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calculus

But no carefully devised calculus can take the place of insight, observation and experience.

95The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.

85Hence, early empiricism makes ethics simply a calculus of pleasures ("hedonism").

73It is not, however, necessary that the notation of the calculus should be employed throughout.

40In 1747 he applied his new calculus to the problem of vibrating chords, the solution of which, as well as the theory of the oscillation of the air and the propagation of sound, had been given but incompletely by the geometricians who preceded him.

42In 1747 he applied his new calculus to the problem of vibrating chords, the solution of which, as well as the theory of the oscillation of the air and the propagation of sound, had been given but incompletely by the geometricians who preceded him.

42In the notation of the calculus the relations become - dH/dp (0 const) = odv /do (p const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure - dp. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.

22From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.

23Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the " calculus of finite differences."

23From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.

23Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the " calculus of finite differences."

23Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.

11For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

11Buchheim, on Extensive Calculus and its Applications, Proc. L.

11By applying the method of the differential calculus, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or analytically that this is a minimum.

11by the methods of the differential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant.

11He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner.

11This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

11In the notation of the integral calculus, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.

11Lacroix's Differential Calculus in 1816.

12This discovery was followed by that of the calculus of partial differences, the first trials of which were published in his Reflexion sur la cause generale des vents (1747).

12Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, and in the solution of which accordingly the calculus is always now employed.

12For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

12In analytical invention, and mastery over the calculus, the Turin mathematician was admittedly unrivalled.

12The calculus of variations is indissolubly associated with his name.

12To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.

12In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful calculus occupies a prominent place.

12A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.

12If therefore we choose a quantity e such that log e I o X X= I, log i oe = X, which gives (by more accurate calculation) e=2.71828..., we shall have lim(loge(I+0))}/0=I, and conversely 'lim' {ex+0 - e x } 143= The deduction of the expansions log e (' +x) = x - Zx 2 + 3x 3 - ..., e x = I +.x+x2/2!+x3/3!-}-..., is then more simply obtained by the differential calculus than by ordinary algebraic methods.

12Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, and in the solution of which accordingly the calculus is always now employed.

12Both these methods, differing from that now employed, are interesting as preliminary steps towards the method of fluxions and the differential calculus.

00Phillips, 1896); A Brief Introduction to the Infinitesimal Calculus (1897); The Nature of Capital and Income (1906); The Rate of Interest 0907); National Vitality (1909); The Purchasing Power of Money (1911); Elementary Principles of Economics (1913); Why is the Dollar Shrinking?

00Sidgwick holds that intuition must justify the claims of the general happiness upon the individual, though everything subsequent is hedonistic calculus.

00The former was professor of mathematics at Bologna, and published, among other works, a treatise on the infinitesimal calculus.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

001 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.

00He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol b gave it the efficacy of a new calculus.

00It was his just boast to have transformed mechanics (defined by him as a "geometry of four dimensions") into a branch of analysis, and to have exhibited the so-called mechanical "principles" as simple results of the calculus.

00General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.

00The idea is utilized in the elementary consideration :of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.

00If He Had Happened To Think Of Them As " Products," He Might Have Anticipated Grassmann'S Discovery Of The Extensive Calculus.

00The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus.

002 enclosing a point B, the pressure p at B is the limit of OP/DA; and p =lt(AP/DA) =dP/ dA, (I) in the notation of the differential calculus.

00Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation S - s =0 (dp /do) (dv/do),.

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

00The same year he went to Geneva, where he gave instruction in the differential calculus to Nicolas Fatio de Duillier, and afterwards proceeded to Paris, where he enjoyed the society of N.

00With the marquis de l'Hopital he spent four months studying higher geometry and the resources of the new calculus.

00Among these were the exponential calculus, and the curve called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density.

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

00He contributed two memoirs to the Philosophical Transactions, one, "Logometria," which discusses the calculation of logarithms and certain applications of the infinitesimal calculus, the other, a "Description of the great fiery meteor seen on March 6th, 1716."

00He also showed that the roots of a cubic equation can be derived by means of the infinitesimal calculus.

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

00of distances between points) as belonging to geometry or trigonometry; while the measurement of curved lengths, except in certain special cases, involves the use of the integral calculus.

00Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus.

00made up of a number of successive elements, so that the analytical treatment involves the ideas and the methods of the infinitesimal calculus.

00The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

00If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.

00The establishment of these formulae involves the use of the integral calculus.

00Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.

00For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

00The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus Of.

00The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.

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

00As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.

00Papers published in 1776 were concerned with quartz, alum and clay and with the analysis of calculus vesicae from which for the first time he obtained uric acid.

00Calculus of differences >>

00The abundant vintage of that year drew his attention to the defective methods in use for estimating the cubical contents of vessels, and his essay on the subject (Nova Stereometria Doliorum, Linz, 1615) entitles him to rank among those who prepared the discovery of the infinitesimal calculus.

00In the pursuit of this inquiry he rashly invaded other departments of science, and much of the Common Place Book is occupied with a polemic, as vigorous as it is ignorant, against the fundamental conceptions of the infinitesimal calculus.

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

00Pascal also distinguished himself by his skill in the infinitesimal calculus, then in the embryonic form of Cavalieri's method of indivisibles.

00There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

00He managed to make practical use of his calculus about his farms, and seems to have been remarkably apt in the practical application of mechanical principles.

00In addition we may also mention his essay on the calculus of variations (Mem.

00Democritean physics without a calculus had necessarily proved sterile of determinate concrete results, and this was more than enough to ripen the naturalism of the utilitarian school into scepticism.

00A mathematico-physical calculus that would work was in question.

00With Hobbes logic is a calculus of marks and signs in the form of names.

005 Leibnitz probably owes to him the thought of a calculus of symbols, and the conception of demonstration as essentially a chain of definitions.

004 In the case of non-identical truths, too, there is a priori proof drawn from the notion of the terms, " though it is not always in our power to arrive at this analysis," 5 so that the question arises, specially in connexion with the possibility of a calculus, whether the contingent is reducible to the necessary or identical at the ideal limit.

00The propositions which deal with actual existence are still of a unique type, with whatever limitation to the calculus.

00There is a formal-symbolic logic engaged with the elaboration of a relational calculus.

001853)4 is qualified by the warning that the real activity of thought tends to fall outside the calculus of relations and to attach rather to the subsidiary function of denoting.

00Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k.

00Hamilton, in fact, remarks, 2 " regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse.

00Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions.

00that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also.

00It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a, chain of given length hanging I I between fixed points must be catenary; it determines the scale of the curve, all cate } stationary (~ 9).

00Examples will be found in textbooks of the calculus and of analytical statics.

00In his desire to bring science home to the imperfectly educated he published anonymously Calculus made Easy by " F.R.S."

00Edwards, Differential Calculus, and for geometrical constructions see T.

00In the years 1815-1817 he contributed three papers on the "Calculus of Functions" to the Philosophical Transactions, and in 1816 was made a fellow of the Royal Society.

00Along 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.

00The length of any arc may be determined by geometrical considerations or by the methods of the integral calculus.

00reference may be made to articles Mechanics and Infinitesimal Calculus.

00At the commencement of his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics.

00He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.

00The Principia gives no information on the subject of the notation adopted in the new calculus, and it was not until 1693 that it was com municated to the scientific world in the second volume of Dr Wallis's works.

00Newton's admirers in Holland had informed Dr Wallis that Newton's method of fluxions passed there under the name of Leibnitz's Calculus Di fferentialis.

00The work on [[Trigonometry]] and Double Algebra (1849) contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic calculus.

00De Morgan's other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the [[Essay]] on Probabilities (1838), forming the 107th volume of Lardner's Cyclopaedia, which forms a valuable introduction to the subject; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus (1837).

