(C. We.)
INFINITE(from Lat.in, not,finis, end or limit; cf.findere, to cleave), a term applied in common usage to anything of vast size. Strictly, however, the epithet implies the absence of all limitation. As such it is used specially in (1) theology and metaphysics, (2) mathematics.
1. Tracing the history of the world to the earliest date for which there is any kind of evidence, we are faced with the problem that for everything there is a prior something: the mind is unable to conceive an absolute beginning (“ex nihilo nihil”). Mundane distances become trivial when compared with the distance from the earth of the sun and still more of other heavenly bodies: hence we infer infinite space. Similarly by continual subdivision we reach the idea of the infinitely small. For these inferences there is indeed no actual physical evidence: infinity is a mental concept. As such the term has played an important part in the philosophical and theological speculation. In early Greek philosophy the attempt to arrive at a physical explanation of existence led the Ionian thinkers to postulate various primal elements (e.g.water, fire, air) or simply the infiniteτὸ ἄπειρον(seeIonian School). Both Plato and Aristotle devoted much thought to the discussion as to which is most truly real, the finite objects of sense, or the universal idea of each thing laid up in the mind of God; what is the nature of that unity which lies behind the multiplicity and difference of perceived objects? The same problem, variously expressed, has engaged the attention of philosophers throughout the ages. In Christian theology God is conceived as infinite in power, knowledge and goodness, uncreated and immortal: in some Oriental systems the end of man is absorption into the infinite, his perfection the breaking down of his human limitations. The metaphysical and theological conception is open to the agnostic objection that the finite mind of man is by hypothesis unable to cognize or apprehend not only an infinite object, but even the very conception of infinity itself; from this standpoint the Infinite is regarded as merely a postulate, as it were an unknown quantity (cf. √−1 in mathematics). The same difficulty may be expressed in another way if we regard the infinite as unconditioned (cf. Sir William Hamilton’s “philosophy of the unconditioned,” and Herbert Spencer’s doctrine of the infinite “unknowable”); if it is argued that knowledge of a thing arises only from the recognition of its differences from other things (i.e.from its limitations), it follows that knowledge of the infinite is impossible, for the infinite is by hypothesis unrelated.
With this conception oftheinfinite as absolutely unconditioned should be compared what may be described roughly as lesser infinities which can be philosophically conceived and mathematically demonstrated. Thus a point, which is by definition infinitely small, is as compared with a line a unit: the line is infinite, made up of an infinite number of points, any pair of which have an infinite number of points between them. The line itself, again, in relation to the plane is a unit, while the plane is infinite,i.e.made up of an infinite number of lines; hence the plane is described as doubly infinite in relation to the point, and a solid as trebly infinite. This is Spinoza’s theory of the“infinitely infinite,” the limiting notion of infinity being of a numerical, quantitative series, each term of which is a qualitative determination itself quantitatively little,e.g.a line which is quantitatively unlimited (i.e.in length) is qualitatively limited when regarded as an infinitely small unit of a plane. A similar relation exists in thought between the various grades of species and genera; the highest genus is the “infinitely infinite,” each subordinated genus being infinite in relation to the particulars which it denotes, and finite when regarded as a unit in a higher genus.
2. 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. The application of these conceptions distinguishes ancient from modern mathematics. Analytical investigations revealed the existence of series or sequences which had no limit to the number of terms, as for example the fraction 1/(1 − x) which on division gives the series. 1 + x + x2+ ...; the discussion of these so-called infinite sequences is given in the articlesSeriesandFunction. The doctrine of geometrical continuity (q.v.) and the application of algebra to geometry, developed in the 16th and 17th centuries mainly by Kepler and Descartes, led to the discovery of many properties which gave to the notion of infinity, as a localized space conception, a predominant importance. A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear coordinates. These subjects are treated inGeometry. A 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. 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 (seeInfinitesimal Calculus).
INFINITESIMAL CALCULUS.1. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis. The name “infinitesimal” has been applied to the calculus because most of the leading results were first obtained by means of arguments about “infinitely small” quantities; the “infinitely small” or “infinitesimal” quantities were vaguely conceived as being neither zero nor finite but in some intermediate, nascent or evanescent, state. There was no necessity for this confused conception, and it came to be understood that it can be dispensed with; but the calculus was not developed by its first founders in accordance with logical principles from precisely defined notions, and it gained adherents rather through the impressiveness and variety of the results that could be obtained by using it than through the cogency of the arguments by which it was established. A similar statement might be made in regard to other theories included in mathematical analysis, such, for instance, as the theory of infinite series. Many, perhaps all, of the mathematical and physical theories which have survived have had a similar history—a history which may be divided roughly into two periods: a period of construction, in which results are obtained from partially formed notions, and a period of criticism, in which the fundamental notions become progressively more and more precise, and are shown to be adequate bases for the constructions previously built upon them. These periods usually overlap. Critics of new theories are never lacking. On the other hand, as E. W. Hobson has well said, “pertinent criticism of fundamentals almost invariably gives rise to new construction.” In the history of the infinitesimal calculus the 17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism.
