69/- AN ELEMENTARY TREATISE ON ALGEBRA. DESIGNED AS FIRST LESSONS IN THAT SCIENCE, 1BY II. N. ROBINSON, A. M., AUTHOR OF AN UNIVERSITY EDITION OF ALGEBRA-AN ELEMENTARY TREATISE ON NATURAL PHILOSOPHY-A WORK ON GEOMETRY, CONTAINING PLANE AND SPHERICAL TRIGONOMETRY; ALSO, AUTHOR OF A TEXTBOOK ON ASTRONOMY, AND SEVERAL OTHER MATHEMATICAL WORKS. THIRD EDITION. CINCINNATI: JACOB ERNST, 183 MAIN STREET. PHILADELPIIIA: E. H. BUTLER & CO., 23 MINOR STREET. ALBANY, N. Y.: E. II. PEASE & CO., 82 STATE ST. 1 8 50. Entered according to Act of Congress in the year 1850, by H. N. ROBINSON, In the Clerk's Office of the District Court of the United States, for the District of Ohio. SIEREOTYPED BY JAMES & CO., CINCINNATI. PREFACE. EVERY teacher is desirous of having as few textbooks in his school as is consistent with efficient and sound instruction, and in accordance with this object, great efforts have been made by several authors to produce a work on Algebra that would be a proper textbook for all grades of pupils. But in this all have failed, and given up the point in despair. The student of adult age, and possessing a passably disciplined mind, requires a different book from the mere lad, who is just commencing the science. If we put a child's book into the hands of a young man, he will, probably, become displeased with the book, and possibly imbibe prejudice and distaste for the science itself; and if we put a logical and philosophical work into the nands of a child, he is sure not to comprehend it, however well and fluently he may be made to repeat the contents of its pages. But, nevertheless, as Algebra is the groundwork of all the mathematical sciences, and is of itself a system of pure logic, it is important that it should be commenced at an early age-eleven or twelve, or if otherwise well employed, thirteen or fourteen is a more suitable age. It is a prevalent impression that Algebra should not be commenced until the pupil has acquired a good knowledge of Arithmetic, but this is a great error. The impression would be well founded, provided Arithmetic was the most elementary science, and Algebra was founded on Arithmetic; but the reverse is the fact-Algebra is elementary Arithmetic, and no one can acquire a knowledge of Arithmetic in an enlarged and scientific sense, without a previous knowledge of Algebra. Beyond notation, numeration, and the four simple rules, Arithmetic is not a science, but a sequel to all sciences, it is numerical computation applied to anything and to everything. Proportion, as a science, is the comparison of magnitudes, and belongs, properly speaking, to Algebra and Geometry; and the rule of three, in Arithmetic, is but little more than some of its forms of application. Problems in mensuration are very properly to be found in books called Arithmetics, but mensuration is no part of the 4 PREFACE. science of Arithlnetic, it is a part of Geometry, and for a good understanding of it, geometrical science must be directly consulted. So it is with many other parts of Arithmetic, the science is elsewhere; and to have a scientific comprehension of many parts of common Arithmetic, we must go to general Arithmetic, which is emphatically Algebra; and in preparing this work, we have given constant attention to this branch of the subject, as may be seen in our treatment of fractions, proportion, progression, the roots, fellowship, and interest. All these subjects can be better illustrated by symbols than by numbers; for numbers apply to everything, and, of course, can be made to show no particular thing; but not so with symbols, at every step the particular elements are all visible, and the logic and the reason is as distinct in every part of an operation as is the result. For these reasons, Arithmetic should be studied by symbols, as it is in many parts of Europe; many of their books, entitled Arithmetics, are as full of signs and symbols as any Algebra that ever appeared. The prominent design of the author has been to adapt this treatise to the wants of young beginners in Algebra, and at the same time not to produce a mere childish book, but one more dignified and permanent, and to secure this end, he has kept up the same tone and spirit as though he were addressing mature and disciplined minds. Great care has been taken in the selection of problems, and all very severe ones have been excluded, and all such as might be difficult when detached and alone, are rendered simple and easy by their connection with other leading problems of kindred character. To bring out the original thoughts of the pupil has been another object which he designed to accomplish, and the illustrations are given in such a way as to command the constant attention of the learner, and if he learns at all, it will be naturally and easily, and what he learns will become a part of himself. In this work, great importance is attached to equations, not merely in solving problems, but they are used as an instrument of illustrating principles, and their application is carried further in this book than in any other known to the author. For instance, we have illustrated the nature of an equation by the aid of simple problems in subtraction and division; and conversely, the simple principle of equality is used to deduce rules for subtraction, division, the reduction of fractions to a common denominator, the multiplication of quantities affected by different fractional exponents, &c. Notwithstanding that this book is designed to be practical, it contains more illustrations, and is more theoretical and scientific as far as it goes, than any other book designed for the same class of pupils. PREFACE. 