GEOMETRICAL THEOREMS AND PROBLEMS WITH THEIR HISTORY BY WILLIAM W. RUPERT, C.E. SUPERINTENDENT OF SCHOOLS, POTTSTOWN, PA. "I am sure that no subject loses more than BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS PREFACE. THE author, having derived much pleasure and inspiration from the brief historical notes in some of the mathematical text-books that he studied when a student in college, has thought that, by giving the history of a few of the most celebrated geometrical theorems and problems, he might place a "light in the window" which may throw a cheerful ray adown the long and sometimes dusty pathway that leads to geometrical truth. In the preparation of this little book most valuable assistance has been derived from Florian Cajori's History of Mathematics, James Gow's History of Greek Mathematics, and G. J. Allman's Greek Geometry from Thales to Euclid. It is, however, to W. W. Rourse Ball's remarkably interesting Short History of Mathematics that Famous Geometrical Theorems and Problems owes the largest debt. To Professor A. D. Eisenhower, Principal of the Norristown High School, George Q. Sheppard, Professor of Mathematics, Hill School, Pottstown, Pa., Dr. George M. Philips, Principal West Chester State Normal School, and Daniel Carhart, C.E., Dean and Professor of Civil Engineering, Western University of Pennsylvania, who have read this book in manuscript, the author is indebted for valuable suggestions and many kind words of encouragement. For the excellent and accurate diagrams, and for proof No. V. in Chapter I., my thanks are due to my young friend and former pupil, Luther D. Showalter, C.E. POTTSTOWN, PA., December 23, 1899. WILLIAM W. RUPERT. FAMOUS GEOMETRICAL THEOREMS. CHAPTER I. THEOREM. 1. The sum of the three angles of every plane triangle is equal to two right angles. The mathematical truth enunciated in the above theorem is not new. It has been known for more than two thousand years. Thales, one of the seven sages of Greece, who was born about 640 B.C., must have been aware that the sum of the angles of a triangle is equal to two right angles. Our reason for believing that Thales was not ignorant of the theorem under consideration is found in the beautiful demonstration by which he proved that every angle in a semicircle is a right angle. This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a |