I.Nature of the Calculus.
2. The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines. This method of representing variable quantities dates from the 14th century,Geometrical representation of Variable Quantities.when it was employed by Nicole Oresme, who studied and afterwards taught at the Collège de Navarre in Paris from 1348 to 1361. He represented one of two variable quantities,e.g.the time that has elapsed since some epoch, by a length, called the “longitude,” measured along a particular line; and he represented the other of the two quantities,e.g.the temperature at the instant, by a length, called the “latitude,” measured at right angles to this line. He recognized that the variation of the temperature with the time was represented by the line, straight or curved, which joined the ends of all the lines of “latitude.” Oresme’s longitude and latitude were what we should now call the abscissa and ordinate. The same method was used later by many writers, among whom Johannes Kepler and Galileo Galilei may be mentioned. In Galileo’s investigation of the motion of falling bodies (1638) the abscissa OA represents the time during which a body has been falling, and the ordinate AB represents the velocity acquired during that time (see fig. 1). The velocity being proportional to the time, the “curve” obtained is a straight line OB, and Galileo showed that the distance through which the body has fallen is represented by the area of the triangle OAB.
The most prominent problems in regard to a curve were the problem of finding the points at which the ordinate is a maximum or a minimum, the problem of drawing a tangent to the curve at an assigned point, and the problem ofThe problems of Maxima and Minima, Tangents, and Quadratures.determining the area of the curve. The relation of the problem of maxima and minima to the problem of tangents was understood in the sense that maxima or minima arise when a certain equation has equal roots, and, when this is the case, the curves by which the problem is to be solved touch each other. The reduction of problems of maxima and minima to problems of contact was known to Pappus. The problem of finding the area of a curve was usually presented in a particular form in which it is called the “problem of quadratures.” It was sought to determine the area contained between the curve, the axis of abscissae and two ordinates, of which one was regarded as fixed and the other as variable. Galileo’s investigation may serve as an example. In that example the fixed ordinate vanishes. From this investigation it may be seen that before the invention of the infinitesimal calculus the introduction of a curve into discussions of the course of any phenomenon, and the problem of quadratures for that curve, were not exclusively of geometrical import; the purpose for which the area of a curve was sought was often to find something which is not an area—for instance, a length, or a volume or a centre of gravity.
3. The Greek geometers made little progress with the problem of tangents, but they devised methods for investigating the problem of quadratures. One of these methods was afterwards called the “method of exhaustions,” andGreek methods.the principle on which it is based was laid down in the lemma prefixed to the 12th book of Euclid’sElementsas follows: “If from the greater of two magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the smaller of the proposed magnitudes.” The method adopted by Archimedes was more general. It may be described as the enclosure of the magnitude to be evaluated between two others which can be brought by a definite process to differ from each other by less than any assigned magnitude. A simple example of itsapplication is the 6th proposition of Archimedes’ treatise On theSphere and Cylinder, in which it is proved that the area contained between a regular polygon inscribed in a circle and a similar polygon circumscribed to the same circle can be made less than any assigned area by increasing the number of sides of the polygon. The methods of Euclid and Archimedes were specimens of rigorous limiting processes (seeFunction). The new problems presented by the analytical geometry and natural philosophy of the 17th century led to new limiting processes.
