LIBRARY 
 
 . OF THE 
 
 University of California. 
 
 GIF^T OF^ 
 
 
A 
 
 KEY 
 
 , CONTAINING 
 
 THE STATEMENTS AND SOLUTIONS OF QUESTIONS 
 PEOFESSOE OHAELES DAVIES' 
 
 NEW ELEMENTARY ALGEBRA. 
 
 FOE THE 
 
 USE OF TEACHERS ONLY. 
 
 V or THF 
 
 UNIVER « 
 
 NEW YORK : 
 
 BARNES & BURR, PUBLISHERS, 51 & 53 JOHN ST. 
 
 CHICAGO : GEORGE SHERWOOD, 118 LAKE ST. 
 
 CINCINNATI: RICKEY <fe CARROLL. 
 
 ST. LOUIS : KEITH <fc WOODS. 
 
 1863. 
 

 Entebed, according to Act of Congress, in the year Eighteen Hundred and Fifty-nine, 
 
 By CHARLES DAYIEB, 
 
 In the Clerk's Office of the District Court of the Southern District of New York. 
 
 "William Denyse, 
 
 Stereotype!' and Electrotyper^ 
 
 188 William Street, N. Y. 
 
 Wood, Printer^ 
 Cor. Fulton and Dutch Sts., N. Y, 
 
P E E F A C E. 
 
 The question, whether a Key to a work on Mathematics 
 facilitates the acquisition of knowledge, is one about which 
 there is much diversity of opinion. If the business of 
 teaching were pursued as a profession — if the teachers in 
 our schools and seminaries looked to no other employment, 
 and gave their entire thoughts and time to the business of 
 instruction, they would have abundant means to prepare, in 
 the best manner, all the exercises for their pupils. 
 
 But, as yet, the case is far different. Teaching, with most 
 instructers, is an occasional and temporary business, and not 
 a permanent profession. Engaged, generally, in prejDaring 
 themselves for other pursuits, and at the same time giving 
 instruction in various branches of education, they have nei- 
 ther the time nor opportunity for that careful preparation 
 which is needed, and must, therefore, avail themselves of 
 all the aids which they can command. 
 
 It was not intended, originally, to prepare a Key to the 
 Elementary Algebra, but the urgent request of many teach- 
 ers has changed that determination. 
 
 iii 
 
IV PREFACE. 
 
 It was not thought best to work out the simple examples 
 which are given as illustrations, nor those which are given 
 to perfect the scholar in the meclianical part of Algebra ; and 
 hence, the work in the Key is limited to the problems, with 
 the exception of two examples, one on page 114, and the 
 other on page 221. The problems, it was supposed, pre- 
 sented difficulties in the statements only, which are fully 
 given, leaving the solution of the equations to be made by 
 the pupil. This will obviate much of the misuse to which a 
 Key may be applied, should it chance to fall into the hands 
 of the student. 
 
 The large figures at the head of each page point out the 
 corresponding page of the Algebra. 
 
KEY 
 
 TO 
 
 DAYIES' NEW ELEMENTARY ALGEBKA. 
 
 Pag^e 114. 
 
 (25.) 
 
 (a + 5) (a; ^ b) _ 3^ ^ 4Mj--J^ _ ^^ a? - hx 
 a — b a + b b 
 
 Multiplying by the least common denominator, 
 
 {a — b) («•+ b)b^ and canceling, we have, 
 
 {a + by {x - b)b - Sab{a - b) {a + b) = {iab - ¥) 
 {a — b)b - 2x{a — b) {a + b)b + {a^ - bx) {a - b) {a +b). 
 
 Performing indicated operations, we have, 
 
 a^bx + 2a52aj + b^x - aW - 2ab'^ — 5^ - da^b + Sab^ = 
 4aW - ctb^ - 4cab^ + b^ - 2a^bx + 2b^x + a* — a^bx 
 
 By transposing, 
 a'^bx + 2ab'^x + ¥x + 2a2^>a; — 2b^x + a^^a; - 5^^ = 
 4a252 - a63 - 4aP + )* + «* - «^^^ + a^b^ + 2ab^ -{- 
 b' -f 3a36 — 3a53. 
 
6 KEY TO DAVIES' NEW ELEMENTAEY ALGEBEA. [125. 
 
 Collecting terms and factoring, 
 
 2bx{2a'^ + ab ^ P) z= a^ + Sa^b + 4a^b^ - eaP + 25*; 
 a^ + Sa^b + ia^^ - 6aP + 25* 
 
 hence, 
 
 25(2a2 + ab - b^) 
 
 Page 125. 
 
 (13.) 
 
 Denote D'^s share by ic. Then, by the conditions of the 
 question, 
 
 aj + 360 = B's share, 
 
 and 2x + 720 — 1000 = A^s share. 
 
 But, X z=: D^s share, 
 
 and 360 — C'>s share; 
 
 hence, 4a; + 440 = 2520, the whole estate; from 
 
 which equation we find cc = 520. 
 
 (14.) 
 
 Let X denote the share of each daughter. Then, by the 
 conditions of the question, 
 
 2x = the share of each son. 
 Also, since there are three daughters and two sons, 
 Sx =z the amount received by the daughters, 
 and 4x = the amount received by the sons ; 
 
 also, 7cc + 500 = the amount received by the widow, 
 and 14aj -{- 500 = 7500, the whole estate ; and from 
 this last equation the value of cc is readily found, equal to 500. 
 
 (15.) 
 
 Let X denote the number of women. Then, by the con- 
 ditions of the question, 
 
 X + S =: the number of men. 
 
126.] EQUATIONS OF THE FIRST DKGKEE. 7 
 
 and 2aj + 8 + 20 = children. 
 
 But, X =z women ; hence, 
 
 4aj + 36 = 180, the whole number ; from 
 which we find cc = 36. 
 
 (16.) 
 Let X denote the share of the youngest brother. 
 Then, x + 40 = 2d son's share, 
 
 X -{- 80 =: 3d son's share, 
 aj + 120 =z 4th son's share, 
 cc + 160 = 5th son's share, 
 and 5x + 400 = 2000, the whole estat^ ; from 
 
 which we find x = 320. 
 
 (17.) 
 
 Let the share of A be denoted by x. Then, since A^s 
 share is to be to JB^s as 6 to 11, it follows that J5'5 shar<? 
 will be -y- of u4'5 ; hence, 
 
 -—X = j5'5 share,' 
 
 and X + —x + 300 = C^s share ; 
 
 22 
 hence, 2x -^ —x + 300 = 2850, the whole estate; 
 
 6 
 
 from which equation we find x = 450. 
 
 (18.) 
 
 Let X denote the number of paces taken by the first 
 person, fi^om the time of starting till the distance between 
 them is 300 feet. Then, the number of paces taken by the 
 second will be represented by 5x. But since the paces of 
 
8 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [126. 
 
 the first are 3 feet, and those of the second 1 J or -^ feet, the 
 distances traveled will be 
 
 Sx = the distcmcG traveled by first, 
 
 and -- X bx =z —x ~ the distance traveled by second ; 
 
 15 
 hence, -—x — ^x — 300, their distance apart ; from which 
 
 we find X = 6Q^ ; that is, the person who steps the longest 
 will have made 66| paces ; and since each pace is 3 feet, he 
 will have traveled • 
 
 66|- X 3 = 200 feet. 
 
 15 
 If, instead of subtracting Sx from — a?, we had written 
 
 o 
 
 the equation, 
 
 15 
 
 3a? — —-cc = 300, 
 
 we should have found, 
 
 X = - 66f, 
 
 which would have shown that the second person traveled 
 farther than the first ; which is indeed proved, when the 
 distance traveled by the second, minus the distance traveled 
 by the first, is positive. 
 
 (19.) 
 
 Let X denote the number of days which they worked. 
 
 Then, 2x =z the number of dollars earned by cai-penters, 
 
 24cc 
 
 -— - = 12a;, the amount earned by journeymen, 
 
 o 
 
 -a; = 2a:, the amount earned by the apprentices ; 
 
 hence, 16a; =: 144, the whole sum earned ; Vhence we find 
 cc = 9. 
 
126.1 EQUATIONS OF THE FJKST DEGREE. 9 
 
 (2O0 
 
 Let the sum at interest be denoted by «j. 
 
 4 
 Then, -x = what bears an interest of 4 per cent., 
 
 and -cc = what draws 5 per cent. 
 o 
 
 Then, since the interest which accrues on any sum for a 
 year, is equal to the sum multiplied by the rate, divided by 
 100, we shall have, 
 
 4 4 4 
 
 -X X — — = T^^t what the first produced, 
 
 15 1 
 
 and ~x X -— - = - — X. what the second produced : 
 
 5 100 100 ' i' J 
 
 then, A«, + _L«,=,2940; 
 
 or, 400a; + 125a; = 36750000 ; or, a; = '70000. 
 
 (21.) 
 
 There are several ways in which this example may be 
 solved : 
 
 First. We see by the conditions, that the first cock will 
 discharge one gallon in a minute, the second cock half a 
 gallon in a minute, and the third cock one-third of a gallon 
 in a minute. Hence, the quantity discharged by the three 
 cocks in a minute, is equal to 
 
 1 +::; + :: = -^ 2jallons. 
 
 Then, — gals. : 60 gals. : : 1 m. : 32y\ minutes. 
 
 But, to resolve the question in a general way, mthout 
 regard to the contents of the cask, let x denote the part of 
 1* 
 
10 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [126. 
 
 the cask which would be emptied in a single minute. Then, 
 as the first cock would empty q\ of the cask, the second 
 •j-^o of it, and the third jl^ of it, in one minute, we have, 
 
 X 
 
 = 
 
 1 
 
 60 
 
 + 
 
 1 
 
 120 
 
 + 
 
 1 
 
 180' 
 
 X 
 
 = 
 
 6 
 360 
 
 + 
 
 3 
 360 
 
 + 
 
 2 
 360' 
 
 
 
 X 
 
 = 
 
 11 
 
 360 
 
 • 
 
 
 or, 
 
 or, 
 
 Now, as JgV of the cask is emptied in one minute, it is 
 evident that to empty the entire cask will require as many 
 minutes as the number 1 contains ^lu ; that is, 32y8y. 
 
 (22.) 
 Let the number of trees in the orchard be denoted by a. 
 
 Then, ^ o^ = apple trees, 
 
 -X = peach trees, 
 
 -X =z plum trees, 
 
 20 = cherry trees, 
 80 = pear trees, 
 
 and X =z -x -^ -X + -X + 120 + 80: 
 
 2 4 6 
 
 from which we find x = 2400. 
 
 (23.) 
 
 Let the number of sheep be denoted by x. Then, by 
 observing the statement of the last problem, we have, 
 
 ^ "^ 4^ "*" 6^ "^ 8^ "^ 12^ "^ ^^^' 
 which, gives, x = 1200. 
 
127.] EQUATIONS OF THE FIRST DEGREE. 11 
 
 (24.) 
 
 Let X = the value of the horse. 
 
 Then, — = the value of the saddle, 
 
 and, ^ "•" 10 "^ -^^^^ 
 
 which gives, a; = 120. 
 
 (25.) 
 Let X denote the amount of rent last year. 
 
 Sx 
 Then, — equal the excess of the present over the past 
 
 year, and the rent of the present year will be expressed by 
 . + - = 1890; 
 
 which gives, 
 
 X =: 1750. 
 
 
 (26.) 
 
 Let 
 
 X = the number. 
 
 Then, 
 
 X — 5 = the difference, and 
 
 
 ? X (aj-5) = 40; 
 
 or 
 
 2a; — 10 = 120; or, x = 61 
 
 
 (27.) 
 
 Let 
 
 X = the length of the post. 
 
 Then, 
 
 '^x + h^+10 = X, 
 
 from which ' 
 
 we have, x = 24. 
 
12 KSY TO DAVIES' NEW ELEMENTARY ALGEBRA. [127 
 
 (2§.) 
 Let X = the number. 
 
 Then, x -- -x — -x = 66 ; 
 
 4 5 
 
 from which we have, x = 120. 
 
 (29.) 
 
 Let the number of beggars be denoted by x. 
 Then, by the conditions, 
 
 3a; — 8 = the amount of money he had ; 
 also, 2cc + 3 = the amount of money. 
 Hence, 3a; — 8 = 2a; + 3, or a; = 11. 
 
 (30.) 
 Let X = the number of shillings which he had. 
 Then, -x = what he first lost ; 
 
 and, a; — -a; + 3 = what he had left after borrowing 
 
 Zx -f- 3 Sx -f- 12 
 Then, - — - — = — -— — = what he lost the second time ; 
 
 o 12 
 
 and this, taken from Ja; + 3, or what he had at the com- 
 mencement of the second game, will give what he had kA; ; 
 that is, 
 
 3 , ^ 3aj + 12 
 
 from which we find, x — 20s, 
 
128.] EQUATIONS OF THE FIRST DEGEEE. 13 
 
 (31.) 
 Let X denote the amount laid out by each. 
 Then, cc + 126 = what A had, after gaining, 
 and, X — 87 =F what B had, after losmg. 
 
 Then, aj -f 126 ■=:: 2{x - 87) = 2iB - 174; 
 from which we find, x = 300. 
 
 (32.) 
 
 Let X denote the sum which he had at first. 
 Then, x — 2 = what he had after spending ; 
 2 
 
 2aj — 4 = what he had after borrowing, 
 and, 2a3 — 6 = what he had after spending at second 
 2 tavern ; 
 
 403 — 12 = what he had after borrowing, 
 and, 4a; — 14 = what he had after spending at third 
 2 tavern ; ' 
 
 8a; — • 28, what he had after borrowing, and 
 then, 8a; — 305. = 0. 
 
 Hence, a; = 3f5. = 35. 9 c?. 
 
 (33.) 
 
 Let X denote the number of yards in the piece. 
 Then, 3a; — 57 = what remained of the first 3 pieces, 
 
 and, a; — 17 = what remained of the fourth piece. 
 
 Hence, 4a; — 74 = 142 yards, what remained in all. 
 
 Therefore, 4a; = 142 + 74 ; or, x — 54. 
 
14 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [128. 
 
 (34.) 
 Let the number of cavany soldiers be denoted by x. 
 
 Then, Sec = the number of the artillery, and 9a; = the 
 number of the infantry : also, 
 
 a; + 3a; + 9a; = 2600 ; or, x = 200. 
 
 (35.) 
 
 Let X denote the number of miles he traveled on horse- 
 back. 
 
 Then, 3Ja; will denote the distance traveled by water, 
 
 no 
 
 and 3|a; x 2J = — a? = what he traveled on foot. 
 
 63 
 Consequently, x + S^x + -—a; = 2970; x = 240. 
 
 o 
 
 (36.) 
 Let X denote the number of pieces JB received. 
 Then, -x will denote the number A received; and, 
 
 o 
 
 5 8a; ^, . 
 
 a; + -a; = -— = the entire sum. 
 
 o o 
 
 But, by the conditions, 
 
 5 5 „ 8 5 40 
 
 -a;-- of 3^ = 3«^~27^ = 50; 
 
 that is, 
 
 45 40 «.^ 5 ^^ 
 
 27^ - 27^ "= ^0; ^^^ 27^ = 50; or, x = 270. 
 
 Hence, A's share is 450 pieces. 
 
129.] EQUATIONS OF THE FIRST DEGREE. 15 
 
 (37.) 
 
 Let the first number be denoted by x ; then, the second 
 will be denoted by mx ; and the third by nx. 
 
 Hence, x + mx + 7ix = «, which gives, 
 
 a ^, an ^^ am 
 
 1st, X =: - — I — ; 2d, — — j — ; 3d, 
 
 I -\- 771 -\- n 1 -\- m -\- n \ -\- m, -\- n 
 
 (38.) 
 
 Let X denote the number of dollars in the younger's 
 portion. 
 
 Then, x -\- -x = the portion of the second ; 
 
 o 
 
 and, 2x = the portion of the thu'd. 
 
 By the conditions, 
 
 ^ + ^ + Q^ + 2a3 = 1170; 
 
 o 
 
 that is, 
 
 3£C + 2aj + cc + 6aj = 3510, and, x = 270. 
 
 ( 39.) 
 
 Let X denote the number to be furnished by the first. 
 
 Then, ~x = that of the second. 
 
 o 
 
 7 5 35 
 and, - of -a; = —x = that of the third, 
 
 5 35 
 and, X + ~x + —x = 594 ; 
 
 that is, 
 
 72aj + 120CC + 105a; z= 42768, and, x = 144. 
 
 (40.) 
 Let X denote the number of dollars in A^s portion. 
 
16 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [129 
 
 Then, x = A'^s portion, in dollars ; 
 
 and, 2x + 200 = B's, 
 
 ^x —'400 = C's, 
 
 2^X + 50 r:= J9'5, 
 
 and, 
 
 Q1.X 150 
 
 8Jaj — 150 + -^ — h 475 = 5600, the estate; 
 
 or, 
 
 17 
 Ux — 600 + —X — 150 + 1900 =: 22400; 
 
 or, 
 
 68a; — 1200 + 17a; — 300 + 3800 = 44800, 
 
 hence, 85a; = 42500; and x = 500. 
 
 (41.) 
 
 Let x denote the number of quarts that will fill the first 
 
 cask. 
 
 4 3 
 
 Then, x — -x =z ro; = the contents of the second : 
 
 7 7 ' 
 
 3 
 
 — -l-a;l ::= —X = contents of third cask; 
 
 9 9 
 
 and, — X =z — , the contents of fourth ; 
 
 or, contents of fourth, = —a; = -x. 
 
 28 7 
 
 rw., '94 
 
 Then, a; = — a; + -a; + 15 ; 
 
 or, 28a; = 9a; + 16a; + 420, 
 
 and, X = 140. 
 
129.] EQUATIONS OF THE FIRST DEGREE. 17 
 
 (42.) 
 
 Let the number of days be denoted by x. 
 
 Then, since the first courier travels four miles a day, we 
 have, 
 
 4a; + 10 X 4 = 453 + 40 =: the distance traveled by the 
 
 first courier ; and 
 9cc = the distance traveled by the second ; 
 
 hence, %x — 4a3 + 40 ; or, cc = 8. 
 
 (43.) 
 
 Rate of first courier, 31i~-5 = 6.3 miles. 
 Rate of second " 22^ -r- 3 = 7.5 miles. 
 
 8 X 6.3 50.4 ^^_ 
 
 X — — T— = — — T- = 42 hours. 
 
 7.5 — 6.3 1.2 
 
 (44.) 
 
 The first will have traveled 3^ x 8 = 28 miles, at the 
 time the second departs; hence they will be 52 miles apart. 
 N"ow, if we denote by x the number of hours after the 
 departure of the second, until they meet, we shall have, 
 
 Z\x ~ distance traveled by the first, after the second starts ; 
 
 and, 
 
 h\x =z the distance traveled by the second ; 
 
 hence, S^x + 5}x = 52 ; 
 
 '^ 31 
 
 or, -X -4 X = 52 ; 
 
 ' 2.6 ' 
 
 42 + 62a; = 624 ; or, x = Q hours. 
 
18 KEY TO DAVIES' NEW ELEMFNTARY ALGEBRA. [135. 
 
 (120 
 
 Let the aumber of dollars possessed by A^ be denoted 
 by cc, and i?'5 number, by y. 
 
 Then, by the first condition, 
 
 X + 40 = 5(y — 40) = 5^ — 200 ; 
 
 by the second, cc + y = 120 ; from wWch we readily find, 
 
 X = 60, and y = 60. 
 
 (13.) 
 
 Let X denote the humber of years in the father's age. 
 
 Then the father's age, twenty years before, will be re- 
 presented by X — 20, and that of the son, by y — 20. 
 Hence, by the first condition, 
 
 a; — 20 =: 4(2/ - 20) := 4y — 80 ; 
 
 by the second, x =z 2y ; from which we find, 
 
 X = 60, and y — 30. 
 
 (14.) 
 
 Let X denote the number of dollars which the elder had, 
 and y the number which the younger had; then, by the 
 first condition, 
 
 aj — -aj = 2/ 4- 1000 ; 
 
 and, by second condition, 
 
 a; — iaj + 2000 = 2{y + 1000 — 500); 
 
 or, 4aj — aj + 8000 — Sy -{' 8000 — 4000 ; 
 
 or, 3a; = 8?/ — 4000; 
 
 from which the values of x and y are easily found. 
 
137.] EQUATIONS OF THE FIRST DEGREE. 19 
 
 I 
 
 (15.) 
 
 Let X denote the number of apples which John had, and 
 y the number Charles had. 
 
 Then, x - \f> == 2/ -f 15, 
 
 and 03 + 15 =: 15(2/ - 15) - 10; 
 
 from which we have x and y. 
 
 ( 16.) • 
 Let X denote the number of dollars in A^s salary, and y 
 that in ^'5 ; then, 
 
 aj + y r= 900 ; 
 
 and by the second condition, 
 
 a, - !«, = y + ia,; 
 from which x and y are easily found. 
 
 Page 137, 
 
 (11.) 
 
 Let X denote the number of dollars which A has, and y 
 the number B has ; then, 
 
 by the first condition, aj + 2/ = 20000 ; 
 by the second condition, y =: Sx; 
 
 whence, x z= 5000, and y = 15000. 
 
 (12.) 
 
 Let X denote the value of the first, and y that of the 
 second. 
 
 Then, by the first condition, x -{- 1 = Sy; 
 
 and, by the second condition, 5x = 2/ + V ; 
 
 from which we have x and y. 
 
20 KEY TO DAVIES' NKW ELEMENTARY ALGEBRA. [146. 
 
 (13.) 
 Let the numbers be denoted by x and y. 
 Then, by 1st condition, 6cc = Sy, 
 
 and by the second, aj — 1 = y — 2 ; 
 
 from which we have, a; = 5, and y z=z ^. 
 
 (14.) 
 Let the numbers be denoted by x and y. 
 
 By first condition, 
 
 aj + 2 = 
 
 Hy; 
 
 or, 
 
 a; + 2 = 
 
 13 
 
 by second condition. 
 
 X 
 
 2 ~" 
 
 2/ + 4; 
 
 from which we find, 
 
 X =z 24, 
 
 and 2/ = 8. 
 
 (15.) 
 
 Let the present ages of the father and son be denoted by 
 X and y; then, 
 
 by 1st condition, a; — 12 = 2y ; 
 
 by 2d condition, 4{y — 12) + 12 =z a; + 12 ; 
 
 from which we have, x = 12^ and y = 30. 
 
 Page 145. 
 
 (4.) 
 Let X and y denote the numbers. 
 Then, x — y z= *Jy 
 
 and a 4- 2/ = ^^ ; 
 
 which equations give, aj =r 20, and y = 13. 
 
146.] EQUATIONS OF THE FIRST DEGREE. 21 
 
 (5.) 
 Let X denote the greater, and y the lesser part. 
 Then, by 1st condition, x ■\- y = 75 ; 
 and by 2d condition, , Sx = 1y + 15; 
 
 which give, aj = 54, and y = 21. 
 
 (6.) 
 
 Let X denote the number of gallons of wine, and y the 
 
 cider. 
 
 X ~\~ 1/ 
 Then, by 1st condition, — -— ^ + 25 = x; 
 
 and by 2d condition, — — ^ — 5=2/; 
 
 from which we have, x = 85, and y = 35. 
 
 ( ^0 
 
 Let X denote the number of guineas, and y the number 
 of moidores used. 
 
 N^ow, it is evident that the number of pieces used, of each 
 kind, multiplied by the number of shillings in the jDiece, will 
 give the number of shillings paid in that particular kind of 
 money. That is, 21i« will be the number of shillings paid 
 in guineas, and 21 y the number paid in moidores. Then, 
 observing that the whole bill, £120 = 2400^., we have, 
 by 1st condition, x + y = 100; 
 
 by 2d condition, 21CC + 21y = 2400; 
 
 which give, a; = 50, and y = 50, 
 
 (8.) 
 
 Let X denote the distance traveled by the first, and y 
 the distance traveled by the second. 
 Then, x + y = 150 
 
22 KEY TO DA VI is' m:w i;lkmkntary algkbra. [146. 
 
 But, since the first travels 8 miles, while the second travels 
 but 7, the distance which they respectively travel will be in 
 the proportion of 8 to V ; that is, 
 
 ce : 2/ : : 8 : 7; 
 or, 1x = ^y\ 
 
 from which we find x = 80, and y = VO; and if the entire 
 distance traveled by each be divided by the distance trav 
 eled each day, the quotient AviU be the time, 10 days. 
 
 (0-) 
 
 Let X denote the number of votes cast for the first, and 
 2/ the number cast for the second. 
 Then, x -{- y — 375, 
 
 and X — y =z 91 ; 
 
 which give, x = 233, and y = 142. 
 
 (lO.) 
 
 Let X denote the value of the poorest horse, and y that 
 of the other. 
 
 Then, by 1st condition, aj + 50 = 2i/, 
 
 and by 2d condition, y -\- 50 = Sx; 
 
 which give, x = £30, and y = £40. 
 
 (XI.) 
 
 In this example, we must bear in mind that the minute 
 hand goes entirely round the face of the clock, while the 
 hour hand passes from one hour to the other ; that is, the 
 minute hand travels twelve times as fast as the hour hand. 
 
 If, then, we suppose the face of the clock to be divided 
 into twelve equal parts, corresponding to the hours, and x 
 and y to represent the distances passed over by the hour 
 
146.] EQUATIONS OF THE FIRST DEGREE. 23 
 
 and minute hands, from the time of separating mitil they 
 are again together, we shall have, 
 
 and 2/ ~ ^ = 12 ; 
 
 since, when the hands come together, the minute hand will 
 
 have gained the entire twelve spaces on the hour hand. 
 
 Multiplying the second equation by 12, and adding them 
 
 together, w^e have, 
 
 122/ = 2/ + 144; 
 
 144 
 or, y = — = 13Jy; 
 
 that is, the minute hand will have gone once around the face, 
 and lyV ^^ ^^ hour spaces in addition ; consequently, the 
 time required will be 1 hour, 5 minutes, Jy of 5 minutes, or 
 y\ of one minute. 
 
 If we subtract the second from the first equation of con- 
 dition, we have, 
 
 12 
 \\x = 12, and x = — = 1tt5 
 
 that is, X is equal to 1 and yV of the hour spaces, which, 
 reduced to ti7ne^ gives 1 hour 5/y minutes, as before. 
 
 (12.) 
 
 Denote by x the portion of the beer which the man would 
 drink in a single day. 
 
 Then, by the conditions of the question, the man and 
 woman together would drink J^ of the cask in a single day, 
 and the woman gV of it ; hence, what the man would drink 
 must be equal to the difference ; that is, 
 
 1 1 30 — 12 18 1 
 
 X = 
 
 12 30 360 360 20' 
 
24 KEY TO DAVILS' NKW j;LI:MKNTAKY ALGEBRA. [146. 
 
 that is, the man will drink gV of the beer in a single day ; 
 and hence, the whole of it in 20 days. 
 
 (13.) 
 
 Let the fresh water to be added be denoted by x. Then, 
 the amount of the mixture will be denoted by cc + 32. But 
 the addition of the fresh water will not increase the quantity 
 of salt in the 32 lbs. of salt water; hence, the cc + 32 
 pounds of the mixture will contain one pound, or 16 ounces, 
 of salt. But, by the conditions of the question, 32 lbs. of 
 this mixture are to contain 2 ounces of salt ; 
 
 hence, a? + 32 : 32 : : 16o^. : 2o2. ; 
 
 consequently, 2cc + 64 = 512 ; 
 
 or, X z=z 224. 
 
 (14.) 
 
 In this example, we must bear in mind that, if the rate of 
 interest be divided by 100, and the quotient multiplied by 
 the principal, the product will always be the amount of 
 interest. 
 
 Let X denote the greater part, and y the lesser. 
 
 Then, x -\- y =: 100000; 
 
 also, — - = what the larger part produced ; 
 
 4v 
 and, -~ = interest of lesser part ; 
 
 5x 4?y 
 
 consequently, -— - + —~ = 4640 ; 
 ^ •^' 100 100 ' 
 
 from which we find, 
 
 X = 64000, and y = 36000. 
 
147.] EQUATIONS OF THE FIRST DEGREE. 25 
 
 (15.) 
 
 Denote the number of votes received by the successful 
 candidate, by x, and the number received by the other, by 
 2/. Then, by the first condition, 
 
 X — y = 1500. 
 
 Had the first received ^ of y in addition, the second 
 would have received y — \y =z f y, and we should have, 
 
 a; + -y zzr 3 X -2/- 3500; 
 
 or 4a3 4- 2/ = 9^ — 14000; 
 
 from which we have, 
 
 X = 6500, and y = 5000. 
 
 (16.) 
 
 Let X denote the value of the gold watch, and y that of 
 the silver watch. 
 
 Then, x + 25 = S^y = \j, 
 
 2/+ 25 =. 1+ 15; 
 
 that is, 2cc + 50 = 72/, 
 
 and 22/ + 50 = cc + 30 ; 
 
 from which we have, 
 
 X = 80, and y = 30. 
 
 (ly.) ^ 
 
 The separate figures which are placed by the side of each 
 other, in order to express any number, are called digits. 
 Now, from the relative value of these figures, resulting from 
 the places which they occupy, we can easily see how the 
 numbers may be expressed. For example, if the number is 
 
 > or THF 
 
 UNIVERSITY 
 
 or 
 
26 KEY TO DA vies' NEW ELEMENTARY ALGEBRA. [147, 
 
 expressed by two digits, then the first figure on the right, 
 plus ten times the second figure, will always give the num- 
 ber. Thus, 
 
 36 = 3 X 10 4- 6; and, 87 = 8 X 10 + 7, &c. 
 
 If the number is expressed by three figures, then one hun- 
 dred times the left-hand figure, plus ten times the middle 
 figure, plus the right-hand figure, will express the number. 
 Thus, 
 246 = 100 X 2 + 10 X 4 H- 6 = 200 -f 40 + 6 = 246. 
 
 Let X z=z the left-hand digit, and y = the other. 
 Then, a; + y = 11 ; 
 
 also, a; -f 13 = 3y; 
 
 from which we have, 
 
 a; = 5, and y z= Q, 
 
 (18.) 
 
 Let X denote the number of gentlemen, and y the num- 
 ber of ladies. Then, y — 15 = the ladies who remained, 
 and X ~ 45 =z the gentleman who remained. And, by 
 the conditions of the question, 
 
 aj = 2(?/ - 15) = 2y - 30, 
 and, 5{x - 45) = 2/ - 15 ; or, 5aj - 225 = y - 15 ; 
 from which we have, 
 
 X = 50, and y =z 40. 
 
 .( 10-) 
 aiicij^^^if, X denote the value of the horse in dollars, and y the 
 number of tickets. If he sells the tickets at $2, he will 
 receive $21/ ; if at $3, he will receive $Sy, 
 Then, 2y = x — 30, 
 
 and, 3y = aj + 30 ; 
 
 which give, aj = 150, and y = 60. 
 
148.] EQUATIONS OF THE FIEST DEOEEE. 27 
 
 (20.) 
 
 Let X denote the number of bushels of wheat purchased, 
 and y the number of bushels of rye. 
 
 Then, lOOoj + I5i/ = 11V50 cents; 
 
 also, 100 X tCC + 75 X -2/ = 2750 cents; 
 
 4 5 
 
 or, 25a; + 15y = 2750 cents; 
 
 from which we have, 
 
 X = 80, and y = 50, 
 
 (21.) 
 
 Let X denote the number A took, and y the number JS 
 took. 
 
 Then, 52 — cc = what A left, 
 
 and, 52 — 2/ == what JB left. 
 
 But, by the conditions, 
 
 X = 2(52 — 2/), and y = 7(52 — x) ; 
 
 that is, X = 104 — 27, and y = 364 — Ix, 
 
 Hence, a; = 48, and y = 28. 
 
 (22.) 
 
 Let X denote the number of dollars which A has, and y 
 the number jB has. 
 
 Then, x + %y z=z 1200, and y -^ ^x — 1200; 
 
 from which we find, 
 
 X = 800, and y = 600. 
 
28 KEY TO DA vies' NEW ELEMENTARY ALGEBRA. [156. 
 
 (23.) 
 
 Let X denote the price of the brandy, and y that of the 
 sherry. 
 
 If, now, we make the first mixture, that is, two dozen of 
 sherry and one dozen of brandy, the mixture itself will 
 contain three dozens, and will, consequently, be worth 
 785. X 3 = 2345. Hence, we have, 
 
 2aj + 2/ = 234, for the first, and, 
 7cc + 2y = 79 X 9 = Vll, for the second; 
 which equations give, 
 
 tc = 81, and y = 72. 
 
 Page 156. 
 
 (3.) 
 Let aj, y, and z denote the separate ages of -4, J?, and C. 
 Then, cc = 2y, y — Sz, 
 
 and, X + y '\' z = 140 ; 
 
 fi'om which we find, 
 
 03 = 84, y = 42, and z = 14. 
 
 Or, this example may be solved ^vith a single unknown 
 quantity. Thus: let x = C^s age; then, 
 
 (7'5 age =z a?, 
 
 ^'5 age = 3cc, 
 
 A^s age = Ccc, 
 and, X + dx -\- 6x = lOaj = 140; 
 
 whence, a; = 14. 
 
156. J EQUATIONS OF THE FIRST DEGREE. ^9 
 
 (4.) 
 
 Let X denote the cost of the horse ; y, the cost of the 
 harness ; and z^ the cost of the chaise. 
 Then, x -\' y ■\- z — £60 ; 
 
 also, X = 2y, and z = 2(cc +y) ; 
 
 from which we find the several answers. 
 
 But we may resolve the question by means of but a single 
 unknown quantity. Thus, let x denote the price of the 
 harness. 
 
 Then, x = the cost of the harness, 
 
 2x = the price of the horse, 
 ilnd, Qx = the cost of the horse and harness ; 
 
 Also, X + 2x + Qx = 9x = £60 ; or, cc = £6 135. 4:d. 
 
 (5.) 
 Let 0?, y, and z, denote, respectively, the three parts. 
 Then, x + y ~\- z = 36 ; 
 
 also, -X = -V, and -y = -z: 
 
 ' 2 3*^' 3^ 4 ' 
 
 or, 3a; == 2y, and 4y =z 3z; 
 
 from which we find, tc = 8, y = 12, and s = 16. 
 
 This example may be resolved by only two unknown 
 quantities. Thus, let x and y represent the first and second 
 numbers, then will 36 — cc — y, denote the thuxl, and we 
 shall have, 
 
 -X = -y, or 3a; = 2y; 
 
 and -y = ■ ^ , or 4y = 108 — 3x— Sy; 
 
 from which we find, x and y, as before, equal to 8 and 12. 
 
30 KEY TO DAVlKd' JSliW KL15MENTAEY ALGEBRA. [156. 
 
 (6.) 
 
 Let X, y, and a?, denote, respectively, the fractions of the 
 work which A^ J5, and C would do in a single day, and let 
 the whole work to be done be denoted by s. Then, in one 
 day, A would do xs of the work ; in two days, 2xs of the 
 work ; in three days, dxs of the work, &c. ; and the same 
 for the others. Hence, by the conditions, 
 
 8XS + 8^/5 r= 5, 
 9xs + 9^5 = 5, 
 
 10y5 + lOys = s; 
 and, by dividing by s, we have, 
 
 8x + 8y = 1, 
 
 9a; + 9z = 1, 
 
 lOy + 10^ = 1 ; 
 
 from which equations we find x = j^-^-^^ y — ^4J_, and z — -^^-L., 
 the parts of the work that each person will do in a single 
 day. 
 
 Then, if each can do in one day the part of the work 
 represented by each of these fractions, it is plain that the 
 number of times which 1 contains each fraction, will express 
 the number of days in which each person would do the 
 whole work ; that is, 
 
 A would do it in — = — — - — 1*4 — days; 
 49 49 49 "^ ' 
 
 ^720 
 
 7? • 1 V20 .23 , 
 
 •^ '"^ 41 = IT = ^^41 '^^^'' 
 
 720 
 
 n ' 1 '^20 ^^7 , 
 
 <7 ^ ^ = IT = 23-- days. 
 
157.] EQUATIONS OF TIU: IIKST DEGREE. 31 
 
 Let cc, 2/, and ^, denote, respectively, the sums mth whicli 
 each began to play. 
 
 Then, x -{: y + z = 600 ; 
 
 first game, a; + - y = 2y, 
 
 second game, x + -7/ — z = x; 
 
 from which we find, x = $300, y = |200, and z = $100. 
 
 (80 
 Let Xj y, and ^, denote, respectively, the sum of each. 
 Then, x + y + z == 3640 ; 
 
 second condition, x + 400 — y — 400 + 320, 
 third condition, 2/ + 140 = 2 — 140; 
 
 from which we have, x = 800, y = 1280, and z — 1560. 
 
 (9.) 
 
 Let X denote the amount of the bill in dollars, y the 
 amount possessed by A^ and z the amount possessed by B ; 
 and let it be remembered that C has |8. 
 
 Then, first condition, 2/ + 7^ = ^ 5 ' 
 
 second condition, z + 1 = x; 
 
 third condition, - + 8 = x: 
 
 ' 2 
 
 from which we have, x = $13, y = $10, and z = $12. 
 
32 KEY TO DAVIES' NEW ELEMENTAET ALGEBEA. [157. 
 
 { 10.) 
 
 We may again remark here, that if the rate of interest be 
 divided by 100, and the quotient multiplied by the principal, 
 the product will always be the amount of interest. 
 
 Let X denote the rate of interest received on the 1st sum ; 
 then, aj + 1 and cc + 2, will be the other rates. Let y de- 
 note the capital of the first ; then, y + 10000 and y -\- 15000, 
 will denote, respectively, the capital of the second and third. 
 
 Then, by 1st condition, 
 
 and, by 2d condition, 
 
 -^ = ^ili X (2/ + 15000) - 1500; 
 100 100 ^^ ^ ^ ' 
 
 that is, xy = xy -\- lOOOOcc + 2/ — TOOOO, 
 
 and xy z=z xy + 15000a; + 2y — 120000; 
 
 or, = 10000a; + y — 70000, 
 
 and = 15000a; + 2y - 120000; 
 
 from which we find, a; = 4, and y — 30000 ; and hence the 
 other two sums are easily found. 
 
 (11.) 
 
 Let X denote a daughter's share. 
 Then, 2a; = what each son received ; 
 
 also, 3a; + 4a; = what the children received ; 
 
 3a; + 4a; + 1 000 = widow's share ; 
 hence, 6a; + 8a; + 1000 — 15000; 
 or, 14a; = 15000 — 1000 = 14000; 
 
 or, X = 1000. 
 
158.] 
 
 EQUATIONS OF THE FIRST DEGEEE. 
 
 33 
 
 12.) 
 Let the sum to be divided be denoted by x, 
 
 X 
 
 Then, 
 
 A's 
 
 share = - ~ 3000, 
 2 
 
 B's 
 
 X 
 
 share = - — 1000, 
 3 
 
 G's 
 
 X 
 
 share = t + 800 ; 
 
 X =z 
 
 l + f + f-3200; 
 
 then the sum, 
 
 or, 24a; = 12aj + Sx + 6x — V6800; 
 
 hence, 2x = ^76800; or x = $38400. 
 
 (13.) 
 
 Let the values of the horses be denoted, respectively, by 
 X, y, and z. 
 
 Then, jc + 220 = 2/ + ^ ; 
 
 2d condition, y + 220 = 2{x + z) = 2x + 2z ; 
 3d condition, z + 220 = S{x + y) = 3aj + 3y ; 
 from which we find, x = 20, y r=: .100, and z = 140. 
 
 (14.) 
 
 Let the number of guns be denoted by cc, the number of 
 soldiers by y, and the sailors by z. Now, as there are 22 
 seamen to every three guns, there will be ^/- seamen to each 
 gun, and 
 
 22 
 
 X cc, for X guns; hence. 
 
 1st condition gives. 
 
 22 
 z =z — X x+ 10; 
 
 or, 
 
 2d condition, 
 2* 
 
 Sz = 22a; + 30 . . . . (1.) 
 
 y + ^ = 5(y + a;) = 5y + 5x. (2.) 
 
34 KEY TO DAVIES' NEW ELEMENTAKY ALGEBRA. [158 
 
 Now, if we denote the number of slain by 5, 
 then, 2/ + ^ — 5 = the survivors, and 
 
 l{y + ^- s) =z s, or s r= -(y + s) ; 
 
 1 4 
 
 then, y ^ z — -(y + z) = ~{y + z) = survivors, and 
 o o 
 
 4 13 
 by 3d condition, -(y + g) 4- 5 == ^x ; 
 
 5 2 
 
 that is, 8y + 82 + 50 = 65a; ; ... (3.) 
 
 from which three equations we find, a; = 90, y = 55, and 
 z ■=: 670. 
 
 (15.) 
 
 Let x^ 2/, and ^, denote the sums which they respectively 
 had at first. 
 
 Then, x — y — z — what A had, 
 
 2y = what JB had, 
 and 2z = what (7 had, 
 
 after the division with A, Also, 
 
 2x — 2y — 2z = what A had, 
 2y •— {x — y — z) — 2^ = what JB had, 
 and Az = what (7 had, 
 
 after the division with JB. Again, 
 
 4x — 4y — 4z = what A had, 
 4y — 2{x — y ^ z) — 4z = what JB had, 
 4s — (2a; — 2y — 2;^) — [2y — (a; — y — s) — 2^] = 
 what C had, after his division with A and i?. 
 
