Chapter IX: Act 1890: , the effect of which is explained in the article Insanity. Any (5)

INFANT SCHOOLS. The provision in modern times of systematized training for children below the age when elementary education normally begins may be dated from the village school at Waldbach founded by Jean Frédéric Oberlin in 1774. Robert Owen started an infant school at New Lanark in 1800, and great interest in the question was taken in Great Britain during the early years of the 19th century, leading to the foundation in 1836 of the Home and Colonial School Society for the training of teachers in infant schools; this in turn reacted upon other countries, especially Germany. Further impetus and a new direction were given to the movement by Friedrich W. A. Froebel, and the methods of training adopted for children between the ages of three and six have in most countries been influenced by, if not based on, that system of directed activities which was the foundation of the type of "play-school" called by him the _Kinder Garten_, or "children's garden." The growing tendency in England to lay stress on the mental training of very young children, and to use the "infant school" as preparatory to the elementary school, has led to a considerable reaction; medical officers of health have pointed out the dangers of infection to which children up to the age of five are specially liable when congregated together--also the physical effects of badly ventilated class-rooms, and there is a consensus of opinion that formal mental teaching is directly injurious before the age of six or even seven years. At the same time the increase in the industrial employment of married women, with the consequent difficulty of proper care of young children by the mother in the home, has somewhat shifted the ground from a purely educational to a social and physical aspect. While it is agreed that the ideal place for a young child is the home under the supervision of its mother, the present industrial conditions often compel a mother to go out to work, and leave her children either shut up alone, or free to play about the streets, or in the care of a neighbour or professional "minder." In each case the children must suffer. The provision by a public authority of opportunities for suitable training for such children seems therefore a necessity. The moral advantages gained by freeing the child from the streets, by the superintendence of a trained teacher over the games, by the early inculcation of habits of discipline and obedience; the physical advantages of cleanliness and tidiness, and the opportunity of disclosing incipient diseases and weaknesses, outweigh the disadvantages which the opponents of infant training adduce. It remains to give a brief account of what is done in Great Britain, the United States of America, and certain other countries. A valuable report was issued for the English Board of Education by a Consultative Committee upon the school attendance of children below the age of five (vol. 22 of the _Special Reports_, 1909), which also gives some account of the provision of day nurseries or _crèches_ for babies.

_United Kingdom._--Up to 1905 it was the general English practice since the Education Act of 1870 for educational authorities to provide facilities for the teaching of children between three and five years old whose parents desired it. In 1905, of an estimated 1,467,709 children between those ages, 583,268 were thus provided for in England and Wales. In 1905 the objections, medical and educational, already stated, coupled with the increasing financial strain on the local educational authorities, led to the insertion in the code of that year of Article 53, as follows: "Where the local education authority have so determined in the case of any school maintained by them, children who are under five years may be refused admission to that school." In consequence in 1907 the numbers were found to have fallen to 459,034 out of an estimated 1,480,550 children, from 39.74% in 1905 to 31%. In the older type of infant school stress was laid on the mental preparation of children for the elementary teaching which was to come later. This forcing on of young children was encouraged by the system under which the government grant was allotted; children in the infant division earned an annual grant of 17s. per head, on promotion to the upper school this would be increased to 22s. In 1909 the system was altered; a rate of 21s. 4d. was fixed as the grant for all children above five, and the grant for those below the age was reduced to 13s. 4d. Different methods of training the teachers in these schools as well as the children themselves have been now generally adopted. These methods are largely based on the Froebelian plan, and greater attention is being paid to physical development. In one respect England is perhaps behind the more progressive of other European countries, viz. in providing facilities for washing and attending to the personal needs of the younger children. There is no _femme de service_ as in Belgium on the staff of English schools. While in Ireland the children below the age of five attend the elementary schools in much the same proportion as in England and Wales, in Scotland it has never been the general custom for such children to attend school.

