His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as quadratic equations (Venice, 1556, 1560).

00Kirchhoff's mathematical teaching that he took up the study of mathematical physics at Konigsberg under F.

00A physicist, however, does more than merely quantitatively determine specific properties of matter; he endeavours to establish mathematical laws which co-ordinate his observations, and in many cases the equations expressing such laws contain functions or terms which pertain solely to the chemical composition of matter.

00In all other cases recourse must be had to a map, a globe or mathematical formula.

00He was he deals with the principles of mathematical geography, map projections, and sources of information with special reference FIG.

00Mercator constructed it graphically, the mathematical principles underlying it being first explained by E.

00p. 43) In music he held that the notes of the scale are to be judged, not as the Pythagoreans held, by mathematical ratio, but by the ear.

00But his great strength lay in metaphysical analysis, as was shown in his answer to the objections raised against the appointment of Sir John Leslie to the mathematical professorship (1805).

00His contributions to the theories of Elasticity and of Waves rank high among modern developments of mathematical physics, although they are mere units among the 150 scientific papers attached to his name in the Royal Society's Catalogue.

00paOiµarLK1), sc. TEXvn or E7rio'7-)µ17; from AecO a, "learning" or "science"), the general term for the various applications of mathematical thought, the traditional field of which is number and quantity.

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.

00It may be noticed that (iv) is the familar principle of mathematical induction.

00The evolution of mathematical thought in the invention of the data of analysis has thus been completely traced in outline.

00The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications.

00Mathematical Logic as based on the Theory of Types,"Amer.

00In calculations the latter hypothesis is made because of its mathematical simplicity.

00A complete classification of mathematical sciences, as they at present exist, is to be found in the International Catalogue of Scientific Literature promoted by the Royal Society.

00For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.

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

00Between them the general theory of the complex variable, and of the various "infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces.

00The applied mathematical sciences of light, electricity and electromagnetism, ' Cf.

00This school of mathematical thought lasted beyond the middle of the century, after which a change and further development can be traced.

00During the same period a brilliant group of mathematical physicists, notably Lord Kelvin (W.

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

00He was the author also of a mathematical work on the use of the astrolabe and of a book (Muhit, " the ocean ") on the navigation of the Indian seas.

00The history of his youth reveals no special predilection for the military service - the bent of his mind was political far more than military, but unlike the politicians of his epoch he consistently applied scientific and mathematical methods to his theories, and desired above all things a knowledge of facts in their true relation to one another.

00Meanwhile the mathematical mind, with its craving for accurate data on which to found its plans (the most difficult of all to obtain under the conditions of warfare), had been searching for expedients which might serve him to better purpose, and in 1805 he had recourse to the cavalry screen in the hope of such results.

00The presentation was accompanied by a kind of mathematical performance, in which Leonardo solved several hard problems proposed to him by John of Palermo, an imperial notary, whose name is met with in several documents dated between 1221 and 1240.

00So also any exhaustive survey of the temperature and salinity of the sea at a great number of points on and below the surface reveals a complexity of conditions that may defy mathematical analysis and could not easily be predicted.

00He took a first class in the final mathematical school in 1854, and the following year was appointed mathematical lecturer at Christ Church, a post he continued to fill till 1881.

00His earliest publications, beginning with A Syllabus of Plane Algebraical Geometry (1860) and The Formulae of Plane Trigonometry (1861), were exclusively mathematical; but late in the year 1865 he published, under the pseudonym of "Lewis Carroll," Alice's Adventures in Wonderland, a work that was the outcome of his keen sympathy with the imagination of children and their sense of fun.

00Dodgson periodically published mathematical works - An Elementary Treatise on Determinants (1867); Euclid, Book V., proved Algebraically (1874); Euclid and his Modern Rivals (1879), the work on which his reputation as a mathematician largely rests; and Curiosa Mathematica (1888).

00Schrenck, Reisen and Forschungen im Amurgebiet (St Petersburg, 1858-1891); Trudy of the Siberian expedition - mathematical part (also geographical) by Schwarz, and physical part by Schmidt, Glehn and Brylkin (1874, seq.); G.

00Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.

00Cayley, " Memoirs on Quantics," in the Collected Mathematical Papers (Cambridge, 1898); Salmon, Lessons Introductory to the Modern Higher Algebra (Dublin, 1885); E.

00His mathematical bent, however, soon diverted him from legal studies, and the perusal of some of his earliest theorems enabled Descartes to predict his future greatness.

00Placed at the university of Cracow in 1491, he devoted himself, during three years, to mathematical science under Albert Brudzewski (1445-1497), and incidentally acquired some skill in painting.

00The determination of the solar parallax through the parallactic inequality of the moon's motion also involves two elements - one of observation, the other of purely mathematical theory.

00Of these thirteen sections, the first contains a simple description of the more prominent phenomena, without mathematical symbols or numerical data.

00The second includes definitions of technical terms in common use, together with so much of the elementary theory as is necessary for understanding the experimental work described in subsequent portions of the article; a number of formulae and results are given for purposes of reference, but the mathematical reasoning by which they are obtained is not generally detailed, authorities being cited whenever the demonstrations are not likely to be found in ordinary textbooks.

00Thomson (afterwards Lord Kelvin) in 1847, as the result of a mathematical investigation undertaken to explain Faraday's experimental observations.

00Regarding it as important that all reasoning with reference to magnetism should be conducted without any uncertain assumptions, he worked out a mathematical theory upon the sole foundation of a few wellknown facts and principles.

00For outlines of the mathematical theory of magnetism and references see H.

00In 1873 James Clerk Maxwell published his classical Treatise on Electricity and Magnetism, in which Faraday's ideas were translated into a mathematical form.

00Induction in Iron and other Metals (3rd ed., London, 2900); Thomson, Recent Researches in Electricity and Magnetism (Oxford, 2893); Elements of Mathematical Theory of Electricity and Magnetism 3rd ed., Cambridge, 1904); H.

00The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers), which Riemann read, mastered and returned within a few days.

00It soon became evident that his mathematical studies, undertaken at first probably as a relaxation, were destined to be the chief business of his life.

00He proceeded in the beginning of 1847 to Berlin, attracted thither by that brilliant constellation of mathematical genius whose principal stars were P. G.

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.

00His mathematical enthusiasm was for the time completely quenched, and during two years the printed volume of his Mecanique, which he had seen only in manuscript, lay unopened beside him.

003 As a mathematical writer Lagrange has perhaps never been surpassed.

00His mathematical discoveries were extended and over shadowed by his contemporaries.

00Among the earlier publications of the academy were the Tudomdnytdr (Treasury of Sciences, 1834-1844), with its supplement Literatura; the KUlfoldi jdtPkszin (Foreign Theatres); the Magyar nyelv rendszere (System of the Hungarian language, 1846; 2nd ed., 1847); various dictionaries of scientific, mathematical, philosophical and legal terms; a Hungarian - German dictionary (1835-1838), and a Glossary of Provincialisms (1838).

00His first distinctions are said to have been gained in theological controversy, but at an early age he became mathematical teacher in the military school of Beaumont, the classes of which he had attended as an extern.

00Here, too, he died, attended by his physician, Dr Majendie, and his mathematical coadjutor, Alexis Bouvard., on the 5th of March 1827.

00By his discovery that the attracting force in any direction of a mass upon a particle could be obtained by the direct process of differentiating a single function, Laplace laid the foundations of the mathematical sciences of heat, electricity and magnetism.

00The theory of probabilities, which Laplace described as common sense expressed in mathematical language, engaged his attention from its importance in physics and astronomy; and he applied his theory, not only to the ordinary problems of chances, but also to the inquiry into the causes of phenomena, vital statistics and future events.

00For the history of the subject see A History of the Mathematical Theory of Probability, by Isaac Todhunter (1865).

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.

00The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter part of the 19th century to an important reaction against the specialization mentioned in the preceding paragraph.

00On the other land, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its intrinsic difficulty.

