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: [Illustration] In the right triangle _OAB_ we shall let _AB_ = _y_, _OA_ = _x_, _OB_ = _r_, thus adopting the letters of higher mathematics. Then, so long as [L]_O_ remains the same, such ratios as _y_/_x_, _y_/_r_, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we call _y_/_r_ the _sine_ of _O_, and we write sin _O_ = _y_/_r_; _x_/_r_ the _cosine_ of _O_, and we write cos _O_ = _x_/_r_; _y_/_x_ the _tangent_ of _O_, and we write tan _O_ = _y_/_x_. Now because sin _O_ = _y_/_r_, therefore _r_ sin _O_ = _y_; and because cos _O_ = _x_/_r_, therefore _r_ cos _O_ = _x_; and because tan _O_ = _y_/_x_, therefore _x_ tan _O_ = _y_. Hence, if we knew the values of sin _O_, cos _O_, and tan _O_ for the various angles, we could find _x_, _y_, or _r_ if we knew any one of them. Now the values of the sine, cosine, and tangent (_functions_ of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows: