At this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, saying that such figures have "their sides proportional and their angles equal." Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of "corresponding angles" we may say "homologous angles," but there seems to be no reason for using the less familiar word.
It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously similar of all figures,—two circles; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points, A1, A2, A3, ..., and B1, B2, B3, ..., as such that A1A2, B1B2, C1C2, ... are concurrent in O, and A1O : A2O = B1O : B2O = C1O : C2O = ... Here the constant ratio A1O : A2O is called the ratio of similitude, and O is called the center of similitude. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilaterals respectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead.
Among the interesting applications of similarity is the case of a shadow, as here shown, where the light is the center of similitude. It is also well known to most high school pupils that in a camera the lens reverses the image. The mathematical arrangement is here shown, the lens inclosing the center of similitude. The proposition may also be applied to the enlargement of maps and working drawings.
The propositions concerning similar figures have no particularly interesting history, nor do they present any difficulties that call for discussion. In schools where there is a little time for trigonometry, teachers sometimes find it helpful to begin such work at this time, since all of the trigonometric functions depend upon the properties of similar triangles, and a brief explanation of the simplest trigonometric functions may add a little interest to the work. In the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection.
It is a mistaken idea that geometry is a prerequisite to trigonometry. We can get along very well in teaching trigonometry if we have three propositions: (1) the one about the sum of the angles of a triangle; (2) the Pythagorean Theorem; (3) the one that asserts that two right triangles are similar if an acute angle of the one equals an acute angle of the other. For teachers who may care to make a little digression at this time, the following brief statement of a few of the facts of trigonometry may be of value:
