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ADVERTISEMENT TO THE FOURTH EDITION. The author has endeavoured to accommodate his Algebra to the present state of science in the United States. Consider- able alterations and improvements have been made in the dif- ferent sections of the original work. There are also intro- duced two new chapters, containing Figurate and Polygonal Numbers, Vanishing Fractions, Indeterminate Coefficients, In- determinate and Diophantine Analysis. The chapters upon these subjects have chiefly been derived from Euler, Bonny- castle, Young, and Bourdon. JAMES RYAN. Neva York, July 4, 1838. til 89/1' ADVERTISEMENT. As Utility is the great object aimed at in this Publication, I have spared no pains to make a careful selection of materi- als, from the most approved sources, which may tend to eluci- date, in a full and clear manner, the Elements of Algebra, both in theory and practice. Those authors of whose labours I have principally availed myself, are Euler^ Clairaut, Lacroix, Garnier^ Bezout, La' grange^ Newton^ Simson^ EmersoUf Woody Bonnycastle, Bridge^ and Bland. To Bland's Algebraical Problems, (a work compiled for the use of Students in one of the first Universities in Europe), I am chiefly indebted for the problems in Simple, Pure, and Quadratic Equations. By permission of the learned Dr. Adrain, I have added, as an Appendix, his method of demonstrating algebraically the pro- positions in the fifth book of Euclid's Elements. JAMES RYAN. New York, July 1, 1824. CONTENTS. ^ INTRODUCTION. VAOI Explanation of the Algebraic method of notation .*— Definitions and Axioms^ ........... 1 CHAPTER I. On the Addition^ Subtraction^ Multiplication, and Division of Algebraic Quantities. SECT. I. Addition of algebraic quantities --- H II. Subtraction of algebraic quantities ...... 18 III. Multiplication of algebraic quantities .._..- 22 IV. Divisionof algebraic quantities ----..-. 33 V. Some general theorems, observations, &c. - - - - 50 CHAPTER II. On Algebraic Fractions, 8ECT. I. Theory of algebraic fractions ---....- 55 n. Method of finding the greatest common divisor of two or more quantities 60 III. Method of finding the least common multiple of two or more quantities -----.--...- 72 rV. Reduction of algebraic fractions '-. 73 V. Addition and subtraction of algebraic fractions - - - 85 VI. M,ultiplication and division of algebraic fractions . - 91 CHAPTER III. On Simple Equations involving only one unknoum Quantity 95 SBCT. I. Reduction of simple equations .------- 96 H. Resolution of simple equations, involving only one un- known quantity -- 101 ni. Examples in simple equations, involving only one un- known quantity .---.. 108 1* vi ^B CONTENTS. CHAPTER IV. PAOB On the Solution of Problems producing Simple Equations 117 SECT. I. Solution of problems producing simple equations, in volving only one unknown quantity ----- 117 * CHAPTER V. On Simple Equations involving two or more unknown quantities -------------- 134 SSCT. I. Elimination of unknown quantities from any number of simple equations ----------- 134 II. Resolution of simple equations, involving two unknown quantities -------------- 140 Examples in which the preceding rules are applied in the solution of simple equations, involving two un- known quantities ----------- 152 m. Resolution of simple equations, involving three or more unknown quantities - 158 IV. Solution of problems producing simple equations, in- volving more than one unknown quantity ----- 164 CHAPTER VI. On the Involution and Evolution of Numbers^ and of Algebraic Quantities. SSCT. I. Involution of algebraic quantities ------- 175 n. Evolution of algebraic quantities 184 ni. Investigation of the rules for the extraction of the square and cube roots of numbers ---- 191 CHAPTER VII. On Irrational and Imaginary Quantities. SECT. I. Theory of irrational quantities -------- 1^ II. Reduction of radical quantities or surds ----- 203 III. Application of the fundamental rules of arithmetic to surd quantities ----- 207 IV. Method of reducing a fraction whose denominator is a simple or a binomial surd, to another that shall have rational denominator ..--- 213 V. Method of extracting the square root of binomial surds 219 VI. Calculation of imaginary quantities ...... 221 CONTENTS. ?a CHAPTER VIII. PAOK On Pure Equations ----------- 225 SECT. I. Solution of pure equations of the first degree by involution ------------- 225 II. Solution of pure equations of the second, and other higher degrees, by evolution 230 ni. Examples in which the preceding rules are applied in the solution of pure equations ------ 231 CHAPTER IX. On the Solution of Problems producing Pure EquMions 236 CHAPTER X. On Quadratic Equations -..-....- 242 SECT. I. Solution of adfected quadratic equations, involving only one unknown quantity 246 n. Solution of adfected quadratic equations, involving two unknown quantities - - 254 CHAPTER XI. On the Solution of Problems producing Quadratic • Equations. «ECT. I. Solution of problems producing quadratic equations, involving only one unknown quantity ----- 261 n. Solution of problems producing quadratic equations, involving more than one unknown quantity - - - 266 CHAPTER XII. On the Expansion of Series .--....-. 272 SECT. I. Resolution of algebraic fractions, or quotients, into infinite series ------------ 272 II. Investigation of the binominal theorem - - - - - 289 HI. Application of the binomial theorem to the expan- . sion of series ---------.._ 296 CHAPTER XIII. On Proportion and Progression. SECT. I. Arithmetical proportion and progression ----- 299 n. Geometrical proportion and progression ----- 302 viii. CONTENTS. raCT. PACK III. Harmonica! proportion and progression ----- 307 IV. Problems in proportion and progression - - - - - 308 CHAPTER XIV. On Logarithms ----- 311 SECT. I. Theory of logarithms .......... 312 n. Application of logarithms to the solution of exponen- tial equations ............ 317 CHAPTER XV. On the Resolution of Equations of the third and higher degrees. SECT. I. Theory and transformation of equations - - - . - 320 II. Resolution of cubic equations by the rule of Cardan, or of Scipio Ferreo 326 in. Resolution of biquadratic equations by the method of Des Cartes 331 IV. Resolution of numeral equations by the method of Di- visors --------------- 333 V. Resolution of numeral equations, by Newton*s me- - thod of approximation --.-.---.- 336 CHAPTER XVI. On Indeterminate Coefficients, Vanishing Fractions^ and Fig- uraie and Polygonal Numbers -------* 339 SECT. I. Indeterminate Coefficients -----..-- 339 II. Vanishing Fractions 342 in. Figurate and Polygonal numbers .-.---- 345 CHAPTER XVn. On Indeterminate and Diophaniine Analysis - - - - 347 SECT. I. Indeterminate Analysis -_-.--.--- 347 n. Diophantine Analysis ----------- 356 APPENDIX. Algebraic method of demonstrating the propositions in the fifth book of Euclid's Elements, according to the text and arrangements in Simson's edition - - 87X AN ELEMENTARY TREATISE ON ALGEBRA. INTRODUCTION. EXPLANATION OF THE ALGEBRAIC METHOD OF NOTATION I— - DEFINITIONS AND AXIOMS. 1. Algebra is aigeneral method of imputation, in which abstract quantities and their several relations are made the subject of calculation, by means of ^ alphabetical lettei* and other signs. 2. The letters of the alphabet may be employed at plea- sure for denoting any quantities, as algebraical symbols or ab- breviations ; but, in general, quantities whose values are known or determined, are expressed hyxhe first letters, a, b, c, &c. ; and uj^nown or undetermined quantities are denoted by the last or final ones, w, v, w, x, &c. 3. Quantities are equal when they are of the same magni-^ tude. The abbreviation a = b implies that the quantity de- noted by a is equal to the quantity denoted by b, and is read a equal to b ; a>6, or a greater than b, that the quantity a is greater than the quantity 6; and a<^b, or a lesslhan b^ that the quantity a is less than the quantity b. 4. Addition is the joining of magnitudes into one sum. The sign of addition is an erect cross ; thus, a-{-b implies the sum of a and b, and is called a plus b, if a represent 8 and b, 4 ; then, a-\-b represents 12, or a4-^ = 8-|-4 = 12. 5. Stibtraction is the taking as much from one quantity as is equal to another. Subtraction is denoted by a single line ; as a~b, or a iniuiis b, which is the part of a remaining, when a part equal to b has been taken from it ; if a=:9, and b = 5 ; a—b expresses 9 diminished by 5, Which is equal to 4, or a-b=9—5=4. 2 2 INTRODUCTION. 6. Also^the difference of two quantities a and h ; when it is not known which of them is the greater, is represented by the sign -w; thus, a-»'6 is a—h, ox h—a\ and a+h signifies the sum or difference of a and b. '*' 7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple appo- sition ; thus, axh^ a . b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted hy b; if « = 5 and b=i7\ then a X 5=5 X 7 = 35, or a . i=-5 . 7 = 35, or ab=5x7 = 35. Scholium. The multiplication of numbers cannot be ex- pressed by simple apposition. A unit is a magnitude consi- dered as a whole complete within itself. And a whole num- ber is composed of units by continued additions ; thus, one plus one composes two, 2-{-lz=z3, 3 + 1=4, &c. 8. Division is the pbtraction of one quantity from another as often as it is contamed in it ; or the finding of that quo- tient, which, when multiplied by a given divisor, produces a given dividend. Division is denoted by placing the dividend before the sign -T-, and the divisor after it ; thus a-^b, implies that the quan- tity a is to be divided by the quantity b. Also, it is frequently denoted by placing one of the two quantities over the other, in the form of a fraction ; thus, j- =: a-^b ; if a^ 12, & = 4 ; then a-7-J=Y=12-^4=— -— 3. b 4 9. A simple fraction is a number which by continual addition composes a unit, and the number of such fractions contained in a unit, is denoted by the denominator, or the number below the line ; tnus, j4-g-+^=l- A nu?nber composed of such sim- ple fractions, by continual addition, may properly be termed a multiple fraction ; the number of simple fractions composing it, is denoted by the upper figure or numerator. In this sense, ^, f, ^, are multiple fractions ; and f = 1,1^=^4-^=1-1— J-=l^. 10. When any quantities are enclosed in a parenih*eses, or have a line drawn over them, they are considered as one quantity with respect to other symbols ; thus a— (6-i-c), or a—b-^c\ implies the excess of a above the sum of b and c. Let a=9, &= 3, and c= 2 ; th en a-(i-f c) = 9— (3+2)=9 — 5=4, or a— 6-fc=9— 3+2 = 9— 5=4. Also, {a-\-b)x INTRODUCTION. {c-\-d), or a-\-bX c-{-d, denotes that the sum of a and b is to be muhiplied by the sum of c and d ; thus, let a=:4, i = 2, c = 3, and ( /=5; then_(a4;^X(c-M) = (4 + 2)x(3-f-5)=:6x8=: 48, ora+bxc-{-d=4 + 2 x 34-5 = 6 X 8=48. And (a—b)-^ (c-\-d), or ; implies the excess of a alJbve 5, is to be di- vided by the sura of c and d; if a=12, 5=2, c=4, and d=l ; then, (a-5)-i-(c+c/)=(12-2)-j-(4-}-l) = 10^5=2,or -^ _ 12-2 _1Q_ ~" 4-f 1 "~5 ~ * The line drawn over the quantities is sometimes called a vinculum. 11. Factors are the numbers or quantities, from the multi- plication of which, the proposi3d numbers or quantities are produced ; thus, the factors of 35 are 7 and 5, because 7x5 = 35 ; also, a and b are the factors o( ab ; 3, a^, b and c^, are the factors of Sa^bc'^ ; and a-f 5 and a—b are the factors of the product (a-\-b)x{a — b). When a number or quantity is produced by the multiplica- tion of two or more factors, it is called a composite number or quantity ; thus, 35 is a composite number, being produced by the product of 7 and 5 ; also, 5acx is a composite quantity, the factors of which are 5, a, c, and x. 12. When the factors are all equal to each other, the pro- duct is called a power of one of the factors, and the factor is called the root of the product or the power. When there are two equal factors, the product is called the second power or square of either factor, and the factor is called the second root or square root of the power. When there are three equal fac- tors, the product is called the third power or cube of either factor, and the factor is called the third root or cube root of the power. And so on for any number of equal factors. 1 3. Instead of setting down in the manner of other products, the equal factors which multiplied together constitute a power, it is evidently more convenient to set down only one of the equal factors, (or, in. other words, the root of the power,) ^nd to designate their number by small figures or letters placed near the root. These figures or letters are always placed at the upper and right side of the root, and are called the indices or exponents of the power. For example : aX«XaXaor aaaa is denoted thus, a* ; yXyXyXyxy or yyyyy, thus, y^\ 4 INTRODUCTION. where a* and y^ are the powers ; a and y the roots, and 4 and 5 the indices or exponents of the powers. Again : ^ax^ X 4aa;2x4aa72, is thus abridged, (4aa;2)3 ; where {^ax^^ is the power, 4aa:2 the root, and 3 the index or exponent of the power. The same method is adopted, whatever be the form of the root: thus, (c^ — x^ — y^)x{a^—x^ — y^)x(a^ — x^—y"^) is written briefly thus, (a^ — x'^—y'^Y, where (a^—x'^ — y'^Y is the power, a^ — x'^—y'^ the root of the power, and 3 its index or exponent. N. B. Care must always be taken, to embrace the root in parentheses, except where it is expressed by a single charac- ter. 14. The coefficient of a quantity is the number or letter pre- fixed to it ; being that which shows how often the quantity is to be taken-; thus, in the quantities Zh and Sx^^ 3 and 5 are the coefficients of h and x"^. . Also, in the quantities 3«y and ba?x~ 3a and ba^ are the coefficients of y and x. 15. When a quantity has no number prefixed to it, the quantity has unity for its coefficient, or it is supposed to be taken only once ; thus, x is the same as Ijr; and when a quantity has no sign before it, the sign + is always under- stood ; thus, Za^b is the same as -\-'da%, and 5a— 36 is the same as -f 5a— 36. . 16. Quantities which can be expressed in finite terms, or the roots of which can be accurately expressed, are rational quantities ; thus, 3a, Ja, and the square root of Aa^, are ra- tional quantities ; for if a=10 ; then, 3a=:3 X 10 = 30 ; fa = |-X 10:::=2_o_4 . ^^^ (\^q squarc root of Aa^=:. the square root of 4 X 102= thg square root of 4x10x10= the square root of 400 = 20. 17. An irrational quantity, or surd, is that of which the value cannot be accurately expressed in numbers, as the square root of 3, 5, 7, &c. ; the cube root of 7, 9, &c. 18. The roots of quantities are expressed by means of the radical sign y, with the proper index annexed, or by fraction- al indices placed at the right-hand of the quantity ; thus, -y/a, 1 1 or a^, expresses the square root of a ; ^ (a + a:), or (a-fa;) , the cube root of {a-\-x)\ ^(a-f-x), or (a+a?)"*, the fourth root of (a-\-x). When the roots of quantities are expressed by fractional indices ; thus, a , (a-\-x)^, (a-f-a?)'* ; they are generally read a in the power (J), or a with (J) for an index ; (a-\-x) in the power (^), or {a-y-x) with (J) for an index; and (a-\-x) in the power (i), or {a-\-x) with (\) for an index. 19. Like quantities are such as consist of the same letters or INTRODUCTION. 5 the same combination of letters, or that differ only in their numeral coefficients ; thus, 5a and 7a ; 4ax and 9ax ; +2ao and 9ac ; — 5ca ; &c., are called like quantities ; and unlike quantities are such as consis*t of different letters, or of diffe't- ent combination of letters ; thus, 4a, 3&, 7ax, daj/"^, &c. are unlike quantities. 20. Algebraic quantities have also different denominations, according to the sign -j-, or — . Positive, or affirmative quantities, are those that are addi- tive, or such as have the sign + prefixed to them ; as, -\-a, -^dab, or 9ax. 21. Negative quantities are those that are subtractivc, or such as have the sign — prefixed to them ; as, — x^ — 3a^, — 4ab, &c. A negative quantity is of an opposite nature to a positive one, with respect to addition and subtraction : the condition of its determination being such, that it must be sub- tracted when a positive quantity would be added, and the re- verse. 22. Also quantities have dif&rent denominations, according to the number of terms (connected by the signs -f- or — ) of which they consist ; thus, a, 3b, —4ad, &c., quantities con- sisting of one term, are called simple quantities, or monomi- als ; a-^-x, a quantity consisting of two terms, a^jjMnomial ; a~x is sometimes called a residual quantity. A trinomial is a quantity consisting of three terms ; as, a-{-2x — Sy ; a quad- rinomial o( four ; as, a-^b-{-3x — 4y ; and a polynomial, or multinomial, consists of an indefinite number of terms. Quan- tities consisting of more than one term may be called compound quantities. 23. Quantities the signs of which are all positive or all negative, are said to have like signs ; thus, -{-3a, 4-4a?, + 5a&, have like signs ; also, —4a, —3b, —4ac. When some* are positive, and others negative, they have unlike signs ; thus, the quantities +3a and — 5ab have unlike signs ; also, the quantities '-3ax, -\-3a'^x : and the quantities —b, -j-b. 24. If the quotients of two pairs of numbers are equal, the numbers are proportional, and the first is to the second, as the third to the fourth ; and any quantities, expressed by such d c numbers, are also proportional; thus, if -7-=-^; then a is to 5 as c to d. The abbreviation of the proportion ; a : b i: c : d ; and it is sometimes written a : b=c : d; if a =8, 6=4 8 12 c=12, and d =6 ; then, -= -^-=2, and 8 : 4 :: 12 ; 6. 4 o 6 INTRODUCTION. 25. A term, is any part or member of a compound quanti- •ty, which is separated from the rest by the signs -|- and — , thus, a and b are the terms of a-f-^ ; and 3a, —'Zb, and -fSac?, are the terms of the compound quantity 3a — 2b -{-bad. In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are compos- ed ; J;hus, a and b are the terms of ab^ or of ^ ; and a, 5, c, 8 ; since 10—6=4, a positive quan- tity ; therefore, a — x is positive. Next, let a=\2, a:=14, and b—20\ then b-\-a—x— 20 -{-12 — 14, which is <20 ; since 12 — 14 = — 2, a negative quantity ; therefore a — x is negative. In like manner, it may be shown that the expression a — b-\-c — d is positive or negative according as a-\-c is > or <^b-\-d\ and so of all compound quantities whatever. 37. The use of these several signs, symbols, and abbrevia- tions, may be exemplified in the following manner : EXAMPLES. Example. 1. In the algebraic expression c-f-ft-j-c— •(?, let a=8, 5=7, c=4, and d=.Q ; then a+&+c— £?=8 + 7-f-4— 6 = 19— 6 = 13. Ex, 2. In the expression ab-^-ax — by, let a=5, J=4 8 INTRODUCTION. a: =8, and y=:12 ; then, to find its value, we have ab-{-aac-^ ^=5X4 + 5X8—4X12 =20+40 — 48 = 60—48 = 12. Ex. 3. What is the value of -^—^jwhere a=4, x=5, y a-f-o ^ nrlO, and b=6 1 Here 3aa:+2y=3x4x 5+2x10=60+20=80, and a+ fc=4+6=10; 3aa;+2y__80_ ■** a+b ~10~ ' Ex. 4. What is the value of a'^+2ab-'C+dy when a=6, i=5, c=4, and £^=1 ? Ans. 93. Ex. 5. What is the value of ab-\-ce—bd, when a=8, i=7, c=6, d=5y and e = l ? Ans. 27. dx-i—bij Ex. 6. In the expression -j— — ^, let a=5, 5=3, x=7, o-{-x and y=5 ; what is its numerical value ? Ans. 5. ax^-\-b^ Ex. 7. In the expression ^^ , let flt=3, 5=5, c=2, 6a: — a'' — c x=6 ; What is its numerical value ? Ans. 7. Ex. 8. What is the value of a2x(a+6)—2a6c, where a=6, 5 = 5, and c=4 ? Ans. 156. Ex. 9. There is a certain" algebraic expression consisting of three terms connected together by the sign plus ; the first term of it arises from multiplying three times the square of a by the quantity b ; the second is the product of a, b and c ; and the third is two thirds of the product of a and b. Required the expression in algebraic writing, and its numerical value, where a=4, 5=3, and c=2 ? Ans. 176. DEFINITIONS. 38. A proposition, is some truth advanced, which is to be de- monstrated, or proved ; or something proposed to be done or performed ; and is either a problem or theorem. 39. A. problem, is a proposition or question, stated, in order to the investigation of some unknown truth ; and which requires the truth of the discovery to be demonstrated. 40. A theorem, is a proposition, wherein something is advanc- ed or asserted, the truth of which is proposed to be demon- strated or proved. 41. K corollary, ox consectary^ is a truth derived from some INTRODUCTION. 9 proposition already demonstrated, without the aid of any other proposition. 42. A lemma, signifies a proposition previously laid down, in order to render more easy the demonstration of some theo- rem, or the solution of some problem that is to follow. 43. A scholium, is a note, or remark, occasionally made on some preceding proposition, either to show how it might be otherwise effected ; or to point out its application and use. 44. An axiom, is a self-evident truth, or proposition univer- sally assented to, or which requires no formal proof. 45. As axioms are the first principles upon which all ma- thematical demonstrations are founded, I will point out those that are necessary to be observed in the study of Algebra, as there will be frequent occasion to advert to them. AXIOMS. 46. When no difference can be shown or imagined between two quantities, they are equal. 47. Quantities equal to the same quantity, are equal to each other. 48. If to equal quantities equal quantities be added, the wholes will be equal. Thus, if a=zh, then a-^-c—b-^-c ; if a — hz=.c, then adding h, a—h-\-h=zc-\-h, or a=:c-^h. 49.*If from equal quantities equal quantities be subtracted, the remainders will be equal. If a — 6, then, a—2=b—2 ; if Z>-4-c = a-{-c, then h=:a. 50." If equal quantities be multiplied by equal numbers or quantities, the products will be equal. Thus, if a=i&, 3a=35; if a=:-, 3a=6; if a=iJ, ca=c5 ; o and if a=zh, a X a=h X h, or a'^=zh'^. 51 . If equal quantities be divided by equal numbers or quan- tities, the quotients will be equal. Ihus, II 5a=106, -— =:— :r-, or a=2o : if ca=:co. — =z , 5 o c c ' or az=zb ; and if c^^=.ba, then — = — , or a^=z.b. a a Scholium. Articles (49), (50), (51), might have been de- duced from Art. (48) ; but they are all easily admitted as axioms. 52. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. Thus, if a=;c, then a-\-b=^c-^b^ and a + 6 — Z>=c-}-& — ^, or^a=c. 10 INTROrrCTION. This might be inferred from Art. (48). 53. If a quantity be both multiplied and divided by another its value will not be altered. Thus, ifa=6; then 3a =36 ,,..,. ^ ^ 3a 3b and dividmg by 3, — =— , or a=o. CHAPTER I. ON THE ADDITIOI^UBTRACTION, MULTIPLICATION, AND DIVISION OF ALGEBRAIC QUANTITIES. ^ 1. Addition of Algebraic Quantities. 54. The addition of algebraic quantities is performed by- connecting those that are unlike with their proper signs, and collecting those that are like into one sum; for the more ready effecting of which, it may not be improper to premise a (ew propositions, from which all the necessary rules may be derived. 55. If two or more quantities are like^ and have like signs j the sum of their coeffi,cents prefixed to the same letter, or letters, with the same sign, will express the sum of these quantities. Thus, 5a added to 7a is = 12a ; And --5a added to — Sa is = — 8a. For, if the symbol a be made to represent any quantity or thing, which is the object of calculation, 5a will represent five times that thing, and 7a seven times the same thing, what- ever may be the denomination or numeral value of a ; and consequently, if the quantities 5a and 7a are. to be incorpo- rated, or added together, their sum will be twelve times the thing denoted by a, or 12a. Moreover, since a negative quantity is denoted by the sign of sulJtraction : thus, if a+& = a—c, b=—c, and c=--i. A debt is a negative kind of property, a loss a negative gain, and a gain a negative loss. Therefore it is plain that the quantities,— 5a and— 3a will produce, in any mixed operation, a contrary effect to that of the positive quantities with which they are conne>ted ; and consequently, after incorporating them in the same man- ner as the latter, the sign — must be prefixed to the result , so that if A be greater than a, it is evident that 5 (a— a) -f- 3(a— a), or (5a— 5a) + (3A— 3a)r=:8A— 8a ; and therefore the sum of the quantities — 5a and — 3a, when taken in their iso-^ lated state, will, by a necessary extension of the proposition be =— 8a. 12 ADDITION. 56. If two quantities are like, hut have unlike signs, the difference of their coefficients, prefixed to the same letter, or letters, with the sign of^tliat which hath the greater coeffcient, will express the sum of those quantities. Thus -\-6a added to— 4a is = 4-2c. Ans. 20a;2— I5Z>c. Ex 6. Required the sum of 3x^-{-4x'^—x, 2x^-^x'^~3xj 7a;3-|-2a;2— 2a:, and4ar"*+2x2 — 3a:. Ans. 1 6a:H 9a:2— 9a:. Ex. 7. What is the sum of 7a^ — 3a^-{-2aP — 3b^, ah'^ — an — b'^-^Aa?,^bh'^^bab'^~4a'^b-{-Q>d\ and —aF'b~\-4ab'^-^ 453+a3? Ans. \8a^-9a^b+\2ab'^—l3¥. Ex. 8. Add together 2a;2y— a:+2, a;2y— 4a:+3, 4x2y— 3af + 1, and 5a;2y_7a;+7. Ans. 12a:2y-15a:4-13. 1 I Ex. 9. Required the sum of 30— 13a:^— 3a?y, 23~10a;^— 4acy, — 14a:^— 7a:y4-14, — 5a;y4-10 — 16a;^,and 1— 2a;^~a:y. Ans. 78 — 55a;^— 20a'y. Ex. 10. Add 3(a: + yf - 4{a—b)\ {x + yf - {a-b)\ — 7(a— 6)3-f.5(a;+y)2, and 2(x+yf-(a—by together. Ans. ll(a:+y)2 — 13(a-6)3. CASE II. When the quantities are like, but have unlike signs. RULE. • 60. Add all the positive coefficients into one sum, and those that are negative into another ; subtract the lesser of these Bums from the greater ; to this difference, annex the common letter or letters, prefixing the sign of the greater^ and the re- sult will be the sum required. ADDITION. 15 EXAMPLE 1, 7a:3— 3a:2+3a; — 4x^-\- x^ — Ax — c^—2x^+7x 9x^+6x'2—9x 3x^-5x^+6x — 5x^-\-3x^ — 6a: 9a;3 * —3a; In adding up the first column, we say 34-9-}-7 =+ 19» and ^(5 + 1+4) = — 10; then, +19 — 10=+9= the ag- gregate sum of the coefficients, to which the common quantity x^ is annexed. In the second column, the sum of the positive coefficients is 3 + 64-1 = 10, and the sum of the negative ones is —(5+2 + 3) = — 10; then, 10 — 10 = 0; consequently, (by Cor. Art. 56), the aggregate sum of the second column is nothing. And in the third column, the sum of the positive coefficients is 6 + 7 + 3 = 16, and the sum of the negative one is — (6 + 9+ 4) = — 19 ; then +16 — 19 = — 3 ; to which the common let- ter is annexed. Ex. 2 Ex. 3. dx"^ — 6a + 4a: — 3 4,ab-\-3xi/—2ax-{- c — 2x^'\- a — 9a;+7 — ab — x7/-\- ax— 5c 7a:2+7a+7a; — 1 5ab—2xy—7ax-i-7c — x^ — 3a—2x-i-3 — 4ai+ a:y+ ax-^- c + 3a;2+ a— 4a;+4 'iab—3xy+4.ax— c — 7a;2 — 4a+3a;— 5 — ab — xy — +%'+3& Ex.* 5. ax^-\-hx'^-\-cx ex^ — dx'^—fx • (a^e)x^+{h-d)x'^-\-{c-^f)x In Ex. 4. The sum of ax and hx, or ax-^-hx^ is expressed by {a-\-h)x ; the sum of -{-hy and -\-dyf or -\-by-\-dy, is = + {h^d)y, 3* 18 SUBTRACTION. In Ex. 5. The sum of arc^ and ea:^, or ax'^-\-ex^is ={a-\-e)x^; the sum of -\-bx^ and —dx^, ov-^-bx"^ —dx^, is =^{b—d)x''^ ; and the sum of -{-ex and —fx, or -\-ex-^fx, is =z-\-(c—f)x. Any multinomial may be expressed in like manner, thus ; the multi- nomial mx^-{-nx^—px''^—'qx'^ may be expressed by {m-^-n—p—q) a?2-; and the m-ixed multinomial ^x^y+^y^ — rxy-\-m7/^ — nxy, by (p—r — n)xy-\-{q-\-m)y'^ \ &c. Ex. 6. Add 2a2+y2_j_9^ Ixy—^ab—x'^, 4xy—y—9, and aj2y — xy-^-'Sx"^ together. Ans. 4x'^i-y^-\-lOxy—3ab—y-\-x^y. Ex. 7. Add together 72a^, 2ibc, 70xy, -18^2, and— 12ic. Ans. 54a--^VZbc-\-70xy. Ex. 8. What is the sum of 43xy, 7a;2, —I2ay, ~4ab, —3x^, End — 4ay ? Ans. 43xy-\-4x'^ — l6ay—4ab. Ex. 9. What is the sum of 7xy, —16bc, —I2xy ; 186c, and lixy ? Ans. 2bc, Ex. 10. Add together 5ax,~60bc, 7ax,-^4xy, —6ax, and — 326c. Ans. 6ax — 72bc—4xy. Ex. 11. Add 8a2a;2_3oa;, Tax— 5a:y, 9a;y— 5aa;, and a:y-f Zd^x^ together. Ans. l0a^x^—ax-\-5xy. Ex. 12. Add 2a;2— 3y2-f-6, 9a;y — 3aa:— a:2, 4y2_y— 6, and w^y—3xy-{-3xr^ together. Ans. 4x^-{-y'^-\-6xy — 3ax—y-{-xy. II 1 Ex. 13. A.M2x^—4x^-\-x^, 5x'^y—ab+x^, 4x^ — x^, and 2ac3— 3+2x2 together. J. 1 Ans. 4x^—x^-\-5x'^-^5x'^y—ab — x^ — 3. Ex. 14. Requiredthesumof4a;2+7(a4-^P,4y2— 5(.a + 6)2, and a^—4x^—3y^—(a-]-b)\ Ans. a3+y2+(a+6)2 Ex. 15. Required the sum of ax^—bx^-\-cx'^, bcx^—acx^^ c^x, and ax^-\-c—bx. Ans. aa:* — (b-\-ac)x^-{-{c-}-bc^a)x^ — {c'^+b)x-{-c. Ex. 16. Required the sum of 5a-f 36— 4c, 2a— 56+6c-|- 2d, a—4b—2c-\-3e, and 7a+46— 3c— 6c. Ans. 15a— 26— 3c+2(i— 3c. § II. Subtraction of. Algebraic Quantities, 62. Subtraction in Algebra, is finding the difference be- xween two algebraic quantities, and connecting those quanti- ses together with their proper signs ; the practical rule for performing the operation is deduced from the following propo' sition. 63. To subtract one quantity from another, is the same thing as to add it with a contrary sign. Or, that to subtract a post' SUBTRACTION. 19 five quantity/, is the same as to add a negative ; and to sub' tract a negative, is the same as to add a positive. Thus, if 3a is to be siibtracted from 8a, the result will be 8a — 3a, whi'ch is 5a ; and if ^ — c is to be subtracted from a, the result will be a—{h — c), which is equal to a~b-\-c : For since, in this case, it is the difference between b and c that is to be taken from a, it is plain, from the quantity b — c, which is to be subtracted, being less than b by c, that if b be onfj' taken away, too much will have been deducted by the quan- tity c ; and therefore c must be added to the result to make it correct. This will appear more evident from the following conside- ration ; Thus, if it were required to substract 6 from 9, the dif- ference is properly 9 — 6, which is 3 ; and if 6—2 were sub- tracted from 9, it is plain that the remainder would be greater by 2, than if 6 only were subtracted ; that is, 9 — (6 — 2) = 9 —6 + 2 = 3 + 2 = 5, or 9-6 + 2 = 9—4 = 5. Also, if in the above demonstration, b—c were supposed ne- gative, or b — c=—d ; then, because c is greater than b by d, reciprocally c—b = d, so that to subtract —d from a, it is ne- cessary to write a+rf. 64. The preceding proposition demonstrated after the man- ner of Gamier, Tl>us, if fi— c is to be subtracted from the quantity a ; we will determine the remainder in quantity and sign, according to the condition which every remainder must fulfil ; that is, if one quantity be subtracted from another, the remainder added to the quantity that is subtracted, the sum will be the other quantity. Therefore, the result will be a — b-\-c, because a—b + C+6 — c=ra. This method of reasoning applies with equal facility to com- pound quantities : in order to give an example ; suppose that from 6a— 3i+4c, we are to subtract, 5a — 5Z>+6c ; designating the remainder by R, we have the equality, R^-Sa— 5i+6c=6a— 3* + 4c: which will not be altered (Art. 49.) by subtracting 5a, adding 5b, and subtracting 6c, from each member of the equality ; therefore the result will be, R=6a— 35+4c— 5a+55— 6c, ^ or, by making the proper reductions, ^ R=a+25— 2c. 65. Another demonstration of the same proposition in La place's manner. 20 SUBTRACTION. Thus we can write, a=za-[-h—h .... (1), a—c=a—c-{-b — b .... (2) ; 80 that if from a we are to subtract -f-^ or — ^, or, which ii the same, if in a we suppress -{-b, or —b, the remainder, from transformation (1), must be a—b in the first case, and a-{-b in the second. Also, if from a — c we take away +^ or — b, the remainder, from (2), will be a — c — b, or a — c-{-b. • 66. Hence, we have the following general rule for the sub- traction of algebraic quantities. RULE. Change the signs of all the quantities to be subtracted into the contrary signs, or conceive them to be so changed, and then add, or connect them together, as in the several cases of addition. Example 1. From ISab subtract I4ab. Here, changing the sign of I4ab, it becomes — 14aJ, which being connected to I8ab with its proper sign, we have I8ab — I4ab=z{l8 — l4)ab — 4ab. Ans. Ex. 2. From lox^ subtract — lOoj^. Changing the sign of — IOj:^^ it becomes ■i-lOx'^, which being connected to ISx^ with its proper sign, we have ISx^-j- 10x^=z2dx'^. Ans. Ex. 3. From 24ab+7cd subtract ISab-^-lcd. Changing the signs of l8ab-{-7cd, we have —I8ab—7cd therefore, 24ab-\-7cd—l8ab—7cd=6ab. Ans. Or, 24ab-{-7cd — I8ab-7cd 6ab Ans. Ex. 4. Subtract7a—55+3aa' from 12a+ 10^4- 13aa:--3aJ, 12a+l0b+l3ax — 3ab\ Changing the signs of J > all the terms of 7a — 5b > — 7a-\-5b — Sax j •f 3aa: ; it becomes, 3 .-. by addition, 5a-{-l5b-i-l0ax—3ab. Ex. 5. From Sab— 7 ax -j- 7 ab-^ Sax, take 4ai— 3aa;— 4ay, Sab ^7 ax ^ 7ab-{-Sax Changinliie si^s of all l_^^j,^Sax+4xy the terms of 4ab—Sax—4xyf > ^ .♦.by addition, 6a5—aa?4-4a;y. Ans. SUBTRACTION. 21 Ex^ 6. From 36a— 125+7C Take 14a— 4+7c— 8 Rem. 22a— 8b+8 Ans. * In the above example, one row is set under the other, that is, the quantities to be subtracted in the lower line ; then, beginning with 14a, and conceiving its sign to be changed, it becomes — 14a, which being added to 36a, we have 36a — 14a=:22a ; also, — 46, with its sign changed, added to — 126, will give Ab — I2b={4 — I2)b=—Sb ; in like manner, 7c— 7c =0, and — 8, with its sign changed, = + 8. The following examples are performed in the same manner as the last. Ex. 7. Ex. 8. From 3a:— 4a4- 6 a+ b Take 2x-{-3a-7b a— b Rem. x—7a-{-8b '+2b Ex. 9. From 3ab—4:cx-\- y Take \ax^2x^ — 3y^ Rem. 3ab—\ax-\-y—^cx—2x?'Ar3'ip- Ex. 11. From bx"^ — 4a;y-{-5 Take 4x2— 4a:y4-9 Rem. —4 Ex. 10. 7x^-\-?,x^—x 6a;3_2a:2-i-8a; x'^-\-bx'^ — 9aj Ex. 12. 7a:2-8 9a;2+5a6-3a;3 3x3— 2a;2 — 5a6 _8 Ex. 13. From ax^ —bx'^ ■\- x Take px"^ — cx"^ -\- ex (a— p)x3 — (6— c)a;2+(l — e)a? Ex. 14. From bx^-\-qx'^^rx-^py'^ Take ax^ — cx"^ + mx — sy"^ (b—a)x^-\-(q-\-c)x'^—{r-\-m)x-\-(p-\-s)y'^ 67. As quantities in a parentheses, or under a vinculum, are 22 MULTIPLICATION. considered as one quantity with respect to other symbols (Art. 10,) the sign prefixed to quantities in a parentheses af- fects them all ; when this sign is negative, the signs of all those quantities must be changed in putting them into the pa- renthaees. Thus, in (Ex. 13), when —cx^ is subtracted from —hoc^, the result is — bx^-\-cx^, or — [h — c)x'^ : because the sign — pre- fixed to {b—c) changes the signs of b and c ; or it may be writ- ten -{-(c—b)x'^. Again, in (Ex. 14), when +ma; is subtracted from — rx, the result is — rx — mx ; and, as this means that the sum of rx and mx is to be subtracted, that negative sum is to be express- ed by — (rx-\-mx)z=z — {r-{-m)x. For the same reason, the multinomial quantity — my'^-\-n^y^ — aby^ — ry^-j-Gy^, when put into a parentheses, with a negative sign prefixed, becomes — {m — n'^-\-ab-{-r — 6)y". Ex. 15. From a — b, subtract a-\-b. Ans. —2b. Ex. 16. From 7a^y— 5y+3a;, subtract 3a:y+3y+3a;. Ans. Axy—Sy Ex. 17. What is the difference between 7aa;^4-5a:y—12ay -j-5^c, and 4ax^-{-5xy — 8ay — 4cd. Ans. Sax"^ — 4ay-\-5bc-{-Acd. Ex. 18. From 8x'^ — 3ax-^5, take 5x^-{-2ax-\-5. Ans. 3x^ — 5ax. Ex. 19. From a-f-i+c, take —a— b—c. Ans. 2a-\-2b-\-2c. Ex. 20. From the sum of 3a;3— 4aa?+3y2^ Ay'^-\-^ax—x^^ yi- — ax-\-bx'^, and 3aa: — 2x^ — y"^ \ take the sum of Sy^ — x^ 4-a;^, ax—x'^-\-A.x^, Zx"^ — ax — ^y"^, and ly'^ — ax-\-l . Ans. 4a;^-l-4aa;— 2y2— 5a;2— 7. Ex. 21. From the sum of x'^y'^—x^y-^xy'^, Oxy"^ — 15 — Sx'^y^, and 70-{'2x'^y'^—3x^y, subtract the sum of 5x^y^—20 ~\-xy^, Sx'^y — x'^y'^-\-ax, and Sxy"^ — Ax'^y'^ — Q + a-rr^. Ans. 2xy'^ — 7x'^y — ax — a'^x'^-\rS4. Ex. 22. From a^x'^y'^ — m'^x^ -f 3cx— 4x^ — 9 : take a^x^y^ —n^x^ + c'^x-\-bx^-i-3. Ans. {a^^a^)x^y^-{m^-n'^)x^+{3c-^c^)x-{4 + h.) a;2-12. ^ III. Multiplication of Algebraic Quantities. In the multiplication of algebraic quantities, the following propositions are necessary to be observed. 68. When several quantities are multiplied continually together, the product will be the same, in whatever order they are muU tiplied. MULTIPLICA^TION. 23 Thus, axh — bXa=ah. For it is evident, from the nature of multiplication, that the product contains either of the factors as many times as the other contains an unit. Therefore, the product ab contains a as many times as h contains an unit, that is, h times. And the same quantity ah, contains b as many times as a contains an unit, that is, a times. Consequently, axb = ba=: ab ; so that, for instance, if the numeral value of a be 12, and of 6, 8, the product ab will be 12x8, or 8x12, which, in either case, is 96. In like manner it will appear that abc=cah = bca, &c. 69. If any number of quantities be multiplied continually tO' getker, and any other number of quantities be also multiplied continually together, and then those two products be multiplied together ; the whole product thence arising will be equal to that arising from the continual multiplication of all the single quantities. Thus, ah x cd=a xbXcX d=abcd. FoT ab — axb, B,nd cd—cxd; if x be put =c(/, then ab X cd=abXoc=zaxbXoc ; but x is =cd=cXd, .-. abxx=abxc X d = aX h X cd—abcd. 70. If two quantities be multiplied together, the product will be expressed by the product of their numeral coe^cients with the several letters subjoined. Thus, 7ax5b=::35ab. For 7a is=:7xa, and 5b=5xb, .'.7ax5b=7 XaX5xb z=z7 X 5 X « X b = 35 X ab—35ab. 71. The powers of the same quantity are multiplied together by addi?ig the indices. Thus, to multiply a^ by a^, it is necessary to write the let- ter a only once, and to give it for an exponent the sum 2-f 3, the exponents of the factors; that is, a'^Xo^ — a^+^ — ^s • because a'^ — axa, and a^=a X « X a ; therefore a- X a^ — « X a XaXaXa = a^. In general, the product of a"* by a" , m and n being always entire positive numbers, is a'"+'» . In fact, a^ is the abbreviation of aXaXa, &c., continued to m factors, and a" is aXaX a, ut ce is equal to cX{a-\-b)=ca-\-cb, because e is equal to a-\-b ; therefore {a~\-b-\-d)xc = ca-{-cb 4-cJ. Also, if {a-\-b) — d be multiplied by c, the product will be ca-\-ch—cd; for let (a-f-^) = e, then (e — d)Xc=ce — cd=z c[a-\-b) — cd=ca-\-cb — cd. Finally, it may be demonstrated in like manner, that if any polynomial, a-\-b — d-\-e—f, &c., be multiplied by c, the pro- duct will be ca-\-cb — cd-\-ce — cf, &c. Also, if a quantity e be multiplied by any polynomial a-{-b — d-\-ej &c., the pro- duct will be ac-\-bc--dc-\-ec, &c. 75. If a compound quantity be multiplied by a compound quan^ tity, the product will be equal to every term of one factor, muU tiplied by every term of the other factor, and the products added together. Let, in the first place, a-\'b be multiplied by c-\-d'. a-\-h taken c times is ca-\-cb, as we have already proved ; but this product is too little by the binomial a-\-b repeated d times, it is necessary then to add to it da-\-db, and we will have ca-{-cb •\-da-\-db for the product sought; in which the term -{-db arises from the multiplication oi -\-bhy -\-d. Suppose, in the second place, that a^b is multiplied by c — d, the product will be ca-\-cb—da — db. Because the product of a-{-b by c, that is, ca-\-cb, is too great by that of a-\-b hy d, which is da-\-db ; we will have therefore the true product equal to ca-\-cb — da-mdb, where the term — db is the product of -\-bhy — d ; in multiplying c~d by a-\-b, we will find that — bd is the product of —d by -\-b. Let, in the third place, a — b he multiplied by c — d ; the product will be ca'—cb—da+db. For, the product of c— & by c, that is, ca—cby is too little by 4 26 MULTIPLICATION. that of a—b by d, which is da—db ; because the multiplier c is too great hy d \ it is necessary then to subtract the second product from the first, and the difference will be (66) ca — cb ~da-\-db. % Here the term -\-bd results from — 6 by — d. Finally, if a^b-\-e be multiplied by c-\-d the product will be ca~\-cb-^ce-\-ad-{-bd-\-de. For, in designating a-\-b by h ; then, {h-^e)x{c-\-d)=zhc-\- ec-{ dh-\-ed, which is equal to hx{C'i-d)-\-cc-\-ed=(a-^b)X [c'i-d)-\-ec-\~ed=ca-{-cb-jrce-{-ad'\-bd-\-del^ The same mode of reasoning may be extended to compound quantities composed of any number of terms whatever. 76. Cor. Hence, in general, if any two terms which are multiplied have different signs, their product must be preceded by the sign — , and if they have the same sign, the product is affected with the sign -j- ; agreeably to what has been de- monstrated (Art. 72.) where simple quantities, or isolated fac- tors, such as, 4- a, -j-^, — «, — b, were only considered. From the division of algebraic quantities into simple and compound, there arises three cases of Multiplication : the practical rules for performing the operation are easily deduced from the preceding propositions. CASE 1. When the factors are both simple quantities. RULE. 77. Multiply the coefficients together, to the product suo- join the letters belonging to both the factors, and the result, with the proper sign prefixed, will be the product required. Ex. 1. Ex. 2. Ex. 3. Ex. 4. Multiply Sab 5x — 6y —Aa^ By 4c — 3a +3x —6x^ Product I2abc Ex.«. 2ax —Sax — 15a.T — 18a;y Ex.7. xY -Ixy —7xY '\-24a^x^ Mul. By Ex. 6. —Sa^c + 5ac2 Ex. 8. '-5a^^c -4a^^-x Pro. - -I6a^x^ — ISaV +20a*b^cx MULTIPLICATION, 27 Ex. 9. Required the product of Aahc and 2a^c. Ans. i2c^bc\ Ex. 10. Required the product of —laxy and —2acx. Ans. -\- ^'^o.^cx^y Ex. 11. Required the product oUx'^y^ and — Sy^a;-*. Ans. — 21a;^y''. Ex. 12. Required the product of a^ and —a^. Ans. — a^. Ex. 13. Required the product of aa?^ and hx'^z. Ans. ahx^z^. Ex. 14. Required the product of — xyz and ahc. Ans. — ahcxyz. Ex. 15. Required the product of —^h'^cd?' and —2a^hc^d. Ans. 8a3^*3c3+3^>2 by a2^,2. Ans. 3a452— 2a3&3+3a2K Ex. 10. Multiply a2a;2_Qa;_|_9by5. K\i%.^d^x'^ — bax'\-\^. Ex. 11. Multiply 2c(f—3a2>— 3 by 4ac. Ans. 8ac2c;— 12a2ic— 12ac. Ex. 12. Multiply 7a7-3 + 3c&— 5y2 ^y —xy. Ai\s. — 7x'^yz—3abxy-{-5xy^. Ex. 13. Multiply a+Z> — c—cZ by aSc-.a&+3)a;2+(6p— 3a)a?+3;) Ex. 14. Mult, ax'^— bx -^-c by x^ — ex -\-\ 1st. ax'^-^ bx^-\- cx^ 2nd. — acx^+bcx^ — c^x 3d. + oa;^ — bx-{-c prod. ax^--(b + ac)x^~\-(c-]-bc+a)x^—(c^-{-b)+c Ex. 15. Required the continual product of a+2a?, a— 2ap, and a2+4a:2. 32 MULTIPLICATION. Multiply a-\-2x by a—2x a^-\-2ax —2ax- 4x^ Multiply by a2_4a:2 a2+4a;2 a4-.4a2a;2 +4a2a;2. -16a;4 Total product a* * -16a;* It may be necessary to observe, that it is usual, in some cases, to write down the quantities that are to be multiplied together, in a parentheses, or under a vinculum, without per- forming the whole operation ; thus, {a-{-2x) X ia—2x) x {a^-\- Ax"^). This method of representing tlie multiplication of com- pound quantities by barely indicating the operation that is to be performed on them, is preferable to that of executing the entire process ; particularly when the product of two or more factors is to be divided by some other quar»tity ; because, in this case, any term that is common to both the divisor and dividend may be more readily suppressed ; as will be evident, from various instances, in the following part of the work. Ex. 16. Required the product of a-f-^ + cby a — b-\-c. Ans. a2_|_2ae — Z>2-|-c2. Ex. 17. Required the product of xH-yH-2^ by a?— y—;^. Ans. x'^ — \p-~2yz—z^. Ex. 18. Required the product of I— a;+a;2— a:^ by l+a;. Ans. 1— a:*. Ex. 19. Multiply a3 4- 3a25-j- 3^524- 53 by a2 4. 2a6-hR Ans. a5^5a*6+10a352_j_i0a2^3_^5g^4_|_^5, Ex. 20. Multiply 4a;2y + 3 j:y — l by 2a;2— a;. Ans. 8a;*y+2a:3y— 2ar2 — 3a:2y+a;. Ex. 21. Multiply a:3-}-a;2y4-iry2-{-y3by a?— y. Ans. x^—y^. Ex. 22. Multiply Zx^—2cP-x^-^'id^ by 2x^ — ^a^x'^-\-ba?. Ans. 6x« — 13a2x5+6tf%*+21fl3x3— 19«^a;2H-15a6. Ex. 23. Multiply 2a2—3aa; + 4x2 by 5a2_ 6arc— 2x2. Ans. lOfli— 27a3a:+34a2a;2— I8ax3— 8x*. Ex. 24. Required the continual product of a-\-x, a— x, c? 4-2aa;-|-a2, and c^ — 2ax-f-a;2. Ans. a^ — 3a'*x24-3a2ar*— x®. Ex. 25. Required the product of «^— aa2-f-Jx— c, and o? — 2a:+3. DIVISION. 33 Ans. a;5-(a+2).r*+(&+2a+3)a;3-(c4-2i + 3a)x2+(2c4.3*) x—2c. Ex. 26. Required the product oi' mx^—nx — r and nx — r. Ans. mnx"^ — (n?--{-mr)x'^-\-r^. Ex. 27. Required the product of px"^ — rx-^-q and x"^ — rx — q. Ans. px^ — {r-\-pr)x^-\-{q-\-r^^pq)x'^—q^. Ex. 28. Multiply3a;2— 2a;y + 5 by a:2-f2a:y— 3. Ans. 3a;'*+4T3y_4a:2 X {\^-y'^)+lQxy—\b. Ex. 29. Multiply a^ + 3a^b + 3a&2 4. 53 by a^-^a^h + 3ab'^—¥. Ans. a^— 30*62 _f-3a254_^6, Ex. 30. Multiply 5a3—4a254-5aZ>2__3^,3 by 4a2_5a^,4.2i^2. Ans. 20a5— 41a'*^>+50a362— 4.5a2J3^25ai*— 655. Ex. 31. Required the continual product of a-\'X, a'^-\-2ax 4-a;2, and a'^-\-^a'^x-\-'^ax'^-\-x^, » Ans. a6-f6a^''x+15a%2_^20a3a;3-f-l5a2a;4-f 6aa;5+a;6. Ex. 32. Required the continual product of a — ar, a^ — 2ax-\- r2 and a^ — 3a2a;4-3aa;2 — x^. Ans. a^—6a^x-{-i5a*x^—20a^x^+\5a*x^—6ax^+x^. ^ IV. Division of Algebraic Quantities. 80. In the Division o( a.]gehrB.ic quantities, the same circum- stances are to be taken into consideration as in their multipli- cation, and consequently the following propositions must be observed. 81 . If the sign of the divisor and dividend be like, the sign of the quotient will be -\- ; if unlike, the sign of the quotient will be — . The reason of this proposition follows immediately from mul- tiplication. Thus, if +ax+b=z+ab ; therefore ^tf_=-f 5 -\-aX—b=—ab; .-. ^^^=—b —aX-\-b=—ab; /. = + J — a ^aX-b=+ab; .-. ±^=_J — a 82. If the given quantities have coefficients, the coefficient of the quotient will be equal to the coefficient of the dividend divided by that of tJie 'divisor. Thus, 4a6-^25, or ~=2a. n, th^exponent of the quotient is m—n ; thus, a^-^a^=a^-^=za^j and a^-^a=a^~^=a^. Also, it may be demonstrated in like manner, that (a+a)*-:- (a + x)^=(a + x)^^=(a-\-xy ; andA^^^3 = (2a:+yr-«= (2x+yf. In the second case, where m=n,yve shall have, From which there results between the *xponents the equality, m = 772Hi- a?, and subtracting m from each of these equals (Art. 49), m—m=.x, or j:=rO .... (2) ; therefore, the exponent of the quotient will be equal to 0, or axz=^a°^ a result which it is necessary to explain. For this purpose, let us resume the division of a"* by a'", which gives unity for the quotient, or — =1 ; and as two quotients, aris- ing from the same division, are necessarily equal ; therefore, a« = l. . Hence, as a may be any quantity whatever, we may conclude that ; any quantity raised to the power zero, must he equal to unity, or 1, and that reciprocally unity can he translated into a°. This conclusion takes place whatever may be the value of a ; which may also be demonstrated in the following man- ner. Thus, let a'>—y\ then, by squaring each member, 0^X0"= yXy, ora°^y2. therefore, (47), y'^^y^ and (51),-^=^, ^ y y ory-1; but a°=;y ; consequently a°=:l. In the third case, where m is less than n ; l^^=m+£?, d being the excess of n above m ; we shall alwsjHpave, and equalising the exponents, because the preceding equality cannot have place, but under this consideration, m=:m-\-d-\-x, subtracting m-\-d from both sides, the final result will be 86 DIVISION. x=—d (3); then the quotient is a-^ . In order to explain this, let us resume the division of a*" by a", or by a^^^zrza^xa^ \ by suppressing the factor a'", which is common to the dividend and divisor, according to what has been demonstrated with regard to the division of letters (Art. 84), we have for the quotient -^ : therefore, °^=i (4); This transformation is very useful in various analytical operations ; in order to see more clearly the meaning of it, we may recollect that a^^ is the same as aXaXa, &c., con- tinued to d factors ; therefore, according to the acceptation and opposition of the signs, a-<^ must represent aXaXa &c., continued to d factors in the divisor. Hence, according to the results (1), (2), and (3), the pro- position is general, when m and n are any whole numbers whatever; thus, a^4-o^=a^~^=a-2, or -^ : because the di- visor multiplied by the quotient is equal to the dividend, a^ X 1 a^ a-'^=a^-^=a^= the dividend, and-^Xa^= ~-r=a^-^=a^= (P- a?' , • 1 the dividend, therefore, a-2=— -. In like manner it may be shown that, — =a-3, — = a-*, &c. But, according to the result (4), in general, — •=:a-<^, where d may be any whole number whatever ; hence the method of notation pointed out, (Art. 32), is evident. 87. If a compound quantity is to be divided by a compound quantity, it frequently occurs that the division cannot be per- formed, in which case, the division can be only indicated, in representing the quotient by a fraction, in the manner that has been already described (Art. 8). 88. But ifltjjk of the terms of the dividend can he produced hy multiply i^T the divisor hy any simple quantity, that simple quantity will be the quotient of all those terms. Then the re- maining terms of the dividend may he divided in the same manner, if they can he -produced hy multiplying the divisor hy any other simple quantity ; and by continuing the same DIVISION. 37 method, until the whole dividend is exhausted ; the sum of all those simple quantities will be the quotient of the whole com- pound quantity. The reason of this is, that as the whole dividend is made up of all its parts, the divisor is contained in the v^^hole as often as it is contained in all its parts. Thus, (ah-\-cb-\-ad-\-cd)-r' (a-\-c) is equal to b-{-d: Yox bx{a-\-c)=zab-\-cb\ and dx(a-\-c)=ad-\-cd] but the sum oi ab-\-cb and ad-\-cd is equal to a6+c^-f arf-j-cc?, which is equal to the dividend ; therefore b-^d is the quotient re- quired. Also, (a24-3a5 + 2i2)4.(a+J) is equal to a+2^». For, it is evident in the first place, that the quotient will include the term a, since otherwise we should not obtain a^. Now, from the multiplication of the divisor a-\-b by o, arises a'^-\-ab ; which quantity being subtracted from the dividend, leaves a remainder 2ab-\-2b'^ ; and this remainder must also be divided by a-^-b, where it is evident that the quotient of this division must contain the term 2b : again, 2b, multiplied by a-\- b, produces 2ab-\'2b^ ; consequently a-\-2b is the quo- tient required ; which, multiplied by the divisor a-^b, ought to produce the dividend a^-f 3a6+262. See the operation at length : a-{-b)a^-^3ab-{-2P{a+2h a^4- ob 2ab+2b^ 2ab-j-2b^ 89. Scholium. If the divisor be not exactly contained in the dividend ; that is, if by continuing the operation as above, there be a remainder which cannot be produced by the mul- tiplication of the divisor by any simple quantity whatever ; then place this remainder over the divisor, in the form of a fraction, and annex it to the part of the quotient already de- termined ; the result will be the complete quotient. But in those cases where the operation will not terminate without a remainder ; it is commonly most convenient to ex- press the quotient, as in (Art. 87). 90. Division being the converse of multiplication^ it also ad- mits of three cases. 38 DIVISION. CASE I. When the divisor and dividend are both simple quantities, RULE. 91. Divide, at first, the coefficient of the dividend by that of the divisor ; next, to the quotient annex those letters or factors of the dividend that are not found in the divisor ; finally, pre- fix the proper sign to the result, and it will be the quotient re- quired. Note. Those letters in the dividend, that are common to it with the divisor, are expunged, when they have the same ex- ponent ; but when the exponents are not the same, the expo- nent of the divisor is subtracted from the exponent of the dividend, and the remainder is the exponent of that letter in the quotient. Example 1. Divide ISax^ by Sax. 18ax2 18 a x^ ^ , o , ^ 3ax Sax Or, — =:—xa^-^Xx^-^ = exa'>Xx=ex. See (Art Sax 3 ^ 86.) Ex. 2. Divide —48a^^c^ by I6ahc, In the first place, 484- 16 = 3= the coefficient of the quo- tient, next, a'^b^c'^-^abcz=a^~^ Xb'^~^ Xc^~^=:abc ; now, an- nexing aic to 3, we have Sabc, and, prefixing the sigr» — ; be- cause the signs of the dividend and divisor are unlike ; the re- sult is —Sabc, which is the quotient required. Or, the operation may be performed thus, —48^.262^2 48 a2 52 c2 — --7 = — ■r^X~X-r-X--=—3XaXbxc= —Sabe. Idabc 16 a c Ex. 3. Divide — 21a;3y%* by —Ix^yH'^. — 21a;V;g* _ 21 —Ix^y'^z^ ~"^ 7 Ex. 4. Divide 280^*^ by — 70252^5. 28 fl* W c^ 7 a^ b^ c^ Xc'-5=:_4xa2X^>3xc2=-4a2^V. In order that the division could be effected according to the above rule ; it is necessary, in the first place, that the divisor contains no letter that is not to be found in the dividend : in the second place, that the exponent of the letters, in the divi- 2 3 - +— a;3-2 X y3-2 x z^-^ = -\-3xi/z. DIVISION. 39 sor, do not surpass at all that which they have in the dividend ; finally, that the coefficient of the divisor, divides exactly that of the dividend. When these conditions do not take place, then, after can- celling the letters, or factors, that are common to the dividend and divisor ; the quotient is expressed in the manner of a frac- tion, as in (Art. 84). • Ex. 5. Divide 48a^b^c^d by Gia^^c^e. The quotient can be only indicated under a fractional form, thus, 48a^^c^d 64a^Ve* But the coefficients 48 and 64 are both divisible by 16, sup- pressing this common factor, the coefficient of the numerator will become 3, and that of the denominator 4. The letter a having the same exponent 3 in both terms of the fraction, it follows that d-^ is a common factor to the dividend and divisor, and that we can also suppress it. The exponent of the letter b is greater in the dividend than in the divisor ; it is ne- cessary to divide b^ by 6^, and the quotient will be b^^ or 55 —-=b^—^=b^f which factor will remain in the numerator. With respect to the letter c, the greater power of it is in the denominator ; dividing c* by c^, we have c^, or —=^c*~^ z=.r?^ therefore the factor c^ will remain in the denominator. Finally, the letters d and e remain in their respective places ; because, in the present state, they cannot indicate any factor that is common to either of them. By these diflerent operations, the quotient, in its most simple ♦orm, is — — . 'Note. The division of such quantities belongs, properly speaking, to the reduction of algebraic fractions. Ex. 6. Divide "iGx^y^ by 9a?y. Ans. 4yy. Ex. 7. Divide 30a^Ay2 by —(Sahy. Ans. —bay. Ex. 8. Divide —^^c-x^y by Ic^x^. Ans. —Gcxy. Ex. 9 Divide — Aax'^y^ by — axy"^, Ans. -{-^xy. Aa'^b"cx Ex. 10. DWidiQ \Qa^¥ ex hy —Aa^bdy. Ans. 7 . EiP 11. Divide — ISo^^V by \2a^¥x. Ans. — — ^. Ex. 12. Divide llxyzw^ by xzyw. Ans. 11 w. Ex. 13. Divide —\'2aWc^ hy —6abc. Ans. 202^2^2. 40 DIVISION. Ex. 14. Divide — ^x^xp-z^ by x^y^'s^. Ans. 2~2~2* Ex. 15. Divide 39a9 by ISa^. Ans. 3a*. CASE II. When the divisor is a simple quantity y and the dividend a (?&m- pound one. RULE. 92. Divide each term of the dividend separately by the simple divisor, as in the preceding case ; and the sum of the resuhing quantities will be the quotient required. Example 1. Divide \Sa^'\-2a^b-\-Qab'^ by 3a. 18a3 ^ , ^d^b , ^ 6ai2 Here, -=6a2, — a&, and ——=2*2 ; 3a 3a 3a therefore, ■8.^-+3a^i+6a^»^g^ ,_ 3a Ex. 2. Divide 20a2a;3 — 12a2a;2+8a3x2—2a*a;2 by 2aaj». 20ff2j:3 Here, _— --=10aa:, -12aV-r2aa;2= — Qa, 8aV-^2aa:2 2aa3^ = 4 4a2, and '-2a^x'^^2ax'^=i — a^ ; 20a2a;3— 12a2a:2 4-8a3a:2_2a*a;2 ,^ hence — =10aa: — 6a4-4a2— as^. 2aa;2 Ex. 3. Divide 20a2a; — 15aa;2-{-30axy2 — 5ax by 5ax. Here 20a'^x-^5ax=4a, —I5ax^-^5ax= —3x, SOaxy^-r- 5ax = 6y'^, and — 5ax-T-5ax=z — 1 ; , . 20a2x — 15aa72-|-30aajy2_5aa; „ . ^ o , therefore, -^ ^ =4a— 3a; + 6y2_l. Oax Ex. 4. Divide 5a^x—25a^x^+50a*x^ — 50a^x^-^25a^x^-~ 5ax^ by 5aa;. Here — — =a5, — = — 5a%, -^ = + lOa^^s oax oax oax -50a^x^ ^. , 3 +25a2x5 -Sarc^ — = — lOa'^a?-^, — = -\-5ax*, and — =— x* ; oaa? oax oax therefore, a^— Sa^a^-l-lOa'-^x^ — 1 Oa^oj^ + 5aa;* — x^ is the quo- tient required. Ex. 5. Divide 3a*a;2— 3a2a;* by — 3a2a;2. Ans. #— a'. Ex. 6 Divide 21a3a;3-7a2a;2-.14aa; by 7 ax. Ans. 3a2a;^— ax--2. DIVISION. 41 Ex. 7. Divide I2abc — ASax'^y^ + 64a26V _ iQa-b^ by 3c 3a:"v^ — 1 6a5. Ans. <>- O- + + 00 00 a a CJi Ol + 4- 1 1 t— • 1— 1 o o ►— 05 ^ ^ OS a ^. ^ *r- C> -J) ^ i" 1 1 " 1 ►f*- 4^ »^ ^ Cr- O" Cr- *■ *. + + + 00 00 00 a a a ^5 ^^ ^3 O CN C?- o> a> o> a a t3 60 60 a a + + I 60 •» Or a ^§ + The sign of the first term 5a'' of the dividend being the same as that of 5a*, the first term of the divisor, the sign of the first term of the quotient is +i which is omitted (Art. 14). Dividing 5a' by 5a*, the quotient is a^, which is written under the divisor. Muhiplying successively the three terms of the divisor by the first term a^ of the quotient, and writing the product under the corresponding terms of the dividend ; sub- tracting 5a'^ — 2a^b-\-4a^b^Apm the dividend, the remainder is —20a^+Sa^b^'-6a*P—4a^b*-^8a^^. Dividing —20aH the first term of this new dividend by 5a\ DIVISION. 43 the result will be —Aa^h, this quotient having the sign — , because the dividend and divisor have different signs Multiplying all the terms of the divisor by —Ad^b ; we hav«» — 20a%-\-Sa^h'^ — IQa^b^ ; subtracting this result from the par* tial dividend, the remainder will be lOa^h^ — Aa^b^-^-Sa^h^, divid- ing the first term of this new partial dividend, 1 0a*A2, by the first term 5a* of the divisor, multiplying all the divisor by the result +2A^, and subtracting the product from the last partial dividend, nothing remains ; therefore the last term of the quotient sought is +26^, and the entire quotient is a^ — Aa?'b-\-2b^. 94. It is very proper to observe that in division, the multi- plications of different terms of the quotient by the divisor, produce frequently terms which are not found in the dividend, and which it is necessary to divide afterward by the first term of the divisor. These terms are such as are destroyed when the dividend is formed by the multiplication of the quotient Hnd divisor. See a remarkable example of these reductions : Ex. 2. Divide a^—b"^ by a—b. Division. Dividend, a^-a^b a^b-- P a^b^ ab^ ab-^ -b^ ab'' -63 Divisor, a-b Quotient. a^ + ab + b^ Multiplication. Mul. a — b by a^-\-ab+b^ a^-a^b -{-a^b—ab^ -\-ab^- P -6' The first term a^ of the dividend divided by the first term a of the divisor, gives a^ for the first term of the quotient ; multiplying the divisor a — b by a^^ the first term of the quotient, the result is a^ — a^ ; subtracting a^—a^b from the dividend, the term a^ destroys the first term of the dividend ; but there remains the term ^-a^b, which is not found at first in the divi- dend ; therefore the remainder is a^b — b^. Because the term a^b contains the letter a, we can divide it by the first term of the divisor, and we obtain -^-ab^ which is the second term of the quotient. Multiplying the divisor by +ai, the product is a^b-^ab"^, which being subtracted from a^b—b^ ; the first term a^b destroys the term a^b which arose from the preceding operation ; but there remains the i€ifa — oJ^, which being not yet in the dividend ; the remainder is therefore ab^—b^. Dividing ab^ by a, the result is b\ which is the third teim 44 DIVISION. of the quotient ; multiplying the divisor by 5^, we have ab^—P ; and subtracting this result from the last remain- der, the terms of both destroy one another ; so that nothing remains. In order to comprehend well the mechanism of the division, it is only necessary to take a glance at the multiplication of the quotient a^-^ab-{-b^ by the divisor a — b, and it will be rea- dily seen that all the terms reproduced in the partial divisions are those which destroy one another in the result of the mul- tiplication. Ex. 3. Divide y^—l by y— 1 Dividend. «/3 «/2 Divisor. y-\ y y y^-y Quotient. y- y- -1 -1 • Ex. 4. Divide a*— a;^ by a—x. Dividend. Divisor, a6-.x« a—x "■" " "'" ■■■'— • Quotient. a^x-x^ a'x—a^x^ a*a,2-a^6 a^x^^a^x^ a^x^-x^ a^x^-a'x*- a^x*-x^ a^x*—ax^ ax'-x^ Ex. 5 Divide sc^+a^ by «+a. DIV Dividend. a:5 + aa;* ISION . Divisor. x-^-a Quotient. xi—ax^-\-a -ax^^a^ ^ax^—a'^oc^ —a^x^-^-a^ — a?x'^ — a'^x 45 •a^oj+o* 95. When we apply the rule, (Art. 93), to the division of algebraic quantities of which one is not a factor of the other, we know it is impossible to effect the division ; because that we arrive, in the course of the operation, at a remainder, of which the first term cannot be divided by that of the divisor. In this case, the remainder is made the numerator of a frac- tion whose denominator is the divisor ; and the fraction thus arising, with its proper sign, is annexed to the other part of the quotient, in order to render its value complete. Ex. 6. Divide a'^-ya:^b^2h^ by a^-^b'^. Dividend. Divisor. a^ + a^b^2h^ a^+ab"^ 1st rem. a?b—ab'^+2b^ a'b-^-b^ Quotient. a + b+ ab^ d'-\-b'^ 2d rem. —ah'^-\-b'^ The first term — a}p- of the remainder, cannot be divided by c^, the first term of the divisor ; thus the division terminates at this point. The fraction — -r^ — tj— , having the remainder for its numerator, and the divisor for its denominator, is an- nexed to the partial quotient a-\-b ; and the complete quotient 96. It is necessary to remark, that the operation of divi* 46 DIVISION. sion may be considered as terminated, when the highest pow- er of the letter, in the first or leading term of the remainder, by which the process is regulated, is less than the first term of the divisor ; as the succeeding part of the quotient, after this, would necessarily become fractional ; and which may be carried on, ad infinitum, like a decimal fraction. This subject belongs to algebraic fractions, and as it is of considerable importance in analysis, we will treat of it with a near attenlion in the next Chapter. 97. In the preceding examples, the product of the first term of the quotient by the divisor, is placed under the dividend ; then the reduction is made by subtraction ; and every succeed- ing product is managed in like manner. In the following ex- amples, the signs of all the terms of the product are changed in placing it under the dividend ; and then the reduction is performed by the rules of addition ; which is the method adopted by some of the most refined Analysts. Ex. 7. Divide fl^+Sa^^'-^+i^—c* by aH ^2+^2. Dividend. a^-\-2a^b'^-\-b^'-c^ 1st. rem. 2d. rem. a'JP-- ,1r1 + 6^. Divisor. Quotient. a2^b'^—c^ .aZb^—b^c^—b^ + aV+52c2+c* Ex. 8. Divide 6a;*— 96 by 3a: — 6. Dividend. Divisor. 6x^ — 96 + 12x3— 96 — 12x3 + 24x2 3x-6 Quotient. 2x3 + 4x2+8x+I6 +24x2—96 — 24x2+48x 48x — 96 48x— 96 DIVISION. 47 Ex. 9. DivideSa^— 4a352+4a3_j_2a3_52_|.iby2a3— 62+1. Dividend. D ivisor. ^a6_4a3^2_|_4a3_|.2a3_62^1 •3a6-f4a362_4a3 2a3_j2 4.i .2a3 -1-62—1 203-62+1 Quotient. 4a3 + l 98. The division of algebraic quantities can be sometimes facilitated by decomposing, at sight, a quantity into its fac- tors ; thus, in the. above example, the divisor forms the last three terms of the dividend, it is only necessary to seek if it be a factor of the first three ; but those have visibly for a common factor 4a^, for 8a'^-'4aW-i-4a^ = 4a^x(2a^—b^i-l), By this observation, the dividend will become or (2a3--62-f-I)x(4a3+l): therefore the division is immediately effected, by suppressing the factor 2a^—b^-\-l equal to the divisor, and the quotient will be 4a3 + l. Experience, in algebraic calculations, will suggest a great many remarks of this kind, by which the operations can be frequently abridged. 99. It sometimes happens that, in arranging the dividend and the divisor according to the same letter, there occur seve- ral terms in which this letter has the same exponent : In this case, it is necessary to range in the same column those terms, observing to order them according to another letter, common to the two quantities. Ex. 10. Divide -.a*h^^b^c*— 0^0^—0^+ 2a*(^i-b^+2Mc^ ■\^a^^ by a'^—b^ — c\ Ordering the dividend according to the letter a, we will place in the same column the terms — a'^b^ and +2a'*c2, in another the terms +0^6* and —a^c* ; finally, in the last column t^je three terms +6^, -{-2b*c^, +6^6'*, ordering them according to the exponents of the letter 6 ; then the quanti- ties, so arranged, will stand thus : 18 DIVISION. Dividend. Divisor. 1st rem. —2a'b"+ar-h^+¥ 4- a4c2— a2c44-2Mc2 4- 62c4 -f-2a*J2_2a2M — 2a262c2 2_i2_c2 Quotie7it. — d^—2a^b'^—¥ 2d rem. +a^c'^— a^b^ -\-b^ —2aH^c^ + 2b^c^ 3d rem. — a^^* +56 — a2^>2c2+2Mc2 + &2c4 + ^254 _J6 -^4^2 4th rem. ^a^^c^-^b^c^ + b^c* J^aWc^—b^c^ -&2c4 * * Ex. II. Divide ffx* - (5+ac)a;3+(c4-Jc+a)a;2—(c2+5)a: 4-c by ax^ — bx-\rc. Dividend. Divisor. ax*—{b + ac)x^ + {c-\-bc-\-a)x'^-'{c^+b)x-{-c -ax* -\-bx^ — ca;2 —acx^-\-(bc-{-a)x^'^{c'^-\-b)x+c ■i-acx^ — bcx"^ -i-c^x ax^ — bx-\-c Quotient. a?2 — cx-{-i ax^ — bx-\-c •ax^-{-bx — c DIVISION. 49 100. The following practical examples maybe wrought ac- cording to either of the methods pointed out, (Art. 93, 97) ; but in complicated cases, the latter should be preferred. See Example 10. Ex. 12. Divide x^—x^-{-x^—x^-[-2x^l by x'^+x. — l. Ans. oc^—x^-^-x^ — x-f 1. Ex. 13. Divide a^ + 5a*x + l0a^x'^—l0a'2x^-\-5ax*—x^ by a^—Sa'^x+Sax^—x^. • Ans. a^ — 2ax-\-x^, Ex. 14. Divide 2x3— 19«2H-26a;-16 by a:-8. Ans. 2x^—3x-j-2' Ex. 15. Divide 483^3— 76ay2_64a2yH-105a3 by 2y — 3a. Ans. 24y2_2ay — 35a2. Ex. 16. Divide a2— 6^ by a— ^. Ans. a+5. Ex. 17. Divide a*— a:* by a^—x"^, Ans. a^-\-x^. Ex. 18. Divide a^ — b^ by a^+2a'^b + 2ab^-\-b^. Ans. a^—2a^b-\-2ab^-~b\ Ex. 19. Divide a^+a^b'^+b^ by a^—ab~\-b'^. Ans. a^i-ab+b\ Ex. 20. Divide 25a;6— a;*— 2x3— 8a;2 by 5x3 — 4x2. Ans. 5x3+4x2+3x+3. Ex. 21. Divide a^+4ab-{-4b-+c^ by a+2b. An8.a-{-2b + -^ a-{-2o Ex. 22. Divide8a*— 2^35 — 130=^^-30 J3 by 4a2+5ai4- 62. Ans. 2a^—3ab. Ex. 23. Divide 20a5—41a464-50a3i2__45a2i3^25a6*—66« by ta^ — 5ab-\-2b^. Ans. 5a3_4a2J^_5aJ2_3j3. Ex. 24. Divide a* + 8a3x+24a2a;2-j-32ax3+16x* by a-f2x. Ans. a3_|_6a2j:-f-l2ax2+8x3. E^ 25. Divide x4—(«—J)x3_|_(^_^^,^3j^2_^(^^__3^)^ 4-3J9 by x2— ax+p. Ans. x24-6x4-3. Ex. 26. Divide ax3-(a2_j_j)a;24.^2by ffa;— 6. Ans. x'^—ax—b. Ex. 27. Divide y6+«2y4^5Y2_^6_2J2y4_a4y2_2a462_ a26* by y*4-2a2y2_|_a4_^>2y2_^^2^2^ Ans. y2 — ^2- Z>2, Ex. 28. Divide 9x6 — 46x^ + 95x2+ 150x by x2—4x~5. Ans. 9x4 — 10x3.4-5x2— 30x. Ex. 29. Divide 6a'^-\-9a'^—l5a by 3^2— 3a. Ans. 2a2 4.2^4- 5. Ex. 30. Divide 2a*— 16a36+31a2i2_38a63^246*by2a2_- 3^+4^^. Ans. a^-^5ab-{-6b\ 6 50 GENERAL THEOREMS. Ex. 31. Divide a^ + Sa^x + 2Sa^x^ + SGa^ar^ -f 70a^x^ + 56a^x^+28a'^x^-\-8ax'^ -^-x^ by a^+4a^x-\-ea'^x'^-\-iax^-\-x^. Ans. a*-\-4a^x+6a^x^-{-4ax^-{-x*. Ex. 32. Divide ««— Ga'^ac-f 15a*a;2— 20a^a'3+i5a2a;4_6aa:S -\-x^ by a^ — 3o%+3aa:2— a:^. ^j^g a^^Sa^x-\-3ax^—x^ ^ V. iSomc General Theorems ^ Observations, &c. |» 101. Newton calls Algebra Universal AritJimetie. This denomination, says Lagrange, in his Traite de la Resolution des Equations numeriques, is exact in some respects ; but it does not make sufficiently known the real difference between Arithmetic and Algebra. Al^gebra differs from Arithmetic chiefly in this ; that in the latter, every figure has a determinate and individual value peculiar to itself; whereas the algebraic characters being ge- neral, or independent of any particular or partial signification, represent all sorts of numbers, or quantities according to the nature of the question to which they are applied. Hence, when any of the operations of addition, subtraction, &€., are to be made upon numbers, or other magnitudes, which are represented by the letters, a, h, c, &c., it is obvious that the results so obtained will be general ; and that any particular case, of a similar kind, may be readily derived from them, by barely substituting for every letter its real numeral value, and then computing the an^unt accordingly. Another advantageUllso, which arises from this general mode of notation, is, that while the figures employed in Arith- metic disappear in the course of the operation, the characters used in Algebra always retain their original form, so as to show the dependence they have upon each other in every part of the process ; which circumstance, together with that of representing the operations of add-on, subtraction, &c., by means of certain signs, renders both the language and al^rithm of this science extremely simple and commodious. Besides the advantages which the algebraic method of no- tation possesses over that of numbers, it may be observed, that even in this early part of the science we are furnished with the means of obtaining several general theorems that could not be well established by the principles of Arithmetic. 102. The greater of any two numhers is equal to half their sum added to' half their difference, and the less is equal to half their sum minus half their difference. Let a and b be any two numbers, of which a is the greater ; let their sum be represented by s ; and their difference by d. Then, GENERAL THEOREMS. 51 a-\-b=sl a^b=dS • .*. by addition, 2a:=s-\-d (Art. 48) ; s d and a—~-\-- (Art. 51) By subtraction, 2&=*— t/ (Art. 49) ; s d 2 • 2 ^ "> ;| and .-. * = o~o C^'^- ^0 2 •<& *Cor."l. Hence if tbe sum and difference of any two nUm bers be given, we can readily find each of the numbers ; thus, if s be equal to the sum of two numbers, and d equal to the s-^-d difference ; then the general expression for the first, is — — - , 2 and for the second 2 Whatever may be the numeral values that we assign to s and d^ or whatever values these letters must represent in a particular question, we have but to substitute them in the above expressions, ia order to ascertain the numbers required : For example. Given the sum of two numbers equal to 36, and the diffe- *rence equal to 8 : Then, by substituting 36 for ^, and 8 for d, in — - — and s~d ^ s-\-d 36-f8 44 „^ ^ s-^d 36—8 , we nave = — - — =-—=22, and 2 2 ' 2 2 2ft —-=14. So that, 22 and 14 are the numbers required. Cor. 2. Also, if it were required to divide the number s into two such parts, that the Jirst will exceed the second by d. It appears evident, that the general expression for the first P I /7 • g fi part is — - — , and for the second — - — ; s and d representing 2 2 any numbers whatever. s-\-d 103. The general expression — - — maybe found after the 2 manner of Gamier. Thus, let x represent the first part ; then according to the enunciation of the question, x—d will be the second ; and, as any quantity is equal to the sura of all its parts, we have therefore, x-\-x — dzzzSj or 2x — d:=:$. 52 GENERAL THEOREMS. This equality will not be altered, by adding the number d to each member, and then it becomes, 2x--d+d=s-\-d, or 2x=zs+d; dividing each member by 2, we have the equality, S= ; in which we read that the number sought is equal to half the sum of the two numbers s and d ; thus the relation between the unknown and known numbers remaining the same, the question is resolved in general for all numbers s and d. 104. We have not here the numerical value of the unl^nown quantity ; but the system of operations that is to be performed upon the given quantities ; in order to deduce from them, ac- cording to the conditions of the problem, the value of the quan- tity sought ; and the expression that indicates these opera- tions, is called a formula. It is thus, for example, that if we denote by a the tens of a number, and the units by b, we have this constant composi- tion of a square, or this formula, a'^-{-2abfh^ ; this algebraic expression is a brief enunciation of the rules ta be pursued in order to pass from a number to its square. 105. From whence we infer that, if a number be divided into any two parts, the square of the number is equal to the square of the two parts, together with twice the product of those parts. Which may be demonstrated thus ; let the number n be di- vided into any two parts a and b ; Then n=:a-\-b, and n=a-\-b; .-.by Muhiplication, n'^=a'^-{-2ab + b^ (Art. 50). 106. If the sum and difference of any two numbers or quan- tities be multiplied together^ their product gives the difference of their squares, observing to take with the sign — that of . the two squares whose root is subtracted.^ Let M and n represent any two quantities, or polynomials whatever, of which m is the greater; then (m + n)x(m — n) is equal to m^-— n^ ; for the operation stands thus ; (M + N)x(M~N)=:rM''+MN J =^2^^2 — MN— N^ S 107. When we put M = a^, and ^=zb^; then, (a3 4-i3)x(«3_j3)_^6_^,6 . (See Ex. 9. page 30). Where a^ is the square of a^, and b^ that of b^, and this last square is subtracted from the first. Reciprocally, the difference of two squares M^ — N^, can he put under the form (m-|-n)x(m— n). GENERAL THEOREMS. 53 This result is ^formula that should be remembered. 108. The difference of any two equal powers of different quanti- ties is always divisible by the difference of their roots, whether the exponent of the power he even or odd. For since x^ — a^ =a;-f a; X — a x^ — a'^ ^^ =a;2-f flta?4-a^ ; ^k X — a ^ aj* — (1% —x^-\-ax'^-\'a'^x-\'a^\ X — a =^x^-{-ax^-\-a'^x'^-{-a^x-\-a^ \ ( X — a x^ — a^ ■.x^-\-ax^-\-a^x^-\-a^x'^-{-a^x-\-a^ ; x—a We may conclude that in general, a;'" — a*» is divisible by a:— a, m being an entire positive number ; that is, X — a 109. The difference of any two equal powers of different quanti' ties, is also divisible by the sum of their roots, tohen the expo- nent of the power is an even number. For since x'^—a^ - — =x—a; x-{-a = a;^ — ax'^-\-a'^x — a^ ; x-^a &c. &LC Hence we may conclude that, in general, .a:2m-i_aa;-'"-2.4- . . -fa2»«-2a:—a2'«-i . (2). x-\-a 110. And the sum of any two equal powers of different quanti- ties, is also divisible by the sum of their roots, when the expo* nent of the power is an odd number. For since x-\-a z=x'^—ax^-{-a'^x'^-\-a^x+c^ x-\-a Hence we may conclude that, in general, ffl!!d!lf!!;!l=:a;2m_oa,2„-i+. .-02»u-la;4-a2«. (3). x-\-a 6* 54 GENERAL THEOREMS. 111. In the formulae (1), (2), (3), as well as in all others of a similar kind, it is to be observed, that if m be any whole num- ber whatever, '2m will always be an even number, and 2m -fl an odd number ; so that ^m is a general formula for even num- bers, and 2m-\-\ for odd numbers. 112. Also, if a in each of the above formulae, be taken =1, and X being always considered greater than a \ they will stand as follows : -—x'^'^-\-x'^-^^x^-^-\- +a:+l . . . (4). X — 1 ^ ' f_!!_Z__a;2m_l_a,2m 2j^^1m-Z^ . . . +3?— 1 . . . (5). _^-a;2m_a;2m_i_|.a,2m-2_, , .— a;+l . . . (6). 113. x\nd if any two unequal powers of the same root be taken, it is plain, from what is here shown, that x^—xn^ or a;"(a;'"-"— 1) (7), is divisible by rr— 1, whether m—n be even or odd ; and that a;"»— a;", or a;"(a;'"-"— 1) (8), is divisible by oj + l, where m—n is ai^ven number ; as also that a">+a;'*, or a:"(a:'«~"+l) (9), is divisible by a:-{-l, when m — n is an odd number. 114. It is very proper to remark, that the number of all the factors, both equal and unequal, which enter in the for- mation of any product whatever, is called the degree of that product. The product a^i^c, for example, which comprehends six simple factors, is of the sixth degree; this, d}}p-c is of the tenth degree ; and so on. Also, that if all the terms of a polynomial, or compound quantity, be of the same degree, it is said to be homogeneous. And it is evident from the rules established in Multiplication. that if two polynomials be homogeneous ; their product will be also homogeneous ; and of the degree marked by the sum of the numbers which desigjiate the degree of those factors. Thus, in Ex. 1, page 29, the multiplicand is of the fourth degree, the multiplier of the third, and the product of the de- gree 4 + 3, or of the seventh degree. In Ex, 12, page 31, the multiplicand is of the third degree, the multiplier of the third, and the prochict of the degree 3+3, or of the sixth degree. Hence, we can readily discover, by inspection only, the er- rors of a product, which might be committed by forgetting some one of the factors in the partial multiplications. CHAPTER II. ON ALGEBRAIC FRACTIONS. 115. We have seen in the division of two simple qtHntities (Art. 84,) that when certain letters, factors in the divisor, are not common to the dividend^ and reciprocally, the division can only be indicated, and then the quotient is represented by a fraction whose numerator is the product of all the letters of the dividend, not common to the divisor, and denominator, all those letters of the divisor, not common to the dividend. Let, for example, abmn be divided by cdmn ; then ahmn ah cdmn cd' It may be observed, that the fraction —z may be a whole number for certain numeral values of. the letters a, b, c, and d ; thus, if we^ had a = 4, 1 = 6^ c=i2, d=z3 ; but that, generally speaking, it will be a numerical fraction which can be reduced to a more simple expression. ^ I. Theory of Algebraic Fractions. 116. It is evident (Art. 103,) that if we perform the same opera' tion on each of the two members of an equality^ that is, upon two equivalent quantities or numbers, the results shall always be equal. It is by passing thus from the fractional notation to the al- gorithm of equality, that the process to be pursued in the researches of properties and rules, becomes simple and uni- form, n 117. Let therefore the equality be a=bxv (1). when we divide both sides b«|| which has no factor common with a, we shall have i=" (2)- Thus V will represent the value of the fraction -r, or the quo- tient of the division of a by 5. 56 ALGEBRAIC FRACTIONS. lis. If the numerator and denominator of a fraction be both mul- tiplied^ or both divided by the same quantity ^ its value will not be altered. For, if we multiply by m the two members of the equality (1), we will have these equivalent results, ma=mbxv (3) ; dividing both by mb, we shall have • ma but T=v > therefore ma a . mb=''=b ('''• m being any whole or fractional number whatever. 119. If the fraction is to be multiplied by m, it is the same whether the numerator be mulliplied by it, or the denominator divided by it. For, if we divide by b, the two members of the equality (3), we obtain the following, ma .. -^=mXv (5). The equality (1) may also be put under the form a=- bxmv (6), m whence we derive, dividing each side by -i, T^='wXv (7). m 120. If a fraction is. to be divided by m, it is the same whether the numerator be divided by m, or the denomi?iator mulliplied by it. For, from the equality (1), we deduce these fl) -=bx-,a=mbx- (9), ^ ' mm m dividing the first by b and the second by mbj in order to have — , they become ^ m (10)....==^;^-!;... .(11). ^ ' b m mb m ^ ' o It is to be observed, that in t, the numerator is — and the ALGEBRAIC FRACTIONS. 57 denominator b, and that we employ the greater line for se- parating the numerator from the denominator. 121. If two fractions have a common denominator, their sum will be equal to the sum of their numerators divided by the common denominator. For, let now the two equalities be (12) a=bXv\ a=bxv' (13), corresponding to the fractions a a' , which have the same denominator ; adding the two equalitio (12) and (13), we shall have a-{-a z:^bv-\-bv ^^b^v-^-W) ; and dividing both members by 6, in order to have the sum sought v\-v\ it becomes —r—^v-Yv' .... (14). Note. In adding the above equalities, the corresponding members are added ; that is, the two members on the left- hand side of the sign =, are added together, and likewise those on the right. The same thing is to be understood when two equalities are subtracted, multiplied, &;c. 122. If two fractions have a common denominator, their differ* ence is equal to the difference of their numerators divided by ^ the Common denominator. For, if we subtract the equality (13) from (12), we shall have a — a=bv — bv'' = b{v — v^) ; dividing each side by b, and we will obtain a — a' -J-^"-" (15)- 123 Let us suppose that the fractions have different de- nominators, or that we have the equalities azzzb . V, a'=zb' . v' ; we will multiply the two members of the first by b\ and those of the second by b, an operation which will give ab' = byv, a'bzzzhb'v' ; then adding and subtracting, we have ab'^ab — hb'(v^v'), ^-''^ "^r^^-^.a^ the double sign JL which we led^A plus or 7wmt/.y, indicating at '^^ the same time both addition and subtraction ; dividing eachy side by bb\ in order to find the sum and difference sought v-^v'j we will have .^ ., i- >« 58 ALGEBRAIC FRACTIONS. from whence we might readily derive the rule for the addi- tion and subtraction of fractions not reduced to the same de- nominator. 124. It would be without doubt more simple to have re- course to property (4) in order to reduce to the same denomi- ivator the fractions a a' but our object is to show, that the principle of equality is suf- ficient to establish all the doctrine of fractions. 125. We have given the rule for multiplying a fraction by a whole number, which will aho answer for the multiplication of a whole number by a fraction. Now, let us suppose that two fractions are to be multiplied by one another. Let the two equalities be a=zb . V, a'=:zb' . v^ ; multiplying one by the other, the two products will be equal ; thus, aa^=by . vv\ and dividing each side by bb', in order to have the product sought vv\ we will obtain i-' 0^); . ... Therefore the product of two fractions, is a fraction having for its numerator the product of the numerators, and for its de- nominator that of the denominators. 126. It now remains to show "how a whole number is to be divided by a fraction ; and also, how one fraction is to be di- vided by another. Let, in the first case, the two equalities be m=^m ; a=zb . v ; if we divide one by the other, the two quotients will be equal, that is, m m a bv ' and multiplying both sides by b, in order to have the expres- m sion — , we shall find ™*=^ (18). a V ALGEBRAIC FRACTIONS. n 59 Therefore, to divide a whole number by a fraction, we must multiply the whole number by the reciprocal of the fraction, or which is the sa/rie, by the fraction inverted Let, in the second case, the two equalities be a=:b . V, a' —b' . v' ; if the first equality be divided by the second, we shall have a b . V multiplying each side by y and dividing by b, for the purpose of obtaining the expression — „ we will arrive at "^^Cjx*: (19). • ab V b a' ^ ' Therefore, to divide one fraction by another, we must multiply the fractional dividend by the reciprocal of the fractional divisor^ or which is the same, by the fractional divisor inverted. 127. These properties and rules should still take place in case that a and b would represent any polynomials whatever. Accordincr to the transformation a-^-=—r, demonstrated (Art. 86), we can change a quantity from a fractional form to that of an integral one, and reciprocally. So that, we have -=bx-=bXar^ —ba-"^, -^=bx-T=h X a-^ z=z ba-^ , and a a a^ a"- a-2i-2^-2-- X-75X-75— -77?i5- In like manner any quan- (P- b^ d^ a^b^d^ tity may be transferred from the numerator to the denominator, and reciprocally, by changing the sign of its index : „, cP'b b bc"^ c-2 a-'^x~'^z-^ c'^y^ ' c^ ~~ a-'^c'^ ~ a-2 a-'^b-^ ' c-^b'^y-^ d^b'^x^z 128. If the signs of both the numerator and denominator of a fraction be changed, its value will not be altered. , — a -\-a a a a — b b — a b +6 b~'b' c — d d—c Which appears evident from the Division of algebraic quan- tities having like or unlike signs. Also, if a fraction have the negative sign before it, the value of the fraction will not be altered by making the numerator only negative, or by changing the signs of all its terms. 60 ALGEBRAIC FRACTIONS. Thus, -_=+ and -_=+-_=__. c-\-d c-f-d c + rf And, in like manner, the value of a fraction having a negative sign before it, will not be altered by making the denominator only negative : Thus, a — b a — b a — b c — d d — c d — c* 129. Note. It may be observed, that if the numerator be equal to the denominator, the fraction is equal to unity ; thus, if az=bj thenT=-=l : Also, if a is >5, the fraction is great er than unity ; and in each of those two cases it is called an improper fraction : But if a is <^b, then the fraction is less than unity, and in this case, it is called a proper fraction. § II. Method of finding the Greatest Common Divisor of two of more Quantities. 130. The greatest common divisor of two or more quanti- ties, is the greatest quantity which divides each of them ex- actly. Thus, the greatest common divisor of the quantities 16^2^2^ \2a^bc and 4abc^, is 4ab. 131. If one quantity measure two others, it will also mea- sure their sum or difference. Let c measure a by the units in m, and b by the units in n, then a=:mcj and b=nc; therefore a-i-b=zmc-\-nc=:(m-\-n)c; and a — b^=mc—nc={m—n)c; or a±6=(m±n)c ; consequently c measures a-{-b (their sum) by the units in m+n, and a—b (their difference) by the units in m — n. 132. Let a and b be any two numbers or quantities, where- of a is the greater ; and let p= quotient of a divided by b, and c= remainder ; q=^ quotient of b divided by c, and d= re- mainder ; r— quotient of c divided by d^ and the remainder —0 ; thus, b)a{p pb c)b(q qc d)c(r rd Then, since in each case the divisor multi- plied by the quotient joZ?/^ the remainder is equal to be dividend ; we have c=rd, hence qc—qrd (Art. 50) ; *b=qc-\-d=qrd-\-d={qr-\-\)d ; and pb=pqrd -\-pd = {pqr-\~p)d (Art. 61.) , a=pb-\- c=pqrd'\-pd-\-rd=.{pqr-\-p-k-r)d. ALGEBRAIC FRACTIONS. 61 Hence, since p, q, and r, are whole numbers or integral quantities^ d is contained in b as many times as there are units in qr-\-l, and in a as many times as there are units in pqr-[-p-{-r \ consequently the last divisor d is a common measure of a and b ; and this is evidently the case, whatever be the length of the operation, provided that it be carried on till the remainder is nothing. This last divisor d is also the greatest common measure of a and b. For let a? be a common measure of a and b \ such that a=mx, and b7=nx, then pb=:pnx ; and Cz=a — pb=mx — pnx=(m—pn)x, also dz=b~qc=.7ix—{qmx — qpnx)—-na—qmx -\-pqnx={n—qm-\-pqn)x ; (because qc=.qmx — qpnx) therefore i» measures d by the units in n — qm-\-pqn^ and as it also measures a, and 6, the numbers, or quantities a, b, and d have a common measure. Now the greatest common measure of d is itself ; consequently d is the greatest common measure of a and b. 133. To find the greatest common measure of three num- bers, or quantities, «, ^, c ; let d be the greatest common measure of a and Z>, and x the greatest common measure of d and c ; then x is the greatest common measure of a, 5, and c. For, as , and d have a common measure ; if d and c have also a common measure, that same number or q^Jhtity will measure a, ^, and c ; and if x be the greatest common measure of d and c, it will also be the greatest common measure of a, b, and c. And, in like manner, if there be any number of quantities ; B. The greatest common divisor of A and B, cannot exceed B ; it could be B itself, which we can readily know, if we perforni^e division of A by B, which gives ^=?+g- . . . . (1), q being the integral quotient, and R the remainder, if A is not exactly divisible by B. The fraction ^ being changed into q Tf R R 4- — cannot be reduced unless that .pr- or its reciprocal -pj- is B • B '^ R reducible, because q is an integral quantity which is always irreducible ; or B being > R, the quantity which ought to re- duce r— , cannot exceed R, it might be R itself, which we will R know in performing the division of B by R, which gives q^ being the integral part of the quotient, and R' the remain- der 3 and Aa^h—^aly^-^-h^, is required. Dividend. Divisor. 3a3-3a2Z»+ ah'^—h^ 4 12a3 — 12a25 + 4a&2— 4&3 12a3-15a2&-|-3a62 (4a25 — 5a62_^Z>3)^5_, 4a2 _5a6 +&2 Partial quot. 3a (3a2Z>+ a52_4Z^3)-ri = 3a2 + ah —452 4 12a24- 4a5— 1662 12a2— 15a6+ 362 19a6— 1962 Divisor. 4a'^—5abi-b^ Partial quot. 3. ALGEBRAIC FRACTIONS. 65 Dividend. Divisor. 4a2— 5ai+i2 4a^ — Aab ab + b^ (I9ab-i9b^)'r-l9b: a — b Quot. 4a — b Here the quantities are already arranged according to the powers of the letter a : the first is taken for a dividend, and the second for a divisor. In the first place, the factor b is found in every term of the divisor, and not in every term of the dividend ; therefore, the divisor is divided by the factor b, and the result is 4a'^ — 5ab-\-b'^ ; but the first term of this re- sult will not divide exactly that of the dividend, on account of the factor 4, which is not in the dividend ; the dividend is therefore multiplied by 4 in order to render the division of their first terms complete. Now, the dividend I2a^ — 12 a'^b-\- 4 ab"^ — 4b^ is divided by the divisor 4u?—bab-\-b'^, and the partial quo- tient is 3a. Multiplying the divisor by this quotient, and sub- tracting the product from the dividend, the remainder is 3a^b -|-a62_4^3^ a, quantity which, according to (Art. 135), must still have with '4a^ — bab-^-b"^ the same greatest common divisor as the first. Suppressing the factor b, common to all the terms of the remainder, or, which is the same, dividing the remainder by 5, and multiplying the result by 4, to render possible the division of its first term by that of the divisor, we have then for the dividend the quantity 12a24.4ai— 1662, and for the divisor the quantity 4o2_5aJ-h62; the partial quotient is 3. Multiplying the divisor by the quotient, and subtracting the product from the dividend, the remainder is 19a6 — 1962, and the question is now reduced to finding the greatest common divisor of 19a6— 19^2 and 4a'^ — bab-\-b'^. But the letter a, according to which the division has been performed, being of the second degree in the divisor, and only of the first in the remainder ; it is necessary therefore to take the last divisor for a new dividend, and the remainder for a new divisor. Having, at the commencement of this new division, divided the divisor I9ab — I9b'^ by the factor 196, common to all its 7* 66 ALGEBRAIC FRACTIONS. terms, and which is not at all common to those of the dividend : therefore the dividend is Aa^— 5 ab-\rb^, the divisor a—b, and the quotient 4a— b ; The operation is completed, because nothing remains ; and consequently, (Art. 135), a—b is the greatest common divisor sought. If we divide the two proposed quantities by a— ft, the quo- tients will be 3a2+i2 and 4ab—b^ : Whence, the two given quantities are thus decomposed as follows : (3a2+62)x(a-&), (4ab-b^)x(a'-b). Ex. 2. Required the greatest common divisor of 30^— .2a— 1 and4a3— 2a2_3a+l. Dividend. Divisor. 4a3_2a2_3^_|_l 3«2_2a_l 3 12a3— 6a2_9a+3 12a3— 8a2_4a 2a2. 3 5a+3 6^2— I5a+g 6a2_ 4a— 2 Partial quot. 4a Divisor. 3a2_2a-l Partial quot. 2 ( — lla+ll)-4-— 11 Dividend. a-l 3«2- -2a- -1 3a2- -3a a- -1 a- -1 Complete quot. 3a 4" 1 In the above operation, the remainder — lla-f-H is divid- ed by —11, (its greatest simple divisor with a negative sign), so as to make the leading term positive : or, which is the same, if any of the divisors, in the course of the operation, become negative, they may have their signs changed, or be taken affirmatively, without altering the truth of the result ; thus, in the above operation, changing the signs of —lla+ 11, it be- comes 11a— 11, and dividing 11a— 11 by its greatest simple divisor 11, we have a— 1, as before. ALGEBRAIC FRACTIONS. 67 Therefore a— I is the greatest common divisor sought, and the two ajiven quantities may be readily decomposed, thus; (3a+l)x(a-l), (4a2+2a-l)x(a— I). Ex. 3. Required the greatest common divisor of a^—P, a^-{-2a^+2ab^+P, and a'^-\-a^b^-\-b\ In the first place, the greatest common divisor of a^—b^ and a^+2aH-{-2ab^-\-b^, is a^-\-ab-{-b^, which is found thus ; Dividend. Divisor. a^+2a'^b+2ab^-\-b^ a3 -63 a3-63 Partial quot. 1 (2a'^b-{-2ab^+2b^)-^2b=z Dividend, a^-b^ a'^+ab-hb^ a^J^a^b-\-ab^ -a^b-ab^-b^ .a^—ab^-b^ Complete quot. a — b Hence, the greatest common divisor of a^—¥ and a^-\-2a'^b -{-2ab^-\-b^, is a^-{-ab-{-b^ ; and the greatest common divi- sor of a'^-{-ab-\-b^ and a^+a^^^-f M, is found to be a'^—ab-\-b^, thus; Dividend. —a^ +b* — a^b — a^^ — aP aW-^ab^^-b^ fl2fe2_^a63+M Divisor. a^-\-ab-{-b'^ Quotient, a^—ab+b"^ Consequently a^-\-ab-\-b'^ is the greatest common divisor which was required ; and dividing each of the given quanti- ties by this divisor, we will thus decompose them as follows : (a-6) [a^J^ab-irb'^), (a+*) (a^-^ab-\-b'^), (a^^ab-^b^) (a^-\- ab+b^y 142. It has been remarked (Art. 136), that if the last divi- sor be unity, and the remainder nothing ; then the fraction is 68 ALGEBRAIC FRACTIONS. already in its lowest terms ; this observation is applicable«to numbers, and as in algebraic quantities, the greatest simple divisor may be readily found by inspection. Now, it only remains to discover, if compound algebraic quantities can admit of a compound divisor. If, by proceeding according to the Rule (Art. 141), no compound divisor can be found, that is, if the last remainder be only a simple quantity ; we may conclude the case pro- posed does not admit of any, but is already in its lowest terms. Ex. 4. Required the greatest common divisor of a^-j-ax-^- sc^ and a^-\-2a'^x-\-3ax'^-\-4x^. It is plain by inspection that they do not admit of any simple divisor ; then the operation according to the rule will stand thus ; Dividend. Divisor. a^-\-2a'^x+3ax^+Ax^ a^x~\-2ax'^-\-4x^ a^aj-j- ax^-\- x^ a^-{-ax-^x'^ Dividend. a^-\- aa?+ x"^ a^-{-3ax — 2ax-{- x^ — 2ax — 6a?2 Partial quot. a-{-x [ax'^-\-3x^)-^x^ = a-\-3x Partial quot. a — 2x * +7x^ Here, the last remainder is found to be the simple quantity 7x^ ; we may therefore conckide that the given quantities do not admit of any divisor whatever. 143. When the quantity which is taken for the divisor con- tains many terms where the letter, according to which we have arranged, has the same exponent ; then every succes- sive remainder becomes more complicated than the preceding one ; in this case, Analysts make use of various artifices, which can only be learned by experience. Ex. 5. Required the greatest common (Jivisor of a^J+oc^ — cZ^ anda6— ac+(?2. Dividend. Divisor. a^J-j-ac^ — d^ a'^b—a^c-\-ad'^ rem. a'^c-\-ac'^—ad^—d^ ah—ac-\-d^ Partial quot. a Dividing at first a% by ah, we find for the quotient, a ; ALGEBRAIC FRACTIONS. 69 multiplying the divisor by this quotient, and subtracting the product from the dividend, the remainder contains a new term, a^c, arising from the product of —ac by a. By proceeding after this manner there will be no progress made in the operation ; for, taking cP-c-\-ac^—ad?'^d^ for a dividend, and multiplying it by 6, to render possibm the divi- sor by a6, we will have Dividend. Divisor. a?bc-\-abc'^ — abd? — hd^ ab — ac-\-d? d^bc—a^c'^-\-acd^ rem. '^c^^abc^—acd?—abd? — bd'^ Partial quot. and the term —ac will still r.eproduce a term aV, in which the exponent of a is 2. To avoid this inconveniency, we must observe that the di- visor ab — ac-\-d?z:za{b—c)-\-d?, reuniting the terms ab — ac into one, and putting, to abridge the calculations, b — c=zm; we will have for the divisor a7n-{-d^ ; it is necessary to mul- tiply all the dividend a'^b-\-ac^—d^ by the factor m, for the pur- pose of finding a new dividend whose first term would be divi- sible by the quantity am forming the first terra of the divisor ; the operation will become, Dividend. Divisor. a^bm-^-ac^m—d^m am-{-d^ a^bm-{-abd^ 1st rem. -^ac^m—abd^—d^m -\-ac^m-\-c^d^ Partial quot. ab + c'^ 2d rem. ~abd^ — c'^d^ — d^m By the first operation, the terms involving a^ are taken away from the dividend, and there remain no terms involving a ex- cept in the first power. In order to make them disappear, we will at first divide the term ac^m by am, and it gives for the quotient c'^ ; multiplying the divisor by the quotient, and sub- tracting the product from the dividend, we will have the second remainder ; taking this second remainder for a new dividend, and cancelling in it the factor d^, which is not a factor of the divisor, it will become — ab — c^ — dm ; multiplying by m, we shall have Dividend. Divisor. — abm- — abm- ■c^m- ■bd:^ ■drr? rem. -{-bd^—c^m—dm^ am-{-d^ Partial quot. — b. 70 ALGEBRAIC FRACTIONS. The remainder, hd? — c-m—dni^, of this last division does not contain the letter a ; it follows, then, that if there exist between the proposed quantities a common divisor, it must be indepen- dent of the letter a. Havin^arrived at this point, we cannot continue the divi- sion with<|fcspect to the letter a ; but observing that if there be a common divisor, independent of a, of the two quantities hd^ — c^m — dm^ and am-^-d^, it may divide separately the two parts am and d^ of the divisor ; for, in general, if a quantity be arranged according to the powers of the letter a, every term of this quantity, independent of a, must divide separately the quantities by which the different powers of this letter are multiplied. In order to be convinced of wFiat has just been said, it is sufficient to observe, that in this case each of the proposed quantities should be the product of a quantity dependent on G, and of a common divisor which does not at all depend on it. Now, if we have, for example, the expression A^^+BflS + Ca^+Da+E, in which the letters A, B, C, D, E, designate any quantities whatever, independent of a, and if we multiply it by a quantity M, also independent of «, the product, MAa4+MBa3 + MCa2+MDa+ME, arranged according to a, will still contain the same powers of a as before ; but the coefficient of each of these powers will be a multiple of M. This being admitted, if we substitute for m the quantity (i— c), which this letter represents, we shall have the quan- tities hd'^^c\h-c)-c{h-c)\ - a(5-c)+c?2; now it is plain that h — c and d^ have no common factor what- ever : therefore the two pro^^osed quantities have not a com- mon divisor. 144. The greatest common divisor of two quantities may sometimes be obtained without having recourse to the general Rule. Some of the methods that are used by Analysts for this purpose, will be exemplified by the following Examples. Ex. 6. Required the greatest common divisor of a^i^-f-^'^^^ -f i*c2— aV— a3Z,c2— ^2^4, and d^b + ab'^-^h^—a''-c—abc-'hH. After having arranged tKese quantities according to the powers of the letter a, we shall have (52_c2)a4_|_(^,3_^,c2)a3-f-J4c2_52c4 (5-c)a2-f(62_Z»c)a+63-J2c; ALGEBRAIC FRACTIONS. 71 it may at first be observed, tbat if they admit of a common di- visor, which should be independent of the letter a, it must di- vide separately each of the quantities by which the different powers of a are multiplied, (Art. 143), as well as the quanti- ties h'^c^—b'^c^ and P — b^c, which comprehend not at all this letter. The question is therefore reduced to finding the common divisors of the quantities b^ — c^ and b — c, and, to verify af- terward, if, among these divisors, there be found some thai would also divide b^—bc^ and b'^—bc, b*c^—b^c^ and b^~b^c. Dividing b'^ — c^ by b — c, we find an exact quotient b-\-c : b—c is therefore a common divisor of the quantities b^—c^ and b — c, and it appears that they cannot have any other di- visor, because the quantity b—c is divisible but by itself and unity. We must therefore try if it would divide the other quantities referred to above, or, which is equally as well, if it would divide the two proposed quantities ; but it will be found to succeed, the quotients coming out exactly, (b+c)a^-{-{b^-^bc)a^'frb^c^+b^c^; and a'^-\-ba-\-b^. In order to bring these last expressions to the greatest pos- sible degree of simplicity, it is expedient to try if the first be not divisibfe by b-{-c ; this division being effected, it succeeds, and we have now only to seek the greatest common divisor of these very simple quantities ; a^-^baHb^C^, and a?-^ba+b^. Operating on these, according to the Rule,- (Art. 141), we will arrive, after the second division, at a remainder contain- ing the letter a in the first power only ; and as this remainder is not the common divisor, hence we may conclude that the letter a does not make a part of the common divisor sought, which is consequently composed but of the factor b — c. Ex. 7. Required the greatest common divisor of (d^ — c^) Xa^+c* — fZV and 4:da:^—(2c'^+4cd)a-i-2c^. Arranging these quantities according to d, we have (a2_c2)£Z2 + c4-a2c2, or (a2_c2)(^2_(^2_c2)c2, and {4a^—4ac)xd—(a—c)x2c^; it is evident, by inspection only, that a^ — c'^ is a divisor of the first, and a — c of the second. But d^ — c^ is divisible by a — c ; therefore a—c is a divisor of the two proposed quantities : Di- viding both the one and the other by a — c, the quotients will be (a+c)x((^— c2), and 4a(?— 2c2; 72 ALGEBRAIC FRACTIONS. which, by inspection, are found to have no common divisor consequently a—c is the greatest common divisor of the pro- posed quantities. , Ex. 8. Required the greatest common divisor of y^—x^ and y^ — y^x— yoj^-j-o?^. Ans. y^ — ^,2^ Ex. 9. Required the greatest common divisor of a*—b^ and a^—¥. Ans. a^—lp-. Ex. 10. Required the greatest common divisor of o^+a^^ — ah'^—h'^ and a'^-^aW^-h'^. Ans. a^-^ab-^-y^. Ex. 11. Required the greatest common divisor of a?-— 'lax ■\-x^ and c? — oP'X — ax^-\'x'^. Ans. a^ — 2ax-\-x^. Ex. 12. Find the greatest common divisor of 6a;^ — %yx^-\- 2y2a; and 12a;2 — 15ya;+3y2. Ans. x — y. Ex. 13. Find the greatest common divisor of 3662a6_ 18^2^5 — 27 J2a4-}- 952^3 and 27^2^5 _i 852^4 _ 952^3. Ans. 9^»2a*— 9i2a3, Ex. 14. Find the greatest common divisor of (c—f- Ex. 3. Required the least common multiple of 8a, 4a2, and I2ab. Ans. 24a2^. Ex. 4. Required the least common multiple of a2— &2^ a+i, and a^-\-b^. Ans. a'^ — b*. Ex. 5. Required the least common multiple of 72a, 1 5b, 9ab, and 3a2. Ans. 135a2^>. Ex. 6. Required the least common multiple of a^+Sa^b-i- 3ab^+b\ a^-j-2ab + b^, a^—b^. Ans. a^-^2a^—2aP-^b^. Ex. ^. Required the least common multiple of a-\-bj a—b, a^-^-ab-^-b^, and a^ — ab-^b^. Ans. a^—¥ ^ IV. REDUCTION OF ALGEBRAIC FRACTIONS. CASE I. To reduce a mixed quantity to an improper fraction. RULE. 147. Multiply the integral part by the denominator of the fraction, and to the product annex the numerator with its pro- per sign : under this sum place the former denominator, and the result is the improper fraction required. Ex. 1. Reduce 3x +^ to an improper fraction. The integral part 3x, multiplied by the denominator 5a of the fraction plus the numerator (2b), is equal to 3a:X5a-|-26 = l5ax+2b', Hence, is the fraction required. 3a? Ex. 2. Reduce 5a to an improper fraction. 8 74 ALGEBRAIC FRACTIONS. Here 5axy=^^ay\ to this add the numerator with its pro- per sign, viz. —3a; ; and we shall have 5ay— 3a;. Hence, — =- is the fraction required. y q2 yZ Ex. 3. Reduce x^ — to an improper fraction. Here, x^Xx^zx^ \ adding the numerator a^ — y^ ^j^A its pro- per sign : It is to be recollected that the sign •— affixed to the q2 J.2 fraction — means that the whole of that fraction is to be X subtracted, and consequently that the sign of each term of the numerator must be changed, when it is combined with x^ hence the improper fraction required is ~. Or, as ^2 «f2 q2 _1_ «»2 |f2 .^ ^2 ^—= :i- — -^ ; (Art. 67), the proposed mixed X X x ^ (jp- j/2 |.2 ^2 quantity a;^ ^, may be put under the from x^-\-- » X X which is reduced as Ex. 1. Thus,x'^Xx-{-i/^—a^=x^-\h/^—a^'f , \ , y^—a^ a;3+y2_Q2 hence, x^+- = . X X 3/p2 a4-7 Ex. 4. Reduce 5a^-\ to an improper fraction. liax Here, 5a'^x2axz=z\0a'^x ; adding the numerator 3a?2— a+7 to this, and we have 10a%-|-3a;^— a+7. __ lOA-f 3a;2-a+7 .,,.,. . , Hence, is the fraction required. 2ax Ex. 5. Reduce Ax"^ to an improper fraction. Here, Ax'^x2ac=Sacx'^,m. adding the numerator with its proper sign ; the sign — • prenxed to the fraction — signi- fies that it is to be taken negatively, or that the whole of that fraction is to be subtracted ; and consequently that the sign of each term of the numerator must be changed when it is combined with Qacx"^ ; hence, is the fraction re- 2ac . , ^ ^ab-{-c . — 3a5— c —Zah—c ,.^ quired. Or, as __= +^__ =_g_- (Art. 108) ; hence the reason of changing the signs of the numera- tor is evident. ALGEBRAIC FRACTIONS. 75 Ex. 6. Reduce x to an improper fraction. X Ans. X d?" '\' c * Ex. 7. Reduce ab — to an improper fraction. Ox . 5abx — a^ — c Ans. . 5x Ex. 8. Reduce ax"^ to an improper fraction. a'^x^-Sb Ans. . a Ex. 9. Reduce a—x-\ to an improper fraction. X . a^—x^ Ans. -. X 4^ 9 Ex. 10. Reduce 3x^ — to an improper fraction. 21aa;2_4a;4-9 Ans. ■ — 7a 2x 5 Ex 11. Reduce 5x — to an improper fraction. o 13i«;+5 Ans. -^— . Ex. 12. Reduce 14-2a? — to an improper fraction. DX a;4-10a!24.4 Ans. , 5a; CASE II. To reduce an improper fraction to a whole or mixed quantity. RULE. 148. Observe which terms of the numerator are divisible by the denominator without a remainder, the quotient will give the integral part ; and put the remaining terms of the nume- rator, if any, over the denominator for the fractional part ; then the two joined together with the proper sign between them, will give the mixed quantity required. 76 ALGEBRAIC FRACTIONS. Ex. 1. Reduce to a mixed quantity. X T-. x^-\-2ax'^ ^ • 1 • 1 , 5 . , Here, =:x-{-2a is the integral part, and — is the fractional part ; • therefore x-\-2a-\ — - is the mixed quantity required. X ^8 I jp4 J.4 J_ y8 Ex. 2. Reduce -— — '-—- — <- to a whole quantity. J »*— 3a:2y2+4aa: . , Ex. 10. Reduce ^-^^r^ to a mixed quantity. x^—Sy^ "^ Ans. x^-{- -3y2 „ , , ,, , x^+Sax^—a^~b Ex. 1 1 . Reduce — — to a mixed quantity. x^-\-a^ ^ "" Sax^—b Ans. a:^— a^-l 5-7— ^• ^ ,« r, J 3a:2— 12aa;+y— 9a: Ex. 12. Reduce — to a mixed quantity. ijX y Ans. a:— 4a— 3+ o^« 8» 76 ALGEBRAIC FRACTIONS. CASE III. To reduce a fraction to its lowest terms, or most simple expression. RULE. 149. Observe what quantity will divide all the terms both of the numerator and denominator without a remainder : Di- vide them by this quantity, and the fraction is reduced to its lowest terms. Or, find their greatest common divisor, accord- ing to thfe method laid down in (Art. 141) ; by which divide both the numerator and denominator, and it will give the frac- tion required. Example 1. o , I4x^-i-7ax^+28x . . Keduce — — to its lowest terms. 21a;2 The coefficient of every terra of the numerator and deno- minator of the fraction is divisible by 7, and the letter x also enters into every term ; therefore 7a? will divide both the nu- merator and denominator without a remainder. _- Mx^+7ax'^-\-28x „ , , . ^ ,21a;2 ^ , Now =:2x^-i-ax4-4,Rr\a—- — —3x: hence 7x 7x , - . . .^ , ^ . 2x'^^-ax-{-4: the fraction m its lowest term is . 3a; T. « r. J SOa^b^c—dabc'^—Ua'^cH . , Ex. 2. Keduce — to its lowest terms. doaocx Here the quantity which divides both the numerator and denominator without a remainder is evidently 6abc ; then 30a262c— 6a6c2— 12a2c2Z> 36abcx =:5ab — c—2ac ; and -—— i — = oa; ; 6abc 6abc ^^ 5ab — c—2bc . , ,, . • • , Hence is the fraction in its lowest terms. 6a; a^ — &2 Ex. 3. Reduce — — -,- to its lowest terms. «*— &4 Here, a* - b^ = {a^ 4 b^) X (a^ - b% (Art. 107.) ; and, consequently, a^—b"^ will divide both the numerator and de- a2-62 , nominator without a remainder ; that is, — — r^ = 1 = new ALGEBRAIC FRACTIONS. 79 ,(a2+&2)x(a2_52) numerator, and ^ r— 7:3 ■'=a--^b^= new denomma- a^ — tf^ tor ; hence, „ . ,., is the fraction in its lowest ternnis. ^ , T. :. a;4— 3aa;3— 8a2a;2^18a3a: — 8a* . , Ex. 4. Reduce — -— -— to its lowest terms. Here, by proceeding according to the method of (Art. 141), we find the greatest common measure of the numerator and denominator to be x'^-\-2ax — 2a2 ; thus, x^—Sax^—Sa'^x'^-i- I8a^x—8a^ ae* — ax^ — 8a'^x'^-\- 6a^x —2ax^-{-l2a^x — Sa* —2ax^+ 2a'^x^+l6a^x—12a^ x^ — ax"^ — 8a^x-{-6a^ Partial quot. x — 2a remaind. . . . — 2a2a;2— 4u"V+4a*; 1 — 2a2a;2_4Q;3^_|_4^4 _ , ^ ^ ^ . then, —-^ =x^ + 2ax — 2a^ z= the next di- — 2a^ visor ; xi-\.2ax—2a^)x'^— ax'^—8a'^x-\-Qa\x—'ia x'^'\-2ax'^ — 20^0? — ^ax"^ —Qa"X-\-Qa^ — 3aa?2 — Ga^aj-j-Ga'^ And, dividing both terms by the greatest common measure, thus found, we have the fraction in its lowest terms ; but the numerator, divided by the greatest common mep,sure, gives x — 3a, as above, equal to the new numerator ; and the denomi- nator, divided by the same, gives x^—5ax-{-4a'^ ; thus, x^^Sax^- 8a'^x^+l8a^x-8a'*' a;*+2aa;3— 2a'2x^ — 5ax^ — 6a'^x'^-{-18a^x — 5aa;3 — 1 Oa^x'^ + 1 Oa^x 4a^x"+ Sa^a;— 8a* 4a2a;24- Sa^aj—Sa* a;2+2aa;— 2a2 Quotient. x'^—5ax-i-4a^ Hence, the fraction in its lowest terms is 80 ALGEBRAIC FRACTIONS. x—3a x^ — 5ax^4a^' 150. In addition to the methods pointed out in (Art. 144), for finding the greatest common divisor of two algebraic quan- tities, it may not be improper to take notice here of another method, given by Simpson, in his Algebra, which may be used to great advantage, and is very expeditious in reducing frac- tions, which become laborious by ordinary methods, to the lowest expression possible. Thus, fractions that have in them more than two different letters, and one of the letters rises only to a single dimension, either in the numerator or in the denominator, it will be best to divide the numerator or de- nominator (whichever it is) into two parts, so that the said letter may be found in every term of the one part, and be to- tally excluded out of the other : this being done, let the greatest common divisor of these two parts be found, which ■will evidently be a divisor to the whole, and by which the division of the other quantity is to be tried ; as in the follow- ing example. -n ^ T^ 1 x^4-ax^-\-bx'^—2a^x4-bax—2ba'^ . , Ex. 5. Reduce — ; — -— —-z toitslow- x^ — ox-^2ax — 2ao est terms. Here the denominator being the least compounded, and b rising therein to a single dimension only ; I divide the same into the parts x^-{-2ax, and — bx — 2ab ; which, by inspec- tion, appear to be equal to (a;+2a)a?, and {x-\-2a)x —b. Therefore x-\-2a is a divisor to both the parts, and likewise to the whole, expressed by (x-\-2a)x(x — b) ; so that one of these two factors, if the fraction given can be reduced to lower terms, must also measure the numerator : but the former is found to succeed, the quotient coming out x^ — ax-\-bx—abi , , 1 r . • 1 1 x'^—ax-\-bx — ab exactly : whence the fraction is reduced to ; , x—b which is not reducible farther hy x—b, since the division does not terminate without a remainder, as upon trial will be found. Ex.6. Reduce -TT — ^ V,^ r-r- —-to its lowest terms. a^b + 20*^/2 ^ 2a?¥ -\- a^b^ Here, the greatest simple divisor of the numerator and de- nominator is evidently, a^b ; Now, ^ =zdc^ cro ALGEBRAIC FRACTIONS. 81 ab^-\-b^. Hence the result is » . » ^7 , » 19 . 79 ; and the a^+2a^b-\-2ab^-\-b^ greatest common measure of this result is a-{-bj which is found thus; a^-\-2a^-\-2ab^'\-b^) da^+lOd^b-^dab^S da^+lOa^+lOab^+ob^ remainder .... —5ab^ — bb^ 5aJ2 5^3 And — =za-\-b, which by another operation is — 50^ found to divide the numerator without a remainder ; and con- sequently dividing both the numerator and denominator of the fraction — -, ,., , ,„ by a-\-b, we have the fraction a^-{-2a^b-{-2ab'^-{-o'^ m Its lowest terms; that is, —7 =5a^+5a6; a^+2^b+2ab''+b^ . , 7 , ,2. and — =a^-\-ab + b^: a-\-b Hence -^^^ — r-T-r7 is the fraction in its lowest terms. a^-}-ab-^b^ ^ „ ^ , 14a:V— 21iKV . •. i Ex. 7. Reduce <^ — ~- to its lowest terms. X Ex. 8. Reduce to its lowest terms. 17a? 3a;2-a?+2 Ans. . a* Ex. 9. Reduce , . ,, to its lowest terms. Ans a^^ab+b^ 4 I fflcjip> I Q^ Ex. 10. Reduce —- = to its lowest terms. a;* + ax^ — o?x — a* x^—ax-Va^ Ans. x'-—a'- Ex. 11. Reduce -^.^lr?M+|.^ to its lowest terms. l a— 2b ALGEBRAIC FRACTIONS. a* J4 Ex. 12. Reduce — — — to its lowest terms. Ans. «'+*= y4 ^4 Ex. 13. Reduce — ^ to its lowest terms. Ans. ^— — . Ex. 14. Reduce — — rr^rr^^j to its lowest terras. a*4-a2^2_|_^4 « — b Ans Ex. 15. Reduce to its lowest terms. Ans. a^+ab-i-b^' !st terms. Ex. 16. Reduce -— — -7-- — -rr-r to its lowest terms. 2a'^—3ba^—5b^a^ Ans. a-\-x jrms. a+2&+362 2a2~3^>a— 562* a J — a^x — ax^ + ic^ Ex. 17. Reduce — ;,-; — - to its lowest terms. Ans. a-\-x Ex. 18. Reduce _ . ^ , , ,^ to its lowest terms. a^'^2ab-\-b^ . a^—ab Ans. — r-v- a-\-b CASE IV. TV? reduee fractions to other equivalent ones^ that shall have a common denominator. RULE I. 151. Multiply each of the numerators separately, into all the denominators, except its own, for the new numerators, and all the denominators together for the common denominator. It is necessary to remark, that, if there are whole or mixed quantities, they must be reduced to improper fractions, and then proceed according to the rule. ALGEBRAIC FRACTIONS. 83 Ex. 1. Reduce -— , — and - to a common denominator. 4 c a 3aXcXa = 3a^c 5bX4x a=z20ab } new numerators ; xxcx Xa = 3a^c J Xa=20ab V X4: = 4:cx y 4 X c X a=4ac common denominator ; TT .!_ r .• J Sa^c 20ab . 4cx Hence the fractions required are - — , — — , and . 4ac 4ac 4ac Ex. 2. Reduce — — j — , and to a common denominator. 3b X {2x-\-\)Xx =z2a:24.a:> ^ 2Jx3b = e>a^b J "ew numerators ; 3b X x = 35a; common denominator ; 2a;^4-a? Qa^b Hence the fractions required are — — ; , and -, — . 36a? 3bx Ex. 3. geduce -, — -, and a-\ — — to a common denomina- tor. . 3a:2 5a4-3a:2 Here a-\- 5 5 3x3x5z=:45 5a: X 4 X 5 r= 1 00a? J- new numerator ; (5 and :r- to a common denomi- a+a? 3 ' 2x nator. 30a ;2 2a'^x—2x^ 3a-f3ar 6aa:-f6^' 6ax + 6x2 ' ?"^ 6aa:+6«2- ALGEBRAIC FRACTIONS. 85 T7 ^ n J 2a?+3 , 5a;4-2 , JiiX. 7. Keduce , and ^ , to a common denomi- X Sab nator. . 6abx-\-9ah . 5a;2+2a; Ans. — -r , and —7—7 — . 3aox 3aox Ex. 8. Reduce -, — - — , and -— ; — to a common denomi- 3 4 1-f-a? nator. 4a?2+ 4a; 3x^+6a;+3 12at— 12 ^''^- 12a;+12' m-hl2 ' *'''* l2^+12' g 20^^ 3a a?2 Ex. 9. Reduce t, -1-, and x-\ to a common deno- a X minator. adx 2hc'^x , Zahd bdx ' bdx ' . bdx 17 in r) J a;2 a;2_5 4flf~15 liiX. 10. Keduce -—-, a — , and 74 ;; — to a com- 5y 3a? 2 mon denominator. 6x^ 30axi/ — l0x^i/-\-50j/ 60axy — 15a;y ■ SOxy" 30xy ' ^" 30^y Ex. 1 1 . Reduce — — -, -— , and — - — to other equiva- a^—x^ 4a— 4x a-\-x lent fractions having the least common denominator. 4a 3ab-\-3bx , 20ax--20x^ Ans. . ' ■ . „ , — -7; — — r, and 4a2_4a,2' 4o2_4a;2' . 4a2_4a;2 * Ex. 12. Reduce -— — — ^, — r, and -r^-r- to the a24-2aa;-|-a;2 a^—x^ a*— a;* least common denominator. a3_^aa;2— aV— a:^ a^-\-ax'^-^a^x-{-ci^ and , ^"3^+^^^^ , . o^— aiK*+a*a;— a* § V. ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS. To add fractional quantities together. RULE. 153. Reduce the fractions, if necessary, to a common deno- minator, by the rules in the last case, then add all the nume- 9 86 ALGEBRAIC FRACTIONS. rators together, and under their sum put the common denomi- nator ; bring the resulting fraction to its lowest terms, and it will be the sum required. Ex. 1 . Add — , — , and - together. 2a;X 7X9 = 126a; 5a;X3x9 = 135a? a;X7x8=: 21a: 3X7X9=189 126a;-i-135x-l-21a? 282a? 189 189 93a; + -— is the sum required. Ex. 2. Add -, -y, and — - together. So 4a l2a^b + 8a^+15P 12a62 aXBhx4a=l2a^^20aH+l5b^ ,^. .^. ^ ,, 2aXbx4a= Sa'b \ r^-Ti = (dividing by b) 5bx3bxb=15b^ I ^^^3 — I ; IS the sum required. bx3bx4a=:l2ab^} 1^«^ Or, the least common multiple of the denominators may be found, and then proceed, as in (Art. 152). It is generally understood that mixed quantities are reduced to improper fractions, before we perform any of the operations of Addition and Subtraction. But it is best to bring the frac- tional parts only to a common denominator, and to affix their sum or difference to the sum or difference of the integral parts, interposing the proper sign. 33.2 Ex. 3. It is required to find the sum of a ^-, and b-\- 2ax c 3j;2 ab—Boc^ , , . 2aa; bc+2ax Here, a j— = 1 — > ^^^ o-\ = . 00 c c Then, (ab-3x^)Xc=abc-3cx'^ Enumerators (bc+2ax) X b = b^c+2abx ] «"n^«^^^o^«- bXc=bcz= denominator. 'abc—3cx'^-\-b'^c-\- 2abx _aI}C-^b^c be be 2abx — 3cx^ , , , 2abx—3cx^ — fc — =«+*+— j^ — is the sum required. ALGEBRAIC FRACTIONS. 87 Or, bringing the fractional parts only to a common deno- minator, Thus, 3x'^xc=3cx^ ) And 5 Xcr=5c common denominator. „,, 3c^2 ^5jc 2a5a?— 3ca:2 , Whence a ; \-b'\ — - — =a-\-b-{ j the sum. Oc be be Ex. 4. It is required to find tho sura of 5pc-{ — -— and 4« 2a:-3 5a3 Here, ( a;— 2) x 5a; =5*2— 10a; ) „„^^^,^,, /o o\vxo c n > numerators, (2ir--3)x3 =6a; —9 > ' And 3 x5a; = 15a; common denominator. Whence 5a;i — h4a; — =9a;-| ^ 15a; 15a; . 15a; 9--6a; ^ . 5a;2— 16a;4-9 , . ' — — — =9a;H — the sum required. i 037 loa; g-ji 9 9 6-j. Here, -— — is evidently = — — — (Art. 128) .; but we loa; loa; might change the fractions into other equivalent forms before we begin to add or subtract ; thus, the fractional part of the 2a; 3 proposed quantity 4a; maybe transformed by chang- tax ing the signs of the numerator, (Art. 128), and the quantity 3 2x itself can be written thus, 4a; H — ; It is well to keep this* oa; transformation in mind, as it is often necessary to make use of it in performing several algebraical operations. T7 .: A J , 3«2 2« , 6 Ex. 5 Add — , — and - together. , 105a2 4-28a5+10J2 ^"" 70^ Ex. 6. Add and — — - together. Ans. -r — ;r. a;— 3 a;+3 a;^— 9 Ex. 7. Add ^ — ~ and ^^^ together. a~b a+b ^ 2a^+2b^ Ans. . a^-^b^ 88 ALGEBRAIC FRACTIONS. Ex. 8. Add and — '^—^ together. a — X a-^-x ° * 4cfa? Ans. Ex. 9. Add 2a;+^ and 3a;^?^ together. a^—x^ Ans. 5a: + 12 Ex. 10. Add 4a;, -— and 2+^ together. Ans. 4a;+24-^^. 45 Ex. 11. Add 5ap — and — 4a? together Ans. 0.+— . Ex. 12. It is required to find the sum of 2a, — ^, and a — X Ans. 2a+24— r — To subtract one fractional quantky from another. RULE. 154. Reduce the fractions to a common denominator, if ne- cessary, and then subtract the numerators from each other, and under the difference write the common denominator, and it will give the difference of the fractions required. Or, enclose the fractional quantity to be subtracted in a parentheses ; then, prefixing the negative sign, and perform- ing the operation, observing the same remarks and rules as in addition, the result will be the difference required. The reason of this is evident ; because, adding a negative quantity is equivalent to subtracting a positive one (Art. 63) ; thus, prefixing the negative sign to the fractional quantity a — h . . /a — b\ a — h h — a . , It becomes — - i ) = = ; to the c \ c J c c Qc I fit y X .li , ftK fractional quantity — , it becomes — f ^^j = , x^-\-a ,. ,^„. 1 /. . 1 . ox — h . •T (Art. 128)^ to the fractional quantity — , it ALGEBRAIC FRACTIONS. 89 becomes — ( J= — - — ; to the mixed quantity y \ y f y 2 X and to the mijjed quantity — 3a+ , it becomes — Ex. 1. Subtract —, from -^. Here 3a;x7=21a:> , ^^^,,^,„ 5 X 7r=35 com. denom.* Ex. 2. Subtract — from -t-t-- 5c 3d Here (2a-4.) X 3J=6a6-12Ja. ) „„„,^^„. (x—y)X5c= 5cx— 5ci/ S 25a;— 21a; 4x. . .'. = -^-is the 35 35 difference required. 5cx35 = 15&c common denominator. , 5ox—5cy Gab — 125a; Scar— 5cy 125a; — 6ab ^^^"''^' "Tsi^ 15b^~ "^ I5bc "^ 156^ 5ca;— 5cv4-126a;— 6n5 . , ,._. . , -^—i IS the difference required. lo6c ^ . n • -, ' ' 1 . 2a — 4x Or, by prefixing the negative sign to the quantity — , DC . , 2a— 4a; 4x — 2a , . , • . ij it becomes = ; then it only remains to add 5c 5c ^ — and -T together, as in addition, .and the result will 5c So be the same as above. Ex. 3. From 2a5H ; — subtract 2a5 ayf-a; a-{-x Here prefixing the tiegative sign to the quantity 2a5— — ; — , we have — (2a5 ) = — 2a5H ; hence the a-{-x \ a-\-x/ a-\-x difference of the proposed fractions is equivalent to the sum of 2a6-f ^^^, and —2ab-{- ^— — ; but the sum of the frac- a-\-x a—x 9* 90 ALGEBRAIC FRACTIONS. - ^ a—x . a-\-x . 2a24-2a;2 tional parts — ; — and , is — — : Therefore the diffe- a-{'X a—x a^—x^ . :, • ^ X o 1 . 2a2+2a;^ 2a2+2a;2 rence required is 2ab—2ab-\ ^ T — — 2 2~* Ex. 4. From subtract — - — . 15 7 Here (I0a;-9)X 7=70a:-63 > „^^ .^^^ . (3a:-5)xl5 = 45x-75r""'''''''''- 15x7=105 common denominator. ^, ^ 70a;— 63 45a;-75 Therefore, 105 105 70a;— 63-45a;+75 25a;+ 12 105 105 is the fraction required. _ ^ _, a-\-h . a—h . Aah Ex. 5. From subtract . ■. Ans. a—b a-\-b a^ — b"^ Ex. 6. From subtract — ; — Ans. ;2— .i?2 a — X fl+a? (v" — X' _ ^ _ 4a;+2 . ^ 2a;— 3 . 4a:2+3 Ex. 7. From — - — subtract — - — . Ans. 3a; 3a; Ex. 8. From 3a; + y subtract x— b c ' cx-\-bx — ab Ans. 2a;+ bc ^ ^ o , 2a;+7 ^ 3x'^+a^ Ex. 9. Subtract — ?-- from ^ 8 3b 24a;2+8a2— 6ia;— 2U Ans. 24^> T, ,« « 1 . 2a;— 3 - ^ , a;— 2 Ex. 10. Subtract 4a; — from 5a; -i — - Ans. x-i r-z — . 15 Ex. 11. Subtract -T rfroma4- a{a—x) a^a-^-x)' 4x Ans. a— Ex. 12. Required he difference of 3a; and a^—x'^' 3g+12a; 5 3a;— 3a Ans. — r ALGEBRAIC FRACTIONS. 91 Ex. 13. From2a;+-^- — subtract 3a; ^J-. . 16ir+23 Ans.-^^ ^ VI. MULTIPLICATION AND DIVISION OF ALGEBRAIC FRAC- TIONS. To multiply fractional quantities together. RULE. 155. Multiply their numerators together for a new nume- rator, and their denominators together for a new denominator ; reduce the resulting fraction to its lowest terms, and it will be the product of the fractions required. It has been already observed, (Art. 119), that when a frac- tion is to be multiplied by a whole quantity, the numerator is multiplied by that quantity, and the denominator is retain- ed : rrn f"' ac , 2a? ^ 10a? i • t . i Ihus, yXc=-j-, and-r-x5=— 7— ; or, which is the same, bob b making an improper fraction of the integral quantity, and then proceeding according to the rule, we have tXt-=-t-, , 2a; 5 10a; Hence, if a fraction be multiplied by its denominator, the product is the numerator ; thus, ^xh =i-r =zh. In like b b manner, the result being the same, whether the numerator be multiplied by a whole quantity, or the denominator divid- ed by it, the latter method is to be preferred, when the de- nominator is some multiple of the multiplier : Thus, let — - be the fraction, and c the multiplier ; then^— Xcr=:-T— = he , ^ be be ad ad ad ad , ^ -rr ; and ^- X c= =-i-, as before. b be bc-^c b Also, when the numerator of one of the fractions to be mul- tiplied, and the denominator of the other, can be divided by some quantity -which is common to each of them, the quo- tients may be used instead of the fractions themselves ; thus 92 ALGEBRAIC FRACTIONS. a-{-b X X rX — — , = J ; cancelling x-\-h in the numerator of the a—b a-\-b a—b ^ one* and denominator of the other. Ex. 1. Muhiply^by ^. 3aX4a=12a2— numerator, ) *i, r *• • j • 5 X 7=35= denominator ,' \ ''' '^^ ^^'^^^^^ ^^^^^^^ ^^ 12a2 35 • Ex.2. Multiply?^ by y. Here, (3x4-2) X 8a;=i24a;2-j- 16a;=: numerator, and 4x7=28= denominator ; 24a;2-{- 16a: Therefore, -— — = (dividing the numerator and de- 28 nominator by 4) — , the product required. ^2 ^2 '7/p2 Ex. 3. Multiply — - — by 3a -^ a—x Here, (a^— x^) X lx'^=(a-\-x) x (a—x)x '7x'^= numerator (Art. 106), and 3ax(a—x)= denominator ; see Ex. 15, (Art. 79.) , . (a-\-x) X {a—x) X 7a;2 . Hence, the product is ^ '- — ^ = (dividing 7a;2 /q _|_ ^\ the numerator and denominator by a—x,) = 7ax^+7x3 3a ' Ex. 4. Multiply a+^ by a—- X 5a+x - X 3a — x Here,a+-=-— , and a^-=-^ : Then, (5a4-a;)x(3a— x)r=15a2— 2ax— x2= new numerator, rm r ISa^ — 2ax— x^ and 5x3 = 15= denominator: Therefore, - 15 2 ftdc "T" jc is the product required. 15 156. But, when mixed quantities are to be multiplied to- gether, it is sometimes more convenient to proceed, as in the multiplication of integral quantities, without reducing them to improper fractions. ALGEBRAIC FRACTIONS. 93 Ex. 5- Multiply a;2- ia;+| by Ja:+2. J^H-2 t 3-^ ■i^+i Ex. 6. Multiply — — by Ans. 14 -^ 3a;3-3a; 3ax — 5a t:^ .r Tv/r n- i 3a;2 15a?-30 ^ 9a: Ex. 7. Multiply 3—^ by -^^. Ans. _. ■I? o TIT u- 1 2a--2a? , 3ax ^ 2x Ex. 8. Muluply -^^ by 3^^-. Ans. ^. Ex. 9. It is required to find the continual product of 3a 2x^ , a~^b ^ 2ax+2bx -5-' -3-' ^""^ ~^' ^"^- 5 ' Ex. 10. It is required to find the continued product of a*— a?* a-\-y . a—y Ex. 11. It is required to find the continued product of a2_a.? ^2_^2 ^ a2_3^, — pv-, — :-- , and -. Ans. . a+o a-\-x ax—x^ x Ex. 12. Multiply a?2—|a?+l by a;2— ^a;. Ans. a:^— 5a;34-ya:2— Jar To divide one fractional quantity by another. RULE. 157. Multiply the dividend by the reciprocal of the divisor, or which is the same, invert the divisor, and proceed, in every respect, as. in multiplication of algebraic fractions; and the product thus found will be the quotient required. When a fraction is to be divided by an integral quantity ; the process is the reverse of that in multiplication ; or, which is the same, multiply the denominator by the integral, (Art. 120), or divide the numerator by it. The latter mode is to be preferred, when the numerator is a multiple of the divisor. 94 ALGEBRAIC FRACTIONS. Ex. 1. Divide — by-. a '' c u c ^t* c 5coc The divisor - inverted, becomes y. hence — Xt = —t is c b a ab the fraction required. T-. ^ T^- -1 3a — 3a: , 5a— 5a? Ex. 2. Divide ;— by — -r— . a-\-b ^ a-\-b The divisor ( — —: — ) inverted, becomes \ a-\-b I 5a — 5a; , 3a— 3a? a-i-b 3a — 3a; 3(a — x) 3 . , hence tt-Xt j-=^ T"— ^7 \='^ ^^ t^® ^^^' a-^-b 5a — 5x 5a— 5x 5[a—x) 5 tient required. Ex. 3. Divide ^ "~ by a+b. X I (j^ ^2 \ The reciprocal of the divisor is — — r ; hence ^X — — r ^ a-\-b X a-\-b (a4-^)(« — ^) a—b . . . ^ . , z=:- p — M ^® *"® quotient required. Or, ~z=a—b ; hence is the fraction required. a-f-6 X Ex.4. Divide — ; — by oH . a+c . « . a:2-.a2 a2_^^2_«2 a;2. , .,/..• a;^— a2 Here, a-\ = — =— ; then, the fraction — ; — a . a a a-tc -f-6=cH-c? — b-{-b ; but b — 6r=0, .