Some commentators add after "one line," definition 15, the words "which is called the circumference," but these are not in the oldest manuscripts. The Greek idea of a circle was usually that of part of a plane which is bounded by a line called in modern times the circumference, although Aristotle used "circle" as synonymous with "the bounding line." With the growth of modern mathematics, however, and particularly as a result of the development of analytic geometry, the word "circle" has come to mean the bounding line, as it did with Aristotle, a century before Euclid's time. This has grown out of the equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the ellipse-_line_, and so on. It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines. Since this is the modern use of "circle" in English, it has naturally found its way into elementary geometry, in order that students should not have to form an entirely different idea of circle on beginning analytic geometry. The general body of American teachers, therefore, at present favors using "circle" to mean the bounding line and "circumference" to mean the length of that line. This requires redefining "area of a circle," and this is done by saying that it is the area of the plane space inclosed. The matter is not of greatest consequence, but teachers will probably prefer to join in the modern American usage of the term. 17. DIAMETER. _A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle._ The word "diameter" is from two Greek words meaning a "through measurer," and it was also used by Euclid for the diagonal of a square, and more generally for the diagonal of any parallelogram. The word "diagonal" is a later term and means the "through angle." It will be noticed that Euclid adds to the usual definition the statement that a diameter bisects the circle. He does this apparently to justify his definition (18), of a semicircle (a half circle). Thales is said to have been the first to prove that a diameter bisects the circle, this being one of three or four propositions definitely attributed to him, and it is sometimes given as a proposition to be proved. As a proposition, however, it is unsatisfactory, since the proof of what is so evident usually instills more doubt than certainty in the minds of beginners.