Having taken up all of the propositions usually given in Book I, it seems unnecessary to consider as specifically all those in subsequent books. It is therefore proposed to select certain ones that have some special interest, either from the standpoint of mathematics or from that of history or application, and to discuss them as fully as the circumstances seem to warrant.
Theorems. In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc, and conversely for both of these cases.
Euclid made these the twenty-sixth and twenty-seventh propositions of his Book III, but he limited them as follows: "In equal circles equal angles stand on equal circumferences, whether they stand at the centers or at the circumferences, and conversely." He therefore included two of our present theorems in one, thus making the proposition doubly hard for a beginner. After these two propositions the Law of Converse, already mentioned on [page 190], may properly be introduced.
Theorems. In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords; and if two arcs are unequal, the greater is subtended by the greater chord, and conversely.
Euclid dismisses all this with the simple theorem, "In equal circles equal circumferences are subtended by equal straight lines." It will therefore be noticed that he has no special word for "chord" and none for "arc," and that the word "circumference," which some teachers are so anxious to retain, is used to mean both the whole circle and any arc. It cannot be doubted that later writers have greatly improved the language of geometry by the use of these modern terms. The word "arc" is the same, etymologically, as "arch," each being derived from the Latin arcus (a bow). "Chord" is from the Greek, meaning "the string of a musical instrument." "Subtend" is from the Latin sub (under), and tendere (to stretch).
It should be noticed that Euclid speaks of "equal circles," while we speak of "the same circle or equal circles," confining our proofs to the latter, on the supposition that this sufficiently covers the former.
Theorem. A line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it.
This is an improvement on Euclid, III, 3: "If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles; and if it cuts it at right angles, it also bisects it." It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. Engineers often check their work by taking three chords, the perpendicular bisectors of the three passing through a single point. Illustrations of this kind of work are given later in this chapter.
Theorem. In the same circle or in equal circles equal chords are equidistant from the center, and chords equidistant from the center are equal.
This proposition is practically used by engineers in locating points on an arc of a circle that is too large to be described by a tape, or that cannot easily be reached from the center on account of obstructions.