This is an OCR edition with typos.
|Publisher:||Creative Media Partners, LLC|
|Product dimensions:||6.14(w) x 9.21(h) x 0.41(d)|
Read an Excerpt
SECTION III. EQUALITY 32. There are three words used in Geometry to denote equality : congruent, equivalent, and equal. 33. Congruence. Two figures are congruent when they can be made to coincide in every point. 34. Equivalence. Closed figures (or figures bounded by closed lines, see §§ 26, 28) are said to be equivalent when their boundaries inclose the same amount of surface. 35. Equality. The word equal is used somewhat in both senses, but in this syllabus it will be used only in those places where there can be no confusion between the ideas of congruence and equivalence. For example, sects will be said to be equal when they can be made to coincide, even though this fulfills the definition of congruence, for since a sect cannot inclose surface, there can be no confusion with equivalence. 36. Congruence includes equivalence, whereas equivalence does not imply congruence; a figure inclosed by a curved line might be equivalent to a figure inclosed by a broken line although it would be impossible to make them coincide. 37. Geometric Equality. In the strictest geometric sense, equality means that coincidence is possible, and in this the test for geometric equality differs from the test for arithmetic equality, for two arithmetic magnitudes are equal if they contain the same unit the same number oftimes. Evidently, then, equivalence is to some extent an arithmetic property, but Geometry is applied so often to calculations of magnitudes in terms of a unit, that it is neither necessary nor desirable to attempt to distinguish too carefully between it and other mathematical subjects. In practical work, Geometry will be found to have many parts that involve Arithmetic andAlgebra, and while the distinctions between the subjects may be kept in mind, their combined use is e...