Introduction to Analysis of the Infinite

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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". 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