# How to use *Infinitesimal* in a sentence

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.In mechanics we are specially concerned with the theory of

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

**infinitesimal**calculus.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.The 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.AdvertisementFrom 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.Linlithgowshire 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.Perfectly 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).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.The chances of individual stars of the two systems colliding are

**infinitesimal**.AdvertisementIt thus appears that an

**infinitesimal**rotation is of the nature of a localized vector, and is subject in all respects to the same mathematical Jaws as a force, conceived as acting on a rigid body.The 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**.Moreover, that an

**infinitesimal**translation is analogous to a couple and follows the same laws.The displacement will consist of an

**infinitesimal**rotation e about some axis through U, whose direction-cosines are, say, 1, m, n.The analytical calculation of the work done by a system of forces in any

**infinitesimal**displacement is as follows.AdvertisementConsidering a rigid body in any given position, we may eontemplate the whole group of

**infinitesimal**displacements which might be given to it.The 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.Conversely, 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.For this purpose the

**infinitesimal**displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.It may however happen that owing to some special relation between the lengths of the bars the frame admits of an

**infinitesimal**deformation.AdvertisementConversely, if this condition be satisfied, an

**infinitesimal**deformation is possible.And a continuous force may be regarded as the limit of a succession of

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

**infinitesimal**calculus.In 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.AdvertisementA 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.It 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.Cavalieri's " indivisibles " into the

**infinitesimal**calculus, all accomplished during the 17th century, immeasurably widened the scope of exact astronomy.The 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.These 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).AdvertisementThe term " derivative, taken from

**infinitesimal**calculus, refers to an isolated aspect, or " function " of a real quantity.Internet players achieved the an

**infinitesimal**of el faro the mobile version.Spike catches the almost

**infinitesimal**start Syd gives on seeing the mummies.These

**infinitesimal**distinctions between man and man are too paltry for an Omnipotent Being.Arithmetic, algebra, and the

**infinitesimal**calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers.AdvertisementThis is a particular case of Taylor's theorem (see

**Infinitesimal**Calculus).Thus, 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.In 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) .Resolving 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.The 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).AdvertisementThe 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**.Leibnitz (see

**Infinitesimal**Calculus).The 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).The 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.A 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).AdvertisementHis 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).The advantages it can offer to its host are, however,

**infinitesimal**when compared with the injury it does it.For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF;

**INFINITESIMAL**CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.When 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.He 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.AdvertisementWith 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.Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the

**infinitesimal**calculus.In 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."Originally they existed in

**infinitesimal**fragments, infinite in number and devoid of arrangement.If 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.The cosmos is filled with energies and we are each one

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

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

**infinitesimal**calculus.Let us conceive the zone in question to be divided into

**infinitesimal**rings of equal area.The 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.A 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.A 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.He also showed that the roots of a cubic equation can be derived by means of the

**infinitesimal**calculus.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.Discussions of the approximate calculation of definite integrals will be found in works on the

**infinitesimal**calculus; see e.g.In 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.Then 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.Pascal also distinguished himself by his skill in the

**infinitesimal**calculus, then in the embryonic form of Cavalieri's method of indivisibles.When 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.For 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,...General aspects of the subject are considered under Mensuration; Vector Analysis;

**Infinitesimal**Calculus.The 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.The 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.When, 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.The 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.'Hence 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.But 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.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."A 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.In 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.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.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.