AusdehnungslehreThe 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 with 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 is the third in an informal sequence of works to be included within the History of Mathematics series, co-published by the AMS and the London Mathematical Society. Volumes in this subset are classical mathematical works that served as cornerstones for modern mathematical thought. |
Contents
| 1 | |
| 3 | |
| 19 | |
Combinatorial Product | 29 |
Inner Product | 93 |
Applications to Geometry | 123 |
Part 2 The Theory of Functions | 191 |
Functions in General | 193 |
Differential Calculus | 249 |
Infinite Series | 263 |
Integral Calculus | 279 |
Common terms and phrases
1)th order a₁ ABCD according to 60 algebraic arbitrary magnitudes assembly assumed assumption Ausdehnungslehre b₁ circles circular evolution coefficients combinatorial product corresponding DEFINITION denote derivation numbers different from zero differential quotients displacements domain of nth E₁ Editorial Note elementary factors elementary magnitudes extensive magnitudes formula fraction function geometry GRASSMANN Hermann Grassmann includes infinitely distant points inner product integration interchangeable length and direction line element linear evolutions magnitudes a1 multiple sum nonzero normal system normal units nth order numerical relation numerical value numerically derivable numerically equal obtains original units orthogonal outer multiplication outer product parallel parallelepiped parallelogram PFAFFian plane planimetric principal domain progressive product PROOF regressive Remark following replaces shadow stand straight line supplement surface element theorem transformed true series units e1 variable numerical magnitudes whence according zeroth order
Popular passages
Page 4 - To add two extensive magnitudes derived from the same system of units means to add their derivation numbers belonging to the same units, that is 7.
Page 3 - I say that a magnitude a is derived from the magnitudes b, c, . . . by the numbers /?, 7, . . . if where /3, 7, . . . are real numbers, rational or irration'al, different from zero or not.
Page xiii - To remove that difficulty thus became an essential task for me if I wanted the book to be read and understood by others as well as myself.