338 



THE POPULAB EDUCATOR. 



lines which have been thus put together at right angles. The tri- 

 angle formed in this manner, as the triangle c E a in Fig. 40, is a 

 right-angled triangle ; and as in the case of this triangle it has been 

 shown practically that the square G E L M described on G E, the 

 side which subtends the right angle E c G, is equal in superficial 

 area to the squares c F K G, c D H E, described on the sides G c, 

 C E, which contain the right angle E c G, so it is true that in any 

 and every right-angled triangle the square described on the side 

 which subtends the right angle is equal to the squares of the 

 sides by which the right angle is contained. 



But there are yet other facts that may be gathered from an 

 examination of Fig. 40, and a consideration of the dotted lines 

 that are drawn in the figure. First, through the point C the 

 straight line c N is drawn parallel to G M or E L, intersecting 

 the straight line E G in the point T, and dividing the square 

 G E L M into the unequal rectangles (see Definition 27, page 53) 

 T E L N, T G M N. Of these the rectangle T E L N is equal to 

 the square c D H E, and the rectangle T N M G equal to the 

 square c G K F, as we will proceed to show. 



The reader will remember that in Problem XXIV (page 308) 

 it was shown that triangles on the same base and between the 

 same parallels are equal to one another, and that triangles on 

 equal bases and between the same parallels are also equal to 

 one another, Now in the trapezoid (see Definition 31, page 53) 

 G D H E, of which the sides G D, H E are parallel, there are two 

 triangles, D H E, G H E. These triangles stand upon the same 

 base H E, and between the same parallels G D, H E, and are 

 therefore equal to one another. But the dotted line D E is a 

 diagonal of the square c D H E, and divides it into two equal 

 parts ; therefore the triangle D H E is equal to the triangle 

 c D E, or, in other words, the square c r> H E is double of 

 the triangle D H E, and as the triangle r> H E is equal to the 

 triangle G H E, the square c D H E is also double of the triangle 

 G H E ; and this brings us to the fact, that when a square and a 

 triangle happen to be on the same base, and between the same 

 parallels, the area of the square is double the area of the triangle. 



Now let us turn to the trapezoid c E L N, of which the sides 

 C N, E L are parallel, and which contains the rectangle, or rect- 

 angular parallelogram E L N T within its limits. In this there 

 are also two triangles, c E L, E L N, standing on the same base, 

 E L, and between the same parallels, the parallel sides E L, c N, 

 of the trapezoid c E L N, and these triangles are consequently 

 equal to 'one another. Now the rectangle E L N T is divided 

 into two equal parts by the diagonal E N, and the triangle E L N 

 is therefore equal to the triangle E T N, or in other words, the 

 rectangle E L N T is double of the triangle E L N, and as tho tri- 

 angle E L N is equal to the triangle GEL, the rectangle E L N T 

 is double of the triangle GEL. And this teaches us that when 

 a rectangle or right-angled parallelogram and a triangle are 

 upon tho same base, and between the same parallels, the area 

 of the rectangle is double the area of the triangle. 



And as it is true that when a square and a triangle, or a rect- 

 angle and a triangle, are upon the same base and between the 

 same parallels, the area of the square or rectangle, as the case 

 may be, is double that of the triangle, so it is equally true that 

 when a square and a parallelogram, or a rectangle and a paral- 

 logram, are upon the same base and between the same parallels, 

 the areas of tho square and rectangle, or the areas of the rect- 

 angle and the parallelogram, thus situated, are equal to one 

 another, as may be seen by drawing the straight line H o through 

 H, parallel to E G. when we have the square c D H E, and the 

 parallelogram o H E G on the same base E H, and between the 

 same parallels H E, G D, equal to one another ; and by drawing 

 the straight line L v through L, parallel to E c, when we get the 

 rectangle E L N T and the parallelogram c E L v equal to one another. 



Parallelograms also on the same base and between the same 

 parallels are equal to one another, and when a parallelogram 

 and a triangle are on the same base, the area of the parallelo- 

 gram is double the area of the triangle ; and more than this, as 

 triangles on equal bases and between the same parallels are 

 equal to one another, so also rectangles and parallelograms on 

 equal bases and between the same parallels are equal to one 

 another. 



