ROBERT E. PETERSON'S CHEAP EdIaVIONAL ^m\lS. 
 
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 ROBERT E. PETERSON'S CHEAP EDUCATIONAL SERIES. 
 
 TREATISE ON ALGEBRA, 
 
 CONTAINING THE 
 
 MOST USEFUL PARTS OF THAT SCIENCE, 
 
 ILLUSTRATED BY 
 
 A COPIOUS COLLECTION OF EXAMPLES: 
 
 DESIGNED FOE THE USB OF SCHOOLS AND COLLEGES. 
 
 BY ENOCH LEWIS. 
 
 FOURTH EDITION, REVISED AND ENLARGED. 
 
 PHILADELPHIA : 
 PUBLISHED BY ROBERT E. PETERSON, 
 
 N. \V. C0RNP:K of fifth and ARCH STREETS. 
 
 1852. 
 
*^ 
 
 Entered according to Act of Congress in the year 1852, 
 
 BY ROBERT E. PETERSON, 
 
 In the Clerk's Office of the District Court of the Eastern 
 
 District of Pennsylvania 
 
 Deacon & Peterson, Printers, 
 66 South Third Street. 
 

 PEEFACE TO THE FOURTH EDITION. 
 
 Since the first edition of this work was issued, a number 
 of years have elapsed, during which several treatises on 
 this branch of science have been given to the public 5 some 
 of which are evidently the productions of able hands ; still 
 the reasons which induced the author of The Practical Ana- 
 lyst to offer his work to public acceptance, appear applicable 
 to the present day. The generality of pupils who undertake 
 to acquire a knowledge of algebra, require a manual which 
 includes the general principles, and processes of the science 
 without needless expansion, or very abstruse speculations. 
 The work has been carefully revised, and such changes in- 
 troduced as were judged needful to adapt it to the present 
 improved state of the science. A number of examples, not 
 contained in the former editions, have been inserted in this, 
 and the original object of smoothing the path of the pupil 
 and diminishing the labor of the instructor, has been kept 
 steadily in view. Though the design of this treatise is 
 rather to meet the wants of students in our schools, than 
 to supply that class who may incline to dive into the most 
 refined intricacies of the analytic art , yet, even to the 
 latter class, this tract will probably be found a safe and 
 correct introduction. 
 
 3 
 
 l^30B114 
 
PREFACE. 
 
 The mathematical sciences have been considered, since 
 the early periods of their existence, among the noblest ob- 
 jects of human inquiry. Of these, next to arithmetic, geo- 
 metry occupies the first place, in regard both to importance 
 and to time. The Grecian philosophers cultivated this 
 branch of knowledge, with an ardor and industry, which 
 manifest their high opinion of its value. In the Ionian 
 and Pythagorean schools, geometry was considered an in- 
 dispensable preliminary to the study of philosophy. When 
 a person, ignorant of geometry, applied for his instructions 
 to the philosopher Xenocrates, he is said to have made this 
 laconic reply : " Thou hast not the handles of philosophy." 
 
 The perspicuity of geometrical reasoning, the accurate 
 and inimitable dependence of its arguments, and the un- 
 faltering certainty of its conclusions, are eminently calcu- 
 lated to form the mind to habits of attention, and to a 
 regular and forcible cbncatenation of its ideas. No won- 
 der, then, that it was so highly prized by that acute and 
 inquisitive people. 
 
 Two modes of procedure, according to the different ob- 
 jects in view, were adopted by the ancient geometers: 
 synthesis and analysis. 
 
 Synthesis, or composition, consists in the direct solution 
 of a problem ; or, the demonstration of a proposition by 
 a series of arguments, regularly deduced, from self-evident 
 truths, or from other propositions previously established. 
 This method is very proper for conveying, with clearness, 
 to another, those truths which are completely understood 
 by the instructor. The works of ihe ancient geometers, 
 which have escaped the ravages of the barbarous ages, are 
 mostly synthetical. The Elements of Euclid, so well 
 
 5 
 
VI PREFACE. 
 
 known in our modern schools of geometry, furnish, proba- 
 bly, the most complete specimen of the ancient synthesis 
 extant. And it is a remarkable fact in the history of 
 science, that the geometry of Euclid, though written two 
 thousand years ago, and some time anterior to any other 
 mathematical tract that has reached us, is still one of the 
 best elementary works on that subject which has ever ap- 
 peared. 
 
 Analysis, or decomposition, on the other hand, assumes, 
 as known, the proposition which is to be examined, or, 
 as already effected, the solution which is to be made, and 
 thence proceeds to examine the consequences necessarily 
 resulting from such supposition, until, in case of a theorem, 
 a conclusion is attained, the truth or falsehood of which is 
 already known, whence the correctness or absurdity of the 
 supposition becomes established ; or, in case of a problem, 
 such relations are determined as prove the possibility or 
 impossibility of the solution. In synthesis, observes Mon- 
 tucla, we proceed from the known to the unknown, from 
 the trunk to the branches 5 in analysis, we proceed from 
 the unknown to the known, from the branches to the 
 trunk. 
 
 The analytical method is frequently indispensable, when 
 new problems are to be solved, or new theorems investi- 
 gated. No doubt, many propositions which the ancients 
 have transmitted to us in the synthetical form, owe their 
 discoveries to analysis. 
 
 Of this method of procedure, the mathematical collec- 
 tions of Pappus^ and the work, De Sectiones Rationis^ of 
 Apollonius, furnish the principal specimens which the 
 ancient geometers have left us. 
 
 In the ancient geometry, the magnitudes under consi- 
 deration, were mostly presented to the mind through the 
 medium of representations, as similar a^practicable to their 
 antitypes. In some instances, however, this analogy was 
 entirely abandoned, as »in the fifth book of Euclid's Ele- 
 ments, where right lines are the only representatives used, 
 yet the reasoning is so conducted, as to be equally applica- 
 ble to magnitudes of every kind, and even to abstract num- 
 
PREFACE. Vn 
 
 bers. In this instance, we may discover a commencement 
 of that species of generalization, which forms so conspicu- 
 ous a feature in our modern mathematics. 
 
 T.he ancient analysis furnished the germ of that branch 
 of mathematics, which, in the hands of the moderns, has 
 become the great master key to all the rest; but it does 
 not appear to have assumed the character of a distinct 
 science, till after the commencement of the Christian era. 
 
 The earliest writer, in whose works the science of algebra 
 is distinctly seen, was Diophantus, a mathematician of the 
 Alexandrian school. The time in which he lived is not 
 precisely known, but it was not later than the fourth cen- 
 tury, as the daughter of Theon, the amiable and accom- 
 plished Hypathia, who died about the beginning of the 
 fifth, wrote a commentary on his w^orks. 
 
 Whether Diophantus was the inventor, or only an im- 
 prover of algebra, cannot now be known ; the latter sup- 
 position, however, is the more probable, as the science in 
 his hands exhibits a degree of maturity, which it can hardly 
 be supposed to have attained in the first period of its ex- 
 istence. 
 
 A part only of the original work of Diophantus is now 
 to be found ; in this he does not explain the first principles 
 of the science, but teaches the solution of a great variety 
 of difficult questions, in that branch of the subject, now 
 called, from him, the Diophantine Algebra, or the indetermi- 
 nate analysis^ applied to equations of the higher orders. 
 The various questions, if original, which he has formed, 
 and the address with which he has conducted their solu- 
 tions, necessarily inspire his readers with a high opinion 
 of his invention and discernment. His work, written in 
 the original Greek, was discovered in the Vatican Library, 
 about the middle of the sixteenth century. 
 
 Though the science of algebra appears to have originated 
 among the Greeks of the Alexandrian school, the inhabi- 
 tants of western Europe derived their knowledge of it from 
 the Arabs, who are by some supposed to have been its in- 
 ventors. 
 
 Dr. Wallis observes, that they differ essentially from 
 
via PREFACE. 
 
 Diophantus in their manner of expressing the powers. 
 The Greek analyst calls the 2d, 3d, 4th, 5th, 6th, etc. 
 powers, the square, cube, squared square, squared cube, 
 cubed cube, etc. ; each power being designated by the two 
 inferior powers of which it is the product. But the Ara- 
 bian algebraists denominate the 5th power the first sur- 
 solid, the 6th the squared cube ; being the square of the 
 cube, and not the product of a square and cube ; the 7th 
 the second sursolid, and so of the other powers. Hence, 
 he infers, that the Greek and Arabian analyses were not 
 derived from a common source. 
 
 With due regard for the opinion of this eminent scholar, 
 it appears quite as rational to suppose, that the Arabian 
 mathematicians may have borrowed from their tutors, the 
 Greeks, this branch, with the rest of the mutilated sciences, 
 and adopted, in the denomination of their powers, a mode 
 of expression peculiar to themselves, as to believe, that 
 while the other parts of Grecian science were sought with 
 avidity, this was permitted to sleep amidst the dust of ne- 
 glected libraries, till the same thing had been reinvented 
 by a people much less advanced in scientific knowledge, 
 and less remarkable for invention than their predecessors.* 
 
 Whether the science of Algebra was invented by the 
 Arabs, or borrowed from the Greeks, the name is unques- 
 tionably of Arabic origin. The names given by the Arabs, 
 for they used a plurality, were algebra valmucahala. These 
 words, according to Lucas de Burgo, signify restauratio et 
 opposition restoration or rebuilding, and opposition. Golius 
 defines the word gebera, or giabera, by religavit consolidavit, 
 it bound or consolidated ; and mocabulat, by comparatio 
 opposition comparison, opposition. 
 
 * In the progress of science, there always appears a regular depend- 
 ence in the successive stages. The discoveries of one age are the re- 
 sults of those made in the former. Logarithms were invented, when 
 the discoveries in astronomy and trigonometry had rendered their use 
 indispensable. The discoveries of Newton could not have been made, 
 even by that gigantic genius, in the time of Copernicus; nor could 
 Columbus have led his trembling companions across the Atlantic, be- 
 fore the invention of the mariner's compass. 
 
PREFACE. IX 
 
 By these words they probably designed to indicate the 
 general objects of the science. The quantity whose value 
 is sought, is commonly interwoven with, or hound to other 
 quantities, in such a manner as to form one or more equa- 
 tions or comparisons of quantities set in opposition to each 
 other. These equations are then transformed, or rebuilt^ 
 till the unknown quantity is brought out in opposition to a 
 given or known one.^ 
 
 The most ancient authors on algebra among the Arabs, 
 are Mohammed ben Musa, and Thebit ben Corah. The 
 former is described by Cardan as the inventor of the me- 
 thod of solving equations of the second degree, a discovery 
 in which he was certainly anticipated by the mathemati- 
 cians of the Alexandrian school. From the title given to 
 his book, it has been inferred, that he flourished during the 
 reign of Almamon, or in the early part of the ninth century. 
 
 Whether the Arabian algebraists proceeded beyond the 
 solution of equations of the second degree, is an unsettled 
 question. An accurate knowledge of the mathematical 
 sciences is seldom combined, in the same individual, with 
 an extensive acquaintance with Arabic literature, and, 
 therefore, little is certainly known on this subject. The 
 Bodleian Library in England, and that of the Escurial in 
 Spain, are said to possess a great number of Arabic works 
 on the subject of algebra. • 
 
 Leonard of Pisa, who lived near the beginning of the 
 thirteenth century, impelled by a thirst for mathematical 
 knowledge, traveled into Arabia and other parts of the east, 
 and on his return, first communicated to his countrymen 
 the science of algebra. I do not find that any of his writ- 
 ings on that subject have ever been published. 
 
 In a treatise upon trigonometry, by Regiomontanus, of 
 Franconia, written about the year 1464, some problems 
 are solved by algebra, in which he refers to the rules, a? 
 though commonly known. 
 
 *The name almncahala was adopted by some of the Italian writers, 
 and it is thus designated in some of the works of Cardan ; but the 
 term algebra appears now irrevocably fixed upon it. 
 
X PREFACE. 
 
 But the earliest European author, whose works, specially 
 on this subject, have been published, was Lucas de Burgo, 
 before mentioned. He was a Franciscan, who traveled in 
 the east, either in pursuit of knowledge, or for some pur- 
 pose not now known, and after his return, taught mathe- 
 matics at Naples, Venice, and Milan*. His work, in which 
 the rules of algebra are laid down, was first printed in 1494. 
 In this the science appears very far below our modern 
 algebra. The rules for the solution of adfected quadratic 
 equations, are given in semi-barbarous Latin verse, and the 
 different cases separately treated. His solutions do not 
 rise to equations above the second degree. 
 
 The solution of cubic equations appears to have been 
 first effected by Scipio Ferrei, professor of mathematics at 
 Bologna, about the beginning of the sixteenth century. 
 
 This solution, however, included but one case, namely, 
 that in which the first and third powers only, of the un- 
 known quantity, were involved. Some questions, including 
 this case, being afterward proposed to Nicholas Tartaglia, 
 an eminent mathematician of Brescia, he discovered a 
 general solution of equations of the third degree. This 
 discovery he communicated to Jerome Cardan, a physician 
 of Milan, under an injunction of secrecy. Cardan, not- 
 withstanding, having found the demonstration, published 
 it in his work, De Arte Magna, in 1545. This is now 
 commonly called the method of Cardan. 
 
 Cardan first remarked the plurality of roots in quadratic 
 equations, and their distinction into positive and negative. 
 The solution of equations of the fourth degree was accom- 
 plished soon after the discovery of Tartaglia, by Lewis 
 Ferrari, a pupil and coadjutor of Cardan. 
 
 Some intricacies belonging to equations of the third de- 
 gree were further unraveled by Raphael Bombelli, of 
 Bologna, and published in 1579. 
 
 The algebraists, whose labors have been noted, expressed 
 known quantities by their proper numerical characters, and 
 therefore their solutions were destitute of that generality 
 which constitutes so prominent a feature of our present 
 analysis. Their modes of solution were applicable to all 
 
PREFACE. XI 
 
 similar problems, but their results were confined to par- 
 ticular questions. Francis Vieta,* by using letters of the 
 alphabet to represent known &.s well as unknown quanti- 
 ties, gave an extent and generality to the science which it 
 did not possess before. By this change, algebraists were 
 enabled to include, in a single solution, a whole class of 
 problems, and to obtain the result in each particular case, 
 by simple substitution. He taught the method of taking, 
 from an equation, its second term, and thus reducing ad- 
 fected quadratics to simple quadratics j and all cases of 
 cubic equations to the case solved by Ferrei. He also 
 taught the solution of cubic equations, having three possi- 
 ble roots, by the trisection of an angle. He made numer- 
 ous improvements in algebra, and furnished the germs of 
 some discoveries which have since grown up under other 
 names. 
 
 Thomas Harriott, an English analyst, followed in the 
 steps of Vieta, and made several important improvements 
 in the science. He first adopted the plan of placing all 
 the terms of an equation on one side of the sign of equality, 
 and zero on the other ; and showed that an equation thus 
 expressed, may be always formed by the multiplication of 
 binomial factors. This naturally led to the discovery, that 
 every equation has as many roots, or values of the unknown 
 quantity, as there are units in the index of its highest 
 power. Harriott, notwithstanding this observation, lay 
 directly in his road, does not appear to have made it. The 
 complete development of negative and impossible roots, 
 was left to exercise the ingenuity of succeeding inquirers. 
 Vieta employed the large letters of the alphabet, and in- 
 dicated their powers by initials placed over them, as ex- 
 ponents are now used ; Harriott substituted small letters, 
 and denoted their powers by repetitions of the letter ; thus 
 instead of ./^% A"^ etc., he wrote aa^ aaa^ etc., a small 
 change, indeed, but still an obvious improvement in the 
 notation. His Artis AnaSyticse Praxis, containing his dis- 
 
 * This eminent mathematician was born at Fontenoy, in Poitou, in 
 1540, and died in 1603. 
 
Xll PREFACE. 
 
 coveries, was published after his death.*" He was born at 
 Oxford, in 1560, and died in 1621. 
 
 The celebrated French philosopher, Eene Descartes, 
 contributed largely to the advancement of algebra. He 
 explained the nature of negative roots, and taught the man- 
 ner of finding their number by the changes of the signs in 
 the general equation. The use of exponents, as now ap- 
 plied, is attributed to him ; as is also the method of inde- 
 terminate co-efficients. 
 
 The first application of algebra to geometry, was long 
 prior to the time of Descartes, yet those sciences are in- 
 debted to him for that intimate union, which has since con- 
 tributed so extensively to the improvement of both. Des- 
 cartes was born in 1596, and died in 1650. 
 
 Soon after the time of Descartes, the analytical science 
 took a flight, which, if followed, would lead me far beyond 
 the bounds of a preface. This sketch of the history will, 
 therefore, be closed with the remark, that this science, as 
 enriched by the discoveries of Newton, Leibnitz, and others, 
 has become, in the hands of our modern philosophers, the 
 torch to guide them through the most intricate labyrinths 
 of science ; that by its light they have traced the motions 
 of the celestial bodies through all their mystic dance, and 
 penetrated many of the recesses of nature, where, without 
 its aid, they must have been bewildered and lost. 
 
 The following work was undertaken from a persuasion, 
 that the books on algebra, used in our schools, were none 
 of them entirely adapted to the wants of a large class of 
 pupils, many of whom do not enjoy the leisure, or the 
 talents requisite for penetrating the depths of science, and 
 yet are desirous of attaining a knowledge of this subject, 
 sufficient to qualify them for studying successfully the com- 
 mon practical branches of mathematics, to which this serves 
 as a key. In the most popular treatise on this science, 
 with which our schools have been furnished, the progress 
 of the student appears to me neecflessly obstructed, by diffi- 
 
 * Bossut says it was published in 1620 ; Montucla and others say in 
 1C31. 
 
PREFACE. XUl 
 
 culties near the commencement, which, to a common in- 
 tellect, are almost insuperable. 
 
 My object has been to present the most useful parts of 
 the science in such order, that no very abstruse process 
 should be required, before the pupil had been sufficiently 
 exercised, to acquire the requisite skill. The expedients 
 demanded for solving the questions are mostly pointed out 
 before they are called for in practice. 
 
 In a popular treatise on a subject which has engaged the 
 attention of so great a number of authors, many of them, 
 unquestionably, endued with talents of the highest order, 
 it would be idle to expect much originality. My object 
 has been to smooth the path of the student, and diminish 
 the toil of the tutor. The work, with all its imperfections, 
 is submitted to the inspection of the public. 
 2 
 
CONTENTS 
 
 Definitions, ... - - - - Page 15 
 
 Addition, 18 
 
 Subtraction, 21 
 
 Multiplication, - 22 
 
 Division, -------- 26 
 
 Involution, or the Raising of Powers, - - 32 
 
 Evolution, or the Extraction of Roots, - - 36 
 
 Algebraic Fractions, 41 
 
 Equations, 50 
 
 Simple Equations, - 51 
 
 Quadratic Equations, 84- 
 
 Promiscuous Examples, 101 
 
 Ratios, 106 
 
 Variations of Quantities, Ill 
 
 Series, - - - - - - - -114 
 
 Summation of Series, 128 
 
 Differential Method, - - - - . 129 
 
 Construction of Logarithms, - - - - 133 
 
 Surds, 142 
 
 Equations in General, 155 
 
 Indeterminate Problems, 166 
 
 Miscellaneous Examples, 176 
 
 xiv 
 
ALGEBRA. 
 
 DEFINITIONS. 
 
 Article 1. Algebra, or specious arithmetic, is the 
 science of computing by symbols or general characters. 
 
 2. Quantities, of whatever kind, are usually denoted by 
 letters of the alphabet. 
 
 3. The relations of quantities, and the operations to be 
 performed on them, are indicated by the following charac- 
 ters, thus : 
 
 4. The sign + plus^ or more^ indicates addition, as 
 a-\-b signifies that b is added to a, 
 
 5. The symbol — minus^ or less^ indicates subtraction ; 
 thus, a — Z>, signifies that b is subtracted from a. 
 
 The characters + and — are called, by way of emi- 
 nence, the signs of the quantities to which they are pre- 
 fixed. 
 
 6. Multiplication is denoted by the sign X , into^ placed 
 between the factors, as axb; or by a period, as a.b; or, 
 more frequently, by joining the letters, like letters in a 
 word, as ab; each of which expressions denotes the pro- 
 duct of a and b, 
 
 7. Division is indicated by the sign -r- %, placed be- 
 tween the terms ; or by writing the dividend above, and 
 
 the divisor below, a horizontal line ; thus a-H^, or -7- de- 
 notes the quotient of a when divided by b, 
 
 15 
 
16 DEFINITIONS. 
 
 8. The difference of two quantities, when it is unknown 
 which is the greater, is indicated by the sign -v, placed 
 between them ; thus, arj)^ or b^a^ denotes the difference 
 of a and b. 
 
 9. : : : : indicates proportion; thus, a : b : : c : d^ may 
 be read, a has to b the same ratio that c has to d, 
 
 10. = equal to, signifies that the quantities between 
 which it is placed, are equal to each other, thus, b-\-c 
 = fl — d+e, is a combination, called an equation, which 
 signifies, that when the operations indicated by the signs 
 are performed on the quantities placed on each side of the 
 sign of equality, the results are equal to each other. 
 
 11. A simple quantity is that which consists of one 
 term only, viz. a quantity denoted by a single letter, or 
 several letters and figures, connected by the sign of mul- 
 tiplication or division, expressed or understood,* as a, abc, 
 
 5cd; J- 
 
 12. A compound quantity consists of two or more sim- 
 ple quantities connected by the signs of addition or sub- 
 traction, as a — 6, ab — ac + bd, 
 
 13. A root is a number or quantity, from which a power 
 is conceived to arise. 
 
 14. A power of a number or root, is the product of a 
 unit, multiplied continually by the given root, any pro- 
 posed number of times ; and the figure or quantity which 
 indicates the number of multiplications thus made, is called 
 the index or exponent; thus, 1x5x5x5x5=6255 and, 
 IXaX axaxaxa, or aaaaa, are the 4th and 5th powers 
 of 5 and a respectively, and are usually expressed by the 
 root with the index of the power set over it: as 5'*, a^; 
 hence a^= 1, whatever value may be assigned to a, 
 
 15. The second power is called the square; the third 
 power, the cube; the fourth power, the biquadrate, etc. of 
 their respective roots. 
 
 * An absolute number, though containing numerous digits, is con- 
 sidered as a simple quantity. 
 
DEFINITIONS. 17 
 
 16. The radical sign^ -/, prefixed to a quantity, indi- 
 cates the square rootj ^, the cube root; -y, the fourth 
 root, etc. Roots are also expressed by fractional expon- 
 
 1 
 ents; thus, y/a, or a"', denotes the square root of a; ^/a, 
 
 1 • .2 
 
 or a^, the cube root of a; and v^ri^, or a^, the cube root 
 of the square of a, 
 
 17. A root which cannot be accurately expressed in 
 numbers, is called a surd^ or irrational quantity, a^ V 5? 
 
 18. A quantity which has no radical sign, or which 
 having a radical sign, admits of an accurate extraction of 
 the root indicated by the sign, is called rational ; thus a, 
 %/16, 4/Z>^, are rational quantities. 
 
 19. When a compound quantity has a line drawn over 
 it, or is enclosed in brackets, the operation indicated by a 
 preceding or subsequent sign, is to be performed on the 
 whole considered as a simple quantity; thus, a-{-b — cXc?, 
 and (b-\'C)x (d — e), signify that the compound quantities 
 connected by the sign X , are multiplied together ; and 
 y/ab-\-dc, '^{ab — cd-{-ef,l signify the sauare root, and 
 the cube root respectively, of the compound quantities 
 which are preceded by the radical signs. 
 
 20. A number prefixed to a letter, or combination of 
 letters, is called the co-efficient; thus, in the expression 
 Sab, 3 is the co-efficient. 
 
 21. A compound quantity, consisting of two terms, is 
 called a binomial, as a + b; one of three terms, a trinomial, 
 as ab-{-ac-{-de. 
 
 22. Like quantities are those which consist of the same 
 letters similarly involved ; as ab, Sab, 5ab, 
 
 23. Unlike quantities consist of different letters, or dif- 
 ferent powers of the same letters, as a, ab, Sa\ 4^ab^. 
 
 24. A simple quantity is usually termed a monomial. 
 
 =*So called from radix^ a root. 
 
18 ADDITION. 
 
 A compound quantity is termed a binomial, a trinomial, or 
 polynomial, according as it consists of two, three or many 
 terms. Thus^ a -\-b, ab + cd, are binomials, a + ^-f^j CLb + 
 cd+ef, are trinomials, etc. 
 
 In the solution of problems, it is usual to denote known 
 or given quantities, by the initial letters, a, b, c, etc., and 
 unknown ones, by the final letters, v, a?, y, etc. 
 
 The following examples are given for the purpose of 
 exercising the student in the application of the algebraic 
 signs. 
 
 Required the numerical values of the following combi- 
 nations, supposing a=7, i=6, c=5, c?=3, e=2. 
 
 a' + Uc—3de=:34^3+ 120—18=445. 
 
 (a5 + cc/)x(35c + 4«6Z) = (42+15)X(90 + 84) = 57xl74 
 = 9918. 
 
 a^ + cd'—3be=:S03. 
 
 b^ + d'd—lce—^bc^ 223. 
 
 {ac^'-^^bd) X 6ac= 15 9425. , 
 
 b' VaH^'-f«5=20 
 
 4 
 
 a'b'hl3cd^ 
 
 -2^^I3S--^^-^^^^ 
 6ad^—2ace^ 
 ~7 bT3d ~'^^^'^^ 
 
 s/4^a^+ld'+63be=zl3 
 
 a^—a^b + 3ad^—4<lce6S 
 
 _ __ — -— o 
 
 ab — cd -f ae 34 
 
 Section I. 
 
 ADDITION. 
 
 Case I. 
 
 25. When the quantities are like, a^d have like signs. 
 
 Add all the co-efficients together, and to their sum 
 
ADDITION. 19 
 
 annex the given literal expression, prefixing the common 
 sign.* 
 
 
 
 EXAMPLES. 
 
 
 3a 
 
 2be 
 
 2a'b 
 
 — 3xy^ 
 
 2a— 3b 
 
 5a 
 
 3be 
 
 3a'b 
 
 — a??/^ 
 
 5a 6b 
 
 la 
 
 be 
 
 Wb 
 
 — 40??/- 
 
 la— 6b 
 
 4a 
 
 5be 
 
 a'b 
 
 — Ixy'' 
 
 8a— 46 
 
 6a 
 
 Sbe 
 
 ba'b 
 
 — 9xy^ 
 
 9a 4^b 
 
 a 
 
 9be 
 
 a'b 
 
 — llxy'' 
 
 Ida— b 
 
 26a 
 
 5x^y — 76c + ab 6xy^ — lac 4fab^ + 3a^c — bc^ 
 
 3a73y— 66c _|_4a5 ^xy; — 3ac bab'' -}- 4<a''c — bc^ 
 
 bx^y^3bc + ab 4a:y^ — 5ac 3a¥ + 6a^c — 2bc^ 
 
 Ix^y— i6c4-4a6 3a??/^— 2ac 8a6^-f Sa^c— 86c3 
 
 Sx^y — 2i6c + 3a6 8xy^ — ac 4^ab^-\-3a^c — Ibc^ 
 
 3x^y—lbc -\-6ab bxyl—lac ab''-\-6a^c—3b(^ 
 
 4,x^y — 36c +2a6 3xy^ — 2ac 5a63 + 5a2c — 46c3 
 
 5a?3y — |5c +3a6 4<xy^ — 13ac 2a63+ a^c — 56c3 
 
 Case 2. 
 26. When the quantities are alike, but have unlike signs. 
 
 Take the difference between the sum of the positive and 
 the sum of the negative co-efficients, prefix the sign of the 
 greater, and annex the literal part. 
 
 =* The sign + or — > is frequently prefixed to quantities which 
 are not preceded by others, to or from which they are required to be 
 added or subtracted ; a preceding quantity, however, may always be 
 supposed. Algebraic quantities are said to be positive, or negative, 
 according as they are preceded by the signs -f or — ; the former 
 being always understood where none is expressed. 
 
20 ADDITION 
 
 2ab^-\- 6cd^ — Sx^ — Sa^^ + 4a?y 
 
 -5ab^ + 4^cd^ 6x'^y^ 4<a-b + 2x^y^ 
 
 ab^ — 7cd^ 7a? Y - — Sa^b-\- x^y^ 
 
 -Sab^ + bcd^ — 4fX^y^ la^b — 3x^y^ 
 
 Qab^ — cd'' — 9xY — a^b + ^x^ 
 
 6ab ■i-4^cd^ 
 
 —Sx^ 
 
 
 Sab^—5c^d^ 
 
 — 5ab^ + 2c^d' 
 
 - a¥ + 4<c^d'^ 
 lab'^—Sc^d^ 
 8ab^—5c^d^ 
 
 —eab^ + lc^-d^ 
 
 S^ay 
 
 — 9\/ay 
 
 — ly/ay 
 
 3 s/ ay 
 
 by/ ay 
 
 — ^y/ ay 
 
 ^Oax — Iby/xy 
 
 — 31ax-\- bby/xy 
 
 21 ax — 6by/xy 
 
 18ax — bby/xy 
 
 — 4aa?+ 13b s/ xy 
 
 —3Qax-\-llby/xy 
 
 Case 3. 
 
 27. When there are unlike quantities, and different signs. 
 
 Collect the like quantities as in the foregoing cases, and 
 connect the results by their proper signs. 
 
 EXAMPLES. 
 
 4.aa:— 130 + 3x^ lax 3a''—'7bc+6ad 
 
 5a?2 + 3ax + 9x^ — 5xy ^xy—lad — 5a» 
 
 Ixy — 4^x^ -h 90 — 3ax 8ad—6xy — 35c 
 
 6x^ — 5s/x-}-3xy 2xy bxy-\-2a^ — 4ac? 
 
 40 — 5ax — lOxy — 3ab 4^bc + 3ad — xy 
 
 2ax + 20x^ —6x^ 4<ax-3xy-3ab 
 
SUBTRACTION. 21 
 
 6x^y — lax — 3bc 14<ax — 20?^ 
 
 26c + 3x^y-{-4<ax 6ax-\- 3xy 
 
 Sax — 56c -\-lad^ Sy^ — 4^ax 
 
 2bc + 3ad^—6ax 3x^+26 
 
 x'^y — 56c +4<ax ^xy — by^ 
 
 Sad^ — 3ax — 4<x^y bx'^ — Sax 
 
 ax —ISad^'—Sbc '7y^—26 
 
 x^y^ + 4^ad^ — 76 V ax'^y^ +3abY — bc^x'^y^ + b^x^ 
 
 Wc^—lad^ +2ac* 3¥xY — 4^aa?y + 4a'crj^3 — 5a 6y 
 
 3a(^ +66V +5a?y 2a¥y' —3b^xY—2axY +2cV/ 
 
 Sad^—5ac^ —66 V 4^c^xY +2ao(^y^—3¥xY +4<abY 
 
 bxhf+ ad^ — 6V 2b^oifyz + 4<abY +2axY — 2c^xY 
 
 353^2 — 5 -j.'?y2 — 2ac* Sc^x^ — 2axY + bb^x^yz + 9^62^3 
 
 QxY — 2ac* — 6ad^ ^ab^ — 3c^xY — «^Y — Ib^x^yz 
 
 1. Add 3« + 6, la — 56, 76 — 3a, 4a — 26, and 8a — 56 
 together. Kesult, 19a — 46. 
 
 2. Add 5aa? + 76c— 3a2, 86c + 4a3— 2aa7, lla^ — 46c— 6a-r 
 and 2aa? — 36c + 6a2 into one sum. 
 
 Result, 86c+18a2 — ax. 
 
 3. Add together the following quantities, labc + ba^d 
 —3xy% 2aH-\-bxy^—3abc, Uc^-^-Sxy^-^-Sa^ Uc^—Sabc— 
 a% and a^d — lOxy^ — a6c — 76c^ 
 
 Kesult, 10a3j+66c^ 
 
 SUBTRACTION. 
 
 28. Change the signs of the quantities which are to be 
 subtracted, or conceive them to be changed, then collect 
 the terms as in addition. 
 
 EXAMPLES. 
 
 From 8xy—3+ 6x — y 3x^ — 2a6 5a62 — 7^20+ ad 
 Take 3xy — 7 — 6x+by x^ — 7a6 3a62 — 2¥c—4^ad 
 
 5a??/+4+ 1207— 6y ^x>/^'- 
 
22 
 
 MULTIPLICATION. 
 
 From 17a5— 3a?y + 552 
 Take llab + 4^xy + '7b^ 
 
 Sax — 55c 
 lax — 76c 
 
 9aey+4<bc 
 2aey-{-lbc 
 
 i: 
 
 -yi 
 
 From l[x''—2a^x—18 + 6bc 
 Take 3x^—1 a y/x— 9— 36c 
 
 4fad — Ixy + Sbc 
 bad — y'^z + 6bc 
 
 - -^ 
 
 1. From a + b take a — b. Result, 26. 
 
 2. From 8a — 12a? take 5a + 3x. Result, 3a — 16x. 
 
 3. From 12a-\-10b-{-13ax — 3ab, take la — 5b + 3ax, 
 
 Result, ba+lbb+10ax—3ab, 
 
 4. From the sum of 3a6 — lax and lab-\-3ax, take 
 /^ab — 3ax — 4<xy, Result, 6ab — ax-{-4<xy, 
 
 5. From 5a?^ — 4a?2/ + 5, subtract 4<x''^ — ^xy+9. 
 
 Result, x^ — 4. 
 
 6. From ax^ — bx^+x subtract po?^ — cx^+ex. 
 
 Result, (a—p) x''+(c—b)x^+(l—e)x. 
 
 1, From bx^ -\- cx'^ — dx + e take px" — ga?^-|-ra? — 5. 
 
 Result, (b—p)x^ +(c + q)x^'^ (d+r)x+e+ s. 
 
 MULTIPLICATION. 
 
 Case I. 
 29. When the factors are both simple quantities. 
 
 Multiply the co-efficients together, and annex all the 
 letters to the product.* 
 
 *From Def. 14, it is obvious, that the product of two or more 
 powers of any letter, may be expressed by that letter with an index 
 equal to the sum of the indices of the factors ; thus, a^a^=aaa,aa=a5» 
 
MULTIPLICATION. 23 • 
 
 J^ote, — When the signs of the factors are like, the pro- 
 duct must be made positive i when unlike, negative. 
 
 EXAMPLES. 
 
 lOah 2xy^ Sla^b bad 
 
 ba^c — Qx'^y Sab^c Sa^b^o 
 
 bOa^bc — 12a?y 
 
 — acx^ Sxyz ba^bc — ^ad^ boo^y'^ 
 
 — bc^x — 2x^y lac^ — ba^b — 4<x^y^ 
 
 Case 2. 
 
 30. When one of the factors is a compound quantity. 
 
 Multiply the simple factor into each term of the com- 
 pound ones, and connect the products by their proper 
 
 a^ 
 
 examples. 
 
 xy+2y^ 2ab—2b^ 
 Sax — 4<¥ 
 
 a*—2a^b+a%^ 
 
 xy^+4bbc^—9 
 bcx 
 
 Sx^y + bay^—lbc^ Sc^+ax 
 — 2a% bax 
 
 
 Case 3. 
 31. When both factors are compound quantities. 
 
 Multiply the whole multiplicand by every term of the 
 multiplier, and collect the several products as in addi- 
 tion. 
 
24, 
 
 4j 
 
 a+b 
 a+b 
 
 MULTIPLICATION. 
 EXAMPLES. 
 
 a^ + ab + b^ 
 a^—ab + b^ 
 
 a^+ab 
 ab+b^ 
 
 a^-\-a^b + a^b^ 
 — a^b — a^b^ — ab^ - 
 a'^b^+ab^ + b^ 
 
 a^+2ab+b'- 
 
 a* ^a^b^ +6* 
 
 a+b x^+xy+y'^ x^+x^y + x^y^+xy^ + y* 
 a — b X — y X — y 
 
 
 2x^+3xy—4t^y^ 
 3x — 5y 
 
 2ab — 3ac+4fbc x^ — ^y-\-y^ 
 ab-\- ac — be x — y 
 
 
 1. Multiply ba^ + ^b^ by ba^—^^bK 
 
 Result, 25a4— 166*. 
 
 2. Multiply a?3 + 3x^y -{•3xy^+ y^ by x^ -r 2xy + y^. 
 
 Eesiilt, x^-\-5x^y-\-10x^y^+ lOx^y^-i- bxy'^+y^ 
 
 3. Multiply 3x^+2xy+6y^ by x^ — ocy+y^ 
 
 Result, 3a7* — x^y-{-6x^y^ — 3xy^ + 5y*. 
 4«. Multiply 6x^+10xy + 6y^ by 5x^ — 10xy+5y^ 
 
 Result, 25a?* — 50x^y^+25y\ 
 
 5. Multiply a^ + 3a^b+3ab^+¥ by a^—3a'b + 3ab^—b\ 
 
 Result, a^ — 3a''b^+3a^b^ — ¥. 
 
 6. Multiply a^ — ab+¥ — be by a'^+ab—b^+bc. 
 
 Result, a^—a-b^ + 2ab^—2a¥c + 2¥c—b^c^—¥. 
 
 7. Multiply 0?* — x^y+x^y^ — ^y'^+y'^ by x^+xy+y\ 
 
 Result, x^+oc*y^ — x'^y'^+x^y*+yK 
 
MULTIPLICATION. 25 
 
 8. Multiply x+y by x — 2/, and the product hyx'^+y^, 
 
 Kesult, X* — y*. 
 
 9. Find the continued product of a^ — 2ab+h% a"^-\-2ab 
 +5^, a^-{.a"b-{-ab'^'{-b^ and a — b. 
 
 Result, a^—2a^b'' + 2a''b^—b^ 
 
 10. Find the continued product of 3a?+6, 3x+2y 3x — 2, 
 and 3x—6, Kesult, Slx^—SQOx'^+lU. 
 
 11. What is the continued product of 3a — 26, 4a — 3b, 
 4<a + 3b, and 3a + '2b? Ans., lUa^—Uda^b^ + 36b\ 
 
 12. Multiply a^—a^b + a"b'' — a'^b^+ab* — b^-, by a+b. 
 
 Result, a«— S'^. 
 
 13. Multiply x'^-^-x'^y+xy^i-y^j by o?^ — x^y+xy^ — y^. 
 
 Result 0?^ + x^y^ — x^y* — y^. 
 
