ths of geometry, and arranged them in logical sequence. Archimedes,
in Syracuse, had attempted the solution of the higher problems by the
method of exhaustions. Such was the tendency of things that, had the
patronage of science been continued, algebra would inevitably have been
invented.

To the Arabians we owe our knowledge of the rudiments of algebra; we
owe to them the very name under which this branch of mathematics passes.
They had carefully added, to the remains of the Alexandrian School,
improvements obtained in India, and had communicated to the subject
a certain consistency and form. The knowledge of algebra, as they
possessed it, was first brought into Italy about the beginning of the
thirteenth century. It attracted so little attention, that nearly three
hundred years elapsed before any European work on the subject appeared.
In 1496 Paccioli published his book entitled "Arte Maggiore," or
"Alghebra." In 1501, Cardan, of Milan, gave a method for the solution of
cubic equations; other improvements were contributed by Scipio Ferreo,
1508, by Tartalea, by Vieta. The Germans now took up the subject. At
this time the notation was in an imperfect state.

The publication of the Geometry of Descartes, which contains the
application of algebra to the definition and investigation of curve
lines (1637), constitutes an epoch in the history of the mathematical
sciences. Two years previously, Cavalieri's work on Indivisibles had
appeared. This method was improved by Torricelli and others. The way was
now open, for the development of the Infinitesimal Calculus, the method
of Fluxions of Newton, and the Differential and Integral Calculus
of Leibnitz. Though in his possession many years previously, Newton
published nothing on Fluxions until 1704; the imperfect notation he
employed retarded very much the application of his method. Meantime, on
the Continent, very largely through the brilliant solutions of some of
the higher problems, accomplished by the Bernouillis, the Calculus of
Leibnitz was universally accepted, and improved by many mathematicians.
An extraordinary development of the science now took place, and
continued throughout the century. To the Binomial theorem, previously
discovered by Newton, Taylor now added, in his "Method of Increments,"
the celebrated theorem that bears his name. This was in 1715. The
Calculus of Partial Differences was introduced by Euler in 1734. It was
extended by D'Alembert, and was follow