But to proceed to show that the rectangle E L N T is equal to 

 the square c D H E, let us look at the triangle G H E, which was 

 proved to be equal to half the square c D H E, and the triangle 

 C E L, which was proved to be equal to half of the rectangle 

 JP U N T, and compare their sides and angles. On inspecting 



them we find that the side E L of the triangle c E L is equal to 

 the side E G of the triangle G H E, each being also a side of 

 the square G E L M, and that the side C E of the triangle c E L 

 is equal to the side E H of the triangle .G H E, each of them 

 being also a side of the square c D H E ; and the angle C E i^ 

 contained by the sides c E, E L of tho triangle C E L, is equal to 

 the angle G E H contained by the sides G E, E H of the triangle 

 G H E, for each of these angles is composed of a right angle and 

 the angle C E G, which is common to both, the angle c E L being 

 composed of the right angle LEG and the angle G E c, while 

 the angle G E H is composed of the angle G E c and the right 

 angle c E H. Here, then, we have two triangles, each having 

 two sides of the one, namely, G E, E H, equal to two sides of the 

 other, L E, E c, and the angles contained by these sides equal, 

 namely, the angle G E H to the angle GEL; and this being true, it 

 is plain that their bases or third sides are also equal, namely, 

 H G to C L ; and the areas of the triangles are equal, as we may 

 prove practically by cutting out the triangle GEL, and turning 

 it, as on a pivot, round the point E, until it rests on the triangle 

 G H E. But the square c D H E has been shown to be double of 

 the triangle G H E, and the rectangle E L N T has been shown to 

 be double of the triangle GEL, and as things which are double 

 of equal things must be equal to one another, the rectangle 

 E L N T must be equal to the square c D H E. In the same way 

 it may be shown that the rectangle G T N M is equal to tho square 

 c F K G, and the learner is recommended to work out the proof 

 of this as a useful exercise. 



READING AND ELOCUTION. XL 



ANALYSIS OF THE VOICE (continued). 

 VII. RIGHT EMPHASIS. 



EMPHASIS distinguishes the most significant or expressive 

 words of a sentence. 



It properly includes several functions of voice, in addition 

 to the element of force. An emphatic word is not uiifrequently 

 distinguished by the peculiar "time," "pitch," "stress," and 

 "inflection" of its accented sound. But all these properties 

 are partially merged, to the ear, in the great comparative force 

 of the sound. Hence it is customary to regard emphasis as 

 merely special force. This view of the subject would not be 

 practically incorrect, if it were understood as conveying tho 

 idea of a special force superaddcd to all the other character- 

 istics of tone and emotion, in the word to which it applies. 



Emphasis is either " absolute " or " relative." The former 

 occurs in the utterance of a single thought or feeling, of great 

 energy ; the latter, in the correspondence or contrast of two 

 or more ideas. 



"Absolute" emphasis is either " impassioned " or "distinc- 

 tive." Tho former expresses strong emotion, as : 

 False wizard, A VAUNT ! * 



But the latter designates objects to the attention, or distin- 

 guishes them to the understanding, as : 



The fall of man is the main subject of Milton's great poem. 

 " Relative " emphasis occurs in words which express compari- 

 son, correspondence, or contrast, as : 



Cowards die many times ; the brave but once. 



Rules on Emphasis. 



Rule 1. Exclamations and interjections usually require 

 " impassioned " emphasis, or the strongest force of utterance, as 

 in the following examples : 



Woe ! to the traitor, WOE ! 



UP ! comrades, UP ! 

 AWAKE ! AEISE ! or be for EVER PALLEI; ! 



Te icefalls ! 



Motionless torrents ! silent cataracts ! 

 Who made you glorious as the gates of heaven, 

 Beneath the keen full moon ? 

 GOD ! GOD ! the torrents, like a shout of nations, 

 Utter : the ice-plain bursts, and answers, GOD ! 

 The silent snow-mass, loosening, thunders, GOD ! 



* Three degrees of emphasis are visually thus denoted in type : the 

 first by Italic letters ; the second, by small capitals ; and the third, by 

 large capitals. Thus, " You shall DIE, BASE DOG ! and that before 

 you cloud has passed over the sun!" Sometimes a fourth, by Ital'C 

 capitals, thus : NEYEE, NEVER, NEVER! 



