FUNCTION . 12. The processes of the integral calculus consist largely in trans-formations of the functions to be integrated into such fndef.+nite forms that they can be recognized as See also:

• DIFFERENTIAL

. . de motibus stellae of Integra- Martis (16og) stated his See also:

• LAWS

In this See also:

• TREATISE

At a later date, and partly in See also:

• ANSWER (derived from and, against, and the same root as swear)

L. See also:

• HEATH
• HEATH, BENJAMIN (1704-1766)
• HEATH, WILLIAM (1737—1814)

The case in which m=—1 was that of the various See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

Ind Kepler in the Stereofnetria doliorum remarked that at the places where the ordinate passes from a smaller value to the greatest value and then again to a smaller value, its variation becomes insensible. Fermat in 1629 was in See also:

• POSSESSION
• POSSESSION (Lat. possessio, possidere, to possess)

Omitting the See also:

• FACTOR (from Lat. facere, to make or do)

17. Among the comparatively few mathematicians who before 1665 could perform differentiations was See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

If the quadrature of a curve y =f(x) is known, so that the area up to the ordinate x is given by F(x), the curve Barrow's y = F(x) can be drawn, and Barrow showed that the theorsion. subtangent of this curve is measured by the ratio of theorem. its ordinate to the ordinate of the original curve. The curve y=F(x) is often called the " See also:

• QUADRATRIX (from Lat. quadrator, squarer)

By these steps the problem of tangents could be solved once for all, and the operation of integration, as we See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

20. The See also:

• TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

This theorem he had discovered; but he did not in this tract See also:

• STATE
• STATE, GREAT OFFICERS OF

In the case of irrational functions, or rational f unctions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only. Thus Newton's rules of differentiation would be in our notation the rules (i.), (ii.), (v.) -of § together with the particular result which we write (Ix"' dx =mx"-1 (m integral). a result which Newton obtained by expanding (x+io)'" by the binomial theorem. The second problem is the problem of integration, and Newton's method for solving it was the method of series founded upon the particular result which we write f x"dx=m+1. Newton added applications of his methods to maxima and minima, tangents and curvature. In a letter to Collins of date 1672 Newton stated that he had certain methods, and he described certain results which he had found by using them. These methods and results are those which are to be found in the Methodus fluxionum; but the letter makes no mention of fluxions and fluents or of the characteristic notation. The rule for tangents is said in the letter to be analogous to de Sluse's, but to be applicable to equations that contain irrational terms. 22. Newton gave the fluxional notation also in the tract De Quadratura curvarum (1676), and he there added to it notation for the higher differential coefficients and for indefinite integrals, as we call them. Just as x, y, z, . . . are fluents of which .x, 9, z, .

. are the fluxions, so z, y, z, . can be treated as fluents of which the fluxions may be denoted by x, 9, 2', . . . In like manner the fluxions of these may be denoted by y, y, z, . and so on. Again x, y, z, . . may be regarded as fluxions of which the fluents maybe denoted by x, y, i and these again as fluxions of other quantities denoted by x, y, z, . . . and so on. No use was made of the notation c, 3, . . . in the course of the tract. The first publication of the fluxional notation was made by Wallis in the second edition of his See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

In the Philosophiae naturalis principia mathematica (1687), commonly called the " Principia," the words " fluxion " and " moment " occur in a lemma in the second book; but the notation which is characteristic of the calculus of fluxions is nowhere used. Nature of the discovery called the tesimal Calculus. by omitting the equal terms z" and c"x5 and dividing the remaining terms by o, tacitly putting o =--o after division. This relation is the same as v=x'". Newton pointed out that, conversely, from the relation v=xm the relation z=x'n+1/(m-F1) follows. He applied his formula to the quadrature of curves whose ordinates can be expressed as the sum of a finite number of terms of the form ax'"; and gave examples of its application to curves in which the ordinate is expressed x,"+1 Publication of the Fluxional Notation. Retarded Publication of the method of Fluxions. 23. It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

25. In See also:

• JULY

In neither of the letters to Oldenburg does the characteristic notation of the fluxional calculus occur, and the words " fluxion " and " fluent occur only in anagrams of transposed letters. The letter of October 1676 was not despatched until May 1677, and Leibnitz answered it in See also:

• JUNE

In connexion with the problem of tangents the differentials are said to be proportional to the momentary the project with regard to fluxions. In 168o Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in x682. Yet it remained unpublished. The See also:

• REASON (Lat. ratio, through French raison)

On returning to See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

A greater novelty was the use of a letter (d), not as a symbol for a number or magnitude, but as a symbol of operation. None of these novelties account for the far-reaching effect which this memoir has had upon the development of mathematical analysis. This effect was a consequence of the simplicity and directness with which the rules of differentiation were stated. Whatever indistinctness might be felt to attach to the symbols, the processes for solving problems of tangents and of maxima and minima were reduced once for all to a definite routine. 27. This memoir was followed in 1686 by a second, entitled De Geometria recondita et analysi indivisibilium atque infinitorum, Develop- in which Leibnitz described the method of using his ment new differential calculus for the problem of quadratures. of the This was the first publication of the notation fydx. calculus. The new method was called calculus summatorius. The See also:

• BROTHERS, RICHARD (1757-1824)

Leibnitz (1695) was the first to differentiate a See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

The memoir was published in 1710. Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans See also:

• SLOANE, SIR HANS (1660-1753)

A second edition of the Commercium Epistolicum was published in 1722. The dispute was continued for many years after the death of Leibnitz in 1716. To translate the words of See also:

• MORITZ, KARL PHILIPP (1757-1793)

Christiaan Huygens, whose opinion carried more See also:

• WEIGHT

See also:

• GIBSON, CHARLES DANA (1867- )
• GIBSON, EDMUND (1669-1748)
• GIBSON, JOHN (1790-1866)
• GIBSON, THOMAS MILNER (1806-1884)
• GIBSON, WILLIAM HAMILTON (1850-1896)

It was not until 1823 that a treatise on the differential calculus founded upon the method of limits was published. The treatise was the Resume des See also:

• MONS (Flemish Bergen)

ItS general tendency has been to show that many theories and processes, at first accepted as of general validity, are liable to exceptions, and much of the work of the analysts of the latter See also:

• HALF

Sometimes additional conditions are imposed upon the functional relations before they are regarded as defining a curve. An aggregate of points which satisfies the two conditions stated R R above is sometimes called a " See also:

• JORDAN (the down-comer; Arab. esh-Sheri'a, the watering-place)
• JORDAN, CAMILLE (1771—1821)
• JORDAN, DOROTHEA (1762—1816)
• JORDAN, THOMAS (1612 ?–1685)
• JORDAN, WILHELM (1819–1904)

If e/fideats. the velocity is changed suddenly the progressive differ- ential coefficient See also:

• MEASURES

R,,_1 a set of intermediate points of the curve, supposed to be described in the sense in which Q comes after P. The points R are supposed to be reached successively in the order of the suffixes when the curve is described in this sense. We form a sum of lengths of chords PRI+RIR2-1- ... +R"-IQ. If this sum has a limit when the number of the points R is increased indefinitely and the lengths of all the chords are diminished indc-Lengths finitely, this limit is the length of the arc PQ. The limit of Curves. is the same whatever law may be adopted for inserting the intermediate points R and diminishing the lengths of the chords It appears from this statement that the differential element cf the arc of a curve is the length of the chord joining two neighbouring points. In accordance with the fundamental artifice for forming differentials (§§ 9, 10), the differential element of arc ds may be expressed by the formula ds = 1(dx)2+(dy)'l, of which the right-hand member is really the measure of the distance between two neighbouring points on the tangent. The square root must be taken to be positive. We may describe this differential element as being so much of the actual arc between two neighbouring points as need.be retained for the purpose of forming the integral expression for an arc. This is a description, not a definition, because the length of the short arc itself is only definable by means of the integral expression. Similar considerations to those used in defining the areas of plane figures and the lengths of plane curves are applicable to the formation of expressions for differential elements of See also:

• VOLUME

In regard to differential coefficients it is an important theorem that, if the derived function f'(x) vanishes at all points of an interval, Constants the function f(x) is constant in the interval. It follows oflnte- that, if two functions have the same derived function gration. they can only differ by a constant. Conversely, indefinite integrals are indeterminate to the extent of an additive constant. 35. The differential coefficient dy/dx, or the derived function f'(x), is itself a function of x, and its differential coefficient is denoted Nigher by f"(x) or d2y/dx2. In the second of these notations Dighea- d/dx is regarded as the symbol of an operation, that of tiaiCe differentiation with respect to x, and the See also:

