plain, watered by numerous rivers flowing southward from the
hills. The coast is fringed for 30 m. from Quarteira to
Tavira, with long sandy islands, through which there are six
passages, the most important being the Barra Nova, between Faro
and Olinao. The navigable estuary of the Guadiana divides
Algarve from Huelva, and its tributaries water the western
districts. From the Serra de Malhao flow two streams, the
Silves and Odelouca, which unite and enter the Atlantic below
the town of Silves. In the hilly districts the roads are
bad, the soil unsuited for cultivation, and the inhabitants
few. Flocks of goats are reared on the mountain-sides. The
level country along the southern coast is more fertile, and
produces in abundance grapes, figs, oranges, lemons, olives,
almonds, aloes, and even plantains and dates. The land is,
however, not well suited for the production of cereals, which
are mostly imported from Spain. On the coast the people
gain their living in great measure from the fisheries, tunny
and sardines being caught in considerable quantities. Salt
is also made from sea-water. There is no manufacturing or
mining industry of any importance. The harbours are bad, and
almost the whole foreign trade is carried on by ships of other
nations, although the inhabitants of Algarve are reputed to be
the best seamen and fishermen of Portugal. The chief exports
are dried fruit, wine, salt, tunny, sardines and anchovies.
The only railway is the Lisbon-Faro main line, which passes
north-eastward from Faro, between the Monchique and Malhao
ranges. Faro (11,789), Lagos (8291), Loule (22,478),
Monchique (7345), Olhao (10,009), Silves (9687) and Tavira
(12,175), the chief towns, are described in separate articles.
The name of Algarve is derived from the Arabic, and signifies
a land lying to the west. The title ``king of Algarve,''
held by the kings of Portugal, was first assumed by
Alphonso III., who captured Algarve from the Moors in 1253.
ALGAU, or ALLGAU, the name now given to a comparatively
small district forming the south-western corner of Bavaria,
and belonging to the province of Swabia and Neuburg, but
formerly applied to a much larger territory, which extended
as far as the Danube on the N., the Inn on the S. and the Lech
on the W. The Algau Alps contain several lofty peaks, the
highest of which is Madelegabel (8681 ft.). The district
is celebrated for its cattle, milk, butter and cheese.
ALGEBRA (from the Arab. af-jebr wa'l-muqabala, transposition
and removal [of terms of an equation], the name of a treatise by
Mahommed ben Musa al-Khwarizmi), a branch of mathematics which
may be defined as the generalization and extension of arithmetic.
The subject-matter of algebra will be treated in the following
article under three divisions:---A. Principles of ordinary
algebra; B. Special kinds of algebra; C. History. Special
phases of the subject are treated under their own headings,
e.g. ALGEBRAIC FORMS; BINOMIAL; COMBINATORIAL ANALYSIS;
DETERMINANTS; EQUATION; CONTINUED FRACTION; FUNCTION;
GROUPS, THEORY OF; LOGARITHM; NUMBER; PROBABILITY; SERIES.
A. PRINCIPLES OF ORDINARY ALGEBRA
1. The above definition gives only a partial view of the scope of
algebra. It may be regarded as based on arithmetic, or as
dealing in the first instance with formal results of the laws
of arithmetical number; and in this sense Sir Isaac Newton
gave the title Universal Arithmetic to a work on algebra.
Any definition, however, must have reference to the state of
development of the subject at the time when the definition is given.
2. The earliest algebra consists in the solution of equations.
The distinction between algebraical and arithmetical reasoning
then lies mainly in the fact that the former is in a more
condensed form than the latter; an unknown quantity being
represented by a special symbol, and other symbols being
used as a kind of shorthand for verbal expressions. This
form of algebra was extensively studied in ancient Egypt;
but, in accordance with the practical tendency of the
Egyptian mind, the study consisted largely in the treatment
of particular cases, very few general rules being obtained.
3. For many centuries algebra was confined almost entirely
to the solution of equations; one of the most important
steps being the enunciation by Diophantus of Alexandria of
the laws governing the use of the minus sign. The knowledge
of these laws, however, does not imply the existence of a
conception of negative quantities. The development of symbolic
algebra by the use of general symbols to denote numbers is
due to Franciscus Vieta (Francois Viete, 1540-1603).This
led to the idea of algebra as generalized arithmetic.
4. The principal step in the modern development of algebra was
the recognition of the meaning of negative quantities. This
appears to have been due in the first instance to Albert Girard
(1595-1632), who extended Vieta's results in various branches of
mathematics. His work, however, was little known at the time, and
later was overshadowed by the greater work of Descartes (1596-1650).
5. The main work of Descartes, so far as algebra was concerned,
was the establishment of a relation between arithmetical
and geometrical measurement. This involved not only the
geometrical interpretation of negative quantities, but also the
idea of continuity; this latter, which is the basis of modern
analysis, leading to two separate but allied developments,
viz. the theory of the function and the theory of limits.
