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The geometry of which we have so far spoken belongs to the Elements. But, before the body of the Elements was complete, the Greeks had advanced beyond the Elements. By the second half of the fifth century B. C. they had investigated three famous problems in higher geometry, the squaring of the circle, the trisection of any angle, the duplication of the cube.
The disciples of Plato invented conic sections, and discovered the geometrical loci. They also attempted to resolve the problems of the trisection of an angle and the duplication of a cube. To Leon is ascribed that part of the solution of a problem, called its determination, which treats of the cases in which the problem is possible, and of those in which it cannot be resolved.
Into this trisection we shall decompose the coarse unity of the question presented by Van Dale and his Vandals, as though the one sole "issue," that could be sent down for trial before a jury, were the likelihoods of fraud and gross swindling.
Yet the problem of the trisection is frequently attacked by men of some mathematical education. I think it was about 1870 that I received from Professor Henry a communication coming from some institution of learning in Louisiana or Texas. The writer was sure he had solved the problem, and asked that it might receive the prize supposed to be awarded by governments for the solution.
In that circle the ratio is 3 1/5; therefore, by the major premise, that is the ratio for all circles. The three famous problems of antiquity, the duplication of the cube, the quadrature of the circle, and the trisection of the angle, have all been proved by modern mathematics to be insoluble by the rule and compass, which are the instruments assumed in the postulates of Euclid.
"As the trisection of the triangle is, perhaps, the simplest of the three problems," said the lecturer, with almost judicial calmness, "we will, if you please, begin with that. I hope that gentlemen who have brought note-books with them will be kind enough to follow my calculations and check any error that I may make."
Now, the three great problems of antiquity which engaged the attention and wonderment of geometricians throughout the Middle Ages, were "the Squaring of the Circle," "the Duplication of the Cube," and lastly, "the Trisection of an Angle," even Euclid being unable to show how to do it; and yet it will be seen that the diagonal of one of the subsidiary figures in the tri-subdivision, together with the diagonal of the whole figure, actually trisect the angle at the corner of the rectangle.
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