# Mathematical Sentence Examples

Space-time in simple terms is a

**mathematical**model that combines space and time into a single continuum.It has been stated that Napier's

**mathematical**pursuits led him to dissipate his means.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.In England, Robert Recorde had indeed published his

**mathematical**treatises, but they were of trifling importance and without influence on the history of science.The fundamental geographical conceptions are

**mathematical**, the relations of space and form.More detailed accounts are to be found in the various

**mathematical**articles.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.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.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).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.AdvertisementPlaced 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.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.Of these thirteen sections, the first contains a simple description of the more prominent phenomena, without

**mathematical**symbols or numerical data.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.Thomson (afterwards Lord Kelvin) in 1847, as the result of a

**mathematical**investigation undertaken to explain Faraday's experimental observations.AdvertisementRegarding 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.In 1873 James Clerk Maxwell published his classical Treatise on Electricity and Magnetism, in which Faraday's ideas were translated into a

**mathematical**form.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.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.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.AdvertisementHis

**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.His

**mathematical**discoveries were extended and over shadowed by his contemporaries.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).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.Here, too, he died, attended by his physician, Dr Majendie, and his

**mathematical**coadjutor, Alexis Bouvard., on the 5th of March 1827.AdvertisementBy 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.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.For the history of the subject see A History of the

**Mathematical**Theory of Probability, by Isaac Todhunter (1865).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.On the other land, the lateness of occurrence of any particular

**mathematical**idea is usually closely correlated with its intrinsic difficulty.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.The following are some further examples of

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

**mathematical**point.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.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.It appears indeed that the purely

**mathematical**question has no definite answer.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.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.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.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.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.But the development of

**mathematical**and physical science soon introduced a fundamental change in the habits of thought with respect to medical doctrine.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**.Mead, a man of great learning and intellectual activity, was an ardent advocate of the

**mathematical**doctrines.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.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.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.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.In 1806 he was appointed

**mathematical**master in the Woolwich Academy, and filled that post for fortyone years.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.This mechanical axiom of the normality of fluid pressure is the foundation of the

**mathematical**theory of hydrostatics.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.).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.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.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.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.For further details of his

**mathematical**investigations see the articles Theory of groups, and Functions Of Complex Variables.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.From 1716 to 1718 he published a scientific periodical, called Daedalus hyperboreus, a record of mechanical and

**mathematical**inventions and discoveries.The same year he published various

**mathematical**and mechanical works.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.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.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.This undertaking, the

**mathematical**and scientific parts of which fell to Hutton's share, was completed in 1809, and filled eighteen volumes quarto.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.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.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.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.The

**mathematical**expression for this potential can in some cases be calculated or predetermined.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.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.This, with other matters appertaining to the calendar, was probably left to be regulated from time to time by the

**mathematical**tribunal.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.A complete summary of the great developments of

**mathematical**learning, which the members of this family effected, lies outside the scope of this notice.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.On his final return to Basel in 1682, he devoted himself to physical and

**mathematical**investigations, and opened a public seminary for experimental physics.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.In 1687 the

**mathematical**chair of the university of Basel was conferred upon Jacques.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.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.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.He was a member of almost every learned society in Europe, and one of the first mathematicians of a

**mathematical**age.On his return to Berlin he was appointed director of the

**mathematical**department of the academy.In 1788 he was named one of its

**mathematical**professors.The second part deals with chronological and

**mathematical**questions, and has been of great service in determining the principal epochs of ancient history.These two simplifying facts bring the properties of the gaseous state of matter within the range of

**mathematical**treatment.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.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.He reproduces and further develops and defends his own views in his

**Mathematical**Memoirs, and in his paper in the Philosophical Transactions for 1785.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.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.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.In 1842 he obtained a

**mathematical**scholarship and graduated as B.A.About this time he became

**mathematical**master at a school at Wimbledon.In 1862 he was made a fellow of the Royal Society, and in 1865 a member of the

**Mathematical**Society of London.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.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.The instrument was a ingenuity, and was called "the

**mathematical**jewel."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.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.The proper treatment of the deviations from

**mathematical**accuracy, in the second and third of the above classes of cases, is a special matter.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.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.The same author's Scientific Papers contains many experimental and

**mathematical**contributions to the science.Thomson, Sound (5th ed., 1909), contains a descriptive account of the chief phenomena, and an elementary

**mathematical**treatment.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.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.The distinctive industry is the manufacture of

**mathematical**and musical instruments.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.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.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.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.He was admitted to the Institute on its organization in 1795, and became, in 1803, perpetual secretary to its

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

**mathematical**physics.Through the influence of Sir Isaac Newton he was elected

**mathematical**master in Christ's hospital.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.He 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.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).Excluding all these, the

**mathematical**works contained in the first and second volumes occupy about 1800 pages.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.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.In 1870 he was elected president of the London

**Mathematical**Society.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.To a mind imbued with the love of

**mathematical**symmetry the study of determinants had naturally every attraction.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."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.There is hardly a branch of

**mathematical**physics which is independent of these functions.For the

**mathematical**investigation see Spherical Harmonics and for tables see Table,**Mathematical**.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.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.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.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.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."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."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.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.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..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**.Graduating at Harvard College in 1829, he became

**mathematical**tutor there in 1831 and professor in 1833.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.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.In respect of

**mathematical**geography, his lack of scientific training was no great hindrance.In the unwritten lectures of his old age, he developed this formal into a

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

**mathematical**from formal numbers.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.If, wrote Aristotle, the forms are another sort of number, not

**mathematical**, there would be no understanding of it.But A System Of 31 Intercalations In 128 Years Would Be By Far The Most Perfect As Regards

**Mathematical**Accuracy.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.After five years spent in

**mathematical**and astronomical studies, he went to Holland, in order to visit several eminent continental mathematicians.In the Zwinger are the zoological and mineralogical museums and a collection of instruments used in

**mathematical**and physical science.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.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.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.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.In a period of general stagnation in

**mathematical**studies, he stands out as a remarkable exception.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.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.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).His theory of bodies involved an idealistic analysis neither into bodily atoms nor into

**mathematical**units, but into mentally endowed simple substances.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.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.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.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."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.He won the King of Sweden's open prize for a

**mathematical**treatise in 1889, and in 1908 was elected to the Academie Frangaise.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.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.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.It can only be contrived by means of complicated

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

**mathematical**axiom.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.Simson's contributions to

**mathematical**knowledge took the form of critical editions and commentaries on the works of the ancient geometers.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.Lie's work exercised a great influence on the progress of

**mathematical**science during the later decades of the 19th century.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.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.His later

**mathematical**papers are published (1794-1816) in the Journal and the Correspondance of the polytechnic school.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.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.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.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.There are five faculties - theological, juridical, medical, philosophical and

**mathematical**.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.Upon the nature of the reasoning by which in

**mathematical**science we pass from data to conclusions, Hume gives no explicit statement.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.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.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.In fact, for a considerable period, the term " theory of heat " was practically synonymous with the

**mathematical**treatment of a conduction.Fourier need not be considered in detail here, as they are in many cases of

**mathematical**rather than physical interest.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.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.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.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.Wallis was to confine himself to the

**mathematical**chapters, and set to work at once with characteristic energy.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.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.But it was no longer a fight over

**mathematical**questions only.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.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.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.He was, however, indefatigable in his

**mathematical**work.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.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.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.The celebrated Rhind

**mathematical**papyrus was coried in the reign of an Apopi from an original of the time of Amenemhe III.His love for

**mathematical**science, geography, &c., in which the Arabs excelled, is noteworthy.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.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.In 1814 Carlyle, still looking forward to the career of a minister, obtained the

**mathematical**mastership at Annan.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.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.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.See C. Hutton,

**Mathematical**and Philosophical Dictionary (1815).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."The " senior wrangler " was the first candidate in order of merit in the first part of the

**mathematical**tripos.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.In

**mathematical**and physical science Cusanus was much in advance of his age.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.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.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.When twenty years of age he entered the army, becoming lieutenant in a regiment of cavalry, and employing his leisure on

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

**mathematical**ability.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.Thus during the session 1778-177 9, in addition to his

**mathematical**work, he delivered an original course of lectures on morals.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.To say, however, that Clarke simply confused mathematics and morals by justifying the moral criterion on a

