analysis in which Diophantus excelled. But whereas Diophantus
aimed at obtaining a single solution, the Hindus strove for
a general method by which any indeterminate problem could be
resolved. In this they were completely successful, for
they obtained general solutions for the equations ax(+ or -)by=c,
xy=ax+by+c (since rediscovered by Leonhard Euler) and
cy2=ax2+b. A particular case of the last equation,
namely, y2=ax2+1, sorely taxed the resources of modern
algebraists. It was proposed by Pierre de Fermat to Bernhard
Frenicle de Bessy, and in 1657 to all mathematicians. John
Wallis and Lord Brounker jointly obtained a tedious solution
which was published in 1658, and afterwards in 1668 by John
Pell in his Algebra. A solution was also given by Fermat
in his Relation. Although Pell had nothing to do with the
solution, posterity has termed the equation Pell's Equation, or
Problem, when more rightly it should be the Hindu Problem,
in recognition of the mathematical attainments of the Brahmans.
Hermann Hankel has pointed out the readiness with which the
Hindus passed from number to magnitude and vice versa.
Although this transition from the discontinuous to continuous
is not truly scientific, yet it materially augmented the
development of algebra, and Hankel affirms that if we
define algebra as the application of arithmetical operations
to both rational and irrational numbers or magnitudes,
then the Brahmans are the real inventors of algebra.
The integration of the scattered tribes of Arabia in the 7th
century by the stirring religious propaganda of Mahomet was
accompanied by a meteoric rise in the intellectual powers of
a hitherto obscure race. The Arabs became the custodians of
Indian and Greek science, whilst Europe was rent by internal
dissensions. Under the rule of the Abbasids, Bagdad became
the centre of scientific thought; physicians and astronomers
from India and Syria flocked to their court; Greek and Indian
manuscripts were translated (a work commenced by the Caliph
Mamun (813-833) and ably continued by his successors); and
in about a century the Arabs were placed in possession of
the vast stores of Greek and Indian learning. Euclid's
Elements were first translated in the reign of Harun-al-Rashid
(786-809), and revised by the order of Mamun. But these
translations Were regarded as imperfect, and it remained for
Tobit ben Korra (836-901) to produce a satisfactory edition.
Ptolemy's Almagest, the works of Apollonius, Archimedes,
Diophantus and portions of the Brahmasiddhanta, were also
translated. The first notable Arabian mathematician was
Mahommed ben Musa al-Khwarizmi, who flourished in the reign of
Mamun. His treatise on algebra and arithmetic (the latter part
of which is only extant in the form of a Latin translation,
discovered in 1857) contains nothing that was unknown to the
Greeks and Hindus; it exhibits methods allied to those of both
races, with the Greek element predominating. The part devoted
to algebra has the title al-jeur wa'lmuqabala, and the
arithmetic begins with ``Spoken has Algoritmi,'' the name
Khwarizmi or Hovarezmi having passed into the word Algoritmi,
which has been further transformed into the more modern words
algorism and algorithm, signifying a method of computing.
Tobit ben Korra (836-901), born at Harran in Mesopotamia, an
accomplished linguist, mathematician and astronomer, rendered
conspicuous Service by his translations of various Greek
authors. His investigation of the properties of amicable
numbers (q.v.) and of the problem of trisecting an angle,
are of importance. The Arabians more closely resembled
the Hindus than the Greeks in the choice of studies; their
philosophers blended speculative dissertations with the
more progressive study of medicine; their mathematicians
neglected the subtleties of the conic sections and Diophantine
analysis, and applied themselves more particularly to
perfect the system of numerals (see NUMERAL), arithmetic
and astronomy (q.v..) It thus came about that while some
progress was made in algebra, the talents of the race were
bestowed on astronomy and trigonometry (q.v..) Fahri des
al Karbi, who flourished about the beginning of the 11th
century, is the author of the most important Arabian work on
algebra. He follows the methods of Diophantus; his work
on indeterminate equations has no resemblance to the Indian
methods, and contains nothing that cannot be gathered from
Diophantus. He solved quadratic equations both geometrically and
algebraically, and also equations of the form x2n+axn+b=0;
he also proved certain relations between the sum of the first
n natural numbers, and the sums of their squares and cubes.
Cubic equations were solved geometrically by determining
the intersections of conic sections. Archimedes' problem
of dividing a sphere by a plane into two segments having a
prescribed ratio, was first expressed as a cubic equation by
Al Mahani, and the first solution was given by Abu Gafar al
Hazin. The determination of the side of a regular heptagon
which can be inscribed or circumscribed to a given circle
was reduced to a more complicated equation which was first
successfully resolved by Abul Gud. The method of solving
equations geometrically was considerably developed by
Omar Khayyam of Khorassan, who flourished in the 11th
century. This author questioned the possibility of solving
cubics by pure algebra, and biquadratics by geometry. His
first contention was not disproved until the 15th century,
but his second was disposed of by Abul Weta (940-908), who
succeeded in solving the forms x4=a and x4+ax3=b.
Although the foundations of the geometrical resolution of cubic
equations are to be ascribed to the Greeks (for Eutocius assigns
to Menaechmus two methods of solving the equation x3=a
and x3=2a3), yet the subsequent development by the Arabs
must be regarded as one of their most important achievements.
The Greeks had succeeded in solving an isolated example; the
Arabs accomplished the general solution of numerical equations.
Considerable attention has been directed to the different
styles in which the Arabian authors have treated their
subject. Moritz Cantor has suggested that at one time there
existed two schools, one in sympathy With the Greeks, the
other with the Hindus; and that, although the writings of the
latter were first studied, they were rapidly discarded for
the more perspicuous Grecian methods, so that, among the later
Arabian writers, the Indian methods were practically forgotten
and their mathematics became essentially Greek in character.
Turning to the Arabs in the West we find the same enlightened
spirit; Cordova, the capital of the Moorish empire in
Spain, was as much a centre of learning as Bagdad. The
earliest known Spanish mathematician is Al Madshritti (d.
1007), whose fame rests on a dissertation on amicable
numbers, and on the schools which were founded by his
pupils at Cordoya, Dama and Granada. Gabir ben Allah of
Sevilla, commonly called Geber, was a celebrated astronomer
and apparently skilled in algebra, for it has been supposed
that the word ``algebra', is compounded from his name.
When the Moorish empire began to wane the brilliant
intellectual gifts which they had so abundantly nourished
during three or four centuries became enfeebled, and
after that period they failed to produce an author
comparable with those of the 7th to the 11th centuries.
In Europe the decline of Rome was succeeded by a period, lasting
several centuries, during which the sciences and arts were
all but neglected. Political and ecclesiastical dissensions
occupied the greatest intellects, and the only progress to
be mcorded is in the art of computing or arithmetic, and
the translation of Arabic manuscripts. The first successful
attempt to revive the study of algebra in Christendom was
due to Leonardo of Pisa. an Italian merchant trading in the
Mediterranean. His travels and mercantile experience had
led him to conclude that the Hindu methods of computing,
were in advance of those then in general use, and in 1202 he
published his Liber Abaci, which treats of both algebra and
arithmetic. In this work, which is of great historical
interest, since it was published about two centuries before the
art of printing was discovered, he adopts the Arabic notation
for nulnbers, and solves many problems, both arithmetical and
algebraical. But it contains little that is original, and
although the work created a great sensation when it was first
published, the effect soon passed away, and the book was
practically forgotten. Mathematics was more or less ousted
from the academic curricula by the philosophical inquiries
of the schoolmen, and it was only after an interval of
nearly three centuries that a worthy successor to Leonardo
appeared. This was Lucas Paciolus (Lucas de Burgo), a Minorite
friar, who, having previously written works on algebra,
arithmetic and geometry, published, in 1494, his principal
work, entitled Summa de Arithmetica, Giometria, Proportioni
et Proportionalita. In it he mentions many earlier writers
from whom he had learnt the science, and although it contains
very little that cannot be found in Leonardo's work, yet it
is especially noteworthy for the systematic employment of
symbols, and the manner in which it reflects the state of
mathematics in Europe during this period. These works are
the earliest printed books on mathematics. The renaissance of
mathematics was thus effected in Italy, and it is to that country
that the leading developments of the following century were
due. The first difficulty to be overcome was the algebraical
solution of cubic equations, the pons asinorum of the earlier
mathematicians. The first step in this direction was made by
Scipio Ferro (d. 1526), who solved the equation x3+ax=b.
Of his discovery we know nothing except that he declared it
to his pupil Antonio Marie Floridas. An imperfect solution of
the equation x3+px2=q was discovered by Nicholas Tartalea
(Tartaglia) in 1530, and his pride in this achievement led him
into conflict with Floridas, who proclaimed his own knowledge
of the form resolved by Ferro. Mutual recriminations led
to a public discussion in 1535, when Tartalea completely
vindicated the general applicability of his methods and
exhibited the inefficiencies of that of Floridas. This contest
over, Tartalea redoubled his attempts to generalize his
methods, and by 1541 he possessed the means for solving any
form of cubic equation. His discoveries had made him famous
all over Italy, and he was earnestly solicited to publish
his methods; but he abstained from doing so, saying that he
intended to embody them in a treatise on algebra which he was
preparing. At last he succumbed to the repeated requests of
Girolamo or Geronimo Cardano, who swore that he would regard
them as an inviolable secret. Cardan or Cardano, who was
at that time writing his great work, the Ars Magna, could
not restrain the temptation of crowning his treatise with
such important discoveries, and in 1545 he broke his oath
and gave to the world Tartalea's rules for solving cubic
equations. Tartalea, thus robbed of his most cherished
possession, was in despair. Recriminations ensued until
his death in 1557, and although he sustained his claim for
priority, posterity has not conceded to him the honour of his
discovery, for his solution is now known as Cardan's Rule.
Cubic equations having been solved, biquadratics soon followed
suit. As early as 1539 Cardan had solved certain particular
cases, but it remained for his pupil, Lewis (Ludovici) Ferrari,