5 We have not given demonstrations of the binomial theorem, nor made any investigations of logarithms, or the higher equations, for these subjects belong exclusively to the higher order of Algebra, and will be found very clear and full in the University Edition of Algebra by the same author. In relation to great generalities, all books on the same science, are, in substance, much alike, yet, in the clearness and distinctness with which they present principles, they may be very different; and to arrive at perfection in this particular, is, and should be, the highest ambition of an author. For peculiarities in this work, the teacher is respectfully referred to abbreviations generally in solving equations, to the philosophical uses made of equations in demonstrating principles-the formation of problems, and the manner of arriving at arithmetical rules, which may be found in various parts of this work. CONTEN TS. PAGE. INTRODUCTION..................9................. 9 Axioms..................................................16 Simple problems for exercises................................ 17 Definition of terms........................................20 SECTION I. Addition..............................25 Subtraction............ X........X.......... 31 Subtraction illustrated by Equations.................35 Multiplication. ^.................... 36 The product of minus by minus illustrated............. 37 Division.......................................45 Negative exponents explained...............................49 Division in tompoand quantities..........................51 Factoring................................................55 Multiple and least common multiple......................... 58 AIGEBRAIC FRACTIONS................................... 62 Complex fractions........................................ 68 Multiplication of fractions................................ 70 Division of fractions.......................................73 Division illustrated by Equations................... 76 Addition of fractions................................ 78 Addition of fractions by Equations.......................... 79 Subtraction of fractions.................................... 85 Subtraction illustrated by Equations........................ 86 SECTION II. Equations................................................89 Transposition........................................... 93 General rule for reducing equations................. 93 Proportion, as applied to equations..98 8 CONTENTS. PAGE. Questions producing simple equations....................... 101 How to propose convenient problems........................ 107 How particular numerals are brought into problems........... 111 Equations having compound fractions....................... 116 Equations containing two unknown quantities................ 120 Three methods of elimination.............................. 122 Equations containing three or more unknown quantities-Rule for elimination....................................... 130 Questions producing equations containing three or more unknown quantities.................................. 134 Negative results, how understood........................... 137 SECTION III. INVOLUTION.............................................. 140 Expansion of a binomial................................... 144 Application of the binomial................................ 148 Evolution.............................................. 150 How to extract roots of polynomials........................ 153 Approximate rule for cube root............................. 163 Product of quantities affected by different fractional exponentsArt, 83.............................................. 167 SECTION IV. Equations of the second degree............................. 170 PURE EQUATIONS.......................................... 172 Problems producing pure equations......................... 174 Rules for completing a square.............................. 180 Resolving a quadratic expression into two factors............. 186 Questions giving rise to quadratic equations.................. 191 Homogeneous and symmetrical equations.................... 193 SECTION V. Arithmetical progression.................................. 