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 / 
 
 
 
 >1 
 
 
 
 ,^ te — ^ 
 
 
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 " L. 
 
 
TREATISE 
 
 ON 
 
 ALGEBRA, 
 
 CONTAINING 
 
 THE LATEST IMPROVEMENTS. 
 
 ADAPTED TO THE USE OF SCHOOLS AND COLLEGES. 
 
 BY 
 
 CHARLES W. HACKLEY, S.T.D., 
 
 1'KOrESSOP OK MATHEMATICS AND ASTRONOMY IN COLUMniA COLLEGE, NEW YOKK. 
 
 NEW YORK: 
 
 HARPER & BROTHERS, PUBLISHERS. 
 
 329 & 331 PEARL STREET, 
 
 i 
 
 FRANKLIN SQUA It E. 
 
 18 6 4. 
 
 i 
 
Entered, according to Act of Congress, in the year 1848 r. 
 
 Harper <fe Brothers, 
 
 n the Clerk's Office of the Southern District of New Yor> 
 
 
PREFACE. 
 
 In the preparation of the folio wing work no pains have been spaied 
 to obtair. from the best sources, such as the later treatise Ip >r 
 
 repute, memoirs of scientific bodies, and mathematical journals in 
 English, French, and German, the materials for a book suited to the 
 present state of mathematical science and the wants of teachers and 
 students. 
 
 The work contains much that has never before appeared in an Eng 
 lish dress, and almost every part will be found to present some new 
 feature. No attempt, however, has been made at originality, unless 
 for the benefit of the student, and in the belief that the existing exposi- 
 tions or processes were inferior. The object has simply been, by any 
 and all means, to make the best book, without aiming so much at indi- 
 vidual reputation as at the author's own convenience and that of other.;, 
 devoted, like himself, to the noble task of guiding the youthful votaries 
 of science. 
 
 The French treatises furnish excellent models of the theory of 
 gebra, the German of ingenuity and brevity of notation and exposi- 
 tion, the English of practical adaptation and variety of illustration and 
 example; and from these, after a careful comparison of many authors 
 in each language, demonstrations have been selected and introdu 
 verbatim when they seemed incapable of improvement ; but when- 
 ever the slightest alteration or amalgamation, or the entire remodeling 
 of them, could give additional clearness or elegance, the limas labor 
 has not been spared. 
 
 The work will be found to contain all that is important in the higher 
 parts of Algebra, upon which usually separate treatises are thought 
 necessary, as well as the elementary expositions suited to beginners. 
 Every variety of symbol and of example has been introduced. 
 
 On page XL those articles of this volume are indicated which con- 
 stitute a minimum coivrse of Algebra requisite for the prosecution of 
 the higher branches of mathematics. A more extended course, such 
 as would ordinarily be advisable, is also pointed out. The rest may 
 very well be reserved for reference, as the student's own discovery of 
 
 o tj ^> *<-» <* J 
 
IV PREFACE. 
 
 his wants, in the advanced stages of mathematical pursuit, shall call it 
 in requisition. 
 
 The author desires to acknowledge the effective assistance which lie 
 nas received, in revising the work and superintending it through the 
 press, from Mr. J. J. Elmendorf, to whom it is indebted for many va 1 
 uable suggestions. 
 
CONTENTS. 
 
 Introdcction . 13 
 
 Definitions and Notation . . 1 
 
 ALGEBRAIC CALCULUS. 
 
 Reduction of Terms . 7 
 
 Addition 8 
 
 Subtraction 11 
 
 Multiplication 13 
 
 Multiplication by detached Coefficients 19 
 
 Division 20 
 
 Division by detached Coefficients 31 
 
 Synthetic Division 32 
 
 Greatest common Measure . 34 
 
 Least common Multiple . 40 
 
 ALGEBRAIC FRACTIONS. 
 
 Reduction 43 
 
 Addition . 47 
 
 Subtraction . 47 
 
 Multiplication 49 
 
 Division ... 4° 
 
 POWERS AND ROOTS. 
 
 Powers and Roots of Monomials 
 
 Addition and Subtraction of Radicals 
 
 Multiplication and Division 
 
 Powers and Roots of Radicals . 
 
 Fractional and negative Exponents 
 
 Square of Polynomials 
 
 Square Root of Polynomials 
 
 Cube Root of Polynomials 
 
 Square Root of Numbers . 
 
 A Square Root of a whole Number can not be a Fraction 
 
 Property of prime Numbers 
 
 Square Root of whole Numbers 
 
 Square Root by Approximation 
 
 S( 1 1 tare Root of Fractions . 
 
 Square Root of decimal Fractions 
 
 Cube Root of Numbers 
 
 Fourth Root of Numbers 
 
 Rationalizing binomial Surds 
 
 Square Root of binomial Surds 
 
 Binomial Theorem 
 
 Higher Roots of Numbers . 
 
 Polynomial Theorem . 
 
 Higher Roots of Polynomials 
 
 Fractional Powers of Binomials 
 
 Degree of Approximation of Series 
 
 Roots of imaginary ExpR-.ssions 
 
 51 
 
 59 
 
 60 
 
 61 
 
 65 
 
 74 
 
 75 
 
 81 
 
 82 
 
 92 
 
 82 
 
 84 
 
 87 
 
 88 
 
 89 
 
 90 
 
 95 
 
 97 
 
 98 
 
 100 
 
 107 
 
 108 
 
 109 
 
 111 
 
 115 
 
 117 
 
VI CONTENDS. 
 
 RATIOS AND PROPORTION. Paj. 
 
 Definitions and general Properties • 119 
 
 Propositions in Proportion 123 
 
 Examples in Proportion 128 
 
 EQUATIONS. 
 Preliminary Remarks 129 
 
 SIMPLE EQUATIONS. 
 
 Simple Equations containing' one unknown Quantity 131 
 
 Examples in simple Equations 134 
 
 Cases of Impossibility and Indetermination in simple Equations containing one unknown 
 
 Quantity 142 
 
 Bimple Equations containing two or more unknown Quantities 143 
 
 Examples 144 
 
 General Formulas of Elimination 155 
 
 Problems producing simple Equations 158 
 
 Negative, indeterminate, and infinite Solutions 173 
 
 Discussion of Formulas furnished by general Equations of the first Degree, with two 
 
 or more unknown Quantities 178 
 
 Problem of the Couriers 181 
 
 Additional Problems in simple Equations . . 183 
 
 Indeterminate Analysis of the First Degree 186 
 
 Problems in indeterminate Analysis . ... 191 
 
 QUADRATIC EQUATIONS. 
 
 Definitions, Divisions • 199 
 
 Pure Quadratics containing one unknown Quantity 200 
 
 Examples in pure Quadratics £01 
 
 Pure Equations of higher Degree 202 
 
 Examples of pure Equations 203 
 
 Complete Quadratics containing one unknown Quantity 204 
 
 Examples in complete Quadratics 205 
 
 Solution of Quadratics by completing the Square 208 
 
 Examples 209 
 
 Quadratics containing two unknown Quantities 218 
 
 Examples 219 
 
 Problems producing pure Equations 224 
 
 Problems producing complete Quadratics -25 
 
 General Discussion of the Equation of the second Degree 228 
 
 Problem of the Lights 232 
 
 Problems solved by Quadratics involving two or more unknown Quantities . . 234 
 
 Decomposition of Trinomials of the second Degree into Factors of the first Degree . 239 
 Indeterminate Analysis of the second Degree .... 240 
 
 Maxima and Minima .... 2-12 
 
 The Modulus of imaginary Quantities 242 
 
 Method of Mourey for avoiding imaginary Quantities 244 
 
 PERMUTATIONS AND COMBINATIONS. 
 
 Definitions 246 
 
 General Formulas . . n - !■? 
 
 Examples 248 
 
 Variations of the general Formulas . 24!) 
 
 Different S] of Notation 250 
 
 Restricted Permutations and Combinations of Numbers .25) 
 
 Calculus of Probabilities ... 259 
 
CONTENTS. Vll 
 
 METHOD OF UNDETERMINED COEFFICIENTS. Fag» 
 
 General Theorem 255 
 
 Decomposition of Fractious 256 
 
 Examples . 257 
 
 LOGARITHMS. 
 
 Definitions and Calculation of Tables 258 
 
 Geueral Properties of Logarithms ... 261 
 
 Description and Use of Tables .... 262 
 
 Examples of the Application of Logarithms 264 
 
 Arithmetical Complement 264 
 
 Exercises in Logarithms 2CG 
 
 Gauss's Logarithms for Sums and Differences 268 
 
 Examples 268 
 
 Effect of different Values of the Baso 26d 
 
 Solution of exponential Equations by Logarithms 269 
 
 Theorems in Logarithms 270 
 
 Exponential Theorem S72 
 
 Series for computing Logarithms 273 
 
 Method of calculating Napierian and common Logarithms 274 
 
 Examples 275 
 
 PROGRESSIONS. 
 
 Arithmetical Progression 27b 
 
 Examples 277 
 
 Ten Formulas in Arithmetical Progression . . ... . 278 
 
 Geometrical Progression , 278 
 
 Examples 280 
 
 Account of the Origin of Logarithms from Progressions 282 
 
 Ten Formulas in Geometric Progression 284 
 
 Harmonical Progression 284 
 
 INTEREST AND ANNUITIES. 
 
 Simple Interest 285 
 
 Present Value and Discount at Simple Interest 286 
 
 Annuities at Simple Interest 287 
 
 Compound Interest 288 
 
 Present Value and Discount at Compound Interest 291 
 
 Annuities at Compound Interest 291 
 
 Reversion of Annuities 292 
 
 Purchase of Estates 292 
 
 Reversion of Perpetuities 293 
 
 Examples for Practice 293 
 
 INTERPOLATION. 
 
 Method of first Differences 294 
 
 Method of second and higher Orders of Differences 295 
 
 Derivation of Formula for higher Orders of Differences by ihe Method of undetermined 
 
 Coefficients 296 
 
 Example of Application to Tables .... 298 
 
 INEQUATIONS. 
 
 Theorems . 298 
 
 Examples in Inequations ■ 800 
 
'Ill CONTENTS. 
 
 GENERAL THEORY OF EQUATIONS. 
 
 NATURE AND COMPOSITION OF EQUATIONS. Page 
 
 Definitions ... 302 
 
 \?f(x) be divided by x — a, tbe Remainder will be /(«) 302 
 
 The first Member of an Equation divisible by tbe Difference between the nuknown 
 
 Quantity and a Root 303 
 
 Every Equation has a Root 303 
 
 An Equation containing one unknown Quantity has as many Roots as there are Units 
 
 in its Degree ... 308 
 
 Relation between the Roots and Coefficients of an Equation 309 
 
 Equations whose Coefficients are whole Numbers; that of the highest Power being 
 
 Unity, can not have Fractional Roots 312 
 
 Changing the Signs of the alternate Terms changes the Signs of the Roots . . . 312 
 
 Surds and impossible Roots enter an Equation by Pairs 313 
 
 All the Roots of an Equation must be of the Form a-\-b \/ — 1 314 
 
 The Roots of two conjugate Equations will be Conjugates of each other . . . 314 
 
 DEPRESSION OR ELEVATION OF THE ROOTS OF EQUATIONS. 
 
 Equations whose Roots are those of the proposed, increased or diminished by a given 
 Quantity 315 
 
 Numbers between the Roots substituted for the unknown Quantity give results alter- 
 nately Positive and Negative 319 
 
 Equation whose Roots separate those of the proposed 320 
 
 Equal Roots ' 321 
 
 NUMBER OF REAL AND IMAGINARY ROOTS IN AN EQUATION. 
 
 Theorem of Sturm 322 
 
 Examples 328 
 
 Horner's Method of resolving numerical Equations of all Orders 332 
 
 Examples 333 
 
 Conditions of Reality of Roots from Sturm's Theorem 340 
 
 Rule of Des Cartes 341 
 
 Theorem of Rolle . ' 342 
 
 Fourier's Method of separating the Roots 345 
 
 Examples 349 
 
 TRANSFORMATION OF EQUATIONS. 
 
 To transform an Equation into another whose second Term shall be removed . . 351 
 To transform an Equation into another whose Roots shall be the Reciprocals of those 
 
 of the proposed 352 
 
 To transform an Equation into another whose Roots shall be any Multiple or Submul- 
 
 tiple of those of the proposed 35. 
 
 To transform an Equation into another whose Roots shall be the Square of those of the 
 
 proposed 355 
 
 To transform an Equation into another wanting any given Term 356 
 
 To transform an Equation into another whose Roots are the Squares of the Differences 
 
 of those of the proposed 357 
 
 Budan's Criterion .... 3.">9 
 
 Degua's Criterion 36 1 
 
 LIMITS OF THE ROOTS OF EQUATIONS. 
 
 Superior and inferior Limits of the Roots 362 
 
 Newton's Method of finding the Limits . . . 305 
 
 Wariug's or Lagrange's Method of separating the Roots . . . .366 
 
 APPROXIMATION TO THE ROOTS. 
 
 Newton's Method 3C9 
 
 Method of Lagrange by continued Fractions . . . r;~a 
 
CONTENTS. 
 
 IX 
 
 BINOMIAL EQUATIONS. 
 
 When the Exponent is a composite Number . 
 Solution of particular Cases 
 Preparatory Propositions .... 
 Trigonometrical Solutions .... 
 Multiple Value of Radicals 
 
 375 
 376 
 
 378 
 37S 
 383 
 
 / 
 
 DETERMINATION OF THE IMAGINARY ROOTS OF EQUATIONS. 
 
 Limits of Moduli of imaginary Roots 334 
 
 Lagrange's Method of determining imaginary Roots by Elimination .... 385 
 Example 38S 
 
 Theory of vanishing Fractions . . 39C 
 
 ELIMINATION. 
 
 Resolution of Eqaat'.ons containing two or more unl.^own Quantities of any Degree 
 
 whatever 392 
 
 SimpliGcation 394 
 
 Method of Labatie 397 
 
 Euler's Method 404 
 
 Degree of the final Equation 403 
 
 EXPONENTIAL EQUATIONS. 
 
 Solution by continued Fractions 407 
 
 By Logarithms or Double Position 407 
 
 Examples . . • 408 
 
 DEMONSTRATION OF BINOMIAL THEOREM FOR ALL CASES. 
 
 When the Exponent is a whole Number 408 
 
 When a Fraction 408 
 
 When Negative either entire or fractional 409 
 
 SERIES. 
 
 RECURRING SERIE8. 
 
 Generation of recurring Series 410 
 
 Return from recurring Series. to generating Fraction 411 
 
 To determine whether a Scries be recurring 412 
 
 To find the general Term of a recurring Series 414 
 
 Summation of Series '. 415 
 
 Difference Series 416 
 
 To separate the Roots of an Equation by Means of difference Seriea .... 417 
 
 The differential Method of summing Series 419 
 
 Powers of the Temis of Progressions ... 420 
 
 Piling of Balls and Shells 422 
 
 Variation 425 
 
 SYMMETRICAL FUNCTIONS. 
 
 Definitions .... 427 
 
 To find the Sums of the like and entire Powers of the Roots of an Equation . . 428 
 
 To determine Double, Triple, &c, Functions 430 
 
 Every rational and symmetric Function of the Root of an Equation can be expressed 
 
 rationally by its Coefficients 431 
 
 Use of symmetric Functions in the transformation of Equations 431 
 
 Solution of the Equation :>f the Squares of the Differences ' . 430 
 
 .An analogous Method fo a great number of Cases . . ... 433 
 
A CONTENTS. 
 
 Page 
 
 Quadratic Factors of Equations . • 433 
 
 Elimination by Symmetric Functions . • • • 436 
 
 GENERAL SOLUTION OF EQUATIONS. 
 
 Method of Tschirnhauseu for solving Equations ... .... 43a 
 
 Method of Lagrange 439 
 
 Examples 441 
 
 GENERAL EQUATIONS OF THE THIRD AND FOURTH DEGREES. 
 
 Resolution of the E C[uation of the third Degree by the Method of Cardan . . 444 
 
 Irreducible Case 446 
 
 Solution of the irreducible Case by Trigonometry 41" 
 
 Solution of the reducible Case by Trigonometry 449 
 
 Woolley's Method of resolving the Cubic Equation 449 
 
 Irrational Expressions analogous to those obtained in the Resolution of Equations of 
 
 the third Degree 452 
 
 Resolution of the Equation of the fourth Degree 455 
 
 / 
 
 THE DIOPHANTINE ANALYSIS. 
 
 Introductory Remarks . . 457 
 
 Examples . ...... 458 
 
 Questions fof Exercise . ... 467 
 
 THEORY OF NUMBERS. 
 
 Elementary Propositions .... 468 
 
 The Forms and Relations of integral Numbers, and of their Sums, Differences, and Prod- 
 ucts 469 
 
 Definitions 470 
 
 Divisibility' of Numbers 471 
 
 To find all the Divisors of any Number whatever 472 
 
 To form a Table of prime Numbers ( 472 
 
 To decompose a Number into prune Factors, and to find afterward all its Divisors . 473 
 To determine how many Times a prime Number is Factor in a Series of natural Num- 
 bers 475 
 
 Determination and expression of perfect Numbers 476 
 
 To find a Pair of amicable Numbers 476 
 
 Congruous Numbers in general. — Definitions v 477 
 
 Theorems with regard to congruous Numbers 477 
 
 No Algebraic Formula can contain prime Numbers only 479 
 
 Other Theorems with regard to prime Numbers 480 
 
 Primitive Roots 483 
 
 Theorem of Fermat . 483 
 
 Table of primitive Roots 484 
 
 The Forms of square Numbers 484 
 
 CONTINUED FRACTIONS. 
 
 Definitions . • .... 486 
 
 Rule for converting an irreducible Fraction into a continued one 487 
 
 Convergents 488 
 
 Periodic continued Fractions 491 
 
 To develop any Quantity in a continued Fraction . 493 
 
 Examples 494 
 
 The Root of a quadratic Equation may be expressed in function of the Coefficients by 
 
 means of continued Fractions 494 
 
 Resolution of the indeterminate Equation of tho first Degree by means of continued 
 
 Fractions ... 496 
 
 Method of resolving in rational Numbers indeterminate Equations of the second Degree 497 
 Gauss's Mo hod of solving binomial Equations . 501 
 
A MINIMUM COURSE OF ALGEBRA. 
 
 Articles* 1-4 inclusive, 6-7, 9-page 17, Art. 15-p. 26, Art. 32-46, 48- 
 p. 60, Art. 63-p. 62, Art. 78-p. 77, Art. 83-90, 105-110, 119-128, 130-133 
 XVI., 134-143, p. 138, 139, Art. 145-p. 150, Arts. 150, 151, 178-186. 
 
 A MORE ENLARGED COURSE. 
 
 Articles 1-93 inclusive, 101-197, 199-238, 244-258, 298-309, 315-321. 
 
 It may be useful to point out in this connection a course of mathematica 
 study. 1°. Algebra; 2°. Geometry :f these two may ba pursued simultane- 
 ously; 3°. Plane Trigonometry, with its applications to Surveying and Navi- 
 gation ; Spherical Trigonometry, with its applications to Practical and Nautical 
 Astronomy and Geodesy ;J 4°. Descriptive Geometry ;§ 5°. Analytical Ge- 
 ometry ;|| 6°. Theoretic Astronomy ;T 7°. Differential and Integral Calculus 
 and Calculus of Variations ;** 8°. Mechanics ;ff 9°. Optics ;%t 10°. Phys- 
 ical Astronomy. §§ 
 
 * The articles are numbered throughout the book at the beginnings of paragraphs. 
 
 t A treatise ou Geometry, compiled from the latest and best foreign sources, has been 
 published by the author. 
 
 t The author has already published a work embracing these subjects, a new and greatly 
 improved edition of which will appear in August. 
 
 § This branch, though it,may be omitted without destroying the connection between those 
 which precede and follow it, is of the highest advantage to the general student, and invalua- 
 ble to the engineer. It may be best taken up in the excellent treatise by Professor Davies. 
 In the French, Monge, the founder of the science, has written extensively upon the sub 
 ject; there is also a treatise by that best of French writers of elementary works, Lefebure 
 de Fourcy. Professor DSvies has published a line volume on the application of descriptive 
 geometry to shadows and perspective. 
 
 || On this subject there are numerous writers, Davies, Pierce, and Young, whose work is 
 republished here, the author of a treatise in the Library of Useful Knowledge; and in the 
 French, among the best, Biot, of whom there is an English translation by Professor Smith, 
 of Virginia, and Lefebure de Fourcy, whose work is most generally preferred. A work on 
 this subject, by the author, may be expected to appear iu the course of the next twelve 
 months. 
 
 1T The authors recommended are Norton, Gummery ; and in the French, Biot, of whom 
 there is a translation in part, known as the Cambridge Astronomy. 
 
 ** This is one of the portions of mathematical science on which the author proposes to 
 put forth a treatise at no distant day. We have at present on the calculus, Church and 
 Davies, in America; Young, O'Brien, and Walton, in England; Lacroix, Duhamel, and 
 Moiguo, who may be mentioned among the numerous writers in France. 
 
 tt Courtenay's Boucharlat; in French, Fraueceur and Poisson. 
 
 XX Bache, Brewster, Bartlett, Biot, and Jackson. This branch may be pursued to some 
 extent immediately after Geometry. 
 
 §§ The authors are Lagrange and Laplace, of whose Mecanique Celeste we have the 
 translation and notes of Bowditch, but for readers of the French, the Systeme du Monde 
 of Pontecouland is to be preferred. 
 
As Greek letters are frequently used in the following treatise, lor 
 the convenience of those unaccostunied to a Greek alphabet, one is 
 here inserted. The names of the letters are given in the last column 
 
 A 
 
 a 
 
 a 
 
 "kXtpa 
 
 Alpha 
 
 B 
 
 P,6 
 
 b 
 
 B7]ra 
 
 Beta 
 
 r 
 
 y 
 
 g 
 
 Tdfifia 
 
 Gamma 
 
 a 
 
 6 
 
 d 
 
 AeXra 
 
 Delta 
 
 E 
 
 e 
 
 e short 
 
 v Et/>tA6v 
 
 Epsilon 
 
 Z 
 
 s 
 
 z 
 
 Zf/ra 
 
 Zeta 
 
 H 
 
 n 
 
 e lonof 
 
 T Hra 
 
 Eta 
 
 e 
 
 $ e 
 
 th 
 
 Qrjra 
 
 Theta 
 
 i 
 
 i 
 
 i 
 
 'iWTft 
 
 Iota 
 
 K 
 
 K 
 
 k 
 
 Kdmra 
 
 Kappa 
 
 A 
 
 X 
 
 1 
 
 \dfi66a 
 
 Lambda 
 
 M 
 
 v> 
 
 m 
 
 Mil 
 
 Mu 
 
 N 
 
 V 
 
 n 
 
 Nv 
 
 Nu 
 
 m*i 
 
 i 
 
 X 
 
 Zl 
 
 Xi 
 
 
 
 
 
 o short 
 
 "0[iiKp6v 
 
 Omicron 
 
 IT 
 
 7T 
 
 P 
 
 m 
 
 Pi 
 
 P 
 
 P 
 
 r 
 
 Tw 
 
 Rho 
 
 2 
 
 o, ? 
 
 s 
 
 1.iyfia 
 
 Si<mia 
 
 T 
 
 T 
 
 t 
 
 Tai) 
 
 Tau 
 
 T 
 
 V 
 
 u 
 
 T T xpadv 
 
 Upsilon 
 
 # 
 
 <P 
 
 ph 
 
 $Z 
 
 Phi 
 
 X 
 
 X 
 
 ch 
 
 Xt 
 
 Chi 
 
 * 
 
 V, 
 
 ps 
 
 i'i 
 
 Psi 
 
 3 
 
 (j 
 
 o long 
 
 'Qfieya 
 
 Omega 
 
INTRODUCTION. 
 
 In every question of numbers there are certain conditions which tne 
 required numbers in connection with the given ones must fulfill, whict 
 conditions are indicated by the question itself. 
 
 The solution has for its object to determine such required quantities 
 as will verify these conditions. It is necessary, therefore, to endeavor 
 first to seize the different relations by which all the quantities; known 
 and unknown, are connected together, and to find afterward, by means 
 of these relations, what opei'ations ought to be performed upon the 
 given quantities to obtain those which are required. Such is the ob- 
 ject proposed in that part of mathematics known by the namo of Al- 
 gebra. 
 
 To show how the use of letters and sisrns arises, let the following 
 simple problem be propose k 
 
 To divide 890 dollars between three persons in such a manner that the 
 second shall have 115 more than the first, and the third ISO more than 
 the second. 
 
 Now let us see by what deductions the values of the unknown num- 
 bers may be derived. 
 
 Since the share of the second is 115 more than that of the first, and 
 the share of the third 180 more than that of the second, it will be 180 
 added to 115, or 295 more than that of the first. 
 
 Then the sum of the tln-ee parts will be formed of 3 times the first 
 part, increased by 115, and also by 295, or, what is the same thing, of 
 3 times the first part inci'cased by 410. 
 
 But this is equal to the sum to be divided, viz., 890. 
 
 Then 3 times the first part, increased by 410, is equal to S90. 
 
 Then 3 times the first part is equal to S90 diminished by 410, or 480 
 
 Then the first part will equal the third of 480, or 160 dollars. 
 
 The first person, therefore, has 160 dollars ; the second, who must 
 nave 115 more, will have 275; and the third, who was to have ISO 
 more than the second, 455 dollai - s. These three sums united make 
 S90 do'lars, which confirms the correctness of the result. 
 
 This example exhibits the kind of reasonings necessary to be em- 
 ployed in the solution of problems in numbers; and it will be per- 
 
XiV INTRODUCTION. 
 
 ceived that, to express these reasonings, it is necessary to repeat tie 
 quently a number of words, designating the quantities, both known and 
 unknown, as xhejirst part, the number to be divided, &c, and other words 
 expressing the relations of these, as increased by, diminished by, &e. 
 
 To obviate the inconvenience of the periphrases, by means of which 
 the quantities which enter into the question are distinguished, it is cus- 
 tomary to represent these quantities by letters. Ordinarily, the giveu 
 quantities are represented by the first letters of the alphabet, a,b,c . . . , 
 and the required or unknown by the last, x, y, z . . . 
 
 The relations are expressed by signs. Thus, increased by is written 
 + ; diminished by is written — ; multiplied by is written X ; or, a mul- 
 
 iplied by b, simply thus, ab ; a divided by b, thus, j; a equal to h, 
 
 vus, a-=b. 
 
 The reasoning of the above example may, with the aid of such 
 
 ibridgments, if x denote the first share, be written briefly thus : 
 
 x 
 
 x+115 
 
 3+115 + 180 
 
 3x + 410 = 890 
 
 3z = 890— 410 
 
 3z = 480 
 
 480 ' 
 x =— =160 
 
 If the numbers had been different in the above problem, the method 
 "proceeding would have been precisely the same. 
 
 Thus, if 1250 had been the sum to be divided, 170 the excess of the 
 second part over the first, and 220 the excess of the third over the sec- 
 ond, the reasoning would have had the same form, as seen below. 
 
 x share of the 1st, 230 
 
 x + 170 170 
 
 .-r+170+220 share of the 2d, 400 
 
 3a;+560 = 1250 220 
 
 3x= 1250— 560 share of the 3d, 620 
 
 3a; = 690 Proof. 
 
 690 230 
 
 x = —r- = 230 
 
 400 
 
 620 
 
 1250 
 
 All these individual cases of the same kind may be generalized, thus : 
 Let a represent the number to be divided ; b the excess of die second 
 over the first share ; c that of the third over the second. The reason 
 mrr will then stand as follows : 
 
INTRODUCTION. *» 
 
 » 
 
 X 
 
 z+b 
 
 x+&-f-c 
 
 3x+2b+c=a 
 
 3x—a — 2b — c 
 
 a — 2b — c 
 
 x- 
 
 The last expression, x=. , shows what operations ought to be 
 
 performed upon the given numbers to produce the required, and may 
 be interpreted into the following rule. 
 
 Subtract double the excess of the second share over the first, together 
 with the excess of the third over the second, from the number to be divided, 
 and divide the remainder by 3. The result will be the first share re- 
 quired. 
 
 Applying this rule to the first case above, we have 
 
 115x2=230 890 and to the 2d, 170 
 
 180 2 
 
 ' " 410 340 
 
 3)480 220 1250 
 
 160 Ans. 560 
 
 3)690 
 ~230 Ans. 
 
 The expression x=. , from which the rule to be applied is 
 
 derived, is called a general formula, or simply a formula from which, 
 instead of from the rule, the answers in the particular cases may be 
 obtained by substitution ; thus, 
 
 in the 1st case, in the 2d case, 
 
 890—230—180 4S0 „ rtrt 1250—2x170—220 690 
 
 z— =——=160, x— = =230 
 
 3 3 3 3 
 
 The nature and utility of algebra being thus briefly indicated, we 
 proceed to give in detail, first, the methods of representing quantities, 
 and all possible relations and combinations of them, and afterward the 
 use of these methods in the solution of questions,, 
 
ALGEBRA, 
 
 DEFINITIONS AND NOTATION. 
 
 1. Algebra is a species of short-hand writing which, by the aid of certain 
 symbols, serves to abridge and generalize propositions relating to numbers.* 
 
 A Proposition is any thing propounded as true. If it express the proper- 
 ties or relations of quantity, it is a mathematical proposition. If it be self- 
 evident, it is called an axiom. If it require demonstration, it is called a theorem ; 
 and if it propose something to be done, or that some required or unknown 
 quantity be found, it is called a problem. 
 
 Symbols may be divided into symbols of quantity, and symbols of relation 
 commonly called signs. 
 
 2. The principal symbols employed in algebra are the following : 
 
 I. The letters of the alphabet, a, b, c, &c, which are employed to denote 
 the numbers which are the object of our reasonings. 
 
 When the Roman letters are exhausted, or when a marked distinction is de 
 sirable between the different classes of quantities employed, the Greek letters 
 are also used as representatives of quantity. If different quantities of the same 
 general nature are used together, it is a common custom to represent tliem by 
 the same letter, distinguishing them from one another by accents, or small 
 numbers written below ; thus, a, a', a", a'", a iv , are representatives of differ 
 ent quantities, and are read a, a prime, a second, &c. ; and a u Og, a 3 , &c, 
 may be read a one subscript, a two subscript, and so on. 
 
 A similar effect is produced by using large and small letters ; thus, the di- 
 ameter of a small circle being represented by d, that of a larger may be by D. 
 
 It is customary, in some cases, to represent quantities by symbols, which 
 indicate distinctly the nature of the quantities represented. Thus, the six 
 trigonometrical quantities, which are known by the names of sine, tangent, 
 secant, cosine, cotangent, cosecant, are represented by the symbols sin, tan, 
 sec, cos, cot, cosec ; and the astronomical quantities, the longitude of the 
 sun, the longitude of the moon, and the longitude of a node, are represented 
 by the symbols O , }) , and t? • 
 
 * In the operations of Arithmetic, with the exception of those which relate to compound 
 numbers, quantities are considered as composed of units, but the kind of unit is not noticed, 
 only the number. In Algebra, neither the kind nor number of units of which a quantity 
 is composed is regarded, and often the quantity is not considered as composed of units at 
 all. The idea of number may, however, always be introduced, and it is best to keep it in 
 mind in the beginning of Algebra. As in Arithmetic the rules of addition, multiplication, 
 proportion, &c, are the same, whatever be the kind of units which the numbers employed 
 represent, so in Algebra these rules are the same, whatever be either the kind or num- 
 ber of units in the quantities employed (upon which the operations are performed). In 
 every part of Algebra, processes analogous to those prescribed by the rules of Arithmetic 
 are in use. Hence, and because of its character of generalization, it was called by New 
 ton General Arithmetic. Algebra, however, presents many relations of quantity of which 
 Arithmetic takes uo cognizance. 
 
 A 
 
2 ALGEBRA. 
 
 These are the symbols of quantity. 
 The following are symbols of relations : 
 
 II. The sign -}-, which is named plus', and is employed to denote the addi 
 tion of two or more numbers. 
 
 Thus, 12-J-30 signifies 12 plus 30, or, 12 augmented by 30. In like manner, 
 a -f- b signifies a plus b, or, the number designated by a augmented by the 
 number designated by b. 
 
 III. The sign — , which is named minus, and is employed to denote the 
 subtraction of one number from another. 
 
 Thus, 54 — 23 signifies 54 minus 23, or, 54 diminished by 23. In like man- 
 ner, a — b signifies a minus b, or, the number designated by a diminished by 
 the number designated by b. 
 
 The sign ~ is sometimes employed to denote the difference of two num- 
 bers, when it is not known which is the greater. Thus, a~& signifies the 
 difference of a and b, when it is not known whether the number designated by 
 a be less or greater than the number designated by b. 
 
 IV. The sign X > which may be read into, is employed to denote the multi- 
 plication of two or more numbers. 
 
 Thus, 72 X 26 is read 72 into 26, or, 72 multiplied by 26. In like manner, 
 a X b signifies a into b, or, a multiplied by b ; and a X b X c signifies the con 
 tinued product of the numbers designated by a, b, c ; and so on for any num- 
 ber of factors. 
 
 The process of multiplication is also frequently indicated by placing a point 
 between the successive factors ; thus, a .b . c . d signifies the same thing as 
 axbxcxd. 
 
 In general, however, when numbers are represented by letters, their multi- 
 plication is indicated by writing the letters in succession, without the interpo- 
 sition of any sign. Thus, ab signifies the same thing as a . b, or a X b ; and 
 abed is equivalent to a . b . c . d, or a X b X c X d. 
 
 Factors expressed by letters are called literal factors, and those expressed 
 by numbers numerical factors. 
 
 It must be remarked, that the notation a . b, or ab, can be employed only 
 when the numbers are designated by letters; if, for example, we wished to rep- 
 resent the product of the numbers 5 and 6 in this manner, 5 . 6 would be con 
 founded with an integer followed by a decimal fraction, and 56 would signify 
 the number ffty-six, according to the common system of notation. 
 
 For the sake of brevity, however, the multiplication of numbers is some 
 
 times expressed by placing a point between them in cases where no ambiguity 
 
 can arise from the use of this symbol. Thus, 1.2.3.4, may represent the 
 
 ,276 
 continued product of the numbers 1, 2, 3, 4 ; and - . - . — may represent 
 
 2 7 6 
 
 the product of -x, -, and — . 
 
 V. The sign -4-, which is named by, and whon placed between two num- 
 bers is employed to denote that the former is to be divided by the latter. 
 
 Thus, 244-6 signifies 24 by 6, or, 24 divided by 6. Id ike manner, a~b 
 signifies a by b, or, a divided by b. 
 
 Two dots without the horizontal line between are also the sign of division. 
 This form of the sign is used in proportions, where either of the two quantities 
 
DEFINITIONS AND NOTATION. 3 
 
 between which it is placed may be regarded as the dividend, and the other tne 
 divisor. It is analogous, in this respect, to the sign ~ in subtraction. 
 
 In general, however, the division of two numbers is indicated by writing the 
 dividend above the divisor, and drawing a line between them. Thus, 21-^-G 
 
 j 7 24 i a 
 
 and a-^-b are usually written — and r. 
 
 J b b 
 
 Every fraction, then, expresses the quotient of its numerator, divided by its 
 denominator. Thus, f of a unit may be regarded as composed of two parts : 
 the one, the third of one unit, and the other, the third of another unit ; or 
 both together, the third of 2 units, or the quotient of 2 divided by 3. This 
 reasoning may be generalized. 
 
 VI. The sign =, called the sign of equality, and read is equal to, when 
 placed between two numbers denotes that they are equal to each other. 
 
 Thus, 56-\-6=62 signifies that the sum of 56 and 6 is equal to 62. In like 
 manner, a = £> signifies that a is equal to b, and a-\-b=c — d signifies that a 
 plus b is equal to c minus d, or that the sum of the numbers designated by a 
 and b is equal to the difference of the numbers designated by c and d. 
 
 VII. The sign <^, which is read is unequal to, and when placed between 
 two numbers denotes that one of them is greater than the other, the opening 
 of the sign being turned toward the greater number. 
 
 Thus, a>6 signifies that a is greater than b, and a<& signifies that a is 
 less than b. 
 
 VIII. The coefficient is a sign which is employed to denote that a number 
 designated by a letter, or some combination of letters, is addnd to itself a cer- 
 tain number of times. 
 
 Thus, instead of writing a+a-j-a-f-a-f-a, which represents 5 a's added 
 together, we write 5a. In like manner, 10a& will signify the same thing as 
 ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab, or ten times the product of 
 a and b. 
 
 The numbers 5 and 10 here are coefficients. 
 
 The coefficient, then, is a number, -written to the left of another number 
 represented by one or more letters, and denotes the number of times that the 
 given letter, or combination of letters, is to be repeated. 
 
 Or the coefficient is the numerical factor written before one or more literal 
 factors. 
 
 When no coefficient is expressed, the coefficient 1 is always understood , 
 thus, la and a signify the same thing. 
 
 In a more enlarged sense, one literal factor may be regarded as the coeffi- 
 cient of another, especially when the former is one of the first, and the latter 
 one of the last letters of the alphabet. Thus, in the expression ax, a may be 
 called the coefficient of x. So, also, in the expression of abxy, ab may be re . 
 garded as the coefficient ofxy. 
 
 IX. The exponent, or index, is a sign which is employed to denote that a 
 number designated by a letter is multiplied by itself a certain number of times. 
 
 Thus, instead of writing ax aXaXaX a, or aaaaa, which represents 
 five a's multiplied together, we write a 5 , where 5 is called the exponent or 
 index of a. Similarly, &X&X&X&X&X< , 'X&X&X&X&» or b .b .b. 
 b . b. b . b . b . b . b, or bbbbbbbbbb ; or the continued product of 10 i's is written 
 more briefly b 10 , where 10 is the exponent or index of b. 
 
 The exponent or index of a number is, therefore, a number *ritten a little 
 
4 ALGEBRA. 
 
 above a letter to the right, and denotes the number of times which the number 
 designated by the letter enters as a factor into a product. When no exponent 
 is expressed, the exponent 1 is always understood ; thus, a 1 and a signify the 
 same thing. 
 
 The products thus formed by the successive multiplication of the same 
 number by itself, are in general called the powers of that number. Thus, a ia 
 the first power of a ; aXa=aa=a 2 is (he second power of a, or the square 
 of a ; aaa=a 3 is the third power, or cube of a ; aaaaa=a 5 is the fifth powet 
 
 of a, and aaaa ton factors = a n , is the nth power of a, or the powei 
 
 of a designated by the number n. 
 
 X. The square root of any expression is that quantity which, when multi- 
 plied by itself, will produce the proposed expression, and is generally denoted 
 by the symbol sf , which is called the radical sign. Thus, the square root 
 of 9 is -v/9=3, and -\/a°=a, is the square root of a"; for in the former case 
 3x3=9, and in the latter aX«=« 2 « 
 
 XI. The cube root of any expression is that quantity which, when multi- 
 Dlied twice by itself, -will produce the proposed expression. The fourth, or 
 biquadrate root of any expression is that quantity which, when multiplier' 
 three times by itself, produces the given expression ; and the nth root of any 
 expression is that quantity which, multiplied (n — 1) times by itself, produces 
 the proposed expression. Thus, the cube root of 8 is 2; for 2x2X2=8, 
 the fourth root of a* is a ; for a . a . a . a=a 4 , and the nth root of x n is x ; for 
 x X x X x . . . . to n factors =x .x.x .x .... ton factors = x n : 
 
 The roots of expressions are frequently designated by fractional or decimal 
 exponents, the figure in the numerator of the fractional exponent denoting the 
 power to which the expression is to be raised or involved, and the figure in 
 the denominator denoting the root to be extracted or evolved. Thus, the 
 symbol of operation for the square root of a is either -y/ a or a- ; for the cube 
 root it is y a, or a*; for the fourth root, $/a, or a*; and y/a, or a 5 , denotes 
 the 7i th root of a. Also, %/a 5 , or a% denotes the sixth root of the fifth power 
 
 m 
 
 of a ; and a a , or V flm > signifies the wth root of the with power of a.* 
 
 XII. A rational quantity is one which can be expressed without a radical 
 sign or fractional exponent, as 3mn, or bx^y*. 
 
 XIII. An irrational quantity is a root which can not be exactly extracted, 
 and is expressed by means of the radical sign -\/, or a fractional exponent, as 
 
 V2 yd*, or x*yK 
 
 XIV. The reciprocal of any quantity is unity divided by that quantity; 
 
 i • , 1 1 1 1 , 
 
 thus, the reciprocals of a?, X s , y 5 , z*, are respectively - 3 , — , -5, —^ ; but the 
 
 following notation is generally used, as being more commodious : thus, the 
 fractions — % , 3 -5, -j, are expressed by a~% x~ s , y~ 5 , z~l* 
 
 It will follow from the above, and from the rule for division of fractions, that 
 the reciprocal of a fraction is the fraction inverted. Thus, the reciprocal of 
 
 a . 1 b 
 
 r is — =-. 
 baa 
 
 * The subject of fractional and negative exponents will lie filly investigated farther in 
 advance. 
 
DEFINITIONS AND NOTATION. 5 
 
 XV. The following characters are used to connect several quantities to 
 gether. viz. : 
 
 vinculum, or bar 
 
 parentheses ( ) 
 
 braces, or brackets ) > or 
 
 Thus, m-\-?i.x, or (m-\-n)x signifies that tho quantity denoted by m-f-n is 
 to be multiplied by .r, and $ 2+jj I . $ £— p - I signifies that ^+? is to be multi- 
 plied by ^—-. The vinculum or bar is sometimes placed vei'ac-ly ; thus, 
 
 -{-ax 
 
 + c 
 signifies that the sum of a, b, and c is multiplied by x. 
 
 XVI. The signs, .-. therefore or consequently, and ••• because, are used to 
 avoid the frequent repetition of these words. 
 
 XVII. Every number written in algebraic language, that is, by aid of 
 algebraic symbols, is called an algebraic quantity, or, an algebraic expression. 
 
 Thus, 3a is the algebraic expression for three times the number a ; 5a 2 is 
 the algebraic expression for' five times the square of the number a ; 7a r 'b 3 is 
 the algebraic expression for seven times tho fifth power of a multiplied by the 
 cube of b. 
 
 3a 2 —6b 5 c 4 is the algebraic expression for the difference between three 
 times the square of a and six times the cube of b multiplied by the fourth 
 power of c. 
 
 2a—3¥c 3 - x -Ad 4 e t f s is the algebraic expression for twice a, diminished 
 by three times the square of b multiplied by the cube of c and augmented by 
 four times the fourth power of d multiplied by the product of the fifth power 
 of e and the sixth power of/. 
 
 XVIII. An algebraic quantity, which is not combined with any other by 
 the sign of addition or subtraction, is called a monomial, or monome, or, a quantify 
 of one term, or simply, term. Thus, 3a 2 , 4b", 6c, are monomials. The de- 
 gree of a term is the number of its literal factors, and is found by adding to- 
 gether the exponents of all the letters contained in the term. Thus, 5a 3 b 2 c 
 is of the sixth degree. 
 
 An algebraic expression, which is composed of several terms, separated 
 from each other by the signs -(- or — , is called generally a polynomial,* or poly- 
 nome. Thus, 3a 2 - r -46 2 — 6c-\-d is a polynomial. A polynomial is said to 
 be homogeneous when all its terms are of the same degree. 
 
 A polynomial, consisting of two terms only, is usually called a binomial ; 
 when consisting of three terms, a trinomial. Thus, a-\-b, 3b*c — xz, are 
 binomials, and a-\-b—c, 3?rc 2 /i 5 — 6p 3 r+9d, are trinomials. 
 
 XIX. Of the different terms which compose a polynomial, some are pre- 
 ceded by the sign -\-, others by the sign — . The former are called additive, 
 or positive tortus, the latter, subtractive, or negative terms. 
 
 The first term of a polynomial is not, in general, preceded by any sign ; in 
 that case the sign -\- is always understood. 
 
 * A polynomial is also called a compound quantity. Polynomials, to save the trouble of 
 writing them repeatedly, are often represented by a single large letter. Thus, if we have 
 two polynomials, x*— i3py-\-ixy*—y* and x*—3xy--\-3x"-y—y 3 , we may represent the first 
 by A and the second by B, and afterward, in referring to them, may call them the poly 
 nomials A and B. 
 
6 ALGEBRA. 
 
 Terms composed of- the same letters, affected with the same exponents, are 
 called similar terms. 
 
 Thus, lab and 3ab are similar terms, so are 6a 2 c and 7a 2 c; also, lOatfrd 
 and 2ab 3 c 4 d ; for they are composed of the same letters, and these letters 
 in each are affected with the same exponents. On the other hand, 8ab 3 c 
 and 3a~b s c are not similar terms, for, although composed of the same letters, 
 these letters are not each affected with the same exponent in each term. 
 
 XX. The numerical value of an algebraic expression is the number which 
 results from giving particular values to the letters which compose the ex- 
 pression, and performing the arithmetical operations indicated by the algebraic 
 symbols. This numerical value will, of course, depend upon the particular 
 values assigned to the letters. Thus, the numerical value of 2a 3 is 54 when 
 we make a =3, for the cube of 3 is 27, and twice 27 is 54. The numerical 
 value of the same expression will be 250 if we make a =5; for the cube of 5 
 is 125, and twice 125 is 250. 
 
 The numerical value of a polynomial undergoes no change, however we 
 may transpose the order of the terms, provided we preserve the proper 
 sign of each. Thus, the polynomials 4a 3 — 3a 2 6+5ac 2 , 4a 3 +5ac 2 — 3a 2 b, 
 5ac 2 — 3a"b-\-4a 3 , have all the same numerical value. This follows mani- 
 festly from the nature of arithmetical addition and subtraction, for it is evident 
 that if the same amounts be added or taken away, it is immaterial in what 
 order. 
 
 Examples of the numerical values of algebraic expressions : 
 
 Let a=4, fe=3, c=2 ; then will 
 
 (1) a + b—c=4< + 3— 2=7— 2=5 
 
 (2) a 2 +a Z>+Z> 2 =4 2 +4 X 3 + 3 2 =16+12+9=37 
 
 (3) ac— a 5 + 6 c=4x 2— 4X3+3X2=8— 12+6=2 
 a 2 +Z> 2 — c 2 4-+3 2 — 2 2 16+9—4 21 
 
 (4) 
 
 ab—ac+bc~4x3— 4x2 + 3x2 12— 8+6~10 
 
 (5) V(« + ^-V(«-^)c 3 =V(4 + 3)X2-V(4-3)x2 3 =Vl4-V8 
 
 = 3-7416574 — 2 = 1-7416574 
 a +& a —c a — b_7_ 2 1 263 
 
 ( 6 ) ^H7+6 + c — a + 6 — 2+5 — 7 — 70 
 
 XXI. Entire quantities are those which are rational and contain no de- 
 nominator; such are 47, 2a 2 6, 3a 2 — be. 
 
 XXII. An algebraic expression containing a quantity is called a function of 
 that quantity. For example, the expression 3.r 2 — ^/x is a function of a: ; the 
 
 expression a(s:-\-y)-\-—{x-{-y) is a function of x-\-y. An entire function of 
 
 a quantity is one in which this quantity does not enter into a denominator. 
 
 A rational function is one in which the quantity does not appear under a 
 radical. 
 
 To express, in a general way, a function of x, we write F(x). Where 
 many different functions of x are to be represented, we vary the form of this 
 initial : thus, F(x), f(x), <p{x), F'(.r), &c, which denote, in a general way, 
 different algebraic expressions containing x. 
 
 To express functions of the same form of different quantities, we use th«» 
 same initial before these quantities; thus, F(.r), F(y). 
 
REDUCTION OF TERMS. 
 
 To express a function like x~-\-2xij-\-if of two quantities, we write F(x,y\,- 
 of three quantities, F(.r, y, z), and so on. 
 What follows to equations may bo called the algebraic calculus. 
 
 REDUCTION OF TERMS. 
 
 3. Reduction of similar terms is the collecting of several similar terms into 
 one. 
 
 The rule may be dividod into two cases : 
 
 (1) When the similar quantities have the same signs. 
 
 (2) When the similar quantities have different signs. 
 
 CASE I. 
 
 When the similar quantities have the same signs. 
 Add the coefficients ; affix the letter or letters of the similar terms, ana 
 prefix the common sign -4- or — .* 
 Thus, a+2a+3a+4a+5a = (l+2+3+4 + 5)a=15a, 
 
 (_a)-f-(— 2a) + (— 3a)?f(— 4a) = — (l + 2+3+4)a = — 10a. 
 It is convenient to write the similar terms to be reduced under, instead of 
 after one another, they being read in the same order in either way. 
 
 (1) (2) (3) (4) (b)_ 
 
 3a abc 9axy — bbx V a-\-x 
 
 la 2abc • 3axy — 2bx 2^/a-\-x 
 
 2a 7abc laxy — bx b^a-\-x 
 
 EXAMPLES. 
 
 
 (3) 
 
 (4) 
 
 9axy 
 
 — bbx 
 
 3axy 
 
 — 2bx 
 
 laxy 
 
 — bx 
 
 baxy 
 
 — bx 
 
 axy 
 
 — Abx 
 
 5axy 
 
 —lObx 
 
 
 \ 
 
 a 3abc baxy — bx -\/a-\-x 
 
 6a abc axy — Abx 7-<Ja-\-x 
 
 8a babe baxy — lObx 4 y/a-\-x 
 
 27a 19abc 
 
 CASE II. 
 
 When the similar quantities have different signs. 
 Collect into one sum the coefficients affected with the sign -f-» and also 
 those affected with the sign — ; to the difference of these sums affix the com- 
 mon literal quantity, and prefix the sign -|- or — , according as the sum of the 
 -f- or — coefficients is the greater.f 
 
 * The truth of this rule is evident ; for suppose the two terms 3a and 5a are to be r©« 
 duced to one, then by the definition of a coefficient we have 
 5a-=a-\-a-\-a-\-a-\-a 
 3a=a-\-a-\-a 
 Hence 5a-\-3a=za-\-a-\-a-\-a-\-a-^-a-\-a-\-a=8a. 
 
 Similarly,— 5«=(— «)+(— «)+(— *)+(—«}+(— «) 
 
 -3a=(-a)-H-a)+(-a). 
 Hence -5a+(-3a)=(-a)+(-a)+(-a)+(-a)+(-a)+(-«)+(-«)+(-a) 
 =8(— a)=— 8a. 
 t The truth of this will be obvious ; for to redaoe 5a and — 3a, we have 
 5a=-a-\-a-\-a-\-a-\-a 
 -3a=(-«)+(-«)+(-a). 
 
ALGEBRA. 
 
 Thus, a — 2a + 3a — 4a + 5a = (l-f3+5)a — (2+4)a = 9a— 6a=3a 
 And, 3x+4y— 2x-\-3y=(3— 2)x+{<L + 3)yz=x+7y. 
 Reduce the terms of the polynomials, 
 
 (6) c+2d— 2c— 3d+3c+ld— 4c— 5d+c-{-d 
 
 (7) 3a— 26+5a — 6e+36 — 9c+a— 6-fl21c 
 
 a m k m a k m 
 
 ( 8 ) *— J — 5 — oo — 7 — 13 — 8 — 9 
 
 (9) 3a— |6-f 6a— 3§6-fl0ia— 22£6— fa 
 
 (10) bxy—A fypqr+4xy— 10a 5 6 3 -f7 Vpqr— 9:n/+3a 5 &*. 
 
 ADDITION. 
 Addition is the collecting of several polynomials into one. 
 
 RULE. 
 
 Write the polynomials one after another, and reduce similar terms.* 
 
 EXAMPLES. 
 
 f) 
 
 20 (a 2 — fr y— 15 -y /x 2 — 3/s 
 
 - /a 2 — 6 2 — 7 Vx*—y 
 
 12 ja° — b' 2 — ^x-—y' i 
 
 4 ( a 2_&2)*_ 3 (x 2 — i/ 2 )* 
 
 2 ( a g__ft3)i_ 5 (x 2 — 3/ 2 ) 1 
 
 (1) 
 
 (2) 
 
 3a 2 -f- 6 2 
 
 2x 2 — xy 
 
 2a 2 + 36 2 
 
 4z 2 — 7x?/ 
 
 6a 2 + 5& 2 
 
 3x-—4xy 
 
 a 2 + 76 2 
 
 x 2 — xy 
 
 a 2 + 66 2 
 
 8a; 2 — Ixy 
 
 13a 2 -f226 3 
 
 
 (4) 
 
 (5) 
 
 a+ 6 
 
 xy — ab 
 
 -2a+36 
 
 2xy-\-3ab 
 
 3a— 46 
 
 — 5xy-\-7ab 
 
 -5a+66 
 
 — xy — 3a6 
 
 7a— & 
 
 8xy — 9a b 
 
 4a -4-56 
 
 
 & 
 
 (6) 
 
 ■\/x"-\-y' i — vi"-\- ri 1 — 2mn 
 
 — 2 Vx^+y + 3m 2 — 3 ?i 2 +5*n7i 
 
 — 5-/a-' 2 +2/ 2 — 4/?i 2 +5 n 2 — Imv 
 
 2 (x 2 +2/ 2 )-4-12»i 2 — 2|n 2 + mn 
 
 8 (x 2 -j-?/ 2 )- — 8m 2 — jra 2 — 6mn 
 
 In example (4), let a=5 and 6=3, then a-\- b= 8 
 
 _2a+36=— 1 
 
 3a— 46= 3 
 
 — 5a+66 = — 7 
 
 7a— 6= 32 
 
 4a+56= 35f 
 
 Hence 5a+( — 3a)=a+a+a-fa+a+( — a)+( — «)+( — a). 
 
 =<z4" a=2a - 
 Similarly, 2a+(—5aJ=a+a+(—o)+(—o)-{-(—o)+(—o)+(— a) 
 
 =+(-«)+(-«)+(-«) 
 s=3{ — a)= — 3a. 
 • For if certain quantities are to be added and subtracted, it is immaterial in what por* 
 tions, or what order. 
 
 t Similar substitutions maybe tried in some of tbe following- examples. Let the learner 
 substitute any other numbers for a and b, and he will find that the sum of the polynomials 
 will be truly expressed by the result 4a-\-5b, the correctness of which does not depend od 
 the values of a and b. This illustrates the general principle stated in the note of Art I 
 
ADDITION 
 (7) (8) 
 
 bax* — \Af+;y -f- ( a — ty 2y /xy-\-xz-\-yz -f ty ax-\-by 
 
 ~- 7aJx+2 (a+y) *— 3(a — b) — 5 -/•"/+•*- + ?/: —3 {ax+by)* 
 12aVx—3 Vx+y + 12{a — 6) 12 (xy+xz+yz)^ + 5 (a.r+ %)* 
 
 — 3a -/a'— 4 Vi-|-i/ — (a— b) — 3 -v/ar^+zz + i/z — 2y<ix-\-by 
 
 — ax* + (ar-j-y)*— 3(a — &) (xy+xr+yz^-f- (ax-t-fcy)* 
 
 (9) (10) 
 
 a+6+c+rf+c— / 4(a+&) Vx 2 — y 3 — 2(a — fc) y /x*+y* 
 
 a+b+c+d — e+f — 3(a+b)Jx*—y 2 + (a — 6) V^+y* 
 
 a+b + c-d+e+f - (a+b) (x*-if)*+3(a-b) (x 2 +y 2 )i 
 a+b-c+d+e+f 6(a+b) (x 3 -y 2 )*- (a-b) (x 2 +y 3 )i 
 
 a — b + c + d+e+f 10(a+Z>)Vz 2 — t —b{a — b) (x 2 +y 2 )* 
 
 —a+b+c+d+e+f — 2(a+6) (x 3 — iff+4{g— b) Vx"*+y* 
 
 4. Dissimilar quantities can only be collected by writing them in succession 
 and prefixing to each its respective sign. Thus, 9xy, — bed, and 3ab are dis 
 Mmilar quantities, and their sum is 9xy-\-3ab — bed. In like manner, 2ab, 
 3aZ> 2 , 4a& 3 are dissimilar quantities, and their sum is 2ab-\-3ab 2 -\-4ab :i ; which, 
 however, admits of another form of expression, as will be explained in the rule 
 of Division. When several polynomials, containing both similar and dissimilar 
 quantities, are to be collected into one polynomial, the process of addition will 
 be much facilitated by writing all the similar terms under each other in verti 
 cal columns. 
 
 This, however, is not absolutely necessary. The similar terms may be col 
 lected together as they stand. 
 
 EXAMPLES. 
 
 (1) Add together ax + 2by + cz; -/•?+ Vy+ V~; 3y- — 2x i -{-2z i ; 4c 
 — 3ax«—2by; 2ax—4-\/y—2z^. 
 
 ax+2by+cz + -/x-f- -/y + Jz 
 — 3ax— 2&y+4cz — 2x*4-3y* +3z 4 
 2ax —4 Vy— 22* 
 
 5c; — \/ .r-|-2-v/ 2= sum required. 
 
 (2) Add together, 
 
 4a 2 6 + 3c s d— 9m"n ; 4m*n + ab" + bcH + 7a"b ; 6m"n—bc 3 d + 'imn i —8ab a ; 
 Imv? -f QcH— bm?n — 6a 2 6 ; IcH — 10a6 2 — 8m 2 n — 10d 4 ; and 12a 2 6 — 6a?* 8 
 -\-2c z d-i r mn. 
 
 A.rranging the similar terms in vertical columns, we have 
 4a 2 6 + 3c 3 d— 9m?n 
 7a 2 6-f bcU-\- 4m 2 n+ a?; 3 
 
 — b<?d-\- §m?n— 8a6 2 -|- 4mn, 2 
 — 6a-b-\- 6c 3 d— bm"n -\- 7?/m 2 
 
 + 7cM— Qm?n— 10a& 2 —10c/ 4 
 
 12a 2 Z> + 2c 3 o 7 — %a b°" -\-mn 
 
 17a 2 6-f 18c 3 d— 12m-n— 23ab 2 + llwm 3 — 10d i +?nn= sum. 
 
 (3) Add ll&c-j^ad— Sac+Sca 7 ; 8ac+7bc—2ad-\-4mn; 2cd—3t:b+bat 
 f a»; and 9an — 2bc — 2ad-\-bcd together. 
 
(7) Add ^_3_^ + 6J^_%±l) and ?a ^__12Vp 4^+r) to 
 v/ y c ' z s y ' c z s 
 
 10 ALGEBRA. 
 
 (4) Add together, without arranging the similar terms in vertical columns, 
 
 2ab 2 -\-3ac z — 8cx 2 + 9b 2 x— 8hy 2 —10ky 
 5a 3 —lab 2 — 7bx 2 — b 2 x— Aky 2 —\bhy 
 Sky— htf+Jlx + 14& 3 — 22ac 2 — 10.E 2 
 19ac 2 — 8& 2 x+ 9z 2 + 6%+ 2/fy 2 + 2ab 2 
 ba 3 — 8c x 2 — x" + 11.T — 9hy 2 +Ub 3 —2ky 2 —bky — 9hy—7bx 2 . 
 
 (5) Add together a 3 — b 3 + 3a 2 Z> — 5a6 2 ; 3a 3 — 4a 2 6 + 3& 3 — 3aZ> 2 ; a 3 + o 3 
 + 3a 2 Z>; 2a 3 — 4b 3 — bab 2 ; 6a 2 b+10ab 2 ; and — 6a 3 — 7a 2 b + 4ab 2 +2b 3 . 
 
 ( 6) Add V^+t/ 2 — j x 2 — y 2 — bxy ; — 3(x 2 — y 2 f + 8.ry — 2 {x 2 + y 2 ) h , 
 2 -y/^+i/ 2 — 3a:2/ — 5 Vx 2 —y 2 ; 7 .1-3/ + 10 jx 2 —y 2 — I2^x 2 +y 2 ; and rrj 
 -j- *J x 2 — y 2 -\- -\/x 2 -{-y 2 together. 
 
 (?) 1 
 gether. 
 
 ABA B 
 
 (8) Add together 4A— 6 — \-7^ and 7-— 2A+3^. 
 
 (9) Add together 3 cos a— A sin 6 + 6 tan c, 2 cos a+2 sin 6 + 7 tan c 
 aDd cos a+3 sin & — 2 tan c. 
 
 (10) Add together 3.290 —2.45 D +1.84 W, 4.560 +0.59 ]) +6.41 tf 
 and 2.220+3.11 D — 4.21 W. 
 
 ANSWERS. 
 
 (3) 16oc+5ac+12ci+4mra— 3a& + 10aw. 
 
 (5) a 3 +a 2 &+a& 2 +& 3 . 
 
 (6) 2 -/a- 3 — 1/ 2 — 10 V* 2 +2/ 2 +8:n/. 
 13a 5m 3 6^/0 (f+r) 
 
 (7) + yjL + nju.. 
 
 v ' y ' c 2 ' s 
 
 A B 
 
 (8) 2A+-+10^. 
 
 (9) 6 cos a+sin 6+11 tan c. 
 (10) 10.07© +1.25 D +4.04 W. 
 
 5. When the coefficients are literal instead of numerical, that is, denoted by 
 Setters instead of numbers, their sum may be found by the rides for the addi- 
 tion of similar and dissimilar terms ; and the sum thus found being enclosed in 
 a parenthesis, and prafixed to the common literal quantity, will express the 
 Bum required. 
 
 EXAMPLES. 
 
 (1) ( 2 ) 
 
 ax+by+cz 3az+ (a+&) (x+y)+2mnz 2 
 
 bx+cy+az _ax+2(a+&) {x+y)—bmnz 2 
 
 cx + ay+bz 4m«z 2 +5(a+Z>) (x+y) + 10ax 
 
 2po2 2 + (p+g) (x+y) + 2px 
 
 (a+6 + c).r 
 
 + (& + c+a)y \ =-«um. {12a+2p)x+ ^8{a + b)+p+q^{x+y) ( 
 + Jc+fl+o)zj_ + ( W m + 2/;ry)z' 
 
 sum. 
 1 
 
SUBTRACTION. 11 
 
 m 
 
 (3) (4) 
 
 {a b) Vz + (»»— n)y/y + J2 (m+n) f — { a— b)x*+axy 
 
 (a+c) x' J — (m— n) 2/ 4 +2-/2 (» — p) I/ 8 — (24+ b)x"-—bxy 
 
 (b—c)Jx +3(m — n) Vy — 3-/2 (p— 2w)# 2 — ( c— 3a)x 3 +cxy 
 
 ( c- a) y,r — 5(m— n) Vy —6 -/2 (?—>») t~ ( c4-2a> 2 — axy 
 
 (5) Add ax 3 +% + c to ii' 2 +%4-^- 
 
 (6) Add together x 2 -f.ry + i/ 2 ; ax 2 — axy+ay* 2 ; and — i?/ 2 4-&x?/4-&x». 
 
 x 2 4-.rw4-' ) / 2 , x-—xy-\-y"- 
 (?) Add K*+2/) » nd \{*—y)' AIs0 ' o and ~ 2 
 
 (8) What is the sum of («4-&)x4-(c— r% — re-/ 2; (a — &)x4-(3c4-2a*)v 
 4-5x-/2; 26x-f3% — 2x-v/2; and —3bx—Jy— 4x^21 
 
 (9) Add ax 4-% 4- cz ; a'x— t'y-f-c'z ,• and a"x-\-b"y— c"z. 
 
 (10) Add together ax-\-by 4- cz; a 1 x4-^ 1 i/ — c x z ; and a 2 x — & 2 2/+ c 2 2 
 
 ANSWERS. 
 
 (3) (a4-c) jx— 2(m — n)^y— 6^/2. 
 
 (4) o?/ 3 — (2c4-2<i)x 3 4-(a-rJ-f c — a^n/. 
 
 (5) («4-4r 2 4-(64-/0 i /+r4-^. 
 
 (G) (i4-a4-fc ) x 3 4-(l_a4-i).n/4-(14-a-%*. 
 
 (7) First part, x. Second part, x 3 4-y 2 . 
 
 (8) (2a — i)x4-(4c4-3o 7 )i/— 2X-/2. 
 
 (9) (aJ-a'+a")x-\-(b — b'+b")y+(c+c , —c")z. 
 (10) a 
 
 -fa, 
 +« 3 
 
 x4-6 
 
 y+c 
 
 + &, 
 
 — c, 
 
 -&„ 
 
 + Co 
 
 SUBTRACTION. 
 
 ROLE. 
 
 6. Place the quantity to be subtracted under that from which it is to be 
 taken ; change the signs of all the terms in the lower line from 4- to — , and 
 from — to 4~) or else conceive them to be changed, and then proceed as di 
 rected in Addition.* 
 
 * The sign — , prefixed to a monomial, serves to intimate that tins monomial ought to en 
 ter subtractively into any combination of which it forms a part. If, for example, it be r© 
 quired to add the subtractive quantity ( — d) to c, the sum c-\-[ — d) is c — d. 
 
 If the difference between two quantities, as m, and s, be required, m and s being both add 
 itive, the expression of the difference is m — g. If the difference be required between m 
 an additive, and ( — s), a subtractive quantity, let the difference =d ; that is, let 
 
 m — ( — s)=d. 
 
 Adding ( — s) to both these equals, there results 
 
 But m — ( — s)-\-{ — s)=m, and d-{-( — s)=d — s. 
 
 Therefore, m=d — s. 
 
 Now vi — ( — s)=d, and m=d — s. 
 
 Hence m — ( — s) is greater than m by the additive quantity s cr is equal to »»4~ s 
 
 The above is the demonstration for isolated temis. 
 
 For polynomials we have the following: 
 
 It is evident, that if all the terms of the quantity to bo subtracted are affected with tha 
 
ly ALGEBRA. 
 
 EXAMPLES. 
 
 (1) (2) 
 
 From4a+36— 2c+8d From :2xy+3y' 1 — m 2 +3 y/2 
 
 Take A+2& + c-\-bd Take — 5xy+7y' 2 -19x' i +2 a/2 
 
 Rem. 3a+ 6 — 3c+3gZ Rem. 17xy— 4y 2 4- 2a: 2 + -y/2 
 
 (3) (4) (5) 
 
 32a+ 36 28aa: 3 — 16a 2 x 2 +25a 3 :r— 13<z 4 2{a + b)+3(a—x) 
 
 5a+17i 18ar 5 +20a 2 x 2 — 2ia 3 x— la* ( a + b)—3{a—x) 
 
 (6) (7) 
 
 Gaby— Zyx+izx ^/x i — 2/' 3 +4(.?: -\-y ) — Sa^H-* 
 
 — 2gfy+6z:r+2;j/3: 3(x + ;/) — 2(.r 2 — ,y 2 )"+3 («+*)* 
 
 (8) (9) 
 
 x 2+2a:7/+i/ a « 2 — 2xy+y <i -\-{x*— y*) + 2(xy— y*) 
 
 x*—2xy-\-y' i x 2 -\-2xy—y"-\-{x i -{-y")—2{xy—y i ) 
 
 (10) 
 2a 2 + ax+ x i —12a i x-\-20ax' 2 — Ax 3 + 6a 2 x 2 — Wax 3 
 g2—3ax-\-2x' 2 —16a' i x-\-12ax -—l2ax 3 —<ix 3 -f- 2a 2 :r* 
 
 (11) 
 
 4y<2-.4yx-\-x' 2 —2a(x+y)-\- 6 -/a 2 — x 2 — 8 Vfc 2 — y 2 
 4a: 2 — 4xy4-y 2 — 4g(a:4-3/)— 10 \7fr 2 — 1/ 2 +4 Va 2 — z 2 
 
 7. In order to indicate the subtraction of a polynomial, without actually per- 
 forming the operation, we have simply to inclose the polynomial to be sub 
 tracted within brackets or parentheses, and prefix the sign — . Thus, 2d 
 
 sign -}-> we must take away, in succession, all tho parts or terms of the quantity to be sub 
 tracted; and this is indicated by affecting all its terms with the sign — . But if some of 
 the terms of the subtrahend are affected with the sign — , as, for instance, if c — d is to be 
 subtracted from a-\-b ; then, if c be subtracted, we shall have subtracted too much by d , 
 hence the remainder a-\-b — c is too small by d; and therefore, to make up the defect, the 
 quantity d must be added, which gives a-\-l — c-\-d ; by inspecting which we perceive thai 
 the signs of the subtrahend have been changed. 
 
 This reasoning may be generalized by supposing c to represent the sum of the additivt- 
 terms, and d to represent the sum of the subtractive terms of the lower line, or quantity tr 
 be subtracted 
 
 Another mode of proving the rule for the signs in subtraction is the following : 
 
 By subtraction we solve the problem, " Given one of two quantities, and their algebraical 
 gum, to find the other." 
 
 Let A be any algebraical quantity, simple or compound, from which it is proposed to 
 eubtract another simple or compound quantity, B. The quantity A may be conceived to be 
 the algebraical sum of B, and some other quantity which it is proposed to discover. Call 
 it X. As A was obtained by annexing to x the polynomial expressed by B, with its prope? 
 signs, the effect of this process will be destroyed by annexing to A the polynomial repro 
 tented by B, with its signs changed. 
 
MULTIPLICATION. J3 
 
 — 3a 2 6-f4a6- — (« 3 +6 3 4-a6 2 ) signifies that the quantity a 3 -f-6 3 +a6' is to bo 
 subtracted from 2a 3 — 3a-b-\-4ab 2 . When the operation is actually perform- 
 ed, we have by the rule, 
 
 2a 3 — 3a 2 6 + 4a6 2 — (a 3 +6 3 +a6 2 )=2a 3 — 3a 2 6 + 4a6 2 — a 3 — 6 3 — ab* 
 
 = a 3 — 3a 2 6+3a6 2 — 6 3 . 
 When, therefore, brackets are removed which have the sign — before them, 
 the signs of all the terms within the brackets must be changed. 
 
 8. According to this principle, we may make polynomials undergo several 
 transformations, which are of great utility in various algebraic calculations. 
 
 Thus, 
 
 a 3 — 3a 2 6+3a6 2 — 6 3 =a 3 — (3a 2 6— 3a6 2 +6 3 ) 
 — a 3 —& 3 — (30*6— 3a& 8 ) 
 
 =a 3 +3a6 2 — (3a 2 6 + 6 3 ) 
 = — (— a 3 +3a 2 6— 3a6 2 -f6 3 ) 
 And x 2 —2xy+y"=X'—(2xy—y-)=y' 2 —(2xy—x' 2 ). 
 
 EXAMPLES OF QUANTITIES WITH LITERAL COEFFICIENTS. 
 
 (1) (2) 
 
 From ax't+byx+cy 2 From (a+i) V^ 2 +?/ 2 +(a + c )( <z + x ) s 
 
 Take dx^—hxy+ky 2 Take (a— b) ■ s /x"+y' i -\- c (a+x)* 
 
 Rem. (a— ^).T 2 4-(6+/i)^3/+(c— % 2 . Rem. 2b ^x'*+ y 3 +a{a+x) 3 . 
 
 (3) From m-nV — 2mnpqx-\-2 ) ' 1< i i ta ^ e P~q 2 J? — 2pqmnx-{-m,"n-. 
 
 (4) From a(x-\-y) — bxy-{-c(x — y) take A{x-\-y)-\-{a-\-b)xy — 7(x— y). 
 
 (5) From (a-f-6) (x+y)—{c — d) (x— y)+h* take (a — b) (x+y) + {c+d\ 
 (r-y)+£ 2 . 
 
 (6) From (2a— 56) ^x+y-\-{a— b)xy— cz 2 take 3bxy— (5+c); 2 — (3a— b\ 
 (x+yf. 
 
 (7) From 2x—y-\-{y—2x) — {x—2y) take y— 2x— {2y— x) + (x+ 2y) 
 
 (8) To what is a+6-fc— (a— b) — (b— c) — (— b) equal? 
 
 (9) From A^+B^+Car+D take A^+B^+C^x+D,. 
 
 ANSWERS. 
 
 (3) (mW— f-q^X'+pY'— mW, or (mV-^jy-^'-pY). or {in*n* 
 
 (4) ( a -4)(x+y)-(a+2b)xij+(c+7) (x-y). 
 
 (5) 2b(x+y) — 2c(x— y)+h' i — Tc*. 
 
 (6) (5a— 66) Vx+3/+(a— 4b)xy+5z*. 
 
 (7) y—x. 
 
 (8) 26+ 2c. 
 
 (9) (A-A 1 )r 5 +(B-B 1 )x 2 +(C-C 1 )x+D-Di. 
 
 MULTIPLICATION. 
 
 9. Multiplication is usually divided into three cases : 
 
 (1) When both multiplicand and multiplier are simple quantities. 
 
 (2) When the multiplicand is a compound, and the multiplier a simple 
 uantity. 
 
 (3) When both multiplicand and multiplier are compound quantities. 
 
14 ALGEBRA. 
 
 CASE I. 
 
 10. Wlien both multiplicand and multiplier are simple quantities, or monomials. 
 To the product of the coefficients affix that of the letters.* 
 
 Thus, to multiply bx by 4y, we have 
 
 5x4=20; xxy=xy; 
 
 . • .bx X 43/ = 20 X xy == 20x?/ = product. 
 
 11. Powers of the same quantity are multiplied by simply adding their id 
 4ices ; for since, by the definition of a power, 
 
 a b —aaaaa : a' r =aaaaaaa, 
 .'.a 5 X a 7 =aaaaa X aaaaaaa=aaaaaaaaaaaa=a l2 =a 5 + 7 . 
 
 Also, a m =aaa .... to m factors ; a°=aaa to n factors ; 
 
 .'.a m X a a =aaa .... to m factors X ctaa to n factors ; 
 
 =aaaaaa to (m-{-n) factors ; 
 
 =a m + n . 
 It is proved, in the same manner, that a m Xa n X« h Xa k =« ra+n+b+k - 
 
 * I. The rule is derived in the following manner : We begin by assuming that when 
 several letters are written one after another without any sign, their continued multiplica- 
 tion is understood, and that the operation proceeds from left to right. Then abed will sig- 
 nify a multiplied by b, that product by c, and that again by d. We shall now prove that in 
 whatever order these letters or simple factors are arranged, their continued product will 
 always be the same ;t and, moreover, that they may be grouped into partial products at 
 pleasure, provided all the letters be employed each time. Thus the above product may be 
 written bade (the multiplication here, as before, going on by each factor successively from 
 left to right), and the result will be the same as before ; or it may be written aXbXcd, un- 
 derstanding the products separated by the sign X as being previously formed and then 
 multiplied together. 
 
 The demonstration depends upon three propositions, which we shall first establish ; 
 
 (1) . . aXb=bXa ^ or * n * ue ^joining table of units let b denote the number 
 
 b of units in each horizontal row, and a the number of rows, 
 
 ' A " v then b multiplied by a, or repeated a times, will give the 
 
 (. . . 1 . . number of units in the table. But a, which is the number of 
 
 111111 horizontal rows, is also the number of units in each column , 
 
 111111 and b is the number of columns ; then a multiplied by b, or 
 
 \1 1 1 1 1 1 repeated b times, will produce the number of units in the 
 
 table again ; whence b multiplied by a is equal to a multiplied by b. 
 
 j.^ In a similar manner, from the adjoining table, it may be 
 
 (a a a a a proved that 
 
 ! a a a a a a.b . c=a ,c.b (2) 
 
 \aaaaa Also that a . £ . c=a . (£c) (3) 
 
 \a a a a a * ' 
 
 II. By (1) abcd=bacd= by (2) bcad= by (2) beda. Thus, we perceive that the factor 
 a has been made to occupy successively every place from the first to the last. The same 
 might now be done with the factor b, and so with all the others. Therefore a product is 
 the same, whatever be the order of its factors. 
 
 III. Again. Take aXbXcXdXe. It may be written by (3) aXbcXdXe or by (3) 
 aXbcdXe, or, instead, by (3) abXcdXe. From which it appears that the factors of a 
 product may be grouped into partial products at pleasure, and then afterward multiplied 
 together or conversely. 
 
 IV. Let us now suppose that the product 3a 3 £ 2 is to be multiplied by the prxluct 5a"b*. 
 Instead of multiplying by the whole product 5a- J<, multiply by its factors separately, and we 
 have 3a3& 2 5a2j4. Since the order may be changed at pleasure, bring the numerical factors 
 together, and the different powers of the same letters; thus, 5 X 3a-a 3 b*b~. Grouping the 
 different powers of the same letters into partial products, as well as the numerical factors, 
 tne result is \5a^b 7 , which has evidently been obtained by multiplying the coefficients and 
 adding the exponents of like letters. 
 
 t Such a relation as that of a product to its factors is called a symmetrical relation. 
 
MULTIPLICATION. J 5 
 
 RULE OF SIGNS IN MULTIPLICATION. , 
 
 The product of quantities with like signs is affected with the sign -{- ; the 
 product of quantities with unliko signs is affected with the sigu — ; 
 
 or 
 -f- multiplied by -f- and — multiplied by — give 4- ; 
 -f- multiplied by — and — multiplied by -f- give — ; 
 
 or 
 like signs produce -f- and unlike signs — . 
 The continued product of an even number of negative factors is positive ; of 
 an uneven number, negative.* 
 
 EXAMPLES. 
 
 (1) 4a"b 2 cd X 3abc*d* = 12a 3 b 3 c 3 d 3 . 
 
 (2) 12 i/ ay X*bx = 48bx-/ay. 
 
 (3) 5£tYz*X6ayz 3 = ZZxhf'z' 1 . 
 
 (4) 13a%»aty X —bdbxif = — 65a 3 b 4 x*y*. 
 
 (5) — 5.r m v/ n X — 4x n y m =+ 20x m + n y m + a . 
 
 (6) — 20aPii X 5a m i n c r = — 100a m +i'6 n -hc r . 
 
 CASE II. 
 
 12. When the multiplicand is a compound, and the multiplier a simple 
 
 quantity. 
 
 Multiply ^ach term of the multiplicand by the multiplier, beginning at the 
 left hand ; and these partial products, being connected by their respective signs, 
 will give the complete product, f 
 
 EXAMPLES. 
 
 (1) (2) 
 
 Multiply a--\-ab + b 2 Multiply a 2 —2ab +6» 
 
 By 4a By 3xy_ 
 
 Product, 4d i - \-4a 2 b-i r 4ab 3 . Product, 3a-xy —6abxy -f Ztfxy . 
 
 (3) Multiply 5mn+3m' 2 — 2n 2 by I2abn. 
 
 (4) Multiply 3ax—5by-\-7xy by —labxy. 
 
 (5) Multiply — 15a 2 o+3ai 2 — 126 3 by — 5ab. 
 
 (6) Multiply ax 3 — bx^+cx— d by — x 5 . 
 
 (7) Multiply •/<*+&+ y/xV—if—Zxy by —2t/x. 
 
 (8) Multiply a m x n -\~b m y n — c*y m —d a x m by x m y n . 
 
 * Let m, mf be two monomial quantities whose product is required. If m, mf are both addi- 
 tive quantities, the product mm' is an additive quantity. This is the case of arithmetic. 
 If the multiplicand m is an additive quantity, and the multiplier m' a subtractive quantity, 
 the expression »«X( — '">■') indicates that the multiplicand m is to be subtracted as many 
 times as there are units in m', or that m' repetitions of the quantity m are to be subtracted, 
 which is expressed by — mm'. 
 
 If m is subtractive and m' additive, — m, taken once is — m ; taken twice is — 2m ; tak- 
 en m' times is — m'm. 
 
 If m and m' are both subtractive, the quantity — m is to be subtracted mf times. Now 
 — m subtracted fence is -\-m, twice is -f-2m; and m' times is -\-m'm. 
 
 t 1st. Suppose the signs to be all plus. The whole multiplicand being to be taken as 
 many times as is denoted by the multiplier, each of its parts or terms must be taken so 
 many times. 2d. For the case where some of the signs are negative, see the demonstra- 
 tion in the • ext note. 
 
 / 
 
1G ALGEBRA. 
 
 t CASE III. 
 
 13. When both multiplicand and multiplier are compound quantities. 
 Multiply each term of the multiplicand, in succession, by each term of the 
 multiplier, and the sum of these partial products will give the complete prod- 
 uct.* 
 
 EXAMPLES. 
 
 (1) 
 
 u-f b 
 a+ b 
 a 2 -f ab 
 
 + ab + b* 
 a i +2ab + b 2 
 
 cW* 
 
 (2) 
 a +b 
 a — 5 
 
 a*-\-ab 
 
 — ab- 
 a^—b 2 
 
 -6 2 
 
 6 2 
 
 (3)f 
 a — b 
 a — b 
 a 2 — ab 
 
 — ab + b* 
 a 2 —2ab+b* 
 
 (4) 
 a b -\-cd 
 a b — cd 
 
 
 a 2 -f i 
 a 2 — 
 
 (5) 
 
 6 2 
 
 aW+abcd 
 —abed — 
 
 a 4 +: 
 
 2a 3 b+ 
 
 a 2 6 2 
 a*b°~—2ab 3 — b* 
 
 a^fr—M* 
 
 a 4 +S 
 
 la s b—l 
 
 2a b 3 —b* 
 
 (6) Multiply 4a 3 — ba?b— 8a& 2 +26 3 by 2a 3 — 3ab— 4& 2 . 
 4 a 3_ r >a i _ 8a6 2 + 2b 3 
 
 2a 2 — 3ab — 4b 2 
 
 8a 5 — 10a*b— 16a 3 6 2 -f- 4a 2 5 3 
 
 — 12a 4 i+15a 3 6 2 -f24a 2 6 3 - 6aZ> 4 
 
 — l6a 3 5 2 -f20a 2 & 3 +32a& 4 — 8& 5 
 
 8a 5 — 22a 4 fc — 17a 3 & 2 -|-48a 2 fc 3 +26a& 4 — 86 5 = product 
 
 * 1st. Suppose all the terms of the multiplier to be affected with the sign -f-. - The mul- 
 tiplicand, being to be taken as many times additively as is denoted by the multiplier, must 
 be taken as many times as is denoted by each term of the multiplier separately, and the 
 separate results added together. Sd. When there are both additive and subtractive terms 
 in the multiplier and multiplicand. The rule for the signs may be thus demonstrated. Let 
 a — b be multiplied by c — d. First multiplying a by c, the product a — b 
 
 is ac ; but b should have been subtracted from a before the multi- c — d 
 
 plication ; b units have, therefore, been taken c times in the a, which ac — be 
 
 ought not to have been so taken ; hence b, taken c times, must be ad — bd 
 
 subtracted, and there results ac — be as the product of a — b by c. ac — be — ad-\-bd. 
 
