University of California Berkeley THE THEODORE P. HILL COLLECTION of EARLY AMERICAN MATHEMATICS BOOKS J, -y i4jj&^3^ /7 0-1 &c\ YOwV \V.WvVV^- ,V >-. ,v\> RECOMMENDATIONS THOMSON'S ABRIDGMENT OF DAY'S ALGEBRA, THE great excellence of DAY'S ALGEBRA has been so fully acknowl- edged by the public, through a period of nearly thirty years, that it would be superfluous to accompany it with any formal recommendation. : All that instructors will require to be assured is, that the present Abridgment, by Mr. James B. Thomson, faithfully presents the spirit and character of the original. I have examined it sufficiently to feel satisfied that such is the fact ; that while it presents to the young learner the science in a simple and attractive form, it surpasses most similar treatises in the aptness of its illustrations, the accuracy of its defini- tions, and the value and copiousness of its principles. DENISON OLMSTED. YALE COLLEGE, June 6, 1843. Speaking of the Abridgment of Day's Algebra, Prof. Silliman says, I have full confidence in the views expressed above bv Prof Olm- sted." We fully concur in the above recommendation of the Abridgment of Day's Algebra, from Prof. Olmsted. Prof. A. D. STANLEY, Yale College. J. L. KINGSLEY, C. A. GOODRICH, T. D. WOOLSEY, C. U. SHEPARD, T. A. THATCHER, Tutor J. NOONEY, D. POWERS, L. J. DUDLEY, P. K. CLARK, HAWLEY OLMSTED, Principal of Hopkins' Grammar School. A. N. SKINNER, Principal of Select Classical School for boys. L. A. DAGGETT, Principal of Select School for boys. NEW HAVEN, June 12th, 1843. Messrs. DURRIE & PECK, I have examined, by request, Thomsons' Abridgment of Day's Algebra, and take pleasure in commending it *to teachers, as having all the well known merits of the original, with such additional illustrations and exercises as adapt it to the capacities of the young, and the method of instruction in schools and acadamies. The fact that this Abridgment has received the approbation of Pres. Day himself, is a sufficient proof of its character. Respectfully yours, STILES FRENCH, Principal of the Collegiate arid Commercial School, at Wooster Place. I have examined, with pleasure, the Abridgment of Day's Algebra, by James B. Thomson, A. M. The accuracy of definition and clear- ness of illustration, which characterize the large work, are faithfully preserved in this ; and the addition of a large number of examples, and some changes in the arrangement, greatly enhance the value of the production. I am acquainted with no elementary work on the sub- ject so well adapted to the purposes of instruction, as this Abridg- ment, and shall gladly introduce it into my school. DANIEL D. TOMPKINS M'LAUGHLIN, Principal of a Classical School in New York city. NEW YORK, June 12th, 1843. Mr. J. B. THOMSON DEAR SIR, The merits of the Algebra, of which yours is an Abridgment, and my confidence in your knowledge of what was needed to adapt it to the use of schools and academies, and in your ability for the task you undertook, led me to expect that you would make the work what I find it to be. one eminently adapted to the wants of that large class of pupils in our schools, whose age and circumstances require a treatise more full in explanation, better furnished with examples for practice, and yet more limited in extent, than those in general use. Yours truly, WM. H. RUSSELL, Principal of New Haven Family School for 'boys. From the New York Tribune, June 12, 1843. Day's Algebra is by far the best work for beginners that has ever been published: and this Abridgment serves to adapt it still more per- fectly and completely to the wants and necessities of the young. The elements of mathematics are very clearly and plainly developed ; the definitions are simple and comprehensive, and the problems well adap- ted . to illustrate the principles taught. We commend this Abridg- ment to the attention of teachers. There is no reason why Algebra should not be much more generally, taught than it is at present ; it is eminently' useful, and this work will greatly facilitate its introduction even into our common schools throughout the country. ELEMENTS ALGEBRA, 1 BEING AN ABRIDGMENT OF DAY'S ALGEBRA, ADAPTED TO THE CAPACITIES OF THE YOUNG, AND THE METHOD OF INSTRUCTION, IN SCHOOLS AND ACADEMIES. BY JAMES B. THOMSON, A. M. NEW HAVEN: DURRIE & PECK. PHILADELPHI A S MITH & PECK. NEW Y o R K R OBINSON, PRATT & Co. B O S T O N C ROCKER & BREWSTER. 1843. Entered according to Act of Congress, in the year 1843, by JEREMIAH DAY and JAMES B. THOMSON, in the Clerk's Office of the District Court of Connecticut. N. B. The Key to this work will shortly be published for the use of teachers. An abridgment of LEGENDRE'S Geometry, by the same au- thor, will also soon be published for the use of schools and academies. Printed by B. L. Hamlen. PREFACE. PUBLIC opinion has pronounced the study of Algebra to be a desirable and important branch of popular education. This decision is one of the clearest proofs of an onward and sub- stantial progress in the cause of intellectual improvement in our country. A knowledge of algebra may not indeed be regarded as strictly necessary to the discharge of the common duties of life ; nevertheless no young person at the present day is considered as having a " finished education" without an acquaintance with its rudiments. The question with parents is, not " how little learning and discipline their children can get through the world with ;" but, " how much does their highest usefulness require ;" and "what are the lest means to secure this end?" It has long been a prevalent sentiment among teachers and the friends of education, that an abridgment of Day's Alge- bra, adapted to the wants of schools and academies, would greatly facilitate this object. Whilst his system has been deemed superior to any other work before the public, and most happily adapted to the circumstances of college students, for whom it was especially prepared ; it has also been felt, that a smaller and cheaper werk, combining the simplicity of language and the uyirivaled clearness with which the princi- ples of the science are there stated, would answer every pur- pose for beginners, and at the same time bring the subject within the means of the humblest child in the land. IV PREFACE. In accordance with this sentiment, such a work has been prepared, and is now presented to the public. The design of the work, is to furnish an easy and lucid transition from the study of arithmetic to the higher branches of algebra and mathematics, and thus to subserve the important interests of a practical and thorough education. Its arrangement, with but few exceptions, is the same as that of the large work. For the sake of more convenient reference, the division by compound divisors, and the bino- mial theorem, both of which were originally placed after mathematical infinity, are brought forward, the former being placed after division of simple quantities, and the latter after involution of simple quantities. The reason for deferring the consideration of compound division in the original, was the fact that some of the terms contain powers which it is impossible for pupils at this stage of their progress to under- stand. To avoid this difficulty in the present work, whenever a power occurs, instead of using an index before it has been explained, the letter is repeated as a factor in the same man- ner as in multiplication, and also in dividing by a simple quantity. (Arts. 80, 94.) Afterwards, under division of powers, copious examples of dividing by compound quantities which have indices, are given. As continued arithmetical proportion and arithmetical pro- gression are one and the same thing, they are placed con- tiguously in the same section. For the same reason continu- ed geometrical proportion and geometrical progression are placed in a similar' manner. Mathematical infinity, roots of binomial surds, infinite series., indeterminate co-efficients, composition and resolution of the higher equations, with equa- tions of curves, are subjects which belong to the higher and more difficult parts of algebra, and it has been thought ad- visable to omit them in the abridgment. Those who have PREFACE. Y leisure and are desirous of acquiring a knowledge of these" subjects, will find them explained with all the author's accus* tomed clearness and ability in his large work, to which they are respectfully referred. The similarity between the ope-* rations in addition, subtraction, multiplication and division of radical quantities, and those of the same rules in powers \ also between involution and evolution of radicals, and of pow-* ers, has been more fully developed, and the rules of both are expressed in as nearly the same language as the nature of the case would admit. It has also been attempted to illus- trate the "Binomial Theorem," on the principles of induc- tion ; the second method of completing the square in quad* ratic equations has been demonstrated ; and other methods of completing the square pointed out, which, so far as the author knows, are original. It was a cardinal point with the distinguished author of the large work, never to use one principle in the explanation of another, until it had itself been explained, a characteristic of rare excellence in school-books and works of science. This plan has been rigidly adhered to, in the preparation of the abridgment. After the principles have been separately ex- plained, and illustrated by examples, they have then been carefully summed up in the present work, and placed in the form of a general rule. This, it is thought by competent judges, will be found very convenient and useful both to teachers and scholars. By this means the peculiar advanta- ges of the inductive and synthetic modes of reasoning have been united, and made subservient alike to the pleasure and facility both of imparting and acquiring knowledge. As a guide to the attention of beginners to the more im portant principles of the science, a few practical questions are placed at the foot of the page. They are intended to be merely suggestive. No thorough teacher will confine himself 1* VI PREFACE. to the questions of an author, however full and appropriate they may be. From a conviction that the answers to prob- lems have a tendency to destroy rather than promote habits of independent thinking and reasoning in the minds of learn- ers, they have nearly all been excluded from the book. For the convenience of teachers and others, who may entertain different views upon this point, the answers are given in a Key, in which may also be found a statement and solution of the more difficult examples contained in the work. The formation of correct habits of study and of thought, together with the extermination or prevention of lad ones, . requires the utmost vigilance and skill on the part of teachers. They must insist upon thoroughness, upon " the why and ivherefore" of each successive step, or in most cases, their pu- pils will fall into superficial and mechanical habits, which are equally destructive of high attainments and future usefulness. To mold the youthful mind right, is an arduous and responsi- ble task ; sufficient to crush the jaded spirit and shattered nerves of a poorly paid teacher. Nevertheless it is a high and noble, as well as indispensable work. Every conscien- tious teacher therefore, who appreciates the importance of his profession, or is worthy to be entrusted with this responsible charge, will cheerfully devote his energies to the work, what- ever may be the sacrifice, or resign his trust to more faithful and able hands. " In mathematics as in war, it should be made a principle," says the author of the large work, " not to advance, while any thing is left unconquered behind. Neither is it suf- ficient that the student understands the nature of the proposi- tion, or method of operation, before proceeding to another. He ought also to make himself familiar with every step, by a careful attention to the examples." It is emphatically true in algebra, that " practice makes perfect." For this reason, the number of problems in the present work, has been nearly PREFACE. Vll doubled ; the most of those added are original, and are calcu- lated to make the principles of the science more familiar. The merits of DAY'S ALGEBRA, are too well known and ap- preciated to require any comment. The fact that it has been adopted, as a text-book, by so many of our colleges and higher seminaries of learning ; that during the last fourteen years more than forty large editions have been called for, affords sufficient evidence of the superior rank, which it holds in pub- lic estimation. With regard to the abridgment, it is fervently hoped, that all who have felt the want of a lucid introductory work upon this subject, will here find the fulfillment of their wishes. Those teachers who have used the large work in their colle- giate course or elsewhere, and who may have occasion to use this, will at least be saved from the inconvenience of unlearn- ing one set of rules, and of learning a new and perhaps an inferior set, a work by no means unfrequent, and of no small magnitude and perplexity. On the other hand, those schol- ars, who chance to use the abridgment in their preparatory course, will avoid the necessity of unlearning its rules and modes of operation in algebra, should they have occasion to use the large work in the subsequent part of their education. It has been the endeavor of the author to divest the study of algebra, once so formidable, of all its intricacy and repul- siveness ; to illustrate its elementary principles so clearly, that any school-boy of ordinary capacity, may understand and apply them ; and thus to render this interesting and use- ful science, more attractive to the young. With what suc- cess these efforts have been attended, it remains for his fellow teachers and an impartial public to decide. J. B. THOMSON. New Haven, May 20, 1843. NOTICE. HAVING been myself prevented, by impaired health, and official engagements, from preparing an abridgment of my Introduction to Algebra, I applied to Mr. J. B. THOMSON, to abridge the work, in such a manner as to adapt it to the demand and use of the higher schools and academies. I had confidence in his mathematical talents and attain- ments, and his practical knowledge, derived from several years' experience in teaching algebra, as qualifying him to make the abridgment proposed ; and I am gratified to find, on examination, that our design has been skilfully and satis- factorily executed. The abridgment, it is hoped, will be fa- vorably received, by those who approve of the original work. J. DAY. Yale College, May 29, 1843. CONTENTS. SECTION I. Page. Introduction, - - 13 Algebra defined and illustrated, - 13,14 Algebraic Notation by letters and signs explained, - - 15-20 Operations stated in common language, translated into algebraic language, - -21 Translation of algebraic operations into common language, 22 Positive and Negative Quantities, ... 23 Axioms, - - . ._*j- : " 25 SECTION II. Addition illustrated and defined, * - 4* .' 26 Adding quantities which are alike and have like signs, - 27 Adding quantities which are alike, but have unlike signs, - 28 General Rule of Addition, " * ' 29 SECTION III. Subtraction illustrated, &c., - 31 General Rule of Subtraction, - - 32 Proof, 33 m SECTION IV. Multiplication illustrated, &c., - - - - 35 Rule for the signs of the Product, 39 The reason why the product of two negatives should be affirma- tive, illustrated, - - - - 41 General Rule of Multiplication, .... 43 CONTENTS. SECTION V. Page. Division illustrated, &c., - - 44, 45 Signs of the Quotient, ---... 47 Division by Compound Divisors, - - 49 General Rule of Division, ..... 51 SECTION VI. Algebraic Fractions explained, &c., - - 53, 54 The eifect of the signs placed before the numerator and denomi- nator, also before the dividing line of a fraction, with their changes, &c., - .... 55^ 55 Reduction of Fractions, - - - 57-60 Addition of Fractions, ..... 61 Subtraction of Fractions, - - - - 63 Multiplication of Fractions, ..... 65 Division of Fractions, ...... 68 SECTION VII. Simple Equations illustrated, &c., .... 71 Reduction of Equations by Transposition, - -73 Reduction of Equations by Multiplication, ... 76 Clearing of Fractions, &c., ----- 76 Reduction of Equations by Division, removing co-efficients, &c., 78 Converting a proportion into an equation, and an equation into a proportion, - - - *.. - - 79,80 Substitution illustrated, &c., - - - - 80 General Rule for solving Simple Equations, 83 SECTION VIII. Involution, powers of different degrees, &c., - -91 Indices, Direct and Reciprocal Powers, - - - 92 To involve a quantity to any required power, - - . ? * 93 To involve a fraction, - :./'.' - - '-*><: 95 To involve a binomial, or residual quantity, - ':*,-*& - 97 CONTENTS. XI Page. Binomial Theorem illustrated, - 98 General Rule for involving binomials to any required power, - 101 Addition of Powers, ... 105 Subtraction of Powers, - - - 106 Multiplication of Powers, - ... JQ7 Division of Powers, ...... 1Q9 Examples of Compound Divisors with Indices, 111 Greatest Common Measure, - - 112 Fractions containing Powers, - - 114 SECTION IX. Roots illustrated, &c., - - ^ - 115 Powers of Roots, - 118 Evolution, the Rule, &c. - - - 120 The Root of a Fraction, - . ... 122 Signs to be placed before roots, ..... 122 : Reduction of Radical Quantities, .... 124 :" Addition of Radical Quantities, - - - 129 The similarity of this and the five following rules to the same rules in Powers, - - " A "^ 129 Subtraction of Radical Quantities, .... 131 Multiplication of Radical Quantities, - - - 132 Division of Radical Quantities, ... 137 Involution of Radical Quantities, - 140 Evolution of Radical Quantities, .... 142 SECTION X. Reduction of Equations*by Involution, .... 144 Reduction of Equations by Evolution, ... 145 Quadratic Equations, Pure and Affected, - - 150, 151 First method of Completing the Square, ... 153 Second Method of Completing the Square, - 156 Demonstration of the second method, &c., ... 157 Other methods of Completing the Square pointed out, 158,159 General Rule for reducing Quadratic Equations, - 163 Xll CONTENTS. SECTION XL Two Unknown Quantities, To Exterminate one of two unknown quantities by Comparison, To Exterminate, &c., by Substitution, - To Exterminate, &c., by Addition and Subtraction, Three Unknown Quantities, Four or more Unknown Quantities, SECTION XII. Ratio and Proportion, - ... Different kinds of Ratio, ... Proportion, its different kinds, &c., SECTION XIII. Arithmetical Proportion and Progression, Various rules, or formulas in Arithmetical Progression, SECTION XIV. Geometrical Proportion and Progression, Case 1. Changes in the Order of the Terms, Case 2. Multiplying, or Dividing by the Same Quantity, - Case 3. Comparing one Proportion with another, Case 4. Addition and Subtraction of equal Ratios, - Case 5. Compounding Proportions, :?.;,. Case 6. Involution and Evolution of the Terms, - Continued Geometrical Proportion or Progression, Various rules, or formulas in Geometrical Progression, SECTION XV. * Evolution of Compound Quantities, Extraction of the Square Root of Compound Quantities, SECTION XVI. Application of Algebra to Geometry, .... Miscellaneous Problems, ..... \ ALGEBRA. SECTION I. INTRODUCTI ON. ART. 1. ALGEBRA is a general method of solving problems, and of investigating the relations of quantities ly means of letters and sign's. I LLU STRATI ON. PROB. 1. Suppose a man divided 72 dollars among his three sons in the following manner : To A he gave a certain number of dollars ; to B he gave three times as many as to A ; and to C he gave the remainder, which was half as many dollars as A and B received. How many dollars did he give I to each ? 1. To solve this problem arithmetically, the pupil would reason thus : A had a certain part, i. e. one share ; B receiv- ed three times as much, or three shares ; but C had half as much as A and B ; hence he must have received two shares. By adding their respective shares, the sum is six shares, which by the conditions of the question is equal to 72 dols. If then 6 shares are equal to 72 dols., 1 share is equal to % of 72, t; viz. 12 dols., which is A's share. B had three times as many, viz. 36 dols., and C half as many dols. as both, viz. 24 dols. QUEST. What is algebra ? How solve Prob. 1 arithmetically ? 2 14 ALGEBRA. [Sect. I. 2. Now to solve the same problem by algebra, he would use letters and signs ; thus, Let x represent A's share ; then by the conditions, o?X3 will represent B's share ; and 4x--2 will represent C's share. Add together the several shares, or #'s ; thus, x-\-3x-\-2x=: 6x. Then will 6x=.72, for the whole is equal to all its parts ; and la?=12 dols. A's share ; 3^=36 dols. B's share ; and 2x=24 dols. C's share. PROOF. Add together the number of dollars received by each, and the sum will be equal to 72, the amount divided. In this algebraic solution it will be observed ; First, that we represent the number of dollars which A received by x. Second, to obtain B's share, we must multiply A's share by 3. This multiplication is represented by two lines crossing each other like a capital X. Third, to find C's share, we must take half the sum of A's and B's share. This division is denoted by a line between two dots. Fourth, the addition of their respective shares is denoted by another cross formed by a horizontal and perpendicular line. Take another example. PROB. 2. A boy wishes to lay out 96 cents for peaches and oranges, and wants to get an equal number of each. He finds that he must give 2 cents for a peach and 4 for an orange. How many can he buy of each ? Let x denote the number of each. Now since the price of one peach is 2 cents, the price of x peaches will be x\2 cents, or 2x cents. For the same reason xX4, or 4x cents will denote the price of x oranges. Then will 2x-\-4x, that is, 6x, be equal to 96 cents by the conditions, and Ix is equal QUEST. How by algebra ? How denote A's share ? How B's and C's ? What is the share of each ? In Prob. 2, how represent the number of each kind ? What represents the price of each kind ? The Ans. ? Arts. 1-7.] INTRODUCTION. 15 to ^ of 96 cents, viz. 16 cents, which is the number he bought of each. 2. QUANTITIES in algebra are generally expressed by letters, as in the preceding problems. Thus b may be put for 2, or 15, or any other number which we may wish to express. It must not be inferred, however, that the letter used, has no determinate value. Its value isjixed for the oc- casion or problem on which it is employed ; and remains un- altered throughout the solution of that problem. But on a different occasion, or in another problem, the same letter may be put for any other number. Thus in Prob. 1, x was put for A's share of the money. Its value was 12 dols. and re- mained fixed through the operation. In Prob. 2, x was put for the number of each kind of fruit. Its value was 16, and it remained so through the calculation. 3. By the term quantity, we mean any thing which can be multiplied, divided, or measured. Thus a line, weight, time, number, &c. are called quantities. 4. The first letters of the alphabet are used to express known quantities ; and the last letters, those which are un- known. 5. Known quantities are those whose values are given, or may be easily inferred from the conditions of the problem under consideration. 6. Unknown quantities are those whose values are not given. 7. Sometimes, however, the quantities, instead of being expressed by letters, are set down in figures. QUEST. How are quantities expressed in algebra ? What does each letter stand for ? Has the letter used no determinate value ? What is meant by quantity ? Give examples. Which letters are used to denote known quantities ? Which unknown ? What are known quantities ? Unknown ? Are figures ever used in algebra ? 16 ALGEBRA. [Sect. I. 8. Besides letters and figures, it will also be seen that we use certain signs or characters in algebra to indicate the re- lations of the quantities, or the operations which are to be performed with them, instead of writing out these relations and operations in words. Among these is the sign of addi- tion (+), subtraction ( ), equality (=), &c. 9. Addition is represented by two lines (+), one hori- zontal, the other perpendicular, forming a cross, and is called plus. It signifies " more," or " added to." Thus a-\-b sig- nifies that b is to be added to a. It is read a plus , or a added to #, or a and I. . 10. Subtraction is represented by a short horizontal line (_ ) which is called minus. Thus a Z>, signifies that b is to be " subtracted from" a ; and is read a minus J, or a less b. 11. The sign -|- is prefixed to quantities which are consid- ered as positive or affirmative ; and the sign , to those which are supposed to be negative. For the nature of this distinction, see Arts. 36 and 37. 12. The sign is generally omitted before ihejlrst or leading quantity, unless it is negative ; then it must always be writ- ten. When no sign is prefixed to a quantity, -|- is always understood. Thus a-\-b is the same as -\-a-\-b. 13. Sometimes both -f- and , (the latter being put under the former, -4-,) are prefixed to the same letter. The sign is then said to be ambiguous. Thus a-\-b signifies, that in cer- Q.UEST. What besides letters and figures are used in algebra? What is the sign of addition ? How read ? What does it signify ? How is subtraction represented ? What called ? What signify ? What sign have positive quantities? What negative? What is said as to the sign of the leading quantity ? When none is expressed, what sign is understood ? When both -J- and - are prefixed to the same letter, what is the sign called ? What does it show ? What are like signs ? What unlike ? Arts. 8-17.] INTRODUCTION. 17 tain cases, comprehended in a general solution, b is to be ad- ded to a, and in other cases subtracted from it. Obser. When all the signs are plus, or all minus, they are said to be alike ; when some are plus and others minus, they are called unlike. 14. The equality of two quantities, or sets of quantities, is expressed by two parallel lines =. Thus a-\-bd, signi- fies that a and b together are equal to d. So 8+4rz:16-~4 10+2=7+2+3. 15. When the first of the two quantities compared, is great' er than the other, the character ^> is placed between them. Thus d^>b signifies that a is greater than b. If the first is less than the other, the character <^ is used ; as a<^b ; i. e, a is less than b. In both cases, the quantity towards which the character opens, is greater than the other. 16. A numeral figure is often prefixed to a letter. This is called a co-efficient. It shows how often the quantity expressed by the letter is to be taken. Thus 2b signifies twice b ; and 9b, 9 times b, or 9 multiplied into b, The co-efficient may be either a whole number or a frac* tion. Thus f b is two thirds of b. When the co-efficient is not expressed, 1 is always to be understood. Thus a is the same as la ; i, e. once a. 17. The co-efficient may be a letter, as well as a figure. In the quantity mb, m may be considered the co-efficient of b ; because b is to be taken as many times as there are units in m. If m stands for 6, then mb is six times b. In 3abc, 3 may be considered as the co-efficient of abc ; 3 the co-effi- cient of be ; or Sab, the co-efficient of c. QUEST. How is equality represented ? How inequality ? What is a co-efficient ? What does it show ? When no co-efficient is express- ed, what is understood ? Js the cp-efficient always a whole number ? Is it always a figure ? fc 2* 18 ALGEBRA. [Sect. I. 18. A simple quantity is either a single letter or number, or several letters connected together without the signs -)- and . Thus a, aft, abd and 8ft, are each of them simple quan- tities. 19. A compound quantity consists of a number of simple quantities connected by the sign -f- or . Thus a-f-ft, d y, bd-{-3h, are each compound quantities. The members of which it is composed are called terms. 20. If there are two terms in a compound quantity, it is called a binomial. Thus a-{-b and a ft are binomials. The latter is also called a residual quantity, because it expresses the difference of two quantities, or the remainder, after one is taken from the other. A compound quantity consisting of three terms, is sometimes called a trinomial ; one of four terms, a quadrinomial, &c. 21. When the several members of a compound quantity are to be subjected to the same operation, they must be con- nected by a line ( ) called a vinculum, or by a parenthe- sis ( ). Thus a ft-f-c, or (ft-j-c), shows that the sum of ft and c is to be subtracted from a. But a -b-\-c signifies that ft only is to be subtracted from a, while c is to be added. 22. A single letter, or a number of letters, representing any quantities with their relations, is called an algebraic ex- pression, or formula. Thus a-\-b-\-3d is an algebraic ex- pression. 23. Multiplication is usually denoted by two oblique lines crossing each other thus X Thus aXft is a multiplied into b : and 6X3 is 6 times 3, or 6 into 3. Sometimes a. point is QUEST. What is a simple quantity? A compound ? If there are two terms, what is it called ? Three ? Four ? When several terms are subjected to the same operation, how is this shown ? What is an al- gebraic expression, or formula? In how many ways is multiplication represented ? First ? Second ? Third ? Arts. 18-26.] INTRODUCTION. 19 used to indicate multiplication. Thus a . b is the same as aXb. But the sign of multiplication is more commonly omitted, between simple quantities ; and the letters are con- nected together in the form of a word or syllable. Thus db is the same as a . b or aXb. And bcde is the same as bXc XdXe. When a compound quantity is to be multiplied, a vinculum or parenthesis is used, as in the case of subtraction. Thus the sum of a and b multiplied into the sum of c and d, is ^+b X c+3, or (a+l) X (c+d). And (6+2) X 5 is 8x5,or40. But 6+2x5 is 6+10, or 16. Whenthemarks of parenthesis are used, the sign of multiplication is frequent- ly omitted. Thus (x+y) (xy) is (x+y) X (*#) 24. When two or more quantities are multiplied together, each of them is called a factor. In the product aJ, a is a factor, and so is b. In the product a?X(a+wO, x is one of the factors, and a-\-m the other. Hence every co-efficient may be considered a factor. (Art. 17.) In the product 3y, 3 is a factor as well as y. 25. A quantity is said to be resolved into factors, when any factors are taken, which, being multiplied together, will pro- duce the given quantity. Thus 3ab may be resolved into the two factors 3a and &, because 3aXb is 3#Z>. And 5amn may be resolved into the three factors 5#, and m, and n. And 48 may be resolved into the two factors 2x24, or 3 X 16, or 4 X 12, or 6X8 ; or into the three factors 2X3X8, or 4X 6X2, &c. 26. Division is expressed in two ways : 1st. By a horizon- tal line between two dots -f-, which shows that the quantity preceding it, is to be divided by that which follows. Thus, a-^-c, is a divided by c. QUEST. What is a factor ? When is a quantity resolved into fac- tors ? Factors of Zab ? 5amn ? 48 ? In how many ways is division ex- pressed ? First ? 20 ALGEBRA. [Sect. I. 2d. Division is more commonly expressed in the form of a fraction, putting the dividend in the place of the numerator, and the divisor in that of the denominator. Thus - is a di- o vided by b. 27. When four quantities are proportional, the proportion is expressed by points, in the same manner as in the Rule of Three in arithmetic. Thus a : b : : c : d signifies that a has to b, the same ratio which c has to d. And ab : cd : : a-\-m : b-{-n, means that ab is to cd, as the sum of a and m, to the sum of b and n. 28. Algebraic quantities are said to be alike, when they are expressed by the same letters, and are of the same power : and unlike, when the letters are different, or when the same letter is raised to different powers.* Thus ab, Sab, ab, and 6ab, are like quantities, because the letters are the same in each, although the signs and co-efficients are differ- ent. But 3a, 3y, 3bx, are unlike quantities, because the let- ters are unlike, although there is no difference in the signs and co-efficients. So, x, xx, and xxx, are unlike quantities, because they are different powers of the same quantity. (They are usually written x, x 2 , and x 3 .) And universally if any quantity is repeated as a factor a number of times in one in- stance, and a different number of times in another, the pro- ducts will be unlike quantities ; thus cc, cccc, and c, are un- like quantities. But if the same quantity is repeated as a factor the same number of times in each instance, the products are like quantities, Thus aaa, aaa, aaa, and aaa, are like quantities, Q,UEST. Second ? The most common ? How is proportion ex- pressed ? What are like quantities ? Unlike f What kind of quanti- ties are 3aJ and Gab ? aa and acia ? na and aa ? xxx and xxx ? * For the notation of poiocrs and roots, see sections VIII, IX. Arts. 27-33.] INTRODUCTION. 21 29. One quantity is said to be a multiple of another, when the former contains the latter a certain number of times with- out a remainder. Thus 10a is a multiple of 2a ; and 24 is a multiple of 6. 30. One quantity is said to be a measure of another, when the former is contained in the latter any number of times, without a remainder. Thus 3b is a measure of 155 ; and 7 is a measure of 35. 31. The value of an expression, is the number or quantity for which the expression stands. Thus the value of 3-|-4 is 7 ; of 3X4 is 12 : of - 1 /- is 2 - 32. The RECIPROCAL of a quantity, is the quotient arising from dividing A UNIT by that quantity. The reciprocal of a is - ; the reciprocal of a-\-b is ; the reciprocal of 4 is -. d CL m I D TC 33. What is the algebraic expression for the following t statement, in which the letters a, 5, c, &c. may be supposed to represent any given quantities ? Ex. 1. The product of , 5, and c, divided by the difference of c and d, is equal to the sum of b and c added to 15 times h. A ( abc , . I-IKT c d 2. The product of the difference of a and h into the sum of Z>, c, and d, is equal to 37 times w, added to the quotient of b divided by the sum of h and b. 3. The sum of a and Z>, is to the quotient of b divided by c, as the product of a into c, to 12 times h. 4. The sum of a, 5, and c, divided by six times their pro- duct, is equal to four times their sum diminished by d. 5. The quotient of 6 divided by the sum of a and 5, is equal to 7 times rf, diminished by the quotient of 5, divided by 36. QUEST. When is one quantity a multiple of another? When a measure ? What is the value of an algebraic expression ? What is the reciprocal of a quantity ? 2 ALGEBRA. [Sect. I. y 34. What will the following expressions become, when words are substituted for the signs ? 6. ?l=abc-6m+ h a-\-c Ans. The sum of a and b divided by A, is equal to the product of , &, and c diminished by 6 times m, and increased by the quotient of a divided by the sum of a and c. 8. 9. a- 771 10 fl-fr , _ ' ' aw 35. At the close of an algebraic process it is often neces- sary to restore the numbers for which letters have been sub- stituted at the beginning. In doing this the sign X must not be omitted between the numbers, as it generally is between factors expressed by letters. Thus if a stands for 3, and b for 4, the product ab is not 34, but 3X4, i. e. 12. Suppose b ; c=2 ; d6 ; m=S ; and n=W. the value of the following algebraic expressions. 11. +a+mn=?^+3+8X 10=9+3+80=92. Ans. 19 ^ d Zed 771 ___ 7) 16. (a+c)X(-w)H -- ;- 77i d (c+b)X(m-d) nd n bc Arts. 34-38.] NEGATIVE QUANTITIES. 23 is. POSITIVE AND NEGATIVE QUANTITIES. 36. A POSITIVE or AFFIRMATIVE quantity is one which is to be added, and has the sign -j- prefixed to it. (Art. 11.) 37. A NEGATIVE quantity is one which is required to be SUBTRACTED, and has the sign prefixed to it. When several quantities enter into a calculation, it is fre- quently necessary that some of them should be added to- gether, while others are subtracted. If, for instance, the profits of trade are the subject of cal- culation, and the gain is considered positive ; the loss will be negative ; because the latter must be subtracted from the former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debt and the credit are distinguished by opposite signs. 38. The terms positive and negative, as used in the mathe- matics, are merely relative. They imply that there is, either in the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. But this opposition is not that of existence and non- existence, nor of one thing greater than nothing, and another less than nothing. For in many cases, either of the signs may be, indifferently and at pleasure, applied to the very same quantity ; that is, the two characters may change pla- ces. In determining the progress of a ship, for instance, her QUEST. What is a positive quantity? What sign has it? What is a negative quantity? What sign has it? In business transactions, how is the gain considered? Loss? How are the terms positive and negative used in mathematics ? Imply what ? 24 ALGEBRA. [Sect. I. easting may be marked -(-? and her westing; or the west- ing may be+, and the easting . All that is necessary is, that the two signs be prefixed to the quantities, in such a man- ner as to show, which are to be added, and which subtracted. In different processes, they may be differently applied. On one occasion, a downward motion may be called positive, and on another occasion negative. 39. In every algebraic calculation, some one of the quan- tities must be fixed upon to be considered positive. All other quantities which will increase this, must be positive also. But those which will tend to diminish it, must be negative. In a mercantile concern, if the stock is supposed to be positive, the profits will be positive ; for they increase the stock ; they are to be added to it. But the losses will be negative ; for they diminish the stock ; they are to be subtracted from it. 40. A negative quantity is frequently greater, than the pos- itive one with which it is connected. But how, it may be asked, can the former be subtracted from the latter ? The greater is certainly not contained in the less : how then can it be taken out of it ? The answer to this is, that the greater may be supposed first to exhaust the less, and then to leave a remainder equal to the difference between the two. If a man has in his possession 1000 dollars, and has contracted a debt of 1500 ; the latter subtracted from the former, not only ex- hausts the whole of it, but leaves a balance of 500 against him. In common language, he is 500 dollars worse than nothing. 41. In this way, it frequently happens, in the course of an algebraic process, that a negative quantity is brought to stand Q,UEST. How determine which quantities are positive ? Negative ? Is a negative quantity ever greater than a positive, with which it is connected? How subtract the former from the latter in such a case ? Give examples. Does a negative quantity ever stand alone ? What denote ? Arts. 39-43.] NEGATIVE QUANTITIES. - 25 alone. It has the sign of subtraction, without being con- nected with any other quantity, from which it is to be sub- tracted. This denotes that a previous subtraction has left a remainder, which is a part of the quantity subtracted. If the latitude of a ship which is 20 degrees north of the equator, is considered positive, and if she sails south 25 degrees ; her motion first diminishes her latitude, then reduces it to noth- ing, and finally gives her 5 degrees of south latitude. The sign prefixed to the 25 degrees, is retained before the 5, to show that this is what remains of the southward motion, after balancing the 20 degrees of north latitude. 42. A quantity is sometimes said to be subtracted from 0. By this is meant, that it belongs on the negative side of 0. But a quantity is said to be added to 0, when it belongs on the positive side. Thus, in speaking of the degrees of a thermometer, O-f-6 means 6 degrees above ; and 06, 6 degrees below 0. 9 AXIOMS. 43. An AXIOM is a self-evident proposition. 1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal. 2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal. 3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal. 4. If equal quantities be divided by the same or equal quantities, the quotients will be equal. 5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered. 6. If a quantity be both multiplied and divided by another, the value of the former will not be altered. QUEST. What is meant by subtracting a quantity from ? Added to 0? What is an axiom ? Name some. 26 ALGEBRA. [Sect. II. 7. Quantities which are respectively equal to any other quantity, are equal to each other. 8. The whole of a quantity is greater than a part. 9. The ivhole of a quantity is equal to all its parts. SECTION II. ADDITION. ART. 44. Ex. 1. John has x marbles and gains I marbles more. How many marbles has he' in all ? In this example we wish to add x marbles to b marbles. But addition in algebra is denoted by the sign +. Hence x-\-T) is the answer : i. e. John has the sum of x marbles add- ed to b marbles. 4 2. What is the sum of 3b dollars added to the sum of c dollars and/ dollars? By algebraic notation, 3b-{-c-\-f dollars is the answer. 45. The learner may be curious to know how many mar- bles there are in x-{-b marbles ; and how many dollars in 3J-J-C-)-/ dollars ? This depends upon the number each let- ter stands for. But the questions do not decide what this number is. It is not the object, in adding them, to ascertain the specific value of x and y, or of b, c, and/; but to find an algebraic expression, which will represent their sum or amount. This process is called addition. Hence 46. ADDITION in algebra may be defined, the connecting of several quantities ivith their signs in one expression. QUEST. How is addition denoted ? Write the sum of a, b, c, and d. What is this process called ? Define addition. Arts. 44-50.] ADDITION. 27 47. Quantities may be added, by writing them one after another, without altering their signs. N. B. A quantity to which no sign is prefixed is always to be considered positive, i. e. the sign -|- is understood. (Art. 12.) What is the sum of a-\-m, and 6 8, and 2h 3m-\-d ? a+m+b-S+Zh-Zm+d. Ans. 48. It is immaterial in what order the terms or letters are arranged. If you add 6 and 3 and 9, the amount is the same, whether you put the 6, 3, or 9 first, viz. 18. But it is frequently more convenient and therefore customary to arrange the letters alphabetically. 49. It often happens that the expression denoting the sum or amount, can be simplified by reducing several terms to one. Thus the amount 2a-)-7a-|-4a, may be abridged by uniting the three terms in one. Thus 2a added to 7 is 9a, and 4 added to 9a make 13. Or 2a-|-7a-t-4a 13a. There are two cases in which reductions can be made. 50. CASE I. When the quantities are alike and the signs alike, as -\-4b-^-5b, or 4?/ 3y, &c. Add the co-efficients, annex the common letter or letters, and prefix the common sign. EXAMPLES. 1. What is the sum of 3a, 4a, and 6a ? 3xy xy 2xy 3. 7b+ xy 8b+Zxy 2b+2xy 6b-{-5xy 4. ry+Sabh 3ry-\- abh 6ry-{-4abh 2ry-\- abh -f-6=13. Ans. 5. cdxy-\-3mg 2cdxy-\- mg 5cdxy-^-7mg 7cdxy-\-8mg N. B. The mode of proceeding is the same, when all the signs are . Thus 3bc be 5Zc = 9bc. QUEST. How add quantities ? When no sign is prefixed to a quan- tity, what is understood ? In what order are the terms or letters generally arranged ? Why ? Can expressions denoting the sum, ever be simplified ? How ? Case first ? 28 ALGEBRA. [Sect. II. 7. ax 8. 2ab my 9. Sack Sidy 3ax db 3my ach My 2ax 7aJ Smy 5ach Ibdy 51. CASE II. When the quantities are alike , but the signs unlike, as -\-9b and 6b ; Take the less co-efficient from the greater ; to the difference, annex the common letter or letters, and prefix the common sign. Suppose a man's loss $500 and his gain $2000. The alge- braic notation is 500+2000, i. e. $500 is to be subtracted from his stock, and $2000 added to it. But it will be the same in effect, and the expression will be greatly abridged, if we add the difference between $500 and $2000, viz. $1500, to his stock. 10. What is the sum of I6ab and *7ab ? Ans. 9ab. 11. 12. 13. 14. 15. To +4J 5c 2hm dy+6m 3h dx Add Qb 7bc 9hm 4dy m 5h+4dx 53. If several positive, and several negative quantities are to be reduced to one term ; first reduce those which are posi- tive, next those which are negative, and then take the differ- ence of the co-efficients of the two terms thus found. 16. Reduce 13b+6b+b 4b 5b 7b, to one term. 13J_|_6J+5=20* ; and 46 55 75=z 16. Then 20 I6b=4b. Ans. 17. Add Sxy xy+2xy Ixy+lxy 9xy-\-*7xy 6xy. 18. Add Sad 6ad+ad+*Iad 2ad+9ad Sad 4ad. 19. Add 2abm abm-}-7abm Sabm-^-labm. 20. Add axy 7axy-\-Saxy axy Saxy-\-9axy. Q.UEST. How may like quantities be reduced when their signs are unlike ? When several positive and several negative quantities are to be reduced to one term, how proceed ? --P Arts. 51-56.] ADDITION. 29 54. If two equal quantities have contrary signs, they destroy each other, and may be cancelled. Thus -f-66 66=0. And (3X6) 18=0, so 76c 76c=0. 55. If the letters, or quantities in the several terms to be added, are UNLIKE, they can only be placed after each other, with their proper signs. (Art. 47.) 21. If 46, and 6y, and 3z, and 17A, and 5d, and 6, be added ; their sum will be 46 6y+3z+l7h 5d+6. 22. Add aa, aaa, to xx, xxx and xxxx. Different letters, and different powers of the same letter, can no more be united in the same term, than dollars and guineas can be added, so as to make a single sum. Six guin- eas and four dollars are neither ten guineas nor ten dollars. 56. From the foregoing principles we derive the following GENERAL RULE FOR ADDITION. Write down the quantities to be added without altering their signs, placing those that are alike under each other ; and unite such terms as are similar, 23. To 3bc 6^+26 3^ These may be arranged thus : Add 3bc,+x 3d+bg Y 3bc6d+2b3y And 2d+y+3x+b J-36c 3d + * 2d + y+3x The sum will be 7d-f26 2y+4x-\-bg+b EXAMPLES FOR PRACTICE, 1. Add ab+ 8, to cd3, and 5ab 4m+2, 2. Add z+3y dx, to 7 x 8+hni. 3. Add aim 3x-\-bm, to y z+7, and 5x- 6y-(-9. QUEST. If two equal quantities have contrary signs, what is the effect ? If the letters in the several terms are unlike, how are they ad- ded ? What then is the general rule for addition f 3* 30 ALGEBRA. [Sect, II. 4. Add 3aro+6 Ixy 8, to lOzy 9+5ai. 6. Add 7ad h-\-8xy ad, to 5ad-\-h 7xy. 7. Add Sab 2ay-\-x, to ab ay-\-bx h. 8. Add 2by 3ax-\-2a, to 3bx by+a. 9. Add ax-\-by xy, to by-\-2xy-\-5ax. 10. Add 4lcdf lOxy 186, to 7 xy -\-24b-\-3cdf. 11. Add 36z 17xy+18a, to 4ax 56x+63cx. 12. Add 8a6 66c+4ed 7xy, to I7mn-\-18fg 2ax. 13. Add 42 a6c-j-10a6c?, to 5Qabc-\-l5abd-}-5xyz, 14. Add ax y+6 df+44, to 4df 20+3az+75y. 15. Add 45 106+4crff, to 826 4c#*+10a 46. 16. Add 12(a+6)+3(a+6), to 2(a+6) 10(a+6). 17. Add xy(a+b)+3xy(a+b), to 2xy(a+b) 4xy(a+b). 18. Add ax-\-aa, x-\-xxx, 4aa-\-2x-\-ax, and 2xxx. 19. Add y yy~\-^y-> 2xx-\-].Qyy, to 4xy-\-6y Sxx. 20. Add aaa-\-4aaa, to lOaaa 14aaa-}-8aaa. 21. Add 12yyyy lOxx, to 2Qxx 8yyyj/-|-2xx-|-3yyyy. 22. Add 4(x y) 13, to (+6) 16(x y) 7(a+6). 23. Add a (x+y) 6y, to 40( 6)+8a(x+y) 36(a 6). 24. Add 10axy--17bcd axy, to 6axy 146cc?. 25. Add x+y+6x(a 6) 7x,tol6y 15x(a-6)+25x. 27. Add 5a6c Qxy-\-mn, a-\-6abc-{-14xy Ila-\-6mn, to 1 5xy 17a6c 1 5a dbc-\-xy 3m-f-a6c. 28. Add a(x+y) - 3b(x+y) - 4a(x+y) - 4(x+y) - (x+y), Arts. 57-59.] SUBTRACTION. 31 SECTION III. SUBTRACTI ON. ART. 57. SUBTRACTION in algebra is finding the difference of two quantities or sets of quantities. 1. Charles has 5a pears, and James has 3a pears. How many more has Charles than James ? In this example we wish to take 3a pears from 5a pears. But subtraction in al- gebra is denoted by the sign . Hence 5a 3 pears rep- resents the answer. But 5 3a=2 pears. Ans. 2. A gentleman owns a house valued at $4500 ; but he is in debt $800. How much is he worth ? $4500- $800=$3700. Ans. 58. Let us now attend to the principle upon which this ope- ration is performed. To illustrate this point, let us suppose that you open a book account with your neighbor. When footed up, the debtor side, which is considered positive, is $500. The credit side, which is considered negative, is $300. You balance the account, and find he owes you $500 $300 $200. Now if you take $50 from the positive or debtor side, it will have the same effect on the balance, as if you add $50 to the negative or credit side. On the other hand, if you take $50 from the negative or credit side, it will have the same effect on the balance, as if you add $50 to the positive or debtor side. 59. Hence universally, taking away a positive quantity from an algebraic expression is the same in effect as adding an equal negative quantity; and taking away a negative quantity, is the same as adding an equal positive one. QUEST. What is subtraction ? On what principle is the rule found- ed f How illustrate this ? What is the rule for subtraction ? 32 ALGEBRA. [Sect. III. 60. Upon this principle is founded the following GENERAL RULE FOR SUBTRACTION. I. Change the signs of all the quantities to be subtracted, 1. e. of the subtrahend, or suppose them to be changed from + to , or from to +. II. If the quantities are ALIKE, unite the terms as in addi- tion. (Arts. 50, 51.) III. If the quantities are UNLIKE, change the signs of the subtrahend, and write its terms after the minuend, as in ad- dition. (Art. 55.) EXAMPLES. 1. From 604-96 ) Change the signs ( 6fl+96 ) . { of the subtra- \ . , \ AttS. Take 3a+46 ) hend thus ; \ 3a 46 ) 2. From 166 3. Uda 4. 28 5. 166 6. Uda Subtr. 126 6da 16 126 6da 7.166 8.126 9. 6da 10, 16 11. 12 6 12. 6da 286 166 Uda 28 166 Uda 14. +166 15. +Uda 16. 28 17. 166 18. Uda 126 6da +16 +126 + 6da 19. From 8a6, take 6xy. Ans. 8a6 6xy. 20. 6aay 21. IGaaxx 22. 6dd+3d4ddd Ylay 2Qax Wdc-\-2dddd+4dy. 62. From these examples, it will be seen that the difference between a positive and a negative quantity, may be greater than either of the two quantities. In a thermometer, the dif- ference between 28 degrees above zero, and 16 below, is 44 degrees. The difference between gaining 1000 dollars in trade, and losing 500, is equivalent to 1500 dollars. Arts. 60-66.] SUBTRACTION. 33 63. PROOF. Subtraction may be proved, as in arithmetic, by adding the remainder to the subtrahend. The sum ought to be equal to the minuend, upon the obvious principle, that the difference of two quantities added to one of them, is equal to the other. 23. From 2xy 1 ") f xy+7 the subtrahend. Sub. %y-\-7 I Proof. < ^ x y 8 remainder. Rem. 3xy 8j (_ 2xy 1 minuend. 24. h+3bx 25. hyah 26. nd 7by 3h9bx 5hy6ah 5nd by 27. 28. 29. 30 3abm xy 17+4ax ax-\- 76 20 ax 65. When there are several terms alike in the subtrahend, they may be united and their sum be used. Thus, 31. From ab subtract 3am-\-am-{-l 'am-{-2am-\-6am. Ans. ab 3am am 1am 2am 6am=ab 19 or by a point. In the latter case, the multiplier is to be written before the other factors without being repeated. The product of 6 X d into a, is abXd, and not &X#e?. For bXd is bd, and this into a, is abd. (Art. 72.) The expres- sion &XeHs not to be considered, like 6-j-e?, a compound quan- tity consisting of two terms. Different terms are always sep- arated by + or . (Art. 19.) The product of bXhXmXy into a, is a X 6 X A X w Xy, or abhmy. But b-\-h-\-m-}-y into a, is ab-{-ah-\-am-\-ay. 78. If both the factors are compound quantities, each term in the multiplier must be multiplied into each in the multipli- cand. Thus (a-\-b) into (c-{-d) is ac-\-ad-\-bc-{-bd. For the units in the multiplier a-\-b are equal to the units in a, added to the units in 6. Therefore the product produced ^by a, must be added to the product produced by b. The product of c-\-d into a is ac-\-ad. ) ^ A t 76 ^ The product of c+d into 6 is 6c+& what does it show ? When , what? When and how is the subtraction performed ? 40 ALGEBRA. [Sect. IV. But this sum is to be subtracted, from the other quantities with which the multiplier is connected. It will then become 4a. (Art. 59.) Thus in the expression b (4Xfl,) it is manifest that 4Xa is to be subtracted from 6. Now 4X is 4a, that is -f-4. But to subtract this from b, the sign + must be changed into . So that 6 (4X) is b 4a. And X 4 is there- fore 4a. Again, suppose the multiplicand is a, and the multiplier (6 4). As (6 4) is equal to 2, the product will be equal to 2a. This is less than the product of 6 into a. To obtain then the product of the compound multiplier (6 4) into a, we must subtract the product of the negative part, from that of the positive part. 33. Multiplying a ) ig ^ game M ( Multiplying a Into 6 4 J t Into 2 And the product 6a 4, is the same as the product, 2a. But if the multiplier had been (6-f-4), the two products must have been added. 34. Multiplying a ) . ^ game ag ( Multiplying a Into 6+4 > ( Into 10 And the product 6a+4a, is the same as the product lOa. N. B. This shows at once the difference between multiply- ing by a positive factor, and multiplying by a negative one. In the former case, the sum of the repetitions of the multipli- cand is to be added to, in the latter subtracted from, the other quantities, with which the multiplier is connected. (Art. 41.) QUEST. What is the difference between multiplying by a positive factor and a negative one ? Art. 85.] MULTIPLICATION. 41 36. Mult, a+b 37. 3dy+te?+2 38. 3A+3 Into b x mr ab ad 6 85. If two negatives be multiplied together, the product will be affirmative : 4X a=+4. In this case, as in the preceding, the repetitions of the multiplicand are to be subtracted, because the multiplier has the negative sign. These repetitions, if the multiplicand is a, and the multi- plier 4, are a a a a 4a. But this is to be sub- tracted by changing the sign. It then becomes -f-4a. Suppose a is multiplied into (6 4). As 6 4=z2, the product is, evidently, twice the multiplicand, that is, 2a. But if we multiply a into 6 and 4 separately ; a into 6 is 6a, and a into 4 is 4. (Art. 83.) As in the mul- tiplier, 4 is to be subtracted from 6 ; so, in the product, 4a must be subtracted from 6a. Now 4a becomes by sub- traction -\-4a. The whole product then is 6a-)-4a, which is equal to 2a. Or thus, 39. Multiplying a ) ig ^ S ^ Q &g ( Multiplying Into 6 4 > ( Into And the prod. 6-|-4, is equal to the product 2a. It is often considered a great mystery, that the product of two negatives should be affirmative. But it amounts to no- thing more than this, that the subtraction of a negative quan- tity, is equivalent to the addition of an affirmative one, (Art. 57 ;) and, therefore, that the repeated subtraction of a nega- tive quantity, is equivalent to a repeated addition of an affirma- tive one. Taking off from a man's hands a debt of ten dol- lars every month, is adding ten dollars a month to the value of his property. QUEST. Explain how into gives -J-. 4* 42 ALGEBRA. * [Sect. IV. 40. Multiply a 4 into 366. 41. Mult. 3ad ah 7 into 4 dy hr. 42. Mult. 2Ay+3m 1 into 4rf 2z+3. 86. Positive and negative terms may frequently "balance each other, so as to disappear in the product. (Art. 54.) 43. Mult, a b 44. mm yy 45. aa-\-ab-\-bb Into a-{-b mm-\-yy a 6 aa ab +abbb Prod, aa * bb 87. For many purposes, it is sufficient merely to indicate the multiplication of compound quantities, without actually multiplying the several terms. Thus (Art. 23,) the product of a+b+c into h+m+y, is (a+b+c)X(h+m+y). 47. What is the product of a-\-m into A-j-x and d-\-y ? By this method of representing multiplication, an important advantage is often gained, in preserving the factors distinct from each other. When the several terms are multiplied in form, the expres- sion is said to be expanded. 48. What does (a-\-b) X (c-\-d) become when expanded ? 89. With a given multiplicand, the less the multiplier, the less will be the product. If then the multiplier be reduced to nothing, the product will be nothing. Thus aXOz=0. And if be one of any number of fellow-factors, the product of the whole will be nothing. QUEST. Is it always necessary actually to perform the multiplica- tion ? What advantage is gained by representing it? When is an expression said to be expanded? When you multiply a quantity by 0, what is the product ? Arts. 86-90.] MULTIPLICATION. 43 49. What is the product of 50. And(a+b)X(c+d)X(h From the preceding principles we derive the following GENERAL RULE FOR MULTIPLICATION. 90. Multiply the letters and co-efficients of each term in the multiplicand, into the letters and co-efficients of each term in the multiplier ; and prefix to each term of the product, the sign required by the principle, that like signs produce -f-, and unlike signs . EXAMPLES FOR PRACTICE. 1. Mult, a+36 2 into 4+c+h). 26. What are the factors of amh-{-amx-}-amy. 27. What are the factors of 4ad+8ah+I2am+4ay. Now if the whole quantity be divided by one of these fac- tors, according to Art. 95, the quotient will be the other factor. Thus, (ab+ad)-a=b+d. 29. (ab+ad)-(b+d)=a. Hence, If the divisor is contained in every term of a compound dividend, it must cancelled in each. 30. 31. 32. 33. Div. ab-\-ac bdh+bdy aah-\-ay drx-\-dhx-\-dxy By a b a dx QUEST. If there are numeral co-efficients, how proceed ? When the divisor is contained in every term of a compound dividend, how proceed ? Arts. 96-100.] DIVISION. 47 34. 35. 36. 37. Div. 6ab+l2ac Wdry+lQd 12Ax+8 35dm+14dz By 3a 2d 4 7d 98. On the other hand, if a compound expression contain- ing any factor in every term, be divided by the other quan- tities connected by their signs, the quotient will be that fac- tor. See the first part of the preceding article. 38. 39. 40. 41. Div. ab+ac+ah amh+amx+amy 4ab+Say ahm+ahy By b+c+h h+x+y b+2y m+y 99. In division, as well as in multiplication, the cau- tion must be observed, not to confound terms with factors. (Art. 77.) 42. Thus (ab+ac)-+a=b+c. (Art. 97.) 43. But (abXac)- : ra=aabc-^-a=^abc. 44. Quot.of(a6+ac)-HH-c). 45 - And abXac+(bXc). 100. SIGNS. In division, the same rule is to be observed respecting the signs, as in multiplication ; that is, if the divi- sor and dividend are both positive, or both negative, the quotient must be positive : if one is positive and the other negative, the quotient must be negative. (Art. 82.) This is manifest from the consideration that the product of the divisor and quotient must be the same as the dividend. 46. If +aX+b= +a&~) 47. " aX+b ab[ 48. +aX-b=- ab> 49. " QUEST. What caution as to terms and factors ? The rule for the signs ? 48 ALGEBRA. [Sect. V. 50. 51. 52. 53. Div. abx Sa Way Sax Gay GamXdh By a 2o 3a 2a ., j 101. If the letters of the divisor are not to be found in the dividend, the division is expressed by writing the divisor under the dividend, in the form of a vulgar fraction. 54. Thus xy-^-a-. 55. (dx)-. h= This is a method of denoting division, rather than an ac- tual performing of the operation. But the purposes of divi- sion may frequently be answered, by these fractional expres- sions. As they are of the same nature with other vulgar fractions, they may be added, subtracted, multiplied, &c. 102. If some of the letters in the divisor are in each term of the dividend, the fractional expression may be rendered more simple, by rejecting equal factors from the numerator and denominator. 56. 57. 58. 59. 60. Div. ab dhx ohm Say ab-\-bx 2am By ac dy ab by 2xy N. B. These reductions are made upon the principle, that a given divisor is contained in a given dividend, just as many times, as double the divisor in double the dividend ; triple the divisor in triple the dividend, &c. 103. If the divisor is in some of the terms of the dividend, but not in all ; those which contain the divisor may be divi- ded as in Art. 92, and the others set down in the form of a fraction. QUEST. If the letters of the divisor are not found in the dividend, how proceed ? If some of the letters in the divisor are found in each term of the dividend ? If the divisor is in some of the terms of the dividend, but not in all ? Arts. 101-105.] DIVISION. 49 61. Thus (ab+d)+a is either , or **+- or b+-. a a a 'a 62. 63. 64. 65. Div. dxy+rx hd 2ah-\-ad-\-x bm-\-3y 2my+dh By x a 6 2m 104. The quotient of any quantity divided by itself or its equal, is obviously a unit. Thus -=1. a 67. Div. . 68. JL. 69. f*=?*. 3az 4+2 a+b 3h 70. 71. 72. 73. Div. ax+x 3bd3d 4axy4 a +8ad 3a6+3 6m By x 3d 4a 3 Cor. If the dividend is greater than the divisor, the quotient must be greater than a unit : But if the dividend is less than the divisor, the quotient must be less than a unit. 74. Divide 25 by 5. Ans. 5. 75. ~=4 Ans. DIVISION BY COMPOUND DIVISORS.* 105. Ex. 1. Divide ac+bc+ad+bd, by a+b. a+b)ac+bc+ad-\-bd(c+d ac-\-bc, the first subtrahend. ad+bd ad-\-bd, the second subtrahend. QUEST. To what is the quotient of any quantity divided by itself, equal ? Corollary ? * The reason for inserting this article in the present place, may be learnt from the preface. 5 50 ALGEBRA. [Sect. V. Here ac, the first term of the dividend, is divided by a, the first term of the divisor, (Art. 92,) which gives c for the first term of the quotient. Multiplying the whole divisor by this, we have ac-{-bc to be subtracted from the two first terms of the dividend. The two remaining terms are then brought down, and the first of them is divided by the first term of the divisor as before. This gives d for the second term of the quotient. Then multiplying the divisor by d, we have ad+bd to be subtracted, which exhausts the whole dividend without leaving any remainder. (Art. 98.) The rule is founded on this principle, that the product of the divisor into the several parts of the quotient, is equal to the dividend. (Art. 91.) 106. Before beginning to divide, the terms should be so arranged that the letter, which is in thejirst term of the divi- sor, shall also be in the Jirst term of the dividend. If this letter is repeated as a factor, either in the divisor, or divi- dend, or in both, the terms should be arranged in the follow- ing order ; put that term Jirst, ivhich contains this letter the greatest number of times ; the term containing it the next greatest number of times, next, and so on. 2. Divide 2aab+bbb+2abb+aaa by aa+bb-\-ab. If we take aa for the Jirst term of the divisor, the other terms must be arranged according to the number of times a is repeated as a factor in each. Thus, aa+ab+bb)aaa+2aab+2abb+bbb(a+b aaa-\- aab-\- abb aab+ abb+bbb aab-\- abb+bbb QUEST. When the divisor and dividend are both compound quan- tities, how arrange the terms ? Arts. 106, 107.] DIVISION, 51 N. B. The strictest attention must be paid to the rules for the signs in subtraction, multiplication, and division. (Arts. 60, 82, 100.) 3. Divide xx 2zy+yyj by x y. 4. Divide aa bb, by a-\-b. 5. Divide bb+2bc+cc, by b+c. 6. Divide aaa-\-xxx, by a-j-z. 7. Divide 2ax 2aax 3aaxy-{-6aaax-\-axy zy,by2#-y. 8. Divide a-\-b c ax bx-\-cx, by a-\-b c. 9. Divide ac+bc+ad+bd+x, by a+b. 10. Divide ad qh+bd bh+y, by d h. 107. From the preceding principles we derive the following GENERAL RULE FOR DIVISION. I. DIVISION, in all cases, may be expressed by writing the divisor under the dividend in the form of a fraction. II. When the divisor and dividend are both simple quanti- ties, and have letters or factors" common to each ; divide the co-efficient of the divisor by that of the dividend, and cancel the ^ factors in the dividend which are equal to those in the divisor. III. When the divisor is a simple, and the dividend a com- pound quantity ; divide each term of the dividend by the divi- sor as before ; setting down those terms which cannot be divi- ded in the form of a fraction. IV. If the divisor and dividend are both compound quan- tities ; arrange the terms according to Art. 106. To obtain thejirst term in the quotient, divide thejirst term of the dividend by thejirst term of the divisor. Multiply the whole divisor by the term placed in the quotient ; subtract the QUEST. What is the general ruje for division? 52 ALGEBRA. [Sect. V. product from the dividend ; and to the remainder, "bring down as many of the following terms, as shall be necessary to con- tinue the operation. Divide again by the jirst term of the divisor, and proceed as before, till all the terms of the divi- dend are brought down. V. SIGNS. If the signs in the divisor and dividend are ALIKE, the quotient will be -f- ; if UNLIKE, the quotient will be . EXAMPLES FOR PRACTICE. 1. Divide 12aby+6abx I8bbm+24b, by 6b. 2. Divide 16a 12+8y+4 2Qadx+m, by 4. 3. Divide (a 2A) X (3m+y) X#, by (a 2/0 X (3 4. Divide ahd-\-4ad-\-3ay a, by hd 4d-\-3y 1. 5. Divide ax ry-\-ad &my 6-f-a, by a. 6. Divide amy-}-3my mxy-\-am d, by dmy. 7. Divide ard 6a+2r M+6, by 2ard. 8. Divide 6ax 8-)-2^+4 6hy, by 4axy. 9. Divide IGabcx l2xyab-{-24abxd 36ahgl>, by 10. Divide 21aaby-\-42cdxaa-\-l4:aaa 35aaaaZ>, by 11. Divide 12abxyz 6hdabxy-\-24:xyalm, by 3alxy. 12. Divide Sax 365^+42 72cz+30aa?, by Sx. 13. Divide 4(kZ> 4(x+y)+72-\-l2(a+b)+48c, by 4. 14. Divide abx cdx-\-8gx-\-x, by ab cd-\-8g~\-l, 15. Divide 24xyz 36cd 48abcd, by 12^2 -I8cd -24abcd. 16. Divide ab ad-^-ax(a-\-b) 42axy-\-ab, by a. 17. Divide 6am -lOa/i+20 12cd+17a, by 2am. 18. Divide xyz-\-6x+2z l-\-2xyz(a-{-b), by 6xyz* 19. Divide 6ac 12Zc Gab 10 2aabbcc, by 6abc. 20. Divide 18abyx+16abx 20JJcm+24a3, by 3^. Art. 108.] FRACTIONS. 53 21. Divide 16* 24+8o+84 20a0 a, by 4. 22. Divide (.r ?/)X(3a+a?)X^ by (a? y)X(3a+x). 23. Divide 41dX(4 a) X (+#) by (4 a)X41cZ. 24. Divide $Qxy-\-7alx Sahmx, by 4Qy-\-7ab 3ahm. 25. Divide 2Q(db+l)--W(ab+l)+50(ab+l), by 5a. Examples of Compound Quantities. 26. Divide 6ax+2xy Sab fy-|-3tfc+c#-f-A, by 3a-|-y. 27. Divide aab3aa+2ab6a 4J+22, by 3. 28. Divide bb+3bc+2cc, by 6+c. 29. Divide SaaaUb, by 2a . 30. Divide xxx 3axx-{-3aax aaa, by x a. 31. Divide 2yyy l9yy-\-26y 16, by y 8. 32. Divide xxxxxx 1, by x 1. 33. Divide 4xxxx 9xx+6x 3, by 2zz+3z 1. NOTE. For examples in dividing compound quantities in which the indices are used, see Art. 194, Exs. 23-40, and Art. 196. SECTION VI. FRACTIONS. ART. 108. FRACTIONS in algebra, as well as in arithmetic, have reference to parts of numbers or quantities. The term is derived from a Latin word, which signifies broken. Thus a . b . 2a . , 4z . - is ; - is %b ; is fa ; and is fc. QUEST. What are fractions ? From what is the term derived ? The meaning of it ? 5* 54 ALGEBRA. [Sect. VI. 109. Expressions in the form of fractions occur more fre- quently in algebra than in arithmetic. Most instances in di- vision belong to this class. Indeed the numerator of every fraction may be considered as a dividend, of which the de- nominator is a divisor. 110. The value of a fraction, is the quotient of the nume- rator divided by the denominator. f\ 7 Thus the value of - is 3. The value of - is a. 2 b 111. From this it is evident, that whatever changes are made in the terms of a fraction ; if the quotient is not altered, the value remains the same. For any fraction, therefore, we may substitute any other fraction which will give the same quotient. 4 10 4ba Sdrx 6+2 each of these instances is 2. 1 12. It is also evident from the preceding articles, that if the numerator and denominator be both multiplied, or loth divided, by the same quantity, the value of the fraction will not be altered. Thus =|., each term being multiplied by 9 ; and = &=f , each term being divided by 3, and that quotient by 3 d bx abx 35x bx -^wjju ~ again, foo - = zr =. -; tor the quotient in b ab ob 4o Ij "* each case is x. 113. Any integral quantity may, without altering its value, be thrown into the form of a fraction, by making I. the de- nominator ; or by multiplying the quantity into any proposed QUEST. Are fractions in arithmetic or algebra the most frequent ? When division is expressed in the form of a fraction, where do you place the divisor ? What is the value of a fraction ? If the numerator and denominator are both multiplied, or both divided by the same quantity, how is the value affected ? How put an integer into the form of a fraction ? Arts. 109-114.] FRACTIONS. 55 denominator, and the product will le the numerator of the fraction required. m, a ab ad-\-ah 6adh .... Thus a -:= = -. The quot. of each is a. 1 6 c?+/t 6dA dx+hx 2drr + 2dr So rf-)-A= . And r+l= ! . x 2dr 114. SIGNS. (1.) Each sign in the numerator and de- nominator of a fraction, affects only the single term to which it is prefixed. (2.) The dividing line answers the purpose of a vinculum, i. e. it connects the several terms of which the numerator and denominator may each be composed. The sign prefixed to it, therefore, affects the whole fraction collectively. It shows that the value of the whole fraction is to be subjected to the operation denoted by this sign. (3.) Hence, if the sign before the dividing line is changed from + to or from to -f-j the value of the whole frac- tion is also changed. The value of - is a. (Art. 110.) But this will become 6 negative if the sign is prefixed to the fraction. Thus, . ab ab y+--=y+> Buty y b. NOTE. There is frequent occasion to remove the denomi- nator ; also to incorporate a fraction with an integer, or with another fraction. In each of these cases, if the sign is prefixed to the dividing line, all the signs of the numerator QUEST. How far does the effect of each sign in the numerator and denominator extend ? How far, the sign prefixed to the dividing line ? What does it show ? When this sign is changed, what is the effect? If the sign is prefixed to the fraction, and you wish to remove the denominator, or to incorporate the fraction with an integer, or with another fraction, what must you do ? 56 ALGEBRA. [Sect. VI. must be changed, as in Art. 66, where a parenthesis, having the sign before it, is removed. Thus b a a (4.) If all the' signs of the numerator are changed, the value of the fraction is changed in the same manner* Thus ^=+a, (Art. 100;) but ^^-a. And^=^ b bo i . ab + be =a c ; but = a-\-c. (5.) Again, if all the signs of the denominator are chang- ed, the value is also changed. ab ab Thus -=+ ; but -~ a, b b 115. If then the sign prefixed to a fraction, or all the signs of the numerator, or all the signs of the denominator be changed ; the value of the fraction will be changed, from positive to negative, or from negative to positive. 116. If any two of these changes are made at the same time, they will balance each other, and the value of the frac- tion will not be altered. Thus by changing the sign of the numerator, ab ab - =-4-a becomes - = a. b b But by changing both the numerator and denominator, it becomes - -|-, where the positive value is restored. b By changing the sign before the fraction, . ab . ab y-\ =y-\-a becomes y T~ y & Q,UEST. If all the signs of the numerator, or of the denominator, or the sign before the fraction are changed, what is the effect ? What is the effect when any two of these changes are made at the same time ? Arts. 115-117.] FRACTIONS. 57 But by changing the sign of the numerator also, it becomes y -- - where the quotient a is to be subtracted from y, or which is the same thing, (Art. 59,) -j-a is to be added, making y-\-a as at first. So 6 -- 6 - ~ 6 - 6 ' 2-^2- ~^ -=2 6-6 6 -6 Hence the quotient in division may be set down in different ways. Thus (a c)-^6, is either I , or - -. o o bo The latter method is the most common. See the examples in Art. 103. REDUCTION OF FRACTIONS. 117. A FRACTION may be reduced to lower terms, by divi- ding both the numerator and denominator, by any quantity ivliich will divide them without a remainder. According to Art. 112 this will not alter the value of the fraction. 1. Reduce to lower terms. Ans. -. cb c 2. Reduce . 3. Reduce . Sdy *lmr . ._ . a-4-bc _, ,., , am-\-ay 4. Reduce ~-^ . 5. Reduce I . (a+bc)Xm bm-^-by N. B. If a letter is in every term, both of the numerator and denominator, it may be cancelled, for this is dividing by that letter. (Art. 97.) Thus, QUEST. How reduce a fraction to lower terms ? 58 .ALGEBRA. [Sect. VI. 6 . Reduce Ans.. 7. Reduce If the numerator and denominator be divided by the great- est common measure, it is evident that the fraction will be re- duced to the lowest terms. For the method of finding the greatest common measure, see Art. 195. 118. To reduce fractions of different denominators to a common denominator. Multiply each numerator into all the denominators except its own for a new numerator ; and all the denominators to- gether, for a common denominator. 8. Reduce -, and -, and to a common denominator. b a y aXdXy=ady } cXbXy=cby ^ the three numerators. mXbXdmbd J bXdXy=bdy the common denominator. The fractions reduced are -, and -, and . bdy bay bay N. B. It will be seen, that the reduction consists in multi- plying the numerator and denominator of each fraction, into all the other denominators. This does not alter the value. (Art. 112.) dr ,2h , 6c , , 9. Reduce , and , and to a common denom. 3m g y 10. Reduce -, and -, and . to a common denom. 3 x d+h 11. Reduce : , and - to a common denom. a *f-o a b Q,UZST. -How to a common denominator ? Does this alter the value of each fraction ? Why not ? Arts. 118-120.] FRACTIONS. 59 An integer and a fraction, are easily reduced to a common denominator. (Art. 113.) 12. Thus a and - are equal to = and -, or and -. c c c c 13. Reduce a, 6, -, -. 14. Redttce , -, 1 my b d f IK T> j 3s y 1 ** -i* T> A -L x c 15. Reduce , j-, -. rr 16. Reduce 6,-, -. a 56 2 y * x b 3c 1 10 T5 j 3x 6 x 17. Reduce -, -, , -. 18. Reduce , -, -. a z y 3 a 4c o a 5 8a 1 .__ _, . 4a , w y c 19. Reduce -, -, , -. 20. Reduce --, 17,-, x, -. o 7 y 2 Jg c 4a 119. To reduce an improper fraction to a whole or mixed quantity. Divide the numerator by the denominator, as in Art. 103. 21. Thus j ~- 7tr . , 22. Reduce , to a mixed quantity .. a 120. To reduce a mixed quantity to an improper fraction. Multiply the integer by the given denominator, and add the given numerator to the product. (Art. 113.) The sum will be the required numerator ; and this placed over the given denominator will form the improper fraction required. N. B. If the sign before the dividing line is , all the signs 4 in the numerator must be changed. (Art. 114, note.) QUEST. How reduce an improper fraction to a whole or mixed quantity ? How reduce a mixed quantity to an improper fraction ? When the sign is before the dividing line, what must be done ? 60 ALGEBRA. [Sect. VI. 23. Reduce a-f~T to an improper fraction. Ans. 7"- . 24. Reduce a . 25. Reduce x c 26. Reduce db-^^-. 27. Reduce x 28. Reduce m+d . <* 29. Reduce 6 _ h d dy 121. To reduce a compound fraction to a simple one. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. 30. Reduce ^of-?-. Ans. .^-rrr- 31. Reduce |of^of - . 32. Reduce of of 3 5 2a m 7 3 b-a EXAMPLES FOR PRACTICE. 1. What is the value of 2. What is the value of abcdf 3. What is the value of X4 ? a Ifi 4. What is the value of -- - ~4x ? a 5. What is the value of when the denom. is X4 ? 6. What is the value of - when the denom. is - : -6ax ? 24ax QUEST. How reduce a compound fraction to a simple one ? Arts. 121, 122.] FRACTIONS. 61 7. What is the value of 5 when both numerator and 34a denominator are X2<#? 8. Reduce - ^- - to a whole or mixed number. 2ab 9. Reduce -- ~- to a whole mixed or number. i^x ab-{-e-\-dx-\-ax-\- a m 10. Reduce ! - j - - - to a whole or mixed No. a Reduce the four next examples to the lowest terms. 11. . 12. _ 13. . aac l%zyy 06+6 x ac-\-abc 15. Reduce and to a common denominator. y d 16. Reduce - ; ^ ; - and - to a common denominator. b' d g y 17. Reduce a -- - to an improper fraction. x 18. Reduce a+6 -j- to an improper fraction. 4i 19. Reduce - of - of - of - to a simple fraction. 3 o a y 20. Reduce - of - of - of of of - to a sim- ple fraction. ADDITION OF FRACTIONS. 122. RULE. Reduce the fractions to a common denomina- tor ; then add their numerators, and place the sum over the common denominator. QUEST. How are fractions added ? 6 62 ALGEBRA. [Sect. VI. EXAMPLES. 1. Add and of a pound. Ans. -X- or . J.O 1O J.O 1O 2. Add - and -. First reduce them to a common denomi- o d nator. They will then be ^- 7 and ~4, and their sum r^ bd ba oa 3. Given - and -- to find their sum. a 3n a b m ~. a , d 4. Given -and -- . 5. Given - and - . ay y m a , b v A jj a * ^ 6. Given and - -. 7. Add , to : = U.1J.U. *.*.****. j -. a-\-b a 6 d m r . b . d , a b 11. Add a+2*> c +^ ; ^ and "4"* tn A JJ An % ^+ C J I H~ c 12. Add 42 -- ; a -- ^ and a -4- { . Add -5=. ; 2c c 2c xy 4c 14. Add 2ax ; - and 123. For many purposes, it is sufficient to add fractions in the same manner as integers are added, by writing them one after another with their signs. (Art. 47.) QUEST. What other way ? Arts. 123-125.] FRACTIONS. 63 15. Thus the sum of - and - and .is 1 . b y 2m b r y 2m 124. To add fractions and integers. Write them one after another with their signs ; or, con- vert the integer into a fraction, (Art. 113,) and then add their numerators. 16. What is the sum of a, and - ? m 17. What is the sum of 3d and m y 18. What is the sum of 5x and c SUBTRACTION OF FRACTIONS. 125. RULE. Change the signs of the fractions to "be sub- tracted from -j- to , and from to -j- ; and then pro- ceed as in addition of fractions. (Art. 122.) 1. From -subtract -. b m First, Reduce the fractions to a common denominator. C aXm=am, the numerator of the minuend. Thus, ? hXb =bh, the numerator of the subtrahend. ( bXmbm, the common denominator. The fractions become and - bm bm Second, Change the sign, before the dividing line of the am bh subtrahend, as . bm bm . QUEST. How are integers and fractions added ? What is the rule for subtraction of fractions ? What sign do you change ? 64 ALGEBRA. [Sect. VI. Third, Unite the terms as in addition of fractions ; thus, 3. From - subtract m y , a+3d 4. From ^- subtract 4 o 5. From - subtract - . m y 6. From ' subtract - . d m 3 4 7. From - subtract -. a b 126. Fractions may also be subtracted, like integers, by setting them down, after their signs are changed, without reducing them to a common denominator. 8. From * subtract-^. Ans. ^ m y my 127. To subtract an integer from a fraction, or a fraction from an integer. Change the sign of the subtrahend, and write it after the minuend ; or, throw the integer into the form of a fraction, (Art. 113,) and then proceed according to the general ride for subtraction of fractions. 10. From - subtract m. Ans. -- m== y y y QUEST. How subtract an integer from a fraction, or a fraction from an integer ? Arts. 126-130.] FRACTIONS. 11. From 4a+- subtract 3a -. c a 12. From 1+ subtract a a 13. From a+3A d -^- subtract 3 A+ o 14. From = take 15. From * take 6 c x y 16. From ^ take . 17. From a -- take -. bx d+y y * 18. From x+y take . 19. From ^ take c 10 20. From,- take MULTIPLICATION OF FRACTIONS. . 128. By the definition of multiplication, multiplying by a fraction is taking a ^ar of the multiplicand as many times as there are like parts of an unit in the multiplier. (Art. 71.) o Suppose a is to be multiplied by -. A fourth part of a is -. a . a a 3a This taken 3 times is 7+7+7 ^T- 444 4 130. Hence ; to multiply a fraction by a fraction. Multiply the numerators together, for a new numerator, and the denominators together, for a new denominator. QUEST. What is meant by multiplying by a fraction ? Rule to mul- tiply a fraction by a fraction ? 6* 66 ALGEBRA. [Sect. VI. 1. Multiply into . Product . c 2m 2cm 2. Multiply into y m 2 o -MT i<- i -- Xh . . 4 3. Multiply . . mto -__. 4. Multiply into 5. Multiply L- into | 6. Multiply together ^, - and . DC? y ,,,,., 2 7t df 6,1 7. Multiply , - , - and - . m y c r 1 8. Multiply ^-LZ, - and -f- 9. Multiply , and -. hy a-\- 1 7 131. The multiplication may sometimes be shortened, by rejecting equal factors, from the numerators and denomin- ators. 10. Multiply - into - and -. Product . ray ry Here a, being in one of the numerators, and in one of the denominators, may be omitted. If it be retained, the product will be -- But this reduced to lower terms, by Art. 117, ary will become as before. Q,UEST. How shorten the process, when the numerators and deno- minators contain equal factors ? Arts. 131-133.] FRACTIONS. 67 11 TI/T I.L- t a d , m , ah 11. Multiply -- into - and - . m 3a %d 12. Multiply into . y ah 13. Multiply am + d into A and ?T. A w 5a 132. To multiply a fraction and an integer together. Either multiply the numerator of the fraction by the inte- ger ; or, divide the denominator by the integer. 14. Thus ^-Xais -. For a=- ; and ^X-=-. Ans. y y l y y 15. Multiply into a. Dividing the denominator by a, we ax m have . x And multiplying the numerator by a, we have \ But ax am 111 , -IP =. , the same as before. ax x 133. A fraction is multiplied into a quantity equal to its denominator, by cancelling the denominator. Thus, _ 16. ^Xb=a. For f Xfi ^-. But the letter 6, being in u ob both the numerator and denominator, may be set aside. 17. Mult. into (a y). 18. Mult, tt^ into (3+m). a y o-j-m N. B. On the same principle, a fraction is multiplied into any factor in its denominator, by cancelling that factor. 19. Mult. into y. 20. Mult. A into 6. QUEST. How multiply a fraction and an integer ? How by a quan- tity equal to its denominator ? 68 ALGEBRA. [Sect. VI. EXAMPLES FOR PRACTICE. 1. Multiply , , and " 2. Mult. * into i * o *1 a a x 3. Mult. ^ into |l 4. Mult. 3x+y into 8. 2 36 24a+32c 5. Mult. Jg 7 Mult a ^ ct * T - into 5z. 25xy 6. Mult, -into ~ into-. x o o 8 Mult intn ^ 3x+y 9. Mult. ~ X - 7 6 - X\ "*" 4 a C X6x x a 6 10. Mult. 2+* into f=5. 4 3 1 Mult 24fl6 X 3x y y 3 o x DIVISION OF FRACTIONS. 134. To divide a fraction by a fraction. Jnvert the divisor, and then proceed as in multiplication. (Art. 130.) 1. Divide? by ' Ans. X - = -. b a o c be To understand the reason of the rule, let it be premised, that the product of any faction into the same fraction inverted, is always a unit. a ab hXy d But a quantity is not altered by multiplying it by a unit. Therefore, if a dividend be multiplied, first into the divisor inverted, and then into the divisor itself, the last product will QUEST. How divide one fraction by another ? Explain the reason of the rule ? Arts. 134-136.] DIVISION. 69 be equal to the dividend. Now, by the definition, (Art. 91,) " division is finding a quotient, which multiplied into the divi- sor will produce the dividend." And as the dividend multi- plied into the divisor inverted is such a quantity, the quotient is truly found by the rule. _ .p.. ., m , 3h A m ^^ y 2. Divide by . Ans. -x = TV -j v i v 3. Divide - by . 4. Divide - by - . r h x a * TV 'A k * TV 'A 5. Divide by - . 6. Divide 5 lO 7. Divide by a+I 135. To divide a fraction by an integer. Divide the numerator by ttie given integer, when it can "be done without a remainder ; but when this can not be done, multiply the denominator by the integer. 8. Thus the quotient of divided by m-, is -. b b 9. Div. -i- -i_fl. Ans ? -. 10. Div. ?^6. a 6 ah oh 4 136. To divide an integer by a fraction. Reduce the integer to the form of a fraction, (Art. 113,) and proceed as in Art. 134. Or, multiply the integer by the denominator, and divide the product by the numerator. Thus, QUEST. How many ways to divide a fraction by an integer ? What are they ? How does it appear, that multiplying the denominator, divides a fraction ? How divide an integer by a fraction ? 70 ALGEBRA. [Sect. VI. 11. Div. - = = . Or, H- = = d 1 d c dec 12. Div. x+ 2* . 13. Div. 14. Div. 3ac z-^- O 136.a. By the definition, (Art. 32,) " the reciprocal of a quantity, is the quotient arising from dividing a unit by that quantity." Therefore the reciprocal of is 1-f- - = 1 X - = -. That is, b b a a The reciprocal of a fraction is the fraction inverted. Thus the reciprocal of - is _JL~lL ; the reciprocal of m+y b . is J or 3y ; the reciprocal of is 4. Hence the recip- &y 1 rocal of a fraction whose numerator is 1, is the denominator of the fraction. Thus the reciprocal of - is a ; of -, is a+6, &c. a a-j-0 EXAMPLES FOR PRACTICE. z y & 10 y X QUEST. What is the reciprocal of a fraction ? Arts. 136.a-137.] SIMPLE EQUATIONS. 71 . .p.. ., a+1 x , , 4. Divide - by a. c TV -j a-4-b , a 5. Divide : by -. x o x v.,, V 6. Divide by 7. Divide -- by . o a o 8 . Divide !Jf by b -. a a 9. Divide T by u 10. Divide 21a6c by . x 11. Divide 8xy by -- . c 12. Divide 18x by ( ^ om . 18(a+x) , 13. Divide ~ '- by O 14. Divide SECTION VII. SIMPLE EQUATIONS 137. MOST of the investigations in algebra are carried on by means of equations. In the solution of problems, for ex- ample, we represent the unknown quantity, or number sought, 72 ALGEBRA. [Sect. VII. by a certain letter ; and then, to ascertain the value of this unknown quantity, or letter, we form an algebraic expres- sion from the conditions of the question, which is equal to some given quantity or number. Thus, A drover bought an equal number of sheep and cows for $840. He paid $2 a head for the sheep, and 812 a head for the cows. How many did he buy of each ? OPERATION. Let x the number bought of each. then 2x = the price of all the sheep. and I2x = " " cows. Hence, 2*+12ff = 840 by the conditions. (Ax. 9.) 14# = 840 by uniting the ff's. and x =. 60, the number bought of each. It will be perceived that the unknown quantity or number sought, is represented by the letter x ; and from the conditions of the problem, we obtain the quantity 14#, which is equal to the given quantity $840. The whole algebraic expression, 14#=$840 is called an equation. 138. An equation, then, is a proposition, expressing in al- gebraic characters, the equality between one quantity or set of quantities and another, or between different expressions for the same quantity. This equality is denoted by the sign , which is read, " is equal to," or " equals." Thus, x-}-ab-\-c ; and 5-|-8= 17 4, are equations in which the sum of x and a is equal to the sum of b and c ; and the sum of 5 and 8 is equal to the difference of 17 and 4. _ QUEST. How are investigations generally carried on in Algebra ? What is an equation? What are the members of an equation ? "-- Arts. 138-141.] SIMPLE EQUATIONS. 73 The quantities on the two sides of the sign are called mem- bers of the equation ; the several terms on the left, constituting the first member, and those on the right, the second member. 139. When the unknown quantity is of the first power, as 3#, the proposition is called a simple equation ; or an equation of the first degree. 140. The reduction of an equation consists, in "bringing the unknown quantity by itself, on one side, and all the known quantities on the other side, without destroying the equality of the members. To effect this, it is evident that one of the members must be as much increased or diminished as the other. If a quantity be added to one, and not to the other, the equality will be de- stroyed. But the members will remain equal, If the same or equal quantities be added to each. Ax. 1. If the same or equal quantities be subtracted from each. Ax. 2. If each be multiplied by the same or equal quantities. Ax. 3. If each be divided by the same or equal quantities. Ax. 4. 140. a. The principal reductions in simple equations, are those which are effected by transposition, multiplication and division. : REDUCTION OF EQUATIONS BY TRANSPOSITION.. 141. In the equation x 7=9, the number 7 being con- nected with the unknown quantity x by the sign , the one is subtracted from the other. To reduce the equation, let I 7 be added to both sides. It then becomes x 7+7=9+7. (Art. 59.) The equality of the members is preserved, because one is as much increased as the other. (Axiom 1.) But on one QUEST. What is a simple equation ? In what does the reduction of an equation consist? How are the principal reductions effected? 7 74 ALGEBRA. [Sect. VII. side, we have 7 and +7. As these are equal, and have contrary signs, they balance each other, and may be cancelled. The equation will then be x9-\-7. (Art. 54.) Here the value of x is found. It is shown to be equal to 9+7, that is to 16. The equation is therefore reduced. The unknown quantity is on one side by itself, and all the known quantities on the other side. In the same manner, if x 6=a , Adding b to both sides x b-\-b a-f-6 And cancelling ( b-\-b) x=a-\-b. Hence, 142. When known quantities are connected with the un- known quantity by the sign -f- or , the equation is reduced by TRANSPOSING the known quantities to the other side, and and prefixing the contrary sign. This is called reducing an equation by addition or subtrac- tion, because it is, in effect, adding or subtracting certain quantities, to or from each of the members. 1. Reduce the equation x-\-3b mh d Transposing -|-36 we have x mh d 36 And transposing m, xh d 36+w. 143. When several terms on the same side of an equation are alike, they must be united in one, by the rules for reduc- tion in addition. (Arts. 50, 51.) 2. Reduce the equation x+56 4A=76 Transposing 56 4A x=^b 56+4A Uniting 76 56 in one term x=2b-}-4h. 144. The unknown quantity must also be transposed, when- ever it is on both sides of the equation. It is not material on which side it is finally placed. Q,UF.ST. Rule to reduce an equation by transposition ? How does it appear that this does not destroy the equality of the members ? When several terms are alike, what must be done ? When the un- known quantity is on both sides of the equation, what? Arts. 142-147.] SIMPLE EQUATIONS. 75 3. Reduce the equation 2z-f 2h=h-\-d-\-3x By transposition 2h k d=3x 2x And h d= x. 145. When the same term, with the same sign, is on oppo- site sides of the equation, instead of transposing, we may ex- punge it from each. For this is only subtracting the same quantity from equal quantities. (Ax. 2.) 4. Reduce the equation x-\-3h-\-d=b-\-3k-{- f Id Expunging 3A x-}-d=b-}-7d And xb+6d. 146. As all the terms of an equation may be transposed, or supposed to be transposed, and it is immaterial which member is written first ; it is evident that the signs of all the terms may be changed, without affecting the equality. Thus, if we have x b=d a Then by transposition d-{-a x-{-b Or, inverting the members x-{-b d-\-a. 147. If all the terms on one side of an equation be trans- posed, each member will be equal to 0. Thus, if x+b=d, then x+6 d=0. 5. Reduce a+2x 8=6 4+x+a. 6. Reduce y-\-ab hm=a-}-2y ab-}-hm. 7. Reduce A+30+7z=:8 6/i+6z d+b. S. Reduce M+21 4x+d=l2 3x+d 76A. 9. Reduce 5x+10+a=25+4a;+a. 10. Reduce 5 c +2x+ 12 3=za:+20+5c. 11. Reduce a+b3x=20+a4x+b. 12. Reduce z+3 2z 4z=34+3z 4 5x. QUEST. When the same term with the same sign is on opposite sides, what ? What is the effect when all the signs of both members are changed at the same time? If all the terms on one side are trans- posed to the other, to what is each member equal ? 76 ALGEBRA. [Sect. VII. REDUCTION OF EQUATIONS BY MULTIPLICATION. 148. The unknown quantity, instead of being connected with a known quantity by the sign -|- or , may be divided by it, as in the equation - z=6. Here the reduction can not be made, as in the preceding instances, by transposition. But if both members be multi- plied by a, the equation will become, x=ab. (Art. 140.) For a fraction is multiplied into its denominator , by re- moving the denominator. (Art. 133.) Hence, 149. When the unknown quantity is DIVIDED by a known quantity, the equation is reduced by MULTIPLYING every term on each side by this known quantity. N. B. The same transpositions are to be made in this case, as in the preceding examples. nr 13. Reduce the equation - -\-a Multiplying both sides by c The product is x-\-ac=J)c-\-cd And z=bc-\-cd ac. y (I 14. Reduce the equation - - +5=20. 15. Reduce the equation - -4-d=h. a-\-b 150. When the unknown quantity is in the denominator of a fraction, the reduction is made in a similar manner, by mul- tiplying the equation by this denominator. Q,UEST. How is an equation reduced by multiplication ? How is a fraction multiplied into its denominator ? How does it appear that this method of reducing equations does not destroy the equality? When the unknown quantity is in the denominator, how proceed ? Arts. 148-153.] SIMPLE EQUATIONS. 77 16. Reduce the equation [-78. 10 x 151. Though it is not generally necessary , yet it is often convenient, to remove the denominator from a fraction con- sisting of known quantities only. This may be done, in the same manner, as the denominator is removed from a fraction, which contains the unknown quantity. 17. Take for example - -+_ a b c TIT i i i dd i f lh Multiplying by a x= -j b c Multiplying by b bxad+ c Multiplying by c bcx=acd-\-abh. 152. An equation may be cleared of FRACTIONS by multi- plying each side into all the DENOMINATORS. Obser. In clearing an equation of fractions, it often happens, that a numerator becomes a multiple of its denominator, (i. e. can be divided by it without a remainder,) or, that some of the fractions can be reduced to lower terms. When this occurs, the operation may be shortened by performing the division, and reducing the fractions to the lowest terms according to Art. 117. 18. Reduce the equation - = [ , a d g m 19. Reduce the equation = [- -ft -, 153. N. B. In clearing an equation of fractions, it will be necessary to observe, that the sign prefixed to any frac- Q.UEST. How clear an equation of fractions ? How prove that this does not destroy the equality ? When a numerator becomes a multiple of its denominator, what may be done ? When a fraction can be reduced to lower terms, what? What must be observed as to the sign before the dividing line ? 7* 78 ALGEBRA. [Sect. VII. tion, denotes that the whole value is to be subtracted, which is done by changing the signs of all the terms in the nume- rator. (Art. 114.) rrt T i a d 20. Reduce =c 21. Reduce - -r=6. 22. Reduce r= - 4- 4- . 5 5 ~ 5 ~ 10 23. Reduce 2x ~ =z -\--. 24. Reduce x ~l~l~l~ir ~^H~ - ~ REDUCTION OF EQUATIONS BY DIVISION. 154. When the unknown quantity is MULTIPLIED into any known quantity, the equation is reduced by DIVIDING every term on both sides by this known quantity. 25. Reduce the equation ax-\-b 3A=^ By transposition ax=.d-\-Sh b TV M- u d+3hb Dividing by a x= ! . 26. Reduce the equation 155. If the unknown quantity has co-efficients in several terms, the equation must be divided by all these co-efficients, connected by their signs, according to Art. 98. QUEST. What does this sign, when thus situated, show ? When the unknown quantity has a co-efficient, how reduce the equation ? If the unknown quantity has a co-efficient in several terms, how ? Arts. 154-157.] SIMPLE EQUATIONS. 79 27. Reduce the equation 3x 6x=a d That is, (Art. 97,) (3b)Xx=ad. Dividing by 3 b x- . 3 b 28. Reduce the equation ax-{-xh 4. 29. Reduce the equation x =- 156. If any quantity, either known or unknown, is found as a factor in every term, the equation may be divided by it. On the other hand, if any quantity is a divisor in every term, the equation may be multiplied by it. In this way, the factor or divisor will be removed, so as to render the expression more simple. 30. Reduce the equation ax-}-3ab=:6ad-\-a Dividing by a x-\-3b=6d-\-l And x 6d+l3b. I 1 7, I .. fj 31. Reduce the equation -21 xxx Multiplying by a?, (Art. 133,) a;+l b h d And x h d+b 1. 32. Reduce the equation x X(a+b) a b=dX(a+b). 157. A proportion is converted into an equation by mak- ing the product of the extremes, one side of the equation ; and the product of the means, the other side. 33. Reduce to an equation axlbl I chid. The product of the extremes is adx The product of the means is bch The equation is, therefore adxbch. 34. Reduce to an equation a-\-blcl ih mly. QUEST. If any quantity is found as a factor in every terra, how ? How convert a proportion into an equation ? 80 ALGEBRA. [Sect. VII. 158. On the other hand, an equation may be converted into a proportion, by resolving one side of the equation into two factors, for the middle terms of the proportion ; and the other side into two factors, for the extremes. ' 35. Convert the equation, adx^^bch^ into a proportion. The first member may be divided into the two factors ax, and d ; the second into ch, and b. From these factors we may form the proportion ax ib 1 1 chid. 36. Reduce ay-\-by=ch cm. 37. Reduce 16z+2=:34. 38. Reduce 4x 8= 3z+13. 39. Reduce lOz 19=7a?+17. 40. Reduce Sx 3+9= 7*+9+27. SUBST I TUTI ON. 159. In the reduction of an equation, as well as in other parts of algebra, a complicated process can often be rendered shorter and more simple, by using letters for the given num- bers when large, (Art. 35 ;) and also by introducing a new letter which shall be made to represent a whole algebraic ex- pression. 160. This process is called SUBSTITUTION. After the ope- ration is completed, the numbers, or the compound quantity for which a single letter has been substituted, must be restored. 41. Reduce - -\ -- =1. Clearing of fractions, o 7 o 375z+3 X 750= 1 X 750 X 375 , 2812502250 .. . and x=i - - - :=744. Ans. 375 QUEST. How can an equation be converted into a proportion ? What is meant by substitution ? What is the advantage of it ? After the operation is performed, what must be done ? Arts. 158-160.] SIMPLE EQUATIONS. 81 By substituting a for 750 ; b for 3 ; and c for 375 ; the equa- tion becomes I =1. a c Clearing of fractions, cz-\-ab=ac : and x=a . 3X750 Restoring the numbers, a?=z750 -- 744. Ans. 42. Reduce + 6 = 84. Substitute a for 3 ; b for 4 ; c for 6 ; and d for 84. y> 43. Reduce s~rH 4500 ; c for 7000 ; and d for 10. r 4500 43. Reduce ^+^=10. Substitute a for 350; b for ooO 7000 44. Reduce -JL_- [-^ &. Substitute d for (ro+w), and the m-\-n c . x , a , equation is - -f- - = o. df c Clearing of fractions, ex + ad = &cd ; and x rz 6c(i-|- restoring (m + n) ; x= - 45. Reduce | =ra5. Substitute h for (l-m-n). I m n c 46. Reduce - - T . a , cd. Substitute h for m b-\- c -j-a EXAMPLES FOR PRACTICE. 1. Reduce 2. Reduce -+--- a ' 6 c 3. Reduce 40 6x 16=120 14x. 82 ALGEBRA. [Sect. VII. 4. Reduce 5. Reduce +f-20 -. 3 5 4 6. Reduce 4=5. x 7. Reduce -4-: 2=8. z+4 8. Reduce -^U=l. z+4 9. Reduce *=ll. 10. Reduce +1-1. n -D J * - 5 284 - X 11. Reduce- 4 12. Reduce 13. Reduce -2= i^ -D j 14. Reduce . 3 n 5x 5 , 977* 15. Reduce Jto- - 4= - -. 4 o 1> 16. Reduce - 17. Reduce ^=^ 53 18. 2 * Art. 161.] SIMPLE EQUATIONS. 83 6r.+7 . 7z 13 iy. Reduce 20. Reduce^- 1 :- -::7:4. 2 4 SOLUTION OF PROBLEMS. 161. For the solution of problems in Simple Equations, we derive from the preceding principles, the following GENERAL RULE. I. Translate the statement of the question from common to algebraic language, in such a manner as to form an equa* tion, i. e. put the question into an equation. (Art. 33.) II. Clear the equation of fractions by multiplying every term on each side by all the denominators. (Art. 152.) III. Transpose all the terms containing the unknown quan~ tity to one side, and all the known quantities to the other, taking care to change the signs of the terms transposed, and unite the terms that are alike. (Arts. 50, 51.) IV. Remove the co-efficients of the unknown quantity, by dividing all the terms in the equation by them. (Art. 154.) PROOF. Substitute the value of the unknown quantity for the letter itself in the operation ; and if the number satisfies the conditions of the question, it is the answer sought. Problem 1. A man being asked how much he gave for his watch, replied ; If you multiply the price by 4, and to the product add 70, and from this sum subtract 50, the remainder will be equal to 220 dollars. To solve this, we must first translate the conditions of the problem, into such algebraic expressions as will form an equation. QUEST. What is the first step in the solution of a problem ? Se- cond ? Third ? Fourth ? Proof? 84 ALGEBRA. [Sect. VII. Let the price of the watch be represented by x This price is to be mult'd by 4, which makes 4x To the product, 70 is to be added, making 4z-|-70 From this, 50 is to be subtracted, making 4x-j-70 50. Here we have a number of the conditions, expressed in algebraic terms ; but have as yet no equation. We must. ob- serve then, that by the last condition of the problem, the pre- ceding terms are said to be equal to 220. We have, therefore, this equation 4x+70 50i=220 Which reduced gives s=50. Here the value of x is found to be 50 dollars, which is the price of the watch. Proof. The original equation is 4z+70 50220 Substituting 50 for x, it becomes 4X^0+70 50=220 That is, 220=220. Prob. 2. What number is that, to which, if its half be added, and from the sum 20 be subtracted, the remainder will be a fourth of the number itself ? In stating questions of this kind, where fractions are concerned, it should be recollected, that ^x is the same as | ; that f x ~, &c. (Art. 108.) 3 5 In this problem, let x be put for the number required. CC T Then by the conditions proposed, x-\-- 20=- Z 4 And reducing the equation or=16. Proof, Prob. 3. A father divides his estate among his three sons, in such a manner, that, The first has $ 1000 less than half the whole ; The second has 800 less than one third of the whole ; Arts. 162, 162.0.] SIMPLE EQUATIONS. 85 The third has 600 less than one fourth of the whole ; What is the value of the estate ? Prob. 4. Divide 48 into two sucH parts, that if the less be divided by 4, and the greater by 6, the sum of the quotients will be 9. Here, if x be put for the smaller part, the greater will be 48 ar. By the conditions of the problem - -I -- - =9. 4 o 162. Letters may be employed to express the known quan- tities in an equation, as well as the unknown. (Art. 159.) A particular value is assigned to the numbers, when they are introduced into the calculation ; and at the close, the num- bers are restored. (Art. 35.) Prob. 5. If to a certain number, 720 be added, and the sum be divided by 125 ; the quotient will be equal to 7392 divided by 462. What is that number ? Let x the number required. Then by the conditions of the problem ' -. b ; h Therefore a= **=?* A (125X7392) -(720X462) Restoring the numbers, a?= - -- r- 5 = 1280. 162. a. When the solution of an equation brings out a nega- tive answer, it shows that the value of the unknown quantity QUEST. When letters are substituted for known quantities, what must be done at the close of the calculation ? When the solution brings out a negative answer, what does it show ? 8 86 ALGEBRA. [Sect. VII. is contrary to the quantities, which in the statement of the question were considered positive. But this being deter- mined by the answer, the omission of the sign before the unknown quantity in the course of the calculation, can lead to no mistake. Prob. 6. A merchant gains or loses, in a bargain, a certain sum. In a second bargain, he gains 350 dollars, and, in a third, loses 60. In the end he finds he has gained 200 dol- lars, by the three together. How much did he gain or lose by the first ? In this example, as the profit and loss are opposite in their nature, they must be distinguished by contrary signs. (Art. 39.) If the profit is marked -)-, the loss must be . Let #1= the sum required. Then according to the statement #-(-350 60=200 And x= 90. Prob. 7. A ship sails 4 degrees north, then 13 S. then 17 N. then 19 S. and has finally 11 degrees of south latitude. What was her latitude at starting ? Prob. 8, If a certain number is divided by 12, the quotient, dividend, and divisor, added together, will amount to 64. What is the number ? Prob. 9. An estate is divided among four children, in such a manner that The first has 200 dollars more than o f the whole, The second has 340 dollars more than | of the whole, The third has 300 dollars more than of the whole, The fourth has 400 dollars more than of the whole. What is the value of the estate ? Prob. 10. What is that number which is as much less than 500, as a fifth part of it is greater than 40 ? Prob. 11. There are two numbers whose difference is 40, and which are to each other as 6 to 5. What are the numbers ? Art. 162.a.] SIMPLE EQUATIONS. 87 b. 12. Three persons, A, B, and C, draw prizes in a lottery. A draws 200 dollars ; B draws as much as A, to- gether with a third of what C draws ; and C draws as much as A and B both. What is the amount of the three prizes ? Prob. 13. What number is that, which is to 12 increased by three times the number, as 2 to 9 ? Prob. 14. A ship and a boat are descending a river at the same time. The ship passes a certain fort, when the boat is 13 miles below. The ship descends five miles, while the boat descends three. At what distance below the fort will they be together ? Prob. 15. What number is that, a sixth part of which ex- ceeds an eighth part of it by 20 ? / ( Prob. 16. Divide a prize of 2000 dollars into two such parts, that one of them shall be to the other, as 9:7.A; )^_/,'< Prob. 17. What sum of money is that, whose third part, fourth part, and fifth part, added together, amount to 94 dollars. / -[ , Prob. 18. Two travellers, A and J5, 360 miles apart, travel towards each other till they meet. A^s progress is 10 miles an hour, and .ZTs 8. How far does each travel before they meet ? Prob. 19. A man spent one third of his life in England, one fourth of it in Scotland, and the remainder of it, which was 20 years, in the United States. To what age did he live ? &,{ Prob. 20. What number is that ^ of which is greater than | of it by 96 ? / ( , Prob. 21. A post is ^ in the earth, ^ in the water, and 13 feet above the water. What is the length of the post ? Prob. 22. What number is that, to which 10 being added, of the sum will be 66 ? 88 ALGEBRA. [Sect. VII. Prob. 23. Of the trees in an orchard, f are apple trees, -fa pear trees, and the remainder peach trees, which are 20 more than of the whole. What is the whole number in the orchard ? Prob. 24. A gentleman bought several gallons of wine for 94 dollars ; and after using 7 gallons himself, sold of the remainder for 20 dollars. How many gallons had he at first ? Prob. 25. A and B have the same income. A contracts an annual debt amounting to nf of it ; B lives upon f of it ; at the end of ten years, B lends to A enough to pay off his debts, and has 160 dollars to spare. What is the income of each ? M Prob. 26. A gentleman lived single of his whole life ; and after having been married 5 years more than ~f of his life, he ha^lva son who died 4 years before him, and who reached only half the age of his father. To what age did the father live ? -^ Prob. 27. What number is that, of which if , , and f be added together the sum will be 73 ? Prob. 28. A person after spending 100 dollars more than -J of his income, had remaining 35 dollars more than ^ of it. Required his income. u \ Prob. 29. In the composition of a quantity of gunpowder, The nitre was 10 Ibs. more than f of the whole, The sulphur 4 Ibs. less than of the whole, The charcoal 2 Ibs. less than ^ of the nitre. What was the amount of gunpowder ? fe '. Prob. 30. A cask which held 146 gallons, was filled with a mixture of brandy, wine, and water. There were 15 gal- lons of wine more than of brandy, and as much water as the brandy and wine together. What quantity was there of each ? /, 'f4f,- v . Art. 162. a.] SIMPLE EQUATIONS. 89 Prob. 31. Four persons purchased a farm in company for 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 ? ] ^ ^/ f Prob. 32. 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. jf_ : Prob. 33. A father divided a small sum among four sons. The third had 9 shillings more than the fourth ; The second had 12 shillings more than the third ; The first had 18 shillings more than the second ; And the whole sum was 6 shillings more than 7 times the sum which the youngest received. What was the sum divided ? j <^ -4 Prob. 34. A farmer had two flocks of sheep, each con- taining the same number. Having sold from one of these 39, and from the other 93, he finds twice as many remaining in the one as in the other. How many did each flock ori- ginally contain ? Prob. 35. An express, travelling at the rate of 60 miles a day, had been dispatched 5 days, when a second was sent after him, travelling 75 miles a day. In what time will the one overtake the other ? Prob. 36. The age of A is double that of B, the age of B triple that of C, and the sum of all their ages 140. What is the age of each ? Prob. 37. Two pieces of cloth, of the same price by the yard, but of different lengths, were bought, the one for ,5, the other for 6. If 10 be added to the length of each, the sums will be as 5 to 6. Required the length of each piece. ; ; .^ 90 ALGEBRA. [Sect. VII. ) jt y * * Prob. 38. ^4. and B began trade with equal sums of money. The first year, A gained forty pounds, and B lost 40. The second year, A lost of what he had at the end of the first, and B gained 40 pounds less than twice the sum which A had lost. B had then twice as much money as A. What sum did each begin with ? Prob. 39. What number is that, which being severally added to 36 and 52, will make the former sum to the latter, as 3 to 4 ? Prob. 40. A gentleman bought a chaise, horse, and har- ness, for 360 dollars. The horse cost twice as much as the harness ; and the chaise cost twice as much as the harness and horse together. What was the price of each ? Prob. 41. Out of a cask of wine, from which had leaked ^ part, 21 gallons were afterwards drawn; when the cask was found to be half full. How much did it hold ? Prob. 42. A man has 6 sons, each of whom is 4 years older than his next younger brother ; and the eldest is three times as old as the youngest. What is the age of each ? Prob. 43. Divide the number 49 into two such parts, that the greater increased by 6, shall be to the less diminished by 11, as 9 to 2. Prob. 44. What two numbers are as 2 to 3 ; to each of which, if 4 be added, the sums will be as 5 to 7 ? Prob. 45. A person bought two casks of porter, one of which held just 3 times as much as the other ; from each of these he drew 4 gallons, and then found that there were 4 times as many gallons remaining in the larger, as in the other. How many gallons were there in each ? Prob. 46. Divide the number 68 into two such parts, that the difference between the greater and 84, shall be equal to 3 times the difference between the less and 40. Art. 163.] INVOLUTION. 91 Prob. 47. Four places are situated in the order of the let- ters A, B, C, D. The distance" from A to D is 34 miles. The distance from A to B is to the distance from C to D as 2 to 3. And J of the distance from A to B, added to half the distance from C to D, is three times the distance from B to C. What are the respective distances ? Prob. 48. Divide the number 36 into 3 such parts, that J of the first, J of the second, and J of the third, shall be equal to each other. Prob. 49. A merchant supported himself 3 years, for ,50 a year, and at the end of each year, added to that part of his stock which was not thus expended, a sum equal to one third of this part. At the end of the third year, his original stock was doubled. What was that stock ? Prob. 50. A general having lost a battle, found that he had only half of his army+3600 men left fit for action ; J of the army+600 men being wounded ; and the rest, who were of the whole, either slain, taken prisoners, or missing. Of how many men did his army consist ? SECTION VIII. INVOLUTION. ART. 163. DEFINITIONS. (1.) When a quantity is multi* plied into itself, the product is called a power. Thus 3X3 =9 ; and dXd=dd. The 9 and dd are powers of 3 and d. (2.) Powers are divided into different orders or degrees, as the first, second, third, fourth, fifth powers, fyc., which are also called the square, cube, Mquadrate, fyc. QUEST. What is a power? How are powers divided? What is the second power called ? Third ? Fourth ? 92 ALGEBRA. [Sect. VIII. They take their name from the number of times the root, or first power, is used as a factor in producing the given power. The original quantity is called the first power or root of all the other powers, because they are all derived from it. Thus, 2X2=4, the square or second power of 2. 2X2X2=8, the cube or third power. 2X2X2X2=16, the biquadrate or fourth power, &c. And X=##? the second power of a. aXaXa=aaa, the third power. aXaXaXa=aaacii the fourth power, &c. (3.) The number of times a quantity is employed as a factor to produce the given power, is generally indicated by a figure or letter placed above it on the right hand. This figure or letter is called the index or exponent. Thus aX& is written a 2 instead of aa ; and aXaXaa 3 . The index of the first power is 1 ; but this is commonly omitted, for a*=a. Obser. An index is totally different from a co-efficient. The latter shows how many times a quantity is taken as a part of a whole ; the former how many times the quantity is taken as a factor. Thus 4a=a-^-a-}-a -f-a; but a*=aXXXa=rt##a. If a=4, then 4a=16; and a 4 =256. (4.) Powers are also divided into direct and reciprocal. Direct Powers are those which have positive indices, as df 2 , c? 5 , &c. and are produced by multiplying a quantity into itself. Thus dXd=d 2 ; dXdXd=d 3 ; and dXdXdXd=d*. A Reciprocal Poiver of a quantity is the quotient arising from dividing a unit by the direct power of that quantity, as i- i' &c - < Art - 33 - } QUEST. From what do they take their name? What is the first power? How are powers denoted? What is this number called? What does it show ? What is the difference between an index and a co-efficient ? What is the index of the first power ? Is it usually written ? How else are powers divided ? What are direct powers ? Reciprocal powers ? Arts. 164-166.] INVOLUTION. 93 It is produced by dividing a direct power by its root, till we come to the root itself ; and then continuing the division, d 3 d 2 we obtain the reciprocal powers. Thus -j=^ 2 ; an d ~r ^ and -1 ; and r-d= ; and _-f-rf -, &c. d d d 2 d- d 3 For convenience of calculation, reciprocal powers are written like direct powers with the sign before the index ; thus znj- 2 , &c. The direct and reciprocal powers of d, are rf*, d 3 , d*,'* 1 , d, d~ l , d~ 2 , d~ 3 , d~, &c. 164. INVOLUTION is the process of finding any power of a quantity by multiplying it into itself. Hence, 165. To involve a quantity to any required power. Multiply the quantity into itself, till it is taken as a factor, as many times as there are units in the index of the power to which the quantity is to be raised. (Art. 80.) N. B. All powers of 1 are the same, viz. 1. For IXlX lXl,-&c.=l. 166. A single letter is involved, by giving it the index of the proposed power ; or by repeating it as many times, as there are units in that index. N. B. If the letter or quantity has a co-efficient, it must be raised to the required power by actual multiplication. 1. The 4th power of a, is a 4 or aaaa. (Arts. 163, 165.) 2. The 6th power of y, is y 6 or yyyyyy. .3. The wth power of x, is x n or xxx . . . n times repeated. 4. Required the 3d power of 3x. QUEST. How are reciprocal powers written ? What is involution ? The rule ? What are all powers of 1 ? How is a single letter involv- ed ? If the quantity has a co-efficient, what must be done with it ? 94 ALGEBRA. Sect. VIII. 5. Required the 4th power of 4y. 6. Required the 7th power of 2a. 167. The method of involving a quantity which consists of several factors, depends on the principle, that the power of the product of several factors is equal to the product of their powers. 7. Thus (ay) 2 =a 2 y 2 . For by Art. 164, (ay) 2 ayXay. But ayXayayayaayya 2 y 2 . 8. What is the 3d power of bmx ? 9. What is the wth power of ady ? In finding the power of a product, therefore, we may either involve the whole at once ; or we may involve each of the factors separately, and then multiply their several powers into each other. 10. What is the 4th power of dhy > 11. What is the 3d power of 46 ? 12. What is the nth power of Gad ? 13. What is the 3d power of 3wzX% ? 168. SIGNS. When the root is positive, all its powers are positive also ; but when the root is negative, the ODD powers are negative, while the EVEN powers are positive. (Art. 82.) 169. Hence any odd power has the same sign as its root. But an even power is positive, whether its root is positive or negative. Thus -\-aX+a=a 2 . And aX a=a 2 . 170. To involve a quantity which is already a power. Multiply the index of the quantity into the index of the power to which it is to be raised. QUEST. On what principle does the method of involving a quanti- ty which consists of several factors, depend ? How then may we find the power of a product ? Rule for signs ? Does this differ from the rule for signs in multiplication ? What sign has every odd power ? Even powers? How involve a quantity which is already a power? Arts. 167-172.] INVOLUTION. 95 14. The 3d power of a 2 , is a 2 '* a 6 . T?ora 2 =aa : and the cube of aa is aaXaaXaa=aaaaaa=a G ; which is the 6th power of a, but the 3d power of a 2 . 15. Find the 4th power of a 3 6 2 . 16. Find the 3d power of 4a 2 x. 17. Find the 4th power of 2a*X3x 2 d. 18. Find the 5th power of (a+b) 2 . 19. Find the 2d power of (+&)". 20. Find the nth power of (x y) m . * 21. Find the nth power of (z+y) 2 . 22. Find the 2d power of (a 3 X& 3 ). 23. Find the 3d power of (a*b 2 h*). 171. A FRACTION is raised to a power, "by involving both the numerator and the denominator. a 2 24. The square of - is . For, by the rule for the mul- tiplication of fractions, ?Xr 77=7^- (Art. 130 bo bo b* 25. Find the 2d, 3d, and nth powers of -. a 2xr 2 26. Find the cube of -= . % x 2 r 27. Find the nth power of . m 28. Find the square of (x+1) 3 172. A compound quantity consisting of terms connected by -f- and , is involved by an actual multiplication of its several parts. Thus, QUEST. How is a fraction involved ? How is a compound quan- ity involved ? 96 ALGEBRA. [Sect. VIII. 29. (a+b) 1 =a+b, the first power a 2 +ab +ab+b 2 ? ... the second power. a +b ab 2 (a-f&)3 a 3 +3 2 H-3a& 2 +& 3 , . . the third power. ^ a + * a+4 3 &-f 6a 2 6 2 -|-4a6 3 +6S fourth power. 30. Find the square of a 6. 31. Find the cube of a+1. 32. Find the square of a-{-b-{-h. 33. Required the cube of a 34. Required the 4th power of 35. Required the 5th power of x+1. 36. Required the 6th power of 1 b. 173. The squares of binomial and residual quantities occur so frequently in algebraic processes, that it is important to make them familiar. If we multiply a-\-h into itself, and also a H into itself, 37. We have a+h 38. And a h a-\-h a h a 2 +ah -\-a7i+h 2 V . Arts. 173- 175. J INVOLUTION. 97 Here it will be seen, that in each case, the first and last terms are sqqares of a and k ; and that the middle term is twice the product of a into h. Hence the squares of bino- mial and residual quantities, without multiplying each of the terms separately, may be found, by the following proposition.* The square of a binomial, the terms of which are both posi- tive, is equal to the square of the frst term, -\- twice the pro- duct of the two terms, + the square of the last term. And the square of a residual quantity, is equal to the square of the first term, twice the product of the two terms, -f- the square of the last term. 39. Find the square of 2a-}-b. 40. Find the square of h+l. 41. Find the square of ab^\-cd. 42. Find the square of 6y-f3. 43. Find the square of 3d h. 44. Find the square of a 1. 174. For many purposes, it will be sufficient to express the powers of compound quantities by exponents, without an actu- al multiplication. 45. Thus the square of a-\-b, is a+6| 2 , or (a-\-b) 2 . 46. Find the nth power of 6c+8+x. In cases of this kind, the vinculum must be drawn over all the terms of which the compound quantity consists. 175. But if the root consists of several factors, the vincu- lum which is used in expressing the power, may either extend over the whole ; or may be applied to each of the factors separately, as convenience may require. QUEST. What is life square of a binomial whose signs are plus ? Of a residual ? Is it always necessary to perform the multiplication ? How far must the vinculum extend when the root contains factors? * Euclid, 2. 4. 9 98 ALGEBRA. [Sect. VIII, 47. Thus the square of (a+6)X(e+d), is either (a+b)X(c+d)\ or ( a +b) 2 X(c+d) 2 . For, the first of these expressions is the square of the pro- duct of the two factors, and the last is the product of their squares. But one of these is equal to the other. (Art. 167.) The cube of aX(b+d), is aX(b+d)\ or a 3 X(H-c/) 3 . 176. When a quantity whose power has been expressed by a vinculum and an index, is afterwards involved by an actual multiplication of the terms, it is said to be expanded. 48. Thus (a+6) 2 , when expanded, becomes a 2 -\-2ab-{-b 2 . 49. Expand (a+b+h) 2 . BINOMIAL THEOREM.* 177. To involve a binomial to a high power by actual mul- tiplication, as in Art. 172, is a long and tedious process. A much easier and more expeditious way to obtain the required power, is by what is called the BINOMIAL THEOREM. This ingenious and beautiful method was invented by SIR ISAAC NEWTON, and has been deemed of so great importance to mathematical investigation, that it is inscribed on his monu- ment in Westminster Abbey. 178. To illustrate this theorem, let the pupil involve the binomial +&, (Art. 172,) and the residual a 6, to the 2d, 3d and 4th powers. Thus, (a 6) 3 a 3 3a 2 b+3ab 2 b 3 . QUEST. What is meant by expanding a quantity? What is the best mode of involving a binomial to a high power? Who is the author of this theorem ? In what light is it regarded ? What is (a-}-b)2 (a+b)3? (-f&)4 ? (a-b)2 ? (ab)s? (a i)4 ? * See Preface. Arts. 176-179.] INVOLUTION. 99 179. By a careful inspection of the several parts of the preceding work, the following particulars will be observed to be common to each power. I. By counting the terms it will be found that the number in each power, is greater by 1 than the index of that power ; e. g. in the 3d power the number of terms is 4 ; in the 4th power, it is 5, &c. II. If we examine the signs we shall perceive when both terms of the binomial are positive, that all the signs in every power are -f- > hut when the quantity is a residual, all the odd terms, reckoning from the left, have the sign -)-, and all the even terms have the sign . Thus in the 4th power, the signs of the first, third and fifth terms are -j-, while those of the second and fourth are . III. Let us now direct our attention to the indices. 1. It will be seen that the index of the first term, or the leading quantity* in each power, always begins with the in- dex of the proposed power, and decreases 1 in each succes- sive term towards the right, till we come to the last term from which the letter itself is excluded. Thus in (adc^) 4 the in- dices of the leading quantity a, are 4, 3, 2, 1. 2. The index of the following quantity begins with 1 in the second term, and increases regularly by 1 to the last term, whose index like that of the first, is the index of the required power. Thus in (arfci) 4 the indices of the follow- ing quantity b, are 1, 2, 3, 4. 3. We shall also perceive, that the sum of the indices, is the same in each term of any given power ; and this sum is QUEST. How many terms are there in each power ? What signs has a binomial? Residual? What are the indices of the leading quantity ? Of the following quantity ? Which is the leading quantity? To what is the sum of the indices in each term equal ? * The first letter of a binomial, is called the leading quantity, an4 the other the following quantity, * 100 ALGEBRA. [Sect. VIII. equal to the index of that power. Thus the sum of the indi- ces in each of the terms of the 4th power, is 4. IV. The last thing to be considered is the co-efficients of the several terms. 1. The co-efficient of the first and last terms in each power, is 1 ; the co-efficient of the second and next to the last terms, is the index of the required power. Thus in the 3d power, the index of the second and next to the last terms, is 3 ; and in the same terms in the 4th power, it is 4, &c. 2. It will be observed also, that the co-efficients increase in a regular manner through the first half of the terms ; and then decrease at the same rate through the last half. Thus, in the 4th power they are 1, 4, 6, 4, 1, in the 6th power they are 1, 6, 15, 20, 15, 6, 1. 3. The co-efficients of any two terms equally distant from the extremes, are equal to each other. Thus in the 4th power, the second term from each extreme is 4 ; in the 6th power, the second term from each extreme is 6, and the third is 15. 4. The sum of all the co-efficients in each power, is equal to the number 2 raised to that power. Thus (2) 4 i=:16 ; also, the sum of the co-efficients in the 4th power, is 16, and (2) 6 =64 ; so the sum of the co-efficients in the 6th power, is 64. 180. If we involve any other binomial or residual to any required power whatever, and examine the result, we shall find the foregoing principles are common, and will apply to QUEST. What is the co-efficient of the first and last term ? What of the second and next to the last? What is peculiar to the first half of them ? To the last half? How do thole equally distant from the extremes compare ? To what is the sum qf all the co-efficients in any power equal ? What is said as to the eatjnt of the foregoing princi- ples ? What then do they furnish ? Arts. 180, 181.] INVOLUTION. 101 all examples. Hence we may safely conclude, that they are universal principles, and may be employed in raising all binomials to any required power. They are the basis, or elements, of what is called the Binomial Theorem. 181. The BINOMIAL THEOREM may be defined, a general method of involving binomial quantities to any proposed power, and is comprised in the following GENERAL RULE. I. SIGNS. If both terms of the binomial have the sign +, all the signs in every power will be -f- ; but if the given quantity is a residual, all the odd terms in each power, reck? oning from the left, will have the sign -]-, and the even terms the sign . II. INDICES. The INDEX of the Jirst term or leading quantity, must always be the index of the required power ; and this decreases regularly by 1 through the other terms. The index of the following quantity begins with 1 in the second term, and increases regularly by 1 through the others. III. CO-EFFICIENTS. The co-efficient of the Jirst term is 1 ; that of the second is equal to the index of the power ; and universally, if the co-efficient of any term be multiplied by the index of the leading quantity in that term, and divided by the index of the following quantity increased by 1, it will give the co-efficient of the succeeding term. IV. The number of terms will always be one greater than the power required. In algebraic characters, the theorem is b+nX *a n ' 2 b 2 , &c. QUEST. What is the Binomial Theorem ? What is the rule for the Bigns? For the indices? For the co-efficients ? The number of terms? 9* 102 ALGEBRA. [Sect. VIII. N. B. It is here supposed, that the terms of the binomial have no other co-efficients or exponents than 1. Other bino- mials may be reduced to this form by substitution. (Art. 159.) 1. What is the 6th power of x-{-y ? The terms without the co-efficients, are z 6 , x 5 y, x*y 2 , z 3 ^ 3 , z 2 y 4 , zy 5 , y 6 . And the co-efficients, are 6X5 15X4 20X3 ' ' ~2~' ~~3~' T~' b ' that is 1, 6, 15, 20, 15, 6, 1. Prefixing these to the several terms, we have the power required ; x 6 +6x 5 y+l5x*y 2 +20x s y*+15x 2 y*+6xy*+y 6 . Ans. 2. What is the 5th power of (d+h) ? 3. What is the wth power of (b-\-y) ? Ans. b n +Ab n -*y+Bb n - 2 y 2 +Cb n -*y*+Db n -*y*, &c. That is, supplying the co-efficients which are here repre- sented by -4, .B, C, &c. X^V, &c. 4. What is the 5th power of z 2 -j-3y 2 ? Substituting a for z 2 , and b for 3y 2 , (Art. 159,) we have And restoring the values of a and ( Z 2^_3 y 2)5_ a .10_|_15 a .8 y 2_|_90 x -f243y 10 . 5. What is the 6th power of (3z 6. What is the 2d power of (a b) ? 7. What is the 3d power of (a b) ? Q,UEST. Can this rule be applied to binomials whose co-efficients exceed 1 ? How ? Arts. 182, 183.] INVOLUTION. 103 8. What is the 4th power of (a 6) ? 9. What is the 6th power of (x y) ? 10. What is the nth power of (a 6) ? 182. When one of the terms of a binomial is a unit, it is generally omitted in the power, except in the first or last term ; because every power of 1 is 1, (Art. 165,) and this when it is a factor, has no effect upon the quantity with which it is connected. (Art. 70.) 11. Thus the cube of (z+1) is z 3+3 x 2 Xl+3xXl 2 +l*, Which is the same as x 3 +3x 2 +3x+l. 12. What is the 4th power of (a 1) ? The insertion of the powers of 1 is of no use, unless it be to preserve the exponents of both the leading and the fol- lowing quantity in each term, for the purpose of finding the co-efficients. But this will be unnecessary, if we bear in mind, that the sum of the two exponents, in each term, is equal to the index of the power. (Art. 179, 3.) So that, if we have the exponent of the leading quantity, we may know that of the following quantity, and v. v. 13. What is the 6th power of (1 y) ? 14. What is the nth power of (1+z) ? 183. The binomial theorem may also be applied to quan- tities consisting of more than two terms. By substitution, sev- eral terms may be reduced to two, and when the compound expressions are restored, such of them as have exponents may be separately expanded. (Art. 159.) 15. What is the cube of a+b+c ? Substituting h for (&+ c )> we have a-{-(b-\-c)=a-}-h. QUEST. When one of the terms of a binomial is a unit, how pro- ceed ? Can the binomial theorem be applied to quantities which hare more than two terms ? How ? 104 ALGEBRA. [Sect. VIII. And by the theorem, (a+h) 3 = a 3 +3a 2 A+3a/i 2 +A 3 . That is, restoring the value of 7t, The last two terms contain powers of (b-}-c) ; but these may be separately involved. 183. a. Binomials, in which one of the terms is a fraction, may be involved by actual multiplication ; or by reducing the given quantity to an improper fraction, and then involving the fraction according to Art. 171. 16. Find the square of z+ ; and x , as in Art. 173. X 2 +$X X 2 %X +1*+* i*+i Or, reduce the mixed quantities to improper fractions. Thus, s-f"2 3 i an( * s *~~o (^ rt * 120.) /2x+l\ 2 4z 2 +4z+l /2z 1\ 4* 2 4z+l ( d ) z:::: d ' ( 9 ) ~ aT * Q 17. Find the square of + o- O 4 + , ' 'JU- ' 18. Find the square of x --. ii'ii'3: * 19. Find the square of f-3zy. w ' 6 20. Find the square of \-2abc. -*% ' -^r^ QUEST. How involve a binomial when one term is a fraction ? Arts. 183..-185.a.] INVOLUTION. 105 EXAMPLES FOR PRACTICE. 1. Expand (x+y) 3 . 2. Expand (a-(-6) 4 . 3. Expand (a 6) 6 . 4. Expand (z+y) 5 . 5. Expand (z y) 8 . 6. Expand (m-\-n) 7 . 7. Expand (a+b) 9 . 8. Expand (x+y) 10 . 9. Expand ( x y) 13 . 10. Expand (a b) 7 . 11. Expand (a+6) 8 . 12. Expand (2+z) 5 . 13. Expand (a 6x+c) 3 . 14. Expand (+36c) 3 . 15. Expand (2a& z) 4 . 16. Expand (4ab 5c) 2 . 17. Expand (3z 6#) 3 . 18. Expand (5-(-3d) 3 . ADDITION OF POWERS. 184. It is obvious that powers may be added, like other quantities, by writing them one after another with their signs* (Art. 47.) 1. Thus the sum of a 3 and b 2 , is 3 +6 2 . 2. And the sum of a 2 b n and A 5 -d*, is a 2 b n +h 5 d*. 185. The same powers of the same letters are like quanti- ti^ (Art. 28 ;) hence their co- efficients ^may be added or sub- tracted, as in Arts. 50 and 51. 3. Thus the sum of 2 2 and 3a 2 , is 5 2 . 4. 5. 6. 7. 8. To Add 185.. But powers of different letters, and different powers of the same letter, are unlike quantities, (Art. 28 ;) hence they QUEST. What is the general method of adding powers? How are the same powers of the same letters added ? How are powers of differ- ent letters^ and different powers of the same letter, added ? 106 ALGEBRA. [Sect. VIII. can be added only by writing them down with their signs. (Art. 55.) 9. The sum of a 2 and a 3 , is a 2 -f-a 3 . It is evident that the square of a, and the cube of a, are neither twice the square of a, nor twice the cube of a. 10. The sum of a*b n and 3 5 i 6 , is 3 n +3a 5 & 6 . 186. From the preceding principles we deduce the fol- lowing GENERAL RULE FOR ADDING POWERS. I. If the powers are like quantities, add their co-efficients, and to the sum annex the common letter or letters with their given indices. 11. If the powers are unlike quantities, they must be added by writing them, one after another, without altering their signs. II. Add 5x(a by+x(a b) 3 to 2z(a 6) 3 +10x(a &) 3 . 12. Add 3(z+y) 4 +5a 2 4(*+y) 4 to 13. Add a 3 6 2 -f-z 6 y 4 +a 2 & 3 and x 14. Add 5 2 6c 3 , 3a 2 6c 3 , a 2 bc* and 15. Add 3 3 +6c 2 +5a 3 +26c 2 and a 3 +56c 2 to 6a+ 2bc 2 . 16. Add %(xy cm) G , 3(xy cm) 6 , ^(xy cm) 6 to SUBTRACTION OF POWERS. 187. RULE. Subtraction of powers is performed in the same manner as addition, except that the signs of the subtra- hend must be changed as in simple subtraction. (Art. 60.) Q,PEST. General rule for adding powers ? How are powers sub- tracted ? Arts. 186-188.a.] INVOLUTION. 107 1. From 2a 4 take 6a 4 . Ans. 8a*. 2. From 36" 3. 3A 2 6 6 4. a s b n 5. 5(a A) 6 Take 46 n 4A 2 6 6 3 6 n 2(a A) 6 6. From 6a(+6) 4 take a(a+6) 4 . 7. From 17a 2 x 3 +5zy 2 take 12a 2 x 3 ixy 2 . 8. From 3 3 (6 2 8) 3 take a*(b 2 8) 3 . 9. From a 2 b 3 +x 3 y i take a b b Q x 2 y 3 . 10. From 5(x 3 +# 4 ) 3 3(a 2 6 3 ) 5 take 3(a 2 & 3 ) 3 + 4(x 3 +^) 3 . 11. From 2x(a i) 3 +3(a 6) 3 take x(a 6) 3 +3(a~6) 3 . 12. From (x+y) 2 +(a+&) 3 take i (x+y) 2 +f(+6) 3 . MULTIPLICATION OF POWERS. 188. Powers may be multiplied, like other quantities, by writing the factors one after another, either with, or without, the sign of multiplication between them. (Art. 72.) 1. The product of a 3 into 6 2 is a 3 6 2 ; and x' 3 into cT is 2. Mult. h 2 b~ n 3. 3a 6 y 2 4. dh^x^ 5. a 2 6 3 y 2 Into a 4 2# 46y* a s b 2 y 188.a. If the quantities to be multiplied are powers of the same root, instead of writing the factors one after another, as in the last article, we may add their exponents, and the sum placed at the right hand of the root, will be the product required. The reason of this may be illustrated thus : a 2 Xa 3 is cz 2 a 3 , (Art. 188.) But a 2 =aa, and a s =aaa. And aaXaaa^aaaaa^a 5 , (Art. 80.) The sum of the ex- ponents 2+3, is also 5. So d m Xd n d m+n . N. B. The same principles hold true with respect to all other powers of the same root. 108 ALGEBRA. [Sect. VIII. 189. Hence we deduce the following GENERAL RULE FOR MULTIPLYING POWERS. I. Powers of the same root may be multiplied by adding their exponents. (Art. 167.) II. If the powers have co-efficients, these must be multiplied, together, and their product prefixed to the common letter or letters. III. Powers of different roots are multiplied by writing them one after another, either with, or without, the sign of multiplication between them. Thus a 2 Xa G =a 2+6 =a 8 . And * 3 X* 2 X*=* 3+2+1 =s 6 . 6. 7. 8. 9. 10. Mult. 4a n 3z* & 2 y 3 a 2 6 3 y 2 (b+k yT Into 2a rt 2z 3 b*y a z b 2 y &+& y 11. Mult. x z +x 2 y-\-xy 2 +y* into z y. 12. Mult. 4x 2 y+3xy 1 into 2x 2 x., 13. Mult. x 3 +x 5 into 2x 2 +x+l. 190. The rule is equally applicable to powers whose ex- ponents are negative, i. e. to reciprocal powers. 14. Thus a- 2 Xa- z =a-*. That is, X = aa aaa aaaaa 15. Mult. y~ n into y~ m into y" 4 . 16. Mult, a" 2 into a' 3 into a' 8 . 17. Mult, a" 2 into a 3 into a" 5 . 18. Mult, a"" into a m into a 2 ". 19. Mult. y~ 2 into y 2 into y~ n y~ s . QUEST. How are powers of the same root multiplied? Of differ- ent roots ? When the powers have co-efficients, what must be done with them ? Is this rule applicable to reciprocal powers ? Arts. 189-192.] POWERS. 109 20. If -|-6 be multiplied into a 6, the product will be a 2 6 2 , (Art. 86;) that is, 191. The product of the sum and difference of two quan- tities, is equal to the difference of their squares. This is an instanca of the facility with which general truths tare demonstrated in algebra. If the sum and difference of the squares be multiplied, the product will be equal to the difference of the fourth pow- ers, &c. 21. Mult, (a y) into (a+y). 22. Mult. (a 2 y 2 ) into (a 2 +y 2 ). 23. Mult, (a 4 y*) into ( 4 +y 4 ). 24. Mult. a 2 +a 4 +a 6 into a 2 -!. 25. Mult. 3(x 2 y 3 ) 3 into 2a(x 2 y 3 ) 4 . 26. Mutt. J(a 2 +6 3 ) 3 into J(o 2 +6 3 ) 2 . 27. Mult, a 3 6 2 into a s +6 2 . 28. Mult. x*+x 2 y+zy 2 +y* into x+y. 29. Mult, a 4 2a 3 &+4rt 2 6 2 8a6 3 +166 4 into a+26. 30. Mult. a 2 +6 into a 2 8. DIVISION OF POWERS. 192. Powers may be divided, like other quantities, by re- jecting from the dividend a factor equal to the divisor ; or by placing the divisor under the dividend, in the form of a fraction. Thus the quotient of a*b 2 divided by 6 2 , is a 3 . 1. 2. 3. 4. By 3a 3 26 3 a 2 ( a h-\-y)* QUEST. What is the product of the sum and difference of two quan- tities equal to ? 10 110 ALGEBRA. [Sect. VIII. 5. The quotient of a 5 divided by a 3 , is ^. But this is equal to a 2 . For, in the series a + S a+ 3 , a+ 2 , o +1 , a, a' 1 , a" 2 , ' 3 , a'S &c. if any term be divided by another, the index of the quotient will be equal to the difference between the index of the divi- dend and that of the divisor. Thus 8 4'= - 2 - And aaa a 193. Hence we deduce the following GENERAL RULE FOR DIVIDING POWERS. I. A power may be divided by another power of the same root, by subtracting the index of the divisor from that of the dividend. II. If the divisor and dividend have co-efficients, the co- efficient of the dividend must be divided by that of the divi" sor. (Art. 96.) III. If the divisor and dividend are both compound quan- tities^ the terms must be arranged, and the operation conduc- ted in the same manner, as in simple division of compound quantities. (Art. 107.) 6. Thus y^-^-y z y^- z y^. That is, ^ y. J \J 7. Divide a n+l by a. 8. Divide z n by x n . 9. 10. 11. 12. 13 Divide y 2m b 6 8a n+m n+3 By y m b 3 4a m a 2 194. The rule is equally applicable to reciprocal powers. Q.UEST. How is a power divided by another power of the same root? If they have co-efficients, how proceed? When the divisor and dividend are both compound quantities, how ? Is this rule appli- cable to reciprocal powers ? ' Arts. 193-194.a. ] POWERS. Ill Thus the quotient of " 5 by a" 3 , is a" 2 . 1 1 1 aaa aaa 1 That is ; = X r~ = aaaaa aaa aaaaa 1 aaaaa aa 15. Divide x" 5 by x~ 3 . 16. Divide A 2 by A' 1 . 17. Divide 6 n by 2a' 3 . 18. Divide ba 3 by a. 19. Divide b 3 by 6 5 . 20. Divide a 4 by a 7 . 21. Divide (a 3 -}-y 3 ) m by (a 3 -\-y 3 )". 22. Divide (&+*)" by (b+x). Examples of compound divisors with indices. (Art. 105.) 23. Divide a 3 +* 3 by a+x. 24. Divide a 4 +4.r 4 by a 2 2ax+2x 2 . 25. Divide x 6 1 by x 1. 26. Divide a 4 +2 3 6-|-2a 2 6 2 +a6 3 J)y a 3 +a 2 b-\-ab 2 . 27. Divide 6 s 16c 8 by b 2 2c 2 . 28. Divide a 6 a*x a 2 x 3 +2x 4 by a 4 x 3 . 29. Divide 4 +4a 3 6+6a 2 & 2 4-46 3 +6 4 by a 2 +2a6+6 2 . 30. Divide 8z 3 # 3 by 2x y. 31. Divide x 3 3ax 2 +3a 2 x a 3 by x a. 32. Divide 2y 3 19^ 2 +26y 16 by y8. 33. Divide X G 1 by x-f-1. 34. Divide 4x 4 9x 2 +6x 3 by 2x 2 +3x 1. 35. Divide 4 -j-4a 2 i-{-36 4 by a-\-2b. 36. Divide a 4 2 z 2 -f 2a 3 x a 4 by x 2 ax+a 2 . 194.a. A regular series of quotients is obtained, by divi- ding the difference of the powers of two quantities, by the difference of the quantities or roots. Thus, 112 ALGEBRA. [Sect. VIII. 37. Divide (y 2 a 2 ) by (y a). Ans. y+a. 38. Divide (y 3 a 3 ) by (y a). 39. Divide (y 4 a 4 ) by (y a). 40. Divide (y 5 a 5 ) by (y a). GREATEST COMMON MEASURE. 195. (1.) A common measure of two or more quantities, is a quantity which will divide, or measure them without a re- mainder. (Art. 30.) Thus 2d is a common measure of I2d, 6d, Sd, &c. (2.) The greatest common measure of two or more quanti- ties, is the greatest quantity which will divide these quantities without a remainder. Thus 6d is the greatest common mea- sure of I2d and ISdj and 8 is the greatest common measure of 16, 24 and 32. 195.cc. To find the greatest common measure of two or more quantities. Divide one of the quantities by the other, and the preceding divisor by the last remainder, till nothing remains ; the last divisor will be the greatest common measure. 196. The greatest common measure of two quantities is not altered, by multiplying or dividing either of them by any quantity which is not a divisor of the other, and which con- tains no factor which is a divisor of the other. The common measure of db and ac is a. If either be multiplied by d, the common measure of abd, and ac, or of ab and acd, is still a. On the other hand, if db and acd are Q,UEST. What is a common measure ? What the greatest common measure ? How found ? How is it affected by multiplying or dividing either of the quantities by any quantity which is not a divisor of the other ? Arts. 195-196.] COMMON MEASURE. 113 the given quantities, the common measure is a ; and if acd be divided by d, the common measure of ab and ac is a. Hence in finding the common measure by division, the divisor may often be rendered more simple, by dividing it by some quantity which does not contain a divisor of the divi- dend. Or the dividend may be multiplied by a factor, which does not contain a measure of the divisor. 1. Find the greatest common measure of 6a 2 -t-llax+3x 2 , and 6a 2 +7ax 3x 2 . 6a 2 +7ax-3x 2 )6a 2 +llx+3x 2 (l 7 that is, 2 multiplied into the root of a, or which is the same thing, twice the root of a. And z\/b, is xX\/6, or x times the root of 6. When no co-efficient is prefixed to the radical sign, 1 is always understood ; \^a being the same as l\/a, that is, once the root of a. 203. The cube root of a 6 is a 2 . For a 2 Xa 2 X<* 2 =a 6 . (Art. 199.) Here the index is divided into three equal parts, and the quantity itself resolved into three equal factors. The square root of a 2 is a 1 or a. For aX=a 2 - By extending the same plan of notation, fractional indices are obtained. Thus, in taking the square root of a 1 or a, the index 1 is divided into two equal parts, and ; and the root is a 2 . On the same principle, the cube root of a, is cF^/a. 1 The wth root, is a n =%/a, &c. 204. Every root, as well as every power of 1, is 1. (Art. 165.) For a root is a factor, which multiplied into itself will produce the given quantity. But no factor except 1 can pro- duce 1, by being multiplied into itself. So that 1", 1, \/l, -J/1, &c. are all equal. 205. Negative indices are used in the notation of roots, as well as of powers. (See Art. 163, 4.) Thus 4-= a ~*; J- = a""*j -L = a 2. a? a a an QUEST. When none is prefixed, what is understood ? What is every root of 1 ? Do roots ever have negative indices ? 118 ALGEBRA. [Sect. IX. POWERS OF ROOTS. 