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A drill-book in algebra / 
 
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The original of tliis book is in 
 tlie Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031252517 
 
DRILL-BOOK 
 
 or 
 
 ALGEBEA 
 
 BT 
 
 Prop. GEOEGE WILLIAM JONES 
 
 CORNELL UNIVERSITY. 
 
 FIRST EDITION. 
 
 ITHACA, N. T.: 
 
 GEOEGE W. JONES. 
 1893. 
 
Entered according to Act of Congress, in the year 1892, by 
 
 GEORGE WILLIAM JONES, 
 jB tbp Office of tbe Librarian of 'Congress, at Washington. 
 
 Robert Druuhond, New York, Pbihtbr. 
 
PREFACE. 
 
 So far as concerns mathematical studies there are always two 
 classes of pupils : those that will use mathematics later in 
 life as one of the tools of their trade — ^the engineers, architects, 
 accountants, teachers, scientists; those that will not so use it. 
 
 To both of these classes the careful study of the elements 
 of geometry and algebra is important : to the first class as lay- 
 ing a sure foundation for work in the higher mathematics and 
 for its professional applications; to the others as giving the 
 power and the habit of exact statement and rigorous proof. 
 In this later day we pride, ourselves on our laboratories, in 
 which the pupil comes face to face with the facts and forces 
 of nature — every mathematical recitation-room under an able 
 teacher is a laboratory in logic, and for sound logic there is 
 always an unlimited demand. 
 
 This book is for use by the more advanced classes in the 
 high schools and academies and the lower classes in the col- 
 leges; and its primary object is to teach young men and 
 women to think. From the beginning the philosophy of the 
 subject is made prominent; and in writing it the author set 
 himself the double task of writing a book whose definitions 
 should be precise and whose proofs should be rigorous, and of 
 writing one so simple that any diligent pupil could read it 
 easily. 
 
 But he has not confined himself to definitions and proofs : 
 a large collection of questions and exercises has been added; 
 and for a good understanding of the fundamental principles, 
 and readiness in their use, quite as much reliance is placed on 
 
 iii 
 
IV PREFACE. 
 
 the questions as on the text. It is hoped that by his effort to 
 answer these questions the pupil will be early taught to think 
 earnestly, to, think independently, and to think hard. 
 
 Believing that the elements of plane geometry are at least 
 as simple as those of algebra, the author has assumed some 
 knowledge of that subject in pupils using this book, and he 
 has not hesitated to use geometric illustrations where their 
 greater concreteness seemed to give greater clearness. 
 
 This book was undertaken as an abridgment of Oliver, 
 Wait, and Jones' Treatise on Algebra; and at first it was 
 hoped that cutting out the more abstruse portions of the text 
 and the harder examples would fit the larger work for the use 
 of the preparatory schools ; but after that excision many al- 
 terations were found necessary, and in the end the order of 
 topics was changed, new lines of proof were adopted, new 
 questions and examples were prepared, and the whole book 
 was rewritten. 
 
 The author is indebted both to the writers on algebra, from 
 whose works he has drawn freely, and to his associates at 
 Cornell Ifciversity, who have been unsparing in their kindly 
 assistance. In particular he returns thanks to Professor 
 Hathaway, who outlined the discussion of the combinatory 
 properties of the simple arithmetic operations, that of meas- 
 ures and multiples, and that of incommensurable numbers; 
 to Mr. John H. Tanner, who made the selections from the 
 Treatise and prepared the first draft of the copy; to Miss Ida 
 M. Metcalf, who spent half a year in giving form to the text 
 and preparing the questions and exercises; and to Professors 
 Oliver and McMahon, who have read the greater part of the 
 book either in manuscript or in proof. 
 
 But with all the care he could exercise, he is conscious 
 that many errors have crept in, and that there are many de- 
 fects that only use in the class-room can bring to light. He 
 will esteem it a great favor, therefore, if his fellow-teachers 
 will tell him freely what they find wrong either in method or 
 matter, in general plan or detail. 
 
PREFACE. V 
 
 SUGGESTIONS TO TEACHERS. 
 
 The author's aim has been to discuss subjects in the order of 
 their logical dependence, so as to construct a continuous and 
 irrefutable line of argument throughout; but it must be re- 
 membered that the logical order is not always the easiest or 
 the most natural, and that not only are the claims of mathe- 
 matical science to be satisfied, but the wants of the individual 
 immature pupil are to be met. It may often happen, there- 
 fore, that deviation from the order of the book will be of 
 advantage; and a subsequent review in the order laid down 
 will then show the drift of the .thought and the links in the 
 chain of reasoning. 
 
 This point should be early and often impressed upon the 
 pupil: that many simple theorems are stated and. proved, not 
 that he may be convinced of their truth, for that conviction 
 can be reached by repeated experiment, but that a firm foun- 
 dation may be laid for a logical structure of ever-increasing 
 height and complexity. That is, they are not ends in them- 
 selves, but only useful tools for future work. 
 
 If parts of the text seem too abstruse for his pupils, or the 
 questions too hard, the wise teacher will reserve such parts 
 for a later reading, and he will choose for himself the order 
 in which the topics shall be presented. P.or example, he may 
 find it well to take up parts of the second chapter before fin- 
 ishing the first, or to set parallel lessons from the two chapters. 
 Theory and practice may thus go hand in hand. 
 
 In this book general principles are stated formally, as in 
 
 text-books on geometry, and illustrations and applications 
 follow; but the living teacher may well reverse this order, 
 and before setting a new topic in the book he may draw out 
 the whole matter from the pupils' own mind, by careful ques- 
 tioning after the Socratic method, first in simple illustrations 
 and then in general principles. Afterwards the pupil may 
 read and explain the text, and answer the questions set down 
 for exercises. The authoj- recognizes the distinction between 
 the office of the text-book and that of the teacher, and he 
 
Tl PREFACE. 
 
 places tlie man aboye the book; but he has been taught by 
 his own experience that a book is yery useful. And what 
 should the book contain ? A treatise on any subject contains 
 the whole body of knowledge on that subject, well digested 
 and well arranged and indexed, so that the reader may find 
 all that he seeks within its pages; but it need contain no 
 exercises for pupils and no questions. A drill-book is more 
 modest in its aims : it leaves out all that is not necessary to 
 the main purpose; it presents the great principles in due order 
 and in simple language, so that the pupil may read them again 
 and again, and it sets him, under his teacher's guidance, to 
 find out their applications; it suggests to him the best methods 
 of work; it oifers him lists of questions on which he may task 
 himself, and. grow strong in the exercise; it helps the teacher 
 to cross-examine him; it serves as a standard to both teacher 
 and pupil; and it saves them endless labor in the giving and 
 taking of notes and in the preparation and copying of exercises. 
 
 It has been the author's good fortune to have a few bright 
 young people come to him every summer for a more complete 
 preparation in elementary mathematics, and he has thus kept 
 fresh in mind the wants of beginners. He has found ihese 
 pupils needing a regular and persistent drill both in the state- 
 ment and proof of the fundamental principles and in their 
 application; and he has written this book to meet their wants. 
 He submits it respectfully to the judgment of his fellow- 
 teachers, only asking that they neither adopt it nor reject it 
 without a very careful examination ; for while in most things 
 it follows well-worn lines, in others it makes radical depart- 
 ures from the common usage. 
 
 An answer-book (not a key) is in preparation for the con- 
 venience of teachers; and the whole list of questions has been 
 printed on cards for use in the class-room. 
 
 George W. Jones. 
 
 Ithaca, N. Y., May 3, 1893. 
 
CONTENTS. 
 
 I. THE PRIMAET OPERATIONS OP ARITHMETIC. 
 
 SECTION PAGE 
 
 1. Number, 2 
 
 2. Multiplication, and division, 6 
 
 3. Positive and negative numbers, 18 
 
 4. Addition and subtraction, 33 
 
 5. Involution and evolution, 30 
 
 6. Questions for review, 34 
 
 II. THE PRIMARY OPERATIONS OF ALGEBRA. 
 
 1. Algebraic expressions, 36 
 
 3. Addition and subtraction, 40 
 
 3. Multiplication, 43 
 
 4. Division, 56 
 
 5. Fractions 64 
 
 6. Questions for review, 68 
 
 III. SIMPLE EQUATIONS. 
 
 1. One unknown element, 73 
 
 3. Two unknown elements, 78 
 
 3. Three or more unknown elements, 88 
 
 4. Questions for review, 94 
 
 IV. MEASURES AND MULTIPLES. 
 
 1. Integers, 100 
 
 3. Entire functions of one letter, 108 
 
 3. Entire factors, 118 
 
 4. The highest common measure 138 
 
 5. The lowest common multiple, 133 
 
 6. Questions for review, 134 
 
VUl CONTENTS. 
 
 V. VARIATIONS, PROPORTION, INEQUALITIES, AND 
 INCOMMENSURABLE NUMBERS. 
 
 SECTION PIGE 
 
 1. Variation, 136 
 
 3. Proportion, . 142 
 
 3. Inequalities, 148 
 
 4. Incommensurable numbers, 152 
 
 5. Questions for review, ^ 160 
 
 VL POWERS AND ROOTS. 
 
 1. The binomial theorem, 162 
 
 2. Fraction powers, . 164 
 
 3. Radicals, 170 
 
 4. Roots of polynomials, 180 
 
 5. Roots of numerals, 186 
 
 6. Roots of binomial surds, 192 
 
 7. Questions for review, 196 
 
 VII. QUADRATIC EQUATIONS. 
 
 1. One unknown element, 198 
 
 2. Two unknown elements, 206 
 
 3. Three or more unknown elements, 212 
 
 4. Questions for review, 214 
 
 VIII. THE THREE PROGRESSIONS, INCOMMENSURABLE 
 POWERS AND LOGARITHMS. 
 
 1. Arithmetic progression, 218 
 
 2. Geometric progression, 222 
 
 3. Harmonic progression, 226 
 
 4. Incommensurable powers, 238 
 
 5. Logarithms, 332 
 
 6. Questions for review, 346 
 
 IX. PERMUTATIONS, COMBINATIONS AND PROBABILITIES. 
 
 1. Permutations, . „ . 250 
 
 2. Combinations, o . . 252 
 
 8. Probabilities, 258 
 
 4. Questions for review, 266 
 
PEIMARY NOTIONS FOR YOUNGER PUPILS. 
 
 While there are many new things to be learned in algebra, 
 there is nothing contradictory to what has been already learned 
 in arithmetic, and there are many points of resemblance be- 
 tween these two sciences. 
 
 For example, the same signs of operation, of grouping, and 
 of equality, are used, +, — , x, :, (), =; and fractions, 
 powers, and roots have the same meaning, and are written in 
 the same form. 
 
 The differences come largely from the frequent 
 
 USB OP LETTEES TO KBPEESENT NUMBBES. 
 
 But there is nothing arbitrary or mysterious about this 
 use; the letters, for the most part, are abbreviations for words, 
 and sometimes they serve to make the statements more general 
 than if numerals alone were used. 
 
 The reasoning is the same whether the numbers be expressed 
 in words, in letters, or in figures. 
 
 For example, if n stand for a certain number, say a man's 
 age, or the number of books in his library; then In stands for 
 the double of this number, 3re for its triple, \n for its half. 
 
 To make this clearer, the pupil may answer these 
 
 QUESTIONS. 
 
 1. What is the meaning of 4w? of 6m? of f«? of 3;i — |m? 
 of i(3ra + 9?»-7w)? of 3V(7« + 8w)? of n:\ni 
 
 If n stand for 60, what are the values of these expressions? 
 
 2. So, if X, y, z stand for three numbers, say the cost of an 
 algebra, a reader, and a grammar, for what does x-\-y stand ? 
 x + y + z'i x + y-z? 5x + 2y-dz? i{2x+y-z)? 5x-4:y-z? 
 
 3. If a; =50, 2/ = 30, 2: = 70, find the values of the ex- 
 pressions written above, and of thes^: 
 
 « x+y XXX z+x z 
 X + -, — -, - + y, — -— , z-\ — , , -+x. 
 
 z' z ' z •" z+y' y y y 
 
S PRIMARY NOTIONS FOR TOUNOER PUPILS. 
 
 4. Show the difference in meaning between the expressions : 
 12a -75 + 3c and 12a - (7b + 3c); 
 {a + b)x{c + d) and a + {ixe) + d; 
 and, if a = 4, 5 = 3, c = 2, d = 6, find their values. 
 
 The use of single letters to stand for words, and of signs to 
 denote operations and relations, constitute 
 
 A KIND OF SHOET-HAKD WRITING, 
 
 and the resulting brevity of expression is of great advantage 
 in many ways. For example, compare these two statements : 
 
 1. I want the result of adding to a number three times 
 itself, subtracting twice the number from the sum, increasing 
 the remainder by five times, and by four times, the same 
 number, and finally subtracting seven times the original 
 number from the last sum. 
 
 2. a + 3a — 2a + 5a + 4a — 7a; wherein a stands for the num- 
 ber, and the additions and subtractions are indicated by the 
 signs + and — . 
 
 In the last form the result, 4a, i.e., four times the original 
 number, is seen at a glance. 
 
 So, let cc, y stand for any two numbers, of which x is the 
 larger, and express in algebraic form : 
 
 the sum of the numbers; their difference; their product; 
 
 the proper fraction got by dividing one by the other; the 
 improper fraction; 
 
 the product of the smaller number by their difference; 
 
 the sum, the difference, and the product of their squares; 
 
 the squares of their sum, their difference, and their product; 
 
 the product and quotient of their sum by their difference; 
 
 the product of the squares of their sum and their difference; 
 
 the product of the sum and difference of their squares; 
 
 the sum and the difference of their cubes; 
 
 the cube of their sum and of their difference. 
 So, tell in words what these algebraic expressions mean : 
 2{x+y); xy(x + y); f(a^-2/'); a?+y^; o^y^; 
 
 {x+yf; l{x+y + &); Id-xy; |(3a; + 3«/); xY; 
 
PRIMARY NOTIONS FOR YOUNGER PUPILS. xi 
 
 The examples above are cases of 
 
 TEANSLATIOK IliTTO ALGEBEAIO FORMS. 
 
 The pupil may also translate the answers to these questions: 
 
 1. At a dollars a pair what will 5 pairs of gloves cost ? 5 
 pairs ? 26 pairs ? a pairs ? 2a pairs ? 
 
 How many pairs can be bought for c dollars ? for I dollars ? 
 
 2. An orchard contains r rows of t trees each, and each tree 
 bears I barrels of apples: how many barrels of apples are 
 raised ? and what is their value at d dollars a barrel ? 
 
 3. A man bought three books, paying a dollars for the first 
 book, 5 times as much for the second as for the first, and for 
 the third c times as much as for the other two : what was the 
 cost of each book ? of all of them ? 
 
 4. A bill of groceries shows t pounds of tea at x cents a 
 pound, s pounds of sugar at y cents, and c pounds of coffee at 
 z cents: what is the whole cost in cents ? in dollars ? 
 
 5. At X dollars a yard, what will c yards of cloth cost ? If 
 the same cloth be sold for y dollars a yard, and something be 
 gained, what relation has y to a; ? How much is gained on 
 one yard ? on the c yards? 
 
 6. A square field is / rods long : how wide is it ? what is 
 its area ? how long a fence is needed to enclose it? how much 
 more to divide it into four equal square fields? how much to 
 divide it into four equal rectangular fields ? 
 
 Note that the numerical values are not determined, only 
 the relations between the length, breadth, area, length of 
 fences, and so on, relations which hold good whatever num- 
 ber/ stands for ; and state these relations in words. 
 
 7. A can do a piece of work in a days, B in 5 days, C in c 
 days: what part of the work can A do in one day ? B ? C ? 
 A and B working together ? B and C ? C and A ? A, B, 
 
 andC? What does -stand for? J-? -? - + ^? ^+-? 
 a c a c 
 
 i + Lv 1_4 + Lv i:(14)p i:(L + l)? i:(l+l)? 
 c a a b c \a b / \b cl \c a} 
 
xu PRIMARY NOTIONS FOR YOUNGER PUPILS. 
 
 \a b cj \a b cl \a b cj 
 
 8. If a = 16, b = 12, c=6, fiud the value of each of the 
 expressions above; and find what part of the work remains 
 undone after A, B, and C have worked together three days. 
 
 9. If the figures used in writing a two-figure number be 
 a, b, the number is written lOa + 5: express half the number; 
 the square root of it; a number that is c units less than this 
 number; a number that is c units less than a fourth part 
 of it ; the number whose figures are b, a. 
 
 10. If a boy be y years old now, how old was he a year ago ? 
 two years ago? how old will he be/ years hence? how old 
 when his age is doubled ? what is a third of his age ? two 
 thirds of what it will be h years hence ? 
 
 11. A rectangular pile of wood is a feet long, b feet wide, 
 and c feet high: how many square feet are there in the top ? 
 in one end ? in one side ? how many cubic feet in the pile ? 
 how many cords ? what is its value at d dollars a cord? 
 
 13. If X be the larger of two numbers and d their differ- 
 ence, what is the smaller number? If x be the smaller num- 
 ber and d their difference, what is the larger ? 
 
 13. If p be the product of two numbers and n be one of 
 them, what is the other ? If g- be the quotient and n one 
 number, what is the other? 
 
 14. If a bushels of wheat cost a dollars, what is the price 
 of one bushel ? If c bushels of this wheat sell for li dollars, 
 what is gained or lost on one bushel? on the whole lot? 
 
 15. At jo cents a yard what will it cost to plaster the walls 
 and ceiling of a room a feet long, b feet wide, c feet high ? 
 
 16. If a man can row w miles an hour in still water, what 
 progress will he make rowing with a tide that runs t miles an 
 hour ? rowing against the tide ? 
 
 17. A man has m miles to walk in h hours; he walks / 
 miles an hour for the first b hours: how fast must he walk 
 the rest of the way? 
 
PRIMARY NOTIONS FOR YOUNGER PUPILS. xiii 
 
 18. A merchant began business with a capital of a dollars; 
 the first year he doubled his money; the second year he 
 gained h times the original capital; the third year he lost 
 I dollars and died, leaving c children; the cost of settling the 
 estate was s dollars: how much did each child receive ? Is it 
 possible that there was nothing to divide ? 
 
 19. A and B start from the same place and walk in the 
 same direction, A at as miles an hour and B, h miles: how far 
 apart are they at the end of an hour ? at the end of h hours ? 
 Do you know from this statement which man walks the faster ? 
 
 So, if they go in opposite directions? 
 So, if starting from two places h miles apart, they walk 
 towards each other ? if away from each other ? 
 
 THE USB OF LETTERS IN SOLVING PROBLEMS. 
 
 Sometimes statements are made of such a nature that the 
 actual value of an unknown number, at first represented by a 
 letter, may afterwards be found. 
 
 For example, to find a number such that if its double and 
 its quadruple be added to it, the sum shall be 63. 
 
 This problem is easily solved by arithmetic, as follows: 
 
 If to a number its double be added the sum is three times 
 the number; and this sum increased by four times the num- 
 ber is seven times the number, which is 63. If then 63 be 
 seven times the number sought, that number is a seventh 
 pai't of 63, i.e., the number is 9. 
 
 The words the number are used seven times in this state- 
 ment, and the process may be shortened by writing some 
 letter, as x, for these words and expressing the operations 
 and relations by signs. In this symbolic language, %x stands 
 for the double of the number and 4a; for its quadruple; and 
 it is easy to translate into algebraic forms what the statement 
 above gives in words: 
 
 a number increased by its double and its quadruple gives 63; 
 X + 2a; + 4a; =68 
 
 Le., x + 2x + 4:X—63, 7a;=63, a;=9, as before. 
 
XIV PRIMARY NOTIONS FOR YOUNGER PUPILS. 
 
 So, if the sum of the ages of father, mother, son, and 
 daughter be 100 years, if the boy be twice as old as his sister, 
 the mother four times as old as her son, and the father's age 
 be three times the sum of his children's ages; these facts may 
 be expressed in algebraic form by writing x for the girl's age, 
 2x for the boy's, four times 3a;, i.e., 8x, for the mother's, 3x 
 for the sum of the children's ages, and three times 3x, i.e., 9x, 
 for the father's age; then 9a; + 8a; + 2a; +a;= 100, 20a;= 100, 
 x = 5; aiul the girl is five years old, the boy ten, the mother 
 forty, and the father forty-five. 
 
 So, if a man pay $45 for a saddle and a bridle, and the 
 saddle cost four times as much as the bridle; then x may stand 
 for the cost of the bridle, 4a; for that of the saddle, and the 
 equation 4a; + a; =45 expresses all the facts set forth in the 
 statement of the problem. From this equation comes 5a' = 45, 
 a; = 9, 4a; =36, the values sought. 
 
 For further practice the pupil may choose some letter or 
 other character (a star or the picture of a dragon-fly would 
 serve just as well if as easily made and read) to stand for one 
 of the unknown elements of the problem, express the other 
 unknown elements as multiples or parts of the letter, trans- 
 late the word-statements into algebraic forms, solve the result- 
 ing equations, and finally determine the values sought. 
 
 1. A and B together have $100, and B has three times as 
 much as A; how much has each of them ? 
 
 3. A man paid $34 for a hat, a vest, and a coat; the vest 
 cost twice, and the coat three times, as much as the hat : what 
 was the cost of each of them ? 
 
 3. Divide 108 into three parts such that the second part 
 shall be twice the first, and the third three times the second. 
 
 4. Three trees together bear 33 bushels of apples; the second 
 tree bears twelve times as much as the first, and the third a 
 fourth part of the yield of the second: how many bushels 
 does the first tree bear ? 
 
 5. John is three times as old as Henry, and the difEerence of 
 their ages is 13 years: how old is each of them ? 
 
PRIMARY NOTIONS FOR YOUNGER PUPILS. XV 
 
 6. The difference of two numlDers is seven times the less, and 
 if four times the less be taken from the greater, the remainder 
 is 24 : what are the numbers ? 
 
 7. B is three times as old, and C four times as old, as A, and 
 the sum of their ages is s years : how old is each of them ? 
 
 THE NATUEB OF EQUATIONS. 
 
 In the problems that have been solved above, some number, 
 or the difference of two numbers, or the sum of two or more 
 numbers, has always been given as equal to some other num- 
 ber. Such expressions of equality are called equations, and 
 their nature and uses should be well understood. The two 
 parts of an equation, separated by the siga of equality, are its 
 members, and the numbers that make up the two members 
 and are joined by the signs + and — are its terms. 
 
 An equation of itself cannot be said to have any sign or 
 value; it is like a balance, one member being the thing weighed 
 and the other member the weights. The sign of equality cor- 
 responds to the pivot. 
 
 Kegarding an equation in this light, it is evident that the 
 same number may be addod to, or subtracted from, both 
 members of an equation, and that both members may be mul- 
 tiplied, or divided, by the same number, and the resulting 
 numbers be still equal. 
 
 For example, if 2a; = 6, then x = 3 and 6a; = 18. 
 
 So, if a;=;6 — 2a;, then a; + 2a;=6 and a; = 2. 
 
 So, if a; + 3 = 24 — 2a;, then 3 may be subtracted from both 
 members, and 2a; be added, and there results the equation 
 3a; = 21; then both members may be divided by 3 and a; = 7. 
 
 So, if ix=3, then a;=12; and if ^z=S, then x = 9. 
 
 In the same manner the pupil may solve the equations below: 
 1. 4-2a; = 8-6a;. 2. 3a;-48 = a; + 12. 3. l = 10-ia;. 
 4. i(a;-6)=20. 5. 5 + 6a; = 2a; + 17. 6. |/a;-3 = 9. 
 
 7. 3a;=i/9. 8. 5(a;-6) =i(17-a;). 9. ^a; + 5 = 24/a; + 3. 
 
 10. |'3a;-2 = l. 11. 4/a;- 1=1^27. 12. -k'a; + 1 = ^16. 
 
XTl PRIMARY NOTIONS FOR YOUNGER PUPILS. 
 
 THE USE OF LETTEKS ASTD EQUATIONS IW STATING RULES 
 AND PRINCIPLES. 
 
 The familiar rules and principles of arithmetic may often 
 be stated in algebraic language. One advantage of such 
 statements is their brevity and clearness, as noted above, and 
 the consequent ability of the pupil to see the whole at once, 
 and to note the relations of all the parts. 
 
 For example, that the value of a fraction is not changed if 
 hoth its terms be multiplied ly the same number, a statement 
 of two lines, is expressed in algebraic form by the use of six 
 
 letters and three signs : it is written -^—^ , wherein t- 
 
 ° b bxiii b 
 
 stands for any fraction whatever and m for any multiplier. 
 
 Ct (i * 171 
 
 So, the equation :r — -r- — means that both terms of a 
 
 ^ b b~m •' 
 
 fraction may he divided by the same number without changing 
 its value. 
 
 So, the proposition — if four numbers be in proportion, the 
 product of the extremes is equal to the product of the means, ia 
 expressed by writing if a:b = c:d, then axd=bxc, 
 wherein a, b, c, d, are any four numbers. 
 
 So, that the remainder is not changed if loth minuend and 
 subtrahend be increased by the same number is expressed by 
 the equation {in + n) — {s + n) = m — s = r, wherein m is the 
 minuend, s the subtrahend, r the remainder, n any number. 
 
 Ct c 
 
 So, J and j may stand for any two fractions, and the rule 
 for their addition may be briefly written ^^ — '-. 
 
 The pupil may in like manner write the rules for subtract- 
 ing one fraction from another; for multiplying two fractions 
 together; for dividing one fraction by anotber. 
 
 So, he may express the relations between the dividend, 
 divisor, quotient, and remainder, in division. 
 
 So, he may write down the well-known properties of pro- 
 portion which are expressed by the words alternation, inver- 
 sion, composition and division. 
 
ALGEBRA. 
 
 I. THE PBTTVTARY OPERATIONS OP ARITHMETIC. 
 
 The pupil who begins the study of algebra is already fam- 
 iliar with the simpler operations, of arithmetic, and has given 
 some thought to the reasons for those operations. As algebra 
 is but a larger arithmetic, he may well stop to review the funda- 
 mental principles, and try to fix in his mind more precise 
 notions of what is to be the basis of his larger knowledge. 
 
 As distinguished from geometry, algebra treats of numbers 
 and of numbers only; and this'chapter, after the definition of 
 number and the distinction between the different kinds of 
 simple numbers, goes on to define and discuss the operations 
 of multiplication and division, of addition and subtraction, 
 and of involution and evolution. 
 
 Algebra difEers from arithmetic chieiiy in this : that while 
 in arithmetic every number has a definite and fixed value, and 
 the numerical expression of value is the thing always sought, 
 in algebra the expression of the relation of numbers, or of 
 some general principle, is often what is aimed at, the nu- 
 merical computation being an arithmetic operation that may 
 be performed later or omitted. 
 
 Algebra differs still further from arithmetic in the use of a 
 set of symbols that constitute a language of its own, and 
 among the advantages of this symbolic language are these: 
 clearness, brevity, and generality of statement; the ability to 
 mass directly under the eye, and thus to bring before the 
 mind as a whole, all the steps in a long and intricate investi- 
 gation; and the facility of tracing a number through all the 
 changes it may undergo. 
 
2 THE PRIMARY OPERATIONS OP ARITHMETIC. [I. 
 
 §1. NUMBER. 
 
 The subject-matter of Algebra is number, and numbers 
 come from counting and measuring ; they answer the questions 
 how many and how much. Anything that is to be measured 
 is a quantity, and the result of the measurement is a number. 
 E.g., if a boy count the apples in a basket the answer is a 
 
 number, say twelve. 
 So, he may measure the side of a room with a yard stick and. 
 find how many times, say twelve, a single yard is con- 
 tained in the whole length. 
 
 INTEGERS AND FRACTIONS. 
 
 If the things counted be entire units, or if the thing meas- 
 ured contain the measuring unit exactly so that there is no 
 remainder, the number is an integer; but if the measurement 
 can be completed only by using some part of the first unit as 
 a new unit of measurement, the resulting number, for the 
 portion so measured, is a simple fraction. 
 E.g., if a side of a room contain a yard twelve times with a 
 remainder that contains a third part of a yard twice, 
 the whole length is twelve yards and two thirds. 
 
 CONCRETE AND ABSTRACT NUMBERS. 
 
 In the examples above, the number is closely associated with 
 the things counted or measured, and the whole answer to the 
 question hotv many is twelve apples, or twelve and two-thirds 
 yards. Such numbers are concrete numbers, and concrete 
 numbers may be defined as measured quantities. 
 E.g., twelve apples, five days, twenty pounds. 
 
 But an abstract mimber implies some operation, and it may 
 be called an operator. The simplest operations are those of 
 repetition, such as doubling, tripling, and quadrupling; and of 
 partition, such as halving and quartering. 
 E.g., if a child pick up twelve blocks, he may pick up one 
 block, then another, and another till he has twelve of 
 them, 
 
S^l NUMBER. 3 
 
 and the operation that the operator twelve calls for is repe- 
 tition of the single act of picking up a block. 
 
 So, he may part his blocks into two equal piles, or cut an apple 
 into two equal parts j the operator is two and the opera- 
 tion is partition. 
 That thing which is acted upon is the unit. 
 
 E.g., the single block in the first case, and the whole group of 
 blocks, or the apple in the other, is the unit. 
 
 The combination of a unit and a numerical operator forms 
 a concrete number. 
 
 QUESTIOKS, 
 
 1. Give some number that answers the question how long a 
 time, the question how much, the question how far. 
 
 3. Name the different standard units of time, and state 
 their relation to each other. 
 
 So, of length, of area, of volume, and of weight. 
 
 Of these units, which are natural ? which artificial ? 
 
 3. Can the question how ill be answered by giving a number? 
 So, of the questions how difficult, how good, how beautiful? 
 So, of the questions how warm, how bright, how strong ? 
 
 4. By what unit can a room that is twelve and two-thirds 
 yards long be measured, so as to give an integer result ? 
 
 5. If the same thing be measured, what is the effect on the 
 number if a smaller unit be used ? if a larger unit ? 
 
 6. Count two dozen eggs with six eggs as the unit; what is 
 the operator ? with one egg as the unit, what is the operator ? 
 
 7. Express the distance six feet with three inches as the 
 unit, then take a smaller unit and a smaller; how far can this 
 process of reducing the unit be carried ? 
 
 So, if the unit be taken larger and larger ? 
 
 8. Draw a line six inches long, and express three quarters 
 of it with half an inch as the unit. How many distinct units 
 are here used ? How many opeiratioris? how many operators? 
 
 WTiich of these operations are repetitions ? which partitions? 
 
4 THE PRIMARY OPERATIONS OF ARITHMETIC. d, 
 
 EQUAL NUMBEES. 
 
 That two concrete numbers be equal, it is necessary that the 
 things counted or measured be of the same kind, and that, il 
 the units be taken of the same magnitude, there shall be as 
 many units in the one group as in the other. 
 
 If there be more units in one group than in the other, the 
 first number is larger than the other ; if fewer, it is smaller. 
 
 That two abstract numbers be equal, it is sufficient that 
 when operating on the same unit they shall give the same 
 result; examples of such numbers appear later. 
 
 That two numbers are equal is shown by the sign = ; that 
 they are unequal by i= ; that the first is larger than the other 
 by the sign > ; that it is smaller by < . 
 
 EXPBBSSION OF NUMBBKS. 
 
 In algebra numbers are represented by Arabic numerals, 
 as in arithmetic; they are also often represented by letters. 
 E.g., in questions about interest, p may stand for principalj r 
 for rate, t for time, i for interest, a for amount. 
 
 If there be four promissory notes, the four principals may 
 be writen p', p", p'", f", read p prime, p second, p third, 
 p fourth, or Pi,Pi, Ps,Pi, read p one, p two, p three, pfour ; 
 and the corresponding rates and times would then be written 
 r', t', r", t", r'", t'", r^, f", or r^,t^, r^,t„ r^,t^, r„ t,. 
 
 Letters or figures attached to other letters are indices. 
 
 It is customary to express a concrete number by writing 
 the operator first, and after it the unit. 
 E.g., twelve blocks, half an apple, two thirds of a yard. 
 
 But the purposes of algebra are sometimes better served by 
 writing the unit first and following it with the sign of opera- 
 tion and the operator. 
 
 E.g., block X 13, wherein the cross means repetition. 
 So, apple : 2, wherein the colon means partition, or apple x 1/2. 
 So, yard : 3 x 2, or yard x 2/3, means that the yard is divided 
 into three equal parts, and two of them are taken. 
 
81] NUMBER. 5 
 
 QUESTIONS. 
 
 1. Find other concrete numbers that are equal to three 
 hundred minutes; to thirty-six inches; to half a mile. 
 
 2. If there be t^o farms each of one hundred acres, are 
 they equal in area ? have they the same shape ? are they 
 equal in value? 
 
 3. If there be two farms, one of fifty acres worth a hundred 
 dollars an acre, and the other of a hundred acres worth fifty 
 dollars an acre, in what respect are the two farms equal ? 
 
 What two concrete numbers are now equal ? 
 
 What is the unit, and what are the opei-ators that give these 
 two equal concrete numbers? are they found by precisely 
 the same, or by different, operations ? 
 
 4. One room is four yards by six and another three yards by 
 eight ; will the same carpet cover both floors? can it be made 
 up in two parts so as to cover either floor at will ? 
 
 In what respect are the two floors equal ? in what unequal? 
 
 5. If the answer to the question how long be six days, which 
 is thought of first, the unit day or the number six ? 
 
 6. If the length of a room be sought, which is found first, 
 the yard stick or the length in yards ? 
 
 7. What does a salesman dO when he measures off 50 yards 
 of carpet ? so, when he cuts this carpet into six equal breadths ? 
 
 In each of these operations what is the unit and what is the 
 operator ? 
 
 8. Following the same system of notation as in interest, 
 how can the area of a rectangular house-lot be expressed if the 
 length and breadth be known in feet ? How can the cost be 
 expressed if the price per square foot be also given? 
 
 So, the cost of a block of marble whose length, breadth, 
 and height are known, and the price per cubic foot ? 
 
 9. Write an expression to show the order of procedure if a 
 grocer sell ten eggs to each of two customers for three succes- 
 sive days; ten eggs to each of three customers for two days. 
 
6 THE PRIMARY OPERATIONS OP ARITHMETIC. [I- 
 
 §2. MULTIPLICATION ANI) DIVISION. 
 
 The product of a concrete number, the mttUiplicancC, by an 
 abstract number, the multiplier, ie the result of the repetition 
 or partition of the concrete number by the other used as an 
 operator. The product is a concrete number of the same kind 
 as the multiplicand. Multiplication thus embraces halving 
 and quartering as well as doubling, tripling, and quadrupling. 
 E.g., $5 X 10 = $50, $50 : 10 = $5, or $50 x 1/10 = $5. 
 
 The product of two concrete numbers is an absurdity. 
 E.g., the product $2x5 days is impossible; 
 but if a man earn $3 a day, in 5 days he earns $10, 
 i.e., $2 X 5 = $10, wherein 5, not 5 days, is the multiplier. 
 
 The solution rests on the well-recognized relation between 
 the time and the wages earned : as five days is five-fold one 
 day, so the wages of five days is five-fold the wages of one day. 
 
 The product of two or more abstract numbers is an abstract 
 number that gives the same result when operating on a unit 
 as is obtained when the unit is multiplied by the first of the 
 given numbers, the product so found multiplied by the second 
 number, and so on till all the numbers are used. The num- 
 bers are factors of the product. 
 
 E.g., if of three men A, B, C, A has $50, B twice as much as 
 
 A, and C three times as much as B; 
 then B has $50 x 2, or $100 ; and C has $100 x 3, or $300, 
 i.e., $50x2x3 = $300; i 
 
 but since $50 x 6 = $300, 
 
 therefore the product of the two abstract numbers 2, 3^ is tlie ' 
 abstract number 6. 
 
 That the product of two or more factors is to be used in- 
 stead of the factors in succession, may be expressed by enclos- 
 ing them in brackets or -placing them under a bar, and these 
 signs indicate that the expression so enclosed is to be first 
 simplified and then used as a single number. 
 E.g., $50 X (2 x 3) or $50 x 2xS. 
 
 The factors so used form a group of factors. 
 
IS] MULTIPLICATION AND DIVISION. 
 
 QUESTIONS. 
 
 1. When is multiplication a process of repetition, and when 
 of partition ? 
 
 2. By what process other than multiplication can the prod- 
 uct of five days by seven be found ? 
 
 3. Can an abstract number be multiplied by a concrete 
 number ? Can an abstract number be made concrete by mul- 
 tiplication ? 
 
 4. If a concrete number be multiplied by an abstract num- 
 ber, of what kind is the product ? Can a concrete product be 
 found without using a concrete multiplicand ? 
 
 5. In finding the area of a rectangle we seem to multiply 
 one concrete number, the length, by another concrete number, 
 the breadth; state what is really done. 
 
 6. What is the test of equality for abstract numbers? 
 
 7. How are we convinced that the product of the abstract 
 numbers 3 and 2 is the abstract number 6 ? 
 
 Does this reasoning apply to this particular pair of numbers 
 only, or is it in the nature of a general proof that applies 
 alike to every pair of abstract numbers? e.g., does it prove 
 that the product of the two abstract numbers 3 and 5 is the 
 abstract number 15 ? 
 
 8. What is the cost of 10 cases of eggs, each containing 6 
 boxes that hold two dozen, at 1^ cents apiece ? Exhibit the 
 factors in the order named. 
 
 Exhibit them in order if the number of dozen be first found 
 and the price per dozen. 
 
 So, if the cost of one case be first found. 
 
 9. What change is made^n the product by multiplying the 
 multiplicand by some number and then using the same mul- 
 tiplier as before ? 
 
 So, by multiplying the multiplier and leaving the multipli- 
 cand unchanged ? 
 
 10. Name the integer factors of 12, of 35, of 315. 
 
8 THE PRIMART OPERATIONS OF ARITHMETIC. [I.Th. 
 
 MULTIPLICATION' ASSOCIATIVE. 
 
 The sign •.■ means since or because/ .'. , therefore/ • • •, and 
 so on; and the letters q.b.d., which was to ie proved. 
 
 Theoe. 1. The product of three or more abstract numbers is 
 the same number, however the factors be grouped. 
 E.g., let 5, 3, 1/3, 7, be four abstract numbers; 
 then are the products (5 x 3) x (1/2 x 7), 5 x (3 x 1/2 x 7), 
 
 (5 X 3 X 1/3) X 7 each equal to 5 x 3 x 1/3 x 7. 
 For, to multiply a unit by the product 5x3 gives the same 
 
 result as to multiply the unit first by 5 and that product 
 
 by 3 ; ' [df. prod, abs, nos. 
 
 So, to multiply this product by the product 1/3 x 7 gives the 
 
 same result as to multiply it first by 1/3 ^nd that 
 
 product by 7; 
 and the result of the four multiplications is the product 
 
 unit X 5 X 3 X 1/3 X 7. 
 
 So, the product unit x (5 x 3 x 1/2 x 7) =unitiX 5x3x1/3 x 7. 
 
 and •.' the abstract products (5 x 3) x (1/3 x 7), 5 x 3 xl/3 x 7 
 do the same work when operating on a unit, 
 .*. these abstract products are equal; 
 
 and so for the other abstract products. q.e.d. 
 
 So, to make the reasoning general, let a, b, 1/c, • • • \/lc, I be 
 any abstract numbers, operators that mean repetitions 
 and partitions; 
 
 then are the products a x (5 x 1/c) x • • • {1/k x Z), 
 
 (a x 6) X (1/c X • ■ . 1/k X I), and all others that may be 
 formed by different grouping of the factors, each equal 
 to the product axbx 1/c x - • • 1/h x I. 
 
 For, to multiply the concrete product unit x a by the ab- 
 stract product bxl/c is to multiply the concrete 
 product unit X a by the abstract number ft, and the 
 consequent concrete product unitxaxi by the ab- 
 stract number 1/c ; [df . prod. abs. nos. . 
 
 i.e., unit x a x (5 x 1/c) = unit xaxbx 1/c. 
 
1,521 MULTIPLICATION AND DIVISION. 9 
 
 So, for the product of this product by the abstract products 
 
 that follow in order, 
 i.e., unit xax{bx 1/c) • • • (1/A x I), 
 = unit X axbx 1/c • • • 1/k x I 
 = unit X (a X 6 X 1/c • • • 1/k x I), 
 So, for the other concrete products. 
 
 .•. the abstract product of these factors is the same, how- 
 ever they be grouped. Q.E.D. 
 The principle proved in theor. 1 is the associative principle 
 of multiplication ; the theorem is sometimes written, multi- 
 plication is an associative operation. 
 
 QUESTIONS. 
 
 1. By actual multiplication, show the equality of the prod- 
 ucts 5 X 1/10 X-Yxll X 1/14 X 2, 5x1/10x2x7x1/14x2. 
 
 2. What is a theorem ? what a proof ? [consult a dictionary. 
 
 3. In the proof of theor. 1, for what purpose are the 
 factors represented by letters ? Which of the factors a, i, 
 1/c, • • • l/Tcy I, indicate repetitions and which partitions ? 
 
 4. Is the caption of theor. 1 a statement of whose truth we 
 are certain at first, or one that must be proved ? 
 
 5. The second paragraph of the proof " then are the prod- 
 ucts • • • " is a restatement of the last line of the theorem; 
 why this restatement ? Are the statements that follow known 
 to be true, or must they be proved ? 
 
 6. May the proof of a theorem rest on statements that 
 " seem reasonable," or must it rest on the authority of defini- 
 tions and axioms, and of other theorems that have been already 
 fully established by means of definitions and axioms ? 
 
 7. Is the theorem true, and the proof conclusive, if there 
 be but five factors ? what is the meaning of the dots ? 
 
 8. State the associative principle of multiplication. 
 
 9. Show that the statement of the theorem and its proof 
 depend directly and wholly upon the definition of the product 
 of abstract numbers. 
 
10 THE PRIMARY OPERATIONS OP ARITHMETIC. lI,Ttt 
 MULTIPLICATION COMMUTATIVE. 
 
 Theoe. 2. Tlie product of two or more abstract numbers is 
 the same number, in whatever order the factors be multiplied. 
 
 (a) Two factors, m, n, both expressing repetition; 
 then the two abstract products mxn, nxm are equal. 
 For let ****...* 
 
 Sp % if! % » a a V 
 
 V 3p "P tP « ■ « 'P 
 
 be a collection of like units, say stars, arranged in rectangular 
 form, m units broad and n units deep, so as to form m 
 vertical columns and n horizontal rows ; 
 then •.•the concrete product star x m is the m stars in one 
 row, 
 .•. the concrete product stav xmxn is the stars in the 
 n rows, 
 i.e., all the stars in the whole collection ; 
 and •.• the- concrete product star xn is the n stars in one 
 column, 
 
 .*. the concrete product star xnxm is the stars in the m 
 columns, 
 i.e., all the stars in the whole collection. 
 
 .•. the abstract product mxn does the same work as an 
 
 operator on a unit as the abstract product nxm; 
 .'. these two abstract products, mxn, nxm, are equal. 
 
 Q.E.D. [df. eq. abs. nos. 
 (b) Two factors, m, l/n, expressing a repetition, and a 
 partition ; 
 
 then the two abstract products m x 1/n, 1/n x m are equal. 
 For, let the unit be divided into n equal parts, and let each 
 part be represented by one of the n stars in any column 
 taken from the block of stars shown above; 
 then*.^ in this block of stars the concrete product (one column 
 X m) is the stars in the m columns. 
 
2.52] MULTIPLICATION AND DIVISION. 11 
 
 i.e., all the stars in the whole collection, consisting of n rows, 
 . •. the concrete product one column xmx 1/n is the stars 
 in one row ; 
 
 and •.• the concrete product one column x 1/n is one star, 
 .•. the concrete product one column x 1/n xm is the stars 
 
 in one row. 
 .'.the two abstract products mxl/n and 1/nxm, 
 when operating on the same unit, do the same work, 
 and are equal. q.b.d, 
 
 QUESTIONS. 
 
 1. In the block of stars used in theor. 2, how many of the 
 m stars in a row are not shown ? of the n stars in a column ? 
 
 2. In the product star x m what is the unit ? what is the 
 operator? why is this product concrete? 
 
 3. What does the product star xmxn represent ? what 
 the product star xnxm ? 
 
 JIow do you know that these two products are equal ? 
 Granting their equality, how does it follow that the two ^.b- 
 stract products mxn, nxm are equal ? 
 
 4. In case (a) it is shown that the product of two abstract 
 integers is the same, in whatever order the factors be taken; 
 what relation does this truth bear to the complete theorem ? 
 
 5. What is the result of the operator m acting on a column 
 of stars as a unit ? of the operator 1/n acting on this product ? 
 
 What is the final result of the two operations ? 
 
 6. What is the result of the operator 1/n acting on a column 
 of stars as a unit? of the operator m acting on this product? 
 
 What is the final result of the two operations ? 
 
 7. What single fact proves that the two abstract products 
 m X 1/n, 1/n x m are equal ? 
 
 8. Make a formal statement of the truth learned in case (b) 
 as if it were a theorem by itself. 
 
 9. Show by a diagram that one seventh of six equal lines 
 is equal to six sevenths of one of tliese lines. 
 
12 THE PRIMARY OPERATIONS OP ARITHMETIC. [I.Th. 
 
 (c) Two factors, \/m, \/n, loth expressing partition ; 
 then the two abstract products 1/mxl/w, 1/nxl/m are 
 equal. 
 
 For, let the unit be divided into mxn equal parts, let each 
 part be represented by a star, and let the whole be ar- 
 ranged in a block of m columns and n rows; 
 
 then ••• in this block of stars, the concrete product 
 
 block X 1/m is the stars in one column of n stars, 
 .•. the concrete product block x 1/m x \/n is one star; 
 and •.• the concrete product block x \/n is the stars in one 
 row of m stars, 
 .'.the concrete product block x 1/n x 1/m is one star, 
 .*. the two abstract products, 1/m x 1/n, 1/n x 1/m, 
 when operating on the same unit, do the same work, 
 and are equal. q.e.d. 
 
 (<?) TTiree or more factors, a, 1/b, c,- • • 1/Jc, I, expressing 
 
 repetitions and partitions ; 
 
 then, however these factors are arranged at first in any 
 product of them, that product is equal to their product 
 when arranged in the order a, 1/b, c, • • • 1/h, I. 
 
 For in any one of these products, if « be not first it may be 
 grouped with the factor before it, may change places 
 with that factor without changing the whole product, 
 and so come to be the first factor. [ths. 1, 2 (a, b,c). 
 
 So, 1/5 may change places in turn with all the factors that 
 stand before it except a, and so come to be the second 
 ~^ factor, and so on ; 
 
 i.e., without changing the product, the factors may chainge 
 places successively, and come to take the order u, 1/b, 
 €,••■ 1/k, I. 
 
 .-. the abstract product « x 1/S x c x • • • 1/hxl, and 
 others that may be formed by different arrangements 
 of the same factors, when acting upon any unit, do 
 the same work, and are equal. q.e. d. 
 
2.S2] MULTIPLICATION AND DIVISION. 13 
 
 The principle proved in theor. 3 is the commutative princi- 
 ple of multiplication, and the theorem is sometimes written, 
 multiplication is a commutative operation. 
 
 QUESTIONS. 
 
 1. After the proof of cases {a, b) what must still be proved 
 before theor. 3 is fully established ? 
 
 2. In case (c) what kind of factors are considered? 
 
 3. What is the result when- the operators 1/m, 1/n act in 
 this order on the group of stars ? when the order is reversed ? 
 
 What is proved by the two products' being the same ? 
 
 4. As a separate theorem, state what is proved in case (c). 
 
 5. What relation does the product (block ■ 1/m) bear to the 
 product (block • 1/n) when m is 4 and w is 3 ? 
 
 6. Show that a third of half a circle is half a third of it. 
 
 7. In what respect do the factors treated in case (c) differ 
 from those in case (a) ? in what ^re they like them ? 
 
 8. How does case {d) differ from all the other cases ? 
 
 9. In the proof of case {d) which of the statements made 
 about the factor a rests on the authority of theor. 1? 
 
 Which of them on that of theor. 2 ? 
 
 With what kind of factors must a be grouped that case {b) 
 may apply ? 
 
 10. Show that a simple fraction with a numerator other 
 than unity could, by the definition of a fraction, be regarded 
 as the product of two factors of the form a and 1/b, and that 
 the proof in case {d) applies to such fractions. 
 
 11. State fully the truth proved in case (d). 
 
 12. AVhat is the commutative principle of multiplication? 
 
 13. Show that the proof of theor. 2 depends mainly on the 
 definition of equal abstract numbers. 
 
 14. By multiplication, show the equality of the products 
 5 X 1/10 X 2 X 7 X 1/14 X 4, 2 x 4 x 1/10 x 1/14 x 5 x 7. 
 
 Arrange and group these factors in other ways, and show 
 that all the products so found are equal. 
 
14 THE PEIMAHY OPERATIONS OF ARITHMETIC. [I.Ths. 
 
 Cob. 1. The product of two or more factors that express 
 
 repetitions and partitions is an abstract. simple fraction. 
 
 For the factors that express partitions may be grouped to- 
 gether at the left, 
 
 and their product, showing how many equal parts the unit is 
 divided into, gives the denominator of a fraction. 
 
 So, the factors that express repetitions may be grouped to- 
 gether at the right, 
 
 and their product, showing how many of these equal parts are 
 taken, is the numerator of the fraction. q.b.d. 
 
 Cob. 2. The product of two or more simple fractions is a 
 simple fraction whose numerator is the product of the numer- 
 ators of the factors; and whose dentfininator is the product of 
 their denominators. 
 
 Cob. 3. If n be any abstract intciftr, then the products 
 '■^i l/'rt, \/n X n are each equal to unity. 
 
 Cob. 4. If the numerator and denominator of a siinple frac- 
 tion be multiplied by the same integer, the value of the fraction 
 is not cfianged thereby. 
 
 For, let n/d be a simple fraction, and multiply n, dhj a; 
 then •.• flxl/a=l, 
 
 .'. na/da = 1/d x (1/a x «) x w = n/d. Q. b.d. [or. 3. 
 
 EECIPBOCALS. 
 
 Two abstract numbers whose product is unity are reciprocals. 
 E.g., 1/4, 4 ; 3/2, 3/3 ; 3i, 2/7 ; 3 x 4, 1/12; n/d, d/n. 
 
 Theob. 3. The product of the reciprocals of two or more ab- 
 stract numbers is the reciprocal of their product. 
 Let a, 5, • • • \/m, \/n, • • • p/q, r/s, • • • be any abstract num- 
 bers; 
 then is the product 
 
 1/a X \/b X — mxnx • • • q/p x s/r • • • 
 the reciprocal of the product 
 
 axbx ■ • ■ 1/m x 1/n x • • • p/q x r/s • • • 
 
2. 3. §21 MULTIPLICATION AND DIVISION. ' 15 
 
 For the product of these two products 
 
 = (a X 1/a) X (J X 1/b) x ■ • ■ (1/m x m) x (1/w x m) x • • • 
 
 {p/q X q/p) X (r/s x s/r) x • • ■ 
 = lxlx •••=!. Q.E.D. [ths. 1, 2. 
 
 QUESTIONS. 
 
 1. What is a corollary ? 
 
 2. Why must the product of factors expressing repetitioii or 
 partition be abstract ? 
 
 3. In what way does cor. 1 rest on theor. 2 ? 
 
 4. Is the product of two or more proper fractions larger or 
 smaller than the several factors ? 
 
 5. What is the oflBce of the denominator of a fraction ? 
 What is the derivation of the word denominator, and what is 
 its jJrimary meaning ? so, of the word numerator ? 
 
 6. On what does cor. 3 rest, directly ? indirectly ? 
 
 7. By cor. 2 prove that ny.\/n=\, then by theor. 2 that 
 l/nxn = l, and so that n x 1/re = 1/n x n. 
 
 8. In proving cor. 4, what use is made of the fact that 
 a X 1/a = 1 ? 
 
 9. What effect is produced by multiplying the numerator 
 of a fraction 5/8 by six ? the denominator by six ? both nu- 
 merator and denominator by six ? 
 
 10. What is the time of a railway journey of 300 miles if 
 the train run 10 miles an hour ? if 20 miles ? if 30 miles ? 
 
 If the speed be multiplied by any number, by what number 
 is the time multiplied ? 
 
 11. If a certain piece of work is to be done, what is the re- 
 lation between the time and the number of men employed ? 
 
 12. What number is its own reciprocal ? 
 
 13. What is the reciprocal of a proper fraction ? 
 
 14. Without going through the whole proof of theor. 3, 
 state why the product of the reciprocals of two or more num- 
 bers is the reciprocal of their product. 
 
 15. Can a concrete number have a reciprocal ? 
 
16 THE PRIMARY OPERATIONS OP ARITHMETIC. [I.Th, 
 
 DIVISION. 
 
 DiviSioif is an operation that is the inverse of multiplica- 
 tion, i.e., when the product of two factors, and one of them, 
 are given, division consists in finding the other factor.' The 
 product is now called the dividend, the given factor is the 
 divisor, and the factor sought is the quotient. 
 
 If the dividend be concrete there are two cases of division: 
 
 1. The divisor abstract and the quotient concrete. 
 
 2. The divisor concrete and the quotient abstract. 
 
 The first is a case of partition, more or less complex, and the 
 other a case of finding how many times one number ia con- 
 tained in another number of the same kind, a ratio. 
 
 E.g., •.• the product of $5 by 4 is $30, 
 .•. the quotient of $20 by 4 is $5, 
 and the quotient of $30 by $5 is 4. 
 
 The sign of division is : or / , and the order of writing is 
 dividend : divisor = quotient, or dividend/divisor = quotient. 
 E.g., $30 : 4 = $5, $30 : $5 - 4, $20/4 = $5, $20/$5 = 4. 
 
 Thbor. 4. If the product of two factors he multiplied by the 
 reciprocal of one of them, the result is the other factor. 
 For, let/, g stand for the two factors, and p for their product; 
 then •.• p =f X g, [df . prod. 
 
 ••• P X Vf=f^9>^ yf=9 xA yf=9- Q-B.D. 
 
 If the divisor be abstract, theor. 4 may be written, the prod- 
 uct of the dividend iy the reciprocal of the divisor is the quotient. 
 
 CoK. If there he a series of multiplications and divisions, 
 the final result is the same, in whatever order they be performed, 
 and however the elements, be grouped. 
 
 For, every division is a multiplication by the reciprocal of the 
 
 divisor, 
 and these multiplications may be performed in any order, and 
 
 the factors be grouped in any way. 
 
4, §2] MULTIPLICATION AND DIVISION. 17 
 
 In applying this corollary, the pupil must take care lest he 
 change the office of any element and multiply by it where he 
 should divide, or divide by it where he should multiply. 
 E.g., 12 : 3 X 2 X 4 : 8 = (12 : 3 X 4) X (2 : 8) 
 :jfc (4 : 3 X 12) : (8 x 2). 
 
 QUESTIONS. 
 
 1. Of the three elements of a division, can all be concrete ? 
 two concrete and one abstract ? two abstract and one con- 
 crete ? all abstract ? 
 
 If a man earn 14 a day, in what time will he earn $20? 
 Are not the dividend, divisor, and quotient all concrete ? 
 
 2. In each of the two cases of division shown on the oppo- 
 site page, to what does the divisor, correspond in the multipli- 
 cation of which the division is the inverse ? 
 
 3. If the dividend be multiplied by some integer and the 
 divisor be unchanged, what is the effect on the quotient ? if 
 the divisor alone be multiplied ? if both be multiplied by the 
 same integer ? if the dividend alone be divided? if the divisor 
 alone be divided ? if both be divided by the same integer ? 
 
 State these principles as applied to the terms of a fraction. 
 
 4. Is a ratio a concrete or an abstract number ? 
 
 5. On what authority is it said, in the proof of theor. 4, 
 that fxgx l/f=g x/x 1//? on what that gxfx '^/f^g ? 
 
 6. Can the second form of stating theor. 4 be used when the 
 divisor is concrete ? does the first form always apply ? 
 
 How is the second form applied in the division of fractions? 
 
 7. Why may the multiplications and divisions spoken of in 
 the corollary of theor. 4, be performed in any order ? 
 
 Why may the factors be grouped in any way ? 
 
 8. Replace the word multiplied by divided in cor. 4 theor. 2 
 and prove the resulting statement. What useful applications 
 has this corollary in reducing fractions ? 
 
 9. Prove that the quotient of the reciprocals of two numbers 
 is the reciprocal of their quotient. 
 
18 THE PRIMARY OPERATIONS OF ARITHMETIC. II 
 
 §3. POSITIVE AND NEGATIVE NUMBEES. 
 
 Sometimes things that are measured by the same unit are 
 
 of opposite qualities. 
 
 E.g., assets and liabilities are both measured by the unit 
 dollar. 
 
 So, dates a.d. and dates B.C. are both given in years. 
 
 So, the readings of a thermometer above and below zero are 
 given in degrees. 
 In all such cases if the measuring unit be taken in the same 
 
 sense as the thing measured, the resulting concrete number 
 
 is positive; if taken in the opposite sense the number is nega- 
 tive. In which sense the unit shall be taken is a matter of 
 
 custom, or of convenience. 
 
 E.g., the unit dollar may be taken either as a dollar of assets or 
 as a dollar of liabilities : 
 
 if as a dollar of assets, then assets are jjositive numbers and 
 liabilities are negative numbers ; 
 
 if as a dollar of liabilities, then liabilities are positive and 
 assets are negative. 
 
 So, if distances measured towards the north be positive, dis- 
 tances to the south are negative, 
 
 i.e., if the measuring unit be a northerly unit, southerly dis- 
 tances are expressed by negative numbers. 
 Some things admit of negatives and some do not. 
 
 E.g., time may be counted backwards as well as forwards from 
 a given date. 
 
 So may distances from a given point. 
 
 So may heat and cold from an arbitrary zero. 
 
 So may money of account, as above., 
 
 But with real dollars, say five of them, the pupil will find 
 when he tries to couut past none — five, four, three, 
 two, one, none — that he is trying to do what is im- 
 possible. 
 
 So, a negative number of persons by itself is an absurdity. 
 
«»] POSITIVE AND NEGATIVE NUMBERS. 19 
 
 The primary notion of a negative concrete number is that 
 of one which, when taken with a positive number of the same 
 kind, goes to diminish it, cancel it altogether, or reverse it. 
 E.g., liabilities neutralize so much of assets, thereby diminish- 
 ing the net assets, or leaving a net liability. 
 
 QUESTION'S, 
 
 1. Show that the measuring unit of longitude may be taken 
 in either of two senses, and that whichever way the unit be 
 taken the two kinds of longitude may be called positive longi- 
 tude and negative longitude. 
 
 How are these two kinds of longitude now distinguished ? 
 
 2. If positive longitude be taken in the direction of the 
 sun's apparent motion and Greenwich be the starting point, 
 is St. Petersburg in positive or negative longitude ? 
 
 So, if the direction of the earth's rotation be taken as that of 
 positive longitude ? 
 
 3. If $1 be taken as the unit of money possessed, how must 
 money spent be represented ? money inherited ? money given 
 away? money owed ? money earned ? inoney staked on a wager ? 
 
 4. If distances toward the north be taken positive, how 
 must the latitude of Morocco be expressed ? of the equator ? 
 
 5. What is the greatest possible positive latitude ? the larg- 
 est negative latitude ? 
 
 6. What effect has a negative concrete number when com- 
 bined with a larger positive number of the same kind ? if the 
 negative number be just as large as the positive ? if larger ? 
 
 7. If a wreck be acted on by a current setting northward 
 and by a north wind, are the two forces of the same nature ? 
 
 What determines the direction in which the wreck moves ? 
 
 ~8. In rowing up a river that flows 4 miles an hour, what 
 
 progress is made by a man who can row 4 miles an hour in 
 
 still water ? What effect then does the man's rowing produce? 
 
 Show that there are two changes, either of which would 
 make his progress visible. 
 
20 THE PRIMARY OPERATIONS OP ARITHMETIC. P. 
 
 EXPEESSIOK OF POSITIVE AND NEGATIVE KUMBEES. 
 
 "When denoted by Arabic numerals, positive numbers are 
 written with the sign + or with no sign, and negative numbers 
 with the sign -, before them. But if a number be denoted by 
 a letter, it is not evident upon its face whether that letter 
 denotes a positive or a negative number. 
 E.g., if the measuring unit be a dollar of assets, then + 100, or 
 simply 100 without the sign, means $100 of assets, and 
 -100 means $100 of liabilities,. 
 But N might stand either for +100 or for -100 at pleasure. 
 E.g., if N stand for + 100, then-N stands for -100 ; 
 and if N stand for -100, then-N stands for + 100. 
 
 In this use of the signs + and -they are signs of quality. 
 
 These signs are also used to indicate the operations of addi- 
 tion and subtraction, and they are then signs of operation. 
 
 To avoid confusion in the two uses of the same signs, the 
 signs of quality may be made small and placed high up. 
 E.g., +100 means $100 of assets, and "100, $100 of liabilities. 
 
 But these small signs are used with the express understand- 
 ing that + is attached only to numbers that are essentially 
 positive, and ~ to those that are essentially negative. In that 
 respect they may have a meaning that differs from the mean- 
 ing of the large signs + , — . 
 
 There is a third sign, ±, made up of the two, and read 
 plus or minus; if written =f , it is read mifius or plus. 
 E.g., ±3 is only an abbreviated way of writing the separate 
 expressions +3 and —3; and * 7 is + 7 or " 7. 
 
 AN ABSTRACT NEGATIVE NUMBEK AS AN OPEKATOR. 
 
 As an operator an abstract negative number has two offices : 
 
 1, The repetition or partition of the multiplicand. 
 
 2. The reversal of its quality; 
 
 and every such number may be regarded as itself the product 
 of two factors : 
 
S3] POSITIVE AND NEGATIVE NUMBERS. 21 
 
 1. The absolute magnitude of the number, the tensor. 
 
 2. " 1, the versor. 
 
 E.g., -10= +10 X -1= -1 X +10. 
 
 So, $1 assets x "10 = $10 debts, and 11 debts x "10 = $10 assets. 
 
 So, 20 north-miles X "10 = 200 south-miles, 
 
 and 20 south-miles x "10 = 200 north-miles. 
 
 QUESTIONS. 
 
 1. If a= -3, what is the value of 3a? of -12fl? 
 
 2. Is it possible, in the course of an example, to have the 
 expression — 12 men ? " 12 men ? 
 
 Explain the difference between these two expressions. 
 
 3. What use is made of the signs + and — iu both arithme- 
 tic and algebra ? what use of them is peculiar to algebra ? 
 
 4. If distances eastward from the point where we stand and 
 time after the present moment be positive, and if a passing 
 train be running eastward at 20 miles an hour : show that 
 
 in 5 hours it will be 100 miles east of ns, +20 x +5 = +100; 
 5 hours ago it was 100 miles west of us, +20 x "5= "100. 
 
 But if the train be running westward, show that 
 in 5 hours it will be 100 miles west of us, "20 x +'5 = ~100; 
 5 hours ago it was 100 miles east of us, "20 x "5= +100. 
 
 5. Show that multiplication by a pogitive integer is a case of 
 addition. What relation exists among the numbers added ? 
 
 Then, with a positive multiplier, what relation has the sign 
 of the product to that of the multiplicand ? 
 
 6. If multiplication by a negative integer be regarded as a 
 case of subtraction, what are the successive subtrahends ? 
 
 How do their signs change in the process ? what relation has 
 the sign of the product to that of the multiplicand? 
 
 7. What two pairs of signs in the factors make the product 
 positive ? what two make it negative ? 
 
 8. Show that the product of any even number of negative 
 factors is positive, and that a product can be negative only 
 when it contains an odd number of negative factors. 
 
22 THE PRIMARY OPERATIONS OP ARITHMETIC. H 
 
 §4. ADDITION AND SUBTRACTION. 
 
 If the concrete numbers added be integers and simple frac- 
 tions, then, at bottom, addition is but counting, either by 
 entire units or by equal parts of a unit : on (forward) if posi- 
 tive numbers be added, off (backward) if negative numbers 
 be added. The result is the sum, and the sign is +, read plus. 
 Kg., 50 cts. + 60 cts. + 90 cts. = |2. 
 
 The sum of two or more abstract numbers, operators, is an 
 operator that, acting on a unit by repetition, partition, or 
 reversion, gives the same concrete number for result as if 
 the several operators acted in turn upon the unit, and their 
 results, like concrete numbers, were then added. 
 E.g., 7 + 5 = 12; 
 for if a man earn $2 a day, then the sum of his earnings for 5 
 
 days and for 7 days is his earnings for 12 days; 
 i.e., $2x5 + $2x7 = 12x12. 
 So, 1/5 = 2/15 + 1/15: 
 
 for if a field be divided into fifteen equal house-lots, a pur- 
 chaser that takes two lots and one lot has a fifth part 
 of the whole field ; and that whether the three lots lie 
 together or apart. 
 In algebra the word addition is used in a broader sense 
 than in arithmetic, and covers negative as well as positive 
 numbers. 
 
 E.g., he that has 110,000 assets and $4000 debts is worth but 
 
 $6000, 
 i.e., $10,000 assets + $4000 debts = $6000 net assets, 
 and + 10,000 + - 4000 = + 6000. 
 
 Though the numbers to be added must always be of the 
 same kind, they are often expressed by letters whose values 
 are not known, or in units whose values are difEerent, or which 
 cannot even be reduced to one sum : such a group is a poly, 
 nomial, and the numbers to be added are its terms. 
 E.g., 5* 33"* 35' + 12* 47'" 25»= 18" 21'»; in interest a=p + i. 
 
S41 ADDITION AND SUBTRACTION. 23 
 
 QUESTIONS. 
 
 1. By dividing up a Jine, find the sum of f and ^ of it. 
 Hence find the sum of the two abstract numbers f and J. 
 Ill what common unit are the two fractions expressed ? 
 
 2. Before two numbers can be added how must they be ex- 
 pressed ? 
 
 3. What is the nature of an operator that acts on a unit by 
 repetition ? by partition ? by reversion ? Name some operator 
 that acts in two of these ways; in all three of them. 
 
 4. If a unit be acted on by two or more operators, why 
 must the results of the several operations be concrete num- 
 bers of the same kind ? 
 
 5. "What two arithmetic operations may be indicated by the 
 word addition in algebra ? If negative numbers be added are 
 the minus signs signs of quality or of operation ? 
 
 6. If 240 men vote for a candidate and 160 vote against 
 him, what is the sum of the votes he receives, or his majority? 
 
 Solve, first using signs of quality; then, stating the question 
 differently, solve, using signs of operation alone, 
 
 7. When can the addition of numbers be indicated but not 
 performed ? 
 
 What is the sum of a, i, c, when their values are not known ? 
 when their values are 2, 3, 5 ? 
 
 8. In the expression 4f, What sign of operation is under- 
 stood between the integer 4 and the fraction f ? 
 
 So, between the dollars and cents of $18.50. 
 
 So, in the compound number 12* 15* 35™ 20"? 
 
 Show how the operations so indicated may be performed. 
 
 9. Can the sum of two numbers be smaller than one of 
 them ? smaller than each of them ? 
 
 10. Draw a straight line and mark A, b, two points taken at 
 random upon it; then show that the sum ab + ba is naught, 
 whichever direction be taken as positive. 
 
 So, take three points A, b, c, in any order upon a straight 
 line, and show that ab + bc = ac and that ab + BC + CA = 0. 
 
24 THE PRIMARY OPERATIONS OP ARITHMETIC. [I.Ths. 
 ADDITION COMMUTATIVE AND ASSOCIATIVE. 
 
 Theor. 5. The sum of two or more abstract numbers is the 
 same number, m whatever order the numbers be added and 
 however they be grouped. 
 
 For, let *a, 'b, c, • • • h/h, *^ be any abstract numbers, posi- 
 tive or negative, 
 let these numbers act as operators upon any unit, 
 and let the results be grouped and added in any way; 
 then •.■ the whole collection of units and parts of units is the 
 same, whichever unit, group of units, part, or group 
 of parts, be counted first, whichever second, and so on, 
 .% the several sums of these operators do the same work 
 and are all equal. [df. sum abs. nos., p. 32. 
 
 The principles here established are the commutative and 
 the associative principles of addition ; and theor. 5 may be 
 written, addition is a commutative and associative operation. 
 If the pupil will cut card-board into any of the common 
 geometric figures, say triangles and squares, and join them all 
 together by placing them edge to edge in various ways, he will 
 find that the figures resultii^g from this geometric addition 
 are quite difEerent in form, and that, in the sense of geometric 
 equality, they are not equal at all ; but he will find the areas, 
 the numerical results of measurement, to be all equal : i.e., in 
 the geometric sense the figures are unequal; in the algebraic 
 sense, and for the purposes of algebra, they are equah 
 
 Theok. 6. 'The sum of two or more simple fractions is a 
 simple fraction, or an integer. 
 For, let a/b, c/d, h/h • • • be simple fractions; 
 then *.• a/b = ad/bd= 1/bd x ad, c/d= bc/bd= 1/bd x be, 
 
 [th. 3, cr. 4, df. sim. frac. 
 
 .'. a/b + c/d = \/ld X ad+l/bd xbc = 1/bd x (ad + be) 
 
 = {ad+bc)/bd. [df. sim. frac, df. ad. 
 
 the sum of this sum and h/k is a simple fraction ; and so on. 
 
 Kg., 2/3 + 3/4 = 17/13; 1/4 + 6/8 = 1. 
 
8.6,841 ADDITION AND SUBTRACTION. 25 
 
 QUESTIONS. 
 
 1. What is the associative principle of addition ? 
 What is the commutative principle ? 
 
 2. Do the letters a,l, c, - • • stand for the same numbers in 
 theor, 5 as in theor. 1 ? Can you tell which of these numbers 
 are positive and which negative ? 
 
 3. In the proof of theor. 5, is the value of the entire collec- 
 tion of units found by adding the results given by the several 
 operators, or by the more detailed process of counting ? 
 
 Which of the operators are to be applied by counting off? 
 Which, by partition and a later counting ? 
 
 4. What other operation is commutative and associative? 
 
 5. If s stand for a square, t tor a triangle, c for a half-circle, 
 and r for a rhombus, is s + r + c-\-t the same as c + s + r + t^ 
 
 If the letters stand for the areas of the figures are the two 
 sums equal ? 
 
 6. Cut a triangle from paper with sides of different lengths, 
 join the mid^points of the sides, and, cutting along these 
 lines^ divide the triangle into four triangles that may be shown 
 to be all equal by placing one upon another; combine these 
 triangles in all possible ways; are the geometric figures so 
 found equal ? are their areas equal ? 
 
 7. If a merchant have various bills to pay from a sum of 
 money lying before him, show that he has the same amount 
 left after the bills are all paid, whatever be the order of 
 their payment. Of what principle is this an example ? 
 
 8. If when two fractions are added the denominator of the 
 sum be a factor of the numerator, how can the fraction be 
 more simply written? 
 
 Is there any number that cannot be written in fraction-form ? 
 
 9. In the proof of theor. 6, if the sum of the operators ad, 
 ic act on a unit, what is the result ? 
 
 What is the result if ad, ic act separately on the unit and 
 the results be then added ? 
 
 What relations have these two results to each other ? 
 
26 THE PRIMARY OPERATIONS OP ARITHMETIC. [I,Ths. 
 
 MULTIPLICATION DISTRIBUTIVE AS TO ADDITION. 
 
 Theoe. 7. The product of the sum of two or more abstract 
 numbers by another number is the sum of the products of the 
 first numbers by the other. 
 Let a, ~b, c,- • • h/k, I, m be any abstract numbers; 
 
 then {a + 'b + c-i \-h/k+l)xm 
 
 = axm+-bxm + exm-{ h/hxm+lxm. 
 
 For the product (fl+"5+cH \-h/h + T)xm 
 
 = mx{a + -b + c+--' h/k + l) [th. 2, th. 6. 
 
 — mxa + mx~b + 7nxc+'-- mxh/k + mxl 
 = axm+-bxm + cxm-\ — • h/kxm + lxm. [th. 3. 
 Cor. Tlie product of two or more polynomials is the sum of 
 the several products of each term of the first factor by each term 
 of the second factor by each term of the third factor, and so on. 
 For the product of two factors is the sum of the partial prod- 
 ucts of each term of one factor by each term of the other, 
 and the product of this product by a third fa(3tor is the sum 
 of the partial products of each term of this product by 
 each term of the third factor, and so on. q.b.d. , 
 The principle here established is the distributive principle 
 of multiplication ; and theor. 7 is sometimes written, mMZii- 
 plication is distributive as to addition, . 
 
 But addition is not distributive as to multiplication. 
 E.g.,(3 + 3)x5 = (3x5) + (2x5); (3x2) + 5=?!=(3 + 5) x(2+5). 
 
 OPPOSITES. 
 
 Two numbers whose sum is are opposites of each other. 
 E.g., +4,-4; -^/3, +3/3; "3 x -4,-13; 3+4,-7; 3-4,4-3. 
 
 Theoe. 8. The sum of the opposites of two or more abstract 
 numbers is the opposite of their sum. 
 
 Let +a, +5, • • •, -m, -n, ■ ■ •, ^-p/q, -r/s,- • •, be any abstract 
 
 numbers, 
 then is the sum -«+-JH — • + +wi++w'+ • • ■ —p/q + r/s- • ■ 
 
 the opposite of the sum 
 
 +a++&+--. + -OT+-raH -Vp/q-r/s-" 
 
V,6,%4] ADDITION AND SUBTRACTION. 27 
 
 For the sum of these two sums 
 
 = {+a+-a) + {*b+,-h)+ + {-»K + +m) + (-« + +«) 
 
 + — I- Ip/q -p/s) + (1 - V* + »■/*) _ 
 
 = + 0+ • • ■ =0. Q.E.D. [df. opp. th. 5. 
 
 QUESTIONS. 
 
 1. In the proof of theor. 7 how many ahstract numbers are 
 used ? are these numbers integers or fractions ? are they pos- 
 itive or negative ? is m abstract or concrete ? 
 
 2. By what principle are the two products equal : 
 {a+-b+c-\ +h/k+l)xm, m x (a + -b+c-i +h/k+l)? 
 
 By what principle is the second product expanded into 
 mxa + mx~b + 7nxc+' •• +mxh/k+mxl? 
 
 By what, is this last product changed to the form sought ? 
 
 3. Find the product of the two polynomials a— &+c+ • • • 
 — h/h + land. m + l/n—p, and show that their product is the 
 sum of the partial products got by multiplying each term of 
 the multiplicand in turn by each term of the multiplier. 
 
 Does the order in which the terms are multiplied affect the 
 final product? 
 
 4. Find the product of a + b, c+d, e +f, and show that each 
 term of this product contains three letters : one from the first 
 factor, one from the second, and one from the third. 
 
 So, for the three factors a — b, c — d, e—f. 
 
 5. In the case of two opposite numbers, what is true of the 
 two measuring units? of the number of times the unit is 
 repeated in each of them? of the quality of the numbers as to 
 that of the unit ? of the tensors in the abstract operators ? 
 of the versors ? 
 
 6. What number is its own opposite? 
 
 7. How is the sum of the opposites of two or mor6 numbers 
 related to the sum of the numbers ? the product of the oppo- 
 sites of two numbers ? that of the opposites of three mimberp ? 
 
 8. Explain the dependence of theor. 8 on theor. 5. 
 
28 THE PRIMAKY OPERATIONS OF ARITHMETIC. [I.Ta 
 
 SUBTRACTION. 
 
 ' SuBTKACTiON is an operation that is the inverse of addition, 
 i.e., when the sum of two numbers, and one of them, are given, 
 subtraction consists in finding the other number. The sum 
 is now called the minuend, the given number is the subtra- 
 hend, and the number sought is the remainder. 
 
 The sign of subtraction is — , read minus and the order of 
 
 writing is minuend — subtrahend = re^aindei" e.g., 
 
 $50- $40= $10, for $40+ $10= $50, +50- +40= +10 
 
 $40- $50= -$10, for $50+ -$10= $40, +40- +50= -10 
 
 -$50- -$40= -$10, for-$40+-$10=-$50, "50- "40= -10 
 
 -$40- -$50= $10, for -$50+ $10= -$40, "40- -50= +10 
 
 $40- -$50= $90, for-$50+ $90= $40, +40- -50= +90 
 
 -$50- $40= -$90, for $40+ "$90= -$50, -50- +40= -90. 
 
 Theoe. 9. If to the sum of two numbers the opposite of one 
 of them be added, the result is the other number. \ 
 For let r, s stand for the two numbers and m for their sum, 
 then •.• m = r+s, [hyp- 
 
 .•. TO + (— s) = r+s + ( — s) = r+(s— s)=n q.e.d. 
 
 Theor. 9 may be written, the sum of the minuend and the 
 opposite of the subtrahend is the remainder. 
 
 Hence the terms of an expression enclosed in a parenthesis 
 and preceded by a minus sign can be added to like terms, pro- 
 vided the sign of every term so enclosed be first reversed. 
 
 CoK. If there be a series of additions and subtractions, the 
 final result is the same, in whatever order they be performed, 
 and however the elements be grouped. 
 
 For every subtraction is an addition of the opposite of the 
 subtrahend, and these additions may be performed in 
 any order and the terms be grouped in any way. 
 
 In applying this corollary the pupil must take care lest he 
 change the oflftce of any element and add it where he should 
 subtract or subtract it where he should add. 
 
 E.g., 12-3 + 2 + 4-8 = (12-3+4) + (2-8) 
 nt(4-3+12)-(8 + 3). 
 
9. H] ADDITION AND SUBTRACTION. 29 
 
 QUESTIONS. 
 
 1. May the three numbers involved in the process of sub- 
 traction be all abstract ? all concrete ? part abstract and part 
 concrete ? 
 
 a all be concrete what else must be true of them ? 
 
 2. Interpret the examples given under the definition of 
 subtraction, when the positive numbers stand for assets or 
 earnings and the negative numbers for debts and expenses, 
 and show what is meant by subtracting a negative. 
 
 3. In arithmetic, how does the remainder compare in size 
 with the minuend ? is this always true in algebra ? 
 
 4. If from a — b, c + d was to be subtracted, and c alone has 
 been subtracted, is d to be added to this remainder or sub- 
 tracted from it ? 
 
 Prove that {a — b) — {o — d + e)=a — b — c + d—e. 
 
 5. If a polynomial be enclosed in a parenthesis and preceded 
 by a minus sign, what changes must be made in removing the 
 parenthesis ? 
 
 So, if a parenthesis is to be inserted after a minus sign, what 
 changes must be made in the signs of the terms included in it ? 
 
 6. Simplify 3a-{a-ib + 2a). 
 
 7. So,x-2y-[-2x+(-y-2z)-4x]. 
 
 8. The rule for algebraic subtraction is : to each term of the 
 minuend add the opposite of the like term of the subtrahend; 
 what is the origin of this rule ? 
 
 9. Prom the remainder subtract the opposite of the subtra- 
 hend; what is found ? 
 
 What other operation will give the same result ? 
 
 10. What two processes have been proved commutative and 
 associative ? 
 
 What other two processes may always be so indicated as to 
 be examples of these two? with what caution ? 
 
 11. Show in what respects theors. 4 and 9 are alike and in 
 what they differ. 
 
30 THE PRIMAEY OPERATIONS OP ARITHMETIC. [I.Th. 
 
 §5. INVOLUTION AND EVOLUTION. 
 
 The continued product of a number by itself is a power of 
 ^that number. The number whose power is sought is the base, 
 and the operatoir that shows how many times the base is used 
 as a factor is the exponerjbt j it is written at the right and above 
 the base. Involution is the process of finding powers. 
 E. g. , 4 X 4 X 4 = 4' = 64, 1/4 X 1/4 x 1/4 = 1/64. [th. 2, cr. 2. 
 
 In both examples the base is 4; in the first the operation is 
 a continued repetition by 4, and in the other it is a continued 
 partition by 4, operation's that tend to neutralize each other; 
 and this relation may be expressed by writing them 4+' and 4"*, 
 wherein the positive exponent shows how many times 4 is used 
 in repetition and the negative exponent- shows how many 
 times 4 is used in partition. 
 
 As an exponent, "1 reverses the quality of the base, i.e., if 
 the base denote repetition, the exponent ~1 changes the opera- 
 tion to partition, so that ?i~^ = l/n. 
 
 The words positive and negative as applied to powers refer 
 to the exponents only; and integer potvers are powers whose 
 exponents are integers. 
 
 E.g., ("4)' is a positive integer power, although its value, ~64, 
 
 is negative. 
 So, 4"* is a negative integer power, although its value, I/^^j is 
 
 positive and a fraction, 
 
 PRODUCT OF INTEGER POWERS OF THE SAME BASE. 
 
 , Theor. 10. The product of two or more integer powers of a 
 
 base is that pvwer of the base whose exponent is the stim of the 
 
 exponents of the factors. 
 
 Let A be any number, I, m,n,-- -any positive integers ; 
 
 then the product a' x a" x A"" • • • is a'+'"-" • • • 
 
 For •.• a'= A X A X • • -7 times, A'"= A x xx •• •m times, 
 
 A"" = 1/a x 1/a X • ■ -n times [df. int. pwr. 
 
 .'. a' X A"" X A-"=: (a X A X • • • l+m-n times) x (a x 1/a) 
 X (a X 1/a) X • • • w times, when l+m^n, 
 
10, §5] INVOLUTION AND EVOLltTION. 31 
 
 and •/ ax1/a=1, 
 
 .'. a'x a'"xa-"=a'+'"-"; 
 
 and a' X A" X a-"= (1/a x 1/a x ■• -n-l+m times) 
 
 (ax1/a) X • • • l + m times when l+m< m = 1/a"-'-'" 
 — ^i+m-n. Q.E.D. [df. neg. pwr. 
 
 and so for more than three factors. 
 
 Cob. The quotient of two integer powers of the same base (s 
 that power of the base whose exponent is the exponent of the 
 dividend less the exponent of the divisor, 
 
 QUBSTIOKS. 
 
 1. By diagram show the square and the cube of 3. 
 
 Can the higher powers of 3 be represented by diagrams ? 
 
 2. Can a concrete number be raised to a power? 
 
 3. Of what numbers are the high powers larger than the 
 low powers ? smaller ? the same ? 
 
 4. What is the value of ( — 3)* and what kind of power is it? 
 So, of ( + 2) -', of ( - \f, of 5', of 5 -» ? 
 
 5. What are the products 5' x 5 -»? ( - 5)' x ( - 5) -', a» x «-'? 
 
 6. What power of I is the product b'^xb^x (l/b)"' ? How 
 many of the f actors, S are cancelled ? how many remain ? 
 
 So, of the products 6™ x S» x (I/S)"; S™ x S« x (l/S)™ x (1/6)" ? 
 
 7. What relation has (l/S)" to J"" ? 
 
 8. What relation has the quotient 5" : b" to the products 
 6'»x6-»and&'»x(l/i)''? 
 
 9. How many times is S a factor of the product i' x S*" ? 
 How many of these factors can be cancelled by the factors 
 
 of S"" ? how many cannot be so cancelled ? 
 
 10. Granted that 5'x J"'x5-"=5'+"'-", what is true of the 
 exponent of the product of integer powers of a number ? 
 
 11. Divide a' by a', a;* by a;-", b-^hjb\ m-'hjm''\ 
 
For 
 
 (A-)" 
 
 So, 
 
 (A-)-» 
 
 So, 
 
 (A-)" 
 
 So, 
 
 (A-")- 
 
 32 THE PEIMARY OPERATIONS OF ARITHMETIC. [I.Ths. 
 
 INTfiGEJi POWER OF AN INTEGER POWER. 
 
 Theor. 11. An integer power of an integer power of a base 
 is that power of the base whose exponent is the product of the 
 two given exponents. 
 
 Let A be any number; m, n any positive integers; 
 then (a"')'' = a"'". 
 
 = A" X A™ X • • • w times 
 
 _jj^m + m + ...»timeB_^mx»i Q.E.D. [th. 10. 
 
 = I/a™ X I/a*" X ■ ■ -n times 
 
 T / .m-hm -i- . . .n times — , 1 / A "*'* :^ A. " *'*"' 
 
 = A"'"xA-"*x ■ • -n times 
 
 . -m~m- . . . n timeB ^ .-mn 
 
 l-"')-" = 1/a~"'x1/a""x • • -w times 
 = A"* X A" X • ■~~-ii times = A"*". 
 
 product of like integer powers of different bases. 
 Thbor. 12. The product of like integer powers of two or 
 more bases vs the like power of the product of the bases. 
 Let A, B, l/o be any numbers and n any positive integer; 
 then A" X Bf X 1/c" = (a x b/c)". 
 
 For va';=axax • •- w times, b'' = bxbx • • • ?i times, 
 1/c" = 1/c X 1/c X • • ■ n times, 
 
 .•. A" X B" X 1/C"= (a X B X 1/c) X (a X B X l/c) X • • ■ 
 
 n times = (a x b x 1/c)". q.e.d. [ths. 1, 3. 
 
 So, V a-" = 1/ax1/ax • • -re times, b-"=1/bx1/bx • ■ ■ 
 n times, l/C"" = c x ex • ■ • n times, 
 .-. A-" X B-" X l/c-"= (1/a X 1/b X c) X (1/a X 1/b X c) • ■ • 
 n times = (c/a/b)''= (a x b/c)"". q.e.d. 
 
 Cor. The quotient of two like integer powers of different 
 bases is the same power' of the quotient of the bases. 
 
 evolution. 
 Evolution is an operation that is the inverse of involution ; 
 i. e., it consists in finding a base that, raised to a given power, 
 is a given number. It is a process of trial and test. The base 
 
11,12,13,151 INVOLUTION AND EVOLUTION. 33 
 
 is now called the root, and the exponent of the power is the 
 root-index. The radical sign,, y', is placed before the number 
 whose root is sought, and the root-index at the left and above 
 this sign; the root-index 3 need not be written. 
 B!g., v'4, orsimply |/4, = +2, or-3; ^'343 = 7; ^^-343= "7. 
 
 Evolution being the inverse of involution, the following 
 converse theorem follows directly. 
 
 Thbob. 13. An integer root of an integer power of a tase is 
 that power of the base whose exponent is the integer quotient 
 of the exponent of the power by that of the root. 
 E.g., V^"'"=A". 
 
 QUESTI02«rS. 
 
 1. What cases are considered in theor. 11 besides a positive 
 integer power of a positive integer power ? 
 
 2. Write (a-™)" so that positive exponents only shall be 
 used; then, so as to express the result as a power of A. 
 
 3. In the product A" x B" x l/o", how many times is A a 
 factor ? B ? 1/c ? how many factors in the entire product ? 
 
 Into how many groups of the fSrm A x b x l/o can these 
 factors be divided ? Indicate this grouping in the simplest way. 
 
 4. Why does l/c-"= c x c x c ■ • • w times ? 
 
 5. Write A""xb""x (1/c)-" with positive exponents. 
 How many times is the group 1/A x 1/b x c a factor of 
 
 this product ? what power is it of the fraction A x b/c ? 
 
 6. Prove that a'"/b'" = {a/b)'". 
 
 7. What powers of numbers and of their opposites are the 
 same ? not the same ? 
 
 8.. To what element in involution does a root correspond ? 
 the root-index ? the number whose root is sought ? 
 
 9. What number has the same effect whether used as a 
 root-index or as an exponent? 
 
 10. |/4=+3, 4/4= -2: is +2, or -3, the square root of -4? 
 
 11. Find the values of (z^, {y-% (5*)"^ («"')"*• 
 13, Find the values of ^.c\ ^/tj-, ^b-^% ^a". 
 
34 THE PRIMARY OPERATIONS OP ARITHMETIC. CI. 
 
 § 6. QUESTIONS FOE EEVIBW. 
 
 Define and illustrate : 
 
 1. Quantity; unit; unity; number; integer; fraction. 
 
 2. Concrete numbers; equal concrete numbers. 
 
 3. Abstract numbers; equal abstract numbers. 
 
 4. Product; multiplication; division; quotient; the prod- 
 uct of two abstract numbers. 
 
 5. Positive numbers; negative numbers. 
 
 6. Sum; addition; subtraction; remainder; the sum of 
 two abstract numbers. 
 
 7. Eeciprocals; opposites; a number larger than another. 
 
 8. Power; involution; evolution; root. 
 State and prove : 
 
 9. The associative principle of multiplication. 
 
 10. The commutative principle of multiplication, four cases. 
 
 11. The associative and commutative principles of addition. 
 13. The distributive principle of multiplication. 
 
 13. The principle on which the theory of division rests. 
 
 14. The principle on which the theory of subtraction rests. 
 
 15. The principle by which the product of two integer 
 powers of the same base is found ? 
 
 How does this principle apply to their quotient ? 
 
 16. The principle by which an integer power of an integer 
 power is found ? 
 
 How does this principle apply to the extraction of roots? 
 
 17. The principle by which the product of like integer 
 power of difEerent bases is found ? 
 
 How does this principle apply to their quotient ? 
 
 18. What two offices has an abstract negative multiplier ? 
 
 19. How can a series of additions and subtractions be made ? 
 a series of multiplications and divisions ? with what caution ? 
 
 20. What is the product of the reciprocals of two or more 
 numbers ? the sum of their opposites ? 
 
S6] QUESTIONS FOR REVIEW. 35 
 
 21. What operation is the inverse of multiplication? of 
 addition ? of involution ? What inverses has the multiplica- 
 tion $5x4? the involution 4' ? 
 
 22. Can two concrete numbers be added together ? with what 
 caution ? Can one such number be subtracted from another ? 
 
 A man may walk a milei then take another step : can a mile 
 and a yard be added together ? 
 
 23. Can two concrete numbers be multiplied together ? Can 
 one such number be divided by another ? with what caution ? 
 
 24. Can a concrete number be raised to a power ? Can a 
 root be taken of such a number ? 
 
 25. What is the meaning of a concrete fraction ? of an ab- 
 stract fraction ? Are both terms of a concrete fraction concrete ? 
 
 26. What effect has it upon a fraction if both terms be mul- 
 tiplied by the same number ? if both be divided by the same 
 number ? if the same number be added to both terms ? if both 
 terms be raised to the same power ? if like roots be taken ? 
 
 27. Group the factors below in such a way as to make the 
 multiplication easiest; hence explain cancellation in the mul- 
 tiplication of fractions : 3/2 • 7/11 • 25/51 • 2/7 • 11/5 • 17/5. 
 
 28. Is involution distributive as to multiplication? is evolu- 
 tion ? Are these two statements true equations ? 
 
 (12 X 6)"= 12' X 6^ ^(27 x 8) = |/27 x ^8. 
 
 29. Is involution distributive as to addition ? is evolution ? 
 Are these two statements true equations ? 
 
 (12 + 6y=12^ + 6\ ^(27±8)^^27±v'8. 
 
 30. Suggest questions that could not possibly be answered 
 by a negative number; not, by a fraction. 
 
 31. Show that multiplication may be defined as the process 
 of doing to one of two numbers that which, when done to a 
 unit, produces the other. 
 
 32. Why is it that, instead of dividing by a composite num- 
 ber, one may divide by one of its factors, that quotient by 
 another factor, and so on, till all the factors are used ? 
 
 33. Why does -/(108 x 27 x 250 x 490) equal 6 x 9 x 5 x 70 ? 
 
36 THE PRIMAEY OPERATIONS OF ALGEBEA. [H. 
 
 II. THE PRIMAEY OPEEATIONS OP ALGEBRA. 
 
 In principle, the operations of algebra differ not at all from 
 the like operations of arithmetic ; the only difEerences arise 
 from the difEerences between the forms of algebraic expressions 
 and the' simpler forms common in arithmetic. 
 E.g., the sum of 4 and 5 is 9, and their product is 20; 
 but the sum and product of a and 6 can only be expressed by 
 
 writing a + b and axh, a-b, or ab, 
 which mean that an addition and a multiplication are in- 
 tended, and that they will be effected when the values 
 of a and b are made known. 
 
 §1. ALGEBRAIC EXPRESSIONS. 
 
 An algebraic expression is a number or combination of 
 numbers written in algebraic form. It is called an expression 
 or a number, according as the thought is of the symbol or of 
 the value that the symbol represents. 
 
 Unless it be a single letter or numeral, an expression is 
 made up of simpler expressions afEected or combined by signs 
 of operation. 
 
 ' The parts of an expression that are joined by the signs 
 + or — are terms. 
 
 An expression of one term only is a monomial, of two terms 
 a binomial, of three terms a trinomial, of four terms a guadri- 
 nomial, of two or more terms a polynomial. 
 
 An expression is numerical if the numbers be expressed 
 wholly by numerals, literal if wholly or in part by letters; 
 finite if the number of operations implied be limited, infinite 
 if unlimited. 
 
 An algebraic expression is entire if it be free from divisors 
 and roots, fractional if not free from divisors. 
 
81] ALGEBRAIC EXPRESSIONS. 37 
 
 When the terms of an expression are so related to each 
 other that each successiTC term is derivable by some fixed law 
 from the previous terms, the expression is a series. 
 
 E.g., l+x+oi!^+a?-\ |-a!''is & finite series, if r be any given 
 
 integer, arranged according to ^rising powers of x; 
 but \+x+a?-\-a?+ • • • +a;'"+ • • -is an infinite series. 
 
 QUESTIONS. 
 
 1. Is a/h an algebraic expression or a number ? 
 If a be the entire cost of I books, what is a/h ? 
 
 3. What name is given to the parts a, I, of the expression 
 a/S ? of the expressions a— I, a + i, a-l, a*, $/a? 
 
 3. What three pairs of contrasting names are applied to 
 algebraic expressions ? define and illustrate them. 
 
 4. In the series \+x-\-a?-\- ••• +x^, how is the third 
 term got from the second ? the fourth from the third ? 
 
 If r be 7, how many terms are in this series ? 
 
 How many operations are performed ? what are they ? 
 
 Is the series finite or infinite? 
 
 5- In the series l+a! + a^+ •••+«"■+••• , how many 
 terms are there after a? ? What is the twelfth term of this 
 series? the twentieth? the wth? 
 
 6. If a==4, 5=1, c=3, d—9, x=5, y=8, find the values of 
 
 V8ac, •7'i/5dx, a^-W-^alc, >/y% >^¥, {M-bcf. 
 
 7. Iffl=l, 5=— 3, c=5, find the value of 
 
 aV + 1 1-aV 2W-iac a^+^ab + W 
 a^+l^ a^-(?^ V-& W-^bc + c"' 
 
 8. If a=25, 6=9, c=-4, d=-l, find the value of 
 , sj —Ic-'r^slaxA—'Lsl — WdA- i\l —&d. 
 
 9. If a^O, l=Al, c=4, <?=-6, find the value of 
 
 3 |'(35^-«) + 2 ^(5«+c'+7)- -^[3(5+c)'+(<Z+S)''+54 
 10. If «= 1/3, how much will the sum of the first three terms 
 
 of the series l + x+a? + a?-\ lack of 3? 
 
 So, the first four terms? five terms? ten terms ? n terms? 
 
38 THE PRIMAEY OPERATIONS OP ALQEBEA. CII, 
 
 SYMMETRY. 
 
 As to any of its letters, an expression is symmetric when its 
 value is unchanged however those letters exchange places. 
 E.g., the expressions xxyy.z and x+y + z are symmetric as 
 
 to X, y, z, or as to any two of them. 
 ^o,w + x-y-z\s symmetric as to w, x, and as to y, z; but not 
 
 as to w, y, as to w, z, as to x, y, or as to x, z. 
 
 DEGEEE — HOMOGElfEOXJS TERMS. 
 
 The sum of the exponents in a monomial is its degree. 
 
 The degree of a polynomial is that of the term whose degree 
 
 is highest of all. 
 A polynomial made up of terms all of the same degree is 
 
 homogeneous. Expressions having the same degree are homo- 
 geneous with each other. 
 
 E.g., a' + 3a°5 + 3aF + V is homogeneous and of the third 
 degree. 
 
 So, a", dP-^l, dP-^'V, • • • a"- '■J'", • • • a5"-', J" are homogeneous 
 with each other and of the mth degree. 
 
 And aa;", V^y, c'y' are of the second degree, and homo- 
 geneous with each other as to x and y,- 
 
 but of the third, fourth, fifth degrees, and not homogeneous, 
 as to all the letters. 
 
 COEFFICIBNTS — LIKE AND UNLIKE TEEMS. 
 
 If an expression have a numerical factor, then usually the 
 numeral alone, together with the sign of the number, + or 
 — , or the numeral and one or more of the letters that follow 
 it, is called the coefficient. 
 E.g., in labc, 7 is the coefficient of dbc, 11a of be, lab of c. 
 
 Terms that differ only in their coefficients are lihe (similar) 
 terms ; other terms are unlihe. 
 E.g., bax, lax are like; but ftax, Iby are unlike. 
 So, bax, 7bx are like if 5a, lb be counted as the coefficients 
 of X ; but unlike if 5, 7 be coefficients of ax, bx. 
 
Sn- ALGEBRAIC EXPRESSIONS. 39 
 
 QUESTIONS. 
 
 1. If x=2, y=3, z=4, w=6, is the expression xxyxz 
 the same as 2x3x4? as 3x2x4? 4x2x3? 2x4x3? 
 3x4x2? 4x3x2? 
 
 So, is w + x-y-z thesameas 6 + 2-3-4? 4 + 6-3-2? 
 2 + 6-4-3? 6+3-2-4? 
 
 2. In the expression axixcxd, what letters may be in- 
 terchanged without changing the value ? 
 
 So, in the expression a + b — c + d? in ai/cd? inai + cd? 
 
 3. What is the degree of a? of a* ? of ah ? 
 
 4. Of the trinomial a^—abc + c, which term is of the 
 highest degree ? 
 
 What is the degree of this expression ? is it homogeneous ? 
 
 Supply the terms that are wanting so as to complete the 
 homogeneous series : 
 
 5. a'+a=5 + a*5'+---+a5=+5'. 
 
 6. a' + a'b + a^lf+'--+ai'' + b''. 
 
 7. cl'-a^b + c^W flS'+S". 
 
 8. a-d?b+aW +aV-U. 
 
 9. a»+a"-»S+a»-''J«+ • • • +a6''-'+J".- 
 
 10. a»-a»-»5 + a»-W a&»-*+6". 
 
 11. a"-ffl"-'6+a'' ^&« +a6"-»-J". 
 
 12. If w he even will the series a"— a""*5+'" end with 
 -S" or with + J" ? if w be odd ? 
 
 13. Supply the terms that are wanting and complete the 
 symmetric series d?+a?b+d'c+ • • • + abc + • • • + c*. 
 
 14. So, a{oi? + ^' + z') + b(pi?y +a^z-] ) + cxyz. 
 
 15. If an expression bevsymmetric as to every pair of letters 
 involved, it is symmetric as to all the letters, and conversely. 
 
 16. In the product ^a^bc^ what is the coefficient of Sc' ? of c"? 
 
 17. Are Sar, ba'x, —^bx, i¥x, like terms? 
 Can they be so taken as to be made like terms ? 
 
 18. As an operator, what process does a coefficient indicate? 
 
40 THE PRIMARY OPERATIONS OP ALGEBRA. [U.Pbs. 
 
 §3. ADDITION" AND SUBTRACTION. 
 
 PbOB. 1. To ABD TWO OR MORE NUMBERS. 
 
 (a) The numbers like : to the common factor, prefix the sum of 
 
 the coefficients. [I th. 7. 
 
 E.g., 10 ft. down + 20 ,f t. up + 60 ft. up = 70 ft. u*p, 
 
 10 ft. up +20 ft down + 60 ft. down = 70 ft. down. 
 So, 10a; - 15a; + 20a; -25a; + 302;= 60a;- 40a; = 30a;, 
 
 Way + 20by - 30cy = (10a + 205 - 30c)«/. 
 {b) The numbers unlike: write the numbers in succession, 
 
 with their signs, in any convenient order. [I th. 5. 
 E.g., 19a;yz-29>nw + 39a-49 is irreducible. 
 So, lOay + 20by — dOey is usually not reduced, but may be 
 
 written (10a + 205 - 30c)«/, as above. 
 (c) Some numbers like and some unlike : unite into one sum 
 
 each set of like numbers ; 
 write these partial stims, together with the remaining terms, 
 
 in any convenient order. 
 E. g., (a' + 3a^ + 3a5' + ¥) + (a' - Sa^c + 3ac^ - c') 
 = 2d' + 3a?(5 - c) + 3a(5^ + c") + (P - c^). 
 
 SUBTRACTION. 
 PrOB. 2. To SUBTRACT ONE NUMBER FROM ANOTHER. 
 
 By trial, or memory, find what number added to the sub- 
 trahend gives the minuend. [df. sub., p. 28. 
 Or, to the minuend add the opposite of the subtrahend. 
 
 [I th. 9. 
 In general, the first rule is best when the numbers are like, 
 and the second, when they are unlike. 
 E.g., 7a - 4a = 3a, 7a- "Sa = 10a. 
 So, ~7a — 3a="10a, ~7a— ~3a=~4a. 
 So, 7a--45 = 7a+45, 7a-+4d = 7a-45. 
 So, [2a' + 3a=(S -c) + 3a{P + c") + (5' - c')] - [a» - 3a'c + 3ac» - c»] 
 = a'+3a'5 + 3a5'' + 5'. 
 
l.S,S21 ADDITION AND SUBTRACTION. 41 
 
 QUESTIONS. 
 
 Add 
 
 1. 7x, 15a;, Sx, Sx, 2x, 12a;, -15a;, 27a:, -31a;. 
 
 2. da^y, ISffl^y, '-25ay, -4:Za% 3da% 25a% 
 
 3. laal', -4:a% 5a% 9a% -15ax\ -26a'x. 
 
 4. '7a+5c-Bx't/, ^a^t/Sb-ia, U+ix^y+2c. 
 
 5. 8mw' + Zx'y^ + 5a, 7a;y + 3m?i'-7fl, 2mw^-17a;^^'+3a. 
 
 6. {a-2p)x', {q-b)a?, (3c-2r)a;, (3i3-ffl)a;', -{b + q)x', 
 — (p — a)x, [arrange the sum to rising powers of x, 
 
 7. a'-a^b+a^b" aP, a^b-a*P+(^P b\ 
 
 8. a'-a'^b+aV +ab% afb-aW+a^iP +¥: 
 
 9. a'-ar-''l^-aP-''W ±aS»-S ffi"--»5-ffl»-^Z>«+ • • • ±5* 
 
 10. From 7a; subtract 3a;, 5a;, 7a;, 9a;, 11a;, in turn, and add 
 the five remainders. 
 
 11. So, from 13ffl«/* subtract 5fl^', Say", ay^, —««/", ~Zay^. 
 Find the value of 
 
 12. (fl + J + c-2 + 5a;)-(a-5-9 + 13a;) + (a + 5-c + 8a;). 
 
 13. (7 + 56)-[(5aa; + 3S-2)-(4m + 3fla;-4&)]. 
 
 14. (^xy + \^^fz - B3?y^) -{3xy- Sa^y^n- 7a?y^). 
 Free from brackets and reduce to the simplest form 
 
 (a) removing first the inner brackets, and going outwards; 
 (J) removing first the^uter brackets, and going inwards; 
 (c) freeing all terms of a kind, from all the brackets: 
 
 15. a-[b-{e-d)]. 16. a-{a+b-[a+b-c-{a-b+c)]}. 
 
 17. -{(l + 2a; + 9a;*) + [(3 + 2a;-a;«)-(-3 + 3a;-3a;«)]}. 
 Introduce brackets, taking the terms two together in their 
 
 present order, and having 
 
 (a) each bracket preceded by-a plus sign; 
 
 (b) each bracket preceded by a minus sign; 
 
 (c) the first term in each bracket positive: 
 
 18. -Bc+id-2e + 2f+2a-5b. 
 
 19. a+b+c-a—b+c+a-b-c-a+b-c—a—i-c. 
 
 30. aic — abd +abe— acd +ace+ade- bed + bee — bde + cde. 
 
4S THE PRIMARY OPERATIONS OE ALGEBRA. [ii.P«- 
 
 § 3. MULTIPLIOATIOlSr. 
 
 PkOB. 3. To MULTIPLY ONE NUMBER BY ANOTHER. 
 
 (a) A monomial by a monomial: to the product of the numer- 
 
 ical coefficients, annex the several letters, each taken as 
 many times as it appears in both factors together ; 
 
 [I ths. 1, 2, 10. 
 
 mark the product positive if the factors be both positive or 
 
 both negative, and negative if one factor be positive and 
 
 the other negative. [df. neg. oper., p. 30. 
 
 E.g., +9a5-'x +7a^=: +63a=Z>-^ -6xz-^ x ■>-7a?z-''= Sbx'z-^ 
 
 +9a-'5=x -7a*=-63a-'^i', "Sas^-'x -7a;-V=+35a;-*. 
 
 (b) A polynomial by a monomial : multiply each term of the 
 
 multiplicand by the multiplier ; 
 add the partial products. [I th. 7. 
 
 E.g., {ixy^-^x-h) X -|a;y-V= -^x'y-'^!^+%x-^y-^s^. 
 
 (c) A polynomial by a polynomial : multiply each term of the 
 
 multiplicand by each term of the multiplier; 
 add the partial products. - ' [I th. 7 cr. 
 
 E.g., (fl« -al + ¥) x{a + b) = a'- a'b + aV + a^b - al? + 5» 
 
 The work takes this form: cp-db +Z>* 
 
 a +b 
 
 a' — a^b + ab' 
 ■Va^b-aV+W 
 
 c? +¥ 
 
 So, to multiply ax^ + 2bxy + cy^ + 2dxz-\-2eyz +f^ 
 ■ by mx + ny +pz : write 
 
 aaf + Hbxy + cy' + 2dxz + 2eyz + fz' 
 
 mx +wy -^pz. \ 
 
 ama;»+36m|a;'y+cm ! xy' +2dm 
 +071 I +26» I -^-cny' 
 
 +ap 
 
 x^z-\-3em 
 +2dn 
 +Sbp 
 
 xyz 
 
 +fm. 
 
 xzf 
 
 +2CI1 
 
 y^z 
 
 +fn yz^ 
 
 +cp 
 
 +mp 
 
 +2ep +fpz' 
 
 Checks. The work is tested by division [prob. 4], and 
 often by the principles of note 1 below. 
 
Sjaj MULTIPLICATION. 43 
 
 QUESTIONS. 
 
 1. In repetition by a positiye multiplier, wliat relation has 
 the sign of the product to that of the multiplicand ? in par- 
 tition ? 
 
 2. Of what two factors is a negative multiplier composed ? 
 Does the tensor change the sign of the multiplicand ? does 
 
 the Tersor ? 
 
 What is the sign of the product of a positive number by a 
 negative number ? of a negative number by a negative number ? 
 Multiply 
 3. {a + b)x{c-d). 4. (2ah-c'+Zx)x{3aI)-c + x). 
 
 5. (3a' - 2a«5 - aV' + 45') x {a' + 2ab - If). 
 
 6. {x'-2xy + y^) x (a^ + «/") X (a;* + "Hxy + y^) x («« - y% 
 
 7. b{a^+a?y+a?y^+xy^+y*')x{x—y). 
 
 8. {4:(?d - IccP + 3<P) X (5c» + 4:(?d - 9cd^). 
 
 9. (iB** + K'y + y^) X (a;" - ^") x (a;** - x'^if + ^*') x (a;" + y"). 
 
 10. (S'-Jc+c') X {W+lc+(^) X (5+c) X (S-c). 
 
 11. {^xy - Zxz + Zyz) x (aa: — 25^ + 3cz). 
 
 13. (3fl-a;+2)x(a-4a;-l). 13. {a?-aa?-\-'7y^)x{x-ay). 
 
 14. (Sa^a; - aa; -Fa;) x (2a - «^) x (3a -1- z), 
 
 15. (4a-'-2a;-«+a;-*-6)x(3a:*-a'). 
 
 16. (3a* + 5aS - 4ac + 85' - c") x (a - 35c + 3c«). 
 
 17. [5a;»i^-(5-3c)]x(5-l)x(a;+c). 
 
 18. (2a;y + a?- Zy^) x (3a;* - a;y -|- %f). 
 
 19. (a*-3a'-l-3a'-3aH-l) x(a*+2a'-H3a*-|-2a-l-l). 
 
 20. (5a;'+4a;«-3-7a;-»-l-10a;-») x (ar'4-a;-2). 
 
 21. (17a'5a;->-l-13a5-W-95c*) x (55'a;-f6a-»5-'). 
 
 22. (3a/+4aV-6-105-i«^-»-5«^-'') x (y^-iiy). 
 
 33. (ia;»-a;*-|-|a;-3)x(3a;'-f|a;*-l-a;-|-i), 
 
 34. (a;*-l-3a;-3)x(a;«-a; + l). 35. [a;*-(5-l-c)a;-l-5c] x (a;-a). 
 
 26. [a;' -h a° - aa;(ar' -I- a')] X [a;' + a' - ax{x + a)]. 
 
 27. (a;'-3a;*-t-3a;-l)x(a;2 + 3a;-|-l)x(a;-|-l). 
 
44 THE PRIMARY OPERATIONS OP ALGEBRA tn,PR. 
 
 FORM OF PRODUCT. 
 
 Note 1. Certain general principles are manifest: 
 
 1. The form of a product is independent of the values of the 
 
 letters that enter into it. 
 
 B.g., {a+h)x {a-V) = d'-V, whatever be the values of a, I. 
 %. If each factor be symmetric as to two or more letters, the 
 
 product is also symmetric as to the same letters. 
 
 E. g., (aH 2ab + 5") x (a + 5) = a' + 3a^5 + 3a5^ + b\ 
 
 3. If any values be given to the letters, the value of the 
 product equals the product of the values of the factors. 
 E.g., if a = 5, 5 = 3, then al+3a5 + 6^ = 25 +20 + 4 = 49, [above. 
 
 a + J = 5 + 2 = 7, a' + 3«H3fl!5H6'=49.7 = 343. 
 
 4. Hie sum of the coefficients of a product is the continued 
 product of the sum of tJie coefficients of the first factor, by the 
 sum of the coefficients of the second factor, and so on. 
 
 E.g., (ia+Sb) X (6a-2b) X (a-35) = 20a'-53a'5-27a5H185». 
 (4 + 3)x(5-2)x{l-3) = 30-53-27 + 18=-42. 
 
 5. The degree of the highest term of a product, as to any 
 letter or letters, is the sum of the degrees of the highest terms 
 of the factors, as to the same letter or letters ; and so of the' 
 lowest term. 
 
 E. g., in {a" + 2ab + i^) x {a + b) = a^ + Za^ + 3a5' + 5' the degree 
 of the highest terms of the factors as to a are 2, 1, and 
 of the highest term of the product, it is 3. 
 
 So, the degrees of the lowest terms as to a are 0, 0, 0. 
 
 6. If each factor be homogeneous as to any letter or letters, 
 the product is homogeneous as to the same letter or letters. 
 E.g., in the example just above, the factors and the product 
 
 are all homogeneous as to the two letters a, b. 
 
 7. The whole number of terms in any product, before reduc- 
 tion, is the corttinued product of the number of terms in the 
 several factors ; and the product of two or more polynomials 
 can never be reduced to less than tivo ter'm.s, that of highest 
 degree and that of lowest degree as to any letter or letters. 
 
3.83] MULTIPLICATION. 45 
 
 QUESTIONS. 
 
 1. For how many different values of a, I, is the statement 
 (a-lf^a^-%db + W true? 
 
 Is the value of a^ — %ab + W the same, whatever values 
 be given to a, J ? Why, then, is the statement always true ? 
 
 2. In the product {a^ + 2ab + W) x (a + &) if every I be re- 
 placed by an a, and every a by a S, is the multiplicand changed ? 
 the multiplier ? the product ? 
 
 As to what letters are these three expressions symmetric ? 
 
 3. As to what letters is the product 
 
 (a' — dbc +W)y.{a + i — c) symmetric ? 
 Eeplace a by 1, 5 by 3, c by 5, and test the product. 
 To make the test doubly sure, replace a, i, c by other 
 numerals and test it again. 
 
 Is the test good whatever values be given to a, b,c? 
 
 Is it a perfect test ? 
 
 What are the most convenient values ? 
 
 4. Find the product (a; — 3) x (2a; — 1) and test it by show- 
 ing that the sum of the coefl&cients of the product is the 
 product of the sums of the coefficients of the factors. 
 
 5. So, {x-2) X (3a;-l) x (a;-4) x (4a!-l) x {x-6) x {6a;-l). 
 How many terms has this product before reduction ? after 
 
 reduction ? 
 
 6. Show that the fourth principle is only another way of 
 stating the third when the value 1 is given to each letter. ' 
 
 7. In the product a^ + da'b + SaV + b^ by a + i, what term 
 contains a with exponent ? what is the value of a" ? 
 
 8. Of what degree, as to x, is the product {x^ + i/)x{a?+y)? 
 as to y ? as to x and y ? 
 
 So, the product {o^y + y^) x (a;y + ^) x {xy^ + y") ? 
 
 9. Of what degree is the product of 
 ar-{^ar-^h+a'"-W+ • • • +a5"-' + 6" by a"+a"-*i+ • • • +5". 
 
 Is this product homogeneous ? symmetric ? 
 
 10. Multiply 45'-3&^c-35c^ + 4c' by 2h + 2c, and show 
 the application of the seven laws just pointed out. 
 
46 
 
 THE PRIMARY OPERATIONS OF ALGEBRA. [H.Pb. 
 
 ARRANGEMENX. 
 
 Note 2. The work is often shortened by arranging the 
 terms of both factors, and of the product, as to the powers of 
 some one letter and grouping together like partial products. 
 E.g., (a' + Bab^ + da^b + ¥) x {¥ + %ab + a') is written 
 a^ + zm + ^aW + V 
 
 a''+2ab +W 
 
 
 
 a= + 3 
 
 + 3 
 
 ft*6 + 3 
 + 6 
 + 1 
 
 aW + 1 
 + 6 
 + 3 
 
 + 2 
 + 3 
 
 a5* 
 
 + V 
 
 a=+5 a!-b + \QaW+lQaW+b a¥+W. 
 
 CROSS MULTIPLICATIOK. 
 
 Note 3. The work is often shortened by picking out and 
 
 adding mentally all like partial products, and writing their 
 
 sum only. In ex. nt. 3 the computer says and writes 
 
 c^xa? is a* 
 
 5a«i 
 
 lQa?W 
 
 Sa'b X a' is 3a*5, a' x 2ab is 2a% whose sum is 
 3a¥ X a' is 3a'^>^ 3a^5 x 2ab is 6aW, a» x P is 
 
 a'5^, whose sum is 
 *' x a^ is aW, Saly' x 2ff5 is 6aW, da'i x S^ is 
 
 daW, whose sum is 
 P X 2a5 is 2a5*, 3a¥ x 5^ is 3aS*, whose sum is 
 5'x6^ is 
 
 and the product is a^ + 5a*b + 1 Oa^b^ + lOa^b^ + 5a¥ + b\ 
 So, to multiply 384 by 287, product 110308, ^ 384 
 
 i.e., 3-10' + 8-10 + 4 by 2-10H8-10 + 7, 287 
 
 the computer says and writes 
 
 28; ' 8 
 
 2; 56, 58; 32, 90; 
 
 9; 21, 30; 64, 94; 8, 102; 2 
 
 10; 24, 34; 16, 50; 
 
 5; 6,11; , 11 
 
 wherein the numbers 28, 56, 33, 21, 64, 8, 24, 16, 6, 
 are the pr'd'ts4.7,8.7,4.8,3-7,8. 8,4-3, 3-8,8-2,3.a. 
 
3,53] 
 
 MULTIPLICATION. 
 
 47 
 
 QUBSTIOKS. 
 
 Multiply, and check the work: 
 
 1. {a?-2x'-^x + l)x{2x'-3x + i). 
 
 2. (l + a^-3a;-2a^)x(4 + 2a;*-3a;). 
 
 3. {a^+y' + z'+xy + yz — zx)x{x—y + z). 
 
 4. {x' — zx+y''+yz + z^+xy)x(z + x—y). 
 
 5. {a?+ax-i'')x(x' + ix-a^)x{x-a + b). 
 
 6. {i^ — x' — ax) X {J)x — a^+3?) x (a+i — x). 
 
 7. l(a-l)x>+ia-iyx'+{a-iyx']x[{a+l)x+{a + iyx-']. 
 
 8. {3:?-ax + a'')x{a + x); {ax—a:^-a^)x{x + a). 
 
 9. (a — i + c) X {b — c + a) x (c — a + b) x {a + b + c). 
 
 10. (l + x+a^+a^ + a^ + st^) x (l-x+x^-x'+x^-a^). 
 
 11. {5x^ + ia^+dx' + 2a? + x)x(5a^-ix'+dx^-2x' + x). 
 
 12. (a" + 36''-3cP)x(a-™-35-" + 3c-*). 
 
 13. (a™-2cP + 3J») X (3J-" + a-'» + 3c-^). 
 
 14. x' + alx + abxx' + c 
 + 51 +tZ 
 
 + b 
 
 x + abxa?—c 
 -d 
 
 x + cd. 
 
 15. of — a 
 
 x + abxa^ + c 
 + d 
 
 x + cd; a? — a\x + abxa? — c 
 -b\ -d 
 
 x+cd. 
 
 16. {x+a)x{x+b)x{x + c)x{x+d). [at one operation. 
 
 17. {x+a)x{x + b)x{x + c); (x—a)x{x — b)x{x—c). 
 
 18. {x — a)x{x + b)x{x—c); {x — a)x{x—b)x{x+c). 
 
 19. 13x15; 35x79; 234x432; 33.4x4.33; 135.7x13.34. 
 
 30. W; 37^; 109^; 163'; 735'; 1881^; 70.23'; 0.205', 
 
 31. Given a + b + c=pi, bc + ca + ab—p^, abc^p^; 
 show that a^+W + (? =p^ — 2pz , 
 
 and that a^b + a¥+i^c + bc' + (?a + a<? =pxp% — SjJs . 
 
 22. So, express d?+}? + & in terms of P\,Pi,p3, using 
 the identity 
 
 a^+V'+&-Zabc=(€?+V+&-lc-ca-ab)[a+b + c). 
 
48 THE PRIMARY OPERATIONS OF ALGEBRA. [U, PR. 
 
 TYPB-FOKMS. 
 
 Note 4. The work is often shortened by the use of certain 
 simple type-forms, which the pupil may prove by actual mul- 
 tiplication and then memorize. He may also translate them 
 into words and read them as theorems. 
 E.g., Form [2] may be read: the product of the sum and the 
 difference of two numbers is the difference of their squares. 
 1] {x + a)x{x + b) = x'' + {a + b)z + ah. 
 E.g., {x + 6) X {x-2) = x' + {6-2)x-12 = x'' + 4x-l2. 
 So, (a-4)x(a-3)=ffl^-7a + 13. 
 3] (a + b)x{a-b) = a^-P. 
 E.g., (a + 5)x(a-5) = a=-35. 
 
 So, (a+F^)x(a-3^) = a^-i&^'=a2-SH65-9. 
 3] {a + bY = a'> + 2ab + b\ 
 
 E.g., {2y + 7y=2y^+2-2y-'r + T=4:f + 28y + i9. 
 So, ' {2x + Saf = ix' + 12ax + 9al 
 4] {a-bY=a''-2ab + b\ 
 
 E.g., (32- 8)^=92^-2 -Sz- 8 + 8^ = 92=- 4824-64. 
 So, (a + b-cy=a^-ir2abJrW-2ac-2bc + d'. 
 5] {a+l+c+'--f=d'+b''+(?+--- 
 
 + 2{ab+acA +bc+---). 
 
 E.g., {a + b-c-dY=a^+W+(?+d'' 
 
 + 2{ab—ac — ad—bc — bd+cd). 
 6] (a-b) X (a— >-ffl«-25 + a''-sd' + a»-*J'+ . . . +«§— a + S""') 
 
 = fl" — b", if n be any integer. 
 E.g., (a;-3)x(a;H32; + 4)=a;»-3' = a;'-8. 
 So, (3c-a») x (8c' + 4aV + 3ffl*c+ffl«) = 16c*-a». 
 7] (a + b) X (a^-'-a"-^b + a"-W-a''-W+ ■ ■ ■ -faJ'-'-S"-*) 
 
 = a" — 6", if « be any even integer. 
 E.g., (a; + 3)x(a^-2a;*H-4a;'-8a;' + 16a;;;-33) = a*-64. 
 8] {a + b) X {a"-^-a''-''b + a''-W-a''-^P+ a-b^-^+b"-^ 
 
 = a" + 5", if n be any odd integer. 
 
 E.g., (m^ + Sl) X {m'-3mn+9mV-27mn' + 8n^)-m''> + 2'k^f. 
 
S.SS1 MULTIPLICATION. 49 
 
 QUESTIONS. 
 
 Name the type-form that applies, find the product, and check 
 
 the work by the principles of note 1. 
 
 1. {x + 2)x{x+d). 2. (a; + 2)x{a;-3). 3. {x-2)x{x + d). 
 
 4. (a;- 2) X (a; -3). 5. {y + a)x{y + b). 6. {y-a)x{y-h). 
 
 '7. {y-a)x{y+b). 8. {i/+a)x{y-b). 9. (a+2)(a + l). 
 
 10. (a;-5)x(a;-6). 11. (a-4) x(a-3). 12. (5+a;) x (5-y). 
 
 13. {x + a + b)x{x+c + d). 14. {x-a + b)x{x-c + d). 
 
 15. {3aV + 5SV) X (3aV + 55V). 
 
 16. (a + 9)x(a-9). 17. (3-a)x(3 + a). 18. (3 + a;) x (3 -a;). 
 19. {& + !) X (i-1). 30. (a^ + 2) x (a;'-2). 21. (a;»+7) x (a?-7). 
 23. (50-5)(50 + 5); 45x55; 78xS3; 47x53; 99x101. 
 23. {ax'+f)x{as^-y'). 24. (2a^ + 3yh)x{2a^-dyh). 
 
 25. {x-a) X {x + a);{x'-a') x {x' + a^);- ••;(«"-«") x (a;" + a"). 
 
 26. (l-a;)x(l + a;)x(l+a^)x(l + a^)x(l + a:»)...(l + a!»»). 
 
 27. Show that the rules that apply to form 1 apply also to 
 
 forms 3, 4. How does the square of the sum of two 
 numbers differ from the square of their difference ? 
 Find the values : 
 
 38. (a; + 6a)'. 29. (3a; -2^)". 30. (a»-l/3)='. 31. (a5-4)^ 
 33. {x + 3yy. 33. {x-ByY. 34. (a; ±3)". 35. (2x'±3yy. 
 36. {a + b^y. 37. (a-b + c)\ 38. {a + b±c^y. 
 
 39. (100-1)2; 992. (61)8. 28^; 73'; 807''; 8.07'; .0807'. 
 
 40. In the square of a polynomial of five terms, how many terms 
 
 are perfect squares ? How are the other terms formed ? 
 Find the values: 
 
 41. {x+y+zy. 42. (2a; +3^ -42)". 43. {xy + yz+zxy. 
 Find the products : 
 
 44. {a^ + ax + x') x (a — x) x (a' — ax+s?) x (a + x). 
 
 45. (a;"-*i+a;"-V+a;»-y+ ■ • • +a;2/«-'+^»-i)'x (a;-jr). 
 
 46. (a;"-'— a;"-''5r + a;"-y ia:^^"-''^^"-') x {x+y). 
 
 47. {p+pr+pt^+pt^i H-^r"-*) x (1— r). 
 
50 THE PRIMARY OPERATIONS OF ALGEBRA. in.Pa. 
 
 BETACHED COBFFICIEKTS. 
 
 ■ Note 5. If both multiplicand and multiplier be such that, 
 when their coefficients are detached, the remaining factors of 
 pairs of successive terms have a constant ratio, the work is 
 shortened by writing down the coefficients only. 
 Take the terms of loth factors in such order that when the 
 coefficients are detached the parts lift, taken two and 
 two in order, have a constant ratio; 
 in place of the given polynomials, write the two groups of co- 
 efficients, with a for the coefficient of any absent term; 
 multiply the coefficients as polynomials; add those partial 
 products that pertain to like terms of the final product; 
 in the final product restore the suppressed letters. 
 E.g., (a' + 3a^5 + 3a¥ + W) x {a^ + 2ab + W) [ratio b : a. 
 
 gives 1 + 3+ 3+ 1 
 1+2+ 1 
 
 1 + 8+ 3+ 1 
 + 2+ 6+ 6 + 2 
 + 1+ 3+3+1 
 
 1 + 5 + 10 + 10 + 5 + 1; 
 and the product is a^ + 5a*6 + lOa'J^ + lOaW + 5a5* + 51 
 
 Check: 1 + 3 + 3 + 1 = 8, 1 + 2 + 1 = 4, 
 
 8-4:=33, 1 + 5+10 + 10+5 + 1 = 32. 
 This method is a familiar one in arithmetic; the ratio is 10. 
 E.g., 1089x237 = 258093, or lth + 0h + 8t + 9u 
 
 237 2h + Zt + '7u 
 
 7623 ' 7th + 6h + 2t+3u 
 
 3267 3tth+2th + 6h + 7t 
 
 2178 2hth + ltth + 7th + 8h 
 
 258093 2hth + 5tth + 8th + 0h + 9t + 3«. 
 
 The first form is a case of detached coefficients, wherein 
 the denominations and the relations of the several figures are 
 shown by their positions, as in the last form they are shown 
 by words and sigiis. 
 
3.8 3] MULTIPLICATION. 51 
 
 , QUESTIONS. 
 
 Multiply and check the work : 
 
 1. (2a; + 3)x(3a;-4). 2. {3!' + 3x + 2)x{x^-3x + 2). 
 
 3. (3y - 5) X {2y + 7) x (2 - 4/) x (1 + 2/). 
 
 4. (x^ + ^x'y + dxy^+y^) x{x'' + 2xy + y') x {x + y). 
 
 5. (23?-3x''y + 2y^)x(2x'+Sxy''+2y^). 
 
 6. (2a; -5)^ 7, (y + S/H-Sf')^. 8. (2-3z-3^« + 2z')^ 
 9. (a» - 3a;^2/' + Sa;?/* - /) x (a;* - 43^^^^ + Sx'y^ - ixf + y'). 
 
 10. (a;'-2a;^ + l)x(2a;^-3a; + 4)x(a; + l). 
 
 H. {x'—mx + m') X {x^ + mx+m') x (a;* + mV + m*). 
 
 12. (a/ + hy^!^—cyh') x {afz^—ifi^ + cy^). 
 Show that 
 
 13. a;x(a; + l) x(a; + 2) x (a; + 3)+l = (ai^ + 3a; + l)=. 
 
 14. (.y-l)x2/x(!i/ + l)x(«/ + 2) + l = (?yH2/-l)^. 
 
 Show that" if y be replaced by a; + l, the equation in ex. 14 
 becomes the equation in ex. 13. 
 
 15. By successive multiplications, and preferably by detached 
 coefficients, find the first five powers of a + b and of a — b, 
 
 16. So, of x+y and of x — y, of h + k and of 7i — k. 
 
 17. From observation, and comparison of the results above, 
 state a general principle that holds good as to: 
 
 1. The number of terms in any power of a binomial. 
 
 2. The exponents of the first letter (a, x, or h) in the 
 successive terms of any power. 
 
 3. The exponents of the last letter (b, y, or k). 
 
 4. The signs of the terms. 
 
 5. The coefficients of the first term, the second, and last. 
 
 18. Show that the terms of any power of a + b are homo- 
 geneous, and that terms equidistant from the ends have the 
 same, or opposite, coefficients. 
 
 What terms of powers of a + b are identical with terms 
 of like powers of a — b, and what are opposites ? 
 
52 THE PRIMAKY OPERATIONS OF ALaEBRA. [n.Pn. 
 
 SYMMETRY. 
 
 Note 6. The work is often shortened by noting any sym- 
 metry that may exist among the factors, singly or in groups. 
 E.g., the product {2a + b + c)x{a + 2b + c)x(a + b+2c), has 
 the terms 2a-a-a = 2al^, 
 
 2a-a-b + 2a-2b-a + b-a-a = la% 
 2a-2b-2c + b-c-a + c-a-b + 2a-c-b + b-a-2c+c-2b-a 
 = IGabcj 
 and •.• every term of the product, being entire and of tht 
 
 third degree, is of like form to one of these, 
 and the product is symmetric as to «, 5, c ; 
 .*. it has the terms 2P, 2& as well as 2a', 
 and Wa, l<?a, laV, '7be', llca^, as well as 7a'5, 
 aud the whole product is 
 
 2a? + 2W + 2(? + Wb + Wc + Id'a + 1a¥ + Ibt? + Ica^ + 16oJc. 
 So, in the product {2a + b-c) x (35 + c-fl) x (3c + a-S) : 
 the terms in a', W, c' have the same coefficient, — 2, 
 those in a^b, Vc, <?a have the same coefficient, 5, 
 those in dl?, b(?, cc? have the same coefficient, — 1, 
 that in abc has the coefficient 2; 
 
 and the product is 
 
 -3(a' + 5» + c») + 5(a^&+5'c + c««)-(fl6« + 5c' + m2)+^a&c. 
 
 So, to find the sum (a + 5-2c)H(5 + c-2a)' + (c + a-2i)': 
 
 by multiplication, or from the type-form (a + S+ • ■ • )^ S^* 
 (a + 5 - 2c)« = aH 5H 4cH 2«5 - 4Jc - 4ca, 
 
 by symmetry, write, 
 
 (5 + c - 2a)^ = S^ + c^ + 4a' + 25c - 4cffl - 4a5, 
 (c + a-25f=c'+aH45' + 2ca-4a5-4Sc, 
 
 and add ; the sum is 6 (c? + W + c' — bc — ca — ab). 
 
 In such symmetric expressions, where three or more letters 
 are involved, these letters may be kept advancing in cyclic 
 order, abc, bca, cab, ab, be, ca, as if they were points on 
 a circle following one another in the same rotary direction. 
 
S,$3] MULTIPLICATION 53 
 
 QUESTIONS. 
 Multiply, add, and check the work: 
 1. {x+yY + {x-yY. 2. ix + yy-{x-yy. 
 
 3. {x+y + zy + {x-y + zy. 4. {x+y+zy-{x-y + zy. 
 
 5. {x+y-zy + {x-y + zy. 6. {x + y-zy-{x-y+zy. 
 
 7. { — x+y + z)x{x-y + z)x{x+y—z). 
 
 8. {-a + b + c + d)-{a-b + c + d)-{a + b-c+d)-{a + b+c-d). 
 
 9. (ax + by) • {bx + ay) + {ax — by) ■ {bx — ay). 
 
 10. {a + b)-{c+d) + {a + b)-{c-d) + {a-b)-{c+d) 
 
 + {a-b)-{c-d). 
 
 11. {a + b + c)-{x+y + z) + {a + b-c)-{x + y-z) 
 
 + {a-b + c)-{x-y+z) + {-a + b + c)-{-x+y+z). 
 
 12. {bc+ad)^+{ca+bdy+{ab+cdy. 
 
 If 2s=a+b+c, prove the identities below, and check the 
 work by putting a=b=c=2. 
 
 13. a^-(6-c)==4(s-5)(s-c). 14. (5 + c)«-a^=4s(s-fl). 
 
 15. (s-a)H(s-&)H(s-c)Hs'=a' + 5' + c«. 
 
 16. (s - ay + (s - J)' + (s - c)» - s' = - 3fl5c. 
 
 17. 2{s-a){s-b){s-c) + a{s-b){s-c)+i{s~^c){s-a) 
 
 + c(js — a){s — b) = abc. 
 
 18. l&s{s-a){s-b){s-c)=2{W<?+(?a''+aW)-{a^+T^+<*). 
 
 19. If a; +y +2 = 0, show that a^+^' + z*— 3a;^2; hence that 
 
 {b-cy-\-{c-ay+{a-by=^b-c)-{c-a)-{a-b). 
 
 20. (a;, + iTj) • {y^ -yi) + (x^+ X3) • (y^ - y^) + {x^ + Xy) ■ («/, - y^) 
 
 21. (ax +by+ cz) • {5a; +cy+ az) • (ex +ay+ bz) 
 
 = aic(a? + ^* + 2") + (a* 4- 5' + (F)xyz + Zdbc • xyz 
 + (a¥+b(?-{- ca^) • (xy^ + yz* +Z3?) 
 + (a'5 + Wc + (?a) • (a?y + y^z + z^a;). 
 
 22. (ax +by+ czy + (ax +cy + bzy +(bx+ay+ czy 
 
 + (bx + cy + azy + (ex +ay + bzy + (ex +by+ az)" 
 =2(a'+P+c')(x'+y^-i-z")+4:(bc+ca+ab)(yz+zx+xy). 
 
54 THE PRIMARY OPERATIOI^S OF ALGEBRA. tH.PR- 
 
 CONTKACTION. 
 
 Note 7. When only the first few terms of a product are 
 wanted, the work is shortened by omitting all partial products 
 that do not enter into the required terms. 
 E.g., to find (l-3a;+5a;^ fas far as the a?-term: 
 
 l-3a) + 5a^ or 1-3 + 5 
 
 l-Zx + ba? 1-3 + 5 
 
 l-3a; + 5£B^ 1-3+5 
 
 -3a: + 92? -3+9 
 
 + 5a? +5 
 
 l-6a; + 19a;'' 1-6 + 19 
 
 So, omitting a;* and higher powers, to find the product, 
 
 {l+x+x'+ . • . ) X (1 - 2a;+3a;^ ) x (l+4a;+9a;=+ • • • ) : 
 
 write 1 1 1 1x1-2 3-4 I'l 2-2x1 4 9 16 
 -2 -2 -2 4-4 8 
 
 3 3 9-9 
 
 -4 16 
 
 1-1 2-3 13 7 13 
 
 and the product sought is l + 3x +11x^ + 130?. 
 
 This method of contracted multiplication may be used, 
 with great profit, with decimal fractions. 
 E.g., to find the product 37.8562 x 14.9716, correct to two 
 places, and .2819 x .3781 x .2148 to three places. 
 
 37.8562 
 
 and 
 
 .2819 
 
 .1065 
 
 14.9716 
 
 
 .3781 
 
 .2148 
 
 378.562 
 
 
 .0846 
 
 .0213 
 
 151.425 
 
 
 197 
 
 11 
 
 34.070 
 
 
 22 
 
 4 
 
 2.650 
 38 
 23 
 
 
 .1065 
 
 1 
 .023 
 
 566.77 
 
 In writing down the partial products, carry what would 
 have been carried had the multiplication been made in full. 
 E.g., the partial product 23 = 3-6 + 5 carried from 78-6. 
 
3. §3] MULTIPLICATION. 55 
 
 QUESTIONS. 
 Find the products (or powers) as far as the a:*-term: 
 
 1. {l + x + a^+---)-{l-x + x' ). 
 
 2. {l+2x + dx^+---)-{l-2x + 53? ). 
 
 3. {l + Sx + 5a?)-{l + ^x)-{l + 5x'). 
 
 4. {3-5x + 9x')-{l-5x)-{l + 9x'). 
 
 5. {l-2x + 3x'-i3^ + 5a^)\ 6. {l-ix + is^-ix^+^x^ 
 7. {3 + 5x + W+---y. ' '?,. {\-\x^\o^ Y. 
 
 9. (a + &a; + ca:^H ) -(a' + S'aj + cVH ). 
 
 10. (a-lx-^co? )-(a'-J'a! + cV ). 
 
 11. (fls + 5a; + ca^ + c?a?+ •••)''• 12. (a-lx^-co?-d7?^- ■ ■ ■)\ 
 13. (a + 5a; + ca;^ + £Za^+ •••)'. 14. {a-lx->rcy?-d3?-\- ■ • -y. 
 
 Find the products (or powers) as far as the a;'-terin, with 
 three-figure decimals : 
 
 15. (l + .5a; + .09a;^)-(l-.5a; + .09a;^). 
 
 16. (3+.5a;-.07a;^)-(3-.5!r + ,07a^). 
 
 17. (l-.07a;=)-(3 + .009a;=). 18. (l + .007ar'').(3-.09a^). , 
 19. (1 + . 167a; + . 014a? + .001a;')l 
 
 30. (l-.167a; + .014a;2-.001a;»)l 
 
 21. (l + .056a;^-.0062;')'. 22. (l-.333a; + .006ar')'. 
 
 23. (3 + . 5a; + .07a;* + .009a;») • (5a; + .073;^) ■ (7 + .009a;*). 
 
 24. (l + .07a;)-(l + .07a;*)-(l + .07a;'). 
 
 25. (l + .07a;^)*. 26. (l + .07a;»)'. 27. (l+.07a;*)*. 
 Find the values correct to thousandths, when a; = .l: 
 
 28. (l + 2a;)». 29. (l + 2a; + 3a;*)'. 30. (l + 22; + 3a;* + 4a;=)l 
 
 31. (a; + 5)». 32. (a;«-a; + 5)». 33. (a;'-a;-5)«. 
 Find the yalues correct to thousandths when a; =.02: 
 
 34. (l + 2a;)'. 35. (l + 2a; + 3a;*)'. 36. (l + 2a; + 3a;* + 4a;»)» 
 37. (a; +5)*. 38. (ar'-a; + 5)*. 39. (a;' -a; -5)*. 
 
 40. \lx=a-\-ly -^cf-^-dy^^ , y-l-^-mz-^n^^-'p^ -^ • • •, 
 
 find the value of x in terms of z as far as the «'-term. 
 
56 THE PEIMARY OPERATIONS OP ALGEBRA. [n,PR. 
 
 §4. DIVISION. 
 PeOB. 4. To DIVIDE ONE NUMBER BY ANOTHER. 
 
 (a) A monomial by a monomial : to the quotient of the nume- 
 rical coefficients, annex the several literal factors, each 
 tahen as many times as the excess of the exponent of the 
 dividend over that of the divisor; [I th. 10 cr. 
 
 marls the quotient positive if the elements be both positive or 
 both negative, and negative if one element be positive 
 and the other negative. ^ [inv. pr, 3. 
 
 E.g., QZa-Wd'' : "Tai^d^^Qa-Wc-^ 
 -35a;*«^-V : 5xy-''^= -7o^y-h-\ 
 
 f|a-«5-«rf-° : — Jac-»(Z-«= -|a-'5-V, 
 — |a;-yz-' : —^-^y-''i?—-^xy^z-\ 
 
 (i) A polynomial by a monomial : divide each term of the divi- 
 dend by the divisor; add the partial quotients. [I th. 7. 
 
 E.g., {i5a?y''z + 105xz-'-lG5x-'>y-^z-';) : -15xy^z-^ 
 = -Zx'^-ly-h + llx-^y-h-K 
 
 (c) A polynomial by a polynomial : arrange the terms of loth 
 polynomials as to the powers of some one letter; 
 
 divide the first term of the dividend by the first term of the 
 divisor; multiply the whole divisor by this partial 
 quotient, and subtract the product from the dividend; 
 
 repeat the work upon the remainder as a new dividend; 
 
 add the partial quotients; their sum is the quotient sought, 
 and the part of the dividend left undivided is the 
 remainder: [I th. 7 cr. 
 
 E.g., to divide a' + 5' by a+S: 
 write a' + 5' \a + b 
 
 -a'b + b^ 
 -a^b-aW 
 
 aV + l? 
 aW + b^ 
 
4,14] 
 
 DIVISION. 
 
 57 
 
 QUESTIONS. 
 
 Find the quotients below, and check the work : 
 1. %a?i:ab. 2. -^ax-.-a?. 3. mn-^-.-mhi. 
 
 i. -r^st-^:2r-Vi-\ 5. blaWc: -lla^b. 6. 231x"+Y:3x''y. 
 7. {cd' + 2ax + h):x. 8. (ia?-|a-^-» + f2/-*): -3a;»jr-l 
 
 9. {af-x-12):{x-4:). 10. (4a,-*-12a;« + 9): (2a^-3). 
 
 11. {15x'+af'y-^+ii/-^):{3x+2y-^). 
 
 12. (aV-2^):(aa^-2/'). 13. {x*-2x'y-3y^):{x>+y). 
 14. (a^» - «=>») : (a;» + a"). 15. (9aV - 49cy ) : (3flV - '7cy'). 
 16. (1-a^) : (1 + a;). 17. (c=-3c* + 3c"-l) : (c«-l). 
 
 18. (a"'+"-a™5" + ffl"6'"-Z>"'+"): (a»-S"). 
 
 19. (a»'»-«):(?--l); (a-flr"):(l-r). 
 
 20. {a''-2ab-2ac + h''+2bc+c'):{a-b-c). 
 
 31. (a«-S*+25c-c»):(a+S-c); (5^+25c + c'-a2):(5+c-a). 
 
 22. (a" — mV + 2TO«a^ — n^a^) : {a — mx + no?). 
 
 23. (a=-a^):(fl-a;); {cf-fy.i^a-y); {pi?-y'^):{p,?-f). 
 
 24. {a^+a?):{a + x); ia'-y'):{a + ij)', {a^ + y'):{a?+y'). 
 
 25. (a;»+« + 3ar'-4a;"-^ + 6a;):(a;^ + 3). 
 
 26. (a;^ + a2; + 6a; + ca; + oJa; + ac + fl<?+Sc + 5<?):(a; + a + J). 
 
 27. (12mV- 17mV+ lOma;- 3) : (4ma;- 3). 
 
 28. (a"+»-3a"+»-a" + 2a"-^-4fl!"-*):(a'-a + l). 
 
 29. {2y^+dy-7 + 15y''-8y-^+5y-^):{2-y-^+y-% 
 
 30. (a;« - 222^ + 60a? - 552;^ + 12a; + 4) : (a;« + 6a; + 1). 
 
 31. (a^y+xy^+x'z+2xyz+yh+xz''+yz'') : (y+z) : {x+y).' 
 
 32. (a;* - 4a?y + 6a?y^ - 4a;«/' + ^*) • (a;^ + 2a;«^ + y") 
 
 : {a^-2x'y + 2xy''-y*). 
 
 33. a;* + a 
 
 x^ + ab 
 
 a?+abc 
 
 x+abcd 
 
 ay'+a 
 
 x-i-ab 
 
 + b 
 
 + ac 
 
 + abd 
 
 
 +b 
 
 
 + c 
 
 +ad 
 
 +acd 
 
 
 
 +d 
 
 + bc 
 
 + bcd 
 
 
 
 
 + bd 
 
 
 
 
 +cd 
 
 
 
 
 
68 THE PRIMARY OPERATIONS OF ALGEBRA. [n, Pb. 
 
 CHECKS. 
 
 The work is tested hj multiplication and sometimes by the 
 principles of prob. 3, note I. 
 
 FORM OF QUOTIENT. 
 
 Note 1. With some changes that are apparent, the princi- 
 ples of prob. 3, note 1 hold true in division. 
 
 AKRANGEMENT. 
 
 Note 3. Unless the division be exact, the quotient and 
 remainder are different for every different choice of trial 
 divisor ; but the complete quotient, including the fraction, is 
 always the same. 
 
 E.g., (x' + l) : (a; + l) gives for quotient and remainder: 
 x — 1, 2 when a; is trial divisor, 
 1—x, 23^ when 1 is trial divisor. 
 
 So, (x'+y'+z^) : {x+y+z) gives for quotient and remainder: 
 
 x—y — z, 2{ij\-\-yz-{-z^) when a; is trial divisor, 
 
 y — z — x, 2{z'' + zx+oi^) when y is trial divisor, 
 
 z — x—y, 2{pi? + xy.+y^) when 2 is trial divisor. 
 
 detached coeeficients. 
 
 Note 3. If both dividend and divisor be such that, when 
 
 their coefficients are detached, the remaining factors of pairs 
 
 of successive terms have a constant ratio, the work is shortened 
 
 by writing down these coefficients. 
 
 In place of the given polynomials, write the two groups of co- 
 efficients, with Ofor the coefficient of any absent term; 
 
 divide these coefficients as in the division of polynomials, and 
 restore the suppressed letters. 
 
 E.g., (fl' + g=) : (a + i) gives 
 
 10 1 
 
 1 1 
 
 1 1 
 
 1-11 
 
 -10 1 
 
 -1 -1 
 
 1 1 
 
 and the quotient u 
 
 a^—ab-i 
 
 + b\ 
 
4.S*] DIVISION. 69 
 
 QUBSTIOlirS. 
 
 1. Show that the rules for the coefficients, exponents, and 
 signs of a quotient are a direct consequence of division's being 
 the inverse of multiplication. 
 
 2. State the first six of the principles in prob. 3, note 1, in 
 such form that they will apply to division. 
 
 3. Show, as a consequence of the seventh principle, that a 
 monomial is not divisible by a polynomial, and that when 
 there is no remainder, the first and last terms of the dividend 
 are divisible by the first and last terms of the divisor. 
 
 4. Find the quotient {l+oif-8y^+6xy):{l + x-'Zy), hav- 
 ing it arranged in the order of rising powers of x, and the 
 coefficients to falling powers of y. 
 
 5. So, {ISxyz + 27i^-af + 8f):{x-3z- 2y). 
 
 6. By detached coefficients, divide a;* + lOa^ + 35a^ + 50a; + 34 
 in turn by x+1, x+2,x+d, x-f4, and check the work. 
 
 7. So,~ a^-ba^ + i by x-2, x-1, x + 1, x + 2. 
 
 8. So, a^-10a? + 35a?-50z+2'ihyx-l,x-2,x-d,x-i. 
 
 9. So, a^—ba^y'+iy* by x—2y, x—y, x+y, x+2y. 
 
 10. So, «* - lOa^y + Sba^y" - 50xy^ +2iy^ hj x-y, x- 2y, 
 
 x—3y, x—4:y. 
 
 11. So, a^+4:a?-6a^+7x-2 by a? + 5x-2, 3?-x + l. 
 13. So, a* + 4a'5 - baW + QaV-b^ by a" + 5«5-5?, a'- db + W. 
 
 13. Divide ^-2x^ + 3ct^-i3? + 5x-6 by a; -3, the quo- 
 •tient by a;— 3, and so on. Write the last quotient and the 
 
 numerical remainders in their order, as coefficient of powers 
 of a;— 3. 
 
 14. So, af^+2a^+33?+io^+5x+6 by x+2, the quotient 
 by a; +3, and so on. 
 
 15. So, !f+2a^y+3ar'y'+4:a?f+5xy*+ey^ by x+2y, the 
 quotient by x + 2y, and so on. 
 
 16. Compare the work in exs. 13-15 with that of reducing 
 seconds to minutes, hours, and days by the ordinary process of 
 reduction ascending: what is the scale in ex. 14 ? in ex. 15 ? 
 
60 THE PRIMARY OPERATIONS OP ALGEBRA. [".Pb- 
 
 TYPE-FORMS. 
 
 Note 4. If the dividend and divisor can be reduced to known 
 type-forms [pr. 3, nt. 4] the quotient may be written directly. 
 
 E.g., {3^+7x + 12) : {x + 4:) = x + 3, [1 
 
 {af-y^):{x+y) = x-y, [3 
 
 SYMMETRY. 
 
 Note 5. If both the dividend and divisor be symmetric as 
 to two or more letters, and if there be no remainder, then the 
 quotient will be a symmetric function of the same letters. It 
 is then often sufficient to get a few characteristic terms, and 
 to write the rest therefrom by symmetry. 
 E.g., (a^y+xy^ + yh + yz^ + z^x+zx' + ^xyz) : {x+y+z), 
 gives xy for one term of the quotient; 
 
 .•. yz and zx are also necessary terms of the, quotient; 
 and •.■ the product {x+y+z)- {xy +yz + zx) is the dividend, 
 
 .*. xy+yz + zx is the quotient sought. 
 The pupil must, however, use great caution if he employ 
 symmetry in division; he may use it as suggestive of the true 
 answer, but hardly ever as conclusive, because he cannot be 
 sure beforehand that there is no remainder. 
 
 CONTRACTION. 
 
 Note 6. When only the first few terms of a quotient are 
 wanted, the work is shortened by omitting all partial products 
 that do not affect the required terms. 
 E.g., (l+x+x' + a?) : {l — Zx + dx' — iar') to four terms gives 
 
 1+ x+ ^+ a? 
 l-2a; + 3a^-4a^ 
 
 l-2a; + 3a;^-4ar' 1111 
 
 l + Sx+'la^'+ia? 1-3 3-4 
 
 Sx-2ce' + 5a? 3 -2 5 
 
 3x-63^ + 9oi^ 3-6 9 
 
 4a^ — 4a^ 4 -4 
 
 isi?-8a^ 4 -8 
 
 4a* 4 
 
 1-2 3-4 
 
 13 4 4 
 
4. 84] DIVISION. 61 
 
 QUESTI0K8. 
 
 Name the type-forms, find the quotients, and check the work: 
 1. (a^-21a;4-104):(a;-13). 2. {?^- f) :[?(?- y^). 
 
 3. (a? -4a; -13): (a; -6). 4. (130 + 31a;^ + a;y): (5+a;,y). 
 
 5. (2 + a; -a?): (2 -a;), 6. (3:i;*-a;-15) : (a;-3), 
 
 7. (4a;« - V) : (2a? + 3jr»),. 8. (aW - 49a;*) : {ab - 7a;»). 
 
 9. (a;*'-2^=»):(a;»±2/"). 10. (1 - 100a*5V) : (1 + lOa^Sc'). 
 
 11. [l-(7a-3S)^:(l + 7ffl-36). 
 
