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ALGEBRA, 
A GENERAL method of refolving mathematical prob¬ 
lems by means of equations. Or, it is a method of 
performing the calculations of all forts of quantities by 
means of general ligns or charafilers. At firft, numbers 
and things were exprefled by their names at full length; 
but afterwards thefe were abridged, and the initials of tile 
words ufed inftead of them; and, as the art advanced 
farther, the letters of the alphabet came to be employed 
as general reprefentations of all forts of quantities ; and 
other marks were gradually introduced, toexprefs all forts 
of operations and combinations; fo as to entitle it to dif¬ 
ferent appellations —univtrfal arithmetic, and literal arith¬ 
metic, and the arithmetic of Jigns. And iince its operations 
in the refolution of problems are performed by taking 
them always backwards in the contrary order, it hence be¬ 
comes a Ipecies of the analytic art, and is called the mo¬ 
dern analvfis, in contradiftinfilion to the ancient analyfs, 
which chiefly refpefiled geometry and its applications. 
The etymology of the name, algebra , is given in various 
ways. It is an Arabic word, and by the Arabians is cou¬ 
pled with the word macabelah , fignifying oppolition and 
companion, to exprefs what we properly call algebra. 
Some derive it from Gebar, a celebrated philofopher, 
chemiu, and mathematician, whom the Arabs call Giabcr ; 
but others, with more probability, derive it from geber, 
by prefixing the article al, which properly fignifies the 
reduction of fractions to a whole number. 
As to the origin of the analytic art, of which algebra is 
a fpecies, it is doubtlefs as old as any fcience in the world, 
being the natural method by which the mind inveftigates 
truths, caufes, and theories, from their obferved eifefils 
and properties. Accordingly, traces of it are obfervable 
in the Works of the earlieft philofophers and mathemati¬ 
cians, the fubjefit of whofe refearches moft of any required 
the aid of fuch an art. The elded treatife, however, 
which has come down to us, is that of Diophantus of 
Alexandria, who flourilhed about the year 350 after Chrift, 
and who wrote, in the Greek language, thirteen books of 
Algebra or Arithmetic, though only fix of them have hi¬ 
therto been printed; and an imperfect book on Multan¬ 
gular Numbers, in a Latin tranflation, by Xilander, in the 
year 1575, and afterwards in 1621 and 1670 in Greek and 
Latin by Gafpar Bachet. Thele books however do not 
contain a treatife on the elementary parts of algebra, but 
only collefilions of difficult queftions relating to fquareand 
cube numbers, and other curious properties of numbers, 
with their folutions. 
But, although Diophantus was the firft author on alge¬ 
bra that we now know of, it was not from him, but from the 
Moors or Arabians, that we received the knowledge of al¬ 
gebra in Europe, as well as that of moft other fciences. 
And it i,s matter of difpute who were the firft inventors of 
it; fome aferibing the invention to the Greeks, while 
others fay that the Arabians had it from the Perfians, and 
thefe from the Indians, as well as the arithmetical.method 
of computing by ten charafilers, or digits; but the Ara¬ 
bians themfelves fay it was invented amongft them by Ma¬ 
homet ben Mufa , or fon of Mofes, who it feems flourilhed 
about the 8th or 9th century. It is more probable, how¬ 
ever, that Mahomet was not the inventor, but only a 
perfon well (killed in the art; and it is farther probable, 
that the Arabians drew their firft knowledge of it from 
Diophantus or other Greek writers, as they i did that of 
geometry and other fciences, which they improved and 
tranfiated into their own language; and from them it was 
that we received thefe fciences, before the Greek authors 
were known to us, after the Moors fettled in Spain, and ■ 
after the Europeans began to hold communications with 
them, and that our countrymen began to travel ambngft 
them to- learn the fciences. And according to the tefti- 
Vol. I. No. 18, 
moriy of Abulpharagius, the Arithmetic of Diophantus 
w r as tranflated into Arabic by Mahomet ben-yahya Baziani. 