6. The great development of all branches of mathematics in
the two centuries following Descartes has led to the term
algebra being used to cover a great variety of subjects, many
of which are really only ramifications of arithmetic, dealt
with by algebraical methods, while others, such as the theory
of numbers and the general theory of series, are outgrowths
of the application of algebra to arithmetic, which involve
such special ideas that they must properly be regarded as
distinct subjects. Some writers have attempted unification
by treating algebra as concerned with functions, and Comte
accordingly defined algebra as the calculus of functions,
arithmetic being regarded as the calculus of values.
7. These attempts at the unification of algebra, and its
separation from other branches of mathematics, have usually
been accompanied by an attempt to base it, as a deductive
science, on certain fundamental laws or general rules; and
this has tended to increase its difficulty. In reality, the
variety of algebra corresponds to the variety of phenomena.
Neither mathematics itself, nor any branch or set of branches of
mathematics, can be regarded as an isolated science. While,
therefore, the logical development of algebraic reasoning must
depend on certain fundamental relations, it is important that
in the early study of the subject these relations should be
introduced gradually, and not until there is some empirical
acquaintance with the phenomena with which they are concerned.
8. The extension of the range of subjects to which mathematical
methods can be applied, accompanied as it is by an extension
of the range of study which is useful to the ordinary
worker, has led in the latter part of the 19th century to an
important reaction against the specialization mentioned in
the preceding paragraph. This reaction has taken the form
of a return to the alliance between algebra and geometry
(\S 5), on which modern analytical geometry is based; the
alliance, however, being concerned with the application of
graphical methods to particular cases rather than to general
expressions. These applications are sometimes treated under
arithmetic, sometimes under algebra; but it is more convenient
to regard graphics as a separate subject, closely allied to
arithmetic, algebra, mensuration and analytical geometry.
9. The association of algebra with arithmetic on the one
hand, and with geometry on the other, presents difficulties,
in that geometrical measurement is based essentially on the
idea of continuity, while arithmetical measurement is based
essentially on the idea of discontinuity; both ideas being
equally matters of intuition. The difficulty first arises in
elementary mensuration, where it is partly met by associating
arithmetical and geometrical measurement with the cardinal and
the ordinal aspects of number respectively (see ARITHMETIC.)
Later, the difficulty recurs in an acute form in reference
to the continuous variation of a function. Reference to a
geometrical interpretation seems at first sight to throw light
on the meaning of a differential coefficient; but closer analysis
reveals new difficulties, due to the geometrical interpretation
itself. One of the most recent developments of algebra is
the algebraic theory at number, which is devised with the
view of removing these difficulties. The harmony between
arithmetical and geometrical measurement, which was disturbed
by the Greek geometers on the discovery of irrational numbers,
is restored by an unlimited supply of the causes of disturbance.
10. Two other developments of algebra are of special
importance. The theory of sequences and series is sometimes
treated as a part of elementary algebra; but it is more
convenient to regard the simpler cases as isolated examples,
leading up to the general theory. The treatment of equations
of the second and higher degrees introduces imaginary and
complex numbers, the theory of which is a special subject.
11. One of the most difficult questions for the teacher of
algebra is the stage at which, and the extent to which, the
ideas of a negative number and of continuity may be introduced.
On the one hand, the modern developments of algebra began
with these ideas, and particularly with the idea of a negative
number. On the other hand, the lateness of occurrence of any
particular mathematical idea is usually closely correlated
with its intrinsic difficulty. Moreover, the ideas which
are usually formed on these points at an early stage are
incomplete; and, if the incompleteness of an idea is not
realized, operations in which it is implied are apt to be purely
formal and mechanical. What are called negative numbers in
arithmetic, for instance, are not really negative numbers but
negative quantities (\S 27 (i.)); and the difficulties incident
to the ideas of continuity have already been pointed out.
12. In the present article, therefore, the main portions of
elementary algebra are treated in one section, without reference
to these ideas, which are considered generally in two separate
sections. These three sections may therefore be regarded as to
a certain extent concurrent. They are preceded by two sections
dealing with the introduction to algebra from the arithmetical
and the graphical sides, and are followed by a section dealing
briefly with the developments mentioned in \S \S 9 and 10 above.
[The intermediate portion of this article is typeset in TeX and
is available elsewhere.]
C. HISTORY Various derivations of the word ``algebra,''
which is of Arabian origin, have been given by different
writers. The first mention of the word is to be found in the
title of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi),
who flourished about the beginning of the 9th century. The
full title is ilm al-jebr wa'l-muqabala, which contains
the ideas of restitution and comparison, or opposition and
comparison, or resolution and equation, jebr being derived
from the verb jabara, to reunite, and muqabala, from
gabala, to make equal. (The root jabara is also met with
