The seragesimal scale is that in common use. According to it, each quadrant or right angle being divided into 90 qegreee, each degree is divided into 60 seconds, and each second into 60 thirds, and so on. According to this scale, 90° represents a right angle; 180', two right angles, ora semicircle; and 360', four right angles, or the NS hole circum ference—the unit in the scale being the of a right angle. As the divisions of the angles at the center, effected by drawing lines from the center to the different points of graduation of the circumference, are obviously independent of the magnitude of the radius, and therefore of the circumference, these divisions of the circumference of the C. may be spoken of as being actually divisions of angles. By laying a graduated C. over an angle, and noticing the number of degrees, etc., lying on the cireumfer ence between the lines including the angle, we at once know the magnitude of the angle. Suppose the lines to include between them 3 degrees, 45 minutes, 17 seconds, the angle in this scale would be written 3' 45'17'.
It is obvious, however, that the division of the quadrant into 90 degrees instead of any other number, is quite arbitrary. We may measure angles by the C., however we graduate it. Many French writers, accordingly, have adopted the Centesimal Division qf the Circle.—In this division, the right angle is divided into 100
degrees, while each degree is divided into 100 parts. and so on. This is a most con venient division, as it requires no new notation to denote the different parts. Such a quantity as 3° 45' 17' is expressed in this notation by 3.4517, the only mark required being the decimal point to separate the degrees from the parts. Of course, in this illus tration, 3' means 3 centesimal divisions of the right angle, and 45' means 45 centesimal minutes, and so on. If we want to translate the quantity 3° of the common notation into the centesimal notation, multiply 3 by 100, and divide by 00. To translate minutes it) the common notation into the centesimal, the rule is to multiply by 100, and divide by 54.
There remains yet another mode of measuring angles, known as the Circular Measure.—The circular measure of angles is in frequent use, and depends directly on the proposition (Enc. vi. 33), that angles at the center dof a C. are propor. tional to the arcs on which they stand. Let PDX be an angle at the center 0 of a C., the radius of which is r; APB a semicircle whose circumference accordingly = angle PO Aa 2 right nr; and let the length of the arc AP = a. Then, by Euclid, --t angles --- = —; and 7rr