_United States of America._--In no country has the kindergarten system taken such firm root, and the provision made for children below the compulsory age is based upon it. In 1873 there were 42 kindergartens with 1252 pupils; in 1898 the numbers had risen to 2884 with 143,720 pupils; more than half these were private schools, managed by charitable institutions or by individuals for profit. In 1904-1905 there were 3176 public kindergartens with 205,118 pupils.

_Austria Hungary._--Provision in Austria is made for children under
six by two types of institution, the Day Nursery
(_Kinderbewahranstalten_) and the Kindergarten. In 1872 as the result
of a State Commission the Kindergarten was established in the state
system of education. Its aim is to "confirm and complete the home
education of children under school age, so that through regulated
exercise of body and mind they may be prepared for institution in the
primary school." No regular teaching in ordinary school subjects is
allowed; games, singing and handwork, and training of speech and
observation by objects, tales and gardening are the means adopted. The
training for teachers in these schools is regulated by law. No
children are to be received in a kindergarten til! the beginning of
the fourth and must leave at the end of the sixth year. In 1902-1903
there were 77,002 children in kindergartens and 74,110 in the day
nurseries. In Hungary a law was passed in 1891 providing for the
education and care of children between three and six, either by asyle
or nurseries open all the year round in communes which contribute from
£830 to £1250 in state taxation, or during the summer in those whose
contribution is less. Communes above the higher sum must provide
kindergartens. In 1904 there were over 233,000 children in such
institutions.

_Belgium._--For children between three and six education and training
are provided by _Écoles gardiennes_ or _Jardins d'enfants_. They are
free but not compulsory, are provided and managed by the communes,
receive a state grant, and are under government inspection. Schools
provided by private individuals or institutions must conform to the
conditions of the communal schools. There is a large amount of
voluntary assistance especially in the provision of clothes and food
for the poorer children. The state first recognized these schools in
1833. In 1881 there were 708 schools with accommodation for over
56,000 children; in 1907 there were 2837 and 264,845 children,
approximately one-half of the total number of children in the country
between the ages of three and six. In 1890 the minister of Public
Instruction issued a code of rules on which is based the organization
of the _Écoles gardiennes_ throughout Belgium, but some of the
communes have regulations of their own. A special examination for
teachers in the _Écoles gardiennes_ was started in 1898. All
candidates must pass this examination before a _certificat de
capacité_ is granted. The training includes a course in Froebelian
methods. While Froebel's system underlies the training in these
schools, the teaching is directed very much towards the practical
education of the child, special stress being laid on manual dexterity.
Reading, writing and arithmetic are also allowed in the classes for
the older children. A marked feature of the Belgian schools is the
close attention paid to health and personal cleanliness. In all
schools there is a _femme de service_, not a teacher, but an
attendant, whose duty it is to see to the tidiness and cleanliness of
the children, and to their physical requirements.

_France._--The first regular infant school was established in Paris at
the beginning of the 19th century and styled a _Salle d'essai_. In
1828 a model school, called a _Salle d'asile_, was started, followed
shortly by similar institutions all over France. State recognition and
inspection were granted, and by 1836 there were over 800 in Paris and
the provinces. In 1848 they became establishments of public
instruction, and the name _École maternelle_ which they have since
borne was given them. Every commune with 2000 inhabitants must have
one of these schools or a _Classe enfantine_. Admission is free, but
not compulsory, for children between two and six. Food and clothes are
provided in exceptional cases. Formal mental instruction is still
given to a large extent, and the older children are taught reading,
writing and arithmetic. Though the staffs of the school include
_femmes de service_, not so much attention is paid to cleanliness as
in Belgium, nor is so much stress laid on hygiene. In 1906-1907 there
were 4111 public and private _Écoles maternelles_ in France, with over
650,000 pupils. The closing of the clerical schools has led to some
diminution in the numbers.

_Germany.___--There are two classes of institution in Germany for
children between the ages of 2½ or 3 and 6. These are the
_Kleinkinderbewahranstalten_ and _Kindergarten_. The first are
primarily social in purpose, and afford a place for the children of
mothers who have to leave their homes for work. These institutions,
principally conducted by religious or charitable societies, remain
open all day and meals are provided. Many of them have a kindergarten
attached, and others provide some training on Froebelian principles.
The kindergartens proper are also principally in private hands, though
most municipalities grant financial assistance. They are conducted on
advanced Froebelian methods, and formal teaching in reading, writing
and arithmetic is excluded. In Cologne, Düsseldorf, Frankfort and
Munich there are municipal schools. The state gives no recognition to
these institutions and they form no part of the public system of
education.

_Switzerland._--In the German speaking cantons the smaller towns and
villages provide for the younger children by _Bewahranstalten_,
generally under private management with public financial help. The
larger towns provide kindergartens where the training is free but not
compulsory for children from four to six. These are generally
conducted on Froebel's system and there is no formal instruction. In
the French speaking cantons the _Écoles enfantines_ are recognized as
the first stage of elementary education. They are free and not
compulsory for children from three to six years of age. (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 infinite [Greek: to ápeiron] (see IONIAN 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. [root]-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 of _the_ infinite 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 articles SERIES and FUNCTION. 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 in GEOMETRY. 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 (see INFINITESIMAL 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._

Geometrical representation of Variable Quantities.

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 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 problems of Maxima and Minima, Tangents, and Quadratures.

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 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.

Greek methods.

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 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.

Differentiation.

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 + [Delta]x, y + [Delta]y those of P´. The
symbol [Delta]x means "the difference of two x's" and there is a like
meaning for the symbol [Delta]y. The fraction [Delta]y/[Delta]x is the
trigonometrical tangent of the angle which the secant PP´ makes with
the axis of x. Now let [Delta]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 [Delta]y/[Delta]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

dy
--.
dx

If the equation of the curve is of the form y = [f](x) where [f] is a
functional symbol (see FUNCTION), then

[Delta]y [f](x + [Delta]x) - [f](x)
-------- = --------------------------,
[Delta]x [Delta]x

and

dy [f](x + [Delta]x) - [f](x)
-- = lim. --------------------------.
dx [Delta]x = 0 [Delta]x

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

[f]´(x),

and is called the "derived function" of [f](x), sometimes the
"derivative" or "derivate" of [f](x). When the function [f](x) is a
rational integral function, the division by [Delta]x can be performed,
and the limit is found by substituting zero for [Delta]x in the
quotient. For example, if [f](x) = x², we have

[f](x + [Delta]x) - [f](x) (x + [Delta]x)² - x² 2x[Delta]x + ([Delta]x)²
-------------------------- = -------------------- = ------------------------ = 2x + [Delta]x,
[Delta]x [Delta]x [Delta]x

and

[f]´(x) = 2x.

The process of forming the derived function of a given function is
called _differentiation_. The fraction [Delta]y/[Delta]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).

Integration.

5. The _problem of quadratures_ leads to a type of limiting process
which may be described as follows: Let y = [f](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 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 + [Delta]x
that of Q, x´ that of P´, the limit in question might be written

_b
lim. \ [f](x´)[Delta]x,
/_a

where the letters a, b written below and above the sign of summation
[Sigma] indicate the extreme values of x. This limit is called "the
definite integral of [f](x) between the limits a and b," and the
notation for it is
_
/ b
| [f](x)dx.
_/ a

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_.

Theorem of Inversion.