 But the multiplier was c—d instead of c; therefore the multiplicand has been taken • 
 times too often; d times the multiplicand, which will be of the same form as c times the 
 multiplicand, viz., ad — bd, must be subtracted, and the rule for subtraction is to change the 
 signs of the quantity to be subtracted. The result is, therefore, ac — be — ad-\-bd ; com- 
 paring which with the given quantities we perceive that like signs have produced -f- and 
 unlike — . To render the demonstration still more general, a may represent the assem- 
 blage of the additive terms of the multiplicand, and b that of the subtractive ; c and d the 
 same for the multiplier. 
 
 t The results in examples (1), (0), and (3) show, 1. That the square of the sum of two 
 cumbers or quantities is equal to the square of the first of the two quantities plus twice 
 the product of the first and second, plus the square of the second. 2. That the product of 
 the sum and difference is equal to the difference of the squares ; and, 3. That the square of 
 the difference is equal to the sum of the squares minus twice the product 
 
MULTIPLICATION. 
 
 (7) Multiply a'b—ab' by h'k—htc'. 
 a'b—ab' 
 h'k—hk' 
 a'bh'k—ab'h'k 
 
 —a'bhk'+ab'hk' 
 
 a'bh'k — ab'h'k—a'bhk' '-{-ab'hk' '= product. 
 (8) Multiply x m -\-x m - 1 y-\-x m --y*-\-z m - 3 y 3 -\- &c, by x+y. 
 
 £m_|_ ;r m-J^_j_ ;r m-2^2_|_ x m-3^3_|_ 
 
 x +2/ 
 
 (9 
 
 (10 
 
 (11 
 (12 
 (13 
 (14 
 (15 
 (16 
 (17 
 (18 
 (19 
 (20 
 (21 
 (22 
 
 (9 
 (10 
 (11 
 (12 
 (13 
 (14 
 (15 
 (16 
 (17 
 (18 
 (19 
 (20 
 (21 
 (22 
 
 x ra +i-|- x m y-\- x m ~ l y 2 -\- x m ~' 2 y 3 -\-. 
 
 _|_ x m y-\- x m ~'i/ 2 -(- x m-2 ;y 3 -f-" 
 
 x m+i _j_ 2x™y -f 2x m ~ 1 y 2 + 2x m ~ 2 y 3 + . 
 
 Multiply x 2 -f y 2 by x 2 —y 2 . 
 Multiply x 2 -\-2xy-\-y 2 by x — y. 
 Multiply 5a 4 — 2a 3 &+4a 2 o 2 by a 3 — 4a 2 &+2o 3 . 
 Multiply x 4 - r -2x 3 +3x 2 +2x+l by x 2 — 2x-fl. 
 Multiply fx 2 +3ax— ^a 2 by 2x 2 — ax— \a?. 
 Multiply a"-\-2ab-\-¥ by a 2 _2ao + Z> 2 . 
 Multiply x 2 -\-xy-\-y 2 by x 2 — xy-\-y 2 . 
 Multiply x 2 -\-y 2 -\-z 2 — xy — xz — yz by x-f-^+z- 
 Multiply together x — a, x — b, and x — c. 
 Multiply together g-{-h, g-\-h, g — h, and g — h. 
 Multiply together 2>-{-q, p-\-2q, p-{-3q, andp-\-4q. 
 Multiply together z— 3, z— 5, z— 7, and z— 9. 
 (a m — a n -\-a 2 ) X {a m —a). 
 {5a 5 x 3 —'ibY)X{5a 5 x 3 +Ab i y 5 ) as ex. 2. 
 
 ANSWERS. 
 
 x 4 — y*. 
 
 ^s-j-a; 2 ?/ — xy 2 — y 3 . 
 
 5a 7 — 22a 6 o+12a 5 o 2 — 6a 4 6 3 — 4a 3 6 4 +8a 2 & 6 . 
 x 6 — 2x 3 -f-l. 
 
 5x 4 +|ax 3 — ^ a 2 x 2 -f f a 3 x+|a 4 . 
 a 4 — 2a 2 6 2 -f6 4 ." 
 a^+x 2 i/ 2 +2/ 4 . 
 x 3 -\-y 3 -\-z 3 — 3xj/z. 
 
 x 3 — (a-\-b-{-c)x 2 -{-(ab-{-ac-\-bc)x — abc. 
 g*—2g 2 h 2 +h A . 
 
 jp 4 +10p 3 5+35/? 2 o 2 +50pa 3 +245 4 . 
 2 4_24z 3 +206z 2 — 744Z + 945. 
 a? m — a m+n +a m+2 — a m+1 + a n+1 — a 3 . 
 25a 10 x 6 — 166Y . 
 
 When the multiplicand and multiplier are each homogeneous, the product 
 will be also ; and the degree of each term of the product will be equal to the 
 sum of the degrees of a term in the multiplier, and a term in the multiplicand. 
 
 This serves conveniently to verify the accuracy of the operation. It is ap- 
 plicable in the above examples to all except the 12th, 20th, 21st, and 22d. 
 
18 
 
 ALGEBRA. 
 
 In multiplying one polynomial by another, there are always two terms of the 
 total product which are not produced by the reduction of similar terms in the 
 partial products. These two terms are the term affected with the highest 
 exponent of any letter, and the term affected with the lowest exponent. If 
 the terms of the multiplicand, multiplier, and product be arranged in the order 
 of the powers of some letter,* as is usual, and as may be seen in the above ex 
 amples, then the two terms in question of the product will be the first and 
 last, the one being produced by the multiplication of the first of the multipli- 
 cand by the first of the multiplier, and the other by the multiplication of the 
 last of the multiplicand by the last of the multiplier. The first of the multi 
 plicand by the second of the multiplier usually produces a terra similar to that 
 which is produced from the multiplication of the second of the multiplicand by 
 the first of the multiplier. The same is the case with the first and third of 
 each, the first and fourth, the second and fourth, the third and fourth, and so on. 
 
 "When a polynomial, arranged according to the powers of some letter, con- 
 tains many terms in which this letter has the same exponent, these terms, 
 after suppressing from them the letter of arrangement, may be placed in a 
 parenthesis, or in a vertical column with a vinculum placed vertically on the 
 right, and the letter of arrangement, with its proper exponent, following after. 
 The polynomial in the parenthesis, or vertical column, is to be regarded as the 
 coefficient of the power of the letter which follows, and is to be operated with 
 exactly as we do with a numerical coefficient; i. e., multiply the coefficient 
 of the letter of arrangement in the multiplicand by the coefficient of the same 
 letter in the multiplier, and afterward add the exponents of this letter. 
 
 26 
 
 Multiplicand <( — 1 
 Multiplier 
 
 26 
 + 1 
 
 EXAMPLE. 
 
 
 a 2 — 46 2 
 -f 26 
 — 1 
 
 a+ 86 3 
 — 46 s 
 
 a — 46 3 
 + 1 
 
 
 Product of the 
 
 multiplicand by 
 
 26 
 
 + 1 
 
 Product of the 
 multiplicand by 
 
 — 46 2 
 
 + 1 
 
 Total product 
 simplified < 
 
 ( 46 2 
 
 a 3 — 16b 3 
 
 a a +326* 
 
 — 1 
 
 -f 46 2 
 
 — 86 3 
 
 + 26 
 
 — 46 2 
 
 - 2 
 
 + 26 
 
 
 
 — 1 
 
 a— 326 5 
 + 166* 
 + 86 3 
 — 46 s 
 
 The letter chosen for this purpose is called the letter of arrangement 
 
MULTIPLICATION. 
 
 19 
 
 WO 
 
 -a 
 
 6 
 
 -o 
 -r 
 
 
 
 + 
 
 + 
 
 
 
 cs 
 
 w 
 
 
 
 iC 
 
 -O 
 
 
 
 co 
 1 
 
 CO 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 rO 
 
 -o 
 
 
 
 CO 
 
 CO 
 
 
 
 1-1 
 1 
 
 1 
 
 
 
 1 
 
 U3 
 
 m 
 
 
 
 HO 
 
 * 
 
 
 
 CN 
 
 CN 
 
 
 
 CO 
 
 CO 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 <3 
 
 
 
 
 
 
 
 i-i 
 
 ■-I 
 
 
 
 1 
 
 1 
 
 
 
 rO 
 
 * 
 
 
 
 CN 
 
 cn 
 
 
 
 + 
 
 + 
 
 1 
 
 
 
 « 
 
 eo 
 
 
 
 r-O 
 
 rO 
 
 
 
 CO 
 I 
 
 co 
 1 
 
 
 
 1 
 
 *»* 
 
 
 
 rO 
 
 HO 
 
 e» 
 
 
 CD 
 
 CN 
 
 HO 
 
 
 i— i 
 
 CO 
 
 -^** 
 
 
 ^— ^ 
 
 
 1 
 
 o 
 
 
 ci 
 
 + 
 
 -Q 
 
 
 r-O 
 
 
 00 
 
 
 -tf 
 
 c* 
 
 + 
 
 
 1 + 
 
 + 
 
 « 
 
 
 ~o <3 
 
 
 
 
 CD *— 
 
 ~© 
 
 I-l 
 
 
 i-H '- 1 
 
 cn 
 
 + 
 
 
 C4 
 
 1 
 
 rO 
 
 ^^ 
 
 -2.* 
 
 -O 
 
 CN 
 1 
 
 l-H 
 1 
 
 r-H CN 
 
 + 1 
 
 1 
 
 t-O 
 
 
 -O ""° 
 
 n 
 
 rO 
 
 -* 
 
 Tt« 
 
 00 ** 
 
 CD 
 
 I 
 
 1 
 
 I I 
 
 1—1 
 
 1 
 
 1 
 
 1 
 
 1 1 
 
 1 
 
 n 
 
 e 
 
 e 
 
 e -o 
 
 « 
 
 ^~- 
 
 ' — > 
 
 <— « CO 
 
 '"■» 
 
 1 ■ 
 
 1 1 
 
 1 
 
 
 
 W 
 
 e* 
 
 ^ 
 
 rO 
 
 -o 
 
 -O . 
 
 OJ 
 
 CN 
 
 3 1 
 
 -S 
 
 
 « 
 
 
 co 
 
 a 
 
 u 
 
 
 CN 
 
 e 
 
 a 
 
 
 ^2 
 
 
 
 
 i-O 
 
 
 
 
 1 
 
 
 
 
 1 
 
 n 
 
 rO 
 
 
 -o 
 
 -O 
 
 *# 
 
 i— ( 
 
 co 
 
 CO 
 
 l 
 
 + 
 
 1 
 
 + 
 
 » 
 
 
 *r 
 
 
 r~."l 
 
 i-O 
 
 -O 
 
 
 CO 
 
 CN 
 
 ■3 
 H 
 
 
 -o 
 -f 
 
 ~3 
 EO 
 
 
 I-l 
 
 
 
 rO rO 
 
 rH 
 
 CN CN 
 
 + 
 
 + 1 
 
 
 rA ta 
 
 i-O 
 
 r-O i-O 
 
 CN i-l 
 
 rC Tji 
 
 1 + 
 
 1 + 
 
 n 
 
 rt 
 
 WO r-O 
 
 rO 
 
 Tt< CN 
 
 CO 
 
 -r 
 
 00 
 
 
 
 
 r-l 
 1 
 
 I-t 
 
 i-l 
 
 CN 
 
 1 
 
 rO 
 
 CN 
 
 1 
 
 -o 
 
 CN 
 
 + 
 
 CN 
 
 1 
 
 + 
 
 -o 
 
 .§1 
 
 
 CO 
 
 
 KO 
 
 
 
 r-O 
 
 n 
 
 HO 
 CO 
 
 
 iJ5 .-O 
 CO M" 
 
 ~o 
 CO 
 
 -O 
 CN 
 
 m 
 
 ^ 1 
 
 .& 
 
 + 
 
 fO wO 
 <N <* 
 
 + 
 
 n 
 -O 
 CO 
 
 EO 
 
 CN 
 
 CO 
 
 
 
 
 1 
 
 i-O 
 
 1 
 
 
 
 
 CN 
 
 CN 
 
 
 
 
 + 
 
 + 
 
 
 
 e» 
 
 
 
 
 
 i-O 
 
 i-O 
 
 
 1-1 
 
 
 •* 
 
 'i' 
 
 
 + 
 
 
 + 
 
 1 
 
 m 
 
 i-O 
 
 
 rO 
 
 
 ~o 
 
 1 
 
 I-H 
 1 
 
 CO 
 
 1 
 
 
 CO 
 1 
 
 1 
 
 
 -*• 
 
 
 1 
 
 HO 
 
 iJD 
 
 -o 
 
 
 -o 
 
 -tfl 
 
 ■<3< 
 
 CO 
 
 
 CO 
 
 
 CN 
 1 
 
 iO 
 
 ~> 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 CO 
 
 
 1 
 
 n 
 
 + 
 
 MULTIPLICATION BY DETACHED COEFFICIENTS. 
 
 14. In many cases the powers of the quantity or quantities in the multipli- 
 cation of polynomials may be omitted, and the operation performed by the co- 
 efficients alone ; for the same powers occupy the same vertical columns, when 
 the polynomials are arranged according to the successive powers of the letters ; 
 and these successive powers, generally increasing or decreasing by a common 
 difference, are readily supplied in the final product. 
 
 EXAMPLES. 
 
 (1) Multiply xt+xty+xyt+y 3 by x— y. 
 Coefficients of multiplicand 1-f- 1-f-l + l 
 multiplier 1 — 1 
 
 1 + 1 + 1 + 1 
 —1—1 — 1—1 
 
 1 + 0+0+0—1 
 
so 
 
 ALGEBIL* 
 
 Since x 3 X£=£ 4 > the highest power of a: is 4, and decreases successively by 
 unity, while that of y increases by unity ; hence the product is 
 ^-fO-^y+O.iy+O.i!/ 3 — 2/ 4 =r» — y i = product. 
 
 (2) Multiply 3a 2 +4ax— 5x 2 by 2a 2 — 6ax+4x 2 . 
 
 3+ 4— 5 
 2— 6+ 4 
 6+ 8 — 10 
 —18— 24 + 30 
 
 + 124-16—20 
 6—10—22+46—20 
 ♦. Product =6a 4 — 10a 3 x— 22a 2 x 2 +46ax 3 — 20X 4 . 
 
 (3) Multiply 2a 3 — 3a6 2 +5Z> 3 by 2a 2 — 56*. 
 
 Here the coefficients of a 2 in the multiplicand, and a in tho multiplier, are 
 each zero ; hence 
 
 2+0— 3+ 5 
 
 2+0— 5 
 
 4 + 0— 6 + 10 
 
 _10_ Q + 15—25 
 4 + — 16+10 + 15—25 
 Hence 4a 5 — 16a 3 fc 2 +10a 2 6 3 +15afc 4 — 256 5 = product. 
 The coefficient of a 4 being zero in the product, causes that term to dis- 
 appear. 
 
 (4) Multiply x 3 — 3x 2 +3x— 1 by x 2 — 2x+l. 
 
 (5) Multiply 3/ 2 — 3/a+^a 2 by yt+ya— |a 2 . 
 
 (6) Multiply ax— 6x 2 +cx 3 by 1— x+x s — xs+x 4 . 
 
 (7) (x 3 — ax 2 +6x— c)x(a: 2 — c?x+e). 
 
 ANSWERS. 
 
 (4) r 5 — Sx^+lOx 3 — 10x 2 +5x— 1. 
 
 (5) ^-ay+ia^-^a 4 . 
 
 (6) ax — a 
 —6 
 
 x 2 +a 
 
 x J — a 
 —6 
 — c 
 
 x'+a 
 6 
 
 x 5 — & 
 — c 
 
 x 5 +cx 7 
 
 Or, ax— (a+5)x 2 +(a+fe+c)x 3 — ( a +6+c)x 4 +(a + &+c)x 5 — (& + c)x« 
 
 + CX 7 . 
 
 (7) x 5 — (a+d)x 4 +(Z>+aa 7 +e)x 3 — (c+6d+ae)x 2 +(a2+e&)x— ce 
 
 DIVISION. 
 
 15 The object of algebraic division is to discover one of the factors of a 
 given product, the other factor being given ; and as multiplication is divided 
 into three cases, so, in like manner, is division. 
 
 (1) When both dividend and divisor are monomials. 
 
 (2) When the dividend is a polynomial, and the divisor a monomial. 
 
 (3) When both dividend and divisor are polynomials. 
 
 CASE I. 
 
 16 When both dividend and divisor are monomials. 
 Write the divisor under the dividend, in the form of a fraction ; cancel like 
 
DIVISION. 21 
 
 quantities in both divisor and dividend, and suppress the greatest factor com- 
 mon to the two coefficients. 
 
 17. Powers of the same quantity are divided by subtracting the exponent 
 of the divisor from that of the dividend, and writing the remainder as the ex- 
 ponent of the quotient.* 
 
 Thus, a 1 ^=aaaaaaa ; a 4 =aaaa 
 
 a 7 aaaaaaa 
 
 ' ' a* aaaa 
 
 Generally, a m =aaaa to m factors ; a"=aaa to n factors ; 
 
 &p =bbbb to p factors ; bi =bbb to q factors ; 
 
 aH? aaa to m factors X bbb to p factors ; 
 
 * ' a n 61 aaa to n factors X bbb to q factors ; 
 
 z=aaa... to (m— n) factors xbbb to {p—q) factors, 
 
 _ =a m-njp-q - 
 
 When a quantity has the same exponent in the dividend and divisor, we have 
 
 a m . a m 
 
 _ a m-m_ a o. Dut — = 1. 
 
 a m a 
 
 .-. a° = l. 
 
 Hence every quantity whose exponent is is equal to 3 . 
 
 a? aaa 1 ' 1 t 
 
 a 5 aaaaa aa a 2 ' 
 
 But we may subtract 5, the greater exponent, from 3, the less, and affect 
 
 the difference with the sign — ; hence 
 
 a 3 a 3 I 
 
 — =a 3 - 5 =a- 2 ; but— =— ; 
 a 5 a 5 a? 
 
 1 
 a 3 
 
 * The rale for division follows from its object, which is, having one of the factors of a 
 product given to find the other. As in multiplication we join together the factors of a prod- 
 uct without any sign, and without regard to order, in division we suppress from the prod- 
 act, i. e., the dividend, one of the factors, i. e., the divisor, to obtain the other, which is the 
 quotient. Note. — The quotient must contain those factors of the dividend which are not in 
 the divisor. Note, also, that dividing 0110 of the factors of a product divides the whole 
 product. Thus, dividing a & bc by a 3 , we divide the single factor aP, and get a-bc ; so to di- 
 vide 16X12 by 8, we divide 16 alone, and get 2X12 for the quotient. 
 
 When there are factors in the divisor which are not in the dividend, the quotient may 
 be expressed in the form of a fraction, as has been previously shown (2, V.). Suppressing 
 the common factors in this case amounts to dividing both numerator and denominator by the 
 same quantity. That such a division does not alter the value of the fraction, will be obvious 
 from the following considerations : 
 
 1. If the numerator of a fraction be increased any number of times, the fraction itself will 
 be increased as many times ; and if the denominator be diminished any number of times, 
 the fraction must still be increased as many times. 
 
 2. If the denominator of a fraction be increased any number of times, or the numerator 
 diminished the same number of times, the fraction itself will, in either case, be diminished 
 the same number of times. 
 
 3. If the numerator of a fraction be increased any number of times, the fraction is in- 
 creased the same number of times ; and if the denominator be increased as many times, tho 
 fraction is again diminished the same number of times, and must therefore have its original 
 value. Hence both terms of a fraction may be multiplied by the same number, and, by 
 similar considerations, it will appear, may be divided by the same number without changing 
 the value of the fraction. 
 
 Corollary. — Rule. To multiply a fraction by a whole number, multiply the numerator of 
 the fraction, or divide its denominator by the whole number. To divide a fraction, divide 
 its numerator, or multiply its denominator. 
 
22 ALGEBRA 
 
 Similarly, -2_ = ( a+ar )-»; ] ——. = {x+ y ) 
 
 -3 
 > 
 
 A-nd -xzzz^+v 2 )- 3 ^ 2 — 2/ 2 )-^, and so on. 
 
 So, also, —=-5^= — -; 
 a 3 a 3-5 a -2 
 
 a 5 
 But -,=a 2 ; 
 
 a 3 
 
 1 
 
 ••• -=S=a a - 
 ar? 
 
 From this it appears that a factor may be transferred from the denominator 
 
 to the numerator, and vice versa, by changing the sign of its exponent. 
 
 EXAMPLES. 
 
 (1) Write a?b 3 c with the factors all in the denominator. 
 
 cPb(? 
 
 (2) Write -rnz with the factors all in one line, and also all in the denomi- 
 
 nator. 
 
 For more of the theory of negative exponents, see a subsequent article. 
 
 18. In multiplication, the product of two terms, having the same sign, is 
 affected with the sign -}- ; and the product of two terms, having different 
 signs, is affected with the sign — ; hence we may conclude, 
 
 (1) That if the term of the dividend have the sign -}-» and that of the di- 
 visor the sign -f-> the resulting term of the quotient must have the sign -|- ; 
 because + X + gives +. 
 
 (2) That if the term of the dividend have the sign -f-, and that of the divisor 
 the sign — , the resulting term of the quotient must have the sign — ; because 
 
 -X— gives +. 
 
 (3) That if the term of the dividend have the sign — , and that of the di- 
 visor the sign -\-, the resulting term of the quotient must have the sign — • 
 because -f- X — gives — 
 
 (4) That if the term of the dividend have the sign — , and that of the di 
 visor the sign — , the resulting term of the quotient must have the sign +• 
 
 RULE OF SIGNS IN DIVISION. 
 
 + divided by +> and — divided by — , give +» 
 
 — divided by -f, and -|- divided by — , give — ; 
 
 or, 
 
 ike signs give -}-, and unlike — , the same as in multiplication. 
 
 + ab , — ab , — ab . 4-ab 
 
 ±— = + b; = + b; - = — b; -2— =— 4. 
 
 -fa * —a ' -f a — a 
 
 EXAMPLES. 
 
 (1) Divide 48a 3 o 3 c 2 a 7 by 12a6 2 c. 
 
 48a 3 6 3 c 2 J AQaaabbbccd , , 
 
 r- — = — r^—Ti =4aaoca=4a 2 ocG 
 
 12ab' 2 c \2abbc 
 
 150a r 'b 8 cd 3 
 
 , — 16a 2 6 2 c 2 , , , 
 
 (3) —, =4a 2 - 1 i 2 - 1 c 2 - 1 =4a6c. 
 
DIVISION. 2J 
 
 I5a im x 3n v 4n 
 
 ( 4 1 •— =5a 2">- m x 3n_2n 7 / 4 "- £n = 5a m x n v -0 . 
 
 v ' Sa-^Y" 
 
 — 48a m 6 n 7 
 
 (5) — r— : =— 8a ra -Pi n -i. 
 
 v ' — 7a"bcd 3 x A y : >z 6 T ^ 
 
 (7) a m i n c r -i-a n & n c=a ra " r, c r - 1 . 
 
 (8) a 3m 6 n+1 c r ^4-a m 6 n c=a 2,n 6c r - 3 . 
 
 (9) 5aP-^3ai , + r 5c- 1 = *a- r i _I c. 
 
 (10) a m-n -J-a p_q =a m ~ n '" p+q - 
 
 (11) ah-±— ab= — 1. 
 
 (12) — a6c-r-a6c=— 1. 
 
 (13) _//n_^_/,m_ 1 . 
 
 a* 
 
 (14) 96a 3 &W-^84aZ> 4 c 7 cZ 5 =|^. 
 
 (15) r- 5 3/- n z-i- 3 -i-a:- 7 3/- m z-P = x 2 2/ m-n z p_<1-3 . 
 
 CASE II. 
 
 19. When the dividend is a •polynomial, and the divisor a monomial 
 Divide each of the terms of the dividend separately by the divisor.* 
 
 EXAMPLES. 
 
 (1) Divide 6a 2 xY— l2a 3 x 3 2/ 6 -|-15a 4 x 5 ?/ 3 by 3a 2 xy. 
 
 Qa"x*y 6 —12aVy 6 -\-15a 4 x 5 y 3 „ ..„«,, 
 ■' 3a3 jy ~ = 2*Y -±axy 4 + bd?x?y. 
 
 t d* 
 
 (2) Divide lba?bc — 20acif-{-5cd 2 by — babe. Ans. — 3a + 4^— ^. 
 
 (3) Divide x n+1 — x^+x 1 ^ 3 — x n+1 by x n . Ans. x— x 2 +x 3 — x 1 . 
 
 (4) Divide 5(a+b) 3 — 10{a+by+15(a+b) by — 5(a + 6). 
 
 Ans. — ( a +Z>) 2 -f 2(a+6)— 3. 
 
 (5) Divide 12a 4 i/ 6 — 16a-y4-20a 6 2/ 4 — 28a Y b y ~ 4a Y- 
 
 Ans. — 37/ 3 4-4a3/ 2 — 5a 2 y+7a 3 . 
 
 CASE III. 
 
 20. When both dividend and divisor are polynomials. 
 
 1. Arrange the dividend and divisor according to the powers of the same 
 letter in both. 
 
 2. Divide the first term of the dividend by the first term of the divisor, and 
 ihe result will be the first term in the quotient, by which multiply all the terms 
 in the divisor, and subtract the product from the dividend. 
 
 3. Then to the remainder annex as many of the remaining terms of the 
 dividend as are necessary, and find the next term in the quotient as before. 
 
 (1) Divide a 4 — 4a 3 x+6a 2 x 2 — 4ax 3 +x 4 by a-— 2ax+x 2 . 
 
 a 2 — 2ax+x 2 ) a 4 — 4a 3 x+6a 2 x 2 — 4ax 3 +x 4 (a 2 — 2ax-\-x i 
 a 4_2a 3 x+ a 2 r 2 
 
 — 2a 3 x-4-5a 2 x 2 — 4ax? 
 — 2a 3 x-f4a 2 x-— 2ax 3 
 
 a-x- — 2ax 3 -\-x 4 
 a 2 x 2 — 2ax 3 -|-x 4 
 
 * This rule follows from that for multiplication, which requires each term of the multipli- 
 cand to be repeated as many times as is expressed by the multiplier. 
 
24 ALGEBRA. 
 
 Arranging the terms according to the descending powers of x, we have 
 i 2 — 2ax+a 2 ) x 4 — 4ax 3 +6a 2 x 2 — 4a 3 r+a 4 (x 2 — 2ax+a 2 
 x 4 -— 2ax 3 -|- a 2 x 2 
 
 — 2ax 3 - r -5a 2 x 2 — 4a 3 x 
 — 2ax 3 +4a 2 x 2 — 2a 3 x 
 
 a 2 x 2 — 2a 3 x+a 4 
 a 2 x 2 — 2a 3 x-|-a 4 .* 
 
 (2) Divide x 4 -\-x 2 y i -\-y* by x 2 +x?/4-2/ 2 . 
 
 x 2 +2:3/+2/ 2 ) x 4 4-x 2 ?/ 2 -|-2/ 4 ( x2 — x y~\-y' i 
 
 tf-^&y -}-x 2 ?/ 2 
 
 — x*y +y* 
 
 — y?y — x 2 3/ 2 — xy s 
 
 x*f+xtf+y* 
 
 x 2 y 2 -\-xy n -\-y*. 
 
 * It has been shown (13) that when the dividend (which is the product of the divisor and 
 quotient) is arranged as directed in the rule, its first term is produced without reduction by 
 the multiplication of the first term of the divisor by the first of the quotient. Hence the 
 rule above for finding the latter. This first term of the quotient being found, and tho di- 
 visor being taken away from the dividend as many times as is expressed by this term, the 
 remainder must contain the divisor as many times as is expressed by the second and re- 
 maining terms of the quotient. Hence the remainder may be regarded as a new dividend, 
 and the object being to find how many times it contains the divisor, it must be arranged in 
 the same manner as was the given dividend, and the first step will be the same as before. 
 Similar reasoning will apply to the rest of the process. 
 
 Note. — The arrangement of the terms is for convenience. The term having the highest 
 or lowest exponent of some letter might be selected from the dividend and remainders with- 
 out any arrangement. The operation must always, however, begin with this term, as a 
 reference to the last example will show ; for if we attempt to commence with the term 
 Scfix 2 , the third of the dividend, for instance, we perceive that this is produced by reduction 
 from the term a^x 2 in the second line, the term Acfix 2 in the fourth line, and the term a 2 x' 1 
 in the sixth. The first of these is produced by the multiplication of the first of the quotient 
 by the last of the divisor, the second by the multiplication of the second of the quotient by 
 the second of the divisor, and the third by the last of the quotient and first of the divisor. 
 It is not till the first and second terms of the quotient have been found by the rule above 
 given, that any portion of the term 6a°x~ presents itself to be divided, or that we can know 
 what part of it is to be used as a dividend. 
 
 In the same manner, it may be shown that it would be impossible to begin with the second 
 term of the dividend 4ax% until the first term of the quotient has been found, which, multi- 
 plied by the second of the divisor, produces — 2ax 3 , a part of — Aax 3 , and the subtraction 
 leaves the other part — 2ax z , which now we know is the product of the first of the divisor 
 by the second of the quotient, which latter we may then find. 
 
 The first of the divisor multiplied by the second of the quotient, and the second of the 
 divisor by the first of the quotient, usually produce the same power of the letter of arrange- 
 ment, and reduce together; the first and third of each, together with the two second terms 
 of each, usually produce the same power, and so on. It is only the first of the divisor and 
 first of the quotient, or last of the divisor and last of the quotient, which always produce a 
 term that does not reduce with any other term. 
 
 N.B. — The arrangement may begin with the lowest as well as the highest power of any 
 letter, and go on increasing instead of decreasing. When either of these arrangements is 
 observed, if the first term of the divisor in any part of the operation is not contained exactly 
 in the first term of the remainder, the division is impossible. By varying the arrangement, 
 therefore, or simply considering which terms would come first, using different letters of ar- 
 rangement, we may often determine beforehand by inspection whether the division is pos- 
 sible or not. 
 
DIVISION. 25 
 
 Another form of the work which has the convenience of placing the quotient 
 which is the multiplier, under the divisor, which is the multiplicand, is the 
 following. 
 
 Dividend; x 4 -\-x 2 y' 2 -}- y 
 x 4 -\-x 3 y -\-x~y- 
 
 i'--\-ry-\-y' 2 , divisor. 
 x- — xy+y*, quotient. 
 
 • x 3 y +y A 
 
 — x*y — x-y 2 — xy 3 
 
 xiyij^xif-^y* • 
 
 x' 2 y' 2 -\-xy s -\-y 4 
 
 (3) Divice a 5 — a 3 Z> 2 +2a 2 & 3 — ab*+b 5 by a 2 — a& + & 2 . 
 
 a*— ab+b*) a 5 — a 37 > 2 +2a 2 6 3 — a& 4 +& 5 (a 3 +a 2 6 — aV+ a? _ ah , 6 , 
 
 a 4 6 — 2a 3 6 2 -f2a 2 6 3 
 a*b — a 3 Z> 2 + a 2 6 3 
 
 — a 3 6 2 + a 2 6 3 — ab* 
 
 — a 3 & 2 4- a?b 3 —ab* 
 
 -f 6 5 
 
 Arranging the tei-ms accoiding to powers of 6, we get 
 
 — a 4 6-fa 5 
 b*—ab + a*) ¥— a b*+2a°-b 3 — a 3 6 2 +a 5 (^ 3 +^+ fc3 _ a6 ■ a » 
 
 a s 6 3 — a 8 6 B +a 6 
 
 a 2 6 3 — a 3 6 2 +a 4 & 
 
 _a 4 fe-fa 5 . 
 
 The results we have obtained in these two arrangements are apparently 
 different ; but their equivalence will be established as follows : 
 (1) (a 2 — ab+b 2 ) (a 3 +a 2 6— a& 2 )=a 5 — a 3 i 2 +2a 2 & 3 — ab* 
 
 Add remainder = -j-^ 5 
 
 Proof a^a 3 Z> 2 +2a 2 6 3 -a& 4 +& 5 
 
 (2) (5 2 — ab-\-a") (b 3 +a 2 b) =¥—a 6<+2a 2 5 3 — a 3 6 2 +a 4 & 
 
 Add remainder = — a 4 fo-}-a 5 
 
 Proof 65— a b 4 +2a°b 3 — a 3 6 s -f a 5 . 
 
 The moment we arrive at a term of the quotient in which the exponent of 
 the letter of arrangement is less than the difference of the exponents of this 
 letter in the last terms of the divisor and dividend, we may be sure that the 
 division will not terminate. If the divisor and dividend be arranged in the re- 
 verse order, that is. beginning with the lowest power of a letter, then the 
 division will not terminate when the exponent of this letter in the term of 
 the quotient is greater than the difference of its exponents in the last terms of 
 the divisor and dividend. 
 
 Thus in tbe following example. 
 
 x 9 + x~ — ax 5 + ax 1 
 3?+ 3*+ as 5 
 
 3* +3?+ a 
 
 .c 5 - 
 
 — x 8 -|- a-' 7 — 2ax 5 4- ax* 
 — x 3 — x 1 — ax 4 
 
 2x — 2ax 5 -\-2ax i . 
 
26 ALGEBRA. 
 
 The last term of the quotient must be x 4 , in order that, multiplied by a, tn« 
 last of the divisor, it may produce the last of the dividend. If, therefoi - e, the 
 division is not completed when this term containing x 4 is obtained, it will not be 
 
 EXAMPLES FOE PRACTICE. 
 
 (1) Divide a 2 — 2ab-\-b 2 by a — b. 
 
 (2) Divide a 2 4-4ax4-4x 2 by a+2x. 
 
 (3) Divide 12x*— 192 by 3.r— 6. 
 
 (4) Divide 63— 6y 6 by 2x' 2 —2y". 
 
 (5) Divide a^Sa^-^Sa^—b 6 by a 3 — 3a 2 6 + 3a& 2 — b 3 . 
 
 (6) Divide x 3 4-5.r 2 2/4-5x,y 2 4-2/ 3 by x^-^-ixy-^-y 2 . 
 
 (7) Divide x 5 — y b by x — y. 
 
 (8) Divide a 4 — 6 4 by a 3 +a 2 6+ai 2 +6 3 . 
 
 (9) Divide x 3 — 9x 2 4-27x— 27 by x— 3. 
 
 (10) Divide x*-|-2/ 4 by x+?/. 
 
 (11) Divide 48x 3 — 76ax 3 — 64a 2 x4-105a 3 by 2x— 3a. 
 
 (12) Divide ix 3 4-x 2 4-fx4-£ by ±x+l. 
 
 (13) Divide 52m. 5 — 93m 4 p— 70m 3 jp 2 + 48m 2 p 3 — 27mp 4 by 13m 3 — InPf 
 -J-3mp 2 . 
 
 (14) Divide 33a 3 6 3 — 77a 2 6 4 +121a 2 6 5 by 3a 2 Z>— 7aZ> 2 4-lla& 3 . 
 
 (15) Divide (6p 4 — 12pa 3 — 6^ 3 o+12 ? 4 ) by (i? — 9). 
 
 (16) Divide (100a 5 — 440a 4 &4-235a 3 & 2 — 30a=& 3 ) by (5a 3 — 2a*£) 
 • (17) Divide (g 4 — 4g- 3 /i+6g- 2 /i 2 — 4gh*+h*) by (h*—2hg+g*). 
 
 (18) Divide (37a 2 m 2 — 26a 3 m4-3a 4 — 14am 3 ) by (3a 2 — 5am+2m-). 
 
 (19) Divide (a 6 — Z> 6 ) by (a— 6) and (a 6 +6 6 ) by (a+6). 
 
 (20) Divide (a 7 — & 7 ) by (a— 6) and (a 7 +6 7 ) by (a+fc). 
 
 (21) (£_62 2 +27z 4 ) -r (i+2z+3z 2 ) = l— 6z+92 8 . 
 
 ANSWERS. 
 
 (!) a — b. 
 
 (2) a+2x. 
 
 (3) 4r 3 +8x 2 + 16x+32. 
 
 (4) 3x*+3x*y*+3y*. 
 
 (5) a 3 +3a 2 Z)+3a& 2 +Z> 3 . 
 
 (6) x+y. 
 
 (7) . r J + X 3 ?/-|-X 2 7/ 2 + X7/ 3 + 2/ 4 . 
 
 (8) a—b. 
 
 (9) x 2 — 6x+9. 
 
 (10) x *-x*y+xy*-y 3 +-^. 
 
 (11) 24x 2 — 2ax— 35a 2 . 
 
 (12) tf+f. 
 
 (13) 4m 2 — 5mp— 9p 3 . 
 
 (14) llai 2 . 
 
 (15) 6p»— 12a 3 . 
 
 (16) 20a 2 — 80a&4-15& 2 . 
 
 (17) g* — 2ghJ r h' ! : 
 
 (18) a?— 7am. 
 
 C a 5 + a 4 2> 4- a 3 i 2 + a 2 i 3 4- a& 4 4- & 5 , and 
 M<N < 2fc" 
 
 ^ ) a 5 — a 4 6 4- a 3 i 2 — a*V> + ab* — fc*+— rz- 
 
DIVISION. 
 
 r as+a^-t-a^+a^+a^+a^+i 6 , and 
 I 20 ) j a e_ G 6 Z>+a<6 2 — a 3 i 3 +a 2 & 4 — ab>+¥— ^-p. 
 
 EXAMPLES WITH LITERAL EXPONENTS. 
 
 (1) Divide 2a 3n — 6a 2n & n +6a. n & 2n — 2& 3a by a a — 6". 
 
 a°— 6") 2a 3n — 6a 2n 6 n + 6a n 6 2B — 26 3n (2a? a —4a a b n +2b a * 
 2a 3n — 2a 2n 6 u 
 
 _4a' 2 "6"+tJa"^ n 
 _4a 2n 6 n -(-4a"^ ;: " 
 
 27 
 
 2a"i 2 "— 26 3u 
 2a n i 2n — 2& 3n . 
 
 (2) Divide x m+1 -\-x m y+xy m +y m+l by x m +2/ m - 
 
 (3) Divide a n — x" by a— x. 
 
 (4) Divide x 4n +a: 2,, ?/ 2n +2/- ,n by x 2n +xy , -f-2/ 2n . 
 
 (5) Divide a m + n 6 n — 4a m + n - 1 & 2n — 27a ra + n - 2 i 3n + 42a m + n - 3 6< n by a n 6» 
 — 7a 1> - 1 b' 2n . 
 
 (6) Divide a 3ra - 2 "6 2 Pc— a 2m + n - 1 6 1_ ' > c n + a- n &- I e m + a3m-nj 3p +2 c n _ a 2ro+2n-i&3 
 
 c 8n-l_j_Jp+l c m+n-l {jy Q-n^-p-l^. fee" -1 . 
 
 ANSWERS. 
 
 (2) x+y. 
 
 (3) (? n - 1 +a n - B a;+aH-«a^+ 
 
 a n - 3 x 3 — x n 
 
 c — x 
 
 (4) a; 2n_ ;r n 2/ n_|_^2n > 
 
 (5) a m +3a m - l b n —6a m - 2 b° n . 
 
 (6) a 3m_n i 3D+1 c— a 2ra+2n - 1 i 2 c n + 6Pc m . 
 
 EXAMPLES WITH LITERAL COEFFICIENTS.* 
 
 (1) Divide ax 5 4-ax 4 +ix 4 4-ax 3 4-6x 3 +c.r 3 +ax 2 +6x 2 +cx 2 +6x+cx4-c by 
 ax^-^-bx-^-c. 
 
 Arrange the terms of the dividend in the following manner, in order to keep 
 the operation within the breadth of the page. 
 
 ax 2 +fcx-J-c) ax 5 -}- a 
 b 
 
 x A -\-a 
 b 
 
 X s 4- a 
 b 
 
 x 2 +?; 
 c 
 
 ax 5 +6 x 4 -f- c x 3 
 
 x-f-c (x 3 -f•x 2 4-• r +l■ 
 
 a x 4 -f-fl 
 6 
 
 b 
 
 x J 
 
 ax*-\-b xs+cx 3 
 
 a x 3 -{-a 
 b 
 
 x 2 -f Z> 
 c 
 
 X 
 
 a x 3 -\-b x 2 -\-c x 
 
 a x 3 -f-6 x-\-c 
 
 a x 2 -f-6 x-\-c 
 
 * The literal multipliers of each power of the letter of arrangement are to be collected 
 together, and regarded as a polynomial coefficient of that power, which is to be treated 
 exactly in the process of division as a numerical coefficient would be, observing only the 
 four ground rules applicable to polynomials instead of numbers. 
 
28 
 
 ALGEBRA. 
 
 46 2 
 
 Divid. 
 
 Product 
 to sub- 
 tract. 
 
 1st rem. 
 or 2d 
 divid. 
 
 Product 
 to sub- « 
 tract. 
 
 2d rem. 
 or 3d 
 divid. 
 
 Product 
 to sub- 
 tract. 
 
