# Calculus Sentence Examples

- But no carefully devised
**calculus**can take the place of insight, observation and experience. - The 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. - In 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. - Hence, early empiricism makes ethics simply a
**calculus**of pleasures ("hedonism"). - This 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). - Reference 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. - In 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. - While 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. - Lacroix's Differential
**Calculus**in 1816. - Amongst 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). - He at once took a leading position in the mathematical teaching of the university, and published treatises on the Di f ferential
**calculus**(in 1848) and the Infinitesimal**calculus**(4 vols., 1852-1860), which for long were the recognized textbooks there. - This latter work included the differential and integral
**calculus**, the**calculus**of variations, the theory of attractions, and analytical mechanics. - For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL
**CALCULUS**; NUMBER; QUATERNIONS; VECTOR ANALYSIS. - For 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. - During 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. - Pp. 8 0 -94, 95112) showed by his
**calculus**of hyper-determinants that an infinite series of such functions might be obtained systematically. - 2 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." - In 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. - The 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. - In analytical invention, and mastery over the
**calculus**, the Turin mathematician was admittedly unrivalled. - The
**calculus**of variations is indissolubly associated with his name. - To 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. - In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful
**calculus**occupies a prominent place. - A 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. - If 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. - From 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. - In 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. - In the extensive
**calculus**of the nth category, we have, first of all, n independent " units," el, e2, ... - All 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. - A 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. - As 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. - If, 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. - For 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. - Buchheim, on Extensive
**Calculus**and its Applications, Proc. L. - By 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. - By the methods of the differential
**calculus**or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant. - He made use of the same suppositions as Daniel Bernoulli, though his
**calculus**was established in a very different manner. - This
**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. - Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the "
**calculus**of finite differences." - In 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. - Substituting for H its value from (3), and employing the notation of the
**calculus**, we obtain the relation S - s =0 (dp /do) (dv/do),. - 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. - The 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. - Among 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. - Meanwhile 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. - He 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." - 1875); 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). - It is not, however, necessary that the notation of the
**calculus**should be employed throughout. - The general method of constructing formulae of this kind involves the use of the integral
**calculus**and of the**calculus**of finite differences. - Roberval 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. - At 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. - The
**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.