204 Examples in arithmetical progression........................ 211 Geometrical progression................................... 213 Examples in geometrical progression...................... 215 General problemis that involve progression................... 220 Proportion, theoretically considered......................... 224 Fellowship, theoretically considered......................... 237 INTRODUCTION. ALGEBRA is the science of computation by means of symbols. Letters of the alphabet are generally used to represent quantities or numbers, and conventional signs are employed to represent operations, and to abridge and generalize the reasoning in relation to propositions or problems. We sometimes meet with persons who can readily solve quite difficult problems, and yet are not able to explain the steps in the process: they call their operations working in the head-and, indeed, their reasoning, properly written out, is Algebra; but not having a knowledge of the signs, and possessing no skill in writing out the thoughts of the mind, they do not know it is Algebra. This natural adaptation of the mind, to solve problems without the aid of writing down the operation, is very essential to success in this science. But the mind can only go a very short distance, unaided by the pen; nor is it important that it should, for the aid given by that instrument is efficient and complete, secures the ground gone over, and leaves the mind free to advance indefinitely. In a purely mental process, the mind must retain all the results thus far attained, and continue the reasoning onward at the same time. And this, carried to excess, breaks down the mind rather than strengthens it; and for this reason, a mere mental Algebra must be regarded as one of the ephemeral efforts of the times. But let no reader construe these sentiments into a disapproval of mental Algebra. Every Algebra, properly understood, is mental Algrebra; for the mental process-the reasoning power —must precede every operation. 10 ELEMENTARY ALGEBRA. To compare the common operations of the mind with the brief and refined language of science, we propose the following problems. But before we use algebraical language, we must explain some of its symbols, and here we insert only those intended for immediate use. THE SIGNS. 1. The perpendicular cross, thus +, called plus, denotes addition. 2. The horizontal dash, thus -, called minus, denotes subtraction.* These signs are written before the quantities to which they are affixed. 3. The diamond cross, thus X, or a point between two quantities, denotes multiplication. For example, 5X 4, or 5'4, shows that 4 and 5 must be multiplied together. 4. A horizontal line with a point above and below, thus. —, denotes division; also two quantities, one above another, as 3 a numerator and denominator, as or a, also indicates division, 7 b and shows that 3 must be divided by 7, and a must be divided by b. 5. Double horizontal lines, thus =, represent equality, and show that the quantities between which it is placed are equal. 6. A number or letter before any quantity shows how many times the quantity is taken, and is called the coefficient of the quantity, thus 3x, shows that the quantity x is taken 3 times, and nx shows that the quantity represented by x is taken as many times as there are units in n. 7. A vinculum, or bar., or parenthesis ( ), is used * The signs plus and minus, in general science, have a far more comprehensive meaning than is here expressed. Here they denote simply what is to be done with the quantities to which they are attached; but in philosophical problems, they may denote the essential value of the quantities, as credit and debt; and in geometry they may represent positions, as north and south, or to the right or left of a zero line, &c. INTRODUCTION. 11 to connect several quantities together. Thus, a+b or (a+b) shows that a and b are there to be considered as connected, or making but one quantity. We now turn our attention to the problems —not for the purpose of finding the answers to them, as the mere arithmetical student might suppose-but for the purpose of teaching the manner of solving them by the science of Algebra. 1. A father divided 120 cts. among his three sons. He gave the youngest a certain number, the second 10 cents more, and the eldest 10 cents more than the second. Wheat sum did each receive? By the use of common lan- By algebraical language, guage, this question may be thus: solved thus: Let x represent the share The youngest son had a of the youngest. Then by share of the money; the sec- the conditions ond son had a like share and x= 3d son's share, 10 cents more; and the eldest X-10= 2d " " had also a like share as the x+20= 1st ".. youngest, and 20 cents more. 3x+30=-120 by add. Therefore the three boys had This expression is called an 3 shares and 30 cents; but equation, and the quantities the three boys had 120 cents, on each side of the sign of hence, 3 shares and 30 cents equality are called members, are equal to 120 cents, or the or sides of the equation. three shares are worth 90 It is an axiom, that equals cents, and one share is worth from equals the remainders 30 cents, which is the sum must be equal; and in this given to the youngest. equation, if we take 30 from both members, we have 3z = 120-30=90 Dividing both members by 3 gives x=30, the share of the youngest. 