 3d rem. 
 
 a 3 — 166 3 
 
 + 
 + 
 
 + 
 
 + 
 
 46 2 c 
 26c 2 
 2c 3 
 
 86 3 
 
 46 2 c 
 
 26c 2 
 
 <z 2 +326 4 
 
 — 86 3 c 
 
 — A¥c 
 + 26c 3 
 
 — c 4 
 
 (2)* 
 
 a— 32b 5 
 + 166 4 c 
 + 86 3 c 2 
 — 46 2 c 3 
 
 Divis. 
 
 — 86 3 
 
 — c 3 
 
 a 2 +326 4 
 
 — 86 3 c 
 
 — 46 2 c : 
 + 26c 3 
 
 — c* 
 
 — 166* 
 
 86 3 c 
 
 46 2 c 2 
 
 46 2 c 2 
 
 26c 3 
 
 c 4 
 
 a— 32b 5 
 + 166 4 c 
 + 86 3 c 2 
 — 46V 
 
 + 
 + 
 
 + 
 
 + 1664 
 — 46 2 c 2 
 
 a— 32b 5 
 + 166 4 c 
 + 86 3 c 2 
 
 — 46 s c 3 
 + 326 5 
 
 — 166 4 c 
 
 — 86 3 e 2 
 + 46 2 c- 3 
 
 o. 
 
 1st Partial Division. 
 4b 2 — c 3 { 26 + c 
 -c 
 
 • c 2 ( 2b-\-c 
 
 • 26c \ 26 -( 
 
 -26c— c 2 
 + c 2 
 
 2<i Partial Division. 
 —8b 3 — c 3 f 26 + c 
 
 + 46 2 c ( — 46 2 +26c— <* 
 
 + 46 2 c— c 3 
 —26c 2 
 
 ■ 26c 2 — c 3 
 + c 3 
 
 3d Partial Division. 
 6 4 _ 46 2 c 2 ( 26+c 
 
 46*7 
 
 l — 46 2 c 2 < 26+c 
 — 86 3 c ( 8b 3 — < 
 
 ■86 3 c— 46 2 c 2 
 + 46 2 c 2 
 
 (3) Divide x 3 +ax 2 +6x+c by x — r. 
 
 x—r) x 3 +ax 2 +6x+c (x 2 +(r+a)x+(r 2 +ar+6) 
 x 3 — rx 2 
 
 (r+a)x 2 +6x 
 (r+a)x 2 — (^-{•arfx 
 
 (r 2 +ar+6)x+c 
 (r 2 +ar+6)x— (rs+a^+ir) 
 
 r 3 +ar 2 +6r+c, remainder 
 
 In the preceding and similar examples, the remainder differs only from the 
 dividend in having r instead of x. That this is always the case when the 
 iivisor is x minus some quantity, will be shown hereafter. (Art. 238, Pr. I.* 
 
 (4) Divide x 3 — <zx 2 +6x — c by x — r. 
 
 (5) Divide x 3 — (a + 6+c)x 2 +(a6 + 6c+ca)x — a6c by x — a. 
 
 (6) Divide x 3 — (a+2).t' 2 +(2a + 6)x— 26 by x— 2. 
 
 (7) Divide lla 2 6— 19a6c+10a 3 — 15a 2 c+3a6 2 +156c 2 — 56 2 c by 5a 2 +3ai» 
 — 56c. 
 
 (8) Divide x 3 — (a+6 + d)x 2 +(ad+6rf+c)x— cd by x 2 — (a+6)x+c. 
 
 (9) Divide x m +j?x m - 1 +gx m - 2 +rx m - 3 +, &c... . + te+« by x— a. 
 
 * N.B. The signs of the products to subtract are actually changed in this example before 
 they are written; a method which is sometimes practised. Their first terms need not bo 
 written, since they aro cancelled by the first terms of the corresponding dividends. 
 
DIVISION. 
 
 29 
 
 (10) Divide 
 
 a 4 
 
 x^+a 5 
 
 x 3 — a B b 
 
 x 2 4-a 4 & 3 
 
 —a*b 
 
 —a 4 b 
 
 —2a 4 b 2 
 
 + 2a 3 Z> 4 
 
 +a 2 6 2 
 
 + a 3 6 2 
 
 — ab 5 
 
 
 — ab* 
 
 -\-a"b 3 
 
 
 
 -a 2 £ 6 by 
 
 a* 
 
 — a& 
 
 x 2 +a 3 x— a 2 6 3 . 
 
 When there are negative exponents of the letter of arrangement, they 
 come after the term containing x", i. e., the term in which x does not appear, 
 those which have the greatest absolute value being placed last. 
 
 (11) Divide — x 3 — x 2 -}-10x+§— '/aT*— ^ar'-f ftr- 9 by x 2 — 2x— 2+1*-* 
 +|x-». 
 
 
 
 
 
 ANSWERS. 
 
 
 (4) x 2 +(r— <z)x+(r 2 - 
 
 -ar-^-b), and remainder 
 
 is r 3 — a? -2 -{-£r — c. 
 
 (5) x 2 — \b-\-c)x+bc. 
 
 
 
 (6) x 2 — ax+6. 
 
 
 
 (7) 2a+6 — 3c. 
 
 
 
 (8) x—d. 
 
 
 
 (9) x^-f-a 
 
 i n - B 4- a 2 
 + ? 
 
 x m-3_|_ a 3 
 
 -fa 2 ?? 
 
 + aq 
 
 x m-4 -f-, &c. 
 
 4-a ra - 2 p 
 
 -\-a m ~ 3 q 
 _j_ a m— i r - 
 
 (10) a 2 
 
 + 6 2 
 
 x 2 — a 2 6 
 — a& 2 
 
 x+ 
 
 V. 
 
 +'t 
 
 (11) -x- 
 
 -3 
 
 + 2.r 
 
 -i_ 
 
 
 
 
 21. In those cases in which the division does not terminate, and the quotient 
 may be continued to an unlimited number of terms, the quotient is termed an 
 infinite teries, and then the successive terms of the quotient are generally reg- 
 ulated bv a law which, in most cases, is readily discoverable. 
 
 EXAMPLES. 
 
 (1) 
 
 Divide 1 
 
 by 1 — x. 
 
 
 1— x) 
 
 1— X 
 
 
 
 +x 
 4-x— X 2 
 
 -fx 2 
 
 -j-X 2 — X 3 
 
 -fx 3 
 
 The quotient in this case is called an infinite series, and the law of formation 
 of this series is, that any term in the quotient is the product of the immedi- 
 ately preceding term by x. 
 
 (2) Divide 1 by 1-f x. Ans. 1— x-f x 2 — x'-fx 4 — ... 
 
 (3) Divide 1-f x by 1— x. Ans. 1-f 2r+2.t--f 2.^+ 2x*+ .. 
 
 Ans. xr 1 — x~ 2 -f-x- 
 
 •-fz-s — 
 
 (4) Divide 1 by x-f 1. 
 
 (5) Divide x — a by x — b. 
 
 Ans. 1 — (a — b)x- x — (a— b)bx~"— (a— o^x -3 — 
 
 (6) Divide 1 by 1— 2x-f x 2 . Ans. 1-f 2x-f 3x 2 -f 4x3+ 5x*-f .... 
 
30 ALGEBRA. 
 
 22. When a polynomial is the product of two or more factors, it is often 
 requisite to resolve it into the factors of which it is composed, and merely to 
 indicate the multiplication. This can frequently be done by inspection, and 
 ly the aid of the following formulas : 
 
 (x+a){x+b)=x"-+{a + b)x-\-ab (1) 
 
 {x+a)(x — b)=x*+ (a — b)x—ab (2) 
 
 (x— a)(x+6)=x 2 — {a — b)x— ab (3) 
 
 (x—a){x—b)=x°— (a+b)x+ab (4) 
 
 (a + b){a — b)=a' i —V (5) 
 
 (n+l)(n+l)=» a +2n+l (6) 
 
 (n— J)(n— l)=n s — 2n-f 1 (7) 
 
 EXAMPLES. 
 
 (1) Resolve ax' 2 -\-bx' 2 — ex 2 into its component factors. 
 
 Here ax 2 +bx 3 — cx 2 =x 2 (a + fc — c). 
 
 (2) Transform the expression n?-\-2n?- t r n into factors. 
 
 Here n?-\-2ri i -\-n=n{n 2 -\-2n+l) 
 
 =n(n+l)(»+l)by (6) 
 =n(n-\-l) 2 . 
 
 (3) Decompose the expression x 2 — x— 72 into two factors. 
 
 By inspecting formula (3), we have — 1 = — 9+8, and —72=— 9X8 
 hence x 2 — x— 72=(x— 9)(x+8). 
 
 (4) Decompose 5a 2 bc-\-lQab-c-{-l5abc 2 into two factors. 
 
 (5) Transform 3m 4 n 6 — 6m 3 n 5 p-\-3m 2 n 4 j) z into factors. 
 
 (6) Transform 3fe 3 c — 35c 3 into factors. 
 
 (7) Decompose x 2 - r -8x-fl5 into two factors. 
 
 (8) Decompose x 3 — 2x 2 — 15x into three factors. 
 
 (9) Decompose x 2 — x— 30 into factors. 
 
 (10) Transform a 2 — 6 2 -f-26c — c 2 into two factors. 
 
 (11) Transform a 2 x— x 3 into factors. 
 
 ANSWERS. 
 
 (4) 5abc(a+2b-\-3c). 
 
 (5) 3m 2 n 4 (mn — pf. 
 
 (6) 3bc{b + c){b — c). 
 
 (7) (x+3)(x+5). 
 
 (8) x(x+3)(x— 5). 
 
 (9) (.r+5)(x-6). 
 
 (10) (a + b— c){a — b+c). 
 
 (11) x(a-\-x)(a — x). 
 
 23. By the usual process of division we might obtain the quotient of a" — 6" 
 divided by a — b, when any particular number is substituted for n; but we 
 shall here prove generally that a n —b n is always exactly divisible by a— b, and 
 exhibit the quotient. 
 
 Tt is required to divide a n — 6 n by a — b. 
 
 a-b)a"-b» (a"-> + ^ -' 
 
 a n — a n ~ x b 
 Rem. a"-ifc_Z> n ; 
 Rem. under another form, b(a n - 1 — & n_1 ). 
 
 Hence, ^T^ + a - (1> 
 
DIVISION. 31 
 
 Now it appears from this result, that a" — b" will be exactly divisible by 
 a — b, if « n_i — b"" 1 be divisible by a — b ; that is, if the difference of the same 
 powers of two quantities is divisible by their difference, then the difference 
 of the powers of the next higher degree is also divisible by that difference. 
 
 But a- — b 2 is exactly divisible by a — i, and we have 
 
 ^=b= a + b < 2 > 
 
 And since a 2 — Z> 2 is divisible by a — b, it appears, from what has been just 
 proved, that a 3 — b 3 must be exactly divisible by a — b ; and since a 3 — b 3 is di- 
 visible, a 4 — b 4 must be divisible, and so on ad infinitum. 
 
 Hence, generally, a n — b n will always be exactly divisible by a — b, and give 
 the quotient 
 
 a n — b n 
 
 — — T-=a' w +a ,1 - s & + a n - 3 & 2 + a'b n ' 3 -\-nb n ~'^.b a ' 1 (5) 
 
 In a similar manner, we find, when n is an odd number, 
 
 a"4-b n 
 
 — ~- r =a n - 1 — a n - 2 6 + a n - 3 6 2 — M a -b a ~ 3 — ab n ~ 3 +b n ~ 1 .... (6) 
 
 a-f-o i i i \ 
 
 And when n is an even number 
 
 a n b n 
 
 — — = a"-i— a°- 2 6+a n - 3 i 2 — _ a 2 i n - 3 +aZ> n - 2 — b n ~ l .... (7) 
 
 a-\-b 
 
 By substituting particular numbers for n, in the formulas (5), (G), (7), we 
 may deduce various algebraical formulas, several of which will be found in the 
 following deductions from the rules of multiplication and division. 
 
 USEFUL ALGEBRAIC FORMULAS. 
 
 (1) a 3 — ¥=(a+b){a — b). 
 
 (2) a 4 — 6«=(a s +& s )(a 3 — i 2 ) = (a 2 +6 2 )(a+6)(a — b). 
 
 (3) a 3 — 6 3 =(a 2 +a&-f-i 2 )(a — b). 
 
 (4) a*+b°={a*— ab+b°~)(a+b). 
 
 (5) a«— bs=(a 3 +b 3 ){a 3 — b 3 ) = {a 3 +b 3 ){ai+ab + b*)(a — b). 
 
 (6) a 6 — b«={a 3 +b 3 )(a 3 — Z> 3 ) = (a 3 — 6 3 )(a 3 — ab + b*){a+b). 
 
 (7) a e — &«=(a 3 +6 3 )(a 3 — & 3 ) = (<z 2 — 6 2 )(a 4 +a 2 & 2 +& 4 ). 
 
 (8) a 6 — i«=(a + 6)(a— &)(a 2 +a&-f-6 2 )(« 2 — a&+6 2 ). 
 
 (9) (a"-—b-)^-{a—b)=a-{-b. 
 
 (10) (a 3 — 6 s )4-(«— &)=a 2 -l-a&+&*. 
 
 (11) (a 3 +6 3 )-i-(a + 6)=a a — a&+Z> 2 . 
 
 (12) (a 4 — & 4 )^-(a+&)=a 3 — a 2 &-fa& 2 — & 3 . 
 
 (13) (a 5 — &*)-i-(a — &)=a 4 +a 3 Z> + a 2 & 2 +a& 3 -|-Z> 4 . 
 
 (14) (a 6 +6 5 )-H(a + ^)=»« 4 — a 3 i+a 2 6 2 — a& 3 -f & 4 - 
 
 (15) (a 6 — i 6 )-^-(a 2 — 6 2 )=a 4 -fa' : i-+6 4 . 
 
 DIVISION BY DETACHED COEFFICIENTS. 
 
 24. Arrange the terms of the divisor and dividend according to the success- 
 ive powers of the letter, or letters, common to both ; write down simply the 
 coefficients with their respective signs, supplying the coefficients of the absent 
 terms with zeros, and proceed as usual. Divide the highest power of the 
 omitted letters in the dividend by that of the omitted letters in the divisor, 
 and the result will be the literal part of the first term in the quotient. The 
 
32 ALGEBRA. 
 
 literal parts of the successive terms follow the same law of increase or de- 
 crease as those in the dividend. The coefficients prefixed to the literal parts 
 will give the complete quotient, omitting those terms whose coefficients are 
 Eero. 
 
 EXAMPLES. 
 
 (i) Divide 6a 4 — 96 by 3a— 6. 
 
 3_6) 6+ 0+0+0—96 (2+4 + 8+16 
 6—12 
 
 12 
 
 
 
 
 12- 
 
 -24 
 
 
 
 
 24 
 
 
 
 
 24- 
 
 -48 
 
 
 
 
 48- 
 
 -96 
 
 
 
 48- 
 
 -96 
 
 But a^-^-a^a 3 , and the literal parts of the successive terms, are, therefore 
 a 3 , a 3 , a 1 , a , or a 3 , a 3 , a, 1 ; hence, 2a 3 +4a 2 +8a+16= quotient. 
 
 (2) Divide 8a 5 — 4a 4 ar— 2a 3 x 3 +a 2 x 3 by 4a 3 — x 3 . 
 
 4 + 0—1) 8 — 4—2 + 1 (2—1 
 8+0 — 2 
 _4 + 0+l 
 _4_0 + l 
 
 Now, a b -±-a?=.a?; hence a 3 and cfiz are the literal parts of the terms in the 
 quotient, for there are only two coefficients in the quotient ; therefore 
 
 2a 3 — a 3 x= quotient required. 
 
 (3) Divided— 3az 3 — 8a 3 x 3 +18a 3 a:— 8a 4 by a: 3 +2a:r— 2a 3 . 
 
 (4) Divide 32/ 3 +3x2/ 3 — 4x*y— 4Z 3 by x-\-y. 
 
 (5) Divide 10a 4 — 27a 3 x+34a 2 x 2 — 18ax*— 8X 4 by 2a 3 — 3ax+4:r*. 
 
 (6) Divide a 4 +5a 3 +a+5 by a 3 +l. 
 
 (3) x 2 — 5aa:+4a 2 . 
 
 (4) _4x 2 +3i/ 3 . 
 
 ANSWERS. 
 
 (5) 5a 2 — 6aa:— 2x*. 
 
 (6) a +5. 
 
 SYNTHETIC DIVISION. 
 
 RULE.* 
 
 25. (1) Divide the divisor and dividend by the coefficient of the first term in 
 
 * The rule here given for Synthetic Division is due to the late W. G. Horner, Esq., of 
 Bath, whose researches in science have issued in several elegant and useful processes, 
 especially in the higher branches of algebra, and in the evolution of the roots of equation of all 
 dimensions. 
 
 In the common method of division, the several terms in the divisor are multiplied by the 
 first term in the quotient, and the product subtracted from the dividend ; but subtraction is 
 performed by changing all the signs of the quantities to be subtracted, and then addir.^ 
 the several tenns in the lower line to the similar terms in the higher. If, therefore, the 
 signs of the tenns in the divisor were changed, we should have to add the product of the 
 divisor and quotient instead of subtracting it. By this process, then, the second dividend 
 would be identically the same as by the usual method. We may omit altogether the 
 products of the first term in the divisor by the successive tenns in the quotient, because 
 in the usual method the first term in each successive dividend is cancelled by these prod- 
 acts. Omitting, therefore, these products, the coefficient of the first term in any dividend 
 
DIVISION. 33 
 
 the divisor, which will make the leading coefficient of the divisor 'ju.ity, and 
 the first term of the quotient will be identical with that of the dividend. 
 
 (2) Set the coefficients of the dividend in a horizontal line with their proper 
 signs, and those of the divisor, with the signs all changed except that of the 
 first, in a vertical column on the right or left, drawing a line under the whole, 
 underneath which to write the quotient. 
 
 (3) Multiply all the terms so changed by tho first term in the quotient, and 
 place the products successively under the corresponding terms of the dividend, 
 in a diagonal column. 
 
 (4) Add the results in the second column, which will give the second term 
 of the quotient ; and multiply the changed terms in the divisor by this, placing 
 the products in a diagonal series, as before. 
 
 (5) Add the results in the third column for the next term in the quotient, 
 by which, again, multiply the changed terms in the divisor, placing the prod- 
 ucts as before. 
 
 (G) This process, continued till the last line of products extends as far to the 
 right as the dividend, will give the same series of terms as the usual mode of 
 division. 
 
 EXAMPLES. 
 
 (1) Divide a 5 — Sa'x+lOa 3 ^— lOaW+Sax*— .r 5 by a 2 — 2ax+x-. 
 
 1 
 
 1—5+10— 10+5 — 1 
 
 + 2 
 
 + o_ 6 _|_ 6—2 
 
 — 1 
 
 — 1+ 3 — 3 + 1 
 
 1 — 3 + 3_ l * * 
 Hence a 3 — 3a 2 .r+3ax 2 — x 3 = quotient. 
 
 In this example the coefficients of the dividend are written horizontally, and 
 those of the divisor vertically, with all tho signs of the latter changed, except 
 the first. Then +2 and — 1, the changed terms in the divisor, are multiplied 
 by 1, the first term of the quotient, which is written in the horizontal line at 
 the bottom, and is the same as the first term of the dividend ; the products 
 + 2 and — 1 are placed diagonally, under — 5 and +10, the corresponding 
 lerms of the dividend. Then, by adding the second column, we have — 3 for 
 the second term in the quotient, and the changed terms +2 and — 1 in the 
 divisor, multiplied by — 3, give — 6 and +3, which are placed diagonally un- 
 der + 10 and — 10. The sum of the third column is +3, the next term in 
 the quotient, which, multiplied into the changed terms of the divisor, gives 
 + G — 3, for the next diagonal column. The sum of the fourth column is — 1, 
 and by this we obtain the last diagonal column — 2+1. The process here 
 terminates, and the sums of the fifth and sixth columns are zero, which shows 
 that there is no remainder. If the last terms did not reduce to zero by addi- 
 tion, their sum would be the coefficients of the remainder ; the quotient is com- 
 pleted by restoring the letters, as in detached coefficients. 
 
 Having made the coefficient of the first term in the divisor unity, that co- 
 will be the coefficient of the succeeding term in the quotient, the coefficient in the first 
 term of the divisor being unity ; for in all cases it can be made unity by dividing: both 
 divisor and dividend by the coefficient of the first term in the divisor. The operation, tbus 
 simplified, may, however, be farther abridged by omitting the successive additions, except 
 so much only as is necessary to show the first term in each dividend, which, as before re 
 marked, is also the coefficient of the succeeding term in the quotient. 
 
 c 
 
34 ALGEBRA. 
 
 efficient may be omitted entirely, since it is of no use whatever in continuin 
 the operation here described. 
 
 (2) Divide x*— hx*+\5x*— 24x 3 +27x 2 — 13x+5 by z 4 — 2r J +4a; 8 — 2x+l 
 l_5_j_15_24 + 27 — 13+5 
 +2 _|_o_ 6-1-10 
 — 4 — 4 + 12—20 
 
 + 2 + 2— 6+10 
 
 —1 — 1+ 3—5 
 
 1_3+ 5 
 Hence .r 2 — 3:r+5=z: quotient required. 
 
 (3) Divide a 5 +2a 4 b+3aW— a°~b 3 — 2ab*— 36 5 by a 2 +2a&+3&». 
 1 + 2+3—1—2—3 
 —2+0 + + 2 
 —3+0+0+3 
 
 —2 
 —3 
 
 1+0 + 0—1 
 Hence a 3 +0-a 2 Z>+0-aZ> 2 — b 3 =a 3 — & 3 = quotient. 
 
 (4) Divide 1— x by 1+z. Ans. 1— 2x+2:e 2 — 2r 3 +, &c. 
 
 (5) Divide 1 by 1—x. Ans. l+.r+^+x 3 -}-. &c. 
 
 (6) Divide x 1 — y 7 by x — y. Ans. x & -\-x 6 y-{-x 4 y' i -\-x z y z -i r x^y i -\-xy' a -\-y t . 
 
 (7) Divide a 6 — 3a 4 .r 2 +3a 2 :r 4 — a* by a 3 — 3a 2 .r+3az 2 — r 5 . 
 
 Ans. a 3 +3a 2 x+3ax 2 +x 3 . 
 
 (8) Divide a 5 — 5a 4 ar+10a 3 :c 2 — IQcPtf+bax*— x 5 by a 2 — 2ax\-x". 
 
 Ans. a 3 — 3a 2 a;+3aa; 2 — 2°. 
 
 (9) Divide 4?/ 6 — 24t/ 5 +60?/ 4 — 802/ 3 +60?/ 2 — 24^+4 by 2i/ 2 — 4?/+2. 
 
 Ans. 2i/ 4 — 8i/ 3 +12i/ 2 — 83/+2 
 
 THE GREATEST COMMON MEASURE. 
 
 26. A measure of a quantity is any quantity that is contained in it exactly, 
 or divides it without a remainder ; and, on the other hand, a multiple of a 
 quantity is any quantity that contains it exactly. Thus, 5 is a measure of 15, 
 and 15 is a multiple of 5 ; for 5 is contained in 15 exactly 3 times, and 15 con 
 tains 5 exactly 3 times, or is produced by multiplying 5. 
 
 27. A common measure, or common divisor, of two or more quantities, is a 
 quantity which is contained exactly in each of them. 
 
 28. The greatest common measure, of two or more quantities, is composed 
 of all the prime* factors, whether numerical, monomial, or polynomial factors, 
 «ommon to each of the quantities ; 3x is a common measure of 12ax and 
 I8bx, and 6x is the greatest common measure of 12a.r and 18bx. The great- 
 est common divisor of 2x7a(b-\-c)d and 2x3a?n(b-{-c) is composed of the 
 common prime factors 2a(b-\-c) ; the factors id of the one and 3 of the other 
 make no part of the common divisor. 
 
 29. To find the greatest common measure of two polynomials. 
 
 Arrange the polynomials according to the powers of some letter, and divide 
 
 that which contains the highest power of the letter by the other, as in division; 
 
 then divide the last divisor by the remainder arising from the first division ; 
 
 consider the remainder that arises from this second division as a divisor, and 
 
 * A prime number or a prime algebraic quantity is one whicb is divisible only by itself 
 or unity. 
 
GREATEST COMMON MEASURE. 35 
 
 the last divisor as tho corresponding dividend, and continue this process of di- 
 vision till the remainder is ; then the last divisor is the greatest common 
 measure. 
 
 Note 1. When the highest power of tho leading quantity is the same in 
 both polynomials, it is indifferent which of the polynomials is made the divisor, 
 the only guide being the coefficients of the leading terms of the polynomials. 
 
 Note 2. If the two given polynomials have a monomial factor common to all 
 the terms of both, it may be suppressed ; but as it forms part of the common 
 measure (28), it must be restored at the end of the process by multiplying it 
 into the common measure which is in consequence obtained. 
 
 Note 3. If any divisor contain a factor, which is not a factor also of the divi 
 dend, that factor may be rejected, as such factor can form no part of the great- 
 est common measure, which is composed of the common factors alone. 
 
 Note 4. If the coefficient of the leading term of any dividend be not divisible 
 by that of the divisor, it may be rendered so by multiplying every term of the 
 dividend by a proper factor, to make it divisible. This new factor thus jntro 
 duced, not being a common factor, does not affect the common measure. 
 
 If it were already a factor of the divisor, it could not be thus used ; the 
 remedy, in this case, would be to suppress it in the divisor, according to Note 3. 
 
 In order to prove the truth of this rule, we shall premise two lemmas.* 
 
 Lemma 1. If a quantity measure another quantity, it will also measure 
 any multiple of that quantity. Thus, if d measures a, it will also measure m 
 times a, or ma; for, let a=hd, then ma=mhd, and, therefore, d measures 
 ma, the quotient being mil. 
 
 Lemma 2. If a quantity measure two other quantities, it will also measure 
 both their sum and difference, or any multiples of them. For, let a=hd, and 
 b=kd, then d measures both a and b; hence a±b=hd^z7cd=d(h-^zJc), 
 and, therefore, d measures both a-\-b and a — b, the quotient being h-\-k in 
 the foi-mer case, and h — lc in the latter : and by lemma 1, d measures any 
 multiples of a-\-b and a — b. 
 
 Now, let a and b be two polynomials, or the terms of a fraction, and let 
 
 a divided by b leave a remainder c 
 
 b c d b) a (m 
 
 c d leave no remainder, as is shown m b 
 
 in the marginal scheme. Then we have, by the c) b (n 
 
 nature of division, these six equalities, viz. : n c 
 
 a—mb=.c .... (1) a=mb-\-c .... (4) d) c (p 
 
 b — nc =d .... (2) b=nc-\-d .... (5) p d 
 
 c—pd=0 .... (3) c=pd .... (6) 
 where the equalities marked (4), (5), (G) aro not deduced from those marked 
 (1), (2), (3), but from the consideration that the dividend is always equal to 
 the product of the divisor and quotient, increased by the remainder. 
 
 Now, by (6) it is obvious that d measures c, since c=pd ; hence (Lemma 
 1) d measures nc, and it likewise measures itself; therefore (Lemma 2) d 
 measures nc-\-d, which by (5) is equal to b ; hence, again, d, measuring b and 
 c, measures mb-\-c by the Lemmas 1 and 2. 
 
 * A lemma is a preparatory proposition, to aid in the demonstration of the main prooosi. 
 tion which follows it. 
 
36 ALGEBRA. 
 
 .•. d measures a, which is equal to mb-\-c by (4). 
 
 Hence d measures both the polynomials a and b, and is consequently a 
 common measure of these polynomials ; but d is also the greatest common 
 measure of a and b ; for if d' is a greater common measure of a and b than d 
 is, it is obvious that by (1) d' measures a — mb, or c ; and d' measuring both b 
 and c, it measures b — nc, or d by (2) ; hence d' measures d, which is absurd, 
 since no quantity measures a quantity less than itself; therefore d is the 
 greatest common measure. Q. E. D.* 
 
 30. If the greatest common measure of three quantities be required, find 
 the greatest common measure of two of them, and then that of this measure 
 and the remaining quantity will be the greatest common measure of all three. f 
 
 31. If the two polynomials be the terms of a fraction, as r» and d their 
 
 greatest common measure, then we may put a=da', and b=db' ; hence 
 
 - = — =— , and, since a', b' contain no common factor (28), by dividing both 
 b db' b' 
 
 numerator and denominator of a fraction by their greatest common measure, 
 the resulting fraction will be simplified to its utmost extent, and thus the pro- 
 posed fraction will be reduced to its lowest terms. 
 
 * These letters stand for the Latin words quod erat demonstrandum, signifying which 
 was to be demonstrated. Another mode of demonstrating the same is as follows : Let A 
 and B represent the two given quanties, D their greatest com m on divisor, Q, the quotient 
 of A by B, and R the remainder. We shall first prove that the greatest common divisor 
 of A and B is the same as the greatest common divisor of B and R. Represent the latter 
 byD'. 
 
 a ^ i w A Ba j R a A Ba . R 
 A=Ba+R, ••• 5=^+5. ^d ^= W + W - 
 
 A and B being divisible by D, B must be, because a whole number can not be equal to 
 a whole number plus a fraction. Again, B and R being divisible by D', A must be, for the 
 sum of two whole numbers can not equal a fraction. Finally, D, a common divisor of B 
 and R, can not be greater than their greatest common divisor D' ; and D', a c . d . of A and 
 B, can not be greater than their g . c . d . D ; i.e., D can not be greater than D', and D' can 
 not be greater than D. 
 
 Or thus : since 
 
 A=BGl+R, 
 the greatest common divisor D of A and B, must divide R. Represent the three quotients 
 by A', B', andR'; then 
 
 A'=B'Q,+R'. 
 B' and R' have no farther common factor, for if they had, it must by this equality divide 
 A; then A' and B' would have still a common factor, and D, the greatest common divisor 
 of A and B, would not contain all the common factors of these quantities, which is contrary 
 to the definition. Since B' and R', which are the quotients of B and R by D, can have no 
 farther common factor, it follows that the greatest common divisor of B and R is equal to 
 D ; then it is the same, as that of the quantities A and B. 
 
 In pursuing the rule for finding the g .c.d., we arrive at a remainder which exactly di- 
 vides the preceding divisor, and which is, therefore, the g.c.d. of itself and this preced- 
 ing divisor ; also by the above demonstration of that divisor and its dividend, and so on up 
 to the given quantities. 
 
 t For suppose we have the three quantities A, B, C; let D be the greatest common di- 
 visor of A and B, and D' that of D and C. According to the definition, D is the product of 
 the common factors of A and B, and D' is that of the common factors of D and C ; then D' is 
 the product of the common factors of the three quantities A, B, C ; therefora D' is their 
 greatest common divisor. 
 
GREATEST COMMON MEASURE. 37 
 
 EXAMPLES. 
 
 (1) What is the greatest common measure of Ax^y^z* and 8x*y 3 z' 2 l 
 
 Here 4 is the greatest common measure of 4 and 8, and x 2 y 3 z 2 is that of the 
 literal parts; hence 4x 2 y 3 z 2 is the greatest common measure required. 
 
 (2) Find the greatest common measure of ^_ . 
 
 %J 
 
 X s — xy 2 
 
 xy 2 -\-y 3 =y 2 (x-\-y) ; rejecting the factor y- 
 x-Ly) x"—y 2 (x—y 
 x 2 -\-xy 
 —xy — y" 
 — xy — y 2 . 
 
 Hence x-\-y is the greatest common measure sought, and 
 a? i ,,3 ( x s_|_j,3w [x+y) x 2 —xy-\-y 2 
 
 n J ==) — 1 -^-+ — ) — L -^= = reduced fraction. 
 
 a 8 — f (a- 1 ' 3 — 2/ 2 )-r(^+2/) *— 9 
 
 (3) Required the greatest common measure of the two polynomials 
 
 6a 3 — 6a 2 y+2ay 2 —2y 3 .... (a) 
 12a 2 — lhay +3y* .... (&)• 
 
 Here 6a 3 — 6a 2 y+2aif— 22/ 3 =2(3a 3 — 3a 2 y+ay 2 — y 3 ) 
 
 12a 2 —15ay +3?/ 3 =3(4a 2 — 5a?/ +r) ; 
 
 And therefore, by suppressing the factors 2 and 3, which have no common 
 measure, and which, not being common factors of the two given quantities, do 
 not affect the common divisor, we have to find the greatest common measure 
 of 
 
 3a 3 — 3a 2 y+ay 2 — y 3 and 4a 2 —5ay-\-y 2 . 
 4a 2 — 5ay+y~) 3a 3 — 3a?y + ay"— y 3 
 
 _4 
 
 12a 3 — 12a 2 ^+4a?/ 2 — 4y 3 (3a 
 
 12a 3 — 15a 2 ?/ -j- 3a?/ 2 
 
 3a"y + ay 2 — 4y 3 
 4 
 
 12a 2 ?/ + 4a?/ 2 — 16y :s {3y 
 I2a 2 y — I5ay 2 + 3y 3 
 
 19ay 2 —19y 3 =l9y 2 ( a— y) 
 Suppressing 19y 2 , by note 3, rule, 
 
 a —y) 4a"—5ay+y" (4a— y 
 4a 2 — 4ay 
 
 — ay+y 2 
 
 Hence a — y is the greatest common measure of the polynomials a and b. 
 
 The factor 4 is introduced into the divkleud in this example to render it di- 
 visible by the divisor. This can be done, because 4 is not a factor of every 
 term of the divisor, and therefore not a factor of the divisor. The quantities 
 employed, after introducing or suppressing factors, are different from the given, 
 but as they have the same greatest common divisor, and as the object is to find 
 this, the circumstance is immaterial. 
 
 (4) Required the greatest common measure of the terms of the fraction 
 
38 
 
 ALGEBRA. 
 aF—ct-x 4 
 
 a 6 -4- a 3 x — a 4 x 2 — aV 
 Here a~ is a simple factor of the numerator, and a 3 is a factor of the denomi- 
 nator; hence a 3 is the greatest common measure of these simple factors, which 
 must be reserved to be introduced into the greatest common measure of the 
 other factors of the terms of the proposed fraction ; viz. : 
 a 4 — x 4 and a 3 +« 2 ^' — ax 2 — x 3 . 
 a 3 4-a 3 x — ax" 1 — x 3 ) a 4 — x 4 (a — x 
 
 a 4 -f- a 3 x — (Px* — ax 3 
 — d?x -j- a 2 .r 3 -|- ax 3 — x 4 
 — a s x — a 2 .r 2 -4- ax 3 -|- x 4 
 
 2a"x 2 —2x 4 =2x' i (a 3 — a: 2 ) ; rejecting 2x 3 
 a?—x' 2 ) d?-\-a*x— a.r 3 — x 3 (a-\-x 
 a 3 — ax 2 
 
 a-x — x 3 
 
 a"x — x 3 
 
 Therefore, restoring a 2 , the greatest common measure, is a 2 (a 2 — .r 2 ). 
 
 gS—cPx 4 (as—a?x 4 )-±a~(a' i —x' 2 ) cP+x* 
 
 • a 6 -\-a 5 x — a 4 x 3 — a¥ (a 6 -\-a 5 x — a 4 a: 3 — a 3 x 3 )-^-a\a <1 — x") a?-\-ax 
 
 ADDITIONAL EXAMPLES. 
 
 (1) Find the greatest common measure of 2a 3 a: 3 , 4a: 3 ?/ 2 , and 63?y. 
 
 (2) Find the greatest common measure of the two polynomials a 3 — c&b 
 + 3aZ> 3 — 3b 3 , and a 2 — 5aZ>+4i 3 . 
 
 (3) What is the greatest common measure of x 3 — xy* and a: 2 -j- 2xy -J- y" 1 
 
 (4) Find the greatest common measure of x 8 — y s and a 43 — y 13 . 
 
 (5) Find the greatest common measure of the polynomials 
 
 (b—c)x^—b(2b— c)a:+& 3 (a) 
 
 lb-\-c)x 3 —b{2b-\-c)x' i +b 3 x (6). 
 
 (6) Find the greatest common measure of the polynomials 
 
 x 4 — 8r 3 +21a: 3 — 20x-f 4 (a) 
 
 2X 3 — 12a; 3 + 21a: —10 (b). 
 
 (7) y 3 —5y 2 z—4yz*+2z 3 and 7;?/ 2 z-f Kh/z 2 4-3z 3 . 
 
 (8) Also of (a^+a 3 a: 3 +a 4 ) and (x 4 -\-ax 3 — a 3 x— a 4 ). 
 
 (9) Also of (7a 3 — 23a&+6& 2 ) and (5a 3 — 18&a 3 +llaZ> 3 — 6b 3 ). 
 
 (10) Also of (5a 5 +10a 4 Z>+5a 3 i 2 ) and (a 3 & + 2a 2 & 2 +2a& 3 +Z> 4 ). 
 
 (11) Also of (6a 5 -fl5a 4 & — 4a 3 c 2 — 10a 2 6c 3 ) and (9a 3 6 — "27a 2 Z>c — 6a&c« 
 4-186C 3 ). 
 
 (12) Also of {a a +Y+aybP-\-a a b s +bP+ l ) and {a a m-\-a a n^m-\-b^i). 
 
 (13) Find the g. c. d. of the three quantities a 3 -f3a 2 & + 3a6 3 +& 3 , N a 2 +2a& 
 + 6 2 , and a 2 — Z> 2 . 
 
 ANSWERS. 
 
 (8) x^+ax+a?. 
 
 (9) a— 36. 
 (10) a-\-b. 
 
 (11) 3a 2 — 2c 2 . 
 
 (12) a a +Z># 
 
 (13) a+b. 
 
 (1) 2x 2 . (5) x—b. 
 
 (2) a— b. (6) x— 2. 
 
 (3) x+y. (7) y+z. 
 
 (4) x—y. 
 A quantity is said to be independent of a letter when it does not contain this 
 
 letter, and, therefore, does not depend upon it for its value. 
 
 Note. — Iu seeking for a common divisor, we find ourselves often working with polynomi- 
 als different from the given, but always with such as have the same common measure with 
 the given polynomials. 
 
GREATEST COMMON MEASURE. 3y 
 
 Proposition. — A divisor of a polynomial, which is independent of the letter 
 of arrangement of that polynomial, must divide separately each bf the multi- 
 pliers of the different powers of that letter. 
 
 Demonstration. — Let Ax m -f-B.r m_1 -f-C;r m_2 , &c, be the polynomial, and 
 D the divisor. The quotient must contain every power of the letter of ar- 
 rangement that the dividend does, since the quotient, multiplied by the divisor, 
 must produce the dividend, and the letter of arrangement is not contained in 
 the divisor. The quotient must, therefore, be of the form A'x m -\-B'x m ~ l 
 -\-C'x m ~ 2 , &c, multiplying which by the divisor gives DA / x m -|-DB'.r m_1 
 -f-DC'i" 1-2 , &c, the original dividend, the multiplier of each power of x in 
 which is evidently divisible by D. Q. E. D. 
 
 N.B. — A' is a different quantity from A, B' from B, &c. 
 
 EXAMPLES. 
 
 (1) Find a common divisor, independent of the letter a, of the two quantities 
 b a-—ca' i -\-b 2 a—c' 2 a-\-b"—2bc-\-c' i and 
 b 3 a*—3b*ca a +3bc*a 3 —c?a 3 +¥a*—c*a*+b 3 a—(?a+b 3 —3b i c+3bc' i —c i . 
 
 Collecting together in the first of these two quantities the multipliers of a 2 and 
 a, and observing that b 2 — 2bc-\-c" is the square oft — c, we hav 
 
 (J_ c )a2_|_(&2_ c 2)a_L.(&_ c )2, 
 
 and from the second by a similar process, 
 
 (b—c) s a*+(b 4 — c*)a 9 +(& 3 — c 3 )a+(6— c) 3 . 
 The multipliers of the different powers of a in the two quantities are, there- 
 fore, b—c, b"—c", (b—c)°, (6— c) 3 , Z> 4 — c 4 , and b 3 —c 3 . The only number 
 which will divide them all is their common divisor b — c, which is, therefore 
 the answer required. 
 
 (2) Find the greatest common divisor of 
 
 (b —c )a 2 — 2b (b —c)a-\-b~{b—c) and 
 (b--c-)a-— & a (& 8 — <■-•). 
 Here the common divisor, independent of a, is b — c ; suppressing which, wo 
 have left the two quantities 
 
 a 2 — 2ba -\-b- and 
 (o +c)(a 2 — Z> 2 ). 
 Suppressing the factor (b-\-c) not common to both, we shall find the common 
 divisor of these last two quantities to be a — 6, and the greatest common 
 divisor of the two original quantities is 
 
 (6 — c) (a — b) or ab — ac — b 2 -{-bc. 
 