• INDEX

Differentials of higher orders may be introduced in the same way as the differential of the first order. In general when y=f(x), the nth differential d"y is defined by the equation d"y=f(")(x) (dx)", in which dx is the (arbitrary) differential of x. When d/dx is regarded as a single symbol of operation the symbol f...dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D-i. D" means that symbols the operation D is to be performed n times in See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of inter-mediate value (" theorem of mean Theorem value," " theorem of finite incre- of Interments," " Rolle's theorem," are mediate other names for it). This theorem value. may be explained as follows: Let A, B be two points of a curve y=f(x) (fig. 9). Then there is a point P between A and B at which the tangent is parallel to the secant AB. This theorem is expressed analytically in the statement that if f'(x) is continuous between a and b, there is a value xi of x between a and b which has the property expressed by the equation f(b) -f(¢).= f (xi). (i.) b-a The value xi can be expressed in the form a+0(b-a) where 0 is a number between o and 1. A slightly more general theorem was given by Cauchy (1823) to the effect that, if f'(x) and F'(x) are continuous between x=a and x = b, then there is a number 0 between o and 1 which has the property expressed by the equation F(h) - F(a) =F'}a+0(b - a)} f(b)-f(a) f'{a+0(b-a)}- The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where f(x) is a rational integral function which vanishes when x=a and also when x=b. The general theorem was given by Lagrange (1997). Its fundamental importance was first recognized by Cauchy (1823). It may be observed here that the theorem of integral calculus expressed by the equation F (b) F(a) =f y' (x)dx follows at once from the definition of an integral and the theorem of intermediate value. The theorem of intermediate value may be generalized in the statement that, if f(x) and all its differential coefficients up to the nth inclusive are continuous in the interval betweenx=a andx fib, then there is a number 0 between o and i which has the property expressed by the equation f(b)=f(a)+(b-a)f'(a)+(b ¢)f"(a) +...

+(b-a)"i f(" "(a) 2 (n- I)! .+(bn) f"){a+0(b-a)l. (i.) 37. This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned Taylor's point are known. The function is expressed by a termin- Theorem. ated series, and, when the See also:

• REMAINDER, REVERSION

See also:

• PRINGSHEIM, NATHANAEL (1823-1894)

The series for log (i+x) was obtained by Nicolaus Mercator (1668) the Expas- division, nand i tegrating the series m e mdbyf term. He slims in regarded his result as a " quadrature of the hyperbola." power Newton (1669) obtained the expansion of sin 'x by ex- serlea, panding (1- x2)-i by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan'x. Newton also obtained the series for sin x, cos x, and ex by reversion of series (1669). The symbol e for the base of the Napierian logarithms was introduced by Euler (1739). All these series can be obtained at once by Taylor's theorem. James Gregory found also the first few terms of the series for tan x and sec x; the terms of these series may be found successively by Taylor's theorem, but the numerical coefficient of the general term cannot be obtained in this way. Taylor's theorem for the expansion of a function in a power series was the basis of Lagrange's theory of functions, and it is fundamental also in the theory of See also:

• ANALYTIC (the adjective of " analysis," q.v.)

. . called Bernoulli's numbers, are the coefficients so denoted in the formula esx = 1 -2+Bix2—4.x4+-at —..., and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type (2n)! I 1 f " 2.n—1,2n12n 22n 32n ~- " The function x"'— 22~ IBix"''2—.. . has been called Bernoulli's function of the nth order by J. L. See also:

• RAABE, HEDWIG (1844-1905)
• RAABE, WILHELM (1831-1910)

This problem is that of reversion of series. It can be shown that provided the See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

This limit is described as the limit of an informs. " indeterminate form." Such indeterminate forms were forms. considered first by de 1'Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y=0(x)/¢(x) near the ordinate x=a, when the curves y=0(x) and y=il,(x) both See also:

• CROSS

Other indeterminate forms are presented in the form o x co , or 1 a0 , or oo /oo , or oo co . The most notable of the forms See also:

• LOO (formerly called " Lanterloo," Fr. lanturlu, the refrain of a popular 17th-century song)

Pringsheim in Ency. d. math. Wiss. Bd. ii., A. , (1890.. Forms of the type 0/0 presented themselves to early writers on analytical geometry in connexion with the determination of the tangents at a See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

39. The nature of a function of two or more variables, and the meaning. to be attached to continuity and limits in respect of such Fenetioes functions, have been explained under FUNCTION. The Functi ", theorems of differential calculus which relate to such variables. functions are in general the same whether the number of variables is two or any greater number, and it will generally be convenient to state the theorems for two variables. 40. Let u or f (x, y) denote a function of two variables x and y. If we regard y as constant, u or f becomes a function of one variable x, and we may seek to differentiate it with respect to x. Partial If the function of x is differentiable, the differential differen- coefficient which is formed in .this way is called the nation. ,, partial differential coefficient " of u or f with respect to z, and is denoted by au or ax. The symbol " a" was appropriated for partial differentiation by C. G. J. See also:

• JACOBI, FRIEDRICH HEINRICH (1743-1819)
• JACOBI, JOHANN GEORG (1740-1814)
• JACOBI, KARL GUSTAV JACOB (1804-1851)

It had before been written indifferently with " d " as a symbol of differentiation. Euler had written " (dx) " for the partial differential coefficient of f with respect to x. Sometimes it is desirable to put in evidence the variable which is treated as constant, and then the partial differential coefficient is written " (4x) " or " (fax) ". This course is often 9 N adopted by writers on See also:

• THERMODYNAMICS (from Gr. Bepµos, hot, Siva trs, force)

The value of F(x, y, z) at P may be denoted shortly by F(P). A limit of the same nature as a partial differential coefficient is expressed by the formula lim.,~~ (P ) As F(P)' in which As is diminished indefinitely by bringing P' up to P, and P' is supposed to approach P along a straight line, for example, the tangent to a curve or the normal to a surface. The limit in question is denoted by aF/Oh, in which it is understood that h indicates a direction, that of PP'. If 1, m, n are the direction cosines of the limiting direction of the line PP', supposed drawn from P to P', then OF aF OF aF Oh -lax +may +naz ' The operation of forming OF/ah is called "differentiation with respect to an axis " or " vector differentiation." 41. The most important theorem in regard to partial differential coefficients is the theorem of the See also:

• TOTAL

We have at once dy of dx= ax f ay' This rule was known, in all essentials, to Fermat and de Sluse before the invention of the algorithm, of the differential calculus. An important theorem, first proved by Euler, is immediately deducible from the theorem of the total differential. If See also:

• FIX, THEODORE (180o-1846)

319). The Jacobian of Ui, us,. . . u„ with respect to xi, x2, . . . x„ is the See also:

• DETERMINANT

, Y„)-a(u1, 142,.. , us) a(yi, y2, ... , ys) a(x1, x2,. .. , "ixs) 0(x1, xs... , xs )' If n functions (u1, us, . . . us) of it variables (xi, x2, . . . , x,,) are not independent, but are connected by a relation f(u1, us, . . . us) =o, then a(ui, 2i2, • , u„) -o a(xi, x2, • , x,,) and, conversely, when this condition is satisfied identically the functions al, 142, . it,, are not independent. 43. Partial differential coefficients of the second and higher orders can be formed in the same way as those of the first order.

For example, when there are two variables x, y, the first partial derivatives of/ax and of/ay are functions of x and y, which we may seek to differentiate partially with respect to x or y. The most important theorem in relation to partial differential coefficients of orders higher than the first is the theorem that the values of such coefficients do not depend upon the order in which the differentiations are performed. For example, we have the equation a (ay) ay (a) (i.) This theorem is not true without See also:

• LIMITATION, STATUTES OF

The differential coefficient anf axPay2azr' in which p+q+r=n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms as r.9.r> Differentials of higher orders are introduced by the defining equation dnf = (dxax+dyay) f anf = (dx)"axa+n(dx)n-idyaxn_iay+ .. . in which the expression (dx -+dya) n is developed by the binomial Y theorem in the same way as if dxax and dyay were numbers, and (a ) r (ay) n f is replaced by ax*af • When there are more than two variables the multinomial theorem must be used instead of the binomial theorem. The problem of forming the second and higher differential coefficients of implicit functions can be solved at once by means of partial differential coefficients. For example, if f (x, y) =o is the equation defining y as a function of x, we have f y_ f 3 2 _2 2 dx2 - ( ay) i \ay) axe ax ay axay+ \ax) aye The differential expression Xdx+Ydy, in which both X and Y are functions of the two variables x and y, is a total differential if there exists a function f of x and y which is such that of/ax=X, of/ay=Y. When this is the case we have the relation aY/ax = aX /ay. (ii.) Conversely, when this equation is satisfied there exists a function f which is such that df =Xdx+Ydy. The expression Xdx+Ydy in which X and Y are connected by the relation (ii.) is often described as a " perfect differential." The theory of the perfect differential can be extended to functions of n variables, and in this case there are ;n(n-1) such relations as (ii.). In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we write az az s 02z p, q, r, s, t for ax' ay axe' axay' aye' Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces.