in the word algebrista, which means a ``bone-setter,''
and is still in common use in Spain.) The same derivation is
given by Lucas Paciolus (Luca Pacioli), who reproduces the
phrase in the transliterated form alghebra e almucabala,
and ascribes the invention of the art to the Arabians.
Other writers have derived the word from the Arabic
particle al (the definite article), and gerber, meaning
``man.'' Since, however, Geber happened to be the name of
a celebrated Moorish philosopher who flourished in about
the 11th or 12th century, it has been supposed that he was
the founder of algebra, which has since perpetuated his
name. The evidence of Peter Ramus (1515-1572) on this point
is interesting, but he gives no authority for his singular
statements. In the preface to his Arithmeticae libri duo
et totidem Algebrae (1560) he says: ``The name Algebra
is Syriac, signifying the art or doctrine of an excellent
man. For Geber, in Syriac, is a name applied to men, and
is sometimes a term of honour, as master or doctor among
us. There was a certain learned mathematician who sent his
algebra, written in the Syriac language, to Alexander the
Great, and he named it almucabala, that is, the book of
dark or mysterious things, which others would rather call
the doctrine of algebra. To this day the same book is in
great estimation among the learned in the oriental nations,
and by the Indians, who cultivate this art, it is called
aljabra and alboret; though the name of the author
himself is not known.,' The uncertain authority of these
statements, and the plausibility of the preceding explanation,
have caused philologists to accept the derivation from al
and jabara. Robert Recorde in his Whetstone of Witte
(1557) uses the variant algeber, while John Dee (1527-1608)
affirms that algiebar, and not algebra, is the correct
form, and appeals to the authority of the Arabian Avicenna.
Although the term ``algebra'' is now in universal use, various
other appellations were used by the Italian mathematicians
during the Renaissance. Thus we find Paciolus calling it
l'Arte Magiore; ditta dal vulgo la Regula de la Cosa
over Alghebra e Almucabala. The name l'arte magiore,
the greater art, is designed to distinguish it from l'arte
minore, the lesser art, a term which he applied to the
modern arithmetic. His second variant, la regula de la
cosa, the rule of the thing or unknown quantity, appears
to have been in common use in Italy, and the word cosa
was preserved for several centuries in the forms coss or
algebra, cossic or algebraic, cossist or algebraist, &c.
Other Italian writers termed it the Regula rei et census,
the rule of the thing and the product, or the root and the
square. The principle underlying this expression is probably
to be found in the fact that it measured the limits of
their attainments in algebra, for they were unable to solve
equations of a higher degree than the quadratic or square.
Franciscus Vieta (Francois Viete) named it Specious
Arithmetic, on account of the species of the quantities
involved, which he represented symbolically by the various
letters of the alphabet. Sir Isaac Newton introduced the term
Universal Arithmetic, since it is concerned with the doctrine
of operations, not affected on numbers, but on general symbols.
Notwithstanding these and other idiosyncratic appellations,
European mathematicians have adhered to the older
name, by which the subject is now universally known.
It is difficult to assign the invention of any art or science
definitely to any particular age or race. The few fragmentary
records, which have come down to us from past civilizations,
must not be regarded as representing the totality of their
knowledge, and the omission of a science or art does not
necessarily imply that the science or art was unknown. It was
formerly the custom to assign the invention of algebra to the
Greeks, but since the decipherment of the Rhind papyrus
by Eisenlohr this view has changed, for in this work there
are distinct signs of an algebraic analysis. The particular
problem---a heap (hau) and its seventh makes 19---is solved
as we should now solve a simple equation; but Ahmes varies his
methods in other similar problems. This discovery carries the
invention of algebra back to about 1700 B.C., if not earlier.
It is probable that the algebra of the Egyptians was of
a most rudimentary nature, for otherwise we should expect
to find traces of it in the works of the Greek aeometers.
of whom Thales of Miletus (640-546 B.C.) was the first.
Notwithstanding the prolixity of writers and the number of the
writings, all attempts at extracting an algebraic analysis
from their geometrical theorems and problems have been
fruitless, and it is generally conceded that their analysis
was geometrical and had little or no affinity to algebra. The
first extant work which approaches to a treatise on algebra
is by Diophantus (q.v.), an Alexandrian mathematician, who
flourished about A.D. 350. The original, which consisted
of a preface and thirteen books, is now lost, but we have a
Latin translation of the first six books and a fragment of
another on polygonal numbers by Xylander of Augsburg (1575),
and Latin and Greek translations by Gaspar Bachet de Merizac
(1621-1670). Other editions have been published, of which
we may mention Pierre Fermat's (1670), T. L. Heath's (1885)
and P. Tannery's (1893-1895). In the preface to this work,
which is dedicated to one Dionysius, Diophantus explains his
notation, naming the square, cube and fourth powers, dynamis,
cubus, dynamodinimus, and so on, according to the sum in
the indices. The unknown he terms arithmos, the number,
and in solutions he marks it by the final s; he explains
the generation of powers, the rules for multiplication and
division of simple quantities, but he does not treat of the
addition, subtraction, multiplication and division of compound
quantities. He then proceeds to discuss various artifices
for the simplification of equations, giving methods which
are still in common use. In the body of the work he displays
considerable ingenuity in reducing his problems to simple
equations, which admit either of direct solution, or fall
into the class known as indeterminate equations. This latter
class he discussed so assiduously that they are often known as
Diophantine problems, and the methods of resolving them as
the Diophantine analysis (see EQUATION, Indeterminate.)
