Page:The New International Encyclopædia 1st ed. v. 04.djvu/861

* CIRCLE. 757 CIRCLE. but IT. beinjr transceiKlental, cannot be repre- Sfntcd by any construction depending solely upon the strai-rht-edge and compasses. It ie(iuiics a transcendental curve, such as the integraph of Abdank-Abakanowicz. Thus, through labors like those of Gauss. Her- mite, and Lindcraann, the true nature of t has been determined, and etTorts at circle-squaring by the instruments of elementary geometry have been proved futile. Jlodern analysis has shown TT to be expressible by certain infinite series; e.g. ,r = 4(l-i + i-i+i- ...) (Leibnitz); or in the form of a continued fraction, as in 1 (Brouncker) ■'■ + 2 +9 2 + 25 2+49 (Vallis). or of a continued product, as r 2-2-4-4-6-6-8 2 l-3'3'5o-7-7 The method of approximating this ratio com- monly used l>efore the introduction of calculus (q.v. ) consisted in computing the perimeters of the circumscribed and inscribed polygons of a circle of diameter 1. For, since the length of the circumference in this case is the desired ratio, the value of -r lies between the values of the perimeters of the given polygons. A history of the development of this important problem of geometry will be found in Kudio. Archimedes, Uui/gens, Lamhert, Lct/endrc: vier Abhandlungen iiber die Ereismessung (Leipzig, 1892). (2) The centre of the circle is a centre of sjinmetry, and any diameter is an axis of sym- metrj' (q.v.). ( .3 ) The perimeter of a circle of radius r is 2irr. and its areajri-. The area is greater than that of any plane figure of the same perimeter. (4) Concentric circles — that is, those having the same centre — never intersect. (5) Circles are similar figures (see SlsnLAB- ITY), and their areas are proportional to the squares of their radii or diameters. (G) Arcs of a circle are proportional to the angles subtended at the centre, and conversely. This property forms the basis of angular meas- ure. ClRCUL.R IMEAsritE. The supposed number of days in the year early led to tlie division of the circle into 300 equal parts, for use in astronomi- cal instruments. A knowledge of the regular hexagon probably led to the further division of .300 degrees into six parts of GO degrees each. The Babylonians divided each degree into 00 equal parts, and each subdivision into GO equal parts, thus producing the sexagesimal .scale. (See XOTATIOX.) Thus the circumference of a circle is divided into 300 equal parts, called degrees: each degree into GO e<iual parts, called minutes; and each minute into GO equal parts, called seconds. Further divisions are better represented by decimal fractions. The circle is also commonly divided into four equal parts of 90 seconds each, called quadrants. By con- necting the centre of a circle with the points of equal division on the circumference, equal unit angles are formed, whose magnitude is in- dependent of the length of the radius : this pro- ducing an angle nu-asure. the basis of the pro- tractor (q.v.). For scientific purposes, however, it would be more convenient to divide a (piadrant into 100 equal parts, called grades, and each of these into 100 equal parts, called centesimal minutes, and each of these into 100 equal jjarts, called centesimal second-s. This plan, attempted in France as part of the metric system, is known as the centesimal division of the circle. For ex- ample, 3S 45' 17" (read 3 grades, 45 centesimal minutes, and 17 centesimal seconds) may be written 3.4517. To translate this into sexa- gesimal notation, 3e equals 3 X fijV = 2.7°, 45 centesimal minutes 45 X j^Stj = 0.405' or 0.00075" ; and so on. T!ie sexagesimal system is, however, so well established that the centesimal has only very recently, in France, come to take important rank. Kadiax ilEA.siRE. In higher especially in anal- jiic trigonometry, another unit of angular measure, called the radian, is in general use. This is defined as the angle subtend- ed at' the centre of a circle by an arc equal in length to the radius ( Fig. 2). The relation of the radian to other angular units is as follows: The radian AOB Fio. 2. arc AB 4 right angles circumference r 2Trr Therefore, angles = radian is the radian equals -— X 4 2Tr X 1 right angle. In degrees 90" 1^ right 2 • — =57. 29° + - V cr, more nearly, 57° 17' 44.0066"+. radians = a quadrant ; ir radians and 2ir radians = 360°. 2 180° Since 1 radian = 180° 180 ,„ TT radians , 1 = — =- and 180 radians. The word radian is commonly omitted in discus- sions of angles: e.g. ir radians = 180° is ex- pressed JT = 180°. A few of the modern theories concerning the circle are suggested by the following: ( 1 ) Vo-axal Circles. — The radical axis XX, (Fig. 3) of two circles of radii r,, r, is the line perpendicular to their centre line C.C., and divid- ing this line so that the dilVerence of the squares on the segments equals the dilVerence of the squares on the radii. The common chord of two intersecting circles is a segment of their radical axis. All circles having a common radical axis pass through two real or two imaginary points, and such a group of circles is called a coaxal system. If two circles are concentric, their