 12. [{aP+a^)-{a^-a^)]:[{x' + ax + a!')-(a?-ax + a'^]. 
 
 13. l{a+by-d']:{a+b-c). 14. [a^-(S-c)^]: (a-5+c). 
 15. (a«+2aa;«+a;*):(a+a:»). 16. (a;«-8a;+16):(a;-4). 
 
 17. (4a? - 12a;^ + 9f) : (2a; - 3^). 
 
 18. (Ada" + 42flS + 95«) : (7a + 35). 
 
 19. (4a'+5'+<?-4aJ + 4ac-2Sc) : (2a-& + c). 
 
 20. (a'6'-343ar'):(a&-7a;). 21. (a" + 216b'): (a + 6b). 
 22. [(a;+2/)»+z»]:(a; + z/ + z). 23. [a?-{y-zy]:{x-y+z). 
 24. (a^ - J*) : (a + b). 25. (a«»+» + 5^+i) : (a + b). 
 26. (a;"'»-l):(a;™-l). 27. (a;"'»-l):(a;"-l). 
 
 By symmetry, find the quotients below, and check the work: 
 
 28. [3abc + a\b + c) + b\c +a)+c'(a + b)'\:{a + b + c). 
 
 29. (a' + S' + c'-3«6c):(a + 5 + c). 
 
 30. (a'-*'+c'+3a6c):(a-S + c). 
 
 31. («» + a'J^ + a*6* + a'5' + 5») : (a* + a»5 + aW + oJ' + J*). 
 
 32. (a^J^-a'S^ + SV-iV + c'ffl^-cW) 
 
 : (a=d - al^ +Wc-b<?+(?a- caF). 
 
 33. [a?(^-z)+^3(«_a;)+2»(a;-2^)] 
 
 . : [ar'(y-2) +2^«(2-a') + 2*(a:-.7)]. 
 Find the quotients to the a;'-term, using three-place decimals ; 
 
 34. (l-.3a; + .04ar' ) : (l + .la; + .01ar'+ • • .). 
 
 35. (l + .2a;-l-.04a?+ • • •): (l-.la; + .01a? ). 
 
 36. (x-:^a?+^\^^ ) : (l-iar' + jj^a^ ). 
 
62 THE PRIMAKY OPERATIONS OF ALGEBRA. [H.Pa, 
 
 SYNTHETIC DIVISION. 
 
 Note 7. If the coefficient of the first term of the divisor be 
 1, the work of division by detached coefficients is shortened 
 by the method of synthetic division. It is a species of short 
 division, in which the same numbers play the part of both 
 remainders and quotients. 
 E.g., to divide a;' + 5a;* + 8a; + 4 by a; + 2: 
 
 write 
 
 2 
 1 
 
 5 8 4 
 3 6 4 
 
 1 3 2, 
 
 The divisor stands at the left in a vertical column, reading 
 upward; the part products 2, 6, 4 are directly below the like 
 terms of the dividend; of the figures 1, 3, 2, 0, in the lower 
 line, 1 is the first quotient figure, 3 is first a remainder stand- 
 ing for %oi?, then a quotient figure standing for Zx, 2 is first a 
 remainder standing for 2a;, then a quotient figure, and is 
 the final remainder. The quotient iso? + '^x + %. 
 
 It is customary to change the signs of the terms of the 
 divisor after the first, and so of the resulting part products; 
 the subtractions are then done by addition, and the work 
 takes the form 
 
 15 8 4 
 
 -2 -6 -4 
 
 1 3 2, 
 
 So, (a;' + 5a;2 + 9a; + 6) : {x' + Zx + 'iS) gives 
 15 9 6 
 -3 -4. 
 -3 -6 
 
 -2 
 
 -3 
 
 1 
 
 1 2, 1 2 
 
 wherein both 3 and 2 have their signs changed, and their 
 products by 1, the first quotient figure, give the "3, "2 in the 
 body of the work; the sum 5 + ~3 is 2, the first remainder 
 and the second quotient figure; and ~6, "4 are the part 
 products of ~3, "2 by this 2. 
 
 The quotient is a; + 2, and the final remainder isx+%. 
 
4,S4] DIVISION. 63 
 
 QUESTIONS. 
 
 1. Show that, with a divisor in which the coeflBcient of th6 
 first term is unity, each term of the quotient has the same 
 coefiBcient as the first term of the dividend used : i.e., that the 
 same number plays a double part, being the coefiBcient of the 
 first term of the remainder and of the new term in the 
 quotient, and that it need, therefore, be written but once. 
 
 2. Show that changing the sign of the divisor and adding 
 the partial products to the dividend does not change the value 
 of the remainder, and therefore not that of the quotient. 
 
 By synthetic division, find the quotients, and check the work: 
 
 3., (a?' -5a;' -46a; -40): (a; + 4). 4. (a;'-40):(a; + 4). 
 
 5. (3i^+a?-5x>-llx-S0):{x-3). 6. (a;*-5a;'-30):(a!-3). 
 
 7. (3a;'-5a;'-7a;+10):(*-3). 8. (Sa;'- 7a; + 10): (a; -2); 
 
 9. (ia^+17a^+9!ir'-20a?-3x + 9):{a; + B). 
 10. (a;'-l):(a;-l). . 11. {3?-f):{x-y). 
 
 12. {icf-f):{x+y). 13. {x'-x'y+xy''-y^):{x-y). 
 
 14. {o^-Ba?y + 3zy''-f):{x-y). 15. {6a;* -96): (a; -3). 
 
 16. {a^-a''l)'+2aP-b'):{al-a6+V). 17. {a^-¥):{a^+P). 
 18. {a;«+10a;-33):(a;2-2a; + 3). 19. (a;« + 10a;-33):(a;« + 3). 
 
 20. (3a;'-4a;*-a;' + 23a;'-28a;+15):(a;2-2a; + 3). 
 
 21. (a* + a'5-8a'6'+19fl5»-155*): (a« + 3a5-5J«). 
 
 22. (4/ - 182» - 162'' - 782 + 54) :{^-2z' + z~9). 
 
 33. (4a;»-5a;' + 8a;*-10a;'-8a;«-5a;-4):(a;*-2ai'+3a;-4). 
 
 34. (a;«+151a;-264):(a;2-4a; + ll). 25. (a;'-364):(a;«+ll). 
 36. (3a;»-83a;-340):(a;2 + 4a!+5). 27. (2a;^-240):(a:H5). 
 38. (a;*+a.-'-4a;^+5a;-3):(a;» + 3a;-3). 29. (a;*-3):(a,^-3). 
 
 30. (a;* - 23a;* + 60ar' - 55a;' + 12a; + 4) : (a;» - 3a: + 3). 
 
 31. Divide 3a,'*-5a;' + 7a;'-9a; + ll by a;-3, the quotient 
 
 by a; — 3, and so on; write the last quotient and the 
 remainders as coefficients of powers of (a; — 3). 
 33. Sodivide'3a;* + 5a,H7a;H9a; + U by a; + 3, by a; + 3,--. 
 
64 THE PRIMARY OPERATIONS OP ALGEBRA. [n,PBS. 
 
 §5. FKACTIONS. 
 
 PeOB. 5. To REDUCE A SIMPLE FKACTIOST TO LOWER TERMS. 
 
 Divide both terms by any entire number that divides them with- 
 out remainder; the quotients are the terms of the re- 
 duced fraction. [I th. 3 cr. 4. 
 
 E.g., ZQa^W(?/2ia^bx = U(?/2ax. [div. 12 a^b. 
 
 For reduction of fractions to their lowest terms, see IV, pr. 4. 
 
 PkOB. 6. To REDUCE A SIMPLE FRACTIOK TO AN EQUAL 
 FRACTIOlf HAVING A GIVEN NUMERATOR OR DENOMINATOR. 
 
 Divide the given new numerator or denominator by the old 
 one, and multiply both terms of the fraction by the 
 quotient. [I th. 2 cr. 4. 
 
 E.g., to reduce ^o^y/ldb to an equal fraction with denom- 
 inator 2c?bc, 
 
 then Ic^bc : 2fl'5 = ac, and the fraction sought is %aco?y /%a^bc. 
 
 So, to reduce "^a^z/Zah to an equal fraction with numerator 
 Q3?yz, 
 
 then &a?yz : 2a?z= Zy, and the fraction sought is &3?yz/Wcy. 
 
 So, entire and mixed numbers are reduced to simple fractions. 
 
 E.g., x + %a={dx + 'ii,ad)/d, x-2a + a^/d -{dx-2ad+ if]ld. 
 
 PrOB. 7. To REDUCE TWO OR MORE SIMPLE FRACTIONS TO 
 EQUAL FRACTIONS HAVING A COMMON DENOMINATOR. 
 
 Over the continued product of the denominators write the 
 product of each numerator into all the denominators 
 except its own, or [I th. 2 cr. 4. 
 
 find some number that can be exactly divided by all the de- 
 nominators; 
 
 divide this number by the denominators in turn and multiply 
 each numerator by the quotient got by using its denom- 
 inator as divisor. 
 
 5«^ 3Jc Z{a- b)_ Sba^y i2ab c 6ax{a-b) 
 ^•^■' 2a'' x' 7 Uax' Uax' Uax ' 
 
 For finding the lowest common denominator^ see IV, pr. 5. 
 
6. 6, 7, S 51 FRACTIONS. 65 
 
 QUESTI0K8. 
 
 Bedace the fractions below to lower terms: 
 
 a?—3x+2 a?— 2a;— 15 aca?+(acl—bc)x—hd 
 
 a?-4x+3' af+2x-35' aV-*« ~' 
 
 . a'-b" , a^-W „ 4a?- 9 
 
 4a?- (3^-42)' (4a?+3a: + 2)'-(2a?+3a! + 4)' 
 
 (2a; + 3«/)*-162?' (3a? + a;-l)i'-(a;''-a;-3)« ' 
 
 m*'—n'\ ' ^—^' ' r"— s'" ' a?"— ^' 
 
 Change the fractions below to equal fractions : 
 
 13. a;/(a; — 3) with denominator a;^ — 5a; + 6. 
 
 14. (4a-3)/(4a-4) with denominator 16a* -28a + 13. 
 
 15. {a-5)/(a''-*+a"-*5+ • • • +5"-*) with denominator a" -5". 
 
 16. (a;— 5)/(a?— 1) with denominator 1— a;*. 
 
 17. (*-a)/(2a;+3)(3-2a;) with denominator 4a? -9. 
 
 18. a/{a-c){b-c), ^h/{a-c){c-i), c/[d'-{a+b)c+ab] 
 
 with denominator {b — c){c — a), 
 
 19. Keduce the fractions below to equal fractions, with the 
 
 common numerator a* — J*: 
 
 ffl'-a' a'+ft' «'+«'&+«&'+&' a^-a'b+a¥-lr' 
 a^+V a'-W a^-a^b+aH'-b''' a'+a^b+aff'+P' 
 Eeduce to equal fractions having common denominators: 
 
 91 ^ ^ - 22 ^ "^ ga; 
 
 1-a?' (1-a;)*' {l + xf a^ + aoi^ a? -ax' a^-x"" 
 
 a 3g 2ga; . 2_ 3 2a;-3 
 
 ^^^^ a+a;* a«-a?' a;' 2a;- 1' i^^TT' 
 
 25. 
 26. 
 
 27. 
 
 a^ — bc V—ca & — db 
 
 (a-6)(a-c)' (*-c)(6-fl)' (c-a)(c-dy 
 
 2 5 3 
 
 (a;-l)(a;-2)' a?-5a; + 6' a?-4a;+3' 
 r'->-- 2 r*+r-l ^-^-l r' + l 1-r r + 1 
 /^r + 1' l-r+r"' 1-r' i + r' r'-i' i"+p* 
 
66 THE PRIMARY OPERATIONS OF ALGEBRA. [II, Pbs. 
 
 PrOB. 8. To ADD FKACTIONS. 
 
 Reduce the several fractions to equal fractions having a com- 
 mon denominator; [pr. 7. 
 
 write the sum of the new numerators over the common de- 
 nominator. 
 
 ■w 350° 3(a-a )_ 21&c'+6fla;(«-5) 
 
 ^^'' 2ax 7 ~ Uax 
 
 Subtraction is but a case of addition ; add the opposite of 
 the subtrahend. 
 
 Uc' 3{a-i )_ 21b(^ - 6ax{a - i) 
 ^^■' 2ax 7 ~ Uax 
 
 PkOB. 9. To MULTIPLY FRACTIONS. 
 
 Write the product of the numerators over the product of the 
 denominators, cancelling any factor that is common to 
 a numerator and a denominator. [I th. 2 crs. 2, 4. 
 
 3Sc' 3(a-&) _ 95c°(a-5 ) 2,i(? Say _ Ay 
 ^^■' %ax^~ 7 " Uax ' 2ax^ 9bV~ Zbx 
 
 Division is but a case of multiplication ; multiply by the 
 reciprocal of the divisor. 
 
 3b^ . 3{a-b) _ 35c' 7__ _ 7b^ 
 
 ^^■' 2ax'' 7 ~ 2ax^3{a-b)~ 2ax(a-b)' 
 
 A complex fraction is an indicated division wherein the 
 dividend, the divisor, or both, are simple fractions. 
 
 E.g., 
 
 ay' + y^ x' — y^ {a^+y'y — {x^ — y^Y Ax'y^ 
 
 x'—y^ af+y'' _ {oi? — y^)-(a?+y^) __ a;*— y* 
 
 x + y x — y ■{x + yy—(x — yy ~ Axy 
 
 x-y x+y iP':-y)-{x+y) al^-y" 
 
 __ 43?^" x^ — y'_ xy 
 x^—y^ Axy x^+y^' 
 
 This example may also be worked by multiplying both 
 terms of the complex fraction by x^—y*'. 
 
8.9.55] FRACTIONS. 67 
 
 QUESTIONS. 
 
 Add, subtract, multiply, and divide as shown below: 
 
 1 + x 1—x 1 + x 1 — x 1+x 1 — x 
 
 . a+b a—h , a+b a—b _ 1 , & a 
 
 H — . 5. — -7 + 
 
 a+x a—x a—x a+x' ' a + b a*—W a*+&*" 
 
 6. 1 1 ^ 1 ^ . 
 {a—b)(a—c) {b—c){J)—a) {c—a)(c—h)' 
 
 7. , l_ , 
 
 {a — b)-{a — c) {b—c)-{f) — a) {c—a)-{c — b) 
 
 a?" a?^ a;" 7? x 3-a 5-a 
 
 11 <» + ! ^ + 1 c + 1 
 
 ' a(a — b){a — c) i{b-c)(b-a) c{c-a)(c — b)' 
 
 1,. ^^.^\^. 13. (i + l\:(:,_L).(i_l]. 
 
 a;— y a — x+y \ xj \ xJ \ x) 
 
 x*-¥ 7?+bx af-Wn? x^-2ba? + Vx^ 
 14. 
 
 15, 
 
 a?-2bx + ¥' x-b a;' + 6' ' 3?-bx + l 
 2^— a? a?— 3? a — x a^ — ax + x^ «^ + 3aa; + c; 
 
 a^+a? a^+3? a + x a^ + ax + x^ a^-%ax + x^' 
 
 ,^ x+1 y' + 2y + l (x-l)y a + b J'' -a' 
 
 y ' a?-l 'iy + iy m ' m^ 
 
 a b c abe c? 
 
 \x a — bl 
 
 ^^' b + c'a + c'a + b " {a? + W){a' + c') ' c'-b" ' 
 
 20. Reduce the complex fractions below to simple fractions: 
 
 x—y m^+mn + n^ f + ^ m + n m-n 
 
 ^~l+xy m^ + n^ p' + if ^ m—n m + n 
 
 ~ x{x — yy m' — n^ ' j>° — g" m — n m + n ' 
 
 1 + xy m'—mn + n" jj'-g' m + n m — n 
 
68 THE PRIMARY OPERATIONS OF ALGEBRA. [H, 
 
 §6. QUESTIONS FOE EEVIEW. 
 Define and illustrate: 
 
 1. An algebraic expression; a binomial; a trinomial; a 
 quadrinomial; a polynomial. 
 
 2. Expressions that are literal; numerical; entire; fractional; 
 symmetric; homogeneous. 
 
 3. A series; a finite series; an infinite series. 
 
 4. The degree of a term, and of a polynomial; a coefficient; 
 like terms; unlike terms. 
 
 Give the general rule, with reasons and illustrations, for : 
 
 5. Adding like numbers; unlike numbers. 
 
 6. Subtracting one number from another. 
 
 7. Multiplying a monomial by a monomial; a polynomial 
 by a monomial; a polynomial by a polynomial. 
 
 8. Dividing a monomial by a monomial; a polynomial by a 
 monomial ; a polynomial by a polynomial. 
 
 9. Eeducing a simple fraction to lower terms; to an equal 
 fraction having a given numerator or denominator. 
 
 10. Eeducing two or more simple fractions to equal frac- 
 tions having a common denominator. 
 
 11. Adding and subtracting fractions. 
 13. Multiplying and dividing fractions. 
 
 State the principles that relate to the form of a product : 
 
 13. As to its independence of the values of the letters. 
 
 14. As to its symmetry. 
 
 15. As to the sum of its coefficients. 
 
 16. As to the degree of its highest and lowest terms. 
 
 17. As to its homogeneity. 
 
 18. As to the number of terms. 
 
 19. Write down the most useful type-forms. 
 
 30. State what arrangement of terms is best in multiplica- 
 tion; in division. 
 
S6] QUESTIONS FOR REVIEW. 69 
 
 31. Explain cross multiplication; and show how it is used 
 in multiplying numerals. 
 
 32. Explain the use of detached coefBcients, in multiplica- 
 tion; in division. 
 
 33. Stow how the symmetry of the factors helps to deter- 
 mine the product; the quotient. 
 
 24. Explain the methods of contraction in multiplication; 
 in division. 
 
 25. Explain the checks used in multiplication; in division. 
 36. Explain synthetic division. 
 
 27. Draw a line whose length equals the sum of two given 
 lines, and show by a diagram that the square on this line is 
 made up of a square on each of the two given lines and two 
 rectangles having these lines as sides: hence illustrate the 
 formula {a + Vf=a?+2ab-\-i\ 
 
 So, the formulae (a -J)»=a^-2a& + J«, {a + l).{a-l) = a^-b\ 
 
 28. How might the""knowing that the product of homoge- 
 neous factors is homogeneous help to find errors in division ? 
 Add, and arrange the sum to falling powers of x and the 
 
 coefficients to falling powers of y. 
 
 29. a;* + ixf - 4a;«' + ^3?y - ix'z + Qoi?y^ + %3?z^ - 13a;j/»« 
 4- \2xyz^ - 12x^yz +y^- 4yh - 4«/«» + z* + 6y V + 2*. 
 
 Expand and add : 
 
 30' {a + b + cy+{a + h-cY + {a-b + cy+{-a + b + cy. 
 
 31. {a + b + cy+{a + b-cy + (a-b + cy + {-a + b + cy. 
 
 32. {a + b + cy + ia + b~cy + (a-b + cy + (-a + b + cy. 
 Given a+^=-b/a, a/3=c/a; then: 
 
 33. a^+/3'^{a + /3y-2a/3 = ir'/a' - 2c/a =(V- 2ca)/a\_ 
 
 34. «» + yS" = (« + /*)' - 3a/3{a + /?) = (- J' + 5abc)/a\ 
 
 35. a*/S' + a'/3'=- {F - 3ac) ■ b(^/a'. 
 
 36. 1/a + 1//? = - b/c. 37. a//3 + fi/a = (6^ - 2ca)/ca. 
 
 38. l/o^ + 1/^ = {P - 2ca)l&. 
 
 39. {a/p-fi/ay = W{V-^ac)/a'(?. 
 
70 SIMPLE EQUATIONS. 
 
 m. SIMPLE EQUATIONS. 
 
 An equation is a statement that two expressions are equal. 
 These two expressions are the members of the equation. 
 
 An identity is an equation that is true for every value of 
 the letters involved; the sign of identity is =. 
 E.g., {x + a)x {x — a)=x' — a' for all values of x and a; 
 but the equation 5a; +3 = 17 is true only when x is 3, 
 and the equation x' — Bx = 4: is true only when a; is 4 or — 1. 
 
 The letter or letters whose values are sought are the un- 
 known elements; the other elements are known elements. The 
 unknown elements of an equation are usually represented by 
 the last letters of the alphabet; and in literal equations the first 
 lettei's then stand for known elements. 
 
 The solution of an equation, or set of equations, consists in 
 making such transformations as shall result in giving the 
 values of the unknown elements. The values so found are 
 the roots of the equation or set of equations; and the test to 
 be applied to them is to replace the unknown elements by 
 these values, and see whether they make the equations true. 
 E.g., of the equation 2x—i, \x unknown], 3 is a root, 
 
 •.•3-3 = 4. [df. root. 
 
 So, of the equation aP—5x + 6 = 0, [a; unknown], 3, 3 are roots, 
 •.• 3^-5-3 + 6 = 0, 3^-5-3 + 6 = 0. 
 
 The degree of an equation is that of its highest term. 
 
 If the unknown element enter an equation by its first 
 power only, the equation is a simple equation. 
 E.g., 3a; =4, [x unknown] is a simple equation. 
 
 AXIOMS OF EQUALITY. 
 
 1. Numbers equal to the same number are equal to each other. 
 
 2. If to equal numbers equals be added, the sums are equal. 
 
 3. If from equal numbers equals be subtracted, the ke- 
 MAINDEBS are equal. 
 
AXIOMS OP EQUALITY. 71 
 
 4. If equal numiers be multiplied ly equals, the products 
 are equal. 
 
 5. If equal numbers be divided by equals, the quotients 
 are equal. 
 
 6. If equal numbers he raised to like integer powers, the 
 POWERS are equal. 
 
 7. If of equal numbers like roots be taken, the KOOiTS are 
 equal. 
 
 QUESTIONS. 
 
 1. Show that 7 is not a root of the equation a; — 4 = 3. 
 What number is a root of this equation ? 
 
 2. Write an equation that is not simple. 
 
 3. Show that ~3 is a root of the equation x'z=d. 
 
 What other root has this equation ? are the two roots equal? 
 
 4. What is the difference between an axiom and a theorem? 
 
 5. If from each member of the equation a; + 5 = ll — 2a;, 
 5 he subtracted, what term in the first member is cancelled ? 
 
 What change is made in the second member? 
 
 So, if 2x be added, what change is made in each member ? 
 
 6. May the equation x+a = b—2x be written x + 2x = b — a? 
 How, then, may a term be transposed from one member of 
 
 an equation to the other, and by authority of what axioms ? 
 
 7. In the equation ^x=ix + l, by what single number may 
 the two fraction-terms be multiplied so as to become integers ? 
 
 What other term must then be multiplied by the same 
 number, and for what reason? 
 
 8. What axiom gives authority for changing the signs of all 
 the terms of an equation ? 
 
 9. When from the equation 5a; =13, we get the value 3| 
 for X, what axiom is applied ? 
 
 So, when from the equation i\/x=3, x=9 is found ? 
 
 10. Write an equation of the second degree with x, y as un- 
 known elements; so, of the third degree. 
 
72 SIMPLE EQUATIONS. [HI, Pa. 
 
 §1. ONE UNKNOWN" ELEMENT. 
 
 PrOB. 1. To SOLVE A SIMPLE EQUATION, ONE UNKNOWN 
 
 ELEMENT. 
 
 Multiply both members of the equation ly some number that 
 contains as factors all of the denominators, if any; 
 
 [ax. 4. 
 transpose to one member all terms that involve the unknown 
 element and to the other member all other terms; 
 
 [axs. 3, 3. 
 reduce both members to their simplest form; 
 
 divide both members by the coefficient of the unknown element. 
 
 [ax. 5. 
 Check. In the original equation, replace the unknown element 
 
 by the result so found. 
 E.g., if"^(a;+12)=4(6+32;)-|a;; [x nnkn. 
 
 then •/ 7a; + 84 = 36 + 18a; - tx, [mult, by 42. 
 
 and Ix - 18a; + 7a;= 86 - 84, [trans. 84, 18a;, - 7a;. 
 
 .%-4a;=-48, and a; =13; [div. by -4. 
 
 and •.• 1(13 + 12) =|(6 + 36) - 2, [repl. x by 13, 
 
 .'. 12 is the root sought. 
 
 If the known elements be wholly or in part literal the pro- 
 cess is essentially the same. 
 E.g., if aa;— (Sa; + l)/a;=«(a;'-l)/a;, 
 
 then a3?—bx—l=aa?—a, [mult, by a;. 
 
 ai^—aa?—bx=—a+l, [trans, aa;^, —1. 
 
 8a;=a— 1, [cancel as?, div. by "1. 
 
 a; = (ffl - 1 )/b, [div. by b. 
 
 So, if {a—b){x—c)—{p—e)(x—a) — (e—a)(x—l)=0, 
 
 then ax—bx—ac+bc—bx-i-cx+db~ac—cx+ax+lc—ab=Q, 
 
 2ax—2bx= 2ac — 2bc, 
 
 x=c. [div. by 3(a-J). 
 
».«!] ONE tJNKNOWN ELEMENT. 73 
 
 QUESTIONS. 
 
 Find values of x that make true equations of the statements: 
 1. 12-5a;=13-a;. 2. l-5a;=7a;+3. 3, Zx + Q-2x=lx. 
 4 8 + 4a; = 13a;— 16. 5. a — Zx=x—h. 6. m — nx—px+q. 
 7. 2a;+K4+a;) = 3|. 8. 'k>:-l=ix + 2x + i. 
 
 9. (a; + 4) (a; -2) = (a; -9) (a; -3). 10. (a; + l)(a;-l) = a;(a;-2). 
 
 11. (a:-2)(a;-7) + (a; + l)(a;-3)-8a;=3(a;-8){a;-7)-2. 
 
 12. 1 +A.= — „. 13. -^„ = 44. 14. ^^-^ «^+^ 
 
 a;+l a; + 2 a; + 3" ' a;-2 * ' 3a; + 4 9a;-10' 
 
 x-a _ 3x-c 3a;-5 _ 6a:+5 1_A A- ^ 
 
 2^-*~6^J' 2 - 7 • X 2i''"7^~28' 
 
 18. i('7x + 5)-i(5-x) = 7i-ix-i{8-'rx). 
 
 19. ^(a: + 10)-|(3a;-4)+J(3a;-3)(3a;-3) = ar'-^. 
 
 20. 
 
 a;— 1 a; — 2 a; — 3 a; — 4 
 
 5 5 = • — T 5. [simp, each mem. separately. 
 
 X — /v X — o X — 4 X'~~0 
 
 _1 1___1 1^ a;-4 a:-5 _ a;-7 a;-8 
 
 ' a; — 3 a; — 4~a; — 5 x — 6' ' x-^5 x — Q~x—8 x — 9' 
 23. Ux-^a)+i{x-ia)+i{x-ia) = 0. 
 
 24. 24 -a;- [7a;- {8a;- (9a;- 3a;*- 6a;) }] = 0. 
 
 25. (a; + 5)«="(4-a;)* + 21a;. . 26. {x-m){x + n) = x{x-q). 
 
 27. -^-^ = 1 28. _^ + a+«?=0. 
 
 a — ca; c 
 
 29. aa;-5(a;-l)-c = 0. 30. (a*-a;)(a«+a;) = fl* + 2«a;-a;«. 
 
 31. (a; + l)^=a;[6-(l-a;)]-2. 32. (a;-l)»=(a;-2)(a; + 3). 
 33. (5a; - 6)/m - (a; - l)/ra = a; - 2. 34. jja; - g'a; = (^ + 2')a; - q^ 
 
 35, i(a;+4)-i(a;-4) = 2+,i5(3a;-l). 
 
 36. i(3a; + 2)-i(12-a;) = ia;. d7.{x + a)(x-b) = {x-c){x + d). 
 
 38. ax-m-Z{hx-n-d[cx-p-4:{dx-q)]]=0. 
 
 39. a;-[2a; + (3a;-4a;)]-5a;-{6a;-[(7a; + 8a:)-9a;]{ = -30. 
 
 40. i(a;-3)(a;+2)-Ka'-4)(2a;+l) = 7. 
 
74 SIMPLE EQUATIONS. [ni.pR. 
 
 SPECIAL PROBLEMS. 
 
 Note 1. If the statement of the problem he in words, that 
 statement must be first translated into algebraic form. 
 E.g., to divide $6341 among A, B,,C, so that B shall have 
 
 $420 more than A, and $560 more than B: 
 put X for A's share, a; +430 for B's, a; + 420 + 560 for C'sj 
 
 then •.• a;+a; + 420 +^+420 + 560 = 6341, 
 
 .■.3a;=6341-420-430-560 = 4941, and a; = 1647; 
 .-. A has $1647, B has $3067, has 12627. 
 
 So, to divide 144 into four parts, such that the first part in- 
 creased by 5, the second decreased by 5, the third multi- 
 plied by 5, and the fourth divided by. 5, are all equal: 
 
 put X for the number to vrhich the several results are equal j 
 
 then •.•x — l> + x + 5+x:5+x-5 = li4:, 
 
 .•.5a;-35 + 5a; + 35 + a; + 35a;=720, [mult, by 5. 
 
 i.e., 36a; = 720, a;=30, 
 
 and the parts are 20 - 5, 30 + 5, 30 : 5, 20 • 5 ; i.e., 15, 35, 4, 100. 
 
 So, to find a number such that if 5, 15, 35 be added to it, in 
 turn, the product of the first and third sums shall be 
 10 more than the square of the second : 
 put X for the number, x + 5, a; + 15, a; + 35 for the three sums; 
 then V (a; + 5) X (a; +35) = (a; +15)'+ 10, 
 .-. a;« + 40a;+175 = a:^ + 30a; + 225 + 10, 
 .•. 10a; =60, a; =6, and the numbers are 11, 31, 41. 
 
 So, the width of a room is two thirds of its length; if the 
 width were three feet more and the length three feet 
 less, the room would be square; find its dimensions: 
 
 put X for a side of the supposed square; 
 
 then the length of the room is a; + 3 and the width a; — 3, 
 
 and •.•a;-3=|(a; + 3), 
 
 .•. 3a; — 9 = 2a; + 6, a; =15, and the room is 12 by 18 feet. 
 In solving problems, it is not sufficient that the result found 
 
 shall satisfy the equation : it must also satisfy the conditions 
 
 of the problem as expressed in words. 
 
>.*»! ONE UNKNOWN ELEMENT. 75 
 
 QUESTION'S. 
 
 1. If to the double of a certain number 14 be added, the 
 sum is 154: what is the number ? 
 
 2. If to a certain number 46 be added, the sum is three 
 times the original number : find the number. 
 
 3. The sum of two members is 20, and if three times the 
 smaller number be added to five times the larger, the sum is 
 84 : what are the numbers ? 
 
 4. Divide 46 into two parts such that if one part be divided 
 by 7 and the other by 3, the sum of the quotients shall be 10. 
 
 5. In a company of 266 men, women, and children, there 
 are four times as many men and twice as many women as 
 children: how many men are there? how many women? 
 and how many children? 
 
 6. Thirty yards of cloth and forty yards of silk together 
 cost $66; the silk is worth twice as much per ya;rd as the 
 cloth: find the cost per yard of each of them. 
 
 7. My purse and the money it contains are together worth 
 $20, and the purse is" worth a seventh part of the money: how 
 much money does the purse contain ? 
 
 8. A shepherd being asked how many sheep he had in his 
 flock, said " if I had as many more, half as many more, and 7 
 sheep and a half, I should then have 500": how many sheep 
 had he ? 
 
 9. A is 58 years older than B, and A's age is as much above 
 60 as B's age is below 50: find their ages. 
 
 10. What number is that whose double being added to 24, 
 the sum is as much above 80 as the number itself is below 100? 
 
 11. What number is that from which if 5 be subtracted, 
 two thirds of the remainder is 40 ? 
 
 13. A and B together can do a piece of work in 8 days, A 
 and in 9 days, B and C in 10 days: in how many days can 
 each man do the work alone ? in how many days can they do 
 it all working together ? 
 
76 SIMPLE EQUATIONS. tm.PB. 
 
 * GENEEAL FORMS. 
 
 Note 2. Erery simple equation with one unknown element 
 may be reduced to the form dx + h = a'x-{-J)', whose solution 
 giyes x={b' — b)/{a — a'), a value that may be positive, nega- 
 tive, zero, infinite, or indeterminate, according to the relations 
 between the elements a, a', t, h'; and there are three cases : 
 (a) a^a'; then x has a single value, positive, negative, or 
 
 zero, that satisfies the equation. 
 {b) a = a', b^b'; then x= ix>, wherein oo, read infinity, de- 
 notes a number larger than can be named. 
 This result may be interpreted by saying that if a and a', or 
 either of them, take gradually changing values, and if a be 
 not equal to a' but approach nearer and nearer to a", then x 
 grows larger and larger without bounds. 
 E.g., if A, A' travel along the same road in the same direction 
 at a, a' miles an hour, and if A be now b miles and 
 A', b' miles from the same starting point; then the 
 quotient (b' — b)/{a — a') is the time that will elapse 
 before they come together. 
 If the hourly gain, a — a', be small, that time is long; 
 if there be no gain, i.e., if « = a', they will never be together, 
 and there is no finite value of x that satisfies the equa- 
 tion, 
 (c) a=a',b = b'; then a; =0/0, and the equation is satisfied 
 by any number whatever. 
 
 These cases may be further illustrated by this question : 
 
 Two men A, A' have 6, b' dollars and save a, a' dollars a 
 
 year: in how many years will they have the same assets? The 
 
 interpretation of the principles in terms of the problem is this: 
 
 If b'>b, the time sought is in the future if a>a', but in the 
 
 past if a<a', 
 and if V < b, these results are reversed; 
 if 5=5t5', a' = a, the answer is never; 
 
 if b=b', the present is the time sought if a=fea', but if a=a', 
 the two men have always the same assets. 
 
1.S11 ONE UNKNOWN ELEMENT. 77 
 
 QUESTIONS. 
 
 1. If the equation a; — 20=— 3a; he represented by the 
 general formula ax + b = a'x + i', for what number does 
 each of the letters a, h, a', I', stand ? 
 
 2. In the fraction {b' — b)/{a—a') what is the sign of the 
 denominator if a>a'? ita<a'? 
 
 So, if b'>b, what is the sign of the numerator? if 5'<5? 
 
 3. What is known about the value ofxitb'>b and a>a' ? 
 So,itb'>b-a,nia<a'? it b' <b ani a> a^ ? itb' = b? 
 
 4. What is the value of a fraction whose numerator is zero ? 
 Show that this value multiplied by the denominator gives 
 
 the numerator, and that no other value will give it. 
 
 5. Eeduce the fractions 6/3, 6/.3, 6/.03, 6/. 003 •• -to 
 whole numbers : what change is going on in the series of de- 
 nominators and what in the quotients ? if the denominator be 
 very small, what is the quotient ? if the denominator be ? 
 
 6. How is an example in division proved? 
 
 Prove that: 0/0 = 2; 0/0 = 10; 0/0 = 5000; 0/0= - .12. 
 What is the value of (&' — b)/{a — a') when b' — b, a=a'? 
 What is meant by an indeterminate expression ? 
 
 7. In the case of the two travelers A, A'; if a = a', b = b', 
 are they now together ? how long have they been together ? 
 how long will they remain together ? 
 
 8. If 6'>&, a<a', is the time of meeting past or future ? 
 i£b'>b, a>a'? iib'<b, a<a'? if b'<b, a>a'? 
 
 9. What is the meaning of the problem if b, V be of 
 opposite signs ? both negative? if a, a' be of opposite signs .? 
 both negative ? 
 
 10. A gives a house worth b dollars and land worth a dollars 
 an acre in exchange for B's house worth V dollars and as 
 many acres of land worth a' dollars an acre: how large is 
 each estate ? 
 
 Discuss the problem for the difEerent relations between 
 a, a', b, V considered before: which of the results interpreted 
 in the last question on page 76 has no meaning here ? 
 
78 SIMPLE EQUATIONS. [III.Pb. 
 
 §2. TWO UNKNOWN ELEMENTS. 
 Equations that involve the same unknown elements^ and 
 are satisfied by the same values of them, are simultaneous 
 equations; and those values are simultaneous values. 
 E.g., if the equations 2a; + 5^ = 19, Qx — Zy — Z, [x, y unknown] 
 be simultaneous, 3, 3 are a pair of roots, 
 •.•2-2 + 5-3 = 19, 6-2-3.3 = 3. 
 
 Elimination is that process by which an unknown element 
 is removed from a pair of equations. 
 
 PkOB. 2. To ELIMISTATB XS UNKNOWN ELEMENT FKOM A 
 PAIE OF SIMPLE EQUATIONS. 
 
 BY ADDITION AND SUBTKACTION. 
 
 Find some number, as small as may be, that exactly contains 
 both the coefficients of the element to be eliminated; 
 
 divide this number, in turn, by these coefficients, and multiply 
 the two equations through by the quotients; [ax. 4. 
 
 subtract one equation from the other, member from member. 
 
 E.g., to eliminate x from the pair of equations 
 &x + 1y-9,f>, 2a; + 3«/ = 33: 
 
 then ■.• 6 contains 6 once and 2 three times, 
 
 .•.&x + ly = 85, Qx + %y = 99, [mult, by 1, 3. 
 
 ' .*. 2y = 14. {subtract. 
 
 BT COHPABISON. 
 
 Solve both equations for the element to be eliminated; [pr. 1. 
 put the two values thus found equal to each other. [ax. 1. 
 
 E.g., to eliminate x from the same pair of equations: 
 then a; = |(85 - ly) = i(33 - Zy), ' [sol. both eq. for x. 
 
 BY SUBSTITUTION. 
 
 Solve either equation for the element to be eliminated; [pr. 1. 
 in the other equation, replace this element by the value so found. 
 E.g., to eliminate x from the same pair of equations : 
 then •.• a;=^(33 - Zy), [sol. 2d eq. for x. 
 
 .: 99 - 92/ + 7«/ = 85. [repl. x in 1st eq> 
 
2-52] TWO UNKNOWN ELEMENTS. 79 
 
 QUESTION'S. 
 
 1. Define elimination; what is the derivation of the word ? 
 
 2. In the equation 2x + 5y = 19 replace a; by —^, yby4: 
 is the equation true for these values ? 
 
 Is the equation 6x — 3y = 3 true for the same values? 
 So, in the second equation replace xhj 3, y hj 5: do these 
 values satisfy the first equation? 
 
 3. Assume any value at random for y m2x + 5y = 19: can 
 a satisfactory value be found for x in that equation ? in 
 6a; — 3^ = 3? with the same value of y, in both equations at 
 the same time ? what is needful to a correct solution ? 
 
 4. What two axioms are applied in elimination by addition 
 and subtraction ? 
 
 5. Multiply the equation 6a; + 7«/ = 85 by 3, 2a; + 3^ = 33 
 by 7; then, subtracting, what letter is eliminated ? 
 
 6. In eliminating x by comparison, how is it known that the 
 two expressions for x, found from the separate equations, are 
 equal ? 
 
 7. To get definite values for two unknown elements, how 
 many independent equations must be used ? 
 
 8. By addition and subtraction, eliminate x from the pair of 
 simultaneous equations 5a; + 6y = 29, 3a; + 2^ = 11. 
 
 9. So, from 2x + 5y = 23, 7a; + 2y = 34. 
 
 10. Eliminate y from 8a; + 13jr = 79, 7x-h2y-4:l. 
 
 11. By comparison, eliminate x from the pair of simul- 
 taneous equations 4a; — 3^=— 10, 7a; + 8y = 62. 
 
 12. So, from Ja; + 4^ = 18, 5x-3y = 17. 
 
 13. Eliminate y from 2x + iy = 20, '7x + 3y = 37. 
 
 14. So, from 4/a; + 7/«/ = If, d/x + 5/y = lh using 1/a;, 
 1/y as the two unknown elements. 
 
 15. By substitution eliminate x from the pair of simul- 
 taneous equations 3a; — 2^ = 1, 5x+3y = di. 
 
 16. So, from 6x + Qy = 15, 8x-16y=ll. 
 
 1 7. Eliminate y from ^a; - f y = - 4, 3a; + 4y = 43. 
 
80 SIMPLE EQUATIONS. [HI, Pes. 
 
 THE SOLUTION OF SIMULTAKEOUS SIMPLE EQUATIONS. 
 
 PrOB. 3. To SOLVE A PAIR OF SIMPLE EQUATIONS, IF ONE 
 HAS TWO UNKNOWN ELEMENTS AND THE OTHER BUT ONE. 
 
 Solve the equation that has hut one unknown element; [pr, 1. 
 replace this element iy its value in the other equation, and 
 
 solve for the other unknown element. [th. 4 cr. 2. 
 
 E.g., to find X, y from the pair of simultaneous equations 
 
 6a; + 72/ = 85, 4a;=24: 
 thena;=6, 36 + 7y = 85, 72/ = 49, y = '1. 
 
 PrOB. 4. To SOLVE A PAIR OF SIMPLE EQUATIONS, IF BOTH 
 HAVE THE SAME TWO UNKNOWN ELEMENTS. 
 
 Comline the two equations so as to eliminate one unknown 
 element, thus forming an equation involving the other; 
 solve this equation for its unknown element; 
 replace this element by its value in either of the given equations; 
 solve the equation so found for the other unknown element. 
 Check. Reploice the two unknown elements by their values in 
 that one of the original equations which was not used 
 in finding the value of the second element. 
 E.g., to find X, y from the pair of simultaneous equations 
 
 Qx + ly = 8b, 2a; + 32/ = 33: 
 then •.• i(85-6a;) = *(33-2a;), [elim. y. 
 
 .-.255 -18a; = 231 -14a;, [mult, by 21k 
 
 .". — 4a;= —24, a;=6; 
 .-.36 + 7^ = 85, y = l. 
 
 DEPENDENT EQUATIONS. 
 
 Note 1. If one of the two equations may be derived from 
 
 the other, there is no single solution, but an infinite number 
 
 of solutions. The -equation is then indeterminate. 
 
 E.g., the equations 2a; + 3«/ = 13, 6a; + 9^ = 39 are but one 
 
 equation in two forms, and any value may be given to 
 
 either of the unknown elements, and the corresponding 
 
 value of the other computed. 
 
3, 4, §8] TWO UNKNOWN ELEMENTS. 81 
 
 QUESTIONS. 
 
 Solve the pairs of equations below, and check the work: 
 
 1. 5x~3y = 15, 2y-10. 2. 3{x + 2y) = 30, |a;=3. 
 
 3. 8x + 3y = U, 5y = l0. 4. 3x-8y = 7, 3iz=5. 
 
 5. 2x-4:-y = x-l, -3y--9. 
 
 6. 2i + y-ix=iy, ix = 3i. 
 
 7. x+y = 9, x~y = l. 8. lbx + 2y-V^, Qx-iy=:b. 
 9. bx + 3y-S, '7x-3y-i. 10. 3x + y = 16, 3y + a; = 8, 
 
 11. 3y = 5x, 16y = 2'7x-l. 12. 8z-5y, 13a: = 8y + l. 
 
 13. x-^y, x-iy = %. 14. lla;-3^ = 0, x-y-—16. 
 
 15. 2x+y = 0, iy-3x = 8. 16. x-y = f, a; + l = |. 
 
 5 
 6' 
 
 _1 
 
 '' 12" 
 
 21. 210a; + 42y + 93 = 0, 22x + Uy + 7 = 0. 
 32. %y-ix + 2i = 0, ^y + ^x+ll = 0. 
 
 23. i{i^-iyH)=ii^-y)> iiiy-\^+i)=i{^+y)- 
 
 91 £iX-q x-3y 5y-x _1 
 ^^•'^^-'^' "~6~+~9"~-2' 
 
 35. TV(80 + 3a;) = 18^-|(4a; + 3«/-8), 
 
 l%+i(6a;-35) = 55 + 10a;. 
 
 36. ^^" = 1, ^:^"=3. 27.4.+-^-, = 2«, ^1=1. 
 
 5x-3y 6i; + 10 a + b a — b iab 
 
 28.1-m=^, ^J^^=3. 
 x—y x—y 33 
 
 2(a; + l)^3(y + l)~ ' x + 1 3{y + l)~'- . 
 30. "Write down any simple equation at will, and then make 
 dependent equations from it by different processes. 
 
 17. 
 
 8_5 
 X y ' 
 
 1 
 ~6' 
 
 7 3 
 X y 
 
 19. 
 
 X y 
 
 7 
 ^13' 
 
 2 3 
 
 X y 
 
 18. 
 
 3 
 
 X 
 
 + ^ = 
 
 lA, 
 
 5 
 
 X 
 
 y 
 
 29 
 ' 12' 
 
 30. 
 
 5 
 
 X 
 
 3 _ 
 
 y~ 
 
 1 
 
 6' 
 
 3 
 
 X 
 
 1 
 y' 
 
 1 
 "30" 
 
82 SIMPLE EQUATIONS. flll.PR. 
 
 SPECIAL PROBLEMS, 
 
 Note 3. In solving special problemB it may be convenient 
 
 to express different unknown elements by different symbols. 
 
 E.g., a vintner at one time sells 30 dozen of port wine and 30 
 dozen of sherry, and for the whole receives |600; and 
 at another time he sells 30 dozen of port and 35 dozen 
 of sherry, at the same price as before, and for the whole 
 receives $700 : what are the prices ? 
 
 put X for the price of a dozen of port, and y for that of a 
 dozen of sherryj 
 
 then •.• 30a; + 30?/ = $600, 30a; + 35?/ = $700, 
 .'.a; =$15, 2/ = $10. 
 
 So, if a certain rectangular bowling-green were 5 yards longer 
 and 4 yards broader, it would contain 113 yards more; 
 but if 4 yards longer and 5 yards broader, it would 
 contain 116 yards more : what are its length and breadth ? 
 
 put X, y for the length and breadth; 
 
 then-.-(a; + 5)-(?/+4)=::a;z/ + 113, (a; + 4)-(y + 5) = a;^ + 116, 
 .•. a; =13 yds., «/ = 9 yds. 
 
 So, if the number of men engaged upon a certain piece of 
 work be made 5 greater, the work can be done in 4 
 days; if 5 less, in 13 days: how many men are at the 
 work, and in how many days can they do it ? 
 
 put X for the number of men and y for the number of days; 
 
 then one man could do the work in xy days, 
 
 and •.•4(a; + 5) = a;y, 12(x — b) = xy, 
 
 .-. 4(a! + 5) = 13(a;-5), a;=10, y = Q. 
 
 So, a certain two-figure number is 6 greater than 6 times the 
 sum of its digits, and reversing the order of the digits 
 makes the number less by 3 times its first figure; find 
 the number: 
 
 put a; for the tens' figure and y for the units' figure; 
 
 then-.-10a; + y = 6(a;4-«/) + 6, lOy + x = 10x + y -^x, 
 .'. x=2, y=6, and the number is 96. 
 
*.S8] TWO UNKNOWN ELEMENTS. 83 
 
 QUESTIOlfS. 
 
 1. Find two numbers such that their sum is 37, and that, 
 if four times the first be added to three times the other, the 
 sum is 93. 
 
 2. Find two numbers such that twice the first and three 
 times the second together make 189 and, if double the second 
 be taken from five times the first, 7 remains. 
 
 3. A flagstaff is sunk in the ground one-sixth part of its 
 height, the flag occupies 6 feet, and the rest of the staff is 
 three-quarters of its whole length : what is the length ? 
 
 4. The diameter of a five franc piece is 37 millimeters and 
 that of a two-franc piece 27 millimeters; thirty pieces laid in 
 contact in a straight line measure one meter: how many of 
 each kind are there ? 
 
 5. A certain number consisting of two figures is equal to 
 four times the sum of its digits, and if 18 be added to it the 
 order of the digits is reversed: what is the number ? 
 
 6. If the tail of a fish weigh 9 lbs., his head as much as his 
 tail and half his body, and his body as much as his head and 
 tail, what is the weight of the whole fish ? 
 
 7. There are two pipes one of which will fill a cistern in an 
 hour and a half, the other in two hours and a quarter: in 
 what time will both fill it ? 
 
 8. Divide 90 into four parts such that if the first be in- 
 creased by 2, the second diminished by 2, the third multiplied 
 by 2, and the fourth divided by 2, the sum, difference, prod- 
 uct, and quotient so found shall all be equal. 
 
 9. A and B engage in play; in the first game 4- wins as 
 much as he had and $4 more and finds he has twice as much 
 as B; in the second game B wins half as much as he had at 
 first and $1 more, when he has three times as much as A: 
 what sum had each at first ? 
 
 10. What fraction is that which becomes equal to | when 
 its numerator is increased by 6, and equal to i when its de- 
 nominator is diminished by 3 ? 
 
84 SIMPLE EQUATIONS. [HI, Pa. 
 
 DISCUSSION OF A PROBLEM. 
 
 Note 3. To discuss a problem whose answer is numerical 
 is to try whether all the conditions of the problem are satisfied 
 by all or any of the numbers that are found to satisfy the. 
 equations into which the problem was translated ; and, if not, 
 to observe what other conditions the unknown elements must 
 satisfy besides those taken account of in putting the problem 
 into equation. 
 
 To discuss a problem whose answer is literal is to observe 
 between what limiting numerical values of the known ele- 
 ments the problem is possible; whether any singularities or 
 remarkable circumstances occur within these limits; and what 
 changes in the statement of the problem would make it possi- 
 ble for the other values of the known elements. 
 E.g., in a certain two-digit number the first digit is half the 
 other, and if 27 be added to the number, the order of 
 the digits is reversed : what is the number ? 
 put X for first digit, y for second digit; 
 then-.-3a;=4/, 10a;+^-|-27 = 10«/ + a;, 
 
 .•.a; = 3, y = 6, the number is 36; and 36 + 27=63. 
 Were this the statement: in a certain two-di^t number, the 
 first digit is half the other, and if 24 be added to the number, 
 the order of the digits is reversed; 
 then-.-2a; = «/, I0x+y + 24, = 10y + x, 
 
 .: x = 2%, «/ = 5J, and the number is impossible. 
 The statement of the problem puts a limitation upon x, y 
 not expressed by the equation: they must be integers. 
 
 And were this the statement: in a certain two-digit number 
 the first digit is half the other, and if a be added to the num- 
 ber, the order of the digits is reversed; 
 then 2x-y, \^x^y-\-a — \Qy + x, x = a/^, y-2a/%. 
 
 Here the special condition is imposed that a shall be a mul- 
 tiple of 9 not greater than 36 nor less than —36; 
 i.e., nia 36, 27, 18, 9, 0, "9, "18, -27, "36, 
 
 and the number is 48, 36, 24, 12, 0, -12, -24, -36, -48. 
 
4,S2] TWO UNKNOWN ELEMENTS. 85 
 
 QUESTIONS. 
 
 1. In a company of a persons each man gave m dollars to 
 the poor, each woman n dollars; the whole sum was ka 
 dollars: how many men were there ? how many women ? 
 
 Show that, if m>n, then in>k>n; and that the example 
 is possible only when (m—k)a, {k — n)a are multiples of m — n. 
 
 2. A is « years old and B b years: when will A be twice as 
 old as B ? AVhat relation between a and b puts the date sought 
 in the future ? what, in the present? w'hat, in the past ? ^ 
 
 3. A laborer receives a dollars a day when he works, and 
 forfeits b dollars a day when idle; at the end of m days he 
 receives k dollars: how many days does he work, and how 
 many is he idle? 
 
 What relation exists between «, b, k, m, if his forfeits just 
 cancel his earnings ? if his forfeits exceed his earnings ? Give 
 numerical illustrations. 
 
 4. A father is now a times as old as his son; ^ years hence 
 he will be b times as old: what are their ages now ? 
 
 Give numerical values to a, b, k, and interpret the results. 
 Show that: *>0,ifffl>Zi; /fc = 0,ifa = &; k<Q, \ia<b. 
 
 5. The sum of two numbers is a, and the difEerence of their 
 squares is P; what are the numbers ? 
 
 Interpret the results : if A*>a'; if P = a'; ifF<a'. 
 
 6. The difference of two numbers is a, and the difference 
 of their squares is k^: what are the numbers ? 
 
 Interpret the results: if F>a^; if ^'=a^; \t¥<a^. 
 
 7. If to the numerator of a certain simple fraction a be 
 added, the result is c/d, and if to the denominator a' be 
 added, the result is c'/d': what is the original fraction ? 
 
 8. In a certain two-digit number the second digit is a times 
 the first, and if b be added to the number, the digits are re- 
 versed : show that a may not exceed 9, be less than 2, or be 
 negative; and show when a may be fractional. 
 
 Show that J is a multiple of 9 or of a — I; and show what 
 bounds b lies between for different values of a. 
 
86 • SIMPLE EQUATIONS. [III. Ph. 
 
 MOKE CajSTDITIONS THAIT UNKNOWN ELEMENTS. 
 
 Note 4. It may happen that the prohlem gives more condi- 
 tions, and so more equations, than unknown elements; such 
 problems can be solved if the conditions be not inconsistent. 
 E.g., to find X, y from the set of three equations 
 3a; + 72/ = 17, 5a;-22/ = l, 82; + «/ = 10: 
 
 take the first two equations and solve; 
 
 then •.• x=\, y = 2, in these two equations, 
 
 and •.• these roots satisfy the third equation, 
 
 .*. this set of equations is possible, and the roots are 1, 3. 
 But not possible is the set of equations 
 
 EEWEK CONDITIONS THAN UNKNOWN ELEMENTS. 
 
 Note 5. It may happen that the problem gives fewer con- 
 ditions, and so fewer equations, than unknown elements; such 
 problems are indeterminate, and the set of equations may 
 have many sets of roots. 
 
 E.g., to find X, y from the single equation 2x + Zy = l%: 
 put 2/= •••-4, -3, -3,-1, 0, n, +2, +3, +4, .■•, 
 then a;=--.+13, +10^,+9, +7i, +6, +4|, +3, +4, 0, • • -, 
 i.e., if to y be given a series of values increasing by 1, there 
 
 results a series of values for x decreasing by 1\; 
 orputa;=----4, -3, -3, -1, 0, +1, +3, +3, +4, • • •, 
 then y=--- +61, +6, +5*, +4f. +4, +3^, +3f, +3, nj,- • •, 
 i.e., if to X be given a series of values increasing by 1, there 
 results a series of values for y decreasing by f . 
 If either of the unknown elements take any value whatever, 
 the corresponding value of the others may be found. 
 E.g., if a;=4i, then y = |; ovity = ^, x=—%. 
 
 If the condition be imposed that the roots shall all be in- 
 tegers, or all positive integers, it may happen that the equa- 
 tions have very few such roots, or even none at all. 
 E.g., 3, 3, is the only pair of positive integer roots in the ex- 
 ample above. 
 
4. §2] TWO UNKNOWN ELEMENTS. 87 
 
 QUESTIONS. 
 
 1. Giyen the three simple equations 
 
 2a; + 3^ = 18, 3a; — 3^ = 1, lx — '^y = b: is it certain, 
 before solving, that values of x and y can be found that will 
 satisfy all the given equations ? 
 
 Solve the second and third equations and see whether the 
 values so found satisfy the first. 
 
 2. Soj for the three equations 
 
 x-^y = 10, 4a; + 10«/ = 14, -2x + 3y = 9. 
 
 3. Find two numbers whose sum is 60, whose difference is 
 34, and one of which is 3 times the other. 
 
 If the results obtained do not satisfy all three conditions, 
 show what change in each condition will make it consistent 
 with the other two. 
 
 4. Solve the equations 6x — 8y = S, 15x = 7^ + 20y, or ex- 
 plain what is the diflQculty with them. 
 
 5. In the example of note 5, why does x decrease when y 
 increases and increase when y decreases ? 
 
 6. In the equation ix — 5y = 1, if increasing values be given 
 to X, will the corresponding values of y increase or decrease ? 
 
 If the values of. x increase by 1, how do the values of y 
 change? if the values of y decrease by 1, how do the values of 
 X change ? 
 
 By what integers may a; increase or decrease so that y also 
 shall change by integers only ? 
 
 What are all the pairs of integer roots smaller than 20 ? 
 
 7. Show that the equation xy = 24: may have an infinite 
 number of pairs of roots, and that the value of one root 
 grows smaller as the other grows larger. 
 
 Show how the relations of x, y differ in this example from 
 their relations in ex. 6. 
 
 Discuss the equation xy = 0, 
 
 8. How many pairs of roots has the equation x = ay? 
 "What relation have the values of x and of y when a is posi- 
 tive ? when a is negative ? 
 
88 SIMPLE EQUATIONS. [Ill, Ph. 
 
 §3. THREE OR MORE UNKNOWN ELEMENTS. 
 
 PeOB. 5. To SOLVE A SET OF n IKDEPENDEN^T SIMPLE 
 EQUATIONS THAT IKVOLTE THE SAME 11 UlfKNOWK ELEMEKTS. 
 
 Combine the n equations, two and two, in n — 1 ways, so that 
 each equation is used at least once, and so as to elimi- 
 nate the same unknown element at each operation; 
 
 thereby form n — \ equations involving the same n — 1 un- 
 known elements; 
 
 so, combine these n—1 equations, and thereby form n-2 
 equations involving the same n — 2 unhnoton elements; 
 
 and so on till there results one equation, involving but one un- 
 known element; 
 
 solve this equation, and replace the tinknoivn elemetit by its 
 value in one of the tioo equations involving two un- 
 known elements; 
 
 solve this equation for the second unknown element, and re- 
 place these two elements by their values in one of the 
 three equations involving three unknown elements; 
 
 and so on till all the roots are found. 
 
 E.g., to find X, y, z from the set of equations 
 
 a; + 2«/ + 32 = 14, 3a; + 2t/ + 2=10, 6x + 92/ + 132 = 63: 
 
 then 
 
 6a; + 12y + 182 = 84 [4] 
 
 [mult, first eq. by 6. 
 
 6a; + 4y + 22; = 20 [5] 
 
 [mult. sec. eq. by 2. 
 
 6a; + 9«/ + 132! = 63 [6] 
 
 
 8^+162 = 64 [7] 
 
 [sub. eq. 5 from eq. 4. 
 
 5y + ll« = 43 [8] 
 
 [sub. eq. 5 from eq. 6. 
 
 40^ + 802 = 320 [9] 
 
 [mult. eq. 7 by 5. 
 
 40,^ + 882=344 [10] 
 
 [mult. eq. 8 by 8. 
 
 82 = 24 and 2 = 3 
 
 [sub. eq. 9 from eq. 10. 
 
 8«/ = 64-162=16 and «/ = 2. 
 
 
 «=14-2«/-32=l. 
 
 
5.8 3] THREE OR MORE UNKNOWN ELEMENTS. 89 
 
 QUESTIONS. 
 
 1. How many independent equations make it possible to find 
 the value of four unknown elements ? of five ? of ten ? of w ? 
 
 Solve the sets of equations: 
 
 2. 3a;-4«/ + 52 = 4, 8x-t/-z=&, Ix-by-'dz- -I. 
 
 3. x-2y-5z = 20, dx~5t/-3z = 22, -8x + lly + 9z= -57. 
 1_1 1_11 2_3 4_71 3 4_1^_20 
 
 X y z~ &' x y •dz~ IS' x y 'lz~ 21 
 5- Ky-2)=i(y~^) = 52 — 4a;, y + z = 2x + l. 
 
 6. x + 2y + Sz + iu = 20, x + 2y + dz-Au = 12, 
 x + 2y-3z + in = 8, x-2y + 3z + iti = 8. 
 
 7. 2x+ y+ Z--16, 8. cy + bz=a, 9. x+.y = 14:, 
 
 x + 2y+ z = 9, az + cx = b, x+z = 16, 
 
 x+ y + 2z = 3. bx + ay — c. y-\-^ = 18. 
 
 X 
 
 y-,_o. 11 l^lA=oo.. 12.^^^ 
 
 10. :^— ^ = 2-2, 11. - + - + -=29> 12. -+t = l, 
 
 w X y z a 
 
 -^^iW + S, 1--- = 9, - + - = 1, 
 
 ■ z X y z a c 
 
 x + y=:iw, i+i_i=_2. ^+? = 1, 
 
 10—1 = z. X y z ' be' 
 
 13. 2(3-4-1) -3(y-l) +2-2 = 2, 14. x+y + z=9, 
 2(a; + l)+40y + l)-5(z-l) = 3, a; + 2.v + 42=l5, 
 
 3(2^ + 2)-2(«/-l) + 3{2 + l) = 20. a; + 3^ + 9« = 23. 
 
 15. (a; + l)(52/-3) = (7a; + l)(2?y-3), 17. ^+^ _? = ,-, 
 {ix-l){z + l) = {x + l){2z-l), I I I 
 
 {y + 3){z + 2) = {3y-Q){3z-l). x~y^~z = '' 
 
 16 «-i=^-^=I + l-^=4. -«+^+?=^. 
 
 X y z X y 2z x y z 
 
 18. x + ay + a'z + aSi + a^^Q, x+by+b^'z+h^u+b" = 0, 
 . x + cy-\-ch+(?u-\-(^=^(^, x + dy + d''z + (Pn + d^ = 0. 
 
 19. u + v + w + x + y = 10, v + w + x + y + z =15, 
 w + x + y + z + it = 13, x + y + z + u+v = 11, 
 y^z-\-u + v + w = 14:, z + u + v + w + x = 12. 
 
90 SIMPLE EQUATIONS. [ni.PB. 
 
 NOT ALL THE UNKNOWN ELEMENTS INVOLVED IN EVERY 
 
 EQUATION. 
 
 Note 1. An unknown element that does not appear in any 
 equation may be considered as already eliminated from it, 
 and the work is shortened by so much; those unknown ele- 
 ments that are in the fewest equations may be eliminated first. 
 E.g., to find X, y, z, t, u from the set of equations 
 
 9x-2z+u =41, (1) ly-bz-t =13, (2) 
 4v-3a; + 3M= 5, (3) 3«/-4m + 3^= 7, (4) 
 7z-5m = 11: (5) 
 of these equations, x appears in two, y in three, z in three, u 
 
 in four, t in two; 
 equations 1, 3 may be combined to eliminate x, and equations 
 3, 4 to eliminate t, and there result two new equations, 
 involving y, z, u; 
 these two equations may be combined to eliminate y, and 
 
 there results one equation, involving z, u; 
 this last equation may be combined with equation 5 to elimi- 
 nate either z or u at pleasure, 
 
 PARTICULAR ARTIFICES. 
 
 N"oTE 3. The equations may have a certain symmetry as 
 to the unknown elements, or functions of them, that permits 
 shorter processes than those of the general rule; sometimes 
 the sum of such unknown elements, or of the functions, may 
 be got first. 
 E.g., to find X, y, z from the set of equations 
 
 1 1 _ 4 1,1 _11 i,i_i. 
 
 x'^y~ 15' y z ~ &0' z x ~ A' 
 
 232 71117 r J,. :,■ r. c. 
 
 then •.• - H }--=—-, - + -+-=—, [add, div. by 2. 
 
 x y z 10 X y z 20 *- •' 
 
 .i_^_ii_]_ i__l_i_i_ l-J.A-Ji. 
 
 ■'■a;~20 ~60~6'y~20 4 ~ 10' z ~ 20 15~13' 
 .•.x=6, «/ = 10, z = 12. 
 
6.88] THHEE OR MORE UNKNOWN ELEMENTS. 91 
 
 QUESTIONS. 
 
 Solve the systems of equations : 
 
 1. 3z-ii/ + 3z + 3v-62i = ll, 2. 3z + 8u = 33, 
 3x-5y + 2z-4:u =11, 7x-2z + 3u =17, 
 
 5z + ^u + 2v-2x =3, 4:i/-2z + v =11, 
 
 10i/-3z + 3u-2v = 2, 4y-3u + 2v = 9, 
 
 6u-3v + 4^-2i/ =6. by-3x-2u = 8. 
 
 3. x + 2y-3z =-1, 4. x+y + z = 0, 
 
 4^-^y-z =8, {l> + c)x+(c + a)y+{a + b)z = 0, 
 
 3z+8y + 2z = —5. bcx + cay + abz = 1. 
 
 5. 5x-2{y+z + v) = -l, -I2y + 3{z + v + x) = 3, 
 4:Z-3{v+x+y)=2, 8v-{x+y+z) =-2. 
 
 6. Three cities. A, B, C, are at the corners of a triangle; 
 from A through B to C is 118 miles; from B through to 
 
 A, 74 miles; from through A to B, 93 miles: how far apart 
 are the three cities ? 
 
 7. The sum of three numbers is 70; the second divided 
 by the first gives 2 for the quotient and 1 for the remainder, 
 and the third divided by the second gives 3 for both quotient 
 and remainder : find the numbers. 
 
 8. A, B, C are three towns forming a triangle; a man has 
 to walk from one to the next, ride thence to the next, and 
 drive thence to his starting point; he can walk, ride, and 
 drive a mile in a, i, c minutes respectively; if he start from 
 
 B, he takes a + c — i hours; if from C, J + fl — c hours; if from 
 K,c+b — a hours: find the length of the circuit. 
 
 9. A number is expressed by three figures, whose sum is 
 19; reversing the order of the first two figures diminishes 
 the number by 180, and interchanging the last two increases 
 it by 18 : what is the number ? 
 
 10. A's money in 9 years at 6 ^ will produce as much in- 
 terest as B's and C's together in 4 yrs. 6 mos. at 4 ^; B's in 
 8 yrs. at 5 ^ as much as A's and C's in 3 yrs. 4 mos. at 6 ^; 
 C's in 7 yrs. at 3 j^ $42 more than A's and B's in 3 yrs. at 4 ^: 
 how much money has each man ? 
 
92 SIMPLE EQUATIONS. Lin,3?B. 
 
 *GENERAL FOEMS. 
 
 Note 6. The two equations ax + by — c, a'x + l'y = c', 
 are the type-forms of, every pair of simple equations that in- 
 volve the same two unknown elements; their solution gives 
 X = {cV~c'b)/{ab'-a'h), y = {ac'-a'c)/{ab'-a'b). 
 
 The values of x, y for a particular pair of equations depend 
 on- the values of a, I, c, a', b', c', and an examination of the 
 possible values and relations of these known elements will 
 determine the possible roots of the pair of equations. There 
 are three general cases : 
 
 (a) ab' i^a'b; then x, y have single values, positive, negative, 
 or zero, that satisfy both the equations. 
 
 {b) ab' = a'b, cb'i=c'b; then ac'^^a'c, x—cc, y = '». 
 Here a/a' — b/b'i=c/o'; the equations are inconsistent, 
 and they can be satisfied by no finite values of x, y. 
 
 The infinite values may be interpreted by saying that, if 
 a, a', b, b' , any of them, take changing valiies, and if ab' =fa'b, 
 but aJ' approach nearer and nearer to a'J, then x, y grow 
 larger and larger without bounds. 
 
 (c) aV — a'b, cb' — c'b; then ac' = a'c, x = 0/0,^ y = 0/0. 
 
 Here a/a' = b/b' = c/c'; the equations difEer by a factor 
 only, and the values sought are indeterminate. : ' : 
 
 The general forms of simple equations involving three un- 
 known elements are ax + by + cz = d, a'x + b'y + c'z = d', 
 a"x + b"y + c"z = d" , whose solution gives 
 
 _ d{b'c"-b"c')^d'{b"c-bc") +d"{bc'-b'd) 
 ^~ a(b'c"-b"c')+a'{b"c-bc") +a"{bo'-b'c)' 
 
 _ d{a'c"-a"c') + d'{a'^c-ac")+d"{ac'-a'c) 
 y~ b{a'c"-a"c') + b'{a"c-ac") + b"{ac'-a'c) ' 
 
 _ d{a'b"-a"b')+d'(a"b-ab") + d"{ab'-a'b) 
 ^ ~ c{a'b"-a"b') +c'{a"b-ab") +c"{ab'-a'by 
 
 and all of these denominators have the same value; but the 
 sign of the second is opposite to that of the first and third. 
 
5. §3] THREE OR MORE UNKNOWN ELEMENTS. 93 
 
 Various relations among the coefficients may be considered: 
 If d, d', d" be all zero, the values of x, y, z are zero, unless 
 the denominator is also zero, and then these values are inde- 
 terminate, and the given equations are not all independent. 
 
 If d, d', d" be not all zero, but the denominator be zero, 
 the equations are inconsistent. 
 
 For if the first equation be multiplied by h'c"—i"c', the 
 second by Vc—tc", the third by bc'—b'c, and 
 the results be added, 
 then the coefficients of y and z vanish identically, and that of 
 a; is a{b'c"-b"c')+u'{b"c-lc")+a"{bc'-b'c), i.e., 
 zero, while the second member is not zero. 
 
 QUESTIONS. 
 
 1. In the pair of equations given in note 6, what relation 
 between the products a'b, aV makes the denominator of the 
 value of X positive ? negative ? zero ? 
 
 So, what relation between the products ac', a'c makes the 
 numerator positive ? negative ? zero ? 
 
 If cb'yc'b and ab'<a'b, is a; positive or negative ? 
 
 3. If the numerator of a fraction be 0, and the denominator 
 not 0, what is the value of the fraction ? if the denominator 
 be and the numerator not 0? if both be 0? 
 
 3. If ab' = a'b and cb' + c'b, find the value of the fraction 
 b/V, and so show that ac'^a'c, and that x, y are both infinite. 
 
 4. If ab' = a'b and cb' = c'b, show that ac' = a'c. 
 
 What then is the numerator of the value of «? what the 
 denominator ? of the value of y ? 
 
 Has the fraction 0/0 any definite value ? 
 
 5. Given the three simple equations 
 
 ax + by = c, a'x + b'y = c', a"x + b"y = c", 
 solve the first two, substitute the values of x, y, so found, 
 in the third, and show that 
 aQ)'c"-b"c') + a'{b"c-bc")^a"{bc'-b'c)=Q, 
 is the equation of condition for the consistency of the 
 three given equations. 
 
94 SIMPLE EQUATIONS. [m. 
 
 §4. QUESTIONS FOE REVIEW. 
 Define and illustrate : 
 
 1. An equation; an identity; an axiom. 
 
 2. Known elements; unknown elements; the solution of an 
 equation ; the roots of an equation. 
 
 3. A simple equation involving one unknown element; two 
 unknown elements; three unknown elements. 
 
 4. A pair of simultaneous equations; elimination. 
 State the axiom : 
 
 5. Of equality; of addition; of subtraction; of multiplica- 
 tion; of division; of involution; of evolution. 
 
 Give the general rule, with reasons and illustrations, for : 
 
 6. Solving a simple equation with one unknown element. 
 
 7. Eliminating an unknown element from a pair of simul- 
 taneous simple equations by addition and subtraction; by 
 comparison; by substitution. 
 
 8. Solving a pair of simultaneous simple equations. 
 
 9. Solving a set of simple equations involving three or 
 more unknown elements. 
 
 Exhibit the general forms, and explain the special cases, for: 
 
 10. A simple equation involving one unknown element. 
 
 11. A pair of simple equations involving" the same two un- 
 known elements. 
 
 What difiBculty arises : 
 
 12. If one of the two equations be dependent on the other? 
 
 13. If there be more conditions than unknown elements ? 
 
 14. If there be fewer conditions than unknown elements? 
 
 Explain what is meant: 
 
 15. By putting a problem into equation; by solving a 
 problem; by checking the work. 
 
 16. By the discussion of a problem whose answer is numer- 
 ical; of a problem whose answer is literal. 
 
S*J QUESTIONS FOR REVIEW. 95 
 
 17. The sum of the three figures of a number is 9; the first 
 figure is an eighth of the number made up of the last two 
 figures, and the last figure is an eighth of the number made 
 up of the first two figures: find the number. 
 
 18. A boatman rows down the river 42 miles in 3 hours; 
 returning, he finds the current only two thirds as strong, and 
 it takes him 10^ hours: find how fast he can row in still water, 
 and how fast the river ran at first. 
 
 19. At 3^ miles an hour, I can walk from p to Q in a certain 
 time; but at the rate of 3 miles going and 4 miles returning, 
 it takes me five minutes longer: how far is it from p to Q ? 
 
 20. A crew that can pull 9 miles an hour in still water takes 
 twice as long to come up a river as to go down it: find the 
 velocity of the current. 
 
 21. If a rectangle be made 3 feet longer and 3 feet bi'oader, 
 the area is 102 square feet greater; but if it be shortened 5 
 feet, and widened 1 foot, the area is 16 square feet less : find 
 the length and breadth. 
 
 32. If a concert room contained 10 more benches, one per- 
 son less might sit on a bench; if it contained 15 fewer, 2 more 
 persons must sit on a bench: how many benches are there 
 and how many people on a bench? 
 
 23. The perimeter of a rectangular field is 70 rods; if the 
 width were increased one rod and the length diminished two 
 rods, the width would be eight ninths of the length: find the 
 area of the field. 
 
 24. The contents of a vessel is 60 per cent alcohol and the 
 rest water; after drawing out 10 gallons of the mixture and 
 filling up tlie vessel with water, 42f- per cent of the contents 
 is alcohol: find the capacity of the vessel. 
 
 25. A cask contains 18 gallons of wine and 12 of water, and 
 another contains 3 gallons of wine and 9 of water; how many 
 gallons must be drawn from each cask to make the mixture 
 contain 7 gallons of wine and 7 of water ? 
 
96 SIMPLE EQUATIONS. C™. 
 
 26. Every year a merchant adds 40 per cent to his capital, 
 but takes out $3000 for expenses; at the end of the third year, 
 after deducting his $3000, he finds that he has doubled his 
 original capital and has $1800 besides: how much had he at 
 first ? how much at the end of each year ? 
 
 27. A lady receives $2100 yearly interest on her capital, but 
 if it were loaned at a half of one per. cent higher, she would 
 receive $240 more: find her capital and the rate of interest. 
 
 38. How much pure copper must be added to 35 pounds of 
 silver, 15 parts pure out of 16, so that the mixture shall con- 
 tain 4 parts of pure silver to one part of alloy ? 
 
 29. In a naval action, a third of a fleet was taken, a sixth 
 was sunk and two ships were burnt; afterwards a seventh of 
 the remaining ships were lost in a storm, and only 24 ships 
 were left: how large Avas the fleet at flrst? 
 
 30. A and B begin a game, both having the same sum of 
 money, and the loser is always to pay the winner a dollar more 
 than half what the loser has; after B has lost one game and 
 won one, he has twice as much money as A: how much had 
 he at first ? 
 
 31. A besieged garrison had bread enough to last 6 weeks, 
 giving each man 10 oz. a day; but, after they lost 1200 men 
 in a sally, the allowance was increased to 12 oz. a day, and 
 the bread lasted 8 weeks: how many men were there at first ? 
 
 33. A man and his family use a barrel of flour in in days, 
 but when the man is away it lasts n days longer: how long 
 would it last the man alone ? 
 