It is difficult to believe that this work of Diophantus
arose spontaneously in a period of general stagnation. It
is more than likely that he was indebted to earlier writers,
whom he omits to mention, and whose works are now lost;
nevertheless, but for this work, we should be led to assume
that algebra was almost, if not entirely, unknown to the Greeks.
The Romans, who succeeded the Greeks as the chief
civilized power in Europe, failed to set store on their
literary and scientific treasures; mathematics was all but
neglected; and beyond a few improvements in arithmetical
computations, there are no material advances to be recorded.
In the chronological development of our subject we have now to
turn to the Orient. Investigation of the writings of Indian
mathematicians has exhibited a fundamental distinction between
the Greek and Indian mind, the former being pre-eminently
geometrical and speculative, the latter arithmetical and mainly
practical. We find that geometry was neglected except in so far
as it was of service to astronomy; trigonometry was advanced,
and algebra improved far beyond the attainments of Diophantus.
The earliest Indian mathematician of whom we have certain
knowledge is Aryabhatta, who flourished about the beginning
of the 6th century of our era. The fame of this astronomer
and mathematician rests on his work, the Aryabhattiyam, the
third chapter of which is devoted to mathematics. Ganessa, an
eminent astronomer, mathematician and scholiast of Bhaskara,
quotes this work and makes separate mention of the cuttaca
(``pulveriser''), a device for effecting the solution of
indeterminate equations. Henry Thomas Colebrooke, one of the
earliest modern investigators of Hindu science, presumes that
the treatise of Aryabhatta extended to determinate quadratic
equations, indeterminate equations of the first degree, and
probably of the second. An astronomical work, called the
Surya-siddhanta (``knowledge of the Sun''), of uncertain
authorship and probably belonging to the 4th or 5th century,
was considered of great merit by the Hindus, who ranked it
only second to the work of Brahmagupta, who flourished about
a century later. It is of great interest to the historical
student, for it exhibits the influence of Greek science upon
Indian mathematics at a period prior to Aryabhatta. After an
interval of about a century, during which mathematics attained
its highest level, there flourished Brahmagupta (b. A.D. 598),
whose work entitled Brahma-sphuta-siddhanta (``The revised
system of Brahma'') contains several chapters devoted to
mathematics. Of other Indian writers mention may be made of
Cridhara, the author of a Ganita-sara (``Quintessence of
Calculation''), and Padmanabha, the author of an algebra.
A period of mathematical stagnation then appears to have
possessed the Indian mind for an interval of several centuries,
for the works of the next author of any moment stand but little
in advance of Brahmagupta. We refer to Bhaskara Acarya, whose
work the Siddhanta-ciromani (``Diadem of anastronomical
System''), written in 1150, contains two important chapters, the
Lilavati (``the beautiful [science or art]'') and Viga-ganita
(``root-extraction''), which are given up to arithmetic and algebra.
English translations of the mathematical chapters of the
Brahma-siddhanta and Siddhanta-ciromani by H. T. Colebrooke
(1817), and of the Surya-siddhanta by E. Burgess, with
annotations by W. D. Whitney (1860), may be consulted for details.
The question as to whether the Greeks borrowed their algebra
from the Hindus or vice versa has been the subject of much
discussion. There is no doubt that there was a constant traffic
between Greece and India, and it is more than probable that an
exchange of produce would be accompanied by a transference of
ideas. Moritz Cantor suspects the influence of Diophantine
methods, more particularly in the Hindu solutions of indeterminate
equations, where certain technical terms are, in all probability,
of Greek origin. However this may be, it is certain that
the Hindu algebraists were far in advance of Diophantus. The
deficiencies of the Greek symbolism were partially remedied;
subtraction was denoted by placing a dot over the subtrahend;
multiplication, by placing bha (an abbreviation of bhavita,
the ``product'') after the factom; division, by placing the
divisor under the dividend; and square root, by inserting
ka (an abbreviation of karana, irrational) before the
quantity. The unknown was called yavattavat, and if there
were several, the first took this appellation, and the others
were designated by the names of colours; for instance, x
was denoted by ya and y by ka (from kalaka, black).
A notable improvement on the ideas of Diophantus is to be found
in the fact that the Hindus recognized the existence of two roots
of a quadratic equation, but the negative roots were considered
to be inadequate, since no interpretation could be found for
them. It is also supposed that they anticipated discoveries
of the solutions of higher equations. Great advances were
made in the study of indeterminate equations, a branch of
