GIFT or J.B, Pelxotto \ "Vv ^ \ J ^ \ \ Digitized by tine Internet Arciiive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/algebraadaptedtoOOtownricli ALGEBRA; ADAPTED rO THE COURSE OF INSTRUCTION USUALLY PURSUED IN THE €o\Up$ miA ^ailf mwis OF THE UNITED STATES. BY P. A. TOWNE, // FORMERLY GENERAL PRINCIPAL OP THE BARTON ACADEMY, MOBILE, ALA. PROFESSOR OF MATHEMATICS, CLINTON LIBERAL INSTITUTE, N. Y. LOUISVILLE, KY.: JOHN P. MORTON & CO. \53 ^^5 cc«: cc c*c %lf y > ^ r^^^^a:-^'^^ Entered, according to Act of Congress, in the year 18C5, by JOHN P. MOllTON & CO. in the Clerk's Office of the District Court of the United States for the District of Kentucky. ZLECTROTTPED BY L. J0IIX30N & CO. PHIIAm'T.pni*. PREFACE. It would be easy to state, by way of preface, the pre- cise reasons which have led the author to add this Algebra to the numberless treatises on the same subject already in existence. If, however, the reader will con- sent to devote a few leisure moments to an examination of the following points, the writer flatters himself with the belief that these reasons v\-ill appear much more forcibly than if stated in the language of an argument Attention is invited to the accuracy of the Definitions; to the brevity and clearness of the demonstrations; the explanation of i^osiiive and negative quantities; the subject oi factoring ; the appropriateness and careful gradation of equations and other problems ; the manner in which the transition from the reduction of equations to the solution of problems is effacted ; the perpetual recurrence of the mind of the pupil, as he advances, to first principles, — as, for instance, compare the subjects of Greatest Common Divisor, Least Common Multiple, Eeduction of Fractions and Equations having equal roots; to the circumstance that many of the problems have been doubled by placing figures in parentheses corresponding to each other; the constant requirement of reducing generalizations to 3 O O «^ v>- "IS: '^ 4 PREFACE. numerical problems previously solved ; the treatment of Quadratic Equations, particularly those involving two unknown quantities; the subject of Logarithms; and, finally, to the practical manner in which the Higher Equations are treated. Many other points might be mentioned ; but, in pass- ing over the above, the reader will not fail to discover them. In the preparation of the work, the author has been occupied some ten or twelve years; and he now feels safe in pledging himself that no material change will be made in any of its discussions during, at least, the same length of time. Every part of it has been repeatedly tested in the class-room. CONTENTS. CHAPTER I. PAGE Definitions and Exercises 9 Symbols of Quantity 9 of Operation 10 of Exponents 10 of Coefficients 11 of Relation 12 Examples 14 Notation 17 CHAPTER 11. Addition 18 Subtraction 24 Multiplication 32 Division 42 CHAPTER III. Factoring 51 Greatest Common Divisor 55 Least Common Multiple 58 General Review 50 CHAPTER IV. Fractions 61 Review of Fractions 75 1* 5 6 CONTENTS. CHAPTER V. PAGE Equations of the First Degree 76 Literal Equations 86 Problems 88 Equations of the First Degree involving Two Unknown Quantities 98 Elimination C3 by Addition or Subtraction 00 by Substitution 104 by Comparison 108 Three Equations involving Three Unknown Quantities Ill Four or more Equations involving a like number of Unknown Quan- tities 112 Symmetrical Equations 113 Problems involving Two or more Unknown Quantities i llo Literal Equations 121 Generalizations 122 Negative Ptesults ••• 125 Interpretation of the Symbols -' ^' -' 129 Indeterminate Analysis 131 CHAPTER YI. Involutions 133 Logarithms 1^2 Multiplication by Logarithms 152 Division by Logarithms 153 Arithmetical Complement 154 Involution by Logarithms 155 Extraction of Roots by Logarithms 156 Evolution and Treatment of Radicals 159 Imaginary Quantities 1"2 CHAPTER VII. Equations of the Second Degree 1"6 Problems producing Incomplete Equations of the Second Degree 178 Complete Equations of the Second Degree ISO CONTENTS. 7 PAGE Trinomial Equutious 187 Literal Equations , 189 Problems involving Equations of the Second Degree 194 Equations with Two Unknown Quantities 197 Review 202 Homogeneous Equations 20'J Equations containing Three Unknown Quantities 211 Problems involving Two Unknown Quantities 212 General Properties of Equations of the Second Degree 218 Application of these Properties 210 Ratio and Proportion 222 Problems 227 Arithmetical Progression 229 Geometrical Progression 234 Problems 237 Indeterminate Coefficients 239 Binomial Theorem 213 The Table of Logarithms 244 Practical Applications 240 CHAPTER VIII. Equations of the Third Degree 251 Law of Derived Polynomials 205 Properties of Derived Polynomials 205 Equal Roots 200 General Solution of the Equation of the Third Degree 208 Numerical Solution of Cubic Equations 200 Higher Equations 270 Questions for Examination 279 '>.-Ai.Ji''0^'^iUj/;i AL GEB E A. CHAPTEE I. DEFINITIONS AND EXERCISES. 1. Algebra investigates the relations of quantities by Symbols. SYMBOLS OF QUANTITY. 2. The symbols of quantity in Algebra are the letters of the alphabet. 3. The first letters of the alphabet, viz., a, 6, c, . . . , usually represent quantities whose numerical values are Icnoivn, 4. The last letters, viz., x, y, z^ represent quantities whose values are unknown, — /. e., unknown before the operations in which they are involved are performed ; after these operations unhnown quantities become known, 5. An algebraic quantity is properly, then, a quantity repre- sented by a letter or letters. 6. An arithmetical quantity is one represented by a figure or figures. •Y. Algebraic quantities are therefore called literal quantities, to distinguish them from numerical quantities. Both kinds of quan- tities are used in Alojebra. 1 I) ]•: F I :; i i i c :.* .s and e x e k c i s e s . SYMBOLS OF OPERATION. 8. The sign +) plus, indicates that the quantity before whicli it is placed is to be taken additively. Thus, a -\-h, a plus h, denotes that the quantity h is to be added to the quantity a. 9. The sign — , minus, indicates that the quantity before whicli it is placed is to be taken suhtractively. Thus, a — h, a minus h, denotes that the quantity h is to be subtracted from the quantity a. When no sign is written, + is understood. Thus, a is the same as + a. 10. There are three ways in which to indicate Multiplication in Algebra, viz., o:Xy, x.y, and xy, all of which indicate that X is to be multiplied by y, 11. There are also three ways in which to indicate Division in Algebra, viz., x~ry, ~, and x\l, all of which signify that x is to be divided by y. 12. The sign ( 1, parenthesis, or , vinculum, whicli may also be drawn perpendicularly, is used to connect several algebraic expressions, and denotes that they are to be treated as a single expression. Thus, -{. h is the same as a -\-h -{- c.x, or (a -\- h -{- c')x. + c all of which signify that the sum of a, h, and c is to be multi- plied by X. Again, (4 + 5) x 6 is the same as 54, but 4 -f 5 x is the same as 34. 13. Of Exponents. — In the expressions a. h c; x y z; m n; etc., each of the letters composing the expression is c.rod a literal factor. If a letter is to occr.r ;i« a factor sevci;;l times, instead of writing a a ; x x x; y y y y; etc., a figure is placed DEFINITIONS AND EXERCISES. 11 at the right hand of the letter and a little above; thus, a^; x^; y^\ etc., signify that ic, y, and ;; have been taken as factors, twice, three times, four times, etc. This figure is called an Ex- ponent. 1. An exponent may be integral, fractional, positive, or negative. 2. An integral •positive exponent of a quantity denotes a ro^VEii of that quantity. 3. A positive fractional exponent of a quantity denotes a koot of that quantity. Thus, x^ is the same as the fourth power of ic, or X X X X. But, x^ is the same as the fourth root of x. 4. The fractional exponent may combine both a power and 3 a root. Thus, a;'^ is the same as the fourth root of x cube. 5. A negative expoiient of a quantity indicates that the rccij)- rocal of the quantity is to be taken with the sign of the ex- ponent changed. Thus, x~'^ is the same as -7 and — - is tlie same as a; 2. {Vide 22, 1.) ^ 6. A letter may represent any exponent: as a:"*, read x wi'* power. 7. Roots are also expressed, as in Arithmetic, by the signs 1 J |/, -j/, etc. Thus, x^ is the same as 1/a;', and av is the V/ — same as V «* . 8. When no exponent is expressed, 1 is understood. Thus, a is the same as a}. 9. Any quantity leaving for an exponent is the same as 1. Thus, a° is 1. {Vide 6S, ex. 1.) 14. Of Coefficients. — Instead of the expression a -\- a, we may write 2a; for a -\- a -{■ a, we may write 3a/ for — x — a*, we may write — 2x ; for — x — x — x — x — x, we may write — ox. In each case the figure standing before the letter shows Jiow many times the letter is taken addltivcly or subtractivcly. This figure is called a Coefficient. 12 DEFINITIONS AND EXEECISES. { 1. A coefficient may be integral, fractional, positive, or nega- i tive. Thus, 5x, Ix, and — ^x. 2. A coefficient may be represented by a letter ; thus, bx^. 3. When no coefficient is written, 1 is understood; thus, a is the same as la. i 4. The expression Ox is the same as 0. | 15. Symbols of Kelation. — The sign = indicates that the quantities between which it is placed are equal; thus, x = i/ \ signifies that x equals y. [ 1. The whole expression of which the sign = is a part, is called an Equation. 2. That part of an equation on the left of the sign = is ] called the First Member. j 3. That part of an equation on the right of the sign = is I 1 called the Second Member. Thus, 2x -{- St/ = a — 5b -{- c is ' an equation of which 2x -f Si/ is the first member, and a — i 56 + c is the second member, and the whole is read thus: 2x \ plus Sy equals a minus 5b plus c; which means that the I numerical value of the first member is the same as the numeri- j cal value of the second member, — thus, 3x4r-f2x5=!!0 — i 4 + 16, or, 22 = 22. ^ 16. The sign >» or <; indicates that the quantities between | which it is placed are unequal, the quantity on the side of the j opening being the larger. Thus, x'^ y indicates that x is ; greater than y\ also, x <^y indicates that x is less than ?/. ; 1. The whole expression of which the sign <1 or ]> forms a part is called an Inequation. Thus, 2x -\- 5y'^ a — b -\- 2J, i is an inequation of which the first member is greater than the second. ; IT. The Signs of I'roportion are thus written, : :: :, ami j a : b : : c : d, is read a. w to b, as c /.s to d. \ DEFINITIONS AND EXERCISES. 13 18. The .sign OC indicates that one quantity varies as an- other. Thus, xozy signifies that x varies as y. 19. An algebraic expression is one involving letters and signs. 1. A Monomial is an algebraic expression consisting of one term. Thus, 6x^y. 2. A Binomial consists of two terms. Thus, 6x'^y + Ahc. 3. A Trinomial consists of three terms. Thus, x -{■ 2y — 4c. 4. A Polynomial consists of many terms. Thus, x •{• Ay — 32+ 6. 5. The tei^m^ of a polynomial are separated by the signs -j- or — . 6. A monomial is positive or negative according as the sign is -}- or — . 20. Sbiilar terms are such as have like letters and exponents. Thus, ^x^y and 2a^y are similar; but ^x^y and 2xy'* are dis- similar. 21. A polynomial is homogeneous when the sum of the ex- ponents in all the terms is the same. Thus, AaPy* + ^^^ — x*z-\-iiXymnp is homogeneous, since the sum of the exponents in each term is 5. 22. The recij^ivcal of a quantity is 1 divided by the quan- tity. Thus, - is the reciprocal of x. 1. The reciprocal of a /inaction is the fraction inverted. Thus, the reciprocal of - is -. 23. The sign /, is the same as the words therefore, hencCy or consequently. 24. The sign '/ is the same as the word because. 25. The letters of a term are usually written alphabetically, though this order is not essential. Thus, ^nhc is the same as 36co. 1 4 I) i: F 1 X I T 1 (J X xS A N I) i: x i: li c i s e s . j i 2S. The tei'Ris of a polynomial are usually arranged with ', reference to the exponent of the leading letter. Thus, x^ -f- ox^y 4- 10a::^y + lOicy -f ^^y^-\-y^i where x is considered the lead- ; infi letter. 1. Of two polynomials involving the same letters, thiit is said ^ to be algebraically the greater whose leading letter has the greater i exponent. Thus, x^ — Sx^y + Zxy"^ — if is greater than x^ — ^xy \ 2V. EXAMPLES. Involving tlie JPrececiiiag Definitions. 1. Convert into algebraic language the square root of seven a square, added to five a multiplied by m. Am. V^ia^ -\- oam^ or (7a^ + 5am)^. 2. Convert into algebraic language three times the cube root of X square, diminished by the square root of five 7??, multiplied by ti square, increased by twice the fifth root of x. Am. 3a;3— {pmn^ + Ix^y, or, ox^ — i^Smn^ + 2 ^^ 3. Convert into algebraic language the fifth root of the sum of X and y. Ans. 4. Convert into algebraic language the square root of x in- creased by the cube root of x square and the square root of x cube. Ans. V x + V x^ + Vx^, or, x^ -\- x'^ -{- x'^ . 5. Convert into common language the algebraic expression 3x' 4- (2x)^. ^,^g S Three times the fflh power of x in- \ creased hy the square root of two x. 6. Write in common language the following algebraic expres- sions : o.r" — ^Ti^-f 12.T7/, x -f y/.T^ + — and x^ .^ •^ -f V.r*+ y'.r^ -f f^~^ .r\ DEFINITIONS AND EXERCISES. 15 -y^' 7. Write in common language the following expressions : ; p= — — - x^ x-\-y X1/+ \4x^ — ^5x^ 4- 'dx., -j= 7x^ -f 8am., -> -^/-la — bh. if ^ J 'J^' 8. Write in common language the following expressions: c^c^^a^ — h', and [x + Qi —py^l(ci^ — 5^2^. 2 ^c Kemark. — The great advantage of algebraic symbols has been seen in the previous examples. By them. are obtained both brevity and perspicuity. 28. The NUMERICAL VALUE of an algebraic expression is the value found by arithmetical reduction on affixing a numerical value to each of the letters composing the expression. , EXAMPLES. — ^ . y - 1. What is the numerical value of the expression x -{• 2a^ when x= 5, Ans. 5 + 2x5^, which is 55. 2. AVhat is the value of the expression 5ab + 3a^c — ??m w'hen a = 2, Z; = 3, c = 4, ?;i = 5, and n = 6. Ans. 5.2.3 + 3.214 — 5.6. = 48. 2 3 5 3. Find the value of a^ + ¥ -\- c^ when a = 8, & = 32, c = 4, (Vide, 13, 4.) Ans. 8^ + 32^' + 4^ = 4 + 8 + 32 = 44. 3 11 4. Find the value of x^ + 5x^ + ^^ when x = 64. Ans, 534. 3 ? 1 5. Find the value of (x^ + 2/^ + ^^) • ^> when rr = 4, ?/ = 8, "~" z = 16, 772 = 2. Ans. 28. 6. Find the value of (x -{- y)(x — y) when .t = 3, ^ = 3. ^4ws. 0. 7. Find the value of (x + y) (x + ?/) when x = 4, 2/ = 3. ^ns. 49. 8. Find the value of (x -{- y)(ci -\- Z>) (x^ — ?/^) when x = 4, 3/ = 4, a = 2, 5 = 3. ^Tzs. 0. 9. Find the value of (x -\- y + Sa + 2h')(x^ — 7/) when x = 2, _y = 1, rt = 4, ?> = 3. ^;?.s. 147. 16 DEFINITIONS AND EXERCISES. ahc 10. Find the value of ^^ _|_ ^^ _^ ^^ when a= 1, Z> = 2, c = 3. 1.2.3 Q_ ^'''* 1.2 + 1.3 + 2.3= 11' 11. Find the value of c^ + 6' _ ^2 ^j^gn ^^=50, h == 50, c = 40. 2c ^^^g^ 20. 12. Find the value of x and ?/ in the equations x = (a -\- h^.c, and ?/=(«+ 6).c, c = 20, a = 1, & = 2. « h Ans. x—QO,y= 30. 13. Find the value of x and y in the equations, x = (a -{-})) x ??2cZ — (c 4- (i) X nh, and ?/ = (c + cQ X «?^ — (a + Z>) X c??/., a(i — 6c ad — he when a = 2, 6=1, m = 78, c = 7, rf = 2, 71 = 79. Ans, a; = 81, 3/ = 72. 14. Find the value of x in the equation » = | (^ + ^6^ — 4^), if a = 4, 6 = 5. Ans. x — 16. 1 1 15. Find the value of x and y in the equations, rB= ^^ ^ 1 1 w^^ — n^ and y = — — when m = 9, ?i = 4. J.?is. a; = 6, ^ = |. 71 16. Find x in the equation a:= (671— VaW4-62wi2__aV), n^ — m^ when cf = 2, 6 = 3, wi = 5, 7i = 3. -^tis. a; = 1|. 17. Find CK and y in the equations x = (^^ — i^?^) X ac ^^^ ac — 6c (am — 7ic)6a _,^ x^ — dxyz Sa 8X2/ IQx^ - 7a3 Zxyz 12a l^xy — Vlx"" — 10^3 — Ixyz — 13a — 20xy — IQixy - la? sJ' — X^xyz , 16a ^j? — ISic^ — XAiXyz Ans. 11. Add together a\v, da'^x, — 7a^.T, Ida^x, — da^x, — 200^0;, and — 3a^x, Ans. — 20a'a7. 12. Add together 5, — 4, 17, — 30, and 80. Ans. 68. 13. Add together Ix^y^, — 9x^i^^, Ax'^f, — x'^y'^. Ans. x^y\ 14. Add together Axy'^^ 7.t?/^, 9xy^, — 11 xy"^. Ans. Sxy*. 15. Add together 15x^y, IGx^y, 18x^y, — Six^y. 16. Add together — 12a:y, — 20xy, and 30xy. 17. Add together — 19xy, — 21a:2^^ ^^j^^j Aoxy, 18. Add together — 30.xy, AOxY, and l^xy. 19. Add together Ax, 5x, — 3x, and — 6x, Ans. 0, 20. Add together 5xyz, Ixyz, and — 12xyz. 34. To add polynomials having in each similar terms, Arrange the iwJynomiaU so that similar terms stand under each other. Add each column of terins hy 33. EXAMPLES. (1.) (2.) (3.) Add x-\- y 3x +2y + z Ix — 5y 4-6^—10 to x-2y 5x — Ay — Qz Ax + 10// — 9^4- 30 2x— y Sx — 2y — 5:: Ux -\- ov — 8.:+20 ADDITION. 21 (4.) (5.) (6.) Add 4x-\-3t/—2z 3x^+2?/'— 42^+10 ic^+ x'^^ 5x}j — 5ic+4z/+62! 4x2— 2^/3+02"— 10 3a;^— 2x2^+ Trr^/ "jx—Sij—^z —5x'^-\-3i/—2z'^+\6 — 90)^—4x2^— 13.T^ 4x— 2/+ z 6.^2—8/ +42"— 16 7x'^+Sx^y-\- xy Am. 10x—2y—4:Z Sx^—5f-{-3z^— 1 2x^—2x2^ 7. Add together the polynomials 3a + 25 — 5c + 12x — 10, — 7a 4- 36 — Cc — 13x + 12, 4a + 36 — 10c — 5x + 8, and 10a — 76 + 3c — x+ 13. Sohitio7i. Sa-\- 2h — 5c + 12x— 10 __7a + 36— 6c— 13ic+12 4a + 36 — 10c — 5x + 8 10a — 76 + 3c — x-j- 13 10a + & — 18c— 7x + 23 8. Add together the polynomials 7a?x + 56' — 7jn^ + 14ri, da^x — 7n-i- 96' — 5??i2^ __ 106' + Am^ + 8?i — lOa'x. Solution, 7c?x + 56' — 7m5 +1471 3a=^x + 96' — 5m2 — 77t — lOa^x — 106' + 4m2 + ^n 46' — 8m2 + 15?i 9. Add the polynomials 2x + 3^ — z and 2x — 3?/ + 2?. Ans. 4x. 10. Add together x + 2^ + c and — x + 2?/ — c. -472S. 4^. 11. Add together x + Sy — 5^ + ^ + 9, 6^ — 2x + 32 — 1 — 5^, 7/7* + 1 — 3z -\- y — X, — g — 8 — 3^ — x, and 3 + 52 — 9?/ — g + Ix. Ans. Ax + 3y + 5g + 4. 22 ADDITION. 12. Add together Zx'^y + 2a;y — Wij + oxij\ \x^i) — Zxy^ — 1 "Ixhf + ^rc^y, — X^xhj + 8xj/' + 5xy + 7x2y and lOx^ — '^W 1 1 i 1 13. Add together 2xy + "Ix^y — Sa;^^ + Zx ^y% 8xy^ — 5xy^ + 2xy^ — ox^^y^^, 15x^y^ — 18x^y^ — ox^y + 7xy and — 2xy — 1 1 "jx^y + 4a;y — Sx^y. Ans. llxy — llx^y — 3xy^ — 3x^y^. 14. Add together dx^y^ — 2xyz + ox, Ixyz — ox — 5x^y^ and — 5xyz + 2x^yK Ans. 0. 15. Add together x^' -\- x^ + re* — x\ ox^ + 7x^ — 8a5* + 5a;^ and •— 2x^ + S.-c^ + 7a;* — lOx^ ^«s. 9a;^ + 6x^ — 6a;^. 16. Add together oxy + 2a2Z>2 _ 5^2?^ + 40, 3mn — 20 — 5rt' ?>=* + 2a:y and 2mu — 10 + Sa^t^ _ 7^;^. 3 5 17. Add together x"^ -{- y^ -\- xy -\- x^y^ -\- x, — 7x — 4x^y^ — 3xy — Sy^ + 2x- and 4.xy — tx^y^ -\- ox — 5y^ + 3x^. 18. Add together 3x* — c + 7m — n -\- 10, -- 5^" — 2c -- 7m + ^ — 15, 7cc^ — 2c + 3??i — 2?i and — lOx'' — 5c + 6m -\- 3x — 3c + 2x\ 19. Add together Ax^y -f- 5.x 4- 3m + 3/ + ^2^1 'h ^^i Zx^y — lOx -f 4m — ^y -\- ^p(l — 82 and — 7m + 7?/ — 7pq -\- 6z -{■ Qx — 7x^y, 20. Add together x^ + S.t:^?^ + 5?/2 + 4^ + 32;^ — lOrc^m — 7y^ + 5;) — 10^2 + 5x2 j^jj^ g^a _ 9^, _|_ 3^ _^ 7^2^ __ 5^2^ 21. Add together 3x^ + 2xy + ?/2, — 2xy + Sj/^ + Sx' and — Ay"^ + 4a:;y + 2a:^ 22. Add together x^ -f 2xy 4- ^' and x^ — 2xy + ?/^ 23. Add together x? + Sx^?/ 4- 3x7/' 4- y^ and x? — ox^y -\- oxy"^ — 2/'. 24. Add together x* 4- ^^^y + Gx-^j/^ + 4x7/' + ?/^ and x"* — A'3i?y -\- 6xy — 4x?/' + ?/*. 25. Add together x? -|- x?/ -\- y"^ and x- — xy -\- _y'. ADDITION. 23 2G. Find the nuiuerical value of the last five examples when a;=2 and ^=2. Ans. 2P'= 5G, 22"^^= 16, 23''^=Gi, 24'" = 256, 25''^= 16. ^ 11 1 L 2/. Add together cc -j- x^y^ + 2/ ^^id re — x^y''^ + 2/- 28. Add together x^ + Gx'y + lorcy + 20.^^+ lox'^y* +Gxy^ + 3/^ and x^ — 6x*y + 15a:''_?/^ — 20x^j/^ + lo.x^ — 6xy^ + 2/^' 29. Find the numerical value of the last two examples, when 05=1, 2/ = 2. Ans. 6 and 730. 30. Add together 3 (x + y), 2 (x + y) and 8 (x + y). Ajis. 13(x -\- y). 31. Add together 2 (x^ + 7/) + 5 (x -}- y -\- z) + 4: (x^ + 2y^ and 6 (aj' + y') — 4(a;+y + ^) — 2 (cc^ + 2?/2), where rc=l, y = 2, ;s = 3. ^7w. 8 (x2 + 2/0 + G^ + 2/ + ^) + 2 (x' + 2y») = 96. 32. Add together x + I (x -\- y + z) -{- 4:, y + m (x + y -\- z) — 3 and z -\- n (x -\- y -j- z) + 5, and find the numerical "value when x= 1, y = 2, s = 3, Z = 4, ?n = o, n = 6, Ans. x + y + z-i-(l+m-{-n)(x + y-\-z') + 6= 102. 33. Add together 3x (1 + 2y') + 9, 5x (1 + 2^/) — 7, —2ix (1 ^ 2?/) + 12 and 7ix (1 + 2y) — 8. ^?zs. 13a; (1 + 2y') + 6, 34. Add together 7 + 4- (2c + d — m) + 3a;2, _ 8 + K'^c + d — 77i) — 5x^ and 8 — -^ (2c -\- d — 7n) + 2x^ ^/?s. 7 + I (2c + c^ — m)- 35. Add together (x + y)^ +7, 4 (a; + t/)^ — 3 and — 8 (a; + y)* — 4. ^«5. (a; + y)^ 36. Add together x ■\- y and x — y. Ans. 2x. 35. By the last example we see that The sum of two numbers added to their difference gives twice the lar-ger number. 24 SUBTRACIION. EXAMPLES. 1. (12 + 5) + 0'2 — 5) = 2 X 12 = 24. 2. (41- 4- 3) + (4i - 3) = 2 X 4^ = 9. 3. (6 + 20 + (6 - 2i) = 12. 4. (17-30 + (17 + 30 =34. 5. (8i - 20 + (Hi + 20 = 17. 6. (3i+10+ (3k- 10 =6i. 7. (4 - 2) + (4 + 2) = 8. (8x + 2?/) + (8x — 2?/) = 16*«. 9. (2ix + Siy) + (2ix - 3^2/) = 10. (5x* + 2y) 4- (5x^ — 27/) = 11. What is the value of the last three examples when x = 5, 7/ = 3 ? Ans. 8"^ = 80, 9^'* = 25, 10'^ = 6250. SUBTEACTION. 36. Subtraction in Algebra consists in finding the simplest expression for the difference of two given expressions^ or, in finding what quantity added to the subtrahend will produce the minuend, 37. Of the two given expressions that which is to be subtracted is called the subtrahend; the other is called the minuend. 3S. To find the difference of two similar monomials : This difference may always be expressed by either a positive or a negative quantity, each result depending upon which of the given expressions is taken for the minuend ; thus, Vide 24, 36. 1. From 5a subtract 3a and we have 2a •.* 3a + 2a = 5a. 2. From 3a subtract 6a and we have — 2a •/ 5a + ( — 2(r) = 3a. SUBTEACTION. 25 3. From 5a subtract — 3a and wc have 8a *.• — 3a -f- 8a = 5a. 4. From — 3a subtract 5a and we have — 8a *.• 5a -f ( — Sa) = — 3a. 5. From — 5a subtract 3a and we have — 8a *.• Sa + ( — 8a) = — 5a. G. From 3a subtract — 5a and ^xe have 8a •.• — 5a -f 8a = 3a. '7. From — 5a subtract — 3a and we have — 2a *.• — 3a -j- ( — oa) = — 8a. 8. From — 3a subtract — 5a and we have 2a •.• — oa -f 2a = — 3a. These may be arranged as follows : (1.) (2.) (3.) (4.) (5.) (6.) (7.) (8.) From f oa oa 5a — oa — oa 3a — oa — 3a take 3a 2a • 5a — 3a 5 a 3a — 5a — 3a — 5 a Ans. -2a 8a — Sa — 8a 8a — 2a 2a By comparing (1) and (2) it will be seen that the difference between 5a and 3a is expressed either by 2a or by — 2a. There is a similar relation between (3) and (4), (o) and (6), (7) and (8). Each of these results may be obtained by the following rule : Consider tJie sign of the subtrahend changed. If the signs are then alike, add the coefficients, prefix the com- mon sign, and amiex the common letters. If the signs arejinlike, subtract the less coefficient from the greater, prefix the sign of tJie larger coefficient, and annex the common letters, EXAMPLES. (1.) (2.) (3.) (4.) (5.) (G.) (7.) (8.) From lOx \Qx — lOx —lOx 2>x — 3.7; Zx — ox take 3x — 3x 3x — ox lOx 10.x —10.x — lOrc Ans. 7x lox — 13.T — 7x —7x—\3x lo.x Ix o O SUBTRACTION. (9.) (10.) (11.) (12.) 1 3 From oci^j^ Iba^x^y \2axyz — Ax^ 1 3 take "Ici^x — IQa^x'^ij 30ax7/z 7x^ 1 3 Ans. — 2a^.» ola~x^f/ — 18axf/z — 11. ^;* \ 13. From 7a\c subtract ^ci'x. Ans. ] 13 13 ! 14. From r- IGa^x'^T/ subtract loa^x'^i/. Ans. ' 15. From SOaxi/z subtract — 12axi/z. Ans. i IG. From 7x^ subtract — 4x*. Ans. ^ 17. From lOx^ subtract lOa;^ Ans. 0. 39. To find the difference of two monomials not similar: ; ! Change the sign of the subtrahend and ivrite it after the i minuend. j EXAMPLES. j (1.) (2.) (3.) (4.) (5.) (6.) I From a 5 a 3.^ 7?/' 4:xy — 12x^ take Z) — 2Z) Ay Am 3x — 3a; Ans. a — h ba-\-2h 3x — Ay 7y^ — Am Axy — 3x — 12x'+3a; 7. From 3a; subtract 2y. 8. From 7a;' subtract 2a;^. 9. From 10a; subtract — Am. 10. From 5a;?/ subtract 12. 11. From 12 subtract — 4a;. 12. From 15 subtract — 2a;^ 40. To find the difference of two polynomials : Write the j^ohjnomial taken as the subtrahend under that tahen as the minuend^ j;/«c?Vi^ similar terms under each other and those not similar in any order. Subtract similar terms by the rule in 38. Subtract terms not similar by 39. SUBTRACTION. 27 EXAMPLES. (1.) (2.) (3.) (-t.) 1 1 1 1 From 3a; — y x — y x^ -{- 2x>/ -j- y^ x -\- x^y^-\- y take 2x — h x -{- 2y x^ — 2xy -{■ y^ x — x^^y^-{- y , y 11 A71S. X — y -\- h — 3?/ Axy 2x''^y^ 5. From 3a + 26 — 3c + % — om + 2;i — 7x — y-\-10 take a — 36 -f 4c — 2g -\- 5m — n -{■ Ax — y — 40 Ans, 2a + 56 — 7c + 6g — 10m + 3n — llx + 50 (6.) (T.) (8.) From x^ -\- oxy -\- y^ x"^ -\- ^U -\- y"^ ^ take — x^y -\- 3xy + xy^ — x^ -{■ xy — y^ x — y Ans. x^ + x^y + 2/^ — ^I/^ 2a;^ 4" ^i/^ y 9. From x — y subtract x — 2y. Ans. y. 10. From x -{■ y subtract x. Ans, y. 11. From x"^ — xy + y"^ subtract x"^ — xy — y"^. Ans. 2y^ 12. From x -\- ^xy -f y subtract x — ^xy -f y. Ans, 2 ^xy. 13. From x^ + ijx^y + ^xy^ + y^ subtract x^ — Sx'^y + 3xy^ — y^. Ans. 6x^y -f- 2_y^ 14. From x'^ -f 4x^y + Gx^y^ + 4^xy^ + y^ subtract x^ — 4x^y + 6x?y^ — 4xy^ -\- y^. 15. From x^ + G.^^y + ISx''^^ ^ 20a:^7/3 + l^x^y^ + Ga:^^ + y\ subtract x^ — Qx^y + 15x^7/= — 20a:^3/3 + 15^2 t/" — Ga^y + ?/^ ,5 Ans. 12x'y -}- 40x' y' -^ 12xy' 16. From 3m + 2;i — ^xy + 4^) — (2 + 2^^ subtract 2m — 3/1 — 4p + 5ary -|- 2a; — q. 17. From ^^3/^ + ^'^ + *^^^^ subtract a;^^^ — xy -^ x^y^. IS. From x^ + 3^:^^ + 3a-2^^ + y^ subtract — x'y — 3x'y^ + 3x2 19. From a;^ -f 3x'y^ + 3.r2^^ + y' subtract x'y + 3.r^j/2 _ 3,-^2^ + a;*'. 28 SUB T R A C T I N . 20. From x -\- 2tj — -1 subtract x — 3y -j- 7. 21. From 4.^: + 7/y — G subtract 2x + 7?/ + 12. 22. From x ^ -f 5x ^ — 7a; 2 subtract 2x ^ — Sx^. 23. From 1 -{' x -\- x"^ -{- x^ subtract 1 — x -\- x^ — x^. 24. From 1 + 2x + Sx^ -j- bx^ subtract 1 — 2a; + Sx^ — 5x». 25. From 1 -[- 5x -{- G.x^ subtract 1 — 5x -J- Gx^ 26. From x -\~ x^ -\- x^ subtract x — x^ — .x^ 27. From x^ -\- 2xy -\- if subtract x? -\- xy. 28. From o^ -\- y^ subtract x^ -j- x'^y. 29. From x^ — ^x'^y'^ -|- Sx^y — y^ subtract x^ — ?»x/'y -j- Zx^y^ — x^y\ 41. From the sum of two or more quantities to subtract any number of quantities: Cliange the signs of the subtrahends and add the columns as in 33. EXAMPLES. 1. From the sum of 4x -\-3y — 2z and — 5a; -{- 4:y -\- Qz take — 7a; -j- Sy -J- 9;3 and — Ax -{- y — z. ( Vide 34, ex. 4.) Ans. lOo; — 2y — 4z. 2. From the sum of Sa;^ -f 2^ — 42"+ 10 and ix^ — 2y^ -\- bz* — 10 take 5a;2 — 3y' -]- 2z* — 15 and — Gx^ -j- Sy^ — 4z' -f 16. Ans. 8a;2 — 5/ 4- 3;s-' — 1. 3. From the sum of Sx — 4x7/ -|- 83/^ and 6xy — 7x -|- 4y^ -]- 8 take 3x + 2xy — by^ + 4 and 5x + 2xy + lOif — 15. 4. From x^ -\- 2xy -\- y^, x^ — 2xy -j- y^ and x^ -\- ^^ -\- 5?/^ take a:2 _ 5.ry _ 23/=, 2x2 _|_ 4,^,^ _j_ 9^2 ^^^ _ ^2 _|_ 2;cy, 5. From 2 (x — 2/) + -5 ^ (x _ 7/) -f 2:; and b(x — y)—z take 4(.^-2/) + ^-2. G. From 3(x — 7/)-2-f 4 and 2(x — 7/)-2-f 6 take 4 (x — 2/)~2-l-8 and find the vakie when x = 8, y = 3. .4«5. to last jioint 2^. S U B T i: A C T I O N . 29 7. From 7 (.x + ?/)^-f 15 and 8 (.x + ^)2 — IG tul;e 12 (.t + y)2 — 1 and find the value when x=lG, y = 9. -<4;i5. ^0 /(cwi />o?V<^ 15. 8. From 8 (x^ -|- t/^j 2 _|_ 2:ry and 4 (x^ -|- t/^)*^ _j_ S.ry take 10 1 (rc^ -|- if) ^ -\- \xy and find the vahae when re = 4, ?/ = 3. ^l«s. 22. 42. In the expressions + (+«), + (- «), - (+ «) and - (- «) the sign before the parenthesis is called the sign of operation. The sign before the letter is called the sign of the quantitij. Thus, (1.) -f- (-}- <^) means that the positive quantity a is to be added. (2.) -j- ( — a) means that the negative quantity — a is to be added. (3.) — (-}- a) means tliat tlie positive quantity a is to be subtracted. (4.) — ( — a) means that tlie negative quantity — a is to be sub- tracted. ]5y performing tlie operations indicated by the sign of operation, we have, (1.) + (+ a) = + a, (3.) - (-1- a) = - a, (2.) + (_ a) = - a, (4.) _(_«) = + a, where the sign in the second member of each equation is called the essential sign. (5.) J5y comparing (1) and (4) it is seen that the addition of a positive quantity is the same as the subtraction of an equal neg- ative quantity ; that is -\- (-f- a) = — ( — a). (6.) By comparing (2) and (3) it is seen that the addition of a negative quantity is the same as the subtraction of an equal positive quantity ; that is -[~ ( — «) = — (-|- «). (7.) It is plain, tlicn, that — ( — ah) = -f ah. 30 S U B T K A C T I N . EXAMPLES. 1. What is the value of 3x — (^-\-^x)"l Ans. — 2x, 2. AVhat is the value of 3x -[- (— 5.x) ? Ans. — 2x, 3. What is the value of 3.x -]- (-|- 5x) ? Ans. 8.x. 4. What is the value of 3x — ( — 5a;) ? Ans. 8.x. 43. The subtraction of a polynomial is indicated by inclosing it in a parenthesis and prefixing the sign — . Thus, x -{- y — (x — y) signifies that x — y is to be subtracted from x -\- y. Performing the operations we have x-\^ y — (x — y^-=.x-\- y — X -{-?/= 2y. Hence, to remove a parenthesis having a nega- tive sign of oj^teration, Change all the signs in the parenthesis and unite the terms as in addition, EXAMPLES. 1. Eemove the parenthesis from a — (h -{- c), Ans. a — h — c. 2. Remove the parenthesis from a — (h — c). Ans. a — h -{- c, 3. Remove the parenthesis from a — ( — h -\- c). Ans. a-\-h — c, 4. Remove the parenthesis from a — ( — h — c). Ans. a-\-h -{- c, 5. Remove the parenthesis from x -\- 2y — 4 — (x — 3?/ -j- 7). (HcZe 40, ex. 20). Ans. 5?/ — 11. 6. Remove the parentheses from a — \h — (c — rf)-f-x]. Ans. a — (h — c -\- d -\- x') = a — h -\~ c — d — x. 7. Remove the parentheses from a — [\h — [c — (d — e) — /] — g]], Ans. a — h -\- c — d.-\-e — f-\- g. 44. By reversing the operations of 43, polynomials may be written in various ways. Thus, 1. X — 5.x2-{-6.x3-j-7.x* — 8x^ is the same as x — (5.x^ — G.x^ — 7.x^ -|- 8.x^), which is the same as x — [5x^ — (Crx^ -f 7.x'') -f 8x*]. AVhat is the value of cither polynomial when x=2? Ans. —-114. SUBTK ACTION. ol 2. Find the value of a — \[—h^lc—{—d—f)-\-(j'] — h]] when a = 1, h = 2, c = 3, c^ = 4, /= 5, ^ = G and h = 7. 45. Since x^ y — (x — ?/) = 2?/, it is plain that Tlie difference of two numbers taken from their sum (jives twice the smaller number, EXAJn'LES. 1. (12 + 5) - (12 - 5) = 2 X 5 = 10. 2. (4i -I- 3) - (4i - 3) = 2 X 3 = 6. 3. (6 + 2i)-(6-2i)= 5. 4. (4 + 2)-(4-2) = 5. (17 + 30 - (17 - 31) = 7. 6. (7 + 20 -(7 -20= 5. 7. (101 + 5|)-(m-5|)= IH. 8. (6| + 2^)-(6|-2A)= 41. 9. (3x^ + 57/) - (3x'' - 57/) = 10. (2x5 _j_ 3^1) _ (2x^ — 3/3) = What is the value of 9'-''^ and lO'^^ examples when x = 256, 2/ = 8? 46. PROBLEMS IX SUBTRACTION. 1. If A is worth 5a dollars and 15 3a dollars, what is the difference between their pecuniary conditions? Ans. -\- 2a or — 2a dollars. 2. If A is worth 5a- dollars and B is in debt 3a dollars, what is the difference between their pecuniary conditions? Ans. -{- Sa or — Sa dollars. 3. If A is in debt 5a dollars, and B is worth 3a dollars, what is the difference between their pecuniary conditions? Ans. — 8a or -|- 8a dollars. 4. If A is in debt 5a dollars, and B is also in debt 3a dollars, what is the difference between their pecuniary condition ? Ans. — 2a or -p 2a dollars. If a = ^3000 what are the answers of each example ? 32 M U L T I P L I C A T I C N . 4'1'. By distinguishing what each one is worth bj -\- and what each one is in debt by — , the connection of these prob- lems with the previous principles is evident, (^Vide 3S.). It is seen that the difference between two quantities can be expressed as well by a negative as by a positive result. In Arithmetic it is not necessary to recognize this fact, but in Algebra it is of the utmost importance to have a correct apprehension of it. 4§. Merely to find a difference it is of no consequence which of two given quantities we call the minuend or which the sub- trahend. After having, however, assumed one of the quantities to be the minuend, the result must always he referred to it, 49. The difference between 7 and 4 is -j- 3 or — 3 ; thus, 7"^ -}- 3 showing that 7 is ! 3 greater than 4, + 3j ^1 ^ » — 3 showing that 4 is I 3 less than 7. _3j 50. The negative sign, then, serves to indicate some peculiar circumstance connected with the quantity before which it is placed. 51. In a given problem, negative quantities have a sense con- trary to that which limits iiositive quantities. MULTIPLICATION. — 52. Multiplication is the operation of finding the product of two quantities. — 53. The multiplicand is the quantity to be multiplied. ^ 54. The luultiplier is the quantity by which to multiply. MULTIPLICATION. 33 55. The multiplicand and the multiplier may be inleichangcd at pleasure. ^ 56. To multiply positive monomials; (^Vide Dcf. 19, 6.): By Def. 10 we have x X y = xy. By Def. 13 we have x.x = x^ and x.x.x = x^ , * , x^ X x^ = xxxxx = x^. By Def. 14 we have 5a x 3 = 5a -J- 5a -|- 5a = 15a . * , bax^ X 3x^y = Ibax^y. Hence, (j:ide 25,) Multijjly the coefficients and add iJie exjioncnis of like letters. EXAMrLES. (1.) (2.) (3.) (4.)^ (5.)^ Multiply 7a26 2a''x IQa^V lOa^^' tJ/ by 5ax 5a3.x2 2ah 2ah^ 3x{f Ans. '6ba'bx lOa'x' d'Za'b' 20ah 2\xy (G.)^ (7.) (8.)^ (9.) (10.) Multiply Ix^y^ ^xyz-'^ Sx'y~^ hx^y^ 3a;^?/-* t 2 13 11, 1 by \x-y^ l^~ y^* 7x^y^ bx^y^ bx'y* Ans. X y ly"^ lbx~y ' .^ 5t. To multiply two monomials with unlike signs : By Def. 14 we have — 5a x 3 = — 5a — 5a — 5a = — 15a. But — 5a X 3 is the same as 3 X — 5a. ( Vide 55.) Hence, Multiply as in 56, and prejix tlie sign — to the jyroduct. EXAMPLES. (1.) (2.) (3.) (4.) (5.) It 3_ 4 1 , 11 Multiply 4a^6 — 2x'^y'^ — 4:m^n'^ — ix'y^* 27n^x'^ by — 2a^b Sx^f 2fnhS 5.-cV° ~2n^'x^ Ans. —8a'b^ —<>.'•?/ —Sm^/i" (8.) (9.) ^x'yh 3xyz — Sax^y — 2ahc 34 MULTIPLICATION. (6.) (7.) Multiply — 2x-^y-'^z-^ — Ax-^y-^z-^ by \^ y ^ Ga^y^^ Ans, — 1 58. To multiply negative monomials: It is plain that — a s= — 1 (-j- a) and — t = -j- 1 ( — h^, Plence Multiply — a = — 1 (_[_ a) Vide Def. 14, 3. by -6= +l(-6) and we have — 1 ( — alj^ = — ( — ah') = ah. ") Therefore ' ^* *^' 7.| Multiply as in 56, a?id 2?r<^a; ^/?e sj^yi -}- io the product. EXAIMPLES. (1.) (2.) (3.) (4.) (5.) Ill Multiply — X — X — 3a^6 — 2a^x^y^ — ^xyz 12 3 by — y ■ — X — ^a^hx — \a?-x^y'^ — Zxyz Ans, xy x^ (Vide Def. 9, last clause.) (6.) (7.) (8.) (9.) Multiply — ^xy^ — bxyz — Ixy — ba^hc by — ^ay^ — ^xyz^ ~2ah — 9axy 59. Hence, to multiply algebraic monomials : Jlidtijjly the coefficients, and add the exponents of like letters. Like signs produce -f- , a7id unlike signs produce — . EXAMPLES. (]-) (2.) (3.) Multiply — hax y z 5 a* h^ c' . 5.t" ?/^ z^ by 4taxy z — 20a''xhfz'' — ?>n' h^c^' ■ — 3.T y z Ans, — Iba^'O'^^c'' i^x'+y+^z'^^ MULTIPLICATION. So 4. Multiply — lOa^x'y by Aa^x^/K Ans. — AOax^f^ , 5. Multiply 7x^fz^ by 3x^i/^z\ 6. Multiply llx^fz^ by — 2x^f^z\ 7. Multiply — ISninj) by Grnnp. r. 111 345 8. Multiply — 2Gx'^}/^z^ by — ^^^x'^i/^zK 9. Multiply 12^3^^^^ by 12xi/z. 10. Multiply 13iCj/^ by — 13xi/z. 11. Find the product of 13x^y X =^^6^^^^ X 14:X}/z X f^jj/^. J. 715. X^ }f Z, 12. Find the product of 7xyz, — Sx^e^ and l^'x^y^^. Ans. — 2x^ifz^. 13. Find the product of la'h^c', ^a}?c^ and —\a^}?c, Ans. —7a^+*h'-'+*c'+\ 14. Find the product of Sa'^h^z" and 4a"* Z)''^^. ^;j5. 12a^+'"Zy»'+^-^'^+-''. .. 60. To multiply a polynomial by a monomial: Multiply each term of the multijilicand by the multijylier, accord- ing to 59. EXAMPLES. Multiply by Ans. Multiply by A7IS. (1.) 3 Sx-\Sy-\-3z (4.) x^-{-x^-\- x^ i ~i 3 1 x^ -f x^^ 4- x^ (2.) ic — y — z 4 (3.) X — y — z — 5 4x — iy — 4cz — bx-{-by-\-bz (5.) x^ — 2x' 4- 3a;y — 77n + 12 ic'^ — 2x*y-{Sx^y^ — 7?7ixy-{-12xy 6. Multiply x^-\-xy-\-y^ by cc^ ^4/^5. re'' -f- a;'y -}~ a:'^^ 7. Multiply x^ -f" 0^3/ -|- y2 by — xy. A71S. — x^y — x^y"^ — xy^. 8. Multiply x^ -\- xy -\- y^ by y^. Ans. x^y^ -4- xy^ -f~ ^^ 9. Multiply x^ -\- x^y -f- xy^ -]- y^ by x. A71S. X* -[" ^'y + ^V ~\~ ^y^' 36 U U L T I P L I C A T I N . 10. Multiply x^ -\- x~^ -{- a'?j-^ -^ 7/^ by — y. Ans. — x^y — x^ij^ — xr/^ — y*. 11. Multi^jly x^ -\~ X* -\- x^ -\- x^ -}- X -\- 1 by x. Alls, x^ + x^ -\- x^ -^ x^ -\~ X. 12. Multiply x' + x' + a;' + x^ + x + 1 by — 1. A^js. — x^ — x'^ — x^ — x^ — x — 1. 13. Multiply X -{■ y by x. Ans. x^ -j- xy. 14. Multiply X -}- y by y. A?is. xy-[-y^. " 61. To multiply one polynomial by another : Multiply every term of the midtij^lkand hy each term of the multij)lier, and add together the several products. 1, Multiply by Product x^' + 2xy-{-7f 2. Multiply ^^ + <'^I/+f by» x'^ — xy-\-y^ x'-\-x'y-\-xY~ — x^y — x^y~ — xy^ ^'y^-^^u'^y' Product X* +a;y- -l-y 3. Multiply x^ J^ x^y -\- xy^ -\- y' by X — y EXAMPLES. x-^y x'^-\-xy (Vide above, ex. 13.) ^y-^-y^ {Vide above, ex. 14.) {Vide 34.) (Vide above, ex. 6.) (Vide above, ex. 7.) (Vide above, ex. 8.) (Vide 34.) Product X* -\- x^y -{- x^y^ -\~ xy^ (Vide above, ex. 9.) — x^y — xhf — xy^ —y^( Vide above, ex. 10.) 4 X -r MULTIPLICATION. 37 4. Multiply x'-^x'^-]-x^-\-x^-{-x-{- 1 by X — 1 x^ -\- x' -\- x^ -\- x^ -{- x"^ -\- X ( Vide above, ex. 11.) — x^ — x'^ — x^ — x~ — X — 1 ( Vide above, ex. 12.) Product x° — 1 5. Multiply by Product 6. Multiply by a;2 -|- 5x + 7 xi — 8x — 3 a;4_}-5a;3_|-7x2 — Sx'—4.0x'- — 3x" — ~ hQx - 15x-21 x^ — Sx' — ^Gx'- x^-{-xy-{-y^ x^ — xy-\- y^ -71a;~21 — x^y — 1 1 ^x^y^ ■ x^y^ — xy^ J^xy^-\-y Pi'oduct X -{-2x^y^—x'^y^ -\- y 7. Multiply X + 5 by a; + G. An%. x^ + 11^^ + 30. 8. Multiply cc + 5 by a; — 6. Am. x^ — x — 30. 9. Multiply X — 8 by a; — 9. An^, x" — llx + 72. 10. Multiply 2^2 4- 3.x — 1 by a; — 5. Ans. 2x' — 7^2 — 16x + 5. 11. Find the product of (x — l) (re — 2) (.-r — 3) (x — 4.). Ans. X* — 10a;' + 35.7)2 _ 50^; + 24. 12. Find the product of (a;^ — 2x+ 5) (x + !)• -^"S- 13. Find the product of (x^ + 2xy -f y^) (x^ + 2xy + if). 14. Multiply a)2 -j- 2ax + 7 by a;' — ax + 5. 15. Multiply x' + 3.xV + 3a'/ + y' by x^ + 2xy + /. 16 INIultiply x2 4- 2.ry + y^ by x^ — 2a-j/ + if. ir. Multiply by Aiis, 18 Multiply by Alls. 38 MULTIPLICATION. x^ — 3x^1/ -j- 3x7/^ — y* x^ — 3xy-{-3xY — y^ x^ 4- ^^^y + Gx^y^ 4- 4.XJ/* + y^ x^ — 4x^y -j- 6x^^^ — 4:X?/^ + y* X* — 4:xy 4- Qx^ — 4:X^y^ -f y* 19. Multiply x^ — X* -\- x^ — x^-{- x — 1 by cc + 1. Ans, x^ — 1. 20. Multiply l—x-{-x^~x^ by 1 -{. x -\- x* -{- xK 21. Find the product of (x — 5) (x — 6) (x — 7) (a; + 8). Ans. x* — 10x» — 37a;2 -|- U6x — 1680. 22. Multiply x^-}-x^ by aj^—a^^. Ans. x—'xK 111 111 23. Multiply x^--x^-\- x^ by x^ -f cc^ — x\ 2 _T 1 I ^ ^?is. X — x^-{- 2x^^ — xK 24. Multiply x^-{-^x^ — x by x^ — ix^-\-x. 25. Multiply a:^ -j- ^x^ -\~ Ix^ by a;^ — 5a;^ — 3a;*. 26. Multiply Ix^-^ix^ by 5aj^-f 6a;^. 27. Multiply a:" + y** by a;" + ^^ ^?is. a:^'' -f 2a:"^'* + ^2". 28. Find the product of x"^ — 3/"*, a:*" + y"* ^"^l a;" — y\ Ans. cc^"'+" — a:"^^'" — x'^"'y'' -\- y^"^\ 29. Find the product of a;" + 2a;'"y» + 3^3/^ by x"' — y\ 30. Find the value of (x^ -j- 3/') (a; + y} (x — ?/) in a single polynomial. 31. Find the value of (x*-^y^) (^^4-^) (x + y) Q^—y)- 32. Find the value of (x^ — xy-\- y^) (x^ -f xy -j- y^) (x + y) (p^—y)' 33. Find the value of (^c^ 4. 5^; -f 7) (a;^ _ 5x + 7). 34. Find the value of (a:^ -f 6a; + 18) (a;^ — Gx -\- 18). 35. Find the value of (a;^ _[_ 4x + 8) (x^ — 4x + 8). 36. Find the value of (a;^ + 2aa; + 2a^) (x^ —- 2ax + 2«2-)^ 37. Find the value of (ar + 3) (x + 4) (x — 5) (;« — 6). 38. Find the value of (a; — 5) (x - 6) (c--4). MULTIPLICATION. 39 39. Multiply x-{- ^ by x -\- ^, x — 5 by a: — 5, x-\-7 byic-f4, and X -{- 7 by x — 4. 62. Since (x + y) (x -\- y) =^ (x -^ y)^ = a;' -f 2xy -\- y\ it fol- lows that The sqtiare of the sum of two quantities is equal to the square of the first + twice their product -f the square of the last, EXAMPLES. 1. (a -\-iy =^ a" -{-^ah -\-h\ 2. (2a + If = 4a2 + 4ah + h\ 3. (a + 26)2 = a2 + 4a& 4- 46^ 4. (3a 4- 2hy = 9a2 + 12a6 + 4b\ 5. (x^ + y^y = x-\- 2x^^y^ + y, 6. (x^ -^-y^y^x^ -{-2x^y^ -\-y^, 7. (2x» + 3x0' = 43;'° + 12x» + 9x». 8. (x^ 4- 5)2 = x* + 10x2 + 25. 9. Find the value of (x^ 4-^)2, (x2 4-^2)2, (x'4-x)2, (x' 4- x)«, (a;3 _|_ y\y^ (x 4- 4)^ (x2 4- x')2 and (x» 4- x^', and the numerical value when x = 8, y = 8. 63. Since (x—y) (x — ^) = (x — y)'= x' — 2xy4-/, it follows that The square of the difference of two quantities is equal to the square of the first — twice their product -\- the square of the last. EXAMPLES. 1. (a — hy^a'^ — 2a'b-\-lK 2. (2a — t)' = 4a' — 4a6 4- &2. 3. (a — 25)2 = a2 — 4a6 4- 4Z>2 4. (x — 2)2 = x2 — 4x 4- 4. 5. (1 — 4x2)2 = 1 __ 8x2 _|_ i6x^ 6. (5 - 4)2 = 25 - 40 4- 16 = 1. 40 ^I U L T I P L I C A T ION. 7. (1 — 1)2 = 1 — 2 + 1 = 0. 8. (ox^ — xy = 25x — 10a;2 -j- xK 9. Find the numerical value of (x — 3)^, (x- — 5)', (1 — xy, (3^2 — 2^3)2, (4x3 _ 2xy, (px^ — 2y)2 and (x — 4)- when X=4:, 7J=1. 64. Since (^ + y) (^ — y) = ^^ — 3/^ i^ follows that T/ie product of the sum and difference of two quantities is equal to the difference of their squares. EXAMPLES. 1. (rt-f-Z>) (a — h) = a^ — l\ 2. (2a + h) (2a — h) = 4a2 — h^ 3. (03 + 4) (x — 4) = a;2 — 16. 4. (x + 5) (x — 5) = x2 — 25. 5. (5 + 2) (5 — 2) = (25 — 4) = 7 X 3 = 21. 6. (4 J- _j_ 4?^t ) (4x^ — 4?/^ ) = 1 6x3 _ I Q^i^ 7. (3x^ + 22/2- ) (3^i _ 2/ ) = 9x — 4y. / 8. (0)^' + /) (x^-/) = x-7/. 9. Find the numerical value of (x^-\-y^) (x^ — y'), (^'+y) (x' — 3/') and (x*-\-y^) (x^ — ^0 when x = 5, y = 5. 65. Since (x + ?/ + 21)2 = x^ + / + 2^ + 2xy + 2x.^ + 23^2;, it follows that The square of a trinomial is equal to the sum of the squares of the three terms united to twice the j^roduct of the terms taken two and tivo. (For the signs observe 59.) EXAMPLES. 1. (a + & + c)2 = a2 ^ 2,2 _|_ c2 _|, 2al + 2ac + 2hc. 2. (a; + 3/ — 2)2 = x2 + y2 _|_ ;j2 _^ 2x3/ — 2o;;2 — 23/^. 3. (-3,_y_^)2^^2 _|_^2 _|_ ^2 _ 2.xy _ 2X^ + 23/-^. 4. (a;2 _|_ a; _}_ 1)2 =: a;4 ^ a;2 _^ 1 _|_ 2x3 _|_ 2x2 _^ 2x = x^ + 2x' + 3x2 + 2x + 1. IM U L T 1 P L I C A T ION. 41 5. (5x2 _ 4a; _j_ 3)2 = 25x^ — 40^^ + 46^^ — 24.c + 0. 6. (3x' 4- 4x2 _j_ 5.,.-) 2 ^ 9,^6 _|_ 24.^:5 _i_ 4Cx^ + 40x^ + 2ox\ 7. Find the value of (Ox^ — 6x — 2)^, (x^ -f y^ _|_ ^y^2^ ^^_|_2^ -\-^z)^ and the numerical value when x = 4, ^=9, 2= 16. 8. Find the values of (^x -\- 4:i/)^, (3x — 4j/)2, (3x + 4y) (3x — 4y), and (1 + 3x — 4?/)^, and the numerical values when X 9 2,^=1. 66. 1. Multiply (x — a), (x — 6) and (x — c) together. (^Vkle Def. 12.) X — a X — h X + ah X? — a — h X — c x^ — a x^ + «c X - - (the ^h 4- he — c -\-ah Ans. 2. AYhat would be the answer to the last example when a = 1, h=2, c = 3 ? Ajis. x' — Gx' + llx — 6. 3. AVhat would be the value Avhen a =2, h = 3, c = 4? Ans. x' — 9x= + 26x — 24. 4. Find ihe product of (x — a) (x — h) (x — c) (x — d), and what will it become when «=1, h == 2, c = 3, f/ = 4 ; also when a = 3, ^> = 4, c = 5, (/=6. (Vide 61, ex. 11.) ^l«s. to last x' — ISx' + 119x= — 342x + 360. 42 DIVISION. DIVISION. 6t. Division in Algebra is the operation o? finding a quotient which multiplied into a given divisor will produce a given dividend. 6S. To divide one monomial by another : By 56 we have 7a^6 X 5ax = 35a' tx .*. ^ha^hx -~7a?b = bax. By 5t we have 4a'Z> X — 2a'Z;r= —^a'h^ .-. —8a*?*' -^4 i 1 2 1 Also, (example 10) 3.x'^y~' X Sx'^y = ISx'^"' .*. l^x'^ y~^ 1 1 -T- 5x'^y = 3x^?/~% Hence, (1.) Divide the coefficient of the dividend hy that of the divisor ^ and annex all the letters, giving to each an eoqjonent found hy subtracting the expo7ient of a letter in the divisor from the exponent of the same letter in the dividend. (2.) If the dividend and divisor have like signs, the quotient is iilus. (3.) If the dividend and divisor have unlike signs, the quo- tient is minus. (4.) If the divisor contains letters not found in the divi- dend, these letters may be inserted in the dividend by giv- ing to each the exponent 0. (Vide Def. 13, 9.) Thus, 4x2 = 4a»?>»c"x^ ahc a h c EXAMPLES. 1. a a 1. 2. 2 a' a^ ■— a ■=■ - = a. a 3 3 . '^ 2 (Vide Def. 13, 9.) DIVISION. 