 The success of the process for finding a greatest common divisor depends upon the fact 
 that the quantities being arranged according to the powers of some letter, each division 
 leads to a remainder of a degree inferior to the divisor. When the polynomials contain 
 many terms of the same degree, a precaution is necessary, without which this reduction 
 would not always obtain, which consists in uniting all these terms under a single multiplier 
 Let there be the two polynomials : 
 
 A=x 3 -\-yx--]-x i — y-x-\-2yx — y 3 -\-y' i 
 'B=yx°~-\-zfi -\-y?x+yx -f x -f-y. 
 I write them thus : * 
 
 A=x 3 +(y+l)x-— (y-— 1y)x— y 3 +yz 
 B= (y +i) x :+(y2+y +l)x+y. 
 The first term, x 3 , not being divisible by (y-\-l)x% on account of the factor y-f-1, I know 
 (Prop, above), that if a quantity is arranged like the preceding, every divisor of this quantity, 
 independent of x, must divide separately the multiplier of each power of x. "From this it 
 follows that y-\-\ has no common factor with B, because, if it had, this factor would be 
 found iu y'2-^-y-^-i and in y ; but it is evident that >j has no factor common with y-\-\ 
 
40 ALGEBRA. 
 
 We can then multiply A by y-{-l without affecting the common divisor sought; and as 
 it would be necessary to multiply again by y+1, we multiply at once by ij/-\-l) 2 . or 
 y" -{-'Zy -\-\* In this manner we arrive at the remaindi r 
 
 R=(— y*— y ■'■+ //-'. ■'■— y b — y*-\-y*- 
 
 Before j>assing- to the second division, it is necessary to suppress in It the factors com- 
 mon to the multipliers of the powers of a-. But the two parts of It are evidently divisible 
 by — y* — y z -\-y-< and after this simplification there remains x-\-y. We can then take 
 x-\-y for a divisor, and as the division is effected exactly, it follows that the common di- 
 visor sought is x-\-y. 
 
 The process is not always so easy. To develop the general method to be pursued in 
 such cases, let us consider the polynomials A and B, which contain two letters, x and y. 
 Take first the greatest monomial common divisor of the terms of A; let a be this divisor, 
 and A' the quotient of A by a : we shall have A=«A'. Arrange A according to the pow- 
 ers of a;, taking care to collect all the terms containing a same power of this letter, and let 
 ns suppose, for example, that we have 
 
 A / =LaB+Ma>|-N. 
 
 All the factors of A', independent of a:, must be factors of the quantities L, M, N, which 
 multiply the different powers of x. These quantities containing only the single letter y, it 
 will be easy to find their greatest common divisor; let us name this divisor a!, and the 
 quotient of A' by a', A"; we shall have A'=a'A", and, consequently, 
 
 A=aa , A". 
 
 a will be the product of the monomial factors of A, a* the product of the polynomial fac 
 tors which do not contain x, and A" the product of the factors which contain x. 
 
 Let us effect the same decomposition of the polynomial B, and let 
 
 B=e?^B". 
 
 Then I determine the greatest common divisor of the monomials a and ,3, as well as that 
 of the polynomials a' and ft', which contain only the letter y ; and if I can also find that 
 of the polynomials A" and B", which contain y and x, I shall have three quantities, the 
 product of which will be the greatest common divisor of A and B. 
 
 But I say that we can iind thc # greatest common divisor of the quantities A" and B", in 
 subjecting them to the same calculus as the preceding examples. It is clear, indeed, that, 
 these quantities having no longer either monomial factors or polynomial independent of x, 
 it will be proper to multiply the partial dividends of the first division by the polynomial 
 which is placed before the highest power of a; in the divisor, and that we shall thus arrive 
 at a remainder of a degree less in x than the divisor. It will be easy to take from this re- 
 mainder all the monomial factors which it contains, as well as the polynomial factors inde- 
 pendent of a - , and then proceed to a second division, taking for a divisor this remainder sim- 
 plified. We operate as in the first ; then we pass to the third, and continuing always in 
 this manner, we are sure of arriving finally at a remainder zero, or independent of x. 
 
 In the first case the quantil id B'' have, for greatest common divisor, the divisor 
 
 last division. 
 
 We have thus seen that the finding of a common divisor, when the polynomials contain 
 two letters, depends upon finding it when they contain one; so ti here they con- 
 
 tain three depends upon that where they contain two, and so on, whatever be the number 
 of letters. 
 
 There is, therefore, no case in which the common divisor con not bo found by the above 
 rules. 
 
 THE LEAST COMMON MULTIPLE. 
 32. We lmvo already defined a multiple of a quantity to be any quantity 
 that contains it exactly ; and a common multifile of two or more quantities i-* a 
 
 ijiiantity tl i;it co!itaiiis t «'uch of litem exactly. 
 
 * Let N be the dh idend, I » the dh isor, e the ooefficii at of the firal term of the di\ 
 Multiplying by the square of this coefficient, the dividend The first term of 
 
 otient will contain the first power of c, Multiplying the whole divisor by this tern 
 of the quotient, every term of the product will contain the lirst power ^i c, and the whole 
 product maj be represented by cP. Subtracting this from the dividend, the remainder is 
 c^N— I term of which contains <•. and, therefore, its lirst term is ready for division 
 
 without multiplying again by c. 
 
LEAST COMMON MULTIPLE. 41 
 
 The least common multiple, of two or more quantities, is, therefore, 'Jie leas' 
 quantity that contains each of them exactly. 
 
 N. B. The least common multiple must, from its nature, contain all the dis- 
 tinct factors in either of the quantities. 
 
 33. To find the least common multiple of two quantities. 
 
 (1) Divide the product of the two proposed quantities by their greatest com 
 mon measure, and the quotient is the least common multiple of these quanti- 
 ties ; or divide one of the quantities by their greatest common measure, and 
 multiply the quotient by the other. 
 
 Let a' and b be two quantities, d their greatest common measure, and m 
 
 their least common multiplo ; then let 
 
 a=hd, and b=kd; 
 
 and since d is the greatest common measure, h and k can have no common 
 
 factor, and hence their least common multiple is hk ; therefore, hkd is the 
 
 least common multiple of lid and kd ; hence, 
 
 hkd 3 hdxkd axb ab 
 
 m=hkd= — r-= ; — = — j—=z-r Q. E. D. 
 
 d d d d ^ 
 
 ^2) Also, the least common multiplo is composed of the highest powers of all 
 
 the factors which enter into the given quantities.* 
 
 EXAMPLES. 
 
 (1) Find the least common multiple of 2a?x and 8a?x?. 
 
 ab 2a 2 x X 8a 5 x 3 
 
 Here m=—rz= — — =8a 3 :r 3 = least common multiple : 
 
 d 2a?x ' 
 
 or, by (2), the two quantities being 2a"x and 2 3 a 3 r 5 , 2 3 a 3 .r 3 is the I. c m. ; be- 
 cause 2 3 is the highest power of 2, a? the highest power of a, and r 5 the 
 highest power of x, in either of the given quantities. 
 
 (2) Find the least common multiple of ±x-(x°-— y-) and 12.r 3 (.r 3 — y 3 ). 
 Here d=4x' 2 (x — y), and, therefore, wo have 
 
 m =^= 4^=^j =l^(x+y) (x*-tf) ; 
 
 or, m=12.r 7 -L.12a; 6 ?/— \2xHf— \2xhf ; 
 
 or, tho two given quantities being 2 2 x 2 (x-{-y)(x — y) and 2 2 . 3.^(3: — y)(x' 2 -\-xy 
 
 +y°), the I. c. m. is 2 2 . S&ix+y^x— y){z""+*y+y~)- 
 
 (3) Find the least common multiple of x 2 -\-2xy-\-y- and x 5 — xy 2 . 
 Here d=x-\-y, and, therefore, we get 
 
 a , x"-\-2xy-\-y 2 , 
 
 d x -\-y 
 
 = (x+y)(x 5 —xy°~) 
 
 =x(x-\-y) (.r 2 — 2/ 2 )= least common multiple 
 or, the two given quantities being (x-\-yY and x(x-^-y) (x — y), the I. c. m. a 
 v(z+y)*(x—y). 
 
 (4) What is the least common multiple of x 4 — 5r'+9.r :! — 7.r+2, and 
 r«— 6.r 2 +8.r— 3? 
 
 By the process for finding the greatest common measure, we find 
 
 d=x 3 —3x"-+3x—l 
 
 = (.r— 2) (ar*— 6a?+8z— 3) 
 
 ~x 5 — 2x* — G.r 3 -f-20x :! — 19.r-f-6, the least common multiple. 
 
 The ordinary arlthmeuc ;t£tn.id depends on this principle. 
 
42 ALGEBRA. 
 
 (5) Find the least common multiple of </- — .V"'< -j- b 9 and a* — b* 
 
 (6) Find the least common multiple off/-' — lr and </'-{-)•'■. 
 
 (7) Find the least common multiple of ./-' — </• and ./• — 
 
 (8) Find the least common multiple of y* — 8y+ 7 a,! ' 1 .'/H-~.'/--8 
 
 ANSWERS. 
 
 (5) (a-b) {a*-b*). (7) {x+y) (r-y ,. 
 
 (G) (a— b) (a 3 + b 3 ). (8) y 3 — 57i/+56. 
 
 34. Every common multiple of two quantities, a anJ b, ?*< a multiple oj in, 
 ifteir least common multiple. 
 
 For let ?n' bo a common multiple of a and &, then, because in' is greater 
 than m, if wo suppose that m' is not a multiple of m, we have, as in the an- 
 nexed scheme, 
 
 m) m' (ft 
 to'=//to-|-Z." ... (1) hm 
 
 m! — hm=k ... (2) k= remainder. 
 
 Now tho remainder k is always less than m the divisor; hence, since a and 
 b measure m and m', it is evident thai a and b measure to' — hm, or, by ( - 2). fc, 
 therefore, % is a common multiple of a and b, and it lias been proved to be less 
 than 7n, tho least common multiple, which is absurd ; henco the supposition 
 that m' is not a multiple of to is false, or to' is a multiple of in. 
 
 35. To find the least common multiple of three or more quantities 
 
 Let a, b, c, d, Sec., be tho proposed quantities ; 
 
 find to, the least common multiplo of a and b ; 
 
 find to', . « c and m ; 
 
 find to", d and m' ; cVc. 
 
 The last, say m", is evidently a multiple of a, b, c, d, &c. 
 Then, since every multiplo of a and b is a multiple of m, their least common 
 multiple (34), the quantity sought, x, is a multiple of to; but x also is by hy- 
 pothesis a multiple of c ; therefore, x is a multiplo of both c and m, and, there- 
 fore, it is (34) a multiple of to' ; but x is a multiplo of d and to', and, therefore, 
 of to" ; hence x can not bo less than m", and, therefore, to" is tho least com- 
 mon multiple. 
 
 EXAMPLES. 
 
 (1) Find tho least common multiple of 2a 9 , 40*6*, and Gab 3 . 
 Hero, taking 2a* and Aa 3 lr, wo find d*=2a 9 , and, therefore, 
 
 ab 2a«X4fl s 6 a 
 
 d 
 
 Again, taking m, or Aa 3 b", and G/z/j 3 , we find /< 7 =2a/V; hence 
 
 cm Gai 3 x4" 
 
 7n,'=—r= — x— r; =12a 3 b 3 = answer required. 
 
 a 2ao" 
 
 Or, the three given quantities beini • and 2.3ao', the 2. cm., by 
 
 <J3.(2), is2*.3</ "/< . 
 
 (2) Find the least common multiple of a— nd </'■—, . 
 
 Taking a — C and r/ : — .;-', we have d = a — X; and henco 
 
 ab a — x 
 
 in=- t = x ((/•—./-') = • — ■ 
 
— =(a-{-x)(a 5 — ar 5 )= answer sought. 
 
 FRACTIONS. 43 
 
 Again, taking a 2 — a: 2 and a 3 — x 3 , we find d=a—x; hence 
 cm (a 3 — 2 J )(a 2 — s 2 ) 
 a! a — a; 
 
 Or, the three given quantities being (a — x), (a—x)(a-\-x), and {a—x^cp+ax 
 +x"-), the least common multiple is {a—x){a-\-x){a?-\-ax-\-x"). 
 
 (3) Find the least common multiple of 15a 2 6 2 , 12ab 3 , and 6a 3 6. 
 
 (4) Find the least common multiple of 6a 2 x"(a— x), 8x 3 (a' 2 — x 2 ), ard 12 
 {a—xy. 
 
 (5) Find the least common multiple of x 3 — x-y— xy*+y 3 , o^—x-y-^xif—y 3 , 
 and x 4 — y 4 . 
 
 (6) Find the least common multiple of {a-\-bf, (a 2 — P), (a — by, and a 3 
 + 3a' 2 b + 3a¥-\-b 3 . 
 
 (7) Find the least common multiple of 45, 50, and 75, or 3 2 . 5, 2 . 5 2 , and 
 3.5 2 . 
 
 ANSWERS. 
 
 (3) 60a 3 b 3 . (6) ( a +b){a*—b*y 
 
 (4) 24a 2 x 3 (a— z)(a 2 — a*). 
 
 (5) x 5 — a:i/ 4 — x 4 ^/+2/ 5 . 
 
 (7) 3 2 .2.5 2 =450. 
 
 OF ALGEBRAIC FRACTIONS. 
 
 36. Algebraic fractions differ in no respect from arithmetical fractions, and 
 all the rules for the latter apply equally to the former. We shall, therefore, 
 merely repeat the rules, adding a few examples of the application of each. It 
 may be proper to remind the reader that all operations with regai-d to frac- 
 tions aro founded upon the three following principles : 
 
 1. In order to multiply a fraction by any number, we must multiply (lit- 
 numerator, or divide the denominator of the fraction by that number. 
 
 2. In order to divide a fraction by any number, we must divide the numera ■ 
 tor, or multiply the denominator of the fraction by that number. 
 
 3. The value of a fraction is not changed, if we multiply or divide both the 
 numerator <lnd denominator by the same number. — See (17, Note). 
 
 REDUCTION OF FRACTIONS. 
 I. To reduce a fraction to its loioest terms. 
 
 37. Rule. — Divide both numerator and denominator by their greatest com- 
 mon measure, and the result will be the fraction in its loioest terms. 
 
 When the numerator and denominator are, one or both of them, monomials, 
 their greatest common factor is immediately detected by inspection ; thus 
 a"bc a"b X c c 
 
 =—r in its lowest terms. 
 
 So, also, 
 
 5a"b~~a i bx5b~5b 
 
 ax" 2 x X ax ax 
 
 -= — ; — in its lowest terms. 
 
 ax-\-x 2 x(a-{-x) a-\-x 
 If, however, both numerator and denominator are polynomials, we must 
 have recourse to the method of finding the greatest common measure of two 
 algebraic quantities, developed in a former article. Thus, let it be required to 
 reduce the following fraction to its lowest terms : 
 
44 ALGEBHA. 
 
 6a 3 — 6a 2 y -4- 2ay- — If 
 
 12a s — 15«i/4-%" 
 The greatest common measure of the two terms of this fraction was found at 
 page 37 to be a — y; therefore, dividing both numerator and dencminator by 
 this quantity, wo obtain as our result the fraction in its lowesl tarns; or, 
 
 Ga-+.' 
 
 12a— 3?/' 
 
 t ri *i- .i e *• 4a<— 4a 5 i 2 4-4«/.— M . 
 
 In like manner, taking tho fraction JL , the greatest 
 
 i_j_4 a 3j_ Q a sjs_ ZaP+ib* 
 
 common measure of tho two terms is found to be 2a?-\-2ab — ft 9 ; and, drridi ^ 
 both numerator and denominator by this quantity, the reduced fraction is 
 
 2a 3 — 2ab+ b* 
 
 3a 2 — ab—2b 2 ' 
 
 EXAMPLES FOR PRACTICE. 
 
 2.r 3 — lG.r— 6 
 
 (1) Reduce ■ _ to its lowest terms. 
 
 48r 5 +36.r 2 — 15 
 
 (2) Reduce — -= — ,., „ , , „ - to its lowest terms. 
 
 v ' 24ar ! — 2l2--f-l&r — 
 
 20.r'4-.r- — 1 
 
 (3) Reduce nr , , . ., to its lowest terms. 
 
 ' 25r'+5r i — x — 1 
 
 / -r. , 3m-/i — m : i) — 6mn*-\-2mnp 
 
 (4) Reduco -— t-t-z — — to its lowest terms. 
 
 ' 12»m — 15n s — 4mp-\-on2> 
 
 m 9 — 2mn 
 
 Ans. — : — . 
 
 4m — on 
 
 (5\ Redu 4aflcx—4aZdx+2ia~bcx—24a*bdx-\-36ab2cx—36al-tljc 
 
 ' labcxS—labdx*+lcufix*—1acdx*—'Zlb-dx* + Klb*afl-{-%\b&x*— % 1 betW 
 
 . , 4a(a + 36) 
 
 to its lowest terms. Ans. — rr-i -■ 
 
 7x*(b-\-c) 
 
 38. It frequently happens, however, that when tho polynomials which form 
 the numerator and denominator of a fraction which can be decomposed are not 
 veiy complicated, wo are enabled by a little practice to detect the common 
 factor and effect the reduction without performing the operation of finding the 
 greatest, common measure, which is generally a tedious process. The resulni 
 to which wo called the attention of the reader at the end of algebraic dh 
 (see pago 30) will bo found particularly useful in simplifications of this nature. 
 
 Thus, for example : 
 
 "■''.'/(•''+.'/) ( ■''+.'/) 
 
 3x»+6xy+^- 3(z+y)« = 2{x+y){x+y)-x+y 
 
 „■_/,-• (a— &)(«+&) a+6 
 
 (G) 
 (7) 
 (8) 
 
 (») 
 
 (10) 
 
 d : — 2ab-{-b 1 (<i — /») J n — V 
 
 oa*+l0a"-h +5ah* 5a(a +2a6+o») 5a(a-fr b) 9 o{a+b) 
 
 8a 3 +8</ ■■/> \-bj~ (<i + l>)~ ~^i ' 
 
 a n — ._L.T-)(a— x) at+ax+x* 
 
 d»— 2ax+:r B= (a— a— x 
 
 ac+bd+ad+bc _ {a+b)c+(a+b)d _ (,■ + ,/)(,/ + h) _ c+d 
 af+ 2bx+2cu + bf~{a + 1 \J'+ 2 1 (a + b)~ (/+ 2 c)(a -f b) -/+ 2 c 
 
 
TRACTIONS. 41 
 
 6ac+Wbc+9ad+15bd 3a(2c+3d)+5b(2c-\-3d) _(3a + y,)(2r+3d) 
 ( U ) 6 c ->+9cd—2c—3d ~ 3c(2c+3d)—(2c+3d) ~(3c—V){^c~^3a) 
 3a+5b 
 
 3c— 1 
 
 ax m — bx m+1 x m (a — bx) x m (a — bx) x m ~ l 
 
 (12) 
 
 (13) 
 
 (14) 
 
 cPbx—bPx 3 bx(a*—b*x*) bx(a-{-bx){a — bx) b(a-\-bx) 
 a 4 —b* _a 2 — & 2 
 a?-\-ab' z a 
 
 2xy + 3f + 2x" + 3xy y + x 
 
 8cx+\2cy—lQdx—lhdij~lc—bd' 
 
 II. To reduce a mixed quantity to an improper fraction. 
 
 39. Rule. — Multiply the integral part by the denominator of the fraction, 
 and to the product add the numerator with its proper sign ; then the result 
 placed over the denominator ivill give the improper fraction required. Thug 
 
 a a4-b 
 
 a 2 — x' i _a?-\-X'-\-a"— x- _ 2a 2 
 ™ 1 "^a 2 4-2- 3== aF+x* = a?+x 2 ' 
 
 , abc—c°d—2cd* abc+c"d+2abd+2cd 2 4-abc—c°~d—2cd i 
 
 (3 > ab + cd + e+2d =~^^ 7+21T 
 
 2abc+2abd 
 — c+2d 
 
 2ab(c+d) 
 = c+2d 
 
 &2_|_ C 2_ a <2 goc-ffo+c 8 — fl 8 (ft-f-c) 8 — a 8 
 (4) ^ 2bc = 26c" = 2bc ' 
 
 40. It is to be remarked that when a fraction has the sign — , it signifies 
 that the whole fraction is to be subtracted; the negative sign, therefore, as 
 will bo shown hereafter, applies to the numerator alone ; when the numerator 
 is a polynomial, the negative sign extends to all its terms; the bar which sepa 
 rates the numerator from the denominator is to be regarded as a vinculum, and 
 if it have the negative sign before it, when removed, all the signs of the numer- 
 ator must be changed. 
 
 (5) 1- 
 
 (6) c 
 
 (7) 1 
 8)1 
 
 b a — b 
 
 a a 
 ef cd — ef 
 ~d = d - 
 
 a s— 2a6 + fr» ar+b 2 — (a 2 — 2ab+b 9 ) Sab 
 a 9 +6» a-+6 2 = a--f-K 
 
 5g4.cs— q» 25c — (fr 2 -f c 2 — a~) 
 2bc ~ 2bc 
 
 a?— (58— 26e-f-c a ) 
 
 "~ 26c 
 
 _a 2 — (b— c) 2 
 "~ 2bc 
 
46 ALGEBRA. 
 
 x 3 — 3x*y + ?>.r if — if 
 
 (9) x*+2xy+if- 
 
 x+y 
 ■t 3 -f 3.r- y + r.rif + if—( x 3 — 3x-y -f 3ry"—f) 
 
 Gx-y + 2y 3 
 
 x+y 
 J>y(3x-+ if) 
 x+y 
 
 2mw 2 — 2pqn m-n — mpq+mn 2 — npq — (2mn' i — 2pqn, 
 
 10) mn—pq— r JLJ - = OI!= r 1 — S LL2 
 
 ' s 1 m-\-n m-\-n 
 
 m-n — mpq — mn-+pqn 
 
 m+n 
 
 mn(m — n) — pq(m — n) 
 
 m + n 
 
 (mn — pq){rn — n) 
 
 m-\-n* 
 
 ill. To reverse this process, or to reduce an improper fraction to a m 
 quantity. 
 
 Rule — Divide the numerator by the denominator ; the quotient obtained m 
 far as practicable, will be the entire part, and the remainder, set over /' 
 nominator, will be the fractional part. Then the two, joined tog' 
 proper sign, will form the mixed quantity required. Thus, 
 
 ay+b b 
 
 11) -^-=0+-. 
 
 a 2 4-x 2 2x 2 
 
 (12) •— - — =a+x+ . 
 
 v ' a — x ' ' a — x 
 
 20X 3 — 10.r+4 4 
 
 „ ., p°+2pq + q°—r —s r+s 
 
 (14) JV, = F+q -¥+S 
 
 mHm*— n*) + 3m 3 — 3mn- 3 
 
 (15) — * ttt — JT = m"+n-+- . 
 
 v ' m-(iit- — ;j 2 ) ' m 
 
 IV. To reduce fractions to others equivalent, and having a common denomi- 
 nator. 
 
 41. Rule. — Multiply each of the numerators, separately, into all the denomi- 
 nators, except its ou-n,for the new numerators, and all tiie dt nominators to 
 gether for a common denominator.] 
 
 a c 
 Thus, reduce r and —. to equivalent fractious having a common denominator 
 
 a X d is the new numerator of the Bret, 
 
 c x b is the new numerator of the second, 
 bxd is the common denominator; 
 
 ad be 
 
 Therefore, the fractions required are , , and —.. 
 
 * Tin; rationale of the above examples is given in the note on the next page. 
 
 t The numerator and denominator of each fraction will thus l>e multiplied by the same 
 number, viz., the product of the other denominatora, nml, oonseqoently, "ta value will be on 
 ■hanged* 
 
FRACTIONS. 47 
 
 ^^accgkm 
 
 .Reduce -r, "?> 7> T> T> — > to a common denominator. 
 b a J hi n 
 
 adfhln cbfhln ebdhln gbdfln Jcbdfhn mbdfhl . _ . 
 
 bdfhln bdfhln bdfhln bdfhln bdjldn bdfhln 
 
 \-i-x 14-x 2 14-ar 5 ' 
 
 Reduce — !— , — JL — , __ L__, to a common denominator. 
 I — x 1 — x 2 1 — x 3 
 
 (!+*)(! -*»)(! -r>) (1 +*"')(! -x)(l—r*) (l+r»)(l -x)(l-x 2 ) 
 
 (l_ a: )(l_x2)(i_r 5 )' (1— x)(l — x 2 )(l — x a )' (l_x)(l— x 2 )(l— x 3 )' ' 
 
 fractions required. , 
 
 ADDITION OF FRACTIONS. 
 42. Rule. — Reduce the fractions to a common dencr.iinalor, add thenumera 
 tors together, and subscribe the common denominator. Thus, 
 a c ad be ad-\-bc 
 ^ 1 + d—bd + bd == bd ' 
 
 a m p x anqy mbqy pbny xbnq 
 ' b'*~n'q*y bnqy*bnqy'bnqy*bnqy 
 anqy -\- mbqy -(- pbny -f- xbnq 
 > bnqy 
 
 ace adfx 5 cbfx* ebdx* 
 ^ bl:^dx^fx s= bdfx^^ r bdfx^ bdftf 
 adfi^-\- befx* + bdex 3 
 
 = bdfi* ' 
 
 1+3* 1-S» (1 + X 2 ) 2 (!-■ *T 
 
 W l _x 3+ 1 + x-~~ (1 — x 2 )(l+x 2 ) + (1 — x 2 )(l +x 2 ) 
 
 (l+X 2 ) 2 +(l-X 2 ) 2 
 2(1 4- X*) 
 
 1 _1 1— X 1 + X 
 
 ^ l+x"*~l—x — (l+x)(l— x) + (l-fx)(l— x) 
 _1— x+l+x 
 -(l+x)(l-x) 
 2 
 
 1— x 2 
 
 SUBTRACTION OF FRACTIONS. 
 43. Rule. — Reduce the fractions to a common denominator, tubtia.** tfu 
 numerator or the sum of the numerators of the fractions to be subtracted, from 
 the numerator or the sum of the numerators of the others, and subscribe the com- 
 mon denominator.* 
 
 a c ad be ad — be 
 (1 ' b~d = bd~bd— bd ' 
 
 am ip x\ anqy mbqy 2 ybn V xbnq 
 ' 6 •" n \q y) bnqy* bnqy bnqy bnqy 
 anqy -\-mbqy — pbny — xbnq 
 bnqy 
 
 * The rales for addition and subtraction of fractions follow from the general principle thai 
 quantities to be added or subtracted must be of the same denomination. 
 
48 ALGEBRA 
 
 a c e g ad/Jufl bcfJufl h d) ' bdfgz? 
 ^ bx+dx 2 ~Jx z ~kx i== bdfhx^ bjfhx 10 ~~ bdfhx l0 ~ bdf t> 
 
 adfhs? -\-hrfh ./•- — bedkx — bdfg 
 — bdfhx* 
 
 \*l 
 
 a — b 
 
 « + 6" 
 
 ~ ( a + b){a—b) 
 Aab 
 
 -a?—b~' 
 
 
 
 
 
 (5) 
 
 1+x 2 
 
 1 — x 2 
 
 (1+x 2 ) 2 
 
 
 (1- 
 
 -X 2 ) 2 
 
 
 1 — x- 
 
 1+x 2 
 
 -(l_a*)(l+a*) 
 
 (1- 
 
 — X 
 
 »)(!+: 
 
 *) 
 
 
 
 
 (l+x 2 ) 2 -(l- 
 
 x 2 ) 2 
 
 
 
 
 
 
 
 ~ (l-x 2 )(l + x 
 
 ') 
 
 
 
 
 
 
 
 4x 2 
 
 
 
 
 
 
 
 
 - 1— X*' 
 
 
 
 
 
 (6) 
 
 1 
 
 a m-n 
 
 1 a' 
 a'" 
 
 ' — 1 
 a m 
 
 
 
 
 
 a 2 frVl a 2 Z r + 2 — Z^o 7 
 
 44. When the denominators of the fractions which it is required to reduce 
 have a common multiple less than their continued product, the result will fre- 
 quently be much simplified by finding this least common multiple, and then 
 reducing the fractions to their least common denominator by multiplying the 
 numerator and denominator of each fraction by the quotient of the least com- 
 mon multiple, divided by the denominator of that fraction. 
 Thus, if we are required to reduce the following fractions : 
 
 a — 3x 3a — 5x 3a — 5x 
 4~ "*" 5 ' 20 ' 
 The least common multiple of 4 and 5 is 20, the denominator of the thirJ 
 fraction ; therefore the fractions, when reduced to their least common denoini 
 uator, are 
 
 5a — 15x 12a— 20x 3a— 5x_5a— 15x-j-12a— 20x+3a— 5x 
 20 "*" 20 ~^~~ 20 = 20 
 
 20a — In 
 
 20 
 
 = a — 2 
 So, also, in 
 
 27— 9x 5x+2 CI 2x-f5 29+4* 5— 37x 
 X ^" 4 G 12"*" 3 "•" 12 12 ' 
 
 tho least common multiple of 3, 4, G is 12, which will be the least common de>- 
 nominator, ;mcl the above fractions become 
 
 12x 81— 27x 10x+4 (11 Rr+20 29+4x 5— 37x 
 12 T 12 L2 12 T 12 T 12 12 
 
 Or, 
 1 2x+81— 27x— lOx— 4— 61 + 8x-f 21 I -f -" 1 4- lx— 5-f 37x_24x-f GO 
 
 12 ~~ ~~ i~2 
 
 =2r+5. 
 
FRACTIONS. 4!» 
 
 MULTIPLICATION OF FRACTIONS. 
 45. Rule. — Multiply all the numerators together for a new numerator, and 
 til the denominators togeOierfor a new denominator. Thus,* 
 a c ac 
 
 W b x d^bd- 
 
 a m v x ampx 
 ' 2 ) aX-X^X-= 
 
 a 
 
 q y bnqy ' 
 
 a+b ■ e-£ k+l p-q (a+b)(e-f)(k+l){p-q) 
 3) c -\-d X g-h X m-n X r+s-{c+d)(g-h)(m-n){r+sy 
 
 abode abcde a 
 U ) bl: X cx^ X a^ X ex iX Jr^ = bcdefx 15= Jx Ti ' 
 
 DIVISION OF FRACTIONS. 
 
 AG. Rule. — Invert the divisor and proceed as in Multiplication.^ 
 a c a d ad c ad acd'. a 
 
 W b+d=b x -c=-b7- Proof ' d x Tc=bTd=b- 
 
 { "> c+d-g-h-c+d X e-f~(c+d){e-fy 
 
 l+g- 3 _ 1— a 8 1+a* 1 + a? (1+x 2 ) 2 
 * 3 ) IZZ^l+x 2- l_a: 2>< l— x 2_ (1— z 2 ) 2 " 
 
 x* — b* x' 2 ^-bx .r 4 — b* x — b 
 
 ^ x q —2bx-\ r b' i ~ r x—b = x 2 — 2bx-\-b iX x^+bx' 
 
 (xi—b*)(x—b) 
 
 '(a?— 2bx+b*)(x*+bx) 
 
 (3*— &3)(a«+6 8 )(j— ft) 
 = ( X —by.x.{x-\-b) 
 
 (x+b)(xg- b)(x"-\-b%x— b) 
 = x (x—b)(x—b)(x+b) 
 
 x 2 +ft 2 
 
 47. Miscellaneous Examples in the operations performed in Algebraic Frae 
 >ions. ^ 
 
 3a hf x 42aey-\-35bfy—8bex 
 ^ Tb^ r 8e~7y~ 56bcy 
 
 2a bdf deg 16abc-\-15cdf—4deg 
 
 (2) 
 
 3bc~8b-c 6b"c-~ 24ft 2 c 2 
 
 /« f g 3 , f m 6c/g(e-/)-3g«+2/W 
 W e -J-2ef + 3eg- 6efg 
 
 a c dg a — cx+G?x r + 8 
 
 T n X n — 1 ~^X n — r — a X n 
 
 * To multiply a quantity by the fraction 3, for instance, is to take it as many times as is 
 expressed by this multiplier, that is, two thirds of a time, or to take two thirds of it, 
 which is done by dividing it by 3, and multiplying by 2. If the multiplicand be a fraction, 
 this is done, as has been before shown (17, Note), by multiplying its numerator by 2, and 
 its denominator by 3, which accords with the rule above given. 
 
 t This rule depends upon the principle that the divisor, multiplied by the quotient, must 
 produce the dividend. 
 
 D 
 
 « 
 
50 ALGEBRA. 
 
 b"c— 5a6 2 c+a 3 2a6 3 — 6c 2 +3a6c s — a* 
 
 6 2 — 6c 6- — be 
 
 (5) c+2a6 — 3ac- 
 
 ( 6 ) -g — I — o~= a 
 a+6 a-b 
 
 (') -Q- Q~ = b - 
 
 13a— 5b 7a— 2b 3a 89a— 556 
 
 (9) 
 
 4 6 5 60 
 
 3a— 46 2a — 6— c 15a— 4c 85a— 206 
 7 ~ 3 "*" 12 = 84 ' 
 
 a a— 36 a 2 — 6 2 — a6 aca 7 — 46 2 +a 3 
 (1 °) 6+~ c~d~ + bed = bc~d * 
 
 a 3 a6 _6 a 3 +a6 2 +6 3 
 
 (U > (a+6) 3_ (a+6) 2 +a+6 _ (a+6) 3 ' 
 
 (12) 3 , 3 1 _ 1-x l + x+X» 
 
 v ' 4(1— x) 2T 8(l — x)^8(l+x) 4(l+x 2 ) - 1- x -x'+x*' 
 
 V 1 ^ a 2 — 6 2X a+6 — a 2 +2a6+6 2 * 
 
 . x 2 — 9x+20 a: 2 — 13x+42 a: 2 — llx+28 
 
 * ' x 2 — 6x X a: 2 — 5.r = 5 * 
 
 x 2 +3x+2 x 2 +5x+4 x+2 ^ 
 V ' x 2 +2x+l X x 2 +7x+12 — x+3* 
 
 a c 
 
 b + d (ad+be)fh 
 
 e ,£~( eh +fs)bd' 
 
 a b 
 
 a+b + a~=b 
 
 a — 6 a+6 
 
 a+x a — x 
 
 a — x ' a+x a 2 +x 2 
 
 (ic\ ! ' 
 
 " ' a+x a — x 2ax 
 a — x a+x 
 
 *-l 
 
 < 19 > — £r 
 i 
 
 < 20 )^ 
 
 71+1 
 
 a 3_a-x+ax 2 — x 3 a 4 — x« 
 
 a 6 — a A x-\-a?x- — a^+ax* — x* a 6 — x 6 
 
 a'+x« 
 
 "o+aV+i*' 
 
 • These examples admit of tho application of the formulas at the top of p.i 
 
EXTRACTION OF HOOTS. 51 
 
 ON THE FORMATION OF POWERS, AND THE EXTRACTION 
 OF ROOTS OF ALGEBRAIC QUANTITIES. 
 
 48. We begin by considering the case of monomials, and, in order to sim- 
 plify the subject as much as possible, we shall first treat of the formation of the 
 square and the extraction of the square root only, and then proceed to gener- 
 alize our reasonings in such a manner as to embrace powers and roots of any 
 degree whatsoever. 
 
 Definition. — The square root of any expression is that quantity which, 
 when multiplied by itself, will produce the proposed expression. Thus, the 
 square root of a~ is a, because a, when multiplied by itself, produces a" ; the 
 square root of (a-\-b)- is a-\-b, because a-\-b, when multiplied by itself, pro- 
 duces (a-\-b) 2 ; in like manner, 8 is the square root of 64, 12 of 144, and so on. 
 The process of finding the square root of any quantity is called the extraction 
 of the square root. 
 
 The extraction of the square root is indicated by prefixing the symbol \f to 
 the quantity whose root is required. Thus, V a 1 signifies that the square root 
 of a* is to be extracted ; -\/a--\-2ab-\-b 2 , or V (a 2 -\-2ab +& 2 )» signifies that the 
 square root of a"-\-2ab-\-b i is to be extracted, &c. 
 
 In order to discover the method which we must pursue to extract the square 
 root of a monomial, let us consider in what manner we form its square. Ac 
 cording to the rule for the multiplication of monomials, 
 
 (5a"b 3 cY=5a'b s c X 5a 2 6 3 c=25aW. 
 
 So, 
 
 (9a6 2 c 3 J- , ) 2 =9a5 2 c 3 ^ X 9ab"-c 3 d 4 =81a-b*c 6 d 8 . 
 
 And, 
 
 (Ax m y n z h - - -)-=Ax m y n z b - - - X Ax m y a z h - - -= A 2 a: 2a yz 2h - - - ; 
 i. e , we add the exponent of each letter of the given monomial to itself. 
 
 49. Hence it appears that, in order to square a monomial, we must square 
 its coefficie/tt, and midtiply the exponents of each of the different letters by 2. 
 Therefore, in order to derive the square root of a monomial from its square, 
 we must, 
 
 I. Extract the square root of its coefficient according to the rules of Arith- 
 metic. 
 
 II. Divide each of the exponents by 2. 
 Thus, we shall have 
 
 V64a 6 6 4 =8a 3 6 2 . 
 This is manifestly the true result, for 
 
 (8a 3 6 2 ) 2 =8a 3 6 2 X 8a 3 6 2 =64a 6 fr 4 . 
 Similarly, 
 
 Here, also. 
 Again, 
 
 • v /625a 2 6 8 c 6 =25a6 4 f 3 . 
 {25ab i c 3 )"z=.25ab 4 c 3 X 25ab 4 c 5 =625a t 't*c e 
 
 V 25a 6 p- 18 c 4 d- 32 =5a 3 p~ 9 c 2 d~ lG —-^ 
 
 Also, 
 
 ■y/8la 2m x im y 6n z*r~~ i =9a m x~ m y 3n zv- 1 . 
 
52 ALGEBRA 
 
 \lso, 
 
 m rk+p — q 1 
 
 \fl6c m d n+ P-ig=4c*d * g i . 
 if the given quantity bo a fraction, extract the square root of its numerator 
 and denominator separately. This rule follows from that for multiplication 
 of fractions. Thus, 
 
 Il9a*be_7a-b 3 
 
 VlGc 2 ^ 4- 4cd*' 
 
 Also, 
 
 l36a- m b Sa 6a m b* n 
 V 64a 2 Pc 4 = 8aPc s " 
 \lso, 
 
 I a^c* a?b 5 c 
 
 V (a + z) -/i 1 Y (a + x) /i 5 3/ a ' 
 Also, 
 
 •>-2ii3 
 
 V \ m 4 X a^ 4 / 8 / "afflW/** 
 
 50. It appears, from the preceding rule, that a monomial can not be the squaie 
 of another monomial unless its coefficient be a square number, and the exponents 
 of the different letters all even numbers. Thus, 98ab* is not a perfect square, 
 for 98 is not a square number, and the exponent of a is not an even number. 
 In this case we introduce the quantity into our calculations affected with the 
 sign VTand it is written under the form */98ab*. Expressions of this nature 
 are called Surds, or Radicals of the Second Degree.* 
 
 51. Such expressions can frequently be simplified by the application of the 
 following principle : The square root of the product of two or more factors i$ 
 equal to the product of the square roots of these factors. Or, in algebraic Ian 
 guage, _ _ _ _ 
 
 y/abed = y/aX y/°X V?X Vd 
 
 In order to demonstrate this principle, let us remark that, according to our 
 definition of the square root of any expression, wo have „ 
 
 ( y/abed )-=abcd 
 
 Again^ 
 
 (V^x Vox Vex Vd---y=(V<i) : x(Vt>Y-x(Vcyx(Vdy~-i 
 
 ssdbcd . 
 
 Hence, since the squares of the quantities \Jabed , and y/a- V° 
 
 y/c y/d — are equal, the quantities themselves must be equal. 
 
 This being established, the expression given above, v98ao*, may be put 
 under the form ^49i«X2a== y/^' T X V~^, but y/Ml>* > 9 by (Art. 49)=75» ; 
 hence 
 
 y/98b*a= yflbT* x i/2a=7&» \J%0. 
 
 Similarly, 
 
 y/TbaW<*ds* y/9a' ! l>-c i Xbt>d= \Z9n"6-c 3 X y/^d 
 
 = 3abcy/5bd- 
 
 * From tho Latin turd/Ut. They arc sometimes called inconimcnsuraWe, having no com- 
 mon measure with unity. They aro also called irrational, because their ratio with unity 
 ota in t be expressed in numbers. Fractions have Kith a common measure and ratio with 
 unity, Tim i the frax don j has } for a rorumon measure w ith unity, and Its ratio with <mi 
 tv is I. t This I m (10, 111., note). 
 
EXTRACTION OF ROOTS. 53 
 
 ^o, also, 
 
 
 Also, 
 Also, 
 Also, 
 Also, 
 Also, 
 
 l/24a"b=! 
 
 -v/54a 3 6 3 c=3a& y/tiabc. 
 
 2 A /8a 2,n + 1 6=4a m y/2ab. 
 
 3 y/75p 9 q 4 = 15 F i f Vty. 
 
 g /48.r 2 i'+ 3 ?y 3 SG-tp+'t/ 2 /3x?/ 18.i't'+ 1 _y 2 /n/ 
 V" 12a 2 6 = 2a V ~3T = a V "6"' 
 
 In general, therefore, in order to simplify a monomial radical of the second 
 degree, separate, those factors which are perfect squares, extract their not (Art. 
 49), place (lie product of all these roots before the radical sign, and place all 
 those factors which are not perfect squares under the radical sign. 
 
 In the expressions, lb' l y/2a, 3abc-\/5bd, 12ab 2 c 5 V 6bc, &c, the quantities 
 7b 2 , 3abc, 12ai 2 c 5 , are called the coefficients of the radical. 
 