206. In the preceding examples of roots, the numerator of the fractional index has been a unit. There is another class of quantities, the numerators of whose indices are greater 2. 3. than 1, as 6 3 , c 4 , &c. These quantities may be considered either as powers of roots, or roots of powers. N. B. In all instances, when the root of a quantity is de- noted by a fractional index, the denominator, like the figure over the radical sign, (Art. 201,) expresses the root, and the numerator the power. Thus a? denotes the cube root of the first power of a, i. e. that a is to be resolved into three equal factors ; for a' 6 Xtf^Xa 5 ^^- On the other hand, c denotes the third power of the fourth root of c, or the fourth root of the third power. One expression is equivalent to the other. 1. What is c? equal to ? 2. What is z* equal to ? 3. What is y^ equal to ? 4. What is b* equal to ? 5. Write the fifth root of the fourth power of a. 6. Write the seventh power of the ninth root of d. 207. The value of a quantity is not altered, by applying to it a fractional index whose numerator and denominator are equal. 2. 2. n Thus a=a 2 a a. For the denominator shows that a is resolved into a certain number of factors ; and the nu- merator shows that all these factors are included in a . QUEST. What is meant by powers of roots ? What does the de- nominator of a fractional index express ? What the numerator ? Explain xT; also i^, cTtf, yTinr. When the numerator and denomi- nator are equal, how does the index affect the quantity ? How sim- plify such an expression ? Arts. 206-208.] ROOTS. 119 On the other hand, when the numerator of a fractional index becomes equal to the denominator, the expression may be rendered more simple by rejecting the index. Instead of a, we may write a. 207.a. The index of a power or root may be exchanged, for any other index of the same value. .2 4 Instead of a 3 , we may put a*. For in the latter of these expressions, a is supposed to be resolved into twice as many factors as in the former ; and the numerator shows that twice as many of these factors are to be multiplied together. Hence the value is not altered. 208. From the preceding article, it will be easily seen, that a fractional index may be expressed in decimals. 7. Thus a^z=a TTr , or a * 5 ; that is, the square root is equal to the fifth power of the tenth root. 8. Express a? in decimals. 9. Express a^ in decimals. 10. Express a 2 in decimals. 11. Express a? in decimals. 12. Express a in decimals. In many cases, however, the decimal can be only an ap- proximation to the true index. 13. Thus ota - 3 nearly, or tt ^ =a o.33333 more nearly, In this manner, the approximation may be carried to any degree of exactness which is required. 14. Express a? in decimals. 15. Express cT* in dec. N. B. These decimal indices form a very important class of numbers, called logarithms. Q,UEST. What is the effect when one index is exchanged for an- other index of the same value ? Can a fractional index he expressed in decimals ? Can it be expressed exactly by decimals in all cases ? What class of numbers are thus found ? 120 ALGEBRA. [Sect. IX. EVOLUTION. f~"~ 209. The process of resolving quantities into equal fac- tors, is called Evolution. In subtraction, a quantity is resolved into two parts. In division, a quantity is resolved into two factors. In evolution, a quantity is resolved into equal factors. Evolution is the opposite of involution. One is finding a power of a quantity, by multiplying it into itself. The other is finding a root, by resolving a quantity into equal factors. A quantity is resolved into any number of equal factors, by dividing its index into as many equal parts. 210. From the foregoing principles we deduce the following GENERAL RULE FOR EVOLUTION. I. Divide the index of the quantity by the number express- ing the root to be found. Or, place over the quantity the radical sign belonging to the required root. II. If the quantities have co-efficients, the root of these must be extracted and placed before the radical sign, or quantity. Thus, To find the square root of d* , divide the index 4 by 2, i. e. d*d 2 . So the cube root of d 6 , is d^^d 2 . Obser. From the manner of performing evolution it is evident, that the plan of denoting roots by fractional indices, is derived from the mode of expressing powers by integral indices. (Art. 203.) Q.UEST. What is evolution ? Into what are quantities resolved in subtraction? Into what, in division ? Into what, in evolution ? How is a quantity resolved into any number of equal factors ? Rule for evolution ? What is the plan of denoting roots by fractional indices derived from ? Arts. 209-210.a.] ROOTS. 121 1. Required the cube root of a 6 . Ans. a 3 . 2. Required the cube root of a or a 1 . Ans. cr or $/a. For a^Xa^Xa^, or ^/aX^/aX^/a=a. (Art. 199.) 3. Required the fifth root of ab. 4. Required the rath root of a 2 . 5. Required the seventh root of 2d z. 6. Required the fifth root of (a z) 3 . 7. Required the cube root of a 2 . 8. Required the fourth root of a" 1 . 9. Required the cube root of a 5 . 10. Required the nth root of x m . 11. Required the third root of a 6 . 12. Required the fourth root of x 8 . 13. Required the second root of z n . 14. Required the fifth root of d 3 . 15. Required the 8th root of a 3 . 2 10. a. The rule in the preceding article may be applied to every case in evolution. But when the quantity whose root is to be found, is composed of several factors, there will fre- quently be an advantage in taking the root of each of the factors separately. This is done upon the principle that the root of the pro- duct of several factors, is equal to the product of their roots. Thus Vafc \/aX \fb. For each member of the equa- tion if involved, will give the same power. When, therefore, a quantity consists of several factors, we may either extract the root of the whole together ; or we may QUEST. What is the root of the product of several factors equal to ? 11 122 ALGEBRA. [Sect. IX. find the root of the factors separately, and then multiply them into each other. 16. The cube root of zy, is either (xy) 1 * or z^y^. 17. Required the fifth root of 3y. 18. Required the sixth root of abh. 19. Required the cube root of 86. 20. Required the nth root of x n y. 211. The root of a fraction is equal to the root of the nu- merator divided by the root of the denominator. 4 i J 21. Thus the square root of J=^T. For X^r ? 6 6* 6* &i 22. Required the nth root of ^. 23. Required the square root of . 212. SIGNS. 1. An odd root of any quantity has the same sign as the quantity itself. 2. An even root of an affirmative quantity is ambiguous. N. B. An even root of a negative quantity is impossible. 213. But an even root of an affirmative quantity may be either positive or negative. For, the quantity may be pro- duced from the one, as well as from the other. (Art. 169.) Thus the square root of a 2 is +, or a. An even root of an affirmative quantity is, therefore, said to be ambiguous, and is marked with the sign -4-. Thus the square root of 36, is \/3b. The 4th root of x, is -4-x^. The ambiguity does not exist, however, when from the nature of the case, or a previous multiplication, it is known QUEST. What is the root of a fraction equal to ? Rule for signs ? What is the even root of a negative quantity ? tarts. 211-216.] ROOTS. 123 (whether the power has actually been produced from a positive jor from a negative quantity. 214. But no even root of a negative quantity can be found. The square root of a 2 is neither -\-a nor a. For +aX+a=+a*. And aX a=+a 2 also. An even root of a negative quantity is, therefore, said to be impossible or imaginary. 215. The methods of extracting the roots of compound quantities are to be considered in a future section. But there is one class of these, the squares of binomial and residual quantities, which it will be proper to attend to in this place. The square of a-\-b, for instance, is a 2 -\-2ab-\-b 2 , two terms of which, a 2 and 6 2 , are complete powers, and 2ab is twice the product of a into 6, that is, the root of a 2 into the root of b 2 . Whenever, therefore, we meet with a quantity of this de- scription, we may know that its square root is a binomial ; and this may be found, by taking the root of the two terms which are complete powers, and connecting them by the sign -f-. The other term disappears in the root. Thus, to find the square root of x 2 -\-2xy-\-y 2 , take the root of x 2 , and the root of y 2 , and connect thcrji by the sign -j-. The bino- mial root will then be z-\-y. In a residual quantity, the double product has the sign prefixed, instead of +. The square of a 6, for instance, is a 2 2ab-\-b 2 . (Art. 173.) And to obtain the root of a quantity of this description, we have only to take the roots of the two complete powers, and connect them by the sign . Thus the square root of x 2 %xy-\-y 2 , is x y. Hence, 216. To extract the square root of a binomial or residual. Take the roots of the two terms which are complete powers, and connect them by the sign which is prefixed to the other term. QUEST. How extract the square root of a binomial, or residual ? 124 ALGEBRA. [Sect. IX. 1. To find the root of ar+2z+l. The two terms which are complete powers, are x 2 and 1. The roots are ff and 1. (Art. 204.) Then z-f-1. Ans. 2. Find the square root of x 2 2x+l. (Art. 173.) 3. Find the square root of a 4. Find the square root of 2 I 2 5. Find the square root of a 2 -\-ab \ . 6. Find the square root of a 2 c c 217. A root whose value cannot be exactly expressed in numbers, is called a SURD, or irrational quantity. Thus \/2 is a surd, because the square root of 2 cannot be expressed in numbers, with perfect exactness. In decimals, it is 1.41421356 nearly. 218. Every quantity which is not a surd, is said to be ra- tional. 219. By RADICAL QUANTITIES is meant, all quantities which are found under the radical sign, or tvhich have a fractional index. REDUCTION OF RADICAL QUANTITIES. 220. CASE I. To reduce a rational quantity to the form of a radical without altering its value. Raise the quantity to a power of the same name as the given root) and then apply the corresponding radical sign or index. QUEST. What is a surd? What a rational quantity? What are radical quantities ? How reduce a rational quantity to the form of a radical ? Arts. 217-221.] RADICAL QUANTITIES. 125 1. Reduce a to the form of the wth root. The wth power of a is a n . (Art. 166.) Over this, place the radical sign, and it becomes ^/a n . It is thus reduced to the form of a radical quantity, with- n out any alteration of its value. For :/"=. 2. Reduce 4 to the form of the cube root. 3. Reduce 3a to the form of the 4th root. 4. Reduce $ab to the form of the square root. 5. Reduce 3Xa x to the form of the cube root. 6. Reduce a 2 to the form of the cube root. N. B. In cases of this kind, where a power is to be reduced to the form of the wth root, it must be raised to the rath power, not of the given letter, but of the power of the letter. Thus in the 6th example, a 6 is the cube, not of a, but of a 2 . 7. Reduce a 3 6 4 to the form of the square root. 8. Reduce a m to the form of the wth root. 221. CASE II. To reduce quantities which have different indices, to others of the same value having a common index. 1. Reduce the indices to a common denominator. 2. Involve each quantity to the power expressed by the nu- merator of its reduced index. 3. Take the root denoted by the common denominator. 9. Reduce a 4 and b^ to a common index. 1st. The indices and reduced to a common denomina- tor, are & and &. (Art. 118.) 2d. The quantities a and b involved to the powers expressed by the two numerators, are a 3 and b 2 . QUEST. How reduce quantities which have different indices to a common index ? 11* 126 ALGEBRA. [Sect. IX. 3d. The root denoted by the common denominator is the T Vth. The answer, then, is (a 3 ) 1 ^" and (6 2 ) T ^. The two quantities are thus reduced to a common index, without any alteration in their values. For by Art. 207.a, *=ia&, which by Art. 206, ^(a 3 ) 1 ^. 1 J!L JL And universally, a n =a mn =(a m ) mn . -i ? 10. Reduce a? and bx* to a common index. Ans. a* and 6z* ora 3 ^ and &%**. 1 11 11. Reduce a 3 and b n . 12. Reduce x n and y m . 13. Reduce 2^ and 3^. 14. Reduce (a+b) 2 and (xyfi. 15. Reduce cfi and b^. 16. Reduce x% and 5^. 222. CASE III. To reduce a quantity to a given index. Divide the index of the quantity by the given index, place the quotient over the quantity, and set the given index over the whole. This is merely resolving the original index into two factors. (Art. 209.) 17. Reduce a^ to the index . By Art 135,' H-i=*xf =f =*. This is the index to be placed over a, which then becomes a*; and the given index set over this, makes it (dv*, the answer. 18. Reduce a 2 and x* to the common index . 2-^-1=2x3=6, the first index. -H=X3=:f, the second index. \ ) i S- 4- Therefore (a 6 ) 3 and (ar) 3 are the quantities required. QUEST. How reduce a quantity to a given index ? Arts. 222, 223.] RADICAL QUANTITIES. 127 19. Reduce 4^ and 3F to the common index |. 20. Reduce x 2 and y 4 to the common index . 21. Reduce cr and 6 3 to the common index . 22. Reduce c 2 and d* to the common index f . - - i 23. Reduce a m and b m to the common index -^. 24. Reduce a 2 , b^ and c* to the common index -j^-. 223. CASE IV. To reduce a radical quantity to its most sim- ple terms ; i. e. to remove a factor from under the radical sign. Resolve the quantity into two factors, one of which is an exact power of the same name with the root ; find the root of this power, and prefix it to the other factor, with the radical sign betiveen them. This rule is founded on the principle, that the root of the product of two factors is equal to the product of their roots. (Art. 210.0.) It will generally be best to resolve the radical quantity into such factors, that one ojf them shall be the greatest power which will divide the quantity without a remainder. N. B. If there is no exact power which will divide the quan- tity, the reduction cannot be made. 25. Remove a factor from \/8. The greatest square which will divide 8 is 4. We may then resolve 8 into the factors 4 and 2. For 4X2=8. The root of this product is equal to the product of the roots of its factors ; that is A/8=V4XV%- But \/4z=2. Instead of \/4, therefore, we may substitute its equal 2. We then have 2X\/2 or 2\/2. QUEST. How reduce a radical quantity to its simplest terms. 128 ALGEBRA. [Sect. IX. 26. Reduce ^/a^x. Ans. */a 2 X*/x aX\fx a*/z. 27. Reduce \/18. 28. Reduce 29. Reduce / 30. Reduce 31. Reduce (a 3 a 2 bft. 32. Reduce (54a 6 6)*. 33. Reduce \/98a 2 x. 34. Reduce v 224. CASE V. To introduce a co-efficient of a radical quan- tity under the radical sign. (Art. 220.) Raise the co-efficient to a power of the same name as the radical part , then place it as a factor under the radical sign. 35. Thus, ayb V~tfb. For a y^ora"". (Art. 207.) And 36. Reduce a(x 6) 3 to the form of a radical. Ans. (o's 8 fc)*. 1 n I 7)2 r \2 37. Reduce 2ab(2ab 2 )*. 38. Reduce |f 24.^2) ' 39. Reduce 2\/2. 40. Reduce 4b$/c. EXAMPLES FOR PRACTICE. 1. Reduce 5\/6 to a simple radical. 2. Reduce \/5a to a simple radical. 3. Reduce 5^ and 6 7 to the common index . 4. Reduce a 2 and a 2 to the common index . 5. Reduce \/98 to its simplest form. 6. Reduce \/243 to its simplest form. QUEST. How introduce a co-efficient under a radical sign ? Arts. 224-225.a.] RADICAL QUANTITIES. 129 7. Reduce ^/54 to its simplest form. 8. Reduce 7\/80 to its simplest form. 9. Reduce 9y81 to its simplest form. 10. Reduce \/x 2 -\-ax 2 to its simplest form. 11. Reduce \^198a 2 x to its simplest form. 12. Reduce \^x 3 a 2 x 2 to its simplest form. ADDITION OF RADICAL QUANTITIES. 225. It may be proper to remark, that the rules for addi- tion, subtraction, multiplication and division of radical quan- tities are essentially the same, and are expressed in nearly the same language, as those for addition, subtraction, multi- plication and division of powers. So also the rules for invo- lution and evolution of radicals, are similar to those for invo- lution and evolution of powers. Hence, if the learner has made himself thoroughly acquainted with the principles and operations relating to powers, he has substantially acquired those pertaining to radical quantities, and will find no diffi- culty in understanding and applying them. 25. . Whe radical quantities have th&^ame radical part) and are under the same radical sign or index, they are like quantities. (Art. 28.) Hence their rational parts or co-effi- cients may be added in the same manner as rational quanti- ties, (Art. 56,) and the sum prefixed to the radical part. Thus, 2v"6+3V6 1. Add y a y to 2. Add 2\/ to 5\/a. 3. Add 4zA* to QUEST. What is said respecting the rules for addition, subtraction, multiplication and division ; also of involution and evolution of radi- cals? How are radical quantities added, when the radical parts are alike f 130 ALGEBRA. [Sect. IX. 4. Add 76A to 5. Add y*/bh to 226. If the radical parts are originally different, they may sometimes be made alike, by the rules for reduction of radi- cal quantities. 6. Add \/8 to \S5Q. Here the radical parts are not the same. But by the reduction in Art. 223, V'S 2\/2, and V50=5\/2. The sum then is 7/y/2. 7. Add V166 to 8. Add \fa 2 x to 9. Add (36a 2 y)* to 10. Add V18a to 3</8. 7. From ?/6 4 y take Q.UEST. How are radical quantities subtracted ? 132 ALGEBRA. [Sect. IX. 8. From %/x take %/x. 9. From 2\/5Q take \/lS. 10. From /320 take -/40. 11. From 5\/20 take 3\/45. 12. From \/8Qa*x take MULTIPLICATION OF RADICAL QUANTITIES. 230. Radical quantities may be multiplied, like other quan- tities, by writing the factors one after another, either with or without the sign of multiplication between them. (Art. 72.) 1. Thus the product of \/ into */b, i g \/aX*/b. 2. The product of $ into y&, is J$y%. But it is often expedient to bring the factors under the same radical sign.' This may be done, if they are first reduced to a common index. (Art. 221.) 231. Hence, quantities under the same radical sign or index, may be multiplied together like rational quantities, the pro- duct being placed under the common radical sign or index.* (Art. 210.) 3. Multiply %/x into %/y, that is, z* into y*. The quantities reduced to the same index, (Art. 221,) are (z 3 )*, and (y 2 )^, and their product is, (x s y 2 )*=$/x*y 2 . 4. Multiply \fa-\-m into \/a m. 5. Multiply \/dx into \/hy. _____ _ ____________________ QUEST. How may radical quantities be multiplied ? How are fac- tors brought under the same radical sign? How multiplied when un- der the same radical sign? * The case of an imaginary root of a negative quantity may be con- sidered an exception. (Art. 214.) Arts. 230-232.] RADICAL QUANTITIES. 133 6. Multiply cfi into x*. 7. Multiply (a+yY into ~ 8. Multiply a m into z n . 9. Multiply \fSxb into \^2xb. Prod. In this manner the product of radical quantities often be- comes rational. 10. Thus the product of \/2 into \/18z=\/36=6. 11. Multiply (a 2 y*)* into (a 2 y)%. 232. Roots of the same letter or quantity may le multi* plied, ly adding their fractional exponents. N. B. The exponents, like all other fractions, must be re- duced to a common denominator, before they can be united in one term. (Art. 122.) 12. Thus c$Xc$ a^=a^=a^. The values of the roots are not altered, by reducing their indices to a common denominator. (Art. 207.a.) Therefore the first factor c$a% ) And the second af^a 5 * But a=aXaX (Art. 206.) And a%=a*Xa*. The product therefore is a*Xa*Xa*Xa%X=Vd>' Hence, 238. Quantities under the same radical sign or index, may be divided like rational quantities, the quotient being placed under the common radical sign or index. 5. Divide (a; 3 ^ 2 )^ by 3^. These reduced to the same index are (x 3 y 2 )^ and (y 2 ) *. And the quotient is (z 3 ) F z x a . QUEST. How is the division of radicals expressed ? How is the radical sign to be placed in this case ? How divide quantities under the same radical sign ? 12* 138 ALGEBRA. [Sect. IX. 6. Divide */6a 3 x by */3x. 7. Divide Vdhx 2 by Vdx. 8. Divide (a*+ax)^ by a*. 9. Divide (a 3 A) by (axf. 10. Divide (a 2 y 2 )* by (ay)%. 239. A root is divided by another root of the same letter or quantity, by subtracting the index of the divisor from that of the dividend. 11. Thus a*-^o*= a*~*=a*~*=a*=fl*. For a^=a*=a?Xd*Xa?, and this divided by a* is i* _l 1 1 _ 1 12. In the same manner, a m -~-a n ~i 13. Divide (3a)^ by 'a*. 14. Divide (ax)* by (ax)*. 15. Divide a by a. 2 . 16. Divide (6-f^) n by 17. Divide (r 2 y s ) r by (r 2 y 3 ) T . 239.a. Powers and roote of the same letter, may also be divided by each other, according to the preceding article. 18. Thus a 2 -a?=a 2 ~ : *=a?. For a*Xcfi= cF=a 2 . QUEST. How divide one root by another root of the same letter ? How powers and roots of the same letter ? Arts. 239-241.] RADICAL QUANTITIES. 139 240. When radical quantities which are reduced to the same index, have rational co-efficients, the rational parts may be divided separately, and their quotient prefixed to the quo- tient of the radical parts. 19. Thus ac\/bd--a\/b=:c\/d. For this quotient multi- plied into the divisor is equal to the dividend. 20. Divide 24x\/ay by 6/v/- 21. Divide 18dh*/bx 1 1 22. Divide by(a*x 2 ) n by y(ax) n . 23. Divide 16\/32 by 8\/4. 24. Divide b\^xy by \fy. 25. Divide a6(x 2 6)* by a(zfi. These reduced to the same index are ab(x 2 b)* and a(x 2 )*. The quotient then is b(b)%=(b b fi. (Art. 224.) To save the trouble of reducing to a common index, the division may be expressed in the form of a fraction. The quotient will then be ^- . (*)* 241. Hence we deduce the following GENERAL RULE FOR DIVIDING RADICALS. I. If the radicals consist of the same letter or quantity, sub- tract the index of the divisor from that of the dividend, and place the remainder over the common radical part or root. II. If the radicals have co-efficients, the co-efficient of the dividend must be divided by that of the divisor. (Art. 96.) QUEST. When the radicals have co-efficients, what is to be done with them ? General rule for dividing radical quantities ? 140 ALGEBRA. [Sect. IX. III. If the quantities have the same radical sign or index, divide them as rational quantities, and place the quotient under the common radical sign. (Art. 193.) EXAMPLES FOR PRACTICE. 1. Divide 2%/bc by 2. Divide 10^/108 by 5^/4. 3. Divide 10^27 by 2*/3. 4. Divide &v/108 by 2V6. 5. Divide (a 2 b 2 d 3 )* by d*. 6. Divide ( 16a 3 I2a 2 x)^ by 2a. 7. Divide 6y removing the index or radical sign. When the radical quantities have rational co-efficients , these must be involved by actual multiplication. 9. Thus the cube of Vb+x is 6-f x. _i 10. And the nth power of (a y) n is (a y.) 11. The square of a^/x is a 21 (/x 2 . For a 12. Required the wth power of a m x m . 13. Required the square of a*/x y. 14. Required the cube of 3a $/y. 244. But if the radical quantities are connected with others by the signs -f- and , they must be involved by a multipli- cation of the several terms, as in Art. 172. 15. Required the squares of a+VV anc * QUEST. How is a root raised to a power of the same name ? If the radicals have co-efficients, how proceed ? If the radicals are com- pound quantities, how ? 142 ALGEBRA. [Sect. IX. a 16. Required the cube of a \fb. 17. Required the cube of 2d-\-\^x. 18. Required the 4th power of \fd. 19. Required the 4th power of v ax 1. _ 20. Required the 6th power of *-0 ; ^ 11. Reduce \/ 5 X Vx+2=2- r.^^^^'-^ #'r&'~~fi 12. Reduce ^=^ -i \/x x ^ QUEST. Wlien the unknown quantity is under the radical sign, how is the equation reduced ? What preparation is it advisable to make before involving the quantities ? 13 15. Reduce x+V^+x^. 16. Reduce x+a=+(62+). ; 17. Reduce V^ +V z= ;7 ^. - ^ ^ *J ' 18. Reduce Vx 32=16 \/^- Af* 19. Reduce /137423+22577 163=237. 251. When an equation is reduced by extracting an even root of a quantity, the solution does not always determine whether the answer is positive or negative. (Art. 212.) But what is thus left ambiguous by the algebraic process, is fre- quently settled by the statement of the problem. Art. 251.] QUADRATIC EQUATIONS. 149 Prob. 3. A merchant gains in trade a sum, to which 320 dollars bears the same proportion as five times this sum does to 2500. What is the amount gained ? | -\ *? ^ % % Prob. 4. The distance to a certain place is such, that if 96 be subtracted from the square of the number of miles, the remainder will be 48. What is the distance ? / ;X Prob. 5. If three times the square of a certain number be divided by 4, and if the quotient be diminished by 12, the remainder will be 180. What is the number ? / (f Prob. 6. What number is that, the fourth part of whose square being subtracted from 8, leaves a remainder equal to 4? Prob. 7. What two numbers are those, whose sum is to the greater as 10 to 7 ; and whose sum multiplied into the less produces 270 ? Q Prob. 8. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128? $ ^ Prob. 9. It is required to divide the number 18 into two such parts, that the squares of those parts may be to each other as 25 to 16. / Q Prob. 10. It is required to drvTde the number 14 into two such parts, that the quotient of the greater divided by the less, may be to the quotient of the less divided by the greater, as 16 to 9. & %Q Prob. 11. What two numbers are as 5 to 4, the sum of whose cubes is 5103 ? ;j /% /, Prob. 12. Two travellers, A and B, set out to meet each other, A leaving the town C, at the same time that B left D, They travelled the direct road between C and D ; and on meeting, it appeared that A had travelled 18 miles more than B, and that A could have gone B's distance in 15 j days, 13 . 150 ALGEBRA. [Sect. X. but B would have been 28 days in going A's distance. Re- quired the distance between C and D. Prob. 13. Find two numbers which are to each other as 8 to 5, and whose product is 360. <^ fy 4*^^f f C s Prob. 14. A gentleman bought two pieces of silk, which together measured 36 yards. Each of them cost as many shillings by the yard, as there were yards in the piece, and their whole prices were as 4 to 1. What were the lengths of the pieces ? A- Prob. 15. Find two numbers which are to each other as 3 to 2 ; and the difference of whose fourth powers is to the sum of their cubes, as 26 to 7. Prob. 16. Several gentlemen made an excursion, each taking the same sum of money. Each had as many servants attending him as there were gentlemen ; the number of dol- lars which each had was double the number of all the ser- vants, and the whole sum of money taken out was 3456 dol- lars. How many gentlemen were there ? f \^ Prob. 17. A detachment of soldiers from a regiment being ordered to march on a particular service, each company fur- nished four times as many men as there were companies in the whole regiment ; but these being found insufficient, each company furnished three men more ; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment ? AFFECTED QUADRATIC EQUATIONS. 252. Equations are divided into classes, which are distin- guished from each other by the power of the letter that ex- presses the unknown quantity. Those which contain only QUEST. Into what are equations divided ? Arts. 252, 253.] QUADRATIC EQUATIONS. 151 the first power of the unknown quantity are called simple equations, or equations of the first degree. Those in which the highest power of the unknown quantity is a square, are called quadratic, or equations of the second degree ; those in which the highest power is a cube, are called cubic, or equa- tions of the third degree, &c. Thus x=ia-{-b, is an equation ofthejirst degree. x 2 e, and z 2 -{-ax=d, are quadratic equations, or equations of the second degree, x*=7t, and x*+ax 2 +bx=d, are cubic equations, or equations of the third degree. 253. Equations are also divided into pure and affected equa- tions. A pure equation contains only one power of the un- known quantity. This may be the first, second, third, or any other power. An affected equation contains different powers of the unknown quantity. Thus, C x 2 =d 6, is a pure quadratic equation. C x 2 -\-bx=d, an affected quadratic equation. C z 3 6 c, a pure cubic equation. C x 3 +ax 2 +&x=rA, an affected cubic equation. In a. pure equation, all the terms which contain the unknown quantity may be united in one, (Art. 185,) and the equation, however complicated in other respects, may be reduced by the rules which have already been given. But in an affected equation, as the unknown quantity is raised to different pow~ ers, the terms containing these powers cannot be united. (Art. 185.a.) QUEST. What are those called which contain only the first power of the unknown quantity ? When the unknown quantity is a square what? When a cube? How else are equations divided ? What is a pure equation ? 152 ALGEBRA. [Sect. X. 254. An affected quadratic equation is one which contains the unknown quantity in one term, and the square of that quan- tity in another term. The unknown quantity may be originally in several terms of the equation. But all these may be reduced to two, one containing the unknown quantity, and the other its square. 255. It has already been shown that a pure quadratic is solved by extracting the root of both sides of the equation. An affected quadratic may be solved in the same way, if the member which contains the unknown quantity is an exact square. Thus the equation x 2 -\-2ax-{-a 2 =b-{-h 1 may be reduced by evolution. For the first member is the square of a binomial quantity. (Art. 173.) And its root is z-j-a. There fore J _> z-(-a=V6+A, and by transposing a, x= Vb+h a. 256. But it is not often the case, that a member of an affected quadratic equation is an exact square, till an addi- tional term is applied, for the purpose of making the required reduction. In the equation x 2 -\-2ax-b, the side containing the un- known quantity is not a complete square. The two terms of which it is composed are indeed such as might belong to the square of a binomial quantity. (Art. 173.) But one term is wanting. We have then to inquire, in what way this may be supplied. From having two terms of the square of a bino- mial given, how shall we find the third ? Of the three terms, two are complete powers, and the other is twice the product of the roots of these powers, or QUEST. What is an affected equation ? How is a pure quadratic equatum solved ? How an affected quadratic, when it is an exact square ? Arts. 254-258.] QUADRATIC EQUATIONS. 153 which is the same thing, the product of one of the roots into twice the other. In the expression x 2 +2az, the term 2ax consists of the factors 2a and x. The latter is the unknown quantity. The other factor 20 may be considered the co-efficient of the un- known quantity ; a co-efficient being another name for a fac- tor. (Art. 24.) As x is the root of the first term x 2 ; the other factor 20 is twice the root of the third term, which is wanted to complete the square. Therefore half 20 is the root of the deficient term, and a 2 is the term itself. The square completed is x 2 -|-20x+0 2 , where it will be seen that the last term a 2 is the square of half 20, and 2a is the co-efficient of x, the root of the first term. In the same manner, it may be proved, that the last term of the square of any binomial quantity, is equal to the square of half the co-efficient of the root of the first term. 257. From this principle is derived the following METHOD FOR COMPLETING THE SQUARE. Take the square of half the co-efficient of the first power of the unknown quantity , and add it to loth sides of the equation. 258. It will be observed that there is nothing peculiar in the solution of affected quadratics, except the completing of the square. Quadratic equations are formed in the same manner as simple equations ; and after the square is com- pleted, they are reduced in the same manner as pure equations. 1. Reduce the equation x 2 -{-6ax=b Completing the square, x 2 -|-6ax-|-9a 2 9 2 -f-& Extracting both sides, (Art. 255,) And Here the co-efficient of x, in the given equation, is 60. _________ _____ QUEST. What is the first method for completing the square ? What is there peculiar in the solution of quadratics ? 154 ALGEBRA. [Sect. X. The square of half this is 9a 2 , which being added to both sides completes the square. The equation is then reduced by extracting the root of each member, in the same manner as in Art. 249, excepting that the square here being that of a binomial, its root is found by the rule in Art. 216. 2. Reduce the equation x 2 8bxh. 3. Reduce the equation x 2 -\-axb-\-h. a 2 a 2 Completing the square, x 2 -\-ax-\ = \ b-\-h. 4. Reduce the equation x 2 x=h d. 5. Reduce the equation x 2 -\-3xd-{-6. 6. Reduce the equation x 2 abxdb cd. 7. Reduce the equation x 2 -\- =h. 8. Reduce the equation x 2 -7h. b 259. In these and similar instances, the root of the third term of the completed square is easily found, because this root is the same half co-efficient from which the term has just been derived. (Art. 257.) Thus in the last example, half the co-efficient of x is -, and this is the root of the third term . 46 2 260. When the first power of the unknown quantity is in several terms, these should be united in one, if they can be by the rules for reduction in addition. But if there are lite- ral co-efficients, these may be considered as constituting, QUEST. How do you know what the root of the third term of the completed square is ? When the first power is in several terms, what is to be done ? If there are literal co-efficients what ? Arts. 259-262.] QUADRATIC EQUATIONS. 155 together, a compound co-efficient or factor, into which the unknown quantity is multiplied. Thus ax+lx+dx(a+b+d)Xx. (Art. 97.) The square of half this compound co-efficient is to be added to both sides of the equation. 9. Reduce the equation x 2 -\-3x-{-2x-\-x=.d Uniting terms x 2 -{-6x=d Completing the square x 2 -\-6x-{-9=:9-\-d And x 10. Reduce the equation x 2 -\-ax-{-bx=h By Art. 120, *'+(+&) X*=A Therefore x 2 +(a+b)X*+ 11. Reduce the equation x 2 -\-ax x=b. \s 261. Before completing the square, the known and un- f known quantities must be brought on opposite sides of the equation by transposition ; the square of the unknown quan- tity must also WJ positive, and it is preferable to make it the first or leading term. 12. Reduce the equation a-}-5x 36=3a: a; 2 Transp. and uniting terms x 2 -{-2x=3b a Completing the square x 2 -j-2a:-f-l l-j-36 a And a; a; 36 13. Reduce the equation - = r-^ x | & 262. If the highest power of the unknown quantity has a co-ef' ficient, or divisor, before completing the square it must be freed from these by multiplication or division. (Arts. 149, 154.) Q,UEST. Before completing the square what preparations is it expe- dient to make? If the highest power has a co-efficient or divisor, what should be done ? 156 ALGEBRA. [Sect. X. 14. Reduce the equation a? 2 +24a 6hl2x 5x 2 Transp. and uniting terms 6x 2 12x 6h 24a Dividing by 6, x 2 2x= hAa Completing the square, x 2 2x-\-Il-\-h 4a Extracting and transp. x=l^-*v \-\-h 4. bx 2 15. Reduce the equation h-\-2x=d -- . a 263. If the square of the unknown quantity is in several terms, the equation must be divided by all the co-efficients of this square. (Art. 155.) 