 But these three last sums are all equal to each other, and 
 
159.] EQUATIONS OF THE FIRST DEGREE. 35 
 
 the sum of a?, y, and s, is equal to 96. Hence, after reduc- 
 ing, we have, 
 
 a; + 2/ + ^ = 96, 
 
 Gx — lOy = 2z, 
 
 and 5x — 3y = llz; 
 
 from which we find, x = $52, y = $28, and s = $10. 
 
 (160 
 
 Let the parts be denoted by cc, y, and z. 
 Then, cc + y + ^ = a, 
 
 and, X : y : : 7n : n; or, ^cc = my ; 
 
 also, y : z :: p : q; or, qy = pz; 
 
 From these equations we find, 
 
 _ amp _ anp __ anq 
 
 '" mp-^-np+nq^ ~~ mp-\-np-\rnq'^ ~~ mp-\-np-\-nq 
 
 ( IV.) 
 
 Instead of denoting the parts of the work done by -4, B, 
 and C, by jc, y, and z, as in Example 6, page 156, let us 
 denote the times in which each would perform the work, 
 respectively, by x, y, and z, and denote the work to be 
 done, by 1. Now, if it takes A, x days to do the work, he 
 
 will, in one day, do a part of the work denoted by - ; hence, 
 
 X 
 
 - = the part A can do in a day, 
 
 X 
 
 - = what B could do in a day, 
 
 y 
 
 - = what G could do in a day ; 
 z 
 
36 KEY TO DAVIES' NEW ELEMENTAET ALGEBRA. [150. 
 
 and these, multiplied by any number of days, would give 
 what each could do in those days. Hence, 
 
 12 12 111 
 
 1st condition, 1 = 1 ; or, - + -=-- : (1.) 
 
 20 , 20 , 11 1 ,^x 
 
 2^' 7+T = ^5 or, - + -^=2-5' (2-) 
 
 3d, l-^ii^l; or, i + i = i. (3.) 
 
 X Z ^ ^ X Z lb ^ ^ 
 
 Subtracting the second equation from the first, we have, 
 
 l_l___l._JL-_^_i. 
 X z '^ 12 20 ~" 240 "" 30* 
 
 Then, adding the third to this, we have, 
 
 2 1,1 3 60 '^ 
 
 ^ = 30 + 15 = 50' "^' ^ ==¥ = '"' 
 
 and y = 30, and z = 60. 
 
 Now, the three together could do in one day, 
 
 20 "*" 30 "^ 60 " 60 60 60 ~ 60 ~ 10 ' 
 
 of the work ; hence, they could do the whole work in ten 
 days. 
 
 Page 204. 
 
 (6.) 
 Let the number be denoted by x. 
 
 *Aien, ~x X ^ = 108; 
 
 that is, j-x^ = 108; or, cc^ = 1296; 
 
 hence, x = ^1296 = 36. 
 
204.] EQUATIONS OF THE SECOND DEGREE. 37 
 
 (7.) 
 
 Let the number be denoted by x. 
 
 Then, ~x x -x -^ 10 =z 3; 
 
 5 6 
 
 x"^ x^ 
 
 that is, —---10 = 3; or, — — = 3 ; 
 
 ' 30 ' ' 300 ' 
 
 hence, x^ — 900, and x ~ -^^900 = 30. 
 
 (8.) 
 
 Let the number be denoted by x. 
 
 x^ 
 Then, aj^ + 18 = -+ 30^; 
 
 hence, 2cc + 36 = cc^ + 61 ; 
 
 consequently, x — -y^ = 5 
 
 (11.) 
 
 Let X denote the greater number ; then, -a; will denote 
 the less. 
 
 Then, x^ - {^x\'' = 28; 
 
 9 
 
 that is, a;2 cc^ = 28 ; 
 
 16 ' 
 
 or, 16x^ — 9x^ = 448, and ^x'^ = 448; 
 
 hence, x^ = 64, and x = S. 
 
 (12.) 
 Let x denote the greater number ; then —x will denote 
 the less. 
 
38 KEY TO DAVIES' NEW ELEMENTAKY ALGEBRA. [208 
 
 25 
 Then, aj2 ^ ^,2 ^^ 534^ 
 
 and, 121cc2 + 25aj2 = 70664; 
 
 hence, 146aj2 =::= 70664, and x^ =484; 
 
 hence, a; = 22. 
 
 Denote the age of the elder by x ; then - will denote 
 
 (13.) 
 
 3 e 
 the age of the younger. 
 
 Then, c«2 - ^^2 _ 240, 
 
 and, 16aj2 — cc^ = 3840; or, x = 16. 
 
 Page 208. 
 
 (5.) 
 Let the two numbers be denoted by x and y. 
 Then, by first condition, 
 
 cc2 + 2/2 = 117; 
 by second condition, 
 
 aj2 _ 2/2 =: 45 ; 
 from which we have, 
 
 a! = 9, and 9/ = 6. 
 
 (6.) 
 
 Let the two numbers be denoted by x and y. 
 Then, aj2 + 2/2 -_. ^^ 
 
 and, aj2 — 2/^ = ^5 
 
 hence. 
 
 '» ^ =" V ^2~' ^^* ^ ^ V~ 
 
208.] EQUATIONS OF THE SECOND DEGREE. 39 
 
 (9'.) 
 
 Denote tlie numbers by x and y. 
 Then, a; : 2/ : : 3 : 4 ; or, 4a; = 3y, 
 
 and, cc2 j^ y^ — 225 ; 
 
 from which we have, 
 
 03 = 9, and y — 12. 
 
 (8.) 
 
 Denote the numbers by x and y. 
 Then, x \ y \\ m \ n\ or, nx =z my^ 
 
 and, «;2 + 2/2 _ ^2^ 
 
 Squaring the first equation, and multiplying the second 
 by w?^ we have, by transposing in the second, 
 
 n^x^ = rn?y'^^ 
 
 and, n%^^ = w?o? — m?y'^ ; 
 
 and then, by addition, we obtain, 
 
 x^(m? + 'n?) = nh^a\ 
 
 T ma - na 
 
 and, X — - _ ; also, y 
 
 (9.) 
 
 Let the largest number be denoted by 2cc ; then the less 
 will be denoted by x. We shall then have, 
 
 4i»2 — a;2 = 75, 
 
 and, 3i«2 = 75, and a;^ = 25 ; 
 
 or, a; = 5. 
 
40 KEY TO DAVIES' NEW ELEMENTAET ALGEBRA. [i^08. 
 
 (10.) 
 
 Let the numbers be denoted by x and y. 
 Then x \ y :: m : n\ or, nx = my^ 
 
 and, x^ ~ y"^ =z b^; 
 
 from which we find, 
 
 mb T nb 
 
 :, and y = 
 
 (11.) 
 
 Let the amount in dollars, placed at interest, be denoted 
 by X. 
 
 Then, — — x cc = the interest for one year, 
 '100 
 
 4 
 and, — - X cc = the interest for six months, 
 
 ^"^^^^ Too ^ ^ ^ 25 "^ 562500, 
 
 and, x^ = 14062500,^ and x = 3750. 
 
 (12.) 
 
 Let x denote the number of women, and y the number 
 of boys. 
 
 Then, x : y : : S : 4; or, 4cc = 3y, 
 
 and, X = -y; 
 
 and, X + y = the number of persons, 
 
 ^ = what the boys receive. 
 
 and, 2y = what the women receive. 
 
221.] EQUATIONS OF THE SECOND DEGREE. 41 
 
 Then, ^ t ^ + 2y r= 138, 
 
 and, aj + 2/ + 4y = 276, 
 
 Then, _ 2/, + 2/ + 4y = 276, 
 
 and, 3y + 4y + 16y = 1104; 
 
 or, 23y = 1104, and y = 48. 
 
 Note. — This, it will be seen, is an equation of the Jirst degree^ and is 
 placed among those of the second degree, to lead the student to have 
 confidence in his own methods, and not to rely too implicitly on the 
 arrangements of the author. 
 
 Page 221.) 
 ( 10.) 
 
 Clearing of fractions, 
 
 a^n^ + b'^91^ — 2bn'^x + n^x'^ =: m'^x'^. 
 Transposing, arranging the terms, and factoring, we have, 
 [n^ — m^)x'^ — 2bn^x = — ahi^ ~ b^-ri^ ; 
 dividing by the coefficient of cc^, we have, 
 
 y.2 
 
 25^2 -— a^nP' — y^n? 
 
 'X = 
 
 w^ — m^ n^ — m 
 
 2 
 
 Completing the square, 
 
 2bn^ lh%^ -d'n'^—'b'^n^ . Ihi"^ 
 
 -,x + 
 
 ^2,2 _ ^2*^ '^ (,2,2 _ ^^2)2 ^^2 _^2 ^ (^^2 _ ,^^2)2' 
 
 Multiplying the numerator and denominator of the first 
 
42 KEY TO DAVIES' NEW ELEMENTAPwY ALGEBRA. [2'f3. 
 
 term of the second member, by n^ — m^, so as to reduce 
 both to a common denominator, we have, 
 
 """ (n^ — rn^Y 
 
 Cancelling like terms, and factoring, 
 
 then extracting the root of both members, 
 
 03 — . _- -J- — I . 
 
 Transposing and factoring, 
 
 
 n' 
 
 n 
 
 rti^ 
 
 n^ 
 
 n 
 
 m^ 
 
 n^ 
 
 — 
 
 w? 
 
 
 a;" = -^ ^ {bn — -y/a^m^ + ^^^2 __ ^2^aj^ 
 
 Page 243, 
 
 (4.) 
 
 Let X denote the greater number ; then 11 — a will 
 denote the less. 
 
 By the conditions of the question, 
 
 x^ + 121 — 22a; -^ x^ z=z 61 ; 
 whence, 1x^ — 22a; = — 60 ; 
 
 or, x^ — 11a; = — 30 ; 
 
 whence, a;' = 6, and a;" = 5. 
 
 Either root will satisfy the conditions of the problem. 
 
243.J EQUATIONS OF THE SECOND DEGREE. 4:3 
 
 (5.) 
 
 Let X denote the greater number ; then, since the differ- 
 ence of the two numbers is 3, cc — 3 will denote the less. 
 
 By the conditions of the question, 
 
 aj2 + cc2 — 6cc + 9 = 89 ; 
 hence, 203^ — 6a? = 80, 
 
 and ic^ — 3a3 = 40; 
 
 hence, cc' = +8, and cc" zz: — 5. 
 
 The plus value belongs to the problem ; hence, the num- 
 bers are 8 and 5. 
 
 (6.) 
 
 Let X denote the number of sheep which he purchased. 
 
 Then, — = the cost of a single sheep ; 
 
 or, z=: the cost of a single sheep in shillings, 
 
 , 1200 X 15 18000 , n.., . 
 
 and = = cost of fifteen sheep. 
 
 XX' ^ 
 
 Then, the number of sheep sold will be represented by 
 a; — 15, and 2(03 — 15) = the amount of profit. Now, 
 had there been no profit, the amount received for the sheep 
 would have been just equal to the cost, less the value of the 
 fifteen unsold ; and, consequently, the amount received, less 
 the profit, must be just equal to this difference. That is, 
 reducing to shillings, 
 
 1080 - 2{x ~ 15) = 1200 - H^; 
 
 X 
 
 hence, 1080aj — 2x'^ + SOx z= 1200i« — 18000; 
 
 and, by reducing and dividing by the coeflicient of x\ we 
 have, 
 
 j»2 + 45a; = 9000 ; 
 
 from which, by taking the positive value, we have, x = 75. 
 
44 KEY TO DAVIES' NEW ELEMENTAEY ALGEBEA. [243. 
 
 {'if') 
 
 Let the number of pieces be denoted by x ; and by re- 
 ducing the cost and the amount received to shillings, we 
 find that he paid 675 shillings, and sold for 48 shillings per 
 piece, 
 
 675 
 
 Then, = the price ]3er piece in shillings, 
 
 and ASx = what he received for the whole. 
 
 But what he received, minus what he gave, must be equal 
 to his -profits ; that is, to the cost of a single piece. That is, 
 
 675 
 
 48a; - 675 = , 
 
 X 
 
 and 48a;2 - 675aj = 675 ; 
 
 675 675 
 
 -^'"--4^^ = I8-' 
 and by completing the square, 
 
 , 675 . (675)2 675 , (675)^ 
 48 ^ (96)2 - 48 ^ (96)2 
 _ 1350 (675)2 
 ~ 96 "^ (96)2 
 _ 1350 X 96 (675)2 
 ~ (96)2 "^ 96 
 585225 
 
 - (96) 
 
 Then, by extracting the square root of both members, and 
 taking the positive root, which answers to the question in 
 its arithmetical sense, we have, 
 
 __ 675 _ 765^ 
 
 ^ "" 96 ~ 96 ' 
 
 675 , 765 1440 
 
 and* X = — H = = 15, 
 
 ' 96 96 96 
 
243.] EQUATIONS OF THE SECOND DEGREE. 45 
 
 (8.) 
 
 Let X denote the greater number ; then, since the differ- 
 ence of the two numbers is equal to 9, the less will be de- 
 noted by cc — 9 ; and their sum, by 2cc — 9. By the con- 
 ditions of the question, 
 
 (x — 9)x = 266 ; or, x^ — 9x = 266 ; 
 
 from which we find, cc = 14 ; and, consequently, the less 
 number is equal to 5. 
 
 (9.) 
 
 Let the number be denoted by x. 
 Then, (10 — x)x =z 21 ; 
 
 hence, lOcc — x- = 21 ; 
 
 or, x^ ■— 10a; = — 21. 
 
 Completing the square, 
 
 i«2 - lOa; + 25 r;=: - 21 + 25 = 4 ; 
 hence, xz=z5±^=:5±:2 = 7^ and 3. 
 
 (10.) 
 