43 4 9 a 1 1 «2 a « (, 5axy y . i . aa; -T-axy = -^=- = y . •^ axy y ^ 6. 10a2x'-^5a-•ic^ = 2a^x-^ 7. — lOOxV^ -T- — SOrcy ^2 = 2a;->^-'2-'. 8. 7xi/-^6?n=:'^-^ = lxi/m-K 9. 21a3Z;5_|_7a2Z>4^ 57a^62_^19a5, l^aHx-^3a, S^a^h^x'' -^7a. 10. 27a^Pc'-^9ac, —33x'f-i-llx7/, — 42a:y -r- 21xy, 4:bx* y^ H- ISary. 11. 7x^y^-T- — 3^xy, —20x^y-^^xy\ 3a?-~2h, 7ahnn-^3aK 12. — 26a;* 3^ -. l^xy, 16xy h- 8xy, — 30xy^ -i- IS.-cy, 72x^y^ -~ 36x'*y^. Ill 111 111 13. x^y^z^ -T-x^y^z^ == x^y^z*^, I t 1 111 i.i_i 14. lOiC^J^^ Z^ -T- 5x6^3 23 _ 2x30^3 g;- 1 2 15. --6lxy'-i--2lxy* = 3y. 16. — 7|rc*3/ -7- — 2-J^xy = Ba;^ 17. 729a;^ -T- — 27ic^^^, —22^x^y^-i- — lbx~^y^, —12ixhji 18. 441a;2y -^ — 21ic/, 361x'y^ -r- — 19a;«/, ISary -i- l^xyz, 69. To divide a polynomial by a monomial: Divide each term of the dividend hy the divisor, according to 68. EXAMPLES. 1. Divide x^ + ^^y + v'^ ^J ^- -^*^s. cc -f- 2y 4- 3/^-'^~'» 2. Divide 3aa;^ + Sa&ic — 9x' by 3a;. -4?is. ax + 2a6 — 3a;^. 3. Divide xy -\- xz -\-yz by xyz. Ans. z~^ -{■ y~^ -\- x~\ 4. Divide xyz -\- xyw + 0:21^ -j- yzw by ccy^zy. 5. Divide — 6x* + SOoj' — 9aa;' -f- Qirix — 3nx by — 3.T. 6. Divide 12a^ + 24a!' — ^Qa" by 12a. Ans, a^-^ + 2ay-^ — 3a'-\ 7. Divide 4:x\x -\- yy -~Sx\x-\- yy -{- 12x\x+ y)' by 4:xXx+yy. Ans, 1 — 2x (x+y') -f Sx^ (re + yy. 44 D I V 1 tj I o X . 8. Divide 7x^ (x + 3^ -j- .) + 4.r ^ (.r + y + ~) by 3x^ (x +y+ 2;). Ans. 2^ + f.'ci •iO. To divide one polynomial by another: (1.) Arrange the polynomials uith reference to the ascendvig or descending poicers of the same Utter, (2.) Divide the first term of the dividend hy the first term of the divisor for the first teim of the quotie?it. Multiply the whole divisor hy this term of the quotient, and subtract the 2)roduct from the dividend for the first remainder. (3.) Z^pon the first remainder repeat the very same operation as vpon the given dividend, for the second re7nainder, and so on till there is no remainder ; or, if the division he not exact, till the first term of the remainder divided hy the first term of the divisor ivould produce a negative exp)on€nt in the quotient. In this case p)lace the remainder over the divisor at the Hght of the quotient, prefixing the j^^^ojJer sign. (4.) It often happens that the division may be carried on indefinitely, giving rise to an infinite seizes, in "which case a few of the leading terms of the quotient will generally be sufficient to determine the rest. EXAMPLES. 1. Divide x^ -f --^j/ + !/^ by x + y- Solution. Dividend x^ + 2xy -j- y'' \ x -j- y = Divisor. (Vide Def. 11.) x^ + xy X -}- y = Quotient, ^'^ + f Kiplanation. The first term, x^, of the dividend is divided by x, the first term of the divisor, and we obtain x, the fir^t tcnn of the quo- DIVISION. 45 tient. Next, multiply the divisor x ~{- y by x, and we obtain the product x^ -\- ccy, which, subtracted from the dividend, gives xy + y"^ for the remainder. Divide its first term xy by £C, and we obtain y for the second term of the quotient. Multiply x-\-y by y, and we have xy -f- y^, which being taken from the remain- der leaves nothing, the quotient being exact. (Vide 62 & 61.) 2. Divide cc^ — 2xy -{• y^ by x — y. Ans. x — y. 3. Divide x^ + '^x^y + Zxy^ + y^ by x^ -\- 2xy -f y^. Ans. x -{- y. 4. Divide x^ — 3x^y ~\- Sxy^ — y^ by x — y. Ans. x^ — 2xy -f y^. 5. Divide x* + 'ix^y + 6x^y^ + 4x/ + ^ by x^ + 2xj/ + y\ Ans. x^ -\- 2xy -j- y^. 6. Divide x"^ — S.-c' — SGx^ — Tla? — 21 by x^ — 8x — 3. Dividend, tc^ — 3x3 _ 3(3^2 _ ^i^ _ 2I | x^ — 8x — 3 Divisor. ic^ + 5x + 7 Quotient. tc^ — 8x3 . -3x2 5x3 — 33x2 — 71x — 21 5x3 -40x2-15x 7x2 _ 5(3^ _ 21 7a;2 _ 56x - 21 7. Divide x3 — x^ -j- 3x 4-5 by x + 1. Ans. x^ — 2x -f 5. 8. Divide 4x« — 25x2 ^ oQx — 4 by 2x3 — 5x + 2. Ans. 2x3 _|. 5,^ _ 2. 9. Divide 2x3 _ 7^2 _ ig^ _|_ 5 by x — 5. Ans. 2x^ + 3x — 1. 10. Divide x^ — 3x^y^ -j- Sx'y* — y^ by x' — 3x2y -j- 3xy^ — y^. 46 DIVISION. Operation. x^ — ?>x'^y'^ -\- Sx^y^ — y^ I x^ — 3x^y -j- 3xy^ — y^ x^ — 3ic*y + ^x^y^ — x^y^ x? + 3x^j/ -f ^ixy"^ -\- y^ Zx^y — Gx"* 2/^ 4" ^y^ + Sx^^** — y^ Zx'^y^ — -8x-^y _|_ 6.xy — 3/« Zx^y"" - - 9x'^3 _^ 9^2y _ 3^^5 x^y^ — 3x^3^^ 4- 3x^^ — y* x^y^ — 3x^y + 3xy' — y^ 11. Divide x^ — 2x^y^ -\- y"" by x^ — 2xy + y"^, Anu. X? -f 2xy 4- 3/^. 12. Divide x^ — ^x^y^ + 6x^ y* — 4x^^^ + ^^ by x" + 4x'y + 6x* 2/^ + 4x2/' + 3/''. ^7is. x"* — 4x^y -f- 6x'^^ — 4x^' + y^, 13. Divide x^ + x^^^ + ^"^ by x* + xy + ?/^ J.?is. x* — xy + y^, i ^ , , ^ 1 14. Divide x + 2x^3/^ _ ^2^2 _j_ y j^y ^^2 __ ^^ _j. ^2^ Operation, 11 11 — x*3/' + ^ + 2x2^2 _j_ y — icy _|_ aj2 _|. y2 33 11- — x^y^ + '^^y + ^3/^ ^y -\- ^^ ~\- y^ i 3 1 j — x^y — xy^ -^ X -}- 2x2^2 _|_ y 3 11 — x^y + a^ + -'^^3/^ 3 1 1 -X3^2 + X2^2 _^ y 3 1 1 ~X^2 + x^y^-i-y 211 11 15. Divide x — x^ by x^ + x^. ^?js. x^ — x', 16. Divide x — x^ + 2x^2 — .x^ by x^ + x^ — xl i 1 1 J.?2S. X^ — X^ -f- X^. 17. Divide x — l^^ajg^ ^x* — fx^ — |xi2 — sx^ by x^ — 5x5 — 3x^. .Ill Ans. x^ -\- |x3 -}- ix'^ DIVISION. 47 1 J 1 A i. x^-\-if — z\ 18. Divide X 2 111 1 _ ^3 _}_ 2^3 2;^ — z- by x^ — 1 ^3 1 19. Divide x' + 5x + 7 by X — 3. Oj~teration, a;2 + 5x + 7 .c — 3 ic2 _ 3a; a; + 8 + 31 z-3 8x + 7 8a; - 24 31 14 20. Divide x' — 6a; — 41 by a; + 3. Ans. a; — 9 — — _^. 21. Divide a;' + 4a;2 + 5a; + 6 by a;' + 3a; — 1. 3x + 7 ^«S. X + 1 +^2 + 3^_i 22. Divide 1 + 3/' by 1 + a;. l_px l__aj_|_ic' — aj'+ &c. — a; + 3/2 — a; — a;' x^-^y^ x^ + a;' -a;' -a;' 23. Divide 1 + 2a; + Sx' by 1 — 4x. Ans. 1 + 6x + 27x2 + lOSx' + 432x^ + &c. 24. Divide 1 — x by 1 + x. Ans, 1 — 2x + 2x2 _ o^z ^^^ 25. Divide 1 — x by 1 + x + x^. Ans. 1 - 2x 4- a;2 + a;' - 2x'' + x* + x« - 2x^ &c.. 26. Divide 1 + 2x by 1 — x — x^. =«-«^ • A71S. 1 + 3X + 4x2 _|, 7^3 _^ ^c. 48 DIVISION. 27. Divide x^" -f x'-'f" -f- i/" by x^" + .T"y -f ?,^'\ Ojjej'alion. ^4rt _j_ ^^2,1 ^2» _|_ ^An I _^2« _^ ^n ^n j^ yin y _J_ ic^^y^ -f- a;2" ?/2,i rf.%1 _ rf-nyn _j_ ^2,1 X _^ ^3}i o.ji ___ /y,2/i o/2;i ^__ ^,»i j/3/i 28. Divide x""^ + 2x2"* y' _|_ ^m+i^p __ ^^^^n __ 2.x"' ^^n _ ^^^-^n ^^^ ^m _ ^n^ ^.,^5^ ^n _^ 2x'"^'' + X^/^ " 71. 1. Divide x' — y^ by x — y. Ans. x^ -\- xy -f ?/*, 2. Divide x^ — y"^ bj x — y. ^7i5. x-\-y. 3. Divide x' -f- ?/^ by x -\- y. Ans. x^ — xy + y^ 4. Divide x^ — y^ by x + y. A?is. x^y 5. Divide x^ -\- y"^ by x — y. Ans. x -\- y -\ ^^ 6. Divide rc^ -{■ y^ '^J x—y. Ans. x^ -\- xy -\- y"^ -{- x — y 2y3 x — y 22/ 7. Di\ide x^ — ^'^ by x + y. Ans. x^ — xy -\- y^ ^ "^ x + y 8. Divide x^ -\- y^ by x + y. Ans. x — y ~\. -^ x + y (1.) For the present we may infer from 1 and 2 that the difference of two quantities ivill divide the difference of the like powers of those quantities, tvithout a remainder. (^Vide I'S.) (2.) From 3 and 4, the sum of two quantities will divide the sum of the like odd or difference of the like even powers of those quantities, loithout a remainder. {Vide ^9,") ' (3.) From 5 and 6, the difference of two quantities will not divide the sum of the like powers of those quantities, witliout a remainder. DIVISION. 49 (4.) From 7 and 8 the sum of two quantities will not divide the differe7ice of the like odd or the sum of the like even powers of those quantities, without a remainder. ' "ys. All the above cases may be solved mentally by observing the following directions: (1.) The exponent of the leading letter aj decreases by one reg- ularly, and this letter disappears in the last term. (2.) The exponent of y increases by one regularly from the second term. (3.) If the divisor contain the negative sign, then the terms of the quotient will all be positive. (4.) If both terms of the divisor are positive, then the odd terms of the quotient are positive, tlie even terms negative, (5.) In the cases which have a remainder, this remainder will always be twice the given power of y, retaining its sign. EXAMPLES. 1. Divide x^ — y^ by x — y. Ans, x* -\- x?y -{• x? y"^ -\- xy^ -f 3/^« 2. Divide x* -- y^ by x — y. Ans. x^ -\- x'^y -\- xy"^ -j- ?/'. 3. Divide x^ -{■ y^ by x -\- y. Ans, x^ — x^y + ^^j/^ — ^y^ + y^. 4. Divide re'' — y^ by x -\- y. Ans. x^ — x^y + ^y"^ — y^. 5. Divide x'^ -\- y^ by x — y, Ans. cc' + x^y + xy^ + j/^ + ~~- • X y 6. Divide x^ -]- y^ hy x — y. Ans. x^ + x^y -\- x-y'^ + ^^^ + 3''' + — ^ • 7. Divide x^ — 3/^ by x + y. Ans. x^ — x^y + x^y^ — xy^ -\- y^ — ^ 8. Divide x^ -\- y^ by x -f- y. Ans. x^ — x'^y + xy"^ — ^^ + 9. Find the value of x + y 2yt x + y X — y x^ — y^ x' — y^ x^ — y^ '^ -\- y X y' X -- y' X — y' x—y' x -\- y ^1±J^ and '^±^. ^-\- y ^ + y 50 DIVISION. 10. Find the value of -^, -^, ^, ^, —T-J-~ '^-\- y •^' + ^ •'^' + ^ ^ + y ^—y and . X ■\- y 11. Divide x"' — y"^ by x—y, Ans. x'''~^ -\- x"'-'^ y -\- x'^'-^y^ &c. . . ^. , , . ^ .-«' — 1 .-c' — 1 0^5 + 32 x' — 2'i 12. Find the vahie ot — , — , ^ , ^, cc — 1 X — 1 X -{- 2 X — 3 , .7;* 4- 2-43 , 81rc< — icy / t ^ ,^ and -^r — ^. f .x + ^ ^^ — ^y To find the answers of examples 12, make in 11, ?/i = 2, 3/ = 1, m = 3, y =1; m = 5, ?/ = 2 ; m = 3, 3/ = 3 ; ?7z = 5, ?/ = 3 ; »?i = 4, and place in the last 3.^; for x and 2y for ?/, using as many terms of 11 as is indicated by the value of m, Ans, to the last 27rc^ + ISx^y + 12xy^ + 8/. 13. Divide a;'"" — ^y*^" by x^ — y\ 14. Find the value of —. ^, ~ -.,—, -, and —^ V- ^ — ^ ^ — ;^ ^ — y ^ — ^ 15. If in ex. 1 cc = y, what does the answer become % Ans. 5x^ or 5^"*, 16. If in ex. 2 a? = y, what does the answer become? Ans. 4x^ or 4^^ 17. If in ex. 11 cc = y, what does the answer become*? Ans. mx^"~^ or wy^^^. 18. If in ex. 16 and 17 £c = 5 and m = 4, what will they be- come? Ans. 500. >^hri'C. fj C> ei t ^ ^' CHAPTER III. FACTORING — GREATEST COMMON DIVISOR — LEAST COMMON MULTIPLE. FACTORING. 73. Factoring is the operation of resolving a quantity into factors. 1. A composite quantity is one whicli may be resolved into factors. 2. A ^)?T?«e factor cannot be resolved into other factors. 3. All the cases of factoring merely reverse the operations of multiplication. 74. To separate a monomial into its prime factors : Separate the coefficient into its prime factors, and annex the literal part, also resolved. EXAMPLES. 1. Find the factors of 12a'^hx. Ans. 2. 2. 3. a. a. h. x. 2. Find the factors of lQa\ l^x^y, IVIx^y^ and 133a;2/. 75. To factor a polynomial when multiplied into a monomial : Divide the polynomial by tJie monomial common to all the tei^ns. The divisor and quotient are the factors. (^Vide 60 and 12.) EXAMPLES. 1. Find the factors of a;^ -f ^^« ^'^^' G'^ + ^) ^« 2. Find the factors o^ xy -{- y-. yins. (^ -h I/) !/• 52 r A C T O 11 1 N G . 3. Find the factors of ax -j- Ijx. Ans. Qi -|- l>) x. 4. Factor ax + ^^ + c.t, a^x^ + Ir x', 5x-j- 2t^.x and ox^ ij -\- Zxip, 5. Factor x^y — x^}f- + x^f^ x^ y^ -\- x^ if and 5ax — 26.T -|- ''^'« 6. Factor .x + ax — 2Jjx^ 6hx + .x — ax and 3orx -j- GZjx — 12cx, 7. Find the factors of x™-'y ^3/"*. ^«s. (x""^' — y""'Oy* 8. Find the factors of x -\- y -{- (I -\- m) (x -\- y). Ans. (1 -|- Z -|- ju) (x -|- y)' 9. (x + a) (x + Z>) = x^ + (a + Z>) X + a6. 10. (x — a) (x — 6) = x^ — (^a-{-h)x-\- al. 11. (x — a) (x -[- &) = x^ — (a — ?>) X — a&. 12. (x -j- «) (^ — ^) = ''^^ H~ (^ — ^) ^ — ^^• •ye. To factor expressions of the form x"^ -}- 2xy -\- y^ : Take the square root of the extreme terms, and place the sign between the roots, that is, hefore the middle term. The result is one of the equal factors. (^Vide 62 and 63.) EXAJIPLES. 1. Find the factors of x' -f 2xy + yl Ans. (x-\-y') (x-\-y). 2. Find the factors of x' -\- 2x^y''-^y\ Ans. {x^-\-y^) (0:= + ^=^). 3. Factor 64x'« — 96x» + 36x^ 1 + 2x + x^, 1 _ 2x + x^ and 1 — 2X3 _^ a;3. 4. Factor 1 + '-^'+^, ^'-2 + 4> 1 + ^' + 1^^' ^nd 1 + 1 + -^. 5. Factor x + 2x^y^ -\- y, x^ — 2x^y'^ -]- y\ 25x — 20x2 -J- 4x' and 2 4 x3 — 2x -]- x\ 6. Factor 1 + - + ^, x^ — Sx+^f, a;^ -f- 14x + 49 and x' — x^ |X + ^ 64 • '1'^. To factor expressions of the form x^ — y": Indicate the product of tlie sum and the difference if the square roots of the quaiitilics. {Vide 64.) L F A C T O III N G . 53 EX A:\irLES. 1. Factor c'^ — ?/'. Ans. (x -{- 1/) (x — t/). 2. Factor dx^~—4i/K Ans. (3x -f 2^/) (?jx — 2>/) 3. Factor 1 — 4a;2, l — 9x% 4 — 1 63/2, Ox'^ — 4i/^ and 1—-. 4. Factor x'^ — a;l u4^?s. a:2 (a:2 _ 1) = a;2 (.x -f 1) (.t — 1). 5. Factor x'^ — y. Ans. (x' + 7/2) (^2 _ j/2) = (.^2 _^ ^.^) (-^ j^ y-^ {x-y). 6. Factor x^— /, x'^—y'% x^'—f% 16x^ — 16^ and IGa:" — 81^. 22 11 11 7. Factor cc — y, x^ — y^, 4^2 — y, a:^ — y^ r^j-,^] ^3 — 7^3^ 8. Factor (x-^-jiy — cj^. A7is, (x -}- j) -\- q) (x -\- ]) — q). •^S. To factor expressions of the form x"" — y'", m being any- positive whole number : /y-m o.TO f. i Ans. axy^ — ^•'^^^' I 5. Find the least common multiple of a — h, a^ — h^, a -\- h and X. Ans. a'^x — h'^x. 6. Find the least common multiple of 2a' — 26^, Sa^ -\- 'd>ah -{■ i 3^^ and 5.T. Ans. SOa^a; — 306'.c. j 86. To find the least common multiple of two quantities ; which cannot be readily factored : j Find their greatest divisor; divide one of tue quantities hj it, and multiply the other ly the quotient, , i EXAMPLES. I 1. Find the least common, multiple of x^ -j- ^'^ -\- 2.x — 4 and x^ — 8. {Vide 84, ex. 3.) Ans. x' — x' — Sx + 8. 2. Find the least common multiple of x^ — 8.x^ + 21.t — 30 and X* — 3x' + ^^^ — 3x4- 6. Ans. x' — Sx' + 22x3 — 38^2 + 21x — 30. 3. Find the least common multiple of 3x^ -f Sx^ -{- 8x -}- d and 2x* + 2x' — 5^2 — 7x — 7. Ans. Gx^ + ISx'' — ox^ — 46^2 — 56x — 35. 4. Find the least common multiple of x'' — ox^ -f- Ox^ — Tx -f 2 and 4x3 _ i5^^2 _l I8x — 7. Ans. 4x' — 27x^ + 71x3 _ 91^2 ^ 57^ _ ^4^ GENERAL REVIEW. 8f. The pupil should not be allowed to proceed to the sub- ject of the fractions till he can easily manage the following EXAMPLES. 1. Add ax to Ix. Ans. (ciA-h^x. 2. Add x^ -f xy to y^ -\- xy. Ans. (x -f yy. 60 GENERAL EEVIEW. 3. Add (x -{- 7/y to —1}f — 'lxy, Am, 0^ + 3/) (•'^ — 3/)« 4. Add x^ — lOx^ — 20 to 3x3 + 12. Am. {x + 1) (x" — x^X){x — 2) (.X- + 2x + 4.) 5. From 1x^ + 4x7/ — 5^/^ take x^ -f 6xj/ — 63^2. ^7js. (x — iff. 6. From the sum of 2a — 2x4-^? 3a — 3x4- 2?/ and 5a — 5x — y take 4a — 4x + y — 2. ^7zs. 6 (a — x) -}- 2/ -f 2. 7. Find the product of a^ + 3 (x + 1) by x — 1. -4«s. a^x — o? -\- o {x? — 1). 8. Multiply 3a' + 3a^x by a — x. Am. ?>a? {ct? — x^). 9. Multiply x^ + hx by x -\- h, Ans. x (x -j- ^)^. 10. Multiply x2 4- x^y^ -\- if by x^ — if, 1 1 Ans,. X — y -\- xy^ — x'^y, 11. Divide 1 4- 2x by 1 — x — x^ Ans. 1 4- 3x 4- 4x2 4- 7^3 _^ 11^4 ^^^^ 12. Divide x by 1 4- ^ + ^'^' Ans. X (1 — X 4- ^^ — x"* 4- •'^^ — x^ 4" ^^ *^c. 13. Divide 1 4- x by 1 — 2x 4- x^. Ans. 1 4- 3x 4- 5x2 _j. 7.^3 _|. 9^^^ ^^^ 1 i i_ 1 14. -Divide x^ -\- xy -\- y- by x — x^?/^ _j_ y^ Ans. x 4- x^y^ -{■ y. 15. Find the factors of x^y 4- 2x'?/2 4- x^'. J.ws. xy (x 4- 5^)^ 16. Find the factors of x^y — xj/'. Ans. xy (x -\- y) (x — ?/), 17. Find the factors of x^y — xy^. Ans. xy (x^ 4- t/S) (x. -{- 7/) (x ^ y). 18. From (a 4- ly take (a — Z>)^ ^77s. 4a6. 19. From x2 (1 4- .x2 4- x") (x^ — 1) take x2 (x' 4- 1) (*' — 1). Ans. 0. 20. From (x* 4- 324) h- (x2 4- Gx 4- 18) take (x — 3) (x — 2). Ans. — X 4- 12. Verify all the above examples when x = 9, y = 4, a = 3, 6 = 1, i. e. insert these numbers in the given example and re- duce ; then insert the numbers in the answers, reduce, and sec if the results ajfree. CHAPTEK IV. TE ACTIONS. 8S. An Algebraic Fraction represents tbe quotient of one quantity divided by another. Thus, -^ is a fraction, of which a is the numerator and 5 the denominator, (1.) An entire quantity is one not involving a fraction. (2.) A mixed quantity is one uniting an entire and a fractional quantity. Thus, cc -f | is a mixed quantity. (3.) A complex fraction is one whose numerator or denomina- a + 6 tor contains a fraction. Thus, -— - and -i=— are complex frac- z + w X tions, and also y . 89. All the propositions in arithmetic in relation to fractions are applicable to algebraic fractions. (1.) The value of a fraction is not changed by multiplying or dividing both terms by the same quantity. (2.) The value of a fraction is multiplied when the numerator is multiplied or denominator divided. (3.) The value of a fraction is divided when the numerator is divided or denominator multiplied. Also tlie following : (4.) The value of a fraction is not changed by changing all — X' the sijrns. Thus, — = = x, and — = — = — x, and o ■ X — X ' X — X ' „o o 2 o 35' — y- y — X- , = r=z X -\- y. 62 F E A C T I N S . (5.) When ii sign is placed before a ffiictlon with a poly- nomial numerator, the sign does not belong to the first term of the numerator but to the whole fraction. Observe, in reducing, the rules of addition and subtraction. ,,,, , — a-\-b — c — a-{-h — c , . — a-\-b — c a — fc + c ihus, A -. = z : but = — ~ — ■, ' ' 5 ' 6 5 90. (i.) To reduce a fraction to its lowest terms: Divide the numerator and denominator hj their greatest common divisor, or cancel the factors common to the riumerator and denominator. EXAMPLES. 1 i OC' V m or, 1. Eeduce -— — to its lowest terms. An^. -— . 85.T y m 5?/ 2. Eeduce -—: — r^, and — — to their lowest terms. Ailxifz^ xyz \\)'i)xyz S.'z:^ -\- 7ax . - . 5.T. -f- 7a 3. lleduce — , _ „ to its lowest terms. Ans. . ox -[- 5^2 3 -[- Dx Sx^ -\- lC)x?y x^ — ?/^ x^ — y"^ 24:X^ -j- i^2x^y ' x'^ -j- 2xy -{-• y^ x^ — 2xy -j- y^ a;2 _ 9x -I- 20 . , ^ x — 5 5. Eeduce —^ -—- to its lowest terms. Ans. — — - . x^ — x — 12 X -{- o a'2_2.x — B5 x^ 4- l^x -4- bQ , x"" — IQ 6. Reduce , , ^ — r-.-^ » -^^-r t-t ^^^ ay^ _|_ 8a; H- 15 ' x^ -{- bx — li x^ — x — 20' x^y — xy^ X? — ?/' o? — y^ 7. Reduce — 0021 T' ~4~^ — 2 2 , "4 ^"^^ ~1 ^4 X y — Zx^y^ -j- xy* x^ -j- X'^y^ -\- y^ x^ — y 27x^ — 04^3 3,^ _ 4^ ^- 1^^'^"^^ SLx^ -1- l^^y'' -I- 2563r^ * ^ "^' 9a;2 _ i2.t3^ -f 16/ ' a? — 4.x x' 4- 2.^^ _ 8.T: — 16 a;^ -f- y\ 0. Reduce .^3 _]_ 6a;2 _j_ 8x ' 3 (r/j2 — 4) ' x' -|- x^'y'' -f- 3/ a;3 _ 8r«2 _L 21.r — 30 , r^; — 5 FRACTIONS. 63 11. Reduce .^ . , ^ „ — — and 2ic'' -j- 2x^ — 5x^ — 7x — 7 rzj^ -j- ox^ — x — 3 ' (Vide §4.) (ii.) To reduce a fraction to an entire or a mixed quantity: Divide the numerator hj the denominator, ivriting the remainder^ if there he any, icith the denominator under it, at tJie right of the quotient, ivith its sign i^rejixed. EXABIPLES. x2x X 1. Keduce — — to a mixed quantity. Ans. x -| . — 2i7x X 2. Reduce — ^-^ — to a mixed quantity. Ans. — 2x — -— -. • 13 x'-i- ax X x" — «= 3. Reduce to an entire quantity. Ans. x -f- a. ~ - . . a* 4. Reduce — — to a mixed quantity. Ans. x . X X K ^ . 2)5 — 9CC+20 . ^ . ^ ^ 8 5. Reduce — z . - to a mixed quantity. Ans. 1 . ic' — X — 12 ^ ^ .-c + 3 {Vide 90, (1), 5.) Sx^* — Qa^ 5x2 1 10^2 5^3 _|_ 7^^ 6. Reduce , =- and -— -r ^r — 6x ox 5x' -f- 3x x^ — y^ x^ — y^ x^ -4- y^ x^ — v^ 7. Reduce ^, ^, — -^-^- and ^. X —y x — y x-^y ^ -\- y % x^ ^ xy + y^ x^ + x^y + xy^ + y^ *8. Reduce i i and x + v X -\- x^y^ -\- y "T* ^ ^3 __ 4^, ^3 _^ g^2 ^ 8x , x^ — 2x — 35 9- I^educe -3 + a^2 + 8x' x3-4a; "^^ ^+-8^+15' X^ +7/2 1 4- w' 10. Reduce ^- to a mixed quantity. Ans. x — 1 4- ~, x -\- 1 '' 1 -f a; ^ ^ _ , 1 + V^ x^ — y"^ 1 — y"^ 1 — X 11. Reduce -p^ , —^, ^ and — --. 1 -f ^- 2-' — 1 ^ — 1 \ -\- X 64 F E A C T 1 N S . 12. Iteduce and u> ; + y x—y 13. Keduce --- — , — - — and — .- , ^ , rr* y -^ X X -{- y x^ -\- 6x^ — x — 6 x^ — Sx^ + 21cc — 30 x~b (ill.) To reduce an entire or a mixed quantity to the form of a fraction : (1.) Multiply the entire quantity by the j^^^osed denominator, and the 2^roduci will he the numerator ; or, (2.) Multiply the entire ^xw^ by the denominator of the frac- tional 2)art, and add or subtract the numerator, according to the sign before the fractional part, then place the result over the given denominator. EXAMPLES. 1. Reduce x to a fraction whose denominator is 1, 2, or 7. X 2x 7x X , /. p -. . 4 12a; 2. Reduce x -{- ~— to the form oi a fraction. A7is. -— - X 1 0^ 3. Reduce x — — — to the form of a fraction. Ans. — — >. T, ^ ,1 1 1 . llri;+l ^ llfl^-l 4. Reduce x -{- —- and a; — — - . Ans. — and — a —^ X 5. Reduce a -{- x -{ . Vide 90, (i.) Ans. 2 (a -f a-) a^ 4- a;2 a^ -f x^ x^ 4- iP 6. Reduce a — x -{ ; , a — x — , x -{• y a -\- x a -\- X ^ -\- y 2xy , a;^ + V^ -, 2a;?/ 7. Reduce x -\- y ; — , x—y-\ and x — y -{■ x-\-y x—y x—y a^ I ?/ a^ —— y^ 8, Reduce x^ + xy + :?/^ + - — — ~ ^"^ ^^ ~ xy -f y^ + * — - — . x—y x-\-y FRACTIONS. G5 X f o ...,.,3 •'«'+^^ :2 ,,2 9. Reduce x^ — crij + f—- , , , , .x= — .r^/ H-^/'- a^'+a^^+y^ " -^^-^^iz+i/ 1 + ^ ...... 1 + .3 10. Eeduce 1 — x — ^ , , 1 -f a; + .t^ ^ and 1 + a; 1 — X 1 + re + x^ ^t^ 1 — x 1 + rt + X «■'' — x^ , 11. Reduce tt + 1 , a -^ x , — ^ and ^2 ][g (5^ 29 12. Reduce 5 + - ^_ _,^^ . (F/t?e 90, (i), ex. 6.) ^«s. -^^^^ - 13. Reduce 1 ^-^ — ^Vr ^^^^ ^ H ^ . . x^ -{■ ax — 14 x^ — 4x (iv.) To reduce fractions having different denominators to equivalent fractions having a least common denominator: (1.) Reduce the fractions to their lowest terms, unless they are so given as to admit of no reduction. (2.) F'ind the least common multiple of all the denominators. (3.) Divide this multiple hj the denominator of the first reduced fraction, and multiply the quotient hj the numerator, and. lurite the product over the mtdtiple. Do the same for all the fractions, and the resulting fractions will be tho?e required, EXAMPLES. X 2ax ^ X -\- x"^ . -, , r 1. Reduce -^- , — r and — to equivalent fractions x^ -j- xy X — xy^ X — xy having a least common denominator. Solution. X 2ax X -\- x^ x^ -\- xy^ x^ — o:y^ ' x^ — xy 1 2a \-\-x y x'—y^ x—y D = fractions given. = fractions reduced. 66 r R A C T I O N s . x — y 2a X + x^ -\- y + ^y ^ ' . . , , — -„ , ~ r = tractions required. ^ — y ^ — y ^ — y ■ ^ rc2 4- 4.-^ + 3 x2 + 8.x + 15 2. Eeduce x + 10, — r and — — to equivalent x^ -\- x — ij a;^ — 25 fractions having a least common denominator. Solution. a; + 10 fc2 + 4.T. + 3 ic^ + 8.T + 15 ^ . -T-' x^ + x-^ ^ -^^325— = f-ctions given. ^±25, (^+3)_^l) (^ + 5)0^+3) ^ 1 ' (.c 4- 3) {x - 2) ' (x + 5) (a) - 5) aj+lO rc+1 cc + 3 X — 2 ' £0 — 5 = fractions reduced. ^3 + 3x2 — 60x+ 100 x2 — 4a; — 5_ x''-\-x— Q fractions ^_7x4- 10 ' a;2 _ 7a; + 10 ' x^^^^Tx+TO required. x a? a; Qx 4x 3x 3. Reduce — , — and - . Ans. — - , — — and — r . jj o 4 ij 1-j ii^ 11,1 . ?/,2 a;^ _ xy 4. Reduce - , - and - . Ans. - — , • and . X y z xyz xyz xyz 11 1 X^ X 1 5. Reduce -, -^ and — • Ans, '—, -^ and -z. X XT X^ X"* X* X* (v.) To add fractional quantities together: (1.) Reduce the fractions to the [east common denominator. (2.) Add the numerators, placing the sum over the least common denominator. In mixed quantities, add the entire parts first, to which annex the sum of the fractional parts by the proper sign, EXAMPLES. 1. Add the quantities x -\ ; — , 3a; , —2x-\ . X -\- y x^ — y^ X — y FRACTIONS. 67 Sohdion. The sura of the entire parts is re + 3a; — 2a; = 2x. = fractions given. ic + y ' x^ — if x—y ^""^„ , -^ — -„ , 4 —1 = fractions reduced, (iv.) a^ — ?/'' ic — y^ re — ?/ 2^ 1 2rc -J — : = the sum as above. X — y . XXX T*^ i i iX \- i X . 2. Add the fractions « ? o > 7 and -. JLns. -— = re + — — . (ii.) or or ^ QC/ 3. Find the sum ofrrr- + ^r-7+-^+-H-r. 11 \6 LKi OJ .11, .111, .1 1 4. Find the sum of - + - , also of - -\ 1 — , also oi — h - o • X y X y z X XT . 1 1 , . 2 3 5 5. Find the sum of -= H — r , also of — — | ^ + -^— _ . X x^ X y xy^ x^y rc+1 , CC+ 1 , (x-^\f 6. Find the sum of 5— and r— . Ans. r-^* 12 3 7. Add the fractions — ■ — , -r ~ and . X -\- y x^ — y^ X — y 2(2rc + 3/ + *l) A71S, ,.2 -,,» Q 1 ^ + y 1 ""^y ^ — y XT -\- '^y -\- y ^ — ^ 1 1 11 ' x'^ — y^ rc^ 4" x^y -\- ^y^ -{" y^ x^ -^ y^ x^ — y^ ' 1 1 10. n — + n — • re 4- x-y^ -j- y x — rc^z/^ -|- y ^^ 4re , re + 2 7re + 26 X — 2^x + 3' * 'rc' + .T — 6 19 ^ + ^ _L ^-'^ x — 7 X + o 13. ^±t + -li^. re — D re — 4 68 F K A C T I N S 14, __1 ^ — _| :• J X- -\- X — 6 x"^ — 25 ,^ a;2— 1 cc2 — 36 15. -^— -^- + ;t2 _|. (j,^ _ 7 ^ ./;2 _ 2.^ _ 48 • x^ + Sx 4- "i X- -\- Ax — 5 1(^. - . o . + ^ ^ a;' + a;^ 4- cc 4- 1 ^' — ^^ — ic 4- 1 17. ^ ' 1 ■ — . x^ -\- x^ — X — 1 x^ — x^ -\- X — 1 ^^ X 4- y X — y ^- x — y x 4- y 18. —LA + __^ . 19. — ^ 4 _^iL r.r. X — y X A- y ^. 2x 1 20. -T— ^ + -rTA. 21., -.4- X* — ?/* x^ -\- y^' * 1 — a;^ a; + 1 * 2 3 4 22. ~r ^^l + T— r^ + ^ X y 23. ^^ — — -^ 4- (14 a)) (cc4-y) (1-y) 0-^ + 3^)* 24 _1_ + — L_ + _JL_. 4 (1 ^ ic) ^ 4 (1 ~ :z^) ^ 2 (1 - ic2) 25. Add 7.T + I ^ 51 to ^ 4- 7y - 99. ^o . ,, 3aj 2v 61 2x 3y 9 ^«.. to last -g? + ^^ - g-. (vi.) To subtract one fraction from another: (1.) Reduce the fractions to the least common denominato7\ (2.) Subtract the numerator of the subtrahend from that of the minuend, l^lacing the difference over the least common denom- inator. Mixed quantities may be reduced to a fractional form, or tlic parts subtracted separately. FRACTIONS. 69 EXAMPLES. 1. From 3x -j- ): take x — . Ojyej'ation. X 4:X ... 3x -\ — . X ^ . = quantities given. Tx on -— . -. = quantities reduced, (ii.j A . 35x 2x 33x _ ox Then zr- = -^rr- = ox -\- -— » Ans. 10 10 10 ^ 10 ,^2 _^ 6a; 4. 8 ^ 0)2 + 3a; — 40 x^ — X — 6 ' x^ -{• X — 30 * Operation. (a;+2)(x + 4) (x + 8) (x — 5) (a; + 2) (a: — 3) (x + 6) (a; — 5) 05+4 a;+8 X — 3 a; -j- 6 ^2 _|_ lOx + 24 a:' 4- 5a; — 24 a;2-j- 3x — 18 x'-fSa; — 18 = fractions factored. = factors cancelled. = fractions reduced, (iv.) 5a; + 48 a;^-j- oa; — 18 \ \ y X 1 1 3. From - take -. Ans. . 4. From 3a;-] — take a;-!--, X y xy X y 4x 11 5. From 7x take -rr-. 6. From - take -^. 5 X x^ 7. From — +r tiike . 8. From — ^-^ take — -^ . X — 1 x-\-l X — y ^-rV y. Irom — !— r: take X — 6 a; -|- 3 * 1^ , .a;2_|_4^_12 x2-f2a; — 15 10. I rem „ , , Y7 t'^kc —7-7—^ L-5r • a;^-fOa; — 14 x^ -{- ox — 18 70 F R A C T I N S . 11. Prom ^^^^^ take -^^ZTsS" * 12. From -^^3 take ii=^. 13. Frdm ^^ take 4=4. •"^ "1^ cc — ?/'' re — i X -\- I 14. From — ^ take »— -^J^. ic — 3/ x^ — y^ ^ „ _ 1 , X — 2 15. Irom — — - take X -\-l o? — x-\-\' 16. rrom - + - take . An^, -. y z y z z , ^ __ , hm-\- c 17. I* rom 711 take — ^ . a-\- -. r. ^. a6 4- <^c 4- ^c , 11 .1 • . , 18. I rom --—, take -, - and - respectively. 19. From 263 _i±^ take h' — l^. 3 4 11a: — l(f , „ . 3a: — 5 20. From 3a: -] — — take 2a: -\ — . JLO • (vn.) To multiply fractional quantities: (1 .) Reduce mixed quantities to the form of a fraction, (iii.) (2.) Factor each numerator and denominator, and cancel such factor's as are found in hath numerator and denominator. (3.) Multiply the remaining factors of the num.eratoj'S for a new ?iumeraio7\ and those of the denominator for a new de- nominator. (4.) Reduce the result, if necessai-y, to a mixed quantity. (11.) 1. Multiply a A bv a — X " X -'- X EXAJIPLES. 2 or.2 2"* FRACTIONS. 71 Operation, ffn^ fl^ ^^ a A . ^-_ . = given quantities. a — X x~\- x^ a? (a 4- x^ (a — x) . - « By II. ■ . -^-~ — r — - . = quantities lactorea. a — X X [1 -\- X) Tiien — ^ r- . = result required. X (1 -{-x) Or*^ qj^ rf* 2/ ^ "^ X? — 2,xy -\-y^ "' x^ -\-xy' Operation, x'^ — y^ X — y = given quantities. = quantities factored. x^ — 2xy^y- x" -\- xy (•^ —y)Q^—y) ' ^ (^ H- y) x^ + y^ , y^ . . , Then —^^ = x -\- '— = quantity required. X X a?bx , Q>m^n^ . 2mn 3. Multiply -^^^ by -;jj^ . Ans. — -. 3mn a^ bx ^ x-\-l . x — 1 oj- 4. Multiply — —- by . An^, 1, X -L X — 1~ X 5. Multiply 2 + -^ by 2 — ^ X — 1 x-\-\ ^' 2 + T by — -r 7. 3 , ^ '^, , by -^H:^. ^'-2'' by 4^. 9.-^^,l>y ''+^ a^ — 2xy-\-y'^ x^-^-y"^' ' x? — ?/^ .r^ — ^y-\~y ^^ x^ — y^ x^4-y^ a; + V x^ — 3/^ X. -\- xy -\- y^ x^ — xy -\- y' ' x' — x— 2 rc^-f a; — 20 11. X '■— = x^ — X — 12 x^ — 6.T — 7 2* 72 I' I>- ACTIONS. \ x^-{-2ax — 'Sa^J \ x^ -{- ax — 6a*/* 16. 1+-^11- X 1 -—). \ a — X J \ a-\-x I 18. a — .T-] 4^— X [a — x 4-- • 20. C^' + x + l) X (i-l + l). 21. (. + 1 + 1) (.x-1 + 1). 22. (a + i^) X -"-t-^ = c + t7. a -T- 6» (viii ^ To divifle fractional quantities : (1.) 'Reduce mixed quantities to the form of a fraction, (2.) Invert the divisor^ and then j^roceed as in multiplication, EXAMPLES. 1 -4- X* \ A-x^ 1. Divide l—x — --^ by 1 + a; — re" — ^ ^ ' . 1 -f- ^ 1 — ^ FRACTIONS. 73 Operatioju 1 — a; 1 — X „,, — 2cc2 1 — X 1 — X . Then nr^ X iri:^^ = np^ ^»'- 7ic?/^ 4:Xv^z 21 2. Divide —^ by -4-- ^^^s- T^a- 2^ + 3/ . ^+3^ 'S- 4. a; — y '^ 5. (a — a; + — ^) -- a — a; ]— -). \ a-j-x/ \ a-{- X / a;2_4 ^ a;2 — 9a^ + 14 - a;2_4 1_^2 l4-3x4-2a;2 1 — 3x + 2a^ ^- 1 _|. 5a; _|_ Ga^^ -^ l_j_.^_6x2* ^^- fipi^qTi^a- a^ — x^ ^ a — a; 2,2 a^ -|- 2aa; -\- x^ a -{- x 11. ''' + ^^y + f ^ , ^ + y _ J«.. X -y. cc^ — ;y2 X — 2x1/ -j- ^2 '^■('+jqn)*('-ST)- 7 74 FRACTIONS ( 14. [l — x 1 -i-x ■)-( l-~x l+x'] ) 15. 16. 17. 18. :.(, 1 -{- X / ' \ 1-j-x / l-\-x^\ /, l-l-a;2\ 2xy x^J^xy^-y x + y x + y )*{ x-\-y x-^y )■ x^^-y^' 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. .'»— 3/ X -\- y } ' \x — y x-\-y)'^ >^ 4-y r ( r A71S. X, X-^y X — 7j \ _^ /X^ —y , »^H-3/ - 4- x—y ' x-^yl \x'—y^ x^-\-y ■?)■ /x-}-! __x — l\ ^ x^ \x — 1 x-j-l/ ^ — 1' \x—y x^ — 2xy-\-y^\ I 1 -<4yj5. a; + l' Divide 1 by Divide 1 by Divide 1 by x-—y^ ah -\-hc -\- ac 2abc ah -\- he -\- ac ah -{- he -\- ac 2ahc + / * \x -{- y * x^ 1 ■y' x — yj 1 6* 1 a (--a - (-')• Ans. x^ A- x -\ 1 — - , X x^ F Jl A C T I X S . 75 REVIEW OF FRACTIONS. 91. Perform the operations indicated in the following EXAMPLES. X — 1 , X — 4 X — 7 , llx-^bb . -4«.s. X. x-\-\ X — 1 X — 2 Q^2 11^1 ft "I 9^2 7.y. J^ '-^ ' ^ 3^2 — 11a: 4- 6 ' 2x^ — 7x + 3 ' x -\- ax= h, where a represents 5 and b 30 of the equation in (3.) (5.) An identical equation is one in which both members are alike, or in which either member is the result of operations indicated by the other. Thus, |— ^ = — ^ , and ^1^ = 1 + 2aj -f 2x'' + &c. (6.) An equation is verified when on the substitution of a quantity for x, it is rendered identical. Thus, if for x in the equation x -f 3x = 24, the number 6 is substituted, it becomes G -f- 18 = 24, where 6 is the only number which will render the given equation identical. (7.) The solution of an equation consists in finding the quan- tity which will verify it. This quantity is called a root of the equation. EQUATIONS OF THE FIRST DEGKEE. 77 (8.) The solution of an equation depends upon one or more of the following self-evident propositions, called axioms. 93. AXIOMS. (1.) Quantities which are equal to the same thing are equal to each other. (2.) If to equal quantities equal quantities be added, the sums will be equal. (3.) If from equal quantities equal quantities be subtracted, tlie remainders will be equal. (4.) If equal quantities be multiplied by the same or equal quantities, the products will be equal. (5.) If equal quantities be divided bj the same or equal quan- tities, the quotients will be equal. (6.) If quantities are equal, their like roots are equah (^Vide Def. 13, 3.) (7.) If quantities are equal, their like powers are equal. (^Vide Def. 13, 2.) SOLUTION OF EQUATIONS OF THE FIEST DEGREE CONTAINING ONE UNKNOWN QUANTITY. 94. The object of every change in equations is, ultimately, to make the unknown quantity x constitute the first member, and the known quantities, reduced to their simjolest form, the second member. The equation is then solved. ( Vide 92, (7.) 9.5. To solve an equation of the form ax = h. Since ax = h, by axiom (5) x = - . Hence, Divide the equation by the coefficient of x. EXAMPLES. 1. Solve the equatiofi 2x = 10. 78 EQUATIONS OF THE FIRST DEGREE. Solution. 2x^= 10 (1) = given equation, x = 5 (2) = required equation. Equation (2) is obtained by dividing (1) by the number 2, which is the coefficient of x. The number 5 will verify the given equation, for 2x5 = 10. In the same way, solve and verify the equations 2. Ix = 21. Aiu. X = 3. 8. 12.x = 156. 3. 5x = 25. 9. 13.x = 169. 4. 4.x = 14-1. 10. ax=a?. An&. x = a. 5. 3x = 15. 11. hx = ah -\- IP. Ans. x= a ~\- h, 6. 10x=20. 12. ax = U. Ans. x = ~ a 7. 9x = 729. 13. 2ax = c. Ans. x = ~. la 96. If both members consist of several terms ; Unite these terms (by 33) and then divide hj the coefficient of x. EXAMPLES. 14. Solve the equation 2x -\- 3.x = 30 -f 15. Solution. 2.x + 3.x = 30 -f 15 (1) = given equation. 5.^: = 45 (2) {Vide 33.) .X = 9 (3) = (2) -- 5. Axiom (5.) 2x9 + 3x9 = 30 -f 15 = verification. 15. Solve the equation 5.x — 2.x = 50 — 20. Ans. .x = 10. 16. Solve the equation 5.x + 3.x — 2.x =50 — 40 + 2. Ans. x = 2. 17. Solve the equation 20x — 18.x + 4.x = 100 — 70 + 30. Ans. X = 10. 18. Solve the equation 11a; + 15.x — 10.x = 100 — 10 + 14 — 8. Ans. .X = 6. 19. Solve the equation ax + hx = r + d. EQUATIONS OF THE FIRST DEGREE. 79 Solution, ax -]- hx ■= c -\- d (1) =i given equation, (a + Z)) x = c + cZ (2) {Vide 75, ex. 3.) c -\- d X a -{- b c -4- d - c -{■ d , .... Then, a x , -. + ^ X — —j = c -j- d = verification. a -{- a-\- If a = 2, h = 3, c = 30 and d=lb, then (ex. 14) x=9. If a = 5, Z) = — 2, c = 50 and d= —20, then (ex. 15) x = 10. 20. Solve the equation ax -\- hx — ex = d — e + /'. Ajis. X = . a -\- — c If a = 5, Z> = 3, c = 2, cZ = 50, e = 40 and /= 2, tlien x = 2. If a = 6, Z/ = 4, c = 5, cZ = 40, e = 20 and /= 10, then x =6. X QK, To solve an equation of the form - = Z>. a X Since - = h, by axiom (4) x = ah. Hence, a Multiply the equation by the denominator of the fraction. i:xa:mples. X 1. Solve the equation - = 7. o 1 = ^ .T = 21 11 = 7 Solution. (1) = given equation. (2) = required equation. = verification. Equation (2) is obtained by multiplying (1) by the number 3. In the same manner solve and verify the following. 2, ^=20. 5 3. ^ = 6. Ans. X = 100. Ans. x= 9. 4.- = X«. 5. 3x 15. 80. EQUATIONS OF THE FIRST DEGREE. 6. ^ = 14. Ans.x= IS. 7. ^=18. Ans.x=24, 11.^ 13' — 99 0.^^=39. 1A ^^ 2 10. — -= a^ o 11 ^^^ 11. = a. € 12.^=26. Ans. X = 3(X, ac Ans. X = — p 46 Ans. X = 130( "36" 13. X a-\-h c. Ans. X = (a -\- Jj) c. 98. When there are several fractions : Multiply the equation hy the least cominon multiple of all the denominates, after which proceed as in 96. EXAMPLES. X X 14. Solve the equation - -f - = 10. X X Solution. (1) s= given equation. (2) (3) {Vide 96. j (4) = required equation. = verification. ?>x-\-2.x= 60 bx = 60 .x = 12 12 12 -2- + X = l« Equation (2) is obtained bj multiplying both sides" of (1) by 6, the least common multiple of 2 and 3. 15. bolve the equation - -f — — = 13. Z 6 '± Solution, X 4iX ox Qx -f- 16a: — dx= 156 13a; = 156 (1) = given equation. (2) = (1) X 12. (3) EQUATIONS OF THE FIEST DEGREE. 81 £c = 12 (4) = required equation. 12 4x12 3 X 12 ^_ .- . : — = 13 = verification. 2^3 4 In the same manner solve the equations, 16 ^ + ^ = 23. Ans. x = 20. 4 5 17. x + ^ + -^ = 11. Ans. a; =;= 6. 2 o 18 ? + '-_- = 7. Ans,x= 12. '2^3 4 ^^ 2x 4:X X ^^ . .- ^, 4.x 3a; 2^7 ^ ^ - o- 21 = 15. ^ Ans. X = 3o. 5 7 ^ 35 22. - 4- 6x= 38. yl»5. cc = 6. 23. - + ^ = c. 3 a b Solution. X X - -] — = c (1) = given equation. a b bx 4- ax = «5c (2) = (1) X ah. (a-\-b)x= abc (3) ( Vide 1'5, ex. 3.) X = 7 (4) = required equation. a -{- b If a = 2, b = 3 and c = 10, then (ex. 14) x = 12. If a = 3, 6 = 4 and c = 14, then x = 24. If a = 5, 6 = 6 and c ^ 11, then x = 30. If a = 7, 6 = 4 and c = 22, then x = 56. mx nx . abc 24. Solve the equation f- -r- = c. Ans. x = ; . a b bni -\- an If w = 3, 71 = 2, a = 4, 5 = 5 c^nd c = 23, then (ex. 16) a: = 20. 82 EQUATIONS OF THE FIEST DEGREE. 99. To solve the equation ax -\- cl= c — hx. By axiom (3) we may subtract d from both sides of the given equation ; thus : ax -\- d — d = c — hx — d. (2) In the first member -f d now cancels — d, giving the equation ax = c — hx — d. (3) By axiom (2) we may now add hx to both sides of equation (3) ; thus ; ax -{- hx = c — hx -{- hx — d. (4) In the second member — hx cancels -f- hx, giving the equation ax -\- hx = c — d. (5) If we now compare (5) with the given equation, we see that — hx has been transposed to the first member of (5), its sign being changed to + . Also, -f- d, of the given equation, has been transposed to the second mem.ber of (5), its sign being changed to — . The value of x is now found by 96, and is c — d x = — — . (6) Hence, Tranq)Qse the terms involving x to the first member, changing the signs. Transpose the terms not involving x to the second memher, chang- ing the signs. After this proceed as in 96. EXAMPLES. 1. Solve the equation Ox — 5 = 40. Solution. 9.x — 5 = 40 (1) = given equation. Transpose - 5, •) 9.^ = 40 + 5 (2) and we have j < v. y 9x = 45 (3) by 96. .'^^ = 5 (4) = required {([nation. 2. Solve the equation 10 + 3.t ~ 20 = x -]- 50. EQUATIONS OF THE FIRST DEGREE. 83 Solution. 10 4- ?yx — 20 = a: + 50 (1) 3x — a; = 50 — 10 + 20 (2) 2x = 60 (3) ^ = 30 (4) 3. Solve the equation 15 + re — 20 = 5a; — T.t + 40. Ans. x — 15. 4. Solve the equation I2x — 50 + GO — 15.x = 8x — 70. Ans. X = Tyy . 5. Solve the equation 14a; — 20 + 5 = IG.r -f 25 — 50. Ans. X = 5. (^ — ^ 6. Solve the equation ax -\- c — Ux -\- d. Ans. x = . ^ a — b If a = 20, ?> = 10, c = 60 and d = 80, then x = 2. If a = 5, 6 = 4, c = 7 and c/ = 8, tlien a; = 1. 26 — 5^ 7. Solve the equation a; + oa = 26 — ca-. Ans. x= — ■, 1 -f- c If 6 = 7, a = 1, c = 2, then x = 3. If 6 = 9, a = 2, c = 3, then a; = 2. IGO. Hence, to solve an equation of the first degree Avith one u nkn \A' n qu an ti ty : (1.) Clear the equations of fractions h/j multij^hjing hj the leabt common multiple of all the denominators. (By 9§.) (2.) Transpose the terms involving x to the first memhcr, and those not involving x to the second member. (By 99.) (3.) U7iite the terms of the first vicmher so as to indicate a single coefficient of x (by 96), and reduce the terms of the second memler to as simp)le a form as possible. (4.) Divide the equation by the coefiicicnt of x. EXAMPLES. X a; 4- 1 9 1. Solve the equation - — ^ — - — = - . 84 EQUATIONS OF THE FIEST D i: G li E E . Solution. 2x — X — 1 = 18 (1) Vide S9, (5). ;t== 19 (2) Vermcation -— = -^ . 2 '± ^ 2. Solve the equation a; -j- a.-r — 20 = 3a; + 80. Ans. .t==33^. 3. Solve the equation x — ^^- \- 20 = 4.r — 29. A o u4/«5. ic = 5. Solve the following equations, and verify the result. X X X ^ \ _ X X X X 4. . + - + -+- = 12-. 5. .-------_-lT. 111_1 J^ _1_ J_ ^- 2"^3"^4""x* • 10 x~ 30* ^^ 3a; — 1 ^ X ,x+35 8. 4a; \- - = x -] {- - . Ans. x=^l. Jo bo a;— 1 a;— 2 x ^, 19 9. . ^ _ + _ = 61 --- . .4,,.. . = Co. X — 2 a; — 4 ^ x — 5 10. X — - — r; 1 p — = '<' H -; — • Ans.x=0. x — 7 a; — 3 a; — 4 ^ ^ 9 11. a; + — ^ . p— = 12-—- . Ans. x = 17. 5 4 10 X — 2 .T — 4 ;/; — 5 ^^3 12. X — ^ -— — = 20-— . Ans. x == GO. 3 o 10 x — 2 a; — 3 a; — 4 ^11 13. X ::, z. — = 6 — - . A71S. X = 20. 4 5 20 x — l?y a; — 40 ^, 3a; — 2 14. X h — ^— = 21 + .,. . ^«5. X = 28A. 5 7 3d !=> x-2 a;-4 a; - 3 ^^29 15. X ^ — - = 19 ^TTT • ^ns. X = GO. o o o oU a;— 2 a; — 4 a; — 7 a; + 3 ^19 7^8 2 ^ 56 56 3;7; — 4 a; a; 1 !'• — r^ — '^2'^'\:~2' Ans.x=2, EQUATIONS or THE Fill ST DEGREE. 85 2 bx 4- 1 18. Solve the equation 7 — x — - (2x -f 3) = 6 . Solution. 2 5a; + 1 7 — a? — - (2x -f 3) = 6 — (1) = given equation. 252 — 36a; — 16a; — 24 = 216 — 45x — 9 (2) = (1) x 36. Transpose & unite } rj 9-1 ^o\ and we have J — ^^ — — -J- C^j X = 3 (4) 19. till _|_ !^zd + 4 To; _ 3^ = 68. Am. x = 17. b 2 ' 4- Tx 4- 1^ 20. - (x + 2) = 3 ^-^-. .^>^5. a; = 7. 21. 10 (.+ 1) -6.(^-1) x4/?5. a; = 2. 4 /T.-r — 9\ 4 /a;+ 17\ , , ^1 22- 3.-5 H^j = 5 (-3--) +4- ^- ^ = ^13- fit.'/; 134 , , , 201k — 268 6Ta; , , 13 -^---(--1) = 15 + -1T- ^«^.- = ley 9 1 129 7 24. -(a:.-0-5(6-9^) = -j^. Ans.x=.^.^. 25. | + 5 = |(.. + 2). 26. |(^+1) = |(:^~1). 101. When the unknown quantity is found in all the terms, involved to powers no two of which differ by more than unity, the equation may be divided by x involved to the lowest power, and thus reduced to an equation of the first degree. EXAMPLES. 1. Solve the equation — — x=. — -. X X Divide by x, and we have -t77 — ^ = "^ • ■^^'^- ^ — ^^' ■^ ' 10 oO 86 LITERAL EQUATIONS. 2. Solve the equation ——= x^ — 5a;. Ans. a; = 5— -. . ^^2 , 498.x' 332a;3 — 166^2 3. Solve the equation 27 - x^ -\ -—- = ~ . O J. D Ans. 0} = 6. 92 115 46 , ,^12 4. Solve the equation _ + — = -^-^^ . ^... a: = 10—. 5. Solve the equation — - = 135.^°. Ans. x = 315. 3?/ 7?/2 1 6. Solve the equation y + r- = — A^is. y = -. 1 — by \ — by 6 . — 4v 8y , A 7. Solve the equation — = -— . ^7i5. y = 4 - ^ 3 — 2y 15— 3/ o LITEEAL EQUATIONS. 102. Literal Equations need be verified only by introducing some number which each letter may be made to represent into the given equation, together with the corresponding value of x, EXAJIPLES. 6a^x Sac 3h^x 4:bac 6hc 9h^x 1. Solve the equation ^j- + -3 ^22^= la" + Tl + "ll"- Solution, 12a^x + 33ac — Sh^x = 45ac + 125c + dh^x. 12a^x — 12h^x = 12ac + 12Z>c. (a2 _ 52) re = (a + h) c. c a — h' If a = 5, 5 = 4 and c = 20, then x = 20. Verification : 6.52.20 3.5.20 3. 42. 20_ 45.5.20 6.4.20 9.4^20 ~~n ^ 2 22~ ~ 22 ^ n ^ ~^2~~ If a = 7, /^ = 5 and c = 30, then x = 15. LITEEAL EQUATIONS. 87 cv ^-~ ij a "— -^ "^h 2. Solve the equation — ^ -^ — \- x= b. Ans. x = — . If a =: anythmg and Z> = 8, then cc = 6. If Z> = 12, a: = 9. If & = 20, a; = 15, &c. 3. Solve the equation ■\ — -\ — - = - . a — 6 a^ — 6"* a -\- b x „ a^ — U' Ans. X = If a = 10 and 6 = 4, then cc = 4. * 2a -j- 1 ' If a = 7 and 6 = 2, then a: = 3. . ah X 4. Solve the equation l\a -\- h) x = a -\- — . 9 6 (a + 6) + 4 If a =: 2 and 6 =: 2, then x = - . 9 If a= 1 and 6 = 2, then x = — ^ . 5. Solve the equation 2x -j \ 1 j — = a -f 6 + c. 6a + 86 + 9c Ans. X = — , If a = 11 J 6 = 11 and c = 11, then x = 23. 5 If a =: 1, 6 = 2 and c = 3, then cc = 4 —— . 6. Solve the equation x — = a — - . Z o o T/. o -, 7 r , /^ ^"s. a? = 3 (a 6). If a = 8 and 6 = 5, then x = 9. ^ ^ If a = 3 and 6=2, then x = 3. v. Solve the equation x — a X — 6 x — c 41a + 456 4- 46c ^ + — n- + — ^ + 6 30 -4ws. a; = a 4- 6 + c If a = 1, 6 = 2 and c = 3, then x = Q. 8. Solve the equation X — a^ X — a^c^ X — c"* 4a'' a^c^ c "^ + ~7 5 35~ = ~ 35 + T~ "^ 17i' Ans. X = —^ — . If a = 3 and c = 1, then x = 2, &c. •^-' 88 PKOBLEMS. r, r. ■, , . x — a^^x — ah 7a^ llah bh^ 9. Solve the equation — - — ■ -{ — == . X — 0'' — . Ans. x=z (a -{■ by. If a = 1, Z> = 2, then x= 9, 1 U- X 1 1 1 0. Solve the equation = 1 -f - , Ans. x = 1 — X a ' 2a -J- 1 * If a= 1, 2, 3, 4, &c., then x= ^, |, -] , J, &c. 11. Solve the equation = = 1 -}- - . Ans. x = — -^ — 1 — a X 2a If a= ^, l, ^, ij &c., then a;= 1, 2, 3, 4, &c. 103. PEOBLEMS. (1.) A Problem is a question proposed for solution. (2.) Any algebraic equation can be considered a statement, in algebraic language, of the conditions of some problem. (3.) Any algebraic problem, if properly expressed, can be converted into one or more algebraic equations, called the equa- tions of the proUem. (4.) The problem is solved by solving the equations. ^1^ ^ (5.) The statement of a problem may generally be effected by considering x as the answer sought, and indicating the oper- ations that would actually he perforfned, if the value of x were known, in the verification of the problem. EXAMPLES. 1. One -third of a certain number is 7. What is the number? Let X = the number. Then I = '^» (!•) and X = 21. {Vide 9':', ex. 1.) (2.) Now if we perform the operation on 21 which is indicated on 21 X in equation (1), the result will be verified, -—=7. o r K B L E M s . 89 2. If Charles had twice as many marbles as lie now has, and also three times as many, lie would have as many as John and William together, the former of whom has 30, and the latter half as many. How many has Charles? Let X = his marbles. Then, 2x + Zx = 30 + 15. Whence, x = 9. (T7t/e 96, ex. 14.) Charles, therefore, has 9 marbles, for 2 x 9 -f- 3 x 9 = 30 -f 15. 3. What number is that whose half and third added together make 10? {Vide 9S, ex. 14.) Ans. 12. 4. * Three-fourths of a number added to two-fifths of it make 23. What is the number? {Vide 9S, ex. 16.) Am. 20. 5. *If a number is added to its half and third, the sum will be 11. What is the number? {Vide 9S, 17.) Ans. 6. 6. '•■ If the fourth of a number be subtracted from the sum of its half and third, the result will be 7. What is the number? {Vid^c 9S, 18.) An$. 12. 7. *If one forty-fifth of all the sheep I have be subtracted from tlie sum of two-fifths and four-ninths of them, the result will be 37. How many sheep have I? (T7c/e9S, 19, also 103, 2.) An^. 45. 8. If from nine times a certain number 5 be subtracted, the remainder will be 40. AVhat is the number? (^Vide 99, ex. 1.) Ans. 5. 9. ''If the fourth of a certain number increased by 1 is sub- tracted from half of the same number, the remainder will be |. What is the number? {Vide lOO, ex. 1.) 10. *Four-t]nrds of a number increased by 2 is the same as three halves of the same number increased by 1. What is the number? {Vide iOO, ex. 20.) Ans. 7. o 90 P li O B L E M 8 . 11. *If 5 be added (o the sixth of a number it Avill make the same thing as tlircc-fourths of the number increased by 2. What is the number? (Vide 1®4>, ex. 25.) A/is. 6. 12. *If from a number its half, its third, and three more be subtracted, the remainder will be 1. What is the number? Ans. 24. 13. *The difference between the fifth and sixth of a number is 4. What is the number? '^ — '- = 4. Aiis. 120. 5 6 14. flf from a number we take 2, and divide the remainder X 2 by 11, the quotient will be 6. -— - — = 6. Ans. 68. 15. * The sum of two-thirds and three-fourths of a number is 68. AVhat is the number? Ans. 48. 16. ■]■ If 4 be added to a number, one-third of the sum will be 5. What is the number. Ans. 11. 17. flf 3 be subtracted from a number, two-thirds of the remainder will be 16. What is the number? Ans. 27. 18. * In one flock a man has one-fourth of all his sheep, in another one-sixth, in another one-eighth, in another one-twelfth, and in another 450 sheep. These five flocks are all he has. How many sheep has he, and how many in each flock ? (1) (2.) (3.) (4.) (5.) Ans. 1200 = 300 -f 200 -f 150 + 100 + 450. 19. A certain nujuber added to ten times itself gives 132. What is the number? Ans. 12, 20. A gold watch is worth ten times as much as a silver watch, and both together arc worth $132. AVluit is each watch worth? Ans. $120 and $12. 21. A man paid $74 for a sheep, a cow and an ox. The cow was valued at 12 sheep, and the ox at 2 cows. What was the price of each? Ans. $2, $24, $48. P B O B L E M S . 91 22. A key winds both a gold and a silver watch. The silver watch is worth twelve times the key, and the gold watch t\venty- five times the key. What is the value of each, if all are worth $342? Ans. key $9, silver watch $108, gold watch $225. 23. A man paid $460 for 20 sheep, 5 cows and a yoke of oxen. A cow was valued at 8 sheep, and an ox at 2 cows. What was the price, per head, of each? Ans. $5, $40 and $80. 24. * Three men and two boys work togetlier. The men get a quarter of a dollar per day, the boys one-fifth of a dollar. How many days must they work to receive 23 dollars? (^Vide 98, ex. 16, also 103, (2) and ex. 4.) Ans. 20 days. The pupil will perceive that any equation may be that of an endless variety of problems, but that these problems are only different methods of expressing the sa^ne conditions, as the uniform statemeiit proves. 25. A starts from a given point, and travels at the rate of one mile per hour. After an absence of 12 hours, B starts after him on the same route, at the rate of twelve miles per hour. How long before A will be overtaken, and how far will B have traveled? o i *- boLution. p Let M N represent the road traveled over. Let X = the number of hours required. Since A goes one mile per hour, in 12 hours he will go 12 miles = M P. Since A goes one mile per hour, in x hours he will go x miles = P N. Since B goes 12 miles per hour, in x hours he will go \2x miles = M N. 92 TROBLEMS. Now M N = M P ■{- P N That is 12x = 12 + x Whence x = 1 j'y = 1 Lour 5 /j- minutes. Now B's distance being 12.t, he will have traveled 13 ^'y miles. 26. The hour and minute hands of a clock are together at 12. "When will they be together again ? Ans. 1 hour, 5 J^ min. 27. Two men start from the same point, and travel in the same direction ; the first steps twice as far as the second, but the second makes five steps while the first makes one. At the end of a certain time they are 300 feet apart. How far has eacl traveled? 2|x= 300 + x. Ans. V 200, 2'"^ 500 feet- 28. Two men start from the same point, and travel in ojiposite directions ; the first steps, each time, two-thirds the distance of the second ; but the second makes only 4 steps while the first makes 7. At the end of a certain time they are 520 feet apart. How far has each traveled? A?is. V* 280, 2*^^ 240. 29. A cistern has three pipes. The first will fill it in 2 (li) hours, the second in 3 (3|) hours, the third in 4 (5) hours. In what time will the cistern be filled when the three pipes are running together. ^^^^^^.^^^^ Let X = the time ; then, I r= the part the 1st will fill in one hour. 1 = " " 2nd " " " " 1 = " " 3rd '* " " " 4 L = the part all will fill in one hour. X Hence, (axiom 1) | + | + i=-j whence x =: i| of an hour. (Vide lOO, ex. 6.) 30. Solve the above problem using the numbers in the ( ) . Ans. 48 minutes. 31. A cistern has three pipes, two at the top and one at the bottom. One of the top pipes would fdl it in 5 hours, the other PROBLEMS. 93 in 6; but the pipe at the bottom empties it in 8| hours. In what time will the cistern be filled when the pipes are running together'? Ans. 4 hours. 32. A can do a piece of work in 3 (2|) days, B in 5 (2) days, and C in 7| (8) days. In what time can they, all do it by working together? A?is. lA days. 33. A and B can do a piece of work in 5 (f^) days. A can do it alone in 7 (Ti) dciys. In what time can B do it alone? Ans. 17^ days. 34. A and B can do a piece of work in 5 days ; A and C in 6a days; B and C in 7 days. In what time would all do it by working together? ^ -|- ^ -j- j^ = i. Ans. 4 days. 35. A man and his wife could drink a cask of beer in 10 days. In the absence of the man it lasted his wife 30 days. How long would the man be occupied in drinking it? Ans. 15 days. 36. A, B and C could do a piece of work in A days; A, B and D in I days; A, C and D in i| days; B, C and D in i| days. In what time could they all do the work, and in what time could each man do it alone? Ans. All in |? days; A in 1 ; B in 2 ; C in 3 ; and D in 4 days. 37. Divide 55 (80) into two parts, so that the less (greater) part divided by the difference (sum) of the parts shall be 2 (|). Ans. 33 and 22. 38. Four places are situated in the order of the four letters A, B, C and D. The distance from A to D is 134 miles. The distance from C to D is | the distance from A to B, and i the distance from A to B added to half the distance from C to D is three times the distance from B to C. What are the distances? 39. A person went to a tavern, where he spent 5 shillings, and then borrowed twice as much as he had left. He does the 94 P 11 O B L E M S . same at a second and a third tavern ; but on spending 21 shil- lings at a fourth tavern he had nothing left. How much had he at first? Arts. 8 shillings. 40. A boy had a number of marbles. He laid aside 2, and then w^oi;;i in play as many as he had left. He then laid aside 3 more, and again won as many as were left. He now adds 4 more to the reserved pile, and wins, as before, as many as he has left. Then counting his marbles he finds 13. How many did he begin with? Ans. 5. 41. Two boys, Charles and John, play marbles. First game, Charles wins 4 marbles. Second game, John wins 12. Charles again wins 4 in the third game, and John wins 6 in the fourth and last game. John now has three times as many marbles as Charles, although each had the same number Avhen the play commenced. How many marbles had each at first ? Ans. 20. 42. A commenced trade, and at the end of the third year found his original stock tripled. Had his gains been $1000 per year more than they actually were, he would have doubled his stock each year. What was his original stock? Ans. $1400. 43. Divide the number 20 into two parts, so that the product of the parts shall be 5 times the greater part. Let X = the greater part, and 20 — x the less. Then 20.^; — x^ = 5.t, whence 15 and 5 are the numbers. 44. Divide the number 40 into two parts, so that the product of the parts may be 35 times the smaller part. 45. A boatman rows 14 miles an hour with the tide. Against a tide two-thirds as strong he rows only 4 miles an hour. What is the velocity of the tide in each case? Ans. G and 4 miles. 46. Three persons, A, B and C, were seen traveling in the same direction. At first A and B were together, and C 12 miles PROBLEMS. 95 in advance of them. A goes 7, B 10, and C 5 miles per hour. In what time Avill B be half vrnj between A and C? How long before C will be midway between A and B? Plow long since A was midway between B and C? Ans. respectively Ih. 30m., oh. 2obn., and 12/i. 104. (1.) It is often much more convenient to represent the unknown quantity by such an expression as will avoid the intro- duction of fractions into the equation of the problem. It is in fact a much better exercise to solve a single problem in several different ways than to be engaged on as many different problems. The shortest method of solution should always be found out, as it leads to the clearest insight into the problem. EXAMPLES. 1. What number is that whose half and third added together make 10? (Vide 103, ex. 3.) Let 6x = the number. Then 3x + 2x =10, whence x = 2 and 6x = 12. In the same way solve those marked * in the preceding sec- tion ; {. e., let the unknown quantity be represented by the least common multiple of the denominators of the fractions in the problems. 2. The rent of an estate is this year 5 per cent, {. e. ^q, greater than it was last year. This year it is 8-10 dollars. What was it last year'? Let 20a; = the rent last year. Ans. $800. 3. If from a number we take 2, and divide the remainder by 11, the quotient will be 6. What is the number? Let llx -f 2 = the number. Then x = 6, and llx -f 2 = 68. (Vide 103, ex. 1-1.) In a similar manner solve those marked f of section 103. 96 PROBLEMS. 4. Wliat number is that from which if 5 be subtracted | of the remainder will be 40? 3x -f 5. Aiis, 65. 5. What number is that to which if 7 be added | of the sum will be 18? 3x — 7. A71S. 20. 6. A teacher spent | of his salary for board and lodging, 1 of the remainder for clothes, i of what remained for books, and saved $120 per annum. AVhat was his salary ? 15.7? = salary. Ans. ^360. 7. In a mixture of wine i the whole, plus 25 gallons, was wine; 1 the whole, minus 5 gallons, was water. What was the quantity of each in the inixture? Let 6x = the whole, then ox + 25 -f 2.^ — 5 = 6x. 8. One-half of a certain number is the same as | another number. But if 5 is added to the first and 10 to the second, then I of the first is the same as | of the second. What are the numbers'? 2x and 3x. Ans. 20 and 30. 9. Divide 00 into four parts so that if the first be diminished by 2, the second increased by 2, the third divided by 2, and the fourth multiplied by 2, the results will be equal. Let 2x = tlie quantity to which they are to be equal. 1st part. 2(1 part. 2d part, ith part. Then 2x + 2 -f 2x — 2 + 4x -}- x = 90, whence x= 10. And 22 18 40 10 are the parts. 10. Divide the number 151 into 5 parts so that twice the 1st, one-half the 2d, one-third the 3d, one-fifth the 4^/?, and three and one-half times the 5th shall be equal. Let 14:X = the quantity to which they are to be equal. 11. A person supported -himself 3 years for $50 a-year. At the end of each year he added to that part of liis stock which was not thus expended a sum equal to I of this part. At the PKOBLE^IS. 97 end of the third year his original stock was doubled. What was the amount of stock at first? Let 27x + 200 = the original stock. Then 27x + 150 = the original stock less $50. And Qx -\- 50 = one-tliird this remainder. 36a: + 200 = stock at the close of first year. In the same way 48x + 200 = " " " second " And 64a; + 200 = " " " third " Therefore 64a; + 200 = 54a; + 400 by the question. Whence a; = 20 And 27a; + 200 = $740 the original stock. 12. From a certain sum of money I took one-third part, and put in its place $50. From this sum I took one-tenth part, and soon replaced it with $37, when the sum amounted to $100. What was there at first? 15a; — 75. 13. A tree 80 feet high was broken by the wind in such a manner that the top reached the ground just 40 feet from the bottom of the tree. How high up was the tree broken? A 40 Let X = the distance from the bottom, and 80 — a; the part broken off. Then (80 — a;)^ == a;^ -f 40= Euclid, Bk. I, 47. Or 6400 — 160a; + x^ = x^ + 1600 a; = 30 = A B. 80 — a; = 50 = B C. 98 EQUATIONS OF THE F 1 11 B T DEGREE. 14. Two trees 80 and 60 feet high stand on the tame hor- izontal plane, 100 feet apart. Where must a person stand to be equally distant from the top of each? (^Vide 121, ex. 4.) Ans. G-4 feet from the shorter tree, or 36 feet from the taller. EQUATIONS OF THE FIRST DEGREE INVOLVING TWO UNKNOWN QUANTITIES. 105. Simultaneous equations are those in which the values of the unknown quantities are the same in both. Thus, X -j- 7/ = ^0 and x — j/ = 6 are simultaneous equations, because either of them can be verified when a; = 18 and ?/ = 12. 106. Simultaneous equations are independent of each other when one is not a mere transformation of the other, or when one equation is not a result of the combination of two or more equations. Thus, a; -f y = 30 and x — j/ = 6 are independent simultane- ous equations ; but, x -^ y = o^ and 3a; = 90 — ?>y are depend- ent, since the first may be easily obtained from the second. Also, 3x + 2j/ + ;i' = 10, X + ?/ + z = 6 and x-]-2y-\-Zz= 14 are dependent, since the second is one-fourth the sum of the other two. ELIMINATION. lOf . Elimination is the operation of combining two equations in such a manner as to cause one of the unknown quantities to disappear in a new equation. There are three principal methods of elimination, by addition or subtraction, by suhstitution, and by comiiarison. ELIMINATION. 99 ELIMINATION BY ADDITION OR SUBTRACTION. 1©§. 1. Eesume the equations, 0^ + 7/ = 30 (1) and X — 3/ = C (2) By axiom 2, we may add these equations together. Doing so we have 2x = 36 (3) = (1) + (2). Vide 34, ex. 36. Whence a; == 18 (4) = (3) -v- 2. By axiom 3, we may subtract (2) from (1). Doing so we have 2y = 24 (5) = (1) — (2). Vide 45. Whence y = 12 (6) = (5) -^ 2. By putting the values of x and 7/ in place of these letters in (1) and (2), we have 18 -f 12 = 30 and 18 — 12 = 6 Vide 105. In the same way find the values of x and y in the followino* sets of equations. iB + ?/=10 X -i- y =12 X — 3/ = 4 X — y=8 2. Again ; take the equations, 3x + 2y = 22 and 2x + 3y = 23 By axiom 4, we may multiply (1) by 2 and (2) by 3. This gives Gx -}- 4y =z 44 (3) = (1) x 2 and 6x+ 9y= 69 (4) = (2) x 3 By axiom 3, subtract (3) from (4), and we have, % = 25 (5) = (4) - (3) Whence y = 5 (6) == (5) -v- 5 By axiom 4, we may multiply (1) by 3 and (2) by 2. This gives 9x + 6y = 66 (7) = (1) x 3 and 4x -f 6y = 46 (8) = (2^) X 2 x-\- y = 20 a; + y = 25} X — y = 15 x—^ = 3l (1) (2) 100 ELIMINATION. By axiom 3, subtract (8) from (7), and we have, 5x = 20 (9) = (7) - (8) Whence a; = 4 (10) = (9) -h 5 By putting the vakies of x and y in phice of these letters in (1) and (2), we have 3.4 + 2.5 = 22 and 2.4 + 3.5 = 23 wliich proves that the values of x and y are correct. We multiplied equation (1) by 2, and equation (2) by 3, simply to make the coefficients of x in these equations alike, and because the signs before the like coefficients of equations (3) and (4) are alike; by subtracting, x disappears in the resulting equa- tion (5), where there is only the letter y, whose value in (6) is obtained in the manner heretofore explained. We now multiply equation (1) by 3, and equation (2) by 2, to make the coefficients of y alike, which leads to the value of x, in the very same way as before. By this process we have eliminated x and found the value of y. We then eliminated y and found the value of x. 3. Again ; take the equations, 5^ + 3j/ = 13 (1) and 3.T — 7?/ = — 1 (2) By ax. 4, 35x + 2ly = 91 (3) = (1) X 7 By ax. 4, 9x — 21y = — 3 (-i) = (2) X 3 By ax. 2, 44.x = 88 (5) = (3) + (4) By ax. 5, x=2 (C) = (5) - 44 By ax. 4, Vox + 9y = 39 0) = (1) X 3 By ax. 4, Ihx - 35?/ = — 5 (8) = (2) X 5 By ax. 3, 44y = 44 (9) = (7) - (8) By ax. 5, 7/=l (10) = (9) _4_ 44 In this example, after making tlie coifficienis of y alike, because the sigiis of these coefficients are unlike, we add equations (3) ELIMINATION. 101 and (4), and y disappears. In other particulars the steps arc the same as in the previous example. 4. Take the er[uations, ' \ \ \ \ ] \ ^ 2 + 3-"^ ^^^ and ^+| = 3| (2) By ax. 4, 3x + 2^ = 22 (3) = (1) x G ( VicU 9S.) By ax. 4, 2x + 3^^ = 23 (4) = (2) x 6 {Vide 9S.) Equations (3) and (4) are the same as (1) and (2) of ex. 2. They should be treated in like manner. 5. Take the equations, X y ^ y - 4- — = 1-3- — - 2 ^ 10 '' 5 0) and X y 1 + X 7 2 14 (2) By ax. 4, 5.x- + ?/ = 13 — 2y (3) = (1) X 10 By ax. 4, 2x — 7^ = — 1 — re W = (2) X 14 Vide 99, hx-\-oy= 13 (5) = (3) transposed. Vide 99, 3x - 7y = - 1 (6) = (4) transposed. The equations (5) and (6) are (1) aiid (2) of ex. 3, and | should be treated in the same manner. Therefore, j 109. Having two equations with two unknown quantities, to find their values by the method of elimination, by addition or subtraction : I (1.) If necessary, clear the equations of fractions. (2.) In each equation collect all the terms involving x into one term, and write this term first in the first member of a new ! equation, prefixing the correct sign. i (3.) In each equation unite all the terms involving y in one 102 E L I ^I I N A T I N . terra, and write this term second in the first member of the new equation, prefixing the correct sign. > . ('4.) Ii' each equation collect the known quantities into one term, and write this term in the second memher of the corres- .pOTiding ,iic 3x — 5^ = — 5 ) nx^Zy= 10^ '^' 8x + 2y=120 5 5x - 4j/ = 7 > 10x4-5^ = 40 5 7x- %=-2^ ^' 8x + 21j/= 29 > 12a: - 7?/ = 12 ^ llx-33/=ll5 ^ 4x + 3j/ = 65> ^- 5a:_23/=41 5 g 3a) + 53/= 15 > 4a; 4- y= 3 ) 7x+ 5^ = 2^ -^^ i4^__10j/=0 5 6. 10. 11. 12. 3x y ^c, 52 "^ 44 " (10) (11) (12) (6) X 24 (10) - (5) (11) -f- 359 13. •"^ _ ^ = _ 90 ^ 7 10 1+ 3^= 131 J 14. ^ = 1 + 20 2 4^ ^ 5 ^3 4^1 15. y_^ 7 6 1 ^ + ^=1^1 16. 2 ! 5.'.-| = 29 ] 17. X — .y-1 J x + y ,^-y r^^^l] ■~2~ "^ ""2~ "^ 3 4 I 3"^4 5 " 7 ^ 19. 2"^3 3^4 1 104 ELI j\I I NATION BY SUBSTITUTION. 20. - + -=.-: X y b 21, 2x Zy 26 3x 4.7; ^ ~ J 22. + ^=20 7 ^ 14 14 7 - ^^ J 1 * f + T^ = 99 I 7x + |=51 ^ 24. 5.7; 3?/ 25. y_ 41) ^ =99 f 49 51 99 * 15 26. 17 27. * 5 ^ 7 09 2.x T+ 5 1631 I 35 J 28: 4x ?>x By T Ay 5 2^*4x + 3j/ = 3> 3;c -}- 4^ == 4 > 30. 4:X 3y 3.7; 4y "8~"^ T 37 42 J * 40^ + 3^ = 7-^ 3a; + 4y = 7 5 ELIMINATION BY SUBSTITUTION. 110. 1. Resume tlie equations, x + y = SO (1) and x—y=6 (2) Transpose y in equation (2), and we have x=6-]-y (3) ELIMINATION BY SUBSTITUTION. 105 Substitute this value of x for x in equation (1), and we have G+y + y = 30 (4) whence ?/=-12 (5) Substitute this value of y for y in equation (1), and we have .T + 12 = 30 (6) ♦vlience re = 18 (7) 2. Take the Cvquations, "^ + | = 6| (1) and I + 1 = ^^ ^^^ Clear (1) of fractions 5x + 3j/ = 95 (3) = (1) x 15 Clear (2) of fractions 3x + 2y = 60 (4) = (2) x 6 60 — 23/ Find a: in (4) x = ^ (o) Substitute this for x in (3) -|- 3y = 95 (6) o whence 3/ = 1^ C'^) and cc = ^^^^^ = 10 (8) =r (5) in which 2^/=: 30 It is evident that these steps may be taken on any two equations, hence, 111. Having two equations with two unknown quantities, to find their values by tlie method of elimination by substitution, (1.) If necessary, clear the equations of fractions, (2.) Find, in either of the equations, the value of one of the unknown quantities in terms of the other, and substitute this value for the same unknown quantify in the other equation. (3.) From the equation thus formed, find the value of the letter involved. (4.) Substitute this last value for the letter to which it is equal in any equation excejjt that from which it was obtained, and find the value of the otlicr letter. 106 ELIMINATION BY SUBSTITUTION. EXAMPLES. 3. Find x and y in the equations, = 4 5x X — y IT T~ and 2x -f 3j/ = 43 20a3 — llx+ lly = 176 ^x + llj/ = 176 43 — 3y X = 387 — 27y ■ 2 whence + lly=17G and 43 — 21 2 = 11 (1) (2) (3) (4) (5) (6) (7) (8) = (1) X 44 = (3) reduced = (2) v{(h above, 2 ^ (4) vide above, 2 vide above, 3 vxde above, 4 3a. + 4y=18^ ^- 2x- y= ly 2x — 3?/ rr 92 9. 5* 4x + 3j/=16^ 3x + 4^ = 19 5 y — 1 . y ^ 6. 10. 7. 8. 4 51 » + |-10 = | i-l^^y-m V ^ ^ 5 5^ ^ V ^ - +^i = 2 5^4 - — ^ = — 1 a;_2 10 — a;_y — 10 ] 11."^ 3---^ ^ 2^ + 4 _ 2:r + y ^ 0^ + 13 j 3 8 4 J 12. 3x + 4^ _ 40 — X 1 5 4 ' o,^ 2.y^ 84-7/ | "■■'' o 6 J ^ ELIMINATION BY SUBSTITUTION. ^ , y 107 13. a^ y o O -J a? + y ■ ^— y ^ q1 r 14. 10 2 • ^ + y , ^ — y ^ -^ ^ 2 2.x — ?/ 3 oy — AiX - - - 15. x-^y 9 2 •^ 3 16. Find a; and y in the equations, (1) 9 9 - + -=10 a; y 'i 1 _3_ X y X -\- y = ^xy y — ox= — 7xy x — bxy = — y (1 ^by)x= —y y X = y + 3y l-5y ,2 (2) (3) W (5) («) (V) = (1) X a-y and -^ by 2 = (2) X xy = (3) transposed = (5) factored = (6) 27V/c^ above, 2 ry 1-57/ 1-5^ whence y = ^, x = ^ 1 (8) = (-15 r?V/e above, 2 4 3^. -+- = 3 I ^^•^ S 5 ^ X = 1 |i 19.t 5 y = -l -+ - a; y 5_6 .X ?/ rzV/e 1®1, ex. 6. 87 18.t -=i ]x=2H x y 6 10 , , X y ' 20.t ' 6 10 , I -+ -=:4 ^ y J 108 ELIMINATION. ELIMIXATIOTT BY COMPARISON. 112. 1. Find the values of x and y in the equations, a; + y=30 (1) and x—y = 6 (2) Transpose y in each of these equations, and we have x=30 — ?/ (3) and x = 6 -j- y (4) Now by ax. 1 tiicse values of x must be equal, that is. By comparison 30 — y =1 Q -\- y (5) whence y = 12 (C) and from (3) or (4) .t=18 (7) 2. Find the values of x and y in the equations, l + ? = 2 (1) and ?-l^=-l (2) Ay + 3.C = 2.ry (3) 8y _ 15.^; == — .tj/ (4) 3x — 2.TJ/ = — 4y (5) lox — .^y = 8y (G) (3 - 2y) X = - 4y (7) (15 - y) rr = 8j/ (8) 3 — 2j/ ^ ^ 15—3/ — 4y 8y By comparison V- = t^- (H) ^"^^^^ ICl, ex. 7. •^ ^ ^V-23/ lo-y ^ whence ?/ = 4i and x = 3i. It is evident that these steps mny be taken on any two equa- tions, hence, 113. Having two equations with two unknown quantities, to find their values by the method of elimination hy comjmrison : (1.) If necessary, clear the equations of fractions. (2.) Find, in both equations, the value of the same unknown quantity, in terms of the other, and make these values equal. (3.) From the equation thus formed, find the value of the let- ter involved. (4.) Same as 110, 4. ELIMINATION BY CO :\I PARI SON. 109 3. 4. 4x + 3y = 7 ^ X -^ y = 2'^ ^ + 2o = x x + y — o y g x + 2 = ^y} • 3/ + 4 = |x 3 Src-f 2 = 14?/ 9. a; + y 2 91 — 4" — i-i^X a; + 25 = ^y 10. 0. X 7/ + 25 = - + 15 10a; + y = 4 (x + 3/) > 10a; + 3^ + 18 = \^y -\-xS 6. :. + |^ = iml ii.f .;c 3y 4"^ 20 ^ 27^ ?+^ = 2 X y - 4 5 a; 3y ^J ' +' =2 ^ 3a; 43/ 1 1 2a; y ■^ 12.t 114. Either of the three methods of elimination may be em- ployed to solve equations consisting of tAvo unknown quantities. Practice, however, and repeated efforts to do. so, will enable the student greatly to abridge the work in almost every case that can happen. To illustrate this remark, we will repeat, by all the metliods, the solution of ex. 16 111. . The equa- tions are 2 2 -+-=10 X y (1) and i_?=-7 X y (2) 1 1 . - + - = X y (3) = (l)-2 1 = 12 y (4) = (3) - (2) whence y = i and a; = 5 . 110 ELIMINATION BY COMPARISON. 2. After equation (3) we may continue thus: 1 1 - = 5 (-1) = (3) transposed X y 13 11 5 = — 7 (5) = (2) since - = 5 y y ^ y whence y =-\ ^^^^^ ^ = -|-' 3. Or, we may continue (1), thus : - = 5 - - (4) X y l = ?-7 (5) X y By comparison 5 = 7 (6) y y whence, again 3/ = -^ and x = -|. In a similar manner, solve those equations marked f in 111, 112, and 113. When the coefficients and signs of the letters x and y inter- change, or if the &igns remain the same in the two equations, we may proceed as follows : Given — -}»»— = 6i (1) i o and vide 3.t. 4?y _ . .^^ ,^. 10"y, ex. 30 8" "^ y ~ ■' ^"^ 32x + 21y = 37 X 56 21ic + 322/ = 42^ X 56 53aj + 53^ = 79i x 56 — llrc + lly= 5^x56 a; + y = | x 56 — a;4-?/ = Jx56 whence, a? = 28' and 3/ = 56 nV/c lOS, ex. 1. In a similar manner, solve the equations marked * in the preceding sections. (3) = (1) X56 (-t) = (2) X 56 (5) = (-1) + (3) (6) = (4) - (3) (') = (5) -=- 53 (8) = (6) - 11 THREE EQUATIONS. Ill THREE INVOLVING THREE 115. 1. Take the equations, EQUATIONS UNKNOWN QUANTITIES. X y +-+-=9 (1) X z 4 + ^ + 8 y a) + 2y-3^=-8J 4x + 22/ + 2 = 36 } 2a; + 8y + ^ = 64 ]" — 2a; + 6y = 28 12a; + 6y + 3^ = 108 ^ a;-f2y — 3^=— 8 5 13a; + 8j/ = 100 — 13a; 4- 39y = 182 47y = 282 Whence, 3/ = 6, and from (6) a; = 4, and from (4) 2; = 8. Hence, having three equations with three unknown quantities, to find their values: (1.) If necessary, clear the equations of fractions. (2.) From any two equations eliminate either of the letters. (3.) From any other two equations eliminate the same letter. (4.) Proceed as in 109, llO, or 111 with the two equations thus obtained. (2) (3) (4) (5) = (1) X 4 = (2) X 8 (6) = (5) - (4) (7) (3) = (4) X 3 (8) = (7) + (3) (9) = (6) X ^ (10) = (8) + (9) EXAIMPLES. 2. 3. x-\- y + 2 = 18 x + ^i/ + 2z = ^S .+ 1 + 1 = 10 J X -\- y -\-2z= 9 X 4- 2y + Sa = 14 6a; -f 5?/ + 3;^ == 25 2z=21-i(ix-\-y) 4. 3a;= 72 38 = 1 (3a; + 3^ - ^) a; + 2 (;y + ^) = 31 5. y + 3 (a; + 2) = 2 + 4 (a; + y) = = 31. = 42 > = 51 ) 112 FOUR OR ?.1 li E EQUATIONS 2 + 34--^" 8x — 9j/ — "iz = — 3G ] 7. 12a; — ?/ — 3<^ = 36 ;> ^_^ + f=12 3 4^2 ^ 6a; — 2y— 5; =10 J 4^2 3 J FOUR OR MORE EQUATIONS INVOLVING A LIKE NUMBER OF UNKNOWN QUANTITIES. 1J6. 1. Given, the four equations, x-\. 2y -\- 2z -f- 2iu = 26 1 ox -\- y -\- ?)Z ■\- 3zt' = 3G I 4a; + 4y + z -f 4?c7 = 44 | 5a; + 5 j/ + 5,^ + zy = 50 J 3cc + 63/ + G;3 + 6w = 78 4a; + 8?/ + 82 + 8z(; = 104 5.^ + lOj/ + IO2 + lOz^ = 130 Three equations, 53/+ 3.- + Siy = 42"] 4y + 7^ + 4i(; = 60 ;> 5^^+52+ 9ii; = 80 J 20y + 12^+ \2w= 168 20y + 35:2 + 20w; = 300 Two 23,2 + 8w; = 132 ^ equations, ^z + ^w = 38 >" 69^ + 2\w = 396 8;2 + 24t6- = 152 (1) (2) (3) (^) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) = (1) X 3 = (1) X 4 = (1) X 5 = (5) - (2) = (6) - (3) = (7) - (4) = (8) X 4 = (9) X 5 = (12) - (11) = (10) - (8) = (13) X 3 = (14) X 4 = (16) - (15) One equation, 6I2; = 244 Whence, ;2 = 4, by (14) w = 5, by (8) 3/ = 3, by (1) a: = 2. It is evident that in the same way we may solve any number of equations involving a number of unknown quantities equal to / SYMMETKICAL EQUATIONS. 113 that of the equations, i. e., we may always find the value of each letter by the following steps : (1.) Eliminate any letter by difl'erent combinations of two equations till that letter entirely disappears, leaving the number of new equations less by one. (2.) By different combinations of these new equations, elimi- nate any other letter, till the number of equations is less by one more. (3.) Continue these operations till an equation is obtained containing one unknown quantity, and find its value. (4.) Find the value of the other letters by successive substi- tutions. It is not nece&sary that every letter he found in all the equations. 2a; + 3?/ + 42 + ^iv = 40 Zx -\- 2y — 4.Z -{■ lu = — I 5x 4- 4y — 2z -{- IV = 11 7x — 5y + 3.:; — tv = 2 EXAMPLES. 3u-i- X -}-2y — z=22 4:x— 7/ -\-3z =35 ' 4u-\-3x — 2y =19 " 2u + 4.y -\-2z =46 SYMMETEICAL EQUATIONS. lit. The preceding principles will solve any set of equations which can occur. Nevertheless there are many short processes which depend upon the nature of the equations involved. Most of these processes occur in connection with what are called symmetrical equations, some examples of which we will now give. 1. Given, to find x, y, and z. '- + '-= ^ (3) 10 114 S Y .AI M E T K I C A L EQUATIONS 999 042 - + - + - X y z i 1^ 1 1 1 121 1_ 49 i~ 72I1 1_ 41 ^~72U 1_ 3]_ z~ 72U 49 = Whence, a-. = 14 (i) = (1) + (2) + (3) (5) = (4) -4- 2 (6) = (5) - (3) (') = (5) - (2) (8) = (5) - (1) y = .17??, . = 23l. In a similar manner solve, 5. x + y=19^ X -]- z = IS y-{-z = 17 1113 -+-+-=0 X y z b 1 11 11 -+-+ -=9l X y to J,4: 1111 X Z IV ^ 1 1 1 13 y Z lU Ji'k 9 O -+- X y 2 2 -+ - X z i 6 16 15 2 2 9 -+- = y z roj x+y -\- z x-\-y -\-w y + z -i- IV X z + iv = 211 22 24 23 ^ + 3 (y + ^) 6. ?/ + 3 (x + e) 2; -f 3 (ic + y) 28 26 rJlOBLEMS. 115 PROBLEMS INVOLVING TWO OR MORE UNKNOWN QUANTITIES. 118. All problems must involve as many independent equations as they contain unknown quantities. EXAMPLES. 1. *The sum of two numbers is 30, and their difference 6 What are the numbers? (^Vide 108, ex. 1.) 2. Three times the money of A added to twice that of B would make §22 ; but twice that of A added to three times that of B would make $23. What is the amount each has in pos- session? (Vide lOS, ex. 2.) 3. *If the first of two numbers be multiplied by 3 and the second by 5, the sum of the products will be 165 ; but if the fiirst be divided by 4 and the second by 7, the sum of the quo- tients will be 8. What are the numbers? Ans. 20 and 21. 4. If 7 be added to the first of two numbers, the sum will be three times the second ; but if 7 be added to the second, the sum will be five times the first. What are the numbers? Ans. 2 and 3. 5. *The sum of two numbers is 100, and their difference 20. What are the numbers? 6. *The sum of two numbers is 40, and the greater is three times the less. What are the numbers? 7. Says A to B, ''Give me $5 and we shall have equal sums.'* Now together they have $50. How much does each possess? 8. If a $25 saddle be placed on a horse his value will be twice a second horse ; but if the same saddle be placed on the second horse his value will still be $25 less than the first horse. The value of each horse is required. 116 PROBLEMS. 9. In a mixture of corn and wlieat ^ the whole -f 5 bushels was corn; but -^ the whole +10 bushels was wheat. What was the quantity of each? 10. Divide 50 into two parts so that twice the first shall be ^ the second. 11. * Divide 72 into two parts so that ^ the first and I the second shall be equal. 12. * Divide 36 into three parts so that ^ the first, ^ the second, and ^ the third may be equal. (^Vide 104, ex. 10.) Let X, y, and z represent the parts, and m the quantity to which they are to be equal when divided by 2, 3, and 4. Then x + y + s = 36 (1) _ X y z and - = m, - = m, - = wz J o 4 Whence x = 2m, y = 3???, z = 4;/i Adding which x -\- y -{■ z ^=. 9??z (2) By comparing (1) and (2), ^m = 36 .•. m = 4, a: = 8, y = 12, and z = 16. 13. *The sum of the first and second of three numbers is 11, of the first and third 12, of the second and third 13. What are the numbers? {Vide IIT, ex. 2.) 14. *The sum of the reciprocals of the first and second of three numbers is 5 ; of the first and third 7 ; of the second and third 8. What are the numbers"? 15. A and B have the same income; A saves \ of his annu- ally ; but B by spending $50 per annum more than A, at the end of six years finds himself §150 in debt. What is the income of each? 16. *If 8 be added to the numerator of a fraction, the value of the fraction will be 2 ; but if 8 be added to tlie denominator, the value Avill be only |. Required the fraction. riiOBLEMS. 117 17. A number expressed by two digits is four times the sum of the digits. If 27 be added to the number, the digits will be interchanged. What is the number? Let X == the left digit, and y the right. Then lOx + y = 4 (x -f 3/) (1) and 10a; + y + 27 = lOy + a: (2) Whence 10x+y = 36 Ans. Vide 29, ex. 1. 18. A number is expressed by three digits. The middle digit is twice the numerical value of the left-hand digit, and is greater by 3 than the right-hand digit. If 99 be subtracted from the number, the right takes the place of the left-hand digit, whilst the middle digit remains the same. What is the number? 19. A number is expressed by four digits. The fourth or left- hand digit is one-half the second. The first or right-hand digit is less than the third by 2. The local value of the fourth is 50 times the local value of the second. If 909 be subtracted from the number, the order of the digits is exactly reversed. What is the number? 20. A number consists of three figures. The left-hand figure is double the right. The sum of the digits is 3. If 81 be sub- tracted from the number, the left-hand figure is found in the middle, the middle figure is removed to the unit's place, whilst the unit figure appears at the left. What is the number? 21. * A and B can do a piece of work in 8 days; A and C in 9 days; B and C in 10 days. In what time can each do the virork alone? And in what time if all work together? (Fide lit, ex. 1.) 22. *A, B, and C can do a piece of work in 2= hours; B, C, and D in lii hours; A, B, and D in 2A hours; A, C, and D in precisely 2 hours. In what time can each do the work 118 PEOBLEMS. alone ? In -what time can all do the work together ? ( Vide 117, ex. 5.) Ans. for all Ihr. oGjiiin. 23. A bill of 85000 was paid in eagles and half-eagles, using of both kinds 560 pieces. What number of each was used? 24. Sajs A to B, '' Ten years ago I was three times as old as you at that time ; now my age is only double yours." "What is the aire of each 1 25. If 5 times A's property is added to | of B's, the sum will be 82700. If 5 times B's is added to | of A's, the sum will be 85100. What is the property of each? 26. I have a gold and a silver watch, and a chain worth $50. If the chain be attached to the silver watch, together they are worth I the gold watch. But when the chain is worn with the gold watch, they are worth 5 times the silver Avatch. The value of each watch is required. 27. At an election the majority was 80 votes. Had | the minority votes and 25 votes more been given to the successful candidate, he would have received in all double the number of his opponent. How many votes were actually given to each candidate ? 28. The crown of Hiero, king of Syracuse, weighed 20 pounds in air and 18| pounds in water. 2now 19|| pounds of gold weigh 18|| pounds in water, and 10^ pounds of silver weigh 9^ pounds in water. How much gold and how much silver did the crown contain? Ans. gold 1-1.77, silver 5.23. 29. *If a grocer mix sherry and brandy in the ratio of 2 to 1, the mixture is worth 78 shillings per dozen. If he mix them in the ratio of 7 to 8, the mixture is worth 79 shillings per dozen. "What is the price per dozen of eacli kind of wine ? P Ji B L E M S . 1 i D 30. In a composition of gunioowder tlie nitre was 10 pouncis more than |- of the whole, the sulphur 4^ pounds less than i of the whole, and the charcoal 2 pounds less than i the nitre. "What was the quantity of powder? 31. A vintner sold at one time 20 dozen of port vrine and 30 dozen of sherry for $;120. At another time, 30 dozen of port and 25 dozen of sherry, by a rise of 81 each per dozen, brought S195. What was the first price of each per dozen? 32. A's property together with -i- B"s and C's is worth $3500; B's with ^ A's and C's is worth $5000; C's with ^ A's and B's is worth $5250. AVhat is the property of each? 33. On examining my watch I find that 2 the time past noon is 4 of the time till midnifxht. What is the time ? 3-4. A farmer has 30 bushels of oats, at 30^ per bushel, which he wishes to mix with corn at 70/^ and barley at 90^ per bushel, making a mixture of 200 bushels, at 80^ per bushel. How much com and barley must he mix with the oats? 35. A farmer has 86 bushels of w^heat, at 4.5. 6d. per bushel. Barley, at 3s. per bushel, and rye, at 3^. 6d. per bushel, are to be mixed with the wheat so as to make a mixture of 136 bushels worth 4s. per bushel. How much rye and barley are to be used? 36. A gentleman left a sum of money to be divided among four servants. The share of the first was -^ the sum of the shares of the other three. The share of the second was -^ the sum of the other three. The share of the third was ^ the sum of the other three. The share of the first exceeded that of the last by $14. What was the amount divided and the share of each? 120 PROBLEMS. 37. A person pays at one time two creditors f 53, giving to one j4j the sum clue him, and to the other $3 over i the sum due him. At anotlicr time he pays the two $42, giving the first I of what remains due him, and the second i of what re- mains due him. How much did he owe each ? 38. Three persons, A, B, and C, have $96 among them. A gives to B and C as much as they already have. Then B gives to A and C as much as they have ; after which C gives to A and B as much as they then have. After this distribution each has $32. How much did each have at first"? 39. *A person has two kinds of money; it takes 10 pieces of one to make a dollar, and 2 pieces of the other to make the same sum. Some one offers him a dollar for 6 pieces, if he could make the change even. How many pieces were used of each kind? 40. *A man had dimes and half-dollars, and paid a debt of $2 with 12 pieces. How many of each kind did he use? 41. A man has eagles and half-eagles, and pays a debt of $65 with 8 pieces. How many of each kind must be used? 42. A man has 100 dimes and half-dimes, out of which he paid a debt of $6.15, and had 27 pieces left. How many dimes and half-dimes were taken? LITERAL EQUATIONS. 121 LITERAL EQUATIONS. 119. 1. Given, the equations, and Whence a h - + - = w X y (1) h c - + - = « X y (2) X y (3) = (1) X 6 ah ac 1 ssr an X y (4) = (2) X a h^ ac y y " - an (5) = (3) - (4) h^ - — ac (6) ^ -hm- - an ac he 1 = cm X y (7) = (1) X c h^ he ^ X y (8) = (2) X i h^ ac X X - cm (9) = (8) - (7) - ac (10) Whence 2. Given, ax -{- hy = m and ex -\- dy = n, to find x and y. dm — hn an — cm Ans. X = — ; — , y = — ■ -— . ad — DC ad — be X y 3. Given, - + r = 1 and x + y = c, to find x and y. . (c — h) a (a — c)h Ans. X = -^ f- , y = ^ ^ , a — b ' ^ a — b 4. Given, — f- - = a, - + - = 5, & - H — = c, to find x, ik z. ^ X y X z y ^ 2 2 2 a + 6 — c -^ a — 6-fc' __ « 4. 5 _j. ^ 122 !•. N E R A L I Z A T I O X S . 112112 112 5. Given, -+- = -, --f-=-, & --f-=-.to lliiJ x, y, z, X y a X z b y z c ahc ahc nhc Ans. X — - — ; , y = hc-\-ac — ah he — ac-\-ah'' — he -]- uc -\- ah' ah c d 6. Given, - -X- - =z m and - -f - = ??, to iincl x and ?/. X y X y ad — he ad — he Ans. X = J- , y = — . dm — bn an — cm X II X' '?/ 7. Given, - + '^ = m and - -f '- = n, to find x and y. a b c d (dii — hni) ae (am — ciC) hd Ans. X = ^ r — ? V = ^ ] r • ad — he ' *^ ad — he 8. Given, ax + hy = m and hx -f ay = ni, to find x and y. m m Ans. X = r , y == 7 . a -^r ^ « + ^ 9. Given, — | — = - and ax = hy, to find x and y, X y c (a+h) e (a + h) c Ans. X = ^^ , y = ^ — j-^ . a b GENERALIZATIONS. 120, 1. A and B together can do a piece of work in c days. The time in whicli A can do the work is to the time in whicli 1* can do it alone as h is to a. In what time can each do it alone? The equations of this problem are those of ex. 9, 119, -rx v . (« + ^')c (a + Z))c Hence, A requires , and L -^^ ; . a b If c == 20, a =1, 6 = 2, then A requires 60, and 1> 30 days. 2. A person has two kinds of money : it takes a pieces of the first to make a dollar, and h pieces of the second to make the same sum. Some one offers him a dollar for c pieces. How many of each kind did it take? The equations of this problem arc those of ex. 3, 119. ITencG it takes ^^ f- and ^— ^~- . a — h a — h GENE R A L I Z A Tl X S . 123 If a= 10, h = 2, c=6, the problem is 39*, IIS. If a = G }, 6 = 13, c = 8, the problem is 41, 118. Heading §05 for a dollar. If a = 61i, h = 123, c = 73, the problem is 42, US. Heading $G.15 for a dollar. 3. If a grocer mix sherry and brandy, in tlie ratio of a to Z>, the mixture is worth m dollars per dozen. If he mix in the ratio of c to d^ the mixture is worth n dollars per dozen. AVhat is the price of each per dozen ? Sherry Brandy (a + V) dm — {c-\- d) hn (c -f d) an — (a -\- h) cm Ans. T , — — — 7 ad — be ad — be If a = 2, 6 = 1, m = 78, c = 7, d= 2, n= 79, the problem is 29, lis. Vide 2S, ex. 13. 4. The sum of two numbers is a, and their difference h. a -{- h a — h AVhat are the numbers ? Ans. x = — —— , ?/ = — - — . 2 2 If a = 30 and Z> =: G, the problem is ex. 1, IIS. If a = 100 and h = 20, the problem is ex. 5, 118. In the same manner generalize the problems marked * in IIS. 5. Find X and y in the equation, x-f y =c (1) and f+l^ = x^-\- «2 (2) ^ince X = c — y, we have y^ -\- h^ = (^c — w)- -f- a,^ (3) that is, 3/2 _|_ ^2 ^ ^2 _|_ 2ry -f y2 4- a2 (4) * C2 -f a2 _ ?,2 C2 + Z>2 _ ^2 Whence, y = and x = . ' •^ 2c 2c 6. A's property, together with I times what B and C are worth, is equal to p dollars. B's property, together with 7n times what A and C are v/orth, is equal to q dollars. C's property, together with qi times what A and B are worth, is equal to r dollars. "What is the property of each ? 124 GENEKALIZATIOKS. Solution: — ^ + Ki/ + ~) = P y 4- m ix + s) = (/ z + n (x -T-y) == r X — Ix -\- Ix -\- ly -\- Iz ■=:i J) x(l-l)-\-l(x^yJ^z)=p X 4- {x-^y -]- z) ? 1 — / m 1 — / (1) (2) (3) (4) (5) (6) ^ + i^(- + ^ + ^~) = rf;;iK') + 71 (x + y + z) (8) 1 91 '^ ' ^ ' ^ 2 ^^ •^ \1 — I \-\-m 1 — 7i/ 1 — / 1 — m 1 — n (9) =(6) + (7) + (8) 1 4- ^ — , + . + -. — • ) (^+y + ^) = i^+r^ + i — 1 — / 1 — 771 1 — n/ 1 — / 1 — 7n 1 — ?i (10) = (9) factored P +_!_+ '' '^ + ^ + - = 1 — ?/z 1 — n 1 + Z 1 — P X = i" 771 n 1 771 1 71 q ?' (11) 1 7/1 1 71 1— / 1 _/ / 1 — 771 11 + 1 + 1 1 771 1 71 (12) q in ^ 1 7/1 1 111 \ ^ I r + + 1 / 1 771 1 71 Z = 7« n 1 -/+1 +1 L — t 1 — 771 1 — 71 7) 7 7' 1 tS + T^— + 1 1 / 1 771 1 — 11 \ !■ (13) 1—71 1—71 / i^ + r^ 111 11 4- ; + (14) 1 _ m. ' 1 — ;/ J NEGATIVI^ KEfciULTS. 125 Equation (4) is obtained by subtracting and adding Ix from the first member of (1). Equation (12) is obtained from (6) by transposition and the substitution of the value of x -^ y -\- z. Equation (7) is obtained from (2) by taking the same steps as were taken on (1), or, what is much better, (7) may be ob- tained from (6) by writing an equation of the exact form of (6), but using the next letters in the alphabet, returning to the Jirst letters when all in the original equations are exhausted. Equation (8) is obtained from (7), and (13) from (12), also (14) from (13) in the same way. (^Vide llf, 1.) If / = 2, 111 = 3. n = 4, p = 56, q = 77, r = 96, then x = 10, ^=11, .= 12. If / = ^, m = ^, 71 = J, ^; = 3500, q = 5000, r = 5250, then X = 2000, y = 3000, z = 4000. ( Vide 118, ex. 32.) NEGATIVE EESULTS. 121. 1. A wishes to pay a debt with 100 dimes and half- dimes, the debt being $4. How many of each are required ! Let X = the dimes, and y = the half- dimes. Then from the question x -{- y = 100 (1) and 10.T-f5y = 400 (2) Whence, x= — 20, and y == 120. Erom this answer we discover that it is impossible to pay $4 by the use of exactly 100 dimes and half- dimes. A must pay 120 half-dimes, and receive back 20 dimes in change. Had the question read, 1.* In paying a debt of $4, A gave 100 more half-dimes than he received dimes in change. What number of ejich passed between the parties? We should then have, y — x = 100 (1) and 5^-10x = 400 (2) Whence, .r = 20, and .y = 120. 126 NEGATIVE RESULTS. The results are now positive, sliowing the problem to be arith- metically possible. (^Vide 51 et ante.^ In general, if in the solution of a problem a negative result is obtained, we conclude that the problem as enunciated involves an arithmetical impossibility. But the results, whatever they may be, are interpreters of the error, and guide to a proper enunciation of the problem. 2. The sum of two numbers is 20, and live times the one added to six times the other makes the sum of 25. What are the numbers'? Ans. a; = 95, y = — 75. The result shows that the problem should have read, 2.* The difference of two numbers is 20, and i\YQ times the greater diminished by six times the less makes a remainder of 25; for if we substitute x and — y into the equations x -\- y s= 20 and hx -\- 'oy =■ 25, they become x — ?/ = 20 and bx — 6y = 25 ; in the last two of which ic = 95 and y = 75. 3. If 1 be added to the numerator of a fraction, its value will be E; but if 1 be added to the denominator, the value will be i. What is the fraction? Let - = the fraction. re + 1 2 , X 1 Then = ^ and — TT = q y 7 y + 1 2 Whence x = — 3 and y = — 7 3 The fraction is therefore — — , which must not be interpreted as — — . The problem should have read, + 7 3.* If 1 be suhtracied. from the numerator of a fraction, its value will be | ; but if 1 be subtracted from the denominator, the value will bo s. Here .r — 3, y—1, and the fraction is I. N E (i A T 1 Y E 11 E )S U L T S . 12: 4. Two trees, a and h feet high, are situated c feet from each other on a horizontal plain. At what point between the trees inust a man stand to be equally distant from the top of each ? E Suppose the point to be C : Let X = tlic distance from A to 6', and y = the distance from B to C. Now, by the Pythagorean Proposition, Euclid, book I, 47, nC~ = BC" 4- BD'^ and EC" = AC- -j- AE\ But by the question, DC = EC, or axiom 7, DC^ =^ EC^. Therefore, axiom 1, BC" -^ BD^ = ylC^ -f AE\ That is, y- -f ?>^ == •'^^ + «^ (1) But, rc + 7/= c (2) Whence, x = + 1' cr- -f- a2 _ ^2 and y = — . rule 12©, ex. 5. I. If a = 80, h = GO, and c = 100, then x = 36, y = 04. ) Vide 104, ex. 14.j" II. If a = 100, h = 40, and c = 50, tlien x = — 59, y = 109. This shows that with these values of a, h, and c it would be impossible to stand between the trees and be equally distant from the top of each. In this case the value of x must be taken to the left of A on the prolongation of BA, as in the following fiuurc. 128 NEGATIVE EESULTS In general, then, if lines to the rigid of a point are considered positive, those to the left will be negative. How should the prob- lem have read? III. If a = 40, 5 ==100, and c = 50, then a; =109 and ^ = — 59. This shows an impossibility similar to the last. Tiiis case causes y to be taken to the right of the point B, on the pro- longation of A B, as shown by the figure : If, then, lines to the left of a point are considered positive, those to the right will be negative. How should the problem have read? Any supposition which makes x ot y negative must be inter- preted in like manner. NEGATIVE RESULTS. 129 122. INTEEPKETATIOIS" OF THE SYMBOLS, J. ^ loo where A represents any finite quantity. The formulae of the last problem are, c2 + 6= — a2 c^ _|. a2 — h^ X = ^ and y = - 2c •2/ I. If a = 100, h = 80, c = 60, then the formulae reduce to, 3600 + 6400 — 1000 X = 120 120 y = and 10000 + 3600 — 6400 l20 = 60 This value of y shows that the point is at the foot of the taller tree, and therefore the expression — — must be infinitely small. This is illustrated by the figure. E D c=60 A B Any supposition which makes (? -\- l/^ = cv^ must make y = c, for if. in the value of y above, we should insert c' -f h^ for a^, we have, y = C2 4_ c2 -f Z)2 _ 52 Zc Hence, we assume that — is the representative of an infinitely small quantity; that is, the following equation is true. ^ = (1) 130 NEGATIVE K E S U L T S . II. If a = 100, h = 80, c = 0, then the formuhie reduce to, + 6400 — 10000 — 3600 X = y md + 10000 — G400 3600 By the supposition the tree A occupies the same spot with the tree J5, differing only in height. It is clear that as the point C recedes from A, the distance to the top of each tree approaches nearer an equality, and the two distances are abso- lutely equal only when C is infinitely removed from A. Hence — 3600 3600 , . . . , the expression — or must be mjimtehj great^ as each of them is the representative of the same distance en different sides of the point A. The supposition of c = and a>» or 18. Find the m'^ power of a"^. Ans, a'"". X 19. Find the I power of a'. {Vide §13,) Ans, a^. 1 X 20. Find the mth power of a*. {Vule §13.) Ans. a^n. If a = 16, X = 3, m == 4, then a'" = 16 * = 8. IST. To raise a monomial fraction to any required power : Observe if the fraction is in its lowest terms; if not, reduce it, and then raise the numerator and denoininator separately to the required power. 32jc^V^ y V^ 1. Find the hth power of / = ^ Ans. ^ Ux'y^ 2x^ 32x- x^y^ y^* 2. Find the StJi power of —~. — . Ans. . x^y x^ x^y — x^ 3. Find the 5^/^ power of — . Ans. ^^ — 'Ixy 6'1 7 — ^ %^ X x'^ 8x* V 4. Find the 4ith powers of — ^^ , , — - , and j . — Sx^ 2y 2x^ 4.7-j/' ^ -r.. -. 1 . 7 z' ^ ITx^y — x^ , 44x^ 5. rind the bin powers oi -, , j, and j.' _ ys Mxy — y' boy^ ■y a^ a^ 6. Find the 2nd power of — . Ans. — r- . 7. Find the ra^^ power of — . Ans. — . 12S. To raise the positive binomial x -\- y to any required power : Multiply the binomial as indicated in §124. « 1. Find the 2nd, 3rd, 4:th, &c. powers of a; + y. 136 INVOLUTION". Solution. Multiply by x-\ry ^ \&t power. ~-? (IX-l, I.) x^-\- xy xy -f 3/' and we have x'^+ 2xy 4- y^ Multiply the 1 ^nd power by \ x-\-y x^-\- 2x^y-\- xy^ xhj-\- 2xy^ + y^ and we have X^+ 3.T=j/+ 3.TJ/2 + 7/3 Multiply the } 3ri power by \ x-\-y x'^-^- ox^y-\- 'ix'^y^ + ^U^ x'y-\- 3a^y-f Zxy' + y' and we have x^4- 4:x^y-\- Qx^y^ + "^^^^ + y* Multiply the ? ^th power by \ x-\-y x^-\- 4:X^y-\- Qx^y"^- -\- ^x^y^ -\- xy* x'^y-\- 4iX^y^ + Qx^y^ -f 4iXy* + y^ ''2nd power. (124, 2.) 'ird power. (124, 3) 5 ^^^ power. — I (124, 4.) and we have x^-{- 5x^j/+10xy +10xy -f ^xy* -\- y^ = ^th power. 1. By the mere observation of either of the above powers, the law which governs the exponents of both letters is readily dis- covered. 2^he exponents of the Jirst and last terms are indicated hy the rOWER ITSELF. The exponents of x decrease towards the right by unity. The exponents of y increase towards the right by unity. Therefore x disappears from the last term, and y cannot be found in the first. (^Vide 12^ 3.) Disregarding the coetiicients, the several powers of .t -}- y may easily be written thus : INVOLUTION. 137 X -\- y = Id puwer. a-? + ^'y -\- y^ = 2//(Z power. x^ + x'^u -f xy^ -\- y^ = 2n-d power. a;" + x^y + x^y^ -\- xy^ -^ y^ = 4:th power. x^ 4- ^V + ^^y"^ -\- 'x^y^ + ^y^ + 3/* = 5^A power. x^ -\- x^y + a^^j/^ 4- x^y^ -}- .T^j/'' + xy^ -\- y^ = 6^/^ power. II. When any coefficient is given, the coefficient of tlie next term can be found by the following K u L E . Multiply the given coefficient by the exjwnent of the leading letter ill the same term, and dinde the product by the number exp)ressing the 2)lace of the term counting from the left. 2. Let us insert the coeHicients of tlie fifth power. The coefficient of the first term must always be one, and we (1) write down x^ as this first term. 1x5 By the rule, the coefficient of the 2nd term is — - — = 5, and (1) (2) 1 we write x^ -\- bx^y. 5x4 Again, by the rule, the coefficient of the SrcZ term is — ^ — (1) J2) (3) = 10, and we write .x* -{- hx'^y -j- l^x^y"^. Again, the coefficient of the 4:th term is — - — = 10, and we «j (1) (2) (3) (4) write x* -f bx^y + Kix^y"^ + \^x''y\ 10 X 2 The coefficient of the bth term is • = 5, and we write (1) (2) (3) (4) (5) x'' + bx^y + 10x3j/2 + l{}xY + ^xy\ 5x1 Finally, the coefficient of the Qth term is — - — = 1, and we o (1) (2) (3) (4) (5) (6) write x"" + 5xV + lOx^j/^ + \i)xY + bxy^ + y\ which corresponds with the bth power as obtained by actual multiplication. 12 138 INVOLUTION. In the same niiinner the coelliclenls of the i!>tii power are, (1) (2) (3) (4) (5) (6) (7) 1x6 6x5 15 X 4 20 X 3 15 x 2 6 x 1 ^' ~T~' "~2~' ~3~' ~~r"' 5 ' "~6~' '^ '"' 1, 6, 15, 20, 15, 6, 1, and v/hen inserted Avith the letters, -we have, (cc 4- y)^ = a;« + 6.xV/ + 15a:y + 2Qx^f -f- 15xy -f 6xj/* + 7j\ 3. Find the 1th power of x + y. Ans. x' + 7ic«y 4- 21x^^2 + 35xy + 350:^^/^ + 21arj/^ + To-/ + y\ 4. Find the 2)id power of x-{-y. (§62.) Ans. x? ■\- %xy -\- y"^ , 5. Find the 3?-cZ power of cc + ^. 6. Find the ^tli power of cc + y. 7. Find the 8>f — 1) 'A?is. (x — ?/)"' = ^"^ -- mx'^'-^y 4 ^- — - — - x'^-Y — I, -J, ,^ („i _ 1) (m - 2) ,3 , jn(m~l)(m-2)(m-3) „, . , , 1, *j, O, J) -> «^i i^. INVOLUTIOX. 139 130* To raise arnj binomial to any required power : Consider each term of the binomial as a single exjn-ession, and proceed as in §128 or §129. Then reduce the result as indicated hij the signs. 1. Find the Vith power of ?>x -f- 2j/. In the first phice, (3x+ 2j/)' = (Sx)^ + 5 (Sa^)^ (2y) + 10 C^xf (2yf + 10 {?>xy Qljjy + 5 (3:.) (2j/)^ + (2^)-\ This, reduced, gives, (3.x + 2j/)- == 2-43.X* + 81 0;/;"^/ + lOSOxy + 720xy -f 24.{)xf +32/,^ 2. Find the Zrd power of 2x -f ^y. First, (2x + 3^)^ = {2xy + 3 (2x)= (3^) -j- 3 (2x) (3j/)= + (3y)', which, on reduction, give?, (2x + 3^)5= 8^3 + 36a;2y -f 54.r3'2 + 27/. 3. Find the Wt power of x — 2y. First (X - 2yy = re" - Ax' (2y) + 6.x^ (27/)^ _ 4.t (2^)-^ + (2^)^ ^'^\^}l'^H(x — 2yy=x^— 8x'y + 24.ry — 32.ry + 1%*. 4. Find the cube of 2x'^ — oyK First, (i2x'-Sy'y=(2xy-S(2xy(^y')-^3(2x')(3yy-iSyy, a^d^this > ^2.x2— 33/2)3= 8x^ — 36xy + 54xy —277/9. 5. Develop (1 + ^)^- J.71S. l + 9x+36x2H-84x2+ 126rc''+ 126a;^-f 84^«4- 3Ga;^ + 9a;''+ a;». 6. Develop (x + 1)*. Ans. x^ + 9x8 ^ ^q-^j^ ^^^ 7. Develop (1 — .x)\ ^«5. 1 — 4.x + 6a;2 — 4x2 _^ ^4^ 8. Develop (x — 1)*. Ans. x^ — 5x< + lOx^ — lOx^ + 5x — 1. 9. Develop {l-x^, (l-x^)^ (xH/)S {x'-{-yy. (2a:+2/)^ and (3x2-2?/)\ 10. Develop (3x— y)^ (5ic + 23/)=, (1 + «^')', {^' — ^\ J^nd 11. Develop (x^ + y^y. Ans. x^ -f 3x^^ -f 3x5y + y\ 12. Develop (ic^ — ^^)''. .4/^s. x^ — 4x^^2 ^ Oxy — 4x%^ + /. 13. Develop (x^ — ?/-)^ ^tw. x — Sx^ + 3x^^' — ;y. 140 INVOLUTION. 131. The two formulae, §128, 8 and §129, 4, may be writ- ten together, giving expression to the BINOMIAL THEOREM. m (m — 1) ^ ^ , "in (m — 1) (m — 2) (x. =b 7jY = a;- ± jnx^'-'y + ^^ .^ ^ x'^'-y ± — ^ l^t , , m (m — 1) (m — 2) (m — 3) . . „ x^f + -^ 1.2,?. A ^^^' ^''' If in this equation x=l and ?/ = 1, we have, . , .. . . in (m — 1) , 7)1 Cm — 1) (in — 2) (1 ± 1)" = 1 ± « + -A^ ± -^-^-i^-^ + m (m — 1) (in — 2) Cm — 3) . ^ — ^ —^ —^ . (a.) I. e. 1.2.3.4 ^ ^ The sum of the coefficients of an?/ power of (a^ + y) is the same as 2 raised to the same power, bj observing the upper signs. If now 7/1 = 3, and we observe the up2^e7' signs, the formula becomes, (1 + 1)' = (2)^ = 1 -f- 3 -f 3 + 1, i. e. the sum of the coefficients of (x + t/Y = 2^ = 8. If m = 4, then (1 + 1)^ = 2^ = 1 -f 4 + 6 + 4 + 1, which are the coefficients of {x -f t/Y =16. If 711 = 5, then (1 + 1)^ = 2^ = 1 H- 5 + 10 + 10 + 5 + 1, tlic coefficients of (x -f i/Y = 32. If m = 10, then (1 + 1)^^ = 2'° = 1 + 10 -f 45 + 120 + 210 A- 252 + 210 + 120 + 45 + 10 + 1. If m = 3, and we observe the lowe7^ signs, the formula becomes, n — 1)' = 1 — 3 + 3 — 1 ; {. e. the positive terms exactly cancel the negative, and in general, from formula (a), above, the sum of the positive coefficients must be exactly the same as the sum of the •negative in any power of x — y. We see also, that in calculating the binomial coefficients^ ive need io actually perform the operations only half way^ hy taking the first half hi the reverse order. The formula above may be used to solve any example. INVOLUTION. 141 132 -(1). To find the cube of a trinomial, {vide 65): Take the trinomial x -\- i/ -\- z. Consider llie terms x -\- 7/ as a single expression, and we may write the whole thus, ((a^ + y) -}- zy. This developed as already explained, gives, Qx + y) + zy = (x + 3/)' + 3 (x + y)2^ + 3 (^x + i/) z' -j- z\ which, reduced, gives (x -\- y -\- zy = x^ -f- 'ix^y -f ?,xy^ + 3/' + S.r^.^ -f ^xyz -f ?>y'^z + ?>xz^ + 83/^2 + 2». This may be arranged thus, (x + 3/ + 2;)3 = a:^ + 3/' + -' + S^i'^J/ + ^x^^ + B^^t ^ 3^2. _^ 3^.2^ + 3x;2y + Qxyz. Hence, to cube a trinomial : Cube the three terms, and to their sum add three times the second 2)oiver of each term into the first jJOiver of each of the others, and also add six times the product of all the terms. 1. Develop (x — y -\- zy, Ans, x^ — y^ -\- z^ — Sx^y + ox^z -} 3y^x + Syh + ^^^x — ^z^y — Gxyz, 2. Develop (x — 2y -\- ozy, Ans, x^ — 83/' + 27z} — Ax^y + 6xh + 83^2^ + 243/^2 + ISzlv — 36z^y — 36xyz. 3. Develop (x^ -i- x + 1)'. Ans. x^ + 'Sx' + 6x^ + 7x^ + Qx^ •f Sx + 1. 4. Develop (py^ — x — 1)'. Ans. x^ — ox^ + ^x^ — 3x — 1. 5. Develop (x' — x"" — l)^ Ans. x'' — Sx'° + 5:^^ — 3x'- — 1. 6. Develop (x^ -f?/^ + z^y, (x^-\- x + x^y and (1 — x + x^y. 1 5 9x-^ 3 9a; 7. Develop (a- — x^ + J)^ Ans. x^ — 3x^ + 4:r^ + — 2 -i 3x' , -X + 4- 132- (2). Useful changes in the form of expressions. l^ .j,2_^f=(x-\-yy- 2xy, or x' -{- y' = (x - yy + 2xy. 2. x^ + 3/' = G'c + yy — 3.T3/ (a? -f 3/). F/cZe § 75. 3, ^3 _ ^3 _ (^,y — 3^)3 — 3.7^ (a: — ^). F?V/e § 75. 142 LOG A p. IT II MS. 4. x' + f = (x + T/Y — 4.rj/ (a; + t/)^ + 2x^f. Vide § 15, § tG. 5. rc^ + ^' = (.« + ^)' — 5^^ 0^ + ;y)^ + 5»y {^ + y). 6. x^ — y^ = (-^ — J/)^ + S.tJj/ (a; — ?/)' -f- 5a:^j/^ (cc — ?/). 8. a;' + 7/5 — a:^' — x^i/ = (;x + ?/) (.-c — ?/)l 132 -(3). To find a term which will make an expression of the form x^ =b 2ax a j^erfect square. "We have (x zhay = x^ dz 2ax -}- a^, v/here a~ = ^ (2a) squared. Hence, Square half the coefficient of the second term, add tJie result to the eorpression, and it ivill be a perfect square. 1. Make x^ -\- 2x a perfect square. Ans. x"^ -\- 2x -\- 1, 2. Make x^ + Ax a perfect square. Ans. x^ -f- 4x -f 4. 3. Make x^ — Gx a perfect square. Ans. x^ — 6^4- 9. 4. Make x^ — ox a perfect square. Ans. x^ — 3a; + £ . 5. Make x"^ -}- 2px a perfect square. Aiis. x^ -\- 2px -{- jp' 6. Make x^ - J^, a perfect square ^^^^^^ ^,^, Ans. x^ — + -J-- — - . tr LOGARITHMS. 133. A logarithm is a number expressing the power to which a given number is to be raised to produce another given number. 134. The number to be raised is called the hose of the system of logarithms to which it gives rise. 135. In the equations, 3°=1, 3'=3, 32=9, 3»=27, 3^=81, 3^=243, 3«=729, &c., three is the base, and 0, 1, 2, 3, 4, 5, 6, are respectively the logarithms of 1, 3, 9, 27, 81, 243, 729. 130. The base of the Cor.rviox System is 10, from AAhich we readily form this table. LOG A KIT II 3[ 8 . lA'o 108= 100000000 10» = 1000000000 101° ^ 10000000000 10" = 100000000000 &c. Since a" = 1, the logarithm of 1 in a^i/ system is 0. 1S1[, Any one of these equations "| _^ ,^ is expressed generally by j And any other by «^ = iV 10° = 1 10^ = 10000 10'- = 10 10* = 100000 10^ = 100 10« = 1000000 10' = 1000 10^ = 10000000 And by multiplying we have a"^^ = 31 X iV. That is, The logarithm of the product of two numhers is equal to the sum of their logarithms. Thus, by the table above, the logarithm of 10000 is 4, and the logarithm of 100000 is 5, and we find the logarithm of ") . -, AAnAnAAnn + i o ^, 1 . ^ xi t >• VIZ : 1000000000 to be i>, the product or these numbers, j which is 5 + 4. 138. Again, a* = M and ay = N M By dividinfr we have a"^" = -— , That is. The logarithm of tlie quotient of two numhers is equal to the difference of their logarithms. Thus, the logarithm of 100000000000 is 11. The logarithm of 100000000 is 8. And the logarithm of quotient, viz: 1000 is o, Avhich is 11 — 8, 139 -(1). By §126, ex. 18, the m*^ power of the equation, a' = M is a*^ = il/'". That is, Tlic logarithm of the w** jiower of a numher i-^ ri times the log- arithii of the numher. 144 L U G A ]l 1 T H M S . Thus, the logarilliin of 100 is 2, and the lofiarithin of ") . i Aa/vnAAAnnA • -< a ^i p.,, "^ ^-.^n /-viz: 10000000000 is 10, the bill power oi 100, j ' which is 5 X 2 139- (2). By §12G, ex. 20, the m'* root of the equation, a* = M « ni. is a"* = i/M, That is. The logarithm of the in*'^ root of a number is the logarithm of the number divided by m. Thus," the logarithm of 10000000000 is 10, and the logarithm of the ") . i nn • o ^th root of 10000000000, j" ^'"^ ' ^^^ '^ "' which is 10 -T- 5. 140. By examining the table § 136, we see that. The logarithms of numbers between 1 and 10 must be greater than and less than 1. The logarithms of numbers between 10 and 100 must be greater than 1 and less than 2. The logarithms of numbers between 100 and 1000 must be greater than 2 and less than 3, &c. Tlie following table illustrates this, Avhere the decimals are carried to 6 phices. 10^ =1 lO-'^^^^'" = 5 lO'-''^^'^!^ 30 1Q4.815098^ 7OQQQ 10.301030^2 lO-^'^'^' = () 10=-^^''»^"= 500 lO^-^^'o^o^ 200000 lO'-l^^^l^g 1Q.903090 ^ 8 102.954243^ QQQ 1 O^-^O^O^O = 8000000 10.602060^4 io'.3oio3o^20 lO^'^o^ceo ^ 4000 10=-^'8'5i= 600000 The integral part of the logarithm is called the characteristic, and in case the number whose logarithm is in question is with- out decimals, (a) The characteristic is always less hy one tlian the number of figures composing it. But if the number has a decimal con- nected with it, then, L G A R I T K JI S . I'Lo (h) The cliaracieristic is less hij one than the number of figures on the left of the decimal. Thus, the characteristic of 2.5 is ; of 25.67 it is 1 ; of 477.3 it is 2, &c. 141. By § 68, Example 4, we may write the following equa- tions : i- = io-'= .1 A_ = io-^=: .0001 i,- = 10-'= .0000001 _L = 10-2= .01 -l- = 10-^= .00001 —- = 10-8= .00000001 10= 10* 10* i^ = 10-« = .001 -^ = 10-''= 000001 ^ = 10-'=. 000000001 Hence, (c) The characteristic of the logarithm of a decimal is a negative number, and is always greater hy one than the number of cyphers at the beginning of the decimal. Thus, The characteristic of .3 is —1 ; of .053 it is —2 ; cf .00057 it is — 4. To save space the sign is usually written above the figure, thus, 3.602060. The decimal part of a logarithm is sometimes called the man- tissa^ and is always positive. 142. The equations of § 140 might be continued, and a com- plete table be formed, including numbers as high as we might choose to go, but such an arrangement would occupy far too much space. We may omit the hase, 10, write the numbers in a column, and the logarithms to the right. Thys, 13 14G L c; A R I T n u s 1 Table of Logarithms from 1 to 100. ' N. 1 Lojr. ! 0.000000 26 Locr. 1 1.414973! N. 51 Log. N. 76 1 Lou. J 1.707570 1.880814 i 2 0.301030 27 1.4313G4| 52 1.716003 77 1.886491 j 3 0.477121 28 1.447158 53 1.724276 78 1.892095 ! 4 0.G02060 29 1.462398 54 1.732394 79 1.897G27 [ 1 5 G 0.698970 30 31 1.477121 I 55 56 1.740363 80 81 1.903090 i 0.778151 1.491362 1.748188 1.908485 i 7 0.845098 32 1.505150 57 1.755875 82 1.913814 i 8 0.903090 33 1.518514 58 1.763428 83 1.919078 ! 9 0.954243 34 1.531479 59 1.770852 84 1.924279 I 10 11 1.000000 35 36 1.544068 60 61 1.778151 85 86 1.929419 1 1.041393 1.556303 1.785330 1.934498 1 12 1.079181 37 1.568202 62 1.792392 87 1.939519 i il3 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.949390 15 16 1.176091 40 41 1.602060 j 65 QQ 1.812913 90 91 1.954243 1.204120 1.612784 1.819544 1.959041 17 1.230449 42 1.623249 i 67 1.826075 92 1.963788' 1 18 1.255273 43 1. 633468 1 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 20 21 1.301030 1.322219 45 1.653213 70 71 1.845098 95 96 1.977724 ! 46 1.662758 1.851258 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 48 1.681241 1.863323 98 1.991226 24 1.380211 49 1.690196 74 1.869232 99 1.995G35 25 1.397940 50 1.698970 175 1.875061 100 m 2.000000 ^.^ EX A:\rPLES. 1. Multiply 2 !)y 28. Always take tlie following- rorm. Logarithm of 2 = 0.301030 I^garitlim of 28 = 1.447158 Logarithm of 56 = 1.748188 We add the logarithms of 2 and 28, and find 1.748188, for which Ave must look in the table. We find it opposite 56. L G A 11 1 T II M S . 147 2. Multiply 11 by 8. Logarithm of 11 = 1.0-41393 Logarithm of 8 = 0.903090 Logarithm of 88 = 1.911483 3. Multiply 2.5 by 3. {Vide §140 Z>.) Logarithm of 2.5 = 0.397910 Logarithm of 3 = 0.477121 Logarithm of 7.5 = 0.875061 4. Multiply .4 by .0023. ^Vide §141 c.) Logarithm of .4 = T. 602060 Logarithm of .0023 = 3". 361728 Logarithm of .00092 = 4.963788 5. Multiply 17 by .0005. Logarithm of 17 = 1.230449 Logarithm of .0005 = 4.698970 Logarithm of .0085 = 3.929419 6. Multiply 3 by 7, .8 by 12, .045 by .02,^.07 by 1.3, &c. 1 14. Before giving other examples we will explain the use of the table at the end of the book, which contains the logarithms of all numbers between 1 and 10000. The characteristic is omitted, as it may be easily supplied by the rules above, marked (a), (6), (c). (1.) If the number consists of tJiree Jigures with cyphers pre- fixed or added: Find these jigures in the column marked N. Opjiosite the num. her ill the next column is the mantissa of the logarithm, to which prefix the cliaracteristic, ly (jci), (?>), or (c). Since the two left-hand figures of the mantissa remain the eaine for several consecutive logarithms, they are printed but 148 L G A II I T n M s . once, in tlie column marked at the top. These figures Iclong to the four figures in all the columns. Thus, The logarithm of 364 is 2.561101. The logarithm of 365 is 2.562293. The logarithm of 739 = 2.868644. Find the logarithms of 201, 453, 510, G20, 729, 841, 934, and 999. The logarithm of 3.61 is 0.561101. The logarithm of .00365 is 3*. 562293. The logarithm of 73900 = 4.868644. Find the logarithms of 28.1, 3.65, .453, .0267, .00384, 765000, and 320. (2.) If the number consists of foia^ figures with cyphers pre- fixed as decimals, or annexed to make up a ivhole iimnher : .Find the first three figures of the number in the column marlccd N. Opposite to these in the column marked iviih a fourth figure at the top are four figures of the mantissa, to which prefix the two left-hand figures of the first column^ and. to the mantissa thus comj)leted prefix the characteristic hy (a), (6), or (c). Thus, The logarithm of 3171 is 3.501196. The logarithm of 3172 = 3.501333. The logarithm of 3173 = 3.501470. Find the logarithms of 3845, 4443, 4552, 6854, 7921, and 9999. If dots are observed in passing to the column marked with the fourth figure at the top, then the two figures of the first column must be taken from the line below. Thus, The logarithm of 3166 = 3.500511. The logarithm of 5014 r= 3.700184. The logarithm of 57.59 = 1.760347. Find the logarithms of 5564, 537.6, 53.76, .5376, and .001234. (3.) If the number consist of more than four figures : Find the logarithms of the first four figures as in (2). Take the figures in the column marked D at the top, found in tJte same line ivifh the yiumhrr, and mnlfiph/ ihrm hy the L O G A 11 1 T H I\I J3 . 149 remaining figures of tJie number given ^ pointing off fom the rigid as many figures as are found in the multiplier. Add the figures remaining at the left to the right of the loga- rithm already founds and the sum ivill be the logarithm of the given number, after 2^^^^fixing the proper characteristic by (a), (6), or (c). 1. Find the logarithm of 246891. First, the Warithm of 246800 = 5.392345. The number in the column marked D on the line of the number 2468 is 176, which multiplied by 91 gives 160.16. We now add the integral part of the product to the right-hand figures of the mantissa already found. Thus, Loirarithm of 246800 = 5.392345 176 X .91 = 160 Which gives Logarithm of 246891 = 5.392505 2. In the same way find the logarithm of 6789532, Logarithm of 6789000 = 6.831806 64 X .532 = .34 Logarithm of 6789532 = 6.831840 3. Find the logarithm of 12.347. Logarithm of 12.340 = 1.091315 351 X .7 = 216 Lonrarithm of 12.347 = 1.091561 We add 246 because 245.7 is nearer 246 than 245. So al- ways when the first figure in the decimal is greater than 5. 4. Verify the following. Logarithm of 67895 = 4.831838 Logarithm of 68707 = 4.837001 Logarithm of 47.306 = 1.674916 Logarithm of 432.156 = 2.635640 150 LOGARITHMS. Logarithm of . 000432 15G = 4.635640 Logarithm of 78.9102 = 1.897133 Logarithm of 4.32195 = 0.635679 Logarithm of .015364 = Logarithm of 123456 = Logarithm of .023967 = Logarithm of .111122 =^ Logarithm of .999999 = 145. To find the number corresponding to a given logarithm: By (a) and (b) if the characteristic be or a j'^ositive number; The number of figures on the left of tlie decimal in the re- quired number must be one greater than is indicated J)y the char- acteristic. By (c). If the characteristic be negative, the required number is a decimal fraction, having the number of cyphers between the decimal point and the first significant figure less hy one than is indicated hy the characteristic of the given logarithm, 146. (1.) If the mantissa of the logarithm can be exactly found : Find the mantissa in ths 'tahle and take out the corresjwnding number. Point off as directed in § 145. 1. Find tlie number corresponding to logarithm 2.928396. Ans. 848, 2. Find tlie number corresponding to logarithm 3.928396. uins. 8480, 3. Find the number corresponding to logarithm 1.928396. Ans. .848. 4. Find the number corresponding to logaritlim 3.962G06. Ans. 9175. 5. Find the number corresponding to logarithm 3.970114. Ans. .000335. LOGARITHMS. 151 (2.) If the mantissa of the logarithm cannot be exactly found: Take from the table the next less logarithm and the number cor- responding to it. Subtract this next less logarithm from the given logarithm^ and divide the remainder hj the number in the column marked. D, found VI the same line. Annex the quotient to the number alreadtj taken out, and point off as directed in § 145. 1. Find the number whose logarithm is 3.123456. Form of the Operation. Logarithm of 1328.78 = 3.123456 Logarithm of 1328 = 3.123198 328)258(.TS The next less logarithm tabulated is 3.123198, and the num- ber corresponding is 1328, which we write opposite 3.123198. Next subtract the logarithms, and we have 258, which we di- vide by 328, found in the column D. Annex the quotient .78 to 1328, and it is the required number, viz: 1328.78. 2. Find the number whose logarithm is 1 . 894325. Operation. Logarithm of 78.40164 = 1.894325 = given logarithm. 1.894316 = next less logarithm. No. in column D = 55)9000(.164 = quotient. Why is the last figure in the quotient 4 and not 3 ? 3. Find the number whose logarithm is 1.910360. Operation, Logarithm of .813504 =T. 910360 pro (last two figures of next "i_ less logarithm. No. in column D = 53)200(.04 = quotient. 4, Find tlic number VN'hoire logarithm is 2.750360. 152 MULTIPLICATION BY LOGARITHMS. Ojieration. Logarithm of .05628078 =Y. 750360 = given logarithm. 54 = next less logarithm. No. in column D 77) 6000 (.078 = quotient. 5. Find the number whose logarithm is 4.700446. Am. .0005017023. 6. Find the number whose logarithm is 2.698971. Am. 500.0011. 7. Find the number whose logarithm is 3.602061. Ans. 4000.009. 8. Find the number whose logarithm is 2.650020. Ans. 446.704. MULTIPLICATION BY LOGAKITHMS. 141^. 1. Multiply 24.6 by 25.3. {Vide §13^.) Operation. Logarithm of 24.6 = 1.390935 Logarithm of 25.3 = 1.403121 Logarithm of 622.38 = 2.794056 2. Multiply 52.74 by 27. Operation. Logarithm of 52.74 = 1.722140 Locarithm of 27 = 1.431364 Logarithm of 1423.98 = 3.153504 3. What is the product of 12 X 34.12 x .0056 x 5.671 X .8123 X .004 X 23.461 DIVISION BY LOGARITHMS. 153 Operation, Logarithm of 12 = 1.079181 Logarithm of 3^.12 = 1.533009 Logarithm of .0056 = 3.748188 Logarithm of 5.671 = 0.753660 Logarithm of .8123 = F. 909716 Logarithm of .001 ="3.602060 Logarithm of 23.46 = T. 370328 Log'm of .009911568 ="3.996142 4. Multiply 23.14 by 5.062. Ans. 117.1347. 5. Multioly 2.581926 by 3.457291. Ans. 8.92648. diyisioj^ by logaeith:ms. 14S. 1. Divide 24163 by 4567. (Vide §13S.) Ojjeration. Logarithm of 24163 = 4.383151 Logarithm of 4567 = 3.659631 Loj^arithm of 5.29078 = 0.723520 2. Divide 2 by 3456. Operation. Logarithm of 2 = 0.301030 Logarithm of 3456 = 3.538574 Ans. .000578704 = T. 762456 3. Divide 1 by 256. 02)eration. Logarithm of 1 = 0.000000 Logarithm of 256 = 2.408240 Ans. .00390625 ="3.591760 4. What is the value of .8697 -j- 98.65? Ans. .008816., 5. Divide 20.76 by 6254. Ans. .00476. 154 LOGAKITHMS. ARITHMETICAL COMPLEMENT. 149. The arithmetical complement of a logarithm is the difference between ten and the logarithm. Thus, The arithmetical complement of 3.6020G0 is 10 — 3.602060 = 6.3979-10. The arithmetical complement of 2^698970 is 11.301030; of 4^477121 it is 13.522879. 15©. To find the arithmetic;^! comj^lement of a logarithm: Take the left-hand figure from 9, and iwoceed towards the right, taJdng each figure from 9 till the last signifcant figure is reached, which must b^t taken from 10. Let X = any logarithm, and 1/ = any other logarithm less than x, and c = the arithmetical complement of v. By definition above 10 — ?/ = c or — ?/ = c — 10. Therefore x — 7/ = x -{- c — 10. From which we see that, The difference between two logarithms is found by adding to tlie first logarithm the arithmetical complement of the other, and diminishing the sum by 10. 1. Divide 24163 by 4567. Operation. Logarithm of 24163 = 4.383151 Arithmetical comp \ ^.^^ _ 6.340369 {Vide §14S, ex. 1.) 01 logarithm ot j . ^ ^ ^ Am. 5.29078 = 0.723520 = sum by rejecting 10. 2. Divide .7438 by 12.9476. Operation. Logarithm of .7438 = T. 871456 Logarithm of 12.9476 = 8.887811 arithmetical complement. An^. 0.057447 — 2". 759267 -- sum by rejecting 10. INVOLUTION BY LOGARITHMS. 155 . , , , 48 X .75 X 72 X .0625 „ .027 X 120 Operation. Logarithm of 48 = 1.681241 Logarithm of .75 = T. 875061 Logarithm of 72 = 1.857332 Logarithm of .0625 = 2'.795880 Logarithm of .027 arith. comp. = 11.568636 Logarithm of 120 arith. com p. = 7.920819 Ans. 50 = 1.698969=5 sum after re- jecting 2 tens. 6.832 .00634 3642 .657 4. Find the values of -^3^, ^^aS ' 2O8' "^^^ 70793 iKA'OLUTION BY LOGARITHMS 151. 1. What is the square of 2.5? (^Vide §139.) Operation, Lofrarithm of 2.5 = 0.397940 o 9 4?is. 6.25 = 0.795880 2. What is the cube of 32.16? Logarithm of 32.16 = 1.507316 3 Ans. 33261.9 = 4.521948 3. Find the square of 6.05987. Ans, 36.72203. 4. I^ind the ^th power of 2.97643. Ans. 233.6031. 5. Find the 1th power of 1.09684. Ans. 1.909864. 156 LOGARITHMS. EXTHACTIOIC or KOOTS BY LOGAEITIIMS. 152. 1. Find the square root of 256. (^Vide §139.) Ojjeration Logarithm of 256 = 2.408240 Logarithm of 16 = 1.204120 = J the logarithm of 256. 2. Find the square root of 2. OjJeration, Logarithm of 2 = 0.301030 Logarithm of 1.41421 = 0.150515 = ^ the logarithm of 2. 3. Find the cube root of 2. Oj)eratwu Logarithm of 2 = 0.301030 Logarithm of 1.2599 = 0.100343 = ^ the logarithm of 2. 4. Find the 4^/i root of 2. Operation. Logarithm oi 2 = 0.301030 Logarithm of 1.1892 = 0.075257.J = I the logarithm of 2. 5. Find the Uh root of 7.0825. Opei^ation. Logarithm of 7.0825 = 0.850187 Logarithm of 1.47923 = 0.170037 = \ the logar'm of 7.0825. 6. Find the cube root of .023. Operation. Logarithm of .023 ="2.361728 ="3+1.361728 Logarithm of .28438 = T. 453909 = -J the logarithm of .023. Here since the characteristic 2 is negative, and the mantissa .361728 positive, we cannot" divide by 3 as it stands. The char- acteristic must be so modified as to be exactly divisible by 3. Kow 2 = 3 -{- 1, and we may write the logarithm thus, 3 -f- 1 .361728, which i.-^ divisible by 3. EVOLUTION BY LOGARITHMS. 157 7. Find the ^oth root of .0621. Operation. Logarithm of .0621 =^.793092 = "5 + 3.793092 = 10 + 8.793092 = 15 + 13.793092 Logarithm of .573612 =T. 758618 = \ the logarithm of .0621. Here '2="5 + 3=To + 8=15 + 13, &c., either of which is exactly divisible by 5, and gives the quotient 1.758618. 8. AVhat is the 25^/z root of 2531000000? Aiis. 2.37756. 9. Find the value of 2}^ Operation, Logarithm of 2 = 0.301030 16 17) 4.816480 Locrarithm of 1.92009 = 0.283322 = \% of the Warithm of 2. 10. Find the lOO/A root of 5. Ans. 1.0162. 11. Find the cube root of 2.987635. Ans. 1.440265. (0-! \ 5 -— jl Ans. .146895. (119 \ ? ■y^Y' -^"«- 1936444. 14. Find the value of \ X (|)^X .012 x (fj)^. Ans. .0011657- iX aO^X .03 X (151)^ 15. Find the value of 7|x(m)^X .19 X (171)^ Ans. .300916. 153. Since the method originally pursued in calculating the mantissa of logarithms is easily understood, ■\ve will insert an exposition of it. If we WTite the two series, Is/, 2??r?, 1 2 3 4 5 &c. 1 10 100 1000 10000 100000 &c. 158 EVOLUTION BY LOGARITHMS. it is at once seen that logarithms are a series of numbers in arith- metical progression corresponding to a series in geometrical pro- gression. To compute the logarithm of any intermediate number, Find the geometrical mean of the two terms of the secojid series between ivhich the given num,ber is found. Find the arithmetical mean of the two corresp>onding terms of the first series. Again, Find the geometrical mean between this new term and the term nearest the given number. Find the corresponding arithmetical 7nean in the first series. Continue this operation till the given number becomes the geometrical mean, when the corresponding arithmetical mean will be the re- quired logarithm. EXAMPLE. Suppose it be required to find the logarithm of 9. The geometrical mean between 10 and 1 is l/lO x 1 == 1/10"= 3.1G22777 . 1 _f The arithmetical mean between and 1 is =rr A = .5 2 Therefore the logarithm of 3.1622777 = .5. Again, The next geometrical mean is 1/8. 1622777 X 10 = 5.6234132 1 + -5 The arithmetical mean between 1 and .5 is '— = .75 2 Therefore the logarithm of 5.6234132 = .75. Sdli/. The next geometrical mean is l/lO x 5.6235132 == 7.4989422 1 + .75 The arithmetical mean is = .875 2 Therefore the logarithm of 7.4989422 is .875. 4thli/. The next geometrical mean is l/ro"xT4989422= 8.6596431 The arithmetical mean is —^ — = .9375 Therefore the logaritlim of 8.6586431 is .9375. TREATMENT OF RADICAL S. 159 bthhj. The next geometrical mean is V 10 X 8.6596431 = 9.3057204 1.9375 The arithmetical mean is = .96875 2 Therefore the logarithm of 9.3057209 is .96875. Qtlil)/. The next geometrical mean is 1/8.6596431 x 9.3057204 = 8.9768713 .-, . 1 • .9375 + .96875 n-mor The arithmetical mean is = .yoolzo 2 Therefore the logarithm of 8.9768713 is .953125. Proceeding in this manner, after 25 extractions, ■v^'e should find that the logarithm of 8.9999998 is .9542425, and that is the logarithm of 9, near enough for ail practical purposes. In this manner Mr. Henry Briggs found the logarithms of all the prime numbers from 1 to 20,000, and from 90,000 to 101,000, carrying the decimal part to 14 places. The student will be glad to learn that in the light of modern analysis all this labor would be lost. EVOLUTION AND TREATMENT OF RADICALS. 154. 1. Evolution investigates the metlwd of finding any root of a quantiti/. 2. A surd is a quantity whicJi requires a radical sign, or index, to exactly express it. 3. A rational quantity requires no radiccd sign to exj>ress it. Thus, 3 is a rational quantity, but l/3 is a surd, X is rational, but f' x is a surd. 4. The coefficient of a. surd is the quantify prefixed to it. Thus, 5x^, where 5 is the coefficient of .t~ or v ^. 5. A rational quantity may have the form of a surd. Thus, 2 = t/4 = 4i IGO TREATMENT OF RADICALS. 6. A surd Is in its simplest form vrlien, from the nature of the root required, the part under the radical sign, or fractional index, is the smallest possible whole number. CASE I. 155. To place a surd involving an integral number in its sim- plest form. Separate the nimiher into two /actors, such that the root of one of them may he exactly talcen. Take this root for a coejjicient of the other factor affected hy the proper sign, 1. Find the simplest form of l/8. v/8"= l/4 X 2 = V^y. V^2"= 2V2. Ans. 2. Find the simplest form of i^ IG. if 16 = f 8"^ = #^8 X 1?'2 = 2 #"2; Ans. 3. Simplify 1/T8, 1/32, V^'SO, VtI, Vm, VWz, 1/2OO, l/242, and t/288. Ans. 3 V^, 4 V% 5 V2, 6 1/2, 8 V% 9 V^, &c. 4. Simplify l/l2, 1/27, l/48, j/tS, i/108, t/147, and l/l92. ^;;s. 2 1/3", 3 i/'3, 4 V% &c. 5. Simplify 1/20, V''28, Vu, V 117, 1/68. ^Tis. 2 1/5, 2 1/7, 2 i/n, 3 1/13, and 2 i/Vf. 6. Simplify i/TG, "/M, t/2048, i/84, l/l89, j/iSO, and t/338. 7. Simplify t/392, t/675, t/1280, i/2023, i/3564, and l/4693. 8. Simplify 1^54, ^128, 1^250, 1^432, #"(386, and if l024. A?is. 3 f2, 4 r 2, 5 1^2, 6 #^2, 7 1^2; and 8 1^2. 9. Simplify 1^81, #"135, 1^189, 1^ 29f, and 1^351 10. Simplify 1^320, ^448, #"704, 1^'lT25, and 1^2376. 11. What is the square root of 8 ? Ans. 21/2 = 2 X 1.4H21 = 2.82842. TREATMENT OF RADICALS. 161 12. What is the square root of 18 ? Ans. 3i/2 = 3 X 1.41421 = 4.24463. 13. What is the square root of 32, 50, 72, 128, 162, and 200? 14. Find the numerical value of all the preceding problems by the tables. CASE II. 156. To place a surd involving a vulgar fraction in its sim- plest form. Multiple/ tlie numerator and denominator of the fraction hy such a nuraher as tcill render the denominator a j)erfect squai'e, cuhe, &c. as the case may require. Simplify the numerator by Case i., and write the required root of the denominator under the coefficient. 1. Find the simplest form of V ^^' VJ^ = l/i^ = t/^^ X l/3 = 1 1/3 2. Find the simplest form of t/^. l/| = l/^ = Vjx^ = l/T X l/6 = 1 1/6 3. Simplify VI Vh y% Vh V~h^ &c. Ans. 1t/3, ii/5, IV\ 4v"7, J,t/11, &c 4. Simplify V%VJ^, V~ff^, l/||, VVi, and l/|l. Ans.J^ V% J- VG, ^ V\ -h V\^~^V\ and -^-^V^. ■ 5. Simplify Vh ^'h -^'h ^h ^h and V^. Ans. 1 1/2; ^ ifi", J- f9, -\ f/% 4 r49, and l fS. 6. Simplify V% Vh Vll -/il, V'i, iZ/y, VJ-,, and VI 'Ans. il-To, |t/14, |i/3, Jl/22, ^Vb, f^V^Z, /.t/B, and \V^. 7. Simplify 9 l/p, 6 l/f|, 5 Vj-^, 10 1/^, and 7 1/^. Jns. 4 1^3", 5 V^, i VlO, V^, and | l/2i. 8. What is the square root of | ? Ans. l/r = ^ 1 '2 = Jl of 1.41421 = .7071. 11 162 T K E A T M E N T T R A D I C A L S. 9. What is the square root of ^ ? Ans. J 1/2 = .35355, 10. What is the cube root of 4 ? Ans. fl= A#r= lof 1.5874 = .7937. 11. What is the yalue of 9 v'p ? Ans. 4l/8 = 4 x 1.73204 = G.0281G. 12. What is the value of VJ^ ? Ans. -g^ Vdd = .20107. 13. What is the value of l/|f ? Ans. f^Vl = .15118. 14. What is the value of V'X^ ? J.ns. %V'b = 4.02492. CASE iir. 15^. To find the root of a positive algebraic monomial. Take the' required root of the coefficient, and divide the expo nent of each letter by the index of the root. 1. Find the square root of 49a*y*'. Ans. 1x-y^ 2. What is the value of T/289xy^ Ans. llxi/. 3. AVhat are the values of VMl^, V 44L^», and •l/2MxV ? 4. What are the values of t/324^^ 1/400^*, and 1/484^^2 ? 5. What are the values of l^8^^ l/64xy^ and i/'lGxY ? 6. What are the values of t/243xy, Vl024:xV, and ^^729:^^' CASE IV. 15S. To find the root of a negative algebraic monomial. (1.) ;}- a X -|- «= + a^, and — « X — a = -\- «*, .-. TJie even root of a negative quantity is impossible. (2.) — a X — « X — a = — a^ .-. 1^— a^ — — a. Hence, The odd root of a negative quantity is negative. (3.) ± a X dca = a^ .-. l/a^ = d= a. Hence, TJie even root of a j^ositive quantity is positive or negative. When the double sign ± occurs two or more times in the same equation, the upper signs must not be confounded with the lower. Thus, db « qp Z> = it c means + a — b = c, ov — a -\- b = — c. 1. Find the cube root of — Sx^^y-*. Ans. — 2x*y^. T R E A T M E N T T 11 A D I C A L S. 163 2. What is the value of ^/— 32xy°? Ans. ^2x7/\ 3. What are the values of t/lQxy, f— 'Zlx^f, and l/l69xy ? 4. What are the values of '^'^V'; V^^^\ and i/Qo^^^ ? 6. W^hat are the values of J/_16a*, ^— 2187x'i*j/^S and l/529^? 6. What are the values of V^^, T^6i^V^^ and T^— 216xy2 ? CASE V. 159. To simplify algebraic monomials whose root cannot be ex- actly taken. Swiplifi/ the numerical part l>y I. or II. Divide each exponent by the index of the root to be taken, and write the letters with the quotient for an exponent on the outside of the radical sign, and the letters with the remainder for an exponent under the radical sign. EXAMPLES. 1. AVhat is the simplest form of j/lSxy? Ans. 3xyi/2. 2. What is the simplest form of l^54xy ? Ans. Zxy 1^2p. 8. Simplify {/Z2^, ^192xy, l/py, and if py. 4. Simplify l/py, V j\x*y\ V^^^xY, and flQxY- 5. Simplify lfl28x^y, ^^OSy^, T^py, and /py. 6. Simplify i/py^ l/py^ 5l/py, and 15l/8^. 7. Simplify l/44xy, l/75xy, l/Sxy^ and 7 l/28xy. 8. Simplify l/50xy, l/200xy, |/243^, and l/|^p: 9. Simplify l/Jxy, l/fxy, l/fa/^; and Vxyz"^. CASE Yl. 160. To add radical quantities. Simplifi/ each expression hy Case V. If the radical parts are then the same, add the coejicients and prefix the stun to the common radical. 164 TKEATMENT OF RADICALS. If the radical parts are not the same, unite the quantities hy the proper sign. EXAMPLES. 1. Add V^TI^\ t/48^^ VI'^xY and Vl^'^xy. Operation. l/75^^ = bx't/ VS \/ld2xY = Sx'^f/V^ 20x^1/ 1/3 = the sum. 2. Add together #^54^, and f^VlSx'y. Operation. •^54^ =3.ry f/2f lfl28x»y= 4:ry "if 25/^ (3^^+4xy) if %^ = the sum. 8. Add together l/320xy, l/lSOxy, l'''245xy, and l/20xy. J.71S. 25:ryi/5. 4. Add together l/28xy, -/CS^y, and T/ll2xy. 5. Add together l/99xy, l/275xy, l/l76xy, and v^Iixy. 6. Add together V py, l^py, l/IiOxy, and y'315xy. 7. Add together V'py, V^PV; V^PV; and V'-^^xy. 8. Add together l/fxy, l/py, i T/84xy, and Jl/l89xy. 9. Add together l/py, l/py, T/|gxy, and l/20^. 10. Add together f'Uxy, if 192^, 1^81^, and #'375xy. 11. Add together ]/32xy, {/l62xy, f/512.ty, and i/l250^«. 12. Add together f^py, J^^V^j f^S^j ^^^ f^^^- TREATMENT OF RADICALS. 165 13. Add together v'^xY, VixY, and \/Jx^^. 14. Add together V-\xYp V^x^tj^, and V'^^x^- 15. Add together fMxY, #^32xy, and f\OxY- 16. Add together l/20xy, l/l^^y, l/86xy, and l/iixy. 17. Add together VlxYz'j V'lx-fz^ and t/^^V^'. 18. Add together l/49xy, l/6ixy, V^lxY, and v'^^y. 19. Add together \'^2x^ + ^.ry + 2^^. and ■l/2x^ — 4:ry + 2_y'. J.71S. 2xl/¥- CASE VII. 161, To subtract radical quantities. Simjplify each expression hi/ Case Y. If the radical parts are then the same^ subtract the coefficients and prefix the difference to the common radical. If the radical parts are not the same, express the difference hy the projyer sign. EXAMPLES. 1. From Vl^xY take V^xY- Operation. VlSxy = 3j7/y2 V^"' = 'IxyV^ xyy2i = the difference. 2. From l/py take l/py. Ans. J^V i/2. 3. From l/63:cy take l/28xy. ^ws. xY i/7. 4. From l/SOxy take l/20xy. 5. From l/fxy take V^xY- 6. From l/py take l/f^y. 7. From l/||xy take i/'if^y. 8. From t/3^« take l/py. 9. From if 27aV take if 8^». 10. From #'l92^iy5 ^^ke f^^i^^. 166 TREATMENT OF H A D I C A L S. 11. From f/lO^y take \/ x^if. 12. From i/82xV^ take V^^y^^- 13. From f/ei^^ take Vxh/''. 14. From "/j^^^ take l/j^xy. 15. From l/^^^y take l/^^cc*/. 16. From 2l/|"take 3l/|. 17 From V% take j/^^. 18. From V'lx' + 4£cy + 2^ take l/2x2 _ 4^^ ^ 4^2^ ^,^5. 2?/i/2, CASE VIII. 162. To multiply radical quantities; multiply tlie coeflicients and also the radicals. Simjolifi/ the result hy Case V. EXAMPLES. 1. Multiply 7 l/5.x-y by 4 l/fx^. Operation. 7 V^x^if 41/ x-y 28 l/2xy = 28xy 1/2 = product. 2. Multiply -I i/pvi^yil/p^ Operation. 3 V-i^xhfz^ = -Jt?/2; i/j^% = -^-^xyz l/l5 = product. 3. Multiply 4-^2 by 2l^4. Ans. 16. 4. Multiply t/.t by l/o^. Ans. x. 5. Multiply 3 V^ by 4 l/20x. ^7is. 120;r l^y. 6. Multiply 7 t/3^ by 8 l/l6pp ^«s. 392xj^» i/^ 7. Multiply 5 Vlxy^ by -| l/27a^. ^ws. ^xy"^ j/y- 8. Multiply 5 l/|^ by y^^ 1/40^ ^«s. 2xy\ TllEATMENT OT 11 A D I C .1 L S. IGT 9. Multiply V'"^^^^~+1 by j/x^-}- 1. Ans. a;^ + 1. 10. Multiply l/:r -f 1 by l/ic — 1. .I'^s. j/ic^ — 1- 11. Multiply Vx^ + a- by V x^ — 12. Multiply "l/x + l/y by l/a: — "j/?/. 13. Multiply i/3 + l/2 by l/3 — V 2. 14. Multiply 1/7 + 1/24 by V7 — VU- 15. Multiply 1/5 + 1^2 by 1 '5 + V2. 16. Multiply V'5.>: + V4:]/ by l^Sx — v'J^. 17. Multiply 1/7+ 1/3 by V7 — VS. 18. Multiply 1/15 + l/'2 by l/l5 + y/'J. 19. Multiply l/2x + V % by t/2x' + |/3y. 20. Multiply 1/^2 + vTS by l/2 + y 18. 21. Multiply i/5x + VSi/ by l/5x + l/Sy. 22. Multiply 4 i/2^ — 3 VT?/ by 5 j/si + 7 T/l8y. 23. Multiply 5 V2x + 3 1/1% by 3 i/ISTc — 5 ;/%. 24. Multiply j/x + y'l/ by l/:£ — V^. 2o. Multiply .- — by ._ Vx — 7 l/rc — 4 26. Multiply ^" + Syj$--'^ l/a: + 7 l^x — 7 27. Multiply X 4- V ^y -f- 3' by :« — V xi/ -\- 1/. Ans. x^-{- xi/ +y^ 28. Multiply — a + l/a^^+l; by —a — Vo^+h. Ans. — h. 29. Multiply a + y a^ + Z> by a — y a'^ + b. Ans. — &. 30. Multiply 6 + l/36 + 2 by 6 — |/36 — 2. Ans. — 2 31. Multiply yx + ^ + 1/ x — 3^ by I'^x -\- y — v x — y. Ans. 2y. 32. Multiply Vx + Z by Vx — o, also Vx + 2 by j/^— 2; and l/o; + 4 by "l/x — 4. 168 TREATMENT OF RADICALS. CASE IX. 163. To divide radical quantities. Simjplift/ each expression hy Case Y. Prefix the quotient of the coefficients to the quotient of the radical parts, and, if necessary, again simplify for the final result. EXAMPLES. 1. Divide i l/28j:y by | V^^xy\ Operation. I l/28xy = |:r^2 1/7^ I VM^ = 2yVTx g^y l/y = quotient. 2. Divide t/20:^ by V^^. Ans. | l/IoZ 3. Divide Vx^ -\- y^xij + Vxy^ by l/^. J.ws. i/^ + |/y -j- y. 4. Divide V'^^xY + t/63^ + VU2^ by j/T^ 5. Divide 4 ]/2^-^ by \ Vpy. Ans. 2 l/2^. 6. Divide 1/I6 + V 4 by 4 |/J. ^7is. |. 7. Divide v''l9 — l/9~by 2. Ans. 2. 8. Divide x'' + o-y -|- y^ by cc + V '-^^ -f y- •^•^'^s- -^ — vscy + y* „ ™ ,^, , ,1/54 81/50 121/28 151/378 l/J ,l/| 9. Innd the values 01 ——3, ;^:::-, — ^, 7=:-; — ^,and — Z.. VQ 41/2 SV7 bVQ V\ Vi Ans. 3, 10, 8, 9 VY, 1 1/6", and |. ia T7- wi 1 f ?l/l8 fV"! -ij/J 1/24-31/1 10. Find the values of ^^ — -^, ^ — -i, -- — p^, and !— = — ^. A 1/2 1 1/3' J. -,/i' J -i/i Ans. 4, 1 1/5, 2, and 10. TREATMENT OF RADICALS. 169 CASE X. 164. To reduce a fraction, whose denominator is a binomial containing radical quantities, to an equivalent fraction without radicals in the denominator. Multiply both the niunerator and denominator of the fraction hy the denominator with one sign changed. EXAMPLES. 3 1. Reduce the fraction — = ;=. to a fraction having no radi- Vb + i/2 cals in the denominator. Operation, 3 ^ i/5-t/2 ^ 31/5-31/2^^- __^^^3^^3^^ l/5 + 1/2 1/5 — 1/2 3 2 2. Find the value of 1/3 + 1/5 Ans. ^^^-^^ = v^ - V^3- = .50401. — 2 3 3. Find the value of ;=. Ans, .31866. 8 + 1/2 5 8 5 4. Find the values of -=, — = 7=, 7^, and 7 _ i/40 1/3 - 1/7 9 - 1/8 1/3 + -/2 1/3 - 1/2' ;; -P- ^ .1, 1 1/3-1/21/5 + 1/3 1/5-1/3 5. Find the values of —= ^, —~ 7-, and -— = 7=. 1/3 + 1/2 1/5 — 1/3 1/5 + 1/3 6. Find the values of — -= 7^, and —^ 1/5 — 1/2 Vh — VS Ans. 3.65028, 2.80588. n 1?' A.i. 1 P 1^7 + 1/5 , 2i/n-3^13 7. Find the values of . 1.. and — — r* 1/7 _ 1/5 21/11 + 3l/l3 15 170 T E E A T M E .N T OF RADICAL S. a \/a^ x'^ 8. Given — • izzzzzzij ^^ ^^^^ ^^^ denominator of radicals 2a2 — a:' — 2al'^a' — x« -4?is. 9. Given , to tree the denominator oi radicals yx — yx — a , 2x — a -{- 2 Vx^ — ax Ans. • ■ • CASE XI. 165. To find the square root of a polynomial. 1. If necessary, arrange the polynomial witli reference to a given letter, and place the square root of the first term to the right of the polynomial, for the first term of the root. Square this term, and subtract it from the polynomial. 2. Double this first term of the root, and place it on the left of the remainder for a part of the divisor; divide the first term of the remainder by this double of the root, and place the quotient in the root as the second term, and also at the right of the divisor. 3. Multiply the wJioIe divisor by the second term of the root, and subtract the product from the remainder. 4. Double the whole of the root, and write the result to the left of the remainder, as a part of the divisor; divide the first term of the remainder by the first term of the partial divisor, and place the result as the third term of the root, and also at the right of the partial divisor. 5. Multiply the whole divisor by the third term of the root, and subtract the product from the last remainder. 6. In a similar manner find other term». T K E A T M E N T OF K A D I C A L S. 171 EXAMPLES. 1. Vfhat is the square root of x^ + 4x3 4. Gx^ -f 4x -f 1 ? Ojieration. x* + 4x3 -i- 6x2 ^ 4.^ ^ X I x2 4- 2x + 1. J[?is. X* 2x» + 2x 4x3 + 6x2 -f 4x + 1 4x3 _|_ 4^2 2x» 4- 4x + 1 I 2x2 4. 4^ _^ 1 I 2x2 4- 4a: 4- 1 2. What is the square root of x* — 2x3 4- Sx' — 2x 4- 1 ? Operation. X* — 2x3 4- 3x2 — 2x 4- 1 I ^2 __ ^ ^ 1 X* 2x2 — X — 2x3 4- 3x2 — 2x 4- 1 — 2x3 4- x2 2x2- -2x4-1 2x2 — 2x 4- 1 2x2 _ 2x 4- 1 3. What is the square root of a:2 4- 2x' 4- 3x* 4- 2x^ 4- x« ? Ans. X (1 4- a; 4- ^')' 4. What is the square root of x2 — 2x^ 4- 3x* — 2x5 4- x"' ? Ans. X (1 — X 4- ^^)« 5. AVhat is the square root of x' 4- 4^2 _^ 9^2 _^ 4^^ ^ q^^ _^ 12^/^? Ans. x + 2i/ + Sz. 6. What is the square root of x* 4- 3x2 _j_ 2x^ _ 2x 4- 1 ? A7IS. X2 4- X 1. 7. What is the square root of x2 4- 2xy 4- ^2 4, 2x 4- 2^ 4- 1 ? -4.715. cc 4- y + !• 8. What is the square root of x^ — 2x^ — x* 4- 4ar' — x' ~ 2a: 4-1? ^«s. x3 __ ^3 __ 2c 4- 1. 172 I M A G I .\ A K Y Q U A X T I T I E S. 9. What is the square root of 1 — x"^ ? X^ T"* QJ^ Ans. 1 _ - - 11 -. _ , &c. MISCELLANEOUS. 166. 1. Complete the square of x- -f ^j^.r, and take the square root. ( Vide 132, 8.) Ans. V x- + 'Zjyx -\- p^ — x -\- p. 2. Complete the square of x- -f 2x, and take the square root. Ans. X -\- 1. o. Complete the square of :c' — 3.r^ and take the square root. Ans. X — 4. 4. Complete the square of x^ j ^, and take the square root. An^. X -' 2r — 71V CASE XII. 1 IMAGINARY QUANTITIES. 167. An imaginary quantity is an indicated even root of a negative quantity. Its general form is — ± A V - 1 where A is either rational or radical. The rules for multiplying imaginary quantities depend upon the fundamental principle that the square root of a quantity multi- plied hy its square root produces the quantify itself. Thus, From this we have the following table of equations. — \/— xX s/—y=--\/z v/HTx v^vZ—l =— v^X— "•= v^. 4. IMAGINARY QUANTITIES. 173 —\/^ X—\/-~i=—\/x\/^ X — >/z\/^ = xX — 1= — 2;.6. \^~z X— v/^= \/?v/^l X — y/z\/^l = — a; X — 1 = a; . 7. '-\/^X v/^=— v^^n/^ X \^y/^ = —z X — 1= X . S. Hence, like signs, on the outside of two imaginary quantities, produce minus, and unliJce signs, p?i/— 3 by — 1 4- |/— 3. Operation. -1-1- /ir3 1- y/zr^ _ 2 — 2 /^^^. ^ns. 2. Multiply — 2 — 2 /^Ts" by — 1 -f- /^^S Operation. _ 2 — 2 /^^^ -1+ /^Ts 2 4-2 /— 3 _ 2 /^I^ + 6 8 = 24-6. Am. lb* 174 IMAGINARY QUANTITIES. By comparing these two examples we see that ( — 1 + •>/ — 3)^ = 8. Wc should also find (— 1 — V^^oJ = 8. 3. Multiply 1/1^2 -f. /^ITl ^ y/'ZT^ by n/^^T^ — V^^ — v'z:5. operation. — 2 — -1/2 — i/ro + 1 + 1/2 + i/lo + 1/6 + 5 + t/5 4 + 2i/5 = A71S. 4. Multiply 5i/7^ + 81/^^28 + 2 i/7I~7 by 61/^^4 — 3l/^9 + St/ITt. 5 i/ZTl + 3 v/Z:28 + 2 i/^^l = 5 i/^Tl + 8 t/^=^ . . . by VI. 61/zn _ 3 i/iTg + 5 T/:r7 = 3 ]/:rT + 5 v":^ . . .by vii - 15 — 24 1/7 — 25t/7--280 — 295 — 49 Vl = ^ris. 6. Multiply V— X + V— xi/ + ;/— ?/ by V— x — i^— aj/ + V — y. Ans. — X -\- xy — y — 2 yxy. 6. Find the value of (l/^^ + V^^y. Ans. — 7 — 2 i/ICh 7. Find the value of (l/^^ + V^^y. Ans. — 13 — 4 /la 8. Find the value of (V' — 27 + 1/^^^)^ JLns. — 48. 9. Find the value of (;/— ~28" — V^^^y. Ans. — 7. 10. Find the value of l/^HJ — •/7r5)l tIjjs. —18 + 2 l/65^ IMAGINARY QUANTITIES. 175 11. Find the value of (v''^^^ -\- V^^) (V-l-V^). Arts. 1. 12. Find the value of (V^^ + V^^) (l/^^ — V^^'). Arts. 2 13. Find the value of (— 1 — V^^f. Ans. (— 1 — V^^^y = (— 1)' — 3 (— 1)^ V^o + 3 (— 1) l/^Ts^ ~ /31;3 = 8. 14. Find the value of (— 1 + V^^f- Ans. 8. 15. Find the different powers of l/ — 1. Ans. V — 1 X V — 1 = — 1 —2.nd power of V — 1. — 1 X t/^^^ = — 1^^^^ = 3rfZ ^^ " i/^^nr. — ]/_ 1 X V— 1 = 1 = 4^/i " " t/— 1. 1 X v^^ = i/^^^"! = 5^A '^ " i/^^nr. And since the quantity and its 5th power are the same, it follows that the 2nd and 6th powers must be the same, so also must the Srd and 7th, the 4th and 8th, &c., be alike. 16. Find the value of (2 + 5 V— 1)* + (2 — 5 v'— 1)*. Ans. 82. 17. Find the value of (1 + V — WJ — (1 — v' — 11)^ An^. = 992. IS. Find the value of 7= 1 7^=^- ^"s- 2. i(l-f/-l) 1(1- /-I) CHAPTER VII. EQUATIONS OF THE SECOND DEGREE. 168, An equation of the second degree is one involving the second power of the unknown quantity. Such equations may be complete or incomp)lcte. A complete equation is one involving both the first and second degrees of the unknown quantity. Thus, x^ + ^^^ == ?• An incomplete equation is one involving only the second degree of the unknown quantity. Thus, 2ax^ = q. 169. To find the value of x in an incomplete equation. Proceed exactly as with simple equations of one unlcnown quantity^ and tahe the square root of the final equation for the value of x. EXAMPLES. 1. Given x' — 192 = -r, to find the values of x. 4' Operation. X ' — 192 = — (1) = given equation. X 4 4x' - 768 = x» (2) = (1) X 4. Sx' = 768 (3) = (2) reduced. x» = 256 (4) = (3) -T- 3. X = d= 16 (5) = l/(4). ( Vide 15S. 3.) ne EQUATIONS or THE SECOND DEGREE. 177 2. Given 8 = f-lO, to find the values of x. 3 9 x"^ 3. Given 8 4- 5x' = \- ^x"" + 28^ to find the values of x. 5 Ans. cc = ± 9. 3 values of x. Ans. £c s= db 5. X ^, oX 4. Given — = — 671 H , to find x. Ans. x = dz 7. 8 ^ ^ 2 /y2 y /y-«2 5. Given — 4- 4 = ■ 15i, to find x. Ans-. a: = db 3. 5 3 ' 6. Given ?^!_±_5 _ '^' + ^^ = 117 - 5x^ to find x. 8 3 Ans. cc = ± 5. 7. Given ^' + 12 = — + 37|, to find x. Ans. x = ±z7. 3 7 8. Given - - 1 = 3 9. Given ^ + ^ ic'^ — 7x 4^2 = 1-5 to find X. Ans. x = da S. 27 ... trt Tinn 'y ^2 ^ 7_^ ^2 _ 73 Ans. ic =s ± 9. 10. Given (x^ -f 1)^ = 25, to find x. Operation, (x^ -f- 1)^ = 25 (1) = given equation, x^ + 1 = ± 5 (2) = 1/(1). x" = 4 or — 6 (3) = (2) reduced. X = ± 2 or ± t/^Tg (4) = 1/(3). p. (^ + 18)' Given -^ 28 = 4^2 — , to find 63' Operation. a;. (x + 18)2 4^2 28 63 (1) = given equation {x -f 18)2 _ 4x' 4 9 (2) = (1) X 7. a- 4- 18 _ ^ 2 2x 3 (3) = v/(2) X = 54 or — 1^ (4) = (3) reduced. 178 EQUATIONS OF THE SECOND DEGREE. 12. Given — - = — , to find x. Ans. cc = 5 or — i|. 80 1 6S ? 13. Given = 5, to find x. Ans. cc = 15 or 4i.f . 5(a; — 7)2 3^2 ^' ^, ^. rz;2 -f 3 2x2 — 13 ^ , 14. Given = , to find x. Ans. x = ± 4. 2x2 — 13 a;2 4- 3 15. Given ax^ = h, to find x. Ans x = =fc ^/- = - ■l/a6. 16. Given x^ -f ai> = 7x2, ^q gj^^j ^ ^^^^ ^ = | T^6a6. 17. Given (x -f a)2 = 2ax + h, to find x. Ans. x = ± i/fe—a^. 18. Given ^ ^ ^ = c\ to find x. (x2 — by Ans. X = =fc J'L±^ or v/ ^^ - ^ . \c - 1 N/c 4- 1 + in n- 5c + a X — a 10«2 19. Given H = , to find x. X— a x4-<^ ^^ — <^^ ^?is. X = dz 2a. rtrt y-.. •'^ — ^ ^ — 2x x2 -f Jx „ - 20. Given = , to find x. a X — a x} — a? Ans. X = ± y/ ah. PROBLEMS PRODUCING INCOMPLETE EQUATIONS OF THE SECOND DEGREE. I'yO. 1. Find a number whose | multiplied by its | will be equal to 2520. Let X = the number: then -- x — = = 2520. 6 7 42 # .-. X = ± 84. 2. Two numbers are to each other as 3 to 7, and the sum of their squares is 522. What are the numbers? Let 3x and 7x = the numbers. The equation is 9x2 ^ i^(^^. ^ 522. .-. X = 3, and 8.r = 9, and 7x = 21. .4ns. 9 and 21. EQUATIONS OF THE SECOND DEGREE. 179 8. Two numbers are to eacli other as | to |, and the differ- ence of their squares is 153. What are the numbers ? Ans. 24 and 27. 4. If 4 be added to a certain numbei^and also subtracted,, the product of the sum and difference will be 609. "What is the number? Ans. 25. 5. If 9 be added to a certain number and also subtracted, | of the product of the sum and difference will be 162. What is the number? Ans. 18. 6. A and B start from different points at the same time, and travel towards each other. On meeting, A has travelled 20 miles farther than B, and A would have gone B's distance in 75 hours, but B would have travelled A's distance in 108 hours. What distance had been travelled by each ? Let X = B's distance; then cc -f 20 = A's distance .-. — = A's progress per hour, and ^±^ = B's " '' " 108 Now A's distance : B's distance : : A's hourly progress : B's hourly progress; or a: X -f 20 75 108 (x_+_20y_ 0^ 108 ~ 75' Ans. A 120, B 100 miles. 7. Two numbers are to each other as a is to h, and the sura of their squares is c. What are the numbers ? The equation is a^x^ -f h^x- = c, whence X = ^ ; then the numbers are —-===. and —-==^ 180 EQUATIONS OF THE SECOND DEGREE. 8. Two numbers are to each other as a to h, and the dif- ference of their squares is c. What are the numbers ? . aye , ^l/c" Ans. : and 9. A man drew from a cask of wine containing a gallons a certain quantity^ and then filled it with water. He then drew of the mixture the same number of gallons as before, and again filled the cask with water. Having done the same a third and fourth time, he has h gallons of wine left in the cask. How many gallons of wine were drawn off each time ? 3J 1211 Ii2j. 131 1 Ans. a4 (a4 _54)^a4 h"^ (a4 —Z>4)^cj4 ^4 (^<^4 _ 2,4)^ 54 (^^4 __ 2,4). If a = 256 and h = 81, then 64, 48, ZQ, and 27 are the answers. If a = 625 and h = 256, then 125, 100, 80, and 64 are the answers. COMPLETE EQUATIONS OF THE SECOND DEGREE. lYlt To solve the equation x2 -f 22)x = q, (1) Vide § 132. (3.) By adding j)^ to both sides of this equation (Ax. IT.), we have x^ + 2j9x + ^2 ^ p2 _j_ ^ ^2) By extracting the square root of both numbers (Ax. VI.), we have X -{-p = ±: Vp' -f q (3) By transposing j^, we have X = — p ± l/f^+q (4) If the upper sign is employed, the answer is called the Jirst root. If the loiver sign is employed, the answer is called the second root. Xli*2» To solve the equation x^ — 2^)0? == q. (V) ; EQUATIONS OF THE SECOND DEGBEE. 181 By taking the same steps as above, we have X = p ±1 Vp' + q. (2) 1'9'3. To solve the equation x^ -\- 2px = — q. (1) By taking the same steps as in § 171, we have X = — P ^ Vp"* — q- (2) X'5'4. To solve the equation . x^ — 2px = — q. (1) By taking the same steps again, we have x =pdci l/p2 _ ^^ (^2) ll'S. The four cases above solved are comprehended in the following general rule for the solution of complete equations of the second degree. RULE. X is equal to half the coefficient of the second term taken with a contrary sign, db the square root of the square of this half coefficient united to the second memhcr of the equation as indicated hy its sign. EXAMPLES. 1. Given x^ + 4x = 21, to find x. Ans. cc = — 2 ± l/4 + 21, a: = 3 or — 7. 2. Given x" -\- ^x = 20, to find x. Ans. cc = — 4 d= t/16 + 20 = 2 or — 10. 3. Given x^ -f lOx = 11, to find x. Ans. a; = — 5 ± t/25 + 11 = 1 or — 11. 4. Given x^ -|- 3ic = 28, to find x. Ans. X = — I db t/| + 28 = 4 or — 7. 6. Given cc^ — 4x = 21, to find x. Ans. x = 2±: 1/4 + 21 = 7 or — 8. 16 182 EQUATIONS or THE SECOND DLGREE. 6. Given x" — 8x == 20, to find x. Ans. a: = 4 ± T/IG 4- 20 = 10 or — 2 7. Given x^ — \0x = 11, to find x. Ans. X = 5 rh t/25 + 11 = 11 or — 1 , 8. Given x^ — ox = 28, to find x. Ans. X = Iziz l^l 4- 28 = 7 or — 4. 9. Given x"^ -{- Qx = — 8, to find x. Ans. X = — S dz V9 _ 8 = — 2 or — 4. 10. Given x"^ -{- 8x = — 15, to find x. Alls. X = — 4 ± v 16 — 15 = — 3 or — - 5. 11. Given rc^ -f lOx = — 16, to find x. Ans. x = — 6 ziz 1/25 — 16 = — 2 or — 8. 12. Given x^ -f 5x =• — 6, to find .t. Ans. X =z — ^ ziz y -f — 6 = — 2 or — 3. 13. Given x"^ — Qx = — 8, to find x. A71S. X = S dz 1-^9 — 8 = 4 or 2. 14. Given x^ — 8x = — 7 to find x. Ans. a:; = 4 =fc 1/ 16 — 7 = 7 or 1. 15. Given x' — Ux = — 28, to find x. Ans. X = u ± i/i|i — 28 = 7 or 4. 16. Given x"^ — 15x = — 56, to find x. Ans. X = \f ± -[/2|5 _ 5(3 == 8 or 7. 17. x^ -{- x = 6. Ans. 2, — 3. 18. x^ + 2x = 8. Ans. 2, — 4. 19. a;2 + 3x = 18. Ans. 3, — 6. 20. a;2 + 4x = 32. Ans. 4, — 8. 21. a:' + 5x = 50.^«s. 5, — 10. 22. a:' -f Gj- = 27. Ans. 3, — 9. 23. x^ + ISx = 68.^ns. 4, — 17. 24. x' + 15x = 154. Ans. 7, — 22. 25. x^ -f 20.r = 125. Ans. 5, — 25. 26. .T^-f 21.r= 196.ylrjs.7,— 28. E Q U A T i :.' S OF TILL S £ C -N I) D L G li E E. 183 27. x'- — X ^ 132. Ans. — 11, + 12. 28. x^ — 4:x = d2.Ans. + 8, — 4. 29. x"" — Gx = 27. Ans. — 3, + 9. 30. x"" — 28x = 29. Ans. — 1,4- 29. 31. X' — ox = — 6. ^ns. + 3, 4- 2. 32. x2 — 7x = — 12. ^«s. 3, 4. 33. cc^ — llx = — 30. Ans. 5, 6. 34. a;2 — 16x = — 63. Ans. 7, 9. 35. a;2 — 20x = — 96. .4ns. 8, 12. 36. x"" — 36x = — 320. Ans. 20, 16. 37. x^ — 38x = — 240. Ans. 8, 30. 38. x-" -f 5x = — 6. ^l;is. _ 2, — 3. 39. x' + 14.r = — 45. Ans. —5,-9. 40. x' + Sx = — 15. J.92S. — 5, — 3. 41. a;2 4- lO.r = — 21. Ans. — 75 — 3. 42. x' 4- 14.r = — 48. Ans. — 8; — 6. 43. .^2 4- 20.1- = — 36. Ans. — 18, — 2. 44. x' 4- 16^ = — 63. Ans. — 9,-7. lt€. All the equations in §§ 171 and 174 inclusive are called reduced equations. Before applying the rule in § 175, the given equation must be brought to the form of one of the reduced equations^ by any process thought to be most convenient; that is, 1. All the terms involving -s? must he united in one terin, ichicJi must stand first. 2. All the terms involving x must he united in one term, icJiich must stand second. 3. All the remaining terms must he placed to the right of the sign of equality , united in as few terms as possible. 4. Divide the whole equation hy the coefficient of x'. 5. If then s? has the sign — , change all the signs of the equation. 184 EQUATIONS OF THE SECOND DEGREE. EXAMPLES. 1. Given - ?^ + 22a: - 15 = — - 2Sx + 30 (1), to find x. 3 3 ' w^ Ojperation. — 5x* 4- ^Ox = 45 (2) = (1) transposed and united. — x^ -}- lQx= 9 (3) = (2) -r- 5. X — 10a: = — 9 (4) = (3) with signs changed, a: = 5 ± l/25 — 9 = 9 or 1. 2. Given 2x=' + 8x + 7 = ^ - |- + 197 (1), to find x. 16x2 + 64a: + 56 = lOo: ^ x^ + 1576 (2) = (1) cleared of fractions. 170:^ + 54x = 1520. (3) , . 54x 1520 x'+ — = (4) 17 17 ^ ^ 27 . ^/729 1520 _ X— db\ (5) 17 ^289 17 x=.-^ ± — =8 or- 11 3 (6) 17 17 3. Given Sx^ -f 3a: = 530, to find a;. (1) x^ + 3a: 5 3 10 =b 106 (2) 9 ^100 + 106 (3) X =s 10 or - 10|. (4) 4. Sx^ — 2x = 8. Ans. 2, — If 6. 2a;2 — 7a: = 72. Ans. S, — 41. 6. 2a: — 5x2 = — 3^ Ans. 1, — -|. 7. 17a: — 4a:2 = 18. Ans. 2, 21. 8. 3a; - 21a:2 = - 78 Ans. 2, — If EQUATIONS OF THE SECOND DEGREE. 185 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 2L. 23. 24. 25. 26. 27. 28. O.^ "Y" ^ — — OX. x^ + 12a: — 16 = 92. ix 5x2 4- — = 7^2 _ 51, 2 a: 5x 4 24 ^^ 5x2 10 X = 78. 2 14 — 2x 6x2 _ ^ ^ 92. 22 "9* 4x - ^tl::^ = 14. X + 1 x2 - ^ = - ^ 6 2x2 __ 14^^ ^ 16, X H- 4 7 — X = -1 + 4x + 7 X 3x - 3x o. X — 3 2 2x2 _ 30a; + 3 = — X2 4- 3y3gX 2 O X 3 X'' — ox = -- — -z. 4 4 x2 + 18x = — 80. X -|- Ox -j— J- ^:= Sx 8x2 + - = - 17.^ _ 34 5 22 — X 15 — X 3 20 X + 3 + X — 6 7x X X + 3 24 = f.3 4" X + X ~ 1 = 3x - 4. Ans. J or |, Ans. 6 or ~ 18. J.ns. 6, 4|. J.?is. ^, ^, ^?is. 4, — 3y9^. J.ns. 3, Ij. ^?zs. 4, — 3|. Ans. 4, — 1|. ^«s. 8, — 1. Ans. 21, 5. ^?is. 4, — 1. ^??s. 11, -^Q. J.71S. 3, J. Ans. — 10, — 8. Ans. — 10, — j^Q. Ans. -2,-1. Ans. 36, 12. Ans. 4, 1. ^Tis. 5, — 2. 16* 186 EQUATIONS OF THE SECOND DEGREE. 29. -^— = '' "^ ^ ■ Ans. 12, - 2. X -\- S 2x + 1 ■ 30. — ^ h ^^±-^ = 21. ^ns. 2, - 3. X -\- 1 X 31. — ' — ' — = — 1. Ans. 21, 0. 3 9 x — n 32. ?^^^ _ ^-±^ = 2. ^«.. 7, |. 8 — cc a^ — 2 83. — ^^ — = -. ^??s. n, - 13. X — 1 X -{- S 35 34. Given cc'^ — 2:c = 7, to find x. Ans. .T = 1 ± 2 v'2 = 3.82842 or — 1.82841. 35. Given x"^ — dx = 12, to find x. Ans. x = 6.772 or — 1.772. 36. Given x'' — Sx = 20, to find x. Ans. x = 6.2169 or — 3.2169. 37. Given x^ + 4x = 10, to find x. Ans. x = 1.7416 or — 5.7416. 38. Given llx"^ — x = 21, to find x A71S. x= 1.1412 or — 1.0824, 39. Given 13x=' + 2x = 100, to find x. Ans. X = 2.6975 or — 2.8514. 40. Given x"" — Sx = 14, to find x. Ans. x = 9.4772 or — 1.4772. 41. Given x^ -j- Sx = 8, to find x. J?is. x = 0.8989 or — 8.8989 42. Given x" — 2x = ~ 10, to find x. A71S. X = i±: Vl — 10 = 1 ± 3 V^^l. 43. Given x^ — 20.^ = — 104, to find x. Ans. X = 10 db 2 V^^^. 44. Given x"^ — lOx = — 26, to find x. Ans. x = 5 ± V — 1, 45. Given x"" — 12^ = — 72, to find x. Ans. a; = 6 (1 =b V — 1). 46. Given x^ — 18x = — 162, to find x. Ans. a; = 9 (1 db V— 1). 8 47. Given a: + - = 4, to find x. Ans. a: = 2 (1 d= i — 1). T K I N M I A L EQUATIONS. 187 TRINOMIAL EQUATIONS. fW* A trinomial equation is of the form where *i is any number whatever. To solve such an equation, apply the Eule under § 175 for the value of x"', after which the value of X is determined as in incomplete equations of the second degree. Thus, a;« = p rt \/p^ + q, and x =V p ± Vp^ + q- EXAMPLES. 1. Given x*^ + 4x2 ^ 32^ to find x. First, x^ == - 2 ± i/4 + 32 = — 2=h6 = 4or-8. Second, x = d= 2 or ± 2 V^^. 2. Given x* — lix" = — 1225, to find x. Ans. X = ± 7 or =t 5. 3. Given x* — 4x2 _ 9^ ^o find x. ^«s. x = ± 3 or db l/— 1. 4. Given x^ — 2x^ = — 1, to find x. Ans. x = 1. 5. Given x^ — 6x* = 160, to find x. Ans. X = rfc 2 or ± 2 l/— 1. I'YS. Sometimes an equation may be solved by considering several of its terms united, as the unknown quantity. EXAMPLES. 1. Given (1 + xy -1- (1 -f x) = 12, to find x. First, 1 4- X = — -J d= \/\ + 12 = 3 or — 4. Then X = 2 or — 5. 2. Given (3 -f x^)* — (3 + x'y = 240, to find x. First, (3 + x'y = i ± Vl + 240 = 16 or — 15. Then 3 -f x« = ± 4 or ± i/^U. Whence x= = 1 or ~ 7, or — 3 ± V— 15. X = ± 1, or ± 1/'- 7, or i l/— 3 d= l/— 15. 188 TRINOMIAL EQUATIONS. 3. Given (1 -f rr + x^ — 2 (1 + x + a:') = 143, to fiad x. First, l + a; + a;2 = ldb Vl + 148 = 13 or — 11. Then x" -{■ x = 12 or — 12. ,♦. cc == 3, or — 4, or a: = -l (— 1 =h j/— 47). 4. Given (x" — 4x)2 + 3 (x^ — 4x) = 0, to find x. Ans. X = 4, 3, or 1. 5. Given (1 + 2x + a;^)^ _ ^ (1 4. 2x + x^) = 254, to find x. Ans. X = 3, or — 5, or — 1 dz 1/ — 15|. 6. Given (x* + Sx^ + 16)* — 2 (x* + Sx^ + 16)^ == 389375, to find X. Ans. X = ± 1, or =h 3 l/— 1, or db 1/ — 4 db t/— 5, or dbl/3T±~P^l^ 7. Given cc* 4- ^' + ^" + ^ + 1 = 0; to find x. Solution. a-.^-f-x-fl +7~H — ^ = given equation divided X X by x"". a^-| --\-x -{■ -- -f 1 = last equation rearranged. a:s -f 2 + — + x-{ 1-1 = 2 2, added to each side x^ X of the last equation. |a;_|. — |_|. lx-\ )= 1 last equation factored. Whence x -\ = — » db 1 t/5 X z ^ a; = - 1(1 rip t/^ q= V - 10 zp 2i/5). 8. Given x* — x' -f- x'' — x + 1 = 0, to find x. Am. X =±: I (1 ± 1/5 d: 1/— lU d= 2 y^5 ). LITERAL EQUATIONS. 189 1*79. LITERAL EQUATIONS. 1. Given x — 2ax = 2ab -f- ^^ to find x. Ans. x= azt: V a' + 'lab + V = a ± (a + 5) = 2a + & or — 6. 2. Given x"^ — 2ax = — a"^ + h"^, to find x. Ans. X = a zt: y a} — d^ •{• 1? = a dcz h. 3. Given x^ — (<^ + ^)-c = — ah, to find x. 2^4 2 a — h , = a or 6. 2 4. Given ct^ — (a — V) x = «, to find x. Ans. x = or, or — 1. 5. Given = r, to find x. Ans. x = h dc ^Z ab -\- I?. X -\- a x — 6 6. Given — h = , to find x. X — a a -\- X aVb + 2 Ans. X = da (J a n -V ff. ! '/* 7. Given -j- 5 = " , to find x. a -\- X a — X 1/6 — 2 Ans. a: = ^ (— 2 ± 1/4 + Z/^). ^I'X 8. Given a? + 6^ _ 9ix + a;^ = ^— (1), to find x. ahi^ + 6V _ 26?i2x 4- ?i'x2 = m^^-, (2) = (1) cleared of fractions. nKz"" — m^x^ — 2bn^x = — ahi'' — bhi" (3) = (2) transposed. ^,__ m^ _ - ^'^^ - f ^' (4) = (3) - (n^ - mO. Then x = -^ i'^'\-n ;^ + X n^ — m^ ^ (?i^ — m^)^ n^ — ni^ bn^ rb n \/ ahn"^ 4- bhu"^ — a^ xz= n^ — iw n And finally, x = — , (bn db \/ a^m''- -f ¥m^ — a^n?'). {Vid^^ S8, ex. 16.) 190 RADICALEQUATIONS. EQUATIONS CONTAINING RADICAL QUANTITIES. 180. Equations containing radical quantities are usually solved by a judicious application of Axiom VII. No rule can be inva- riably followed in such equations. Some general directions will be better understood after tlie solution of a few EXAMPLES. 1. Given 7 l/x + 5 = 10 + 4 i/^ (1), to find x. 3 t/x = 5. (2) == (1) transposed and united. Vx = |. (3) = (2) - 3. X = V. (4) = (3)3. (Hc?e Ax. VII.) 2. Given Vx — 8 = Vx — V'2> (1), to find x. a: _ 8 = X - 2l/2^+ 2. (2) = (1)^ Ax. VII l/2x"= 5. (3) = (2) reduced. 2x = 25. (4) = (3)^ Whence x = 12i- 3. Given - + ^ ^l^^' == - (1), to find x. ^ X 4 |/16 - x^ _ ^ __ 4 ^2) = (-1) transposed. X 4 X Whence x = ± 4. (4) = (3) reduced. 4. Given }^E±1 + ,_!_ = -A^ (1). ^o find x. l/x _ l/x + 9 |/x + 9 X + 9 + 6 l/x = 4x (2) = (1) cleared of fractious. 6i/x = 3x — 9. (3) = (2) transposed. 36x = 9.x2 — 54x + 81. (4) = (3) squared. Whence x = 9 or 1. (5) = (4) reduced. _ „. x — 9 X — 4 4Cx — 16) .-,. . . , 5. Given ,- , » H 7= ^ = - , / (1), to find x. ■y/x + 3 ^/x — 2 |/x H- 4 ^ '^' ( Vide § 163, ex. 32.) RADICAL E Q U A T I ^* S. 191 |/^ __ 3 4- y'x + 2 = iV'x ~ 16. (2) = (1) with each term reduced. 2 y'x = 15. (3) = (2) with terms united. Whence x = 56| (4) = (3)^ and reduced. ^ 2a2 6. Given x + V a' + x' = .--- -„ (1), to find a:. V « + it- £C i/a^ + a;2 + «2 ^ .^2 _ 2a2. (2) = (1) cleared of fractions. X Va" + x^ = «' — x^ (3) = (2) transposed. aKv"" + x^ = «^ — 2a2x2 + x' (4) = (3)2. Sa^x^ = «* (5) = (4) reduced. Whence x = ± - -1/3. . (G) o *^ 1. Girea V^+f^ + -r^^ = -^^ (1), to find x. -j/x V X -\- a V X -\- a " + %2V^ = J' (2) = (1) X ^^ + ~" X X i/x 1 _!. 11 _{_ 2^/" = Z/2 (3^ = (2) modified. X \ X ^ 4. 2-^/" + 1 = 6' (4) = (3) modified. X \ X 142 + 1 = ± ^ (5) = 1^(4). Wx = J^. (6) = (5) reduced. •^ 6 zp i W hence x ^ ^. l/x + l/x — a ^r-^a /IN , £ J 8. Given ^ /- =^ = " ^1)? to find x. y X — y X — a X — a (y^ -i-l/x- ay ^ ^^'^^ .(2) = (1) modified (TWe a X — a § 164, ex. 9). y^ _^ y^x — a = — =i^. (3) = (2) modified, and y^x — a root taken. 192 RADICAL EQUATIONS. l/x^ — ax -{- X — a = dz na (4) = (3) cleared of fractions. l/x^ — ax = a (1 ziz n) — X (5) = (4) transposed. x^ — ax = a\l ± 7iy — 2ax (1 ± n) + x^ (6) = (5)^ Whence x = —z. — -, — r^-^- 1 ± Zn From the exam^Dles now given, it will have been seen that the object has been, in every instance, to relieve x from its radical sign, after which its value is obtained in the usual way. To effect the object, the terms of the equation must be so arranged that, on squaring, as many of the radicals as possible will disappear. If, on squaring, radical terms still remain, re-arrange, and square the equation a second time. Examples 7 and 8, above, exhibit anomalous methods of solu- tion. They should be carefully studied, — that is, studied until the reason for each change is clearly perceived. The pupil will find in the following examples ample oppor- tunity to improve his powers of analysis ; and we take this occa- sion to remind both teacher and piqnl, that a day occupied in the investigation of a single equation is discreditable to no one desirous of obtaining a familiar acquaintance with the various operations of algebra. Indeed^ such examinations are absolutely necessary/. Complicated equations can generally be solved in a variety of ways, but the best method can be learned only from practice. As a further illustration, we will resume example 4 above, and then leave the pupil to exercise his own ingenuity. 9. Given "^^^^ + ±_ = _±g^ n^ to find x. y^X Vx -\-d Vx + 9 ^ RADICAL EQUATIONS. 193 ? -I- _£_ 4_ 1 = 4. (3) = (2) modified. Whence a; = 9 or 1. From (4) we liave l/x = 3, or l/x = — 1 ; and it is with this limitation that the value 1 satisfies the original equation. 10. Given 17 + 2 Vx^ + 9 = 27, to find x. Ans. x = ± 4. 11. Given 5 — l/25 — x'^ = Sx, to find x. 12. Given \/x — 82 = 16 — Vx, to find x. 13. Given Vx + 40 = 10 — Vx, to find x. 14. i/x — 16 = l/x — 2. 15. i/^+~8 — t/^^^:^ = 2 i/2. / / 9 16. yx-i-yx — d= ^ - y a: — y *17. l/l + X Vx"" — 1 = 1 — a:. l/x + 1/^10 — x 18. i/x — i/lO — X = 2 ^«s t. X = : 3. J.71S. X = 81. J.ns. X = = 9. Ans. 25. Ans. 10. Ans. 12. Ans 5 • 4* Ans t. 9. 9x — 1 , . l/9x — 1 . o 19. — ,= = 4 + Ans. 9. l/9x + 1 2 20. Sv'g+lO _ l/ 2x + 16 ^^^^^ 4, 3 i/2x — 10 l/'2x — 4 21 V:^+^ = VUl^. Ans. 4. 1/:^ + 4 i/x + 6 22. - + ^""^ ~~ ^' = ^. ^71.9. ± V2ab - h\ X X ^^„ 2t/x + a 2a — i/x ^ 2 ^ ^^«' *23. —i^ ■ = Ans. a* or —77-- yx + 2a -j/x 9 a — Va' — x2 , . _;_ 2a l/t" 24. — =z b. Ans. ± a -f- 1 o2 __ ^2 1 + 6 n 1 D4 P R L L E M ?. ^ - Va + X V a — X \/x , . ^ / yx . yx yi) _^ a + .T -f V'2>ax + a;2 ± a (1 ± i/26 — Z;^) ZD. = 6. ^?2S. ^^ i^- 5 -f i/25 — a;2 ^ l/8 + a^ l/8 - ;:c Vx *28. j — — = — -. Ans. =b 8. yx y X J *29. 1^^ + V5 = l^-ii^. ^«.. 2 or _ 10, l/20 4- .T l/20 — :c ,„ l/4x 4- 20 4 - t/x 64 dU. -i — , -^ = y= — . A?is. 4 or ^• 4 + \/x -/x 3 l/x 4- 1 t/x — 1 ^ . « ol. 4 . = a. ^?is. ± • y a; — 1 yx 4- 1 va^ — 4 32. l/a^ -\- ax == a — ya^ — ax. Ans. db 9 1/3. 33. l/a 4- t/x = l/a^;. ^?2S. J a (Va-iy PROBLEMS INVOLVING EQUATIONS OF THE SECOND DEGREE. 181. 1. Three times the square of a number added to four times the number is equal to 64. What is the number? 8a;2 4- 4x = 64. Ans. 4 or — 5J. 2. A man bought a number of sheep for $200, and, reserving 20, he sold the remainder for $150, gaining $1 on the price of each sheep. What number was purchased? Let X = the number. Then — = price per head of those bought. and p-pr = price per head of those sold. X — 20 P R B L E M S. 195 Now, by the question tlie latter price is $1 more than the former. Hence, — -f 1 = — -. An^. 50 sheep. X X — 20 3. Divide 50 into two parts so that their product may be 621 An^. 23 and 27. 4. The difference of two numbers is 6, and their product is 216. What are the numbers ? Ans. 12, 18. 5. A man sold a watch for $75, and gained as much per cent, as the watch cost him. What did he pay for it? Aiu. $50. 6. A man sold a watch for $24, and lost as much per cent, as the watch cost him. What did he pay for it ? Ans. $40 or $60. 7. If 7 be added to a certain number and 3 be subtracted, the product of the sum and difference will be 119. What is the number? Ans, 10 or — 14. 8. A merchant bought a quantity of flour for $72. Had he bought 6 barrels more for the same sum, the price per barrel would have been $1 less. How many barrels did he buy, and at what price per barrel ? Ans. 18 barrels, at $4 per barrel. 9. If a certain number is subtracted from 12 and the remainder is multiplied by the number, the product will be 35. What is the number ? Ans. 5 or 7. 10. If a certain number be divided by 10, and this quotient be added to the quotient of 10 divided by the number, the sum will be 34. What is the number? Am. 30 or 34. 11. A man travelled 105 miles, and then found that if he had gone 2 miles less per hour he would have been 6 hours longer on his journey. At what rate did he travel per hour? Ans,. 7 miles. 12. Divide 40 into two parts so that the sum of their squares may be 1000. Ans. 30 and 10. 13. Two fields differing in quantity by 10 acres were each 196 PROBLEMS. sold for $2800, one bringing $5 per acre more than the other. What was the number of acres in each ? Ans. 70 and 80 acres. 14. The product of two numbers is 120. If 2 be added to the less and 3 be subtracted from the greater, the product of the sum and difference will still be 120. What are tlie numbers ? Ans. 8 and 15. 15. Two men are travelling towards each other. On meeting, B has travelled 20 miles farther than A. A, bj preserving his rate of travel, will go the distance B has already travelled in 20 hours; but B will be only 15 in passing over A's distance (at his former rate). What is the rate per hour of each ? Ans. A, 7.464, and B, 8.61 jf. 16. Two merchants sold the same kind of stuff, and together received $35. The second sold 3 yards more than the first. Had the prices per yard been interchanged, the first would have received $24 and the second $12J, gaining thereby $1^. How many yards were sold by each, and at what price per yard? Ans. 15 at $1-J and 18 at $|, or 5 at $3 and 8 at $21. 17. Divide the number 10 into two parts so that 10 times the second part may be the square of the first part. Ans. 5 (— 1 + 1/5) and 5 (3 — l/5.) 18. Divide the number a into two parts so that the square of the second part may be the first multiplied by a. Ans. - (3 ± i/5) and ^ (- 1 =f= i/5)- 19. A and B travel at the same rate towards Washinoton. At the 50th mile-stone from Washington, A overtakes a flock of geese travelling 1^- miles an hour, and two hours afterwards meets a coach travelling 2\ miles per hour; B overtahes the geese at the 45th mile-stone, and meets the coach 40 minutes before reaching the 31st mile-stone. What is the distance between A and B? Ans. 25 miles. EQUATIONS VriTIl TWO U N K N W X Q U A X T I T IE S. 1P7 EQUATIONS WITH TV.'O UNKNOWN QUANTITIES. 182. 1. Given .r + 7/ = 8 (1), ] ^ . , , y to nnd x and y. and cci/ = Id (!^), j x^ + 2x^ + ^^ = 64, (3) = (1)=. 4r^ = 60, (4) = (2) X 4. x^ - 2x1/ + if = ^, (5) = (3)^ (4). x - y = 2, (6) == Vip). X = 5, (7) = ((!) + (6) ) - 2. y =3, ((8) = (l)-(6))-2. The above operation v/ill be readil}^ understood, and tlie object of eacli step. In the same way solve and verify the following: — ^ (x-+^=10.) ^ (a;+^=12.) ^ (^+i/=20.) ^ (x+y=50. ) •^' ( ^-^=16.) ""• ( .T^=32.) • ( .r^=:64.) • ( ^-^'=400.) (6.) (7.) (8.) (9.) (a:+^=3-i.) (.r+^=-l. ) {x+I/= 14.) (a;+y= 2. ) ( ^i.'=U-) ( .'^•J/=-56.) ( .r^=+45.) ( .Ty=~63.) 1S3. (1.) Given x -\- y = a (1), ] ^ _ , y to find X — y. and .ry = h (2), j -^ X' + 2^-y + ^= = «^ (3) = (1)^ \xy = 4Z>,(4) = (2) X 4. x^ - 2xy + ^^ = a^ - 45, (5) = (3) - (4). X — y = Va^ — 46, (6) = l/'(5). Hence, when the sum and product of two numbers are given, take the square root of four times the product subtracted from the square of the sum, and this root will be the difference of the numbers. (2.) Given x -\- y = 10, and xy = 24, to find x and y. By the rule x — y = j/lO^ — 4 x 24 = 2. Hence, x = 6 and y = 4. 198 EQUATIONS WITH (3.) (4.) (5.) (6.) (x+y = 13.) Cx--f^ = n.) (x+y = 20.) (a) + 3. = 5^.) ( a-y=24.) ( 2;y = 28.) ( xi/ = dQ.) ( ^3/ = 2^0 184. 1. Given x - 7/ = S (1), -) , . , , V to una X and y. and xi/ = 48 (2), J "^ x^ - 2:.-y +f^ 64, (3) - (1)1 4xy = 192, (4) = (2) X 4. :,2 j_ 2;,.^ + y = 256, (5) = (4) + (3). X + 7J =16, (6) = i/(5). Hence cc = 12 and ^ = 4. In the same manner solve the following equations. (2.) (3.) (4.) (5.) (x — ?/ = 4. ) (.X — ?/ = 3. ) (x — y = 5. ) (x — ?/ = 3^) ( x^ = 21.) ( x^ = 70.) ( xj/ = 300.) ( x^ = 2. ) (6.) (7.) (8.) (9.) (.T - y = - 2.) (X - y = 7~.) (x - y = 1. ) (x -y = 11.) C a^ = 24.) ( x^ = 2f .) ( .;y = 3|.) ( :ry = 26.) 10. Given x — y =. a (1), and xy ■= h (2), I to find .T + y- X 2xy -\-f = a\ (3) = (1)^. ^xy = \h, (4) = (2) X 4. x' + 2x2/ 4- 2/' = «^ + 45, (5) = (4) ± (3). X + ;?/ = /ct^ + 46, (6) = T/C5). Hence, when the difference and product of two numbers are given, take the square root of four times the product added to the square of the difference, and this root will be the sum of the numbers. 11. Given x — _?/ = 3, and xy = 28, to find x and y. By the ride, x + y = V 9 + 28 X 4 = 11. Hence x = 7 and y = 4. TWO U N K N W N QUANTITIES. 199 (12.) (13.) (14.) (15.) (a: - y = 10. ) (.x - y = 5. ) (x - y = 21.) (x - ^ = 5^.) ( x^ = 119.) ( X2/ = 24.) ( x^ = U.) ( rry = 3. 3 185. 1. Given a;^ +f = 25 (1), ] ^ . . , ^ to nnd X and y. and X -\- ^ = 7 (2), ) ^2 _|_ 2^.^ + ^2 _ 49^ ^3^ _ ^2)^ 2:ry =24, (4) = (3)-(l). x^ - 2xy + y -= 1, (5) = (1) - (4). x-y =1, (6) = 1/(5). Hence a: = 4 and ?/ = 3. In the same way solve the following equations. (2.) (3.) (4.) (5.) x" + if ^ 50. x^ + ?/2 = 5. .t- + y' = 29. x"" -\- if =^ 40. a: + y = 8. a: + y = 3. a- + y = 7. x -\- y = ^. 6. Given a; + y = a (1), ) , ^ , , y to nnd X and ?/. and a;=^ + / = c (2), j ^2 _f_ 2xij +f = a\ (3) = (1)^ 2xy = a^ - c, (4) = (3) - (2). X' - 2xy + f=2c- a\ (5) = (2) - (4). x—y = V-lc — a\ (6) = 1/(5). Hence, when the sum of two numbers is given, and also the sum of their squares, take the square root of the square of the sum subtracted from twice the sum of their squares^ and this root will be the difference of the numbers. 7. Given x"^ -{- y"^ = 89. and x -\- y ^ 13, to find x and y. By the rule, x — y = l/89 X 2 — 13*^ = 3. Hence a: = 8 and y = b. (8.) (9) (10.) (11.) (x3+y2=50|.) (x^^y^=2b.) (^2+/= 58.) (x^-^y^= 41.) (x+y=.10. ) {x+y=7.) (x-{-y=^10.) (.x+y=-l. ) [ to find X and y. 200 E Q U A T I N S AV I T 11 (12.) (13.) (14.) (15.) (x'+r=nOb.) (...2-f/=34.) (x^+?/2=65.) (^•2+y=10.) (.r -i-y =47. ) (r. +^ =2. ) (X +y =9. ) (x +j/ =3. ) 186. 1. Given x-" + f = 52 (1), and X — 9/ = 2 (2), X' + 2xy +2/^ = 4, (3) = (2)^ 2xij = 48, (4) = (1) - (3). x-^ 4- 2.r^ + ?/^ = 100, (5) = (1) + (4). + y = rh 10, (6) = t/(5). Hence a; = 6 or — 4, and y = 4 or — 6. In the same wr.y solve the following equations. (2.) (3.) (4.) (5.) (a;=+/ = 25.) (.x-^ + 7/^ = 41.) (x2+y^ = 65.) (x2 + y^=61.) (03 —y = 1. ) (:c — 3^ = 1. ) (a? — y = 3. ) (x — y c= 1. ) C. Given :r — i/ = d (1), and x^ -}- ?/2 = c (2), .t'' — 2X2/ -\- 1/"^ = iV 2xy ^ c — iV" X- 4- 2x\j -^ if ^2c ^ d"" X + y = ± -l/2c — c?2. Hence, when the difi'erence of two numbers is given, and also the sum of their squares, take the square root of the square of the difference subtracted from twice the sum of their squares, and this root will be the sum of the numbers. (7.) (8.) (9.) (10.) (.,;= -f 3/2 ^ 74.) (x2 + 7/2 = 45.) (x2 4-?/2 = 65.) {x^ -\-y' =^^h:) {^x -7/ =2. ) {x -y =3. ) (X -y =3. ) {x -y =11.) 187. 1. Given x' - y» = 17 (1), ) ^ . , r. ,r^ ^ to find X and ?/. and X -y ^2 (2), j *^ ^-^+^ = 8^,(3) = (1) -?-(2.) Hence .t = 5|, and y = 3|. >■ to find X and y. T W U ^* K N 6 W N QUANTITIES. 201 In the same \Yay solve tlie equations — (2.) (3.) C'l-) (^0 (,,^ -f=. 55.) {X- - 3/^ = 12.) {x^ - y = IB.) {X- -f=^ 14.) (x -rj =11.) (x -y =3. ) (X -y =U.) (X -y =5. ) (6.) (7.) (8.) (9.) (^,.2 _y.^ 15.) (x^ _ ^2 = 26.) (x^ - y = 15.) (x^ - 3/' = 30.) (a; -y =10.) (X -y =13.) (x -i/ =3. ) (x -y =60.) 188. 1. Given x^ - r == ^^ (1),, | ^^ ^^^ ^^ ^^^ ^^ and X + y = 15 (2), j a-_y = l, (3) = (1) -^ (2). Hence cc = 8, and ?/ = 7. In the same way solve the equations — (2.) (3.) (!•) C^O (^2_y_18.) (x2~7/-^= 27. ) (x^-/=53. ) (x^-7/^=500.) (:,+y=9. ) (x+y=-13^.) (:r+j/=17f.) (x -y =125.) 6. Given x^ - f = -. | ,, g.^ , ,,a y. and X — y = d .) in + d^ wi — fZ2 7. Given x^-,/=^ m, | ^^ ,^^ ^ ^_^^ ^^ and X -{■ y = a, ) ^2 -f m «' --WI 189. 1. Given x^ + ^ = 5 (1), | ^^ ^^^ ^ ^^^ ^^ and x?/ = 2 (2), j 2xy = 4, (3) = (2) X 2. ^2 + 2xy + y^ = 9, (4) = (1) + (3). cc^ - 2xy + 3/' = 1> (^) = Cl)_r ^'^^* cc + ?/ = ± 3, (6) = 1/(4). X _ y = ± 1, (7) = l/(5). Hence a: = ± 2 and 3/ = ± 1. 202 EQUATIONS WITH In the same way solve the equations — (2.) (3.) (4.) - (5.) (0^2+^=10.) (^x'-\-f=is.') (x'^f=.m:) (xH^'= 13 ) ( icy=3. ) ( Ty=6. ) ( ^^=8|. ) ( o-y=~6. ) (6.) (7.) (8.) (9.) ( a:^=-24|.) ( cr?/=-7|.) ( o^y =25.) ( ^3/ =12.) 10. Given x^ + j/'^ = c, and .t^ = h, to find a; and y. ±i/c- -f 26 ±t/c— 2& rbl/c + 26=b-i/c — 2i ^ws. a = ^ , 7/ =t — . 190. 1. Given x^ — y = 7 (1), to find X and y. rry = 12 (2), x^ — 2xy + y = 49, (3) = (ly. 4:xy = 576, (4) = 4 X (2)2. x'^ + 2x^2 + y* = 625, (5) = (3) + (4). x^ + f = zh 25, (6) = t/(5). Hence a: = ± 4, or rt 3 V — 1, and y ±4i/=rT. In. the same way solve and verify the following equations. (2.) (3.) (4.) (5.) (£c2— y2^24.) (:x«— y2 = 21.) (^2— y»=16.) (a;2— y2^40.) ( ^_y = 5. ) ( ay = 10.) ( xi/ = 15.) ( xi/ = 21.) (6.) (7.) (8.) (9.) (x2_y=60.) (^;2— y2=80.) (x"—y=l. ) (x^— y2=3. ) ( XT/ =16.) ( :ry =9. ) ( x?/ =t/6.) ( xij =l/lO.) i2i57 7/^Tr. 191. (1.) x"" — f = 19. X2 (2.) - y = 16. X + y =19. Ans. X = 10, y = 9. 3; -y =2. Ans. X = '^;y r-3. TWO U N K N Vv' N QUANTITIES. 203 (3.) (4.) ic2 — / = 15. c«2 _ ^2 ^ 120. a: —y = 3. cc — y = 10. Ans. X = 4, 1/ = 1. Ans. X = 11, y = 1. (5.) (6.) x^ ^7f = 39. x^ — ^^ = 6. ^ + y =13. X -\- y =^. A71S. X = S, y = 5. Ans. cr = 31, y = 2|. (7.) (8.) a;2 — ?/2 ^ 40. _^2 _ y =^ 45. X -\- y =10. .^ + y =9. J.WS. a; = 7, 3^ ^ 3. J.7ZS. cf = 7, y = 2. (9.) (10.) x+y = 8. cc — y = 1. cry = 15. xy = 6. Ans. X = 5, ?/ = 3. Ans. .X = 3, y = 2. (11.) (12.) a;2 -f y2 = 169. a-2 — y2 = 16. X -\- y =Vl. a; + y = 8. Ans. X = 12, 7/ = 5. Ans. cc = 5, y = 3. (13.) (14.) ^2 _ y ^ 21. x' + y' = 325. ir — y =1. c^y = 150. J.7iS. CC = 11, y = 10. Ans. cc = 15, y = 10. (15.) (16.) x^ —y"" == 33. x" -\- y"" = 1300. cry = 272. cc — y = 10. ns. cr = db 17, or =h 16 V — 1, ^7is. CC = 30, y = 20. y = ± 16, or ± 17 V— 1. • (17.) (18.) x'-^f = lyi^. X' -f=- 9. ^y = - i- X — y = — 1. ^??s. a; = 1, y = — |. Ans. X = 4, y = 5. 204 EQUATIONS AV I T n (19.) X — y =1. Am. X = \ -^ l/5, ^ = 1/5 — i- (21.) 0^ -1- ^2 ^ 44. .Ty = 3 1/51. J.71S. X = ± 3 V 3, or ± j/lL t/ = ± i/lT^ or rt 3 VZ. (23.) a;2 + 7/2 ^ 123. X — y 1/3. ^ = 4 1/3; 192. 1. Given x"" -j- y = 9 (1), and :« + 7/ = 3 (2), (20.) o;^ -3/^ = 1. xy = 9. 1/3. u.4?is. a- = =fc 2, or d= y — 3 2^ = =fc 1/ 3, or ± 2i/— 1. (22.) . a;2 + 7/2 = 26. •-c — y = 1/2. J.?is. X = 3 1/2. X = 2 1/2^ (24.) 0^2 _ y = 45^ X — y = |/^5. Ans. X = D Vb. y to find X and 2''' = 4i/5, FIRST METHOD. X» + Zx-^y + Sx/ + 3,'3 _ 27, 3xV 4- 3x^2 ^ i§^ ^y (^ + ^) = 6. xy = 2. X ^ y x (3) = (2)^ W = (3) - (1). (5) = (4) -f- 3 and factored. (6) = (5) - (2). Vide 5 183. Hence x = 2 and ?/ = 1. In the same M'^ay solve and verify tlie following equations. (2.) . (3.) (4.) • (5.) (x'+3/' = 35.) (a;'+y'' = 91.) (x^ + ^' = 341.) (x»+y = 65.) (x -\-y =5. ) (x +7/ =7. ) (x +y =11. ) (x + y =5. ) X = 3, y = 2. X = 4, ?/ = 3. x = G, y = 5. x = 4, ?/ = 1. TWO UNKNOWN QUANTITIES. 205 193. 1. Given x' + f=^d (1), } , . , , ^ to una X and y. and X -\- y =3 (2), ) SECOND METHOD. a;3_ xy-^f=^-o, (3) = (1) - (2). x^ + 2:r?/ +3/^ = 9; (4) = (2)^ xy = 2, (5) = (4) - (3) - 3. X — 7/ = 1, Vide § 183. i Hence x = 2 and y = 1. In the same way solve and verify tlie following equations. (2.) (3.) (4.) (5.) (x'-f-/=133.) (a;3+/=217.) (:fc^+ 7/^=520.) {x^-\-7f=nO.) (x+y=7. ) {x+y=l. ) (a: H y =10. ) (:c +7/ =iO. ) \ Answers a: = 2, ^ = 5. x = 6, y = l. a:; = 8, ?/ = 2. a: = 9, y=sl and a: + y ^ to tind X and ?/. = 3 (2), J 194. 2. Given a;^ + ^ = 9 (1), y = 3 (2), THIRD METHOD. x^ J^y^=.{x+ yy - ^xy (x + y), (3). Vide § V4, ex. 1, and § 132 (2). ... (^ 4. y)3 _ 3:^y (a; + y) = 9, (4). That is, 27 — 9a:y = 9, (5) since x -\- y = Z. xy = 2, (6). x-y = l, (7). §183. Hence x = 2, and y = 1. In the same way solve the equations of § 192, and § 193, and also — (2.) (3.) (4.) (5.) (a.-»-y = 61.) (^»-3/» = 342.) (a;'-y = 485.) (.t3-/ = 7.) (a: -3/ =1. ) (.r -^^ =6. ) (.r-y =5. ) (.r -y = 1.) IS ^GG EQUATIONS "WITH 195. 1. Given a^ ^ f = d (1), and •'» + y y to nnd X and ?/. 3 (2), j FOURTH METHOD. X = o — 1/. (3) = (2) transposed. a:3 = 27 — 27j/ + 9f — f (4) = (3)2. .-. 27 — 27?/ + 9j/2 — ?/^ + ?/ = 9. (5) by substitution. Hence x = 2, and y == 1. In the same way solve the equations — (2.) (3.) (4.) (5.) (a;3 H- / = 35.) Gx^ + r = ^1-) C-^' +f = 341.) (a;» — ?/' = 19.) (X H-^ =5. ) (x +y =7. ) (X +?/ =11. ) (X -^ =1. ) ^96. 1. Given x^ -\- f = 9 (1) and x -f y = 3 (2) ^' I to fin 2) J d X and y. FIFTH METHOD. Let X == a -{- h, and y =. a — h. Then x -f ?/ = 2a = 3, and a = ^. By addition and (2). x' == (a + Z.)3 = a^ + 3«2Z> 4- 3«Z;'^ + Z/^ ,f = (a _ Z>)3 = a? — 3«2^ + 3aZ>2 — Z.3. .x^ -j- y = Id^ -f 6«Z>2 = 9. By addition. Y + 9^2 = 9. Since « = |. Hence h = \. But .r. = a + 5 = I + ^ = 2, and ?/ = a — ^^ = -J — J == 1. In the same way solve the equations in the preceding sections, and also (2.) (3.) , (4.) (5.) Cx»+y=28.) (a;3— y=26.) (.x^— y=7000.) (x3+y=9000.) (x -fy =4. ) (X -y =2. ) (x -y =10. ) (x ^y =30. ) 19^. 1. Given :i^ + y = a (1), and .r -}- 3^ = h (2), to find X and y. TWO U X K X W X Q U A X T I T I E S. 207 - ^•J/ + / = r, (3) = (1) -^ (2). I 0} + 2.ry + ;?/^ = ^'^ (4) = (2)^ l^ — a:y = -^' (5) = (-i) - (o) ^ 3. a:_y = ±^-^ Hence :. = ^^^ ± \— SX" )' '^^ '^ = i(^ - V-W j" Vide § 28, ex. 19. 2. Given x^ — f = a,) V to find X and ?/. and a: — i/ = h, ) Apply tliese formulas to all tlie equations in § 192 and § 196, inclusiYe, 19S. 1. Given x* -f t/* = 17 (1), ) , . , , V to find X and i/. and X + 7/ = o (2), ) :,* + 3/^ = C:c + y)^ - 4^y (x + i/y + 2x^y^ = 17. (3) Vide § 132 (2). 81 — o6x7/ + 2xy = 17. (4) Since x + 3/ = 3. Hence a:^/ = 2, or 16. (0) = (4^ reduced. Then x — 7/ = 1, ot V — 55. .-. X = 2 or i (3 + 1/— 55), and ?/ = 1 or ^ (3 — y^— 55). In the same way solve the equations — ■ (2.) (3.) (4.) (5.) (x*+y=82.) (x*+3/^=626.) (x*+7/*=1297.) (x^+^'^=2Vg-) (X 4-y =4. ) (:^ +y =6. ) (x +3/ =T. ) C^ +y =f ) 6. Given x* + ?/* = cr, and x + ?/ = Z>, to find x and y. X = i (6 db t/- 3Z.-^=|=2T/2a + 25*). J.ns. "" y = A (h =F ■/- 362 _p 2 T/2a + 26*). Apply these formulas to the above examples, ( Vide 28, ex. 24."^ 208 E Q U A T I N S W I T II iS9. 1. Given x^ — / = 992 (1), and X — 7/ =2 (2), I to find X and ^. 0:5 — y = (^^ _ yj _^ 5^^^. ^:c _ y-y _{_ 5^^2y (^^ — y) = 992. (3) Tide § 132 (2). 32 -f 40.Ty + lO^y = 992. (4) Since :r — y = 2. Then xy = 8, or — 12. Hence a; = 4 or — 3, or 1 rt y — 11, and ?/ = 3 or — 4, or — 1 rt V— 11. Fif^e § 16Y, ex. 17. In the same way solve the equations — (2.) (3.) . (4.) (5.) (,xH/=33.) (:^'— /=7/^.) (a;5-fy5^1056.) (x5— /=781.) (2^ +y =3. ) (:c —y =1. ) (x +y =6. ) {x —y =1. ) 6. Given x^ -\- y^ = a, and x -\- y =■ h, to find a; and y. 200. 1. Given x^ J^ 7/ = ^ (1), > CO unu ; 8 and xy = 2 (2), y to find X and y. From (2) x^y^ = 8, hence x^ = —3. o Then - -^ y^ =. 9. yZ Hence .t = 2 or 1, and ?/ = 1 or 2. In the same way solve the equations — (2.) (3.) (4.) (5.) (x'+2/'=351.) (x*-/=240.) (rz;5 +3/5= 1267.) (x'+y'=\l.) ( :ry=14. ) ( xy =8. ) ( xy =12. ) ( xy =2. ) ^?? sixers. x = 2or7. x = ±4ordb2 \/ ^ 1. x = 4 or 3. x = 2 or 1. y = 7 or 2. y = =b 2 or ± 2 {/— 1. y = 3 or 4. y = 1 or 2. [■ to find X and y. T "\V U N K N \S X Q U A K T I T I E S. 209 2©1. In an equation in whicli the terms are liomcgeneous, we may, with great advantage, introduce an auxiliary unknown quantity, ?jy letting x = my. The value of m can easily be found, and from this x and y. Thus, 1. Given X (x + ?/) = 24 (1), and y {:c — ?/) = 4 (2), These equations may be written thus, by multiplying (2) by 6 : x^ -f- xy = 24; and ^xy — 6y^ = 24. o:"^ -\- xy =■ ^xy — 6j,/^, or "x- = bxy — 6y'. Now let :*" -= wy, whence x- = m-y-^ and we have, by substi- tution in the last equation, nry"^ = bmy" — G^^. Divide this by y'\ and vre have r:i^ — bin = — 6. Hence in = 3 or 2. X = St/ or x = 1y. Substitute the lust value in (1), and we have 4/ + 2y = 24. Hence y = =h 2, and x = zt 4. Substitute the first value of x in (1), and we have Hence y = dsz. l/2, and x = =i= o V' 2. In the same way solve the equations — 2. X (x -f ^) = 77, and y (^x ■- y) = 12. Ans. X = 1 or V i/2, ?/ = 4 or | l/2. 3. j.^y + .xy = G, and :c^ + j^ = 10. Alls, .r = 2 or 1, y = 1 or — 3. 4. x'^ -f a,y = 12, and xy — 2y^ = 1, Ans. .T = ± 3 or I v^G, y = ± 1 or J |/6. 5. ^.?^/2 _|_ y ^ 5^ j^^(j .^.4 ^ .^-zy ^ or)_ ^ns. X = ± 2 or =b 2 i/— 1, ^' = d= 1 or ± V — 1. 13 210 E (-i U ATI N S vV I T H 6 x^u"^ -\- ?z' = 20^*, and x~ -f- }/' = 'IS. Ans. a: = rfc 6 or rp ^rf , y = ± 3 or rh ^ V'^ — 5. 7. x^ -\- if' =. 61, and a;^ — xy = 6. J.n.s. .t = 6, ^ = 5. 202. Sometimes we may introduce tiro auxiliary unknown quantities, one of wliicli represents the sum, and the other the product^ of x and y. Thus, 1. Given x^y + xy^ = 6 (1), and x^ + l/^ = 9 (2), to find x and y These equations may easily he written as follows : — ^2/ (.^ + y) = 6 (3), and (x + yf ~ ^xy (x + y) = 0, (4). Now let x -\- y = a, and xy = 5, and the equations become ah = 6 (5), and a^ — 3ah = 9, (G). d^ = 27 or a= 3, and h = 2. Hence a; -|- ^ = 3, and xy = 2, from which w^e have X = 2, and 7/ = 1. 2. Given x-* + J/^ — -^y = 7, and x-"* ~\- y^ = 35, to find c?- and y. Ans. X = 3, and 7/ = 2. 3. Given x^ -\- y''- -\- xy = 28, and x^ — ^^ = 56, to find x and y. J[?«.s. X = 4, and y = 2. 4. Given a;^ + 3/^ _ 2^^-^ — 2^7,/^ + ^^ "^ =15^, and ^'^ -f- / x- -j- ^ = 244, to find x and y. Ail the values of x and y in these equations are as follows — .T = 8, and y = 1. x = 2 zh 3 V^^^, and y = 2 ^ 3 V^^l. a: = 1, and ?/ = 3. / a:==2 (- 1 ± V -(1 + 5^0 l/l5), and 7/= 2 (-lq= i/_(l + ,.i_|/l5.) ^T=2 (- 1 ± i/-(l_^L 1/15), andy= 2 (_l=p i/_(l -,1^1/157) 203. Sometimes it is of much advantage to introduce two auxiliary quantities, one of which represents the sum, and the other the difference, of x and y. Thus, 1. Given x^ — y^ — xy"^ -\- x^y ■== 25 (1), and x^ + V^ — '^IJ^ ~ xhj = 5 (2). THREE U X K N V,' X QUANTITIES. 211 These equations are easily transformed into Qvide § 132 (2), ex. 7) (X - ^) (X + ff = 25 (3), and (.x + y) Qx - yj = 5 (4). Now, let X + 3/ = cf, and x — y = h. By the substitution of these values in (3) and (4), we have a^h = 25, and ah'' = 5. By the multiplication of which, we have a^J/ = 125, or ah = 5. • Hence a = 5, and h =. \. X -f y = 5, and x — ?/ = 1. From which x = 3, and y ^= 2. 2. Given x^ + ^x"" (?/ — 1) + 3^^ (a: -f- 1) + yS ^ 80, and ic' -}- rr (2y -f 3) = IG — y (y -\- 3), to find a:; and y. Ans. a: = 4, y == 1. 3. Given x* — ?/* 4- 2x^j/ — 2.t^ = 27, and x^ — y^ = 3, to find X and y. A71S. x = dt=: 2, y = zh 1. 204. If the preceding sections, commencing at § 182, have been studied with sufiicient care, the student will easily overcome all the difficulties attending equations of the kind we have been examining. We will finish this subject by adding a few EQUATIONS CONTAINING THREE Ui^'KNOWN QUANTITIES. 1. Given X + y + z -=6 (1), >v ^^ + y' + 2;' = 14 (2), y to find X, y, and z. and x^ + z- = 10 (3), J Ans, X = 1, y = 2, z =1 3. 2. Given Ax^ -f 4/ = 2z' + 2a'- (1), >v 4^2 _|- 4^2 = 2^2 + 2h^ (2), I to find x, 1/, and 2;, Ax^ + 4^2 == 2?/2 4- 2c2 (3), J ^«s. .T = ± A l/2a-^ + 2c2 — b', ?/ = db ^ 1/252 4- 2a^ — c^ £ = ± J- l/2c^ + 21' — a\ 212 PROBLEMS IN VOL TING 3. Given xy -f ^ = 5 (0;") xijz + s' = 15 (2), y to find cc, y^ and z. xy^ -f a:2y — 2x + 2^ = 8 (3), J J.?is. X = 2^ ^ = 1^ 2; = 3. PEOBLEMS. INVOLVING TWO UNKNOWN QUANTITIES. 205. 1. The ^m of two numbers is 100, and the dif- ference of the square roots of the numbers is 2. What are the numbers ? Let x^ = one, and y"^ = the other number. Then xJ -^ y"^ :=■ 100, and x — y =z 2. Hence, a;2= 64, andy'= 36. 2. The property of A and B together amounts to $13,000, and each receives the same income. But if A should let his money at B's rate per cent., his income would be $360, while B's in- come at A's rate per cent, would be $490. What is the property of each ? Let X = A's rate per cent., and y = B's. „. 36000 ., ^ ^ 49000 ^, Ihen = As property, and = B s. y X Therefore, by the question, y X f \ y 36000x 49000?/ and = ^. (2^ y X ^ ^ 36x + 49y = I2>xy. (3) = (1) reduced. 6x = 7y. (4) = (2) reduced. 42y + 49y = -^-. (5) :=* (3) combined with (4). ?/ = 6, and x = 7. n^neo '-%^ ^ S6000, and '^ ^ ?7000. TWO UNKNOWN QUANTITIES. 213 8. The sum of two numbers is 2-i, and their product is 35 times their diflference. AVhat are the numbers ? Let X = the greater, and y = the less number. Then x -\- y =■ 24, and xy = 35 (x — ?/). Find the value of x in the first equation, and substitute it in the other. Ans. 14 and 12. 4. A number divided by the product of its digits gives a quotient of 2^. ^ If 18 be added to the number, the digits are lOx + y inverted. What is the number ? The equations are = 2|, and 10.r + y + 18 = 10^^ + x. Ans. 35. 5. The sum of the digits of a certain number is 10, and if the product of the digits be increased by 40, the sum is the number inverted. What is the number ? Ans. 46. 6. The sum of two numbers is 7-^, and the sum of the third powers is 343^. Yv'hat are the numbers ? Ans. 7 and i. 7. The sum of two numbers is 47, and their product is 546. What is the sum of their squares ? Ans. 1117. 8. The sum of two numbers is 20, and the product is 99. What is the sum of their cubes? (T7^el3*2 (2).) Ans. 2060. 9. The sum of two numbers is 8, and the product is 15. What is the sum of their fourth powers? Ans. 706. 10. The sum, product, and difference of the squares of two numbers are all equal. What are the numbers? Ans. i (3 ± y/5) and A (1 ± |/5). 11. The sum of the squares, the product of the squares, and the difi'erence of the fourth powers of two numbers are all equal. What are the numbers? Ans. 1.27203, and 1.61808. 12. The sum of the fourth powers of two numbers is a, and the product h. What are the numbers ? Ans. V\ (a ± j/cr - 46^ V\ (a =F \/a^ — 4i* 214 PROBLEMS INVOLVING 13. Divide GO into two parts so that the product of the parts shall be to the difference of their squares as 2 to 3. Ans. 40 and 20. 14. There are two numbers whose product is 77, and the difference of their squares is to the square of the difference as 9 to 2. What are the numbers ? Ans. 11 and 7. 15. The product of two numbers is 48, and the difference of their cubes is to the cube of the difference as 37 to 1. What are the numbers? ^l^is^S and 6. IG. The difference of the fourth powers of two numbers divided by the difference of the numbers is 2336, and the product of the difference of their squares by the difference of the numbers is 576. What are the numbers ? Ans. 11 and 5. 17. The product of two numbers is 320, and the difference of their cubes is equal to 61 times the cube of their difference. AVhat are the numbers ? Ans. 20 and 16. 18. Divide a number a into two parts, so that the greater part may be a mean proportional between the whole number and the less part. Let x = the greater part, and ^ = the less. Then X -{- 7/ = a, and a : x : : x : t/. Ans. I (3 - 1/5), and ^ (~ 1 + V^.) lit Li If a = 20, then the numbers are 12.36 and 7.64. 19. The sum of two numbers is a, and the sum of their re- ciprocals is h. What are the numbers ? , a \a} a , a \a? a ^«- 2 + Vi - J' ^""^ 2 - Vl - %■ 20. The sum of the squares of two numbers is a, and the sum of the reciprocals of the numbers is h. What are the sum and product of the numbers ? 1 , 1 / Ans. Sum, - (1 ± l/l + ah')', product, ^ (1 ± T/l + al?). If a = 5 and 5 = l^i, then the numbers are 1 and 2. TWO UNKNOWN QUANTITIES. 215 , I 21. A mercliivut bought 51 gallons of j^Iadcira wine, aud a | certain quantity of Teneriffe. For the former he gave half as many shillings by the gallon as there were gallons of Teneriffe, and for the latter 1 shillings less by the gallon. He sold the mixture at 10 shillings by the gallon, and lost £28 10s. by his I bargain. Required the price of the Madeira and the quantity : of Teneriffe. Ans. Madeira, 18 shillings; Teneriffe, 36 gallons. 22. The side of one square garden exceeds the side of an- other by 5 rods, and both gardens contain 1025 square rods. ■ What is a side of each ? Ans. 20 and 25. : 23. A farmer has a field 16 rods long and 12 rods wide. He wishes to enlarge the field so that it may contain twice as much area, and not change the proportion of the sides. What will be the sides of the field? Ans. Length, 16 l/2'; breadth, 121/2". 24. A rectangular grass-plat has its sides in the ratio of 4 to .; 3. A walk outside the plat, 6 feet wide, contains ^tt as much [ ground as the plat itself. AThat is the length and breadth of , the plat ? Ans. Length, 342.72; breadth, 257.04. 25. A grocer sold 80 pounds of mace and 100 pounds of cloves for £66, and finds that he has sold 60 pounds more of cloves for £20 than of mace for £10. What was the price of each per pound ? Ans. 10 shillings and 5 shillings. 26. Find two numbers whose sum multiplied by the second i is 84, and whose difference multiplied by the first is 16. Ans. d= 8 and db 6, or zp l/2 and ± 7 "l/2 27. The square of the sum of the squares of two numbers is 169, and the product of the squares is 36. What are the num- bers ? Ans. =h 3 and ± 2, or ± 3 V^^ and ± 2 V^^^ 28. The difference of the fourth powers of two numbers, mul- tiplied by the product of the squares, is 147,600. The sum of ! 216 PROBLEMS INVOLVING tlie squares multiplied by the product of the numbers is 820 AYhat are the numbers ? Ans. db 5 and ± 4, or db 5 V — 1 and it 4 l/— 1. 29. A and B bought a farm containing a acres, each paying m dollars. A paid h dollai"S per acre more than B, in considera- tion of taking his share from the best portion. What does each one take, and at what price per acre ? Ans. A takes ^''' ' at 2m + c.^ + l/4m- + ^^^ l 2m -\-ab -{- V'^m'' -\- a'U' 2a B takes g^^^! i- at 2m-a5 + T/4m- + a-^-. 2m — ah + l/4m^ -f a'b'' 2a Tc ^1, A * 1 2a 2 4- h + l/4 + b^ If j)i = a, then A takes ^ at ~ ~ — , 2 + i + l/4 -f 6^ 2 and J3 takes r:=r at - ' > 2 _ ^, -j- v 4 + ^' 2 In this case — that is, when each pays as many dollars as there are acres — the price per acre does not depend upon the number of acres purchased. If m = a = 200 and & = | or $.75, then A takes 81.867 acres at S2.443, and B takes 118.133 acres at $1,693. If m = a = 200 and b = $2, then A takes 58.579 at $3.41421 per acre, and B takes 141.421 at $1.41421 per acre. If m = a = 300 and b = $1.50, then A takes 100 acres at $3.00 per acre, and B takes 200 acres at $1.50 per acre. 206. Every equation must be regarded as the algebraic con- ditions of some problem. If, therefore, on solving the equation the value of x is imaginary, it is absolutely impossible to fulfil the conditions of the problem. 80. Divide 10 into two parts, so that the product shall be 26. Let X = one, and 1/ = the other part. Then X 4- y == 10, and X2/ = 26. Hence x— r -- 2 ]/— 1, and a; =:^ 5 4- i/— 1, 7; = 5 — l/— 1 TWO UNKNOWN QUANTITIES. 217 Therefore it is impossible to divide 10 into two parts so that the product shall be 26. This is readily seen on trial, thus : 10 = 9 + 1, and 9 X 1 = 9. 10 = 8 + 2, and 8 X 2 = 16. 10 = 7 + 3, and 7 X 3 = 21. 10 = 6 + 4, and 6 X 4 = 24. 10 = 5 + 5, and 5 X 5 = 25. We see that the product is greatest when the pai-ts are equal. That this is generally the ease may readily be shown. Let X = one part, y the other, 2s the sum, and 2d the diflference. Then x -\- y = 2s and x — y =■ 2d. Whence x = s + c?"| (1). and ^ == s — d ) (2). The product of which is xy = s^ — d^. (3). It is plain that the second member of (3) increases as d diminishes, and therefore that it is the greatest when d = 0, i.e. when there is no difference between the parts. 31. Divide 20 into two parts, so that the product shall be 150. Ans. a: = 10 + 5 V^^^, ?/ = 10 — 5 V^^- 32. The sum of two numbers is 1, and the sum of their re- ciprocals 2. What are the numbers? (^Vide IG'T, ex. 18.) Ans. i(l -{- V^^), and ^ (I — V"^^^). 33. If 4 is added to a certain number, and the sum is divided by the number itself, the quotient is the same as that obtained by dividing three times the number by the number diminished by 4. What is the number? Ans. ± v — 2. 19 218 EQUATIONS OF THE SECOND DEGREE. GENERAL PROPERTIES OF EQUATIONS OF THE SECOND DEGREE. Dejinitlons. 20'?'. 1. A root of an equation is a quantity which, being sub- stituted for X in the given equation, satisfies it. 2. An imaginary root is one involving the expression y — 1. 3. A real root is one not involving an imaginary quantity. 4. Equal roots are where the roots are the same quantity, EQUATIONS OP THE SECOND DEGREE HAVE TWO ROOTS, AND ONLY TWO. Demonstration. 20S. Every equation of the second degree can be reduced to the form of x^ -{- 2/^-^ = 9.' Add 'p^ to both members, and we have x^ 4" 2px ~{- p^ = p^ -\- q^ or {x -\- jyy =2^ + Q> and, by transposition, (X + py - (p2 + 2) = 0, or (x + p + Vp^~+~q) (x -{-p — y'pn^Tg^ = 0. (Vide "YT.) Divide this equation first by one factor, and then by the other, and wc have 33 4- p _ >/^7T7= .-. X = ~^9 + l/pM^ (1)^ j.nd [q.E.]). X — p -{■ V'f + q = .-. X = — V — Vp^ + q. (2) ' 209. If J) = 0, then x = \/q, and x =: — \/q^ which are the roots of the incomplete equation a.'^ -f 2 x 0. x = q, or x^ = q. ( Vide 158 (3).) EQUATIONS OF THE SECOND DEGREE. 219 210. The aJQehraic sum of the fico roofs is equal to the co- efficient of the second term icith its sign changed ', for The roots are .r = — p -f V i^^ -f 5', and x = — jf — l/p'' + q^ the sum of Avhich is — 2j?. 211. 77ie product of the two roots is equal to the second term icith its sign changed ; for The roots are x = — ^) -]- |/j>^ -j- q, and x = — p — \/p^ -f q^ the product of which is — q. APPLICATION OE THESE PROPERTIES. 212. 1. What is the equation whose roots- are 5 and — 9 ? By 210. 2p = 4, and by 211^ q = 45. Therefore the equation is x^ + 4x = 45. 2. What is the equation whose roots are 4 and 1 ? Ans. x^ — ^x = — 4. 3. What is the equation whose roots are 9 and — 1 ? Ans. x^ — Sx == 9. 4. What is the equation whose roots are — 2 and — 2 ? Ans. x^ -f 4x =: — 4. 5. What is the equation whose roots are 1 and — 1^^? Ans. x'^ 4- /-^ = 1^^,. 6. What is the equation whose roots are 7 and — 8 ? Ans. x^ -\- X = 56. 7. What is the equation whose roots are — 7 and — 7 ? Ans. x' 4- 14x = — 49. 8. What is the equation whose roots are 7 and 7? Ans. x^ — 14:x == — 49. 9. What is the equation whose roots are a and h ? Ans. x^ — (^a -\- h) X = — ah, 220 EQUATIONS OF THE SECOND DEGREE. -1^ iTTi , • .1 • 1 a\/b 4- hi/a , 10. What IS the equation whose roots are ; , and a — 6 a-[/b — h\/a 2al/l>x — ab ■ J . Ans. x^ — = 7' a — a — b a — 6 213. Advantage may be taken of these general properties in solving any equation of the second degree. 1. Given x^ — Sx = — 15, to find the two roots. Let X = one root, and i/ = the other. Then x + ?/ = 8, By 210. and xy = 15. By 211. AYhence x = ^, and ^ = 3. Vide 182, ex. 1. 2. Griven a;^ -f 6.c = 187, to find the two roots. Here a; -}- y = — 6, and ^y = — 187. Hence x = 11, and i/ = — 17. 3. Griven x^ + 4x = — 4, to find the roots. Here x -{- ?/ == — 4, and x?/ = 4. Hence x = — 2, and y = — 2. 7x 4. Given Sx^ + 1 .^ 5 = 4|, to find the roots. By reductio n rf.2 . "^-^ _ 2,2 ^ + 15 - ^^• Then ^ + ^ = - t\, and ^!/ == — f f • Hence X = 1, and y — — \J^ 5. Find the roots of ox'^ _ 2x = 8. (^Y'ule 1^76, ex. 4.) Ans. 2 and — 1^. 6. Find the roots of — ;; j ;— ^ = 5|. Ans. 4 and 1. X -\-^ 7x ~x ^ X -f 8 7. Find the roots of x"" — 20:/; = — 104. Am. 10 + 2]/~in and 10 — 2l/^^ EQUATIONS OF THE SECOND DEGREE. 221 8. Find the roots of cc^ — (a — V) x = a. Ans. a and — 1. 9. Find the roots of 6 = — —-- X — a a -\- X Ans. ( Vide ll'O, ex. 6.) 10. Find the roots of x'' — IOjc = — 26. Vide li^G, ex. 44, and 206, ex. 30. 314. Every equation of the second degree, as we have stated, may be reduced to the form x"^ -f- 2px = q (1), the signs not being considered. This form is obtained when p and q are hoth j)ositive. If p is negative and q positive^ we have x^ — 2px = q (2) If p is p)ositive and q negative, we have x' + 2i)x = — q (3) If p is negative and q negative^ we have x^ — 2px = — q (4) And these are all the combinations that can be made in the signs; for if x"^ is negative, we may multiply, or divide, the whole equation by — 1, and it will be found in one of the above forms. (Vide 1^6, 5.) The roots of these equations are respectively (^ide IT'l, l'y4) X = — p d= Vp' + q. (1) x=: p± Vp' + q. (2) X ■= — p dtz V'p'^ — q. (3) X = 2> ^ V p"^ — q- (4) From these roots we easily deduce the following facts: — 1. The roots of (1) and (2) are always real. 2. |/p2 -f ^ > p. .-. The first root of (1) and (2) is positive^ the second negative. 3. The negative root of (1) is numerically the larger. 222 RATIO AND PROPORTION. 4. The positive root of (2) is numerically the larger. 5. If 5" <^ jj^i the roots of (3) and (4) are hotli real. 6. If 2' p^, the roots of (3) and (4) are imaginary. {Yide 9. If p = 0, the roots of (1) and (2) are numerically equal. but of contrary signs. 10. If p = 0, the roots of (3) and (4) are imaginary and equal, but of contrary signs. 11. If 2' = 0, the first root of (1) and (3) is 0, the second -2p. 12. If 2' = 0^ the first root of (2) and (4) is 2j), the second 0. 13. If 2? = 0; and ^ = 0, the roots of (1), (2), (3), and (4), are all 0. RATIO AND TROPORTION. 215. Ratio is the quotient which is obtained hy dividing one quantity hy another of the same hind. Thus, The ratio of a to Z> is -, commonly expressed by a : h. 1. The two quantities forming a ratio are together called terms. 2. The first term alone is called the antecedent. 3. The second term alone is called the consequent. Thus, a : h are the terms, a is the antecedent, and h the consequent. 216. A proportion is an equality of ratios. Thus, - = -, commonly written a : h : : c : d. 1. The first and last terms of a proportion arc called extremes. RATIO AND PROPORTION. 223 2. The second and third terms are called means. I i 8. Tlie first and second terms form tlic first coujilet. 4. The third and fourth terms form the second couplet. \ 5. The last term is a fourth proportional to the other three. Thus, I a and d are extremes^ h and c are means, a and h the first i couplet, and h and d the second couplet. I 6. If the means of a proportion are the same quantitjj, that j quantity is called a mean proportional between the other two j ' and the last term is a third proportional to the first term and one of the means. Thus, in the proportion a \h \'.h \ Cj 6 is a mean proportional between a and c. and c is a fourth pro- portional to a and h. I 7. A continued pronortion is one in which several ratios are I equal. Thus, i a : b : : c : d : : ni : n : : p> : q, &c. I 8. Three or four quantities are in harmonical jij-ojyorfion when the first is to the last, as the difference between the first two is to the difference between the last two. Thus, a, h, c, are in harmonical proportion when a: c '.: a — Tj :h — c. , a, &,c,andf?, '' " '' '' a'.d::a — h:c — d. \ 21f . Quantities are in proportion b}'' alternation, when ante- cedent is compared with antecedent and consequent with con- sequent. I 21S. Quantities are in proportion by inversion, when ante- ; cedents are made consequents and consequents are made ante- cedents. 224 RATIO AND PROPORTION. 219. Quantities are in proportion by composition, when the Bum of antecedent and consequent is compared with either ante- cedent or consequent. 220. Quantities are in proportion by division, when the dif- ference of antecedent and consequent is compared with either antecedent or consequent. 221. Two varying quantities are reciprocally or inversely pro- portional when one is increased as many times as the other is diminished. Thus, V y y icx^=2xx -=3xX o= ^^^ X -^ = xy, the product hQ\n^ fixed. Li O Ifth 222. Equimultiples of two quantities are the results obtained by multiplying both by the same quantity. Thus, ma and mh are equimultiples of a and I). 223. Proposition I. If four quantities are in proportion, the product of the extremes is equal to the product of the means. For, since a : h : : c : d, we have - = _. (1) ct o Clear (1) of fractions, and we have ad = he. (2) Cor 1. If 6 = c, then (2) becomes ad = h"^. (3) That is. The product of the extremes is equal to the square of the means. Cor. 2. If both members of (2) be divided by ac, we have d h . h d - = -, I.e. - = -, or a : :: c : d. (iZ) c a a c That is, If the product of two quantities he equal to the pro- duct of two other quantities, the first two may he made the ex- tremes, and the second two the means, of a proportion. RATIO AND PROPORTION. 225 Prop. II. If four quantities are in proportion^ they will he in j proportion hy alternation. For, since a : 6 : : c : cZ, we have ad ^ he. (2) (Prop. 1.) i Dividing both members of (2) by dc^ we have — i ad he . a h z j /^n^ I -—=.—-, I.e. - = -, whence a '. c w b \ d. (lU) I dc dc c d I Prop. III. If four quantities are in proportion^ they will he I in proportion hy inversion. \ For, since a : 5 : : c : cZ, we have ad = he. or -— = — ;. (4") i he ad -' ^ Multiply both members of (4) by ac, and we have ' ac ac . a c , , y ^--. -— = — •, I.e. 7 = -» whence b : a : : d : c. (11) be ad b d ^ , Prop. IV. If four quantities are in proportion, they will he in proportion hy composition or division. i For, since a : h : : c : d, 'we have ad = he. (2) I Add or subtract hd according to AX. I., and we have — ad db hd = he ztz hd, or (« zh h) d = (c zh d') h, whence a±:h :h :: c± d : d. (5) (Prop. 1, Cor. 2, and Prop. 3.) From (2) we also have ac dz ad = ac db he, or (c db (?) a = j (a ± h) c, whence | adzh :a::cdzd:d. (6) (Prop. 1, Cor. 2). Prop. V. If four quantities are in proportion, the sum of the I first and second icill he to their difference, as the sum of the third and fourth is to their difference. For, by (5) and Prop. 2, we have — \ 7 7 7 7 . c -f tZ cZ 1 a -\- b '. c -\- d '.'. b '. d, i.e. r = T- ' c — d d and a — b : c — d :: b : d, i.e. j = t^- I ' a — h o whence c d- d ^ cj-_^, ^^^^ is, a + Z; : a — Z) : : c -j-dic—d. (7) (Prop. 2.) a d- h a — b 226 RATIO AND PROPORTION. Prop. VI. Equimultiples of two quantities are proportional to the quantities themselves. For — = -, or ma : mh :: a : h. (8) ma a mo izo Cor. Since — = — , we have ma : mh wna: nh. (9) ma na Prop. VIT. If four quantities are in proportioiij the like powers or like roots loill he in prop)ortion. For, since a : h : : c : d, we have ad = he, or a'^d'^ = h'^c"', in which m is a whole number or a fraction. Restore the propor- tion, and we have a'" : h"" :: c"^ : : : c : c?, we have ad = he; and since m : n w p : q^ ViQ have mq = np ; whence am X dq == hnXcp>, or am : hn : : cj) : dq. (14) Prop. IX. In a. continued proportion, the sum of all the ante- cedents is to the sum of cdl the consequents as any one ante- cedent is to its consequent. (^Vide § 2, def. 7.) For, since a '. h :: c '. d, ^e. have ad = he ; and since a : 6 : : m : w, we have an = hm; and since a : h :: a : h, we have ah = ah. The sum of these equations is (jj^cl-\-n) a = (a-]-c-^m) h, or a-{-c-\-m : h-j-d-{-n :: a :h. (15) Prop. X. If the first couplets of two proportions are the same^ the second couplets will form a prop>ortion. For, since a : h : : c : d, and a : h :: e :f, we have - = — , and — = — . a c a e .'. — = ^—1 whence c : d :: e : f (10) c e \ PROBLEMS. 227 Cor. By alternation, If tlie antecedents of tico 2:)roportions are the same, the consequents v:ill he proportional. ILLUSTRATION OF THE PRECEDING PROPOSITIONS. 4:3::20:15. By Prop. 1st, 4 x 15 = 3 x 20, i.e. GO = 60 2nd, 4 : 20 : : 3 : 15. 3d, 3 : 4 : : 15 : 20. 4t]i, 4 ± 3 : 4 : : 20 ± 15 : 20, 7 : 4 : : 35 : 20, 4:3: :20:15. u 4:3: : 20 : 15. il 4:3: :20:15. a I.e. 1:4:: 5 : 20. 4:3::20:15 4:3:: 20: 15 5th, 4+3 : 4-3 :: 20 + 15 : 20-15, i.e. 7 : 1 :.35 : 5. " 7tli, 4--^ : 3^ : : 20^ : 25^ i.e. 16 : 9 ::400 : 225. 4:3::20:15::44:33. By Prop. 0th, 4 + 20+44:3 + 15 + 33::4:3, i.e. 68 : 51 ::4 : 3. 4 :3 5 :8 4 :3 4 : 3 20 : 15 10 : 16. j 20 : 15, 44 : 33. I By Prop. 8th, 20 : 24 : : 200 : 240. By Prop. 10th, 20 : 15 : : 44 : 33. PROBLEMS. 224. 1. Any three terms of a proportion being given, to find the other term. In the proportion a : h :'. c : cl, let X take the place of «, 5, c, and d, in succession. In each case we are to find the value of x. X : h :: c : d, whence dx = he, i.e. x = — • (1) (2) a : X :: c : d, ^' ex = ad, i.e. x^ a : h :: X : d, " hx = ad, i.e. x = d ad c ad (3) he a : h : : c : Xj " ax = he, i.e. x — — • (4) 228 PROBLEMS. (1) and (4) show that either extreme is equal to the j)rod>ict of the means divided hy the other extreme. (2) and (3) show that cither mean is equal to the product of the extremes divided hy the other mean. 2. Find the value of x in the proportion x : 2 : : b : 1. Ans. 10. 3. Find the vakie of x in the proportion 12 : x :: 4 : 7. Ans. 21. 4. Find the value of x in the proportion 8 : 5 : : x : 10. Ans. 16. 5. Find the value of x in the proportion 14 : 12 : : 7 : x. Ans. 6. 6. To find a mean proportional be'twcen two quantities. In the proportion a : x : : x : cZ, we have x"^ = ad .*. x == y ad. Hence, the mean 'proportional hcticccn two quantities is the square root of the product of the quantities. 7. Find the mean proportional between 9 and 4. Ans. "]/ o6 = 6. a : X :: y : h,') to find the relations 8. Given the proportions 7 t p a : m: : p) '• l>: J oi x^ y, m, and p. Ans. X '. m '.'. p '. y. 9. Given x" : (14 — xj : : 16 : 9, to find x. X : 14 — X : : 4 : 3. Prop. 8th. - X : 14 : : 4 : 7. Prop. 4th. 7x = 56. Prop. 1st. X = 8. 10. Given xy = 24 and x^ — y^ : (x — yy : : 19 : 1, to find x and y. We have a;' — y^:x^ — ^x^y-j-Sxy"^ — y^: : 19 : 1. 3xV — 3x3/2 : (^x _ ^)3 : : 18 : 1. Prop. 4. xy (x — y) : (^ — 3/)^ '•' 6:1. Dividing antecedents by 3. xy : (x — yy :: 6:1. Dividing 1st couplet by (x — ?/). 24 : (x — 3/)2 : : 6:1. Since xy = 24. 4:(x — ?/)2:: 1:1. Hence x — y = 2 and xy = 24, whence x = 6 and y = 4, 11. Given (a + '-^T • (« — ^0^ : : x + y : x — y, to prove that a : X :: l/2a — y : ^/y. ARITHMETICAL PROGRESSION. 229 a^-{-2ax-^x^ : a^ — 2ax-\-x'^ :: x-\-y : x — i/. By expanding. 2a3 + 2x2 . 4^^ : : 2x : 2y. Prop. 5th. a^ -\- x^ : 2ax : : x : y. Dividing by 2. Transferrins; the fac- a? -{■ X . 2a :: x^ '. y. ( Trt I 1 o tor X. a? -\- x^ : x^ \\2a'.y. Prop. 2nd. a^ : x^ : : 2a — y '• V- Prop. 4th. a : X : : l/2a — ^ : l/y. Prop. 7th. 12. Giren xy = 135 and x^ — ^^ : (x —- ?/)2 : : 4 : 1, to find X and y. ^?is. x = 15; y = 9. (X — 3/:x + ?/::2:3;) 13. Given \ o r r ^^ ^"'^ ^ ^^^ V- ( X + ?/ : xj/ : : 3 : 5, j J.?is. X = 10, y = 2. 14. Given x + y = 24 and xy '. x^ •\- y"^ \\ ?i -. 10, to find x and ?/. -47ZS. X = 18, ?/ = 6. 15. Given x : y : : 3 : 2 and x + 6:y~6::3:l, to find x and y Ans. x = 24, y = 16. 16. Given xy — 320 and x^ — y : (x — y)^ : : 61 : 1, to find X and y. ^»s. x = 20, y = 16. ARITHMETICAL PROGRESSION. 225. A Series is a succession of terms, each of which is de- rived from one or more of the preceding terms, by a Jixed law. 1, 3, 5, 7, 9, &c. is a series in which any term is derived from the preceding one by adding 2. 3, 6, 12, 24, 48, &c. is a series in which any term is derived from the one preceding by multiplying by 2. 1, 3, 4, 7, 11, 18, &c. is a series in which any term is found by adding the two preced- 'ng it, after the second term. 230 ARITHMETICAL PEOGRESSION. 22€i. An Aritlimetical Progression is a series whose law is that any term is found h)/ adding a constant qiiantify to the 'pre- ceding term. The common difference is the constant quantity to be added. The progression is increasing when the common difference is positive. The progression is decreasing wlien the common difference is 7iegative. The number of terms of a series may be limited or infiuite. The first term is that with which the progression commences. The last term is that with which the progression is supposed to terminate. The sum of tJte terms is the amount of all the terms of the progression. 3, 7, 11, 15, 19, is an increasing arithmetical progression, in which 3 is the frst term^ 4 is the common difference, 19 is the last term, 5 is the numher of terms, and 55 is the simi of the terms. 19, 15, 11, 7, 3, is a decreasing arithmetical progression, in which 19 is the first term, — 4 is the common difference, 3 is the last term, 5 is the number of terms, and 55 is the siim of the terins. 4, 3, 2, 1, 0, ~ 1, - 2, - 3, - 4, is a decreasing arithmetical progression, in which 4 is the first term, — 1 is the common difference^ — 4 is the last term, 9 is the number of terms, and the sum of the terms is 0, 22*T, To find the last term, when the first term, number of terms, and the common difference, are known. Let I = last term, a = first term, n = number of terms, and d :^ common difference', then the progression will be a, a -f d, a + 2<^7, a + Zd, a + 4c7, a -j- hd, &c. ARITHMETICAL PROGRESSION. 231 In wliich any of the numerical coefficients is represented by ti — 1. Therefore, I = a -{■ (?i — 1) cl (1) If d is negative, then I = a — (u — 1) d. (2) 2. To find the sum of the terms, when the first term, the last term, and the number of terms, are known. Let s = sum of the terms, I = last term, a = JiJ-st term, and 71 = number of terms; then, writing the progression in a direct and reverse order, we have the equations — s = a -\- a -{- d -{- a -{- 2d -\- a -}- Sd -{■ /. s = ^ + I — d + I — 2d + I —M + a. By addition, 2s = a-{-l-{-a-]-l-\-a-{-l-{-a-{- I -{• a -\- I, in which a -{- I is taken as many times as is indicated by n, the number of terms. Therefore, 2s = (a + Z) ??, or s = (a + I) n (3) From equations (1) and (3) the following table is easily made : — No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Given. Unknown. Values of the Unknown Quantities. a, d, n, I, s, I =1 a-\. {ji — l) d; s = \n\2a-\-{n — l)d\ a, d, I, 71, S, n = \{l-a)^l; . = 1 (Z+«) [l-a+d). a, d, s, 71, I, 71 = ^— — '^ ; Z = a + (n — 1) d. a, n, I, s, d. -^j 2' 4' 8? is a decreasing geometrical progression, whose ratio is i, first term 4, last term |, number of terms 6, and sum of terms 7|. 1. To find the last term, when the first tei-m, number of terms, and ratio are known. Let I = last term, n = number of terms, and r = ratio, a = first term. In this case the progression will be — a, ar, ar'^^ ar^, ar^, a?-^, &c. ; in which any exponent is represented by n — 1. Therefore, I = av^-i. (1) 2. To find the sum of the terms, when the first term, number of terms, and ratio are given. If, in addition to the above nota- tion, s = sum of the terms, then s = a -\- ar -}- ar"^ -{-.... + «^'"~"-^- Multiplying by r, rs = or -f «^'^ + ar'^~^ + ar\ whence rs — s = ar'^ — a, ar"" — a a (r» — 1) and . = -^— ^ = V-l • (2) GEOMETRICAL PR OGRES SIGN. 235 From (1) and (2) the following tabic is readily formed No. Given. a, r, n, a, r, s, a, ??, s, r, n, s, a, r, n, a, r, I, a, n, Z, r, n, I, Required. 9. r, ??, I, 10. r, n, s, 11. r, I, s, 12. 13. n, I, s, a, 71, I, 14. a, n, s, 15. a, ?, s, IG. 17. n, I, s, a, r, I, 18. a, r, s, 19. «, I, s, 20. r, I, s, Formulas. a -f (r l{s — 1 = 1) — a (s — a)"'~^ (r — 1) s?-"-i o-J* — 1 = 0. a (7-^ — 1) r — 1 — 1 n — a«-i Z«— 1 — a« — 1 I (r^ — 1) (r — 1) r**-^' |.M 1 (;■ - 1) !■"■ — 1 a = r/ — (j- — 1) s. a (s — «)'»-i ^ l(s — ly-'^ : 0. Qj^.ti — ^.g _j_ 5 — a = 0. (s — r'^ — sr"-i 4- ? = 0, ?• = ~ , • ^ s — 6 (s — Z) r'* — sr"-l + ? = 0. loo- ^ __ loe: a n n = n Iog: ?' + 1- log [a + (r — 1) s] — iog a log r loo; I — I02: <^ + 1- log (s — a) — log (s — log Z — log [rl — (r — 1) s] ^ ^^ log >• 236 GEOMETRICAL PROGRESSION. EXAMPLES. 1. If « = 5, T = 10, and n == 7, what is the sum of the series? Ans. 55^555, 555. 2. If rt = 1, r = 2, and n = 7, what is the last term? Ans. 64. 3. If a = 1, r = 2, and s = 127, what is the number of terms? Ans. 7. 4. If a = l, 1= 27, and s = 40, what is the number of terms? Ans. 4. 5 Insert 4 geometrical means between 2 and 486. 1 Since r = i — j , we have r = 1/243 = 3. 2, 6, 18, 54, 162, 486, is the series. 6. Insert 5 means between 1 and q\. Ihe series is, 1, -^, ^5 g, yg, -g^, g^. -r^ , ,«. . *"^ — « cs — rl 232. Formula (6) is s = .- = -^ ^ y — i 1 — r If r is a proper fraction, the progression is decreasing; and if the series be carried to infinity, the last term becomes 0. The formula will then become s = r . (21) 1 — r ^ ^ EXAMPLES. 1. Find the sum of the series 4 + i + g + y g + 32) ^^- *^ /= 0. i Here a = ^, »' = 2 : hence s = :| — ^^ = 1. Ans. 1. 2. Find the sum of the series J + ^ + ^V + bV» <^°- ^^ infinity. 1 3 1 i-i 2. Ans. ^. 3. Find the sum of the series 1 4- 4 _{_ ^j^ ^c. to infinity. s r=^ r -^ 5. Ans. 5. 1 — i GEOMETRICAL PROGRESSION. 237 4. Find the sum of the series | + 2 + |j &c. s — -, _ 3 — Y — ^3. ^ns. ^3. 5. What is the sum of the series 1 + — + -^ H — ^' &c. ? *x!y Cc/ fcC- S = r = ^. Alls. \ i- X — 1* X — 1. X 6. What is the sum of the series 1 + -. + . ...^ + (x4- 1) _ 1 rc + l X X + 1 7. What is the sum of the series x -\- xf/ -\- xi/^ + x^^, &c. whenv/' - r^) ^' +; &c. =0. (2) S35. From the nature of an identical equation (vide 92, 5), equations (1) and (2) must be true for all possible values of x; that is, we may take a; = ^ = 0, 1, 2, 3, &c. ] and what is true of the coefficients when x equals any one of these values, is true when any other value is taken for x. 236. Because the coefficients of the different powers of the unknown quantity in equation (2) are coefficients of indeter- minate quantities, they are called Indeterminate Coefficients, 240 INDETERMINATE COEFTICIENTS. ' , I 23'^. In any identical equation, containing but one unknown \ quantity, the coefficients of tlie like powers of this quantity in the i two members are equal to each other. For, if a: = in equa- tion (1), the equation reduces to ] P = i^'- I If now these quantities are cancelled, we have j qx + 1'^^ +; &c. = q^x -f- T^x"^ -j-; &c. Or, by dividing by x, i q + rx -[-, &c. = q^ -{- r^x -{-, &c. Now make x = 0, and we have i In the same way we may obtain i ■■ 1 238. To develope an expression by means of the principle of indeterminate coefficients. j 1. Assume the expression to be equal to a series of the form p -\- qx -\- rx^ + sx^ -f , &c. i 2. Clear the equation of fractions, or raise it to the required power. 3. Equate the coefficients of the like powers of the unknown j quantity. 4. Find from these equations the values of p>, q, r, &c. 5. Substitute these values in the assumed development. i I EXAMPLES. I , ^ , 1 + 2x . . I 1. Develope :; : into a series. 1 1 — X — x^ Operation. 1 + 2x = p> -\- qx -f ^'-^'^ -^ sx'^ -f ^1^, &c. INDETERMINATE COEFFICIENTS. 24:1 Clear this of fractious^ and we have 1 -{- 2x = p -\- q X -{- 7 — P — 9. — P X- + s x"" -\- t — r — 9. — s — r X*, &c. Equating the like powers of x, we have (\ — p^ 3.0 _ Q^ whence p = 1. = r — q — Pj " r = 4. = s — r — 2-, ^^ s = 7. = i _ s — 7-, " ^ = 11 1 + 2x Hence 2. Develope 1 — X — x^ 1 — X == 1 4- 3x + 4^2 -f 7x' + llo;*, &c. ( Vide 70, ex. 26.) 1 + ^ + ^ - into a series. (FtVZe TO, ex. 25.) 3. Develope — "-^ -r into a series. {Vide 70, ex. 23.) 2 4- 3x 4. Develope 7^ — ~ -. — ^-1; into a series. 8 4- 4:^; + bx' X 34^2 121a;^ „ Ans. 1 + - _ -^ + -^. &c. 5. Develope y 1 — x^ into a series. Operation. |/l — x^ = p -\- qx -\- rx^ + sa;' + ^-^S &C- Square both sides, and we have 1 — x^ = p^ + pq pq X + pr x^ +_ps a;2 -{- pt + 2^ -f qr + ^s + pr + 2^' _j_ ,.2 + i^s + 2S .^*+, &c. 21 242 INDETERMINATE COEFFICIENTS. Equating the proper coefficients^ Ave have jp2 _. 1^ whence p = 1. 2pq = 0, 2pr -\- q- =i — 1, 2ps + 2qr = 0, 2pt + 2qs + 7-2 = 0, 2 ^.■t ^,.6 cc 5 = 0. u r = — 1 u s = 0. u t = — 1 8 X' a. Therefore, V\ — x^ = 1 — tt — V — tt^ — ? ^^• ^ 2 8 lo {Vide 165, ex. 9.) 6. Develope yl — x into a series. ^4ns. 1 _ - _ -^-^ _ ^-^-^ - 0-747678 -' •^^- 7. Develope y 1 -f- x into a series. .7: x^ 3x^ S . 5x* Ans. 1 + 2 - "274 + 27176 ~ 2.4.6.8' 8. If X =^ \ in the last example, to what does the answer reduce? Ans. i/2 = 1.41421. 339. 1. Develope the expression (a -f ^)'* ii^^o a series. We have (a + Z>)»" = a*" (1 + — )"'. For convenience make x = — , and the expression becomes a *- cC^ (1 + xY, If now we develope (1 -f xy^, and then restore the value of x and multiply by a"*, we shall have the development of (a + ?>)"•. Let (1 + xY = p 4- ^'^ 4- ^'-f' + sx^ + ^-^S &c. (1) If now X = 0, this equation becomes l""- = p ; that is, p = 1 ; whence (1 + x)"" = 1 + IZ-c + ^'-^^ + s.r' + ^.r^ (2) INDETERMINATE COEFFICIENTS. 243 Since (2) is identical^ we may have x == y ; whence (1 + ^)- = 1 + ^^ + rf + sif + tyK (3) Subtract (3) from (2), and divide by x — y, and we have (14 - z)"» — (1 -f vY _q [x — ?/) r (3;2 — .v^) s (x^— ?/3) ^ (a:*— .V*) (l + zj — (1 4-?/) ~ x — y x — y x — y x—y Let now x = y, whence \ -]- x = \ -\- y, and we have m (1 + cr)"^-! = ^ 4- 2ra: + Ssa;^ 4- 4;x', &c. (5) Mukiply both sides of (5) by 1 4- -^7 and we have (4) m (1 4- a:)"* = ^ 4- 2r :c2 4- 4i I x\ &c. (6) + 8s x -\- Zs + 2r Multiply both sides of (2) by m, and we have m (1 4- ^)'" = m -j- ??i2x -{- mrx'^ -\- msx^ -\- mtx^j &c. (7) x-j-3s Zr a;2+4^ 3s a;' . . . (8) Hence ??i+m2a:-|-W2rx^4-w^^^°4-- ■ •■==(l-{-2>r Equate the coefficients of the like powers of x, and we have m = q, whence q = m. _ m (in - 1) mq = 2r + q, mr = 3s 4" 2r, qns = -it -\- OS, 1.2 m (ill — 1) (m — 2) 1.2.3 ' 7?i (m — 1) (m — 2) (m — 3) 1.2.3.4 Substitute these values in (2), and we have m(m—l) ,, mhn—l)(m—2) , , m(m—l)(m—1)(m—^ . (l4-x)-=l+?«x4--A__Jx2+_L__^A_ _^,;34.^ r2T3T4 ' &c. (9) Substitute for x its equal — , and reduce, and we have a ((/4i/" = rt*"+?/ia'"-iZ-^ \ >' am-2^,2_^ _!^.^^ — A_ y a^-sis^ &c. (10) after multiplying both sides by a"*. This last equation is the Binomial Theorem, and it is true for any value of m whatever. 244 INDETERMINATE COEFEICIENTS. I 2. Develoi3e (1 -{- a-)^ into a series. Here m = ^. Substitute 1 for a, and x for Z», in equation (10). ^«.. (1 + :.). = 1 + - _ ^-^ 4- ^^-^ _ ^-^-^, &c. 3. Develope (1 -\- x)—"^ = into a series. Hero m = — 1. ^?2S. (1 + a:;)~i = 1 — X -{- x"^ — x^ -}- x*" — x^, &c. 4. Develope (l-f-x) — ^ == ^^ into a series. Here 011= — 2. (1 + xy Ans. (1 -]- x)-^ = 1 — 2x + 3x2 — 4x3 _j_ 5^4^ ^^^ 5. Develope (1 — x^—^ = y^ ~3 i^^^ ^ series. Here m= — 3. ^?2s. (1 — x)-3 = 1 + 3x + 6x2 _|_ 10x3 + 15x*, &c. 6. Develope (1 + x)- into a series. Here 711 = |. ^ ^3 ox ox X ox p ^ws. (1 + x)^ = 1 + — + -g To + "128" ""' 7. Develope (a -f x)^ into a series. ^1 j/, . X 2x2 2.5x3 „ \ ^ns.(a + x)3 = a3^1+___^ + ____ ^e.j Ifa=l,andx=l,thenTr2=l + ]-^+^-^^,&^^ Ti. -, ^ o .1 3/:7 T , o 2.4 2.5.8 2.5.8.16 , If a=l,and x=2, then V — 1 + 3-^ +.^ir9-3X9J2-' ^^• If a=8,and x=l, then ^9=2 ( l + ..,--J^+^-^, &e. ) 8. Develope (1 + ^)~' into a series. Ans. (1 + x)-^ = 1 — - + _- -f.^ &c. THE TABLE OF LOGARITHMS. ^40. AVe propose to show in the present chapter how the table of logarithms has been constructed. For this purpose it TABLE 0¥ L G A 11 I T 11 M S. 245 will be sufficient if we actually calculate the logaritliuis of a few of the lower prime numbers. In the equation cv' = iV (1) a is the base of the system, and x is the logarithm of the num ber A^. Assume a = 1 -f- m, and N = 1 -{- n-, Then (1 + mf = 1 + «, (2) and (1 + myy = (1 + ^0" (3) By the binomial theorem, (l+r.O-i/==l + xym+^y^"-^^-^^..^+^^^"V.^^ j'^^ -"-\ m^-{.,&c. (4) 1.2 1 . 2 . o (1 + »)v = 1 + y« + -^^ n^ + "'-"''l':^ "' +' S'- (6) Equate the right-hand members of (4) and (5), reject the unity, and divide by ^, and we have . ( „ -f »^ . „,. + (^!^-m^;>-^-) . ,„, +, &,. _ „ + .vzd . „, + iiszm;^„ls.o.) (0) If now y = 0, we have «; = log A^= log (1 + n) = fVri-^ — , 4T r- (7) Since m = a — 1, and n = N — 1, we have ^^ ' ~ (a -l)-i(« -1)'+K« -l)'-i(« -iy+;&c. (8) This equation contains the logarithm of N in terms of N and the base 3 but, for actual computation, it is necessary to modify its form. Let the reciprocal of the denominator of (8) be represented by My and replace n for N — 1, and we have log (1 + 7i) = 31 Qi — hi" + \ii? — {ii^ +, &c.) (9) 21* 246 TABLE OF LOGARITHMS. The factor M is called the modulus of the system. If n were negative, we should have log (1 _ n) = M (_ n — hi" — h^ — ]n* — , &c.) (10) Subtract (10) from (9), and we have 1 _i_ 1) ( Vide 138 .) log ^^^ = 2M(^n + In' -f hi' + ]7i' -\-, &c. (11) W e may now assume n = -, whence z, = -^ -i? + 1 1 — 7i p Then ^ ^^^ ^ " ^ ^ \ (2;.+ l)^3(2^+ir 5(2^+1)^7(2^+1)^' / (12) In (12) make 31 = 1, and p = 1. Then, since log p = log 1 = 0, we have The method of summing: this is as follows : — '^ o 82 = 9 9 9 9 9 9 9 9 0.66666686 ~ 1 = 0.66666666. 0.07407407 -~ S = 0.02469136. 0.00823045 0.00091449 0.00010161 0.00001129 0.00000014 5 == 0.00164609. 7 = 0.00013064. 9 = 0.00001129. 11 = 0.00000103. 0.00000125 -^ 13 = 0.00000010. 15 2 0.00000001. log 2 = 0.69314718. In (12) make 31 = 1 and ^; = 2, and we shall find log 3 = 1.098612. 2 log 2 = log 4 == 1.38629436. If^ = 4, then, log 5 = 1.60943790. Log 5 + log 2 = log 10 = 2.30258508. TABLE OF L G A 11 1 T II M S. 247 Logaritlims calculated as above are known as Napierian Loga- rithms, in honor of Lord Napier, their inventor. It is usual to distinguish them by the contraction aYa^). Thus, Nap. log 10 = 2.30258508. We are now to show how common logarithms may be calcu- lated from them. 241. Since Napierian logarithms assume the modulus to be 1, it follows that if the Xap. log. of any numhcr is multiplied hy the modulus of any other system, the result will he the log. of the same numher in that system. 242. To find the modulus of the common system cf loga- rithms. Since log (1 -f n) = il/ | « — ^ -f ^ — j, &c. j n^ u^ ?{■* and Nap. log (1 4- ''■) = '^ — t," "^ ~ 1' ^^^' iU O "x . ir ^'"^ (^ + ^0 we have Jl = vj -. —. — -, -• ^ap. log (1 -T- n) That is, the modulus of any system is the log. of any number in that system, divided by the Nap. log. of the same numher. The log. of the base of any system is L Therefore, the base of the common system being 10, we have .r log 10 L_ ~ Nap. log lU 2.30258508 ' Hence, the modulus of the common system is, 0.4342944:819. 243. If now the Nap. logs, of 2, 3, 4, 5, &e. are multiplied by 0.4342944819, we shall have Common log. 2 = 0.301030. " '^ 3 = 0.477121. " " 4 = 0.602060 = 2 log 2. " " 5 = 0.698970. c^vC. &C. 248 TABLE OF LOGARITHMS. In this -vvaj tte whole table may be constructed. In practice, the modulus may be retained in the formula to save the trouble of multiplication; and the decimals may be carried to any de- sirable extent. ( Vide 153.) 244. To find the Napierian base. From 242 we must have the following property. The logs, of the same number in different systems are to each other as the moduli of the respective systems. If a represent the base of the Napierian system, then, since its log. is Ij and the modulus of the system 1, we have com. log a : 1 : : 0.4342944819 : 1. Multiply extremes and means, and we have com. log a = 0.4342944819. The number in the tables corresponding to this log. is 2.718281828459, which is the base of the Napierian system. S45. To construct a table of logarithms according to any system whatever. In the equation cv' = JSf, assume a equal to the desired base, and N equal to the consecutive numbers, and resolve the result- ing equations. Thus, If we desire the base to be 2, make 2' = 1, 2== = 2, 2' = 3, 2=^ = 4, 2* = 5, &c. and resolve the equations. Thus, 1. X log 2 = log 1, whence x = J-g = gjj^ = 0.000000. 2. .log 2 = log 2, " . = g = |lg= 1.000000. PRACTICAL APPLICATIONS. 249 log 3 477121 . r,Qif;^Q 3. rr log 2 = log 3, whence ^ = j^ = 301080 = ^'^^^^^^ o &C. &C. By continuing tlie process, we should form a table of logs, with 2 as a base. It is evident that the base cannot be a negative numler. PRACTICAL APPLICATIONS. 246. 1. Solve the cr|uation 7^ = 13. log 13 01Q1 log ( log 2 - lo g 3 log: o — loo; 2. Solve the equation ^ 8. Solve the equation onn'' = a. I02: ct — log m Ans. X =-- — ^= — -, log n 4. Find the value of n in the equation I = «r"-i. log I = log a + (« — 1) log r. whence n — 1 = log I — iVg '«- I02: r Therefore, n = ' "" '~ '°^ " + 1. ( Vide 231, form. 17.) y log r CT (?•" — 1) 5. Find the value of n in the equation s = — ^TZTl — We have s (r — 1) = ar"" — a, or, a -\- (r — 1) s = ar'", whence log [a + (r — 1) .s] = log ^y + ^^ log r. log la + (/• — 1) -s] — log « Therefore, n = j^^^; ' (Fi(7e 231, form. 18.) 250 PRACTICAL APPLICATIONS. 6. Find tlie value of n in the equation a (s — a)^~^ = \Ye have log a -f 0' — 1) log (s — l) = log l-^Cn—l) log {s — I). mi ^ I02: ^ — log a Therefore, n = . ^^^ .-^— -f 1. log C« — a) — log is - I) Vide 231, form. 19.) 7. Find ?i. in the equation (s — ^) ?■" — fe-r"— ^ -f ? = 0. This is the same as r (s — l}?-''-^ — s/'"-i -f ? == 0, which is [r/ — (r — 1) .s] 7'' — 1 = /. The log of this is log [rl— (r — 1) .s] + (n + 1) log r = log^. losr I— log [rZ — (?' — 1) .si whence n = -^ ^ !^ ^— ^ + 1 log r Vide 231, form. 20.) CHAPTER VIII. EQUATIONS OF THE THIKD DEGREE. a47. The general form of equations of tliis degree i^ cc' -}- px^ -{- 2'X = m. (1) In which p, q, and m may be positive or negative. 24S. If p = 0, (7 = 0, and m = «% we have a:^ = a^j or a.^ — a^ = 0. By 78, this equation may be written thus: (.r — «) (.i'^ + ax + «") = 0. Divide by each factor of the first member, and we have X — a = and x- + ax -f a- = 0. From the first of these we have X = a; and from the second, a: = ^ ( -- 1 ± |/_ y). ( FiWe 167.) Li Hence the equation has three roots, two of which are imagi- nary. 249. If p = 0, (^ = 0, and m = — a^ wc have a.-^ = — (x\ or .>:' -f <^^^ = ^• By 79. this may be written thus : {x -f a) 0^2 — ax -f- rr) = 0. 251 252 EQUATIONS OF THE THIRD DEGREE. Hence x = — a, and x = ^ (1 dz y" _ ^S). ( Vide 167.) 250. 1. Given x^ = 1, to find the values of x. Here x^ — 1 = 0; Hence (x — 1) (x^ + rz: + 1) = 0; Whence x=l and a: =H. — 1 ± V — 3). 2. Given x^ = 8, to find the values of x. Ans. X =: 2 and — 1 it V — o. 3. Given a;^ == 27, :r« = 64, x^ = —125, and x^ = —216, to find, &c. 4. Given x^ = 10, to find the values of x. Alls. X = i^Tu and #" y^ ( — 1 ± l/"^^^), or X = 2,1544 and — 1.0772 it 1.8657 V^^. 251. If, in the equation x^ -\- px^ -\- qx -{■ m = 0, a is an EXACT ROOT, THEN THE FIRST MEMBER IS EXACTLY DIVISIBLE BY X — a. For, since a is an exact root, wc may substitute it for x in the given equation, and write a' + i^"^ + S'^ + *'^ = 0. (2) Now let us proceed to divide the given equation by x — a. 0? -j- pj^ -\- qx -\- in\x — a x^ — ax^ x^ -|- (p + «)^ + 2' + <^ (p + <^) (p 4" ci^x"^ — a (p -j- a)x {q -{• a {p + «) )'^ + ^^ (<7 + a (^3 + a) )x — o'5' — a^ {p -\- a) Tlie last remainder is aq -\- a? (^p -\- a) -\- m =■ d? -\- pa} -\- qa -{- m. But, by (2), this remainder is 0. Therefore the quotient is exact. EQUATIONS OF TUE THIRD DEGREE. 253 252. It is evident that the same reasoning applies to any equation of the form ^n _j_ ^^^n _ 1 _^ ^^^n - 2 &c. + fX + 111 = 0. (3) For, since the dividend is equal to the divisor multiplied by the quotient plus the remainder, if we denote the quotient by /, and a supposed remainder, on division, by r, we should have X^ -f px'' - 1 -f- qx"" - 2 &C. + tX -\- 7)1 = (^x — a) I -\- v. (4) Now, if a is substituted for x, the first member of this equa- tion is 0. But the first term of the second member is 0, and .-. r = 0. Hence, if a is a root of any equation of the form of (3), THAT EQUATION IS DIVISIBLE BY X — a. EXAMPLES. 253. 1. One root of the equation x^ — 6^^ + llx = 6 is 1 : what are the other roots? We may write x^ — 6x^ -f llx — G = 0, which, by 251, is ex- actly divisible by x — 1. On dividing, the factors of the equa- tion are found to be {x — 1) {pc^ — bx -\- Of) = 0. Hence x^ — bx = — 6 ; From which a; = 3, or x =■ 2. 2. One root of x^ -\- 4x- — 7x = 190 is 5: what are the other roots? Ans. J ( — 9 =h /^fi). 3. One root of x^ — 4a;^ — llx = — 30 is 2 : what are the other roots? Ans. x =■ b and x = — 3. 4. One root of x^ -\- x^ — 22a; = 40 is 5 : what are the other roots ? Ans. x = — 4 and x = — 2. 5. One root of x^ -\- 2x^ =: IG is 2 : what are the other roots? Ans. 2 (_i±i/^in:). G. One root of x^ — 2x^ — 2x = — 4 is 2: what are the other roots? Ans. ± 1.41421. 22 25-i EQUATIONS OF THE THIRD DEGREE. 7. One root, of jc^ — 2x = — 3 is I : what arc the other roots? Ans. 1.0962 and — 1.5962. 8. One root of x^ — 6x- + x = — 28 is 4 : what are the other roots? .1;^.'. 3.82841 and — 1.82841. 9. One root of x^ + ^x- -f 26x- = — 24 is — 4 : what are the other roots? Ans. —2 and —3. 254. If the equation x^ + jjx^ -\- qx -\- m =1 is divisi- ble BY X — a, THEN a IS A ROOT OF THE EQUATION. For, in this case, the remainder is 0, and, by § 251, we may "Write x^ -f po6^ -\- qx 4- 'in = (cc — a) (x^ -f (p ~\- a^x + ^ + « (p + ^0 )• lfx = a^ the second member reduces to ; .•. a substituted for X will reduce the first member to 0. Hence a is a root of the given equation. 255. The same troposition is true of the equation aj" H- jj.z,"-i -f qx''-- &c. -f tx 4- on = 0. For in this case we may write, by § 252, a" + px" -1 4- q.r''-- kc. -f fx -f 711 = (.r — a)l If x=-a, the second m.ember reduces to 0; .•. a substituted for X will reduce the first member to 0. Hence a is a root of the given equation. 256. Hence, to ascertain if a polynomial containing x is ex- actly divisible by x — a, substitute a for x, and if the polijnomial reduces to 0, the division is exact. EXAMPLES. 1. The equation x^ — 12x2 -{- 47.'c — 60 = has three loots less than 10: what are they? Ans. 3, 4, and 5. 2. The equation x^ -f- ^-^^ + 26x + 24 = has three negative rojts less than —10: what :;re thov? Auft. —2, — '1, and —4. EQUATIONS OF THE T II 1 R ]) DEGREE. 255 / 3. The equation x^ — Qx^ +ll:c — 6 = lias three roots less than 10: what arc they? Ans. 1, 2, and 3. 4. The equation x^ + Ux^ + 44:c + 32 = has three nega- tive roots less than — 10: what are they? Ans. — 1, — 4, and — 8. 25T. The equation x^ -f-p.r- -\- qx -f ??i = has three ROOTS, and only THREE. For, if a is one root of the equation, its factors, by § 254, are (x _ a) Oi-2 + (j) + a)x -f 5 + a (p + cv)) = 0. Dividing first by one factor and then by the other, we have X — a = 0, and x"^ + (j^ + «)-^ + 2' + <^ (P + ^) = ^• The roots of which are x = a, and ^ = - K(P + «) ± V (p - ay - 4 C^r + ^j). (5) 258. The equation :/;" A-jpx''-'^ -f ^x" — -&c. -f ^j? + ??i has Qi roots, and only n. For^ if the equation were of the 4th degree, i.e. n = 4, on dividing by a: — «, a being one root, it would be reduced to an equation of the 3d degree, which, by §257, has 3 roots; .-. the equation itself has 4 roots. Since the same reasoning applies to an equation of the 5th, 6th, 7th, &c. degree, we conclude generally that an equation OF the 7?TH degree has n ROOTS. examples. 1. One root of x^ — llx^ + 1^-^ -f 84 = is — 2 : what are the other roots? Here a = — 2, j:» = -— 11, and 5 = 16, and formula (5) 'gives a: = 6, and a: = 7. 2. One root of x^ + 7^^ — 4x — 28 = is — 7 : what are the other roots? Ans. x = ± 2. 256 EQUATIONS OF THE THIRD DEGREE. 3. Oue root of x^ + IZj~ -\- 44.r + 32 = is — 1 : Tvhat are the other roots? Ans. — 4 and — 8. 4. One root of x^ — 12:c' + 2Sx'- -}- 6Sx — 84 = is 1 : what are the other roots? Ans. . 5. One root of x' + 4^^ — 25x2 — 16x + 84 = is 3 : what are the other roots? Ans. . 6. One root of x= + Ox* — bx^ — 141x2 + 4x + 420 = is — 5 : what are the other roots? Ans. . 259. If a, h, and c are the roots of x^ +/>x2 -j- qx -\- m = , then we shall have x^ -\- px^ -^ qx -\- m = (x — a) (x — Z^) (x — c). (6) For, since the given equation is divisible by either of the expressions (x — a), (x — ?y), or (x — c), it must be composed of these three factors, and none others, considering, also, that it can have only three roots. 260. In the same manner, if a, h, c, .... Ic, and I are the roots of the equation x"- -\- px'"' — '^ -]- qx"- — ^ kc. -\- tx -\- m^ then we shall have x'* -f px'' — ^ + g-x" — 2 &c. + ^x -f m = (x — a) (x — l>) (x — r) {x-h) {x- I). (7) EXAMPLES. 1. Find the equation whose roots are 1, 2, and 3. Ans. (x — 1) (x — 2) (x _ 3) = x3 — Gx^ + llx — 6 = 0. 2. Find the equation whose roots are 1, 2, and — 5. Ans. x' + 2x2 _ 13^ -f- 10 = 0. 3. Find the equation whose roots are l/ 2, — V^, and 2. 2x2 _ 2x = — 4. EQUATIONS OF THE T II 1 K D DEGIIEE. 257 4. Find the equation Tihosc roots arc — G, — i/8, and l/o. Alts, x^' + 6.t^ — ox == 13. 5. Find the equation Avhosc roots arc 3, 1 + yd, and 1 — V o. Ans. x^ — hx^ -f- 4x = — G. 261. If the second member of (6) is developed bj actual mul- tiplication, -^e have .i? -f- "px- -f ^x -f- "^ == ^-'^ — (ci -j- ^ + 0^^" "i- {ah -}- ac -\- hc)x — ahc. (S) Whence p = — {ci -\- h -{- c), q= ah -{- ac -{- he, and m= — ahc. ( Vkh 237.) That is, — 1. The coefficient of the first term is 1. 2. The coefficient of the second term is the algebraic sum of the roots with a contrary sign. 3. The coefficient of the third term is the algebraic sum of the roots taken in products, as many times as there are different sets of trco roots in each set. 4. The term independent of x is the product of the roots with a contrary sign. EXA^IPLE^. 1. Find the equation whose roots are 1, 2, and 3. Here j; = - (1 + 2 + 3), 2= 1 X 2 -f 1 x 3 + 2 x 3 and m = — 1x2x3. Ileuce the equation is x^ — Gx- + H-'' — G = 0. (T7(7e§ 260, Ex. 1.) 2. Find the equation whose roots arc 3, 3 + v'o, and 3 — ] o. Here p = — 9, q = 24, and m = — IS. Hence the equation is x^ — 9,/- + 24x — 18 = 0. 3. Find the equation whose roots arc 1, 2, and — 3. Here p = 0^ q = — 7, and m = G. Hence the equation is x' — Ix -\~ G = 0. 4. Find the equation whose roots are — ^, — |, and — ]. 22-' 258 EQUATlO:sS 01-' TiiE T II 1 11 D DEGllEE. i;^' \L — ij7 ""^ '"■ — ^4 Here p Hence the efjiuition is x^ A — T~r-{- — r -f Ti = ^• 5. Find the equation whose roots are 5, 5 + ]/ — 1, and 5 — \ l/'^TT. Ans. x^ — lox- H- 76.x- — 130 = 0. j 262. If the second member of equation (7) were developed by actual multiplication; we should find the law for the coefficients j of the first three terms the same as before, while the coefficient j of the fourth term iciU he the algehraic sum of the j^^'oducfs of the \ roots talcen os many times as there are different sets icith three \ roots in each set, the sign of the final residt being changed. \ The coefficient of the fifth term icill he the algehraic sum of all ' the p?*oc7wc^s of tJie roots, taken as many times as there are sets icith four roots in each set. \ And, in general, the cocfiicient of the ni\\ term will he the ' algehraic sum of all the products of the roots, taken as many times as there are sets with n — 1 roots in each set, the sign of the final result being changed when n is even and retained when n is odd. This n must not be confounded with the n which denotes the degree of the equation. EXAMPLES. 1. Find the equation whose roots are — 2, — 2, 4^ and — 4. Here p = 4, (/ = — 12, t = — 04, and m = — 64. Hence the equation is x'^ -j- 4:x^ — VZx"^ — 64 x — 64 = 0. 2. Find the equation whose roots are 3, 4, — 1, and — 6. A?is. x* — 31.t'^ + 42x + 72 = 0. 3. Find the equation whose roots are 1, 2, 3, 4, and 5. Ans. a.-^ — 15.^* + Sbx"" — 225x2 + 274x- — 120 = 0. iJ6J$. The equation x^ +7>'" + qx -j- ??i = may be trans- formed INTO another of the same form, in v/iiicii the EQUATIONS OF THE THIRD DEGREE. 259 ROOTS ARE ANY GIVEN MULTIPLE OF THOSE OF THE GIVEN i EQUATION. y 1 For^ in the equation x" + i^t} + qx + m = 0, substitute j for Xy 1 and we have 73 ^T^ + T" + '^^ = ^' which multiplied ! by U gives y^ + ^i^^/^ + J^^9.y + ^^^^^ =0; (9) which is the equation it was proposed to obtain. The roots of ' (9) are h times the roots of the given equation; since y = kx. I EXAMPLES. I 1. Transform the equation aj^ — 6x- + 11^ — 6 = into an- other whose roots are 3 times as large. ?/ Here ic='^, i.e. y = 3x'. Make k of (9) = 3, and we have ! y _ 18^2 _^ 99^ __ x(52 = 0, where p = — G, ^ = 11, m = — 6. : 2. Transform the equation x^ — Gx^ + llx — 6 = into an- , other whose roots are A as large. Here 7c = ^. Ans. y - 3/ + -^ - I = 0. I 264. In THE EQUATION X^ -\- l^x"^ -\- qx -\- 77ix'^ — 0, IF THE i COEFFICIENTS ARE FRACTIONAL, THE EQUATION MAY BE TRANS- | FORMED INTO ONE WHOSE COEFFICIENTS ARE ENTIRE. | For, in equation (9) it is manifest that k may be so taken 1 that all the coefficients may be entire if either j^} 9.) ^^ *^^ ^^® ' fractions. I EXAMPLES. I 1x^ Ix 1 I 1. Transform the equation a:^ + -^ 4- ^ + ^ = into one ^ whose coefficients are entire. Here j; = |, q=i ■^^, and m = q\. Make 7c = 8, and equation (9) becomes // + 7^^ + 1^^ -f 8 = 0, the roots of which arc 8 times those of the given equation. 230 EQUATIONS OF THE THIRD DEGREE. 2. Transform ^^ -h -j "f~Tp~f'FT'^^ ^°^^ ^" equation wtose coefficients are entire. Make 7c = 4. ^?IS. f -^ if ^ 1/ ^ I =, 0. lice 3. Transform a;' — ox^ -J — ^ | == into one whose coeffi- cients are entire. Make k = 2. Ans. f — Qf -f lly — 6 = 0. 4. Transform x^ — -rp- + ,^^ — — - = into one whose coeffi- 25 30 40 cients are entire. Make k = 150. Ans. f — 12/ 4- 750y — 84375 = 0. 265. The propositions of §§ 263 and 264 are equally ap- plicable TO THE EQUATION X"" -f jrx''—'^ -\- qx"" — ^ . . . -\- tx -}• m =0. For, make x = y, and we have ^ 4- p ^"~ -J- y ^"*"^ ... + ^ ^ -f 771 = 0. Multiply this by ^'*, and we have j y^ _}_y,y^-l -f. qkY~^- • • + ^^^"''"^ + ^nk"" = 0. (9) ! In this equation, 7c may always be so taken that the fractions will all disappear. EXAMPLES. 5.r3 bx-" 7x 13 1. Transform .x* — -j- .pr- — ^r^r^-x into one whose b 12 loO 9000 coefficients are entire. Make 7c = 30, and vre have / _ 25/ + 375/ — i:i60y —1170 = 0. The roots of this equation are 30 times those of the given equation. 2. Transform :,.- _4- _- — -_- ^ = 0. EQUATIONS OF THE THIED DEGREE. 261 Make h = GO, arrd we have f ^ 65/ + 1891V — 30720/ — 928800y + 072000 = 0. 2S®. The equation x^ + px^ -\- qx -{• m = may be TRANSFORMED INTO ANOTHER EQUATION OF THE SA3IE FORM, WHOSE ROOTS differ FROM THOSE OF THE GIVEN EQUATION BY A GIVEN QUANTITY. For, let X = y + ''• Substitute this value of x in the given equation, and we have (y j^ ry 4- P (i^ + '0^ + 2 (y + ^0 + ^^^ = 0. By expanding the terms, we have ,f _|- 3^v _j. Sf/r'- 4- r^ -\- j^if -f liwy + p>^ + 5^ + 'F + ^'^ = ^■ Arrange the terms according to the powers of y, and we have / + (3r + p) / + (3r2 + 2pr + q)u + r^ +p?-2 -f ^r + m = (10). Which is the equation it was proposed to obtain, since y ^=- x — r. If we write the coefficients of this equation above each other, commencing with the last, we have 3r2 + 2pr + ^, (D^). If the term independent of r be considered as the coefficient of r", these coefficients are derived in the following way: — The first coefficient is what the given equation becomes when r is substituted for x. The second coffcient is derived from the first by multiplying the coefficient of each term by the exponent of r in that term, and then diminishing the exponent by 1. It is called the First Derived Polynoviiah The third coefficient is derived from the second in the same way, except that each product is divided by 2. It is called the Second Derived Polynomial. 262 EQUATIONS OF THE T HI 11 1) DEGREE. EXAMPLES. 1. Transform the equation x^ — 6x^ -j- llx — G = into one whose roots are less than those of the given equation by 4. Rere 7-' -}- pr' + qr -\- m = (4)3— (3(4)2 -f 11(4) —6= q 3r2 + 2^r 4- (2 = 3(4)2 _ i2(4; +11 =11. 3r+p==3(4) - ^3^ = 6. Hence the equation is j/^ + Qy^ -]- 1^}/ -\- Q = 0. The roots of the given equation are 1, 2, and 3. Those of the transformed equation are — 3, — 2, and — 1. 2. Transform the equation x^ — 12it;2 -f 47a: = 60 into one vrhose roots are less than those of the given equation by 1. Ans. f — 9/ _f_ 26j/ = 24. 3. Transform :i^ — Ox^ -f 26./: = 24 into one whose roots are less than those of the given equation by 1. Ans. y^ — 6^2 _j_ \\y s=: G. 4. Transform x^ — Gj.- -f llx = 6 into one equation whose roots are less than those in the given equation by 1. Ans. y^ — 3^^^ -j- 2y = 0. 5. Transform x:' -f 13x- -f 44x -f 32 = into an equation whose roots are greater than those in the given equation by 10. {Vide 256, Ex. 4.) Ans. / — 17/ + 84^ = 108. SG^y. The proposition of § 266 is equally applicable to the equation x" + px'^ — ^ + qx^ — "^ ...-{- tx -\- m •= Q. For, make x = 1/ ~{- Vj substitute in the given equation, develope the several terms by the binomial theorem, and arrange the terms according to the powers of y, and the coefficients will become as follows: — Of / it is ?•" + ^jy«-i -f qr^-'^ tr -{- m (D^). Of y it is 7?r"-i+(?i — l)pr"-2 + (n — 2)qr''-^.... -\-t (D^). Of / it is «(»-^ ^ ,„_2 , 0—l)C«-2 ) „_, ( «-2)(«-3) 1.2 "^ 1.2 "'"1.2 ■■\^2)- EQUATIONS or THE T II I K D DEGREE. '26o Of / it is ,(,,-1)0,^2) 3 (. - 1) (n - 2) (. - S) ,^ 1.2.3 . 1.2.3 ^'^''- ^-^^^• „ - ^ . . ?i(?i — 1) (n — 2) ( 71— 3) , „ .^ . Oh' It IS ^ '^"-^ ^ , ^^^ --^ v--^ &c. (DJ. Each of these coefficients is derived .from the one immediately preceding, according to the following taw of derived jiol^iio- mials : — Multiply each term in succession by the exponent of r in that term, divide the product by the number which designates the place of the coefficient^, and diminish the exponent of 7' by unity. examples. 1. Transform the equation 3x* — 4:X^ + ^-^^ + 8'' — 12 = into one whose roots are less than those in the given equation by 3. Here (D„) = 3(3^- 4(3y> + 7(3)-^ + 8(3) -12 = 210, (D^) = 12(3/ - 12(3)-^ + 14(3) + 8 = 266, (D,) = 18(3)-^- 12(3) + 7 r= 133, (DJ = 12(3) - 4 =32, (DJ= 3 = 3. Hence the equation is, 3/ + 32/ + 133^^ -f 266y + 210 = 0. 2. Transform x^ — 31.r^ -f 42j; + 72 = into one whose roots are greater by 5. Ans. / — 20/ + 119/ — 148y= 216. 26S. The equation x^ -^ j)x^ -\- qx -j- m = may be trans- formed INTO another equation WHOSE SECOND OR THIRD TERM IS WANTING. For, to make the second term disappear, make, in (10), 3r + p = 0, or r = ^ . 261 E Q U A T I X 5 OF THE T 11 1 L D DEGREE. To make the third term disappear, make; in (10), S/- -f 22^ + ^ = ; i.e. r = |(— p dr V p' — oq)- EXAMPLES. 1. 3Iake the second term disappear in the equation dS' — C..-.^ -f 11./: -6 = 0. Here /• = {' = 2. Therefore the coefficients of (10) become r3 ^ pr- -f qr + m = (Tf — 6(2)2 ^ ii(2) — 6 = 0. Sr^ + 2pr 4- q = 3(2)^ - 12(2) +11 = - 1. 3/-+ ^9 = 3(2) — 6 =0. Hence the equation u ]f — y =^ 0. The roots of this equation are, of course, less than those of the given equation by 2. The roots of the given equation are 1, 2, and 3. Those of the transformed equation ought to be — 1, 0, and -fl. And -^e really have y^ — ,y = ,y (if — 1) = ^; whence y = and y = = i. It may frequently happen that more than one term disappears at a time, as in the example above. 2. 31ake the third term disappear in the equation x^ — ^j} -f- 11.,- — = 0. Here r =: 2 ± J i/o. The coefficients of (10) become (2 ±L I Vl'f - 6(2 ± \ v'y^y + iir2 ± i V^) - 6 = zp 1 1/3, 3(2 ziz I V%f — 12(2 ± \ 1/8) -f 11 = 0, 8(2 dz \ VVj) — 6 = ± 1/3. Hence the equation is y' =t y Ij. y- zp | y 3 = 0. 3. Make the second term disappear in the equation x^ — 14x^ -fGlx = 84. , , 13y o 4. Make the second term disappear in the equation :>^ — 12.i''' -f 47jc = GO. Ans. y — y = 0. 2'iX?Ii2i:i£ IJ it2iiIX23> JliILT3riJtlJLl.5- i^ s:?5SCKr JT3- i,. . : - - - 7.,.- 2 — — xU SBB£. — _ -1 ;k 4 % i^^. -w^- r" — 3::r" — ez ^ » = {r — 3* — 3r *2 — If* ,7 — -» wi-=zji — - — r n^ — - — xi r — ? / — " — aur^ — Si — js — -;r- — _7«^ — ' — — ■ — XL r — 2; — _[ — ff" — ~ — — -^-i;»fT— 5; — » 266 L Q U A L ' ROOT 3. ?•' -f jyr- + qi' -f m = (r — a) (r — Z>) (r — c). 3,-2 4_ 2p>- + q= 0' - «) 0' - ^) + (^- - a) (r - c) + 0' - ^) (^ - c). 3r -f ^9 = (r - «) + (^- -h)-\- (r - c). 1 = 1. Since r is arbitrary, we may make it equal to re, and thus obtain x^ -\- px'^ -{- qx -\- m z:= {x — a) (2; — b) (x — c) (G ), (Di) = 3z2 + 2i.x- + g = (.T-a)(.r-i) + (:t--«)(z-c) + (a:-i)(2--c)(f/,), Equation (d^ proves that the first derivative is equal to the sum of the products of the several factors of (6) taken as many times as there are different sets with two factors in each set. Equation (^d^ proves that the second derivative is equal to the sum of the several factors of (6). By taking the same steps with the equation x^-\-px'^^^-\-qx'"—'^. . . -\-tx-^m=^(^j:—a)(x — 6)(ic — c) (x — I) (12), "we should find the following relations : — The first derived polynomial, that is, D^, is equal to the sum of the products of the n factors op (12) taken as many times as there are sets with n — 1 factors in EACH SET. The SECOND derived polynomial, that is, i>2, IS EQUAL TO THE SUM OF THE PRODUCTS OF THE n FACTORS OF (12) TAKEN AS MANY TIMES AS THERE ARE SETS WITH 71 — 2 FAC- TORS IN EACH SET. The DERIVED POLYNOMIAL, i>„ _ 1, IS EQUAL TO THE SUM OF THE n FACTORS OF (12). EQUAL ROOTS. 2'YO. If, in (6) and (t?^), § 269, we make a = b, the equations become x^ -\- px^ -\- qx -\- m = (x. — o,)^ (x — c), and 3x^ + 2px -\- qz=z (x — a)^ -\-2 (x — a) {x — c), the second members of which arc each divisible by x — a. Hence, EQUAL R T S. 267 If the equation a.^ -\- jpx^ -{- qx -\- m=.^ contains equal roots, the equation and its derived polynomial will have a common divisor containing that root; and, conversely, If the equation and its derived polynomial have a common divisor, the equation has equal roots. If we make a-=h = c, the equations become a;' -f px^ •}- qx -\- m = (x — a)^, and 3x^ -f- 2j)X -\- q = ^{x — ay, the second members of which are each divisible by (x — ay. Hence, If the equation has three equal roots, the greatest common di- visor of the equation and its derived polynomial contains two of these equal roots. And, conversely, if the greatest common divisor contains two equal roots, the given equation contains three roots of the same value. And, generally. If the equation x" -f px''—'^ + qx'' — '^ . . . . -\- tx -^ m = CONTAINS EQUAL ROOTS TO THE NUMBER OF S, THE EQUATION AND ITS DERIVED POLYNOMIAL HAVE A COMMON DIVISOR CON- TAINING S — 1 OF THESE EQUAL ROOTS ; or. If the GREATEST COMMON DIVISOR OF AN EQUATION AND ITS DERIVED POLYNOMIAL CONTAINS S — 1 EQUAL ROOTS, THE EQUATION CONTAINS S ROOTS OF THE SAME VALUE. EXAMPLES. 1. Find the roots of the equation x^ — llx^ -f 32:c — 28 = 0. The FIRST DERIVATIVE is Sx^ — 22x -f 32 = ; the roots of which are x = 2 and Ls. Of the factors x — 2 and x — \^ the former is that which will divide the given equation (^Vide §256); .-. the roots are 2, 2, and 7. And, in fact, (.r — 2"/ {x — 7) = x^ — Ux^ -f 32a: ~ 28, 268 GENERAL SOLUTION OF THE 2. Find the roots of the equation x^ — lOx^ -{- 33x — 36. (Vide 84, Ex. 8.) Ans. 3, 3, and 4. 3. Find the roots of the equation x^ — 13x^ + ^6x — 80. (Vide 84, Ex. 9.) Ans. 4, 4, and 5. 4. Find the roots of the equation cc* — 5^^ + 9x^ — 7x + 2. (Vide 84, Ex. 6.) Ans. 1, 1, 1, and 2. 5. Find the roots of the equation x* — lOx^ + 37x2 — 60x + 36. (Vide 84, Ex. 7.) Ans. 3, 3, 2, and 2. GENERAL SOLUTION OF THE EQUATION OF THE THIRD DEGREE. flKl, By § 268, any equation of this degree may take the form x^ •}• qx -\- m = 0. (1) If X = y + r, we have x^ = ?/^ + *^ + ^P' (.y + ^') J Or, which is the same thing, x^ = ^^ + /^ + o^r.x. Hence x' — Si/r.x — 7/^ — r^ = 0. (2) Therefore x^ + g-x + ^^^ = ^'^ — 3^r.x — i/^ — r^. (3) Hence, by § 237, i/^ -{- r^ = — m, and 3?/7' = — q. Whence y =^ - - + ^^ + |^, and r=\ ~ ^ - \-^ + |^- Therefore x =;/-f+V?+| +^/-f-^/f^• W If we suppose this root to be equal to a, the factors of (1) become x^ '\- qx ■{- m = (x — ci) {;x^ + ox + «^ + 2"); Therefore x"^ -\- ax -\- a^ -\- q == 0, whence x = ^J (— a db V^'Sa^ — 4^). (5) Equations (4) and (5) contain the roots of (1). EXAMPLES. What are the roots of the equation x' — ISx^ + lOlx = 180? By making the second term disappear, we have f—7i/ = 6. Then q = -^7 and m = — 6. EQUATION OF THE THIRD DEGREE. 269 By substituting these values of q and m in equation (4)^ we have X = ..3+ io;/_3.|_ v>_io^/_3. Now, each of these terms may be expanded by the binomial theorem, and added together. The terms involving j/— 3 will cancel, leaving x a real quantity. This is called the irre- | DUCIBLE CASE, and always arises when t- "^ 97 ^^ ^ negative quantity. Its occurrence renders formula (4) entirely useless in | practice. The roots of the given equation are 4, 5, and 9, and ] yet the formula will not reveal them without the use of an infinite series. ' SI'S. Numerical SGlution of Cubic Equations. ' Take the equation | x^ -\- px"^ -\- qx = m. (1) ! I Find by trial a number which, on being substituted for x in I the given equation, will produce a result less than m, but such I that if it is increased by uniti/^ and again substituted, the result ! will be greater than m. Let r be such a number. Then, if we ! regard it temporarily as the exact root, we may write | i f 3 _^ pyi _^ ^^. __ ^11 . (^2) I Whence 111 q + p'' + '''^ Having found ?*, denote the remaining figures of the root by n, wheace ^ = /• + >J- (3) Substitute this value for ./: in equation (1), and we have (>• + if + p {>• -V u'y + 2 ('• + :j) = '»• W Expand, and arrange in reference to y, and we have ;/ 4. (3r + iOy + (3'' + 2i)r + g) y + (/' + P'-^ + qr) = m. (5) 23- 270 GENERAL SOLUTION OF THE 3Iake i^ = or -f /:>, (j^ = Sr^ -f 2p^ + 2^? and m^ = ??i — (r^ 4- V'^ ^" !?'')? and we have y^ -}- p^j/^ + cj^y = ??i^ (6) The first figure in tlie root of equation (6) which may be found in precisely the same way as in equation (1), is the second figure in the root of (1). By repeating the process, the third, fourth, fifth, &c. figures of the root of (1) may be found. In applying the preceding principles, we proceed as folloAvs : — 1. Arrange the coefiicients with their signs in a line, and to the right of them place the right-hand member of the equation. 2. Having found the first figure of the root, multiply it into the first coefiicient and add the product to the second coefficient, which sum multiply by the same figure of the root, and add the product to the third coefficient, multiply this sum by the same first figure and subtract the product from the term constituting the second member of the equation. The remainder is the first dividend. 3. Multiply the first coefficient by the same first figure of the root, and add the product to the last number under the second coefficient; which sum must be multiplied by the same figure, and the product added to the last number under the tJiird co- efficient. This last sum is the first tried dicisor. Multiply the first coefficient by the first figure of the root, and add the product to the last figure under the second coefficient. 4. Divide the first dividend by the first trial divisor, and the (juotient is the second fi< ''^^ 4- .y 4- '^v = '^h I IG. Given ' , , ' y U> find y:, y, z, and ip. ( v; + d (x +y -h z) = y, J . m ^^/ 1 — a \ — n 1 — c 1 — a 17. Given —1^—^ => !t^^, to find the value of :/:. yl/i«. :/: = 5. 1 l'^ Given ^' „ , = ^~ -. to find the values of x. 2x* — 6 X* 4- a vln«. X = =t ya 4- ^• J'^ Givf;n x'' — :/, — '/'), U) find the valuea of x. 1 AriM. X = H and — 7. j 20. Given — ~ ^ — = 7, to find the valuen of x. X x^ Ana. X = 'J or — ^. ' 2J. Giv<;n lOx — x^ = 05, to find tlje values of x. ' yfw;?. X = 8 db 1/^^. ■ ..o ^.. - 1017x 2071^- /.,,./. J 22. (fi\'<:f) '// — , == — , , to ijn'i til'; vaJuCrt of x. 1 2i 2J ' ; Am. x = 00.72 or 10.27. 1 1 2'i. GIvf;n X — x^ = 20, to find tlio valu^iH of x. vl//,;,'. x = 25. i 21 Given (x 4 \()/ — ( j. 4 10/ = 2, to find x. yl««. x = 6. j Q V i; >> T [ X ;> r o u i: x a m i n a t i o :;. 281 25. Given (7.r= — .'• H- !>' — ("••''' — •'" + 1) = '^'^-^ ^'^ ^''^'^ '^^'^ values of .r. A 11^. .r = /j (1 ± 1 '^r^f), .r = 'J or — 'j-'- 20. ("livea .<" + 2.;- = 2:"!, io ilnd ,r. .l;rs-. .r = 3.808070r> and .r -= — r).S0S07!^r). 27. (Uvon //.s. .(• = 5, 1 — . -I- I 2.r// 28. Given .M-.V--:^y .'^' +.'/ ^ 1 .,„a 7.,.,, :=^ -S, lo 1111,1 .r a.ul y. x^ + y' '' An.. .r = 2,// = 2. 2!>. (Jlveu ■-- ..•''^-_:i-21:j^ == (/ ami .-y = A, (o lin.d .r ar.d //. X + ,y a; — y X — 11 X -}- ,y ,!;,.•. ,7= \ [^=ir^ (.'/;4-26=Fl/(7/> — 2A], .,. ^ ), [riz ^ ,//, _|. •_'/, =h 1 V//> — -/.]. ^JO. G ivoii .r -f- // -f 1 "".»; + // = aii.l .r' + if- = 1 0, to liiul ,r ami//. J.;n. .(• = - or 1, or U ± ^1 '—Ol, // = 1 or :l, or 1] qz jl — 1)1. ai. Given J ^ ^. — „' , . = l.J\, U) lind .r. 4/?.s-. .r = 8, or 1 t o2. Given 1. ~.r- f 1 ^ •-'" *" ^'"'^ •'' .u/s. .r = ±3, or .r = ii= \\ — 1 33. One root of .<^ — in.Vr- + 4(>^r = 2i) is 8: what arc the other roots? J/zs. .r = 5 and .r = ^. 84. One root of ./•' -f Iy\ y^ — \~^x = — 1 Is — (> : what are the other roots? yi;?.9. \ and \ O 1 282 Q U E iS T I N S F K E X A M I N A T I N. 35. Has the equation ;/.^ -}- 2^.'-^ — 15x = 36 eqnal roots ? If so, find all the roots. Ans. It has; and — 3^ — 3^ and — 4 are the roots. 36. What are the roots of the equation x^ — 20^2 -f 142./2 — 420x = - 441 ? Ans. 7, 7, 3, and 3. 37. Find one value of x in the equation x^ — 2x = 50. Ans. 3.8648854. 38. If a certain number is divided by 7, and if, then, the quo- tient is taken from the sum of the dividend and divisor, the re- mainder will be 73 : what is the number? Ans. 77. 39. To find three numbers in arithmetical progression, of which the first is to the third as 5 to 9, and the sum of all three is 63. Ans. 15, 21, 27. 40. A sets out from C toward D, and travels 8 miles a day. After he had gone 27 miles, B set out from D toward C, and goes every day 379^^^ ^^ ^^^® whole journey, and after he had traveled as many days as he goes miles in one day, he met A. Required the distance of tlie place C from D. Ans. 180 or 60 miles. 41. Two post-boys, A and B, set out at the same time from two cities, 500 miles apart, in order to meet each other. A rides 60 miles the first day, 55 the second, 50 the third, and so on, decreasing 5 miles every day. B goes 40 miles the first day, 45 the second, 50 the third, and so on, increasing 5 miles every day. In what number of days will they meet? Ans. In 5 days. 42. A tree, 100 feet high, stands just at the water-line on the bank of a river 200 feet wide. The tree broke in a gale of wiud, and the upper part was found to point exactly to the water-line of the opposite bank, the top being within 20 feet of the surface of the water. How high from the ground did the tree break ? Ans. 30.472 feet. TABLE OF SOUAIIK J'.OOTS. r: No. fc?q\. j !S4U;ii-e Itoot. L J14JI , VaMaWJUi^ESrU.!^ ' , * g. w t:m i'. .j a *.p; T ^g».-- <.JttJ-g T gL,*!iJgg. 121 II • 122 1 1 • 123 1 1 • 124 1 1 • 120 1 1 • 126 11 • 127 1 1 • 128 1 1 • 129 II • i3o 1 1 • i3i ii> l32 1 1 • i33 II • 1 34 1 1 • i35 i { • i36 II- 137 II • i38 II • ,39 11 • 140 1 1 • 141 1 1 • 142 11 • 143 II • 144 12- 145 12- 146 12- 147 12- 148 12- 149 12- !0O 12- i5i 12- l52 12- !53 12- 104 12- i55 12- 1 56 12- i57 12- i58 12- 159 12- i6o 12- 161 12- 162 12- 16J 12- 164 12- i65 12- 166 12- 167 12- 168 12- 169 i3- 170 i3- 171 i3- 172 i3- 173 i3. 174 i3- 175 i3- 176 i3- 177 i3- 178 i3- 179 i3- 180 i3- 0000000 0453610 ■0905360 1355387 1803399 2249722 2694277 3 70S 5 •3578167 4017543 ■445523i •4891253 532 5626 5708369 ■6189500 ■6619033 •7046999 ■7473444 •7898261 •8321096 •8743421 •9163753 ■9582607 •0000000 •0410946 •o83o46o 1243557 i65525i 2o65o56 2474487 •2882057 •3288280 ■3693169 1096736 4498996 4899960 ■5299641 56980O1 •6095202 ■6491 106 •6885775 ■7279221 •7671453 •8062485 ■8452326 •8840987 •9228480 ■9614814 ■0000000 •0384048 ■0766968 1148770 1529464 1909060 2287566 2664992 •3o4i347 •3416641 •3790S82 •4I6407Q JJMiJUlda Mg'IT IJJI'S- ' igg A TABLE OF LOCiAlllTIiMS FKOil 1 10 10,000. N. I 2 3 4 5 j 6 |- 7 1 8 9 i D. 100 000000 0434 0S68 1 3oi 1734 2166 2398 3029 3461 3891; 432 !0J 432 1 4731 5i8i 5609 6o38 6466' 6894' 732 1 j 7748 8174' 428 I02 8600 0026 3239 945 1 9876 e3oo •724 1147 1570 1993 24 1 5 424 io3 012837 368o 4100 4521 4940| 536o; 6779 6197 66} 6 419 104 7033 7431 7868 8284 8700 91 16 9532I 9947 •36j *773 416 io5 021 189 i6o3 2016 2428 2841 3232 i 3664 4075 44S6 4896 412 io6 53o6 5715 6125 6533 6o42 735o 7737 8164 8571 8978 408 107 ^84 9789 •195 •600 1004 1408 1812 2216 2619 3o2i 404 io3 o3J424 3826 4227 4628 5029 543o 583o 6?3o 6629 7028 400 i '09 7426 7823 8223 8620 9017 9414 9811 •207 ®6o2 •998 396 no 041393 1787 2182 2576 2969 6883 3362 3755 4148 4540 4q32 393 III 5323 5714 6io5 6495 7275 7664 8o53 8442 883o 389 I !2 9218 05J078 9606 99^)3 •38o *766 n53 1 538 1924 2309 2694 386 !l3 3463 3S46 423o 4615 4996 83o5 5378 5760 6142 6524 382 M4 6905 7 2 86 7666 8046 8426 9185 0563 9942 •320 379 ii5 060698 1075 1452 1829 2206 2582 2953 3333 3709 4o83 370 116 4458 4832 0206 558o 5953 6326 6699 7071 7443 7815 372 U7 81S6 8557 8928 9298 9668 ••33 •407 •776 1145 i5i4 369 i ii3 071882 3230 2&17 2985 3352 3718 4o85 445 1 4816 5i82 3t)6 I!9 &547 5912 9543 6276 6640 7004 7368 773i 8094 8457 8819 363 120 0791 8 1 9904 •266 *626 -■c;S7 4376 1 347 1707 2067 2426 36o 121 082785 3i44 33o3 386 1 4219 4934 5291 5647 6004 357 122 636o 6716 7071 7426 77H1 8i36 8490' 8845 9198 9552 355 12,3 9905 •258 •611 ®963 i3i5 1667 20{3: 2370 2721 3^371 35 1 124 093422 3772 4122 447 » 4820 5169 55 rS! 5866 62 1 5 6562 34a 125 6910 7237 7604 795 1 8298 8644 8990 ; 9335 9681 ••26 346 126 1 0037 1 0715 1039 i4o3 1747 2091 2434 2777 3119 3462 343 127 3So4 4146 4487 4828 5169 55 10 535 1 6191 653 1 6871 340 12S 7210 7549 7888 82 27 8565 8903 9241 9579 9916 •253 338 129 1 10590 0926 12&3 1599 1934 2270 26o5 2940 3273 3609 335 i3o 1 1 3943 4277 461 1 49 i4 5278 56ii 5943 6276 6608 6040 333 i3i 7271 7603 7934 8265 8595 8926 9256 95^6 99 1 5 •245 33o l32 120374 0903 123l i56o 18S8 2216 2544 2871 3198 3525 323 b33 3852 4178 45o4 4S3o 5i56 5481 58o6 61 3 1 6456 6781 325 1 34 7ia5 7429 7753 8076 8399 8722 9045 9368 9690 ••12 323 j35 i3o334 0635 0977 1 29'^ 1619 1939 2260' 23S0 2900 32IQ 321 i35 3539 3858 4177 4496 4814 5 1 33 545 1 : 5769 6086 64o3 3i8 137 672 1 7037 7354 7671 7987 83o3 86 1 S; 8934 9249 9^64 3i5 1 38 f79 143015 •iQi ®5o8 «822 ii36 1 45o 1763; 2076 2389 2702 3i4 i39 3327 3639 3931 4263 4574 4885' 3196 55o7 58 18 3ii 140 146128 6438 6748 7o58 7367 7676 7983; 8294 86o3 8911 309 141 921a 1 52288 9527 9835 ei42 «449 •756 io63 1370 1676 10'^2 307 142 2594 2900 32o5 35io 38 J 5 4120 4424 47 28 5o32 3o5 143 5336 5640 5943 6246 6549 6852 7 J 54' 7457 7759 8061 3o3 144 8362 8664 8965 9266 9367 986S 2863 <»i63 •469 •769 1 06S 3oi 145 i6i368 1667 1967 2266 2564 3 1 61 3460 3758 4o55 299 146 4353 4630 4947 5244 5541 5838 6i34 643 6726 7022 297 J 47 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 14S 170262 o533 0848 1141 1434 1726 2019 23ll 26o3 2895 293 149 3i86 3478 3769 4060 435i 46.',! 4932 5222 55 1 2 58o2 291 i5o i-Ooqi 63 M, 6670 6939 9S39 7248 7336 7825 8ii3 8401 868^ 289 i5i 8977 9264 9552 •126 •4i3 •699 •985 1272 1 55b 287 132 181 S44 2129 24 1 5 2700 2985 3270 3555 3839 4 1 23 4407 283 !53 4691 4913 7803 5259 5342 5825 6108 6391 6674 6956 7239 283 1 54 732 1 8084 8366 8647 8928 9209' 9490 2010^ 2289 9''7' ••31 281 i55 190332 0612 0892 368 1 1171 i45i 1730 2367 2846 279 1 56 3123 34o3 3959 4237 45 1 4 4792, 5069 5346 5623, 278 07 5899 6176 6453 6729 7005 7281 7536, 7832 8107 8382 276 i58 8657 8932 9206 9481 9755 ••29^ ^303 •577; •830 1124' 274 139 201397 1670 1943 2 2216 3 ' 2488 4 2761: 3o33 33o5 3377 3848} 272 1 N. I 5 1 6 ! 7 ' 8 1 1 D. A TABL'Z OF LOG A "iirii: !S h'l CM ] TO 10,000. 3 N. 1 X 2 3 -1 4 5 6 7 8 1 9 D. i6o 2041 20 i 4391 4663 4934 5204 5475 5746 6016 6286 65561 27> i6i 6826' 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 162 95i5 9783 ••5i •3 1 9 "586 •853 1121 1 388 1 654 1921 267 1 63 212188 2454 2720 2986 3252 35i8 3783 4049 43 1 4 45791 266 164 4844 5109 5373 5638, 5902 6i66 643o 6694 6957 7221 264 i65 7484 7747 8010 8273; 8536 8798 9060 9323 9585 9846 1 262 ! 166 220108 0370 o63j 0892! 1 1 53 1414 1675 1936 2196 24561 261 167 2716 2976 3236 3496: 3755 4oi5 4274 4333 4792 5o5i 25o 168 5309' 5568 5826 60S4 6342 6600 6858 7u5 7372 7630 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 •193 236 170 230449 0704 0960 12l5 1470 1724 '979 2234 2488 2742 254 I 171 2996 5328 3230 3304 3757 4011 4264 4517 4770 5o23 5276 253 172 5781 6o33 6285; 6537 6789 7041 7292 7544 7795 252 173 8046 8297 8348 8799' 9049 9299 9550 9800 ••30 •3oo| 25o 174 240549 0799 1048 1297 1346 1793 2044 2293 2541 2790, 240 173 3o38 3 2 86 3534 3782 4o3o 4277 4525 4772 5019 5266, 240 176 55i3 5739 6006 6252 6499 6745 6991 7237 74S2 7728; 246 177 7973 S219 S464 8709 8954 9198 9443 9687 9932 •176! 245 178 230420 0664 0908 1 i5i 1395 i638 1 88 1 2125 2363 2610 243 179 2853 3096 3338 358o 3822 4064 43o6 4548 4790 5o3il 242 180 255273 5314 5755 5996 6237 6477 6718 6958 7193 7439 241 181 7679 7918 8i58 8398 8637 8877 9116 9355 9394 9S33 239 ■ 182 260071 o3ic 0348 0787 1025 1263 i5oi 1739 1976 2214 238 , i83 2431 2688 2925 3162 3399 3636 3S73 4109 4346 4582 237 184 4818 5o54 5290 5525 57&I 5996 6232 6467 6702 6937 235 1 85 7172 7406 7641 7875 8IIO 8344 8578 8812 9046 9279! 234 186 9513 9746 9980 •2l3 •446 •679 •qI2 1 144 i377 1609 233 187 271842 2074 23o6 2538 2770 3ooi 3233 3464 3696 39271 232 188 4i58 43 89 4620 4S5o 5o8i 53ii 5542 5772 6002 6232 23o 189 6462 6692 6921 7131 7380 7609 9896 7838 8067 8296 8525; 229 190 278754 8982 9211 943o 9667 •I23 •35i •578 •806 22S 191 281033 1261 1488 1715 1942 2169 23^6 2622 2849 3075 227 192 33oi 3527 3753 3979 42o5 443 1 4636 4882 5 1 07 5332 226 193 5557 5782 6007 6232 6456 6681 6905 7i3o 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9143 9366 9389 98121 223 195 290035 0257 0480 0702 0925 1147 1369 1591 i8i3 2o34 222 196 2236 2478 2699 2920 3i4i 3363 3584 38o4 4023 4246 221 197 4466 4687 4907 5 1 27 5347 5567 5787 6007 6226 6446 220 198 6665 6334 7104 7323 7542 7761 7979 8198 84:6 8635 210 199 8853 9071 9289 9307 9725 9943 ®i6i •378 •595 •8i3 218 200 3oio3o 1247 1464 1681 1898 2114 233i 2547 2^64 2980 217 201 3196 3412 3628 3844 4o5g 4275 4491 4706 4921 5i36 216 202 535i 5566 5781 5996 62! I 6425 6639 6854 7068 7282 2l5 2o3 7496 7110 7924 8i37 8331 8564 8778 8991 9204 9417 2l3 204 963o 9843 •«56 •268, •481 •693 •906 1118 i33o 1 542 212 2o5 311754 1966 2177 2389 2600 2812 3o23 323i 3445 3656 211 206 3867 4078 4289 4499 4710 4920 5i3o 5340 555i 5760 210 207 0970 6180 6390 6599 6809 8689; 889S 7018 7227 7436 7646 7854 200 208 8o63 8272 8481 9106 9314 9522 9730 9938 j 208 209 320146 o354 o562 0769' 0977 1184 1391 1598 i8o5 20I2j 207 210 322219 2426 2633 2S39' 3046 3252 3458 3665 3871 4077: 206 211 4282 4488 4694 48q9 5io5 53io 53i6 5721 5926 6i3i| 2o5 212 6336 6541 6745 6950 7i55 7359 7563 7767 7972 8176 204 ai3 833o 8583 8787 8991 9194 9398 9601 9805 •••8 •211 2o3 214 330414 0617 0S19 1022 1225 1427 i63o i832 2o34 2236 202 2l5 2438 2640 2842 3o44 3246 3447 3649 3B5o 4o5i 4253 202 2t6 4454 4655 4856 5o57 5257 5458 5658 5859 6039 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 9849 8o5S 8257 200 218 8456 8656 8855 9054 9253 945 1 965o ••47 •246 199 219 : 340444 0642 0841 1039 1237 1435 i632 i83o 2028 2235 198 i "N. i 1 2 3 4 5 6 7 8 9 D. i A T ABLE OF LOG A III 711 ".■ iri TR OM 1 TO iO,0( "iO N. i 1 i 2 i ' 1 4 1 5 1 6 1 7 1 8 9 D." 220 342423; 2620 2M,7 1 3oi4 3212 3409- 36o6^ 33o2 3999 4196 197 221 4392 1 4589 ■ 47S5 4981 51781 53-74 5570! 5766 5962, 6107 196 222 6353 6349 6744 ' 6939 7i35; 7330, 7525; 7720 7915 8110 195 223 S3o5 85oo 8694 8889 9083: 9278 9472' 9666 9860! •»34 194 224 350248 0442 ' o636 0829 1023 1216: 1410, i6o3 1796, 1989 193 225 2i83 2375 2568 2761 ' 2954' 3147: 3339! 3532 3724I 3916 193 226 4108 43oi 4493 4685 4S76! 5o68i ^260: 5452 5643, 5834 19? 227 6026 6217 6408 65g9 ' 6790' 6q8i| 7172' 7363 7554! 7744 igr 228 7935 8125 83 16 85o6 8696 8886 9076 9266 9456 9646 100 229 9835 ©025 ®2l5 ®4o4 •593 •783 •972 I i6i i35o 1539 1^9 23o 361728 1917 2io5 2294 2482 2671 2859 3048 3236 3424 188 23l 36i2 3800 3988 4176 4363 455i 4739 4926 5ij3 53oi 188 232 5488 5675 5862 6049 6236 6423 6610 1 6796 1 8659 6983 7169 187 233 7356 7542 7729 7915 8101 8287 8473 8845 go3o 186 234 9216 940 r 9587 9772 9958 •143 •328 •5i3 •698 •883 i85 235 371068 1253 1437 1622 1806 199! 2175 236o 2544 2728 184 236 2912 3096 3280 3464 3647 383 1 4oi5 4198 4382 4565 184 237 4748 4932 5i i5 529S 5481 5664 5846 6029 6212 6394 I S3 238 6577 6759 858o 6942 8943 7306 7488 7670 7852 8o34 8216 182 23g 8398 8761 9124 93o6 9487 9668 9849 ••3o 181 240 3302 1 1 0392 0573 0754 0934 ui5 1296 1476 i656 1837 181 241 2017 2197 2377! 2557 2737 2917 3097 3277 3456 3636 180 242 38i5 3995 4174' 4353 4533 4712 4891 5070 5249 5428 179 243 56o6 5785 5964} 6142 6321 6499 6677 6856 7034 7212 .'■78 244 7390 7568 77461 7923 8101 8279 8456 8634 8811 8989 i78 245 9166 9343 9520" 9698 9875 ®»5i •228 •40 5 •582 •759 177 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521 176 247 2697 2873 3o48 3224 3400 3575 3731 3926 4101 4277 176 248 4452 4627 4802 4977 5i52 5326 55oi 5676 585o 60 2 5 173 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 25o 397940 8114 8287 8461 8634 8808 8981 9!54 9328 gSoi 173 201 9674 9847 ©©20 «I92 •365 •538 •711 •883 io56 122S 173 232 401401 1573 1745 I9I7 2089 2261 2433 26o5 2777 2949 172 233 3l2I 3292 3464 3o35 3807 3978 4149 4320 4I92 4663 111 254 4834 5oo5 5176 5346 55i7 5688 5858 6029 6199 6370 171 255 6540 6710 6881 7o5i 7221 739, 7561 7731 7901 8070 170 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 257 9933 •102 •271 •440 •609 •777 •946 1114 1283 i45i 169 258 41 1620 1788 1956 2124 2293 2461 2629 2796 2964 3i32 168 259 33oo 3467 3635 38o3 3970 4137 43o5 4472 4639 4806 167 260 414973 5i4o 5307 5474 5641 58o8 5974 6141 63o8 6474 167 261 6641 6807 6973 7139 7306 7472 7633 7804 7970 8i35 166 262 83oi 8467 8633 8798 8964 9129 9295 9460 9625 9791 1 65 263 9956 •121 6286 •45 1 •616 •781 •945 mo 1275 1439 1 65 264 421604 1788 1933 2097 2261 2426 2^90 2754 2918 3o82 164 265 3246 3410 3574 3737 3901 4o65 4228 4392 4355 4718 164 266 4882 5o45 5208 5371 5534 5697 5860 6023 6186 6349 1 63 267 65ii 6674 6836 6999 7161 7324 7486 7648 7811 7973 162 268 8i35 8297 845q 8621 8783 8944' 9 '06 9268 9429 9591 162 269 9752 9gi4 ••73 8236 •398 •559 ^720 •881 1042 I203 161 270 43 1 364 i525 1 685 1846 2007 2167 2328 2488 2649 2809 161 271 2969 3i3o 3290 345o 36io 3770 3(^30 4090 4249 4409 160 272 4569 4729 4888 5o48 5207! 5367 5d26 5685 5844 6004 1 59 273 6i63 6322 6481 6640 6798, 695/ 71 16 72-'5 7433 7592 159 274 775i 7909 8067 82 26 8384! 8542 8701 8859 9017 9175 1 58 275 9333 9491 9648 9806 9964 •122 •279 •437 •594 •752 1 58 276 440909 1066 1224 i38i i538 1695 i852 2009 2166' 2323 1 57 277 2480 2637 2793 2950 43131 3 1 06 3263 3419 3576 3732: 3SS9 1 57 278 4045 4201 4357 4669' 4825 4981 5i37 5293 5449 1 56 279 56o4 5760 5915 6071 62261 63821 6537 6(592 6848 7003 i55 N. I 1 2 1 3 4 1 5 6 7 1 3 9_ T>, j A TABLE OF I.OC; A HI T iIM3 FflOM 1 TO 10.000. "n:" I 2 1 3 i ' 1 ' 1 6 i 7 8 j 9 1 D. '280" 447158 73 1 3 7463 7623 : 77 73 7933 So38; 8242! 83,77 i 8552 1 55 23 1 8700 8861 9015 9170 : 9324 9^78 9633; 9787! 99 i I •»95 1 54 282 450249 04:3 0)57 07 i 1 ! o355 ., 1018 1172; 1 326 1479 1 633 1 34 2 S3 n86 I9=i0 2093 2247 i 2400 2)')3 2706; 2339 3oi2 { 3i65 1 53 2S4 33 18 347 1 362 \ 3777 1 3930 4082 4235; 4337 4540 ; 4692 1 53 285 4845 4997 5i5o 53o2 '■ 5454 56o6 5753: 5910 6062 6214 132 2S6 6366 65 1 8 6670 ' 6321 i 6973 7125 72761 7428 7379 7731 l52 i 2^7 1 , ' 7S82 8o33 81S4 8336 8487 8633 8789' 8940 0091 9242 i5i 2VS 9392 9543 9604 9845 ; 9995 •i46 •296 •447 •597 •748 i5i 2 '^9 460S98 1048 1198 1 3 ',8 1499 i6j9 f799' 1918 2098 2248 i5o 29) 462398 2548 2697 2347 2997 3 146 3296, 34 P 3594 3744 i5o 291 3393 4042 4191 43 io 4490 4639 4788, 4936 5o85 5234 149 1 292 5383 5532 563o 5329 5977 6126 6274 6423 6571 6719 140 293 6868 7016 7! 64 73i2 7460 7608 7756; 7904 8o52 8200 148 294 8347 8493 8643 8790 8933 ■ 9035 9233 9.330 9527 9675 148 293 9822 9969 •116 •263 •4!0 •537 •704; ®35i •998 1145 147 296 471292 1433 1 535 1732 1878 2025 2171' 23 1 8 2464 2610 146 297 2756 2903 3 049 319') 3341 3487 3633; 3779 3925 4071 146 298 4216 4362 45o3 4653 4799 4944 5090 5235 533i 5526 146 299 5671 58i6 5962 6107 6252 6397 6542 6687 683 2 i 6976 145 3oo 477121 7266 74 1 1 7535 7700 7844 79^^.9; 8r33 8278, 8422 145 3oi 8566 8711 8355 8909 9143 9287 9 '.3 1 9575 9719 9863 144 302 480007 Ol5£ 0294 043 3 o5S2 0723 0869 10 1 2 1 1 56 1299 144 3o3 1443 1 586 '729 1872 2016 2! 59 23o2 2443 2583 273. 143 3 04 2S74 3oi6 3i59 33o2 3445 3537 3730; 3372 40 i 5 4r57 143 3o5 43oo 4442 4585 4727 4869 Soil 5i53| 5295 5437 5579 142 3o6 5721 5863 6oo5 6t47 628q 643o 6572! 6714 6855 6997 142 3o7 7t38 7280 7421 7563 7704 7845 7986; 8127 8269 8410 141 3o3 855 1 8692 8333 S974 9114 9255 9396' 9537 9677 9818 141 3 09 9958 ••99 •239 •33o •520 •661 •3oi •941 1081 1222 140 3io 491362 l502 1642 1782 1922 2062 2201 2341 2481 2621 140 3ii 2760 2900 3o4o 3179 33 19 3453 33o7 3737 3376 4oi3 139 3l2 4i55 4294 4433 4072 4711 435o 49S9 5 1 23 5267 5406 139 3i3 5544 5683 5822 6960 6099 6238 6376, 65i5 6653 6791 i39 3 14 6q3o 7058 7206 7344 74S3 7621 7759 7897 8o35 8173 1 38 3i5 83ii 8448 8586 8724 8862 8999 9137 0275 9412 9550 i38 3i6 . 9687 9324 9962 **99 •236 *374 •5u 0548 •783 •922 137 3i7 Doio59 1196 i333 1470 1607 1744 1880 2017 21 54 2291 137 3i8 2427 2 564 2700 2337 2973 3109 3246 3382 35i8 3655 i36 319 3791 3927 4o63 4199 4335 4471 4607 4743 4873 5oi4 1 36 320 5o5i5o 5286 5421 5557 5693 5828 5964 6099 6234 6870 1 36 321 65o5 6640 6776 6911 7046 7.S, 73i6: 745i 7536 7721 i35 322 7856 7991 8126 8260 8395 S33o 8664| 8799 8934 9068 i35 323 9203 9337 9i7i 9606 9740 9'^74 •••9 •143 •277- •411 i34 324 516545 0679 08 1 3 09 17 1081 I2l5 1349- 1482 1616 i75o !34 323 1 833 20[7 2l5l 2284 2418 2 35 r 2684 2818 2951 3084 i33 326 3218 335i 3484 3617 3750 3383 4016 4149 4282 4414 i33 327 4548 463 1 48 1 3 4916 5o79 521 I 5344 5476 5609 5741 ]33 323 5874 6006 6139 6271 6403 6333 6668 6800 6932 7064 1 32 3^9 7196 7328 7460 7592 7724 7355 7987 8119 825i 8382! l32 33o 5!85i4 8646 8777 8909 9040 9171 93 o3 9434 9566 96971 i3i 33 1 9828 9959 ••90 ®22I •353 •48' •6i5 •743; •876 1007; i3i 332 D2II3S 1269 1 400 i53o 1661 1792 192-2 2d53; 2i83 23i4l i3i 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 36i6; i3o 334 3746 3876 4006 41 36 4266 4396 4526 4636 4785 4915; i3o 335 5o45 5i74 53o4 5434 5563 5693 5822 5951 60S I 6210' 129 336 6339 6469 6593 6727 6856 6935 7114 7243] 7372 75oi| 129 337 7630 7759 7888 8016 8145 8274 8402 853 1 ! 8660 8788 120 338 8917 9045 9'74 9302 943o 9559 9687 93,5| 9943 ••72 12B 339 N. 530200 o328 0456 o584 0712 0840 0968 1096I 1223 1 i35i: 128 I 3 4 5 6 7 1 8 1 9 ! 8 A T \j3LE OF LOG a; :r;ii:.3 5 Fli JM 1 TO 10,000. "nT" 1 I j 2 3 4 5 ( 6 7 8 9 ■— 1 340 531479 1607* 1/34 i862 1990 2117, 2245 2372 25oo 2627 128 341 2754 28821 3009 3x36 3264 3391I 35i8 3645 3772 3899 127 342 4026 41 53 4280 4407 4534 4661! 4787 4914 5o4i 5167 127 343 0294 5421 5547 5674 58oo 5927 6o53 6180 63o6 6432 126 344 6558 6685 6bii 6937 7063 7189 73i5 8574 7441 7567 7693 126 345 7819 7945 8071 8197 8322 8448 8699 8823 8951 126 346 9070 9202 9327 9452 9578 9703 9829 9954 •*79 ®204 123 347 540029 0455 o58o 0705 o83o 0955 1080 12o5 i33o 1454 123 34« 1 579 1704 1829 1953 2078 220} 2327 2452 2576 2701 123 349 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 35o 544068 4192 43i6 4440 4564 4688 4812 4936 5o6o 5i83 124 35i 5307 543 1 5555 5678 58o2 5925 6049 6172 6296 6419 124 3D2 6543 6t)66 6789 6913 7o36 7159' 7282 74o5 7529 7652 123 353 7775 789S 8021 8144 8267 8389 85i2 8635 8758 8881 123 354 9003 9126 9249 9371 9494 9616 9739 9S61 9984 •106 123 355 550228 o35i 0473 0095 0717 0840 0962 1084 1206 1 328 122 356 1430 1572 1694 1816 1938 2060 2181 23o3 2425 2547 122 357 266S 2790 2911 3o33 3i55 3276 3398 35i9 3640 3762 121 358 3883 4004 4126 4247 4368 4489 4610 4731 4832 4973 121 359 5094 52 1 5 5336 5457 5578 5699 5820 5940 6061 6182 121 36o 5563o3 6423 6544 6664 6785 600 5 7026 7146 7267 7387 120 36i 7007 7627 7748 7868 7988 8ioS 8228 8349 8469 8589 120 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 363 9907 «e26 «i46 «265 •385 ®5o4 •624 •743 0863 '982 119 364 56i 101 I22I 1 340 1459 1578 1698 1817 1936 2o55 2174 119 365 2293 2412 253i 2600 2769 2887 3 006 3125 3244 3362 119 366 3481 36oo 3718 3837 3953 4074 4192 43ii 4429 4548 119 367 4666 47f^4 4903 0021 5 139 0257 5376 5494 56 1 2 5730 118 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 iiS 369 7026 7144 7262 7379 7497 7614 7732 7S49 7967 8084 118 370 568202 83 1 9 8436 8o54 8671 8788 8905 9023 9140 9257 117 371 9374 9491 9608 9725 9842 9959 ««76 •193 e309 •426 117 ^72 570543 0660 0776 0093 1010 1 1 26 1243 1339 1476 1592 117 373 1709 1825 1942 2o58 217^ 2291 2407 2523 2639 2755 116 374 2872 29S8 3 1 04 3220 3336 3452 3568 3684 3Soo 3915 116 375 4o3i 4147 4263 4379 4494 4610 472-6 4841 49^7 5072 I!6 376 5i88 53o3 5419 5534 565o 5765 588o 0996 61 1 1 6226 ii5 377 6341 6457 6J72 6687 6002 6917 7032 7147 7262 7377 ii5 37b 7492 7607 7722 7830 7931 8066 8181 8295 8410 8525 ii5 379 8639 8754 8868 8983 Q007 9212 9326 9441 9555 9669 114 3bo 579784 9898 e«i2 ®I26 ^241 •355 •469 •583 •697 •811 114 38i 580925 1049 11 53 1267 i38i 1495 1608 1722 1 836 igSo 114 382 2o63 2177 2291 2404 25i8 263 1 2745 2858 2972 3o85 114 383 3199 33 12 3426 3539 3652 3765 3879 3972 4io5 4218 ii3 384 433 1 4444 4557 4670 4783 4896 5009 3122 5235 5348 ii3 385 5461 5574 5686 5799 5912 6024 6i37 6230 6362 6475 ii3 386 6587 6700 6812 6925 7037 7149' 7262 7374 74S6 7599 112 387 7711 7823 7935 8047 8160 8272: 8334 8496 8608 8720 112 388 8832 8944 9o56 9167 9279 9391 95o3 9615 9726 9838 112 389 9950 ««6i •173 •284 •396 »5o7 •619 •730 •842 •953 112 390 591065 1 176 1287 1399 i5io 1621 1732 1843 1955 2066 1 1 1 391 2177 2288 2399 25lO 2621 2732 2843 2954 3o64 3175 III 392 3286 3397! 35o8 36 18 3729 3840 3930 4061 4171 4282 III 3c;3 4393, 45o3| 4614 472-1 4834 4945 5o55 5i65 5-276 5386 no 394 5496 56o6 5717 5827 5937 6047 6.57 6267 6377 6487 no 395 6597' 6707 6817 6927 7037 7146 7256 7366 7476 7586 no 396 7695 7805 7914 8024 8i34 8243 8353 8462 8372 8681 no 397 8791: 8900 9009 9119 9228 9337 9446 9556 9663 9774 109 398 9S83. 9992 •lOI •210 •319 •428: •537 •646 •755 •864 109 399 600973 1082 1191 1299 1408 i5i7i 1625 1734 1843 1951 109 N- 1 I ,2 3 4 5 i 6 7 8 9 T). A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 1 I 1 2 2277 3 4 j 5 6 7 8 9 1 I>- 400 1 602060 2169' 2386! 2494' 2603' 2711! 2819 2928 3o36 108 401 3i44 3253 336i 3469' 3577! 36861 4658' 4766I 3794 3902 4010' 4118 108 402 4226I 4334' 4442 455o 4874 4982 5089 0197 108 4o3 53o5 54i3 5521 5628 5736! 58441 5951 6059 6166 6274 108 404 638i 6489' 6596 6704 681 1 6919I 7026 7133 7241 7348 107 40 5 7455 7562 7669 8740 7777 7884 7991 8098 8205 83i2 8419 107 406 8526 8633 8847 8954 9061 9167 9274 9381 9488 107 407 9594 9701 9808 9914 ••21 •128 •234 •341 •447 •554 107 408 610660 0767 0873 0979 1086 II92 1298 i4o5 i5ii 1617 106 409 1723 1829 1936 2042 2148 2234 236o 2466 2572 2678 106 410 612784 2890 2996 3l02 3207 33i3 3419 3525 363o 3736 106 411 3842 3947 40D3 41 59 4264 4370 4473 458i 4686 4792 106 412 4897 5oo3 5io8 52i3 5319 5424 5529 5634 5740 5845 103 4i3 SgDo 6o55 6160 6265 6370 6476 658i 6686 6790 6895 io5 4i4 7000 7io5 7210 8257 73i5 7420 7525 7629 7734 7839 7943 io5 4i5 8048 8£53 8362 8466 8571 8676 8780 8884 8989 io5 416 9093 9198 9302 9406 9311 9615 9719 9824 9928 ••32 104 4«7 620136 0240 o344 0448 o552 o656 0760 0864 0968 1072 104 418 1176 1280 1 3 84 1488 1592 i6q5 1799 1903 2007 2110 104 419 ' 2214 23i8 2421 2525 2628 2732 2833 2939 3o42 3i46 104 420 623249 3353 3456 3559 3663 3766 3869 3973 4076 4179 io3 421 4282 4385 4488 4591 4695 4798 4901 5oo4 5io7 52IO io3 422 53i2 54i5 55i8 5621 5724 5827 5929 6o32 6i35 6238 io3 423 6340 6443 6546 6648 6751 6853 6956 7o58 7161 8i85 7263 io3 424 7366 7468 7571 7673 7775 7878 7980 8082 8287 102 425 8389 8491 8093 8695 8797 8900 9002 9104 9206 9308 102 426 9410 9512 9613 971 5 9817 9919 ••21 •123 •224 •326 102 427 630428 o53o 063 1 0733 o835 0936 io38 1139 I24f 1 342 102 428 1444 1 545 1647 1748 1849 1951 2052 2i53 2255 2356 lOI 429 2457 2559 2660 2761 2862 2963 3064 3i65 3266 3367 lOI 43o 633468 356q 3670 3771 3872 3973 4074 4175 4276 4376 100 43 1 4477 4578 4679 4779 4880 4981 5o8i 5i82 5283 5383 100 432 5484 5584 5683 5785 5886 5986 6087 6187 6287 6388 loo 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 Ih"" loo 434 7490 7390 8589 7690 7790 7890 8888 7990 8090 8190 8200 8389 99 435 8489 86«9 8789 8988 9088 9188 9287 9387 99 436 94S6 9586 9686 9783 9885 99S4 ••84 •i83 •283 •382 99 437 640481 o58i 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2939 3o58 3i56 3255 3354 9? 440 643453 355i 365o 3749 3847 3946 4044 4143 4242 4340 •^o 44 1 4439 4537 4636 4734 4832 4931 5029 5i27 5226 5324 ^l 442 5422 5521 56i9 5717 58i5 5913 601 1 6110 6208 63o6 9^ 443 6404 65o2 6600 6698 6796 6894 6992 7089 7187 7285 9^ 444 7383 7481 8458 7379 7676 7774 7872 7969 8067 8i65 8262 98 445 836o 8555 8653 8750 8848 8945 9043 9140 9237 97 446 9335 9432 9530 9627 9724 9821 9919 ••16 •u3 •210 97 447 65o3o8 o4o5 o5o2 0399 0696 0793 0890 0987 1084 1181 97 448 1278 1375 1472 1569 1666 1762 1839 1936 2o53 2130 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3019' 3ii6 97 430 6532i3 3309 34o5 35o2 3398 3693 3791 3888 3984 4080 t 431 4177 4273 4369 4465 4562 4658 4754 485o 4946 5o42 ^i 432 5i38 5235 533 1 5427 5523 5619 5715 58io 5906 6002 9^ 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 454 7o56 7i52 7247 7343 7438 7534 7629 7723 7820 79tC)j 90 ^774! 8870 95 455 80!I 8107 8202 8298 9230 1 8393 8488 8584 8679 456 8965 9060 9i55 9346 9441 9336 9631 97261 9821 95 •676; •771! 95 437 9916 ••11 •106 •201 ! •296 •391 *4S6 •58 1 458 66o865 0960 1033 ii5o 1243 i339 1434 i529 i623! 1718 95 459 1 j8i3 1907 2002 2096 1 2191 2286 1 2380 1 2475 ; 2569J 2663 95 I 2 1 3 1 4 1 5 ! 6 i 7 1 8 i 9 i D. 23" y 8 A TABLE OF LOGARITllN IS FROM 1 TO 10,000. 460 I 2 3 4 5 6 1 7 8 1 9 D. 94 662708 3701 2852 2947 3o4i 3i35 323o 3324i 3418 3512' 3607 461 3795 3889 3983 4078 4172 4266: 436o 4454 4D48 94 462 4642 4736 4830 4924 5oi8 5i 12 5206 5299 5393 5487 94 463 558i 5675 5769 5862 5956 6o5o 6143 6237 633 1 6424 94 464 65i8 6612 6705 6799 6892 6986 7079 7173 7266 7360 9j 465 7453 7546 7640 7733 7826 7920 8oi3 8106 8199 8293 93 466 8386 8479 8572 8665 8759 8852 8945 903 8 9i3i 9224 93 467 9317 9I10 93o3 9596 96S9 9782 9875 9067 ••60 •i53 93 468 370246 o339 6431 o524 0617 0710 0802 0895 0988 1080 9? 469 1 173 1265 i358 i45i 1543 1636 1728 1821 1913 2005 93 470 672098 2190 2283 2375 2467 256o 2652 2744 2836 2929 92 471 302I 3ii3 32o5 3297 3390 3482 3574 3666 3758 385o 90 472 3942 4o34 4126 4218 43 10 4402 4494 4586 4677 4769 92 473 4861 4953 5045 5i37 0228 5320 5412 55o3 5595 5687 92 474 5778 5870 5962 t)o53 6145 6236 6328 6419 65ii 6602 92 475 6694 6785 6876 6968 7059 7i5i 7242 7333 7424 7516 91 476 7607 7698 8609 -77 S9 8700 7881 7972 8S82 8o63 8i54 8245 8336 8427 91 477 S5i8 8791 8973 9064 9t55 9246 9337 91 47» 9428 9519 9610 9700 979' 9882 9973 ••63 •i54 •245 91 479 68o336 0426 o5i7 0607 0698 078Q 0879 0970 1060 I i5i 91 480 681241 i332 1422 i5i3 i6o3 1693 1784 1874 1064 2o55 90 481 2145 2235 2326 2416 25o6 2596 2686 2777 2867 2957 90 482 3047 3i37 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3947 4o37 4127 4217 4307 4395 4486 4576 4666 4756 90 484 4845 4935 5o35 5ii4 5204 5294 5383 5473 5563 5652 00 485 5742 583 1 5921 6010 6100 6189 6279 6368 6458 6547 486 6636 6726 68 1 5 6904 6994 7083 7172 7261 735i 7440 ?9 487 7529 7618 7707 7796 7886 7975 8064 8i53 8242 833 1 \^ 488 8420 85o9 8598 8687 8776 8865 8953 9042 91 3 1 9220 «9 489 9309 9398 9486 9575 9664 9753 9841 9930 ••19 •107 «9 490 690196 0285 0373 0462 o55o 0639 0728 0816 090D '^93 89 491 1081 1170 1258 1 347 1435 i524 1612 1700 1789 1877 88 492 1965 2o53 2142 223o 23i8 2406 2494 2583 2671 27D9 88 493 -2847 2935 3o23 3iii 3199 3287 3375 3463 355i 3639 88 494 3727 38i5 3903 3991 4078 4166 4254 4342 443o 4517 88 495 4605 4693 4781 486S 4956 5o44 5i3i 5219 5307 5394 88 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 t^ 497 6356 6444 653 1 6618 6706 679J 6880 6968 7o55 7142 87 498 7229 7317 74o4 7491 7578 7665 7752 7839 7926 8014 V 499 8101 8188 S275 8062 8449 8535 8622 8709 8796 8883 87 5oo 698970 9057 9144 9231 93 1 7 9404 9491 9578 9664 975i •617 87 Do I 9838 9924 ©«,i «»«98 •184 *27I e358 •444 •53 1 87 5o2 700704 0790 0877 0963 io5o ii36 1222 i3o9 1395 1482 86 5o3 1 568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 5o4 243 1 2317 26o3 2689 2775 2861 2947 3o33 3ii9 32o5 86 5o5 3291 3377 3463 3549 3635 3721 3 Ho 7 38^3 3979 4o65 86 5o6 4i5i 4s36 4322 4408 449'^ 4579 4665 4731 4837 4922 86 507 5oo8 5094 5.79 5265 5350 5436 5522 5607 5693 5778 86 5o8 5864 5949 6o35 6120 6206 6291 6376 6462 6547 6632 85 509 6718 680 3 6888 6974 7059 7144 7229 73 1 5 7400 7485 85 5io 707570 7655 7740 7826 7911 7996 8081 8166 825i 8336 85 5ii 8421 85o6 859 1 8676 8761 8846 8931 9015 9100 oi85 85 5l2 9270 9355 9440 9524 9609 9694 9779 9863 9948 ••33 85 5i3 710117 0202 0287 0371 0436 0540 0625 0710 0794 0879 1723 85 5i4 0963 1048 Il32 1217 i3oi 1 385 1470 1 554 1639 84 1 5i5 1807 1892 1976 2060 2144 2229 23 1 3 2397 2481 2566 84 1 5i6 265o 2734 2818 2902 2986 3070 3 1 54 3238 3323 3407 84 1 5,7 3491 3575 3659 3742 3826 3910 399'! 4078 4162 4246 84 i 5i8 433o 4414 4I97 458i 4665 4749 4833 4916 5ooo 5o84 84 i ^19 1 N, 5167 525i 5335 5418 55o2 5586 5669 5753; _7_l 5836 5920 84 "1j I 2 3 _J 5 6 8 ! 9 A TABLE OF LOGARirii: ,IS FI iOil L 10 10,000. g N. I 2 1 3 j 4 1 5 "~6"'j 7 1 8 1 9 i). 520 716003 6087 6170 6254 6337 ^421 65o4. 6588 6671! 6754 83 521 6838 6921' 7004 7088 7171, 7254 7338! 7421 7304 : 7587 11 522 7671 7754 7837 792c 8oo3 8086 8169I 8253 8336 9000' 9083 9165 8419 83 523 85o2 8585, 8668 8751 8834' 8917 1 9248 83 534 9331 9414 9497 9580 9663 9745 9828' 991 1 9994 ••77 83 5^5 720139 0242 0323 0407 0490 0573 o655' 0738 0821 0903 83 525 09S6 1 008 u5i 1233 i3i6| 1398 1481 i563 1646 1728 ?' 527 1811 189] 1975 2o58 2140 2222 23o5 2387 2469 2552 82 52y 2634 2716 279« 28811 2963: 3045 3127 3209: 3291 3374 82 529 3456 3538 3620 3702! 3784 3866 3948 4o3o 4II2 4194 82 5Jo 724276 ' 4358 4440 45221 4604 4635 4767 4849 4931 5oi3 82 53 1 5095 5176 3258 534o[ 5422' 55o3 5585 56671 5748 583o 82 532 5912 5993 6075 6i56: 6238. 6320 6401 6483! 6564 6646 82 533 6727 6809 6890 6972} 7053 7134 7216 7297 7379 7460 8i 534 7 Ml 7623 7704 7785; 7866 7948 8029 8iiO| 8191 8273 81 535 8354 8435 85i6 8597! 8678 8739 8841 8922 9003 9084 81 536 9165 9246 9327 94081 9489 9370 9651 9732 9813 9893 81 537 9974 ••55 •i36 •217! ^298 •378 •4^9 •540 1 •621 •702 81 538 730782 086 3 0944 1024I iio5 1186 1266 1 347 1428 i3o8 81 539 1589 1669 1730 i83o 1911 1991 2072 2l52 2233 23i3 81 540 732394 24741 2555 2635 2715 2796 2876 2956 3o37 3ii7 80 541 3197 3278 335S 3438 35i8 3398 3619 3739 3839 3919 80 54'? 3999 4079 4160 4240 4320 4400 4480 4360 4640 4720 80 543 4800 4880 4960 5o40| 5 1 20 3200 5279 5359 5439 55 1 9 80 544 5599 5679 5739 5838 3918 5998 6078 6i57 6237 63i7 80 545 6397 6476 6556 6635 6713 6795 6874 6934 7034 7ii3 80 546 7193 7272 7352 7431 73 1 1 7390 8463 7749 7829 7908 79 547 19^1 80671 8146 8225 83o5 8384 8543 8622 8701 79 548 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 •284 79 549 9572 9631 9731 9810 9889 9968 «®47 •126 •205 79 55o 74o363 0442 0321 0600 0678 0757 o836 0913 099 i 1073 79 55i Il52 I23o i3o9 i388| 1467 1546 1624 1703 1782 i860 79 552 1939 2018 2096 2i75| 2254 2332 2411 2489 2568 2647 79 553 2720 2804I 2882 2961 3o39 3ii8 3196 3980 32751 3353 343 1 -8 554 35io 3588 3667 3745 3823 3902 4o58 4i36 42i5 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5i5j 523' 53o9 5387 5465 5543 5621 5699 5777 78 557 5855 5933 601 1 6089 6167 6245 6323 6401 6479 6556 78 558 6634 67 1 2 679O 6868 6945 7023 7101 7179 7236 7334 78 559 7412 7489 7367 7645 7722 7800 7878 7953 8o33 8110 78 56o 748188 8266 8343 8421 8498 8376 8653 8731 8808 8885 77 56 1 8963 9040 9II8 9193 9272 935o 9427 9304 9382 9639 77 562 9736 98.4 989 1 9968 0043 •123 •200 •277 •354 •43 1 77 563 75o5o8 0386 0663 0740 0817 0894 0971 1048 II25 1202 77 564 1279 i356 1433 i5ioj i587 1664 1741 1818 1895 1972 77 565 2048 2125 2202 22791 2356 2433 25o9 2586 2663 2740 77 566 2816 2893 2970 3o47| 3 1 23 3200 3277 3353 3430 35o6 77 567 3583 366o! 3736 38i3 3889 3966 4042 4119 4193 4272 77 568 4348 4423 45oi 4578] 4634 4730 4807 4883 4960 5o36 76 569 5lI2 5189 3265 5341 5417 5494 5370 5646 5722 5799 76 570 755875 5951 6027 6io3 6180 6236 6332 6408 6484 656o 76 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 7472 7548 7624 7700 7775 785i 7927 8oo3 8079 76 573 8i55 823o 83o6 8382 8458 8533 8609 8685 8761 8836 -6 574 89.2 8988- 9o63 9139 9214 9290 9366 9441 9517 9392 76 575 9668 9743! 98,9 9894 9970 ••43 •121 •196 •272 •347 7-'> 576 760422 0498, 0373 0649 0724 0799 C875 0950 1025 noi 75 577 1 1 76 I25i| 1326 1402 1477 i552 1637 1702 1778 1 853 75 578 1928 2oo3| 2078 2i53 22:8 23o3 2378 2453 2529 2604 75 D79 2679 2754J 2829 2904 297 8 3 033 3128 32o3 3278! 3353; 75 ■ = , 3 4 1 5 1 6 7 1 8 1 9 1 D. ^'^ 10 A TABLE OF LOGARITHMS FROM J TO 10,000. N. ' 7 3 1 4 5 6 7 8 9 D. 58o 763428 35o3 3578 3653 : 3727 38o2 3877 3952! 4027 4101 ~~w 58 1 4176 425i 4326 4400 4475 455o' 4624 4699' 4774 4848 75 582 4923 499B 5072 j 5 147 i 5221 5296J 5370 5445 5520 5594 73 583 5669 5743 58 1 8 0892 5966 6041 1 611 5 6190 6264 6338 74 584 6413 6487 6562 6636 6710 6785 j 6859 6933 7007 7082 74 585 7 1 56 723o 7304 7379 7453 75271 7601 7675 7749 7823 74 586 7898 7972 8046 8120 8194 8268! 8342; 8416 8490 8564 74 ^^7 8638 8712 8786 8860 8934 9008 9082 9i56 9230 93o3 74 588 9377 945 1 ,9525 9599 9673 9746 9820 9S94 9968 ••42 74 589 770115 0189 0263 o336 0410 0484 0557 o63i 0705 0778 74 590 770852 0926 0999 1073 1146 1220J 1293 1 367 1440 i5i4 74 591 1587 1661 1734 1808 1881 1955! 2028 2102 2175 2248 73 592 2322 2395 2468 2542 26i5 2688! 2762 2835 2908 2981 73 593 3o55 3 1 28 3201 3274 3348 3421 3494 3567 3640 3713 73 594 3786 386o 3933 4006 4079 4i52 4225 4298] 4371 4444 73 595 45i7 4590 4663 4736 4809 4882 4955 5o28 0100 5173 73 596 5246 5319 5392 5465 5538 56io! 5683 5756 5829 5902 73 597 5974 6047 6120 6193 6265 6338 641 1 6483 6556 6629 73 5q8 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 72 600 778151 8224 8296 8368 8441 85i3 8585 8658 8730 8802 72 60 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 9596 9669 9741 9813 9885 9957 ••29 •ici •.73 •245 72 6o3 780317 0389 0461 o533 o6o5 0677 0749 0821 0893 0965 72 604 io37 1 109 1181 1253 i324 1396] 1468 1540 1612 1684 72 6o5 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2688 2759 283 1 2902 2974 3046 3ii7 72 607 3189 ?>26o 3332 3403 3473 3546 36 1 8 36S9 3761 3832 71 608 3904 3975 4046 4118 4189 4261 4332 44o3 4475 4546 71 609 4617 4689 4760 483 1 4902 4974 5o45 5ii6 5187 5259 71 610 785330 5401 5472 5543 56i5 5686 5757 5828 5899 5970 71 611 6041 61 12 6 1 83 6254 6325 6396 6467 6538 6609 6680 7« 612 6701 6822 6893 6964 7035 7106 7177 7248 7319 7390 7J 6i3 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 7» 614 8i68 8239 83io 838 1 8451 8522 8593 8663 8734 8804 71 6i5 8875 8946 9016 9087 9157 9228 9399 9369 9440 95io 71 616 958 1 9651 9722 9792 9863 9933 •••4 •<»74 •144 •2l5 70 617 790285 o356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1 129 1 199 1269 1 340 I4J0 1480 j55o 1620 70 619 1691 1761 i83i 1901 1971 2041 2111 2181 2252 2322 70 620 792392 2462 2533 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3162 323i 33oi 3371 3441 35 1 1 358i 365i 3721 70 622 3790 386o 3930 4000 4070 4139 4209 4279 4349 5045 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5ii5 70 624 5i85 5254 5324 5393 5463 5532 56o2 5672 5741 58n 70 625 588o 59i9 6019 6088 6i58 6227 6297 6366 6436 65o5 69 626 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 627 7268 7337 7406 7475 7545 7614 7683 7752 7S21 7890 69 628 7960 8029 8098 8167 8236 83o5: 8374 8443 85 1 3 8582 69 629 865i 8720 8780 8858 8927 8996! 9065 9134 9203 9272 69 63o 799341 9409 947S 9547 9616 9685 9754 9823 1 9802 9961 69 63 1 8D0029 OOQO 0167 0236 o3o5 0373 0442 o5ii! o58o 0648 69 632 0717 0786 o854 0923 0992 1061 1129 1 198 1266 i335 69 633 1404 1472 i54i 1609 1678 1747 i8i5 1884I 1952 2021 69 634 2089 2i58- 2226 2295 2363 2432 25oo 25681 2637 2705 60 635 2774 2842 2910 2979 3o47 3ii6 3i84 3252 3331 3389 68 636 3457 3525 3594 3662 3730, 3798 3867 1 3935 4oo3 4071 68 637 4139 4208 4276 4344 4412: 4480 4548 4616 4685 4753 68 638 4821 4889 4957 5o25 5093' 5i6i 5220 5297 5365 5433 68 639 55oi 5569 5637 5705 5773: 5841 5908 5976 6044 6112 68 N. " I , 1 3 4 5 6 7 8 9 D. A TABLK OF LOGARITHM 3 FROM 1 TO 10,000. 11 N. I j 2 3 4 5 6 7 8 9 6790 D. 640 806180 6248: 63i6 6384 645 1 65i9 6587 6655 6723 68 641 6858 6926' 6994 7061 7129 71971 7264 7332 7400 7467 68 642 7535 7603 : 7670 7738 7806 7873 7941 8008 8076 8143 68 643 8211 8279: 8346 8414 8481 8549 8616 8684 8751 8818 67 644 8886 8953 9021 9088 9 1 56 9223 9290 9358 9425 9492 67 645 9560 9627 9694 9762 9829 9896 9964 ••3 1 ••98 •i65 67 646 810233 o3ool 0367 0434 o5oi 0569 o636 0703 0770 0837 67 647 0904 0971 1039 1106 1173 1240 i3o7 1374 I44I i5o8 67 648 1575 1642 1709 1776 1843 1910 1977 2044 21II 2178 67 649 2245 23l2 2379 2445 25l2 2579 2646 2713 2780 2847 67 65o 812913 2980 3047 3i 14 3i8i 3247 33i4 338i 3448 35i4 67 65i 358 1 3648 3714 3781 3848 3914 3981 404S 4II4 4181 67 652 4248 43i4 438 1 4447 45i4 458 1 4647 4714 4780 4847 67 653 4913 4980 5o46 5ii3 f/79 5246 53i2 5378 5445 55ii 66 654 5578 5644 5711 5777 5843 5910 5976 6042 6109 6173 66 655 6241 63 08 6374 6440 65o6 6373 6639 6705 6771 6838 66 656 6904 6970 7o36 7102 7169 7235 7301 7367 7433 7499 66 6j7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 659 8S85 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 9610 9676 974 1 9807 9873 9939 o«®4 ••70 •i36 66 66 1 820201 0267 o333 0399 0464 o53o 0395 0661 0727 0792 66 662 o858 0924 0989 io55 1120 1 186 I23I i3i7 i382 1448 66 663 i5i4 1379 1645 1710 1775 1841 1906 1972 2o37 2io3 65 664 2168 2233 2299 2364 243o 2495 256o 2626 2691 2756 65 665 2S22 2887 2952 3oi8 3o83 3i48 32i3 3279 3344 3409 65 666 3474 3539 36o5 3670 3735 3 800 3865 3930 3996 4061 65 667 4126 4191 4256 4321 4386 445 1 45i6 458i 4646 4711 65 668 4776 4841 4906 4971 5o36 DIOI 5i66 3231 3296 536i 65 669 5426 5491 5556 5621 5686 5751 58i5 588o 5945 6393 6010 65 670 826075 6140 6204 6269 6334 6399 6464 6528 6658 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 73o5 65 672 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 673 80 ID 8080 8144 8209 8273 8338 8402 8467 853 1 8395 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 675 9304 9368 9432 9497 956 1 9625 9690 9754 9818 9882 64 676 9947 ••11 ••75 •139 •204 •268 •332 •396 •460 •525 64 677 83o589 o653 0717 0781 0845 0909 0973 io37 1102 1166 64 678 I230 1294 i358 1422 i486 i35o i6i4 1678 1742 1806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 238i 2445 64 680 832509 2573 2637 2700 2764 2828 2892 2956 3020 3o83 64 681 3i47 32II 3275 3338 3402 3466 353o 3593 3657 3721 64 682 3784 3848 3912 3975 4o39 4io3 4166 423o 4294 4357 64 683 4421 4484 4348 4611 4675 4739 4802 4866 4929 4993 64 684 5o56 5l20 5i83 5247 53 10 5373 5437 55oo 5564 5627 63 685 5691 5754 5817 588 1 5944 6007 6071 6i34 6197 6261 63 686 6324 6387 6451 65i4 6577 6641 6704 6767 6830 6894 63 687 6957 7020 -083 7146 7210 7273 7336 7399 7462 7325 63 688 7D88 7652 77i5 7778 7841 7904 7967 8o3o 8093 8i56 63 689 8219 8282 8345 8408 8471 8334 8597 8660 8723 8786 63 690 838849 8912 8975 903 8 9101 9164 9227 9289 9352 941 5 63 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 ••43 63 692 840106 0169 0232 0294 0357 0420 0482 o545 0608 0671 63 • 693 0733 0796 o85g 0921 0984 1046 1 109 1172 1234 1297 63 694 1359 1422 1485 1 547 1610 1672 1735 1797 i860 IQ22 2547 63 695 iq85 2047 2110 2172 2235 2297 236o 2422 2484 62 696 2009 2672 2734 2796 2859 2921 2983 3046 3io8 3170 62 697 3233 3295 3357 3420 3482 3544 36o6 3669 373 r 3793 62 698 3855 3918 3980 4042 4104 4166 4220 4291 4353 44i5 62 699 4477 4^39 4601 4664 4726 4788 485o 4912 4974 5o36 62 N. I 2 3 4 5 6 7 8 9 D. 12 A TAULE OF LOG A [UllIMS FROM I TO 10,000. N. 9 1 I 1 2 3 1 4 1 5 6 1 7 M 9 D. 700 845098 5i6o 3222 5284 5346 5408', 5470 j 5532 5594 5656 62 701 5718 5780' 5842 5904 5g66 6028 60901 61 5i 62i3 6273 62 702 6337 6399 6461 6323^ 6585 6646 6708 6770 6832 6894 62 703 6955 7017 7079 71411 7202 7264 73261 7388 7449 75ii 62 704 7573 7634 7696 7738 7819' 7881 7943 8559 8004 8066 8128 62 7o5 8189 8231 83i2 8374| 8435 8497 8620 8682 8743 62 706 88o5 8866 8928 8989 9o5i 91 12 9174 9235 9297 9358 61 707 9419 9481 9542 9604 9665 9726 9788 9849 991 1 9972 61 708 85oo33 0095 01 56 0217 0279 0840 0401 0462 0324 o585 61 709 0646 0707 0769 o83o 0891 0952 ioi4 1075 ii36 1 197 61 710 85i258 l320 i38i 1442 i5o3 1364 1625 1686 1747 1809 61 711 1870 I93I 1992 2o53; 2114 2175 2236 2297 2358 2419 61 712 2480 2541 2602 2663 2724 2783 2846 2907 2968 8029 6i 713 3090 3i5o 321 I 3272 3333 3394 3455 35i6 3577 3637 61 714 3698 3759 3820 388ii 3941 4002 4o63 4124 4i85 4245 61 7i5 4306 4367 4428 4488 1 4549 4610 4670 473i 4792 4852 61 716 4913 4974 5o34 5095 5 1 56 52i6 5277 5337 5398 5459 61 717 5519 558o 5640 5701 3761 5822 5882 5943 6oo3 6064 61 7.8 6124 6i85 6245 63o6 6366 6427 6487 6548 6608 6668 60 719 6729 6789 685o 6910. 6970 703 1 7091 7i52 7212 7272 60 720 857332 7393 7453 75i3 7374 7634 7694 7755 7815 7875 60 721 7935 7995 8o56 8116 8176 8236 8297 8357 8417 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 723 9i38 9,98 9258 9318 9379 9439 9499 9559 9619 9679 60 724 9739 9799 9859 9918 9978 ••38 ••98 •i58 •218 •278 60 720 86o338 0398 0458 o5i8 0578 0637 0697 0757 0817 0877 60 726 0937 0996 io56 1116 1 1 76 1236 1295 1355 I4i5 1475 60 727 1 534 1394 1634 1714 1773 i833 1898 1032 2012 2072 60 728 2l3l 2191 225l 23l0 2870 243o 2489 2549 2608 2668 60 729 2728 2787 2847 2906 2966 3o25 3o85 3 1 44 3204 3263 60 730 863323 3382 3442 35oi 356i 3620 36So 8789 3799 3858 59 73i 3917 3977 4o36 4096 41 55 4214 4274 4333 4392 4452 59 732 4011 4570 463o 4689 4748 4808 4867 4926 4985 5o45 59 733 5io4 5i63 5222 5282 5341 5400 5459 5319 5578 5637 59 734 5696 5755 58i4 5874 5933 5992 6o5i 6110 6169 6228 59 735 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7326 7585 7644 7703 7762 7821 7880 7939 7998 59 738 8o56 8ii5 8174 8233 8292 835o 8409 8468 8527 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9o56 9114 9173 59 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 39 741 9818 9877 9935 9994 ••53 •hi •170 •228 •287 •345 59 742 870404 0462 o52( 0379 o638 0696 0755 o8i3 0872 0980 58 743 0989 1047 1 106 I 164 1223 1281 i339 1898 1456 I3l5 58 744 1673 i63i l6qo 1748 1806 1 865 1928 1981 2040 2098 58 745 2 1 56 22l5 2273 233i 2389 2448 23o6 2564 2622 2681 58 746 2739 2797 2855 2913 2972 3o3o 3o88 3146 3204 3262 58 747 3321 3379 3437 3495 3353 36ii 3669 2727 3785 4366 3844 58 748 3902 39601 4018 4076 4i34 4192 4250 43o8 4424 58 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5oo3 58 700 875061 5119 5.77 5235; 5293 335i 5409 5466 5524 5582 58 75i 5640 5698 1 5756 58i3: 5871 5929 5987 6045 6l02 6160 58 732 6218 6276! 6333 6391 6449 65o7 6564 6622 6680 6737 58 753 6795 6853 1 6910 6968 7026 7083 7141 7199 7256 7314 58 754 7371 7429 7487 7344 7602 7659 7717 7774 7832 7889 58 755 7947 8004 8062 8119 8177 8234 8202 8349 8407 8464 57 756 8022 8579 8637 8694 8752 8809: 8866 8924 898 1 9089 57 7?7 9096 9i53 9211 9268 9825 9383' 9440 9497 9555 9612 P 758 9669 9726 9734 9841 9898 9936 ••i3 ••70 •127 •i85 ?7 759 880242 0299 o356 o4i3 0471 o528| o585 0642 0699 0756 57 N. " 1 2 3 ! 4 1 5 6 7 1 S 9 D. A TABLE OF LOGARITHMS FU OM 1 TO 10,000 13 N. I 2 3 4 5 6 7 8 9 D. 760 880814 0S71 0928 0985 1042 1099 1 1 56 I2l3 1271 i328 57 761 i385 1442 1499' ^556 i6i3 1670 1727 1784 1841 1898 57 762 1955 2012 2069 2126 2i83 2240 2297 2354 2411 2468 57 763 2325 25Si 2638; 2695 2732 2809 2866 2923 2980 3o37 57 764 3093 3i5o 3207 3264 3321 3377 3434 3491 3548 36o5 57 765 366i 3718 3773 3832 3888 3945 4002 4059 4ii5 4172 57 766 4229 4285 4342 4399 4455 4312 5i33 4623 4682 4739 57 761 76^ 4795 4852 4909 4963 5o22 5078 5(92 5248 53o5 57 536i 5418 5474 553 1 5587 5644 5700 5757 58i3 5870 57 769 5926 5983 6547 6039 6096 6i52 6209 6265 6321 6378 6434 56 770 886491 6604 6660 6716 6773 6829 6885 6942 6998 56 771 7054 7111 7167 7223 7280 ''l^i 7392 7449 75o5 7561 56 772 7617 7674 7730 7786 7842 7898 85i6 801 1 8067 8123 56 773 8179 8236 8292 8348 8404 8460 8573 8629 8685 56 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 775 9302 93d8 9414 9470 9326 9582 9b38 9694 9750 9806 56 776 9862 9918 9974 ••3o ••86 •141 •197 •253 •309 •365 56 777 890421 °477 0333 o589 0643 0700 0736 0812 0868 0924 56 778 0980 I033 logi 1 147 I203 1259 i3i4 1370 1426 1482 56 779 1637 1593 1649 1705 1760 1816 1872 1928 1983 2039 56 780 892095 2i5o 2206 2262 23i7 2373 24:9 2484 2340 2595 56 781 265i 2707 2762 2818 2873 2929 298D 3o4o 8096 3i5i 56 782 3207 3262 33i8 3373 3429 3484 30 io 3595 365i 3706 56 783 3762 3817 3873 3928 3984 4039 4094 4i5o 42o5 4261 55 784 43i6 4371 4427 44S2 4338 4593 4648 4704 4759 4854 55 ■^L^ 4870 4925 4980 5o36 5091 5146 5201 5257 53i2 5367 55 786 5423 5478 5533 5588 5644 5699 5734 5809 5864 5920 55 787 Ull 6o3o 6o85 6140 6195 6231 63o6 636i 6416 6471 55 788 658 1 6636 6692 6747 6802 6837 6912 6967 7022 55 789 7077 7i32 7187 7242 7297 7352 7407 7462 1317 7572 55 790 897627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 791 8176 823i 8286 8341 8396 8944 8451 85o6 856 1 86i5 8670 55 792 8725 8780 8835 8890 8999 9054 9109 9164 9218 55 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 794 9821 9873 9930 9983 •♦39 ••94 •140 •203 •258 •3l2 55 795 900367 0422 0476 033l o586 0640 0695 1240 0749 0804 0859 55 796 0913 0968 1022 1077 ii3i 1186 1295 1349 1404 55 797 1458 I3l3 1 567 1622 1676 1731 1783 1840 1894 1948 54 798 2oo3 2037 2112 2166 2221 2275 2320 2384 2438 2492 54 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3o36 54 800 903090 3i44 3199 3253 33o7 3361 3416 3470 3524 3578 54 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 8o3 4716 4770 4824 4878 4932 4986 5o4o 5094 5i48 5202 54 804 5256 53io 5364 5418 5472 5326 558o 5634 5688 5742 54 8o5 6796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 806 6335 6389 6443 6497 655i 6604 6658 6712 6766 6820 54 807 6874 6927 6981 7035 7089 7143 7196 725o 7304 7358 54 808 7411 7463 7519 7573 7626 7680 7734 7787 7841 7895 54 809 7949 8002 8o56 8110 8i63 8217 8270 8324 8378 843 1 54 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 54 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 95o3 54 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 ••37 53 8i3 91 0091 0144 0197 0231 o3o4 o358 041 1 0464 03I» 0571 53 814 0624 0678 073 1 0784 o838 0891 0944 0998 io5i 1 104 53 8i5 ii58 1211 1264 i3i7 1371 1424 1477 i53o i584 1637 53 816 1690 1743 1797 i85o 1903 1936 2009 2o63 21 16 2169 53 818 2222 2275 2328 238i 2435 2488 2541 2594 2647 2700 53 2753 2806 2839 2913 2066 3019 3072 3i25 3178 323i 53 819 [N. 3284 3337 3390 3443 3 3496 4 3549 36o2 3655 3708 3761 53 I 2 5 6 7 8 9 D- 11 A TABLE OF LOGARITHMS FROM I TO 10,000. N. I 1 2 3 4 1 5 1 ^ 1 7 8 9 D. 820 9i38i4 3867 3920 3973 4026, 4070 4i32j 4184 4237 4290 53 821 4343 4396 4449 45o2 4555I 4608 4660! 4713 4766 4819 ^=3 5294 5347 53 822 4872 4925 4977 5o3o 5o83: 5 1 36 6189! 5241 823 5400 5453 55o5 5558 56ii| 5664 5716J 5769 5822 5875 53 824 5927 5980 6o33 6o85 6i38. 6191 6243 6296 6349 6401 53 825 6454 65o7 6559 6612 6664 6717 6770 6822 6875 6927 53 826 6980 7033, 7085 7i38 7190 7243 72951 7348 7400 7453 53 827 7306 755s; 761 1 7663 7716: 7768 7820 7873 7925| 7978 52 828 8o3o 8o83 8i35 8188 8240' 8293 8345, 8397 8450 85o2 52 829 8555 8607 8609 8712 87641 8816 8869 I 8921 8973 9026 52 83o 919078 9i3o' 9183 9235 9287 1 9340 9392! 9444 9496 9549 52 83 1 9601 9653; 9706 9758 9810, 9862 9914 9967 •«ig, ••^, 52 832 920123 0176 0228 0280 0332; o384 0436 0489 o54i 0593 52 833 0645 0697' 0749 0801 o853; 0906 0958 lOIO 1062 1114 52 834 ■ 1166 1218 1270 l322 1374; 1426 1478, i53o i582 i634 -52 835 1686 1738 1790 1842 1894I 1946 1998 2o5o 2102 2 1 54 52 836 2206 2258^ 23l0 2362 2414 2466 23i8| 2570 2622 2674 52 837 2725 2777 2829 2881 2933 2985 3o37| 30S9 3i4o 3192 52 838 3244 3296 3348 3399 345i 3do3 3555 3607 3658 3710 52 839 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 4331 4383 4434 4486 4538 4589 4641 4693 4744 52 841 4796 4848 4899 49^51 5oo3 5o54 5io6 5i57 5209 5261 52 842 53 1 2 5364 54i5 5467 55i8 5570 5621 5673 5725 5776 52 843 5828 5879 5931 5982 6o34 6o85 6i37 6188 6240 6291 5i 844 6342 6394 6445 6497 6548 6600 665i 6702 6754 68o5 5i 845 6857 6908 6959 747^ 701 1 7062 7114 7i65 7216 7268 7319 5i 846 7370 7422 7524 7576 7627 7678, 7730 7781 7832 5i 847 7883 7935 7986 8037 8088 8140 9191 8242 8293 8345 5i 848 8396 8447 8498 8549 8601 8652 8703 8754 88o5 8857 5i 849 8908 8959 9010 9061 9112 9163 92i5 9266 9317 9368 5i 85o 929419 9470 9521 9572 9623 9674 9725 9776 9827 9879 5i 85i 9930 9981 ••32 ••83 •i34 •l83 •236 •287 •338 •389 5i 852 930440 0491 o542 0592 0643 0694 0745 0796 0847 0898 5i 853 0949 1000 io5i no2 ii53 I204 1254 i3o5 i356 1407 5i 854 1438 1 509 i56o 1610 1661 I7I2 1763 1814 i865 1915 5i 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 5i 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 5i 857 2981 3o3i 3o82 3i33 3i83 3234 3285 3335 3386 3437 5i 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 5i 859 3993 4o44j 4094 4145 4195 4246 4296 4347 4397 4448 5i 860 934498 4549 4599 465o 4700 .4751 4801 4852 4902 4953 5o 861 5oo3 5o54 5io4 5i54 52o5 5255 53o6 5356 5406 5457 5o 862 5507 5558 56o8 5658 5709 5759 58o9 586o 5910 5960 5o 863 6011 6061 6111 6162 6212 6262 63 13^ 6363 64 1 3 6463 5o 864 65i4 6564 6614 6665 6715 6765 68i5 6865 6916 6966 5o 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 5o 866 7518 7568 7618 7668 7Ti8 7769 7819 7869 7919 7969 5o 867 8019 8069 8119 8169 8219 8269 832o: 8370 8420 8470 5o 868 8520 8570 8620 8670 8720 8770 8820! 8870 8920 8970 5o 869 9020 9070 9120 9170 9220 9270 9320 9369 941Q 9469 5o 870 939519 9569 9619 9669 0168 9719 0218 9769 9819 9869 9918 9968 5o 871 9400 1 8 0068 0118 0267 o3i7 0367 0417 0467 5o 872 o5i6 o566 0616 0666 0716 0765 081 5 o865 0915 0964 5o 873 1014 1064 iii4 ii63 I2l3 1263 i3i3 i362 1412 1462 5o 874 i5i I i56i 1611 1660 1710 1760 1809 1859 1909 1958 5o 875 200S 2o58 2107 2157 2207 2256 23o6 2355 24o5 2455 5o 876 25o4 1 2554 26o3 2653 2702 2752 2801 285i 2901 2950 5o 877 3 000 3 040 3oQ9 3148 3198 3247 3297 3346 3396 3445 59 878 3495 3544 3593 3643 36q2 3742 3791 3841 38oo 3939 l") 879 3989' 4o38 4088 1 413-? 4186 4236 4285 4335 4384 4433 59 N. I ! 2 3 1 4 5 1 6 1 7 1 8 1 9 1). (sy A TABLE OF LOGARITflMS FROM 1 TO 10,000. 15 N. 1 1 1 i 2 3 j 4 5 6 7 1 8 1 9 49 88o 944483, 4532', 458i 463i 4680 4729' 4779 4828; 4877; 4927 88i 4976 5o25 5469, 55i8| 5074 5i24 5173, 5222| 5272, 532i| 5370 5419 49 882 5567 56i6 56651 5715; 5764' 58 1 31 5862 1 5912 49 883 596 1 j 6010 6059 61081 6157; 6207! 6256 63o5| 6354' 6403 49 884 6452 65oi 655i 6600 6649' 6698, 6747 i 6796 6845i 6894 49 885 6943 6992 704I 7090' 7140; 7189I 7238 7287 7336: 7385 49 886 7434 7483 7532 758i 763o 7679 7728; 7777' 7826: 7875 49 887 7924 7973 8022 8070 8119I 8i68i 8217 8266' 83i5! 8364 49 888 8413 8462 85ii 856o: 8609! 8657! 8706 1 8755I 88041 8853 49 889 8902 8951 8999 90481 9097 9146I 9195I 9244 9292 9341 49 890 949390 9439 94S8 9536: 95851 9634' 9683; 9731 9780. 9829 49 891 9878 9926 9975 ••24' ••73 •l2l| •170 •219 ^267 •3 16 49 892 95o365 0414 0462 o5iij o56o 0608: 0657 0706 0754' o8o3 49 893 o85i 0900 0949 0997 i 1046 10951 1 143 1192 1240 1289 49 894 1 338 i3S6 1435 1483 i532 i58oi 1629 1677 1726 1773 49 ?95 1823 1872 1920 1969 2017 2066 2114 2i63 2211 2260 48 896 23o8 2356 24o5 2453 25o2 255o 2599 2647 2696 2744 48 M 2792 2841 2889 2938; 2986 3o34! 3o83 3i3i 3i8o| 3228 48 898 3276 3325 3373 3421I 3470 35i8 3566 36 1 5 36631 3711 48 899 3760 38o8 38561 3905! 3953 4001 4049 4098 4146 4194 48 900 954243 4291 4339 4387 4435' 4484 4532 458o 4628 4677 48 901 4725 4773 4821 4869 4918 4966 5oi4! 5o62 5iio 5i58 48 902 5207 5255 53o3 535i 5399 5447 5495 5543 5G92 5640 48 903 5688 5736 5784 5832 588o 5928 5976 6024 6072 6120 48 904 6168 6216 6265 63i3 636 1 6409 6457 65o5 6553 6601 48 905 6649 6697 6745 6793 6840 6888 6936 6984 7082 7080 43 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 907 7607 7655 7703 775i 7799 7847 7894 7942 7990 8o38 48 908 8086 8i34 8181 8229 8277 8325 8373 8421 8468 85i6 48 909 8564 8612 8659 8707 8755 88o3 885o 8898 B946 8994 48 910 959041 9089 9137 9185 9232 9280 9328 9375 9423 947 « 48 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 912 9995 ••42 ••90 •i38 •i8d •233 •280 •328 •376 •423 48 913 960471 o5i8 o566 o6i3 0661 0709 0756 0804 o85i 0899 48 914 0946 0994 1041 1089 ii36 1 184 I23l 1279 1826 1874 47 giD 1421 1469 i5i6 1 563 1611 1 658 1706 1753 1801 1848 47 916 1895 1943 1990J 2o38 2o85 2l32 2180 2227 2275 2822 47 9'I 2369 2417 2464; 25tl 2559 2606 2653 2701 2748 2795 47 918 2843 2890 2937} 2985 3o32 3o79 3i26 3174 3221 3268 47 919 33i6 3363 34ioi 3457 35o4 3552 3599 3646 3693 3741 47 920 963788 3835 3382 3929 3977 4024 4071 4118 4i65 4212 47 921 4260 4307 4354 4401 4448 4495 4542 4590 4687 4684 47 922 473 1 4778 4825 4872 4919 4966 5o!3 5o6i 5io8 5i55 47 923 5202 5249 5296 5343 5390 5437 5484 553i 5578 5623 47 924 5672 5719 5766 58i3 586o 5907 5954 6001 6048 6095 47 926 6142 6189 6236 6283 6329 6376 6423 6470 65i7 6564 47 926 661 I 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 7267 73i4 7861 7408 7454 7501 47 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 929 8016 8062 8109 8i56 8203 8249 8296 8343 8890 8436 ! 47 930 968483 853o 8576 8623 1 8670 ! 8716 8763 8810 8856 8903 47 931 8950 8996 9043 9090 9i36 9'83 9229 9276 9323 9369 47 932 9416 9463 9509 9556 96o:> 9649 9690 9742 9789 9835 47 933 9882 9928 j 9975 ••21 ••63 i •la •161 •207 •254 •3oo 47 934 970347 o3q: 0440 0486 o533 0579 0626 0672 0719 0765 46 935 0812': o858 1 0904 0951 0997 1044 1090 1187 ii83 1229 46 936 1276 1 l322 i36g i4i5 1461 i5o8 i554 1601 1647 1693 46 937 1740 1786 i832 i 1879 1925 1971 2018 2064 :iio 21D7 46 938 2203 2249 2295 2342 2383 2434 2481 2527 2578; 26iq 46 939 2666 2712 2758 2804 285i 2897 2943 2989 3o35 3082 46 N. I I 2 3 4 5 6 7 8 9 16 A TABI.K OF LOOAHITHMS FROM 1 T«J 10,000. 940 973128 I 3i74 2 3 "3266 4 5 6 7 8 9 D. 46 \ 3220 33i3 3359 34o5 345i 3497 3543 4 941 3090 3636 3682: 3728 3n4 3820 3866 3913 3959 4oo5 46 ''^ 942 40 5 1 40Q7 4143 4189 4235 4281 4327 4374 4420 4466 46 943 45 12 4538 4604' 465o 4696 4742 4788 4834 4880 4926 46 \ 944 4972 5oi8 5064^ 5 1 10 5 1 56 5202 5248 5.>94 5340 5386 46 , , 9/45 5432 5478 5524' 5570 56i6 5662 5707 5753 5799 5845 46 946 5891 5937 5983 _ 6029 6075 6121 6167 6212 6258 63o4 46 947 635o 6396 6442; 6488 6533 6579 6625 6671 6717 6763 46 948 6808 6854 6900' 6946 6992 7037 7083 7129 7175 7220 46 949 7266 7312 7358 7403 7449 7495 7541 7586 7632 1678 46 ] 930 977724 7769 7815' 7S61 7906 7932 7998 8043 8089 8i35 46 9J1 8iSi 8226 8272 83 1 7 8363 8409 8454 85oo 8546 8591 46 902 8637 8683 8728 8774 8819 8865 891 1 8956 9002 9047 46 953 9093 9i38 9 184 9230 9275 9321 9366 9412 9457 95o3 46 954 9548 9594 9639' 9685 9730 9776 9821 9867 99-2 9958 46 V V ^ 955 980003 0049 0094 0140 oi85 023l 0276 0322 0367 0412 45 - ^-^ 956 0458 o5o3 o549 0594 0640 o685 0730 0776 0821 0867 45 957 0912 0957 ioo3. 1048 1093 1 139 1 184 1229 1275 l320 45 958 i366 1 4 1 1 i456' i5oi 1 547 1592 1637 i6S3 1728 1773 45 959 1819 1864 1909; 1954 2000 2045 2090 2i35 2181 2226 45 960 9S2271 23i6 2062; 2407 2452 2497 2543 2588 2633 2678 45 961 2723 2769 2814' 2859 2904 2949 2994 3o4o 3o85 3i3o 45 ', 062 3175 3220 3265 33io 3356 3401 3446 3491 3536 358i 45 963 3626 3671 3716 3762 3807 3852 3897 3o42 3987 4o32 45 \ 964 4077 4122 4167 4212 4257 43o2 4347 4392 4437 4482 45 > 965 4527 4572 4617 4662 4707 4732 4797 4842 4887 4932 45 966 4977 5022 5067 5ll2 5i57 5202 5247 5292 5337 5382 45 967 5426 5471 55i6 556i 56o6 565i 5696 5741 5786 583o 45 968 5875 5920 6369 5965 6010 6o55 6100 6144 6189 6234 6279 45 969 6324 64i3 6458 65o3 6548 6593 6637 6683 6727 45 970 986772 6817 6861 6906 6951 6996 7040 7085 7i3o 7175 45 -^ 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 40 972 7666 8ii3 7711 77D6 7800 7845 7890 7934 7979 8024 8068 43 973 8157 8202 8247 8291 8336 838i 8425 8470 85i4 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 973 9003 9049 9094 9i38 9i83 9227 9272 9316 9361 94o5 45 976 q45o 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 9895 9939 9983 ••28 ••72 •117 •161 •206 •25o •294 44 978 990339 o383 0428 0472 o5i6 o56i o6o5 o65o 0694 0738 44 979 0783 0827 0871 0916 0960 1004 1049 1093 ii37 1182 44 980 991226 1270 i3i5| i359 i4o3 1448 1492 1536 i58o 1625 44 N 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 982 21 II 2 1 56 2200 2244 2288 2333 2377 2421 2465 25o9 44 \ 983 2554 2598, 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3o39 3u83 3127 3172 32x6 3260 33o4 3348 3392 44 ■ 985 3436 3480 3524 3568 36i3 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4o53 4097 4141 4 1 85 4229 4273 44 987 4317 436i 44o5 4449 4493 4537 458i 4625 4669 47i3 44 ,, 988 4757 4801 4845 4889 4933 5^72 4977 5o2I 5o65 5io8 5i52 44 989 5196 5240 5284 5328 5416 5460 55o4 5547 5591 44 99c J95635 5679 5723 5767 58ii 5854 5898 5942 5986 6o3o 44 991 6074 6117 6161 62o5 6249 6293 6337 638o 6424 6468 44 . 992 65i2 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 \ '■ 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7736 7343 44 994 7386 7430 7474 7517 7561 7605 7648 7692 7779 44 995 7823 7867 7910 7954 7998 8041 8o85 8129 8172 8216 44 \. . 996 8259 83 o3 8347 8390 8434 8477 8521 8564 8608 8652 44 -^ 997 8695 8739 8782 8826 8869 8913 8o56 9392 9000 9043 9087 44 998 9i3i 9174 9218 9261 93o5 9348 9435 9479 9522 44 i 999 N. 9565 9609 9652 9696 9739 9783 9826 9870 _99i3 9957 43 'i I 2 3 4 5 6 7 8 J VALIABLE SCHOOL AND MISCELLANEOUS BOOKS i¥10RT0iy & CC LOUISVILLE, KY. F H I ]M E 12^ S . Goodrich's Small Primer — Child's First Book, or Illustrcated Gradations in Bead- ing and Spelling. 32 passes, IGmo. Mother's Primer— 21 pages, 12nio. Elegant new ensraviuss. Common-School Primer— 9G pages, iGmo. Cloth backs. Vf'ebster's Speller and Deflner— A Sequel to Webster's Elementary Spelling Book. OOOI>I?,ICH'S IN^ETV I?,Ej?>.r>EHS. Edited by Noble Butler, A. M. Goodrich's New First Reader— 72 pages, large IGmo. Cloth backs. Elegantly il- lustrated. Goodrich's New Second Header— 144 pages, large ICnio. Half-bound. Elegantly il- lustrated. Goodrich's New Third Reader— 21G pages, Embossed backs. Elegantly illustrated. Goodrich's New Fourth Reader— 276 pages, large 12mo. Embossed backs. Elegantly illustrated. Goodrich's ^tyi Fifth Reader— 384 pages, large 12nio. Half roau, embossed. Ele- gantly illustrated. Goodrich's New Sixth Reader— For High- Schools. 652 pages. Half roan, embossed. OOOIDIIICH'S SER^IES OW HISTORIES. Primary Series. 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