 When the factors consist of several terms with the 
 powers regularly ascending, the multiplication may be 
 effected by the co-efficients alone, and the literal part be 
 afterwards supplied. Thus in the 5th example, we may 
 take 
 
 1+3+3+1 
 l_3 + 3— 1 
 
 1+3+3+1 
 —3—9—9—3 
 
 3+9+9+3 
 _1_3_3— 1 
 
 1 0—3 0+3 0—1. Then to supply the literal 
 part, we observe, that a^, a^=a^, the first term; and as the 
 powers of a descend and those of b ascend in a regular 
 order, the terms, without their co-efficients, must be, a^, a^6, 
 a*Z)% a^b^, a^b\ ab\ b\ But the 2d, 4th and 6th co-efficients 
 are each =0, therefore the terms are a^ — 3a^b^ -\'3a'^b^ — b^. 
 
 In like manner, the 7th example may be performed 
 thus : 
 
 3 
 
26 DIVISION. 
 
 1—1+1-1+1 
 1+1+1 
 
 1—1+1^1+^ 
 
 1_14-1._1 + 1 
 1^1+1—1 + 1 
 
 1 + 1—1 + 1 + 1. In which the 2d and 6th co- 
 efficients are =0, hence the 2d and 6th terms x^y and xy^^ 
 do not appear in the product. 
 
 DIVISION. 
 
 Case I.* 
 
 32. When the divisor and dividend are both simple 
 quantities. 
 
 Divide the co-efficient of the dividend by that of the 
 divisor; expunge from the dividend such letters as are 
 common to it and the divisor, when they have the same 
 exponents ; when the exponents are not the same, subtract 
 the exponent of the divisor from that of the dividend, and 
 use the remainder as the index of the letter in the quotient ; 
 write in the quotient the letters which have not been ex- 
 punged, with the co-efficient above determined. 
 
 JVb^e. — In division, as in multiplication, like signs in the 
 divisor and dividend require the positive sign in the quotient j 
 unlike signs, the negative. 
 
 * If the student, who is just entering into this science, should 
 express the powers by repetitions of the letters, the process will be 
 more simple, and the grounds of the method indicated by the rule be 
 rendered obvious. 
 
3ab)9a^bcd 
 Sacd 
 
 DIVISION. 
 EXAMPLES. 
 
 '5x^y^)15x*y^z^ 
 
 —3x^z^ 
 
 27 
 
 2ad)6a^d^c 
 
 llab^pla^^ 
 
 3a^c) 12a*bc^ la^b)—% la^bc^ 
 
 — 3^0:2:) — ISa^xy^z^ 
 
 lbabc^)—4<5a^b^c'^ 
 
 33. When the dividend is not a multiple of the divisor. 
 
 Set them down as a vulgar fraction, the divisor being 
 the denominator; and divide the terms by such numbers 
 and quantities as are common to both ; the result will be 
 the fractional answer. 
 
 
 Iba^bc Sac 
 10aZ>2~26 
 
 ISaxy^ — 9y 
 —Sax^y~ 4j7^ 
 
 1. 
 
 Divide 2Sxy^ by 7a? Y- 
 
 Result, -^ 
 
 2. 
 
 Divide — lOabc by 15abd 
 
 Eesult, 3^ 
 
 3. 
 
 Divide limn by 22m^ 
 
 Kesult, 2^, 
 
 4. 
 
 Divide 19xyz by 2bx^yz^ 
 
 Case 2. 
 
 ^^^""'25^ 
 
 34. PFAe/i ^Ae dividend is a compound quantity, and the 
 divisor a simple one. 
 
 Divide every term of the dividend by the divisor, as in 
 
28 DIVISION. 
 
 the former case, and connect the results by their proper 
 signs, 
 
 EXAMPLES. 
 
 3a 
 
 -z=z6x''—5ay + 9a^ 
 
 2. Divide 15axy^ — SOb'^xy — lOx'^y^ by 6xy. 
 
 3. Divide 30a^c— ISac^+lS^c by Sac, 
 
 4. Divide 15a^bc — 15acx'^-{-5ad^ hj — 5ac, 
 
 5. Divide 21a^x'^ — la^x^ — 14<ax by lax. 
 
 6. Divide 72a2Z>Y— 12a%+36a% ^7 ^^%- 
 
 7. Divide 2Sx^y^-\-3bxY — 1(oc^y^z by 7a?y. 
 
 ^ Case 3. 
 
 35. When the divisor and dividend are compound 
 quantities. 
 
 Arrange the terms, both of the divisor and dividend, in 
 such manner that the higher powers of some one letter 
 shall always precede the lower; then find the first term of 
 the quotient, by using the first of the divisor and dividend, 
 as in the first case. Multiply the whole divisor by the 
 quotient thus found; subtract the product from the divi- 
 dend, as in common arithmetic; and proceed till the divi- 
 dend is exhausted. The remainder, if any thing remain, 
 with the divisor for a denominator, must be annexed, with 
 its proper sign, to the quotient, 
 
 EXAMPLES. 
 
 a+b)a^ + 3a^b + 3ab^+b\a^ + 2ab+b'' 
 a3+ a^b 
 
 'Ha-b + Sab^ 
 2a''b + 2ab^ 
 
 ab^+b^ 
 a¥-\-b^ 
 
DIVISION. 29 
 
 ^^' 
 
 — ^a^b -f 9a^Z>2 — lOa'^b^ a^^2ab + 6^ 
 
 3a3^3 — la^b^-^bab* 
 3^353 — 6a2^>3^3fij54 
 
 — a^b^-i-^ab^ — b^ 
 —265 
 
 3. Divide a?* — 4-a7^y + 6a?y — 4a?y3 4-y* by x — y ^ 
 
 Eesult, x^ — 3x-y + 3xy^ — y, 
 
 4^. Divide a^ — b^ by a — b. Result, a + b, 
 
 5. Divide a^ + b^ by a+b. Result, a'^-r-ab+b^ 
 
 6. Divide y^^3y^—4^y + 12 by y^—4<. Result, y—3, 
 
 7. Divide b^—3b^x''+3b^x^—x^ by ¥—3b^x + 3bx''—x\ 
 
 Result, 53 + 36^0? + 3bx'' + a?^. 
 
 8. Divide 24a*— 6* by 3a— 65. 
 
 (^03^4 
 
 Result, 8a^+ 16a^b + 32ab'^+6W+\ ^ 
 
 oa — K)o 
 
 9. Divide a*-r4^a^b'''—32b^hj a + 2b. 
 
 Result, a^—2a^b + Sab''—16b^ 
 
 10. Divide oj^+y^ by x+y. 
 
 Result, 0?^ — a?*y4-a?^y^ — x^^+xy* — j/^H — ^ — 
 
 11. Divide x^ — y^ by x — y. 
 
 Result, 0?^ + x'^y + x^y^ + x^y^ + a:*2/* 4 x'^y^ 4- a?^^^ 4- a^y^ 4 y^. 
 
 12» Divide a* -4- 4a^6 + Ga^^^ + 4a6^ + 6* + ^a^c 4- 12a26c + 
 12ab^c + U^c-\'6a''c^-\-12abc^+6b^c''-{-^ac^ + 4<bc^-\-c^ by a^-f 
 2a5+2ac+62+2k-}-c2. 
 
 Result, a^+2ah + 2ac+b^+2bc + c^. 
 o 
 
30 DIVISION. 
 
 PROMISCUOUS EXAMPLES. 
 
 1. Eequired the sum of 5ac-f7^e, Sac — ibe, lac — Sbe, 
 8ac4-46e, and 2ac — She, Ans 26aC'\-be, 
 
 2. Collect 3ax+5xy — lz% Sxy — 2ax+4^z% lax + Sz^ — 
 xy^ and 6z^ — 4^ax+2xy, into one sum ; and subtract 
 therefrom the sum of the following quantities, 3ax — 5a:y4- 
 4}Z% Ixy — bax — 62?% and bax — 2xy-\-^z'^, 
 
 Result, ax-\'^xy-\-z^, 
 
 3. Required the product of o?^ +37^?/ + 0:2/24-2/3 by x — y, 
 
 Ans. 0?* — ?/*. 
 
 4. What is the sum of the products of a'^+Sx^y-fSo?^'^ 
 +y3 by 07+2/, and x'^ — 2xy+y^ by x'^ — 2xy+y^1 
 
 Ans. 2x'+12xY+2y\ 
 
 5. Divide x^ — y^ by x — y, and from the quotient sub- 
 tract the product of x^-{-y^ by a?^ — xy. 
 
 Result, 2x^y + 2xy^+y\ 
 
 6. Multiply 3a^—9a^b-{-9ab^—3b^ by a''+2ab+b% and 
 a^-\.3a''b-\-3ab^.+b^ by 3a''—6ab~\-3b% and find the sum of 
 the products. Result, 6a^—12a^b'^-\'6ab\ 
 
 7. If a* — 6*, be divided by a — 5, and the product of 
 a^+ab + b^ by a — b, subtracted from the quotient, what 
 will the remainder be 1 Ans. a'^b-]-a¥-\-2b'^, 
 
 8. Divide x^+Sx7y+28x^y''+5t6xY+l[0xY+^Q^Y+ 
 2Sx'^y^ + Sxy^ + y^ hvx^ + 2xy +y^: and multiply ar* — 4<x"y 
 +6x^y^ — 4<xy^-\'y^ by"a?'^ — 2xy+y^, and find the sum and 
 difference of the results. 
 
 J. , ( Sum, 2x^-\^30xY+^0xY+2y^ 
 
 '^Difference, 12x^y-{-4<0xY+12xy\ ^ 
 
 9. Divide a^— 6^ by a^-ha^b^ + b^-, and multiply a""— 
 ah+b^ by a + b; and find the sum and difference of the 
 results. Sum, 2a\ Diff. 2bK 
 
 When a co-efficient, either numerical or literal, is com- 
 mon to every term of the divisor or dividend, the process 
 
DIVISION. 31 
 
 maybe shortened, by first dividing by that co-efficient, then 
 using the quotient instead of the given quantity, and mul- 
 tiplying or dividing the result by such co-efficient, accord- 
 ing as it was found in the dividend or divisor. 
 
 13. Divide Sax"^ — 9ax^y-{'9axy~ — 3ay^ by x^ — 2a7^ + y^ 
 Here the dividend =3a (x^ — 3x^y+3xy^ — y^). Hence if 
 we divide x^ — 3x^y-{-3xy^ — y'^ by x^ — 2xy+y^, and multi- 
 ply the resulting quotient x — y by 3a, we shall have 
 3ax — 3ay, the true quotient required. 
 
 14. Divide 5a* — 56* hja^+b\ Result, 5a^—5b\ 
 
 15. Divide '7a^—28a^b + 4<2a^b^—28ab^^7b^ by 3a''— 
 6ab+3b^ y Result, J(a^—2a6 +62.) 
 
 When the dividend and divisor contain no more than 
 two letters, with regularly ascending and descending 
 powers, the operation may frequently be abridged by 
 omitting the letters, and proceeding with the co-efficients. 
 Thus in the third example : page 29. 
 
 -3 + 3 — 1 And we perceive 
 from the first term 
 of the dividend and 
 divisor that the first 
 term of the quo- 
 tient is a?^5 hence 
 the result is readily 
 discovered. 
 
 1- 
 
 -1)1—4 + 6—4+1(1. 
 1—1 
 
 
 —3+6 
 —3 + 3 
 
 
 3 4 
 
 
 ,3-3 
 
 
 —1 + 1 
 
 
 —1 + 1 
 
 In the 10th example, the dividend, when the powers 
 are regularly arranged, is x^+0x^y-{-0x^y^-{-0x^y^-{'0x'^y*-\- 
 Oxy^ — y^. Hence we may proceed in this manner. 
 
92 INVOLUTION. 
 
 1 + 1)1 + 0+0 + + 0-1-0—1(1—1 + 1—1 + 1—1 
 
 1+1 
 
 —1 + 
 —1—1 
 
 And as— =a?55weread- 
 
 1 + ^ 
 
 ^ J] J ily discover the powers of 
 
 X and y which must be 
 annexed to these co-effi- 
 cients. 
 
 1 + 1 
 
 —1 + 
 —1—1 
 
 1 + 
 
 1+1 
 
 —1—1 
 —1—1 
 
 Section II. 
 INVOLUTION, OR THE RAISING OF POWERS. 
 
 36. To involve a given quantity to any power,* 
 
 Multiply the quantity by itself as many times as there 
 are units in the index of the given power diminished by 
 one. 
 
 EXAMPLES. 
 
 1. Required the 5th power of a^b^c, 
 (a^b^cfz^aH^^cK 
 
 *When the quantity to be involved is a simple one, multiply its 
 index, or indices, by that of the power proposed, observing that the 
 even powers of negative quantities are positive, and the odd ones 
 negative. 
 
INVOLUTION. 33 
 
 2. Required the 4th power of 2b — c. 
 
 2b —c 
 2b —c 
 
 U^—2bc 
 — 2bc +c3 
 
 453 — 45c _^c2 
 2b— c 
 
 863— 862c + 2bc^ 
 — 463c 4- 46c2 — c3 
 
 26— c 
 
 166*— 246"c +126V— 26c3 
 — 86'^c + 1262c2— 66c3+c4 
 
 166*— 326'^c 4-2462C2— 86c3+c* 
 
 3. Find the 3d power of ba^, 
 
 4. Required the 5th power of — 2a. 
 
 5. Required the 4th power of a +07. 
 
 6. Required the square of 3a — 26. 
 
 7. Required the 6th power of 5a +6. 
 
 8. Required the 3d power of a +26+ c. 
 
 37. A binomial is raised to any power, with great fa- 
 cility, by the following method.* 
 
 Set down, for the first term of the power, the first term 
 of the root involved to the given power. 
 
 The succeeding terms, without the co-efficients, consist 
 of the successive powers of the first term of the root, regu- 
 larly descending, joined to the powers of the second, regu- 
 
 * I'he reasons for this process are given in a subsequent part of this 
 ^^ork. -See Art. 98. 
 
34^ INVOLUTION, 
 
 larly ascending from the first ^ the common difference of 
 the indices being one. 
 
 The co-efficient of the second term is the index of the 
 given power; and if a co-efficient, already found, be multi- 
 plied by the exponent of the leading quantity, or first term 
 of the root, contained in the same term of the power, and 
 divided by the number of terms to that place ,- the quotient 
 will be the co-efficient belonging to the next term.* 
 
 JSTote, — To determine the signs of the different terms, 
 it must be remembered that the even powers of negative 
 quantities are positive, and the odd powers negative. 
 
 EXAMPLES. 
 
 Eequired the 5th power oi a-\-x, 
 
 a^ a^x a^x"^ a^x^ ax^ x^ 
 
 The co-efficients, 1, 5, —^— = 10, — - — = 10, etc. 
 
 Hence, (a-{-xy=a^+6a*X'^ 10a^x^-\- lOa^x^-^ bax^+x^. 
 
 Eequired the 6th power of 2x — 3y, 
 
 (2xy (2xy.3y (2x)\(3yy (2xy,(3yy (2a?)3.(3y)* 
 (2xX3yy (3yy 
 
 Or, 640^6 96a?5y 14<4<x^y^ 216xY 324a?Y 4^S6xy^ 729/. 
 
 6x5 15x4 
 
 The co-efficients 1, 6,--^r-=15,— ^— =20, 15, 6, 1. 
 
 Hence, (2x—3yy=64<x^—616x^y+2160xY-^4^320x'y^ 
 +4860a?2y*— 2916a:/+7292/«. 
 
 2. Eequired the 3d power of a+y, 
 
 Ans. a^ + 3a^y+3ay'^+y^ 
 
 =* The co-ellicients of any two terms, equally distant, the one from 
 the beg;inning, and the other from the end, are alike ; hence, the com- 
 putation of the first half of them will supply the whole. 
 
INVOLUTION. 35 
 
 3. Required the 4th power of a — 5. 
 
 Ans. a^—^a^h+^a^b^—^ah^^h^ 
 
 4. Required the 7th power oi x+y. 
 
 Result, a?7+7a?sy+21a?y+35a?*y3^, etc. 
 
 5. Required the 8th power of a — c. 
 
 Result, a«— 8a7c+28«6c2— 56a5cH'70a*c4, etc. 
 
 6. Required the 4th power of 2 + ^. 
 
 Result, 16+32a?+24a?H8^+^- 
 
 7. Required the 3d power of a — 3Z>. 
 
 Result, a^—^a''b-\'Tlah''—2W, 
 
 This method of involution is easily extended to trino- 
 mials, quadrinomials, etc., by considering two or more of 
 the terms as a single compound one. 
 
 8. Required the 4th power of a-\'h — c. 
 
 Here considering b — c as the second term of the bino- 
 mial, we have {a-\-b — c)*=a*+4a^(6 — c)+Qa\b — c)^-}- 
 4a(6 — c)^+(6 — c)*j and involving b — c, by the same me- 
 thod, to the powers indicated, we have (b — cy=b^ — 26c + 
 c2 ; (6 _ cy = b^—Sb^c 4- 3bc^—c^ ; (b — cy= ¥ — ^¥c 4- 
 6Z>2c3 — 45c^ + c* j consequently, (a-\-b — c)*= a* + ^a^b — 
 4a3c+6a262 — 12a2^c-f 6a2c2+4a63 — l2abH-\-12abc^ — ^ac^J^ 
 
 ^4 4,^3^2 _|. 6^2c2 ^J)c3, _J_ c4, 
 
 9. Required the 3d power oi x-\-y-\-z. 
 
 Result a?3 -)- 3a:2y ^ ^xy'^^y^ + 307^2: + 6a?yz+ 
 3/2; + 3a:2;2+ 33/2:24-03. 
 
 10. Required the second power of a 4-6 — c — d. 
 
 Result, a2^.2a64-62— 2ac— 26c4-c2— 2a6?— 
 ^bd+'Zcd-ird^ 
 
 11. Required the 4th power oi x-\-y — 32r. 
 
 Result, a?*4-4a?^2/4-6a?'^2/2_j.4a?y3 4_/ — 12a?'^2: — 
 3<6x''yz—3Qxy'-z—11y^z 4- ^4^x^z^+ lOSxyz^ 
 +b4<y^z^—lOSxz^—10Syz^+81z\ 
 
36 EVOLUTION. 
 
 12. Required the 5th power of a+6+c. 
 
 Result, a^ + 5a45+10a^62+l0«253+5aZ>*+55 
 
 I0a''c^+30a^bc^+30ab^c^+10b^c^+10a^c^ 
 + 20abc^+l0b^c^+6ac^ i-bbc^+c\ 
 
 13. Required the 3d power of a+b — c — d. 
 
 Result, a^ + 3a^b + 3ab^+b^ — 3a''c — 3a^d—6abc 
 — Qabd—3b''c—3b^d+ 3ac^ + 3bc^ +6acd+ 
 6bcd+3ad^+3bd^—c^ — 3c^dr—3cd^—d\ 
 
 EVOLUTION, OR THE EXTRACTION OF ROOTS. 
 
 Case I. 
 38. To extract the root of a simple quantity. 
 
 Extract the root of the co-efficient, for the numeral part ; 
 and divide the index or indices of the literal part, for the 
 exponents of the root.* 
 
 When the root cannot be extracted, it must be indi- 
 cated as in definition 16. 
 
 EXAMPLES. 
 
 1. Required the ^th root of 266a*x^, 
 
 4 8 
 
 v/256 = db4, the co-efficient; a'^x'^^ax^ the literal part. 
 Hence, the root required is 4}ax% or — 4aa?^. 
 
 2. What is the 5th root of—baH^'^l 
 
 The 5th root of 5 is a surd, and must be indicated 
 1 
 thus: y 5 or b^. 
 
 Hence, the root required is —5'^a^b^ or — a^b^^6. 
 
 * The odd roots have the same signs as their powers ; but the even 
 roots of positive quantities may be either positive or negative. The 
 even roots of negative quantities are impossible. These impossible or 
 imaginary roots frequently become the subject of important investiga- 
 tions, as will appear in the sequel of this work. 
 
EVOLUTION. 37 
 
 3. Required the square root of 16a'^b\ 
 
 Eesult, ±:4^abK 
 
 4. What is the square root of 626oc^y^1 
 
 Ans. d[=25a?y. 
 
 5. Required the cube root of — 12oa^b", 
 
 Result, —5a^b. 
 
 6. What is the 5th root of—3'2aH'c^^1 
 
 Ans. — 2a^bc^, 
 
 7. What is the 3d root of 7^y? Ans. x^^Ky^ ■ 
 
 8. What is the 4th root of Sla'b^l 
 
 Ans. ±i3as/b% or dzSab^b. 
 
 9. Required the 5th root of —24<3a'b^ 
 
 Result, — 3aby/b. 
 
 Case 2. 
 
 39. To extract the square root of a compound quantity. 
 
 Arrange the terms according to the dimensions of some 
 letter, beginning with the highest. 
 
 Take the square root of the first term for the first term 
 of the root, and subtract its square from the given quan- 
 tity. Double the root thus found, for a defective divisor, 
 divide the first term of the above remainder by this defec- 
 tive divisor, and annex the result both to the root and to 
 the divisor. 
 
 Multiply the divisor thus completed by the term of the 
 root last obtained, and subtract the product from the for- 
 mer remainder. 
 
 Divide the first term of the remainder as before, for the 
 next term of the root ; add this last and the preceding 
 term of the root to the last complete divisor for a new 
 divisor. Multiply, subtract, and proceed as before. 
 
 J^ote, — In these additions, regard must always be paid 
 to the signs of the quantities. 
 4 
 
38 EVOLUTION. 
 
 4^3 + 1 2a5 + %^—20ac—30bc + 2bc%^a + 36—5c 
 
 4a+36 \12a5 + 9Z>2 
 36— 5cll2a6 + 962 
 
 4a+6^>— 5c \— 20«c— 306c + 25c2 
 
 /— 20ac— 306c+25c3 
 
 2. Required the square root of x'^'\-2xy+y^. 
 
 Result, x~{-y. 
 
 3. Required the square root of a*4"4*^^+^^^+'^<^4■l• 
 
 Result, a2+2a+l. 
 
 4. Required the square root of x* — '^ax^+Qa-x^ — i^a'x 
 4- a*. Result, x^ — ^ax+a"-, 
 
 5. What is the square root of 16 x' + 24^x^ + 89 x^+ 60 x 
 + 100. Ans. 4a?2+3a? + 10. 
 
 6. What is the square root of 4a^ — 12a^x+ba^x^+6ax'^ 
 +a?*'? ' Ans. 2a'—3aX'^x'', 
 
 7. What is the 4th root of a'+12a''b+54^a'b^+lOSab^ 
 + 816*'?^ Ans. a+3b. 
 
 8. What is the 4th root of a^^4<a^b^ + 6a'b^-^4^a^b^ 
 +6«'? Ans. a^—b\ 
 
 Case 3. 
 40. To extract any root of a compound quantity. 
 
 Arrange the terms as before directed 5 take the root of 
 the first term for the first term of the root, and subtract 
 its power from the given quantity. 
 
 Take for a divisor twice this root, three times its square, 
 four times its third power, five times its fourth power, etc., 
 according as the 2d, 3d, 4th, 5th, etc. roots are required 
 to be extracted 5 divide the first term of the remainder, and 
 annex the result to the root before found. 
 
 *The 4th, 8th, 16th, etc. roots, may be obtained by 2, 3, 4, etc. 
 extractions of the square root. 
 
EVOLUTION. 
 
 39 
 
 Involve the whole root, thus obtained, to the given 
 power, and subtract from the given quantity. 
 
 Divide the first term of the last remainder by the former 
 divisor, annex the result to the root, and proceed as before. 
 
 1 
 
 « 
 
 1 
 
 
 ^ 
 
 
 
 
 CO 
 
 ^ 
 
 CO 
 
 
 
 
 + 
 
 00 
 
 H- 
 
 
 
 
 e 
 
 o 
 
 53 
 
 
 
 
 CM 
 
 1—1 
 
 C^ 
 
 
 
 
 Jl 
 
 1 
 
 V--' 
 
 
 
 
 A 
 
 '^ 
 
 
 
 
 o 
 
 t<5 
 
 ^ 
 
 
 
 
 r* 
 
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 *^ 
 
 
 
 
 CO 
 
 ^ 
 
 
 
 
 
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 -•a^ 
 
 
 
 
 
 
 tH 
 
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 ^ 
 
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 rH 
 
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 n 
 
 
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 r<5 
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 rH 
 
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 00 
 
 
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40 E\rOLTJTION. 
 
 The reason of this process, as well as of that laid down 
 under the former case, is readily seen if we consider that 
 in the involution of powers, the second term of the power 
 consists of the continued product of the index of the 
 power, the first term of the root raised to the next inferior 
 power, and the second term of the root. Thus, if we 
 raise a-{-b to the 2d, 3d, 4th, etc. power, we find the second 
 term 2ab, Sa^b^ 4a-^Z>, etc., and the first term of the power 
 consists of the first term of the root involved to the power 
 in question. 
 
 2. Required the 3d root of 21x^—54^x''y + 36xy"—Sy\ 
 
 Ans. 3a? — 2y. 
 
 3. What is the 3d root of a?'^— 60?^+ 15a?*— 20a?^+15a?'2 
 —6a? -f- 1'? Ans. a?^— 2a? + 1 . 
 
 4. Required the 5th root of 32a?5— 80a?*+80a?^— 40a?2-f 
 lOo?— 1. Result, 2a?— 1. 
 
 5. Required the 4th root of 16a^—96a^x+216a^x''- 
 216«a?3 + 81a?*. Ans. 2a— 3a. 
 
 6. Find the 5th root of a?^^+10a?9+35a?3+40a?7— 30a?«— 
 68a/5+30a?*+40a?3— 35a?^+ lOo?— 1. 
 
 Result, a?H2a?— 1. 
 
 41. J^ote, — The roots of compound quantities may 
 sometimes be obtained, with great facility, by a method of 
 trial. To effect which, it may be observed, that the power 
 of a binomial consists of as many terms as there are units 
 in the exponent, increased by one. Hence, if the number 
 of terms of a quantity, whose root is to be extracted, ex- 
 ceeds this sum, we may conclude that the root consists of 
 more terms than two. 
 
 When the root is judged to be a binomial, take the roots 
 of the extreme terms, and connect the results by the sign 
 + or — 5 but when the root appears to contain more terms 
 than two, take also the roots of one or more of the other 
 given terms, and connect the roots thus obtained by the 
 signs + or — , as may be judged proper. 
 
 Involve this supposed root to the given power, and if 
 
ALGEBRAIC FRACTIONS. 4fl 
 
 the result agrees with the quantity given, the root is mani- 
 festly correct. 
 
 This expedient, however, is not likely to abridge the 
 labor of a student in the early periods of his course. 
 
 Section III. 
 
 ALGEBRAIC FRACTIONS. 
 
 42. Algebraic fractions depend upon the same princi- 
 ples as those in common arithmetic, and are managed by 
 similar rules, proper regard being paid to the signs and 
 algebraic modes of operation. 
 
 EXAMPLES. 
 
 1. Required the greatest common divisor of cc^ — b^Xy 
 and x'^+^bx+bK* 
 
 x^—b''x)x^+2bx+ b\\ 
 
 25a: +263 
 x-\-b)x'^ — h\x — h 
 x^-{-bx 
 
 —bx--b^ 
 —bx—b] 
 Hence, x+b is the divisor required. 
 
 * In examples of this kind, we are frequently obliged to adopt 
 expedients which are not usual in common arithmetic. When every 
 term of the divisor has a factor, which does not contain a letter or 
 number common to every term of the dividend, we divide by it, and 
 use the quotient as a divisor. When the divisor has been thus re- 
 duced, it often happens, that the division is still impracticable ; in 
 4* 
 
42 ALGEBRAIC FRACTIONS. 
 
 3^2 2a 1 
 
 2. Required to reduce ^ ^ ^g ' , -i ^^ its lowest 
 
 terms."^ 
 
 First to find a common divisor. 
 
 3a2_2a— -1) 4a^— 2a3— 3a+ 1 
 3 
 
 12a^— 6^2— 9a.+ .3(4a 
 
 2a3_ 5a + 3 
 3 
 
 6a^—15a+ 9(2 
 6a3_ 4a_ 2 
 
 — lla+11 
 
 which case, we multiply the dividend by such a number as will make 
 its first term a multiple of the first of the divisor. 
 
 By these processes we make no change in the common divisor, un- 
 less we divide by a number or quantity which is common to the given 
 terms ; or introduce into one of them a divisor of the other. 
 
 * Solutions may sometimes be facilitated, by resolving the quanti- 
 ties concerned into their component factors. In this example the 
 numeratoris evidently =3^^ — 3a-^a — l=3a (a — 1) -f a — l=(3a-|- 1) 
 (a — 1); and the deuomin?i.tor=^Aa^—4a^-\-2a^—2a^a'\-l=4a^{a—l) 
 Hr2«(a— l)--l(o— l)=(4a24-2a— l)(a— 1). Hence a— 1 is mani- 
 festly the common divisor, and ' ^ . ^ : the required result. 
 
 In example 3, we have the numerator= o3(a2 — l!^), and from ex- 
 ample 8, page 25, it is seen that a'^—'b^=(^a^—b^){a^J^h2), Hence, 
 
 ai — J2 IS the common divisor ; and — ;; — — •= „ , -- 
 
ALGEBRAIC FRACTIONS. 43 
 
 Com. div. a — l/Sa^ — 3a 
 
 3. Reduce— — rr— to its lowest terms. Result, r- '^■- 
 
 4. Reduce ; — to its lowest terms. Result, 
 
 xy+y y 
 
 P,a^+10a*b +5a^b^ . , 
 
 5. Reduce --j—, — o ^i:^ ■ n ?,■. . j.^ to its lowest terms. 
 a^b-^ 2a^b^-{'2ab^+b'^ 
 
 ^ , 6a^+da^b 
 Result, ■ 
 
 ^a^b^ab^+b^ 
 
 6. Reduce — -to its lowest terms. 
 
 a^ — a^x — ax^ + x^ 
 
 Result, 
 
 ^ ^ , 6a'' + lax—-3x^ . , 
 
 7. Reduce^— ——r rrrrto its lowest terms. 
 
 6a^+llax-\-3x^ 
 
 Result, TT — ; — 
 ^ 3a+x 
 
 8. Reduce ^ - , ^ „ . ^ to its lowest terms. 
 
 Result, ^ 
 
 '3a?2+2a?+l 
 
44? ALGEBRAIC FEACTIONS. 
 
 (I 2c 3x 
 9. Reduce oT? q3 7~> to other equivalent fractions 
 
 having a common denominator.* 
 
 6a^d Sabc . %dx 
 '12abd' nabd' T2M 
 
 10. Reduce t-^-tt'-, and ^r- to a common denominator. 
 
 9b Sac 6ad 
 ^^^ '12^2' 12a^' 12^ 
 
 1 3a^ 2a^4-5'^ 
 
 11. Reduce TT,— 7- , and 7-; to a common denomi- 
 
 nator. 
 
 4a 4- 45 9a^ + 9a25 24^a''+12b^ 
 
 Result, 
 
 12a+126' 12a + 126' 12a +126 
 
 * The least common denominator should always be foand in exam- 
 ples of this kind. 
 
 To find the least common multiple of two or more numbers or quanti- 
 ties, divide such of them as have a common divisor, by the greatest 
 prime number they contain. Repeat the process on the quotients and 
 undivided quantities as long as a divisor common to two of them can be 
 found ; then multiply these divisors, the several final quotients, and 
 undivided quantities continually together, the product is the common 
 multiple required. For such product must contain all the factors in- 
 cluded in the given quantities, and none of them are repeated oftener 
 than in the given terms. Thus to find the least common multiple of 
 4a^, Sab, 66^, IQbc. The only common measures of the co-efficients 
 are 2 and 3, and the literal divisors, common to two or more, are a 
 and b. Arranging them thus : 
 
 2)4a3, 3ab, 652, 105c 
 
 2a3 Sab Sb'^ 5bc 
 2^3 ab b^ 5bc 
 2^3 b 53 5bc 
 2«3X 1 XiX 5c X 2X3c5=60a3i% the quantity 
 
 sought. 
 
ALGEBRAIC FRACTIONS. 45 
 
 12. Reduce ttt^-tt j -, to a common denommator. 
 
 40 3 a ' 
 
 93a 80aa? 120a^ + 240a? 
 •^^^" ' 120^' 120^ 120a 
 
 2a7 
 
 13. Reduce a — Trrto an improper fraction. 
 
 3^—207 
 
 Result, — 
 
 3a 
 
 2a7 8 
 
 14. Reduce 10 -| — ^ — to an improper fraction. 
 
 32x— S 
 Result, — ^ • 
 
 15. Required an improper fraction equivalent to 
 l+2a:— ^ . Result, ^^ 
 
 16. What is the equivalent improper fraction to a — 
 
 .22 — a^ 2a- — z^ 
 
 7 Ans. 
 
 a ' a 
 
 3a3? ' 1 4^?^ 
 
 17. Reduce — — to a whole or a mixed quantity. 
 
 Result, 3a? -4 
 
 a-^x 
 
 2a3+252 
 
 18. Reduce -. — to a whole or mixed quantity. 
 
 4Z'2 
 
 Result, 2a +25+ =. 
 
 a — b 
 
 ^^ ^ , 27a3— 36^—40? +9a2 ^ 
 
 19. Reduce ^— to a mixed quantity. 
 
 Result, 3«+l-?^^ 
 
46 ALGEBRAIC FRACTIONS, 
 
 20. Eeduce — — to a whole or mixed quan- 
 
 tity. 
 
 _, , '^ax^' — h 
 Result, x^ — a^ -\ 
 
 21. Add ^ 3^r and ^.^ together. Sum, -^^^— 
 
 22. Required the sum of a — -r- and h-\ ■. 
 
 _ , _ 2ahx — 3c^2 
 Result, a-\-b-\ ^ • 
 
 ^ _i_ 3 2a 5 
 
 23. What is the sum of 2a+ — ^, and ^a-\ — ^ 
 
 14«— 13 
 Ans.GaH ^^j — 
 
 24. Add — 1-, to — —,. bum — - — 7— or 2+-; — ji 
 
 a — b a-\-b a^—b^ a\ — 6-, 
 
 a a — X 
 
 25. What is the sum of 2a, and — --1 
 
 ^ a — X a-\-x 
 
 Sx^ — ax 
 Ans. 2a+2+ ^^3^ 
 
 ^ Sa 4)a -n. . 1 ^^ 
 
 26. From-x- subtract-y- Remainder,f)g 
 
 ^ Sax , 2ax ^ ax 
 
 27. From-^- take -tt . Rem. j^ 
 
 2a — b Sa — 46 
 
 28. From — -. — subtract — -^t — . 
 
 Qab—3¥—12ac+16bc 
 Kem. j23^ 
 
ALGEBRAIC FRACTIONS. 47 
 
 29. From 4a -{ — - — subtract 2a — ■^-— — ^ 
 
 lOa^ — 5a6 — 7ac + 1 lie 
 Eem. 2«+ ^^^^^^ 
 
 a-\'CO a — X 4a? 
 
 30. Subtract^-; . from a-{- — — — r Kem. a- 
 
 a(a — x) a(a + xy a^ — x^ 
 
 31. Required the continued product of -r -q- and 
 a — X _, - , 2a — 2a? 
 
 32. Required the product of . ^ and — g^v. 
 
 Prod. "^ 
 
 ' «6+c2 
 
 33. Required the continued product of a r "TTT 
 
 3a? , 76 ^ , 216 
 
 75 and 7- r, Frod.-y — 
 
 a — ox — a?^. b — x 
 
 34. Required the continued product of— 7-7-7 — r^' 
 
 fi> -J" c? ax ~Y~ X 
 
 ax 
 
 and 0?+ Prod. 2a^ — ax — 2a64-6a? 
 
 a— —a?* 
 
 3D. Divide ^ by ^^-. Quotient,g- 
 
 36. Divide o"- — by-7r^ — . Quot. —^ - 
 
 Soay ^ 2oyz la 
 
 37. Divide a by a+- . Quoi. - 
 
 a + x ^ a — X a^ -i~ax 
 
 38. Divide -— - by -— -. — --7, Quot. 1 
 
48 ALGEBRAIC FRACTIONS. 
 
 on n- 'A 3^— li. 1-^+^ n r^^ 
 
 39. Divide ^^-by-^^. Quot. -^^^ 
 
 40. Divide -vrT^ by-Tj-r Quot. —pr — . 
 
 12cd ^ bd. ^ I0ac2 
 
 ^1- ^^^^^^ 2a^-4aH~2^^ ^^ "4-^-4T-* 
 
 Quot. 2a + ' 
 
 a 
 
 42. Divide ^z:^^-j^-^^y -z:^. Quot. a:+~ 
 
 Quot. a3+2a2a?+2aa?2+.r'^ 
 
 13 4 4 
 
 44. Divide a:* — -^x'^'\-x^'\--kx — 2byg^ — 2. 
 
 3 1 
 
 Quot. jo?"^ — 2^^+l. 
 
 * By inverting the divisor, and resolving the terms into their com- 
 ponent factors we have — 
 
 2{a—b) (a—b) ^ 5a(^b) 
 
 "Whence by expunging the corresponding factors, from the numerator 
 
 and denominator we find the fraction reduced to — 2a + 
 
 a ' a 
 
 In like manner — 
 
 a:^— 3^ a:~h __{x^ + b^Xx + h)(x—b).(a;—b) _ x2 + b^ 
 
 x^—2bx + b^Xc^-\-bx '(x—b){x^b),x(^x'fb) ~ ^ 
 
 = X-^ 
 
 ' X 
 
ALGEBUAIC FK ACTIONS. 49 
 
 45. Eequired the 4th power of -3^. Result, ^^ 
 
 a — b 
 
 46. Required the 3d power of -^r^ 
 
 Result 1- 
 
 '^a^ + 3a^b-{'3at^-\-b^ 
 
 2x 
 47. What is the 4th power of 1 — —7 
 
 a 
 
 Sx 24a?2 32a:3 I6x^ 
 Ans. 1--+-^--^+-^ 
 
 64a?^ 4 J? 
 
 48. What is the cuhe root of ^7^'^ '^^^'"Sa^ 
 
 49. Required the square root of J^il^- 
 
 3a^ 
 Result, 
 
 2xy^ 
 
 11 3 
 
 50. Required the square root of x^—Sx^^-^x^—^x 
 
 1 3 1 
 
 +_. Result, a^'— 2^4-j 
 
 1 1 41 43 
 
 51. What is the 3d root of go^s— -a:^ +— a:*— — a:^ 
 
 41 1 1 .1^1 
 
 52. Required the square root of a^+2b. 
 
 b b^ b" 55* 
 Result, a+~^3+^^g^, etc. 
 
 5 
 
50 EQUATIONS. 
 
 Section IV. 
 EQUATIONS. 
 
 43. An equation is an algebraic expression, indicating 
 that one quantity, or combination of quantities, is equal 
 to another. The quantity, or combination, which stands 
 on either side of the sign of equality, is called a member 
 of the equation. 
 