 33. A man was engaged to work 48 days at two dollars a 
 day and his board, which was estimated at a dollar a day; at 
 the end of the time he received $42, his employer having de^ 
 ducted the cost of board for the days he was idle: how many 
 days had he worked ? 
 
 34. Out of a certain sum of money, a man paid his creditors 
 $432, lent a friend a third of the remainder, and spent a 
 quarter of what was then left; after this he had a fifth of the 
 original sum: how much money had he at first? 
 
SS] QUESTIONS FOR REVIEW. 97 
 
 PROBLEMS OF PUESUIT, 
 
 35. A is 12 miles behind B, and gains 3 miles an hour; when 
 will they be together? When, if A have m miles to gain, and 
 gain h miles in k hours? 
 
 36. A is 180 miles east of B, A's train runs 35 miles an hour 
 and B's 30 miles: when will they be together if A goes west 
 and B east? A east and B west? both west? both east? when, 
 if A be TO miles east of B and they travel at a, b miles an hour? 
 
 Interpret the results if m be positive, negative; a, positive, 
 negative; 5, positive, negative; a>b; a = b; a<b. 
 
 37. B has a start of h hours, A goes c miles an hour faster 
 than B, and overtakes him in k hours: what is each man's rate ? 
 
 Interpret the results if c be negative; h negative; k negative. 
 
 38. A, B starting together walk around a track; at the end 
 of half an hour A has made 3 circuits and B 4^: when will B 
 next pass A? when, if A has made a circuits and B, b circuits? 
 
 39. The circle of a clock face is divided into 13 hour-arcs: 
 how many arcs does the hour hand pass over in an hour ? in 
 two hours ? — in twelve hours ? the minute hand ? 
 
 How many arcs does the minute hand gain, over the hour 
 hand, in one hour ? in two hours ? • • • in twelve hours ? 
 
 40. How many hour-arcs has the minute hand gained, over 
 the hour hand, when they are together between one and two ? 
 between two and three? • • • between ten and eleven? 
 
 41. At what time are the hour hand and minute hand op- 
 posite to each other between twelve and one? between one 
 and two ? • • • between five and seven ? • • • between eleven and 
 twelve ? At what times between twelve and twelve are the two 
 hands at right angles to each other ? 
 
 43. If A, B, C starting together at noon walk around a 
 track, A ten times in an hour, B twelve times, and C fifteen 
 times, when will A and B be together for the first time ? B 
 and C? C and A? When will they be all together? 
 
 When will A be midway between B and C ? B midway be- 
 tween C and A ? midway between A and B ? 
 
98 SIMPLE EQUATIONS. [HI. 
 
 43. If the hour, minute, and second hands of a clock all turn 
 upon the same centre, at what times ■will the minute and 
 second hands be together ? at what times will each of the three 
 hands be midway between the other two ? 
 
 44. A passenger train 200 ft. long passes a freight train 
 680 ft. long in 30 seconds when they are running in opposite 
 directions, and in one minute when in the same direction: find 
 the rate of each train. 
 
 PEECENTAaB AND SIMPLE INTEEBST. 
 
 45. Put I for the basis of percentage, p for the percentage, 
 r for the rate, a for the amount of the basis and the percent- 
 age, V for the net value of the basis less the percentage; then, 
 by definition, r=p/b, a = b+p, v — i—p. 
 
 From these three fundamental equations show that: 
 h=p/r, p = b-r, a=h-{l-\-r), b = a/{l+r), 
 v=b{l-r), b = v/{l — r). 
 Translate these six formulae into theorems and into rules. 
 
 46. Put p for the principal, r for the yearly rate, t for the 
 time in years, i for the simple interest, a for the amount; then, 
 by definition, i—p-r-t, a =p + i. 
 
 From these two fundamental equations, show that 
 
 p = i/rt, r = i/pt, t=i/pr, a=p-{l + rt), p = a/{l+ri). 
 Translate these five formulae into theorems and into rules. 
 
 47. Bank discount is simple interest prepaid, and the bank 
 present worth, or proceeds, is the principal less the discount. 
 
 Put V for the present worth, then, by definition, v—p-^i. 
 
 Show that v=p-{l — rt), p = v/{l—rt). 
 
 Translate these two formulae into theorems and into rules. 
 
 Show the relation between the true present worth and the 
 bank present worth for a given principal, rate, and time; and 
 that between the true rate earned and the bank rate. 
 
 ATEEAGES. 
 
 48. If the foreman gets $5 a day, two sub-foremen $3 each, 
 and forty men $3 each, what is the average daily wages for the 
 whole party? 
 
 Eeplace the numerals by letters and make a general formula. . 
 
:3] QUESTIONS FOR REVIEW. 99 
 
 49. A grocer mixes 20 lbs. of tea worth 50 cents a pound 
 with 30 lbs. worth 40 cents, and 50 lbs. worth 30 cents, and 
 sells the whole at 45 cents; what is the real value ? the profit 
 on one pound ? the rate of profit ? Make a general formula. 
 
 50. With debts ^1, Pi,Ps,- • -due at times ti, 4, t^- • -from 
 a fixed date, find a single date when the whole may be paid 
 without loss to debtor or creditor. 
 
 Discuss the problem if some of the ^'s or p's be negative. 
 
 WORK. 
 
 51. A man can do a piece of work in n days: what part of 
 it can he do in one day ? in two days ? in three days ? in ten 
 days? in n days? in 3w days ? 
 
 53. A man can do n units of work in one day: in what part 
 of a day can he do one unit? two units? three units? ten 
 units ? 11 units ? 2« units ? 
 
 53. A can do a piece of work in a days, B in S days, in 
 c days : what part of the work can A and B together do in 
 one day? B and C ? C and A ? A, B and C ? 
 
 54. A can do a units of work in a day, B, I units, C, c units: 
 in what part of a day can A and B together do a unit of work? 
 BandC? CandA? A, Band C? 
 
 55. A can do a units of work in a' days, B, h units in 
 h' days, C, c units in e' days : in how many days can they to- 
 gether do a + S + c units ? 
 
 66. To do a certain piece of work A needs m times as long 
 as B and C together; B, n times as long as and A; C, ^ 
 times as long as A and B : what relation have m,n,p? 
 
 57. A reservoir is filled by pipes A, B in c hours, by pipes 
 B, C in ffl hours, by pipes 0, A in i hours : in what time is it 
 filled by each pipe running alone ? by the three pipes together ? 
 
 58. A reservoir holding m gallons is filled by two pipes, 
 running a, b gallons an hour, and emptied by two pipes run- 
 ning c, d gallons an hour : what is the relation between a, l, 
 c, d so that, with all the pipes running, the reservoir, if empty, 
 shall be filled in h hours ? if full, emptied in h hours? 
 
100 MEASURES AND MULTIPLES. [IV, Th. 
 
 IV. MEASUEES AKD MULTIPIiES. 
 
 §L INTEGERS. 
 
 The product of two integers is a multiple of either integer; 
 and either integer is a meastire of the product. 
 E.g., +15, -15, are multiples of +5, -5, +3, "3; 
 and +5, "5, +3, "3, are measures of +15, and of -15. 
 
 Every integer is a multiple of itself, its opposite, and *1. 
 
 COMMOK MULTIPLES ASB MEASUBES. \ 
 
 If the same number be a multiple of two or more integers, 
 it is a common multiple of them; and a measure of two or 
 more integers is a common measure of them. 
 E.g., +30, "30, are common multiples of 1, "5, 10, "15, 30; 
 and +3, -3, are common measures of 3, -6, 9, -15, 30. 
 
 The smallest of all the common multiples of two or more 
 integers is their lowest common multiple; and the largest of 
 all their common measures is their highest common measure. 
 E.g., *30 is the lowest common multiple of 3, 10, -15, but 
 
 not of 3, -15; 
 and *3 is the highest common measure of 6, -9, 13, but not 
 
 of 6,12. 
 
 As. 1. Tlie sum, the difference, and the product of two in- 
 tegers are integers. 
 
 Theor. 1. A common measure of two or more integers is a 
 measure of their sum. 
 
 Let A, B, • ■ • be any integers and M a common measure of them; 
 then is M a measure of the sum A + B+ • • • 
 For •.• A = M • a, B = M • 5, ■ • • , and a, 5, • • • are integers, [hyp. 
 
 .•. A + BH — ■ =M-a + M-b-\ 
 
 = M-{a + b+---); [I,th. 7. 
 
 and •.• the sum a + b+ • ■ • is an integer, [ax. 1., 
 
 „ .•. M is a measure ot a + b-{ q.e.d. [df. msr. 
 
1.S11 INTEGERS. 101 
 
 Cor. a common measure of hoo integers is a measure of the 
 sum and of the difference of any multiples of them. 
 
 QUESTIONS. 
 
 1. Explain the difference between a product and a multiple; 
 between a divisor and a measure. 
 
 2. Name some multiples of 6, of ~3, of 0, of 1. 
 
 3. Name some measures of 24, of 60, of ~64, of 1, of 0. 
 
 4. Name some common multiples of 3, 3, "4, 1; of 5, 7, 11. 
 
 5. Name some common measures of 24, 64, 120. 
 
 6. Name the lowest common multiple of 6, 15, 10 : what is 
 their highest common multiple ? 
 
 7. Name the highest common measure of ~120, ~45, "60: 
 what is their lowest common measure? What common 
 measure of two of these numbers is not a measure of the third ? 
 
 8. In the proof of theor. 1, why must a, S, • • • be integers ? 
 Explain the dependence of the last statement of the proof 
 
 on that just before it. 
 
 9. Prove that a common measure of two integers is a meas- 
 ure of their difference and of their product. 
 
 '10. Prove that a measure of the sum of two or more integers, 
 if it measure all of them but one, measures that one also. 
 
 11. Prove that a measure of ajQ integer is a measure of any 
 multiple of that integer; that a multiple of an integer is. a 
 multiple of all measures of that integer. 
 
 12. What two common measures have all integers ? 
 What other two measures has any given integer ? 
 
 13. Why is any positive integer a multiple of its opposite ? 
 Why is any measure of a negative integer a measure, of the 
 
 same integer with the positive sign ? 
 
 14. In finding the highest common measure of integers, 
 will the answer be affected if all the integers be taken positive ? 
 
 15. Prove that the sum, the difference, and the product of 
 two even numbers are even. 
 
102 MEASURES AND MULTIPLES. [IV.Th. 
 
 Euclid's process" for finding the highest commoit 
 
 MEASURE. 
 
 Theor. 2. If the larger of two integers be divided by the 
 smaller, the common measures of the divisor and the remainder 
 are the common measures of the two integers. 
 
 If the smaller integer be divided by the remainder, this 
 divisor by the second remainder, and so on, then some re- 
 mainder is zero. 
 
 The last divisor is the highest common measure of the kuo 
 integers. 
 Let A, B be two integers, Qi the quotient of A by b, and Ei , 
 
 Ej , • • • R„ the successive remainders; 
 then •,• El = A — B • Qi , [df , rem. 
 
 .*. the common measures of A, B are measures of R, , 
 
 [th. 1, cr. 
 i.e., the common measures of A, B are common measures of 
 
 So, •.•A = B-Qi + Ei, 
 
 .*. the common measures of B, Rj are measures of A, 
 i.e., the common measures of B, Ei are common measures 
 of A, B ; 
 .•. the common measures of A, B are the common meas- 
 ures of B, El, and there are no others. q.e.d. 
 
 3. •.• El, R^- ■ -are all integers, and grow smaller and smaller, 
 .'. some one of them, say E„ , is 0. q.e.d. 
 
 3. '.' the common measures of b, Ei are the common meas- 
 ures of Ei, Eg, [above. 
 .*. the common measures of A, B are the common meas- 
 ures of El , E2 ; and so on; 
 .*. the common measures of A, B are the common meas- 
 ures of E„_2, B„_i; 
 and ".• R,i-i is the" highest common measure of E„_2, E„,i , 
 .*. E„_i is the highest common measure of A, B. q.e.d. 
 
3.S1] INTEGERS. 103 
 
 If Ea-i be 1, then A, b are usually said to have no common 
 measure; for unity, being a measure of all integers, is not a 
 characteristic common measure of A, B. 
 
 Cor. JEvery remainder, Ei , E^ • • • E„ is the difference of two 
 multiples of A, B. 
 
 E.g., Ei = A-B-Qi, 
 
 Ejj = B-Ki-Q2 = B-(A-B-Q,)-Q2=-A-Q2+B-(1+Qi.Q2), 
 Es = Ki-B2'Qs = A-(1 + Q^Q3)-B-(Qi + Q3 + Qi.Q2.Q3). 
 
 QUESTIONS. 
 
 1. In division, how does the remainder compare, in size, 
 with the divisor ? if the divisor be divided by the remainder, 
 how does the second remainder compare with the first ? 
 
 In Euclid's process how do the remainders change ? 
 Can any remainder be a fraction or a negative number? 
 Why must an exact divisor be reached at last? 
 
 2. If E„ be the last remainder, for what two things does 
 E„_i stand? for what three things does E„_g stand ? 
 
 3. Show that the following statements are true: 
 
 any measure of B and Ei is a measure of A, 
 any measure of Ej and Eg is a measure of b, 
 
 any measure of E„_2 and E„_i is a measure of B„_j, 
 any measure of E„_i and e„ is a measure of B„_2. 
 
 4. What measure of e„_2 is found by Euclid's process ? 
 Is this measure also a measure of e„_i ? 
 
 Of what other successive pairs of numbers is B„_i a measure ? 
 
 5. What is the highest measure of E„_i ? the highest com- 
 mon measure of E„_i, E„_a? of E„_8, e„_3 ? • • • of A, b ? 
 
 6. Find the highest common measure of 14637, 51306. 
 
 7. Draw any two unequal lines and call them a, i; lay off b 
 on a as many times as it can be repeated; lay off the re- 
 mlBfler, c, on i; lay off the remainder, d, on c, and so on, till 
 there is no remainder : what is true, of the last line used ? 
 
 8. Prove that every common measure of two integers is 
 the difference of two multiples of the integers. 
 
104 MEASURES AND MULTIPLES. [IV.Thb. 
 
 PRIME NUMBERS. 
 
 All integer that has no measures but itself and unity is a 
 prime number; and two integers that have no common 
 measure except unity are prime to each other. 
 E.g., 3, "3, 5, ~7, 11 are prime numbers; and 9, "25 are prime 
 to each other, though not prime numbers. 
 
 An integer that can be measured by another integer is a 
 composite number. 
 E.g., 15, "35, whose prime measures are 3, 5; 5, 7. 
 
 Theoe. 3. If two integers be prime to each other, then two 
 multiples of them can be found such that their difference is 
 unity; and conversely. 
 
 For, let A, B be two integers that are prime to each other, and 
 
 El, Kg, ■ • • , E„ the remainders as in theor. 2; 
 then •.• A, B are prime to each other, [hyP' 
 
 .'. K„_i, their highest common measure, is 1; 
 and •.•E„_i is the difEerence of two multiples of A, b, say 
 m-A, n-B, . [th. 2 cr. 
 
 /. m-A — m-B = l. Q.E.D. 
 
 Conversely, let m • A — re • b = 1 ; 
 then •." every common measure of A, b is a measure of 1, 
 .'. A, B are prime to each other. q.e.d. 
 
 Theoe. 4. If an integer be prime to two or more integers, it 
 is prime to their product. 
 Let A, B, c, • • • be any integers and p an integer that is prime 
 
 to each of them; 
 then is p prime to the product a ■ b • c • • • 
 For take m, n; p, q°, r, s;- • • integers such that 
 
 m-p — «'A = 1, ^-p — g'-B = l, r-p — s-c = l,- • •, [th. 3. 
 then •.• (wj-p — W'A) x (^-p — g-^B) x {r-v — s-d)- • • =1, ^| 
 i.e., A-p + /fc-A-B-C' • • =1, wherein h-v is the sum of all 
 the terms that contain p, and k= ±n-q-s- • • 
 .'. p is prime to the product a • b • c • • • q.e.d. [th. 3 conv. 
 
3.4.811 INTEaERS. 106 
 
 Cob. 1. If two integers he prime to each other, so are their 
 positive integer powers. 
 
 QUESTIONS. 
 
 1. Name seven prime integers; the only even prime integer. ' 
 
 2. If all the integers from 1 to 1000 be written in order, 
 and all the even numbers be struck out, every third number 
 from 3, every fifth from 5, and so on, what numbers remain ? 
 
 3. Name five integers, no one of which is prime, but which 
 are all prime to each other. 
 
 4. Is it possible for an integer to have but one factor ? 
 Separate 36 into sets of two factors, of which one shall be 
 
 2, 3, 4, 6, 9, 12, 18, 36, in turn : were all the possible sets 
 obtained before the entire series of divisors were tried ? 
 
 In trying all possible factors of an integer in the order of 
 their size, how far need the process be carried ? 
 
 5. If two integers that are prime to each other be subjected 
 to Euclid's process, what is the last remainder ?j the last 
 divisor ^the last remainder but one ? 
 
 6. Find two multiples of 9, 17, whose difference is 1; so, of 
 5, 13; of 11, 13; of 13, 17; of 17, 19; of 13, 19. 
 
 ^ 7. What is the converse of a theorem ? 
 State the converse of theor. 3 as a separate theorem. 
 
 8. In the converse of theor. 3, why have A, B no common 
 measure but 1 ? 
 
 9. What statement in theor. 4 is a direct application of 
 theor. 3 ? what, of the converse of that theorem ? 
 
 10. Are all prime numbers prime to each other ? 
 
 If an integer be resolved into its prime factors, what rmp'' 
 tion have other prime numbers to these factors separately ? 
 
 What relation have these numbers to the original number ? 
 
 .11. Prove that an even number can not measure an odd 
 
 number; that a number having any even factor is even; and 
 
 that the product of any number of odd factors is an odd 
 
 number. Is the sum of two odd numbers odd or even ? 
 
106 MEASURES AJ!ID MULTIPLES. [IV.Ths. 
 
 Cor. 2. If an integer measure the product of two integers 
 and be prime to one of them, it measures the other. 
 Let A, B be two integers and p an integer that measures the 
 
 product A-B and is prime to A; then p measures b. 
 For, let m • A, w ■ p be two multiples of A, p such that 
 
 mA — nv = l; [th. 3. 
 
 then (m-A — W'P)-b = ?m-A'B — Ji-P'B = B, 
 and ■.• p measures both A-B, and p-b, [hyp., df. msr. 
 
 /. p measures B. Q.E.D. [th. 1 cr. 
 
 CoE. 3. If a prime number measure the product of two or 
 more integers, it measures at least one of them, 
 
 OoE. 4. If a prime mimber measure a positive integer power 
 of an integer, it measures the integer; and if the integer, then 
 the power. 
 
 Cob. 5. If an integer be measured by two integers that are 
 prime to each other, it is measured by their product. 
 For, let A be an integer and p, q integers prime to each other 
 
 that measure a; and let a = b-p; 
 then •.• Q measures the product b • p and is prime to p, [hyp. 
 .*, Q measures b, [cr. 3. 
 
 i.e., W'Q = B, wherein m is some integer; [df. msr. 
 
 .•. m-p-Q = B-p, and the product p-q measures the 
 product B • p, 
 i.e., the product p • Q measures A. q.b.d. 
 
 Theor. 5. An integer can be resolved into prime factors in 
 bu^one iuay. 
 I^ the integer N be the prdduct a^-b'-c" • • • wherein A, b, c, 
 
 • • • are primes, and, a, b, c, ••• are positive integers; 
 then N has no other prime factor. 
 
 For •.• any other prime, f, is prime to each of the prime factors 
 
 A, B, c,---, [hyp. 
 
 and to their powers A", b", C, • • •, [th. 4 cr. 1. 
 
 /. F is prime to the product K. q.e.d. [th. 4. 
 
4, 5, SI] INTEGERS. 107 
 
 And A can be a factor of n not more than a times, B not more 
 
 than b times, and so on. 
 For vn'=a''-b''-c''- • •, 
 .•• sr/A''=B''-c''---; 
 and •.' A is prime to b'-c"- • •, [th. 4. 
 
 .'. A is a factor of n not more than a times. 
 So, A can be a factor of N not fewer than a times; and so with 
 B, with c, and with the other prime factors. q.e.d. 
 
 Cob. a common measure of tiuo or more integers can con- 
 tain no factor that is not in all of them. 
 
 QUESTIONS. 
 
 1. In the proof of theor. 4 cor. 3, are m, n given numbers ? 
 Are they numbers assumed at random ? 
 
 Why does p measure otab ? why does p measure B ? 
 
 2. So, if P measure ab, what kind of number is ab/p ? 
 If p be prime to A, is a/p an integer ? what then is b/p ? 
 
 3. If an even number be measured by an odd number, the 
 quotient is even. 
 
 4. In the proof of cor. 3, if p measure the product A • B • c • D 
 • • • and be prime to A, of what is p a measure ? 
 
 So, if P be prime to A, b, c, d, of what is p a measure ? 
 
 5. Cor. 4 is found by inserting the word equal in cor, 3. 
 
 6. A composite number may measure the product of two or 
 more integers and not measure either of them separately. 
 
 7. If A be measured by the integers p, q, b, prime to each 
 other, and if a = »1'P-q, then k measures m, and the product 
 p • Q • K measures A. 
 
 What application is here made of theor. 4 ? l-i 
 
 8. In the proof of theb»i5, if p be prime to A, why is it 
 prime to a"? why prime to if?. If the factors of n be all 
 different, what are the values of a, 5, • • • ? 
 
 9. If two integers haiVe several common prime factors, the 
 product of these factors is a common measure of the integers. 
 
 What factors are found in their highest common measure ? 
 
108 MEASURES AND MULTIPLES. [IV. 
 
 §2. ENTIEE FUNCTIONS OP ONE LETTER. 
 
 Au algebraic expression whose value depends on the value 
 of a single letter is a function of that letter; the letter is the 
 letter of arrangement. 
 
 E.g., the value of the expression o^—2a?+Zx-\-b depends 
 on the value of x, and it is known when x is known. 
 If the value of an expression depend on the values of two 
 or more letters the expression is a function of those letters. 
 E.g., the value of x' — 2x'y-^ + Zxy-^-ir^y-^ is known when 
 the values of x, y are known. 
 The letter /stands for the word function. 
 E.g., /a; means a function of x, and fa is what fx becomes 
 when X is replaced by a. 
 If there be more than one function of x, such functions 
 may be distinguished as /a;, vx, 0a;, • • • oxf\x,f^,fsX,- • • 
 
 An entire function of one letter is the sum of positive inte- 
 ger powers of that letter, with or without coefficients. 
 E.g., a;' + 3a;^ + 3a; + l is an entire function of x of the third 
 
 degree, whose coefiScients are integers.*" 
 go, \y^ — y^+y — ^ is an entire function of y. 
 
 So, if w be a positive integer, a!a;" + 5a;''-* + ca;"-^ hhx + l 
 
 is an entire function of x of the nth. degree, with literal 
 coefficients, which may be either integers or fractions. 
 
 In general, the definitions and principles established for 
 integers apply with slight changes to entire functions of one 
 letter, and, for the most part, they "are here stated, proved, 
 and illustrated in the same wordt. ^ 
 
 The product of an entire function of one letter by another 
 such function is a multiple of either function, as to that Ifttter, 
 and either function is a measure of the product. 
 E.g., a' — 3a^+3a — 1 is a multiple of a"— 2a + l, a — 1, 
 and a' — 3a + l, a — 1 are measures of 
 
88] ENTIRE FUNCTIONS OP ONE LETTER. 109 
 
 So, 8{as — a){y — b) isameasureof 2m(x'—a^){i/ + b) asto 
 
 a, m, X, 
 and a multiple of it as to the numerals; 
 but neither measure nor multiple as to y, nor as to h. 
 
 Every entire function of one letter is both a multiple and 
 a measure of itself, and a multiple of all expressions that are 
 free from this letter; and every such expression is a measure 
 of any entire function. 
 
 QUESTIONS. 
 
 1. Write an entire function of y of the 4th degree, whose 
 coefficients are all negative integers. 
 
 So, an entire function of a?, and one of a;+^. 
 
 2. If w be a fraction, is no? —{n — l)a^ + {n— 2)a? —{n — 3)a^ 
 an entire f unctioh of a; ? 
 
 If n be a fraction and m an integer, is 
 
 nx'"+2nx'"-^ + inx^-''-{ an entire function of a; ? 
 
 a;" + ama;"-H5mV-''? a + bx'" + cx"' + dx"'+"'+cx"'"? 
 
 3. If n be an integer, on what two conditions is 
 
 aa;" + 5a;<"-"'l/2 + ca;(»-"^ + «Za;<''-«''« an entire function of x? 
 a:" -aa;"-H5a;"-'^+.ca;"''? x^ + ax'^-^^^^ + ix'^^-^Vi + caf'^-'V'! ? 
 
 4. If /a;=ar' + 3* + 6, write an expression for /a, f{-y); 
 and find the values of / 1, / 0, / - 2, / 5, / - 5, /. 167. 
 
 5. When functions of a single letter are under discussion 
 and an expression is said to be free from the letter of arrange- 
 ment, what do you know about the nature of that expression ? 
 Can you tell whether it is an integer or a fraction ? whether 
 it is positive or negative? whether it involve powers? or roots? 
 
 6. If an entire function of a single letter be divided by any 
 expression free from the letter of arrangement, how do the 
 exponents of the letter of arrangement in the several terms of 
 the quotient compare with those of the given function ? 
 
 Why, then, is every such expression a measure of every such 
 entire function ? 
 
 What integer is, for a like reason, a measure of all integers? 
 
110 MEASURES AND MULTIPLES. [IV.Th. 
 
 COMMON MULTIPLES AND MEASURES. 
 
 If the same entire function be a multiple of two or more 
 entire functions, it is a common multiple of them; if a meas- 
 ure of them all, it is their common measure. 
 E.g., 7a%*^-l), i(^'-l)' are common multiples of 
 
 y+1, y-1, y^+l, y^-1, y^-y + 1, y^+y+l-, 
 and a; — a is a common measure of a? — a^, a^— a', af—aK 
 
 That common multiple of two or more entire functions which 
 is of the lowest degree is their lowest common multiple; and 
 that common measure which is of highest degree is their high- 
 est common measure. 
 ^•Sv ^° — 1 is the lowest common multiple of 
 
 y + l,y-\, f-\, y^'-y+l, y^+y + l, f+X,^-!, 
 
 and s? — a^ is the highest common measure of 3?^ a*, 
 
 a;'— a', a;* — a', x' — d'; but not of a;*— a', a^ — «*. 
 
 Ax. 2. The sum, the difference, and the product of two en- 
 tire functions of a letter are entire functions of that letter. 
 
 Theok. 6. A common measure of two or more entire func- 
 tions of a letter is a measure of their sum. 
 For,letaa;" + &a;"-i+ . . . +fe + ?, a'a;'" + 5'a;'"-'+ • • • +¥x+l', 
 be two entire functions of x, and m a common measure 
 of them ; 
 then-.- fl!a;" + 5a;"-*+'"+fe + Z=M-p, 
 
 a'a;'" + 5'a;"'-'+ • • • +Fa; + r = M-Q, 
 wherein p, q, are entire functions of x, [hyp. 
 
 ■.-. aa;" + 5a;"-»+ • • • +fe + Z + a'a'" + &'a;'"-'+ • • • +h'x+l' 
 = m-p + m-q = m-(p + q); 
 and -.- the sum p + Q is an entire function of x, [ax. 2. 
 .-. M is a measure of 
 
 aa;'' + &a;''-H • - - +*a;+?+a'a;"' + 5'a;™-*+ • • • +k'x + l', 
 and so for three or more functions. q.e.d. [df. msr. 
 
 If the entire functions aa;" + 5a;"-^H \-hx + l, 
 
 a'x^■\■Vx'^-'■^ +Fa; + /',•• • be represented by A, b,- • •, 
 
 the proof of theor. 1 applies word for word to theor. 6. 
 
e. S2]' ENTIRE FUNCTIONS OP ONE LETTER. Ill 
 
 Cob. a common measure of two entire functions of a letter 
 is a measure of the sum and the difference of their multiples. 
 
 QUESTIONS. 
 
 1. A common multiple of two or more entire functions of 
 one letter is of at least what degree ? 
 
 A common measure cannot exceed what degree ? 
 
 2. In what three ways can two entire functions of a letter 
 be combined with absolute certainty that all the exponents 
 of the result will be positive integers ? 
 
 3. What is the lowest common multiple of a + b/c, a — b/c, 
 as entire functions of a ? as entire functions oi b? 
 
 4. Following the line of argument of theor. 6, show that 
 a" — a? is a measure of {a" —!(?) + (a° - a;') + (a* - a;*). 
 
 What is p ? Q ? K? What is the letter of arrangement? 
 
 5. Kepresent a'— a? hy A, a" — of by B, a*— ic* by c, 
 and write out the proof of theor. 6. 
 
 6. In the proof of theor. 6, if m be a function of the nth 
 degree, what is p ? 
 
 if M be of (m— 2)nd degree, of what degree is p? 
 if n>.m, what is the highest degree possible to m ? 
 if Q be free from x, what is the degree of m ? 
 if p, Q be both free from x, what relation have m, n'? 
 What is the degree of p + Q ? 
 
 7. A common measure of two or more entire functions of. 
 a letter is a measure of their sum, their difference, and their 
 product: is it also a measure of their quotient? 
 
 Illustrate these principles by aid of the functions in ex. 3. 
 
 8. A multiple of a multiple of a'number is a multiple of 
 the number; a measure of a measure of a number is a measure 
 of the number; and a common measure of two or more num- 
 bers is a common measure of any multiples of them. 
 
 9. A common multiple of two or more numbers is not 
 necessarily a multiple of their sum; but the sum of two such 
 common multiples is a common multiple of the numbers. 
 
112 MEASURES AND MULTIPLES. [IV, Tb 
 
 Euclid's process foe finding the highest common 
 
 MEASURE. 
 
 Theoe. 7. If the higher of two entire functions of a letter 
 be divided oy the lower, the common measures of the divisor 
 and the remainder are the common measures of the functions. 
 
 If the lower function be divided ly the remainder, this 
 divisor by the second remainder, and so on, then some re- 
 mainder is zero. 
 
 Tlie last divisor is the highest common measure of the two 
 entire functions. 
 
 Let A, B be two entire functions of a letter, Qi the quotient of 
 
 A by B, and Ei, Ej,- • • E„ the successive remainders; 
 then •.•Ri = A — B'Qi, [df. rem. 
 
 .'. the common measures of A, B are measures of Ej ; 
 
 [tb. 6 cr. 
 i.e., the common measures of A, B are measures of B, R,. 
 
 So, ■.■A = B-Qi + Ri, 
 
 ,•. the common measures of b, Ri are measures of a; 
 i.e., the common measures of B, Ej are measures of A, b; 
 .'. the common measures of A, b are the common meas- 
 ures of B, Ej, and there are no others. Q.B.D. 
 
 And •/ El, Eg, ■ • • are all entire functions of one letter, and 
 of lower and lower degree, 
 /. some one of them is of zero degree, as to that letter, 
 and, if not itself zero, the next remainder, R„, is zero. 
 
 Q.E.D. 
 
 And •.• the common measures of b, Ej are the common meas- 
 ures of Ej , Ea , [above. 
 .*. the common measures of A, B are the common meas- 
 ures of El, Ej, and so on; 
 ,•. the common measures of A, b are the common meas- 
 ures of E„_a, E„_i ; 
 and •.• E„_i is the highest common measure of E„_a, E„-i, 
 ,•. B„-jis the highest common measure of a, b. q.e.d. 
 
7, §2] ENTIBE FUNCTIONS OF ONE LETTER. 113 
 
 If E„_i be free from the letter of arrangement, then a, b 
 are usually said to have no common measure; for expressions 
 free from the letter of arrangement, being measures of all 
 entire functions of that letter, are not characteristic common 
 measures of A, b. 
 
 Cob. Every remainder, Ri , Ka , • • ■ e„ is the difference of 
 two multiples of the entire functions, a, b. 
 
 QUESTIONS. 
 
 1. In the proof of theor. 7, if the higher function be of the 
 With degree and the lower of the nih, of what degree is the first 
 quotient ? the product of the divisor by this quotient ? 
 
 What is the highest degree possible to the first remainder ? 
 to the second quotient ? to the second remainder ? 
 Why can no remainder have a negative exponent ? 
 
 2. The common measures of b, Kj are common measures, 
 and the only common measures, of Bj , k^ : state the proof. 
 
 3. If two entire functions of a letter have an integer as 
 a common measure, is this measure a common factor of all 
 the coeflBicients ? 
 
 4. What common measure free from x have the functions 
 
 8aoi^ — \%aa? + 34aa;^ - l%ax — 4a, 
 34aa;* - 44aa;' + Q^ao? - 2iax -20a? 
 Divide out this common factor and find by Euclid's process 
 the highest common measure of the quotients. 
 
 5. If all common monomial factors of the two functions 
 have been divided out before applying Euclid's method, what 
 does a remainder free from the letter of arrangement show 
 about the original functions? 
 
 6. Show that A = B-Q,4-Ri, B = Ei'Qa + B2, Ei = R2-Q3 + E3. 
 Counting Ej, Ej, B3 as unknown elements, solve these 
 
 equations for E3 , and show that its value is the difference of 
 two multiples of A, b. 
 
 Can all other remainders be expressed as such differences ? 
 
 7. In ex.4 express Ei, Eg, E3 as the differences of two 
 multiples of the given functions. 
 
114 MEASURES AND MULTIPLES. [IV.Ths. 
 
 PRIME FUNCTIOKS. 
 
 An entire function of a letter that has no measures but 
 itself, and expressions that are free from that letter, is a prime 
 function; and two entire functions of a letter that have no 
 common measure except expressions that are free from that 
 letter are prime to each other. 
 
 An entire function of a letter that has another entire func- 
 tion of that letter as a measure is a composite function, and 
 the measuring functions are its factors. 
 E.g., a? + x + l, 4a;'— 7 are prime functions of a;, 
 and a? + 7?—'iiiX is prime to each of them, though itself com- 
 posite, having the factors X, x + 2, x — 1. 
 
 Theok. 8. If two entire functions of a letter he prime to 
 each other, then ttvo multiples of them can be found such that 
 their difference is free from that letter, and conversely. 
 For, let A, B be two functions of a letter that are prime to each 
 
 other, and E„_i be their highest common measure ; 
 then '.• B„_i is the difEerence of two multiples of a, b, say 
 wiA, MB, and is free from the letter of arrangement, 
 .•. w-A-w-B is free from the letter of arrangement. 
 
 Conversely : let »i • A — « • B be free from the letter of arrange- 
 ment, and not 0; 
 then •.• every common measure of A, B is a measure of an ex- 
 pression that is free from the letter of arrangement, 
 .".A, B are prime to each other. q.e.d. 
 
 Theok. 9. If an entire function of a letter he prime' to two 
 
 or more such functions, it is prime to their product. 
 
 Let A, B, c, • • ■ be any entire functions of a letter, and p an 
 entire function that is prime to each of them; 
 
 then is p prime to the product a-b-C- • • 
 
 For, take m, n, p, q, r, s, — entire functions of the 
 letter such that m-F — n-A, p-F — q-B, r-p — s-G,--- 
 are free from the letter of arrangement; [th. 8. 
 
8, 9. §2] ENTIRE FUNCTIONS OP ONE LETTER. 115 
 
 then"/the product {m-v — n-A)-{p-v — q-B)-{r-p — s-c)--' 
 
 is free from the letter of arrangement, 
 i,e., h-v + k-x-B-G • • • is free from the letter of arrange- 
 ment; wherein A-p is the sum of all the terms that 
 contain p, and k= ±n-q-s- • •, 
 .*. p is prime to the product A-B-c- • • Q.E.D. [th. 8 conv. 
 
 QUESTIONS. 
 
 1. Are «'+8, a'— 8 prime functions of a ? are they prime 
 to each other ? what is the letter of arrangement ? 
 
 2. What two common measures belong to all integers ? how 
 then can integers be called prime to each other ? 
 
 3. All entise functions have many common measures that 
 are ignored in the definition of functions prime to each other: 
 what are these measures ? why are they ignored ? 
 
 4. In Euclid's process for" finding the highest common 
 measure a remainder of unity corresponds, in integers, to an 
 expression free from the lettei of arrangement, in entire 
 functions. 
 
 5. In the proof of the converse of theor. 8, if the expression 
 wi-A — w-B be free from the letter of arrangement, every com- 
 mon measure of A, b is free from that letter: state the proof; 
 
 6. In the proof of theor. 9, what relation has /t • p to p ? 
 ^•A-B-c- • -to the product a-b-c- • •? 
 
 How does the last statement depend on the one before it ? 
 
 7. If there be two entire functions of a letter such that one 
 of them is a measure of the other, an entire function of this 
 letter that is prime to both of them is prime to their quotient. 
 
 8. If an entire function of a letter be prime to another such 
 function, it is prime to any integer power of that function. 
 
 9. Prom the fact that a common measure of two numbers 
 measures their sum, prove that 3 is a factor of any number 
 ending in 0, 2, 4, 6, 8; that 4 is a factor, if a factor of the last 
 two figures, and 8, if of the last three figures; and that the re- 
 mainder got by dividing the last figure of a number by 5 is 
 the same as that for the entire number. 
 
116 MEASURES AND MULTIPLES. [IV.Ths. 
 
 Cor. 1. If two entire functions of a letter be prime to each 
 other, so are their positive integer powers. 
 
 Cor. 3. If an entire function of a letter measure the product 
 of two such functions and be prime to one of them, it measures 
 the other. 
 Let A, B be two entire functions of a letter, and p an entire 
 
 function that measures their product, and is prime to A, 
 and let mA, nv be multiples of A, p such that w(A-wp = Z, 
 
 wherein I is free from the letter of arrangement ; [th. 8. 
 then m/l-A. — n/l-p = l and m/l-A-B — n/l-p-B~B. 
 and •.• p measures both a-b and p-b, [hyp., df. msr. 
 
 .-. p measures b, q.e.d. [th. 6 cr. 
 
 Cor. 3. If a prime function of a letter measure the product 
 of tioo or more entire functions of that letter, it measures at 
 least one of them. 
 
 Cor. Ai. If a prime function of a letter measure a positive 
 integer power of an entire function of that letter, it measures 
 the function; and if the function, then the power. 
 
 Cor. 5. If an entire function of a letter be measured by two 
 such functions that are prime to each other, it is measured by 
 their product. 
 
 Thbor. 10. An entire function of a letter can be resolved 
 into factors that are prime functions in but one way. 
 Let N, an entire function of one letter, be the product 
 A^-B^-C ■ ■ • wherein A, B, C, • • • are prime functions 
 of that letter and a,b, c, • • • are positive integers; 
 then N has no other prime factor. 
 
 For '.' any other prime function, s, is prime to each of the 
 
 prime factors A, B, c, • • • [hyp- 
 
 and to their powers A", b*, c°, • • • [th. 9 cor. 1. 
 
 ,-. E is prime to the product N. [th. 9. 
 
 So, A can be a factor of N not more, and not fewer, than a 
 times; and so with B, and with the other prime factors. 
 
9,10,S2] ENTIRE FUNCTIONS OF ONE LETTER. 117 
 
 Cob. a common measure of two or more entire functions of 
 a letter can contain no factor that is not in all of them, 
 
 QUESTION'S. 
 
 i. In the proof of theor. 9 cor. 2, if m/l, n/l be fractions, 
 how can b be a multiple of p ? 
 
 2. Show that cor. 4 is only an application of cor. 3 to factors 
 having a special relation to each other. 
 
 3. In cor. 5, let A be an entire function of a letter, and let 
 p, Q be entire functions prime to each other, but measures 
 of a: if a = «-q, what relation has w to p? 
 
 If n = m-T, what is A in terms of m,n,v? 
 
 4. A measure of an entire function of a letter contains no 
 factor that is not a factor of the function; and a common 
 measure of two or more such functions contains no factor 
 that is not in all the functions. 
 
 What are the factors of the highest common measure ? 
 
 5. If B be a factor of an entire function of a letter, f" is a 
 factor of the wth power of the function. 
 
 6. An entire function of a letter, if measured by any num- 
 ber of such functions that are prime to each other, is measured 
 by their product; and the product of all the prime factors 
 common to two or more such functions is a common measure 
 of the functions. 
 
 7. If a, b, c, " • I be any entire numbers and m another, 
 if a', V, c', • • • V be the remainders when a,}), c,---l are 
 divided by m, and if the sum a' + l' + c'-\- • • • +1' be meas- 
 ured by ot, so is the sum a + b + c+ ••• -\-l. 
 
 If 3 be a factor of the sum of the digits of an integer, it is 
 a factor of the integer; and so with 9. 
 
 8. In the last part of the proof of theor. 10, suppose the 
 factor A to occur fewer than a times : what are the only factors 
 by which it can be replaced? Can a power of one of the 
 other factors take the place of a power of a ? If a were used 
 more than a times, what change must occur among the other 
 factors ? Is that possible ? 
 
118 MEASURES AND MULTIPLES. [IV.Pbs 
 
 §3. ENTIRE FACTORS. 
 
 Integers and entire functions of a letter, or letters, may be 
 classed together as entire numbers. 
 
 The measures of an entire number are its entire factors. 
 
 The prime factors of a composite entire number are the 
 prime entire numbers whose product is the given number; 
 and to factor a number is to find all its prime factors. 
 E.g., 600a V - 600a V = 3' • 3 • 5^ • a" • a;* • (a + a;) • (a - a;). 
 
 PbOB. 1. To FACTOR AN INTEGEE. 
 
 Divide the number, and the successive quotients in order, by 
 the primes 3, 3, 5,- • •, using each divisor as many times 
 as it measures the successive dividends. 
 The successful divisors, and the last undivided dividend, 
 
 are the prime factors sought; and no divisor larger than the 
 
 square root of the dividend need be tried. 
 
 For the dividend is the product of the divisor and quotient, 
 
 and if the divisor be larger than the square root of the divi- 
 dend, then the quotient is smaller; 
 
 i.e., if there be a factor larger than tbe square root of the 
 dividend, there is also a smaller factor; 
 
 and all the possible smaller factors have been already found. 
 
 If a number have no factor smaller than its- square root, it 
 is prime. 
 
 E.g., of 11908710, 2 is a successful divisor once, 3 twice, 5 
 once, 11 once, 33 once, and the square root of the quo- 
 tient, 523, is smaller than 23; 
 .•• the prime factors are 2, 3, 3, 5, 11, 23, 533. 
 
 PrOB. 2. To FACTOR A POLTWOMIAL OF KKOVTN TTPE-FOEM. 
 
 Express the number in one of the type-forms, and write its 
 
 factors directly in the form of the factors of the type. 
 E.g., oi?+2ax+a'- 25mV ={x + ay- {pmnf, 
 
 = (a; -I- a + 5mw) -{x+a — 5mn). [II. pr. 3 nt. 4. 
 
1. 2, S 3] 
 
 ENTIRE FACTORS. 
 
 119 
 
 QUESTIONS. 
 
 1. Make a table of the prime numbers from to 100. 
 Factor, or prove to be prime : 
 
 2. 30; 37; 73; 120; 333; 367; 1331; 1683; 8279; 15635. 
 
 4. m'' + 2»» + l. 
 7. 4a'' + 2a -20. 
 
 10. a?-y'\ 
 
 13. a;"-y". 
 
 16. a«-2565-«. 
 
 19. 37c' + l. 
 
 23. a;*-10a;H9. 
 
 35. a* -81. 
 
 38. {x±yY-A 
 30. 13a* +«'»"- a;*. 31. (?-10cd-\-2bd\ 33. a'^ + ^aH + l^. 
 33. a;-« + 6a;-' + 9. 34. x'^-bx-^ + Q. 35. a-*'»-3a-'»5 + ^i«. 
 36. 9x' + 3ax+6x + 2a. 
 
 8. a' + 2ab + b''. 
 6. a^ + 5a; + 6. 
 
 9. 4m^-9a^. 
 13. a^f-Uy^K 
 15. e^-itS + e-*". 
 18. a'-d'. 
 
 21. 2a^ + 6a;-8. 
 34. a^-4a-32. 
 27. 4a«-4a5 + S^ 
 
 5. «*— 2a;^+^^ 
 
 8. m' — re*. 
 11. a^-ly>\ 
 14. e^-e-^. 
 17. a*a^ + a^x — a\ 
 20. a;* -13a; + 43. 
 23. y^-y -30. 
 26. 4y*-2«/ + l. 
 29. a^-a^x-Gaa^. 
 
 38. a?+2xy + y^ + 5x + 5y + 6. 
 40. (a+a;)'-(a-a;)». 
 42. a;' + «/'+3a;«/(a; + v/). 
 44. 5(a;'-.y') + 3(a; + 2/)^ 
 46. 3(fl» + a=6 + aJ^)-(a'-6'). 
 48. 2a;'y + 5a;*/ + 3a;«/'. 
 50. a^-W-(?-2lc. 
 
 53. a;^ + (a + S)a;p + a6. 
 
 54. ««^ + (a-S)a?'-aS. 
 
 56. ar'+y*+2*-2a;y±2a;«±3y2. 
 
 57. 2a*6'+3aV+35V-a*-&*-c*, 
 
 58. a» + 45'' + 9c*+-«-+4a5 + 6ac+ 
 
 37. a; + r + a + 3-a; + l + 3a. 
 39. (a-a-»)«-(i-J-»)». 
 41. aV-3a'a; + 3a*. 
 43. a'-6'-3a5(a-S). 
 45. 3(a;*-2^^)-5(a;-^)2. 
 47. a*-6* + (a2-SY. 
 49. 6^*-3a;^'-9a;«2^*. 
 51. ac + ad+id + lc. 
 53. ^-{a + b)xP+ah. 
 55. a;^-(a-5)a;''-a&. 
 
 [observe the signs. 
 lia^W-{a^ + W-(?f. 
 + 135c+- •• + ••• 
 
 59. m^-n^-m{m^—n^)+n{p,-nY+{ni^+n^){m-n), 
 
 60. a''-a5-3(a5-S^) + 3(a2-J')-4{a-6)l 
 
 61. {x-\-yy+2{oi? + xy)-3(p^-y^) + 4:{xy+y^). 
 
120 MEASURES AND MULTIPLES. tlV.Ps. 
 
 PkOB. 3. To FACTOR AN ENTIEE FUNCTION OF ONE LETTER. 
 
 Find the common factors of the coefficients and divide them 
 
 out; [th. 6 cr. 
 
 by trial, or by comparison with known type forms, find a factor 
 
 of degree not higher than half the degree of the function. 
 If all the coefficients be integers, try no factor unless its first 
 
 and last coefficients measure the first and last coefficients 
 
 of the function; 
 try no factor unless its value measures that of the function 
 
 when the letters have convenient values given to them ; 
 if all the coefficients in the function be positive, try no factor 
 
 whose first and last coefficients are not both positive. 
 
 E.g., of 4f)aa?+\^0axy + 15ay^ the factors are 
 5, a, 8a;H26a;«/ + 15/; 
 
 and •.• 1, 2, 4, 8 are the measures of 8, and 1, 3, 5, 15, of 15, 
 and all the coefficients are positive, 
 .■. the possible factors of 8a;^ + 26a:^ + 15^' are 
 
 ^x+y, 8x+y, 
 
 ix + Sy, 8x+2y, 
 
 ix+by,' 8x + hy, 
 
 4x + 15y, 8x+15y. 
 
 In 8x^ + 23xy + 15y^, and in these sixteen possible measures 
 
 put x=l, y = l', 
 then 8x' + 26xy + 15y''=4:9, whose measures are 1, 7, 49, 
 and only ix + dy, =7, and 2x + 5y,='7, pass this test; 
 and, by multiplication, ix + ^y, 2x + 5y are found to be 
 the factors sought. 
 
 So, of 7a^ — 30a;' + 62a; — 45, the possible linear factors are 
 a;±l, a;±3, x±5, x±9, a;±15, a;±45, 
 7a;±l, 7a;±3, 7a;±5, 7a:±9, 7a:±15, 7a;±45. 
 In 7a:' — 30a^ + 62a; — 45, and in these factors, put a;=l; 
 then the function is — 6, and the only possible fetors of it are 
 a; + l, a;-3, a; + 5, 7a;-l, 7a;-5, 7a:-9. 
 
 x + y. 
 
 2x+y, 
 
 x + Zy, 
 
 2x\Zy, 
 
 x + by. 
 
 2x + 5y, 
 
 x + lby. 
 
 2x + 15y, 
 
8. S3] ENTIRE FACTORS. 131 
 
 So, put x=2; then the function is 15, and out of the six 
 possible factors above the only ones still possible are 
 x + 1, x-d, 7x-9; 
 
 and •/ of these three factors 7a; — 9 succeeds, and the others fail, 
 .•.7a; — 9, a? — 3x + 5 are the fectors sought. 
 
 QUESTIONS. 
 
 1. In factoring an entire function of one letter, why need no 
 factor of degree higher than half that of the function be tried ? 
 what direction in the rule for factoring integers is like this ? 
 
 2. Of what two terms is the first term of the dividend the 
 product ? the last term of the dividend ? 
 
 3. Is a^+x+1 a factor of a^—1 for certain values 
 only of X or for all values? 
 
 In these functions replace a; by 1, 3, 3, • • • in turn, and 
 show that the first result measures the second in every case. 
 
 4. Let a? + baf^ + cx + d be an entire function of x, and 
 5, c, d be all positive; divide by x — a; show that a positive 
 remainder is left at every step of the process; and that, there- 
 fore, a function whose coefficients are all positive has no 
 measure of the form x — a. 
 
 Factor, or prove to be prime: 
 5. 6x^ + x'y-12y\ 6. 6lr'aP-'7h3?-3xK 
 
 7. 6a?+{2a + l)x~{a + 2). 8. 15x' + 8a?y-32xy^-15f. 
 
 9. abx'^ + a'x + i'x + ab. 10. aV-by + e'zK 
 
 11. a^^ + 2a'ba?+2ab^x+¥. 12. a*a;* + a^JV + 6*. 
 
 13. da'+6abz-4:acz-8bcz\ 14. ix^-(9¥ + Wa^)af' + S6aW. 
 
 15. a^+{a~i + c — d)af+( — ab + ae — ad—bo + bd — cd)x' 
 
 + (—abc+ dbd — acd + bcd)x + abed. 
 
 16. abf + (a' + aW + ¥)f + (a5 + a'5 + aW)y + a^l?. 
 
 17. 45a;» + 83a;«2^-100a;2/^-49«/». 18. 73:^-250;= + 11a; + 3. 
 19, 5a;' + 172; + 3. 20. 7a;'-10a;» + 9a; + 5. 21. a^±ab + ¥. 
 32. a^±a''b + ab^±b^ 23. 18a'-34a2-19a + 18. 
 
 24. 8a;'-26a;' + 39a;-13. 25. 12a;*-16a;»-lla;»-8a;-42. 
 
122 MEASURES AND MULTIPLES. [IV, Th 
 
 LINEAR FACTORS. 
 
 First degree factors are linear factors. 
 'E.g.,x — a,y + z,z — d + c. 
 
 Theor. 11. If fx be an entire function of x, then x—a 
 measures fx—fa. 
 For, let /a;=A + Ba;+ca;^H Vlx^, wherein A, B, c-"L 
 
 are free from x, but may contain other letters and 
 
 numerals; 
 
 then /a = A + Ba + caH 1- La", 
 
 and /a;-/a = B(a;-a) + c(a--'-a^)H +L(a;''-a"), 
 
 which is measured by x — a. q.e.d. 
 
 Go^. \. If fx be divided iy x—a, the remainder is fa. 
 For fx=-&(x-a)+G{x^-a') +••• +L(a;"-a") +fa, 
 and each term of the function in this form is divisible by 
 x — a, except /fl, which is free from x, and is the re- 
 mainder. Q.E.D. 
 
 Cor. 3. Iffa=0, then x—a measures fx, and conversely. 
 
 From this corollary comes a new rule for finding linear 
 
 factors of a function of one letter: 
 
 In the function, replace the letter of arrangement by any num- 
 ber; if the function is thereby made zero the letter of 
 arrangement less this number is a factor. 
 
 E.g., to factor a:* - 8a^ + 9a;» + 38a; - 40 : 
 
 put 1 for a;; then /a;=l-8 + 9 + 38-40 = 0, and x-\ is 
 a factor, with the quotient c^ — 7x'' + 2x + iO; 
 
 put 1 for X in this quotient; then /ia;=: 1 — 7 + 2 + 40 = 36, 
 and x—1 is not again a factor; 
 
 put 2 for x; then /xa;-8-28 + 4 + 40 = 24, and x-2 is 
 not a factor; 
 
 put -2 for a;; then /ia;= -8-28-4 + 40 = 0, and a;+2isa 
 factor, with the quotient a^ — 9a; + 20, 
 
 whose factors are a; — 4, a; — 5. 
 
 .•.CB*-8ar' + 9a:H38a;-40 = (a;-l)-(a; + 2)-(a;-4)-(a;-5). 
 
".83] ENTIRE FACTORS. 123 
 
 QUESTIONS. 
 1. If in x^ — a^ X be replaced by a, what does the value 
 of the expression a;" — a" become ? what does this prove ? 
 So, if X be replaced by 'a, when n is even ? 
 So, if X be replaced by ~a, when m is odd ? 
 So, if in a;"+a", a; be replaced by "a when w is odd? 
 Is a;" + a" divisible by either x — a or a; + « when n is even ? 
 3. State, as theorems, all the conclusions reached in ex. 1. 
 
 3. Divide a;* — 6a;' + 10a; — 8 by a; — 3, and compare the 
 remainder with/2. 
 
 So, divide by a;— 4 and compare with/4. 
 
 4. If fx be when x is replaced in turn by a,b, €,••• , 
 then fx is divisible by (a; — a) • (a; — S) • (a; — c) • • • 
 
 5. x — a is a factor of 
 
 (a? + 3a; + 3) • (a» + a) - (a^+ 2a + 3) • (a;' + a;). 
 Find the linear factors of: 
 
 6. 6a;'-7a?-8a; + 16. 7. a* + 4a»+4fl» + 4a + 8. 
 8. a;* + 4a;'-25a:'-16a;+84. 9. a;»-8a;« + 19a;-13. 
 
 10. a'-7a* + 14fl-8. 11. y*^-^f-ty^ + \Oy. 
 
 13. a;' + 8a? + 20a; + 16. 13. ^-Zx^+lOz-S. 
 
 14. c*-13c' + 36. 15. ^-Il2^»+ 18^-8. 
 16. a;* - a? - 39a;* + 34a; + 180. 17. ai^^ 5a;* -9a; -45. 
 
 18.. a' - 8a' + 13a -6. 19. a;*-a;»-lla;'+9a; + 18. 
 
 30. af-3a? + 6a!^-3a?-Sx + 2. 21. Sa" - 6afe + iacz - Sbcz". 
 33. a;»-2a;*-15a;»+8a;' + 68a; + 48. 
 
 23. S{a + by + 8{a + bY + a + b-2. 
 
 24. a^—{a — b + e)'x' + {ac — ab-bc)x + abc. 
 
 35. afly''+y^z''+z'x''—afy'—j/''z^—z''xf is measured by 
 (x—y)-{y—z)- (z — x) if q, r be any positive integers. 
 
 By the light of ex. 25, find the factors of: 
 
 26. a?y+y^z-\-z^x—xy*—yz^-za?. 
 
 27. v?y+y^z+^x—xy^—yi? — zijc'. 
 
 28. a?y^-\-i^z^+^oi?—m?y'—y^i^—z^3?. 
 
124 MEASURES AND MULTIPLES. [IV.Ths. 
 
 FRACTIONS IW THEIR LOWEST TERMS. 
 
 Theor. 12. If the terms of a simple fraction he prime to 
 each other, the fraction can be reduced to no equivalent simple 
 fraction in lower terms. 
 
 For, let a/b be a fraction such that A is prime to B, and p/q 
 
 an equal fraction; 
 then •.• a/b = p/q, [typ- 
 
 .-. A-Q = B.p; 
 and •.■ B measures A- Q, and is prime to A, [hyP- 
 
 .'. B measures Q. [ths. 4, 9. 
 
 So, A measures p. 
 and p/q is not in lower terms than a/b. q. b. d. 
 
 Cor. 1. If a fraction he in its lozoest terms, so is every 
 integer potoer of it. 
 
 Cor. 2. A fraction in its lowest terms can he resolved into 
 but one set of factors and divisors, a", b*" • • ■ g", h'' ■ • •, 
 wherein A, b, ■ • ■ G, h — are different prime numbers, and 
 a, b, — g, h,- • • are integers, some of them negative. 
 
 factors of THE HIGHEST COMMON MEASURE OF TWO NUMBERS. 
 
 Theor. 13. TJie prodtict of all the common prime factors 
 of two or more numbers, each taken with the least exponent it 
 has in any of the numbers, is the highest common measure 
 of the numbers. 
 
 CoR. 1. Every common measure of two or more numbers is 
 a measure of their highest common measure. 
 
 CoR. 2. If each of two or more numbers be multiplied or 
 divided by the same number, their highest common measure is 
 multiplied or divided hy this number. 
 
 Cor. 3. The highest common measure of two or more num- 
 bers is not changed by multiplying or dividing either number 
 by a number prime to any of the others. 
 
12. 13. S3] ENTIRE FACTORS 125 
 
 (iUKSTIOKS. 
 
 I. When is it desirable to change a given fraction to higher 
 terms ? how can it be done ? 
 
 3. If a fraction be in its lowest terms, what is true of its 
 numerator and denominator ? of any integer powers of them ? 
 
 3. Is the fraction 3289/3335 in its lowest terms ? 
 
 4. If the fraction a/b be not in its lowest terms but A, b 
 have the single common factor F, by what process can (a/b)" 
 be reduced to its lowest terms ? 
 
 5. The entire factors of a fraction are factors of which 
 part of it ? the divisors are factors of which part? 
 
 In how many ways can either of these parts be factored ? 
 What kind of numbers are the factors of a fraction ? 
 What, the reciprocals of the divisors ? 
 Why are some of the integers a, b, — hoi theor. 13 cor. 3 
 negative? why not all of them ? 
 
 6. Prove theor. 13, by showing that such a product measures 
 each of the numbers, and that no higher number can measure 
 them all. 
 
 7. How many of the common factors of several numbers 
 are found in their highest common measure ? 
 
 8. In how many of the numbers must a factor be found in 
 order to be a factor of their highest common measure ? 
 
 9. If a common factor be rejected from two or more num- 
 bers and the highest common measure of the quotients be 
 found, what has been done to the highest common measure of 
 the given numbers? 
 
 How must the highest common measure that has been 
 found be changed to give that of the original numbers ? 
 
 10. If' two fractions, when reduced to their lowest terms, 
 have different denominators, their sum can not be an entire 
 number. 
 
 II. If the denominator of a fraction in its lowest terms 
 have other factors than 3 and 5, the fraction can not be exactly 
 expressed as a decimal- 
 
126 MEASURES AND MULTIPLES. [IV,Thr 
 
 FACTORS OF THE LOWEST COMMON MULTIPLE OF TWO OB 
 MOEE IfUMBEBS. 
 
 Theoe. 14. The product of all the different prime factors 
 of two or more entire numbers, each with the greatest exponent 
 it has in any of the numbers, is their lowest common multiple. 
 
 [df. 1. c. msr., th. 4 cr. 5, th. 9 cr. 5, 
 
 Cor. 1. Every common multiple of two or more numbers is 
 a multiple of their lowest common vniltiple. 
 
 Cor. 3. If each of two or more numbers be multiplied or 
 divided by the same number, their lowest common multiple is 
 multiplied or divided by this number. 
 
 Cor. 3. The product of two numbers is the product of their 
 highest common measure and lowest common multiple. 
 
 COMMON MEASURES AND MULTIPLES OF THREE NUMBERS. 
 
 Theor. 15. The highest common measure of three numbers 
 is the highest common measure of the highest common measure 
 of any two of the numbers and the tjiird number j and so for 
 the lowest common multiple. 
 
 common measures and MULTIPLES OF TWO FRACTIONS. 
 
 Theor. 16. The highest common measure of two fractions 
 in their lowest terms is the quotient of the highest common 
 measure of the numerators by the lowest common multiple of 
 the denominators; and their lowest common multiple is the 
 quotient of the loivest common multiple of the numerators by 
 the highest common measure of the denominators. 
 For, let a/b, c/d be two fractions in their lowest terms, 
 and let f/m be a measure of them in its lowest terms; 
 then •.• a/b : f/m, c/d : f/m, i.e., am/bf, cm/dp are entire, 
 
 .". F is a common measure of A, c, and m a common mul- 
 tiple of B, d; [th. 4 cr. 2, th. 9 cr. 2. 
 and f/m is highest when F is the highest common measure 
 of A, c, and M the lowest common multiple of b, d. 
 
14, 15, 16, §3] ENTIRE FACTORS. 127 
 
 So, let m/f be a common multiple of a/b, c/d; 
 
 then ■/ m/f : a/b, m/p : c/d, i.e., bm/af, dm/cf are entire^ 
 
 .'. F is a common measure of B, D, and m a common mul- 
 tiple of A, c, [th. 4 cr. ?, th. 9 cr. 2. 
 
 and m/f is lowest when F is the highest common measure 
 of B, D, and M the lowest common multiple of A, o. 
 
 QUESTIONS. 
 
 1. In the proof of theor. 14, show that each of the numbers 
 is a measure of such a product, and that, if any factor were 
 omitted from this product or taken fewer times, some one of 
 the numbers would no longer be a measure of it. 
 
 3. In the lowest common multiple of A, B, what factors of 
 A are found, and what factors of B are among them ? 
 
 What other factors must be added ? 
 
 3. Let A, B be any two entire numbers, H their highest 
 common measure, L their lowest common multiple, and let 
 A=aH, b = 5h; let a have the factor c" and B the factor c", 
 and m>n: how many times is the factor c in h ? in a ? in L ? 
 
 If m<n, how many times is c in H ? in 5? in L? 
 
 4. A highest common measure found as in theor. 15 contains 
 all the factors common to all the numbers and no others; and 
 a lowest common multiple contains all their factors. 
 
 5. In the proof of theor. 16, if the fractions be multiplied 
 by the lowest common multiple of their denominators, what 
 do the fractions become ? How is the highest common measure 
 of these products related to that of the given fractions ? 
 
 If the fractions be in their lowest terms, can the lowest 
 common multiple of the denominators contain any factor in 
 both numerators? Are the multipliers of the numerators 
 prime to each other? 
 
 6. Let a/b, c/d be two fractions in their lowest terms, and 
 let B=:5'F, d = ^-f: what is f? what is S-t?-F? 
 
 Multiplying both these fractions by J • <? • F gives two integers 
 whose lowest common multiple is hd times the lowest common 
 multiple of a, c, and Idv times that of the fractions. 
 
128 MEASURES AND MULTIPLES. [IV.Pb. 
 
 § 4. THE HIGHEST COMMON MEASUEB. 
 
 PkOB. 4. To EIITD THE HIGHEST COMMON MEASURE OF 
 TWO OB MOEE ENTIRE NUMBERS. 
 
 (a) The prime factors of all the numbers known: rmiUiply 
 together all the different pritne factors, each with the 
 greatest exponent it has in all of the numbers, [th. 13. 
 E.g., of 9a'5^c, Sabred, 15ab'(f-12ab'', the common factors 
 are 3, a, W; and the highest common measure is Zab^. 
 So, ••• 23?y + Qxy* -<ooi?-%y^-2{x- y) (x +y){y- 3a;) 
 16ase'y — 12aa?—4:axy''—4:ax{x—y)(y — 3x) 
 lOy - 502;^^ + lOx'y - 303? = 10(a; - yY{y - 3a;), 
 
 .'. the h. c. msr. of these expressions is 2{x-y){y-3x). 
 (J) The prime factors not known, two entire numbers: divide 
 the higher number by the lower, the divisor by the 
 remainder if any, that divisor by the second remainder, 
 and so on till nothing remains; [ths. 2, 7. 
 
 at pleasure, suppress from any divisor any entire factor that 
 is prime to the dividend corresponding, and introduce 
 into any dividend any entire factor that is prime to 
 the divisor; [th. 13 cr. 3. 
 
 at pleasure, suppress from any divisor and the corresponding 
 dividend any common measure of them, but reserve this 
 measure as a factor of the final result. [th. 13 cr. 2. 
 The product of the last divisor, as above, by the reserved fac- 
 tors if ajiy;^lift» highest common measure sought, [ths. 2, 7. 
 E-g.,"^© find the h. 0. msr. of a.''+a;-13, s^-lOaj + ^l: 
 
 write a?-\-x-12)x'-10x + 2l{X or 1 1 "12)1 -10 +21 
 
 a?+ a:-12 1 1 -12 
 
 -11 ) -11a; + 33 -11 )- 11 33 
 
 i^-3x a;-3(a;+4 1 -3 1 "3 
 
 4a;- 12 4-12 
 
 and a;— 3 is the measure sought. 
 
 So, of 4aa;^ + 4aa;-48a, 4aa;* - 40aa; + 84a, 4fl is a common 
 factor, a; — 3 is the h.c.msr. of the remaining factors, 
 and 4a(a; — 3) is the highest common measure. 
 
4,8 4] THE HIGHEST COMMON MEASURE. 129 
 
 QUESTIONS. 
 
 Find the highest common measure of: 
 1. x-1, a^-l, a^-l. 2. l-x', (l+xf, 1 + a?. 
 
 3. ^{a^-a'x), 4:{x' + ax), 5(£B*-a*), 6(a^-a«). 
 
 4. a^+2x-3, a?-7x'+6x. 5. 3a^-34a;-9, 2a;'-16a;-6. 
 
 6. l-3a;, 1-ia^, I-Sjb'. l-16a;*, l-SSa;^ 
 
 7. a^+a^ + a^ + a^ + x + 1, x'-x + l. 
 
 8. i + 5x + x% 8-2X-X', n + 7x + x% 20 + x-a?. 
 
 9. 529(a;Ha;-6), 782{3a;^ + 7a; + 3), 935(aa^-3a;-2). 
 10. a;' + 32;^ + 4a; + 12, a^-5x^ + 'ix-2Q. 
 
 Reduce these fractions to their lowest terms : 
 
 x^.-6x + 5 l + 3a;-4a:^-12a;' 
 
 72^-12a; + 5* 8a^ - 4a;2 - 2a; + 1 ' 
 
 l + g:'' + 25a;* 3ffl^-18a' + 33a-18 
 
 1 + 3a; -15a;* -25a;* 12«' - 84a + 72~" 
 
 15. Given 3a^-10a + 3, 65*+7&-30, m' + w», all prime 
 
 to each other: find the highest common measure of 
 (3a2-10a + 3)'-(65H75-20)*-(»i'+w'')* 
 (3aV 10a + 3)* • (66* + 75 - 20)' • (m' + w») , 
 Find the highest common measure of: 
 
 16. a*+4a' + 4a^ a^b-iab, a^b + 5a^b + 6a^b. 
 
 17. a^-{y+iy, y^-{x+iy, l-{x+yf. 
 
 18. a^-b', a^+ab + W, a^+a'b^ + bK 
 
 19. a;'+4a;* + 4a-+3, a;*-a;'-a;-3. 
 
 20. 4a;* + 9a;' + 2a;*-3a;-4, 3ci? + 5x>-x+2. 
 
 21. 2a;'+(2a-9)a;^-(9a + 6)a;+27, 2a;*-13a; + 18. 
 
 22. 4a''-4aa;-15a;^, 6ffiH7fla;-3a;'. 
 
 23. no? + Znoi?y — 27ixy^ — 2n'f, ima?+ma?y~2mxy'—dmy\ 
 
 24. a;*-;?a;' + (5'-l)a;''+joa;-g', x^-qx'+(p-l)x''+qx-p. 
 
 25. 0?+ {4:a + b)x'+ {Sa' + 4rab)x + 3a% 
 
 a;' + (2a-b)x'- (3a* + 3a5)a; + 3a*5. 
 
 26. a'e*^ + e*^-a'-l, (a-2 + a-i)-(e'"-2 + 6-"'). 
 
130 
 
 MEASURES AND MULTIPLES. 
 
 [IV.Pb. 
 
 Note 1. The arrangement of terms may be as to the rising 
 powers of the letter of arrangement, or as to the falling powers. 
 B.g., 3a^ + ll£B^ + 20a; + 21 and af-x-G; 
 or 21+20a; + lla^ + 3a;' and 6 +x-a?. 
 
 That arrangement is commonly best which makes the trial 
 divisor smallest; and at any step of the work the highest or 
 the lowest term of the divisor may be used as trial divisor. 
 Kg., to find the h. c. msr. ot a? + 37^ + 5x+3, x^ + Gce'+Qx+i, 
 write x'^ + 33f+ 5a; + 3 
 
 92;H12a; + 3 
 
 a?- Gx^-lx 
 -21a? -28a? -7x 
 
 22^+22^(22a;« 
 
 x + 1 
 
 1 
 
 x' + 6x'' + 2x+4: 
 
 
 a^ + Sx' + bx + S 
 
 Z-7x 
 
 3a^+4a;+l 
 
 3a; + l 
 
 33?+3x 
 
 x+1 
 
 (c) TJie prime factors not known, three or more entire num- 
 
 bers : find the highest common measure of any two of 
 them (preferably the lowest), the highest common meas- 
 ure of this measure and the next number, and so on. 
 
 E.g., to find the highest common measure of 
 
 ar'+a;-12, a;''-10a; + 21, a;' -Gk^- 19a; + 84: 
 
 then-.'of a?+x—\2, a;'-10a;+21, the h. c. msr. is a;-3, 
 
 and x — 3 measures a;' -6a;*— 19a; + 84, 
 
 .'. a; — 3 is the measure sought. q.e.e. 
 
 (d) Some or all of the numbers fractions . divide the highest 
 
 common measure of the entire numbers and the numer- 
 ators by the lowest common multiple [pr. 5] o/ the 
 denominators. [th. 16. 
 
 E.g., to find the highest common measure of the fractions 
 (a;^ + a;-12)/(a;-5), (a;^-10a; + 21)/(a: + 5): 
 
 then •.■ the h. c. msr. of the numerators is a; — 3, [above. 
 
 and the 1. c. mlt. of the denominators is a;* — 25, [inspection. 
 .-. (a; — 2)/(a-'^ — 25) is the measure sought. 
 
4,54] THE maHEST COMMON MEASURE. 131 
 
 QUESTIONS. 
 
 Eeduce to lowest terms by means of the highest common 
 measures of their numerators and denominators: 
 
 x^ + a^+af + 1 a^-a*^ 
 
 l-x'+a^-x^' a^+a'^x-ace'-s^ 
 
 c^ + 3^ ^ 0* —afl a;y-' + 8 + a;-'y 
 
 x^^ + a^o' a»">-a?»" xy^+x-^y 
 
 a?+y-^. „ <i?—y-^ a:-'+lla;-' + 30 
 
 WTy~^' x^-y-"^ 9a;-»+53a!-«-9a;-^-18' 
 
 Find the highest common measure of: 
 9. 15iB'-9a^ + 47a;»-21a; + 28, 
 
 20a^-12a;'+16a;*-15a;*+14a;»-15a;+4. 
 
 10. a^-a?-^3?+bx-2, a?-2x^^-a?+5x'-4x+l. 
 
 11. Baf+{'La-2b)x-2ab + a'', 3?+{2a-b)a?-{2ai-a')x-d?b. 
 
 12. 2a^ + 3a;+l, %3?+5x+2, 23?+ba?-ix-Z. 
 
 13. 13r'-2a;*-7a;+2, 183?-^a?-Sx + A, 36a!*-25a;«+4. 
 
 14. da''x + 6aix+db% 12a'-12ab + ZP, 10a'' + 5ab-ol^. 
 
 15. a^y + 12x'y + S5xy, x'+3c(^-283^, s^z-a^z-56xz. 
 
 16. Prove that the highest common measure of two or more 
 
 numbers is the reciprocal of the lowest common mul- 
 tiple of their reciprocals. 
 
 Find the highest common measure of: 
 
 i^- ^4^ 64' 16' 128' ^' y'^ y y ' 2y y 2^y' 
 x'-ix a?-8x+16 a?-2x-8 
 
 20. 
 21. 
 
 a?+y^'' a^-xY+y^' af^+y" 
 
 6a;'+10a;-24 2a;'-2a;-24 8a;^ + 22a;-6 
 
 a^+5a^+6x ' a?y — a?y — Qxy' a?-y — 2y 
 
 a?+2cx+(!' ' a?+{c-b)x-bc 3a;«-3c« 
 
 {x+a){x+bY x^+{a + c)x+ao' x^+{b+c)x+bo' 
 
132 MEASURES AND MULTIPLES. [rv.PR. 
 
 §5. THE LOWEST COMMON MULTIPLE. 
 
 PeOB. 5. To FIND THE LOWEST OOMMOIT MULTIPLE OF TWO 
 OB MOEE NUMBERS. 
 
 (a) The prime factors of all the numbers known : multiply 
 
 together all the different prime factors, each with the 
 greatest exponent it has in any of the numbers, [th. 14. 
 
 E.g., of Qabh-\ 12aWd\ Ua' + ^Wbd, the prime factors, 
 in their highest powers, are 2", S% a", V, c", d^, ba^' + lbd, 
 
 and the lowest common multiple is IQWb^d^ + 2b2a^V'd^. 
 
 (b) TJie prime factors not Tcnoton, two entire numbers: divide 
 
 either number by iheir highest common measure and 
 
 multiply the quotient by the other number, [th. 14 or. 3. 
 E.g., to find the lowest common multiple of 
 
 a?+x-12, 3?-lOx-\-21: 
 then their h. c. msr. is a; — 3, and the multiple sought is 
 
 (a? +x-\2):{x-3) • {x^ -10x + %l)-a?-&7? -l^x + S4^. 
 
 (c) The prime factors not known, three or more entire num- 
 
 bers : find the lowest common multiple of any two of the 
 numbers (preferably the highest), the lowest common 
 multiple of this multiple and the next number, and so on. 
 
 E.g., to find the lowest common multiple of 
 
 a;' + 6a;= + 9a; + 4, a? + Zoi?-llx-12, 3?-1lx-Q: 
 
 then the 1. c. mlt. of a;' + 6a^ + 9a; + 4, a? + %x^- 11a; - 12 is 
 (a; + l)^(a; + 4)(a;-3), 
 
 and the 1. c. mlt. of (a; + l)2(a; + 4) (a; - 3), a;' - 7a; - 6 is 
 (a; + l)=(a; + 4)(a;-3)(a; + 2). 
 
 {d) Some or all of the numbers fractions : divide the lowest 
 common multiple of the entire numbers and numerators 
 by the highest common measure of the denominators. 
 