4. In theproblem of tangentsthe new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x + Δx, y + Δy those of P′. The symbol Δx means “the difference of twoDifferentiation.x’s” and there is a like meaning for the symbol Δy. The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted bydy.dxIf the equation of the curve is of the form y = ƒ(x) where ƒ is a functional symbol (seeFunction), thenΔy=ƒ(x + Δx) − ƒ(x),ΔxΔxanddy= lim.Δx = 0ƒ(x + Δx) − ƒ(x).dxΔxThe limit expressed by the right-hand member of this defining equation is often writtenƒ′(x),and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x2, we haveƒ(x + Δx) − ƒ(x)=(x + Δx)2− x2=2xΔx + (Δx)2,ΔxΔxΔxandƒ′(x) = 2x.The process of forming the derived function of a given function is calleddifferentiation. The fraction Δy/Δx is called the “quotient of differences,” and its limitdy/dxis called the “differential coefficient of y with respect to x.” The rules for forming differential coefficients constitute thedifferential calculus.The problem of tangents is solved at one stroke by the formation of the differential coefficient; and the problem of maxima and minima is solved, apart from the discrimination of maxima from minima and some further refinements, by equating the differential coefficient to zero (seeMaxima and Minima).Fig. 3.5. Theproblem of quadraturesleads to a type of limiting process which may be described as follows: Let y = ƒ(x) be the equation of a curve, and let AC and BD be the ordinates of the points C and D (see fig. 3). Let a, b be the abscissae of theseIntegration.points. Let the segment AB be divided into a number of segments by means of intermediate points such as M, and let MN be one such segment. Let PM and QN be those ordinates of the curve which have M and N as their feet. On MN as base describe two rectangles, of which the heights are the greatest and least values of y which correspond to points on the arc PQ of the curve. In fig. 3 these are the rectangles RM, SN. Let the sum of the areas of such rectangles as RM be formed, and likewise the sum of the areas of such rectangles as SN. When the number of the points such as M is increased without limit, and the lengths of all the segments such as MN are diminished without limit, these two sums of areas tend to limits. When they tend to the same limit the curvilinear figure ACDB has an area, and the limit is the measure of this area (see § 33 below). The limit in question is the same whatever law may be adopted for inserting the points such as M between A and B, and for diminishing the lengths of the segments such as MN. Further, if P′ is any point on the arc PQ, and P′M′ is the ordinate of P′, we may construct a rectangle of which the height is P′M′ and the base is MN, and the limit of the sum of the areas of all such rectangles is the area of the figure as before. If x is the abscissa of P, x + Δx that of Q, x′ that of P′, the limit in question might be writtenlim.Σbaƒ(x′) Δx,where the letters a, b written below and above the sign of summation Σ indicate the extreme values of x. This limit is called “the definite integral of ƒ(x) between the limits a and b,” and the notation for it is∫baƒ(x) dx.The germs of this method of formulating the problem of quadratures are found in the writings of Archimedes. The method leads to a definition of a definite integral, but the direct application of it to the evaluation of integrals is in general difficult. Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute theintegral calculus.Fig. 4.6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by theTheorem of Inversion.integral∫ξaƒ(x)dx, and it is a function of ξ. Let BD be displaced to B′D′ so that ξ becomes ξ + δξ (see fig. 4). The area of the figure ACD′B′ is represented by the integral∫ξ+Δξaƒ(x)dx, and the increment ΔA of the area is given by the formulaΔA =∫ξ+Δξξƒ(x) dx,which represents the area BDD′B′. This area is intermediate between those of two rectangles, having as a common base the segment BB′, and as heights the greatest and least ordinates of points on the arc DD′ of the curve. Let these heights be H and h. Then ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we havedA= ƒ(ξ).dξThe introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F(x) which has ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove thatA =∫baƒ(x) dx = F(b) − F(a).When we recognize a function F(x) which has the property expressed by the equationdF(x)= ƒ(x),dxwe are said tointegratethe function ƒ(x), and F(x) is called theindefinite integralof ƒ(x)with respect tox, and is written∫ƒ(x) dx.7. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derivedDifferentials.function ƒ′(x). In general we can write down an equation of the formΔy = ƒ′(x) Δx + R,in which R is different from zero when Δx is different from zero; and then we have not onlylim.Δx=0R = 0,but alsolim.Δx=0R= 0.ΔxWe may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called thedifferentialof ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we havedƒ(x) = ƒ′(x) dx.Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differential of a dependent variable such as y, or of a function of the independent variable x, is the product of the differential of x and the differential coefficient or derived function. It is important to observe that the differential coefficient is not to be defined as the ratio of differentials, but the ratio of differentials is to be defined as the previously introduced differential coefficient. The differentialsare either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients.Again let F(x) be the indefinite integral of a continuous function ƒ(x), so that we havedF(x)= ƒ(x),∫baƒ(x) dx = F(b) − F(a).dxWhen the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x1, x2, ... xn−1be the abscissae of the points in order. The integral is the limit of the sumƒ(a) (x1− a) + ƒ(x1) (x2− x1) + ... + ƒ(xr) (xr+1− xr) + ... + ƒ(xn−1) (b − xn−1),every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences{F(x1) − F(a)} + {F(x2) − F(x1)} + ... + {F(xr+1) − F(xr)} + ... + {F(b) − F(xn−1)},for this sum is F(b) − F(a). Now the difference F(xr+1) − F(xr) isnotequal to the differential ƒ(xr) (xr+1− xr), but the sum of the differences is equal to thelimitof the sum of these differentials. The differential may be regarded as so much of the difference as is required to form the integral. From this point of view a differential is called adifferential element of an integral, and the integral is the limit of the sum of differential elements. In like manner the differential element ydx of the area of a curve (§ 5) is not the area of the portion contained between two ordinates, however near together, but is so much of this area as need be retained for the purpose of finding the area of the curve by the limiting process described.8. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letterNotation.Fundamental Artifice.“d” is the initial letter of the worddifferentia(difference) and the symbol ∫ is a conventionally written “S,” the initial letter of the wordsumma(sum or whole). The notation was introduced by Leibnitz (see §§ 25-27, below).9. The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x + Δx for x and y + Δy for y. If there is a differential coefficient of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx=0(R/Δx) = 0, as in § 7 above. The artifice consists in rejectingab initioall terms of the equation which belong to R. We do not form R at all, but only φ·Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form theelementof an integral in the same way as the element of area y·dx is formed. In fig. 3 of § 5 the element of area y·dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we havelim.MN = 0MN · QR= 0.MNThus the artifice by which differential elements of integrals are formed is in principle the same as that by which differentials are formed without first forming differential coefficients.10. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which areOrders of small quantities.connected together by any relation, and if when x tends to zero y also tends to zero, the fraction y/x may tend to a finite limit. In this case x and y are said to be “of the same order.” When this is not the case we may have eitherlim.x=0x= 0,yorlim.x=0y= 0,xIn the former case y is said to be “of a lower order” than x; in the latter case y is said to be “of a higher order” than x. In accordance with this notion we may say that the fundamental artifice of the infinitesimal calculus consists in the rejection of small quantities of an unnecessarily high order. This artifice is now merely an incident in the conduct of a limiting process, but in the 17th century, when limiting processes other than the Greek methods for quadratures were new, the introduction of the artifice was a great advance.11. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain theRules of Differentiation.rules by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as follows:—(i.) The differential coefficient of aconstantis zero.(ii.) For asumu + v + ... + z, where u, v, ... are functions of x,d(u + v + ... + z)=du+dv+ ... +dz.dxdxdxdx(iii.)For a product uvd(uv)= udv+ vdu.dxdxdx(iv.)For a quotient u/vd(u/v)=(vdu− udv) /v2.dxdxdx(v.) For afunction of a function, that is to say, for a function y expressed in terms of a variable z, which is itself expressed as a function of x,dy=dy·dz.dxdzdxIn addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table:—ydy/dxxnnxn−1for all values of nlogaxx-1logaeaxaxlogeasin xcos xcos x−sin xsin−1x(1 − x2)−1/2tan−1x(1 + x2)−1Each of the formal rules, and each of the particular results in the table, is a theorem of the differential calculus. All functions (or rather expressions) which can be made up from those in the table by a finite number of operations of addition, subtraction, multiplication or division can be differentiated by the formal rules. All such functions are calledexplicitfunctions. In addition to these we haveimplicitfunctions, or such as are determined by an equation containing two variables when the equation cannot be solved so as to exhibit the one variable expressed in terms of the other. We have also functions of several variables. Further, since the derived function of a given function is itself a function, we may seek to differentiate it, and thus there arise the second and higher differential coefficients. We postpone for the present the problems of differential calculus which arise from these considerations. Again, we may have explicit functions which are expressed as the results of limiting operations, or by the limits of the results obtained by performing an infinite number of algebraic operations upon the simple functions. For the problem of differentiating such functions reference may be made toFunction.12. The processes of the integral calculus consist largely in transformationsIndefinite Integrals.of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we have those in the following table:—ƒ(x)∫ƒ(x)dxxnxn+1/ (n + 1)for all values of n except −11/xlogexeaxa−1eaxcos xsin xsin x−cos x(a2− x2)−1/2sin−1(x/a)1 / (a2+ x2)(1/a) tan−1(x/a)The formal rules of § 11 give us means for the transformation of integrals into recognizable forms. For example, the rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms. The rule (iii.) for a product leads to the method of integration by parts. The rule (v.) for a function of a function leads to the method of substitution (see § 48 below.)
