Proportionality in Similar Triangles: A Cross-Cultural Comparison - Conclusion

While the ratio of corresponding sides in similar triangles may be equal, and this equality is often justified by an appeal to reason, such an appeal relies on the Euclidean parallel postulate. Many scholars in the 2000 years following Euclid have attempted to prove the parallel postulate, and in doing so, unsuspectingly clarified for modern mathematics exactly what theorems or constructions are equivalent to this axiom of parallels. John Wallis (1616-1703) "proved" the parallel postulate beginning from a statement about the existence of similar triangles with proportional sides. Both Girolamo Saccheri (1667-1733) and Johann Heinrich Lambert (1728-1777) sought inherent contradictions in hyperbolic geometry by studying the implications of "rectangles" with angle sum less than 360^° [2 ]. Today the work of these two geometers can be interpreted as demonstrating that in non-Euclidean geometry, there are no rectangles, squares, nor a true "square unit" of measure.

In the non-Euclidean case, part 4(a) above would fail, since it cannot be the case that line AJ and FL when extended meet in a right angle and lines AB, FE also meet in a right angle. Moreover, in hyperbolic geometry, the angle sum of a triangle is less than 180^°, with larger triangles having a smaller angle sum. In spherical geometry, a triangle has angle sum greater than 180^°, with larger triangles having greater angle sum. In non-Euclidean geometry, if two triangles have all three corresponding angles equal, the triangles are in fact congruent [2, p. 190]. In this sense, the property of similarity is more restrictive in the non-Euclidean world. In teaching the module, the instructor should identify where in the proof of Euclidean similarity theorems the concept of area occurs, and, if time permits, explain that the construction of a rectangle depends on the Euclidean parallel postulate. This could be used as a spring board to study hyperbolic geometry or to motivate attempts to prove the parallel postulate, since many appealing theorems fail without this axiom. For further results in other geometries, see Euclidean and Non-Euclidean Geometries [2 ], or the chapter on the parallel postulate in Mathematical Expeditions [6].

The instructor may assign all parts (a)-(e) as a one-week individual or group project, or simply assign the parts one-by-one as in-class problems or out-of-class exercises. Also, part (e) may be omitted if the Pythagorean Theorem has already been covered, or part (e) may be assigned independently of the other parts.

Acknowledgment: The work for this article has been partially supported by the National Science Foundation's Course, Curriculum and Laboratory Improvement Program under grant DUE-0717752, for which the author is most appreciative. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.