•.x=c-\-d. 162. If every term on each side of an equation be multiplied by the same quantity, the results will be equal : because, in multi- plying every term on each side by any quantity, the valiJfe of the whole side is multiplied by that quantity ; and, (Art. 50), if equals be multiplied by the same quantity, the products will be equal. Thus, if x=:^-\-a, then 6a?=30-|-6a, by multiplying every X term by 6. An3,if -=4, then, multiplying each side by 2, 2 we have -X2=4x2, or a;=8, because, (Art. 155),- X2=x. Also, if- — 3=a — i, then, by multiplying every term by 4, we shall have x — 12= 4a— 46. •. . 3 Again, if 2x — --\-\z=zx ; then, 4a; — 3 4-2 =20; ; and Ax — 2a;i=3— 2, or 2x—\. Cor. 1. Hence, an equation of which any part is fraction- al, may be reduced to an equation expressed in integers, by multiplying every term by the denominator of the fraction ; but if there be more fractions than one in the given equation, it may be so reduced by multiplying every term by the pro- duct of the denominators, or by the least common multiple of 10_ 98 SIMPLE EQUATIONS. them ; and it will be of more advantage, to multiply by the least common multiple, as then the equation will be in its lowest terras. CC *T C£ ^ Let 2 + q + I"-'-^ ' ^^®"' ^^ every term be multiplied by 24, which is the product of all the denominators ; we have ^X24 + ^X24-1-^X24 = 11X24; and 12a;+8a;+6a?=264 ; "^ d 4 or, if every term of the proposed equation be multiplied by 12, which is the least common multiple of 2, 3, 4, (Art. 146) ; we shall have 6a:+4:c + 30?= 132, an equation in its lowest terms. Cor. 2. Hence also, if every term on both sides have a common divisor, that common divisor may be taken away ; , ..3a; , a4-6 2x-\-1 . , . , . ^ , thus, It — -j — r— = — r — , then, multiplymg every term by 5, o o o we shall have dx-\-a-\-Q—2x+l, ox x=zl—a. Also, if 1 — = , then multiplying by c, we shall have ax — b-\-2=:7 — a?, or Ga?+iP=^+4. 163. If every term on each side of an equation he divided hy the same quantity, the results will be equal : Because, by divid- ing every term on each side by any quantity, the value of the whole side is divided by that quantity; and, (Art. 51), if equals be divided by the same quantity, the products will be equal. Thus, if 6a'^+3x=z9', then, dividing by 3, 2o2-f a?=3. Also, if ax^-\-bx=acx; then, dividing every term by the common multiplier x, we shall have 4- — = — , or ax-jro ^ XXX =ac. Cor. 1. Hence, if every term on both sides have a common multiplier, that common multiplier inay be taken away. Thus, if ax-{-ad=ab, then, dividing every term by the com- mon multiplier a, we shall have x-{-d = b. (IOC (lb 4fl 3? Also, if 1 — = ; then dividing by th€ common multi- c c c plier -, or (which is the same thing) mwltiplying by -, we shall have x-\-b=zAax. Cor. 2. Also, if each member of the equation have a com- mon divisor, the equation may be reduced by dividing both Bides, by that common divisor. SIMPLE EQUATIONS. 99 Thus, Uax^-~a'^x=abx — a^b, or {ax~a'^)x={ax—a'^)b ; then it is evident that each side is divisible by ax — a^, whence x=b. Again, if x"^ — a'^=:x-\-a ; then, because x"^ — a^=(x-\-a) . {x—a), it is evident that each side is divisible by x-{-a ; and hence we have = — i — , or x—a=l, and x=a-{-l. x-f-a x-jra 164. The unlinown quantity may be disengaged from a divi- sor or a coejficient, by multiplying or dividing all the terms of the equation by that divisor or coejficient. Thus, if 2a;+4=5, then x-\-2= and a;=- —2. 2 2t X Also, let -4-9=17 ; then, multiplying by 2, we shall have |x2+18 = 17X2, 4> or a; + 18 = 34, .-. a;=34 — 18. Again, let ax-\-bx:=c—d, or, which is the same, let {a-\-b)x =c—d ; then, dividing both sides by a+^, the coefficient of X, and we shall have c — d X=z -. a+b Finally, let Y=c-\-d ; then, the equation may be put under this form, \a bJ' ■c+d; and dividing each side by j-, we shall have x={c-{-d)-r' I t) ; which may be still farther reduced, because j- = — 7— ; therefore ao or x=(c+d)x 7 , — a abc-\-abd .•.X=z — . — a 165. Any proportion may be converted into an equation ; for the product of the extremes is equal to the product of the means. 100 SIMPLE EQUATIONS Because, i( a : b : : x : d ; then t= jj (Art. 24), and .*. (Art. 162), ad=zbx, by clearing effractions. Let 3a; : 5a: : : 2a; : 7 ; then 7x3x=2xX5x, or 21a:=10a;2 : and .•.21=:10a;. Again, let 5a;+20 : 4x+4: : : 5 : a;+] ; then, (5X+20) x{x-hl) = 5x (4a:+4) ; or, 5a;2+25a;+20=20a; + 20 ; and (Art. 161), 5a;24-25a;=20a; ; .-.(Art. 163), 5a:+25=20r 166. When an unknown quantity enters into, or forms a part of an equation ; and if the equation can be so ordered, that the unknown quantity may stand by itself on. one side, with its simple or first power, and only known quantities on the other, the quantity that was before unknown, will then become known. Thus, suppose 3a;-|-18=:5a; — 2; then, by transposing 3a; and — 2, we shall have 184-2=5a;— 3a;, or 20=:2a; ; 20 ,^ thereiore, a;=r— =10. Here, in the above equation, the value of the unknown quantity a;, becomes known, and 10 is the value of x that ful- fils the condition required, which we can readily see verified, by substituting this value of x in the givjen equation ; thus, 3a;=3 X 10 = 30, and 5a; = 5 X 10=50 ; hence, 3a;+ 18 = 30+18 = 48, and 5a;— 2 = 50 — 2 = 48 ; therefore 10 is the true value of x, which answers the condi- tion required, and this value of x is called the root of the equa- tion. 167. Hence the root of an equation is such a number or quantity, as, being substituted for the unknown quantity, will make both sides of the equation vanish or equal to each other : Thus, in the simple equation 3a;— 9 + 6=0; the value of x must be such, that if substituted for it, both sides must vanish, because the right-hand side is ; but this value is found to be 1, for by transposition 3a; = 9 — 6=3, and dividing by 3, we shall have 3a; 3 -=-,ora.= l; therefore 1 is the root of the given equation, which can be easily verified by substituting it for x ; thus, SIMPLE EQUATIONS. 101 3a;— 9 + 6 = 3x1—9 + 6 = 3— 94-6=9— 9=0. Hence, the value of the unknown quantity being substitu- ted in the equation, will always reduce it to 0=0. ^ II. RESOLUTION OF SIMPLE EQUATIONS, Involving only one unknown Quantity . 168. The resolution of simple equations is the disengaging of the unknown quantity, in all such expressions, from the other quantities with which it is connected ; and making it stand alone, on one side of the equation, so as to be equal to such as are known on the other side, or, which is the same thing, the value of the unknown quantity cannot be ascertain- ed till we transform the given equation, by the addition, sub- traction, multiplication, or division of equal quantities, so that we may fully arrive at the conclusion, a;=n, n being a number, or a formula, which indicates the opera- tions to be performed upon known numbers. This number n being substituted for x in the primitive equation, has the pro- perty of rendering the first member equal to the second. And this value of the unknown quantity, as has been already ob- served, is called the. root of the equation, this word has not here the same acceptation as in (Art. 15.) 169. In the resolution of simple equations, involving only one unknown quantity, the following rules, which are dedu- ced from the Articles in the preceding Section, are to be ob- served. RULE I. When the unknown quantity is only connected with known quan- tities hy the signs plus or minus. 170. Transpose the known quantities to one side of the equation, so that the unknown may stand by itself on the other ; and then the unknown quantity becomes known. Ex. 1. Given a;+8 = 9, to find the value of a?. By transposition, a:=9 — 8, .-. j;=l. Ex. 2. Given 3a;— 4=2a; + 5, to find the value of ap. By transposition, 3a;— 2a;=5 + 4, .*. a;^9. Ex. 3. Given x\-a=ia-\-^^ to find the value oi x, , By taking a from both sides, we have a;=5 ; or by transposition, a;=a— a+5 ; but a—a=iO .'. a?=5. 10* 102 SIMPLE EQUATIONS. Ex. 4. Given 9 — x=:2, to find the value of a?. By changing the signs of all the terms, we have — 9 + a:=— 2, by transposition, aci^Q— 2, .•.x=7. It may be remarked, that it is the general practice of Ana- lysts, to make the unknown quantity appear on the left-hand side of the equation, which is principally the reason for changing the signs. Ex. 5. Given — h — x=a — c to find x in terms of a, b, and c. (161. Cor. 1), by changing the signs of all the terms, we have 5-f-a: — c — a ; .'. by transposition, a?z=c — b—a. Ex. 6. Given 2x—4:-\-7 = 3x—2, to find the value of a;. (161.) by transposition, 2x—3x=4:—7—2, and (161. Cor. 1), by changing the signs, 3a; — 2a;=:74-2— 4 ; but 3a; — 2a;= a:, and 74-2 — 4=:5 ; .-. x=:5. Ex. 7. Given 7a;-|-3— 5= 6a?— 2 + 7, to find the value of x. Ans. a: = 7. Ex. 8. Given 3a;+5— 2— 2a?— 7=0, to find the value of x. Ans. a?=4. Ex. 9. Given a?— 3-f4 — 6 = 0, to find the value of a?. Ans. x==5. Ex. 10. Given 7-|-a7ni2a?-f-12, to find the value of a?. Ans. a;=— 5. Ex. 11. Given 12 — 3a?=9— 2a?, to find the value of a?. Ans. a?=:3. Ex. 12. Given x — a-i-b — c=:0, to find the value of a? in terms of a, b, and c. Ans. x = a — b-\-c. Ex. 13. Given x—a-\-b=2x — 2a+5, to find the value of x in terms of a and b. Ans. x=a. Ex. 14. Given 2x-{-a:=x-{-bj to find a? in terms of a and b. Ans. x=zb—a. RULE n. 171. Transpose the known quantities to one side of the equation, and the unknown to the other, as in the last Rule ; then, if the unknown quantity has a coefficient, its value may be found by dividing each side of the equation by the coeffi- cient, or by the sum of the coefficients. Ex. 1. GiVen 3a?-f-9 = 18, to find the value of x. By transposition, 3a?=18— 9, or 3a?=:9 ; dividing both sides 3a; 9 of the equation by 3, the coefficient of a;, we have-^=-, .*. x o o =3. SIMPLE EQUATIONS. 103 Ex. 2. Given 2a— 3=i9— a;, to find the value of ar. By transposition, 2a:+a?=i9+3, by collecting the terms, 3a?=:12, 1 1- • . 3a; 12 by division, — =— ; .*. x=4. Ex. 3. Given 7—4x=z3x—7, to find the value of «. By transposition, —4x — 3a: = — 7 — 7, by collecting the terms, — 7£c= — 14, by changing the signs, 7a:=14, , ,. . . 7x 14 - by division, -— = — - ; /. a;=2. Ex. 4. Given 6a:+10 = 3a:+22, to find the value of ar. By transposition, 6x — 3aj=:r22 — 10, by collecting the terms, 3a: =12, u A- ■ 3a; 12 . by division, — =— ; .-. a:=4. o o Ex. 5. Given aa?+^=--c to find the value of x in terms of a, b, and c. By transposition, ax=c—b, , ,. . , ax c — b c—b by division, — = ; .*. x= . a a a The value of x is equal to c— 5 divided by a, which may be positive or negative, according as c is greater or less than 9 5 b\ thus, if c=r 9, ^=5, 0=2, then a;=— -— =2 ; if c=12, &= 2 16, and a=i2, then, — "^ — =^=—2. Ex. 6. Given 3a;— 4 = 7a;— 16, to find the value of x. Ans. a =3. Ex. 7. Given 9— 2a;=3a;— 6, to find the value of ar. Ans. a?=3. Ex. 8. G'lYen' ax"^ -\-hxz=z9x'^ -\-cx, to find the value of x in c— i terms of a, b, &c. Ans. x=z -. Ex. 9. Given a;— 9 = 4a;, to finj the value of a?. • Ans. x=. — 3. Ex. 10. Given 5aa;— c=i— 3 to find the value of a;. 5 5 5 -_,.,. , ^ 5aa; 5bx 5x2a Multiplying by 5, — —=-—~^ .'. ax — bx=2a, by collecting the coefficients, (a— i)a;=2a, .•.by division, x= r. •^ a—b Ex. 6. Given 1 — --= — [-3, to find the value of a;. c 2 a Here 2ac, the product of 2, a, and c, being the least common multiple, Multiplying by 2ac, 4a2a;+3a6ca;=10ca;4-6ac, ♦ by transposition, and collecting the coeflicients, we shall have (Aa^-^-Sabc — 10c)a:=6ac, .-. by division, xz= — -. •^ 4a2+3aoc — 10c Ex. 7. Given 3a;— ^^^^^ ^="~t^ A' *o ^"^ ^^® ^^' me of X. Multiplying by 12, the least common multiple, we have 36a;— 3a;+12— 48=:20a;-h56— 1, by transposition, 36a;— 3a;— 20a;=56 — 1+48— 12, or 13a; = 91, , ,. . . 13a; 91 by division, Y3-=j3 ; ••• a?=7. 106 SIMPLE EQUATIONS. Ex. 8. Given—— —=— -, to find the value of a:. Ans. x=l. Ex.9. Given ^+^ = 16-^, to find the value ofjc. Ans. a; =13. Ex. 10. Given ^1^+^=20—^^^ — , to find the value of sc. Ans. x=z22\. XI X 19 X Ex. 11. Given x-\ — = — - — , to find the value of x. o 2 Ans. xz=b. T« ,r, ^- a?— 5 , ^ 284 — X ^ , , Ex. 12. Given — \-Qx= , to find the value of x. 4 5 Ans. a:r=9. Ex. 13. Given 3x+^^^=5+li^^=:5I, to find the value of x. Ans. a?=7. Qx 4 18 4ar Ex. 14. Given — 2= — \-x, to find the value of o o X. Ans. a;=4. T. ,= r.- «^— 3 hx-^2 2aj— 9 a:— 1 ^ ^ , Ex. 15. Given — — = — -— , to find the o o ^ «J 8T value of a?. Ans. x=z 106 + 20— 6a* T^ ,^ r^- ^—1 ^+3 2x+l a;— 3 ^ , , Ex. 16. Given — i~=~T7 X~'*^ ^"^ ^^® ^^" lue of a;. Ans. a;=: — 9^. RULE IV. 173. If the unknown quantity be involved in a proportion, the proportion must be converted into an equation (Art 165); and then proceed to resolve this equation according to the foregoing Rules. Ex. 1. Given 3a; —2 ; 4 : : 5a;— 9 : 2, to find the value of at. Multiplying extremes and means, we have 2(3a;— 2) = 4(5a;— 9), or 6a;— 4 = 20a;— 36, by transposition, 6a?— 20a: =—36 +4 ; or — 14a;=— 32, by changing the signs, 143*= 32, SIMPLE EQUATIONS. 107 by division, ^-= j^ ; •*• a?=2f . ' Ex. 2. Given 3a : x : : b-{-5 : a:— 9, to find the value of «. Multiplying extremes and means, we have 3a . (x—9)=x . (&4-5), • or 3ax—27a=bx-\-5x, by transposition, 3ax—bx — 5ir=27a, collecting the coeff's, (3a— b — 5)x=27a, /. by division, x=- — -. 3a — — o Ex. 3. Given — -- : a;— 5 : : 7: '• -:, ^o find the value of x, 4 3 4 Multiplying extremes and means, we have 3 /x-5\ 2 . .. 4 -(^ = 3 •("-")' 3j;— 15 2a;-10 °'-T6-=-F-' by clearing of fractions, 9a?— 45 = 32a; — 160, by transposition, 9a: — 32a7=45 — 160, collecting and changing signs, 23a:=115, ..... 23a: 115 . by division, — =_- ; .-. a:=5. Ex. 4. Given 2a:— 3 : a:— 1 : : 4a: : 2a:4-2, to find the value of a:. Multiplying extremes and means, we shall have (2a:— 3) . (2a:+2)==4a:(a— 1), or 4a:2— 2a?— 6 = 4a:2— 4af, by transposition, &c., 2a:=6, .-. by division, a?=:3. Ex. 5. Given a-^x : b : : c— a? : d, to find the value of x in terms of a, b, c, and d. Multiplying extremes and means, ad-\-dx=bc — Ja;, by transposition, bx-\-dx = bc-^adj or {b + d)x=bc-^ad, , ,. . . be— ad .'. by division, x= . , -j-. o-\-a /p ] ^ Ex. 6. Given — -— : a:+2 : : - :.l, to find the value of a?. 3 4 Multiplying extremes, &c., — — — — , • 3 4 clearing of fractions, 4a?— 4 =9a:4- 18, by transposition, 4a?— 9a; =18+4, 108 SIMPLE EQUATIONS changing the signs, &c., 5a;=:— 22, 22 /.by division, x= = --4f. Ex. 7. Given 2a?— 1 : x4-l : : -r- • t> ^o fij^cl the value of a;. 2 4 •Ans. ic= — IJ. Ex. 8. Given x-\-3 : a : : b : - to find the value of x. X Ans. a;: ab — 1 1 3x Ex. 9. Given -:—-:: 5 : 2a:— 2, to find the value of a?. 2 4 Ans. a?= — j^. 7 4 4 4 3 2a: 1 Ex. 10. Given - : - : : a;— 1 : — : — , to find the value of x. Ans. a;=lj-^. Ex. 11. Given — - — : — - — ; ; 6 : 3, to find the value of a?. Ans. a?=3. 3a 3a ^ III. EXAMPLES IN SIMPLE EQUATIONS, Involving only one unknown Quantity. 174. It is necessary to observe that an equation express- ing but a relation between abstract numbers or quantities,»may agree with many questions whose enunciations would differ from that of the one proposed : but the principles of the reso-. lution of equations being independent of any hypothesis upon the nature and magnitude of quantities ; it follows, therefore, that the value of the unknown quantity substituted in the equation, will always reduce it to = 0, although it may not agree with the particular question. This is what will hap- pen, when the value of the unknown quantity shall be nega- tive ; for it is evident that when a concrete question is the subject of inquiry, it is not a negative quantity which ought to be the value of the unknown, or which could satisfy the question in the direct sense of the enunciation. The negative root can only verify the primitive equation of a problem, by changing in it the sign of the unknown ; this equation will therefore agree then with a question in which the relation of the unknown to the known quantities shall be different from that which we had supposed in the first enuncia- SIMPLE EQUATIONS. 109 tion. We see therefore that the negative roots indicate not an absolute impossibility, but only relative to the actual enuncia- tion of the question. The rules of Algebra, therefore, make not only known certain contradictions, which may be found in enunciations of problems of the first degree ; but they still indicate their rectification, in ren- dering sub tractive certain quantities which we had regarded as additive, or additive certain quantities which we had regarded as subtractive, or in giving for the unknown quantities, values affect- ed with the sign — . Hence, it follows, that we may regard as forming, properly speaking, but one question, those whose enunciations are not connected to one ^another in such a manner, that the solution which satisfies one of the enunciations, can, by a simple change of the sign, satisfy the other. We must nevertheless observe that we can make upon the signs and values of the terms of an equation, hypotheses which do not agree with the enunciation of a concrete question, whereas the change which we will make in this enunciation might be always represented by the equation. These principles, which will be illustrated by examples, are applicable to equations of all degrees, and to determinate equa- tions containing many unknown quantities. The question which conducts to the equation, ax-^b^=cx-\-d, is not well enunciated for a>c, and by-d, since the first mem- ber is greater than the second. Thus the formula d-b x= , a — c gives for x a negative value ; but by rendering the unknown X negative, the equation is changed into the following, b^ax=:id—cx, which is possible under the above relations between a and c, b and d, and which gives then for x an absolute value. If we have b^d and c^a, the two subtractions become im- possible in the formula d-b x=z ; a — c but in order to resolve the equation, let us subtract cx-\-h from both members, which would be impossible, because that cx-\-b is greater than each of the two members : we must 11 110 SIMPLE EQUATIONS. therefore, on the contrary, take away ax-\-d from both sides, and it becomes b — d=.cx — ax ; from whence we deduce _b-d c~a This formula compared to the preceding, differs from it in this, that tlte signs of both terms of the fraction are changed. We may therefore conclude, that we can operate on negative isolated quantities, as we would do if they had been positive. These principles will be clearly elucidated, when we come to treat of the solutions of Problems producing simple Equa- tions : we shall now proceed to illustrate the Rules in the pre- ceding Section, by a variety of practical examples. T? in- 01. 3a:-ll 5a: -5 97 -Tar _ , , Ex. 1. Given 21-f — r^ — =—x 1 — , to find the lo o Z value of X. Multiplying both sides of the equation by 16, the least com- mon multiple of 16, 8, and 2, we shall have 336 + 3a:— ll = 10j:-I0 + 776-56a7; /. by transposition, 3a?- 10a?4-56ar=ll- 10 + 776—336, or 49a?=441 ; 441 by division, ^—-7^^ .'.x=9. ^/p, 5 2a: 4 Ex. 2. Given a; -I — =12 — , to find the value of r. Multiplying both sides of the equation by 6, the product of 2 and 3, which is the least common multiple, we have 6a:H-9a^-15 = 72— 4a:-f 8 ; /.by transposition, 6a?-|-9a:H-4a7— 72-f 8+15,- or 19a:=95 ; 95 by division, a?=:— , .'. x=b. 2x 4 In this example, when the fraction — , is multiplied o I2x 24 by 6, the result is =— (4a:— 8)=— 4a?-|-8, or, which is the same thing, when the sign — stands before a fraction, it may be transformed, so that the sign -f- may stand before it, by changing the sign of every term in the numerator ; therefore, we make the above step —4a; +8, and not 4a;— 8. SIMPLE EQUATIONS. lU Ex. 3. Given 4x i" =«H i" — (-24, to find the value of a?. Multiplying by 10, the least common multiple, and we have, 40a;— 5a: -j-5 = 10a; -f4jc— 4+240, by transposition, 40a; — 5a;— 10a; — 4a;=:240— 4 — 5, or, 40a;— 19a;=231 ; and 21a;=z231, * by division, x=—-, .-. a;=ll. Ex. 4. Given 2a;----|-l=5a;— 2, to find the value of a?. Multiplying by 2, we have, 4a;— a;-f2 = 10a;— 4, .-. by transposition, 4«— a;— 10*= —4—2, or — 7x=— 6, by changing the signs, 7a; =6, .-.by division, x=-. Ex. 5. Given ^ax—2bx=3b—'a, to find the value of a;. Here, 3aa;— 2fta;=(3a— 2^)a;, by collecting the coefficients of X. Therefore, {3a—2h)x—3b — a, . ,. . . 3b = a by division, x= -r, Ex. 6, Given bx-{-xz=:2x-{-3a, to find the value of a;. by transposition, Z>a;-4-a:— 2a;=3a, or (b — l)x=3ay .*. by division, x=- — -. 3a; X 2x Ex. 7. Given c+Y=4a;-| — -, to find the value of a;. a a Multiplying by abd, we have, 3bdx — abcd-\- adx =z 4abdx -\- 2abXf by transposition, 3bdx-\-adx—4abdx — 2abx = abcd, or {3bd-{-ad—4:abd—2ab)x=abcd, . ,. . . abed •. by division, x=—rr-, — j — 7-7-5 — ;l— ?• •^ 3bd-\-ad—4:abd—2ab Ex. 8. Given c— 7^+7^ = ^+<^j to find the value of x. 5 6 6 Multiplying by 30, the product of 5 and 6, the product be- comes 112 SIMPLE EQUATIONS. 6x--5x-{-5a=30b + 30c; by transposition, 6x—5x=30b-\-30c—5a, and .-. x=z30b+30Q—5a. •« « r>.- 12 — a; ^ 144-a; , « ,. -, i , Ex. 9. Given — -— : 5x — : : 1 : 8, to find tne value y o of a;. Multiplying extremes and means, we have 96—80; ^ l4: + x 9 3 ' Multiplying by 9, the least common multiple, 96— 8a:=45a:— 42 — 3ar, by transposition, — 45a7 — 8a;-f 3a;= — 96 — 42, by changing the signs, 45a!+ 8a;— 303=96+42, or 50o;=138, ^ ,. . . 138 ^19 • .-.by division, a;=--=2-. in ,^^' 0.x — h , a bx bx—a ^ , , . Ex. 10. Given — ro~~^ o — » ^^ """ ^^® value of X. Multiplying by 12, the least common multiple of the deno- minators, and the equation will become, 3ax—3b-{-4a=z6bx—4bx+4a, , . (1), by taking away 4a from each member, we shall have 3ax—3b=z6bx—4bx=2bxy by transposing —3^ and 2bx, it becomes 3ax—2bx=3b, by collecting the coefiicients of x, we shall have {3a—2b)x=3b, by division, a.=3-?^. Ex. 11 . Given 2ax-\-b=3cx-\-4a, to find the value of x. by transposition, 2ax—3cx = 4a — by by collecting the coefficients, (2a— 3c)a:=4a — i, .*. by division, a:= —. ^ ' 2a — 3c Ex. 12. Given 19a?+ 13=59— 4ar, to find the value of x. by transposition, 1 9a; -f- 4a; =59 — 13, or, 23a; = 46 ; .- by division, a?=2. X Ex. 13. Given 3a;+4— -=46--2a:, to find the value of op. Multiplying both sides by 3, 9a;4-12— a;=138— 6a?, SIMPLE EQUATIONS. 113 by transposition, 9x-\-6x'—x=\38—'l2, or I4a;=126 ; by division, a;=-— -, .*. a?=9. Ex. 14. Given a;2_|_i5a._35a:_3a;2^ to find the value of x. Dividing every term by x, 074-15 =35— 3a:, by transposition, ar-f3x=35 — 15, or 4a;=20 ; .•.x=z5. Ex. 15. Given-— j+ 10=^— ^4- 11, to find the value of a?. Here 12 is the least common multiple of 6, 4, 3, and 2 ; .*. multiplying both sides of the equation by 12, 2aj— 3a;+120=4a;— 6a;+132; by transposition, 2a?— 3aj— 4a? + 6a; =132 — 120, or 8j?— 7a:=12 ; .-. a?=12. f-. -« ^- a?— 1 , 23 — A _ 4 + a; - , , t Ex. 16. Given — -— + — - — -=7 — , to find the value 7 5 4 of X. Ans. x=8. ■r. ,^ r.' 7a:4-5 164-4a: , ^ 3a?+9 ^ , , Ex. 17. Given — ^ ^ \-6= , to find the value of a:. , Ans. a?=l. ^ ,o r^- 17-3a; 4a;4-2 ^ . 7a;H-14^ ^ , Ex. 18. Given ■ — =5— 6a:H j- — ,tofind 5 3 o the value of x. Ans. a? =4. 17 ion- 3a?-3 , , 20-« 6a;-8 4a:-4 Ex. 19. Given x h— +4 = —^ ^ ^—r~» 5 2 7 5 to find the value of x. Ans. a; =6. T^ «^ r.- 4a:— 21 . „, . 57— 3a; „,, 5a; — 96 Ex. 20. Given ^ +3j+ ^^"=241 12~ "" 11a:, to find the value of a:. Ans. a:=21. 17 o, n- 6a:+18 ^, ll-3a: ^ ._ 13-a: Ex. 21. Given— j^ 4f _-=5a;-48 ^^ — — , to find the value of a?. Ans. a: =10. a^—Sbx , , ebx—5a^ Ex. 22. Given ax ah^ =z bx -] a ^ict 5a;+4a . . v. , r a 4g&'^-10a — - — , to find the value of x. Ans. a;= . __g, . 11* 114 SIMPLE EQUATIONS. Ex. 23. Given -^^^ _Z__^_^ to find the value of ar. Ans. a;=8. Ex. 24. Given -^-f-^-___=:_^, to find the value ^^^- Ans. a:==4. T?^ OK o- 4a;+3 , 7ir— 29 8a:4-19 Ex. 25. Given—^— +— -__:=_J^,to find the value ^^^' Ans. a:=6. Ex. 26. Given 12-a; : ^ : : 4 : 1, to find the value o{ x. Ans. a;=:4. Ex. 27. Given -^±i : i^Zlf : : 7 : 4, to find the value of a;. Ans. a:=2. Ex. 28. Given (2a;+8)2=4a;2+l4a:+172, to find the value o^^- Ans. a = 6. Ex. 29. Given— ^—+20?=—.^+ 16, to find the value °^^- Ans. a; =7. Ex. 30. Given I^+4 = ?i^ii+?f!+L^ to find the va- ^"6 0^^- . Ans. a:=::3. 2 "^2~~2 Ex. 31. Given ^+^r=r-^-, to find the value of a;. . 1 Ans. a:= ■3a— 1 Ex. 32. Given 2a:--^ + 15=i?^,tofindthevalue r d 5 °*^- Ans. a:=12. Ex. 33. Given 5aa;--26 + 4^a:=2a;+5c, to find the value of^- ■ Ans. cc=- ^'^^^ . 5a+46 — 2 T?^ -iA n 2a: — 5 , 19 — x lOar— 7 5 ^ , , tiX. 34. Criven = _ to find the 18 ^ 3 9 2' value of a;. Ans. a; =7. Ex. 35. Given x- ?^±l=^±i, to find the value of x. 3 4 Ans. a;=13. Ex. 36. Given -^ l±fl^3g^5^^ to find the value <>f«- Ans. a:=9. SIMPLE EQUATIONS. , /^^ 115 Ex. 37. Given4a;-^l5±5f^l5. lue of X. Ex. 38. Given value of X: 21— 3a? 4a;+6 =6— 7a; 4- 11 4 5a;+l Ex. 39. Given 7?+^- o 4 6 ~ 4 7a:+3 8a;+19 , to find the va- Ans. 07=3. , to find the Ans. a;=3. , to find the value of X. Ex. 40. Given the value of x. Ex. 41. Given a; 6a; -f 8 16 8 Ans. a;=7. 5a;+3 27— 4a; 3a;+9 ^ , ^ - — ^, to find 11 2 3 27-9a; 5a;4-2 61 Ans. a;=6. 2a;+5 29+4a; 11 5a:-l to find the value of x. Ex. 42. Given value of X. Ex. 43. Given of a;. Ex. 44. Given of a;. Ex. 45. Given the value of x. 6 15a;+8_ 12 13 7a;— 2 :3a;- 31 12 ' Ans. a:=5. to find the Ans. a;=:9. 2 10+a; 5 17-4a; Ex. 46. Given 16a;+5 the value of a;. 10 4a;— 9 7 * * 15+2a; 3 4a;-f 14 9a; 4-31 Ex. 47. Given the value of x. 4a; + 3 6a;— 43 =6f — -, to find the value Ans. a;=3. : : 14 : 5, to find the value Ans. a;=4. 2a; : ; 5 : 4, to find Ans. a;=3. : 36a;-f 10 : 1, to find Ans. x=z5. 2a;+19 : 3a;— 19, to find Ans. a;=8. 7a;_L9 lOa;^ 18 Ex. 48. Given 5a:4- ^ 7^ = 9+ - — r-r:—, to find the va- lue of X. Ex. 49. Given Ex. 50. Given lue of X, 4a;+3 9a;4-20 4a; 2a;+3 Ans. a;=3. 36 12 X --4-tj to find the value of a;. 5x — 4 4 Ans. a;=8. 20a;4-36 , 5a; 4- 20 4a; , 86 25 9a;- 16 5 ' 25 \-—-j to find the va- Ans. a;=4. 116 SIMPLE EQUATIONS. Ex. 51. Given — ;= — - — , to find the 18 13a:— 16 9 ' value of a;. Ans. a;=::4. Ex. 52. Given — -r- f--^ — rTT^ — t-a — , to find the va- 28 6a?+14 14 lue of a?. Ans. x=z7. Ex. 53. Given -^^-T =ac+-^, to find the value of a?. ox b Ans. «=-. c Ex. 54. Given r— = ^-, to find the value of x. a-{-ox e+Jx . ad — ce Ex. 55. Given -; — \-- — f-^ — f--r-=^j to find the value of x. ox ax jx hx A«« ^_ adfh-\-bc fk+b deh+ bdfg Ans. X- -^^ . find the value of x. Ans. x= Ex. 56. Given (b-^x).(b-{-x)—a.{b+c)z=—--^x'^, to ac T' T? :.^ n- 3a;-3 3a:-4 _. 27+40; . . ,. Ex. 57. Given — — =5J — , to find the value of a;. Ans. a; =9. T- .o r.- 4a;-34 258-5a: 69-a; , . , . Ex. 58. Given — — = — - — , to find the va- lue of a;. Ans. a;=51. 4a:-2 2a;+ll 7— 8a; ^ , , Ex. 59. Given 2a; tT~= — \ ^~"' ^^ ^"^ ^^® value of a;. Ans. a; =7. ^ .^ r.' 2a;4-l 402-3a; _ 471 -6a; ^ ... Ex. 60. Given —-^; — — =9 , to find the value of X. Ans. a;=:72. 3 = 14 ; then a:=16-f 14 = 30 ; that is, 30 is a number, from which 16 being subtracted, the remain- der is 14. Prob. 3. To find a number which, being subtracted from a, leaves b for a remainder. Designating the unknown number by x, we shall have this translation, a — a?=6, .'. x=a — b. 178. If we suppose a= 10, 5 = 4, we shall have a? = 6 ; then the subtraction is arithmetically performed. But if we had a=10, 5 = 14, we must subtract 14 from 10, which cannot be done except in part, or that with respect to the portion of 14 equal to 10. The excess, in as much as it exists subtractively, will indi- cate that the number x of which it is the representation must PRODUCma SIMPLE EQUATIONS. 119 enter negatively in the enunciation where it is already sub- tracted from the number a, so that the enunciation of the pro- blem is corrected and brought to these terms : to find a num.' her which being added to 10, the sum will be 14; a problem whose translation is, designating the unknown quantity by a:, 10 + a:=:14; .-. a;=14 — 10=4 ; whereas, the translation in the former case would be 10 — a;=14; .-.aj^lO — 14, or £c=--4. The negative root —4, satisfies the equation of the problem, besides it announces a rectification in the enunciation ; this is what appears evident, since the subtraction of a negative quantity is equivalent to the addition of a positive, (Art. 63). In fact, as has been already observed, (Art. 174), it makes known that the enunciation ought to be taken in an opposite sense to that which we first proposed in the problem. Prob. 4. A person lends at interest for one year a certain capital at 5 per cent ; at the end of the year, according to agreement, he is to receive a sum b, besides the principal and interest, and the whole sum he receives must be equal to the capital. I demand what is the capital ? Let the capital be designated by x : Since 100 dollars becomes at the end of the year 105 dollars, we shall have the capital at the same time by this proportion, 105a; 100 : 105 : : a; : -r^= the capital. 105a; The sum +&, by the problem, must be equal to x, we have therefore the equation 105;r -— +Z>=a;; .-. 105a;+ 1006 = 100a; ; by transposition, 5x= — l00h; .;. by division, x=—20b. 179. Thus the capital shall be — 20b. This answer does not agree with the problem, and still if this value —206, be substituted for x in the equation found, we obtain and, performing the operations indicated in the first member, it becomes —20b=—20b, which is true. This value of x, although it is negative, satis- fies the equation of the problem, as has been already observed (Art. 174), since its two members become identicalli/ eqaal by making the proper substitution . 120 SOLUTION OF PROBLEMS If we return again to the enunciation, we discover that it ia impossible that a capital augmented by the interest would re- main equal to itself, and that much more this impossibility takes place, if, besides the interest, we add to it a sum h ; it is necessary therefore that one of these two parts, namely, the interest at 5 per cent, and h, be subtracted. In fact, if we carry into the first equation this circumstance — X, which is but x=z — a number, we find 105 , , 105a; ^ -Ioo^+^=~^'-"--io^~^=^' a translation of the enunciation, by supposing the interest ad- ditive to the capital, in which case, the sum b ought to be subtracted. This equation, treated as the preceding, shall give x = 20b, If the interest at 5 per cent be subtracted from 100, in which case 100 reduces itself to 95, we have the capital x at the end of the year, by the proportion 95a; 100 : 95 : : a; : j7r^= the capital, consequently, T7)(\'^ ' multiplying by 100, and transposing, we shall 'have 100^>=:5a:, .-. x=2Qb. The negative isolated result, that is, the negative value of X, would announce a rectification or a correction in the terms of the enunciation, and the problem proposed could be re-es- tablished in two ways. Prob. 5. What number is that, the double of which exceeds its half by 6 ? Let a;= the number ; Then by the problem, 2x— -=6, .-.multiplying by 2, 4a:— a; = 12, or3a: = 12, .'. by division, a; = 4. Prob. 6. From two towns which are 187 miles distant, two travellers set out at the same time, with an intention of meet- ing. One of them goes 8 miles, and the other 9 miles a day In how many days will they meet ? Let x— the number of days required ; then 8a: = the number of miles one travelled, and 9a: = the number the other travelled ; PRODUCING SIMPLE EQUATIONS. 121 and since they meet, they must have travelled together the whole distance, consequently, 8a;4- 9a:= 1 87, or 17a?=187, .-. by division, a; = 1 1 . Prob. 7. What number is that, from v^rhich 6 being sub- tracted, and the remainder multiplied by 11, the product will be 121 ? Let x= the number required ; Then by the problem (a;— 6) X 11 = 121, by transposition, lla;=121 + 66, or lljr=187, .'.by division, a: = 17. Prob. 8. A Gentleman meeting 4 poor persons, distributed five shillings amongst them : to the second he gave twice, the third thrice, and to the fourth four times as much as to the first. What did he give to each ? Let a?= the pence he gave to the first, .■.2x=: the pence given to the second, and 3a; =: to the third, 4t= to the fourth. .-.by the problem, a;-|-2a:-f 3ir+4a;=5 X 12 = 60, or 10a; --60, by division, x=zQ^ and therefore he gave 6, 1 2, 18, 24 pence respectively to them. Prob. 9. A Bookseller sold 10 books at a certain price ; and afterwards 15 more at the same rate. Now at the latter time he received 35 shillings more than at the former. What did he receive for each book ? Let a;=r the price of a bool^. then 10a;= the price of the first set, and 15a;= the price of the second set , but by the problem, 15a; = 10a;+35 .-.by transposition, 5a;=:35 and by division, a;=:7. Prob. 10. A Gentleman dying bequeathed a legacy of 1400 dollars to three servants. A was to have twice as much as B ; and B three times as 'much as C. What were their re- spective shares ? Let a;=zG's share, .-. 3a;=:B's share, and. 6a; =A's share • i2r 122 SOLUTION OF PROBLEMS. them by the problem, a^-f 3x+6a;=1400, or 10a: =1400, .'.by division, a;=:140r=:C's share. .-. A received 840 dollars ; B-, 420 dollars ; and C, 140 dol- lars. Prob. 11. There are two numbers whose difference is 15, and their sum 59. What are the numbers 1 As their difference is 15, it is evident that the greater num- ber must exceed the lesser by 15. Let, therefore, x= the lesser number; then will a?+15= the greater ; .'.by the problem, a;+a:4-15=:59, or 2a:-i-15=z59, by transposition, 2a?= 59 — 1 5 =: 44, .'.by division, a:=:22 the lesser number, and a: 4- 15=22 4- 15 = 37 the greater. Prob. 12. What two numbers are those whose difference is 9 ; and if three times the greater be added to five times the lesser, the sum shall be 35 ? Let x= the lesser number ; then a:+9= the greater number. And 3 times the greater =:3(a?+9) = 3a;+27, 5 times the lesser =5a:. .-. by the problem, (3a;-f27)-f-5a?=35 ; by transposition, 3a? + 5a:=35— 27, or8a:=8; .•.by division, x=l the lessernumher, and a:+9= 1 + 9 = 10 the greater number. Prob. 13. What number is that, to which 10 being added, |ths of the sum will be 66 ? Let x= the number required ; then a; 4- 10 = the number, with 10 added to it. Now fths of (.+ 10)=f(.+ 10)=?(^>=^^. But, by the problem, fths of (a;+10)=66 ; 3a:+30 .-.-^—=66; by mukiplication, 3x4-30 = 330; by transposition, 3a;=300; .-. by division, a:= 1 00. Prob. 14. What number is that, which being multiplied by PRODUCING SIMPLE EQUATIONS. 123 6, the product increased by 18, and that sum divided by 9, the product shall be 20 ? Let x:= the number required ; then 6x=: the number multiplied by 6 ; 6j;+18= the product increased by 18 ; and — - — = that sum divided by 9, y 1. 1 1.1 6rr+18 ^^ .'. by the problem, — - — =20 by multiplication, 6x4-18=20x9 by transposition, 6a?=180 — 18 or6a;=162 /, by division, a;=27. Prob. 15. a post is Jth in the earth, fths in the water, and 13 feet out of the water. What is the length of the post ? Let x=^ the length of the post ; then -= the part of it in the earth, 3a: -— -= the part of it in the water, and 13= the part of it out of the water. But by the problem, part in the earth + part in water + part out of water = whole part ; y(I)+(T)+-=- and ^ X35 + ^X35+I3x35 = 35a;; 5 7 or 7a: 4- 15a: +455 = 3 5a:; by transposition, 455 = 35a: — 7a:— 15a: = 13a:, or 13a:=455 ; .*. by division, a:=35, length of the post. Prob. 16. After paying away Jth and ith of my money, I had 850 dollars left. What money had I at first ? Let X— the money in my purse at first ; then-4--= money paid away. But money at first — money paid away = money remaining ; .-. by the problem ^—(7+7) =850> era:— -—-=850. 4 7 124 SOLUTION OF PROBLEMS Multiplying by 28, the product of 4 and 7 , which is the least common multiple, and28a:— I X 28— 1x28 = 850x28, or 28j;— 7a;— 4a?=:23800, .-. 17a;=:23800 ; and by division, a;=1400 dollars. Prob. 17. What number is that, whose one half and one third, plus 12, shall be equal to itself? Let x= the number required ; then, by the problem, a!=-+-+12 ; Now to clear this of fractions, multiply by 6, and 6x=3x+2x-\-72 ; by transposition, 60?— 50?= 72 ; /.a;=72. It can be readily proved that 72 is the number required ; 72 72 thus, — +— -|-12 = 36 + 24 + 12=72. 2 o All other problems in this Section may be proved in like manner. Prob. 18. To find a number, whose half, minus 6, shall be equal to its third part, plus 10. Let 3?=: the number required ; then by the problem, -—6 = -+ 10, .-. clearing of fractions, Sa? — 36=2ji: + 60, by transposition, 3a; — 2a:=60-(-36, .-. a;=96. Prob. 19. Two persons, A and B, set out from one place, and both go the same road, but A goes a hours before B, and travels n miles an hour ; B follows, and travels m miles an hour. In how many hours, and in how many miles travel, will B overtake A ? Let a;= the hours that B travelled ; then x-{-a= the hours that A travelled. Also mx=z the nusnber of miles travelled by B ; and n(x-\-a)=nx-\-na= the miles travelled by A ; .-. by the problem, mx=nx-^na ; by transposition, mx—nx=.nay or {m—'n)x—na ; PRODUCING SIMPLE EQUATIONS. 125 ,. . . (m~n)x na .'. by division, -^ —= , m—n m — n net .'. x= , the hours that B travelled. m — n ^, . na , na-\-ma — na ma . . Then x-\-a = f-<3f= = , the hours m — n m — n m — n that A travelled ; and mx= = the miles travelled. m — n 180. This is a general or literal solution, because m, n, a, may be any numbers or quantities taken at pleasure ; for ex- ample, Let a=9, n=5, and m=:7 ; Then, A travels 9 hours at the rate of 5 miles an hour, be- fore B sets out ; and B follows after at the rate of 7 miles an hour. Now, by putting these values of a, n, and m, in the formula found above ; we have, na 9x5 45 „^- , , , _, ,, , x= =- — -= — =22^, the hours that B travelled ; m—n 7—5 2 ^ , ma 9X7 63 ^,, , , ,, j , * and x= =- — -=-—=311, the hours travelled by A. m—n 7—5 2 ^ -^ And ma;= 7x22^= 157^, the miles travelled by each. pROB. 20. Four merchants entered into a speculation, for which they subscribed 4755 dollars ; of which B paid three times as much as A ; C paid as much as A and B ; and D paid as much as C and B. What did each pay ? Here, if we knew how much A paid, the sum paid by each of the rest could be easily ascertained ; Let, therefore, «= number of dollars A paid ; 3ir= number B paid ; 4a;= number C paid ; and 7a; = number D paid ; .-. (a;+3a;+4a;+7a;=)15a:=4755, and a?=-317. .-.they contributed 317, 951, 1268, and 2219 dollars re- spectively. Prob. 21. Let it be required to divide 890 dollars between three persons, in such a manner, that the first may have 180 more than the second, and the second 115 more than the third. Here, it is manifest that if the least or third part were known, the remaining parts could be easily ascertained ; therefore, . 12* 126 SOLUTION OF PROBLEMS Let the least or third part . . z=x. Then the ^econc? part . . . =j;4-115. .'. the ^reaic^^ or ^r^i part . . =:a?4"115 + 180. But the sum of the three parts . =890. .-. 3ir+115 + 115-f 180=890, or 3a:+410 = 890; /.by transposition, 3a;=890— 410, or 3a;=480, .'. a;=:160=: least part. .•.a;4-115 = 160+ 115=275= second pan. and a;4-115+ 180=160+115 + 180 = 455=: greatest pait. Prob. 22. A prize of 2329 dollars was divided between two persons A and B, whose shares therein were in proportion of 5 to 12. What was the share of each ? Let 5jc=:A's share ; then 12a;=B's share ; .-. 5a.'+12a:=2329, or 17a:=2329 ; and a;=137 ; .-.their shares were 685 and 1644 dollars respectively. Prob. 23. A fish was caught, whose tail weighed 91bs. ; his head weighed as much as his tail, and half his body ; and his body weighed as much as his head and tail. What did the fish weigh 1 Let 2x= the number of lbs. the body weighed ; then 9-\-x = the weight of the tail ; .-. 9 + 9 + ir=2ir ; by transposition, a; = 1 8 ; .-. the fish weighed 36+27 + 9=72lbs. Prob. 24. A hare, 50 of her leaps before a greyhound, takes 4 leaps to the greyhound's three ; but two of the grey- hound's leaps are as much as three of the hare's. How many leaps must the greyhound take to catch the hare ? Let 3x= the number of leaps the greyhound must take ; /. 4x= the number the hare takes in the same time, .-. 4a;+50= the whole number she takes, and 2 : 3 : : 3j; : 4a;+50 ; .♦. 9a;=8x+100 ; by transposition, a;=100, and the greyhound must take 300 leaps. Prob. 25. The number of soldiers of an army is such, that its triple diminished by 1000, is equal to its quadruple aug- mented by 2000. What is this number ? PRODUCING SIMPLE EQUATIONS. 127 Let X designate the number required ; then, we are conducted to this equation, 3^5—1000=4^:4-2000, whence «:=— 3000, which gives an absurd answer with respect to the terras of the question, since that a number of soldiers cannot be negative. 181. We shall render this impossibility very plain, by ob- serving that the triple of a number being less than the quadru- ple of thp same number, the triple diminished by 1000' is much less than the quadruple augmented by 2000. But by writing — X in the place of -|-^ in the equation of the problem, then changing the signs of both sides, we find 3a;+1000 = 4a;— 2000; .-. a;:=3000. We can from the equation 3a;+1000=:4a;— 2000, re-establish the enunciation of the problem in such a manner that there results from the solution an absolute number, that is, a;=3000. If in place of taking x for the representation of the unknown number, we had taken 3a; — 6000, or a:=a;'— 6000 we should find for the equation a:' — 19000==4a;'— 22000 ; .-.by transposition, 22000 — 19000 = 4a;' — 3a;', and .-. a;' = 3000 as before. A I M I A' Thus the value a; = — 3000 being represented, on a line, by the length A^M, counted from A' towards M, or to the left of A^, we pass by the substitution a;=a;^— 6000 from the origin A^ to the origin A, to the left of A', and distant from A' by 6000=2A'M ; then the length AM=a/ is positive. Prob. 26. A Courier sets out from Trenton for Washington, and travels at the rate of 8 miles an hour ; two hours after his departure another Courier sets out after him from New-York, supposed to be 68 miles distant from Trenton, and travels at the rate of 12 miles an hour. How far must the second Cou- rier travel before he overtakes the first 1 N ■■ W T R M Let X represent the number of miles which the second Cou- irier travels before he overtakes the first ; then, by a little at- tention, we discover that this distance should be equal to the distance from New-York to Trenton, or NT =68 miles, plus 128 SOLUTION OF PROBLEMS the distance travelled by the first Courier in two hours which his departure preceded that of the second, together with the number of miles which the first travels whilst tjie second Courier is on route ; that is, NM, or a;=NT + TR+RM. Let us translate the two last distances, that is, TR and RM ; in the first place, 2 x 8zi= 16=TR= the number of miles which the first Courier travels before the second sets out ; then, in order to find an expression for MR ; we shall say, gince the distances passed over in an hour are as 8 : 12, or 2 : 3 ; as, 2 : 2x 3 : : MR: a;; and consequently MR =—. So that we obtain for a translation of the enunciation, ^=,68+16+1=84+1^; by multiplication, docz='2^2-\-2x ; .*. a:=i:252, that is to say, the two Couriers would meet when the second shall have travelled 252 miles. In fact, while the second 2x travelled 252 miles, the first travelled 168 miles ; since — is o the expression for the number of miles which the first travelled while the second was on route ; that is, substituting 252 for a;, 2x 2X252 504 ^^^ .. -=-3— ^-3-^168 miles. Now, the place from whence the first Courier departed, be- ing 68 miles distant from New- York, besides he has the ad- vantage of having travelled 16 miles before the other set out. Consequently 684-16-f-168 must be equal to the number of miles which the second Courier travels before they meet ; that is, 68 + 16 + 168=252. We see here an example of verification of the value of the unknown ; it is a proof which the student can, and should al- ways make. 182. In order to have a general solution of this problem, let us therefore represent in general, by a the distance between the two places of departure, which was 68 miles in the preced- ing question, by b the number of hours which the departure of the first precedes that of the second, by c the number of miles that the first Courier travels per hour, and by S the number which the second travels in the same time. Let x= the dis- tance which the second Courier must travel before they meet ; then, we shall have the distance travelled by the first Courier during the time that the second has been travelling, by calcu- lating the fourth term of a proportion that commences thus ; PRODUCING SIMPLE EQUATIONS. 129 cXx ex a : c : : X : — r— or — . a a The first Courier travelling c miles an hour, he will have tra- velled cxb miles before the second set out. Therefore by the condition of the problem, we shall have ca? , , , , d{cb+a) x= -—-{'hc-{-a ; whence a:=— -j a a — c which gives the solution of all questions of the same kind. In order to show the use of this formula, let us resume again the preceding enunciation, and by recollecting that we must replace a by 68, 6 by 2, c by 8, and d by 12. Then the value of x becomes x= — ^=252 miles as before. 12--6 Prob. 27. What two numbers are those, whose difference is 10, and if 15 be added to their sum, the whole will be 43 ? Ans 9 and 19. Prob. 28, What two numbers are those, whose difference is 14, and if 9 limes the lesser be subtracted from six times the greater, the remainder will be 33 ? Ans, 17 and 31. Prob. 29. What number is that, which being divided by 6, and 2 subtracted from the quotient, the remainder will be 2 ? Ans. 24. Prob. 30 What two numbers are those, whose difference is 14, and the quotient of the greater divided by the lesser 3 ? Ans. 21 and 7. Prob. 31. What two numbers are those, whose sum is 60, and the greater is to the lesser as 9 to 3 ? Ans, 45 and 15. Prob, 32. What number is that, which being added to 5, and also multiplied by 5, the product shall be 4 times the sum ? Ans. 20. Prob, 33. What number is that, which being multiplied by 12, and 48 added to the product, the sum shall be 18 times the number required ? Ans. 8. Prob. 34 What number is that, whose -J part exceeds its J part by 32 ? Ans, 640. Prob. 35. A Captain sends out ^ of his men, plus 10 ; and there remained J, minus 15 ; how many had he ? Ans. 150. Prob, 36. What number is that, from which if 8 be sub- tracted, three-fourths of the remainder will be 60 ? Ans. 88. Prob. 37, What number is that, the treble of which is as much above 40, as its half is below 51 ? Ans 26. Prob, 38. What number is that, the double of which ex- ceeds four-fifths of its half by 40 ? Ans 25. 130 SOLUTION OF PROBLExMS • Prob. 39. At a certain election,' 946 men voted, and tho candidate chosen had a majority of 558. How many men voted for each ? Ans. 194 for one, and 752 for the other. Prob. 40. After paying away i of my money, and then i of the remainder, I had 140 dollars left : what had I at first ? Ans. 180 dollars; Prob. 41. One being asked how old he was, answered, that the product of Jj of the years he had lived, being multiplied by f of the same, would be his age. What was his age ? Ans. 30. Prob. 42. After A had lent 10 dollars to B, he wanted 8 dollars in order to have as much money as B ; and together they had 60 dollars. What money had each at first ? Ans. A 36 and B 24. Prob. 43. Upon measuring the corn produced by a field, being 48 bushels ; it appeared that it yielded only one third part more than was sown. How much was that ? Ans. 36 bushels. Prob. 44. A Farmer sold 96 loads of hay to two persons. To the first one half, and to the second one fourth of what his stack contained. How many loads did that stack contain ? Ans. 128 loads. Prob. 45- A Draper bought three pieces of cloth, which together measured 159 yards. The second piece was 15 yards longer than the first, and the third 24 yards longer than the second. What was the length of each ? Ans. 35, 50 and 74 yards respectively. Prob. 46. A cask which held 146 gallons, was filled with a mixture of brandy, wine, and water. In it there were 15 gallons of wine more than there were of brandy, and as much water as both wine and brandy. What quantity was there of each ? Ans. 29, 44, and 73 gallons respectively. Prob. 47. A person employed 4 workmen, to the first of ■whom he gave 2 shillings more than to the second ; to the se- cond 3 shillings more than to the third ; and to the third 4 shillings more than to the fourth. Their wages amounted to 32 shillings. What did each receive ? Ans. 12, 10, 7, and 3 shillings respectively. Prob. 48. A Father taking his four sons to school, divided a certain sum among them. Now the third had 9 shillings more than the youngest ; the second 12 shillings more than the third ; and the eldest 18 shillings more than the second; and the whole sum was 6 shillings more than 7 times the sum which the youngest received. How much had each ? Ans. 21, 30, 42, and 60 shillings respectively PRODUCING SIMPLE EQUATIONS. 131 Prob. 49. It is required to divide the number 99 into five such parts, that the first may exceed the second by 3 ; be less than the third by 10 ; greater than the fourth by 9 ; and less than the fifth by 16. Ans. 17, 14, 27, 8, and 33. Prob. 50. Two persons began to play with equal sums of money ; the first lost 14 shillings, the other won 24 shillings, and then the second had twice as many shillings as the first. What sum had each at first ? Ans. 52 shillings. Prob. 51. A Mercer having cut 19 yards from each of three equal pieces of silk, and 17 from another of the same length, found that the remnants together were 142 yards. What was the length of each piece ? Ans. 54 yards. Prob. 52. A Farmer had two flocks of sheep, each con- taining the same number. From one of these he sells 39, and from the other 93 ; and finds just twice as many remaining in one as in the other. How many did each flock originally contain? , Ans. 147 Prob. 53. A Courier, who travels 60 miles a day, has been dispatched five days, when a second is sent to overtake him, in order to do which he must travel 75 miles a day. In what \irae will he overtake the former ? Ans. 20 days. Prob. 54. A and B trade with equal stocks. In the first year. A tripled his stock, and had $27 to spare ; B doubled his stock, and had $153 to spare. Now the amount of both their gains was five times the stock of either. What was that ? Ans. 90 dollars. Prob. 55. A and B began to trade with equal sums of mo- ney. In the first year A gained 40 dollars, and B lost 40 ; but in the second A lost one-third of what he then had, and B gained a sum less by 40 dollars, than twice the sum that A had lost ; when it appeared that B had twice as much money as A. What money did each begin with ? Ans. 320 dollars. Prob. 56. A and B being at play, severally cut packs of cards, so as to take off more than they left. Now it happened that A cut oft' twice as many as B left, and B cut off seven times as many as A left. How were the cards cut by each ? Ans. A cut off 48, and B cut oft" 28 cards. Prob. 57. What two numbers are as 2 to 3 ; to each of which if 4 be added, the sums will be as 5 to 7 ? Ans. 16 and 24. Prob. 58. A sum of money was divided between two per- sons, A and B, so that the share of A- was to that of B as 5 to 3 ; and exceeded five-ninths of the whole sum by 50 dollars. What was the share of each person ? Ans. 450. and 270 dollars. 13a SOLUTION OF PROBLEMS Prob. 59. The joint stock of two partners, whose particu- lar shares differed by 40 dollars, was to the share of the les- ser as 14 to 5. Required the shares. Ans. the shares are 90 and 50 dollars respectively. Prob. 60. A Bankrupt owed to two creditors 1400 dollars ; the difference of the debts was to the greater as 4 to 9. What were the debts ? Ans. 900, and 500 dollars. Prob. 61. Four places are situated in the order of the four letters. A, B, C, D. The distance from A to D is 34 miles, the distance from A to B : distance from C to D : : 2 : 3, and one-fourth of the distance from A to B added to half the dis- tance from C to D, is three times the distance from B to C. What are the respective distances ? Ans. AB = 12, BC = 4, and 00 = 18 miles. Prob. 62. A General having lost a battle, found that he had only half his army plus 3600 men left, fit for action ; one-eighth of his men plus 600 being wounded, and the rest, which were one-fifth of the whole army, either slain, taken prisoners, or missing. Of how many men did his army consist ? Ans. 24000. Prob. 63. It is required to divide the number 91 into two such parts that the greater being divided by their difference, the quotient may be 7. Ans. 49 and, 42. Prob. 64. A person being asked the hour, answered that it was between five and six ; and the hour and minute hands were together. What was the time ? Ans. 5 hours 27 minutes 16^ seconds. Prob. 65. Divide the number 49 into two such parts, that the greater increased by 6 may be to the less diminished by 11 as 9 to 2. Ans. 30 and 19. Prob. 66. It is required to divide the number 34 into two such parts that the difference between the greater and 18, shall be to the difference between 18 and the less : : 2 : 3. Ans. 22 and 12. Prob. 67. What number is that to which if 1, 5, and 13, be severally added, the first sum shall be to the second, as the second is to the third. Ans. 3. Prob. 68. It is required to divide the number 36 into three such parts, that one-half of the first, one-third of the second, and one-fourth of tlie third, shall be equal to each other. Ans. 8, 12, and 16. Prob. 69. Divide the. number 116 into four such parts, that if the first be increased by 5, the second diminished by 4, the third multiplied by 3, and the fourth divided by 2, the result in each case shall be the same. Ans. 22, 31, 9, and 54. PRODUCING SIMPLE EQUATIONS. •iSS Prob. 70. A Shepherd, in lime of war, was phmdered by a party of soldiers who took i of his flock, and ^ of a sheep ; another party took from him ^ of what he had left, and J of a sheep ; then a third party took ^ of what now remained, and J of a sheep. After which he had but 25 sheep left. How many had he at first ? Ans. 103. Prob. 71. A Trader maintained himself for 3 years at the expense of 50/. a year ; and in each of those years augmented that part of his stock which was not s'o expended by J thereof. At the end of the third year his original stock was doubled. What was that stock ? Ans. 740/. Prob. 72. In a naval engagement, the number of ships ta- ken was 7 more, and the number burnt 2 fewer, than the num- ber sunk. Fifteen escaped, and the fleet consisted of 8 times the number sunk. Of how many did the fleet consist ? Ans. 32. Prob. 73, A cistern is filled in twenty minutes by three pipes, one of which conveys 10 gallons more, and the other 5 gallons less, than the third, per minute. The cistern holds 820 gallons. How much flows through each pipe in a minute 1 Ans. 22, 7, and 12 gallons. Prob. 74, A sets out from a certain place, and travels at the rate of 7 miles in five hours ; and eight hours afterwards B sets out from the same place, and travels the same road at the rate of 5 miles in three hours. How long, and how far, must A travel before he is overtaken by B ? Ans. 50 hours, and 70 miles. Prob. 75. There are two places, 154 miles distant, from which two persons set out at the same time to meet, one tra.- velling at the rate of 3 miles in two hours, and the other at the rate of 5 miles in four hours. How long, and how far, did each travel before they met ? Ans. 56 hours i and 84, and 70 miles. 13 134 CHAPTER V. ON SIMPLE EQUATIONS, INVOLVING TWO OR MORE UNKNOWN aUANTITIES. 183. It has been observed (Art. 159), that an equation was the translation into algebraic language of two equivalent phrases comprised in the enunciation of a question ; but this question may comprehend in it a greater number, and if they are well distinguished two by two, and independent of one an- other, they furnish a certain number of equations. Thus, for example, let us propose to find two numbers, such that double the first added to the second, gives 24, and that five times the first, plus three times the second, make 65. We find here two phrases, which express the same thing in different terms ; 1st, the double of an unknown number, plus another un- knoimt number, then the equivalent 24 ; 2d, five times the first un- known number, plus three times the second, then the equivalent 65. The translation is easy, and it gives these two determinate equations : 2a:+y:=i24 ; 5ar+ 3y=65. When two or more equations, involving as many unknown quantities, are independent of one another, they are called c?e- terminate. But if for the second of these two conditions we had substituted this : and such that six times the first number, plus three times the second, make 72 ; these two phrases ex- press nothing more than the first two, since that we have only tripled two equal results ; we should have but one translation, and consequently a single equation. It can therefore happen tha^ we may have less equations than unknown quantities, and then the question is said to be indeterminate ; because the number of conditions would be insufiicient for the determina- tion of the unknown quantities, as we shall see clearly illus- trated in the following section. § I. ELIMINATION OF UNKNOWN QUANTITIES FR03V ANY NUM- BER OF SIMPLE EQUATIONS. 184. Elimination is the method of exterminating all the un- known quantities, except one, from two, three, or more given SIMPLE EQUATIONS. 135 equations, in order to reduce them to a single, or final equa- tion, which shall contain only the remaining unknown, and certain known quantities. 185. In order to simplify the calculations, by avoiding frac- tions, we shall here make use of literal equations, which will modify the process of elimination : And also, to avoid the in- convenience arising from the multitude of letters which must be employed in order to represent the given quantities, when the number of equations involving as many unknown quanti- ties surpasses two, we shall represent by the same letter all the coefficients of the same unknown quantity ; but we shall affect them with one or more accents, in order to distinguish them, according to the number of equations. 186. In the first place, any two simple equations, each in- volving the same two unknown quantities, may, in general, be written thus : ax-\-by=:c (A), a'x-\-b'y=zc' ...... (B). The coefficients of the unknown quantity x are represent- ed both by a ; those of y by i ; but the accent, by which the letters of the second equation are affected, shows that we do not regard them as having the same value as their correspond- ents in the first. Thus a! is a quantity different frOm a, b' a quantity different from b, 187. We can readily see, by a few examples, how any two simple ecjuations, each involving the same two unknown quan- tities, may be reduced to the above form. Ex. 1. Let the two simple equations, 5x-\-2y^b—y—2x+7, 9a:— 2y4-3=:a;— 7y4-16, be reduced to the form of equations (A) and (B). By transposition, these equations become 5a;+3y— y-f2a;=7-f-5, Qx—2y—x + ly — \6 — ^; by reduction, we shall have 7a;4-2y=12, 8a; + 5y=13; equations which are reduced to the form of (A) and (B), and which may be expressed under the form of the same literal equations, by substituting a, 6, and c, for 7, 2, and 12 ; and a\ b', and c^, for 8, 5, and 13. Ex. 2. Let the two simple equations, ma? 4- 6y — 7 3= pa: — 2y -f 3 , ra;— 9y+6=:3y — 3a;+12, be reduced to the form of equations (A) and (B). 136 SIMPLE EQUATIONS. By transposition, these equations become mx-{-6y — px-\-2i/ = 3-^7, rx—9y—3i/-{-3x=l2—6 ; by reduction, we shall have (m—p)x-{- 8y=10, (r+3)a:-12y=6; which are reduced to the form required, and which may be expressed under the form of the same literal equations, by substituting a for m—p, b for 8, c for 10, a^ for r-\-3, b' for — 12, and c' for 6. In like manner any two simple equations may be reduced to the form of equations (A) and (B) ; hence we may conclude that a, b, c, a, b', and c', may be any given numbers or quan- tities whatever, positive or negative, integral or fractional. It is to be always understood, that when we make use of the same letters, marked with different accents, they express different quantities. Thus, in the following equations, a, 5X4-6X5=i:-10; J °^ ^ -10 = — 10. J Therefore, by transposition, , 14 — 14=0, or = 0; and —10+10=0, or 0=0. Since (Art. 56) 14 — 14=0, and 10—10=0. If these conditions do not take place, it is evident that there must be an error in the calculation : therefore, the student, whenever he has any doubt respecting the answer, should al- ways make similar substitutions. Ex. 5. Given \ ll^+3y = 100, } ^^ ^^^ ^^^ ^^j^^^ ^^ ^ ^ 4a;— 7y= 4, S and y. Mult, the 1st equation by 7, then 77aj+21y=700, 2d ... 3, . 12a,— 21y= 12; .-. by addition, 89a;=712, 712 • by division, x=-^ ; and .-. a;=8 ; whence 3y=100-lla;=100-ll X8=100-88=12 ; .•.y=-3-=4. 144 SIMPLE EQUATIONS. -+7y=99, Ex. 6. Given to find the values of x and y Multiply each equation by 7, .-. a;+49y=693, andy4-49a:=357; .-. by addition 50a:+50y=1050, 1 ,- J- . • . 1050 ^, and by division, x-\-y=———=21 ; oU but since a;+49y=693, Bubtracting the uppet equation from the lower, we have 48y=693— 21=672, whence a;=z21— y=21 — 14=7. x-\-2 \ ■n F^ /-<• _ 7 3 ^~ ' f to find the values of « Ex. 7. Given < ,^ > „„ j v4-5 C and y. 4 y Clearing the first equation of fractions, , a;4-2+24y==:93 ; .-. by transposition, a;+24y=91 . . . (1) Clearing the second equation of fractions, y+5 + 40a;=768; .-. by transposition, 40a;+y=763 . . . (2). Multiplying equation (1) by 40, and subtracting equation (2) from it, 40a;+960y=3640; 40a;- y= 763; .-. 959y=2877, and by division, y=3 ; From equation (1), a; = 91— 24y, # .*. by substitution, a;zz:91 — 24x3, or a;=91— 72, .-. a;=19. If from equation (2), multiplied by 24, equation (1) had been subtracted, an equation would have arisen involving only a?, the value of which might be determined, and this being sub- stituted in either of the equations, the value of y might also be found. SIMPLE EQUATIONS. 145 Ex. 8. Given \ ^+y=f I to find the values of x and y. f a? — y — 0, > ^ By addition, 2x=a-\-b ; .-. a;=— ^- — By subtraction, 2y=za — 5, .-. yn: . 2 Ex. 9. Given H^+g^^^^' ^ to find the values of a; and y Multiply the 1st equation by 2, then a?+4y=24 ; 9 2nd '; . . . 2, . ic— 4y=: 8 ; by addition, 2a; =32, 32 .'. by division, x—-—=\Q. 4> By subtraction, 8y=!l6 ; , .