00We then obtain a set of equations, and by means of these equations we establish the required result by a process known as mathematical induction.

00The following are some further examples of mathematical induction.

00A period of mathematical stagnation then appears to have possessed the Indian mind for an interval of several centuries, for the works of the next author of any moment stand but little in advance of Brahmagupta.

00English translations of the mathematical chapters of the Brahma-siddhanta and Siddhanta-ciromani by H.

00Although Pell had nothing to do with the solution, posterity has termed the equation Pell's Equation, or Problem, when more rightly it should be the Hindu Problem, in recognition of the mathematical attainments of the Brahmans.

00About the beginning of the 17th century various mathematical works by Franciscus Vieta were published, which were afterwards collected by Franz van Schooten and republished in 1646 at Leiden.

00See also John Wallis, Opera Mathematica (1693-1699), and Charles Hutton, Mathematical and Philosophical Dictionary (1815), article " Algebra."

00In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make.

00We readily find (with substitution for k of 27r/X) a2b S n J s in fl „2a2E2 „2b2n2 f2X2 f2X2 as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.

00The following table gives the actual values: - In both cases the image of a mathematical point is thus a symmetrical ring system.

00The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point.

00Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest.

00The position of the middle of the bright band representative of a mathematical line can be fixed with a spider-line micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument.

00It appears indeed that the purely mathematical question has no definite answer.

00The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded as a merely mathematical one, without reference to the physical properties of actual screens.

00Ann., 18 95, 47, p. 317), with great mathematical skill, has solved the problem of the shadow thrown by a semi-infinite plane screen.

00The point at issue was, that neither in the polyphonic school, in which Zarlino was educated, nor in the later monodic school, of which his recalcitrant pupil, Vincenzo Galilei, was the most redoubtable champion, could those proportions be tolerated in practice, however attractive they might be to the theorist in their mathematical aspect.

00Didymus, writing in the year 60, made the first step towards establishing this pleasant-sounding scale upon a mathematical basis, by the discovery of the lesser tone; but unhappily he placed it in a false position below the greater tone.

00(e) His mathematical works consist of a Regula de abaco cornputi, of which a 12th-century MS. is to be found at the Vatican; and a Libellus de numerorum divisione (11thand 12th-century MSS.

00The mathematical discussion of Airy showed that the primary rainbow is not situated directly on the line of minimum deviation, but at a slightly greater value; this means that the true angular radius of the bow is a little less than that derived from the geometrical theory.

00His mathematical researches were also concerned with the theory of equations, but the question as to his priority on several points has been keenly discussed.

00But the development of mathematical and physical science soon introduced a fundamental change in the habits of thought with respect to medical doctrine.

00The medicine of the i 8th century is notable, like that of the latter part of the 17th, for the striving after complete theoretical systems. The influence of the iatro-physical school was by no means exhausted; and in England, especially through the indirect influence of Sir Isaac Newton's (1642-1727) great astronomical generalizations, it took on a mathematical aspect, and is sometimes known as iatro-mathematical.

00Mead, a man of great learning and intellectual activity, was an ardent advocate of the mathematical doctrines.

00"It is very evident," he says, "that all other means of improving medicine have been found ineffectual, by the stand it was at for two thousand years, and that, since mathematicians have sot themselves to the study of it, men already begin to talk so intelligibly and comprehensibly, even about abstruse matters, that it is to be hoped that mathematical learning will be the distinguishing mark of a physician and a quack."

00Before he was sixteen he attended lectures at Owens College, and at eighteen he gained a mathematical scholarship at Trinity College, Cambridge, where he graduated in 1871 as senior wrangler and first Smith's prizeman, having previously taken the degree of D.Sc. at London University and won a Whitworth scholarship. Although elected a fellow and tutor of his college, he stayed up at Cambridge only for a very short time, preferring to learn practical engineering as a pupil in the works in which his father was a partner.

00Further, it is certain that Hero used physical and mathematical writings by Posidonius, the Stoic, of Apamea, Cicero's teacher, who lived until about the middle of the 1st century B.C. The positive arguments for the more modern view of Hero's date are (1) the use by him of Latinisms from which Diels concluded that the 1st century A.D.

00He early distinguished himself as professor of mathematical and mechanical philosophy in the college of Ragusa; but after residing there for several years he returned to his native city, where he became a professor in the Collegio Nazareno, and began to form the fine mineralogical cabinet in that institution.

00In his Principes de la theorie des richesses (1863) he abandoned the mathematical method, though advocating the use of mathematical symbols in economic discussions, as being of service in facilitating exposition.

00In 1806 he was appointed mathematical master in the Woolwich Academy, and filled that post for fortyone years.

00Barlow's principal works are - Elementary Investigation of the Theory of Numbers 0810; New Mathematical and Philosophical Dictionary (1814); Essay on Magnetic Attractions (1820).

00u3po ajXavuta), the science of the mechanics of water and fluids in general, including hydrostatics or the mathematical theory of fluids in equilibrium, and hydromechanics, the theory of fluids in motion.

00At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.

00This mechanical axiom of the normality of fluid pressure is the foundation of the mathematical theory of hydrostatics.

00The practical problems of fluid motion, which are amenable to mathematical analysis when viscosity is taken into account, are excluded from treatment here, as constituting a separate branch called "hydraulics" (q.v.).

0010 Published and discussed by Hilprecht, " Mathematical, Metrological and Chronological Texts " (Bab.

00In 1785 appeared his Recherches theoriques et experimentales sur la force de torsion et sur l'elasticite des fils de metal, &c. This memoir contained a description of different forms of his torsion balance, an instrument used by him with great success for the experimental investigation of the distribution of electricity on surfaces and of the laws of electrical and magnetic action, of the mathematical theory of which he may also be regarded as the founder.

00He was also the author (1852) of the "Dissertation on the Progress of Mathematical and Physical Science," published in the 8th edition of the Encyclopaedia Britannica.

00Munich has long been celebrated for its artistic handicrafts, such as bronze-founding, glass-staining, silversmith's work, and wood-carving, while the astronomical instruments of Fraunhofer and the mathematical instruments of Traugott Lieberecht von Ertel (1778-1858) are also widely known.

00In 1815 he entered the cathedral school at Christiania, and three years later he gave proof of his mathematical genius by his brilliant solutions of the original problems proposed by B.

00Further state aid enabled him to visit Germany and France in 1825, and having visited the astronomer Heinrich Schumacher (1780-1850) at Hamburg, he spent six months in Berlin, where he became intimate with August Leopold Crelle, who was then about to publish his mathematical journal.

00For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.

00CuroaaKTLK6s, capable of demonstration), a logical term, applied to judgments which are necessarily true, as of mathematical conclusions.

00So Comte remained in Paris, living as he best could on something less than 80 a year, and hoping, when he took the trouble to break his meditations upon greater things by hopes about himself, that he might by and by obtain an appointment as mathematical master in a school.

00From 1716 to 1718 he published a scientific periodical, called Daedalus hyperboreus, a record of mechanical and mathematical inventions and discoveries.

00The same year he published various mathematical and mechanical works.

00"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.

00His own account of his school and college training, given in a letter to the same correspondent (6th August 1767), is: "I employed almost my whole time at Oxford in the mathematical and classical knowledge, but more particularly in the latter, so that I understand Latin and Greek tolerably well.

00This account appeared in the Philosophical Transactions for 1778, was afterwards reprinted in the second volume of his Tracts on Mathematical and Philosophical Subjects, and procured for Hutton the degree of LL.D.

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.

00This undertaking, the mathematical and scientific parts of which fell to Hutton's share, was completed in 1809, and filled eighteen volumes quarto.

00His name first appears in the Ladies' Diary (a poetical and mathematical almanac which was begun in 1704, and lasted till 1871) in 1764; ten years later he was appointed editor of the almanac, a post which he retained till 1817.

00The pay of his rank was small, and his appointment on the quartermaster-general's staff made it necessary to keep two horses, so that he had to write mathematical school-books in his spare time to eke out his resources.

00On returning to Prussia he became mathematical instructor at the school of military engineering, leaving this post in 1792 to take part as a general staff officer in the war against France.

00It must be noted, however, that potential is a mere mathematical concept, and has no objective existence like difference of level, nor is it capable per se of producing physical changes in bodies, such as those which are brought about by rise of temperature, apart from any question of difference of temperature.

00The mathematical expression for this potential can in some cases be calculated or predetermined.

00The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges ql, q2, q3, &c., distributed in any manner, is the sum of them separately, or qi/xl+q2/x2+q3/x3+&c. =F (q/x) =V (17), where xi, x2, x 3, &c., are the distances of the respective point charges from the point in question at which the total potential is required.

00Thomson, Elements of the Mathematical Theory of Electricity and Magnetism (Cambridge, 1895); J.

00Atkinson (London, 1883); Watson and Burbury, The Mathematical Theory of Electricity and Magnetism (Oxford, 188.5); A.

00The 0, 4) diagram is useful in the study of heat waste and condensation, but from other points of view the utility of the conception of entropy as a " factor of heat " is limited by the fact that it does not correspond to any directly measurable physical property, but is merely a mathematical function arising from the form of the definition of absolute temperature.

00It is the basis of the famous Canon of kings, also called Mathematical Canon, preserved to us in the works of Ptolemy, which, before the astonishing discoveries at Nineveh, was the sole authentic monument of Assyrian and Babylonian history known to us.

00This, with other matters appertaining to the calendar, was probably left to be regulated from time to time by the mathematical tribunal.

00The mathematical tribunal has, however, from time immemorial counted the first year of the first cycle from the eighty-first of Yao, that is to say, from the year 2277 B.C.

00A complete summary of the great developments of mathematical learning, which the members of this family effected, lies outside the scope of this notice.

00More detailed accounts are to be found in the various mathematical articles.

00At the conclusion of his philosophical studies at the university, some geometrical figures, which fell in his way, excited in him a passion for mathematical pursuits, and in spite of the opposition of his father, who wished him to be a clergyman, he applied himself in secret to his favourite science.

00On his final return to Basel in 1682, he devoted himself to physical and mathematical investigations, and opened a public seminary for experimental physics.

00Jacques Bernoulli cannot be strictly called an independent discoverer; but, from his extensive and successful application of the calculus and other mathematical methods, he is deserving of a place by the side of Newton and Leibnitz.

00In 1687 the mathematical chair of the university of Basel was conferred upon Jacques.

00Jacques Bernoulli wrote elegant verses in Latin, German and French; but although these were held in high estimation in his own time, it is on his mathematical works that his fame now rests.

00There, in addition to the learned lectures by which he endeavoured to revive mathematical science in the university, he gave a public course of experimental physics.

00He had declined, during his residence at Groningen,' an invitation to Utrecht, but accepted in 1705 the mathematical chair in the university of his native city, vacant by the death of his brother Jacques; and here he remained till his death.

00He was a member of almost every learned society in Europe, and one of the first mathematicians of a mathematical age.

00constitution that he continued to pursue his usual mathematical studies till the age of eighty.

00ep. 199), held for a time the mathematical chair at Padua, and was successively professor of logic and of law at Basel, where he died on the 29th of November 1759.

00On his return to Berlin he was appointed director of the mathematical department of the academy.

00His writings consist of travels and astronomical, geographical and mathematical works.

00In 1788 he was named one of its mathematical professors.

00The second part deals with chronological and mathematical questions, and has been of great service in determining the principal epochs of ancient history.

00These two simplifying facts bring the properties of the gaseous state of matter within the range of mathematical treatment.

00The kinetic theory of gases attempts to give a mathematical account, in terms of the molecular structure of matter, of all the non-chemical and non-electrical properties of gases.

00Louis Charles d'Albert (1620-1690), duke of Luynes, son of the constable, was an ascetic writer and friend of the Jansenists; Paul d'Albert de Luynes (1703-1788), cardinal and archbishop of Sens, an astronomer; Michel Ferdinand d'Albert d'Ailly (1714-1769), duke of Chaulnes, a writer on mathematical instruments, and his son Marie Joseph Louis (1741-1793), a chemist; and Honore Theodore Paul Joseph (1802-1867), duke of Luynes, a writer on archaeology.

00He reproduces and further develops and defends his own views in his Mathematical Memoirs, and in his paper in the Philosophical Transactions for 1785.

00But Landen's capital discovery is that of the theorem known by his name (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two elliptic arcs.

00(1 754, 1760, 1768, 1 77 1, 1 775, 1 777, 1785); Mathematical Lucubrations (1755) A Discourse concerning the Residual Analysis (1758); The Residual Analysis, book i.

00(1764); Animadversions on Dr Stewart's Method of computing the Sun's Distance from the Earth (1771); Mathematical Memoirs (1780, 1789).

00A large collection of such curious information is contained in the Bibliotheca of Apollodorus, a pupil of Aristarchus who flourished in the and century B.C. Eratosthenes was the first to write on mathematical and physical geography; he also first attempted to draw up a chronological table of the Egyptian kings and of the historical events of Greece.

00The founder of the mathematical school was the celebrated Euclid (Eucleides); among its scholars were Archimedes; Apollonius of Perga, author of a treatise on Conic Sections; Eratosthenes, to whom we owe the first measurement of the earth; and Hipparchus, the founder of the epicyclical theory of the heavens, afterwards called the Ptolemaic system, from its most famous expositor, Claudius Ptolemaeus.

00In 1842 he obtained a mathematical scholarship and graduated as B.A.

00About this time he became mathematical master at a school at Wimbledon.

00In 1862 he was made a fellow of the Royal Society, and in 1865 a member of the Mathematical Society of London.

001875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).

00It is chiefly distinguished for its mathematical and philosophical studies, and possesses a famous observatory, established in 1811 by Frederick William Bessel, a library of about 240,000 volumes, a zoological museum, a botanical garden, laboratories and valuable mathematical and other scientific collections.

00His devotion to mathematical science seems to have interfered alike with his advancement in the Church and with the proper management of his private affairs.

00The instrument was a ingenuity, and was called "the mathematical jewel."

00As a result of the importance both of the formulae obtained by elementary methods and of those which have involved the previous use of analysis, there is a tendency to dissociate the former, like the latter, from the methods by which they have been obtained, and to regard mensuration as consisting of those mathematical formulae which are concerned with the measurement of geometrical magnitudes (including lengths), or, in a slightly wider sense, as being the art of applying these formulae to specific cases.

00In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.

00The proper treatment of the deviations from mathematical accuracy, in the second and third of the above classes of cases, is a special matter.

00Goursat, A Course in Mathematical Analysis (1905; trans.

00His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.

00The mathematical investigation of forced vibrations (Rayleigh,Sound, i.

00The standard treatise on the mathematical theory is Lord Rayleigh's Theory of Sound (2nd ed., 1894); this work also contains an account of experimental verifications.

00The same author's Scientific Papers contains many experimental and mathematical contributions to the science.

00Thomson, Sound (5th ed., 1909), contains a descriptive account of the chief phenomena, and an elementary mathematical treatment.

00These antinomies are four - two mathematical, two dynamical - connected with (I) the limitation of the universe in respect of space and time, (2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist), (3) the problem of freedom in relation to universal causality, (4) the existence of a universal being - about each of which pure reason contradicts the empirical, as thesis and antithesis.

00In mathematical geography the problem of representing the surface of a sphere on a plane is of fundamental importance; this subject is treated in the article MAP.

00- If the lengths of the links be assumed indefinitely short, the chain under given simple distributions of load will take the form of comparatively simple mathematical curves known as catenaries.

00The distinctive industry is the manufacture of mathematical and musical instruments.

