The American Mathematical Monthly: The Official Journal of the Mathematical Association of America, Volume 22
Mathematical Association of America, 1915 - Mathematicians
Registers of officers and members were issued as supplements to some vols.
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Achilles algebra American angle appeared applied arguments calculus called chapter Chicago circle College common condition consideration considered contains continuous course curve definition determined direction discussion distance divisible EDITED equal equation existence expressed fact formula functions geometry give given Hence high school idea infinite infinity integral interest limit mathe mathematics matter means meeting method Michigan MONTHLY motion moving nature obtained original pass philosophy plane positive possible present problem Professor Proposed prove published question reach reference relation respectively Science sides simple Society solution solved space sphere student teachers teaching theorem theory things triangle University volume York Zeno Zeno's
Page 144 - These, I say, assert there are infinitesimals of infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinilum of parts. Others there be who hold all orders of
Page 145 - nay, it will be evident this is never done, it being impossible. And, whatever mathematicians may think of fluxions, or the differential calculus and the like, a little reflection will show them that, in working by those methods, they do not conceive or imagine lines or surfaces less than what are perceivable to sense.
Page 143 - those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits toward which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.
Page 76 - to the privileges and feelings of a votary is only to be gained by one means—sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, without which no man can ever make such advances in this or
Page 2 - The second argument is the famous puzzle of Achilles and the tortoise. Achilles must first reach the place from which the tortoise started. By that time the tortoise will have got on a little way. Achilles must then traverse that, and still the tortoise will be ahead. He is always nearer, but he never makes up to it.
Page 144 - And, as this notion is the source from whence do spring all those amusing geometrical paradoxes which have such a direct repugnancy to the plain common sense of mankind, and are admitted with so much reluctance into a mind not yet debauched by learning.
Page 113 - Je suis tellement pour Vinfini actuel, qu'au lieu d'admettre que la nature l'abhorre, comme l'on dit, vulgairement, je tiens qu'elle l'affecte partout, pour mieux marquer les perfections de son auteur. Ainsi je crois qu'il n'ya aucune partie de la matière qui ne soit, je ne dis pas divisible, mais actuellement divisée, et par conséquent, la moindre particelle doit estre considérée comme
Page 2 - which is a free paraphrase of Aristotle's statements. We shall find it convenient, for future reference, to use the names "Dichotomy," "Achilles," "Arrow," and "Stade" for the four arguments against motion, respectively. 1. "DICHOTOMY": You cannot traverse an infinite number of points in a finite time. You must traverse the
Page 143 - Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.