 Thus, ax-]-bx=ah-\-cd, is an equation of which ax-\-bx 
 is the first member, and ah-\-cd the second. 
 
 44. In the application of algebra to the solution of 
 problems, the quantities concerned, both those which are 
 given, and those which are required, are usually repre- 
 sented by symbols, and the conditions of the problem 
 translated into the language of algebra. One or more 
 equations are thus formed, including unknown as well as 
 known quantities. 
 
 45. The number of independent equations required for 
 the solution of a problem, is equal to the number of un- 
 known quantities employed. 
 
 46. A simple equation is one which includes only the 
 iirst or single power of the unknown quantity or quanti- 
 ties, as ax+h=^cx — d, 
 
 47. A quadratic equation is one whiclr includes the 
 square of the unknown quantity. An equation which 
 contains no power of the unknown quantity, but the 
 square, is called a simple qnadraticy as ax^=^h; but when 
 the square and simple power of the unknown quantity are 
 both included, it is called an adfected quadratic equation; 
 as x^ — ax=bc, 
 
 48. A cubic equation is one which contains the cube of 
 the unknown quantity; as o(f^-{-aX'-i-bx=c, 
 
 49. A biquadratic equation is one which contains the 
 fourth power of the unknown quantity; ns x^-\'aV-{-c'''X 
 =^bH\ 
 
 50. Equations, in general, are said to be of as many 
 
SIMPLE EQUATIONS. 51 
 
 dimensions as there are units in the index of the highest 
 power of the unknown quantity. 
 
 51. In general, an equation which contains but one 
 power of the unknown quantity is called a pwe equation; 
 thus, x'^—-b-\-c^ is called a pure quadratic equation; x^=b, 
 a pure cubic equation, etc. 
 
 Great part of the business of algebra consists in de- 
 ducing, from given equations, the values of the unknown 
 quantities which they contain. This is effected by making 
 correspondent changes in both the members of the equa- 
 tion, so as to obtain, eventually, the unknown quantity on 
 one side of the sign of equality, and known quantities on 
 the other. 
 
 SIMPLE EQUATIONS. 
 
 Containing one unknown quantity. 
 
 To facilitate the solution of equations generally, the fol- 
 lowing principles or modes of operation must be observed. 
 
 ♦ Mode I. 
 
 52. Any quantity may be transposed, that is, removed 
 from one member of the equation to the other, by chang- 
 ing its sign.* 
 
 EXAMPLES. 
 
 Given '7x-\-5a=:6x-{-3b to find x. 
 By transposing ba and 6x, we have Ix — 6x=3b — 5a, 
 or x=3b — Sa, 
 
 * Hence, if any term be contained in both members of the equa- 
 tion with the same sign, it may be taken from both; and the signs of 
 all the terms msy be changed, without destroying the equation. 
 
52 SIMPLE EQUATIONS. 
 
 2. Given Sy— 3=:7i/+5 to find y. Result, 2/=8. 
 
 3. Given 15a?— 6 =140? + 20, to find x. Result, a?=:26. 
 
 4. Given 20— 132^=45— 14y+ 4, to find y. 
 
 Result, ?/=29. 
 
 5. Given 8a?+ 15=15+907—17, to find cc. 
 
 Re :ilt, a:=17. 
 
 6. Given 25a? — 5=24a? + 26 — c, to find a?. 
 
 Result, a?= 36 — c. 
 
 7. Given 36— 9a?=76— 8a?— 4J, to find a?. 
 
 Result, a?=4c^ — 46. 
 
 Mode 2. 
 
 53. The CO- efficient of an unknown quantity may be 
 taken away, by dividing every term of the equation by it. 
 
 EXAMPLES. 
 
 Given ax-{-ba=ad, to find a?. 
 By dividing by a, we have x-{-b=d; hence by transpo- 
 sition, x=d — b. 
 
 2. Given 5a?== 25, to find a?. Result, a?=5. 
 
 3. Given 30a?=180, to find a?. Result, x=6, 
 
 4. Given 6a?+ 17=44 — 3a>, to find a?. Result, a?=3. 
 
 5. Given 3aa? — 4a6=2aa? — 6ac, to find a?. 
 
 Result, a?=46 — 6c. 
 
 6. Given 3a?- — 10a?=8a?+a?2, to find a?. 
 
 Result, a?=9. 
 
 7. Given 6y^—Sy''=22y^—Sy^, to find y. 
 
 Result, y=3i 
 
 Mode 3. 
 
 54. An equation may be cleared of fractions by multi- 
 plying all its terms by any common multiple of the deno- 
 minators. 
 
SIMPLE EQUATIONS. 53 
 
 _,. X X X 6 ^1 
 
 Given - — r + -=l7T5to find x. 
 2 3 5 o 
 
 Multiply by 30, and then we have 15a? — 10x-\-6x=b5; 
 whence lla?=55, and a?=5. 
 
 XXX 
 
 2. Given —+—-==: — + 7, to find x. Result, x=12. 
 
 ^ ^. ^—5 ^ 284—07 ^ , 
 
 3. Given —r — [-Qx= — ^ , to find x. 
 
 Eesult, x=9, 
 
 ^. 11 — X 19 — X ^ _ T-, , 
 
 4. Given a? -1 ^ — = — ^ — , to find a?. Kesult, a7=::5. 
 
 . ^. o 2a?+6 , llo;— 37 , , 
 
 5. Given 3a? H — =5-^ , to find x. 
 
 Ai 
 
 Result, a?=7. 
 
 Mode 4. 
 
 55. When the unknown quantity is under a radical 
 sign, such sign may be taken away, by first transposing 
 the terms, so as to leave the radical quantity alone on one 
 side of the equation, and then involving each member of 
 the equation to the proper power. 
 
 EXAMPLES. 
 
 Given ^^+40=10— ^/a?, to find a?. 
 Squaring both sides, a?+40=100 — 20 v/a?+a?. 
 By transposition, 20^/^=60, or ^/a?=3. 
 Squaring both members, a:=9. 
 
 2. Given x/ a? + l— 2:rr3, to find x. Result, a?=24. 
 
 3. Given 4^3^+4 + 3=6, to find x. Result, a?=7f . 
 
 4. Given sf x — 16 = 8 — sf x^ to find x. Result, a?=25. 
 
 5* 
 
54? SIMPLE EQUATIONS. 
 
 5. Given v/4a4-a7=2V^+^ — V^j to find a?. 
 
 Eesult, 0?= — ^ i — 
 
 2a — b 
 
 6. Given 's/4>a^+x^=:i^U^+x*, to find x. 
 
 Result, a:=V-2^ 
 
 7. Givena? + a+\/2aa?-[-^^=^ to find x, 
 
 b^—2ab+a^ 
 
 Result, X= pry 
 
 Mode 5. 
 
 56. When the member of an equation, which contains 
 the unknown quantity, is a complete square, cube, etc., 
 the equation may be reduced by extracting the square root, 
 cube root, etc. of both its members.* 
 
 EXAMPLES. 
 
 Giv^n x^—6x^+ 12a?— 8= 343, to find x. 
 Extracting the cube root, x — 2=7, or x=9, 
 
 2. Given a?24.4a?4-4=81, to find x. Result, a?=7» 
 
 1 81 
 
 3. Given ^H^+T=-r' ^^ ^^^ ^* Result, a?=:4?. 
 
 4. Given x^+Sax'^+Sa^x+a^^b^^ to find x. 
 
 Result, x=b — a 
 
 5. Given 9a?2+17=:98, to find x. Result, x=3. 
 
 6. Given a?*— 4a?^-f 6a?2~-4a?+l = 256, to find x. 
 
 Result, a? =5 
 
 7. Given 3a?5=96, to find x. Result, x=% 
 
 * These modes of reducing equations depend upon a few self-evi- 
 dent truths. When two quantities are equal, and equal quantities are 
 added to or subtracted from both, or both are divided cr multiplied by 
 the same number, or the same power or the same root of each is taken, 
 the results in each case are equal to each other. 
 
SIMPLE EQUATIONS. 55 
 
 PEOMISCUOUS EXAMPLES. 
 
 1. Given -0? — 2;a?+-a7=-, to find x. Result, x=l^ 
 
 4a 
 
 2. Given s/x-^- ^2a+x= — , to find a?. 
 
 Result, x=-^a 
 o 
 
 Result x=-xy/ — r— 
 Z 
 
 4". Given a + x= ^/ a^+x s/ ^^^-i- x\ to find x. 
 
 Result, x-= a 
 
 5. Given 2+ v'3a?=\/4<+5a7, to find a?. 
 
 Result, x—\% 
 
 6. Given a — a?= , to find a?. Result, x=~ 
 
 a — X ' 2 
 
 7. Given V«+3?+V«^ — ^= \/aa:, to find a:. 
 
 4a^ 
 Result, 0?=-- — 7 
 
 a a 5 — 2a 
 
 8. Given --; — l-:; =5, to find x. Result, a:= V — t — 
 
 ^ ^. 7a?+16 a?+8 a? , , 
 
 9. Given -jj ^___^^, to find a.. 
 
 ^_ ^. 4a? + 3 , 7a?— 29 8a?+19 ^ . , 
 10. Given^^+g^-^=-^g-,tofinda?. 
 
 Result, a?=8. 
 
 ind X, 
 Result, a?=6. 
 
56 SIMPLE EQUATIONS. 
 
 11. Given — ^r— : — - — : :7 : 2, to find a?.* 
 
 <w 
 
 Eesult, a:=3/g 
 
 ijto find 0?. 
 
 12. Given \/5+a7+\/a?=:\/5+a? 
 
 Result, a?=4'. 
 
 13. Given a?*4-2a:2+ 1=4225, to find x. Result, a?=8. 
 
 14. Given a?s_9a?*+ 270-2— 27=2197, to find x. 
 
 Result, a?=4. 
 
 15. Given a?^— 17=130— 20?^, to find x. Result, a?=7. 
 
 16. Given a?^— 100=412— 7a:3, to find x. Result, a?=4. 
 
 17. Given a?*— 4a?^+6a?2—4a?+ 1=4096, to find x. 
 
 Result, a?=9. 
 
 18. Given x^—^ax''-\'l^a^x—'^a?—h% to find x. 
 
 Result, a?=62_|.2«. 
 
 19. Given 625a:4— 1000a?3+ 600072— 160a? +16 = 256a?S 
 to find X, Result, a?=2. 
 
 20. Given ^la^x''+b^ha^x'^-\-^^b^ax-\'^¥=12bc^xS to 
 find X. 
 
 Result, X—-Z — 
 
 ' 5c — 3a 
 
 21. Given x-\- s/a — x=^\/ a- — x to find x. 
 
 Result, x=a — 1. 
 
 22. Given V4^+ Va?*— ^^=a?— 2, to find a?. 
 
 Result, x=2\ 
 
 * If four numbers are proportional, the product of the extremes is 
 equal to that of the means ; hence an analogy is readily converted into 
 an equation. Vide Article 62. 
 
SIMPLE EQUATIONS. 57 
 
 4 . 
 
 23. Given V24-a?+v^a?=N/2+a? to find x. 
 
 Result, a?=S 
 
 Section V. 
 SIMPLE EQUATIONS. 
 
 Containing two unknown quantities. 
 Rule I. 
 
 57. Multiply or divide one or both equations, by such 
 numbers or quantities as will make the co-efficients of one 
 of the unknown quantities the same in both equations. 
 Then take the sum or difference of these equations, accord- 
 ing as the signs of the corresponding terms are different, or 
 the same ; the result will be a new equation, containing 
 but one unknown quantity.* The value of the remaining 
 unknown quantity may then be determined by the methods 
 in the last section. 
 
 EXAMPLES. 
 
 Given 4r+2/=34«, and ^y-{-x=lQ^ to find the values of 
 X and y» 
 
 First, to eliminate a?, we multiply the second equation 
 by 4^ ; whence, 16y+4a?=64'. From this we take the first 
 equation and 15y=30; whence y=2. 
 
 The value of y being now obtained, we find x— 16 — 4?/ 
 =rl6— 8=8. ' 
 
 Or to eliminate a?, divide the first equation by 4, and 
 1 34 
 we shall have ^+"T'3^~X' which subtracted from the 
 
 second equation, leaves —y=—5 whence y= 2, as before. 
 
 *When, by this or any other process, a quantity or letter is caused 
 to vanish from the equations, it is said to be eliminated. 
 
58 SIMPLE EQUATIONS. 
 
 2. Given ax-\-byt=m, ex — dyz=n^ to determine the 
 values of x and y» 
 
 To eliminate a?, multiply the first equation by c, and the 
 second by a, whence acx-\-bcy=^cm^ and acx — ady=^an; 
 by subtraction, bcy-^ady=cm — an; and by dividing by 
 cm — an 
 
 Or, to eliminate y, multiplying the equations by d and b 
 
 respectively, adx-{-bdy=dm, bcx — bdy=bn; adding these 
 
 " dm + bn 
 
 equations, adx-\-bcx=^dm-\-bn; whence a?= . , 
 
 * 
 oc' 
 
 6. uiven ^ g^^3y^39^ to find the values of a? and y. 
 
 Result, a:=3, y=5. 
 
 , p. ( 15a?+34y=241, 
 
 ^- ^iven ^ 18a?— 172/= 58, to find the value of x and y. 
 
 Result, 07=7, 2/=4. 
 p. p. C 6a7 — 4^y=34, 
 
 0. uiven ^ 9^_2a?=27. Required the values of x 
 and y. Ans. a?=9, y=5. 
 
 ^'^^^ ^ 4^ — lyz=^^ to find the values of a? and y. 
 
 Result, a?=8, 3/=4<. 
 
 7. Given . 
 
 X y 
 2^3 
 
 X y 
 -o' + "o"=8) to find X and y. 
 
 Result, a:=6, y=12. 
 
 *The known quantities being expressed in general terms, this 
 solution is applicable to the following exanaples. 
 Thus if in the 6th example we make — 
 
 700 + 12 
 af=ll, ^=3, m=^100, c=4, d=l, w=4, «=rz— ^77^=8, 
 
 77 -7- 1-* 
 ___400— 44__ 
 
 ^"~77 4-12""^* 
 
SIMPLE EQUATIONS. 
 
 59 
 
 S. Given. 
 
 15 20 
 
 y_ 
 1 10 
 
 30 
 
 = 1, to determine x and y. 
 
 Result, 3?=: 30, ?/=20. 
 
 Rule 2. 
 
 58. Find, from each equation, the value of one unknown 
 quantity in terms of the other quantities* 
 
 Assume the values thus obtained, as the different mem- 
 bers of a new equation. The value of the unknown quan- 
 tity contained in this equation may then be determined as 
 in the last section. 
 
 EXAMPLES. 
 
 Given 
 
 707+^=51, 
 
 9x — -|- = 61i, to find the values of a? and y. 
 
 o 
 
 From the first equation y= 357 — 49a7. 
 From the second equation y='72x — 490, 
 Hence, 72a:— 490=357— 49a7, 
 72a:+49T=357+490, 
 
 _847_ 
 
 ^-121-^' 
 
 y= 357— 49a:= 357—343 = 14. 
 
 2. Given 
 
 and y. 
 
 y + 72/=99 
 
 7a7+-^=51. 
 
 Required the values of x 
 Ans. a?-^7, y=14. 
 
60 
 
 SIMPLE EQUATIONS. 
 
 3. Given , 
 X and y. 
 
 ^+2 ^ 
 
 2/ + 5 
 
 + 10a?= 192. Eequired the values of 
 Ans. 07=19, ^=3. 
 
 4. Given 
 
 5. Given , 
 
 
 X y 
 -=r — oT~'^* Quere the values of x and y 1 
 
 Ans. a?=20, 3/=:18. 
 
 3 "^ 5 ~^^' . 
 
 3a? 2y 67 ^ . , 
 
 -^+-o-="3'* Quere the values of x and y ? 
 
 Ans. 07=15, y=20. 
 
 an- ( ^ — ^y=^i 
 
 ^^^^ ^ x:y::a:b. Quere the values of o? and y? 
 
 bd 
 
 ad 
 Ans. 0?= ;tt, y 
 
 a_^26' ^"~«— 26 
 
 7 Given 
 and y? 
 
 (x—2y X _ 3x-^y 
 4 "^ 5 "~ 5 ~^^' 
 
 0? -4- 1/ T 1] 
 
 — ~A — =r^=8|. Quere the values of x 
 4 ' 7 'J ^ 
 
 Ans. 07=20, 2/= 8. 
 
 Rule 3. 
 
 59. Find in either equation, the value of one unknown 
 quantity, and substitute the value thus found in the other 
 equation j whence will arise a new equation, containing 
 but one unknown quantity, whose value may be determined 
 as already taught. 
 
SIMPLE EQUATIONS. 61 
 
 EXAMPLES. 
 
 C V ^ 
 
 \ 0?+— =12 f 
 Given < 2 > to determine x and y. 
 
 ( 7a?— 3y=:58 ) 
 
 From the first equation x— 12^—-^. This value being 
 substituted for x in the second equation, we have 
 
 xMultiplying by —2, Ty + G^/— 168=— 116. 
 13 
 
 Whence ?/= =4, 
 
 a?=:12— 1-=12— 2=10. 
 
 2 Given ^ ^^— %=15^' 
 
 z. uiven^ 10a?+152/=825. Quere the values of a? 
 
 and yl Ans. a?=45, y=25. 
 
 3. Given, 
 
 3 5"^' 
 
 — -4.^=:31j to find 07 and y, 
 
 Eesult, a?= 30, 3^=25 
 
 ^- ^^^^^ ^ 0? : 2/ :: 5 : 3, to find 07 and 3/. 
 
 Result, 07=15, y=9 
 
 5. Given 
 
 -^ + 3o?=29, to find x andy. 
 
 Result, 07=9, y=:6. 
 
62 SIMPLE EQUATIONS. 
 
 Rule 4. 
 
 60. Multiply one of the equations by an indefinite quan- 
 tity, and to the equation thus formed, add the other given 
 equation. 
 
 Assume the sum of the co-efficients of one required 
 quantity equal to 0, and thence determine the value of 
 the assumed multiplier. The new equation will then 
 contain but one unknown quantity, which may be found 
 as before. 
 
 iSXAMPLES. 
 
 Given | o Xfi^— 1 S2 ( ^^ ^^^^ ^^^ values of x and y. 
 
 Multiply the second equation by the indefinite quantity 
 m, and we shall have 3mx-\-8my=132m, 
 
 Adding this equation to the first, 
 
 Smx+Hx+Smy — 9y=132m+69. 
 
 7 
 Assume 37?^+7=0, whence m= — —^ 
 
 _ 132^+59 _ — 308 + 59 _ 924<— 177 _ 
 ^^^y- Sm—9 " 56 ^ - 56 + 27 ~^- 
 
 -3-^ 
 
 132— 8y 132—72 60 ^^ 
 Hence x= ^—^= — ^ =—=20. 
 
 9 
 
 Or assuming Sm — 9=0, 7n=—^ 
 
 1188 ^„ 
 132;7^+59 _ 8 ^ 1188+472 _^^ 
 ^'^^^- 3^2+7 "27 -" 27+56 "~^^- 
 
 8 "^ 
 
SIMPLE EQUATIONS. 
 
 63 
 
 ^ p. ( 15a?— 171/= 12, 
 
 Z. triven I i7^_i0y.:z.97, to find x and 3/. 
 
 Result, a?=ll, 2/=9. 
 
 f 3a74-72/=79, 
 3. Given j o^_ -i.T=:9, to find x and 3/. 
 
 Result, a?=10, 2/=7. 
 
 , p. ( 807 — 5i/=60, 
 
 *• vjiven ^ 102/— 3a?=75, to determine x and 3/. 
 
 Result, X— 15, 2/= 12. 
 
 5. Given 
 
 ^ V ^ 
 
 2 3 6 
 
 _-a:4-_y=l0— , to find a? and y. 
 
 Result, a?=lO, ^=16. 
 
 Examples to exercise the foregoiivg rules, ^ 
 
 p. ( X — ?/=c/, 
 
 ^ xy=^p, to find a: and y. 
 Squaring the first equation, x'^ — 2xy+y'^=^d^. 
 Multiplying the second by 4, 4a?y=4p. 
 Taking the sum, x^-{-2xy-\-y^=d^-{-4^p. 
 Extracting the root, x+y= ^/ d'^+4}p,(^.) 
 
 d-h\/d^-\-^P 
 
 Adding the first, 2x=d-{- y/d^ + 4<pj or x= ^ 
 
 Subtract the first equation from equation j^j and 
 
 V> + 4^ — d 
 
 2?/= V c?^ + ^P — d, or y = 
 
 *In the solution of these and other similar problems, the ingenious 
 student will find expedients which are not clearly indicated in any of 
 the preceding rules, by which his labor may be frequently abridged. 
 
64f SIMPLE EQUATIONS. 
 
 2. Given ^ ^+2/=^» 
 
 I xy=py to find x and y. 
 
 Subtracting 4 times the second from the square of the 
 
 first, we have o?^ — 2xy'{-y'^=s^ — 4j9. 
 
 Whence x — y = zh >/ s^ — 4/?/ * and _ 5± s/ 5^ — 4p; 
 
 y=szf.^s^ — 4p. 
 
 ^ 0?^ — y^=n, to find a? and t/. 
 
 Dividing the second by the first, a? + 2/=—, 
 
 Hence, taking the half suiji and half difference as before 
 n-i-m^ n — m^ 
 
 x=—^ and 2/=-o — • 
 
 ^. Lriven I ^3_ ^3^^^ ^^ ^^^ ^ ^^^ ^^ 
 
 Subtracting three times the first equation from the se- 
 cond, x^ — 3x^y + 3xy^ — y^=b — 3a, 
 
 Hence, extracting the root, x — y=4^b — 3a, which 
 put =c, (c/^.) 
 
 Divide the first equation by this, whence xy= — (jB.) 
 
 From equations ^ and J5, by proceeding as in the first 
 example, we find 
 
 , ( 4a > , ( 4a > 
 
 oc=h < ^c^^ +c hy=i I VC-] c I 
 
 ( c ' ) ( c ) 
 
 Or divide the sum of the given equations, by equation 
 
 a+b 
 ^ and a?3 4-2a?y+2/2=:— — , 
 
 * As the square of a quantity whether positive or negative is al- 
 ways positive, the square root is often ambiguous; and therefore sus- 
 ceptible of the sign -f or — . In this example, the ambiguity arises 
 from the uncertainty which re or y is the greater. 
 
SIMPLE iiQUATlOiXS. 65 
 
 Whence by extracting the square root, 57+1/=^/ , 
 
 Then taking the half sum and half difference of this 
 equation and equation A^ 
 
 5. Given) oT^~^?'a i: j j 
 
 I x^-\-y^=Oj to find x and y. 
 
 From the first a?^+3a?22/+3a:?/2+2/3=:a3. 
 
 By subtraction, 3a?^y 4- 3a?^2=a^ — b. 
 
 Dividing by three times the first, a?y=~r — , which 
 put =^c, ^^ 
 
 From this and the first equation, we find as in example 2d. 
 
 00= 7^ 
 
 y= 2 
 
 Otherwise, dividing the second equation by the first, 
 
 b 
 
 4}b 
 Hence, 4<x'^ — 4<xy+4<y^= — , 
 
 Squaring the first equation, x^+^xy+y^^a"^. 
 
 Subtracting the latter from the former, we have 
 
 45 — a^ 
 3x''—6xy+ 3y^= 5 
 
 4<b——(i^ 
 Consequently, a?2 — 2xy+y^= — - — -, 
 
 4:b—a^ 
 And by extracting the root, x — y=dzy/ — ^ — -, 
 
 ^ ^~3^~ «±V— -3— 
 
 Whence, a?= and y= ^r 
 
 2 ^2 
 
 6* 
 
Q6 SIMPLE EQUATIONS, 
 
 If we put -r — =c, the values of x and y will be found 
 the same as before. 
 
 ^iven ^ ^__y_^^ ^Q £jj^ ^ g^j^jj y -jj ^i^g terms of s 
 and c?. 
 
 Kesult, a?=-^, ^==-^ 
 
 7. Given | ^3_^,^ J^ ^^ ^^^ ^^ before. 
 
 8. Given | ^.1^2/^1,26, to find a? and y. 
 
 Kesult, 07=65 y=4. 
 ro?— 2 10— a7_2/— 10 
 
 9. Given . 
 
 4 ' 
 
 2y+^_2^+^^+17^ Eequiredthenu- 
 3 8 4 ^ 
 
 merical values of x and y. Ans. a?=7j y= 10. 
 
 r a?+l:2/— 1::5:2, 
 
 10. Given } ^o? 5—2/ 41 2a7— 1 , . . , 
 
 JT-"T'=12 -4— to find.: and 3/. 
 
 Kesult, a?=4, 3/= 3. 
 r 2/ 4a? — 1__ 4+y a? — y 
 
 11. Given] T"""!^""" 3~"^~6~' 
 
 ( 07 : 3?/ : : 4 : 7, to find x and y. 
 
 Kesult, 07=12, 2/= 7. 
 4_ _5^__ 9 
 
 ^ y'~' y~ ' 
 
 12. Given 
 
 I A+A^:^ I A, to find 07 and y. 
 [ X '^ y 0? "•" 2 
 
 Kesult, 07=4, y=2. 
 
SmFLlS. EQUATIONS. 67 
 
 13. Given ^^'2/+^y^= 120, 
 
 I x^ + y^=162. What are the values of x 
 and y ? Ans. x=.o or 3 . . 2/1=3 or 5. 
 
 14 Given ^ x+y=12, 
 
 I a?^+2/^=468, to find x and y. 
 
 Result, a: =7 or 5, 2/ =5 or 7. 
 
 15 Given ^ x'+a:y=in, 
 
 ( ^y+y'^ =:84<, to find X and y. 
 
 Result, a?=8, y=6. 
 
 16.* Given ^5^+*^='"' . , , 
 
 I dx-^-ey = ?^, to find x and y. 
 
 _, ,, em — bn an — md 
 
 Result, a?=: ■ir,y= tt 
 
 ae — do ^ ae — do 
 
 17 Given ^"'*-2'- 12240, 
 
 II. uiven ^ ^_^y,^ J70, to find x and y. 
 
 Eesult, a;=ll, yr=7. 
 
 ( x^ — y^=9S, to determine x and y. 
 
 Result, 07=5, 2/ =3. 
 
 19 Given r^+*^2/= 30, 
 
 ly. uiven ^ 5^ ^32^^34^ to find x and y 
 
 Result, a:=2, y=8. 
 
 '^ 1 1 o 
 
 2^+-3^==^ 
 
 -o-a^— "^2/= 1 to find X and y . 
 
 Result, x= 12, y=6. 
 
 * When we have two equations, and two unknown quantities, 
 neither of which rises above the first power, the required values may- 
 be obtained, from the results of the 16th example, by simple substi- 
 tution. 
 
 20. Given « 
 
68 SIMPLE EQUATIONS. 
 
 {Ix + ^y 
 
 I A — 
 
 21. Given ^ 
 
 ^ 54— 8y 
 
 2a?= — ^ — to find x and ^. 
 
 Eesult, a?=:l, 2/— 6. 
 0/" arithmetical progression, 
 
 61. When the successive terms of a series of numbers 
 are formed by the addition or subtraction of a constant 
 number, the series is called an equidifFerent one j or the 
 numbers are said to be in arithmetical progression ; and 
 the constant number is termed the common difference. 
 Thus 1, 3, 5, 7, 9. a, azhc?, ad=2G?, a=l=3c?, are series in 
 arithmetical progression, the common differences being 2 
 and c?, respectively. 
 
 In any series in arithmetical progression, it is obvious : 
 
 1. That the last term is equal to the first increased or 
 diminished by the common difference multiplied by the 
 number of "terms less one. 
 
 2. That the sum of the extremes is equal to the sum of 
 any other two which are equally distant from them, or to 
 twice the mean when the number of terms is odd. 
 
 3. That the whole sum of such series is equal to the 
 sum of the extremes multiplied by half the number of 
 terms. 
 
 1. What is the sum of a descending arithmetical series, 
 of which the first term is 10, the common difference ^, and 
 the number of terms 211 
 
 ^^ 20 30 20 10 , ^^ 
 10——=— ^=— =last term. 
 
 /lO 10\ 21 40 21 ^^. _. 
 
 i — + — ••— =-^X-^ = 140=sum of the series. 
 
 2. Suppose a car descending an inclined plane moves 5 
 feet during the first second j 15 feet the second 5 25 feet the 
 
SIMPLE EQUATIONS. 69 
 
 third; increasing the distance 10 feet each second of time; 
 how far will it move in a minute '? 
 
 Ans. 18000 feet. 
 
 3. Suppose 100 loads of wood placed in a straight line 
 at intervals of 10 perches each, and that a wagoner is en- 
 gaged to transport them to a place in the same line con- 
 tinued, 20 perches from the nearest pile ; how far must he 
 travel in performing the service, his journey being com- 
 menced at the place of deposit 1 
 
 Ans. 160 771. 300 p. 
 
 4. What is the sum of the odd numbers 1, 3, 5, 7, etc., 
 continued to n terms 1 , Ans. n^. 
 
 Of geometrical progj'ession. 
 
 62. When a series of numbers, are such that any term 
 multiplied by a constant number, either integral or frac- 
 tional, produces the next following term, the numbers are 
 said to be in geometrical progression, thus 2, 6, 18, 54 ; 
 216, 144, 96, 64; «, «r, ar% ar^, ar% are in geometrical 
 progression, the common multipliers being, respectively 
 3, t and r. 
 
 The constant multiplier is usually termed the ratio. In 
 these series we readily perceive that the last term is equal 
 to the product of the first by the ratio involved to the 
 power denoted by the number of terms less one : 
 
 n 1. 
 
 Thus, the n term of the series «, ar, ar% etc. is ar 
 
 To find the sum of a series in geometrical progression 
 assume, the series a + ar-^ar^ ar°~2+^r°~^=5, then 
 
 multiplying by r, ar + ar^ ar""^ + «r°~^ + ar° = rs^ 
 
 whence by subtraction — a 4-flr"= or (r° — l)a=^rs — s or 
 
 7-° 1 
 
 (r — l).s, wherefore -a=s. If r is a proper fraction 
 
 1 — r" 
 
 .a=zs, 
 
 1 — r 
 
70 SIMPLE EQUATIONS. 
 
 EXAMPLES- 
 
 1. What is the sum of, 2 + 6 + 18, etc. to 10 terms'? 
 
 2.^g—-j)=3^«— 1 = 59048. 
 
 2. The first term of a geometrical series being «, the 
 
 m 
 ratio, the proper fraction — , required the sum of the series 
 
 continued to infinity. 
 
 As — is a proper fraction, or n more than m^ — raised 
 
 to an infinite power, may he considered =0. 
 
 Hence, the sum required 
 
 m n — m 
 
 1 
 
 n 
 
 2 4 
 
 3. Required the sum of I + 0+9+ etc. to infinity. 
 
 Result, 3. 
 
 4. The series 1 — ^ + q — ^^, etc., being continued to 
 infinity, required the amount % 3 
 
 5. What is the sum of I+h + t? continued 20 terms] 
 
 524287 
 
 ^'''' ^524288' 
 
 Of harmonical progression, 
 
 63. If we have a series of numbers in arithmetical pro- 
 gression, their reciprocale, taken in the same order, form 
 
 a series in harmonical progression. Thus, if — , -y , — ,-y 
 
 are in arithmetical progression, «, 5, c, d are said to be in 
 harmonical progression. 
 
SIMPLE EQUATIONS. 71 
 
 Let — . -7-. — be in arithmetical progression, then Art. 
 
 112 
 
 61 1 ==="T" •*• bc-{-ab—2ac, or (a-\-c)b=z2ac. From 
 
 the former of these equations, be — ac=ac — ab, whence 
 arc:: b — a : c — b. Hence we have the relations of the 
 terms, by which any two of three numbers in harmonical 
 progression being given, the third may be found. 
 
 EXAMPLES. 
 
 1. The first and third terms of an harmonical series are 
 6 and 12, required the mean. 
 
 2X6X12 lU ^ _,. , . 
 
 -^ --r — =-— -=8. 1 he number souo;ht. 
 
 6+12 18 ° 
 
 2. The first and second terms of an harmonical series 
 are 12 and 16 , what is the third '? Ans. 24. 
 
 SIMPLE EQUATIONS. 
 
 Containing three or more unknown quantities. 
 
 6'!'. The unknown quantities may be eliminated one 
 after another, by the first three rules laid down in the 
 last section. Or multiply each given equation, except one, 
 by an indefinite quantity 5 add the remaining given equa- 
 tion, and all the new equations together; and assume the 
 sum of the co-efficients of each required quantity, except 
 one, equal to 0; whence, as many equations as there are 
 assumed multipliers will be formed; and thence the values 
 of those multipliers may be determined, and consequently, 
 the last unknown quantity. 
 
 There are other methods which may often be advan- 
 tageously applied, but which can be learned only from 
 practice. 
 
72 SIMPLE EQUATIONS. 
 
 EXAMPLES. 
 
 (00+ y+ 2r=29, >j 
 
 I 0^+22/ + 32^=62, to find the values of 
 
 Given a: 1 1 [o^, 2/, and z. 
 
 2 1 
 Multiply the last equation by 2, a?+-^y-i--^2;=20. 
 
 Subtract this from the first equation, — y+— 2;=9. 
 
 3 
 
 Multiply by 3, and y+—z=2'7. 
 
 Subtract the first equation from the second, and 
 
 2/ + 2z=33. {A,) 
 From this equation subtract the former, and we find 
 
 -2;z=6^ or 2;= 12. 
 
 Take double the last equation from equation A^ and 3/= 9. 
 Hence y+2:=21, and this equation subtracted from the 
 first leaves a? =8. 
 
 Otherwise, — From the given equations, 
 
 a? =29 — y — z, 
 
 a?=62— 2?/— 3;^. 
 
 on ^ 1 
 
 Whence 29— y— 2^=62—23/— 32?. 
 
 2 1 
 
 And 20— — 2/— -2r=:29— y— 2r. 
 
 From the former of these, y=33 — 22:. 
 
 3 
 
 And from the latter, y=27 —z. 
 
SIMPLE EQUATIONS. 73 
 
 Therefore, 'jll—~z=33—2z, 
 /u 
 
 Or, 54—3^=66—4^^; r,z=12.'^ 
 
 Buty=33—2z=9. 
 
 And x=29 — y—z=:8. 
 
 Or multiplying the second and third equations by m and 
 n respectively, mx-{-2my-i-3mz=62m,, 
 
 ■^nx-\--ny+-nz=10n. 
 
 Hence, by adding these equations to the first, 
 
 x-{-mx-\—^nx-\-y-{'2my-\-—ny-\-z + 3mz-{—j-nz=: 
 29 + 62m +1071. 
 
 1 1 
 
 Assume l+2;?^ + -?^=:0, and l^-3;?^4--7^=0 
 o 4* 
 
 3 1 
 
 From the former of these equations - + 37?^4--7^=rO. 
 
 -^ Z 
 
 Hence m= — -. 
 o 
 
 29-^-20 
 29 + 62m + -i0n ^^ 6 
 
 Whence, a:= ^j =- 
 
 1 1 '• 
 
 l+m + —n 1— 1 
 
 Again making l+7?z+— 7i=0, and l4-37?^+— 7i=0, 
 We have n= —.B.ndm= — p-. 
 
 
 
 *The sign .«. is used in place of the word therefore. 
 7 
 
74« SIMPLE EQUATIONS. 
 
 29+62m+10n ^ 
 Consequently, y= — =9, 
 
 Hence, 2:= 12. 
 
 2. Given] a7+22/+32r= 105, , 
 
 ( 07+3^+42;= 134, to find a?, y, and z. 
 
 Subtract the first from the second, y-{-2z=^2. 
 Second from third, ?/ + 2:=29. 
 Subtracting the last, 2r=23. 
 r.y=:6. 
 
 And a?=24. 
 
 C x+y=20, 
 
 3. Given < a?-f-2r=24, 
 
 ^ y+2:=30, to find 0?, y, and z. 
 
 Eesult,a?=7, ^=13, 0=17. 
 
 C 7a?+53/+22:=79, 
 
 4. Given? 8a7-|-7i/ +9^=122, 
 
 ( a?+4y + 52:=55, to determine the values of 
 a?,'y, and z. Result, a?=4, y=9, z=3, 
 
 ( ^— y=2, 
 
 5. Given < a? — z=3, 
 
 ( 2/ + 2r=9, to find a?, 3/, and z. 
 
 Result, 07=7, y=5, z=4f. 
 
 - C 2a:+3y+4;^=38, 
 
 6. Given < 5a?+ 72/4- 112:= 98, 
 
 ( 7a? -f 9y + 152:= 132. to find a?, y^ and z. 
 
 Result, 07=3, 2/ =4, z=5 
 
 C aa7+5^+C2r=77?, 
 
 7. Given < dx+ey +fz = w, 
 
 ( gX'\'hy-\-kz=p^ to find a?, y, and z. 
 
SIMPLE EQUATIONS. 
 
 75 
 
 Kesult,* 
 
 8. Given 
 
 9. Given < 
 
 _ kem — hfm -f hen — hkn + hfp — ecp 
 aek — ahf-\- dhc — dbk-\-gbf — gee 
 akn — afp -f- dcp — dkm -f- gfm — gen 
 ^ ~~ aek — ahf-\- dhe — dbk +gbf—gec 
 
 aep — ahn + d/im — dbp -\-gbn — gem 
 
 ~ aek — afif-\-dhc — dbk +gbf' — gee 
 
 ,^4-y+2:, = 33, to find z?, a:, y, and z. 
 
 Result, ?;=9, a?=10, y=ll, 2:= 12. 
 f4a7+3y+-2: 22/+22r— a7+l_ a?— z— 5 
 10 15 '' 5 ' 
 
 9a? + 5y— 2 2f 2a? + y— 32r_7y+2r + 3 . 1 
 
 12 5 - n 
 
 + 6^ 
 
 5y4-32r 2a7+3y— 2r 
 
 + 22r=y— 1 + 
 
 3a:+2y4.7 
 
 4 12 
 
 Required the values of a:, y, and 2^. 
 
 Ans. 07=9, y=7, 2?= 3, 
 
 f 3v + 4a?-f-5y-h62r=:102, 
 
 ( 62;-!- 5a: — 7^ + 2^=7, to find v, a?, y, and z. 
 
 Result, 2;= 5, a?=4, y=7, 2^=6. 
 
 Examples producing simple equations. 
 
 Required to find two numbers whose difference shall be 
 4, and the difference of the squares 64. 
 
 Let 07 = the greater number, and y = the less. 
 
 Then, by the question, | ^^^^Jq^^ 
 Dividing the 2d equation by the 1st, a?+y=16 
 
 * From these general results, the values of x, y, z^ in the preceding 
 examples, may be found by simple substitution. 
 
76 SIMPLE EQUATIONS. 
 
 Taking the sum and difference of this equation and the 
 first, 2x=20, and 2y=12. 
 
 .•. a?=10, and y=6. 
 Otherwise. Let x = the less number, then x+4} = the 
 greater : 
 
 Whence, by the question, a7-|-4]^ — a:^=64<. 
 
 Or 807+16 = 64. 
 .'. by transposition and division, x=6, and a:-h4=10. 
 
 2. A person having bought three loads of grain, the 
 first, consisting of 30 bushels of rye, 20 of barley, and 10 
 of wheat, for 230 francs ; the second, containing 15 bushels 
 of rye, 6 of barley, and 12 of wheat, for 138 francs; the 
 third consisting of 10 bushels of rye, 5 of barley, and 4 of 
 wheat, for 75 francs ; required the price per bushel of each 
 of these kinds of grain. 
 
 C a bushel of rye cost a?, 1 
 Suppose < a bushel of barley, y, > francs 
 
 ( and a bushel of wheat, z, ) 
 
 C 3007+203^+ 10^=230,(^.) 
 
 Then by the question, } 1507+6^+122;= 138, 
 
 ( 10o7+5y+42;=75. 
 
 Multiplying the 2d equation by 2, and the 3d by 3, 
 
 30o7+122/+24;^=276, 
 
 3007+151/ + 12^=225, (B,) 
 
 Taking the difference of these equations, 
 
 32/— 12z=— 51. (C.) 
 
 Take the difference of equations (^d,) and (B,) and 
 
 5y— 22;=5. 
 
 Multiplying this by 6, 30y—12z=30. 
 
 And subtracting equation (C) 
 
 27y=81. Whence, y= 3. 
 
 15__5 23—6—5 , 
 .•. z= — - — = 5, and 07= =4. 
 
SIMPLE EQUATIONS. 77 
 
 3. Required to find two numbers, such that their dif- 
 ference multiplied by their product, shall be 308, and the 
 difference of their cubes 988. 
 
 Let X — the greater, and y = the less. 
 Then from the question, 
 
 (x — y).a?y, or x^y — xy^=30S, 
 And 0?^— i/3^988. 
 
 From the latter taking thrice the former, 
 
 We have x'^ — 3x''-y + 3xy'^ — 2/3=64. 
 And taking the cube root, x — y—4<, (^A,) 
 
 Dividing the sum of the two primary equations by this 
 last, 
 
 x^ -f x'^y — j?y^ — y^ 
 
 =a?2-f2a?2(+2/2=324. 
 
 Hence, by extracting the square root, a7+y=18. (J5.) 
 Taking the half sum and half difference of equations 
 {A.) and (jB.) 
 
 18+4 18-— 4 
 
 07=— ^=11, and 2/=— ^=7. 
 
 4. The sum of two numbers multiplied by their differ- 
 ence is 20, and the difference of their 4th powers 1040 ; 
 what are the numbers 1 
 
 Let X = the greater, y = the less. 
 
 Then from the data, \ ^y^y^irolol 
 
 Dividing the latter by the former, x'^-\-y'^=52. 
 Taking the sum and difference of the first and last equa- 
 tions, 2a?2=72, and 2y^=32. 
 
 .'. x=VS6 = 6. 
 And y= ^16=4. 
 
 5. The difference of two numbers is 5, and the differ- 
 ence of their cubes 1685 ; what are the numbers '? 
 
 Let X = the greater, y = the less. 
 7* 
 
78 SIMPLE EQUATIONS. 
 
 Then, per dataA^T''^^'7^\r,oK 
 ' -r J ^ ^> — y.— X685. 
 
 By division, ^;;;:::^=a?2 4- ^y+S''^ 337, {A.) 
 
 Squaring the first equation, a?^ — 2a:y4-2/2=25. 
 
 By subtraction, 3a?2/=312. 
 
 Adding i of this equation to equation (.^.) 
 
 Extracting the square root, 07+3/= 21. 
 
 From this and the first equation, a?=:13, and 2/~8. 
 
 Or from the second equation subtract the cube of the 
 first, whence So^^y — 30:3/2= 1560; and dividing this equa- 
 tion by three times the first, ccy= 104' ; v^rhence as in page 63, 
 
 ^25+416 + 5 ^25 + 416— 5 
 a?= o =13. and y— ^ = S* 
 
 6. Required to find two numbers, such that half the first 
 added to a third of the second shall make 9, and a fourth of 
 the first, added to a fifth of the second, shall make 5 
 
 Ans. 8 and 15. 
 
 7. A testator bequeaths to his widow § of his estate, to 
 his son i, and the remainder, which is found to be 756 dol- 
 lars, to his daughter; what was the estate left] 
 
 Ans. 2835 dollars. 
 
 8. There are two numbers in the ratio of 3 to 5, and 
 their sum is one fourth of the difference of their squares; 
 what are the numbers'? Ans. 6 and 10. 
 
 9. The ages of a man and his wife, at the time of mar- 
 riage, were in the ratio of 7 to 6, and at the end of 30 
 years, they are found to be as 17 to 16. Quere their ages 
 at the former period 1 Ans, 21 and 18. 
 
 10. A post being i of its length in the earth, i in the 
 water, and 10 feet above the water; what was its whole 
 length 1 Ans. 24 i^^L 
 
SIMPLE EQUATIONS. 79 
 
 11. Divide 1100 dollars among A. B. and C, so tha4 
 B. shall have 100 dollars more than A., and C. 150 more 
 than B.J what is the share of each'? 
 
 Ans. A. 250, B. 350, C. 500. 
 
 12. There are 21 persons, whose ages form an equi-dif- 
 ferent series ; the age of the eldest is 4 times that of the 
 youngest, and the sum of all their ages is 525 years 5 how 
 old is the youngest 1 Ans. 10 years. 
 
 13. At a certain election, 1296 persons voted, and the 
 successful candidate had a majority of 120 ; how many 
 voted for eachl 
 
 Ans. 708 for one, and 588 for the other. 
 
 14. A servant agreed to serve his master for ^8 a year 
 and his livery, but was turned away at the end of 7 months, 
 and received only £2 13s. M. besides has livery; wlmt is . 
 the price of his livery'? Ans. £4} 16s, 
 
 15. A servant having eloped from his master, travels 14? 
 hours in the day, at the rate of 3^ miles an hour; at the ^ 
 end of two days, a courier is sent in pursuit, who rides 9 
 hours in the day, at the rate of 7 miles an hour; in what 
 time, and at what distance will he overtake him'? 
 
 Ans. 7 days, and 441 miles. 
 
 16. The sun's mean daily motion in the ecliptic is 59' 8", 
 and that of the moon 13^ 10' 35"; what is the time of a 
 mean synodic revolution of the moon, viz. a revolution from 
 conjunction with the sun to conjunction again '? 
 
 Ans. 29 days, 12 hours, 44 minutes nearly. 
 
 17. A farmer, having hired a laborer, on condition, that 
 for every day he wrought he should receive 50 cents, and 
 for every day he was idle he should forfeit 20 cents, finds 
 at the end of 420 days, that neither is indebted to the 
 other; how many days did he labor. Ans. 120. 
 
 18. A certain number, consisting of two places of figures, 
 is equal to the difference of the squares of its digits ; and if . 
 36 be added to it, the order of the digits will be inverted; 
 what is the number 1 Ans. 48.^ 
 
 19. A vintner proposing to mix three softs of wine, viz. 
 
80 SIMPLE EQUATIONS. 
 
 at 65 cents per gallon, at 45 cents, and at 35 cents; the 
 number of gallons at 65 cents, to the number at 35 cents, 
 being as 5 to 3, so as to compose a hogshead worth 50 
 cents per gallon. Quere the number of each 1 
 
 Ans. 22iat 65, 27 at 45, and 13iat 35. 
 
 20. There are 4 equi-difFerent numbers, whose sum is 56, 
 and the sum of whose squares is 864 ; what are the numbers'! 
 
 Ans. 8, 12, 16, and 20. 
 
 21. If 38 federal dollars and 11 English crowns be 
 given for 271 French francs, and 76 dollars and 33 crowns 
 for 610 francs; at how many francs were the dollar and 
 crown respectively estimated '? 
 
 Ans. the dollar 5||, the crown 6f-y. 
 
 22. There is a number consisting of three digits in arith- 
 metical progression, whose sum is 21 ; and if to the number 
 396 be added, the sum will be expressed by the same digits 
 in an inverted order; what is the number 1 Ans. 579. 
 
 23. Eequired to find four numbers, such that the con- 
 tinued product of the 1st, 2d, and 3d shall be 24 ; the pro- 
 duct of the 1st, 2d, and 4th, 30 ; of the 1st, 3d, and 4th, 
 40 ; and of the 2d, 3d, and 4th, 60. Eesult, 2, 3, 4, 5. 
 
 / 24. It is required to find three numbers, such that the 
 first multiplied by the' sum of the other two shall be 96 ; 
 the second multiplied by the sum of the other two shall be 
 105 ; and the third multiplied by the sum of the former two 
 shall be 117. The numbers are 6, 7, and 9. 
 
 25. What fraction is that, to the numerator of which, if 
 1 be added, the value will be |, but if to the denominator 
 1 be added, the value will be i 1 Ans. y4_. 
 
 26. A person bought a chaise, horse, and harness, for 
 150 dollars; the price of the chaise was twice the price 
 of the harness, and the price of the horse as much and 
 half as much as the price of the chaise and harness ; what 
 was the cost of each % 
 
 Ans. harness 20 dollars, chaise 40, horse 90. 
 
 27. Two persons, A. and B., have the same income; A. 
 saves ^ of his yearly ; but B., by spending as much in 3 
 
SIMPLE EQUATIONS. 81 
 
 years as A. does in 4, finds himself, at the end of ^ye years, 
 200 dollars in debt 5 what was the income 1 
 
 Ans. 600 dollars. 
 
 V 28. A. and B. put equal sums in trade: A. gained a sum 
 equal to i of his stock, and B. lost 450 dollars; when A.'s 
 money was double of B's; what was the sum laid out by 
 each 1 Ans. 1200 dollars. 
 
 29. Two persons comparing their ages, find them, at 
 present, in the ratio of 7 to 5, but that 30 years ago, they 
 were in the ratio of 2 to 1 ; what are their ages '? 
 
 Ans. 70 and 50. 
 
 ^ ^^ 30. A. and B. began trade with equal sums of money. 
 In the first year A. gained J240, and B. lost .£40; but in 
 the second, A. lost one third of what he then had, and B. 
 gained a sum which was £40 less than twice what A. had 
 lost ; when it appeared that B. had twice as much as A. ; 
 what sum had each of them at first 1 Ans. £320. 
 
 ^ ^ 31. A farmer sold 96 loads of hay to two persons. To 
 the first one half, and to the second one fourth of what his 
 stack contained ; how many loads were in the stack 1 
 
 Ans. 128. 
 
 32. If 116 be divided into four parts, in such manner, 
 that the first being increased by 5, the second diminished 
 by 4, the third multiplied by 3, and the fourth divided by 
 2, the results will all be the same ; what are the parts % 
 
 Ans. 22, 31, 9, and 54. 
 
 /•^ v^ 33. A shepherd, in time of war, was plundered by a 
 party, who took from him i of his flock, and i of a sheep 
 more ; another party took from him ^ of what he had left, 
 and ^ of a sheep more ; afterward, a third party took \ of 
 what remained, and \ a sheep more, when he had but 25 
 left ; how many had he at first \ Ans. 103. 
 
 J 34. A person has two horses and a gig ; the gig is worth 
 150 dollars. When the first horse is attached to the gig, 
 the value of the two is twice that of the second ; but when 
 the second horse is put to the gig, the value is three times 
 
82 SIMPLE EQUATIONS. 
 
 that of the first horse ; what were the horses respectively 
 worth ] Ans. the first 90 dollars, the second 120. 
 
 / 35. When a company at a tavern came to pay their 
 ^ reckoning, they found, that if there had been three persons 
 more, they would have had a shilling a piece less to pay ; 
 but if there had been two less, they would have had a shil- 
 ling a piece more to pay 5 required the number of. persons, 
 and the quota of each 1 
 
 Ans. 12 persons 5 quota of each 5s. 
 36. There are three equi-different numbers, whose sum 
 is 324, and the first is to the third as 5 to 7 ; what are the 
 numbers'? Ans. 90, 108, and 126. 
 
 J 37. A man and his wife usually drank a cask of beer in 
 12 days ; but when the man was from home, it lasted the 
 woman 30 days ; how long would the man alone be in 
 drinking it 1 Ans. 20 days. 
 
 J 38. A. and B. can perform a piece of work in 8 days, A. 
 and C. in 9 days, and B. and C. in 10 days ; how many 
 days would they severally require to perform the same 
 work'? Ans. A. 14|| days, B. 17|f, and C. 233^. 
 
 / 39. The hypothenuse of a right angled triangle is 13, 
 "^ and the area 30 5 what are the other two sides 1 * 
 
 Ans. 12 and 5. 
 40. There are four numbers in geometrical progression, 
 the sum of the extremes is 18, and the sum of the means 
 12 j what are the numbers if Ans. 2, 4, 8, and 16. 
 
 * To solve this problem, it must be recollected that the square of 
 the hypothenuse is equal to the sum of the squares of the other two 
 sides, and that the area is half the product of those sides. 
 
 t To solve this question by simple equations, put x, y to denote the 
 
 means, then since, when three numbers are in geometrical progression, 
 
 the square of the mean is equal to the product of the extremes, the ex- 
 
 x^ yn 
 
 tremes will be expressed by — , and — . Hence rr4-y=12, and 
 
 — [-— =1^' From the last ics-f-j^s^riSariy. Subtracting this equa- 
 y ic 
 
 tion from the cube of the first, 3rr2y-}-3a;y3 _ 1728 — ISxy, But 
 
 2x^ H- 3a;y3==3(a;-f-y). xy = ^Qxy. Hence 5^xy = 1728 or ir^=32. 
 
 Whence x and y are found as in page 64. 
 
# 
 
 SIMPLE EQUATIONS. 83 
 
 41. Required to find four numbers in geometrical pro- 
 gression, such that the difference of the means shall be 100, 
 and the difference of the extremes 620. 
 
 The numbers are 5', 25, 125, and 625, 
 
 42. There are three numbers in harmonical progression, 
 whose sum is 26, and the difference between the second, and 
 third exceeds the difference between the first and second 
 by 2 5 what are the numbers % Ans. 6, 8, and 12. 
 
 43. A person having £21 6s. sterling in guinea^/- and 
 
 1/ crowns, pays a debt of il4 175., and then finds the Humber*.'' ^ 
 of guineas left equal to the number oi*^crownstpaid awS,y; 
 and the number of crowns left equal to the number of 
 guineas paid , how many of each had he at first 1 
 
 Ans. 21 of each. 
 
 ^ 44. There is a number consisting of two digits, which is 
 equal to four times the sum of those digits j but 18 being 
 added, the order of digits will be inverted. Quere the 
 number 1 Ans. 24. 
 
 45. Required to find three such numbers, that i of the 
 first, i of the second, and i of the third added together 
 shall make 46 ; i of the first, i of the second, and J of the 
 third shall make 35 ; and i of the first, | of the second, and 
 1 of the third shall make 28^. Result, 12, 60, and 80. 
 
 46. A person having £22 14^. sterling in crowns, gui- 
 V neas and moidores, finds that if he had as many guineas as 
 
 he has crowns, and as many crowns as he has guineas, the 
 whole sum would be ^36 6s, ; but if he had as many moi- 
 dores as he has crowns, and as many crowns as he has 
 moidores, his money would amount to JG45 16s, How 
 many had he of each 1 
 
 Ans. 26 crowns, 9 guineas, and 5 moidores. 
 . 47. Required two numbers, such that their sum, differ- 
 ence, and product may be as the numbers 3, 2 and 5 re- 
 spectively. Result, 10 and 2. 
 
 48. The sum of the first and third terms of four num- 
 bers in geometrical progression is 74, and the sum of the 
 second and fourth 444 ; what are the numbers '? 
 
 Ans. 2, 12, 72, and 432. 
 
 €^^' 
 
 
84 QUADRATIC EQUATIONS. 
 
 ^ 49. A. and B. enter into trade with different sums, A. 
 gains 750 dollars, and B. loses 250 dollars; when their 
 stocks are found to he as 3 to 2. But if A. had lost 250 
 dollars, and B. had gained 500 dollars, their stocks would 
 have been as 5 to 9. What was the original stock of each \ 
 Ans. A. 1500 dollars, B. 1750 dollars. 
 
 Section VII. 
 
 QUADEATIC EQUATIONS. 
 
 65. When an equation contains the square and simple 
 power of an unknown quantity ; or in general, two powers, 
 one of w^hose indices is double the other, of an unknown 
 quantity, whether that unknown quantity be simple or 
 compound, it is called an adjected quadratic equation. 
 
 Rule I. 
 
 Arrange the terms, by transposition or otherwise, so that 
 the highest power of the unknown quantity shall be con- 
 tained in the first, and the other power in the second term 
 on the left side of the equation ; and the term or terms con- 
 sisting of known quantities, on the other side. 
 
 When the highest power of the unknown quantity con- 
 tains a co-efficient, divide the equation by that co-efficient. 
 Then add, tp each member of' the equation, the square of 
 half the co-efficient of the unknown quantity, (or the nu- 
 merical part of the second term,) and the first member of 
 the equation will be a complete square. 
 
 Extract the root and find the value of the unknown 
 quantity by the former methods. 
 
 Rule 2. 
 
 ^^, Arrange the terms as before. When the highest 
 power of the unknown quantity has a co-efficient, multiply 
 the equation by four times that co-efficient, and to each 
 
QUADRATIC EQUATIONS. 85 
 
 member of the resulting equation, add the square of the co- 
 efficient of the unknown quantity contained in the primary- 
 equation 5 then the first member of the resulting equation 
 will be a complete square. 
 
 Extract the root, and proceed by the former methods, 
 to determine the value of the quantity sought. 
 
 EXAMPLES. 
 
 Given 2x^ — 9a?=266, to find the value of a?. 
 
 9 
 By rule 1. — ^Dividing by 2, a?^— -x— 133. 
 
 9\2 , , 9 81 2209 
 
 Adding V-) to each member, x^ — -o?^ _ 
 
 9 47 
 Extracting the root, x--:=~ 
 
 4r 4b 
 
 .-. a?=14. 
 By rule 2.— Multiplying by 8, 16a?2-^-72a7=2128. 
 Adding 81 to each member, 160?^— 72a? +8 1=2209 
 Extracting the square root, 4a: — 9=47. 
 
 r,x= — z — = 14, as before. 
 
 4 
 
 2. Given ax^+bx=c, to find the value of a?. 
 
 b c 
 By rule 1. — Dividing by a, x^-\--x—-» 
 
 52 ^ b ¥ b^-h4<ac 
 
 Adding^, to each, c^+^+-=-^- 
 
 By evolution, 0;+-=—^—- 
 
 By transposition, a?= ^ 
 
 By rule % — Multiplying by 4cj, ^a'^x^+^ahx^iac. 
 Adding 5^ to each, ^a''x^-\-^abx-\'h'^^^ac-^h^. 
 8 
 
86 QUADRATIC EQUATIONS. 
 
 By evolution, 2ax-\-b=^y/4^ac-\-b^, 
 
 •D ^ '.' ;i J- • • VUc + b^'—b 
 J3y transposition and division, x= 
 
 3. Given 20?"" + 3a?'' = 2, to find the value of x. 
 
 # 3 - 
 Byrule 1. ^ +x^' = l. 
 
 r .1. f 3 A 9 25 
 
 Comp. the sqr. x -\-^x +15=15 
 
 - 3 5 
 
 By evolution, a7^+j=j 
 
 ^_5— 3__1 
 
 •'•^ ■"~T~~2 
 
 Whence, a?=:(^) =g 
 
 By rule 2. 16 a?" +24a?' = 16. 
 Adding 9 to each, 16 x^ +24>x +9 = 25. 
 
 By evolution, 4 a:" +3=5. 
 
 "2 
 
 i 1 
 By transp. and division, x =-^ 
 
 Therefore. oc=- 
 o 
 
 4. Given a:* — 12a?3+44a?2— 48a?=9009, to determine x. 
 This equation is equivalent to to the following: 
 
 a?*—12a:3+36a?'2+8a?2— 48^=9009, 
 Or to this, (a?2— 6a?)2+8.(a?2_6a?)=9009. 
 In which, x^ — 6x may be considered as the unknown 
 quantity. 
 
 Hence, proceeding as in the former examples, 
 
QUADRATIC EQUATIONS. 87 
 
 (a^3__6a?)2+8.(a?2— 6^) + 16=9025. 
 .'.x^ — 6a?4-4 = 95, or a:^ — 6a7=91. 
 Whence, x^—6x + 9 = 100. 
 
 ... 0?— 3=10, or 07=13. 
 
 5. Given ^(x — -) + v^ ( 1 )=^) to find the value 
 
 of X, 
 
 Multiplying by \/a?, ^/x"^ — l + y/x — l=Xs/x, 
 
 bquaring this, x'^ — 1 + 2 \/ ^^ — ^^ — ^ + 1 +^ — 1 = ^^• 
 By transposing and changing the signs, 
 
 a?^ — x'^ — X + 1 — 2 \/ 0?^ — X" — X + 1 = — 1 . 
 
 Here y/x^ — a?^ — ^ + 1? niay be considered as the un- 
 known quantity. 
 
 Hence, adding 1 to each side, 
 
 0.3 — ^2 — a? 4- 1 — 2 s/ a?3 — x'^—x 4-1 + 1 = 0. 
 
 ... ^x'l — x'2 — ^4-1 — 1=0, or v/a:^ — a:^ — a? +1=1. 
 
 .^x^—^x^ — a?+l=l; 
 Whence a?^ — a:=l. 
 15 1 5 1^ 
 
 1 1 , 
 
 •••^=2+2^ 
 6. Given 3a^2"— 2a7"+3=ll, to find x. 
 
 Sa?'^"— 2a?"=8. 
 
 2 8 
 By rick 1. a?^" — -a7"= - 
 
 2 1 25 
 
 ^ 3^ +9" 9 
 
 0?"—^=^, or a? =2". 
 
88 QUADRATIC EQUATIONS 
 
 By rule 2. 36a?'2° — 24a?°=96. 
 36j:3°— 24a7° + 4=100. 
 .6a?°— 2=10. 
 
 12 
 
 a?°=-7r-=2, or 07= V 2, as before. 
 
 7. Given a?*°— 2a:3'^ + a?°=132, to find the value of a? 
 This equation may be changed to 
 
 a?4°_2a?^° 4_a;2°_a72n + 37^== 1 32. 
 Which is (o?^"— a?")^— (a?^"— a?°)=132. 
 Complete the square, 
 
 1 529 
 
 (a?2°_a?°)'2— (a?3°— 0?")+-= 132i=^ 
 
 1 23 
 
 Extract the root, a?^° — a?° — o=ir 
 
 /O At 
 
 .-. a?2»— a?''=12. 
 
 1 49 
 
 Complete the square, a?^° — 0?° +^=12i=-^ 
 
 1 7 
 Extract the root, x"" — 9 = 9 
 
 .«. a?°=4, or 07= V 4. 
 
 8. Given a?^4-12a7=64. Kequired the value of a?. 
 
 Ans 07=4. 
 
 9. Given 7o?2+5o7 + 4=82, to find o?. Ans. 07= 3. 
 
 10. Given^o?^— -07+71 = 8, to find 07. 
 
 Ans. 07=14 
 
 11. Given o?H20o7^=224, to find 07. Ans. 07= 2. 
 
 12. Given 507^ — 4o7+3=159, to find 07. Ans. 07=6. 
 
 35 3^ 
 
 13. Given 607 -| =44, to find 07. 
 
 Ans. 07=7, or |.* 
 
 * This example falls under the ambiguous case, which is explained 
 in the next note. 
 
QUADaATIC EQUATIONS. 89 
 
 14. Given 4-0? -—=14, to find the value of a:. 
 
 Ans. x=4>, 
 
 1221 4^x 
 
 15. Given 3a? =2, to find the value of a?. 
 
 X ' 
 
 Ans. 07=19. 
 
 .r r.' n ^O?— 3 ^ 3x—6 ^ , 
 
 16. Given 5x ^=^oc-i ^— - to find x. 
 
 X — o z ' 
 
 Ans. a?=4. 
 67. The foregoing solutions have been effected by exe- 
 cuting the whole process in each case \ but it is more con- 
 venient in practice, as v^ell as more elegant, to solve all 
 such equations by a few general formidcR. In order to 
 which it is to be observed, that all adfected quadratic 
 equations are reducible to one of the following forms. 
 
 2. x''—1ax=b, C (^.) 
 
 3. x" — 2ax=^ — h, ) 
 
 These equations, when the squares are completed, be- 
 come, 
 
 1. a?^-}-2aa?+a2_.^3_j_^ 
 
 2. 0?^ — 2ax + a^=a^+b 
 
 3. a:^ — 2ax-i-a^=a^ — b. 
 
 Hence, by evolution and transposition, we have 
 
 1. x= — a-\"^a^-\-b. 
 
 2. x=a'\'^/a^-rb, 
 
 3. x=a±is/ a^—bJ^ 
 
 Adfected quadratic equations are sometimes expressed 
 by one of the following forms. 
 
 1. o^x'^-\-bx=^c, ^ 
 
 2. ax^'-bx=c. >(B.) 
 
 3. ax''- 
 
 See note, page 90. 
 8* 
 
90 QUADRATIC EQUATIONS. 
 
 Multiplying these equations severally by 4a, and adding 
 b^ to each member, they become 
 
 1. 4<a''x^+4!abx+b^=b^+4^ac. 
 
 2. 4a V — ^abx+b'^=b'^+4<ac: 
 
 3. 4a2a?2 — ^^abx+b^^b'' — 4ac. 
 
 The roots being extracted, the values of a? are easily 
 found. 
 
 —b+^b^+4^ac 
 
 1. x==. 
 
 2a 
 
 b+s/b^+4<ac 
 ^- ^= 2a ■ 
 
 bzt:s/b^—4<ac* 
 3. x= 
 
 2a 
 
 From these formulce^ any adfected quadratic equation 
 may be resolved, by assuming for a and 5, in the equations 
 (j1,) and for a, b, and c, in the equations (5,) such values 
 as will render the general equation identical with the par- 
 ticular one under consideration, and substituting the values 
 thus assumed in the final equation. 
 
 Simple quadratic equations are evidently solvable by the 
 SQ,me formulcB if we assume the co-efficient in the second 
 term =0. 
 
 ♦The value of Xj in the third case, is ambiguous; because, 
 Vx^ — 2ax-{-az=x — a^ or a — x, and \/ 4a^x^ — 4abx'{-b^—2ax — 5, or 
 b — 2ax ; and there is nothing in the general equation to show whether 
 
 X is greater or less than a in the former case, or than — - in the 
 latter. The ambiguity does not exist in the second case, because 
 V a^ + b must be greater than «, and y/b2-\'4ac greater than b, and 
 therefore, cannot be equal respectively to a — .r, and b — 2aXf unless x 
 be negative. When the sign of x is doubtful, the first and second 
 cases are also ambiguous. 
 
QUADRATIC EQUATIONS. 91 
 
 17. Given 9a?3— 7^^=116. 
 
 Dividing by 9, x^ — -a?=-^ 
 
 Comparing this equation with the 2d form, (./^,) we have 
 
 2a=- and £>=-^j 
 
 7 49 116, 72 , 
 
 Hence, x=^+^(^^+—)=-=^. 
 
 Or comparing the given equation with the 2d form, (jB.) 
 a=9, 6=7, c=116. 
 
 74-s/ 49+4176 7+65 , 
 Hence, a:= jg ==-l8-=^- 
 
 18. Given ^/10 + a?— -Vl0+x=2, to find the value 
 of X, 
 
 Considering VlO + ^ ^s the quantity sought, and com- 
 paring the equation with the 2d form, (^,) we have 
 2^=1,6=2. 
 1 
 
 ...VlO + a^=^+ V2-=2. 
 10 + 0?= 2*= 16, and therefore, x=6. 
 
 Or VlO+a?— 2=VlO+a7. 
 
 Hence by involution, 10+a? — 4-v/10+j?+4=v/l0+ir. 
 
 Hence, 14 + 07= 5 V 10+ a:. 
 By involution, 196 + 28a? + 0?^= 250 + 250?. 
 Hence, a?2+3a?=54. 
 Comparing this with the 1st form, (^,) 
 2a=3, and Z>=54. 
 
 19. Given a?+ \/5a?+ 10=8, to determine the value of a?. 
 
92 QUADRATIC EQUATIONS. 
 
 Multiplying by 5, and adding 10 to each member, 
 
 5a? -1-10 + 5 v/ 5^+10=: 50. 
 
 Considering \/5a?+10 as the quantity sought; and com- 
 paring the equation with the first form, (*/?,) we have 
 
 2«=5, and 6:^=50. 
 
 . 5 
 
 Hence V^oo-\-lO^—^+^(60\^)=z5. 
 
 By involution and transposition, 5x=26 — 10. 
 
 Therefore, x=3. 
 Otherwise. — By transposition, ^/5J?+10 = 8 — x. 
 By involutio-n, 5a?4-10=64' — 16x+x'^j 
 
 Or ^^—2107=— 54. 
 Comparing this with the 3d form, (j5,) 
 a=l, Z>=:21, and c=54. 
 
 -_ 21dbv/441— 216 _ ^^ 
 Hence, 07= =18 or 3.* 
 
 20. Given > ^ %« ^ ^ j j 
 
 ^ x + y=12j to find 07 and y. 
 
 Comparing the 1st equation with form 1, (.^.) 
 
 a=i, b=12Q0. 
 
 Hence, 07^=— -i + \/(^ + 1260)=:35. 
 
 Tf in this equation, We substitute for y, its value, 12 — x, 
 found from the 2nd we have 
 
 1207—07^=35 or 07^— 12o7=— 35. 
 Comparing this with third form (j1) 
 
 a=6, and 6=35. 
 Hence, o?=6dbl = 7 or 5. 
 
 One of which being taken for o?, the other will be the 
 value of y, 
 
 * The ambiguity is introdnced into this solution by the involution 
 of the given equation; for (8 — xY=^{p^ — 8)^>* and one of the values 
 of a; corresponds to the supposition, that 52;-f-lO=(a; — 8)^, and the 
 other to the true one, that 52;+ 10= (8 — rr)^. 
 
QUADRATIC EQUATIONS. 93 
 
 21. Given } 'Z2 ~Z' V to find the values of x and y. 
 
 t xy=S ) 
 
 Comparing the 1st equation with form 2d, (J?,) 
 
 a=3, h=b, c=2. 
 
 „ X 5+V/25+24 ^ 
 
 Hence, -= -7; = 2. 
 
 y 6 
 
 .-. x=2y, and 2^2=8. 
 
 o 
 
 ... y—^-z=^1^ and a?=4'. 
 
 22. Given \ ^'+^0^7 ■^?' ^ to find x and y. 
 Assume vy=x, then v-y^=x^, and vy^=xy, 
 
 .•. 'z;2y'3_|_^2/^=12, or2/2=:-— — 
 
 ^^2__23/2=l, ory2=^j— ^ 
 
 12 1 
 
 Hence,— --—= — ^, or 122? — 24.=2?2^v. 
 
 Therefore, v^ — llv= — 24. 
 
 Comparing this with form' 3d, (^,) a=-^,5=24. 
 
 Whence, 2;=^±v'(^— 24)=8 or 3. 
 
 Otherwise. — Comparing the first equation with 12 times 
 the second, x'^-{-xy=12xy — 2%^, 
 .'. x"^ — llxy=^'24^y^. 
 Hence, considering 3/ as a given quantity. 
 
 11±5 
 By form 3. (B.) x=—^y=Sy or 3y. 
 
94 
 
 QUADRATIC EQUATIONS. 
 
 Whence, y=iy/6 or 1, and x=j^/6 or 3. 
 
 23. Given } y x > to find co and y, 
 
 (x+y—12, ) 
 Assume x=vy, 
 
 Then v''y-\-^-=:28=a. 
 
 vy-\-y=l2—b, 
 a b 
 
 Multiplying by z^^-h 1. 
 
 By form 3. (J5,) v- 
 
 5+«±V«^+2a6— 36^ 
 
 26 
 
 :3 or §. 
 
 Hence, y=3 or 9. 
 Wherefore a:=9 or 3. 
 
 24-. Given < 
 
 x^ ^2 
 
 Assume a?+2/=5, xy—p, 
 -a — 5. 
 
 Then ^V^^ 
 
 And multiplying by xy=p, x'^-{-y^=ap — sp. 
 But, x^+y^=s^^3sp.* 
 
 ^x-\-y=Sf xy=p. 
 From the square of the first equation subtract twice the second, 
 
 "From this equation subtract the second, and 
 
 x"^ — xy -\- jr2=: ^3 — 2p. 
 
 Multiply by the first, and x^-\^y'=s^ — 3s/^. 
 
QUADRATIC EQUATIONS. 95 
 
 Again, x'^-\-y^=zs^ — 2/?. 
 Also, ~Jr^^=(a-^sy—'lp = a^—^as^s^—^p. 
 
 Hence, — + x'^ + y^ ^^^=a^ — 2as-i-2s^ — 4<p=b. 
 
 By substitution, (C,) a^ — 2as + 2s^ ?r=^- 
 
 Clearing the equation of fractions, «^- — 2as~=ab'{'2bs, 
 b a^ — b 
 
 a 2 
 
 — b ¥ a?—b^ 
 
 Consequently, ^=0^ + v^(4^3 + -2~) 
 
 53 
 Whence, p-=^ becomes known. 
 
 Now, {x — y )^ = {x + yy — 4a?2/ = 5^ — 4p . 
 Or X — 3/==±=a/52 — 4p^ 
 
 and 2/— 
 
 2 ' ^ 2 
 
 Questions producing quadratic equations, 
 
 1. Required to find two numbers, such that their sum 
 added to their product shall be 47 j and the sum of the 
 squares shall be 74. 
 
 Let X and y denote the numbers. 
 
 Then a? + v + ^V=4^7, ) , ., -, , 
 A.r.Ax4}Lll, '5 by the data. 
 
 Add twice the first equation to the second, and 
 a?24. 2a?y + 2/2^ 2a? + 23/= 168. 
 Or, (a7 + 2/)^+2.(a? + 2/)=168. 
 Hence form 1, (^.) a;+j^=--l + 13=12. 
 
96 QUADKATIC EQUATIONS. 
 
 This equation subtracted from the first, leaves 
 
 By subtracting twice the last equation from the 2d, 
 
 X- — 2xy-\-y^=4f 
 
 Hence by evolution, x — y=d=:2. 
 
 .•.07=7 or 5. 
 
 2. A. and B. sold 130 ells of silk, for 42 crowns ; 40 ells 
 belonged to A. and 90 to B. Now, A. sold for a crown i 
 of an ell more than B. did. How many ells did each sell 
 for a crown 1 
 
 Let X = the number B. sold, > r 
 
 Then a: + ^ = the number A. sold, \ ^"^^ ^ ^^^^^' 
 
 90 
 .*. — =number of crowns received for B.'s silk. 
 
 X 
 
 * n 40 120 , .,..,.„ 
 
 And — r-r=-^ 7= number received for A.'s silk. 
 
 a?4-3 3a:+l 
 
 90 120 _ 
 
 •'•^"^3¥+l""^^' 
 
 270a?+ 90 + 120a?= 126a?3 4-42a?. 
 
 By transposition and division, 21a:- — 58a?=15. 
 
 Comparing this with form 2, (B^) we find 
 
 a=21, 6=58, c=15. 
 
 V 1260 + 3364 -4-58 126 ^ 
 Hence a? = — : ==.——=3, number of 
 
 42 42 ' 
 
 ells B. sold for a crown. 
 
 .'. 3 J, number A. sold for a crown. 
 
 3. There are two square yards, paved with stone, each 
 stone being a foot square ; the side of one yard exceeds 
 that of the other by 12 feet, and the number of stones in 
 the two is 2120 ^ what are the sides 1 
 
 Ans. 26 and 38 feet. 
 
QUADRATIC EQUATIONS. 97 
 
 4. A laborer dug two trenches, one of which was 6 
 yards longer than the other, for £11 16s., and the digging 
 of each cost as many shillings per yard as there were yards 
 in its length ', what was the length of each '? 
 
 Ans. 10 and 16 yards. 
 
 5. A. and B. set out from two towns, which were dis- 
 tant from each other 247 miles, and traveled till they met. 
 A. traveled 9 miles a day, and the number of miles walked 
 by B. in a day, increased by 3, was equal to the number of 
 days occupied by the journey. Kequired the number of 
 days they were traveling, and the number of miles passed 
 over by each 1 
 
 Ans. 13 days; and A. went 117 miles, and B. 130. 
 
 6. A. and B. having bought 41 oxen, for which each of 
 them paid 420 dollars, find A.'s worth a dollar a head more 
 than B.'s; how must they divide themi 
 
 Ans. A. 20, B. 21. 
 
 7. Divide the number 14 into two parts, whose product 
 shall be 48. Ans. the numbers are 6 and 8. 
 
 8. Given the sum of two numbers=9, and the sum of 
 the squares =45 5 what are- the numbers'? 
 
 Ans. 6 and 3. 
 
 9. What two numbers are those, whose sum, product, 
 and the difference of their squares, are equal to each other I 
 
 Ans. | + 5\/5, and ^ + W5. 
 
 10. There are four numbers in arithmetical progression, 
 of which the product of the two extremes is 22, and that 
 of the means 40 ; what are the numbers 1 
 
 Ans. 2, 5, 8, and 11. 
 
 11. There are three numbers in geometrical progression, 
 whose sum is 7, and the sum of their squares 21 ; what are 
 the numbers 1 Ans. 1, 2, and 4. 
 
 12. Required to find two numbers, such that the less 
 may be to the greater, as the greater is to 12 ; and the sum 
 of the squares may be 45. Ans. 3 and 6. 
 
 13. What two numbers are those, whose difference is 2, 
 and the difference of their cubes 98 1 Ans. 3 and 5. 
 
 9 
 
98 QUADRATIC EQUATIONS. 
 
 l^. What two numbers are those, whose sum is 6, and 
 the sum of whose cubes is 72 1 Ans. 2 and 4. 
 
 15. Required to find two such numbers, that their pro- 
 duct shall be 20, and the difference of their cubes 61 1 
 
 Ans. 4 and 5. 
 
 16. Required to divide the number 5 into two such 
 parts, that if each part be divided by the other, the sum of 
 the quotient shall be 2J. Ans. 3 and 2. 
 
