As regards the analytical notes about games or Openings, I have tried to be short and to the point. Analytical detail is therefore not abundant, but I think, reliable. The method of enumerating all the variations thought possible, or probable, has been laid aside, and in its place an analysis has been given, which makes use of both the consideration of the leading variations and general principles. The diction and style of the work are those of a lecturer. Feeling that I have not been able to make them as perfect as I should have desired, I must ask for the lenient judgment of the reader. I take this opportunity for expressing my hearty thanks to Professor Villin Marmery for his kind assistance in looking over the proofs.