218 



THE POPULAE EDUCATOR 



and railed off, while sometimes this part is separated from the 

 rest of the building by a handsome carved screen. In parish 

 churches the chancel, consisting only of the central portion 

 here marked in the ground-plan, often terminates in a semi- 

 circular recess called the apse, which is occupied by the altar. 



The lectern (L in the plan), situated almost in the centre of 

 the transept in this instance, as in some others, is the place 

 from which the lessons are read ; and the letter P marks the 

 position of the pulpit. It must, however, be noticed that in 

 cathedrals, where the service is almost invariably performed in 

 the chapel (the position of which is marked in our present 

 ground-plan by the "choir"), the pulpit and lectern are often 

 placed in the chancel, near the spot marked thus +. 



The division marked C, which to a certain extent mars the 

 symmetrical arrangement of the ground-plan, is for convenience' 

 sake commonly occupied either, as in this case, by the vestry, 

 or by a chapter-house. Here, also, are frequently found the 

 cloisters, or covered ways which in olden times were used for 

 the purpose of exercise by the persons engaged in the service cf 

 the church. 



LESSONS IN ALGEBRA. VII. 



DIVISION. 



ART. 91. (1.) A man divided 48 apples among 6 boys. How 

 many did each receive ? 



Here, if 6 boys receive 4-8x apples, it is manifest that 1 boy 

 will receive a of 48x apples ; but | of 48 = 8x apples ; for 

 48x -f- 6 = 8x. Whence 8;c apples is the answer. 



(2.) If 8 hats cost 24a shillings, what will 1 hat cost ? 



Here, reasoning as before, 1 hat will cost | of 24a shillings, 

 but 24a -T- 8 = 3a ; therefore 3a shillings is the answer. 



The process followed in these examples is called DIVISION. It 

 consists in finding liow many times one quantity contains another, 

 and is the reverse of multiplication. The quantity to be divided 

 is called the dividend ; the given factor, the divisor ; and that 

 which is required, the quotient. 



92. DIVISION, therefore, is finding a quotient, which multiplied 

 into the divisor will produce the dividend. As the product of 

 the divisor and quotient is equal to the dividend, the quotient 

 may be found by resolving the dividend into two such factors, 

 that one of them shall be the divisor. The other will, of course, 

 be the quotient. 



Suppose, for instance, that dbd is to be divided by a. The 

 factors a. and bd will produce the dividend. The first of these, 

 being a divisor, may be set aside as the one factor. The other 

 factor is the quotient. 



93. When the divisor therefore is found as a factor in the 

 dividend, the division is performed by cancelling this factor. 



EXAMPLES. (1.) Divide ex by c. Ans. x. 

 (2.) Divide dh by d. Ans. h. 

 (3.) Divide drx by dr. Ans. x. 

 (4.) Divide limy by hm. Ans. y. 

 (5.) Divide dhxy by dy. Ans. hx. 



94. PROOF. Multiply the divisor aid the quotient together, 

 and the product will be equal to the dividend, if the ivork is right. 



Thus ax -4- a gives the quotient x. Proof. Here x X a gives 

 bhe dividend ax. 



95. If a letter is repeated in the dividend, care must be taken 

 that the factor which is rejected be only equal to the divisor. 



EXAMPLES. (1.) Divide aab by a. Ans. ab. 



(2.) Divide bbx by b. Ans. bx. 



(3.) Divide aadddx by ad. Ans. addx. 



(4.) Divide aammyy by amy. Ans. amy. 



(5.) Divide aaaxxxh by aaxx. Ans. axh. 



(6.) Divide yyy by yy. Ans. y. 



In such instances as the preceding, it is obvious that we are 

 not to reject every letter in the dividend which is the same with 

 one in the divisor. 



96. If the dividend consists of any factors whatever, expunging 

 one of them is dividing by that factor. 



EXAMPLES. (1.) Divide a (b -\- d) by a. Ans. b -j- d. 



(2.) Divide a (b +d) by b + d. Ans. a. 



(3.) Divide (b + x) (c + d) by b -f x. Ans. c -f d. 



(4.) Divide (6 + y) x (d h) x by d h. Ans. (b -f- y) x. 



97. If there are numeral co-efficients prefixed to the letters, the 

 co-efficients of the dividend must be divided by the co-efficients qf 

 the divisor. 



EXAMPLES. (1.) Divide Gab by 2b. Ans. 3a. 



(2.) Divide IGdxy by 4dx. Ans. 4>y. 



(3.) Divide 25d7w by dh. Ans. 25r. 



(4.) Divide I2xy by 3. Ans. 4on/. 



(5.) Divide 34:drx by 34. Ans. drx. 



(6.) Divide 20/wn by m. Ans. 2Qh. 



98. When a simple factor is multiplied into a compound one, 

 the former enters into every term of the latter. [Art. 76.] Thus 

 a into b -f- d, is ab -f- ad. Such a product is easily resolved again 

 into its original factors. Thus ab -f- ad = a X (6 -f- d). 



EXAMPLES. (1.) Resolve ab + ac -|- ah into its factors. 



Here ab + ac + ah = a X (b + c -f- h). Ans. 



(2.) Eesolve c 9 n + c*dx + c z y 2 into its factors. Ans. c 2 X 

 (n + dx + y 2 ) or c 2 (n + dx + y*). 



(3.) Eesolve bd + Wed? + btfd into its several factors. Ans. 

 bd (1 + bed + c 2 ). 



(4.) What are the factors of amh -f- amx -f- amy ? Ans. am 

 + y). 



(5.) What are the factors of 4acl + 8ah 4- 12am -i- 4ay ? Ans. 



In these examples, if the whole quantity be divided by one of tho 

 factors, according to Art. 96, the quotient will bo the other factor. 

 Divide (ab -f- ad) by a. 

 Here ab + ad -f- a = b + d. Ans. 

 Divide ab + o-d by b -f- d. 

 Here (ab -J- ad) 4- (b +d) = a. Ans. 



Hence, if the divisor is contained in every term of a compound 

 dividend, it must be cancelled in each. 



(6.) Divide ab + ac by a. Ans. b + c. 



(7.) Divide bdh + bdy by b. Ans. dh + dy. 



(8.) Divide aah-+- ay by a. Ans. ah + y. 

 . (9.) Divide drx + dhx -f- dxy by dx. Ans. r -J- h + y. 

 (10.) Divide Gab + I2ac by 3a. Ans. 2b + 4c. 

 (11.) Divide IQdry + IGd by 2d. Ans. 5ry + 8. 

 (12.) Divide 12hx + 8 by 4. Ans. 3hx + 2. 

 (13.) Divide 35dm + 14ifc by 7cZ. Ans. 5m + 2x. 



99. On the other hand, if a compound expression, containing any 

 factor in every term, be divided by the other quantities connected 

 by their signs, the quotient will be that factor. [See Art. 98.] 



EXAMPLES. (1.) Divide ab + ac + ah by b + c + h. Ans. a. 

 (2.) Divide amh + amx + amy by h + * + y- Ans. am. 

 (3.) Divide 4ab + Say by b + 2y. Ans. 4a. 

 (4.) Divide ahm + ahy by m + y. Ans. ah. 



100. In division, as well as in multiplication, the caution must 

 be observed, not to confound terms with factors. [See Art. 76.] 



EXAMPLES. (1.) Divide (ab + ac) by a. 



Here (ab + ac) -j- a = b + c by Art. 98. 



(2.) Divide (ab X ac) by a. 



Here (ab X ac) -j- a = aabc -f- a = abc by Art. 95. 



(3.) What is the quotient of (ab -f ac) -~ (b + c) ? ' Ans. a. 



(4.) What is the quotient of ab X ac-j- (b X c) ? Ans. aa. 



BULB FOR SIGNS IN THE QUOTIENT. 



101. In division, the same rule is to be observed respecting 

 the signs as in multiplication; that is, if the divisor and 

 dividend are both positive, or both negative, the quotient must 

 bo positive : if one is positive and the other negative, the 

 quotient must be negative. [Art. 82.] 



This is manifest from the consideration that the product of 

 the divisor and quotient must be the same as the dividend. 

 For if + aX+b = + ab, then + ab -- (- b = + a; 



If a X + b = ab, then ab -- \-b= a; 



If + a X b = ab, then ab -- b = + 

 And if a X b = -j- ab, then + ab -- b = a. 

 EXAMPLES. : (1.) Divide abx by a. Ans. b. 

 (2.) Divide 8 a Way by 2a. Ans. 5y 4. 

 (3.) Divide 3aa; 6ay by 3a. Ans. x 2y. 

 (4.) Divide 6am X dh by 2a. Ans. 3mdh. 



102. If the letters of the divisor are not to bo found in the 

 dividend, the division is expressed by writing tJie divisor under the 

 dividend in the form of a vulgar fraction. 



NOTE. This is a method of denoting division, rather than an 