00The first investigates mathematical facts relating to the earth as a whole, its figure, dimensions, motions, their measurement, &c. The second part considers the earth as affected by the sun and stars, climates, seasons, the difference of apparent time at different places, variations in the length of the day, &c. The third part treats briefly of the actual divisions of_the surface of the earth, their relative positions, globe and map-construction, longitude, navigation, &c. Varenius, with the materials at his command, dealt with the subject in a truly philosophic spirit; and his work long held its position as the best treatise in existence on scientific and comparative geography.

00The laborious enterprise of drawing up the famous Tables du Cadastre was entrusted to his direction in 1792, and in 1794 he was appointed professor of the mathematical sciences at the Ecole Polytechnique, becoming director at the Ecole des Ponts et Chaussees four years later.

00The university of Modena, originally founded in 1683 by Francis II., is mainly a medical and legal school, but has also a faculty of physical and mathematical science.

00In this belief he differed from his pupil, Roger Cotes, and from most of the great mathematical astronomers of the 18th century, who worked out in detail the task sketched by the genius of Newton.

00He was admitted to the Institute on its organization in 1795, and became, in 1803, perpetual secretary to its mathematical section.

00The introduction of the coefficients now called Laplace's, and their application, commence a new era in mathematical physics.

00Although teaching was uncongenial to him he took pupils in mathematics, and held for a time the position of mathematical coach for Woolwich and Sandhurst.

00Through the influence of Sir Isaac Newton he was elected mathematical master in Christ's hospital.

00Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v 2 /k or v3/k.

00He studied medicine at GÃ¶ttingen, 1 7771 7 80, attending at the same time Kaestner's mathematical course; and in 1779, while watching by the sick-bed of a fellow-student, he devised a method of calculating cometary orbits which made an epoch in the treatment of the subject, and is still extensively used.

00The mathematical works are published, some of them in a small 4to volume (Oxford, 1657) and a complete collection in three thick folio volumes (Oxford, 1693-1699).

00Excluding all these, the mathematical works contained in the first and second volumes occupy about 1800 pages.

00His mathematical capacity was early noticed; he pursued his studies at Gottingen under Abraham Gotthelf Kastner (1719-1800), and in 1787 he went to Berlin and studied practical astronomy under E.

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.

00This was his first publication of original mathematical work; and from this time scarcely a year passed in which he did not give to the world further mathematical researches.

00In 1870 he was elected president of the London Mathematical Society.

00The mastery which he had obtained over the mathematical symbols was so complete that he never shrank from the use of expressions, however complicated - nay, the more complicated they were the more he seemed to revel in them - provided they did not sin against the ruling spirit of all his work - symmetry.

00To a mind imbued with the love of mathematical symmetry the study of determinants had naturally every attraction.

00The effect of the study on Mr Spottiswoode's own methods was most pronounced; there is scarcely a page of his mathematical writings that does not bristle with determinants."

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.

00BESSEL FUNCTION, a certain mathematical relation between two variables.

00There is hardly a branch of mathematical physics which is independent of these functions.

00For the mathematical investigation see Spherical Harmonics and for tables see Table, Mathematical.

00Beside the equivalence of the hon to 5 utens weight of water, the mathematical papyrus (35) gives 5 besha = (2/3)cubic cubit (Revillout's interpretation of this as 1 cubit cubed is impossible geometrically; see Rev. Eg., 1881, for data); this is very concordant, but it is very unlikely for 3 to be introduced in an Egyptian derivation, and probably therefore only a working equivalent.

00His mathematical lectures roused so much enthusiasm that they were discontinued by order of the authorities, who disliked the disturbance of the university routine which they involved.

00Chalmers then opened mathematical classes on his own account which attracted many students; at the same time he delivered a course of lectures on chemistry, and ministered to his parish at Kilmany.

00The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2.71828 18284 ...

00The mathematical function log x or log x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin-' x,&c., log x and e x which are universally treated in analysis as known functions.

00It has been thought necessary to give in detail the facts relating to the conversion of the logarithms, as unfortunately Charles Hutton in his history of logarithms, which was prefixed to the early editions of his Mathematical Tables, and was also published as one of his Mathematical Tracts, has charged Napier with want of candour in not telling the world of Briggs's share in the change of system, and he expresses the suspicion that " Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms."

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 invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's Descriptio.

00For more detailed information relating to Napier, Briggs and Vlacq, and the invention of logarithms, the reader is referred to the life of Briggs in Ward's Lives of the Professors of Gresham College (London, 1740); Thomas Smith's Vitae quorundam eruditissimorum et illustrium virorum (Vita Henrici Briggii) (London, 1707); Mark Napier's Memoirs of John Napier already referred to, and the same author's Naperi libri qui supersunt (1839); Hutton's History; de Morgan's article already referred to; Delambre's Histoire de l'Astronomie moderne; the report on mathematical tables in the Report of the British Association for 1873; and the Philosophical Magazine for October and December 1872 and May 1873.

00In the years1791-1807Francis Maseres published at London, in six volumes quarto " Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton's historical introduction to his new edition of Sherwin's mathematical tables..

00Those engaged upon the work were divided into three sections: the first consisted of five or six mathematicians, including Legendre, who were engaged in the purely analytical work, or the calculation of the fundamental numbers; the second section consisted of seven or eight calculators possessing some mathematical knowledge; and the third comprised seventy or eighty ordinary computers.

00See Tables, Mathematical: Exponential Functions.

00For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the article Tables, Mathematical.

00A full and valuable account of these methods is given in Hutton's " Construction of Logarithms," which occurs in the introduction to the early editions of his Mathematical Tables, and also forms tract 21 of his Mathematical Tracts (vol.

00The first attempt at a mathematical theory of dispersion was made by A.

00Graduating at Harvard College in 1829, he became mathematical tutor there in 1831 and professor in 1833.

00He had already assisted Nathaniel Bowditch in his translation of the Mecanique celeste, and now produced a series of mathematical textbooks characterized by the brevity and terseness which made his teaching unattractive to inapt pupils.

00Magini in the mathematical chair of Bologna.

00He became director of the music-school at Pforten in 1572, was transferred to Leipzig in the same capacity in 1594, and retained this post until his death on the 24th of November 1615, despite the offers successively made to him of mathematical professorships at Frankfort and Wittenberg.

00" Strabo indeed appears to be the first who conceived a complete geographical treatise as comprising the four divisions of mathematical, physical, political and historical geography, and he endeavoured, however imperfectly, to keep all these objects in view."

00In respect of mathematical geography, his lack of scientific training was no great hindrance.

00In the unwritten lectures of his old age, he developed this formal into a mathematical metaphysics.

00In order to explain the unity and variety of the world, the one universal form and the many individuals, and how the one good is the main cause of everything, he placed as it were at the back of his own doctrine of forms a Pythagorean mathematical philosophy.

00Identifying the form of the good with the one, he supposed that the one, by combining with the indeterminate two, causes a plurality of forms, which like every combination of one and two are numbers but peculiar in being incommensurate with one another, so that each form is not a mathematical number (pa077pa-1.6s apt°pos), but a formal number (EDBnTLKOS apiepos).

00Xenocrates as president from 339 onwards taught that the one and many are principles, only without distinguishing mathematical from formal numbers.

00Aristotle's critics hardly realize that for the rest of his life he had to live and to struggle with a formal and a mathematical Platonism, which exaggerated first universals and attributes and afterwards the quantitative attributes, one and many, into substantial things and real causes.

00If, wrote Aristotle, the forms are another sort of number, not mathematical, there would be no understanding of it.

00containing his report of Plato's lectures on the Good, he was dealing: with the same mathematical metaphysics which in his dialogue on.

00This is obvious enough in the Metaphysics: it has two openings (Books A and a); then comes a nearly consecutive theory of being (B, F, E, Z, H, 0), but interrupted by a philosophical lexicon A; afterwards follows a theory of unity (1); then a summary of previous books and of doctrines from the Physics (K); next a new beginning about being, and, what is wanted to complete the system, a theory of God in relation to the world (A); finally a criticism of mathematical metaphysics (M, N), in which the argument against Plato (A 9) is repeated almost word for word (M 4-5).

00Mathematical Philosophy, about quantitative things in the abstract.

00the mathematical.

00But A System Of 31 Intercalations In 128 Years Would Be By Far The Most Perfect As Regards Mathematical Accuracy.

00In these last Mayer sought to explain the magnetic action of the earth by a modification of Euler's hypothesis, and made the first really definite attempt to establish a mathematical theory of magnetic action (C. Hansteen, Magnetismus der Erde, i.

00Recorde published several works upon mathematical subjects, chiefly in the form of dialogue between master and scholar, viz.: - The Grounde of Artes, teachinge the Worke and Practise of Arithmeticke, both in whole numbers and fractions (1540); The Pathway to Knowledge, containing the First Principles of Geometry.

00In 1707 Berkeley published two short mathematical tracts; in 5709, in his New Theory of Vision, he applied his new principle for the first time, and in the following year stated it fully in the Principles of Human Knowledge.

00Of less interest nowadays are Robins's more purely mathematical writings, such as his Discourse concerning the Nature and Certainty of Sir Isaac Newton's Methods of Fluxions and of Prime and Ultimate Ratios (1735), "A Demonstration of the Eleventh Proposition of Sir Isaac Newton's Treatise of Quadratures" (Phil.

00After five years spent in mathematical and astronomical studies, he went to Holland, in order to visit several eminent continental mathematicians.

00In the Zwinger are the zoological and mineralogical museums and a collection of instruments used in mathematical and physical science.

00But in the incessant travelling, drawing, collecting specimens and composition in prose and verse he had gained but a very moderate classical and mathematical knowledge when he matriculated at Oxford; nor could he ever learn to write tolerable Latin.

00The Copernican theory of the solar system - that the earth revolved annually about the sun - had received confirmation by the observations of Galileo and Tycho Brahe, and the mathematical investigations of Kepler and Newton.

00Rigaud, Memoirs of Bradley (1832), and in Charles Hutton, Mathematical and Philosophical Dictionary (1795); a particularly clear and lucid account is given in H.

00Though the experimental and theoretical developments were not necessarily dependent on each other, and by far the larger proportion of the subject which we now term " Spectroscopy " could stand irrespective of Gustav Kirchhoff's thermodynamical investigations, there is no doubt that the latter was, historically speaking, the immediate cause of the feeling of confidence with which the new branch of science was received, for nothing impresses the scientific world more strongly than just that little touch of mystery which attaches to a mathematical investigation which can only be understood by the few, and is taken on trust by the many, provided that the author is a man who commands general confidence.

00This result, which, accepting the possibility of having an absolutely opaque enclosure of uniform temperature, was clearly proved by Balfour Stewart for the total radiation, was further extended by Kirchhoff, who applied it (though not with mathematical rigidity as is sometimes supposed) to the separate wave-lengths.

00not quite keep up with the mathematical expression but tend to become more equal.

00In a period of general stagnation in mathematical studies, he stands out as a remarkable exception.

00How far he was above his contemporaries, how little appreciated or understood by them, is shown by the absence of references to him in other Greek writers, and by the fact that his work had no effect in arresting the decay of mathematical science.

00From these introductions we are able to judge of the style of Pappus's writing, which is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions.

00Other works are A Discourse concerning a New Planet (1640); Mercury, or the Secret and Swift Messenger (1641), a work of some ingenuity on the means of rapid correspondence; and Mathematical Magick (1648).

00His theory of bodies involved an idealistic analysis neither into bodily atoms nor into mathematical units, but into mentally endowed simple substances.

00He also maintained throughout the book that physical and psychical energy do not interfere, but that the psychical is, like a mathematical quantity, a function of the physical, depending upon it, and vice versa, only in the sense that a constant relation according to law exists, such that we may conclude from one to the other, but without one ever being cause of the other.

00Proceeding on this suggestion, and misled by the mathematical expression which he had given to Weber's law, Fechner held that a conscious sensation, like its stimulus, consists of units, or elements, by summation and increments of which conscious sensations and their differences are produced; so that consciousness, according to this unnecessary assumption, emerges from an integration of unconscious shocks or tremors.

00But he had also to endure countless objections to his mathematical statement of Weber's law, to his unnecessary assumption of units of sensation, and to his unjustifiable transfer of the law from physical to physiological stimuli of sensations, involving in his opinion his parallelistic view of body and mind.

00It retains some relics of Fechner's influence; first, the theory of identity, according to which the difference between the physical and psychical is not a dualism, but everything is at once both; and secondly, the substitution of mathematical dependence for physical causality, except that, whereas Fechner only denied causality between physical and psychical, Mach rejects the entire distinction between causality and dependence, on the ground that " the law of causality simply asserts that the phenomena of Nature are dependent on one another."

00Now, Mach applies these preconceived opinions to " mechanics in its development," with the result that, though he shows much skill in mathematical mechanics, he misrepresents its development precisely at the critical point of the discovery of Newton's third law of motion.

00He won the King of Sweden's open prize for a mathematical treatise in 1889, and in 1908 was elected to the Academie Frangaise.

00It is also more particularly applied to a mathematical instrument ("pair of compasses") for measuring or for describing a circle, and to the mariner's compass.

00If an iron ship be swung when upright for deviation, and the mean horizontal and vertical magnetic forces at the compass positions be also observed in different parts of the world, mathematical analysis shows that the deviations are caused partly by the permanent magnetism of hard iron, partly by the transient induced magnetism of soft iron both horizontal and vertical, and in a lesser degree by iron which is neither magnetically hard nor soft, but which becomes magnetized in the same manner as hard iron, though it gradually loses its magnetism on change of conditions, as, for example, in the case of a ship, repaired and hammered in dock, steaming in an opposite direction at sea.

00Instead of observing the deviation solely for the purposes of correcting the indications of the compass when disturbed by the iron of the ship, the practice is to subject all deviations to mathematical analysis with a view to their mechanical correction.

00In addition to a large number of publications in the Proceedings of the Royal Society and the Philosophical Magazine, he has published A Treatise on the Motion of Vortex Rings (1884); The Application of Dynamics to Physics and Chemistry (1886); Recent Researches in Electricity and Magnetism (1892); Elements of the Mathematical Theory of Electricity and Magnetism (18 95, 5th ed.

00It can only be contrived by means of complicated mathematical analysis.

00The mathematical treatment of the subject from this point of view by Lagrange (1736-1813) and others has afforded the most important forms of statement of the theory of the motion of a system that are available for practical use.

00Salusbury's Mathematical Collections and Translations (1661-1665); Mechanics and Local Motion, by T.

00The law of nature is unalterable; God Himself cannot alter it any more than He can alter a mathematical axiom.

00The contents of the remaining books are as follows: ii., mathematical and physical description of the world; iii.

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.

00Simson's contributions to mathematical knowledge took the form of critical editions and commentaries on the works of the ancient geometers.

00In 1749 was published Apollonii Pergaei locorum planorum libri II., a restoration of Apollonius's lost treatise, founded on the lemmas given in the seventh book of Pappus's Mathematical Collection.

00Trail, Life and Writings of Robert Simson (1812); C. Hutton, Mathematical and Philosophical Dictionary (1815).

00Lie's work exercised a great influence on the progress of mathematical science during the later decades of the 19th century.

00Lie was a foreign member of the Royal Society, as well as an honorary member of the Cambridge Philosophical Society and the London Mathematical Society, and his geometrical inquiries gained him the muchcoveted honour of the Lobatchewsky prize.

00Monge contributed (1770-1790) to the Memoirs of the Academy of Turin, the Memoires des savantes strangers of the Academy of Paris, the Memoires of the same Academy, and the Annales de chimie, various mathematical and physical papers.

00His later mathematical papers are published (1794-1816) in the Journal and the Correspondance of the polytechnic school.

00Monge's various mathematical papers are to a considerable extent reproduced in the Application de l'analyse a la Geometrie (4th ed., last revised by the author, Paris, 1819); the pure text of this is reproduced in the 5th ed.

