After the Russian edition of this book appeared some of my fellow Lecturers asserted that many of the Lectures in the book are far too long to be physically delivered during the allowed two teaching periods. By the right of friendship I had to remind them that a Lecturer must prepare for his Lectureseven if he has been lecturing for over a dozen years and make in advance an elaborate, practically minute-by-minute plan of every Lecture. It is necessary to consider eforehand the rhythm of the Lecture to be deliveredwhat portions of it are to be read slowly, almost at dictation speed, and what may be said quickerand its pattern of intonationwhere to raise the voice and where to lower it. One also needs a joke somewhere about the middle of the Lecture to rouse the tired students and it should be prepared yet at home, and in every detail, up to a play of facial muscles. It goes without saying that one must plan in advance what to write on the blackboard and in what order and where, and when to delete anything, and coordinate all this with everything else. It is surprising how all this extends the limits of Lecture time and how much it is then possible to say in an outwardly unhurried and thorough manner, with numerous repetitions and explanations.
Some reviewers have reproached me for a systematic use of bivectors and trivectors saying that one may well do without them. Some well-known physicist, Max Planck, I think, once said that new ideas (he meant scientific ideas but this can be fully applied to methodical ideas as well), could win only when their opponents have retired from the stage as a result of a natural change of generations. An ex- cellent example illustrating this thesis is the introduction of vectors into the courses of analytic geometry half a century ago. Now only a few people remember the fierce discussions concerning this matter and the present generation does not know how many a lance was broken and how much ink split in attempts to prove that vectors were a harmful thing because replacing three equations in coordinates by one vector equation they saved paper but proportionally hampered comprehension. The last of the authoritative opponents of vectors in the USSR died soon after the war but some ten years more passed before diffidently excusatory reservations disappeared altogether from vectorial presentations of geometry (as well as from mechanics and physics where, however, this happened a little earlier). Now bivectors and tri-vectors are awaiting their turn.
I have taken the opportunity to introduce some minor improvements in the text. The most serious one is perhaps a simpler construction of the complexification of an affine space in Lecture 19. It is true that it contains a certain element of arbitrariness (which was what restrained me at first) but experience has shown that this arbitrariness is perfectly harmless. Besides, the last Lecture has been divided into two since two versions of the concluding Lecture were combined in it for purely technical, internal editorial reasons. So the book contains 29 Lectures now.
As far as I can judge with my poor knowledge of English the translation is well done and conveys all the nuances of my thought. M.M. Postnikov May 1, 1980
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