 E.g., to find the lowest common multiple of the fractions 
 {x" - y^)/{a' -a- 30), (a;« - «/*)/(«' + 6a + 8) : 
 
 then •.• the 1. c. mlt. of x^—y^, x^—y^ is af'—y^, 
 
 and the h. c. msr. of a^ — a — 20, a^ + 6a + 8 is a + 4, 
 .*. the multiple sought is («° — y*)/(a + 4). 
 
6,85] THE LOWEST COMMON MULTIPLE. 133 
 
 QUESTIONS. 
 
 Find the lowest common multiple of: 
 
 1. ea^-x-l, 2a;^ + 3a;-2. 3. a + 1, 1-a, a' + a + l. 
 
 3. a^-5», a^ + 2ab + iK 4. x'-6x^-x + Z0, a? + Zx^-L 
 
 Reduce to their lowest common denominators: 
 
 v(?-\-y^ a?—y^ a? + xy+y^ 3? — xy + y^ x—y x + y 
 
 6. 
 
 a?-y^ ar'+y^ a?+y'' ' 
 a + b a-b a^+¥ €?-¥ 
 
 a-b' a + b' a^-W a^-\-l 
 Add the fractions: 
 
 13 5 
 
 7?-f 
 
 a?-¥' 
 
 ' x + if x — y 
 aJ' + W 
 
 sX 
 
 3 2a;-3 
 
 2(a+a;)' 4:{a-x)' G{a'-a?)' 
 9. 5/6(a^ + a^), l/Siff-x"), 9/10{a'' + ax+a?). 
 
 10. l/4:a^{x+y), l/23?{a?+y% \/i3?{x-y), l/%a?{a?-y% 
 Find the lowest common multiple of: 
 
 11. a?-^a^, x' + ^ax' + Wx + M^ a?-2aa^+4:a^x-8a\ 
 13. 4:{x>-y''){x-yy, \2{a*-y%x-y)\ 20{3?-yf. 
 
 13. ^-'^c?, a? + 6ax' + 12aJ'x+8a\ a^-GasP + Ua'x-Sa^ 
 
 14. mV — 2TO?i' + Ji*, a^ + 2xy+y\ mnx — nx-ny + mny. 
 
 15. «*—«/*, ^ — y^> ^—y^> a? — 2xy+y^, a? + 2xy-\-y^. 
 
 16. a^m^ + Sam + lQ, a^-4, aH3a + 2, a*-3a + 2. 
 
 17. a?-6x> + llx-6, a?-9x' + 26x-24:, a;'-8a;^ + 19a;-13. 
 
 18. a*+aW + b^, a*+3a^b + 4aW + 3aP + b\ 
 
 19. 4ffi»-4a'-29a«-21, 4a« + 24a* + 41aH21. 
 
 X' + X 7? — X X^ + X 0^ — X 
 
 20. 
 21. 
 22. 
 23. 
 
 2-2a;^' S + Sa;' 13 + 133;*' l + 2a; + 2«*+a;'" 
 
 d' + bx + A o? + 2x-% o?-\-lx + \2 a;*-3a;-4 
 
 ic^-4a; + 3' ar'-a;-12' a*+a;-20' a? + 2x-l&' 
 x^—y^—z^ + 2yz ci?—y^ + z^—2xz 
 
 x^ + 2xy + y^-z^' -Sx^ + Sy^-Gyz + dz"' 
 27?+7?-bx-3 23?+bx'~x~Q a?-\ 
 
 a^+2a? + 23?+xf a?-2x-l ' a^-a?-2x-l' 
 
134 MEASURES AND MULTIPLES. t^, 
 
 § 6. QUESTIONS FOR REVIEW. 
 
 Define and illustrate: 
 
 1. An integer; an entire function of one letter; an entire 
 number. 
 
 2. A multiple of an integer; a measure. 
 
 3. A multiple of an entire function of a letter; a measure. 
 
 4. An entire factor of an entire number; a linear factor. 
 
 5. A common multiple of two integers; a common measure. 
 
 6. A common multiple of two entire functions of a letter; 
 a common measure. 
 
 7. The lowest common multiple of two or more entire num- 
 bers; the highest common measure. 
 
 8. A prime integer; a prime entire function of one letter. 
 
 9. Two numbers prime to each other; a composite number. 
 
 10. State the axioms that relate to the sum, difference, and 
 product of two entire numbers. 
 
 11. Show how a common measure of two entire numbers is 
 related to their sum; to their difference; to the sum and the 
 difference of any multiples of them. 
 
 12. State Euclid's process for finding the highest common 
 measure of two entire numbers, with proofs and illustrations. 
 
 Show that every remainder so found is the difference of two 
 multiples of the given numbers; and that, if the numbers be 
 prime to each other, two multiples of them can be found 
 whose difference is either a unit or some expression that is 
 free from the letter of arrangement. 
 
 State and prove the converse of this proposition. 
 
 Prove that these statements are true: 
 
 13. If an entire number be prime to two or more such num- 
 bers, it is prime to their product. 
 
 14. If two entire numbers be prime to each other, so are 
 any positive integer powers of them. 
 
 15. If an entire number measure the product of two such 
 numbers and be prime to one of them, it measures the other. 
 
S«l QUESTIONS FOR REVIEW. 13^ 
 
 16. If an entire prime nuni'ber measure the product of two 
 Or more entire numbers, it measures at least one of them, 
 
 17. If an entire prime number measure a positive integer 
 power of an entire number, it measures the number, and if 
 the number, then the power. 
 
 18. If an entire number be measured by two such numbers 
 that are prime to each other, it is measured by their product. 
 
 19. An entire number can be resolved into prime factors in 
 i-but one way. 
 
 20. A common measure of two or more entire numbers can 
 contain no factor that is not in all of them. 
 
 21. If /*•, an entire function of x, be divided by x — a, the 
 remainder is /a; and if /a = 0, a; — a is a measure of /a;. 
 
 22. If the numerator and denominator of a fraction be 
 prime to each other, the fraction is in its lowest terms j and 
 so is every integer power of it. 
 
 23. "Which factors of two entire numbers are factors of their 
 highest common measure ? of their lowest common multiple ? 
 
 What effect has it upon their highest common measure to 
 multiply or divide either of the numbers by a number that is 
 not a factor of the other number ? by a number that is a fac- 
 tor of the other ? upon the lowest common multiple ? to 
 multiply or divide both numbers by the same number? 
 
 34. What is the product of the highest common measure 
 and the lowest common multiple of two numbers ? 
 
 Give the general rule, with reasons and illustrations, for: 
 
 35. Factoring an integer; factoring a polynomial of known 
 type-form; factoring an entire function of one letter; finding 
 the linear factors of an entire function of one letter. 
 
 26. Finding the highest common measure and the lowest 
 common multiple of two or more entire numbers whose prime 
 factors are known; of two entire numbers whose prime factors 
 are not known; of three or more entire numbers whose prime 
 factors are not known ; of fractions. 
 
136 VARIATION, PROPORTION, INEQUALITIES. [V, 
 
 V. VARIATION, PROPOETION, INEQUALITIES, AMD 
 INCOMMENSURABLE NUMBERS. 
 
 Hitherto concrete numbers, integers or fractions, have been 
 found by counting like entire units, — apples, horses, guests — 
 or by measuring by some definite unit, or part of a unit, and 
 counting the number of times the unit is contained in the 
 thing measured. Abstract units and simple fractions express 
 repetitions and partitions and their combinations. 
 
 Such numbers are conmiensurable numbers. 
 
 Hitherto also a number has been thought of as something 
 fixed and definite, and if a letter were used to denote a num- 
 ber, it was some fixed and definite number. 
 
 Such numbers are constants. 
 
 But now come two new notions : that of changing values, 
 variables; and that of numbers, definite and distinct, which 
 however cannot be expressed by repetitions and partitions, 
 incommensurable numbers. 
 
 §1. VAEIATION. 
 
 If a boy count apples — one, two, three • ■ • , as he drops them 
 into a basket, the number of apples in the basket changes and 
 increases ; or, if he have twelve apples at the start and take 
 them out one by one, the number left changes and decreases — 
 twelve, eleven, ten,- • •, and the number of apples in the basket 
 is a variable. Or, if he count the horses that pass through a 
 gate into a field — one, two, three, ■ • • , or the guests as they 
 rise from table, or the 3's he gets when he reduces the simple 
 fraction 1/3 to a decimal : in all these cases the numbers so 
 found are variables. 
 
 So, if the cross-section of the trunk of a growing tree be a 
 circle, it is a circle whose radius, circumference, and area are 
 all variables; and a growing peach is a sphere whose radius, 
 circumference, surface, and volume are variables. 
 
 So, with a sum of money at interest, the principal and rate 
 are constants; the time and accrued interest are variables. 
 
Sn VARIATION. 187 
 
 QUESTION'S. 
 
 1. Show that a variable may increase or decrease by regular 
 additions or subtractions, or by irregular ones. 
 
 2. In annexing successive 3's to the decimal expression of 
 the fraction 1/3, are the successive additions to the value of 
 the variable the same or different ? 
 
 3. Is the number 1/9 a variable ? is its decimal expression 
 a variable ? can the same number, then, be both a variable 
 and a constant, or is 1/9 not equal to its decimal expression ? 
 
 4. Is the reciprocal of a variable a constant or a variable? 
 How does the reciprocal of ^n increasing variable change ? 
 
 the opposite ? the opposite of the reciprocal ? 
 
 5. In general, is the sum of two variables a constant or a 
 variable? their difference? their product? their quotieiit ? 
 
 Show by examples that the sum, the difference, the prod- 
 uct, and the quotient, of two variables may be constants. 
 
 6. In the case of an express train running ov6r a long, level, 
 straight track, with the same pressure of steam, which of the 
 following elements are variables: the time si nce t he train 
 started, the distance it has run, the~speed, the relation of the 
 distance to the speed, its'^relation to the time ? 
 
 7. If the wine in a full cask run into an empty one, what 
 two variables are there during the process? what constant? 
 
 Do these two variables vary in the same way ? 
 
 8. The diagonal of a square whose side is a is \/2a'; what 
 is the diagonal of a square whose side is 5 ? 
 
 What relation does the diagonal of a square bear to a side ? 
 If the length of a side change, does the diagonal change ? 
 Does the relation between the side and diagonal change ? 
 
 9. What is the area of a square whose side is a? of one 
 whose side is 5? What is the relation of the area to the side ? 
 
 Is the area of a growing square a fixed number of times the 
 side ? What definite law connects these two magnitudes? 
 
 If the length of a side can not be exactly expressed, what 
 effect has that on this law ? 
 
138 VARIATION, PROPORTION, INEQUALITIES. [V, 
 
 CONTINUOUS AND DISCONTINUOUS VAEIABLBS. 
 
 If a concrete variable, in passing from one value to another, 
 pass through every intermediate value, it is a continuous 
 variable; otherwise it is a discontinuous variable. 
 
 So, the abstract ratios of these concrete variables to the 
 constant measuring unit are continuous or discontinuous 
 variables. 
 
 E.g., if a man has waited two hours for an incoming steamer, 
 he has also waited an hour, an hour and a quarter, an hour 
 and a'half, and every other portion of time less than two hours 
 that can be named or conceived of; and he has been conscious 
 of the continuous passage of time. 
 
 So, if he run along the street he knows that he cannot get 
 from one fixed point to another without going through every 
 intermediate point of some path, and that the distance run 
 has a continuous growth. 
 
 But, of the regular polygons inscribed in a circle, one may 
 have three sides, another four, another five, and so on; but 
 no one can have four and a half sides, nor five and a quarter 
 sides. The number of sides, the perimeter, and the area- are 
 all discontinuous variables. 
 
 EELATED VARIABLES. 
 
 If two variables be so related that the value of one of them 
 depends upon that of the other, the first variable is a junction 
 of the other. [compare IV, § 3. 
 
 E.g., with a given principal and rate, the accrued interest is 
 a function of the time. 
 
 So, the circumference and area of a growing circle are 
 functions of the radius. 
 
 So, of the regular polygons inscribed in a given circle, the 
 perimeter and area are functions of the number of sides. 
 
 A variable may be a function of two or more variables. 
 
 E.g., the volume of a stick of timber is a function of its 
 length, breadth and thickness; and the cost is a function of the 
 length, breadth, thickness, and cost per foot. 
 
SI] VARIATION. 139 
 
 QUESTIONS. 
 
 1. Can a variable be continuous, and its reciprocal be dis- 
 continuous ? its opposite ? its square ? its square root ? 
 
 2. State whether the numbers below are constants or vari- 
 ables, and if variables, whether continuous or discontinuous: 
 
 the length of a line revolving about the centre of a circle as 
 a pivot and reaching to the circumference; 
 
 such a line revolving about any other point than the centre; 
 the principal at simple interest; at compound interest; 
 the size of an angle if the bounding lines be lengthened; 
 the number of telegraph poles passed by the electric current. 
 
 3. If a falling body has at one moment a velocity of 30 ft. 
 a second, and later a velocity of 40 ft. ; how many other dif- 
 ferent velocities has it had between these times ? 
 
 4. What relation does the perimeter of a square bear to the 
 length of one side ? If the side, for any reason, can not be 
 measured, does that fact affect this relation ? 
 
 5. The circumference of a circle is readily seen to be some- 
 what more than three times the diameter; the exact relation 
 is found by geometry, and it is usually represented by the 
 Greek letter it, read pie, whose value is nearly 3.1416: what 
 is the relation of the circumference of a circle to its radius f 
 does this relation depend in any way on the length of the 
 radius ? on the length of the circumference ? 
 
 6. If the radius of a circle increase, does the circumference 
 increase at th^^ same rate or at a greater or less rate ? the area ? 
 
 7. As interest is computed in business, for days and not for 
 fractions of a day, thus making the time discontinuous, is the 
 accrued interest a continuous or a discontinuous function of 
 the time ? the amount ? 
 
 8. If a; be a continuous variable, is a? — 2af' + x + o a con- 
 stant, or a continuous, or a discontinuous, function of a;? 1/x? 
 
 9. If the diameter of a peach grow continuously, does the 
 circumference grow continuously ? the surface ? the volume ? 
 the weight ? the value ? 
 
140 VARIATION, PROPORTION, INEQUALITIES. [V, 
 
 RELATIVE VARIATION. 
 
 One number varies directly as another if when the second 
 is doubled so is the first, when the second is tripled so is the 
 first, when the second is halved, so is the first, and so on. 
 E.g., with a constant principal and rate the interest accrued 
 
 varies directly as the time. 
 So, with men of equal efficiency, the amount of work done in 
 a day varies directly as the number of men employed. 
 The sign of variation is oc, read varies as. 
 E.g., if i stand for simple interest and t for time, then icxi. 
 So, if w stand for the work done in a day and m for the num- 
 ber of men, then wocm. 
 
 If two abstract numbers, or two concrete numbers of the 
 same kind, vary in such wise that some relation between their 
 magnitudes shall continue to hold true, this relation may be 
 expressed by an equation. 
 
 E.g., if d be the diameter of a circle and r its radius; 
 then doer, and d=2r. 
 
 One number varies inversely as another, if when the second 
 is doubled the first is "halved, when the second is tripled the 
 first is trisected, when the second is halved, the first is doubled, 
 and so on. If the numbers be abstract one varies directly as 
 the reciprocal of the other. 
 E.g., the time of running a fixed distance varies inversely as 
 
 the speed. 
 So, the illuminating power of a light varies inversely as the 
 square of its distance. 
 
 So, if X, y, be two abstract numbers and x vary inversely as y; 
 then x<xl/y, and a; = ^/t/, wherein ^ is some constant. 
 
 One number varies jointly as two others if it vary directly 
 as their product. 
 
 E.g., if w stand for the wages earned, m. for the number of 
 men and t for the time, then wonm-t, or w = lc-m-t. 
 
 [k, & constant. 
 
§1] VABIATION. 141 
 
 A numlber may vary directly as one number and inversely as 
 another. 
 E.g., tlie time of running varies directly as the distance run 
 
 and inversely as the speed. 
 So, the value of a fraction varies directly as the numerator 
 
 and inversely as the denominator. 
 
 QUESTIONS. 
 
 1. What is meant by saying that the circumference of a 
 circle varies as the diameter? that the radius varies as the 
 circumference ? that the area varies as the square of the radius ? 
 
 If a stone be dropped into a pond of water, how many of 
 these relations will be true at each and every stage of the 
 growing circular wave ? 
 
 2. Does the light received from a lamp vary as its distance ? 
 how then ? the entire cost as the number of like articles bought? 
 the time needed for a piece of work as the number of men ? 
 
 3. If a lady spend the same sum for gloves every year, but 
 the price increase, how does the number of pairs bought vary ? 
 
 4. Of rectangles having the same area, how does the width 
 vary? the length? the perimeter? the'diagonal ? 
 
 5. If the sum of two numbers be constant, will one vary as 
 the other ? inversely as the other ? if the product be constant ? 
 the quotient ? the difference ? 
 
 6. If at different times varying numbers of like things be 
 bought, and at varying prices, how do the bills vary ? 
 
 7. If the dividend be constant and the divisor vary, how 
 does the quotient vary ? if the divisor btf constant and the 
 dividend vary? if the dividend and divisor both vary? 
 
 May the dividend and divisor both vary and the quotient 
 remain constant? 
 
 8. In simple interest is the principal a constant or a varia- 
 ble ? with a fixed principal and rate, how do the interest and 
 time vary? that a fixed principal, at interest for various 
 periods of time, may bring in the same interest for each 
 period, how must the rate be related to the t'.me ? _, 
 
142 VARIATION, PROPORTION, INEQUALITIES. [V, 
 
 §3. PEOPOETION. 
 The ratio of one quantity to another quantity of the same 
 kind is the number found by measuring the first quantity by 
 the other as a unit; i.e., it is the multiplier that, acting on 
 the second quantity as a unit, gives the first quantity as result. 
 The ratio of two like numbers is the quotient of the first by 
 the other, and it is an abstract number. 
 E.g., the ratio of 13 bushels of wheat to 4 bushels of wheat is 
 3, and the ratio of 4 bushels to 13 bushels is' 1/3. 
 The ratio of a to & is written a:b or a/b. 
 No ratio is possible between unlike numbers. 
 The ratio a:l is the direct ratio of a to b, and b: a is 
 the inverse ratio. 
 X In a direct ratio, the dividend is the antecedent and the 
 divisor is the consequent. 
 
 The equation of two ratios is a proportion, and the four 
 numbers forming the two ratios are the terms of the propor- 
 tion, or the Iqvli: proportionals. 
 
 E.g., if the ratio of a to 5 equals that of c to d, this fact may 
 be stated in a proportion in three ways : 
 a:h::c:d, a/b = c/d, a:b = c:d. 
 The first form is read, a is tob asc is to d; and the others 
 more briefly, ato b equals c to d. 
 
 A proportion between concrete numbers is but an expanded 
 statement of direct variation. ' 
 
 E.g., 3 years : 3 years = 3 years' interest : 3 years' interest 
 means that, the principal and rate being constant, the 
 interest varies as the time. 
 
 The first and last terms of a proportion are the extremes, 
 the second and third the means, the first and third the ante- 
 cedents, the second and fourth the consequents. 
 
 If the means of a proportion be the same number, the com- 
 mon term is the mean proportional, and the last term is the 
 third proportiotial. 
 
52] PROPORTION. 143 
 
 E.g., in the proportion a:l = b:c, J is the mean propor- 
 tional between a, c, and c the third proportional to a, b. 
 If three or more ratios be equal, they may be written in 
 succession, with the sign = between them. Such an expres- 
 sion is a continued proportion. 
 E.g., a:l — c:d=e:f=g:h. 
 
 QUESTIOlirS. 
 
 I. Can a ratio be concrete ? can it be negative ? 
 
 3. If two numbers be equal, what is their ratio ? opposites ? 
 if the antecedent be the larger ? the consequent ? 
 
 3. What two relations between the antecedent and conse- 
 quent make the direct and inverse ratios equal ? 
 
 4. If a ratio be zero, which term is zero ? if infinite ? 
 
 5. Name two other numbers that have to each other the 
 same ratio as 10 has to 5, and write these four numbers as a 
 proportion. Find other pairs of numbers having the same 
 ratio, and of them all, make a continued proportion. 
 
 6. As a statement of variation, interpret the proportion 
 
 3 hrs. : 5 hrs. = distance run in 3 hrs. : distance run in 5 hrs. 
 
 7. As a proportion write the statement that the areas of 
 circles vary as the squares of their radii. 
 
 8. If the antecedents of a proportion be equal, what is true 
 of the consequents? if one antecedent be ten-fold the other? 
 
 9. If the first antecedent be the square of its consequent, is 
 the second antecedent likewise the square of its consequent ? 
 
 10. Of the continued proportion at the top of the page, make 
 six different simple proportions. 
 
 II. What effect is produced on a ratio by doubling the 
 antecedent ? the consequent ? both terms ? 
 
 13. Regarding a proportion as an equation between two 
 fractions, show why it is allowable to multiply or divide both 
 antecedents by the same number ? both consequents ? both 
 terms of one ratio ? all four terms ? to multiply one antece- 
 dent and divide the other consequent by the same number ? 
 
144 VARIATION, PROPOETION, INEQUALITIES. [V.Tna 
 
 PROPBKTIES OB PROPORTIONS. 
 
 A proportion is but aa equation whose members are frac- 
 tions; and the principles already established for fractions and 
 for equations apply directly to the proportions used in the 
 proof of the theorems below. 
 
 In these proofs, the proportionals are abstract numbers, 
 
 Theor. 1. In any proportion, the product of the extremes 
 equals that of the means. 
 Let a:b = c:d, then will ad=bc. 
 
 For •.• a/i = c/d, [hyp. 
 
 .'.ad=ic. Q.E.D. [mult, by St?; ax. mult. 
 
 Cor. 1. Either extreme is the quotient of the product of the 
 means when divided ly the other extreme; and so for the means. 
 
 Cor. 2. A mean proportional is the square root of the prod- 
 uct of the extremes. 
 
 Theor. 2. If the -product of two numbers equal the product 
 of two others, the four numbers may form a proportion, in 
 which the factors of one product shall be the extremes and those 
 of the other product the means. 
 Let ad=bc; then a:b = c:d, a:c=b:d. 
 For the first proportion, divide both members of the equation 
 ad^ic by bd; for the other, divide by cd. 
 
 Theor. 3. In any proportion, the two means may change 
 places. [alternation. 
 
 Let a:b = c:d; then a:c=b:d. 
 
 Prove by aid of theors. 1, 2. 
 
 Theor. 4. In any proportion, the consequents may change 
 places with their antecedents, [inversion. 
 
 Let a:b = c:d', then b:a = d:c. 
 Prove by aid of theors. 1, 2. 
 
 Theor. 5. In any proportion the sum of the first two term,s 
 is to the first term or the second as the sum of the last two terms 
 is to the third term or the fourth, [composition. 
 
1, 2, 3, 4, 5, § 2] PROPORTION. 145 
 
 QUESTIONS. 
 
 1. What is the test of the correctness of a given proportion ? 
 
 2. Supply the missing tei-ms in the proportions below : 
 
 6:4 = 9: (>; 7:;=-35:-5; -9:-.= :16; 
 
 :3 = 0:10; :16 = 4: ; :-9 = 16: . 
 
 Is there more than one way of completing the third of these 
 proportions ? how many ways? could the two means be equal ? 
 
 3. How is theor. 2 related to theor. 1 ? 
 
 4. In proving theor. 3, what multiplier will change the given 
 ratio a/b to the desired one, a/a ? 
 
 5. How is the reciprocal of a number found ? Prove theor. 4. 
 
 6. Clear the equation a/b = c/d of fractions, and see what 
 multiplier will make the first member d/c; and so make a new 
 proof of theor. 4. 
 
 7. If the extremes of a proportion be to each other as the 
 means, the ratios are each unity. 
 
 8. If two terms of one proportion be the same as the two 
 like terms of another, the four terms that are left may form 
 a proportion. 
 
 9. The direct ratio of two numbers is the inverse ratio of 
 their reciprocals. 
 
 10. May the reciprocals of any four proportionals form a 
 proportion ? their opposites ? the opposites of their reciprocals ? 
 
 11. May any four proportionals be written in reverse order ? 
 their opposites ? the opposites of their reciprocals ? 
 
 13. If m:a=n:b and c :m^d:n, then a:b = c:d. 
 So, if a :m = b:n and c :d = m:n. 
 
 13. If a :m = n:b and c :m=n:d, then a:l/b=c:l/d. 
 So, if a :m = n:b and m:c =d:n. 
 
 14. No single number can be added to each of the propor- 
 tionals a, b, c, d and leave the sums in proportion. 
 
 15. If there be a single number, not 0, that may be added 
 to each antecedent without destroying the proportion, what 
 relation have the terms of the proportion 't 
 
146 INEQUALITIES, INCOMMENSURABLE NUMBERS. [V, Ths. 
 
 Theok. 6. In any proportion the difference of the first two 
 terms is to the first term or the second as the difference of the 
 last two terms is to the third term or the fourth. [division. 
 
 CoK. 1. The sum of the first two terms is to their difference 
 as the sum of the last two terms is to their difference. 
 
 CoE. 3. The sum of the antecedents is to their difference as 
 the sum of the consequents is to their difference. 
 
 Theor. 7. If two proportions be multiplied together, or if 
 one be divided by the other, ter'm by term, the results are pro- 
 portional. 
 
 Theoe. 8. Like powers and liTce roots of the terms of a pro- 
 portion are proportional. 
 
 Theor. 9. In a continued proportion, the sum of all the 
 antecedents is to the sum of all the consequents as any antece- 
 dent is to its consequent. 
 For, let «:5 = c:(^ = e:/= • • ■ 
 then •.• a/a = b/b, c/a=d/b, e/a—f/b, — 
 
 .-. (fl + c + eH ):a = {f)-Vd-\-f^ • • •):b, [add. 
 
 .•. (a + c + e+ • • •): (& + «:?+/+ ■••) = «: 5. [alternation. 
 
 Of these nine theorems, only thedrs. 4, 5 are always directly 
 applicable to concrete numbers ; if, however, all the terms be 
 of the same kind, theors. 3, 8 also apply. 
 
 Moreover, though the terms of a proportion be all concrete, 
 their ratios are abstract, and so are the products, quotients, 
 powers and roots of the.se ratios; and these results may be used 
 as operators on concrete units. 
 
 E.g., if a days : b days = %c : %d, and e men :/ men = %g : $h, 
 then ae days' labor : &/" days' labor = $c^: $dh. 
 For the proportions give the abstract equations 
 
 a/b = c/d, e/f=g/h, ae/bf=cg/dh, 
 and •.• ae days' labor : bf days' \ahov = ae : bf, 
 and $cg : $dh — eg : dh, 
 
 .: ae days' labor : bf days' labor = $cg: %dh. 
 
6, 7, 8, 9, § 2 ] PROPORTION. 147 
 
 QUESTION'S. 
 
 1. To each side of the equation a/h = c/d add 1, and re- 
 duce these two mixed numbers to fractions; from the resulting 
 equation find a hint for the proof of theor. 6. 
 
 Write the proportion by inversion, then prove theor. 5. 
 
 2. Divide the equation {a + b)/a—(c + d)/c by the equation 
 {a — h)/a = {c — d)/c, member by member, and prove theor. 6 
 cor. 1. 
 
 So, from the equation a/h = c/d get a/c—i/d, and 
 prove theor. 6 cor. 2. 
 
 3. One fraction can be divided by another by dividing the 
 numerator and denominator of the first by the like terms of 
 the other; and such a process is equivalent to the usual one 
 of multiplying by the divisor inverted; why is the latter rule 
 of tener used than the other ? 
 
 Apply this method in proving theor. 7. 
 
 4. If the first two terms of a proportion be squared, by 
 what is the first side of the equation multiplied ? if the other 
 two be also squared, is the same multiplier used or a different 
 one ? Hence prove theor. 8. 
 
 5. Why may the two antecedents of a proportion be multi- 
 plied by one number and the two consequents by another ? 
 
 6. Show why theors. 1, 7, 8 are of no direct use when all the 
 terms are concrete numbers; and why theor. 2, but not its 
 converse,,may apply to concrete numbers. 
 
 1l.lt{a+i+c-\-d){a-l-c+d) = (a-i-\-c-d){a + b-c-d), 
 then a, h, c, d are proportionals. 
 
 8. If a:b=c:d=e:f, and h, h, I be any numbers, then 
 
 a:i = {ha-\-hc-\-le):{Jil + M+lf), 
 and a» : 6" = (Aa» + A;c» + Ze") : (A5» + hd" + If). 
 
 9. The distance fallen varies as the square of the time; a 
 body falls 16 feet in one second: how far does it fall in two 
 seconds? in three seconds? in the third second? in five 
 seconds ? in the last two of the five seconds ? How high is a 
 tower from whose top a stone falls in 3| seconds ? 
 
148 INEQUALITIES, INCOMMENSURABLE NUMBERS. [v. 
 
 §3. INEQUALITIES. 
 
 LAKGBB-SMALLEK INEQUALITIES. 
 
 One concrete number is larger than another of the same 
 kind if it contain more units than the other; and smaller if 
 it contain fewer units. The positive or negative quality of the 
 numbers is not thought of, but only their magnitudes. 
 
 One abstract number is larger than another if it give a larger 
 result when acting on the same unit. 
 
 E.g., if A have |50 and owe |30, and B have $60 and owe $80; 
 then a's assets are smaller than b's, and so are his debts, 
 a's assets are larger than his debts, and b's are smaller, 
 a's net assets are as large as b's net debts: 
 and +50<+60; -30<-80; +50^-30; +60<-80. 
 
 AXIOMS. 
 
 1. If of three numlers the first he larger than the second, and 
 the second he equal to or larger than the third, then is the first 
 number larger than the third. 
 
 2. If one nuniber ie larger than another, and if each of them 
 be multiplied ly the same number or by equal numbers, then is 
 the first product larger than the other. 
 
 3. If one number be larger than another, and if each of them 
 ie divided by the same number or by equal numbers, then is the 
 first quotient larger than the other. 
 
 4. If one number be larger than another, and if the same 
 number or equal numbers be divided by each of them, then is 
 the first quotient smaller than the other. 
 
 5. If one set of numbers be larger than another set of as 
 many more, each than each, then is the product of the first set 
 larger than the product of the others. 
 
 6. If one number be larger than another, and if liJce positive 
 powers or roots of them be tahen, i' en is the power or root of 
 the first larger than that of the other. 
 
 7. If one number be larger than another, and if like nega- 
 tive powers or roots of them be taken, then is the power or rant 
 of the first smaller than that of the other. 
 
SS] INEQUALITIES. 149 
 
 Not all of these axioms are axioms in the sense of truths 
 too elementary to admit of proof, for some are directly deriv- 
 able from others, but they are all self-evident. 
 
 QUESTIONS; 
 
 1. When a number is made larger, what efEect is produced 
 on its reciprocal ? its opposite ? the opposite of its reciprocal ? 
 
 2. Which is the larger, zero or a negative number ? 
 
 3. Explain the axioms, and illustrate each of them by an 
 inequality between known numbers. 
 
 Show which of them are deducible from the others. 
 
 4. Prove the following statements, with reference to larger- 
 smaller inequalities, giving axioms as authority: 
 
 both members of an inequality may be multiplied or divided 
 by the same number, or raised to the same positive power; 
 
 the products of the corresponding membei's of several in- 
 equalities maybe taken without changing the sign of inequality; 
 
 but if the same operations be performed on the reciprocals 
 of both members, the sign of inequality must be reversed. 
 
 5. Show by trial that adding the same number to both 
 members of a larger-smaller inequality will sometimes reverse 
 the sign of inequality. 
 
 So, subtracting the same number from both members. 
 So, dividing two such ineqiialities member by member. 
 
 6. Of what numbers are negative powers larger than the 
 like positive powers? of what numbers are positive powers 
 smallest when the exponents are largest ? 
 
 7. Show that if both terms of a proper fraction be positive 
 and to both the same positive number be added, the fraction 
 is made larger thereby. 
 
 So, that an improper fraction is thus made smaller. 
 
 Which number is the smallest: 
 .8. ^2, ^4, -v^8? 9. 1^3, p6, 1^9? 10. ^5, iJ^lO, y'15? 
 11. (l/2)»/^, (1/4) V*, (1/8)V8? 12. {l/Bf/^, (1/6)V«, (l/9)'/»? 
 13. (l/5)Vf, {1/loy/^, (1/1 5y/^ ? 14. 3 • 5 • 7 . 9, 4« . 8^ 6* ? 
 
150 INEQUALITIES, INCOMMENSURABLE NUMBERS. [T, 
 GEBATBK-LESS INEQUALITIES. 
 
 One number is greater than another number of the same 
 kind if the remainder be positive, when from the first the 
 other is subtracted, and less if the remainder be negative. 
 
 Of positive numbers the larger is also the greater, but of 
 negative numbers the smaller is the greater; and any positive 
 number, however small, is greater than any negative number 
 of the same kind, however large. 
 
 The signs are >, greater than ; <, less than. 
 E.g., +50<+60, -30>-80, +50>-30, +60>-80. 
 
 AXIOMS. 
 
 8. If of three nunibers the first be greater than the second, 
 and the second be equal to or greater than the third, then is the 
 
 , first number greater than the third. 
 
 9. If one number be greater than another, and if the same 
 number or equal numbers be added to them, then is the first 
 sum greater than the other. 
 
 10. If one number be greater than another, and if the same 
 number or equal numbers be subtracted from them, then is the 
 first remainder greater than the other. 
 
 11. If one number be greater than another, and if they be 
 subtracted from the same number or from equal numbers, then 
 is the first remainder less than the other. 
 
 12. If one set of numbers be greater than another set of as 
 many more, each than each, then is the sum of the first set 
 greater than the sum of the others. 
 
 13. If one number be greater than another, and if they be 
 multiplied or divided by the same or equal positive numbers, 
 then is the first product or quotient greater than the other. 
 
 14. If one number be greater than another, and if they be 
 multiplied or divided by the same or equal negative numbers, 
 then is the first product or quotient less than the other. 
 
 Note. — The pupil may compare these axioms with those on 
 page 148, taken in order: he will find that adding and sub- 
 tracting in greater-less inequalities are analogous to multiply- 
 
SS] INEQUALITIES. l6l 
 
 ing and dividing in larger-smaller inequalities, and that mul- 
 tiplying and dividing in the one are analogous to finding 
 powers and roots in the other. The note at the top of page 
 149 applies also to this set of axioms. 
 
 QUBSTIOKS. 
 
 1. Can a positive number be less than a negative number ? 
 can it be smaller ? larger ? greater ? 
 
 2. N'ame two numbers equally large, but unequal. 
 
 3. Which of the pair — (a + b), {a + b) is the larger if a, b 
 be both positive ? which is the greater ? ii a, b be both nega- 
 tive ? if a be positive, b negative, and a larger than b ? 
 
 4. In the pair x, a? which is the larger if x>l? if a; be a 
 positive proper fraction ? if a; be a negative fraction ? 
 
 What two values of x make a^ equal to a; ? 
 
 5. What axiom proves that in a greater-less inequality a 
 term may be transposed from one member to the other by 
 changing its sign ? that if the signs of every term be changed 
 the sign of inequality must be reversed? that two like in- 
 equalities may be added without changing the sign ? 
 
 6. Why do not axioms 6, 7 apply to inequalities of this kind ? 
 
 7. If a>b and a'>b', the elements may hg,ve such re- 
 lations that a — a'>b — b', or a — a' = b — b' or a — a'<b — b'. 
 
 8. If 3a;-7>29, 3a;-5<2a; + 16, then 17<a;<18. 
 
 9. If 16 more than three times the number of sheep exceeds 
 twice their number by 37, and four thirteenths of their num- 
 ber less one be less than 3, how many sheep are there ? 
 
 10. Twice a number less 3 is less than the number plus 6; 
 and 11 plus twice the number is less than 3 times the number 
 plus 5: what is the number? is the number definite? 
 
 11. If a; be any positive number, a; + l/a;<2. 
 
 12. If x-^y = s, 4:xy = s^ — {x — yy, and xy is greatest 
 when x—y = 0; hence show that the product of two numbers 
 whose sum is constant is greatest when the numbers are equal. 
 
 13. Show which of axs. 8-14 are derivable from others. 
 
152 INEQUALITIES, INCOMMENStJEABLE NUMBERS. tv, 
 
 §4. INCOMMENSUEABLE NUMBBES. 
 
 Kumbers that arise from the effort to measure any quantity 
 by a unit that has no common measure with it are incommen- 
 surable numlers. 
 
 E.g., a side and a diagonal of the same square are both definite 
 lines, but the length of the diagonal caunot be ex- 
 pressed exactly by any simple fraction in terms of the 
 side, nor that of the side in terms of the diagonal; 
 i.e., neither line can be got from the other by partition and 
 repetition. 
 
 So, the circumference of a circle is incommensurable with its 
 diameter. 
 
 So, a period of time expressed exactly either in days, in lunar 
 months, or in solar years, can be expressed exactly in 
 neither of the other units. 
 
 An absti-act number is incommensurable if it can be written 
 approximately, but not exactly, as a fraction. Such numbers 
 arise from attempts to find operators that, used in a specified 
 way, shall yield certain given results. 
 
 E.g., to find an operator that, used twice in succession as 
 multiplier, will double a unit. 
 
 1. No integer can be such operator. 
 
 For the operator *1 used twice as multiplier gives the unit as 
 result; *2 gives four times the unit; and other integer 
 operators give still larger results. 
 
 2. No simple fraction can be such operator. 
 
 For, let n/d be any simple fraction in its lowest terms; 
 then the concrete product unit x n/d x n/d is unit x n^/d'', 
 and •.• the fraction n^/d^ is irreducible, [IV, th. 12 cr. 1. 
 
 .-. n''/d^=^2. 
 
 3. Sut a simple fraction can be found that shall very closely 
 approach the operator sought. 
 
 For •••unit X 1.4^ = unit X 1.96 and unit x 1.5^ = unit x 2.35, 
 
 .*. the operator sought lies between 1.4 and 1.5. [ax. 2. 
 
§4] INCOMMENSURABLE NUMBERS. 153 
 
 So V unit X 1.41* = unit X 1.9881, 
 and unit x 1.42^ = unit x 2.0164, 
 
 .•. the operator sought lies between 1.41 and 1.42. 
 So it lies between 1.414 and 1.415, between 1.4142 and 1.4143, 
 and so on. 
 But such operator exists, and is a definite number, which 
 is perfectly expressed in the language of geometry by saying 
 that it is the ratio of the diagonal to the side of the same 
 square, and in the language of algebra by y'2. 
 
 QUESTIONS. 
 
 1. If two squares have a side of the same length, how do the 
 two perimeters compare ? the two areas ? the two diagonals ? 
 
 If one square be placed on the other, how will the diagonals 
 lie ? If the sides of the two squares increase continuously at 
 the same rate, how do the two diagonals increase ? 
 
 2. If a side of a square be one inch, can the lengths of the 
 diagonals be expressed in inches ? 
 
 If the side increase continuously to two inches, has its 
 diagonal been at any time measurable in inches ? was the 
 side measurable in inches at that time ? 
 
 But, through all changes of value, and from commensur- 
 ability to incommensurability, what relation holds true be- 
 tween the two diagonals ? between a side and a diagonal ? 
 
 3. Why could not the decimal expression for 4/2 end with 
 the figure 1? 2? 3P---9? with what figure must it end so 
 that the square of it may be 2.0 ? 
 
 Show that no decimal fraction ends in 0. 
 What is thus proved as to 4/2 ? 
 
 4. Is the fraction 2/3 a commensurable number? is the 
 corresponding decimal .666 • • • a commensurable number? 
 
 Have these two expressions the same or different values ? 
 Is one of them more definite than the other ? 
 
 5. Are the approximate decimal expressions for incom- 
 mensurable numbers variables or constants ? If variables, do 
 they grow continuously or discontinuously ? 
 
154 INEQUALITIES, INCOMMENSURABLE NUMBERS. [V,Th. 
 MULTIPLICATION AND DIVISION. 
 
 The product of a concrete number by an abstract number is 
 a number that bears the same relation to the multiplicand 
 that the multiplier bears to unity. 
 
 This definition covers the earlier and simpler ones. 
 E.g., if the multiplier be 2, the product is the double of the 
 
 multiplicand; if ~2, it is the opposite of the double. 
 So, if the multiplier be 3/4, the product is the triple of a 
 fourth part of the multiplicand. 
 It also defines multiplication by an incommensurable mul- 
 tiplier; and later it defines multiplication by an imaginary. 
 E.g., the product six sq. ft. x 4/2 is a number that has the same 
 relation to six sq. ft. that the length of the diagonal of a 
 square has to that of one of its sides. 
 So, the product 6 hours x ;r is a period of time that bears the 
 same relation to six hours as the length of the circumfer- 
 ence of a circle bears to that of its diameter. 
 So, if simple interest be of continuous growth, the interest 
 varies as the time and i=p-r-t, wherein the product 
 p-r is the interest for one year, and the product p-r-t 
 is the interest accrued at any given time, t, whether t 
 stand for a length of time that is commensurable with 
 the unit year or not. 
 The definition of the product of two abstract numbers is 
 identical with that given on page 6; and the definitions of 
 division and of a quotient are identical with those given on 
 page 16. 
 
 MULTIPLICATION ASSOCIATIVE. 
 
 Theoe. 9. TJie product of three or more abstract numiers 
 (commensurable or incommensurable) is the same number, 
 however the factors be grouped. [I, th. 1. 
 
 Let a, b, G • ■ • he three or more abstract numbers; 
 then is their product the same number, however the numbers 
 be grouped. 
 
9,S4] INCOMMENSURABLE NUMBERS. 155 
 
 For, to multiply the concrete product unitx a by the ab- 
 stract product ixc is to multiply the concrete prod- 
 uct unit xa by the abstract number I, and the conse- 
 quent concrete product unit xaxb by the abstract 
 number c, [df. prod. ab. nos. 
 
 and so for other factors, and for other groupings of them. 
 
 Q.E.D. 
 
 QUESTION'S. 
 
 1. Draw a square; then another square whose side is a diag- 
 onal of the first square: how does the diagonal of the second 
 square compare with the side of the first ? 
 
 What operator, acting as multiplier on the side of the first 
 square and again on that result, has doubled the given side ? 
 Is that operator a commensurable number ? 
 
 Show that the definitions of multiplication and of a multi- 
 plier apply here. 
 
 2. Of what two numbers is the diagonal of a square the 
 product ? Can the product, or the quotient, of two incom- 
 mensurable numbers be a commensurable number ? 
 
 3. How long is the diagonal of a square whose side is one 
 inch? of a square whose side is a inches? What number 
 bears the same relation to unity as the diagonal of a square 
 bears to its side ? 
 
 4. What two lines are proportional to tt and unity ? 
 Is TT a commensurable number ? 
 
 What perfectly definite meaning has it ? 
 What line is exactly expressed by 4;r ? 10;r ? 
 If the circumference of a circle be 10, what is IO/tt ? 5/;r ? 
 If the radius of a circle be 4, what is its area, the area of a 
 circle being half the product of its radius and circumference ? 
 
 5. If a side of a square be two feet, and if a circle be 
 described upon its diagonal as diameter, what is the length of 
 the circumference of the circle ? what is its area ? 
 
 6. By what shall a unit be multiplied three times in succes- 
 sion to quadruple it ? four times, to triple it ? 
 
156 INEQUALITIES, INCOMMENSURABLE NUMBERS. tV, 
 
 MULTIPLICATION COMMUTATIVE. 
 
 Lemma. Two rectangles that have the same altitude are to 
 each other as their bases. 
 
 {a) The bases commensurable. 
 Let A, B be two rectangles of the same height, whose bases, 
 a, b, have a common measure, c ; 
 
 A 
 
 let c be contained in a m times and in b n times; and at the 
 points of division draw perpendiculars to a, b, thus 
 dividing the rectangle A into m parts and B into n 
 parts that may be shown to be all equal by placing one 
 upon another; 
 
 then •.■ A: B = m: M and a:b~m:n, [df. ratio. 
 
 .". A:B = a:&. Q.e.d. [ax. equal. 
 
 (&) The bases incommensurable. 
 
 Let A, B be two rectangles of the same height, whose bases, 
 a, b, have no common measure; 
 
 A B 
 
 then if b be not the fourth proportional to A, B, a, that pro- 
 portional is some line shorter or longer than b. 
 
 If possible, let that proportional be CD, shorter than b by the 
 part de, 
 
 and find a measure, c, of a, that is shorter than de. 
 
 Apply c repeatedly to b; then at least one point of division, f, 
 falls between d and e; 
 
 draw perpendiculars to a, b as in (a); 
 
 then •.• a, CF are commensurable, [constr. 
 
 .'. A : rectangle on GV = a: CF. (a) 
 
 .But A:B = a:CD; [hyp. 
 
th.io§4] incommensurable numbers. 157 
 
 and •.• A : rectangle on of > A : b and a : or < a : CD, [ax. 4. 
 
 .*. these two proportions are not both true, 
 i.e., the hypothesis that A:B = a:some line shorter than b 
 
 leads to a false conclusion, and is itself false. 
 So, as may be proved in like manner, the hypothesis that 
 A : B = a : some line longer than b is false, 
 .•. it is only left that A : b = « : 5. q. e.d. 
 
 QUESTIONS. 
 
 1. What is a lemma? [consult a dictionary. 
 
 2. In case (a), let C be one of the small rectangles into which 
 A, B are divided ; express a, b in terms of c, and show why 
 A:'B = m:n, and why a:b = m:n. 
 
 3. If a be an incommensurable line, can c be an exact 
 measure of a ? and if c be a measure of a, how can a be called 
 incommensurable ? 
 
 4. In case (b) why must F fall between d and e ? Give a 
 reason for the two inequalities stated. How do these in- 
 equalities prove that both proportions can not be true ? 
 
 How is it known which is the false proportion ? 
 
 5. Such a proof as that of this lemma is called an indirect 
 proof, because, instead of proving directly what we wish to 
 establish, we prove that every other possible supposition leads 
 to a false conclusion, and is therefore itself false : why is this 
 proof also called a proof by exclusion ? 
 
 6. Construct two rectangles whose areas are proportional to 
 a side and a diagonal of a square; two, whose areas are pro- 
 portional to the diameter and circumference of a circle. 
 
 7. By aid of the well-known theorem of geometry, "the 
 square of tbe hypothenuse of a right triangle is the sum of 
 the squares of its legs," construct a line whose length is 
 
 4/3, 4/5, 4/6, 4/7, 4/8, 4/10, 4^11, V13, 4/13. 
 
 8. Construct a square; a square on the diagonal of this 
 square; a third square on the diagonal of the second square; 
 and a fourth square on the diagonal of the third: what rela- 
 tion have the sides of these squares ? what, their areas ? 
 
158 INEQUALITIES, INCOMMENSURABLE NUMBERS. [V,Th. 
 
 Theor. 10. The product of two or more abstract numbers 
 (commensurable or incommensurable) is the same number, in 
 whatever order the factors be multiplied. 
 
 (a) two factors a, b. 
 For, let u be a square whose side is of unit length, p a rectan- 
 gle of height 1 and breadth a, q a rectangle of breadth 
 1 and height 5, r a rectangle of breadth a and height b\ 
 
 1 
 
 1 
 
 then*."P=uxa, Q=ux5, R = Px5 = QXfl!, [lem. 
 
 wherein a is the ratio of line a to the unit line, 
 and so of b, 
 
 .•.n = vxaxb = vxbxa, 
 
 .: the abstract products axb, bxa, do the same work 
 and are equal. 
 (6) three or more factors a, b, c. 
 The proof is identical with that of I theor. 2 {d). 
 
 AXIOMS AND THEOREMS THAT APPLY TO INCOMMENSURABLE 
 
 NUMBERS. 
 
 The quality of numbers as denoted by the positive and 
 negative signs applies alike to commensurable numbers and 
 to incommensurable numbers. 
 
 The axioms of equality and of inequality, and the definitions 
 and principles of division and reciprocals, are the same for in- 
 commensurable numbers as for commensurable numbers; and 
 I theors. 3, 4, relating to reciprocals and division, apply to 
 incommensurable numbers without change in their statement 
 or proof. 
 
 So, theor. 5, that addition is commutative and associative, 
 has the same statement, and the proof is as follows : 
 For, let a, b, c • • • he any abstract numbers, commensurable 
 
 or incommensurable, 
 let these numbers act as operators on any unit, 
 and let the results be grouped and added in any way; 
 
10, §4] INCOMMENSURABLE NUMBERS. 159 
 
 then •.• the quantities so found are of the same kind, 
 and their aggregate is the same, in whateTer order they be 
 arranged and however they be grouped, 
 .'. the several sums of these operators do the same work 
 and are all equal. q.e.d. 
 
 So, theor. 7, that multiplication is distributive as to addi- 
 tion; theors. 8, 9, relating to opposites and subtraction; and 
 theors. 10, 11, 13, relating to integer powers, apply to incom- 
 mensurable numbers without change. 
 
 QUESTIONS. 
 
 1. Show that the proof of theor. 10 could not have been 
 complete without the aid of the preceding lemma. 
 
 2. State the proof of theor. 10 for three or more factors. 
 
 3. What modification is made in I theor. 5, to make it ap- 
 plicable to incommensurable numbers ? 
 
 4. Eeview all the theorems that apply without change to 
 incommensurable numbers and give the proofs, remembering 
 that they include such numbers. 
 
160 INEQUALITIES, INCOMMENStJHABLE NUMBERS. [V. 
 
 § 5. QUESTIONS FOR EBVIBW. 
 
 Define and illustrate: 
 
 1. A constant; a variable; related variables; a function of 
 a variable; continuous variables; discontinuous variables. 
 
 2. Variation; direct variation; inverse variation; joint 
 variation. 
 
 3. The ratio of one quantity to another quantity of the 
 same kind ; the ratio of two abstract numbers. 
 
 4. A direct ratio; an inverse ratio. 
 
 5. Proportion; proportionals; a continued proportion. 
 
 6. An antecedent; a consequent; the extremes; the means. 
 
 7. A mean proportional; a third proportional. 
 
 8. An inequality; larger-smaller inequalities; greater-less 
 inequalities. 
 
 9. A product where the multiplier is an incommensurable 
 number; a quotient where the elements are incommensurable. 
 
 10. A lemma; an indirect proof. 
 
 State, with any necessary explanations or illustrations : 
 
 11. The axioms of larger-smaller inequality. 
 
 12. The axioms of greater-less inequality. 
 
 13. How to find a missing proportional. 
 
 14. The theorems that apply to proportions involving con- 
 crete numbers. 
 
 Prove that : 
 
 15. The product of the extremes of a proportion equals 
 that of the means. 
 
 16. The factors of two equal products form a proportion. 
 
 17. A proportion may be written by alternation; by inver- 
 sion; by composition ; and by division. 
 
 18. The terms of a proportion may be multiplied or divided 
 by the like terms of another proportion. 
 
 19. The terms 'of a proportion may be multiplied or divided 
 by the same number. 
 
551 QUESTIONS FOK REVIEW. 161 
 
 20. The antecedents of a. proportion may be multiplied or 
 divided by the same number, and so may the consequents. 
 
 21. The first and second terms of a proportion may be 
 multiplied or divided by the same number, and so may the 
 third and fourth terms. 
 
 23. The terms of a proportion may be raised to the same 
 power, and the same root.may be taken of them. 
 
 23. In a continued proportion the sum of the antecedents 
 is to the sum of the consequents as any antecedent is to its 
 consequent. 
 
 24. Two rectangles of equal altitude are as tlieir bases. 
 
 25. Variation by a fixed law is expressed by an equation. 
 
 26. A proportion is an expression of direct variation. 
 
 27. Addition is a commutative and an associative operation 
 with incommensurable, as with commensurable, numbers. 
 
 28. What operations may, and what may not, be performed 
 on the two members of 'a larger-smaller inequality? of a 
 greater-less inequality ? 
 
 29. State and prove the associative and commutative prin- 
 ciples of multiplication, with incommensurable numbers. 
 
 30. The sum of the greatest and least of four positive pro- 
 portionals is greater than the sum of the other two. 
 
 What is the law if some of the proportionals be negative ? 
 
 31. If X, y be such numbers that a + x : i + y = a : b, 
 then X : y = a : h. 
 
 32. The sum of a real positive number and its reciprocal 
 cannot be less than 3. 
 
 ,33. If X : y = {x + zY + {y + z''), z is a mean proportional 
 between x and y. 
 
 34. If a :b = c : d and 5 be a mean proportional between c 
 and d, then c is a mean proportional between a, and h. 
 
 35. IfaaJ", b on c\ c on d\ then « of £?"". 
 
 36. If a; a?/" ^ and a: = 1/2 when «/ = 4, what is a; when ^=2/3? 
 
 37. If a:b-c:d,i\iQ\i a:d-a' : b\ b-^a'd, c=^ad\ 
 
163 POWERS AND ROOTS. tVI,TH. 
 
 VI. POWERS AND BOOTS. 
 
 The words power, root, base, .exponent, root4ndex are defined 
 in I, § 5. A root-index is always assumed to be a positive 
 integer J but an exponent may be any number whateyer. 
 
 §1. THE BINOMIAL THBOEEM. 
 
 Thboe. 1. If a binomial be raised to any positive integer 
 power, that power is symmetric as to the two terms of the 
 binomial, and consists of a series the number of vihose terms 
 is one greater than the exponent of the binomial; and the suc- 
 cessive terms of this series are the products of three factors : 
 
 1. Coefficients that come from the exponent of the binomial: 
 the first, one; the second, the exponent; the third, the product 
 of the second by half the-exponent-less-one; the fourth, the prod- 
 uct of the third by a third of the-exponent-less-two; and so on. 
 
 2. Falling powers of the first term, beginning with that 
 power whose exponent is the exponent of the binomial. 
 
 3. Rising powers of the second term, beginning with the 
 zero-power. 
 
 Let a, b be any two numbers, then the theorem is written 
 
 +|«(w-l)(w-2)a"-»5'+ • • • +5", 
 and it is proved by induction, as shown below. 
 
 (a) The laiv is true for the second power. 
 For ••• {a + by = a''+ 2ab + W, [multiplication. 
 
 .-. (a + 6)"=a" + wa"-'6+---+5", when n = 2. 
 
 (6) If the law be true for any one power it is true for the 
 next higher power. i 
 
 For, assume {a-¥bf=a^+ka^-^b-^^k{h-l)a^-%+ • • • +V, 
 and multiply both members of this equation by a + b, 
 then (a + 5)*+i=a'^+H(>fc+lK5+i(* + l)*a*-iZi^+ ■ • • +5*+i 
 a series of the same form as that for the Mh power, k + 1 tak- 
 ing the place of h, and conforming to the law of development. 
 
1,S1] THE BINOMIAL THEOREM. 163 
 
 (c) The law is true, whatever the power. 
 
 For •.• it is true for the second power, [(a) 
 
 ,•. it is true for the third power j [(&) 
 
 and •.• it is true for the third power, [above 
 
 /. it is true for the fourth power, and so on. q.e.d. 
 
 QUESTIONS. 
 
 1. In any power of {a — i) what terms are negative ? 
 In what powers is the last term negative ? positive ? 
 Expand : 
 
 3. {x+yf. 3. (a -4)*. 4. {a + bf. 5. {^x-yf. 
 
 6. {a?+Zy)K 7. (a-2Sc)*. 8. («±S)«. 9. (m-w)'. 
 10. \d/{m+n)]K 11. l-2/{x-y)f. 12. {ix + 2yy. 
 13. {2x/3-B/2zy. 14. {ix+iay. 15. {iax + Sy-lf. 
 
 16. {3c + ddy. 17. {x+y-izf. 18. {x-y+4:zY. 
 
 19. [(2a-&) + (c-«Z)]«. 20. [(2x+y)-{x-2y)f. 
 
 21. A proof by induction consists of three steps: name them. 
 
 22. In the expansion of {a + i)^, and of (a — b)^, write 
 the literal part of the third term^ of the 50th term; of the 51st. 
 
 23. What term of {a + bY is 
 w(ra-l)---(w-/-+l)-a''-'--SVr!? [r! =l-2.3---n 
 
 How may the laat factor rn the' numerator of a coefficient 
 be found from the number of the term ? in the denominator ? 
 
 24. Of what term of (a + b)^ is 50- 49 -48 •• -43/8! the 
 coefficient ? 50 • 49 • 48 • • • 9/42 ! ? 
 
 25. What other term has the same coefficient as the 49th ? 
 the 50th ? the 41st ? What is the general principle ? 
 
 Make use of the general formula in £x. 23 to write: 
 
 26. The fourth term of (a;- 5)'=; the T'th term; the 12th. 
 
 27. The 12th term of (l-i«^)"; the third term; the 8th. 
 
 28. The middle term of (a/x-kx/aY"; the third; the 7th. 
 
 29. In (x+yY" the sum of the coefficients of the odd 
 terms equals the sum of the coefficients of the even terms. 
 
 30. What term of {x+l/xy is free from x ? 
 
164 POWERS AND ROOTS. CVL 
 
 §2. FRACTION POWEES. 
 
 K fraction power of a number is either a root of the num- 
 ber or some integer power of a root. The record of the oper- 
 ation begins by naming the base, then the number of factors 
 it is resolved into, then the number of such factors that are 
 multiplied together. The two numbers last named appear as 
 tho denominator and the numerator of a simple fraction. 
 E.g., 64^''^=( y'64)«=4-4=16, wherein 64 is resolved into 
 the three equal factors 4, 4, 4, and the product of two 
 of them is taken. 
 So, 64-^-"= ( 1^64) -^=1/4^= 1/16, wherein 64 is resolved 
 into the three equal factors 4, 4, 4, and two of these 
 factors are used in partition. 
 
 Note the new use of the fraction form 3/3 : As an exponent, 
 it means that 64 is resolved into three equal factors, and two 
 of them are multiplied together; in the ordinary usage, some 
 unit is divided into three equal parts, and two of them are 
 added together. Later it appears that these fraction exponents 
 are subject to the laws already established for fractions; but 
 this must not be assumed without proof. 
 
 Integer powers and fraction powers are classed together as 
 commensurable powers; incommensurable powers appear later. 
 
 The value of a fraction power is often ambiguous'. 
 B.g., 100i'''=±10; 9-'''2=±l/27; (ay/'^±a; («*)'/*=*«'. 
 
 Different powers of a base are in the same series if they be 
 integer powers of the same root. An integer power of a base 
 belongs to all series alike. 
 
 B.g., 
 
 9-S 
 
 9-V2, 
 
 9», 9'/% 
 
 9S 
 
 gSA 
 
 9^ .. 
 
 . are the 
 
 
 -2d, - 
 
 -1st, 
 
 0th, 1st, 
 
 2d, 
 
 3d, 
 
 4th,- 
 
 • . powers of |/9, 
 
 i.e., 
 
 1/9, 
 
 -1/3, 
 
 1, -3, 
 
 9, 
 
 -27, 
 
 81,. • 
 
 •, powers of "3 
 
 and 
 
 1/9, 
 
 1/3, 
 
 1, 3, 
 
 9, 
 
 27, 
 
 81,.. 
 
 ., powers of +3 
 
 When several powers of the same base occur together, they 
 are assumed to be all taken in the same series. 
 
 Powers that have the same exponent are like powers. 
 E.g., 4/a, ^b, n/ab; a", ¥, aF; 2", 3", 6"; «-»/*, ^-»/*, z"'/*. 
 
«2] FRACTION POWERS. 165 
 
 QUESTIONS. 
 
 Explain the meaning of each part of the expressions below. 
 1. -37^ 2. -37-«/». 3. -37*^. 4. sm 
 
 5. 81-v*. 6. 243'^. 7. -343^/«. 8. 243''^. 
 
 9. -2431-^. 10. 343-*/^. 11. SI"*/'. 12. 1738^'^. 
 
 13, What operation is indicated by the denominator of a 
 simple fraction ? what by the numerator? what by the terms 
 of a fraction exponent ? Can it be assumed that fractions, as 
 exponents, are operated with just as other fractions ? 
 Express with fraction exponents: 
 
 14. i/a^. 15. I'a*. 16. ^o^Vc. 
 17. ( y/y^K 18. 4 ^abc'/x^ 19. i^{xyz/W). 
 
 20. . (3y'&'cxy. 21. ^2a^{x/y). 22. y'{x+y)/(x-yy. 
 Express with radical signs: • 
 
 23. {a^/^y. 24. (&-«/*)»/». 25. (c-d)*'*. 
 
 26. {a?y^^)-^^. 27. (5aa!-''^)-«'''. 38. Z/x'f/^/4:m}/K 
 
 39. a'^/b-"'*. 30. 8a->/' + S-^/=. 31. x'-'^^f'^/^'K 
 Express with positive exponents : 
 
 33. x-^. 83. 6ffl-*a;-l 34. a-»&-V. 
 
 ^;^2»^ 3a;-V «-«(a;-y )-v* 
 
 aa/3y-»/4-" ^''- 3-Wa;-*' '' a-Xa; + sr)-'/*' 
 
 41. Is the square root of an incommensurable number a 
 commensurable or an incommensurable power ? what is 3 V" ? 
 
 43. What kind of powers give rise to two values? what 
 powers of these powers are alike in both series ? 
 
 Find the first five powers of 16'''^ in both series. 
 Find the value, or values, of: 
 
 43. 37-'''^-37''"+(-37)'''^ 44. 35i''H25-''«+25«. 
 
 45. 16'/*+16»''*-16~v*-16-n 46. S-vs + s-^-S^/'+B-^/*. 
 
 47. 32*''*-32^''H32-»'*+32-*''^ 48. 36»/^-36»''H36-'/«. 
 
166 POWEES AND ROOTS. Ivr.lBa. 
 
 A COMMEKSUEABLB POWER OP A COMMENSURABLE POWER. 
 
 Theoe. 2. A commensurable power of a commensurable 
 potver of a base is tliat power of the base whose exponent is the 
 product of the two given exponents. 
 
 Let m, n be any two commensni*able numbers, and A any base; 
 then (a™)»=a™». 
 
 (a) m, n integers, positive or negative. [I, th. 11. 
 
 (J) m, n reciprocals of positive integers, \/p, 1/q. 
 
 For, resolve A into p equal factors B, B, • • • , commensurable or 
 
 incommensurable, so that b=a*'*, 
 and resolve B into q equal factors c, 0, • • •, so that c = (A**)'''''; 
 then •.' A is thus resolved into jag' equal factors c, o, • • • , 
 
 .•.,(a»'*)'''«=aVm. Q.E.D. [df. fract. pwr. 
 
 (c) m, n, simple fractions, p/q, r/s, and q, s positive. 
 
 For, resolve A intb q equal factors B, B, • • • and take p of them, 
 
 and resolve B into s equal factors c, c,' • -and take r of them; 
 
 then •.• A is thus resolved into qs equal factors 0, 0, • • • and pr 
 
 of them are taken, 
 
 ,.. (AP/9)r/«_^pr/g»_ q j, j,^ [df. fract. pwF. 
 
 And so if three or more powers be taken in succession. 
 
 COR. 1. (a'»)'''» = A'"''». . 
 
 Cor. 2. (a»)v« =(aV»)»=a- 
 
 equal feaction powees. 
 
 Theoe. 3. The value of a fraction power of a base is the 
 
 same, whether the fraction exponent be in its lowest terms or not. 
 
 For, let Jc, p, q be any positive Integers, A any base, 
 
 then a'^''*9={[(AV9)V*]»}i» [th, 3. 
 
 = (aV9)' [th. 2 cr. 3 
 
 = A.f\ Q.E.D, [df. fract. pwr. 
 
8,8,88] FRACTION POWERS. 167 
 
 QXTESTIOirS. 
 
 1. Explain the proof of theor. 2, and tell how it applies 
 when A is not a perfect power of the pqth degree. 
 
 Find the values of: 
 
 2. (8»/»a-')*''. 3. iix-"/^"/^. 4. {6Af)-^^. 
 
 5. ^^{a'bcl^a'bcy. 6. (9^^)'. 7. (125 -«/»)-»/«. 
 
 8. [(4a?-12a;+9)V«]». 9. (IGa-yslJ')-'/*. 
 
 10. {9a*/25f)-^^. 11. (256/625) -»/*. 
 
 12. ^{a^b/^abf. 13. ^^o^f/x-^'Y'^. 
 
 14. Resolve 64 into six equal factors and indicate the con- 
 tinued^product of twenty-five of them; then resolve 64 into 
 two equal factors, resolve the product of five of these factors 
 into thr-ee equal factors, and indicate the prime factors in the 
 product of five of them : hence show that (64='^)''^= 64^/«, 
 
 15. In the expression a"*'**, can the exponent Tcp/kq be 
 replaced by the equal fraction f/q without further proof ? 
 
 16. Resolve 729 into three equal factors and take the product 
 of two of themj then resolve 739 into six equal factors and take 
 the product of four of them: hence show that 729«'^=729*'". 
 Simplify the radicals: 
 
 Yt.^'K 18. (a + 6)-*'^. Y^.^x-yf"!^, 20. ajiv-si^ 
 
 21. ^a«. 22. ^(a-S)*. '2.Z.'^fq-\ 'il4..^{a?/f). 
 Zb.ix^yf. 26. {x-'/^/y-^/»y. 21. {a?/^/ pay. 28. (8a-»)-«/», 
 
 29. How can fraction powers of different degrees be reduced 
 to powers of like roots ? are their values changed thereby ? 
 Reduce to powers of the twelfth root : 
 
 30. a;"/', 3a^^ a^.- 31. {x+yf/\ f'K 32. -5-v», c^, (^l\ 
 33. x-\ y"^, z-y\ 34. (a; + 2/)"'^, y-^'*". 35. 6'/», c-\ «-»'«. 
 36. i/x, py, y/^. 37. ( i^x ^yf'% z'^i^. 38. («»/«) -V*, («-»/»)»/*, 
 
 . Reduce to powers of like roots : 
 
 '&9.'yx,yy. 40. ip'a^Va^'Ta- 41.-^^,^3. 
 
 42. |/rt&, yc. 43. a;V2, fl^. 44. x-'^'^, y-^^. 
 
168 POWERS AND ROOTS. tVI,THS. 
 
 PRODUCT OF COMMBlirSUEABLB POWEES OF THE SAME BASE. 
 
 Theoe. 4. The product of two or more commensurable powers 
 of a base is that power of the base wJiose exponent is the sum 
 of the exponents of the factors. 
 
 Let m, n be two commensurable numbers, and A any base; 
 then A'"-A"=A"'+". 
 
 (a) m, n integers, positive or neaative. [I, th. 10. 
 
 (5) m, n simple fractions, p/q, r/s^n^LS positive, 
 
 AT tP/Q. \'-/t~ kPa/qa. ,gr/qB W 
 
 For iP/a.Ar^'=ii"/v-MP'''i' ,^ 
 
 ' [th. 3. 
 
 = (^/i'Y' : (a*/**) ^ 
 
 [df. fract, pwr. 
 
 — /^Vas^PS+QT 
 
 [(«). 
 
 zz^^'/l'+vM' 
 
 [th. 3. 
 
 — ^P/g+r/'_ 
 
 Q.E,D. [th. 3. 
 
 And so for three or more powers. 
 
 
 CoE. a"'/a"=a'"-". 
 
 
 PEODUCT OF LIKE COMMENSUEABLE POWEES OF DIFFERENT 
 
 BASES. 
 
 Theoe. 5. The product of like commensurable powers of two 
 or more bases is the like power of their product. 
 Let n be any commensurable number, and A, B, 0, • • • any 
 
 then A"-B''-c" ••• = (a-b-c •••)"• 
 (a) n an integer, positive or negative. [I, th. 13. 
 
 ■ {b) n a simple fraction, p/q. 
 For-.*A*-tf'=(A-B)^, [{a). 
 
 and a*=:(ap/«)«, bp=(bp/9)9, [th.3. 
 
 .% (a"/*)* . (bp/')* = (A • b)'', 
 and ••• (A^/a)' • (b^/«)9 = (a^/9 ■ ^'^)% ' [(«). 
 
 .•.(ap/9-bp/«)«=(a-b)^; 
 
 ... AP/9 . 32-/9 _ (A . B)J>/9. [Ill, ax. 7. 
 
 Cor. aVb"=(a/b)". 
 
*.5,S9] FRACTION POWERS. 169 
 
 The g'th roots involved in the last statement must be used 
 with some caution as to the signs: the equation means that 
 the real positive roots, if any, are equal,, and that all the roots 
 have the same magnitude. 
 
 QUESTION'S. 
 
 1. Does the proof of theor. 4 difEer from that given, when A 
 is an incommensurable number ? 
 
 2. Prove theor. 4 for the product of three or more powers. 
 
 3. Assume a^/a^^a"-", and multiply both members by 
 A" : if the result be contradictory to theor. 4, what is proved ? 
 
 What is such a method of proof called ? 
 Write the expressions below in their simplest forms, using only 
 positive exponents: 
 
 4. 81'/*-81-V2. 5_ 8ivy81-v* 6. 2b6'/»-256-^/^/256'^\ 
 
 10. [{(a-'")-"}^]«:[{ (a™)"} -*]-«. 11. a;»/y''V''ya;-»''y"'^* 
 
 12 [&T .[^J^^\[t}\'\ 13 (a;-y)-^-(a:+yr 
 
 \xl \y^l \aV) ' ' {x+y):{x—y)-^' 
 
 14. Prove that A -"''^ ■ B -"'^ ■ c --p/' = (a • b • c) -^/a. 
 
 15. How many signs has A*/*, or B*''*, when q is odd ? even ? 
 In each case what is the sign of the product a'/* • b'^' ? 
 Show that in any case/avalue of (a- b)'/' has the same sign. 
 
 16. Multiply the product of two of the three e'qual factors 
 of 8 by the product of two of the three equal factors of 135, 
 and compare the result with the product of two of the three 
 equal factors of the product 8 • 125 : write the conclusion, 
 using fraction exponents. 
 
 Find the values of : 
 
 17. 81V*. 256^/*. 18. Q^y^.l2b^/\ 19. 64-*/'/125-2/'. 
 20. {axy'''y-y\ 21. /^a?- \/y\ 22. i\fy?/ it/f. 
 
 W+ab) 'W-abJ ' \ x-y'l Hi 
 
 23. 
 
 -y I \x-yl 
 25. If fl'' = S«, then («/&)«/* = «"/"-»; and if fl = 25, then 5 = 2. 
 
170 POWERS AND ROOTS. [VI. 
 
 §3. EADICALS. 
 
 A radical is an indicated root of a number. There may be 
 a coefficient; and then the whole expression is called a radical, 
 and the indicated root is the radical factor. 
 
 Any expression that contains a radical is h radical expression. 
 
 A radical is rational if the root can be found and exactly 
 expressed in commensurable numbers; irrational, if the root 
 cannot be so found. It is real if it do not involve the even 
 root of a negative; imaginary, if it involve sach root. 
 
 An expression that contains an irrational radical is a surd. 
 
 E.g., -(S'ase, ^8, ^-8, M -)/(«' +3«6 + S') are radicals 
 
 with the rational values *3', 3, "3, a, *(a + 5); 
 but \^x, ^a\ ^a\ fa-a-v*, ^{d'+Wf/^ are irrational; 
 and, while all these radicals are real, 
 
 -(/-I, y-fls', ^-3a', {-ay^\ a + i^-l are imaginary. 
 
 Eoots of rational bases, and integer powers of such roots, 
 with rational coefficients, if any, are simple radicals. 
 
 The degree of a simple radical is shown by its root index. 
 
 A simple radical is quadratic, cubic, quartic, (biquadratic) 
 • • • when the root index is 3, 3, 4, • • • 
 
 E.g., |(«2+5y/^ dab"- p{a^-b(r'), a^-a^'\ are simple quad- 
 ratic, cubic, and quartic surds in their simplest forms ; 
 
 but * |/a', y/a"-, 4/8, 4/-8, -|(aV + 5V)i'"', are simple radicals 
 not in their simplest forms; they may be written: 
 
 ^a\fa, a^a, *3|/3. *3V-3. ''Ic^a^-^Wfl^. 
 Two radicals are like (similar) if they have the same radical 
 factor; they are conformable if they can be made like; non- 
 conformable if they cannot be made like. 
 
 E.g., '2,x{a''+Wfl\ 8{x-y)ia^ + bY' are like radicals, 
 and ^18, ^33, 4/98 are conformable. 
 
 The sum of two non-conformable simple surds, or of a 
 rational expression and a simple surd, is a binomial surd; the 
 sum of three non-conformable simple surds, or of two such 
 surds and a rational expression, is a trinomial surd. 
 
13] BADICALS. 171 
 
 QUESTIONS. 
 
 Are these radicals rational or irrational ? real or imaginary ? 
 
 1. |/5. 2. ^343. 3. |/-256. 4. |^-343. 
 
 ^- M^-y% 6. 4/(1-24/2; + a;). 7, v'256 4/-I. 8.-^-216. 
 
 9. What power of a radical is sure to be rational ? 
 Tell why the statements below are true : 
 10. x'/'=x''-a^/\ 11. v'-4=-4?'4. 12. 4/-4^ - 4/4. 
 13. 4^3= 4/48. 14. 3 4/-4= 4/-36. 15. -34/4= 4/86. 
 
 16. 4/(2/5)= 4/(10/25)= 4/(1/35). 4/10= ±14/10. 
 
 17. -3a;^5a^J= v'-27ic'- -^^5«2d= "^-135a«5a;». 
 
 18. 4/420fl»=±2a4/105a. 19.^t'(V48) = Vsl^80. 
 Reduce the radicals below to their simplest forms. 
 
 20. ^288. 21. 4/- 169. 22. ^16(a + 5). 23. ^^729. 
 24. 81-'/*. 25. 49=/". 26. 500'/». 27. 900^/« 
 
 28. Change the quadratic surds on the opposite page to 
 
 quartio surds, and the cubic surds to sextic surds. 
 
 29. 4/30, 4/45, 4/(4/5) are conformable surds. 
 
 30. Change \/2, p3 to surds of the same degree. 
 
 If possible make like the radicals below, and reduce the ex- 
 pressions to their simplest form : 
 
 31. ^32 + 6^3. 32. 4/3+ 4/^. 33. ^"'■\-^l\ 34. 4/a;- ^y. 
 35. 5 4/982; + 104/23;. 36. (3a«5)V2_ (27^25)1/2 
 
 37. 4/(5a;-5)- 4/(2a;-2). 38. y'(8«'6 + 16a*)- ^(5*+2aS'). 
 39. (36a'«/)'/*-(25.y)n 40. 1/x-^'^-\y y/x. 
 
 41. Separate 592704 into its prime factors: which of them 
 
 occur three times ? what is the value of v'592704 ^ 
 
 42. So, 4/78400; -^50635; 4/27235; -^3111696. 
 
 43. Are i^[a — V) and y^ rational or irrational ? 
 Replace a by 22, S by 6,.y by -^2: what changes are made? 
 Can a literal expression be imaginary and its numerical value 
 
 be real, or the reverse ? 
 