But whoever were the inventors and firft cultivators of al¬ 
gebra, it is certain that the Europeans firft received the 
knowledge, as w-ell as the name, from the Arabians or 
Moors, in confequence of the clofe intercourfe which fub- 
fifted between them for feveral centuries. And it appears 
that the art was pretty generally known, and much culti¬ 
vated, at lead in Italy, if not in other parts of Europe 
alfo, long before the invention of printing, as many wri¬ 
ters upon the art are ftill extant in the libraries of rnanu- 
feripts; and the firft authors, prefently after the inven¬ 
tion of printing, (peak of many former writers on this 
fubjefit, from whom they learned the art. 
It was chiefly among the Italians that .this art was firft 
cultivated in Europe. And the firft author whofe works 
we have in print, was Lucas Paciolus, or Lucas de Burgo, 
a Cordelier, or Minorite Friar. He wrote feveral treatifes 
of Arithmetic, Algebra, and Geometry, which were 
printed in the years 1470, 1476, 1481, 1487, and in 1494. 
His principal work, intitled Summa de Arithmetica, Geome- 
tria, Proportioni, et Proportionalita, is a very mafterly and 
complete treatife on thofe fciences, as they then flood. In 
this work Ire mentions various former writers, as Euclid, 
St. Auguftine, Sacrobofcoor Halifax, Boetius, Prodocimo, 
Giordano, Biagio da Parma, and Leonardus Pifanus, front 
whom he learned thofe fciences. 
After Paciolus appeared Stifelius and Scheubelius, Ger¬ 
man authors; and after them came Scipio Ferreus, Car¬ 
dan, Tartalea, and feveral others, whofe works reached as 
far as the folution of fome cubic equations. Ramus and 
Bombelli followed thefe, and went a little farther. At 
■laft came Nunnius, Stevinus, Schoner, Salignac,.Clavius, 
See. who all of them took different courfes, but none of 
them went beyond quadratics. 
In 1599, Vieta introduced what he called Iris “Specious 
Arithmetic,” which confifts in denoting the quantities, 
both known and unknown, by fymbols or letters, ftle.fdfo 
introduced an ingenious method of extrafiling’the roots of 
equations by approximations; fmee greatly improved and 
facilitated by Raphfon, Halley, Maclaurin, Simpfon, and 
others. 
Vieta was followed by Oughtred, who, in his Clavis 
Mathematica, printed in 1631, improved Vieta’s method, 
and invented feveral compendious charafilers, to fnew the 
fums, differences, rectangles, fquares, cubes, Ac. 
Harriot, another Engli(hman, cotemporary with Ough¬ 
tred, left feveral treatifes at his death; and, among the 
reft, an Analyfis, or Algebra, which was printed in 1631, 
where Vieta’s method is brought into a ftill more commo¬ 
dious form, and is much efteemed to this day. 
In 163-7, Des Cartes publiflied his geometry, wherein he 
made life of the literal calculus and the algebraic rules of 
Harriot; and as Oughtred in his Clavis, and Marin. Ghe- 
taldus in his books of mathematical compofition and re¬ 
folution publiflied in 1630, applied Vieta’s, arithmetic to 
elementary geometry, and gave the conftrufition of Ample 
and quadratic equations; fo Des Cartes applied Harriot’s 
method to the'higher geometry, explaining the nature of 
curves by equations, and adding the conftrufilions of cubic, 
biquadratic, and other higher, equations. 
Des Cartes’s rule for cdnftrufiting cubic and biquadratic 
equations was farther improved by Thomas Baker, in his 
Clavis Geometrica Catholica , publiflied in 1684; and the 
foundation of■ fuch conftrufilions, with the application of 
algebra to the quadratures of curves, queftions de maximis 
et minimis, the centrobaryc method of GuldVmis, &c. was 
given by R. Slufius, in 1668; as alfo by Fermat in his 
Opera Mathematica, Roberval in the Mem. de Malkem. et de 
Phyfique, and Barrow in his I.cEt. Geomet. In 1708, algebra 
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