From the preface of the author: "
Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra, even of all the algebraic technique so far discovered, still there are topics whose consideration prepares a student for a deeper understanding. However, in the ordinary treatise on the elements of algebra, these topics are either completely omitted or are treated carelessly. For this reason, I am certain that the material I have gathered in this book is quite sufficient to remedy that defect. I have striven to develop more adequately and clearly than is the usual case those things which are absolutely required for analysis. More- over, I have also unraveled quite a few knotty problems so that the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite. There are also many questions which are answered in this work by means of ordinary algebra, although they are usually discussed with the aid of analysis. In this way the interrelationship between the two methods becomes clear".
Author(s): Leonard Euler
Edition: 1
Publisher: Springer
Year: 1990
Language: English
Pages: 515
City: New York, NY
Tags: Analysis, Differential & Integral Calculus, Algebraic Geometry
CONTENTS
Preface------------------------------------------------------------------------------------------v
Translator’s Introduction-----------------------------------------------------------------------ix
Book II------------------------------------------------------------------------------------------1
I. On Curves in General----------------------------------------------------------------------3
II. On the Change of Coordinates-------------------------------------------------------------12
III. On the Classification of Algebraic Curves by Orders--------------------------------------26
IV. On the Special Properties of Lines of Any Order------------------------------------------35
V. On the Second Order Lines----------------------------------------------------------------44
VI. On the Subdivision of Second Order Lines into Genera-------------------------------------81
VII. On the Investigation of Branches Which Go to Infinity-----------------------------------104
VIII. Concerning Asymptotes-------------------------------------------------------------------121
IX. On the Species of Third Order Lines----------------------------------------------------137
X. On the Principal Properties of Third Order Lines----------------------------------------151
XI. On Fourth Order Lines-------------------------------------------------------------------165
XII. On the Investigation of the Configuration of Curves-------------------------------------176
XIII. On the Disposition of Curves------------------------------------------------------------183
XIV. On the Curvature of a Curve-------------------------------------------------------------194
XV. Concerning Curves with One or Several Diameters-----------------------------------------212
XVI. On Finding Curves from the Properties of the Ordinate-----------------------------------226
XVII. On Finding Curves from Other Properties-------------------------------------------------247
XVIII. On the Similarities and Affinities of Curves--------------------------------------------274
XIX. On the Intersection of Curves-----------------------------------------------------------287
XX. On the Construction of Equations--------------------------------------------------------312
XXI. On Transcendental Curves----------------------------------------------------------------330
XXII.On the Solution to Several Problems Pertaining to the Circle-----------------------------353
Appendix---------------------------------------------------------------------------------------375
I. On the Surfaces of Solids---------------------------------------------------------------377
II. On the Intersection of a Surface and an Arbitrary Plane---------------------------------394
III. On Sections of Cylinders, Cones, and Spheres--------------------------------------------406
IV. On the Change of Coordinates------------------------------------------------------------428
V. On Second Order Surfaces----------------------------------------------------------------437
VI. On the Intersection of Two Surfaces-----------------------------------------------------452
Figures----------------------------------------------------------------------------------------465