 Let X denote the distance traveled in one hour; then, 
 X — 2 will denote the distance when he travels 2 miles an 
 hour slower. 
 
 105 
 Then, — = the number of hours, under the first sup- 
 position ; 
 
 J.05 
 
 and, = the number of hours, under the second 
 
 X — 2 ' 
 
 supposition. 
 
46 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. 1 244 
 
 But, by tlie conditions, 
 
 
 105 105 
 a; - 2 X ~ ' 
 
 
 whence, 105a; — 105aj + 210 — Qx^ — 
 
 12a;; 
 
 that is, 6a;2 — I2x = 210, 
 
 
 a;2 - 2x =: 35. 
 
 • 
 
 X = 1 
 
 (11.) 
 
 Let X denote the number of sheep ; then, 
 
 — = what was paid for each ; and, 
 
 448 
 
 [- 2 = what he would have paid, under the 
 
 X 
 
 second supposition, 
 llien, by the conditions of the question, 
 
 — + 4 = x; 
 
 whence, 896 + 4a; = a;^; 
 
 from which we find, a; = 32. 
 
 (12.) 
 
 Let x denote the greater number. Then, since the differ- 
 ence of the numbers is 7, x — 1 Avill denote the lesser, 
 and their sum will be denoted by 2a; — 7. By the condi- 
 tions of the question, 
 
 (2a; — 1)x = 130; or 2a;2 — 1x = 130; 
 
 whence, we find the plus value of a;, which belongs to the 
 problem, to be 10; hence, the numbers are 10 and 7. 
 
244,] EQUATIONS OF THE SECOND DEGKEE. 47 
 
 (13.) 
 
 Let X denote the greater number ; then, since their sum 
 is 100, 100 — a; will denote the less. By the conditions 
 of the question, 
 
 a;2 + 10000 - 200a3 + x^ = 5392; 
 
 hence, 2aj2 — 200a; = 5392 - 10000 — — 4608; 
 
 or, x^ — lOOcc = — 2304 ; 
 
 whence, we find the positive value of cc, which belongs to 
 the problem, to be 64. 
 
 Let X denote the number of feet in the side of the smaller 
 court yard; then, x ■\- \1 will denote the side of the 
 larger, in feet. Then, since each stone contains 1 square 
 foot, 
 
 (x + 12) ~ cc2+ 24aj + 144 = number of stones in larger, 
 
 and, Qi? z= number in smaller ; 
 
 hence, 2cc2+ 24cc + 144 = 2120; 
 
 or, x'^-h 12x z=z 1060 - 12 - 988 ; 
 
 whence, x = 26. 
 
 (15.) 
 
 Let X denote the number of persons. 
 
 240 
 Then, = what each received in the first distribution ; 
 
 X 
 
 and, = what each received in the 2d distribution. 
 
 X'\- 4 
 
48 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [244. 
 
 By the conditions of the question, 
 
 240 240 
 — - = 10; 
 
 X X -[- 4: 
 
 whence, 240a; + 960 — 240a; = lOx^ + 40a;; 
 whence, lOa;^ + 40a; = 960 ; 
 
 or, x^ -\- 4:X = 96 ; 
 
 whence, cc' = 8, and a;" = — 14. 
 
 The plus value alone belongs to the problem ; and the 
 numbers of persons were 8 and 12. 
 
 (10.) 
 
 Let X denote the number of dollars in A's capital. Then, 
 since he received 520 dollars as capital and profits, he 
 gained, in dollars, 520 — a;, in 12 months; hence, the 
 number of dollars which he gained in 1 month, will be 
 
 denoted by — • 
 
 ^ 12 
 
 Now, B^s gain was the whole gain diminished by A^s ; 
 
 hence, B''s gain is denoted by 360 — (520 —a;) = a;— 160; 
 
 since B'^s capital was in trade 16 months, his gain in 1 month 
 
 will be denoted by 
 
 ^ 16 
 
 But the gains of each in 1 month must be proportional to 
 the capitals they employed ; hence, 
 
 520 — a; x — 160 
 
 X : 600 
 
 12 * 16 ' 
 
 X — 160 ^^^ 520 — X 
 or, X X — -^ — = 600 X j^— • 
 
 Clearing of fractions, and reducing, 
 
 a;2 — 160a; = 416000 — 800a;; 
 or, x^ + 640a; =: 416000; 
 
 whence, we find the plus value of a; = 400. 
 
255.] EQUATIONS OF THE SECOND DEGREE. 49 . 
 
 * Page 255. 
 
 (3.) 
 Let X denote one number, and y the other. 
 Then, by the conditions, 
 
 aj + y = 8 (1.) 
 
 and, x^+ y^ = S4: . . \ . . (2.) 
 
 From the first equation, we have, 
 
 X =z S — y, and cc^ -_ 54 _ jgy + y^. 
 Substituting for aj^, in Equation ( 2 ), we have, 
 
 64 _ 162/ + 2/^+ 2/^ = 34; 
 whence, 2y^ —1^2/ = — 30, 
 
 2/2 - 82/ =: - 15. 
 
 Hence, 2/' = ^? ^^^ 2/" = ^• 
 
 Hence, cc' = 3, and a;" = 5. 
 
 (4.) 
 
 Let X denote the first number, and y the second. 
 Then, by the conditions, 
 
 X : y :: y : IQ; 
 
 whence, 16x = y^ (1.) 
 
 cc2+ 2/2 = 225 (2.) 
 
 Substituting the value of 2/% from Equation ( 1 ), we have, 
 
 x^ + Ux = 225; 
 whence, a? = 9, and y z= 12, 
 
 L 
 
 3 
 
50 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [256. 
 
 (5.) 
 Let X denote the less, and y tlie greater. 
 Then, by the conditions, 
 
 2/ :: 3 ; 5; 
 
 whence, bx = Zy , . . . . . (1.) 
 
 2d condition, x^ ■\- y'^ = 1666 (2.) 
 
 From Equation (1), we have, 
 
 25 
 Substituting this value in Equation ( 2 ), 
 
 -~ + y^=- 1666 ; 
 
 whence, 342/^ = 41650; 
 
 and 2/2 = 1225 ; or, y = 35. 
 
 Hence, the numbers are 21 and 35. 
 
 (6.) 
 
 Let X denote the less number, and y the greater. 
 Then, by the conditions, 
 
 y — X = 7 ; whence, y = jc + 7, 
 
 and, ^ + 30 = x\ 
 
 Substituting for y, its value, we have, 
 
 ^^^ ^ "^^ + 30 = ic2; or, ic^ _^ 7aj + 60 = 17?\ 
 whence, x^ — ^x — 60; 
 
 or, X — 12, and y — 19. 
 
256.] EQUATIONS OF THE SECOND DEGKEE. 51 
 
 Let the numbers be denoted by x and y, 
 
 X + y = 5 (1.) 
 
 aj3 + 2/3 = 35 (2.) 
 
 x^ + Zx^y + 3aj?/2 -\- y^ = 125, by cubing (1), 
 
 Zx^y + 3CCI/2 — 90, by subtraction, 
 Zxy(x -\- y) = 90, by factoring, 
 15xy — 90, fromEq. (1); 
 
 hence, ^2/ = 6 (3.) 
 
 Combining (1) and (3), we find, 
 
 a; — 2, and y = 3. 
 
 (8.) 
 
 Let X denote the greater, and y the less. 
 Then, by the conditions, 
 
 cc + y : cc : : 11 : 7; 
 whence, 1x + 1y — lice; 
 
 or, aJ = ^2/; or, aj^ ^ _y2. 
 
 2d condition, a;^ __ ^/^ _ 132. 
 
 whence, by substitution, 
 
 -^-y'^= 132; 
 
 or, 491/2 - 162/2 = 2112, 
 
 332/2 — 2112, 
 
 2/2 = 64 ; 
 
 and, 2/ = ^/^ = ^> 
 
 and X is found to be equal to 14. 
 
52 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [256. 
 
 (9.) 
 
 Let X and y denote the numbers. 
 Then, by the conditions, 
 
 a? + 2/ = 100 (1.) 
 
 ajy : cc2 + 2/^ : : 6 : 13 ; 
 hence, 13ajy = ea:^ + 6^/^ . . . . (2.) 
 
 From Equation ( 1 ), we have, 
 
 cc = 100 ~ 2/, and x^ = 10000 — 200y + y^. 
 
 Substituting these values of x and aj^, in Equation ( 2 ), we 
 shall have, 
 
 13y(100 — y) — 6(10000 — 200y + y'^) + 6y2. 
 Perfoi:ming the operations indicated, we have, 
 
 1300y — 13y2 — 60000 - 1200?/ + 6?/2 + e*/'-^ ; 
 or, 25^/2 _ 2500y — — 60000 ; 
 
 or, 2/^ — lOOy = — 2400 ; 
 
 whence, y = 60, and a; = 40. 
 
 (10.) 
 
 Let X denote the distance, in miles, traveled by -4, and 
 y, the distance traveled by JB, 
 
 Then, by the conditions, 
 1st condition, cc — 2/ = 18 (1.) 
 
 Now, the rate of travel, or the distance traveled in a 
 single day, will be found by dividing the distance by the 
 number of days ; hence, 
 
 -^ = what A traveled in 1 day ; 
 15|- 
 
 and, — = what JB traveled in 1 day. 
 
 28 
 
256.] EQUATIONS OF THE SKCOND DEGREE. 53 
 
 [N'ow, the e^itire distance traveled by A^ divided by what 
 he traveled in 1 day, will give the time he Avas traveling ; 
 and the same for J?. But these times are equal ; hence, 
 
 ^ __ y 
 
 (y\ ~ (-)' 
 
 \15}/ \28/ 
 
 and, by reducing, 
 
 15faj2 _ 282/2; 
 
 or, reducing to 4ths, 
 
 63aj2 ~ 112?/2. 
 
 Substituting the value of x^^ from Equation (1), we have, 
 
 63(324 + 36?/ + y^) =^ lUy^ ', 
 
 or, 20412 + 22682/ -f 63y2 = U2y^; 
 
 492/2 — 22682/ = 20412 ; 
 
 2 2268 __ 20412 
 
 ^ igT^ ~" 49 ' 
 
 Completing the square. 
 
 2268 (2268)2 20412 , (2268)2 
 
 ^ 49 *^ (98)2 - 49 ^ (98)2 ' 
 
 2 ^^^^ I (2268)2 _ 40824 X 98 (2268)^^ 
 and y - -—y + -j^^ - — (^p + 798)^ ' 
 
 which, after performing the operations indicated, gives, 
 2/ = 54, and, consequently, x = 72, 
 
 (11.) 
 
 Denote the less number by x. Then the greater will be 
 denoted by cc + 15 ; and we shall have, 
 
 (^ + 15)a? ^ 
 
 2 ' 
 
 and, dividing by a?, a; + 15 = 2a;2; 
 
54 KEY TO DAVIKS' NEW ELEMENTARY ALGEBRA. [256. 
 
 and, by transposing, x'^ — -x = 7.5 ; 
 
 and, completing the square, 
 
 x^ — .5x + .0625 =r 7.5625 ; 
 hence, aj := .25 ± V^5625 = 3, 
 
 by taking the positive root. 
 
 (12.) 
 
 Let the greater number be denoted by x, and the less 
 ^7 2/- 
 
 Then, by first condition, (x + y)^ = 77 .... (1.) 
 
 by second condition, (x — y)y == 12 .... (2.) 
 
 that is, aj^ -f ccy r= 77 . . . . (3.) 
 
 and, ccy — 2/^ =: 12 . . . . (4.) 
 
 By adding, we have, x^ — y"^ + 2aji/ = 89 ; 
 
 and, by transposing, x^ — y'^ = 89 — 2xy, 
 
 K we multiply the first and second equations together, 
 we obtain, 
 
 {x^- — y'^)xy =z 924 ; 
 
 and hence, x"^ — y"^ = • 
 
 '' xy 
 
 Placing this value of cc^ — ^/^ equal to that found above, 
 and Ave have, 
 
 924 
 
 89 — 2xy = ; 
 
 xy 
 
 or, S9xy — 2a;y = 924 ; 
 
 and, placing xy = z, we obtain, 
 
 89s - 2^2 = 924 ; 
 
257.] EQUATIONS OF THE SECOND DEGEEE. 65 
 
 and hence, by changing the signs and dividing, we have, 
 
 a;2 __ 44.5^ — _ 462. 
 Then, by completing the square, 
 
 ^2 - 44.5^ + 495.0625 = 33.0625 ; 
 
 hence, z = 22.25 ± -v/53.0625 ; 
 
 or, z z=z 22.25 ± b.lb ; 
 
 or, s'rr 28; and, z^' — 16.5. 
 
 Snbstitutmg the first value of z for xy^ in Equations ( 3 ) 
 and ( 4.) , gives cc r= 7, and y = 4 ; and substituting the 
 second value, s" = 16.5, for xy^ in the same equations, we 
 find, 
 
 (13.) 
 Let the numbers be denoted by x^ and y'^. 
 
 Then, cc^ + ^/^ = 100 (1.) 
 
 and, X + y = 14 (2.) 
 
 From the second equation, we have, by transposing, 
 
 aj = 14 - 2/; 
 and, by squaring, x^ = 196 — 28y -{- y\ 
 
 Substituting this value in Equation ( 1 ), we have, 
 
 196 - 2Sy + y^ + y^ = 100; 
 and, by reducing, y^ — liy = — 48. 
 
 Completing the square, we have, - 
 
 y2 __ i4y _|. 49 _ _ 48 + 49 = 1, 
 and, 2/=z:+7±l = 8; 
 
66 KEY TO DA vies' NEW ELEMP:NTARY ALGEBRA. [257. 
 
 or, if we take the minus sign, then 7/ =z Q, If we take 
 2/ = 8, we find x .— 6, and if we take y = 6, we find 
 X = 8 ; hence, the numbers are 64 and 36. 
 
 (14.) 
 
 Let the numbers be denoted by x and y. 
 Then, x -{- y = 24^ 
 
 and XT/ = 35(cc — y) = S5x — 35y. 
 
 From the first equation, we have, 
 x — 24 — 9/, 
 Substituting this value in the second, we obtain, 
 2/(24 - y) = 35(24 - 2/) - 35^; 
 that is, 24y — y'^ — 840 — 351/ — 35y ; 
 
 hence, j/^ — 94^/ = — 840. 
 
 Completing the square, 
 
 2/2 __ 94y j^ 2209 = 1369, 
 and, y = 47 ± 37 = 84, and 10. 
 
 If we take the first root, 84, the value of x will be — CO, 
 and these two numbers will satisfy the two equations of 
 condition. But the enunciation of the question required 
 the number 24 to be divided into two parts, and this re- 
 quired that neither x nor y should have a value exceeding 
 24 ; hence, we must take the second value of y = 10. This 
 gives jc = 14. 
 
 (15.) 
 Let the numbers be denoted by x and y. 
 
 Then, a; + 2/ = 8 (1.) 
 
 and, a;3 + 2/^ = 152 .... (2.) 
 
257.] EQUATIONS OF THE SECOND DEGREE. 57 
 
 By cubing both members of Equation ( 1 ), we have, 
 
 x^ -f 3cc2y + 3CC2/2 + 2/3 = 512 . . (3.) 
 
 and, by subtracting the second equation from the third, we 
 have, 
 
 3aj2y + 3032/2 :^ 360 ; 
 
 and, dividing by 3, ive obtain, 
 
 x^y + xy"^ = 120; 
 
 factoring, xy{x + y) = 120 ; 
 
 but, since, in Equation ( 1 ), 
 
 X + y =2 8, 
 
 we have, 8xy = 120 ; or, xy = 15. 
 
 Combining this with Equation ( 1 ), we readily find, 
 
 cc =: 3, and y =: 5. 
 
 (16,) 
 
 Let the number of yards sold by the first, be denoted by 
 x, and the number sold by the second, by y. 
 
 N"ow, if the whole amount received, for any number of 
 things solely be divided by the number of things, the quo- 
 tient will be the cost of each thmg. Hence, if 24 dollars 
 be divided by the number of yards of stufi* sold by the 
 second, the quotient will be the amount per yard received 
 by the first; and, for a like reason, 12^ divided by cc, will 
 be the amount per yard received by the second. 
 
 24 
 That is, — = what the first received per yard. 
 
 12^ 
 and, — ~ = what the second received per yard. 
 
 But, the first sold x yards, and the second y yards ; and, 
 3* 
 
58 KEY TO DAVIES' KEW ELEMENTAKY ALGEBRA. [257. 
 
 if the amount per yard be multiplied by the number of 
 yards, the product will be the amount received. Hence, 
 
 24 , 124- 
 
 — X X -\ ~ X 2/ = 35 ; 
 
 2/ ^ . 
 
 and, by the second condition, 
 
 2/ — X = S; or, y = x + S, 
 
 Then, by clearing the first equation of fractions, we have, 
 
 24a;2 + 12^2/^ = SBxy ; 
 
 and, by substituting for y its value, a: + 3, we obtain, 
 
 24a;2 + 12i(aj2 + 6a; + 9) =z 35a;(a; + 3); 
 
 that is, 
 
 24cc2 + 12iaj2 4- "jBx + 1121 = S5x'^ + 105a;; 
 
 and reducing, l-^a;^ — 30a; := — 112J; 
 
 and, dividing by 1^, we have, 
 
 a;2 — 20a; = — 75 ; 
 
 which gives, a; = 10 ± 5 = 15, and 5 ; 
 
 ^i'om which we have the corresponding values of y = 18, 
 or y =1 8, 
 
 (17.) 
 
 Let the highest rate of interest be denoted by y, and the 
 lowest by z. Now, as the incomes are to be equal, it is 
 plain that the first sum put at interest will be the least, 
 which let us denote by x. Then, the larger, or second part 
 will be denoted by 13000 — x. Then, since the amount of 
 interest on any sum is equal to the sum, multiplied by the " 
 rate, divided by 100, we have, 
 
 by first condition, x X ~~ = ^13000 — a^) X -— ; 
 
257.] EQUATIONS OF THE SECOND DEGREE. 69 
 
 Z 
 
 by second condition, x x — — -— 360 ; , 
 
 by third condition, (13000 — ^-^ = 490. 
 
 Clearing of fractions,, we have, 
 
 xy — 130003 — CCS .... (1.) 
 
 xz =2 36000 (2.) 
 
 and, 13000y — xy = 49000 (3.) 
 
 If, now, we substitute the value of xz from Equation (2), 
 in Equation (1 ), we shall have, 
 
 xy =z 13000s - 36000 .... (4.) 
 
 then, adding together Equations (3) and (4), we have, 
 
 13000y =: 13000s + 13000; 
 
 or, ISy = 13s + 13; 
 
 or, y = z + 1 . (5.) 
 
 l^ow, to elimmate x from Equations (2) and (3), mul- 
 tiply the first by y, and the second by s, and we have, 
 
 xyz — 36000y, 
 
 and, 13000ys — xyz = 49000s, 
 
 and by adding, 13000ys = 36000y + 49000s; 
 
 or, 13ys = 36y + 49s. 
 
 Now, substituting for y its value in Equation ( 5 ), we 
 have, 
 
 13s(s + 1) = 36(s + 1) + 49s; 
 
 that is, 13s2 + 13s = 36s + 36 + 49s; 
 
 A ■ '72 36 
 
 and, ,2___, ^_. 
 
z^ 
 
 72 , 
 
 -r3^ + 
 
 (36)^ 
 (13)^ 
 
 = 
 
 36 /3( 
 13 "^ UJ 
 
 
 
 
 
 
 36 X 13 
 
 
 
 
 
 
 (13)2 
 
 
 z = 
 
 36 
 13 
 
 ~ 13 
 
 = 
 
 13 
 
 60 KEY TO DAVIKS' NEW ELEMENTARY ALGEBRA. [257. 
 
 by completing the square, we have, 
 
 4- 
 
 Hence, 
 
 The negative value of z is not applicable to the. ques{ion. 
 
 (18.). 
 
 Let the second number be denoted by a;, and the differ- 
 ence between the second and first by y. 
 
 Then, ^ — V = 1st number, 
 
 ic = 2d number, 
 
 and, a; + 2/ + 6 = 3d number. 
 
 Then, 3cc + 6 == 33, and, hence, x = 9; 
 
 also, {x - yY -\- x^ + (x + y + 6)^ = 467 ; 
 
 that is, 3aj2 + 120? + 12?/ + 2y2 = 431 ; 
 
 or, substituting for cc, its value, 9, 
 
 351 + 12y + 22/2 = 431; 
 
 hence, y^ + ^y = 40, 
 
 and, y = 4, or — 10. 
 
 Hence, the numbers are 5, 9, and 19. 
 
 (19.) 
 
 Let X denote the digit which stands in the ten's place, 
 and y the digit which stands in the unit's place. 
 
 Then, lOx -{- y ~ the number. 
 
257.] EQUATIONS OF THE SECOND DEGREE. 61 
 
 By the first condition, 
 
 10^ + y ^ 3 
 xy ' 
 
 and, by the second, 
 
 lOcc + y + 18 = 102/ + ^5 
 
 from which we readily find, 
 
 £c = 2, and y = 4. 
 
 (20.) 
 Let X and y denote the numbers. 
 Then, x : y : i m : n\ or, y = — • • (1.) 
 
 and, cc2 + 2/2 _ 52 (2.) 
 
 Substituting the vahie of y"^ from Equation (1 ), we have, 
 9 . ^^^^ J. 
 
 
 or, w?x^ + 
 
 n^x^ - 
 
 = m?h 
 
 5 
 
 or, 
 
 {m? + n^)x^ - 
 
 = . m% 
 
 
 
 2 m^h 
 
 and 
 
 X — 
 
 m^/b 
 
 and, from Equation ( 1 ), 
 
 
 y = 
 
 n^/h 
 
 
 
 (21. 
 
 ) 
 
 
 Let 
 
 X and y denote the numbers. 
 
 
 Then, 
 
 by the conditions. 
 
 
 
 
 7lX 
 
 X : y : : m : n: or v = — * • (!•) 
 
 and, cc^ — 2/^^ = ^* 
 
62 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [258, 
 
 Substituting the value of y^ from Equation ( 1 ), we have, 
 
 aj2 = 5 ; 
 
 'in^/h 
 
 hence, x = 
 
 and, y = 
 
 myfb 
 
 (22.) 
 
 Let the numbers be denoted, respectively, by a?, y, and z. 
 
 Then, ccy = 2, xz — 4, y^ + ^^ _. 20. 
 
 From the first equation, we have, 
 
 2 2^ 
 
 aj = - ; and hence, — = 4 ; or, ^ = 2.v. 
 2/ y 
 
 Substituting this value of s in the third condition, we have, 
 
 y1 + 4i/2 :::. 20 ; 
 
 whence, 5^/^ =20, 2/^=4, and y = 2, 
 Hence, the numbers are 1, 2, and 4. 
 
 (23.) 
 Let the numbers be denoted by cc, y, and z. 
 Then, j» + y + s = 38 .... (1.) 
 
 a;2+ 2/2^ a:2 = 634 .... (2.) 
 and, 2/--cc = ^ — 2/ + 7 . . . (3.) 
 
 Adding the first and third equations, we have, 
 
 3y = 45; or, y = 15 ; 
 from which we easily find, 
 
 cc =: 3, and z = 20. 
 
258.] EQUATIONS OF THE SECOND DEGREE. 63 
 
 (24.) 
 Let the numbers be denoted by x^ y, and z. 
 Then, ccy = a, xz = 5, y^ -{- z^ = c. 
 
 From the first equation, we have, . 
 
 a -, - az ^ 
 
 X = - . and hence, — = o: 
 
 y y 
 
 or, az = 5y, and y^ = —z^ ; 
 
 hence, |V + ^^ ^ c; or, z = h^J —^—^, 
 y = ^\/5^5 or, X =^/~r 
 
 + b^' ' V a2 4_ 52 
 
 (25.) 
 Let the greater number be denoted by cc, and the less by y. 
 Then, {x + y)x = lU . . . . ( 1.) 
 
 {x-y)y=zU .... (2.) 
 and multiplying the equations together, we obtain, 
 
 (x2 - y'^)xy = 2016 .... (3.) 
 But Equations ( 1 ) and ( 2 ) may be put under the forms, 
 
 x'^ -\- xy = 144, 
 and, xy — y^ =z 14 ; or, xy = 14 + 2/2. 
 
 and subtracting, 
 
 aj2 + 2/2 _ 130; or, x'^ = 130 — y\ 
 
 Substituting in Equation (3), the value of xy = 144 — a;^, 
 and then for x^, its value, 130 — y^, and we obtain, 
 
 (130 - 2^2) (14 _^ 2/2) = 2016; 
 that is, 1820 — 28.y2 4. i30?/2 — 2y^ = 2016. 
 
64: KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [264. 
 
 Hence, y* — ol^/^ = — 98. 
 
 Then, placing z = y\ and z^ for y*^ we have, 
 
 which gives, by taking the j)ositive root, which is the one 
 corresponding to the arithmetical enunciation, 
 
 z^ = 49, and, consequently, y^ = 49, or y = 7. 
 
 The value of x is easily found, and is equal to 9. 
 
 Page 264. 
 
 ARITHMETICAL PROGRESSION. 
 
 (2.) 
 
 I z= a — {n — l)r. 
 Make, « = 90, r = 4, and n = 15 ; 
 then, I = 90 — (15 — 1)4 = 90 — 56 = 34. 
 
 (3.) 
 
 I = a — (n — l)r. 
 Make, a = 100, n — 40, and r = 2; 
 then, I = 100 - (40 - 1)2 = 100 — '78 = 22. 
 
 (4.) 
 
 I = a — {n — l)n 
 Make, a = 80, n = 10, and r = 4; 
 then, I = 80 - (10 — 1)4 = 80 - 36 = 44. 
 
 (5.) 
 
 I ~ a — {n — l)r. 
 Make, a == 600, n = 100, and r = 5; 
 then, I = 600 — (100 - 1)5 = 600 — 495 = 105. 
 
266.] AKmiMETICAL PEOGEESSION. 66 
 
 (6.) 
 
 ^ = a — {n — l)r. 
 Make, a = 800, 7i = 200, and r = 2; 
 then, I = 800 — (200 - 1)2 = 800 — 398 = 402. 
 
 Page 266. 
 
 (2.) 
 
 =m 
 
 Make, a = a, I = 21^ and ^i = 12; 
 
 Q I 2'7 
 
 then, aS = "^ X 12 = 180. 
 
 (3.) 
 
 Make, a = 4, I = 20, and n = 10. 
 
 then, S = (—2^) X 10 = 120. 
 
 (4.) 
 
 S = (-^} X n. 
 
 Make, a = 100, 5 = 200, and n = 80. 
 
 then, S = (1^^°) X 80 = 12000. 
 
66 KEY TO DAVIES' NEW ELEMENTAEY ALGEBEA. [269. 
 
 (5.) 
 
 Make, a = 500, 3 = 60, and n = 20; 
 
 /500 + 60\ 
 then, S = ( J- — j X 20 = 5600. 
 
 (6.) 
 
 ^ == (-2-) ^ ^- 
 
 Make, a = 800, b ~ 1200, and 9i =-. 50. 
 
 '800 + 1200\ ^^ ^^^^^ 
 1 X 50 == 50000. 
 
 then, S = \^ 
 
 Page 267. 
 
 (2.) 
 
 I — a 
 
 r = -• 
 
 w — 1 
 
 Make, ^ = 22, a = 4, and ?i = 10; 
 
 22 — 4 18 
 
 then, r = 
 
 10-1 9 
 
 Page 269. 
 
 (2.) 
 
 z= a + {n — l)r. 
 Make, a = 2, n = 100, and r = 7; 
 
 then, Z = 2 + (100 - nV == 2 + 693 = 695. 
 
270. 1 ARITHMETICAL PROGRESSION. 67 
 
 (3.) 
 
 First, to find the last term. We have, 
 I — a -\- {n — l)r ; 
 and, making a = 1^ ti = 100, and r = 2, 
 we have, 1=1 + (100 - 1)2 := 1 + 198 =: 199; 
 
 then, S = (^^) xn = ^— X 100 = 10000. 
 
 (4.) 
 To find the last term. We have, 
 
 I = a — (n — l)r\ 
 
 and, making a == '70, ;z = 21, and r = 3, we have, 
 
 ^ = 70 - (21 - 1) X 3 = 70 — 60 = 10; 
 
 then, 8 = \ T ) X ^^l 
 
 and, making a = 70, ^ = 10, and n = 21, 
 
 we have, S = (— ~^) X 21 = 840. 
 
 (5.) 
 To find the last term. We have, 
 
 I = a + {n — l)r; 
 and, making a = 4, ^ = 8, and r = 8, 
 we have, ^ = 4 + (8 — 1)8 = 4 + 56 = 60; 
 
 then, S = \ T ) X n; 
 
 and, making a = 4, ^ =i 60, and ^ = 8, 
 
 we have S = i—^) X 8 = 256. 
 
68 KEY TO DAVIES' NEW ELiaiENTARY ALGEBKA. [270. 
 
 (6.) 
 
 b — a 
 
 T = -. 
 
 n — 1 
 
 Make, b = 20, a = 2, and n =z 10, 
 
 , , 20 — 2 18 
 and we have, r = -— = — = 2. 
 
 b — a 
 r = . 
 
 n + 1 . 
 
 Make, 5 = 19, a = 4, and n = 4; 
 
 .-u 1. 19 - 4 15 
 
 then we have, r = — - — =z -— — Si 
 5 5 
 
 hence, 4. 7. 10. 13. 16. 19, form the series. 
 
 (8.) 
 First, to find the last term. We have, 
 I = a — {71 — l)r. 
 
 Make, a = 10, n = 21, and r — ^\ 
 
 then, I =: 10 - (21 - l)\ == 10 - 6| = 3i; 
 
 then, 
 
 a /10 + 3J-\ ^^ 30,10 ^^ 40 ^, 840 ,,^ 
 
 2 
 (9.) 
 We have the equations, 
 
 S = y I X ^^, and I =z a + {7i — l)r. 
 
270.] AEITHMKTTCAL PKOGRKSSION. 69 
 
 In these equations all the quantities are known, except a 
 and I, Substituting the numbers for the known quantities, 
 we have, 
 
 2945 
 
 = (^-H!yx n, and 185 = a + {n - 1)6. 
 
 ( 10.) 
 We have, from Art. 183, 
 
 h — a 
 
 m + 1 
 
 Making 5 = 5, and a =: 2, and m = 9, 
 
 5 2 
 
 we have, r = = 0.3 ; 
 
 from which the terms are easily found. 
 
 (11.) 
 We have the formula, 
 
 S = (-f-) X n. 
 
 Now, (^ = 1, and I =: n; hence, 
 
 (1 + n\ (n + 1\ 
 
 ^=(-2-)><^^ = H-2-)- 
 
 (12.) 
 The formula for the last term, 
 
 I = a -{- {n — 1) X r. 
 'Making a = 1, and r = 2, we have, 
 
 Z = 1 + (?i — 1)2 =: 1 + 2?2 — 2 = 2/^ — 1. 
 Then, in the formula, 
 
70 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [285. 
 
 substitute for Z, its value, and for a, its value, 1, and we have, 
 
 ^ n + 2n — 1\ 
 
 /3 = I 1 X n = n X n zzz n^. 
 
 (IS.) 
 
 In this example, we know that the person must travel 
 four yards to place the first stone in the basket, and that he 
 must travel four yards in addition for each additional stone 
 which he brings. Hence, we have the first term, the com- 
 mon difference, and the number of terms, to find the sum 
 of the series. 
 
 First, to find the last term. We have, 
 
 I = a -{- {n — l)r. 
 
 Making a = 4, ^ = 100, and r = 4, 
 
 we have, 1 = 4= + (100 — 1)4 = 4 + 396 = 400; 
 
 then, 
 
 S = (^^ xn = (^— ) X 100 = 20200 yards, 
 
 which, divided by 1760, the number of yards in a mile, gives 
 11 miles and 840 yards. 
 
 Page 285. 
 
 (6.) 
 
 Here we have given the first term, the common ratio, and 
 the number of terms, to find the last term. 
 
 Hence, i: = 1 x 2^ = 1 x 512 = 512 cents. 
 
 Page 287. 
 
 (4.) 
 In this example, we have the first term, the common ratio, 
 
288.] GEOMETRICAL PROGRESSION. 71 
 
 and the number of terms given, to find the last term and 
 the sum of the series. 
 
 ^ = 1 X 2^1 = 1 X 2048 = 2048. 
 
 Then, to find the sum of the series, we have, 
 
 ^ ^ Iq - a 
 q -1' 
 
 in which I = 2048, q = 2, and a = 1; 
 
 4096 - 1 ,^^, 
 
 hence, S = = 4095. 
 
 (5.) 
 
 In this example, we have given the first term, the common 
 ratio, and the number of terms, to find the sum of the series. 
 
 First, to find the last term, we have, 
 
 Z = 1 X 2" = 2048. 
 
 r^, rY Iq — a 4096 — 1 ,^^^ , .,,. 
 
 Then, S = -^ r= • = 4095 shilhngs, 
 
 which is equal to £204 155. 
 
 (6.) 
 
 In this example, we have the first term, the ratio, and the 
 number of terms, to find the last term and the sum of the 
 series. We have, 
 
 l:=zlxZ^ — IX 19683 =: 19683 cents. 
 
 ^ Iq — a 19683 x 3 — 1 ^^^^, 
 
 S = = = 29524 cents 
 
 9 — 1 2 
 
 In this example, we have the first term, the ratio, and the 
 number of terms, to find the last term and the sum of the 
 series. 
 
72 KEY TO DAVIES' NEW ELEMENTARY ALGEBRA. [289. 
 
 I— 4 X 8^5 r= 4 X 35184372088832 - 140737488355328. 
 
 For the sum of the series, we have, 
 
 140737488355328 X 8 — 4 
 ^= • 8-~I 
 
 that is, S — 160842843834660 
 
 (3.) 
 First, to find the last term, we have, 
 I = aq^j 
 and making a = 512, and q — ^^ we have, 
 
 1 
 
 ^ = 512 X HY == 512 X 
 
 1024 
 
 rrn ^ a — ZO' 512 — -i 
 
 Then, S = ^ = — ^ = 682^. 
 
 I - q i 2 
 
 (4.) 
 First, to find the last term, we have, 
 
 I — aq^ = 2187 X (iY = 3. 
 
 Then, 'S = ^^ = ^-^ = 3279. 
 
 (5.) 
 First, to find the last term, we have, 
 
 I = aq^ = 972 X (i)^ = 4. 
 Then, g^a-lq _^12-^ _ ,,,, 
 
 Then, S 
 
 
 1 — 
 
 !Z 
 
 (6.) 
 
 
 
 I zz 
 
 : aq'^ 
 
 = 
 
 147456 X 
 
 {\r 
 
 = 9. 
 
 a 
 
 -Iq 
 
 
 
 147456 - 
 
 ± ^ 
 
 196605, 
 
 UNIVERSITY 
 
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 All the geographies in use in our common schools have received from ir.e a 
 careful and critical examination. The National Series was one of two series 
 that received my full approbation. The opinion that I formed of their gieat 
 merit is justiGed by their extensive use in the public schools of this city. I hHve 
 found, by examination of the Book of Supply of our Board, that considerably the 
 largest number of any series now used in our public schools, is the National, by 
 Monteith and McNally. — B. A. Adams, Chairman of ^'^ Conimittee on Course of 
 Studies and School-hooJfs,^' and Member of " Committee of Supplies" of Board of 
 Education of New York. 
 
 During an experience of ten years in teachinsj, I have found no series of 
 Geographies so well calculated, in matter, arrangement, and system, to facilitate 
 the progress of the learner as Monteith and McNally's.— Solomon ili/erl, Prof. 
 of English Grammar and Geography in the York Co. Normal School, Pa. 
 
 This s;nies was adopted after a careful examination of the best works in this 
 branch of study, and a year's experience makes us better and better satisfied with 
 our choice. — Josiah T. Read, Priii. JIarshall (Mich.) Union SchooL 
 
 We have used McXally's Oeogrnphy since its publication, deetning it the best 
 class-book in the market. It not only is the equal of its rivals in positive merits, 
 bnt is superior to most of them as to icJiat it omits. It is both practical and prac- 
 ticahle as a text-book. — Jiev. Joseph E. King, A.M., Principal of Fort Edward 
 Institute, N. Y. 
 
 We have used McNally's and Monteith's Geographies for three years, and 
 would not exchange them for any others In the maiKet. — Rev. B. St. James Fry, 
 A. M., President of Worthington Female College, Ohio. 
 
A. S. ISarnes <£• Burr's Publications. 
 
 ptioual 3ixiu. 
 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 KEY TO 
 DAVIES' 
 KEY TO 
 DAVIES' 
 DAVIES' 
 KEY TO 
 DAVIES' 
 DAVIES' 
 
 DAVIES' 
 KEY TO 
 DAVIES' 
 KEY TO 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 DAVIES' 
 
 By CHARLES DAVIES, LL.D. 
 
 Elementarg (Course. 
 
 PRIMARY ARITHMETIC AND TABLE-BOOK $ 
 
 FIRST LESSONS IN ARITHMETIC 
 
 INTELLECTUAL ARITHMETIC 
 
 NEW SCHOOL ARITHMETIC 
 
 DAVIES' NEW SCHOOL ARITHMETIC 
 
 NEW UNIVERSITY ARITHMETIC 
 
 DAVIES' NEW UNIVERSITY ARITHMETIC... e 
 
 GRAMMAR OF ARITHMETIC 
 
 NEW ELEMENTARY ALGEBRA 
 
 DAVIES' NEW ELEMENTARY ALGEBRA 
 
 ELEMENTARY GEOMETRY AND TRIGONOMETRY 
 
 PRACTICAL M ATHEM ATICS 
 
 ^Ttj&anceti Louise, 
 
 UNIVERSITY ALGEBRA 
 
 DAVIES' UNIVERSITY ALGEBRA 
 
 BOURDON'S ALGEBRA 
 
 DAVIES' BOURDON'S ALGEBRA 
 
 LKGENDRE'S GEOMETRY 
 
 ELEMENTS OF SURVEYING 
 
 ANALYTICAL GEOMETRY 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 DESCRIPTIVE GEOMETRY 
 
 SHADES, SHADOWS, AND PERSPECTIVE 
 
 LOGIC OP MATHEMATICS 
 
 MATHEMATICAL DICTIONARY 
 
 Mathematical Chart (Sheet) 
 
 EECOMMENDATIONS. 
 
 We have tested the completeness of tliis dictionary by looking: for a consider- 
 able number and variety of titles, under all of which we have found statements and 
 discussions, succinct without being obscure, and sufficiently thorough to render the 
 work a reference-book for proficients as well as for pupils in Mathematics. It is 
 just such a book as we have needed for a score of years. — North Amer. Review. 
 
 Each treatise serves as an introduction to the next higher by the similarity of 
 its reasonings and methods, and the student is carried forward by easy and 
 gradual steps over the whole field of mathematical inquiry, and that, too, in a 
 shorter time than is usually occupied in mastering a single department. I sin- 
 cerely and heartily commend them to the attention of ray fellow-teachers in 
 Canada. — John McLean Bell, B. A., Prin. Lower Canada College. 
 
 The undersigned has examined with care, and taught some time since, several 
 volumes of Davies' Mathematics, and is of the opinion that, as a whole, it is the 
 most complete and best course for academic and collegiate instruction with which 
 he is acquainted. — Horace Webster, LL.D., Fres. N. Y. Free Academy. 
 
A. S, Barries S Burros Publications. 
 
 i»ti0ma Mxm. 
 
 tm 
 
 By EMMA WILLARD. 
 
 School History of the United States $0.75 
 
 Large History of the United States 1.50 
 
 The author^s long experience as a teacher at the 
 head of one of the most noted Seminaries of the 
 country, gave her great advantage in preparing a 
 book on our National History adapted to use in 
 schools. The unqualified commendation bestowed 
 upon these books by the leadmg statesmen of the 
 age is a sufficient guarantee of their merit. 
 
 EECOMMENDATIONS. 
 
 I can not better express my sense of the value of your History of the United 
 States, than by saying I keep it near me as a book of reference accurate in facts 
 and dates. — Daniel Webster. 
 
 Similar testimonials have been received from Henry Clay, John McLean, 
 Elijah Willard, Senator Dickinson and many other eminent men. 
 
 This is a noble and well-'w'ritten book, whi;h it is both a pleasure aiid a profit 
 to read. The sty .6 is lucid, and varies with the impulse of the subject. Mrs. 
 Willard should 1 e considered as a benefactress, not only by her own sex, of 
 whom 6h« became in early years a pyorainent and permanent educator, but by 
 the country at large, to whose good she has dedicated the gathered learning and 
 faithful labor of life's later periods. The truths that she has recorded, and the 
 principles that she has impressed, will win from a future race gratitude that can 
 not grow old, and a garland that will never fade.— Jfrs. Z. //. Sigoumey. 
 
 18 
 
A. S. Barnes & Burros Puhlicaiions. 
 
 1. Principles of Chemistry, embracing the most recent 
 
 discoveries " in the Science, and the outlines of its 
 application to Agriculture and the Arts. Illustrated 
 bj numerous experiments newly adapted to the sim- 
 plest apparatus. By John A. Poetek, A. M., M. D., 
 Professor of Agriculture and Organic Chemistry in 
 Yale College. Price $1.25. 
 [A Box OF Apparatus prepared expressly for this Work, will 
 
 COST ONLY $8.00.] 
 