 17. Divide 12 into two parts, so that their product may 
 be equal to 8 times their difference. Ans. 8 and 4. 
 
 18. There are two numbers, the sum of whose squares is 
 89, and their sum, multiplied by the greater, is 104 j what 
 are the numbers % 
 
 Ans. \^s/2, and -|V2, or 8 and 5. 
 
 19. What number is that, which being divided by the 
 product of its two digits, the quotient is 5^ ; but when 9 is 
 subtracted from it, the remainder is expressed by the same 
 digits in an inverted order 1 Ans. 32. 
 
 20. Required to divide 13 into three such parts, that 
 their squares may be equi-different, and the sum of those 
 squares may be 75. Ans. 1, 5, 7. 
 
 21. The sum of three equi-different numbers is 12, and 
 the sum of their 4th powers 962; what are the numbers "{ 
 
 Ans. 3, 4, and 5. 
 
 22. There are three equi-different numbers, such that 
 the square of the least, added to the product of the other 
 two, makes 28; but the square of the greatest, added to 
 the product of the other two, makes 44; what are the 
 numbers 1 Ans. 2, 4 6. 
 
 23. Three merchants, A. B. C, on comparing their 
 gains, finds they amount, collectively, to 1,444 dollars; 
 that B.'s gain added to the square root of A.'s makes 920 
 dollars; and that B.'s gain added to the square root of C's 
 makes 912 dollars; how much did they severally gain 1 
 
 Ans. A. 400, B. 900, C. 144. 
 
QUADRATIC EQUATIONS. 99 
 
 24. What two numbers are those, whose sum, added to 
 the square of the sum, makes 702, and whose difference, 
 subtracted from the square of the difference, leaves 56 1 
 
 Ans. 17 and 9. 
 
 25. The sum of two numbers is 10, and the sum of 
 , their 4th powers 1552; what are the numbers 1* 
 
 Ans. 6 and 4. 
 
 26. There are four numbers in arithmetical progression, 
 the common difference of which is 4, and their continued 
 product 9945 , what are the numbers 1 
 
 Ans. 5, 9, 13, 17. 
 
 27. The sum of three numbers in harmonical proportion 
 is 191, and the product of the first and last is 4032,* what 
 are the numbers 1 Ans. 56, 63, 72. 
 
 28. The sum of two numbers added to their product 
 makes 31; but the sum subtracted from the sum of the 
 squares, leaves 48. Quere the numbers 1 
 
 Ans. 7 and 3. 
 
 29. Required to find three numbers m continued pro- 
 portion, whose sum shall be 26, and the sum of their 
 squares 3641 Ans. 2, 6, and 18. 
 
 30. Required to find two numbers whose product shall 
 be 320, and the difference of their cubes to the cube of 
 their difference as 61 to 1 '? Ans. 20 and 16. 
 
 31. If 700 dollars be divided among four persons, so 
 that their shares may be in geometrical progression, and 
 the difference of the extremes to the difference of the 
 means as 37 to 12, what will be the several shares 1 
 
 Ans. 108, 144, 192, and 256 dollars. 
 
 * In examples of this nature, we may assume, letters to denote the 
 half sum and half difference of the required numbers ; whence, expres- 
 sions for the numbers themselves are readily obtained, and these being 
 involved, and the powers added together, the odd powers of the un- 
 known quantity will disappear. Hence, if the unknown quantity does 
 not rise higher than the 5th power, the solution may be effected by 
 qnadratic equations. 
 
100 QUADRATIC EQUATIONS. 
 
 32. The sum of two numbers is 11, and the sum of 
 their 5th powers 17831 5 quere the numbers 1 
 
 Ans. 7 and 4. 
 
 33. The difference of two numbers is 8, and the differ- 
 ence of their 4th powers is 14560 5 required the numbers 1^ 
 
 Ans. 11 and 3. 
 
 34. What number is that, from the square of which, if 
 9 be subtracted, and the remainder be multiplied by the 
 number itself, the product will be 80 1 Ans. 5. . 
 
 35. Required a number, from the square of which, 30 
 being deducted, and the remainder multiplied by the nmn- 
 ber itself, the product shall be 56 1 Ans. 2+3 x/ 2. 
 
 36. The continued product of five equi-different num- 
 bers is 945, and their sum 25 , what are the numbers'? 
 
 Ans. 1, 3, 5, 7, and 9. 
 
 *In this example, assuming for the half sum and half difference, 
 we obtain a cubic equation, whence it appears that questions of this 
 nature are not generally solvable by quadratics. When, however, 
 we have an equation of the form x^-\-ace=b ; if b=mri, and^^-j- 
 ff=n, the equation may be reduced to a quadratic. For multiplying 
 by a:^, and adding m'^^^ to each member, the given equation becomes 
 cc^-\'ax^^m2x^=m^x'2-\-bXj or cc^'^-nx^=m'^x'2'\-?imx, whence by 
 
 completing the squares and extracting the roots, x^-^-='mx ^~ , or 
 
 In the above example, x being made = half the sum of the re- 
 quired numbers, we find x^-j;-l6x=4S5=l X ^^i where 7^+16= 
 65, whence, x=l. If x^ — ax=mn, and m^ — a=7i ; or if .-^^ — ax 
 = — m?f., and a — m^=7i, the equation may be reduced to a quadratic, 
 and X found =m, as before. But if x^ — ax=miif and a — m^=n, or 
 x^ — ax= — 7n?tf and m^ — a=n; the equation may be changed to a 
 quadratic, from which a second quadratic x^ — mx=z^n, will arise, 
 
 whence x= : — . 
 
j.e^ 
 
 v^ 
 
 PROMISCUOUS EXAMPLES. 101 
 
 37. The sum of three numbers in geometrical progression 
 is 35, and the mean is to the difference of the extremes as 
 2 to 3 ; what are the numbers \ Ans. 5, 10, and 20. 
 
 38. The sum of three numbers in geometrical progres- 
 sion is 13, and the product of the mean by the sum of the 
 extremes is 30 5 required the numbers % Result, 1, 3, 9. 
 
 39. There is a number consisting of three digits in geo- 
 metrical progression ; the number itself is to the sum of 
 its digits as 124 to 7 ; and if 594 be added to it, the order 
 of the digits will be inverted 5 what is the number!* 
 
 Ans. 248. 
 
 Promiscuous examples to exercise the foregoing rules, 
 
 1. What number is that, to the double of which, if 44 
 be added, the sum will be equal to 4 times the number 
 proposed! Ans. 22. 
 
 2. A gentleman, meeting 4 poor persons, divided a dol- 
 lar among them in such manner that their several shares 
 composed an equi-different series ; and the^sum given to 
 the last was 4 times that given to the first. Quere their 
 several shares % Ans. 10, 20, 30, and''40 cents. 
 
 3. A sum of money being divided among 6 poor per- 
 sons, the second received 10c/., the third 14c?., the fourth 
 
 * Adfected quadratic equations maybe reduced to simple quadratics, 
 and solved without completing the square, in the following manner : 
 
 If the highest power of the unknown quantity has a co-efficient, 
 divide the whole equation by it ; then assume the unknown quantity, 
 equal to another unknown quautity with half the co-efficient united by 
 the opposite sign. Substitute this new value, and a simple quadratic will 
 arise. Thus, if x^J^2ax=h, assume z — a=x, then z^ — 2az + a^=x'2, 
 2az — 2a^=2ax, Yience z"^ — a^=x'^--\-2axz=b, and z=\/ bJ^a^y hence 
 
 x=zz — a= — a-\- \/ b J^a'^. 
 
 In like manner, the second term of an equation of a higher order 
 may be taken away by assuming a new unknown quantity with |, f , 
 etc., of the co-efficient of the next term united by the opposite sign, 
 in place of the quantity sought. 
 
102 PROMTSCUOUS EXAMPLES. 
 
 25c?., the fifth 28c?., and the sixth 33d. less than the first. 
 The whole sum divided was 10c?. more than three times 
 what the first received. How much did they severally re- 
 ceive 1 Ans. 40c/., 30c?., 266/., 15c/., 12c/., and Id. 
 
 4. A mercer, having cut 19 yards from each of 3 equal 
 pieces of silk, and 17 yards from another of the same 
 length, finds the four remnants together measure 142 
 yards 3 what was the original length of each piece 1 
 
 Ans. 54. 
 
 5. A grazier, having two flocks of sheep, containing the 
 same number, sells 39 from one, and 93 from the other, 
 and then finds one flock twice as numerous as the other 5 
 what number did each of them contain at first '? 
 
 Ans. 147. 
 
 6. From each of 16 coins an artist filed the value off 20 
 cents, when the coins, being examined, were found worth 
 only 11 dollars 68 cents 5 what was the original value of 
 each] Ans. 93 cents. 
 
 7. The hold of a ship, containing 442 gallons, is emptied 
 in 12 minutes by two buckets ; the greater of which, hold- 
 ing twice as much as the less, is emptied twice in three 
 minutes, and the less is emptied three times in two minutes. 
 Quere the number of gallons held by each bucket 1 
 
 Ans. 26 and 13. 
 
 8. A trader maintained himself for three years at an 
 annual expense of ^50, and in each of those years aug- 
 mented that part of his stock which was not expended by 
 i thereof. At the end of the third year, his original stock 
 was doubled ; what was that stock % Ans. ^740. 
 
 9. A gentleman having a rectangular yard, 100 feet by 
 80, purposes to make a gravel walk of equal width lialf 
 round it, so as to occupy one fourth of the ground. What 
 must be the width of the walkl Ans. 11.8975 feet. 
 
 10. Eequired to find a fraction, to the numerator of 
 which, if 4 be added, the value will be h. ; but if 7 be 
 added to the denominator, the value will be J 1 
 
 Ans. j%. 
 
PROMISCUOUS EXANPLES. 103 
 
 11. What two numbers are those, whose difference, sum, 
 and product are as the numbers 2, 3, and 5 respectively % 
 
 Ans. 10 and 2. 
 
 12. A vintner, having mixed a quantity of brandy and 
 water, finds that if he had mixed 6 gallons more of each, 
 he would have bad 7 gallons of brandy for every 6 of 
 water 5 but if he had mixed 6 gallons less of each, he 
 would have had 6 gallons of brandy for every 5 of water. 
 Quere the number of gallons of each 1 
 
 Ans. 78 of brandy, and 66 of water. 
 
 13. A. and B. together, are able to perform a piece of 
 work in 15 days; after working jointly 6 days, B. finishes 
 it alone in 30 days ; in what time would each of them 
 singly effect itl Ans. A. in 21| days, B. in 50 days 
 
 14. A company of smugglers found a cave which would 
 exactly hold their cargo, viz: 13 bales of cotton, and 33 
 casks of rum ; but while they were unloading, a revenue 
 cutter appeared, on which they sailed away with 9 casks 
 and 5 bales, having filled f of the cave ; how many bales, 
 or how many casks would the cave contain 1 
 
 Ans. 24 bales, or 72 casks. 
 
 15. There are two numbers, whose sum is to their dif- 
 ference as 8 to 1, and the difference of whose squares is 128 5 
 what are the numbers 1 Ans. 18 and 14. 
 
 16. Eequired those two numbers, whose sum is to the 
 less as 5 to 2 ; and whose difference multiplied by the dif- 
 ference of their squares is 1351 Ans. 9 and 6. 
 
 17. A merchant laid out a certain sum upon a specula- 
 tion, and found, at the end of a year, that he had gained 
 £69. This he added to his stock, and at the end of the 
 second year found he had gained as much per cent, as in 
 the first. Continuing in this manner, and each year add- 
 ing to his stock the gain of the preceding, he found, at the 
 end of the fourth year, that his stock was to the sum first 
 laid out as 81 to 16. Quere the sum first invested 1 
 
 Ans. ^138. 
 
I04f PROMISCUOUS EXAMPLES. 
 
 18. There are two numbers, whose sum is to the greater 
 as 40 is to the less, and whose sum is to the less as 90 to 
 the greater 5 what are the numbers'? Ans. 36 and 24. 
 
 19. The area of a rectangular parallelogram is 960 
 yards, and the length exceeds the breadth by 16 yards ; 
 what are the sides 1 Ans. 40 and 24. 
 
 20. The area of a rectangular parallelogram is 480, and 
 the sum of the length and breadth 52. Quere the sides ] 
 
 Ans. 40 and 12. 
 
 21. The sides of a right angled triangle form an equi- 
 different series, whose common difference is 3 ; what are 
 the sides 1 Ans. 15, 12, and 9. 
 
 22. There are three numbers, the difference of whose 
 differences is 5, their sum is 20, and their continued pro- 
 duct 130 5 required the numbers! Ans. 2, 5, and 13. 
 
 23. The sum of three numbers is 21, the sum of the 
 squares of the greatest and least is 137, and the difference 
 of the differences is 3. Quere the numbers '? 
 
 Ans. 4, 6, and 11. 
 
 24. There is a number consisting of two digits, which, 
 divided by the sum of its digits, has a quotient greater by 
 2 than the first digit. But, the digits being inverted, and 
 divided by the sum of the digits increased by unity, the 
 quotient is equal to the first digit increased by 4. Quere 
 the number 1 Ans. 24. 
 
 25. Eequired to find four numbers in geometrical pro- 
 gression, whose sum shall .be 15, and the sum of the 
 squares 85 1* Ans. 1, 2, 4, and 8. 
 
 a;2 y'2 
 
 * Note* — If we assume x,y to denote the means, — . and — will re- 
 
 y X 
 
 present the extremes ; the problem will then be the same as example 
 21, page 94. The following solution includes an expedient which may 
 be sometimes used with advantage. 
 
 Let X denote the first term, and y the common multiplier. 
 
 Then x-Yxy-i^xy'^-\-xy^=,l6=a; x^-^-x^-y^ -\- x'^y^ -{•x'^y^=S5=^b. 
 
PROMISCUOUS EXAMPLES. 105 
 
 26. There are five numbers in geometrical progression, 
 whose sum is 242, and the sum of their squares 29524 5 
 what are the numbers! Ans. 2, 6, 18, 54, 162. 
 
 27. There are six numbers in geometrical progression, 
 the sum of the extremes is 99, and the sum of the other 
 four terms is 90. Quere the numbers % 
 
 Ans. 3, 6, 12, 24, 48, and 96. 
 
 o2 
 
 Then 2:2=- 
 
 But from the nature of progressionals, 
 
 y^ + y^+7/+l=^-^', and 2/6 + 3'4 + S'' + l=^^; 
 The above eqjiation may therefore be expressed thus, 
 
 . ^^(y— i) ^ ^(y+i) 
 
 And dividing the numerator and denominator of the first member 
 by (y — i)j and clearing the equations of fractions, 
 
 .•.a2(y4+l)==i(y4 4. 22/3 + 2^24-2^ + 1.) 
 .-. (a2— i)y4— 2iy3__25y2_25y -f- (a2— 3)=0. 
 
 ... (a2_5)y2-«23y— 25— 2i.- + (aS— 3)— =0. 
 
 y y^ 
 
 Now (3/4-^)2=3^24.24-^3 
 
 .•.(a2— J)(y4-_)3;=(a2— 5)y2 f 2(a2— 5) + (o^— 5)— 
 
 y y^ 
 
 Hence, {a^—1)){yJ^—Y—2h{y-\'-)=:2a% a quadratic equation. 
 
 y y 
 
 1 1 .1 174-53 5 
 
 22/2— 5y=— 2, 3/=^li-— -=2ori. 
 
 Hence 1, 2, 4, 8, or 8, 4, 2, 1> are the numbers sought. 
 
 The 26th and 27th examples may be solved in a similar way- 
 
106 RATIOS. 
 
 28. The American dollar consists of 1485 parts by 
 weight of pure silver, and 179 of copper. Quere the 
 specific gravity of this dollar, the specific gravity of pure 
 silver being 11092, and that of copper 9000] 
 
 Ans. 10821. 
 
 29. A person having set out from a certain place, travels 
 one mile the first day, two the second, etc., in arithmetical 
 progression. In six days another sets out from the same 
 place, and travels in the same direction at the rate of 15 
 miles a day. In how many days will they come together 1 
 
 Ans. In 9, and in 20 days after the first sets out. 
 
 30. Heavy bodies near the earth's surface are known to 
 fall 16yi^ feet in the first second of time ; and to pass over 
 S'paces, which, reckoned from the commencement of the fall, 
 are as the squares of the time. Now, a haiJU stone, being 
 observed to descend 595 feet during the last Second ; quere 
 how long was it falling, and from what height did it de- 
 scend] Ans. 19 seconds, and 5806 feet. 
 
 31. The arithmetical mean of two numbers exceeds the 
 geometrical mean by 13 ; and the geometrical mean ex- 
 ceeds the harmonical mean by .12. What are the num- 
 bers 1 Ans. 234, and 104. 
 
 32. The fore wheel of a carriage makes six revolutions 
 more than the hind wheel in going 120 yards ; but if the 
 circumference of each wheel be increased one yard, it will 
 make only four revolutions more than the hind wheel in 
 going the same distance. Eequired the circumference of 
 each ] Ans. The fore wheel 4 yards, the hind 5. 
 
 Section VIU. 
 ON EATIOS. 
 
 68. Ratio is the relation that two quantities of the same 
 kind bear to each other. 
 
RATIOS. 107 
 
 Ratio is estiiilated by the quotient arising from the divi- 
 sion of the first term by the second. Thus, if 7=-, ^ a is 
 
 said to have the same ratio to h that c has to d; or the 
 quantities a, b^ c, c?, are termed proportionals ; which is 
 briefly expressed thus, a : b : : c : d. 
 
 The first and third terms are called antecedents, the 
 second and fourth consequents,^ 
 
 69. When four quantities are proportionals, the product 
 of the first and fourth is equal to the product of the second 
 and third ; and reciprocally. 
 
 a c 
 If-=-, by multiplying by bd, ad=bc, 
 
 ^, .J, y T cid be a c 
 
 Eeciprocally, if aa= be, -r-j=-r-j, or 'j^-j* 
 
 70. When four quantities are proportionals, the sum of 
 the first and second is to the second, as the sum of the third 
 and fourth is to the fourth. 
 
 ^^a c a ^ c _^ a+b c + d 
 
 Or, a+b : b : : c+d : d. 
 
 If the word difference be substituted for sum, the propo- 
 sition will still be a true one. 
 
 a c a c a^b c^s^d 
 
 Or, a^b : b : : C'^d : d, 
 
 71. If four quantities are proportionals, the sum of the 
 first and second is to their difference as the sum of the third 
 and fourth is to their difference. 
 
 * The terms are similarly designated when more then four are con- 
 cerned. 
 
108 RATIOS. 
 
 For by the last article, —j — =—7—, and -^=—7—, 
 
 And dividing the former by the latter, 
 
 a + b c + d 
 
 7= ,. or a 4-0 : a^o : : cA-d: c^d, 
 
 a^b C'^d 
 
 72. When any number of quantities are proportionals, 
 as one antecedent is to its consequent, so is the sum of all 
 the antecedents to the sum of all the consequents. 
 
 Let a:b :: c : d: : e:f: : g : h, etc., then (art. 69.) 
 
 ad=bc, af=be, ah=^bg^ etc., also ab=^ba. 
 .*. ah-\-ad-\-af+ah^ etc. =zba-\-bc-\'be-\-bg^ etc. 
 Or, aX fZ) + (/+/+ A, etc.} =6x Ja-f-c+e+g*, etc.} 
 .•.(art. 69,) a : 6 :: a+c + c+g, etc. : 6 + c?+/+A, etc. 
 
 73. When four quantities are proportionals, if the first 
 and second be multiplied or divided by any quantity, and 
 likewise the third and fourth, the resulting quantities will 
 be proportionals. 
 
 ^ ci c a ma 
 
 If a : 6 : : c : a, -r = -7, but 7= — -,: 
 
 ^ b d^ b mV 
 
 ^ .c nc ma nc ' . . 
 
 And -,= — ,, .'. —7= — "75 or ma : mb : :nc: nd, 
 d nd mb nd 
 
 The demonstration is manifestly applicable when m and 
 n are, one or both, fractional numbers. 
 
 74. When the first and third of four proportionals are 
 multiplied or divided by any number, and also the second 
 and fourth, the resulting quantities are proportionals. 
 
 a c ma mc ma mc 
 b~~ d ' ' b d nb nd"* 
 
 Or ma :nb ::mc ind, m and n being any numbers, either 
 integral or fractional. 
 
 75. If four quantities are proportionals, the like powers 
 or roots of these quantities will be proportionals. 
 
If a:b :: c: d. then i = -7, andr- = 
 ' h d^ 0"" 
 
 RATIOS. 109 
 
 a c a"" e 
 
 1 
 
 Also, —=— 5 or a" : 6" : : c" : c?^, 
 6^" d- 
 
 1 I X 1 
 
 And aF:b'':i6^:d'', 
 
 76. A ratio compounded of several ratios is indicated by 
 the continued product of the quantities which denote the 
 component ratios. Thus, the ratio compounded of the 
 ratios of a : 6, of c : c?, and of e : /*, is indicated by 
 
 ace ace 
 
 y^T^/'''^ Vdf 
 
 The ratio which the first of a series of quantities, of like 
 kind, has to the last, is the same as the ratio compounded 
 of the ratios of the first to the second, of the second to the 
 third, etc. to the last. 
 
 Let «, 6, c, c?, e,/*, be quantities of a like kind, then 
 a a b c d e 
 f b c d e f 
 
 77. In tw^o ranks of proportionals, if the corresponding 
 terms be multiplied together, the products will be propor- 
 tionals. 
 
 Let a:b:: c: d^ and e :f: : g : h, 
 
 ^^^^' b~ d' f~h '''bfdh' 
 
 Or, ae, :bf::cg: dh. 
 
 The demonstration may be easily applied to any number 
 of proportions. 
 
 78. A ratio compounded of two, three, four, etc., equal 
 ratios, is called the duplicate, triplicate, quadruplicate, Qic, 
 of one of the component ratios. 
 
 10 
 
1 10 RATIOS. 
 
 The ratio compounded of any number of equal ratios is 
 the same as the ratio of such power of the first term, as is 
 indicated by the number of component ratios to a like 
 power of the second. 
 
 a h 
 
 Let a : h : : h \ c. then -7-= — 
 
 ^ b c 
 
 a^ ah a 
 
 0^ be c ' ' 
 
 Again, let a : b : : b : c : : c : d : : d : e, etc. 
 
 aaabacadae 
 Then, y=y, y=y» T"^i:' T^^T' y=-yr,etc. 
 
 to 7^, equations 5 and multiplying the first and second mem- 
 bers respectively together, 
 
 a^ a b c d e a 
 
 Or, a'':b''::a :/. 
 
 79. When the ratio of the first of three quantities to the 
 second is the same as the ratio of the second to the third, 
 the ratio of the first to the second is termed the sub-dupli- 
 cate of the ratio of the first to the third. 
 
 Whec^four quantities are continued proportionals, the 
 ratio of the first to the second is called the sub-triplicate 
 of the ratio of the first to the fourth. 
 
 A ratio, compounded of a simple and sub-duplicate ratio, 
 is called a sesquiplicate ratio. 
 
 80. The sub-duplicate ratio is equivalent to the ratio of 
 the square roots ; the sub-triplicate, to the ratio of the cube 
 roots ; and the sesquiplicate, to the ratio of the square roots 
 of the cubes. 
 
 Let a : b : : b : c : : c : d. Also, b:e:: e:c, 
 
 h 
 a (L a a 
 
 First, (art. 78,) y=y , •-. T^l ^^^ ^ ' ^ • * ^ 
 
 c^ 
 
VARIATIONS. Ill 
 
 i 
 
 a 
 b """ 
 
 In like manner, -^=-7, ora:5: : a"^ : d 
 
 1 
 
 Also, since 5 : e : : e : c, and a : b : : c : d, 
 
 e'^=bc=ad. (art. 69.) But r— =-7-= — -7=-t-> 
 ' ^ ^ b"^ d ad e^ 
 
 3 
 
 .•. -— = — , or a^ :b^ ::a: e, in which the ratio of 
 
 a : e, is compounded of the ratios of a:b, and of 6 : e, or of 
 the ratio of a:b, and the sub-duplicate of the ratio of 5 : c. 
 
 ON THE VARIATIONS OF QUANTITIES. 
 
 81. In the investigation of the relation which varying 
 and dependent quantities bear to each other, the conclu- 
 sions are more readily obtained, by expressing only two 
 terms in each proportion, than by retaining the four. But 
 though, in considering the variations of such quantities, 
 two terms only are expressed, it must be remembered that 
 four are supposed ; and that the operations, by which our 
 conclusions are obtained, are in reality the operations of 
 proportionals. 
 
 82. One quantity is said to vary directly, as another, 
 when one is such a function * of the other, that, if the for- 
 mer be changed, the latter will be changed in the same 
 ratio. Thus, if B be such a function of .5, that by chang- 
 ing Jl io a, B shall be changed to b ; making A i a i : B :b^ 
 
 =* The ///7zc/io/j of any variable quantity ic, is an algebriac expres- 
 sion, in which x, combined with invariable quantities, is involved. 
 Thus, l + ic, (l-fa;)^, ax, iP", «^, etc. are functions of x. Analysts 
 sometimes use the Greek letter 0, to denote a function. Thus^ (^x may 
 represent any function of x. 
 
112 VARIATIONS. 
 
 A is said to vary directly as J5. This relation is designated 
 thus, Ji^B, 
 
 83. One quantity is said to vary inversely as another, 
 when the latter is such a function of the former, that the 
 one being increased or diminished, the other will be dimi- 
 nished or increased in the same ratio. Thus, if B be such 
 a function of ^, that, by changing A io a^ B becomes 
 changed to h ; making A \a::h i B, Then A is said to 
 
 vary inversely as ^. Indicated thus, Aac -^ , 
 
 84*. One quantity is said to vary as two others jointly, 
 when the two last are such functions of the first, that the 
 ratio, which any two values of the first bear to each other, 
 shall be the same as the ratio compounded of the ratios of 
 the corresponding values of the other two. Thus, A varies 
 as B and C jointly, (^AccBCj) when A being changed to 
 
 ABC 
 
 a, B chang-es to 5, and C to c, so that — =-7-X — . 
 ° a c 
 
 85. One quantity is said to vary directly as a second, 
 and inversely as a third, when the ratio which any two 
 values of the first bear to each other, is the same as the 
 ratio compounded of the direct ratio of the corresponding 
 values of the second, and the inverse ratio of those of the 
 third. 
 
 Thus, A varies directly as B, and inversely as C, 
 (Aoc YTf) when A, B^ C ; a, Z>, c, being corrresponding 
 
 A B c 
 values, — =-7-X77-. 
 a C 
 
 In the following articles. A, J5, C, etc. represent cor- 
 responding values of any quantities, and a, 6, c, etc. any 
 other corresponding values of quantities. 
 
VARIATIONS. 113 
 
 86. If A^B^ and m^ n denote any given numbers, 
 
 -n 
 
 Then A^xmBo: — , 
 
 ^ . A B B mB ^B A mB ^B 
 
 For since — = i-, and -r-= — 7=7-" 5 — = — i:=^~r'* 
 a mo ^if a mo ^5 
 
 87. liAxB, then ^°x^%- and A^azB\ See art. 75. 
 
 88. If AazB, and CazD, then AC^BD. See art. 77. 
 
 89. \iA:f:BC, then -Soc^, and Coc-g-, 
 
 ^ A B C Ac B B A c ^^ A^ 
 
 a c aC ^ a C ^ C^ 
 
 ,, Ab C C A b ,^ A ^^ 
 
 Also, -^=— , or— =— X-H-, (C'gc^.) 
 ^ aB c' c a B ' ^ B "^ 
 
 From the three preceding articles it appears, that quan- 
 tities connected by the sign oc, may be treated as the mem- 
 bers of an equation, as far as multiplication, division, in- 
 volution, or evolution is concerned. 
 
 90. If AccC, and J9xC, A and B being quantities of the 
 same kind, then (^Ad:zB)QfCj and y/ABccC. 
 
 ^ . A C B C A B A a 
 
 Forsince — = — , and --,--= — : — =-t-. .*. 7r=Tj 
 a c' b cab B b 
 
 ^ A a ^ Ad^B a±b A±:B B C 
 
 And-^d=l=T=bl, or — ^— =— ^— . .-. -— -^=— ==— 
 B b ^ B b adub b c 
 
 Again, since AccC, and B ocC, (art. 88,) ABocC^, 
 
 .-.(art. 87,) s/AB^C. 
 
 91. If while A and B vary, AB=b, constant quantity, 
 
 A cx-g , and B oc-^ 
 For since AB=ab^ — ~~»' (•^^c">) 
 
 10* 
 
114 SERIES. 
 
 Sectiojt IX. 
 OF SERIES. 
 
 92. From the nature of powers, (art. 14,) we readily 
 discover that any two powers of the same root, multipli(:»d 
 together, produce a power indicated by the sum of the ex- 
 ponents of the factors ; thus a^.f^S—^S. ^m^^n__^ni4-n. 
 
 _ , , , , fl^^ la,a,a.a,a aJ^ 
 
 On the other hand, — =— ^ = fla=a^--=ia"~'' 
 
 ,, a"^ a"^ 1 
 Suppose n==m + v, then —=-^=- 
 
 But in this case m — n= — v^ if then— can be always de- 
 noted by a""""; a""" must be equivalent to — 
 
 Considering the consecutive powers of a given root as 
 the terms of a geometrical series, extending at pleasure, 
 above and below unity. 
 
 111. 
 
 - i ~z-> "f 1? ^5 ^^ ^^ etc. 
 
 Or the equivalent series, a~^, a-^, a"^, a^, a\ a% a^, etc. 
 
 The index denotes the order of the term beginning with 
 unity, and making the index positive or negative, accord- 
 ing as the quantity is above or below the point of com- 
 mencement. 
 
 93. Between any two of these terms, let n — 1 mean pro- 
 portionals be interposed, ex. gr. between a^ (=1?) and a, 
 calling the first x, then (art. 78,) 1 : a: : V : x"" : : 1 : a?". 
 
 Our series then becomes, (x'^ being = 1 =a^,) 
 
 'r~2Il ^— n— 1 r^—U rp—2 rn—1 /y)0 -yil vt2 
 
 ... a?", a?°+S ^""^^ ^'^"j ^^"■^'S etc. 
 
SERIES. 115 
 
 Now, the product of any two terms of this series is mani- 
 festly indicated by x, Avith an exponent equal to the sum 
 of the exponents of the factors, as a?°X a^2''+^=a;5"+^5 
 
 ^.+1 X 0?-°= - - - X — — =: 0?^= 0?"+-^-", 
 
 1 a?" 07° 
 
 ^2a+ 3 
 
 =: a:'"+ i = ^2n-h3— (n-f- 2) 
 
 If for X and its powers we write a^ and its powers, the 
 series, though changed as to the form of its exponents, 
 evidently retains the same essential character, and the 
 operations of multiplication and division are performed by 
 taking the sum and difference of the exponents of the fac- 
 tors, whether those exponents are positive or negative, in- 
 tegral or fractional. By thus arranging the powers as the 
 terms of a geometrical series, we perceive that the expo- 
 nent, whether integral or fractional, serves not only to 
 designate the power, but to indicate the situation of the 
 term, in relation to the unit's place. 
 
 94. From these principles we readily infer, that any 
 letter or quantity may be removed from the denominator 
 of a fraction to its numerator, and vice versa, by changing 
 the sign of its exponent. 
 
 Thus, ~^=—x— = ax-^. 
 x^ 1 x^ 
 
 a-\-b , , , , a 
 
 ^ , \„,, ==(a + b).(a''—b^)-k. ab''= ,— 
 
 95. It frequently happens in the division or evolution 
 of algebriac quantities, as well as in common arithmetic, 
 that to whatever extent the process may be continued, a 
 remainder will still occur; in which case the resulting 
 quotient or root mostly assumes the form of an infinite 
 series. 
 
116 SERIES. 
 
 Tlius^ ~ ■ — l-\-x+x^-\-x^+x^-{-j etc. to infinity. 
 
 b^ h^ b^ b¥ 
 
 ^«^+A»=:« + 2--8^3+i6^-T28^.+. etc.* 
 
 In some instances, when a few terms of the series are 
 obtained, the law of continuation^ or the relation of the 
 successive terms to those which precede them, becomes 
 manifest. The first of the series above given may be 
 readily continued to any proposed extent. The law of con- 
 tinuation in the last is not obvious on first view 5 it will 
 however, be shown further on. 
 
 96. In the investigations connected with series, the me- 
 thod of indeterminate co-efficients is often found particularly 
 convenient. It depends upon the following theorem. 
 
 Let .^a7+5a?'2+Ca?^ + Da?*, etc. =«a?4-^a?^ + ca?^+^a?*, etc., 
 the series being both infinite, or, if finite, extending to the 
 same number of terms; and A^ B^ C, a, 5, c, invariable; 
 if then the above equation be true, whatever value may be 
 assigned to a?, A will be equal to «, B=b, etc. 5 for dividing 
 by a?, we have 
 
 A + Bx+Cx'^+Dx^^ etc.=a'\-bx+cx'^-\-dx'^, etc. 
 
 Now, the equation being true for all values of a?, must 
 hold if a?=0, in which case the equation becomes A=a; 
 subtracting this equation from the given one, 
 
 Bx + Cx^'+Dx^, eic. = bx-^cx^+dx'^, etc. 
 Dividing by x, B-{-Cx-\-Dx\ eic, = b-{-cx+dx% etc. 
 Whence, if a?=0, B = b, 
 
 In the same manner C=c, D=d^ etc. m 
 
 This equation becomes by transposition, 
 
 Ax-\-Bx^'\'Cx''-\-I)x'-{', etc. ) _^. 
 — ax — bx'^ — cx^ — c?a?*-i-) etc. ) "~ 
 
 * When the successive terms of an infinite series continually 
 decrease, it is called a converging series ; in which case, the sum of the • 
 series may be approximated by collecting a finite number of terms. 
 
SERIES. / 117 
 
 Hence it appears, that when all the terms of a general 
 equation are brought to one side, the co-efficients of the 
 several powers of the unknown quantity are respectively 
 equal to 0. 
 
 This principle is applied in the following examples. 
 
 1. Required to express - — ^ : in a series. 
 
 ^ ^ 1 — 2a? +07^ 
 
 It is easy to perceive that the first term must be 1. 
 
 Assume then - — ^ — : — -=!-{■ ax ^bx^-\-cx^-\-dxK etc. 
 1 — 2x-\-x^ ' ' ' 
 
 Multiply by 1 — 2a? +a:^, and bring the terms to one side 
 of the equation 
 
 Then, l-\-ax-{'hx'^ -fca?^ +dx^ +, etc. 1 
 
 — 2a7 — 2aa?^ — 2Z)a?'^— 2ca7*— , etc. V =0 
 — 1 +0:2 _pc5^3 _j_5^i _p^ etc. ) 
 
 Hence, «— 2=0, h—1a-V 1 = 0, c— 25-f «=rO, c?— .2c + 6=0. 
 ... a=.2, 5=4—1 = 3, c=6— 2=4, c^=8— 3=5. 
 
 Whence, ^ =l+2a?+3a?^+4a?'^+5a?^+, etc. in 
 
 which the law of continuation is manifest. 
 
 2. Required to develope ^/a^'{-x^^ in a series. 
 
 V«^+a?^=V(«^Xl + -)=ax/(l+-)=aVl+2^% 
 
 a?2 
 Puttino; - = 2;2. 
 
 Assume -,/ l-\-z''—l+az'^-\-hz^+cz^-\'dz^-\-, etc.* 
 
 * If the series iJ^az-^-bz^ J^cz^^ etc. had been assumed, we should 
 have found a=0, ^=0, etc. 
 
1 ] 8 SERIES. 
 
 By involution and transposition, 
 
 l-{.2az''+2bz^+2cz^'\-2dz^+2ez^''+, etc. ^ 
 
 a''z^ + 2abz^+2acz^+2adz^^, etc. V = 0. 
 6^2:8 +26c2r^o, etc. 3 
 — 1 — z"". 
 
 .-. 2a— 1 =0, 2b+a^=0, 2c+2ab=0, 2d+2ac + b'^=0, etc. 
 
 Whence, «=2' ^^"S'^^Tg' ^^"128' '"^256 
 11 1 „ 5 
 
 ... Vl+-^=l+5^--§-^+ je^^-j^8-^+256^^^ ^^^- 
 Consequently, 
 
 >• /p3 'j^ oa8 ^^^ ^ 
 
 Va^+x^ = a J 1 + 2^-8^.+ i6^B-li8^s' etc. j = 
 
 07^ o:* 0?^ So;'^ • 7a?^o 
 ' ^+2^~8^3+i6^— 128^7+256^9' ^*^- 
 
 To find, if possible, the law of continuation, we observe 
 that 
 
 1__1 —1 ^_1 —1 —3 
 
 ~B~2^T"' 16~2^ir^ 6"' 
 
 5 1 _i -_3 _5 7 1—1 —3 —5 
 
 X-J->^-7r-X-77-, K^ — nX-j-.X-^X- 
 
 128""2 4 6^8' 256~2 4 6 
 
 —7 
 X -T^» Hence the law is evident 
 
 97. Eequired to express ^ in a series, n being an in- 
 tegral number. 
 
 0?— y 0? 1- -2r 
 
SERIES. 119 
 
 y 
 
 Assuming 2:=-. 
 cc 
 
 1 2^n 
 
 Now, if be resolved into a series, it is manifest 
 
 1 — z ' 
 
 the first term will be 1. 
 
 And therefore, — = o?""^ X -z resolved into a 
 
 X — y 1 — z 
 
 series, will have its first term a?°~^ 
 
 a?" 
 
 a:" — y"^ y^ — a?** 2/^ ^ _ 1 — ^" 
 
 Again, • 
 
 ^—y y—^ y r i— ^ ' 
 
 1— - 
 y 
 
 making v=-^ which, resolved into a series, will have its 
 first term y"~^. 
 
 •vi^ y^ 
 
 Whence it appears, that if be resolved into a se- 
 
 X — y 
 
 ries, the first and last terms will be x^~^^ and y""-^ respec- 
 tively. 
 
 1 — z"" 
 Assume then = l+az-^-bz^ -{',.,, jp z""'^ -\- qz^'-K 
 
 Multiply by (1 — 2:,) and transpose 1 — z"", whence, 
 
 1+az+bz^ + cz^j pz^'-^+qz''-^ ^ 
 
 — z — az- — bz^, — mz""-^ — pz""-^ — qz% > =0 
 —1 +z% ) 
 
 .-. (art. 96,) a— 1=0, 5—^=0, c— 5=0, p^m=:0, 
 ^— p=0, 1 — g=0. Or a=l, b=l, c=l, etc. 
 
 .-. — -5- = a?^-'X {l+z+z^-\-z\ 2r^-3+2r°-'| = 
 
120 SERIES. 
 
 Whence 03^—^"=: (a? — y)X [x^-^+x^'-^y+x^'-^y'^. 
 
