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infinitesimal

Good and evil, right and wrong, become infinitesimal, ephemeral matters.

8133The advantages it can offer to its host are, however, infinitesimal when compared with the injury it does it.

1312In mechanics we are specially concerned with the theory of infinitesimal displacements.

1210The former was professor of mathematics at Bologna, and published, among other works, a treatise on the infinitesimal calculus.

1110Originally they existed in infinitesimal fragments, infinite in number and devoid of arrangement.

910Let us conceive the zone in question to be divided into infinitesimal rings of equal area.

88While 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.

811When two legs of my level were on the shore and the third on the ice, and the sights were directed over the latter, a rise or fall of the ice of an almost infinitesimal amount made a difference of several feet on a tree across the pond.

811He 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.

813In 1883 he was chosen to succeed Henry Smith in the Savilian chair of geometry at Oxford, and there he produced his theory of reciprocants, largely by the aid of his "method of infinitesimal variation."

77If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point 0 in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs.

77It 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.

77From the use of gold and silver as a medium of exchange, it followed that they should approximate in all nations to a common degree of fineness; and though this is not uniform even in coins, yet the proportion of alloy in silver, and of carats alloy to carats fine in gold, has been reduced to infinitesimal differences in the bullion of commerce, and is a prime element of value even in gold and silver plate, jewelry, and other articles of manufacture.

54Linlithgowshire yields nearly three-fourths of the total output, Midlothian produces nearly one-fourth, a small quantity is obtained from Lanarkshire, and there is an infinitesimal supply from Sutherland.

54Reference 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.

55The acreage devoted to any other crop is practically infinitesimal, though in the eastern part more attention is paid to fruit-growing than perhaps in any other part of South Wales.

559 is not, strictly j speaking, fixed, but admits of infinitesimal displacement, when ever the directions of the three links are concurrent (or A

43Roberval 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.

44For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.

44For 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.

44Perfectly pure distilled sea-water dissociates, to an infinitesimal degree, into hydrogen (H) and hydroxyl (HO) ions, so that one litre of such water contains 1 X 10 7, or 1 part of a gram-molecule of either hydr010,000,000 gen or hydroxyl (a gramme-molecule of hydrogen is 2 grammes, or of hydroxyl 17 grammes).

44When the applied electromotive force is diminished by an infinitesimal amount, the cell produces a current in the usual direction, and the ordinary chemical changes occur.

44He therefore employed the corresponding expression for a cycle of infinitesimal range dt at the temperature t in which the work dW obtainable from a quantity of heat H would be represented by the equation dW =HF'(t)dt, where F'(t) is the derived function of F(t), or dF(t)/dt, and represents the work obtainable per unit of heat per degree fall of temperature at a temperature t.

44With this definition of temperature 0, if the heat H is measured in work units, the expression of Carnot's principle for an infinitesimal cycle of range do reduces to the simple form dW/d9=H/0.

44Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus.

44made up of a number of successive elements, so that the analytical treatment involves the ideas and the methods of the infinitesimal calculus.

44They afford yet another great advantage in that the derivation of the results requires only the analytic operations of the infinitesimal calculus.

30Under 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.

12A simple approximate calculation of the pressure exerted by a gas on its containing vessel can be made by supposing that the molecules are so small in comparison with their distances apart that they may be treated as of infinitesimal size.

14He also showed that the roots of a cubic equation can be derived by means of the infinitesimal calculus.

14In 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.

14Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.

14In 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.

14Then let us heat both ice and solution through the infinitesimal temperature range dT to the freezing point T of the solvent, melt the ice by the application of an amount of heat L, which measures its latent heat of fusion, and allow the solvent so formed to enter the solution reversibly through a semi-permeable wall into an engine cylinder, doing an amount of work Pdv.

14In these solid solutions, as in aqueous ones, the ratios in which the different chemical substances are present are not fixed or definite, but vary from case to case, not per saltum as between definite chemical compounds, but by infinitesimal steps.

14Pascal also distinguished himself by his skill in the infinitesimal calculus, then in the embryonic form of Cavalieri's method of indivisibles.

14The available energy A is the amount of external work obtainable by an infinitesimal, reversible change in the system which occurs at a constant temperature T.

15For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...

15General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.

15The 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.

15The 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.

15The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.

15When, in order to apply Huygens's principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies O in the plane of reference, and may be considered to be represented thereby.

15The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.'

15Other works of Cournot's were Traite elementaire de la theorie des fonctions et du calcul infinitesimal (1841); Exposition de la theorie des chances et des probabilLites (1843); De l'origine et des limites de la correspondance entre l'algebre et la geometrie (1847); Traite de l'enchainement des idees fondamentales dans les sciences et dans l'histoire (1861); and Revue somma.ire des doctrines economiques (1877).

15Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero.

15of temperature (o' - o") is small, the figure ABCD may be regarded as a parallelogram, and its area W as equal to the rectangle BE XEC. This is accurately true in the limit when (0' - 0") is infinitesimal, but in practice it is necessary to measure specific heats, &c., over finite ranges of temperature, and the error involved is generally negligible if the range does not exceed a few degrees.

15But the line of constant energy on - the diagram does not represent the path of the transformation, unless it be supposed to be effected in a series of infinitesimal steps between each of which the substance is restored to an equilibrium state.

15He 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."

15A minute fraction is always separating out of the water, and as a prodigious length of time may be accepted for the accomplishment of all the chemical and physical processes in the deep sea, we must take account of the gradual accumulation of even this infinitesimal precipitation.

