Many writers state this proposition so that it reads that "central angles are _measured by_ their intercepted arcs." This, of course, is not literally true, since we can measure anything only by some thing, of the same kind. Thus we measure a volume by finding how many times it contains another volume which we take as a unit, and we measure a length by taking some other length as a unit; but we cannot measure a given length in quarts nor a given weight in feet, and it is equally impossible to measure an arc by an angle, and vice versa. Nevertheless it is often found convenient to _define_ some brief expression that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus we _define_ "area of a circle," even when we use "circle" to mean a line; and so we may define the expression "central angles are measured by their intercepted arcs" to mean that central angles have the same numerical measure as these arcs. This is done by most writers, and is legitimate as explaining an abbreviated expression. THEOREM. _An inscribed angle is measured by half the intercepted arc._ In Euclid this proposition is combined with the preceding one in his Book VI, Proposition 33. Such a procedure is not adapted to the needs of students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle standing on the same arc. Since Euclid never considered an angle greater than 180 deg., his inscribed angle was necessarily less than a right angle. The first one who is known to have given the general case, taking the central angle as being also greater than 180 deg., was Heron of Alexandria, probably of the first century A.D.[68] In this he was followed by various later commentators, including Tartaglia and Clavius in the sixteenth century. One of the many interesting exercises that may be derived from this theorem is seen in the case of the "horizontal danger angle" observed by ships. [Illustration] If some dangerous rocks lie off the shore, and _L_ and _L'_ are two lighthouses, the angle _A_ is determined by observation, so that _A_ will lie on a circle inclosing the dangerous area. Angle _A_ is called the "horizontal danger angle." Ships passing in sight of the two lighthouses _L_ and _L'_ must keep out far enough so that the angle _L'SL_ shall be less than angle _A_. To this proposition there are several important corollaries, including the following: 1. _An angle inscribed in a semicircle is a right angle._ This corollary is mentioned by Aristotle and is attributed to Thales, being one of the few propositions with which his name is connected. It enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board, a pencil being held at the vertex. [Illustration]