Geometry of the Diric theory

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Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

Largely considered as a vintage of contemporary arithmetic, this extended model of Felix Klein's celebrated 1894 lectures makes use of modern thoughts to envision 3 well-known difficulties of antiquity: doubling the quantity of a dice, trisecting an perspective, and squaring a circle. modern scholars will locate this quantity of specific curiosity in its solutions to such questions as: below what situations is a geometrical building attainable?

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Presenting the Investigation If possible, construct Napoleon's figure on the chalkboard before class begins or prepare a construction to use on an overhead projector. Discuss the figure in class, asking students to explain how each construction is done. Although the next relation uses only elementary concepts, most students find it challenging. What seems to be most perplexing is the selection of the correct pair of triangles to prove congruent. If, after a few minutes, students do not find them, tell them to name triangles that use the required segments BD and CF as sides.

Do you' remember the ratio of length to width in agolden rectangle? It's called

_1_ = VS - 1 cp ~ 2 Rectangle CDEF is also a golden rectangle since the ratio oflength to width is cp as shown below. DE=ME -MD EF I DE = ,g - N K The relation between cp and its reciprocal cp' is developed next. 1- ~ = , ....... rs 2+ I DE = H "" 4, AE = AM + ME = Lf \ Presenting the Investigation Briefly discuss the two rectangles at the top of the student page. Point out that the rectangle at the bottom requires a scanning motion in viewing, while the one at the top can be appreciated at one glance.

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