How to use Mathematical in a sentence

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

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

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  • The 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.

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

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

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  • All 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.

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  • The mathematical fragments are collected by Fr.

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  • He 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.

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

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

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  • He 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.

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  • The 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.

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  • Meanwhile 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.

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  • The 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.

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  • So 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.

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  • He 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.

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  • His 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.

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  • Dodgson 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).

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  • His 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.

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  • Placed 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.

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  • The 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.

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

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  • The 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.

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

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  • Regarding 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.

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

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  • The 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.

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

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  • This 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.

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  • His 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.

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

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  • Among 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).

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  • His 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.

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

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  • By 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.

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  • The 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.

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

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  • The 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.

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

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

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  • The following are some further examples of mathematical induction.

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  • A 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.

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  • Although 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.

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  • About 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.

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  • In 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.

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  • We 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.

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

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  • Apart 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.

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  • The 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.

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

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  • The 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.

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  • The 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.

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  • Didymus, 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.

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  • The 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.

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

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

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  • The 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.

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

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  • Before 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.

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  • Further, 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.

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  • He 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.

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  • In 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.

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

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  • At 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.

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

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  • The 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.).

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  • In 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.

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  • He 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.

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  • Munich 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.

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  • Further 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.

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

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  • So 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.

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

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

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  • This 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.

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  • After 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.

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  • His 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.

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

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  • His 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.

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  • The 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.

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  • On 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.

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  • It 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.

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

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  • The 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.

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  • It 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.

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

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  • The 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.

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

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

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  • At 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.

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

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  • Jacques 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.

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

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  • Jacques 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.

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  • There, 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.

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  • He 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.

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

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

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

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

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

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  • The 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.

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  • Louis 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.

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

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  • But 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.

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  • A 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.

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  • The 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.

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

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

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

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  • It 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.

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

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

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  • As 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.

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  • In 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.

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

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  • His 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.

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  • The 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.

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

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

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  • These 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.

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  • In 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.

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

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  • The 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.

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  • The 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.

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  • The 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.

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  • In 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.

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

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

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

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  • Now 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.

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  • He studied medicine at Göttingen, 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.

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  • The 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).

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

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  • At 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.

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  • This 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.

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

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  • The 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.

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

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  • The 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."

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  • His 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.

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

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

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  • Beside 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.

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  • His 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.

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  • Chalmers 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.

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  • The 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.

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  • It 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."

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  • Now 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."

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  • The 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.

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  • For 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.

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  • In 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..

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  • For 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.

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

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  • He 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.

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  • He 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.

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

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

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  • In 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.

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

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  • Aristotle'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.

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

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

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  • In 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.

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

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

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  • But 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.

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  • The 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.

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  • Though 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.

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  • This 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.

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

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  • How 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.

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  • From 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.

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  • Other 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).

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

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  • He 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.

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  • Proceeding 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.

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  • But 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.

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  • It 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."

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  • Now, 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.

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

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  • It 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.

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  • If 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.

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  • Instead 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.

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

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  • The 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.

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

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  • He 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.

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

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  • In 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.

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

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  • Lie 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.

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  • Monge 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.

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

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  • Monge'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.

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  • The 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.

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

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  • Here 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.

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

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  • No 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.

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

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  • A 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.

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  • It 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.

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  • The 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.

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

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

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  • This 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.

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  • He 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.

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  • In 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.

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  • His 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.

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

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  • Wallis'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.

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  • The 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.

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

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  • Arguing 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.

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  • Molesworth 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.

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  • Wallis 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.

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

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  • There 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.

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  • The 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.

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  • Whether 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.

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

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

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  • The 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.

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  • At 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.

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

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  • He 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.

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  • Jeffrey 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.

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  • Having 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.

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

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  • The 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."

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

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  • The 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.

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

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  • We 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.

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  • At 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.

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  • Sylvester 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.

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

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  • Perhaps 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.

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

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  • After 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.

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

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  • In 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.

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

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  • It 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."

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  • On 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.

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  • On 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.

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

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  • All 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.

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  • Its 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.

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

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  • Plato is full of the faith of mathematical physics.

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

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  • For 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.

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

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  • It was otherwise with the mathematical instrument of Galilei.

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

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

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

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  • Mathematical 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.

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  • Quaternions (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.

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

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  • Again, 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.

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  • Statics 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.

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  • It 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.

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  • It 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.

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  • The 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.

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  • The 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.

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  • For 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.

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

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  • For 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.

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

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

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  • A 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.

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  • The 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.

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

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  • By 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.

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  • Ohm (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.

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  • Weber 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.

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  • Helmholtz 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.

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  • His 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.

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  • He 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.

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  • A 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.

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  • Maxwell'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.

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  • Coulomb 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.

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

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  • Oliver 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).

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  • The 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.

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  • The 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.

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  • The 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.

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  • He 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.

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

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  • In 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.

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  • Elected 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.

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  • He 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.

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  • During 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.

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  • The 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.

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  • In 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.

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  • As 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.

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  • This 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.

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  • Leslie'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.

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  • His 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.

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  • In this way he produced a great many of the forms of equilibrium of a liquid under the action of surfacetension alone, and compared them with the results of mathematical investigation.

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  • A good account of the subject from a mathematical point of view will be found in James Challis's " Report on the Theory of Capillary Attraction," Brit.

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  • It is also practically independent of the curvature of the surface, although it appears from the mathematical theory that there is a slight increase of tension where the mean curvature of the surface is concave, and a slight diminution where it is convex.

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  • In Egypt he settled for seven years, during which he studied the mathematical and physical systems of the ancient schools.

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  • Though he did not compete in the mathematical tripos, he acquired a great reputation at the university.

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  • Along with Sir John Herschel and George Peacock he laboured to raise the standard of mathematical instruction in England, and especially endeavoured to supersede the Newtonian by the Leibnitzian notation in the infinitesimal calculus.

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  • Davy on the application of machinery to the calculation and printing of mathematical tables, he discussed the principles of a calculating engine, to the construction of which he devoted many years of his life.

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  • He received his education at an ordinary school, and afterwards at the Albany Academy, which enjoyed considerable reputation for the thoroughness of its classical and mathematical courses.

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  • He soon showed that he was a boy of great capacity, and in particular that he was possessed of remarkable mathematical ability.

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  • From the time he went first to Cambridge till his death he was constantly engaged in mathematical investigation.

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  • At various times he was president of the Cambridge Philosophical Society, of the London Mathematical Society and of the Royal Astronomical Society.

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  • He also received the De Morgan medal from the London Mathematical Society, and the Huygens medal from Leiden.

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  • Schering the Disquisitiones arithmeticae, (2) Theory of Numbers, (3) Analysis, (4) Geometry and Method of Least Squares, (5) Mathematical Physics, (6) Astronomy, and (7) the Theoria motus corporum coelestium.

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  • Kirchhoff's contributions to mathematical physics were numerous and important, his strength lying in his powers of stating a new physical problem in terms of mathematics, not merely in working out the solution after it had been so formulated.

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  • The discovery of ether brought with it a reconstruction of our ideas of the physical universe, transferring the emphasis from the mathematical expression of static relationships to a dynamic conception of a universe in constant transformation; matter in equipoise became energy in gradual readjustment.

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  • The mathematical demonstration of its truth was left by Augustine for his disciple,.

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  • The mathematical influence of Monge had two sides represented respectively by his two great works, the Geometric descriptive and the Application de l'analyse a la geometrie.

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  • Induced by the encouragement of his mathematical friends in England, Plucker in 1865 returned to the field in which he first became famous, and adorned it by one more great achievement - the invention of what is now called "line geometry."

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  • His father, Jerome Quinet, had been a commissary in the army, but being a strong republican and disgusted with Napoleon's usurpation, he gave up his post and devoted himself to scientific and mathematical study.

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  • In the same year, 1490, Leonardo enjoyed some months of uninterrupted mathematical and physical research in the libraries and among the learned men of Pavia, whither he had been called to advise on some architectural difficulties concerning the cathedral.

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  • Pacioli was equally amazed and delighted at Leonardo's two great achievements in sculpture and painting, and still more at the genius for mathematical, physical and anatomical research shown in the collections of MS. notes which the master laid before him.

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  • The mathematical investigation of this subject was worked out by Gaspard Monge.

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  • Xenocrates indeed, identifying ideal and mathematical numbers, sought to ' That Plato did not neglect, but rather encouraged, classificatory science is shown, not only by a well-known fragment of the comic poet Epicrates, which describes a party of Academics engaged in investigating, under the eye of Plato, the affinities of the common pumpkin, but also by the Timaeus, which, while it carefully discriminates science from ontology, plainly recognizes the importance of the study of natural kinds.

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  • His son, Lucien De La Rive, born at Geneva on the 3rd of April 1834, published papers on various mathematical and physical subjects, and with Edouard Sarasin carried out investigations on the propagation of electric waves.

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  • In thus reverting to the crudities of certain Pythagoreans, he laid himself open to the criticisms of Aristotle, who, in his Metaphysics, recognizing amongst contemporary Platonists three principal groups - (1) those who, like Plato, distinguished mathematical and ideal numbers; (2) those who, like Xenocrates, identified them; and (3) those who, like Speusippus, postulated mathematical numbers only - has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideas are numbers made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations.

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  • The undulatory theory of light, first founded upon experimental demonstration by Thomas Young, was extended to a large class of optical phenomena, and permanently established by his brilliant discoveries and mathematical deductions.

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  • The offer in 1795 of a mathematical chair in one of the schools of Paris was declined on account of his infirm health, and he was still in straitened cirumstances in 1798, when he published a second edition of the first part of his Histoire.

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