This volume of the author's mathematical discoveries makes its appearance at the request of many able mathematicians, teachers, and scholars throughout the United States, among a few of whom are: J. M. Greenwood, Superintendent of Kansas City Schools and President of the National Association of Education; N. B. Newsom, Professor of Mathematics and Languages at the Kansas State University, Lawrence; R. P. Baker, Professor of Mathematics and Languages, Lamar, and J. C. Shelton, President of Scarritt College, Neosho, Missouri; F. С Colburn, Principal of High School, Texarkana, Texas; and L. D. Harvey, State Superintendent of Wisconsin.