The author feels confident that this work will be of assistance to his co-workers in the field of mathematics and allied fields. If in addition it should aid in a better appreciation of mathematicians and their work on the part of laymen and students in other fields, the author’s foremost aim in the preparation of this work will have been achieved. ROBERT EDOUARD MORITZ, _September, 1913_. CONTENTS CHAPTER PAGE I. DEFINITIONS AND OBJECT OF MATHEMATICS 1 II. THE NATURE OF MATHEMATICS 10 III. ESTIMATES OF MATHEMATICS 39 IV. THE VALUE OF MATHEMATICS 49 V. THE TEACHING OF MATHEMATICS 72 VI. STUDY AND RESEARCH IN MATHEMATICS 86 VII. MODERN MATHEMATICS 108 VIII. THE MATHEMATICIAN 121 IX. PERSONS AND ANECDOTES (A-M) 135 X. PERSONS AND ANECDOTES (N-Z) 166 XI. MATHEMATICS AS A FINE ART 181 XII. MATHEMATICS AS A LANGUAGE 194 XIII. MATHEMATICS AND LOGIC 201 XIV. MATHEMATICS AND PHILOSOPHY 209 XV. MATHEMATICS AND SCIENCE 224 XVI. ARITHMETIC 261 XVII. ALGEBRA 275 XVIII. GEOMETRY 292 XIX. THE CALCULUS AND ALLIED TOPICS 323 XX. THE FUNDAMENTAL CONCEPTS OF TIME AND SPACE 345 XXI. PARADOXES AND CURIOSITIES 364 INDEX 385 Alles Gescheite ist schon gedacht worden; man muss nur versuchen, es noch einmal zu denken.--GOETHE. _Sprüche in Prosa, Ethisches, I. 1._ A great man quotes bravely, and will not draw on his invention when his memory serves him with a word as good.--EMERSON. _Letters and Social Aims, Quotation and Originality._ MEMORABILIA MATHEMATICA MEMORABILIA MATHEMATICA CHAPTER I DEFINITIONS AND OBJECT OF MATHEMATICS =101.= I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.--SYLVESTER, J. J. _Presidential Address to the British Association, Exeter British Association Report (1869); Collected Mathematical Papers, Vol. 2, p. 659._