172 POWERS AND ROOTS. [Vl.Piw. 
 
 OPERATIONS ON RADICALS. 
 PeOB. 1. To REDUCE A RADICAL TO ITS SIMPLEST FORM. 
 
 Resolve the number whose root is sought into two factors: 
 one the highest possible perfect power of the same degree 
 as the radical, and the other an entire number; 
 
 write the root of the first-named factor as a coefficient before 
 the indicated root of the other. [th. 5. 
 
 E.g., v'48a'6* = )/{8w'W ■eb) = 2ab ^65. . 
 
 PrOB. 2. To FREE A RADICAL FROM COEFFICIEIirTS. 
 
 Raise the coefficient to a power whose exponent is the root-index 
 of the radical; 
 
 multiply this power by the expression under the radical sign, 
 and put the same radical sign over the product, [th. 5. 
 
 E.g., 2ab^Ql= |'(8a'5».65) = ^48a% 
 
 So, a Hifijr - oP) = f [a» • (S™ - c")] = ^(a^S" - ««c*'). 
 
 PeOB. 3. To REDUCE RADICALS TO THE SAME DEGREE. 
 
 Write the radicals as fraction 'powers; [df. fract. pwr. 
 
 reduce the fraction exponents to equivalent fractions having a 
 
 common denominator; [th. 3. 
 
 restore the radical sighs, using the common denominator as the 
 
 root-index and the new numerators as exponents. 
 
 B.g., ax, yby, ^ib+c)=ax, {byf", {b+cy^ 
 
 = {axY>/'^, {byY''', {i + cY^, 
 
 = ^(«^r, "^{hyr, ^{b+cf. 
 
 PrOB. 4. To ADD RADICALS. 
 
 Reduce the radicals to their simplest form; [pr. 1. 
 
 add lihe radicals by prefixing the sum of their coefficients to 
 the common radical factor; [add. incom. num. 
 
 write unlike radicals in any convenient order. 
 E.g., 3|/8 + 5 |/2- 10 1/33 = 64/2 + 5 4/2 -40 4/2 =-29 4/3. 
 So, a^b + d' y-S* - a* ^V =apb + a^b pb - a^^W y/b 
 = {a+a^-a^¥)^b. 
 
l-*-§3] RADICALS. 173 
 
 QUESTIONS. 
 
 Eeduce the radicals below to their simplest forms : 
 1. lasv*. 2. 567*/*. 3. 392V2. 4, loOSV^. 
 
 5. 216V». 6. 731/=. 7. 1621'". §_ ^giA 
 
 9. 160V5. 10.' (llfl)*/'. 11. (joUY^- 12. (lOi)'/*- 
 
 13. 25001/*. ;^4_ ^296352. 15. i/l4:7x-^yz\ 16. f56a*SV_ 
 17. t'llSa-^d-V. 18. y'64a»*-V. 19. pl6a'b'c^\ 
 
 20. |/50a*S»c«d^ 21. i V(3/7). 22. (a + b) y[{a-b)/{a + b)~\. 
 
 ,3. 2/,-^-,. 24. ^^^5¥. 25. ^i;^-^,. 
 
 ' 4(a + a;) 3a;' «^ 6' a"" 
 
 26. (ff./6) V(*«''Va^-')- 37. (a-»/V4c*)-^. 28. v'(a;-^'«3^i'«)^ 
 29. i/{'72a'b-72b + 18a-^b). 30. l/[a^y-^-xf-3x\x-y)]. 
 31. 3 4/14'^-! V'(V3)- i/(l/27). 32. 5 4/24-2 4/54- 4/6. 
 
 33. 
 
 a-& /«Vh-2^+^ fli»-Z>'^/3a* + 6a& + 3&^ 
 
 !/f fl + 6 " '«4-6' 
 
 a;-(y»' fl + 6 ■ ■ a + 6 f' 9(a^-6^) " 
 
 Free the radicals below from coefficients: 
 35. 6 4/5. 36. 2 4/a;. 37. 3a; 4/2. 38. 4a 4^55. 
 
 39. 4 -^^6. 40. 5a ^y. 41. | 4/9'/'. 43. ,^ 4/35, 
 
 ^Q.a^a-Wxy/xy. 47. (a;^-?/')'/'- \/i{x-y)/{p^ + %xy+y^)']. 
 
 Eeduce the radicals below to the same degree : 
 
 48. a^'\ a^'\ 49. a}'^, ¥/K 50. 3^2, 4^*. 51. aVs, J*'". 
 
 53. ^«5, ^ac, ^bc, ^{b + c). 53. a;V^ a;'/», 3;=/*, a;*/=, a;^/'. 
 
 54. 3a;V^ 22/^/=, 42?/*, 5y*/^ 55. ^^a?, ^l?, ^^&, 'y'tZ", ^e*. 
 Which is the larger: 
 
 56. (1/3)1/^ or (2/3)''/»? 57. |'3 or ^^3? 58. y/^ or v'18? 
 
 59- (i)''". (i)*^»(i)*^(i)'''°? 60. 4/(a + 5 + c) or 4/^+ 4/5+ 4/c? 
 
 Find the sum of: 
 
 61. 4/I8-4/8. 63. 6 4/(3/4) -3 4/(4/3). 
 
 63. 3 4/(2/5) + 4 4/(1/10). 64. 2 ,^(1/5) + 3 1^(1/40). 
 
 65. V128-3 4/50 + 7 4/73. • 66. «W/H2a5*/» + 5'/'. 
 
 67. 9 4/8O-2 ^125- 5 4/245+ 4/320. 
 
174 POWEES and roots [VI,Pbs. 
 
 PeOB. 5. To MUIiTIPlT EADICALS. 
 
 If all the radicals have the same iase, add the fraction ex- 
 ponents of the factors for the exponent of the product. 
 
 If the bases be different, but the radicals be of the same degree, 
 write their product under the common radical sign. 
 
 If the radicals be not of the same degree, make them like. 
 
 If there be coefficients, prefix their product to that of the radical 
 factors. [V, th. 10. 
 
 E.g., 3 V8-5 v'3--10|/32=-3-5-10- 4/(8-2-33) 
 
 = -150- 4/513= -2400- |/3, 
 
 So, aSi-" • aW^^ . a-35-r/3^ ^1+2-3. 51/3+4/3-7/3= j-2/3. 
 
 Note 1. The product and quotient of two conformable sim- 
 ple quadratic surds are rational; of two such non-conformable 
 surds they are surds. 
 
 For in the first case, the surd factors occur in pairs in the 
 product and vanish in the quotient; and in the other, they 
 are single, and the square root cannot be taken. 
 E.g., 4/6 is conformable with 4/(2/3), but not with 4/6,, 
 and 4/(6-2/3), 4/(6:2/3), 4/(2/3:6) are rational, 
 but 4/(6-5), 4/(6:5), 4/(5:3/3) are surds. 
 
 Note 2. The square of a binomial quadratic surd is a surd. 
 E.g., if 4/«, 4/5 be non-conformable surds, 
 then is ( ^/a + ^/bf =a + b + 2 /^ab, a surd. 
 
 PkOB. 6. To GET A POWBB, (OR KOOT) OF A RADICAL. 
 
 Multiply the exponent of the given radical by the exponent of 
 the power sought. [th. 2. 
 
 E.g., (3 -8'/^)' ' = 27-8=''^ = 432-21'* _ 433 ^3, 
 
 So, y'(3- 4/8) = v'4/72 = v'73; 
 
 (a=- 4/5')^ = a"- 4/P= a^-¥'- 4/5; 
 (tt3.j7/2)i/6 = a'/W/i". 
 
 Two quadratic binomial surds are conjugate if they differ 
 only in the sign of one term. 
 E.g., a+ ^b,a- \/b; lOV^ + S, 10^2-3; 4/2;+ ^y, \fx- 1,/y. 
 
8. 6, S3] RADICALS. 175 
 
 QUESTIONS. 
 
 Eeduce the expressions below to their simplest forms: 
 1. 3 V3-2 yS. 2. 8 |/6:2 >>/2. 3. 5 4/7-2 V7. 
 
 4 S*-" •21'^. 5. 4 4/3 • 3 4/5 • 5 v'3. 6, Si-^ • 3V» • i^K 
 
 7. (|)'''*:(f)*'^. 8. aV3.5i/3ci/ya-iA5-2/3c-V4. 
 
 9. 3 4/6 ■ 2 4/3 4 4/5 : 12 4/IO. 10. ^ ■ (D^/' : (|) • mf/'. 
 
 11.53/4.42/3.33/2.606/8. 13. (^^i-CD^/'i^CD-i/^ 
 
 13. 4/(a^ - S^) : ^{a - b) : ^{a - b). 14. (5 + 3 f 2) • (5 - 2 4/2). 
 15. |«,j^6'-i5 4/a».|a-»^&-i/». 16. (8 4/2 + 2 4/3). (2 4/2+ yS). 
 
 17. (4+ 4/3).(l- 4/3)-(4- 4/2). (5- 4/3)-(l+ 4/3).(5+ 4/3). 
 
 18. (a + 5)i'"» • (ffl + 5)^'" • {a - 5)»'"" • (« - S)V» . (a^ + 52)o» +»)/m» 
 
 19. V-a- 4/-J- ^-a- ^-b- ^-a- ^-b- ^-a- p-b. 
 
 20. (4^2+ 4^3+ i/5)-{- 4/2+ 4/3+ 4^5) 
 
 •(4/2- 4/3+ 4/5)-(4/2+ 4/3- 4/5). 
 Find the powers and roots: 
 
 21. (3 4/3)*. 22. (2 f 5)«. 23. ( 4/3- 4/3)^ 
 24. (4/10- 4/5)^ 25. (S'-^-S-'/y. 26. ( v'l- 4/i)^ 
 37. (2»/»-3-«/')». 28. (3i'^-3-»/»)». 29. (4'/»+4-»'^)«. 
 
 30. |'(fa;-iy)*. 31. [a'5(a»5c)»''»]»'^. 32. 4'-2»'"»a'»S'"c^. 
 33. (ai/»5-i/»+a-»/35V3)s. 34. [(a + S)»/2_(a-5)i/2]«. 
 
 3^. Find the first five powers of 4/"!. 
 
 Are the even powers of y~l real or imaginary? are the 
 fourth, eighth, twelfth powers positive or negative? the 
 second, sixth, tenth powers ? 
 Write the surds that are conjugate to : 
 36. 3-24/a;. 37.4^3-24/0;. 38. i/{a + b)+ ^(a-b). 
 39.3 + 2 4^5. 40.4/3 + 34/5. 4:1. x-3y-^/^. 43. x^/^-3y-^/\ 
 
 43. What terms of the square of 4/ffl + 4/5 are products of 
 
 conformable factors ? of non-conformable factors ? 
 
 44. If the product of four simple quadratic surds be rational, 
 
 the product of any two of them is conformable with 
 that of the other two. 
 
176 POWERS AND ROOTS. [VI, Pb. 
 
 Two surds are complementary if their product be rational. 
 E.g., a'/', a^/'; 5'/^, 5-^/^; V{o^ + W), i/id' + V). 
 
 So, any two conjugate binomial surds are complementary; 
 and three or more surds whose product is rational form a 
 group of complementary surds. 
 
 E.g., a + i\/b, a-)/l; S + S-jZ-l, a-SiZ-lj 
 
 PkOB. 7. To REDUCE A FRACTION WITH A SURD DENOMI- 
 NATOR TO AN EQUIVALENT FRACTION WITH A RATIONAL 
 DENOMINATOR. 
 
 (a) The denominator a monomial: multiply both terms of the 
 fraction by some complement of the denominator. 
 
 {b) The denominator a simple binomial quadratic surd: mul- 
 tiply both terms of the fraction by the conjugate of the 
 denominator. 
 
 E.g., a/¥'*^=a-b'"^/b; a/{^b- )/c) = a-{yfb + )/c)/{b-c). 
 
 (c) The denominator a binomial quadratic surd containing a 
 
 complex radical: multiply both terms- of the fraction by 
 a group of conjugate radicals that, taken together, are 
 complementary to the denominator, 
 
 a _a-^/{b + \/c) _a-y'{b + ^c)-{b — ^c) 
 ^^■' V{b+^c)~ b + i/c~~~ ^^ ■ 
 
 (d) The denominator any binomial surd: multiply the two 
 
 fraction exponents of the denominator by the loiuest 
 common multiple of their denominators, and attach the 
 products as exponents to the two bases; 
 
 divide the sum, or difference, of the powers so found by the de- 
 nominator, and multiply the numerator by the quotient. 
 
 B.g., to rationalize the denominator of 6Vy(3«/s4.33/i) . 
 
 then '.■ 12 is the lowest common multiple of 3, 4, 
 
 and 13-3/3 = 8, 13-3/4 = 9,' 
 
 and v(3'-3»):(22/»+3'/*) 
 
 = 322/3 _ 320/3 . 33/1 j^ 218/3 . 3«/i |. 2S/3 . 330/* _ 333/4 
 
7.83] RADICALS. 177 
 
 QUESTIONS. 
 Prove these surds complementary: 
 
 1. c»/», c*/^ 2. |/5-6,V5 + 6. 3. a;»/«, a;"/*. 4. f'^y-^"-. 
 5. 2a;-jfv'-l, 2a; + «/|/-l. 6. a-irl — i^c, a + J+4/c. 
 7. (4-3Va;)/(5^+6V2), (4 + 34/a:)/(5-64/3). 
 
 8. ^/a+ys-^s, 4/2+4/3+4/5, 4/6. 
 
 9. /^a — ^h — ^c, 4^« — 4/5 + 4/c, a + 5— c + 34/a5. 
 
 10. The product of any two conjugate quadratic binomial 
 surds is rational: what other complement has such a surd 
 besides its conjugate ? 
 
 11. The product of two surds differing only in the sign of 
 one term, but of higher degree than the second, is not rational. 
 
 Write three complements of each of the surds : 
 
 13. wi-'A 13. {^x^Y'K 14. {fl-Vf'K 15. «J->/«. 
 
 Reduce to equivalent fractions with rational denominators: 
 16. 1/4/3. 17. 6/4^2. 18. 3^/8/2^2. 19. 2xl%y"\ 
 20. Ax/%f'\ 21. {a/yfi^. 23. (m/ny^^'K 33. 2/(4/3 + 1). 
 
 ''*• 4/2 + 1" 4/3 + 4/3' "^ 4/10-4/3" '''• 4/5-4/3' 
 
 28. !-+^):^;+;-!)!!. ' 29. 
 
 (a + 5)V3_(a-5)va- '-•'• a + 4/[i + 4/(c+4/£i)]' 
 
 30. |/2- (4/2-3) g^ a' + C^-ir 
 
 (4/3 + 8) -(4/3 -4/5)" aJ-Ca^-l)'/'' 
 
 „„ x-i^^^-rf^ _„ a;-(a;^+l)'/^ 
 
 '*''• a;+(a^-l)'/«- '^^- :c + (a^+l)V«' 
 
 _ . («+4/-l)''-(g-4/-l)'' „, «4/(l-g')-Ml-«') 
 (ffl+4/-l)'-(a-4/-l)''' 4/(1 -5'') + 4/(1 -a") * 
 
 4/6-4/5-4/3 + 4/3 1 + 34/2-84/3 
 
 4/6 + 4/5-4/3-4/2' 4/3+4/3 + 4/6' 
 
 38. (3 + 4/3) . (3 + 4/5) ■ (4/5 - 2)/(5 - 4/5}/(l + 4/3) = I4/I5. 
 
178 POWERS AND ROOTS. [VI, 
 
 EQUATIONS THAT CONTAIN SUKDS. 
 
 An equation that contains surds is rationalized when it is 
 replaced by an equivalent equation free from surds. 
 
 E.g., the equation a; = y3, i.e., x= \^2 or x—'\/2, 
 when rationalized, becomes x^—%. 
 
 So, if 4/3; + 4/^=0: 
 
 then x — y=0; [mult. ea. mem. by |/a; — yy. 
 
 or V^=-Vl/> [ax. add. 
 
 and x=y. [squaring. 
 
 So, if \/x+^^^/ + /^/z=0•. 
 
 then x+y—z + 2Yxy = 0, [mvlt.hy (\/x + i^i/ — j/z\, 
 
 2 i^xy =z^x-y, [ax. add. 
 
 and 4,xy = z'+a?+y^+Zxy — 2zx — 2zy. [squaring. 
 
 So, if 7?/^ +3^/^-1 = 0: 
 then X - 20^1^ +1 = 0, [mult, by (a;'^ - 1). 
 
 2a^^=x + l, 8x=oi? + 33?+3x + l, [cubing, 
 
 and a^ + 3a:'-5:B + l = 0. 
 
 ., x-i/'x — 12 _ \'x + 2 
 ^°' 'x + i^x-Q ~ i^x + l' 
 
 then — ?. = ■--, — r-j [red. first mem. to lowest terms. 
 
 i^x — 2 ^i^x->r\ "- 
 
 4a; — 15 y'a; — 4 = a; — 4, [clear, of f ract. 
 
 and x—^i^x, a?=25x, a; =25 or 0. 
 So, if V(3 + a;) + 4/a;=6/|/(3+a;): 
 'then 3 + a; + )/{3x + a?) = 6, [clear of f ract. 
 
 4/(3a! + a?) = 3-a;, 
 and 3x+a?=9 — Sx + a?, 9a;=9, a;=l. [squaring. 
 
 So, if a;>^-[a;-(l^a;)V7A=l: 
 then [x-il-xy/^f/^ =0^/^-1, x-{l-x)^=x-2x^/'+l, 
 
 {l-xf/' =2x^/^-1, (l-a:) = 4a;-4a;>''Hl, 
 and 4ic'''*=5a;, lto=25a;*, a;= 16/25 or Q, 
 
§8] EADICALS. 179 
 
 QUESTION'S. 
 
 Solve the equations: 
 
 1. |/(3-a;) + 6 = 7. 2. \/{x+^) = ^-^x. 3. jr»/' + 5 = ll, 
 4. xJra=^[a' + xi^{l^+oi?)']. 5. ^(20 -4/22;) = 2. 
 
 6. -^ — — = 4h . [red. first fract. to lower terms. 
 
 V3a;-V 3 jk + Z |/(4a; + l) + f4a; 
 
 ' V2a;-4/2 2 * **" -|/(4a; + l)-|/4a; ~ ' 
 
 9. a;'''«+(a+a;)*'"'=3a(a+a;)-v«. 10. (a^-l)'/2=8(a;'-l)->'^. 
 11. (a?-3a; + 4)»'^=a;-3. 12. (8-4a;)V2 + {13-4a;)''«=5. 
 
 13. 18(2a; + 3)-v^=(2a;-3)»^ + (2a;+3)V2. 
 
 14. 3v'(a:-|)+74/(a;+A) = 104/(a; + .03). 
 
 15. ^(4/3 + a;V7) + -^(4/8 - a;^/?) = 4^12. 
 
 16. |/(a + a;) + 4/(a - a;) = 5[>/(a + a;) - !/(« - «)]• 
 Rationalize the equations: 
 
 Vll.H/a + j\/i + c=0^ 18. 3Va;-24/5^=-l. 
 
 19. i^x-i^y=zi^y-i^x. 20. -4^ar'-^a; + l=0. 
 
 21. y'a;«-^a;-l = 0. 22. y'a^-v'a;+l = 0, 
 
 23. What effect is produced on the degree of an equation 
 by rationalizing it ? 
 
 24. Divide 18 into two parts whose squares shall be to each 
 other as 25 to 16. In solving this problem four square roots 
 are found : what is the effect if one of them be taken negative ? 
 
 25. If to a certain number 22577 be added, the square root 
 of the sum be taken, and from this root 163 be subtracted, 
 the remainder is 237: what is the number? 
 
 26. The length of the side of a square whose area is 1 square 
 inch less than that of a given square, increased by the side of 
 a square whose area is 4 square inches more than that of the 
 given square, equals the side of a square whose area is 5 square 
 inches more than 4 times that of the given square: find the 
 area of the given square. 
 
180 POWERS AND ROOTS. [VI, Pb. 
 
 §4. EOOTS OF POLYNOMIALS. 
 
 Evolution is an inverse operation : the work is an effort to 
 'retrace the steps taken in getting the power whose root is now 
 sought; it is a process of trial, by progressive steps, like divi- 
 sion and other inverse operations, and its success is established 
 by raising the root to the required power and finding the 
 result identical with the given polynomial. 
 
 SQUAEB BOOT. 
 PkOB. 8. To FIND THE SQUARE BOOT OF A POLTNOMIAL. 
 
 Arrange the terms of the polynomial in the order of the powers 
 
 of some one letter, apcrful square first; 
 taJce the square root of the first term of the polynomial as the 
 
 first term of the root; 
 divide the remainder iy double the root so found, and make 
 
 the quotient the next term of the root and of the divisor; 
 multiply the complete divisor by this term and subtract the 
 
 product from the dividend; 
 
 double the root found for a new trial divisor, and proceed as 
 before. 
 The rule is based on the type form for the expansion of the 
 square of a polynomial; when any complete divisor is mul- 
 tiplied by the new term of the root, and the product is sub- 
 tracted from the last remainder, the whole root thus far found 
 is thereby squared and subtracted from the polynomial. 
 E.g., a'' + 2ab + b^ + 2ac + 2bc + (f{a + b + c. 
 
 2a+b)2ab + W 
 
 2a + 2b+c ) 2ac+2bc+<? 
 
 So, {Zx-^-ay^^ + a-''^ 
 
 9a;-*-6aa;-y'^+afj?*'^+6a-;a!-V-22r^''V+a-»2» 
 
 6a;-^ - ayV^ ) -6aa;-y/°+ay/» 
 
8.54] ROOTS OF POLYNOMIALS. 181 
 
 QUESTIOlffS. 
 
 Find the square root, preferably by detached coefiQcients, of : 
 1. 16a?-4:0xy + 25y\ 2. H-2a; + W + 6a:' + 9a^. 
 
 3. a*-2a*+3ffl*-a+i. 4, x-*-2x'^ + Z-2a^+x\ 
 
 ' y^ a? y X 4:' ■' V b a aF 
 
 7. 9a!«-30aa;-3a^a; + 25aH5a'+ia*. 
 
 8. ^a^-\2ab+4:ax + m-Ux + x\ 
 
 9. ga" - 12a^ - 26a* + 44a' + 9a^ - 40a + 16. 
 
 10. 4a^+8aar'+4aV + 16&W + 16a&^a; + 16S*. 
 
 11. ^a^T? - ^aioi?z + ^aj'bx'z' + 6%« - 4aS^ar'z' + 4aWa^2*. 
 
 12. 4a?+^a;-= + 16a!-*-i + 16a;-*-fa;-'. 
 
 13. a + 4a''/' + Ga'/* + 4a=''« - 6a'/* - 12a''^. 
 
 14. 1 + 4/a; + Id /a? + 30/a;' + 25/iB* + 24/a;= + 16/a;«. 
 
 15. 1 - 4/2 + 10/2^ - 20/2' + 25/2* - 24/2« + 16/2». 
 
 16. How many terms has the square of a binomial ? 
 Supply a third term to a*+2a6 that shall make the result- 
 ing trinomial a perfect square ; so, to a' + 5'; to 3a5 + 5^. 
 
 Make a general rule for completing such a square. 
 Complete a square from each of the expressions : 
 
 17. 7? + &x. 18. a?-^x. 19. !ii? + bx. 30. x'-'Ux. 
 21. a^-^oi?. 33. a;'+8a?. 23, y*-12y\ 24. 92^122. 
 25. 16ci?-2ix. 26, 25a^-302;2, 27, 92*-122', 38. 4a^-10a;. 
 29. {y''+y-6y-Z{y^+y-6). 30. {x+yY + 2{x+y). 
 
 31. {a-'+2x-lY+iaP + 8x-4>. 33. {a-3yy+Q{a-dyY. 
 Find four terms of the square root of: 
 
 33. 1 + a?. 34. x' + l. 35. ci?-aK 36. a»-a?. 
 
 37. l-a?. 38. a;«-l. 39. a?+a*. 40. a'+«*. 
 
 ^how that the expressions below are perfect squares: 
 
 41. {x'-yzf+^^-zxy+iz^-xyf 
 
 -Z{oi?-yz)-{f-zx)-{z^-xy). 
 
 42. 4\{(f-W)cd+{(?-d'')alY+ [{a?-W)(c'-d^)-4Mhc^\ 
 
182 POWERS AND ROOTS. .[VI, Pb. 
 
 CUBE KOOT. 
 PeOB. 9. To FIND THE CUBE ROOT OF A POLYNOMIAL. 
 
 Arrange the terms of the polynomial in the order of the powers 
 
 of some one letter, a perfect cube first; 
 take the cube root of the first term of the polynomial as the first 
 
 term of the root; 
 divide the first term of the remainder ly three times the square 
 
 of the root found, and make the quotient the next term 
 
 of the root; 
 to the divisor add three times the product of the first term of 
 
 the root ly the second term, and the square of the 
 
 second term; 
 multiply the complete divisor iy this term and subtract the 
 
 product from the dividend; 
 
 take three times the square of the root found for a new trial 
 
 divisor and proceed as before, treating the root so far 
 
 found as the first term and the new term as the second. 
 
 The rule is based on the type-form for the expansion of the 
 
 cube of a polynomial 
 
 and when any complete divisor is multiplied by the new term 
 of the root and subtracted from the remainder, the whole root 
 so far found is thereby cubed and subtracted from the poly- 
 nomial; 3a^ is the trial, %a^+Zab-\-¥ the complete, divisor. 
 E.g., (a+S +c 
 
 a? + Za?b + SflsS' + J' + 3a'^c + 6a6(!+3oc'+36'c + 3Sc''+e' 
 
 ^ 
 
 3a'' + 3aJ+ ft' ) Sa'S + 3fflft' + S° 
 
 3a' + 6a& + W+ 3ae + 35c + c' ) Za?c + 6aSe+3a<!'+3SV + Sic'Hc^ 
 
 The rule given and illustrated under prob, 11 is perhaps 
 the better rule for getting cube roots; the cubing of the root 
 takes the place of the less familiar process of completing the 
 divisor. 
 
9. Ml ROOTS oy POLTNOMIALS. 188 
 
 QUESTIOKS. 
 Find the cube root, preferably by detached coeflBcients, of: 
 1. H-6a; + 12a^ + 8a;'. 3. d'-6a:^y + l2xY-8f. 
 
 3. a*+3a + 3a-Ha-=. 4. a^y-^-Qa^ + Uce^f-Sy''. 
 
 8"^2"^3a;'"^27a^' "'■2c^a^"*'4cV"^8cV 
 
 7. a;= + 6a:'-40a? + 96a;-64. 8, x^-6af'+'i0a?-96x-6i. 
 ,9. a«-9a*5»c+27a'W-37^iV. 
 
 10. l-6a; + 21a?-44a? + 63a!*-54a^+27a;'. 
 
 11. (« + l)'"a? - 6cfl!^(a + l)*"x' + 12<f a^{a + l)^a; - Sc'a*'. 
 13. 8a^ + 48a;5^ + GOa^?/' - 80:r'«/= - 90a;y + lOSa;^^ - 27/. 
 
 13. a?-9a^ + 30a;-45 + 30a;-i-9a;-Ha;-'. 
 
 14. a'/* - Ga^/* - 9a»''^ + 12flV* + 36aH 19a'''* - 36a^/' - 54a"''* 
 
 -37a». 
 Find the cube root, preferably by the type-form, of: 
 
 15. c^-6y\a^ + 12y'' 
 
 + 92 1 -36yz 
 + 27z^ 
 
 16. Arrange the terms in ex. 15 to rising powers of y andthen 
 
 find the cube roots; so, to falling powers of z. 
 In what are these cube roots alike, and how do they differ ? 
 
 17. a= + 6a^5 + 12«S'^ + 85' 18. a? + 6x'y + 12xy"+8y'' 
 -a^ + 6a!'b-12ai^ + 8P -a^ + 6ix?y-12xtf+ 8/ 
 
 8a^ + 12a'S + 6ai^ + b\ 8a;» + 12a:^y + Qxy^+f. 
 
 19. In the cube of a + b, how many terms are of the form 
 
 a' ? of the form Sa'b ? how many terms in all ? 
 
 Without finding the root, show that a? + 6x'y + 12xy^ + 8y' 
 
 and x'—6a^y + 12xy' — 8y^ are perfect cubes. 
 
 20. Find the cube of a + b + c. How many terms are of the 
 
 form «' ? of the form 3a'6 ? of the form 6abc ? in all? 
 Can a polynomial of two terms be a perfect cube ? of three 
 terms? of four? of five ? of eight ? of nine? of ten? 
 
 X- 8f 
 
 ^ + Qy 
 
 3? + 12y^ 
 
 x+ 8f 
 
 + S6yh 
 
 -92 
 
 -36yz 
 
 -3Qyh 
 
 — b^yz^ 
 
 ' +272* 
 
 + 54y2« 
 
 + 272» 
 
 
 
 -272» 
 
184 POWERS AND ROOTS. t VI, Pus. 
 
 PbOB. 10. To FISTD THE ROOT OF A POLYNOMIAL, WHEN 
 THE BOOT-INDEX IS A COMPOSITE NUMBER. 
 
 Resolve the root-index into its prime factors; 
 
 find that root of the polynomial whose index is the smallest of 
 
 these factors; [th. 2. 
 
 of this root find that root whose index is the next larger factor, 
 
 and so on. 
 E.g., the fourth root is the square root of the square rootj 
 
 the sixth root is the cube root of the square root; 
 
 the eighth root is the square root of the square root of 
 the square root. 
 
 So, |/(««-8a;« + 28a;*-56:s^+'J'0-56a;-f+38a;-*-8a;-«+a;-») 
 = |/(tB*-4a^+ 6 -4a:-''+a;-*) 
 = 4/(a;«-2 + a;-«) 
 = x—x~^. 
 
 Pros. 11. To find the wth root of a polynomial 
 
 WHEN n IS PRIME. 
 
 Arrange the terms of the polynomial in the order of the powers 
 of some one letter, a perfect nth power first; 
 
 write the nth root of the first term as the first term of the root; 
 
 divide the second term ly n times the {n — l)th power of the 
 first term of the root, as a trial divisor, and maJce the 
 quotient the second term of the root; [th, L 
 
 from the given polynomial subtract the nth power of the bi- 
 nomial root already found; 
 
 divide the first term of the remainder iy the same trial divisor 
 as before, and so continue, always subtracting the nth 
 power of the root so far found from the entire polynomial. 
 
 E.g. , a;=/27 - a^/3 + 2a; - 7 + 18/a; - 27/a;^ + 27/a;^( a;/3-l + 3/a; 
 a:^/27-ig'/3+ x-\ 
 a;y3)a;-6 
 ^J%1 - ^/Z + 2x-7 + 18/x-27/x'+ 27/rK». 
 
10.11,84] ROOTS OP POLYNOMIALS. 185 
 
 QUESTIONS. 
 
 Find the fourth root of: 
 1. ^- 4xf + 10a;« - 16a:= + 19a;* - IGa;' + IW - 4a; + 1. 
 3. a;* + 4.7?y + ^o?f + ^.x'f + «/* + '^oi?z + 1 2a; V + \%xyH 
 + 4^2 + 6a;V + 12a;«/2' + 6 jrV + 4a!z' + 4«/2' + z*. 
 
 3. Expand (a + 5)*. How many terms of the form a*? of 
 
 the form 4a'6 ? of the form ^Wt in all ? 
 
 4. Expand (a + J + c + <^)*. How many terms of the form 
 
 a*? of the form 4a^6? of the form 6a^6*? of the form 
 12a^6c? of the form 24a6cf?? in all? 
 
 5. Can an expression of four terms be a perfect fourth 
 
 power ? of five terms ? of nine? of ten? of fifteen ? 
 Find the sixth root of: 
 
 6. ««-13a*+60a''-160 + 240a^«-193a-*+64a-». 
 
 7. a= - %a'l + 4a*6» - ff a^d' + ^a«5* - ^\aS« + ^i^S'. 
 
 8. a;^ - 12a;^" + 66a;' - 220a;' + 495a;* - 792a;* + 924 - 792a;-« 
 
 + 495a;-*-220a;-H66a;-»-12a;-i» + a;-". 
 By the process of prob. 11 find the cube root of: 
 
 9. «« - low'l + 15a*5* - 20«'5» + 15a'6* - 6a5= + 5«. 
 
 10. a;' - Sjr' + 27«= - 6a;*^ + Vixy"^ + %^z + 27a;z« - %ioyH + 54?/2* 
 
 — 36xyz. 
 
 Find the fourth root of: 
 
 II.' a;*- 12a;» + 62a,^- 180'?; + 321 - 360a;-' + 348a;-*- 96a;-» 
 + 16a;-*. 
 
 12. (a;* + a;-y-4(a;+a;-i)«+13. 
 Find the fifth root of: 
 
 13. a^ + 10a^b + 4OaW + 80aW+80a¥ + 32¥-5a^c-A0a!'bc - 
 
 - 120a*6*c - IGOaPc - 805*c + lOasV + %Oa^b(? + 120«&V 
 + 805V - lOaV - 40a5c' - 405*c' + 5ac* + lOSc* - c». 
 
 Find the sixth root of: 
 
 14. 64a;'-384a;*+960a;»-1280 + 960a;-«-384a;-* + 64a;-«. 
 Find these roots correct to three terms : 
 
 15. ^iX-^x). 16. ^(l-Sar^). 17. |^(l-4a;«). 18. ^(l-5a:*). 
 
186 POWERS AND ROOTS. [VI.Pr. 
 
 §5. ROOTS OF NUMEKALS. 
 
 PkOB. 13. To FIlfD THE SQUARE ROOT OF A KUMERAL. 
 
 Separate the numeral into periods of two figures each, both to 
 
 the left and to the right of the decimal point; 
 take the square root of the largest perfect square in the left- 
 hand period; 
 subtract this square from the period, and to the remainder 
 
 annex the next period to form the first dividend; 
 double the root already found and use it as a trial divisor, 
 
 omitting the last figure of the dividend; 
 annex the quotient to the root and to the trial divisor; 
 multiply the complete divisor by this root figure, subtract the 
 product from the dividend, bring down the next period, 
 and proceed as before. 
 Note 1. Numerals are polynomials, but polynomials in 
 whicli the terms overlie and hide each other; and virtually 
 the rule for finding roots is the same for both. 
 
 The separation into periods is a matter of convenience only; 
 it comes from this : that the figures of the root, of different 
 orders, are best found separately, and that, since the square of 
 even tens has two O's, therefore the first two figures, counting 
 from the decimal point to the left, are of no avail in getting 
 the tens of the root, and are set aside and reserved till wanted 
 in getting the units' figure. So the square of even hundreds 
 has four O's, and the first four figures, two periods, are set 
 aside and reserved till wanted in getting the tens ; and so on. 
 So, in getting roots of decimal fractions, the square of 
 tenths has two decimal figures, and the first two figures, one 
 period, are used in getting the tenths' figure of the root; the 
 square of hundredths has four decimal figures, and so on. 
 The same thing appears from this : that the root of a fraction 
 is most easily found if the denominator be a perfect square; 
 and this it is, with decimal fractions, only when it consists of 
 1 with two O's, or four O's, or six O's, and so on; that js, when 
 the number of decimal figures used is even. 
 
12.85] ROOTS OF NUMERALS. 187 
 
 QUESTIONS. , 
 
 Find the square root of: 
 
 1. 494209. 2. 3345341. 3. 135457.64. 
 
 4. 3533.1136. 5. 17.75358325. 6. 11090466. 
 
 7. 1733.333601. 8. 576864334. 9. 1771561. 
 
 Find the square root, correct to three decimal places, of: 
 10. 144. 11. 14.4. 13. 1.44. 13. .144. 
 
 14. .0144. 15. .00144. 16. .000144. 17. .0000144. 
 Find the fourth root, correct to three decimal places, of; 
 
 18. 16.0001. 19. 160.001. 30. 1600.01. 31. 16000.1. 
 Find the square root, correct to a twelfth, of : 
 
 33. 49/144. 33. 49/73. 34. 49/36. 35. 49/18. 
 
 Find the square root, correct to a ninth, of: 
 
 26. 17/9. 37. 17/37. 28. 17/6. , 29. 17/36. 
 
 30. Find the square of 15; then find (10 + 5)^ (9 + 6)^ 
 using the type-form, and show that the powers and products 
 of, 10, 5, 9, 6, overlie and hide each other in the results. 
 
 31. In getting the square root of 235, show that 10 is not 
 necessarily the first guess. Make the computation, using 9 
 for the first number; then, in turn, using 13, 16, 20, for the 
 first number. 
 
 32. How many figures in 1^ ? in g^^ ? in 10^ ? in 99^ ? in the^ 
 square of the smallest three-figure number ? the largest ? 
 
 How many figures in the square of a number of n figures ? 
 
 33. How many figures are there in the square root of an 
 integer of four figures? of three figures? of 2n figures? of 
 3ra — 1 figures ? 
 
 34. What is the object of separating a numeral into two- 
 figure periods? Why could not the process begin with the 
 first figure of the number, irrespectiye of the decimal point? 
 
 35. Why must a decimal fraction that is a perfect square 
 be expressed by an even number of figures ? 
 
 36. How many decimal figures has a perfect nth. power ? 
 
188 
 
 POWERS AND ROOTS. 
 
 [VI, P». 
 
 GEOMETRIC ILLUSTKATION. 
 
 Note 2. The reason of the rule for finding the square root 
 is made clear by a geometric illustration. 
 E.g., to find the side of a square of 6064 square inches. 
 
 The length of the side lies between 70 and 80 inches. 
 
 B 
 
 C 
 
 A 
 4900 sq. 111. 
 
 B 
 
 The square A, 70 inches on a side, contains 4900 square 
 inches, leaving 1184 square inches, to be so added that the 
 resulting figure shall be a square. This is done by adding 
 equal rectangles, b, on two sides and a square, c, to complete 
 the figure. i 
 
 70 in. 
 
 70 in. 
 
 These additions, placed end to end, form a rectangle whose 
 length is a little greater than 140 inches, and whose area is 
 1184 square inches. The breadth of this rectangle is a little 
 less than the quotient of 1184 by 140. Try 8 inches : then 
 the entire length of b + B + o is 148 inches, and the area 
 (148 X 8) square inches, is the desired addition. 
 
 The side of the square is therefore 78 inches. 
 
 COKTKACTION". 
 
 Note 3. When the first n figures of a square root have 
 been found by the rule above, then n — 1 more figures may be 
 got by dividing the remainder by double this root. 
 
 In principle, this process is the same as using a trial divisor; 
 only the possible error in the quotient needs consideration. 
 
 The whole root is divided into two parts, the first n figures, 
 already found, and the n — 1 figures that follow. 
 Put p, Q, for the values of these two parts; 
 then-.-p>Q-10"-', 
 
12, §5] ROOTS OP NUMERALS. 189 
 
 /. 2p differs from 2p + Q, the complete divisor, by less 
 than one part in 2-10""* parts, of this divisor, 
 and the resulting quotient, q', differs from the true quo- 
 tient, Q, by less than one part in 2 • 10""' parts, of Q. 
 AndvQ^lO"-, 
 
 .•. q' is in error by less than half a unit in' the last figure. 
 
 QUESTIONS. 
 By contraction find the value, correct to three decimal places, of : 
 
 I. V185. 2. |/912. 3. ^739. 4. ^'1008. 5. -{^8000. 
 
 6. If the square root of a number contain three figures and 
 two of them have been found, the part of the root still to be 
 found is less than a tenth of the whole root, and less than a 
 twentieth of the next divisor. 
 
 7. The quotient found by using a divisor between twenty 
 twenty-firsts and twenty-one twenty-firsts of -the true divisor 
 gives a quotient whose error, if any, is less than a twentieth 
 of the remaining figure. Can this error be half a unit ? 
 
 8. If three out of five root figures have been found, what 
 part of the whole root is the value of the remaining figures ? 
 
 The error in the contracted method is less than what frac- 
 tion of the remainder of the root ? what fraction of a unit ? 
 
 9. If n figures of a root have been found, what is the 
 greatest possible error that can be introduced by finding the 
 next n — 1 figures by division ? 
 
 10. If a true quotient be 27365.7 and the division be carried 
 to but five figures, shall the fifth figure written be 5 or 6 ? 
 
 II. If the true root be 27365.7, and if, after finding three 
 figures, the next two be got by division, what danger is there 
 that the fifth figure be written 5 and not 6 ? 
 
 If the true root be 27365.49, what is the danger ? 
 
 In writing down the last figure of the root, in which direc- 
 tion should an allowance be made ? 
 
 12. Find the square root of 40297104, and illustrate by a 
 diagram showing the rectangular additions that give the 
 hundreds' figure, the tens' figure, and the units' figure. 
 
190 POWERS AND ROOTS. [TI.Pr. 
 
 PkOB. 13. To FIND THE CUBE BOOT OF A NUMERAL. 
 
 Separate the numeral into periods of three figures each, loth to 
 
 the left and to the right of the decimal point; 
 take the cube root of the largest perfect cube in the left-hand 
 
 subtract this cube from the period, and to the remainder annex 
 the next period to form the first dividend; 
 
 to three times the square of the root already found annex two 
 ciphers and use the result as a trial divisor, placing 
 the quotient as the next figure of the root; 
 
 to the trial divisor add three times the product (with one 
 cipher annexed) of the first part of the root by the new 
 root figure, and add the square of this figure; 
 
 multiply the complete divisok- by this figure, subtract the prod- 
 uct from the dividend, bring down the next period, and 
 proceed as before. 
 
 Note 1. The reason for separating the nnmher into periods 
 of three figures is made eyident by considering the number of 
 figures in the cubes of numbers having one digit, two digits, 
 three digits, and so on, and the number of zeros in the cubes 
 of exact tens and hundreds. [comp. pr. 13 nt. 1. 
 
 CONTEACTION. 
 
 Note 3. When the first n figures of a cube root have been 
 found as above, then n — % more figures may be got by divid- 
 ing the remainder by three times the square of this root. 
 Put p for the value of the figures already found, and Q for 
 
 that of the n — % figures that follow; 
 then-.-p>Q-10"-S 
 
 .•. 3p^, the trial divisor, differs from Sp' + Spq + q", the 
 complete divisor, by less than one part in 10""* parts, 
 and the resulting quotient, q', differs from the true quo- 
 tient, Q, by less than one part in 10"~' parts, of q. 
 And -.-Q^ 10"-^, 
 
 .-. q' is in error by less than a tenth in the last figure. 
 
13, §5] ROOTS OP NUMERALS. 191 
 
 If the first figure of the root be 3, or larger, then to— 1 
 figures may be found by division. 
 
 QUESTIONS. 
 Find the cube root of: 
 
 1. 148877. • 2. ,007821346625. 
 
 3. 2439656.927128. -4. 836.803326004904. 
 
 Find the cube root, correct to three decimal places, of: 
 
 5. 1738. 6. 172.8. 7. 17.28. 8. 1.728. 
 
 9. .1728. 10. .01738. 11. .001728. 12. .0001728. 
 
 By contraction, find the value correct to three decimal places, of: 
 
 13. ^625. 14. ^587. 15. f 1728. 16. ^18.625. 
 
 17. How many figures in 1'? in 9'? in 10'? in 99»? in the 
 cube of any three-figure number ? of any »t-figure number ? 
 
 18. How many figures in the cube root of a number ex- 
 pressed by 3ra figures? 3w — 1 figures? 3re — 2 figures ? 
 
 19. If three out of four figures of a cube root have been 
 found; if this part be called a, and the remaining part t; then 
 %ab + }? is the part of the divisor omitted in the contracted 
 process, 5 is smaller than a hundredth part of a, 3fl5 + S' is 
 smaller than 'a hundredth pairt of the true divisor, and the 
 error in the quotient is smaller than" a tenth part of a unit. 
 
 20. If 262144 cubic inches are to be arranged in the form 
 of a cube, and if a cube be formed whose edges are 60 inches, 
 how many cubic inches are so used and how many are left ? 
 
 If additions 60 inches square made to the top, front, and 
 one side face, would complete a perfect cube, how thick could 
 these additions be ? 
 
 21. Eegarding the integer part of the result in ex. 20 as the 
 second figure of the root, make four other additions, three of 
 them with a length of 60 inches, and a width and thickness 
 each equal to the thickness of the first additions. 
 
 What are the dimensions of the final addition required ? 
 How many cubic inches have thus been added ? 
 Draw the original cube, and the several additions. 
 
192 POWERS AND ROOTS. [V1,Ths.6,7. 
 
 § 6. EOOTS OP BINOMIAL SUEDS. 
 
 Thbor. 6. If two simple surds, of the same degree, in their 
 simplest forms, be equal, their coefficients are equal and their 
 radical parts are equal. 
 
 Let ay'A, Sy'B be equal simple surds in their simplest forms; 
 then will a = h, A = B. 
 
 For, fls:5=^(B:A), a true equation, but true only when 
 
 ^(b : a) is rational, [th. 5 cr. 
 
 i.e., when a = b; and in that case a = t also. Q.B.D. 
 
 -CoE. 1. Two non-conformable simple surds cannot be equal. 
 
 Theor. 7. The sum of two simple non-conformable quadratic 
 surds cannot be rational. 
 For, let |/A, 4/B be two simple non-conformable quadratic 
 
 surds, and a, b, G, he rational numbers, 
 and if possible let a^A + 54/B = ; 
 
 then •,' 2ab^AB = (f — a^A. — l^B, [sqr. both mem. and transp. 
 and y'AB, 2ab\/A'B, are surds, [pr. 5, nt. 1. 
 
 .% a surd equals a rational number, which is impossible, 
 .:a'^A + b\/B^G. Q.E.D. [df. surd. 
 
 Cob. 1. If a, X be rational numbers, ^b, n/y simple quad- 
 ratic surds, and x + ^y = a+ i^b, then x = a, ^y — ^b. 
 For, if possible, let x = a + c, and c be rational; 
 then \' a + c + /^y = a+ ^b, [%?• 
 
 .\i\/b — i\/y = c, which is impossible, [th. 7. 
 
 .'.x—a, i^y—^b. 
 Cob. 2. If x + /^y — a + ^b, then x — i\/y=a—/^b. 
 Cob. 3. If i^x + /^y = ^(ja + \/b), then /^x—^y = /^{a — /^b). 
 For •.•x + y + 2 ^xy — a + \/b, " [sqr. both mem. 
 
 .•.x+y = a and 2\/xy = n/b, [cr. L 
 
 .'.x + y — 2^xy = a — ^b, 
 and ^x-^y = ^{a-Yb). q.b.d. 
 
PB.HS6] ROOTS OP BINOMIAL SURDS. 193 
 
 PbOB. 14. To FIND A SQUARE BOOT OE A BINOMIAL SUBD. 
 
 Let «+ 4/5 be a binomial surd and let i^x + yfy = |/(a + 4/5) ; 
 then %• i^x—n/y= i^(a - 1/5), [th. 7 cr. 3. 
 
 .-. 4/a; = i [ y (a + 4/6) + |/(a - 4/J)], 
 ^y = i[i/{a + i/b) - 4/(a - yS)], 
 x=i[a+^{a'>-b)l y^i[a-i/ia>-h)l 
 
 .-. 4/X+ v'«/ = |/^[a + V(a«-S)] + yi[a- VCa'-*)]. 
 
 QUESTIONS. 
 
 I. In the proof of theor. 6, |/A, |/B, being surds, ■|/(b/a) 
 can be rational only when b/a=1: state the proof. 
 
 3. The equation a\/A=i\/B is satisfied when a= —i, 
 a=b; but not when a—i. A— —b, nor when a— —I, A= — B. 
 
 3. If a + h^c=a' + l'/\/c', then a = a', h=V, c=c'. 
 
 4. If |/a + 4/5=4/0 + -(/<?, the surds are equal in pairs. 
 
 5. A quadratic surd cannot equal the sum of two other 
 non-conformable quadratic surds, nor their difference. 
 
 6. If 5 - 24/6, 34/6-5 be the two square roots of 49 - 2O4/6, 
 and 4/3-4/3, 4/3-4/3, (4/3-4/3)/-l, (4/2-4/3)4/-l be 
 its four fourth roots, what is the relation between the two 
 square roots ? between the four fourth roots ? 
 
 7. If 4/5 + 4/— 3 be one of the fourth roots of a binomial 
 surd, what are the others ? what are the square roots ? 
 Find » square root of: 
 
 8. 7 + 34/10. 9. 7 + 44/3. 10. 3-4/3. 
 
 II. 16 -64/7. 12. 4/I8-4/I6. 13. 9 ±24/14. 
 14. 84/3-64/5. 15. 75-124/31. 16. 4/37 + 4/15. 
 
 17. -13+64/3. 18. 34/[l + (l-c)^. 19. l-3a4/(l-a«). 
 30. a5 + c^+(6^-c^)4/(a'-c«). 31. xy-2x{xy-a?f/\ 
 
 33. 9+34/3 + 34/5 + 34/15. 33. 3<3- 4/3 -4/3 + 4/6). 
 Find a fourth root of: 
 
 34. 38-I64/3. 25. 49 + 2O4/6. 26. 137-364/14. 
 37. aHJ^ + 6a5-4(a«/2JV3+flV3^,3A). 28. -8+84/-3. 
 
194 POWERS AJSTD ROOTS. [VI.Pbs. 
 
 Note 1. If a*— 6 be a perfect square, so are a-\-^l, a — i/b. 
 For %• y'a; + y?^ = |/(a + i^V), and ^x — i^ij = ^{a — yS), 
 
 .%a;— ^ = 4/(a'— 5), 
 and -/(a*— 5) is rational if the root sought be possible. 
 E.g., 7 + 4-/3 is a perfect square; 
 
 for a = 7, & = 48, a^— 5 = 49— 48 = 1, a perfect square. 
 But 7 + 2|/3 is not a perfect square; 
 for a = 7, J = 12, a^— 6 = 49— 12 = 37, not a perfect square. 
 
 Note 2. Since (|/a; + 4/j^)^=a; + «/ + 2y'a;y, it appears that 
 the square root of a + 3-/5 is the sum of the square roots 
 of two numbers whose sum is a and whose product is h. This 
 consideration often makes it possible to find such roots by 
 inspection. 
 
 E.g., 4/(-4 + 4/-84) = |/(-4 + 2y-21); 
 and V -7 + 3= -4, -7x3= -21, 
 
 .•.|/(-4 + V-84) = |/-7 + y3. 
 So, 4/(1 + i'S) = ■/(* + 3 t'i) = l + i|/5._ 
 
 So, ^-(4/32 - ^24) = |/(4|/2 - 2^6) = 4/(3^2) - 1/(|/2) 
 = V'18-t^2. 
 
 PeOB. 15. To FIIfD A CUBE KOOT OF A BINOMIAL SUBD. 
 
 Let a + 1^6 be a binomial surd, and a; + /(/«/ = y'(a + |/J); (1) 
 then '.'^■\- Zxy + (3a^ + «/) 4/y = « + ^b, (2) [cube both mem. 
 .\of-V'?>ocy- (3a^ + «/) y'J/ = a - 1/6, (3) [th. 7 cr. 2. 
 
 and x—i^y= y/(a — 1/5), (4) 
 
 .% x^ — y— ^{a^ — h), [mult. eq. 1 by eq. 4. 
 
 and the root is possible only if y'(a^-6) be rational. 
 Put TO for ^{a' — h); then y = Q^ — m, 
 and •." a;* + 3a;«/ = a, [add eqs. 2, 3. 
 
 .'. 4ar — 3ma; = a. [elim. y. 
 
 From this point on there is no general solution, but partic- 
 ular examples may be solved by finding a value of x, by in- 
 spection, from the equation 42;" — 3TOa;=a. 
 
14,15,56] ROOTS OF BINOMIAL SURDS. 195 
 
 E. g. , to find a cube root of 10 + 6^3: 
 thenva = 10, S = 108, m= -^(100- 108) =-3; 
 
 /. 4'c' + 6a;=10, 
 
 .•,x=l, t/=3; and l + f'S is the root sought. 
 
 QUESTIONS. 
 
 1. Any coefficient may be prefixed to a radical, if the num- 
 ber under the radical sign be divided by that power of the 
 coefficient whose exponent is the index of the radical. 
 
 By inspection find a square root of: 
 
 2. 8-21/7. 3. 3y5+|/40. 4. 2i--|/2. 
 
 5. x'+x+2xi/x. 6. -5-2^-6. 7. 2a-24/(a^-5»). 
 8. 2a-b-2)/{a'-ab). 9. 2a + b + 2\/{a^ + ab). 
 
 rind a fourth root of: 
 
 10. 17 + 12|/2. 11. 49-20|/6. 13. 14 + 84/3. 
 
 13. 89 + 28^/10. 14. -221-60-,/-l. 15. 4a*. 
 Find a cube root of : 
 
 16. 7+5i/2. • 17. 16 + 8^5. 18. 45-29^/2. 
 
 19. 22 + 10|/7. 20. 38 + 174/5. 21. 2l4/6-23|/5. 
 
 22. 3fl-2a' + (l + 2a')V(l-«')- 23. l + 3fl + (3 + a)|/a. 
 
 Of the binomial surds below, which are perfect squares ? ■ 
 
 24. 4i-|/5J. 25. 3-4/2. 26. ^xy-i^{x/y). 27. 9-84/5. 
 
 28. If 4/a;+4/«/ + 4/2=|/(ffl + 24/5 + 24/c + 24/«?), a;, «/, z must 
 satisfy the four conditions 
 
 x-\-y-\-z=a, xy=b, xz=c, yz=d. 
 Find a square root of 6 + 34/2 + 2^/3 + 34/6. 
 
 29. The square root of IO + 24/6 + 24/I4 + 34/3I cannot 
 be expressed in the form 4/a; + 4/y + 4/^. 
 
 Find a square root of : 
 
 30. 10 + 24/6+24/10 + 24/15. 31. 8 + 24/2 + 24/5 + 24/10. 
 32. 15-24/15-24/21+24/35. 33. 11 + 24/6+44/3+64/3. 
 34. 15 -21^3 -24/15 + 64/3 -24/6 +24/5 -24/30, 
 
196 POWERS AND BOOTS. [VI, 
 
 §7. QUESTIONS FOE KEVIEW. 
 Define and illustrate : 
 
 1. A power; a root; a base; an exponent; a root index. 
 
 2. A fraction power; a commensurable power. 
 
 3. Powers of a base in the same series; like powers. 
 
 4. A radical; a radical factor; a radical expression. 
 
 5. A rational radical; a real radical; a surd; an imaginary. 
 
 6. A simple radical; a quadratic, a cubic, a quartic, and 
 a quintic radical. 
 
 7. Like radicals; conformable radicals. 
 
 8. A binomial surd; a trinomial surd. 
 
 9. A pair of conjugate quadratic surds; a pair of comple- 
 mentary surds; a group of complementary surds. 
 
 State and prove: 
 
 10. The binomial theorem. 
 
 11. The principle by which a commensurable power of a 
 commensurable power is found. •* 
 
 13. .The principle of equal fraction powers. 
 
 13. The . principle by which the product of two commen- 
 surable powers of the same base is found. 
 
 How does this principle apply in finding their quotient ? 
 
 14. The principle by which the product of like commen- 
 surable powers of different bases is found. 
 
 How does this principle apply in finding their quotient? 
 
 15. The principle of the equality of the like parts of two 
 equal simple surds. 
 
 16. The principle of the inequality of two non-conformable 
 simple surds. 
 
 17. The principle of the equality of the like parts of two 
 equal simple binomial quadratic surds. 
 
 18. What is the product of two conformable simple quad- 
 ratic surds? their quotient? the product of two such non- 
 conformable surds ? their quotient ? 
 
fl QUESTIONS FOR REVIEW. 197 
 
 Give the general rule, with reasons and illustrations, for: 
 
 19. Reducing a simple radical to its simplest form. 
 
 20. Freeing a simple radical from coefficients. 
 
 21. Adding and subtracting radicals. 
 
 22. Multiplying and dividing radicals. 
 
 23. Getting powers and roots of radicals. 
 
 24. Eeducing a fraction with a surd denominator to an 
 equivalent fraction with a rational denominator when the surd 
 denominator is a monomial; when it is a simple quadratic 
 surd; when it is a binomial quadratic surd containing a com- 
 plex radical; when it is any binomial surd. 
 
 25. Eationalizing an equation that contains surds. 
 
 26. Finding the square root of a polynomial. 
 
 27. Finding the cube root of a polynomial. 
 
 28. Finding the root of a polynomial when the root index 
 is a composite number. 
 
 29. Finding the wth root of a polynomial when n is prime. 
 
 30. Finding the square root of an integer, and of a decimal 
 fraction. 
 
 Explain the principle of dividing the numeral into periods; 
 and that of contraction. 
 
 31. Finding the cube root of an integer, and of a decimal 
 fraction. 
 
 Explain the principle of dividing the numeral into periods; 
 and that of contraction. 
 
 32. Finding a root of a fraction. 
 
 33. Finding a square root of a binomial quadratic surd. 
 
 34. Finding a cube root of a binomial quadratic surd. 
 
 35. If a^+6a^+73^-6x+m be a perfect square, what is m? 
 
 36. If ix'' + 12a:^ + 5ai^-2x' + mx'' + nx+p be a perfect 
 square, what are the values oim,n,p? 
 
 37. Apply the square root process to factoring 
 ix' + Vixy + Sy' + 16xz + 22yz + 15z\ 
 
198 QUADRATIC EQUATIONS. [Vn.PiiB. 
 
 Vn. QUADRATIC EQUATIONS. 
 
 An equation that is of the second degree as to its nnknowa 
 elements is a quadratic equation. 
 
 E.g., a^=9, a;*+3a;=18, aa?+5a;+c=0, [x nnkn. 
 
 So, xy = 13, aa? + 2hxy + hy^ + %gx + 2fy + c = 0, [a;, y, nnkn. 
 So, as? + %* +02^+ 3/yz + S^'za; + 2hxy = 0. [a;, ^, z, nnkn. 
 
 §1. ONE UNKNOWN ELEMENT. 
 
 An equation of the form a?= 9, is an incomplete, or pure, 
 quadratic equation/ one of the toim a? + Zx=i8 is complete. 
 
 PbOB. 1. To SOLVE AK INCOMPLETE QITADKATIC 'EQUATIOIT. 
 
 Seduce the equation to the type-form a?=q, and take the 
 square root of each memier; then «= ± i^q. [Ill, ax. 7. 
 
 E.g., if i(a;^-10)+^(6a^-100) = 3a;»-65: 
 
 then 10a?-10O+18a?-300=90a?-195O, [mnlt.by30. 
 
 -62a?=-1550, a:«=25 and x=±f>. 
 There are two square roots, opposites; they are' both real if 
 
 the absolute term be positive, and both imaginary if it be 
 
 negative. It may seem that the last equation should be 
 
 ±x= ±5; but this gives no new roots. 
 
 PkOB. 3. To SOLVE A COMPLETE QUADRATIC EQUATION. 
 
 Reduce the equation to the type-form, a?+px+q=0; 
 
 transpose the absolute term, and to each memier of the equation 
 add the square of half the coefficient of the first-degree 
 term; [II, pr. 3 nt. 4 frm. 3, 4. 
 
 take the square roofs of these sums, and solve the two simple 
 
 
 equations so found. 
 
 
 
 [III, ax. 7. 
 
 The result is of the form 
 
 x=- 
 
 ■ip^Wijf- 
 
 -iq). 
 
 E.g., 
 
 ifa;= + 3a;=40: 
 
 
 
 
 then 
 
 a? + 3a; + 3i = 43i, 
 a;=-H±6J±=5 or 
 
 -8. 
 
 [add (3/3)' 
 
 ' to each mem. 
 [take sqr. rts. 
 
1.8. St] ONE UNKNOWN ELEMENT. 199 
 
 QTTESTIOlfS. 
 
 1. Make a quadratic equation to state that: the area of a 
 square ifl^4225 square yards; the area of a rectangle is 1200 
 square rods ; the sum of the squares of two numbers is three 
 times their product; the product of the sum and difEerenoe of 
 two numbers is 33 ; the product of two numbers, one 5 less 
 than the other, is 24; the sum of the squares of three numbers 
 increased by twice their products, two by two, is 36. 
 
 Which of these equations are complete? 
 
 2. How can two independent simple equations be obtained 
 from one quadratic equation ? Write the forms for the two 
 roots separately, and find their sum and their product. 
 Solve the pure quadratic equations: 
 
 3. (a?+l)(ic«+2) = (a«+6)(a,-»-l). 
 
 4. i(a^-K)-i(a^-i«'')+i(a^-T^r«') = 0- 
 
 6. 3(5a;«-'?)(35-2a;)+27(5a«+7) = 9(52?-7)(17-|a;). 
 
 7. Why is an incomplete quadratic equation called a pure 
 quadratic ? a complete equation, an affected quadratic ? 
 Solve the complete quadratic equations: 
 
 8. iB«-5a; + 6 = 0. 9. a? -8a; + 15=0. 10. a? + 10a; =-34. 
 11. a?-5x+4: = 0. 13. Ga;"- 19a; +10=0. * 
 
 13. 7a;'-3a;=160. 14. 110a;«-21a;+l=0. 
 
 15. (a;-2)-'-2(a; + 2)-'=3/5. 16. 4/(2a;+5)=a;+l. 
 
 <L Zx-2 2a;-5 _8 - x + a x + b Jg + c^g 
 2a;-5 3a;-3~3" ' x-a x-b x-e ' 
 
 10 a; + 3 a;-8_ 2a;-3 x-Z x _'i 
 
 ^^- ^+2'^^:^~ x-1' 2x^x + 5 10' 
 
 X a;-l_ 3 __l__-l,l.l 
 
 21. ^+^n- -r ^^- 7^^+x-a ^b ^x 
 
 _ffl_ _6_^_2c_ 2^ a+cja+ x) ^ a+x_ a 
 x-a x-b x-c' ' a+c(a-x) x a-2cx' 
 
200 QUADRATIC EQUATIONS. tvn,pR. 
 
 SPBCIAI. OASES. 
 
 Note 1. The roots of the equation 3?+pz+q=0 ' are 
 
 -iP+Wif-^Q)> -^P-Wif-H)> [above, 
 
 whose values depend upon the values of ^, q. 
 
 There are four special cases : 
 (a) p positive, q negative : 
 
 two real roots, the larger negaflve, the smaller positive. 
 (5) p, q loth negative : 
 
 two real roots, the larger positive, the smaller negative, 
 (c) p, q loth positive : 
 
 two real roots, both negative, if j?—4:q he positive; 
 
 two real roots, both negative and equal to — |^, if p'— 4g'=0; 
 
 two imaginary roots, conjugates, if p^—^q he negative. 
 ((?) p negative, q positive : 
 
 two real roots, both positive, if ^—iq be positive; 
 
 two real roots, both positive and equal to — Jp, if ^*— 4g' = 0; 
 
 two imaginary roots, conjugates, if p^—iq be negative. 
 
 THE SUM AND THE PEODTJCT OF THE ROOTS. 
 
 Note 2. The sum of the two roots is —p, and their product 
 is g'. A quadratic equation can have not more than two roots. 
 
 THE ABSOLUTE TERM ZERO. 
 
 Note 3. If q=0, then of the equation a?+px=0 the 
 two roots are 0, —p, both real. 
 
 SOLUTIOK BY FACTORING. 
 
 Note 4. If the expression af'+px+q be readily factored, 
 then each factor maybe put equal to zero, and the two simple 
 equations so found give two values for x. 
 B.g., to solve the equation a?—5x+6=0i 
 then:- a?-5x+6 = {x-2){x-Z), 
 
 and this product is whether »— 3=0 or a;— 3=0, 
 /. the roots are 3, 3. 
 
 The equation is found by subtracting the roots in turn from 
 X and equating the product of the remainders to 0. 
 E.g., if the roots be 2, 3, the equation is {as— 2) (x — S)=0. 
 
8.8n ONE UNKNOWN ELEMENT, 201 
 
 QUBSTIOKS. 
 
 1. Whatever be the sign Qtp, what is that of ^'p 
 
 2. If q be negative, what is the sign of '^p^ — ig? 
 What does this show about the character of the roots ? 
 Is IViP' — ^Q) then larger or smaller than |jo ? 
 
 3. If p be positive and q negative, which is the larger, 
 
 -iP+W{p'-^Q) or -ip-^^/^p'^-iq)? 
 
 4. If q be positive and p^=^q, what are the roots ? 
 
 5. If p^>i:q, are the roots real ? if p^<Aq^ 
 
 6. If q be 0, what is the product of the two roots ? 
 
 7. If p be 0, of what kind is the quadratic ? 
 
 Write the sum and the product of the roots of the equation : 
 
 8. a?-ix=QO. 9. Za?+Gx = 24:. 10. 5a;»-15a;=140. 
 11, 2ar»-3a; = 13, 12. a?-ax-0. 13. 5a;* -15a! =200. 
 
 14. Suppose the quadratic equation a;' +;?a! +5' =0 to have 
 three different roots, r, r', r"; what is the value of rr'? of 
 rr" ? Prove that two of the supposed three roots are the same, 
 
 15. If r, r' be the roots of the equation a?—px+q=0, 
 find the value of r/r'+r'/r; of r^+r'"; of r'+r'K 
 
 16. Write the expression x'—{a + b)x+ab in the form 
 {x—a){x—i) : what is the value of this product if x—a=0? 
 if a; — 5 = ? What are the roots of the equation ? 
 
 Form a quadratic equation whose roots shall be: 
 
 17. 4, -5. 18. 2|, 2. 19. -f, -8. 30. a + b, a-b. 
 By factoring, solve the equations : 
 
 21. 2/«+132^=14. 32.a;*+7a!=30. 23. 4a;«+12a;+9 = 0. 
 
 24, a;*-a*=0. 25, a? -5a! =14. 26. a;*+a!«-a!-l = 0. 
 
 27. 9a;*- 30a; + 35 = 0. 38. a;«+6a!«-4a;=24. 
 
 Put the functions below equal to 0, solve the equations so 
 
 formed, and by aid of their roots factor the functions: 
 
 39. 6ar'-19a;+15. 30. (a;-a)*-S*, 31. a?-2mx+m*-n\ 
 
 33. s?-{m + n)x+{m+p){n-p). 33, a?+a;+l. 
 
 34. a!»-oa;-3a»-S*+3a5=0. 35. a;«-a;+l. 
 
202 QtTADHATlC EQtJATlONS. [Vll,Pii. 
 
 GENEEAL FOEMS. 
 
 Note 5. The rule for solving complete quadratic equations 
 may be stated in a more general form: 
 Reduce the equation to the type-form aoi? + bx + c=0; 
 multiply the equation by any number that makes the coefficient 
 of the first term a perfect square; 
 
 mahe the first member a perfect square, tahe the square root 
 of each member, and solve the simple equations so found. 
 
 With a proper multiplier fractions may be avoided. 
 E.g., if b be even, multiply by «; if odd, by 4a. 
 
 Both rules rest on the form assumed by the square of a 
 binomial, as does that for finding the square root. 
 
 The roots are [ - S + 1^{¥ - 4:ac)y2a, [ - 5 - 1/(5= - iac)]/2a. 
 
 There are three special cases: 
 
 (a) czero: then x = 0, x= —b/a, two real roots. 
 
 (b) bzero: then x= ±y'{ — c/a), 
 
 two real roots, opposites, if a, c be of contrary signs; 
 
 two imaginary roots, conjugates, if a, c be of the same sign. 
 
 (c) « «ero; then x={~b + b)/0, x={ — b — b)/0, 
 
 i.e., a; =0/0, x=co, an indeterminate and an infinite root. 
 
 But this indeterminate root can be determined. 
 For if a=0, and a;#oo, the equation becomes bx + c=0, 
 whose single root is —c/b. 
 This case is specially important as showing the values of 
 a; if a be thought of as taking changing values and growing 
 smaller and smaller; for, then, as a grows very small, one of 
 the roots grows very large, and the other approaches —c/b. 
 This is also evident if the equation be written in the form 
 x-^{b + cx~^)= —a. [div. eq. bx+c= —ax' by a?. 
 
 For, if a grow very small, 
 
 then either x~^ grows very small and x very large, 
 or b + cx-^ grows very small and a; approaches —c/b; 
 and both oo and —c/b satisfy the equation when « = 0, 
 
2.61] ONE UNKNOWN ELEMENT. 203 
 
 QUESTIONS. 
 
 Solve the equations: 
 
 1. af +1x^-10/9. 2. 6a;^ + 9a;=81. 3. 8a^-31a;+|=0. 
 4. a?-2mx + m' = w'. 5. 6a? -11a; = 10. 6. ^a;^-3^a; + 30 = 0. 
 7. a'a? + 2ajbx+¥-c'-0. 8. abx^ + {3b -4:a)x -12. 
 
 ■ 9, Find the sum and the product of the roots of ax' + bx + c. 
 If b = 0, of what kind is the quadratic? what is the sum 
 of the roots? what relation to each other have they? 
 
 If c/a be negative, what relation have the roots ? if c = « ? 
 
 10. If in aa^ + bx + c, a = 0, what is one value of a; ? 
 Show that the product of the roots is then infinite, and 
 
 hence find the other root. 
 
 11. Discuss the equation a3^+bx+c=0, after the manner 
 of note 1, and show that the two roots are: real and unequal, 
 if W>^ac; real and equal, if 6^ = 4ac; imaginary, if lf<^ac. 
 
 Of the real and unequal roots which is the larger? What 
 conditions make the real roots both positive ? both negative ? 
 
 13. If r, r' be the two roots of aa? + bx + c, then aoi?+bx-\-c 
 may be factored and written in the form a{x — r){x — r'). 
 
 13. What form has a quadratic equation whose roots are 
 opposites ? reciprocals ? 
 
 14. For what value of c will the equation 2x' + Gx + c = 
 have equal roots? reciprocal roots ? 
 
 15. If ?•, r' be the roots of the equation i!'+px + q=0, find 
 the equation, whose roots are — r, —r'; 1/r, l/r'. 
 Without solving, show that the equation 
 
 16. af'zt2{p + q)x + 2{p'' + q^)=0 has imaginary roots. 
 
 17. x'±2{p + g)x + {p + qy — has equal roots. 
 For what value of m will the equation 
 
 18. a?— 15— ffi(3a; — 8)= have equal roots ? 
 
 19. (s'— 5a;)/(aa; — c) = (m-l)/(m + l), opposite roots ? 
 Without solving, tell the signs of the roots of the equations: 
 
 20. a^-5x=S6. 31. a;«+5a;=14. 33. a;^+^a; = l/5. 
 23. a? + 5a;=-6, 34. a?- 5a; = -6. 35. a;' + 5a; = -7. 
 
204 QTJADRATIC EQUATIONS. [\TI,Pb. 
 
 EQUATIONS SOLVED AS QUADRATICS. 
 
 Note 6. Equations of the form aa^" + 5a;"+c=0, or 
 (aa^ + 5a;" + c)*" +p {ax"'+hx''-\-cy + q = 0, are solved as quad- 
 ratic equations. 
 E.g., if 9«*-52a?+64=0: 
 
 then •.• 81a:* - 468a:= + 676 = 100, [mult, by 9, add 100. 
 
 .". 9a;^ — 26 = ± 10, [t&ke sqr. rts. 
 
 .-.0^=4 or 16/9, 
 
 .*. a;=: ±2 or ±4/3, four real roots. 
 So, if (9a:;*-52««+80)H9(9a;*-52a:^ + 80)-400 =0: 
 then ••• 4(9^:* - 52ar' + 80)« + 36(9jK*-52a? + 80) + 81 = 1681, 
 .•.2(9a:*-52a?+80)= -9±41 = 32 or -50, 
 .•. 9a;*-52ar'+80=16 or^-25, 
 and a;=±2, ±4/3, ±J4/(26±4/-269), eight roots. 
 So, if a:*-6ar'+4a;« + 15a;=14: 
 then (a;«-3a;)'-5(a;^-3a;) = 14, 
 
 a;* - 3a; = 5/2 ± 1/(81/4), [solve for a;*- 3a;. 
 
 a?-3a;=7 or -2, 
 
 a;=3/2±^4/37 or 3/2 ±1/2, [solve for a;. 
 
 a;= ^(3 ± |/37), 2, or 1, four roots. 
 So, if 53?-2i^(3a?+2x-'7) = 10-2x: 
 then 3a;'+2a;-7-2-v/(3a?+2a;-7) = 3, 
 4/(3a;2+2a;-7)=l±2 = 3or -1, 
 3a;2+2a;-7=9orl, 
 a?+|a;=16/3 or 8/3, 
 a;=-l/3±7/3 or -1/3 ±5/3, 
 x=2, -8/3, or 4/3, -2, four roots. 
 
2,§1] 
 
 Solve 
 
 1. 
 
 3. 
 
 5. 
 
 1. 
 
 9. 
 11. 
 13. 
 15, 
 17. 
 19. 
 21. 
 23. 
 25. 
 27, 
 
 29. 
 
 30. 
 
 31, 
 32, 
 33. 
 
 34, 
 35, 
 36. 
 
 ONE UNKNOWN ELEMENT. 
 
 QUESTIONS. 
 
 206 
 
 the equations: 
 
 |/{2a; + 7) + v'(3iB-18) = 4/(7a; + l), 3, 3a; + 24/a;-l = 0. 
 
 a;^ + 3 = 2 |/(a^ - 2a; + 2) + 2a;. 
 ^{a;'-2a;+9)-|a;^=3-a;, 
 3a;^ + 15a; - 2 4/(a;^ + 5a; + 1) = 2. 
 (a;» + a;-6)^-4(ai»+a;-6) = 12 
 (a;» + a;)'-3(a?+a;) = 108, 
 a;+5-V'(a; + 5) = 6, 
 2a;^+6 + 34/(3a;'+6) = 10. 
 (a;'+2)'' + 198 = 29(a;^+2). 
 V'(a; + 4)-i/a;=4/(a;+|). 
 {x + 1/xy - 4(a; + 1/a;) = H- 
 2(|a;'-|)H5(fa?-|) = 63, 
 5y(3/a;) + 7|/(a;/3) = 22f, 
 a? + l/a*+2(a; + l/a;) = 9i, 
 (x-af , {x-lf 
 
 4. a;»''»-18a;'''*'=14, 
 
 6. a;*-14a;= + 40=0. 
 
 8. a;»/' + fa;-V3=3Vt. 
 
 10, y'2a;-7a;--52. 
 
 12, na?+x + n + l = Q. 
 
 14, ^a;^+3^a;=18, 
 
 16, a;»(19 + a?) = 216, 
 
 18, 4 = 52;" -a;*, 
 
 20, 3a;' + 8a;*-8a;^=3. 
 
 22. a;'+a;^-4a;-4 = 0. 
 
 24, s^ + d'W/a?=d'+W. 
 
 26. a;*+2a;'-3a;'-4a;=96. 
 
