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By E. Mach
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Extra resources for Space and Geometry - In the Light of Physiol., Psych. and Physical Inquiry
Example text
Also M II L. So L' II L. If the plane is magnified by a factor of congruent angles. This is best seen by a picture. L goes to L' and M goes to L' and M is parallel to M': Suppose that, under the magnification, M'. We just saw that L is parallel to By the Z-principle, 0 = ~, and, again by the Z-principle, ~ = 0'. So 0 = 0' .
So eventually, these distances will have to go past zero, meaning that the line M will have to meet the line L. Since M does not meet L, our assumption that a < 90' must have been wrong. Going to the right instead of to the left, we see in the same way that assuming that possibility is that a = ~ ~ < 90' also leads to a contradiction. So the only = 90', which means that Land M are parallel. In t u i t ion 42 1. One thing we did not check when we were doing 114 is that, when we s1ide the rectang1e PQRS down the 1ine M, the copies we make as we go fit together side-by-side as shown in the picture in 114.
1 < - - - 1_ C/2 As you cut more and more finely, the figure gets closer and closer to a rectangle of base C/2 and height 1. It still takes 1r cans of paint to paint it. So C/2 = 1r. So: C = (circumference of circle of radius 1) = 21r Now the circle of radius r is obtained by magnifying the circle of radius 1 with magnification factor r. Since length is one-dimensional, the Magnification Principle says that: C = (circumference of circle of radius r) r·21r 21rr Intuition: 52 I18 When are triangles congruent?