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Every right line, mn, fig. % pi. 3, terminating at bothends in the circumference of a circle, is called a chord(cord). Every portion of the circumference of a circlemqn, is called an arc (bow). The chord is also called thesubtense of an arc. The part pq of the radius C pq inclu-ded between the chord and the arc, and perpendicular tothe chord is called the sine (arrow).
These names are borrowed from the use among theancients of a piece of wood bent, by a piece of cord, nearlyinto the form of a portion of the circumference of the circlefig. 3, pi. 8, which they called an arc, and which was in-tended to propel the arrows (sines) placed on the middle ofthe chord, and in a direction perpendicular to it. This isone instance of practice having preceded science and sup-plied it with terms.
The radius Cpq, fig. 2, perpendicular to the chord mn,divides both the chord and the arc into two equal parts.
Let us draw the radii Cm, Cn, which are oblique lines formingequal angles with the perpendicular C p. Therefore, first, mp = np.The chords mg ng are also oblique lines equal to one another, and ifCgn is folded on Cqm, the point n will fall on the point m, and thearc nsg on the arc mrq ; for no point of the former arc can fall eitherwithin or without the latter, unless it be nearer to or further fromthe centre C. Therefore, secondly, the two arcs mrq, nsq are equal.
Application to linear design. —The property of the cir-cle just demonstrated is very usefully applied in the art ofdesign, and in most of the arts in which exact measures areto be taken and combined together.
It serves in the first place to divide an arc of a circle mqn, fig. 4,pi. 3, in two equal parts. Take a pair of compasses and open themmore than the half of mn ; placing one leg of the compasses in m,with the other describe an arc of a circle rst ; then, fixing one pointin n, describe another arc vsu, taking care that the compasses areneither opened nor closed during the operation. The point s, wherethese two arcs intersect each other, will be equally distant from mand n ; and therefore it will fall on a line perpendicular to the rightlin u «zra, which passes through the middle of this line, and throughthe centre of the circle. This perpendicular will divide the chordmn, as well as the arc mqn, into two equal parts.
If the exact position of the centre is not known, draw on the side