Geometric Calculus: According to the Ausdehnungslehre of H. GrassmannThe 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. |
Contents
Geometric Formations | 15 |
Formations of the First Species | 5 |
Formations of the Second Species | 19 |
Formations of the Third Species | 29 |
Formations on a RightLine | 33 |
Formations in the Plane | 37 |
Formations in Space | 57 |
Derivatives | 83 |
Transformations of Linear Systems | 95 |
Editorial Notes | 121 |
Subject Index | 123 |
Other editions - View all
Geometric Calculus: According to the Ausdehnungslehre of H. Grassmann Giuseppe Peano Limited preview - 2013 |
Geometric Calculus: According to the Ausdehnungslehre of H. Grassmann Giuseppe Peano No preview available - 2011 |
Common terms and phrases
A U B a₁ AB.CD ABCD algebraic analogous angle Ausdehnungslehre b₁ barycenter bivector called categorical proposition coefficients conditional proposition construct contains correspond cos(I cos(U deduces Definition derivative dihedra dihedron entities equal in magnitude equivalent expression fixed formulas geometric calculus geometric formations given lines Grassmann homogeneous function homothetic identity indicated integers linear system locus logical equations m₁ monomial multiplying numerical function osculating plane parallel perpendicular point at infinity point of intersection polygonal position quaternions recovers reducible regressive product represents respect right-line S₁ satisfied says second species sense separated form sin(a sin(I sin(U sine space SS₁ substitution suppose surface symbol taking at random tan² tetrahedron Theorem third species transformation triangle trihedron u₁ unit of measure unit trivector variable vectors equal vice-versa volume whence zero
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