 52. We have not hitherto considered the sign with which the radical may 
 be affected. But since, as will be seen hereafter, in the solution of problems 
 we are led to consider monomials affected with the sign — , as well as the 
 sign -J-, it is necessary that we should know how to treat such quantities. 
 Now the square of a monomial being the product of the monomial by itself, it 
 necessarily follows that, whatever may be the sign of a monomial, its square 
 must bs affected with the sign +. Thus, the square of -\-ba?b 3 , or of — 5a 2 b 3 , 
 is + 25a 4 6 6 . 
 
 Hence we conclude that, if a monomial be positive, its square root may be 
 either positive or negative. Thus, ■\/9a 4 =-\-3a! i , or — 3a 2 , for either of these 
 quantities, when multiplied by itself, produces 9a 4 ; we therefore always affect 
 the square root of a quantity with the double sign rt , which is called plus or 
 minus. Thus, •/9a*= ± 3a 2 , 71141^6= ±12aZ> 2 c 3 .* 
 
 53. If the monomial be affected with a negative sign, the extraction of ita 
 square root is impossible, since we have just seen that the square of every 
 quantity, whether positive or negative, is essentially positive. Thus, sf — 9, 
 
 * The double sign may be omitted, being always understood before y/ . An important 
 proposition, not usually noticed, should be demonstrated here ; it is, that the quantity A has 
 no otber square root than the two, -\-\/ h. and — -j/A. To prove this, let us observe that 
 the different square roots of A are the values of x in the equation x 2 :=A, or what is the 
 same, 
 
 x'i—K—Q. 
 
 Instead of x 2 — A, we may write x" — (1/A) 2 ; then, decomposing this difference into two 
 factors, we have 
 
 x 2 — A= (x— V A) (x-f-i/A). 
 
 Under this fonn we perceive that every value of x which is not either +V'A or — \ZA, 
 will fail to render either of these two factors zero ; then it will not render the product x- — A 
 •zero ; therefore the quantity A has no other square root than ^y/A. 
 
 The square root of a quantity lias, therefore, two values, which are equal with contrary 
 *ign$ and it has no otlier values. 
 
54 ALGEBRA. 
 
 •\/ — Ad z , -/ — 5, are algebraic symbols which represent operations which it is 
 impossible to execute. Quantities of this nature are called imaginary or im- 
 possible quantities, and are symbols of absurdity which we frequently meet 
 with in resolving quadratic equations. 
 
 By an extension of our principles, however, we perform the same opera 
 tions upon quantities of this nature as upon ordinary surds. Thus, by (Art 
 51), 
 
 V~^9 = y /9X— 1 == -/?■ V~l =3 V"l. 
 
 V— 4£_= V^a- X — 1 = yfia? V~—i =2a -/ — 1 _ 
 
 V — 8a 2 6= -/^X4a-X^X— 1= i/ia= X V~2b X V— 1 =2a \f 26 V —1 
 
 54. Let us now proceed to consider the formation of powers and extraction 
 of roots of any degree in monomial algebraic quantities. 
 
 Definition. — The cube root of any expression is that quantity which, mul 
 tiplied twice by itself, or taken three times as a factor, will produce the pro 
 posed expression. The fourth, or biquadrate, root of any expression is that 
 quantity which, multiplied three times by itself, or taken four times as a fac 
 tor, will produce the proposed expression ; and in general, the n & root of any 
 expression is that quantity which, multiplied (n — 1) times by itself, or taken 
 n times as a factor, will produce the proposed expression. Thus, the cube 
 root of a?b 3 is ab, because ab, multiplied by itself twice, or taken three times 
 as a factor, produces a 3 Z> 3 ; for the same reason, {a-\-b) is the 6 th root of 
 (a-\-b) 6 , 2 is the seventh root of 128, and so on. 
 
 55. Let it be required to form the fifth power of 2a 3 b 2 . 
 
 i2a 3 b-y=2a 3 b"- X 2a*b 2 X 2a 3 b i X 2a 3 i' : X 2a 3 6 2 
 =32a 15 6 l °. 
 Where we perceive, 1°. That the coefficient has been raised to the fifth 
 power ; 2°. That the exponent of each of the letters has been multiplied by 5 
 Tn like manner, 
 
 (8a 2 i s c) 3 =8a 2 6 3 c X 8a 2 6 3 c X 8a 2 i 3 c 
 
 = 8 3 a 2+2 +2£3+3+3 c l +1+1 
 
 =512u 6 i 9 c 3 . 
 So, also, 
 
 (2ab°chl i ) n =2ab"-cWx2ab 2 c 3 d- i X to n factors 
 
 =2 n a n b- n c 3 "d in . 
 Hence we deduce the following general 
 
 RULE TO RAISE A MONOMIAL TO ANT POWER. 
 
 Raise the numerical coefficient to the given power, and multiply the exponents 
 of each of the letters by the index of the power required.* 
 And hence, reciprocally, wo obtain a 
 
 R0LE TO EXTRACT THE ROOT, OF ANY DECREE, OF A MONOMIAL. 
 
 1°. Extract the root of the numerical coefficient according to the rules of 
 arithmetic. 
 
 2°. Divide the exponent of each letter by the index of the required root 
 Thus, 
 
 J/Gla-'^c 6 =4a 3 6c 9 
 
 Vl6aft 1 W= ' 
 
 • When ii quantity is positive, nil ll tivej bat it" it is u D its 
 
 ■ven powen will be positive, and its uneven qi I 
 
EXTRACTION OF ROOTS. 54 
 
 EXAMPLES. 
 
 (1) {2abc) 5 z=32a 5 ¥c 5 . 
 
 (2) (3a' 2 m 3 n i ) 3 =27a 6 tnPri 12 . 
 
 (3) (z m ifzr) s =x 8m y m z 8 P. 
 
 (^.m+l^n— 3\ 7 ^7m+Jy7n— 21 
 2 n-p+l / == 2 7u-7p+7 * 
 
 (5) (m - 432 X p 4 ' 23 * X o 3,789 X r° ,(M | 013 =m 005618 p°- 66,>42 g - 49257 r a00<i8 . 
 
 /a k \ m a kra 
 
 (6) ^j =55T. 
 
 p pji 
 
 (7) (x^^xi. 
 
 m pq[ 
 
 (8) (y»)°»=y»». 
 
 (A 4 & kr c mn \ * A 4 ^5 kp *c mn ^ 
 
 V v 
 
 m s n 10 2mn 2 
 
 pisqio - ~ p3q\ 
 
 (12) i I™** 
 (13)^1 
 
 < 14 ) x/i 
 
 21q B 2 &PZ 3 1 
 
 1042 — a 6/SG 
 
 032^16^24 ©4])2t3 3 
 
 256 = 2 * 
 sin 2 0cos 2 i sin0cos0 
 
 ' tan 4 i/> sec 6 ^ tan 2 i/' sec 3 ^ 
 6 K 3'Q<r-5(«+&)i5(a:+y)- 10 (&+g— x ) 20 \ _ &ctr-i{a+b)3(x-\-y)-*(b+c—x)* 
 
 When the root to bo extracted is of an uneven degree, its sign should be that 
 of the given quantity ; when of an even degree, it should be i . (See last note.) 
 
 56. By the rule for extracting a root, we perceive that, in order that a 
 monomial may be a perfect power of that degree whose root is required, its 
 coefficient must be a perfect power of that degree, and the exponent of each 
 letter must be divisible by the index of the root. 
 
 When the monomial whose root is required is not a perfect power of the re- 
 quired degree, we can only indicate the operation by placing the radical sign 
 ■/ before the quantity, and writing within it the index of the root. Thus, 
 if it be required to extract the cube root of 4a-b s , the operation will be indi- 
 cated by writing the expression, 
 
 y~4aF&>. 
 Expressions of this nature are called surds, or, irrational quantities, or radi- 
 cals of the second, third, or n' h degree, according to the index of the root re- 
 quired. 
 
 57. We can frequently simplify these quantities by the application of the 
 following principle, which is merely an extension of that already proved in 
 (Art. 51). 
 
56 ALGEBRA. 
 
 The n" 1 root of the product of any number of factors is equal to the produc 
 of the n' A roots of the diffi rent factors. Or, in algebraic language, 
 
 yabcd = y~a xVbXVcxVdX • 
 
 Raise each of these expressions to the power of the degree n, then 
 
 {"Veiled ) n =abcd 
 
 And, 
 
 (V«x Vox VcX Vd- ")"=( V«)°X(Vi) n X( Vc) n X( W) n 
 
 =abcd . 
 
 Hence, siuco the n' h powers of the quantities V abed, and y a . \f b . "y/c 
 Vd are equal, the quantities themselves must be equal. Q. E. D. 
 
 This being established, let us take the expression y 5ia*b 3 c", whose roct 
 can not bo exactly extracted, since 54 is not a perfect cube, and the exponents 
 of a and c aro not exactly divisible by 3. 
 
 We have, 
 
 (1) y'oAaWc-— V 27 X 2 X a 3 X a X ^ X c 3 
 
 = V~'7X Va :j X tytPx V2ac* 
 by the principle just proved, 
 
 = 3aby/'2ac 2 . 
 So, also, 
 
 (2) V WaWc* = V16 X'3 X a* X ax b^Xc^X* 
 
 = vTgxV^x V&x V*x V"sx V«x V* 
 
 =2ab"cy3ac 2 . 
 
 (3) f i/ld2a'bc li =y6ix3xa 6 Xa_XbXc u 
 
 = V64 Xj/jfiX V<P X y/~3 X VaX V~b 
 =2ac"fy3ab. 
 
 (4) Vl92=4^3.* 
 
 (5) 5 y 56u* 6 5 =10aiV '? <**> 2 - 
 
 (G) V x u> y~ 6 z $m + l =x a y- 1 2 ra ^/z. 
 ( 7) bS2 ^*i-S3 
 
 *)^/ 
 
 =B b C-s7 — . 
 
 ?/l V 7/1 
 
 In the above expressions, the quantities 3ab, 2ab 2 c, 2ac a , <5cc, placed before 
 the radical sign, are called the coefficients of the radical. 
 
 58. There is another principle which can frequently bo employed with ad- 
 vantage in treating these quantities; this is, 
 
 Tlie m"' power (fth < a of any quantity is equal to the nan 1 * power of 
 
 Oiat quantity. Or, in algebraic language, 
 
 |a"} m =a mn . 
 
 * A good way of separating a number into factors, some of which arc perfect powers, is 
 to try perfect powers upon it as divisors, beginning with powers of the lowest nunbaai 
 Tims, in the -nil example, B, the cube of 9, will divide 199, and the quotient ia 94 1 again, 8 
 will divide 94, and the original number, 199, may be pot under the form ^X8X3:=G4X3, 
 and the cube tool will be "X-X ^3,0x4^3. The cube root of IPSO may be found by first 
 tiriding by 93, and that quotient by 3 s , or 97. The remit N <5=9X3^5==64<5 
 
EXTRACTION OF ROOTS. 57 
 
 For we have, 
 
 \a s \ i =a 3 Xa :i Xa 3 Xa 9 
 
 And, in general, 
 
 \a n \ m ^a n Xa n Xa"xa n ---tom factors; 
 
 /-fn-l-n-f-n + n — to m terms > 
 
 =a m ". 
 AeJ, recipi-ocall^, 
 
 The mn'* root of any quantity is equal to the m" root of the n* root of that 
 quantity. Or, in algebraic language, 
 
 mn/ — / „ r— 
 
 V«=™/ tya. 
 For, let 
 
 ^Jya—p ; 
 
 Raise the two quantities to the power m, 
 
 y~a=p m ? 
 Again, raise both to the power n, 
 
 a — 2)'"" • 
 
 Extract the mn" 1 root, 
 
 mn# 
 
 But, by supposition, 
 
 4 1 
 
 yVa=p, 
 
 .-. m ya = ^jVa. 
 
 Hence, as often as the index of the root is a number composed of two or 
 more factors, we may obtain the root required by extracting, in succession 
 the roots whose indices are the factors of that number. Thus, 
 
 (1) V4^= 3 * Vla~", 
 
 =* h/4a" by the above principle, 
 
 (2) V36a 2 b-= V36a 2 b' 2 
 
 = -J Gab. 
 
 (3) ^256=4/^256 = ^/16=2. 
 
 (4) ^/32oJb 5 = y/2ab. 
 
 (5) ^16a 4 x 2 T/ 2m 2- t "- 4 = V 4.a?xy m z° a - 2 . 
 
 (6) In general, 
 
 m ya n =™Jya a 
 
 = l/«- 
 That is to say, When the index of the radical is multiplied by a certain 
 number n, and the quantity under the radical sign is an exact n' h power, we 
 can, without changing the value of the radical, divide its index by n, and ex 
 tract the n' /l root of the quantity under the sign. 
 
58 
 
 ALGEBRA. 
 
 Thus, 
 
 
 
 i/25a*b-c 6 = */oa"bc 3 , 
 
 
 ty27 m l Wp 6 = ^/3m 6 n 3 p' i , 
 
 
 ^/27a :i x 3n 'y 3 ^-^ =?/3ax m yP-*, 
 
 ty q^^r 5 " = -\Zqj)- l r a . 
 
 59. This last- proposition is the converse of another not less important, 
 wnich consists in this, that we may multiply the index of a radical by any num.' 
 ber, provided we raise the quantity under the sign to the power whose degree is 
 marked by that number, or, in algebraic language, 
 
 y~a= m ya™. 
 For, if the last rule be applied to the second of these quantities, it will pro- 
 duce the first. 
 
 60. By aid of this last principle, we can always reduce two or more radi- 
 cals of different degrees to others which shall have the same index. Let it be 
 required, for example, to reduce the two radicals \/2a and ^/'3bc to others 
 which shall be equivalent, and have the same index. If we multiply 3, the 
 index of the first, by 5, the index of the second, and, at the same time, raise 
 2a to the 5th power; if, in like manner, we multiply 5, the index of the 
 second, by 3, the index of the first, and, at the same time, raise 3bc to the 3d 
 power, we shall not change the value of the two radicals, which will thus 
 become 
 
 • ty2a~ =«*^(2a) 5 = ty32a 6 
 
 y3bc= 3 *fy{3bc) 3 = l i/27b 3 c 3 . 
 We shall thus have the following general 
 
 RULE. 
 
 In order to reduce two or more radicals to others which shall be equivalent 
 and have the same index, multiply the index of each radical by the product of 
 the indices of all the others, and raise the quantity under the sign to the power 
 whose degree is marked by that product. 
 
 Thus, let it be required to reduce -\f2a, $/3b n -c 3 , %/4d*e 6 f 6 to the same 
 index, 
 
 2/7,7 —3X6X »/f.;> /7 \3X 6 — 27 915/715 
 
 y 2a = 3X6xy(o (l)8 x5 _y 
 
 ty3b-c 3 =- x 5 x s/(36' 2 c 3 ) 2X5 =*y3 lo b' x >c 30 
 
 ■v/4d-'e 5 / 6 = 3X3X y{id*ef»)-* 3 = ^A 6 d^e 30 j - 
 The above rule, which bears a great analogy to that given for (he reduction 
 of fractions to a common denominator, is susceptible of the same modifications. 
 
 RULE. 
 
 To reduce radicals to their least common index, find the least common multi- 
 ple of all the indices, divide it by the index of each radical, and raise the 
 quantity under the radical to thepowt r expressed by the quotient.* 
 Tins rulo, applied to the radicals {/a. fybb, \/2c, gives 
 
 Va= 5 v / ^'. VTb= , yG2!Jb\ V 3c= V'STc 3 - 
 l \ UUPLES. 
 
 (1) Reduce \/a m , \/b", and %/<-' ii> die same index. 
 
 (•J) Reduce 'y/a, \/l>, and ^/.- to the same index. 
 
 (3) Reduce fyefi, fyb\ V<\ ;11 " 1 $/<P to the aame index. 
 
 * This is, in effect, multiplying the index of cadi radical, and the- exponents under that 
 radical. l»v the quotient- 
 
CALCULUS OF RADICALS. ' 59 
 
 (4) Reduce W— , \J-ri and W— to the same index. 
 
 (5) Reduce -J _, , ll . ■, and $/- to the same index 
 
 ANSWERS. 
 
 (1) >9 a 4ra , '-yi 3n , and ^c 3 p. 
 
 (2) m 7"^; mn ^"FF, and ran -(/T™. 
 
 (3) MfyrtW, ^W, "tofyfrfi, and "Pifd&Br. 
 
 /O /Dp 7 ////.pi 
 
 < 4 > "V^ T*r. and "V^s- 
 < 5 ) v^zs? VfcFip and v? 
 
 61. Let us now proceed to execute upon radicals the fundamental opera- 
 tions of arithmetic. 
 
 ADDITION AND SUBTRACTION OF RADICALS. 
 
 Definition. — Radicals are said to be similar when they have the same in- 
 dex, and when, also, the quantity under the radical sign is the same in each ; 
 thus, 3-\/a, I2ac-\/a, 156 -\/a, are similar radicals, as are, also, Aa-b^mn^jP, 
 bltymnFjP, tydynm*]?, &c. 
 
 This being premised, in order to add or subtract two similar radicals we 
 have the following 
 
 RULE. 
 
 Add or subtract their coefficients, and place the sum or difference as a coeffi- 
 cient before the common radical. For example, 
 
 (1) 3^/bJ t -2^T=5^/b. 
 
 (2) 3^6—2^6 = ^/6. 
 
 (3) 3pq\fim+4ltyrm=:(3pq+4l)tymn.* 
 
 (4) Sicds/a — 4cd-\Za=5cd-Ja. 
 
 If the radicals are not similar, we can only indicate the addition or subtrac- 
 tion by interposing the signs -f- or — . 
 
 It frequently happens that two radicals, which do not at first appeal - similar, 
 may become so by simplification ; thus, 
 
 (5) V48a6 3 + 6 V75a= V3Xl6xaX6 3 -f6-v/3X25xa 
 
 =46 V3a-f-56 </3a 
 = 96 -/3a. 
 
 (6) 2 V45— 3 V5=2 -/o X 9— 3 y/Z 
 
 s=3-/5. 
 
 (7) ^/8a 3 6+16a*— ^6<+2a6 3 =^8a 3 (6+2a) — #6 ;! (6+2a) 
 
 = (2a — 6)^2a+6. 
 
 * When two products, consisting each of several factors, have any common factors, the 
 other factors may be regarded as the coefficients of these, since they show how many times 
 the common factors are repeated, and the addition may be performed by adding the coeffi- 
 cients, and annexing the common factors to the sum; thus, abcd-\-m ncd=iab-\-mn)cd, and 
 5abi/x-{-4cb\/x=(oa-\-4c)b-\/x, on the same principle as 8a-\-ia=12a. 
 
60 ALGEBRA. 
 
 (8) 2VTa*+2y2a=3V2a+2V2a 
 
 =5ty2a. 
 
 (9) V8+ V50— -/18=4V2. 
 
 (io) iy ab"+cy~^—dy ad< n ={i--\-c"-— ( r-)y a. 
 
 (11) 2 V:;+ ^60- V15+ V$=n V15-* 
 
 (12) 4a^a 3 i-»4-i^/ri^6 — ^125a*'6' = a-i^/i. 
 
 (13) V(3a-c -|- 6a6c+3Z/ 2 c)=:(a+5) -y/3c. 
 
 (14) v'45c- 3 — V«0c 3 + v '5a 2 c=(a— c) V '5c. 
 
 MULTIPLICATION AND DIVISION OF RADICALS. 
 62. Li the first place, with regard to radicals which have the same index, 
 let it bo inquired to multiply or divide y a by yh, then we shall have 
 
 V'aX V~b= Vol, and y~a+- yh=yj"r. 
 
 For, if we raise y/aX yh, an ^ V ah, each to the 7i th power, we obtain the 
 same result, ah ; hence these two expressions are equal. The same principle 
 is demonstrated in (57). 
 
 y<i fa a 
 
 In like manner, — = and \Jr, when raised to the n th power, give t ; hence 
 
 the two expressions are equal. We shall thus have the following 
 
 RULE. 
 
 In order to multiply or divide two radicals which have the same index, mul- 
 tiply or divide the quantities under the sign by each other, and affect the result 
 with the common radical sign. If there he any coefficients, we commence by 
 multiplying or dividing Oicm separately. The latter part of this rule depends 
 upon the principles set forth and alluded to in 17, note ; the coefficients, or ra- 
 tional parts, and the radical parts being regarded as factors composing a product 
 
 »)-#^x-3#3S=-c,#!S 
 
 cd 
 6a-(a--\-b-) 
 
 Vcd 
 
 (2) 3aV8a-x2hyia-c=Gab i/32a<c 
 
 = l2a°bty-2c. 
 
 (3) 2a y/Tc X 2b y/abc X a -/2a = 6a : h -J-Ja : h : c- 
 
 = Ga :1 i-c-/2. 
 5ay/b 5a K 
 
 (4) 
 
 2by/c 2iVt- 
 
 (5) 
 
 25a-b Vm 3 n o 5a ^ f~^ 
 
 bah^^mn 1 5alr\tnn* 
 _ba Im? 
 ~b\ n 
 
 ■ Tin' numerator and denominator of each of tin 1 two fraction* in tins example are multi- 
 plied by its denominator. Tin.' ilenominator becomes thus n perfect square, ami may bo set 
 
 le ili'' rad 
 
CALCULUS OF .RADICALS. gj 
 
 Va 3 6 2 4 b* _ 9 fobjaW+b*) 
 
 fiCp V a 2 — ¥ 
 
 \f~ 
 
 =26 a@ 
 
 86 
 
 • 6 2 
 
 (7) (a+6-/^l)X(«-Z'\/ :Z l)=a 3 +6'. 
 
 (8) '/flXyixV^yaic. 
 
 (9) atyxxWyX^Vz=ateyxyz. 
 
 (10) 4X^V~3X V72=8V6. 
 
 (11) c*\/axd^a=acd. 
 
 (12) 5- v /8x3v f 5=30V , 10. 
 
 (13) ^18X5^4 = 10^9. 
 
 (14) »V6X T 3 3 /9f=TVV6. 
 
 (15) 2 ^ X 3 Va$ X 4 ^6<5=24 V^+^R 
 
 (17) (V-15+V-12-V-21)4-V-3 = 2+ V5- -v/7- 
 If the radicals have not the same index, we must reduce them to others 
 having the same index, and then operate upon them as above ; thus, 
 
 (1) 2aylx r ob\/2c=Za^'b i X^h^/S^ 
 =15abfy8b*c?. 
 
 (2) jbab<?x V 2a*bc*— yvZbcP&d* X tylaWc* 
 
 = V500a 7 6 5 c 13 
 
 =ac 2 V500a6 5 c. 
 (3) my a X y/b x n %/c=mn ty a 15 6 20 c 12 . 
 
 a jc x jz ax IcV 
 ^ by]d X y\u~by^d^' 
 
 (5) x$/m s X y^tm=xy ym^i*. 
 
 ( 6 ) V^T^ X ^"- 6x V(a*-6*)»=V~^=6T- 
 
 (7) VaX V^X Vc= m 7a n P6 m Pc mn . 
 
 (8) AV«' r 'XB'{/i"XCVc r =ABC a V« m ^ Da ''c r ^- 
 
 /a m i /a m - J C 3 / a m(3n-2)+i2j3n^l0-3n 
 
 (10) c y/a?—x*-±- 1/a+x=cV(a— xf(a"— x 2 ). 
 
 (11) ^Z?4.( a -z)=^E. | 
 
 Ira <rv A l,.an— im >^ 
 
 (12) A,™ - + A„J--^J x --z-r 
 
 \y p ~\'y s A 2 ~<JyP a -° m 
 FORMATION OF POWERS AND EXTRACTION OF ROOTS OF RADICAL8. 
 63. Let it be required to raise tya to the nth power; then, 
 
 (Vaj n = yfaX V«X ^a to n "factors, 
 
 = tya n , according to the rule for multiplication just established. 
 Hence we have the following 
 
62 ALGEBRA. 
 
 RULE. 
 
 In order to raise a radical quantity to any gicen powet raise the quantity 
 under the sign to that power, and place over the result the radical sign wiOi its 
 original index. If there he any coefficient, we must raise die coefficient sepa 
 rately to the required power. Thus, 
 
 (1) {\/Ta?y= Vl6a 6 
 =2a \TcP. 
 
 (2) (3V2a) 5 =3 5 V32a 6 
 
 =243-^ 32a 5 
 3=486aV4a». 
 When the index of the radical is a multiple of the exponent of the power 
 which we wish to form, the operation may be simplified. 
 
 Let it be required, for example, to square ty2a ; we have seen (Art. 58) that 
 
 ^/2a=yJ V~ a i Dut m order to square this quantity, it is sufficient to sup- 
 press the first radical sign ; hence, ( y/2a)-= <J '2a. Again, let it be required 
 
 to raise y/abc to the 5th power; now, 1 tyabc=\! ^abc; but in order to raise 
 this quantity to the 5th power, it is sufficient to suppress the first radical sign • 
 hence, {^abcf=z -J abc, and, in general, 
 
 (^) ra =W W =V~a; 
 that is to say, 
 
 If the index of the radical be divisible by the index of the required power, we 
 may divide the index of the radical by Oie index of the power, and leave the 
 quantity under the sign unchanged.* 
 
 64. With regard to the extraction of roots, either by virtue of the principle 
 established in (Art. 59), or by reversing the last rule, we shall manifestly have 
 the following 
 
 RULE. 
 
 In order to extract any root of a radical quantity, multiply Oie index of the 
 radical by the index of the root required, and leave the quantity under the sign 
 unchanged. If there be a coefficient, we must extract its root separately. 
 Thus, 
 
 (1) V i/Tc= i y'3c~. 
 
 (2) V V5«"= V5a\ 
 
 ( 3 ) yj8<?\/a r b=2cy~l 
 
 If the quantity under tho sign bo a perfect power of the samo degree as the 
 
 root required, we may simplify. Thus, 
 
 — — . — — , * M 
 
 • It may !»■ w ell to noto hero thai the b\ en pon er of a radios] of the second degree is 
 rational, and the uneven power Irrational, the tatter being farmed by the multiplication of 
 tho proponed radical by a rational quantity. 
 
CALCULUS OF RADICALS- 63 
 
 = yza ; 
 that is, we may extract the root of the quantity under the radical sign. 
 
 MISCELLANEOUS EXAMPLES. 
 
 V24+ V54— \/6= 4 VG- 
 V"l2+2 V27 + 3 V^5+9 V48=59 y/5. 
 V81 — 2 V24+ V28+2 V63=8 </7 — ¥$- 
 V45c 3 — V r 80c 3 + ■)/~ba T c'={a— c) </bc. 
 
 ( 
 
 Vl8a 5 6 3 + V50a 3 6 3 =(3a 2 6+5a6) V2a6. 
 
 V 2 H a 13 6 5 c— V4 X 5 4 a 5 6 9 c 5 + 4/4 X 6 4 a5 5 c 
 
 = (8a 3 6 — 5a6 2 c + 66)y4a6c. 
 
 8 f27a B x_ 3 g£ = (3 a _i) 
 V~2T V26 
 
 /y2i 
 
 26 
 
 ^54a ra + 6 6 3 — ^/16a m - 3 6 6 + ^2a 4m+9 + ^/2c 3 a m 
 
 26 2 
 
 = (3a 2 6 — — + a m+3 +c)^/2a m . 
 
 ^3X2¥/ 4 #2 3 ^ f / # 2 j /3X2V^ 
 
 ^^ y3 5 c 3 ^/ 2 < « 3crf > v / 2 g 
 
 d^3cd\-\/~ 
 
 *M 
 
 8a* 16a 3 \ 2a.r 
 
 m 3 +276 2 i=^^+ 26 - 
 V"4"a 3 2/+8a6i/+46 2 ?/=2(a4-6) ■/#■ 
 A /4a 5 6 2 — 20a 3 6 3 +25a& 4 =(2a 2 — 56)6 >' al 
 
 •v/a 2 x — 2ax 2 -f-^' 3 a — x ^_ 
 
 y/a 2 +2ax-\-x' i a-\-x 
 
 a — 6 -\f ac -\fac 
 
 a+b" 1 / a 2_2a6+P~ «+ & ' 
 
 a-j-6 /a — 6 /a+6 
 ^6-V^+6 == V^36' 
 
 V2x\|xV3= 2 </^. 
 
 V4 X 7~3 X ^6= ^3981312 
 
 atyxxb y/y X c \Tz=ahc mn ty x B Py m Pz ma . 
 
 \a™ 2 Ja 3m + 2 
 ~b~~\ 6 5 c 9 ' 
 
 "£* 
 
 It is manifest that, in gnneral, V ■£/<*= Vv/a; for, by (Ait. 58), each of these expre# 
 eions is = v '«• 
 
C4 ALGEBHa. 
 
 ,o 0) <" 3 lM Wtt l.&V. 
 
 65. Let us now inquire with what sign a moncmial root is to be affected. 
 We have seen (Art. 52) that, whatever may be the sign of a monomial, 
 
 its square is always positive ; and it is evident that, in like manner, every even 
 power must be positive, whatever may bo the sign of the original monomial, 
 and that every uneven power will be affected with the same sign as the origina 
 monomial. 
 
 Thus, — a, when raised to different powers in succession will give 
 
 —a, -{-a 2 , — a '> + «'S — <* 5 » + q6 > — a 7 i &c. 
 A.nd -\-a, in like manner, will give 
 
 -\-a, +a 2 , -{-a 3 , -\-a\ -\-a\ + a<5 » +a 7 , &c. 
 In fact, every even power 2n may be considered as the square of the ?i th powei 
 or a 2n = (a n ) 2 , and must, therefore, be positive; and, in like manner, every 
 power of an uneven degree (2«-|-l) may be considered as the product of the 
 2« tU power by the original monomial, and must, therefore, have the same sign 
 with the monomial. 
 Hence it appears, 
 
 I. Tliat every root of an uneven degree of a monomial quantity must be 
 iffected with the same sign as the quantity itself. Thus, 
 
 V + 8a 3 =2a; y — 8a 3 = — 2a; V~ 32a 10 6 5 = — 2a-b. 
 
 II. That every root of an even degree of a positive monomial may be affected 
 with the sign +, or (he sign — , indifferently. Thus, 
 
 V81a<6 : -=±3a& 3 ; ^/6Aa ls =±2a 3 . 
 
 III. That every root of an even degree of a negative monomial is an impos- 
 sible root ; for no quantity can be found which, when raised to an even power- 
 can give a negative result. Thus, V — a» V — c, . . . are symbols of opera 
 tions which can not be performed, and are called impossible, or imaginary 
 quantities, as V — a, •/ — b, in (Art. 53). 
 
 66. The different rales which have been established for the calculation oi 
 radicals are exact so long as we treat of absolute numbers; but are subject to 
 some modifications when we consider expressions or symbols which are 
 purelytalgebraical, such as the imaginary expressions just mentioned. 
 
 Let it be required, for example, to determine the product of 1/ — a by 
 ■/ — a ; by the rule given in (Art. 62), 
 
 V— «X V — a= V —aX —a 
 
 = V + cr. 
 
 But V+" : =i a i so that there is apparently a doubt as to the sign with 
 which a ought to bo affected in order to answer th>' question. However, the 
 true result is — a ; because, in general, in order bo Bqnare •/>/>, it is sufficient 
 to suppress the radical sign ; but \t — <tx V — " is the same thing as ( ^ — </)* 
 
 and, consequently, is equal to — a- 
 
 Next, let it be required t" determine the product of -y/ — a by y*~l>; by 
 
 the rale (Art. 62) 
 
 V^-aX ■/ — ''= V —aX —b 
 = V + «6 
 
 Pot farther i-xaniples of traoifbrmatiooa, sec Appendix. 
 
FRACTIONAL AND NEGATES EXPONENTS. 6fc 
 
 The true result, however, is — V 'ab, so long as we suppose the radicals 
 y/ —a, V— & to be each preceded by the sign -j- ? for we have, according 
 to (Art. 53), 
 
 Hence, 
 
 
 V -a X V -b~ Vab( V -I) 2 
 = \/«^X —1 
 = — -\/ab. 
 According to this principle, we shall find for the different powers of ■/ — 1 
 the following results : 
 
 & 
 
 V-l =V-i 
 
 (v-ir=-i 
 
 
 (V-i)*=(V-i) 2 x(V-i) 2 
 
 =-ix-i 
 =+i. 
 
 Since the four following powers will be found by multiplying +1 by the 
 first, the second, the third, and the fourth, we shall again find for the four new 
 powers -f- -/ — 1, — 1, — •/ — 1, -f-1 ; so that all the powers of -y/ — 1 will 
 form a repeating cycle of four terms, being successively, -/ — 1, — 1, — ■/ — 1, 
 
 -H-* 
 
 Finally, let it be required to determine the product of 4/ — a by </ — b, 
 which, according to the rule, would be ty-\-ab. To determine the true result. 
 we must observe that 
 
 V— a = V a . V — 1 
 
 V— b =V6.V — 1- 
 
 And .-. 
 But, 
 
 Hence, 
 
 
 = y-i. 
 
 y—aX \/—b=yab.j—\. 
 The above principles will enable the student to operate upon these quanti- 
 ties without embarrassment. 
 
 THEORY OF FRACTIONAL AND NEGATIVE EXPONENTS. 
 67. This is the proper place to explain a species of notation which is found 
 extremely useful in algebraic calculations. 
 
 * This may be expressed in its most general form thus, if n be any whole number : 
 (a\f—i.)* a =a* n X+l =a*° 
 
 (aV^)4a+l=a-«n+lX-fV--- 1=:a 4 D +l . V~i 
 
 (a\/— l)-»°+2=a<n+2X— 1 =— a< n + 2 
 
 Tbe first in the note corresponds to the last in the text, the second in the note to the first 
 in the text, and the third in the note to the second in the text. 
 
 E 
 
 » 
 
66 ALGEBKA. 
 
 ] Let it be required to extract the 71 th root of a quantity such as a m . We 
 have seen by (Art. 55) that, if m is a multiple of n, wo must divide m, the 
 index of the power, by n, the index of the root required. But if m is not 
 divisible by n, in which case the extraction of the root is algebraically impos- 
 sible, we may agreo to indicate that operation by indicating the division of the 
 exponents. We shall thus have 
 
 m 
 
 ya™=a°, 
 
 m 
 
 the expression a a being understood to signify the 71 th root of a m , by a conven 
 tion founded upon the rulo for the extraction of roots of monomial quantities. 
 According to this convention or definition, wo shall have 
 
 tya-=J; !/iv=aJ. 
 
 It may be observed that the denominator of the fractional exponent is the 
 index of the radical, and the numerator the exponent of the quantity under the 
 radical. 
 
 II. Let it be required to divide a m by a n . According to the rule in (Art. 
 17), we must subtract the index of the divisor from the index of the dividend ; 
 so that 
 
 a m 
 
 — z=a m ~° ; 
 a a 
 
 it is to be remarked, however, that here it is supposed that m > n. But if 
 
 m < n, in which case the division is algebraically impossible, we may agree to 
 
 indicate the division by the aid of a nogativo index equal to the excess of n 
 
 over m. Let p be the absolute difference of m and n, so that n=m-{-p ; we 
 
 shall then have 
 
 a m a m 
 
 __ fl m— (m+p) 
 
 =arr. 
 
 a m 1 
 
 But -^jr may also be put under tho form — , by suppressing the factor a a 
 
 common to both terms of the fraction ; we shall then have 
 
 1 
 
 o-p=— . 
 a? 
 
 The expression or? is then tho symbol of a division which can not be exocuted ; 
 and tho true value of the expression is unity divided by the same U-tter a 
 affected with tho exponont^>, takon positively. According to this convention, 
 we shall have 
 
 3 l -» 1 At 
 
 a~ 3 =:— : a ■=— , <Vc. 
 
 Again, by supposing the exponent of the numerator to be larger by j> than 
 tho exponent of the denominator, it may bo proved in B similar maunor that 
 
 1 
 aP=— . 
 
 a i 
 
 From these expressions it appears that a factor may bo transferred from the 
 denominator to the numerator of a fraction, or vice versa, by changing tbo pi^n 
 of its exponent 
 
FRACTIONAL AND NEGATIVE EXPONENTS. 67 
 
 EXAMPLES. 
 
 aW 
 
 Write -ttj in one line. Ans. a 2 6*c _s d"^. 
 
 2a m C a 
 
 Write —5 — in one line. Ans. 3a m c a d~fe' <*. 
 
 d v e^ 
 
 Write - - is m one ^ ae ' ■^■ ns ' 2 x 3 _I g n 7i'»-r&*. 
 
 a 5 6 4 1 
 
 Write -rr„ all in the lower line. Ans. 
 
 A«B^C 1 
 
 Write — tt — all in the lower lino. Ans 
 
 A 5 C- 4 A 6 B6 
 
 Write v ;_ 6r>3 with all positive exponents. Ans. T^ffr 
 
 a a b—$ a a d£ 
 
 Write y ,_ 5 with all positive exponents. Ans. -zttj- 
 
 III. By combining the last two conventions, we arrive at a third notation, 
 which is the negative and fractional exponent. 
 
 Let it be required to extract the n lh root of — . 
 
 1 PT _J2 
 
 In the first place, — =a~ m ; hence y—=y/a~ m =a n > substituting the 
 
 fractional exponent for the ordinary sign of the radical. 
 
 As in words, a m is usually enunciated a to the poioer m, m being a positive 
 
 m m 
 
 integer ; so by analogy, a", a~ m , a n are usually enunciated, a to the power m 
 by n, a to the power minus m, and a to the power minus m by n. 
 
 All that has been hitherto said, with regard to fractional and negative ex 
 ponents must be considered as a mere matter of definition ; in short, that by a 
 
 m 
 
 convention among algebraists a a is understood to mean the same thing as 
 
 1 _iH Jl 
 
 V a m , a~ m to be the same as — , and a n as a /— . We shall now proceed to 
 
 prove that the rules already established for the multiplication, division, forma- 
 tion of powers, and extraction of roots of quantities affected with positive in- 
 tegral exponents, are applicable without any modification, when the exponents 
 are fractional or negative. We shall examine the different cases in succession. 
 
 3 j| 
 
 68. Multiplication. Lot it bo required to multiply aJ by a* ; then it ia 
 asserted that it will be sufficient to add the two exponents and that 
 
 3 2 3 I 3 
 
 
 
 19 
 
 For, by our definition, 
 
 
 
 
 3 
 
 a* 
 
 = y~a~\ 
 
 And, 
 
 
 
 
 2 
 
 a? 
 
 = vV; 
 
 .•. a5Xa^=V« 3 X Va' 
 
 S3 V« 19 
 
 =aT3 by definition in (Art. 67, I.), 
 
orf ALGEBRA. 
 
 3 s 
 
 Again, lot it be required to multiply a * by a 7 ; then it is asserted mat 
 
 3 5 _ s • « \ . 
 
 8 110 
 
 =a' 
 
 For, 
 
 _3 /r & 
 
 a f =\J—, and a*= V« 5 
 
 _3 * A fl 
 
 •.a *xa«=\J- 3 xVa* 
 
 i 
 
 =a 12 by definition in (Art. 67, 1.) 
 
 m p 
 
 Generally, let it be required to multiply a n by a* ; ther 
 
 p «« j p 
 
 a n xa q =a n q 
 
 np — mq 
 
 =a n i 
 For, 
 
 i T =V5= 
 
 p 
 
 and a^= V flP 
 
 m p 
 
 np— mq 
 
 =a n i by definition. 
 69. Hence we have the following general 
 
 RULE FOR EXPONENTS IN MULTIPLICATION. 
 
 In order to multiply quantities expressed by the same letter, add the ex 
 ponents of that letter, whatever may be the nature of the exponents. 
 
 This is the same rule as was established in (Art. 11) for quantities affected 
 with integral and positive exponents. According to this rule, we shall find 
 
 3 _3 , 2 3 It 5 2 
 
 a f b 2 c xa-Pc s =a i b *c ■ 
 3a-*b* X 2cT~h*& =6a~ T iV. 
 
 i 
 
 70. Division. Let it be required to divido a- by a T ; then it is asserted 
 that it will bo sufficient to subtract the index of tho divisor from the index of 
 the dividend, and that we shall thus have 
 
 a- ?,-\ 
 sra*. 
 
FRACTIONAL AND NEGATIVE EXPONENTS. 69 
 
 For, 
 
 3 1 _ 
 
 a 2 = "y/a*, and a* = \/c, 
 
 3 
 
 .•. a 2 V a? 
 
 =V- by (Art. 62, 
 = V^ 
 
 5 
 
 =a 4 by definition. 
 In like manner, we can prove that 
 
 o 
 <* T 5 (_3\ 
 
 _3—a 
 a * 
 
 13 
 
 m p 
 
 Generally, let it bo required to divide a a by a"). 
 Then, 
 
 m p m p 
 
 mq — np 
 
 For, 
 
 m p 
 
 a n = V"a™, and a^= V"cp, 
 m p V"a" 
 
 ■•■ a "+ a * = vaT 
 
 Va"P 
 
 mq— np 
 
 =a "i by definition. 
 
 71. Hence we have the following general 
 
 RULE FOR EXPONENTS IN DIVISION. 
 
 In order to divide quantities expressed by the same letter, subtrac the ex- 
 wonent of the divisor from the exponent of the dividend, whatever may be the 
 nature of the exponents. 
 
 This is the same rule as that established in (Art. 17) for quantities affected 
 with integral and positive exponents. According to this rule, we have 
 
 2 __3 2 / 3\ 
 
 a 3 + a 4 =a 3 * *' 
 
 17 
 
 =a 12 . 
 
