READING AND ELOCUTION. 



direction and aim aa the two radii proceeding from the ccntro of 

 a cirolo that together form a diameter of that circle. 



Any portion <>i the circumference of a circle, no matter how great 

 or how small, is called an arc of that circle. Thus the portion A c 

 of tlm dri'uiuferenoe of the cirolo A c B E is an arc of that circle. 

 r'or i In* sake of clearness we will call this portion tho arc AUG. 

 r rii<> remainder of the circumference, c B A, is also an arc of tho 

 :ia are also the portions A c B, B E A, E A c, c B E, etc. 

 etc. Tho Btraitfht lino that joins tho extr. ;mtii-s of on arc is 

 called the clionl of tluit arc,. Thus A c in tho chord of tho aro 

 A M c, A K is the chord of tho aro A N E, and A B is tin- chord of 

 the aro A c B or tho arc A E B. The student may ask why a 

 portion of tho circumference of a circle is called an arc, and to 

 this wo may reply by two counter-questions Why is a man who 

 ahoota with a bow and arrow called an archer ? and, What does 

 the aro and its chord resemble P It may not be easy to give a 

 reply to our first counter-query, but it will suggest a reply to 

 our second, namely, that tho arc and its chord look very much 

 like a bent bow and its string tightened to its utmost tension. 

 Let it now bo said that tho Latin word for a bow is arcus, and 

 the meaning of the words arc and archer becomes clear enough. 

 The word c/umZ is derived from the Latin cliorda, the string of 

 a harp or lyre, and hence any kind of string. It is from this 

 word that wo have "cord," a term applied to any small rope, 

 or thick or closely-twisted string. From this it may be seen 

 how much a letter more or less disguises an otherwise familiar 

 word. 



A segment of a circle is any part of the surface of tho entire 

 circle enclosed by an aro and its chord. The word segment is 

 derived from the Latin segmenlum, & piece, shred, or paring. 

 Thus in Fig. 49, A M c represents a small segment of the circle 

 A c B E contained by the arc A M c and the chord A c. In like 

 manner, tho portion of tho circle c B E A, contained by tho arc 

 c B E A and tho chord c A, is a segment of tho circle A c B E, 

 as is also A c B, contained by the aro A c B and the chord A B. 

 This chord, however, is a diameter of the circle A c B E ; and 

 when a diameter is tho chord of an arc that encloses a segment 

 in conjunction with it, tho portion of the surface thus enclosed 

 is called a semicircle or half-circle. 



The figure enclosed between any two radii of a circle and tho 

 arc of the circumference intercepted between the extremities of 

 the radii that touch or meet tho circumference, is called a sector, 

 from scctum, a part of the Latin verb seco, to cut. Thus o c E, 

 the portion of the circle A c B E bounded or enclosed by tho radii, 

 o c, o E, and tho arc c A E, is the sector of that circle, as is also 

 the remainder of the surface ; namely, the portion bounded by 

 the radii o c, O E, and the arc c B E. 



We shall have occasion in the course of future problems to 

 epeak of circles touching or meeting one another, and of straight 

 lines that aro tangents to a circle. In Fig. 50, the smaller circles 

 A D E, c F G touch the larger circle ABC, A 



the former in tho point A, and tho latter 

 in the point c. One circle can touch 

 another in only one point. The circum- 

 ference of tho touching circles meet in 

 one point only. If, however, the surface 

 of one circle overlaps the surface of the 

 other in the slightest degree, contact is 

 destroyed, and the circumferences of the 



circles are said to cut ouo another. Thus in Fig. 51, the circum- 

 ference of the circle BCD, which overlaps tho circle ACE, cuts 

 the circumference of tho hut-named circle in the points c and F. 

 When, therefore, two circles touch ono another, they touch or 

 meet in one point only. They cannot touch 

 each other by any possibility whatever in 

 more points than one. When one circle is 

 said to cut another circle, the circumference 

 of the ono cuts the circumference of the 

 other in two points only ; it cannot by any 

 possibility whatever cut the circumference 

 of the other in more points than two. 



In Fig. 50, tho straight line x Y which 

 touches the circle A B c in tho point B is called a tangent to that 

 circle. Now in speaking of a lino in plane geometry as tho 

 tangent of or to a circle, nothing more is meant than this, that 

 it touches the circle. Tho word tangent is derived from the 

 Latin tango, I touch. In Fig. 52, the straight lines p Q, B s, are 

 tangents to the circle A B c, P Q touching it in the point B, and 



Fig. 50. 



B 8 in the point c. Straight lines drawn at right angles to thfe 

 tangents of a circle intersect in the centre of that circle. Thus 

 in the annexed figure, the straight lines B o, C o drawn at right 

 angles to the tangents p Q, It a respec- 

 tively, intersect in tho point o, the cen- 

 tre of the circle ABC. The straight 

 lines B o, c o are radii of the circle ABC; 

 and thus, conversely, if a straight line be 

 drawn through the extremity of a radius 

 that meets tho circumference of any 

 circle, at right angles to that radius, the 

 straight lino thus drawn at right angles 

 to the radius is a tangent of that circle. 



The ratio of tho diameter of a circle to 

 its circumference is aa 1 to 3$, or very 

 nearly so, expressed in figures in the 

 simplest way possible that is, the circumference of a circle 

 is equal in length to three times the length of the dianxitflr, 

 and one-seventh of its length ; or it may be written as 7 to 22, 

 which means that if a diameter of a circle be divided into 7 

 parts, the length of the circumference is equal to 22 of those 

 parts. Expressed in decimals, the ratio is as 1 to 3'14159. 

 Of course a greater degree of exactness may be obtained by 

 increasing the number of decimal places in the above, bnt the 

 number given is sufficient for all practical purposes. 



The following remarks have been made on the ratio of the 

 diameter of a circle to tho circumference by General T. Perronet 

 Thompson, whoso name has been already mentioned in our 

 Lessons in Music : 



" If it is asked, what after all is the proportion between the 

 diameter of a wheel and its circumference? it is as 1 to 

 3*14159, etc. etc., to as many figures of decimals as anybody 

 shall think it worth while to discover and add ; but, as in the 

 case of the square root of 5, coming to the end, No ! As the 

 Irish sailor said of the rope, the end is cut off. It is not quite 

 so easy to add figure to figure as in the case of the square root 

 of 5, but tho conclusion is the same The simplest pro- 

 portion for common purposes is as 7 to 22. The next, which 

 there is very seldom any occasion to go beyond, is as 113 to 355, 

 on which may be given a useful piece of what is called ' artificial 

 memory.' The ablest man it was ever my chance to know, 

 professor of mathematics in the University of Cambridge, wanted 

 this proportion one day, and was observed to be fidgeting with 

 a pen and a piece of paper. At hist he broke out, ' There it is, 

 sir. Write down the first three odd figures in pairs, and cut 

 them in two 



113 | 355.' 



I remember telling this to the driver of a French cabriolet on 

 the Pont Neuf, to his great delight. He will never forget it 5 

 nor should any working man to whom it may be ever likely to 

 be of use." 



READING AND ELOCUTION. XIV. 



ANALYSIS OP THE VOICE (continued). 

 EXERCISES ON INFLECTIONS. 



Rising Inflection. 



Rule 1. " High rising inflection." 

 3<i .'say you t6 1 



JHiut .' confer a crtfien on the author of the public calamities ? 

 Indeed ! acknowledge a trditor for our sovereign ? 



Rule 2. " Moderate rising inflection." 



In every station which Washington was called to fill, be acquitted 

 himself with honour. 



As the evening was now far advanced, the party broke up. 



Where your treasure is, there will your heart be I also. 



Though we cannot discern the reasons which regulate the occur, 

 renoe of cvonts, we may rest assured that nothing can happen without 

 the cognisance of Infinite Wisdom. 



Despairing of any way of escape from the perils which surrounded 

 him, he abandoned his struggles, and gave ii'm^if up to what seamed 

 his inevitable doom. 



Had I suffered such enormities to pass unpunished, I should hare 

 deemed myself recreant to every principle of justice and of duty. 