 2. First Book of Chemistry, and Allied Sciences, in- 
 
 cluding an outline of Agricultural Chemistry, by 
 Prof. John A. Poetee. Price 60 cents. 
 
 RECOMMENDATIONS. 
 
 I am greatly pleased with Porter's Chemistry. It does not give a mere outline, 
 but exhibits the atomic proportions in all combinations so as to make the student 
 scientific in his processes. It is fully up with the progress of the age in this most 
 rapidly advancing science ; and last, not least, in its behalf, it is so simple, easy, 
 and beautiful, that a teacher can make himself a chemist while teaching a class. — 
 Prof. J. Young, of the North- Western. ChrUtian University. 
 
 1 put a class into it last winter, which nearly finished it. I think I never used & 
 text-book on any subject that pleased me so well. It seems just what our acade- 
 mies require. I feel sure that this book meets a great and general want, and shall 
 be much mistaken if it does not prove one of the most successful ever issued. — 
 M. Sfurges, Monroe, Butler County, Ohio. 
 
 I have examined Porter's Chemistry with no little attention. As an elementary 
 Text-book it is a model. The bork must greatly extend the study of this important 
 science.— JS. A. Grant, Franlcfort, Ky. 
 
 I do not hesitate to pronounce it a work most admirably adapted to convey to 
 any inquiring mind a clear idea of the elements of chemistry. — M. L. Morse, Prin- 
 cipal of High School, Dover, N. H. 
 
 We have read this book with great pleasure, and can recommend it as the sim- 
 plest, most concise, and most comprehensive School Chemistry known to us. Its 
 simplicity is its great va.QXit.— Massachusetts Teacher. 
 
A. S. Barnes & Burr'' 8 Publications. 
 
 "§'Ximm\ 3txm. 
 
 INTRODDCTORY COUKSE OF NATURAL PHILOSOPHY. 
 
 FROM THE FRENCH OF M. GANOT. 
 
 By professor W. G. PECK, M. A. 
 Price $1.25. 
 
 One of the most attractive volumes ever published is 
 this new work on Natural Philosophy. The original edi- 
 tion is regarded as the standard text-book in France, and 
 emanates from one of her ablest and most popular writers. 
 It has been Americanized and adapted with remarkable 
 skill to the peculiarities of our systems of instruction, by 
 Professor Peck. One of the most striking features of the 
 book is its profuse and magnificent system of illustration. 
 By special arrangement with M. Ganot, fac-similes of all 
 the original engravings are presented, rendering it un- 
 equalled in point of beauty, accuracy, abundance and 
 general adaptation of illustration. The volume embraces 
 a full and logical development of all the elementary prin- 
 ciples of Physics. 
 
 RECOMMENDATIONS. 
 
 I have examined Peck's Ganot's Natural Philosophy with some care, and know 
 not which to admire most, the lucid, compact, and charming style of the composi- 
 tion, or the beautiful and faultless mechanical execution of the book. The typo- 
 graphical execution seems to be unsurpassed, and the illustrations are numerous, 
 accurate, and elegant. It is just such a book as students should have in their hands, 
 as a model of excellent scholarship and good taste. We have adopted it in this 
 Institution.— /rri W. AUetiy Smtior rrofeasor of Union Chrintian College. 
 
 The Natural Philosophy of M. Ganot, edited by Prof. Peck, is, in my opinion, 
 the best work of its kind, for the use intended, ever published in this country.— 
 D. C. Van Norman, Principal of the Van Norman Inslitule, New York. 
 
A. S. Barnes & Burr's Publications. 
 
 %ixXm\\\ ^n'm. 
 
 EMBRACING 
 
 Grammar, Conversation, Literature, and Commercial Cor- 
 respondence, together with an Adequate Dictionary. By 
 Louis Pujol, A. M., of the University of France, and D. C. 
 Van Norman, LL.D., of N. Y. Young Ladies' Institute. 
 Price $1.50. 
 
 Part I. inchides an elaborate Treatise on Pronunciation; u 
 Theoretical and Practical Grammar, on a plan entirely new, with 
 Exercises in both French and English, and a series of Vocabula- 
 ries, forming an adequate dictionary. 
 
 Part IL presents a complete and orderly development of French 
 Syntax, with Exercises in both French and English, and Polite 
 Conversations ilhistrative of the idiomatical rules. It includes also 
 a general Course of Versions, which embraces an instructive series 
 of biographies, and a very useful abstract of the history of France. 
 ^ Part IlL contains a new system for teaching French Conversa- 
 tion. The exercises embrace a great variety of topics, and are 
 illustrative of more than four thousand words and phrases in 
 common use. 
 
 Part IV, includes— 1. Progressive Lessons in Translating, as 
 an introduction to French Literature. — 2. Selections in prose and 
 verse, from the French Classics and the best modern writers. 
 Also, an Appendix of Polite and Commercial Correspondence, 
 and translations of all the more difficult words and phrases, with 
 figures of reference, together with an Adequate Dictionary, 
 
 RECOMMENDATIONS 
 
 I take great pleasure in recommending Pujol and Van Norman's French Class- 
 book, and there is no French Grammar or Class-book which can be compared with 
 it in completeness, system, clearness, and general utility. — Elias Peissner, Prof, 
 of German, and Lecturer on Political Economy, Union College. 
 
 I have been examining with increasing admiration your Course of Instruction in 
 the French L8,Tiguage. It furnishes all that is needful to qualify the student not 
 only to speak and write the French with fluency, I ut also to read any French 
 author.— i?eu. A. L. Lindsey, A. 31., Prin. Parsonage I;,st., South Salem, iV. Y. 
 
 The plan of your Complete French Class-book is as ingenious as it is unique, and 
 appears to me admirably adapted tojthe purposes of instruction. — Bev. W. Ormistan, 
 D.D., of Hamilton, Canada West. '" 
 
 Having been engaged, occasionally, for forty years in teaching the French lan- 
 guage, I have ca»-efully examined nearly all the works on the subject that have 
 been published in this country, and am free to say that I consider thic, on the 
 whole, decidedly superior to any other. I have already adopted it in my class.— 
 — A. Curtis, M.M., M.D.^ Pres. Lit. and Sci. Institute, Cincinnati. 
 
A. S. Barnes & Burros Publications. 
 
 %^imxdX ^nm. 
 
 CLASS-BOOK OF BOTANY; being Outlines of the 
 Structure, Philosophy, and Classification of Plants, with a 
 Flora of the United States and Canada. By Alphonso Wood, 
 A.M., Principal of Female Academy, Brooklyn. 
 832 pp. 8vo. Price $2 25. 
 
 Teachers, pupils, and amateurs, ^vill find in the new Class-book 
 of Botany the following peculiar advantages : 
 
 First The Scientific Treatise embraced in Parts I., II., and III., is re- 
 markable for its comprehensiveness, clearness, and brevity; is divided into 
 short paragraphs, suited to the learner's convenience, and each paragraph, 
 with the topic prefixed in capitals, is adapted to recitation and reviews. 
 
 Second. The Flora comprehends a wider territory than that of any School 
 Botany now in use, extending from the Atlantic to the Mississippi, and 
 from the St. Lawrence to the Gulf of Mexico. 
 
 Third. The Flora comprehends not only our native, spontaneous vege- 
 tation, but also one thousand species of cultivated plants — almost our entire 
 exotic flora, growing in the field, the garden, and the conservatory; thus 
 rendering the study of Botany as practicable in the city as in the country. 
 
 Fourth, The descriptions of species are unusually full, graphic, and 
 popular in style. 
 
 Fifth. The Tables for analysis are far in advance of those in the former 
 editions, both in simplicity and extent, reaching now from the Grand Divi- 
 sion to the Species, and by thoir peculiar form adapted to class exercise 
 throughout the entire route ; thus rendering the pursuit at once a vigorous 
 discipline and an exciting amusement 
 
 RECOMMENDATION. 
 
 Professor Wood is well known among Botanists, and it wonld be presumption in 
 us to affect to praise him. His Class-book was first issued fifteen years ago, and 
 has had extensive sale and use. It is now thoroughly revised ; we might say, re- 
 written. The results of the author's repeated and far-extended journeys and re- 
 searches are here given. Ue says that most of the descriptions are verified by his 
 own personal observation. The list of plants includes every species known to 
 exist from the St. Lawrence to the Gulf of Mexico, and from the Atlantic to the 
 Mississippi, and also the exotics which have obtained any considerable culture in 
 this country. The illustrations are abundant (numbered to 745), and well cut. 
 The first part of the book, nearly 200 pages, treats at length of the structure *of 
 plants, of ibo uses of their parts, and of classification. The information given is 
 full, and the style is clear and direct ; with excellent tables, index, and a glossary. 
 The book is excellent, both in substance and in style. — lUinoia Teacher. 
 
A. S. Barnes & Burr's Publications. 
 
 patwiral Mnu. 
 
 POR TEACHERS IN PiaMAlU' SCHOOLS AND rRIMARY CLASSES. 
 BY A. S. WELCH, 
 
 PRINCIPAL OF MIOrilGAN STATE NORMAL SCHOOL. 
 
 From Pestalozzi: — ^''Observation is the al)solute lasts of all knowl- 
 edge. The first object, then, in education^ must be to lead a child to 
 observe with accuracy; the second^ to express with correctness the 
 result of Jiis observations.'^'' 
 
 Principles, without application, have been harbored and sus- 
 tained by many of the so-called disciples of Comenius and Pesta- 
 lozzi, but we look in vain for a development, in systematic ob- 
 servation, at all commensurate with the practical results that 
 might have been expected to follow the teachings of these two 
 educators. 
 
 In commending this book to the general examination of teach- 
 ers, the publishers would remark, that the author brings to its 
 preparation great and varied experience in our schools, and ex- 
 tended observation in those schools of England and Germany 
 where a systematic presentation of the subject has been secured. 
 
 The following Text-books will be found to contain principles 
 involving the method of Obiect-teaching, and are warmly com- 
 mended to the attention of educators : • 
 
 J. — Object Lessons ; designed for the use of Teachers in Primary Classes and 
 Primary Schools. By A. S. Welch. Price 50 cents. 
 
 2.— Exercises for Dictation and Pronunciation ; embracing a numerous col- 
 lection of difficult words, including nearly 300 military and war terms, together 
 with a variety of useful lessons. By Charles Northend. Price 50 cents. 
 
 3. — Juvenile Definer : a collection and classification of familiar 'words and 
 names, correctly spelled, accented, and defined. By Wm. W. Smith. Price 
 30 cents. — The words are grouped with referenc'e to similar signification or use : 
 as the several kinds of huildings compose one class, vessels another, cloths 
 another, &c., &c. 
 
 4.— First Book in Composition. By F. Brookfield. Price 30 cents.— This 
 little book is a successful attempt to aid thought, by a series of illustrations and 
 suggestions of topics calculated to inspire interest in a study heretofore repulsive 
 to a child. Subjects have been selected upon which the thoughts of all children 
 exercise themselves spontaneously. 
 
 5.— The Child's Book in Natural History ; illustrating.the Animal, Vegetable, 
 and Mineral Kingdoms, with application to the Arts. By M. M. Carl. Price 
 35 cents, 
 
 5. — Treasury c/Knowledge : embracing Elementary Lessons in common things! 
 —Practical Lessons on common objects— Introduction to the Sciences. By Wil- 
 liam and Robert Chambers. Price 75 cents.— This last-named book contains 
 a vast amount of information, and deserves a place in every school-room. 
 
A. S. Barnes & Burr'^s Publications. 
 
 §ati0»»I 3mt^. 
 
 BY CHAKLES'NORTHEND, A.M. 
 250 pp. 18mo. Price 50 cts. 
 
 IsTo work has been prepared for teachers and scholars 
 that will excite more interest and pleasure, than a small 
 book by Charles ISTorthexd, of the State Normal School 
 of Connecticut, entitled "Exekoises for Dictation and 
 Pronunciation;" containing a large numberof the most 
 difficult words in the language, including nearly three 
 hundred military and war terms, and two thousand words 
 which are frequently mispronounced as well as misspelled ; 
 alphabetically arranged, pronunciation indicated, and mean- 
 ing given ; together with a variety of other useful lessons. 
 
 This work will prove a valuable Hand-book for both teacher 
 and scholar, and should have a place in every school. 
 
 It has not been the aim of the author to furnish a substitute for 
 the spelling-book, but rather to prepare an accompaniment to it, 
 for the use of higher classes. In the several collections or group- 
 ings of words, the most prominent in each department have been 
 brought together. The various miscellaneous exercises in the 
 book will, it is believed, readily commend themselves to teachers, 
 and open a wide field for nnich general instruction in every-day 
 matters — making a good basis for useful Object Lessons. 
 
 The following table of contents will, perhaps, best exhibit the 
 main features of the work, viz. : 
 
 Hints on Spelling— Rnles for the use of Capitals— Rules of Spelling— Words simi- 
 lar in pronunciation, but dissimilar in spelling and meaning- Words pronounced 
 nearly alike, but differing in spelling and dcUning — Words of two pronunciations — 
 Synonyms- Words of special resemblance- Words varying in use— Words liable 
 to be misspelled— Cluistian names of males and females— Occupations— Professions 
 — Auiraals—Birds— Fishes— Trees— Flowers— Productions of the farm and garden — 
 Agricultural implements— Furniture and articles of household-Arithmetic— Geog- 
 raphy— Grammar— Philosophy— Botany— Physiology— Forms— Books— Wearing 
 apparel— Architective origin of words — Military and government terras — Hardware 
 —Boats and parts of a ship— Marine journal— Review of tho market— Monetary 
 affairs— Exports— Prefixes— State mottoes— Abbreviations— Proof-marks, &c. 
 
-.'."l'*™"^^^ CENTS 
 
 DAY AND TO $Uo! oTthbV^ ''°^'^rH 
 OVERDUE. " ^"^E SEVENTH DAY 
 
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