*. by division, y= 2. Or, the values of x and y may be found thus : From the first equation subtract the second, and we have 4y=8, .•.y=2. Add the first equation to the second, and .*. a;=16. Ex. 10. Given 4rr+3y=31, and 3a:+2y=22 ; to find the values of x and y. Ans. a:=4, y=:5. Ex. 11. Given 5a;— 4y=19, and 4a?-f2y=36, to find the values of x and y. Ans. x=7, y=:4. Ex. 12. Given ^_2y=2, and ^-^Ij^yJ^ ; to find the values of^ and y. Ans. a;=ll, y=l. Ex. 13. Given 5^-^+14=:18, to find the values of x Ans. a;=5, and y=2. Ex. 14. Given?^±5y +^=8,)^^ ^^^ ^^^ ^^1^^^ ^^ ^ , 7y— 3a? , , (and y. and-i-^ y=ll») 14 Ans. a; =6, and yac8. 146 SIMPLE EQUATIONS. Ex. 15. Given 3a:+-^=22 4i and lly — ^=20, 2x \ ^^ ^"^ ^^® values of x andy. Ans. a:i=5, and y=2. Ex. 16. Given a?+l : y : : 5 : 3, \ , 2a? 5— y _41 2a;— 1 > to find the values ^" 3 2 ~i2 r~'i of X and y. Ans. a?=:4, and y=3. ^ \» r^- ^—2 10— a? y— 10 \ Ex. 17. Given — =^ , J 2y+4 2a:+y a,+ 13, > to finS the values and-^ ^=-4-3 of X and y. Ans. a; = 7, and y = 10. Ex. 18. Given a:+15y=:53, > , ^ j ,1 , r 1 and y + 3^=27, ] ^"^ ^^ ^^^ ^^^"^' ^^ ^ ^"^ y- Ans. a;=8, and y=3. Ex.19. Given 4a;4- 9y=51, ) ^ , ,, , n , and 8x- 1 3y = 9, J ^^ ^"^ ^^^ ^^^"^" «^ ^ ^^^ y- Ans. a:=6, and y=3. Ex. 20. Given |+t=6, ) o 4 r ^ ,, > to find the values of x and v- a„d|+|=5f,^ Ans. a;=12, and y=16. RULE II. 199. Find the value of one of the iinknown quantities in terms of the other and known quantities, in the«more simple of the two equations ; and substitute this value instead of the quantity itself in the other equation ; thus an equation is ob- tained, in which there is only one unknown quantity ; the va- lue of which may be found as in the last Rule. Ex. 1. Given J 3^+ ^^^^M to find the values of a? andy. From the first equation, a?r=17— 2y ; Substituting therefore this value of x in the second equation, 3.(17-2y)-y=32, or 51 — 6y — y=2 ; by changing the signs, and transposing ; SIMPLE EQUATIONS. 147 7y=51— 2 = 49, .'. by division, y—^ ; whence a:= 17— 2yr= 17—14=3. Here a value of y might be determined from either equa- tion, and substituted in the other ; from which woukl arise an equation involving only a:, the value of which might be found ; and therefore the value of y also might be obtained by sub- stitutio)!, thus ; From the second equation, 3/ = 3a;— 2 ; substituting there- fore this vahie of y in the first equation ; we have, a: + 2 . (3x— 2) = 17, or a;-f-6a; — 4 = 17 ; .-.by transposition, 7a; =17 +4=21 ,. V • • 21 by division, a: = -— , .•.a;=3 ; and .-.3^ = 30;— 2 = 3x3— 2 = 9— 2=7 Ex. 2. Given \ ^^t^^^^L . \ to ^^^ ^he values of x > 5.r-f 10 = 78 +y, S and y. From the first equation, y=60 — Sa? ; Let the value of y be substituted in the second equation, and it becomes, 5a:+10=78-f-(60-3a:). Then, by transposition, 8a;=78 + 60 — 10 ; 128 and by division, a:=-— -= 16. Whence, y=60 — 3x=60 — 3x16 = 60-48 Ex. 3. Given ^ ^ "" """' ^ ^^ ^"^ '^^ ^^^"^^ ^^ ^ .•.y=12. 3 Mult, the 1st equation by 3, then a:-hy=198-6y . . . (1), 2nd by 3, then a;— y= 186 — 6x . . . (2); From equation (1), we have a:=198 — 7y ; (2), 7a;-y=186; By substituting the above value of a:, in the last equation, it becomes 7(198- 7y)-y=186, or, 1386— 49y—y= 186; by transposition, —50y=186 — 1386 = — 1200, by changing the signs, 50y = 1200, 148 SIMPLE EQUATIONS. ..... 1200 „^ . . by division, y=— — -=24. Whence, a;=:198-7y=198— 7x24=198-168, .-. a;=30. Ex. 4. Given < , ^~c^' > to find the values of a; and y. From the second equation, a; =60 — y : By substituting this value of a? in the 1st equation, we have, 60— y4-2y=80, by transposition, y=80— 60, .•.y=20. And a;=60— y= (by substitution) 60—20, .-. 07 = 40. Ex. 5. Given J q^_ ^'Z. o' M^ find the values of a; and y. From the 1st equation, a;=17— 2y. And this value substituted in the second, . 3(17-2y)-y=2, or 51— 6y — y=2, by transposition, &;c., 7y=49, .-. by division, y=7, whence, a;=17-2y=17-2 x7 = 17-14, Ex. 6. Given < ^2_ 2Z5' ( ^^ find the values of a; and y. From the first equation, a: =5 -y, squaring both sides, a?2— .(5_yj2 And by substituting this value for x^ in the second equa- tion, it becomes, (5-y)2-y2 = 5, by reduction, 25 — 10y=5, by transposition, 10y=20, .-. by division, y=2. Whence, a;=5— y = 5 — 2 = 3. Ex. r Given < > ^nd y. ^|+8x=131,) ^ Multiplying the first equation by 8, a?-|--64y=1552, .'.by transposition, a;=1552— 64y. And substituting this value for a;, in the second equation, it becomes, SIMPLE EQUATIONS. 149 |+8(1552-64y)=:131, o by reduction, y+99328— 40963/=1048, by transposition, 4095y=z: 98280, ^ ,. . . 98280 by division, y=^^^; .•.y=24. Whence a;=1552 — 64y= 1552 — 64x24, or 0:= 1552 — 1536; .•.a;=16. The value of y might be found from the second equation, in terms of x and the known quantities ; vi^hich value of y substi- tuted for it in the first, an equation would arise involving only a?, the value of which might be found ; and therefore the value of y also may be obtained by substitution. Ex. 8. Given ?^±^=27, and ^^=6, to find the va- lues of a; and y. Ans. jc=9, and y=:6. Ex. 9. Given 15y4-4^a;=300, and a?+15y=36,to find the values of x and y. Ans. a:=6, and y=:2. Ex. 10. Given 3a;+y=:60, and 5a;+10=78H-y, to find the values of a? and y- Ans. a:=::16, and y=12. Ex. 11. Given 10a;— 3y=38, and 3a?— y=ll, to find the the values of x and y. Ans. a;— 5, and y=4. Ex. 12. Givena;+y=:198 — 6y,anda;— y=186 — 6a;, tofind the values of x and y. Ans. a;=30, and y=24. Ex. 13. Given ^+^=26, and |4-8a;=131, to find the va- lues of X and y. Ans. a; =16, and y=24. X 11 X 11 Ex. 14. Given --1-^:^7, and --+^=8, to find the values of 2 3 o 2 X and y. Ans. a;r=6, and y=:13. Ex. 15. Given 4a;4-y=34, and 4y+aj=16, to find the va- lues of X and y. Ans. a;=:8, and y=2. Ex. 16. Given 3a;+2y=54, and a: : y : : 4 : 3, to find the values of x and y. Ans. a;=12, and y=9 Ex. 17. Given ^±?-f6y=21, and ^^+5a;=23, to find the values of x and y. Ans. a;=4, and y=3. 14* 150 SIMPLE EQUATIONS. RULE in. 200. Find the value of the same unknown quantity in terms of the other and known quantities, in each of the equations ; then, let the two values, thus found, be put equal to each other ; an equation arises involving only one unknown quantity ; the va- lue of which may be found, and therefore, that of the other un- known quantity, as in the preceding rules. This rule depends upon the well-known axiom, (Art. 47) ; and the two preceding methods are founded on principles which are equally simple and obvious. Ex. 1. Given | 2^t^^^lOo' ( ^^ ^"^ ^^® ""^^"^^ *^^ ' and y. From the first equation, a:=100— 3y, ^^ ^ , 100— y and from the second, x= — - — - ; •••^=100-3,. Multiplying by 2, 100— y=200-6y, by transposition, 6y— y=200 — 100, or, 5y= 100; .'. by division, y=:20, whence, a;= 100— 3y=100 — 3x20; .•.a;=40. Here, two values of y might have been found, which would have given an equation involving only x ; and from the solu- tion of this new equation, a value of ar, and therefore of y, might be found. Ex. 2. Given ^aj+Jy =7, and ia;4-iy= 8, to find the values of X and y. Multiplying both equations by 6, and we shall have 3a:+2y=42, and 2a;-|-3y=48, 42— 2y From the first of these equations, x= — - — -^ o and from the second, x= — - — -; 42— 2y _ 48— 3y ^ •■ 3 ~" 2 * Multiplying each member by 6, we shall have 84— 4y=114— 9y; by transposition, 9y— 4y=144— 84, or5y=60; .•.y=12 SIMPLE EQUATIONS. 151 And, by substituting this value of y, in one of the values of Xi the first, for instance, we shall have 42— 24_ 18_^ 3 3 Ex. 3. Given 8a;+18y=94, and 8a;— 13y=l, to find the values of x and y. From the first equation, x= — ^—^ ; 4 and from the second, x=z — - — ^ : 8 47— yy__H-i3y •■•^T~~"~8 ' And multiplying both sides of this equation by 8, 94 — 18y=14-13y; .-.by transposition, —18y—13y=— 944-1 ; Changing the signs, or what amounts to the same thing, multiplying both sides by — 1 , and we shall have 18y+13y=94-l,or31y=93; , l + 13y 1 + 39 40 ^ whence x= -= — ■ — = — =5. 8 8 8 From the first equation, x=a-^y ; and from the second, a;= — - — ^ ; b de — cy •••«-y=— j-^; and multiplying by b, we shall have ab — by=:de — cy ; by transposition, cy — by=^der—ab ; by collecting the coefficients, {c-'b)y=de—ab ; , J. . . de — ab .*. by division, y= — ; , de — ab whence x=a—y=a r-; ^ c — b ca—ab — de-^-ab _ca — de e—b "^ c—h that is, X: Ex. 5. Given 3/r-|-7y=79, and 2y—\x=9, to find the va- lues of a: and y. Ans. «=10, and y=7. 162 SIMPLE EQUATIONS. Ex. 6. Given^i^-f 1=6, and^^4-3=4, to find the values of a? and y. Ans. a:=:ll, and y=4. 2/p 3 57 Ex. 7. Given — hy=7, and 5a:— 13y=r— ,to find the values of a? and y. Ans. a?=:8, and y=^ Ex. 8. Given !^=?fijH:l_ and 8 - ^ = 6, to 3 5 5 find the values of x and y. Ans. a; =13, and y=3. Ex. 9. Given a:+y=10, and 2a;— 3y=5, to find the values of X and y. Ans. a; =7, and y=3. Ex. 10. Given 3a?— 5y=13, and 2a:4-7y=:81, to find the values of a? and y. Ans. a:= 16, and y=7. Ex. 11. Given -^^+8^=31, and ?^ -t 10a;=192, to find the values of x and y. ' Ans. a;=19, and y=3 Ex. 12. Given ?^3^+14=18, and?^^=3, to find the values of x and y. Ans. a:=5, and y=2. Ex. 13. Given— ^-^=8--, i , ^ , , , , 6 3 r to find the values of x 7y— 3a; ( and y. and-^- — =ll+y,\ ^ Ans. a;=6, and y=8. 201. Examples in which the preceding Rules are applied, in the Solution of Simple Equations, Involving two unknown Quantities. r. . r^- .-. «^+3 r, . 3a;— 2y n Ex. 1. Given 2y ^=7H -^, j ^ , , ^ 4 5 ' f to find the va- 1 . 8— y „,, 2a;4-l i lues of a; and y. and 4a;- -^=24 J ^, ^ ^ Multiplying the first equation by 20, 40y— 5a;— 15 = 140+ 12a;-8y; .". by transposition, 48y— 17^=155. Multiplying the second equation bv 6, 24a;— 16+2y=147-6a;-3 ; « .-.by transposition, 2y+30a;=160 . . . (A). Multiplying this by 24, we have SIMPLE EQUATIONS. 153 48y+720a?=3840; but48y— 17a;= 155; I .*. by subtraction, 737x=3685, and by division, x=5. From equation (A), 2y=160 — 30a;; .-. by substitution, 2y= 160 — 150. by division, y=-r- ; .*. y=5. 2 The values of x and y might be found by any of the methods given in the preceding part of this Section ; but in solving this example, it appears that Rule I, is the most expeditious method which we could apply. Ex. 2. Given ^- —-=1 — 3J!+-^, and a? : 3y : : 4 : 7, to find the values of x and y. Reducing the first equation to lower terms, y ^^~^_i 4+y , « — y 9 18^" S"""* 6~' and therefore, multiplying by 18, 2y— 4x4- 1 = 18— 24 — 6y+3a?— 3y ; .-. by transposition, 7=7a; — 1 ly But from the second equation 7a; = 12y. Substituting therefore this value in the preceding equation, it becomes 12y— lXy=7, or y=7, 12y 84 ,^ and.-.ir=: J^z=z— -=12. 7 7 15x4-^ Ex.3.Given.-?^^.l + -^. 3a;-f2y y— 5 _ 11j;+1,52 3y+l ^""^ ~~6 r- 12 2"' to find the values of x and y. Multiplying the first equation by 33, 33a;— 9y+6— 3a:=33 + 15a:4-^; multiplying again by 3, and transposing, we shall have 45ic^ 31y=81. Multiplying the second equation by 12, 6a;+4y—3y4-15=--lla; 4-152 — 18y-6 ; .-. by transposition, 19y— 5a:=131. 154 SIMPLE EQUATIONS. Multiplying this by 9, 171y— 45a;=1179 ; but 45a;— 31y= 81 ; .'. by addition, 140y=1260 ; and by division, y=9. Now, 5a;=19y-131 = 171-131=40; /.by division, x=8. Ex. 4. Given — -^=zl8i ^^ ,i ^ , , 15 3 7 'f to find the va- j iA 6a;— 35 __ , .^ I lues of a; and y. and 10y-| — z=55 + 10a;, \ ^ o ^ Multiplying the first equation by 105, the least common multiple of 3, 7, and 15. 560+21a: = 1925— 60a?-45y4-120 ; .*. by transposition, 81a;4-45y=1485 , and dividing by 9, 9a;4-5y=165 From the second equation, 50y-f-6a;— 35=r275+50a;, .'. by transposition, 50y — 44a;=310 ; and dividing by 2, 25y— 22a;=155 ; but multiplying the equation i o5„4.45^_«25 • found above, by 5, \ ^5y-f-45a;_-B^5 , .*. by subtraction, 67a; =670, and by division, a; =10. Now 5y=165-9a;=165-90=z:75, .•.y=15. Ex. 5. Given i^+5|^--l,) , . , . , ^ y y \ to fi^d the values of x ,5.4 7,3 i and y. and -+-=-+-, ) a; y a? 2 J Reducing the first equation to lower terms, X y y u ..44, .'. by transposition, = — 1 ; X y 2 4 3 from the 2nd equation, by transposition, 1 — =- ; X y 2 .-. by addition, -=-. X 2 and, consequently, a; =4. 4 4 Now -=—[-1=2 ; .-. 2y=4, and a;=2 y a? SIMPLE EQUATIONS. 155 b a b \ -4— =n, \ Ex. 6. Given find the values of x and y. = 71, \ Multiplying the first equation by c, and the second by a, we shall have ac . Jc =7nc, « y . ac ad, and — j — =na, by subtraction, {^c^adi) . -z=imc-'na ; Jc — ad ... y_ . mc — na . , a b mbc—nab And -=m =m— ^c — ac? fnbc — mad — mbc-^-nab nab — mad be — ad ~ be — ad * 1 nb — md , -be — ad •••-=1 Ty and x=-T -z. X be — ad no — ma 2x 74-— Ex. 7. Given 3 -^=5-^^, 5 3y ' 107 , 44-15y ^^y— 8- to find the values of x and y. Multiplying the first equation by 15y, .-. 45y-21y— 6rK=:75y-25a;— 45 ; and by transposition, 51y — 19a;=45. Multiplying the second equation by 2a;+5, 2.y+5y -—^ ^-2.y- _; ^ , 107 8a;4-20+30a:y4-75y •••^y+-r= 6^^ ' and multiplying by 6a;— 2, tvre shall have 30a;y-10y+?Hl^.Ili^=8a:+20+30a?y4-75y; 156 SIMPLE EQUATIONS. 321iK-107 ^ . „, . „^ •'• Z =8j;+85y+20, and 321a;— 107=32ir4-340y+80 ; .-. by transposition, 340y— 289a;= — 187. The coefficients of y in this case, having aliquot parts ; multiplying the first by 20, and the last by 3, 1020y— 380a:= 900, and 1020y— 867a:=— 561 ; .*. by subtraction, 487a;z^l461, and a;=3 ; consequently, 51y=45 + 19a;=45 + 57=102 ; .•.y=2. ^ o r.. o 164-60a; 16a;y-107 N Ex. 8. Given 8a; ^ , = /, „ ,i x- j ,-. 3y — 1 54-2y f to find the va- , , 27a;2 — 12y2+38 (luesofa;andy. Multiplying the first equation by 5-|-2y, .« . .. 80 + 300a; + 32y4-120a;y _ ,^„ 40a;4-16a;y — _^ ^^=16a;y-107 ; « . ,^« 80 + 300a;+32y+120a;y .-.by transposition 40a;+107= __.^ and multiplying by 3y— 1, we shall have 120:ry— 40a:+321y— ]07=:80 + 300a:+32y+120a;y; .-.by transposition, 289y— 340a; = 187. And from the second.equation, 27a:2— 12y2+15a;+2y+2=r27a;2-12y2-l-38; .'. by transposition, 15a;+2y=36 ; whence, the coefficients of x having aliquot parts, multiplying the first equation by 3, and the second by 68, 867y — 1020a;=561, and 136y4-1020a;=2448; .-.by addition, 1003y=:3009, and y=3 ; consequently, 15a;=36— 2y=z36-6=:30 ; and .-.by division, a;=2. fir. 9. Given ar- 1^=20- 5£=?^,^^^ ^^^ ^^^ ^^_ , . V— 3 „^ • 73— 3y (lues of a: and y. Ans. a:=21, and y=20. SIMPLE EQITATIONS. 167 3^ 1 Ex. 10. Given h3y-4=15, to find the values of J 3y— 5 , o o ^o V a; and y. Ex. 11. Given 9a:+^=70, 5 and 7y --=44, Ans. a;=:7, and y=5. to find the values of x and y. Ans. a;=:6, and y=10. Ex. 12. Given ^ - ?^-3y-5, 5 4 and ^L_+___-18-5jc, to find the values of x and y. Ans. a;=3, and y=2. X. ,o ^. . , 3y+4a; „ 9y4-33 Ex. 13. Given a: 4-1 ^ — =7 ^TJ-"' „ 5a;— 4y lly— 19 and y-3 2"^=^ 4 ' to find the values of x and y. Ans. a:=6, and r/=5. Ex. 14. Given 4a:4-i^-=2y4-5+^^^^^, 4 lb to find the values of x and y. Ans. a;=3, and y=4. Ex. 15. Given x-5^+17=5y+lf±I, 22— 6y 5x-7 a;+l 8y+5 and -^ rr =-6 18~' to find the values of x and y. Ans. a?=:8, and y=2. Ex. 16. Given — ^- — =4H — , 6^3 2 ' , 2a;+y 9a; — 7_3y+9 4a;+5y __ ______ 16 » to find the values of x and y. Ans. a;=9, and y=4. „ ,„ r.- 7a;4-6 , 4y— 9 „ 13— a; 3y— a? Ex. 17. Given -— f — {—^ — =3a; ^ — ,and 113 2 o 3a?+4 : 2y— 3 : : 5 : 3, to find the values of a; and y. Ans. a;=7, and y=9. 15 158 SIMPLE EQUATIONS. Ex. 18. Given — ^ ^— ^ =9+ ^ ,and 2d o fjL_ : -^ U4a? : : 4 : 21, to find the values of x and y. 3 4 ^ Ans. a=5, and y=4. Ex. 19. Given —-^^^-^^^ 1^—-^ = 5+^--, and 10 . 15 5 9y+5a?— 8 jr-f y 7a!-f 6 , ^ , ^-u i r j -^ ^= — -— , to find the values oi x and y. 12 4 11 ' ^ Ans. Of =7, and y=9. Ex. 20. Given 3a;—2y=: 15, > .^ a ♦!, i r a J , , ^ 1 K n' o ^ tohnd the values of x and y. and y+iO ^ «— 15 : : 7 : 3, > ^ Ans. af=i45, and y=60. Ex. 21. Given aj+lSQ : y— 50 : : 3 : 2, > to find the va- and a?— 50 : y+100 : : 5 : 9, > lues of a? and y. Ans. a; = 300, and y=:350. Ex. 22. Given (a:+5). (y+7)=(a;+l){y—9) + 112, and 2ic4-10 = 3y+l, to find the values of x and y. Ans. x—'^j and y=5. 2a; — 4y+3 151— 16j: 9a:y— 110 to find the values of a; and y. Ans. a; =9, and y=2. ,^ . ^ , 128«2_i8ya-f217 Ex.24. Gwenl6.+6y-l^ 8.-3y%2 — - 10a:4-lQy-35 _ 54 ^ 2a;-f-2y-i-3 "~" 3a;+2y-r to find the values of x and y. Ans. x—^, and y=5. \ III. RESOLUTION OF SIMPLE EQUATIONS, Involving three or more unknown Quantities. 202. When there are three independent simple equations involving three unknown quantities. RULET, From two of the equations, find a third, which involves only two of the unknown quantities, by any of the rules in the pre- ceding Section ; and in like manner from the remaining equa- Uon, and one of the others, another equation which contains the SIMPLE EQUATIONS. 159 same two unknown quantities may be deduced. Having therefore two equations, which involve only two unknown quantities, these may be determined ; and, by substituting their values in any of the original equations, that of the third quantity will be obtained- 203. If there be four unknown quantities, their values may be found from four independent equations- For from the four given equations, by the rules in the last Section, three may be deduced which involve only three unknown quantities, the va- lues of which may be found by the last Article ; and hence the fourth may be found by substituting in any of the four given equations, the values of the three quantities determined. If there be n unknown quantities, and n independent equa tions, the values of those quantities may be found in a similar manner- For from the n given equations, n—\ may be de- duced, involving only n — I unknown quantities ; and from these n — 1, n— 2 may be obtained, involving only n—2 un- known quantities ; and so on, till only one equation remains, involving one unknov/n quantity ; which being found, the va- lues of all the rest may be determined by substitution. Ex. 1. Given x-^y^z=:29, \ x-|-2y-i- 35^=62, f to find the values of a:, y, X , y , z , ^ C and z. Subtracting the first equation from the second, y-l-2^=^33 . . . (A). Multiplying the thircL^quation by 12, the least common multiple of 2, 3, and 4, 6a; -l-4y-f 3:^=120 multiplying the I st equation by 6, 6a; -|- 6y -f- 6^^ = 1 74 ; .-. by subtraction, 2i/-\-3z=54 ; but, multiplying equation (A) by 2, 2^4-4^=66 ; .*. by subtraotion, ;s-=:12- From equation (A), by transposition, y = 33— 2,^ ; .-. by substitution, y=33— 24, or y = 9. From the first equation, by transposition, a;=:29—y—z; .-. by substitution, a:=29— 9 — 12, and a:i:=29— 21, .■.x=8. In like manner, had the first equation been multiplied by 2, and subtracted from the second, an equation would have re- sulted, involving only x and z ; and had it been multiplied by 4, and subtracted from the third when cleared of fractions. 160 SIMPLE EQUATIONS. another equation would have been obtained, involving also x and z ; whence by the preceding rules, the values of x and z could be found, and consequently the value of y also, by sub- stitution. Or if the first equation be multiplied by 3, and the second subtracted from it, an equation would arise, involving only x and y ; and if the first when multiplied by 3, be subtracted from the third when cleared of fractions, another would arise involving only x and y ; whence the values of x and y might be determined. And hence the third, that of ^, might be found. SECOND METHOD. From the first equation, x=29—y—z; substituting this value of x in the second equation, 29— y— 2+2^+3^=62 ; .-.by transposition, y=33—2z. Also substituting, in the third equation, the value of x found from the first, 29-y-z y z 2 ^3^4~ ^ multiplying this equation by 12, the least common multiple (rf 2, 3, and 4, 174--'6y—ez-^4y+3z=120, and by transposition, 2y-{-3z=z54: ; in which, substituting the value of ^ound above, 2(33-2;^)+32=54 ; or 66— 4^+3;^=:54 ; .*. by transposition, 2:= 12 ; whence y=33— 2^ = 33— 24 = 9, and a;=.29— y— 2=29 — 9 — 12 = 8. It may be observed, that there will be the same variety of solution, as in the last case according as x, y, or 2, is exter- minated. THIRD METHOD. The values of x, found in each of the equations, being compared, will furnish two equations each involving only y and z ; from which the values of y and z may be deduced by any of the rules in the preceding Section, and hence, the va- lue of X can be readily ascertained. SIMPLE EQUATIONS. 161 The same observation applies to this method of solution, as did to the last. In some particular equations, two unknown quantities may be eliminated, at once. Ex. 2. Given x+y-{-z=3l _ ' to find the values of a;, y, & z. x+y-{-z=3l \ x+i/—zz=25 > X — y—z=:9 } Adding the first and third equations, 2a;=40 ; .♦. a?=20. Subtracting the second from the first, 2z=6; ,:z=3; and subtracting the third from the second, 2y=16; .-.3^=8. Ex. 3. Given ^ x—z=3, J to find a:, y, and z. rx-y=2,) ^ X — Z=:3, > to (y-^ = l,) Here subtracting the first equation from the second, we have t/—z — l ; which is identically the third. Therefore, the third equation furnishes no new condition ; but what is already contained in the other two ; and, conse- quently, the proposed equations are indeterminate ; or, what is the same, we may obtain an infinite number of values which •will satisfy the conditions proposed. 204. It is proper to remark, that in particular cases. Ana- lysts make use of various other methods besides those pointed out in the practical rules ; in the resolution of equations, which greatly facilitate the calculation, and by means of which, some equations of a degree superior to the first, may be easily resolved, after the same manner as simple equations. We shall illustrate a few of those artifices by the following examples. Ex. 4 Given -+-=1, 1 X y 8 \ -+-=-, J> to find the values of a?, y, and s. X Z i) and— 1— =— , y z 10 15* 162 SIMPLE EQUATIONS. By adding the three equations, we i x^y^z 8^9^10 Or, dividing by 2, xy^z 720 shall have 121 •■~360' From this subtracting we shall have 1_ 31 ;^~720' each of the three first equations, and 720 ^„ 7 or .=:--;.. ..==23-; 1 41 y~720' 720 — '!f 1 49 a;~720' 720 "' ="= 49 ■' • ■■'< Ex. 5. Given 2a;=:y4-;sr-|-w,^ di/=x-{-z-\-u, f to find the values of x, y, z, 4:ZT=zx-\-y-\-u, I and u. and u=zx — 14, ) By adding x to each member of the first equation, y to the second, and z to the third, we shall get x-^y-\-z-\-u=z'dx=.Ayz=:^z \ 3a; 3a; and from thence, ^--r^ and y=r— - ; which values being substituted in the first equation, we have 3a? . 3a; . 13a; but, by the fourth, equation, w = a; — 1 4 ; 13a; .♦.a;— 14=— -, or 20a;— 280=13ap; — 3a; whence a;=40 : consequently y=— = 30, ;?=24, and «=» —14=26. Ex. 6. Given 4a; — 4v—43:=24, "i , ^ i ^r ^ c c o o o ^ Mo find the values of x. y, 6y— 2a;— 2;?=24, > i '^' and 7z-- y— x=24, J By putting x-\-y-{-z^S, the proposed equations become 8a;— 4S = 24, 8y-2S=24, 8;^-S=24 ; .-. a;=3 + iS, y==3+iS, ;?=3+ JS. By adding these three equations, we have a;+y+;^=9+|S ; whence S=72. SIMPLE EQUATIONS. 163 Substituting this value for S, in a;, y, and jsr, we shall find a'=39, y— 21, and z=12. Ex.7. Given oc-\- y-\-z =90, \- /. , ,, ■, ■ e 2a:+40i3y+20; i '^ ^""^ '^^ ^^^^^^ °^ ^' ^^ and 2x-4z+40=^l0, ) ^^"^ ^• Ans. x=35, y=30, and z=25. Ex. 8. Given a?4-y ) to and z-\-a=Zx-\-^y, ) ^" a 5ct , 7a Ans. a;=— ' ^^JY' ^"^^ ^= n Ex. 9. It is required to find the values of x, y, and Zf in the following equations ; a;+y=13, x-{-z=14j and i/-\-z=15. Ans. x=6, y=7, and z=8, Ex. 10. In the following it is required to find the values of X, y, and z. X y , z 3+1+5=94, X , y z r a?=48, n-=120, ( ;S=240. Ex.11. Given a:+y+;sr=26, ^ , c a .x. i r ^ ^ f to find the values of x, y. and X — z = 6, ^ Ans. a;=12, y = 8, and z=zQ. Ex. 12. Given 0?+ y4- ;3-= 9, ) , ^ , ,, , . ^ ' ' to find the values of a?, y, a;+ y+ ;3-= y, ^ a;+2y+3^=16, > , and ^4- y-2^= 3^^"^^ Ex. 13. Given a;+ y+ z=l2, \ , . . ,. ^ c a,+2v+3;^=20 J ^^ ?"^ ^^^ ^*^"®^ ^'^ ^' y» Ans. a:=4, y=3, and z=l2. = 12 =20 and \oc-\-\y-\- z- Ans. a;=6, y=4, and zz=z2. Ex. 14. Given a;+y— ^=8, a;+;^— y=:9, and y-\-z—x=^ 10 ; to find the values of x, y, and z. Ans. xz=.%\, y = 9, and ;^=9^. Ex. 15. Given a;+^y = 100, y4.i;j=lOO, and ;?+Ja;= 100 ; to find the values of a;, y, and z. Ans. a?=:64, y=72, and ;y=84. 164 SOLUTION OF PROBLEMS Ex. 16. Given a:4-iy=357,y-|-i^=476, if 4-iM=595, and u-\-^x=z7l4j to find the values of cc, y, z, and u. Ans. a:=190, y = 334, z=A26, and w = 676. § IV. SOLUTION OF PROBLEMS PRODUCING SIMPLE EQUATIONS, Involving more than one unknown Quantity. 205. The usual method of solving determinate problems of the first degree, is, to assume as many unknown letters, name- ly, X, y, z, &c., as there are unknown numbers to be found ; then, having properly examined the meaning and conditions of the problem, translate the several conditions into as many distinct algebraic equations ; and, finally, by the resolution of these equations according to the rules laid down in Chapter IV, the quantities sought will be determined. It is proper to observe that, in certain cases, other methods of proceeding may be used, which practice and observation alone can sug- Problem I. There are two numbers, such, that three times the greater added to one-third the lesser is equal 36 ; and if twice the greater be subtracted from 6 times the lesser, and the remain- der divided by 8, the quotient will be 4. What are the num- bers ? Let X designate the greater number, and y the lesser num- ber. Then 3a:-f^=36;) ^^ , ,n«/A\ '3 f . < 9a;+ y = 108 (A), . 6y— 2a; , ( '' \ Gy-2x= 32 (B) ; and -^— — =4 ; \ o y Multiplying equation (A) by 6, Oy-}- 54a: = 648 ; but 6y— 2x= 32 ; .-. by subtraction, 56a:rr:616, and by division, a:=ll. From equation (A), y=108— 9a? ; .-. by substitution, y= 108—99, or y— 9. PRODUCING SIMPLE EQUATIONS. 165 Prob. 2. After A had won four shillings of B, he had only- half as many shillings as B had left. But had B won six shil- lings of A, then he would have three times as many as A would have had left. How many had each 1 Let x= designate the number of shillings A had, and y= the number B had ; then y— 4=:2a?+8, and ■i/-\-6=z3x—l8', .-. by subtraction, 10=a?— 26, and by transposition, 36 = x, or a; =36 by substitution, y+6=3x 36 — 18 and by transposition, y = 84 .-. A had 36, and B 84. Prob. 3. What fraction is that, to the numerator of which if 4 be added, the value is one-half, but if 7 be added to the denominator, its value is one-fifth ? Let ^= its numerator, ) ,^^„ ^^^ f^^^^j^^ x yz=z denommator, ) y Add 4 to the numerator, then =i, .-. 2a;+8=y ; y Add 7 to the denominator, then — — =J, .-. 5a?=y+7; by subtraction, 3x— 8=7 ; by transposition, 3a;=:15 ; .". xz^b ; and y=2a?+8 ; .-.by substitution, y=10-f-8=18, 5 and the fraction is -— . 18 Prob. 4. A and B have certain sums of money, says A to B, give me 15/ of your money, and I shall have 5 times as much as you have left : says B to A, give me 5/ of your money, and I shall have exactly as much as you will have left. What sum of money had each ? Let x=i A's money, > then x-\-\b=. what A would have, y=z B's, S after receiving 15Z from B. y — 15= what B would have left. Again,y-f 5= what B would have after receiving 5/ from A. x — b— what A would have left. Hence, by the problem, a;+15=5 x(y— 15) = 5y--75, and y+5=a?— 5. 166 SOLUTION OF PROBLEMS by transposition, 5y— a?=i90, and y — a;z= — 10 ; /. by subtraction, 4y=100, and by division, y=25 B's money. From the second equation, a:=y-f 10 ; .'. by substitution, a;=25 4-10=35 A's money. Prob. 6. A person was desirous of relieving a certain num- ber of beggars by giving them 2s. 6d. each, but found that he had not money enough in his pocket by 3 shillings ; he then gave them 2 shillings each, and had four shillings to spare. What money had he in his pocket ; and how many beggars did he relieve ? Let x=. money in his pocket {in shillings) ; t/z= the number of beggars. Then 2^ Xy, or -^= number of shillings which, would have been given at 2s. 6d. each ; and 2 Xy, or 2y=: .... at 2s. each. Hence, by the problem, -^=:a?-j- 3(A), and 2yz=a?— 4(B). V .-. by subtraction, ^=7, lit or y=14, the number of beggars. From equation (B), a;=2y4-4=2 x 14-^-4, by substitution, .-. a;zr:32, the shillings in his pocket. Prob. 6. There is a certain number, consisting of two digits. The sum of those digits is 5 ; and if 9 be added to the number itself, the digits will be inverted. What is the number ? Here it may be observed, that every number consisting of two digits is equal to 10 times the ;ligit in the tens place, plus that in the units ; thus, 24=^2 X 104-4x=20-f 4. Let x= digit in the units place ; y=. that in the tens. Then 10a;-4-y= the number itself, and 10y-f-a;= the number with its digits inverted. Hence, by the problem, a:-|-y=5(A), and 10a;-|-y+9=:10y4-^» or by transposition, 9a;— 9y=— 9 ; .'.by division, a;— y = — 1(B). Subtracting equation (B) from (A), 2y=sQ ; PRODUCING SIMPLE EQUATIONS. 167 .•.y=3, and a:=5— y=5— 3=2 ; .-. the number is (10a;+y)=23. Add 9 to this number, and it becomes 32, which is the num- ber with the digits inverted. pROB. 7. A sum of money was divided equally amongst a certain number of persons ; had there been four more, each would have received one shilling less, and had there been four fewer, each would have received two shillings more than he did : required the number of persons, and what each received. Let X designate the number of persons, • y the sum each received in shilUings ; then xy is the sum divided ; .^^%t.%l%-l^)lZ\\ by the question; .•.xy-\-A:y — X — 4 = a?y, or 4y — a:=4, and xy—Ay-^2x—S=ixy, or — 4y-|-2j: = 8 ; .*. by addition, a?=12 ; and 4y=44-a?=4 + 12 ; .'.^=4. Prob. 8. A man, his wife, and son's years make 96, of which the father and son's equal the wife's and 15 years over, and the wife and son's equal the man's and two years over. What was the age of each ? Suppose X, y, and z =: their respective ages. 1st condition x-{-y-\-z=:96, ^ 2nd . . . x-\-z=y-^l5, > by the problem. 3d ... y-\-z=x-\- 2,) Subtracting the 2nd from the 1st, y=96— y— 15 ; .'.2y=z81, and y — 40J by division. Subtracting the 3d from the 1st, x=96—x—2 ; .•.by transposition and division, x=47, And from the 1st, z=z96—y—x ; .-. ^^=8^. And their ages are 47, 40J, and 8J respectively. Prob. 9. A labourer working for a gentleman during 12 days, and having had with him, the first seven days, his wife and son, received 74 shillings ; he wrought afterwards 8 other days, during 5 of which he had with him his wife and son, and he received 50 shillings. Required the gain of the labourer per day, and also that of his wife and son. Let x= the daily gain of the husband, yzz: that of the wife and son ; 12 days work of the husband would produce 12a:, 7 of the wife and son would be 7y ; 168 SOLUTION OF PROBLEMS .-.by the first condition, 12a? -f ly= 74 and by the second, 8a;-|- ^y— 50 Multiplying the 1st equation by 2, 24a;4-14y=rl48 2nd . . by 3, 24x4" 15y= 150 /.by subtraction, y=2. And from the 2nd, 8a;=50— 5y=50 — 10 ; .'.by division a:=5. Consequently the husband would have gained alone 5s. per day, and the wife and son 2 shillings in the same time. 506. Let us now suppose that the first sum received by the workman was 46s, and the second 30s, the other circumstances remaining the same as before ; The equations of the question would be 12a;+7y=46, and 8a;-f-5y=30. From whence we find, by proceeding as above, a:=5, and y = — 2. By putting in the place of a? its value 5, in the above equa- tions, they become 604-7^=46, and 404-53/ = 30. The inspection alone of these equations show an absurdity. In fact, it is impossible to form 46 by adding- an absolute num- ber to 60, which is already greater than it, and in like man- ner it is impossible to form 30 by adding an absolute number to 40. Consequently what we attributed as a gain to the labour of the wife and son, must be an expense to the husband, which is also verified by the result y=—2. 207. The negative value of y makes known therefore a rectification in the enunciation of the problem ; since that, in- stead of adding 7y to 12a: in the first equation, and 5y to 8a: in the second, y being considered a positive or an absolute num- ber, we must subtract them in order to have the sum given for the common wages of these three persons ; or, what is the same thing, if, in place of considering the money attributed to the wife and son as a gain, we would regard it as an expense made by them to the charge of the workman ; then we must subtract this money from what the man would have gained alone, and there would be no contradiction in the equations, since they would become 60-7y = 46, and 40-5y=30 ; from either of which we would derive y=2 ; and we should therefore conclude that if the workman gained 5s. per day, his wife and son's expense is 2s., which can be otherwise verified thus : PRODUCING SIMPLE EQUATIONS. 169 For 12 days work, he receives 5 x 12 or 60s. ; the expense of his wife and son for 7 days, is 2 X 7 or 14s. ; and there re- mains 46 shillings. Again, he receives for 8 days work 5 X 8 or 40s. ; the ex- pense of his wife and son during 5 days, is 2x5 or 10s. ; therefore his clear gain is 30 shillings. 208. It is very evident that, in place of the enunciation-of (Prob. 9), we must substitute the following, in order that the problem proposed may be possible, with the above given quantities : A labourer working for a gentleman during 12 daySy having had with him, the first 7 days, his wife and son, who occasion an expense to him, received 46 shillings ; he has wrought, after- wards, for 8 other days, on 5 of which he had with him his wife and son, whose expenses he must still defray, and he received 30 shillings. Required the salary of the workman per day, and also the expense of his wife and son in the same time. Designating by x the daily wages of the workman, and by y the expense of his wife and son, for the same time ; the equations of the problem shall be \2x—ly — AQ, and 8a;— 5y = 30 ; which, being resolved, will give a; = 5s., and yz=2s. 209. Although negative values do not answer the enuncia- tion of a concrete question, as has been observed (Art. 174), yet they satisfy the equations of the problem, as may be rea- dily verified, by substituting 5 for x, and —2 for y, in the equations (Art. 206), since they would then become identi- cally equal. Prob. 10. Two pipes, the^vater flowing in each uniformly, filled a cistern containing 330 gallons, the one running during 5 hours, and the other during 4 ; the same two pipes, the first running during two hours, and the second three, filled another cistern containing 195 gallons. The discharge of each pipe is required. Let X represent the discharge of the first in an hour ; y that of the second in the same time. And in order to have a general solution, put a=zb, 5=4, c=330, a'=2, 5' — 3, c' = 195 ; then by the conditions of the problem we shall have these two equations, ax-\-hy=c, and a^x-\-h'y=zc' ; which, being resolved (Art. 190), will give h'c — hc^ - ac' — a'c aa^ab ^ al/—ao 16 170 SOLUTION OF PROBLEMS Now, by restoring the values of a, i, c, cScc, we have 990—780 210 975-660 ,^ .Thus, the first pipe discharges 30 gallons per hour, and the second 45. 210. Let us now suppose that the first pipe running during 3 hours, and the second during 7, filled a cistern containing 190 gallons ; that afterwards, the first running 4 hours, and the second 6, filled a cistern containing 120 gallons. In this case, a=3, 5=: 7, c=190, a' — i, b' = 6, c'z=:120; and, consequently, &'c—^c'= 1140 — 840 = 300, ab'^a'b=l8 —28=: — 10, ac'-a^c=360— 760zz:-400, which will give x = —30, andy = 40. In order to understand the meaning of these results, we must return again to the conditions of the problem, or, what amounts to the same, we must try how these values of x and y satisfy the equations of the problem : Thus, if we substitute —30 for x, and 40 for y, in the equations 3a;4-7y=:190 and 4a;-+-6y = 120, resulting from the above problem, we find first, that 3a:= — 90, and 7y = 280, consequently 3a;+7y= — 90 + 280, which in efiect is equal to 190. In like manner 4a;+ 6y is found to be — 120+240, which is equal to 120. Having, therefore, discovered how the values — 30 and +40 of X and y answer the equations 3ir+7y = 190 and 4a; + 6y = 120, we perceive at the same time how they would answer the conditions of the problem ; for*since the use that has been made of the quantities 3a; and 4.x, which express the quanti- ties of water discharged by the first pipe in the first and se- cond operation, was to subtract them from 7y and from 6y, which express the quantities furnished in the same operations by the second pipe. The first pipe must be considered in this case as depriving the cisterns of water instead of fur- nishing any, as it did in the preceding problem, and as it was supposed in expressing the conditions of this problem. 211. Hence, in almost every question solved after a gene- ral manner, we may always conclude that when the value of the unknown quantity becomes negative, the quantity ex- pressed by it should be considered as being of an opposite kind from what it was supposed in expressing the conditions of the problem. What has been said with respect to unki\pwn quantities, is PRODUCING SIMPLE EQUATIONS. # 171 equally applicable to known quantities, that is, when a gene- ral solution is applied to any particular case, if any of the gi- ven quantities a, 6, c, &c. in the problem, are negative. 212. Let it be proposed, for example, to find what should be, in the foregoing problem, the discharges of two pipes, that the first furnishing water during 3 hours, and the second 4, may fill a cistern containing 320 gallons, and that the se- cond pipe afterwards furnishing water during 6 hours, whilst the first discharges it during 3 hours, may fill a cistern con- taining 180 gallons. We have only to put in the general solution (Art. 209), a— 3, b — 4, c=320, a' = — 3, b' = 6, c'=zl8(J, and there will result a;=40, and y=:50. From whence it appears that the discharge of the first j)ipe is 40 gallons per hour, either to carry away the water as in the second operation, or to furnish it as in the first, and the discharge of the second, *50 gallons an hour, which it furnishes in both operations. Prob. 11. a certain sum of money put out to interest, amounts in 8 months to 297/. 12.y. ; and in 15 months its amount is 306Z. at simple inter^t. What is the sum and the rate per cent ? Ans. 288/. at 5 per cent. Prob. 12. There is a number consisting of two digits, the second of which is greater than the first, and if the number be divided by the sum of it« digits, the quotient is 4 ; but if the digits be inverted, and that number divided by a number greater by 2 than the difference of the digits, the quotient be- comes 14. Required the number. Ans. 48. Prob. 13. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes I ; but the denominator being doubled, and the numerator increased by 2, the value becomes |- ? Ans. ^. Prob. 14. A farmer parting with his stock, sells to one person 9 horses and 7 cows for 300 dollars : and to another, at the same prices, 6 horses and 13 cows for the same sum. What was the price of each ? Ans. the price of a cow was 12 dollars, and of a horse 24 dollars. Prob. 15. A Vintner has two casks of wine, from the great- er of which he draws 15 gallons, and from the less 11 ; and finds the quantities remaining in the proportion of 8 to 3. Af- ter they became half empty, he puts 10 gallons of water into each, and finds that the quantities of liquor now in them are as 9 to 5. How many gallons will each hold ? Ans. the larger 79, and the sm^l^^r ,35 gallons \ OF ■. K J^ 172 SOLUTION OF PROBLEMS Frob: 16. A person having laid out a rectangular bowling- green, observed that if each side had been 4 yards longer, the adjacent sides would have been in the ratio of 5 to 4 ; but if each had been 4 yards shorter, the ratio would have been 4 to 3. What are the lengths of the sides ? Ans. 36, and 28 yards. Prob. 17. a sets out express from C towards D, and three hours afterwards B sets out from D towards C, travelling 2 miles an hour more than A. When they meet it appears that the distances they have travelled are in the proportion of 13 to 15 ; but had A travelled five hours less, and B had gone 2 miles an hour more, they would have been in the proportion of 2 : 5. How many miles did each go per hour, and how many hours did they travel before they met 1 Ans. A went 4, and B 6 miles an hour, and they travelled 10 hours after B set out. Prob. 18. A Farmer hires a farm For 245Z. per annum, the arable land being valued at 2l. an acre, and the pasture at 28 shillings : now the number of acres of arable is to half the excess of the arable above the pasture as 28 : 9. How many acres were there of each "? • Ans. 98 acres of arable, and 35 of pasture. Prob. 19. A and B playing at backgammon, A bets 3s. to 2s. on every game, and after a certain number of games found that he had lost 17 shillings. Now had A won 3 more from B, the number he would then have won, would be to the num- ber B had won, as 5 to 4. How many games did they play ? Ans. 9. Prob. 20. Two persons, A and B, can perform a piece of work in 16 days. They work together for 4 days, when A being called off, B is left to finish it, which he does in 36 days more. In what time would each do it separately ? Ans. A in 24 days, and B in 48 days. Prob. 21. Some hours after a courier had been sent from A to B, which are 147 miles distant, a second was sent, who wished to overtake him just as he entered B ; in order to which he found he must perform the journey in 28 hours less than the first did. Now the time in which the first travels 17 miles added to the time in which the second travels 56 miles, is 13 hours and 40 minutes. How many miles does each go per hour ? Ans. the first goes 3, and the second 7 miles an hour. Prob. 22. Two loaded wagons were weighed, and their weights were found to be in the ratio of 4 to 5. Paris of their loads, which were in the proportion of 6 to 7, being taken out, PRODUCING SIMPLE EQUATIONS. 173 their weights were then found to be in the ratio of 2 to 3 ; and the sum of their weights was then ten tons. What were the weights at first ? Ans. 16, and 20 tons. Prob. 23. A and B severally cut packs of cards ; so as to cut off less than they left. Now the number of cards left by A added to the number cut off by B, make 50 ; also the num- ber of cards left by both exceed the number cut off, by 64. How many did each cut off? Ans. A cut off 11, and B 9. Prob. 24. A and B speculate with different sums ; A gains 150/, B loses 50Z, and now A's stock is to B's as 3 to 2. But had A lost 50/, and B gained lOOZ, then A's stock would have been to B's as 5 to 9. What was the stock of each ? Ans. A's was 300/, and B's 350/. Prob. 25. A Vintner bought 6 dozen of port wine and 3 dozen of white, for 12/. 12^. ; but the price of each after- wards falling a shilling per bottle, he had 20 bottles of port, and 3 dozen and 8 bottles of white more, for the same sum. What was the price of each at first 1 > Ans. the price of port was 2s. and of white 3s. per bottle. Prob. 26. Find two numbers, in the proportion of 5 to 7, to which two other required numbers in the proportion of 3 to 5 being respectively added, i\\e sums shall be in the propor- tion of 9 to 13 : and the difference of those sums =16. Ans. the two first numbers are 30 and 42 ; the two others, 6 and 10. Prob. 27. A Merchant finds that if he mixes sherry and brandy in quantities which are in the proportion of^ to 1, he can sell the mixture at 78s. per dozen ; but if the proportion be as 7 to 2, he must sell it at 79 shiUings a dozen. Required the price of each liquor. Ans. the price of sherry was 81s., and of brandy '72s. per dozen. Prob. 28.- A number consisting of two digits when divided by 4, gives a certain quotient and a remainder of 3 ; when di- vided by 9 gives another quotient and a remainder of 8. Now the value of the digit on the left-hand is equal the quotient which was got when the number was divided by 9 ; and the other digit is equal Jyth of the quotient got when the number was divided by 4. Required the number. Ans. 71. • Prob. 29. To find three numbers, such, that the first with J the sum of the second and third shall be 120 ; the second with 1th the difference of the third s^nd first shall be 70 ; and •| the sum of the three numbers shall be 95. Ans. 50, 65, and 75. Prob. 30. There are two numbers, such, that J the greater 16* 174 SOLUTION OF PROBLEMS, &c. added to i the lesser is 13 ; and if ^ the lesser be taken from J the greater, the remainder is nothing. What are the numbers ? Ans. 18, and 12. Prob. 31. There is a certain number, to the sum of whose digits if you add 7, the result will be three times the left-hand digit; and if from the number itself you subtract 18, the digits wiirbe inverted. What is the number ? Ans. 53. Prob. 32. A person has two horses, and a saddle worth 10/ ; if the saddle be put on the Jirst horse, his value becomes double that of the second ; but if the saddle be put on the se- cond horse, his value will not amount to that of the Jirst horse by 13/. What is the value of each horse ? Ans. 56 and 33. Prob. 33. A gentleman being asked the age of his two sons, answered, that if to the sum of their ages 18 be added, the re suit will be double the age of the elder ; but if 6 be taken from the difference of their ages, the remainder will be equal to the age of the younger. What then were their ages ? Ans. 30 and 12. Prob. 34. To find four numbers, such, that the sum of the 1st, 2d, and 3d, shall be 13 ; the sum of the 1st, 2d, and 4th, 15 ; the sum of the 1st, 3d, and 4th, 18 ; and lastly the sum of the 2d, 3d, and 4th, 20. * Ans. 2, 4, 7, 9. Prob. 35. A son asked his father how old he was. His father answered him thus. If you take away 5 from my years, and divide the remainder by 8, the quotient will be i of your age ; but if you add 2 to your age, and multiply the whole by 3, and then subtract 7 from the product, you will have the number of the years of my age. What was the age of the father and son ? Ans. 53, and 18. Prob. 36. Two persons, A and B,had a mind to purchase a house rated at 1200 dollars ; says A to B, if you give me ^ of your money, I can purchase the house alone ^ but says B to A, if you will give me |th of yours, I shall be able to pur- chase the house. How much money had each of them ? Ans. A had 800 and B 600 dollars. Prob. 37. There is a cistern into which water is admitted by three cocks, two of which are exactly of the same dimen- sions. When they are all open, five-twelfths of the cistern is filled in 4 hours ; and if one of the equal cocks be stopped, seven-ninths of the cistern is filled in 10 hours and 40 minutes. In how many hours would each cock fill the cistern ? Ans. Each of the equal ones in 32 hours, and the other in 24. Prob. 38. Two shepherds, A and B, are intrusted with the charge of two flocks of sheep. A's consisting chiefly of ewes, INVOLUTION. 175 many of which produced lambs, is at the end of the year in- creased by 80 ; but B finds his stock diminished by 20 : when their numbers are in the proportion of 8 : 3. Now had A lost 20 of his sheep, and B had an increase of 90, the numbers would have been in the proportion of 7 to 10. What were the numbers ? Ans. A's 160, and B's 110. Prob. 39. At an election for two members of congress, three men offer themselves as candidates ; the number of voters for the two successful ones are in the ratio of 9 to 8 ; and if the first had had 7 more, his majority over the second would have been to the majority of the second over the third as 12 : 7. Now if the first and third had formed a coalition, and had one more voter, they would each have succeeded by a majority of 7. How many voted for each ? Ans. 369, 328, and 300, respectively. CHAPTER VI. ON INVOLUTION AND EVOLUTION OF NUMBERS, AND OF ALGEBRAIC aUANTITIES. 213. The powers of any quantity, are the successive products, arising from unity, continually multiplied by that quantity. Or, the power of the order m of a quantity, m being a whole pos- itive number, is the product of that quantity continually mul- tiplied 772 — 1 times into itself, or till the number of factors amounts to the number of units in that given power. 214. Involution is the method of raising any quantity to a given power ; Evolution, or the extraction of roots, being just the reverse of Involution, is the method of determining a quantity which, raised to a proposed power, will produce a given quantity. Note. — The term root has been already defintfi, (Art. 12). § I. involution of algebraic quantities. 215. It has been observed, (Art. 13), that the powers of al- gebraic quantities are expressed by placing the index or expo- nent of the power over the quantity. 176 INVOLUTION. Hence, if a proposed root he a single letter, and without a cO' eff,cient, any required power of it will he expressed hy the same letter with the index of the power written over it. Thus, the nth power of a is z=a", n being any positive number whatever. 216. If the proposed root he itself a povjer, the required power will he obtained hy multiplying the index of the given power into that of the required power. Thus the mXh power of a^, or {a'')'"=a'"P ; for since, (Art. 213), (a'')'"=:a^XaPXa'', &c. = ^p+p+p+etc._^^m ^ ^jj where the number of factors a'' is equal to m. 217. Also, if a simple quantity he composed of several factors, it can he raised to any power by multiplying the index of every fac- tor in the quantity hy the exponent of the power. Thus the mth power of {a''h''c'), or (a''^>V)^ is =: a'""6'''V" ; for since (Art. 274), (a^hfc')'" = (a'h'c') X (a^^'c''), 2? Here, {-^a^h'^Y= (Art. 220), -C^aWf^-^l Xa^'^ Xb"^-^ z=~-27a%^. Ans. Ex. 3. What is the 4th power of —2a^x^ ? Here, (—2a3x^Y= (Art. 219), +{2a^cc^Y=l6xa^''^x^-'^= Ida^^x^. Ans. Ex. 4. What is the cube, or third power of abc ? < Here, abc XabcXabcz= a X a X a X bxbxbXcXcXcz:^ a^b-^c^. Ans. 222. When the quantity to be involved is a fraction, raise both the numerator and denominator to the power proposed. Ex. 5. Required the 4th power of — — . tia 178 INVOLUTION. ( b\_ 6* _ b* _ ¥ 2a) """'~(2a)*~2*Xa*~16a*' Ex. 6. What is the 4th power of —7—? Ans. . ^ 3a? 81a;* Ex. 7. What is the 8th power of 2a^ ? Ans. 256a^^. Ex. 8. What is the 7th power of —x ? Ans. —x''. Ex. 9. What is the 6th power of ? Ans. — . x^ x^^ C -5 Ex. 10. What is the 5th power of -? Ans 5 * ■ 3125* 5a; 625a:* Ex. 11. What is the 4tli power of -— ? Ans. . ^ 7 2401 Ex. 12. Required the cube of -z- ? Ans. —. So 27 0-^ Ex. 13. Required the square of %a2^2? Ans. a*K Ex. 14. Required the 9th power of —xyl Ans. — a;^^^. Ex. 15. Required the 0th power of a;y ? Ans. 1. Ex. 16. Required the 4th power of a-2 ? Ans, a- or V a? CASE II. To involve a compound algebraic Quantity. RULE I. 223. Multi-ply the given quantity continually into itself as many times minus one as is denoted by the index of the power, as in the multiplication of compound algebraic quantities (Art. 79), and the last product will be the power required. Ex. 1. What is the square of a -J- 25? a+2b a-{-2b a^+2ah +2a6+4J2 Square =a2+4«6 4-462 INVOLUTION. 179 Ex. 2. What is the cube of a2— a2 ? a*— 2aV+a;* f£l — ^2 Cube =a6— 3a*a;24-3a2a;'t— a;6 Ex. 3. Required the fourth power of a-{-3b. Ans. a*+i2a^-{-54aH^-\-108aP-^8lhK Ex. 4. Required the square of 3a;2-|-2ir4-5. Ans. 9a:'t + 12a;3 + 34a;2+20a;+25. Ex. 5. Required the cube of 3a;— 5. Ans. 27^3_i35^2_|_ 225a;— 125 Ex. 6. Required the cube of x^— 2a;+l. Ans. x^—6x^-\-l5x*--20x^+l5x^—ex+l Ex. 7. Required the fourth power of 2 + 3a;. Ans. 16 + 96a;+216a;^+216a3+81ar* Ex. 8. Required the fifth poiver of 1 —2a;. Ans. 1 — 10>r+40a;2 — 80a;34-80a'*— 32a;*. Ex. 9. Required the square of a+^+c-j-J. Ans. a'2i-b^+c'^-^d^ + 2(ab + ac-\-ad+bc-^bd+cd). 224. In the invokition of a binomial or residual quantity of the form a-^-b, or a — b ; the several terms in each successive power are found to bear a certain relation to each other, and observe a certain law, which the following Table is intended to explain. 180 INVOLUTION. TABLE OF THE POWERS OF « + J. Powers. Mode of ex- pressing them. Powers expanded. Square. {a + hf. a^-\-2ah-\-b'^. Cube. {a^hf. a3 + 3a25-}-3«52_|_^,3. 4th power. [a+hf. a^-\-^a?h-^6a?b'^-{-4.ah^-\-bK 5th power. (a-\-bf. a^-\-5a^b-{-\0aW+\0a^¥^5a¥ -\-b\ 6th power. (a + hf. ^6 ^ 6a5^ 4- 15^4 ^>2_|_ 20^3^3 -\-\5a%^+Qab^-\-¥. The successive powers of a— 6 are precisely the same as those of a-\~b, except that the signs of the terms will be al- ternately + and — . Thus, the ffth power of a — b is a^— 5fl!*5-|-10a3^,2„l0a;262-f-5a6*— Z/5. 225. In reviewing that column of the above Table which contains the powers of a+6 expanded, we may observe, I. That in each case, the first term is raised to the given power, and the last term is b raised to the same power ; thus, in the square, the first term is a^, and the last b"^ ; in. the cube, the first term is a^, and the last b^ ; and so on of the rest. II. That, with respect to the intermediate terms, the pow- ers of a decrease, and the powers of b increase, by unity in each successive term. Thus, in the fifth power, we have In the second term, a'^b \ third, a^b"^ ; fourth, a2^,3 . fifth, a¥ ', and so on in other powers. III. That in each case, the coefficient of the second term is the same with the index of the given power. Thus, in the square, it is 2 ; in the cube, it is 3 ; in the fourth power, it is 4 ; and so on of the rest. IV. That if the coefficient of a in any term be multiplied by its index, and the product divided by the number of terms to that place, this quotient will give the coefficient of the next term. Thus, in the fifth power, the coefficient of a in the second INVOLUTION. 181 4 v> 5 oQ to that place =— - — =—-=10= coefficient of the third term. 2 3 term multiplied by its index, and divided by the number of terms —=10= coefficient of the third ten i Tv, *!,« „;^+l, «^..r«« CoefF. of a in the 4th term . its index 20 X 3 In the sixth power, r — tt r— ti.— :— i = — : — = ^ number of terms to that place. 4 fin —-=15= coefficient of the fifth term. Hence, we are furnished with the following general ruler for raising a binomial or residual quantity to any power, without the process of actual multiplication. • * RULE II. 226. Find the terms without the coefficients, by observing that the index of the first, or leading quantity, begins with that of the given power, and decreases continually by 1, in every term to the last ; and that, in the following quantity, its indices are 1, 2, 3, &c. Then, find the coefficients, by observing that those of the first and last terms are always 1 ; and that the coefficient of the second term is the index of the power of the first ; and, for the rest, if the coefficient of any term be mul- tiplied by the index of the leading quantity in it, and the pro- duct be divided by the number of terms to that place; it will give the coefficient of the term next following. Ex. 1. Required the 8th power oi a-^b. Here the terms, without the coefficients, are «8, a^b, a%-\ a^b\ a^b\ a?b\ a?b\ ab\ b^. And the coefficients, according to the rule, will be. 1, 8, ^=28,?5>M. "^"-^^"^rxsxir/'^'""' -"-"^- The last, b\ INVOLUTION. • 183 Or (V=g"+ng"-'54- ''^'^~^W &^+ n{n-l)x{n-2) n(n-l)x(n-2)x(n-3) 2.3 "•" 2.3.4 ar-^b\SLc +&". By the same process, {u—bY=a'' — 7ia''~'^-i- n(7i — 1) ^ „, n{n—l)x{n—2) ^.,. , -J ^z"~2J2 J^ i — i i a»-3p ^ 4) 4i.O •"^ o o o iflr~*5* — &c. ; the signs of the terms .^.0.-3 being alternately + and —1 ; and the sign of the last term is + or — 1 , according as n is even or odd ; we have the last term in the former case^ -\-h\ and in the latter — &". This general and compendious method of raising a binomial quantity to any given power, is called from the name of its ce- lebrated inventor, Sir Isaac Newton's " Binomial Theorem." The demonstration of this Theorem, with its application to the finding the powers and roots of compound quantities, forms the subject of another Chapter. Its present use will appear from the following Example. Ex. 2. Required the fifth power of x'^-\''^y^. Substituting these quantities for *a, 5, w, in the foregoing general formula, it appears that ''):£' V'^) ■ ■ ■ ■ M' • - - • • -"• 2nd,. . {naJ^-^b) . is 5 X (a:^)^ x 3y2 . . , =15a:y. 3d, . ^ ^(^~^) ^n_2^2\ . 5x|x(^2)3x(3/)2=90a;y. 4th, . (±^±Z-^^-.b.) , is5x|x|x<^)^X (3y2)3 =27Wy\ /n(w— l)(n— 2)(n — 3) ,, \ . ^432 5th, . (J \-^ -V-3^)is5x-X-X-x^^X (3y2J4 =405^y. Last, . (6") is (3y2)5 , ^243y»o. So that (a:2 + 3y2)5 = jpio ^ I5a;8y2 _j_ 80a;y + 270a?*/ 4- 405a;y + 243yi'^. 229. By means of this Theorem, we are enabled to raise a trinomial, or quadrinomial quantity to any power, without the process of actual multiplication. Ex.3. Required the square of a+^ + <'- Here, including a-\-b in a parentheses {a-\-h), and consider- ing it as one quantity, we should have {a-\-h-{-cf-=z\{a-\'b\ -f cp ; and comparing them with the general formula ; 184 • EVOLUTION. we have (a") = (a+bY=za'^+2ab-{-b^ ) (na''-^Y=2{a-{-l))xc=2ac-{-2bc l {b")=:c^ - =c2 ) Hence, {a+b + cf=(a+bf-\-2(a-]-b)Xc-^c^=a^+2ab + Ex. 4. Required the seventh power of a — b. Ans. a^—7a^^+ 2la^b'^—35a^b^ + 35a^*—2la'^b^ + 7ab^ Ex. 5. Required the sixth power of 3a; + 2y. Ans. 729a;6+2916a:5y4-4860a:y+4320a;y4-2160a;y-f 576xy^-{-e4f. Ex. 6. Required the square oi x-\-t/-\-3z. Ans. oe^-{-2ocy-^y'^-^6xz-\-6yz-\-9z\ Ex. 7. Required the fifth power of 14-2«. Ans. l-i-10a;+40a^+80ar3 + 80ir*+32ar^. Ex. 8. Required the cube of x^—2xy-{-y^. Ans. x^—6x^7/i-l5x*f—20x^y^-^15x'^y*-'6xi/^+y^. ^ II. EVOLUTION OF ALGEBRAIC QUANTITIES. 230. The quantity which has been raised to any power is call- ed the root of that power ; thus the with root of a power, is that quantity which we must continually multiply into itself, till the number of factors be equal to m, m being a positive whole number, in order to produce the power proposed. We may conclude from this definition, and from the Articles in the pre- ceding section. 231. That the mih root of a quantity such as a^,pm being a multiple of p, is obtained by dividing the exponent pm of this quantity, by the index of the required root. Thus the mth root of pm g a^=a'" =a^ ; the square root of a^=a^ = a?, and the cube 6. ropt.of a^=a^=a^. 232. Also that the mih. root of a product such as ai^b^, is equal to the mih. root of each of its factors multiplied together. Thus, the mth root of a^P'" is = the mth root of a^ x the mth root of P'"=a~^ X b'^= a^^. 233. And that the mth root of a fraction such as -r-, is equal to the mih root of the numerator divided by the mih root of its denominator. Thus the mth root of t- = — = -. b"* "L b EVOLUTION. 185 234. The square^ the fourth root, or any even root of an affir- mative quantity may he either •{■ or — 1 . Thus the square root of a^ — ^^r _fl5 J for -|-ax+«=^-a^ and —ay.—a=i-\-(jP-. In fact, the 2 with root of cP-^l^ equal to + a or —o J for {^-^af^ 235. Any odd root of a quantity, will have the same sign as the quantity itself. Thus the (2m4- l)th root of ia^*" +^ is equal to ±a; for (±a)2'"+i is equal to ia^^+i. 236. Evolution, or the rule for extracting the root of any algebraic Quantity whatever, is divided into the four following Cases. CASE I. To find any root of a simple algebraic Quantity. RULE. 237. Extract the root of the coefficient for the numeral part, and the root of the quantity subjoined to it for the literal part, by the methods- pointed out in the above propositions ; then, these, joined together, will be the root required. Ex. 1. It is required to find tl;e square root of x^. 4 Here, the square root of a:^= J;-\/a;*z:r J-a:2 = _[-a;2. Ex. 2. Required the cube root of — 21x^a^. Here, the cube root of —21x^a^=—^ 21x^a^=: — ^ 27 X ^ a;3 X 3/ a6_ _3 X a; X a2= _3a2a;. Ex. 3. Required the square root of -r--. o^c^ Here, the square root of a'^x^=:\/a'^x-\/x^:=ax, and the square root of ¥'c'^=i^h'^x \/c^=bc ; .*. i^— is the root re- bc quired. Ex. 5. It is required to find the square root of Sia^x*. Ans. &ax^, or —8ax\ Ex. 6. It is required to find the cube root of 729a^x^^. Ans. 90%*. Ex. 7. Required the fourth root of 25*6a*5^. Ans. 4ah^, or —4ab^. Ex. 8. Required the fifth root of 32a^x^^. Ans. 2ax\ Ex. 9. Required the sixth root of tttttx-t^. Ans. 4- — -. 4096a;^2 4a:* . 17* 186 EVOLUTION. Ex 10. Required the ninth root of — rf^. Ans. v. a^o^ ah Ex. 11. -Required the square root of -— . Ans. 4-- . * ^ 4a;2y2 =»= 2xy o4^ 4/17 Ex. 12. Required the cube root of r—-rrT' Ans. —-^. CASE 11. To extract the square root of a compound .Quantity. RULE 238. Observe in what manner the terms of the root may be derived from those of the power ; and arrange the terms ac- cordingly ; then set the root of the first term in the quotient ; subtract the square of the root, thus found, from the first term, and bring down the next two terms to the remainder for a dividend. Divide the dividend, thus found, by double that part of the root already determined, and set down the result both in the quotient and divisor. Multiply the divisor, so increased, by the term of the root last placed in the quotient, and subtract the product from the dividend, and to the remainder bring down as many terms as are necessary for a dividend, and continue the operation as be- fore. Ex. 1. Required the square root of a'^-\-2ab-^b^, a^+2'ab-\-b^ 2a 4-^ 2aJ+62 2aJ+52 On comparing a-\-b with c^-\-2ab-{-b'^, we observe that the first term of the power (a-) is the square of the first term of the root (a). Put a therefore for the first term of the root, square it, and subtract that square from the first term of the power. Bring down the other two terms 2ab'\-b'^^ and double the first term (a) of , the root ; set down 2a, and having divi- ded the first term of the remainder (2ai) by it, we have 5, the other term of the root ; and since 2ab-\-b'^ — {^a-\-h) xb, if to 2a the term b is added, and this sum multiplied by 5, the re- sult is 2ab-\-b'^ ; which being subtracted from the terms brought down, nothing remains. EVOLUTION. 187 Ex. 2. Required the square root of a2+2a5+52+2ac4-2dc a^-{-2ab+b^-\-2ac+2bc-{-c\a-{-b+c a2 2a+i 2ab+b^ 2ab+b^ 2a-\-2b-\-c 2ac+25c4-c2 2ac4-2ic+c2 On comparing the root a-j-^+c, thus found with its power, the reason of the rule for deriving the root from the power is evident. And the method of operation is the same as in the last example. Thus, having found the first two terms of the root as before, we bring down the remaining three terms 2ac 4-26c+c2 of the power, and dividing 2ac by 2a, it gives c, the third term of the root. Next, let the last term {b) of the pre- ceding divisor be doubled, and "add c to the divisor thus in- creased, and it becomes 2a-{-2b-\-c \ multiply this new divisor by c, and it gives 2ac-\-2bc-{-c'^, which being subtracted from the terms last brought down, leaves no remainder. In like manner the following Examples are solved. 