THE POPULAR EDUCATOR. 



divided into regular and irregular ; the former having 1 all their 

 sides and angles equal to each other ; and tho latter having any 

 variation whatever in these respects. Tho sum of all tho sides 

 of a polygon is called its perimeter, and when viewed in position 

 its contour. Irregular polygons are also divided into convex and 

 non-convex ; or, thoso whose angles are all salient, and those 

 of which one or more are re-entrant. Tho irregular polygon 

 (Fig. 20) has its angles at B, c, and D, salient ; and its angles 

 at A and E, re-entrant. 



34. Polygons are also divided into classes, according to the 

 number of their sides ; as, tho pentagon (Fig. 21), having five 

 sides ; the hcxayon (Fig. 22), having six sides ; the heptagon 

 having seven aides ; the octagon having eight sides ; and so on. 

 According to this nomenclature, the triangle is called a trigon, 

 and the quadrangle a tetragon. 



LESSONS IN ARITHMETIC. IV. 



MULTIPLIC ATION. 



1. THE repeated addition of a number or quantity to itself is 



called multiplication. Thus, tho result of the number 5, for 



instance, added to itself 6 times, is said to bo 5 multiplied by 6. 



5 + 5 + 5 + 5 + 5 + 5 = 30, or 5 multiplied by 6 is 30. 



When the numbers to be multiplied are large, it is evident that 

 the process of addition would be very laborious. The process 

 of multiplication which we are going to explain is therefore, in 

 reality, a short way of performing a series of additions. Let it, 

 then, be borne in mind, that multiplication is, in fact, only 

 addition. 



2. Definitions. The number to be repeated or multiplied is 

 called the multiplicand. The number by which we multiply is 

 called the multiplier: it, in fact, indicates how many times the 

 multiplicand is to bo repeated, or added to itself. The number 

 produced by the operation is called the prodiwt. The multiplier 

 and multiplicand are also called the factors of which the product 

 is composed, because they make the product. 



Thus, since 5 multiplied by 6 is 30, 5 and 6 are called 

 factors of the number 30. 



The sign X placed between two numbers means that they 

 are to be multiplied together. 



3. Before proceeding farther, the learner must make himself 

 familiar with the following table, which gives all products of 

 two numbers up to 12 : 



MULTIPLICATION TABLE. 



To determine the product of any two numbers by the above 

 table, find one of the numbers in the top line reading across the 

 page, and then find the other in the line on tho left hand which 

 runs down the page. Follow the column down the page in 

 which the first number stands, and the column across the pago 

 fn which the second number stands. The number standing in 

 thei square where these two columns meet is the product of the 

 two numbers. 



Thus, to find the product of 4 multiplied by 6 ; 4 in the top 



line and 6 in the left-hand line stand in lines which meet in a 

 square containing 24, which is therefore the product of 4 mv 

 plied by 6. 



It may bo observed that 6 in tho top lino and 4 in the left- 

 hand side line stand in lines which meet in a square also cor., 

 taining 24. The reason of this is that when the product of twc 

 numbers is required, it is indifferent which we consider to be the 

 multiplier and which the rmdtiplicand. Thus, 4 added to itself 6 

 times, is tho same as G added to itself 4 times. The truth of 

 this may bo seen, perhaps, more clearly as follows : 



If we make four vertical rows containing six dots each, as 



represented in the figure, it is quite evident that tho 



whole number of dots is equal cither to the number 



" .' of dota in a vertical row (6) repeated 4 times, or to 



' tho number of dots in an horizontal row (4) repeated 



' six times. And the same is clearly true of any other 



' two numbers. 



Hence we talk of two numbers being multiplied 

 together, it being indifferent which we consider to be the multi- 

 plier and which the multiplicand. 



4. If several numbers bo multiplied together, the result is 

 called the continued product of the numbers. Thus, 30 is the 

 continued product of 2, 3, and 5, because 2 X 3 X 5 = 30. 



N.B. On learning the multiplication table, let the following 

 facts be noticed : 



The product of any number multiplied by 10 is obtained by 

 adding a cipher to the number. 



The results of multiplying by 5 terminate alternately in 5 and 0. 

 The first nine results of multiplying by 11 are found by merely 

 repeating the figure to be multiplied. Thus, 11 times 7 are 77. 



In the first ten results of multiplying by 9 the right hand 

 figure regularly decreases, and tho left hand figure increases by 

 1 ; also, the sum of the digits is 9. Thus, 9 times 2 are 18, 

 9 times 3 are 27. 



5. It is evident that [as 2 X 3 X 5 = 30, and 2x3 = 6, and 

 6x5 = 30] in multiplying any number, 5, for instance, by 

 another, 6, for instance, it will be the same thing if we multiply 

 it successively by the factors of which tho second is composed. 

 Thus, the product of any number multiplied by 28 might be got 

 by multiplying it first by 7, and then multiplying the result 

 by 4. 



The product of ar.v number multiplied by 10 is obtained by 

 annexing a cipher to the number. The product of any number, 

 therefore, multiplied by 100 will be obtained by adding two 

 ciphers, because 10 x 10 = 100; first multiplying by 10 adds 

 one cipher, and then multiplying the result by 10 adds another 

 cipher. Similarly a number is multiplied by any multiplier 

 which consists of figures followed by any number of ciphers, by 

 first multiplying by the number which is expressed by the figures 

 without the ciphers, and then annexing the ciphers to the result. 

 Thus, 5 times 45 being 225, we know that 500 times 45 is 22500. 



G. The process of multiplication which we now proceed to 

 explain, depends upon tho self-evident fact that if tho separate 

 numbers of which a number is made up bo multiplied by any 

 factor, and the separate products added together, tho result is 

 the same as that obtained by multiplying the number itself by 

 that factor. Thus 



5 + 4 + 2 = 11 



7x5 = 35, 7 X 4 = 28, 7X2 = 14. 

 35 + 28 + 14 = 77 = 7 x 11. 



7. We shall take two cases : first, that in which tho multiplier 

 consists only of one figure ; and, secondly, when it is composed 

 of any number of figures. 



Case 1. Required to multiply 2341 by 6. 



2311 = 2 thousands + 3 hundreds + 4 tens + 1 unit. 



Multiplying these parts separately by 6, wo get 6 units, 24 

 tens, 18 hundreds, and 12 thousands, which, written in figures 

 and placed in lines for addition, are 



6 



240 

 1SOO 

 12000 



Giving as the result 14040 



The process may be effected more shortly, as follows, in one 

 line ; the reason for the method will bo sufficiently apparent 

 rr,om the preceding explanation : 