00Among these may be mentioned the Treatise on the Differential and Integral Calculus (1842); the Elementary Illustrations of the Differential and Integral Calculus, first published in 1832, but often bound up with the larger treatise; the essay, On the Study and Difficulties of Mathematics (1831); and a brief treatise on Spherical Trigonometry (1834).

00p. 118.) Two of his most elaborate treatises are to be found in the [[Encyclopaedia]] metropolitana, namely the articles on the Calculus of Functions, and the Theory of Probabilities.

00Late in 1847 De Morgan published his principal logical treatise, called Formal Logic, or the Calculus of Inference, Necessary and Probable.

00A notion related to that of infinitesimals is presented in the Greek " method of exhaustion "; the more perfect conception, however, only dates from the 17th century, when it led to the infinitesimal calculus.

00Infinitesimal Calculus >>

00At the age of twenty-five he published a treatise on the integral calculus, as a supplement to De l'Hopital's treatise, Des infiniment petits.

00A certain common agreement has been reached concerning the impossibility of regarding pleasure as the sole motive criterion and end of moral action, though different opinions still prevail as to the place occupied by pleasure in the summum bonum, and the possibility of a hedonistic calculus.

00It must be admitted that any intelligent comprehension of the subject requires at least a grasp of the fundamental conceptions of analytical geometry and the infinitesimal calculus, such as only one with some training in these subjects can be expected to have.

00They afford yet another great advantage in that the derivation of the results requires only the analytic operations of the infinitesimal calculus.

00Cavalieri's " indivisibles " into the infinitesimal calculus, all accomplished during the 17th century, immeasurably widened the scope of exact astronomy.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the differential calculus.

00He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.

00Starting from simple elementary propositions, Steiner advances to the solution of problems which analytically require the calculus of variation, but which at the time altogether surpassed the powers of that calculus.

00He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus.

00Combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate calculus.

00Propensity theories obey the probability of calculus.

00He sent his results on making the differential calculus into a purely algebraic theory to the Royal Society.

00Keywords: duration calculus, real-time systems, probabilistic automata, satisfaction probability.

00calculus of variations considers the notion of optimizing an integral where the quantity to be varied is a function.

00To calculate accurately the motion of the Earth around the Sun, Isaac Newton was inspired to invent calculus.

00In 1702 he applied the calculus to clocks driven by a spring.

00For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered.

00It enables us to extend matrix algebra calculus in an easy way.

00Have you suggestions for the best way to introduce calculus?

00Semantic analysis or normalization by evaluation for typed lambda calculus.

00Recently, I started working on X, a calculus based on the sequent calculus for Classical Logic.

00The satisfiability problem for the propositional calculus; Cook's theorem.

00Such a question is irrelevant at the level of predicate calculus.

00The term " derivative, taken from infinitesimal calculus, refers to an isolated aspect, or " function " of a real quantity.

00Overview of the thesis Chapter 2 is an introduction to stochastic calculus.

00John has worked on a range of topics related to Hamiltonian mechanics and variational calculus, and in particular their application to water waves.

00Adams calculus is aimed at the upper end of the three semester calculus course.

00The theory is a strictly smaller theory than the lambda calculus.

00pi calculus systems are evolving to become the new wave of software infrastructure to manage business processes.

00They constitute the first step toward the conception of tensor calculus.

00We extend their result to more general structures of first order predicate calculus.

00The combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate calculus.

00In Morgan's refinement calculus it appears with respect to initial variable values.

00Renal cell carcinomas can present with ' clot ' colic similar in nature to ureteric colic due to renal calculus.

00differential calculus against attacks by Berkeley.

00His statement in his essay on Descartes that he invented the differential calculus is probably typical of many other statements he makes.

00In this approach we use differential calculus and differential geometry both to filter and analyze multi-dimensional images.

00elementary differential calculus, and understood its range of application.