6. The chief of these rules is obtained by regarding the extreme
ordinate BD as variable. Let [xi] now denote the abscissa of B. The
area A of the figure ACDB is represented by the integral [int] {a to
[xi]} [f](x)dx, and it is a function of [xi]. Let BD be displaced to
B´D´ so that [xi] becomes [xi] + [delta][xi] (see fig. 4). The area of
the figure ACD´B´ is represented by the integral [int] {a to [xi] +
[Delta][xi]} [f](x)dx, and the increment [Delta]A of the area is given
by the formula

_[xi]+[Delta][xi]
/
[Delta]A = | [f](x) dx,
_/ [xi]

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 [Delta]A is
intermediate between H[Delta][xi] and h[Delta][xi], and the quotient
of differences [Delta]A/[Delta][xi] is intermediate between H and h.
If the function [f](x) is continuous at B (see Function), then, as
[Delta][xi] is diminished without limit, H and h tend to BD, or
[f]([xi]), as a limit, and we have

dA
----- = [f]([xi]).
d[xi]

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 [f](x) as its differential coefficient.
If [f](x) is continuous between a and b, we can prove that
_
/ b
A = | [f](x) dx = F(b) - F(a).
_/ a

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

dF(x)
----- = [f](x),
dx

we are said to _integrate_ the function [f](x), and F(x) is called the
_indefinite integral_ of [f](x) _with respect to_ x, and is written
_
/
| [f](x)dx.
_/

Differentials.

7. In the process of § 4 the increment [Delta]y is not in general
equal to the product of the increment [Delta]x and the derived
function [f]´(x). In general we can write down an equation of the form

[Delta]y = [f]´(x)[Delta]x + R,

in which R is different from zero when [Delta]x is different from
zero; and then we have not only

lim. R = 0,
[Delta]x=0

but also

R
lim. -------- = 0.
[Delta]x=0 [Delta]x

We may separate [Delta]y into two parts: the part [f]´(x)[Delta]x and
the part R. The part [f]´(x)[Delta]x alone is useful for forming the
differential coefficient, and it is convenient to give it a name. It
is called the _differential_ of [f](x), and is written d[f](x), or dy
when y is written for [f](x). When this notation is adopted dx is
written instead of [Delta]x, and is called the "differential of x," so
that we have

d[f](x) = [f]´(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
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
[f](x), so that we have
_
dF(x) / b
----- = [f](x), | [f](x) dx = F(b) - F(a).
dx _/a

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, ... x_(n-1) be the abscissae of the points in order. The integral
is the limit of the sum

[f](a)(x1 - a) + [f](x1)(x2 - x1) + ... + [f](x_r) [x_(r+1) - x_r]
+ ... + [f] [x_(n-1)] [b - x_(n-1)],

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

{F(x1) - F(a)} + {F(x2) - F(x1)} + ... + {F[x_(r+1)] - F(x_r)}
+ ... + {F(b) - F[x(n-1)]},

for this sum is F(b) - F(a). Now the difference F(x_(r+1)) - F(x_r) is
_not_ equal to the differential [f](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.

Notation.

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 [int] is a conventionally written "S," the initial letter of
the word _summa_ (sum or whole). The notation was introduced by
Leibnitz (see §§ 25-27, below).

Fundamental Artifice.

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 [Delta]x and [Delta]y, by substituting x + [Delta]x for x and y +
[Delta]y for y. If there is a differential coefficient of y with
respect to x, then [Delta]y can be expressed in the form
[phi].[Delta]x + R, where lim.{[Delta]x = 0} (R/[Delta]x) = 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
[phi].[Delta]x, or [phi].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

MN·QR
lim. ------ = 0.
MN=0 MN

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.

Orders of small quantities.

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 same order." When this is not the
case we may have either

x
lim. --- = 0,
x=0 y

or
y
lim. --- = 0,
x=0 x

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.

Rules of Differentiation.

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 follows:--

(i.) The differential coefficient of a _constant_ is zero. (ii.) For a
_sum_ u + v + ... + z, where u, v, ... are functions of x,

d(u + v + ... + z) du dv dz
----------------- = -- + -- + ... + --.
dx dx dx dx

(iii.) For a _product_ uv

d(uv) dv du
----- = u -- + v --.
dx dx dx

(iv.) For a _quotient_ u/v

d(u/v) / du dv\ /
------ = ( v -- - u -- ) / v².
dx \ dx dx/ /

(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,

dy dy dz
-- = -- · --.
dx dz dx

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:--

+---------+---------------------+
| y | dy/dx |
+---------+---------------------+
| x^n | nx^(n-1) |
| | for all values of n |
+---------+---------------------+
| log_a x | x^-1 log_a e |
+---------+---------------------+
| a^x | a^x log_e a |
+---------+---------------------+
| sin x | cos x |
+---------+---------------------+
| cos x | -sin x |
+---------+---------------------+
| sin^-1 x| (1 - x²)^-½ |
+---------+---------------------+
| tan^-1 x| (1 + x²)^-1 |
+---------+---------------------+

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.

