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The geometric calculus, in general, consists in a system of
operations on geometric entities, and their consequences, analogous
to those that algebra has on the num bers. It permits the
expression in formulas of the results of geometric constructions,
the representation with equations of propositions of geometry, and
the substitution of a transformation of equations for a verbal
argument. The geometric calculus exhibits analogies with analytic
geometry; but it differs from it in that, whereas in analytic
geometry the calculations are made on the numbers that determine
the geometric entities, in this new science the calculations are
made on the geometric entities themselves. A first attempt at a
geometric calculus was due to the great mind of Leibniz (1679);1 in
the present century there were proposed and developed various
methods of calculation having practical utility, among which
deserving special mention are 2 the barycentric calculus of Mobius
(1827), that of the equipollences of Bellavitis (1832),3 the
quaternions of Hamilton (1853),4 and the applications to geometry 5
of the Ausdehnungslehre of Hermann Grassmann (1844). Of these
various methods, the last cited to a great extent incorporates the
others and is superior in its powers of calculation and in the
simplicity of its formulas. But the excessively lofty and abstruse
contents of the Ausdehnungslehre impeded the diffusion of that
science; and thus even its applications to geometry are still very
little appreciated by mathematicians."
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