00epsilon calculus and consistency proofs in Hilbert's Program ", Synthese.

00equalize outcomes for equally deserving individuals first, and then applies a utilitarian calculus.

00Students are only required to have some training in basic calculus and some familiarity with first-order difference equations.

00Language goes no further in this direction, but the mathematical symbolism of the differential calculus allows of an indefinite extension of this procedure.

00Systematic analysis of dental remains focuses on caries, ante-mortem tooth loss, abcesses, calculus, and enamel hypoplasias.

00infinitesimal calculus, refers to an isolated aspect, or " function " of a real quantity.

00Definite and indefinite integrals and the Fundamental Theorem of Integral Calculus.

00integral calculus.

00invented the calculus.

00invention of calculus, he begins a bitter conflict over priority.

00lambda calculus.

00In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic.

00These include complete sets of curriculum material for various college and university level courses, including calculus, linear algebra and engineering mathematics.

00pi calculus systems are evolving to become the new wave of software infrastructure to manage business processes.

00When you are doing calculus you always use radians.

00refinement calculus for real-time systems.

00Database languages: relational algebra, relational algebra, relational calculus, SQL.

00Database languages: relational algebra, relational calculus, SQL.

00ultrasonic scalers are used to remove calculus rapidly from the tooth surface.

00sonic scalers remove calculus from the surface of the tooth.

00dental scaler Removing the hard calculus from teeth helps eliminate the places where bacteria can lurk in the mouth.

00sequent calculus for Classical Logic.

00stochastic calculus.

00Calculus, sometimes called tartar, is hardened calcified plaque.

00As plaque ages, it hardens and calcifies to form tartar (called dental calculus ).

00tensor calculus.

00theorem of calculus.

00ultrasonic scalers are used to remove calculus rapidly from the tooth surface.

00utilitarian calculus in this " your child or your dog?

00vector calculus that have wide application in physics.

00Course Description This module will extend the vector algebra of the first year to the calculus of three dimensional vector algebra of the first year to the calculus of three dimensional vectors.

00- The range and importance of the scientific labours of Archimedes will be best understood from a brief account of those writings which have come down to us; and it need only be added that his greatest work was in geometry, where he so extended the method of exhaustion as originated by Eudoxus, and followed by Euclid, that it became in his hands, though purely geometrical in form, actually equivalent in several cases to integration, as expounded in the first chapters of our text-books on the integral calculus.

00Both these methods, differing from that now employed, are interesting as preliminary steps towards the method of fluxions and the differential calculus.

00The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.

00Speculations concerning a calculus of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities (1854), should alone be considered as containing a mature statement of his views.

00While still an undergraduate he formed a league with John Herschel and Charles Babbage, to conduct the famous struggle of "d-ism versus dot-age," which ended in the introduction into Cambridge of the continental notation in the infinitesimal calculus to the exclusion of the fluxional notation of Sir Isaac Newton.

00Lacroix's Differential Calculus in 1816.

00Peacock powerfully aided the movement by publishing in 1820 A Collection of Examples of the Application of the Differential and Integral Calculus.

00Phillips, 1896); A Brief Introduction to the Infinitesimal Calculus (1897); The Nature of Capital and Income (1906); The Rate of Interest 0907); National Vitality (1909); The Purchasing Power of Money (1911); Elementary Principles of Economics (1913); Why is the Dollar Shrinking?

00Hence, early empiricism makes ethics simply a calculus of pleasures (" hedonism ").

00Sidgwick holds that intuition must justify the claims of the general happiness upon the individual, though everything subsequent is hedonistic calculus.

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

00The former was professor of mathematics at Bologna, and published, among other works, a treatise on the infinitesimal calculus.

00This discovery was followed by that of the calculus of partial differences, the first trials of which were published in his Reflexion sur la cause generale des vents (1747).

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

00This latter work included the differential and integral calculus, the calculus of variations, the theory of attractions, and analytical mechanics.

