CIRCLE 



259 



then r (circumference) = 2r x r. Archimedes, in 

 lii.- I ..... k /'i I'tini'iixione Circuit, showed that the 

 i alii i is nearly that of 7 to 22. VarioiiH closer 

 Approximations in large numbers were afterwards 

 m:i'lc. as, for instance, tlie ratio of 181. r > to 5702, 

 Ktiu to 3173; or the excellent one of Adrian Metin> 

 viz. 113 to 355. Vieta in 1579 showed that if tlie 

 diameter of a circle be 1000, then the circumference 

 will \*> greater than 3141-5926535, and less than 

 314r.'ii)-_ > i>r>37. This approximation he made by 

 ascertaining tlie perimeters of the inscribed and 

 riivuinsciibed polygons of 393,216 sides. By in- 

 creasing th- number of the sides of the polygons, 

 their perimeters are brought more and more 

 nearly into coincidence with the circumference of 

 the circle ; but this operose method was long 

 ago superseded by easier modes derived from the 

 higher mathematics. Suffice it to say that vari- 

 ous series were formed expressing its value ;^ by 

 taking more and more of the terms of which into 

 account, a closer and closer approach to the value 

 can be obtained by ordinary arithmetic. We sub- 

 join some examples : 



where n = = and m = 



O 



Series (3), or one of its modifications, is the 

 most expeditious mode yet known of extending 

 the approximation to the ratio ir. It has now 

 been calculated to 707 places of decimals, and 

 verified to over 600. The number ir, though fixed 

 in value, cannot be exactly expressed in figures, 

 being Incommensurable (q.v.). Finally, the multi- 

 plier (3 - -008 - -000,007) (1 + -fa) gives a close 

 approximation, useful to the practical arithmeti- 

 cian. 



4. The area of a circle is equal to IT multiplied 

 by the square of the radius ( = irr 2 ) ; or to the 



square of the diameter multiplied by -g ; i.e. by 



7854. Archimedes proved this 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. 



5. 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 circle is almost always employed in measur- 

 ing or comparing angles, from the fact demon- 

 strated in Euclid (Book vi. Prop. 33), that angles 

 at the centre of a circle 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 centres, the arcs inter- 

 cepted l>etween the sides of the angles are always 

 proportional to the angles. The easiest subdivi- 

 sion of a circumference is into six equal parts, 

 because then the chord of the arcs is equal to the 

 radius. Divide one of these arcs into sixty equal 

 parts, and we thus obtain the unit of the sexa- 

 gesitnal scale, called a degree. 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, 



or a semicircle ; and 360, four right angle*, or the 

 whole circumference the unit in the scale being 

 the j/.-.th of a right angle. AH the division* of the 

 angles at the centre, effected by drawing lines 

 from the centre to the different point* of gradua- 

 tion of the circumference, are obviously inde- 



tiendent of the magnitude of the radius, and there- 

 ore of the circumference, these divisions of the 

 <-i i -IMI inference of the circle may IKJ Hiioken of a* 



les. B 



actually divisions of angles. By laying a 

 graduated circle over an angle, and noticing the 

 number of degrees, &c. lying on the circumference 

 between the Hues including the angle, we at once 

 know the magnitude of the angle. Suppose the 

 lines to include between them 3 degrees, 45 minute*, 

 17 seconds, the angle in this scale would be written 

 3 45' 17". 



The sexagesimal measurement of circumferences 

 and angles is the most ancient, and still recom- 

 mends iteelf universally to practical mathema- 

 ticians. A second mode was proposed at the 

 French Revolution, but though adopted by La- 

 place in the Mecanique Celeste, has long been 

 abandoned even in France. By this scale, called 

 the centesimal, the right angle is divided into 100 

 degrees, while each degree is divided into one 

 hundred parts, and so on. 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 ordi- 

 nary degrees into the centesimal notation, we must 

 multiply by 100, and divide by 90. To translate 

 minutes in the same way, multiply by 100, and 

 divide by 54 ; and for seconds, multiply by 250, 

 and divide by 81. 



There is also a theoretical method of measuring 

 angles, which, though indispensable in advanced 

 trigonometry and other branches of analysis, is 

 scarcely required in elementary mathematics. For 

 the circular measure, as it is called, the unit angle 

 is thus found : Let POA be an angle at the centre 

 of a circle, the radius of which is r ; APB a 

 semicircle whose arc accordingly = irr ; and let the 

 length of the arc AP = a. Then, by Euclid, 



angle POA a 



- = ; and 

 2 nght angles irr 



Now, supposing and r to be given, although the 



angle POA will be deter- 



mined, yet its numerical 



value will not be settled 



unless we make some con- 



vention as to what angle 



we shall call unity. We 



therefore choose such a 



one as will render the 



preceding equation the 



most simple. It is made 



... . , 2 right angles , w 

 most simple if we take JJ-VTT^Q = * >>e 



" . . 



pnA _2right^8 a 

 ~ 3-141597&C. ' r' 



shall then have (denoting the numerical value of 

 the angle POA by 6 ) = ". The result of our con- 



vent ion is, that the numerical value of two right 

 angles is IT, instead of 180, as in the method of 

 angular measurement first alluded to ; and the 

 unit of angle, instead of being the ninetieth part 



, . 2 right angles __ K 7i7' AA." A&'" 

 of a nght angle, is or6 ' 17 4 



nearly. Making = 1 in the equation 6 = , 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 