00The same year the executors of Henry Lucas, who, according to the terms of his will, had founded a mathematical chair at Cambridge, fixed upon Barrow as the first professor; and although his two professorships were not inconsistent with each other, he chose to resign that of Gresham College, which he did on the 20th of May 1664.

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.

00The most important are :- Euclid's Elements; Euclid's Data; Optical Lectures, read in the public school of Cambridge; Thirteen Geometrical Lectures; The Works of Archimedes, the Four Books of Apollonius's Conic Sections, and Theodosius's Spherics, explained in a New Method; A Lecture, in which Archimedes' Theorems of the Sphere and Cylinder are investigated and briefly demonstrated; Mathematical Lectures, read in the public schools of the university of Cambridge.

00Here he continued his multifarious labours; but the church seems to have decreased, and his many engagements and bulky correspondence interfered seriously with his pulpit work, and with the discipline of his academy, where he had some 200 students to whom he lectured on philosophy and theology in the mathematical or Spinozistic style.

00There are five faculties - theological, juridical, medical, philosophical and mathematical.

00For a complete treatment of this portion of the theory of knowledge, there require to be taken into consideration at least the following points: (a) the exact nature and significance of the space and time relations in our experience, (b) the mode in which the primary data, facts or principles, of mathematical cognition are obtained, (c) the nature, extent and certainty of such data, in themselves and with reference to the concrete material of experience, (d) the principle of inference from the data, however obtained.

00(b) How then are the primary data of mathematical cognition to be derived from an experience containing space and time relations in Hume, in regard to this problem, distinctly separates geometry from algebra and arithmetic, i.e.

0097.) (c) In respect to the third point, the nature, extent and certainty of the elementary propositions of mathematical science, Hume's utterances are far from clear.

00No question arises regarding the existence of the fact represented by the idea, and in so far, at least, mathematical judgments may be described as hypothetical.

00Upon the nature of the reasoning by which in mathematical science we pass from data to conclusions, Hume gives no explicit statement.

00A somewhat detailed consideration of Hume's doctrine with regard to mathematical science has been given for the reason that this portion of his theory has been very generally overlooked or misinterpreted.

00It is also noted for its bleach and dye works, its engine works, foundries, paper factories, and production of silk goods, watches, jewelry, mathematical instruments, leather, chemicals, &c. Augsburg is also the centre of the acetylene gas industry of Germany.

00The mathematical theory of conduction of heat was developed early in the 19th century by Fourier and other workers, and was brought to so high a pitch of excellence that little has remained for later writers to add to this department of the subject.

00In fact, for a considerable period, the term " theory of heat " was practically synonymous with the mathematical treatment of a conduction.

00But later experimental researches have shown that the simple assumption of constant coefficients of conductivity and emissivity, on which the mathematical theory is based, is in many respects inadequate, and the special mathematical methods developed by J.

00Fourier need not be considered in detail here, as they are in many cases of mathematical rather than physical interest.

00- The experimental law of conduction, which forms the basis of the mathematical theory, was established in a qualitative manner by Fourier and the early experimentalists.

00This is generally assumed to be the case in mathematical problems, but the assumption is admissible only in rough work, or if the temperature difference is small.

00He was educated at the City of London school and at St John's College, Cambridge, where he took the highest honours in the classical, mathematical and theological triposes, and became fellow of his college.

00In other words, if we could know exactly all these conditions, we should be able to forecast with mathematical certainty the course which the agent would pursue.

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.

002 So hopeless, meanwhile, was he growing of being able to return home that, later on in the year, he was on the point of leaving Paris to take up his abode in the south with a French friend, 3 when he was engaged " by the month " as mathematical instructor to the young prince of Wales, who had come over from Jersey about the month of July.

00Wallis was to confine himself to the mathematical chapters, and set to work at once with characteristic energy.

00Wallis's Elenchus geometriae Hobbianae, published in 1655 about three months after the De corpore, contained also an elaborate criticism of Hobbes's whole attempt to relay the foundations of mathematical science in its place within the general body of reasoned knowledge - a criticism which, if it failed to allow for the merit of the conception, exposed only too effectually the utter inadequacy of the result.

00The consequence was that, when not spending himself in vain attempts to solve the impossible problems that have always waylaid the fancy of self-sufficient beginners, he took an interest only in the elements of geometry, and never had any notion of the full scope of mathematical science, undergoing as it then was (and not least at the hands of Wallis) the extraordinary development which made it before the end of the century the potent instrument of physical discovery which it became in the hands of Newton.

00But it was no longer a fight over mathematical questions only.

00Arguing in the Lessons that a mathematical point must have quantity, though this were not reckoned, he had explained the Greek word UTCy v, used for a point, to mean a visible mark made with a hot iron;; whereupon he was charged by Wallis with gross ignorance for confounding artypii and o - y,ua.

00Molesworth reprints the Latin, not from the first edition of 1655, but from the modified edition of 1668 - modified, in the mathematical chapters, in general (not exact) keeping with the English edition of 1656.

00Wallis having meanwhile published other works and especially a comprehensive treatise on the general principles of calculus (Mathesis universalis, 1657), he might take this occasion of exposing afresh the new-fangled methods of mathematical analysis and reasserting his own earlier positions.

003 The Problemata physica was at the same time put into English (with some changes and omission of part of the mathematical appendix), and presented to the king, to whom the work was dedicated in a remarkable letter apologizing for Leviathan.

00He was, however, indefatigable in his mathematical work.

00There are wonderful stories on record of his precocity in mathematical learning, which is sufficiently established by the well-attested fact that he had completed before he was sixteen years of age a work on the conic sections, in which he had laid down a series of propositions, discovered by himself, of such importance that they may be said to form the foundations of the modern treatment of that subject.

00The mathematical theory of probability and the allied theory of the combinatorial analysis were in effect created by the correspondence between Pascal and Fermat, concerning certain questions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Mere.

00Whether we look at his pure mathematical or at his physical researches we receive the same impression of Pascal; we see the strongest marks of a great original genius creating new ideas, and seizing upon, mastering, and pursuing farther everything that was fresh and unfamiliar in his time.

00It has four sections: physical, mathematical, philosophical and historical.

00The celebrated Rhind mathematical papyrus was coried in the reign of an Apopi from an original of the time of Amenemhe III.

00His love for mathematical science, geography, &c., in which the Arabs excelled, is noteworthy.

00(Vitruvius names Cicero and Lucretius as post nostram memoriam nascentes.) The subjects of the eight chapters are - (1) the signs of the zodiac and the seven planets; (2) the phases of the moon; (3) the passage of the sun through the zodiac; (4) and (5) various constellations; (6) the relation of astrological influences to nature; (7) the mathematical divisions of the gnomon; (8) various kinds of sundials and their inventors.

00The principal text is the Rhind Mathematical Papyrus in the British Museum, written under a Hyksos king c. 1600 B.C.; unfortunately it is full of gross errors.

00Their domination must have lasted ~s~s a considerable time, the Rhmn.d mathematical papyrus period, having been copied in the thirty-third year of a king Apophis.

00At the same time he was more taken up than ever, as is proved by the contents of a sketch-book at Dresden, with mathematical and anatomical studies on the proportions and structure of the human frame.

00In 1814 Carlyle, still looking forward to the career of a minister, obtained the mathematical mastership at Annan.

00He received £50 for a translation of Legendre's Geometry; and an introduction, explaining the theory of proportion, is said by De Morgan to show that he could have gained distinction as an expounder of mathematical principles.

00Jeffrey naturally declined to appoint a man who, in spite of some mathematical knowledge, had no special qualification, and administered a general lecture upon Carlyle's arrogance and eccentricity which left a permanent sense of injury.

00Having laid the foundation of his mathematical studies in France, he prosecuted them further in London, where he read public lectures on natural philosophy for his support.

00See C. Hutton, Mathematical and Philosophical Dictionary (1815).