 Here the indices of x descending regularly from n — 1, 
 to 0, or n — 7^, it appears the number of terms in the series 
 is n, 
 
 98. Required to develope (l+o?)'' in a series, n being a 
 whole positive number. 
 
 It is easy to perceive that the first term of the series is 
 1 ] and that the powers of x regularly ascend. 
 Assume then, 
 
 (1 + 0?)"— l + aa?+Z)a?2+ca?'^ + c/a?H-, etc. {A.) 
 Consequently, {l-\-yy=::l-\-ay-\-hy'^-\-cy'^-\-dy*^ etc. 
 By subtraction, (l+a?)° — (l-^-yY—a^x — y)+h{x'^ — y^) 
 + c(x^ — y^) 4- d(x^ — y%) etc. 
 
 Divide by 1-f a? — l+i/=a? — y, and we shall have 
 
 X — y ^ X — y ^ 
 
 e(^±=?(!)+<Z(^-=i:) etc. 
 
 ^ X — y ' ^ X — y ^ 
 
 Or (l+a:)"-i4-(l+^r-^X(l+i/) (1+2/^"^ 
 
 =ia-\-h{x-Yy)~\-c[x^~'-\-^y-\-y^) 
 +(f(a?'^ + a?'^2/ + a?^2+?/3)+, etc. 
 This equation being true, whatever value may be as- 
 signed to X and y, let us suppose a:=y, then this equation 
 becomes 
 
 (art. 97,) n\\-\-xy-^z=za-\-'lhx-\-'^cx''-\-Ux^-\-bex', etc. 
 Multiply by (1 + 0?) 
 
 .-. 71.(1 + 0?)°— a+2^a7+3ca;2+4ffir^ + 5eo?*, etc. 
 ao?+26o?2+3co?^ + 4c?o?*, etc. 
 But (eq. ^,) multiplied by tz, 
 
 71.(1 +o?)°=7z+wao?+7z5o?'2+7ico?3+wc?o?*, ctc. 
 
 _, , , ^ — 1 ^-""1 
 
 Consequently, a=7z, 6)=«.— ^=7?.-^, 
 
SERIES. 121 
 
 , n — 2 n — 1 n — 2 , n — 3 n — 1 n — 2 n — 3 
 n — 4 n — 1 n — 2 n — 3 n — 4 
 
 Hence, (a+J)''=a°X(l+-)''=: 
 
 a«.Sl+«-+n.-^-+,etc.}= 
 
 n — 1 , n — 1 n — 2 
 
 aJ'-\-na^-'h-\-n,—^a''~^y^-^n,—^ -_a»-353 4-, etc. 
 
 When n is B, whole positive number, as here supposed, 
 
 the series is finite , the number of terms being n-\- 1 ; for 
 
 the co-efficient of term n-{-2is 
 
 n — 1 n — 2 n — n 
 n. — ^r—' — 5— •••• , 1=0. 
 
 If (a — by be required, we may put — b and its powers 
 for +6 and its powers. 
 
 n — 1 
 Whence, (a — Z>)°=a° — na''-^b+n—^a''"^b^ 
 
 n — 1 n — 2 , n — 1 n — 2 n — 3 
 ^n. —^ —a^-^¥-\-n,—^ g ^a"-*i*,etc. 
 
 99. Required to develope (l+a?)° in a series. 
 Assume (l+x)"=l+aa?+Ja7^+ca?3+(Za7*+ea?^, etc. (^.) 
 Then (l+^)^=l+«2^+^y'^+c?/34-c?2/+e?/5, etc. (J5.) 
 Put (l+a?)i='y, (l+2/)r=^5 then l+a?=t;°, \^y^w'\ 
 (l+a:)"=2;-, (l+y)^=t^-. 
 
 Substituting these expressions in equations A and J?, and 
 subtracting, we have 
 
 if^ — yf^ — aix^-y) + b(x''—y^) + c(a?*^— y^) ^ 
 (/.(a?* — 2/*)+, etc. 
 
122 SERIES. 
 
 .'.(art. 97,) {v — w)X [v'^-^+v'^-^w. w'^-^l^ 
 d^x-'+x^y + xy^+y^ etc. | (C.) 
 
 But X — y=l+x — 1+2/=:?;°— 2^°= (v — w)X 
 .«. Dividing equation C by this, we have 
 
 c(a?2 + a7y + y^) + d(x^ + a?^y ■{■ xy^ + y^), etc. 
 
 Now, suppose x=y^ whence v-^w and our equation be 
 comes (art. 97,) 
 
 — =(-.—)= a + 2507+ 3ca?2 + 46Za?3 + 560?*, etc. 
 
 712;°""^ '71 V 
 
 Multiply by ?;°=l+07, and we have 
 
 711 W> m 
 
 — v'°=: — ,(l'i'Xy=a+2bx+3'cx^+4}dx^ + 6exS etc. 
 
 aa? + 2bx^ + Sco?^ + ^c^o?*, etc. 
 But (equation ^,) 
 
 — .(l+a?)n= j ax A hx'^A cx^-\ dxf^. etc. 
 
 Hence, by equating the homologous co-efficients, 
 
 m m ^ 
 
 -1 
 
 771 n m m — n n 
 
 n* 2 n In ^ \ 
 
 " -3 
 
 m m — n m — 2n , n m m — n m — 2/i m — 3n 
 
 n 2n 3n ^ ' * 4f n 2n 3n 4^n 
 
SERIES. 123 
 
 Hi ta O m 
 
 m m b m m — n b^ m m — n m — 27i b^ 
 
 l(y). In the preceding Investigation m has been supposed 
 to be aflSirmative ; if we now take m negative, and make 
 the same assumptions as before, we have 
 
 1 1 w'"" — v"^ 1 
 
 1 
 
 V~^ W~^ = = > = • 7)™ 7/»^ = 
 
 (x—y).\a+b(x+y) + c.{x^+xy+y^)-{- 
 
 d{x^ + x"y + X2f'\'y^) + , etc. | = 
 
 (o — w). { ^;"~^ 4- v''"-w. Tif-^ \ 
 
 |a+6(a?+2/) + c(a?2 + a?3^4-2/2) + , etc. | 
 
 Whence, dividing by n,v — w^ and assuming x=y^ 
 
 1 7n,v^~^ 
 i^X ='y'^-^|a + 26a?+3ca?2-}-4c?a?3-{-, etc. I 
 
 .*. Multiplying by ?;, and putting 1-f-a? for v% 
 
 m 
 
 v-'^=z 1 4- /rj a4-2&a:+3ca:H4£fa73-f , etc. | = 
 
 a'\-^'bx+^cx^ + ^dx\ etc. 
 
 aa?+26a?2 + 3ca:3, etc. 
 
 — m, 
 Now, multiplying equation A by , and equating the 
 
 homologous co-efficients, 
 
 — m — m — m — n 
 
 7^ ' n 27i ' 
 
 — m — m — n — m — 2n , 
 c= X — T. X — 5 , etc. 
 
124* SERIES. 
 
 Whence it appears that the formula is correct whether 
 the index is positive or negative. This is Newton's cele- 
 brated binomial theorem. 
 
 101. By the aid of this theorem, a binomial maybe 
 raised to any power, or evolved to any root by simple 
 substitution. 
 
 Several examples of the use of this theorem, when the 
 index is a whole positive number, are given in section 
 second. 
 
 1. Required to develope V , sr^ in a series. 
 
 ^ a 
 Here m— — 2, 7i=3. 
 
 . x.-i ,, 2 a? 2— 2— 3a:^ 2—2—3 
 
 .•.a(l + -) =ajl — -. -5-. — ^ — .— — ?r. — ^ — •. 
 
 ^ a^ < 3a3 6a^3D 
 
 —2—6 x^ 2 —2—3 —2—6 —2—9 ^ , 
 2a? 5a?2 40a?' llOa?^ 
 
 2. What is the value of Va^+6 in a series 1 
 
 b ¥ 563 105* ^ 
 Ans. a+__— +— _^j3^-,etc 
 
 3. What is the value of V— in a series. 
 
 a^ — a?2 
 
 1 a?^ 3a?* 5a?9 35a?^ ^ 
 
SERIES. 125 
 
 2 
 
 4. Required the value of (a — b)^ in a series. 
 Result, a^ ,\1 
 
 5a 25a^ 12oa^ 
 26b* 4686^ 
 
 625a* 15625^5' ^ 
 
 Ct ■ I /K 
 
 5. Required V in a series.* 
 
 Result, 1-] |-7r-a+2r^+5-7+5-Tj^'tc. 
 
 6. What is(7 )^ in a series 1 
 
 ^b — X 
 
 .c^.l < . 3a? 12a?2 52a?3 234a?* 
 Ans.y .|l4.gjH-^,+^^3 + g-^, etc,} 
 
 REVERSION OF SERIES. 
 
 102. Given ^=a?+| + g+^-f^^, etc. to in- 
 finity, to find x in terms of z. 
 
 * This quantity is reducible to 
 11* 
 
126 
 
 SERIES. 
 
 Assume x^az 4 hz'^ + cz^ -\'dz^ + , etc. ' 
 Then -h= -^-{-ahz^+acsi^^ etc. 
 
 , etc. 
 
 2.3"" 
 
 a?* 
 
 2.3.4'' 
 
 — ^r. 
 
 2 
 2:3 +T^*' etc. 
 
 2.3.4' 
 
 = 
 
 a=l, 5=—-^, 0=-^^—-^=-^, c^=— -^ — -^ + 
 
 8^4 
 
 "2i^ 
 
 4.- 
 
 j^3 -j3 ^4 
 
 Whence, a?=2r— ^+^ — 7-+) e*^* where the law of 
 
 continuation is manifest. 
 
 z^ z^ 
 
 2. Given a?=2r- 
 z in terms of x. 
 
 2.3^2.3.4.5 2.3.4.5.5.7' 
 
 ;, etc. to find 
 
 Let 2r=aa7-f 5a?3+c^5+c^^ 
 z^ 
 
 Then — . 
 
 2.3" 
 
 a3 0^6 ah^ ^ 
 
 2:3A5'^'+2:3:4^'- 
 
 a7 
 
 2.3.4.5.6.7 
 
 0' 
 
 — X. 
 
 =0 
 
SEEIES. 127 
 
 __, , J_ __1 1 _ 1»3 
 
 •'• ^"'■^' ^""2.3'^~2.3.2 2.3.4.5~2.4.5 
 
 1 1.3 1 1 1.3.5 
 
 ^~^0 A. Q'O Q F; a. Q a.* 
 
 2.4.9"^ 2.8.5 4.9.4, "^2.3.4.5.6.7 2.4.6.7 
 
 cc^ 1.3a:5 i.3.5a;7 i.3.5.7a?9 
 Hence, z=x+^^+^^^+^-^+^^^-^^, etc. 
 
 3. Required to revert the general series x=ay'\-hy^-\' 
 cf-^-dy*^ etc. 
 
 Assume y^Ax+Ex'^-^-Cx^-^-Dx*^ etc. 
 
 ay=Aax+Bax^'\-Cax'^'\-Dac[^^ etc. 
 Z>y2= 6^3a>2^25^j5^^ ( j52^ ^. 2ACh)x^, etc. 
 cy3= c./73^+3^2j?ca7*, etc. ^ =0 
 
 c;y*= dA^x^^ etc. 
 
 (art. 96,)^=-, B=--,C=^—— 
 
 D=- 
 
 a 
 5^3 — ^ahc^aH 
 
 a? 
 . , X hx^ 263 — fljc 5^3 — 5a5c+a2c^ 
 
 4. Given 2r=w — -q-+-e- — ~7+"q" + > ^*^' *^ infinity, to 
 find u in terms of z. 
 
 Result, ?i=2r+§2;3+/g2;5+^Yg2r7, etc. 
 
 «i2 1^3 7.4 «i5 
 
 5. Givena?=2^— ^ + |- — T^r\ — j etc. to infinity, to 
 find y in terms of x. 
 
 Result, y:=^ + -+-+^^ + -^^^, etc. 
 
128 SERIES. 
 
 SUMMATION OF SERIES. 
 
 103. Infinite series are sometimes of such nature, that 
 a quantity can be found, to which the series continually 
 approximates, and to which, without attaining perfect 
 equality, it arrives more nearly than by any assignable dif- 
 ference. The quantity to which the series, by continued 
 extension, thus approximates, is called the sum of the in- 
 finite series. Thus we say, .3333, etc. to infinity, =^5 for 
 no number less than | can be assigned, which the series 
 may not, by extension, be made to exceed. 
 
 Required the sum of l-{-x+x'^-{'X% etc. to infinity, x 
 being supposed <^ 1."^ 
 
 Put y—l+x-\-x^'{'X^, etc. to infinity. 
 
 .\ xy=x-\-x^+x'^j etc. to infinity. 
 
 Andy—xy=l. .'.y^j^;^. 
 
 2, Required the sum of l + 2x+3x^+4!X^'\-6x*+, etc. 
 to infinity, supposing x<^ 1. 
 
 Put 2/=l+2a?+3a?2+4a?3+5a?*, etc. 
 ... — 2xy= — 2a?— 4073 — g^s — ga?*, etc. 
 And x'^y=x^-\-2x^ + 3x\ etc. 
 Adding these three equations together 
 y — ^xy-^-x^y^l, 
 
 _ 1 _ 1 
 
 •'• ^"" 1— 2a7+a?3""(l— a?)2* 
 
 3. Required the sum of r^+"9~q" + o~4~5 ^^^' *^ infinity. 
 
 *The character <^ is used to express inequality, the opening being 
 presented to the greater quantity. 
 
SERIES. 129 
 
 Assume ^=l-4--rt-+-o--f-T- + "K"+j ^tc. to infinity. 
 
 Then x-l=:-i+i-+-l+l+l, etc. 
 
 Subtract the latter from the former, and 
 ■^^lS "^273+31^ "^475' ^*^' *^^ ^""^ required. 
 4. Kequired the sum of jy3+2Ti"'"3X5'*^^^^"^^^"^' 
 By last example, l = j;2+2:3 + 3:5"^4:5'^*^* 
 Subtract^. ...^=^+^+^^+^,etc. 
 
 By subtraction, i=.^3+^^+3|;5+j^^ 
 
 Whence, ^=ji-3 + ^^ + 3^+ji^^ etc. the sum 
 
 required. 
 
 The student who desires to pursue this subject, may con- 
 sult Wood's Algebra, Article 411, etc., or Young's Alge- 
 bra, page 251 etc. 
 
 DIFFERENTIAL METHOD. 
 
 104. In any series of quantities, «, 6, c, J, e, etc., if each 
 term be subtracted from the next following one, and each 
 term of the series of differences be taken from the next, 
 and so on, the following series will be obtained. 
 
 1st differences, h — a, c — 5, d — c, e — <f, etc. 
 2d diff. c— 25+a, cU-2c+6, e— 2c?+c, etc. 
 
130 SERIES. 
 
 3d difF. (f— .3C+35— a, e—3d+Sc—b, etc. 
 
 4th diff. e—4^d+6c—4^b+a, etc. 
 
 Or these, 1st diff. — «+^, — ^4-^, — c+c?, etc. 
 
 2d difF. a — 25 + c, b—2c+d, etc. 
 
 3d difF. —a-\-3b—3c+d, —b + 3c—3d-^€, etc. 
 
 4th difF. a — ib + 6c — 4c?+e, 
 
 5th diff. —a + bb—10c^l0d—5e+f, etc. 
 
 By a little attention to these expressions, and a com- 
 parison of their formation with the powers of a binomial, 
 we perceive that if .^ be taken to denote the first term of 
 any (the ^th) order of differences. 
 
 , _ _ n — 1 n-->ln — 2_ , 
 
 dtz^=^a — nb+n.—^c — n. , d, etc. 
 
 the sign + being used when n is an even number, and — 
 when 71 is odd. 
 
 If the differences of any order vanish, any one of the 
 terms may be found by means of the others. 
 
 Suppose the 4th difference a — 45 + 6c — 4c^+e=0, and c 
 was not known, we should find, 
 
 '- 6 ' 
 
 105 Let Z), D, D, D, etc. denote the first terms of the 
 
 12 3 4 
 
 1st, 2d, 3d, 4th, etc. orders of differences, viz : D=: — a+b, 
 
 1 
 
 D=a—2b'\'C, JD= — a+3b—3c+d, D=a^4<b-\-6c—4^d 
 
 2 3 4 
 
 +e, etc.; then we find b=a-\'D, c=a-\-2D + D^ d=a+ 
 
 1 12 
 
 3D + 3I)+D, c=a + 4D+6D+4i)+D. 
 
 12 8 12 8 4 
 
 =* By this method the computers of the Nautical Almanac verify 
 their calculations of the moon's longitude, latitude, etc. 
 
SERIES. 131 
 
 Hence, the n+lth term of the series 
 
 ^ n — 1 n — In — 2_ 
 
 =a+wD+7i.-7c— D + w.-»— .— ^— I), etc. 
 
 1 -^2 Z O 3 
 
 Consequently the Tith term 
 
 ^ — 2 ^ n — 2 n — 3 , 
 
 =a+n—l.D+n—l.—^D +n—l.-^.-^D, etc. 
 
 1. Required the Tzth term of the series of odd numbers, 
 1, 3, 5, 7, etc. 
 
 • Here the 1st diff. are 2, 2, etc. ; 2d difF. 0. 
 
 .•.the nth term =l + 2.n — 1 =2n — 1. 
 
 106. Required the sum of n terms of the series a, b, c, c?, 
 etc. 
 
 This is manifestly the same as the n+lih term of the 
 following : 
 
 0, 0+a, 0+a+b, 0+a+b + c, etc. 
 Hence, by the last article, the required sum 
 
 n — 1 _ n — 1 n — 2 ^ n — 1 n — 2 n — 3 ^ 
 
 etc. 
 
 1. Required the sum of n terms of the square numbers, 
 1, 4, 9, 16, etc. 
 
 Here a=l, D=3, D = 2, D=0. 
 
 12 3 
 
 .'. the sum 
 
 , n — 1^ n — In — 2^ n.n+l.^n+l 
 
 2. Required the sum of the cube numbers, 1, 8, 27, 64^, 
 etc. continued to n terms. 
 
 3. Required the sum of 25 terms of the series, 1, 3, 5, 
 7, etc. Ans. 625. 
 
...32 SERIES. 
 
 4. Required the sum of 15 terms of the series 1, 16, 81, 
 256, 625. 1296, ate. Ans. 178312. 
 
 When the differences at length vanish, any term of the 
 series, or the sum of any number of terms, may be accu- 
 rately determined by the methods used in this and the pre- 
 ceding articles : when the differences become small, but 
 do not vanish, a near approximation can be made. 
 
 107. In article 105, the number n is supposed to be an 
 integer, in which case, if the differences vanish, the rule is 
 demonstrably correct 5 the same formula is, however, ap- 
 plied to the case where n is fractional. 
 
 Suppose a series ^, ^, r, 5, etc., of equi-different quanti- 
 ties, and another series, a, 6, c, c/, etc., such that the terms 
 of the latter shall be similar functions of the correlative 
 terms of the former. It is proposed to find a term y, in 
 the latter series, corresponding to v in the former 5 v — p 
 being given. 
 
 Let q — p : v — p : : 1 : a?, and take as before, /), D, D, etc. 
 
 13 3 
 
 the first of the 1st, 2d, 3d, etc. differences. 
 Then assuming the series, (art. 105.) 
 
 ^ a: — 1 ^ 0:^1 X — 2^ . 
 
 y=a-\-3cB-\-x.—^D +x.—^,—^D, etc. 
 
 This gives, when a?=0, y=a, when a?= 1, y^=a-\-'D^ when 
 
 1 
 
 a?=2, 2/=a+2D + D. etc.; but these expressions equal Z>, 
 1 3 
 
 c, etc. as they ought to do. This formula, being demon- 
 strably correct when x is integral, is assumed as a conve- 
 nient and useful approximation when a: is a fraction. 
 
 Required to find the 6ith term in the series 1, 4, 9, 16. 
 
 Herea:=-^ a=l, D=3, D=2, D=0. 
 
 -^ 12 3 
 
LOGARITHMS. ' 133 
 
 , 11 3 11 9 2 169 
 
 2. Le\ a, b, c, c?, be arcs of a great circle intercepted be- 
 tween a fixed star and the moon's centre at noon and mid- 
 night of two successive days, it is required to find the dis- 
 tance at 15 hours from the first noon. 
 
 Here 12: 15 : : 1 :a?=^ 
 
 4. 
 
 ...dist. (3^)=«+|i)+|x^Z) + Jx^x7|l)= 
 
 CONSTRUCTION OF LOGARITHMS. 
 
 108. From article 92, it is obvious that) a being any 
 
 number, if a°=JV, and d^=M, a^'^'^^J^M^ and «''-"= ^. 
 
 If, therefore, all the numbers used in calculation were ex- 
 pressed in powers of a, multiplication and division might 
 be performed by the addition and subtraction of the expo- 
 nents of a. These exponents, thus employed, are termed 
 logarithms. When a°=^Jf^ n is the logarithm of JV. 
 
 Let a? be such number, that a^ may denote any proposed 
 number. 
 
 Put a=l-f5, then (art. 98, 99, 100,) 
 
 X — 1 X — 1 X — 2 0?- — 1 
 
 X — 2 X — 3. , f x'^ X \ 
 
 __.__6S etc. = l+6x + 6^ J 2 -2 S + 
 
 *i6-2+3i+*i2i— +^— l^*-^- 
 12 
 
134? LOGARITHMS. 
 
 If, therefore, we collect the terms containing the succes- 
 sive powers of x, and denote the above equation thus, 
 
 (l-\-bY=l + ^x + Bx^+Cx^+Dx% etc. 
 
 We readily find that 
 
 ^ ^ ¥ b^ ¥ h^ 
 
 But the values of B, C, etc. remain to be determined. 
 
 If for X in the above equation we substitute 2a?, we shall 
 have 
 
 (l4-i)2x=i+2^a7 + 45a?2+8Ca?3+16i)a?S etc. 
 Squaring the same equation, 
 
 (1 + ^)3^= l + 2./^a? + 25a72+2Ca?34.2Z)a?S etc. 
 
 A^x^'\'2ABx'+2ACx^, etc. 
 
 B^x\ etc. 
 
 Comparing the co-efficients of like powers of x. 
 
 A^ A^ A^ 
 
 ^=^' ^=213-^=2X1' '**= 
 
 A^x'^ A^x^ A*x* 
 ...a-=(l+6)'=l+^a?+-^ + -^+2:3;-^, etc. (R.) 
 
 In like manner, 
 
 etc. 
 
 A^x^ A^x^ A^x^ ^ . , . . 
 
 = 1 4-^07 + -^+-^ +2:3:4;' ^^^- ('^•) '^ ^^'^^ 
 
 52 b^ b^ ^ 
 
 •^=^+2 +3-+4:' ^^^- (^) 
 
 109. Let 1 + ^ denote a number whose logarithm is re- 
 quired; then l+2/=(l+^r' 
 
LOGARITHMS. 135 
 
 .'. (l'\-yY={l-hbY% z being any number whatever. 
 Here x is the logarithm of (1+b)'', or of a% or 1+y. 
 
 But (l+y)^=i + F2r+-^+-^, etc. (J?. 108.) 
 Fbeing ==y-|V|--?J + f , etc. (Q. 108.) 
 
 Also (1+6)"= 1 4- ^^^H 2~"^T3~' ^*^' ^^' ^^^'^ 
 
 Whence, by comparing the co-efficients of 2r, 
 A.= V, and ,^^^y-4y-+irzJy!±Ly-, etc. = 
 
 y<2. y^ y\ yS J 
 
 Again, putting = a number whose logarithm is 
 
 required, and making ^j =(14-6)^, we find 
 
 V (the log. of (1 +5') or -^—)= 
 
 -.3 t/2 y* 7® 
 Jlfx 12/+%"+^ + 4 + 5 5 ^*^' I ^y ^ process exactly ana- 
 logous to the former. 
 
 Now the logarithm of ■- (or of 1 + yX-— — = lo- 
 garithm of (1+2/)+ logarithm of ■— - = 
 
 (r+t;=2Jlf|y + ^+^+^, etc.} {U.) 
 
 As is manifest by adding together the values of x and v 
 above obtained. 
 
136 LOGARITHMS. 
 
 110. As A^ and consequently -^ or JIf, is given in terms 
 of 5, {a — 1) and a may be assumed of any value whatever, 
 (art. 108j) it follows that M is not limited to one particu- 
 lar value. 
 
 If the value of a be assumed, M may be thence deter- 
 mined, or if we take any number at pleasure as the value 
 of JkT,*' a will be limited ; but in this case the formula TJ 
 may be applied to the construction of logarithms, without 
 computing the value of a. M is termed the modulus, and 
 a the radix of the system. Instead of fixing the value of 
 M^ or G5, by arbitrary assumption, we may take 1 as the 
 logarithm of any proposed number, and thence deduce the 
 value of M, 
 
 Since a^=l, the logarithm of 1 must in every system, 
 = 0. If we make 1= logarithm of 10, the radix will be 
 10, and M might have been found from equation /, (art. 
 108,) by assuming 6=9, if that series or its reciprocal had 
 been a converging one. As that is not the case, a different 
 expedient must be used. 
 
 jo+ 1 
 Let^^- -denote a number whose logarithm is to be 
 
 \^y 7? + l 1 
 
 found, and put - — -^= : then ^=p; — —r : this value 
 
 ' ^ 1 — y p ' ^ 2p4-l 
 
 substituted for y in equation Z7, (art. 109,) will make the 
 
 seri.es always converge, and more rapidly the greater the 
 
 value of j9. 
 
 * When Mis assumed =1, the logarithms thence obtained are termed 
 Naperian, from th« name of the inventor, or hyperbolical, from their 
 relation to the areas contained between the curve and asymptote of an 
 equilateral hyperbola. 
 
LOGARITHMS. 137 
 
 Now since X --=p-f-l, it is evident that the log. of 
 
 p + 1= log. i?+ log. of -— . 
 
 First, let p=l, then =2, and -^ — -——-' also log. 
 
 ^=.3333333333 
 -i=^= 370370370 
 
 (|)'=^of (|)3= 41152263 
 
 (|)' = ^of (g)5= 4572474(^0 
 
 (3)^ = 6272 (C.) 
 
 (J)« = 697 
 
 (|)"=) 77 (i).) 
 
 12* 
 
138 LOGARITHMS, 
 
 If then, we divide these terms by the co-efficients, 1, 3, 5, 
 7, etc., we shall have — 
 
 3 ' 
 
 .3333333333 
 
 1 
 
 3.33"" 
 
 123456790 
 
 1 
 5.3^"" 
 
 8230453 
 
 1 
 
 7.37~" 
 
 653211 
 
 1 
 
 9.39"' 
 
 56450 
 
 1 
 11.3^" 
 
 5132 
 
 1 
 13.3^" 
 
 483 
 
 1 
 
 15.3«" 
 
 46 
 
 1 
 17.3^7- 
 
 4 
 
 
 .3465735902 
 
 V- 
 
 Hence, log. of 2= .6931471804 Jf, and log. of 8 (or 2^) 
 =2.0794415412^1/: J 
 
 Second, let ;?=4, then ^^:^=^ 
 
 5 
 
 Hence to find the log. of ^ 
 
LOGARITHMS. 
 
 139 
 
 .1111111111 
 
 g of {~f=^= 45724^74 
 
 I of ilY=.6B. 
 
 33870 
 
 1 . 1 C 
 
 7 ^^ (9) -21^ ^^^ 
 
 .1115717757 
 
 Whence, log. of -=223 14355 14 JIf. 
 Consequently the log. of (tX j) 10 
 
 But log. of 10 
 Hence, M, 
 
 =2.3025850926JJf. 
 1. 
 1 
 2.3025850926 
 
 ,4342944822 
 
 Substituting for M the number last found, we obtain the 
 log. of 2=. 30102999599, and thence the log. of 4, 8, 16, 
 32, etc. may be had by simple multiplication. Also the 
 log. of 5= log. of 10— log. of 2=. 69897000401; whence' 
 may be found the logarithms of all the powers of 5. 
 
 The numerical value of JIf being thus ascertained, if we 
 make p denote a number whose log. is known, we have 
 
 log p + 1 =:log.j?+log.' 
 
 p + 1 
 
 P 
 
 :l0g.;7 + 
 
 2^+1"^ 3.(2p + l)2 "^5.(2^ + 1)^+7.(2^ + 1)2' ^^^* 
 
140 LOGARITHMS. 
 
 Where Jl^ B, C, etc. denote the preceding term exclu- 
 sive of the divisor, 3, 5, etc. ; and thus the logarithms of 
 all the prime numbers may be computed. But as the above 
 series, when the number p is small, converges but slowly, 
 and therefore requires a considerable number of terms to 
 be used, the labor may be abridged by proper expedients, 
 a few of which are subjoined. 
 
 Let the log. of 3 be required j putp=80, then 
 
 2p+l 161 
 
 And log. of ^=^+3^^921- -0053950319 
 To which add log. of 80=^1+3 log. of 2== 1.9030899880 
 
 Hence, log. of 81, or 4 log. of 3= 1.9084850199 
 
 And log. 3=. 4771212549. 
 
 Again, let the log. of 7 be required. Put J3=27, then 
 
 1___ 1 
 
 2^+1 ~^55 
 
 28 2M Jl 
 And log. ^=:^ +^^ = .0157942671 
 
 Adding log. of 3^ or of 27 = 1.4313637647 
 
 Log. of 28= 1.4471580318 
 Subtracting log of 4, .6020599920 
 
 Log. of 7 = .8450980398 
 
 * As the logaritlims of 10 and its powers are wholly integral, it ia 
 manifest the logarithnn of a number is changed only in the integral 
 part; by varying the position of the decincial point in the number itself^ 
 
LOGARITHMS. 141 
 
 In general, suppose a?4-2 to denote a prime number 
 whose logarithm is required, those of the inferior numbers 
 being known. 
 
 (x—lYx(x+2) x^—Sx + 'i , , , 
 
 femce , ^-z.rr-1 rr,='^ — is ?»? where both terms 
 
 (a? + 1)2 X [x — 2) x^ — 307—2 ' 
 
 are divisible by 4, because x — 1, and 07+1 are even num- 
 bers, it follows that this fraction in its lowest terms, may 
 
 p+1 
 be expressed by , and therefore the log. of 
 
 (07—1)2.(07+2) 
 
 (a;+ 1)2.(07—2)' 
 be obtained from the formulse above given ; to which add- 
 ing 2 log. (o7+l) + log. {x — 2) — 2 log. (07 — 1) the result will 
 be the log. of 07+2. 
 
 Let the log. of 13 be required. Here 07=11, and our 
 fraction becomes 
 
 10U3_ 1300 __ 325 
 T2"2;^'~1296~35i 
 
 Hence, .?.=324, and g^j=^ 
 
 325 
 
 Whence, log. ^^ = .0013383507. To which add 4 log. 
 
 3+log.of 4-2, logof5=. 1126050039 
 
 Log. of 13 = 1.1139433546 
 
 In this manner the logarithms may be derived from 
 those already obtained, to any proposed extent. The dif- 
 ferential method may be advantageously applied to the 
 completion of a logarithmic table. But the labor of those 
 computations, being already finished, and not likely to be 
 renewed, a further elucidation of the subject is deemed 
 unnecessary. 
 
142 SURDS. 
 
 Section X. 
 
 SURDS. 
 
 111. It has been remarked (note, art. 38,) that the even 
 roots of negative quantities are impossible ; hence when- 
 ever, in the solution of a problem, the square' root of a 
 negative quantity appears in the result, such result is im- 
 possible, or imaginary. But instead of abandoning, as 
 hopeless, every example in which such expressions appear, 
 they are found conformable to the general principles of 
 the science, and sometimes connected with the most refined 
 analytical processes. 
 
 Impossible roots may be introduced into a problem in 
 two ways, quite distinct from each other. First, by ad- 
 mitting incompatible assumptions into the data of the pro- 
 blem ; in which case the impossible root serves to detect 
 that incompatibility ; and its appearance or disappearance 
 marks the limits of the problem. 
 
 Thus in the equation x'^ — 2aa?= — 5, we have (by art. 
 67,) a?=azfc^/a2 — b as the general expression of the value 
 of X, If now a^^b,^ a^ — b is positive, and therefore, 
 's/a^ — bj and consequently x, a possible quantity. But 
 if a^<Cp^ ^^ — ^ is negative; and therefore, ^a^ — b im- 
 possible ; whence x is, in this case, an imaginary quan- 
 tity. Here the terms of the original equation are com- 
 patible, whenever a^ is equal to, or greater than b; but 
 incompatible when a^<^b ; for x^ — 2ax= — b is equivalent 
 to 2a — x.x=b, but 2« — x.x cannot be greater than a^. 
 
 Put x=aztc, then 2a — x=azpc, and 2a — x.x=a^ — c^. 
 
 When c=0, x=a, or 2a — x.x=(b=^a^' this is there- 
 fore the limit of b, 
 
 *a'^b is read a is greater than h, and a<Zb, and a is less than b. 
 
SURDS. 143 
 
 Suppose a?+- = 2«, to find a?. 
 
 Multiplying by a?, a?^4-l = 2«a?, or o?^ — 2«a?= — 1. 
 
 Whence, x=azri v/a^ — 1, which value of a? is imaginary 
 when «<^1. Hence the least possible sum of a number 
 and its reciprocal is 2.*" 
 
 112. Impossible roots may be sometimes obtained, when 
 the data are compatible, by the admission of inconsistent 
 suppositions into the solution. 
 
 Suppose we have x'^-^ny'^=a, and xy=h, to find x and 
 y ; a, b, and n being any given numbers. 
 
 To and from the first equation add and subtract 2^/71 
 times the second, whence, x'^'{-2xy\/n-{-ny^=a + 2by/ny 
 and x'^ — ^xy^/n-^ny^^a — 2by/n; and by evolution, 
 
 x-^ys/n—s/ a-Y'lbs/n^ (M.) 
 
 X — y s/n=s/ a — 2b s/ n^ ( JV.) 
 
 Va + 2Z>v'^+N/« — 2by/7i 
 .x= :^ 
 
 * These principles may be applied to determine the maxima 
 and minima of geometrical quantities. Let b = the base, « = the 
 altitude of a plane triangle, x = altitude of its inscribed rectangle ; 
 
 thenbx = the area of the rectangle, which put =c ; 
 
 a 
 
 whence we find x=-zi^K/ ('- ), which is impossible when 
 
 2 4b 
 
 ^ab 
 
 If c——, x=z~, this is, therefore, the greatest possible value of x, 
 4 2 
 
 and the greatest rectangle is — ^ - the triangle. 
 
 4r 2 
 
14i^ SURDS. 
 
 And y= — ^c—. 
 
 Which are general expressions for the values of a? and y. 
 
 If now we take n= — 1, or a?^ — y^=a, the above expres- 
 sions become 
 
 X— ^ 7Z ■ 
 
 N/a+26v/— 1— \/a— 26v' — 1 
 2/= 27=1 
 
 These values, though expressed by imaginary quantities, 
 are real ones. For putting — 1 instead of n^ in equations 
 JkT, JV, and multiplying, we have 
 
 a?2+y3_^^3_|.4^3^ (P.) 
 But from the first equation, a?^ — y^=.a. 
 Consequently, by addition and subtraction, 
 
 2x''=y/a^-\-^b^-\'a. 
 
 And 2y2=v/aH4^^'— «, 
 
 ora^=:( g f; y=( g ) 
 
 The same conclusions may be obtained by squaring the 
 values of x and y first found, and extracting the square 
 roots of the results. 
 
 The impossible surd V — 1, was evidently introduced 
 into the result, by adopting a process in the general solu- 
 tion, which was not applicable to the particular equation 
 x^ — y'^—a; yet the values of x and y, when cleared of 
 imaginary surds, are the true ones , as may be shown by 
 a different solution of the problem. 
 
SURDS. 145 
 
 To the square of the first equation adding 4 times the 
 square of the second, x^-\-2x'^y^'^y~=^a'^ + 4:b^. 
 
 Whence, by evolution, x'^-\-y^= y/a^-{-4^b^j the same as 
 equation P, deduced from imaginary surds. 
 
 113. From what is shown in the foregoing article, we 
 readily infer, that when, in the solution of a problem, the 
 value of the quantity sought appears in terms of imaginary 
 surds, we are not thence immediately to conclude, that the 
 data are inconsistent ; as the adoption of an inapplicable 
 process may produce such a result ; yet in this case the 
 imaginary surd may be eliminated by the use of proper 
 expedients. When, however, the data are inconsistent, no 
 analytical address can clear the final equation of its im- 
 possible quantities. 
 
 Imaginary surds differ from real ones in this important 
 particular. Keal surds, however complex, admit of an 
 approximation to their value ; but imaginary surds admit 
 of no approximation, and must either be eliminated, when 
 practicable, or remain the intractable indications of incon- 
 gruous assumptions. 
 
 The following cases exhibit the most useful applications 
 of algebra to surd quantities. 
 
 Case I. 
 
 114. To reduce a rational quantity to the form of a surd. 
 
 Raise the given quantity to the power denoted by the 
 index of the surd, and to this power apply the radical sign 
 or index proposed. 
 
 EXAMPLES. 
 
 1. Reduce 5 to the form of a square root, and a to that 
 of a 4th root. 
 
 5=v^5-=v/25, and a=(a^y=i/a\ 
 
 2. Reduce 3 to the form of a cube root. 
 
 Result, -727. 
 13 
 
146 SURDS. 
 
 3. Express — -^a in the form of a cube root. 
 
 o 
 
 I A. 
 
 Result, (—^a^)'. 
 
 4. Reduce 2\/5 to the form of a square root. 
 
 Result, v'20. 
 
 5. Reduce 3 \/ 2 to the form of a ^th root. 
 
 Result, (324)*. 
 
 6. Express a-^b in the form of a square root. 
 
 7. Express — ^ — in the form of a quadratic surd 
 
 Result, v/a'^+2a6 + 63. 
 
 2 surd. 
 6+2v/5 
 
 Result, s/ - . 
 
 Case 2. 
 
 115. To reduce radical quantities^ having different in- 
 dices^ to other equivalent quantities with a common 
 radical sign. 
 
 Reduce the fractional exponents to a common deno- 
 minator, involve the given quantities to the powers de- 
 noted by their respective numerators ; and to the results 
 apply the reciprocal of the common denominator as the 
 common exponent. 
 
