UC-NRLF 
 
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 EBEA 
 
 BY THE LATE 
 
 |PK<>f. OF MATItKiTAVrCS AND Is'AT. PHlIvOSOPilY i* T»tB EAST INDIA COLL., H^RTFORii. 
 
 THOMAS ATTUNSON, M. A., 
 
 LATS nciii.LAJi. oir co'::.r. en, cot.t... oa-MBUtdge, 
 
 JP^wm tfjt jTourt^ ILon"tJon Etiition, iuttlj |t>T5(t{ona 63 tfjr ISrotfterjs of IJic 
 
 R-& 1 SADl.^h _.:i,^ I 0:FPANY, 164 WIlllAM STREKi' 
 
 i. J B T ( ) N : - ^. t Z S I' ■ L D E li" A I. S T P. E L T . 
 
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 £i^\^nrW^ 
 
 IN MEMORIAM 
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** 
 4 
 
 ^ftMENTARY TREATISE 
 
 ON 
 
 / 
 
 ALGEBEA. 
 
 BY THE LATE 
 
 EEY. B. BKLUGE, B.D., xA E. S., 
 
 PBOr. or VATDLBMATlOfl AND NAT. POTLOSOFHT IN TUB EAST INDIA COLL., ITEBTFOBDw 
 REVISED, IMPROVED, AND SIMPLIFIED, 
 
 By THOMAS ATKINSON, M. A., 
 
 LATE BOHOLAB OF COKP. Cn. COLL., CAMBUIBaE. 
 
 jrom t^e JFourt]^ Hontjon EtiitCon, feit^ a"&"Dit(ons fig t^e 33rot6cr» of i^t 
 Cf)ri8tian Retools. 
 
 NEW YORK: 
 n & J. SADLIER AND COxAIPANY, 164 WILLIAM STRECT. 
 
 BOSTON: — 128 FEDEKAL STPwEET. 
 MONTREAL, C. E.: — 179 NOTRE DAME STREET. 
 
 1861. 
 
% 
 
 
 Entered, "jf/OordLO,^ to Act of Coug^ew, 'd the year WW, 
 
 By D. & J. Sadueb and Compant, 
 
 Bl Mm Clerk's Office of tbe District Court ot tne united Siales foi tho Soudiein iAtidak d 
 
 New York. 
 
 Uim 
 
 stereotyped by Biixin A Bbothbss, 20 North William street, N. T. 
 
ADVERTISEMENT. 
 
 The excellence of "Bridge's Algebra," as an ele- 
 mentarj treatise, lias long been well known and 
 extensively recognised. In the Preface to the Second 
 Edition the author expressly states, that "great pains 
 were taken to give to it all the perspicuity and simpli- 
 city which the subject would admit of, and to present 
 it in a form likely to engage the attention of young 
 persons just entering on their mathematical studies." 
 The design, which he thus proposed to himself, was 
 accomplished with singular felicity, — for not one of 
 the many publications on Algebra, which have during 
 a period of forty years issued from the press, with the 
 professed object of producing a more simple and ap- 
 propriate introduction to the study of the science, has 
 evinced such merits as justly entitle it to be placed- 
 in comparison with the performance of Mr. Bridge. 
 These publications have accordingly failed to secure 
 for themselves the same measure of public approba- 
 tion. 
 
 This small compendium embraces all which is com- 
 prised in the former large and expensive editions, that 
 
IV ADVERTISEMENT. 
 
 is either practically useful or theoretically valuable. 
 By introducing Equations and Problems at the ear- 
 liest stage possible, a novel and instructive feature — 
 which the editor is persuaded Oannot fail to excite the 
 curiosity and stimulate the ardour of the young alge- 
 braist, so as to induce him to pursue his studies with 
 more than usual alacrity, intelligence, and success — 
 has been given to the work. A great variety of new, 
 easy, and interesting problems, which are not con- 
 tained in former editions, have thus been interspersed 
 in the several chapters. Besides these additions many 
 alterations have been made, either for the sake of uni- 
 formity of arrangement, or of rendering the subject 
 still more easy and accessible to youthful minds. Mr. 
 Bridge, nearly forty years ago, "was not without a 
 hope that his ' Elementary Treatise on Algebra^ would 
 find its way into our Puhlic Schools; where, it was 
 very well known, this branch of education was [then] 
 but little attended to:" and it is now confidently 
 hoped, that this new edition will (in consequence of its 
 cheap and improved form) find its way into many 
 schools, where this science is not yet sufficiently at- 
 tended to, and thus be the means of rendering this 
 instructive study a subject of general education. 
 
 T. A. 
 
 Guilford, December^ 1847. 
 
f 
 
 CONTENTS. 
 
 Chapter I. Definitions 1 
 
 Chap. II. On the Addition, Subtraction, Multiplication, and Divi- 
 sion of Algebraic Quantities 6 
 
 Addition 6 
 
 Simple Equations. . . .• 9 
 
 On the Solution of Simple Equations 10 
 
 Problems 14-17 
 
 Subtraction 18 
 
 - On the Solution of Simple Equations 19 
 
 Multiplication 24 
 
 On the Solution of Simple Equations 28 
 
 Problems 29-35 
 
 Division 85 
 
 Problems producing Simple Equations 41-43 
 
 Chap. III. On Algebraic Fractions 43 
 
 On the Addition, Subtraction, Multiplication, and Divi- 
 sion of Fractions 49 
 
 On the Solution of Simple Equations 65 ^ 
 
 Problems 61-66 
 
 « On the Solution of Simple Equations, containing two or 
 
 more unknown Quantities 66 
 
 Problems '72-'75 
 
 Chap. IV. On Involution and Evolution 76 
 
 On the Involution of Numbers and Simple Algebraic 
 
 Quantities 76 
 
 On the Involution of Compound Algebraic Quantities. ... 78 
 On the Evolution of Algebraic Quantity 78 
 
VI CONTENTS. 
 
 PAoa 
 On the Investigation of the Rule for the Extraction of the 
 Squai-e Root of Numbers 81 
 
 Chap. V. On Quadratic Equations 83 
 
 On the Solution of pui'e Quadratic Equations 84 
 
 On the Solution of adfected (^adratic Equations 86 
 
 Problems 
 
 On the Solution of Problems producing Quadratic Equa- 
 tions 92-98 
 
 On the Solution of Quadratic Equations containing two 
 unknown Quantities 98 
 
 Chap. VI. On Arithmetic, Geometric, and Harmonic Progression... 102 
 
 Problems , . .105-109 
 
 On Geometric Progression lOQ 
 
 On the Summation of an infinite Series of Fractions in 
 Geometric Progi*ession ; and on the method of finding 
 
 the value of Ch*culating Decimals 113 
 
 Problem 116 
 
 On Harmonic Progression ,... 11*7 
 
 Chap. VII. On Permutations and Combinations 119 
 
 Appendix. On the Different Kinds of Numbers 125 
 
 On the Four Rules of Arithmetic 126 
 
 On the Two Terms of a Fraction 127 
 
 On Ratios and Proportions 128 
 
 On the Squares of Numbers and then* Roots 128 
 
 On the Factors and Submultiples of a Number 130 
 
 On Odd and Even Numbers 131 
 
 On. Progressions 131 
 
 On Divisible Numbers without a Remainder IdS 
 
 Properties and various Explanations 134 
 
 Miscellaneous Problems 137* 
 
BRIDGE'S lL(}E:BRi. ; 
 
 CHAPTER I. 
 
 DEFINITIONS. 
 
 1. Algebra is a general method of computation, in which 
 number and quantity and their several relations are ex^ 
 pressed by means of written signs or symbols. The symbols 
 used to denote numbers or quantities are the letters of the 
 alphabet. 
 
 2. Known or determined quantities are generally repre- 
 sented by the^r^^ letters of the alphabet ; as, a, 6, c, &c. 
 
 3. Unknown or undetermined quantities are usually ex- 
 pressed by the last letters of the alphabet ; as, x^ y, ^, &c. 
 
 4. The multiples of quantities, that is, the number of times 
 quantities are to be taken, as twice a, three times b, jive times 
 aar, are expressed by placing numbers before them-, as 2a, 
 35, hax. The numbers, 2, 3, 5, are called the coefficients of 
 the quantities, a, 5, ax^ respectively. When there is no 
 coefficient set before a quantity, 1 is always imderstood : thus 
 a is the same as la. 
 
 5. The symbol = (read is equal to) placed between two 
 •quantities means that the quantities are equal to each other. 
 
 Thus 12 pence = 1 shilling ; 3 added to 5 = 8 ; 2a added to 
 4a = 6a. This symbol is called the sign of equality/, 
 
 6. The sign + (read plus) signifies that the quantities be- 
 fore which it is placed are to be added. Thus 3 + 2 is the 
 same thing as 5 ; and a + b + x, means the sum of a, b, and 
 Xy whatever be the values of a, 6, and x, 
 
 ^ 1. Define Algebra. What symbols are used to denote numlers or quan- 
 tities ? — 2. What letters of the alphabet are employed to represent IcTwwn 
 or determined quantities ? — 3. How are unhnown or undetermined quantities 
 represented ? — 4. What are coefficients ? When is the coefficient omitted f 
 — 6. What is the sign of equality ?— 6. What is the use of the sign+ ? 
 
2 ALGEBRA. 
 
 7. . The sign. — . (reg4 mtnus) signifies that the quantity to 
 which it is prefixed^' is ifo be subtracted. Thus 3 — 2 is the 
 same thin^ as 1 ; a — , b means the difference of a and b, or b 
 t^ken from; (t ; aird a -*f- J' — ar, signifies that x is to be sub- 
 tracted from me sum oi a and b. 
 
 8. Quantities which have the sign + prefixed to them are 
 called positive, and those which have the sign — set before 
 them are termed negative quantities. When there is no sign 
 before a quantity + is understood : thus a stands for + «. 
 
 9. The symbol X (read into) is the sign of multiplication, 
 and signifies, that the quantities between which it is placed 
 are to be multiplied together. Thus, 6x2 means that 6 
 is to be multiplied by 2 ; and a X b X c, signifies that 
 a, b, c, are to be multiplied together. In the place of this 
 symbol a dot or full-point is often used. Thus, a. b ,c, means 
 the same as a X b X c. The product of quantities repre- 
 sented by letters is usually expressed by placing the letters 
 in close contact, one after another, according to the position 
 in w^hich they stand in the alphabet. Thus, the product of a 
 into b is denoted by ab ; of a, b, and x, by abx ; and of 3, a, x, 
 and y, by Saxg. 
 
 10. In algebraical computations the word therefore often 
 occurs. To express this word the symbol .*. is generally 
 made use of Thus the sentence " therefore a + 6 is equal 
 to c + c?," is expressed by " .*. a + 6 = c + d" 
 
 EXAMPLES. 
 
 F % 1. In the algebraical expression, a + 5 — c, let a = 9, 
 5 = 7, c = 3 ; then 
 
 a + 5-c= 9 + 7--3 
 = 16-3 
 = 13 
 Ex. 2. In the expression ax -\- ay — xy, let a = 5, a; = 2, 
 y = 7 ; then, to find its value, we have 
 
 ax ■\- ay — xy — 5x2 + 5x7 — 2x7 
 = 10 + 35-14 
 = 45-14 
 = 31 
 
 Y. IIow is the symbol — read ?— 8. What is meant by 'positive and what 
 by negative q_iiantities ?— 9. Write down the sign of multiplication ? Is any 
 other warZ: used to denote multiplication ? When is no symbol used?— 
 10. What symbol is used to denote the word therefore f 
 
Ans, 
 
 13. 
 
 Ans, 
 
 3. 
 
 Ans, 
 
 65. 
 
 Ans, 
 
 29. 
 
 Ans, 
 
 176. 
 
 DEFINITIONS. 3 
 
 Ex. 3. If a = 5, 5 = 4, c = 3, c? = 2, a; = 1, y = 0, find 
 the numerical values of the following expressions : 
 
 (1.) a-{'b + c + x. 
 
 (2.) a — b + c — x^ y. 
 
 (3.) ab + 3ac — be + 4ccx -— xy. 
 
 (4.) abc_ — abd + bed -— acx, 
 
 (5.) Zabc + ^cx — 85c/^ + axy, 
 
 11. The symbol -r- (read divided by) is the s^^^ o/ c^zVi- 
 5^o^l, and signifies that the former of two quantities between 
 which it is placed, is to be divided by the latter. Thus, 
 8 -r 2 is equivalent to 4. But this division is more simply- 
 expressed by making the former quantity the numerator, 
 
 and the latter the denominator of a fraction : thus -r- means 
 
 o 
 
 a divided by b, and is usually, for the sake of brevity, read 
 
 a by b, 
 
 EXAMPLES. 
 
 Ex. 1. If a = 2, 6 = 3 ; then, we find the values of 
 , . 3a _ 3X2 _ ^ _ ^ 
 
 ^ '^ 56 "" E~x^ ■" 15 ~ y 
 
 \H. 8a~36 8X2-3X3 16-9 7 
 
 Ex. 2. If a = 3, 6 = 2, c = 1, find the numerical values of 
 
 ,..Sa + c .10 
 
 (1.) jr-. ^ns, ~. 
 
 ^ ^ 46 + a 11 
 
 , . ab + ac^bc . 7 
 
 ^^■> 2ab-2ac + bc' ^"'- 8" - 
 
 12. When a quantity is multiplied into itself any number 
 of times, the product is called a power of the quantity. 
 
 11. By what symbol is division denoted? What is its name? Is 
 division ever expressed in any other manner ? — 12. What is meant by the 
 power of a quantity ? 
 
•4 ALGEBKA. 
 
 13. Powers are usually denoted by placing above the quan« 
 tity to the right a small figure^ which mdicates how often the 
 quantity is multiplied into itself. Thus, 
 
 a - - - - i]iQ first power of a is denoted by a {a})» 
 a X ct ' - - the 2d power oAsquare of a " a^ 
 a X « X « - the 3d power or cube oi a " a^. 
 a X « X « X « the 4th power of a " a*. 
 
 The small figures ^, ^, \ &c., set over a, are respectively 
 called the index or exponent of the corrdfeponding power of a. 
 
 14. The roots of quantities are the quantities from which 
 the powers are by successive multiplication produced. Thus, 
 the root of the square number 16 is 4, because 4 X 4 = 16, 
 and the root of the cube number 27 is 3, since 3x3x3 = 
 27. 
 
 15. To express the roots of quantities the symbol -^Z, (a 
 corruption of r, the first letter in the word radix,) with the 
 proper index, is employed. Thus, 
 
 V^ or -y/cp, expresses the square root of a. 
 ^a " '• " the cube root of a. 
 
 V« " " " the fourth root of a. 
 
 &;c. &c. 
 
 EXAMPLES. 
 
 Ex. 1. If a = S,b = 2] then a^ = 3 X 3 = 9, a^ = 3 X 
 3 X 3 = 27, ^>^ = 2 X 2 X 2 X 2 = 16. 
 
 Ex. 2. If a = 64 ; then ^a = -y/64 = 8, V« = V64 = 
 
 V4 X 4 x"4 = 4, V« = v/64 = 2. 
 
 ax"^ -4- <^* 
 Ex. 3. In the expression — - , let a = 3, 6 = 5, c = 2, 
 
 ox '~'~ a "^~ c 
 
 m 
 
 = 6. What is the numerical value ? 
 
 Here arr' + 5^ = 3 X 6 X 6 + 5 X 5 == 108 + 25 = 133, 
 andJa; — a' — c = 5x6 — 3x3 — 2 = 30 — 9--2=19 
 
 «a;2 + ^' _ 133 _ ^ 
 
 6:r— a^ — c 19 
 
 18. How Sirepotoers denoted? — 14. What are the roots of qnnntities ? — 
 15. By what symbol are the roots of quantities expressed ? What is tho 
 origin of this symbol ? 
 
DEFINITIONS. 5 
 
 Ex. 4. If a = 1, 6 = 3, f = 5, c? = 0, find the values of 
 (1.) a^ -f 26 — c. Ans. 2. 
 
 (2.) a^ + Sb' - c\ Ans. 3. 
 
 (3.) a" + 2b^ + 3c^ -f^c?'. ^ns. 94. 
 
 (4.) Sa'b - 25V + 4c^ - 4aU ^?i5. 19. 
 
 (5.) a^ + b\ Ans. 28. 
 
 «3 A^ c^ 
 (6.)|-+|-+3-. ^«^-51- 
 
 Ex. 5. Let a = 64, 6 = 81, c = 1 : find the values of 
 
 (1.) ^a-\-^b, Ans, 17. 
 
 (2.) Va+'v/^+'v/^- ^/i5. 18. 
 
 (3.) V^ ^^s. 72. 
 
 16. When several quantities are to be taken as one quan- 
 tity^ they are enclosed in brackets^ as ( ), j (, [ ]. 
 Thus, (a ■\- b — c) ,{d — e) signifies that the quantity repre- 
 sented by a + 5 — c, is to be multiplied by that represented 
 by c? — e ; if then a = 3, 6 = 2, c=l, o? = 5, ^ = 2, a + S 
 _ c = 4, c? — e = 3, and .-. (a + 6 — c) . (c? — e) = 4 X 3 
 = 12. 
 
 Great care must be taken in observing how brackets are 
 employed, and what effects arise from the use of them. Thus 
 (a + 6) . (c + (i), (a + 6) c + G?, a + 6c + cZ, are three very 
 different expressions ; for if a = 3, 6 = 2, c = 3, c? = 5, 
 
 (1.) {a+b) . (c+cZ) = (3+2) . (3+5)=5x8=40. 
 
 (2.) (a+6)c+c?=(3+2) 3+5=5x3+5=20. 
 
 (3.) a+6c+cZ=3+2x3+5=3+6+5=14. 
 
 17. Instead of brackets, a line called a vinculum is some- 
 times used, and is drawn over quantities, which are taken col- 
 lectively. Thus, a—b—c is the same as a---(6— c). 
 
 The line which separates the numerator and denominator 
 of a fraction may be regarded as a sort of vinculum, cor- 
 
 16. When are Jr«<;^6^5 employed ? — 17. What is a vinculum? May tho 
 line which separates the numerator and denominator of a fraction be re- 
 garded as a vinculum ? 
 
 1* 
 
6 ALGEBRA. 
 
 responding, in fact, in Division to the bracket in Multiplic(u 
 tion. Thus, — implies that the whole quantity a+5— c 
 
 is to be divided by 5. 
 
 18. Like quantities are such k^ consist of the same letter^ or 
 the sa7ne combination of letters ; thus, 5a, and 7«, 4a6 and 
 9a6, 2bx^ and Qhx^, &;o., are called like quantities ; and utu 
 like quantities are such as consist of different Utters^ or of dif- 
 ferent combinations of letters ; thus, 4a, 36, lax^ bbz^^ <Sjc., 
 are unlike quantities. 
 
 19. Algebraic quantities have also different denominations 
 according to the number of terms (connected by the sign + 
 or — ) of which they consist: thus, 
 
 a, 26, 8a:r, &c., quantities consisting of one term, are called 
 simple quantities. 
 
 a+x^ a quantity consisting of two terms, is called a 
 binomial, 
 
 bx-^-y—z^ a quantity consisting of three terms, is called a 
 trinomiaL 
 
 CHAPTEE II. 
 
 ON THE ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVI- 
 SION, OF ALGEBRAIC QUANTITIES. 
 
 ADDITION. 
 
 20. Addition consists in collecting quantities that are like 
 into one sum, and connecting by means of their proper signs 
 those that are unlike. From the division of algebraic quan- 
 tities into 2^<^sitive and negative^ like and unlike^ there arise 
 three cases of Addition. 
 
 Case I. 
 To add like quantities with like signs. 
 
 21. In this case, the rule is " To add the coefficients of the 
 several quantities together, and to the result annex the com- 
 
 18. What are like and what are unlike quantities ? — 19. What is a simph 
 quantity ? What is a binomial and what a trinomial ? — 20. In what doea 
 addition of algebra consist ? Into how many cases is it divided ? 
 
ADDITION. 7 
 
 mon sign, and the common letter or letters ;" for it is evident 
 from the common principles of Arithmetic, if +2^, 4-3a, and 
 -f-5a, be added together, their sum must be 4-10«; and if 
 — -36\ — 45\ and —Sb\ be added together, their sum must 
 be -1561 . 
 
 Ex. 1. 
 
 Ex.2. 
 
 Ex.3. 
 
 2x+ 3a — 
 
 46 
 
 7x'-{- Sx2/- 
 
 56c 
 
 4a'- 
 
 3a'+ 1 
 
 Sx+ 2a- 
 
 56 
 
 9^^+ 2:ry— 
 
 76c 
 
 2a'— 
 
 a^+17 
 
 4:r+ 8a- 
 
 7b 
 
 ll;r^-f 5.ry- 
 
 46c 
 
 5a'- 
 
 2a^+ 4 
 
 9x-\- 4a— 
 
 66 
 
 a;'+ 4a:y— 
 
 6c 
 
 Sa'- 
 
 7a^+ 3 
 
 5^4- 7a-_ 
 
 96 
 
 x'-h 9X7/- 
 
 26c 
 
 a^- 
 
 a^+10 
 
 23:r+24a— 316 29^'+23^y— 196c 15a^— 14a^+35 
 
 Ex.4, 
 
 3;r»+4;c'^— X- 
 2x'+ x'—Sx 
 7x'-\-2x''—2x 
 Ax'+x''— X 
 
 Ex, 5. 
 
 7a^-3a^6+2a6^-36' 
 
 4a«- a^6+ ab'- 6* 
 
 a^-2a^6-f3a6^— 56» 
 
 5a8~3(f&+4a6^-26« 
 
 Ex. 6. 
 
 2x^7/— Sx+ 2 
 4x^7/— 2x+ 1 
 3a:V— 5.r4-10 
 
 - In these Examples it may be observed that some of the 
 quantities have 7io coefficient In this case, unity or I t* 
 always understood. Thus, in adding up the first column of 
 Ex. 2, we say, 1 + 1 + 11+9+7=29; in the third, 2+1+4 
 +7+5=19; and so of the rest. 
 
 Case IL 
 
 To add like quantities with unlike signs^ 
 
 22. Since the compound quantity a+6— c+cf— c, &c,, is 
 positive or negative, according as the sum of the positive 
 terms is greater or less than the sum of the negative ones, the 
 aggregate or sum of the quantities 2a — 4a+7a — 3a will be 
 +2«, and of the quantities 76*— 56*+26^— 86* will be —46*; 
 for ill the former case the excess of the sum of the positive 
 
 22. State the rule in the Ist ease. 
 
8 
 
 ALGEBRA. 
 
 terms above the negative ones is 2a, and in the latter 4b\ 
 Hence this general rule for the addition of like quantities with 
 unlike signs, "Collect the coefficients of the positive terms into 
 one sum, and also of the negative ; subtract the lesser of these 
 sums from the greater ; to this difference^ annex the sign of 
 the greater together with the common letter or letters, and the 
 result will be the sum required." 
 
 If the aggregate of the positive terms be equal to that of 
 the negative ones, then this difference is equal' to ; and con- 
 sequently the sum of the quantities will be equal to 0, as in 
 the second column of Ex. 2, following. 
 
 Ex. 1. 
 
 E^^2. 
 
 Ex. 3. 
 
 4:X'-^X+ 4 
 
 — 7a5+35c-- xy 
 
 ~5^^+13a:« 
 
 -2a;^+ X- 5 • 
 
 — . ah+2hc+4:xy 
 
 . -'2x'- 4x^ 
 
 Zx''-^x+ 1 
 
 Sab— hc+2xy 
 
 7x'+ x' 
 
 7a^+2a?- 4 
 
 —^ah+Uc-Sxy 
 
 9x^-Ux' 
 
 - rr^-4a;+13 
 
 5ab — Sbc+ xy 
 
 -J3a:^~2a;« 
 
 lla;^-9^4- 9 
 
 -2ab 
 
 +Sxy 
 
 — 4x^-'Qx'' 
 
 Ex. 4. 
 
 4ir^— 2x+Sy 
 
 - x^+ 4:X— y 
 
 7x^-' x-{-9y 
 
 9x^+2lx-2y 
 
 Ex.5. 
 
 5a^-2a5+ b^ 
 
 -a3_|_ ab-2b' 
 
 4a'-Sab+ b' 
 
 2a^+4:ab—4:b^ 
 
 )L'' 
 
 Ex. 6. 
 
 ^4xy+2xyS 
 
 — x^— xy—1 
 
 Sxy+4xy-5 
 
 '-9xY—2xy-\-9 
 
 Case III. 
 
 23. There now only remains the case where unlike quan- 
 tities are to be added together, which must^e done by col- 
 lecting them together into one line, and annexing their proper 
 signs ; thus, the sum of 3a?,— 2a,+5J,— 4y, is 3;r— 2a+5S— 
 Ay ; except when like and unlike quantities are mixed to- 
 
 23. State the rule in the 2d and 8d cases. 
 
SIMPLE EQUATIONS. 
 
 9 
 
 gether, as in the following examples, where the expressions 
 may be simplified, by collecting together suclf quantities as 
 will coalesce into one sum. 
 
 Ex. 1. 
 
 Sah + X — y Collecting together like quan- 
 
 4c — 2y + a; titles, and beginning with 3a6, 
 
 6ab — 3c + c? we have Sab -f bab = Sab ; + x 
 
 4y + o;^ — 2y + a; = + 2a; ; — y — 2y + 4y 
 
 — 2y = — • y; 4c — 3c = + c ; 
 
 besides which there are the two 
 quantities + d and + x^, which do not coalesce with any of 
 the others ; the sum recpiired therefore is Sab '\-,2x — y 
 + c + d-\'x\ 
 
 8a6 + 2a; — y + c + (^ + a;^ 
 
 Ex.2. 
 
 ^x^—2xy+l — 3y+4a;» 
 4y -\Sx^^y'^+xy— x^ 
 5a;^ — 2 a; +y —15+ y' 
 Zx'—xy — 14+2y + 12a;^— 2.r. 
 
 Here 4a;'— a;^=3a;' 
 — 2a;y+a;i/=— a;y 
 + 1_.15=_14 
 
 -3?/+4y+y=+2y 
 +4a;^+3a;^+5a;^= + 12a?* 
 
 -2a;=-2a;. 
 
 SIMPLB EQUATIONS. 
 
 24. When two algebraic quantities are connected together 
 by the sign of equality (=), the expression is called an 
 equation. Equations, in their application to the solution of 
 problems, consist of quantities, some of which are known and 
 others unknown. Thus 2a;+3=:a;+7 is an equation in which 
 X is an unknown quaiitity^ and its value is such a number as 
 wil] make 2^+3 and x-\-l equal to each other. The number 
 which here satisfies the equation, or is the value of a?, is man- 
 ifestly 4, since 2x4+3=11, and 4+7=11. The value of 
 the unknown quantity, which has in this example been found 
 by inspection, is usually obtained by a direct calculation 
 which is call'ed the solution of the Equation. 
 
 24. What is an equation? What is meant by the solution of an equation*' 
 
10 ALGEBKA. 
 
 25. In effecting the solution, the several steps of the pro- 
 cess must be conducted by means of the following axioms, 
 and in strict accordance with them : — 
 
 (1.) 'Things which are equal to the same thing are equal to 
 one another. ^ 
 
 (2.) If equals be added to the same or to equals, the sums 
 will be equal. 
 
 (3.) If equals be subtracted from the same or from equals, 
 the remainders will be equal. 
 
 (4.) If equals be multiplied by the same or by equals, the 
 products will be equal. 
 
 (5.) If equals be divided by the same or by equals, the 
 quotients will be equal. 
 
 (6.) If equals be raised to the same power, the powers will 
 be equal. 
 
 (7.) If the same roots of equals be extracted, the roots 
 will be equal. 
 
 These axioms, exclusive of the first, may be generalized^ 
 and all included in one very important principle, which should 
 in every investigation in which equations are concerned be 
 carefully borne in mind ; viz., that whatever is done to one side 
 of an equation the same thing must be done to the other side, 
 in order to keep up the equality, 
 
 26. If an equation contain no power of the unknown quan- 
 tities higher than the Jirst, or those quantities in their simplest 
 form, it is called a Simple liquation. 
 
 ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONLY ONE 
 UNKNOWN QUANTITY. 
 
 27. The rules which are absolutely necessary for the solu- 
 tion of simple Equations, containing only one unknown quan. 
 tity, may be reduced to four, each of which will in its proper 
 place be formally enunciated and exemplified. 
 
 25. What are the axioms employed in the solution of equations, and 
 state the general principle which is based upon them ? — 26. What is a sim- 
 ple equation ? 
 
SIMPLE EQUATIONS. U 
 
 Rule I. 
 
 " If the unknown quantity has a cbefRcient, then its value 
 may be found by dividing each side of the equation by that 
 coefficient ;" and the foui^ation of the Eule is, that " If 
 equals be divided by the same, the quotients arising will be 
 equal." 
 
 Ex.1. Let 2:r=14; then dividing both sides of the 
 
 2x 14 2x 14 
 
 equation by 2, we have --=— ; but 77=^, and 77-= 7, 
 
 -r^ ^ X T . , ax h-\-c , ax 
 
 Ex.2. Let ax=zb+ci then — = ; but — =a; ; .'. « 
 
 a a ^ a ' 
 
 b-i-c 
 *" a 
 Ex. 3. Let x+2x+4x+ex=62. 
 Collecting the terms, 13a; =52. 
 Dividing both sides of the equation by 13, 
 ar=4. 
 
 Ex. 4. Let 6x—4:X+Sx—x=S6, 
 The terms being added as in Case 2d of Addition, 
 4ar=36. 
 Dividing each side of the equation by 4, 
 x=9. 
 
 Ex. 5. 10^=150. Ans. a?=15. 
 
 Ex. 6. Sx+4x+7x=zS4:. Ans. x— 6. 
 
 Ex. 7. 8a;— 5a;-|-4a;— 2a;=25. Ans. a?= 5. 
 
 Ex. 8. \2x—Zx'-4:X—x=z2A. Ans. x= 6. 
 
 28. Arithmetical questions may with great ease Be ex- 
 hibited under the form of an equation, and it will be seen hy 
 the subjoined examples in what relation arithmetical and alge^ 
 braic operations stand to each other : as for instanccj 
 
 If 3O5. be given for 5 lbs. of tea^ what is the price of 1 lb.? 
 
 (1.) Tho, price of} lb. is that which is to be found.. 
 
12 ALGEBRA. 
 
 (2.) Then it is clear that the price o/*l lb. X5 must give the 
 price of 5 lbs. 
 
 (3.) But the cost of 5 lbs. by the question=305. 
 
 (4.) Therefore, the price of 1 lb. in shillings X 5=305. 
 
 (5.) And therefore by dividing by 5, we obtain the price 
 ofllb.=:65. 
 
 The several steps of this solution expressed algebraically 
 would take the following more compendious form : 
 
 (1.) Let ir=the price of 1 IJb. in shillings. 
 
 (2.) Then 5a;=the price of 1 lb. in shillings X 5. 
 
 (3.) But the cost of 5 lbs. is by the question=305. 
 
 (4.) .•.5a;=:305. 
 
 (5.) and /. x=Qs. which is the price of 1 lb., as was re- 
 quired. 
 
 It will be seen by steps (2) and (3) of this example, that 
 there are two distinct expressions for the same thing, and 
 that in step (4) these expressions are made equal to each 
 other. In framing equations from problems, this will in all 
 cases take place. As a second example let this problem be 
 taken: — 
 
 A house and an orchard are let for £28 a year, but the rent 
 of the house is 6 times that of the orchard. Find the rent of 
 each. 
 
 The rent of the house is equal to that of G orchards; we 
 may therefore change the house into 6 orchards, and we shall 
 have 
 
 Bent of the orchard+6 times rent of the orchard=£2S, 
 Taking the sum of the rents of the orchard, we get 
 
 7 times the rent of the orchard z=z £28, 
 ftnd the 7th part of each side of the equation being taken, 
 
 The rent of the orchard=£A ; 
 .and .•. the rent of the house=6 times th6 rent of the orchard 
 
 =6 times £4 
 ^ =£24. _ . . • 
 
 Now to give to these operations an algebraical shape, 
 Let x=:the rent of the orchard in £ 
 then 6x= " " house " 
 
SIMPLE EQUATIONS. 13 
 
 But by the condition of the question, 
 
 Rent of the orchard-{-6 times rent of the orchard=£2S, 
 
 or, lfx=£2S. 
 and, dividing each side of t?^ equation by 7, 
 a: =£4, the rent of the orchard, 
 and .-. 6x=6x£^=£'24, the rent of the house. 
 
 Again, suppose the following arithmetical question was 
 proposed for solution ; viz., " To divide the number 35 into 
 two such parts, that one part shall exceed the other part 
 by 9." A person unacquainted with algebra might with 
 no great difficulty solve this question m the following 
 manner : — 
 
 (1.) It appears, in the first place, that there must be a 
 greater and a less part. 
 
 (2.) The greater part must exceed the less by 9. 
 
 (3.) But it is evident that the greater and less parts added 
 together must be equal to the whole number 35. 
 
 (4.) If then we substitute for the greater part its equivalent, 
 viz., " the less part increased hy 9''," i^ follows, that the less 
 part increased by 9, with the addition of the said less part is 
 equal to 35. 
 
 (5.) Or, in other words, that twice the less part with the 
 addition of 9, is equal to 35. 
 
 (6.) Therefore, twice the less part must be equal to 35, 
 with 9 subtracted from it. 
 
 (7.) Hence, twice the less part is equal to 26. 
 
 (8.) From whidh we conclude, that the less part is equal to 
 26 divided hy 2; i. e., to 13. 
 
 (9.) And consequently, as the greater part exceeds the less 
 Dy 9, it must be equal to 22. 
 
 But by adopting the method of sUgebraiG notatio7i, the differ- 
 ent steps of this solution may be much more briefly expressed as 
 follows : — 
 
 (1.) Let the less part - - - - - z=:x. 
 
 (2.) Then the greater part ==x-\-9, 
 
 (3.) But greater part + less part - - - =35. 
 2 
 
14 ALGEBRA. 
 
 (4.) .-. x-h9+x ... ... =35. 
 
 (5.) or 2a;+9 - =35. 
 
 (6.) .\2x =35-9. 
 
 (7.) or 2a; - - - =26. 
 
 ^ 26 
 
 (8.) ,' . X (less -psLTt) =—=13. 
 
 {9.) SLXid x+9 {greaier ^SLTt) .... =13+9=22. 
 
 29. Having thus explained the manner in which the several 
 steps in the solution of an arithmetical question may be ex- 
 pressed in the language of Algebra, we now oroceed to its ex- 
 emplification. 
 
 PROBLEMS. 
 
 Prob. 1. A dessert basket contains 30 apples and pears, 
 but 4 times as many pears as apples. How many are there 
 of each sort ? 
 
 Let a;=the number of apples; 
 then, as there are 4 times as many pears as apples, 
 4x= the number of pears. 
 But by the question the apples and pears together =30. 
 
 .-. a;+4;r=30. 
 Adding the terms contaming x, 
 
 5a; =30. 
 Dividing each side of this equation by 5, 
 
 ir=6, the number of apples, 
 /. the number of pears =4a: =4x6 =24. 
 
 Prob. 2. In a mixture of 16 lbs. of bla«k and green tea, 
 there was 3 times as much black as green. Find the quantity 
 of each sort. 
 
 Let a;=the number of lbs. of green tea, 
 then3a;= " " " black. 
 
 But the black tea -f- the green tea=16 lbs. 
 
 /.a;+3a; = 161bs. 
 Collecting the terms which contain .r, 
 4a;=161bs. 
 Dividing each side of this equation by 4, 
 
 ^ a? =4 lbs. of green tea, 
 
 .-. the black tea -3ar=3x4=121bs. 
 
SIMPLE EQUATIONS. 15 
 
 Prob. 3. An equal mixture of black tea at 5 shillings a lb. 
 and of green at 7 shillings a lb. costs 4 guineas. How many 
 lbs. were there of each sort ? 
 
 Let a;=the number of lbs. of each sort; 
 then 5x=zthe cost of tke black in shillings, 
 and 7x=z " " green " 
 But cost of black + the cost of green =4 guineas =845. 
 
 /. bx-\-l[X=zS4: 
 
 12x=zS4 
 
 .*. x=7 lbs. of each sort. 
 
 Prob. 4. The area-of the rectangular floor of a school-room 
 is 180 square yards, and its breadth 9 yards. What is its 
 length ? 
 
 Let a: = the length in yards; then since the area is the 
 length multiplied by the breadth, we have 
 
 irX9=:the area of the floor, 
 .\9a;=:180 
 and .'. a; =20 yards, the length required. 
 
 Prob. 5» Divide a rod 15 feet long into two parts, so that 
 the one part may be 4 times the length of the other. 
 
 Let rr=the less part, i i ^^^- i 
 
 then 4^= the greater part, x 4x 
 
 Now these two parts make together 15 ft. 
 .\x-^4:Xz=16fL 
 5;r = 15ft. 
 /,x=z 3 ft. the less part, 
 and the greater part =4 times 3 ft. =: 12 ft. 
 
 Prob. 6. A horse and a saddle were bought for £40, but 
 the horse cost 9 times as much as the saddle. What was the 
 price of each ? Ans. £36 and £4. 
 
 Prob. 7. Divide two dozen marbles between Richard and 
 Andrew, so that Richard may have three times as many as 
 Andrew 1 Ans, Richard's share = 18. 
 
 Ans, Andrew's share = 6. 
 
 Prob. 8. A boy being asked how many marbles he had, 
 - f ^ fl.idj If I had twice as many more, I should have 36^ How 
 many had he 1 Ans. 12. 
 
 Prob. 9. A bookseller sold 10 books at a certain price, and 
 
16 ' ALGEBRA. 
 
 afterwards 15 more at the same rate,- and at the latter time 
 received 355. more than at the former : what was the price 
 per book ? 
 
 Let rr=the price of a book in shillings ; 
 then 10a:=:the price of thp 1st lot in shillings, 
 and I5x= " " ^ 2d " 
 
 Now, if the price of the 1st lot be taken from that of the 
 2d, there remains a difference in price of 355. 
 ,\15x—l0xz=S5s, ^ 
 
 Subtracting the 10a; from the 15:r, we have ' 
 5a; =355. 
 
 .-. x=7s. the price of a book. 
 • Prob. 10. Divide £300 amongst A, B, and C, so that A 
 may receive twice as much as B, and C as much as A and B 
 together. 
 
 Let a;=B's share in £ 
 then 2a;=z:A's share in £ 
 and x-\-2x or 3j;=rC's share in £. 
 But amongst them they receive £300 ; 
 .\ x-{-2x-\-Sx=£S00, 
 6a:=£300 ; 
 .-. a; =£50 B's share ; 
 .-. A's shaT-e=£100, and C's share=£150. 
 Prob. 11. If to nine times a certain number, three times 
 the numl^er be added, and four times the same number be 
 taken away, there will then be obtained the number 48. 
 What is the number ? 
 
 Let a; = the number; 
 then 9a;=nine times the number, 
 
 3a: = three times the number, 
 and 4a; = four times the number; 
 
 .-. 9a;+3a;—4a;=48, 
 ' • . 8a;=48; 
 
 .'. x=6, the number required. 
 
 Prob. 12. The sum of £100 is to be divided among 2 men, 
 3 women, and 4 boys, so that each man shall have twice as 
 much as each woman, and each woman three times as much 
 as each boy. Find the share of each. 
 • Let each boy's shares a;; 
 then each woman's share =r 3a;, 
 and each man's share =2 times 3a;=:6a;; 
 
SIMPLE EQUATIONS. • 17 
 
 Hence we have, 
 
 the share of the 4 boys =4rr, 
 the share of the 3 women =3 times Sxz=z9x, 
 and the share of the 2 menr=:2 tunes 6^ = 12^ : 
 But the sum of all these shares is to amount to £100 ; 
 /.4.^+9a;+^2a;==£100, 
 25:r=£100 ; 
 
 ,\ each boy's share =£4 ; each woman's =3 times £4 
 p ^ A =£12 ; and each man's=6 times £4=:£24. 
 
 Prob. 13. a gentleman meeting 4 poor persons gave five 
 shillings amongst them ; to the second he gave twice, to the 
 third thrice, and to the fourth four times as much as to the 
 first. What did he give to each ] 
 
 Ans, 6d., 12c/., 18d, 24c?., respectively. 
 
 Prob. 14. Divide a line 12 ft. long into three parts, such 
 that the middle one shall be double the least, and the greatest y/L 
 triple the least part. Ans, 2, 4, 6. 
 
 Prob. 15. Divide 40 into three such parts, that the first 
 shall be 5 times the second, and the third equal to the differ- 
 enee between the first and second. A7is. 20, 4, 16. 
 
 Prob. 16. A grocer mixed three kinds of tea, Bohea at 35. 
 per lb., Twankay at 5^., and Souchong at 7^. per lb. The 
 mixture contains the same quantity of each, and Cost £6. 
 How many lbs. are there of each kind 1 Ans. 8 lbs. 
 
 Prob. 17. A bill of £700 was paid in sovereigns, half- 
 sovereigns, and crowns, and an equal number of each was used. 
 Find the number. Ans, 400. 
 
 Prob. 18. Two travellers set out at the same time from 
 Guildford and London, a distance of 27 miles apart ; the one 
 walks 4 miles an hour, and the other 5 miles. In how many 
 hours will they meet ? Ans. S hours. 
 
 Prob. 19. A person bought a horse, chaise, and harness, 
 for £120 ; the price of the horse was twice the price of J;he 
 harness, and the price of the chaise twice the price of both 
 horse and harness ; what was the price of each ? 
 
 ( Price of harness =£13 6 8 
 Answer, \ " " horse z= 26 13 4 
 ( " " chaise = 80 
 2* 
 
18 ALGEBRA. 
 
 SUBTRACTION. 
 
 30. Subtraction is the finding the difference between two 
 algebraic quantities, and the connecting them by proper signs, 
 so as to form one expression^* thus, if it were required to 
 subtract 5—2 (^^ e., 3) from 9, it is evident that the remainder 
 would be greater by 2 than if 5 only were subtracted. For 
 the same reason, if 6— c were subtracted from a, the remain- 
 der would be greater by c, than if h only were subtracted. 
 Now, if b is subtracted from a, the remainder is a — 5 ; and 
 consequently, if 5— c be subtracted from a, the remainder 
 will be a—b-\-c. Hence this general Rule for the subtraction 
 of algebraic quantities, " Change the signs of the quantities to 
 he subtracted^ and then place them one after another, as in 
 Addition." 
 
 Ex. 1. From 5cH-3a;— 25 take 2c --4y. The quantity to 
 be subtracted w;i7A its signs changed, is-— 2c -{-Ag ; therefore 
 the remainder is 5a+3a;— 25— 2c+4y. 
 
 Ex.2. From 7x^-^2x +5 tsike Sx'^+^x-^l -, 
 The remainder is 7x^ — 2a; + 5 —Sx^—5x-{-l] 
 
 or 7x^-2x'^2x -^x +5 +1=4^;'^- 7a; +6. 
 
 But when like quantities are to be subtracted from each 
 other, as in Ex. 2, the better way is to set one row under the 
 other, and apply the following Rule ; " Conceive the signs of 
 the quantities to be subtracted to be changed, and then proceed 
 as in Addition." 
 
 Ex. 3. Ex. 4. I Ex. 5. , 
 
 YTom7x'-2x+5 l2a^-~Sa+ b-l 5y^-4y+3a 
 
 Subtract Sx'+6x-^l 6a'-\- a-2b+S 6/-4 y- a 
 
 Remainder 4:x^—lix-{-6 6a^— 4a+35— 4 —y^ * +4a 
 
 *Ex. 6. Ex. 7. Ex. 8. 
 
 From 7xg+2x-'Sg Ux+g—z— 5 lSx^-2x''^7 
 Subtract 3a:?/— x-\- y x-^y+z—\\ — aj^-f- x^—Q 
 
 80. What is suUraction ? State the rule for the subtraction of algebraie 
 quantities, and explain the principle on which it rests. 
 
SIMPLE EQUATIONS. 19 
 
 OP THE SOLUTION OF SIMPLE EQUATIONS, CONTAININ(i ONLY ONE 
 UNKNOWN QUANTITY. 
 
 ]^LE 11. 
 
 81. "Any quantity may be transferred from one side of 
 the equation to the other, by changing its sign ;" and it is 
 founded upon the axiom, that "If equals be added to or 
 subtracted from equals, the sums or remainders will be 
 equal." 
 
 Ex. 1. Let a;+8 = 15; subtract 8 from each side of the 
 equation, and it becomes ir-[-8— 8 = 15— 8; but 8— 8=0, 
 .•.a:=15-8=7. 
 
 Ex. 2. Let a:— 7=20; add 7 to each side of the equation, 
 thena:-7+7=20+7; but -7+7=0; .-. a:=20+7=27. 
 
 Ex.3. Let 3a;— 5=2a;+9; add 5 to each side of the 
 equation, and it becomes 3a;— 5 + 5=2a;+9 + 5, or 3a;=2a;+ 
 9+5. Subtract 2x from each side of this latter equation, 
 then Sx—2x=:2x—2x-{-9-^5', but 2x—2x=z0, .'.Sx—2x=:9 
 + 5. Now 3a;— 2a:=a;, and 9 + 5 = 14; hence a; = 14. 
 
 On reviewing the steps of these examples, it appears 
 
 (1.) Thatir+8=15 is identical with a;=15— 8. 
 
 (2.) " a;-7=20 " with a;=20+7. 
 
 (3.) " 3a;-5=2a;+9 " with 3a; -2a; =9 +5. 
 
 Or, that " the equality of the quantities on each side of the 
 equation, is not affected by removing a quantity from one side 
 of the equation to the other and changing its sign;"* 
 
 From this rule also it appears, if the same quantity with 
 the same sign be found on'hoth sides of an equation, it may 
 be left out of the equation; thus, ifa;+a=c+a, then a;=c+ 
 a — a ; but a — a=0, .*. a;=c. 
 
 It further appears, that the signs of all the terms of an equa- 
 tion may be changed from + to — , or from — to +, without 
 altering the value of the unknown quantity. For let a;— 6=c 
 — ^ • then bv the Rule. a;=c— a+6; chano-e the si"-ns o^ all 
 
20 ALGEBRA. 
 
 the terms, then 5— irs=:a— c, in which case b—a-{-c=x, or x^=. 
 c— a+6, as before. 
 
 Ex.4. 2a;+3=a;+n. Ans, x=U, 
 
 Ex. 5. 5x—4:z=4tx+26. Ans, x=29. 
 
 Ex. 6. 7^— 9=r6a;— 3. C Ans, x^(S, 
 
 Ex. 7. 4:X+2a=Zx+^h, Ans. ir=96— 2a. 
 
 Ex.8. 15:r+4=34. Ans. x=2. 
 
 Ex. 9. 8^+7=6^+27. -4^5. a;=10. 
 
 Ex. 10. 9^—3=4^+22. Ans. x—b. 
 
 Ex. 11. 17a;— 4a;+9=:3ar+39. Ans. x=^. 
 
 Ex. 12. ax—cz=zh+2c, Ans. x= . 
 
 a 
 
 Ex. 13. 5;r-(4a;— 6) = 12. 
 
 The sign — before a bracket being the sign of the whole 
 quantity enclosed, indicates that the quantity is to he sub- 
 tracted ; and therefore, according to the Rule, when the brack- 
 ets are removed the sign of each term must be changed. 
 Thus, the signs of 4a; and of 6 are respectively + and — , but 
 when the brackets are removed they must be changed to — 
 and + respectively. The equation then becomes 
 5a;-4a;+6 = 12. 
 By transposition, 5a; — 4a; = 1 2 — 6 ; 
 
 .•.a;=6. 
 Ex. 14. 6a;-(8+a;)=4a;-(a?-10). 
 
 By removing the brackets, and changing the signs of the 
 terms w^hich they enclose, the equation becomes 
 6a;— 8— a;=4=4.r— OJ+IO. 
 Transposing, 6a; — a; — 4a; + a; = 1 + 8 ; 
 
 .•.2a;=18. 
 Dividing both sides of the equation by 2, 
 
 a;=9. 
 Ex.15. 4a;-(3a;+4)=8. Ans. x=z\2. 
 
 Ex. 16. 8a;— (6a;— 8)=9— (3— a;). ^^5. a; =—2. 
 
 e Ex. 17. 4a;-(3a;-6)-(4a;-12) = 12-(5a;-lG). 
 
 Ans. x=2. 
 Ex. 18. 5a;— (3-4-aai)=8 — (— a;— 1). Ans. a;=12. . 
 
SIMPLE EQUATIONS. 21 
 
 PROBLEMS. 
 
 Prob. 1. There are two numbers whose difference is 15 and 
 their sum 59. What are the numbers ? 
 
 As their difference is 15, itjs evident that the greater num- 
 ber must exceed the less by 15. 
 
 Let therefore ir=: the less number; 
 then will a; +15=: the greater : 
 But their sum=:59 ; 
 .-. a;+a;+15=59, 
 or2a;+15=59. 
 And, transposing 15, 2a; =59 — 15, 
 or 2a; =44; 
 .•. a:=22 the less number, 
 and a;+15=22+15=37 the greater. 
 
 Prob. 2. I gave to Richard and James 27 marbles, l&ut to 
 Eichard 5 more than to James. How many did I give to 
 each? 
 
 Let ir=the number I gave to James ; 
 thena;+5= " " " " Richard: 
 
 But together they receive 27 ; 
 .•.a;+a;+5=27, 
 or2a;+5=27. 
 Transposing, 2a; =27— 5, 
 
 or 2a; =22; 
 .*. a;=ll, the No. James received, 
 anda;+5=16 " " Richard received. 
 
 Prob. 3. Four times a number is equal to double the num- 
 ber increased by 12. What is the number? 
 Let a;=the number ; 
 then 4a; =4 times the number, 
 2a; = double tlie number, 
 and 2a;+12=double the number increased hy 12. 
 Therefore, by the equality stated in the question, 
 
 4a;=2a;+12. 
 
 By transposition, 4a;— 2a; =12. 
 
 2a; = 12; 
 
 ,\xt=l 6. 
 
 Prob. 4. At an election 420 persons voted,''and the success- 
 
22 ALGEBRA. 
 
 ful candidate had a majoi^ity of 46. How many voted for 
 each candidate ? Aris, 187 and 233. 
 
 Prob. 5. One of two rods is 8 feet longer than the other, 
 but the longer rod is three times the length of the shorter. 
 What are their lengths ? | Ans. 4 ft. and 12 ft. 
 
 Prob. 6. Five times a number diminished by 16 Is equal 
 to three times the number. What is the number ? Ans. 8. 
 
 Prob. 7. A horse, a cow, and a sheep, were bought for 
 £24; the cow cost £4 more than the sheep, and the horse £10 
 more than the cow. What was the price of the sheep 1 
 Let a:=the price of the sheep in £ ; 
 then a;+4= " " cow " 
 
 and a;+4-f 10= " " horse " 
 
 But these three prices taken together amount to £24 ; 
 
 ... rrH-(a;+4) + (a;+4+10)=24. 
 Ad^ng together like terms, 
 
 3a;+18=z24. 
 By transposition, 3;r = 24 — 1 8, 
 Sx=6; 
 .'. x=z£2, the price of the sheep. 
 Prob. 8. A draper has three pieces of cloth, which togethei 
 measured 159 yards; the second piece was 15 yards longei 
 than the first, and the third 24 yards longer than the second. 
 What is the length of each piece ? 
 
 Ans. 35, 50, and 74 yds. 
 Prob. 9. Divide £36 among three persons. A, B, and C, in 
 such a manner that B shall have £4 more than A, and C £7 
 more than B. Ans. £7, £11, and £18. 
 
 Prob. 10. A gentleman buys 4 horses ; for the second he 
 gives £12 more than for the first ; for the third £5 more than 
 for the second ; and for the fourth £2 more than for the third. 
 The sum paid for all the horses Was £240. Find the price of 
 each. Ans. £48, £60, £65, and £67. 
 
 Prob. 11. What number is that whose double is as much 
 above 21 as it ie itself less than 21 ? 
 Let ir = the number ; 
 then 2.r=: double the immber, 
 2:r— 21i=:what double the number is above 21, 
 and 21— a: = what the number is less than 21 : 
 
SIMPLE EQUATIONS. 23 
 
 But bj the question these two values are equal to each 
 other ; /. 2^—21 r=21 — ar. 
 
 By transposition, 2a;+ic=21+21, 
 3a: =42; 
 .-..^ = 14. 
 The answer may easily be%)roved to be correct, for 2a;— 21 
 =28-21=7, and 21-a:rz:^l-14=7 ; that is, twice 14 is 
 as much above 21, as 21 is above 14, namely 7. 
 
 Prob. 12. In dividing a lot of oranges among a certain 
 number of boys I found that by giving 4 to each boy I had 
 6 to spare, and by giving 3 to each boy I had 12 remaining. 
 How many boys were there ? 
 
 Let ir=:the number of boys ; 
 then, if I gave to each boy 4 oranges, I should give away 4 
 times X oranges ; 
 
 .*. 4:r=number of oranges distributed at first ; 
 But the total number of oranges is 6 more than this num. 
 ber; 
 
 /, Total number oi OT2iT\gQs-=4,x-\-^: 
 Again, if each boy received 3 oranges, there were 12 
 oranges left ; 
 
 .'. Total number of 0V2ingQS=^x-\-\2, 
 These two values for the number of oranges expressed in 
 terms of a; must necessarily be equal ; Axiom (1.) / 
 
 .-.4^+6=3^+12. MJ/^t^) 
 
 By transposition, 4a; — 3a: = 1 2 — 6 ; ' ^""^^ 
 
 .*. a; =6, the number of boys. 
 
 Prob. 13. An express set out to travel 240 miles in 4 days, 
 but in consequence of the badness of the roads, he found that 
 he must go 5 miles the second day, 9 the third, and 14 the 
 fourth day less than the first. How many miles must he 
 travel each day ? 
 
 Let a;=:the number of miles on the 1st day ; 
 thena:-5= " " " 2d " 
 
 a:— 9= " " " 3d " 
 
 and a: -14= « " " 4th " 
 
 Now the number of miles which he travels in 
 =240. 
 
 ... a;+a;—5 + a:—9+a: — 14=240. 
 Collecting the terms, 4a:— 28=240. 
 
24 ALGEBRA. 
 
 By transposition, 4x = 240 -h 28, 
 4.r=268; 
 .*. re =67, the number of miles he goes on 1st day, 
 a:-5 = 62 " " " " " 2d " 
 
 a:-9=58 " " " " " 3d " 
 
 and :^-14r=:53 " " i " * 4th " 
 
 pROB. 14. It is required io divide the number 99 into five 
 such parts that the first may exceed the second by 3, be less 
 than the third by 10, greater than the fourth by 9, and less 
 than the fifth by 16. 
 
 Ans. The parts are 17, 14, 27, 8, 33. 
 
 Prob. 15. Tv70 merchants entered into a speculation, by 
 which A gained £54 more than B. The whole gain was £49 
 less than three times the gain of B. What were the gains ? ' 
 A71S. A's gain=r:£157 ; B's=£103 
 
 Prob. 16. In dividing a lot of apples among a certain num- 
 ber of boys, I ^und that by giving 6 to each I should have too 
 few by 8, but by giving 4 to each boy I should have 12 re- 
 maining. How many boys were there? Ans, 10. 
 
 MULTIPLICATION. 
 
 32. Multiplication is the finding the product of two or 
 more algebraic quantities ; and in performing the process, the 
 four following rules must be observed. 
 
 (1.) When quantities having like signs are multiplied to- 
 gether, the sign of the product will be + ; and if their signs 
 are unlike, the sign of the product will be — .* 
 
 * This rule for the multiplication '"of the Signs may be thus ex- 
 plained : — 
 
 I. If +a is to be multiplied by -|-6, it means, that -fa is to be added 
 to itself as often as there are units in b, and consequently the product 
 will be -f <*^- 
 
 II. If — a is to be multiplied by -\-b, it means, that — a is to be 
 added to itself as often as there are units in b, and therefore the product 
 is — ab. 
 
 82. What is multiplication, and what are the Kules to be observed m 
 multiplication ? 
 
MULTIPLICATION. 25 
 
 (2.) The ooefficients of ikiQ factors must be multiplied to- 
 gether, to form the coefficient of the product, 
 
 (3.) The letters of which they are composed must be set 
 l-'>wn, one after another, according to their order in the 
 Alphabet, ^ 
 
 (4.) If the same letter is found in both factors, the indict 
 of it must be added together, to form the index of it in the 
 product. 
 
 Thus, +a multiplied by +6 is equal to +ab^ and —a mul- 
 tiplied by —b is also equal to -\-abi +3a;X —% = — 15a;y ; 
 -SabX+4:cd=-12abcd ; -4:a'b^X -Sabd'=-hl2a'b'd* ; 
 &c., &c. 
 
 From the division of algebraic quantities into simple and 
 compound, there arise three cases of Multiplication. In per- 
 forming the operation, the Rule is, "To multiply Jlrst the 
 signs, then the coefficients, and afterwards the letters." 
 
 Case I. 
 
 33. When both factors are simple quantities ; for which the 
 Eule has been already given. 
 
 III. If + a is to be multiplied by — 6, it meana, that +« is to be 
 subtracted as often as there are units in b, and consequently the product 
 is — ab. 
 
 IV. If — a is to be multiplied by — b, it means, that — a is to be 
 subtracted as often as there are units in b ; and, since to subtract <i 
 negative quantity is the same as to add a positive one, the product will 
 be -f- <^b. 
 
 Or, these four Rules might be all comprehended in one ; thus, 
 
 To multiply a — 6 by c — c?, is to add a — 6 to itself as often as there 
 are units in c — d] now this is done by adding it c times, and subtract- 
 ing it d times ; 
 
 But a — b, added c times . . . =ac — be, 
 and a — b, subtracted d times = — «c? -f- bd, 
 
 ,'. a — b X c — d =ac — be — ad-i-bd. 
 
 i. e., -{- a X -{- c =:^ -\- ac 
 — b X +c = — bc 
 -{-aX — d = — ad 
 ^bX — d = -\-bd, 
 
 3 
 
26 
 
 ALGEBRA. 
 
 Ex. 1. 
 
 4ab 
 Sa 
 
 Ex. 2. 
 
 2axy 
 ~3y 
 
 Ex. a. 
 
 — Sahc 
 ba% 
 
 Ex. 4. 
 
 — ^a^bc 
 
 - 2h''x' 
 
 12a^b 
 
 —6axy' 
 
 Ex.6. 
 
 ~2y 
 
 -^l^a'b'c 
 
 + 10a%'cx^ 
 
 Ex.5. 
 
 4abc 
 Sac 
 
 Ex. 7. 
 
 — 4cdx 
 
 2c 
 
 Ex.8. 
 '-2ac^x 
 
 Case II. 
 
 84. When one factor is compound and the other simple, 
 " Then each term of the compound factor must be multiplied 
 by the simple factor as in the last Case, and the result will 
 be the product required." 
 
 Ex.1. 
 
 Multiply Sab—2ac-\-d 
 by 4a 
 
 Ex.2. 
 
 Sx^ 2x^+A 
 — 14aa; 
 
 Product \2a%^Sa''c+^ad 
 
 —A2ax^-\-2Sax^—bQa^ 
 
 Ex. 3. 
 
 Multiply Ix"" —2a: +4a 
 by — Sa 
 
 Ex.4. 
 
 12a3-2a2+4a-l 
 Sx 
 
 Product — 21aa;^ + 6aa;— 12a^ 
 
 
 Ex. 5. 
 
 Multiply ^a'^x+Sa—x+l 
 by — x^ 
 
 Product 
 
 Ex. 6. 
 
 Ax^y+Sx—2y 
 Sxy 
 
 Case IH. 
 
 35. When both factors are compound quantities, each term 
 of the multiplicand naust be multiplied by each term of the 
 
\ 
 
 MTJXTIPLICATION. 27 
 
 multiplier ; and then placing like quantities under each othet 
 the sum of all the terms will be the product required. 
 
 Ex. 1. Ex. 2. Ex. 3. 
 
 Multiply a + b (P+ b a^+ab+b^ 
 
 hy a -{• b a — b a — b 
 
 1st, by a a^+ ab a^+ab a^+a^b-{-ab^ 
 
 2d,'byi ab-hP -ab^b'' ^a ^h-ab''-^b^ 
 
 Product a'^+2a^>+62 a^ * -^^ "Z~^ *~~^» 
 
 Ex. 4. . Ex. 5. 
 
 3^*+ ^x 3a:2- 2a; +5 
 
 4.^+7 6a; - 7 
 
 12a:^+ 8a;« 18a;^-12a;^+30a: 
 4-21a;^+14a; ~21a;'+14a;--35 
 
 12^^+29^+14^ 18a:«-33a:2+44a:-35 
 
 Ex. 6. 
 
 14a c — 3a 5 + 2 
 ac — a 6 + 1 
 
 14a V- 3a^6c+~2^ 
 
 -14a^6c +3a'»62-2aJ 
 
 +14ac — 3a6+2 
 
 14aV-17a'^6c+16ac+3a^62-5a6+2 
 
 Ex. 7. 
 
 Ex. 8. Multiply a3+3a^6+3a5^+62 by a+ft. 
 
 ^n5. a*+4a^5+6a262+4a6M i^*- 
 
28 ALGEBRA. 
 
 Ex. 9. Multiply 4a:V+3^y— 1 - - by ^x^—x, 
 
 Ans, ^x^'y+^x^y-^^x^-'Zx^y+x. 
 
 Ex. 10. " (x^-^x^+x-^ by ^x'^+x+l. 
 
 Arts. 2^'— icHS.'i;^— lOo;^— 4a;— 5. 
 
 Ex. 11. « 3a«+2a6-52 ^^ ^a?^'^ah+h\ 
 
 Ans. 9a*-4a262+4a63-6^ 
 
 Ex. 12. " oi^+x^y+xy^+y^ by x^y, 
 
 Ans, a;*— 2/\ 
 
 Ex. 13. « x^-'lx+l - - - . by x'^^lx. 
 
 /Ans. x^—\x^+^-^x^--\x. 
 
 OH THE SOLUTION OP SIMPLE EQUATIONS CONTAINING ONLY ONE 
 UNKNOWN QUANTITY. 
 
 Ex. 1. 3a;+4(a;+2)=36. 
 
 The term 4 (a; +2) means, that a: +2 is to be multiplied by 
 4, and the product by Case 2d is 4a;+8 ; 
 .-. 3a:+4a?+8=36. 
 Adding together the terms containing ar, and,transposing 8, 
 7ir=36-8, 
 7a;=:28; 
 /, ii;=4. 
 
 Ex.2. 8(a?+5)+4(a;+l)=8a 
 Performing the multiplication, 
 
 8a;+40+4a;+4=80. 
 Collecting the terms, 12a; +44 =80. 
 Transposing, 1 2a; = 80 — 44, 
 
 12a;=36; 
 
 .-. a?=3. 
 Ex.3. 6(a;+3)+4a;=58. Ans, x=^4. 
 
 Ex.4. 30(a;-3)+6=6(a:+2). Ans. a;=4. 
 
 Ex.5. 5(a;+4)~3(a;-5)=49. Ans. x:=l 
 
 Ex.6. 4(3+2a;)-2(6-2a;)=60. Ans. x-^b 
 
 Ex.7. 3(a;-2)+4=4(3-a:). Ans. x==^2. 
 
 Ex.8. 6(4-~.a;)— 4(6— 2a;)-12=0. Ans. X'-=.Q. 
 
SIMPLE EQUATIONS. 29 
 
 PROBLEMS. 
 
 Prob. 1. What two numbers are those whose difference is 
 9, and if 3 times the greater be added to 5 times the less, the 
 •eum shall be 35 ? -I 
 
 Let ar=the less number; 
 then a; +9= the greater. 
 And 3 times the greater =8 (a; +9)= 3a; +27, 
 
 5 times the less =bx. 
 But by the problem, 3 times the greater + 5 times the less 
 =35; 
 
 /. Sx+27+5x=S5, 
 8a;+27=35. 
 Transposing, 8a;=35— 27=8; 
 
 .'. aj=l, the less number, 
 and a;+9=10, the greater. 
 
 Prob. 2. A courier travels 7 miles an hour, and had been 
 dispatched 5 hours, when a second is sent to overtake him, 
 and in order to do this, he is obliged to travel 12 miles an 
 hour. In how many hours does he overtake him ? 
 Let a;=the number of hours the 2d travels; 
 thena;+5= " " " " 1st " 
 
 .•. 12a;=:the number of miles the 2d " 
 and7(rr+5)= " " " " 1st " 
 
 But by the supposition the couriers both travel the same 
 number of miles ; 
 
 .-. 12a;=7 (rr+5), 
 12a;=7a:+35. 
 Transposing, 12a;-- 7a; =3 5, 
 5a;=:35, 
 • a; =:7, the number of hours the second 
 
 coirier is in overtaking the first. 
 
 Prob. 3. In a railway train 15 passengers paid £3 12*.; 
 the fare of the first class being 6.5., and that of the second 4«. 
 How many passengers were there of each class ? 
 
 Let a;=the number of passengers of the 1st class, 
 then 15— a;= " " ^ " " 2d " 
 
 .•. 6a; = sum in shillings paid by 1st class passengers, 
 and4(15-a;)= " "^^ " 2d « '' 
 
so ALGEBEA. 
 
 But these two sums amount to £3 125., or to 12s, 
 /. 6x+4{15-x)=:72, 
 6a;+60-4a;rr:72. 
 By transposition, 6a; — 4a; = 72 — 60, 
 2a;r=12! 
 
 ,'. x~6 No. of 1st class passengers ; 
 /. the number of 2d class passengers=15— a;=9. 
 
 Prob. 4. What number is that to which if 6. be added twice 
 the sum will be 24 ? A7is, 6. 
 
 Prob. 5. What two numbers are those whose difference is 
 6, and if 12 be added to 4 times their sum, the whole will be 
 60 ? Ans, 3 and 9. 
 
 Prob. 6. Tea at 6s. per lb. is mixed with tea at 4.5. per lb., 
 ' and 16 lbs. of the mixture is sold for £3 18^. How many 
 lbs. were there of each sort ? Ans, 1 lbs. and 9 lbs. 
 
 Prob. 7. The speed of a railway, train is 24 miles an hour, 
 and 3 hours after its departure an Jm)ress train is started to 
 run 32 miles an hour. In how m/nySiours does the express 
 overtake the train first started 1 ' Ans, 9 hours. 
 
 Prob. 8. A mercer having cut 19 yards from each of three 
 equal pieces of silk, and 17 from another of the same length, 
 found that the remnants taken together measured 142 yards. 
 What was the length of each piece 1 
 
 Let a;=the length of each piece in yards ; 
 .*. a;— 19=the length, of each of the 3 remnants, 
 and a;— ]7=the length of the other remnant ; 
 then 3 (a;-19)+^-n=:142, 
 or3a;-57+a;-17=:142, 
 
 4a;-74=:142. • 
 
 Transposing, 4a; = 1 42 + 74, 
 
 4a;=z216; 
 /. a; =54. 
 
 Prob. 9. Divide the number 68 into two such parts, that 
 the difference between the greater and 84 may equal 3 times 
 the difference between the less and 40. 
 Let a;=the less part, 
 then 68— a;=the greater; 
 
SIMPLE EQUATIONS. 31 
 
 .'. 84 — (68— a;)= difference between 84 and the greater, 
 and 3 . (40— a;) =3 times the difference between the less and 
 40. 
 
 But by the question the differences are equal to each other ; 
 ...84-(68-ir)=J8 . (40-^), 
 or 84-684-^=120-3^;. 
 By transposition, a;+3a;z=120-{-68— 84, 
 
 .-. rr=26, the less part ; 
 and .'. the greater z= 42. 
 
 Prob. 10. a man at a party at cards betted three shillings 
 to two upon every deal. After twenty deals he won five shil- 
 lings. How many deals did he win ? 
 
 Let x=zthe number of deals he won ; 
 .*. 20— a;=zthe number he lost; 
 .-. 2^ — the money won ; 
 and 3 . (20— .1;)=: the money lost. 
 
 But the difference between the money won and the money 
 lost was 55. 
 
 .'.2x"-S.{20—x)z= 5, • 
 
 2a;— 60+3a;= 5, 
 
 6x-60= 5, 
 
 5.r— 65; 
 
 .-. x=lS. 
 
 Prob. 11. A and B being at play cut packs of cards so as 
 to take off more than they left. Now it happened that A cut 
 off twice as many as B left, and B cut off seven times as many 
 as A left. How were the cards cut by each 1 
 
 Suppose A cut off 2x cards ; 
 then 52— 2a: =: the number he left, 
 and a;=the number B left; 
 .*. 52— or =: the number he cutoff. 
 But the number B cut off was equal to 7 times the number 
 A left; 
 
 .-. 52-.t;=:7 . (52-2a;) 
 52~-x=:S64-Ux. 
 Transposing, 1 4a; — a: =: 364 — 52, 
 13a;z:z312; 
 .-. a;=i24; 
 .-. A cut off 48, and B cut off 28 cards. 
 
/ 
 
 32 * ALGEBRA. 
 
 Prob. 12. Some persons agreed to give sixpence each to a 
 waterman for carrying them from London to Greenwich ; but 
 with this condition, that for every other person taken in by 
 the way, threepence should be abated in their joint fare. 
 Now the waterman took in thret more than a fourth part of 
 the number of the first passengers, in considerr.tion cf which 
 he took of them but fivepence each. How many persons were 
 there at first 1 
 
 Let 4x represent the number of passengers at first ; 
 then S more than a fourth part of this number = a; +3, and 
 they paid 3 (x+S) pence. 
 
 .-. the original passengers paid 6x4rr~3 (x+S) pence. 
 But the original passengers paid 5x4a; pence ; 
 .*. by equalizing these two values, we get 
 6x4:r~3(a;+3)z=:5x4(r, 
 24:r — 3:?;— 9=20;^. 
 Transposing, 24x — 3^ — 20:r =r 9 ; 
 
 and .'. the No. of passengers were=:4x9=36. 
 
 Prob. 13. There are two numbers whose diflTerence is 14, 
 and if 9 times the less be subtracted from 6 times the greater, 
 the remainder will be 33. What are the numbers ? 
 
 Ans. 17 and 31. 
 
 Prob. 14. Tw^o persons, A and B, lay out equal sums of 
 money in trade ; A (/ains £120, and B loses £80 ; and now 
 A's money is t7'eble of B's. What sum had each at first ? 
 
 Ans. £180. 
 
 Prob. 15. A rectangle is 8 feet long, and if it were two feet 
 broader, its area would be 48 feet. Find its breadth. 
 
 A71S, 4 feet. 
 Prob. 16. William has 4 times as many marbles as Thomas, 
 but if 12 be given to each, William will then have only twice 
 as many as Thomas. How many has each ? 
 
 Ans. 24 and 6. 
 Prob. 17. Two rectangular slates are each 8 inches wide, 
 but the length of one is 4 inches greater than that of the other. 
 Find their lengths, the longer slate being twice the area of the 
 other. 
 
 Let a;— the length in inches of the less ; 
 then a; -I- 4= " " " " greater. 
 
SIMPLE EQUATIONS. 33 
 
 Now the area of a rectangle is its length multiplied by its 
 breadth ; 
 
 .'. Sx and 8 (rr+4) are the areas of the slates. 
 But the larger slate is twice the area of the less ; 
 .'.SxX2=:S{x+4:),^ 
 
 .-. 8a:=32; 
 .-. x=z 4, the length of the less slate, 
 and 0:4-4= 8, " " " greater slate. 
 Prob. 18. Two rectangular boards are equal in area; the 
 breadth of the one is 18 inches, and that of the other 16 
 inches, and the difference of their lengths 4 inches. Find the 
 length of each and the common area. 
 
 Ans. 32, 36, and 576. 
 
 Prob. 19. A straight lever (without weight) supports in 
 equilibrium on a fulcrum 24 lbs. at the end of the shorter arm, 
 and 8 lbs. at the end of the longer, but the length of the longer 
 arm is 6 inches more than that of the shorter. Find the 
 lengths of the arms. 
 
 Let a;= length in inches of the shorter arm ; 
 then a; 4- 6= " " " longer " 
 
 Now the lever will be in equilibrium, when the weight at 
 one end multiplied by the length of the corresponding arm is 
 equal to the weight at the other end, multiplied by its corres- 
 ponding arm ; 
 
 .\24x=S{x-\-Q), 
 24:r=i8x+48, 
 ^16a:=48; 
 .'. x=S inches, the length of the shorter arm ; 
 and a: 4- 6 =9 " " " " longer « 
 Prob. 20. A weight of 6 Jbs. balances a weight of 24 lbs. on 
 a lever (supposed to be without weight), whose length is 20 
 inches ; if 3 lbs. be added to each weight, what addition must 
 be made to each arm of the lever, so that the fulcrum may 
 preserve its original position, and equilibrium still be re- 
 tained 1 
 
 This problem resolves itself into two other problems : — 
 
 (1.) To find the lengths of the arms in the original posi- 
 tion : 
 
 Let a:=the length in inches of the shorter arm ; 
 then 20— a: = " " " " longer " 
 
34 ALGEBRA. 
 
 Now, in order that there may be equilibrium, 24a; and 6 (20 
 — x) must be eq^ual to each other ; 
 
 30a: ==120; 
 
 ,\ xz= 4, the ^ngth of the shorter arm; 
 and20-a;= 16, " ^ " " longer " 
 
 (2.) To find the addition to be made to each arm, so that 
 there may again be equilibrium on the fulcrum in its original 
 position, after 3 lbs. have been added to each weight : 
 
 Let X = .number of inches to be added to each arm ; 
 then the lengths of the arms become 4+a? and 16+a; inches 
 respectively : and the weights at the arms have been respect- 
 ively increased to 27 lbs. and 9 lbs. 
 
 But by the principle of the equilibrium of the lever, 27 (4 
 +ar) and 9 (16+^) niust be equal to each other; 
 /. 27(4+0;) =9(16+0:). 
 Divide each side of the equation by 9, and 
 3(4+0:) = 16+0:, 
 12+3o: = 16+o;, . 
 3o:-o: = 16-12 
 2o;=i 4; 
 ,\x=: 2. 
 
 Prob. 21. The condition being the same as in the last 
 problem, how many inches must be added to the shorter arm 
 in order that' the lever may in its original position retain its 
 equilibrium 1 Ans, 1^ inch. 
 
 Prob. 22. A garrison of 1000 men were victualled for 30 
 days ; after 10 days it was reinforced, and then the provisions 
 were exhausted in 5 days ; find the number of men in the re- 
 inforcement. Alls, 3000. 
 
 Prob. 23. Two triangles have each a base of 20 feet, but 
 the altitude of one of them is 6 feet less than that of the other, 
 and the area of the greater triangle is twice that of the less. 
 Find their altitudes. Ans. 6 and 12. 
 
 N. B. The area of a triangle = ^ base X altitude. 
 
 Prob. 24. A and B began to play with equal sums; A 
 won 125. ; then 6 times A's money was equal to 9 times B's. 
 What had each at first 1 Ans, £S. 
 
^ SnVIPLE EQUATIONS. 85 
 
 Prob. 25. A company settling their reckoning at a tavern, 
 pay 4 shillings ea^h, but observe that if thpre had been 5 
 more they would only have paid 3 shillings each. How 
 many were there ? Ans. 1^, 
 
 Prob. 26. Two persons, A and B, at the same time set 
 out from two to wn^^a^ ja ilea apart, and meet each other in 
 5 hours, but B walks ^N|^te|^U||^fl|||ore than A. How 
 many miles does A walk inaSHKH^ Ans, 3 miles. 
 
 DIVISION. 
 
 36. The Division of algebraic quantities is the finding their 
 quotient, and in performing the operation the same circum- 
 stances are to be taken into consideration as in their multipli- 
 cation^ and consequently the four following Rules must be 
 observed. 
 
 (1.) That if the signs of the dividend and divisor be liJce^ 
 then the sign of the quotient will be + ; if unlike^ then the 
 sign of the quotient will be -— .* 
 
 (2.) That the coefficient of the dividend is to be divided by 
 the coefficient of the divisor^ to obtain the coefficient of the 
 quotient. 
 
 (3.) That all the letters common to both the dividend and 
 the divisor must be rejected in the quotient, 
 
 (4.) That if the same letter be found in both the dividend 
 and divisor with different indices, then the index of that letter 
 
 * The Rule for the signs follow immediately fi-om that in Multipli- 
 cation; thus, 
 
 If Ji^aX-\'h—-\-ah, then ^ =-f 6, and 4^ =+a 
 
 4 « X — 6= — ah, - - - - — = — 6, and — - =+a 
 -}- a — 
 
 — aX — 6=+a6, - - - — — = — 6, and X^ = — a 
 — a — b 
 
 i.e^ like signs 
 " produce^-, and: t 
 unlike signs — i, > I 
 
 86. "What is meant by the division of algebraic quantities ? State th» 
 Rules in Division. 
 
36 
 
 ALGEBKA. 
 
 in the divisor must be subtracted from its index in the dividend, 
 to obtain its index in the quotient. Thus, 
 
 (1.) +abc divided by +ac - - 
 
 -\-ac 
 (2.) +6a6c -2a - - or -^- 
 
 — lOxyz 
 
 (3.) --lOxj/z 
 
 ' +5y- - or- 
 
 + 5y 
 
 =+b, 
 = -35c. 
 
 ■=z—2xz. 
 
 20a'^xV 
 
 (4.) '-20a^xy - - - — 4aary or——- =+5ax]/\ 
 
 Of Division, also, there are three Cases : the same as in 
 Multiplication, 
 
 Case I. 
 37. When dividend and divisor are both simple terms. 
 Ex. 1. 
 Divide 18ax^ by Sax, 
 
 =6a?. 
 
 18ax'' 
 Sax 
 
 Ex. 3. 
 
 Divide — 28a;^2/^ by — 4a:y. 
 
 Ex.5. 
 
 Divide — 14a^6^{j by 7ac 
 ^UaWc _ 
 +7ad " 
 
 Ex.2. 
 
 Divide ISa'i^ by— 5a. 
 
 + 15a^62 
 
 — oa 
 
 Ex. 4. 
 
 Divide 25a^c^ by — 6a^c, 
 + 2 5aV_ 
 — 5a^c """ 
 
 Ex.6. 
 
 Divide — 20^;^^^ by — 4y2r. 
 
 ~4y^. 
 
 Case II. 
 
 88. When the dividend is a coinpoiind quantity, and the 
 -divisor a simple one ; then each term of the dividend must be 
 .divided separately, and the resulting quantities" will ic»e the 
 -quotient required. 
 
 State the rule for Case 2d. 
 
 ule 
 
DIVISION. 8? 
 
 Ex. 1. 
 
 Divide 4:2a^+Sab+l2a^ by 3a. 
 
 —- = lAa+b+4a, 
 
 ,3a 
 
 Ex.2. 
 
 Divide 90aV — 18a:z;'+4a'^—2a^ by 2aa?. 
 
 ma^x'-lSax^+4a'x-2ax ^^ , ^ , o i 
 
 J z=z4,6ax^—9x+2a-^i. 
 
 2ax 
 
 Ex. 3. 
 
 Divide Ax^ -2x'' -\-2x by 2a;, 
 4^— 2:r^+2a; 
 
 2x 
 
 Ex. 4. . 
 
 Divide —24aVy'-^axy+QxY by — 3ary. 
 — 24a'a: V — ^axy + 6a; V^__ 
 
 Ex. 5. 
 Divide 14a6^+7a^6^-21a^5H35a^6 by ^ah, 
 14a53^ 7^252_21a^6^+35a^6_ 
 
 Case III. 
 
 39. When dividend and divisor are both compound quan- 
 tities. In this case, the Eule is, " To arrange both dividend 
 and divisor according to the powers of the same letter, begin- 
 ning with the highest ; then find how often the first term of 
 the divisor is contained in the first term of the dividend, and 
 place the result in the quotient ; multiply each term of the 
 divisor by this quantity, and place the product under the cor- 
 responding (i. e., like) terms in the dividend, and then subtract 
 it from them ; to the remainder bring down as many terms 
 of the dividend as will make its number of terms equal to 
 
 39. When dividond and divisor are both compound quantities, what is 
 the Kule ? 
 
33 ALGEBRA. 
 
 that of the divisor; and then proceed as before, till all the 
 terms of the dividend are brought down, as in oommon 
 arithmetic." 
 
 Ex. II 
 
 Divide a^Sa'b+Sab''--b' by a—b. 
 
 a^b] a'^Sa'b+Sab'-'b' (a^^2ab+b^ 
 
 * -2a^6+3a5^ 
 
 •^2a^-\-2ab' 
 
 ab'-P 
 ab'-b^ 
 
 In this Example, the dividend is arranged according to the 
 powers of a, the first term of the divisor. Having done this, 
 we proceed by the following steps : — 
 
 (1.) a is contained in a\ a^ times; put this in the quo- 
 tient. 
 
 (2.) Multiply a— 5 by a\ and it gives a^—a^b. 
 
 (3.) Subtract a^—a'b from a^—Sa'b, a 
 ■2a'b. 
 
 (4.) Bring down the next term +3a5^ 
 
 (5.) a is contained in —2a^b, — 2a5 tij 
 idtient. 
 
 (6.) Multiply/ and subtract as before, and the remainder is 
 
 (3.) Subtract a^—a^b from a^-~3a^^, and the remainder is 
 
 -2a^6. 
 
 (5.) a is contained in — 2a^i, — 2a5 times ; put this in the 
 quotient. 
 
 aJb\ 
 
 (7.) Bring down the last term — 6^ 
 
 (8.) a is contained in a6^, +S^ times ; put this in the quo- 
 tient. 
 
 (9.) Multiply and subtract as before, and nothing remains ; 
 the quotient therefore is a^—2ab'\-b'^. 
 
DIVISION. 39 
 
 Ex. 2. 
 
 )a'-\-2a^x-{- aV -.^ \ 
 
 ^ 
 
 3a'ai+9aV4-10aV 
 
 "* 3a V+ la'x' + 5aar^ 
 3a V+ 6aV +3aa;^ 
 
 aV+2aa:^+a:^ 
 
 /12^^~21rz;* \ 
 
 Ex.3. 
 
 *~-20a;3+39a;2 
 
 -20a:^+35a;^ 
 
 * +~4? 
 
 Ex. 4. 
 
 3a:-6\6a;*-96 /2;r^+4a;^+8a;+16 
 /6^^--22^\ 
 
 + 12 ^^— 24a;-^ 
 
 *~+24?-96 
 +24a;^-48a; 
 
 * "+48^-96 
 +48a;-96 
 
 • "When there is a remainder, it must be made the numerator of a 
 Fraction whose denominator is the divisor ; this Fraction must then be 
 plac(^d in the quotient (with its proper sign), the same as in common 
 arithmetic. 
 
40 ALGEBRA. 
 
 Ex. 5. 
 
 a^+x-l)x'-x'+x^---x'+2x'-l(x^-^x''+x'-^x+l 
 /x'+x'-x' V 
 
 ^—x^-\-x' 
 —x^—x' 
 
 '■\-x^ 
 
 
 % 
 
 -x^-^2x—\ 
 —x^— x^+x 
 
 * + x'+x- 
 + x'+x- 
 
 -1 
 -1 
 
 
 * * 
 
 * 
 
 A 
 
 Ex. 6. 
 
 -ix]x'-^x'+'-lx'--ix(x'-lx+ 1 
 
 + x'-ix 
 
 ^V 
 
 Ex. 7. Divide a^+4a^b+6a%^+4:ab^+b^ by a+5. 
 
 Ex. 8. Divide a'— 5a^^+10aV — 10aV+5aa:*— a;^ 
 by a^—Sa'x+Sax^-x^ 
 
 Ans, a^—2ax+x\ 
 
 Ex. 9. Divide 25x'—x*—2x^Sx^ by 5;z;^— 4a;l 
 
 ^W5. 5x^+4x^+3x+2. 
 
 Ex. 10. Divide a*+8a^ar+24aV+32a^^+ 16^* by a+2ar. 
 
 u4n5. a^-}-(ja'x-^l2ax'' + 8x\ 
 
DIVISION. • ^ 1 
 
 Ex. 11. Divide a^—x^ by a— ar. 
 
 Ex. 12. Divide Qx'^+9^—20x by Zx^'-^x, 
 
 ^Ans. 2x^+2x-^^- ^"^ 
 
 Zx^'-Zx. 
 
 Ex. 13. Divide 9.c«-46a;^+ 95:^2 +150.r by a;^-4^-5. 
 
 Ans. 9;?;"— 10.r^+5a;'— 30^. 
 
 Ex. 14. Divide aj^~^a;^+a;^+|a?-2 by ^x-2. 
 
 Ans, f.i!^— ^rr^ + l. 
 
 PROBLEMS PRODUCING SIMPLE EQUATIONS, CONTAINING ONLT 
 ONE UNKNOWN QUANTITY. 
 
 Prob. 1. A fish was caught, the tail of which weighed 9 
 lbs. ; his head weighed as much as his tail and half his body, 
 and his body weighed as much as his head and tail. What 
 did the -fish weigh ? 
 
 Let 2ir= weight of the body in lbs. ; 
 .'. 9+^= weight of tail+l- body = weight of head. 
 But the body weighs as much as the head and tail ; 
 
 ,\2a;=(9+a;)+9, ^ 
 
 2a;=rr+18; 
 .•.a;=18, 
 and .*. 2a;i=36, the weight of body in lbs., 
 9+.'i:=27, the weight of head in lbs., 
 and the weight of fish=3G4-27+9=721b§. 
 
 Prob. 2. A servant agreed to serve for £8 a year and a 
 
 livery, but left his service at the end of 7 months, and received 
 
 only £2 135. 4i and his livery ; what was its value? 
 
 Let 12a;=the value of the livery in d. 
 
 But £8 = 1920c?., and £2 13^. 4c^.=:640d ; 
 
 then, the wages for 12 months =12^* +1^20 ; 
 
 ^ . 1 12a;+1920 
 .'. the wages for 1 month = — =:a;+loO. 
 
 and .'. the wages for 7 months = (a; +160) 7. 
 4* 
 
42 ' ALGEBRA. 
 
 But the wages actually received for 7 months =12a?-f-640 ; 
 /. 12:r+640=:7ar + 1120; 
 
 .-. 5a:=480, 
 
 and .-. 12x=zUb2d.=£4: 16s., value of livery. 
 
 From the solutions of the two preceding problems, it will 
 be seen, that by assuming /or the unknown quantity^ x with a 
 'proi^r coefficient, an equation free from fractions will be 
 obtained. It is frequently not only convenient to make such 
 an assumption, but a more elegant solution is generally 
 thereby obtained. The coefficient of x must be a multiple 
 of the denominator of all the fractions involved in the 
 problem. 
 
 Prob. 3. A cistern is filled in 20 minutes\ by 3 pipes, one 
 of which conveys 10 gallons more, and another 5 gallons less, 
 than the third per minute. The cistern holds 820 gallons. 
 How much flows through each pipe in a minute ? 
 
 Ans, 22, 7, and 12 gallons, respectively. 
 
 Prob. 4. A and B have the same income : A lays by \ of 
 his ; but B spending £60 a year more than A, at the end of 4 
 years finds himself £160 in debt. What did each annually 
 receive 1 Ans, £100, 
 
 Prob. 5. A met two beggars, B and C, and having a certain ' 
 sum in his pocket, gave B -^ of it, and C f of the remainder: 
 A now had 20d left; what had he at firsts Ans, 5s. 
 
 Prob. 6. A person has two horses, and a saddle worth 
 £60 : if the saddle be put on the first horse, his value will be- 
 come double that of the second ; but if it be put on the second, 
 his value will become triple that of the first. What is the 
 value of each horse? A7is, £36 and £48. 
 
 Prob. 7. A gamester at one sitting lost I of his money, 
 and then won ISs. ; at a second he lost -J of the remainder, 
 and then won Ss., after which he had 3 guineas left. How 
 much money had he at first 1 
 
 Let 15.r=the number of shillings he had at first; 
 having lost I of his money, he had f of it, or 12.r remaining ; 
 he then won I85., and therefore had 12a; -f 18 in hand ; losin,^ 
 
ALGEBRAIC FRACTIONS. 43 
 
 J of this, he had f of it, or 8^' + 12 left ; he then won Ss., and 
 so had {Sx-{-12)-{-S sliillings, which was to be equal to 3 
 guineas, or 63^. 
 
 .-. (8a;+12)+3=63, 
 8a;+12=r6(f; 
 
 x= 6', 
 Hence 15x=9ps.=£4: lOs. 
 
 Prob. 8. A person at plaj; lost a third of his money, and 
 then won 45. ; he, again lost a fourths of his money, and then 
 won 135. ; lastly, he lost an eighth of what he then had, and 
 found he had 285. left. What had he at first ? 
 
 <^ ^^ ^ ^""^ 
 
 -K ^ TN OHAPTEE III. 
 
 ^- \ ij* ON ALGEBRAIC FRACTION. 
 
 40. The Rules for the management of Algebraic Fractions 
 are the same as those in common arithmetic. The principles, 
 on which the rules in both sciences are established, are the 
 following : — 
 
 (1.) If the numerator of a fraction be multiplied, or the de- 
 nominator divided by any quantity, the fraction is rendered 
 so many times greater in value. 
 
 (2.) If the numerator of a fraction be divided, or the de- 
 nominator multiplied by any quantity, the fraction is rendered 
 80 many times less in value. 
 
 (3.) If both the numerator and denominator of a fraction 
 be multiplied or divided by any quantity, the fraction re- 
 mains unaltered in value. 
 
 40. Are the same rules employed in the management of algebraic frac- 
 tions as in those of common arithmetic? Enuraerato the^principles which 
 are the foundation of the rules in both sciences. 
 
44 ALGEBRA. 
 
 ON THE REDUCTION OF FRACTIONS. 
 
 41. To reduce a mixed Quantity to an improper Fraction, 
 
 Rfle. " Multiply the integMl part by the denominator of 
 the fraction, and-to the product annex the numerator with its 
 proper sign ; under this sum place the former denominator, 
 and Ihe result is the improper fraction required." 
 
 Ex. 1. 
 
 2x 
 Eeduce 8«^+^2 to an improper fraction. 
 
 The integral part X the denominator of the fraction + the 
 nwnerator = 3a X 5a^ + 2a; = 1 5a^ + 2x ; 
 
 Hence, — ^ — is the fraction required. 
 
 Ex. 2. 
 
 4a? 
 
 Reduce ^x— — to an improper fraction. 
 
 Here 5a;X6a^=30a^ir ; to this add the numerator with its 
 proper sign, viz., —4a:; then -—^ — is the fraction re- 
 quired. 
 
 ^ Ex. 3. 
 Reduce 5a; ~ — to an improper fraction. 
 
 Here 5a; X "7= 35a:. In adding the numerator 2.2:— 3 
 
 with its proper sign^ it is to be recollected, that the sign — 
 
 2a: 3 
 
 affixed to the fx-action — — — means that the whole of that 
 
 7 
 
 fraction is to be subtracted^ and consequently that the signs 
 of each term of the numerator must be changed when it is 
 combined with 35a? ; hence the improper fraction required is 
 35a:-2a:+3 33a:+3 
 
 41 . How is a mixed quantity reduced to an improper fraction ? 
 
ALGEBKAIC FRACTIONS. 45 
 
 2c 
 Ex. 4. Reduce 4a5+— to an improper fraction. 
 
 , A2a'b-{'2c 
 
 4a ^ 
 Ex. 5. Reduce 35^— — to an improper fraction. 
 ox 
 
 . 155^;r— 4a 
 Ans, . 
 
 DX 
 
 d^ — ax 
 Ex. 6. Reduce a—x-{ to an improper fraction. 
 
 a'-x^ 
 Ans, . 
 
 X 
 
 ^^ 9 
 
 Ex. 7. Reduce 3a;^ --— to an improper fraction. 
 
 30^'-4^+9 
 
 Ans, 
 
 10 
 
 42. To reduce an improper Fraction to a mixed Quantity. 
 
 Rule. " Observe which terms of the numerator are divisi- 
 ble by the denominator without a remainder, the quotient 
 will give the integral part ; to this annex (with their proper 
 signs) the remaining terms of the numerator with the denom- 
 inator under them, and the result will be the mixed quantity 
 required." 
 
 Ex. 1. 
 
 Reduce to a mixed quantity. 
 
 Here =a+6 is the integral part, 
 
 and — is the fractional part ; 
 
 b^ , 
 
 /. a+^H — is the mixed quantity required. 
 
 Si . 
 
 42. What is the rule for reducing an improper fraction to a mixed quan" 
 
 tity? ^ 
 
46 ALGEBRA. 
 
 Ex. 2. 
 
 Keduce to a mixed quantity. 
 
 Here — — =Sa is the i?itegrS -pa,rtf 
 
 2^ g^ 
 
 and — - — is the fractional part ; 
 oa 
 
 2x—Sc . , . , 
 /. 3a H — IS the mixed quantity required. 
 
 Ex. 3. Reduce — to a mixed quantity. 
 
 Ans. 2x—^r-, 
 2x 
 
 Ex. 4. Eeduce ^ ^^ * mixed quantity. 
 
 ^a 
 
 3c 
 
 ^n5. 3a+l— —-. 
 4a 
 
 Ex. 5. Reduce — -^ to a mixed quantity. 
 
 X 
 
 2b^ 
 Ans, 10y4-3a; ^, 
 
 43. To reduce Fractions to a common Denominator, 
 
 Rule. " Multiply each numerator into every denominator 
 hut its own for the new numerators, and all the denominators 
 together for the common denominator." 
 
 Ex. 1. 
 
 2'X ^x 4a 
 
 Reduce -^j -y, and — , to a common denominator. 
 o o o 
 
 Hence the frac- 
 tions required are 
 lObx 75^ 12a^ 
 1X56"' 156' 156* 
 
 2xXbX^ = lObx^ 
 
 5:c X 3 X 5 = 15x y new numerators ; 
 
 4aX3x6 = 12a6j 
 
 3 X6X5=156 common denominator ; 
 
 43. How are fractions reduced to a common denominator ? 
 
ALGEBRAIC FRACTIONS. 47 
 
 Ex.2. 
 
 Keduce — - — , and -— , to a common denomuiator. 
 5 4 
 
 Hence the frac- 
 tions required 
 are 
 8:^^+4 , Ibx 
 -20"' ^^^ 20-' 
 
 f 
 
 Here (2^+1) X4= 8^+4) new nume- 
 3a; X 5 = 1 5^ ) rators ; 
 
 5X4=20 common denomi- 
 nator ; 
 
 Ex.3. 
 
 Eeduce — ; — , -^r— , and r-, to a common denominator; 
 a-\-x 3 2x 
 
 Here 5xX Sx2x=S0x^ ] •*• the new frac- 30x ^ 
 
 {a-^x) {a-{-x)x2x=2a''x—2x^[ tions are 6ax+6x^' 
 
 1 X(a+^)X3 =Sa +3.r 1 2a'x-2 x^ Sa+Sx 
 
 {a-\-x) X3 X2:i: =6ax^6x''J Qax -i-6?' 6^+0?' 
 
 337 4:bx 5x 
 
 Ex. 4. Reduce -— . -^r-, and — , to a common denominator. 
 o oa a 
 
 9a'x 20abx , 75ax^ 
 
 ■n^ n -n ;, 2a; + 3 ^ 5a:+l 
 
 JbiX. 5. Keduce , and — — — , to a common denommator. 
 
 X o 
 
 . 6a;+9 , 5a;2+a; 
 
 Ans, —- — , and -— , 
 
 Sx Sx 
 
 Ex. 6. 
 
 4:X^4"2x Sx^ 2x 
 
 Reduce — , -— , and — , to a common denominator. 
 
 o 4a oo 
 
 . 4:Sabx''+24ahx 45bx'' , 4.0ax 
 
 Ex. 7. 
 
 7x'^—l 4x^—x-{-2 
 
 Reduce — - — , and — ^ — , to a common denominator. 
 
 2x 2a^ 
 
 14aV~2a2 8^^-2a:^+4^ 
 
i8 ALGEBRA. 
 
 44. To reduce a fraction to its lowest terms. 
 
 Rule. " Observe what quantity will divide all the term3 
 
 both of the numerator and denominator without a remainder ; 
 
 Divide them by this quantity, |nd the fraction is reduced to 
 
 its lowest terms." 
 
 Ex. 1. 
 
 ^ , \4:r^-\-lax+2lx' . , 
 
 Keduce --—- to its lowest terms. 
 
 The coefficient of every term of the numerator and denomi- 
 nator of this fraction is divisible by 7, and the letter x also 
 enters into every term ; therefore Ix will divide both nume- 
 rator and denominator without a remainder. 
 
 Now '—2x^-{-a+^x, 
 
 Ix 
 
 ^ Zbx^ , 
 and-— — =5^; 
 
 Hence, the fraction in its lowest terms is . 
 
 5a; 
 
 Ex. 2. 
 
 ^ , 20a5c— 5a^+10ac . , 
 
 Keduce — -r to its lowest term. 
 
 ha^c 
 
 Here the quantity which divides both numerator and de- 
 nominator without a remainder is 5a; the fraction therefore 
 
 . . , . 46c— a+2c 
 
 m its lowest terms is . 
 
 ac 
 
 Ex. 3. 
 
 Reduce — — r-„ to its lowest terms. 
 a^—¥ 
 
 Here a—h will divide both numerator and denominator, 
 
 for by Ex. 2. Case III. page 27. a^^y'^{a+b) (a-5); 
 
 hence — — r- is the fraction in its lowest terms. 
 a-\-b 
 
 \()x^ 
 
 Ex. 4. Reduce ——-^ to its lowest terms. 
 Ibx^ 
 
 2x 
 
 ^^^- 3 ' 
 
 4A. Show how fractions are reduced to their lowest terms. 
 
ALGEBRAIC FRACTIONS. 49 
 
 Ex. 5. Eeduce -r: — to its lowest terms. 
 Qax 
 
 . hx 
 Ans. -. 
 
 UxV 21:c^y^ 
 
 Ex. 6. Eeduce r— ^ to its lowest terms. 
 
 "Ix^y 
 
 Ans. ^I^:^. 
 
 X 
 
 Ex. 7. Reduce — — to its lowest terms. 
 
 Yix^ 
 
 Sx''-x^\-2 
 Ans, -, . 
 
 ON THE ADDITION, SUBTRACTION, MULTIPLICATION, AND 
 DIVISION, OF FRACTIONS. 
 
 45. To add Fractions tog-ether. 
 
 Rule. " Reduce the fractions to a common denominator 
 and then add their numerators together; bring the result- 
 ing fraction to its lowest terms, and it will be the sum re- 
 quired." 
 
 Ex. 1. 
 
 Add —,-=:, and -, together. 
 
 O 7 o 
 
 3a:X7x3=63a;] 
 
 2a:X5x3=30a; ( 6SxA-S0x+S5x 128.i; . , ^ . ^ 
 .X5X7:=35. \.-- ro-5— = -105- ^^ ^^T^ 
 
 5x7x3=105. 
 
 ^ ^ . ^ ^ a 2a .6b . 
 
 Ex. 2. Add -, — -, and -— , together, 
 
 6 36 4a ° 
 
 aX35x4a=12a^6'' 
 2aX6X4a= Sa^b 
 56X^X36 = 156^ 
 6x36x4a = 12a62j 
 
 12a'^6 + 8a^6 + 156^ _ 20a^6 + 156» 
 
 •*• 12^6^ ~ 12^2 
 
 //I- •;,• T. 7^ 20a^+156^ . . 
 = (dividmg by 6) — 12^— is the 
 
 sum required. 
 
 45. Stftte the rule for adding fractions. 
 
50 ALGEBRA. 
 
 ^ « A -,-. 2^+3 3^—1 ^4x ^ . 
 Ex. 3, Add — - — , —^ — , and — , together. 
 
 (2.t' + 3)x2.rX 7=28^2 +42:c^ . 28x^+42 a :+105a:— 3 54-40a:« 
 (3^-l)x5 x7=:105:r-35 U* 70ar 
 
 5X2^7^70^ J """" 70^ ^^ ^ ^ 
 
 » . sum required. 
 
 _ ^ . ^ ^ 3:c 5a; _ 4a; 
 
 Ex. 4. Add — , — , and — , together. 
 
 934a; 
 
 Arts, 
 
 603' 
 
 ^ ■ . , , 3a2 2a ,35 
 
 Ex. 5. Add ^j "H"' ^^^^ iT"' together. 
 
 105a^+28a^6+306« 
 
 70a6 
 
 -n ^ Ajj^^+1 4a;4-2 _ a; ^ . 
 Ex. 6. Add — ——^ — -— , and -, together. 
 
 16ar+77 
 
 ^^^^- -T05- 
 
 Ex. 7. Add — ■— — , and — — — , together. 
 
 37a.'+115 
 
 Ex. 8 Add — - — , and -^r-^ together. 
 
 ^''"- —- 6^' 
 Ex. 9, Add -, and — -^, together. 
 
 X — *i X-\-0 
 
 2a;2 
 Ans. 
 
 x'-9' 
 
 Ex. 10. Add L, and — — ,, together. 
 
 a — a-f-o 
 
 2a^+26* 
 
ALGEBRAIC FRACTIONS. 5i 
 
 46. To Subtract Fractional Quantities, 
 
 BuLE. " Reduce the fractions to a common denominator ; 
 and then subtract the numerators from each other, and under 
 the difference write the common denominator." 
 
 Ex. 1. 
 
 Subtract — from -~r-. 
 5 15 
 
 3;rXl5=45:r) 70a;— 450- 25a: x, , ..^ 
 
 UxX 5=70a; V •'• 75 — = "75" = 3 '^ ^^^ difference 
 
 5 Xi5=75 ) required. 
 
 Ex.2. 
 
 ^_ 2x+l^ 5x+2 
 
 Subtract — -— from — - — . 
 
 (2a:+l)x7=14a;+7) l5x+6-Ux-1f x-1 , , 
 
 (5a;+2)x3z:rl5a:+6 V 21 "^ l^T '^ ^^ 
 
 3x7=21 ) fraction required. 
 
 Ex. 3. 
 
 lOo;— 9 ,^ 3a;— 5 
 
 From — - — subtract — — — . 
 
 (10a:-9) x7=70a;-63 ) 70a; -63 -24a; +40 _ 46a;-23 
 ( 3a;-5)x8=24a;-40 V •*• 56 - 56 
 
 8 X 7=56 ) is the fraction required. 
 
 Ex. 4. 
 
 _ a+b . ^a—b 
 
 Erom subtract — tt. 
 
 a— 6* a-{-b 
 
 ^ , A^r . n 2.0 a.aO . a'-^'^ab-\-b'-a'+2ab--b' 
 {a-{-b) {a-]-b)=a^+2ab-}-b' 
 
 {a-b) {a-b)=a'^2ab+b 
 
 (^ct-b){ai-b)=:a'-b- 
 
 a^-b'' 
 -5 — 7j IS the fraction required. 
 
 Ex. 5. Subtract — from — . Am. -^^ 
 
 46. (5ive the rule for subtracting fractions. 
 
52 
 
 ALGEBRA. 
 
 Ex, 6. Subtract — —— from : — . 
 
 Ex. 7. Subtract — -— - from ---. 
 
 Ex. 8. Subtract?:?;^ from ^~. 
 Sx 3 
 
 Ex. 9. Subtract — --7 Irom — -. 
 a-f-o a—o 
 
 Ex. 10. Subtract — -^ — from — . 
 
 \21x+ 17 
 
 U^ltO, 
 
 28 • 
 
 
 
 4a:2_ll^_ 
 
 -5 
 
 
 50^+5 
 
 
 Ans, 
 
 4x' + S 
 
 
 Ans. 
 Arts. 
 
 2b 
 a'-b'' 
 
 lla;+49 
 
 
 56 
 
 8 
 
 47. To Multiply Fractional Quantities* 
 
 KuLE. " Multiply their numerators together for a new 
 numerator, and their denominators together for a new de- 
 nominator, and reduce the resulting fraction to its lowest 
 terms." 
 
 Ex. 1. 
 
 2xx4:X=zSx'^ 
 7 x9 
 
 Multiply -^ by -|. 
 =63 i •*• ^^^ fraction required is 
 
 Sx^ 
 63* 
 
 Ex. 2. 
 
 ,^ , .I 4x+l , 6^ 
 Multiply — -— by y. 
 
 Here* 
 
 {4.x+i)x6xz=z24:x^+6x ^ 
 and 
 
 3x7 =21 
 
 24:x^+6x 
 21 
 
 = (dividing the nu- 
 
 merator and denominator by 3) 
 — - — is the fraction required. 
 
 47. State the rule for the multiplication of fractio:is. 
 
ALGEBRAIC FRACTIONS. 53 
 
 Ex. 3. 
 
 Multiply — — ,— Dv — — ;. 
 
 By Ex. 2. Case III. p^e 27, {a'—b')xSa'={a + b) 
 
 i 7x o 9 1 1 1 . 3a'X (a+^>)(«-^) 
 (a— 6)Xoa'': hence the product is -rr — . \ -t — ^ = 
 
 (dividing the numerator and denominator by a +5) —^ - 
 
 Ex. 4. 
 
 Multiply — — — by 
 
 14 ' '^x^'—Zx 
 
 21a.r^— 35aa; 
 
 Here 
 
 (3;i;'— 5:r) X7a=21aa;^— 35aa; 
 
 and 
 (2.r^-3aj) X 14=28^-^-42a; 
 
 •• 28.3-42. -^(^^-^^-g 
 
 the numerator and denonii- 
 
 ^ - ^ . 3a.— 5a . , 
 nator by 7.) — — ; — - is the 
 
 fraction required. 
 
 Ex. 5. Multiply by — . ^4/15. 
 
 .T-i -^ 7* • 7.-r 
 
 T. ^ n^r 1 . , 3.2 — . ,10 , , 8.— 1 
 
 Ex. 6. Multiply -^ by ^-^,-^. Ans. — -. 
 
 Ex. 7. Multiply by — -— . Arts. — - — 
 
 Ex. 8. Multiply ^— ^ by -^_ . Ans. -. 
 
 48. 0/i //i^ Division of Fractions, 
 
 Rule. " Invert the divisor, and proceed as in Multiplioa 
 tion." 
 
 48. Enimciate the rule for division of fractions. 
 
54 
 
 ALGEBRA. 
 
 Ex. 1. 
 Divide -^ bj — . 
 
 Inter t the divisor, and it becomes -- : hence -— - x ;r- 
 
 2x' 9 2a? 
 
 42ic'^ 7x 
 = r-^— z=z — (dividing the numerator and denominator by 6x) 
 
 is the fraction required. 
 
 Ex. 2. 
 
 ^. ., Ux—S , 10x^-4: 
 
 Divide — z — by 
 
 Ux-^S 25 
 
 — ^ — X 
 
 5 ^' 25 ' 
 (14^_3)X5 70a;--15 
 
 10a;-4 " 
 
 lOx-4: 
 
 lOx-^4. 
 
 Ex. s; 
 
 Divide — by — -- — . 
 
 2a -^ 6b 
 
 ba'-6b^ 5x{a+b){a-b) 
 
 2a "" 2a 
 
 4a 4-46 4x(a + ^') 
 
 5x{a+b){a-b) ^ 
 
 m 
 
 6b 
 
 66 
 
 2a ''4x(a+6) 
 
 i SObx{a-b ) 15a6-156^ . 
 8a 4a 
 
 the fraction required. 
 
 Ex. 4. Divide -x- by -^. 
 
 7 -^ 5 
 
 Ex. 5. Dunde —7; — by — - — . 
 3 '^ 5x 
 
 Ex. 6. Divide ^- by ^. 
 
 Ex. 7. Divide ?^^ by f-. 
 3 / 5 
 
 ^/15. 
 
 ^725. 
 
 Ans. 
 
 Ans, 
 
 20 
 
 63* 
 
 
 lOo; 
 
 
 3 * 
 
 
 4.^- 
 
 12 
 
 5 
 
 
 9.r- 
 
 3 
 
SIMPLE EQUATIONS. 55 
 
 ON THE SOLUTION OF SIMPLE EQUATIONS, CONTAINING ONLY 
 ONE UNKNOWN QUANTITY. 
 
 Rule III. 
 
 49. An equation may be ckared of fractions by multiply- 
 ing each side of the equation Dy the denominators of the frac- 
 tions in succession. 
 
 Or, an equation may be cleared of fractions by multiplying 
 each side of the equation by the least common multiple of the 
 denominators of the fractions. 
 
 This Rule is derived from the axiom (4), that, if equal 
 quantities' be multiplied by the same quantity (or by equal 
 quantities), the products arising will be equal. 
 
 Ex. 1. Let ^=6. 
 Multiply each side of the equation by 3 ; then (since the 
 
 multiplication of the fraction ^ by 3 just takes away the de- 
 
 o 
 
 nominator and leaves x for the product) we have 
 
 ir=6x3 = 18. 
 
 Ex. 2. Let 1+1=1. 
 
 Multiply each side of the equation hy 2, and we have 
 
 a.+-=14. 
 
 Again, multiply each side of this equation by 5, an<J '^r 
 comes 5a;+2ar=70, 
 
 7a;=70 ; 
 v.'P=10. 
 
 Ex.3. Let f + 1=13- 1. 
 
 Multiply each side by 2, then a;+— • =26— — . 
 
 Multiply each side by 3, and 3a:+2a;=78 — — . 
 Multiply each side by 4, and 12a;+8ir=312— Oa:. 
 
56 ALGEBEA. 
 
 By transposition, 1 2a; + 8a: -f 6a; = 3 1 2, 
 
 26a:r=312; 
 
 .-. x= 12. 
 
 - This example might have been solved more simply, by mul- 
 tiplying each side of the equatmn by the least common multi- 
 ple of the numbers 2, 3, 4, which is* 12. 
 
 Multiply each side by 12, —^ — f--^ = 156 -r^, 
 
 or, 6a;+4a;=156— 3a;. 
 By transposition, 6a; + 4a; + 3a;— 156, 
 13a;=156; 
 .\x=z 12. 
 
 Ex. 4. Let *^+|=22. Ans, x=24. 
 
 ~ Ex.5. Let T—-::^-?:' ^^s. a;=10. 
 
 4 6 6 
 
 Ex. 6. Let |+^=31-|. Ans. a;=30. 
 
 Z o 5 
 
 Ex. 7. Let ~ -^+^=44. Ans. a;=60. 
 
 5 o 2 
 
 60. In the application of the Rules to the solution of simple 
 t/t|uations in general containing only one unknown quantity, it 
 will be proper to observe the following method. 
 
 (1.) To clear the equation of fractions by Rule III. 
 
 (2.) -To collect the unknown quantities on one side of the 
 equation, txid iliie known on the other, by Rule II. 
 
 (3.) To liiid the value of the unknown quantity by di- 
 viding each R^le of the equation by its coefficient, as in 
 Rule L 
 
 _ 
 
 50 Y.)j r ,-•-•; the three steps by which a simple equation containing 
 only onw urf y> .wn quantity may be solved. 
 
SIMPLE EQUATIONS. 57 
 
 Ex. 1. 
 
 ^. -. , , ^ . . . So; , a; 13 
 
 Find the value of a; m the equation -— — 1=-+-—. 
 
 t o o 
 
 7x 91 
 Multiply b^ 7, then Sx-\- 7=—+--. 
 
 Multiply by 5, then Ibx-^Sb-lfx+dl. 
 
 Collect the unknown quantities on ) 
 
 one side, and the known on the >• 15a: --7a: =9 1—35, 
 
 other ; ) 
 
 or 8a: =56. 
 
 56 
 Divide by the coefficient of a:, a:=— • =-7. 
 
 o 
 
 Ex.2. 
 
 0:4-3 X 
 
 Find the value of a: in the equation — 1=2—-. 
 
 5a: 
 Multiply by 5, then x+ 3— 5 = 10——; 
 
 Multiply by 7, then 7a: +21 -35=70-50:. 
 Collect the unknown quantities ) 
 
 on one side, and the known >• 7o:+ 5o:= 70— 21+35. 
 on the other ; ) 
 
 or 12a:=84; 
 __84 
 
 •*'^""12 
 Ex. 3. 
 
 Find the value of x in the equation 
 
 ■ x-l 2a:-2 , _ . 
 
 4a: ^— =^H ^ — [-24. 
 
 2 5 
 
 Multiply by the to< I 40^-5^+5=10.r+4^-4+240. 
 common multiple (10), j 
 
 By transposition, 40a;— 5a;— lOo:— 4a;=240— 4— 5. 
 or40a:— 19.1— 231, 
 i. e. 21a:=231 ; 
 
 .•.o:=_ = ll. 
 
 As the first step in this Example involves the case " where 
 the sign — stands before a fraction," when the numerator of 
 
58 ALGEBRA. 
 
 that fraction is brought down into the same line with 40^*, the 
 signs of both its terms must ,be changed^ for the reasons as- 
 signed in Ex. 3, page 44; and we therefore make it — 5:r-|-5, 
 and not 5a;— 5. 
 
 Ex*4. 
 
 X 
 
 Find the value of a; in the equation 2^—- +1=5:^7— -2. 
 
 Multiply by 2, then Ax—X'\-2—\()x—4:. 
 
 By transposition, 4+2=rl0a;--4a;+^j 
 or 6= 7;r; 
 6 t 
 
 6 
 . or^=-. 
 
 Ex. 5. 
 
 What is the value of a; in the equation 3aa;+25:r=3c+a? 
 
 Here 3a^+26^=:(3a+26) X:r ; 
 
 .-. (3a+25)x^=3c+a. 
 Divide each side of the equation by 3a +26, which is the 
 
 coefficient of a; ; then 07=- — — -,. 
 
 3a -{-26 
 
 Ex. 6. 
 Find the value of a; in the equation Sbx+az=2ax-\-4c, 
 Bring the miTcnown quantities to one side of the equation, and 
 the known to the other ; then, 
 
 Sbx—2axz=z4:C—a ; 
 but Sbx'-2ax={Sb—2a)xx; 
 ,\ {Sb—2a)x=4c—a. 
 
 ^Q ^ 
 
 Divide by 36— 2a, and x=— — --. 
 -^ ' 36— 2a 
 
 Ex.7. 
 
 Find the value of a; in the equation bx+x=z2x+Sa, 
 
 Transpose 2x, then bx-{-x—2x=Sa, ^ 
 
 or bx — x=3a ; 
 
 but 6a:— x=z{b — l)xi 
 
 .-. (6--1) x=Sa, 
 
 3a 
 and x=- — - . 
 — 1 
 
Si^MPLE EQUATIONS. 69 
 
 Ex. 8. x+l+lz=U. Ans. x=6. 
 
 Ex. 9. f+|+f=|+17^ Am. x=QO. 
 
 Ex. 10. 4a;-20=y+^. Ans. a;=10. 
 
 -p ,, a; , a; a; 1 . 6 
 
 ^'^•"•2+3-4=2- Ans..=-. 
 
 Ex. 12. Sx+l^"^. Ans. x=l. 
 
 Sx 
 
 ' Ex. 13. Y-5=29--2:r. Ans, x=U. 
 
 Sx 
 
 Ex.14. 6x—--—9z=5x. Ans. x=B(j. 
 4 
 
 Ex.15. 2x ^+15= J- — Ans. a;=12. 
 
 o 5 
 
 ^ Ex. 16. ^+^=20-^^. A,is. a;=18. 
 Ex.17. 5a!-?^^ + l=3;r:+^+7. .4»s. x=8. 
 
 Ex. 18. 2ax+b=2cx+4a. Ans. x=f^''^ 
 
 "2a— 3c 
 
 Ex. 19. 
 
 , . . 4 7a;-9 4/ ^a;-l\ . , 
 30.-4— .^-=-(6+— j; find ^. 
 
 Multiply by 15, 45^-60-28^4-36=: 72+4^;— 4. 
 
 45:i;-28a?-4.r— 72-4+60-36, 
 13:r=:92; 
 
; find a?. 
 
 60 ALGEBRA. 
 
 Ex. 20. 
 4a;+3 7a;— 29 _ 8a:+19 
 9 '^5a;-l^:~~18^ 
 
 Mult, by 18, 8a;+6+i^^^=8:r+19. 
 
 126^522 _ 
 ~5^:=32"-"-^^- 
 Multiply by 5(c-12, 126a; -522== 65a: -156, 
 126a:-65a;r=522-156, 
 61a;=366; 
 .-. x=6, 
 
 Ex.21. 
 
 Mult.by7(a:-1), 7-l^ij=i^=l. ^ 
 
 14(a;-l) 
 ^"— ^7~- 
 7(a:-l) 
 
 Divide by 2, 3= ^^^ .^ 
 
 3a;+21=7a;-7, 
 7a;-3a;=21+7, 
 4a; =28; 
 • - .•.a;=7. 
 
 Ex. 22. 
 ^ 8a;+5 , 7a;-3 16a;+15 , 2i . , 
 
 ^'' -r4-+6-.+2=— 28- +y ' ^^^ ' 
 
 Mult.by28,16a;+10+^-— = 16a;+15+9. 
 •' ba;+2 
 
 196^-84_ 
 
 6a;+2 ~ ' 
 196a;-84=84a;+28, 
 112a;=:112; 
 .•. a; = l. 
 
SIMPLE EQUATIONS. * 61 
 
 Ex.23. — 3- =-54 J—- ^^•^• 
 
 •r. c.^ 9^+20 4:r-12 x ^ a ^ 
 
 Ex. 24. -^^^^-^-t^. ^n.. 8. 
 
 Ex. 25. -^^ +^_^^=^+^^. ^n,. 4. 
 
 Ex.26. _^^-__+-^-^-_-. ^..4. 
 
 Ex.27. 4(5a;-3)-64(3-a:)-3(12a;-4)=96. Ans.Q. 
 
 Ex.28. 10(a:+|)-6^^~i^=23. ' ^W5. 2. 
 
 T. or. 30+6a; , 60+8a; ,,,48 . ^ 
 
 Ex. 29. -^;^+-^:p3-=14+— . An.Z. 
 
 PROBLEMS. 
 
 pROB. 1. What number is that to which 10 being added, 
 |ths of the sum shall be 66 ? 
 
 Let a;=:the number required; 
 then a;+10=the number, with 10 added to it. 
 
 Now Iths of (.+10)=|(.+ 10)=i(^tH)=?^. 
 
 But, by the question, |ths of (a; +10)= 66 ; 
 
 Hence, — - — =66. 
 5 
 
 Multiply by 5, then 3iir+30=330; 
 
 .•.3a:=330-30=300; or a; =^=100. 
 
 Prob. 2. What number is that which being multiplied by 
 
 6, the product increased by 18, and that sum divided by 9, 
 
 the quotient shall be 20 ? 
 
 Let a:=the number required ; 
 
 then 6a; = the number multiplied by 6; 
 
 6a; +18= the product increased by 18, 
 
 , 6a;+18 , -...-, 1 n r. 
 
 and * — - — =that sum divided by 9. 
 
 6 
 
62 * ALGEBRA. 
 
 Hence, by the question, — - — =20. 
 
 Multiply by 9, then 6^ + 1 8 = 1 80, 
 
 or6a;=180-fpl8 = 162; ov x=~=:21. 
 
 6 
 
 pROB. 8. A post is Jth in the earth, fths in water, and 13 
 
 feet out of the water. What is the length of the post? 
 
 Let a;=length of the post in ft. ; 
 
 then ^=the part of it in the earth, 
 
 —-= the part of it in the water, 
 
 13= the part of it out of the water. 
 But part in earth + part in water + part out of water =: 
 whole post ; 
 
 ... (I) + f^) + 18 =.. 
 
 15a: 
 Multiply by 5, then x-\ — — + 65= 5a;; 
 
 Multiply by 7, then 7x+l5x+455=:S5x, 
 
 or 455=350?— 7rr—15a;=13ar. 
 
 455 
 Hence a; =-—-=35 length of post in ft. 
 
 Prob. 4. After paying away J-th and -^-th of my money, I 
 had £85 left in my purse. "What money had I at first ? 
 Let ir= money in purse ^t first ; 
 
 then -+-= money paid away. 
 
 But money at first— money paid away = money remaining. 
 
 Hence x — (1+7I = ^^j 
 
 1. e., x—-—~: = 85. 
 > 4 7 
 
 Multiply by 4, then 4:X—x — --=340; 
 
 Multiply by 7, then 28a;— 7ir—4.r= 2880. 
 .-. 17a;=2380; 
 
 or 0:=-— -=£140. 
 
 \ % 
 
SIMPLE EQUATIONS. 63 
 
 Prob. 5. What number is that, to which if I add 20, and 
 from |ds of this sum I subtract 12, the remainder shall be 10 ? 
 
 Ans, 13. 
 
 Prob. 6. What number is that, of which if I add -Jd, ^Jth, 
 and f ths together, the sum sl^ll be 73 '? Ans. 84. 
 
 Prob. 7. What number is that whose Jd part exceeds its 
 ithby72? Ans, 540. 
 
 Prob. 8. There are two numbers whose sum is 37, and if 
 S times the lesser be subtracted from 4 times the greater, and 
 this difference divided by 6, the quotient will be 6. What 
 are the numbers ? Aris, 21 and 16. 
 
 Prob. 9. There are two numbers whose sum is 49 ; and if 
 ^th of the lesser be subtracted from ^th of the greater, the re- 
 mainder will be 5. What are the numbers 1 ^ 
 
 Ans, 35 and#iy 
 
 Prob. 10. To divide the number 72 into three parts, so 
 that -J- the Jlrst part shall be equal to the second, and |ths of 
 the second part equal to the third. 
 
 Ans, 40, 20, and 12. 
 
 Prob. 11. A person after spending Jth of his income plus 
 £10, had then remaining ^^ of it plus £35. Required his 
 income. A7is. £150. 
 
 Prob. 12. A gamester at one sitting lost J-th of his money, 
 and then won 10 shillings; at a second he lost -J^ of the re- 
 mainder, and then won 3 shillings ; after which he had 3 
 guineas left. What money had he at first ? Ans, £5. 
 
 Prob. 13. Divide the number 90 into four such parts, that 
 the first increased by 2, the second diminished by 2, the third 
 multiplied by 2, and the fourth divided by 2, may all be equal 
 to the same quantity. Ans. 18, 22, 10, 40. 
 
 Prob. 14. A merchant has two kinds of tea, one worth 
 9s. 6(f. per lb., the other 13^. 6d. How many lbs. of each 
 must he take to form a chest of 104 lbs., which shall be worth 
 £56? Ans. 33 at 13^. 6d. 
 
 71 at 95. 6d. 
 
64 ALGEBRA. 
 
 Prob. 15. Three persons, A, B, and C, can separately reap 
 a field of corn in 4, 8, and 12 days respectively. In how 
 many days can they conjointly reap the field ? 
 
 Let x = No. of days required^ by them to reap the field ; 
 then if 1 represent the work, or me reaping of the field, 
 -J-=the part reaped by A in 1 day, 
 
 1__ u a u B " 
 
 1 __ a a u Q U 
 
 ... |+-i.4-?=: " " « all three " 
 
 But the part reaped by all three in 1 day multiplied by the 
 number of days they took to reap the field, is equal to the 
 whole work, or 1 ; 
 
 •••(i+KTV)«' = l: 
 Clearing of fractions by multiplying by 24, 
 (6+3+2) a;=24, 
 lla;=24; 
 .*. x=2^^ days. 
 
 Prob. 16. A man and his wife usually drank a cask of 
 beer in 10 days, but when the man was absent it lasted the 
 wife 80 days ; how long would the man alone take to drink 
 it 1 Ans. 15 days. 
 
 Prob. 17. A cistern has 3 pipes, two of which will fill it in 
 8 and 4 hours respectively, and the third will empty it in 6 
 hours ; in what time will the cistern be full, if they be all set 
 a-running at once ? Ans, 2h. 24m. 
 
 Prob. .18. A person bought oranges at 20d. per dozen ; if 
 he had bought 6 more for the same money, they would have 
 cost 4d. a dozen less. How many did he buy ? 
 Let ojnrthe number of oranges; 
 then.T+6= " " " " at 4c?. less per dozen. 
 
 Price of each orange in 1st case=f f =:fd 
 and " " " " " 2d " =i|=|cZ. 
 
 .*. the cost of the oranges =—. 
 o 
 
 But we have also 
 
 the cost of the oranges =| (a; +6). 
 
 Two independent values have therefore been obtained for 
 
SIMPLE EQUATIONS. 65 
 
 the cost of the oranges ; these values must necessarily be equal 
 to each other ; 
 
 Multiplying each side of thft equation by 3, 
 '^5a;=4 (:rH-6), 
 5a;=4a;+24; 
 .-. a; =24, the No. of oranges. 
 Prob. 19. A market-woman bought a certain number of 
 apples at two a penny, and as many at three a penny, and 
 sold them at the rate of five for twopence ; after which she 
 found that instead of making her money again as she expected, 
 she lost fourpence by the whole business. How much money 
 had she laid out ? Ans. 86\ Ad. 
 
 Prob. 20. A person rows from Cambridge to Ely, a dis- 
 tance of 20 miles, and back again, in 10 hours, the stream 
 flowing uniformly in the same direction all the time ; and he 
 finds that he can row 2 miles against the stream in the same 
 time that he rows 3 with it. Find the time of his going and 
 returning. 
 
 Let 3iP=zNo. of miles rowed per hour with the stream ; 
 .-. 2x— " " " " " " against " 
 Now the distance divided by the rate per hour gives the time ; 
 
 20 
 
 .*. — =:-the No. of hours in going down the river, 
 
 20 
 
 and ^ == " " " " coming up " 
 
 But the whole time in going and returning is 10 hours ; 
 
 Dividing by 10, ~ +1=1. 
 
 Multiplying each term of the equation by 3a?, 
 2+3=:3:r; 
 
 and .'. 3.r=5, miles per hour down, 
 
 20 
 /. the time in going down the river=:— =4 hours, and con- 
 
 scquently the time of returning=10— 4=6 hours. 
 
 6* 
 
68 ALGEBRA. 
 
 Prob. 21. A lady bought a hive of bees, and found that the 
 price came to 2s, more than f ths and |-th of the price. Find 
 the price. Ans, £2. 
 
 Prob. 22. A hare, 50 leap^, before a greyhound, takes 4 
 leaps for the greyhound's 3 ; btrc two of the greyhound's leaps 
 are equal to three of the hare's. How many leaps will the 
 greyhound take to catch the hare ? 
 
 Let X be the No. of leaps taken by the greyhound ; 
 
 then --- will be the corresponding number taken by the hare. 
 
 Let 1 represent the space covered by the hare in 1 leap ; 
 
 then- " " " " " greyhound" 
 
 .*. ~ X 1 or — will be the whole space passed over by the 
 o o 
 
 2 Sx 
 
 hare before she is taken ; and a; X - or -~ will be the space 
 
 2 ii 
 
 passed over in the corresponding time by the greyhound. 
 Now, by the problem, the difference between the spaces 
 respectively passed over by the greyhound and hare is 50 X 1, 
 or 50 leaps ; 
 
 Sx Ax 
 
 •'- 2 - ¥ =^^' • 
 
 9a;— 8a::=300; 
 .-. ir=300 leaps. 
 
 ON THE SOLUTION OF SIMPLE EQUATIONS, CONTAINING TWO 
 OR MORE UNKNOWN QUANTITIES. 
 
 51. For the solution of equations containing two or more 
 unknown quantities, as many independent equations are re- 
 quired as there are unknown- quantities. The two equations 
 necessary for the solution of the case, when two unknown 
 quantities are concerned, may be expressed in the following 
 manner : 
 
 ax-\-hy=:^c 
 a'x-\-h'y^^c\ 
 Where a, 6, c, a\ 6', c\ represent known quantities, and x^ y, 
 
SIMPLE EQUATIONS. 67 
 
 the unknown quantities whose values are to be found in terms 
 of these known quantities. 
 
 There are three different methods by which the value of 
 one of the unknown quantities may be determined. 
 
 FIRST METHOD. 
 
 Find the value of one of the unknown quantities in terms 
 of the other, and the kno-svn quantities by the rules already 
 given. Find the value of the same unknown quantity from 
 the second equation. 
 
 Put these two values equal to each other ; and we shall 
 then have a simple equation, containing only one unknown 
 quantity, which may be solved as before. 
 
 Ex. 1. Given x+yz=z^ -(l))i.i;j j 
 
 . ^ x^y=4. U} to find a: and 3^. 
 
 From (1) y=S—x--'- (a) 
 
 " (2) y=x-4: 
 
 Putting these twd values equal to each other, we get 
 
 a;— 4=8— a;, 
 
 2^ = 12; 
 
 .'.x=z6, < 
 
 By (a) y=8-a;=8-6=2.. 
 
 Ex. 2. Let x+4y=l6 (1) 
 
 4:x+ y=34 (2) 
 
 From ^nation (1), we have x=16—4t/, 
 
 (2) « « rr=?^, 
 
 Hence by the rule, —j — =16— 4y, 
 
 34— y=64— 16y, 
 15^=30; 
 
 .-. 2/ =2. 
 It has already been shown that ii;=16 — 4y= (since y=2; 
 
 and .-. 4y=8) 16—8=8. 
 
 <i . 
 
 51. For the solution of equations containing two or more unknown quan- 
 tities, how many independent equations are necessary ? State the ^r«< 
 method of solution. 
 
68 ALGEBRA, 
 
 Ex.3. 
 
 5a:+3y=38) ^n. i^=^ 
 
 -2y=6 f ^^'- •|2/=3. 
 
 Ex.4. 2rr-3y=~l 
 Sx 
 
 SECOND METHOD. 
 
 From either of the equations find the value of one of tha 
 unknown quantities in terms of the other and the known quan- 
 tities, and for the same unknown quantity substitute this valu© 
 in the other equation, and there will arise an equation which 
 contains only one unknown quantity. This equation can be 
 solved by the rules already laid down. 
 
 ~ Ex. 1. y-x^^ (1) 
 
 x+y=^ (2) 
 
 From (1) y—2+x, {a) 
 This value of y being substituted in (2), gives * 
 
 x+2+x=S, 
 2.^=6; 
 
 .\Xz=^, ^ 
 
 And by (a) y=:2+a:=2+3=5- 
 ^'^' ^4?+8.=31 (1) 
 
 Lti_pl0;r = 192 (2) 
 
 Qeariiig equation (1) of fractions, 
 
 ir+2+24y=93, or a;+24y=:91 (a). 
 Gearing equation (2) of fractions, ^ 
 
 2/+5+40a;=768, or ?/+40a:=763 (/5). 
 From (a) ir=91— 24?/. 
 Substitute this value of a:, accordmg to the rule in equation 
 ifi)) and 
 
 2/+40(91-24y)rr:763, 
 or, y+3640~960y=763; 
 
 .-. 959y=3640-763=:2877, 
 and y=3. 
 By referring to equation (a) we have 2;=91—24y= (since 
 y =3 ; and .-. 24y=72) 91 -72=19. 
 
 Enunciate ilnQ second metTwd of solution. 
 
SIMPLE EQUATIONS. 69 
 
 4a;+3y=31) .^ i x=4 
 
 THIRD METHOD. 
 
 Multiply the first equation by the coefficient of a in the 
 second equation, and then multiply the second equation by the 
 coefficient of a; in the first equation ; subtract the second of 
 these resulting equations from the Jirst, and there will arise an 
 equation which contains only y and known quantities, from 
 which the value of y can be determined. 
 
 It must be observed, however, that if the terms, which in the 
 resulting equations are the same, have unlike signs, the re- 
 sulting equations must be added, instead of bemg subtracted, 
 in order that x may be eliminated (^. e., expelled from the 
 equations). 
 
 Ex. 1. Given 5:r+4y=55 - - - (1) 
 3a;+2?/=:31 - - - (2) 
 To find the values of x and y 
 
 Mult. (1) by 3, then 15^+12y=165 
 " (2) by 5, " 15a;+lQy:^155 
 
 /. by subtraction, we have 2y= 10 * . 
 
 .-. y= 5. 
 Now from equation (1) we have 
 
 55— 4y 
 
 _ 55-20 
 
 ~" 5 
 _35 
 
 ■~5 
 
 =7. 
 
 Ex. 2. Let the proposed equations be 
 
 ax+h y=zc - - - (1) 
 
 a'x+b'x=c' (2) 
 
 Mult. (1) by a', and adx-\-a%yz=za^c 
 " (2) by a, " aa'x+ah'y=.ac' ; 
 
 How are equations solved by the third metlwd f 
 
70 ALGEBRA. 
 
 .•, by subtraction, (a'h — ab ')y = a'c — ac' 
 
 a'c — ac' 
 
 Mult. (1) by h\ and ah'x-\-hh'yz=:b c 
 Mult. (2) by b, and a'bx-^b'y=^bc' 
 By subtraction, (ab' —a'b^x-=b'c—bd \ 
 
 b'c—bc' 
 
 .'. X- 
 
 Ex. 3. Let 3a?+4y=29 (1^ 
 
 ab'—a'b' 
 
 ' (1) 
 17a;-3y=:36 (2) 
 
 Mtilt. (1) by 3, then 9;r+12y=87 
 Mult. (2) by 4, then 68a;--12yr=144. 
 
 The signs of 12y in the two equations are unlike ; .'. to 
 eliminate y from them, the two last equations must be added 
 together ; and then 
 
 7707=231 ; 
 .-. ;r=3. 
 From (1) we have 4y=29~3^, 
 
 =29—9, (since a;=3 ; and .-. 3:r=9) 
 =20; 
 
 - Ans, \ ' 
 
 Ex.4. Let 6.r4-3y=33) ^„_ ^ x=S 
 
 ;y=5. 
 
 Ex.5. /<)4a:+3y=31) ^ ^ ^^^] J^J 
 
 Ex.6. r//3^+2y=40) rAnsA''=l^ 
 
 7 
 4. 
 
 IC 
 
 7. 
 
 Ex.7. rA.\ 5^-4y=19) . . . . ^^^ < ^=7 
 
 Ex.8. /^l 8^+7y=79) .... ^^,^ 5^=10 
 
 ix= 
 
 Ans. 
 
 =4. 
 
SIMPLE EQUATIONS. 
 
 7X 
 
 Ex. 10. 
 
 x+i/ 
 
 -2y=2 
 
 Ex. 11. 
 
 2x—4i/ 
 
 5 
 2a:— 3 
 
 
 67 
 
 2 
 
 5a;— 13?/= 
 
 Ans. 
 
 Arts. 
 
 Ex. 12. 
 
 Sx-'7y=2x+i/-\'l 
 
 
 - - Ans, 
 
 (x=U 
 
 x=S 
 
 y=i 
 
 a;=13 
 
 52. When three unknown quantities are concerned, the 
 most general form under which equations of this kind can be 
 expressed, is ax-rby+cz=d (1) 
 
 a'a;+6V+c'^=^' (2) 
 - a"a;+6'V+c"2=^" (3), 
 and the solution of these equations may be conducted as in 
 the following example : 
 
 Ex. 1. Let ^-+%+4.=29 (1) ) ^^ ^^ ^^^ ^^^^^^ 
 3a;+2y+5^=32 (2) V of x v z 
 
 I Multiply (1) by 3, then 6a;+%+12^=87 (4) 
 Multiply (2) by 2, then' 6a:+4y+ 10^=64 (5). 
 
 Subtract (5) from (4) then 5y+ 2^=23 (a). 
 
 Multiply (2) by 4, then 12a;+8y+205:=128 
 Multiply (3) by 3, then 12a;+92/+ 6^=75 
 
 Subtract ... - -2/+14^=53 (/?)• 
 
 II. Hence the given equations are reduced to, 
 
 5?/+ 2^=23 (a) 
 -y+14^=53 (/3). 
 
72 ALGEBRA. 
 
 / 
 
 Again - - 5y+ 2z—2S 
 Mult. (i8) by 5,then-5y+7O0=265 
 
 By addition - - - 72^=288, or ^=^=4 
 
 From equation (/3) '- - ^^=14^— 53=56— 53=3. 
 
 trr T. .. /ix 29—3^—4^ 29—25 
 
 111. From equation (1) - x= ^ — 
 
 2 2 
 
 Ex.2. x+y+z=90 ) (x=S5 
 
 2a;+40=3y+20 L - - Ans.\y=SO 
 
 2^+40=4^+10 ) (2=25. 
 
 Ex. 3. x+ y+ z= 6S ) . ( :r=24 
 
 a;+2y+32=105 V Ans.\y= 6 
 
 a:+3y+42=134 )' (2; =23. 
 
 PROBLEMS. 
 
 Prob. 1. There are two numbers, such, that 3 times the 
 
 greater added to ^d the less is equal to 36 ; and if twice the 
 greater be subtracted from 6 times the less, and the remain- 
 der divided by 8, the quotient will be 4. What are the 
 numbers ? 
 
 Let a;=the greater number, 
 2/= the less number; 
 
 Then 3ar+|=36 
 6y— 2a?___ 
 
 ^^^» 6y-2a;= 32; 
 
 8 
 
 Or,y+9a:=108 (1) 
 6y-2a:= 32 (2) 
 Mult, equation (1) by 6, then 6?/+54:r=648 
 .Subtract " (2) then 6y— 2x= 32; 
 
 then 56:i;=616; 
 ..x-^^-U. 
 
 JFrom equation (1) 2/=108—9a;= 108—99=9. 
 
 ^IProb. 2. There is a certain fraction, such, that if I add 8 
 ttg the numerator, its value will be ^d ; and if I subtract ons 
 
SIMPLE EQUATIONS. 73 
 
 from the denominator, its value will* be |th. What is the 
 fraction ? 
 
 Let a;=its numerator, ) , , .i /. ^. - x 
 ' ^ then the fraction is _. 
 
 i:\' 
 
 y= denominator, J y* 
 
 Add 3 to the numerator, then — 
 
 y 3 
 
 X 1 
 
 ' 5a;=2/— I. 
 
 Subtract one from denom'., and , — ^ 
 
 y— 1 5 
 
 By transposition^ y—Sx=9 (!) 
 
 y-5r=l (2). 
 
 'Subjtract equation (2) from (1), and we have 
 
 2x=S; 
 
 8 
 /. a?=-=4, the numerator. 
 
 From equation (1) y=9+3a;=9+12=21, the denominator. 
 
 4 
 
 Hence the fraction required is — . 
 
 Prob. 3. A and B have certain sums of money ; says A 
 to B, Give me £15 of your money, and I shall have 5 times 
 as much as you will have left ; ^ys B to A, Give me £5 of 
 your money, and I shall have Exactly as much as you will 
 have left. What sum of money had each ? 
 
 Let ir=A's money ) ,, 4, i f^_ i ^^^^ ^ would have after 
 y=B's money ) ( receiving £15 from B. 
 
 ♦ y— 15= what B would have left. 
 
 ■ ' -A ' , K ( what B would have after 
 Again, y+ 5= < - - n- e. \ 
 
 =* ' ^ ' ( receiving £d ii'om A. 
 
 X— 5= what A would have left. 
 
 Hence, by the question, a?+ 15=5 X(y— 15) =5y— 75, ) 
 F and y-j- 5=rc— 5. 
 
 Mf By transposition, 5?/— a:=90 (1),) 
 
 f andy— x— — \0 (2). f 
 
 Set down equation (1) 5y-— rr=90. 
 
 Multiply eq^ (2) by 5, 5y-5x=-50. 
 
 Subtract (2) from (1) 4a:=140; 
 
 7 
 
<N^ ALGEBRA. 
 
 140 ^^ ., 
 .'. x=—-=Sd, As money. 
 
 From equation (1) 5y=90+a;=90+35=125; 
 
 .•.^=-—=25, B's money. 
 o 
 
 Prob. 4. What two numbers are those, to one-third the 
 sum of which if I add 13, the result shall be 17 ; and if from 
 half their difference I subtract one, the remainder shall be 
 two ? Ans. 9, and 3. 
 
 Prob. 5. There is a certain fraction, such, that if I add one 
 to its numerator, it becomes -^ ; if 3 be added to the denojni- 
 nator, it becomes J. What is the fraction 1 Ans. j^. 
 
 Prob. 6. A person was desirous of relieving a certain 
 number of beggars by giving them 2s. 6d. each, but found 
 that he had not money enough in his pocket by 3 shillings; he 
 then gave them 2 shillings each, and had four shillings to 
 spare. What money had he in bis pocket ; and how many 
 beggars did he relieve 1 
 
 Let a; = money in his pocket (in shillings) ; 
 y= number of beggars. 
 
 Then 2^Xv or— = M^^' ^^ ^^^'^^^' which would have 
 ^ 2 ( been given at 25. 6d. each, 
 
 and 2Xy, or 2y= " " at 2s, each. 
 
 Hence,by the question ,-^=rrr+ 3 (1) \ 
 
 and 2y=x—4 (2). 
 Sub*. (2) from (1), then |=7,. or y=14, the No. of beggars. 
 From eq". (2), a:=22/+4=28+4=32 shillings in his pocket 
 
 Prob. 7. A person -has two horses, and a saddle worth 
 £10; if the saddle be put on the j^rs^ horse, his value be 
 comes double that of the second; but if the saddle be put on 
 the second horse, his value will not amount to that of the 
 first horse by £13. What is the value of each horse? 
 
 Ans, 56, and 33. 
 
 Prob. 8. There is a certain number, consisting of two 
 digits. The sum of those digits is 5 ; and if 9 be added to 
 
SIMPLE EQUATIONS. 75 
 
 the nun^ber itself, the digits will be inverted. What is the 
 number ? 
 
 Here it may be observed that every number consisting of 
 two digits is equal to 10 times the left-hand digit plus the 
 right-hand digit : thus, 34 = 1 >>3 + 4. 
 Let x=:left-hand digit. 
 y=iright-hand digit. 
 Th^n 10j;+y=the number itself, 
 and 10y+a:=:the number w^ith digits inverted. 
 Hence, by the question, x-{-y=zb (1), 
 and 10a;+y+9z=:10y-fa;,or 9^— 9y=— 9,or a;—y=— 1 (2). 
 Subtract (2) from (l),.then 2y=6, and 2/= 3, 
 
 a;=i5— y=:5 — 3=2; 
 .'. the number is (10.r+2/)=23. 
 Add 9 to this number, and it becomes 32, which is the 
 number with the digits inverted, 
 
 Prob. 9. There are two numbers, such, that -J- the greater 
 added to \ the less is 13 ; and if -J- the less be taken from 
 ^ the greater, the remainder is nothing. What are the num- 
 bers ? ^715. 18 and 12. 
 
 Prob. 10. There is a certain number, to the sum of whose 
 digits if you add 7, the' result will be three times the left- 
 hand digit; and if from the number itself you subtract 18, 
 the digits will be inverted, Wlijfc is the number 1 
 
 Ans. 53. 
 Prob. 11. A merchant has two kinds of tea, one worth 
 95. 6(f. per lb., the other 13s. 6c?. How many pounds of each 
 must he take to form a chest of 104 lbs. which shall be worth 
 £56? . ^?^s. 33 at 13.S'. 6c?. . 
 
 71 at 95. Qd, 
 Prob. 12. A vessel containing 120 gallons is filled in 10 
 minutes by two spouts running successively ; the one runs 14 
 gallons in a minute, the other 9 gallons in a minute. For 
 what time has each spout run ? 
 
 Ans, 14 gallon spout runs 6 minutes. 
 9 gallon spout runs 4 minutes. 
 Prob. 13. To find three numbers, such, that the Jlrst with 
 J the sum of the second and thii^d shall be 120 ; ^Q^seco7id 
 with \\h the difference of the third and first shall be 70 ; and 
 \ the sum of the three numbers shall be ^5. 
 
 Ans. 50, 65, 75. 
 
76 
 
 ALGEBRA. 
 
 CHAPTER IV. 
 
 ON INVOLUTIOI|,AND EVOLUTION. 
 
 ON THE INVOLUTION OF NUMBERS AND SIMPLE ALGEBRAIC 
 QUANTITIES. 
 
 53. Involution^ or " the raising of a quantity to a given 
 power," is performed by the continued multiplication of that 
 quantity into itself till the number of factors amounts to the 
 number of units in the index of that given power. Thus, the 
 square of a=aXci^=cb^ \ the cube of b=bX^Xl>=b^ ', the 
 fourth poiver of 2=2x2x2x2=16; the Jiftk poiver of 3 
 
 =3x3x3x3x3=243; &;c., &;c. 
 
 54. The operation is performed in the same manner for 
 simple algebraic quantities, except that in this case it must be 
 observed, that the powers of negative quantities are alter- 
 nately + and — ; the ev^/i powers being positive, and the odd 
 powers negative. Thus the square of +2a is +2aX +2a or 
 +4a^; the square of —2a is — 2aX — 2a or +4a^; but the 
 cube of — 2a = --2aX— 2aX— 2a=+4a2x— 2a=— 8al 
 
 The several powers of - 
 are, ^ 
 
 a a Oj^ 
 
 a a a_a^ 
 
 Cube =lXlX^-^' 
 
 4th _ a a a a a^ 
 
 p,^er--X^X^X^=P 
 
 &;c.=&c. 
 
 Mi( 
 
 d the several powers of— ;r-, 
 2c 
 
 Squ. 
 
 ~ 2c^ 2c~"'"4c^' 
 
 4th 
 power - 
 
 2c 
 h 
 
 '2'c 
 b 
 
 X 
 
 2c 
 
 2c 
 
 X- 
 
 b_ 
 
 b 
 
 2c 
 
 X- 
 
 "Sc^' 
 b^ 
 
 "2c 
 
 = + 16?' ^^•^^^- 
 
 ON THE INVOLUTION OF COMPOUND ALGEBRAIC QUANTITIES. 
 
 55. The powers of compound algebraic quantities are 
 
 53. What is involuti(5^? How is it performed? — 54. In what manner ia 
 invohiti on performed for simple algebraic quantities? — 55. How are- the 
 powers of compound quantiti 3S raised ? 
 
INVOLUTION. 77 
 
 raised by the mere application of the Rule for Compound 
 Multiplication (Art. 34). Thus, 
 
 Ex. 1. What is the square Ex. 2. What is the cube of 
 
 ofa+26? a'-xl 
 
 a +26 ^ a^-x 
 
 a 4-26 a^^x 
 
 Square =a^+4a6+ 46^ Square = a*— 2a^a;-fiu^ 
 
 _-^_a^+2a'^^--^ 
 Cube=a«-3a^ar+3aV-a;8 
 
 Ex. 3. 
 
 Whft is the 5th power of a+6 1 
 a +6 
 
 a+5 
 
 + ah +6^ ^^ 
 
 a^ + 2a6 + 6^ = Square 
 
 ' ___^ « 
 
 a^+2a^6+ ay 
 
 + a^6+ 2«^>' +6 ^ 
 
 aH 3a^6+ 3a6^ +63=Cube 
 a+ 6 
 
 a*+3a^6+ 3a'6^+ a6^ 
 + a^6+ 3a^6^+ 3a6^+ 6^ 
 
 ^*+4^^+ 6a'Z»'+ 4a5^+^>'*=4th Power 
 a_jr_b 
 
 a^J^^a'h + ^aW-\- Wh''-\-ah^ 
 + a^6 + 4a^6^+ 6a^6^+4 a5^+6^ 
 
 gs ^ 5a46 + 10ag6^ + 10a^6^+5a6'+6^=:5th Power , 
 7* 
 
78 * ALGEBRA. 
 
 Ex. 4. The 4:'^ poiver of a+Sb is a*+12a^6+54a^Z>'^+108a5» 
 + 816^ 
 
 Ex.5. The square of Sx'+2x+5 is 9x'+12x^+S4.x^+2(h 
 +25. 
 
 Ex. 6. The cw6e of Sx-5 is W-135a;^+225a;--125. 
 
 Ex. 7. The cube of a;^-2a:+l is ic«-6aj^+15^*-20;r^+15a?« 
 -6x+l. 
 
 ' Ex. 8. The square of a+5+c is a2+2a5+62+2ac+25c+d 
 
 ON THE EVOLUTION OF ALGEBRAIC QUANTITIES. 
 
 56. Evolution^ " or the rule for extracting the root of any 
 quantity," is just the reverse of Involution ; and to perform 
 the operation, we must inquire what quantity multiplied into 
 itself, till the number of factors amount to the number of 
 units in the index of the given root, will generate the quantity 
 whose root is to be extracted. Thus, % 
 
 (1.) 49=7 X7 ; .-. the 5^. roo^of 49 (or by Def^ 15,y'49)=:7. 
 
 m i^^j-b^^-bX-bX-b) ,\cube rooto^-¥ {^Z:h') = '-'^' 
 •^ _ 16a*_2a 2a 2a 2a ^ 4 /16a*_2a 
 
 ^ '^ 8iJ*~36 ^36^36 ^S^"*'* V 8l6"*~'36' 
 
 (4.) 32=2X2X2X2X2;. -.^32=2, 
 
 (5.) a'^^a^ Xa'Xa^', /. ^a'-a\ 
 
 Hence it may be inferred, that any root of a simple quart,* 
 tity can be extracted^ by dividing^^ its index, if possible, by the 
 index of the root, 
 
 57. If the quantity under the radical sign does not admit 
 of resolution into the number of factors indicated by that 
 sign, or, in other words, if it be not a complete power, then its 
 exact root cannot be extracted, and the quantity itself, with 
 
 . the radical sign annexed, is called a Surd. Thus 'y/37, J/a\ 
 V/>^, ^47,- &c., &c., are Surd quantities. 
 
 . » 
 
 56. What is EvolMtion ? How is ifc performed ?— 57. What is a Surd quantity f 
 
EVOLUTION. 79 
 
 58. In the involution of negative quantities, it was observed 
 that the even powers were all +, and the odd powers — ; 
 there is consequently no quantity which, multiplied into itself 
 in such manner that the number of factors shall be even^ can 
 generate a negative quantity. •Hence quantities of the form 
 
 -/^^ -^=ao~y^^ V-^^ -V^^^ '^^•' ^^'^ ^^^^ ^^ 
 real root, and are therefore called impossible, 
 
 59. In extracting the roots of compound quantities, we 
 must observe in what manner the terms of the root may be 
 derived from those of the power. For instance (by Art. 55, 
 Ex. 3), the square of a+5 is a^-f-2a/^+^^ where'the terms are 
 arranged according to the powers of a. On comparing a-^b 
 with a^-|-2a6+Z>^, we observe that the first term of the power 
 (a^) is the square of the first term of the 
 
 root {a). Put a therefore for the first o}-\-2ah-\-b^ la-^-b 
 term of the root, square it, and subtract o? \ 
 
 that square from the first term of the %^ hA.}fi 
 
 power. Bring down the other two terms 2^+^ Io^7,Ia2 
 2ah-\-'t\ and double the first term of the |2aH-6_ 
 
 root; set down 2a, and having divided ^ ^ 
 
 the first term of the remainder (2a6) by " 
 
 it, it gives 5, the other term of the root ; 
 
 and since 2ab-\'b'^=^{2a-\-b)b^ if to 2a the term b is added, 
 
 and this sum multiplied by 5, the result is 2a5-j-6'^; which 
 
 being subtracted from the two terms brought down, nothing 
 
 remains. 
 
 60. Again, the square of a+^ + c (Art. 55, Ex. 8.) is a^+I 
 2ab-\-b'^-{-2ac-{-2bc-\-c^\ in this case the root may be? 
 
 continuingthe 2a+i|2«J+6« 
 process m the L z, 72 
 
 last Article. \}±^ 
 
 2ac+2bc+c'' 
 2ac-\-2bc—c^ 
 
 Thus, having 2a+2b-{-c 
 
 tound the two 
 
 first terms {a-\-b) * % * 
 
 of the root as '"^ 
 
 before, we bring down the remaining three terms 2ac-\-2bc 
 
 58. Explain the nature of an impossible quantity. — 59. How are the roots 
 of compound quantities extracted ? 
 
80 ALGEBRA. 
 
 + c* of the power, and dividing 2ac by 2a, it gives c, the 
 third term of the root. Next, let the last term (J) of the 
 preceding divisor be doubled, and add c to the divisor thus 
 increased, and it becomes 2a+26-|-c; multiply this new 
 divisor by f, and it gives 2ac-t-26c+c^, which being subtracted 
 from the three terms last brought down, leaves no remainder. 
 In this manner the following Examples are solved.^ 
 
 Ex. 1. 
 
 ^x' 
 
 4:X' + 
 
 6x'+^x' 
 4 
 
 A 8 . Q IK \ 20aj^+15a:+25 
 4:X'\-Sx-i-i) j 20^;^ 4- 15^^+25 
 
 Ex.2. 
 
 x^+4:x'+2x^+9x^-4x+4:{x^+2x^-^x+2 
 
 2x^+2x')4:x'+2x* 
 4x' +4.x' 
 
 2x^+4.x^-x)-2x^-{-9x''-4x 
 -2x^-4x^+ x'' 
 
 2x^+4x'^2^A^+4:x'+Sx'-4x+4t 
 +4aj'+8a;'— 4^+4 
 
 Ex. 3. The square root of 4x^+4x7/+y'' is 2^+y. 
 
 Ex. 4. The square root of 25a^+30a5+95' is 5a +85. 
 
 Ex. 5. Find the square root of 9a;*+12^^+22a?^+12;r+9. 
 
 Alls. Sx'-{-2x+S. 
 
EVOLUTION. 81 
 
 Ex. 6. Extract the square root oMx^—Wx^+24:X^-'l6x+4. 
 
 Ans. 2x'—4x-y2. 
 
 4 
 "9 
 
 Ex. 7. Find the square root of S6x'~S6x'+l7x'-4x'^^' 
 
 ' 2 
 
 A71S. 60;"^— 3.1; -f-. 
 o 
 
 Of) 1 /» 
 
 Ex. 8. Extract the square root of a;^+8^'+24-| — --\ — 5. 
 
 X X 
 
 4 
 
 •4^5. a;^+4H — 5. 
 
 ON THE INVESTIGATION OF THE RULE FOR THE EXTRACTION 
 OF THE SQUARE ROOT OF NUMBERS. 
 
 Before we proceed to the investigation of this Rule, it will 
 be necessary to explain the nature of the common arithmeti- 
 cal notation. 
 
 61. It is very well known that the value of the figures in 
 the common arithmetical scale increases in a tenfold propor- 
 tion from the right to the left ; a number, therefore, may be 
 expressed by the addition of the iinits^ tens^ hundreds^ &c., of 
 which it consists. Thus the number 4371 may be expressed 
 m the following manner, viz., 4000 + 300+70 + 1, or by 4X 
 1000+3x100+7x10 + 1 ; hence, if the digits* of a number 
 be represented by a, ^, c, d^ e, &c., beginning from the left 
 hand, then, 
 
 A No. of 2 figures may be expressed by 10a +5. 
 . " 3 figures " by lOOa+105 + c. 
 
 " 4 figures " by 1000a + 1006 + 10c+cf. 
 
 &;c. &c. &c. 
 
 62. Let a number of three figures (viz., lOOa+105+c) be 
 
 * By the digits of a number are meant the figures which compose 
 it, considered independently of the value which they possess in the arith- 
 metical scale. Thus the digits of the number 537 are simply the num- 
 bers 5, 3, and 7 ; whereas the 5, considered with respect to its place in 
 the numeration scale, means 500, and the 3 means 30. 
 
 61. Explain the common arithmetical scale of notation. What is a 
 digit? — 62. Show the relation between the algebraical and numerical 
 method of extracting the square root, and that they- are identical. 
 
82 ALGEBRA. 
 
 squared, and its root extracted according to* the Rule in Art. 
 60, and the operation will stand thus ; 
 
 I. 10000a'+2000ah + 100b'+200ac+20bc+c\100a+l0b+c 
 IQQO Oa^ ^ 
 
 200a + 106) 2000a5 + 1006^ 
 2000a6+100^>^ 
 
 200a+206+c) 200aH-205c+e* 
 200ac+20bc-\-c' 
 
 , _o / and the operation is transformed into the fol- 
 ^ZiC lowing one; 
 
 40000+12000+900+400+60+lf200+30+l 
 4 0000 V 
 
 400 + 30^ 1 2000 + 900 + 400 
 7 12000+900 
 
 4OO+6O+1W0+6O+I 
 7400 + 60+ 1 
 
 III. But it is evident that this operation would not be 
 affected by collecting the several numbers which stand in 
 the same line into one sum, and 
 leaving out the ciphers which are ^qqai/oqi 
 
 to be subtracted in the several parts oddblf 2dl 
 
 of the operation. Let this be done ; 
 
 and let two figures be brought down 431133 
 at a time, after the square of the 1129 
 
 first figure in the root has been sub- 461 
 tracted; then the operation may be 
 exhibited in the manner amiexed; : 
 
 from which it appears that the square ::::;::= 
 root of 53,361 is 231. ' 
 
 461 
 461 
 
QUADRATIC EQUATIONS. 83 
 
 63. To explain the division of the given r umber into 
 periods consisting of two figures each, by placing a dot over 
 every second figure beginning with the units (as exhibited in 
 the foregoing operation), it must be observed, that, since the 
 square root of 100 is 10 ; of n),000 is 100 ; of 1,000,000 is 
 1000, (fee, &c. ; it follows tharthe square root of a number 
 less than 100 must consist of one figure ; of a number be- 
 tween 100 ancif 10,000, of two figures, of a number between 
 10,000 and 1,000,000, of three figures, &;c., &c. ; and conse- 
 quently the number of these dots will show the number of 
 figures contained in the square root of the given number. 
 Thus in the case of 53361 the square foot is a number con- 
 sisting of three figures. 
 
 Ex. 1. Find the square root of 105,625. Ans. 325. 
 
 Ex. 2. Find the square root of 173,056. Ans, 416. 
 
 Ex. 3. Find the square root of 5,934,096. Ans, 2436. 
 
 CHAPTEE V. 
 
 ON QUADRATIC EQUATIONS. 
 
 64. Quadratic Equations are divided into pure and adfected. 
 Pure quadratic equations are those which contain only the 
 square of the unknown quantity, such as a:'^36; a;^+5= 
 54; ax^—h:=c\ &c. ^^ec^eo? quadratic equations are those 
 which involve both the square and simple power of the un- 
 known quantity, such as x^+4x=46 ; 3a;^— 2ar=21 ; ax'^-{- 
 ''2bx = c-\-d', &c., &c. 
 
 63. Explain the principle of the rule and the object of pointing off in 
 extracting the square root of numbers. — 64. How are quadratic equations 
 divided ? What is an adfected quadratic equation ? 
 
 \ 
 
84 ALGEBRA. 
 
 ON The solution of pure quadratic equations. 
 
 65. The Rule for the solution of pure quadratic equations 
 is this : " Transpose the terms of the equation in such a man- 
 ner, that those which contain x^ may be on one side of the 
 equation, and the known qud^iities on the other ; divide (if 
 necessary) by the coefficient of x^ ; then extract the square 
 root of each side of the equation, and it will give the values 
 of x:' 
 
 Ex. 1. 
 
 Leta;^+5=54. 
 By traiRposition, 0^^=54—5=49. 
 Extract the square root ) 
 
 of both sides of the [ then x=±^4:9=:±7. 
 equation, ) 
 
 Ex.2. 
 Let Sx'-4.=zlfh 
 
 By transposition, 3.c^=71 +4=75. 
 
 75 ' ' 
 
 Divideby3, .1:^=— =25. 
 
 Extract the square root, a;= + ^/25=="l"5. 
 Ex.3. 
 Let ax^—b=:c; 
 then ax'^=zc-\-bj 
 
 and x'^= 
 
 V a 
 Ex. 4. 5a;*— 1 =244 - - Ans. x=+7. 
 
 Ex. 5. 9^^+9 =3a;^+63 Ans. x=±S. 
 
 Ex. 6, — ^ =45 - . Ans. x= + 10, 
 Ex. 7. bx'+c+S =2bx'+l Ans. x= + ^'±3, 
 65. State the rule for solving pure quadratic equations. 
 
QUADRATIC EQUATIONS. 
 
 85 
 
 ON THE SOLUTION OF ADFECTED QUADRATIC EQUATIONS. 
 
 66. The most general form under which an adfeeted 
 
 quadratic equation can be exhibited is ax^-{-bx=:c', where 
 
 a, 6, c may be any quantities whatever, positive or negative, 
 
 integral or fractional, Dividff each side of this equation by 
 
 be be 
 
 a, then x^+-x=-» Let -=p, -—q\ then this equation is 
 a a a a 
 
 reduced to the form x^+px=^q, where p and q may be any 
 quantities whatever, positive or negative, integral or frac- 
 tional. 
 
 67. From the twofold form under which adfeeted quad- 
 ratic equations may be expressed, there arise two Rules for 
 their solution. 
 
 EULE I. 
 
 Let x'^'^ pxr=zq. 
 Add |- to each side) ,4, ^^^i>^ ^ ^^+4^ 
 
 of the equation, then 
 
 Extract the square root 
 of each side of the 
 equation, then 
 
 x'X.P^+r 
 
 ^+r- 
 
 —2- 2 
 
 and.=.±^±±^±^. 
 
 Hence it appears, that " if to each side of the equation 
 there be added the square of half the coefficient of x, there 
 will arise, on the left-hand side of the equation, a quantity 
 which is a complete sqimre ; and by extracting the square root 
 of each side of the resulting equation, we obtain a simph 
 equation, from which the value of x may be determined.'' 
 
 * Since the square of -{-a is -\-a\ and of — a is also +a*, the 
 square root of ■\-aP' may be either -\-a or — a; henc»^e square root 
 of p2.^4^ may be expressed by X -y/p^-j-^^'* 
 
 66. What is the most general form of a quadratic -equation ? Can it be 
 reduced to another form ? — 67. Enunciate the 1st Rule. 
 
86 ALGEBRA. 
 
 68. From the form in which the value of x is exhibited in 
 each of these Rules, it is evident that it will have tivo values ; 
 one corresponding to the sign +, and the other to the sign—, 
 of the radical quantity. 
 
 El4 1. 
 
 Let ir'+8a:=65. 
 Add the square of 4 (i. e, 16) to each side of the equation, 
 then - - - a;^+8a:+16=:65 + 16=:81. . 
 Extract the square root of each side of the equation, then 
 
 ir+4=+v/81 = +9, -► 
 and a; = 9 — 4 = 5 ; 
 
 OVX =— 9— 4:zi— 13. 
 
 Ex.2. 
 
 Let a;*— 4a; =45. 
 
 Add the square of ) ^'i_a^a.a-ak^a-ao 
 2 (^. e. 4), then [ ^ 4a;+4_4D+4_4y. 
 
 Extract the square root, and a;— 2= +y^49— +7, , 
 andrr=7+2=:9; 
 or, a;=:2— 7= — 5. 
 Ex.3. x''+\2x=im - - - - Ans. x=z 6 or —18. 
 Ex.4. x'^—\4:X=i 51 - - - - Ans, ir=17 or — 3. 
 Ex.5. x^— 6ar= 40 - - - - Arts, rr=10 or — 4, s 
 
 Ex. 6. a:^— 5a:=6. 
 
 In this example the coefficient of x is 5, an odd number. 
 
 5 
 
 Its half is -; and .*. adding to each side of the equation 
 
 /5\* 25 
 
 we get 
 
 ^ , 25 24+25 49 
 
 zQ + -—= =— -. 
 
 O I 7 
 
 Extracting the square root, x—- =_^ > 
 
 ^4-7 ^ 
 
• QUAI>KATIO EQUATIONS. 87 
 
 Ex.7. x'^x^zQ, 
 
 Here the coefficient of a: is 1 ; adding therefore {^Y or ^ 
 to both sides, we get / 
 
 Extracting the square root, x—^='^~ ; 
 
 /. a;r=^+-=:3or— 2. 
 
 <w 
 
 Ex. 8. a;^+7a;=78. ^?i5. x=6 or —13. 
 
 Ex. 9. x^+Sx=2S. Ans. x—4: or — 7. 
 
 Ex; 10. a;^— 3a:=40. Ans. a;=8 or — 5. 
 
 Ex. 11. x^+ ir=30. Ans, a;=5 or — 6. 
 
 Ex.12. Let7a;2-20^=32; fin^rr. 
 
 Dividing by 7, ic^— -y^=y. 
 
 CompF.) 3^20 /10y32 100 224 * 100 324 
 
 the sq. P ~ 7 ^+\ 7 j - 7 + 49 - 49 + 49 - 49 • 
 
 Hence, x-—=±^^-^=:±—', 
 
 . 10, 18 . ^, . 
 
 and ir=— -lL--=4 or —If. 
 7 7 ^ 
 
 Ex. 13. 5^^+4a:=z273. 
 
 Dividing by 5, a;^+-.r=— — . 
 
 To each ) /2\= 4 ^,4,4 273 , 4 1369 
 side add \ (5J°^25""'^" +5^+25=^+25=^5- 
 
 2 37 
 
 Extracting the square root, a;+r — "I"— » 
 
 o o 
 
 _37_2 
 Ex.14. 3a;H2:r=161 - - - - ^^5. a; = 7 or -7|. 
 
 , ^=+^_^^7, or -71. 
 
88 ALGEBRA. 
 
 Ex.15. 2x^—5x=lllf - . - Ans. x= 9 or — ^^, 
 
 Ex.16. So;^— 2a:=:280 - - - Ans. x =10 or - 9^, 
 
 Ex.17. 4x'-7x=:492y ' - - Ans. xz=zl2 or -lOi. 
 
 A quadratic equation seldom appears in a. form so simple 
 as those of the preceding examples ; it is therefore generally 
 found necessary to employ in the solution of a quadratic the 
 following reductions. 
 
 (1.) Clear the equation of fractions. 
 
 (2.) Transpose the terms involving x'^ and x to the left- 
 hand, and the numbers to the right-hand side of the equation. 
 
 (3.) Divide all the terms of the equation by the coefficient 
 of x'^, 
 
 (4.) Complete the square. 
 
 (5.) Extract the square root of both sides, and there will 
 arise a simple equation, from which the value of x may be 
 found. ^ 
 
 Ex.1. 3--"4- 
 
 Multiply by 3, and 4^^—33=^. 
 By transposition, 4^;^— a; =33. 
 
 ^. .. . . , , 1 33 
 
 Divide by 4, and x^—-:X=-—: 
 J •> 4 4 
 
 Complete the ) ^_\ 1 _33 1 _528 1 _529 
 
 square, j ^ 4^"^64~ 4 ''"64"" 64 "^64~ 64 
 
 i 
 1 23 
 Extracting the sq. root, x—-=^—' 
 o o 
 
 .,.4+1^3 or -21. 
 
 Q 4- 
 
 Ex. 2. _^+*=5. 
 
 4a;+4 
 9-) =5a;+5. 
 
QUADRATIC EQUATIONS. 89 
 
 6x^-Sx=4, 
 
 2 8 , 16_4 16_36 
 
 4 ,6 
 
 and x = -J2.-z=2 or — --. 
 5 5 5 
 
 Ex.3. —-Izzrar+ll. ... - ^715. a:=:12or-a 
 o 
 
 xLx. 4. -^+-=0 Ans. a;=:3 or L 
 
 3 ic 3 ^ 
 
 Jix. 5. -——-=9. Ans, ar=6 or — -. 
 
 Ex. 6. -4t+?=3 .... - ^7i.s. a;=2 or -1 
 
 Ex. 7. a;'— 34=^0;. ^ns. a;=6 or — 5|. 
 
 Ex. 8. f +- =51. ^/i5. ir=25 or 1. 
 
 5 .T 
 
 24 
 
 Ex.9. a;H -=3a;-4. - - - Ans. a;=5 or — 2, 
 
 x — \ 
 
 Ex. 10. -4_+^±l=^. - . . ^n^. x=2 or -3. 
 a;+l ^ ^ 
 
 Ex.11. — -—=.^—9.- - - Ans. a;=10 or — — -. 
 
 a;+2 6 7 
 
 Ex. 12. Given x^+^x=-U ; find x. 
 
 ic^H-8a?+16=16-31 = -15, 
 0:4-4 = + ^"^^^; 
 .% a;= ~4+'v/-15, & a.— -4 - 'v/-15, 
 
 both of which are impa ssihle or imaginary values of x. 
 
90 ALGEBRA. 
 
 Ex, 13. x"— 2x=— 2. -. . - Ans. a;=l + V~l. 
 Ex.14. a;^— 16.^=— 15. - - - Ans, ir=:15 or 1. 
 Ex. 15. Let lSx'+2x=60. 
 
 Divide by 13, ^^+?^=g. 
 
 Add the 1 2^ 1 _60 1 __780 1 _781 
 square of J3 p +13+169~13"^169~"169"^169~169* 
 Extract the I 1 __|_ y781 _ ^ 27.94 
 square root f ^ ""is""" — 13 ~~ — ^3 
 
 +27.94-1 26.94 ^ ^^ ^ ..n^ 
 
 /. x==^ — =:— — -=2.07 or —2.226. 
 
 13 lo 
 
 Ex. 16. rr^-6^+19==13. - Ans. a;=4.732 or 1.268. 
 
 Ex. 17. 5^^+4:r=25. - - Ans. ic=1.871. 
 
 Any equation, in which the unknown quantity is found only 
 in two terms, with the index of the higher power double 
 that of the lower, may be solved as a quadratic by the pre- 
 ceding rules. 
 
 Ex. 18. Let ir«-2aj^=48. 
 
 Complete the square, x^—2x^-{- 1 =49 ; 
 Extract the square root, .'T^— l = + 7; 
 
 .-. ^^3=8 or — 6; 
 and .-. X =2 or^__^^ 
 Ex. 19. 2a;-7-y/a7=99. 
 
 7 , 99 
 
 ^--2^-"=¥' 
 7 , , ny 99 , 49 841 
 
 . 7 ,29 . 11 
 
 V^=j±-j=9or--. 
 
 121 
 
 •. by squaring both sides, ^=81 or — — . 
 
QUADRATIC EQUATIONS. 91 
 
 Ex.20. x^-\-4x''=l2, - - Ans. x= + ^2oT±jy/^^. 
 Ex. 21. a;«-8^«=513. - A71S. x=S or^^^^oT 
 
 Rui# II. 
 
 Let ax^^bx=Cy 
 Multiply each side of ) ^j^^^ 4„v+4a6^=.4ac. 
 
 the equation by 4a, j — 
 
 Add 6^ to each side, 1 4^,^,+ ^^^^ 6^=4ac+6^ 
 we nave \ — 
 
 Extract the square root as before, 2aa;_+5 == + V4ac+6^ 
 
 .-. 2aa;= + 'v/4ac+62q:5. 
 
 and.=±V;.5!±?±*. 
 2a 
 
 From which we infer, that " if each side of the equation 
 be multiplied by four times the coefficient of .r^, and to each 
 side there be added the square of the coefficient of x, the 
 quantity on the left-hand side of the equation will be the 
 square of 2ax~^b. Extract the square root of each side of 
 the equation, and there arises a simple equation, from which 
 the value of x may be determined."* 
 
 If a=l, the equation is reduced to the form x^^px=q', 
 in this case, therefore, the Rule may be applied, by " multi- 
 plying each side of the equation by 4, and adding the square 
 of the coefficient of a;." 
 
 From the form in which the value of x is exhibited in this 
 Rule, it is evident that it will have two values ; one corre- 
 sponding to the sign +, and the other to the sign — , of the 
 radical quantity. 
 
 Ex. 1. 
 
 Let3a;^+5a:=42. 
 Multiply each side of the ) 
 
 equation by (4a) 12 ; [• 36a:'+60a;=504. 
 then ) 
 
 * The principle of this Kule will be found in the Bija Ganita^ a 
 Hindoo Treatise on the Elements of Algebra. For a full account of that 
 work, as translated by Mr. Strachey, see Dr. Hutton's Tracts^ vol. IL 
 Tract 33. 
 
92 ALGEBRA. 
 
 Add (h^) 25 to each side } oa i t an i ok ka^ . oc cor. 
 of the equation, we have } 36^^+60^+^5=504+25=529. 
 
 Extract the square root of each side of the equation, which 
 
 gives 6a;+5=+V529=+23; 
 
 .-.^^=+23-5=18 or -28; 
 
 and rr=— -=3, 
 
 D 
 
 28 14 
 
 Ex. 2. 
 
 Let a;*— 15a:=— 54. 
 Multiply by 4, then 4x^—60x=—216, 
 
 ^lich s?de \ ^^^ 4:r«-60a:+225=225~216=9. 
 Extract the square root, 2ar---15=+ v/9=+3; 
 
 .-. 2a:=15+3=18or 12, 
 
 18 12 ^ ^ 
 and x=— or -77-= 9 or 6. 
 
 ON THE SOLUTION OF PROBLEMS PRODUCING QUADRATIC 
 EQUATIONS. 
 
 69. In the solution of problems which involve quadratic 
 equations, sometimes both, and sometimes only one of the 
 values of the unknown quantity, will answer the conditions 
 required. This is a circumstance which may always be very 
 readily determined by the nature of the problem itself. 
 
 Problem 1. 
 
 To divide the number 56 into two such parts, that their 
 product shall be 640. 
 
 Let x-=one part, 
 then 56 — ii:=the other part, 
 and X (56— rr)= product of the two parts. 
 Hence, by the question, x (56— a;) =640, 
 ' or 56a:-a:^=:640. 
 
QUADRATIC EQUATION'S. 93 
 
 By transposition, x^—56x— -640. 
 
 By completing the square, ) ^«_56;,+784=784-640=144 • 
 (Rule I.) j 
 
 .\ a;~28 = + V144=±12, 
 and ^=28+12=40 or 16. 
 In this case it appears that the two values of the unknown 
 quantity are the two parts into which the given number was 
 required to be divided. 
 
 Prob. 2. There are two numbers whose difference is 7, and 
 half their product plus 30 is equal to the square of the less 
 number. What are the numbers 1 ^ 
 
 Let a;=the less number, 
 then .r-}-7=the greater number, 
 
 and ^r ^+30— half their product ^^W5 30. 
 
 Hence, by the question, ^ — ^ + 30 =a;'^ (square of less)^ 
 
 Multiply by 2 - - x'-\-lx-{-m-1x\ 
 By transposition - o;^— 7a:=:60. 
 
 ™9 ^rJle no ""^^ \ 4^'~28.:+49=240+49=289, 
 .-. 2a;-7=y289=17 
 
 2a;=17+7=24, or x^l2 less number; 
 hence ir+7=:12-|-7=19 greater number. 
 
 Prob. 3. To divide the number 30 into two such parts, 
 that their product may be equal to eight times their dit 
 ference. 
 
 Let a;=the less part, 
 then 30— -a: = the greater part, 
 and 30— or— a; or 30— 2^:= their difference. 
 Hence, by the question, .tX(30— a;)=8x(30— 2.r), 
 or30.r-ir^=:240-16ar. 
 
94 ALGEBRA. 
 
 By transposition ,x'^—4:6x=-^ 240. 
 ^■^"^^(W^^^ } ^^-46^+52i=529-240=280; 
 
 V- ^'^-23=±y'289=:±17, 
 , and a;=''23^117=40 or 6=less part ; 
 
 30 —^=30— Qr=:24:=:greafer part. 
 
 In this case, the solution of the equation gives 40 and 
 for the less part. Now as 40 cannot possibly be a part of 
 80, we take 6 for the less part, which gives 24 for the greater 
 part ; and the two numbers, 24 and 6, answer the conditions 
 required. 
 
 Prob. 4. A person bought cloth for £33 155., which he 
 sold again at £2 8s., per piece, and gained by the bargain 
 as much as one piece cost him. Required the number of 
 pieces. 
 
 Let rr=the number of pieces. 
 
 675 
 
 Then =the number of shillings each piece cost, 
 
 and 48:r=the number of shillings he sold the whole for; 
 .*. 48aj— 675= what he gained by the bargain. 
 
 675 
 
 Hence, by the problem, 48a;— 675= . 
 
 By transposition ) ^ ^^^ _225 
 and division, | ^ 16"^ "16"* 
 
 Complete the ) ,__225 50625 _225 50625 _ 65025 
 sq. (Rule I.) J ^ Jq^'^ 1024 "' 16 "^1024 ~ 1024* 
 
 225 /65025 255 
 .'. X — 
 
 V 
 
 32 ~V 1024" 32' 
 
 255+225 ,, 
 and x= ~- =15. 
 
 Prob. 5. A and B set off at the same time to a place at 
 the distance of 150 miles. A travels 3 miles an hour faster 
 than B, and arrives at his journey's end 8 hours and 20 
 minutes before him. At what rate did each person travel 
 per hour ? 
 
QUADRATIC EQUATIONS. 95 
 
 Let rr=rate per hour at which B travels. 
 Theiia;+3=: " " A " 
 
 150 
 
 And = number of hours for which B travels. 
 
 X 
 
 ir+3 
 
 But A is 8 hours 20 minutes (8^ hours) sooner at his jour« 
 ney's end than B ; 
 
 150 ,oT_150 
 
 Hence ^ , ^ +8|-= — ^, 
 
 150 25_150 
 
 By reduction, ic^4-3a;=54. 
 
 9 9 225 
 
 Complete the square, x^+^x+--=b4:+-z=z—- (Rule I.) : 
 
 :.x+^ 
 
 4~" 4 
 3 
 
 _ / 225_15 
 ~ V 4 ~ 2 ' 
 
 25 3 
 
 and x= — —^=6 miles an hour for B, 
 
 rr+3=9 " A. 
 
 Prob. 6. Some bees had alighted upon a tree ; at one 
 flight the square root of half of them went away ; at another 
 f ths of them ; two bees then remained. How many alighted 
 on the tree ?* 
 
 Let 2a;^=the No. of bees 
 
 then x+ ——4-2=2^^ 
 
 or'9a;+16a;^+18=18x2 
 .-. 18ir'-16^2~9a:=:18, 
 . or 2a;'— 9^:^:18. 
 (Rule II.) Multiply by 8 
 
 16^^-72^=: 144. 
 
 * This question is taken from Mr. Strachey's translation of tlie 
 Bija Ganita ; and the several steps of the operation will, upon com- 
 parison, be found to accord with the Hindoo method of solution, as it 
 stands in that translation, p. 62. 
 
96 - ALGEBRA. 
 
 Add 81; then 16a;^~72rr+81=225, i 
 
 or 4:X— 9 ±=15 ; 
 .-. 4x=15+9 =24, 
 
 24 
 
 and x=—-z=6i 
 
 .-. V=72, No. of bees. 
 
 Prob. 7. To divide the number 83 into two such parts that 
 their product shall be 162. Ans. 27 and 6. 
 
 Prob. 8. What two numbers are those whose sum is 29, 
 and product lOO ? Ans. 25 and 4. 
 
 Prob. 9. The difference of two numbers is 5, and ;|th part 
 of their product is 26.^ What are the numbers 1 
 
 A71S, 13 and 8. 
 
 Prob. 10. The difference of two numbers is 6 ; and if 47 
 . be added to twice the square of the less, it will be equal to the 
 square of the greater. What are the numbers ? 
 
 Ans. 17 and 11. 
 
 Prob. 11. There are two numbers whose sum is 30; and 
 Jd of their product jpZw5 18 is equal to the square of the less 
 number. What are the numbers'? Ans. 21 and. 9. 
 
 Prob. 12. There are two numbers whose product is 120. 
 If 2 be added to the less, and 3 subtracted from the greater, 
 the product of the sum and remainder will also be 120. 
 What are the numbers'? Atis. 15 and 8. 
 
 Prob. 13. A and B distribute £1200 each among a certain 
 number of persons : A relieves 40 persons more than B, and 
 B gives £5 apiece to each person more than A. How many 
 persons were relieved by A and B respectively 1 
 
 Ans. 120 by A, 80 by B. 
 
 Prob. 14. A person bought a certain number of sheep 
 for £120. If there had been 8 more, each sheep would 
 have cost him 10 shillings less. How many sheep were 
 there ? Ans. 40. 
 
 Prob. 15. A person bought a certain number of sheep 
 for £57. Having lost 8 of them, and sold the remainder at 
 8 shillings a-head profit, he is no loser by the bargain. How 
 many sheep did he buy '? Ans. 38. 
 
QUADRATIC EQUATIONS. 9T 
 
 pROB. 16. A and B set off at the same time to a place at 
 the distance of 300 miles. A travels at the rate of one mile 
 an hour faster than B, and arrives at his journey's end 10 
 hours before him. At what rate did each person travel per 
 .our ? Ans, A^ravelled 6 miles per hour. 
 
 Br " 5 " 
 
 Prob. 17. To divide the number 16 into two such parts, 
 that their product shall be equal to 70. 
 
 Let a;=one part, 
 then 16— a; = the other part 
 Hence x {16— x) or 16a;— a;^=70. 
 
 Transpose, and x'^—lQx=— 70. ♦ 
 Complete the square, 
 
 a;2_i6a;+64=-70+64=-6, 
 
 .-. x-S=±V^, or x=S+V—6* 
 
 Prob. 18. To divide the number 20 into Wo such parts, 
 that their product shall be 105 - - - a;=10+.\/— 5. 
 
 Prob. 19. To resolve the number a into two such factors, 
 that the sum of their nth. powers shall be equal to b. 
 
 Let ar=one factor,f 
 
 then -==:the other factor. 
 
 x 
 
 a" 
 Hence a;"H — -=5, 
 
 or ic*'*4-a"=^"; 
 .•. a;**— 5i»**=— a*. 
 
 * It is very well known that the greatest product which carj arise 
 ifrom the multiplication of the two parts into which any given number 
 may be divided, is when these two parts are equal ; the greatest pro- 
 duct therefore, which could arise from the division* of the number 16 
 into two parts, is when each of them is 8 ; hence, in requiring " to 
 divide the number 16 into two such parts" that their product should be 
 *70," the solution of the question is impossible. 
 
 f By factors are here meant the two numbers which being multi- 
 plied together shall generate the given number ; if therefore x = on« 
 
 factor, - must be the other faot3r, for xX-=a. 
 
 X ^ X 
 
 9 
 
98 ALGEBRA. 
 
 By Rule II. 
 
 and 2a;"— 6 = ^6^— 4a», 
 
 h + y/¥^Aa* 
 
 or 2a;"= 5 + v^6^-4a\ and a;"= „ 
 
 ... x=\/bJ-/_^-^ 
 
 Prob. 20. To resolve the number 18 into two such fac- 
 tors, that the sum of their cubes shall be 243. 
 
 # Ans, 6 and 3. 
 
 ON THE SOLUTION OF QUADRATIC EQUATIONS CONTAINING TWO 
 UNKNOWN QUANTITIES. 
 
 The solution of equations with two unknown quantities, in 
 which one or both these quantities are found in a quadratic 
 form, can only, in particular cases,* be effected by means of 
 the preceding Rules. Of these cases, the two following are 
 very well known. 
 
 Case I. 
 
 70. " When one of the equations by which the values of 
 the unknown quantities are to be determined, is a simple 
 equation;" in which case, the Rule is, " to find a value of one 
 of the unknown quantities from that simple equatioUj and 
 then substitute for it the value so found, in the other equation ; 
 the resulting equation will be a quadratic, which may be 
 solved by the ordinary Rules." 
 
 * The most complete form under which quadratic equations con- 
 taining two unknown quantities could be expressed, is this : 
 
 a x'^-\-h y'^-\-c xy-\-d ar-j- e y=m 
 
 a'x^-\-h'y'^-\-^xy-\-d'x-\-e''y^=m' \ but the general 
 solution of these equations can only be efifected by means of equations 
 of higher dimensions than quadratics. 
 
 70, There are two well-known cases, which adinit of solution by the 
 preceding rules ; state them, aud the rules employed for reducing the two 
 equations to one quadratic of the usual form. 
 
QUADKATIC EQUATIONS. 99 
 
 Ex. 1. 
 
 Let aj+2y=7, [ to find the values of x and y. 
 
 and x^+^xy—y^=^o J 
 
 From 1st equation, a;=7— 2y, /. a;^=49— 28^+4?/* ; 
 
 Substitute these values for x anfx'^ in the 2d equation, 
 
 then 49-.28y+4y^+.2l2/-6y^-/=23, 
 
 or 32/^+7y==49~23=:26. 
 
 By Rule II. 36/+84y+49=312+49=:361, 
 
 /. 6y+7=19 
 
 6?/=19-7=12, ory=:2 
 
 a;=7-2y=7-4=3. 
 
 Ex. 2. 
 
 2^+y_ ) 
 
 "3 ~ >• to find the values of x and y. 
 
 and 3a;y=210 ; 
 
 From 1st equation, 2x+y=21l ; 
 
 .-. 2a;=27-.y. 
 
 # ^ 27-y 
 
 and x= — - — 
 
 27 —v 
 Hence, 3a;y=3 X — ^ Xy =210, 
 
 or3x (27-y)xy=420 
 81y-3?/2=420 
 27y- 2/^=140; 
 or 2/^-27?/= -140. 
 By Rule II., 42/'-108y +729=729-560= 169 r ^M 
 
 .-. 2y-27=13, or y=?I±l?=20, * 
 
 . 27-20 _, 
 and a;= — - — =3^. 
 
 Ex.3. 
 
 There is a certain number consisting of two digits. The 
 left-hand digit is equal to 3 times the right-hand digit ; and 
 if 12 be subtracted from the number itself, the remainder 
 
 / 
 
100 * ALGEBKA. 
 
 will be equal to the square of the left-hand digit. What is 
 the number ? 
 
 Let X be the left>hand digit, ) then, by Art. 61, 10^+y 
 and y the other ; j is the number. 
 
 Hence, ir=3v ) , ., ,. 
 
 and 10a:+y-12= x^ \ ^^ *^^ ^^^^^^^^\ 
 
 'st&on" \ 30y+y-12=9y^ (for 10a.=30y, and a:'= V) ; 
 V-31y=-12; 
 ,31 12 
 
 „ ^ ^ , 31 . 961 961 12 961-432 529 
 By Rule I., 2,^_2,+_=_-_=_^^^^_. 
 
 TT 31 23 54 . 
 
 Hence, y~jg=-; or y=-=3, 
 
 rr=3y=9; 
 and consequently the number is 93. 
 
 Ex 4. Let 2x—Sy= 1 ) . ^ . ^, i % . 
 
 Ans. x=5^ 3/=3. 
 
 Ex. 5. There are two numbers, such, that if the less be 
 taken from three times the greater, the remainder will be 
 35 ; and if four times the greater be divided by three times 
 the less plus one, the quotient will be equal to the less num- 
 ber. What are the numbers? Ans, 13 and 4. 
 
 Ex. 6. What number is that, the sum of whose digits is 
 15, and if 31 be added to their product^ the digits will be 
 inverted? Ans, 78. 
 
 Case II 
 
 71. When a;', y^, or xy^ is found in every term of the two 
 equations, they ajsume the form of 
 
 ax^'\-h xy'\-cy^z=zd^ 
 
 a'x^-\-Vxy-\-c'y'^z=zd' \ and their solution 
 may be effected :— as in the following Examples : 
 
QUADRATIC EQUATIONS. 101 
 
 Ex. 1. 
 
 Let2a;^+3a;y+3/^=20 
 
 ^ 20 
 
 Assume x=vi/, then22;y +3v3/'+y'=20, or y^= ^y'^-i-Sv+V 
 
 41 
 
 and 5i;y +4y2=41, or y''=^:j^^', 
 
 which reduced is, 62;^— 41 v= — 13; 
 
 41v 13 
 
 „ ^ ^ , 4I1; . 1681 1369 
 ByRuLEl.,.'-- ^+-j5^=-j45:, 
 
 41 +37 41+37 13 1 
 
 -^-12=l2-'"'^=-^r"=2"'3- 
 
 ' 1 ^ , 41 41 369 ^ 
 
 Let .=-, thenf^^-^=^^=—=^9, or y=3, 
 
 a;=i;y=ix3=l., , 
 
 Ex. 2. ,/'-^ •^^' ' ' • ' ' 
 
 What two numbers are those\ who§e; suni ^n/ultipU'^d b^r > ^ 
 the greater is 77? and whose diffefen'ce ^tdtiplie'd by fcteiess ' 
 is equal to 12? 
 
 Let ir= greater number, 
 yr=less. 
 Then {x+y)Xx=x^+xi/==17, 
 and (a;— y)Xy=^y— y*=12. 
 Assume x=v2/; 
 
 Then vy+vy^=17, ( ^^ ^ v^-\-v ' 
 
 and V— y'=12 C , 12 
 ^ ^ 3 or y^= -. 
 
 12 77 
 Hence, — -^-—-^ 
 
 9* 
 
 
102' ALaEBRA. 
 
 OT l2v^+l2v=lf7v-77 ', 
 
 which gives v^——v = ——- 
 
 ■ . 65 . 4225^.529 
 
 and V* v-^ :^ — : 
 
 12 ^576 576' 
 
 65+23 88 or 42 11 7 
 
 -or-. 
 
 24 24 3 4 
 
 Either value of v will answer the conditions of the question ; 
 
 but take z;=-; then y^=^^--^=^_-^=^r:-^=:-=16, 
 
 and y=4, 
 
 7 
 a;=:vy=- x4=7. 
 
 Hence, the numbers are 4 and 7. 
 
 Ex. 3. Find two numbers, such, that the square of the 
 greater minus the square of the less may be 56 ; and the 
 square of the less plus Jd their product may be 40. 
 
 Arts, 9 and 5. 
 Ex. 4. There are two numbers, such, that 3 times the 
 square of the greater ^Zw5 twice the square of the less is 110; 
 ■ and half their pio4uct ^Zw5 the square of the less is 4. What 
 afe- tfie nunibetst ^ Ans, 6 and 1. 
 
 CHAPTER VI. 
 
 ON ARITHMETICAL, GEOMETRICAL, AND HARMONICAL 
 PROGRESSIONS. 
 
 72. If a series of quantities increase or decrease by the 
 continual addition or 5w6^rac^{o;i of the same quantity, then 
 those quantities are said to be in Arithmetical Progression. 
 
 * For a great variety of questions relating to quadratic equations 
 which contain two unknown quantities, see Bland's Algebraical Proh^ 
 kms. 
 
ARITHMETICAL PROGRE^biO^, 103 
 
 Thus the numbers, 1, 2, 3, 4, 5, 6, 6oii. (which increase by 
 the addition of 1 to each successive teim), and the numbers 
 21, 19, 17, 15, 13, 11, &c. (which decrease by the suhtrao 
 Hon of 2 from each successive term), are in arithmetical pro- 
 gression. 
 
 73. In general, if a represents the first term of any arith- 
 metical progression, and d the cominon difference^ then may 
 the series itself be expressed by a, a-\-dj a+2c/, a-f 3c^, a-^^d^ 
 &c., which will evidently be an increasing or a decreasing one, 
 according as c? is positive or negative. 
 
 In the foregoing series, the coefficient of d in the second 
 term is one; in the third term it is two; in i\iQ^ fourth it is ^^ree, 
 &c., i. e., the coefficient of d in any term is always less by 
 unity than the number which denotes the place of that term 
 in the series. Hence, if the number of terms in the series 
 be denoted by (7^), the nth. or last term in the progression 
 will be a+(n— l)c/; and, if the nth term be represented bj 
 I ; then 
 
 ^=a+(n — l)c?. 
 
 Ex. 1. Find the 50th term of the series, 1, 3, 5, 7, &c. 
 Here a= 1 ) .-. Z=l-|-(50-l) 2 
 d= 2\ =1+49x2 
 n=50 ) =99. 
 
 Ex. 2, Find the 12th term of the series 50, 47, 44, &o. 
 Here a= 50 ) .-. ^=50-f (12-1)X -3. 
 d=- SV =50-11x3 
 n= 12) =17. 
 
 Ex. 3, Find the 25th term of the series, 5, 8, 11, &c. 
 
 Ans, 77. 
 Ex. 4. " 12th « « « 15, 12, 9, &c. 
 
 Ans, —18. 
 Ex. 5. Find 6 arithmetic means (or intermediate terms) 
 between 1 and 43. 
 
 Here the number of terms is 8, namely, the 6 terms to be 
 inserted, and the 2 given terms, and consequently 
 
 73. What is an arithmetiG progression? Give an example of a series of 
 quantities in arithmetical progression. 
 
104 ALGEBRA. 
 
 a= 1 ) Buta+ {n^l) d=l 
 Z=43V .•.1+76^=43; 
 
 n= 8 ) .-. d=6. 
 
 And the means required are 7, 13, 19, 25, 31, 37. 
 
 Ex. 6. Find 7 arithmetic means between 3 and 59. 
 
 Ans. 10, 17, 24, 31, 38, 45, 52. 
 
 Ex. 7. Find 8 arithmetic means between 4 and 67. 
 Ex. 8. Insert 9 arithmetic means between 9 and 109. 
 
 74. * Let a be the ^rst term of a series of quantities in 
 arithmetic progression, d the common difference, n the num- 
 ber of terms, I the last term, and S the sum of the series : 
 Then 
 
 S=^a-\'{a+d)-\-{a+2d)-^ - - - +^ 
 and, writing this series in a reverse order, 
 
 S=l+{l-d)+{l-2d)+ --.+«. 
 These two equations being added together, there results 
 2 S={a+l) + {a+l) + {a+l)+ - - - +(a+0 
 = (a+Z)X^, since there are n terms; 
 ... S={a+l)l .... (1). 
 
 Hence it appears that the sum of the series is equal to the 
 sum of the first and last terms multiplied by half the number 
 of terms : 
 
 And since l=a+{n—l) d; 
 
 .\S=ha+{n^l)dl^ (2). 
 
 From this equation, any three of the four quantities a, rf, 
 n, 8, being given, the fourth can be found. 
 
 Ex. 1. Find the sum of the series 1, 3, 5, 7, 9, 11, &c. 
 
 continued to 120 terms. 
 
 Herea= 1 Y... ^= | 2a+(n-l) c?[ x| 
 
 = i2xl+(120~l)2J. X 
 
 d= 2 r , ^ 120 
 
 7i=120j - ] •'*''" *^'^f'^2 
 
 (2+ 1 19 X 2) X 60=240X60= 14400. 
 
ARITHMETICAL PROGRESSION. 
 
 105 
 
 Ex. 2. Find the sum of the series 15, 11, 7, 3,-~l, — 5, 
 
 &;c., to 20 terms. 
 
 = |2xl5 + (20~l)x-4 Ix 
 
 20 
 2 
 
 = (30-19x4)xl0 
 = (30-76) X 10 
 = —46X10= -460. 
 
 Ex. 3. Find the sum of 25 terms of the series 2, 5, 8, 11, 
 14, &c. Ans. 950. 
 
 Ex. 4. Find the sum of 36 terms of the series 40, 38, 36, 
 34, &c. Ans, 180. 
 
 Ex. 5. Find the sum of 150 terms oif the series ^, |^, 1, j, 
 f , 2, I, &c. 
 
 Herea=i 1 .'. >^= | 2a+(.-l) c^[ ^ 
 
 ^=1 I =|2XK(150-I)xij^ 
 
 n=150j = g+l|?)75=l|lx75=3775. " 
 
 Ex. 6. Find the sum of 32 terms of the series 1, 1^, 2, 
 2^, 3, &c. Ans. 280. 
 
 PROBLEMS. 
 
 pROB. 1. The sum of an arithmetic series is 1240, common 
 difference — 4, and number of terms 20. What is the Jirst 
 term ? 
 
 Here 5= 1240 1 .-. S= | 2a+ (ti-I) cZl ^ 
 
 ^=-" '*hl240=i2a+(20-l)x ' '^\^ 
 
 w= 20 J = (2a-19x4)10 
 
 124=2a-76; 
 .-. 2a= 124+76=200, 
 and.-. a=100. 
 Hence the series is 100, 96, 92, &o. 
 
106 ALGEBRA. 
 
 Prob. 2. The sum of an arithmetic series is 1455, \hQ first 
 term 5, -and the number of terms 30. What is the common 
 difference ? 
 
 Here .S^ 1455 ' 
 
 a= 5 
 nz=z 30 J 
 
 |<Ja+(n-iyj.^=>Sf - 
 '.-. i2x5+(30~l)cZl^=1455 
 
 a= 7 
 d= 2) 
 
 (10+29^)15=1455; 
 Dividing both sides by 15, 10+29c?=97, 
 
 29cZ=87; 
 .-. c?=3. 
 Hence the series is 5, 8, 11, 14, &c. 
 
 Prob. 3. The sum of an arithmetic series is 567, ihe first 
 term 7, the common difference 2. Find the number of terms. 
 
 Here 5=567l .-. since i 2a+ (n-1) d 1 1=5 
 
 ■" '^ i2x7+(7i-l)2l|=567 
 
 7i^+6?i=567. 
 Completing the square, ^'+6^+9=576, 
 
 . Extracting the square root, 7i+3=_+24; 
 
 .-. w=21 or —27. 
 
 Prob. 4. The sum of an arithmetic series is 950, the com- 
 mon difference 3, and number of terms 2b. What is theirs/ 
 term? Ans, 2. 
 
 Prob. 5. The sum pf an arithmetic series is 165, the^rs^ 
 term 3, and the number of terms 10. What is the common 
 difference? Ans. 3. 
 
 Prob. 6. The sum of an arithmetic series is 440, first 
 term 3, and common difference 2. What is the number of 
 terms? Ans. 20. 
 
 Prob. 7. The sum of an arithmetic series is 54, \hQ first 
 term 14, and common difference —2. What is the number of 
 terms ? Ans. 9 or 6. 
 
ARITHMETICAL PROGRESSION. 107 
 
 Prob. 8. A traveller, bound to a place at the distance of 
 198 miles, goes 30 miles ihejirst day, 28 the second, 26 the 
 third, and so on. In how many days will he arrive at his 
 journey's end ? 
 
 Here is given a= SO ) ^ 
 
 d=: — • 2 >- to find the number of terms. 
 S= 198 . 
 
 Now i.2a+ {n-l)dl'^=S, 
 
 i2x30+(7i-l)x~2l^=198, 
 (31-^)^=198, 
 
 or, 71^—3171=: — 198, 
 
 31 , 13 
 
 "-2=±2-' 
 
 31 , 13 ^^ ^ 
 /. 7i=— +— =22or9. 
 
 To explain the apparent difficulty arising from the two 
 positive values of n, which gives us two different periods of 
 the traveller's arrival at his journey's end, we must observe, 
 that if the proposed series 30, 28, 26, &c., be carried to 22 
 terms, the 16th term will be nothing, and the remaining six 
 terms ^ill be negative ; by which is indicated the rest of the 
 traveller on the 16th day, and his return in the opposite 
 direction during the six days following ; and this will bring^^ 
 him again, at the end of the 22d day, to the same point at 
 which he was at the end of the 9th, viz., 198 miles from the- 
 place whence he set out. 
 
 Prob. 9. How much ground does a person pass ov^ ih' 
 gathering up 200 stones placed in a straight line, at intervals 
 of 2 feet from each other ; supposing that he brings each stone^ 
 singly to a basket standing at the distance of 20 yards from, 
 the first stone, and that he starts from the spot where the- 
 basket stands ? 
 
 It is evident that the space passed over by this person; will^ 
 be twice the sum of an arithmetic series, whose j'^rs^ term is 
 
108 ALGEBRA. 
 
 20 yards (i. e. QfOfeet)^ common difference 2 feet, and number 
 of terms 200. • 
 
 Herea=60),^^|2,+ („_l),J^« 
 
 ^ ^=200) =(12i^+398)Xl00. 
 
 ==518x100=51800 feet. 
 
 feet, miles, furlongs, feet. 
 
 Hence the distance required^ 103,600= 19 - 4 - 640. 
 
 Prob. 10. a person bought 47 sheep, and gave 1 shilling 
 for the^r^^ sheep, 3 for the second^ 5 for the' third, and so on. 
 What did all the sheep cost him? Ans, £110 9^. 
 
 Prob. 11. A gentleman began the year by giving away a 
 farthing \he first day, a halfpenny the second, three farthings 
 the third, and so on. What money had he disposed of in 
 charity at the end of the year? Ans, £69 11 5. 6fc?. 
 
 Prob. 12. A travels uniformly at the rate of 6 miles an 
 
 hour, and sets off upon his journey 3 hours and 20 minutes 
 
 before B ; B follows him at the rate of 5 miles the first hour, 
 
 , 6 the second, 7 the third, and so on. In how many hours will 
 
 B overtake A? Ans, In 8 hours. 
 
 Prob. 13. There is a certain number of quantities in 
 arithmetic progression, whose common difference is 2, and 
 whose sum is equal to eight times their number ; moreover, 
 If 13 be added to the second term, and this sum ba> divided 
 by the number of terms, the quotient will be equal to \h.e first 
 term. What are the numbers % 
 
 Jn the expression 2a-\-(ii—\)(i)<,-, substitute x for a, 2 for 5, 
 
 ■and y for^i, and it becomes 2a:+(y — l)2X^(=^y+2/^^y), 
 ibr the sum of the series. 
 
 By the problem, xy+y^—y=Sy, or y=9— a?, 
 
 ^ a;4-2+13 
 and =ix. 
 
GEOMETEIC PROaRESSION. 109 
 
 Hence, — ■ — =x, or ic^— 8^= — 15: 
 
 9 — X 
 
 .\ x'—Sx+16=l6-16 = l, 
 
 andic— -4= + l ; .*. x=5 or 3, 
 ^ y=9— ir —4 or 6. 
 From which it appears that there are two sets of numbers 
 which will answer the conditions required; viz., 5, 7, 9, 11, or 
 3, 5, 7, 9, 11, 13. 
 
 Prob. 14. There is a certain number of quantities in 
 arithmetic progression, whose ^rs^f term is 2, and whose sum 
 is equal to 8 times their number ; if 7 be added to the third 
 term, and that sum be divided by the number of terms, the 
 quotient will be equal to the common difference. What are 
 the numbers] Ans. 2, 5, 8, 11, 14. 
 
 ON GEOMETRIC PROGRESSION. 
 
 75. If a series of quantities increase or decrease by the 
 continual multiplication or division by the same quantity, then 
 those quantities are said to be in Geometrical Fvogression. 
 Thus the numbers, 1, 2, 4, 8, 16, &;c. (which increase by the 
 continual multiplication by 2), and the numbers 1, i i ^, 
 &c. (which decrease by the continued division loj 3, or 
 multiplication by ^), are in Geometrical Progression. 
 
 76. In general, if a represents the Jirst term of such a 
 series, ^nd r the common multiple or ratio, then may the 
 series itself be represented by a, ar, ar"^, ar^, ar\ &;c., which 
 will evidently be an increasing or decreasing series, according 
 as r is a whole number or 2i. proper fraction. In the foregoing 
 series, the index of r in any term is less by unity than the 
 number which denotes the place of that term m the series. 
 Hence, if the number of terms in the series be denoted by 
 (n), the last term will be ar"^"^, 
 
 77. From the series given in the two preceding articles it 
 is evident, by mere inspection, that the common ratio can be 
 found by dividing the second term by Xhe first, or by dividing 
 any term by that which precedes it. 
 
 75. Define a geometrical progression, and give an example. — 77. How is 
 the common ratio of a series of numbers in geometrical progression found ? 
 
110 ALGEBRA. 
 
 Ex. 1. Find the common ratio of the geometrical progression 
 1, 2, 4, 8, &c. 
 
 Here the common ratio =~=2. 
 
 ^ 2 4 8 
 
 Ex. 2. Find the common ratio of the series -, -, — , &;c. 
 
 4 2 2 
 
 In this example the common ratio = -H--=^. 
 
 y o o 
 
 5 3 9 
 
 Ex. 3. Find the common ratio in the series -, 1, -, -— , &c. 
 
 o 5 25 
 
 Ans. -. 
 5 
 
 78. Let S be the sum of the series a, ar^, ar^, (fee, then 
 a-{-ar+ar'^+ar^+k>Q,. - - - ar""'"^ + ar'^'^ =. S, 
 
 Multiply the equation by r, and it becomes 
 
 Subtract the upper equation from the lower^ and we have, 
 ar''—a=zrS^S^ox{r—\) S=ar''—a', 
 
 ar^ cCr 
 
 and therefore, S= — . 
 
 r— 1 
 
 If r is a proper fraction, then r and its powers are less 
 
 than 1. 
 
 For the convenience of calculation, therefore, it is better 
 
 a '~~ ar^ 
 in this case to transpose the equation into S = — , by" 
 
 multiplying the numerator and denominator of the fraction 
 
 ar^'^a , 
 
 T- by —1. 
 
 79. If I be the last term of a series of this kind, then 
 
 l=ar*^^, ,\rlz=:ar'^\ hence 0=1 7-)= r-. xrom this 
 
 \r— 1/ r—l 
 
 equation, therefore, if any three of the four quantities >S', a, r, 
 
 Z, be given, the fourth may be found. 
 
 78. What is the expression for the sum of n terms of a series of mimbera 
 in geometrical progression? 
 
GEOMETRIC PROGRESSION. Ill 
 
 Ex. 1. 
 
 Find the sum of the series 1, 3, 9, 27, &c. to 12 terms. 
 ar--a lx3^'-l 
 
 ,s=:- 
 
 _8P-1 
 "" 2 • 
 531441-1 531440 
 
 =265720. 
 
 ~ 2 2 
 
 Ex. 2. 
 
 2 4 8' 
 Find the sum often terms of the series l+o+5+^, <^c. 
 
 o y i4 / 
 
 a= 1 
 
 2 
 
 '= 3 
 
 n=10 
 
 1-r ~ ^_2 3-2 W 
 
 3 
 
 ,, ,^V' 2'' 1024 
 
 N0W(-; 
 
 ■•■'-©'■= 
 
 3'^~59049' 
 
 1024 _ 58025 
 "59049" 59049' 
 3 X 58025 _ 174075 
 ^"""^ '^"~59049~~"59049"- 
 
 Ex. 3. Find the sum of 7 terms of the series, 1, 3, 9, 27, 
 81, &c. Ans. 1093. 
 
 Ex. 4. Find the sum of 1, 2, 4, 8, 16, &c. to 14 terms. 
 
 Ans. 1638eS. 
 
 Ex. 5. Find the sum of 1, -, -, — , &;c. to 8 terms. 
 
 O 9 y&7 
 
 , 3280 
 
 ^"^•2187- 
 Ex. 6. Find three geometric means between 2 and 32. 
 Here a= 2 ) And ar"-^=Z 
 Z=32 ^ .-. 2r^ =32, 
 71= 5) r* =16, 
 
 .-. r = 2. 
 And the means required are 4, 8, 16. 
 
112 ALGEBBA. 
 
 Ex. 7. Find two geometric means between 4 and 256. 
 
 Ans. 16 and 64. 
 
 Ex. 8. Find three geometric means between ^ and 9. 
 
 ^. Ans. ^, 1, 3. 
 
 Ex, 9. Find a geometric mean between a and I. 
 Let X be the geometric mean required ; 
 Then a, x, Z, are three terms in geometric progression, 
 
 - X I 
 
 and -= - 
 a X 
 
 or x^=:al 
 .*. xziz-y/aL 
 
 EiT 10. What is the geometric mean between 16 and 64? 
 
 Ans. 32. 
 Ex. 11. Insert four geometric means between ^ and 81. 
 
 Ans. 1, 3, 9, 27. 
 
 PROBLEMS. 
 
 Prob. 1. Find three numbers in geometric progression; 
 such that their s^im shall be equal to 7 ; and the sum of their 
 squares to 21. 
 
 Let -, 0?, xy^ be the numbers. Then by the problem, 
 -+x+xy =7 - - (1) 
 ^--^+x''+xY=2\ . - (2) 
 
 From equation (1), xi — l-l+yj = 7 
 
 /I 2 \ 
 
 •. by squaring, a;'/— +-+3+2y+2/M=49 
 
 From (2) x'O-^ +1 +y')==21 
 
 .-. by subtraction, x4--+2+2y\ =28, 
 
 or 14a: =28 
 
 .-. X=: 2. 
 
GEOMETKIC PROGRESSION. 113 
 
 This value ofx being inserted in (1), 
 1 7 
 
 •••^-22'+l2J =16-1=16 
 
 5±3 „ 1 
 
 Hence, the numbers are 1, 2, 4; or 4, 2, 1. 
 
 Prob. 2. There are three numbers in geometric progression 
 whose product is 64, and sum 14. What are the numbers ? 
 
 Ans. 2, 4, 8 ; or 8, 4, 2. 
 
 Prob. 3. There are three numbers in geometric progression 
 whose sum is 21, and the sum of their squares 189. What are 
 the numbers? Ans. 3, 6, 12. 
 
 Prob. 4. There are three numbers in geometric progression; 
 the sum of the Jirst and last is 52, and the square of the mean 
 is 100. What are the numbers? Ans. 2, 10, 50. 
 
 Prob. 5. There are three numbers in geometric progression, 
 whose sum is 31, and the sum of thej^rs^and last is 26. What 
 are the numbers? Ans. 1, 5, 25. 
 
 ON THE SUMMATION OF AN INFINITE SERIES OF FRACTIONS IN 
 GEOMETRIC PROGRESSION ; AND ON THE METHOD OF FINDING 
 THE VALUE OF CIRCULATING DECIMALS. 
 
 79. The general expression for the sum of a geometric 
 
 series whose common ratio (r) is a fraction, is (Art. 78) 
 
 a ~~ ar^ 
 S z=— . Suppose now n to be indefinitely gi-eat, then 
 
 r* (r being a proper fraction) will be indefinitely small,'^ so 
 
 * When r is a proper fraction, it is evident that r" decreases as n 
 
 increases; let r=:i for instance, then r«=ji^, r^=^^, ri=^^M., 
 
 and when n is indefinitely great, the denominator of the fraction 
 becomes so large with respect to the numerator, that the value of the 
 fraction itself becomes less than any assignable quantity. 
 
 10* 
 
114 ALGEBRA. 
 
 that ar"" may be considered as nothing with respect to a in 
 the numerator a—ar"" of the fraction expressing the value 
 of /S^; the limit^ therefore, to which this value of /S approaches, 
 
 when the number of terms i^infinite, is . 
 
 1 — r 
 
 Ex. 1. 
 
 Find the sum of the series 1 +0+7+0? ^^' ^^ infinitum, 
 Herea = l) c^ ^ 1 ^ ^ 
 
 Ex. 2. 
 
 Find the value of k+ 97 +79^ + ^0. ad infinitum. 
 
 Here a=- 
 5 
 
 5 ' 25 ' 125 
 
 1 
 
 5 1 1 
 
 ,=i 
 
 ,'.S=: 
 
 1 5-1 4 
 5j 5 
 
 Ex. 3. Find the value of 1 +— +— +^+&c. ad^ infinitum, 
 
 Ans, —. 
 
 3 9 27 
 
 Ex. 4. Find the valueof l+--r-+ 777+^-7 +&c. ad infinitum. 
 
 4 Id d4 
 
 An$^ 4. 
 • 2 4 8 
 
 Ex. 5. Find the value of -=-+^+t^+&;c. ad infinitum. 
 
 80. These operations furnish us with an expeditious me- 
 thod of finding the value of circulating decimals^ the num- 
 bers composing which are geometric progressions, w^hose 
 
 80. What is the expression for the sum of a geometric series, when the 
 number of terms is iunnite ? 
 
GEOMETRIC PROGRESSION. 
 
 115 
 
 1 1 1 
 
 common ratios are — , — — , , &c., according to the num- 
 
 ber of factors contained in the repeating decimal. 
 
 Ex. if 
 
 Find the value of the circulating decimal .33333, &;c. This 
 decimal is represented by the geometric series 
 3 3 3 .3 
 
 Tfi"^Tno"^ inna "^^^'' ^^^^® f''^^^ ^^^^ ^® Ta' ^^^ common 
 
 . 1 
 ratio 
 
 10* 
 Hence «=j^, 
 
 10' 
 
 S=z 
 
 10 
 
 l-r' 
 
 1- 
 
 10-1 
 
 10 
 
 Ex. 2. Find the value of .32323232, &;c. ad infinitum. 
 
 Here a = 
 
 32 "^ 
 100' 
 
 1 
 100' 
 
 •. s=- =■ 
 
 32 
 100 
 
 32 
 
 1- 
 
 2^ 
 
 100 
 
 32 
 
 'l00-l~99* 
 
 Ex. 3. Find the value of .713333, &c. ad infinitum. 
 
 The series of fractions representing the value of this 
 
 71 3 3 
 
 decimal are TTr^+ (geometric series) 7777^+77^7:7777+ &c. 
 
 Here a=: 
 
 100 
 
 1000 
 1 
 
 1000 ' 10000 
 
 1000 
 
 1- 
 
 2 
 
 "10 J ^ 10 
 
 Hence the value of the decimal = ( 
 
 _107 
 ""150' 
 
 1000-100 900 300 
 
 100 
 
 +>S 
 
 A2L. J_. 
 
 7100 "^300"^ 
 
 214 
 
 300 
 
116 • ALGEBRA. 
 
 Ex. 4. Find the value of .81343434, &c. ad infinitum. 
 
 34 
 a 10000 34 34 
 
 Here a=i 
 
 ^ 10000 
 
 . Cf, 
 
 J (" ~T^~, 1 10000- 100-9900 
 
 '■-100 J ^ 100 
 
 K A ^ f,u a'- ^ «1 , c 81 , 34 8053 
 And value of the decimal =—+S=~+^^^=^^. 
 
 ^x. 5. Find the value of .77777, &c. ad infinitum. 
 
 , 7 
 Ans.-. 
 
 Ex. 6. " « .232323 &o. ad infijiitum. 
 
 A 23 
 
 Ex. 7. « « .83333, &c. ad infinitum. 
 
 Ans.^. 
 
 Ex. 8. « " .7141414, &c. ad infinitum. • 
 
 . 707 
 ^'^^•990- 
 Ex. 9. " " .956666, &c. ad infinitum. 
 
 A 287 
 
 ^'^^•soo- 
 
 The value of a circulating decimal -may also be found as 
 follows : — In Ex. 4 above, 
 
 Let S= .813434 .... 
 .-. 10000 5=8134.3434 .... 
 and 100 S== 81.3434 .... 
 • .-.9900/^=8053 
 
 ^ 8053 , ^ 
 •'•^=9900'"^^"^"^"- 
 
 pROB. 1. A body in motion moves over 1 mile theirs/ 
 second, but being acted upon by some retarding cause, it 
 only moves over \ a mile the second second, \ the third^ 
 
HARMONIC PROGRESSION. 117 
 
 and so on. Show that, according to this law of motion, the 
 body, though it move on to all eternity^ will never pass over 
 a space greater than 2 miles. 
 
 ON HARMONIC PbI^GRESSION. 
 
 81. A series of quantities, whose reciprocals are in arith- 
 metic progression, are said to be in Harmonic Progression. 
 Thus the numbers 2, 3, 6, are in harmonic progression^ since 
 their reciprocals |-, -J, |-, are in arithmetic progression (—-J 
 being the common difference), 
 
 2 
 
 Ex. 1. Find a harmonic mean between 1 and — . 
 
 Let a? be the mean required : 
 
 1 3 
 
 Then 1, — , — , are in arithmetic progression, 
 
 And 
 
 1 
 
 -1= 
 
 3 
 
 1 
 
 X 
 
 
 "2"" 
 
 X 
 
 
 2 
 
 
 3 
 
 • 
 
 
 = 1 + 
 
 
 
 X 
 
 
 2 
 
 
 
 5 
 
 
 
 
 '2 
 4 
 
 
 
 \ X- 
 
 '~h' 
 
 
 Ex, 2. Find a third number to be in harmonic progression 
 with 6 and 4. 
 
 Let * be the number required : 
 
 111 .^ . 
 
 Then -— , — -, — , are in arithmetic progression. 
 6 4 a; 
 
 A :i 1 1 1 1 
 
 • ' a; ~ 2 6 
 ~6 "6" 
 
 .•.ar=3 
 
118 
 
 ALGEBKA. 
 
 Ex. 3. Insert three harmonic means between 9 and 3. 
 
 The reciprocals of 9 and 3 are — - and — , which are Xhe first 
 
 9 o 
 
 and last term of an aritkrr^ic progression, between which 3 
 
 arithmetic means are to be inserted. We have therefore, 
 
 according to Art. 73 — 
 
 And a+(n-'l)d=l, 
 
 
 1 
 
 a=z 
 
 — 
 
 
 9 
 
 
 1 
 
 1= 
 
 
 
 ¥ 
 
 X=z 
 
 5 
 
 • 
 
 i+(5-l)fcl 
 
 ^ 3 9 
 
 "~ 9 9 
 ""9 
 
 Hence, — , — , ~, are the arithmetic means to be inserted 
 
 between — and •— , and therefore their reciprocals 6, — , — , 
 
 are the three harmonic means required. 
 
 Ex. 4. Find a harmonic mean, between 12 and 6. 
 
 A71S, 8. 
 
 Ex. 5. The numbers 4 and 6 are two terms of a harmonic 
 progression; find a third term. Ans. 12. 
 
 Ex. 6. Find two harmonic means between 84 and 56. 
 
 Ans, 72 and 63. 
 Ex. 7. Insert three harmonic means between 15 and 3. 
 
 Ans.~,5,-. 
 
 82. Let c, b, c, d, e, &c., be a series of quantities in 
 
 1 1 1 1 1 p 
 
 harmonic progression ; then —, —, — , --7, — , &c., are m ant/i- 
 
 a - c a e 
 
HARMONIC PROGRESSION. 119 
 
 rnetic progression, and according to the definition of an 
 arithmetic progression (Art. 72), we have 
 
 ' ' ' ' (1) 
 
 (2) 
 (3) 
 
 From (1) 
 
 b 
 
 a 
 
 c 
 
 b 
 
 1 
 
 1 
 
 1 
 
 f 
 
 c 
 
 b' 
 
 " d 
 
 c 
 
 1 
 
 d 
 
 1 
 
 c 
 
 _ 1 
 
 ~ e 
 
 1 
 d 
 
 (fee. 
 
 
 = 
 
 &c. 
 
 a- 
 
 -b 
 
 b- 
 
 -c 
 
 ab 
 
 c 
 
 a- 
 
 -b 
 
 b- 
 
 -c 
 
 a 
 
 c 
 
 .', 
 
 a 
 c 
 
 a- 
 
 -b 
 
 -c' 
 
 or, converting this equation into a proportion-, 
 a'.cwa — b'.b—c 
 Similarly from (2) b:d::b—c:c^d 
 " " (3) c:e::c—d:d'-e 
 and so on for any number of quantities. 
 
 These proportions are frequently assumed as the deJinitio7\ 
 to quantities in harmonic progression, and may be thus 
 expressed in words: — if any three quantities in harmonic pro- 
 gression be taken, .the first is to the third as the difference 
 between the first and second is to the difference between the second 
 and third. 
 
 Prob. 1. Given a^=b^ =z(f , where a, 5, c, are in geometric 
 progression. Prove that ar, yf z, are in harmonic progression. 
 
 y 
 a^^by\ ,\a=b^ - - - (1). 
 
 y 
 <f=h'^'^ ,\c=b^ (2). 
 
 By multiplying (1) by (2) acz=zb^ « 
 
120 ALGEBRA. 
 
 But by geometric progression ac^zh^ 
 
 y X z 
 
 y ^~2 y' 
 
 CHAPTEE VII. 
 
 ON PERMUTATIONS AND COMBINATIONS. 
 
 83. By Permutations are meant the number of changes 
 which any quantities, a^ b, c, d, e, &c., can undergo with re 
 spect to their order, when taken two and two. together, three 
 and three^ (fee, &c. Thus a6, ac, ad^ ba, be, bd, ca, cb, cd, da, 
 db, dc, are the different permutations of the four quantities a, 
 b, c, d, when taken two and two together; abc, acb, bac, bca, 
 cab, cba, of the three quantities a, b, c, when t'aken three and 
 three together, &;c., &c. 
 
 84. Let there be n quantities, a, b, c, d, e, &c. : then, by 
 Art. 83, it appears that there will be (^—1) permutations in 
 which a stands first ; for the same reason there will be (?^ — 1) 
 permutations in which b stands first ; and so of c, r/, e, &c. 
 Hence there will be 7i times [n — 1) permutations of the form 
 ab, ac, ad, ae, &c. ; ba, be, bd, be^ &c. ; ca, cb, cd, ce, &c. ; i. e. 
 "/A^ number of permutations ofn things taken two and two is 
 n(n-l)." 
 
 85. If these n quantities be taken three and three together, 
 then there will be n (n — \) (?i— 2) permutations. For if 
 (n — \) be substituted for n in the last article, then the number 
 of permutations of n—\ things taken two and two together 
 will be (^ — 1) ('i— 2) ; hence the number of permutations of 
 5, c, d^ e^ &c., taken two and two together, are (n — \) (?i— 2), 
 
PERMUTATIONS. 121 
 
 and consequently there are (^ — 1) (n— 2) permutations of the 
 quantities a, 6, c, c/, e, &c.^ taken ^Ar^g ant/ tliree together, in 
 which a may stand first ; for the same reason there are {ji — V) 
 (n— 2) permutations in which h may stand first; and so of c, 
 «?, e^ &;c. The numbers of the j^rmutations of this kind will 
 therefore amount to ti (/i — 1) (n— 2). 
 
 86. To find the number of 'permutations of n things taken 
 r together. 
 
 By Arts. 85 and 86— 
 The No. taken two together z=zn{7i—\) 
 
 " " three " =n\n—\){n—2) 
 
 Similarly " four " =n \n-\) {n-2) (/i-3) 
 
 If the law, which is observed in these particular cases, be 
 supposed to hold generally, that is, if the number of permu- 
 tations of /I things «, 6, c, d^ &;c., taken r — 1 together, be 
 
 n (/i — 1) (^—2) (7i-r+2) 
 
 Then, by omitting «, it is equally true that the number of 
 permutations of n — 1 things 6, c, d^ &;c., taken r—\ together, 
 will be by putting n — \ iov n in this last expression 
 
 {n — \) {n—2) (?i— r+1) 
 
 Now, if a be placed before each of these permutations, there 
 will be 
 
 (n-1) {n-2) ..... {n-r+1) 
 
 permutations of things taken r together, in which a stands first. 
 It is clear that there will in like manner be the same number 
 of permutations of things taken r together, in which each of 
 the other things 6, c, d, (fee, stand respective Ig first; and as 
 there are n things, the entire number of permutations of ?» 
 things taken r together, will be the sum of the permutations 
 taken r together in w^hich the n things a, 6, c, d, &c., respect- 
 ively stand first, that is, n times (w—1) [n—2) 
 
 (n-r+1), 
 
 or n {n — 1) [n—2) [n—r+l) 
 
 It has thus been proved, that if the law by w^hich the ex- 
 pression for the number of permutations of n things taken 
 r— 1 together is found, be true, it is also true for the next 
 superior number, or when n things are taken r together; 
 but the law of the expression has been found to hold for 
 
 11 
 
122 ALGEBRA. 
 
 the number of permutations of n things taken two together, 
 and for the number of permutations of n things taken three 
 together ; it is therefore true by the theorem just demonstrated 
 when the n things are taken four together, and if true when 
 taken four together, it is tr^ also when taken j^yg together, 
 and so on for any number not greater than n^ which may be 
 taken together. 
 
 This proof affords an excellent example of demonstrative 
 induction^ a method of reasoning of great iniportance in the 
 mathematical sciences. 
 
 87. If rr=7i, i. e. if the permutations respect all the quan- 
 tities at once, then (since n — r=^ri) the number of them will 
 be n (n — \) (?i— 2) &c. ... 2. 1. Thus, the number of per- 
 mutations which might be formed from the letters composing 
 the word ''virtue''' are 6x5x4x3x2x1=720. 
 
 88. But if the same letter should occur any number of times, 
 then it is evident that we must divide the whole number of per- 
 mutations by the number of permutations which would have 
 arisen if different letters had occurred instead of the repetition 
 of the same letter. Thus if the same letter should occur twice^ 
 then we must divide by 2x1 ; if it should occur thrice^ we 
 must divide by 3x2x 1 ; if p times by 1.2.3. . . ^; and so for 
 any other letter which may occur more than once. Hence 
 the general expression for the number of permutations of n 
 things, of which there are jp of one kind, r of another^ q of 
 
 . n (n — l) in— 2) (w— 3) . . 2.1 - 
 
 another, &c., &c., is -. \ ^ . ^ o r^r^ • J-^^s 
 
 ' ' ' 1.2.3. .j(?X 1.2.3.. rX 1.2.3.. g 
 
 the permutations which may be formed by the letters com- 
 posing the word ''easiness'''^ (since s occurs thrice^ e twice) are 
 ^.7.6.5.4.3.2.1^33 
 1.2.3.x 1.2 
 
 Ex. 1. What is the number of different arrangements which 
 can be made of 6 persons at a dinner-table? 
 
 The number=lx2x3x4x5x6=720. 
 
 Ex., 2. Required the number of changes which can be rung 
 upon 8 bells. 
 
 The No. of Changes=lx2x3x4x5x6x7x8=40320. 
 
COMBINATIONS. 123 
 
 Ex. 3. With 5 flags of difterent colours, how many signals 
 can be made ? 
 
 The number of signals, when the flags are taken — 
 Singly^ are ---^---nr 5 
 Two together =5.4- - - =20 
 Three - - - =5.4.3 - - = ^60 
 Four - . - =5.4.3.2 - =120 
 Five - - - =5.4.3.2.1 =120 
 
 .'. the total number of signals = 325 
 
 Exl^. How many permutations can be formed out of 10 ' 
 letters, taken 5 at a time 1 Ans. 30240. 
 
 Ex. 5. How many permutations can be formed out of the 
 words Algebra and Missippi respectively, all the letters being 
 taken at once? Ans, 2520 and 1680. 
 
 ON COMBINATIONS. 
 
 89. By Combinations are meant the number of collections 
 which can be formed out of the quantities, or, 6, c, c?, ^, &c,, 
 taken two and two together, three and three together, &;c., (fee, 
 without having regard to the order in which the quantities 
 are arranged in each collection. Thus ab, ac, ad, be, bd, cd, 
 are the combinations which can be formed out of the four 
 quantities a, b, c, d, taken two and two together; abc, abd, 
 acd, bed, the combinations which may be formed out of the 
 same quantities, when taken three and three together; &c., &c. 
 
 90. From the expression (in Art. 86) for finding the num- 
 ber of permutations of n things taken r and r together, we 
 immediately deduce the theorem for finding the number of 
 combinations of n things taken in the same manner. For the 
 permutations of n things taken two and two together being n 
 {n — \), and as each combination admits of as msiuj permutatiofis 
 as may be made by two things (which is 2x1), the number 
 of combinations must be equal to the number o^ permutations 
 divided by 2 ; i. e. the number o^ combinations o^n things taken 
 
 two and two together is — —- — -. For the same reason, the 
 
124 ALGEBRA. 
 
 combinations of n things, taken three and three together, must 
 
 71 (n-—l) (n^2) , . -, , , . 
 
 be equal to — i ^ \j \ ^^<^ ^^ general, the combma- 
 
 tions of n things taken r ^d r together must be equal to 
 n (n-1) (^-2) . . . {n-r+1) 
 
 rT2". S , . ,r 
 
 Ex. 1. Find the number of combinations which can be 
 formed out of 8 letters, when taken 5 at a time. 
 
 _,, . 8X7X6X5X4 _ 
 
 Ihe number =- — - — - — - — -=56. 
 1X2x3x4x5 
 
 Ex. 2. What is the total number of combinations which can 
 be formed out of 6 colours taken in every possible way? 
 No. of combinations when the colours are taken — 
 
 1 at a time ----'---r=6 
 
 2 -^-^ -15 
 
 6.5.4 
 
 = 20 
 = 15 
 = 6 
 
 ^ ' " " 1.2.3.4.5.6 ■ ^ 
 
 1.2.3 
 
 
 6.5.4.3 
 
 
 1.2.3.4 
 
 
 6.5.4.3.2 
 
 
 1.2.3.4.5 
 
 
 6.5.4.3.2. 
 
 _1 
 
 Hence the total number = 63 
 
 Ex. 3. Eind how many different combinations of 8 letters, 
 taken in every possible way, can be made. 
 
 Ans. 255. 
 
 *^* Several other useful and interesting subjects of an elementary 
 character yet remain to be treated of. The Editor is preparing 
 for publication a Second Part, which will embrace these subjects 
 
APPENDIX. 
 
 GENERAL PRINCIPLES, PROPERTIES OF NUMBERS, AND OTHER 
 EXPLANATIONS IMPORTANT, NOT ONLY FOR THE UNDERSTAND- 
 ING OF THE CALCULATION, BUT FOR SIMPLIFYING THE OPER- 
 ATIONS IN AN INFINITY OF WAYS. 
 
 ON THE DIFFERENT KINDS OF NUMBERS. 
 
 1. A number expresses single units, or parts of a single 
 unit, or at the same time, units and parts of units. 
 
 By unit we understand a whole 1, and by a part of a unit, 
 or Si fraction^ all that is below the value of 1. 
 
 2. The number which expresses units only is called a whole^ 
 a simiyle^ or a7i incomplex number. That which expresses 
 units and parts of units taken together, is called ?i fractionary^ 
 a compound^ or a complex number. That which expresses on 
 ly parts of a unit is called diffraction, 
 
 3. The number which does not specify any particular kind 
 of units, such as 1, 2, 3, 4, is called an abstract number. 
 That whose kind is specified, such as 5 cents, 6 bushels, is 
 called a concrete number. 
 
 4. The number terminated by 2, 4, 6, 8, or 0, is called an 
 even number. That which ends with 1, 3, 5, 7, or 9, is called 
 an odd number, 
 
 5. The number which has no exact divisors, in whole num- 
 bers, but itself or the unit, such as 1, 2, 3, 5, 7, 11, 13, &c., 
 is called a prime number. That which has other exact di- 
 visors than itself or the unit, such as 4, 6, 8, 9, 10, 12, 14, 15. 
 &c., is called a complex number. 
 
 11* 
 
126 APPENDIX. 
 
 The exact divisors of a complex number (or multiple) are 
 called submultiples. Thus, for example, 1, 2, 3, 4, 6, 8, and 
 12, are submultiples of 24. 
 
 ON THE FOUR RULES OF ARITHMETIC. 
 
 6. By Addition we add two or more numbers so as to 
 make but one. The result is called sum or total. 
 
 By Subtraction we cut off a number from another num- 
 ber. The result is called -the remainder^ surplus^ or difference. 
 
 By Multiplication we take a number called multiplicand 
 as many times as there are units in another number called the 
 multiplier. The result is named the product. The multipli- 
 cand and the multiplier are called the two factors of the 
 product. 
 
 By Division* we ascertain how many times a number called 
 the divisor is contained in another called the dividend. The 
 result is named the quotient. The dividend and the divisor 
 are named the two terms, 
 
 7. The unit neither multiplies nor divides. 
 
 8. To multiply by a number less than the unit is to di- 
 minish the number multiplied ; hence it follows that to multi- 
 ply is not always to increase. 
 
 To divide by a number less than the unit, is to increase the 
 number divided ; hence it is that to divide is not always to 
 diminish. 
 
 9. Of the two factors of a product, the one being multi- 
 plied an»t the other divided by a like number, the result re- 
 mains the same. 
 
 10. The iwo terms of a divisor being multiplied or divided 
 by a like number, the quotient remains the same. 
 
 1 1. The general product of several numbers is always the 
 same in whatever order they are multiplied. 
 
 12. A quantity multiplied or divided by a number, gives 
 the same product or the same quotient multiplied or divided 
 successively by the factors of that nymber. 
 
 13. If the quotient be greater than the unit, the dividend 
 is greater than the divisor, and vice versa. If it be the unit, 
 the divisor is equal to the dividend. 
 
APPENDIX. 127 
 
 14. Jf the product of two factors be less than their sum, 
 it is because that one of them is necessarily the unit. 
 
 15. To double^ treble^ quadruple, centriple, &c., a number, 
 is to multiply it by 2, 3, 4, 100, &c. 
 
 16. A quantity multiplied and divided in turn by a like 
 number, becomes again what it was at first. In this case, 
 therefore, the shortest way is to dispense with both multiplying 
 and dividing. 
 
 17. The product of two numbers divided by one of 
 them, gives the other. 
 
 18. The quotient multiplied by the divisor gives the 
 dividend ; the dividend divided by the quotient gives the 
 divisor. 
 
 ON THE TWO TERMS OF A FRACTION. 
 
 19. Every fraction is composed of two terms : the f.rst 
 mentioned is called the numerator^ the second denominator. 
 The numerator indicates how many equal parts of the unit 
 are contained in the fraction : the denominator gives the name 
 of those parts. 
 
 These two terms of a fraction are assimilated to the two 
 terms of a division. The numerator represents the dividend ; 
 the denominator the divisor. 
 
 20. If the numerator be equal to the denominator the frac- 
 tion is equal to 1. If the numerator is less than the denom- 
 inator, the fraction is less than 1. If the numerator is 
 greater than the denominator, the fraction is greater than 1. 
 
 21. Of two fractions having the same denominator, the 
 greatest is that which has the greatest numerator. Of two 
 fractions having the same numerator, the greatest is that 
 which has the smallest denominator. 
 
 22. To render a fraction greater, the numerator is multi 
 plied without touching the denominator, or the denominator is 
 divided without touching the numerator: 
 
 To render a fraction smaller, the numerator is divided with- 
 out touching the denominator, or the denominator is multi- 
 plied without touching the numerator. 
 
 23. The two terms of a fraction being multiplied or divi- 
 ded by a like number, its value remams the same. 
 
128 APPENDIX. 
 
 24. Any whole number may always be put indifferently 
 under the form of a fraction : the only thing is to give it the 
 unit for denominator. 
 
 25. To take any part or fraction of a number, is to multi- 
 ply it by that fraction. 
 
 Thus to take the f of 12=12xf= V=Q- 
 
 Then if the fraction to take has the unit for numerator, we 
 have only to divide by the denominator. Thus to take the 
 ■^, ■^, or ^, (fee, of a number, is to divide tliat number by 2, 
 3, or 4. 
 
 26. To increase a number by any fraction of itself, is to 
 multiply it by a new fraction whose denominator equals the 
 sum of the two gtven terms, and whose denominator remains 
 the same. 
 
 Thus to increase 60 by j5^=60xi| = ^H^=^^- 
 
 ON RATIOS AND PROPORTIONS. 
 
 27. The ratio of two numbers is the quotient of the first 
 number di^dded by the second. Thus the ratio of 15 to 5 is 3. 
 
 The connection of two equal ratios is called the geometrical 
 proportion. Thus 15 : 5 : : 6 : 2 is one proportion, as the ratio 
 of 15 to 5=3, and the ratio of 6 to 2= also 3. 
 
 The first term of a ratio is named antecedent ; and the 
 second consequent. 
 
 The first and the fourth term gf a proportion are called 
 the extremes ; the second and third are called the mediums. 
 
 The mediums may always exchange places without dis- 
 turbing the proportion. 
 
 28. The product of the extremes always equals that of the 
 mediums. 
 
 29. We determine the fourth term of a proportion when 
 unknown, by dividing the product of the mediums by the first 
 term. 
 
 ON THE SQUARES OF NUMBERS AND THEIR ROOTS. 
 
 The square number or second power of a number is that 
 number multiplied once by itself. 
 
 The square root or second root of a number is the very 
 number which has been raised to the square. 
 
APPENDIX. 129 
 
 This is the natural series of squares up to 100. 
 Squares. 1 4 9 16 25 36 49 64 81 100. 
 Roots. 12345 6789 10. 
 Observe that the successive difference between the squares 
 always exceeds itself by two unit# Thus from 1 to 4 the 
 difference is 3 ; from 4 to 9 the difference is 5 ; from 9 to 16 
 the difference is 7, &c. 
 
 30. A number which is not a perfect square is called a 
 surd, an irrational or an incommensurahle number^ consequent- 
 ly the root of those numbers is never more than approx- 
 imative. 
 
 31. To increase the square root of a given number by a 
 unit, add to that number the double of its root+1. Example : 
 
 Let the number be 25 with the v/=5. To have 6 at the 
 root, add to 25 the double of 5 + 1 = 11, and you will have 
 25 -f 11 = 36, with the v/=6. 
 
 If the root of the given number have a remainder, cut it 
 off from the double-j-l of the root, the new remainder will 
 be the number to add to the given number. Example : 
 
 Let the number be 53, of which the v/=:74-the remainder 
 4. To have 8 at the root, from the double of the root 7+1 
 = 15, cut off the remainder 4, you will have 11. Then 53 + 
 11=64, of which the v/=8. 
 
 32. To diminish the square root of a number by a unit, 
 take from that number the double of its root— 1. Example : 
 
 If the number be 64, of which the \/ = 8, to have only 7 at 
 the root, take from 64 the double of 8 — 1 = 15, you willl^ave 
 64-15=49, of which the x/=^l. 
 
 If the root of the given number have a remainder, add it 
 to the double — 1 of the root, and take the total from the 
 given number. Example : 
 
 Suppose the number is 86, of which the \/=9 + the remain- 
 der 5. In order to have the root = 8, to the double of the 
 root 9 — 1 = 17, add the remainder 5, you will have 17 + 5 = 
 22. Then 96—22=64, of which the v/=8. 
 
 33. The proof of the extraction of the root of a number 
 is done by multiplying that root by itself and joining to the 
 product the remainder, if there be one ; the total ought to 
 produce again the number whose root has been extracted. 
 
 34. Knowing the difference of two numbers and that of 
 
130 APPENDIX 
 
 their squares, the quotient of the second difference divided by 
 the first, gives the sum of the two numbers, which enables us 
 to determine each of them immediately. 
 
 ON THE FACTORS ANdS^UBMULTIPLES OF A NUMBER. 
 
 35. Every number has at least two factors ; it may have 
 more. If it have but two factors, it is necessarily a prime 
 number, (5,) and those two factors, consequently, are that 
 number itself and the unit. 
 
 36. To reproduce by multiplication a number which has 
 more than two factors, the greater factor must be multiplied 
 by the lesser ; 
 
 Or the factor immediately inferior to the greater by the 
 factor immediately superior to the lesser ; so on, all through, 
 always following the same order. Example : 
 
 Suppose the number be 24, which has the eight factors, 1, 2, 
 3, 4, 6, 8, 12, and 24 ; to reproduce that number we shall have : 
 
 24X1=:24, or 12x2^=24, or 8x3=24, or 6x4=24. 
 
 In this example the quantity of the factors is even: if it be 
 odd^ then it is the medium factor which, multiplied by itself, 
 produces the number in question. Example : 
 
 Suppose the number be 64, which has the seven factors, 1, 
 2, 4, 8, 16, 32, 64 ; to reproduce that number we shall have: 
 
 64Xl==64, or 34x2=64, or 16x4=64, or 8x8 = 64. 
 
 From this article we shall deduce the three following prin- 
 ciples : 
 
 37. The greatest factor of a^ number is that number itself 
 and its least factor is the unit. 
 
 38. The greatest factor of an even number (that number 
 itself excepted) is always the half of that number, and its 
 least factor (the unit excepted) is always 2. 
 
 39. Every number which has an uyieven quantity of fao 
 tors is a square number whose root is the mean factor. 
 
 40. Observe that every number has always a suhmultiple 
 less than it hdiS factors. Indeed a number, whatsoever it be, 
 figures itself amongst its factors, but never amongst its sub- 
 multiples. 
 
 Observe, further, that to produce a number by the multi- 
 plication of two of its sub multiples, the unit will never be 
 
APPENDIX. ISl 
 
 one of those submultiples, seeing that it cannot in any way 
 contribute to that result. 
 
 Thus then, excluding the unit f7om the submultiples of a 
 number, we shall deduce that the principles established, Arti- 
 cles 36, 38, and 39, regarding tb^ factors of a number, apply 
 without exception to its submultiples. 
 
 ON ODD AND EVEN NUMBERS. 
 
 41. The sum of two even numbers is an even number. — 
 That of two odd numbers is also an even number. That of 
 an even number and an odd number is an odd number. 
 
 42. The difference between two even numbers is an even 
 number. That between two odd numbers is also an even 
 number. That betw een an even number and an odd number 
 is an odd number. 
 
 43. The product of two even numbers is an even number. 
 That of two odd numbers is an odd number. That of an 
 even number and an odd number is an even number. 
 
 44. Every odd number has only odd factors and submul- 
 tiples. 
 
 45. Every prime number, with the single exception of the 
 number 2, is an odd number. 
 
 ON PROGRESSIONS. 
 
 46. The arithmetical progression is a series of terms suc- 
 cessively increased or diminished by a like quantity. In the 
 former case the progression is called increasing, and in the 
 latter decreasing. The difference between the terms is called 
 2yroportion. 
 
 The geometrical progression is a series of terms successive- 
 ly multiplied or divided by a like quantity. In the former 
 case the progression is said to be increasing, and in the latter 
 deci-easing . The quantity which multiplies or divides is called 
 p)roportion. 
 
 There are five quantities to be considered in progressions ; 
 the first term, the last term, the number of terms, the sum of 
 the terms, and the proportion or ratio. 
 
 47. The last term of an arithmetical progression is com- 
 
132 APPENDIX. 
 
 posed of the first term, more as many times the ratio as there 
 are terms before it. 
 
 Example. — Suppose —2, 5, 8, 11, 14, 17, of which the ratio 
 is 3. Remark that the last term 17, which has^i'e terms be- 
 fore it, is composed of the fet term 2+ (the proportion 3x 
 5) = 17. 
 
 48. The sum of the terms of an arithmetical progression 
 is composed of the sum of the first and last terms, multiplied 
 by half the number of terms. 
 
 Example, — Suppose -^5, 7, 9, 11, 13, 15, of which the half 
 of the number of terms =r 3. Then the first terms 5 -{-the 
 last 15=20 w^hichx3=:60^ the sum of the six terms. 
 
 49. In all arithmetical progre||ions the sum of the first 
 and last terms equals that of the second and last but one, or 
 that of the third and the antepenultima, and so on all through, 
 still observing the same order. 
 
 Example.— ^Vi^^o^Q ~4, 7, 10, 13, If^, 19, 22, 25. Remark 
 that the 1st and the 8th term =4 +25 =29 ; 
 that the 2d and the 7th " =7+22=29 ; 
 that the 3d and the 6th " =10+ 19=29 ; 
 that the 4th and the 5th " =13 + 16=29. 
 In the example just given the number of terms x^even. If 
 it be odd^ it is then the double of the iniddle term, w^hich equals 
 th^ sum of the other taken two by two as above. 
 Example. — Suppose -f-11, 16, 21, 36, 31. Remark 
 that the 1st and the 5th term = ll+31=42; 
 that the 2d and the 4th " =16 + 26=42; 
 that the 3d doubled =21 +21 =42.* 
 
 50. If the number of the terms of an arithmetical pro- 
 gression be odd^ the middle t-erm always equals the sum divi« 
 ded by the number of the terms, that is to say, that if that 
 number be 3, or 5, or 7, &c., the middle term=^ or \ or i 
 &c., of the sum of the terms. 
 
 Example, — Suppose -f-10, 20, 30, 40, 50, of which the sum 
 = 150. 
 
 Then that sum divided by 5, the number qjf the terms 
 = i|o=30. 
 
 Hence, as we see, the naiddle term =: 30. 
 
 If the number of the terms be even, the two means taken 
 
APPENDIX. 133 
 
 together, equal the sum divided by half the number of the 
 terms. 
 
 Example. — Suppose -f-5, 10, 15, 20, 25, 30, of which the 
 'uni = 105. Now 105 divided bv 3, the half the number of 
 terms=r^§^i=35. Hence the mo middle terms=15 + 20 
 =35. 
 
 51. The last term of a geometrical progression equals the 
 first, multiplied by the ratio raised to a power of a degree 
 equal to the number of the terms — 1. 
 
 Exam-pie, — Suppose -f^ 3 : 6 : 12 : 24 : 48, of which the ratio 
 is 2. Now the progression having -^vg terms, the ratio 2 
 raised to the fourth power wdll give us 2x2x2x2=16. 
 Then, observe that the last ^erm= the first term 3x16=48. 
 
 . 52. To find out the sum of all the terms of a geometrical 
 progression, the last term is multiplied by the ratio : the first 
 term is taken from the product, and the remainder is divided 
 by the ratio, lessened by a unit. 
 
 Example. — Suppose : : 2 :-6 : 18 : 54 : 162, of which the 
 ratio is 3, and the sum 242. Hence, the last term 162x3 
 =486. From 486 subtract the first term 2, remainder 484. 
 Now this remainder 484 : (the ratio 3 — 1) = '*|-^=242, sum 
 of the terms. 
 
 Remark. — All these difi'erent principles on progression ap- 
 ply textually to increasing progressions, but to make them 
 applicable to decreasing progressions, it is only necessary to 
 change the words ^rs^J into last and last into Jirst. 
 
 ON DIVISIBLE NUMBERS WITHOUT A REMAINDER. 
 
 53. It is important, for abridging the calculation in a mul 
 titude of cases, to know the characters which render one num- 
 ber exactly divisible by another. Thus, we may always di- 
 vide without any remainder : 
 
 By 2, all even numbers without exception. 
 
 By 3, all even or odd numbers the sum of whose figures, 
 considered as simple units, is 3 or a multiple of 3. 
 
 By 4, all even numbers whereof the two last figures on the 
 right are divisible by 4. 
 
 By 5, all numbers terminated by a 5 or an 0. 
 
 By 6, all even numbers already divisible by 3. 
 12 
 
134: APPENDIX. 
 
 By 8, all even numbers whereof the three last figures on 
 the right are divisible by 8. 
 
 By 9, all even or uneven numbers the sum of whose figures 
 is 9 or a multiple of 9. 
 
 By 10, 100, 1000, &c., \^\ numbers ending with 0, 00, 
 000, &c. 
 
 By 11, all even or uneven numbers whereof the sum of the 
 1st, 3d, 5th, 7th figures, &c., is equal to the sum of the 2d, 
 4th, 6th, 8th, (fee, or whose diflTerenceis 11 or a multiple of 11. 
 
 By 12, all even numbers already divisible by 3 and 4. 
 
 By 25, all even or uneven numbers whereof the two last 
 figures on the right are divisible by 25. 
 
 PROPERTIES AND VARIOUS EXPLANATIONS. 
 
 54. Of two unequal numbers the greatest equal (the sum 
 4-the difference) divided by 2, and the smallest equal (the 
 sum —ih.Q diiference) divided also by 2 ; whence it follows 
 that the sum increased by the difference equals twice the 
 greater number, and that, diminished by the difference, it 
 equals tiuice the smaller. 
 
 55. To equal two unequal numbers, without altering their 
 sum^ the greater is diminished by half the difference, and the 
 lesser increased by the other half. 
 
 56. The difference between two numbers cannot be superior 
 nor even equal to the greater number, but it may equal or 
 exceed the lesser. 
 
 57. The least difference possible between two whole num- 
 bers, even, or between two whole numbers, odd, is 2, 
 
 58. Knowing how many times A is greater than B, we at 
 once perceive how many times B is smaller than A, preserving 
 the same numerator to the given fraction, and forming a new 
 denominator from the sum of the two terms. JExamples : 
 
 A being > B by i, B is < A by J. 
 A being > B by J, B is < A by /j. 
 
 59. Knowing how many times A is smaller than B, we in- 
 stantly determine how many times B is greater than A, pre- 
 serving the same numerator to the given fraction, and forming 
 a new denominator of the given denominator, from which tho 
 numerator is subtracted. Examples : 
 
APPENDIX. 135 
 
 Abeing<Bby I, Bis> Abyi. 
 A being < B by I, B is > A by |. 
 
 60. Knowing that the number A is equal to such a part of 
 the number B, we instantly detennine what part of the num- 
 ber A is equal to the number B, by reversing the two terms 
 of the given fraction. Examples : 
 
 A being I of B, B=:f of A. 
 A being I of B, B=:f of A. 
 
 61. Knowing how many times the sum of two numbers 
 contains their difference, we determine the proportion of one 
 number to the other in the following way : 
 
 Place that number of times under the form of a fraction, 
 and with that first fraction form a second, whose numerator is 
 equal to the numerator of the first, less its denominator, and 
 w^hose denominator is equal to the numerator of the first, 
 more its denominator. This second fraction will express the 
 proportion of the lesser number to the greater, and reversed 
 it will express the proportion of the greater to the lesser. 
 
 Example, — Take the numbers 5 and 7,, of which the sum = 
 12 and the difference 2 ; the one contains, therefore, 6 times 
 the other. Then 6 in a fraction [24] = J, and forming our 
 second fraction such as it is, we shall have |-, that is to say 
 that the lesser number ir:|- of the greater. 
 
 Another example. — Take the numbers 60 and 35, of which 
 the sum =95 and the difference 25 ; the one contains, there- 
 fore, 3 times |- the other. Then 3-| = y, which, according to 
 what is prescribed, will give for second fraction -^=xV • ^^ 
 fine, 45 = j^ of 60. 
 
 62. Every number having two unequal figures, lohen read 
 backwards^ differs from what it is by 9 or a multiple of 9. If 
 it differ by 9, the difference between the two figures is 1. If 
 it differ by twice 9, by three times 9, &c., the difference between 
 the two figures is 2, 3, &c. Thus the number 81, read back- 
 wards, =18 ; from 81 to 18 the difference =63. Now 63= 
 seven times 9. So the difference of the figures 8 and 1=7. 
 
 63." The smaller the difference is between two numbers 
 
 forming a like sum, the greater their product is. Examples : 
 
 riX7= 7 riX8= 8 
 
 No. 1. 12x6 = 12 No. 2. J 2x7 = 14 
 
 Sum 8. I 3X5 = 15 Sum 9. 13x6 = 18 
 
 [4x4=16 [4X5=20 
 
J.36 , APPENDIX. 
 
 Now, the two examples here given furnish us with the fol- 
 lowing remarks : 
 
 1st. The one of the two numbers which, multiplied one by the 
 other, give the smallest product, is always the unH, whether 
 the sum be odd or eveii ; ' ^> 
 
 2d. If the sum be even^ No. 1, each of the two numbers 
 which give the greater product is equal to half the sum ; 
 
 3d. If the sum be odd^ No. 2, the two numbers which give 
 the greater product differ between themselves' by the unit. 
 
 64. There are some quantities which, according to the 
 nature of the question, can only be fractionary. Thus, for 
 instance, when, in speaking of workmen^ birds^ ^99^-, <^c., we 
 mention the half^ thirds^ quarters^ &;c., it is necessarily sup- 
 posed that those quantities are exactly divisible by 2, 3, 4, &;c. 
 
 But, to know, with accuracy, the number whereby a quan- 
 tity of this kind becomes divisible, if increased by one of its 
 parts, add the two terms of the fraction which expresses that 
 part, the sum will be the answer. 
 
 Thus, for example, if 1 increase by \ the contents of a bas- 
 ket of eggs, I conclude that those contents, at first exactly di- 
 visible by 7, is now divisible by 7 -|- 5 = A. 12. 
 
 If, on the contrary, the quantity of one of its parts be di- 
 minished, you will determine the number by which it becomes 
 divisible, taking the numerator from the denominator. The 
 remainder will be the answer. 
 
 Thus, for example, if I diminish by ^ the birds of an aviary, 
 I conclude that their number, at first exactly divisible by 7, is 
 now divisible by 7— 5= A. 2. 
 
 65. This is an operation which often occurs in my solutions. 
 In order that the reader may perfectly understand it, I am 
 about to give here an explanatory example. 
 
 Let us suppose that the question is to divide (.t+4) into 
 two' parts, one of which =: the | of the other. Make the sum 
 of the two terms of the given fraction, you will have 3-f-5 = 
 8, which indicates that the lesser part ought to have the \ of 
 the number (aT+4) and the greater the |-. 
 
 Operation : (a? +4) f = j — - — the lesser number. 
 
 (a? +4) -1= — ^ — the greater numbeF. 
 
APPENDIX. 137 
 
 MISCELLANEOUS PROBLEMS. 
 
 66. Divide 46 into two such ^arts that the sum of the quo- 
 tients obtained after dividing one by 7 and the other by 3 
 may be 10. Ans. 28 and 18. 
 
 67. Divide $1170 between three persons, A, B, C, propor- 
 tionally to their ages : B's age is one third greater than that 
 of A, who is but half of C's. What is the share of each 1 
 
 Ans. A, 1270 ; B, $360 ; C, $540. 
 
 68. A capital is such, that, augmented its simple interest 
 for 5 years at 4 per cent,, it raises to the amount of $8208. 
 What is the capital ? Ans, $6840. 
 
 69. A capitalist placed the | of his stock at 4 per cent, 
 and the remaining J at 5 per cent. : it produces in all $2940. 
 What was the whole sum lent ? Ans, $70,000. 
 
 70. If two sums of money be placed at interest, one of 
 $5500 at 4 per cent., and, 4|- years afterwards, $8000 be 
 placed at 5 per cent., in what time will the sums produce the 
 same interest 1 
 
 Ans, 10 years from the time the first sum was placed. 
 
 71. 1 had a certain sum in my purse : I took out the third 
 of its contents : I then put in $50 ; some time afterwards I 
 took out the fourth of what it then contained, and put in $70 
 more, after which it contained $120. How 'many were in ii 
 first ? Ans. $25. 
 
 72. Says A to B, Give me $100, and we shall have equal 
 sums. Give- me, says B to A, $100, and I shall have double 
 what thou hast. How many had each? Ans, $500 and $700. 
 
 73. Find two numbers whose difl?erence, sum, and product, 
 may be to one another as the numbers 2, 3, and 5. 
 
 Ans, 2 and 10. 
 
 74. The sum of two numbers is 13 and the " difference of 
 their squares is 39. What are the numbers ? Ans. 5 and 8. 
 
 75. A and B together have but the f of C ; B and C to- 
 gether 6 times A, and if B had $680 more than he really has, 
 he would have as much as A and C together. How many dol- 
 lars has each? Ans, A, $200 ; B, $360 ; C, * 
 
 12* 
 
138 APPENDIX. 
 
 76. What is that number whose seventh part multiplied by 
 its eighth and the product divided by 3 would give 298 1 for 
 result ? Ans. 224. 
 
 77. What are two numbers whose product is 750 and 
 whose quotient is 3^ ? <i Ans. 50 and 15. 
 
 78. A person being questioned about his age, replied : My 
 mother was twenty years of age at my birth, and the number 
 of her years multiplied by mine exceed by 2500 years her 
 age and mine united. What is his age ? Ans. 42. 
 
 79. A gentleman bought some furniture and sold it shortly 
 after for $144; by which he gained as much per cent, as it 
 cost. Required the first cost. Ans. $80. 
 
 80. Determine two numbers whose sum shall be 41, and 
 the sum of whose squares shall be 901. Ans. 15 and 26. 
 
 81. The difference between two numbers is 8, and the sum 
 of their squares 544. What are the numbers ? 
 
 A71S. 12 and 20. 
 
 82. The product of two numbers is 255. and the sum of 
 their squares 514. What are the numbers? Ans. 15 and 17. 
 
 83. Divide the number 16 into two such parts, that if to 
 their product the sum of their squares be added, the result 
 will be 208. Ans. 4 and 12. 
 
 84. What is the number which added to its square root the 
 sum shall be 1332? Arts. 1296. 
 
 85. What is«the number that exceeds its square root by 
 48 J? Ans. 56^. 
 
 86. Find two numbers, such that their sum, their product, 
 and the difference of their squares, may be equal ? 
 
 Ans. 3+^5 1±^5 
 2 ' 2 * 
 
 87. Find two numbers, whose difference multiplied by the 
 difference of their squares gives for product 160, and whose 
 sum multiplied by the sum of their squares gives 580 for pro- 
 duct. Ans. 7 and 3. 
 
 88. What is the ratio of a progression by difference of 22 
 terms, the first of which is 1 and the last 15? Ans. f. 
 
 89. There is a number of two figures, such, that if you di- 
 . vide it by the sum of its figures, then inverting the number 
 
APPENDIX. 139 
 
 and dividing this new number by the sum of its figures, the 
 difference of the two quotients is equal to the difference of its 
 figures, and the product of the two quotients to the number 
 itself. What is the number ? Ans. IS. 
 
 90. Triple Louisa's age is as ngpch above 40 as the third of 
 her age is under 10. What is her age? Ans, 15. 
 
 91. Being questioned how many eggs I had put in an ome- 
 let, I answered that the | of the whole augmented by the |- 
 of an egg exceeded the f of the whole by the square root of 
 all the eggs employed. A^is, 16. 
 
 92. Fortune was against me this morning, but it favoured 
 me this evening : I doubled the remainder of my^oney, and 
 instead of £5 that I lost, I have now £10 gain. How 
 many pounds have I now ? Ans, £30. 
 
 93. If my salary were doubled, said a comedian, it would 
 be 96 shillings more than the square of the twenty-fifth part 
 of what it is. What was if? Ans. 1200 shillings. 
 
 94. There are two numbers whose sum is 63, and |- of one 
 is equal to quintuple the other. What are they ? 
 
 Ans. 56 and 7. 
 
 95. Add 5 to the number of my children, double the re- 
 sult, and you will have triple my family. How many chil- 
 dren have n Ans. 10. 
 
 96. Two numbers are such, that triple the less is 3 units 
 more than the greater ; and if the greater be augmented its 
 ■A and the less its f , the former becomes double the latter. 
 What are they ? Ans. 33 and 12. 
 
 97. A brother said to his sister : I would want ^ of your 
 pounds to make £30 ; give me them. I would want, replied 
 the sister, | of yours to have £40 ; will you give me them ? 
 How many pounds had each 1 Ans. £25 and £20. 
 
 98. By.^neans of a legacy that quintupled his revenue, 
 Michael can spend $2 a day and lay up yearly the |^ of the 
 legacy. What was it? Ans. $800. 
 
 99. A fruiterer bought pomegranates at 5 cents for 6, and 
 by selling them at 3 cents for 2 she gained $1.20. Ho^ 
 many did she buy? Ans. 180. 
 
 100. Eobert buys a horse and sells him immediately. If 
 he had gained ^ more, his gain would have been douljle, and 
 
140 APPENDIX. 
 
 would have been but 10 shillings less than half the first cos^ 
 Required the first cost and selling price. 
 
 A71S, 100 and 120 shillings. 
 
 101. Determine two numbers whose diflference shall be 
 equal to one of them, and^ whose product shall be 18 more 
 thantriple their sum. Ans. 12 and 6, 
 
 102. Of three numbers the mean is l greater than the less, 
 and the former is ^ less than the greater ; now if each was re- 
 duced i their sum would be reduced 19 units. What are 
 the three numbers ? Ajis. 24, 18, and 15. 
 
 103. Three times a number, less 20, is as much above 
 -double the same number as its fourth part, plus 2, is undei 
 its half. What is the number ? Ans. 24. 
 
 104. Louisa buys 2^ lbs. of sugar, at 6d. per lb., and gives 
 in payment a piece of silver such that the square of the piece 
 returned exceeds the triple of the expense by a sum equal to 
 the return. Required the value of the piece given hy Louisa. 
 
 , Ans. Is. Sd. 
 
 105. By what number should 3 be multiplied in order that 
 the ^2 of the product may be equal to the sum of the two 
 fkctors? Ans. By 12. 
 
 106. When the granddaughter was born, the grandfather 
 wag 3|- times the granddaughter's present age, and 10 years 
 after, the latter was but -J of the grandfather's aforesaid age. 
 Required their respective ages. Ans. 90 and 20. 
 
 107. Determine three numbers, of which the greater, equal 
 to the sum of the two others, is also equal to the | of their 
 product, and of which the less is but the ^ of the other two 
 together. Ans. 24, 18, and 6. 
 
 108. An uncle claims the ^^ of an inheritance, the nephiew 
 J» and the niece the remainder. The product of the parts of 
 the two latter is 13 millions less than the square of the un- 
 cle's part. What was the inheritance ? Ans. $12 fiOO. 
 
 109. I drew two numbers, whose sum is equal to 4 times 
 their difference, and the greater of which, plus the difference, • 
 exceeds the less by 48 units. What are the numbers ? 
 
 A?is. 60 and S^. 
 llj}. My Fatch is very methodical in its time. Were I to 
 
APPENDIX. 141 
 
 let it go, it would mark the exact time every two months reg- 
 ularly. What is the yariation of my watch per hour ? 
 
 Ans. "I minute fast or slow. 
 
 111. Four years ago, the sister was | of the brother's years 
 older than the brother : four year^^ence the brother will be 
 J of the sister's years less than the sister. What is the age 
 of eachi Ans^ Sister, 16; brother, 14. 
 
 112. Had I double my gain, said a gamester, I would have 
 squared the number of my pounds ; were it but -J- what it is, 
 I would have tripled them only. What was the gain ? 
 
 A71S. £36. 
 
 113. My uncle spent J of his lifetime bachelor, ^ married, 
 and ^ a widower. When he married my aunt, she was i of 
 his years younger than he,. and 8 years after his years were J 
 greater than hers. At what age did each die ? 
 
 Ans. At 72 and at 40. 
 
 114. I have one sister and two brothers, the two latter are 
 twins : now my sister's age is equal to the sum or the prod- 
 uct, just as you please, of my brothers' ages. Please find 
 my sister's and brothers' ages separately. 
 
 Ans. Sister's age, 4 ; twins, 2 years each. 
 
 115. Subtract ^ of my brother's age, and ^ of my sister's 
 age, you shall have made them twins and the sum of their 
 years will have lessened 2. What is the age of each ? 
 
 A71S. 24 and 18. 
 
 1 16. A gamester being questioned about his gain, answered : 
 One of the factors of my gaih is but half the other, and their 
 sum is but half my gain. How many dollars did he win ? 
 
 Ans. 18. 
 
 117. Find two numbers, such, that the square of the lesser 
 may be equal to | of the greater, and the greater, diminished 
 its f , may exceed the lesser by ^. Ans. 20 and 4. 
 
 118. The product of two numbers equals three times their 
 sum, and their quotient is 3. What are the numbers ? 
 
 « Ans. 12 and 4. 
 
 119. Square ^ plus 1 of the years that I am short of a 
 quarter of a century and you will produce my age. What is 
 it? Ans. 16. 
 
 li^O. A gamester lost at the first game the square of ^ of 
 
142 APPENDIX. 
 
 the dollars he had about him ; but at the second round he 
 quintuples his remainder, and withdraws neither gainer nor 
 loser. How many dollars had he ? Ans, 80. 
 
 121. I am going to add %^new shelves to my library, each 
 of which will hold 20 volumes more than the ten already ex- 
 isting, and so I shall have 1000 volumes in all. How many 
 have I now ? A71S. 600. 
 
 122. Multiply half the father's age by half the son's age and 
 you will have the square of the son's age ; this square is 
 equal to double the sum of their ages. How old is each ? 
 
 Ans. 40 and 10. 
 
 123. The breadth of my room is but the f of its length. — 
 As broad as it is long, it would contain 144 square feet more. 
 Required its dimensions. Ans. 24 by 18. 
 
 124. The difference between the |- of my age, less 5, and 
 its I, plus 3, is the square root of my age. "What is it 1 
 
 Ans. 36 years. 
 
 125. The product of two numbers is 220. If from the 
 greater you subtract the difference, their product will lessen 
 99 units. Find the two numbers by one unknown term. 
 
 Ans. 20 and 11. 
 
 126. What is the number of your house ? The sum of its 
 digits, considered as units, is equal to -^ of the number. Find 
 it. ' Ans. 54. 
 
 127. The square of the difference of two numbers is equal 
 to its sum, and I of the former is equal to ^ of the latter. 
 What are the two numbers 1 Ans. 10 and 6. 
 
 128. An officer gave the following indication of the number 
 of his regiment : One of its factors is to the other : : 1:5, 
 and ,their sum is to their product : : 6 : 25. What was the 
 number? Ans. 125. 
 
 129. My age is composed of two figures, and read back- 
 wards it makes me I older. What is it ? Ans. 45. 
 
 130. The product of two ii umbers is J more than their 
 sum, and is equal to triple their difference. What are they ? 
 
 A71S. 2 and 6. 
 
 131. When the brother's age was the square of the sister's, 
 she was ^ of fhe brother's present age, and 8 years hence me 
 
APPENDIX. 143 
 
 sum of their ages will be augmented its |. WhaC is the age 
 of each? Ans. 21 and 15. 
 
 132. My garden, longer than broad, contains 900 square 
 yards ; if with the same superfici^ it were a perfect square, 
 its length would diminish the |, how much should its breadth 
 be augmented ? Ans, Its f. 
 
 133. Eight years ago, the brother's age was equal to his two 
 sisters' ages together ; 8 years hence it will be but the f of it, 
 and at the same period the age of the youngest is exactly the 
 ^ of the three ages united. What are the ages ? 
 
 Ans. 24, 20, and 12. 
 
 134. Divide a basket of pears among three sisters, so that 
 the part of the older may be to that of the second :: i : ^', 
 and that of the second to the part of the youngest : : -g- : i ; 
 then the sum of their squares will be 549. How many pears 
 had each'? Ans, 18, 12, and 9. 
 
 135. A gambler questioned about his gain, replied : Divided 
 by its least sub multiple, it diminishes £10 ; divided by its 
 greatest submultiple, it is diminished £12. How many 
 pounds did he gain ? Ans. £15. 
 
 136. The number of my house has seven sub-multiples be- 
 sides Unit. The fourth, in the order of magnitude, is ^ of the 
 number. What is it^ Afis. 36. 
 
 137. 1 sold ^ of my basket of eggs in one house, and 25 
 in another. Triple the remainder, and you shall reproduce 
 the primitive contents. What was it ? Ans. 60 eggs. 
 
 138. I have in my right hand double the contents I have in 
 my left ; but if I put in my left hand the square root of the 
 number in my right, each will contain the same number. How 
 many did each contain ] Ans. 16 and 8. 
 
 139. If I had paid ^ more for my watch, its price would 
 have been less by £4 than double what it cost me. What. 
 did I pay for it ] Ans. £6. 
 
 140. One of the factors of a multiplication is 5, and the- 
 other is 8 less than the product. What is the other factor!: 
 
 Ans. %. 
 
 141. A gentleman looking at his watch, was asked thetmie ;; 
 he reflected an instant and then replied: The square of the 
 
144 APPENDIX. 
 
 preseiit time, plus its root, is half greater than less its root. 
 What o'clock was it 1 A71S. 5. 
 
 142. The sum of the four terms of an arithmetical pro- 
 gression is 44 ; that of the^wo first, 18. What are the four 
 terms? • Ans. S, 10, 12, M. 
 
 143. There is 4 difference between two numbers, and their 
 sum is less than their product. Required the two numbers. 
 
 A71S. 2 and 6. 
 
 144. The difference of two numbers equals |- of the greater, 
 and represents the square of the less. What are the num- 
 bers? Ans, 30 and 5. 
 
 145. There are two unequal numbers ; the less is equal to 
 I of their sum, and their sum equals 3^ of their product. 
 
 Ans. 6 and 4. 
 
 146. The sum of two numbers, plus their difference, makes 
 100, and their difference joined to their quotient equals 45. 
 What are the numbers? Ans. 50 and 10. 
 
 147. I received this morning a basket of peaches : I laid 
 ■|- by for myself, and the remainder, a prime number, I divided 
 amongst my children in equal parts. How many children 
 have I? Ans. 11. 
 
 148. The f of the numerator equals the |- of the denomi- 
 nator, and the sum of the two terms is 1 1 more than the pro 
 duct of ^ of the denominator by J of the numerator. What 
 is the fraction? Ans, |. 
 
 149. I bought a horse yesterday and sold him at a profit 
 equal to the f, less £11, of my outlay, by which I gained 20 
 per cent. Required the first cost and selling price. 
 
 Ans. £20 and £24. 
 
 150. The four terms of a proportion make, together, 100. 
 The first is equal to the third and the ratio is 4. What are 
 die terms ? • . Ans. 40 : 10 : : 40 : 10. 
 
 151. The dividend is equal to the square of the divisor, and 
 their sum added to their quotient, is equal to 840. What is 
 the dividend? what is the divisor ? Ans. 784 and 28. 
 
 152. One square is quadruple another, and their sum added 
 to the sum of the two roots is 530. What are the two 
 squares ? Ans. 400 and 100. 
 
APPENDIX. 145 
 
 153. A milkmaid sells hens' eggs and ducks' eggs. Their 
 mean price is 16 cents the dozen. Now 6 dozen of the latter 
 are worth 10 dozen of the former. Required the price of each 
 dozen. ^ Afis. 12 and 20 cents. 
 
 154. What cost these six pounds of sugar? If they were 
 worth 3 cents per lb. more, my outlay would have been J 
 more. What did it cost per lb. 1 Ans. 15 cts. 
 
 155. The four terms of a proportion equal 100. Add 1 to 
 each term and they shall be still proportional. What are 
 they ? Ans. 25 : 25 : : 25 : 25. 
 
 156. The sum of two numbers exceeds their difference by 
 12, and their product exceeds their sum by 24. What are the 
 two numbers? Ans. 6 and 6. 
 
 157. A number squared is such, that if 25 be taken from 
 it, the remainder will be 7^ times its root. Required the 
 square ? Ans. 100. 
 
 158. An annuity placed at 13 J per cent, per annum, pro- 
 duces monthly a rent equal to its square root. What is the 
 annuity? Ans. $8100. 
 
 159. The product of two numbers is 120. Add 1 to each, 
 and their product shall be 150. What are the numbers'? 
 
 Ans. 24 and 5. 
 
 160. Two numbers are equal. If 3 be added to each, their 
 product will increase 51. What are the two numbers? 
 
 Ans. 7 and 7. 
 
 161. The sum and difference of two numbers taken together, 
 is 24 ; the product and the quotient, 51. What are the num- 
 bers? Ans. 4 and 12. 
 
 162. The number of my years is divisible by 5 and 7; but 
 the quotient by 7 is the lesser by 4. What is my age ? 
 
 Ans. 70. 
 
 163. The two terms of a division, added to their quotient, 
 make 109. Take 8 from the dividend, and the quotient w^ll 
 be less by 2. What are the three terms ? 
 
 Ans. 84, 4, and 21. 
 
 164. The square of the numerator is 1 more than the de- 
 nominator, and the sum of the two terms is 1 more than twice 
 their difference. What is the fraction ? Ans. f. 
 
 13 
 
146' APPENDIX. 
 
 165. The two terms of a division sum stre the same as the 
 two terms of another, but in an inverted order. The sum of 
 the four terms is 80, and that of their quotients 2^. What 
 are the terms of these division sums? Ans. 24 and 16. 
 
 166. Half the dividend is equal to the square of the divi- 
 sor; half the divisor equals the square of the quotient. Ke- 
 quired the dividend and divisor. Ans. 128 and 8. 
 
 167. Two numbers are : : 8 : 5. Subtract .5 from the first 
 to add to the second, and they will be : : 7 : 6. What are the 
 two numbers ? Ans. 40 and 25. 
 
 ] 68. How many miles from A to B ? said an inquisitive per 
 son. He received for answer that their number had but -two 
 factors, whose sum was 20. What is that number? 
 
 Ans. 19. 
 
 169. The product of the two terms of a fraction is 120. — 
 Add 1 to the numerator, subtract 1 fi'om the denominator, their 
 quotient will be 1. What is the fraction? Ans.^. 
 
 170. The failure of an insolvent debtor took away the | of 
 the capital that I had placed in his hands. The interest of the 
 remainder placed at 5 per cent, is equal to the square root of 
 the first capital. What was it ? A^is^. $10,000. 
 
 171. If you find me too old, read my age backwards, and 
 you will make me the |- younger. What was the person's 
 age? Ans. 81. 
 
 172. Find me three numbers in arithmetical progression, 
 the third of which shall be equal to the square of the first, 
 and the second triple the ratio. Ans. 2, 3, 4. 
 
 173. I travelled 6 miles an hour, said a pedestrian : had I 
 travelled 7|-, I should have arrived 8 hours sooner. Required 
 
 the length of the journey. Ans. 240 miles. 
 
 174. If you double the denominator of a fraction, the sum 
 of its two terms will be 22, and if you triple its numerator, 
 the sum will only be 21. Determine the fraction. Ans. J. 
 
 175. Increase the contents of my purse £3, and it will be- 
 come a perfect square. Jf, on the contrary, you lessen it £3, 
 it will contain but the root of the aforesaid square. What are 
 the contents? Ans. £6. 
 
 176. A lady questioned about hei- age, answered : Increase 
 
APPENDIX. 147 
 
 it the f , and in that state lessen it the f , and you shall have 
 made me 25 years younger. What was the lady's age ? 
 
 A71S. 60. 
 
 177. The figure 9 is not to be found among those that rep- 
 resent my wife's age ; but if yo^read backwards the two 
 figures that do, she will be, to her great displeasure, older by 
 a number of years between 55 and 70. Please tell me my 
 wife's age. Ans, 18 
 
 178. Well, Tom., what's the time ? Square the § of it and 
 you shall have the hour that shall strike 6 hours hence. What 
 o'clock was it? Ans. 10 o'clock. 
 
 179. How many children have you, sir? Their number is 
 equal to y ; the square of y is equal to 2y ' plus 9x, and y ex- 
 ceeds X the I". Find the number of my childi»en with one 
 equation, only and one unknown term. A7is, 9. 
 
 180. Find two numbers whose sum : their difference : : 7:3, 
 and whose product increased 10 is equal to the square of f of 
 the greater number. Ans. 15 and 6. 
 
 181. Part of my capital is placed at 5 per cent., the remainder 
 at 6 ; their sum is $35,000. I of the revenue of the first 
 part is equal to -^ that of the second. Required the two parts. 
 
 Ans. $15,000 and $20,000. 
 
 182. Four numbers are in geometrical proportion; the ra- 
 tio is ^. The fourth term is square of the first, the quotient 
 of the second by the third is 1. Bequired the four terms. 
 
 Ans. 4: 8 :: 8 : 16. 
 
 183. The number x exceeds the number y by the whole 
 root of X, and ^ of x equals the ^q of y. Determine the two 
 numbers with one unknown term. Ans. 36 and 30. 
 
 184. Yesterday I bought 5 yds. of black cloth and 6 yds. 
 of blue, ^his morning I bought 3 yds. of blue and 9 of black, 
 and my expense was the same as that of yesterday. Their 
 mean price was 85 shillings. Required the price of a yard of 
 each. Ans. 30 an(^0 shillings. 
 
 185. Six years hence, said a lady, my daughter's age will 
 be the square of what it was 6 years ago. \^hat is her age 
 to-day? Ans. 10 years. 
 
 186. The sum of two numbers is 4 times their difference, 
 
148 APPENDIX. 
 
 and the difference 5^^ of their product. What are the two num- 
 bers? Afis. 10 and 6. 
 
 187. An officer being questioned about the number of men 
 in his detachment, answered : Their number has but 3 factors, 
 whose sum is 31. What ^fas their number? Ans, 25. 
 
 188. Waiter, your bill of fare amounts to so much, does it 
 not] Yes, sir. Well, here are so many dollars, and receipt 
 it. Oh, sir, 1 off cannot be. Well, then, here is another sum 
 that lacks but 1 dollar of the I of the first; and let's say no 
 more about it. It is, nevertheless, hard, sir, to lose the double, 
 plus 1, of the square root of my bill. What was the amount '^ 
 
 Ans, 144 dollars. 
 
 189. The ages of the grandpapa and grandson are such, 
 that their quotient is equal to J of their product, and the sum 
 of the aforesaid quotient and product is 320. What is the age 
 of each ? Ans. 96 and 3 years. 
 
 190. I mixed two pipes of wine ; one cost 180 shillings and 
 the other 140 shillings. The first contained 20 bottles more 
 than the second, and cost 5d. less per bottle. What is the value 
 of a bottle of the mixture? Ans. 2s. 5d. 
 
 191. The quotient exceeds the 'divisor by half plus 1, and 
 the sum of the divisor and quotient exceeds double the square 
 root of the dividend plus 1. Required the dividend, divisor^ 
 and quotient. . Ans. 400, 16, and 25. 
 
 192. Two faucets running together, filled a basin in 3 hours. 
 If the first had run but 2 hours, it would have taken the second 
 6 hours to do the remainder. What time would it take each 
 running alone ? Ans, 4 and 12 hours. 
 
 193. The father's age has two factors, of which one repre- 
 sents his daughter's age, and the other is 18 less. Square this 
 other factor, add to it ^ of the daughter's age, and this result 
 to the sum of both ageis, and the general result wil^ be 100. 
 Required the age of each. Ans. 63 and 21. 
 
 194. The product of two numbers exceeds their sum by 14, 
 and their difference is 2. What are the two numbers ? 
 
 Ans. 6 and 4. 
 
 195. A number of three figures is a multiple of 11, and the 
 anits is quadruple the hundreds. What is the number? 
 
 Ans. 154. 
 
APPENDIX. 149 
 
 196. A number of two figures is such, that its root placed 
 at the right or left of the aforesaid number gives two results, 
 such, that the latter exceeds the former by 252. Ans. 16. 
 
 197. A bird-seller, forgetting t9 shut his coop, the greater 
 part of the prisoners availed themselves of the opportunity, 
 and flew away. Being questioned about the number of fugi- 
 tives, the bird-seller answered ; If J^ moje flew away, the prim- 
 itive number, which , was over 130 and under 180, would 
 now be reduced ^. What was the primitive number 1 What 
 is their present number? Ans. 156 and 84. 
 
 198. One square is quadruple another, and their sum is 4 
 more than the next square above the greater of the two. What 
 are the two squares'? Ans. 2p and 100. 
 
 199. Lawrence swops his flute for Edward's violin, and gives 
 to boot a number of pounds equal to ^ of the price of the 
 flute 'plus the square root of the price of the violin. What is 
 the price of each instrument, if the sum of the two prices is 
 quadruple the pounds given to boot by Lawrence ? 
 
 Ans. £15 and £25. 
 
 200. A number of three figures is such, that the sum of its 
 digits is 16; and by inverting the number, then adding it to 
 the number inverted, the sum will be 1211 and the diflerence 
 297. What is the number ? Ans. 457. 
 
 201. An oflTicer gave the following indication of the number 
 of his regiment : One of its factors is to the other : : 1 : 5, and 
 their sum is to their product : : 6 : 25. What was the num- 
 ber ? Ans. 125. 
 
 202. Two sisters have unequal sums for their purchases. — 
 The elder has the most. The sum of their pounds is equal to 
 ^ of their product, which latter would have been ^ less if half 
 the pounds of the younger were given to the elder. How many 
 pounds had each? Ans. £15 and £10. 
 
 203. What is the numb.er, whose square, reduced to its 
 quarter, excneds by | three times the |- of the number ? 
 
 Ans. 12. 
 
 204. Aunt wants to have a room squared, which is 2J 
 times as long as it is broad. She asked the mason how many 
 squares of marble of a certain dimension would be requisite. 
 The mason answered, that if the length was but double the 
 
 13* 
 
150 APPENDIX. 
 
 breadth, it would have taken 800 less. What do you con- 
 clude from this answer ? 
 
 Ans. That it would have taken 4000. 
 
 205. A merchant gave ^ of his profit to the poor. At the 
 year's end his alms amounted to $390. I demand what was 
 the amount of his sales, if half was at 10, ^ at 15, and the 
 remainder at 18 per cent, profit? Ans. $60,000. 
 
 206. If you were sent to a house whose number was rep- 
 resented by three figures, knowing that the digit representing 
 the hundreds was double that of the tens, and that the sum 
 of the three digits was but ^ of the number, at what num- 
 ber would you rap? " Ans. At No. 216. 
 
 207. The sum of the four terms of a proposition is 63 ; 
 the first is 4 more than the second ; the quotient of the third 
 by the second is 8^ ; and the product of the means is 136. 
 Required the four terms. Ans. 8 : 4 : : 34 : 17. 
 
 208. The product of the four terms of a proportion is 576 ; 
 the difference between the first and the fourth term is 10, 
 and the quotient of the third by the second is f . What are 
 the four terms ? Ans. 2 : 6 : : 4 • 12. 
 
 THK EHD. 
 
BOOKS PUBLISHED BY D. 4 J. SADLIER <fe COMPANY. 
 GREAT SUCCESS OF THE 
 
 The Volumes of the Library publvihed are the most interesting as 
 
 well as the most useful Books yet^sued from the American 
 
 Press, and supply a want long'y'elt by Catholic Parents. 
 
 Books which they can safely place in the hands of 
 
 their children. 
 
 (EIGHTH THOUSAND JUST PUBLISHED.) 
 
 T 
 
 he Popular liibrary 
 
 OF HISTORY, BIOGRAPHY, FICTIOIN^, and MIS- 
 CELLANEOUS LITERATURE. A Series of Works 
 by some of the most eminent writers of the day ; Edited 
 by Messrs. Capes, Northcote, and Thompson. 
 
 The " Popular Library " is intended to supply a desideratum which 
 has long been felt, by providing at a cheap rate a series of instructive 
 and entertaing publications, suited for general use, wr'tten expressly 
 for the purpose, and adapted in all respects to the circumstances of 
 the present day. It is intended that the style of the wcrk shall be 
 such as to engage the attention of young and old, and of all classes 
 of readers, while the subjects will be so varied as to render the series 
 equally acceptable for Home use. Educational purposes, or railway 
 reading. 
 
 The following are some of the subjects which it is proposed to in- 
 clude in the " Popular Library," though the volumes will not neces- 
 sarily be issued in the order here given. A large portion of the Series 
 will also be devoted to works of Fiction and Entertaining Literature 
 generally, which will be interspersed with the more solid publications 
 here named. 
 
 F» 
 
 abiola ; or the Church oFthe Catacomhsi. 
 
 By his Eminence Cardinal Wiseman. 
 
 Price. — Cloth, extra. 12mo., 400 pages, $ 15 
 Gilt edges, - - - 1 12 
 
 The Press of Europe and America is unanimous in praise of this 
 work. We give a few extracts below : — 
 
 " Eminently popular and attractive in its character, * Fabiola ' is in 
 many respects one of the most remarkable works in the whole range 
 of Modern Fiction. The reader will recognise at once those charac- 
 teristics which have ever sufficed to identify one illustrious pen." — 
 Dublin Review. 
 
 " We rejoice in the publication of ' Fabiola,' as we conceive it the 
 commencement of a new era in Catholic literature." — Telegraph. 
 
 *' Worthy to etafld among the highest in this kind of literature." — 
 C. Standard. 
 
BOOKS PUBLISHED BY D. & J. SADLIER <fe COMPANY. 
 
 " Were we to speak of ' Fabiola ' in the strong terms our feelings 
 would prompt, we should be deemed extravagant by those who have 
 not read it. It is a most charming book, a truly popular work, and 
 alike pleasing to the scholars and general reader." — Brownson's Re- 
 view. 
 
 " A story of the early day^f Christianity, by Cardinal Wiseman, 
 is a sufficient notice to give of this volume, lately published in Lon- 
 don, and re-published by the Sadlier's in a very neat and cheap vol- 
 ume." — ^iV. Y. Freeman^ s Journal. 
 
 " As a series of beautifully wrought and instructive tableaux of 
 Christian virtue and Christian heroism in the early ages, it has no 
 equal ia the English language." — American Celt. 
 
 " We think that all who read * Fabiola ' will consider it entirely 
 
 •juccessful We must do the Messrs. Sadlier the justice to say, 
 
 that the book is beautifully printed and illustrated, and that it is one 
 of the cheapest books we have seen." — Boston Pilot. 
 
 " We would not deprive our readers of the pleasure that is in store 
 for them in the perusal of * Fabiola ; ' we will therefore refrain from 
 any further extracts from this truly fascinating work. We know, in 
 fact, no book which has, of late years, issued from the press, so wor- 
 thy of the attention of the Catholic reader as ' Fabiola.' It is a most 
 charming Catholic story, most exquisitely told." — True Wit7iess (Mon 
 treal.) 
 
 " It is a beautiful production — the subject is as interesting, as is the 
 ability of the author to treat of it unquestioned — and the tale itself 
 one of the finest specimens of exquisite tenderness, lofty piety, great 
 erudition, and vast and extended knowledge of the men and mannersi 
 of antiquity, we have ever read." — Montreal Transcript. 
 
 '^ As a faithful picture of domestic life in the olden time of Roman 
 splendor and prosperit}^ it far exceeds the Last daj^s of Pompeii ; and 
 the scenes in the arena, where the blood of so many martyrs fertilized 
 the soil wherein the seed of the Christian faith was fully planted, are 
 highly dramatic, and worthy of any author we have ever read." — 
 N'em York Citizen. 
 
 *^* This volume is in process of translation into the French, Ger- 
 man, Italian, Flemish and Dutch language. 
 
 Ti 
 
 2d VOL. POPULAR LIBRARY, SER. 
 
 he liife of St. Francis of Rome ; 
 
 Blessed Lucy of Narni ; Dominica of Baradiso ; and Anne 
 of Montmorency, Solitary of the Pyrenees. By Lady Ful- 
 LERTON. With an Essay on the Miraculous Life of the 
 Saints, by J. M. Capes, Esq. 
 
 Price. — Cloth, extra. 12nio. - - 50 cents. 
 
 Gilt edges, - - - - YS *' 
 
 '' Life of St. Francis of Rome, by Lady Georgiana FuUerton, together 
 
BOOKS PUBLISHED BY D. &. J. SADLIER <fe* COMPANY. 
 
 with some lesser sketches, is the second volume of the series, and one 
 that will be read with delight by those whose faith is untarnished, 
 and whose imagination desires to be fed by what is pure and holy. 
 Those whom the Apocalypse describes " without," " those that serve 
 idols, and every one that loveth and maketh a lie," or who feed their 
 imaginations on what is of earth or o^hell, had better not read the 
 book for they will not find it to their taste." — N. Y. Freeman's Journal. 
 
 C3d VOL. POPULAR LIBRARY, SER. 
 atliolic liCg^ends. 
 
 Containing the following : — The Legend of Blessed Sadoc 
 and thij Forty-nine Martyrs ; the Church of St. Sabina ; 
 The Vision of the Scholar ; The Legend of the Blessed Egedius ; 
 Our Lady of Chartres ; The Legend of Blessed Bernard and his 
 two Novices ; The Lake of the Apostles ; The Child of the Jew ; 
 Our Lady of Galloro ; The Children of Justiniani ; The Deliv- 
 erance of Antwerp ; Our Lady of Good Counsel ; The Three 
 Knights of St. John j The Convent of St. Cecily ; The Knight 
 of Champfleury ; Qulima, the Moorish Maiden ; Legend of the 
 Abbey of Ensiedeln ; The Madonna della Grotta at Naples ; The 
 Monks of Lerins ; Eusebia of Marseilles ; The Legend of Placi- 
 dus ; The Sanctuary of Our Lady of the Thorns ; The Miracle of 
 Typasus ; The Demon Preacher ; Catharine of Rome ; The Legend 
 of the Hermit Nicholas ; The Martyr of Roeux ; The Legend of 
 St. Csedmon ; The Scholar of the Rosary ; The Legends of St. 
 Hubert ; The Shepherdess of Nanterre. 
 
 Price. — Cloth, extra, - 50 cents. 
 
 Gilt edges, - - '75 " 
 
 " A varied and beautiful volume, containing thirty-one of the choic- 
 ^t specimens, for the most par^new to English readers." — Tablet. 
 
 H 
 
 4th VOL. POPULAR LIBRARY, SER. 
 
 eroines of Charity. 
 
 With a Preface, by Aubrey De Yere, Esq. 12mo. 300 pp. 
 
 1. TPTE SISTERS OF PROVIDENCE, founders of St. Mary's 
 
 of the Woods, Indiana. 
 
 2. JEANNE BIS COT, Foundress of the Community of St. 
 
 Agness, 
 
 3. ANNE DE MULEN, Princess of Epinoy, Foundress of the 
 
 Religious of the Hospital at Beauge. 
 
 4. LOUISE DE MARILLAC, (Madame la Gras,) Foundress of 
 
 the Sisters of Charity; DUCHESS OF AIGUILL0I7, 
 niece of Cardinal Richelieu ; MADAME DE POLLAIJOKr 
 and MADAME DE LAMOIGNON, her Companions; 
 MADAME DE MIRAMION. 
 
BOOKS PUBLISHED BY D. & J. SADLIER <fe COMPANf. 
 
 5. MES. ELIZA A. SETON, Foundress of the Sisters of 
 
 Charity at Emmittsburgh. 
 
 6. JEANNE JUGAN, the Foundress of the Little Sisters of 
 
 the Poor. ^ ^ 
 
 Price. — Cloth, «t^tra. 12nio. 60 cents. 
 Gilt edges, - - 76 " 
 
 M. Diipin, President of the French Academy, remarked, when award- 
 ing her the prize for virtue : — 
 
 ** But there remains a difficulty, which has doubtless suggested itself 
 to the mind of each amongst you, how is it possible for Jeanne alone 
 to provide the expenses of so many poor ? What shall I reply to you ? 
 God is Almighty I Jeanne is indefatigable, Jeanne is eloquent, Jeanne 
 has prayers, Jeanne has tears, Jeanne has strength to work, Jeanne 
 has her basket, which she carries untiringly upon her arm, and which 
 she always takes home filled. Saintly woman, the Academy deposits 
 in this basket the utmost placed at its disposal, decreeing you a prize 
 of 3,000 francs." 
 
 " This is another of the Popular Library that the Sadliers are doing 
 such good service by .re-publishing in this country. At the earnest 
 advice of a gifted friend, whose opinion has every weight with us, 
 we publish entire the preface to this volume by Aubrey de Yere. It 
 is one of the most interesting and valuable essays that has been given 
 to the world for a long time. For the remaining contents of the book 
 we must refer our readers to its own pages." — N. Y. Freemari's Journal. 
 
 ^^^ 5th VOL. POPULAR uBRARY SER. 
 
 1 he IVitch of miltoii Hill. 
 
 A Tale, by the Author of *' Mount St. Lawrence," " Mary, 
 Star of the Sea."- 
 
 Price. — Cloth, extra. 12mo. 60 cents. 
 Gilt edges, - - 75 " 
 
 "The Tales of this Author are distinguished by a skill in the deline- 
 ation of character, and life-like pictures of domestic society. The 
 present, though bearing clear marks of the same hand, is more astory 
 of movement and incident. It is in no sense a 'religious' novel and 
 still less a controversial one. But the progress of the story is through- 
 out made to depend upon the practical adoption or reception of cer- 
 tain principles of right and wrong, so that the 'poetical justice' 
 awarded to every body in the end is not a mere chance consequence 
 of circumstances, but the natural result of their character and con- 
 duct. This it is which gives their chief value to books of fiction as a 
 means for influencing the reader's mind, and this it is which will make 
 *The Witch' a volume fit for general circulation to an extent which 
 
 •can be asked for few novels The story is worked out with 
 
 considerable ingenuity and a rapid succession of events, increasing in 
 
 interest as it proceeds ' The Witch ' is a very clever story, 
 
 and, will suit the young as well as the old." — Tablet. 
 
p 
 
 BOOKS PUBLISHED BY D. <fe J. SADLIER <fe COMPANY. 
 
 eth VOL. POPULAR LIBRARY SER. 
 
 ictures or Christian Heroism. 
 
 With a Preface by the Rev. H. E. Manning. 
 
 CONTENTS. 
 
 ^n The Martyrs of the Carmes. 
 
 8. Gabriel de Naillac, 
 
 9. Margaret Clitherow, 
 
 10. Geronimo at Algiers, 
 
 11. Martyrdoms in China, 
 
 12. Father Thomas of Jesus. 
 
 1. Father Azeved:>, 
 
 2r Sister Honoria Magaen, 
 
 3* Blessed Andrew Bobola, 
 
 4. Blessed John de Britto, 
 
 5. The Nuns of Minsk, 
 
 6. A Confessor of the Faith, 1793. 
 
 Price. — Cloth, extra. 12nio. 60 cents. 
 Gilt edges, - - 75 " 
 
 " The record of the Martyrdom of the lovely Father Azevedo and 
 his Companions, who suffered most of them in the bloom of their 
 years, is one of the most delightful pictures of Christian heroism that 
 
 ever came under our eyes FeW- will read the touching narrative 
 
 of the Massacre of the Carmes without feeling their hearts melt .... 
 "We have seldom seen so much information within so short a compass 
 on the state of religious belief in the Chinese Empire as is given in 
 
 the introductory chapter to the account of the Chinese Martyrs 
 
 . . , .We take leave of this most interesting and edifying work, and we 
 recommend it most heartily to the Catholic reader who loves — and 
 who does not? — to contemplate those pictures of heroism which the 
 Church alone has fed and called forth." — Tablet. 
 
 --^ 7th VOL. POPULAR LIBRAPY SER. 
 
 JtSlakes and Flanagans. 
 
 A Tale of the Times, Illustrative of Irish Life in the United 
 States. By Mrs. J. Sadlier, Author of " ^N'ew Lights, or 
 Life in Galway.'^ " Willy Burke," "Alice Kiordan," &.c. 
 12mo., 400 pages. 
 
 Price.^-Cloth, extra, 75 cents. 
 Gilt edges, $1 12 " 
 
 In a recent visit to Montreal, the Rt. Rev. Mgr. Charbonnell, preach- 
 ing on the education of children, paid the following high compliment 
 to our Tale — " The Blakb6 and Flanagans" — which turns mainly ou 
 that topic. His Lordship remarked : — 
 
 " On coming to town I called at a Bookstore to purchase a number 
 of copies to bring with me to Quebec, but found that it was not yet 
 printed in book-form. It is now being published in the American 
 Celt, and I would like to see it circulated by the hundred thousand." 
 
 " This compliment coming from the Bishop of Toronto, cannot be but 
 highly valued by the gifted writer, Mrs. Sadlier. 
 
 *' We cannot forbear assuring the Authoress, contemporaneous with 
 
BOOKS PUBLISHED BY D. & J. SADLIER & COMPANY. 
 
 announcement, and thus publicly, that her story has been read with 
 the liveliest interest by our readers, and that it has elicited the most 
 earnest expressions of respect for herself from persons distinguished 
 in every walk of life. 
 
 "Before parting with our favorite contributor, let us add the fol- 
 lowing opinion recently expresf'^d to us by a gifted Western clergy- 
 man 1 * Mrs. Sadlier's story,' he said, * has done more to bring hoine 
 to the hearts of parents the importance of Catholic education, than 
 any other, and all other advocacy combined.' This is extreme praise, 
 but we will say, not undeserved." — American Celt. 
 
 " The style is excellent, thoroughly natural and unaffected, the nar- 
 rative flowing, the conversations full of vivacity, and the characters 
 
 well sustained We cannot but wish it the widest circulation 
 
 that a book can have." — >S'^. Zouis Leader, 
 
 p— ^ 8th VOL. POPULAR LIBRARY SER. 
 
 X he liife and Tiiiiei§i of St. Bernard. 
 
 Translated from the French of M. L'Abbe IIatisbonne. 
 With a Preface by Henry Edward Manning, D. D., and 
 a Portrait. 1 vol., 12mo., 500 pages. 
 
 Price. — Cloth, extra, $1 00. 
 Gilt edges, 1 50. 
 
 " St. Bernard was so eminently the saint of his age, that it would 
 be impossible to write his life without surrounding it with an exten- 
 sive history of the period in which he lived, and over which he may 
 be truly said, to have ruled. The Abbe Ratisbonne has, with this 
 view, 'very ably and judiciously interwoven with the personal narra- 
 tive and description of the ^aint, the chief contemporaneous events 
 and characters of the time. 
 
 " There seems to have been in this one mind an inexhaustible abund- 
 ance, variety, and versatility of gifts. Without ever ceasing to be 
 the holy and mortified religious, St. Bernard appears to be the ruling 
 will of his time. He stands forth as pastor, pr cache"", mystical writer, 
 controversialist, reformer, pacificator, mediator, arbiter, diplomatist, 
 and statesman. He appears in the schools, at the altar, in the preach- 
 er's chair, in councils of the Church, in councils of tlie State, amid 
 the factions of cities, the negotiations of princes, and the contests of 
 anti -popes. And whence came this wondrous power of dealing with 
 affairs and with men ? Not from the training and schooling of this 
 world, but from the instincts, simplicity, and penetration of a mind 
 profoundly immersed in God, and from a will of which the fervour 
 and singleness of aim were supernatural." — Extracts from Preface. 
 
BOOKS PUBLISHED BY D. & J. SADLIER «fe COMPANY. 
 
 -^^ 9th VOL. POPULAR LIBRARY SEP. 
 
 J. he ffiiie and Victories of the Karly ilJar- 
 
 TYRS. By Mrs. Hope. Written for the Oratorian's 
 Schools of our Lad3^'s Compassion. 1 vol., 12mo., 400 
 pages. •« 
 
 Price. — Cloth, extra. $15. 
 Gilt edges, 1 12. 
 
 " Tlie interesting Tale of * Fabiola ' has made most readers familiar 
 with the sufferings of the Early Martyrs, and desirous to know more 
 of their history, and of the Victories which they achieved over the 
 world. Every age, every clime, has its martyrs, for it is a distinctive 
 mark of the Catholic Church, that the race of martyrs never dies out. 
 And since her earliest times, a single generation has not passed away 
 without some of her children shedding their blood for the name of 
 Jesus. Other religious bodies may have had a few individuals here 
 and there and at distant intervals, who have died for their opinions. 
 But it is in the Catholic Church alone the spirit of martyrdom has 
 ever been alive. Nor is it difficult to account for this, as the Catholic 
 Church is the only true Church, the devil is ever ready to raise up 
 persecutions against her, and as the Lord ever loves his spouse, the 
 Church, he bestows upon her all graces, and among them the grace of 
 martyrdom, with a more lavish hand than on others." — Extract from 
 Jo\trodaction. 
 
 10th VOL. POPULAR LIBRARY, SER. 
 
 History of the War in lia Vendee, and 
 
 the Little Chouannerie. By G. J. Hill, M. A. With 
 
 Two Maps and Seven Engravings, 12mo. 
 
 Cloth, extra, 75 cents; cloth, extra, gilt edges, $1 12. 
 
 Til is is a number of the popular Catholic Library now in course oP 
 publication by Messrs. Burns and Lambert, London, and Messrs. 
 Sadlier <fe Co., in this city, and is one of the most interesting num- 
 bers of that valuable series that has as yet a]ipeared. It is full of 
 romantic incident, and is as exciting as any romance ever written. 
 The author has grouped his incidents with much skill, and told his 
 story with much grace and feeling. The public can hardly fail to ap- 
 preciate it highly ; and the intelligent reader will find it full of in- 
 
BOOKS PUBLISHED BY D. & J. SADLIER & COMPANY. 
 
 struction as well as interest. The war in La Vendee was the war of 
 the peasants, and a war chiefly for the freedom of religion. For the 
 freedom of religion, the freedom of conscience, the freedom to have 
 their own priests, and to worship God according to their own faith, 
 they took up arms, and they di€rnot lay them down till they had se- 
 cured it. They were the true French patriots in the time of the Re- 
 public ; the men who preserved fresh and living the France of St. 
 Louis and the Crusaders, historical and traditional France, and it is 
 not too much to say that they saved France in her darkest da3"s, and 
 prevented the continuity of her life from being broken by the madness 
 and excesses of the revolution. They prove how much religion warms 
 and strengthens patriotism, and that it can make undisciplined peas- 
 ants able to cope successfully with the best drilled and appointed 
 armies. When fighting for their religion, these half-unarmed peasants 
 were invincible, and almost always victorious. When from a Catholic 
 army they became a Royalist army, and acted under the direction of 
 chiefs, who thought only of restoring fallen monarchy, they melted 
 away before their enemies, as wax at the touch of fire, or were scat- 
 tered as the morning mist before the rising sun. — BrownsorCs Review. 
 
 The period of the war was from *91 to '97, from the first proclama- 
 tion of "the Civil Constitution of the Clergy" to the pacification, and 
 consequent restoration of public worship under the Consulate. The 
 scene of the war was the most Celtic region in the west of France, on 
 both banks of the great Celtic river, the Loire. All Bretagne, parts 
 of Maine, Anjou and Poitou, were engaged at various p-eriods of the 
 struggle, the lai*gest number in arms being not less than 100,000 in 
 every quarter of this country, while against them were sent in succes- 
 sion the armies of Generals Kleber, Westerman, and Hoche. On the 
 side of the insurgenjts, the successive commanders were Cathelineau, 
 D'Elbee, Henri de Larochejacquelein, Charette, and Stofilet; St. Malo, 
 ^iuiberon, JSTantes, Laval, Moulins, Fontenay, Thouars, and Angers, 
 were the towns and cities lost and taken on both sides, but the citadel 
 and nursery of the war was La Vendee. — ^American Celt. 
 
 11th VOL. POPULAR LIBEAEY, SEE. 
 
 X ales and liegends iroin Hiistory. 
 
 Contents. — 1. Gonsalvo of Amaranta. 2 The Victory 
 of JMiiret. 3. The Dominicans in Ghent. 4. The Martyrs 
 of Stone. 5. The Abbey of Premontre. 6. Legenis of 
 St. Winifrile. 7. The Feast of the Immaculate Ccmcep- 
 
BOOKS PUBLISHED BY D. <b J. SADLIER A COMPANY. 
 
 tion. 8. The Consecration of Westminster Abbey. 9. 
 The Monk's Last Words. 10. The Martyr Maidens of 
 Ostend. 11. The Loss of the '* Conception.'^ 12. Foun- 
 dation of the Abbey of Anchin. 13. Our Lady of Mercy. 
 14. John de la Cambe. 15. Th? Carpenter of Roosendael. 
 16. The Widow of Artois. 17. The Yillage of Blanken- 
 berg. 18. St, Edward's death. 19. The Windows of 
 San Petronio. 20. The Vessels of St. Peter. 
 
 12mo., cloth, extra, 63 cents; cloth gilt, 88 cents. 
 
 12th VOL. POPULAK LIBPwAEY, SEE. 
 
 The Missions in Japan and Paraguay. 
 
 By Cecilia Caddell, author of " Tales of the Festivals/' 
 " Miser's Daughter," " Lost Jenoveffa," &c., &c. 
 
 12mo., cloth, extra, 63 cts. ; cloth, extra, gilt edges, 88 cts. 
 These are two additional numbers of the Catholic Library already 
 referred to, and are well adapted to its purpose. The History of the 
 Missions in Japan and Paraguay, by Miss Caddell, is peculiarly inter-' 
 esting and instructive. It is written with simplicity and taste. — 
 Brownson^s Revieio. 
 
 13th VOL. POPULAR LIBRAEY, SEE. 
 
 Calllsfa; A TaJe of the Third Century. 
 
 By Yery Rev. John Henry Newman, D. D., Rector of 
 
 the Catholic University, Dublin. 
 
 12mo., cloth, extra, 75 cents; cloth, extra gilt, $1 12. 
 
 The following extract is from a very long notice of the work in the 
 Dublin Tablet: 
 
 *' The story is partly interwoven with historical facts, but its author 
 professes, at the outset, that as a whole, it is 'a simple fiction from 
 beginning to end.' However that may be, as an instrument of convey- 
 ing a real and genuine historical knowledge of the days of which it 
 treats, in their aspect towards Christianity, it will probably remain 
 without a rival in the literary world." 
 
 Dr. Newman is peculiarly felicitous in portraying the exceeding dif- 
 ficulty an old Roman Pagan had in understanding the principles and 
 motives w^hich governed the Christian. He could not enter into his 
 way of thinking, could not seize his stand-point, and comprehend ht)W 
 a man well to do in the world, intelligent, cultivated, respectable, in 
 
BOOKS PUBLISHED BY D. &l J. SADLIER <fe COMP^Nr. 
 
 the road to wealth, pleasure, distinction, honors, under Paganism, 
 shonld forego all his advantages, join himself to a proscribed sect, re- 
 garded as almost beneath contempt, professing, as it was assumed, a 
 mo^it debasing and disgusting miperstition, without intelligence, com- 
 mon sense, and common decency, and that too when to do so was sure 
 persecution, and almost certain death, as a traitor to Caesar. It was 
 a sore puzzle to the wise heathen, it is a sore puzzle also, to the proud 
 an4 self-sufficient non-Catholic even to-day, and will remain so to the 
 end of the world. It is hard even for worldly-minded Christians to 
 comprehend how youth and beauty can forego the world, and wed 
 themselves to an unseen Lover, and live and suffer only for an in- 
 visible Love. The solution is found only in giving our hearts to God, 
 and'living for heaven alone." — BrownsorCs Heview." 
 
 14th VOL. POPULAR LIBEARY, SEE. 
 
 Manual of Modern Hii^tory. 
 
 Bj Matthew Bridges, Esq. 600 pages. 
 
 12mo., cloth, or half roan, $1 00. 
 
 This volume, containing, as it does, a large amount of matter, with 
 complete indexes, tables of chronology, <fec., &c., will be found equally 
 useful for popular reading, as a Student's Text-book, or as a manual 
 for schools. 
 
 "Mr. Bridges' excellent Popular Modern History." — Card'mal Wise- 
 man^ s late Lecture on Ro7ne. 
 
 "The author dates this history from the irruption and settlement of 
 the barbarous nations on the ruins of the Western Empire, and brings 
 it down, noting events as they transpired in the succeeding ages, to 
 the present. The author has carefully compiled the work from such 
 resources as were within his reach, and he considered were of a re- 
 liable character." — Pittsburgh Catholic.^^ 
 
 15th VOL. POPULAR LIBRARY, SEB. 
 
 Ancient History. 
 
 By Matthew Bridges, Esq. 
 
 Price, ^5 cents. 
 
 This volume, containing, as it does, a large amooit of matter, with 
 complete indexes, tables of chronology, <fcc., <fec., will be found equally 
 useful for popular reading, as a Student's Text-bock, or as a manual 
 for schools. 
 
BOOKS PUBLISHED BY D. <fe J. SADLIER «fe COMPANY. 
 
 16th VOL. POPULAR LIBRARY SER. 
 
 Life an» i.abor8 of st. vincewt 
 
 ©E PAUIi. A new, complete, and careful Biography. 
 
 By H. Bedford, Esq. 12mo. ^ 
 
 Clotli^ extra 50 cU. 
 
 Cloth^ extra gilt 75* " 
 
 " The loving warmth which pervades this book — the accuracy 
 of detail with which it is written — and, above all, the subject — 
 one of God's greatest saints and the world's greatest benefactors — 
 must ensure it a wide circulation in America. For the rest, the 
 publishers have got it out in good style, on good paper, with clear 
 type. " — Rami ler. 
 
 17th VOL. POPULAR LIBRARY SER. 
 
 Ai 
 
 XICE SHERWI?¥. 
 
 An Historical Tale of the Days of Sir Thomas More. 
 
 12mo. 
 
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 " Alice Sherwin". — This able and interesting historical novel is 
 reprinted from the English, and has been ascribed, we know not 
 whether justly or not, to the distinguished author of Sunday in 
 London — a convert from Anglicanism, who deserves the thanks 
 of every English speaking Catholic for the valuable contributions 
 he has made since his conversion, and is still making, to English 
 Catholic literature. But by whomsoever written, Alice Sherwin 
 is, so far as we know, the most successful attempt at the genuine 
 historical novel by a Catholic author yet made in our language, 
 and gives goodly premise that, in due time, we shall take our 
 proper rank in this department of literature, rendered so popular 
 by the historical romances of Sir Walter Scott. The author has a 
 cultivated mind, a generous and loving spirit, and more than usual 
 knowledge of the play of the passions and the workings of the 
 human heart. He has studied with care and discernment the 
 epoch of Sir Thomas More, or, as we prefer to say, of Henry YI|I., 
 and has successfully seized its principal features, its costume and 
 manners, and its general spirit, and paints them in vivid colors, 
 and with a bold and free pencil, though after, all with more talent 
 and skill than genius in its highest sense." — Brownson's Review, 
 
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 mind of St. Francis of Sales that union of sweetness and strength of manly 
 power and feminine delicacy, of profound knowledge and practical dexter- 
 ity, which constitute a character formed at once to win and subdue minds 
 of almost every type and age. As the rose among flowers, so is he among 
 Saints. From the thorny, woody fibre of the brier comes forth that blossom 
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 hot and vehement nature of the young Savoyard, came a spiritual bloom 
 whose beauty and fragrance were perfect in an extraordinary degree." 
 
 " When we say that the author of the work before us has done this, it is to 
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BOOKS PUBLISHED BY D, & J. SADLIER & COMPANY. 
 
 -wwy- NOW PUBLISHED-THE COMPLETE 
 
 V? orks aiKl Life of Gerald Griffin, * 
 
 New Edition of the LIFE AND WORKS OF GER- 
 ALD GRIFFIN, revi^d and corrected by his Brother. 
 Ilkistrated with splendid Steel Engravings, and printed on 
 the finest paper. Complete in thirty- three weekly parts, 
 at Twenty -five Cents each. Comprising the following 
 Tales :— 
 
 VOL. 1.— THE COLLEGIANS. A Tale of Garryowen. 
 2.— CARD DRAWING. A Tale of Clare. 
 
 THE HALF SIR. A Tale of Munster. ^-^ 
 
 SUIL DHUV. A Tale of Tipperary. 
 S.^THE RIVALS. A Tale of Wicklow ; and TRACY'S 
 
 AMBITION. 
 4— HOLLAND TIDE,^ THE AYLMERS OF BALLYAYL- 
 MER, THE HAND AND WORD, and BARBER OF 
 BANTRY. 
 5.— TALES OF THE JURY ROOM. Containing: SIGIS- 
 MUND, THE STORY-TELLER AT FAULT, THE 
 KNIGHT WITHOUT REPROACH, &g., &c. 
 6.— THE DUKE OF MONMOUTH. A Tale of the English 
 
 Insurrection. 
 7.— THE POETICAL WORKS AND TRAGEDY OF GYS- 
 SIPUS. 
 " 8.— INVASION. A Tale of the Conquest. 
 
 9.— LIFE OF GERALD GRIFFIN. By his Brother. 
 '' 10.— TALES OF THE FIVE SENSES, and NIGHT AT SEA. 
 ^^" The Works will also be bound in eloth extra, and issued in ten 
 monthly Volumes, at ONE DOLLAR per volume. Sent free by post 
 to any part of the United States. 
 
 In presenting to the American public, a first edition of the Works 
 of Gerald Griffin, the Publishers may remark that it will be found to 
 be the only complete on£. Neither in the London nor Dublin editions, 
 could the Publishers include the historical novel of *' The Invasion," 
 and the celebrated tragedy of '* Gj'ssipus." As Ave are not subject to 
 any restriction arising from the British copyright, we have included 
 the former with the prose, and the latter with the poetical works of 
 the Author. 
 
 We are also indebted to near relatives of Mr. Griffin, residing in this 
 country, for an original contribution to this edition; which will be 
 found gratefully acknowledged in the proper place. 
 
 As the life of the Author forms the subject of one entire volume,' 
 we need say little here, of the uncommon interest his name continues 
 to excite. Unlike the majority of writers of fiction, his reputation 
 has widely expanded since his death. In 1840, when he was laid in 
 his grave, at the early age of seven and th'irty, not one person knew 
 
BOOKS PUBLISHED BY D. & J. SADLTER & COMPANY. 
 
 the loss a pure lii-erature had sustained, for fifty who now join vener- 
 ation for his virtues, to admiration for his various and delightful tal- 
 ents. The goodness of his heart, the purity of his life, the combined 
 humor and pathos of his writings, all promise longevity of reputation 
 to Gerald Griffin. O 
 
 " He had kept 
 The whiten/»ss of his soul, and so men o'er him wept." 
 
 He united all the simplicity and cordiality of Oliver Goldsmith to 
 OQueli of the fiery energy and manly zeal of Robert Burns. His life 
 docs not disappoint the reader, who turns from the works to their 
 author ; it is, indeed, the most delightful and harmonious of all his 
 works. From his childish sports and stories by the JShannon until his 
 solemn and enviable death beside " the pleasant waters " of the Lee, 
 a golden thread of rectitude runs through all his actions. A literary 
 adventurer in London at nineteen, with a Spanish tragedy for his sole 
 capital, famous at thirty, a religious five years later, a tenant of the 
 Christian Brothers' Cemetery at thirty-seven — the main story of his 
 life is soon told. Over its details, we are confident, many a reader 
 will fondly linger, and often return to contemplate so strange and so 
 beautiful a picture. Out of his secret heart they will find sentiments 
 issuing not unworthy of St. Frances de Sales, while from his brain 
 have sprung creations of character which might have been proudly 
 fathered by Walter Scott. 
 
 Canvassers^ wanted in every part of the United States and Canadas 
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 The Works are bound in the following styles of Binding: — 
 
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 « " " " full gilt, 15 00 
 
 " " Sheep, Library style, or half Roan, cloth sides,- 12 50 
 " " Half morocco, marble edges, - - - - 15 00 
 
 " " " calf, antique, 20 00 
 
 " " " " " or morocco, gilt, - - 80 00 
 
 NOTICES OF THE PRESS. 
 
 The Complete Works of Gerald Griffin. — We welcome this new 
 and complete edition of the works of Gerald Griffin, now in the course 
 of publication by the Messrs. Sadlier & Co.^ We read The Collegians, 
 when it was first published, with a pleasure we have never forgotten, 
 and which we have found increased at every repeated perusal. Ire- 
 land haS' produced many geniuses, but rarely one. upon the whole, 
 superior to Gerald Griffin. When we have his life, and the publica- 
 tion of the edition is completed, we shall endeavor to render our tri- 
 bute of gratitude to the memory of the gifted author. — [Browjison's 
 Review. 
 
 It is not surprising that comparatively little is known in this coun- 
 try, save among the more intelligent American citizens from the 
 ''Emerald Isle," of Gerald Griff'n, when we consider that he was 
 
BOOKS PUBLISHED BY D. & J. SADLIER & COMPANY. 
 
 born in^the beginning of the present centur\', and closed his brief, 
 but reall}' bright and brilliant, career of authorship at the early age 
 of 37. He however achieved a reputation as a writer of no ordinary 
 power, and as has been remarkffi, united all the simplicity and cord- 
 iality of Oliver Goldsmith, to much of the fiery energy and manly 
 zeal of Robert Burns. We have now before us four volumes, the 
 commencement of a complete edition of his works, embracing the 
 *' Collegians,^^ and the fii^et series of his " MunUer Tales." The na-^ 
 tionality of these tales, and the genius of the author in depicting the 
 mingled levity and pathos of Irish character, have rendered t'frem ex- 
 ceedingly popular. The present edition, the first published in Amer- 
 ica. The style in which the series is produced, is highly creditable 
 to the enterprise of the American publishers, and we are free to say, 
 that the volumes are worthy of being placed in our libraries, public 
 or private, alongside of Irving, Cooper, or Scott. — Hunt's Merchant^ 
 Magazine. 
 
 "Whoever wishes to read one' of the most passionate and pathetic 
 novels in English literature Avill take with him, during the summer 
 vacation, The Collegians, by Gerald Grifhn. He was a young Irish- 
 man, who died several years since, after writing a series of works — 
 novels and poetry, which gave him little reputation during his life, 
 but since his death have given him fame, as, in our judgment, the best 
 Irish novelist. The .picture of Irish character and manners a half 
 century since, in The CollegianSy is masterly, and the power with 
 which the fond, impetuous, passionate, thoroughly Celtic nature of 
 Hardress Cregan is drawn, evinces rate genius. Griffin died young, 
 a disappointed man. But this is one story, if nothing else of his, will 
 surely live among the very best novels of the time. It is full of in- 
 cident, and an absorbing interest allures the reader to the end, and 
 leaves him with a melted heart and moistened eye. It is a very con- 
 venient and attractive edition. It will contain all his novels, dramas, 
 and lyrics. The latter have a thoroughly Irish flavor, and will, we 
 sincerely hope, be the means of making the talented young Irishman 
 widely known, and, consequently, admired in this country. — Putnam's 
 Monthly. 
 
 The works of the author in question are written in an easy, flowing 
 style, and his sketches of character exhibit a rare knowledge of hu- 
 man nature — Irish human nature especially — while the descriptive 
 portions carry with them a truthfulness and fidelity which are pecu- 
 liarly charming. Many of his narratives are founded upon real inci- 
 dents in Irish history which are ingeniously worked up, and rendered 
 thrillingly exciting and absorbingly interesting. They are interspers- 
 ed with scenes of the deepest pathos, and the most genuine humor — 
 at one moment we are convulsed with laughter, at the next affected 
 to tears — while an atmosphere of sound moralit}^ containing many 
 touches of true philosophy and deep reasoning, surrounds the whole, 
 showing that the author well knew the way to the human heart. 
 While every one who possesses true literary taste will be charmed 
 with Griffin's works, they will be peculiarly fascinating 
 
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 J. lie Life of our Lord and Saviour Jesus 
 Christ; or, JESUS REVEALED TO YOUTH. By 
 
 the ^BBE F. Lagrange. Tr^slated from the French by 
 Mrs. J. Sadlier ; with the approbation of the Most Rev. 
 John Hughes, D. D., Archbishop of New York. Illus- 
 trated with 10 plates. 16mo. 
 
 . Cloth extra, 60 cents ; Cloth gilt, '76 cents. 
 
 " Suffer little children to come unto me ! " said our Lord himself, 
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 knowing, they may love him ; and loving, they will be sure to serve 
 him all their life long. Let them behold him as he was Avhen, clothed 
 with our frail humanity, he walked amongst men, scattering blessings 
 as he went. This little volume is, I think, well calculated to make 
 Jesus known to the younglings of his flock. It is written in a simple, 
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 eternal kingdom. 
 
 Montreal, 
 
 Feast of the Annunciation of B. V. Mary, 
 March 25th, 1^57. 
 
 T 
 
 lie Prophecies of 8§. Columbkille^ 
 
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 the Life of ST. COLUMBKILLE. By the Rev. Thos. Walsh. 
 16mo. Cloth, Price 38 cents. 
 
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 JLhe liife of St. Joseph. 
 
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 bations accorded to the great work from which it is derived, by the supreme 
 Pontiifs, during the two last centuries, cannot be too highly recommeDded 
 to the Catholic reader. 
 
 M 
 
 SUPPLEMENT TO ** THE FOLLOWING OP CHRIST." 
 
 ethodieal and Explanatory Table of 
 
 THE CHAPTERS OF THE FOLLOWING OF CHRIST. 
 Arranged for every day in the year in the order best 
 calculated to lead to perfection. Compiled from the 
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 wish to meditate with fruit. 
 
 The order thus introduced into this estimable book will, we hope, still 
 more enhance its ''nlue. 
 
 . he liittle Catechiisiii 5 or, Questions and 
 
 ANSWERS on those Truths which are the most neces- 
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 FIFTH TJHOUS.lJrn. ^ 
 
 FROM THE SECOND REVISED EDITION. 
 
 _KoME— Its Ruler^ aiifl its Institutions. 
 
 By John Francis Maguire, M. P. 
 
 At the earnest request of several of t je clergy and laity, we have concluded to 
 publish this most important work. To the Catholic it is invaluable, and it is just 
 the book for those outside the Church. All will be satisfied with what the Church 
 has done and is doing for the spiritual as well as temporal welfare of man. 
 
 It is not necessary to say anything further, when the highest authority in the 
 Church has spoken. We append the following : — 
 
 Mr. Maguire has been honored with a letter from his Holiness, of which thefollowxng 
 is a translation. 
 " Well-beloved Son • Health and Apostolical Benediction — ^We have lately receiv- 
 ed your most dutiful letter, dated the 17th day of November last, in which you 
 have TOen pleased to present to Us a work composed by you in the English lan- 
 guage, and published this year in London, with the title, 'Rome, its Ruler, and 
 its Institutions.' Of this work we have been unable to enjoy the perusal, owing to 
 our extremely imperfect acquaintance with the language in which it is written. 
 Yet, from the statement of persons of the highest character for competency and 
 trustworthiness, who have perused the same woi'k. We learn with peculiar satis- 
 faction that it contains many evidences of your singular devotedness. attachment, 
 and reverence to Us, and towards this Holy See ; a circumstaYice which could not 
 fail to be most gratifying to Our feelings. Therefore, while bestowing Our heart-- 
 felt recommendation on this noble expression of your sentiments to Us, We return 
 you thanks for the gift ; and, as a testimony of Our fatherly love towards you, We 
 impart to you affectionately from Our heart tte Apostolic Benediction. 
 
 Given at Rome, at St. Peter's, the 14th day of December, 1857. in the twelfth 
 year of Our Pontificate." PIUS IX. 
 
 " My Dear Sir : — London, August 28, 1857. 
 
 According to your desire, I have delayed acknowledging the receipt of your 
 ' Rome ' till I had read it through. This I have now done, taking it up at every 
 leisure moment with renewed pleasure, till I had finished it. Having myself had 
 to go over a great pai-t of the ground, whether personally or by the study of docu- 
 ments, I think I am qualified to form a just judgment of the work. It is most 
 truthful, accurate, and une^aggerated picture of the Holy Father, of his great 
 works, and of his most noble and amiable character, drawn with freshness, elegance 
 and vigor — with admiration, and even, if you please, enthusiasm ; but not greater 
 than is sliared by every one who has drawn near the person of the Holy Father. 
 There is not a trait in your portrait which I do not fully recognize ;. not an action 
 or a speech which I could not easily imagine to have been performed or spoken in 
 my presence— so like are they to what I have myself seen and heard. In estimat- 
 ing what has been done during the late years of quiet rule for the prosperity of 
 the Pontifical States. I think you have prudently kept rather below than gone 
 above what might have been stated. The result will be more manifest in time — 
 to the confusion, one may hope, of those who, dishonestly or ignorantly misrepre- 
 sent every measure of the Sovereign Pontiff. I feel sure that your work is calcu- 
 lated to do much good wherever it is read ; and I cannot help hoping that the very 
 novelty of daring to speak the bold truth, the abundance of information which is 
 communicated, and the eloquence of the style, will obtain for j^our book all the 
 popularity which it deserv^. I need not say fhat by this work you have nailed 
 your colors to the mast, and become the Pope's champion, in the House as well as 
 out of it ; and I am sure you Will not allow him to be vilified by any one, howevei 
 lofty. I am ever, my dear sir. 
 
 Your affectionate servant in Christ, 
 John Francis Maguire, Esq., M. P." N. Card. Wiseman. 
 
 Cloth, extra $1 25 
 
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 NEW EDITION. 
 
 istory of ihc l¥ar in the Peninsula 
 
 AI^D IN THE SOUTH OF FRANCE. From the 
 year 1807 to the year 1814. By VV. F. P. Napier, C. B., 
 Colonel H. P. Porty-Hiird Regiment, Member of the 
 Royal Swedish Academy of Military Sciences. 
 
 Royal 8vo. 800 pages. Cloth, extra, $2.50;' half raor., $3.00. 
 
 " The aiithor*s style is exceedingly vi^orows and terse. It has a plain 
 and simple vehemence, which is both captivating and imposing. He knows 
 nothing of amplification. He tells his story in the fewest words possible ; 
 marches straight to a fact, seizes it, and places it before the reader without 
 any useless preliminaries, takes some fortress of error built up with care 
 and pains by some precedent, interested or partial historian, by a coup de 
 rmi'ut.^ and razes it to the ground. His narrative is full of life, motion and 
 impulsiveness ; his portraiture of character, bold, striking aud impressive. 
 His honest and chivalrous nature enabled him to rise above the then popu- 
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 Sir,— We have examined the manuscript copy of the Anthnietic which you scut us, and 
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 your stating that we have adopted it for our schools. 
 
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 I am, with profound respect, Sir, 
 
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 r; 
 Extract from the Pw'ace. 
 Treatises on Arithmetic are so numeroim^and excellent, that it might 
 seem superfluous to add another to the number; yet it was thought tnat 
 room might still be found for one, which, discarding, to a great degree, 
 the multitude of complicated rules which, at the present day, encumber 
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