150 



THE POPULAR EDUCATOR. 



might expect from a man who had seen a great deal of the 

 world, yet who had retained his geniality and his benevolence. 

 The story of Dr. Primrose and his family has furnished many 

 commonplaces to the language ; and Hazlitt has remarked 

 that we are the better for the well-known allusion to the 

 guinea which the two Miss Primroses kept unchanged in their 

 pockets, for being told about the picture of the vicar's family 

 and that of the Flamborough family, the latter all painted with 

 oranges in their hands, and for the adventure of the gross of 

 green spectacles. Another of the charms which even the least 

 critical perceive in "The Vicar of Wakefield" is the beauty of 

 the style. On analysis, this proves to be a curious compound 

 of Addison and Johnson; yet the two elements are mingled 

 with an art which belongs to Goldsmith himself. The some- 

 what forced and self-conscious simplicity of the vicar reminds 

 us of Sir Roger de Coverley ; his axiomatic sentences are mani- 

 festly caught from the Rambler. 



LESSONS IN ALGEBRA. Y. 



MULTIPLICATION. ' 



66. EXAMPLES. (1.) What will 4 oranges cost at x pence 

 each ? 



Here we say, if one orange costs x pence, 4 oranges will cost 

 4 times as much ; they will therefore cost 4c pence ; and this is 

 the answer. 



(2.) How much can a man earn in 5 months at a. pounds per 

 month ? Reasoning as before, we have a X 5 = 5a pounds for 

 the answer. 



Now 4# is equal iox-\-x-\-x-^-x; and 5a = a-}-a + a-j-a-f-a. 



67. Hence the repeated addition of a quantity to itself is called 

 MULTIPLICATION. From this definition of multiplication, it is 

 manifest that the product is a quantity of the same kind as the 

 multiplicand, 



68. It is plain, therefore, that multiplying by a ivhole number 

 is toMng the multiplicand as many times as there are units in the 

 multiplier. Thus multiplying a by 1, is taking the multiplicand 

 once, as a. 



Multiplying a by 2, is taking the multiplicand twice, as 

 a -f- a, etc. 



69. On the other hand, multiplying by a FRACTION is taking a ' 

 certain PORTION of the multiplicand as many times as there are ' 

 like portions of a unit in the multiplier. Thus : 



Multiplying a by |, is taking i of the multiplicand once, 

 a* Jo. 



Multiplying a by , is taking of the multiplicand twice, as 



7, a + \ a - 



70. Multiplying two or more letters togetJier, is writing them, 

 one after the other, either with or unthout the sign of multipli- 

 cation between them [see Art. 23> page 21]. Thus b multiplied 

 into c is b X c, or b . c, or be ; and the product of x into y, into 

 z, is x X y X a, or x . y . z, or, as it is more commonly written, 

 xyz. Also the product of am into xy is amxy ; and of abc into 

 xyz, is abcxyz. 



71. There will be no difference as to the result in whatsoever 

 order the letters are arranged. Thus the product of ba is the 

 same as that of ab ; and 3 times 5 is equal to 5 times 3. In 

 like manner, the product of a, b, and c, is abc, cab, bac, or cba. 

 It is more convenient, however, to place the letters in alpha- 

 betical order. 



72. When the letters have numerical CO-EFFICIENTS, these 

 must be multiplied together, and prefixed to tlie product of the 

 letters. 



EXAMPLES. (1.) Multiply 3a into 26. 



Here the answer is Gab. For if a into b is ab, then 3 times a 

 into b is evidently Sab and if, instead of multiplying by b, we 

 multiply by twice b, the product must be twice as great, that is 

 2 X Sab, which is Gab. 



(2.) Multiply 12% (3.) 3dh (4.) 2ad (5.) 76d7i (6.) Say 

 By 2rx my 13ghm x 8mx 



factors are to be multiplied, the product will be the same in 

 whatsver order the operation is performed. 



74. If the multiplicand be a compound quantity, each of its 

 terms must be multiplied into the multiplier. Thus the product 

 of b -f- c + d into a, is ab + ac -f- ad. For the whole of the 

 multiplicand is to be taken as many times as there are units in 

 the multiplier. 



EXAMPLES. 



(1.) Multiply d + Zxy (2.) 2h + m (3.) 3hl + 1 (4.) 2hm + 3 

 By 36 Gd>j my 46 



Product: 36d-t-6&ai/ I2dhy + 6dmy 3hlmy + my 8bhm+12b 



75. It must be carefully observed that the preceding instances 

 are not to be confounded with those in which several factors are 

 connected by the sign X , or by a point. In the latter case, the 

 multiplier is to be written before the other factors without being 

 repeated. The product of b X d into a, is ab X d, and not 

 a6 X ad ; for b X d is bd, and this into a is abd [Art. 70]. 

 The expression b X d is not to be considered like b -f- d, a com- 

 pound quantity consisting of two terms. Different terms a're 

 always separated by + or [Art. 19]. The product of 

 b X h X m X y into a, is a X b X h X m X y, or abhmy. But 

 6 -j- h -f- m + y into a. is ab -f- ah -f- am + ay. 



76. If both the factors are compound quantities, each term vn. 

 the multiplier must be multiplied into each term in tlie multipli- 

 cand. Thus (a -\- b) into (c + d) is ac + ad -f- & c + id. For the 

 units in the multiplier a + b, are equal to the units in a, added 

 to the units in b. Therefore the product produced by a must 

 be added to the product produced by b. Whence, the product 

 of c + d into a + b, is ac -f- ad -f- be -}- bd. 



For the product of c + d into a is ac -\- ad ; and the product 

 of c + d into 6 is be + bd [Art. 75] ; therefore the product of 

 c -f- d into a -f- b is ac + ad -\- be + bd. 

 EXAMPLES. 



(1.) Multiply 3x + d (2.) 4ay + 26 



By 2a + hm 3c + rx 



Product: 



(3.) Multiply 

 By 



-\- dhm 12acy + 66c + 4cmn/ + 2brx 

 a + 1 (4.) 26 -f- 7 



3x + 4 6d + 1 



Product : 24shrxy 3dhmy 2Gadghm fbdhx 24*amxy 



73. If either of the factors consist of figures only, these must 



be multiplied into the co-efficient of the other factor, and the 



letters annexed. Thus 3ab into 4, is 12a6 ; 36 into 2x, is 72x ; 



and 24 into hy, is 247ii/. 



From the preceding rules we have the general one, that when 



Product: 3ax + 3x -(- 4a -f 4 126d + 42d + 26 + 7. 



(5.) Multiply d + rx + h by 6m -f 4 + 7y. Ans. 6dm + Gmrx 

 + 6hm + 4d + 4)-a; + 47i + Idy + 7rxy + "7hy. 



(6.) Multiply 7 + 66 + ad by 3r + 4 + 27i. Ans. 21r + 186r 

 + 3adr + 28 + 246 + 4ad + 14h, + I2bh + 2adh. 



77. When several terms in the product are alilce, it will be 

 expedient to set one under the otlier, and then unite them by the 

 rules for reduction in addition, as in the following examples : 



(1.) Multiply 6 + a 

 By 6 + a 

 66 -f- ab 



+ ab + aa 

 Product : 66 + 2a6 + aa 



(2.) 6 + c + 2 

 b + c + 3 

 bb + bc + 26 

 + be + cc -f 2c 



+ 36+3c+6 



66 -f- 26c + 56 + cc + 5c + 6 



(3.) Multiply a + y + 1 

 By 36 + 2x + 7 



3a6 + 3by + 36 



+ 2ax + 2xy + 2x 

 + 7a 



Product : 3a6 + 36y + 36 + 2cw; + 2xy -f 2x -f 7a + 7y f 7 



(4.) Multiply 3a + d -f 4 by 2a + 3d + I. Ans. 6a 2 + Had 

 + Ha+3cl 2 +-13d+4. 



(5.) Multiply 6 + cd -f 2 by 36 + 4ccl + 7. Ans. 36 2 -f 76cd 

 + 136 +4c 2 d 2 + 15cd + 14. 



(6.) Multiply 36 + 2x -j- h by a X d X 2x. Ans. Gabdx + 4adr 8 

 -f Zadhx. 



78. It is plain that when the multiplier and multiplicand 

 consist of any quantity, repeated as a factor, this factor will be 

 repeated in the product as many times as it is in the multiplier 

 and multiplicand together. 



EXAMPLE. Multiply a X a X a 

 By a, X a 



Product : axaxaxaxa= aaaaa, or a 3 . 