 EXAMPLES. 
 2 3 1 
 
 1. Reduce a^^ b"^, and c^ to equivalent quantities, 
 having a common exponent. 
 
 2 3 1^ ^ . , _ 40 45 12 
 
 3, -, g reduced are equivalent to g^, g^, - 
 
 2 1 4C 
 
 Hence, a^ = (a'^)^^=a^^', 
 
 3 1 45 1 1 12 
 
SURDS. 147 
 
 1 JL 
 
 2 Keduce 3 , and V to a common mdex. 
 
 Eesult, 27^, 16^. 
 
 3. Keduce a^, b^, to a common radical sign. 
 
 Result, aT\ b'f~5. 
 
 4. Reduce {a-\-x)^, (a — x)^ to a common index. 
 Result, {a''+2ax-{'X^)^, and (a^ — 3a^x-^3ax^—x^)^. 
 
 IX 1 
 
 5. Reduce a^, and a?* to the common index —.* 
 
 I/O 
 
 Result, (a*)i"5, (a?3)T2. 
 
 1 i 
 
 6. Reduce 4^, and 5"* to a common index. 
 
 Result, ( 16)7^2 and ( 125)^5. , 
 
 Case 3 
 116. To reduce surds to their most simple terms. 
 
 Resolve the surd, if possible, into two factors, one of 
 which shall be the greatest power it contains. Extract the 
 root, and thereto annex the other factor with its proper 
 radical sign. 
 
 EXAMPLES. 
 
 1. Reduce V250 to its simplest terms. 
 
 250:^125X2=5^^X2. .-. ^250=5^2. 
 
 2. Reduce x/32 to its simplest terms. 
 
 Result, 4 v^ 2. 
 
 * When the common index, to which the fractional exponents are to 
 be reduced, is given, divide each of the given exponents by that com- 
 mon index, and involve the given quantities to the powers indicated 
 by the quotients. 
 
148 SURDS. ^ 
 
 3. Reduce V243 to its simplest form. 
 
 Result, 3 Vs. 
 
 4. Reduce V^IS^ to its simplest terms. 
 
 Result, 12 V 3. 
 
 5. Reduce (144)* to its simplest terms. 
 
 Result, 2 V 3. 
 
 6. Reduce [a^b — a^x)^ to its most simple terms. 
 
 Result, al/b — x, 
 
 18 
 
 7. Reduce \^t^ to its simplest terms.^ 
 
 3 
 Result, ^VIO. 
 
 8. Reduce V 5 to its simplest terms. 
 
 9 
 
 Result, I Vs. 
 
 9. Reduce — -^ j^- to its simplest form. 
 
 Result, 8+2v/ 15. 
 
 Case 4. 
 
 117. To add or subtract surd quantities. 
 
 Reduce the quantities, when fractional, to a common 
 denominator; and when the exponents are different, to a 
 common radical sign. Also express the surds in their 
 simplest terms. If then the reduced surds are alike, they 
 may be added or subtracted as other algebraic quantities. 
 
 * When the given surd is fractional, the denominator nnay generally 
 be made rational, without changing the value of the fraction, by mul- 
 tiplying both terms by proper numbers or quantities. When the de- 
 nominator consists of two quadratic surds connected by the sign + or 
 — 5 the proper multiplier consists of the same surds connected by the 
 opposite sign. 
 
SURDS. 149 
 
 When the surds are unlike, the operation must be merely 
 indicated. 
 
 1. Required the sum and difference of 1/ 128 and V 16. 
 
 V128=4V2, V 16 = 2^2, 4V2+2V2=6V2, 
 4^2— 2V2=2V2. 
 
 2. What is the sum of ^/27 and >/48'? 
 
 Ans. Tv/S. 
 
 3. What is the difference between \/50 and v/18'? 
 
 Ans. 2v/2. 
 
 4. Required the sum of ^56 and Vl89. 
 
 Ans. 5V7. 
 
 5. What is the sum oi ^^/a'^b and 5 V 16a*5 '? 
 
 Ans. (3a +20^2) ^5. 
 
 6. What difference is there between ^2^a^h^ and 
 V 54a6* 1 Ans. (2a6— 36^) ^Z 6a. 
 
 7. Required the sum of Vt and V 09 '^ 
 
 Ans. J V 2. 
 
 ^/14+^/12 
 8. Required the difference between — j^ j-^ and 
 
 ^3~^^ ^ Ans. 872+4.x/21 + 4x/3. 
 
 Case 5. 
 118. To multiply or divide surd quantities. 
 
 Like quantities, with different exponents, are multiplied 
 or divided by adding or subtracting their indices. 
 13* 
 
150 SURDS. 
 
 When the quantities are different reduce them, if neces- 
 sary, to their equivalent ones with a common radical sign. 
 Then the product or quotient, with a common radical sign 
 applied, will be the quantity sought.* 
 
 EXAMPLES. 
 
 1. Eequired the product of 5V2 by 7^3. 
 
 5V2=5V4, 7v/3=7V27, 5 V^^X 7V27=r:35Vl08. 
 
 2. Divide x/21+v/15 by ^7— v/5. 
 
 x/21+n/15_ x/21 + \/15 V7 +v/5 _ 12v/3 + 2v/105 
 v^7_^5~- ^7—^5 ^ V7+v/5 ~ 7—5 
 
 = 6v/3+'>/105, the quotient required. 
 
 3. Required the product of 5 v/ 2 by 7 V 3. 
 
 Result, 35 v/ 6. 
 
 4. What is the product of 3 v/ 5 by 4V25 % 
 
 Ans, 60. 
 
 5 3 9 2 
 
 5. What is the product of^v/^ by tkn/c'? 
 
 ^^^•20^^^- 
 
 6. What is the product of 2V U and 3 V^^ '^ 
 
 Ans. 12 V 7. 
 
 7. What is the quotient of 6^972 by 31/21 
 
 Ans. 6Vl8. 
 
 • In the division of surds, as well as of rational quantities, it is fre- 
 quently convenient to set down the terms as a vulgar fraction, and to 
 reduce that fraction to its simplest forijp. 
 
SURDS. 151 
 
 gs/^ divided by |v/g 
 
 5 1 3 2 
 
 8. What quotient will -fV^ divided by -VT make % 
 
 Ans.QV5. 
 
 9. What is the product of (a+b)^ and (a + b)^ 1 
 
 Ans. (a+byk 
 
 10. What is the product of{a + 2Vb)^ and (a—^Vby^l 
 
 Ans. (a^ — 4.6)2". 
 
 11. What is the product of 3+ v" — -2 by 5— x/— 2'? 
 
 Ans. 17 + 2V— 2 
 
 12. What is the product of (a + V—b)^ by (a--'2 s/ —b)i 1 
 
 Ans. (a3+2&— av/— 5)i 
 
 4 2 6 ^ 
 
 13. Divide ^Vfl^ by- V«' Quotient, -a^. 
 
 XX 3 ^r? 
 
 14. Divide 3a ° by ^a"' Quot. ^a ™ ' 
 
 15. Divide 5+3^^—3 by 7— 2v/— 3. 
 
 17 31 
 Quot.g^+g3V-3. 
 
 As involution is effected by the multiplication of equal 
 factors, this case evidently includes the involution of surd 
 quantities. 
 
 16. What is the 3d power of 4 v" 2 1 
 
 Ans. 128 v/ 2. 
 
 X n 
 
 17. Required the nth power of a"" 1 Ans. am, 
 
 18. What is the 3d power of 3+2n/— 1 1 
 
 Ans. 46 x/— 1—9. 
 
152 SURDS. 
 
 Case 6. 
 
 119. To extract the roots of surd quantities. 
 
 When the quantity consists of one term only, the root 
 is obtained by dividing the index of each quantity by the 
 index of the power. 
 
 When the quantity is a compound one, the root may 
 sometimes be extracted, as in article 39. 
 
 When the given quantity is a binomial, one term, at 
 least, of which is a quadratic surd, the square root may 
 sometimes be extracted by the following formula. 
 
 Let y/adt:y/b=^x, put s/a^ — 6=c?, then 
 
 * This formula is thus obtained. 
 
 Let \/a-±:'^h—'sJv:^s/w, 
 Then a±is/h=v±i2s/vW'\-w, 
 
 Now, as a rational quantity cannot be equal to a surd, we must 
 take 
 
 a=v+w^ (Jf.) and s/h—^s/vw. 
 By squaring these equations, 
 
 a2=2;3^2vw+wj2, and b=4tvw. 
 Whence, by subtraction and evolution, 
 
 V «^ — b = s/ v^ — 2vw -^w^y or d= v — w. 
 From this and equation N, we have 
 
 v=o^ 4 o«) a^d w=-^a — -^d. 
 
 X— y/vdzs/wz= ^ (-a + ^d) ±:y/ (^a — -d.) 
 
 Hence, it appears that this method applies only to the case where 
 a^ — h is a complete square. 
 
SURDS. 153 
 
 EXAMPLES. 
 
 1. Eequired the square root of 16^/6, 
 
 ^/16=4, and ^6^=6^, .-. ^1l67Q=4<\/6. 
 
 2. What is the square root of 6^ 1 
 
 Ans. 6^ or 6v^6. 
 
 3. What is the cube root of ^x/SI 
 
 Ans.gV3 
 
 4. What is the square root of a^ — 6a v/ — b — 9^? 
 
 Ans. a — 3\/ — b, 
 
 5. Required the square root of 6+2\/5. 
 
 Herea=6, n/6=:2v'5= V20. 
 
 .'.b=20, and ^=v/ 36— 20=4. 
 
 1 6+4 6 — 4 
 Whence (6 + 2v/5)^= V'-^^ N/-y- = v^ 5 + 1, the 
 
 root required. 
 
 6. Whatisthesquareroot of 3— 2v/2'? 
 
 Ans. V2— 1. 
 
 7. What is the square root of 7— 2n/ 10 '? 
 
 Ans. v'5— v'2. 
 
 PROMISCUOUS EXAMPLES. 
 
 120. Required the difference between 
 
 V5+2n/5 3+ n/5 n/5 + 2v/5 
 
 2"" ^'''^2V(5 + '2V5)' 2 ' 
 
154 SURDS. 
 
 5 + 2V5 5 + 2v/5 3 + v/5 
 
 2v/(5 + 2n/5)' 2v^(5+2n/5) 2v/(5 + 2v/5)" 
 
 2v/(5+2V5r2^/(5+2v/5r ^^^^^^^^^^^^ ^^^- ^^^ 
 
 V5+2V5 1 
 
 denom. by ^/5-2^/5) ^ ^^^" =oV^ (l + 5>/5) 
 the difference sought. 
 
 372+2 
 
 2. Kequired the sum of 5^2 — 1, and ^ ^ y-^ 1 
 
 Ans. 8^2 + 3. 
 
 3. What is the sum of 12V j and ^V'^'l 
 
 27 
 Ans. "-tn/J^. 
 
 4i. Required the difference of 3V3 and X/121 
 
 Ans. v/9. 
 
 5. What difference is there between 2 n/ a^6* and a/ Sa^i^ 1 
 
 Ans. 2ab(a—b)l/b. 
 
 6. Required the product of 4 + 2 V 2 by 2— v/ 2 '? 
 
 Ans. 4. • 
 
 7. Multiply a+bs/—l by a— 5s/— 1. 
 
 Product, a2+53. 
 
 44- V— 3 , 1— n/— 3 
 
 8. Divide —^-^ by ^ . 
 
 . l + 5v/— 3 
 
 Quotient, j^ . 
 
 9. Required the square root of 51 — lOv/21 
 
 Ans. 5^2— 1. 
 
EQUATIONS IN GENERAL. 155 
 
 10. Required the 3d power of a — 6y— 1 % 
 
 Axis, a?^— 3a2^V — 1 — SaS^+^V — 1. 
 
 11. Required the square root of 7 — 24x/ — 1 \ 
 
 ^ Ans. 4— 3v/— 1. 
 
 12. Divide a?'\-h^ by a—hsf—X^ 
 
 Quotient, a-^hsf — 1. 
 
 13. Add —-7—; — 17 to r-, r. 
 
 2a3— 2^2 
 Sum, — T-nir* 
 
 \/7 a/ 3 
 
 14. Add -^^-^and-^-^--^ 
 
 Sum, g. 
 
 15. Divide 18 + 26V— Iby 3 + >/— 1. 
 
 Quot. 8 + 6 y—l. 
 
 16. Required the square root of 13 — 20 x/ — 3 % 
 
 Ans. 5— 2v/— 3. 
 
 Imaginary surds are of great importance, in the investi- 
 gation of several valuable formulss, in the arithmetic of 
 sinesc 
 
 Section XI. 
 EQUATIONS IN GENERAL. 
 
 121. It has been remarked, page 90, that quadratic 
 equations sometimes admit of more answers than one. 
 The principles on which the ambiguity of quadratic equa- 
 tions depends, are productive of similar results in equations 
 of the higher orders. 
 
156 EQUATIONS IN GENERAL. 
 
 In the general -equation a?° — •j9a?''~^+5'a?"~^......dbr=0, 
 
 where p^ q, r, are supposed to be given, the different values, 
 of which X is susceptible, are called the roots of the equa- 
 tion. 
 
 122. If two binomials, as x — a. a?-W, be multiplied to- 
 gether, the product x"^ — ax — bx-^ab, or a?^ — (a-^b)x-{-ab, is 
 manifestly a quadratic; if now, x — a— Of or x — ^=0, we 
 have evidently 0?'^ — (a-\-b)x+ab—0. On the other hand, 
 the given quadratic equation x^ — px-\-q—0, may be re- 
 solved by making a-{-b—p^ ab=q^ and determining a and 
 b. For, on these assumptions being made, the equations 
 become identical, and their conditions are fulfilled by 
 taking a?=a, or x=b; the equation x" — px-\-q=0, has, 
 therefore, two positive roots, which are both possible when 
 {p^ is greater than q^ and both impossible when q exceeds 
 i/>2. This expression corresponds to form 3d, ^, (art. 67.) 
 
 ^ 123. Again, a? — axx-^b=x^ — ax-i-bx — a5, which being 
 supposed =0, corresponds to x'^-i-px — ^'=0, or to x'^ — px 
 — ^=0, according as b is greater or less than a. 
 
 The conditions of this equation are answered by taking 
 X — a=0^ or x-{-b=0; the equation has, therefore, two 
 roots, a, and —b. These expressions correspond to forms 
 1, 2, c/f, (art. 67.) These equations consequently admit of 
 a negative and a positive root, which are both possible ; be- 
 cause when the square is completed, the second member 
 consists of positive quantities. 
 
 124^ Moreover, {x-\-a),{x-^b)=x"-{-ax-{-bx-{-ab, which 
 being supposed =0, agrees with x'^-\-px-{-q=^0. 
 
 Hence an equation of this form is resolved by making 
 a-\-b=p^ ab=zq^ and determining a and b. The roots of 
 this equation are manifestly both negative; and (as in art. 
 122,) both possible or both impossible. 
 
 125. It is worthy of remark, that in the equation 
 a?2 — px+q=::0^ the signs are alternately -|- and — , that is, 
 they are twice changed ; and the equation has tivo positive 
 roots. But in the equations (art. 123,) the same sign is 
 
EQUATIONS IN GENERAL. 157 
 
 once continued, and the sign once changed ; and these equa- 
 tions have one affirmative and one negative root. In the 
 equation (art. 124,) the sign is twice continued^ and this 
 equation has two negative roots. 
 
 It also appears, from what is above shown, that every 
 quadratic equation has two roots, which are both possible, 
 or both impossible.* 
 
 126. Assuming three factors, (a? — a).(x — b).(x — c)= 
 x'^ — (a-i-b + c)x''^+(ab'^ac + bc)x — abc, which being sup- 
 posed = 0, will be identical with a?^ — px'^ + qx — r=0, if 
 p = a-{'b -{-c^ q=ab-\-ac-j'bc^ and r=abc. But the condi- 
 tions of the former are answered by making x — a=0, 
 X — 6=0, or X — c=0; therefore, the latter has three posi- 
 tive roots, a, bj and c. It has likewise three changes of the 
 signs. 
 
 127. If instead of x — c we take a?+c, our equation 
 (x — a).(x — b).(x-\^c) = x'^' — (a 4-5 — c)x''^ + (ab — ac — 6c)a? + 
 abc=0, will, manifestly, have two positive roots, a and b, 
 and one negative root, — c. In this case, if a+b'^c, the 
 sign of the second term is — , the equation may, therefore, 
 be expressed x^ — px^dzqx-^r=0, in which there are two 
 changes of the signs, and one continuation of the same sign. 
 If c^a4-6, the second term will have the sign +, but the 
 third — , because in that case {a-\-b), c^{a-^bY, and 
 therefore, ^ab. The equation then becomes x'^-^px^ — 
 qx-{-r=0 ; having as before two changes of the signs, and 
 one continuation of the same sign. 
 
 128. A cubic equation, composed of the factors (x — a), 
 (a? + 6) . (a? + c) = o?"^ + (6 + c — a)x'^ — (a6.4- ac — bc)x — abc= 0, 
 has plainly one positive root, «, and two negative roots, 
 — &, and — c. But in this case, if 6 + 6-^a, the sign of 
 the second term is +5 and the equation may be expressed 
 x^-^px^ztzqx — r=0, having one change of the signs, and 
 
 *In form 3, A, (art. 67,) when h=a^, the roots are equal to each 
 other. In forms 1 and 2, if 2a=0, the equation becomes a simple 
 quadratic, and the roots are equal quantities with contrary signs. 
 Every simple quadratic has, therefore, a positive and negative root. 
 
 u 
 
158 EQUATIONS IN GENERAL. 
 
 two continuations of the same sign. . If a^h-\-c^ the second 
 and third terms are both negative, because (h-\'C)d^(b-\-cf 
 ^bc; the equation, therefore, may be expressed x'^ — px'^ — 
 qx — r=0, having, as before, one change of the signs, and 
 two continuations of the same sign. 
 
 129. If we use the factors x-\-a, x-\-b, ^+c, their pro- 
 duct a?^+(«-|'6-fc)a?'2+(a6+ac+Z>c)a7+a5c, being put —0, 
 the equation may be expressed, a?^+j9^-+ga?4-r— 0, in 
 which there are three continuations of the same sign. 
 The equation has likewise three negative roots, — a, — 6, 
 and — c. 
 
 By pursuing this inquiry it will be found, that any 
 equation of this kind admits of as many roots as there are 
 units in the index of the highest power of the unknown 
 quantity ; that the number of positive and negative roots 
 will be, respectively, equal to the number of changes in 
 the signs, and the number of continuations of the same 
 sign. It likewise appears, that the last term, or absolute 
 number, is the continued product of all the roots with their 
 signs changed. 
 
 130. It may be observed, that as every cubic equation 
 is composed of three factors, and every quadratic of two ; 
 a cubic equation may always be considered as the product 
 of a simple and a quadratic equation. But (art. 125,) 
 every quadratic has, either two possible, or two impossible 
 roots ; hence, a cubic equation, the terms of which are 
 possible, having one impossible root, has two; and, as in 
 the multiplication of compound quantities, containing im- 
 possible parts, those impossible parts can disappear only 
 when two like roots are multiplied together, it follows, 
 that every cubic equation, consisting of possible quantities, 
 has, at least, one possible root. 
 
 131. In like manner, it appears that every equation con- 
 sisting of possible quantities, having an odd number of 
 roots, has, at least, one of those roots possible. And that 
 
EQUATIONS IN GENERAL. 159 
 
 every equation which is made up of possible quantities, 
 has all its roots possible, or an even number of impossible 
 roots. 
 
 132. When the roots of an equation are integral, they 
 may sometimes be found with great facility, by seeking 
 the divisors of the last term, and substituting them in place 
 of the unknown quantity, till one or m.ore are found which 
 answer the conditions of the equation. (See art. 129.) 
 
 When one root has been found, the equation may be 
 depressed by connecting that root, with its sign changed, 
 with the unknown quantity, and dividing the given equa- 
 tion by the sum, 
 
 EXAMPLES. 
 
 1. Given x^ — Saj^-f 5a? — 15 = 0, to find the value of a?. 
 (Art. 126,) the roots, if possible, are all positive.* 
 Also, the divisors of 15 are 1, 3, 5, 15. 
 
 Now, by substituting these for a:, 
 
 1_3.|-5_15^_-12. ... 1 is not a root. 
 
 27— 27-f 15—15=0. .-. 3 is a root. 
 
 « ^ , 0?^— 3a?24-5a;— 15 ^ ^ 
 
 a?— 3 = 0, and —, =a?2+5=0. 
 
 X' o 
 
 Whence, a:=zt:V — 5. 
 
 .-. the roots are 3, ■+ -s/ — 5, and — v/ — 5. 
 
 2. Given a?^— 2a?2— 5a7+6 = 0, to find x. 
 
 Eesults, 1, 3, —-2. 
 
 * The rules, in the foregoing articles, for determining the signs of 
 the roots from the changes in the signs of the terms composing the 
 equation, being founded on the supposition that each root has but one 
 sign, do not apply to impossible roots ; because the negative signs 
 under the radicals, when developed by multiplication, are combined 
 with those of the roots, and, therefore, change the signs of the terms 
 of which the equation is composed. 
 
160 EQUATIONS IN GENERAL. 
 
 3. Given a?*^+6a?2— 7a?— 60=0, to find x 
 
 Results, 3, — 4, — 5 
 
 4. Given a?^+3a:'-— 6a?— 8=0, to find a?. 
 
 Results, 2, — 1, — 4-. 
 
 5. Given a?^— 2a?+4=0, to find x. 
 
 Results, —2, l+\/— 1, l--v/— -1. 
 
 6. Given a?*— 10a?3 + 35a?^— 50a7+24=0, to find x. 
 
 Results, 1, 2, 3, 4. 
 
 7. Given a?*-— So'H ^Sa?^— 64a? +120=0, to find a?. 
 
 Results, 5, 3, 2^—2, — 2n/— 2. 
 
 133. When the roots are not integral, they may gene- 
 rally be determined by approximation.* For this purpose, 
 various rules have been investigated. Among these the 
 following is probably the most convenient in practice. 
 The demonstration is given in the subsequent article. 
 
 To find the root of a general equation. 
 
 1. If all the terms of the equation are not on one side, 
 by transposition, place them so j and arrange them accord- 
 ing to the powers of the unknown quantity, placing the 
 highest power on the left hand. If any of the lower 
 powers are not contained in the equation, consider each 
 one omitted, as having a cipher for its co-efficient, 
 
 2. Place the co-efiicients and the absolute number with 
 their proper signs, in order, in a horizontal line. 
 
 *If all the roots of the equation are impossible, this method is not 
 applicable, (see art. Ill,) but possible roots may always be approxi- 
 mated. 
 
EQUATIONS IN GENERAL. 161 
 
 3. Find by trial the first root figure, attending to its 
 value as being units, tens, tenths, or hundredths, etc and 
 place it to the right of the absolute number. 
 
 4. Multiply the first co-efficient by the root figure, and 
 add the product to the second co-efficient ; multiply the 
 sum by the root figure, and add the product to the third 
 co-efficient ; proceed thus to the end of the line, adding 
 the last product to the absolute number. Again, multiply 
 the first co-efficient by the root figure, and add the pro- 
 duct to the sum under the second co-efficient ; multiply 
 the resulting sum by the root figure, and add the pro- 
 duct to the sum under the third co-efficient ', and so on, 
 stopping under the last co-efficient. Repeat the process, 
 stopping each succeeding time, one term nearer to the 
 left hand, till the last sum falls under the second co- 
 efficient. 
 
 5. Try how often the last sum under the last co-effieient 
 is contained in the sum- under the absolute number, and 
 take the result for the next root figure. 
 
 6. Using the first co-efficient, and the last sum in each 
 column, instead of the co-efficients and absolute number, 
 proceed with this new root figure as with the preceding 
 one. 
 
 7. Obtain another root figure in the manner last men- 
 tioned, and thus continue the operations as far as neces- 
 sary. 
 
 Jfote 1. — In multiplying by each root figure, attention 
 must be given to its value. Thus, if it is of the order of 
 tens, the multiplication must be made by the number of 
 tens which it represents ; and so, for other values. Also, 
 in the multiplications and additions, regard must be had to 
 the signs of the numbers. 
 
 2. The signs of the successive sums under the absolute 
 number and last co-efficient, must continue the same through- 
 out the operation, or both change by the same root figure. 
 14* 
 
162 equatio3n:s in general. 
 
 If the operation for any of the root figures causes only one 
 sign to change, another value must be taken. This will 
 not unfrequently occur with regard to the second root 
 figure, but it will seldom be the case for the others. 
 
 3. After two or three root figures have been obtamed, 
 and the multiplications and additions corresponding to 
 them have been completed, the succeeding parts of the 
 operation may be contracted in the following manner. 
 Cut off the right hand figure of the sum, in the column 
 under the last co-efficient; the two right hand figures 
 of the sum in the preceding column ; the three right 
 hand figures of the sum in the column preceding that, and 
 so on. 
 
 If either of the figures next to the right of the marks 
 of separation is 5, or more than 5, add, mentally, a unit 
 to the first figure on the left of the mark, when using 
 it in the succeeding multiplication and addition. Re- 
 peat the same contraction for each of the following root 
 figures. 
 
 These contractions may commence, in cubic equations, 
 after the second or third decimal figure in the root is ob- 
 tained ; in biquadratic equations, after the first or second 
 decimal figure ; and in higher equations, after the first de- 
 cimal figure. And if the operation is closed when the sum 
 under the absolute number is reduced to two figures, all 
 the figures in the root will be true.* 
 
 EXAMPLES. 
 
 1. Given 3a?*--4a?3+ 2a:— 1000=0, to find the value 
 ofx. Ans. 4.342447603. 
 
 •This rule, improved from Young's Algebra, was communicated by 
 my friend John Gummere, of Burlington. 
 
EQUATIONS IN GENERAL. 
 
 163 
 
 
 12 
 
 12 
 12 
 
 —4 
 48 
 
 44 
 96 
 
 140 
 144 
 
 284 
 14.67 
 
 + 2 
 176 
 
 178 ■ 
 560 
 
 738 
 89.601 
 
 —1000(4.342447603 
 712 
 
 —288 
 248.2803 
 
 24 
 12. 
 
 —39.7197 
 37.39678208 
 
 36 
 12 
 
 827.601 
 94.083 
 
 —2.32291792 
 1.89781666 
 
 48.9 
 .9 
 
 49.8 
 .9 
 
 50.7 
 .9 
 
 51.6 
 .12 
 
 51.72 
 12 
 
 51.84 
 12 
 
 298.67 
 14.94 
 
 313.61 
 15.21 
 
 328.82 
 2.0688 
 
 921.684 
 13.235552 
 
 934.919552 
 13.318496 
 
 948.23804,8 
 .67028 
 
 330.8888 948.90833 
 2.0736 .67048 
 
 332.9624 949.5788,1 
 2.0784 .1340 
 
 335.04,08 949.7128 
 10 1340 
 
 51.96 
 .12 
 
 335.14 949.846,8 
 10 12 
 
 5,2.08 
 
 335.24 
 10 
 
 949.859 
 12 
 
 —.42510126 
 
 37988512 
 
 —4521614 
 3799436 
 
 —722178 
 664909 
 
 -57269 
 56994 
 
 —275 
 285 
 
 3,35.34 94,9.8,7,1 
 2. Given a?5+2a7^+3a?3 + 4372+ 5a?— 54321— 0, to find the 
 value of 07. Ans. 8.41445475. 
 
164 
 
 1 2 
 
 8 
 
 3 
 
 80 
 
 83 
 144 
 
 227 
 208 
 
 435 
 
 272 
 
 707 
 16.96 
 
 EQUATIC 
 4 
 
 664 
 
 668 
 1816 
 
 2484 
 3480 
 
 5964 
 
 289.584 
 
 )NS IN GENERj 
 
 5 
 3344 
 
 5349 
 
 19872 
 
 25221 
 2501.4336 
 
 -54321(8.41445475 
 42792 
 
 10 
 
 8 
 
 -11529 
 11088.97344 
 
 18 
 8 
 
 —440.02656 
 304.11052 
 
 26 
 
 8 
 
 27722.4336 
 2620.0064 
 
 —135.91604 
 122.02904 
 
 34 
 
 8 
 
 6253.584 30342.440,0 
 296.432 68.612 
 
 —13.88700 
 12.21504 
 
 42.4 723.96 
 .4 17.12 
 
 6550.016 30411.052 
 303.344 68.690 
 
 —1.67196 
 1.52700 
 
 42.8 741.08 
 .4 17.28 
 
 6853.3,60 30479.74,2 
 
 7.8 27.52 
 
 —14496 
 12216 
 
 43.2 758.36 
 .4 17.44 
 
 6861.2 
 
 7.8 
 
 6869.0 
 
 7.8 
 
 30507.26 
 
 27.52 
 
 —2280 
 2135 
 
 43.6 
 .4 
 
 77,5.80 
 ( 
 
 30534.7,8 
 
 2.8 
 
 —145 
 153 
 
 44.0 
 
 5,87,6.8 
 
 30537.6 
 
 2.8 
 
 
 30,5,4,0.4 
 
 3. Given a?^ + 9a?H4a?=80, to find x. 
 
 Result, 07=2.4721359. 
 
 4. Given x^-\-x'^-\-x—^0^ required the value of a?. 
 
 Ans. 07=4.1028323. 
 
 5. Given a^^-f lOa?^-]- 5^7=: 2600, required the value of x, 
 
 Ans. 11.00679934. 
 
 6. Given 2a?* +1607^+40073+3007= 4500, required the 
 value of 07. Ans. 07=5.03770809. 
 
EQUATIONS IN GENERAL. 165 
 
 134. To show the rationale of the process directed in the 
 last article, I begin with the cubic equation 
 
 ax'^-{-bx'^ + cx + d=0. 
 
 Let the first figure in the root be indicated by r, regard 
 being paid to its local value, and the remaining part of the 
 root by y; then x=y -^r. 
 
 Hence, ax^'=ay'^-\-3a7'y'^-{-3ar~y-}-ar^ ) 
 
 hx'^ = by^ + 2bry + br^ I _ ^ 
 
 cx= cy-\-cr 
 
 d^ d 
 
 Collecting the co-efficients of like powers of y, 
 ay^+b'y'^-{-c'y-\-d'-=0. 
 
 It is obvious that d'=\(ar-]-b)r-{-c\r'\'d; 
 
 dz={ar'\-b)r-\-{^ar-\-h)r-\-c; b' z=ar -\- ar '\-ar -{-b . 
 
 But these are the quantities found by the fourth precept. 
 
 Again, since r^y, c'y-\-d' approximates to 0, or y= 
 
 — — nearly ; but this is the mode prescribed in the fifth 
 
 precept for finding the next figure of the root. 
 
 Denoting the number obtained by the last operation by 
 5, and the remaining part of y by z, so that y=^z-\-s^ we 
 shall obtain a new equation az^-{'b"z^ + c"z + d"=^0, in 
 which d'z=\(as-{-b')s + c'ls+d'; c"=(as+b')s-{-(2as+b')s 
 -|-c'; b"=as-\-as+as+b'; whence an approximate value of 
 2:, or a new figure of the root, is manifestly deducible from 
 this new equation, as before. 
 
 Assuming now the general equation 
 
 ax''-\-bx''~^ mx^-\-nx+p=^0. 
 
^==0 
 
 166 INDETERMINATE PROBLEMS. 
 
 And denoting as before, the first figure of ^the root by r, 
 and making x=y-\-rj we have 
 
 ax''=^ay''-{-nary^~^ etc. nar^~^y'\-ar^ 
 
 bx''-^= by""-^,, etc. n — l.^r^-^y + Z^r"-^ 
 
 etc. ... etc. 
 
 mx'^= 2mry-^mr^ 
 
 nx— ny-^nr 
 
 P= P) 
 
 Or ay'' + bY~^+" etc. n'y+p'=Oy in which the quantities 
 ^'5 ^'> ?'•> ^re composed of the co-efiicients a, b, etc. and the 
 powers of r combined, as directed in the fourth precept. 
 
 P' 
 And here as before y= — =-j nearly. 
 
 Hence it is obvious, that the successive figures of the 
 root may be obtained by the same kind of process ; what- 
 ever may be the index of the highest power of the un- 
 known quantity. 
 
 INDETERMINATE PEOBLEMS. 
 
 135. When a problem is given, in which the number 
 of unknown quantities employed, is greater than the num- 
 ber of independent equations furnished by the conditions 
 of the problem 5 one, at least, of those quantities, may be 
 assumed at pleasure. Such problems, therefore, generally 
 admit of an indefinite number of answers. There are, 
 however, certain conditions sometimes annexed, by which 
 the number of answers is partially limited. For example, 
 the answers are required to be whole positive numbers, or 
 they are required to be square or cube numbers. Such 
 problems are termed indeterminate, or unlimited, though, 
 in some instances, each unknown quantity admits of but 
 one value. If simple powers only of the unknown quan- 
 
INDETERMINATE PROBLEMS. 167 
 
 titles are included in the equations, the problem is said to 
 be of the first decree. 
 
 Case I. 
 
 To find the values of x and y, in whole positive numbers,, 
 from the equation ax=by-{-c; a, b, c, being given num- 
 bers, positive or negative,* 
 
 by-\-c 
 Here a?=-^ — , and as x is to be a whole number, 
 
 by -\-c 
 
 must also be a whole number. 
 
 a 
 
 Now, if this quantity be multiplied by a whole number, 
 
 the product must, evidently, be a whole number; also the 
 
 sum or difference of this quantity, or either of its multiples, 
 
 and any whole number, must necessarily be a whole num- 
 
 bv -\-c 
 ber. Let, therefore, — — be thus chano-ed till we obtain 
 
 V -4~ c 
 
 =2^A,f which put =J9; then y=ap — c', a quantity 
 
 that must be a whole number, because n, p, and c' are 
 
 whole numbers. And this value being substituted for y in 
 
 by-\'C 
 the equation x= , the value of a? will be obtained in 
 
 terms of j?, and given numbers. 
 
 Assuming then j9=0, 1, 2, 3, etc. successively, (omitting 
 such numbers as make x or y negative,) the various nu- 
 merical values of x and y become known. 
 
 The number of answers will be limited when the signs 
 of p in the values of x and y are unlike : but unlimited 
 when p has the same sign in both. 
 
 =* If ff and b have a common divisor, it must also be a divisor of c, 
 or the problem is impossible. 
 
 fl'his expression is used to designate any trholc vumlcr. 
 
168 INDETERMINAAE PROBLEMS. 
 
 EXAMPLES. 
 
 1. Given l9a?=14?/4-15, to find the values of x and y 
 in whole positive numbers. 
 
 14y+15 , Uy + lb , 
 
 56?/+60 ^ ^ 18^ + 3 
 
 _, 19y ^ 19y 18y-f 3 y— 3 • , 
 But --|=z.A. ... ^--A^^y—^^h^p 
 
 ...2/=19;? + 3, and a?=— ^-i^-j^ — l—^=Up + 3. 
 
 If now, we assume p successively =0, 1, 2, 3, etc. 
 
 2/= 3, 22, 41, 60, etc. 
 
 a?=3, 17, 31,45, etc. 
 Here the number of answers is evidently unlimited. 
 
 2. Given lla?+l'73/=987, to find a? and y in whole posi- 
 tive numbers. 
 
 987— 17y ^^ 8— 6v , 8— 6v 
 
 x= jj-^=-89— y + -^^=t^A. .-. — ^^=u'^, 
 
 8—61/ ^ 16— 12i/ 5— y V— 5 
 
 y — ^ 7 
 
 Whence y = 1 Ip + 5, 
 
 987— (11;)+5)X17 ^^ ^^ 
 And a?:=- i—Y^-^^ =82— 17p. 
 
INDETERMINATE PROBLEMS. 169 
 
 Assuming ^=0, 1, 2, 3, 4, 
 
 2^=5, 16, 27, 38,49. 
 a:=82, 65, 48, 31, 14 
 
 Which are all the possible values in whole positive 
 numbers. 
 
 3. Given 7a: + 93^= 2342, to find the number of values 
 of X and y in whole positive numbers. 
 
 2342— 9 V ^^, 2y— 4 , 2y— 4 
 x= = — ^=334 — y ^ — =wL r.~^ — =wh. 
 
 %-i..._8y-i6_ ,v-2_, y-2 
 
 ■X4 
 
 =y—2+^-Y-=wL .'.^=wk=p. 
 
 7 
 .•.3/=7p+2, and a?=332~9i?. 
 
 From the first of these expressions we perceive that 
 the least value of p is 0, and from the second, that the 
 greatest value of p is 36. Hence the required number 
 is 37. 
 
 4. Given 5a? + 73/ +92: =337, to find the number of val- 
 ues of X, y, and z^ in whole positive numbers. 
 
 3^7— 7y— 9z ^^ , 2— 2y— 4^ , 
 x=:^ -^ =67 — y — z+ 1 z=zwh. 
 
 ^ 2— 2y— 42r_ 
 * * 5 
 
 . , 2— 2v— 42? ^ 4— 4v— 82: 
 And g X 2= 1 =wk. 
 
 52/— 5 + 102? , 
 
 But -^ — ~ =wk. 
 
 5 
 
 52/-_5 + 102? 4— 4v— 82? V— l + 22r 
 
 r,y=^p — 22? + 1, x=^66 — 7/? + 2-. 
 15 
 
170 INDETERMINATE PROBLEMS. 
 
 Assuming now z successively equal to 1, 2, 3, etc. we 
 shall have the corresponding values of x and ?/, the greatest 
 and least values of jp, and the number of answers as fol- 
 lows: 
 
 z 
 
 X 
 
 y 
 
 P 
 
 P 
 
 
 
 
 
 Gr. 
 
 Least. 
 
 No. of Ans. 
 
 1 
 
 Ql—lp 
 
 5p—l 
 
 9 
 
 1 
 
 9 
 
 2 
 
 6S—7p 
 
 6p—S 
 
 9 
 
 1 
 
 9 
 
 3 
 
 69— Ip 
 
 bp — 5 
 
 9 
 
 2 
 
 8 
 
 4 
 
 10— Ip 
 
 bp—1 
 
 9 
 
 2 
 
 8- 
 
 5 
 
 71— Ip 
 
 5;?— 9 
 
 10 
 
 2 
 
 9 
 
 6 
 
 12— Ip 
 
 5jo— 11 
 
 10 
 
 3 
 
 8 
 
 7 
 
 13— Ip 
 
 5p— 13 
 
 10 
 
 3 
 
 8 
 
 8 
 
 1^—lp 
 
 5j9— 15 
 
 10 
 
 4 
 
 7 
 
 9 
 
 Ib—lp 
 
 5/7—17 
 
 10 
 
 4 
 
 7 
 
 10 
 
 16— Ip 
 
 5;?-~19 
 
 10 
 
 4 
 
 7 
 
 11 
 
 11— Ip 
 
 5;?— 21 
 
 10 
 
 5 
 
 6 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 Here we may observe, there are seven successive values 
 of^r, (beginning with the 5th,) which produce no change 
 in the greatest value of p; and there are alternately two 
 and three equal least values of p; ajid this order will evi- 
 dently continue as long as successive values of z are as- 
 sumed. 
 