00The standard of examinations was raised in Cambridge at an earlier date than at Oxford, and in the 18th century the tripos " established the reputation of Cambridge as a School of Mathematical Science."

00The " senior wrangler " was the first candidate in order of merit in the first part of the mathematical tripos.

00of writing a letter or a report on a particular subject with a particular object in view, of translating from or into a foreign language, of solving a mathematical problem, of criticizing a passage from a literary work, of writing an essay on an historical or literary subject with the aid of books in a library, of diagnosing the malady of a patient, of analysing a chemical mixture or compound; and (the highest form under the rubric) of making an original contribution to learning or science as the result of personal investigation or experiment.

00The advantage of numerical marks is that they are more easily manipulated than symbols; the disadvantage, that they produce the false impression that merit can be estimated with mathematical accuracy.

00Rouse Ball, Origin and History of the Mathematical Tripos (Cambridge, 1880); Adolf Beier, Die hoheren Schulen in Preussen and ihre Lehrer (1902-1906) (in progress); Cloudesley Brereton, " A New Method of awarding Scholarships," School World, 1907, p. 409; G.

00In mathematical and physical science Cusanus was much in advance of his age.

00We marvel at the obstinacy with which he, with inadequate mathematical knowledge, opposed the Newtonian theory of light and colour; and at his championship of "Neptunism," the theory of aqueous origin, as opposed to "Vulcanism," that of igneous origin of the earth's crust.

00At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for algebraical equations up to and including the fifth degree.

00Sylvester was a good linguist, and a diligent composer of verse, both in English and Latin, but the opinion he cherished that his poems were on a level with his mathematical achievements has not met with general acceptance.

00The first volume of his Collected Mathematical Papers, edited by H.

00When twenty years of age he entered the army, becoming lieutenant in a regiment of cavalry, and employing his leisure on mathematical studies.

00Having given promise of mathematical talent he was sent to the Ecole Centrale of Fontainebleau, and was fortunate in having a kind and sympathetic teacher, M.

00Perhaps the most original, and certainly the most permanent in their influence, were his memoirs on the theory of electricity and magnetism, which virtually created a new branch of mathematical physics.

00His father, a wealthy soapboiler, placed him at St Paul's school, where he was equally distinguished for classical and mathematical ability.

00After a single session in Glasgow, Dugald Stewart, at the age of nineteen, was summoned by his father, whose health was beginning to fail, to conduct the mathematical classes in the university of Edinburgh.

00Thus during the session 1778-177 9, in addition to his mathematical work, he delivered an original course of lectures on morals.

00In this were found large numbers of inscribed clay tablets (it is estimated that upward of 40,000 tablets and fragments have been excavated in this mound alone), dating from the middle of the 3rd millennium B.C. onward into the Persian period, partly temple archives, partly school exercises and text-books, partly mathematical tables, with a considerable number of documents of a more distinctly literary character.

00To say, however, that Clarke simply confused mathematics and morals by justifying the moral criterion on a mathematical basis is a mistake.

00It had also the mathematical meaning of ratio; and in its use for definition it is sometimes transferred to essence as the object of definition, and has a mixed meaning, which may be expressed by " account."

00On the other hand, the demonstrations of mathematical sciences of his time, and the logical forms of deduction evinced in Plato's dialogues, provided him with admirable examples of deduction, which is also the inference most capable of analysis.

00On the whole, however, Aristotle, Bacon and Mill, purged from their errors, form one empirical school, gradually growing by adapting itself to the advance of science; a school in which Aristotle was most influenced by Greek deductive Mathematics, Bacon by the rise of empirical physics at the Renaissance, and Mill by the Newtonian combination of empirical facts and mathematical principles in the Principia.

00Now, there is no doubt that, especially in mathematical equations, universal conclusions are obtainable from convertible premises expressed in these ways.

00All M is P. Proceeding from one order to the other, by converting one of the premises, and substituting the conclusion as premise for the other premise, so as to deduce the latter as conclusion, is what he calls circular inference; and he remarked that the process is fallacious unless it contains propositions which are convertible, as in mathematical equations.

00Sigwart declares himself in agreement with Jevons; except that, being aware of the difference between hypothetical deduction and mathematical analysis, and seeing that, whereas analysis (e.g.

00Its axioms, such as the law of contradiction, belong to first philosophy, but the doctrine as a whole falls neither under 'this head nor yet, though the thought has been entertained, under that of mathematics, since logic orders mathematical reasoning as well as all other.

00The form which a mathematical science treats as relatively self-subsistent is certainly not the constitutive idea.

00Plato is full of the faith of mathematical physics.

00The mathematical sciences, at least, had not proved disappointing.

00For those of the school of Plato whc refused the apostasy of the new academy, there was hope either in the mathematical side of the Pythagoreo.

00But consider Bacon's own doctrine of forms. Or watch the mathematical physicist with his formulae.

00It was otherwise with the mathematical instrument of Galilei.

00It is concurrently with signal success in the work of a pioneer in the mathematical.

00Though he makes his bow to mathematical method, he, even more than Hobbes, misses its constructive character.

00The clue of mathematical certainty is discarded in substance in the English form of " the new way of ideas."

001 Locke's logic comprises, amid much else, a theory of general terms 2 and of definition, a view of syllogism 3 and a declaration as to the possibility of inference from particular to particular,4 a distinction between propositions which are certain but trifling, and those which add to our knowledge though uncertain, and a doctrine of mathematical certainty.

00Mathematical knowledge is not involved in the same condemnation, solely because of the " archetypal " character, which, not without indebtedness to Cumberland, Locke attributes to its ideas.

00ideas and Hume's change of front as to mathematical certainty.

00In employing such a word to denote a new mathematical method, Sir W.

00Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space.

00The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.

00Clifford in his paper of 1873 (" Preliminary Sketch of Bi-Quaternions," Mathematical Papers, p. 181) seems to have come from Sir R.

00The learned societies of Washington are to a large degree more national than local in their character; among them are: the Washington Academy of Sciences (1898), a "federal head" of most of the societies mentioned below; the Anthropological Society (founded 1879; incorporated 1887), which has published Transactions (1879 sqq., with the co-operation of the Smithsonian Institution) and The American Anthropologist (1888-1898; since 1898 published by the American Anthropological Association); the National Geographic Society (1888), which since 1903 has occupied the Hubbard Memorial Building, which sent scientific expeditions to Alaska, Mont Pelee and La Souffriere, and which publishes the National Geographic Magazine (1888 sqq.), National Geographic Monographs (1895) and various special maps; the Philosophical Society of Washington (1871; incorporated 1901), devoted especially to mathematical and physical sciences; the Biological Society (1880), which publishes Proceedings (1880 sqq.); the Botanical Society of Washington (1901); the Geological Society of Washington (1893): the Entomological Society of Washington (1884), which publishes Proceedings (1884 sqq.); the Chemical Society (1884); the Records of the Past Exploration Society (1901), which publishes Records of the Past (1902 sqq.); the Southern History Association (1896), which issues Publications (1897 sqq.); the Society for Philosophical Inquiry (1893), which publishes Memoirs (1893 sqq.); the Society of American Foresters (1900), which publishes Proceedings (1905 sqq.); and the Cosmos Club.

00Thus in studying the flight of a stone through the air we replace the body in imagination by a mathematical point endowed with a masscoefficient.

00Again, the conception of a force as concentrated in a mathematical line is as unreal as that of a mass concentrated in a point, but it is a convenient fiction for our purpose, .owing to the simplicity which it lends to our statements.

00Statics of a Particle.By a particle is meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point.

00It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force.

00It thus appears that an infinitesimal rotation is of the nature of a localized vector, and is subject in all respects to the same mathematical Jaws as a force, conceived as acting on a rigid body.

00if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimais f, 77, i, X,, s, v, say A~+Bi7+Ci~+FX+GfL+Hv=O.

00The analogy between the mathematical relations of infinitely small displacements on the one hand an-d those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other.

00The mathematical properties of a twist or of a wrench have been the subject of many remarkable investigations, which are, however, of secondary importance from a physical point of view.