4. In theproblem of tangentsthe new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x + Δx, y + Δy those of P′. The symbol Δx means “the difference of twoDifferentiation.x’s” and there is a like meaning for the symbol Δy. The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted by
If the equation of the curve is of the form y = ƒ(x) where ƒ is a functional symbol (seeFunction), then
and
The limit expressed by the right-hand member of this defining equation is often written
ƒ′(x),
and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x2, we have
and
ƒ′(x) = 2x.
The process of forming the derived function of a given function is calleddifferentiation. The fraction Δy/Δx is called the “quotient of differences,” and its limitdy/dxis called the “differential coefficient of y with respect to x.” The rules for forming differential coefficients constitute thedifferential calculus.
The problem of tangents is solved at one stroke by the formation of the differential coefficient; and the problem of maxima and minima is solved, apart from the discrimination of maxima from minima and some further refinements, by equating the differential coefficient to zero (seeMaxima and Minima).
5. Theproblem of quadraturesleads to a type of limiting process which may be described as follows: Let y = ƒ(x) be the equation of a curve, and let AC and BD be the ordinates of the points C and D (see fig. 3). Let a, b be the abscissae of theseIntegration.points. Let the segment AB be divided into a number of segments by means of intermediate points such as M, and let MN be one such segment. Let PM and QN be those ordinates of the curve which have M and N as their feet. On MN as base describe two rectangles, of which the heights are the greatest and least values of y which correspond to points on the arc PQ of the curve. In fig. 3 these are the rectangles RM, SN. Let the sum of the areas of such rectangles as RM be formed, and likewise the sum of the areas of such rectangles as SN. When the number of the points such as M is increased without limit, and the lengths of all the segments such as MN are diminished without limit, these two sums of areas tend to limits. When they tend to the same limit the curvilinear figure ACDB has an area, and the limit is the measure of this area (see § 33 below). The limit in question is the same whatever law may be adopted for inserting the points such as M between A and B, and for diminishing the lengths of the segments such as MN. Further, if P′ is any point on the arc PQ, and P′M′ is the ordinate of P′, we may construct a rectangle of which the height is P′M′ and the base is MN, and the limit of the sum of the areas of all such rectangles is the area of the figure as before. If x is the abscissa of P, x + Δx that of Q, x′ that of P′, the limit in question might be written
lim.Σbaƒ(x′) Δx,
where the letters a, b written below and above the sign of summation Σ indicate the extreme values of x. This limit is called “the definite integral of ƒ(x) between the limits a and b,” and the notation for it is
∫baƒ(x) dx.
The germs of this method of formulating the problem of quadratures are found in the writings of Archimedes. The method leads to a definition of a definite integral, but the direct application of it to the evaluation of integrals is in general difficult. Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute theintegral calculus.
6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by theTheorem of Inversion.integral∫ξaƒ(x)dx, and it is a function of ξ. Let BD be displaced to B′D′ so that ξ becomes ξ + δξ (see fig. 4). The area of the figure ACD′B′ is represented by the integral∫ξ+Δξaƒ(x)dx, and the increment ΔA of the area is given by the formula
ΔA =∫ξ+Δξξƒ(x) dx,
which represents the area BDD′B′. This area is intermediate between those of two rectangles, having as a common base the segment BB′, and as heights the greatest and least ordinates of points on the arc DD′ of the curve. Let these heights be H and h. Then ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we have
The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F(x) which has ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove that
A =∫baƒ(x) dx = F(b) − F(a).
When we recognize a function F(x) which has the property expressed by the equation
we are said tointegratethe function ƒ(x), and F(x) is called theindefinite integralof ƒ(x)with respect tox, and is written
∫ƒ(x) dx.
7. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derivedDifferentials.function ƒ′(x). In general we can write down an equation of the form
Δy = ƒ′(x) Δx + R,
in which R is different from zero when Δx is different from zero; and then we have not only
lim.Δx=0R = 0,
but also
We may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called thedifferentialof ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we have
dƒ(x) = ƒ′(x) dx.
Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differential of a dependent variable such as y, or of a function of the independent variable x, is the product of the differential of x and the differential coefficient or derived function. It is important to observe that the differential coefficient is not to be defined as the ratio of differentials, but the ratio of differentials is to be defined as the previously introduced differential coefficient. The differentialsare either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients.
