Carl Gauss a inovat cartografia si topografia, un lucru mai putin cunoscut despre el fiind faptul ca acesta este, de fapt, inventatorul unei versiuni timpurii a telegrafului, in ciuda faptului ca aceasta inventie a fost perfectionata si patentata ulterior in Statele Unite de catre Samuel Morse. Gauss a pus bazele calculului cu numere complexe, tot lui datorandu-i-se si denumirea acestor numere. Tot el a dat interpretarea geometrica a numerelor complexe si a stabilit corespondenta biunivoca dintre numerele complexe si punctele planului. Tot Gauss este reponsabil si pentru introducerea seriei hipergeometrica, aceasta avand un rol important in teoria ecuatiilor diferentiale. In geometria diferentiala a gasit formulele fundamentale ale suprafetelor si a elaborat o teorie a liniilor geodezice. In jurul anului , Gauss s-a cunoscut cu mai tanarul Wilhelm Weber si au lucrat impreuna la problemele asociate cu electromagnetismul, supus, in acea perioada, unui proces nou de conceptualizare de catre Michael Faraday.
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I offer it to YOU as a sacred token of my filial devotion. Were it not for YOUR unceasing benefits in support of my studies, I would not have been able to devote myself totally to my passionate love, the study of mathematics. It has been YOUR generosity alone which freed me from other cares, allowed me to give myself to so many years of fruitful contemplation and study, and finally provided me the opportunity to set down in this volume some partial results of my investigations.
And when at length I was ready to present my work to the world, it was YOUR munificence alone which removed all the obstacles that threatened to delay its publication. Now that I am constrained to acknowledge YOUR remarkable bounty toward me and my work I find myself unable to pay a just and worthy tribute.
I am capable only of a secret and ineffable admiration. And everyone knows that You have never excluded from YOUR patronage those sciences which are commonly regarded as being too recondite and too removed from ordinary life.
Therefore I present this book as a witness to my profound regard for You and to my dedication to the noblest of sciences. Most Serene Prince, if YOU judge it worthy of that extraordinary favour which You have always lavished on me, I will congratulate myself that my work was not in vain and that I have been graced with that honour which I prize above all others. I will rarely refer to fractions and never to surds. The Analysis which is called indeterminate or Diophantine and which discusses the manner of selecting from an infinite set of solutions for an indeterminate problem those that are integral or at least rational and especially with the added condition that they be positive is not the discipline to which I refer but rather a special part of it, just as the art of reducing and solving equations Algebra is a special part of universal Analysis.
However what is commonly called Arithmetic hardly extends beyond the art of enumerating and calculating i. It often includes some subjects which certainly do not pertain to Arithmetic like the theory of logarithms and others which are common to all quantities. As a result it seems proper to call this subject Elementary Arithmetic and to distinguish from it Higher Arithmetic which properly includes more general inquiries concerning integers.
We consider only Higher Arithmetic in the present volume. The celebrated work of Diophantus, dedicated to the problem of indeterminateness, contains many results which excite a more than ordinary regard for the ingenuity and proficiency of the author because of their difficulty and the subtle devices he uses, especially if we consider the few tools that he had at hand for his work.
They opened the door to what is penetrable in this divine science and enriched it with enormous wealth. I shall give them their due praise in the proper places in these pages. The purpose of this volume whose publication I promised five years ago is to present my investigations into the field of Higher Arithmetic. Lest anyone be surprised that the contents here go back over many first principles and that many results had been given energetic attention by other authors, I must explain to the reader that when I first turned to this type of inquiry in the beginning of I I was unaware of the more recent discoveries in the field and was without the means of discovering them.
What happened was this. Engaged in other work I chanced on an extraordinary arithmetic truth if I am not mistaken, it was the theorem of art. Since I considered it so beautiful in itself and since I suspected its connection with even more profound results, I concentrated on it all my efforts in order to understand the principles on which it depended and to obtain a rigorous proof.
When I succeeded in this I was so attracted by these questions that I could not let them be. Thus as one result led to another I had completed most of what is presented in the first four sections of this work before I came into contact with similar works of other geometers.
Only while studying the writings of these men of genius did I recognize that the greater part of my meditations had been spent on subjects already well developed. But this only increased my interest, and walking in their footsteps I attempted to extend Arithmetic further. After a while I began to consider publishing the fruits of my new awareness. And I allowed myself to be persuaded not to omit any of the early results, because at that time there was no book that brought together the works of other geometers, scattered as they were among Commentaries of learned Academies.
Besides, many of these results are so bound up with one another and with subsequent investigations that new results could not be explained without repeating from the beginning. Since this book came to my attention after the greater part of my work was already in the hands of the publishers, I was unable to refer to it in analogous sections of my book.
I felt obliged, however, to add some observations in an Appendix and I trust that this understanding and illustrious man will not be offended. The publication of my work was hindered by many obstacles over a period of four years. During this time I continued investigations which I had already undertaken and deferred to a later date so that the book would not be too large, but I also undertook new investigations.
Similarly, many questions which I touched on only lightly because a more detailed treatment seemed less necessary e. Finally, since the book came out much larger than I expected, owing to the size of Section V, 1 shortened much of what I first intended to do and, especially, I omitted the whole of Section Eight even though I refer to it at times in the present volume; it was to contain a general treatment of algebraic congruences of indeterminate rank.
All the treatises which complement the present volume will be published at the first opportunity. In several difficult discussions I have used synthetic proofs and have suppressed the analysis which led to the results. This was necessitated by brevity, a consideration that must be consulted as much as possible. The theory of the division of a circle or of a regular polygon treated in Section VII of itself does not pertain to Arithmetic but the principles involved depend uniquely on Higher Arithmetic.
This will perhaps prove unexpected to geometers, but I hope they will be equally pleased with the new results that derive from this treatment.
These are the things I wanted to warn the reader about. It is not my place to judge the work itself. My greatest hope is that it pleases those who have at heart the development of science and that it proposes solutions that they have been looking for or at least opens the way for new investigations.
Despre Carl Friedrich Gauss, denumit neoficial si „Printul Matematicii”
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