Extension Theory (History of Mathematics, 19.)

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. 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.--jacket. The Elementary Conjunctions Of Extensive Magnitudes -- Addition, Subtraction, Multiples And Fractions Of Extensive Magnitudes -- Concepts And Laws Of Calculation -- Connection Between The Magnitudes Derived From A System Of Units -- Number As The Quotient Of Extensive Magnitudes And The Replacement Of The Equations Between Extensive Magnitudes By Numerical Equations -- The Product Structure In General -- Product Of Two Magnitudes -- Product Of Several Magnitudes -- The Various Types Of Product Structure -- Combinatorial Product -- General Laws Of Combinatorial Multiplication -- The Combinatorial Product As Magnitude -- Outer Multiplication Of Magnitudes Of Higher Order -- Supplement Of Magnitudes With Respect To A Principal Domain -- Product With Respect To A Principal Domain -- Interchange Of Factors And Removal Of Parentheses In A Pure And In A Mixed Product -- Shadow And Replacement -- Elimination Of The Unknowns From Algebraic Equations By Combinatorial Multiplication -- Inner Product -- Fundamental Laws Of Inner Multiplication -- Concept Of The Normal And Its Correlates -- Laws Of The Inner Product Associated With The Concept Of The Normal -- Special Theorems On The Inner Multiplication Of Two Magnitudes Of First Order -- Introduction Of The Angle -- Applications To Geometry -- Addition, Subtraction, Multiples And Fractions Of Points And Displacements -- Spatial Domains -- Combinatorial Multiplication Of Points -- Addition Of Lines And Surfaces -- Planimetric And Stereometric Multiplication. Hermann Grassmann ; Translated By Lloyd Kannenberg. Includes Bibliographical References And Index. "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."--BOOK JACKET.