 ' 28. 25/a;»-10/a; = 3, 
 
 {^-cf ^3 
 
 (a;-S)(a;— c)^(a;— a)(a; — c)^(a;— a)(a! — i) 
 
 '^/IV{^ - 6) + 4] +12/[4/(a; - 6) + 9] + l/[y(a; - 6) - 4] 
 + 6/[i/(a;-6)-9] = 0, 
 
 m\x -\-m+ 11 n) {x—m+ Iny 
 
 =n''{x+17m+n){x+7m—ny. 
 
 (x-a + I>y-(x-ay+{x-bY-a?+c^-{a-iY-¥ 
 
 = {a-by. 
 \x - ^{af- (f)\/i^\x + 4/(3;" - a^)] 
 
 = ^(a^-a^li/ix'+ax) - ^{a?-ax)]. 
 
 d'-Sx+4: + 2^{a?-3x + 6)^6. 
 as^ + 5a;" + c)^ ± (ea;» +/)'^ = 0, 
 {ax^ + ix" + cf^ ± {dx^" + ea;")^= 0. 
 
 \+-. 
 
 + 
 
206 
 
 QUADRATIC EQUATIONS. 
 
 tvn, Pa 
 
 §2. TWO UNKNOWN ELEMENTS. 
 
 PkOB. 3. To SOLVE A PAIR 01' EQUATIOlfS INVOLTIlirG THE 
 SAME TWO UNKNOWN ELEMENTS, ONE EQUATION SIMPLE, THE 
 OTHER QUADRATIC, 
 
 Eliminate one of the unhnown elements ffbm the quadratic 
 
 equation; [III, pr. 2. 
 
 solve the resultant for the other unhnown element and replace 
 
 this element hy its value in the simple equation; 
 solve this equation for the first unhnown element. 
 E.g., if Zx + 2y- 20, 3a;^ + bxy + 7«/^ = 425 : 
 then •.• a; = J(20 — 2«/), [sol. first eq. for x. 
 
 .: 1(20 -2iy)H|«/(20- 2?/) +72/^=425, [repl. x in sec. eq. 
 /. \by^ + 20«/ = 875, y = t or -S\, [sol. quad, for y. 
 
 .'. 3a; + 2 • 7 = 20, x = 2, [repl. y in first eq. 
 
 and 3a; -2 -8^ = 20, a!=12f, 
 and the two pairs of roots are 2, 7; 12f, — 8^. 
 
 Check. Both pairs of roots satisfy the quadratic equation. 
 If the two equations be such that they can be combined in 
 one of the familiar forms x'±'ixy+y^, 3?—y^, the work is 
 shortened by the use of such form. 
 
 E.g., if a; + .v = 13, a;^^ = 12 : 
 then •.• a;^ + Ixy +y^= 169, 
 and ^xy = AS, 
 
 .■.a^ — 2xy + y^—121, x—y= ±11, 
 .•. from the equations x + y = lS, 
 x-1%, «/=l; 
 from the equations x+y—lZ, 
 x=l, y — 12. 
 
 x''+y''=ib, x—y=—9: 
 
 • a? — 2xy + y^—81, [sqr. sec. eq. 
 
 .:2xy=-36, x^+_2xy+y^=:9, x+y=±3, 
 .•.x=—6, y=B; x=—3, y=^6. ^ 
 
 and 
 
 So, if 
 then ' 
 
 [sqr. first eq. 
 [mult. sec. eq. by 4. 
 [sub. and get sqr. rts. 
 x—y = ll, come 
 
 x—y=:—ll, come 
 
8>S3] TWO UNKNOWN ELEMENTS. 207 
 
 QXTESTIONS. 
 
 Find the values of x, y from the pair of equations: 
 
 1. x+y = 1l, iB»+2j/«=34, 2, x-y=12,, a;«+2^«=74. 
 3. x+y=a, xy=W. 4. x-y=a, xy-b\ 
 
 5. 3x-5y=2, xy = \. 6. x+y=\{iO, xy = 2i00. 
 
 7. x+y = a, a?+f=V>. 8. a;»/«+2^-»'^=4, x-y-'^=S. 
 9. a;+^ = 4, a;-»+^-*=l. 10. 2x + dy=B7,x-^+y-^=\^. 
 
 11. iB+2^=:2, a?- 2a;«/ -«/«=!. 13. a;+y=18, a;'+y=4914. 
 
 13. x + y^'72, ^x + py=6. li. x-y = 18, a^-y^=4:9U. 
 
 15. x'y-^ + y''x-^=9, x-^+y-^-3/4:. 
 
 16. In the pair of equations a; +^=13, a;y = 12, beware 
 the results affected if x, y exchange places ? 
 
 Show why either x ov y may be 12, and the other be 1. 
 
 17. In the pair of equations x^+y^=4:b, x—y=—Q, can 
 X, y exchange places ? x, —y? What relation have x, y? 
 Solve the pair of equations : 
 
 18. x-y = 5, xy=126. 19. x+y=8, a^-/=16. 
 
 20. x-y=4:, a?~y^=Z2. 21. x + y=ll, !s'+/ = 407. 
 22. x-yz=i,a^-y^=988. 23. 3a; -42^=4, 9a;* -162/'= 176. 
 24.0^-2/^=21, x{x + yy=4t5. 25. x-y=2, a^-f=992. 
 26. 3x-2y = 10, Zx'-ixy-y^=80. 
 
 21. x/y-y/x~%/2, x-y=\. 28. a;+2/=2, a;^+2/^=992. 
 
 29. l/a;» + 1/2/*= 126/125, l/a; + l/2/ = 6/5. 
 
 30. x+iy = ll, a^+2x''y + ixy^+-i^f = ldSl. 
 
 31. 5x-y=B, y^-G3?=2b. 32. x-y = 2, a^+f=82. 
 33. x+y^l012,a^^+y^^=Q. 34. x-y=a, x^+f=l^. 
 
 35. 3a;-22/ = 13, {x+yY'^+2{x-yf^'=Z{a?-yf/^. 
 
 36. 7a; + 52/ = 29, {2x+y)/{Zx-y)-{x-y)/{x+y) = B8/lh. 
 
 37. 5a;-72/=4, (a?+2r')/(a:+2/)''+(a^-y')/(a;-^)*=43a;/8, 
 
 38. x+y=2, l%{7^+f)=12Hp?+f). 
 
 39. a;+2/=4 41(3;=+2/') = 122(a;*+3/*). 
 
 40. x+y=a, x/(i-y) + (b-y)/x=o. 
 
208 QUADRATIC EQUATIONS. [vn, Pb. 
 
 PeOB. 4. To SOLVE A PAIE OF QUADRATIC EQUATIONS IN- 
 VOLVING THE SAME TWO UNKNOWN ELEMENTS. 
 
 No one rule is best for all cases; many special devices may 
 be used, and the examples given below suggest methods. 
 
 If by combining the old equations, new equations can be 
 found that involve some of the familiar type-forms, such as 
 x'^'Zxy+y', a?—y^, then very often either the square roof 
 may be found, or by factoring and division a quadratic equa- 
 tion may be replaced by a simple one. 
 E.g., if 3a;y-4a;-4?/ = 0, a;*+/+a;-h^-26=0: 
 put {x+yY—'Sixy for a?+y^, and write the equations 
 
 Zxy-4k{x+y)=0, {x+yf+{x-{-y)-'Zxy-2&=Q, 
 eliminate xy, and solve the quadratic in {x+y)\ 
 then a;-l-2/=6, xy=Q, or x+y=-iyz, a!^=-52/9; 
 solve these two pairs of equations for x, y, 
 So,it x-y=ixy, af'+y^-^xy: 
 subtract the square of the first equation from the second and 
 
 solve the resulting equation to find the values of xy; 
 join each of these equations with the first equation to find 
 
 values of x, y. 
 
 So, if s^+xy-\-y^=l^, x^-xy-\-y^=n: 
 
 find the value of xy, then of x-\-y, x—y, then of x, y. 
 
 There are four pairs of roots. 
 So, if \^{x+y) + ^(x-y)=^a, i^{a?+f) + )/{a?-y^ = b: 
 then 2x + 2^{a?-f) = a, 2af+2^{a^-y^) = b\ [squaring, 
 and a?-y^=ia''-ax+a?, a^-/=i5*-SV-f-a^ 
 i.e., f=ax-\a\ ^=1^3?-^', 
 
 .•. 5^a*— i5*=(ffla;— ia*)', whence x is found; then ^. 
 So,if {x+yY-2xy= -{x+y) + 2&, &xy=S{x+y)'. 
 then {x+yy-i{x+y)=2Q; [elim. xy. 
 
 and x + y=Q or -13/3, xy=% or -53/9; 
 
 X, y=4:, 2; 2, 4; i(-13±|/377), K-13=f 1/377). 
 
«.§2] TWO UNKNOWN ELEMENTS. 209 
 
 QUESTIONS. 
 Solve the pair of equations : 
 
 1. 4a;' + 7/=148, Saf-y^^ll. 2. x+y=a?, Zy-x=f. 
 3. a?+y^=^xy, x-y=izy. 4. a^+xy=6, aj^+y^=5. 
 5. ci?+y^=a% xy=i\ 6. a?+y^=9, xy = 2, 
 
 '7. 3? + xy^=10, y^+oi?y = b. 8. ci?+y = ix,y^+x=ii/. 
 9. af{x+y)^ 80, oi?{2x-Sy) = 80. 
 
 10. x^+a?y''+y^=ldS, a?-xy+y^=';i. 
 
 11. a?=ax+iy, y^=ay+ix, 
 
 12. x-\-y=10, /^/xy-^ + i^yx-'^=f>/^. 
 
 13. ix+ay=ab, bx+ay=4:xy. 
 
 14. 8a?/^-y^=U, a?/^f/^=2y\ 
 
 15. 81a^- 162/*= 1296, 9a;^ + 4/=36. 
 
 16. a^+xy=6d, y^+xy = 18. 17. x-y=9, a?-^=243. 
 18. a?-xy + y''=25, a?+f=125. 19. 3(a!+«/) = 15, a;2/=6. 
 20. 4a;+43r=12, a^+2^'=63. 31. x-y=d, xy=4:. 
 
 23. a?+f=d(x+y), x-y=l. 23. «*+^=13, -xy=6. 
 
 24. a;*+/=25, a;^-a!+3/= -5. 
 
 35. 4:{x+y) = '3xy, x+y+a?+y*=26. 
 
 36. a;+2/ + *^«^=14, s^+y''+xy=8i. 
 
 37. xy+6x+7y=50, 3xy+2x+5y=72. 
 
 38. a;*+a^^^ + «/*=243, a?-xy+y''=27. 
 
 39. a!*-a?y+a?/-a!2^+/=31, a?+/=31. 
 
 30. a?+/+a;+^=4, 3a;^ + 3a; + 3i/ = 8. 
 
 31. a;-2/-3v'(:r-2/)=-l, a?-2/*+4|/(a?-2^')=60. 
 
 32. a;+2^=5/6, a^2'''+l/a;'2/H4(a;«/ + l/a;^) = 60f^. 
 
 33. 8x/y=50y/x, xy+x-y=ld. 
 
 34. af'/y'+{2x+y)/j^y=20-{y''+x)/y, x+8=iy. 
 
210 QUADRATIC EQUATIONS. [VU, Pb. 
 
 CHAlfGB OF THE UNKSTOWN ELEMENTS. 
 
 Sometimes the solution of a pair of equations may be sim- 
 plified by changing the unknown elements. 
 E.g., if x+y-A, a;*+«/*=83: 
 put u + v for a;, u — v ioi y; 
 then {u + v) + {u — v) = 4, u = 2, 
 and {u + vy+{u-vy-82, u^ + 6uV + v^=41, 
 
 1^—1 or —25, v=±l or ±5;j/-l, 
 
 x=3, 1, 3 + 5|/-l, .2-5|/-l, 
 
 y=l, 3, 2-5^-1, 2 + 5|/-l. 
 So, if «*-a;'+«/*-y«=84, a^+a^^''+/=49: 
 then {a^+y^y-2a?y''-{x' + y^) = 84:, {x'+y^)-i-xY^49; 
 put M for a?+y^, v for a??/^, and solve. 
 There are eight pairs of roots. 
 
 BOTH EQUATIONS OF THE FORM aOS' + hxy-lfCy^-h. 
 
 If both equations be of the form a3?-\-ixy+cy'^—k, then: 
 replace y by vx, eliminate a? hy comparison, and solve the 
 resultant for v; 
 
 replace v by its values in either of the vx-equations and solve 
 for xj 
 
 get the products of corresponding values of v, xfor values ofy. 
 Check. Replace x, yby their pairs of values in the other 
 original equation, and see whether it be satisfied thereby. 
 
 E.g., if 2a;2/ + 5/=195, Z3?-4:xy=1l: 
 
 then V 2wa;:^ + 5wV= 195, 3a;^— 4z;a^=7, [repl. ^z by va;. 
 
 .-. 7(2w + 5«;^ = 195(3 -4t;) and v=h/l or -117/5; 
 
 .:Z7?-S^o^=':i, x=±1, y=±5, if «;=5/7. 
 and 3a?+^^a?=7, a; = ±4/(5/69), «/= =f117/|/345 if 
 v= —117/5, four pairs of roots. 
 
 Check. 2-7.5+5.5''=195, 2--7--5+5.(-5)''=195; 
 and so for the other pair of roots. 
 
*.S2] TWO UNKNOWN ELEMENTS. 211 
 
 QUESTIONS. 
 
 Solve the pair of equations : 
 
 1. x + y = 5, a^+y^ = 97. 2. x^ + 3xy = 54:,xy + iy''-115. 
 3. x-y = 3, a?-f=3093. A. x^/^+y^/^=i, x''+y=17. 
 
 5. x' + xy + iy''=6, 3x' + 8y''=14:. 
 
 6. a;* + ^'=14a;y, x+y = d, 7. a?—y^=a\ xy = W. 
 
 8. 0:^+^=45, x + y-{-\^%xy = lb.' 
 
 9. a?+y^=l + xy, a? + y^=^Gxy — l. 
 
 10. a?-xy<[/2-y^=2, x^+y^=20. 
 
 11. xy{x+y)-30, a?+f-35. 
 
 12. x'y+xy^= 4:8, a^y-xy^ =16. 
 
 13. x+y + l = 0, a^ + y^+211 = 0. 
 
 14. ic»+/+a;«/ = 15i, x^-y''=2i. 
 
 15. l/a;V'-l/«/»'"=l, l/x-l/y=37. 
 
 16. x'/y + yyx=Q/2, 3/{x+y) = l. 
 
 17. a:« + a;?/ + 2/=74, 2a^ + 2a;2/ + «/^=73. 
 
 18. a^-a^+y^-/=84, a^ + a^«/^+2/^=49. 
 
 19. a^ + iy*+3a; + 3^=378, a;'+ 3^- 3a; -3.^ = 324. 
 
 20. a?+y^+x+y = bO, xy+x+y = 29. 
 
 31. tB*-a;y+2/*=16, 2a^ + 3a;*/-3/=32. 
 
 22. ()f+a''=y''+b''={x+yy+(a-iy. 
 
 23. a;*«/ + a;«/^=30, a:*«/^+a;y=468. 
 
 24. x'+xy+f=84:, x-yxy+y = 6. 
 
 25. iB*-/=48, (a; + 2/)^ = 36. 
 
 26. a;"'«^»=(3/2)'»-", a;"^'»=(2/3)'»-«. 
 
 37. vWy)+v{yM=W^> ;»+«/= 10. 
 
 28. x'/y''+y/x+x/y=27/4-yya^, x-y=2. 
 
 29. (a:+2/)(a?-«/») = 819, (a; ~y) (a? + «/') = 399. 
 
 30. a;*+a;*i^H^=931, 3;^-a;2/+/=19. 
 
212 QUADRATIC EQUATIONS. [vn.Pn- 
 
 §3. THESE OE MOEB UNKNOWN" ELEMENTS. 
 
 EbOB. 5. To SOLTE A SYSTEM OE Vb QUADEATIO EQUATIONS, 
 INVOLVING THE SAME n UNKNOWN ELEMENTS. 
 
 The examples given below suggest methods. 
 B.g.,if a;(a; + «/ + 2) = 18, y{x+y+z) = 12, z{x+y + z) = 6: 
 then {x + y + z){z+y+z) = d6, [add. 
 
 x+y + z= ±6, [take sqr. rt. 
 
 and x=±3, y=±2, z= ±1. [div. eqs. 1, 2,3 by a; + 2/ + 2;. 
 So, if xyz = a\y + z) — h\z + x) — (?{x + y): divide by xyz; 
 then 1 = a\l/zx + 1/xy) = F{l/xy + 1/yz) = o\l/yz + 1/zx). 
 and l/yz^ii-l/d' + l/P + l/^, 
 1/zx = i(l /a^ - 1/5^ + 1/c'), 
 1/xy = i(l/a^ + 1/5^ - l/(f)', 
 and •.■ iK^ = 1/yz : {1/zx • 1/xy), [identity. 
 
 .'. x'=2{-l/a'' + l/P + l/(^) 
 
 : [1/a' - 1/V + 1/<?) ■ (1/a^ + 1/S^ - l/(f)] ; 
 and so for y", z\ 
 
 So, if y^ + yz+z'='1, z^ + zx + x^=U, a^ + xy + y^-'S: 
 then (x—y){x + y + z) = 6, [sub. eq. 1 from eq. 2. 
 
 {z—y){x+y + z) = 10. [sub. eq. 3 from eq. 3. 
 
 .'. x—yiz — y = Z:5, 5x — 5y=:dz — dy, x=\{2y + 3z); 
 .■.z'+i{2y + 3z)-:^{2y + 8z) = U; , 
 
 [repl. X by ^{2y + 3z) in eq. 2. 
 combine this equation with eq. 1, and solve for y, z; 
 then y-±2, ±1/|/19; z=q=3, ±11/^19; 
 ahd a;= T 1, ± 7/4/I9. [repl. y in sq. 3. 
 
 So, if a?—yz = a, y^ — zx^i, ^~xy=c: 
 from the square of each equation subtract the product of the 
 
 other two; 
 then '.°<c{a?+y'+^ — Zxyz) = a^— Ic, 
 y{!ii?+y' + ^— Sxyz) = 1^—00, 
 
B,S3] 
 
 THREE OR MORE UNKNOWN ELEMENTS. 
 
 213 
 
 and z{!Ji? + i^-\-z^-^xyz)=^(?—al. 
 
 .-. a;/(a' - 5c) = y/(6' - ca) = %/(<? - ab), 
 
 .: x= ±(a*-5c)/y(a' + 6'+c'-3aZic). [repl. y, z in eq. 1. 
 So, y=± {W - ca)//^{a!' + W + (?-%abc), 
 z=±{(?-ab)/\f{a'' + W+<?- Zabc). 
 
 QUESTIONS. 
 
 Find the values of x, y, z from the set of equations: 
 
 1. yz —Ic, 2. xyz =10, 3. x-^y-\-z =31, 
 
 ix+ay—ai, yz/x=10, x-^ + y-^ + z-^ = di, 
 
 cx+az—ac. xy/z =10. xyz =1. 
 
 4. 9x+y — 8z =0, 5. xy = a{x + y), 6. x+y+z =6, 
 4a;-8jf + 7z=0, xz =b{x + z), ix+y-%z = Q, 
 
 yz + zx + xy=^4:7. yz =c{y+z). a?+y^+z' =14:, 
 
 7. x+y+z =13, 8. x + 2y-z =11, 9. 3x+y-2z = 0, 
 
 a?+y''+!^ = 65, 
 xy = 10< 
 
 10. x + y+z =a, 
 a? + y^ + z^ = a^, 
 x'+'f + !?=a^, 
 
 12. {x+y){y+z)=^0, 
 {y + z){y+x) = lh, 
 {z+x){z+y)=lQ. 
 
 14. x^yh^w=12, 
 o^y\v?=i, 
 ^y^v?=\, 
 a;^V«o*=4/3. 
 
 16. xyz/{x + y)=-%, 
 xyz/(y+z)=24:, 
 xyz/{x + z) =12. 
 
 19. 3a?+2y''+5z''=0, 
 
 a;8_4/ + 22=37, 4:Z-y-3z = 0, 
 xz =24. a;=+^»+«*=467. 
 
 11. x + y =z, 
 
 Sx-2y + l'}!z = 0, 
 a^ + df + 2s^ =167. 
 
 13. (y-z){z + x) =22, 
 {z + x){x-y) = dS, 
 (x-y){y-z) = 6. 
 
 15. (a;+v)(y + «)=-«+J+c, 
 {v+y){x+z)=a-i+c, 
 {v+z){x + y)=a + b + c, 
 o^+y''+z^+v'=3{a + b + c). 
 
 17. a;y«»=108, 18. a^yh =12, 
 yz'/x = 18, s?y!? =54, 
 
 y3^/z=2/3. xyz'>=72. 
 
 7ci?-Zf-l^z^ = 0, 5x-4y + 7z=6. 
 
214 QtTADRATlC EQUATIONS. fW, 
 
 §4. QUESTIONS FOR REVIEW. 
 Define and illustrate: 
 
 I. A complete quadratic equation. 
 
 By what other name is such an equation known ? 
 3. An incomplete, or pure, quadratic equation. 
 Give the general rule, with reasons and illustrations, for: 
 
 3. Solving an incomplete quadratic equation. 
 
 4. Solving a complete quadratic equation of the form 
 
 x'+px + q — O; of the form ax' + bx + c=0. 
 
 5. Solving a pair of quadratic equations involving the same 
 two unknown elements, when one equation is simple and the 
 other quadratic; when both equations are quadratic. 
 
 6. Discuss the equation x^+px + g — O : what are the roots 
 Up be positive and q be negative ? it p, q be both negative? 
 if jo, q be both positive ? if ^ be negative and q be positive ? 
 
 Find the sum of the roots, and their product. 
 Show what relation between ^ and q makes the two roots 
 equals; opposites; reciprocals; opposite reciprocals. 
 
 7. Discuss the equation aa? + 'bx-\-c=0: what are the 
 roots if c be zero ? if & be zero ? if « be zero ? 
 
 What relation have a, b, c, if the two roots be real and 
 equal ? if real and unequal? if imaginary? 
 Can one root be real and the other imaginary ? 
 
 8. Show how to form a quadratic equation that shall have 
 two given numbers for its roots. 
 
 9. If a, /5 be the roots of the equation ax^ + hx + c—O, 
 find the value, in terms of a, i, c, of a— /S, a^ + ^, o? — /?*, 
 o?^-^, «»-/?», a/^ + ^/a, ay/?+/Sy«. 
 
 10. So, find the equation whose roots are : 
 
 «//?, /?/«; a/A -/?/«; «', /S^ IM l/ySl 
 
 II. Show how to solve a quadratic equation by factoring. 
 12. Show how to factor a quadratic function by solving the 
 
 quadratic equation formed by putting the function equal to 0. ' 
 
5 4] QUESTIONS FOR REVIEW. 215 
 
 13. Show how a pair of equations that inTolve the same two 
 unknown elements may sometimes be solved by changing the 
 unknown elements. 
 
 14. Solve the equation 2a;* - Qar" + lia? - 9a; + 3 = 0. 
 [Divide by a;^; then 2(a;'' + a;-')-9(a; + a;-') + 14 = 0; 
 write y for x+x~^; then 2(2/'— 3) — 9«/ + 14 = ; 
 
 solve this equation for y, then the equation a; + a;"*=^ for x. 
 
 Such equations, in which terms equidistant from the ends of 
 the function have equal coefiBcients, are reciprocal equations, 
 and their roots come in reciprocal pairs. 
 
 THE KOOTS OF +1 AND OF "1. 
 
 15. The three cube roots of +1 ^.re 1, ^( — lzt4/— 3). 
 [Write a;=^l; then a;'-l = = (a;-l)(a;' + a; + l); 
 
 put these two factors, in turn, equal to 0, and solve the equa- 
 tions so formed for x. 
 
 16. The sum of the three cube roots of +1 is 0, and so is the 
 sum of their products in pairs; the product of the last two 
 roots is "1, and each of them is the square of the other. 
 Express the three cube roots of 1 by 1, r, r^, and show that: 
 
 17. l + r + r^=0. 18. (l-r + »-«)(l + r-r') = 4. 
 
 19. (l + r^)' = r^ 20. (l-r)-(l-r')-(l-r*)-(l-r=) = 9. 
 
 21. (l-»-+»^)-(l-r' + »-*)-(l-r*+r«)- • •2w factors::=2^, 
 
 22. Find the three cube roots of ~1. 
 
 The sum of these three roots is 0, and so is the sum of their 
 products in pairs; and either of the last two roots is the oppo 
 site of the square of the other. 
 
 23. What are the cube roots of +a' and of ~a^ ? 
 
 24. Find the four fourth roots of +1. 
 
 The sum of these four roots is 0, and so are the sums of 
 their products in pairs and in threes; and the product of the 
 four roots is "1. 
 
 25. The four fourth roots of " 1 are (1 + i) • \^h (1 - «') • Vh 
 (\ + i)--Vh (l-*>-Vi. [i=4/-l. 
 
 26. What are the fourth roots of ■^a* and of ~a*? 
 
216 QUESTIONS FOR KEVIEW. (VH, 
 
 27. The five fifth roots of +1 are 
 
 1, i[-l±4/5±iV(10 + 24/5)]. 
 [Write a^-l = 0=:{x-l)-{x' + x' + x^ + x + l); 
 then a; — 1=0 and x=l: 
 
 and V x^ + x + l+x-^ + x-''=0; [put sec. fact. = 0, div. by a?*. 
 .-. a? + 2 + x-'' + x + x-'^ = l, {x + x-''y + {x + x-^)+i = ^, 
 .: x + x~^= — ^±^|/5; and these two quadratic equa- 
 tions, when solved, give the last four values above. 
 
 The sum of these five roots is 0, and so are the sums of 
 their products in pairs, in threes, and in fours; and the prod- 
 uct of the five roots is "l. 
 
 38. Find the five fifth roots of "1. 
 
 29. What are the fifth roots of +«" and of "a"? 
 
 30. The six sixth roots of +1 are the three cube roots of +1 
 and the three cube roots of "1. 
 
 Group these six roots in three pairs of opposites. 
 
 31. The six sixth roots of ,"1 are ±i, ±i{j/3±i). 
 [Write a;' + 1 = = (a;* + 1) (a;* - k' + 1) ; put these two factors, 
 
 in turn, equal to 0, and solve the equations so formed.] 
 
 32. What are the sixth roots of +«' and of "a' ? 
 
 MAXIMA AND MIITIMA. 
 
 33. If X vary, but remain real, what is the least possible 
 value of x''—4:X + S? 
 
 [Write x'-ix+3=y; then {x—2y=y + l, 
 and •.• {x — 2y is always positive, 
 
 .'. y + 1 is positive and ~1 is the value sought. 
 
 34. Find the least value possible of a?+px+q, and find 
 the value of x which makes this function least. 
 
 35. If X vary, but remain real, what bounds has the frac- 
 tion {x' + 2x — ll)/{x — d) ? what are the like values of a;? 
 
 [ Write (a?' + 2a; - ll)/(a; - 3) = «/; then a;^ + (3 - «^)a; 4- 3?/ - 11 = 0, 
 and %• X can be real only when (2 — ?/)^<12y — 44, 
 
§4] QTIADRATIC EQUATIONS. 217 
 
 i.e., /.when ^*— 162/+48<0 and ((5^— 4) (2/ — 13) is positive, 
 .'. the fraction has no value between 4 and 13, but has all 
 lother values. 
 Equate the fraction to 4 and 12, in turn, and solve for x. 
 
 36. The fraction {x + d)/(a? + lx + (?) lies always between 
 two fixed bounds if a^ + (?>ab and 5' < 4c'; it lies always 
 beyond two fixed bounds if a^ + <?>db and h^>^(?; and 
 it takes all values if a^ + (f'<ai. 
 
 In the light of ex. 36, find the bounds of these fractions : 
 
 37. (a; + 4)/(a^ + 3a! + 8). 38. (a; + 2)/(2;H8» + 4). 
 39, (a; + 8)/(a;H4a; + 3). 40. (a; + 4)/(a;H8a; + 2). 
 41. {x + %)/(a? + ix + S). 42. (a; + 8)/(a^ + 2a;+4). 
 48. Find the condition that the quadratic function 
 
 aa? + '^hxy + hy^+ 2gx + 2fy + c 
 may be resolved into two linear factors. 
 [Put the function equal to and solve the equation for x; 
 then ax + hy+g=± \f\_y\h^— ai) + "^yijig - of) + {g^ - «c)]; 
 and this radical is rational only it {hg — aff = (A* - ab) (/ — ac), 
 i.e., if aic + 2fgh - af - hg^ - elf = 0. 
 In the light of ex. 43, show whether these functions can be 
 
 factored, and, if possible, factor them. 
 
 44. 5a?+i2a;y-V-24a;-18y-5. 
 
 45. 5a;^ + 12a;2/-9^'-34:B + ]8?/ + 5. 
 
 46. 5x^-18xy + 9y'' + 24:X-l2y-5. 
 
 47. 6x^-18xy + 9y^ + 24:X + 18y + 5. 
 
 48. Divide a given number a into two parts, («) whose 
 product shall be a maximum, (6) the sum of whose squares 
 shall be a minimum. 
 
 49. Of all the rectangular fields that have the same area, the 
 square has the shortest perimeter, and the shortest diagonal. 
 
 [Use the equations 
 
 {x+yy—{x—yy + ixy, x''+y''={x— yf + 2xy. 
 
 50. Of all the rectangles that have the same perimeter, the 
 square has the greatest area. Which of them has the least ? 
 
218 PEOGEESSIONS, INCOM. POWERS, LOGARITHMS. [vm.Tks. 
 
 Vni. THE THREE PROGRESSIONS, HfCOMMEN- 
 SURABLE POWERS, AND LOGARITHMS. 
 
 A series is an expression such that each term is connected 
 with the preceding terms by some law. [comp. df. series, p. 34. 
 
 §1. ARITHMETIC PROGRESSION. 
 
 A series is in arithmetic progression if each term after the 
 first be found by adding a constant to the term before it. ■ The 
 constant added is the common difference. A series in arith- 
 metic progression is an arithmetic series. 
 
 The abbreviations are: a first term, I last term, d common 
 difference, n number of terms, s sum of all the terms. 
 
 If d be positive, the series is a rising series; if d be nega- 
 tive, it is & falling series. 
 E.g., 1, 3, 5, 7, 9 is a rising series, 
 
 wherein d=*2, a=l, ^=9, « = 5, 5 = 25. 
 So, 9, 7, 5, 3, 1, "1, "3, is a falling series, 
 
 wherein d=~%, a=Q, Z=~3, w = 7, s=2\. 
 
 Theoe. 1. In an arithmetic series, l=a + {n — l)d. 
 For '.• a, a-^d, a + ^d, a + %d,-'', a + (Jc-l)d are the 
 first, second, third, fourth, — Ath terms, [df. 
 
 .". a + {n — l)d= I, the last of a series of n terms, q.e. d. 
 
 Thbor. 2. 7m an arithmetic series, s=in{a + l). 
 For V s=a + {a + d) + {a + 2d)+- • • +{l —d) + l, w terms, 
 and s = l +{l—d) + {l —2d)+ • ' ■ +{a + d) + a, w terms, 
 .-. 2s-{a + l) + {a + l) + {a + l)+---+{a + l), m times, 
 
 = n-{a + l), 
 .: s=in{a+l). q.e.d. 
 
 IfoTB. The numbers a, d, I, n, s are the_/?fe elements of an 
 arithmetic series. In theors. 1, 2 these elements are connected 
 by two equations; hence, any three of them being given, 
 the other two can be found, and in all there are twenty equa- 
 tions : four that contain no a, four that contain no d, and so on. 
 
^•^■8J1 AHITHMETIC PROGRESSION. 219 
 
 QUESTIONS. 
 
 1. Write an arithmetic series in which a = 3, d=~2, to=5, 
 
 2. "What two conditions would make negative all the terms 
 of an arithmetic series ? Can an arithmetic series be an end- 
 less rising series and have some terms negative ? all negative ? 
 
 3. From the equation l=a+{n- l)d, find the value of a 
 in terms of I, n, d. 
 
 4. So, solve this equation for n, and for d. 
 
 5. Solve the equation s = \n(a-\-l), in turn for n, a, I. 
 
 6. In the series of integers 1, 3, 3, • • •, find the last term 
 and the sum gf 5 terms; of 30 terms; of n terms; of 2» terms; 
 of 3w+l terms. 
 
 7. In the series of odd positive integers, find the sum of 35 
 terms; of 50 terms; of n terms; of 2w terms; of 2m + 1 terms. 
 
 8. In the series of even positive integers, find the sum of n 
 terms; of 3m terms; of 2« — 1 terms; of 2w + l terms. 
 
 9. Find the five elements of the arithmetic series 
 
 1,3, 5,.. -99; 1, 3, 5,-.-3*-l; 4 + 5+ •.• =5350. 
 
 10. In an arithmetic series, the sum of two terms equidis- 
 tant from the extremes is constant. 
 
 11. What multiple of the common difference must be added 
 to the fifth term of an arithmetic series to give the ninth ? to 
 the twelfth to give the twentieth ? 
 
 12. If a = l, d=2, then s = n\ 
 
 13. A three digit number is 26 times the sum of its digits, 
 which are in arithmetic progression; if 396 be added to the 
 number the order of the digits is reversed : find the number. 
 
 14. A hundred stones are laid in a row, a meter apart, and 
 a basket is placed a meter from the first stone: how many 
 kilometers will a man walk who, starting from the basket, 
 picks up all the stones, one by one, and brings them separately 
 to the basket ? 
 
 15. The sum of three numbers in arithmetic progression is 
 37 and the sum of their squares is 293 : find the numbers. 
 
220 PROGRESSIONS. INCOM. POWERS, LOGJARITHMS. [Vin,PB. 
 
 ABITHMETIC MEANS — INTEEPOLATIOlir. 
 PbOB. 1. To INTERPOLATE ??l ARITHMETIC MEANS BETWEEN 
 TWO NUMBERS «, I. 
 
 Divide the remainder, I— a by m + 1 for the common differ-: 
 ence; to a add one, two, three- • -times this difference. 
 E.g., to interpolate 5 means between 13 and 48: 
 then •.• (48 — 12) : (5 +.1) = 6, the common difference, 
 .'. the series sought is 12, 18, 34, 30, 36, 42, 48. 
 Note. By aid of this problem, from every arithmetic series 
 a new arithmetic series may be formed by interpolating the 
 same number of arithmetic means between every pair of con- 
 secutive terms ; and the common difference of this new series 
 is the quotient of the common difference of the other divided 
 by one more than the number of terms so interpolated. 
 E.g., if two means be interpolated between pairs of consecu- 
 tive terms, 
 then the series 6, 13, 18, 34, 
 
 becomes the series 6, 8, 10, 13, 14, 16, 18, 3Q, 33, 24. 
 
 geometric illusteation. 
 Let ox, OT be two straight lines at right angles to each other; 
 take points A, b, c, • • • such that they are 1, 3, 3, • • • inches 
 above ox, and 6, 13, 18, • • -inches to the right of OT; 
 
 "" 6 ' ' ' iSii Mr- X 
 
 then A, B, c, • • • lie on a straight line through o, and their 
 distances from ox are in -arithmetic progression, and 
 so are their distances from ot, and from o; 
 
 and the common differences of these three series are 1,6, i^ZI-. 
 
1.11] ARITHMETIC PROGRESSION. 221 
 
 Between a, b, C, • • • interpolate other equidistant points; 
 then three new arithmetic series are formed, however close 
 together the points of division may lie. 
 
 CONTINUOUS PEOGEESSION. 
 
 Lay a pencil on the figure and move it slowly to the right, 
 keeping it always parallel to OY and letting it cut AB 
 at a moving point p; 
 
 then three series are formed, in continuous arithmetic progres- 
 sion : the growing distances of P from ox, from OY, and 
 from o. 
 
 QUESTIONS. 
 
 Find the five elements of the arithmetic series: 
 1. 5, 6 means, 75. 2. 3, 6 means, —11. 
 
 3. 2i, 4 means, 20. 4. 5 terms, 19, 7 means, 67. 
 
 5. When m arithmetic means are interpolated hetween two 
 given terms, these two terms and the means make a series of 
 how many terms ? What are a and I in this new series ? 
 
 6. Form a new arithmetic series by interpolating three 
 means between every pair of terms of the series 4, 8, 13, • • • 
 
 What is the new common difEerence ? 
 Form a new series by taking every fifth term of the series 
 last formed, beginning with 'the second. 
 What is now the common difEerence ? 
 
 7. Show that, at simple interest, the given principal and its 
 amounts for successive years form an arithmetic series, wherein 
 n is one more than the number of years. What are a, d, I? 
 
 Write such a series for five years, and show that the final 
 amount agrees with the formula l=a+{n — l)d. 
 
 8. If the interest be reckoned half-yearly, how many means 
 must be interpolated between every two terms ? if quarterly? 
 if monthly? What is d in each of these new series? 
 
 If the interest be reckoned instantly, what kind of a series 
 results ? what is the number of terms ? the common difference ? 
 What two elements of the series are unchanged ? 
 
222 PROGRESSIONS, INCOM. POWERS, LOGARITHMS, [vni, Ths. 
 
 §2. GEOMETEIO PEOGEESSION. 
 
 A series is in geometric progression if each term after the 
 first be found by multiplying the term before it by a constant 
 multiplier. This multiplier is the common ratio. A series in 
 geometric progression is a geometric series. 
 
 The abbreviations are : a first term, I last term, r common 
 ratio, n number of terms, s sum of all the terms. 
 
 If r be larger than 1, the series is a rising series; if smaller, 
 & falling series. 
 E.g., 1, 2, 4, 8, 16 is a rising series, 
 
 wherein ?-=+2, a = \, Z=16, n = b, s=31. 
 So, 1, ~2, 4, "8, 16 is a rising series, 
 
 wherein r="2, a = l, Z=16, 7i = 5, s = ll. 
 But 16, 8, 4, 2, 1, 1/2, 1/4 is a falling series, 
 
 wherein r = l/2, a = 16, Z=l/4, ^ = 7, s = 31i. 
 
 Theoe. 3. In a geometric series, l=ar^~^. 
 For •.• a, ar, ar^, • • • «/•*"* are the first, second, third, • • • 
 Hh terms, [df. 
 
 .*.«»•""* = ?, the last of a series of w terms, q.e.d. 
 Theoe. 4. In a geometric series, s — {a—rT)/{l—r). 
 For V s -a +ar +ar^+ ■ ■ ■ +^.-"+^-^ + 1, [df. 
 
 .•. rs — ar + ar^ + ar^+ • ■'• +Zr~* + Z +lr, 
 .: s — rs = a — lr, and s={a — rl)/{l — r). q.e.d. 
 
 CoE. 1. In an infinite falling geometric series, I is indefi- 
 nitely small; and the value of s is indefinitely near to the 
 quotient a/{l—r). 
 
 CoK. 3. A repeating decimal equals a common fraction whose 
 numerator consists of the repeating figures, and the denomi- 
 nator of as many Q's as there are repeating figures. 
 
 E.g., the decimal .45, i.e., .454545 • • •, is the sum of the 
 
 geometric series 45/100 + 45/100* + 45/100' + • • • 
 and 5=45/100/(1-t|t^)=:45/99. 
 
3> ■«•«»] GEOMETRIC PROaEESSION. 223 
 
 QUESTIONS. 
 
 1. If in a geometric series the first term be positive and the 
 ratio a positive proper fraction, what signs have the terms ? 
 how do they change ? if the ratio be a negative proper fraction ? 
 
 2. Solve the equation l=ar''-^ for a and for r. 
 
 3. In a geometric series a=-3,l=-i8, r=-2: what is ?i? 
 
 4. Solve the equation 5= (a — rZ)/(l — r) in turn for a, r, I. 
 
 5. Find the last term and the sum of 10 terms of the series 
 of integer powers of *3; of « terms; of 3w terms. 
 
 Find the 13th term and the sum of 13 terms of the series: 
 6.1- + -3/4+-.. 7. 3/5'+3/5''■^-3/5'+3/5*^-••• 
 8. If »• be a proper fraction, how do rising powers of rvary? 
 What is the value of a very high power of such a ratio ? 
 
 Find the value of: 
 
 9. .313131- •• 10. .673. 11. .36*84. 13. .153737- • • 
 
 13. The geometric mean between two positive numbers lies 
 between them, and is their mean proportional. 
 
 14. By what power of the common ratio must the fifth 
 term of a geometric series be multiplied to give the ninth 
 term ? the twelfth term to give the twentieth term ? 
 
 15., In a geometric series the product of any two terms 
 equidistant from a given term is the square of that term. 
 State and prove the like truth about an arithmetic series. 
 
 16. If all the terms of a geometric series be multiplied (or 
 divided) by the same number, the products (or quotients) 
 form a geometric progression with the same ratio as before. 
 
 17. From the two fundamental equations l—ar^-^, s= 
 {a — rl)/{l—r), eliminate a, Z, r, in turn. 
 
 18. A man invests $100 in stocks that pay 3 per cent half- 
 yearly dividends, and invests the dividends, as received, at the 
 same rate : how much has he invested at the end of 5 years ? 
 
 19. The sum of three numbers in geometric progression is 
 13, and the product of the mean by the sum of the extremes 
 is 30 ; what are the numbers ? 
 
224 PROGRESSIONS, INCOM. POWERS, LOGAEITHMS. [Vra.Pa. 
 
 GBOMBTKIC MEANS. 
 PkOB. 2. To INTERPOLATE m GEOMETEIC MEANS BETWEEN 
 TWO NUMBERS, fl, I. 
 
 Take the (m + 1) th root of the quotient l/a for the' common ratio; 
 multiply a by thejirst, second, • • • powers of this ratio. 
 E.g., to interpolate three means between 3 and 48: 
 then •.• ^(48 : 3) = 3, the common ratio, 
 .'. the series sought is 3, 6, 12, 24, 48. 
 
 Note. By aid of this problem, from every geometric series 
 a new geometric series may be formed by interpolating the 
 same number of geometric means between every pair of con- 
 secutive terms; and the common ratio of this new series is 
 that root of the common ratio of the other whose index is one 
 more than the number of means so interpolated. 
 E.g., if two means be interpolated between pairs of consecu- 
 tive terms, 
 then the series 3, 6, 13 • • • 
 
 becomes the series 3, 3^2, 3v'4, 6, 6^2, 6 1^4, 13- •• 
 
 geometric illustration. 
 Let ox, OT be two straight lines at right angles to each other; 
 take points a, B, c, • • • such that they are 0, 1, 3, • • • inches 
 to the right of oy and 3, 6, 12, • • • inches above ox; 
 
 
 
 
 
 
 
 =4 
 
 
 
 
 -^\ 
 
 w 
 
 
 18 
 
 
 
 
 
 
 
 6 
 
 
 
 3 
 
 24 
 
 3 X 
 
 then A, B, c, • • • lie on a curve; their distances from oy are 
 in arithmetic progression, with a common difference 1, 
 
 but their distances from ox are in geometric progression, with 
 a common ratio 3. 
 
"•§2] GEOMETRIC PROGRESSION. 225 
 
 Between a, b, c, • • • interpolate other points whose distances 
 from OT are arithmetic means between the terms of 
 the series 1, 2, 3, • • • 
 
 and whose distances from ox are the like geometric means 
 between the terms of the series 3, 6, 12, • • • 
 
 COKTINUOUS PBOGBESSIOlir. 
 
 Lay a pencil on the figure and move it slowly to the right, 
 
 keeping it always parallel to ot, and letting it cut the 
 
 curve at a moving point p; 
 then the growing distance of p from oy forms a series in 
 
 continuous arithmetic progression, 
 and the growing distance of P from ox forms a series in 
 
 continuous geometric progression. 
 
 QUESTIONS. 
 
 1. Prom the formula r=n~^{l/a) find the new ratio when 
 7n geometric means are interpolated between every two terms. 
 Insert geometric means : 
 
 2. Four between 1 and 32. 3. Two between 1 and 1000. 
 4. Three between 1/9 and 9. 5. Three between 2 and 1/8. 
 
 6. Form a new geometric series by interpolating three terms 
 between each pair of terms of the series 3, 9, 37, • • • 
 
 What is the new common ratio ? 
 
 Form a new series by taking every fifth term of this series, 
 beginning with the second. What is now the common ratio ? 
 
 7. In compound interest the principal and its amounts at 
 the ends of successive years form a geometric series. 
 
 Show that «=^(l+rate)' agrees with the formula ?=«?•""'. 
 
 8. If the interest be compounded half-yearly, but in such a 
 way as not to change the final amount, how many means are 
 inserted between every two terms ? if quarterly ? if monthly ? 
 
 What is r in each of these new series ? 
 How must the interest be compounded to make the amount 
 a continuous variable ? What is then the value of m ? of r ? 
 AVhat two elements of the series are unchanged ? 
 
226 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [VIII, Th. 6. 
 
 §3. HARMONIC PEOGRESSIOK 
 A series is in harmonic progression if the reciprocals of the 
 terms be in arithmetic progression. 
 E.g., hhh,h---; 3,4, 6,13, ■■• 
 
 The last term of a harmonic series is found by computing 
 the last term of the arithmetic series of reciprocals and invert- 
 ing it; the sum of a harmonic series can be found only by the 
 actual addition of the terms. 
 
 Theor. 5. Of three consecutit)e terms of a harmonic series 
 the first is to the third as the excess of the first over the 
 second is to the excess of the second over the third. 
 Let p, q, r be three consecutive terms of a harmonic series; 
 
 then p •.r=p — q:q — r. 
 
 For •.• 1/q - 1/p =.l/r - 1/q, [df. 
 
 •■• {p-q)/pq^ig!-r)/qr 
 
 and {p-q)/iq-r)=pq/rq=p/r. Q.E.D. 
 
 PkOB. 3. To INTEEPOLATB m HAKMONIC MEANS BETWEEN 
 TWO NUMBERS, fl, I. 
 
 Find m arithmetic means between the reciprocals of a, I, and 
 
 take the reciprocals of these means. 
 E.g., to interpolate two harmonic means between 12 and 48; 
 then •.• 1/13 - 1/48 = 3/48, and 3/48 : 3 = 1/48, 
 
 .-. the arithmetic series is 1/13, 1/16, 1/34, 1/48, 
 and the harmonic series is 13, 16, 34, 48. 
 
 THE ANALOGIES OF THE THREE PROGRESSIONS. 
 
 Note. The analogies and relations of 'the three progres- 
 sions may be thus stated : if p, q, r be three numbers 
 
 in arithmetic progression, then p—q:q—r =p :p; 
 
 in geometric progression, then p — q:q — r =p : q ; 
 
 in harmonic progression, then p — q:q — r =j» : r. 
 
 The three means are i{,p + r), ^pr, 2pr/{p + r); and 
 
 the geometric mean of p, r is the geometric mean of their 
 
 arithmetic and harmonic means. 
 
PB-3.S3] HARMONIC PROGRESSION. 227 
 
 QUESTIONS. 
 
 Continue the harmonic series for three terms in each direction: 
 1. 2, 3, 6. 2. 3, 4, 6. 3. 1, If, If. 4. 1|, If, If. 
 
 5. The harmonic mean of two numhers is twice the prod- 
 uct of the numbers divided by their sum. 
 
 Insert harmonic means as follows : 
 
 6. Pive between 1/3 and 1/5. 
 
 7. Three between 7/5 and 7/13. 
 
 8. Five betweeh 4/5 and -8/11. 
 
 9. Three between l/(4a + 5) and l/5b. 
 
 10. Whatever be the values of p, r, {p — ry is positive, 
 and p^+r^>2pr; i(p+r)>^pr, 4p^r^/{'p + rf<pr. 
 
 11. Prove the analogies of the three progressions as above. 
 13. The geometric mean between two numbers is 8, the 
 
 harmonic mean 6f : find the numbers. 
 
 13. The difference of two numbers is 8 and their harmonic 
 mean is If: what are the numbers ? 
 
 14. The arithmetic, geometric, and harmonic means of two 
 numbers greater than unity are in descending order of •mag- 
 nitude. 
 
 15. The arithmetic mean between two numbers is 3 and the 
 harmonic mean 2f : find the numbers. 
 
 16. If % be the harmonic mean of a, 5, then 
 
 1/(2 - a) + \/{z -b) = \/a + 1/6. 
 
 17. What number must be added to each of three given 
 numbers a, b, c, that the three results may be in harmonic 
 progression ? 
 
 18. If a, b, c be in harmonic progression, then 
 
 l/{a -b)+ 1/(5 - c) + 4/(c - a) = 1/c - 1/«. 
 
 19. If to each of tliree consecutive terms of a geometric 
 progression the second of the three be added, the sums are in 
 harmonic progression. 
 
 20. If G be the geometric mean of A/ B, then the harmonic 
 means of A, G, and g, b are equal. 
 
228 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [vra,LEM. 
 
 §4. INCOMMENSURABLE POWBES. 
 
 If r', r^, ?V ■ • • ^" 136 a series in geometric progression, 
 the exponents 1, 2, d, • • • n form a series in arithmetic pro- 
 gression; and the numbers 1, 2, 3, ■ • • n serve also to define 
 the position of the terms of the geometric series, r' being the 
 first term, Tj, r'' the second term, Tj, /•' the third term, 
 T3, • • • r" the ?ith term, t„ . 
 
 If two geometric means be interpolated between every pair 
 of consecutive terms of this geometric series, the exponents of 
 r in the new series form the arithmetic series 1, 1^, If, 3, 2^, 
 2 1, 3, • • • ; and here, too, the exponents serve to define the 
 position of the terms of the geometric series, the terms of the 
 original series being the major terms, or integer terms, and 
 the others the minor terms, or fraction terms. 
 
 So, for other means that may be interpolated between pairs 
 of terms, however great their number; and r" is the nth term, 
 T„ , whether n be an integer or a fraction, positive or negative. 
 The exponents form an arithmetic series of commensurable 
 numbers, and the corresponding terms of the geometric series 
 are commensurable powers of r. 
 
 But if the arithmetic series be made continuous, then also 
 the geometric series is a continuous series of the powers of r. 
 Among the terms of this continuous arithmetic series are 
 included incommensurable numbers, and the corresponding 
 terms of the continuous geometric series are incommensurable 
 powers of r, incommensui-able powers being distinguished 
 from commensurable powers as powers whose exponents are 
 incommensurable numbers. 
 
 Lemma. 77ie ratio of two terms of a geometric series is that 
 230wer of the common ratio whose exponent is the excess of the 
 number which defines the position of the first term over that 
 which defines the other. [df. geom. prog. 
 
 E.g., Ts:T6=rS T2o:Ti3=r% Ti/2:Ti/3 = »•'''«, T^rT^^r*"*, 
 
84] INCOMMENSURABLE POWERS. 229 
 
 QUESTIONS. ^ 
 
 1. In the geometric series • • • 1/64, 1/8, 1, 8, 64, • • • find 
 the common ratio. What is To? Tj? t_i? t^? t_2? Tj? t_3? 
 
 Interpolate two means between every pair of consecutive 
 terms, and name the terms so interpolated. 
 
 2. In the geometric series • ■ • , 16/81, 4/9, 1, 9/4, 81/16, 
 • • • , find the common ratio, and name the terms. 
 
 Interpolate three means between every pair of consecutive 
 terms, and name the terms so interpolated, 
 
 3. On a horizontal line as axis take a point o, and on this 
 axis lay ofE distances to the right and left from o proportional 
 to the positive and negative exponents of the series in ex. 2 ; 
 at the points so found draw vertical lines and lay off distances 
 upward proportional to the terms themselves; join the upper 
 ends of these vertical lines, in their order, by straight lines, 
 thus forming a plat of the series that rises faster and faster. 
 
 4. If a dollar be put at compound interest at the annual 
 rate of 10 per cent, find the amount at the end of 1 year; 2 
 years; 3 years; • • • 8 years. These amounts form a true geo- 
 metric series with the common ratio 1.1. 
 
 5. In ex. 4, the amount would be the same at the end of 
 any period of years if the interest were compounded half 
 yearly at the ratio |/1.1, quarterly at the ratio .^'l.l, 
 monthly at the ratio y'l.l, and so for any shorter periods. 
 
 What relations have the arithmetic series of exponents, the 
 growing time, and the geometric series of amounts ? 
 
 6. In ex. 4, as the time grows continuously, so may the in- 
 terest and the amount, i.e., just as soon as any interest is 
 earnedj.that interest may itself become principal and begin to 
 earn interest; and the plat of the growing amount is a con- 
 tinuous curve, rising faster andJaster from the axis. 
 
 7. Growing continuously, the amount in ex. 4 becomes 
 double the principal at some time between -seven and eight 
 years. This time is definite and distinct; it is not an integer, 
 and not a simple fraction; hence it is incommensurable, and 
 2 is an incommensurable power of 1.1. 
 
230 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [Vm.Tna. 
 THE PRODUCT OF POWERS OF THE SAME BASE. 
 
 Theor. 6. The product of two or more powers of a base is 
 that power of the base whose exponent is the sum of the expon- 
 ents of the factors. 
 
 Let m, n be any two numbers, A any base, then a"'-a"=a"+". 
 
 (a, b) m, n both commensurable : [VI, th. 4. 
 
 (c) m, n either or both of them incommensurable. 
 
 For, let the geometric series A^, a^, a', • • • , whose common 
 
 ratio is A, be made continuous by letting the exponent 
 
 grow continuously through all the intermediate values, 
 
 commensurable and incommensurable; 
 
 then '.• A^^T^, whatever m may be, and a'"+''=Tot+„, [df. 
 
 and •.•Tot+„ is the wth term beyond T„, and its value is 
 
 A" -A", Pem. 
 
 •'. A*"* a" = A*" "*"**■ 
 
 and so for three or more powers. Q.E.D. 
 
 A POWER OF A POWER, 
 
 Theor. 7. A power of a power of a base is that power of the 
 base whose exponent is the product of the given exponents. 
 Let m, n be any two numbers, A any base; then (a"')"=a'"'". 
 («, b) m, n both commensurable : [VI, th. 3. 
 
 (c) m, n either or both of them incommensurable. 
 For, let the geometric series a^, a*, a*, ■ • • , whose common 
 ratio is a, be made continuous, and, in this series, 
 mark A™, (A")", (a™)', • • • as the principal terms of 
 a series whose common ratio is A*"; 
 then •.• (a"*)" is both the nth. term of this series and the mnth 
 term of the original series, whose value is a*"", [df . 
 .-. (a"')'' = a"'". Q.E.D. 
 
.6.'',8,S4]- INCOMMENSURABLE POWERS. 231 
 
 THE PRODUCT OF LIKE POWERS OF DIFFERENT BASES. 
 
 Theor. 8. The product of like powers of two or more bases 
 is the like power of their product. 
 
 Let n be any number and A, : 
 
 B, c, • • 
 
 • any 
 
 bases; 
 
 then A^-B^-c"- • • :=(a-b-c- • 
 
 •)«. 
 
 
 
 {a, b) n commensurable : 
 
 
 
 [VI,th.5. 
 
 (c) n incommensurable. 
 
 
 
 
 For, let BrzA"; 
 
 
 
 
 then A"-?" = A" -A™-" 
 
 
 
 [th. 7. 
 
 . 1 n + nt-n 
 
 
 
 [th. 6. 
 
 = A»+'"'-"=(A-A™)" 
 
 
 
 [th. 7. 
 
 = (A-B)"; 
 
 
 
 [above. 
 
 and so for three or more bases. 
 
 
 
 Q.E.D. 
 
 QUESTIONS. 
 
 1. If the interest be compounded instantly at a rate that is 
 equivalent to the annual rate 10, and if the principal be 
 doubled in m years, and this double be tripled in n years more, 
 the whole time is m + w years, and the final amount is six- 
 fold the first principal: 
 
 I.e., if 2^=i?-(l.ir, and 6p = 2p-(l.iy, 
 then v6i?=i?-(l.l)"'-(l.l)", and 6^=^-(l.l)"'+», 
 .-. (1.1)'»-(1.1)«'=(1.1)'"+". 
 
 2. If the interest be compounded instantly at a rate that is 
 equivalent to the annual rate 10, and if the principal be 
 doubled in m years, and this double be tripled in n periods of 
 m years, the whole time is m-n years, and the final amount is 
 sixfold the first principal : 
 
 i.e.,it 2jo=jo-(l.ir and &p=p-T; 
 then v6p=p-[(l.l)'"]", and &p=p- {1.1)^^. 
 .: [(LI)™]" =(1.1)""*. 
 
232 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [VIU.Ths. 
 
 §5. LOGAEITHMS. 
 
 The logarithm of a number is the exponent of that power to 
 which another number, the iase, must be raised to give the 
 number first named. 
 
 E.g., in the equation a"'=n, a is the base, n is the number, 
 and X is the exponent of A and the logarithm of w. 
 
 Operations upon or with logarithms are therefore operations 
 upon or with the exponents of the powers of the same basej 
 and the principles established for such powers apply directly 
 to logarithms, with but the change of name noted above. 
 
 The three equations A"'=if, a; = log^iir, N=logj'a; are 
 equivalent equations. The second is read, x is the logarithm 
 of's to the Iase A, or x is the K-logarithm of's; and the last 
 means that N is the number whose logarithm to the base A 
 is x; it is read, N is the anti-logarithm of x to the Iase A. ; 
 E.g., 0=logAl and l = logr*0, whatever A may be. 
 So, l = log32, 2=loga9, 3 = log464, 4 = logs625, 
 
 and 2 = log2-il, 9 = log3-'2, 64 = logr'3, 625 = log5-^4, 
 So, -l = log2l/2, -2 = log3l/9, -3=log4l/64, -4 = log5l/625, 
 and -l=logv22, -3=logv39, -3 = logv464, -4 = logv6635, 
 
 If the base be well known it may be suppressed, and these 
 two equations may then be written a; = log ~s, K=log~*a;. 
 
 If while A is constant K take in successicm all possible 
 values from to oo, the corresponding values of x constitute a 
 system of logarithms to the iase A. 
 
 Theor. 9. The logarithm of unity to any iase is zero. 
 
 Theoe. 10. The logarithm of the iase itself is unity. 
 
 Theor. 11. If the iase ie positive and larger than unity, the 
 logarithms of numiers greater than unity are positive, tohile 
 of numiers positive and smaller than unity they are negative; 
 and if the iase ie positive and smaller than unity, the log- 
 arithms of numbers greater than lonity are negative, while of 
 . numiers positive and smaller than unity they are positive. 
 
9.l«.i'.85] LOGARITHMS. 233 
 
 QUESTIONS. 
 
 With 4 as base, find the logarithms of : 
 
 1. 16; 8; 1; 64; 32; 256; 4"^; 16"'; Z2~^/\ 
 
 2. .35; 16V2; 1/16; 1/8; 128/1024; ^2/256. 
 
 3. Are the numbers below commensurable or incommen- 
 surable powers of 4 ? 
 
 5; 25; 125; 625; .5; .25; .125; .0625. 
 Between what commensurable powers do the incommen- 
 surable powers lie ? 
 
 4. What is the logarithm of 144 to the base 2^/3 ? 
 
 5. What effect is produced on the logarithm of a number 
 by making the base smaller ? by making it larger ? 
 
 6. Find logs 3125; log^ 343"^; logi/, 81; log,/, 343. 
 
 7. Find, to the base a, Jog y'^-'^/^; log [(a-^/')-'/']-'/'. 
 
 8. With 9 as base find the anti-logarithm of: 
 
 i; -1; 5/2; "3/3; 2; "2; 0; 1; 3/4; -3/4. 
 
 9. With 8 as base, write a series of six logarithms. 
 What base makes : 
 
 10. log 64=2? log 6i = 2? Iog-i3=-1000? log 125 = 3? 
 
 11. log-»f = 343? log-'-| = 32? log2i=i? log 64= "3? 
 
 12. A number has different logarithms to different bases, 
 and it may have the same logarithm to two different bases, 
 but with a given base, it can have but one logarithm. 
 
 13. With a negative base what positive numbers and what 
 negative numbers have logarithms? What negative numbers 
 have logarithms to positive bases ? 
 
 In making computations by logarithms, when may the signs 
 of the numbers be disregarded ? 
 
 14. With a base smaller than unity, what is the logarithm 
 of a very large number? of unity? of a very small number? 
 of zero ? of the base itself? 
 
 15. With a positive base larger than unity, what is the 
 logarithm of a very small fraction ? of zero ? of a very large 
 number? of unity? of the base itself ? 
 
334 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [VIII.Ths. 
 LOGARITHMS OF PEODUCTS AND QUOTIENTS. 
 
 Theor. 12. The logarithm of a product is the sum of the 
 logarithms of the factors; and the logarithm of a quotient is 
 the excess of the logarithm of the dividend over that of the 
 divisor. [th. 6. 
 
 E.g., logi(B-c:D) = log,B+logiC-log^D. 
 
 LOGARITHMS OF POWERS AKD ROOTS. 
 
 Theor. 13. The logarithm of a power of a numier is the 
 product of the logarithm of the number iy the exponent of the 
 power sought; that of a root is the quotient of the logarithm 
 by the root-index. [th. 7. • 
 
 E.g., log,(B'.-^c) = 2 log,B+i log,o. 
 
 CHANGE OF BASE. 
 
 Theor. 14. If the logarithm of a number be fahen to two 
 different bases, the first logarithm is the product of the second 
 logarithm into the logarithm of the second base taken to the 
 first base, and vice versa. 
 
 For, let N be any number, a, b two bases, and ?/ = logBN; 
 then-.- N = B'', [df.log. 
 
 .-. log, N = log^ B" = 2/ • log^ B, [th. 13. 
 
 = logBN-logiB. Q.E.D. 
 
 So, logBN^logiN-logBA. Q.E.D. 
 
 Cor. 1. The. logarithms of tivo numbers, each taken to the 
 other number as base, are reciprocals. 
 For, let B = A^ 
 
 then -.- a = b'''"', \og^^ — x, logBA = l/a:, [df.log. 
 
 .-. logi B-logs A = a;-1/»=1. " Q.E.D. 
 
 Note. There are two systems of logarithms in use: 
 natural logarithms, whose base is e [3.71828- • •], and com- 
 mon logarithms, whose base is 10. Their relations to each 
 other are expressed by the equations 
 
 logioN = logeN • logio«, logi„e = . 424294. 
 
12, la, 14, § 5] LOGARITHMS. 235 
 
 QUESTIONS. 
 
 If the logarithms of x, y, a, h be known, show how to find : 
 1. log yx^ • i^xf). 2. log ya-^h ■ ^ah~% 3. log abxy. 
 4. \og{al\mjf.^Aog{ax:lyyy^. 6.1og(«V2j-V3a;-V4.j^V6), 
 Given logi, 2 = .3010, log^ 3 = .4771, logi„ 7 = .8451, find: 
 7. log 5. 8. log 6. 9. log 8. 10. log 9. 
 
 11. log 10. 12. log 12. 13. log 14. 14. log 15. 
 
 15. log 16. 16.. log 20., 17. log 18. 18. log 21. 
 
 19. log 24. 20. log 25. 21. log 28. 22. log 30. 
 
 23. log 4/72. 24. log 300'/*. 25. log 161 26. log 1728. 
 
 37. log 2/5. 28. log 3^. 29. log 4^ 30. log 12^. 
 
 31. log If. 32. log i4/15. 33. log ^'(32. 5*: 4/2). 
 
 34. log f (729 • |'9 -' • 37-*'"). 35. log (27 -^/= : 64"''"'). 
 
 36. What is the logarithm of the arithmetic mean of 15, 21 ? 
 of the harmonic mean ? of the geometric mean ? 
 
 37. The logarithms of two given numbers bear a constant 
 ratio to each other, whatever the base. 
 
 38. Given log, 49 = 2, and logi„ 7=:.8451, find ]og,„ 49, 
 
 39. Given logi« 64 = 3/2, logi, 16 = 1.2040, find logi„ 64. 
 
 40. If loga:y = a, loga;/« = 5, find logs; and logy. 
 From the logarithms of 2, 3, 5, 7 to the base 10, above, find : 
 
 41. log, 10. 42. logs 10. 43. log, 10. 44. log^ 10. 
 45. logj 700. 46. logs 7. 47. logs 4/30. 48. logs li- 
 49. logs 2. 50. logs 800. 51. log, 3^. 52. logs 270, 
 53. log^S. 54. logs 343. 55. logs 28. 56, log, 14f. 
 
 57. log 75/16 - 2 log 5/9 + log 32/243 = log 2. 
 From logi„2 = .30103 and logm 7 = .84509, find: 
 
 58. log, 4/2. 59. logva 7. 60. log^ 4/7. 61. log,,, 2. 
 
236 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [Vin,TH. 
 
 SPECIAL PEOPBBTIES OF THE BASE 10. 
 
 The logarithm of an integer power of 10 is an integer. 
 E.g., of 1000, 100, 10, 1, .1, .01, .001, 
 the logarithms to the base 10 are 
 
 +3, +2, +1, 0, -1, -3, -3. 
 But of any other number the logarithm is fractional or in- 
 commensurable; and if incommensurable, it consists of a 
 whole number, the characteristic, and an endless decimal, the 
 mantissa. , 
 
 As a matter of convenience the mantissa is always taken 
 positive; and the characteristic is the exponent, positive or 
 negative, of that integer power of 10 which lies next below 
 the given number. 
 
 A. negative characteristic is indicated by the sign — above it, 
 E.g., the logarithms of the numbers 
 
 2000, 20, .3, .003, 
 
 are 3.30103- ••, 1.30103- ••, 1.30103---, S'.SOIOS---, 
 whose characteristics are 3, 1, 1, 3, and whose common 
 mantissa is .30103- - -. 
 
 Theok. 15. // a number he multiplied {or divided) by any 
 integer power of 10, the logarithm, of the product {or quotient) 
 and the logarithm of the number have the same m,antissa. 
 For •.• the logarithm of a product is the sum of the logarithms 
 of its factors, [th. 13. 
 
 and •.• the logarithm* of the multiplier is an integer, [hyp. 
 .*. the mantissa of the sum is identical with the mantissa 
 of the logarithm of the multiplicand. q.e.d. 
 So, if a number be divided by an integer power of 10. 
 
 Cob. The logarithms of all numbers expressed by the same 
 figures in the same order have the same mantissa, but different 
 characteristics. 
 E.g., the logarithms, correct to four figures, of the numbers 
 
 79500, 795, 7.95, .0795, .000795, 
 
 are 4.9004, 2.9004, 0.9004, 2.9004, 4.9004. 
 
15. S 5] LOGARITHMS. 237 
 
 QUESTIONS. 
 
 1. What kind of power of 10 is a number whose logarithm 
 is a simple fraction ? 
 
 Can such a power of 10 be a commensurable number ? 
 Conversely, what kind of logarithm has a commensurable 
 number that is not an integer power of 10 ? 
 What is the mantissa of such a logarithm ? 
 
 2. What kind of series is formed by the powers of 10 on 
 the opposite page ? what by their logaritlims ? 
 
 So, with the series 2000, 20, .2, .002 and their logarithms ? 
 
 3. Name some base that gives a rising series of logarithms 
 for a falling series of numbers. 
 
 4. What relation has the difference in the series of log- 
 arithms to the ratio in the corresponding series of numbers ? 
 
 5. Explain why the logarithms of 9520, 95.2, .952, 
 .00952, have the same mantissa. 
 
 What are the characteristics of these logarithms ? 
 
 6. Moving the decimal point one place to the left in a 
 number has what effect on the characteristic of its logarithm? 
 
 So, moving the point three places to the right ? 
 
 Given log 4096000 = 5.6124, find: 
 
 7. log 4096. 8. log 40.96. 9. log 6.4. 10. log 8. 
 11. log 4. 12. log 513. 13. log .016. 14. log .0002. 
 
 15. How many figures in 10^ ? in any number between 10" 
 and 10'? in a number between 10"~' and 10"? 
 
 What is the characteristic of the logarithm of an integer 
 expressed by three figures? by n figures? How many figures 
 are there in the anti-logarithm if the characteristic be 5 ? 
 
 16. Given log 2 = .30103, how many figures are there in 2^°? 
 
 17. The logarithm of a decimal fraction has a negative 
 characteristic, a unit larger than the number of ciphers that 
 follow the decimal point. 
 
 18. If (1/2)'°° be reduced to a decimal, how many ciphers 
 follow the decimal point ? 
 
238 PKOGEESSIONS, INCOM. POWERS, LOGARITHMS. [VIII.Pb. 
 TABLES OF LOGARITHMS. 
 
 The logarithms of any set of consecutive numbers, arranged 
 in a form convenient for use, constitute a table of logarithms. 
 A table to the base 10 need give only the mantissas ; the char- 
 acteristics are evident. The dirference of two consecutive 
 mantissas is their tabular difference. 
 
 Tables may be carried to any number of decimal places; 
 but the four-place table on pages 244, 245 is sufficiently accu- 
 rate for ordinary use. The fourth figure is in' error by less 
 than half a unit. The first two figures of each number are 
 printed at the left of the page, and the third figure at the top 
 of the page, over the mantissa of the corresponding logarithm. 
 
 PeOB. 4. To TAKE OUT THE LOGARITHM OF A NUMBER. 
 
 (a) A three-figure number : take out the tabular mantissa that 
 lies in line with the first two figures and tinder the third; 
 
 the characteristic is the exponent of that integer power of 10 
 which lies next below the given number. 
 
 If a number have one or two figures, make it three-figured by 
 annexing zeros. 
 
 E.g., log 567 = 2.7536; log 5.6 = 0.7482; log ,05 = 2.6990. 
 
 {b) A number of more than three figures: take out the tabular 
 
 mantissa of the first three figures; ^ 
 
 subtract this mantissa from the next greater tabular mantissa; 
 multiply the difference so found by the remaining figures of 
 
 the number as a decimal; 
 add this product, as a correction, to the mantissas of the first 
 
 three figures. 
 E.g., to find log 500.6: 
 
 then •.' log 500 = 2. 6990, log 501 = 2. 6998, [tables, 
 
 and log 501 -log 500 = .0008, 500. 6 -500 = .6, 
 .-.log 500.6 = 2.6990 + 6 tenths of .0008 = 2.6995. 
 The rule for interpolation rests upon this property of log- 
 arithms, that their differences are nearly proportional to the 
 differences of the numbers when the differences are small. 
 
4.S51 LOGARITHMS. 239 
 
 QUESTIONS. 
 
 1. How does a table of logarithms of prime numbers make 
 it possible to find the logarithms of all other numbers ? 
 
 2. Why do not tables of common logarithms contain char- 
 acteristics ? must they be given in tables to other bases ? 
 
 3. Find log 2 — log 1000, giving a wholly negative 
 •logarithm as the result, and show that it is the same as 
 log 1/1000+ log 3, or 3.30103. 
 
 '4. In getting logarithms from a table, by what right and 
 for what purpose are zeros annexed to numbers having fewer 
 than three significant figures ? 
 From the table take out the logarithms of: 
 •^ 5. 13. 6. 130. 7. 133. 8. 124. 
 
 9. 133.4. 10. 1.234. 11. 12350. 12. .001235. 
 
 13. In finding log 73265, to how large a difference in the 
 number does the tabular difference correspond ? 
 " What part of this difference is the rest of the number ? 
 Take out the logarithms of : 
 
 ^ 14. 9033. 15. .00064. 16. 75.15. 17. 6.873. 
 
 18. .35. 19. 2496000. 20. .00854. 31. 1000000. 
 
 23. 246.3. 23. .9467. 24. .007009. 25. 1463. 
 
 ' 26. Show that the last paragraph in case (a) of prob. 4 
 might read : make the characteristic one less than the number 
 of figures in the integer part of the numler. 
 By use of the table on pp. 244, 245, find the logarithm of: 
 ' 27. 43962. 28. 521.6701. 29. .004381. 30. 134365QQ0. 
 '31. 3.76314. 33. .4580013. 33. 6309.25. 34. .000519328. 
 '35. Show how the logarithm of a number lying between two 
 tabular numbers may be found from the larger of the two 
 tabular logarithms. 
 
 36. Find in the table log 60, log 60.5, log 61, and show that, 
 while the second of these logarithms is almost half-way between 
 the other two, in numbers differing more widely, e.g., 60, 70, 
 80^ this proportion does not hold true. 
 
240 PROGRESSIONS, INCOM. POWERS, LOGARITHMS, [vm, Pa. 
 PrOB. 5. To TAKE OUT A NUMBER FROM ITS LOGARITHM. 
 
 (a) The mantissa found in the table: join thefigtire at the top 
 that lies above the given mantissa to the two figures 
 upon the same line at the extreme left; 
 
 in the three-figure number thus found so place the decimal 
 point that the number shall lie next above that power of 
 10 tuhose expooient is the given characteristic. 
 
 E.g., log-i 2.7536 = 567; log"' 0.7483 = 5,6; log-^ 2.6990 = .05. 
 
 {b) Tlie mantissa not found in the table : take out the three^ 
 figure anti-logarithm of the tabular mantissa next less 
 than the given mantissa; 
 
 and to it join the quotient of the difference of these two man- 
 tissas by the tabular difference. 
 
 E.g., to take out log-^ 2.6995: 
 
 then •.• the next less tabular mantissa is . 6990, and the next 
 greater .6998, 
 
 .•. the tabular difference is .0008; 
 i.e., an increase of .0008 in the mantissa .6990 corresponds to 
 
 an increase of 1 in the number. 
 But ■.■ the given mantissa differs from .6990 by .0005, 
 
 .*. this difference corresponds to an increase in the anti- 
 logarithm of .0005/. 0008, i.e., of .6. 
 and -.-log-i 2.6990 = 500, 
 
 .•. the figures in the number sought are 5006, and the 
 characteristic 2 shows that the number is 500.6. 
 
 So, to find log-' 3.4986: 
 
 then •.• the given mantissa lies between .4983 and .4997, 
 and .4986 -.4983 = .0003, .4997 -.4983 = .0014, 
 .0003/. 0014 = .2, 
 ,-. log-' .4986 has the figures 3152, 
 and log - ' 3. 4986 = .003152. 
 
 The process is but the inverse of that for taking out log- 
 arithms, and the reason of the rule is the same for both. 
 
5.S5] LOGAEITHMS. 241 
 
 LIMITATIONS IN THE USB OF THE TABLES. 
 
 The possible error of any logarithm, as printed in this table, 
 is half a unit in the fourth place, and the possible error of 
 any tabular difference is a unit; but the probable error is 
 much less. The fourth figure of the anti-logarithm, the first 
 got by division, is generally trustworthy; the fifth is rarely to 
 be used. The possible error in the result is nearly ten times 
 greater if the logarithm be near the end of the table than if 
 near the beginning, for then the tabular difference, the divisor, 
 is much smaller, and an error either in it or in the dividend 
 has greater effect. If greater accuracy be desired, larger 
 tables must be used. 
 
