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PREFACEThe mathematical essays and recreations in this volume are by one of the most successful teachers and text-book writers of Germany. The monistic construct- mathematical essays and recreations in this volume are by one of the most tion of arithmetic, the systematic and organic development of all its consequences from a few thoroughly established principles, is quite foreign to the general run of American and English elementary text-books, and the first three essays of ProfessorSchubert will, therefore, from a logical and esthetic side, be full of suggestions for elementary mathematical teachers and students, as well as for non-mathematical readers. For the actual detailed development of the system of arithmetic heresketched, we may refer the reader to Professor Schubert's volume Arithmetik und Algebra, recently published in the G]oschen-Sammlung (G]oschen, Leipsic), -an extraordinarily cheap series containing many other unique and valuable text-books inmathematics and the sciences.The remaining essays on Magic Squares, The Fourth Dimension, and TheHistory of the Squaring of the Circle, will be found to be the most complete generally accessible accounts in English, and to have, one and all, a distinct educational and ethical lesson.In all these essays, which are of a simple and popular character, and designed for the general public, Professor Schubert has incorporated much of his original research.

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