 The last three columns of the above table may, there- 
 fore, be continued without the former ones. 
 
 Gr. val. oi p, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 
 Least do. 11222334445566 
 No. of Ans. a 9889 8 8777676 6 
 
 Gr. val. of^, 11 11 11 11 12 12 12 12 12 12 12 13 13 13 
 
 Least do. 6 7 7 8 8 8 9 9 10 10 10 11 11 12 
 
 No. of Ans. 65545544333332 
 
 Gr. val. of p, 13 13 13 13 14 14 14 14 
 Least, 12 12 13 13 14 14 14 15 
 
 No. of Ans. 2 2 111110 
 
 The number of answers, or sum of the numbers in the 
 last column, is 169. 
 
INDETERMINATE PROBLEMS 171 
 
 Otherwise — 
 
 irom the equation z— ^ , since neither a?, 
 
 nor 3/ can be less than 1, it is obvious that 2: cannot exceed 
 36. Then in the equation 07=66 — Tp+ar, make 2:= 36 
 and x=^ 102 — 7^ ; where the greatest possible value of 
 j3=14. Put p=14f in the equation y=op — 22r+l, and 
 2^=71— -22r. From which we perceive that the greatest 
 value of 2;= 35. Assuming then z= 1, 2, 3, etc. to 35, and 
 taking all the values of p, from 14 to the lowest, which 
 will give a positive value to y; and combining the results 
 five by five, we shall have a series in arithmetical progres- 
 sion, 67 -1-57+ . . . 7=259, which is the number of whole 
 positive values of y. But some of the corresponding values 
 of X will be or negative. 
 
 From the equation x=66 — Ip+z, we find the first four 
 values of z, 1, 2, 3, 4, render a?=0, or negative if p^9. 
 Hence, we have 20 inadmissible values of x. For the 
 next 7 values of z^ we have x=0^ or negative if p^lO ; 
 which give 28 inadmissible values. Proceeding until 
 z=35, we have the series 20 + 28-}- 21+ . .7=90. Hence 
 259 — 90=169, the number required. 
 
 4. Given 14a?=5?/+19, to find the least possible values 
 of X and t/, in whole positive numbers. 
 
 Ans. x=6^ y=l3. 
 
 5. Given 11a? + 5^=254, to find all the values of x and 
 y, in whole positive numbers. 
 
 Ans. a?=19, 14, 9, 4. 
 y=9,20,31,42. 
 
 6. Given 9a?+133/=2000, required the number of values 
 of X and y. Ans. 17. 
 
 7. Required to divide 100 into two parts, so that one of 
 them may be divisible by 7, and the other by 11 1 
 
 The parts are 56 and 44. 
 
172 INDETERMINATE PROBLEMS. 
 
 8. Given 17a?4-193^+2l2r=400, required the number of 
 values of ^, y, and z, Ans. 10. 
 
 9. Required to pay 1000 dollars, in French crowns, and 
 five franc pieces, so that the number of coins used shall be 
 the least possible \ what number of each kind will be ne- 
 cessary, and how many ways can that sum be paid in those 
 coins, the French crown being 1 10 cents, and the five franc 
 piece 93 cents 1 
 
 Ans. 833 crowns, 90 five franc pieces, and 9 different 
 ways. 
 
 10. How many gallons of liquor at 12 cents, 15 cents, 
 and 18 cents per gallon, may be mixed, to compose 300 
 gallons at 17 cents per gallon 1* 
 
 C at 12 cents, 1, 2, 3, etc. to 49. 
 Ans. \ at 15 cents, 98, 96, 94, etc. to 2. 
 I at 18 cents, 201, 202, etc. to 249. 
 
 11. In how many ways can £1053 sterling be paid 
 without using any coins besides guineas and moidores ; the 
 guinea being 2 J 5. sterling, and the moidore 271 
 
 Ans. 112 ways. 
 
 12. A foreigner having a bill of $1 75 to pay at a hotel, 
 offers napoleons in payment, the landlord agrees to receive 
 them on condition that Turkish sequins shall be taken as 
 change ; how many pieces must be used, a napoleon being 
 worth $7 25, and a sequin rated at $2 10 \ 
 
 Ans. 35 napoleons, and 120 sequins. 
 
 13. Given 7a7 4-92/-|-232r=9999 ; of how many values 
 will 0?, 2/, and z admit % 
 
 Ans. 34365. 
 
 , * When more equations than one are given, one, at least, of the 
 unknown quantities may be eliminated, and the equations reduced to 
 
indeterminate problems. 1v3 
 
 Case 2. 
 
 136. To find a whole number^ which ^ being divided by any 
 given numbers^ shall leave given remainders. 
 
 Denoting the required number by a?, the given divisors 
 by a, Z>, c, etc. and the remainders by /, g, A, etc. we 
 
 shall have , — j — , , etc. severally equal to whole 
 
 numbers. 
 
 /p f 
 
 Put — - =p. find the value of x. and substitute it for x 
 a ^' 
 
 in the second fraction ; reduce this new fraction, as in the 
 former case, so that the co-efficient of p may be a unit, and 
 put the fraction thus obtained =q; find the value of p, 
 and thence of x, in terms of q and given numbers. Sub- 
 stitute the value of x in the third equation, and so proceed 
 The last value of x will be the number sought. 
 
 EXAMPLES. 
 
 1. Required a number, which, being divided by 7, 9, 
 10, and 11, shall leave the remainders 5, 4, 7, and 9, re- 
 spectively. 
 
 Let x=i the number sought. 
 
 _, X — 5 X — 4 X — 7 X — 9 , . , 
 
 Then -y-, -^, -jg"' TF' ^^^ numbers. 
 
 —--=:wh—p, .-.O^rrrTp + S. 
 
 7^+5-4 7^+1 , 
 Whence, • ^ = -f^-^—.-=wh. 
 
 7p-fl , 28;?+4 ^ P-f4^ . 
 
 15* 
 
174? INDETERMINATE PEOBLEMS. 
 
 79+4 
 
 9 
 
 -=wk=q, nnd p~9q — 4. r,x=63q — 23. 
 
 And consequently. 
 
 ^* =wh-=r, .-.^^lOr, anda?=630r— 23. 
 
 630r— 32 ,^ ^ 3r+l , 3r + l 
 
 .-. ==51r—3+-jY-=wL .-. —j—=zwh. 
 
 ^ ,3r+l ^ 12r + 4 r— 7 
 
 And-jPx4=--^j-=r+l+--^==2^A. 
 
 r— 7 
 .•.— — -='i^A=5. .-.r— 115 + 7. 
 
 And a:= 69305 4- 4387= (if 5=0)4387. 
 Assuming 5= 1, 2, etc. other values would arise. 
 
 2. Eequired a number, which, being divided by 2, 3, 4, 
 6, shall leave no remainder; but being divided by 7, the 
 remainder shall be 6 1 
 
 The least common multiple of 2, 3, 4, 6,t is 12. 
 
 • If a fractional expression - ri=i^?A, and we divide the terms into 
 
 
 
 iheii primes, it is plain that all the prime numbers contained in 
 the denominator, must be also contained in the numerator ; thus, 
 
 ^- z=:^'^z=wh, where the primes 2 and 5 not being contained in the 3> 
 10 2.5 
 
 must be contained in <?, and therefore, — =<»/;^. 
 
 10 
 
 fSee Aritheraeticai Expositor, Part 1, page 90. 
 
INDETERMINATE PROBLEMS. 175 
 
 Let, therefore, 12x— the number sought. 
 
 12a?— 6 , 12a?— 6 36a:— 18 
 .-. — y-==w;A. .-. — ,^— x3= ==:5a?— 2-i- 
 
 =iwh; and -——z=wh=p, .•.a?=7p + 4. 
 
 And 12a7=:84p + 48=48, 132, 216, etc. 
 
 3. Required the least whole number, which, being di- 
 vided by 3, 5, 7, and 8, shall leave the remainders 1, 3, 5, 
 and 0, respectively % Ans. 208. 
 
 4. Required the least whole number, which, being di- 
 vided by 9, 10, 11, and 12, shall leave the remainders 4, 
 5, 6, and 7, respectively 1 Ans. 1975. 
 
 5. What is the least whole number, which, being di- 
 vided by each of the nine digits, shall leave no remainder ; 
 but divided by 17, the remainder shall be 13 1 
 
 Ans. 40320. 
 
 6. In what year of the Christian era was the solar cycle 
 15, the lunar cycle 3, and the Roman indiction 141 * 
 
 Ans. 1826. 
 
 7. Required the least whole number, which, being divided 
 by 2, 3, 4, and 5, shall leave a remainder of 1 ,• but divided 
 by 7, there shall nothing remain 1 Ans. 301. 
 
 * The solar cycle is a period of 28 years, at the end of which, in 
 case a centurial year has not intervened, the days of the week always 
 return to the same days of the month. To the year of the Christian 
 era add 9, and divide by 28, the remainder will be the number of the 
 cycle. ^ ^ 
 
 The lunar cycle is a period of 19 years, at the expiration of which 
 the new and full moons return nearly to the same time of the year. 
 To the year of the Christian era add 1, and divide by 19, the remain- 
 der is the lunaf- cycle. 
 
 The Roman indiction is not an astronomic period, it consists of 15 
 years. To a given year add 3, and divide by 15, the remainder is the 
 number of the indiction. 
 
 If, in either of these cases, no remainder occurs, the divisor must 
 be taken as the number of the cycle. 
 
17G MISCELLANEOUS EXAMPLES. 
 
 MISCELLANEOUS EXAMPLES. 
 
 137. — 1. A bankrupt owes A. twice as much as he owes 
 B., and he owes to C. as much as to A. and B. together; 
 the sum to be divided is 600 dollars ; how much must each 
 receive'? Ans. A. 200, B. 100, C. 300. 
 
 2. A. can perform a piece of work in 7 days, and B. in 
 9 days , in what time would they jointly effect it '? 
 
 Ans. 3 51 days. 
 
 3. A laborer being employed on condition, that for every 
 day he worked he should receive 50 cents, and for every 
 day he was idle, he should forfeit 20 cents, finds at the qnd 
 of 500 days only 89 dollars due ; how many days did he 
 work'? Ans. 270. 
 
 4. There are two numbers in the ratio of 4 to 5, and the 
 sum of their squares is 1476 ; what are the numbers '? 
 
 Ans. 24 and 30. 
 
 5. A Greek epitaph, designed for the tomb of Diophan- 
 tus, is said to have stated that he passed one-sixth of his 
 life in childhood ; one-twelfth in adolescence ; that after 
 one-seventh and five years more had been passed in a 
 married state, he had a son who lived to half his own age, 
 and whom he survived four years ; what then was the age 
 of Diophantus '? Ans. 84 years. 
 
 6. A person's age in years is a number, consisting of two 
 digits ; ^ of this number is a mean proportional between 
 these digits ; and two years hence his age will be a third 
 proportional to those digits, beginning with the tens ; what 
 IS the age '? Ans. 14 yearsi 
 
 7. Givena7 + y + 2r=9, xy -^xz -\-yz=26, xyz =24^, to ^nd 
 the values of a?, y, and z* Result, a?=2, 3, or 4. 
 
 * If we assume the equation v^ — 9^3^26^ — 24=0, then art. 126, the 
 roots of this equation will be the values required ; and those roots are 
 easily found by art. 132. 
 
MISCELLANEOUS EXAMPLES. 177 
 
 8. By selling a piece of muslin at a certain price per 
 yard, I gained the prime cost of 9 yards, which was just as 
 much per cent, as the number of yards in the piece ; what 
 was that number '? Ans. 30 yards. 
 
 9. There are four numbers in continued proportion, the 
 sum of the first and last 1728, and the sum of the other two 
 1152 5 what are the numbers 1 
 
 Ans. 192, 384, 768, and 1536. 
 
 10. The hypothenuse of a right angled triangle is 3?^% 
 and the other sides a?^^, and x^; what is the areal 
 
 Ans. W(2+V5). 
 
 }.l. There is a number, consisting of three digits in 
 arithmetical progression, which number being .divided by 
 the sum of the digits, the quotient will be 48, but if from 
 the number 198 be subtracted, the remainder will be ex- 
 pressed by the same digits in an inverted order. Quere 
 the number 1 Ans. 432. 
 
 12. The sum of the squares of two numbers being mul- 
 tiplied by the quotient arising from the division of the less 
 by the greater, produces 83.2, and the difference of the 
 squares multiplied by the quotient of the greater divided 
 by the less, produces 1920 ; what are the numbers 1 
 
 Ans. 20 and 4. 
 
 13. A ball falling from" the top of a tower, is observed 
 to descend one-fourth of the distance in the last second of 
 the time ; required the height of the tower, heavy bodies 
 being known to fall 16 J^ feet during the first second, and 
 to describe spaces, which, reckoned from the beginning of 
 the fall, are as the squares of the times "l 
 
 Ans. (28+16^3)16^2 feet. 
 
 14. What are the values of x and z, from the equations 
 x'z+xz^=b4^6bQ0, x^+z^ =10869921 
 
 Ans. a?=32, 2r=14. 
 
 15. GiYen x+y+z+v=:56, x''+y^'^z^+v^=910, xv-\- 
 22/2— 2^2=6, and z=2y^ to find the values of a?, y, z, and v 
 
 Result, x=S, y=:9, 2:= 18, 2;=21. 
 
178 MISCELLANEOUS EXAMPLES. 
 
 16. Required to find a number, which, being any way 
 divided into two unequal parts, the greater part added to 
 the square of the less, shall be equal to the less part added 
 to the square of the greater. Ans. 1. 
 
 17. Given ccy=n5x+300y, and 2/3— 0?^= 90000, to find 
 X and y by a quadratic* Result, a?=400, y=500. 
 
 18. Given (a?-^+l).(a?^+l).(a?+ l) = 30a?^ to find the 
 value of X by a quadratic equation.! 
 
 Result, a:=i(3±v/5). 
 
 19. The sum of three numbers in harmonical proportion 
 is 26, and their continued product 576 ; required the num- 
 bers by a quadratic equation. Ans. 12, 8, and 6j^ 
 
 * This is readily solved by quadratics, by finding x in terms of y 
 from the first equation, and substituting the value in the second. 
 
 fThis question requires expedients not readily found. 
 By multiphcation and transposition. 
 
 x^-j^x^+x* — 280^3 4- a?2+a?+ 1 = 0, whence 
 
 oo 1 1 1 ^ 
 
 But 2/=a?-f-, whence y^-fy2 — 2y — 30=0 
 
 X 
 
 1 , ' 7 790 79 10 . 
 
 Assume z=y+p wherefore z^^-z=-^=— ,— 
 
 '7 '79 10 ,^^. (lO^V 
 And z* — ^z^=^ "q"'"^^- Adding ^-^ — to each member 
 
 79 10' 79 10 
 
 ^*+T^^=-(T")^+y-3" 
 
 Hence 2r=-jr-, and y=3. Whence x^ — 30?= — 1. 
 
 3zfcs/5 
 
 and x = — tr — 
 
MISCELLANEOUS EXAMPLES. 179 
 
 20. The sum of two numbers is 152, and the cube root 
 of the square of their difference multiplied by the square 
 root of the cube of the same difference, produces 8192; 
 what are the numbers 1 Ans. 44 and 108. 
 
 21. The sum of two numbers, added to the sum of their 
 squares, is 120, and the product of the same numbers 45 ; 
 what are they % Ans. 9 and 5. 
 
 22. Given x^+xy^=z4<64}0y^ and x'^y — 2/3= 537.6a?; what 
 are the numerical values of x and y ? 
 
 Ans. X — 40, y=16, 
 
 23. GiYen x-\-y=:z, z^ — x^+y^=4}10, z^ — x^ — y'^—^Q'^^ 
 to find 0?, y, and z. Result, x= 12, y= 1, z— 13. 
 
 ( .3 
 
 24. Given < x-\-y' +a?+2/=30. 
 
 ( X — y= 1. Required the values of x and y, 
 Ans. a?=2, y=l. 
 
 25. Given x^ — 2a:3-f a7=«, to find a? by a quadratic equa- 
 tion. Result, a:= 4 + ^/ (I + s/~^i), 
 
 26. Given c(f^+x[y-\-z)=a^ y^'{-y(x+z)=b, z^+z{x-{-y) 
 = c, to find X, y, and z. 
 
 Results, ^ 
 
 b 
 
 c 
 ' y/(a+b+c) 
 
 27. Given 2a:3 — x* — x'^'\-2x^y — y^=^2xy— s/x^y — s/x^y, 
 to find X and y. Result, x=5, y—20, 
 
 28. Required the roots of the equation 4fOC^-\-Sx^ — 
 89a?2+28a?+49=0, by quadratics only. 
 
 7 —13+ V 1 13 — 13— v/ 113 
 Ans. 1, ^, ^ , J 
 
180 MISCELLANEOUS EXAMPLES. 
 
 29. There are three numbers in geometrical progression, 
 whose continued product is 4096, and the sum of the ex- 
 tremes is 68 j what are the numbers % 
 
 Ans. 4", 16, and 64. 
 
 30. There are two numbers expressed by the same two 
 digits, and the difference of their squares is 1485 ; quere 
 the numbers ] Ans. 14 and 41. 
 
 Or 78 and 87. 
 
 31. What two numbers are those whose product, differ- 
 ence of their squares, and quotient of their cubes, are all 
 equan Ans. 3 + ^^5^ and i + W5. 
 
 32. The product of two numbers is 10, and the product 
 of their sum by the sum of their squares is 203 , required 
 the numbers found by a quadratic equation. 
 
 Ans. 5 and 2, 
 
 33. There are three numbers in geometrical progression, 
 the difference of whose differences is 6, and the sum of the 
 numbers 42, what are the numbers] 
 
 Ans. 24, 12, and 6. 
 
 34. There are three numbers in harmonical proportions, 
 the difference of whose differences is 1, and the product of 
 the extremes 18 -, what are the numbers 1 
 
 Ans. 3, 4, and 6. 
 
 35. Required three equi-different numbers, such that if 
 the first be increased by 1, the second by 2, and the third 
 by the first, the sums may constitute an harmonical pro- 
 gression ; but if 3 be added to the second, the sum may be 
 a mean proportional between the sum of the numbers and 
 the first diminished by k* Ans. 5, 6, and 7. 
 
 36. Required two such squares, that their difference 
 shall be to the square root of the less as 3 to 7, and the 
 square roots of the numbers to each other as 5 to 2. 
 
 Ans. (loy, and (^%y. 
 
 37. Given the sum of the cubes of two numbers =35, 
 and the sum of their 9th powers =20195; what are the 
 numbers 1 Ans. 3 and 2 
 
MISCELLANEOUS EXAMPLES. 181 
 
 38. Given 9 -. + 36^=85, -^+-^ = — ^16, 
 required the values of x and y. Ans. a:==3^, y~2. 
 
 39. What two numbers are those whose difference is 4, 
 and their product multiplied by the sum of their squares 
 480 1 Ans. 6 and 2. 
 
 40. Giyen x^y'^xi/=a, xz^'s/xz=b,y^Zs/zy=c, to find 
 the values of x, y, and z. 
 
 
 41. Required the values of a?, y, and 2r, from the equa- 
 tions yz=a, xz=.h^ xy=c, 
 
 . be ac ah 
 
 42. There are three numbers in harmonical proportion, 
 the sum of the first and third is 54, and their continued 
 product 15552 ; what are the numbers'? 
 
 Ans. 18, 24, and 36. 
 
 43. In how many different ways is it possible to pay 
 JBIOOO, without using any other coins than crowns, guineas, 
 and moidoresj a crown being 55., a guinea 2l5., and a 
 moidore 275. % Ans. 70734. 
 
 44. Given x''+y^-^z^=2QQ.^, a?'2+y+;s2^ 176.5, 
 
 {a?-fy+2:)y=286. Required the values of x 
 2/, and z, Ans. 07=10.5, 2/= 10, 2r=7.5. 
 
 45. Required two cube numbers, such that the first mul- 
 tiplied into the product of their roots, shall be equal to the 
 second ; but the second multiplied into the product of their 
 roots, shall be equal to 64 times the first. 
 
 Ans. 8 and 64. 
 16 
 
182 MISCELLANEOUS EXABIPLES. 
 
 46. What number is that which being added to, and sub- 
 tracted from 36, the sum of the cube roots shall be 6 1 
 
 Ans. 28. 
 
 47. Given ^2+3/2— a?—i/=249740, a??/ + a?+?/=;:8516, to 
 determine the values of x and y. 
 
 Result, x=zb00,"y=:16. 
 
 48. Given x'-j-y'-\-x''+y^=^23S63'2—2xY, 
 
 x^+y^ + z^=: '.y (216100— x^—2f—z%) 
 to find the values of a?, y, and z, 
 
 Ans. a:=22, ^=2, z=6, 
 
 49. The sum of five numbers in geometrical progression 
 is 242, and the fourth difference is 32, required the num- 
 bers. Ans. 2, 6, 18, 54, and 162. 
 
 50. Given a?-f^/ + 2r+i;+w=12.15=a, 
 
 X'\-y+z-\-vw=9.16 = bj 
 x-i-y-\-zvw=^6.6S=c, 
 ^ w -\- y zvw =6. 09 =-d, 
 
 xyzvw=A5—e, Required the values of 
 a?, y, z, V, and w, 
 
 , Ans..=^^'^(|=^i) = 5,or.09. 
 
 _ c—x±^ sf \ {c—xy—^{<i—x) \ 
 
 y- 2 
 
 a 
 
 b — 0?— 2/=h s/ \ (b — X — yf — 4(c — a? — y \ 
 ; — X — y — 2r-_4= V \ (a — X — y — zy — 4(5 — x — y — z) j 
 
 2 
 w=a — X — y — z — V. 
 
 From the ambiguity of the signs, it is manifest that the 
 number of numerical values of the unknown quantities go 
 on increasing from x to v. 
 
 THE END. 
 
ROBERT E. PETERSON, 
 BOOKSELLER AND PUBLISHER, 
 
 N. W. CORNER OF FIFTH AND ARCH STREETS, 
 
 PHILADELPHIA, 
 
 Invites attention to the following valuable works, 
 just published. 
 
 FAMILIAR SCIENCE, 
 
 OR 
 
 THE SCIENTIFIC EXPIINATION OF COfflON THINGS. 
 
 EDITED BY R. E. PETERSON, 
 
 MEMBER OF THE ACADEMY OF NATURAL SCIENCES, PHILADELPHIA. 
 
 12mo. Sheep, 550 Pages. Price 75 Cents. A deduction 
 made to Teachers. 
 
 The attention of Teachers is invited to the following opinions of 
 the work : 
 
 From Professor W. H. Allen, President of Girard College for Orphans, 
 Philadelphia. 
 
 Girard College, May 6, 1851. 
 Robert E. Peterson. — Dear Sir — I beg leave to tender my 
 thanks for your courtesy in sending me a copy of "Familiar 
 Science." I have read parts of each division of the work, and 
 have been pleased with the precision of the questions, and the ac- 
 curacy of the answers. The book is not merely a volume of /«• 
 vniliar knowledge^ but a volume in which much rare and profound 
 knowledge is made familiar to the common mind and applied to 
 common things. I consider the book a valuable contribution to 
 our means of instruction in schools, and hope to see it generally 
 introduced and used by teachers. Fathers of families also, who are 
 now frequently puzzled by the questions of the young philosophers 
 of their households, will do well to procure a copy and avoid saying 
 so often, " I do not know.'''' I remain truly yours, etc., 
 
 Wm. H. Allen. 
 
2 
 
 From the Rev. Lyman Coleman, D. D., Principal of the Presbyterian In- 
 stitute, Philadelphia. 
 
 "Familiar Science" embodies a vast amount of facts and princi- 
 ples, relating to the several branches of natural science, judiciously 
 selected and arranged, and very useful to awaken inquiry in the 
 young, and form a taste for such studies. For this purpose it is 
 used as a text book in our school. 
 
 Lyman Coleman, Prin. 
 
 PniLADELrHiA, May 15, 185L 
 
 From the Right Rev. Bishop Potter. 
 
 Broad Street, June 2, 1851. 
 Dear Sir — Absence from the city and urgent engagements have 
 prevented me from acknowledging, as promptly as I wished, your 
 note of last month. The same causes have prevented ray giving 
 to your work, entitled "Familiar Science," the thorough examina- 
 tion to which its apparent merit richly entitles it. So far as I have 
 been able to look through it, it fully justifies your selection of it 
 and the laborious revision which you have bestowed upon it. It 
 contains a vast amount of useful information on subjects which 
 force themselves upon the attention of both old and young, and it 
 is likely to cultivate in those who read it habits of inquiry and re- 
 flection. The mechanical execution is entitled to unqualified 
 praise. Very respectfully yours, 
 
 Alonzo Potter. 
 
 Mr. R. E. Peterson. — Dear Sir— I am much pleased with the 
 volume you left me for perusal, entitled " Familiar Science, or The 
 Scientific Explanation of Common Things." It treats, in a familiar 
 and interesting manner, of a large variety of subjects, which all 
 children should understand, but many of which are indifferently un- 
 derstood by persons of mature age. The book cannot fail to inter- 
 est the inquisitive pupil, and become a valuable auxiliary in the 
 diffusion o{ practical science, both in schools and families. 
 
 Yours respectfully, 
 
 J. Simmons, 
 Principal of " The Locust Street Institute for Young Ladies." 
 Philadelphia, May 17, 1851. 
 
 FromN. C. Brooks, A. M., Principal of the Baltimore Female College. 
 Late Principal of the Baltimore High. School. 
 
 Baltimore, May 28, 1851. 
 Having examined the American edition of Dr. Brewer's Fa- 
 miliar Science, with additions by R. E. Peterson, I take pleasure 
 in commending it to teachers and parents, and all others interested 
 m the cause of education, as a most excellent work. It is worthy 
 of very extensive circulation. I shall most prooably have it used 
 as a text book in our Institution. 
 
 N. C. Beooks 
 
From the Hon. Joel Jones, Ex- president of Girard College for Orphans 
 Philadelphia * 
 
 1 take great pleasure in concurring in the foregoing recommenda- 
 tion. JoEii Jones. 
 
 _ Having carefully exammed the book entitled "Familiar Science," 
 i accord to it my decided approbation. 
 
 As a class book, in the hands of a judicious teacher, it cannot fail 
 to strengthen the intellect and enlarge the useful knowledge of all 
 young persons that may study it. So much am I impressed with 
 its value, that I think it must speedily find its way into all good 
 schools. 
 
 Edmund Smith, A. M. 
 "Franklin Hall," Newton University, Baltimore, 
 
 June 2, 1851. 
 
 Burlington, May 24, 1851. 
 Robert E. Peterson, Esq. — Dear Sir — It gives me great plea- 
 sure to acknowledge the receipt of a late reprint of yours entitled 
 "Familiar Science." It affords another of the many examples, 
 which are constantly being furnished nowadays, of the increased 
 facilities of instruction. Such books as these implant in the youth- 
 ful mind a desire and determination to inquire more extensively 
 into science, and thus no one can calculate their value, for we are 
 daily called upon to acknowledge the immense advantages which 
 an insight into chemical action brings to bear upon mechanical, as 
 well as agricultural fields of labor. I can well imagine the plea- 
 sure which a perusal of this little book of yours will afford to those, 
 who first glean from its pages a knowledge of the subjects on which 
 it treats, from a recollection of the feelings with which I used to 
 read a much more unpretending volume of the same nature, now 
 out of print, entitled " Common Things." Assuring you that it will 
 give me great pleasure to recommend your book as an elementary 
 one in the Institution with which I am connected, 
 I have the honor to subscribe myself 
 
 Very respectfully yours, George H. Doane, 
 Instructor in Chemistry, and Lecturer on Physiology at Burlington Col- 
 lege. 
 
 From Miss Phelps, Philadelphia. 
 Y"our excellent book, "Familiar Science," has been received, 
 and, after having carefully examined it, I am prepared to concur 
 entirely with the numerous testimonials you have already received 
 of its merit as a text book for schools or families. It is eminently a 
 book for the times and for the people, as a vast amount of general 
 information on useful and practical subjects may be acquired with- 
 out exploring multiplied abstruse matter for learning, compara- 
 tively, a few facts. I predict for it a ready and rapid sale, and con- 
 
gratulate educators in having so able an auxiliary in the business (jt 
 instruction. Thank you for the presentation of a volume, and 
 Am, very truly, H. M. Phelps, 
 
 Prin. St. Mary's School. 
 
 Philadelphia, May 13, 1851. 
 Mr. Robert E. Peterson. — Dear Sir — Many thanks to you for 
 your present of " Familiar Science ;" I have given it a careful pe- 
 rusal, and believe it to be better calculated to foster in the mind of 
 3'^outh a spirit of philosophic and scientific inquiry than any work 1 
 have ever seen. The questions are precise and the answers accu- 
 rate ; and the common things of every day life receive a profound 
 and luminous explanation. The work is admirably arranged, and 
 the index remarkably complete. It should be in all the schools in 
 the country. I shall introduce it into mine at the next term. 
 With great respect, your friend, 
 
 D. R. ASHTON, 
 
 Female Institute, Philadelphia. 
 
 Weccacoe Boy's Grammar School, May 15, 1851. 
 
 Sir — From an examination of the book entitled "Familiar 
 Science," which you were pleased to send me, I feel convinced 
 that it would be a valuable addition to the list of text-books now in 
 use in our schools. 
 
 The importance of the subjects, and the familiar manner of treat- 
 ing of them, should make it a popular family, as well as school 
 book. 
 
 John Joyce, Prin. 
 
 Mr. Robt. E. Peterson. 
 
 May 16, 1851. 
 Dear Sir — The pages of "Familiar Science" are its best recom- 
 mendation. The common phenomena of life are treated of in a 
 simple and intelligible manner, which renders it both pleasing and 
 instructive. In the family circle, as a text book, it will form the 
 basis of an hour's interesting conversation, and in the hands of the 
 pupil it will be a valuable aid in the acquisition of useful know- 
 ledge. Respectfully, 
 
 Wm. S. Cleavenger, 
 Prin. Boys' Grammar School, Locust St. 
 To Robert E. Peterson, Esq. 
 
 Philadelphia, May 17, 1851. 
 Dear Sir — I am highly pleased with the treatise on " Familiar 
 Science, or the Scientific Explanation of Common Things," of 
 which you sent me a copy for examination. I have examined the 
 work and find it contains, within a comparatively small compass, a 
 great amount of useful information. It is a work that should be in 
 
the hands of every person, both young and old, who has any desire 
 to become acquainted with the common phenomena of life. 
 Respectfully yours, etc. 
 
 Samuel F. Watson 
 Prin. of Mount Vernon Boys' Grammar School. 
 EloBERT E. Peterson, Esq. 
 
 Robert E. Peterson, Esq. — ^Dear Sir — I have been much gratified 
 by an examination of your book entitled "Familiar Science," and 
 believe its introduction into schools and families would do more in 
 imparting a general knowledge of scientific principles than any trea- 
 tise ever used. 
 
 The cause of every day phenomena, such as evaporation, con- 
 densation, the formation of cloudsj rain, dew, etc., are so familiarly 
 explained, that all classes of persons may readily comprehend 
 them, and I believe the book has only to be known to be appre- 
 ciated by teachers. I am truly yours, 
 
 Wm. Roberts, 
 Prin. of Ringgold Grammar School. 
 
 Philadelphia, May 19, 1851. 
 
 Philadelphia, May 13, 1851. 
 Robert E. Peterson, Esq. — ^Dear Sir — I am highly pleased with 
 your "Familiar Science." As a teacher, I am constantly called 
 upon to answer questions upon almost every conceivable subject, 
 and children wish, and should have correct and intelligent answers. 
 Your two thousand questions and answers, which I consider most 
 admirable, will much facilitate the labors of parents and teachers. 
 Yours truly, 
 
 J. H. Brown, A. M. 
 Prin. of Zane St. Boys' Grammar School 
 
 From Professor James C. Booth, A. M,, M. A., P. S. Author of the Ency- 
 clopedia of Chemistry; Melter and Refiner in the U. S. Mint; Pro- 
 fessor of Applied Chemistry in the Franklin Institute. 
 
 Analytical Laboratory, College Avenue, ) 
 Philadelphia, July 3, 1851. { 
 Dear Sir — I have examined "Familiar Science" with some care, 
 and must express a hearty approval of the manner in which the most 
 "common things" of life are familiarly and clearly explained, 
 without sacrificing the correctness of science. Embracing such 
 questions as are usually put by the developing mind of children, with 
 clear and precise answers, it will relieve parents and teachers of 
 the unhappy necessity of crushing youthful inquiry, while it will 
 tend to nourish a spirit of reflection and investigation in young and 
 old. I commend it as a valuable catechism for schools, and for 
 amusement and instruction at the fire-side. 
 
 Respectfully yours, Jas. C. Booth. 
 Me. R. E. Peterson, Phila. 
 
From Mrs. Phelps, favorably known as the author of various popular 
 Bchool books on Chemistry, Botany, Philosophy, Geology, etc. etc. 
 
 Mr. Peterson. — Sir — In thanking you for the book called 
 "Familiar Science," permit me to express the satisfaction with 
 which it is received by myself and ihe teachers and pupils of this 
 Institution ; having- labored for many years to render Science 
 Familiar, I am glad to meet with this unpretending assistant. It is 
 recommended by the fact that if adopted in a school, it will crowd 
 out no other book, since it is sui generis and may be studied as an 
 amusement. 
 
 With respect, yours, etc. 
 Almira Lincoln Phelps, 
 Patapsco Institute, near Ellicott's Mills , Md. 
 June 30, 1851. 
 
 From the Rev." Father Sourm. 
 
 Philadelphia, July 4, 1851. ) 
 St. John's Cathedral. ) 
 Dear Sir — I comply most willingly with your request, though after 
 the high and well-deserved praise which your valuable work has al- 
 ready received, further commendation would appear unnecessary. 
 An examination of its contents will "satisfy every intelligent reader 
 that it is a treasure of various, useful, and entertaining knowledge, 
 well calculated to promote one of the chief purposes of true 
 education, viz. to make us more familar with those works of God 
 ''so often full of singular beauty and power When, seemingly, most 
 common) in the midst of which " we live and move, and have our 
 being." It will afford me much pleasure to aid in its circulation. 
 Kespectfully, your obed't serv., 
 
 Edw. J. Sourin. 
 fl. E. Peterson, Esq. 
 
 From T. S. Arthur, Editor of the Home Gazette. 
 
 "Familiar Science, or the Scientific Explanation of Common 
 Things," edited and published by Robert E. Peterson, of this city, is 
 one of the most generally useful books that has lately been printed. 
 This work, or a portion of it, came first from the pen of the Rev. 
 Doctor Brewer, of Trinity Hall, Cambridge, but, in the form it first 
 appeared from the English press, it was not only unsuited to the 
 American pupil, but very deficient in arrangement. These defects, 
 the editor has sought to remedy. To give not only to the parent 
 a ready means of answering inquiries, but to provide a good book 
 for schools, is the object of this volume. About two thousand ques- 
 tions, on all subjects of general information, are answered in lan- 
 guage so plain that all may understand it. We had marked a num- 
 ber of these for publication, but must defer their insertion to another 
 number. 
 
 The book is very strongly recommended, and not without good 
 
reason, by competent judges of its merits, and we add our own tes 
 timony in its favor. 
 
 From the Episcopal Recorder. 
 We are here furnished with one of those time and labor-saving 
 volumes which, by judicious arrangement, simple illustration, and 
 accurate definition, greatly facilitate the important effort to impart 
 a familiar knowledge of the facts and principles of natural science. 
 To all interested in the education of youth, it will be found a valu- 
 able auxiliary, while its study, in families and schools, cannot fail 
 to enlarge the domain of Science, by fixing in the mind the elemen- 
 tary principleis of the every day phenomena of life. 
 
 From the Saturday Gazette. 
 One of the most valuable "family library" books we have ever 
 met. The title explains the object of the work, which embraces a 
 wonderful variety of topics cleverly treated. Twenty-five thou- 
 sand copies of the English edition v/ere sold in two years, and the 
 present publisher, also the editor, has made it worthy of as large a 
 circulation, by the clear type and paper and the durable binding. 
 We commend it to all parents or teachers of the young. 
 
 From Fitzgerald's City Item. 
 Occasionally a book is published that fills a void that seems to 
 have been left for it — that exactly furnishes what has been ofteu 
 wished for. Such a work is "Familiar Science." It gives con- 
 cisely, yet clearly, about two thousand explanations of the scientific 
 principles involved in the most frequently recurring operations of 
 life. It is an invaluable book for all whose relation with the young, 
 whether as parent or teacher, demand a facility in explaining what 
 so often attracts the curiosity of childhood. To the general reader, 
 too, it opens an interesting field for observation ; and even to the 
 scholar accustomed to philosophical investigation, it may present 
 familiar applications of what are often regarded as abstruse and un- 
 practical researches. 
 
 From the Saturday Evening Post. 
 Here is a book which might find an appropriate place in every 
 school, and in every house where there are young inquiring minds, 
 athirst for knowledge. It contains nearly two thousand questions, 
 with scientific answers, simply arranged and clearly expressed, con- 
 veying, in a small compass, a great amount of information respect- 
 ing the every-day wonders which puzzle the restless brains of 
 children, and sometimes their elders, too. The publisher and editor 
 merits, and will no doubt receive, the thanks of all parents and 
 
teachers who have experienced (and what instructor of youth has 
 not?) the difficulty of meeting, with satisfactory explanations, the 
 innumerable questions of those under their charge. The book 
 needs only to be known, to be pronounced by a discerning public a 
 rich fund of useful entertainment for the household circle, as well as 
 a valuable aid to instruction in schools. We are not surprised at 
 the statement in the preface, that "Twenty-five thousand copies of 
 the English edition of this work were sold in London in less than 
 two years;" neither shall we be astonished if double that number 
 are sold in the United States in even a shorter space of time. 
 
 From the Banner of the Cross. 
 It is an exceedingly interesting, as well as instructive book for 
 children. It teaches them the reasons of many things, which 
 though common, are yet mysterious; and it also gives many practi- 
 cal hints upon matters which relate to the experience of every-day 
 life. * We can heartily unite with the many strong recommendations 
 from those whose opinion in such matters are especially valuable, 
 in their favorable judgment of this work, as a most excellent book 
 for school or family use. 
 
 From the Christian Observer. 
 This is an excellent and valuable book for the young in families 
 and schools. It opens to the reader a rich repository of science re- 
 specting the most common things — facts and lucid explanations of 
 them which cannot fail to interest and instruct those for whom it is 
 intended. We commend it to the public as a valuable book for the 
 home library and the school. 
 

 
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