FOR BEGINNERS BY A. R. HORNBROOK, A.M. TEACHER OF MATHEMATICS IN HIGH SCHOOL, EVANSVILLE, IND. NEW YORK: CINCINNATI.:. CHICAGO PREFACE. "If you will put that into figures for me, perhaps I can understand it," was said to the author of this book by a pupil, who, having tried in vain to grasp a general proposition in demonstrative geometry, sought a kind of assistance which he had found to be useful. In his demand for figures (by which he meant numbers) he was expressing the universal demand of the learning mind for the concrete and the particular, as stepping stones to the abstract and the general. Much of the material of this book was prepared to meet this demand on the part of the author's own pupils during several years of teaching, and all of it has been subjected to the test of the schoolroom under plans which encourage on the part of the pupils the freest disclosures of the mental processes induced by the work. The aim of the work is to awaken gradually, by simple and natural methods, the mathematical consciousness of the child and to guide his perceptions in such a way as to lead him to lay a firm foundation for demonstrative geometry by means of his own observations and inventions. The recognition of different geometric forms and of their numerical relations, and the practicing of the geometric imagination, constitute for children a useful and delightful exercise, which this book is intended to promote. The author has selected from standard mathematical works the statements of some important facts and principles that are capable of concrete demonstration from the standpoint of the child, and has presented them in various ways and in different relations. Emphasis is placed upon the number relations of geometric forms, as a means of arousing those definite and exact ideas of form that are induced by actual measurements and computations. As a step towards the correlation of studies which is acknowledged to promote economy in educational effort, the algebraic equation in its simplest forms and uses has been introduced, as |