Extension Theory / Edition 1

by Hermann Grassmann, Lloyd C. Kannenberg (Translator)

Overview

The Ausdehnungslehre of 1862 is Grassmann's most mature presentation of his ''extension theory''. The work was unique in capturing the full sweep of his mathematical achievements. Compared to Grassmann's first book, Lineale Ausdehnungslehre, this book contains an enormous amount of new material, including a detailed development of the inner product and its relation to the concept of angle, the ''theory of functions'' from the point of view of extension theory, and Grassmann's contribution to the Pfaff problem. In many ways, this book is the version of Grassmann's system most accessible to contemporary readers. This translation is based on the material in Grassmann's ''Gesammelte Werke'', published by B. G. Teubner (Stuttgart and Leipzig, Germany). It includes nearly all the Editorial Notes from that edition, but the ''improved'' proofs are relocated, and Grassmann's original proofs are restored to their proper places. The original Editorial Notes are augmented by Supplementary Notes, elucidating Grassmann's achievement in modern terms. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, ''Sources'', are classical mathematical works that served as cornerstones for modern mathematical thought.

Product Details

ISBN-13:	9780821820315
Publisher:	American Mathematical Society
Publication date:	03/01/2000
Series:	History of Mathematics Series , #19
Pages:	411
Product dimensions:	6.69(w) x 9.84(h) x (d)
Language:	German

Table of Contents

	Translator's Note	ix
	Foreword	xiii
Part 1.	The Elementary Conjunctions of Extensive Magnitudes	1
Chapter 1.	Addition, Subtraction, Multiples and Fractions of Extensive Magnitudes	3
1.	Concepts and laws of calculation	3
2.	Connection between the magnitudes derived from a system of units	8
3.	Number as the quotient of extensive magnitudes and the replacement of the equations between extensive magnitudes by numerical equations	14
Chapter 2.	The Product Structure in General	19
1.	Product of two magnitudes	19
2.	Product of several magnitudes	22
3.	The various types of product structure	24
Chapter 3.	Combinatorial Product	29
1.	General laws of combinatorial multiplication	29
2.	The combinatorial product as magnitude	36
3.	Outer multiplication of magnitudes of higher order	45
4.	Supplement of magnitudes with respect to a principal domain	49
5.	Product with respect to a principal domain	52
6.	Interchange of factors and removal of parentheses in a pure and in a mixed product	69
7.	Shadow and replacement	80
8.	Elimination of the unknowns from algebraic equations by combinatorial multiplication	85
Chapter 4.	Inner Product	93
1.	Fundamental laws of inner multiplication	93
2.	Concept of the normal and its correlates	98
3.	Laws of the inner product associated with the concept of the normal	103
4.	Special theorems on the inner multiplication of two magnitudes of first order	115
5.	Introduction of the angle	117
Chapter 5.	Applications to Geometry	123
1.	Addition, subtraction, multiples and fractions of points and displacements	123
2.	Spatial domains	131
3.	Combinatorial multiplication of points	138
4.	Addition of lines and surfaces	151
5.	Planimetric and stereometric multiplication	157
6.	Special laws for a planimetric {and stereometric} product set to zero. Plane {algebraic} curves. {Algebraic surfaces}	162
7.	Inner multiplication in geometry	176
Part 2.	The Theory of Functions	191
Chapter 1.	Functions in General	193
1.	Concept of a function, and reduction of several functions of several variables to a single function of a single variable	193
2.	Complete functions and their representation by open products	196
3.	Algebraic multiplication	201
4.	Complete functions of first degree. Quotient	207
5.	Functions as extensive magnitudes	226
6.	Relations considered from the standpoint of functional conjunctions	231
7.	Normal units of functions, continuity of the latter	239
Chapter 2.	Differential Calculus	249
1.	Differential of first order	249
2.	Differential quotient of first order	252
3.	Differentials of higher order	258
Chapter 3.	Infinite Series	263
1.	Infinite series in general	263
2.	Series as functions of a numerical magnitude	265
3.	Development of functions of several numerical magnitudes or a single extensive magnitude in series	273
Chapter 4.	Integral Calculus	279
1.	Integration of differential expressions	279
2.	Integration of differential equations when the independent variable is a numerical magnitude	288
3.	Integration of differential equations when the independent variable is an extensive magnitude	294
	Index of Technical Terms	327
	Editorial Notes	331
	Grassmann's investigations into Pfaff's problem	378
	Supplementary Notes	391
	Subject Index	399

Editorial Reviews

A translation of Grassmann's (1809-77) fullest and most mature presentation of his mathematical thought, (1862). It is based on a later edition in his collected works, and includes nearly all the editorial notes but relocates the improved proofs and restores the original proofs to their proper places. Additional notes explain the significance of his work in modern terminology. He develops in detail the inner product and its relation to the concept of angle, the theory of functions from the perspective of extension theory, and his contribution to the Pfaff problem. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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