

1. Introduction 2. Pavilion  angles of reflected rays 3. Pavilion  limits of reflected rays 4. Pavilion angle limits to reflect all incident rays 5. Best pavilion angle 6. Table size 7. Bezel angle 8. General method of determining bezel angle 9. Light entering the table 10. Central group of light rays 11. Group oblique from left 12. Group oblique from right 13. Mean light angle = bezel angle 14. Final angles and proportions 15. Girdle facets 16. Star facets 17. Comparison to actual cuts 18. Comments 19. Editors note 

Introduction  
In 1919 Marcel Tolkowsky published "Diamond Design" in which he developed the angles and proportions of the brilliant cut which have become widely accepted as ideal. Copies of this book are hard to find ^{2}, so Tolkowsky's analysis is reiterated here for those who are curious to know how these 'magic numbers' came about. This review follows Tolkowsky's logic but uses less awkward and more meaningful mathematics. It shows errors, omissions, and inconsistencies in his work, but none of these have a significant effect on the final results. Tolkowsky was intimate with diamondcutting and implies that he knew the answers before developing the theory. This is a common danger in research and it appears that some of his logic is tainted in an effort to arrive at these results. Tolkowsky's first step involves a compromise of brilliance to get good dispersion, and all subsequent calculations are based on the results of this initial step ^{3}. For this reason, and others, this method cannot be applied directly to studies of other gemstones; however, it is a good source of concepts for such efforts^{ 4}. Certain limitations of this analysis must be pointed out:

* ^{1} Never published. Submitted to GIA but declined by Richard Liddicoat, Pres. because it could undermine faith in the established cutgrading standards. Copy sent to Shinichi Minawa, Tokyo, Japan, 1976 Dec. 24. * ^{2} Most of this book has been reproduced by Moscow State University at Internet sites www.gemology.ru (Russian) and www.cutstudy.com (English). * ^{3} This analysis neglects partial reflections as rays are refracted. It cannot, therefore, be used for quantitative analysis of light returned. * ^{4} At that time this author was trying to discover criteria which governed the faceting of all transparent gemstones; this effort led to "Faceting Limits", published by GIA in 1975, presently being reproduced by Moscow State University at their web site. 
Author:
Bruce L. Harding,
Holden, MA 1975 May 15 with footnotes by BLH 2001 Mar 19 

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