![]() ![]() Moreover, the argument founded on Chinese principles does not require a comparison of possibly two incommensurable lengths for the bases in similar triangles, as Euclid must consider. The ease by which similarity results are then proven (as a modern exercise) is appealing. While Euclid follows a step-by-step model of deductive reasoning with every statement justified by a previous proposition, definition, or postulate, the Chinese method is a bit more intuitive, particularly when identifying what today would be called congruent triangles. See Mathematics in China at David Joyce's Mathematics History website for further information about the history of Chinese mathematics. Strictly speaking gou refers to base or shadow and gu refers to height or gnomon, although there apparently was no word per se for the concept of a triangle in ancient China. Another engaging use of the inclusion-exclusion principle is the proof of the gou-gu theorem itself, borrowing an idea from the text Zhou bi suan jing ( Mathematical Classic of the Zhou Gnomon), compiled between 100 BCE and 100 CE. ![]() The principle is easily applied when the excluded triangles are right triangles, and to account for all possible pairs of corresponding sides in a right triangle, further application of the gou-gu (Pythagorean) theorem is used. When applied to a rectangle, the principle identifies certain (non-congruent) sub-rectangles of equal area that remain after the exclusion of congruent triangles. This article offers curricular materials for the proof of similarity theorems, based on an ancient Chinese principle of area known as the "in-out" or "inclusion-exclusion" principle. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |