to devise a general method. His solution, which is sometimes
erroneously ascribed to Rafael Bombelh, was published in the
Ars Magna. In this work, which is one of the most valuable
contributions to the literature of algebra, Cardan shows
that he was familiar with both real positive and negative
roots of equations whelher rational or irrational, but of
imaginary roots he was quite ignorant, and he admits his
inability to resolve the so-called ``irreducible case'' (see
EQUATION.) Fundamental theorems in the theory of equations
are to be found in the same work. Clearer ideas of imaginary
quantities and the ``irreducible case'' were subsequently
published by Bombelli, in a work of which the dedication is
dated 1572, though the book was not published until 1579.
Contemporaneously with the remarkable discoveries of the
Italian mathematicians, algebra was increasing in popularity
in Germany, France and England. Michael Stifel and Johann
Scheubelius (Scheybl) (1494-1570) flourished in Germany, and
although unacquainted with the work of Cardan and Tartalea,
their writings are noteworthy for their perspicuity and the
introduction of a more complete symbolism for quantities and
operations. Stifel introduced the sign (+) for addition or
a positive quantity, which was previously denoted by plus,
piu, or the letter p. Subtraction, previously written as
minus, mone or the letter m, was symbolized by the sign
(-) which is still in use. The square root he denoted by
(sqrt. ), whereas Paciolus, Cardan and others used the letter R.
The first treatise on algebra written in English was by
Robert Recorde, who published his arithmetic in 1552, and
his algebra entitled The Whetstone of Witte, which is the
second part of Arithmetik, in 1557. This work, which is
written in the form of a dialogue, closely resembles the
works of Stifel and Scheubelius, the latter of whom he often
quotes. It includes the properties of numbers; extraction of
roots of arithmetical and algebraical quantities, solutions of
simple and quadratic equations, and a fairly complete account of
surds. He introduced the sign (=) for equality, and the terms
binomial and residual. Of other writers who published
works about the end of the 16th century, we may mention Jacques
Peletier, or Jacobus Peletarius (De occulta parto Numerorum,
quare Algebram vocant, 1558); Petrus Ramus (Arithmeticae
Libri duo et totidem Algebrae, 1560), and Christoph Clavius,
who wrote on algebra in 1580, though it was not published until
1608. At this time also flourished Simon Stevinus (Stevin) of
Bruges, who published an arithmetic in 1585 and an algebra
shortly afterwards. These works possess considerable
originality, and contain many new improvements in algebraic
notation; the unknown (res) is denoted by a small circle,
in which he places an integer corresponding to the power. He
introduced the terms multinomial, trinomial, quadrinomial,
&c., and considerably simplified the notation for decimals.
About the beginning of the 17th century various mathematical
works by Franciscus Vieta were published, which were afterwards
collected by Franz van Schooten and republished in 1646 at
Leiden. These works exhibit great originality and mark an
important epoch in the history of algebra. Vieta, who does
not avail himself of the discoveries of his predecessors--the
negative roots of Cardan, the revised notation of Stifel
and Stevin, &c.--introduced or popularized many new terms
and symbols, some of which are still in use. He denotes
quantities by the letters of the alphabet, retaining the
vowels for the unknown and the consonants for the knowns;
he introduced the vinculum and among others the terms
coefficient, affirmative, negative, pure and adjected
equations. He improved the methods for solving equations, and
devised geometrical constructions with the aid of the conic
sections. His method for determining approximate values
of the roots of equations is far in advance of the Hindu
method as applied by Cardan, and is identical in principle
with the methods of Sir Isaac Newton and W. G. Horner.
We have next to consider the works of Albert Girard, a Flemish
mathematician. This writer, after having published an edition
of Stevin's works in 1625, published in 1629 at Amsterdam a
small tract on algebra which shows a considerable advance on
the work of Vieta. Girard is inconsistent in his notation,
sometimes following Vieta, sometimes Stevin; he introduced the
new symbols ff. for greater than and sec. for less than; he
follows Vieta in using the plus (+) for addition, he denotes
subtraction by Recorde's symbol for equality (=), and he had
no sign for equality but wrote the word out. He possessed
clear ideas of indices and the generation of powers, of the
negative roots of equations and their geometrical interpretation,
and was the first to use the term imaginary roots. He also
discovered how to sum the powers of the roots of an equation.
Passing over the invention of logarithms (q.v.) by John
Napier, and their development by Henry Briggs and others, the
next author of moment was an Englishman, Thomas Harriot, whose
algebra (Artis analyticae praxis) was published posthumously
by Walter Warner in 1631. Its great merit consists in the
complete notation and symbolism, which avoided the cumbersome
expressions of the earlier algebraists, and reduced the art
to a form closely resembling that of to-day. He follows
Vieta in assigning the vowels to the unknown quantities and
the consonants to the knowns, but instead of using capitals,
as with Vieta, he employed the small letters; equality he
denoted by Recorde's symbol, and he introduced the signs
> and < for greater than and less than. His principal
discovery is concerned with equations, which he showed to be
derived from the continued multiplication of as many simple
factors as the highest power of the unknown, and he was thus
enabled to deduce relations between the coefficients and
various functions of the roots. Mention may also be made
of his chapter on inequalities, in which he proves that the
arithmetic mean is always greater than the geometric mean.
William Oughtred, a contemporary of Harriot, published an
algebra, Clavis mathematicae, simultaneously with Harriot's
treatise. His notation is based on that of Vieta, but he
introduced the sign X for multiplication, @ for continued
proportion, :: for proportion, and denoted ratio by one
dot. This last character has since been entirely restricted
to multiplication, and ratio is now denoted by two dots
(:). His symbols for greater than and less than (@ and
@) have been completely superseded by Harriot's signs`
So far the development of algebra and geometry had been
mutually independent, except for a few isolated applications
of geometrical constructions to the solution of algebraical
problems. Certain minds had long suspected the advairages which
would accrue from the unrestricted application of algebra to
geometry, but it was not until the advent of the philosopher
Rene Descartes that the co-ordination was effected. In his
famous Geometria (1637), which is really a treatise on the
algebraic representation of geometric theorems, he founded
the modern theory of analytical geometry (see GEOMETRY),
and at the same time he rendered signal service to algebra,
more especially in the theory of equations. His notation is
based primarily on that of Harriot; but he differs from that
writer in retaining the first letters of the alphabet for
the known quantities and the final letters for the unknowns.
The 17th century is a famous epoch in the progress of science,
and the mathematics in no way lagged behind. The discoveries
of Johann Kepler and Bonaventura Cavalieri were the foundation
upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz
erected that wonderful edifice, the Infinitesimal Calculus
(q.v..) Many new fields were opened up, but there was still
continual progress in pure algebra. Continued fractions, one
of the earliest examples of which is Lord Brouncker's expression
for the ratio of the circumference to the diameter of a circle
(see CIRCLE), were elaborately discussed by John Wallis and
Leonhard Euler; the convergency of series treated by Newton,
Euler and the Bernoullis; the binomial theorem, due originally
to Newton and subsequently expanded by Euler and others, was
used by Joseph Louis Lagrange as the basis of his Calcul
des Fonctions. Diophantine problems were revived by Gaspar
Bachet, Pierre Fermat and Euler; the modern theory of numbers
was founded by Fermat and developed by Euler, Lagrange and
others; and the theory of probability was attacked by Blaise
Pascal and Fermat, their work being subsequently expanded by
James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and
others. The germs of the theory of determinants are to be
found in the works of Leibnitz; Etienne Bezout utilized them
in 1764 for expressing the result obtained by the process of
elimination known by his name, and since restated by Arthur Cayley.
In recent times many mathematicians have formulated other
kinds of algebras, in which the operators do not obey the laws
of ordinary algebra. This study was inaugurated by George
Peacock, who was one of the earliest mathematicians to recognize
the symbolic character of the fundamental principles of
algebra. About the same time, D. F. Gregory published a paper
``on the real nature of symbolical algebra.'' In Germany the
work of Martin Ohm (System der Mathematik, 1822) marks a step
forward. Notable service was also rendered by Augustus de
Morgan, who applied logical analysis to the laws of mathematics.
The geometrical interpretation of imaginary quantities had a
far-reaching influence on the development of symbolic algebras.
The attempts to elucidate this question by H. Kuhn (1750-1751)
and Jean Robert Argand (1806) were completed by Karl Friedrich
Gauss, and the formulation of various systems of vector analysis
by Sir William Rowan Hamilton, Hermann Grassmann and others,
followed. These algebras were essentially geometrical, and it
remained, more or less, for the American mathematician Benjamin
Peirce to devise systems of pure symbolic algebras; in this
work he was ably seconded by his son Charles S. Peirce. In
England, multiple algebra was developed by James Joseph
Sylvester, who, in company with Arthur Cayley, expanded the
theory of matrices, the germs of which are to be found in the
writings of Hamilton (see above, under (B); and QUATERNIONS.)
The preceding summary shows the specialized nature which algebra
has assumed since the 17th century. To attempt a history
of the development of the various topics in this article is
inappropriate, and we refer the reader to the separate articles.
REFERENCES.---The history of algebra is treated in all
historical works on mathematics in general (see MATHEMATICS:
References.) Greek algebra can be specially studied in
T. L. Heath's Diophantus. See also John Wallis, Opera
Mathematica (1693-1699), and Charles Sutton, Mathematical and
Philosophical Dictionary (1815), article ``Algebra.'' (C. E.*)
[The article on Algebraic Forms is typeset in TeX and is available
elsewhere.]
ALGECIRAS, or ALGEZIRAS, a seaport of southern Spain in the
province of Cadiz, 6 m. W. of Gibraltar, on the opposite side of
the Bay of Algeciras. Pop. (1900) 13,302. Algeciras stands at
the head of a railway from Granada, but its only means of access
to Gibraltar is by water. Its name, which signifies in Arabic
the island, is derived from a small islet on one side of the
harbour. It is supplied with water by means of a beautiful