00For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short.

00mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction.

00W~ Crofton read a paper on Stress-Diagrams in Warren and Lattic Girders at the meeting of the Mathematical Society (April I3~

007 (1904), the qfiestion is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above.

00He afterwards entered at Clare College, Cambridge, where he applied himself to mathematical study, and obtained a fellowship in 1693.

00For several years Whiston continued to write and preach both on mathematical and theological subjects with considerable success; but his study of the Apostolical Constitutions had convinced him that Arianism was the creed of the primitive church; and with him to form an opinion and to publish it were things almost simultaneous.

00The rest of his life was spent in incessant controversy - theological, mathematical, chronological and miscellaneous.

00It is sometimes known as the "expectation of life," a term, however, which involves a mathematical hypothesis now discarded.

00A study of his works reveals an unusual combination of skill and originality in the mathematical treatment of many of the most difficult problems of astronomy, an unfailing patience and sagacity in dealing with immense masses of numerical results, and a talent for observation of the highest order.

00The accuracy of his measurement, by which he established within 2% the above law, was only limited by the sensibility, or rather insensibility, of the pith ball electrometer, which was his only means of detecting the electric charge.2 In the accuracy of his quantitative measurements and the range of his researches and his combination of mathematical and physical knowledge, Cavendish may not inaptly be described as the Kelvin of the 18th century.

00Accordingly the close of the 18th century drew into the arena of electrical investigation on its mathematical side P. S.

00Respecting this achievement when developed in its experimental and mathematical completeness, Clerk Maxwell says that it was " perfect in form and unassailable in accuracy."

00By a series of well-chosen experiments Ampere established the laws of this mutual action, and not only explained observed facts by a brilliant train of mathematical analysis, but predicted others subsequently experimentally realized.

00Ohm (1787-1854) rendered a great service to electrical science by his mathematical investigation of the voltaic circuit, and publication of his paper, Die galvanische Kette mathematisch bearbeitet.

00Weber at the same time deduced the mathematical laws of induction from his elementary law of electrical action, and with his improved instruments arrived at accurate verifications of the law of induction which by this time had been developed mathematically by Neumann and himself.

00Helmholtz gave at the same time a mathematical theory of induced currents and a valuable series of experiments in support of them (Pogg.

00Helmholtz brought to bear upon the subject not only the most profound mathematical attainments, but immense experimental skill, and his work in connexion with this subject is classical.

00His early contributions to electrostatics and electrometry are to be found described in his Reprint of Papers on Electrostatics and Magnetism (1872), and his later work in his collected Mathematical and Physical Papers.

00He added definiteness to the idea of the self-induction or inductance of an electric circuit, and gave a mathematical expression for the current flowing out of a Leyden jar during its discharge.

00Green's Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, published in 1828, contains the first exposition of the theory of potential.

00A second relation connecting magnetic and electric force is 3 The first paper in which Maxwell began to translate Faraday's conceptions into mathematical language was " On Faraday's Lines of Force," read to the Cambridge Philosophical Society on the 10th of December 1855 and the I ith of February 1856.

00Maxwell's electric and magnetic ideas were gathered together in a great mathematical treatise on electricity and magnetism which was published in 1873.1 This book stimulated in a most remarkable degree theoretical and practical research into the phenomena of electricity and magnetism.

00Coulomb experimentally proved that the law of attraction and repulsion of simple electrified bodies was that the force between them varied inversely as the square of the distance and thus gave mathematical definiteness to the two-fluid hypothesis.

00The formulation of electrical theory as far as regards operations in space free from matter was immensely assisted by Maxwell's mathematical theory.

00Oliver Heaviside after 1880 rendered much assistance by reducing Maxwell's mathematical analysis to more compact form and by introducing greater precision into terminology (see his Electrical Papers, 1892).

00The subject was pursued by Thomson and the Cambridge physicists with great mathematical and experimental ability, and finally the conclusion was reached that in a high vacuum tube the electric charge is carried by particles which have a mass only a fraction, as above mentioned, of that of the hydrogen atom, but which carry a charge equal to the unit electric charge of the hydrogen ion as found by electrochemical researches.

00Burbury, The Mathematical Theory of Electricity and Magnetism (2 vols., 1885); Lord Kelvin (Sir William Thomson), Mathematical and Physical Papers (3 vols., Cambridge, 1882); Lord Rayleigh, Scientific Papers (q.

00The excavations at Senkereh were peculiarly successful in the discovery of inscribed remains, consisting of clay tablets, chiefly contracts, but including also an important mathematical tablet and a number of tablets of a description almost peculiar to Senkereh, exhibiting in basrelief scenes of everyday life.

00It can be shown, however, that Newton was not ignorant of Bacon's works, and Dr Fowler explains his silence with regard to them on three grounds: (1) that Bacon's reputation was so well established that any definite mention was unnecessary, (2) that it was not customary at the time to acknowledge indebtedness to contemporary and recent writers, and (3) that Newton's genius was so strongly mathematical (whereas Bacon's great weakness was in mathematics) that he had no special reason to refer to Bacon's experimental principles.

00The main school is divided into two parts - the Latin school, corresponding to the classical side in other schools, and the mathematical school or modern side.

00He soon showed mathematical powers, but these were not fostered by the careful training mathematicians usually receive, and it may be said that in after years his attention was directed to the higher mathematics mainly by force of circumstances.

00It became at once the text-book of the chief mathematical schools of Europe, though its critical notes were of little value.

00According to the mathematical theory of the instrument,' if V and V' are the potentials of the quadrants and v is the potential of the needle, then the torque acting upon the needle to cause rotation is given by the expression, C(V - V'){v-2(V-{-V')}, where C is some constant.

00In the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations.

00Rogers in the Proceedings of the London Mathematical Society (series 2, vol.

00In the mathematical tripos three years later he was senior wrangler, beating J.

00Elected a fellow of his college, he devoted himself to teaching, and quickly proved himself one of the most successful mathematical "coaches" ever known at Cambridge.

00He was educated at home and at Aberdeen University, where he attained the highest academic distinctions, winning among other things the Ferguson mathematical scholarship, which is open to all graduates of Scottish universities under three years' standing.

00During this period he was not only most successful as a teacher, but produced much original work - especially in the experimental and mathematical treatment of electricity - which is still regarded as standard.

00The theory of utility above referred to, namely, that the degree of utility of a commodity is some continuous mathematical function of the quantity of the coin modity available, together with the implied doctrine that economics is essentially a mathematical science, took more definite form in a paper on "A General Mathematical Theory of Political Economy," written for the British Association in 1862.

00In 1864 he published a small volume, entitled Pure Logic; or, the Logic of Quality apart from Quantity, which was based on Boole's system of logic, but freed from what he considered the false mathematical dress of that system.

00Next comes the mathematical world of space, then the corporeal world, and finally the empirical world with its ]imitations of space and time.

00As a mathematician Salmon was a fellow of the Royal Society, and was president of the mathematical and physical section of the British Association in 1878.

00His published mathematical works include: Analytic Geometry of Three Dimensions (1862), Treatise on Conic Sections (4th ed., 1863) and Treatise on the Higher Plane Curves (2nd ed., 1873); these books are of the highest value, and have been translated into several languages.

00This explanation of the action of the solid is equivalent to that by which Gauss afterwards supplied the defect of the theory of Laplace, except that, not being expressed in terms of mathematical symbols, it does not indicate the mathematical relation between the attraction of individual particles and the final result.

00Leslie's theory was afterwards treated according to Laplace's mathematical methods by James Ivory in the article on capillary action, under "Fluids, Elevation of," in the supplement to the fourth edition of the Encyclopaedia Britannica, published in 1819.

00His essay contains the solution of a great number of cases, including most of those afterwards solved by Laplace, but his methods of demonstration, though always correct, and often extremely elegant, are sometimes rendered obscure by his scrupulous avoidance of mathematical symbols.

00His results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations.

00

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