The American Mathematical Monthly: The Official Journal of the Mathematical Association of America, Volume 7

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Mathematical Association of America, 1900 - Electronic journals
 

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Page 6 - ... in some most general results of experience, which the power of executing certain parts of the operation has already taught us practically. In drawing any subsidiary line for the sake of his demonstration, the well-trained geometer always asks if it is possible to draw such a line. It is well known that problems of construction play an essential part in the system of geometry. At first sight, these appear to be practical operations, introduced for the training of learners; but in reality they...
Page vii - YEARS' EXPERIENCE TRADE MARKS • DESIGNS ... COPYRIGHTS &c. Anyone sending a sketch and description may quickly ascertain our opinion free whether an invention is probably patentable.
Page 6 - LET it be granted that a straight line may be drawn from any one point to any other point.
Page vi - And this is the condemnation, that light has come into the world, and men loved darkness rather than light, because their deeds were evil.
Page 1 - this demonstration seems to have been generally overlooked by writers of geometrical text-books, though apparently exactly what was needed to put the theory upon a perfectly sound basis.
Page 1 - And that, if a straight line falling on two straight lines make the angles, internal and on the same side, less than two right angles...
Page 1 - But was courage likely to fail the man who, asked by King Ptolemy if there were no shorter road in things geometric than through his Elements ? answered, ' ' To geometry there is no special way for kings!" In the brilliant new light given by Bolyai and Lobachevski we now see that Euclid understood the crucial character of the question of parallels. There are now for us no better proofs of the depth and systematic coherence of Euclid's masterpiece than the very things which, their cause unappreciated,...
Page 7 - The foundation of all proof by Euclid's method consists in establishing the congruence of lines, angles, plane figures, solids, &c. To make the congruence evident, the geometrical figures are supposed to be applied to one another, of course without changing their form and dimensions.
Page 2 - At this moment it is not yet finished, but the way which I have followed promises me with certainty the attainment of the goal, if it in general is attainable. "It is not yet attained, but I have discovered such magnificent things that I am myself astonished at them. It would be damage eternal if they were lost. When you see them, my father, you yourself will acknowledge it. "Now I cannot say more, only so much: that from nothing I have created another wholly new world.

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