# 1911 Encyclopædia Britannica/Infinitesimal Calculus/Nature of the Calculus

Infinitesimal Calculus (§1) | Infinitesimal Calculus I. Nature of the Calculus (§2-12) |
II. History (§13-22) |

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,
when it was employed by Nicole Oresme, who studied
Geometrical representation of Variable Quantities.
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.

Fig. 1. |

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 of determining the area of the curve. The relation of The problems of Maxima and Minima, Tangents, and Quadratures. 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,” and
the principle on which it is based was laid down in the
Greek methods.
lemma prefixed to the 12th book of Euclid’s *Elements* as 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 its
application is the 6th proposition of Archimedes’ treatise *On the*
*Sphere 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 (see Function). The new problems
presented by the analytical geometry and natural philosophy
of the 17th century led to new limiting processes.

4. In the *problem of tangents* the 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*+Δ*x*, *y*+Δ*y* those
of P′. The symbol Δ*x* means “the difference of two
*x*’s” and there is a like meaning for the symbol Δ*y*.Differentiation.

Fig. 2. |

The fraction Δ*y*/Δ*x* is the trigonometrical tangent of the angle
which the secant PP′ makes with the
axis of *x*. Now let Δ*x* 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 Δ*y*/Δ*x* 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 (see Function), then

.

and

.

The limit expressed by the right-hand member of this defining equation is often written

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 Δ*x* can be performed, and the limit is found by substituting zero for Δ*x* in the quotient. For example, if ƒ(*x*) = *x*^{2}, we have

and

The process of forming the derived function of a given function
is called *differentiation*. The fraction Δ*y*/Δ*x* is called the “quotient
of differences,” and its limit *dy*/*dx* is called the “differential coefficient
of *y* with respect to *x*.” The rules for forming differential
coefficients constitute the *differential 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 (see Maxima and Minima).

5. The *problem of quadratures* leads 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 these
points. Let the segment AB be divided into a numberIntegration.
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

Fig. 3. |

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*+Δ*x* that of Q, *x*′ that of P′, the limit in question might be written

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

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 the *integral 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 the integral , and it is a function of ξ. Let BD be displaced to B′D′ so that becomes (seeTheorem of Inversion. fig. 4). The area of the figure ACD′B′ is represented by the integral and the increment ΔA is given by the formula:

which represents the area BDD′B′.

Fig. 4. |

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 *h*Δξ, 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

When we recognize a function F(*x*) which has the property expressed
by the equation

,

we are said to *integrate* the function ƒ(*x*), and F(*x*) is called the
*indefinite integral* of ƒ(*x*) *with respect to x*, and is written

7. In the process of § 4 the increment Δ*y* is not in general equal
to the product of the increment Δ*x* and the derived
function ƒ′(*x*). In general we can write down an equation
of the formDifferentials.

,

in which R is different from zero when Δ*x* is different from zero;
and then we have not only

,

but also

We may separate Δ*y* into two parts: the part ƒ′(*x*)Δ*x* and the
part *R*. The part ƒ′(*x*)Δ*x* alone is useful for forming the differential
coefficient, and it is convenient to give it a name. It is called the
*differential* of ƒ(*x*), and is written *d*ƒ(*x*), or *dy* when *y* is written for
ƒ(*x*). When this notation is adopted *dx* is written instead of Δ*x*,
and is called the “differential of *x*,” so that we have

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 differentials
are 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 *x*_{1}, *x*_{2}, ... *x*_{n−1} be the abscissae of the points in order.
The integral is the limit of the sum

every term of which is a differential of the form ƒ(*x*)*dx*. Further the integral is equal to the sum of differences

for this sum is F(*b*) − F(*a*). Now the difference F(*x*_{r+1}) − F(*x*_{r}) is
*not* equal to the differential ƒ(*x*_{r}) (*x*_{r+1} − *x*_{r}), but the sum of the
differences is equal to the *limit* of 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 a *differential 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 letter “*d* ” is the initial letter of the word *differentia* (difference)
and the symbol “∫” is a conventionally written “S”, the
initial letter of the word *summa* Notation.(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 Δ*x* and Δ*y*, by substituting
*x*+Δ*x* for *x*
and *y*+Δ*y* for *y*. If there is a differential coefficient Fundamental Artifice.of *y*
with respect to *x*, then Δ*y* can be expressed in the form
φ.Δ*x* + R, where lim._{Δx = 0} (R/Δx) = 0, as in § 7 above. The artifice
consists in rejecting *ab initio* all terms of the equation which belong
to R. We do not form R at all, but only φ.Δ*x*, 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 the *element* of 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 are
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 Orders of small quantities.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 the 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 Rules of Differentiation. follows:—

(i.) The differential coefficient of a *constant* is zero

(ii.) For a *sum u*+*v*+ . . ., where *u*,*v*,... are functions of *x*,

(iii.) For a *product uv*

(iv.) For a *quotient u*/*v*

(v.) For a *function 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 called *explicit* functions. In addition to these we have
*implicit* functions, 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 to
Function.

12. The processes of the integral calculus consist largely in transformations 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 Indefinite Integrals. 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).