00(See Infinitesimal Calculus.) He also discovered a method of deriving one curve from another, by means of which finite areas can be obtained equal to the areas between certain curves and their asymptotes.

00But no carefully devised calculus can take the place of insight, observation and experience.

00Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers.

00For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

00Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

00During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.

00pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically.

001 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.

00At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations.

00He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol b gave it the efficacy of a new calculus.

002 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem."

00In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions.

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

00By means of this "calculus of derived functions" Lagrange hoped to give to the solution of all analytical problems the utmost "rigour of the demonstrations of the ancients"; 6 but it cannot be said that the attempt was successful.

00The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements.

00In analytical invention, and mastery over the calculus, the Turin mathematician was admittedly unrivalled.

00The calculus of variations is indissolubly associated with his name.

00To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.

00It was his just boast to have transformed mechanics (defined by him as a "geometry of four dimensions") into a branch of analysis, and to have exhibited the so-called mechanical "principles" as simple results of the calculus.

00In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful calculus occupies a prominent place.

00A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.

00is now obsolete owing to the more extended facilities afforded by the calculus of operations.

00Some writers have attempted unification by treating algebra as concerned with functions, and Comte accordingly defined algebra as the calculus of functions, arithmetic being regarded as the calculus of values.

00General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.

00This is a particular case of Taylor's theorem (see Infinitesimal Calculus).

00The idea is utilized in the elementary consideration :of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.

00If therefore we choose a quantity e such that log e I o X X= I, log i oe = X, which gives (by more accurate calculation) e=2.71828..., we shall have lim(loge(I+0))}/0=I, and conversely 'lim' {ex+0 - e x } 143= The deduction of the expansions log e (' +x) = x - Zx 2 + 3x 3 - ..., e x = I +.x+x2/2!+x3/3!-}-..., is then more simply obtained by the differential calculus than by ordinary algebraic methods.

00Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.

00If He Had Happened To Think Of Them As " Products," He Might Have Anticipated Grassmann'S Discovery Of The Extensive Calculus.

00In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

00In the extensive calculus of the nth category, we have, first of all, n independent " units," el, e2, ...

00All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

00A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.

00As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.

00If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative.

00For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

00Buchheim, on Extensive Calculus and its Applications, Proc. L.

00The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus.

00By applying the method of the differential calculus, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or analytically that this is a minimum.

00by the methods of the differential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant.

00the catenary solves the problem: to find a curve joining two given points, which when revolved about a line co-planar with the points traces a surface of minimum area (see Variations, Calculus Of).

00He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner.

00This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

002 enclosing a point B, the pressure p at B is the limit of OP/DA; and p =lt(AP/DA) =dP/ dA, (I) in the notation of the differential calculus.

00The same work contained the celebrated formula known as " Taylor's theorem (see Infinitesimal Calculus), the importance of which remained unrecognized until 1772, when J.

00The quest of a solvent for calculus in the bladder and kidneys was pursued by him as by others at the period, and he devised a form of forceps which, on the testimony of John Ranby (1703-1773), sergeant-surgeon to George II., extracted stones with "great ease and readiness."

00Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that vV = o.

00In the notation of the calculus the relations become - dH/dp (0 const) = odv /do (p const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure - dp. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.

00Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation S - s =0 (dp /do) (dv/do),.

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

00The same year he went to Geneva, where he gave instruction in the differential calculus to Nicolas Fatio de Duillier, and afterwards proceeded to Paris, where he enjoyed the society of N.

00With the marquis de l'Hopital he spent four months studying higher geometry and the resources of the new calculus.

00Among these were the exponential calculus, and the curve called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density.

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

00He contributed two memoirs to the Philosophical Transactions, one, "Logometria," which discusses the calculation of logarithms and certain applications of the infinitesimal calculus, the other, a "Description of the great fiery meteor seen on March 6th, 1716."