 QUESTIONS. 
 
 From the table take out the anti-logarithms of : 
 
 I. 1.0792. 2. 2.0793. 3. 2.0899. 4. 2.0934. 
 * 5. 2.0913. 6. 0.0913. 7. 4.0917. 8. 1.9652. 
 
 9. Show that the last paragraph of case {a) in prob. 5 might 
 read: make the number of figures in the integer part of the 
 number one more than the characteristic of the logarithm. 
 
 10. After dividing the difference between a given mantissa 
 and the next less tabular mantissa by the tabular difference, 
 why is the (juotient annexed instead of added to the figures 
 given by the table ? 
 
 Find the anti-logarithms of: 
 
 II. 2.9053. 12. 1.7126. 13, .3402. 14. 1.4612. 
 15. 3.0024. 16. 2.5832. 17. 3.7368. 18, .5505. 
 19. 2.5337, 20. 2.6193. 21. 1.8000. 32. .0971. 
 
 23. If the logarithms of a series of multipliers and divisors 
 be added, what logarithms must be regarded as negative ? 
 
 If the sum of any column be negative it may be made posi- 
 tive by adding one or more tens to it, and subtracting the same 
 number of units from the sum of the next column at the left. 
 
 24. Write under each other the logarithms of 3426, 4.003, 
 .00163, 324.5, 1.64, and then so add them as to find the log- 
 arithm of 3436 X 4.003 x ,00162 : 334.5 ; 1,64. 
 
242 PROGRESSIONS, INCOM. POWERS, LOGARITHMS. [VIII, Pus. 
 
 PrOB. 6. To DIVIDE A LOGARITHM WHOSE CHARACTERISTIC 
 IS NEGATIVE. 
 
 Write down as the- characteristic of the. quotient the number of 
 times the divisor is contained in that negative multiple 
 of itself which is equal to, or next larger than, the nega- 
 tive characteristic; 
 
 carry the positive remainder to the mantissa and divide. 
 
 E.g., 4.1234: 3 = (-6 + 2.1234): 3 = 3.7078. 
 
 So, 3.4770-3/2=8.4310/2 =4.2155. 
 
 Pros. 7. To compute by logarithms the products, quo- 
 tients, POWERS, AND ROOTS OE NUMBERS. 
 For a product: add the logarithms of the factors, and take out 
 
 the anti-logarithm of the sum. 
 For a quotient : from the logarithm of the dividend subtract 
 
 that of the divisor, and take out the anti-logarithm. 
 For a power : multiply the logarithm of the base by the expo- 
 nent of the power sought, and take out the anti-logarithm. 
 For a root: divide the logarithm of the base by the root-index, 
 
 and take out the anti-logarithm. 
 E.g., to find the value of (.01519 x 6.318 : 7.354)='^: 
 
 NUMBERS. LOGARITHMS. 
 
 .0151? 2.1815 
 
 X 6.318 +0.8006 
 
 : 7.254 -0.8605 
 
 2.1216 X 3/2 
 3.1824 
 and the numher sought is 0.001522. 
 
 PrOB. 8. To SOLVE THE EXPONENTIAL EQUATION A*=B. 
 
 Divide the logarithm of b by the. logarithm of the base A. 
 
 The quotient is x, the exponent sought. 
 For •.■ a'" = b, 
 
 .•. a; -log A = log B, whatever system of logarithms be used, 
 and a; = log s/log A. Q.e.d. 
 
6. 7, 8, § 5 ] LOGARITHMS. 243 
 
 QUESTIONS. 
 
 By logarithms, find the value of : 
 
 1. 575.25 X l.Oe"". 2. 675.25 x 1.03*". 
 
 3. ^'•00010098. 4. [76=/^-45V*]V6_ 
 
 2' •5" -85^ ^(97^- 9 ^) V13j^5 ^83.64 
 
 S"-?' • • 81 •1^572' VS-y'-ls"" -OSUS'- 
 
 9. v'. 000000003^591. 10. 4/236 • 140 ■ f 96: 215. 
 
 11. What power is 2 of 1.05 ? 3 of 1.04 ? 4 of 1.03 ? 5 of 1.02 ? 
 
 12. Given the logarithms of a, b, c, d, show how to find 
 the value of ( - a)^ ■ ( - b)V - 4c/ - d. 
 
 13. Divide the logarithm 3.2614 by 2, by 5, and by 6, in turn. 
 
 14. Find the cube root of log-^ 7.3550. 
 
 15. Find the cube of the fourth root of log-' 6.5448. 
 Find by logarithms : 
 
 16. The simple interest of $23.65 for 25 yr. 3 mo. at 7i ^. 
 
 17. The amount of $246 for 12^ years if the interest be 
 compounded annually at 8 per cent; if half-yearly at 4 per 
 cent; if quarterly at 2 per cent. 
 
 18. The 20th term of a geometric series if a — 5, r — l^. 
 
 19. If the number of births each year be one in forty-five 
 of the population, and of deaths one in sixty, in how many 
 years will the population double, taking no account of other 
 sources of increase or decrease? triple? quadruple? 
 
 20. From the formula for compound interest, a =p ■ (1 + r)*, 
 find an expression for t in terms of a, p, r. 
 
 21. How long must $1000 be at compound interest to 
 amount to $1191.02 at 6 per cent a year ? at 3 per cent half- 
 yearly ? at li per cent quarterly ? 
 
 22. In a geometric series, given ?=«/•"-*, then 
 
 n-l = logr?/a, n = l+ log^? - log,.*? = 1 + ( log Z - log «) • log r. 
 
 23. In a geometric series, a= 5, Z= 1280/6561, /• = 2/3:flnd«. 
 34. Solve the equations (5^)'' = 13^|; a^+yj'^-i^c*", 
 
244 INCOMMENSURABLE POWERS, LOGARITHMS. [Vm, 
 
 w 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Dif. 
 
 lO 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0394 
 
 0334 
 
 0374 
 
 43 
 
 11 
 
 0414 
 
 0453 
 
 0493 
 
 0531 
 
 0569 
 
 0607 
 
 0645 
 
 0683 
 
 0719 
 
 0755 
 
 38 
 
 12 
 
 0793 
 
 0838 
 
 0864 
 
 0899 
 
 0934 
 
 0969 
 
 1004 
 
 1038 
 
 1073 
 
 1106 
 
 35 
 
 13 
 
 1139 
 
 1173 
 
 1206 
 
 1239 
 
 1371 
 
 1303 
 
 1335 
 
 1367 
 
 1399 
 
 1430 
 
 33 
 
 U 
 
 1461 
 
 1492 
 
 1533 
 
 1553 
 
 1584 
 
 1614 
 
 1644 
 
 1673 
 
 1703 
 
 1732 
 
 80 
 
 15 
 
 1761 
 
 1790 
 
 1818 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 3014 
 
 38 
 
 16 
 
 3041 
 
 2068 
 
 3095 
 
 2122 
 
 3148 
 
 2175 
 
 2301 
 
 2327 
 
 3358 
 
 3379 
 
 36 
 
 17 
 
 3304 
 
 3880 
 
 3355 
 
 2880 
 
 2405 
 
 2430 
 
 3455 
 
 2480 
 
 3504 
 
 2529 
 
 25 
 
 18 
 
 3553 
 
 3577 
 
 3601 
 
 2625 
 
 3648 
 
 2673 
 
 3695 
 
 2718 
 
 3742 
 
 2765 
 
 34 
 
 19 
 
 2788 
 
 2810 
 
 2833 
 
 3856 
 
 3878 
 
 2900 
 
 3933 
 
 3945 
 
 2967 
 
 3989 
 
 33 
 
 30 
 
 8010 
 
 3033 
 
 3054 
 
 8075 
 
 3096 
 
 8118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 21 
 
 21 
 
 8222 
 
 8343 
 
 3263 
 
 8384 
 
 3304 
 
 3334 
 
 3345 
 
 3365 
 
 8385 
 
 3404 
 
 20 
 
 22 
 
 8434 
 
 3444 
 
 3464 
 
 3483 
 
 3503 
 
 8533 
 
 3541 
 
 3560 
 
 3579 
 
 3598 
 
 19 
 
 23 
 
 8617 
 
 3635 
 
 3655 
 
 8674 
 
 3693 
 
 8711 
 
 3729 
 
 8747 
 
 8766 
 
 3784 
 
 19 
 
 24 
 
 3802 
 
 3830 
 
 3838 
 
 3856 
 
 3874 
 
 8893 
 
 3909. 
 
 3937 
 
 3945 
 
 3962 
 
 18 
 
 25 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4083 
 
 4099 
 
 4116 
 
 4183 
 
 17 
 
 26 
 
 4150 
 
 4166 
 
 4188 
 
 4300 
 
 4316 
 
 4333 
 
 4349 
 
 4365 
 
 4381 
 
 4398 
 
 16 
 
 27 
 
 4314 
 
 4330 
 
 4346 
 
 4362 
 
 4878 
 
 4893 
 
 4409 
 
 4425 
 
 4440 
 
 4456 
 
 16 
 
 38 
 
 4472 
 
 4487 
 
 4502 
 
 4518 
 
 4538 
 
 4548 
 
 4564 
 
 4579 
 
 4594 
 
 4609 
 
 15 
 
 39 
 
 4624 
 
 4639 
 
 4654 
 
 4669 
 
 4683 
 
 4698 
 
 4713 
 
 4728 
 
 4743 
 
 4757 
 
 15 
 
 30 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4839 
 
 4848 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 14 
 
 81 
 
 4914 
 
 4938 
 
 4943 
 
 4955 
 
 4969 
 
 4983 
 
 4997 
 
 5011 
 
 5024 
 
 5038- 
 
 14 
 
 32 
 
 5051 
 
 5065 
 
 5079 
 
 5093 
 
 5105 
 
 5119 
 
 5133 
 
 5145 
 
 5159 
 
 5172 
 
 13 
 
 33 
 
 5185 
 
 5198 
 
 5311 
 
 5234 
 
 5237 
 
 5350 
 
 5363 
 
 5276 
 
 5289 
 
 5302 
 
 13 
 
 34 
 
 5315 
 
 5338 
 
 5840 
 
 5353 
 
 5366 
 
 5378 
 
 5391 
 
 5403 
 
 5416 
 
 5438 
 
 13 
 
 35 
 
 5441 
 
 5453 
 
 5465 
 
 5478 
 
 5490 
 
 5503 
 
 5514 
 
 5527 
 
 5539 
 
 5551 
 
 12 
 
 36 
 
 5563 
 
 5575 
 
 5587 
 
 5599 
 
 5611 
 
 5628 
 
 5635 
 
 5647 
 
 5658 
 
 5670 
 
 13 
 
 37 
 
 5682 
 
 5694 
 
 5705 
 
 5717 
 
 5729 
 
 5740 
 
 5753 
 
 5763 
 
 5775 
 
 5786 
 
 12 
 
 38 
 
 5798 
 
 5809 
 
 5821 
 
 5833 
 
 5843 
 
 5855 
 
 5866 
 
 5877 
 
 5888 
 
 5899 
 
 11 
 
 39 
 
 5911 
 
 5932 
 
 5933 
 
 5944 
 
 5955 
 
 5966 
 
 5977 
 
 5988 
 
 5999 
 
 6010 
 
 11 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 11 
 
 41 
 
 6128 
 
 6138 
 
 6149 
 
 6160 
 
 6170 
 
 6180 
 
 6191 
 
 6301 
 
 6212 
 
 6333 
 
 10 
 
 42 
 
 6283 
 
 6243 
 
 6253 
 
 6263 
 
 6274 
 
 6384 
 
 6294 
 
 6304 
 
 6814 
 
 6335 
 
 10 
 
 43 
 
 6835 
 
 6345 
 
 6355 
 
 6365 
 
 6375 
 
 6385 
 
 6395 
 
 6405 
 
 6415 
 
 6435 
 
 10 
 
 44 
 
 6435 
 
 6444 
 
 6454 
 
 6464 
 
 6474 
 
 6484 
 
 6493 
 
 6503 
 
 6513 
 
 6533 
 
 10 
 
 45 
 
 6533 
 
 6542 
 
 6551 
 
 6561 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6609 
 
 6618 
 
 10 
 
 46 
 
 6628 
 
 6687 
 
 6646 
 
 6656 
 
 6665 
 
 6675 
 
 6684 
 
 6693 
 
 6702 
 
 6712 
 
 9 
 
 47 
 
 6721 
 
 6780 
 
 6739 
 
 6749 
 
 6758 
 
 6767 
 
 6776 
 
 6785 
 
 6794 
 
 6803 
 
 9 
 
 48 
 
 6813 
 
 6821 
 
 6830 
 
 6839 
 
 6848 
 
 6857 
 
 6866 
 
 6875 
 
 6884 
 
 6893 
 
 9 
 
 49 
 
 6902 
 
 6911 
 
 6920 
 
 6928 
 
 6987 
 
 6946 
 
 6955 
 
 6964 
 
 6972 
 
 6981 
 
 9 
 
 50 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 9 
 
 51 
 
 7076 
 
 7084 
 
 7093 
 
 7101 
 
 7110 
 
 7118 
 
 7136 
 
 7135 
 
 7148 
 
 7153 
 
 9 
 
 52 
 
 7160 
 
 7168 
 
 7177 
 
 7185 
 
 7198 
 
 7202 
 
 7210 
 
 7318 
 
 7236 
 
 7385 
 
 8 
 
 53 
 
 7343 
 
 7251 
 
 7259 
 
 7267 
 
 7275 
 
 7384 
 
 7293 
 
 7300 
 
 7808 
 
 7316 
 
 8 
 
 54 
 
 7324 
 
 7832 
 
 7340 
 
 7348 
 
 7356 
 
 7864 
 
 7873 
 
 7380 
 
 7388 
 
 7396 
 
 8 
 
85] 
 
 LOGARITHMS. 
 
 346 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 
 Dif. 
 
 55. 
 
 .7404 
 
 7413 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 8 
 
 56, 
 
 .7482 
 
 7490 
 
 7497 
 
 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 7551 
 
 8 
 
 57 
 
 .7559 
 
 7566 
 
 7574 
 
 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 8 
 
 58 
 
 7634 
 
 7643 
 
 7649 
 
 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 7701 
 
 7 
 
 59 
 
 7709 
 
 7716 
 
 7733 
 
 7731 
 
 7738 
 
 7745 
 
 7753 
 
 7760 
 
 7767 
 
 7774 
 
 7 
 
 60 
 
 7783 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 7 
 
 61 
 
 7853 
 
 7860 
 
 7868 
 
 7875 
 
 7883 
 
 7889 
 
 7896 
 
 7903 
 
 7910 
 
 7917 
 
 7 
 
 62 
 
 7934 
 
 7931 
 
 7938 
 
 7946 
 
 7953 
 
 7959 
 
 7966 
 
 7973 
 
 7980 
 
 7987 
 
 7 
 
 63 
 
 7993 
 
 8000 
 
 8007. 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 7 
 
 64 
 
 8062 
 
 8069 
 
 8075 
 
 8083 
 
 8089 
 
 8096 
 
 8102 
 
 8109 
 
 8116 
 
 8123 
 
 7 
 
 65 
 
 8129 
 
 8136 
 
 8143 
 
 8149 
 
 8156 
 
 8163 
 
 8169 
 
 8176 
 
 8183 
 
 8189 
 
 7 
 
 66 
 
 8195 
 
 8303 
 
 8209 
 
 8315 
 
 8323 
 
 8228 
 
 8235 
 
 8341 
 
 8248 
 
 8254 
 
 7 
 
 67 
 
 8261 
 
 8367 
 
 8374 
 
 8380 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 6 
 
 68 
 
 8325 
 
 8331 
 
 8338 
 
 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8883 
 
 6 
 
 69 
 
 8388 
 
 8395 
 
 8401 
 
 8407 
 
 8414 
 
 8430 
 
 8436 
 
 8433 
 
 8439 
 
 8445 
 
 6 
 
 70 
 
 8451 
 
 8457 
 
 ^63 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 6 
 
 71 
 
 8513 
 
 8519 
 
 8535 
 
 8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 6 
 
 72 
 
 8573 
 
 8579 
 
 8585 
 
 8591 
 
 8597 
 
 8603 
 
 8609 
 
 8615 
 
 8631 
 
 8627 
 
 6 
 
 73 
 
 8633 
 
 8639 
 
 8645 
 
 8651 
 
 8657 
 
 8668 
 
 8669 
 
 8675 
 
 8681 
 
 8686 
 
 6 
 
 74 
 
 8692 
 
 8698 
 
 8704 
 
 8710 
 
 8716 
 
 8722 
 
 8737 
 
 8733 
 
 8739 
 
 8745 
 
 6 
 
 75 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 6 
 
 76 
 
 8808 
 
 8814 
 
 8820 
 
 8825 
 
 8831 
 
 8837 
 
 8843 
 
 8848 
 
 8854 
 
 8859, 
 
 6 
 
 ,7.7 
 
 .'78 
 
 8865 
 
 8871 
 
 8876 
 
 8882 
 
 8887 
 
 8893 
 
 8899 
 
 8904 
 
 8910 
 
 8915 
 
 6 
 
 ^93t 
 
 -8937 
 
 8932 
 
 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 6 
 
 79 
 
 8976 
 
 8982 
 
 8987 
 
 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 5 
 
 80 
 
 9031 
 
 9036 
 
 9042 
 
 9047 
 
 9053- 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 5 
 
 81 
 
 9085 
 
 9090 
 
 9096 
 
 9101 
 
 9106 
 
 9112 
 
 9117 
 
 9123 
 
 9128 
 
 9133 
 
 5 
 
 82 
 
 9138 
 
 9143 
 
 9149 
 
 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175 
 
 9180 
 
 9186 
 
 S 
 
 83 
 
 9191 
 
 9196 
 
 9201 
 
 9206 
 
 9213 
 
 9317 
 
 9222 
 
 9337 
 
 9232 
 
 9388 
 
 5 
 
 84 
 
 9243 
 
 9248 
 
 9353 
 
 9258 
 
 9363 
 
 9269 
 
 9274 
 
 9379 
 
 9284 
 
 9289 
 
 5 
 
 85 
 
 9294 
 
 9399 
 
 9304 
 
 9309 
 
 9315 
 
 9330 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 5 
 
 86 
 
 9345 
 
 9350 
 
 9355 
 
 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 5 
 
 87 
 
 9395 
 
 9400 
 
 9405 
 
 9410 
 
 9415 
 
 9430 
 
 9435 
 
 9430 
 
 9435 
 
 9440 
 
 5 
 
 88 
 
 9445 
 
 9450 
 
 9455 
 
 9460 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 , 5 
 
 89 
 
 9494 
 
 9499 
 
 9504 
 
 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 5 
 
 90 
 
 9542 
 
 9547 
 
 9553 
 
 9557 
 
 9563 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 5 
 
 91 
 
 9590 
 
 9595 
 
 9600 
 
 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9634 
 
 9628 
 
 9683 
 
 5 
 
 92 
 
 9638 
 
 9643 
 
 9647 
 
 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 5 
 
 93 
 
 9685 
 
 9689 
 
 9694 
 
 9699 
 
 9703 
 
 9708 
 
 9713 
 
 9717 
 
 9722 
 
 9737 
 
 5 
 
 94 
 
 9731 
 
 9736 
 
 9741 
 
 9745 
 
 9750 
 
 9754 
 
 9759 
 
 9763 
 
 9768 
 
 9773 
 
 5 
 
 05 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 5 
 
 96 
 
 9823 
 
 9827 
 
 9832 
 
 9836 
 
 9841 
 
 9845 
 
 9850 
 
 9854 
 
 9859 
 
 9863 
 
 4 
 
 97 
 
 9868 
 
 9872 
 
 9877 
 
 9881 
 
 9886 
 
 9890 
 
 9894 
 
 9899 
 
 9903 
 
 99C8 
 
 4 
 
 98 
 
 9912 
 
 9917 
 
 9921 
 
 9926 
 
 9930 
 
 9934 
 
 9939 
 
 9943 
 
 9948 
 
 9953 
 
 4 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 4 
 
246 PROGRESSIONS, I-NCOM. POWERS, LOGARITHMS. [Vm, 
 
 § 6. QUBSTIOTSrS POK REVIEW. 
 Define and illustrate: 
 
 1. A series; an arithmetic series; a geometric series; a 
 harmonic series. 
 
 2. A rising sej-ies; a falling series ; a continuous series. 
 
 3. The five elements of an arithmetic series ; of a geometric 
 series; the four elements of a harmonic series. 
 
 4. Arithmetic means ; geometric means.; harmonic means. 
 
 5. The major terms of a series; the minor terms. 
 
 6. A commensurable power ; an incommensurable power. 
 
 7. A base ; a logarithm ; an anti^ogarithm. 
 
 8. A system of logarithms ; a table of logarithms. 
 
 9. The characteristic of a logarithm; the mantissa. 
 
 10. Ifatural logarithms ; common logarithms. 
 
 11. Tabular numbers; tabular logarithms; tabular dif- 
 ferences. 
 
 Write and prove the formulae for : 
 
 12. The last term of an arithmetic series; the sum. 
 
 13. Tlie last term of a geometric series ; the sum. 
 
 14. The last terra of a harmonic series. 
 
 15. The sum of an infinite falling geometric series; the 
 value of a repeating decimal. 
 
 16. The arithmetic mean of two numbers; the geometric 
 mean; the harmonic mean. 
 
 State the analogies of the three progressions. 
 
 17. The difEerence of any two terms of an arithmetic series. 
 
 18. The ratio of any two terms of a geometric series. 
 Give a general rule, with reasons and illustrations, for : 
 
 19. Finding the other two elements of an arithmetic series, 
 when any three of them are given; of a geometric series. 
 
 20. Inserting means in an arithmetic series; in a geometric 
 series ; in a harmonic series. 
 
16.1 QUESTIONS FOR REVIEW. 247 
 
 State the principle, with proof, that relates to : 
 
 31. The product of incominensurable powers of the same 
 base; the quotient of two such powers. 
 
 32. An incommensurable power of an incommensurable 
 power of a base. 
 
 23. Tlie product of like incommensurable powers of difPer- 
 ent bases ; the quotient of two such powers. 
 
 24. What is the logarithm of unity to any base? of the 
 base itself ? of zero ? ,, 
 
 35. If the base be positive and larger than unity what is 
 the logarithm of a number smaller than unity ? of one larger 
 than unity ? How does the logarithm change as the number 
 increases ? 
 
 36. If the base be positive and smaller than unity what is 
 the logarithm of a number smaller than unity ? of one larger 
 than unity ? How does the logarithm change as the number 
 increases ? 
 
 37. Wh%J; is the logarithm of the product of two numbers ? 
 of the quotient of two numbers ? of the reciprocal of a 
 number? 
 
 38. What relatian has the logarithm of a power of a number 
 to that of the number ? the logarithm of a root ? 
 
 29. What relation have the logarithms of two numbers, 
 each taken to the other as base ? 
 
 30. What relation have the natural and the common 
 logarithms of the same number ? 
 
 Give the general rule, with reasons and illustrations, for : 
 
 31. Taking out a logarithm from the table, when the 
 number is found in the table ; when not so found, 
 
 33. Taking, out a number, from its logarithm, when the 
 logarithm is found in the table ; when not so found. 
 
 33. Define the characteristic and the mantissa of a loga- 
 rithm, and show what relation the characteristic has to the 
 position of the decimal point in the number. 
 
248 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX 
 
 IX. PERMUTATIONS, COMBINATIONS, AND 
 PROBABILITIES. 
 
 The different groups that can be made of two or more 
 things, without regard to order, are their comhinations; the 
 different groups that can be made of them, their order being 
 considered, are \he\Y permutations. 
 
 Two permutations are different when either the things them- 
 selves are different or their order of arrangement is different; 
 but two combinations are different only when at least one of 
 the things contained in one of them is not found in the other. 
 E.g., ab, ba, ac, ca, he, cb, are the six permutations of 
 
 the three letters a, b, c, taken two at a time ; 
 but ab, ba are the same combination, ac, ca are the same, 
 
 and be, cb are the same, 
 and in all, there are but three combinations. 
 So, abc, bac, acb, cab, bca, cba are the six permutations of 
 
 the three letters a, b, c. taken all together; 
 but there is only one combination. 
 So, of four letters, a, b, c, d, taken three at a time, there 
 
 are four combinations, abc, abd, ac4, bed; 
 and of each of them can be made six permutations, as above. 
 Nearly all investigations as to permutations and combina- 
 tions depend upon the following self-evident principle: 
 
 Ax. If there be a'group of m things and a group of n things 
 so related that either of the m things may be taken at random, 
 and then either of the n things may be joined to it, there are 
 mn ways in which such pairs may be made up, and no more. 
 E.g., if a boy have 5 apples and 6 peaches, he may form 30 
 
 different pairs of them, an apple and a peach; 
 for the first apple may go with either of the 6 p«aches, and so 
 
 with the others. 
 So, if 5 men enter a room with 6 chairs the first man has choice 
 
 of 6 chairs, the second man of the other 5 chairs, the 
 
 third man of the other 4 chairs, and so on; 
 
AS.] PERMUTATIONS AND COMBINATIONS. 249 
 
 and, while the first man can be seated in but six ways, the 
 first two men can be seated in 6 • ^ ways, the first three 
 men in 6 -5 -4 ways, and so on.' ♦ 
 
 QUESTIONS. 
 
 ^ 1. Which of the groups below are permutations and which 
 are combinations ? 
 
 the three-figure numbers that can be made from n figures, 
 the products of four factors, taken from ten given factors, 
 the parties of four that can be made up from six men, ■ 
 the ways twelve men can stand in a row, or in a ring, ■ 
 the ways of dividing six things among three men. . 
 " 2. What are the three combinations, in pairs, of a, b, c? 
 Make two permutations of each of these combinations. 
 So, of each of the combinations abc, abd, acd, bed make 
 six permutations, thus forming the twenty-four permutations 
 of a, b, c, d, taken in groups of three. 
 . 3. If permutations are to be made of the letters a, b, c, d, 
 how many choices are there for the first place ? 
 ' Make all possible permutations of two by annexing to each 
 of these first letters the other three letters in turn : how does 
 the number of choices compare with that for the first place ? 
 'To each of the couplets annex the two remaining letters 
 in turn, and form all the possible groups of three : how many 
 letters remain to annex to each group of three ? 
 ' ^4. How does the axiom apply in finding the number of 
 couplets? in finding the number of threes? 
 
 ' 5. Of the letters of the word thing, make all the possible 
 permutations of two letters; of three letters; of four letters. 
 
 6. Why can more permutations be made with the same 
 letters arranged in a row than- in a ring ? 
 
 7. Find the sum of all the four-figure numbers that can be 
 expressed by the figures 1, 3, 3, 4. 
 
 If all these numbers be written one under another, how 
 many times is each figure found in each column ? 
 
 Show that the si^m of the numbers formed as above is 
 divisible by the sum of the four figures involved. 
 
260 PERMUTATIONS, COMBINATIONS, PROBABILITIES. pX-Tna. 
 
 § 1. PERMUTATIONS. 
 
 Theoe. 1. TJie nitmker of permutations ofn tilings, all dif- 
 ferent, taken r at a time, is' n{n — l){n — 2)---{n — r + l). 
 For of n things taken singly, there are n choices and no more ; 
 i.e., 7jn = n; < 
 
 so, if each of the n things be followed in turn by each of the 
 
 n — 1 things that remain, there are formed n{n — l) 
 
 conplets, all different, 
 i.e., p^n = n{n — l); 
 so, if each of these w(w— 1) couplets be followed in turn bj 
 
 each of the n — 2 things thalj regaain, there are formed 
 
 n{n — l){n — 2) threes, all different, 
 «.e., P3W = w( w — 1) (w - 2) ; 
 and so for thi^ groups of four, of five,- • -otr, 
 i.e., yrn = n{n — l){n — 2)- • -{n-r + l). Q.B.D. 
 
 CoE. 1. The number of permutations of n things all differ- 
 ent, taken all together, is the product of all the integers from 
 1 to n, inclusive. 
 
 For ',• here r is n, and ra— r + 1 is 1, 
 .-. p„w=w(w-l)(?j-2). • .3-2-1. 
 
 This product is indicated by the symbol Iw or «1, and it is 
 TQa,dL_factonal n. 
 
 CoE. 2. If in each group of r things some, or all, may be 
 alike (permutations with repetition), thpn the number of such 
 permutations is nZ. 
 
 For •.• there is a choice of n things for the first place, then of 
 n things for the second place, and so on, 
 .'. Fr,rep. n = n-n-n- ■ -r times = re''. 
 Theoe. 2. The number of permutations of n things, taken 
 all together, tohen p things are alike, q things alike, r things 
 alike, and so on, is n\/p\ q\ r\- • • 
 
 For •.•^ ! permutations of the n things are formed by inter- 
 changing any^ things among themselves, while the 
 other things stand fast, if the p things be all diflerentj 
 but only one permutation if they be alike. 
 
».2,ii] PERMUTATIONS. 251 
 
 /. the whole number of permutations is ^! times larger if 
 the p things be all different than if they be alike; 
 and •.•v„n~n\ if the n things be all different, [tb. 1 cr. 1. 
 
 .•. the number of permutations of n things, jb alike, is « \/p\. 
 So, if q things be alike, r things alike, and so on, 
 
 •■■ PH.paUke,aaIJke,7-aUke...»i = »»!/^J §'! ^!- •• Q.E.D. 
 
 QUESTIONS. 
 
 ' 1. In making up permutations, why are there fewer choices 
 for each successive place to be filled ? at what rate does tlie 
 number of such choices decrease? 
 
 What relation has the number of places filled, at any stage 
 »of the process, to the number of choices for the next place? 
 
 ■ 2. Why is the number of permutations of n things tajjen 
 ra — 1 at a time the same as the number taken n at a time ? 
 
 ■ 3. Find the number of permutations of 10 things, all 
 different, 3 at a time; 5 at a time; 7 at a time; all together. 
 
 4. How many permutations, 3 letters at a time, can be made 
 up from the word mucilage ? from the word formula? 
 
 5. Of how many unlike things, taken all together, are there 
 720 permutations ? 5040? 40320? 
 
 6. What is the value of m!/(w-1)!? of w!/(w-2)!? 
 of m!/(w-3)!? ot nl/{n-r)l? ot nl/rl 
 
 7. The number of permutations of n things, r at a time, 
 plus r times their number r — 1 at a time, is 
 
 n{n — l)'--{n-r + 2){n + l), i.e., Vrn + r-Vr.in = Fr{n + l). 
 '8. In how many ways can 8 men stand in a row ? n men ? 
 In how many ways can 8 men sit around a table? n men? 
 
 9. In how many ways can 5 prizes be given to 5 boys ? 
 
 10. A man has three ways of going to his place of business; 
 in how many different ways can he plan his route for six days ? 
 
 11. Find the number of permutations of ten flags if three 
 be red and seven blue ? if two be red, three white, five blue? 
 
 12. How many permutations can be formed from the word 
 London? Washington? Mississippi? Constantinople? 
 
252 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX, Th- 
 
 §2. COMBINATIONS. 
 
 Thbob. 3. The number of combinations of n things all 
 different taken r at a time, is the quotient of the number of 
 such permutations divided by r! 
 
 For •.• any combination of r different things gives-rl permu- 
 tations of r things, [th. 1 cv. 1. 
 .-. o^n — Vrn/rl q. b. d. [th. 1. 
 
 Cor. 1. The number of combinations of n things taken n—r 
 at a time, is the same as their number taken r at a time. 
 For each group of r things leaTes a, group of n — r things. 
 
 Cor. 3. The number of combinakons of n things is greatest 
 when r is the integer nearest in value to in. 
 ^ n n n — l \' n n — 1 n — 2 
 
 n n — l n—r+1 n n — r + 1 n — r 
 
 Orn = - ^ , C,+iW = - 
 
 "13 r ' -•■+i'"-i r r + l' 
 
 .'. G^n is greatest when {n — r + l )/r >l>{n — r)/{r + l), 
 i.e., when i{n + l)>r>i{n — l), if w be evenj q.e.d. 
 and c^n, Or+i« are equal to each other, and greater than 
 the other terms of the series when 
 {n-r + l)/r>l = (n-r)/(r+l), 
 t.e., when r=i{n — l), if ra be odd. q.e.d. 
 
 Cor. 3. If in each group of r things some, or all, may be 
 alike (combinations with repetitions) then the number of such 
 combinations of n things is n(n + l){n+2)- • •{n+r — l)/rl 
 For, let a,b, c, — I, «i, be any 71 + 1 different things, form 
 
 all possible groups of two, (w + 1) • n/2 ! in all, 
 and replace am by aa, bm by bb, cm by cc, • • • Im by U; 
 then m vanishes, and there result n{n + 1)/3 ! combinations, 
 
 with repetitions, of the n things a, b, €,• • • I. 
 So, let a, b, c, ■ • • I, m, n be any « + 3 different things, form 
 
 all possible groups of three, «(» + l)(« + 3)/3! in all. 
 
3.S2] COMBINATIONS. 253 
 
 and replace amn by aaa, bmn by Ibl, •", , Imn by III, 
 dbm by aba, acm by aca, • • -, Ikm by Ikl, 
 abn hj abb, acn hj ace, •••, Ikn hj Ikk; 
 then m, n vanish, and there result n{n + l)(n + 2)/3l com- 
 binations, with repetitious, of the n things, a,b,c,--- 1, 
 and so for groups of four things, of five things, • • • of r things. 
 
 QUBSTiosrs. 
 
 1. How many things are taken at a time, if the number of 
 permutations and of combinations be the. same? 
 
 2. Take the letters a, b, c, d, e and join to each of them 
 every letter that follows it in the list, thus making all the 
 groups of two; form the threes by joining to each couplet 
 every letter that follows all its elements; so, the fours; the fives, 
 
 3. How many triangles can be formed by joining three ver- 
 tices pf a polygon of n sides? how many pentagons by joining 
 five vertices ? With six points construct the fifteen pentagons. 
 
 4. If an indefinite line be cut at four points, how many seg- 
 ments are formed ? at six points ? at ra points ? 
 
 5. Show that the formula for the number of combinations 
 of n things taken /• at a time,. may be written n\/r\{n — r)\ 
 
 Hence prove that CrW = o„_rW. 
 
 6. Find the number of combinations of 10 things all differ- 
 ent taken 3 at a time; 5 at a time; 7 at a time. 
 
 The number of groups 3 at a time is the same as their num- 
 ber 7 at a time; and their number 5 at a time is greatest of all. 
 
 7. If there be four straight lines in a plane, no two parallel 
 and no three meeting in a point, how many triangles are 
 formed ? if five lines ? if to lines ? 
 
 8. If there be six points in a plane, no three colinear, and 
 lines join them so as to form the greatest possible number of 
 figures of the same kind, what will those figures be? 
 
 9. If there be seven points in a plane, and they be joined so 
 as to make triangles and quadrangles, of which sort is there 
 the greater number ? Draw the quadrangles. 
 
 10. Apply cor. 3 to find the number of terms in {a+h)\ 
 
254 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX,Th. 
 
 Theok. 4. If there be n tilings all different, and if p, q, he 
 any positive integers whose sum is n, then there are n\/p\q\ 
 ways in which these n things- can be made up into sets of p 
 things and sets of q things. 
 
 For a group of q things is left for every group of p things taken, 
 and CpM = n]/p\ {n -p) l = n\/plq] Q. e. d. [th. 3 cr. 1. 
 
 Cob. 1. If n =p -{-q + r, the number of sets of ji things, q 
 things, and r things, is n\/p\q\r\ [above. 
 
 For •.• n things give " n]/pl{q + r)l sets of p things and the 
 
 same number of sets of q + r things, 
 and •.■ any set of q + r things gives (q + r)l/q\r\ sets of 
 q things and the same number of sets of r things, 
 .'. the whole number of sets is 
 
 n\/pl{q + r)lx{q + r)l/qlrl, i.e., nl/plqlrl q.e.d. 
 So, when n is the sum. of more than three integers. 
 B.g., if 12 recruits be divided into squads of three, four, and 
 five, the number of such squads is 12!/3!4!5!; 
 if divided into three equal squads and sent to different 
 
 companies, their number is 13!/(4!)'. 
 if simply divided into three equal squads, 13!/3!(4!)'. 
 
 CoE. 2. The number of combinations of n different things 
 
 in sets of p things, q things, and so on, when n=p+q-\ , 
 
 IS the number of permutations of the n things taken all to- 
 gether, when p things are alike, q things alike, and so on. 
 For the j!?! permutations of ^ different things are but a single 
 
 combination, 
 and, if the p things be like things, but a single permutation. 
 
 CoK. 3. The value of the quotient n\/p\q\r\- • • is greatest 
 when no one of the numbers p,q,r,'-- exceeds any other of 
 them by more than a unit. 
 
 For, let p = q + 2i', [hyp. 
 
 ih.Qn:-p\q\=^{q+2)\q\={q + \)\{q + ^)q\>{q + l)\{q + \)\, 
 .: the divisor is smallest and the quotient largest when 
 p, q, r • • - are nearest to equality. 
 
4.83] COMBINATIONS, 255 
 
 QUESTIONS. 
 
 1. There are as many combinations of n things taken ?• at a 
 time as there are permutations of n things, all taken, when r 
 things are of one kind and n—r things of another. 
 
 2. A cent, a dime, a quarter of a dollar, a half dollar, and a 
 dollar are each claimed by two boys : in how many different 
 ways can the coins be divided between them ? 
 
 3. In the proof of theor. 4, why is the number of combina- 
 tions in sets of p things and q things the same as the number 
 when p things are taken at a time ? 
 
 4. Pi'ove theor. 4 cor. 1 when n=p-\-q + r + s^t. 
 
 5. Whatever n may be, three numbers p, q, r can be found 
 whose sum is n and which difEer from each other by not more 
 than a unit; but if w be a multiple of 3 and^, q, r be not 
 taken all equal, then the difference between some two of them 
 is at least 3. Generalize this proposition. 
 
 6. Show that the number of permutations that can be made 
 from all of 2n things that are of two kinds is greatest when 
 there are n things of each kind. ' * 
 
 1. How can 18 thmgs, be divided among 5 persons in the 
 greatest number of ways, each person receiving the same num- 
 ber of things at each distribution ? 
 
 8. In the expansion of {x+yY*^, what term has the greatest 
 coeificient ? what, term of {x + y)^ ? 
 
 9. Of the combinations of eight letters a, b,c,---, taken 
 four at a time, how many contain a ? not a ? both a and b ? 
 a and not 5? neither a nor b? a, b, and c? a or b or <?? 
 neither a nor b nor c ? 
 
 10. In finding the product (a;+a) •(« + «)•• -w factors, the 
 various terms in the several partiaV products are all the possi- 
 ble permutations of n letters taken all together, wherein part 
 are u'b and the rest afs. 
 
 The coeflBcient of a;" is P„»i„aiike; that of x"-% P„w„_i ^j^,; 
 that of x"-V, P»M«-2 alike, 2auke; and so on. 
 Hence prove the binomial theorem. 
 
256 PERMUTATIONS, COMBINATIONS, PROBABILITIES. PX. Th. 
 
 Thuor. 5. If there le n sets of things, the first set containing 
 p things, the next q things, and so on, and if combinations of 
 n things be made uj} by taking one thing from each set, then 
 the number of such combinations is the product p-q-r- •• 
 Foi each of the p things may be joined to each of the q things, 
 each of these pq pairs to each of the r things, and so on. 
 
 Q.E.D. [ax. 
 
 OoK. 1. The number of combinations made by taking h oj 
 the p things, j of the q things, h of the r things, and so on, is 
 the product of the number of combinations of p things taken 
 h at a time, of q things taken j at a time, and so on, 
 
 CoE. 3. With the data of cor. 1, the number of permutor 
 tions possible is c^p •Gjq-G^.r- • •{k+j + k-\ ) ! 
 
 OoE. 3. The number of possible comtinations of some or all 
 of p + q-^r+ •• • things, of which p things are alike, q filings 
 
 alike, and so on, is {p + l)-{q + \)-(r + l) 1. 
 
 For thejj things may be treated in jo + 1 ways; 
 i.e., none, or one, or two, • • • or^ of them may be talten, 
 and each of these (^ + 1) dispositions of the p things may be 
 joined to each of the {q + 1) dispositions of the q things, 
 each of these pairs may be joined to each of the {r + \) dis- 
 positions of the r things, and so on, 
 and the whole number is ( j9 + 1) ■ (g 4- 1) • (?• + 1) . • • ; 
 but ".• this includes the case when no thing is taken from any 
 group, and this case can not be counted, 
 .\c={p + l).{q + l)-{r + l) 1.' 
 
 CoE. 4. The number of possible combinations of n different 
 things, taken some or all at a time, is 2" — 1. 
 For •.• each thing may be either taken or left, 
 and either disposition of one thing may be followed by either 
 disposition of every other thing, 
 ••. the whole number of combinations including that where 
 no thing is taken is 3 • 3 • 3 • • • m times, = 2". [ax. 
 .•.0 = 3"-l. Qj, jj 
 
5'^^] COMBINATIONS. 357 
 
 OTESTIONS. 
 
 I. Write out all the measures, prime and composite, of 6; 
 of 30; of 3310; of abc; of abed; of a*-a;*. 
 
 3. Out of 12 democrats and 16 republicans how many comr 
 mittees can be made up, each consisting of 3 democrats and 4 
 republicans? how many committees of seven can be made up 
 with the condition that each committee shall contain at least 
 two men of each party ? 
 
 3. How many different signals can be made by hoisting 6 
 differently colored flags one above another, when any number 
 of them may be raised at once ? 
 
 4. If a, 5, c, • • • be w different prime numbers : find the num- 
 ber' of different measures of the product a^-V^-^'C^-*- • . 
 
 5. If all groups of letters were words, how many words 
 composed of two consonants and one vowel could be made 
 from our alphabet of five vowels and twenty-one consonants ? 
 
 6. The number of permutations of n things of two kinds 
 taken « at a time, when some or all are alike, is the same as 
 the number of combinations of n different things, some or all 
 at 'a time; hence the sum of the coefficients of {a + xf is 3". 
 
 7. From six apples, five pears, and four plums, how many 
 selections of an apple, a pear, and a plum can be made ? 
 
 8. From a party of six ladies and seven gentlemen, how 
 many groups each of four ladies and four gentlemen can be 
 formed ? how many sets of four couples for a quadrille ? 
 
 9. Given m things of one kind and n things of another, find 
 how many permutations can be made by taking r things of the 
 first set and s things of the second. 
 
 10. How many different sums of money can be formed from 
 a cent, a half -dime, a dime, three half-dollars, five dollars ? 
 
 II. A signal staff has five arms, each of which may assume 
 four positions : how many signals can be made ? 
 
 13. If of p+q + r things, p things be alike, q things alike, 
 and the rest all different, the whole number of combinations 
 possible is (p -)- 1) . (g + 1) • S' - 1. 
 
258 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX. 
 
 §3. PKOBABILITIES. 
 
 The theory of probabilities is concerned with classes of 
 
 things about whose indiyiduals there is uncertainty; it might 
 
 well be called the doctrine of averages. It seeks to show what 
 
 likelihood there is that a particular event may happen or fail, 
 
 the basis of computation being such known facts as these : 
 that the event considered must happen once, or a fixed 
 
 number of times, in a given number of possible cases; 
 that it cannot happen more than such number; 
 that in the past Such events have happened with such, or 
 
 such, a degree of frequency. 
 
 E.g., if the twenty-six letters of the alphabet be written singly 
 on cards of the same size and shape, and these cards be 
 thrown into a box; 
 
 then, in twenty-six successive drawings, each letter will be 
 drawn out once, and but once, and at the first drawing 
 one letter is as likely to come out as another. 
 
 So, as shown by the Institute of Actuaries' tables, of 100,000 
 healthy boys of ten 96,323 have reached the age twenty, 
 and 609 have died between twenty and twenty-one; 
 1460 have reached ninety, and 408 have died within a 
 year thereafter; and what has happened in the past 
 may be expected in the future with ratios very slightly 
 changed. 
 The probability of an event is the ratio of the number of 
 
 cases in which the event happens, /a«;o;'a6fe cases, to the whole 
 
 number of cases considered. 
 
 E.g., in the example above, the probability that the letter A be 
 drawn out, at the first drawing, is the ratio 1/26; that 
 A be not drawn, 25/26 ; that one of the five vowels be 
 BO drawn, 5/26; that neither of the vowels be drawn, 
 21/26; that one of the consonants be drawn, 21/26,. 
 
 So, barring special conditions of health and occupation, the 
 probability that a certain boy of ten live to the age 
 of twenty is .96223; to the age of ninety, .0146; that 
 he die before twenty, .03777; before ninety, .9854. 
 
J«l PROBABILITIES. 259 
 
 QtTESTIONS. 
 
 1. If the twenty-six letters of the alphabet be written on 
 separate cards, what is the probability 
 
 that X be first drawn ? that x be not drawn ? 
 that either x or y be drawn ? that neither x nor y be drawn ? 
 that either a; or ^ or « be drawn ? that neither x nor y nor 
 z be drawn ? 
 
 2. How many different pairs of letters are there ? [th. 3. 
 If two letters be drawn at a time what is the probability of 
 
 drawing a particular pair ? of not drawing that pair ? 
 
 3. How many Ways are there of drawing two letters in suc- 
 cession ? three letters ? four letters ? [th. 1. 
 
 What is the probability of drawing first x, then b ? a, B 
 without regard to order ? A, then b, then o ? A, b, c without 
 regard to order? A, then B, then c, then d ? A, b, c, D with- 
 out regard to order? 
 
 4. If of men of A's age one in eight die in five years there- 
 aiter, and of men five years older one in seven die, what is the 
 probability that A will die within five years ? that he will live 
 at least five years ? that he will die within ten years? that he 
 will live ten years? that he will die within fifteen years ? 
 
 5. If of 588 men of sixty, 17, on an average, die in a year, of 
 the 571 men of sixty-one 18 die; of the 553 men of eixty-two 
 19 die; of the 534 men of sixty-three 30 die; and of the 514 
 men of sixty-four 21 die, what is the probability that a man of 
 sixty lives one year? two years? three years? four years ? five 
 years ? that a man of sixty-one dies within a year ? two years ? 
 three years ? four years ? that a man of sixty-two lives till he 
 is sixty-five ? that he dies before he is sixty-five ? 
 
 6. In throwing a single die what is the probability of a 
 six ? not a six ? a six or an ace ? neither a six nor an ace ? 
 
 7. If a boy with three red marbles in his pocket, five white, 
 and seven blue ones, take out one at random, what is the 
 probability that it is blue ? not blue ? a particular blue one ? 
 not that blue one ? red ? white ? either red or white? 
 
260 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX, Th. 
 SIMPLE PROBABILITIES. 
 
 If the probability of a single event be considered, it is the 
 simple probability of the event; and the sum of the probabil- 
 ities for and against such event is always unity, i.e., certainty. 
 E.g., that the letter A be drawn out at the first drawing the 
 
 probability is 1/36, that it be not drawn 25/36, 
 and 1/26 + 35/26 = 1. 
 
 Thboe. 6. If there be two or more events that are mutu- 
 ally exclusive, the probability that some one of them shall 
 happen is the sum of their separate probabilities. 
 E.g., the probability that some one of the five vowels shall be 
 drawn is five-fold that of one of them. 
 The probability of an event is not the same to every man; 
 to each one it depends on his knowledge of the facts. 
 E.g., To one who knows, of a horse, only that he has been a 
 frequent winner, the probability of his winning at a 
 race to-day may be high; 
 but to the groom, who knows that he has gone lame, his de- 
 feat is almost certain. 
 
 PEOBABLE VALUES. 
 
 The probable value of a sum of money payable at some 
 future time on conditions whose fulfilment is uncertain is 
 the product of the sum due by the probability of receiving it. 
 
 This principle is of special importance in life-insurance. 
 E.g., to find the cost of insuring a man of twenty for a year: 
 Were death during the year certain and payment made at its 
 end, the premium would be the present worth for a 
 year, at an agreed rate of interest, of the face of the 
 policy; and at four per cent the premium on $1000 
 would be $961.54; 
 
 but •.* on an average, 609 men out of 96223 die between 
 twenty and twenty-one, 
 
 ,". the probability of death, and of the consequent pay- 
 ment of the money, is 609/96223, 
 and $961.54 x 609/9C323 =$6,086, the net cost for $1000. ' 
 
6, S3] PROBABILITIES. 261 
 
 QUESTIONS. 
 
 1. If each letter of the alphabet when drawn out be replaced 
 before another drawing, how many possible ways are there of 
 drawing two letters in succession? three letters? the same 
 letter twice ? three times ? [th, 1 cr. 2. 
 
 2. With two dice what is the probability of throwing a four 
 and a three ? a double four ? a three, a four, and a five, with 
 three dice ? three fours ? 
 
 3. What is the probability of throwing exactly 10 in a 
 single throw with three dice ? 12 ? 15 ? 18 ? 20 ? less than 8 ? 
 
 Are the probabilities the same to a bystander as to a player 
 who has honest dice ? loaded dice ? 
 
 4. If a man know that he is to receive a sum of money that 
 is expressed in dollars by a three-figure number made up of 
 the digits 3, 5, 7, but know not the order of the digits, what 
 is the value of his expectation ? 
 
 5. A friend is one of tw:o hundred passengers on a ship that 
 carries a crew of a hundred men, and it is reported that one 
 man was lost during the voyage; what is the probability, to me, 
 that it was my friend ? later it is reported that it was a pas- 
 senger; what is the probability now ? and still later the name 
 Johnson is given, my friend's name, but there were three 
 Johnsons aboard; what now? what, to the ship's surgeon ? 
 
 6. If of 1000 boys of ten 956 live to be twenty-one, what 
 is the present value of 110,000 to be paid on his twenty-first 
 birthday to a boy now ten, the amount of $1 at compound 
 interest for eleven years, at four per cent, being $1,53945 ? 
 
 So, if the money earn five per cent, the amount of $1 for 
 eleven years being 11.71034 ? 
 
 So, if it earn but three per cent, the amount being $1.38423 ? 
 
 7. If p stand for one payment, r for the rate of interest, 
 h,k>h>h> for tlis probabilities of living one, two, three, four 
 years; find v, the present value of an annuity to run four 
 years, or till previous death. 
 
 Make the problem general by writing n years, and Z,, ?„ • • ■!„ 
 for the probabilities of living one, two, • • •, m years. 
 
263 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [IX Ta. 
 JOINT PROBABILITIES. 
 
 The probability of the simultaneous occurrence of two or 
 more independent events is their Joint probability. 
 E.g., that the letter A be drawn and an ace be thrown. 
 
 Thboe. 7. If there be two or more independent events such 
 that the simple probability of the first is m/n, that of 
 the second, m' /n', and so on, then their joint probability 
 is the product m/n • m'/n' • • • 
 
 For •.• the first event happens m times out of n, the second 
 m' times out of n', and so on, [hyp. 
 
 .*. of any nn' joint events, mn' are favorable to the first 
 event, 
 
 and of these mn' joint events favorable to the firat event, 
 
 mm' are favorable to the second event, 
 i.e., of 7in' joint events, mm' are favorable to both events, 
 
 /. the joint probability of the two events is mm'/nn'; 
 and so if there be three or more events. q.e.d. 
 
 B.g., that the letter a be drawn and an ace be thrown, the 
 probability is 1/26 • 1/6, 
 that A be drawn and an ace not thrown, 1/26-5/6, 
 that A be not drawn and an ace be thrown, 25/26-1/6, 
 that A be not drawn and an ace not thrown, 25/26-5/6; 
 
 and the sum of all these products is unity. 
 
 So, the probability that a be first drawn and then b is 1/26' 
 if A be replaced after the first drawing, 
 
 and it is 1/26-1/25, i.e., 1/650, if A be not replaced. 
 
 So the probability that A, aged ninety, and B, aged twenty, 
 shall both die within a year is 408/1460 - 609/96323, 
 that both live the year, 1052/1460-95614/96223, 
 that A lives and B dies, 1052/1460-609/96223, 
 that A dies and B lives, 408/1460-95614/96223; 
 
 and the sum of all these products is unity. 
 
6,18] PEOBABILITIES. 263 
 
 QUESTIONS. 
 
 1. If a bag hold three red balls, five white, and seven blue 
 ones, find the probability of drawing three red balls in suc- 
 cession. Show that this probability is the same as that of 
 drawing the three red balls all at once. 
 
 3. A bag holds m white balls and n black ones; the proba- 
 bility of drawing first a white ball and then a, black one, and 
 so on till all the balls left are of one color, is the same as that 
 of getting all the white balls in a single drawing of m balls. 
 
 3. A man on a journey must make four connections to get 
 through in time: if the probability of making each connection 
 be 3/4, what is the probability of making them all ? 
 
 4. A man of thirty marries a wife of twenty-five: if of 93 
 persons of twenty-five 90 reach thirty, 26 reach seventy-five, 
 and 14 reach eighty, what is the probability of their celebrate 
 ing a golden wedding ? 
 
 5. On an average A speaks the truth three times out of 
 four, and B nine times out of ten: what is the probability that 
 both will assert a fact known to them both ? that both will 
 deny it? that their statements will be contradictory? that one 
 or the other of these cases will occur ? ^ 
 
 6. The probability that A can solve a certain problem is 
 2/5, that B can solve it 2/3: if both try it, what is the prob- 
 ability of its being solved? what, that A succeeds and B 
 fails ? that A fails and B succeeds ? that both succeed ? 
 
 7. In one purse are ten coins, a spvereign and nine shillings; 
 in another purse are ten coins all shillings; nine coins are 
 taken from the first purse and put in the other, then nine coins 
 are taken from the second purse and put in the first: what is 
 the probability that the sovereign is still in the first purse ? 
 
 8. If the probability that a ship will not meet a gale be 3/4; 
 that, if it meet one, it will not be disabled, 5/6; that, if dis- 
 abled, it will be kept afioat b.y the pumps, 1/2; that, if the 
 pumps fail, the passengers will all escape by the boats, 1/3; 
 find the probability of loss of life by shipwreck. 
 
264 PERMUTATIONS, COMBINATIONS, PROBABiATIES. [IX, Ta 
 
 Cob. 1. If pie the simple proldbility of the occurrence of 
 an event in one trial, then p^ is the probability that it occurs 
 in all of n successive trials. 
 
 E.g., the probability that the letter A be drawn twice in suc- 
 cession, being replaced after the first drawing, is 1/36^ 
 
 CoK. 3. If the probability of the occurrence of an event in 
 one trial be p and of its failure q, then the probability of its 
 occurrence r times in n trials is the {n — r + l)th term of the 
 expansion of {p + q)\ 
 
 For the probability that the event occurs n times in succession 
 
 iSjjj", [cr. 1. 
 
 that it occurs n — 1 times and fails once, is the product ^""'•g 
 
 taken as many times as there are permutations of n 
 
 things with w — 1 of them alike, i.e., n-p^~^q, [th. 6. 
 
 that it occurs 7i — 2 times and fails twice, is the product 
 
 pn-:i.g2 ^;aten as many times as there are permutations 
 
 of n things with n — 2oi them alike and 2 alike, 
 
 i..e., ^n{n — l)-p'^~''-q^,' and so on; 
 
 that it occurs r times and fails n — r times, is the product 
 pT.q^-'' taken as many times as there are permutations 
 of n things with r of them alike and n — r alike, 
 «.e., n(n-l)---in-r+l)/rl-p''-q^-''. q.e.d. 
 
 OoE. 3. The probability that an event occurs at least r times 
 in n trials is the sum of the first n—r+1 terms in the expan- 
 sion of {p+q)". 
 
 Cob. 4. That value of r for which the probability is greatest 
 
 is the largest integer in q{n + \). 
 
 Sot the expression n{n — l){n — 2)--'{n — r+X)/r\-p'^~^-q'^ 
 
 J. 4. I, n-r + 1 q ^^^n-r q _,, „ „ 
 
 IS greatest when > 1 > — -r • -, [tn. 3 cr. 2. 
 
 o r p r + 1 p *■ 
 
 i.e., when nq — rq + q> rp, and nqJrq>r{p + q), 
 
 and when rp+p>nq-rq, and r{p + q)>nq-p; 
 
 mdL:-p + q = l, [hyp. 
 
 .•. it is greatest when q{n + l)>r>q{n + l) — l. q.e.d 
 
6. §31 PROBABILITIES. 265 
 
 QUESTIONS. 
 
 L If the probability that a man of fifty live to be eighty be 
 1/5, what is the probability that of six men of fifty, three at 
 least reach eighty ? four at least ? five at least ? all of the six ? 
 
 2. If on an average, of the ships engaged in a certain trade, 
 nine out of ten return safely, find the probability that out of 
 five ships expected three come into port. 
 
 3. In how many trials is there an even chance of throwing 
 double sixes with two dice ? a single six with one die ? 
 
 4. Two players A, B, of equal skill are interrupted in a game 
 when A wants two points of winning and B three: show that 
 the prize should be divided in the ratio 11/5. 
 
 5. The probability that a man will die within a year is 1/8; 
 that his wife will die, 1/10; that his son will die, 1/60: if all 
 three, or any two of them, be living at the end of the year 
 they are to receive $10,000 in equal shares; what is the value 
 of the expectation of each of them, interest at five per cent ? 
 
 6. At simple interest, five per cent, find the present value 
 of an annuity of $200 to run two years, and contingent on 
 the joint lives of two persons whose probabilities of living for 
 the next two years are 76/77, 74/75 for the first person, and 
 66/76, 65/75 for the other, the annuity being payable only if 
 both be living; payable if either be living. 
 
 7. Three men A, B, C, throw a die alternately in the order 
 of their names, and whoever first throws a five wins $182; 
 show that their expectations are $72, $60, $50. 
 
 8. It is a question whether A has been elected; B tells C 
 that D told him that A was elected, but thinks it an even 
 chance whether D said elected or-not elected. A is elected if B 
 and D both spoke truly or both falsely: find the probabilities. 
 
 9. If of thirteen aldermen at dinner the probabilities of 
 living a year be 13/14, 14/15, 15/16, 16/17, 17/18, 18/19, 
 19/20, 20/21, 21/22, 22/23, 23/24, 34/25, 25/26; what is the 
 probability that all of them live a year? what, that some one 
 of them dies within a year ? what, if there be but twelve men? 
 
266 PERMUTATIONS, COMBINATIONS, PROBABILITIES. PX, 
 
 § 3. QUESTIONS FOE EEVIEW. 
 Defiae and illustrate : 
 
 1. Permutations; permutations of n different things taken 
 r things at a time ; permutations with repetitions. 
 
 2. Combinations; combinations of n different things taken 
 r things at a time ; combinations with repetitions. 
 
 3. A factorial number. 
 
 4. The probability of an event ; simple probability. 
 
 5. Probable values ; the joint probability of two events, 
 
 6. State the fundamental principle of permutations and 
 combinatfons. 
 
 7. Show how to write out the permutations of n letters in 
 groups of two letters; of three letters; • ■ • of r letters. 
 
 8. Show how to write out the combinations of n letters in 
 groups of two letters; of three letters; • • • of r letters. 
 
 State and prove the general rule for finding : 
 
 9. The number of permutations of n things, all different, 
 in groups of r different things; 
 
 all together; 
 
 in groups of r things with repetitions allowed. 
 
 10. The number of permutations of n things, all together, 
 with'^ things alike, q things alike, r things alike, and so on. 
 
 11. The number of combinations of n things, all different, 
 in groups of r different things ; 
 
 in groups of r things with repetition allowed ; 
 
 in groups of p things, q things, r things, and so on. 
 
 13. The number of combinations, in groups: 
 
 of one thing from each of n sets of things that contain p 
 things, q things, r things, and so on; 
 
 of h things out of a set of jo things, / things out of a set ol 
 q things, Ic things out of a set of r fhings, and so on; 
 
 of some or all of j3 + g' + r+ • • • things, when p things are 
 alike, q things alike, r things alike, and so on; 
 
 of some or all of n different things. 
 
14.] QtJESTIONS EX)R REVIEW. 2&7 
 
 13. The value of r that makes CyW the greatest. 
 
 14. The relations of p, q, r,- • • that give the greatest 
 number of combinations of n things in sets of p things, q 
 things, r things, and so on. 
 
 Prove that: 
 
 15. Of n different things there as many combinations' in 
 groups ot n — r things as in groups of r things. 
 
 " 16. Of n different things there are as many combinations 
 in sets of p th'ings, of q things, of r things, and so on, as 
 there are permutations of n things taken all together, when 
 p things are alike, q things alike, r things alike, and so on. 
 
 As deductions from the principles established in this chapter: 
 
 17. Prove the binomial theorem. 
 ' 18. Find the number of terras in the expansion of a power 
 of a binomial. 
 
 19. Find the term of the expansion whose value is greatest. 
 State and prove the rule for finding the probability: 
 
 20. That some one of n mutually exclusive events will 
 occun 
 
 21. That two or more events of known probability will 
 occur jointly. 
 
 23. That an event will occur in all of n successive trials. 
 
 23. That it will occur exactly r times in n trials. 
 
 24. That it will occur at least r times in n trials. 
 
 25. State and prove the rule for finding the most probable 
 number of successes in n trials of things of known proba- 
 bility. 
 
 2 6. Show how far the doctrine of probabilities can be applied 
 in any individual case; and where it fails, 
 
 27-. Three works, one of two volumes, one of three, and 
 one of four, stand side by side:, what is the probability that 
 the volumes of each work stand in their proper order ? 
 
 38. If C;i8=C,+iil8, find Cin 
 
268 PERMUTATIONS, COMBINATIONS, PROBABILITIES. [XI. 
 
 29. Find the whole number of combinations of p + q + r 
 things of which p things are alike, q things alike, and the 
 rest all different. 
 
 30. Find the odds against A's winning four games before B 
 wins two at a game where A is twice as good a player as B. 
 
 31. If a, 5, c, • • • w be different prime numbers, what is the 
 number of measures of the product a"- J^'-c"^- • •P-m^-n^. 
 
 33. Find the chance of throwing at least eight in a single 
 throw with two dice; with three dice. 
 
 33. A and B play a set of games in which A's chance in 
 each game is p, and B's q: show that the probg-bility of A's 
 winning m games out ot m + n games is 
 
 p'"-[;+np + n{n + l)f/2l +■■■ 
 
 + n{n + l)- • -{n-hm -2)p'^-^/{ni -1)1], 
 
 34. From a bag that holds n balls a man draws out a ball 
 and replaces it n times : what is the probability of his having 
 drawn every ball in the bag? 
 
 35. Into a box having three equal compartments four balls 
 are thrown at. random: show that there are eighty-one- pos- 
 sible arrangements; and find the probability that the four 
 balls are all in one compartment; that three of them are in 
 one compartment and one in another; that two of them are 
 in one compartment and two in another; that two of them 
 are in one compartment, and one in each of the others. 
 
 36. The number of combinations of n different things . in 
 groups of r things, with repetition, is the number of combi- 
 nations of n + r — 1 things in groups of r things without 
 repetition. 
 
 37. The number of possible combinations of n things in 
 groups the number of whose elements is even differs by one 
 from the number of such combinations in groups the number 
 of whose elements is odd. 
 
INDEX. 
 
 Abstract numbers, definition of, 3.- 
 ~ equal, 4. 
 negative, as operators, 2, 20. 
 product of, 6. 
 sum of, 23. 
 Addition, definition and sign of, 33. 
 associative and commutative, 34; 
 multiplication distributive as to, 
 
 36. 
 of fractions, 34, 66. 
 of radicals, 173. 
 rules for, 40, 66, 173. 
 Algebra, as distinguished from 
 arithmetic and geometry, 1. 
 primary; operations of, 36-69 
 Algebraic expressions, 86-39. 
 Alternation, proportion by, 144. 
 ■ Anti-logarithm, definition of, 333t 
 
 rule for finding, 340. 
 Arrangement, 46, 58. 
 
 letter of, 108. 
 Associative operations, 8, 34, 154. 
 Axioms, of equality, 70, 71. 
 of inequality, 148, 150. 
 relating to combinations and 
 permutations, 348. 
 to products, sums, and di£Eer- 
 ences of integers, 100. 
 of entire functions of one 
 letter, 110. 
 Base, of logarithm, 333. 
 change of, 334. 
 of power, 30. 
 Binomial, definition of, 36. 
 surds, 170, 173, 174, 193, 194. 
 theorem, 163. 
 Coefficients, definition of, 38. 
 
 use of detached, 50, 58. 
 Combinations, definition "of, 348. 
 
 Combinations, fundamental prin- 
 ciple of, 348. 
 
 maximum number of, j 353, 354.' 
 
 of n things all difEei;ent, 353-356. 
 some or all alike, 353, 356. 
 Commutative operations, 10, 84, 38, 
 
 110, 158. 
 Composition, proportion by, 144. 
 Constants, definition of, 136. 
 Continuous and discontinuous va- 
 riables, 138. 
 Contraction, in division, 60. 
 
 in fljiding roots, 188, 190. 
 
 in multiplication, 54. 
 Cross multiplication, 46. 
 Decimal, value of repeating, 333. 
 Degree, of equation, 70. 
 
 of term or expression, 38. 
 
 of terms of product, 44. 
 Detached coefficients, their use, 
 
 in division, 58. 
 
 in multiplication, 50. 
 Discussion of a problem, 84. 
 Division, definition and signs of, 16. 
 
 arrangement of terms in, 58. 
 
 checks in, 58. 
 
 contraction in, 60. 
 
 of fractions, 66. 
 
 rules for, 56-63, 66, 174, 343. 
 
 proportion by, 146. 
 
 symmetry in, 60. 
 
 synthetic, 63. 
 
 use of detached coefficients in, 58. 
 
 use of type-forms in, 60. 
 Elimination, 78. 
 Equations, definition, of, 70. 
 
 dependent, 80. 
 
 elements of, known and un- 
 known, 70. 
 
 369 
 
270 
 
 INDEX. 
 
 Equations, exponential, 243. 
 indeterminate, 80, 86. 
 involving surds, 178. 
 quadratic, 198-317. 
 complete and incomplete, 198. 
 formation of, from roots, 300. 
 general forms of, 303. 
 higher equations solved as, 304. 
 maxima and minima deter- 
 mined by, 316. 
 of one unknown element, 198. 
 properties of roots of, 300. 
 simultaneous, 306-313. 
 solved by factoring, 300. 
 special cases of, 300. 
 reciprocal, 315. 
 roots of, 70. 
 simple, 70, 99. 
 fewer conditions than un- 
 known elements, 86. 
 general forms of, 76, 93. 
 more conditions than unknown 
 
 elements, 86. 
 of one unknown element, 73. 
 of three or more unknown 
 
 elements, 88, 90. 
 of two unknown elements, 78. 
 special problems, 74, 83. 
 Euclid's process for finding highest 
 
 common measures, 103, 113. 
 Evolution, definition of, 33. 
 an inverse process, 180. 
 geometric illustration of, 188. 
 rules for, 180-186, 190. 
 Exponents, definition of, 30. 
 fraction, 164. 
 
 work with, 30, 33, 166, 168, 330, 
 381. 
 Expressions, definition of, 36. 
 ' degree and kinds of, 38, 76, 170. 
 Factors, definition of, 6. 
 entire, 118. 
 linear, 133. 
 
 Factors, of highest common meas- 
 ures, 134. 
 of lowest common multiples, 136. 
 prime, 106, 116. 
 rules for finding, 118-133. 
 Fraction powers, 164r-169. 
 Fractions, as exponents, 164^169. 
 complex, 66. 
 
 highest common measures and 
 lowest common multiples of, 
 136, 130, 133. 
 operations on, 14, 34, 66. 
 rational denominator of, "176. 
 reductioti of, 14, 64, 134. 
 simple, 3. 
 Functions, definition of, 108. 
 entire, of single letter, 108. 
 measures and multiples of, 
 108-117. 
 
 Identity, definition of, 70. 
 Incommensurable numbers, defini- 
 tion of, 136, 153. 
 
 multiplication by, 164. 
 
 multiplication of, associative and 
 commutative, 154, 156, 158. 
 
 previous theorems applied to, 158. 
 Incommensurable powers, 338-331. 
 Induction, proof by, 163. 
 Inequalities, 4, 148;- 
 
 greater-lesB, 150. 
 
 larger-smaller, 148. 
 Insurance, cost of, 360. 
 Integers, definition of, 3. 
 
 measures and multiples of, 100- 
 106, 118. 
 Integer powers, 30-33. 
 Interpolation of terms, 330-336. 
 Inversion, proportion by, 144. 
 Involution, definition of, 30. 
 
 general principles of, 33, 164, 
 174, 234. 
 
 Logarithms, 233-245. 
 
INDEX. 
 
 271 
 
 liOgarithms, base of system of, 332. 
 change of, 234. 
 characteristic and mantissa of/ 
 
 236. 
 of products, qxiotients, powers, 
 
 and roots, 334, 342. 
 properties of, to base ten, 236. 
 systems of, 234. 
 table of, 244, 245. 
 limitations in the use of, 241. 
 rules for using, 288-243. 
 Measures, definition of, 100, 108. 
 highest common, 100, 110. 
 composition of, 106, 117, 124. 
 Eviclid's process for, 103, 112. 
 rules for finding, 128, 130. 
 Monomials, definition of, 36. 
 Multiples, definition of, 100, 108. 
 lowest common, 100, 110. 
 composition of, 126. 
 rules for finding, 132. 
 Multiplication, definition and signs 
 of, 6. -^ 
 
 arrangement of terms in, 46. 
 associative, 8, 154. 
 checks in, 43. 
 comm.utative, 10, 156. 
 contraction in, 54. 
 cross, 46. 
 
 distributive as to addition, 36. 
 of fractions, 66. 
 
 of powers, 30, 33, 166, 231, 334. 
 of radicals, 174. 
 rules for, 43-54, 66, 174, 243. 
 use of detached coefficients in, 50. 
 of symmetry in, 53. 
 of type-forms in, 48. 
 Multiplier always abstract, 6. 
 Numbers, definition and kinds of, 
 3,4. 
 commensurable and incommen- 
 surable, 136. 
 entire, 118. 
 
 Numbers, expression of, 4, SO. 
 " positive and negative, 18-31. 
 
 prime and composite, 104, 106, 
 114. » 
 
 prime to each other, 104, 105, 
 106, 114, 116. 
 Operations, inverse, 16, 38, 33, 180. 
 
 repetitions and partitibns, 3. 
 Operator, definition of, 3. 
 
 office of negative, 20. 
 
 position of, 4. 
 Opposites, 26. 
 Parentheses, use of, 6, 38. 
 Partition, definition and sign of, 4. 
 Permutations, definition of, 248. 
 - fundamental principle of, 348. 
 
 of n things all different, 250. 
 not all difEerent, 250. 
 Polynomials, definition of, 22, 36. 
 
 degree of, 38. 
 
 operations on, 40, 42, 56. 
 
 roots of, 180-184. 
 Positive and negative numbers, 18. 
 
 expression of, 30. 
 Powers, definition and sign of, 30. 
 
 and roots, 162-197. 
 
 conmiensurable, 164. 
 
 fraction, 164-169. 
 equal, 166. 
 
 operations on, 166-169. 
 series of, 164. » 
 
 incommensurable, 164, 228-331. 
 
 integer, 30. 
 
 like and unlike, 164. 
 
 of powers, 33, 33, 166, 280. 
 
 operations with, 230, 331. 
 
 products and quotients of, 30, 
 81, 33, 168, 330, 331. 
 Prime numbers, 104, 106, 114, 116. 
 Probabilities, 358-365. 
 Problems, discussion of, 84. 
 Product, definition of, 6, 154, 
 
 form of, 44, 
 
273 
 
 INDEX. 
 
 Product, logaritlim of, 333, 334. 
 
 of abstract numbers, 6. 
 
 of numbers expressing partition 
 *or repetition, 14. 
 
 of polynomials, 26. 
 
 of powers, 30, 32, 168, 230, 331. 
 
 of product of two numbers by the 
 reciprocal of one of them, 16. 
 
 of simple fractions and of re- 
 ciprocals, 14. 
 Progressions, see Series. 
 
 analogies of the, 336. 
 Proportion, definition of, 143. 
 
 continued, 143, 146. 
 
 properties of, 144, 146. 
 
 transformations in, 144, 146. 
 Proportionals, 142, 144. 
 Quadratic equations, 198-317. 
 Quadrinomials, definition of, 36. 
 Quantity, definition of, 3. 
 Qiiotient, definition of, 16. 
 
 form of, 58. 
 
 logarithm of, 234. 
 
 of powers, 81, 32, 168, 380-334. 
 Eadicals, definition of, 170. 
 
 kinds of, 170. 
 
 operations on, 173-176. 
 Ratio, definition of, 142. 
 
 direct and inverse, 143. 
 
 in geometric progression, 332. 
 Eationalization, of fractions, 176. 
 
 of an equation, 178. 
 Reciprocals, definition of, 14. 
 
 product of, 14. 
 Repetition, definition and sign of, 4. 
 Review questions, 34, 68, 94, 184, 
 
 160, 196, 214, 246, 266. 
 Roots, of binomial surds, 192, 194. 
 
 of equations, 70. 
 
 of numbers, 33. 1 
 
 of numerals, 186-190. 
 by contraction, 188, 190. 
 
 of polynomials, 180-184. 
 
 Roots, square, geometric illustra- 
 tion of, 188. 
 Root-index, 83. 
 
 Series, definition of, 37, 318. 
 
 arithmetic, 218-221. 
 continuous, 221, 238. 
 interpolation of terms in, 220. 
 
 finite and infinite, 37. 
 
 geometric, 222-225. 
 continuous, 335, 238. 
 interpolation of terms in, 324. 
 
 harmonic, 336. 
 
 major and minor terms of, 338. 
 Signs," of aggregation, 6. 
 
 of continuance, 8. 
 
 of equality and inequality, 4, 150. 
 
 of inference, 8. 
 
 of infinity, 76. 
 
 of operation, 4, 16, 30, 33, 33. 
 
 of quality, 30. 
 
 of repetition and partition, 4. 
 
 of variation, 140. 
 
 radical, 33. 
 Subtraction, theory of, 28. 
 
 of fractions, 66. 
 
 rules for, 40, 66, 172. 
 Sum of abstract numbers, 22. 
 
 of concrete numbers, 23. 
 Surds, definition of, 170. 
 
 equations containing, 178. 
 
 kinds of, 170, 174, 176. 
 
 properties of, 174, 193. 
 
 roots of binomial, 193, 194. 
 Symmetry, definition of, 88. 
 
 as an aid in division, 60. 
 in multiplication, 53. 
 Synthetic division, 62. 
 
 Tensors and versors, 31. 
 Trinomials, definition of, 86. 
 Type-forms, their use, 48, 60, 120. 
 
 Unit, definition of, 8. 
 
 Variables and variation, 186-140. 
 
mt'