which never inquires what the actual facts are. It is "the science which draws necessary conclusions." This acute argument is, I think, at fault in its contention that construction is employed in philosophical reasoning, but is otherwise sound. It fails, however, to point out clearly these facts: 1. The human mind is so constructed that it must see every perception in a time-relation-in an orderand every perception of an object in a space-relation as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics. In the present book, the purpose of which is to present in simple and succinct form to teachers the results of mathematical scholarship, to be absorbed by them and applied in their class-room teaching, the author has wisely combined the genetic and the analytic methods. He shows how the elementary mathematics has developed in history, how it has been used in education, and what its inner nature really is. may safely be asserted that the elementary ma matics will take on a new reality for those who st this book and apply its teachings. NICHOLAS MURRAY BUTLE It e- y HISTORICAL REASONS FOR TEACHING ARITHMETIC. — Impor- tance of the question. The evolution of reasons. The beginning utilitarian. Early correlation. Utilitarian among trading peoples. Tradition and examinations. The cul- ture value. As a remunerative trade. As a mere show of PAGE WHY ARITHMETIC IS TAUGHT AT PRESENT. - Two general The utility of arithmetic overrated. The culture value. Teachers generally fail here. Recognition of the culture value. What chapters bring out the culture value. remainder theorem. The quadratic equation. Equiva- lent equations. Extraneous roots. Simultaneous equa- tions and graphs. Methods of elimination. Complex THE GROWTH OF GEOMETRY.-Its historical position. The - Geometry defined. Limits of plane geometry. The reasons for studying. Geometry in the lower grades. TYPICAL PARTS OF GEOMETRY. - The introduction to demon- strative geometry. Symbols. Reciprocal theorems. Con- verse theorems. Generalization of figures. Loci of points. |