12 ELEMENTARY ALGEBRA. 2. If 75 dollars be added to a share in a certain bridge company, the sum will be the value of 4 shares. What is the value of a share? BY COMMON LANGUAGE. ALGEBRAICALLY. One share and 75 dollars is Let x represent the value the same as 4 shares, there- of a share. fore 75 dollars is 3 shares, Then x —75-4x. and one share is 25 dollars, Taking x an equal quantity the third part of 3 shares. from both members and 75 —3x. Dividing by 3 gives 25=x. 3. A gentleman purchased a horse, a chaise and a harness, for,$230. The harness cost a certain sum, the chaise 3 times as much as the harness, and the horse $20 more than the chaise. Required the price of each. BY COMMON LANGUAGE. ALGEBRAICALLY. The harness cost a certain Let x= the value share of the money, and the of the harness, chaise cost 3 such shares, and Then 3x = the value the horse cost 3 such shares of the chaise, and 20 dollars more; there- And 3x+20= the value fore the whole cost 7 shares of the horse. and 20 dollars, which must Sum 7x+20=230. make 230 dollars. Take the Taking equals from both 20 dollars away, and the 7 members 7x=210 shares is the same as 210 dol- Or x=30 by division. lars. Therefore 1 share is 30 dollars, the value of the harness, and.$90 is the value of the chaise, and $110 the value of the horse. 4. In a certain school - of the pupils are learning geometry, -- are learning Latin, and 10 more, which comprise all in the school, are learning to read. What was the whole number? INTRODUCTION. 13 BY COMMON LANGUAGE. ALGEBRAICALLY. One-third and one-fourth Let x- the number in added together make ~-!. school. Therefore -A is the number Then by the conditions learning to read, which, by x x the problem, is 10; hence'-t- 3 4 10-x. of the number in the school is 2, and the whole number is 24, the number required. This equation may be troublesome on account of the fractions; but in due time we shall give rules to clear equations of fractions; however, fractions here, are just the same as fractions elsewhere. One-third and one-fourth of anything is i-7 of that thing; therefore, 721+10- = \'. Now from the two equals take -,; and 10=52; dividing by 5, 2=; multiplying by 12 gives 24=x, the final result. From these examples it will be perceived that Algebra is but an artificial method of briefly writing out our mental operations when we solve mathematical problems, and as such, it may be extended and applied to almost every branch of the mathematics; and, therefore, the value of this science cannot be over estimated. The three following problems are extremely simple when algebraic language is applied, but would be rather difficult by common language. 5. On a certain day, a merchant paid out $2500 to three men, A, B, and C; he paid to A a certain sum, to B $500 less than thle sum paid A, and to C he paid $900 more than to A. Required the sum paid to each. Let..= the sum paid to A. Then.... x- 500- " " B. And.... x+900- ~' C. By addition,.. 3x+400=2500, the whole sum paid. 14 ELEMENTARY ALGEBRA. Subtracting 400 from both members-that is, equals from equals-and we have 3x=2100. Take -I of each member, or divide both sides by 3 and x=700, the sum paid to A. From this we determine that $200 was paid to B, and $1600 to C. In this example the sign before 500 shows what is to be done with that quantity; it shows that it is to be taken out, and accordingly it was taken out, diminishing the 900 to 400. The operation is called addition, but it is algebraic addition, that is, writing the quantities according to their signs, and finding the result. 6. it is required to divide the number 99 into five such parts that thefirst 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. Let.. x= the first part, then..... x- 3= the second, x+10= the third, x- 9= the fourth, x+16= the fifth, Sum is... 5x+ 14, but the sum of all the parts is 99; therefore, 5x+-14=99. By subtracting 14 from both members, and dividing the remainders by 5, we have x=17, the first part; and, therefore the parts are 17, 14, 27, 8, and 33. 7. Divide $1000 among four men, giving the second twice as much as the first, minus $200; the third double the sum of the second, plus $400; the fourth three times as much as the first, plus $100. Required the share of each. Let.... = the sum paid to the 1st, then.... 2x-2-00-' " " 2d, 4. —400+400 " " " 3d, 3x+100= " " " 4th, The sum is lOx-100= 1000. INTRODUCTION. 