16. Reduce the equation bx 2 -{-dx 2 4z=b h Dividing by b+d, * 2 17. Reduce the equation az 2 -\-x=h-\-3x x 2 . Given ax 2 -{-bx=d, to find x. t If this equation is multiplied by 4a, and if b 2 is added to both sides, it will become, the first number of which is a complete square of the bino- mial 2ax-\-b. 264. From the foregoing principle is deduced A SECOND METHOD OF COMPLETING THE SQUARE. Multiply the equation by 4 times the co-efficient of the highest power of the unknown quantity, and add to both sides the square of the co-efficient of the lowest power. The advantage of this method is, that it avoids the intro- duction of fractions, in completing the square. QUEST. If the square of the unknown quantity is in several terms, how proceed ? What is the second method of completing the square ? What advantage has this method ? Arts. 263, 264.] QUADRATIC EQUATIONS. 157 DEMONSTRATION. 1. The object of multiplying the equation by the co-effi- cient of the highest power, is to render the first term a per- fect square without removing its co-efficient, and at the same time to obtain the middle term of the square of a binomial. But we must multiply all the terms of the .equation by this quantity to preserve the equality of its members. (Ax. 3.) The equation above when mult, by a becomes a 2 x 2 +abx=ad. That the first term will, in all cases, be rendered a complete square when multiplied by its co-efficient, is evident from the fact, that it will then consist of two factors, each of which is a square, viz. x 2 , and the square of its co-efficient. But the product of the squares of two or more factors, is equal to the square of their product. (Art. 167.) 2. It will be seen that one term is still wanting in the first member, in order to make it the square of a binomial, viz. the square of the last term. (Art. 173.) This deficiency may be supplied by adding to both sides the square of half the co-efficient of the lowest power, as in the first method of completing the square. But in taking half of this co-efficient, the learner will often be encumbered with fractions which it is desirable to avoid. Thus in the equation above, half of the co-efficient of the lowest power is -, the I 2 square of which is -. Adding this to both sides, the equa- b 2 I 2 tion will become, a 2 x 2 '-\-abx-\- ad-\- , the first mem- ber of which is a complete square of the binomial, ax-\- ^. QUEST. Why is the equation multiplied by the co-efficient of the highest power ? How does it appear that this will make the first term an exact square ? Why multiply the equation by 4? 14 158 ALGEBRA. [Sect. X. Now it is obvious, that multiplying the equation by 4, is the same as removing the denominator 4 from the third term. Hence multiplying the equation by 4 will avoid the introduc- tion of fractions, and also leave the square of the whole of the co-efficient of the lowest power to be added to both sides ac- cording to the rule. The first term evidently continues to be a square after it is multiplied by 4, for it is still the product of the powers of certain factors. (Art. 167.) 3. It will be perceived at once, that the second term is composed of twice the root of the first term into the co-effi- cient of the last term, which constitutes the middle term of a binomial square. (Art. 173.) Obser. It is manifest from the preceding demonstration, that multiply- ing by 4 is not a necessary step in completing the square, but is resorted to as an expedient to prevent the occurrence of fractions. When therefore the co-efficient of the lowest power is an even number, so that half of it can be taken without a remainder, we may simplify the operation by multiplying by the co-efficient of the highest power alone, and ad- ding to both sides the square of half the co-efficient of the lowest power of the unknown quantity. Take the equation, 7z 2 -f-40z=7lf-. Multiplying by 7, it becomes 49z 2 -f-280z=500 Adding the square of half the co-efficient, 49z 2 -f-280z-|-4 00=900 By evolution and transposition, x=40. 265. From the preceding principles we may also deduce OTHER METHODS OF COMPLETING THE SQUARE. Multiply the equation by 16 times the co-efficient of the highest power of the unknown quantity, and add to both sides 4 times the square of the co-efficient of the lowest power. QUEST. Why add the square of the co-efficient of the lowest power to both sides ? How may the operation be simplified when the co- efficient of the lowest power is an even number? Arts. 265, 265.a.] QUADRATIC EQUATIONS. 159 And universally, multiplying the equation by the product of any square number, as n 2 , into the co-efficient of the high' est power, and adding to both sides the square of half the root of this number into the square of the co-efficient of the lowest power, will render it a complete square. Take the equation x 2 3z 4 Multiplying by 16, &c. 16z 2 48z-f-36=100 By evolution and transposition, x=4 Or, take the equation ax 2 -}-cxd. Mult, by n 2 , &c. n 2 a 2 x 2 +n 2 acx+^-=n 2 ad+^- ; the first member of which is the square of the binomial, naz-f- -. 4> There is an obvious advantage, however, in employing 4 in preference to any other square number. For multiplying the equation by 4 times the co-efficient of the highest power, will produce the middle term of a binomial square, the third term of which is the square of the co-efficient of the lowest power. 18. Reduce the equation ax 2 -\-dx=^h. 19. Reduce the equation 3x 2 -\-5x=42. 20. Reduce the equation x 2 15zi=z 54. 265.a. In the square of a binomial, the first and last terms are always positive. For each is the square of one of the terms of the root, and all even powers are positive. (Arts. 168, 173.) If then x 2 occurs in an equation, it cannot with this sign form a part of the square of a binomial. But if all the signs in the equation be changed, whilst the equality of the sides will be preserved, the term x 2 will become positive, and the square may then be completed. (Art. 146.) QUEST. What other ways of completing the square are mentioned? In the square of a binomial, what sign have the first and last terms ? If the square of the unknown quantity has the sign before it, what must be done ? 160 ALGEBRA. [Sect. X. 21. Reduce the equation x 2 -}-^x=d h Changing all the signs x 2 2xr=A d. 22. Reduce the equation 4x x 2 = 12. 266. In a quadratic equation, the first term x 2 is the square of a single letter. But a binomial quantity may consist of terms, one or both of which are already powers. Thus x 3 + is a binomial, and its square is x 6 -\-2ax* +a 2 , where the index of x in the first term is twice as great as in the second. When the third term is deficient, the square may be completed in the same manner as that of any other binomial. For the middle term is twice the product of the roots of the two others. So the square of o^+a, is x 2n +2ax n +a 2 . 1 2. J. And the square of # n -)-a, is x n -\-2ax n -}-a 2 . Therefore, 267. Any equation which contains only two different pow- ers or roots of the unknown quantity, the index of one of which is twice that of the other , may be solved in the same manner as a quadratic equation, by completing the square. N. B. It must be observed, that in the binomial root, the letter expressing the unknown quantity may still have a frac- tional or integral index, so that a farther operation may be necessary. (Art. 250.) 23. Reduce the equation x* x 2 ~6 a Completing the square x- Extracting and transposing, x 2 =j Extracting again, (Art. 249,) x: 24. Reduce the equation x 2n 46x n =. QUEST. How solve equations which contain only two different powers or roots of the unknown quantity, when the index of one is twice that of the other ? Arts. 266-269.] QUADRATIC EQUATIONS. 161 25. Reduce the equation z-[-4\/z=A . 2 JL 26. Reduce the equation x n -\-8z n =a-{-b. 268. The solution of a quadratic equation, whether pure or affected, gives two results. For after the equation is re- duced, it contains an ambiguous root. In a pure quadratic, this root is the whole value of the unknown quantity. Thus the equation z 2 =:64, Becomes when reduced, a; ;t\/64. (Art. 249.) That is, the value of x is either +8 or 8, for each of these is a root of 64. Here both the values of x are the same, except that they have contrary signs. This will be the case in every pure quadratic equation, because the whole of the second member is under the radical sign. The two values of the unknown quantity will be alike, except that one will be positive, and the other negative. 269. But in affected quadratics, a part only of one side of the reduced equation is under the radical sign. When this part is added to, or subtracted from, that which is without the radical sign ; the two results will differ in quantity, and will have their signs in some cases alike, and in others unlike. 27. The equation Becomes when reduced, x=. That is, Here the first value of x is 4-j-6z=-f- 2 I one positive, and And the second is 4 6 10 ) the other negative. 28. The equation x 2 8x= 15 Becomes when reduced a?=:4\/16 15 That is, QUEST. How many results does the solution of a quadratic give ? In pure quadratics, iu the whole value ambiguous ? Is this the case in affected quadratics ? 14* 162 ALGEBRA. [Sect. X. Here the first value of x is 4+1^+5 ) both ^.^ And the second is 4 lzr:-f-3 * That these two values of x are correctly found, may be proved by substituting first one and then the other, for x itself, in the original equation. (Art. 161.) Thus 5 2 8X5:^25 40 15 And 3 2 8X3= 9 24= 15. 270. In the reduction of an affected quadratic equation, the value of the unknown quantity is frequently found to be imaginary. 29. Thus the equation x 2 8x= 20 Becomes, when reduced, a:=:4 = h'\/16 20 That is, x=4\/4. Here the root of the negative quantity 4 cannot be as- signed, (Art. 214,) and therefore the value of x cannot be found. There will be the same impossibility, in every in- stance in which the negative part of the quantities under the radical sign is greater than the positive part. 271. When one of the values of the unknown quantity in a quadratic equation is imaginary, the other is so also. For both are equally affected by the imaginary root. Thus in the example above, The first value of x is 4-+V 4 And the second is 4 \/ 4 ; each of which contains the imaginary quantity \/ 4. 272. An equation which when reduced contains an ima- ginary root, is often of use, to enable us to determine whether QUEST. Is the value of the unknown quantity ever imaginary ? When one of the values is imaginary, what is true of the other? Are equations containing an imaginary root of any use ? What use ? Arts. 270-274.] QUADRATIC EQUATIONS. 163 a proposed question admits of an answer, or involves an ab- surdity. 30. Suppose it is required to divide 8 into two such parts that the product will be 20. If x is one of the parts, the other will be 8 x. By the conditions proposed (8 z)Xz 20 This becomes when reduced, x=4dzV f 4. Here the imaginary expression \/ 4 shows that an an- swer is impossible ; and that there is ,an absurdity in suppo- sing that 8 may be divided into two such parts, that their product shall be 20. 273. Although a quadratic equation gives two results, yet both these may not always be applicable to the subject pro- posed. The quantity under the radical sign may be produced either from a positive or a negative root. But both these roots may not, in every instance, belong to the problem to be solved. (Art. 251.) 31. Divide the number 30 into two such parts, that their product may be equal to 8 times their difference. If z= the less, then 30 z the greater part. By the supposition, zX(30 x)= 8X(30 2z). This reduced, gives x 23 17=40 or 6= the less part. But as 40 cannot be a part of 30, the problem can have but one real solution, making the less part 6, and the greater part 24. 274. The preceding principles may be summed up in the following GENERAL RULE. I. Transpose all the unknown quantities to one side of the equation ; and the known quantities to the other. Q,UF.ST. Are both of the results of a quadratic always applicable to the problem under consideration ? What is the general rule for the solution of quadratic equations ? 164 ALGEBRA. [Sect. X. II. Make the square of the unknown quantity positive (if it is not already) by changing the signs of all the terms on both sides ; and place it for thejirst or leading term. (Art. 261.) III. To complete the square, 1. Remove the co -efficient of the second power of the un- known quantity, and add the square of half of the co-efficient of thejirst power of the unknown quantity to loth sides of the equation. (Art. 257.) Or, 2. Multiply the equation by four times the co-efficient of the highest power of the unknown quantity, and add to both sides the square of the co-efficient of thejirst power of the unknown quantity. (Art. 264.) IV. Reduce the equation by extracting the square root of both sides ; and transpose the known part of the binomial root thus obtained to the opposite side. (Art. 255.) -W/ EXAMPLES FOR PRACTICE. 1. Reduce 3x 2 9x 4=80. Ol? 2. Reduce 4x -- =46. x 3. Reduce 4x ll 14. x+l 4. Reduce 5x- =2x+ 16 100 9x Q 5. Reduce --- - =3. x 4x 2 6 . Reduce Q , X 3 10x 2 -fl 8. Reduce ^-J x 2 6x-{~9 Art. 274.] QUADRATIC EQUATIONS. 165 9. Reduce 10. Reduce -_ =0: 9. 6 11. Reduce - + - = -. ax a 12. Reduce x*+ax 2 =b. 13. Reduce _= 14. Reduce 15. Reduce 16. Reduce 2x 4 x 2 +96=99. 17. Reduce (10+x)* 18. Reduce 3x 2n 2x n =8. 19. Reduce 2(l+x x 2 ) Vl+x x 2 =z . 20. Reduce 21. Reduce 22. Reduce 23. Reduce 24. Reduce 25. Reduce x+16 7-V: 26. Reduce . 3z 7 27. Reduce --- - = x 3x+7 13x 166 ALGEBRA. [Sect. X. 28 . Reduce -l--" X- K - 29. Reduce (z 5) 3(z 5)*=40. ^ 30. Reduce *4V^=2+3V7p. #** *- i v ,4 PROBLEMS IN QUADRATIC EQUATIONS. Prob. 1. A merchant has a piece of cotton cloth, and a piece of silk. The number of yards in both is 110: and if the square of the number of yards of silk be subtracted from 80 times the number of yards of cotton, the difference will be 400. How many yards are there in each piece ? Let #= the yards of silk. Then 110 z=the yards of cotton. By supposition 80 X ( 1 10 a) a: 2 =400. Therefore x= 40\/ 10000= 40;= 100. The first value of x, is 40-|- 100=60, the yards of silk ; And 1 10 a?= 1 10 60=50, the yards of cotton. The second value of x, is 40 100= 140 ; but as this is a negative quantity, it is not applicable to goods which a man has in his possession. t Prob. 2. The ages of two brothers are such, that their sum n is 45 years, and their product 500. What is the age of each ? *" Prob. 3. To find two numbers such, that their difference shall be 4, and their product 117. Prob. 4. A merchant having sold a piece of cloth which cost hin^O dollars, found that if the price for which he sold it were multiplied by his gain, the product would be equal to the cube of his gain. What was his gain ? !* Prob. 5. To find two numbers whose difference shall be 3, and the difference of their cubes 117. Art. 274.] QUADRATIC EQUATIONS. 167 Prob. 6. To find two numbers whose difference shall be 12, and the sum of their squares 1424. J), (,) Prob. 7. Two persons draw prizes in a lottery, the differ- ence of which is 120 dollars, and the greater is to the less, as the less to 10. What are the prizes ? *f > ' Prob. 8. What two numbers are those whose sum is 6, and the sum of their cubes 72 ? / / J r<~~~ Prob. 9. Divide the number 56 into two such parts, that their product shall be 640. q j | L Prob. 10. A gentleman bought a number of pieces of cloth for 675 dollars, which he sold again at 48 dollars by the piece, and gained by the bargain as much as one piece cost him. What was the number of pieces ? J A~~ Prob. 11. A and B started together, for a place 150 miles distant. A's hourly progress was 3 miles more than B's, and ' .-., he arrived at his journey's end 8 hours and 20 minutes before B. What was the hourly progress of each ? $ $ i vi/ rK/ Prob. 12. The difference of two numbers is 6 ; and if 47 be added to twice the square of the less, it will be equal to the square of the greater. What are the numbers ? // Prob. 13. A and B distributed 1200 dollars each, among a certain number of persons. A relieved 40 persons more than B, and B gave to each individual 5 dollars more than A. How many were relieved by A and B ? /jduO ^ ! * ^0 (*&* Prob. 14. Find two numbers whose sum is 10, and the sum of their squares 58. y f- ji Prob. 15. Several gentlemen made a purchase in company for 175 dollars. Two of them having withdrawn, the bill was paid by the others, each furnishing 10 dollars more than would have been his equal share if the bill had been paid by the whole company. What was the number in the company at first ? 168 ALGEBRA. [Sect. X. Prob. 16. A merchant bought several yards of linen for 60 dollars, out of .which he reserved 15 yards, and sold the remainder for 54 dollars, gaining 10 cents a yard. How many yards did he buy, and at what price ? Prob. 17. A and B set out from two towns, which were 247 miles distant, and travelled the direct road till they met. A went 9 miles a day ; and the number of days which they travelled before meeting, was greater by 3, than the number of miles which B went in a day. How many miles did each travel ? Prob. 18. A gentleman bought two pieces of cloth, the finer of which cost 4 shillings a yard more than the other. The finer piece cost <18 ; but the coarser one, which was 2 yards longer than the finer, cost only <16. How many yards were there in each piece, and what was the price of a yard of each ? Prob. 19. A merchant bought 54 gallons of Madeira wine, and a certain quantity of Teneriffe. For the former, he gave half as many shillings by the gallon, as there were gallons of TenerifFe, and for the latter, 4 shillings less by the gallon. He sold the mixture at 10 shillings by the gallon, and lost ^28 16s. by his bargain. Required the price of the Madeira, and the number of gallons of Teneriffe. Prob. 20. 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 be multiplied by 2, and the product divided by the number itself, the quotient will be 4. What is the number? Prob. 21. A person being asked his age, replied, If you add the square root of it to half of it, and subtract 12, the remainder will be nothing. What was his age ? ^, /. Prob. 22. Two casks of wine were purchased for 58 dol- lars, one of which contained 5 gallons more than the other, and the price by the gallon, was 2 dollars less than J of the Art. 274.] QUADRATIC EQUATIONS. 169 number of gallons in the smaller cask. Required the number of gallons in each, and the price by the gallon. Prob. 23. In a parcel which contains 24 coins of silver and copper, each silver coin is worth as many cents as there are copper coins, and each copper coin is worth as many cents as there are silver coins ; and the whole are worth 2 dollars and 16 cents. How many are there of each ? Prob. 24. A person bought a certain number of oxen for 80 guineas. If he had received 4 more oxen for the same money, he would have paid one guinea less for each. What was the number of oxen ? Prob. 25. It is required to divide 24 into two such parts that their product shall be equal to 35 times their difference. Prob. 26. The sum of two numbers is 60, and their product is to the sum of their squares as 2 to 5. What are the num- bers ? Prob. 27. Divide 146 into two such parts, that the difference of their square roots may be 6. Prob. 28. What two numbers are those whose difference is 16, and their product 36 ? Prob. 29. Find two numbers whose sum shall be 1J and the sum of their reciprocals 3. Prob. 30. Required to find two numbers whose difference is 15, and half of their product is equal to the cube of the less number ? Prob. 31. A company incurred a bill of 8 15s. Two of them absconded before it was paid, and in consequence, those who remained had to pay 10s. a piece more than their just share. How many were there in the company ? Prob. 32. A gentleman bequeathed <7 4s. to his grandchil- dren ; but before the money was distributed two more were 15 170 ALGEBRA. [Sect. X. added to their number, and consequently the former received one shilling a piece less than they otherwise would have done. How many grandchildren did he leave ? Prob. 33. The length added to the breadth of a rectangu- lar room makes 42 feet, and the room contains 432 square feet. Required the length and breadth. Prob. 34. A says to B, " the product of our years is 120 ; and if I were 3 years younger, and you were 2 years older, the product of our ages would still be 120." How old was each ? Prob. 35. Should the square of a certain number be taken from 89, and the square root of their difference be increased by 12, and the sum multiplied by 4, and the product divided by the number itself, the quotient will be 16. What is the number ? Prob. 36. A mason laid 105 rods of wall, and on reflection found that if he had laid 2 rods less per day, he would have been 6 days longer in accomplishing the job. How many rods did he build per day ? Prob. 37. The length of a gentleman's garden exceeded its breadth by 5 rods. It cost him 3 dols. per rod to fence it ; and the whole number of dollars was equal to the number of square rods in the garden. What were its length and breadth ? Prob. 38. What number is that, which being added to its square root will make 156 ? Prob. 39. The circumference of a grass-plot is 48 yards, and its area is equal to 35 times the difference of its length and breadth. What are its length and breadth ? Prob. 40. A gentleman purchased a building lot, and in the center of it, erected a house 54 feet long and 36 feet wide, Art. 274.] QUADRATIC EQUATIONS. 171 which covered just one half his land. This arrangement left him a flower border of uniform width all round his house. What was the width of his border, what the length and breadth of his lot, and how much land did he buy ? Prob. 41. A general wished to arrange his army, which consisted of 1200 men, in a solid body, so that each rank should exceed each file by 59 men. How many must he place in rank and file ? Prob. 42. A man has a painting 18 inches long, and 12 inches wide, which he orders the cabinet maker to put into a frame of uniform width, and to have the area of the frame equal to that of the painting. Of what width will the frame be ? Prob. 43. A and B together invest 8500 in business, of which each put in a certain share. A's money continued in trade 5 months, B's only two months, and each received back $450 for his capital and profit. What share of the stock did each contribute ? Prob. 44. A merchant sold a quantity of goods for <39, and gained as much per cent, as the goods cost him. How much did he pay for the goods ? Prob. 45. A farmer bought a flock of sheep for ,60. Af- ter selecting 15 of the best, he sold the remainder for ,54, and gained thereby 2 shillings a head. How many sheep did he buy, and what was the price of each ? Prob. 46. A and B started from two cities 247 miles apart, and travelled the same road till they met. A*s progress was 9 miles per day, and the number of days before they met was greater by 3 than the number of miles B went per day. How many miles did each travel ? Prob. 47. Two persons, A and B, invest $900 in business. A's money remained in trade 4 months, and he received $512 172 ALGEBRA. [Sect. XI. for his share of the profit and stock ; B's money was in trade 7 months, and he received $469 for his share of the profit and stock. What was each partner's stock ? Prob. 48. A merchant bought a piece of cloth for $54 ; the number of shillings which he paid per yard was f of the number of yards. Required the length of the cloth, and the price per yard. Prob. 49. There was a cask containing 20 gallons of wine ; a quantity of this was drawn off and put into another cask of equal size, and then this last was filled with water ; and af- terwards the first cask was filled with the mixture from the second. It appears that if 6f gallons are now drawn from the first and put into the second, there will be equal quanti- ties of wine in each cask. How much wine was first drawn off? Prob. 50. A man bought 80 Ibs. of pepper and 100 Ibs. of ginger for <65, at such prices that he obtained 60 Ibs. more of ginger for <20 than he did of pepper for .10. What did he pay per pound for each ? SECTION XI. TWO UNKNOWN QUANTITIES 275. IN the examples given in the preceding sections, each problem has contained only one unknown quantity. Or if, in some instances, there have been two, they have been so re- lated to each other, that they have both been expressed by means of the same letter. Arts. 275-277.] UNKNOWN QUANTITIES. 173 But cases frequently occur, in which two unknown quanti- ties are necessarily introduced into the same calculation. Suppose the following equations are given. 1. z+y=14 2. xy2. If y be transposed in each, they will become 2. x=2+y. Here the first member of each of the equations is z, and the second member of each is equal to x. But according to Axiom 7, quantities which are respectively equal to any other are equal to each other ; therefore, 2+y 14 y, and y=6. By substituting the value of y in the 1st equation, (Art. 159,) we have x+6z=14 ; then x=S. 276. In solving the preceding problem, it will be observed that we first found the value of the unknown quantity #, in each equation ; and then by making one of the expressions denoting the value of x equal to the other, (Axiom 7,) we formed a new equation, which contained only the other un- known quantity y. This process is called extermination or elimination. There are three methods of extermination, viz. by com- parison, by substitution, and by addition and subtraction. EXTERMINATION BY COMPARISON. 277. CASE I. To exterminate one of two unknown quan- tities by comparison. QUEST. How are problems solved which contain two unknown quantities ? What is this process called ? How many methods of ex- termination ? Name them. . 15* 174 ALGEBRA. [Sect. XI. Find the value of one of the unknown quantities in each of the equations, and form a new equation by making one of these values equal to the other. Prob. 1. Given x+y = 36 ) TQ find ^ ^ f and And xy = 12 f 1. In the first equation a;-{-yz=:36 2. In the second equation x y 12 3. Transposing y in first equation x=36 y 4. Transposing y in second equation x=12-\-y 5. Making 3d and 4th equal, (Ax. 7,) I2-\-y36 y 6. Transposing, &c. y 12 Substituting value ofy in 4th (Art. 159,) z= 12+ 12 24. Prob. 2. Given 2x3=28 To find the value of x and And Prob. 3. Given 4x+y=43 Given 4x+y=43 ) ^ And 5x+2 ^56) Tofindthevalueofxand ^ Prob. 4. Given 4z 2w= 16 ) And 6x=9 y r To find the value of x and y Prob. 5. Given 4x 2/=20 And To find the value of x and Prob. 6. Given And the Prob. 7. To find two numbers such, that their sum shall be 24 ; and the greater shall be equal to five times the less. Let z= the greater ; and y= the less. Prob. 8. To find one of two quantities, whose sum is equal to h ; and the difference of whose squares is equal to d. Prob. 9. Given ax+by=h ) To find And x4-v d ) QUEST. What is the rule to exterminate one of two unknown quantities by comparison ? Arts. 278, 279.] UNKNOWN QUANTITIES, 175 278. When the value of one of the unknown quantities is determined, the other may be easily obtained by substituting, in one of the previous equations, the value of the one found for the quantity itself. (Art. 159.) The rule given above, may be generally applied for the extermination of unknown quantities. But there are cases in which other methods will be found more expeditious. Prob. 10. Suppose z=% And ax-\-bx=y 2 As in the first of these equations z is equal to hy, we may in the second equation substitute this value of x instead of z itself. The second equation will then become, ahy-\-bhy=y 2 . The equality of the two sides is not affected by this altera- tion, because we only change one quantity x for another which is equal to it. By this means we obtain an equation which contains only one unknown quantity. This process is called extermination by substitution. Hence, 279. CASE II. To exterminate an unknown quantity by substitution. Find the value of one of the unknown quantities, in one of the equations ; and then in the other equation, SUBSTITUTE this value for the unknown quantity itself. (Art. 159.) Prob. 11. Given find ^ And 4z-|-5y=32 Transposing 3y in the 1st equation, zrrlS 3y. Substituting the value of x in the 2d equation, (Art. 159,) we have 60 I2y+5y=32 Then y And zr=15 12=3. QUEST. After the value of one unknown quantity is found, how obtain the other ? What is the second method of extermination called ? What is the rule ? 176 ALGEBBA. [Sect. XI. Prob. 12. Given 8*+y=42 ) T fi d h rf f and And 2z+4y=18 J Prob. 13. Given 2H-8y=84) Tofindthevalueofxand And 4z+6y=68> Prob. 14. Given 3*+3y=72 1 Tofind thevalueofxandy . And 4z5:rrll6 ) Prob. 15. Given And Prob. 16. A privateer in chase of a ship 20 miles distant, sails 8 miles, while the ship sails 7. How far will each sail before the privateer will overtake the ship ? Prob. 17. The ages of two persons, A and B, are such that seven years ago, A was three times as old as B ; and seven years hence, A will be twice as old as B. What is the age of each ? Prob. 18. There are two numbers, of which the greater is to the less as 3 to 2 ; and their sum is the 6th part of their product. What are the numbers ? 280. There is a third method of exterminating an unknown quantity from an equation, which in many cases, is preferable to either of the preceding. Prob. 19. Suppose that x-\-3y=a And x 3y 6 If we add together the first members of these two equa- tions, and also the second members we shall have 2x=a+b, an equation which contains only the unknown quantity #. The other, having equal co-efficients with contrary signs, has disappeared. (Art. 54.) The equality of the sides is pre- served because we have only added equal quantities to equal quantities. Arts. 280, 281.] UNKNOWN QUANTITIES. 177 Again, suppose 3x-\-y=h And 2x+y d If we subtract the last equation from the first, we shall have x=hd, where y is exterminated, without affecting the equality of the sides. Again, suppose z 2y a And a;-}-4y & Multiplying the 1st by 2, 2x 4y=2a, Then adding the 2d and 3d, 3x=b+2a. This process is called extermination by addition and sub- traction. Hence, 281. CASE III, To exterminate an unknown quantity by addition and subtraction. Multiply or divide the equations, if necessary , in such a manner that the term which contains one of the unknown quan- tities shall be the same in both. Then subtract one equation from the other, if the signs of this unknown quantity are alike, or add them together, if the signs are unlike. N. B. It must be kept in mind that both members of an equation are always to be increased or diminished alike, iri order to preserve their equality. Prob. 20. Given 2z-Hy=20 > ^ J* > To find the value of x and y. And 4z--5:=r28 ) 1. Mult, the 1st equation by 2, 4z+8y=40 2. The 2d equation is 4z+5y=:28 Subtracting the 2d from the 1st, 3^=12 Dividing, &c. y 4 ; and z 2. QUEST. What is the third method of extermination called ? What is the rule? What is the object of multiplying the equation by a certain quantity ? How do you know when to add and when to subtract ? 178 ALGEBRA. [Sect. XI. Prob. 21. Given 2x+y=l6 ) ~ , ,, And 3z - %=6 I T find the ValUG f X and y ' Prob. 22. Given 4x+3^50 | TQ fim] ^ yalue of ^ And 3x %= 6 J Prob. 23. Given 3z4-2y 38 ) c , ., , f A y > To find the value of x and y. And 5z+4y=68 J Prob. 24. Given 4z 40= 4y ) ,. . , 2* > To find the val. of x and y . And 6x 63= 7# f Prob. 25. The numbers of two opposing armies are such, that the sum of both is 21110 ; and twice the number in the greater army, added to three times the number in the less, is 52219. What is the number in each army ? Prob. 26. A boy purchased 8 lemons and 4 oranges for 56 cents. He afterwards bought 3 lemons and 8 oranges for 60 cents. What did he pay for each ? Prob. 27. The sum of two numbers is 220, and if 3 times the less be taken from 4 times the greater, the remainder will be 180. What are the numbers ? 282. In the solution of the succeeding problems, either of the three rules for exterminating unknown quantities may be used at pleasure. N. B. That quantity which is the least involved should be the one chosen to be exterminated first. The pupil will find it a useful exercise to solve each ex- ample by each of the several methods, and carefully observe which is the most comprehensive, and the best adapted to different classes of problems. Prob. 28. The mast of a ship consists of two parts : one third of the lower part added to one sixth of the upper part, is equal to 28 feet ; and five times the lower part, diminished by six times the upper part, is equal to 12 feet. What is the height of the mast ? Art. 282.] UNKNOWN QUANTITIES. 179 Prob. 29. To find a fraction such that, if a unit be added to the numerator, the fraction will be equal to J ; but if a unit be added to the denominator, the fraction will be equal to J. Let the numerator, And y the denominator. 1. By the first condition, = By the second, TT^J 3. Therefore x=4, the numerator. 4. And y 15, the denominator. Prob. 30. What two numbers are those, whose difference is to their sum, as 2 to 3 ; whose sum is to their product, as 3 to 5 ? Prob. 31. To find two numbers such, that the product of their sum and difference shall be 5, and the product of the sum of their squares and the difference of their squares shall be 65. Prob. 32. To find two numbers whose difference is 8, and product 240. Prob. 33. To find two numbers, whose difference shall be 12, and the sum of their squares 1424. Prob. 34. A certain number consists of two digits or figures, the sum of which is 8. If 36 be added to the num- ber, the digits will be inverted. What is the number ? Prob. 35. The united ages of A and B amount to a certain number of years consisting of two digits, the sum of which is 9. If 27 years be subtracted from the amount of their ages, the digits will be inverted. What is the sum of their ages ? Prob. 36. A merchant having mixed a quantity of brandy and gin, found, if he had put in 6 gallons more of each, that the compound would have contained 7 gallons of brandy for - 180 ALGEBRA. [Sect. XL every 6 of gin ; but if he had put in 6 gallons less of each, the proportions would have been as 6 to 5. How many gal- lons did he mix of each ? THREE UNKNOWN QUANTITIES. 283. In the preceding examples of two unknown quanti- ties, it will be perceived that the conditions of each problem have furnished two equations independent of each other. It often becomes necessary to introduce three or more unknown quantities into a calculation. In such cases, if the problem admits of a determinate answer, there will always arise from the conditions as many equations independent of each other, as there are unknown quantities. 284. Equations are said to be independent when they ex- press different conditions. They are said to be dependent when they express the same conditions under different forms. The former are not con- vertible into each other ; but the latter may be changed from one form to the other. Thus b %y ; and b=y-^-x, are dependent equations, because one is formed from the other by merely transposing x. Obser. Equations are said to be identical when they express the same thing in the same form ; as 4x 6 = 4r-6. Prob. 37. Suppose x+y+z=l2 ) 4 - , O 1A f are S lVen t0 filld *1 And x+2y 2%=10 > V y, and z. And x-\-y z=4 } From these three equations, two others may be derived which shall contain only two unknown quantities. One of the three in the original equations may be exterminated, in Q,UEST. How many independent equations does a problem of three or more unknown quantities furnish ? What are independent equa- tions ? What are dependent ones ? What identical ones ? Arts. 283-285.] UNKNOWN QUANTITIES. 181 the same manner as when there are at first only two, by the rules already given. In the equations given above, if we transpose y and z, we shall have, In the first, xn=12 y z. In the second, x= 10 2y-f-2z. In the third, x=: 4 y-\-z. From these we may deduce two new equations, from which x shall be excluded. By making the 1st and 2d equal, 12 y z=W 2y-f-2z. By making the 2d and 3d equal, 10 2y-f-2zr=4 y+z. Reducing the first of these two, y=3z 2. Reducing the second, y z-[-6. From these two equations one may be derived containing only one unknown quantity, Making one equal to the other, 3z 2 z-f-(> And z~ 4. Hence, 285. To solve a problem containing three unknown quan- tities, and producing three independent equations. First, from the three equations deduce two, containing only two unknown quantities. Then, from these two deduce one, containing only one un- known quantity. For making these reductions, the rules already given are sufficient. (Arts. 277, 279, 281.) Prob. 38. Given z+5y+6z=53 ^ 2. And r-f 3y+3z 30 > To find a?, y and ^ 3. And arz 12 ) From these three equations to derive two, containing only two unknown quantities, QUEST. Rule for solving problems with three unknown quantities? 16 182 ALGEBRA. [Sect. XI. 4. Subtract the 2d from the 1st, 2y-\-3z=23. 5. Subtract the 3d from the 2d, 2y+2z=l8. From these two, to derive one, 6. Subtract the 5th from the 4th, z=5. To find x and y, we have only to take their values from the third and fifth equations. (Art. 278.) 7. Reducing the fifth, y=9 2=9 5=4. 8. Transposing in the third, #=12 z y=12 5 4=3. Prob. 39. Given z+y-|-z=12 ^ And z-f2y+3z=20 (TO find x, y and z, And z4-y-|-z=6 } 286. In many of the examples in the preceding sections, the processes given might have been shortened. But the object has been to illustrate general principles, rather than to furnish specimens of expeditious solutions. The learner will do well, as he passes along, to exercise his skill in abridging the calculations which are here given, or substituting others in their stead. Prob. 40. Given z-f-y=a \ And z-j-z =6 > To find x, y and z. And y-j-^ c ' Prolx 41. Three persons, A, B, and C, purchase a horse for 100 dollars, but neither is able to pay for the whole. The payment would require, The whole of A's money, together with half of B's ; or The whole of B's, with one third of C's ; or The whole of C's, with one fourth of A's. How* much money had each ? 287. The learner must exercise his own judgment, as to the choice of the quantity to be first exterminated. It will QUEST. How do you know which unknown quantity to extermin- ate first ? Arts. 286-288.] UNKNOWN QUANTITIES. 183 generally be best to begin with that which is most free from co-efficients, fractions, radical signs, &c. Prob. 42. The sum of the distances which three persons, A, B, and C, have travelled, is 62 miles ; A's distance is equal to 4 times C's, added to twice B's ; and Twice A's added to 3 times B's, is equal to 17 times C's. What are the respective distances ? ^ Prob. 43. Given jx+Jy+Jz 62 ) And Jz+Jy+i*= 47 > To find x, y and z. SSJ And Prob. 44. Given xy = 600 \ And xz= 300 I To find z, y and z. And z=20oJ FOUR UNKNOWN QUANTITIES. 288. The same method which is employed for the reduc- tion of three equations, may be extended to 4, 5, or any num- ber of equations, containing as many unknown quantities. The unknown quantities may be exterminated, one after another, and the number of equations may be reduced by successive steps from five to four, from four to three, from three to two, &c. Prob. 45. To find 10, z, y and z, from 1. The equation 2. And x+y+u>9 I ,, 1rt r F ur equations. 3. And x+y+z=l2 4. And x+w+z=10 } 5. Clear, the 1st of frac. y+2z-f-w=16" 6. Subtract 2d from 3d, z w=r3 Three equations. 7. Subtract 4th from 3d, y w=2^ QUEST. How are problems solved containing four or five unknown quantities ? 184 ALGEBRA. [Sect. XI . 8. Adding 5th and 6th, y +3z^:19 ) ^ tions< 9. Subtract 7th from 6th, y+zl ) 10. Adding 8th and 9th, 4 2 =20. Or z=5^ 11. Transposing in the 8th, y 19 3z=4 I Quantities 12. Transposing in the 3d, x=l2 y z=^3 \ required. 13. Transposing in the 2d, w= 9 x y=:2 j Prob. 46. Given w+ 5Q= And And And z+195:=3w> Answer. w= Prob. 47. There is a certain number consisting of two digits. The left-hand digit is equal to three times the right- hand digit ; and if twelve be subtracted from the number itself, the remainder will be equal to the square of the left- hand digit. What is the number ? Prob. 48. If a certain number be divided by the product of its two digits, the quotient will be 2 ; and if 27 be added to the number, the digits will be inverted. What is the number? Prob. 49. There are two numbers, such, that if the less be taken from 3 times the greater, the remainder will be 35 ; and if 4 times the greater be divided by 3 times the less-f-1, the quotient will be equal to the less. What are the numbers ? Prob. 50. There is a certain fraction, such, that if 3 be added to the numerator, the value of the fraction will be J ; but if 1 be subtracted from the denominator, the value will be I. What is the fraction ? Prob. 51. A gentleman has two horses, and a saddle which is worth ten guineas. If the saddle be put on thejirst horse, the value of both will be double that of the second horse ; but if the saddle be put on the second horse, the value of both Art. 288.] UNKNOWN QUANTITIES. 185 will be less than that of the first horse by 13 guineas. What is the value of each horse ? Prob. 52. Divide the number 90 into 4 such parts, that the first increased by 2, the second diminished by 2, the third mul- tiplied by 2, and the fourth divided by 2, shall all be equal. If x, y, and z, be three of the parts, the fourth will be 90 x y z. And by the conditions, &c. Prob. 53. Find three numbers, such that the first with J the sum of the second and third shall be 120 ; the second with i the difference of the third and first shall be 70 ; and J the sum of the three numbers shall be 95. Prob. 54. What two numbers are those, whose difference, sum and product, are as the numbers 2, 3, and 5 ? Prob. 55. A vintner sold at one time, 20 dozen of port wine, and 30 dozen of sherry ; and for the whole received 120 guineas. At another time, he sold 30 dozen of port and 25 dozen of sherry, at the same prices as before ; and for the whole received 140 guineas. What was the price per dozen of each sort of wine ? Prob. 56. A merchant having mixed a certain number of gallons of brandy and water, found that, if he had mixed 18 gallons more of each, he would have put into the mixture 8 gallons of brandy for every 7 of water. But if he had mixed 18 less of each, he would have put in 5 gallons of brandy for every 4 of water. How many gallons of each did he mix ? Prob. 57. What fraction is that, whdse numerator being doubled, and the denominator increased 'by 7, the value be- comes ; but the denominator being doubled, and the nume- rator increased by 2, the value becomes f ? Prob. 58. A person expends 30 cents in apples and pears, giving a cent for 4 apples and a cent for 5 pears. He after- 16* 186 ALGEBRA. [Sect. XI. wards parts with half his apples and one third of his pears, the cost of which was 13 cents. How many did he buy of each ? i 289 j If in the algebraic statement of the conditions of a problem, the original equations are more numerous than the unknown quantities ; these equations will either be contradic- tory, or one or more of them will be superfluous. Thus the equations 3^=60 > And 4*=20 f a For by the first #=20, while by the second, 3=40. But if the latter be altered, so as to give to x the same value as the former, it will be useless, in the statement of a prob- lem. For nothing can be determined . from the one, which cannot be from the other. Thus of the equations 3x6Q \ And ^=10 i ne 1S su P erfluous - 290. But if the number of independent equations produced from the conditions of a problem, is less than the number of unknown quantities, the subject is not sufficiently limited to admit of a definite answer. If for instance, in the equation #-|-y 100, x andy are required, there may be fifty different answers. The values of x and y may be either 99 and 1, or 98 and 2, or 97 and 3, &c. For the sum of each of these pairs of numbers is equal to 100. But if there is a second equation which determines one of these quantities, the other may then be found from the equation already given. As i-|-y=:100, if zz:46, y must be such a number as added to 46 will make 100, that is, it must be 54. No other number will answer this condition. QUEST. When the equations are more numerous than the unknown quantities, what is said of them ? Arts. 289-292.] UNKNOWN QUANTITIES. 187 291. For the sake of abridging the solution of a problem, however, the number of independent equations actually put upon paper is frequently less, than the number of unknown quantities. Prob. 59. To find two numbers whose sum is 30, and the difference of their squares 120. 292. In most cases also, the solution of a problem which contains many unknown quantities, may be abridged, by particular artifices in substituting a single letter for several. Prob. 60. Suppose four numbers, M, z, y and z, are re- quired, of which the sum of the three first is 13, the sum of the two first and last 17, the sum of the first and two last 18, the sum of the three last 21. Then 1. it+x+y=lB 2. u+x+zW 3. u+y+z=I8 4. *+y+z=21. Let S be substituted for the sum of the four numbers, that is, for w+z+y+z. (Art. 159.) It will be seen that of these four equations, The first contains all the letters except z, that is, S z 13 The second contains all except y 1 that is, S y 17 The third contains all except x, that is, S x=:18 The fourth contains all except w, that is, S u=21. Adding all these equations together, we have 4S z y x u 69 Or, 4S (z+y+z+n)z=69. (Art. 67.) But S=i(z-\-y-\-x-\-u) by substitution. Therefore, 4S Sz=69, that is, 3S=69, and 8=23. Then putting 23 for S, in the four equations in which it is first introduced, we have 188 ALGEBRA. [Sect. XII. 232=13^ fz=:23 13 10 =23 17=6 23 w=2l U 23 21=2. N. B. Contrivances of this sort for facilitating the solution of particular problems, must be left to be furnished for the occasion, by the ingenuity of the learner. They are of a nature not to be taught by a system of rules. SECTION XII. RATIO AND PROPORTION. ART. 293. The design of mathematical investigations, is to arrive at the knowledge of particular quantities, by comparing them with other quantities, either equal to, or greater or less than those which are the objects of inquiry. This end is most commonly attained by means of a series of equations and proportions. When we make use of equations, we deter- mine the quantity sought, by discovering its equality with some other quantity or quantities already known. We have frequent occasion, however, to compare the un- known quantity with others which are not equal to it, but either greater or less. 294. Unequal quantities may be compared with each other in two ways. QUEST. What is the design of mathematical investigations? How is this end commonly attained ? In equations how is the value of the unknown quantity determined ? In how many ways are unequal quan- tities compared ? What are they ? Arts. 293-298.] RATIO. 189 First, We may inquire how much one of the quantities is greater than the other : or, Second, We may inquire how many times, one contains the other. 295. The relation which is found to exist between the two quantities compared, is called the ratio of the two quantities. RATIO is of two kinds, arithmetical and geometrical. It is also sometimes called, ratio by subtraction, and ratio by division. 296. ARITHMETICAL RATIO is the DIFFERENCE between two quantities or sets of quantities. The quantities themselves are called the terms of the ratio, that is, the terms between which the ratio exists. Thus 2 is the arithmetical ratio of 5 to 3. This is sometimes expressed, by placing two points between the quantities thus, 5 3, which is the same as 5 3. Indeed the term arithmetical ratio, and its notation by points, are almost needless, and are seldom used. For the one is only a substitute for the word difference, and the other for the sign . 297. If both the terms of an arithmetical ratio be multiplied or divided by the same quantity, the ratio will in effect, be multiplied or divided by that quantity. Thus if abr Then multiply both sides by h, (Ax. 3,) ha hb=hr And dividing by h, (Ax. 4,) - T^T* ft ft /^ 298. If the terms of one arithmetical ratio be added to, or subtracted from, the corresponding terms of another, the ratio QUEST. What is ratio ? Of how many kinds is it ? What are they called ? What is arithmetical ratio ? What are the quantities them- selves called ? If both the terms are multiplied, or divided, by the same quantity, how is the ratio affected ? If the terms of one ratio are added to the corresponding terras of another, how is the ratio affected ? 190 ALGEBRA. [Sect. XII. of their sum or difference will be equal to the sum or differ- ence of the two ratios. T/fc j .* . ** > are the two ratios, And d A J Then ( a +d)-(b+h)=(a-b)+(d-h). Foreach = a+d-b-h. And (a d) - (6 h)(a b) - (d h). For each a-d-b+h. Thus the arithmetical ratio of ll-4 is 7, And the arithmetical ratio of 5-~2 is 3. The ratio of the sum of the terms 166 is 10, which is also the sum of the ratios 7 and 3. The ratio of the diff. of the terms 62 is 4, which is also the difference of the ratios 7 and 3. 299. GEOMETRICAL RATIO is that relation between quanti- ties which is expressed by the QUOTIENT of the one divided by the other. Thus the ratio of 8 to 4, is f or 2. For this is the quo- tient of 8 divided by 4. In other words, it shows how often 4 is contained in 8. So a I b expresses the ratio of a to b. 300. The two quantities compared, are called a couplet. The Jirst term is the antecedent , and the last, the consequent. 301. GEOMETRICAL RATIO is expressed in two ways. 1. In the form of a fraction, making the antecedent the numerator, and the consequent the denominator ; thus the ratio of a to b is -. And b 2. By placing a colon between the quantities compared ; thus, alb expresses the ratio of a to b. Obser. The French mathematicians put the antecedent for the de- nominator ; and the consequent for the numerator. Some American QUEST. What is geometrical ratio ? What is a couplet? The an- tecedent? The consequent? In how many ways is geometrical ratio expressed ? The first ? Second ? What is the French mode ? What are the comparative advantage* of the English and French methods ? Arts. 299-303.] RATIO. 191 authors have followed their example. It is believed however that the English method, which is adopted in the larger work, is most in ac- cordance with reason; while the French mode may perhaps have some advantage in practice. 302. Of these three, the antecedent, the consequent, and the ratio, any two being given, the other may be found. Let az= the antecedent, c= the consequent, r the ratio. By definition r= - ; that is, the ratio is equal to the ante. cedent divided by the consequent. Multiplying by e, a cr, that is, the antecedent is equal to the consequent multiplied into the ratio. Dividing by r, c= -, that is, the consequent ig equal to the antecedent divided by the ratio. Cor. 1. If two couplets have their antecedents equal, and their consequents equal, their ratios must be equal. ( Euc. 7. 5.) Cor. 2. If in two couplets, the ratios are equal, and the antecedents equal, the consequents are equal ; and if the ratios are equal and the consequents equal, the antecedents are equal. (Euclid, 9. 5.) 303. If the two quantities compared are equal, the ratio is a unit, or a ratio of equality. The ratio of 3X6:18 is a unit, for the quotient of any quantity divided by itself is 1. If the antecedent of a couplet is greater than the conse- quent, the ratio is greater than a unit. For if a dividend is greater than its divisor, the quotient is greater than a unit. Q.UEST When the antecedent and consequent are given, how is the ratio found ? When the consequent and ratio are given, how find the antecedent? When the antecedent and ratio are given, how find the consequent ? Wljat is the first corollary ? The second ? If the two quantities compared are equal, what is the ratio ? If the antece- dent is the largest, what is the ratio ? What called ? 192 ALGEBRA. [Sect. XII. Thus the ratio of 18:6 is 3. (Art. 103, Cor.) This is called a ratio of greater inequality. On the other hand, if the antecedent is less than the con- sequent, the ratio is less than a unit, and is called a ratio of less inequality. Thus the ratio of 2:3, is less than a unit, because the dividend is less than the divisor. 305. INVERSE or RECIPROCAL ratio is the ratio of the re- ciprocals of two quantities. (Art. 32.) Thus the reciprocal ratio of 6 to 3, is to , that is The direct ratio of a to b is -, that is, the antecedent divided b by the consequent. The reciprocal ratio is - : -, or - -f- r = - X 7 = - ; a b a b a 1 a that is, the consequent b divided by the antecedent a. Hence a reciprocal ratio is expressed by inverting the frac- tion which expresses the direct ratio ; or when the notation is by points, by inverting the order of the terms. Thus a is to 6, inversely, as b to a. 306. COMPOUND RATIO is the ratio of the PRODUCTS of the corresponding terms of two or more simple ratios. Thus the ratio of 6:3, is 2 And the ratio of 12:4, is 3 The ratio compounded of these is 72 : 12=6 Here the compound ratio is obtained by multiplying to- gether the two antecedents, and also the two consequents, of the simple ratios. Hence it is equal to the product of the simple ratios. QUEST. If the consequent is the largest, what is the ratio ? What called ? What is inverse ratio ? How expressed ? What is com- pound ratio ? Does it differ from other ratio in its nature ? Arts. 305-308.] RATIO. 193 Compound ratio is not different in its nature from any other ratio. The term is used, to denote the origin of the ratio, in particular cases. 307. If in a series of ratios the consequent of each pre- ceding couplet, is the antecedent of the following one, the ratio of the first antecedent to the last consequent, is equal to that which is compounded of all the intervening ratios. (Euc- lid, 5th B.) Thus, in the series of ratios a: b blc eld dlh the ratio of alh, is equal to that which is compounded of the ratios of : b, of 6:c, of c!rf, of dlh. For the compound ratio by the last article is , =- or alh. (Art. 117.) bcdh h 308. A particular class of compound ratios is produced, by multiplying a simple ratio into itself, or into another equal ratio. These are termed duplicate, triplicate, quadruplicate, &c. according to the number of multiplications. A ratio compounded of two equal ratios, that is, the square of the simple ratio, is called a duplicate ratio. One compounded of three, that is, the cube of the simple ratio, is called triplicate, &c. In a similar manner, the ratio of the square roots of two quantities, is called a subduplicate ratio ; that of the cube roots a subtriplicate ratio, &c. Thus the simple ratio of a to V, is alb The duplicate ratio of a to b, is a 2 lb 2 QUEST. What is it equal to ? When the consequent of each pre- ceding couplet is the antecedent of the next, what is the ratio of the first antecedent to the last consequent equal to ? What is a duplicate ratio? Triplicate? Sjibduplicate ? Subtriplicate? 17 194 ALGEBRA. [Sect. XII. The triplicate ratio of a to b, is a 3 : 6 3 The subduplicate ratio of a to b, is \Sal\/b The subtriplicate ratio of a to 6, is %/al$/b, &c. The terms duplicate, triplicate, &c. ought not to be con- founded with double, triple, &c. The ratio of 6 to 2 is 6:2=3 Double this ratio, that is, twice the ratio, is 12:2=6 Triple the ratio, i. e. zAree fa'mes the ratio, is 18:2=9 The duplicate ratio, i. e. the square of the ratio, is 6 2 :2 2 = 9 The triplicate ratio, i. e. the cube of the ratio, is 6 3 :2 3 =27 309. That quantities may have a ratio to each other, it is necessary that they should be so far of the same nature, as that one can properly be said to be either equal to, or greater, or less than the other. Thus a foot has a ratio to an inch, for one is twelve times as great as the other. 310. From the mode of expressing geometrical ratios in the form of a. fraction, (Art. 301,) it is obvious that the ratio of two quantities is the same as the value of a fraction whose numerator and denominator are equal to the antecedent and consequent of the given ratio. Hence, 311. To multiply, or divide both the antecedent and conse- quent by the same quantity, does not alter the ratio. (Art. 112.) To multiply, or divide the antecedent alone by any quantity, multiplies or divides the ratio ; to multiply the consequent alone, divides the ratio ; and to divide the consequent, multi- plies the ratio. (Arts. 132, 135.) That is, multiplying and dividing the antecedent or consequent, has the same effect on the ratio, as a similar operation, performed on the numerator or denominator, has upon the value of a fraction. QUEST. What effect does it have on the ratio to multiply or divide both the antecedent and consequent by the same quantity? To multi- ply or divide the antecedent only ? The consequent ? Arts. 309-313.] RATIO. 195 312. If to or from the terms of any couplet, there be add- ed, or subtracted, two other quantities having the same ratio, the sums or remainders will also have the same ratio. (Eu- clid 5 and 6. 5.) Thus the ratio of 12:3 is the same as that of 20:5. And the ratio of the sum of the antecedents 12+20 to the sum of the consequents 3+5, is the same as the ratio of either couplet. That is, or ^ *. 5 So also the ratio of the difference of the antecedents, to the difference of the consequents, is the same. That is, 20 -12:5 3:: 12: 3 = 20:5, 313. If in several couplets the ratios are equal, the sum of all the antecedents has the same ratio to the sum of all the consequents, which any one of the antecedents has to its cow- sequent. (Euclid 1 and 2. 5.) 12:6=2 Thus the ratio or 8-. 6 ::4-- 2 ) are arithmetical ( a b c*'d, or a b : : c d J proportions. And f 12:6=r8:4, or 12 : 6 :: 8 : 4 ) are geometrical t a'.bdlh, or alblldlh? proportions. The latter is read, ' the ratio of a to b equals the ratio of c? to h ;' or more concisely. l a is to b as d to A.' 318. The first and last terms are called the extremes, and the other two the means. Homologous terms are either the two antecedents or the two consequents. Analogous terms are the antecedent and consequent of the same couplet. QUEST. What is the difference between ratio and proportion f In how many ways is proportion expressed ? How is the latter read ? Which are the extremes ? Which the means ? What are homologous terms ? What analogous terms ? 17* 198 ALGEBRA. [Sect. XII. 319. As the ratios are equal, it is manifestly immaterial which of the two couplets is placed first. If alb: Icld, then cldl lalb. For if ^=-^ then %=^. b a do 320. The number of terms must be at least four. For the equality is between the ratios of two couplets ; and each couplet must have an antecedent and a consequent. There may be a proportion, however, among three quantities. For one of the quantities may be repeated, so as to form two terms. In this case the quantity repeated is called the mid- dle term, or a mean proportional between the two other quan- tities, especially if the proportion is geometrical. Thus the numbers 8, 4, 2, are proportional. That is, 8:4: : 4:2. Here 4 is both the consequent in the first couplet, and the antecedent in the last. It is therefore a mean pro- portional between 8 and 2. The last term is called a third proportional to the two other quantities. Thus 2 is a third proportional to 8 and 4. 321. Inverse or reciprocal proportion is an equality between a direct ratio and a reciprocal ratio. Thus 4:2: : J: ; that is, 4 is to 2, reciprocally, as 3 to 6. Sometimes also, the order of the terms in one of the couplets, is inverted, without writing them in the form of a fraction. (Art. 305.) Thus 4:2:: 3: 6 inversely. In this case, the first term is to the second, as the fourth to the third ; that is, the first divi- ded by the second, is equal to the fourth divided by the third. 322. When there is a series of quantities, such that the ratios of the first to the second, of the second to the third, of QUEST. Which couplet must be placed first ? How many terms must there be ? Can there be a proportion with three quantities ? What is the middle term called ? The last term ? What is inverse proportion ? Arts. 3 19-326.] ARITHMETICAL PROGRESSION. 199 the third to the fourth, &c. are all equal ; the quantities are said to be in continued proportion. The consequent of each preceding ratio is then the antecedent of the following one. Continued proportion is also called progression. 323. In the preceding articles of this section, the general properties of ratio and proportion have been defined and illus- trated. It now remains to consider the principles which are peculiar to each kind of proportion, and attend to their prac- tical application in the solution of problems. SECTION XIII. ARITHMETICAL PROPORTION AND PROGRESSION. ART. 324. If four quantities are in arithmetical proportion, the sum of the extremes is equal to the sum of the means. Thus if a b : : h ro, then a-\-m=b-\-h For by supposition, a b =A m And transposing b and m, a-{-m=b-{-h. So in the proportion, 12 10 : : 1 1 9, we have 12+9= 10+ 1 1 . 325. Again if three quantities are in arithmetical proportion, the sum of the extremes is equal to double the mean. If a b : I b c, then, a b=b c And transposing b and c, a+c=26. 326. Quantities, which increase by a common difference, as 2, 4, 6, 8, 10 ; or decrease by a common difference, as QUEST. When four quantities are in arithmetical proportion, what ia the sum of the extremes equal to ? When there are but three terms in the proportion, what is the sum of the extremes equal to ? What is continued arithmetical proportion ? 200 ALGEBRA. [Sect. XIII. 15, 12, 9, 6, 3, are in continued arithmetical proportion. (Art. 322.) Such a series is also called an arithmetical progression ; and sometimes progression by difference, or equidijferent se- ries. 327. When the quantities increase, they form what is call- ed an ascending series, as 3, 5, 7, 9, 11, &c. When they decrease, they form a descending series, as 11, 9, 7, 5, &c. The natural numbers, 1, 2, 3, 4, 5, 6, &c. are in arithmet- ical progression ascending. 328. From the definition it is evident that, in an ascending series, each succeeding term is found, by adding the common difference to the preceding term. If the first term is 3, and the common difference 2 ; The series is 3, 5, 7, 9, 11, 13, &c. If the first term is a, and the common difference d', Then a-\-d is the second term, a-\-d-\-d=a-}-2d the third, a -^2d+d= a+3d the 4th, a+3d+d=a+4d the 5th, &c. 123 4 5 And the series is a, a-f-df, a-}-2d, a-}-3d, a-\-4d, &c. If the first term and the common difference are the same, the series becomes more simple. Thus if a is the first term, and also the common difference, and n the number of terms, Then a-^-a^a is the second term, 2a-\-a=3a the third, &c. And the series is a, 2, 3a, 4a, na. 329. In a descending series, each succeeding term is found, by subtracting the common difference from the preceding term. QUEST. What else is this series called ? When the series increases, what is it called ? When it decreases, what ? How is each successive term found in an ascending series ? How in a descending series ? Arts. 327-330.] ARITHMETICAL PROGRESSION. 201 If a is the first term, and d the common difference, the se- 123 4 5 ries is a, a d, a 2d, a 3d, a 4J, &c. In this manner, we may obtain any term, by continued ad- dition or subtraction. But in a long series, this process would become tedious. There is a method much more expeditious. By attending to the series a, a+d, +2d, a+3d, a+4d, &c. it will be seen that the number of times d is added to a, is one less than the number of the term. Thus, The second term is a-\-d, i. e. a added to once d ; The third is a-\-2d, a added to twice d ; The fourth is a-\-Sd, a added to thrice d, &c. So if the series be continued, The 50th term will be +49rf, The 100th term a+99df. If the series be descending, the 100th term will be a 99d. In the last term, the number of times d is added to o, is one less than the number of all the terms. If then 7z=the common difference, the first term, zzrrthe last, 9i=zthe number of terms, we shall have, in all cases, za(n l)Xd; that is, 330. To find the last term of an ascending series. Add the product of the common difference into the number of terms less one to the first term, and the sum will be the last term. If the series be descending. From the first term subtract the product of the common dif- ference into the number of terms less one, and the remainder will be the last term. QUEST. How is the last term of an ascending series found ? How the last of a. descending series ? 202 ALGEBRA. [Sect. XIII. N. B. Any other term may be found in the same way. For the series may be made to stop at any term, and that may be considered, for the time, as the last. Thus the roth term=ai;(ro l)Xd. Prob. 1. If the first term of an ascending series is 7, the common difference 3, and the number of terms 9, what is the last term? Ans. z a +(n l)d= 7+(9 1)X3=31. Prob. 2. If the first term of a descending series is 60, the common difference 5, and the number of terms 12, what is the last term? Ans. z=a (nl)d=6Q (12 1)X5=5. Prob. 3. If the first term of an ascending series be 9, and the common difference 4, what will the 5th term be ? Ans. z= a-Kro l)Xfc9+(5 1)X4=25. 331. There is one other inquiry to be made concerning a series in arithmetical progression. It is often necessary to find the sum of all the terms. This is called the summation of the series. The most obvious mode of obtaining the amount of the terms, is to add them together. But the na- ture of progression will furnish us with a more expeditious method. i Let us take, for instance, the series 3, 5, 7, 9, 11, And also the same inverted 11, 9, 7, 5, 3, The sums of the terms will be 14, 14, 14, 14, 14. Take also the series a a-\-d, a-\-2d, a-\-3d-> And the same inverted a-j-4J, a-J-3d, a-\-2d, a-\-d, The sums will be Hence, it will be perceived that the sum of all the terms in the double series, is equal to the sum of the extremes repeat- ed as many times as there are terms. Thus, The sum of 14, 14, 14, 14 and 1414X5. Arts. 331-333.] ARITHMETICAL PROGRESSION. 203 And the sum of the terms in the other double series is (2+4d)X5. But this is twice the sum of the terms in the single series. If then we put the first term, nnzthe number of terms, z=the last, s the sum of the terms, we shall have this equation, s X- Hence, 332. To find the sum of all the terms in an arithmetical progression. Multiply half the sum of the extremes into the number of terms, and the product will be the sum of the given series. Prob. 4. What is the sum of the natural series of numbers 1,2, 3,4, 5, &c. up to 1000? Ans. 5 =Xn=X 1000=500500. At til 333. The two formulas, z=a(n l)d, (Art. 329,) and s~ -- Xw, (Art. 331,) contain five different quantities; Z viz. a, the first term ; d, the common difference ; n, the num- ber of terms ; z, the last term ; and s, the sum of all the terms. From these two formulas others may be deduced, by which, if any three of the jive quantities are given, the remaining two may easily be found. The most useful of these formulas are the following. By the first formula, 1. The last term, z=.a(n l)d, in which a, n and d are given. QUEST. How is the sum of all the terms found? When the first term, the common difference, and the number of terms are given, how is the last term found? 204 ALGEBRA. [Sect. XIII. Transposing (n l)d, 2. The first term, a=z(n l)d, z, n and d being given. Transposing a in the 1st, and dividing by n 1, 3. The common difference, d= -, a, z and n being given. Transposing and dividing, 4. The number of terms, n = \- l,a,z and d being given. d By the second formula, a-4-z 5. The sum of the terms, s=z~- Xn,a,z and n being given. Or by substituting for z its value, s - Xn, in which a, n and d are given. Reducing the preceding equation, 6. The first term, a=. - - , s, d and n being given. Zn 7. The common difference, d S ~ an , s, a and n being given. n* n (2a d) 2 +8ds 8. The number of terms n -^ -^ cf and s being given. QUEST. When the last term, the common difference, and the num- ber of terms are given, how find the first term ? When the first, and last, and the number of terms are given, how find the common differ- ence ? When the first and last terms, and the common difference are given, how find the number of terms ? When the first, and last, and the number of terms are given, how find the sum of all the terms ? When the sum, difference, and number of terms are given, how find the first term ? When the first term, the sum, and the number of terms are given, how find the common difference ? When the first term, the common difference, and the sum of the terms are given, how find the number of terms ? Arts. 334, 335.] ARITHMETICAL PROGRESSION. 205 Obser. A variety of other formulas may be deduced from the pre- ceding equations, the investigation of which will afford the student a a pleasing and profitable exercise. 334. By the third formula, e. g. may be found any num- ber of arithmetical means, between two given numbers. For the whole number of terms consists of the two extremes and all the intermediate terms. If then m number of means, w+2z=:n, the whole number of terms. Substituting m+2 for n in the third equation, we have, The common difference, c? , in which , z and m fw-j 1 are given. Prob. 5. Find 6 arithmetical means, between 1 and 43. . C The common difference is 6. I The series 1, 7, 13, 19, 25, 31, 37, 43. 335. It is obvious from the illustration in Art. 331, that the sum of the extremes in an arithmetical progression, is equal to the sum of any other two terms equally distant from the ex- tremes. Thus, in the series 3, 5, 7, 9, 11, the sum of the first and last terms, of the first but one and last but one, &c., is the same in each case, viz. 14. The same is true of every series. Prob. 6. If the first term of an increasing arithmetical se- ries is 3, the common difference 2, and the number of terms 20 ; what is the sum of the series ? Prob. 7. If 100 stones be placed in a straight line, at the distance of a yard from each other ; how far must a person travel, to bring them one by one to a box placed at the dis- tance of a yard from the first stone ? QUEST. How find any number of arithmetical means between two given numbers ? In a series of arithmetical progression, what is the sum of the extremes equal to ? 18 206 ALGEBRA. [Sect. XIII. Prob. 8. What is the sum of 150 terms of the series 3' 3' *' ? 3' 2 ' 3' &C ' ' Prob. 9. If the sum of an arithmetical series is 1455, the least term 5, and the number of terms 30 ; what is the com- mon difference ? Prob. 10. If the sum of an arithmetical series is 567, the first term 7, and the common difference 2 ; what is the num- ber of terms ? Prob. 11. What is the sum of 32 terms of the series 1, 1J, 2, 2J, 3, &c. ? Prob. 12. A gentleman bought 47 books, and gave 10 cents for the first, 30 cents for the second, 50 cents for the third, &c. What did he give for the whole ? Prob. 13. A person put into a charity box, a cent the first day of the year, two cents the second day, three cents the third day, &c. to the end of the year. What was the whole sum for 365 days ? Prob. 14. How many strokes does a common clock strike in 24 hours ? Prob. 15. The clocks of Venice go on to 24 o'clock ; how many strokes do they strike in a day ? Prob. 16. Required the sum of the odd numbers 1, 3, 5, 7, 9, &c. continued to 101 terms. Prob. 17. Required the 365th term of the series of even numbers 2, 4, 6, 8, 10, 12, &c. Prob. 18. The first term of a series is 4, the common differ- ence 3, and the number of terms 100 ; what is the last term ? Prob. 19. A man puts $1 at interest at 6 per cent. What will be the amount in 40 years ? Art. 336.] ARITHMETICAL PROGRESSION. 207 Prob. 20. The extremes of a series are 2 and 29 ; and the number of terms is ten. What is the common difference ? Prob. 21. The extremes of a series are 3 and 39, and the common difference 2. What is the number of terms ? Prob. 22. Find 5 means between 6 and 48. Prob. 23. Find 6 means between 8 and 36. . 336. Problems of various kinds, in arithmetical progression, may be solved by stating the conditions algebraically, and then reducing the equations. Prob. 24. Find four numbers in arithmetical progression, whose sum shall be 56, and the sum of their squares 864. If # the second of the four numbers, And y= their common difference : The series will be x y, a?, #-f-y, x-\-2y. By the conditions, (x y)-\-x-}-(x-\-y)-{-(x-\-2y)=56 And (x -y)2 + That is, And 4x 2 -\-4xy +6y 2 = Reducing these equations, we have x=12, and y 4. The numbers required, therefore, are 8, 12, 16, and 20. Prob. 25. The sum of three numbers in arithmetical pro- gression is 9, and the sum of their cubes is 153. What are the numbers ? Prob. 26. The sum of three numbers in arithmetical pro- gression is 15, and the sum of the squares of the two ex- tremes is 58. What are the numbers ? Prob. 27. There are four numbers in arithmetical progres- sion : the sum of the squares of the two first is 34 ; and the sum of the squares of the two last is 130. What are the numbers ? 208 ALGEBRA. [Sect. XIV. Prob. 28. A certain number consists of three digits, which are in arithmetical progression, and the number divided by the sum of its digits is equal to 26 ; but if 198 be added to it, the digits will be inverted. What is the number ? Let the digits be equal to x y, #, and z+y, respectively. Then the number=lW(x-y)+Wx+(x+y)=lllx-99y, &c. Prob. 29. The sum of the squares of the extremes of four numbers in arithmetical progression is 200 ; and the sum of the squares of the means is 136. What are the numbers ? Prob. 30. There are four numbers in arithmetical progres- sion, whose sum is 28, and their continued product is 585 ? What are the numbers ? SECTION XIV. GEOMETRICAL PROPORTION AND PROGRESSION. ART. 337. If four quantities are in geometrical proportion, the product of the extremes is equal to the product of the means. Thus, 12:8::15:10; therefore 12X10=8 X 15. Hence, 338. Any factor may be transferred from one of the means to the other, or from one extreme to the other, without affect- ing the proportion. Thus if almb: Ixly, then alb: Imxly ; for the product of the means in both cases is the same. So if nalbllxly, then albllxlny. QUEST. If four quantities are in geometrical proportion, what is the product of the extremes equal to ? Arts. 337-341.] GEOMETRICAL PROGRESSION. 209 339. On the other hand, if the product of two quantities is equal to the product of two others, the fpur quantities will form a proportion if they are so arranged, that those on one side of the equation shall constitute the means, and those on the other side the extremes. Thus since 6X 12=8x9, then 6:8::9:12. Cor. The same must be true of any factors which form the two sides of an equation. Thus if (a-\-b)Xc=(d w)Xy, then a-\-b:d mllylc. 340. If three quantities are proportional, the product of the extremes is equal to the square of the mean. For this mean proportional is, at the same time, the consequent of the first couplet, and the antecedent of the last. (Art. 320.) It is therefore to be multiplied into itself, that is, it is to be squared. Thus, 4:6: :6:9; therefore 4 X9z=6X 6. If albl'.b'.c, then mult, extremes and means, ae~ ft 2 . Hence, a mean proportional between two quantities may be found, by extracting the square root of their product. If alx: :xic, then x 2 =ac, and z=\/ac. (Art. 249.) 341. It follows, from Art. 338, that in a proportion, either extreme is equal to the product of the means, divided by the other extreme ; and either of the means is equal to the pro- duct of the extremes, divided by the other mean. 1. If albllcld, then adlc. 2. Dividing by cZ, a=bc-^-d. 3. Dividing the first by c, b=ad-r-c. 4. Dividing it by 6, c~ad--b. 5. Dividing it by a, d=bc-~a. QUEST. How is an equation put into a proportion ? If three quan- tities are in proportion, what is the product of extremes equal to? How is a mean proportional between two quantities found ? When the means and one extreme are given, how find the other extreme ? When the extremes and one of the means are given, how find the other? 18* 210 ALGEBRA. [Sect. XIV. That is, the fourth term is equal to the product of the se- cond and third divided by the first. N. B. On this principle is founded the rule of simple pro- portion in arithmetic, commonly called the " Rule of Three" Three numbers are given to find a fourth, which is obtained by multiplying together the second and third, and dividing by the first. 342. The propositions respecting the products of the means and of the extremes, furnish a very simple and convenient criterion for determining whether any four quantities are pro- portional. We have only to multiply the means together, and also the extremes. If the products are equal, the quan- tities are proportional. If the products are not equal, the quantities are not proportional. 343. It is evident that the terms of a proportion may un- dergo any change which will not destroy the equality of the ratios ; or which will leave the product of the means equal to the product of the extremes. These changes are numerous, but they may be reduced to a few general principles. CASE I. Changes in the order of the terms. 344. If four quantities are proportional, the order of the means, or of the extremes, or of the terms of both couplets, may be inverted without destroying the proportion. Thus if alb: Icld, and 12:8: :6:4, then, , r J7 * ( al cl ibid ) the 1st is to the 3d as 1. Inverting the means,* < \ 12:6: :8:4 ) the 2d to the 4th. QUEST. What rule is founded on this principle? How can you tell whether four quantities are proportional ? What alterations can be made on the terms of a proportion ? When the means are invert- ed, what is it called ? When the terms of each couplet are inverted, what.? If the terms of only one couplet are inverted, what is the ef- fect on the proportion ? * This is called alternation, (Euclid 16. 5.) Arts. 342-345.] GEOMETRICAL PROGRESSION. 211 2. Inverting the extremes, \ ' ' * c ' ( 4:8::6: 12 i as the 3d to the 1st. 3. Inverting the terms of ( bl a I ldi2 \ the 2d is to the 1st as each couplet* I 8: 12: :4:6 J the 4th to the 3d. 4. We may change the order of the two couplets. (Art. 319.) Cor. The order of the whole proportion may be inverted. N. B. If the terms of only one of the couplets are inverted, the proportion becomes reciprocal. (Art. 321.) If alb: I eld, then a is to 6, reciprocally, as d to c. CASE II. Multiplying or dividing by the same quantity. 345. If four quantities are proportional, the two analogous or two homologous terms may be multiplied or divided by the same quantity, without destroying the proportion. Thus, If alb : I eld, then, if analogous terms are multiplied, or divided, the ratios will not be altered. (Art. 311.) 1. malmbllcld. 2. albllmclmd. a b . c d 3. -l-llcld. 4. albll-l-. mm mm If homologous terms be multiplied or divided, both ratios will be equally increased or diminished. 5. malhllmcld. 6. almbllclmd. a , c b d 7. ~lbll-ld. 8. al-llcl-. mm mm Cor. All the terms may be multiplied, or divided by the rnu T -abed same quantity, ihus, malmbl Imclmd, or : :::. mm mm Q.UEST. If two analogous terms are multiplied or divided by the same quantity, what is the effect ? If two homologous terms are mul- tiplied or divided, what ? * This is technically called inversion. 212 ALGEBRA. [Sect. XIV, CASE III. Comparing one proportion with another. 346. If two ratios are respectively equal to a third, they are equal to each other. (Euclid 11. 5.) ' This is nothing more than the 7th axiom applied to ratios. ' , [ then a'.bllc'.d, or alcllbld. (Art. 344.) And clal Imln " 2. If albllmln) , And mini Iclb* then albl lcldl r alcl lbld ' Cor. If alb: ^nln ) ^ a . b>c . d (Eudid 13 5 } fft Tt^^C % CL ' For if the ratio of mln is greater than that of eld, it is manifest that the ratio of alb, which is equal to that of 01:91, is also greater than that of eld. 346.a. In these instances, the terms which are alike in the two proportions are the twojirst and the two last, and the re- sulting proportion is uniformly direct. But this arrangement is not essential. The order of the terms may be changed, in various ways, without affecting the equality of the ratios. (Art. 344.) The proposition to which the instances of equality belong, is usually cited by- the words, " ex aquo" or " ex cequali." (Euclid 22. 5.) 347. Any number of proportions may be compared, in the same manner, if the two first or the two last terms in each preceding proportion, are the same with the two first or the two last in the following one. Thus if albllcld^ And c'.dllhll I , A , , 7 >then alb: Ixly. And hllllmln I And mini Ixly J QUEST. When two ratios are each equal to a third, how are they to each other ? How is this proposition cited in geometry ? When may any number of proportions be compared in this manner ? Arts. 346-349.] GEOMETRICAL PROGRESSION. 213 That is, the two first terms of the first proportion have the same ratio, as the two last terms of the last proportion. For it is manifest that the ratio of all the couplets is the same. 348. But if the two means, or the two extremes, in one proportion, be the same with the means, or the extremes, in another, the four remaining terms will be reciprocally pro- portional. If almllnlb). 1 1 A , _ ? then ale::-:-, or alclldlb. And clmllnld J b d For ab=mn ) ^ Art 33? j Therefore a & =c rf, an d ale: :d:b. And cd=mn * In this example, the two means in one proportion, are like those in the other. But the principle will be the same, if the extremes are alike, or if the extremes in one proportion are like the means in the other. If mlallbln > A i .7. rihen alclldlb. And *:c: :alw > Orif :m::n:6 ) , Anri ,/*, c tnen zc::rf:6. Ana ml c. lalw > The proposition in geometry which applies to this case, is usually cited by the words " ex aquo perturbate" (Euc. 23. 5.) CASE IV. Addition and Subtraction of equal ratios. 349. If to or from two analogous or two homologous terms of a proportion, two other quantities having the same ratio be added or subtracted, the proportion ivill be preserved. (Euclid 2. 5.) QUEST. If the two means or two extremes in one proportion be the same as the means or extremes in another, how are the remaining terms ? How is this proposition cited in geometry ? When two anal- ogous or homologous terms are added to or subtracted from two other quantities having the same ratio, how is the proportion ? 214 ALGEBRA. [Sect. XIV. For a ratio is not altered, by adding to it, or subtracting from it, the terms of another equal ratio. (Art. 313.) If albllcld, and albumin, Then by adding to, or subtracting from a and b, the terms of the equal ratio mln, we have a-\-mlb-\-nl Icld, and a mlb nllcld. And by adding and subtracting m and n, to and from c and d, we have albl :c+w:e?+n, and albllc mid 71. Here the addition and subtraction are to and from analo- gous terms. But by alternation, (Art. 344,) these terms will become homologous, and we shall have a-{-mlcl lb-^nld, and a mlcllb nld. Cor. 1. This addition may, evidently, be extended to any number of equal ratios. (Euclid 2. 5. cor.) fc: ^ < a-\-m I c-\-n I ibid And mlnllbld) or a-\-mlbllc-{-nld. 350. Hence, if two analogous or homologous terms be add- ed to, or subtracted from the two others, the proportion will be preserved. Thus, if albllcld, and 12:4: :6;2, then, 1. Adding the two last terms, to the two first. a+clb+dllalb 12+6: 4+2:: 12:4 zuda+clb+dllcld 12+6: 4+2:: 6:2 or a+claiib+dlb 12+6:12: :4+ 2:4 12+6: 6::4+2:2. Arts. 350, 351.] GEOMETRICAL PROGRESSION. 215 2. Adding the two antecedents to the two consequents. a +b:b::c+d:d 12+4: 4::6-f2:2 a+blall c+dlc, &c. 12+4:12: : 6+2: 6, &c. This is called composition. (Euclid 18. 5.) 3. Subtracting the two first terms from the two last, c alalld 6:6, or c alclld bid, &c. 4. Subtracting the two last terms from the iwojirst. a clb r/:::6, or a clb dllcld, &c. 5. Subtracting the consequents from the antecedents, a 6:6: :c did, or ala bllclc d, &c. The alteration expressed by the last of these forms is call- ed conversion. 6. Subtracting the antecedents from the consequents, b alalld cic, or 6:6 alldld c, &c. 7. Adding and subtracting, a-\-bla bllc-{-dlc d. That is, the sum of the two first terms, is to their differ- ence, as the sum of the two last, to their difference. Cor. If any compound quantities, arranged as in the pre- ceding examples, are proportional, the simple quantities of which they are compounded are proportional also. Thus, if -(-6:6: lc-}-dld, then alb 1 1 eld. This is called division. (Euclid 17. 5.) CASE V. Compounding Proportions. 351. If the corresponding terms of two or more ranks of proportional quantities be multiplied together, the product will be proportional. QUEST. What is composition ? Conversion ? Division ? If the corresponding terms of two or more ranks of proportionals are multi- plied together, how will the product be ? 216 ALGEBRA. [Sect. XIV. This is compounding ratios, (Art. 306,) or compounding proportions. It should be distinguished from what is called composition, which is an addition of the terms of a ratio. (Arts. 350, 2.) If aibucid 12:4::6:2 And hllllmln 10:5: :8:4 Then ahlbl: Icmldn 120:20: :48:8. For from the nature of proportion, the two ratios in the first rank are equal, and also the ratios in the second rank. And multiplying the corresponding terms is multiplying the ratios, (Art. 311,) that is, multiplying equals by equals, (Ax. 3 ;) so that the ratios will still be equal, and therefore the four products must be proportional. The same proof is applicable to any number of proportions. fa:&::c: 4 , &c. 363. If several quantities are in continued proportion, they will be proportional when the order of the whole is inverted. This has already been proved with respect to four propor- tional quantities. (Art. 344, cor.) It may be extended to any number of quantities. Between the numbers, 64, 32, 16, 8, 4, The ratios are, 2, 2, 2, 2, Between the same inverted, 4, 8, 16, 32, 64, The ratios are, J, J, J, J. So if the order of any proportional quantities be inverted, the ratios in one series will be the reciprocals of those in the other. For by the inversion each antecedent becomes a con- sequent, and v. v., and the ratio of a consequent to its antece- dent is the reciprocal of the ratio of the antecedent to the consequent. (Art. 305.) That the reciprocals of equal quan- tities are themselves equal, is evident from Ax. 4. 364. To investigate the properties of geometrical progres- sion, we may take nearly the same course, as in arithmetical progression, observing to substitute continual multiplication and division, instead of addition and subtraction. It is evi- dent, in the first place, that, Q.UEST. When several quantities are in continued proportion, what ratio has the first to the last ? If the series is inverted, what effect has it ? How are the ratios in one series, compared with those of another, when the order is inverted ? Arts. 362-368.] GEOMETRICAL PROGRESSION. 223 365. In an ascending geometrical series, each succeeding term is found, by multiplying the ratio into the preceding term. If the first term is a, and the ratio r, Then aXr^ar, the second term, orXr=or 2 , the third, ar 2 Xr=ar 3 , the fourth, or 3 Xr=ar 4 , the fifth, &c. And the series is a, or, or 2 , ar 3 , ar 4 , ar 5 , &c. 366. If the first term and the ratio are the same, the pro- gression is simply a series of powers. If the first term and the ratio are each equal to r, , Then rXr=r 2 , the second term, r 2 Xrrr:r 3 , the third, r 3 X r=r 4 , the fourth, r 4 X r=r 5 , the fifth. And the series is r, r 2 , r 3 , r 4 , r 5 , r 6 , &c. 367. In a descending series, each succeeding term is found by dividing the preceding term by the ratio, or multiplying by the fractional ratio. If the first term is ar 6 , and the ratio r, 6 The second term is , or ar 6 X^ ; And the series is ar 6 , ar 5 , ar 4 , ar 3 , or 2 , ar, a, &c. If the first term is a, and the ratio r, The series is a, , -, , &c. or a, ar" 1 , ar~ 2 , &c. r r 2 r 3 123 4 5 6 368. By attending to the series a, ar, ar 2 , ar 3 , ar 4 , ar & , &c., it will be seen that, in each term, the exponent of the power of the ratio, is one less t than the number of the term. If then azr the first term, r= the ratio, z the last, n the number of terms ; we have the equation z ar 71 "" 1 , that is, QUEST. In an ascending geometrical series how is each succeeding term found ? When the first term and ratios are the same, what is the progression ? How is each term found in a descending series ? 224 ALGEBRA. [Sect. XIV. 369. In geometrical progression, the last term is equal to the product of the first, into that power of the ratio whose index is one less than the number of terms. When the least term and the ratio are the same, the equa- tion becomes z=^rr n ~ 1 =r n . (Art. 366.) 370. Of the four quantities a, z, r, and n, any three being given, the other may be found. 1. By the last article, z=rar n ~ 1 =the last term. 2. Dividing by r""" 1 , -^Y =a= the first term. 3. Dividing the 1st by a, and extracting the root, /z\-i u =r= the ratio. 371. By the last equation may be found any number of geometrical means, between two given numbers. If m= the number of means, m-|-2 n, the whole number of terms. Substituting m+2, for n, in the equation, we have i fz\ mJ *-i =zr, the ratio. \a] When the ratio is found, the means are obtained by con- tinued multiplication. Prob. 1. Find two geometrical means between 4 and 256. Ans. The ratio is 4, and the series is 4, 16, 64, 256. Prob. 2. Find three geometrical means between and 9. 372. The next thing to be attended to, is the rule for find- ing the sum of all the terms. QUEST. What is the last term equal to ? What is the first term equal to ? How find the ratio ? Arts. 369-372.] GEOMETRICAL PROGRESSION. 225 If any term, in a geometrical series, be multiplied by the ratio, the product will be the succeeding term. (Art. 365.) Of course, if each of the terms be multiplied by the ratio, a new series will be produced, in which all the terms except the last will be the same, as all except the first in the other series. To make this plain, let the new series be written under the other, in such a manner, that each term shall be removed one step to the right of that from which it is pro- duced in the line above. Take, for instance, the series 2, 4, 8, 16, 32 Multiplying each term by the ratio, 4, 8, 16, 32, 64 Here it will be seen at once, that the four last terms in the upper line are the same, as the four first in the lower line. The only terms which are not in loth^ are the first of the one series, and the last of the other. So that when we subtract the one series from the other, all the terms except these two will disappear, by balancing each other. If the given series is a, ar, ar 2 , ar 3 , .... ar"""" 1 . Then mult, by r, we have ar, ar 2 , ar 3 , ar*" 1 , ar n . Now let s= the sum of the terms. Then snra+ar+ar^-ar 3 , +ar n ~ 1 , And mult, by r, rs= ar+ar 2 +ar 3 , .... +ar* 1 +ar n . Subt. the first equation from the second, rs s=ar n a ar n a And dividing by (r 1,) (Art. 98,) s= r 1 In this equation, ar n is the last term in the new series, and is therefore the product of the ratio into the last term in the given series. Therefore, s= , that is, r 1 ALGEBRA. [Sect. XIV. 373. To find the sum of a geometrical series, Multiply the last term into the ratio, from it subtract the first term, and divide the remainder by the ratio less one. Obser. From the above formula, in connexion with the one in Art. 368, there may be the same variety of other formulas deduced as in Art. 333. The others however involve principles with which, it is presumed, the pupil is not yet acquainted. Prob. 3. If in a series of numbers in geometrical progres- sion, the first term is 6, the last term 1458, and the ratio 3, what is the sum of all the terms ? rz a 3X14586 Ans. s - = - - 2184. r 1 o 1 Prob. 4. If the first term of a decreasing geometrical se- ries is J, the ratio J, and the number of terms 5 ; what is the sum of the series ? Prob. 5. What is the sum of the series, 1, 3, 9, 27, &c. to 12 terms ? Prob. 6. What is the sum of ten terms of the series 1, f , |, jfr, &c. Prob. 7. If the first term of a series is 2, the ratio 2, and the number of terms 13 ; ^hat is the last term ? Prob. 8. What is the 12th term of a series, the 1st term of which is 3, and the ratio 3 ? Prob. 9. A man bought a horse, giving 1 cent for the first nail in his shoes, three for the second, and so on. The shoes contained 32 nails ; what was the cost of the horse ? 374. Quantities in geometrical progression are propor- tional to their differences. QUEST. How is the sum of a geometrical series found? If quan- tities are in geometrical progression, what is true of their differ- ence ? Arts. 373-375.] GEOMETRICAL PROGRESSION. 227 Let the series be a, ar, ar 2 , ar 3 , ar*, &c- By the nature of geometrical progression, alar', \ar\ar 2 '. Iar 2 lar 3 l :ar 3 :ar 4 , &c. In each couplet let the antecedent be subtracted from the consequent, according to Art. 350, 6. Then alarl lar alar 2 arl lar 2 ar:ar 3 ar 2 , &c. That is, the first term is to the second, as the difference between the first and second, to the difference between the second and third ; and as the difference between the se- cond and third, to the difference between the third and fourth, &c. Cor. If quantities are in geometrical progression, their dif> ferences are also in geometrical progression. Thus the numbers 3, 9, 27, 81, 243, &c. And their differences 6, 18, 54, 162, &c. are in geometrical progression. 375. Problems in geometrical progression, may be solved, as in other parts of algebra, by means of equations. Prob. 10. Find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Let the three numbers be z, y, and z. By the conditions, xly I lylz, or xz=y 2 And x-\-y-{-z= 14 And X 2_|^2^ 2 _. 8 4 Ans. 2, 4 and 8. Prob. 11. There are three numbers in geometrical progres- sion whose product is 64, and the sum of their cubes is 584. What are the numbers ? Prob. 12. There are three numbers in geometrical progres- sion : the sum of the first and last is 52, and the square of the mean is 100. What are the numbers ? 228 ALGEBRA. [Sect. XV. Prob. 13. Of four numbers in geometrical progression, the sum of the two first is 15, and the sum of the two last is 60. What are the numbers ? Prob. 14. A gentleman divided 210 dollars among three servants, in such a manner, that their portions were in geo- metrical progression ; and the first had 90 dollars more than the last. How much had each ? Prob. 15. There are three numbers in geometrical progres- sion, the greatest of which exceeds the least by 15 ; and the difference of the squares of the greatest and the least, is to the sum of the squares of all the three numbers as 5 to 7. What are the numbers ? Prob. 16. 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. What are the numbers ? SECTION XV. EVOLUTION OF COMPOUND QUANTITIES. 376. RULE. Arrange the terms according to the powers of one of the letters, so that the highest power shall stand first, the next highest next, &c. Take the root of the first term, for the first term of the required root : Subtract the power from the given quantity, and divide the first term of the remainder, by the first term of the root in- QUEST. How should the terms be arranged to extract the root of a compound quantity ? What are the other steps ? Art. 376.] EVOLUTION OF COMPOUND QUANTITIES. 229 volved to the next inferior power, and multiplied by the index of the given power ;* the quotient will be the next term of the root. Subtract the power of the terms already found from the given quantity, and using the same divisor, proceed as before. This rule verifies itself. For the root, whenever a new term is added to it, is involved, for the purpose of subtracting its power from the given quantity : and when the power is equal to this quantity, it is evident the true root is found. 1. Extract the cube root of a 6 , the first subtrahend. 3 4 )* 3a b , &c. the first remainder. a 6 +3a 5 +3a 4 +a 3 , the 2d subtrahend. 3a 4 ) * * 6a 4 , &c. the 2d remainder. 2. Find the 4th root of a 4 +8a 3 +24a 2 +32a+16. 3. Find the 5th root of a 5 + 5 4 6 + 10a 3 6 2 + 10a 2 6 3 4. Find the cube root of a 3 6a 2 b+2ab 2 86 3 . 5. Find the 2d root of 4a 2 12ab+9b 2 + IGah 24bh + 16A 2 . N. B. In finding the divisor in the 5th example, the term 2 in the root is not involved, because the power next below the square is the first power. * By the given power is meant a power of the same name with the required root. As powers and roots are correlative terms, any quantity is the square of its square root, the cube of its cube root, &c. 20 230 ALGEBRA. [Sect. XV. 377. The square root may be extracted by the following RULE. Arrange the terms according to the powers of one of the letters, take the root of the first term, for the first term of the required root, and subtract the power from the given quantity. Bring down two other terms for a dividend. Divide by double the root already found, and add the quotient both to the root, and to the divisor. Multiply the divisor thus in- creased, into the term last placed in the root, and subtract the product from the dividend. Bring down two or three additional terms and proceed as before. PROOF. Multiply the root into itself, and if the product is equal to the given quantity, the work is right. 6. What is the square root of a 2 +2ab+b 2 +2ac+2bc+c 2 (a+b+c a 2 , the first subtrahend. 2a+b) * 2ab+b 2 Into 6=r 2ab-\-b 2 , the second subtrahend. 2a+2b+c) * * 2ac+2bc+c 2 Into c= 2ac-{-2bc-\-c 2 , the third subtrahend. Proof. The square of the root a-\-b-\-c, is equal to the given quantity. For (a+b) 2 =ia 2 +2ab+b 2 =:a 2 +(2a+b)Xb. (Art. 97.) And substituting h=a+b, the square h 2 =a 2 +(2a+b)Xb. And ( a +b+c) 2 =(h+c) 2 =h 2 +(2h+c)Xc ; that is, restoring the values of h and h 2 , (a+b+c) 2 =a 2 +(2a+b)Xb+(2a+2b+ c )Xc. In the same manner, it may be proved, that, if another term be added to the root, the power will be increased, by the product of that term into itself, and into twice the sum of the preceding terms. QUEST. What is the rule for extracting the square root? Arts. 377, 378.] EVOLUTION OF COMPOUND QUANTITIES. 231 The demonstration will be substantially the same, if some of the terms be negative. 7. Find the square root of 1 46+46 2 +% ^ty+y 2 - 8. Find the square root of a 6 2a 5 +3a 4 2a 3 +a 2 . 9. Find the square root of 4 +4a 2 6+46 2 4a 2 86+4. 378. It will frequently facilitate the extraction of roots, to consider the index as composed of two or more factors. Thus i~a* X *. And a^=a^ X ^. That is, The fourth root is equal to the square root of the square root ; The sixth root is equal to the square root of the cube root ; The eighth root is equal to the square root of the fourth root, &c. To find the sixth root, therefore, we may first extract the cube root, and then the square root of this. 10. Find the square root of x 4 4x 3 -|-6x 2 4x-)-l. 11. Find the cube root of x 6 6x 5 +15x* 20x 3 +15z 2 6x+l. 12. Find the square root of 4x 4 4x 3 +13x 2 6x+9. 13. Find the 4th root of 16a 4 96a 3 x+216 2 x 2 216ax 3 14. Find the 5th root of x 5 +5x 4 +10x 3 +10x 2 +5x+l. 15. Find the 6th root of a 6 6a 5 6 + 15a6 2 20a 3 & 3 QUEST. How may the extraction of roots be facilitated ? What is the fourth root equal to ? The sixth ? The eighth ? How then ma/ we find the sixth root ? 232 ALGEBRA.' [Sect. XVI. SECTION XVI. ART. 379. It is often expedient to make use of the alge- braic notation, for expressing the relations of geometrical quantities, and to throw the several steps of a demonstration into the form of equations. By this, the nature of the reason- ing is not altered. It is only translated into a different lan- guage. Signs are substituted for words, but they are intend- ed to convey the same meaning. A great part of the de- monstrations in geometry, really consist of a series of equa- tions, though they may not be presented to us under the alge- braic forms. Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc. 32, 1,) may be demonstrated, either in common lan- guage, or by means of the signs used in algebra. Let the side AB, of the triangle ABC, (Fig. 1,) be continued to D ; let the line BE be parallel to AC ; A and let GHI be a right angle. The demonstration, in words, is as follows : 1. The angle EBD is equal to the angle BAG, (Euc. 29, 1.) 2. The angle CBE is equal to the angle ACB. 3. Therefore, the angle EBD added to CBE, that is, the angle CBD, is equal to BAG added to ACB. 4. If to these equals, we add the angle ABC, the angle CBD added to ABC, is equal to BAG added to ACB and ABC. * This section is to be read after the Elements of Geometry. Arts. 379-381.] APPLICATION TO GEOMETRY. 233 5. But CBD added to ABC, is equal to twice GHI, that is, to two right angles. (Euc. 13. 1.) 6. Therefore, the angles BAG, and ACB, and ABC, are to- gether equal to twice GHI, or two right angles. Now by substituting the sign -f-, for the word added, or and, and the sign =, for the word equal, we shall have the same demonstration in the following form. 1. By Euclid 29. 1. EBDz=BAC 2. And CBE^ACB 3. Add the two equations EBD+CBEz=BAC+ACB 4. Add ABC to both sides CBD-f ABC=BAC+ACB+ABC 5. But by Euclid 13. 1. CBD+ABC=:2GHI 6. Make the 4th& 5th equal B AC+ACB+ ABC=2GHI. By comparing, one by one, the steps of these two demon- strations, it will be seen, that they are precisely the same, ex- cept that they are differently expressed. 380. It will be observed that the notation in the example just given, differs, in one respect, from that which is general- ly used in algebra. Each quantity is represented, not by a single letter, but by several. In common algebra when one letter stands immediately before another as ab, without any character between them, they are to be considered as multi- plied together. But in geometry, AB is an expression for a single line, and not for the product of A into B. Multiplication is denoted, either by a point or by the sign X The product of AB into CD, is AB.CD, or ABxCD. 381. There is no impropriety, however, in representing a geometrical quantity by a single letter. We may make b stand for a line or an angle, as well as for a number. 20* 234 ALGEBRA. [Sect. XVI. If, in the example above, we put the angle , ACB= found by dividing the area by the breadth. If a be put for the area of a square whose side is AB, Then by Art. 392, a=(AB) 2 , And extracting both sides \/a=AB. That is, the side of the square is found, by extracting the square root of the number of measuring units in its area. 396. If AB be the base of a triangle and BC its perpen- dicular height ; Then by Art. 393, And dividing by JBC, - AB. That is, the base of a triangle is found, by dividing the area by half the height. 397. As a surface is expressed, by the product of its length and breadth ; the contents of a solid may be expressed, by the product of its length, breadth and depth. It is necessary to bear in mind, that the measuring unit of solids, is a cube ; and that the side of a cubic inch, is a square inch ; the side of a cubic foot, a square foot, &c. Let ABCD (Fig. 3,) represent the base of a parallelepiped, five inches long, three inches broad, and one inch deep. It is evident there must be as many cubic inches in the solid, as there are square inches in its base. And, as the product of the lines AB and BC gives the area of this base, it gives, of course, the contents of the solid. But suppose that the depth of the parallelepiped, instead of being one inch, is four inches. Its contents must be four times as great. If, then, the length be AB, the breadth BC, and the depth CO, the expression for the solid contents will be, ABxBCxCO. 398. By means of the algebraic notation, a geometrical demonstration may often be rendered much more simple and . 396-399.] APPLICATION TO GEOMETRY. 241 concise, than in ordinary language. The proposition, (Euc. 4. 2.) that when a straight line is divided into two parts, the square of the whole line is equal to the squares of the two parts, together with twice the product of the parts, is demon- strated, by involving a binomial. Let the side of a square be represented by s ; And let it be divided into two parts, a and b. By the supposition, s=a-{-b And squaring both sides, s 2 =a 2 -}-2ab-\-b 2 . That is, s 2 the square of the whole line, is equal to a 2 and 6 2 , the squares of the two parts, together with 2a6, twice the product of the parts. 399. The algebraic notation may also be applied, with great advantage, to the solution of geometrical problems. In doing this, it will be necessary, in the first place, to raise an algebraic equation from the geometrical relations of the quantities given and required ; and then by the usual reduc- tions, to find the value of the un- Fig. 9. known quantity in this equation. Prob. 1. Given the base, and the sum of the hypothenuse and per- pendicular, of the right angled tri- angle ABC, (Fig. 9,) to find the perpendicular. Let the base AB=b The perpendicular I$C=x The sum of hyp, and perp. x-\-AC=a Then transposing z, AC=a x 1. By Euclid 47. 1, (BC) 2 +(AB) 2 :=(AC) 2 2. That is, by the notation, x 2 -f6 2 (a-x) 2 = 2 -2ax+x 2 . a 2 2 And, x= =BC, the side required. Hence, 21 242 ALGEBRA. [Sect. XVI. 4 In a right angled triangle, the perpendicular is equal to the square of the sum of the hypothenuse and perpendicular, diminished by the square of the base, and divided by twice the sum of the hypothenuse and perpendicular.' ' It is applied to particular cases by substituting numbers for the letters a and b. Thus if the base is 8 feet, and the sum of the hypothenuse and perpendicular 16, the expression 2 7*2 1 f\2 Q2 , becomes ?- ~6, the perpendicular ; and this lid . <^X lu subtracted from 16, the sum of the hypothenuse and perpen- dicular, leaves 10, the length of the hypothenuse. Prob. 2. Given the base and the difference of the hypothe- nuse and perpendicular, of a right angled triangle, to find the perpendicular. Fig. 10. Fig. 11. C DC Let the base, (Fig. 10,) The perpendicular The given difference, Then will the hypothenuse 1. Then by Euclid 47. 1, 2. That is, by the notation, 3. Expanding (x+d) 2 , 4. Therefore AB 6rr AC:rr; (AC) 2 =(AB) 2 +(BC) 2 b 2 d 2 Prob. 3. If the hypothenuse of a right angled triangle is 30 feet, and the difference of the other two sides 6 feet, what is the length of the base ? Art. 399.) APPLICATION TO GEOMETRY. 243 Prob. 4. If the hypothenuse of a right angled triangle is 50 rods, and the base is to the perpendicular as 4 to 3, what is the length of the perpendicular ? Prob. 5. Having the perimeter and the diagonal of a par- allelogram ABCD, (Fig. 11,) to find the sides. Let the diagonal AC:r=7i 10 The side AB=z Half the perimeter BC+AB=BC-\-x=b= 14 Then by transposing z, BC=6