Again let F(x) be the indefinite integral of a continuous function ƒ(x), so that we have
When the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x1, x2, ... xn−1be the abscissae of the points in order. The integral is the limit of the sum
ƒ(a) (x1− a) + ƒ(x1) (x2− x1) + ... + ƒ(xr) (xr+1− xr) + ... + ƒ(xn−1) (b − xn−1),
every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences
{F(x1) − F(a)} + {F(x2) − F(x1)} + ... + {F(xr+1) − F(xr)} + ... + {F(b) − F(xn−1)},
for this sum is F(b) − F(a). Now the difference F(xr+1) − F(xr) isnotequal to the differential ƒ(xr) (xr+1− xr), but the sum of the differences is equal to thelimitof the sum of these differentials. The differential may be regarded as so much of the difference as is required to form the integral. From this point of view a differential is called adifferential element of an integral, and the integral is the limit of the sum of differential elements. In like manner the differential element ydx of the area of a curve (§ 5) is not the area of the portion contained between two ordinates, however near together, but is so much of this area as need be retained for the purpose of finding the area of the curve by the limiting process described.
8. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letterNotation.Fundamental Artifice.“d” is the initial letter of the worddifferentia(difference) and the symbol ∫ is a conventionally written “S,” the initial letter of the wordsumma(sum or whole). The notation was introduced by Leibnitz (see §§ 25-27, below).
9. The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x + Δx for x and y + Δy for y. If there is a differential coefficient of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx=0(R/Δx) = 0, as in § 7 above. The artifice consists in rejectingab initioall terms of the equation which belong to R. We do not form R at all, but only φ·Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form theelementof an integral in the same way as the element of area y·dx is formed. In fig. 3 of § 5 the element of area y·dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we have
Thus the artifice by which differential elements of integrals are formed is in principle the same as that by which differentials are formed without first forming differential coefficients.
10. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which areOrders of small quantities.connected together by any relation, and if when x tends to zero y also tends to zero, the fraction y/x may tend to a finite limit. In this case x and y are said to be “of the same order.” When this is not the case we may have either
or
In the former case y is said to be “of a lower order” than x; in the latter case y is said to be “of a higher order” than x. In accordance with this notion we may say that the fundamental artifice of the infinitesimal calculus consists in the rejection of small quantities of an unnecessarily high order. This artifice is now merely an incident in the conduct of a limiting process, but in the 17th century, when limiting processes other than the Greek methods for quadratures were new, the introduction of the artifice was a great advance.
11. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain theRules of Differentiation.rules by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as follows:—
(i.) The differential coefficient of aconstantis zero.
(ii.) For asumu + v + ... + z, where u, v, ... are functions of x,
(iii.)For a product uv
(iv.)For a quotient u/v
(v.) For afunction of a function, that is to say, for a function y expressed in terms of a variable z, which is itself expressed as a function of x,
In addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table:—
Each of the formal rules, and each of the particular results in the table, is a theorem of the differential calculus. All functions (or rather expressions) which can be made up from those in the table by a finite number of operations of addition, subtraction, multiplication or division can be differentiated by the formal rules. All such functions are calledexplicitfunctions. In addition to these we haveimplicitfunctions, or such as are determined by an equation containing two variables when the equation cannot be solved so as to exhibit the one variable expressed in terms of the other. We have also functions of several variables. Further, since the derived function of a given function is itself a function, we may seek to differentiate it, and thus there arise the second and higher differential coefficients. We postpone for the present the problems of differential calculus which arise from these considerations. Again, we may have explicit functions which are expressed as the results of limiting operations, or by the limits of the results obtained by performing an infinite number of algebraic operations upon the simple functions. For the problem of differentiating such functions reference may be made toFunction.
12. The processes of the integral calculus consist largely in transformationsIndefinite Integrals.of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we have those in the following table:—
The formal rules of § 11 give us means for the transformation of integrals into recognizable forms. For example, the rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms. The rule (iii.) for a product leads to the method of integration by parts. The rule (v.) for a function of a function leads to the method of substitution (see § 48 below.)
II.History.
13. The new limiting processes which were introduced in the development of the higher analysis were in the first instance related to problems of the integral calculus. Johannes Kepler in hisAstronomia nova ... de motibus stellae MartisKepler’s methods of Integration.(1609) stated his laws of planetary motion, to the effect that the orbits of the planets are ellipses with the sun at a focus, and that the radii vectores drawn from the sun to the planets describe equal areas in equal times. From these statements it is to be concluded that Kepler could measure the areas of focal sectors of an ellipse. When he made out these laws there was no method of evaluating areas except the Greek methods. These methods would have sufficed for the purpose, but Kepler invented his own method. He regarded the area as measured by the “sum of the radii” drawn from the focus, and he verified his laws of planetary motion by actually measuring a large number of radii of the orbit, spaced according to a rule, and adding their lengths.
He had observed that the focal radius vector SP (fig. 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the auxiliary circle, and he had further established the theorem which we should now express in the form—the differential element of the area ASp as Sp turns about S, is equal to the product of SZ and the differential adφ, where a is the radius of the auxiliary circle, and φ is the angle ACp, that is the eccentric angle of P on the ellipse. The area ASP bears to the area ASp the ratio of the minor to the major axis, a result known to Archimedes. Thus Kepler’s radii are spaced according to the rule that the eccentric angles of their ends are equidifferent, and his “sum of radii” is proportional to the expression which we should now write∫φ0(a + ae cos φ) dφ,where e is the eccentricity. Kepler evaluated the sum as proportional to φ + e sin φ.
He had observed that the focal radius vector SP (fig. 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the auxiliary circle, and he had further established the theorem which we should now express in the form—the differential element of the area ASp as Sp turns about S, is equal to the product of SZ and the differential adφ, where a is the radius of the auxiliary circle, and φ is the angle ACp, that is the eccentric angle of P on the ellipse. The area ASP bears to the area ASp the ratio of the minor to the major axis, a result known to Archimedes. Thus Kepler’s radii are spaced according to the rule that the eccentric angles of their ends are equidifferent, and his “sum of radii” is proportional to the expression which we should now write
∫φ0(a + ae cos φ) dφ,
where e is the eccentricity. Kepler evaluated the sum as proportional to φ + e sin φ.
Kepler soon afterwards occupied himself with the volumes of solids. The vintage of the year 1612 was extraordinarily abundant, and the question of the cubic content of wine casks was brought under his notice. This fact accounts for the title of his work,Nova stereometria doliorum; accessit stereometriae Archimedeae supplementum(1615). In this treatise he regarded solid bodies as being made up, as it were (veluti), of “infinitely” many “infinitely” small cones or “infinitely” thin disks, and he used the notion of summing the areas of the disks in the way he had previously used the notion of summing the focal radii of an ellipse.
14. In connexion with the early history of the calculus it must not be forgotten that the method by which logarithms were invented (1614) was effectively a method of infinitesimals. Natural logarithms were not inventedLogarithms.as the indices of a certain base, and the notation e for the base was first introduced by Euler more than a century after the invention. Logarithms were introduced as numbers which increase in arithmetic progression when other related numbers increase in geometric progression. The two sets of numbers were supposed to increase together, one at a uniform rate, the other at a variable rate, and the increments were regarded for purposes of calculation as very small and as accruing discontinuously.
15. Kepler’s methods of integration, for such they must be called, were the origin of Bonaventura Cavalieri’s theory of the summation of indivisibles. The notion of a continuum, such as the area within a closed curve,Cavalieri’s Indivisibles.as being made up of indivisible parts, “atoms” of area, if the expression may be allowed, is traceable to the speculations of early Greek philosophers; and although the nature of continuity was better understood by Aristotle and many other ancient writers yet the unsound atomic conception was revived in the 13th century and has not yet been finally uprooted. It is possible to contend that Cavalieri did not himself hold the unsound doctrine, but his writing on this point is rather obscure. In his treatiseGeometria indivisibilibus continuorum nova quadam ratione promota(1635) he regarded a plane figure as generated by a line moving so as to be always parallel to a fixed line, and a solid figure as generated by a plane moving so as to be always parallel to a fixed plane; and he compared the areas of two plane figures, or the volumes of two solids, by determining the ratios of the sums of all the indivisibles of which they are supposed to be made up, these indivisibles being segments of parallel lines equally spaced in the case of plane figures, and areas marked out upon parallel planes equally spaced in the case of solids. By this method Cavalieri was able to effect numerous integrations relating to the areas of portions of conic sections and the volumes generated by the revolution of these portions about various axes. At a later date, and partly in answer to an attack made upon him by Paul Guldin, Cavalieri published a treatise entitledExercitationes geometricae sex(1647), in which he adapted his method to the determination of centres of gravity, in particular for solids of variable density.
Among the results which he obtained is that which we should now write∫x0xmdx =xm+1, (m integral).m + 1He regarded the problem thus solved as that of determining the sum of the mth powers of all the lines drawn across a parallelogram parallel to one of its sides.
Among the results which he obtained is that which we should now write
He regarded the problem thus solved as that of determining the sum of the mth powers of all the lines drawn across a parallelogram parallel to one of its sides.
At this period scientific investigators communicated their results to one another through one or more intermediate persons. Such intermediaries were Pierre de Carcavy and Pater Marin Mersenne; and among the writers thusSuccessors of Cavalieri.in communication were Bonaventura Cavalieri, Christiaan Huygens, Galileo Galilei, Giles Personnier de Roberval, Pierre de Fermat, Evangelista Torricelli, and a little later Blaise Pascal; but the letters of Carcavy or Mersenne would probably come into the hands of any man who was likely to be interested in the matters discussed. It often happened that, when some new method was invented, or some new result obtained, the method or result was quickly known to a wide circle, although it might not be printed until after the lapse of a long time. When Cavalieri was printing his two treatises there was much discussion of the problem of quadratures. Roberval (1634) regarded an area as made up of “infinitely” many “infinitely” narrow strips, each of which may be considered to be a rectangle, and he had similar ideas in regard to lengths and volumes. He knew how to approximate to the quantity which we express by∫10xmdx by the process of forming the sum
and he claimed to be able to prove that this sum tends to 1/(m + 1), as n increases for all positive integral values of m. The method of integrating xmby forming this sum was found also by Fermat (1636), who stated expressly that heFermat’s method of Integration.arrived at it by generalizing a method employed by Archimedes (for the cases m = 1 and m = 2) in his books onConoids and Spheroidsand onSpirals(see T. L. Heath,The Works of Archimedes, Cambridge, 1897). Fermat extended the result to the case where m is fractional (1644), and to the case where m is negative. This latter extension and the proofs were given in his memoir,Proportionis geometricae in quadrandis parabolis et hyperbolis usus, which appears to have received a final form before 1659, although not published until 1679. Fermat did not use fractional or negative indices, but he regarded his problems as the quadratures of parabolas and hyperbolas of various orders. His method was to divide the interval of integration into parts by means of intermediate points the abscissae of which are in geometric progression. In the process of § 5 above, the points M must be chosen according to this rule. This restrictive condition being understood, we may say that Fermat’s formulation of the problem of quadratures is the same as our definition of a definite integral.
The result that the problem of quadratures could be solved for any curve whose equation could be expressed in the form
y = xm(m ≠ −1),
or in the form
y = a1xm1+ a2xm2+ ... + anxmn,
where none of the indices is equal to −1, was used by JohnVarious Integrations.Wallis in hisArithmetica infinitorum(1655) as well as by Fermat (1659). The case in which m = −1 was that of the ordinary rectangular hyperbola; and Gregory of St Vincent in hisOpus geometricum quadraturae circuli et sectionum coni(1647) had proved by the method of exhaustions that the area contained between the curve, one asymptote, and two ordinates parallel to the other asymptote, increases in arithmetic progression as the distance between the ordinates (the one nearer to the centre being kept fixed) increases in geometric progression. Fermat described his method of integration as a logarithmic method, and thus it is clear that the relation between the quadrature of the hyperbola and logarithms was understood although it was not expressed analytically. It was not very long before the relation was used for the calculation of logarithms by Nicolaus Mercator in hisLogarithmotechnia(1668). He began by writing the equation of the curve in the form y = 1/(1 + x), expanded this expression in powers of x by the method of division, and integrated it term by term in accordance with the well-understood rule for finding the quadrature of a curve given by such an equation as that written at the foot of p. 325.
By the middle of the 17th century many mathematicians could perform integrations. Very many particular results had been obtained, and applications of them had beenIntegration before the Integral Calculus.made to the quadrature of the circle and other conic sections, and to various problems concerning the lengths of curves, the areas they enclose, the volumes and superficial areas of solids, and centres of gravity. A systematic account of the methods then in use was given, along with much that was original on his part, by Blaise Pascal in hisLettres de Amos Dettonville sur quelques-unes de ses inventions en géométrie(1659).
16. The problem of maxima and minima and the problem of tangents had also by the same time been effectively solved. Oresme in the 14th century knew that at a point where the ordinate of a curve is a maximum or a minimumFermat’s methods of Differentiation.its variation from point to point of the curve is slowest; and Kepler in theStereometria doliorumremarked 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 possession of a method which he then communicated to one Despagnet of Bordeaux, and which he referred to in a letter to Roberval of 1636. He communicated it to René Descartes early in 1638 on receiving a copy of Descartes’sGéométrie(1637), and with it he sent to Descartes an account of his methods for solving the problem of tangents and for determining centres of gravity.