00He also showed that the roots of a cubic equation can be derived by means of the infinitesimal calculus.

00== Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852, 6th ed., 1873), Treatise on Analytical Statics (1853, 4th ed., 1874); Treatise on the Integral Calculus (1857, 4th ed., 1874); Treatise on Algebra (1858, 6th ed., 1871); Treatise on Plane Coordinate Geometry (1858, 3rd ed., 1861); Plane Trigonometry (1859, 4th ed., 1869); Spherical Trigonometry (1859); History of the Calculus of Variations (1861); Theory of Equations (1861, 2nd ed.

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

00of distances between points) as belonging to geometry or trigonometry; while the measurement of curved lengths, except in certain special cases, involves the use of the integral calculus.

00Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus.

00made up of a number of successive elements, so that the analytical treatment involves the ideas and the methods of the infinitesimal calculus.

00It is not, however, necessary that the notation of the calculus should be employed throughout.

00In the notation of the integral calculus, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.

00The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

00in terms of ug m and its even central differences (see Differences, Calculus Of).

00Starting from any ordinate ue,o, the result of integrating with regard to x through a distance 2h is (in the example considered in § 61) the same as the result of the operation 3h(I + 4E + E 2), where E r denotes the operation of changing x into x+h (see Differences, Calculus oF).

00If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.

00The establishment of these formulae involves the use of the integral calculus.

00The formula of § 76 may (see Differences, Calculus Of) be written f x udx = h .µxu + h(- 2 µ6u + 2 0 = (hu-- f 2 6 hu +.; 2 0 6 hu - udx = h.

00Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.

00For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

00In mensuration, "cubature" is sometimes used to denote the volume of a solid; the word is parallel with "quadrature," to determine the area of a surface (see Mensuration; Infinitesimal Calculus) .

00The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus Of.

00The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.

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

00Formulae of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.

00The Arithmetica infinitorum relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Bonaventura Cavalieri in 1629 (see Infinitesimal Calculus).

00As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.

00Papers published in 1776 were concerned with quartz, alum and clay and with the analysis of calculus vesicae from which for the first time he obtained uric acid.

00Calculus of differences >>

00The abundant vintage of that year drew his attention to the defective methods in use for estimating the cubical contents of vessels, and his essay on the subject (Nova Stereometria Doliorum, Linz, 1615) entitles him to rank among those who prepared the discovery of the infinitesimal calculus.

00In the pursuit of this inquiry he rashly invaded other departments of science, and much of the Common Place Book is occupied with a polemic, as vigorous as it is ignorant, against the fundamental conceptions of the infinitesimal calculus.

00(See Infinitesimal Calculus.) His Sermons have long enjoyed a high reputation; they are weighty pieces of reasoning, elaborate in construction and ponderous in style.

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

00Pascal also distinguished himself by his skill in the infinitesimal calculus, then in the embryonic form of Cavalieri's method of indivisibles.

00There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

00Leibnitz (see Infinitesimal Calculus).

00The equations to the tangent and normal at the point x' y are yy' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal Calculus).

00The length of a parabolic arc can be obtained by the methods of the infinitesimal calculus; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram.

00(See Infinitesimal Calculus.) REFERENcEs.

00He managed to make practical use of his calculus about his farms, and seems to have been remarkably apt in the practical application of mechanical principles.

00In addition we may also mention his essay on the calculus of variations (Mem.

00Democritean physics without a calculus had necessarily proved sterile of determinate concrete results, and this was more than enough to ripen the naturalism of the utilitarian school into scepticism.

00But from the new point of view its method was inadequate too, its contentment with an induction that merely leaves an opponent silent, when experiment and the application of a calculus were within the possibilities.

00A mathematico-physical calculus that would work was in question.

00With Hobbes logic is a calculus of marks and signs in the form of names.

005 Leibnitz probably owes to him the thought of a calculus of symbols, and the conception of demonstration as essentially a chain of definitions.

004 In the case of non-identical truths, too, there is a priori proof drawn from the notion of the terms, " though it is not always in our power to arrive at this analysis," 5 so that the question arises, specially in connexion with the possibility of a calculus, whether the contingent is reducible to the necessary or identical at the ideal limit.

00The propositions which deal with actual existence are still of a unique type, with whatever limitation to the calculus.

00we have as Leibnitz's remaining legacy to later logicians the conception of Characteristica Universalis and Ars Combinatoria, a universal denoting by symbols and a calculus working by substitutions and the like.

00Leibnitz, fresh from the battle of the calculus in the mathematical field, and with his conception of logic, at least in some of its aspects, as a generalized mathematic,' found a fruitful inspiration, harmonizing well with his own metaphysic, in Bacon's alphabet of nature.

00There is a formal-symbolic logic engaged with the elaboration of a relational calculus.

001853)4 is qualified by the warning that the real activity of thought tends to fall outside the calculus of relations and to attach rather to the subsidiary function of denoting.

00Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k.

00Hamilton, in fact, remarks, 2 " regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse.

00Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions.

00that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also.

00It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a, chain of given length hanging I I between fixed points must be catenary; it determines the scale of the curve, all cate } stationary (~ 9).

00Examples will be found in textbooks of the calculus and of analytical statics.

00In his desire to bring science home to the imperfectly educated he published anonymously Calculus made Easy by " F.R.S."

00Edwards, Differential Calculus, and for geometrical constructions see T.

00In the years 1815-1817 he contributed three papers on the "Calculus of Functions" to the Philosophical Transactions, and in 1816 was made a fellow of the Royal Society.

00Along 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.

00The length of any arc may be determined by geometrical considerations or by the methods of the integral calculus.

00reference may be made to articles Mechanics and Infinitesimal Calculus.

00At the commencement of his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics.

00He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.

00The Principia gives no information on the subject of the notation adopted in the new calculus, and it was not until 1693 that it was com municated to the scientific world in the second volume of Dr Wallis's works.

00Newton's admirers in Holland had informed Dr Wallis that Newton's method of fluxions passed there under the name of Leibnitz's Calculus Di fferentialis.

00The work on [[Trigonometry]] and Double Algebra (1849) contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic calculus.

00De Morgan's other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the [[Essay]] on Probabilities (1838), forming the 107th volume of Lardner's Cyclopaedia, which forms a valuable introduction to the subject; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus (1837).

00Among these may be mentioned the Treatise on the Differential and Integral Calculus (1842); the Elementary Illustrations of the Differential and Integral Calculus, first published in 1832, but often bound up with the larger treatise; the essay, On the Study and Difficulties of Mathematics (1831); and a brief treatise on Spherical Trigonometry (1834).

00p. 118.) Two of his most elaborate treatises are to be found in the [[Encyclopaedia]] metropolitana, namely the articles on the Calculus of Functions, and the Theory of Probabilities.

00Late in 1847 De Morgan published his principal logical treatise, called Formal Logic, or the Calculus of Inference, Necessary and Probable.

00A notion related to that of infinitesimals is presented in the Greek " method of exhaustion "; the more perfect conception, however, only dates from the 17th century, when it led to the infinitesimal calculus.

00A curve came to be treated as a sequence of infinitesimal straight lines; a tangent as the extension of an infinitesimal chord; a surface or area as a sequence of infinitesimally narrow strips, and a solid as a collection of infinitesimally small cubes (See Infinitesimal Calculus).

00Infinitesimal Calculus >>

00At the age of twenty-five he published a treatise on the integral calculus, as a supplement to De l'Hopital's treatise, Des infiniment petits.

00A certain common agreement has been reached concerning the impossibility of regarding pleasure as the sole motive criterion and end of moral action, though different opinions still prevail as to the place occupied by pleasure in the summum bonum, and the possibility of a hedonistic calculus.

00It must be admitted that any intelligent comprehension of the subject requires at least a grasp of the fundamental conceptions of analytical geometry and the infinitesimal calculus, such as only one with some training in these subjects can be expected to have.

00They afford yet another great advantage in that the derivation of the results requires only the analytic operations of the infinitesimal calculus.

00Cavalieri's " indivisibles " into the infinitesimal calculus, all accomplished during the 17th century, immeasurably widened the scope of exact astronomy.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the differential calculus.

00He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.

00Starting from simple elementary propositions, Steiner advances to the solution of problems which analytically require the calculus of variation, but which at the time altogether surpassed the powers of that calculus.

00He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus.

00His great work De maxims et minimis brought him into conflict with Rene Descartes, but the dispute was chiefly due to a want of explicitness in the statement of Fermat (see Infinitesimal Calculus).

00When you are doing calculus you always use radians.

00TAM (Temporal Agent Model) is a formal refinement calculus for real-time systems.

00Database languages: relational algebra, relational calculus, SQL.

00Ultrasonic scalers are used to remove calculus rapidly from the tooth surface.

00Sonic scalers remove calculus from the surface of the tooth.

00Dental scaler Removing the hard calculus from teeth helps eliminate the places where bacteria can lurk in the mouth.

00Calculus, sometimes called tartar, is hardened calcified plaque.

00As plaque ages, it hardens and calcifies to form tartar (called dental calculus).

00Of course, the two are linked through the fundamental theorem of calculus.

00If one used a utilitarian calculus in this your child or your dog?

00The course is also a vehicle for the introduction of theorems in vector calculus that have wide application in physics.

00Course Description This module will extend the vector algebra of the first year to the calculus of three dimensional vectors.

00In calculus class, I learned how to integrate a function.

00Whether you want to learn all about macrame or understand how calculus can change your everyday life, there are online courses for you.

00They offer help in Math, Algebra, Calculus, Statistics, Economics, Finance, Accounting, Physics, Programming, Biology, Chemistry, and Psychology.

00No, sweater vests are the tacit signature look of the chess team captain, the calculus expert, the valedictorian and a young lad who prefers Stephen Hawking to Curious George.

00Among the specific courses offered are options in botany, finite math, calculus, astronomy, geology, classics, and political science.

00Dental calculus is a very hard substance and, as such, is difficult to remove.

00In dentistry, calculus refers to a hardened yellow or brown mineral deposit from unremoved plaque, also called tartar.

00In dentistry, calculus refers to a hardened yellow or brown mineral deposit from unremoved plaque, also called tartar.

00In fact, for some students, the field of poetry is the equivalent of a verbal calculus.

00Bingo games that you create at home can help your children learn everything from basic colors to math, even calculus.

00While still an undergraduate he formed a league with John Herschel and Charles Babbage, to conduct the famous struggle of "d-ism versus dot-age," which ended in the introduction into Cambridge of the continental notation in the infinitesimal calculus to the exclusion of the fluxional notation of Sir Isaac Newton.

01Amongst the most important of his works not already mentioned may be named the following: - Mathematical Tracts (1826) on the Lunar Theory, Figure of the Earth, Precession and Nutation, and Calculus of Variations, to which, in the second edition of 1828, were added tracts on the Planetary Theory and the Undulatory Theory of Light; Experiments on Iron-built Ships, instituted for the purpose of discovering a correction for the deviation of the Compass produced by the Iron of the Ships (1839); On the Theoretical Explanation of an apparent new Polarity in Light (1840); Tides and Waves (1842).

01He 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.

01This latter work included the differential and integral calculus, the calculus of variations, the theory of attractions, and analytical mechanics.

01For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.

01During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.

01pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically.

01At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations.

012 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem."

01In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions.

01The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions.

01The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements.

01In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

01In the extensive calculus of the nth category, we have, first of all, n independent " units," el, e2, ...

01All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

01A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.

01As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.

01If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative.

01

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