89 Ex. 3. Required the square root o{ Ax^-\-&x^-{-~x'^-{- Ibx -^25. 89 / 3 4«4-f6aj3-f— a;2+15a;-f-25^2a;2-f--a:+5 4x* 4 2 / 4 6x^-\--x^ 4 4a:2+3a;+5)20.T24-l5a;+25 20a:2-f 15a;+25 Ex. 4. Required the square root of a;^-|- 4a^-}- 2a;* +9i»^— 4a; +4. Ans. a:3+2a;2— a;+2. Ex. 5. Required the square root of x'^-\'Aax^+&a'^x^'\-^a^x -f-a*. Ans. a;2-}-2aa;-|-a2. Ex. 6. Required the square root oia'*'—2a^-{-^a'^—\a+ ^. • Ans. a^ — «+^' 188 EVOLUTION. Ex. 7. Required the square root of Aa^+l2a^x-\-l3a'^x^-\' eax^-^xK Ans. 2a'^-\-3ax-{-x'^. Ex. 8. Required the square root of 9a?* +12a;3+34a;2+ 20a: +25. Ans. 3a?2^2a:+5. Ex. 9. Required the square Voot of a^-\-2ab+b^-\-2ac-\- ' 2ic+c2+2a(f4 2bd-\-2cd+d^. Ans. a-]-b+c+d. Ex. 10. Required the square root of a*+12a3^-f54a262+ I08ab^-^8lb*. Ans. a^-\-6ab + 9b\ Ex. 11. Required the square root of a^ — da^x+lda'^x^— 20a^x^+l5a^x'^—6ax^-\-jic^. . Ans. a^—Sa^x+Sax^—x^. Ex. 12. Required the square root of a'^—2a'^x^-\-x*. Ans. a^—x^ CAS-E III. To extract the cube root of a compound Quantity, RtJLE. 239. Arrange the terms as in the last case ; and set the root of the first terms in the quotient ; subtract the cube of the root, thus found, from the first term, and bring down three terms for a dividend. Next, divide the first term of the dividend by 3 times the square of that part of the root already determined, and set the result in the quotient ; then, to .3 times the square of that part of the root, annex 3 times the product of the same part and the last result, and also the square of the last result, with their pro- per signs ; and it will give the divisor, multiply the divisor by the term of the root last placed in the quotient, and subtract the product from the dividend, bring down three terms or as many as may be necessary for a dividend, and proceed as be- fore. Ex. 1. Required the cube root of a3 4- 3a2J+ 30^24.^3. a^-i-Sa^+Sab^-hP a3 (a+& 3a^-\-3ab-^b^)3a^b+3ab^+b^ 3a25-f3a62_j_J3 The reason of the rule may be made evident from a com- parison of the roots with its cube. Or, thus, if the quantity whose root is to be extracted, has an exact root, the root of* the leading term must be one term EVOLUTION. 189 of its root ; that is, the cube root of a^, which is a, is one term of the root, and the remaining terms being brought down, the root of the last term P is consequently another term of the root ; but as the root may consist of more terms than two ; the next term (b) of the root is always found by dividing (——=b\ the first term of the dividend by three times the square of the divisor, and the two remaining terms of the di- vidend 3aP-\-b3={3ab-{-b^)b ; hence 3ab-{-b^ must be added to 3a^ for a divisor ; and so on. Ex. 2. Required the cube root of x^-[-6x^—40x^-\- 96a;— 64. a;6_|_6^5_40a;3-f 96a;— 64 (a;2+2a;— 4 a;2 3a;*+ 6a;3 + 4a;2)6a;'^ — 40a;3 6a;5+12a;*+8a;3 3a;*+12.'r3— 24a;+ 16) — 12a;4— 48x3+960;— 64 — 12a;*— 48a;3 + 96a;— 64 ^ Ex. 3. Required the cube root of (a'\-bY-{-3{a-\-bYc+ 3(a+&)c2-fc3. Ans. a+6+c. Ex. 4. Required the cube root of a;^— 6a;^+15a;*— 20a;3-|- 15a;2— 6a;+l.. Ans. a;2— 2a;+l. Ex. 5. Required the cube root o( x^-\-6x^i/-\-l5oc^y^-{-20x^^^ + 15acVH-6a;yHy6. Ans. a;24-2a;y+y2, Ex. 6. Required the cube root of 1— 6a;+12a;2— Sa;^. Ans. 1— 2a;. CASE IV. To find any root of a compound Quantity. RULE. 240. Find the root of the first term, which place in the quo- tient ; and having subtracted its corresponding power from that term, bring down the second term for a dividend. Divide this by twice the part of the root above determined, for the square root ; by three times the square of it, for the cube root ; by four times the cube of it, for the fourth root, &c. and the quotient will be the next term of the root. Involve the whole of the root, thus found, to its proper power, whi^ subtract from the given quantity, and divide the first term of the remainder by the same divisor as before. 190 EVOLUTION. Proceed in the same manner for the next following term of the root ; and so on, till the whole is finished. 241. This rule may be demonstrated thus; {a-{-bf=(f •^na"-^b+, &c. Here the nth root of a" is a, and the next term na"-^b contains b, (the other term of the root) na"-'^ times ; hence, if we divide na"-'^b by na"-^, we have 5, or na ;j-^ =b ; and so on, for any compound quantity, the root of which consists of more than two terms. Now, if n=:2 ; then, the divisor na'*-^=2a, for the square root; if 71=3; then, . , . . na"-^=3a^, for the cube root ; ifn=4; then, .... na"-^=4a^, for the 4th root; ifw=5; then, .... na"--^z=5a*, for the 5th root. And so on for any other root, that is, involve the first term of the root, to the next lowest power, and multiply it by the index of the given power for a divisor. Ex. 1. Required the square rootof a*— 2a^a:-f'3a^a;^ — 2ax^ + a:*. 2a'^)—2a^x (a'^—ax)^=a^—2a^x-\-a'^x^ 2a2)-f.2a2a;2 {a^ — aX'\-x^Y=a^~2a^x+3a'^x'^—2ax^+x^. Ex. 2. Required the 4th root of l6a* — 96a^x-\-2l6a^x^— 2ieax^-{-Slx\ l6a\—96a3x-\-2l6a2x^—2l6ax^ + 8lx^(2a—3x 16a4 4x{2ay = 32a^) — 96a^x (2a'-3xY = l6a*—9ea^x-\-2l6a^x^^2ieax^-{-8lx^. 242. As this rule, in high powers, is often found to be very laborious, it may be proper to observe, that the woots of cer- tain compound quantities may sometimes be easily discovered ; EVOLUTION. 191 thus, in the last example, the root is 2a— 3a?, which is the difference of the roots of the first and last terms ; and so on, for other compound quantities. Hence, the following method in such cases ; extract the roots of all the simple terms, and connect them together by the signs + ^^ — , as may be judged most suitable for the purpose ; then involve the compound root thus found, to its proper power, and if it be the same with the given quantity, it is the root required. But if it be found to differ only in some of the signs, change them from + to — , or from — to + , till its power agrees with the given one throughout. How- ever, such artifices are %ot to be used by learners, because the regular mode of proceeding is more advantageous to them ; besides, a knowledge of those artifices which are used by ex- perienced Algebraists, can only be acquired from frequent practice. Ex. 3. Required the square root of a'^-\-2ah-\-h'^-\-2ac-\-2hc Here, the square root of a'^=za ; the square root of Ir^zzih ; and the square root of c'^^:zc. Hence, a-\-b-\-c, is the root re- quired, because {a-\-b-{-cY=ia'^-\-2ah-\-b'^-\-2ac-{-2bc-fc'^. Ex. 4. Required the fifth root of 32a;5— SOar^+SOx^ — 40a;2 + 10a:— 1. Ans. 2a;— 1. Ex. 5. Required the cube root oi x^—Qx^-\-\bx^'—20x^-{^ \bx^--bx^\-\. * Ans. ar^— 2a;-l-l. Ex. 6. Required the fourth root oi a^~A.a^x-\-Qa'^x'^-^Aa!x^ 4-a:*. Ans. a—x Ex. 7. Required the square root o{ x^-{-2x^y^-\-y^. Ans. a:*4"y* Ex. 8. Required the square root of a;^— 2a;y-f-y^. Ans. x^—y^. Ex. 9. Required the cube root of a^-. Ga^a? 4-12 aa;2—8a;3. Ans. a—2x, Ex. 10. Required the sixth root of a;^— 6a;S4-15a;'*— 20a;3+ 15a:2 — 6a?+l. Ans. a; — 1. Ex. 11. Required the fifth TOOt of x^^-\'\bx^y'^-\-9Qx^y^-\- 270a:y+405a:y+243yio. Ans. a;2 + 3y2. Ex. 12. Required the square root of x^-^2xy-\-y'^-\-Qxz-\' Gyz-{-92^. Ans. x-{-y+3z. ^ HI. INVESTIGATION OF THE RULES FOR THE EXTRACTION OF THE SQUARE AND CUBE ROOTS OF NUMBERS. 243. It has been observed, (Art. 104), that, a denoting the tens of a number, and b the units, the formula a'^-\-2ab-\-b^ would represent the square of any number consisting of two 192 EVOLUTION. figures or digits ; thus, for example, if we had to square 25 put a=20 and 5=5, and we shall find ^2^400 2ab=200^ b^= 25 {a+bf = (25f=:625. 244. Before we proceed to the investigation of these Rules, it will be necessary to explain the nature of the common arithmetical notation. It is very well known that the value of the figures in the common arithmetical scale increases in a tenfold proportion from the right to ftie left ; a number, there- fore, may be expressed by the addition of the units, tens, hun- dreds, &LC. of which it consists ; thus the number 4371 may be expressed in^the following manner, viz. 40004-300-|-70-|-l, or by 4x 10004-3 X 100+7X 10-j-l ; also, in decimal arith- metic, each figure is supposed to be multiplied by that power of 10, positive or negative, which is expressed by its distance from the figure before the point : thus, 672..53 = 6 X 102+7 X 10^+2x100+5x10-1 + 3x10-2 = 6x100 + 7x10+2x1 + — + —-=672+— -+-^- = 672-^1. Hence, if the digits ^10^100 ^100 100 100 ' ^ of a number be represented by a, h, c, d, e, &c. beginning from the left-hand ; then, A number of 2 figures may be expressed by \Oa-\-h. 3 figures ... by lOOa+106+c. 4 figures . by 1000a+100Z>+10c+d &c. &c. &c. By the digits of a number are meant the figures which com- pose it, considered independently of the value which they possess in the arithmetical scale. Thus the digits of the number 537 are simply the numbers 5, 3 and 7 ; whereas the 5, considered with respect to its place, in the numeration scale, means 500, and the 3 means 30. 245. Let a number of three figures, (viz. lOOa+105 + c) be squared, and its root extracted according to the rule in (Art. 288), and the operation stands thus ; I. 10000a2 + 2000a&+ \0Qb'^^200ac-\-20hc-\-c'^ 10000^2 (lOOa+106+c 200a+ 1 0b)2Q00ab + 1 005^ 2000a5+100& 200a+20J+c)200ac+205c+c2 200ac+206c+c2 EVOLUTION. 193 * ^ ^— s V ^^^ ^^® operation is transformed into the _ following one , 40000+12000+9004.400+60+1(200+30+1 40000 400+30)12000+900 400+60+1)400+60+1 400+60+1 III. But it is evident that this operation would not be af- fected by collecting the several numbers which stand in the same line into one sum, and leaving out the ciphers which are to be subtracted in the operation. 53361(231 4 43 461 133 129 461 461 Let this be done ; and let two figures be brought down at a time, after the square of the first figure in the root has been sub- tracted ; then the operation may be exhibited in the manner annexed ; from which it appears, that the square root of 53361 is 231. 246. To explain the division of the given number into pe- riods consisting of two figures each, by placing a dot over every second figure beginning with the units, as exhibited in the foregoing operation. It must be observed, that, since the square root of 100 is 10 ; of 10000 is 100 ; of 1000000 is 1000 ; (fee. &c. it follows, that the square root of a number less than 100 must consist of one figure ; of a number between 100 and 10000, of two figures ; of a number between 10000 and 1000000, oi three figures ; &c. &c., and consequently the num- ber of these dots will show the number of figures contained in the square root of the given number. From hence it fol- lows, that \hefi.rst figure of the root will be the greatest square root contained in the first of those periods reckoning from the left. Thus, in the case of 53361 (whose square root is a num- 18 194 EVOLUTION. ber consisting of three figures) ; since the square of the figure standing in the hundred^s place cannot be found either in the last period (61), or in the Jast but one (33), it must be found in the first period. (5) ; consequently the first figure of the root will be the square root of the greatest square number contained in 5 ; and this number is 4, the first figure of the root will be 2. The remainder of the operation will be readily understood by comparing the steps of it with the several steps of the pro- cess for finding the square root of (a + &4-c)2 (Art. 238) ; for, having subtracted 4 from (5), there remains 1 ; bring down the next two figures (33), and the dividend is 133 ; double the first figure of the root (2), and place the result 4 in the divisor ; 4 is contained in 13 three times; 3 is therefore the second figure of the root ; place this both in the divisor and quotient, and the former is 43 ; multiply by 3, and subtract 129, the re- mainder is 4 ; to which bring down the next two figures (61), which gives 461 for a dividend. Lastly, double the last figure of the former divisor, and it becomes 46 ; place this in the next divisor, and since 4 is contained in 4 once^ 1 is the third figure of the root ; place 1 therefore both in the divisor and quotient ; multiply and subtract as before, and nothing remains. 247. The method of extracting the cube root of numbers may be understood by comparing the process for extracting the cube root of (a-j-^+c)^, (Art. 239), with the following operations, in which is deduced the cube root of the number 13997521. EVOLUTION. 195 13997521(200+40+1 o3=(200)3=8000000 1st remainder 5997521 3a2=3x(200)2.= divisor, •.• 3a2^> = 3(200)2 X 40=4800000 3a62=3x200x(40)2= 960000 63=40x 40x40= 64000 5824000 2nd remainder 173521 3(a+&)2c=3(200 + 40)2x 1 = 172800 3((2+i)c3=3(200 + 40) X 1 = 720 c3=lxlXl^ 1 173521 3d remainder 000000 Omitting the superfluous ciphers, and bringing down three figures at a time, the operation will stand thus ; 13997521)241 23= 8 5997 300x2^X4= 4800 30X2X42= 960 43= 64 5824 173521 300 X (24)2x1 = 172800 30x24x12= 720 13= 1 173521 196 EVOLUTION. 248. These operations may be explained in the following manner ; I. Since the cube root of 1000 is 10, of 1000000 is 100, &c. ; it follows, that the cube root of a number less than 1000 will consist of one figure ; of a number between 1000 and 1000000 oitvoo figures, &;c. &c. ; if, therefore, the given num- ber be divided into periods, each consisting of three /figures, by placing a dot over every third figure, beginning with the units, the number of those dots will show the number of figures of which the cube root consists ; and for the reason assigned in the preceding Article, (respecting the fir^ figure of the square root), the first figure of the root will be the cube root of the greatest cube number contained in the first period. II. Having pointed the number, we find that its cube root consists of three figures. The first figure is the cube root of the greatest cube number contained in 13 ; this being 2, the value of this figure is 200, or a =200, consequently 0^= 8000000 ; subtract this number from 13997521, and the re- mainder is 5997521. Find the value of ^x'^, and divide this latter number by it, and it gives 40 for the value of a, the se- cond number of the root ; put this in the quotient, and then calculate the value of 3aV}-\-^ah'^-\-P, and subtract it, and there remains 173521. Find now the value of 3x(«^-^)^ and divide 173521 by it, and it gives 1 for the value of c, the third member of the root ; put this in the quotient, and then calculate the amount of 3(a+^)2c4-3(a+Z>)c2-f-c^ which sub- tract, and nothing remains. III. In reviewing the first of these two operations, it is evident that six ciphers might have been rejected in the va- lue of a^, and three in the value oi^d^b-\-3ali^-^h'^, without af- fecting the substance of the operation ; having therefore sim- plified the process as in the second operation, we are fur- nished with the following rule, for extracting the cube root of numbers. RULE. 249. Point off every third figure, beginning with the units ; find the greatest cube number contained in the first period, and place the cube root of it in the quotient. Subtract its cube from the first period, and bring down the next three figures ; divide the number thus brought down by 300 times the square of the first figure of the root, and it will give the second figure ; add 300 times the square of the first figure, • EVOLUTION. 197 30 times the product of the first and second figures, and the square of the second figure together, for a divisor ; then mul- tiply this divisor by the second figure, and subtract the result from the dividend, and then bring down the next period, and so proceed till all the periods are brought dovi^n. The rules for extracting the higher powers of numbers, and of compound algebraic quantities, are very tedious, and of no great practical utility. Examples for practice in the Square and Cube Roots of Numbers. Ex. 1. Required the square root of 106929. 106929(327 9 62 169 124 647 4529 4529 Ex. 2. Required the cube ) root of 48228544. 48228544(364 27 3276(21228 19656 Divide by 300X32=2700 30x3x6= 540 6X6= 36 393136) 1572544 1572544 1st Divisor =3276 Divide by, (36)2x300=388800 30x36x4= 4320 4x4= 16 2d Divisor 393135 Ex. 3. Required the square root of 152399025. Ans. 12345. Ex. 4. Required the square root of 5499025. Ans. 2345. Ex. 5. Required the cube root of 389017. Ans. 73. Ex. 6. Required the cube root of 1092727. Ans. 103 18* ' 198 CHAPTER VII.. ON IRRATIONAL AND IMAGINARY QUANTITIES. § I. THEORY OF IRRATIONAL QUANTITIES. 250. It has been demonstrated (Art. 231), that the mih. root of a^f the exponent p of the power being exactly divisible by the index m of the 'root, is a'". Now in case that the expo- nent p of the power is not divisible by the index m of the root to be extracted, it appears very natural to employ still the same method of notation, since that it only indicates a division which cannot be performed : then the root cannot be obtained, but its approximate value may be determined to any degree of ex- actness. These fractional exponents will therefore denote im perfect powers with respect to the roots to be extracted ; and quantities, having fractional exponents, are called irrational quantities, or surds. It may be observed that the numerator of the exponeit shows the power to which the quantity is to be raised, and the denominator its root. Thus, a" is the wth root of the mih power of a, and is usually read a in the power ( — j. 251. In order to indicate any root to be extracted, the ra- dical sign y is used, which is nothing else but the initial of the word root, deformed, it is placed over the power, and in the opening of which the index m of the root to be extracted is written. p_ We have therefore Y a^ =a"'. For the square root, the sign -/ is used without the index 2 ; thus, the square root of a^ is written •v^a^ as has been already observed, (Art. 18). Quantities having the radical sign ^ prefixed to them, are called radical quantities: thus, J/ a, -/^, {/ c2,y/ a;"',&c, are radical quantities ; they are, also, commonly called Surds. 252. From the two preceding articles, and the rules given in the second section of the foregoing Chapter, we shall, in general, have IRRATIONAL QUANTITIES. 199 p_ t- L y (a^.h''.c') = y aPxyb''X'yc'z=a'^xb^Xe^\ f^la/'.h'' _ y {aK¥) _ y ¥ x"y/ b^ _ a^ Xh^ V7F~ ycd' ~Yc'xyd''~ '- '' Therefore, ^ aH=y a^X^ bz=aX^ b=ay b ; ^j a^^c^ ^ J/ a%^c'^ _ ^ a^X^b'^xyc^ _ a^by c^ ^a^b ^jc^ exy xz ex'y xz' 253. Two or more radical quantities, having the same in- dex, are said to be of the same denomination, or kind ; and they are of different denominations, when they have different indices. In this last case, we can sometimes bring them to the same denomination ; this is what takes place with respect to the twofollowing, yaWar\dya%^ = a^xb*=a^ . b^-^a^^b^ = -y/a^b"^. In like manner, the radical quantities ^ 2a^b and y 160^6, may be reduced to other equivalent ones, having the same radical quantity ; thus, ^ 2a^b=zy a^ Xy^ 2b=a'^ y 2b, and y I6a^=zy 8a^ . 2b=y S.^aK^ 2bz=2ay 2b ; where the radical factor y 2b is common to both. 254. The addition and subtraction of radical quantities can in general be only indicated : Thus, y d^ added to, or subtracted from yfb, is written -y/h ■^y a^, and no farther reduction can be made, unless we as- sign numeral values to a and b. But the sum of -y/a^, -y/a^b, and ^^a^b is =a^/b-\-a^/b-{■2a^b—^a^/b^, Sy ab—^ ab —2yab', and ^/ ab'^-^y a^'^—byf a^aby a^^^b^ a^ab^/ a — {b-^ab)ya. 255. Hence we may conclude, that the addition and sub- traction of radical quantities, having the same radical part, are performed like rational quantities. Radical quantities are said to have the same radical part when like quantities are placed under the safhe radical sign ; in which case radical quantities are similar or like. It is some- times necessary to simplify the radical quantities, (Art. 252), in order to discover this similitude, and it is independent ot the coefficients. Thus, for example, the radical quantities 35 y/ 2a^b'^, Sa^ 2a^b^, and ^laby 2a%'^, become, by reduction, 3o6^ 2^262, Saby 2a%'^, and —laby 2aW ; which are similar quantities, and their sum is =4^6^/ 2a262. 200 IRRATIONAL QUANTITIES. 256. We have demonstrated, (Art. 252), this formula, v^ aJ'Vc^—y dPy."^ V'y.'y c^ \ from which the rule for the multiplication of radical quantities, under the same radical sign, may be easily deduced. 257. Let us pass to radical quantities with different indices, and suppose that we had to find, for instance, the product of ± ± 'y aPhy%/ h\ or that of oT by 5 *"' : we can bring this case to the preceding, by reducing to the same denominator, (Art. 152), the fractions^ and^ ; and we shall have V a^X V Z>* ' mm p 1- f""' . ?"* = a"^6"''=a'""-' X b"""'="""-l^ a'""' X"™V' b'"''=z"""y oT'b''"'. 258. The rule for dividing two radical quantities of the same kind, may be read in this formula (Art. 233). y a''__"'jaP and it only remains to extend it to two radical quantities of different denominations. Let therefore y a'' be divided by "^ b'' : by passing from radical signs to fractional exponents, we have y aP a'" a"""' ^^y a'""' _"""' /a'"'" We may likewise suppose, under the radical signs, any number of factors whatever, and it shall be easy to assign the quotient, (Art. 252). Let now a = 6 in the formula y aJ'xyb'' = y aP . b"] it becomes, by passing from radical signs to fractional expo- nents, a™ X a'" =y a^^^=a "" =a'^ "» . Therefore the rule demonstrated (Art. 71), with regard to whole positive exponents, extends to fractional exponents. 259. In the same hypotheses 5= a, the quotient ^j^be- p_ comes „ m l^p p-1 p-q_ -=^-^=yaP-''=ra'^=a'^ - ; or . another extension of the rule given (Art. 86), to fractional positive exponents. IRRATIONAL QUANTITIES. 201 260. We may, in the preceding formula, suppose p=o ; and 1 ± 1 -i. it becomes, (since a'"=a'"=a''=l)—=a •", a transformation or demonstrated, (Art. 86) in the case of whole exponents, and which still takes place when the exponents are fractional. 261. If we now admit the two equalities, 1 -?- 1 -^ p_ a , ,_— a J a™ or and if we multiply them member by member, we shall hare the equal products, 111 1 i i_L ~l^~i— T^' or a^Xa'^o"* "•* a *" oT a"" ^ '" It appears therefore evident, that exponentials with frac- tional negative exponents, follow the same rule in their mul- tiplication, as those with whole positive exponents. __£ _-? [ 262. The division of oT by a", gives for the quotient, a"* or Now the exponent of the quotient, namely-^ 4. J^, is the expo* nent of the dividend, minus that of the divisor, which is still a generality of the rule (Art. 86), relative to the division of exponentials. 263. The rules that have been demonstrated in the pre- ceding articles may be extended to radical quantities having irrational exponents : For instance, — 7— 7— rr, &c. since that ay Z, yf 6 the roots of -^1 and -v/S might be obtained with a sufficient degree of approximation, and such that the error may be ne- glected ; so that these exponents shall be terminated decimal fractions, which can be always replaced by ordinary fractions. 264. The formation of the powers of radical quantities, is nothing else but the multiplication of a number of radical quantities of the same denomination, marked by the degree of the power ; so that it is sufficient to raise the quantity ^ under the radical sign to the proposed power, and afterwards 202 IRRATIONAL QUANTITIES to affect this power with the common radical sign. If the in- dex of the radical sign is divisible by the exponent of the power in question, the operation then is performed by dividing that index by the exponent of the power. Let us give two examples for these two cases, (^ a''b'')'=y aP'b'" ; Cy a^^)' =y aPb". 265. If the exponent of the power is equal to the index of the radical sign, the power is the quantity under the radical sign. In fact, the indication y a'', shows that a^ is the mth. power of a certain number y a", which we can always assign, either rigorously, or by an approximation, so that the mth. power of y a'' is a^. In like manner, the square of -y/a is a ; the cube of ^ a is a ; the 5th power of ^ (— «^) is ~d^ ; and so on. 266. A rational quantity may he reduced to the form of a given surd, hy raising it to the power whose root the surd expresses, and prefixing the radical sign. Thus a^=z^ a^^=zy a^=^ a^, &c. m and a-\-x=^{a-\-x)"'. In the same manner, the form of any radical quantity may be altered ; thus, '^{a-\-x)=iy (a+^)^ = ^ {afxY, &LC. or (a4-a;)2 — («-f a;)4 — («-f a;)6, Slc. Since the quantities are here raised to certain powers, and the roots of those powers are again taken ; therefore the values of the quantities are not altered. Also, the coefficient of a surd may he introduced under the radical sign, hy first reducing it to the form of the surd, and then multiplying as in (Art. 257). Thus, a^x — ^/a^X^/xz=z^a'^x', 6^2 = ^36 Xy^2 = 'v/'''2; and ir(2a -.ir)^=(a;2)ix(2a-xP= y/ (20x^-0?). 267. Conversely, any quantity may be made the coefficient of a surd, if every part under the sign be divided by this quantity, raised to the power whose root the sign expresses. Thus, Vl"^ — a2^)= ■^/a'^X^/(a—x)=a^{a—x)\ ^60= -v/(4Xl5)=i y'4x •v/15=2-v/15 ; and 'y (a'"" — a-^a:") =y [a"'X («"— a^")] =y a'» X"/ (a"— x") = ay (a"— a;"). 268. Let us pass to the extraction of roots of radical quan- tities, and let the mth root of y a* be required, which we in- dicate thus, y y a*. We shall put y y/a^ — x, or y a'=.x, by making y a*=ia'. Involving both sides to the power m, we find a' or y a*=x'", raising again to the power n, we obtain a'=x"'". If the mnih root of both sides be extracted, we have another enunciation of x ; namely, '"ya'=:x=yya'. IRRATIONAL QUANTITIES. 203 We shall find, by a like calculation, And, in fact, we make lst,y {/{/ a'=a', whence y a'=:x, and a'=y(/y a'=:a;'"; 2d, by putting {/{/ a' = a", whence ^ a" = 07'", and a^'z^a;'"'' ; 3d, making {/ a*=ia"', whence {/ a^"=a;"'", and a"'=y a'=o(r"P ; and finally a'=a;'^w^ .-. a:= ya'. Thus, for example, the 12th root of the number a can be trans- formed into ^ ^ y/ «, 169. It is to be observed, that radical quantities or surds, when properly reduced, are subject to all the ordinary rules of arithmetic. This is what appears evident from the preced- ing considerations. It may be likewise remarked, that, in the calculations of surds, fractional exponents are frequently more convenient than radical signs. § II. REDUCTION OF RADICAL QUANTITIES OR SURDS. CASE I. To reduce a rational quantity to the form of a given Surd. RULE. 270. Involve the given quantity to the power whose root the surd expresses ; and over this power place the radical sign, or proper exponent, and it will be of the form required. Ex. 1 . Reduce a to the form of the cube root. Here, the given quantity a raised to the third power is a^, and prefixing the sign ^ , or placing the fractional exponent (^) over it, we have a=^ a^={a^)'^ (Art. 251). 271 . A rational coefficient may, in like manner, be reduced to the form of the surd to which it is joined ; by raising it to the power denoted by the index of the radical sign. Ex. 2. Let 5-v/a=y'25X'/« = V25a. Ex. 3. Reduce — da'^b to the form of the cube root. Here, { — 3aHy = -27a^P ; .-. — ^ 27a6J3 is the surd re- quired. Ex. 4. Reduce —4x1/ to the form of the square root. Here, (—4xy)'^=zl6xY ; .-. — 4a;y= — Vl^^V- Ex. 5. Reduce 4a; to the form of the cube root. ^ 1 Ans. (ia;3)^. 204 IRRATIONAL QUANTITIES. Ex. 6. Reduce a-\-z Xo the form of the square root. Ans. {a'^-^2az+z'^)^ i. Ex. 7. Reduce 4a;* to the form of the cube root, Ans. (^ 64a;* ) or (64a;*)^. Ex. 8. Reduce —x^y^ to the form of the square root, Ans. — -y/ocy. Ex. 9. Reduce —ah to the form of the square root. Ans. — ■y/oP-h'^, CASE II. To reduce Surds of different indices to other equivalent ones, having a common index. RULE. 272. Reduce the indices of the given quantities to fractions having a common denominator, and involve each of them to the power denoted by its numerator ; then 1 set over the com- mon denominator will form the common index. Or, if the common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. Then over the said quantities, with their new indices, set the given index, and they will make the equivalent quantities sought. Ex. 1. Reduce ^/a and ^5 to surds of the same radical sign. Here, -yjazzza^, and ^ h — h^ . Now, the fractions \ and \ reduced to the least common denominator, are \ and | ; .-. ^=:a^^{a?f^ya^, and h^ =^ =:{h''f ^^ V^ . Consequently ^ a^ and ^ h^ are the surds required Ex. 2. Reduce -y/a and ^ x to surds of the same radical sign ^ , or to the common index 4. 1 X (Art. 251), ^Ja—c?, and y a;=a:* ; then ^4-^=^X6 = 3 ; and i-r- J=i X 6=3| ; .-. |/ a^ and ^ o? , or {a^^ and {i^f, are the quantities required. Ex. 3. Reduce a^ and i^ to the same radical sign ^ . Ans. y a®, and ^ b^» IRRATIONAL QUANTITIES. 205 Ex. 4. Reduce a* and x^ to surds of the same radical sign Ans. ^^ x^ and ^ ^ aj*. Ex. 5. Reduce y a and v^ y to surds of the same radical sign. Ans. ""^ a'" and ""^ y". 2. 1 Ex. 6. Reduce a^ and i* to surds of the same radical sign. Ans. i^a^ and^y R Ex. 7. Reduce 3^ 2 and 2-y/5 to the same radical sign. Ans. 3^4 and 2^ 125. Ex. 8. Reduce \/ xy and ^/ ax to the same radical sign. Ans ^^ x'^y^ and ^^. a^x"^, CASE III. To reduce radical Quantities or Surds, to their most simple forms. RULE. 273. Resolve the given number, or quantity, under the ra- dical sign, if possible, into two factors, so that one of them may be a perfect power ; then extract the root of that power, and prefix it, as a coefficient to the irrational part. Ex 1. Reduce ■\/a'^b to its most simple form. Here Va25_yfo2xy'^>=aXV'^=«'/*• Ex. 2. Reduce Y ^"^ ^^ i^^ most simple form. ni Uerey a'^x^y a'"X'y x=za^xy x=zaxy X. Ex. 3. Reduce ■v/'^2 to its most simple form. Here ^72 = V(36 X2)=r -/SSx ^2=6^2. 274. When the radical quantity has a rational coefficient prefixed to it ; that coefficient must be multiplied by the root of the factor above mentioned ; and then proceed as before. Ex. 4. Reduce 5^ 24 to its simplest form.# Here 5^ 24 =: 5y (8 x 3) = 5^/ 8 x V 3^ 5 X 2 X 3/ 3 = 10V3. Ex. 5. Reduce -ya^hc and -^^^d^x to their most simple form. Ans. a'^-^hc and '7a^/2x. Ex. 6. Reduce 1/ 243 and V 96 to their most simple form. Ans. 3y 3 and 2^ 3. Ex. 7. Reduce ?/ (a^-ira^"^) to its most simple form. Ans. aV(l + ^'). Ex.8. Reduce /( ^ J to its most simple a—2h , , form. Ans. yab, 19 206 IRRATIONAL QUANTITIES. Ex.9. Reduce {a-{-b)\/ [(a—bYxx"^] to its most simple form. Ans. (fl2_i2)3^^2, 275. If the quantiti; under the radical sign he a fraction, it may he reduced to a ivhole quantity^ thus : Multiply both the numerator and denominator by such a quantity as will make the denominator a complete power cor- responding to the root ; then extract the root of the fraction whose rmmeralor and denominator are complete powers, and take it frorn under the radical sign. c a'- Ex. 1. Reduce - X i/-T- lo aa intes^ral surd in its most d ^ b ^ simple form. Tx c ,(P c ,a'^b c ,d^ , c a ,. ca ,, Ex. 2. Reduce i^ ij to an integral surd in its simplest form. / H V 2 \ v> V ^2 nere, ^ / -gr— ^ / \^27 x 3 j ~ a v 2T ^ / a — » X ^v — ^j— Ex. 3. Reduce J-v/f ^"^ ^^ integral surd in its most simple form. An&. ^^g-y/K. b c^ Ex. 4. Reduce x-y/ - and «?/ — to inteo-ral surds in their y ^ a ^ most simple forna. Ans. - -/^^ and ^ e^a^, Ex. 5. Reduce ^ i and f -/J icy integral surds in their most simple form. *" Ans. ^\^ 21 and 3'\/2. Ex. 6. Reduce ?/ -— - and -\/ —-- t€> their most simple form. ^ 125 ^ Sx* ^ • Ans. "V 2 and — -r\/2a. 5 *^ ^ 4a;2 '^ 276. The utility of reducing surds to their most simple forms, especially when the surd part is fractional, will be readily per- ceived from the 3d example above given, where it is found that f -v/f^-g^V^M, in which case it is only necessary to extract, the square root of the whole number 14, (or to find it in some of the tables that have been calculated for that purpose), and then multiply it by ^ ; whereas we must, otherwise, have first divid- ed the numerator by the denominator, and then have found the root of the quotient, for the surd part ; or else have determined the root of both the numerator and denominator, artd then divide the one by the other ; which are each of them troublesome pro- IRRATIONAL QUANTITIES. 207 cesses ; and the labour would be much greater for the cube and other higher roots. 277. There are other cases of reducing algebraic Surds to simpler forms, that are practised on several occasions ; for in- stance, to reduce a fraction whose denominator is irrational, to another that shall have a rational denominator. But, as this kind of reduction requires some farther elucidation, it shall be treated of in one of the following sections. § III. APPLICATION OF THE FUNDAMENTAL RULES OF ARITH- METIC TO SURD (QUANTITIES. CASE I. To add or subtract Surd Qiuzniities, RULE. 278. Reduce the radical parts to their simplest terms, as in the last case of the preceding section ; then, if they are similar, annex the common surd part to the sum, or difference of the rational parts, and it will give the sum, or difference required. Ex. 1. Add 4 v^ a?, -/a;, and 5 -y/ a; together. Here the radical parts are already in their simplest terras, and the surd part the same in each of them; .'.4:-)/x-{--\/x ■i-5i/x={44-l-\-^)X y'jr=:10y'a; the sum required. Ex. 2. Find the sum and difference of '\/\&a'^x and -y/Aa^x. '\/lQa^x=:-y/\Qa'^X -y/xzizAa^Xf and -/4a%= ■y/i.a^ X ^xz=z2a-}/x ; .'. the 57iwi =:(4<2-j-2a) X -y/jp^Say^a? ; and the difference ={4« — 2a) X ^/x\=:2a^/x. Ex. 3. Find the sum and difference of ^ 108 and 9^/ 32. Here^l08=:^27X^4 = 3x|/4= 3^/4, and 9^ 32:r39y 8X^ 4 = 18 X^ 4 = 18^ 4, the 5wm =(18+3)x^4=2l|/4; and the difference =(18 — 3) X'V 4=15^/ 4. 279. If the surd part be not the same in each of the quan- tities, after having reduced the radical parts to their simplest terms, it is evident that the addition or subtraction of such quantities can only be indicated by placing the signs + or ^ between them. Ex. 4. Find the sum and difference of 3^ a% and hs/c^d. Here 3J/ a'^h=^'iy a^ X 3/ ^>=3« X^ hr^Za]/ h, and hyj c^d—hA/ c'^ X -y/d—hc X -y/d^hc yfd ; the sum =z3al/ b-\-bc-y/d- and the diff'erence z=3a^ h^hc-^/d. 208 IRRATIONAL QUANTITIES. Ex. 5. Find the sum and difference of -/^^y and -/J. Ans. The sum =z:^j^6, and difference =Jg.y^6. Ex. 6. Find the sum and difference of ■\/27a^x and -/3a*a:. Ans. The ^wm =4a'^^3x, and difference —2a?-^3x. Ex. 7. Find the ^w»i and difference of J -/a^ft and ^^-/^a;*. A rr^u /2a:24-3a\ .. ^ ^.^ /'2x'^2a\ Ans. The ^wm =» — — jy^, and difference i ) Ex. 8. Required the sum and difference of 3^/ 62.5 and 2|/ 135. Ans. The ^wwi =21|/ 5, and difference =9^ 5. Ex. 9. Required the ^mw and difference of y a^i^ ^nd ^ a;^^^. Ans. The 5«m z=.a^/ab-\-xy x^y"^, and difference z^a-^ah^ x^ x^y^, CASE II. To multiply or divide Surd Quantities. RULE. 280. Reduce them to equivalent ones of the same deno- mination, and then multiply or divide both the rational and the irrational parts by each other respectively. The product or quotient of the irrational parts may be re- duced to the most simple form, by the last case in the preced- ing section. 1 jL Ex. 1. Multiply -y/a by ^ b, or a^ by b^. The fractions J and ^, reduced to a common denominator^ are J and ^. .-. a^=^a^=^ «3 ; and b^=b^=^ b\ Hence V*=^ a^x^/ J2_6/ ^352, Ex. 2. Multiply 2^3 by 3iJ/ 4. 3 By reduction, 2^/'6=2 x 3« =2 x «/ 33=2^ 27 ; and 3^ 4^:3 X 4^=3^42=^3^ 16. .-. 2 V3 X 3^ 4=2?/ 27 X 3^ 16=:6f/ 432. Ex. 3. Divide 8^ 512 by 4^/ 2. Here 8-^4=2, and ^ 512-^-^ 2=^ 256=4^ 4. .-. 83/ 512-r43/ 2=2 X43/ 4 = 8^ 4. Ex. 4. Divide 2 3/ be by 3 Vac. Now 2^ &c=2 X (icf =2 X {hcf =2^ ^2^2 IRRATIONAL QUANTITIES. 209 1 ^ and 3 7ac=3 X (ac)2 =zz3 X (ac)^ = 3^ aV . •'• 37^~3^ V^3c"^~3V^~3V a6c6 ~3acV "" "^ * 281. If two surds have the same rational quaMity under the radical signs, their product^ or quotient, is obtained by making the sum, or difference, of the indices, the index of that quantity. Ex. 5. Multiply ^ a^ hy ^ a^ or a^ by a^ . Here, a^ X cP =^ ^=a^=a^. Ov^a^X^ 0^=^ (a^Xa^) = ?/ a^^a"^, as before. Ex. 6. Divide y a^ by |/ a\ or a* by «=^ 3 4 ji_4 9 16 Here, aT-ra' — " »-- .TJ-W 282. If compound surds are to be multiplied, or divided, by each other, the operation is usually performed as in the multi- plication, or division of compound algebraic quantities. It fre- quently happens that the division of compound surds can only be indicated. Ex. 7. Multiply ^S—ya'^hy^S + ya. -/S-^^' ^ Since V-'^X^ 3=3^X 3^ = ys-h^a iy (3^X3^)=:^ (27X9)=^ . n/ 243 y243-y{3«2) +y{27a^)-a Product =^ 243-y (Sa^)^^/ 27a^-a. Ex. 8. Divide ^b'^ca-{-'\/a'^b — be— ^abc by ^bc+ y/a. •\/b'^ca -{--yj a-b — bc — -^abc ^Jb'^ca-{^^/d^b — be — -y/abc — be — ^/abc Ex. Ex. Ex. Ex. Ex. Ex. ^bc-\- ^a Quot. — -v/6a— 'Jbc. 9. Multiply 3/ 15 by -/ 10. 10. Multiply 13/ 6 by|3/ 18. 11. Multiply 3/18 by ^4. 12. -Multiply \y 6 by -f^y 9. 13. Divide 4-/50 by 2'/5. 14. Divide IVI by W\. Ex. 15. Divide |/ 0^(^352 by yrf: 19* Ans. y 225000. Ans. ^ 4. Ans. 23/ 9. Ans. -^y 2. Ans. 2v^l0. Ans f -y/lO. Ans.^ai. 210 IRRATIONAL QUANTITIES. a i 1 JL IJL 2 Ex. 16. Multiply a^ x^ by a* x^. Ans. a^^ x^ 5. Ex. 17. Multiply y aH^c^ by y a^PcK Ans. a^^c^. Ex. 18. Divide (a^+Pfhy (a'^+b^f Ans.(/(a*+&^). Ex 19. Multiply 4+2 -v/a by 2-^2. Ans. 4. Ex. 20. Multiply V(« - V(* — V^)) by V(« + Vl*— V3)). Ans. V(a2-6+'/3). Ex. 21. Divide a35_aJ2c by a^-\-ay^bc. Ans. ab—b^bc. Ex. 22. Divide a*+a;* by a2+aa;-/2 + a:2 Ans. a^ — ax'^2-\-x'^. 283. It is proper to observe, sin-ce the powers and roots of quantities may be expressed by negative exponents, that any quantity may be removed from the denominator of a fraction into the numerator ; and the contrary, by changing the sign of its index or exponent ; which transformation is of frequent occurrence in several analytical calculations. 1 a2 Ex. 1. Thus, (since — =5-^), — may be expressed by 1 a^ \ a2j-3 ; and (since a^= — -), we have 7-^=71 — "o- a%^ Ex. 2. The quantity —^ may be expressed by a^Pc-*e-^, c e i 2 ^2^3 Ex. 3. Let the denominator of — r^r- be removed into the c b^ 1. Z numerator. Ans. a^x^c-'^b—^. Ex. 4. Let the numerator of -7— be removed into the deno- o minator. Ans. a-^x-^b Ex. 5. Let x^y^a^ be expressed with a negative exponent. Ans. v-^ir-^a * IRRATIOxNAL QUANTITIES. 211 CASE III. To involve or raise Surd Quantities to any power RULE. 284. Involve the rational part into the proposed power, then multiply the fractional exponents of the surd part by the index of that power, and annex it to the power of the rational part, and the result will be the power required. Compound surds are involved as integers, observing the rule of multiplication of simple radical quantities. Ex. 1. What is the square of 2^ a 1 The square of 2^a—(2a^f=2'^ X a^' =4a. Ex. 2. What is the cube of ^ (a^—h'^-\-^/^) ? The cube of ^ {a^-h'^-\-^^) = (a'^-h'^^-^/^f^=a^^b'^ + V3. 285. Cor. Hence, if the quantities are to be involved to a power denoted by the index of the surd root, the power re- quired is formed by taking away the radical sign, as has be^a already observed. Ex. 3. What is the cube of ^ y/2ax 1 " 1.3 3 Here (\Y=^, and {^2axf=i{2axY' z={2axY =z(2ax)x{2axy^ ; .-. \x2axX{2axY = \ax^/2ax is the power required. Ex. 4. It is required to find the square of -y/a— V^- 'y/a — ^/b -y/a — ^/b a — -y/ab — \/ab-\-b The square a~~2-\/ab-\-h. Ex. 5. It is required to find the square of 3 ^Z 3. Ans. 93/9 Ex. 6. Find the cube of y/a. Ans. a^^/a Ex. 7. Find the 4th power of — ^ a-. Ans. a^^/ ^2 Ex. 8. Find the 5th power of —\^ab. Ans. —ab Ex. 9. Required the cube of a — -y/b. Kl\s.a^—^a^^/b-\■^ab—h^b 212 IRRATIONAL QUANTITIES. Ex. 10. Required the square of 3+ V^* Ans. 14+6V5. Ex. 11. Required the cube of — ^ W^~ V^^)- Ans. -y/bc— 'sja. CASE IV. To evolve or extract the Roots of Surd Quantities. RULE. 286. Divide the index of the irrational part by the index of the root to be extracted ; then annex the result to the proper root of the rational part, and they will give the root required. If it be a compound surd quantity, its root, if it admits of any, may be found, as in Evolution. And if no such root can be found, prefix the radical sign, which indicates the root to be extracted. Ex. 1. What is the square root of 81 ^/a ? Here -/81=9, and the square root of -y/a or a^z=a^-^2 = Jx^=ai^\/a ; .-. '/(81/a) = 9V a, or 9 A Ex. 2. What is the square root of a^ — 6a-\/b-\-9b. a^—6a-^b + 9b{a—3^/b 2a-3y/b)-6a^/b+9b -6aV^ + 9i Ex. 3. Find the square root of 9^ 3. Ans. 3 V 3. Ex. 4. Find the 4th root of |}^ a^. Ans. f (/ c. 3 Ex. 5. Find the cube root of (5a'^—3o^)^. Ans. ^(5a'^-3x^). Ex. 6. Required the cube root of ^a^. Ans. ^a^ b. Ex. 7. What is the fifth root of 32^ a^* ? Ans. 2 3/ x. Ex. 8. What is the 4th root of 16a^ x ? Ans. 2^ a*x. Ex. 9. What is the nth root of 7/ a''x^ ? LI. Ans. a'^x^' Ex. 10. It is required to find the cube root of a^—3a^^x-{- Sax—x-^x. Ans. a—^x. IRRATIONAL QUANTITIES. 213 § IV. METHOD OF REDUCING A FRACTION, WHOSE DENOMI- NATOR IS A SIMPLE OR A BINOMIAL SURD, TO ANOTHER THAT SHALL HAVE A RATIONAL DENOMINATOR. 287. A fraction, whose denominator is a simple surd, is of the form - — ; where x may represent any rational quantities y X whatever, either simple or compound ; thus, be a c — d „ are fractions, whose denominators are simple surd quantities. 288. It is evident that, if a surd of the form y x he multi- plied by y a?"-^, the product shall be rational ; since y xX y x"—^=y (xxx'^~^)=y x"=x ; in like manner, if ^ (a+oc) be multiplied by ^ (a+a;)^, the product will he a-^x. 289. Hence, if the numerator and denominator of a fraction of the form - — be multiplied by y x^~^j the result will be a 'Y X fraction, whose denominator shall be rational. Thus, let both the numerator and denominator of the frac- tion be multiplied by ^x, and it becomes ; and by ^x X multiplying the numerator and denominator of the fraction ,,y . , , by y (a-\-x)^, it becomes yLlA-^-l-^ = ^ / ' -, y{a-{-xy ^^^ '' y {a^^f a+a? Or, in general, if both the numerator and denominator of a fraction of the form ~ — be multiplied by y a;"~^ it becomes y X a\/ a:" — ^ — , a fraction whose denominator is a rational quan- X tity. 290 Compound surd quantities are such as consist of two or more terms, some or all of which are irrational ; and if a quantity of this kind consist only of two terms, it is called a binomial surd ; and a fraction whose denominator is a binomial surd, is, in general, of the form —. y a±y b 291. If a multiplier be required, that shall render any bi- nomial surd, whether it consist of even or oc?c? roots, rational, it may be found by substituting the given numbers, or letters, of 214 IRRATIONAL QUANTITIES. which it is composed, in the places of their equals,, in the fol- lowing general formula : Binomial, y/ aJti\/ b. Multiplier, y a^-^^^ ^"""^* + V «"~^*^ f V a^-^b^-{-, &c., where the upper sign of the multiplier must be taken with the upper sign of the binomial, and the lower with the lower ; and the series continued to n terms. This multiplier is de- rived from observing the quotient which arises from the actual division of the numerator by the denominator of the following 'fractions : thus, I. ^■=x^—'^ + x^—^y + x^—'^y'^ +, (fee. . 4-^*"^ to ^ X — y terms, whether n be even or odd, (Art. 108). a;" 1/" II.. ^-=a;"— ^— a;"— 2w + a:"— ^w2 — , &c. . . — t/"— ' to x-\-y y y ^ J n terms, when » is an even number, (Art. 109). x" -4~ if* III. — -^—a:"— 1— a;"— 2y-f-a;"— V— J &c. . . +v»— ^ x-\-y :} 3 • :/ to n terms, when n is an odd number, (Art. 110). 292. Now let a;"=a, y"=i ; then, (Art. 116), x^'^^ a, y=.y h, and these fractions severally become ;^ — l, ■^ a — y^ — -, and 7 ; and by the application of the rules Ya-\-yb' ya+yb' ^ ^^ in the preceding section we have a;"— ^ =y a"— ^ ; a^— 2=^ a"— 2, a:"— 3— y a»— 3, &c. also, y'^=y b"^ ; y'^=y/ b^ \ &c. ; hence, y,n-2y—ny an-^xy b=Y a^-H ; x'^-^y^^y a^-^xy b^=y a n—^3 . ^Q gy substituting these values of a;**— ',a;'»— ^y, a;"— y, &c., in the several quotients, we have — —--=y a—^-\- y d — y y a''—'^b-\-y a«— 3Z>2_j-, "— 1 to w terms ; where n may be any whole number whatever. And — — \-m = ^ y a-\-y b y gn—^—y a^-2b-{.y an-H"^ — , &c. . . . 4zy b"^^ to n terras ; where the terms b and y 5" -^ have the sign +, when n is an odd number : and the sign — , when n is an even number. 293. Since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations that, if a binomial surd of the form y a—y b be multiplied by y a"— ^-f- y a"— 2& + , &c. . -{-y b^—'^ (n being any whole number what- ever), the product will be a — 6, a rational quantity ; and if a binomial surd of the form^ a+y b be multiplied by y a"— ^ —y a"— 26+5/ »— 3i2_^ (5^c. . . . ±y b"^^, the product will be IRRATIONAL QUANTITIES. 213 a-\-h or a—b, according as the index n is an odd or an even number. 294. Hence it follows, that, if the numerator and denomina- tor of the fraction (Art. 290), be multiplied by the multiplier, (Art. 291), it becomes another equivalent fraction, whose deno- minator shall be rational. There are some instances, in which the reduction may be performed without the formal application of the rule, which will be illustrated in the following examples. Ex. 1. Reduce ^^—7- — ^r- to a fraction with a rational yo — y o denominator. To find the multiplier which shall make -/5 — y^S rational, we have n=z2, a = 5, b = 3 ; .'. (Art. 291), y a"-^-\- y a"-% = (since a"-2 = a2-2=a'':zzl) ^5 + ^3; .-. :^0g±^X 295. This multiplier, ^^■\- v/3, could be readily ascertain- ed, without the application of the formula, by inspection only ; since the sum into the difference of two quantities gives the difference of their squares ; also the multiplier that shall render -y/a-f-'v/^ ''^cttional, is evidently -y/a — -y/b. In like manner, a trinomial surd may also be rendered rational, by changing the sign of one of its terms for a multiplier ; and a quadrinomial surd by changing the signs of two of its terms, &c. 2 Ex. 2. Reduce . . — — 7- to a fraction with a rational V5+^3 — -v/2 denominator. • In the first place, ,. '1 , X ^^^^^^^^S ^ VS+VS — V2 y5 + '/3 + V2 2(1/54-1/34-^2). . •V/54-V34-V2 -34-Vl5_ 64-2v'15 ' 34-VI5 _34-'/15"~' (V5+/3+V2) 2^3tVl5) .^ ^^^ ,^^^,.^„ ^^^^.^^^ Ex. 3. Reduce ^ — - — 5 — - to a fraction with a rational de- ^3 — ^2 nominator. To find the multiplier which shall make ^3 — ^2 rational, we have w=r3, anz3, b=2 ; .'. V a"-! 4- y a"-'^b-\- U 5"-' = 3/94.3/6 + 3/4. Now (3/3-V2)(V9 + ^64-^4) = a-6=3-2 = l; 216 IRRATIONAL QUANTITIES. . the denominator is 1, and the fraction is reduced to ?/ 9-{- 296. Hence for the sum, or difference, of two cube roots, which is one of the most useful cases, the muhiplier will be a trinomial surd consisting of the squares of the two given terms, and their product, with its sign changed. Ex. 4. Reduce V, J ■ to a fraction with a rational denominator. Ans. -—. 2 3 Ex. 5. Reduce —r- y- to a fraction with a rational de- V ^ — y ^ nommator. Ans. —^ — 5 — X o Ex. 6. Reduce . , — . , ^ to a fraction whose denomi V3+V2 + 1 nator shall be rational. Ans. 4-f-2'y/2— 2'y/6. Ex. 7. Reduce ^- — T-h-r- to a fraction whose denominator shall be rational. 2 Ex. 8. Reduce ^ ^ , 4 , q *^ ^ fraction whose denominator shall be rational. Ans. ^125 — '*/ 75-f ^ 45-^ 27. 297. It may not be improper to take notice here of another transformation which binomial surd quantities may undergo by equal involution, and evolution. Ex. 1. To transform -v/24-'y/3 to a universal surd. Its square =5 + 2 -/e ; .-.the root = -/(5-h2-/6). Ex. 2. To reduce ^21 -\- -v/48 to a universal surd. Here ( v'274-'v/48)2=27+2V'l296+48=147 ; .'.-^21 4.'/48 = 'v/147= V49 X 3=:7-v/3. Ex- 3. To transform ^ 320 — ^ 40 to a general surd. Here (^ 320-3/ 40)3^320-33/ 4096000 + 3 ^ 512000 —40=40; .-.^320-^40=2^5. 298. This transformation is very useful, since, by means of it, we can always reduce the sum or difference of any two surd quantities, if they admit of the same irrational part, to a single surd. This may be proved, in general, thus ; if y/ a and y/ b admit of the same irrational part, they must be of the form 7 a'"m and y b'"m ; and ( 7 a"'wi+ y b"'mY=za"'m+n IRRATIONAL QUANTITIES. 217 b"'m=a"'m-^na"^^'Xmb"'-^&c b'^'m .-.y a-\-y b=y (a'"m-{-nma'"—'^y'*-\-&,c i^''m)= the nth root of a rational quantity. Hence the product of -y/a by -x/b is rational if -y/a and -y/b admit of the same irrational part; also, ^ a^X y b, or y axy b'^, is rational, if ^ a and y b admit of the same irrational part ; and, in general, y a -'^ xy 6, or ^ c a y 5*^^, is rational, if y a and y b admit of the same irrational part. 299. It is propel to observe, that, for the addition or sub- traction of two quadratic surds, the following method is given in the Bija Ganita, or the Algebra of the Hindoos, translated by Strachey. Thus, to find the sum or difference of two surds, ^a and -y/b, for instance. RULE. Call a-^b the greater surd ; and, if aX& is rational, (that is, a square)^ call 2 ^/ab the less surd, the sum will be '\/(a-{-b ■\-2^/ah)^ (nr ( V^iV^)^)' ^"^^ ^^® difference '\/{a-\-b — 2'v/a5). If a X & is irrational, the addition and subtraction are impossible ; that is, they can only be indicated. Example. Required the sum and difference of -y/2 and -y/S. Here 2 + 8 = 10=> surd; 2 x8z=16, .-. 'v/16=:4, and2Vl6 =2x4 = 8=] be of the forma;+y, it is ^■\/(2n-i-s-\-t)-{-^-\/(s-\-t—-2n) ; and the cih root of A+B is -5-^ ■ — ' 'If^ -'. ^ \/ Q. Ex. 1. Required the cube root of 10 + -/ 108. In this case, ^108 is >10; .-. A= ^108, B = 10, A2— B2 = 108— 100 = 8, and 8Q = n^. Now, since 8 is a cube number, Q may be taken equal to 1 ; then8Q = 8=n^; .-. n=2. Also, 3/[(A+B)2]=7+/; ^[(A-B)2] = l-/, where /is some fraction less than unity ; .-. ^=7, t=l ; and a7+y=-^^^ — -V3 + 1. If therefore the cube of 10+ -y/ 108 can be expressed in the proposed form, it is -y/S+l ; which on trial is found to suc- ceed. Ex. 2. Find the cube root of 26+ 15 ^3. Ans. 2 + -v/3 Ex. 3. Find the cube root of 9 -/ 3 — 1 1^2- Ans. '/3 — 'v/2. Ex. 4. Find the cube root of 4^5 +8. A V5+1 Ans. „^^ . 308. In the operation, it is required to find a number Q, such that (A^ — B)2xQ may be a perfect cth power ; this will be the case, if Q be taken equal to (A^ — l^^y-'^ ; but to find a less number which will answer this condition, let A^ — B^ be divisible by a, a, ... (m) ; 6, 6, . . . (n) ; d, d, ... (r) ; &c. in succession, that is, let A'^-^B'^z=a'"b"d'', &c ; also, let Q = a'b^d^ &c. Then {A'2^B^).Q = ar*^ xb"*^ X d"*', &c. which is a perfect cth power, if x,i/, z, &c., be so assumed that m+x, w+y? ^+^> c, or in the hypotheses, for instance, b=c-^\, be- comes -/ — IX V'^='/— « ; and, in general, 2^ —azzz'^y aX It may be demonstrated, in a similar manner, that j IMAGINARY QUANTITIES. 223 313. Hence, in the calculation of imaginary radicals, it is sufficient to demonstrate the rules for multiplying and involving the imaginary radical -y/ — 1 ; since imaginary quantities can be always resolved into factors ; so that — 1 only shall remain under the radical sign. 314. In the first place, then it may be observed, when a^ is considered abstractedly, or without any regard to its gene- ration, then -y/a^ maybe either -{-a or — a there being no- thing in the nature of the quantity so taken, to denote from which of these two expressions it was derived. 315. But this ambiguity, which, in the above mentioned case, arises from our being unacquainted with the origin of the quantity whose root is to be extracted, will not take place when the sign of the quantity from which it was produced is known ; as there can, then, be only one root, which must evidently be taken in plus or minus, according to the state it existed in be- fore it was involved. 316. Thus, V[( + «) X(4], or -/[C + a^)] cannot be of the ambiguous form Jka, as it would have been if a^ had been un- conditionally assumed, but it is simply a ; and, for a like rea- son, V[( — a)x{ — a)], or ^( — aY is = —a, and not J-a ; since the value of the equivalent expression -(--v/^^' ^^ — V^^ in these cases, is determined, from the circumstance of its be- ing known how a^ is derived. 317. Hence the product o/ -y/— 1 by -y/— 1, or which is the same, (■\/ — l)2is = — -\/l=z — 1. This is what appears, evi- dent from, since that in squaring a quantity with the radical sign ■;/, we have only to take it away, that is, to pass the quantity from under the radical sign, 318. Also, if the factors, in this case, be both negative, the result will be the same as before ; since — (-y/ — l)x — (-\/ — 1) =:+(\/ — 1)^=— 1 ; but if one of the factors be positive and the other negative, we shall have +(V — 1)X — (V — 1) = — (V-l)'^= + l. 319. All whole positive numbers are comprised in one of these fourformul(Bi 4», 4n-M, 4n+2, 4ra+3, n being a whole positive number ; since that, if any whole num- ber be divided by 4, the remainder must be 0, 1, 2, or 3. 224 IMAGINARY QUANTITIES. If we designate y' — 1 by ar, the several powers of-/ — 1 shall be therefore represented by one of these four formulae : (^_l)4»+i=:a;*"+i=:a;*".ir=a;= + V — 1 ; Thus, in order to know any given power of '^ — I, it is sujp.- cient to divide the exponent of the power proposed by 4, anct the power of ^/ — \ indicated by the remainder shall be that which is required. 320. When one imaginary quantity is to be multiplied by an- other, the result whether they be both positive or both negative, is equal to minus the square root of the product, taking them as real quantities. Thus, (-f/ — a)x(4-/ — ^) = — /«& ; since, ( + / — a) X(+^/ ~b)=^ ax^/ -IX/ix/ — l=/aX/6X (-/ — 1 )2 — — 1 X ./ a5 = — / a5. And, in a similar manner, it may be proved that ( — / —a) X ( — / —b)— —^/ ab. 321 . And if one of the imaginary radicals be positive, and the other negative, the result arising fromtheir multiplication will be plus the square root of their product, taking them as before. Thus, {^\-^/—a)x( — ^—b = -{-'^ab•, since + ^— a= + y/aX y/ — l, and —^ — 5 = (-.y — ljx -/* ; -'-{V^X y'_l)X(-^-l)xV*)=[( + V-l).(-V-l)]V'«6 = + 1 x-}/ab = + -\/ab. 322. When one imaginary radical is to be divided by another, the result, whether they be both positive or both negative, will be equal to plus the square root of their quotient, taking them as real quantities. rri + V — <2 — y/ —a a -f -/ — a 323. And if one of the imaginary radicals be positive and the other negative, the result arising from division, will be minus the square root of their quotient, taking them as before. Th"«. Z^-i 0' +7—4= - Vl ; and -^— or --/-« = 1. + -/-« 324. If an imaginary radical is to be divided by a real radi- cal, or a real radical by an imaginary one, the result will be equal to plus or minus the square root of their quotient, according as the radical is affirmative or negative. IMAGINARY QUANTITIES. 225 Thus, -i^-T- or -y—T= + / — T ' ^^^ ^—7- or -7^=^^= The several powers of imaginary radicals can be readily derived from the formulae (Art. 319) ; it only now remains to illustrate the preening rules by a few practical examples. Ex. 1. It is required to multiply a—^ —hhy a—-/ —5, or to find the square of a — ■/ — b. ' ' a-^-b dp- — a^ — h —a\/ — b—h a^—2a{^/~-b)—h Ans Ex. 2. It is required to find the- quotient of 1 + -y/"" ^ divid- ed by 1 — y — 1. Ans. Ex. 3. It is required to multiply l + y' — 1 by l + y'— 1 ; or to find the square of \■\•^/ — \. Ans. 2-\/ — 1. Ex. 4. It is required to find the product arising from mul- tiplying 1 +-/— 1 by 1 — V — 1 Ans. 2. Ex. 5. It is required to find the square, or second power of a^\■^/—W■. Ans. a2_^2_j.2a6^_l. Ex. 6. It is required to multiply 5+2^—3 by 2 — -/— 3. Ans. 16—/ —3. Ex. 7. It is required to find the cube, or third power, ol a-/ ~y^. Ans, o?~'^a\P'^{ly^~^d^b)y/ — L Ex. 8. It is required to find the quotient of 3+y' —4 di vided by 3-2/ —1. Ans. Jg.(5+12/ —1) Ex. 9. It is required to find the square of / (a-j-J/ —1)4- / (a— V —1)- Ans. 2a+2/ {a^^V^) 226 CHAPTER VIII. ON PURE EQUATIONS. 325. Equations are considered as of two kinds, called sim- ple or pure, and adfected ; each of which are differently de- nominated according to the dimensions of the unknown quan- tity. 326. If the equation, when cleared of fractions and radical signs or fractional exponents, contain only the first power of the unknown quantity, it is called a simple equation. 327. If the unknown quantity rises to the second power or square, it is called a quadratic equation. 328. If the unknown quantity rises to the third power or cube, it is called a cubic equation, &lc. 329. Pure equations, in general, are those wherein only one complete power of the unknown quantity is concerned. These are called pure equations of the first degree, pure quadratics, pure cubics,pure biquadratics, Sic, according to the dimension of the unknown quantity. Thus, x — a + b is a. pure equation of the first degree; cc^=:a^-{-ab is a. pure quadratic ; x^=:a^-^a'^b-\-c is a pure cubic ; x'^z=a^-\-a^b-{-ac'^-\-d is 3, pure biquadratic ; &c. 330. Adfected equations are those wherein different powers of the unknown quantity are cohcerned, or are found in the same equation. These are called adfected quadratics, adfected cubics, adfected biquadratics, &c., according to the highest dimension or power of the unknown quantity. Thus, x'^-\-ax = b, is an adfected quadratic x'^-\-ax^-{-bx=c, an adfected cubic ; x'^-^ax^-^bx'^-{-cx=:d, an adfected biquadratic. In like manner other adfected equations are denominated ac- cording to the highest power of the unknown quantities. § I. SOLUTION OF PURE EQUATIONS OF THE FIRST DEGREE BY INVOLUTION. 331. We have already delivered, under the denomination of Simple Equations, the methods of lesolviiig pure equations of the PURE EQUATIONS. 227 first degree, in all cases, fexcept when the quantity is affected with radical signs or fractional exponents, in which case the following rule is to be observed. RULE. 332. If the equation contains a single radical quantity^ transpose all the other terms to the contrary side ; then in- volve each side into the power denominated by the index of the surd ; from whence an equation will arise free from radi- cal quantities, which may be resolved by the rules pointed out in Chap. III. If there are more than one radical sign over the quantity, the operation must be repeated ; and if there are more than one surd quantity in the equation, let the most complex of those surds be brought by itself on one side, and then proceed as before. Ex. 1. Given -^ (4a?4-16)=12, to find the value of a;. Squaring both sides of the equation, 4a;+16=:144 ; by transposition, 407=144 — 16 ; .-. a;=32. Ex. 2. Given y (2a;+3) + 4=::7, to find the value of x. By transposition, ^ (2a:-|-3)rr7— 4=3 ; cubing both sides, 2x-f 3=27; by transposition, 2a:=27 — 3 ; .-. a;=12. Ex. 3. Given -/(12+j;) = 2-f -/a:, to find the value of x. Bysquaring, 124-a:=4-}-4'/a;+a;; by transposition, 8=4 ^Jx^ or \/x=2 \ .'. by squaring, «=4. Ex. 4. Given V(^+ 40) = 10 — ^0:, to find the value of x. By squaring, x-\-A:0=:l00—20^/x-\-x ; by transposition, 20-y'a;=60, or -v/a:=3 ; .-.by squaring, a;=9. Ex. 5. Given -/(a:— 16)=8 — -/ar, to find the value of a:. By squaring both sides of the equation, a;— 16 = 64 — 16-/^:+^ ; .'. 16ya:=64 + 16 = 80 ; by division, ^/x=b ; .*. a;=25. Ex, 6. Given ^(x—a)— y/x—^ -y/a, to find the value of x. Squaring both sides of the equation, x — a=x — '\/{ax)-^^a; .-.by transposition, ^(ax) = ^a; 25a2 25a by squaring, ax=-j^ ; .'. ^^-^q- Ex. 7. Given -y/SX 'y/(a;+2) = -/5a;+2, to find the value of a?. 238 PURE EQUATIONS. By squaring, 5a;+10=:5a;4-4'v/5a;+4 ; by transposition, 6 = 4 v^5a? ; .-. ^5x=^ ; by squaring again, 5x=z^ ; .-. x=^. Ex. 8. Given — 7 — =-^, to find the value of a;. yx X Multiplying both sides of the equation by -/a;, X 1 X — ax=z-=l, or (l—a)x^=l ; .-.a-rr- . a? 1 — a T^ « ^- V^4-28 -v/x+38 ^ , ,, , c Ex. 9. Given — — ■ =-^ , to find the value of x. Va:+4 yx-f-6 Multiplying both sides by ( ^/x•\•4:) x ( ^x-\-6), we have x-[-34:yx-{-l68=x+42^x-^152 ; by transposition, I6=z8^x, or 2 = -/ a?; .-. by squaring, a:=r4. T, ,^ ^. \/ax—b 3^ax—2b , ^ , ,, , ^ Ex. 10. Given ~^, --r:z=~-. -7, to find the value of x. yax-\-o Syax-j-DO Multiplying both sides by (-/aa;+6)x(3-/aa;-f-56), Sax +21 y/ax—5b^ — 3ax + b-)/ax~2b^y .-.by transposition, by^axz=3b^ ; by division, ^ax = 3b; *.• by squaring, ax=9b'^f and x= — . Ex. 11. Given -/(a;4- V^?) — -/(a?— 'v/^)= I / ( 7-r l» to find the value of x. Multiply both sides of the equation by y^{x+ ^x), x+ v'(«)-^(x^-«)=i|^, .'. by transposition, x ^r— = \^{x^~x) ; and dividing by -y/a:, -/a:— J= -/(a:— 1) ; .♦.by squaring, x—'^x-{-^=x—l ; .-. -y/aj^-J, 25 and by squaring, aj^rT^. Ex. 12. Given / (a;— 24) = -v/a;— 2, to find the value of x. Ans. a?=49. Ex. 13. Given ^ (4a 4- a:) =2/ {b+x) — \/x, to find the va- lue of X. Ans. x='-^^---^. Ex. 14. Given x+a-\-y/ (2ax+x^) = bf to find the value of a;. PURE EQUATIONS. 229. Ex. 15. Given ^ ^ = —. — 7-777, to find the value of a;. Ans -=(^)' Ex. 16. Given—— — —-=14-^—— , to find the value y^dx-|-l 2 of X, Ans. a: = 3. Ex. 17. Given x=y[a^+x\^(b^+x^)]—a, to find the va- lue ol X, Ans. x= — . 4a 4 Ex. 18. Given y/{2+x)+'i/x=-j- -. , to find the va- y{2-\-x) lue of a;. Ans. x=-. Ex. 19. Given ^ (10a;+35) — 1=4, to find the value of x. Ans. a;=:9. Ex. 20. Given ^ (9x—4) + 6=8, to find the value of x. Ans. x=4, Ex. 21. Given y/{x-\-i6)=z2-{-^^, to find the value of x. Ans. a;=9. Ex. 22. Given '^(x — 32) = l6 — ^x, to find the value of ar. Ans. a:=:81. Ex. 23. Given y(4a;+21)=2Va;-f 1, to find the value of X. Ans. a:=25. Ex.24. Given ^[l+a;'v/(a;2 4- 12)3=1+3?, to find the va- lue of a:. Ans. a;=2 36 Ex. 25. Given ■^x+x^{x—9) = —r x-,, to find the va- y(a; — 9) lue of a?, Ans. a? =25. Ex. 26. Given 7 (a-f a;)=27 [x'^+5ax-^h^),to find the va- lue 01 X. Ans. x=—- — . 3a Ex. 27. Given ^^—r^—-= — +V ^ ^^ ^^^^j ^-^^ ^ j^^ of ^^ V«+2 -v/a;+40' Ans. a;=4. Ex. 28. Given \ f~ -= ^ f~~„ , to find the value of x. -/ 6x4-2 4/6a;4-6' Ans. x=6. Ex. 29. Given f"!"^^ / 5^-3 ^^ ^^^ ^^^ ^^j^^ /5a?4-3 2 ' of X. Ans. a;=5 . 21 230 PURE EQUATIONS. Ex. 30. Given --=zc-{-- , to find the value of or. -/ ax-\-b c Ans..=i.