15 By adding 100 to both members we have Ox=-1 100 By adding 100 to -100 in the first member of the equation, makes 0, and then 10x only is left in that member, which must be equal to 1100, or x= 110, the sum paid to the first, and the several sums are $110, $20, $440, and $430. The preceding remarks and problems serve to show, only in some small degree, the advantage of Algebra over common language, and the learner should examine every problem, and the reason of every step in the process of its solution, until all is thoroughly understood; then he will have no difficulty in solving the examples that follow in this introduction. But before we give additional problems, let us call the student's mind to the precise idea of an EQUATION. An equation is simply what the word implies; equality as to value, weight, or measure; and can be best understood by comparing it to a pair of scales delicately balanced. The balance can be preserved by adding equal weights to both sides; by taking equal weights from both sides; by multiplying both sides by the same number, or by dividing both sides by the same number, or by taking like roots or like l)owers of the. weights in both sides. The object of working an equation is to bring the unknown quantity to stand alone as one member of the equation, equal to known quantities in the other member. The unknown quantity thus becomes known; and we may do anything to accomplish this end, that the nature of the case may seem to require, only taking scrupulous care to preserve equality through every change. It is usual to represent known quantities by their numerical values, or by the first letters of the alphabet, as a, b, c, d, &c.; and unknown quantities by the last letters, as u, t, x, y, &c. 16 ELEMENTARY ALGEBRA. AXIOMS. Axioms are self-evident truths, and of course are above demonstration; no explanation can render them more clear. The following are those applicable to Algebra, and are the principles on which the truth of all algebraical operations finally rests. Axiom 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 by 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. 7. Quantities which are respectively equal to any other quantity are equal to each other. 8. Like roots of equal quantities are equal. 9. Like powers of the same or equal quantities are equal. Now suppose we have the following equation x +-a=b in which x is the unknown quantity, and a and b known quantities. Before x can become known, a must be disengaged from it, that is, a must be subtracted from both members. It must be subtracted from the first member, because it is our object to have x stand alone, and we must subtract it from the other member, to preserve equality. The equation the,stands x=b —a Here we find the quantity a, whatever it may be, on, th. other side of the equation, with the contrar'y sign. INTRODUCTION. 17 Now let us suppose we have an equation like x-a=b In this equation we perceive that x is diminished by a; therefore to have the single value of x we must add a to the first member; and, of course, to preserve equality, we must add a to the second member, then we shall have x-a+a=b+a But -a and +a destroy each other, and the equation is in brief x=b-+a Here, also, we find a on the opposite side of the equation, wtth its sign changed; and from these investigations we draw the following rule of operation. R u L E.- We may change any quantity from one member of ar7 equation to the other, if we change its sign. The operation itself is called transposition. For examples, transpose the terms so that the unknown quantity x shall stand alone in the first member of the following equations: x+c —d=4g.... Ans. x=4gy —+ —c. x+3-a+t-m 30.... A. 30- n-m+a-3. SIMPLE PROBLEMS FOR EXERCISES. 1. A man bought a saddle and bridle for 45 dollars; the saddle cost four times as much as the bridle. What was the cost of each? Ans. Bridle $9; saddle $36. 2. Three boys had 66 cents among them; the second had twice as many as the first, and the third three times as many as the first. How many had each? Ans. 1st boy had 11; 2d, 22; 3d, 33 cents. 3. Two men had 100 dollars between them, and one had 3 times as many as the other. How many had each? Ans. One had $25, the other $75. N. B. This last problem may be enunciated thus: Two men had 100 dollars between them; the first had one-third as many as the other. How many had each? Ol 18 ELEMENTARY ALGEBRA. 4. Three men had 880 dollars among them; the first had', the second had I as many as the third. How many had each? Let 6x= what the third had. Ans. 1st had 160; 2d, 240, and the 3d, 480 dollars. 5. There are three numbers which together make 72, the second is twice as much as the first, and the third is as much as both the others. What are the numbers? Ans. 1st is 12; 2d, 24; 3d, 36. 6. Two men built 90 rods of fence in 3 days. The second built twice as many rods in a day as the first. How many rods did each build per day? Ans. 1st built 10 rods, 2d, 20. 