 3 4 1 
 
 a T -i-a~ s =a 20 . 
 
 2 3 1 7 9 1 
 
 a H*+a s b*=a 10 b *. 
 
 72. Formation of powers In order to raise a monomial to any power, 
 
 the rule given in the case of positive and integral exponents was, to multiply 
 the index of the quantity by the index of the power sought. We have now 
 to prove that this holds good, whatever may be the nature of the exponent. 
 
70 ALGEBRA. 
 
 5 
 
 Let it be required to raise a 1 to the 4 lb power. 
 Then, 
 
 (a 1 ) =a^ 
 
 For, 
 But, 
 
 20 
 
 =a * . 
 
 J= Va\ and (o"V = ( Va 6 ) 4 - 
 
 ( V« 5 ) 4 = Va 20 , by (Art. 63) 
 
 20 
 
 Generally, let it be required to raise a n to the power p. 
 Then, 
 
 (#=a> 
 
 Dip 
 
 =a~. 
 
 For, 
 But, 
 
 m / m\p 
 
 a° = V~cF, and \o°/ =.( V"a™) r 
 
 ( V« ra ) p = Va mp 
 
 rap 
 
 The demonstration will manifestly be precisely the same if we suppose one 
 or both of the indices to be negative. 
 
 73. Hence we have the following general 
 
 RULE FOR RAISING A MONOMIAL TO ANT POWER. 
 
 Multiply the exponent of the monomial by the exponent of the power required, 
 whatever may be the nature of the exponents. 
 
 This is the same rule as that established in (Art. 55) for quantities affected 
 with positive integral exponents. According to this rulo, we have 
 
 • (a 4 ) =a* X 
 
 15 
 
 = 
 
 l- :: 
 
 =a' 
 
 (a*) =a c ' X3 
 
 :i ' .fX8 
 
 =a 
 
 (2a~~^> 4 ) =2 b a _ -^A 1 
 
 =64a ' 
 
 74. Extraction of Roots. — Tn order to extract the «"' root of any quan- 
 tity nccordiug to the rulo in (Art. 55), we must divide the exponent of each 
 letter by tho index n of tho root. Let us examine the case of fractional ex- 
 oononts. 
 
 Lot it bo required to extract tho cube root of a'-'. 
 
FRACTIONAL AND NEGATIVE EXPONENTS. 71 
 
 Then, 
 
 =a*. 
 
 For, 
 
 But, 
 
 5 3/53/ 
 
 a 3 = Va\ and .-. V« 3 =V V<* 
 
 3 rz 
 
 V V" 5 
 
 ' ya?= Va\ 
 
 =a°, by definition. 
 
 m 
 
 Generally, let it bft required to extract the jp ,h root of a". 
 Then, 
 
 ^ 
 
 ra m 
 
 For, 
 But, 
 
 a a =\a m , and .•.ya a = y tya m . 
 
 V , « ra = n v / a m » (by Art. 58), 
 
 m 
 
 =a"P, by definition. 
 75. Hence we have the following 
 
 RULE FOR THE EXTRACTION OF ANT ROOT OF AN ALGEBRAIC MONOMIAL. 
 
 Divide the exponent of the monomial by the exponent of the root required*, 
 
 whatever 
 
 may 
 
 be the nature of the exponents. Thu9, 
 
 3/ _! _^ 3 
 
 \Ja °=a 5 
 
 
 
 =a~~ T X 
 
 
 ■\Ja 1 b~ 2 =a^~ 1 b 
 
 
 
 3 2 
 =a 3j b *. 
 
 76. We shall close this discussion by an operation which includes the demon, 
 stration of eveiy possible variety of the two preceding rules. 
 
 m y 
 
 Let it be required to raise a" to the power of — ; we must prove that 
 
 s 
 
 m r m r 
 
 (a°) — *=o° ~~ 7 
 
 mr \ 
 n "3 
 
J2 ALGEBRA. 
 
 If we recur to the origin of this notation, wo find that 
 
 m r . / 
 
 _.n~ 
 
 V (Va m ) r 
 
 V \J a mr 
 
 x a mt 
 
 mr 
 
 =a "", by definition. 
 77. The notation above explained can be extended to polynomials, by in- 
 cluding them within brackets, in the same manner as was explained in the case 
 of integral exponents. 
 
 Thus, (x-f-a) 2 signifies the same thing as ■y/x-\-a, or tlie square root of 
 
 x-\-a. 
 
 i 1 
 
 So, (x+fi) 2 is equivalent to , or unity divided by the square rovl 
 
 ■y/x-\-a 
 
 ofx-\-a. 
 
 In like manner, (:r-|-a-|-o)* will be the samo as ty{r-\-a-\-b)*, or the fourth 
 
 3 
 
 root of the third poicer of the quantity x-\-a-\-b, and (x-j-a-r-^) 4 will be 
 
 anity divided by the last-mentioned quantity. Since unity is always under- 
 
 itood to be the exponent when no other is expressed, (x-\-a)~ l is the same as 
 
 — ; — , and so on. The samo rules which have been established for the treat 
 x-|-« 
 
 ment of monomials affected with exponents will also manifestly apply to poly- 
 nomials under the samo restrictions.* 
 
 EXAM 11 
 _3 7 13 1 
 
 (I) a *xa *=a s =—=. 
 
 4, J- ' — 3 c. n la 
 
 a y& 
 
 (2) a~*b~ " Xa t b*c=a rs b~*K= : £y / 
 
 7 
 
 , . o. a^b i S 1 , 1-7— 
 
 (3)— x— =</»>',-!= s/Z 
 
 &V C " 2 
 
 ,r\ ,ir- 
 
 * The calculus of fractional exponents, says Lacroix, is one of the most remarkable 
 unpies of the utility of signs, when they are wi a. 'J" h . ■ analogy which eacists be- 
 
 tween traotiona] and entire exponents renders the roles to be followed in the calculus of 
 the 1 ii, r applicable to the former, while particular roles are requisite for the calculus of 
 '-'. The farther we advance in al cobra, the more we perceive the nnmerooa a<lv:m- 
 which have resnlted to that science tram Uic Dotation of exponents Invented ly 
 Descartes. 
 
(4 
 (5 
 
 (G 
 (7 
 
 (8 
 (9 
 
 (10 
 
 (11 
 (12 
 (13 
 
 (14 
 
 (15 
 
 (16 
 (17 
 
 lie 
 
 (19 
 (20 
 
 FRACTIONAL AND NEGATIVE EXPONENTS. 73 
 
 m p p m np — mq 
 
 a a ~a ^=ai n =a "i 
 
 3 5 c I 
 
 ca 7i ^rdo?=2- a 12 ' 
 
 3 
 3 1 7 1 „27,T 
 
 _n 3 _29 II 11 34 
 
 a 2 6 3 , a * d 3 _ a*b l11 
 
 c*d 3 6 5 c*<2 ' 
 
 (3 2\J. 12 
 
 a*b 3 ) 3 =a 7( b 5 . 
 
 (I _2\_1 1 1 I 
 
 a*b 2 c 5 j *=a 2 6V° 
 
 S c"d f 3 c 3 d 3 
 
 <(a+6)^ (a + 6) 3 
 
 (1 112 1 4 S\ / JL 1\ 
 
 a' 3 '+a s 6 3 +a 8 6 T +a6+o 2 6 3 + fc 3 ) X (a 2 — i 3 )=a 3 — 4». 
 
 (1 l i i\ / ' U 3 3 
 x a +x*y*+y s ) x Vc T — y*J — ^ — 2/ 4 - 
 
 (.r 2 +7 / 2 ) X (.r 2 + t/ •)=««, 2 + 2+x 2 2/ 2 . 
 a 3 — b 3 
 
 9 3 3 3 3 O 
 
 | 3= a T_ a ^4 +a 4 6 2_ & 4 > 
 
 =a 2 — 6. 
 
 1 2 
 
 a 3 — b 3 
 
 (3 3\ / 1 1\ I 1 I 
 
 B*.— &*) : (a 4 — Z/ 4 )=a 2 + & 2 + (aZ>)<. 
 
 3 5 1 13 » 99 3_3_18 27 13 
 
 m?p i q 2 r i XP 'V r 4 m 13 X.p 32 ? 4 = m *.P 105 ? 2 r * • 
 
 1 Sfl 
 
 2 3 2 17 
 
 a 2 b T c- 5 d 3 ^-a i b 2 c s a- 8 = 
 
 a*d 3 
 
 5 37 
 
 4 
 
 (z' + 6Z'a* + 9a^) . -/Z& 8 . ( 7 i/Z + 3 Vfl s ) = (z*+3a ; ) 3 -i V5* 8 
 
 It may be asked here whether the rules for the calculus of exponents apply to incom- 
 mensurable and imaginary exponents. 
 
 With regard to incommensurable exponents, it may be said that they have not absolutely 
 of themselves any signification, and that, in order to give them one, it is necessary to con 
 ceive them in imagination, replaced by their approximate commensurable values. A forrnu 
 la, therefore, into which incommensurable exponents enter, should be considered as repre- 
 senting the limit toward which the values deduced from it tend by the substitution of 
 commensurable numbers for the exponents, differing from them by as small a quantity as 
 we choose to assign ; in this way we perceive that the proposed expression will represent 
 exactly this same limit, when the same operations shall have been executed upon the in- 
 commensurable exponents which it contains, as would be if they were commensurable. 
 
 Thus, for example, m and n being incommensurable quantities, we shall always have 
 
 a m X« n =a tn + n - 
 For, if ir.' and nf represent their approximate commensurable values, we have 
 
 a m ' X a n '=a m ' + "'. 
 
 * For a variety of examples in transformations, see Appendix. 
 
74 ALGEBRA. 
 
 The first members of this equality tend toward the same limit as the second. But 
 a m Xa n represents the limit of the one, and a m +" that of the other; hence, a m Xa n =a m + n 
 
 With regard to imaginary exponents, there is necessary here, as every where, a tacit 
 admission that the general relations of real quantities, represented by letters, hold good when 
 these letters are replaced by symbol titiea which are imaginary. 
 
 This subject will be better understood after the student has been over that of extrac- 
 tion of roots by approximation. 
 
 78. Having thus discussed the formation of powers, and the extraction of 
 roots in monomial quantities, wo shall now direct our attention to polynomials ; 
 and, in the first place, let it be required to determine the square of x-\-a , 
 then, 
 
 (x+a)*=(x+a)x(x+a) 
 
 =x°-\-2xa-\-a- by rules of multiplication. 
 
 By inspection of this result, it is perceived that the square of a binomial con 
 tains the square of each term together with twice the product of the two. 
 
 Next, let it be required to form the square of a trinomial (x-\-a-\-b). Let 
 us represent, for a moment, the two terms, x-\-a, by the single letter z 
 Then, 
 
 (x+a+by=(z-\-by 
 
 =z*+2zb + b' i (1). 
 
 But, 
 
 And, 
 
 z"={x-\-ay 
 =x*+2xa + a": 
 
 2zb=2b{x+a) 
 =2xb-\-2ab. 
 Therefore, substituting for z" and 2zb their values iu (1), we find 
 (x+a + b)-=x--\-a' i +b~+2xa+2xb + 2ab. 
 
 Hence it appears that the square of a trinomial is composed of the sum of die 
 squares of all the terms, together with the sum of twice the products of all Oie 
 terms multiplied together two and two. 
 
 We shall now prove that this law of formation extends to all polynomials, 
 whatever may be the number of terms. In order to demonstrate this, let us 
 suppose that it is true for a polynomial consisting ^>f n terms, and then en- 
 deavor to ascertain whether it will hold good for a polynomial composed of 
 (n-\-\) terms. 
 
 Let x-\-a^-b-\-c-\ M L "+^ D0 a polynomial consisting of n-f-1 terms, 
 
 and let us represent the sum of the first n terms by the single letter Z ; then 
 
 (*+«-£&+«+.— +*+*) =(*+*), 
 
 and .•■.(i+a+&+c4---- + /j+/) ; = (r + i 
 
 or. putting for z its value, =(x-f-a-f-6+ e H \-k)'* + 2(x+a + 
 
 + c+ --+/)/ + /--. 
 
 But the first part of this expression, being the square of a polynomial con 
 sisting of n terms, is, by hypothesis, composed of the sum (4' the squares of 
 all the terms, together with twice the sum of the products of all the tonus 
 multiplied two and iw<>; the Becond pari of the above expression is equal to 
 twice thi fthe products of all the first n terms of the proposed poly- 
 
 nomial, multiplied by the (/t-f-l)''' term /; and tin' third part is the square of 
 tho (n+1 ) term I 
 
SaUARE OF A POLYNOMIAL. 75 
 
 Hence, if the law of formation already enounced holds gcod foi a poly- 
 nomial composed of n terms, it will hold good for a polynomial composed of 
 {n-\-\) terms. 
 
 But we have seen above that it does hold good for a polynomial composed 
 of three terms ; therefore it must hold for a polynomial composed of four terms, 
 and therefore for a polynomial of five terms, and so on in succession. There- 
 fore the law is general, and we have the following 
 
 RULE FOR THE FORMATION OF THE SQUARE OF A POLYNOMIAL. 
 
 The square of any 'polynomial is composed of the sum of the squares of all 
 the terms, together loith twice the sum of the products of all the terms multiplied 
 together two and two. According to this rule, we shall have, 
 
 (1) (a + 6 + c +rf+e) s =a s +6 8 +c s +d 3 +c 9 + 2ab + 2ac+2ad+2ae+2bc 
 +2bd+2be-\-2cd-\-2ce+2de. 
 
 (2) (a — b— c+dy=a?+b*-\-c' 2 -\-d~— 2ab— 2ac-\- 2ad-\-2bc— 2bd— 2cd. 
 If any of the terms of the proposed polynomial be affected with exponents 
 
 ~>v coefficients, we must square these monomials according to the rules already 
 established. 
 
 (3) (2a— 4Z> 2 c 3 ) 2 =4a- s 4-16Z> 4 c6— 16aZ> 2 c 3 . 
 
 (4) (3a*_ 2a6+46 2 ) 2 =9a 4 +4a 2 6 2 -f 166 4 — 12a 3 6 
 
 +24a 2 l 2 — 16ab 3 
 =9a* — 12a 3 Z>+ 28a 2 i 3 — 16aZ> 3 -f- 16b*, arranging ac 
 cording to powers of a, and reducing. 
 
 (5) {5a i b—4abc-\-6bc"—3a"cfz=z2oa t b"-+16a-b 2 c"+36b-c i -{-9a 4 c' 2 
 
 —i0a 3 b"-c-[-60a i b-c i —30a*bc 
 —48ab°-c 3 +24a 3 bc~—36a-bc 3 . 
 =25a 4 Z> 3 — 40a s b-c+76a°-b-c~— 48a6 2 c 3 
 4. 36b 2 c 4 — 30a 4 Z>c+ 24a 3 Z>c 3 
 — 36a-bc 3 +9a*c°-. 
 79. Let us now pass on to the extraction of the squaro root of algebraic 
 quantities. 
 
 Let P be the polynomial whose root is required, and let R represent the 
 root which for the moment we suppose to be determined ; let us also suppose 
 the two polynomials, P and R, to be arranged according to the powers of 
 6ome one of the letters which they contain ; a, for example. 
 
 If we reflect upon the law just given of the formation of the square of a 
 polynomial, it will be seen that the first two terms of the polynomial P, when 
 thus arranged, are formed without reduction, and will enable us at once to de- 
 termine the first two terms of the root sought ; for, 
 
 1°. The square of the first term of R must involve a, affected with an ex- 
 ponent greater than any that is to be found in the other terms which compose 
 the square of R ; because this exponent is double the highest exponent of a in 
 R, and must be greater than the doublo of any lower exponent, or than the re- 
 sult produced by adding it to one of the lower exponents, or by adding any 
 two of them together. 
 
 2°. Twice the product of tho first terrn of R by the second must contain a, 
 affected with an exponent greater than any to be found in the succeeding 
 terms ; for it will be the sum of tho highest, and the next to the highest ex 
 ponent of a in R. 
 
76 ALGEBKA. 
 
 It follows from this, that if P bo a perfect square, 
 
 I. Tho first term roust be a perfect square ; and the square root of this 
 terra, when extracted according to the rule for monomials I Art. 49), is the first 
 term of R. 
 
 II. The second term must be divisible by twice the first term of R thus 
 found, and the quotient will be the second term of R. 
 
 III. In order to obtain the remaining terms of R, square the two terms ofR 
 already determined, and subtract the result from P ; we thus obtain a new 
 polynomial, P', which contains twice the product of the first terra of R by the 
 third term, together with a series of other terras. But twice the product of 
 the first term of R by the third must contain a, affected with an exponent 
 greater than any that is to be found in the succeeding terms, and hence this 
 double product must form the first term of P'.* 
 
 IV. The first term of P' must be divisible by twice the first term of R, and 
 the quotient will bo the third term of R. 
 
 V. In order to obtain the remaining terms of R, square tho three terms of 
 the root already determined, and subtract the result from the original poly- 
 nomial P;f wo thus obtain a new polynomial, P", concerning which we may 
 reason precisely in tho same manner as for P', and continuing to repeat the 
 operation until we find no remainder, we shall arrive at the root required. 
 
 Tho above observations may be collected and imbodied in the following 
 
 ROLE FOR THE EXTRACTION OF THE SQUARE ROOT OF ALGEBRAIC POLT 
 
 NOMIALS. 
 
 1°. Arrange the polynomial according to tiie powers of some one letter. 
 
 2°. Extract the square root of the first term according to the rule for monomi- 
 al, and the result will he the first term of the root required. 
 
 3°. Square the first term of the root thus determined, and subtract it from the 
 orig inal polynomial. 
 
 4°. Double the first term of the root, and divide by it the first term of the re- 
 mainder, and annex the result (which will be tiie second term of the root), with 
 its proper sign, to the divisor. 
 
 5°. Multiply the whole of this divisor by tJie second term of the root, and sub- 
 tract the product from the first remainder. 
 
 G°. Divide this second remainder by twice the ram of the first two terms of 
 the root already found, and annex the result (which will be Oie third term of 
 tfie root), with its proper sign, to the divisor. 
 
 7°. Multiply the whole of this divisor by the Oiird term of the root, and sub- 
 tract the product from the second remainder ; continue the operation in this 
 ■manner until the whole root is ascertained. 
 
 Tho above process will be readily understood by attending to the following 
 examples: 
 
 r\ MirLE 1. 
 Extract tho squaro root of 10r»— 10* 3 — l?r'+ ">r : 4-P. ---.V-f-1. 
 Or, arranging according to tho powers of r, 
 
 * Ti of the second term of 11 usually contains tin- tame exponent of khe Ictlur 
 
 of an at, but this is already subtracted from V, and not left Id 1' . 
 
 t Ii. . this operation is dispensed with by following the p In the fol- 
 lowing rulo, which evidently come to the same thing. 
 
SQUARE ROOT OF POLYNOMIALS. 77 
 
 Ox 6 — lSx 6 -^ 10x* — 10x 3 -f 5x 2 — 2x+l 3X 3 — 2x 2 +x— 1 
 
 9X 6 
 
 6X 3 — 2x ; 
 
 — 12x 5 + 10x< —lO-^+Sx 2 — 2x+l 
 —12.^+ 4x* 
 
 6r» — 4x 2 + x 
 
 6x* —10x i -\-5x 2 —'2x-{-l 
 Qx* — 4r*+ a* 
 
 63? __ 42; 2 +2z— 1 
 
 — Gx 3 -^ 2 — 2x+l 
 
 — 6x 3 +43: 2 — 2x+l 
 
 0. 
 Having arranged the polynomial according to powers of x, we first extract 
 the square root of Ox 6 , the first term ; this gives 3x? for the first term of the 
 root required ; this we place on the right hand of the polynomial, as in division ; 
 squaring this quantity, and subtracting it from the whole polynomial, we ob- 
 tain for a first remainder, — 12x B +10x 4 — 10x 3 -f-5x 2 — 2x+l ; we now double 
 3a?, and place it as a divisor on the left of this remainder, and dividing by it 
 — 12.1- 5 , the first term of the remainder, we obtain the quotient — 2x 2 (the 
 second term of the root sought), which we annex, with its proper sign, to the 
 double root 6X 3 ; multiplying the whole of this quantity, 6X 3 — 2X 2 , by — 2x s 
 (which produces twice the product of the first term of the root by the second, 
 together with the square of the second), and subtracting the product from the 
 first remainder, we obtain for a second remainder, Gx 4 — lOx'+Sx 2 — 2x+l. 
 Next, doubling 3x? — 2x 2 , the two terms of the root thus found, and dividing 
 6x*, the first term of the new remainder, by Gx 3 , the first term of the double 
 root, wo obtain x for a quotient (which is the third term of the root sought), 
 and annex it to the double root 6X 3 — 4x 2 , multiplying the whole of this quan- 
 tity Gx 3 — 4x 2 +x by x (which produces twice the first by the third, twice the 
 second by the third, and the square of the third), and subtracting the product 
 from the second remainder, we obtain a third remainder, — 6x 3 - r -4x 2 — 2x+l ; 
 we now double 3x 3 — 2x 2 -f-:r> the three terms of the root already found, and 
 dividing — Gx 3 , the first term of the new remainder, by 6X 3 , the first term of 
 the double root, we obtain — 1 for the quotient (which is the fourth term of 
 the root sought), and annex it to the double root 6X 3 — 4x 2 -|-2x; multiplying 
 the whole of this quantity 6X 3 — 4x 2 -f-2x — 1 by — 1, and subtracting it from 
 the third remainder, we find for a new remainder, which shows that the 
 root required is 
 
 3X 3 — 2x 2 +x— 1. 
 
78 
 
 ALGEBRA. 
 
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SQUARE ROOT OF POLYNOMIALS. 
 
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80 ALGEBEA. 
 
 80. If the proposed polynomial contain several terms affected with the same 
 power of tho principal letter, we must arrange the polynomial in the manner 
 explained in division (Art. 20) ; and in applying the above process we shall be 
 obliged to perform several partial extractions of the square roots of tiie coeffi- 
 cients of the different powers of the principal letter, before we cau arrive at the 
 root required. 
 
 Extract the square root of 
 (a-—2ab + b i )x*+2{a—b)(c—d)r i +\2{a — b){f-\-g)-\-(c—dy\3?+2{c—d) 
 
 Ans. (a—b)x-+(c—d)x+f+g. 
 
 Such examples, however, veiy rarely occur. 
 
 Before quitting this subject, we may make the following remarks : 
 
 I. No binomial can be a perfect square ; for the square of a monomial is a 
 monomial, and the square of the most simple polynomial, that is, a binomial, 
 consists of three distinct terms, which do not admit of being reduced with 
 each other. Thus, such an expression as a--\-b 2 is not a square ; it wants the 
 term ±2a6 to render it the square of (a± b). 
 
 II. In order that a trinomial, when arranged according to the poire s of 
 some one letter, may be a perfect square, the two extreme terms must he perfect 
 squares,* and the middle term must be equal to twice the product of the square 
 roots of the extreme terms. When these conditions are fulfilled, we may obtain 
 the square root of a trinomial immediately, by the following 
 
 RULE. 
 
 Extract the square roots of the extreme terms, and connect the tico terms thus 
 found by the sign -\-, when the second term of the trinomial is positive, and by 
 the sign — , when the second term of the trinomial is negative. Thus, the ex- 
 pression 
 
 9a 6 — 48a*b 2 +64a?b* 
 is a perfect square ; for the two extreme terms are perfect squares, and the 
 middle term is twice the product of the square roots of the extreme terms; 
 hence the square root of the trinomial is 
 
 i/thF— y /~64a 1 b*. 
 Or, 
 
 3a?—8ab-. 
 
 An expression such as 4a 2 -\-12ab — db 2 can not be a perfect square, although 
 4a 2 and Ob", considered independently of their signs, are perfect squares, and 
 I2ab =2(2a . 5b) ; for — Ob- is not a square, sinco no quantity, when multi- 
 plied by itself, can have the sign — 
 
 III. In performing the operations required by the general rule, if we find 
 that tho first term of one of tho remainders is not exactly divisible by twice 
 the first term of the root, wo may immediately conclude that the polynomial 
 is not a perfect square ; and when we arrive nt a term in the root having a 
 power of the letter of arrangement of a degree less than half that of this letter 
 in (he last term of tho giren polynomial, wo ma\ be sure that the operation 
 will not terminate. This is on tho supposition that the given polynomial is ar- 
 
 * In order that any polynomial maybe n perfect square, tin' two extremo terms must be 
 perfect squares, If it be nrr:> oordlng to the powi ra of some letter. 
 
CUBE HOOT OF POLYNOMIALS. 81 
 
 ranged according to the decreasing powers of the letter. If it be according to 
 the increasing powers, substitute the word greater for " less" in the above 
 precept. 
 
 IV. We may apply to the square roots of polynomials which are not per- 
 fect squares the simplifications already employed in the case of monomials 
 (Art. 51). Thus, in the expression 
 
 VV& + 4a 2 6 3 - r .4a& 3 . 
 
 The quantity under the radical sign is not a perfect square, but it may be 
 put under the form 
 
 ^/ab{a--{-4ab- r -Ab'~). 
 
 The factor within brackets is manifestly the square of a -\-2b; hence 
 
 V r a 3 &-f4a 2 6 3 +4a6 3 = Vab{a--i r Aa b-i r 4b-) 
 
 = Vabla+^f 
 = (a+2b) j~ab. 
 
 81. Let us next proceed to form the cube of x-\-a. 
 
 (x-f a) 3 =(.r+a) X (x+a) X (x+a) 
 
 =x 3 -{-3x' 2 a-{-3xa 2 -{-a 3 by rules of multiplication. 
 
 Let it be required to form the cube of a trinomial (x-\-a-\-b)', represent 
 the last two terms a-\-b by the single letter s ; then 
 
 {x+a+b)*=:(x 4-s) 3 
 
 =x*+3x' 2 s-f-3xs 2 4-s 3 
 =x s +3x°{a+ Z,)-f-3.r(a+Z>) 2 +(a + 6) 3 
 =.r 3 +3.r 2 a-f3x 2 6-f 3.ra 2 +6.ra& + 3:r& 2 -fa 3 
 -j-3^ 2 b + 3ab*+ V. 
 This expression is composed of the sum of the cubes of all Hie terms, together 
 with three times the sum of the squares of each term, multiplied by the simple 
 power of each of the others in succession, together with six times the product of 
 the simple power of all Hie terms- 
 
 By following a process of reasoning analogous to that employed in (Art. 78), 
 we can prove that the abovo law of formation will hold good for any polynomial 
 of whatever number of terms. We shall thus find 
 (a+6+c+d) 3 =a 3 + i 3 4-c 3 +rf 3 +3a 2 6 + 3a 2 c+3a 2 J+36 2 a+3t-c+36 2 rf 
 
 + 3c*a+3c-b + 3cH+3d i a-\-3d%+3d~c-\-Gabc+Gabd-\-6acd+Gbcd 
 (2a'—4ab-\-3b''Y= 8a 6 — 64a 3 5 3 + 27b 6 — 48a 5 i + 36a 4 b- -4- 96a 4 /r + U-ia-b* 
 
 _|_54„:/,i_108a/; 5 — 144a 3 6 3 
 —8a 6 —48a 5 b-l r 132a*b i —208a s b 3 -\-198a 2 b- i — 108ab 5 -\-27b\ 
 
 In a similar manner, we can obtain the 4th, 5th, &c, powers of any poly- 
 nomial. 
 
 For more upon this subject, see a subsequent article (105). 
 
 82. We shall now explain the process by which we can extract the cube 
 root of any polynomial, a method analogous to that employed for the square 
 root, and which may easily be generalized, so as to be applicable to the ex- 
 traction of roots of any degree. 
 
 Let P be the given polynomial, R its cube root. Let these two poly- 
 nomials be arranged according to the powers of some one letter, a, for example. 
 It follows, from the law of formation of the cube of a polynomial, that the cubo 
 of R contains two terms, which are not susceptible of reduction with any 
 others ; these are, the cube of the first term, and three times the square of 
 
 F 
 
12 ALGEBRA 
 
 tho first term multiplied by the second term; for il is manifest that these fw D 
 terms will involve a affected with an exponent higher than any that is to oe 
 found in the succeeding terms. Consequently, these two terms must form 
 the first two terms of P. Hence, if we extract the cube root of the first term 
 of P, we shall obtain the first term of R, and then, dividing the second term 
 of P by three times tho square of the first term of R thus found, the quotient 
 will be the second term of R. Having thus determined the first two terms of 
 R, cube this binomial, and subtract it from P. The remainder, P', being ar- 
 ranged, its first term will be three times the product of the square of the first 
 term of R by tho third, together with a series of terms involving a, affected 
 with a less exponent than that with which it is affected in this product. 
 Dividing tho first term of P' by three times the square of the first term of R, 
 the quotient will be the third term of R. Forming the cube of the trinomial 
 root thus determined, and subtracting this cube from the original polynomial 
 P, we obtain a new polynomial, P", which we may treat in the same manner 
 as P', and continue the^operation till the whole root is determined.* 
 
 EXAMPLES. 
 
 (1) Extract the cube root of 27Z 3 — 135a: 2 +225x— 125. 
 
 (2) V(8^+48zx 5 +60; 2 r 4 — mz*J?— 90r 4 .c-+108c 5 x— 27z 6 ). 
 
 ANSWERS. 
 
 (1) 3r— 5. | (2) 2x 2 +4;.r— 3z 3 . 
 
 EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 
 83. Rules are given in Arithmetic for extracting Ac square and cube roots ol 
 any proposed number; we shall now proceed to explain the principles upon 
 which these rules are founded. 
 The numbers 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000, 
 
 when squared, become 
 
 1, 4, 9, 1G, 25, 3G, 49, 64, 81, 100, 10000, 1000000, 
 
 and reciprocally, the numbers in the first line are the square roots of the nura 
 bers in the second. 
 
 Upon inspecting these two lines we perceive that, among numbers expressed 
 by one or two figures, there are only nine which arc the squares of other 
 whole numbers; consequently, the square root of all otht>r numbers consisting 
 of one or two figures must be a whole Dumber pins a fraction. 
 
 Thus, the square root of 53, which lies between 19 and 64, is 7 plus a frac- 
 tion. So, also, tho square root of 91 is 9 plus a fraction. 
 
 84. It is, however, very remarkable Omt the square root of a whole number, 
 which is not a perfect square, can not be , r t >r, m d art fraction, and is, 
 
 therefore, incommensurable with unity. 
 
 To prove this, lot -r, a fraction in its lowest terms, 1"-. if possible, the square 
 
 a ./-' 
 
 root of some whole Dumber; thou the square of j, or . must b.« equal to tins 
 
 whole iiunih.-r. But since a and l> are, by supposition, prime to each other 
 
 • Tliis Bttbject wi'l be resumed a f.'u pa ei farther an. 
 
SQ.UARE ROOT OF NUMBERS. 83 
 
 {i. e., have no common divisor), a 2 and b 2 are also prime to each other;* there- 
 at 
 
 fore 7- is an irreducible fraction, and can not be equal to a whole number. 
 b* 
 
 85. The difference between the squares of two consecutive whole numbers 
 is greater in proportion as the numbers themselves are greater ; the expres 
 sion for this difference can easily be found. 
 
 Let a and a-|-l be two consecutive whole numbers ; 
 Then, 
 
 (a+1) 2 =a 3 +2a+l. 
 Hence, 
 
 (a+1) 2 — a 3 =2a+l; 
 
 that is to say, the difference of the squares of two consecutive whole numbers u 
 equal to twice the less of the two numbers plus unity. 
 
 Thus, the difference between the squares of 348 and 347 is equal to 
 
 2x347 + 1, or 695. 
 
 • 
 
 * This depends upon the principle that, if any prime number, P, will divide the product 
 
 of two numbers, it must divide one of them, which may be demonstrated as follows : 
 Let A and B be the two numbers, and let it be sujTposed that P will not divide A, we 
 
 are to prove that it must divide B. 
 
 Dividing A by P, and denoting the quotient by Q, and the remainder by P', we have 
 
 AB P'B 
 
 A=PQ+P' .-.multiplying by B, AB=PQ.B+P'B ,\ dividing by P, — = Q.B+— 
 
 Since by hypothesis AB is divisible by P, P'B must be, else we should have a whole 
 number, equal to a whole number plus a fraction, which is impossible. Proceed now with 
 P and P' after the method for finding a common divisor, and let P", P'", &c, be the suc- 
 cessive remainders, which can none of them be zero, because P is by hypothesis a prime 
 number (i. e., a number divisible only by itself and unity) : these remainders must go on di 
 minishing till the last becomes ujiity, and we shall have the series of equalities, 
 
 P=P'a'+P". P'=P" Q/'+P'", &C. ; 
 or, multiplying by B and dividing by P, 
 
 _ P'Q'B , P"B P'B P"Q/'B , P'"B 
 
 B =-p-+-p-- -r=~p — i"p-« &c - 
 
 The first of these equalities shows that if P'B is divisible by P, P"B must also be divisi- 
 ble ; and if both these are divisible, the second equality shows that P'"B is divisible by 
 P, and so on. But the remainders, P", P"', &c, diminish till the last becomes unity, and 
 we shall thus have, finally, 1XB, or B divisible by P. Q_. E. D. 
 
 Now, since a 2 is the product of a and a, any prime number which divides a 2 must divide 
 a, or which divides b 2 must divide b, so that any prime number which divides both a- and 
 b 2 must divide a and b. 
 
 Every number is either prime or composed of prime numbers as factors, and if this nura 
 ber will divide the two terms of a fraction, its prime factors will successively divide them 
 This follows from (10, I., 2). 
 
 As an addition to this note may be demonstrated the following theorem : A literal quart 
 lily can not be decomposed into prime factors in different ways. 
 
 Let ABCD... be a product of prime factors, and suppose that it could be equal to an 
 other product, abed . . ., the factors a, b, c, d. . . being also prime. The factor a, dividing 
 abed, must divide the equal ABCD . . . ; but if the prime quantity a is different from each 
 of the quantities A, B, C, D, &c, it can not divide any of them. Not dividing either A or 
 B according to the above theorem, it can not divide the product AB. Not dividing either 
 AB or C, it will not divide the product ABC. and so on. The factor a must, therefore, 
 necessarily be equal to one of the factors A, B, C, <Scc. Suppose «=A. Dividing the two 
 products by A, the remaining products, BCD . . . and bed . ■ ., are still equal, and applying to 
 them the preceding reasoning, we conclude that b ought to be equal to one of the factors of 
 the product, BCD..., and so on. The two products, ABCD... and abed..., must, there- 
 fore, be composed of the same prime factors. Q_ E. D 
 
7fc 
 
 34 ALGEBRA- 
 
 Tlio square of a number will always consist of twice as many digits, or om 
 iess than' twice as many, as the number itself. Thus, the square of 10 is 100, 
 and the square of any number less than 10 must be less than 100, or contain 
 not more than two 6gures. The square of 100 is 10000, and the square of all 
 numbers between 10 and 100 must be between 100 and 10000; i. c, consist 
 of 3 or 4 figures. In the same way it may be shown that the square of a 
 number containing three figures must be one containing five or six figures, and 
 so on; i. < .. the square of a number consists of twice as many digits as the 
 number itself, or one less than twice as many. 
 
 Let us now proceed to investigate a process for the extraction of the square 
 root of any Dumber, beginning with whole numbers. 
 
 EXTRACTION OF THE SdUARE ROOT OF WHOLE NUMBERS 
 
 86. If the number proposed consist of one or two figures only, its root ma> 
 oe found immediately by inspecting the squares of the nine first numbers in 
 Art. 83). Thus, the square root of 25 is 5, the square root of 42 is 6 plus a 
 fraction, or G is the approximate square root of 42, and is within one unit of 
 the true value ; for 42 lies between 36, which is the square of G, and 49, which 
 is the square of 7. 
 
 Let us consider, then, a number composed of more than two figures, 6084 
 for example. 
 
 Since this number consist of four figures, its root must 60'84 
 
 necessarily consist of two figures, that is to say, of tens 49 
 
 and units. Designating the tens in the root sought by a, 148 
 and the units by b, we have 
 
 6084 = (a+t)-=a 2 - r -2a&4-& 2 , 
 which shows that the square of a number consisting of tens and units is com- 
 posed of the square of the tens, plus twice the product of the tens by the ui 
 phis the square of the units. 
 
 This being premised-, since the square of a certain number of teus must be 
 a certain number of hundreds, or have two ciphers on the right, it follows that 
 the squares of tho tens contained in the root must be found in the part 60 (or 
 60 hundreds), to the left of the last two figures of 6081 (which written at full 
 length is 6000 -(-804-4), tho 81 forming do part of the square of the tens; we, 
 therefore, separate the last two figures from the others by a point. The part 
 60 is comprised between the two perfect squares 19, and 64 f the roots of which 
 are 7 and 8; hence 7 is tho figure which expresses the number of tens in the 
 root soufiht : for 6000 is evidently comprised between 1900 and 6400, which 
 are the squares of 70 and 80, and tho root of 6084 must, therefore, be com- 
 prised between 7(1 and 80; hence, the rOOl BOUghl 18 Composed of 7 tens and 
 u certain number of units less than ten. 
 
 The figure 7 being 'bus found, we place it on the righl of the given Dumber, 
 in the place often*, separated by e vertical line as in division; we then sub- 
 tract in, whieb is the square of 7. from 60, which leaves as remainder 11 
 
 (which is 11 hundreds), after which we write the remaining figures, B4« 
 Having taken away the Bquare of the tens, the remainder, 1184, contains, as 
 w»« have Been above, twice the product of the tens multiplied by the units 
 plus the square of the units. But the product of the tens multiplied by the 
 
 units must be tens, CT have one cipher on the right, iind, therefore, the last 
 
 L18'4 
 
 118'4 
 
 0. 
 
SQUARE ROOT OP NUMBERS. 85 
 
 figure 4 can not form any part of the product of the tens by the units; we, 
 therefore, separate it from the others by a point. 
 
 1 1' we double the tens, which gives 14, and divide the 118 tens by 14, the 
 quotient 8 is the figure of units in the root sought, or a figure greater than the 
 one required. It may manifestly be greater than the figure sought, for 118 
 may contain, in addition to twice the product of the tens by the units, other 
 tens arising from the square of the units, which may exceed the denomination 
 units. In order to determine Whether 8 expresses the real number of units 
 in the root, it is sufficient to place it on the right of 14, and then multiply the 
 number 148, thus obtained, by 8. In this manner we form, 1", the square of 
 the units ; 2°, twice the product of the units by the tens. This operation 
 being effected, the product is 1184; subtracting this product, the remainder is 
 0, which shows that 6084 is a perfect square, and 78 the root sought. 
 
 It will be seen, in reviewing the above process, that we have successively 
 subtracted from G084, the square of 7 tens or 70, plus twice the product of 70 
 by 8, plus the square of 8, that is, the three parts which enter into the com- 
 position of the square of 70-f-8, or 78 ; and since the result of this subtraction 
 is 0, it follows that G084 is the square of 78. 
 
 The quotient obtained from dividing by double the tens is a trial figure ; it 
 will never bo too small, but maybe too great, and on trial may require to be di- 
 minished by one or two units. 
 
 Take as a second example the number 841. 8'41 29 
 
 This number being comprised between 100 and 10000, its 
 
 44'1 
 441 
 
 0. 
 
 root must consist of two figures, that is to say, of tens and 49 
 units. We can prove, as in the last example, that the root 
 af the greatest square contained in 8, or in that portion of the 
 number to the left of the last two figures, expresses the number of tens in the 
 root required. But the greatest square contained in 8 is 4, whose root is 2, 
 which is, therefore, the figure of the tens. Squaring 2, and subtracting the 
 result from 8, the remainder is 4 ; bringing down the figures of the second 
 period 41, and annexing them on the right of 4, the result is 441, a number 
 which contains twice the product of the tens by the units, plus the square of 
 the units. 
 
 We may farther prove, as in the last case, that if we point off the last figure 
 I, and divide the preceding figures 44 by twice'the tens, or 4, the quotient 
 will be either the figure which expresses the number of units in the root, or a 
 figure greater than the one sought. In this case the quotient is 11, but it is 
 manifest that we can not have a number greater than 9 for the units, for other- 
 wise we must suppose that the figure already found for the tens is incorrect. 
 Let us tiy 9 ; place 9 to the right of 4, and then multiply this number 49 by 
 9 ; the product is 441, which, when subtracted from the result of the first 
 operation, leaves a remainder 0, proving that 29 is the root required. 
 
 Let us take, as a third example, a number which is not a perfect square, 
 such as 1287. 
 
 Applying to this number the process described in the pre- 12'87 35 
 
 ceding example, we find that the root is 35, with a remainder 9 
 
 62. This shows that 1287 is not a perfect square, but that 65 
 it is comprised between the square of 35 and that of 36. 
 rims, when the number is not a perfect squai e, the above 
 
 38'7 
 325 
 
 62 
 
56'£2'14'44 
 49 
 
 145 
 
 78'2 
 725 
 
 1503 
 
 571 '4 
 4509 
 
 15068 12054'4 
 120544 
 
 
 0. 
 
 86 ALGEBRA. 
 
 process enables us at least to determine the root of llie greatest square job 
 tained in the number, or the integral part of the root of the number. 
 
