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I was curious to know whether synthetic geometry is more powerful than analytical geometry. Take as an example, the conic sections. Can one describe conic sections without the co-ordinate system, without the equations of conic sections? How difficult would it be?

codetalker
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    You are aware that all conic sections are sections of a cone with a plane, right? – xyzzyz Jul 30 '16 at 07:27
  • @xyzzyz, yes, I am – codetalker Jul 30 '16 at 07:29
  • What kind of descriptions are you looking for, then? – xyzzyz Jul 30 '16 at 07:29
  • @xyzzyz, by descriptions, I mean can we study conics using synthetic geometry? Most of the modern books and resources talk of them through equations, which is extremely simple and useful, but synthetic geometry has its own taste, isn't it? So, can they be studied using synthetic geometry? – codetalker Jul 30 '16 at 19:08

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I havent read it but I think that this book might be what you are looking for : https://archive.org/stream/conicsectionstre00besarich#page/n12/mode/1up

In the introduction we have

The object of the following pages is to discuss the general forms and characetistics of these (conic sections) curves and to determine their most important properties by help of the methods and relations developed in the first six books , and in the eleventh book of Euclid's Elements and it will be found that for this purpose a knowledge of Euclid's Geometry is all that is necessary.

Also, I am also looking for books of this type so if you have found some yourself please post them as an answer or as a comment.

alexgiorev
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Yes, synthetic geometry works well for plan sections in conics in finding the focus points, the axes, the directors line positions. Danut Dragoi, PhD

Danut Dragoi
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    Welcome to the site! I think your answer would be greatly improved by some detail, like an example or explanation. – The Count Jan 26 '17 at 02:38