15are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, v, w lie within ranges such that (i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds: (ii) E2mu2 E2mv 2 E2mw2 ?2aie12 s S s s s s - s E s' S' s' - - s' ' See Jeans, Dynamical Theory of Gases (1904), ch.

15Phillips, 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?

00Since the area of the triangle FPP' is one half the product of FP into the perpendicular p from P on FP', it follows that if these perpendiculars were equal all round the orbit, the areas described during the infinitesimal time would be smallest at the pericentre and continually increase during the passage of the body to B.

00The chances of individual stars of the two systems colliding are infinitesimal.

00Moreover, that an infinitesimal translation is analogous to a couple and follows the same laws.

00The displacement will consist of an infinitesimal rotation e about some axis through U, whose direction-cosines are, say, 1, m, n.

00Thus in the case of fig 36 it may be required to connect, the infinitesimal rotations f, i1, l about OA, OB, OC with the variations of the angular co-ordinates 0, ~, ~.

00The analytical calculation of the work done by a system of forces in any infinitesimal displacement is as follows.

00These may be, for instance, the five ratios f: ~: l: X: u v of the six quantities which specify an infinitesimal twist about the screw.

00Considering a rigid body in any given position, we may eontemplate the whole group of infinitesimal displacements which might be given to it.

00The velocities referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements.

00Conversely, we can show that if the, virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium.

00For this purpose the infinitesimal displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.

00It may however happen that owing to some special relation between the lengths of the bars the frame admits of an infinitesimal deformation.

0052, if an infinitesimal deformation is possible without removing the bar CF, the instantaneous centre of CF (when AB is fixed) will be at the intersection of AF and BC, and since CC, FF represent the virtual velocities of the points C, F, turned each through a right angle, CF must be parallel to CF.

00Conversely, if this condition be satisfied, an infinitesimal deformation is possible.

00(13) The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.

00(i6) The factors of ~, i~, ~, X, if, v on the right-hand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on, the same scale) to the work done by the given system of impulsive forces in such a displacement, As in 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse.

00And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.

00Again, if (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is 2~(X~x+Y&y+Z~z) = Qi~gi+Qi~qi+...

00If in (21) we imagine that x, y, I denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of in are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.

00with infinitesimal objects, images and lenses; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

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.

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

00In mathematics, the term " infinite " denotes the result of increasing a variable without limit; similarly, the term " infinitesimal," meaning indefinitely small, denotes the result of diminishing the value of a variable without limit, with the reservation that it never becomes actually zero.

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 >>

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.

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

00- The limiting value, dE/dt, of the coefficient, p, for an infinitesimal difference, dt, between the junctions is called the Thermoelectric Power of the couple.

00The chemical action between metals in the solid state must be infinitesimal, and could only suffice to produce small charges analogous to those of frictional electricity; it could not maintain a permanent difference of potential at a metallic junction through which a current was passing.

00The worker-bee is incapable of egg-production and can therefore take no part in the perpetuation of its species, so that individually its value to the community is infinitesimal.

00These are mainly embodied in his three great treatises, Cours d'analyse de l'Ecole Polytechnique (1821); Le Calcul infinitesimal (1823); Lecons sur les applications du calcul infinitesimal a la g'ometrie (1826-1828); and also in his courses of mechanics (for the Ecole Polytechnique), higher algebra (for the Faculte des Sciences), and of mathematical physics (for the College de France).

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

00Internet players achieved the an infinitesimal of el faro the mobile version.

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

00infinitesimal geometry were equivalent to those used by the ancient Greeks.

00infinitesimal amount of all His good qualities.

00infinitesimal change in the outside temperature.

00infinitesimal operation of the bodily differential.

00Spike catches the almost infinitesimal start Syd gives on seeing the mummies.

00infinitesimal in comparison to e.g. the incidence of Campylobacter - which can actually kill.

00infinitesimal in relation to it.

00mechanizethis to investigate some new geometric relations as well as ways of rigorously mechanizing geometric proofs that involve infinitesimal and infinite arguments.

00These infinitesimal distinctions between man and man are too paltry for an Omnipotent Being.

00potent carcinogen, causing an increase in cancer risk with infinitesimal doses.

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.

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.

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

00Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are 1, m, n, k=lu+mv+nw; (2) and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u', v', w')dt; so that Dk dl, do dt dt dt dt + dtw +1 (?t +u, dx +v, dy +w, dz) +m (d +u dx + v dy +w' dz) dw, dw +n (dt ?dx+v?dy +w dz) But as 1, m, n are the direction cosines of a line fixed in space, dl= m R-n Q, d m = nP-lR an =1Q-mP dt dt ' dt ' so that Dk __ du, du, du, du dt l (dt -vR+ wQ+u + v dy + w dz) +m(..

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.

00Thus, subject to a probability of error which is infinitesimal in the limit, we may state as general laws that A system starting from an abnormal state tends to assume the normal state; while A system starting from the normal state will remain in the normal state.

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) .

00Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially, (18) v(di/dt) =g cos i, where di denotes the infinitesimal decrement of i in the infinitesimal increment of time dt.

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).

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.

00The large depots of stores at Assuan, Halfa and Dongola could only be cursorily supervised by British officers, and yet when the stores were received at the advance depot the losses were infinitesimal.

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.

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).

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).

00The cosmos is filled with energies and we are each one infinitesimal part of that universe.

01Formaldehyde is a known carcinogen, but it's usually present in such small doses that the amount in nail polish is infinitesimal.

01The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements 3t into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.

02A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates.

02

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