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by showing that the area is equal to that of a triangle whose base is the circumference, and perpendicular height the radius of the circle.

4. It follows that different circles are to one another as the squares of their radii or diameters, and that their circumferences are as the radii or diameters.

The C. is almost always employed to measure angles, from its obvious convenience for the purpose, which depends on the fact demonstrated in Euclid (book iv. prop. 33), that angles at the center of a C. are proportional to the arcs on which they stand. It follows, from this, that if circles of the same radii be described from the vertices of angles, as centers, the arcs intercepted between the lines, including the angles, are always proportional to the angles. The C. thus presents us with the means of compar. ing angles. It is first necessary, however, to graduate the C. itself; for this purpose its circumference is divided into four equal parts, called quadrants, each of which obviously subtends a right angle at the center, and then each quadrant is divided into degrees, and each degree into minutes, and so on. The systems of graduation adopted are various, and will now be explained.

The sexagesimal scale is that in common use. According to it, each quadrant or right angle being divided into 90 degrees, 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 whole circum. ference—the unit in the scale being the oth 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 circumfer: 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 O., however we graduate it. Many French writers, accordingly, have adopted the

Centesimal Division of 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 potation 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 illustration, 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, we must multiply 3 by 100, and divide by 90. To translate minutes in 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 (Euc. vi. 33), that angles at the center of a C. are proportional to the arcs on which they stand. Let POA be an angle at the center o of a C., the radius of which is p; APB a semicircle whose circumference accordingly =

angle POA ar; and let the length of the arc AP = a. Then, by Euclid,

- . and , POA _ 2 right Zsa

" Now, supposing a and r to be given, 1

r although the angle POA will be determined, yet its numerical value will not be settled unless we make some convention as to what angle we shall call unity. We are free to make any con. vention we please, and therefore choose such a one as will render the preceding equation the most simple. It is made most sim.

if tot 2 right angles ple if we take

ugles = 1. We shall then have (denoting

TE the numerical value of the angle POA by 018 = . The result of our convention is, that the numerical value of two right angles is a, instead of 180°, as in the method of angular measurement first alluded to; and the unit of angle, instead of being the nine

;. 2 right angles or 57° 17' 44' 48'"' nearly. Making 0 = 1 in tieth part of a right angle, is

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the equation 0 =“, we have a (or AP) = r (or AO), which shows that in the circular measure, the unit of angle is that angle which is subtended by an arc of length equal to radius. It is frequently a matter of indifference which mode of measuring angles is adopted; the circular measure, however, is generally the most advantageous, as being the briefest. It is easy to pass from this mode of measurement to the sexagesimal. If

[graphic][graphic][graphic]

CIRCLE, MERIDIAN, ETC.-1. Universal instrument. 2. Transit instrument. 3. Passage iastrumeut.

flecting circle. 9. Box-chronometer. 10. Pendulum with mercury compensation. 11. Astror

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4. Equatorial. 5. Meridian circle. 6. Lassel's reflecting telescope. 7. Reflecting sextant. 8. Reomical clock. 12. Micrometer.

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