In general the present editor has refrained from expressing any opinions that would strike a discordant note in the reading of the text as De Morgan left it. The temptation is great to add to the discussion at various points, but it is a temptation to be resisted. To furnish such information as shall make the reading more pleasant, rather than to attempt to improve upon one of the most delicious bits of satire of the nineteenth century, has been the editor's wish. It would have been an agreeable task to review the history of circle squaring, of the trisection problem, and of the duplication of the cube. This, however, would be to go too far afield. For the benefit of those who wish to investigate the subject the editor can only refer to such works and articles as the following: F. Rudio, _Archimedes, Huygens, Lambert, Legendre,--mit einer Uebersicht ueber die Geschichte des Problemes von der Quadratur des Zirkels_, Leipsic, 1892; Thomas Muir, "Circle," in the eleventh edition of the _Encyclopaedia Britannica_; the various histories of mathematics; and to his own article on "The Incommensurability of [pi]" in Prof. J. W. A. Young's _Monographs on Topics of Modern Mathematics_, New York, 1911. The editor wishes to express his appreciation and thanks to Dr. Paul Carus, editor of _The Monist_ and _The Open Court_ for the opportunity of undertaking this work; to James Earl Russell, LL.D., Dean of Teachers College, Columbia University, for his encouragement in its prosecution; to Miss Caroline Eustis Seely for her intelligent and painstaking assistance in securing material for the notes; and to Miss Lydia G. Robinson and Miss Anna A. Kugler for their aid and helpful suggestions in connection with the proof-sheets. Without the generous help of all five this work would have been impossible. DAVID EUGENE SMITH. TEACHERS COLLEGE, COLUMBIA UNIVERSITY. * * * * * A BUDGET OF PARADOXES {1} INTRODUCTORY. If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavor to persuade me that he was larger than the elephant, I might by possibility be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be unable wholly to reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others--I should feel quite at my ease. I should certainly say, My little friends, the case of each one of you is destroyed by the rest. I intend to show flies in the swarm, with a few larger animals, for reasons to be given.