Indefinite Integrals.

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 have those in the following table:--

+----------------+-------------------------------+
| [f](x) | [int][f](x)dx |
+----------------+-------------------------------+
| | x^(n+1) |
| x^n | ------- |
| | n + 1 |
| | for all values of n except -1 |
+----------------+-------------------------------+
| 1 | |
| --- | log_e x |
| x | |
+----------------+-------------------------------+
| e^(ax) | a^-1 e^(ax) |
+----------------+-------------------------------+
| cos x | sin x |
+----------------+-------------------------------+
| sin x | -cos x |
+----------------+-------------------------------+
| | x |
| (a² - x²)^-½ | sin^-1 --- |
| | a |
+----------------+-------------------------------+
| 1 | 1 x |
| ------- | --- tan^-1 --- |
| a² + x² | a 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.)

II. _History._

Kepler's methods of Integration.

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 his _Astronomia nova ... de motibus stellae Martis_ (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
ad[phi], where a is the radius of the auxiliary circle, and [phi] 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
_
/ [phi]
| (a + ae cos [phi]) d[phi],
_/ 0

where e is the eccentricity. Kepler evaluated the sum as proportional
to [phi] + e sin [phi].

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.

Logarithms.

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 invented 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.

Cavalieri's Indivisibles.

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, 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 treatise _Geometria 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 entitled _Exercitationes 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
_
/ x x^(m+1)
| x^m dx = -------, (m integral).
_/ 0 m + 1

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.

Successors of Cavalieri.

Fermat's method of Integration.

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 thus 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 [int] (0 to 1) x^m dx by the process of forming the sum

0^m + 1^m + 2^m + ... (n - 1)^m
-------------------------------,
n^(m+1)

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 x^m by forming this sum was found also by Fermat (1636), who stated expressly that he arrived at it by generalizing a method employed by Archimedes (for the cases m = 1 and m = 2) in his books on _Conoids and Spheroids_ and on _Spirals_ (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.

Various Integrations.

The result that the problem of quadratures could be solved for any curve whose equation could be expressed in the form

y = x^m (m [Not Equal] -1),

or in the form

y = a1 x^m1 + a2 x^m2 + ... + a_n x^m_n,

where none of the indices is equal to - 1, was used by John Wallis in his _Arithmetica 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 his _Opus 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 his _Logarithmotechnia_ (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.

Integration before the Integral Calculus.

By the middle of the 17th century many mathematicians could perform integrations. Very many particular results had been obtained, and applications of them had been 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 his _Lettres de Amos Dettonville sur quelques-unes de ses inventions en géométrie_ (1659).

Fermat's methods of Differentiation.

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 minimum its variation from point to point of the curve is slowest; and Kepler in the _Stereometria 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 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's _Géométrie_ (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.

Fermat's method for maxima and minima is essentially our method.
Expressed in a more modern notation, what he did was to begin by
connecting the ordinate y and the abscissa x of a point of a curve by
an equation which holds at all points of the curve, then to subtract
the value of y in terms of x from the value obtained by substituting x
+ E for x, then to divide the difference by E, to put E = 0 in the
quotient, and to equate the quotient to zero. Thus he differentiated
with respect to x and equated the differential coefficient to zero.

Fermat's method for solving the problem of tangents may be explained
as follows:--Let (x, y) be the coordinates of a point P of a curve,
(x´, y´), those of a neighbouring point P´ on the tangent at P, and
let MM´ = E (fig. 6).