**mathematical**basis is a mistake.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."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.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.Now, there is no doubt that, especially in

**mathematical**equations, universal conclusions are obtainable from convertible premises expressed in these ways.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.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.The form which a

**mathematical**science treats as relatively self-subsistent is certainly not the constitutive idea.Plato is full of the faith of

**mathematical**physics.The

**mathematical**sciences, at least, had not proved disappointing.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.But consider Bacon's own doctrine of forms. Or watch the

**mathematical**physicist with his formulae.It was otherwise with the

**mathematical**instrument of Galilei.It is concurrently with signal success in the work of a pioneer in the

**mathematical**.Though he makes his bow to

**mathematical**method, he, even more than Hobbes, misses its constructive character.The clue of

**mathematical**certainty is discarded in substance in the English form of " the new way of ideas."**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.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.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.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.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.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.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.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.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.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.He afterwards entered at Clare College, Cambridge, where he applied himself to

**mathematical**study, and obtained a fellowship in 1693.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.The rest of his life was spent in incessant controversy - theological,

**mathematical**, chronological and miscellaneous.It is sometimes known as the "expectation of life," a term, however, which involves a

**mathematical**hypothesis now discarded.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.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.Respecting this achievement when developed in its experimental and

**mathematical**completeness, Clerk Maxwell says that it was " perfect in form and unassailable in accuracy."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.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.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.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.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.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.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.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.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.The formulation of electrical theory as far as regards operations in space free from matter was immensely assisted by Maxwell's

**mathematical**theory.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).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.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.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.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.It became at once the text-book of the chief

**mathematical**schools of Europe, though its critical notes were of little value.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.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.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.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.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.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.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.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.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.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.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.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.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.In Egypt he settled for seven years, during which he studied the

**mathematical**and physical systems of the ancient schools.Though he did not compete in the

**mathematical**tripos, he acquired a great reputation at the university.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.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.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.He soon showed that he was a boy of great capacity, and in particular that he was possessed of remarkable

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

**mathematical**investigation.At various times he was president of the Cambridge Philosophical Society, of the London

**Mathematical**Society and of the Royal Astronomical Society.He also received the De Morgan medal from the London

**Mathematical**Society, and the Huygens medal from Leiden.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.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.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.The

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

**mathematical**investigation of this subject was worked out by Gaspard Monge.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.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.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.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.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.His exemplary diligence and unusual

**mathematical**capacity were soon noticed.The name of " abacus " is also given, in logic, to an instrument, often called the " logical machine," analogous to the

**mathematical**abacus.His

**mathematical**genius gained for him a high place in the 'esteem of Jean Bernoulli, who was at that time one of the first mathematicians in Europe, as well as of his sons Daniel and Nicolas Bernoulli.At the same time, by the advice of the younger Bernoullis, who had removed to St Petersburg in 1725, he applied himself to the study of physiology, to which he made a happy application of his

**mathematical**knowledge; and he also attended the medical lectures at Basel.It was in these circumstances that he dictated to his servant, a tailor's apprentice, who was absolutely devoid of

**mathematical**knowledge, his Anleitung zur Algebra (1770), a work which, though purely elementary, displays the**mathematical**genius of its author, and is still reckoned one of the best works of its class.Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by

**mathematical**principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit.He stated that it wa g practically impossible to determine the stresses by purely

**mathematical**means.It is obvious that experiments of the kind referred to cannot take into account all the conditions of the problem met with in actual practice, such as the effect of the rock at the sides of the valley and variations of temperature, &c., but deviations in practice from the conditions which

**mathematical**analyses or experiments assume are nearly always present.His

**mathematical**studies seem to have been undertaken and carried to their full development without any assistance whatever, and the result is that his writings belong to no particular " school," unless indeed we consider them to form, as they are well entitled to do, a school by themselves.Nothing could be better fitted to call forth such

**mathematical**powers as those of Hamilton; for Laplace's great work, rich to profusion in analytical processes alike novel and powerful, demands from the most gifted student careful and often laborious study.Their report, while acknowledging the novelty and value of its contents, and the great

**mathematical**skill of its author, recommended that, before being published, it should be still further developed and simplified.