7. A man bought 3 oxen, 4 cows, and 6 calves, for 260 dollars. He paid twice as much for an ox as he did for a cow, and twice as much for a cow as for a calf. How much did he give for each? Ans. For a calf, $10; cow, $20; and for an ox, $40. 8. A man bought a boat load of flour for 132 dollars, onehalf at 5 dollars per barrel, the other half at 6 dollars per barrel. How many barrels did the boat contain? Let x= half the number of barrels. Ans. 24. 9. A boy bought an equal number of apples, oranges and pears, for 96 cents: the apples at 3 cents apiece, the oranges at 4, and the pears at 5. How many of each kind did he buy? Ans. 8. 10. Two men bought a carriage for 86 dollars; one paid five times as much as the other, and 26 dollars more. What did each pay? Ans. One paid 10, the other 76 dollars. 11. If from 5 times a certain number we subtract 24, the remainder will be 196. What is the number? Ans. 44. 12. To the double of a certain number, if we add 18, the sum will be 96. What is the number? Ans. 39. 13. What number is that whose double exceeds its half by 78? Let 2x- the number. Anqs. 52. 14. A man had six sons, to whom he gave 120 dollars, giving INTRODUCTION. 19 to each one 4 dollars more than to his next younger brother. How many dollars did he give to the youngest? Ans. $15. 15. Three men received 65 dollars; the second received 5 dollars more than the first, and the third 10 dollars more than the second. What sum did the first receive? Ans. $15. 16. A man paid a debt of 29 dollars, in three different payments; the second time he paid 3 dollars more than at first, and the third time he paid twice as much as at the second time. What was the amount of his first payment? Ans. $5. 17. A man bought 6 pounds of coffee, and 10 pounds of tea, for 360 cents, giving 20 cents a pound more for the tea than for the coffee. What was the price of the coffee? Ans. 10 cents. 18. A man bought 6 barrels of flour, and 4 firkins of butter, for 68 dollars. He gave 2 dollars more for a barrel of flour than for a firkin of butter. What was the price of flour? Ans. $7.60. 19. A pound of coffee cost 5 cents more than a pound of sugar, and for 3 pounds of sugar or for 2 pounds of coffee you must pay the same sum. What is the price of sugar? Ans. 10 cents. 20. A person in market selling apples, peaches, and oranges, asked 1 cent more for a peach than for an apple, and 2 cents more for an orange than for a peach, and the prices were such that 10 apples and 5 peaches cost as much as 5 oranges. What was the cost of an apple? Ans. 1 cent. 21. One-half of a post stands in the mud, one-third in the water, and the remainder, which is 3 feet, is above the water. What is the whole length of the post? Ans. 18 feet. 22. One-third and one-half of a sum of money, and two dollars more, make the whole sum. What is the sum? Ans. 60 dollars. 20 ELEMENTARY ALGEBRA. 23. Divide 25 cents between two boys, and give one four times as much as the other. Required the share of each. Ans. 5 and 20 cents. 24. Divide 15 cents between two boys, and give one double of the other. Required the share of each. Ans. 5 and 10 cents. Similar problems to the preceding might be framed indefinitely, but it would be improper to propose any that involve any difficulty until the pupil is better prepared to meet difficulties. We only give the preceding to convince the learner that he can find real utility in the science; but before he can go into the subject to advantage, he must learn the nature of algebraic expressions, and acquire the art of adding, subtracting, multiplying, and dividing algebraic quantities, both whole and fractional. We now assure the young beginner that we will conduct him through the elements of this science with as little delay and trouble as possible; and neither remarks nor examples will be given which are not, in the judgment of the author, essential to the progress of the pupil. With this assurance we close this introduction, and comm4ence ALGEBRA, by giving more extended definitions of terms. ALGEBRA DEFINITION OF TERMS. THE signs for addition, subtraction, multiplication, division, and equality, have already been explained. We have also explained coefficient and vinculum. The word coefficient can hardly be understood by a mere definition. It means any factor connected with another, and may be simple or compound, thus, ax; a is the coefficient of x, and in the term 3ax, 3a is the coefficient of x, and 3 is the coefficient of ax. 1. When. a letter stands alone, as b, y, or any other letter, one or unity may be considered its coefficient. In the expression (3a+-2b —c)x, (3a+2b —c) is a compound coefficient to x. It is also a factor, and x is another factor. The word factor has the same signification as in Arithmetic. 2. When we wish to note that two quantities are unequal, we write this sign > between them. The opening of the sign is always put toward the greater quantity, thus, a>b, signifies that a is greater than b, and a