(*+-il)3. § 11. SOLUTION OF PURE EQUATIONS OF THE SECOND, AND OTHER HIGHER DEGREES, BY EVOLUTION. RULE. 333. Transpose the terms of the equation in such a man- ner, that the given power of the unknown quantity may be on one side of the equation, and the known quantities on the other ; then extract the root, denoted by the exponent of the power, on each side of the equation, and the value of the un- known quantity will be determined. In the same way any adfected equation, having that side which contains the un- known quantity, a complete power, may be reduced to a sim- ple equation, from which the value of the unknown quantity will be ascertained, by the rules in Chap. III. Ex. 1. Given a;2— 17=130— 2a;2, to find the values of a:. By transposition, 3ic2=:147 ; .-. by division, j:2_49^ and by evolution, 0:=^ 7. 334. It has been already observed, that ^y a may be either -h or — . where n is any whole number whatever ; and, con- sequently, all pure equations of the second degree admit of two solutions. Thus, 4-7 X 4-7, and — 7 x —7, are both equal to 49 ; and both, when substituted for x in the original equation, answer the condition required. Ex. 2. Given x^-\-ab = 5x'^^ to find the values of x. By transposition, 4x'^—ab ; .-. 2a;=4z'/«^, and x=±^^/ab. Ex. 3. Given x'^ — 6x-{-9=a^, to find the values of a:. By evolution, ar — 3 :=:!:« ; '. oc = 3:^a. Ex. 4. Given 4x'^—4ax-\-d^=x^-\-i2x-\-36, to find the va- lue of X. By extracting the square root on both sides, we have 2x — azzix-jrG ; .-. by transposition, x=a-{-6. Ex. 5. Given ..>+/=13, ) ^^ g^^ ^^^ ^ and x^—p^=5j y ^ By addition, 2a;"'^=l 8; .-. x—:i^^9=z^3. By subtraction, 2y2_8 ; ... y — ±J^4 — ^2. Ex. 6. Given 81x^=256 to find the values of x. PURE EQUATIONS. / 231 By extracting the square root, 9a;2 = -|-,16 ; By extracti:i>g again, 80?=:^ -y/rt 16 = ^:4, or ±4^^ — 1 ; .-. x= if, or oc= if-/— 1. Ex. 7. Given a;^ — 3x*+3a;^ — 1=27, to find the values of a?. By evohition, a;^ — 1=3 ; .-. a;2=4, and xzzz-^2. Ex. 8. Given 36x'^=za'^, to find the values of a;. Ans. x=:^:^^a. Ex. 9. Given a;^=r27, to find the value of a;. Ans. x=3. Ex. 10. Given a;"H-6a;-t-9=25, to find the values of a;. Ans. xz=2, or —8. Ex. 11. Given Sa:^ — 9=21 + 3, to find the values of a:. Ans. a;=i-v/ll. Ex. 12. Given x^—x'^-\-lx—^Y=a^, to find the values of x. Ans. x=za-i-^. Ex. 13. Given x^-\-^x-\'^=aH^, to find the values of x. Ans. x=^ah—^. Ex. 14. Given x^-\'bx-{-jb^=a\ to find the values of a:. Ans. a;= j^a— ^&. Ex. 15. Given a;*— 2a;2 4-l=9, to find the values of a;. Ans. .rz=_l-2, or ±V— 2. Ex. 16. Given a;* — 4a;2 4-4=:4, tofind the values of a;. Ans. a;= J^2, or 4: v'O. Ex. 17. Given 5a;2—27=3a;2+215, to find the values of a:. Ans. a^=:-|-ll. 'Ex. 18. Given 5a;2--l =244, to find the values of x. Ans. xz=4~7. Ex. 19. Given 9a;24-9 = 3a;24.63, to find the values of x. Ans. a;= J;3. Ex. 20. Given 2ax'^ + b—4=cx^—5+d—ax'^, to find the values of x. Ans. a;= J- . / 2-. V 3a — c Ex. 21. Given x^-\-y'^^a and a;*— y^^^, to find the values of X and y. Ans. a; = ±-/(±i-/{2a + 2^')) and y = ±y^(±^^(2a — 2b)). § III. EXAMPLES IN WHICH THE PRECEDING RULES ARE AP- PLIED IN THE SOLUTION OF PURE EQUATIONS. 335. When the terms of an equation involve powers of the unknown quantity placed under radical signs. Let the equation be cleared of radical signs, as in Sect. I ; then, the value of the unknown quantity will be determined by extracting the root, as in Sect. IL. And by a similar process, any equation containing the pow 232 PURE EQUATIONS. ers of a function of the unknown quantity, or containing the powers of two unknown quantities, may frequently be reduced to lower dimensions. # Ex. 1. Given y x^-^y («-{-&), to find the values of x. Cubing both sides, a;2 = a-f-5 ; .•.a:=iV(a + i). Ex. 2. Given^(a;2— 9)= ■v/(a;-3); tofindthevaluesofar. Here, the given quantity may be exhibited under the form JL ' ' A X2 (ir2— 9)4-_(a;_3)'? ; then, by squaring both sides, (a:^— 9)^ = (a:-3)^^^' or (a:2_9)^zra:-3 ; by squaring again, x^—^:=zx^ — 6x-t-9; .-.by transposition, 6a;=18 ; and ar=:3. Ex. 3. Given a;^— ^2^9^ and x—y=L\ ; to find the values of a; and y. Dividing the corresponding members of the first equation by those of the second, we have a?-f-y=9 ; adding this equation to the second, 2j:=10 ; .•.af;=5, and y= 9—07 ; .-. y=z4. Ex. 4. Given ^x->r^fy=^, ) ^ ^^^ ,j^^ ^^j^^^ „f ^ ^^^ and ya:— vy = Ij S Adding the two equations, 2y'a;=:6, .-. ■/a::=3, and by involution, a:=:9. Subtracting the two equations, 2-v/y=4, and y'^=2 ; .-.by involution, y=4. Ex. 5. Given »^+«y=12. > ^ g„j ,j,, ^,,„,, „, ^ ^^^ and y2H-a:y=:24, 5 ^ By addition, a;24-2a:y-}-y2— 36 ; .*. extracting the square root, a?+y= dt^. Now x^-^xy=.x . (a?+3(r)=^6a: ; /. -t-6a:=: 12, and a;=:J::2 ; Ex. 6. Given a;+V(a24.a;2)=:-—— -., to find the values y(a2 -f-a:2) of X. Multiplying by -1/(0^4- «^), we have x^/((a?■'\■x'^)-\-a^^\^x'^ — 2a2; by transposition, x-y/{c?-\-x'^')^^€?- — a;^, and squaring both sides, c^x'^-\-s^=:a'^—^(P-x^-^x^ ; .-. 3a2»2=a*, and ac=;i— 7-. 13 Ex. 7. Given a:2+y2=_- ^ g ^ ^ to find the values of x and y. and a;y= PURE EQUATIONS. 233 From the 1st equation subtracting twice the 2d. a;2__2a:y+y2=(a:— y)2=— — , .-. (a?-.-y)3=l, and x—y = l ^ .•.x'^-\-y'^=:l^ ; and 2a:y=12 ; /.by addition, x'^-\-2xy-\-y'^=25, .'. by extracting the square root, a:+y=i5 ; but x-%y=l ; Ex. 8. Given a,--rv--^io, f , /• i ,u i r j to find the values of x and y .'. by addition, 2a:=:6, or— 4 ; and a; = 3, Or —2 ; by subtraction, 2y=4, or— 6 ; and y=2, or —3. and a;3-j-y^=: 5, ) 2 1. J. a Squaring the second equation, x^ -{-2x^y^ -\-y^ =i25 but j;^ +y^=13 1 1 .'. by subtraction, 2x^y^=zl2. Subtracting this from the 1st equation, x^—2x'^y^-{-y^=l .'. extracting the square root, x^—y^=z^l 1 L but a?3-fy3-_5 i .-. by addition, 2a!;3 =6, or 4 ; JL andx3=3, or2; .•.a;r=27, or 8; 1. .-.by subtraction, 2y3— 4^ or 6, 1 and y3— 2, or 3 ; .'.^=8, or 27. Ex. 9. Given x^+x^y^-\-y^=273, > to find the values of x and x^-j-x"y^-\-y'^—2l, ) and y. Dividing the first equation by the second, a;* — x^y^-{-y* ^13; subtracting this from the second iequation, 2x'^y^=8 ; .'. x^y'^ = 4: ; by adding this equation to the second, x*-\-2x'^y^-{-y^ =25; .'.x'^+y^=:^5. Subtracting the equation x'^y^z=4:, from x^^x^y^-\-y^=l3, a;*— 2a;2y2+y*=9; .:x^-'y^=±3, 21* 234 . PURE EQUATIONS. .-.by addition, 2a;2=-4-8, and x—:^2, or ±2-/+! ; and by subtraction, 2y2— ±2, and y = ±l, or rtV"!- Ex. 10. Given ^^^±yp:f}-i, ,o find the value of a?. Multiply the numerator and denominator by -v/(^+^)"f" V .'. y/(a'^—x^)=hx—ai and squaring both sides, a^—x^=b^x'^ — 2a5a?^a2, .-. b^x^-{-x^=2abx, and a?=Y^— — . Ex. 11. Given (x^—y^)x(x^y) = 3xy, K ^ j *v i and (x^--y^) x(x^-y^)=45xY, I of a; and y. Dividing the second equation by the first, (oi^-¥y^) . (x-{-y)=l5xy ; .-. x^-{-x^y-{-xy^-l-y^—l6xy ; but from the first, oc^ — x'^y—xy^-\-y^= 3xy ; .-. by addition, 2x^+2y^=l8xy, and x^-{-y^=9xy. But by subtraction, 2x'^y-{-2xy'^=l2xy, and a;+y = 6 ; .-. by cubing, j^ +30^2^^33,^24.^3 =^216 , x^ -{-y^z^dxy ; .-. by subtraction, 3x^y-\-3xy^=2l6 —9xy, or 3 .{x+y).xy=z3x6 . xy=2l6-9xy ; .-. 27a;y=216, and Now x^-i-2xy+y^ = 36, and 4a:y =32 ; .-. by subtraction, x^—2xy-{-y^=4j and by extracting the square root, x—y=^2, by x-{-y= 6, .*. by addition, 2a;=8, or 4 ; and x=4, or 2 ; and by subtraction, 2y = 4, or 8 ; .•.y=2, or 4. Ex. 12. Given -+ ^^ ~"'^ — ^, to find the values of a?.* ac a? 6 Ans. x=:^^(2ab — b^). 1 8 Ex. 13. Given a:2+3a;—7=a:+2H , to find the values X of X. Ans. a; =3, or —3. V (a?4-J' PURE EQUATIONS. 235 to find the values of ar. Ans. x=— — -rr Ex. 15. Given x-j-y : x : : 5 : 3, and xi/ = 6, to find the va- lues of a: and y. Ans. a;=:i3, and y=:jt2. Ex. 16. Given x—y : x : : 5 : 6, and xy^ = 384, to find the values of x and y. Ans. a;=24, and y=i4. Ex. 17. Given x-\-i/ : x : : 7 : 5, and a:y +3/^=126, to find the values of x and y. Ans. a;— 4:15, and y=±6. Ex. 18. Given xf+i/=:2l, and a'2y*4-y2 — 333^ to find the values of x and y. Ans. a:=2, or jig- ; and y = 3, or 18. Ex. 19. Given a;2y+a:y2= 180, and a;3+y3= 189, to find the values of x and y. Ans. x=5, or 4 ; and y — 4, or 5. Ex. 20. Given x-^- ^xy-^y=l9, and a:2+a:y + y2r= 133, to find the values of x and y. Ans. a;=9, or 4 ; and y=4, or 9. Ex. 21. Given x'^y-{-xy'^=6, anda;y+*y=12, to find the values of a; and y. Ans. a:=:2, or 1 ; and y=:l, or 2. Ex. 22. Given {x^-\-y^) X (a:-f y) =2336, and (a:^— y2) (a:— y)=576, to find the values of x and y. Ans. a;=:l 1, or 5 ; and y=:5, or 11. Ex. 23. Given x^-{-y'^z=z(x-\-y) . xy, and a;2y-|-a?y2=:4a:y, to find the values of x and y. Ans. a:=2, and y = 2. Ex. 24. Given 2 . (a:2-f y2) . (a^+y) = 15a:y, and 4 {x'^—y^) (a:2-|-y^) = 75a;2y2j to find the values of x and y. Ans. a:=:2, and y = l Ex. 25. Given x~y : y : : 4 : 5, and a:2-f-4y2=:181, tq.find the values of x and y. Ans. x= ±9, and yrr -[-5. Ex. 26. Given x'^+y^ : x^—y^ : : 17 : 8, and a:y2=:45, to find the values of x and y. Ans. a:— 5, and y = 3. Ex. 27. Given ^x— yyz=3, and ^ x-\-{^ y=z7 ; to find the values of a? and y. Ans. a: = 625, and y=:16. Ex. 28. Given -y/a^ + '/y : \/x — -\/y : : 4 : 1, and x—y~ 16, to find the values of x and y. Ans. a;:=25, and y=:9. Ex. 29. Given a:3+y3 : x^—y^:: 559 : 127, and a;2y=294 ; to find the values of x and y. Ans. a:=7, and y=6. Ex. 30. Given x^-\-y^=20, and x^+y'^=6 ; to find the values of x and y. Ans. a;=-l-8, or i -^8, and y=32, or 1024 Ex. 31. Given a;4+2a:2y2-f-y4 := 1296— 4a:y(a;2+a:y+y2), and X — y=4 ; to find the values of x and y. Ans. 5, or —1, and y=l> or —5. Ex. 32. Given V(^^+1)+V4a; ^ ^^ ^^^ ^^^ ^^^^^ ^^ ^^ y (4a;+l) — V 4aj Ans. x=^f 236 SOLUTIQN OF PROBLEMS Ex. 33. Given xy — d^, and a^^+y^— .^2 . ^o find the values of a; and y. Ans. x^±\\^J{s'^-\.^a^)J^ y'(^2_2a2)], and y=rl=Mv^(^2_{_2a2)_y(^2_2a2)]. Ex. 34, Given x^-\-xy xy^z=:208, and y^+y^/ x^y=^l053, to find the values of a? and y. Ans. a:=i8, andy=i27. Ex. 35. Given x'^+x'^y^-{'y'^= 1009, 3. 3. and x^ + x^y^-i-y^ = 582l93, to find the values of x and y. Ans. a;=81, or 16 ; and 5/= 16, or 81. CHAPTER IX. ON THE SOLUTION OF PROBLEMS, PRODUCING PURE EaUATIONS. 336. In addition to what has been already said, with re- spect to the translation of problems into algebraic equations, it is very proper to observe, that, when two quantities are re- quired which are in the given proportion of m to n, the un- known quantities are represented by mx and nx ; then the values of a:, found from the equation of the problem by the methods in the preceding chapter, being multiplied by m and n respectively, will give the numbers required. If three quantities are required, which have given ratios to one another, assume wia?, nx, and px, m to n being the ratio of the first to the second, and n to p being that of the second to the third ; then proceed as before. Problem 1. There are two numbers in the proportion of 4 to 5, the diflference of whose square is 81. What are those numbers ? Let 4a; and bx= the numbers ; then (25a:2_i6a;2 = ) 9a:2=81 ; .*. x'^=9, and a:= ±3. Conse- quently the numbers are ^^12 and ±15. Prob. 2. It is required to divide 18 into two such parts, that the squares of those parts may be in the proportion of 25 to 16. Let a;= the greater part ; then 18— a; = the less ; .-. a;2 : (18— a:)2 : : 25 : 16, and 16x2=25(18— a:)^ ; PRODUCING PURE EQUATIONS. 237 .*. extracting the square root, 4a: = 5(18— ac), and 9a; = 90 ; .'. a; =^10, and the parts are 10 and 8. Prob. 3. What two numbers are those whose difference, multiplied by the greater, produces 40, and by the less 15 ? Let xz=. the greater, and y= the less ; /. a:2—a:y = 40, and a-y— y^ — 15 ; .-. by subtraction, x^ — 2a:y4-y2— 25, and x—yz^^b. .'. from the first equation, x[x — y) = J::5a;=40, and x=.-^S. From the 2d, y(a:— y)=r±5y=±15 ; .-. y = ±3. * Prob. 4. What two numbers are those whose difference, multiplied by the less, produces 42, and by their sum 133 ? Let x=: the greater, and y= the less ; /. {x—y) . y=:42, and {x^y) . (a:4-y) = 133 ; /.iy subtracting twice the first from the second, a;2 — 2j:y+y2=49 ; .-. a:— y=-l-7 ; whence -l-7y = 42, and y = ±6 ; buta;=yi7; .-. a;=i6±7 = d=13. Prob. 5. What two numbers are those, which being both multiplied by 27, the first product is a square, and the second the root of that square ; but being both multiplied by 3, the first product is a cube, and the second the root of the cube ? Let X and y be the numbers ; then -^'llx — Tly, and .*. x=21y'^ , also -y 3a; = 3y ; and .-. x—9y^ ; whence 9y3=:27y2, and y=3 ; .-. a;=9 X 27=243 ; .'. the numbers are 243, and 3. Prob. 6. Two travellers, A and B, set out to meet each other ; A leaving the town C at the same time that B left D. 'rfiey travelled the direct road, CD : andy on meeting, it ap- peared that A had travelled 18 miles more than B : and that A could have gone B's journey in 15|^ days, but B would have been 28 days in performing A's journey. What was the dis- tance between C and D 1 Let a;= the number of miles A has travelled ; .'.ar— 18= the number B has travelled ; and a; — 18 : a; : : 15J : the number of days A travelled, = 63a; — — - ; also a; : a;— 18 : : 28 : to the number of days B tra- 18)2=9a;2; .-.4.(3;— 18) = i3a?, and a;=72, or lOf ; whence 238 SOLUTION OF PROBLEMS A travelled 72, and B 54 miles ; and, the whole distance, CD 126 miles. Prob. 7. Two partners, A and B, dividing their gain (60/), B took 20/. A's money continued in trade 4 months ; and if the number 50 be divided by A's money, the quotient will give the number of months that B's money, which was 100/., con- ^ tinned in trade. What was A's money, and how long did B's money continue in trade ? 50 Suppose A's money was x pounds ; .*. — = the number of months B's money was in trade ; and since B gained 20/., A gained 40/. , 50X100 ^ , , , 10000 /. 4:X : : : 2 : 1, and 4x=: ; X X .: 4a;2=:10000, and a:2=2500 ; .-. a;=±50. .*. A's money was 50/., and B's money was one month in trade. Prob. 8. A detachment from an army was marching in re- gular column, with 5 men more in depth than in front ; but upon the enemy coming in sight, the front was increased by 845 men ; and by this movement the detachment was drawn up in five lines. Required the number of men. Let x= the number in front ; .•• a:+5= the number in depth, and x{x-^5)=z the whole number of men ; also, (a:-f-845)x5=: the whole number of men ; .-. a:2-f 5a;=5a: + 4225, and a;2 = 4225 ; .-. a;=±65. And, consequently, 5a:4-4225 — 325+4225 = 4550, the number of men. Here, although the negative value of x will not answer the conditions of the problem, yet it will satisfy the above equation ; for, if we substitute — 65 for x, we shall have ( — 65)2 + 5( — 65)=5( — 65) + 4225 ; that is, or 4225 — 325 = — 325 + 4225 ; .-. 4225=4225, or 4225—4225=0, that is, 0=0. Prob. 9. It is required to divide the number a into two such parts, that the squares of those parts may be in the proportion of m to n. Let x= one of these parts ; then a — x=z the other ; and ac- cording to the enunciation of the problem, we shall have the equation, x^ m X , /^ / ,r ^ f\ t. — =— : .-. = +^/ — , or (puttmg — =m),x = ± (a-x)2 n' a-x "^^jn' ^^ ^n " {a—x)^/mf. PROTDUCING pure equations. 239 By resolving separately the two equations of the first de- gree comprised in the above formula, namely, x=-j-{a—x)^m', and x=—-(a~-x)-\/m'j we shall have, from the first, — and from the second x: 1 + Vm" """ """' """ """""" — 1- ^m' • By the first solution, the second part of the proposed num- ber IS a- -^— —--——— ; and the two parts, ^ 1+ym \-\-y'm 1 + ym' and -— — — -, are, as was required in the enunciation of the 1+ yjmf question, both less than the number proposed- By the second solution, we have (~a\/m'\ , a'J mf x , , ; --—r\=a-\-^ — , ■ = - ;— 7 ; and the two parts l — ^m/ l—y/m X—^nv aWm - ^ are — -— ^-— — -, and 1 — ^m^ 1 — -y/m'* Their signs being contrary, the number a is not, properly speaking, their sum, but their difference. Now, if a=18, m=25, and 7i=l6 ; then substituting these values in the formula — - — ; — -, and -—; — ; — -., we shall find 10 l-{-y/m^ l+V^ and 8 equal to the two parts required, the same as in Ex. 2., which is a particular case of this general problem. • Prob. 10. What two numbers are those, whose sum is to the greater as 10 to 7 ; and whose sum, multiplied by the less, produces 270? Ans. ±21 and ^9. Prob. 11. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128? Ans. ^18 and ±14. Prob. 12. A mercer bought a piece of silk for 16/. 4^. ; and the number of shillings which he paid for a yard was to the number of yards as 4 : 9. How many yards did he buy, and what was the price of a yard ? Ans. 27 yards, at 126-. per yard. Prob. 13. Find three numbers in the proportion of ^, |-, and J : the sum of whose squares is 724. Ans. ±12, i 16, and ±18. Prob. 14. It is required to divide the number 14 into two 240 SOLUTION OF PROBLEMS. such parts, that the quotient of the greater part, divided by the less, may be to the quotient of the less divided by the greater as 16 : 9. Ans. The parts are 8 and 6. Prob. 15. What two numbers, are those whose difference is to the less, as 4 to 3 ; and their product, multiplied by the less, is equal to 504 ? Ans. 14 and 6. Prob. 16. Find two numbers, which are in the proportion of 8 to 5, and whose product is equal to 360. Ans. ±24, and ±15. Prob. 17. A person bought two pieces of linen, which, together^ measured 36 yards. Each of them cost as many shillings per yard, as there were yards in the piece ; and their whole prices were in the proportion of 4 to 1. What were the lengths of the pieces ? Ans. 24 and 12 yards. Prob. 18. There is anumber consisting of two digits, which being multiplied by the digit on the left hand, the product is 46 ; but if the sum of the digits be multiplied by the same digit, the product is only 1 0. Required the number. Ans. 23. Prob. 19. From two towns, C and D, which were at the distance of 396 miles, two persons, A and B, set out at the same time, and met each other, after travelling as many days as are equal to the difference of the number of miles they tra- velled ^er day; when it appears that A has travelled 216 miles. How many miles did each travel ■per day ? Ans. A went 36, and B 30. Prob. 20. There are two numbers, whose sum is to the greater as 40 is to the less, and whose sum is to the less as 90 is to the greater. What are the numbers ? Ans. 36, and 24. Prob. 21. There are two numbers, whose sum is to the less as 5 to 2 ; and whose difference, multiplied by the dif- ference of their squares, is 135. Required the numbers. Ans. 9, and 6. Prob. 22, There are two numbers, which are in the pro- portion of 3 to 2 ; the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Required the numbers. Ans. 6, and 4. Prob. 23. A number of boys set out to rob an orchard, each carrying as many bags as there were boys in all, and each bag capable of containing 4 times as many apples as PRODUCING PURE EQUATIONS. 241 • there were boys. They filled their bags, and found the num- ber of apples was 2916. How many boys were there ? Ans. 9 boys. Prob. 24. It is required to find two numbers, such that the product of the greater, and square of the less, may be equal to 36 ; and the product of the less, and square of the greater, may be 48. Ans. 4, and 3. Prob. 25. There are two numbers, which are in the pro- portion of 3 to 2 ; the difference of whose fourth powers is to the difference of their squares as 52 to 1. Required' the num- bers. Ans. 6, and 4. Prob. 26. Some gentlemen made an excursion, and every one took the same sum. Each gentleman had as many ser- vants attending him as there were gentlemen ; and the num- ber of dollars which each had was double the number of all the servants ; and the whole sum of money taken out was $3456. How many gentlemen were there ? Ans. 12. Prob. 27. A detachment of soldiers from a regiment, be- ing ordered to march on a particular service, each company furnished four times as many men as there were companies in the regiment ; but those becoming insufficient, each com- pany furnished 3 more men ; when their number was found to be increased in the ratio of 17 to 16. How many compa- nies were there in the regiment ? Ans. 12. Prob. 28. A charitable person distributed a certain sum among some poor men and women, the numbers of whom were in the proportion of 4 to 5. Each man received one- third of as many shillings as there were persons relieved ; and each woman received twice as many shillings as there were women more than men. Now the men received all to- gether 185. more than the women. How many were there of each ? Ans. 12 men, and 15 women. Prob. 29. Bought two square carpets for 62/. 1^. ; for each of which I paid as many shillings per yard as there were yards in its side. Now had each of them cost as many shillings per yard as there were yards in a side of the other, I should have paid lis. less. What was the size of each ? Ans. One contained 81, and the other 64 square yards. Prob. 30. A and B carried 100 eggs between them to mar- ket, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them ; and if B had only taken as many as A, he would have received 8 pence. How many had each 1 Ans. A 40, and B 60 22 242 QUADRATIC EQUATIONS. Prob. 31. The sum of two numbers is 5 (s), and their pro- duct 6 (p) : What is the sum of their 5th powers ? Ans. 275 = {s^ — 5ps^-{-5p^s), CHAPTER X, ON QUADRATIC EQUATIONS. 337. Quadratic equations, as has been already observed, are divided into pure and adfected. All pure equations of the second degree are comprehended in the formula x^=ny where n may be any number whatever, positive or negative^ integral or fractional. And the value of x is obtained by ex- tracting the square root of the number n ; this value is dou- ble, for we have, x=z^-y/n, and in fact, {^-y/nYz^n. This may be otherwise explained, by observing, (Art. 106), that x'^ — n=z(x-\-^/n).{x — ^/n):=^o, and that any product consist- ing of two factors becomes nought, when there is no restric- tion in the equality to zero of that product, by making each of its factors' equal to zero. We have, therefore, x=z — ^/ny x:=:z-\--y/n, or x=i^-\/n. 338. Now, since the square root is taken on both sides of the equation, x'^=-n, in order to arrive at x=^ -/n ; it is very natural to suppose tkat, x being the square root of a;^, we should also affect x with the double sign ^ ; and, therefore, in resolving the equation x^=n, we should write iaj^iy'n ; but by arranging these signs in every possible manner, namely; -{-x=-\- -^n, -]-x=z — -y/n, — x= — -\/7i, — X— -f--v/n, we would still have no more than the two first equations, that is, 4-^=^1/^ i for if we change the signs of the equations ■^x=-~'\/n and — a:=4- y/n, they become +a;= + ^» and -{■xc=z — '}/n, or x=-^ y/71. 339. If, in the formula x'^=n, n be negative, or, which is the same thing, if we have x^=—n, where n is positive ; then, «=±V— n=±VwX V — 1» and in fact(± V'»)^X('/— 1)* QUADRATIC EQUATIONS. 243 =nX — 1 = — n ; therefore, the two roots of a pure equation are either both real or both imaginary. 340. All adfected quadratic equations, after being properly reduced according'to the rules pointed out in the reduction of simple equations, may be exhibited under the following general forms; namely, a;2-j-na:=:o, and a;2-}-na:=:n' ; where n and w' may be any numbers whatever, positive, or negative, integral ox fractional. 341. The solution of adfected quadratic equations of the ioxmx^+nx^o, is attended with no difficulty ; for the equation a;2-|-na;=o, being divided by x, becomes x-\-n=^o, from which we lind a;= — n, though we find only one value of x, according to this mode of solution, still there may be two values of x, which will satisfy the proposed equation. In the equation, a;2=:3a?, for example, in which it is required to assign such a value of x, that x"^ may become equal to 3a?, this is done by supposing a:==3, a value which is found by di- viding the equation by a? ; but besides this value^ there is also %nother which is equally satisfactory ; namely, x=.o ; for then ti^-=.o, and 3a: = 0. 342. An adfected quadratic equation is said to be complete, when it is of the form x'^-\-nx-=zn' ; that is, when three terms are found in it ; namely, that which contains the square ol the unknown quantity, as x ; that in which the unknown quan- tity is found only in the first power, as nx ; and lastly, the term which is composed only of known quantities ; and, as there is no difficulty attending the reduction of adfected quadratic equations to the above form by the known rules : the whole is at present reduced to determining the true value of x from the equation x^-\-nxz=^n' . We shall begin with remarking, that if a^^+na: were a real square, the resolution would be attended with no difficulty, be- cause that it would be only required to extract the square root on both sides, in order to reduce it to a simple equation. 343. But it is evident that x^-^-nx cannot be a square ; since we have already seen, that if a root consists of two terms, for example, x-\-a, its square always contains three terms, namely, twice the product of the two parts, besides the square of each part, that is to say, the square of x-{-a \%^x^-\-1ax-\-a^. 344. Now, we have already on one side x^-{-nx ; we may, therefore consider x^ as the square of the first part of the root, and in this case nx must represent twice the product of x, the first part of the root, by the second, part : consequently, this second part must be \n, and in fact th*e square oix\i\n is found to be x^-\-nx-\-\n^. 244 QUADRATIC EQUATIONS. 345, Now x'^-{-nx-\-^n'^, being a real square, which has for •its root x+ Jti, if we resume our equation x'^-j-nx=n', we have only to add ^ri^, to both sides, which gives us x^-\-?ix-\-^n'^ = n 4-Jw2, the first side being actually a square, and the other containing only known quantities. If, therefore, we take the square root of both sides, we find x-\-^n= ^ {^ri^-^n) ; and as every square root may be taken either affirmatively or ne- gatively, we -shall have for x two values expressed thus ; x=-^in4:v^ (in^ + n^y 346. This formula contains the rule by which all quadratic equations may be resolved, and it will be proper, as Euler justly observes, to commit it to memory, that it may not be necessary to repeat, every time, the whole operation which we have gone through. We may always arrange the equation in such a manner, that the pure square x^ may be found on one side, and the above equation have the form x^=L—jix-\-n'y where we see immediately that x=:z — Jn±y/ {^n^-^-n'). 347 The general rule, therefore, which may be deduced from that, in order to resolve x'^=—nx-\-n\ is founded on thin consideration. That the unknown x is equal to half the co- efficient or multiplier of x on the ^ther side of the equation, plus or minus the root of the square of this number, and the known quantity, which forms the third term of the equation. Thus, if we had the equation x^=6x-{-7, we should imme- diately say, that a?=:3±y'(9-|-7) = 3±4 ; whence we have these two values of a; ; namely, x-—7, and x= — l. 348. The method of resolving adfected quadratic equations will be still better understood by the four following forms ; in which n and n' may be a^ny positive numbers whatever, integral or fractional. I. In the case x^-\-nx=n', where x=:—^n-\- ■^{^n'^-\-7i'),or — J'l — V{i^^~^^')^ ^^® ^^^^ value of x must be positive, be- cause '\/{\n'^-{-n') is >-v/j«^, or its equal Jn ; and its second value will evidently be negative, because each of the terms of which it is composed is negative. II. In the case x'^ — nx=:n\ from which we find x=^n-{-^ (Jn^-j-n'), or -Jn— -/(i^^ + ^O' ^^® first value of a:, is manifest- ly positive, being the sum of two positive terms ; and the se- cond value will be Jiegative, because •\/(i'*^"i~'* ) ^^ ^ V(i'*^)» or its equal ^n. III. In the case x'^—nx=: —n\ we have x — \n-\- -/(in^— n'), or Jn — -v/(Jn2— n^) ; both the values of x will be positive, when 1^2 is >n' ; for its first value is then evidently positive, being composed of two positive terms ; and its second value will also be positive, because -/(i^^—^O is less than/ (J'*")* QUADRA.TIC EQUATIONS. 245 or its equal Jn. But if J-n^, in this case, be less than n% both the values of x will be imaginary ; because the quantity, \n^ — n' under the radical sign, is then negative ; and consequently '^(\ji^ — n')'wi\\ be imaginary, or of no assignable value. IV. Also, in the fourth case, x^-^nx^i — n\ where x =. — \n-{- ^J[\n'^—n'), or — J«— VCi^^""^')' ^^^ ^^^ values of x will be both negative, or both imaginary, according as ^n^ is greater or less than n'. 349. Hence wt#may conclude, from the constant occur- rence of the double sign before the radical part of the preced- ing expressions, that every quadratic equation must have two roots ; which are both real, or both imaginary ; and though the latter of these cannot be considered as real quantities, but merely as pure algebraic symbols, of no determinate value,_ yet when they are submitted to the operations indicated by the equation, the two members of that equation will be always identical, or which is the same, it shall be always reduced to the form — 350. It may here also be further observed, that, in some equations involving, radical quantities of the form -y/i^ax-^-b) both values of a;, found by the ordinary process, nvill not an- swer the proposed equation, except that we take the radical quantity with the double sign -j^. Let, for example, the values of X be found in the equation x^\-^/{6x-{-\0)zzzS. Here, by transposition, y/ (bx-\-lO)^=zS—x , therefore by squaring, 5a;+10=:64 — IQx-^-x"^, ox x^—2\x= — 54; and •-. a:=18, or 3. Now, since these two values of x are found from the reso- lution of the equation a;^— 2]a;= — 54 ; it necessarily follows, that each of them, when substituted for x, must satisfy that equation ; which may be verified thus ; in the first place, by substituting 18 for x, in the equation x'^—2\x=z — 54, we have (18)2 — 21 Xl8ri:— 54, or 324— 378=— 54 ; that is, —54 = —54, or = 0. Again, substituting, 3 for x, we have (3)2— 21 X3 = — 54, or 9-63 = — 54; — 54= -54, or 54— 54=0 ; .-.0 = 0. 351. And as the equation ot^- 21a;=— 54, may be deduced from the equation +-/ (5a;-|-10) = 8— a?, or — /(5a:+10) = 8— a? ; it is evident that the radical quantity-/ (5a:-(-10), must be taken, with the double -sign ^^ , in the primitive equation, in order that it would be satisfied by the values, 18 and 3, oi X, above found ; that is, 18 answers to the sign — , and 3 to the sign + • But if one of these signs be excluded by the nature of the question ; then only one of the values will sa- 22* 246 QUADRATIC EQUATIONS. tisfy the original equation ; for instance, if in the equation x 4--/ (5a; +10) = 8, the sign — be excluded from the radical quantity, then the square root of 5a;-f 10 must be considered as a positive quantity ; and because it is equal to 8 — x; the value of oc, since both are positive, which will answer the pro- posed equation, must be less than 8 ; therefore, 3 is the value of X, whioh will satisfy the equation x-\-y^ (5a:+10)z=8, which can be readily verified thus ; substituting 3 for a:, we have 3 + V(15-|-10)=:8, or 3 + 5=18. And for a stnilar reason, 18 is the value of x, which will answer the equation a;— y'^5a;+10) =8; for 18- -v/(90 + 10) = 18 — 10=:8; /. 8 = 8, or 0=0. 352. It is proper to take notice here of the following me- thod of resolving quadratic equations, the principle of which is given in the Bija Ga?iita, before mentioned : thus, if a quadratic equation be of the form 4a'^x^4^4abx=J^4ac, it is evident that, by adding b^ to both sides, the left-hand member will be a complete square, since it is the square of 2«a: + Z> ; and, therefore, by extracting the square root of both sides, there will arise a simple equation, from which the values of x may be determined. 353. Now, any quadratic equation of the form ax^dcbx=: + c, (to which every quadratic may be reduced by the known rules), by multiplying both sides by 4a, will become 4rtV"^i 4a6a;=+4ac. From which we infer, that if each side of the equation be muUipUed by four times the coejfficient of x^^ and to each side there be added the square of the coefjicient of x, the quan- tity on the left-hand side of the resulting eqtiation will always be a complete square ; from vihich, by extracting the square root, the values of X will be determined. If the coefficient a=l, then both sides of the equation is multiplied by 4, and the square of the coefficient of x is added, as before. § I. SOLUTION OF ADFECTED QUADRATIC EQUATIONS, INVOLV- ING ONLY ONE UNKNOWN QUANTITY. 354. Rule I. Let the terms be arranged on one side of the equation, according to the dimensions of the unknown quantity, beginning with the highest ; and the known quanti- ties be transposed to the other ; then, if the square of the un- known quantity has any coefficient, either positive or negative, let all the terms be divided by this coefficient. If the square of half the coefficient of the second term be now added to both sides of the equation, that side which involves the unknown quantity will become a complete square ; and extracting the QUADRATIC EQUATIONS. 247 square root on both sides of the equation, a simple equation will be obtained, from which the values of the unknown quan- tity may be determined. 355. Rule IL The terms of the equation being arranged as above, let each side be multiplied by four times the coeffi- cient of x^, and to each side add the square of the coefficient of X ; then the left-haHd member, being a complete power, ex- tract the square root on each side of the equation, and there arises a simple equation, from which the values of x may be determined. 356. It may be observed, that all equations may be solved as quadratics, by completing the square, in which there are two terms involving the unknown quantity, or any function of it, and the exponent of one is double that of the other- Thus, x^-\-px^=q, x^''—paf=q, x^-\-x* = a, a^x'^-\-ax=b, x -\-ax"^=ih, /jV*"— J9x2''=fECTE© QUADRATIC EQUATIONS, IN- VOLVING TWO UNKNOWN QUANTITIES. 360. When there are two equations containing two un- known quantities, a single equation, involving only one of the unknown quantities, may sometimes be obtained, by the rules laid down for the solution of simple equations ; from which equation the values of the unknown quantity may be found, as in the preceding Section. Whence, by substitution, the values of the other may also be determined. In many cases, however, it may be more convenient to solve one or both of the equations first ; that is, to find the values of one of the un- known quantities, in terms of the other and known quantities, as before ; when the rules for eliminating unknown quantities, (^ I. Chap. IV), may be more easily applied. The solution will sometimes be rendered more simple by particular artifices ; the proper application of which shall be illustrated in the following examples. i' 9 Vt^ ^ 9^~^o' f to find the values of x and y. &na x^+3xi/—y^=23,S ^ From the 1st equation x-=l — 2y ; ...j:2_49_28y+4y2; Substituting these values for x and a;2 in the 2d equation, then 49— 28y+4y2+21y— 6y2_y2_23, or 3y2+7y:::349-23=26. 36y2+84y + 49==312 + 49 = 361 ; .'.extracting the square root, 6y+7=19, and 6^=19—7=12; y=3, and a?=7— 2y=7— 4 = .^ QUADRATIC EQUATIONS. 255 Ex. 2. Given 4ry=96— aj^, and :c+y=6, to find the va- lues of X and y. From the first equation a;y 4-4ary+4=100, and extracting the root, a:y-|-2=: ±10 ; .-. a?y=8, or —12. Now squaring the second equation, «24.2j:y-f y2 = 36 ; but 4a:y =32, or —48. .*. by subtraction, *2— 2a;y4-y^ — 4, or 84 ; and extracting the root, a:— ^==±2, or ^^ -1/84 ; but X'\-y=^ 6; .-.by addition, 2a:=8, or 4, or 6rt ■v/84 ; whence, «=:4, or 2, or 3 ^^-^^21 ; and by subtraction, 2y=4, or 8, or 6^-^/84 ; .•.y=2, or 4, or 3=PV21. Ex. 3. Given ac2-^a;-fy=18—y^ and a:y= 6, to find the va- lues of X and y. By transposition, x^-\-y'^-\-x-\-y—\^\ and from the second equation, 2a;y 1=12 ; .-. by addition, a:2-f-2a:y-f y2-|_a?-|-y=30 ; and completing the square, (»:+#+(*+y)+-l=30+i=^ ; .•.extracting the root, .-K-f-y+i^iy* and a;4-y=:5, or — 6 ; whence, from the first equation, a;2-|-y''i=13, or 24; but 2a:y=12; .-.by subtraction, a:^— 2a7y + y2:=il, or 12 .*. extracting the root, x — y = drl,or ::t2y'3 Now aj+y = 5, or — 6 .-. by addition, 2r=z6, or 4, or — 6±2-/3 .•.a;=3, or 2, or — SJ^-y/S and by subtraction, 2y=4, or 6, or — 6 =F 2 1/3 .•.y=2, or 3, or — 3=F \/2. Ex. 4. Given*— 2'v/a:y4-y — -v/^-fVy^^' ^"^ \f^-\-y/y =r5, to find the values of x and y. Completing the square in the first equation, and extracting the root, y/ x — -/y — \z=:i-^\ ; 256 QUADRATIC EQUATIONS. .-. ^/x— -y/y, =1 or but from the second equation, ^/x-^ ^Jy~^ > /. by addition, 2'/a:=z6, or 5, 5 25 and ^aj=3, or -, .-.icnrQ, or — . 2 4 25 By subtraction, 2-v/y=4, or 5 ; .•.y=:4, or — . 2. 3 \ \ Ex. 5. Given a;^ y'^ =2y2, and Sa-^ — y^ = 14, to find the va- lues of X and y. 2 1 2 1. From the 1st equation, a?3=2y2 ; and :.\x^z=iy'^ ; substi- tuting this value in the second equation, Sa;^— Aa;3 = l4 ; and .-. 16a;^— a:^:=28 ; 2. X or, by changing the signs, x^ — \%x^=i — 28 ; 2. X completing the square, a?3 — 16a;3 + 64=36 ; and extracting the root, x^^%=l^^ ; J, .•.a;3 = 14, or 2, and ac=i2744, or 8. JL 2 \ \_ Ex. 6. Given a;^+y^ = 3a:, and a;^+y^=:a?, to find the va- lues of X and y. 11. 2 By squaring the second equation, x-\-2x^y^ ■\-y'^ =0^- ; 3 2 but a; 2 -fy3=:3a;; 3 11 .-.by subtraction, x-^o^ -\-2x^y^ ^x^ —"^x ; \ 1 but from the second equation, y^-=ix~3^\ Let this value be substituted in the preceding equation ; then X — o[-^-\-2x^—2x=x^ — 3j; ; 3 .*. by transposition, 2x=x^—x^ ; and dividing by x, 2=x—x^; 1 completing the square, a;— ;c^4-J=:2-f -1=1- ; and extracting the root a:^— J=-|-J ; ,-. x'^—2, or — 1 ; and a;=4, or 1. By taking the former value of x, we have y^=zx — x^ =4 -2=2; .'.y = 8. 11 and by taking the latter value, y^z^a;— a;^=l-|-l=2, t8inceaj^=: — 1, — a;i=-i-l); /. y=8 QUADRATIC EQUATIONS. 257 Ex.7. Given y2_64=8a;2y,andy— 4=2y^a:^, tofind the values of x and y. From the first equation, y^— 8x^2/ =64 ; completing the square, y^ — 8x^y-{-l6x=16x-{-64 ; extracting the root, y— •4a;2 = -|-4'v/(a:+4) ; • and .•.y=:4a;2_j_4-y/(a;-|-4). Also, from the second equation, y—2y^x^=4; .'.completing the square, &;c., y'^=zx^di '\/{x-\-4) ; multiplying by 4, 4y2=a;2_|^4y(a:-f 4) ; .-. y=4y^, and y=16. But, from the second equation, x^=- — j=—-=- ; .-. by involution, a;=-, 4 361. When the equations are homogeneous, that is, when flc^j y2, or a:y, is found in every term of the two equations, they assume the form of ax"^ -\- bxy -\- cy"^ = d, a'x'^-{-h'xy-\-c'y^=d' ; and their solution may be effected in the following manner : Assume xz^vy, then x^z^zv^y"^ ; by substituting these values for x"^ and x in both equations, we have avY+bvy^-^cy^=d I /.y2^_____ . . . (1), a'vY+b^vy^ + cY==d^; r.f=-^^-^, .. . . (2). ^ d d^ Hence av^-{-bv-i-c a'v'^-\-b'v-{-c' .'.(a'd—ad')v'^-\-{b'd—bd')v=cd' — c'd; which is a quadratic equation, from whence the value of v may be determined. Having the value of v, the value of y may be found from ei- ther- of the equations (1) or (2) ; and then the value of x, from the equation x = vy. Ex. 8. Given 2a;2+3a;y+y2=:20, and 5a;24-4y2=41, to find the values of x and y. Let x=vy, then 2v^y^'i-3vy^-\-y^=20 ; 23* 258 QUADRATIC EQUATIONS. 20 •'•y = A 2 L ^ > Hence . ^ — ;: 9 , . > o^ 6t;2— 41t;= -13; .•.by division, completing the square, &c. u=^2^ or ^. • 41 369 Let t;=i then y2^___=___^9 . ...y^g^ or -3, and xz=zvy=z\, or —1. A • 1 13 , , ,164 , . 13 ,164 Again, let «=—; theny=±V— , and a;=±— -/-— . Consequently there are four values, both of x and y, which satisfy the proposed equations. 362. When the unknown quantities in each equation are similarly involved, the operation may sometimes be shortened, by substituting for the unknown quantities the sura and differ- ence of two others. Ex. 9. Given-+^-18,) . , , . . , y X > to find the values of x and y. and a;+y=12, ) Assume x=zz-{-v^ and y=.z—v ; .•.a;+y=2;?=12 ; or 0=6; .-. a;=6-j-t;, and y=6 — v. Also, since }-^=18, x^-\-y^=^\Sxy ; y x J J .-. (6 + vV + {6-t^)3 = 18(64-«) X (6-t;) ; or 432 + 36^2-648 — 18i;2 ; and by transposition, 54i;2=216 ; ...v2=4 . and t;=±2 ; .-. a?=6i2 = 8, or 4 ; and y=6±2=:4, or 8. 363. In all quadratics of this kind, in which x may be changed for y, and y for a;, in the original equations, without altering their form, the two values of one of the quantities may be taken for the values of the two quantities sought. Ex. 10. Given x-\-y=2a, and x'^^y^—h, to find the values of X and y. Let x—yr=i2z ; then a;=:a+«, and y=a — z ; .-.by substitution, (a+;?)5+(cr— 0)-^=J, or, by involution and addition, 2a^^-1^a^z''-^-\^az'' = h ; .'.x^a^^Jl-a^±^/^^^)\ and y=.a^./\-a^^ ^("lor"- QUADRATIC EQUATIONS. 259 ' Now, let x-\-y = 6, and a;5-|-y5=1056 ; then by substituting 3 for a, and 1056 for b, in the formula of roots, the values of a?andy will be found ; that is, a? — 3 J::!, or Si-yZ — lO ; and y=:3=f 1, or 3=p-v/ — 19- Or, by substituting the above va- lues of a and b in the equation lOaz"^ '-20a^z^-\-2a^ — b, it be- comes 30;2'*+540z+486 = 1056 ; from which the values of z may be found ; whence, by substitution, the values of x andy will be determined, as before. Ex. 11. Given a;+4y = i4i and y24-4a?=2y+ll, to find the values of x and y. Ans. a;=— 46, or 2 ; and y=15, or 3. Ex. 12. Given 2a;-f-3y=118, and 5a;2— 7y2=:4333, to find the values of x and y. ' 3899 _, ,^ 3268 Ans. a;=35, or : and y=16, or-——. 17 -^ 17 Ex. 13. Given x^+4i/^=256—4x7/, and 3y2-.a;2 = 39, to find the values of x and y. Ans. a?=i6, or il02 ; and y=:i5, or ±59. Ex. 14. Given a^-f y''=2a'', and xy=c'^, to find the values of X and y. ]_ • a;=[a''-l-V'(o«"--c"')]''; Ans. to find the values of a; and a;* — 2x^y^'{-y'^—x^+y'^=20, > and y. Ans. a;=-t3, or iV^, or :kWi^^:^^V^) i andy=2, or -1, or i(l±3 ^5).* * There are four other values, both of* and y, which are all imaginary. %0 QUADRATIC EQUATIONS i 1 Ex. 19. Given 4— a;^=:3— y, and 4— aj^ry— y^jtofindthe values of x and f. Ans. x=4, or i ; and y = l, or 2^. 3 i Ex. 20. Given a: 2 + a:— 4x2 =y2 4- y 4- 2, and xi/=y^-{-3y, to find the values of x and y. Ans. a:=-4, or 1 ; and y = l, or — 2. Ex. 21. Given a:^+a:y=56, and a:y + 2y2=60, to find the values of x and y. Ans. x= dL4'v/2, or 4= 1 4 ; ' and yr=i3^2, or ±10. Ex. 22. Given a;— y = 15, and xyz=.2y^, to find the values of a: and y. Ans. a;=18, or 12j ; and y=r3, or — 2^. Ex. 23. Given 10a;-j-yr:z3a;y, and 9y — 9ar=18, to find the values of a; and y. Ans. a:=2, or —\ ; and y = 4, or J. Ex. 24. Given .x+y : x—y : : 13 : 5, > to find the values and y''*+a:r^25, J of a; and y. Ans. a?=9, or —14^^; and y=4, or — 6^^. Ex. 25. Given a?2y*— 7a'y2=rl710, and a^y—y = 12, to find the values of x and y. — 19 Ans. a:=5, or J, or ■ ^ — - ; and y=3, or —15, or — 6iV-2. • Ex. 26. Given a:y-|-a:y2 = 12, and a;+a:y^=18, to find the values of a? and y. Ans. a;=2^ or 16 ; and y=:2, or ^. Ex. 27. Given a:4-y + 'v/(a'+y)=6, and a;2+y2.^10, tofind the values of x and y. Ans. iK— 3, or 1 ; or 4^^\^/ —Ql ; and y=l, or, 3, or 4iTiV-6i. Ex. 28. Given a;2-|-4^(a:2+3y+5) = 55— 3y, and 6a;— 7y .= 16, to find the values of x and y. -53 — 9±V5072 a;=5, or -y- ; or ^ . ^^^' ^ „ 430 -166^6^5072 y=2,or-— ;or . Ex. 29. Given a;24-2a:3y=441~a:y, and a:y=34-a:, to find the values of a; and y. . Ja:=:3,or -7; or -2^^-17, A"«-^y=2,or|; or|T|V-17. Ex. 30. Given(a:H-y)^ — 3yr=28H-3ar, and2a;y+3a:=35,to find the values of x and y. < a;=5, or 5, or -^^J^\^{-^2b5\ ^"'- \ y=2,or I, or -U T V'i(-255.) Ex. 31. Given a:24-3a;+y=73— 2a:y, andy2+3y4-ar=44, to find the values of x and y- . 5a:=4, or 16; or — 12T'v/58, ^^^' \ y=5, or -7 ; or -li -v/SS. SOLUTION OF PROBLEMS, &c. 261 Ex. 32. Given^+^=136J— 2a:y,anda;+y=10, tofind y ^ the values of x and y. , (a:z=6,or4; ox 5^b^(-\\,) ''^"^- ^y=4,or6; or 5 ={=5 ^ -(H)- Ex. 33. Given y*-432 = 12a:y2, and y^^-l^ + ^xy, to find the values of x and y. Ans. x=2, or 3 ; and y=:6, or-v/(21)+3. CHAPTER XL THE SOLUTION OF PROBLEMS, PRODUCING aUADRATIC EaUATIONS. § I. SOLUTION OF PROBLEMS PRODUCING QUADRATIC EQUA- TIONS, INVOLVING ONLY ONE UNKNOWN QUANTITY. 364. It may be observed, that, in the solution of problems which involve quadratic equations, we sometimes deduce, from the algebraical process, answers which do not correspond with the conditions. The reason seems to be, that the alge- braical expression is more general than the common language, and the equation, which is a proper representation of the con- ditions, will express other conditions, and answer other suppo- sitions. Prob. 1. A person bought a certain number of oxen for 80 guineas, and if he had bought four more for the same sum, they would have cost a guinea a piece less. Required the number of oxen and price of each. Let x-=. the number ; then —zzz the price of each ; x =r — ^ — 1, by the problem, a:+4 X and by reduction, a;2+4a:==320 ; .-. a?2-}-4a;+4:=;324, and a:+2 = il8; .•.a;=16, or —20. 262 SOLUTION OF PROBLEMS A ^80 80 . ^ • r ^ And — =——5 guineas, the price of each. The negative value (—20) of x, will not answer the con- dition of the problem. Prob. 2. There are two numbers whose difference is 9, and their sum multiplied by the greater produces 266. What are those numbers ? Let xzzz the greater; .-.aj— y= the less. 9 2fifi and x.{2x — 9)=266 ; .-. x^— -.x=~~. ^ w v , 9 • 47 completing the square, &c. x — -=: db-r » .*. a:r=:14, or — 9J; and a? — 9 = 5, or — 18J. Here both values answer the conditions of the problem. Prob. 3. A set out from C towards D, and travelled 7 miles a day. After he had gone 32 miles, B set out from D to- wards 0, and went every day one-nineteenth of the whole journey ; and after he had travelled as many days as he went miles in one day, he met A. Required the distance of the places C and D. Suppose the distance was x miles. .*. Tq= the number of miles B travelled per day ; and also = the number of days he travelled before he met A. a;2 7x .-. l-32-j =a;: by transposition and completing the square, extracting the root, ^3—6=^2; 1 y .-. Y3=8, or 4 ; and a!:=152, or 76, both which values an- swer the conditions of the problem. The distance therefore of C from D was 152, or 76 miles. Prob. 4. To divide the number 30 into two such parts, that their product may be equal to eight times their difference. Let a;=: the lesser part ; .-.30— x=: the greater part, and 30 — x—x, or 30—20!;= their difference. Hence, by the problem, a(30 — a;) = 8(30— 2a;), or 30a:— «2 =240- 16a;; .•.a;^— 46a;=— 240. PRODUCING QUADRATIC EQUATIONS. 263 .-. completing the square, rr2__46a?+ 529=289 ; .-. a: — 23^17=40, or 6= lesser part; and 30 — a;z=:30— 6=r24= ^rea^er part. In tliis case, the solution of the equation gives 40 and 6 for the lesser part. Now as 40 cannot possibly be a part of 30, we take 6 for the lesser part, which gives 24 for the greater part ; and the two numbers, 24 and 6, answer the conditions required. Prob. 5. Some bees had alighted upon a tree ; atone flight the square root of half of them went away ; at another eight- ninths of them ; two bees then remained. How many then alighted on the tree ? 16x2 Let 2jc2= the number of bees ; x-\ — - — |-2=2a;2, or 9x-f 16a;24-18 = 18a;2; .-. 2a;2— 9a:=18 ; Multiplying by 8, 16^2 — 72a; = 144 ; adding 81 to both sides, 16a?2_72a;+81 =225 ; .-. 4a;=9i 15=24, or —6 ; and j;=6, or — 1^. .-. 2a;2 = 72, or4f But the negative value — 1^ of a?, is excluded by the na- ture of the problem ; therefore, 72= number of bees. 365. If, in a problem proposed to be solved, there are two quantities sought, whose sum, or difference, is equal to a given quantity, for instance, 2a ; let half their difference, or half their sum, be denoted by x ; then a:-}-« will represent the greater, and x—a the lesser, (Art. 102). According to this method of notation, the calculation will be greatly abridg- ed, and the solution of the problem will often be rendered very simple. Prob. 6. The sum of two numbers is 6, and the sum of their 4th power is 272. What are the numbers 1 Let a?= half the difference of the two numbers ; then 3 + a:= the greater number, and 3 — a:= the lesser. .-. by the problem, (3 + a:)++(3-a;)4 = 272, or 162 + 108a;2-|-2a;*=272 ; from which, by transposition and division, a!*+54a;2=55 : .-. completing the square, a;*+54a;2 4-729 = 784, and extracting the root, a;2-f 27= Jr28 ; .-. a;2==— 27±28, and a:=il, or J- V-55. Now, by taking the positive value, +1, for x, (since in this case, it is the only value of x which will answer the problem) ; we shall have 3+1-4= the greater, and 3 — 1=2= the lesser. 264 SOLUTION OF PROBLEMS Prob. 7. To divide the number 56 into two such parts, thai their product shall be 640. Ans. 40, and 16. Prob. 8. There are two numbers whose difference is 7, and half their product plus 30, is equal to the square of the lesser number. What are the numbers ? Ans. 12, and 19. Prob. 9, A and B set out at the same time to a place at the distance of 150 miles. A travelled 3 miles an hour faster than B, and arrives at his journey's end 8 hours and 20 minutes before him. At what rate did each person travel per hour 1 Ans. A 9, and B 6 miles an hour. Prob. 10. The difference of two numbers is 6 ; and if 47 be added to twice the square of the lesser, it will be equal to the square of the greater. What are the numbers ? Ans. 17, and 11. Prob. 11. There are two numbers whose product is 120, if 2 be added to the lesser, and 3 subtracted from the greater,, the product of the sum and remainder will also be 120. What are the numbers? Ans. 15, and 8. Prob. 12. A person bought a certain number of sheep for 120Z. If there had have been 8 more, each would have cost him ten shillings less. How many sheep were there ? Ans. 40. Prob. 13. A Merchant sold a quantity of brandy for 39/. and gained as much per cent as the brandy cost him. What was the price of the brandy ? Ans. 30/. Prob. 14. Two partners, A and B, gained 18/. by trade. A's money was in trade 12 months, and he received for his principal and gain 26/. Also, B's money, which was 30/. was in trade 16 months. What money did A put into trade ? Ans. 20/. Prob. 15. A and B set out from two towns which were at the distance of 247 miles, and travelled the direct road till they met. A went 9 miles a day ; and the number of days, at the end of which they met, was greater by 3 than the number of miles which B went in a day. How many miles did each go ? Ans. A 117, and B 130 miles. Prob. 16. A man playing at hazard won at the first throw, as much money as he had in his pocket ; at the second throw, he won 5 shillings more than the square root of what he then had ; at the third throw, he won the square of all he then had ; and then he had 112/. l^y. What had he at first ? Ans. 18 shillings. OK To- ;; UNiVERsr: PRODUCING QUADRATIC EQUATir^^^^'^'"*'^' Prob. 17. If the square of a certain number be taken from 40, and the square root of this difference be increased by 10, and the sum multiplied by 2, and the product divided by the number itself, the quotient will be 4. Required the number. Ans. 6. Prob. 18. There is a field in the form of a rectangular parallelogram, whose length exceeds the breadth by 16 yards ; and it contains 960 square yards. Required the length and breadth. Ans. 40 and 24 yards. Prob. 19. A person being asked his age, answered, if you add the square root of it to half of it, and subtract 12, there will remain nothing. Required his age. Ans. 16. Prob. 20. To find a number from the cube of which, if 19 be subtracted, and the remainder multiplied by that cube, the product shall be 216. Ans. 3, or —2. Prob. 21. To find a number, from the double of which if you subtract 12, the square of the remainder, minus 1, will be 9 times the number sought. Ans. 11, or 3^. Prob. 22. It is required to divide 20 into two such parts, that the product of the whole and one of the parts, shall be equal to the square of the other. Ans. 10v^5 — 10, and 30—10^5. Prob. 23. A labourer dug two trenches, one of which was 6 yards longer than the other, for 111. I6s., and the digging of each of them cost as many shillings per yard as there were yards in its length. What was the length of each ? Ans. 10, and 16 yards. Prob. 24. A company at a tavern had 8/. 15^. to pay, but before the bill was paid, two of them sneaked off, when those who remained had each lOs. more to pay. How many were there in the company at first ? Ans. 7. Prob. 25. There are two square buildings, that are paved with stones, a foot square each. The side of one building ex- ceeds that of the other by 12 feet, and both their pavements taken together contain 2120 stones. What are the lengths of them separately ? Ans. 26, and 38 feet. Prob. 26. In a parcel which contains 24 coins of silver and copper, each silver coin is worth as many pence as there are copper coins, and each copper coin is worth as many pence as there are silver coins, and the whole is worth 18 shillings. How many are there of each ? Ans. 6 of one, and 18 of the other. Prob. 27. Two messengers, A and B, were despatched at 24 266 SOLUTION OF PROBLEMS the same time to a place 90 miles distant ; the former of whom riding one mile an hour more than the other, arrived at the end of his journey an hour before him. At what rate did each travel per hour ? Ans. A went 10, and B 9 miles per hour. Prob. 28. A man travelled 105 miles, and then found that if he had not travelled so fast by 2 miles an hour, he should have been 6 hours longer in performing the journey. How many miles did he go per hour ? Ans. 7 miles. Prob. 29. Bought two flocks of sheep for 65/. 135., one containing 5 more than the other. Each sheep cost as many shillings as there were sheep in the flock. Required the number in each flock. Ans. 23, and 28. Prob. 30. A regiment of soldiers, consisting of 1066 men, is formed into two squares, one of which has 4 men more in a side than the other. What number of men are in a side of each of the squares ? Ans. 21, and 25. Prob. 31. What number is that, to which if 24 be added, and the square root of the sum extracted, this root shall be less than the original quantity by 18 ? Ans. 25. Prob. 32. A Poulterer going to market to buy turkeys, met with four flocks. In the second were 6 more than three times the square root of double the number in the first. The third contained three times as many as the first and second ; and the fourth contained 6 more than the square of one-third the number in the third ; and the whole number was 1938. How many were there in each flock ? Ans. The numbers were 18, 24, 126, and 1770, respectively. Prob*. 33. The sum of two numbers is 6, and the sum of their 5th powers is 1056. What are the numbers ? ^ Ans. 4, and 2. §11. SOLUTION OF problems PRODUCING QUADRATIC EQUA- TIONS, INVOLVING MORE THAN ONE UNKNOWN QUANTITY. 366. It is very proper to observe, that the solution of a problem, producing quadratic equations, involving two un- known quantities, will sometimes be very much facilitated by assuming x equal to their half sum, and y equal to their half diflference ; then, (Art. 102), x-\-y will denote the greater, and x—y the lesser. The solution, according to this method of notation, will, in general, be more simple than that which would have been found, if the two unknown quantities were represented by x and y respectively. PRODUCING QUADRATIC EQUATIONS. 267 Problem I. Required two' numbers, such, that their sum, their product, and the difference of their squares, may be all equal. Let x-{-i/= the greater ; and x—y-=. the lesser ; and2a:=:(.x+y)2— (a:— y)2 = 4a:y, S ^ ^ From the 2d equation, y = 4 ; ••• y^— -J '• Now, by substituting this value of y"^, in the first we have 2x=x^-\\ .-.0:2-20:=^, and aj^liJV^- 367. The preceding problem leads also to the solution of the following. pROB. 2. To find two numbers, such, that their sum, their product, and the sum of their squares, may be all equal. Let, as in the last problem, x-\-y=^ the greater, and x—y=z the lesser ; then, by the problem, 2a;=a;2-y2, and 2xz^(x-\-yf-\-{x—yf=2x'^-^2y'^ ; but 2xz=x'^—y'^ ; .-. by addition, 3x=2a:2, and a;— ^ ; .-. by substitution, ^=| + 3/^ ^ and y= :\zW — 3 ; .-. a;;f-y=JJti V — 3, and x — y^iTiV — 3. Hence it follows, that no two numbers can be found to answer the conditions ; and therefore the problem is impossible : Al- though the above values of x and y are imaginary, still they will satisfy the equations, ^x^zx^—y"^, and 2x^=z2x^-{-2y'^y which may be readily verified by substitution. 368. It is sometimes more expedient to represent one of the unknown quantities by x, and the other by xy- The utility of this method of notation for eliminating one of the unknown quantities, will appear evident, from the solution of the following problem. Prob. 3. What two numbers are those, whose sum multi- plied by the greater is 77 ; and whose difference, multiplied by the lesser, is equal to 12 ? Let xyz=. the greater, and x^=i the lesser ; then by the pro- blem, x'^y''--{-xy~ll, and x'^y—x'^ — \2 ; 2 77 , , 12 12 77 •• a;^=~- — , and x^z=z- y-'-\-y y-1' "y-i y^^y' and clearing of fractions, 12y2-|-i2y=77y— 77 ; by transposition and division, y^ _y_ ; 268 SOLUTION OF PROBLEMS .'.completing the square, and extracting the root, y=y, or J. Either value of y will answer the conditions of the prob- 12 lem ; Lety=zJ; then a?=: --=16; .•.a?=:^4, and a:y=: ±7. Hence the numbers, by taking the positive values, are 4 and 7. Let also y = V ; then x^—\\ .-.0?=: -tf .^2, and a:yr=y X ifV^ — i V V 2- Hence the irrational numbers, \^'l and y -^2, will also answer the conditions of the prob- lem. 369. When a problem expresses more than two distinct conditions, which require to b^ translated into as many equa- tions ; the solution cannot be obtained by means of quadra- tics, unless that some of the equations are of the first degree ; for the final equation resulting from the elimination of the imknown quantities will, in general, be of a higher degree than the second. There are, however, some particular cases in which the unknown quantities may be eliminated by certain artifices, (which are best learned by experience), so as to leave the final equation of a quadratic form. Prob. 4. It is required to find three numbers, such, that the product of the first and second, added to the sum of their squares, shall be equal to 37 ; the product of the first and third added to the sum of their squares, shall be equal to 49 ; and the product of the second and third added to the sum of their squares, shall be equal to 61. Let a;= the first number, 3/= the second, and z= the third Then, x^+y'^-{-xy='^l ; \ x'^-\-2:^-\-xz=49 ; > by the problem. and y2_|_2:2_[_y;3 — 61 ; ) By subtracting the first equation from the second, x^—y^-\- 12 (z-'y)x = l2;.'.z-^y-\-x=^— (a). By subtracting the second equation from the third, y^—x^+ 12 (y^x)z=l2;.'.y+x-[-z=-— (b); •••jz:^=^'^"^y-^=^-y; .•.2y=^+^. By substitutrng 2y for x-\-z, in equations (a) and (i), we ^ j« 12 ^ „ 12 find 3y= , and 3y= y y-x zy — y'^=:4, and y^—yx=4 ; y2+4 - y^— 4 n ly^ z=/- , and a;=^^ ; /. -^—z:' y r^): PRODUCING QUADRATIC EQUATIONS. 269 Now, by substituting these values of x and x"^ in the first of the original equations, it becomes (7/2 — 4v2 v^ — 4 ) ■\-y'^-\-y'- =37; .-.by reduction, 49 y* — — y2=_16 ; and, by completing the square, o ^ 49 , , /49\2 2401-192 , 49,47 y -yy +(t) =—36 — = •■•* =-6-^-6-' and, by taking the positive sign, y=-]-4 ; y2_4 16—4 .'.by taking y=:4, x=.- — = =3, and _yH-4__16-H_20__ ~ y ~~ 4 ~~ 4 ~ * Hence the three numbers sought are 3, 4, and 5, which are in arithmetical progression. This relation appears also evi- dent from the result 2y^=.x-\-z^ found in the beginning of the solution. Prob. 5. There are three numbers, the difference of whose differences is 8 ; their sum is 41 ; and the sum of their squares 669. What are the numbers ? Let a;=: the second number, and y= the difference of the second and least ; .-. x~y, X, and x+y + S are the numbers, and their sum =:3a:-f 8 = 41 ; .*. 3a:=33, and a;=rll ; .•.(ll-y)2 + r214-(19 + y)2 = 669, or y2_|_ 8^=48 ; .-. completing the square* and extracting the root, y + 4= ±8, .and y = 4, or — 12, both which values answer the conditions ; and the numbers are 7, 11, and 23. Prob. 6. What number is that, which being divided by the product of its two digits, the quotient is 2 ; and if 27 be added to it, the digits will be inverted ? Ans. 36. Prob. 7. There are three numbers, the difference of whose differences is 5 ; their sum is 44 ; and continual product is 1950. What are the numbers ? Ans. 6, 13, and 25. Prob. 8. A farmer received 71. 4s. for a certain quantity of wheat, and an equal sum at a price less by 1^. 6c?. per bushel, for a quantity of barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of each ? Ans. 32 bushels of wheat, and 48 of barley. Prob. 9. A poulterer bought 15 ducks and 12 turkeys for five guineas. He had two ducks more for 18 shillings, than 24* 270 SOLUTION OF PROBLEMS he had of turkeys for 20 shillings. What was the price of each ? Ans. the price of a duck was 3^. and of a turkey 5^. Prob. 10. There are three numbers, the difference of whose differences is 3 ; their sum is 21 ; and the sum of the squares of the greatest and least is 137. Required the numbers. Ans. 4, 6, and 11. Prob. ] 1. There is a number consisting of 2 digits, which, when divided by the sum of its digits, gives a quotient greater by 2 than the first digit. But if the digits be inverted, and then divided by a number greater by unity than the sum of the digits, the quotient is greater by 2 than the preceding quotient. Required the number. Ans. 24. Prob. 12. What two numbers are those, whose product is 24, and whose sum added to the sum of their squares is 62 ? Ans. 4, and 6. Prob. 13. A grocer sold 80 pounds of mace, and 100 pounds of cloves, for 65/. ; but he sold 60 pounds more of cloves for 201. than he did of mace for 10/. What was the price of a pound of each 1 Ans. the mace cost 10^. and the cloves 5^. per pound. Prob. 14: To divide the number 134 into three such parts, that once the first, twice the second, and three times the third, added together, may be equal to 278 ; and that the sum of the squares of the three parts may be equal to 6036. Ans. 40, 44, and 50, respectively. Prob. 15. Find two numbers, such, that the square of the greater minus the square of the lesger, may be 56 ; and the square of the lesser plus one third, their product may be 40. Ans. 9, and 5. Prob. 16. There are two numbers, such, that three times the square of the greater plus twice the square of the less is 110 ; and half their product, plus the square of the lesser, is 4. What are the numbers? Ans. 6, and 1. Prob. 17. What number is that, the sum of whose digits is 15 ; and if 31 be added to their product, (he digits will be in- verted 1 Ans. 78. Prob. 18. There are two numbers, such, that if the lesser be taken from three times the greater, the remainder will be 35 ; and if four times the greater be divided by three times the lesser plus one, the quotient will be equal to the lesser number. What arc the numbers ? Ans. 13, and 4. Prob. 19. To find two numbers, the first of which, /)/u5 2, multiplied into the second, minus 3, may produce 110 ; and PRODUCING QUADRATIC EQUATIONS. 271 the first minus 3, multiplied by the second plus 2, may pro- duce 80. Ans. 8, and 14. Prob. 20. Two persons, A and B, comparing their wages, observe, that if A had received per day, in addition to what he does receive, a sum equal to one-fourth of what B receiv- ed per week, and had worked as many days as B received shillings per day, he would have received 48.y. ; and had B received 2 shillings a day more than A did, and worked for a number of days equal to half the number of shillings he re- ceived per week, he would have received-4Z. \8s. What were their daily wages 1 Ans. A's 5 shillings, and B's 4. Prob. 21. Bacchus caught Silenus asleep by the side of a full cask, and seized the opportunity of drinking, which he continued for two-thirds of the time that Silenus would have taken to empty the whole cask. After that Silenus awoke, and drank what Bacchus had left. Had they drunk both together, it would have been emptied two hours sooner, and Bacchus would have drunk only half what he left Silenus. Re- quired the time in which they could have emptied the cask separately. Ans. Silenus in 3 hours, and Bacchus in 6. Prob. 22. Two persons, A and B, talking of their money, A says to B, if I had as many dollars at ^s. 6rf. each, as I have shillings, I should have as much money as you ; but, if the number of my shillings were squared, I should have twice as much as you, and 12 shillings more. What had each ? Ans. A had 12, and B 66 shillings. Prob. 23.'^t is required to find two numbers, such, that if their product be added to their sum it shall make 62 ; and if their sum be taken from the sum of their squares, it shall leave 86. Ans. 8, and 6. Prob 24. It is required to find two numbers, such, that their difference shall be 98, and the diff'erence of their cube roots 2. Ans. 125, and 27. Prob. 25. There is a number consisting of two digits. The left-hand digit is equal to 3 times the right-hand digit ; and if 12- be subtracted from the number itself, the remainder will be equal to the square of the left-hand digit. What is the number ? Ans. 93. Prob. 26. A person bought a quantity of cloth of two sorts for 11. 18 shillings. For every yard of the be,tter sort he gave as many shillings as he had yards in all ; and for every yard of the worse as many shillings as there were yards of the bet- 272 EXPANSION OF INFINITE SERIES. ter sort more than of the worse. And the whole price of the better sort was to the whole price of the worse as 72 to 7, How many yards hadr he of each ? Ans. 9 yards of the better, and 7 of the worse. Prob. 27. There are four towns in the order of the let- ters, A, B, C, D. The difference between t!ie distances, from A to B, and from B to C, is greater by four miles than the dis- tance from B to D. Also the number of miles between B and D is equal to ,two-thircl3 of the number between A to C. And the number between A and B is to the number between C and D as seven times the number between A and C : 26. Required the respective distances. Ans. AB=42, BC=6, and CD=26 miles. CHAPTER XII. ON THE EXPANSION OF INFINITE SERIES. § I. RESOLUTIONS OF ALGEBRAIC FRACTIONS. 370. An infinite series is a continued rank, or^ogression of quantities, connected together by the signs + or — ; and usu- ally proceeds according to some regular, or determined law. Thus, i + i+J+i+l+^V+. ^^' In the first of which, the several terms arc the reciprocals of the odd numbers, 1, 3, 5, 7, &c. ; and in the latter the recipro- cals of the even numbers, 2, 4, 6, 8, &c., with alternate signs. 371. We have already observed (Art. 96), that if the first or leading term of the remainder, in the division of algebraic quantities, be not divisible by the divisor, the operation might be considered as terminated ; or, which is the same, that the integral part of the quotient has been obtained. And it has also been remarked, (Art. 89), that the division of the remain- der by the divisor can be only indicated, or expressed, by a fraction: thus, for example, if we have to divide a' by a-f-1, i EXPANSION OF INFINITE SERIES. 273 we write for the quotient — -— : This, however, does not pre- a-f- 1 vent us from attempting the division according to the rules that have been given, nor from continuing it as far as we please, and we shall thus not fail to find the true quotient, though under different forms. 372. To prove this, let us actually divide a" or 1, by 1— a, thus ; • I -a I I— a remainder a Quot. 1+- ^ 1 + a Therefore- =1+:; ;but- =a-\-- ; ■- = 1 — a I— a I— a I — a 1 — a 1 — a 1 — a 1 — a 1 — a 1 — a This shows that the fraction may be exhibited under 1 — a "^ all the following forms : la d^ 1— a 1 — a I — a 1-a' ' ^ ^1— a' 1 — a Now, by considering the first of these formulae, which is l+i , and observing that 1 =:- , we have 1+- =: 1 — a ° 1 — a 1 — a 1— a a _1— a-f-a_ 1 1 — a l—a~ 1— a 1 — a If we follow the %ame process with regard to the second ex- pression, that is to say, if we reduce the integral part l + a to the same denominator, 1 —a, we shall have the fraction , 1 — a to which if we add , we shall have =- • 1 — a \—a i—a . In the third formula of the quotient, the integers l+a+«^ \ flS reduced to the denominator 1 —a make , and if we add 1 — a a^ 1 to it the fraction the sum will be I— a I 274 EXPANSION OF INFINITE SERIES. Therefore each of these formula is in fact the value of the proposed fraction . ■^ ^ 1 — a 273. This being the case, we may continue the series as far as we please, without being under the necessity of performing any more calculations ; by observing, in the first place, that each of these formulae is composed of an integral part which is the sum of the successive powers of a,*beginning with a^=z\ inclusively ; * Secondly, of a fraction which has always for the denomi- nator 1 — a, and for the numerator the letter c, with an ex- ponent greater, by unity, than that of the same letter in the last term of the integral part. This constant formation of the successive formulae, is what Analysts call a law. And the manner of deducing general laws by the consideration of certain particular cases, is usu- ally called induction ; which, though not a strict method of proof, says Laplace, has been the source of almost all the discoveries that have hitherto been made, both in analysis and physics, of which all the phenomena are the mathematical re- sults of a small number of invariable laws. It is thus that Newton, by following the law of the numeral coefficients, in the square, the cube, the fourth power, &c. of a binomial, arrived soon at the general law, that is to say, at the general formula that bears his name, and which will be demonstrated in one of the following Sections : This Geometer has carefully added, that in following this mode of investigation, we must not generalize too hastily ; as it often happens, that a law, which appears to take place in the first part of a process, i^ not found to hold good throughout. Thus, in the simple in- , , . 531251 ^ . , . stance of reducmg to a decimal, its equivalent value is 17174949, &c., of which the real, repeating period is 49, and not 17, as might, at first, be imagined. 374. From what has been observed with regard to the suc- cessive quotients, we can, in general, put --i- = H-a + «2:|-a3+a* . """-^TI^^ I— a l-ha n being a whole positive number, which augmented by unity, gives the place of the term. In fact, making w=:3, a" becomes* a^, which is the fourth term of the quotient, for n = 4, cr" becomes c*, which is the fifth term. But as nothing hinders us from removing indefinitely the fractional term which terminates the series, that is, of adding always a term to the integral part ; EXPANSION OF INFINITE SERIES. 275 so that we might stiil go on without end ; for which reason it may be said that the proposed fraction has been resolved into an infinite series ; which is, l-}-a-\-a'^-^a'^-{-'^-^a^-\-a^-{-a''-\-a^ -\-a^-{-a^^-\-a^^-\-a^'^-\-, &c.4o infinity: and there are sufli- cient grounds to maintain that the value of this infinite series is the same as that of the fraction I— a Orthat,-i-=14-a+a2-j-a3_j_a4_|- . &c. 1— a 375. What has been just observed may at first appear strange ; but the consideration of some particular cases will make it easily understood. Let us suppose, in the first place, a = l ; the general quo- tient above will become a particular quotient corresponding to the fraction - — -. The series taken indefinitely, shall be ^=14-1 + 1 + 1 + 1 + 1+, &c. In order to seie clearly the meaning of this result, let us suppose that we have to divide unity or 1 successively by the numbers 1. ^, ^-, J^, ^, &c., we will have the quotient, 1, 10, 100, 1000, 10000, &c., continually and inde- finitely increasing ; because the divisors are continually and indefinitely decreasing ; but these divisors tend towards zero, which they cannot attain, although they approach to it con- tinually, or that the difference becomes less and less ; and at the same time the value of the fraction increases continually, and tends to that which correspo.nds to the divisor zero or ; and it is as much impossible that the fraction in its successive augmentations, attains -, as it is that the denominator in its successive diminutions arrives at zero. Thus - is the last term or limit of the increasing values of the fraction : at this period, it has received all its augmentations : - is not therefore a number, it is the superior limit of numbers ; such is the no- tion that we must have of this result -, which the analysts call, for abbreviation, infinitt/, and which is denoted by the character 00, (Art. 35). It is frequently given as an answer to an im- 276 EXPANSION OF INFINITE SERIES. possible question, (which will be noticed in a subsequent part of the Work) ; and in fact, it is very proper to announce this circumstance, since that we cannot assign the number denoted by this sign. It may still be remarked, that if we would take but the first six terms of the series, we must close the development by the corresponding remainder divided by this divisor, which gives, :^4=l + l + l + l + l + l+i; this equality, absurd in appearance, proves that six terms at least do not hinder the series from being indefinitely conti- nued. And in fact, if after having taken away six terms from this series, it would cease to be infinite, or become terminat- ed, in restoring to it these six terms, it should be composed of a definite or assignable number of terms, which it is not. Therefore the surplus of the series must have the same sum as the total. We can yet, say that -, inasmuch as it is not a magnitude, can receive no augmentation, so that 1 -\-.l -}- 1 + , Slc. +- must remain equal to -. Hence, we might conclude that a finite quantity added to, or subtracted from infinity, makes no akeration. Thus, QO-l-azz: c». However, it may be'necessary in this place to observe, that, although an infinity cannot be increased, or decreased, by the addition, or subtraction, of finite quantities ; still, it may be in- creased or decreased, by multiplication or division, in the same manner as any other quantity ; Thus, if- be equal to infinity, 2 3 - will be the double of it, - thrice, and so on. See Euler's Algebra, Vol. I. Note. — -, -j-, — j— , — ^ — , &c. are considered to be frac- 1 10 Too" lOoo tions, in which the denominators are 1, -y-, -^, — j — , &c. To TWU ToTTTT Now, as 1 divided by any assignable quantity, however great it may be, can never arrive completely at 0, consequent- ly the fractions in their successive augmentations can never arrive at infinity, except that unity or 1, be divided by a quantity infinitely great ; that is to say, it must be divided by EXPANSION OF INFINITE SERIES. 27T infinity ; hence we may conclude that -^ is in reality equal to nothing, or oo-=0. 360. It may not be improper to take notice in this place of other properties of nought and infinity. I. That nought added to or subtracted from any quantity, makes it neither greater nor less ; that is, a-J-0=a, and a — 0=a. II. Also, if nought be multiplied or dirided by any quantity, both the product and quotient will be nought ; because any number of times 0, or any part of 0, is : that is, Ox«, or ax = 0, and -=0. a III. From the last property, it likewise follows, that nought divided by nought, is a finite quantity, of some kind or other. For since X a=0, or 0=^0 x a, it is evident from the ordinary rules of division, that = ^- IV. Farther, if nought be multiplied by infinity, the pro- .1 a duct will be some infinite quantity. For since - or -=oo ; therefore, x oo =a. 361. It may be also remarked, that nought multiplied by produces ; that is, OxOr^O. For, since x « = 0, whatever quantity a may be, then, sup- posing a = 0, 0X0 = 0. From this we might infer, according to the rules of division, that the value of --=:0, or that nought divided by nought is nought, in this particular case. Also, that 0, raised to any power, is ; that is, 0'«=0 ; it Q^ w^ follows that -— =- : but if in <2"»-'»=— (Art. 86), we suppose c=:0, which may be allowed, since a designates any number, we have 0®=-. If we really effect the division of by 0, we could put for the quotient any number whatever, since any number, multi- plied by zero, gives for the product zero, which is here the dividend. 25 278- EXPANSION OF INFINITE SERIES. This expression, O**, appears therefore to admit of an infi- nity of numerical values ; and yet such a result as - can, in many cases, admit of a finite and determined value. It is thus, foi example, that the fraction , in the hypothesis of a=0, , Kxo 6 becomes — - — =-. But, if at first we write this fraction under the form Ka"*-"*, and that we put a=0, we find that it becomes KxO*"~", which is for wi>7» ; in case of m<^n, or m=n — d, we shall rr rr have (Art. 86), -frj=-^ ; which is equal to infinity, as has been already observed ; finally, for m=n, we can divide above and below by a*", and the fraction is reduced to K, which is a finite quantity. 362. If we suppose, in the fraction (Art. 358), a=2, we find ,-^ = 1+2 + 4 + 8+16 + 32 + 64 + , &c., which at first sight it will appear absurd. But it must be re- marked, that if we wish to stop at any term of the above se- ries, we cannot do so without joining the fraction which re- mains. Suppose, for example, we were to stop at 64 ; after havmg written 1+2+4+8+16+32 + 64; we must join the 128 128 fraction - — -, or — -, or —128 ; we shall therefore have for 1 —2 — 1 the complete quotient 127 — 128, than is in fact —1. Here, however far the fractional term may be extended, its numerical value, which is negative, will always surpass, by a unit, that of the integral part, so that this is totally destroyed ; and as in the hypotheses of a>l, we shall always subtract more than what we will add, we shall never meet with the result -. 363. These are the considerations which are necessary when we assume for a numbers greater than unity ; but if we now suppose a less than 1, the whole becomes more in- telligible ; for example, let a=^, and we shall have = - — r-=Y=2, which will also be equal to the following se- 1— ^ 7 EXPANSION OF INFINITE SERIES. 279 ries, l+i+i+i:4-TV+3^2+^+Tl8. &c., to infinity (Art. 358). Now, if we take, only two terms of the series, we shall have 1+i, and it wants -J .of being equal to 2 ; if we take three terms, it wants 1, for the sum is 1|- ; if we take four terms, we have i-J, and the deficiency is only J : There- fore, we see very clearly that the more terms of the quotient we take, the less the difference becomes ; and that, conse- quently, if we continue to take successive portions of this series, the differences between those consecutive sums and the fraction f =2, decrease, and end by becoming less than any given number, however small it may be. The number 2 is therefore still a limit, according to the acceptation of this word. Now, it may be observed, that if the pieceding series be continued to infinity, there will be no difference at all between its sum and the value of the fraction j-, or 2, 364. A limit, according to the notion of the ancients, is some fixed quantity, to whioh another of variable magnitude can never become equal, though, in the course of its variation, it may ap- proach nearer to it than any difference that can be assigned ; always supposing that the change, which the variable quantity undergoes, is one of contimced increase, or continued diminution Such, for example, is the area of a circle, with regard to the areas of the circumscribed and inscribed polygons, for, by in- creasing the number of sides of these figures, their difference may be made less than any assigned area, however small ; and since the circle is necessarily less than the first, and greater than the second, it must differ from either of them by a quantity less than that by which they differ from each other. The circle will thus answer all the conditions of a limit, which is included in the above definition. 365. The preceding considerations are very proper to de- fine the nature of the word limit ; but as algebra, which is the subject we are treating of here, needs no foreign aid to demonstrate its principles, it is necessary, therefore, to explain the nature of the word limit, by the consideration of algebraic expressions. For this purpose, let, in the first place, the very simple fraction be , in which we suppose that x may x-\-a be positive, and augmented indefinitely ; in dividing both terms of this fraction by x, the result -r-j;T^ evidently shows that 280 EXPANSION OF INFINITE SERIES. the function remains always less than a, but that it approaches continually to a, since that the part ^, of its denominator, di minishes more and more, and can be reduced to such a degree of smallness as we would wish. 366. The difference between a and the proposed fraction be- ing in general expressed by a — -= — ; — , becomes so much x-\-a x-\-a smaller, according as x is larger, and can be rendered less than any given magnitude, however small it may bo ; so that the proposed fraction can approach to a as near as we would QX wish : a is therefore the limit of the fraction , relatively to , x-\-a '' the indefinite augmentation which x can receive. It is in the characters which ^e have just expressed, that the true accep- tation, which we must give to the v/ord limit, consists, in order te comprehend every thing which can relate to it. . 367. If we had remarked in the preceding example, that by carrying on, as fa#as we would wish, the augmentation of a?, we could never regard, as nothing, the fraction — ; — ; therefore x-\-a QX we would reasonably conclude, that the fraction — - — , though x-\-a it would approach indefinitely to the limit a, could never at- tain a, and, consequently, cannot surpass it ; but it would be wrong to insert this circumstance as a condition in the gene- ral definition of the word limit ; we would thereby exchide the ratios of vanishing quantities, ratios whose existence is incontestable, and from which we derive much in analysis. 368. In fact, when we compare the functions ax and ax-\- a?^, we find th&t their ratio, reduced to its most simple ex- pfession, is — ■ — , and that it approaches nearer and nearer to a-\-x ^'^ unity, according as x diminishes. It becomes exactly 1 , when x=.Q ; but the quantities ax and ax-^-x"^^ which are then rigor- ously nothing, can they have a determinate ratio ? This is what appears diflficult to conceive ; and we cannot give a clear idea of it but by presenting the quantity 1 as a limit to which .the ratio of the functions ax and ax-^-x"^ can approach as near a x as we would wish, since the difference, 1 ; — = — ; — , can a + x a-{-x be rendered less than any assignable magnitude, however small this magnitude may be. EXPANSION OF INFINITE SERIES. 281 On the other hand, the ratio — ; — ,.of the quantities ax and a-\-x ax-^-x"^ can not only attain unity when we make x=0, but surpass it when we suppose x negative, since it -becomes then , a quantity which is greater than 1, when x3 /*A x^ 1 ,1. • . . . — a a?3 a?* a a2 4 X ^ X* x^ fl2 a3 J.- The terms in the quotient are found thus ; dividing the first remainder rc^, by a, the first term of the divisor a— a:, a;2 we shall have — for the second term of the quotient, because a the division can be only indicated ; multiplying the divisor by — , and subtracting the product from x"^, the remainder is — , again, dividing this remainder by a, the result will be — , which is the third term in the quotient ; and, in like manner, we might continue the operation as far as we please : But the law of continuation is evident, because the powers of or increase by unity in each successive term of the quotient, and the powers of a increase by unity in the denominator of each of the terms after the first. EXPANSION OF INFINITE SERIES. 283 And the sum of the terms infinitely continued is said to be equal to the original fraction . Thus we say that thfe numerical fraction f , when reduced to a decimal, is equal to .6666, &c., continued to infinity. Ex. 2. It is required to convert into an infinite se- nes. a a — X X X" X a a- Quotient. , , X . x"^ . a? . . a a'' a'' a a a -2-, &C. • In this example, if x be less than a, the series is convergent, or the value of the terms continually diipinishes ; but, when a; is greater than a, it is said to diverge : Thus, let a=3 and a;=2,then 1+^+^+^+, &c. =1 + 2 + 4_f_^_|.^ &c. ; O/ Q, CI where the fractions or terms of the series grow less and less, and the farther they are' extended the more they converge or approximate to 0, whichis supposed to be the last termor limit. But if a=2, and a;=3, then 1+-+^+^+, &c. =1-F a a^ a^ |+f4-V + » ^^-j ^^ which the terms become larger and larger. This is called a diverging series. Ex. 3. It is required to convert -— ; — into an infinite series. 1+a 284 EXPANSION OF INFINITE SERIES. 1+a Quotient. 1— a4-«^— «^+«*— o*4-«®~, &c. 1 1+a — a — a — a^ a'^ + a^ — a5, &c. Whence it follows, that the fraction • is equal to the l-j-a series, l~'a-\-a^—a^-^a'^—a^-{-a^ — a^-\-j &c. 372. If we make a=zl, we have this remarkable compari- son : —-— =1 — 1 + 1 — 1 + 1 — 1 + 1— j&c. to infinity; which 1+a appears rather contradictory; for if we stop at — 1, the se- ries gives ; and if we finish at +1, it gives +1. The real question, however, results from the fractional parts, which (by division) is always + J when the sum of the terms is 0, and —J when the sum is +1 : because the complete quotient is found by placing the remainder over the divisor, in the form of a fraction, and annexing it to the terms in. the quotient with its proper sign ; but the remainder in the present case is + 1 , or — 1 ; hence the fraction to be added is +^, or —J ; and, consequently,! is the trae quotient in the former case, and 1 — ^, or i in the other. This will appear evident by taking successive portions of the series ; thus, for six terms, we shall ^ve 1 — 1 + 1 — 1-f 1 — l+i=r^, and for seven terms, 1 — 1 + l-l + l-l + l-i=f Scholium. Here we might infer, by conversion, that the sum of an infinite series is found, \^en we know the fraction which would produce such a series by actual division ; but, although it is a fact that the fraction is a value of the series, still it may not be the only one which would produce the same series : Thus, the above series, 1 — 1 + 1 — 1 + 1 — 1 + 1 — l + i &c., to infinity, can be produced by several other fractions besides the fraction ^. EXPANSION OF INFINITE SERIES. 285 Let, for example, ^ be converted into an infinite series by actual division : Now, it is plain that 4= , . , . , , and the ^ ^ 1 + 1 + 1* operation will stand thus : 1 1+1+1 1+1+1 Quotient. 1-141— 1 + 1— l+.&c —1 77 +1 +1+1 +1 • • • -1-1 _i— 1— 1 + 1,&C. In like manner, J will produce the above series, and so on. 374. Let us now make a^^; and the preceding develop- ment shall be ^f=i-i+i-i+A-A+. &"• = \+i The sum of two terms is J, which is too small by J ; three terms give -J, which is too much by -^ ; for the sum of four terms, we have |-, which is too small by gV? ^^' We see here that the successive portions of the series are alternately greater and less than the fraction f , which repre- sent it ; but that the difference, whether it be in excess or deficiency, becomes less and less. 375. Suppose again a=:^, and we shall have Now, by considering only two terms, we have J, which is too small by -f^ ; three terms make |-, which is too much by ■^Q ; four terms give |^, which is too small by j^, and so on. 376. The fraction may also be resolved into an infi- 1 + a nite series another way ; namely, by dividing 1 by a+1, as follows : . 286 EXPANSION OF INFINITE SERIES. 1 1 + a+i Quot. 1 l+i,_l + l_.&e. -9~r _9 It is however unnecessary to carry the actual division any farther, as we are enabled already to continue the series to any length, from the law which may be observed in those terras we have obtained ; the signs are alternately plus and mi- nuSy and each term is equal to the preceding one multiplied by-. It is thus by changing the order of the terms of the deno- minator, we obtain the quotienfunder different forms, and that we pass from a diverging series, for certain values of a, to a converging series for the same values. It may also be here observed, that in the division of the two polynomials, if we deviate from the established rule (Art. 93), we arrive at quotients which do not terminate : Thus, for example, a^—h"^, divided by a-\-b, according to the rule above quoted, gives for the quotient a — b; but if we divide c^—.]P- by i-j-a. we shall arrive at a quotient which does not terminate : thus, EXPANSION OF INFINITE SERIES. 287 62 Here, we can clearly see that the quotient will not termi- nate, however far we may continue the operation, because we have always a remainder. In this case, by taking 6 -fa for a divisor, we must, in order to find the quotient a—h, divide the whole dividend by all the divisor, that is to say, a^ — b"^ or [a-\-b)x(ci—h) by a-{-h. 377. When there are more than two terms in the divisor, we may also continue the division to infinity in the same man- ner. 1 Ex. 4. It is required to convert :; ; — l into an infinite \—a+a^ series. 1 l-a + a^ a — a^ — «3 a^ + a^ Quot. l+a — a^—a^+a^-i-a^j &c. We have therefore a^ -a' + dS a7 -084.^9 &c. 1 _,^. 288 EXPANSION OF INFINITE SERIES, to infinity: where, if we make a=rl, we have 1-1 + 1 1 = 1 + 1 — 1 — 1 + 1 + 1, &c., which series contains twice the series found, (Art. 372), 1 — 1 + 1 — l + l, hav- Tfl ing the fractional index — ; where m and n are whole positive gl^umbers. Then, since (l+a;)'" is the nth power of (l+a;)" ; and, as above shown, {\-\'x)"'=:i-\-ax-\-b'^-{-cx^->rdx'^-\-, &c., such a series must be assumed for (1-f a;)" ,. that, when raised to the nth power, will give a series of the form \-\-ax-\-hx'^-\-cx'^-\- dx^^, &c. But the nth or any other integral power of the series 1 + px-\-qx'^--\-rx'^-\-sx^-\-, &c. will be found, by actual multiplica- tion, to give a series of the form here mentioned ; whence^ in this case, also, it necessarily follows, that m (\-\-xY z=\-[-px-\-qx'^-\-rx^-{-sx^-\-, &c. And if each side of this last expression be raised to the nth power, we shall have {l-\-x)'"z=[l-\-{px-{-qx'^-^rx^-\-sx'^-\-, &,c.)]" ; or, by actual involution, * l-{-mx-\-bx'^-\-cx^-{-, Sic. =:l-\-n{px-\-qx'^-{-, (fee.)-}-, &;c. Whence, by comparing the coefficients of x, on each side of this last equation, we shall have, according to (Art. 380), np=:m, or p=~ ; so that, in this case, n (i+xY =l-^'^x+qx^-\-rx^-\'Sx^'\-, See (2); n where the coefficient of the second term, and the several powers of x, follow the same law as in the case of integral powers 292 BINOMIAL THEOREM. 386. Lastly, if the index — be negative, it will be found n by division as above, that (1 + a;) »" or the equivalent expres- sions, —^=——-} = i_^a:-/^2-, &c. (3). (l-fa?)" 1-j — x-\-qx^, &c. where the series still follows the same law as before. 387. x\nd as the several cases, (1,2, 3), here given, are of the same kind with those that are designed to be expressed in universal terms*, by the general formula ; it is in vain, as far as regards the first two terms, and the general form of the se- ries, to look for any other origin of them than what may be derived from these, or other similar operations. 388. Hence, because {a-{-!ic)"'=a"' (l-\ — V, if there be as- sumed {a-^-xyzzza"" -{■ma'^-^x -j- Bx'^ + Cx^ + T>x\ &c. ; or which will be more commodious, and equally answer the de- sign proposed, 0+-:)"=^+^;©+M^hM^)'+"^'= w- it will only remain to determine the values of the coefficients Aj , Aj, A3, &c. and to show the law of their dependence on the index (m) of the operation by which they are produced. 389. For this purpose, let m denote any number whatever, whole or fractional, positive or negative ; and for -, in the above formula, put y-{-z ; then, there will arise ( IH — )'" = [1 4- (y+j»)]'"=[(l+y)-f- ;?]'", which being all identical expres- sions, when taken according to the above form, will evidently be equal to each other. 390. Whence, as the numeral coefficients, Aj, A^, A3, &c. of the developed formulae, will not change for any value that can be given to a and x, provided the index (m), remains the same, the two latter may be exhibited under the forms [l+(y4-^)]'"=l + A, {y+^)+A, (y+^)2+, &c.^ [(l+y)+^]'" = (l+y)"'+Ai^(14-y)'"-^ + A202(l+y)m-2_|,&c. And, consequently, by raising the several terms of the first of these series to their proper powers, and putting l+y=|)in the latter, we shall have &c. =p'»+Ajp'"-iz-f-Aap'«-2^2^A3i)«-3;23-f-, &c. BINOMIAL THEOREM. 293 l+A, A.y^ + ^k^'^ A3y3 + 4Ay + 3A3y + 6Ay H-lOAgyS &c. ;23+&C. (5). V+ &C. 391. Or, by orderinglhe terms, so that those which are af-| fectei with the same power of z may be all brought together, and arranged under the same head, this last expression will stand thus : ^2+ A3 + 4A,y 4- 10 Ay + 30A,y3 &c. In which equation it is evident, that both y and z are inde- terminate, and independent of the values Aj, A^, A3, &c. ; since the result here obtained arises solely from the substitu- tion of the sum of these quantities for - in equation (4). 392. Hence, as the first terms and the coefficients, or mul- tipl^rs of the like powers of z^ in these two expressions, are, in this case, identical, we shall have, by comparing the first column of the left-hand member with the first term of that on the right, H-Ajy + A2y24.A3y3+A4y*+ &:c. =;?'", which is an identity that verifies itself; since, by hypothesis, (l+y)'"=jo'", and, according to the general formula, (l+y)*" = 1 + A,y+A,y2+A3y3+ &c. 393. Also, if the second of these columns be compared in like manner, with the second on the right, there will arise the new identity, Aj-f2A2y-f3A3y2-f.4A4y3=A,p"»-i; which will be suffi- cient, independently of the rest of the terms for determining the values of the coefficients A^, Aj, A3, &;c. o"* A For since k^f^-^^k^—=:—^ (1 + k^y + k^'^-\-k^^ &;c.), the equating this series with the last, and multiplying the left-hand side by 1+y, will give ■[Aj + 2A,y+3A3y2-f&;c.] (1 + y)=A^+AjA,3/+AjA,y24.A, k^"^ -f &c. And, therefore, by actually performing the operation, and arranging the terms accordingly, we shall have A,+2A, y+3A3 + 2A, y2+4A, + 3 A3 y3+ &c. = Aj4-A,A,y + A,A,y2+A,A3y3+ &c. 394. From which last identity, there will obviously arise, by equating the homologous terms of its two members, the fol- lowing relations of the coefficients : 26* 294 BINOMIAL THEOREM. \ A,=A, 2A2=A,Aj— Aj uAg — AjA2"^2A2 4A^=AjA3— 3A3 nA =AjA _-("-!) A or A,: A,= A,=A. A.(A.-1) 2 A,(A,-2) A3(A,-3) A_-[A-(»-l)) And, consequently, as the coefficient Aj of the second term of the expanded binomial, has been shown to be equal, in all cases, to the index {m) of the proposed binomial, the last of these expressions will become of the form Aj=:m m{m — 1) • ^^=—2 _m{m — l).{m—2) _m(m — l).(m— 2).(m — 3) ^*- 2":3:4 m{m—l) .{m — 2).(m—3) .... [m—{n —l)] "" 2.3.4.5 7~^ ' where the law of the continuation of the terms, from A^ to the general term A„, is sufficiently evident. 395. Whence it follows, that, whether the index m be in- tegral or fractional, positive or negative, the proposed binomial (a+a:)'", when expanded, may always be exhibited under the form a"'(l'\--Y= &c.]; or (a-f 3?)'" = , . , , m(m — 1) - _ or la — j:)'" = a'" — ma^—^x-t—^—x a"*— ^a;^ — 2.4 Where it is to be observed, that the series, in each of these cases, will terminate at the (m+l)th term, when m is a whole positive number ; but if m be fraction|4|or negative, it will proceed ad infinitum ; as neither the facrors m— 1, m — 2, m— 3, &c. can then become =rO. 396. To this we may add, that in the two last instances here mentioned, the second term ( -) of the binomial must be less than 1, or otherwise the series, after a certain number of terms, will diverge, instead of converging. 397. It may also be farther remarked, that when a and x in these formulae, are each equal to 1, we shall have, agree- ably to such a substitution, (a4-n)"'=(l + l)'"=2'" = l+7n-f- m{m—\) m(m-'\) . {m—2) m{m — l) . (m— 2) . (m— 3) 2 "^ 2^3 ' 2.3.4 ' * &c., and (a—x)"'=.{l — l)'^ = 0'" = 0—l—m-\' m{m—l) __ m(m— 1) . (m—2) m{m^l) . (m— 2) . (m —3) 2 2^3 ' 2.3.4 — , &c. From which it appears, that the sum of the coefficients arising out of the development of the mth power, or root of any binomial, is equal to 2'" ; and that the sum of the coeffi- cients of the odd terms of the mth power, or root of a resi- dual quantity, is equal to the sum of the coefficients of the even terms. m 0—1 398. Finally, let m=0 ; then (a-|-a;)=a4'0Xa x -{• ?(t±V"V4-, &c., =a+0 . --f-0 . ^+, &c. 2 a a^ where it is evident that the series terminates at the first term (a°)i since the coefficient of every successive term involves for one of its factors ; therefore {a-{-xy=a^=l, (Art. 86). And, if a=a: ; then (a-^x)^=a^=l, that is, 0°=1. Hence, it 296 BINOMIAL THEOREM. follows, that any quantity, either simple or compound, raised to the power 0, is equal to unity or 1 ; and also that 0^ is in all cases equal to unity or 1 . 399. Although it has been observed, that O*' appears to admit of an infinity of numerical values ; because it is equal to ■§-, which is the mark of indetermination ; yet it is plain, from what is above shown, that 0'^ is only one of the values of 0™ g, which, in that particular case, where — =0*^=:-, is equal to unity. The intelligent reader is referred to Bonnycastle's Algebra, 8vo. vol. i^ Also, Lagrange's Theorie des FonC' tions Analytiques^ aWrLecons sur le Calcul des Fonctions. § III. APPLICATION OF THE BINOMIAL THEOREM TO THE EXPANSION OF SERIES. 400. The method of expanding any binomial of the form (a^^xY^ when m is any whole number whatever, has been already pointed out, and it has also been observed, that the series will always terminate, when m is a whole number : But when m is a negative number, or k fraction, then the se- ries expressing the value of (a-\-xY does not terminate. Let m='^, and substitute •" for m in the series then »■ , nor =za H r jiifi ri/ofi\ + ^ ^ ^ -( -g-j + J <&^c., which is a general expression for find- » ing the approximate value of any binomial surd quantity, •■ being either positive or negative, n and r any whole numbers whatever. Ex. 1. Find the approximate value of 3 ^ (i^+c^) or (i^-l-c^)^. Here a=^h^ n=l r=3 7W""3\P/ 363' n{n-\)(x\_\{-^)(c^\_ 3r2 W/ 2.32 w)~' 32^8' BINOMIAL THEOREM. 297 ¥.3r^ ~Wy~ 2.3.33 W)~3^l 5c» 69 &c. = &c. -3 y.6 5^9 1 Ex. 2. Find the value of (b+cf &c.). or (64-c)~^in a series. Here a=b~ c=x 71: 1 1 nix 2c (D=-f= J 2r ^__2_l)/c2> 2 <^c., and the sum of n terms of this series, is evidently equal to na ; there- ^0 ^ . or"— a r"— 1 1— r" fore S=-=:n«. Or, since ;— — a • r— «X— — = r— 1 r— 1 1— r a, [rn-i + r'^-z+r'—^.. +r+ l]=;tf X [l+r+r2-}«r3 . . ^n— i]^ which, in the case of 7 = 1, becomes a. [i + l + l + j &c.],and the sum of w terms of the series l + l + l-f-j for r, and changing the signs of the numera- 7* 306 PROPORTION AND PROGRESSION. tor and denominator, will become — ^ — - — - ; where it is r — 1 plain, that the term -j^ will be indefinitely small when n is incTefinitely great ; and consequently, by prolonging the se- ries, S may be made to differ from — — - by less than any as- signable quantity. 433. Whence, supposing the series to be continued indefi- nitely, or without end, we shall have in that case, Sr=— — - ; which last expression is what some call the radix, and others the limit of the series ; as being of such a value, that the sum of any number of its terms, however great, can never exceed it, and yet may be made to approach nearer to it than by any given difference. 434. If the ratio, or multiplier, r, be negative, in which case the series will be of the form a—ar-\-ar^ — ar^-\- _|-ar"—'^, where the terms are -f and — alternately, we, shall have S = — -z — . And if r be a proper fraction, -, as before, we shall have, for the sun^f an indefinite number of terms of the series a— a , a a , . _, ar" Ex. 1. Find the sum of the series, 1, 3, 9, 27, &c. to 12 terms. Here a=l,x ar" - a _lx3^^ — 1 _8V-' I r=3j''- ^--^—i- 3_i -—2— -=12 ; > 531441- 1_ 531440 ^ ^^,^ ; ~ 2 ~ 2 Ex. 2. Find three geometric means between 2 and 32. Here a=zi wi=3 ; ) and the means required are 4, 8, 16. Ex. 3. The first term of a geometrical progression is 1, the ratio 2, and the number of terms 10. What is the sum of the series? • Ans. 1023 PROPORTION AND PROGRESSION. 307 Ex. 4. In a geometrical progression is given the greatest term =1458, the ratio =3, and the number of terras =7, to find the least term. Aus. 2: Ex. 5. It is required to find two geometrical proportionals between 3 and 24, and four geometrical means between 3 and 96. Ans. 6 and 12 ; and 6, 12, 24, and 48. Ex. 6. Find two geometric means between 4 and 256. Ans. 16, and 64. Ex. 7. Find three geometric means between ^ and 9. Ans. J, 1, 3. Ex. 8. A gentleman who had a daughter married on New- year's day, gave the husband towards her portion 4 dollars, promising to triple that sum the first day o. every month, for nine months after the marriage ; the sum paid on the first day of the ninth month was 26244 dollars. What was the lady's fortune ? Ans. 39364 dollars. Ex. 9. Find the value of l+i4-7-f-J+ &C. ad infinitum. Ans. 2. Ex. 10. Find the value of H-J+t\+It4- <^c. ad infini- Ans. 4. § III. HARMONICAL PROPORTION AND PROGRESSION. 435. Three quantities are said to be in harmonical propor- tion, when the first is to the third, as the difference between the first and second is to the difierence between the second and third. Thus, a, h, c, are harmonically proportional, when a : c :-. a—h : h — c, or a : c : : h — a : c — h. And c, [since a(6— c)=c(a— 6) or ah={2a—h)c\, is a third harmonical proportion to a and h, when c=z -. 436. Four quantities are in harmonical proportion, when the first is to the fourth, as the difference between the first and second is to the difference between the third and fourth. Thus, a, 6, c, df, are in harmonical proportion, when a : d :: a—b : c—d, or a : d : : b — a : d—c. And d, [since a{c—d) = d(a'—b) or ac={2a—b)d], is a fourth harmonical proportional to a, b, c, when d=- 7. In each of which cases, it is obvious, that twice the first term must be greater than the second, or otherwise the pro- portionality will not subsist. 308 PROPORTION AND PROGRESSION. 437. Any number of quantities, c, h, c, d^ e, «fcc. are in har- monical progression, if a : c : : a — b : 6— c; b : d : : b—c : c — d ; c : e : : c— (f : d — e, &c. 438. The reciprocal of quantities in harmonical progression, are in arithmetical progression. For, if a, A, c, d, e, &c. are in harmonical progression ; then, from the preceding Article, we shall have ^c-\-ab^=i2ac ; dc-\-bc=2db ; ed-[-cd—2ec, &c. Now, by dividing the first of these equalities by abc ; the second by bdc: the third by cde ; &c., we have, — 1-- = a c 2112112. ^. . 11111- T ; t+:j=- ; -+-=:7 ; <^c. Therefore, -, -, -, -, -, &c. o b a c c e d abode are in arithmetical, progression. 439. An harmonical mean between any two quantities^ is equal to twice their product divided by their sum. For, if a, x, b, are three quantities in harmonical proportion, then, a : h : : a — X : X — b ; .'.ax — abz=ab — bx, and x= r. a-\-o Ex. 1. Find a third harmonical proportional to 6 and 4. Let x= the required number, then 6 : x :: 6—4 : 4 — x; .-. 2^ — 6x=2x, and x=2. Ex. 2. Find an harmonical mean between 12 and 6. Ans. 8. Ex. 3. Find a third harmonical proportional to 234 and 144. Ans. 104. Ex. 4. Find a fourth harmonical proportional to 16, 8, and 3. Ans. 2. ^ IV. PROBLEMS IN PROPORTION AND PROGRESSION. Prob. 1. There are two numbers whose product is 24, and the difference of their cubes : cube of their difference : : 19 : 1. What are the numbers ' Let x= the greater number, and y= the lesser. Then, a;y=24, and x^—y^ : {x—yY : : 19 By expansion, x^ — y^ : x^ — 3x'^y-^3xy^—y^ : : 19 .-. 3x^y—3xy^ : (x—yY : : 18 and, dividing by x^y, 3xy : {x—y)'^ : : 18 but xy=24: i .: 72 : {x-y)^ : : 18 PROPORTION AND PROGRESSION. 309 Hence, 18 (x-yf=12, or (a;— y)2=4 ; .•. X — y=2. Again, x'^^2xy-\-y'^=i 4, and Axy =96, .•.aj2+2ry+y2=i00, and a:-fy=10, but x—y=: 2, .•.a:=6, and y=4. Prob. 2. Before noon, a clock which is too fast, and points to afternoon time, is put back five hours and forty minutes ; and it is observed that the time before shown is to the true time as 29 to 105. Required the true lime. Ans. 8 hours, 45 ra-inutes Prob. 3. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6. Ans. 32, and 24. Prob. 4. What two numbers are those, whose difference, sum, and product, are as the numbers 2, 3, and 5, respectively ? Ans. 10, and 2. Prob. 5. In a court there are two square grass-plots ; a side of one of which is 10 yards longer than the other ; and their areas are as 25 to 9. What are the lengths of the sides ? Ans. 25, and 15 yards. Prob. 6. There are three numbers in arithmetical progres- sion, whose sum is 21 ; and the sum of the first and second is to the sum of the second and third as 3 to 4. Required the numbers. Ans. 5, 7, 9. Prob. 7. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean ex- ceeds the harmonical mean by 12. What are the numbers ? Ans. 234, and 104. Prob. 8. Given the sum of three numbers, in harmonical proportion, equal to 26, and their continual product =576 ; to find the numbers. Ans. 12, 8 and 6. Prob. 9. It is required to find six numbers in geometrical progression, such, that their sum shall be 315, and the sum of the two extremes 165. Ans. 5, 10, 20, 40, 80, and 160. Prob. 10. A number consisting of three digits which are in 310 PROPORTION AND PROGRESSION. arithmetical progression, being divided by the sum of its di- gits, gives a quotient 48 ; and if 198 be subtracted from it, the digits will be inverted. Required the number. Ans. 432. Prob. 11. The difference between the first and second of four numbers in geometrical progression is 36, and the diffe- rence between the third and fourth is 4 ; What are the num- bers ? Ans. 54, 18, 6, and 2. Prob. 12. There are three numbers in geometrical pro- gression ; the sum of the first and second of which is 9, and the sum of the first and third is 15. Required the numbers. Ans. 3, 6, 12. Prob. 13. There are three numbers in geometrical pro- gression, whose continued product is 64, and the sum of their cubes is 584. What are the numbers 1 Ans. 2, 4, 8. Prob. 14. There are four numbers in geometrical progres- sion, the second of which is less than the fourth by 24 ; and the sum of the extremes is to the sum of the means as 7 to 3. Required the numbers. Ans 1, 3, 9, 27. Prob. 15. There are four numbers in arithmetical progres- sion, whose sum is 28 ; and their continued product is 585. Required the numbers. Ans. 1, 5, 9, 13. Prob. 16. There are four numbers in arithmetical progres- sion ; the sum of the squares of the first and second is 34 ; and the sum of the squares of the third and fourth is 130. Required the numbers. Ans. 3, 5, 7, 9. CHAPTER XIV. ON LOGARITHMS. 440. Previous to the investigation of Logarithms, it may not be improper to premise the two following propositions. 441. Any quantity which from positive becomes negative, and reciprocally, passes through zero, or infinity. In fact, in order that m, which is supposed to be the greater of the two quantities m and n, becomes n, it must pass through n ; that is to say, the difference m — n becomes nothing ; therefore p, being this difference, must necessarily pass through zero, in order to become negative, or —p. But if p becomes —p, the fraction jf will become — ^ ; and therefore it passes through ^, or in- finity. 442. It may be observed, that in Logarithms, and in some trigonometrical lines, the passage from positive to negative is made through zero ; for others of these lines, the transition" takes place through infinity : It is only in the first case that we may regard negative numbers as less than zero ; whence there results, that the greater any number or quantity a is, when taken positively, the less is —a ; and also, that any ne- gative number is, a fortiori, less than any absolute or positive number whatever. 443. If we add successively different negative quantities to the same positive magnitude, the results shall be so much less according as the negative quantity becomes greater, abstract- ing from its sign. For instance, 8 — 1>8— 2>8— 3, &c. It is in this sense, that 0> —1 > — 2> ~3, &c. ; and 3> 0>-l>-2>-3>-4, &c. 444. Any quantity, which from real becomes imaginary, or reciprocally , passes through zero, or infinity. This is what may easily be concluded from these expressions, considered in these three relations, 312 ON LOGARITHMS. ^ I. THEORY OF LOGARITHMS. 445. Logarithms are a set of numbers, which have been compnted and formed into tables, for the purpose of facilitat- ing arithmetical calculations ; being so contrived, that the ad- dition and subtraction of them answer to the multiplicatioa and division of the natural numbers, with which they are made to correspond. 446. Or, when taken in a similar, but more general sense, logarithms may be considered as the exponents of the pow- ers, to which a given, or invariable number, must be raised, in order to produce all the common, or natural numbers. Thus, if af—y^ a"z=zy\ a"" =y^' , &c. ; then will the indices a:, X', x'\ &c. of the several powers of a, be the logarithms of the numbers y, y, y", &c. in the scale or system, of which a is the base. 447. So that, from either of these formulae, it appears, that the logarithm of any num.ber, taken separately, is the index of that power of some other number, which, when it is involved in the usual way, is equal to the given number. And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural numbers. 448. Let us suppose, in the equation a'z^y, at first, a:=0, we shall havey=l, since a^z^l ; to a?=z:l, corresponds yz=a. Therefore, in every system^ the logarithm of unity is zero ; and also, the base is the number whose proper logarithm^ in the sys- tem to which it belongs, is unity. These properties belong es- sentially to all systems of logarithms. 449. Let -{-X he changed into — x in the above equation, and we shall have 1 Now, the exponent x augmenting continually, the fraction — , if the base a be greater than unity, will diminish, and may be made to approach continually towards 0, as its limit ; to this limit corresponds a value of x greater than any assignable number whatever. Hence it follows, that, when the base a is greater than unity, the logarithm of zero is infinitely negative. 450. Let y and y' be the representatives of two numbers, X and x' the corresponding logarithms for the same base : we ON LOGARITHMS. 313 shall have these two equations, a'—y, and a"—y', whose pro- duct is a'-a^'^y-y', or a"^^' =yy\ and consequently, by the de- finition of logarithms, a;+,r'=:log. yy\ or log. yy' = \og. y-|- log. y\ And, for a like reason, if any number of the equations (f=y, af'—y', a'"—y'^^ &c. be multiplied together, we shall have a:-^+^^+^^''+^"'- =yy'y', &;c. ; and, consequently, a;+aj''4-^'^ &LC. = log. yy'y'\ &c. ; or log. yy'y'\ &c.=:log. y+log-y'+ log. y'\ &c. The logarithm of the product of any number of factors isj therefore, equal to the sum of the logarithms of those factors. 451. Hence, if all the factors y, y\ y'\&Lc. are equal to each other, and the number of them be denoted by m, the pre- ceding property will then become log. {y"') = m, log. y. Therefore, the logarithm of the mth power of any number is equal to m times the logarithm of that number. 452. In like manner, if the equation a'=y, be divided by a" ), and —^a~^ 494. An adfected cubic equation is said to be complete, when, after being properly reduced by the known rules, it is o[ the form x^-i-a'x'^-jrb'x-\-c' = 0. And it has already been shown, that every cubic equation of the above form, whose roots are r\ r", r", may be transformed into another deficient in its second term, by substituting y— Ja' for x in the given equa- tion ;- in which case the roots of the transformed equation will be r— ia' / — ^' r"—-^a' ; if, therefore, the roots of the transformed equation be known, the roots of the given equation will be known also. Hence the resolution of a cubic equation complete in all its terms will be -effected, if we can arrive at the resolution of it in the form x'^-^ax^=.b. In which a and b may be any positive or negative numbers whatever. 495. For this purpose, let there be taken x=.y-\- z, and the above equation, by substitution, will become y^4-3y^^ + 3y^'^ -\-z'^-]rai/-]raz=ib. Or, because 3i/^z-\'3i/z^r:z3i/z{y-{-z),a.r\dai/-\-az=za{i/-\-2)y it will be y^-{-z''^-\-{3i/z-\-a){y-^z)=:b. Now, as another unknown quantity has been introduced into the equation, another condition may be annexed to its solution. Let this condition be, that 3yz-\-a=:0, or z=——, in which case the transformed equation becomes y^'\'Z^=b, or by substitution y^ — ———b ; .*. y^ — by^—^a^ ; which equation solved, gives y—V [^^+ v(i^^+iT^^)l ' •■• ^^"^® z'^=:b—y'^, we have ;.=^[i6-V'(i&2+^i_a^]; and x=y+z=^\^b^^/(\b'^^ RESOLUTION OF EQUATIONS. 329 where by taking a and i in -f or —1, as they may happen to be, we have always one root of the transformed equation ; and this is the formula which is called the Rule of Cardan. 496. And since one value of x is now determined, the equa- tion may be depressed to a quadratic, from which the other two roots may be readily found. Ex. 1. Given x^ + 2xz=zl2, to find the values of a; Comparing this with the general equation, x^-\-axz=b, we have a = 2, and b— 12 ; therefore, by substituting these values for a and b in the above formula (I), x=l/ [GJrV[{^Q+M+V [6- V(36+^V)] =^ (6 + 6.024633)+^ (6 — 6.024633) =|/ (12.024633) + ;/ ( — .024633)zz:2.29 — .29=2. One root of the equation, therefore, is 2 ; divide x^-{-2x — 12 by x — 2, and the quotient is aj^ — 2a: + 6 ; '.'.x^ — 2a:+6=0, whose roots are Id: V — 5. Hence, the three roots of the equation are 2, 1 + -^/ — 5, 1 — ■\/ — 5, the two last of which are imaginary. Ex. 2. Given a;^— 48a;=128, to find the values of x. Here, by comparing this with the equation, (Art. 494), we have a=— 48, and ^>=^128 ; .•.a;=^ [64+ ^(4096 -4096)] + ^ [64 — -v/(4096-4096)] =^ (64 + 0)+^ (64-^0)=4+4 = 8. One root of the equation, therefore, is 8 ; divide x^ — 48a; — 128 by y — 8, and the quotient is x"-\-Sx-{-\Q\ .-.x'^-^r^x-^ 16 = 0, whose roots are — 4 + ; the three roots of the pro- posed equation are 8, —4, — 4, the two last of which are equal. 497. Hence we may infer, if a be negative, and 2Y<^^> taken with a positive sign, equal to 16^, or i&2_|__i_^3_-() . then two roots of the proposed equation are always equal. 498. But if a be negative, and -j^^^, taken with a positive sign, greater than ^6^ . \\\^QXi \b'^-\--^ja^ is a negative, quantity ; and consequently, ^/ {\i'^ -\- yjO?) is imaginary. Although the value of x cannot be obtained from Cardan's formula, (Art. 495), by the ordinary method, we are not, how- ever, to conclude, that the value of x, in this case, is imagi- nary ; since it may be proved to-be a real quantity after the following manner. 499. For this purpose, let \b be represented by a\ and ^/{\b'^•\•■^a'^), supposed imaginary, by b'-y/ ~\ ; then x=^^ (a^+5V-l)+^ (a'-^V-l)- Now, let ^ {a' -^b' ^/ -~\) and ^^ [a' — b' ■}/ — \) be expanded by means of the binomial theorem ; and since, by adding the resulting series together, 29* 330 RESOLUTION OF EQUATIONS. the terms involving the imaginary quantity -y/ — 1 destroy one another, we shall have 4 h"^ 105'* 1545's which is a real expression. When a' is greater than h' ; the above series converges rapidly, and a few of the first terms will give a near value of the root required. But if a' is less than h\ h'^ — 1 must be put for the first term of the hinomialy and a' for the second : See Clairaut's Algebra, Vol. II. Ex. 3. Given a;^ — 6a;=:5.6, to find the values of x. Com- paring this with the equation a;3+aa;=5, we have a=: — 6, and Z>z=5.6 ; therefore, x=^ [2.8+ V(7.84-8)]-f^ [2.8- V(7-S4-8)] =^ (2.8-f .4V-1)+^ (2.8-..4V-1.) Now, by comparing this value of x, with ^ (a''-|-5'y' — l)-f- ^ {of — y-y/ — 1), we have a'=2.8, and h'=iA ; .-. substituting these values for a' and h' in the above formula (2), xz=2'^ 2-8 (^ + 7^ - I49SS08' '^'=) = ^•«2(1+ .00227-.00002, &c.) =2.826345 nearly. Here, three terms of the series are sufficient, on account of its converging so rapidly, to give an approximate value of x, which is exact enough for all practical purposes. And, in fact, the value maybe still found more accurate by continuing the serieS'to five or six terms. Ex. 4. Given ;?6—3;s4— 2^^ — 8=0, to find the values of z. Let z^ = x-{-l, and the equation will be transformed into x^ — 5a:=12 ; .•. since «:= — 5, and5=rl2. x:=^ [6 + V(36- W)] + i/ [6- V(36- W)] =^ (6 + 5.6009)+^ (6~5.6009)==2.26376 + .73624=3. And, consequently, z'^=x-\-l=4:, or z=:^2. 500. Two roots of the proposed equation, therefore, are 2' and —2 ; divide z^ — 3z^—2z^—8 by z'^—4:, and the quotient is z*-{-z^-{-2 ; .•.;&* + ^^ + 2 = 0, whose roots are z = dtz ■\/{ — J± J-y/ — 7). Hence four roots of the proposed equation are imaginary. It may be observed that, in general, all equations, as z^"*-^ az^'"-{-bz'"-{-c=Oy may be reduced to one of the third degree, by putting z"'=x—^a. Ex. 5. Given a;3 + 30a:=117, to find the values of aj. Ans. x = 3, or — IJ-J-/ — 3. Ex. 6. Given a:3+9a!:=270, to find the values of x. Ans. x = Q, or — 3^6-/ — 1. Ex. 7. Given x^ — 36a;=91,to find the values of ar. Ans. a:=7, or -J + J-v/-^. RESOLUTION OF EQUATIONS. 331 Ex. 8. Given x^ — 6x'^-i-l0x—8=0, to find the values of a?. Ans. a;=:4, or l±-v/— I. Ex. 9. Given x^ — Sx — 4 = 0, to find the values of a?. Ans. a?=2.2; 1.1-4--/— .63 ; — 11— -v/~-^3, very Ticar/y. Ex. 10. Given a;3-|-24a;=250, to find the value of a:. Ans. 3;= 5.05. Ex. 11. Given ^3— 602+13^—12=0, to find the values of 2. Ans. ;?=3, or — |jt:i\/ — 7. Ex. 12. Given 2a;3 — 12x24-36a—44, to find the value of a:. Ans. 2.32748, &c. § III. RESOLUTION OF BIQUADRATIC EQUATIONS BY THE METHOD OF DeS CaRTES. 501. The same observation may be applied to biquadratic equations as was applied to cubic equations in (Art. 494), that, since the equation x^-\-a'x^-\-b'x'^-\-r'x-\-&'=.0, may be trans- formed into another which shall be deficient in its second term, and whose roots shall have a given relation to the roots of the given equation, the complete solution of a biquadratic equation will be eftected, if we can arrive at the solution of it in the form x^i-ax'2 + bx+c = ..*..(!); where a, b, c, may be any numbers whatever, positive or ne- gative. 502. In the solution of a |j|^quadratic equation, after the manner of Des Cartes, the formula x^-\-ax^-{-bx-{-cis suppos- ed to be the product of two quadratic factors, x^-i-px-\-q and x'^-{-rx-\-s, in which p, q, r, s, are unknown quantities. Or, which is the same, the biquadratic equation x'^-i-ax^-\-bx'\-c=0 is considered as produced by the multiplication of the two quadratics, (2) •. . . . x'^-hpx-\-q=0; x^-{-rx-{-s=0 . . . (3). 503. Hence, by the actual multiplication of the above two factors, we shall have x'^-{-{p-\-r)x^-{-{s-\rq-hpr)x^-]r{ps-\-qr)x-\-qsz=: a;* -\-ax^ -{-bx-\-c. And, consequently, by equating the coefficients of the like powers of a? in this last equation, we shall have the four fol- lowing equations, , p-\-r=zO ; s-{-q-{-prz=za ; ps-\-qr-=ib ; qs=zc. Or, if — p, which is the value of r in the first of these, be substituted for r in the second and third, they will become, s-\-q=a+p'^ ; s—q=- ; qs=c. * p Whence, subtracting the square of the second of these from 332 RESOLUTION OF EQUATIONS. that of the first, and then changing the sides of the equation, we shall have a2-j-2ap2_|_p4 _— 4^^j or 4c. And, therefore, by multiplying by "p^, and placing the terms according to the order of their powers, the result will give, ;)«-f2a;>4+(a2— 4c)p2-&2. ^ . (4)^ From which last equation, if there be put p^=^, we shall have,;^3+2a;52^(a2^4c)^z=62 (5). Hence, also, since ^+7=a+»2 and s—q= — , there will P arise, by addition and subtraction, ^=la+ip24._. q^la+lp'^-—; where j9 being known, s and q are likewise known. And, consequently, by extracting the roots of the two as- sumed quadratics, (2) and (3) ; or of their equals, x^-{-px-^ q—0, and x"^ — px 4- s z=zO ; we shall have «=-iP±V(ip'-«) (6); =o=},P±V{ip''-^) • • .• ■ (7); which expressions, when taken in + and — , give the four roots of the proposed biquadratic, as was required. 504. It may be observed, that whichever of the values of the unknown quantity, in the fpbic or reduced equation (5), be used, the same values of x will be obtained. 505. To this we may further add, that when the roots of the cubic, or reduced equation (5), are all real, then the roots of the proposed biquadratic are all real also. But if only one root of the cubic equation (1) be real, and, therefore, the other tivo imaginary ; then the proposed biquadratic will have two real and two imaginary roots. Ex. 1. Given the equation a:"^ — 3a;2-}-6a?4-8 = 0, to find its roots, or the values of x. Comparing this equation with a;*4-«^^+^^+c = 0j we have a= — 3, 6 = 6, and c — Q ; therefore, 23+20224- (fl2_4c)^_52^^3_622_|.23^ — 36 = 0. Let 2=y+2, and substitute y+2 for z in the latter equa- tion ; the»resulting equation is y^ — 3by — ^S = 0. Now, by comparing this last equation with x'^'\-ax=zb, we have a= — 35, and ^> = 98 ; therefore, (Art. 495), y=^ [49+iVl65856)]+|/ [49-1^(65856)] =^ (49 + 28.514)-!- ^ (49-28.514)=^ (77.514)+^ 20. 466) =4.264+2.736=7. RESOLUTION OF EQUATIONS. 333 Hence, z—y-\-2 — 9, and p^=z=9, or p= ±3 ; .-.(Art. 504), taking p:=3, s= — f +f +1==3 + 1=4, and 5"=— 14-|— 1=2. Consequently, by substituting these va- lues for p, q, and s, in the equations (2), (3), we shall have x2+3a:4-2 = 0, and a:2 — 3a;-h4^0; .-. a;= — |±i, and a^^fij^— 7 ; so that the four roots of the given equation are —1, —2, |-|- iV-7, ^-W-^- Ex. 2. Given x^ — 6x^—l7x-\-2l =0, to find the values of x. Ans. a::=3, or 1 ; or — 2±'\/ — 3. Ex. 3, Given the equation a:* — 4a;3 — 8x4-32=0, to find its roots, or the values of x. Ans. 4, or 2 ; or — l±y — 3. Ex. 4. Given the equation x'^—6x^-\^3x^-\-2x—\0 — 0, to find its roots, or the values of x. Ans. —1, or +5 ; or IrtV — 1- Ex. 5. Given a;* — 9a:''+30a;2— 46a: + 24=0, to find the roots, or values of a;. Ans. a^^l, or 4 ; 2 j^-y/— 2. Ex. 6. Given a;*H-16a;34-99x2-|-228a:4- 144 = 0, to find the roots, or values of x. Ans. a:= — 1, —3 ; or — Gi^ — 12. Ex. 7. What two numbers are those, whos^ product, mul- tiplied by the greater, is equal to 1 ; and if from the square of the greater, added to six times the lesser, the cube of the lesser be subtracted, the remainder shall be 8. Ans. -V2iV(l + V2), +V2iV(l-V2). § IV. RESOLUTION OF NUMERAL EQUATIONS BY THE METHOD OF DIVISORS. 506. Since the last term (v) of the equation {a)z=zx"'-\- Ax'^—^-i-Bx'^—^ .... Ta;+v=o, is equal to the product of all its roots, it is evident, that if any of those roots be whole numbers, they will be found among the divisors of that term. To discover, therefore, whether any of the roots of a given equation be whole numbers, we have only to find all the divi- sors of its last term, and substitute each of them, first with the sign -}- 3-i^d then with the sign — , for x, in the given equa- tion, such of them as reduce the equation to 0=0, will be roots of the equation. 507. Or, if the divisors of the last term should be too nu- merous, the equation may be transformed into another, that shall have its last term less than that of the former ; which is done by increasing or diminishing the roots by 1, or some other quantity. Ex. 1. Given ir^— a;^— 2a; + 8r=:0, to find the roots of the equation, or values of x. 334 RESOLUTION OF EQUATIONS. Here the divisors of its last term, are 1, 2, 4, 8 ; substitute 1, 2, 4, 8, and — 1, —2, — 4, — 8, for x in the given equation, and —2 will be found to be the only one of these numbers which gives the result ; — 2 therefore is the only integral root of the equation. Hence, oc-\-2 will divide x'^—x'^—2x-{- 8 without a remainder ; let this division be made, and the quotient being put equal to 0, we shall have x^ — 3a?+4 = 0, a quadratic equation which contains the other two roots. The solution of this quadratic gives xz=.^J^^y^^7 ; the three roots of the given equation, therefore, are ~2, f-hiV — 7, 508. The integral roots of any numeral equation of the Ivind above mentioned, may also be found, by Newton's Me- thod of Divisors, which is founded upon the following prin- ciples. Let one of the roots of the equation (a) = 0, be — a, or, which is the same, let the proposed equation be represented under the form (a'+«)p=0, where the binomial x-\-a denotes one of the divisors, or factors, of which the equation is com- posed, and p the product of the rest. Then, if three or more terms of the arithmetical series, 2, 1, 0, —1, — 2, be succes- sively substituted for x, the divisors of the results, thus ob- tained, will be a-\-2, a+1, a, a — 1, and a—^. Arid as these are also in arithmetical progression, it is plain that the roots of the given equation, when integral, will be some of the numbers in such a series. Whence, if a progression of this kind, whose common dif- ference is 1, can be found among the divisors of the results above mentioned, by taking one number out of each of the lines, that term of it which answers to the substitution of for X, taken in + or — » according as the series is increasing or decreasing, will generally be a root of the equation. Ex. 2. Givena;'^H-a;* — 14a;3 — 6a;2-f20a:+48 = 0, tofindthe roots of the equation, or values of x. Divisors. 1,2, 5, 10, 25, 50, 1 1, 2, 3, 4, 6, 8, 12, 24, 48, 2 1,2, 3, 4, 6, 9, 12, 18, 36, 3 Here the numbers to be tried are 2, 3, —4, all of which are found to succeed ; so that the equation has three integral roots ; namely, 2, 3, —4. The equation whose roots are 2, 3, —4, is {x-2) . (x-S) . {x-{-4) = x^--x^ — \4x-\-24==0Jet the given equation be divided by it, and the quotient is x'^-{- 2 =24. ( —8x=—8a — 8z } And by rejecting the terms z^-\-3az'^-{-2z^, (Art. 514), as be- ing small in comparison with z, we shall have a^-^2a'^ — 8a-\-3a'^z+4az-'8z=24: ; 24— a^— 2a2+8a 3 •••^= — ^-rrz — ^=^=-09 ; 3a^-t-4a — 8 31 and consequently x—a-{-2=:3.09, nearly Again, if 3.09 be substituted for a, in the last equation, we shall have _ 24-a^—2a^-\-8a _ 24—29.503629 — 1 9.0962 + 24.72 ^- 3a2+4a— 8 "" 28.6443-f 12.36-8 = .00364 ; and, consequently, x=:a + z=3.09 + .00364 == 3.09364, for a second approximation INDETERMINATE COEFFICIENTS. 339 And, if the first four figures, 3.093, of this number, be sub- stituted for a in the same equation, an approximate value of x will be obtained to six or seven places of decimals. And by proceeding in the same manner the root may be found still more correctly. ^ , , , r i Ex 2 Given 3a;5 + 4a;3— 5a;r=140,tofindthevalue ofa:by approximation. ^ Ans. .=2.07264 Ex 3 Given a:*-9a;3 + 8a;2-3x+4 = 0, to find the value of X by approximation. Ans. a:^ 1.1 14789. Ex. 4. Given a:3 + 23 3a:2-39x-93.3=0, to find the va- lues of X by approximation. Ans. a:=2.782; or -1.36; or -24.72 ; very nearly. Ex 5 Find an approximate value of one root of the equa- tion a:3+.x2 + ^^90. Ans. a:=.4.10283. Ex. 6. Given a;3+6.75a:2-f-4.5a:-10.25=0, to hnd the va- lues of X by approximation. Ans. a;z=:.90018 ; or -2.023 ; or -5.627 ; very nearly CHAPTER XVL ON INDETERMINATE COEFFICIENTS, VANISHING FRAC- TIONS, AND FIGURATE AND POLYGONAL NUMBERS. § I. ON INDETERMINATE COEFFICIENTS. 519. This is a species of investigation, which is frequently used for obtaining the development of certain fractional and other expressions, without having recourse to the operations of division, or the extraction of roots ; the method of per- forming which is as follows : RULE. Assume a series, or other expression, with unknown coeffi cients, for that which is required to be found ; then, having multiplied it by the denominator of the given fraction, or raised it to its proper powers, find the value of each of these 240 INDETERMINATE COEFFICIENTS. coefficients, by equating the homologous terms of the two ex- pressions, or putting such of them as have no corresponding terms, equal to 0, as the case may require. Example 1. Let it be required to find the development of , according to the above method. Assume , = A + Ba:+Ca:2+Da:3 + Ea;*, &c. Then, multiplying the right hand side of the equation by a'-^-Vx^^ and, transposing a, we shall have = Aa'-}-Ba' x^, &;c. x-\-Ca' + B6' And by putting the first term, and the coefficients of the several powers of x, each =0, there will arise the following equations : Hence, Aa^ — a=0 Ba^+AZ>'=0 Ca'-i-Bb' = Da'+C6'=0 &c. a a a'-\-b'x'^ a' or A=r B = -— A a a &c. ■A a; -Bx"^ a y -Ca:3, &c Where it is obvious, that each coefficient, in parting from the second inclusively, is equal to that which precedes it, multi- 7/ plied by — : which law renders it unnecessary to take a greater number of equations, or to push the calculation far- ther. Ex. 2. Required the development of ing to the same method. Assume a'-i-b^x-\-c'x^ accord- A + Bx-^Cx^-hDoc^ &c. a'-\-b'x-{-c'x^ Then multiplying the right hand side of the equation by a^'\-b'x-\-c^x^, and transposing a-{-bx, we shall have =Aa'4-Ba' x+Ca' 0:2+ Da' - a-\-Ab' + By + Cb' - b + A; 344 VANISHING FRACTIONS. whicli is the true value of the expression, in the case pro- posed. Ex. 3. Let there be taken, as another example of this kind, the equation • _ P(a:— g)" where P and Q are supposed to be certain functions, or com- binations of X, which do not become for the same value of x. Then taking x = a, the expression, according to this hypo- thesis, will become of the form P_xO__0 QxO"0" But by considering the indices wi, n, of the proposed frac- tion, under each of the relations m'^n, m = n, m<^n, we shall have, by division, the three following results : _ P(a: -«)>»-" _P P_ ^~ Q '^"Q'^-QCo^-a)"— * And consequently, by now taking x-=a, there will arise _PX0 _P P ^~~Q~'^~Q'^""QX0' "Whence, the value of the symbol --, in this case, will be no- thing, finite, or infinite, according to the conditions above mentioned. Ex. 4. It is required to find the value of the fraction * when X is equal to 1. Ans. 4. X — X Ex. 5. It is required to find the value of the fraction »P«» fi"* , when xz=:a. Ans. ma"^—^. Ex. 6. It is required to find the value of the fraction when X is equal to a. Ans. Sa^. x—a *.The value of this fraction was the cause of a violent controversy between Waring and Powell, in 1760, when these gentlemen were candidates for the mathematical professorship at Cambridge ; Waring maintaining that the value X — x^ of the fraction -rzi — ^^ equal to 4 when a:=l, and Powell, (or rather Maseres, who is commonly thought to have conducted the dispute,) that it was equal to-O. The idea of vanishing fractions first originated about the year 1702, in a contest between Varignon and Rolle, two French mathematicians of consider- able eminence, concerning the principles of the Differential Calculus, of which Rolle was a strenuous opposer. FIGUKATE AND POLYGONAL NUMBERS. 345 Ex. 7. It is required to find the value of ^ ^, when X is equal to a. Ans. (2a) . 1 -p» Ex. 8. It is required to find the value of the expression , 1 — X when X is equal to 1 . Ans. n. Ex. 9. It is required to find the value of the expression a '\/ doc ^'^x —-^ =:r , when X is equal to a, Ans. 8a. a— ^ax Ex. 10. It is required to find the value of the expression nx"^^-{n-[-\)x"+\ , . , — ^ — ^ , when X is equal to 1. Ans. n{n+\) Ex. 11. Tt is required to find the value of the expression '\/x—-y/a-\r-\/{^ — (^) I, • 1 . A 1 yr— — -^^ -y when X is equal to a. Ans. — :=.. V'(a:2-a2) ^ ^2a ^ III. ON FIGURATE AND POLYGONAL NUMBERS. 522. Figurate Numbers, are such as arise from taking the successive sums of the series of natural numbers 1, 2, 3, 4, 5, &c. ; and then the successive sums of these last, and so on : and polygonal numbers, are those which are formed of the successive sums of the terms of any arithmetical pro- gression beginning with unity ; each of ihem being usually divided into orders, according to the scale of their generation, which, as far as regards those of the first class, may be shown as foHows : Order. Fisurate Nurrvbers. Gen. Terms. 1 Figurate Nurrvbers. 1, 2, % 4, 5, 6, &c. 1, 3, 6, 10, 15, 21, &c. 1, 4, 10, 20, 35, 56, &c. 4 1, 5, 15, 35, 70, 126, vtc. &c. &c. Where it is to be observed that the general terms, here given, are so called, because if 1, 2, 3, 4, &c. be respectively sub- n n(n+l) TTT2- n(n +l)(n + 2) 12 3 ;^(/^+l )( n + 2)(/^ + 3) 1.2.3 4 &c. 346 FIGURATE AND stituted in each of them, for n, we shall obtain the several terms of the series. And if, instead of the natural numbers 1,2, 3, 4, &c. which give triangular numbers, an arithmetical series be taken, the common difference of which is 2, the sum of its successive terms will be the series of square numbers ; if the common difference be 3, the series will be pentagonal numbers ; if 4, hexagonal ; and so on : thus, Arith. Series. 1, 2, 3, 4, &LC. 3, 5, 7, &c. 10, &c. 1, 5, 9, 13, &c. 1, 4, 7, Ord. 1 Polygonal Numbers. Gen. Terms. Tri. 1, 3, 6, 10, &c. «(n-f 1) Sqrs. 1, 4, 9, 16, &Lc n(2n-f0) 2 Pent. 1, 5, 12,22, &c. n{Zn — l) 9 Hex. 1, 6, 15, 28, &c. n(4n-2) 2 &c. &c. &c. Where the number denoting any order, is the common differ- ence of the arithmetical series, from which the polygonal num- bers, belonging to that order, are generated. In like manner, if w^e take the successive sums of the se- veral polygonals thus obtained, and then the successive sums of these last, and so on, a great variety of other orders of series of this kind may be readily obtained. Hence, also, in general, if n be made to denote the number of terms of the series, a figurate of any m, may be expressed by the following formula. n n-fl n + 2 n4-(m — 1) 1^ 2 ^~3~ m • And supposing n to be the number of terms of the series, as before, a polygonal number of the order m — 2, or one that has the number of its sides denoted by w?, may be expressed {m—2)n'^~{m'-^)n by w-^ . So that figurate numbers, of any order, may be always de- termined, without computing those of the preceding orders, by taking as many factors, in the first of these formulae, by •substituting the number denoting that order for»j— 2, or the number of sides of the polygon, for w, and taking n equal to the term required. Example 1. Required the 15th term of the second order of figurate numbers, 1, 3, 6, 10, 15, &c. POLYGONAL NUMBERS. 347 Here m being =2, and n = 15, we shall have hy the first formula, !^l)=l£iH±i)=H2il^ =15 X8 =120. o X6 = 360 — 18 = 342, the term required. Ex. 3. It is required to find the 20th term of the 5th order of figurate numbers. Ans. 42504 Ex. 4. It is required to find the 1 3th term of the 9th order of figurate numbers. Ans. 293930. Ex. 5. It is required to find the 36th term of that order of polygonal numbers, which is denoted by a figure of twenty- five sides. Ans. 14526 CHAPTER XVII. ON INDETERMINATE AND DIOPHANTINE ANALYSIS. ^ I. ON INDETER-AIINATE ANALYSIS 523. When the enunciation of a question does not furnish as many equations as there are unknown quantities to be de- termined, the question is said to be indeterminate, being usually such as admit of a great variety of solutions ; although, when the answers are required only in whole positive numbers, they are generally confined within certain limits : the determina- tion of which forms a particular branch of Algebra, called Indeterminate Analysis, 348 INDETERMINATE ANALYSIS. To begin with one of the easiest questions ; let there be required two positive integer numbers, the sum of which is equal to 10. Let us- represent them by x and y ; then we have, or+y = 10, and a;=10 — y, where y is so far only determined that it must represent an integer and positive number. We may therefore substitute for it all integer and positive num- bers from 1 to infinity ; but since x must likewise be a posi- tive number, it follows, that y cannot be greater than 10 ; be- cause X must be positive ; and if we also reject the value a; = 0, we cannot make y greater than 9 ; so that only the following solutions can take place : If y = l,2, 3, 4, 5, 6, 7, 8, 9, then x:rr9, 8, 7, 6, 5, 4, 3, 2, 1. But the four last of these nine solutions being the same as the four fir.st, it is evident, that the question really admits only of five different soiutions. 524. As we have found no difficulty in this question, we may proceed to others, which require different considera- tions. Problem 1. To find the values of the unknown quantities X and y in the equation aa:4;5yr=c, or ax-{-by=c, where a and b are given numbers which admit of no common divisor, except when it is, also, a divisor of c. RULE. 525. I. Let wh. denote a whole or integral number, and reduce the equation to the form 3?=::; -^^^ z=wh. 2. Make — = -^ , by throwing all whole numbers out of it, till d and e be each less than a. 3. Find the difference, or sum, of -~ , or some mul- a OLV tiple of it, and — , or any other multiple of it that comes near the former, and the result will be a whole number. 4. Take this, or anymuliiple of it, from one of the fore- going fractions, or from any whole number which is nearly equal to it, and the result, in this case, will also be a whole number. INDETERMINATE ANALYSIS. 349 5. Proceed in the same manner with this last result, and 80 on, till the coefficient of y becomes equal to 1, or a ^ 6. Then will y^ap^^r^ where p may be any whole num- ber whatever, that makes y positive; and as the value of y is now known, that of a: may be found from the given equa- tion. Example 1. Given 2a:+3y=25, to determine x and y in whole positive numbers. • Herex=Hi^?-?^=12-y+i^. Hence, since x must be a whole number, it follows that must also be a whole number. 2 Let, therefore, ^ =wh.z=p ; then 1 —y—2p, or y = 1 — 2/>. And since a:=:12-y-h^^=12-(l-2j9)-f;) = 12+3;i-l, we shall have jr^i^ll-fSp, andy=l— 2p ; where p may be any whole number whatever, that will render the values of x and y in these two equations positive. But it is evident, from the value of y, that p must be either or negative, and, consequently, from that of x, that it must be — 1,-2, or — 3. Whence, j9 = 0,/) — — 1,J3 = — 2,p= — 3 ; , i Xz=\\, X=zS, X-=:zd. Xz=z2 \ then < 1 o c ^ \y— l,y=3,y = 5,y = 7; which are all the answers in whole positive numbers that the question admits of. Ex. 2. Given 21a;-4-17y=r2000, to find all the possible va- lues of X and y in whole numbers. Here x=i ^=n95H ---^ — wh.\ or omitting the 95, — —~-=ich. ; Zl 1 u i:,- . 21y . 5-17y 4y+5 , consequently, by addition, -^,-H oT 21 — ^^^'^^ *i "^y-^^ « 20y4-25 , , 4 + 20y , Also. JL__x5 = -J^.= l+-^=.A.; 31 92 75 58 41 24 4 25 46 67 88 350 INDETERMINATE ANALYSIS. or, by rejecting the whole number 1, -~zzz.wh. . ^ » y. . 21y 4 + 20y w— 4 And, by subtraction, —f —-^—^———wh.^-p ; Z\. 21 21 whence i/ = 21p-{-4, . 2000-17y 2000-17(21^+4) ^„ ,^ Whence, if p be put = 0, we shall have the least value of y =: 4, and the corresponding, or greatest, value, of a; —92. And the rest of the answers will be found by adding 21 continually to the least value of y, and subtracting 17 from the greatest value of x ; which being done, we shall obtain the six following results : 7 109 These being all the solutions the question admits of. Ex. 3. Given 19a; = 14y— 11, to find x and y in whole numbers. _j 14y-ll J , 19y . Here x= — ^— — =zwh., and -~-=vm. ; V u t, • 19y 14y-ll 5y+ll , whence, by subtraction, — -^- ^-— — = — z=wn. 5y-f 11 20y-|-44 „ , y + 6 Also, J^X4=-^-=y+2+^=»A.; and by rejecting y+2, which is a whole number, ^ =wh.=p\ .-.^==19^— 6, and _ 14y--ll _ 14(19;?- 6)-ll_ 266jP-9 5_^^ ''-' 19 - r9 - 19 -i4p-5. Whence, if p be taken =1, we shall have a? = 9 and y=13, for their least values ; the number of solutions being obviously indefinite. 526. When there are three or more unknown quantities, and only one equation by which they can be determined, it will be proper first to find the limit of that quantity which has the greatest coefficient, and then to ascertain the diff'erent va- lues of the rest, by separate substitutions of the several values of the former, from 1 up to the extent required, as in the fol- lowing example. Ex. 4. Given 3x + 5i/-^7z = \00, to find all the different values of x, y, and z, in whole numbers. Here each of the least integer values of x and y are 1, by the question ; whence it follows, that INDETERMINATE ANALYSIS. 351 100-5-3 100 — 8 92 ,„, Consequently z cannot be greater than 13, which is also the limit of the number of answers ; though they may be con- siderably less. By proceeding, therefore, as \i\ the former rule, we shall have 100— 5y-7;2 ^^ ^ , l-2y-^ X — -^ r=:33— y— 2^H f r=to/<.; and by rejecting 33— y — 2^, \—2y—z , ^y \—1y—z y-\-\~z — 3 — =^^-' "^Y+— T— =^^-3— =«'^-=^- Whence, y = 3j9 + ^ — 1 ; and, putting J9 = 0, we shall hav« the least value of y=-z — 1 ; where z may be any number from 1 up to 1 3, that will answer the conditions of the ques tion. When, therefore, 2^=1, we have y=0, 100-7 ^, and a; = — - — =31- And by taking ^ = 2, 3, 4, 5, y=:9, 20, 31,42. Ex. 10. Given 17x-f-19y+2l2r.T=400, to find all the an- swers in whole numbers which the question admits of. Ans. 10 different answers. Ex. 11. Given 5ar-f-7y-h 11^^=224, to find all the possible values of a:, y, and z, in whole positive numbers. Ans. The number of answers is 59. Ex. 12. A person bought as many ducks and geese, to- gether, as cost him 28^. ; for the geese he paid 4s. Ad. a piece, and for the ducks 2s. 6d. a piece ; what number had he of each? Ans. 3 geese and 6 ducks. Ex. 13. How many gallons of spirits, at 12^., 15^., and 18 jr. a gallon, must a rectifier of compounds take to make a mixture of 1000 gallons, that shall be worth 17 shillings a, gallon? Ans. lllj at 12.?., 111^ at 15^., and 777J at 18;?. PROBLEM. 528. To find such a whole number, as, being divided by other given numbers, shall leave given remainders INDETERMINATE ANALYSIS. 353 1. Call the number to be determined a:, the numbers by which it is to be divided a, &, c, &c., and the given remain- ders/, g, h, &LC. 2. Subtract each of the remainders from a?, and divide the differences by a ; and there will arise ^, —^ ' , 6lc a a a = whole numbers. x—f 3. Put the first of these fractions — —=p, and substitute a the value of x, as found from this equation, in the place of x in the second fraction. 4. Find the least value o{ p in this second fraction, by. the last problem, which put =zr, in the place x in the third frac- tion. 529. Find, in like manner, the least value of r, in this third fraction, which put =s, and substitute the value of a;, in terms of s, in the fourth fraction, as before ; and so on, to the last ; when the value of x thus found,* will give the whole number required. Example 1. It is required to find the least whole number, which, being divided by 17, shall leave a remainder of 7, and when divided by 26, shall leave a remainder of 13. Let x=z the number required. Then and — — -= whole numbers. jp 7 And, putting =p, we shall have x=llp-\-l ; which value of a:, being substituted in the second fraction, gives wh'. 17;? 4-7-13 _17p-6 26 26 „ . . ^ . , 6o 17p — 6 9»4-6 But It IS obvious that -—- ^— — =-%^ — =^wh. ; 9p^& ^ 27/;+18 ,/?+18 , And by rejecting p, there remains --— — =t/?A.=r ; therefore p=26r— 18 ; where, if r be taken =1, we shall liave p=iQ. And consequently a-— 17;?-^7=17x8 + 7=143, the num- ber required. 31* 354 INDETERMINATE ANALYSIS. Ex. 2. To find a number, which, being divided by 6, shall leave the remainder 2, and virhen divided by 13, shall leave the remainder 3. Ans. 68. Ex. 3. It is required to find the least whole number, which, being divided by 39, shall leave the remainder 16, and when divided by 56, the remainder shall be 27. Ans. 1147. Ex. 4. It is required to find the least whole number, which, being divided by 11, 19, and 29, shall leave the re- mainders, 3, 5, and 10. Ans. 4128. Ex. 5. It is required to find the least whole number, which, being divided by each of the nine digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, shall leave no remainder. Ans. 2520. PROBLEM. On Compound Indeterminate Equations. 530. Equations of this kind, not higher than the second degree, which admit of answers in whole numbers, are chiefly such as consist of the products, or squares, of two unknown quantities, together with the quantities themselves ; being, usually, one of the four general forms given in the following rule. RULE. 1. If the equation be of the form cvi/=ax-\-bi/-^c, we shall have, for its solution in whole numbers, y=a-\ ; where X — X — b must be a divisor of ab-^c. 2. If the equation be of the form x'^-\-xi/ = ax-\-bi/-\-c, we shall have c+b{a^b) ^ y=_a;+a_J-t— — -^ ; where x — b must be a divisor o( c-\-b{a — b). 3. If the equation be x'^=y^-\-ay-j-by we shall have y— ^2 — 4j j^ — a a , , , , — f- , and x=--\-y — n ; where a and n must be even numbers, and n be so taken that 8w may be a divisor of a^ — Ab. 4. If the equation be a;2=ay2-f5y+c2, we shall have y = — , and a:=c4-ny ; where n must be some whole num- n^ — a # ber between -y/a and — -. ^ 2c INDETERMINATE ANALYSIS. 350 Example 1. Given a;y =42— 2a;— 3y, to find tlie several va- lues of X and y in whole numbers. Here, by the first form, a— —2, b=—3, and c==42, o . 6+42 „ , 48 whence y=-2 + -^3-=-2+-p3. Where it is plain, that x must be such a number, that, when added to 3, it shall be a divisor of 48. But the divisors of 48, that will give quotients greater than 2, are 16, 12, 8, 6, 4, and 2. And consequently the integral values of the two unknown quantities are a;=16 — 3, or 13 | =12—3, or 9 | =8—3, or 5 | =6—3, or 3 I =4—3, or 1. s-.- -l?-2 12 ,or 2 =?-..o.4 48 ~"6 " -2, or 6 _48 4 ■ -2, or 10. Which are all the answers in whole positive numbers that the question admits of. Ex. 2. Given x^ = ij'^-\-207/, to find the values of x and y in whole positive numbers. Here, by the third form, a =20, and b =0, 400 . n— 20 50 n , • whence, y=~^ — I ^ — = ho""!^' ^^^ x=lO + i/—n. Where it is plain, that n must be some even number which is a divisor of 50. But the only number of this kind, that will give positive re- sults, is 2. •.y=5^+l_10=16, and a:=10 + 16-2=24. Ex. 3. Given x^ = 5y^-~l2i/-\-64f to find the values of x and y in whole positive numbers. Here, by the 4th form, a=5, 5 = — 12, and c=8. -12-16n 16(n— f) , ^ , Whence, y= — - — = — -^ r^, and x=8-\-ni/. ^ TL^ — 5 5 TV" Where it is plain, that n must be less than the -v/5, and great- er than \ ; which numbers are only 1 and 2. — 12-3 2 4—5 and :c=8+lx 7=15 | =8+2x44=96. -12-16 ^ ^ 1-5 =44. 356 DIOPHANTINE ANALYSIS. Ex. 4. Given x^-\-xi/=2x-\-3i/-\~29, to find the values of « and t/ in whole positive numbers. ^^^•^y=21,7. Ex. 5. It is required to find two numbers, such, that their product, added to their sura, shall be 79. Ans. ^3^^ 19, 15, 19. Ex. 6. Given x'^+orr/=z4x+3f/-{-27f to find the several va- lues of X and y in whole numbers. Ans J^= ^' 5, and 6, ^''^•2_19 Let 5a:4-l9=n2, or x= — - — ; 5 where n may be any number whatever greater than ^\ 9. Whence, if n be taken =5, 6, 7, respectively, we shall have DIOPHANTINE ANALYSIS. 357 25-1^ ,, 36-19 ^2 49-19 ^ the latter of which is the least value of jc, in whole numbers, that will answer the conditions of the questioti ; and conse- quently 5x4-19=5x6+19 = 304-19=49, a square number, as was required. 533. Ex. 2. Find a number such, that if it be multiplied by 5, and the product increased by 2, the result shall be a square. n^— 2 • Put 5x^-2=11^, then x = ; 5 if we assume n=2, then a;=f ; and by assuming other values for «, different values of x may be obtained. 534. Case 2. When the expression is of the form ■\/(hx-{' cx^) ; that is, when a — 0. Put y^(bx-\-cx^)=:nx; .'. bx+cx'^z=n'^x'^^ then b-\-cx=:n^x; whence x=— , and whatever value may be given to n in this expression, there will result a value of x that will make ^^(bx-^-cx^) rational. Example 1. .It is required to find an integral number, such, that it shall be both a triangular number and a square. It is here first to be observed, that all triangular numbers X" -\- X are of the form — - — ; and therefore the question is reduced to the making — — — , or it equal — a square. But since a square number, when multiplied, or divided, by a square number is still a square ; it is the same thing as if it were required to make 2x2-[-2a: a square. • TTl^X^ Let therefore 2x^-\-2x=z — ^, then dividing by x, andmul- n^ " tiplying the result by n^, the equation will become 2vP'x-\-2r? ^=.n^x ; and consequently __ 2n ^ Where, if n be taken =2 and wi=3, we shall have , x'+x 64 + 8 72 „^ *==8, and --2L-=_L_=: =36, 358 DIOPHANTINE ANALYSIS. for the least integral triangular number that is at the same time a square. 535. Ex. 2. Find a number such, that if its half be added to double its square, the result shall be a square. Let X denote the number, then we must have 'Zjc^-^-^x^l a square ^^ri^x"^, or 2x-\-\=n'^x \ therefore, a: = — , n be- 2n2__4 ing any number whatever : if n—2, then xz=: _.-, a 8 — 4 4 square number. 536. Case 3. When a is a square number, put it equal to <^2, and make 'y/(d'^-\-bx-\-cx'^) = d-\-nx\ i\i^.i\ d'^-\-dx-{-cx'^=z d'^-\'2dnx-\-n^x'^,ox b^cx = 2dn-\~n'^x ; and consequently, x=: 2dn—b ^ .. , ^ 2dn -. Or, if 5 = 0, x= -. c—n^ c—n?- Example L It is required to divide a given square number into two such parts, that each of them shall be a square number. Let c^— . the square to be divided, x'^'=. one of its square parts, a^ — x^zzz the other ; which is also to be a square. Put a^ — x'^={nx—af':=n^x'^—2anx-\-a^i and we shall have 2anx =. n^x^-^x"^, or n'^x-\-x=^2an ; and consequently x= 2an 2an 2an^ an'^-^a an^—a — „-r--. and nx—a= „ . . a= Hence, (-y— j^and ( 21 ?)^ ^^® ^^*® parts required; where a and n may be any numbers whatever, provided n be greater than unity. 537. Ex. 2. Find two numbers, whose sum shall be 16, and such, that the sum of their squares shall be a square. Let x=: one of the numbers, then 16— a; denotes the other, and we have to m^e x'^+{x — l6)^, or 2a.2— 32a?+256, a square. Put 2x'^ — 32x+256 = {nx—l6y = n^x'^ — 32nx-j-256 ; hence, 2x'^—32x=n^x^ — 32nx, and 2a;— 3'2=:»2a;— 32n ; . , 32(71—1) consequently x= — ^ — -— . n^—2 If we take n = 3, we shall have a'5=9^ ; therefore the two numbers are 9i and 6^. 538. Case 4. When c is a square number, put it =6^, and '\/{a+bx-\-e'^x'^)z=n-\-ex ; then, a-{-bx-\-e^x^ = n^ + 2cna?4- e^x^, or DIOPHANTINE ANALYSIS. 359 a — n^ 2en — b' Or,i{b=0,x: '""''' 2en Example 1. It is required to find the least integral number such, that if 4 times its square be added to 29, the result shall be a square. This being the same thing as to make 4a:2-f 29 a square • let 4a;2_|.29=(2x+n)2=4a:2 + 4na + n2. Then, 4na:+n2=29; or4na:=29-n*''; .-. a?=— ^^ ; where, if n be taken equal to 1, we shall have a?=— ^=^ = 7, which is the only in- tegral number that answers the conditions of the question. 539. Ex. 2. Find a number such, that if it be increased by 2 and 5 separately, the product of the sums shall be a square. Let x= the number, then we have to make (a:+2) (a:+5), or a:2+7a;-fl0, a square, which denote by lx—nY\ then! a;2+7rr4-10 = a:2-2«a'+w2, or ^2— 10 7a:+10=— 2na:+n Xrr- 7 + 2n' If we take n=4, we shall have x=z-. 5 540. Case 5. When neither a nor c are square numbers, yet if the formula can be resolved into two simple factors, (which it always can when b^ — 4c is a square, but not other- wise), the irrationality of it may be taken away, by putting V{a+bx-^cx-^)= ^\(d-^ex) {f^ gx)\=.n{d+ex) ; in which case we shall have {d-{-ex){f+gx)=:n^(d-\-ex)\ or f+gxz=n^{d+ex) ; J , dn^-f and consequently x= -. g — en^ Or, if (f=0, x= -4—, and if /=:0, x-. ^'^^ en^—g -" g-en^' The two factors above mentioned will be found by putting c+iaj4-ca?2=0 ; and solving this equation, we shall have '=-i+yyF^-4ac), and a:=-A--i V(62-4ac); or putting ^(b'^—Aac)=5^, the values of a; are 360 DIOPHANATINE ANALYSIS. and, consequently, (cx-\ — - — ), and (x-\ — - — ), are the factors required. Example. It is required to find such a value of a?, that •\/(6+13a:+6x2) shall be rational, and consequently 6+ 13a; 4-6a;2 a square. Let 6a:2+13a;-f-6=0; and solving this equation, we shall have x= — f , and x=z —J : therefore the two factors are 2x-f-3, and 3a?4-2. TTI? 7n^ Put (2a:4-3)(3a;+2)=— (3a:4.2)2, or 2a:+3=— (3a;+2), n n and consequently, by reduction, a;-=— — ^ — — j. Where it appears, that, in order to obtain a rational answer, m — must be less than f , and greater than f . Whence, if 7n = 6, and n=5, we shall have 3X25-2X36 75-72- 3 , . . , a;= =7777^ 7:=—;^^ the value required. 3X36-2X25 108—50 58' ^ Case 6. When neither of the foregoing will apply, if the formula can be resolved into two parts, one of which is a square, and the other the product of any two simple factors. Put '^(aA-hx-\-cx'^) ^ 'y/\{d+exY ^ (f-^gx){h+kx)\ = (d-{-ex)-\-n(f-\-gx) ; in which case we shall have {d-\-exY-\r (f+gx){h+kx) = {d+exf+2n{d-{-ex){f^gx) + n\f^gx)\ or h'{-kx=2n(d+ex)-\-n\f-\-gx) ; and consequently, xz=. -^ — tJ—t — r- k—n{2e-tgn) Or, if d=0, x=z^-—-- r. k — n{2e-{gn) Or, if the part in this case, which is found to be a square, be a known quantity, put ■y/{a-\-hx-]rCx^) = V\(^'^(^'^f^) (g-\-hx)=d-\-n{e-\-fx)\ ; then we shall have d;^^(e^fx){g+hx)=d'^-ir2dn{e-{-fx)-\-(e-^fxf or g-\-hxz=2dn-\-{e-{-fx), and consequently, by transposing and uniting the different e-\-2dn—g terms, a;= — r — -^ — . A-/ Example 1. It is required to find such a value of a:, that V(13a;2-f. 15a?+7) shall be rational, or 13d:2+15a;+7 a square. DIOPHANTINE ANALYSIS. * 361 Let this formula be separated into the two parts (1 —xf and 6+17a;+12a:2. Then, since 1 72—4(6 X 12), which is equal to 1, is a square, the latter part may be divided in the %tors 3a; +2, and Ax-\- 3 ; and consequently the original fprmula maybe represented ^ (l-x)2 + (2 + 3a:)x(3 + 4a:). Hence, putting V(13a;24-15a:+7) = y^j(l_a:)2-|-(2 + 3a:)x(34-4^)i=(l-a^) + «(2 + 3a:), we shall have (1 -xY+{2-\-'ix) X (3 + 4)=(l -xf+2n{\ -a:) X(2 + 3) + 7i2(2 + 3a:)2, or 3 + 4a' = 2n(l-a:) + n2(2 + 3a:) ; ^ . 2n4-2n2— 3 and consequently, by reduction, ^^ Ay^n—^n^ ' 2+2-3 1 Where, taking n=l, we have a?= ^ _ g = g ; 13 15 13 . 45 . 63 121 and 13a;2+15a:+7 = — 4-^-f7=-^4-9-+-9-=^- a square number, as required. Ex. 2. Find a value of x, such, that 2j:2 + 8a;+7 shall be a square. This expression, after a few trials, is found to be equiva- lent to (a; + 2)2+(a:+l)x(a; + 3), which being equated with j(^_f_2)-4t+l)|2 = (a:2_|-2)2-2n(x + 2) X (x+l) + n2(x+ 1)2, there results xj-^ = -2n{x-ir'^)'\-n\x-ir\) \ n2 — 4rt — 3 whence, a:r= . , ^ 5« i-f2n — w2 If we take n=3, we shall have a:=3, and 2x'^-\'Sx-\rl— 49, a square, as was required. PROBLEM II. 541. To render surd quantities of the form ^{a-]rhx-\-cx'^ -\-dx^) rational, or to find such values of x as will make a-^hx -fca:2-fda;3 a square. This problem is much more limited, and difficult to be re- solved, than the former, there being but a few cases of it that admit of answers in rational numbers. The rules for obtain- ing them are of such a confined nature, that when the un- known quantity has more than one value— which, however, is not often the case— the rest can onlv be determined one at 32 362 DIOPHANTINE ANALYSIS. a time, by repealing the operation with the value last obtain- ed, as often as may be found necessary. « RULE. 542. Case 1. When a=^, and 5=0, put the remaining part '^(cx'^'\-dx^) = nx, or cx^-\-dx^ = n'^x^ ; then we shall have 71— -C c-{-dx=n^ ; .'.x = — - — . a Where n may be any number whatever greater than the square root of c. Example 1. It is required to find such a value of a? that y(3x24-lla:^) shall be rational, and consequently Saj^+H*' a square. Let ■x/{3x'^-{-\lx^) = nx, or 3x^+11 x^ = n'^x^. Then, by dividing, we shall have 3-j- IIx^ti^. fl2 3 And consequently x=—- — ; where n may be any number, positive or negative, that is greater than -y/S. Taking therefore, n=2, 3, 4, 5, &c. respectively, we shall 1 6 13 22 have ^=T-r> tt, tji ttj or 2, the last of which is the least in- tegral answer which the question admits of. Ex. 2. Find a number such, that if three times its cube be added to twice its square, the sum shall be a square. Here we must make 3x^-\-2x'^ a square ; let n'^x^ be the square, then 3a:-|-2 = n-; 2 If we take n=3, we have x=:3, the number required. 543. Case 2. When a is a square number, put it equal to e^, and make ■\/(e'^-\-bx-\-cx^-\-dx^)z=€-{-—-Xf or e^-{-bx -\-cx'^-\- b b^ dx^={e+---)x = e'^-{'bx+ -t-%^' Hence, cx^-\-dx?=-—;rx. and by division and reduction Note. The assumed root e-\-—x is determined by first tak- ing it in the form c+nar, and then equating the second term DIOPHANTINE ANALYSIS. 363 of it, when squared with the second term of the original for- b inula; in which case n will be found^:^^' Example 1. It is required to find such a value of x, that l_|_2a;--j;2-f a;3 shall be a square. Here, 1 being a square, let i+2x-x'^'^x^ = (l+xf=l-\' 2x+x'^ ; then, we shall have x^-x'^ = x'^, or 3:3=2^2 ; and con- sequently a:=32, and l+2x — x'^-\-x^=z\ +4 — 4 + 8=r9, a square as required. Ex. 2. Find such a value of x as will make the expression 3x^ — 5x'^ + 6x+4: a square. Put 3a;3-5a:2+6a:+4 = (fx-f-2)2 = |a;2 + 6x+4, then, 3x3-5a;2=:fx2, or 3x-5 = |; .-. a:=?f , which being substituted in the proposed expression, makes it equal to [-^j ^ PROBLEM III. 544. To render surd quantities of the form y/{a'\-bx'\-cx'^ -\-dx^-{-ex'^), rational, or to find such values of x as will make a-\-bx-\'Cx'^-{'dx^-{-ex^ a square. RULE. Case 1. When o is a square number, put it =/2, and make p + bx + cx''-\-dx^-i-ex^=.{f+yx+^^^-^^x^) =P+bx-{- ^^ + 87^"" "^^ + ""64/^ ' then since the first three terms on each side of the equation destroy each other, we shall have — ^-pr^ — a;*i ^g^^^ x'^ = ex^-{-dx'^ ; and therefore, by division and reduction, 64dp-8bpi4cp -b^ ^~ {'icf^—bY — ^'^cp ' which form fails when any two of the* coefficients b, c, d, are each =0. Example 1. It is required to find such a value of x, that l—2x-\-2x- — Ax^-\-bx^ shall be a square. Here, the first term being a square number, let 1— 2a;4-3a;2 4a:3+5a;*=(l-a;— 1:2)2=. 1—2x4-3x2 -2a:3 + x*. Then, since the first three terms on each side of the equa- tion destroy each other, we shall have 5a:*— 4a;=^=x*— 2x3; 364 DIOPHANTINE ANALYSIS. .'.x=^, and consequently l—2x-\-3x^—4x^-^5x^=:zl — l-^^ — 5+tV— tV » which is a square number, as was required. Ex. 2. Find such a value of x that we may have 22x*— 40a;3 o — 40a;^+64a:4-16 a square. Ans. ~. 545. Case 2. When c is a square number, put it =_g-2, and mdkeg^x*-\-dx^-\-cx'^-\-bx+a=(gx'^-{ x+ %T ) =g^^* +c?a;3 4-ca;2+^^-!p^!l^+il!^^=^!l! ; then, we shall have (4c^2_^2y2_64a^6 64^6—+ Qj. 646o-6_8%2(4c^2_^2)- which form also fails in the same case as the former. Example 1 . It is required to find such a value of Xj that — 2-\-2x — x'^—2x^-\-4:X^ shall be a square number. Let4a:*-2a:3— a;24-3a:— 2=(2aj2_ia,__5_)2^4^4_2a;3-a;2 + T6^+2¥¥ » ^^^"^ '^^^ shoW. have "6x—2=^x-\-^-fQ\ .'.x 537 ' — 688* Ex. 2. It is required to find such a value of x, that 4a;*+ 4x^-{'4x'^-\-2x — 6 shall be a square. Vut 4x^-\-4x^'\-4x^-h2x-e = {2x^ + x-h^f = 4x'^-{-4x^-\-4x^~\-%x^-fQ, and we "have 2x-6^?,x+j%; .'.x=.l3l. 546. Case 3. When the first and last terms are both squares, put a=f^ and e^ig"^, and make f^-{-bx-{-cx'^-\-dx^-\-g^x*= (/ + ^a:+^a:2)2^y-2+j^+(2/o-+^^)2^24.^^34.^,2^4. then, since the second terms, as well as the first and last, on each side of the equation, destroy each other, we shall have f(bg-fd) '■ And because g is found in the original formula only in its second power, it may be taken either positively or negatively ; and consequently we shall also have ,_i*!-/W^_±£). ■o that this mode of solution furnishes two diflferent answers. DIOPHANTINE ANALYSIS. 365 Example 1. It is required to find such a value of a?, as shall make l-{'3x-\-7x'^—2x^-\'4x* a square. Let l + 3a;+7x2—2x3 + 4ic4 = (l+fx-f 2x2)2== 25 l + 3i:+— a:2-f6a;3+4a;*; 4 25 3 .-. 6a:3+— a:2=:7a;2-.2a;3, and x=—. 4 o<^ Ex. 2. It is required to find such a value of of, as shall make 16— 24a? +4 Jc^ — 6x3 4- a;* ^ square. Letx*— 6a:3 + 4a;2-24a;-}-16 = (a:2— 3a:— 4)2= x*—6x^-\-x^+24x-{-\6, and there results 4x2— 24x=a:2^24x, or 4x— 24=a?+24 ; .•.x=l6. PROBLEM IV. 547. To render surd quantities of the form ^ {a-\-bx-{-cx'^ -\-dx^) rational, or to find such values of x as will make a-\- bxi-cx'^-\-dx^ a cube. Case 1. When a is a cube number, put it =e^, and take h ^2 ^3 e3j^bx+cx^-\-dx^={e+—-xy=e^-{-bx-\-:^x^ + ^;^x^;ihen b^ ^2 we shall have 6?a?3-f-cx2=— -x^-f-— -x2 ; or, by dividing by x2, and reducing the terms, 9e3(3ce3— 52) 21[de^x-^21ce^=.¥x-\-9h'^e^ : whence x=.-^ — -'. Example 1. It is required to find such a value of x, as will make the formula l+x-4-x2 a cube. Let l-fx-|-x2 = (l+^x)3 = H-x-|-lx2+^Lx3, or x2z=:ix24-2V^^ » •'• 3: = 18, and consequently, l+a?+a:2=l + 18+324 = 343 = 73, a cube number, as was required. Ex. 2. It is required to find such a value of x that will make the formula 2x-^ + 3x2 — 4x+8 a cube. Let2x3 + 3x2-4x+8 = (— Jx+-2)3= — 2T*3+fx2— 4x+8, and we have 2x3-f 3x2=:— Jyx34-|x2, or 2x4-3= -Jy^+f; .-. x=-fj. Ex. 3. It is required to find such a value of x, as to make the formula 3x34-2x4-1 a cube. Ans. ^. 548. Case 2, When c? is a cube number, put it =/3, and 32* 366 DIOPHANTINE ANALYSIS. take a^hx+cx^-\-px^ = (_i_+/,,)3^_^+^^ 4. ,^2 ^ j^ac^ ; then we shall have a + ^a;= — -—-\-—x ; .•. a;=: 27/^ 3/3 27ff/6-c3 9/3(c2-3i/^)' Example 1. It is required to find such a value of a: as will make ISS + Sx^-fa:^ a cube. Let 1334-3x2+a:3 = (14-a:)3=l-f 3a: + 3a;2-f.a;3; and since the two last terms of this equation destroy each other, there will remain 1 + 3j:=:133, or 3a;=133 — 1 = 132 ; whence a; = i|2r=44, and consequently 1334-3a;2+a;3=92025 = (45)3, a cube number, as was required. Ex. 2. It is required to find such a value of a; as will make the formula Sa:^ — 4a;2-|-2a — 12 a cube. Let Sx^ - 4a:2+2a; - 12 = (2a: - ^f^^x'^-Ax'^+lx-^^, and we have 2x—\2 — '^x-\--^-j\ . -, 325 . . X— 3g^ . Ex. 3. It is required to find such a value of a? as will make the formula a:^ — 3a;2-f a; a cube. Ans. x=^. 549. Case 3. When a and d are both cube numbers, let them be put z=ze^ and/^^ and make e^-{-bx-\-cx^-\-f^x^=:[e-\- fxYz=ie'^-\-3fe'^x-{-^ef'^x^'-\-f^x'^ \ then, we shall have hx^cx"^ = ^fe^x-r^ef^x'^ ; /. xz=i- — ^^2» which formula may be also c — 6ej resolved by either of the two first cases. Example 1. It is required to find such a value of a?, that 8 + 28a; + 89x2 — 125a;3 shall be a cube. Let 8 + 28a:-f89x2-125a:3z:z(2-5a:)3z=8-60a:+150a;2— 125a:3 . and, since the first and last terms of this equation destroy each other, there will remain 28a:4-89a:2= — 60a:-f- 150a;2; .-. 150a;— 89a:=28 + 60, or, 61a:=88, and a:=-||, the value required. And as this formula can, also, be resolved by the first or second case, other values of x may be obtained, that will equally answer the conditions of the question. Ex. 2. It is required to find such a value of x, that the for- mula S-\-Ax-\-^x'^-\-x'^ shall be a cube. Let 8 + 4a;+9a:2+a:3=(2H-ar)3=^8+12a:+6a;2+a:3, and we shall then have 9x2 + 4a:=6a:2+12a; ; .-. a: = 2f. PROBLEM V. On the Resolution of Double and Triple Equalities. 550. When a single formula, containing one or more un- known quantities, is to be transformed to a perfect power, DIOPHANTINE ANALYSIS. 367 such as a square or a cube, this is called in the Diophantine Analysis, a simple equality ; and when two formulae, con- taining the same unknown quantity, or quantities, are each to be transformed to some perfect power, it is then called a double equality, and so on ; the methods of resolving which, in such cases as admit of any rule, are as follows. Prob. 1. When the unknown quantity does not exceed the first degree, as in the double equality a-\-hx:= a square, and c^dxz=z a square. Let the first of these formula a-f-^a?— ^^, and the second c-\-dx=.n^ ; then, by eliminating x from each of these equa- tions, we shall have hu'^-\-ad — bc=zdt^, or bdu'^-{-{ad—bc)d = d'^t^ ; and since the quantity on the right hand side of this last equation is now a square, it is only necessary to find such a rational value of m, as will make bdu^-\-{ad — bc)d a square, which being done according to one of the methods already explained, we shall have xz= — - — . Example. It is required to find a number a;, such, that x-\- 128 and a:+192 shall be both squares. Here, let a:+ 128=: w^ and a:-f 192 = ^2 . • then, by eliminating x, we shall have w^ — 128 = ^2 — ^92 ; or w2-j-64z=^2. and, as the quantity on the right-hand side of the equation is now a square, it only remains to make tt2-f-64 a square ; for which purpose, put u'^-{-64=:{u-\-nY = 64 — n^ u'^-^2nu-{-n^, 2nu+n'^ = 64 ; whence, u=—— — ; or, taking 64 — 4 n, which is arbitrary, =2, we shall have u= — - — =15 ; and consequently, x = 7i2— 128=225 — 128 = 97, the number re- quired. 551. Prob. 2. When the unknown quantity does not ex- ceed the second degree, and is found in all the terms of the two formulae, as in the double equality, ax'^-\-bx= a square, and cx^+dx= a square. Let x=- ; then, by multiplying each of the two resulting equations by y"^, we shall have a-\-byz=i a square, and c-{-dy= a square ; from which the value of y, and consequently that of x, may be determined, as in Problem L But if it were required to transform the two general expres- 368 DIOPHANTINE ANALYSIS. sions then r/ 1, > and by multip. rB/B, ) and rB=:B, ) and rB/B. > Q. E. D. PROP. B. THEOR. If four magnitudes are proportionals, they are proportionals also when taken inversely. DEMONSTRATION. Let rA, A, rB, B be any four proportionals, we are to prove that A,.rA, B, rB will also be proportionals. A 1 The ratio of A to rA is —r-=- rA r t) 1 and the ratio of B to rB is -7^=- ; rtJ r and therefore A : rA : : B : rB. Q. E. D. PROP. C. THEOR. If the first be the same multiple of the second, or the same part of it that the third is of the fourth ; the first is to the second as the third is to the fourth. DEMONSTRATION. 1 . Supposing m to be any integer greater than unity, let mA the first be the same multiple of the second A, that mB the third is of the fourth B ; we are to prove that mA, A, wiB, B are proportioi^ls. The ratio of mA to A is —-—m, A TJ and the ratio of mB to B is — rj^=m, B therefore mA : A : : mB : B. 2. The letter m still denoting an integer greater than unity, let A the first be the same part of mA the second, that B the third is of mB the fofurth ; then we are to show that A, mA, B, mB are proportionals. A 1 The ratio of A to mA is — :-= — , mA m T> 1 and the ratio of B to mB is ^5^= ; mo m therefore A : mA : : B : mB. Q. E. D. PROP. D. THEOR. If the first be to the second as the third to the fourth, and 378 APPENDIX. if the first be a multiple, or part of the second ; the third is the same multiple, or the same part of the fourth. DEMONSTRATION. Any four proportionals being expressed by rA, A, rB, B ; 1. Let the first rA be a multiple of A, then it is to be proved that rB is the same multiple of B. Because rA is a multiple of A, it is evident that r is an in- teger greater than unity, and r being such an integer, rA,and rB are manifestly equimultiples of A and B. 2. If r A be a ^art of A, w^e are to show that rB is the same part of B. A 1 Because rA is a part of A, therefore — -=- must be an in- ^ rA r teger greater than unity ; but — , when reduced, is also equal to ", that is, to the same integer, and therefore rA, rB, are the same parts of A and B. Q. E. D. PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magni- tude ; and the same has th^ same ratio to equal magnitudes. DEMONSTRATION. Let A and B be any two equal magnitude, and C any other, we arfe to prove that A and B have each the same ratio to C, and that C has the same ratio to A and B. Because by hypothesis A=:B, therefore by division 7^=7^- *» that is, A : C : : B : C. Again, since by hypothesis A=B, C C therefore by division -7-= 5- ; A x5 that is, C : A : : C : B. Q. E. D. PROP. VIII. THEOR. Of unequal magnitudes the greater has a greater ratio to the same, than the less has : and the same magnitude has a greater ratio to the less, than it has to the greater. DEMONSTRATION. Let A and B be two unequal magnitudes, of which A is the greater, and let C be any magnitude whatever of the same Kind with A and B : it is to be shown that the ratio of A to C APPENDIX. 379 is greater than the ratio of B to C : and also that the ratio of C to B is greater than the ratio of C to A. 1 Because by hypothesis A>B, therefore, by division 7r>7i 5 that is, the ratio of A to C is greater than the ratio of B to C. 2 Because by hypothesis A7B, therefore B^A, C C and therefore by division we have ^7-jrt because the less the divisor of C is, the greater is the quo- tient ; and therefore the ratio of C to B is greater than the ratio of C to A. Q. E. D. PROP. IX. THEOR. Magnitudes which have the same ratio to the same magni- tude are equal to one another ; and those to which the sarnie magnitude has the same ratio, are equal to one another. DEMONSTRATION. 1. Let A and B have the same ratio to C, it is to be proved that A is equal to B. Because A and B have, by hypothesis, the same ratio to C, A B therefore we have the equality 7^=7^-1 and therefore by mul- tiplication A=B. 2. Because by hypothesis, C has the same ratio to A as to C C B, therefore we have the equality— =.g-, therefore, by divid- ing by C, and multiplying by A and B, we have A=B. Q. E. D. PROP. X. THEOR. That magnitude which has a greater ratio than another has to the same magnitude, is the greater of the two : and that magnitude to which the same has a greater ratio than it has to another, is the less of the two. DEMONSTRATION. 1 . Let A have to C a greater ratio than B has to C, it is to be proved that A is greater than B. A B Since the ratios of A and B to C, are 7^ and 7^, thereftJre by supposition jt^t^i ^^^ therefore by mul" tiplication A>B. 2. Here the ratio of C to B is greater than he ratio of C to A, and we have to prove that B is less than to A : 380 APPENDIX. C C We have, therefore, by hypothesis -ttZ-t-. li A. Since then C contains B oftener than C contains A, it is manifest that B must be less than A. Q. E. D. PROP. XI. THEOR. Ratios that are the same to the same ratio, are the same to one another. DEMONSTRATION. Let A be to B as C to D, and also E to F as C to D ; it is to be shown that A is to B as E is to F. A C Because A is to B as C to D, therefore, ^^r^^yrt a u ■p p for the same reason r=p=:—-; therefore r D ^~, that is, A : B : : E : F. Q. E. D. D r PROP. XII If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antece- dents taken together be to all the consequents. DEMONSTRATION. By Cor. 2. Def. 5. any number of proportionals may be expressed by rA, A ; rB, B ; rC, C ; Where rA, rB, rC, are the antecedents, and A, B, C, the consequents ; and we are to prove that as rA is to A, so is rA+rB+rC to A+B+C. rA The ratio of rA to A is expressed by -T-=^>and the ratio of A rA+rB+rC to A+B+C, by ^^^^i^^^=r ; and therefore A-j-Jb>-f-L' rA : A : : rA-frB+rC : A+B+C. Q. E. D. PROP. XIII. THEOR. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth has to the sixth. DEMONSTRATION, Let A, B, C, D, E, F be the first, second, third, fourth, fifth, and six magnitudes respectively. The ratios of A to B, of C to D, and of E to F ACE "®B~ ET'F' 4 APPENDIX. 381 .AC and since by hypothesis -o=|t-» and also-^>p-, A E therefore we have -u>^r-. ^ t^ t^ B r Q. E.D. Cor. And if the first have a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth ; it may be de- monstrated, in Hke maimer, that t^he first has a greater ratio to the second than the fifth has to the sixth. PROP. XIV. THEOR. If the first has to the second the same ratio which the third has to the fourth ; then, if the first be greater than the third, the second shall be greater than the fourth ; if equal, equal, and if less, less. DEMONSTRATIOISr. Let rA, A, rB, B, be any four proportionals. 1. Suppose rA/rB, then by division A/ B ; next, suppose rA=::rB, then by division A= B ; lastly, suppose rA / rB, then by division A/ B. Q. E. D. PROP. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have. DEMOrrfSTRATION. Let A, B, be any two magnitudes of the same kind ; and m being any integer greater than unity, let mA, mB, be equimul- tiples of A, B ; it is to be proved that A, B, mA, mB, are proportionals. The ratio of A to B is the numerical quotient rjr-, and the JD ratio of mk. to mB is — ^r-, which is reducible to -^\ therefore ma J3 the two ratios .rr-, —z^ are equal, and therefore Jb> mt) 1^ A : B ; : mA : mB. PROP. XVI. THEOR. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. 382 APPENDIX. DEMONSTRATION. We may express any four proportionals by rA, A, rB, B, and we are to demonstrate that the four rA, rB, A, B, will also be proportionals. rk The ratio of rA to rB is -j-, which, because the factor r is in both numerator and denominator, is evidently reducible to A ^ A ^ ; again the ratio of the third A to the fourth B is also -^ ; D D therefore, the two ratios, viz. of r A to rB, and of A to B, be- ing equal, we have rA : rB : : A : B. Q. E. D. PROP. XVII. THEOR. If magnitudes taken jointly be proportionals, they shall also be proportionals when taken separately ; that is, if two mag- nitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remain- ing one of the last two has to the other of these. DEMONSTRATION. By hypothesis we have A + B : B : : C + D •* D, and we are to prove that A : B : : C : D. A-f-B A Now the ratio of A+B to B is —-^-=^+\, D X> C-l-D C and the ratio of C + D to D is —L -=^+1 ; and since by hypothesis these two ratios are equal, therefore A . , C , , , A C , . we have ^ + 1 = |^ + I, consequently, — =^ ; that is, A : B : : C : D. Q. E. D. PROP. XVIII. THEOR. If magnitudes taken separately be proportionals, they shall also be proportionals when taken jointly ; that is, if tl^first be to the second as the third is to the fourth, the first ami se- cond together shall be to the second as the third and fourth together to the fourth. APPENDIX. 383 DEMONSTRATION. By hypothesis we have A : B : : C : D, and we are to demonstrate that A+B:B::C-i-D:D. Since the ratio of A to B is the same with that of C to D, therefore jt-=yti to each side of this equation add unity, and we have A , . C ^, . . A + B C + D j3-+l=^ + l, that IS, -j^=--^-; and therefore A-fB : B : : C + D : D. Q. E. D. PROP. XIX. THEOR. If a whole magnitude be to a whole, as a magnitude taken from the first is to a magnitude taken from the other, the re- mainder shall be to the remainder as the whole to the whole DEMONSTRATION. Let A, B, be the two whole magnitudes, and C, D, the mag- nitudes taken from them. So that by hypothesis A : B : : C : D, we are to prove that A : B : : A — C : B — D. A B By Prop. XVI. we have ^=Yr-, .A ^ B , ^ . A-C B-D consequently ^ — 1 =-- — 1, that is, — p— = — t=^ — ; By this last divide the first equation, and the equal -quotients are - — pr=T5- — f7 A — O D — D and therefore by mult, and div. ^=^3 — f^, r> B — U' thatis, A : B :: A~C : B— D. Q. E. D. ANOTHER DEMONSTRATION. Since by hypothesis A : B : : C : D, therefore by alternation, prop. XVI. A : C : : B : D, and by division, prop. XVII. A — C : C : : B — D : D, and by alternation. A — C:B — D::C:D, and therefore by prop. XL A — C : B — D :: A : B. Q. E. D. ANOTHER DEMONSTRATION. J?-! .\+C, and B-j-D, be the whole magnitudes, and C, D, iiifc magnitudes taken away, so that by hypothesis 384 APPENDIX. A+C : B + D :: C : D. And we are to show that A+C : B + D :: A : B. Since by hypothesis A+C : B + D : : C : D, therefore by prop. XVI. A + C:C::B + D:D, consequently by prop. XVII. A : C : : B : D, and therefore by prop. XVI. A : B : : C : D, therefore by prop. XI. A+C : B + D : : A : B Q. E. D ANOTHER DEMONSTRATION. Supposing r greater than unity, let rA, rB, be the two wholes, and A, C the magnitudes taken away, so that by hy polhesis, we have rA : rB : : A : C ; of course we have -^ = -, or- =-, whence C = B, and we have therefore only to show that rA : rB :: rA— A : rB— B ; rA A Now the ratio of rA to rB is -fr=T^ ; rB B , , /. . . T^ T^ • *'A — A (r— 1).A and the ratio of rA— A to rB— B, is -jr — -—- 77-5= r^ — o (r— 1).B =^ and therefore rA : rB : : rA-A : rB-B. Q.E.D PROP. E. THEOR. If four magnitudes be proportionals, they are also propor- tionals by conversion ; that is, the first is to its excess above the second as the third is to its excess above the fourth. DEMONSTRATION. Let rA, A, rB, B, be the four proportionals, we have to demonstrate that rA : rA— A : : rB : rB— B. r A r The ratio of r A to r A— A is rA-A T'-V and the ratio of rB to rB— B is - — rr-=: -^ rn — a T — 1 therefore rA ; rA— A : : rB : rB— B. Q.E.D APPENDIX. 385 PROP. XX. THEOR. If there be three magnitudes, and other three, which taken two and two have the same ratio ; if the first be greater than the third, the fourth will be greater than the sixth ; if equal, equal ; and if less, less. • DEMONSTRATION. Let the three first magnitudes be A, B, C, and the other three be D, E, F ; so that by hypothesis, A is to B as D to E, and B to C as E to F ; and it is to be proved that if A be greater than C, D will be greater than F ; if equal, equal ; and if less, less. A D Because A : B : : D : E, therefore r=r- = -^, i E and because B : C : : E : F, therefore ^=^ : , therefore by multiplication of fractions, AB_DE , . A _D^ FG ""EF' ^ ^^ ^^ C "F ' from which it is evident that when the quotient -- is greater than unity, the quotient — is also greater than unity ; that is, if A be greater than C, D is also greater than F ; in a similar manner it is shown that when A is equal to C, D is equal to F; and if less, less. Q. E. D PROP. XXI. THEOR If there be three magnitude^, and other three, which have the same ratio taken two and two, but in a cross order ; if the first be greater than the third, the fourth shall also be greater than the sixth ; if equal, equal ; and if less, less. DEMOr^STRATION. Let the three first magnitudes be A, B, C, and the other three be D, E, F, SO that A is to B as E to F, and B to C as D to E ; it is to be shown that if A be greater than C, D will be greater than F ; if equal, equal ; and if less, less. A F Since A : B : : E : F, therefore we have :r— =:=rj3^^™dbe- rJ r B D cause B : C : : D : E, therefore also 7^==- ; and therefore O Jb4 by multiplication, 3^ 386 APPENDIX. AB_DE A __D from which it is manifest, that according as the quotid .= -^ is greater than, equal to, or less than unity, the quotient ^ must also be greater than, equal to, or less than unity, and therefore if A be greater than C, D will be greater than F ; if equal, equal ; and if less, less. PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first shall have to thegfast of the first rank of magnitudes, the same ratio which the first of the others has to the last. N. B. This is usually cited by the words ex aquali, or ex (Bquo. DEMONSTRATION. Let the first rank of magnitudes be A, B, C, D, and the second rank be E, F, G, H, so that by hypothesis A is to B as E to F, B to C as F to G, and C to D as G to H ; we are to show that A : D : : E : H. A E Since A : B : : E : F, therefore we have — =r-^, o F R F in like manner we have —-=:—-, O G , C G ABC now multiply the quotients — , — , =- together, and also the E F G _, ^ ^ . ABC EFG quotients j^, ~, ^, and we have the equation g^=:— ^, which by reduction becomes rp— =— -, JJ rl and therefore A : D : : E : H. In like manner the truth of the proposition may be shown, whatever be the number of magnitudes. Q. E. D. PROP. XXIII. THEOR. If there be any number of magnitudes, and as many others which, taken two and two in a cross order, have ihe same ra« A G B ^H' B F C ~G* C E D -F' APPENDIX. 387 tio ; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words ex equali in pro- portione perturbata, or ex cequo perturbato ; that is, by equality in per tur bate proportion. • DEMONSTRATION. Let the first rank of magnitudes be A, B, G, D, and the other rank E, F, G, H, so that, by hypothesis, A is to B as G to H ; B to C as F to G, and C to D as E to F ; we are to prove, that A : D : : E : H. Since A : B : : G : H, therefore and because B : ^F : G, therefore, and because C : D : ; E : F, therefore ABC now multiply the quotients ^^ p-, ^ri together, and also the G F E ^ ^ , 1 . ABC GFE quotients — ' 7T-» p-, and we have the products uq^= hqF* which reduced, becomes .p-=:=j-, D Jti and therefore A : D : : E : H. In like manner we may proceed for any number of magni- tudes. Q. E. D. PROP. XXIV. If the first has to the second the same ratio which the third has to the fourth ; and the fifth to the second the same ratio which the sixth has to tke fourth ; the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth. • DEMONSTRATION. By hypothesis we have rA : A : : rB : B, and r'A : A : : r'B : B, in which rA is the first, A the second, rB the third, B the fourth, r'A the fifth, and r^B the sixth : r' denoting each of the two equal ratios when the fifth is divided by the second. and the sixth by the fourth ; and we have to show, that rA+r'A : A::rB + r'B : B. rA + r'A The ratio of rA-i-r'A to A is =r-{ /, 388 APPENDIX. rB-f-r'B and the ratio of rB+^'B to B is — =r-\-r^ ; D therefore, rA-fr'A : A : : rB + r'B : B. Q.E.D Cor. 1. If the same hypothesis be made as in the propo- sition, the excess of the first and fifth shall be to the second as the excess of the third and sixth to the fourth. CoR. 2. The prop, holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio which the corres- ponding one of the second rank has to a fourth magnitude. PROP. XXV. THEOR. If four magnitudes of the same kind be proportionals, the greatest and least of ihem together ar^reater than the other two together. DEMONSTRATION. Let the proportionals be rA, A, rB, B ; and let the iirst rA be the greatest : then since by hypothesis rA is the greatest, rXy A, therefore r/l. Again, since by hypothesis rA is the greatest, therefore rA 7" rB, and consequently A /^ B ; since then r is greater than unity, and A is greater than B, it is manifest that B is the least ; and we are to show that rA-(-B7^B-f-A Now because A — B — A— B, and »*Z1» therefore, by multiplication rA — rB/^A — B ; .to each side of this equation add rB+B, and we shall have rA-fB/A + rB. A similar mode of demonstration may be adopted, which- ever of the four proportionals be the greatest. ^ Q. E. D. PROP. XXVf. THEOR. If there be any number of magnitudes of the same kind, the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is equal to the ratio of the first to the last. DEMONSTRATION. Let the magnitudes of the same kind be A, B, C, D ; we are to prove that the ratio compounded of the ratios of A to B, of B to C, and of C to D, according to the definition of com- pound ratio, is equal to the ratio of A to D. APPENDIX. 389 Take any magnitude P, and let A be to B as P to Q, and A, B, C, D, B to C as Q to R, and C to D as P, Q, R, S ; R to S ; then by the definition of compound ratio, the ratio of P to S is the ratio compounded of the ratios of A to B, B to C, and of C to D ; and it is to be proved that the ratio of A to D is the same with P to S. Now because A, B, C, D, are several magnitudes, anj P, Q, R, S, as many others, which, taken two and two. in order, have the same ratio ; that is, A is to B as P to Q ; B to C as Q to R, and C to D as R to S ; therefore ex equali, prop. XXII. A : D : : P : S. In like manner the proposition is proved for any number of magnitudes. Q.E.D. PROP. XXVII. THEOR. If four magnitudes be proportionals according to the com- mon algebraic definition, they will also be proportionals ac- cording to Euclid's definition. DEMONSTRATION. Let the four rA, A, rB, B, be the proportionals according to our fifth definition ; that is, according to the common algebraic definition ; it is to be proved that the same four rA, A, rB, B, are proportionals by Euclid's fifth def. of the fifth book. Let m and n be any two integer^, each greater than unity, so that mrA, mrB, are aliy equimultiples whatever of the first and third ; and nA, wB, are any whatever of the second and fourth ; and the four multiples are therefore mrA, nA, mrB, wB ; Now the thing to be proved is, that according as the multiple mrA is greater than, equal to, or less than nA ; the multiple mrB will also be greater than, equal to, or less than nB. First let TwrA/wA, - then by division mryn^ and by multiplication mrBynB. Secondly, if nirA = nA, then mr=n, and therefore wirB=nB. Lastly, if ftrA^nA, then mr/^rij therefore mrB / nB. Q. E. D. 390 APPENDIX PROP. XXVIII. THEOR. If four magnitudes be proportionals by Euclid's fifth defini- tion, they will also be proportionals by the common algebraic definition. DEMONSTRATION. Let A', A, B', B, be any four magnitudes, such that m, n, being any integers greater than unity, and the equimultiples, mA^, mB\ being taken, and likewise the equimultiples nA, nB ; making the four multipyes mA', nA, mB', nB ; the hypothesis is, that if twA' be greater than nA, mB' is also greater than nB ; if equal, equal ; and if less, less : and it is to be proved that A' : A : : B' : B ; A'' B'' or, which is the same thing, that — =-p-. A' B' If — be not equal to ^g- , one of these quolients must be the A' B' greater ; first, let — be the greater, so that if :5-=:r, we may A' have j-=:zr+r' ; then the four quantities A', A, B', B, are equal to rA + rA, A, rB, B. Now, let m be such an integer greater than unity, that mr and mr' may be each greater than 2 ; and take n the next in- teger greater than mr, of course n will be less than mr~{-mr' ; and the four multiples mA', nA, mB', nB, become mrA+mr'A, nA, mrB, nB, By construction mr-\-mr'yn, and therefore mr A -f mr' A 7 nk ; But by construction mr<:^n, and therefore mrBnA, therefore, by hypothesis, also mh^ynB ; so that mB' is both greater and less than nB, which is impos- sible # A' B' It is manifest therefore that -j- cannot be greater than fr- ; A ii APPENDIX. 391 and in like manner it is shown that jr cannot be greater than ■J-; and tnereCore — =p-, that is A' : A . : B' : B. Q. E. D. Scholium. Thus we have shown, that if four quantities be proportionals by the common algebraic definition, they will also be proportionals according to Euclid's definition ; and con- versely, that if four quantities be proportionals by Euclid's de- finition, they will also be proportionals by the common algebraic definition ; and by a similar method of reasoning we may easily show, that when four quantities are not proportionals by one of these two definitions, they cannot be proportionals by the other definition. Thus it appears, that the two definitions are altogether: equivalent ; each comprehending, or excluding, whatever is comprehended, or excluded, by the other. THE END Ul o> O > 3 m CD J3 -2 (D O l>0 CD _A C CD H 00 D fi) C C r 3 3 (/) f^ . fi> c O 3r 0) c r T ex. ' i 11189 ^ mm^' r43F?*r>-tJ>:.;\'5?;*¥^,.:-->v^- =■ ^3*->i«