 87. Let us pass on to consider the extraction of the square root of a num 
 ber composed of more than four figures. 
 
 Let 56821444 be the number. 56'ir2'14'44 7538 
 
 Since the number is greater than 10000, its root 
 must be greater than 100 ; that is to say, it must 
 consist of more than two figures.* But, whatever 
 the number may be, we may always consider it as 
 composed of units and of tens, the tens being ex- 
 pressed by one or more figures. (Thus, any num- 
 ber such as 37142 may be resolved into 37140 + 2, 
 or 3714 tens, plus two units.) 
 
 Now the square of the root sought, that is, the proposed number, contains 
 the square of the tens, plus twice the product of the tens by the units, plus 
 the square of the units. But the square of the tens must give at least hun- 
 dreds ; hence the last two figures, 44, can form no part of it, and it is in the 
 portion of the number to the left hand that we must look for that square. 
 But this portion containing more than two figures, its root will consist of units 
 and tens ; it will, therefore, be necessary to commence the process for finding 
 the root of this portion by cutting off its two right-hand figures, 14, and the 
 square of the tens of the tens is to be sought in tho figures now remaining at 
 the left, 5682. This number being the square of two figures, we again separate 
 82, and seek for the square of the tens of the tens of the tens in the two re- 
 maining figures, 56. The given number is thus separated into periods of two 
 figures each, beginning on the right. We then go on to extract the root of 
 the number 5682, as in the previous examples; this will give the tens of the root 
 of the number 568214. We then double these tens for a divisor, and take the 
 remainder after the last operation, with 14 annexed for a dividend ; we divide 
 this dividend, after cutting off the right-hand figure, and the quotient will be 
 the units of the root of 568214. All the figures now found of the root will 
 constitute the tens of the root of the given number, and we find the units by 
 the rule previously given. The detail of the whole operation is as follows : 
 
 Extracting tho root of 56, we find 7 for the root of 49, the greatest square 
 contained in 56; we place 7 on the righl of the proposed Dumber, and squaring 
 it, subtract 49 from 56, which gives a remainder 7, to which we annex the fol- 
 lowing period, 82. Separating the last figure to tin- right of 782, and then 
 dividing 78 by 14, which is twice the root already found, we have 5 for a quotient, 
 which we annex to 14; we then multiply the whole Dumber 115 by 5, and 
 subtract tho product 725 from 782. We next bring down the period 11, an- 
 nex it to the second remainder 57, ami point off the last BgUTO of this number 
 5714. Dividing 571 by 150, which is twice tho root already found, the quotient 
 is 3, which we place to the right of 150, and multiplying the whole nuiiher 
 1503 by 3, we subtract the product 4509 from . r >? 1 I. 
 
 Finally, we bring down the last period 1 1, annex it to the third remaindei 
 1205, and point elf the last figure of this Dumber 12054 I. Dividing L2054 Uy 
 
 * Wo liPfvo Been in tin: lust article that it will lialf as many as tl)« 
 
 given comber. Bad die given number contained bat i an i, the root would still he 
 
 composed of four 
 
SQUARE ROOT BY APPROXIMATION. 87 
 
 150G, which is twice the root already found, the quotient is 8, which we place 
 yn the right of 1506, and multiplying tho whole number 15068 by 8, we sub- 
 tract the product 120544 from the last result 120544. The remainder is ; 
 hence 7538 is the root sought. 
 
 From what has been said above, it is easy to deduce the rule, ordinarily 
 given in Arithmetic, for the extraction of the square root of a number consist- 
 ing of any number of figures, and which it is unnecessary here to repeat. 
 
 EXTRACTION OF THE SQUARE ROOT BY APPROXIMATION. 
 
 88. When a whole number is not tho square of another whole number, we 
 have seen (Art. 84) that its root can not be expressed by a whole number and 
 an exact fraction ; but although it is impossible to determine the precise value 
 of the fraction which completes the root sought, we can approximate it as 
 nearly as we please. 
 
 Suppose that a is a whole number which is not a perfect square, and that 
 
 we are required to extract the root to within — , that is, to determine a number 
 
 which shall differ from the true root of a, by a quantity less than the fraction — . 
 
 To effect this, let us observe that the quantity a may be put under the form 
 
 an* 
 
 —5- ; if we designate the integral, or whole number, portion of the root of an 1 
 
 an 1 
 by r, this number an? will be comprised between r 2 and (r+1)- ; hence, — - 
 
 r"- (r+1) 2 
 is comprised between — and — , and consequently, the root of a is com- 
 
 r 2 (r+1) 2 r r+1 
 
 prised between the roots of — and : — , that is, between - and . Thus, 
 
 r n 2 n 2 n n 
 
 r 1 
 
 it appears that — represents the square root of a within - of the true value 
 
 From this we derive the following 
 
 RULE. 
 
 To extract the square root of a whole number to within a given fraction, mul- 
 tiply the given number by the square of the denominator of the given fraction ; 
 extract the integral part of the square root of the product, and divide this in- 
 tegral part by the given denominator. 
 
 Let it be required, for example, to find the square root of 59 within T V of 
 the true value. 
 
 Multiply 59 by the square of 12, that is, 144, the product is 8496 ; the in- 
 tegral part of the root of 8496 is 92. Hence f § or 7^ is the approximate root 
 of 59, the result differing from the true value by a quantity less than -i. 
 
 So, also, 
 
 Vll = 3 T 4 j true to-Jj, 
 -/223=142J true to ^. 
 
 89. The method of approximation in decimals, which is the process most 
 frequently employed, is an immediate consequence of the preceding rule. 
 
 In order to obtain the square root of a whole number within J^, T L, jJ^ . . . 
 of the true value, we must, according to' the above rule, multiply the proposed 
 number by (10) 2 , (100) 2 , (1000) 2 , or, which comes to the same thing, 
 
88 ALGEBRA. 
 
 place to the right of the number, two, four, six, ciphers, then extract 
 
 the integral part of the root of the product, nod divide the result by 10, 100, 
 1000 
 
 Hence, in order to obtain any required number of decimals in the root, we 
 must 
 
 Place on the right hand of Oie proposed number twice as many zeros as we 
 loish to have decimal figures ; extract the integral part of the root of this new 
 number, and then marlc off in the result the required number of decimal places 
 
 EXAMPLES. 
 
 (1) Extract the square root of 3 to six places of decimals. 
 
 Ans. 1.732050. 
 
 (2) Extract the square root of 5 to six places of decimals. 
 
 Ans. 2.236068. 
 
 (3) Extract the square root of 12 to six places of decimals. 
 
 Ans. 3.464101. 
 When half, or one more than half, the figures are found, the rest may be 
 found by division. 
 
 (4) Extract the square root of 2 to nine places of decimals. 
 
 The first five figures of the root found by the ordinary method are 1.41 1J . 
 with the remainder, 3836. The next divisor is 28284. Dividing 3836 by 
 28284, according to the ordinary method of division, produces 1356 for a quo- 
 tient, which, annexed to 1.4142, befoi-e found, gives for the root required 
 1.41421356.* 
 
 Extract the square root of 11 to six places of decimals. 
 
 Ans. 3.316624. 
 
 EXTRACTION OF THE SQ.UARE ROOT OF FRACTIONS. 
 
 la V« 
 
 We have seen (Art. 62) that - T =~ 7r» hence, in order to extract tne 
 
 V o y 
 
 square root of a fraction, it is sufficient to extract the square roots of the numer- 
 ator and denominator, and then divide the former result by the latter. This 
 method may be employed with advantage when either one or both of the terms 
 of the proposed fraction aro. perfect squares; but when this ia not the case, it 
 will bo found inconvenient in practice. If, for example, we take the fraction 
 
 /3 V~3 
 |, although .. /-=~= (since each ot these expressions, when multiplied by it- 
 self, produces the same quantity. '•;), we must find an approximate value both 
 
 for <J?> anu> a ' s0 f ur V&i il,,, l' il ' u ' 1 ' il "' Av< ' shall not be able to determine at 
 
 once tho degree of approximation in the result. Under such circumstances 
 
 the following process may bo emplo] ed : 
 
 a al> 
 
 Let tho proposed fraction bo j, this may be put under the form y-; this 
 
 feeing premised, let r represent the integral part of the root of the numeratos 
 
 * The reason tor this rulo may be •■\\>-u thus : Let k be the part <>f the root already 
 found, and z the remaining part Then /.--(-- will be the whole root, and (A--f-r)-=A-'-f-2£; 
 -j-:- the given number; aa i if hut a small fraction of k, :■ w ill be a still smaller fraction, 
 ami may be di looted, so that the given camber nay, without sensible error, he considered 
 eqaal to /.--f--/.r. Bat k* baa bean taken 1 away, and the remainder, ••'A:, divided i>\ 
 i .-. 
 
SQ.UARE HOOT OF FKACTIO: 89 
 
 ab; hence tj, <tr y, is comprised between -j- and — .-, — ; consequently, the 
 
 d y T-l— 1 T 
 
 root of j is comprised between t and — ?— . Thus, it appears that r repre- 
 
 a 1 
 
 Bents the root of -r within 7- of the true value. Hence, in order to obtain the 
 b 
 
 square root of a fraction, 
 
 Make the denominator of the fraction a perfect square, by multiplying botli 
 terms of the fraction by the denominator ; extract the integral part of the root of 
 Uie numerator, and divide the result by the denominator. 
 
 Let it be required to extract the square root of T 7 j. 
 
 7 X 13 91 
 
 This fraction is the same as , or . But the integral part of the 
 
 9 
 square root of 91 is 9 ; hence — is the root so, lit, a result within ^ of the 
 
 true value. 
 
 A greater degree of approximation may, perhaps, bo required. In this case, 
 
 91 
 returning to the number , extract the root of 91 to any required degree 
 
 of approximation. Suppose, for example, we wish to find the root of 91 within 
 
 — of the real value, it will become by (Ait. 88) -\/91=9 . 53 . . . . Hence 
 
 7 91 9.53 1 
 
 the root of — , or , will be — - — , equal -73 within of the true value. 
 
 13 (13)* 13 1300 
 
 Remark. — It frequently happens that the denominator of the fraction, al- 
 though not a perfect square, has a perfect square for one of its factors, in 
 which case the above operation may be simplified. 
 
 23 
 Let the fraction, for example, be — . 48 is equal to 16x3, or (4) 2 x3; 
 
 23 X 3 
 hence, multiplying both terms of the fraction by 3, it becomes 7- - — j—, or 
 
 69 
 tt— — ; and the denominator is thus made a perfect square. Extracting the 
 
 1 R 3 83 
 
 root of 69 to — , which eives 8 . 3, we find — '— , or for the root required, a 
 
 10 s 12 120 H 
 
 result within — ■ of the true value. 
 
 In general, therefore, whenever the denominator of the fraction involves a 
 
 factor which is a perfect square, multiply both terms of the fraction by the factor 
 
 which is not a perfect square. 
 
 _ 5 1 
 
 .hxrract the square root of- to within — . 
 1 6 48 
 
 5 = 5x6x8" =i 920_ — =43 «« 
 
 6 6 s X 8 J 6 Z X 8 2 V 6 48 
 
 EXTRACTION OF THE SOUARE ROOT OF DECIMAL FRACTIONS. 
 
 90. This process is an immediate consequence of the preceding remark. 
 Required, for example, the square root of 2 . 36. 
 
90 ALGEBRA. 
 
 236 
 This fraction is the same as y— ; in this case the denominator is a perfect 
 
 square ; extracting, therefore, the integral part of the root of the numerator, we 
 
 15 1 
 
 have — , a result within — of the time value. 
 10 10 
 
 Again, let it be required to extract the square root of 3.425. 
 
 3425 
 
 This fraction is the same as . But 1000 is not a perfect square ; it is, 
 
 however, equal to 100x10, or (10) : Xl0; thus, in order to render the de- 
 nominator a perfect square, it is sufficient to multiply both terms of the frac- 
 
 34250 34250 _ 
 tion by 10, which gives , or , .. . Extracting the integral part of the 
 
 185 
 root 34250, we find 185; hence the root required is — -, or 1.85, a result 
 
 which is within — - of the true value. 
 
 It appears from the above that the number of decimal places must always 
 be made even before the operation commences. 
 
 If we wish to have a greater number of decimal places in the root, we must 
 add on the right of 34250 twice as many zeros as we wish to have additional 
 decimal figures. 
 
 We thus deduce for the extraction of the square root of a decimal fraction 
 the following 
 
 RULE. 
 
 Annex ciphers till there arc twice as many decimal places as are required in 
 the root, and then*proceed as in whole numbers ; or, beginning at the decimal 
 point, point off both ways the usual periods of tivo figures each. 
 
 By which we obtain 
 
 V 0799. 6516=82. 46, V?3 .5=8.5, .y/Tl)T=2. 81. 
 
 EXTRACTION OF THE CUBE ROOT OF NUMBERS. 
 
 i)i. The numbers ' 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000, 
 
 when cubed, become 
 
 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1000000, 1000000000 
 and, reciprocally, the numbers in the first line are the cube roots of the nuiu 
 bers in the second. 
 
 Upon inspecting the two lines, we perceive that, among tho number- 
 pressed by one, two, or three figures, there are only nine which are perfect 
 cubes ; consequently, tho cubo root of all the rest mast be :i whole number plus 
 u fraction. 
 
 92. But we can prove, in tho same manner as in the case of tho squarn 
 root, that the cube root of a whole number, which is not the /" rfeel ruin- of I 
 other whole number, run not be expressed by an cruet fraction, and, 
 
 ijin nil y, its cube root is incomnu nsurable with unity. 
 
 93. The difference between the cubes of two consecutive whole numbers 
 reater in proportion as the numbers themselves are greater; tho expression 
 
 for this difference can easily be found. 
 
Let 
 
 Then, 
 Hence, 
 
 EXTRACTION OF THE CUBE ROOT. 9\ 
 
 a and a-\-l be two consecutive whole numbers , 
 (a+l) 3 =a 3 +3a 2 4-3a4-l ; 
 
 (a + l)3— a 3 =3a 2 +3a+l ; 
 that is to say, the difference of the cubes of two consecutive whole numbers it 
 equal to three times the square of the less of the two numbers, plus three timet 
 tiie si?nple poiver of the number, plus unity. 
 
 Thus, the difference between the cube of 90 and the cube of 89 is equal to 
 3X(89) 2 +3X 89 + 1=24031. 
 
 Let us now proceed to investigate a process for the extraction of the cube 
 root of any number. 
 
 EXTRACTION OP THE CUBE ROOT. 
 
 94. The cube root of a proposed number, consisting of one, two, or three 
 figures only, will be found immediately by inspecting the cubes of the first 
 nine numbers in (Art. 91). Thus, the cube root of 125 is 5, and the cube root 
 of 54 is 3 plus a fraction, for 3 X 3 X 3=27, and 4x4 X 4=64 ; therefore 3 is 
 the approximate cube root of 54, within one unit of the true value. 
 
 For the purpose, of investigating a new and simple rule for the extraction of 
 the cube root, it will be necessaiy to attend to the composition of a complete 
 power of the third degree. Now, since we have 
 
 (a4-o) 3 =(a + o)(a + o)(a + Z>)=a 3 +3a 2 o+3aZ> 2 +Z< 3 , 
 it is obvious that the cube of a number, consisting of tens and units, will be al- 
 gebraically indicated by the polynomial 
 
 a 3 -f3a 2 6+3aZ> 2 +o 3 , 
 where a designates the number of tens, and o the number of units in the root 
 sought. The number in the tens' place will evidently be found by extracting 
 the cube root of the monomial a 3 , for 3 /a :! =ra, and removing a? from the poly 
 nomial <? 3 +3a 2 6- r -3a6 2 - r -i 3 , we have the remainder, 
 
 3a 2 5 + 3a6 2 +6 3 =(3a 2 +3ct64-6 2 )6 ; 
 and the difficulty that has been hitherto experienced in the extraction of the 
 cube root entirely consists in the composition of the expression Sfi^-j-Sai-f-o'. 
 which is obviously the true divisor by which to divide the remainder, aftei 
 subtracting a 3 , or the cube of the tens, for the determination of o, the figure 
 of the root in the place of units. The part 3a 2 of the expression 3a 2 4- 3<z&+o 2 , 
 being independent of b, the yet unknown part of the root, is employed as a 
 trial divisor for the determination of 6 ; but since the expression 3a 2 -{-3ab-\-b* 
 involves the unknown part of the root in its composition, it is obvious that the 
 trial divisor 3a 2 , which does not contain 6, will, at the first step of the opera- 
 tion, give no certain indication of the next figure of the root, imless the figure 
 denoted by b be very small in comparison with that denoted by a ; for the 
 trial divisor 3a 2 will be considerably augmented by the addend 3ab-{-b" when 
 6 is a large number, while the augmentation, when 6 is a small number, will 
 not so materially affect the trial divisor. 
 
 When the figure in the tens' place is a small number, as 1 or 2, it is hence 
 obvious that little or no dependence can be placed on the trial divisor; but if a 
 
9-2 ALGEBRA. 
 
 reat and b small, the trial divisor, 3a 8 . will generally point out the value 
 of l>. All this will be evident if we consider that the relative values of a and 
 ft materially affect the true divisor, 3a*-\-3ab-\-b*. In the successive steps, 
 li >wever, of the cube rool this uncertainty diminishes; for, conceiving a to 
 designate a number consisting of tens and hundreds, and b the number o 
 units, then the value of b being small in comparison with a, the amount of the 
 effect of b in ilia addend '.jab-\-b- will be very inconsiderable ; hence the trial 
 divisor, 3a 5 , will generally indicate the next figure in the root. 
 
 To remove, iu some measure, the difficulty which has hitherto been ex 
 perienced in the extraction of the cube root, we shall proceed to point out two 
 methods of composing the true divisor, 3a--\-3ab-{-b -'. and leave the student 
 to select that which ho conceives to possess the greater facility of operation.* 
 
 95. First method of composition of3a' 2 -\-3ab-\-b' 1 . 
 axa = a 2 a?+3a-b-{-3ab-+b* (<z + b= root sought. 
 
 a a- X a = « 3 
 
 a a 2 
 
 — 3a 2 6 + 3ai 2 +i 3 
 
 3a 2 
 (3a+h)xb= 3aZ>-f- 6 2 
 
 b 
 
 b (3a 2 +3a5+ b-) X b = . . . . 3a-b+3ab 2 +b* 
 
 b* 
 
 3a+3fc 3a 2 -f Gai-r-C//-'. 
 
 Distinguishing the three columns from left to right by first, second, and 
 third columns, we writo a in the root, and also three times vertically in the 
 first column; then aXa produces a-, which write, also, three times vertically 
 in the second column; multiply the second a? by a, placing the product, « 3 , 
 under a 3 in the third column ; then, subtracting a 3 from the proposed quantity, 
 we have the remainder, ?M-b-\-3oJ>'-\-}>'. The sum of the three quantities in 
 the second column gives 3a 2 for the trial divisor, by v, ' i !i find b, the next 
 figure of the root, and to 3</, the sum of the last three written quantities in 
 the first column, annex b ; then the sum, 3a-{-b, is multiplied by b, and the 
 product, 3ab-\-b 2 , is placed m the second column; thru the trial divisor. 
 and ihe addend, 3a&+6 9 , being collected, give tbe true divisor, 3a 9 -f-3afr-|-&*i 
 which multiply by b, and place the product, :\,, : b-\-3(tli : -\-b : \ under the re- 
 mainder, 3a-b-\- 3ao s '-\-l}\ When there is a remainder after this operation, 
 the process may bo continued by writing b twice in the first column, under 
 ?„i-\-b, and h'- once m the second column, under the last true divisor ; thei 
 _J_ r;,,/,_i_ ,*)/,-, the sum of the last written three lines in the second column, will 
 bo another trial divisor, with which procoed as above. We have written a- 
 in the second column three times in succession, to assimilate the first step in 
 the operation 1o the other successive steps, hit the fust trial divisor, :\n : . may 
 lie written at once, ami the symmetry of the disposition of the quantities in 
 the Brsi steps disregarded.! 
 
 * These methods may be nnsscd over by the itadent, u will as that Lven for tliu bi- 
 i ! the in, -ill,., I employed, which is .1. jcribed at (Art 112), which ia appli 
 naMe t" i In' extraction of the root of the third and fourth, ai well aa ofanj other 
 i Three quantities are added each ti ; in die method on ne: two. 
 
EXTRACTION OF THE CUBE ROOT. 93 
 
 96. Second method of composing 3a"-{-3ab-\-b 2 , the true aivisor. 
 
 a 
 
 a 2 
 
 
 a 3 +3a 2 Z> + 3a& 2 +& 3 (a+6 
 . a 3 
 
 
 
 
 
 3a 2 6 + 3a£ 3 +Z> 3 
 
 a 
 
 3a % 
 
 
 
 3a + b . . . 
 
 
 3ab+ i 2 
 
 tm 
 
 b 
 
 3a 2 +3a& + 6 s . . . 
 3aZ>+2& 3 
 
 f?/72_l_fW)_]_. r> ,7|2 Rfi 
 
 3a 2 6 + 3aZ> 2 +& 3 
 
 3«+2& . . . 
 b 
 
 nnnd trinl rlivisnr. 
 
 3a+3& 
 In this method we write a under a in the first column, and the sum 2a 
 being multiplied by a, gives 2a 3 to place under a 2 in the second column, and 
 the sum of 2a 2 and a- is 3a 2 for the trial divisor. Again, under 2a in the first 
 column write a, and the sum of 2a and a gives 3a. Now, having found b by 
 the trial divisor, annex it to 3a in the first column, making 3a-\-b, which, mul- 
 tiplied by b, and the product placed in the second column, gives, by addition, 
 the true divisor, 3a 2 +3a& + & 2 , as before. We shall exhibit the operation of 
 extracting the cube root by both these methods. 
 t 
 
 EXAMPLES. 
 
 (1) What is the cube root of x s — 9a*+39:c*— 99a«+15Ga: s — I44.r+G4? 
 
 By the first method, 
 afl x* afi— 9x5+39x4— 99x3+156x2— 144x+64 (x— 3x-H 
 
 #2 x* .X 6 
 
 X* X* 
 
 3x* 
 
 3^2 — 3x . . — 9x3+ 9x 2 
 
 —9x5+39x4— 99x-3 
 
 — 3x 3xi— 9x3+ 9x" .... —9x5+27x4—27x3 
 
 _3x 12xi— 72x3+156x 2 — 144x+64 
 
 3x4— 18x3 +27a; 2 
 
 3^3 — 9x+4 . . . 12j2— 36a+16 
 
 3^4 — 18x3+39x2— 36x+ 16 . . . 12*4— 72*3.1. 156*2— 14J3+64. 
 
 (2) What is the cube root of x 6 +6^— 40r*+96a:— 64 ? 
 
 By the second method. 
 x2 x6+6x5— 40x3-f-96x— 64 (x2-f 2a;— 4 
 
 X 2 X* X6 
 
 2.r» 2ar* 6x 5 — 40x3 
 
 X2 
 
 3_£-t 
 
 3x3+2x ... 6x3+ 4x 2 
 
 2x 
 
 3x»+ 6x3+ 4x2 ... . 6x5+12x4+ 8x3 
 
 3x2+4x . . 6x3+ 8x2 
 
 2a; — 12aH — 48x3+96x— 64 
 
 3x4+12x3+12x2 
 
 3x2+6.1— 4 . . — 12x2— 24x+lG 
 
 3x4+12x3— 24x+16 — 12.r4— 48x3+96x— 64. 
 
94 ALGEBRA. 
 
 (3) What is the cube root of a 3 +3a=i + 3aZ; +6 3 +3a-c4-6a6c+3fc 3 c+3ac' 
 + 36c 2 +c 3 ? Ans. a + b + r - 
 
 (4) Extract the cube root of &—Gx !i +15x*—20x 3 -\-15x-—6x-\-l. 
 
 Ans. x-— 2x-r-l. 
 
 97. The same process is employed in the extraction of the cube root of 
 numbers, as in the subsequent examples. 
 
 EXAMPLES. 
 
 (1) Extract the cube root of 403583419. 
 
 • • 
 
 7 49 403583419 (739 = root 
 
 7 49 343 
 
 7 49 
 
 60583 
 
 147 
 213 639 
 
 3 
 
 3 15339 46017 
 
 9 
 
 15987 
 
 1456G419 
 
 2199 19791 
 
 1618491 14566419. 
 
 (2) What is the cube root of 115501303 ? 
 
 • ■ • 
 
 115501303 (487 = root 
 4 16 64 
 
 4 51501 
 
 3 32 
 
 4 48 
 
 128 1024 
 
 8 
 
 5824 46592 
 
 136 1088 
 
 8 4909303 
 
 — 6912 
 
 1447 1012 I 
 
 701329 4909303. 
 
 98. The local values of the figures in the rool determine the arrangement 
 of tin' figures in the several columns, as i> exemplified by working the la 
 ample as on nexl page; by omitting the terminal ciphers, the arrangement is 
 precisely the same as m the preceding example. 
 
EXTRACTION OF THE FOURTH ROOT. 9S 
 
 115501303 (400+80 + 7 
 
 400 160000 64000000 
 
 400 
 
 51501303 
 
 800 320000 
 
 400 
 
 480000 
 
 1200 
 80 
 
 1280 102400 
 
 80 
 
 582400 46592000 
 
 1360 108800 
 
 80 4909303 
 
 691200 
 
 1440 
 7 
 
 1447 10129 
 
 701329 4909303 
 
 99. Extraction of the fourth root of whole numbers. 
 
 1 he investigation of a method for extracting tho fourth root of any numbei 
 is similar to that employed for the cube root. Thus, since 
 
 (cr+ &) 4 =a 4 + 4a?b + 6aW+ 4a& 3 + b\ 
 we may conceive a to denote the number of tens, and b the number of units 
 in the root of the number expressed by a 4 +4a 3 &+6a 2 6 2 +4a& 3 +& 4 . Then 
 i/a i =a, the figure in the tens' place, and the remainder, when a 4 is removed, is 
 4a 3 6+6« 2 i 2 +4a6 3 +6 4 =(4a 3 +6a 2 &+4a6 2 +£ 3 )i. 
 The method of composing the divisor 4a 3 +6a 2 6 + 4a2> 2 +& 3 , for the deter- 
 mination of b, the figure in the units' place, may be illustrated as follows : 
 
 ax a = a 2 a 4 +4a 3 Z> + 6a 2 & 2 +4afc 3 +Z> 4 (a+6 
 
 a a'X« = a 3 
 
 2a x a =2a 2 a?xa =a 4 
 
 3a 2 X a =3a 3 4a 3 6+6a 2 £ 2 +4a& 3 +6 4 
 
 3a X a =3a 2 4a 3 
 
 a 
 
 6a* 
 
 (4a+6)6=4a&+6 2 
 
 (6a 2 +4a&+6 2 )&=6a 2 i+4at 2 +& 3 
 
 (4a 3 +6a 2 6 + 4a& 2 +& 3 )&=4a 3 Z> + 6a 2 Z> 2 +4a& 3 +& 4 . 
 
 100. From this mode of composing the complete divisor we easily derira 
 the following process for the extraction of the fourth root of an 7 number. 
 
9G ALGEBRA. 
 
 1 \ \MI>LE. 
 
 What is tho fourth root of 1185921 ? 
 
 3X3 = 9 
 
 
 3 9X3 = 
 
 27 
 
 6X3 =18 
 
 27X3 
 
 3 27 X 3 = 
 
 81 
 
 9X3 =27 
 
 108 .. . 
 
 t 
 
 
 51 
 
 
 123X3 = 369 
 
 
 5769X3 = 
 
 17307 
 
 1185921 (33 = root 
 = 81 
 
 375921 
 
 125307X3 = 375921 
 
 in tho same manner, tho student may readily investigate rules for the m 
 traction of the higher roots of numbers, simply observing to use an additionn' 
 column for each successive root. 
 
 101. To represent a rational quantity as a surd. 
 
 Let it be required to represent a in the form of a surd of the n\\x order, 
 
 then, by (Art. 63), the form will bo V« n . or (a") n 5 for by raising a to the ?ith 
 power, and then extrasting the nth root of tho nth power of a, we must evi 
 dently revert to the proposed quantity, a. Hence we have 
 
 a— -/« 2 = Vd? = Va 1 = !$Ta u = ^a m = Va° 
 
 1 B ' — - 
 
 a = {a-) 2 = {a i -)*=(a' s f=(a") m . 
 102. When tho given quantity is the product of a rational quantity and a 
 surd, we must represent the rational quantity in the form of- the given surd, 
 and then express tho product with a single radical sign, or fractional index 
 Tb"S, we have 
 
 ay/b = -v/""X Vb= V a "b 
 
 \/7>Z= V 3a X 3a X Vltb = ■y/9d : Xob = -^IbcTb 
 ay~xi)= V ax ax aX V^/= V^X Vx y= Va*xy 
 
 lJ-v/7 = VUlX >/"! =-/ll*X7 =-v/1008 
 
 i i 
 
 a{l—a--.L-y s ={a : )- {I— = (a 9 — = ■/«*— . 
 
 (1) Represent a s in the form of a surd, whose index is 5. 
 
 (2) Represent 2 — -/3 in the form of a quadratic snrd. 
 
 (3) Transform 6 ^11 into the form of a quadratic snrd. 
 (1) Transform a-)/ a — b into the form of a quadratic surd. 
 
 (5) Represent as a surd tho mixed quantity (a " + .'/)/' 
 
 • r +.V* 
 
 (6) Represent as a surd tin* mixed quantity (r-f- 1) / 
 (I)*/" 17, m* (fi" i". 
 
 \ sc+4* 
 
 \N \\ | 
 
 (2) V7-4V3- 
 
 (3) J 
 
 (I) v'- — </•'/« or (<r 1 — fl'o) 
 (6) /^ -;/')* 
 
 . I i or(k+4)J. 
 
 '• 
 
BINOMIAL SURDS. 97 
 
 103. To find multipliers which will render binomial surds rational. 
 The product of two irrational quantities is, in many instances, a rational 
 quantity, and, therefore, an irrational quantity may frequently be found, which, 
 employed as a factor to multiply some other given irrational quantity, will 
 produce a rational result ; thus, 
 
 y/aX V a = a 
 yxx Vx" =x 
 
 i/yxi/y m - l =y> 
 
 Again, since the product of the sum and difference of two quantities is equal 
 to the difference of their squares, we have, evidently, 
 ( y a — y/b){ ■/«+ y/b)=a —b 
 (x+ yfy){x -y y )=x"-y 
 ( Va-'— y){ V*+ y) =* — 2/ 2 - 
 
 Hence it is obvious that, in these and similar equalities, if one of the factors 
 be given, the other factor or multiplier is readily known, and the proposed 
 irrational quantity is thus rendered rational. By a double operation of this 
 kind, multiplying ( V«+ Vp-\- V'l) by ( V w + Vl>— Vq)> we have ( y/n 
 _f_ y/py—q, or n-\-p — q+2y/np; and multiplying this by n-\-p— q— 2 y/np, 
 the given expression, \/n-\- \/p-\- V(?> i3 rationalized. In the same manner, 
 since 
 
 {x ± y) (x~ ^ xy + y") —x 3 i y* 
 .'. (yx±yy)(yx^¥xy+¥y*)=x±y, 
 and the expression V x i y/y ma y> therefore, be rationalized by multiplying it 
 by fy&^=f/xy+tyy*-, and fyx*=f fyxy -{• fyy*, multiplied by tyx^yy, will 
 produce a rational result. 
 
 Again, by division [see Art. 23 (5), (6), (7)], 
 
 x ~~y — x n - 1 ^-x"- J2 y-{-x n - 3 y i -\-x n ~ i y 3 -\- +2/ n_1 
 
 x y 
 
 n — 1 „.n— 5». I ^.n — 3 17 2__T n — *1/ 3 -L — I/" -1 
 
 x+y 
 
 x n -\-y n 
 
 = x J '- 1 —x n - 2 y+x a - 3 y' 2 —x*- 4 y 3 -{- .... — y n 
 
 x+y 
 
 z= x n - 1 —x n - 2 y^-x n ~ 3 y i — x°— 1 y 3 + • • • • +2/ n ~ 
 
 Put x n —a ; theD x= Va ; x n ~ ] = V« n_1 ! z n_2 = V a n ~ 2 , &c. ; 
 y n =b ; then yz= \Vb ; nf = yV ; y 3 = yb 3 , &c. ; 
 hence, by substitution in the three preceding equalities, we have 
 
 a a ~ yb = V^ =I + • v / «^+ V« n- ^' 2 + Va^M h Vb^ 1 • (1) 
 
 a — 6 
 
 a+b 
 
 Va"- 1 — Va"~ 2 *+ ya n ' 3 b 2 — Va"^ 4 t 3 + • • • — Vb"' 1 ■ (2) 
 
 = Va - 1 — Va n ^+ Va^V— Va a -*b 3 ^ \- Vb°~ l . (3) 
 
 V«+ Vb' 
 
 Now, the dividend being the product of the divisor and quotient, it is obvi- 
 ous that a binomial surd of the form y/a—yb will be rendered rational by 
 multiplying it by n terms of the second side of equation (1), and a binomial 
 surd of the form V a-\- yb will be rationalized by employing n terms of the 
 second side of equality (2) or (3), according as n is even or odd, the product 
 in the former case being a — b, and in the latter a — b or a-\-b. 
 
 G 
 
98 ALGKBKA. 
 
 Note. — When n is an even number, employ equation (2), and when it is aa 
 odd number, equation (3), in order to rationalize y a+ yb. 
 
 EXAMPLES. 
 
 (1) Find a multiplier to rationalize v/11 — ^/7. 
 
 Employing equation (1), we have a = ll, 6=7, and 72=3 ; hence required 
 multiplier =^li»+ 101^4-^=^121+^77+^49. 
 
 And, VIS +V77+V49 
 
 Vll -V7 
 
 Vl331+\/847+V539 
 
 _ ^647— V539— V343 
 11 * * — 7 =4, a rational product. 
 
 (2) Rationalize the binomial surd -v/5+^/4. 
 
 Hero we have a=5, 6=4, n=3, an odd number; hence by equation (:J) 
 we have multiplier required, = ^25 — s/20-\-\/liJ; and, by multiplication, 
 (^5+ £/4) (^25 — ^/20 + s/T6)=5+4 = 9= a rational number. 
 
 (3) What multiplier will render the denominator of the fraction -^p. — irp, 
 
 a rational quantity ? 
 
 5 
 
 (4) Change .. . - , into a fraction that shall have a rational denominator 
 
 j/x- 
 
 (5) Change • -.- , ., , .. „ into a fraction that shall have a rational de 
 
 nominator. 
 
 Va+£+ V# — x 
 (6) Change , . into a fraction that shall have a rational de- 
 
 nominator. 
 
 ANSWERS. 
 
 (3) V7 4 + V7 3 .2+^/7-.2-+ V7.2 3 + V2 4 - 
 5(yiG+ffS+V4) 
 
 ( 4 ) o • 
 
 (5) V^Vx^^y) _ x^Vl^ 
 1 x ±y x^y 
 
 (G) • 
 
 104. To extract the square root of a binomial surd. 
 
 x>eforo commencing the investigation of the formula for tin* extraction of 
 
 the square root of a binomial surd, it will be necessary to premise two or three 
 lemmas. 
 
 Lemma 1. Tho square root of a quantity can not be partly rational and partly 
 irrational. 
 
 For, if i/a=b-\- -y/r, then, by squaring, we b 
 
 (1 — t 3 — c 
 a=b--\-c- T -2b V 1 ' ; therefore, ■/.-= — ^r — ; 
 
 that is, an irrutional equal to a rations! quantity, which is absurd. 
 
BINOMIAL SURDS. 99 
 
 Lemma 2. If «i -y/&=- r =t *Jy be an equation consisting of rational and ir- 
 rational quantities, then a=x, and •;/&= Vy ! *• «•> the rational ana iiTational 
 parts of the two members of an equation must be separately equal. 
 For, if a be not equal to .r, let a — x = d ; then we have 
 
 i Vy-f- Vb=a — x; but a — x=.d; therefore 
 
 i Vz/T Vb=d, which is impossible ; 
 
 .•. a=x, and, taking away these equals, -\/b = -yjy. 
 
 Lemma 3. If -/«+ V^ =;r +1/) then V 'a — *Jb=x — iy ; where r and y 
 are supposed to be one or both irrational quantities. 
 
 For, since a-\- iJb=x--\-y"-\-2xy ; and since x 2 and y" are both rational, 
 2xy must be irrational. By Lemma 2, we have 
 
 a=x 2 +2/ 2 ; ^b=2xy 
 .•. a — *Jb=X" — 2xy-\-y* 
 
 and -\/a — -\/b=x — y. 
 
 Let it now be required to extract the square root of a-\- -y/6. 
 
 Assume V a -\- Vb=x-{-y ; then ^J a — <\/b=x — y 
 
 .'. a-{- -\/b=x"-\-y :2 -\-2xy 
 a — -\/b=x' 2 -\-y' 2 r — 2xy 
 
 .-. By addition, 2a =2(x"+y"), or a=x"+y*. 
 
 Again, y/~a~+V~ b X Va— Vb=x°—y' 2 , or y/a i —b=:x' i —y 3 . 
 Hence x*J r y' 2 =a 
 
 x " — y~= V^ — b=c, suppose. 
 Therefore, by addition and subtraction, we have 
 
 a-^-c , a — c 
 
 X2= 2 and 2/2==_ 2~" 
 
 \a-\-c la — c 
 
 ••• x =V - 2~ y ~V~2~' 
 
 . j aJ r c l a — 
 
 Hence V«+ ^ b== \~2~^"\~2 
 
 \a-\-c la — 
 
 V«-V^=V~2 - ~V~2" 
 
 ! — C 
 
 ! C 
 
 (1) 
 
 (2) 
 
 where c= V^ 2 — & » an d> therefore, a 2 — Z» must be a perfect square ; and this 
 is the test by which we discover the possibility of the operation proposed.* 
 
 * When the quantity a 2 — b is not a square, the values of a and b are no longer rational 
 but it is clear that the formulas (1) and (2) will still give true results. As, howe7er, these 
 are more complicated than the original expressions themselves, they are rarely employed 
 yet, when -\/b is imaginary, the result merits attention. 
 
 In order to examine this case, change b into — b"; a-\-\/b becomes a-\-b\/ — 1. The re- 
 markable circumstance just alluded to is this, that the square root of a-\-b\/ — 1 has tho 
 same form as this quantity itself. 
 
 This is shown from the formula (1), for since c=\/ a*-\-b' 2 , when b is changed into — #», 
 
 the second member becomes J a-\--[/a--i-b : ^_ l a \Za--\-b~ ^ The quantity under the 
 first radical is positive, and that under the second negative, since -\/a--\-b- is greater thai 
 
100 ALGEBRA. 
 
 EXAMPLES. 
 
 (1) "What is the square root of 11+ \f72, or 11 + G y/2 ? 
 
 Here c = ll ; b=72 ; c= Va 2 — 6 = Vl^l — 72=7 
 . — /a-l-c /a — c 
 
 .-. 7n+(i v~ , =v^ : -+v^ _=3 + ^~- 
 
 (2) What is the square root of 23— 8 -y/7 ? 
 
 Here a=23 ; b = 8 2 X 7=448; c= -/a 5 — b = Vo29 — 448=9 
 
 . r- /a+c /a — c 
 
 V23-8V7=j^--yJ—=4-V7. 
 
 (3) What is the square root ofl4 + 6-/5? Ans. 3 + y/'o 
 
 (4) What is the square root of 18 ±2 y/T7 ? Ans. -/lT± V? 
 
 (5) What is the square root of 94 + 42 y/5 ? Ans. 7 + 3 V 5 
 
 (6) To what is '\np-\-2m 2 — 2my/np-\-m i equal ? Ans. y/np-\-7nr — nt 
 
 (7) Simplify the expression\/ 16+30 -y/ — 1+V 16 — 30 V — 1- Ans. if». 
 
 (8) To what is • v /28+10-/3 equal? Ans. 5+ y/Z. 
 
 (9) \lbc+2by/bc—b 2 —\Jbc—2bi/bc — b' i =±2b 
 
 v/ 
 
 (10) Vafc+4c 2 — ^+2 V4a6c- — «W-= ■/<*&+ • v /4c :! — c? 3 . 
 
 (11) What is the square root of — 2y/ — 1? Ans. 1 — \/ — 1. 
 
 (12) What is the square root of 3 — 4 ■/— 1 ? Ans. 2— V -1 
 
 „„ . 3-/3 + 2-V/0 112+20 -/T^, 
 
 (13) What is square root of — — J~ — . / 3 — * 
 
 Ans. (1+ y/2) • (5+ y/3) 
 
 BINOMIAL THEOREM. 
 
 105. It is manifest, from what has been said above, that algebraic polynomials 
 may be raised to any power merely by applying t ho rules of multiplication. 
 We can, however, in all cases obtain the desired result without having rocourse. 
 to this operation, which would frequently prove exceedingly tedious. When 
 a binomial quantity of the form x+a is raised to any power, the successive 
 terms are found in all cases to bear a certain relation to each other. This law, 
 when expressed generally in algebraic language, constitutes what is called the 
 "Binomial Theorem." It was discovered by Sir Isaac Newton, who seems 
 to have arrived at the general principle by examining the results of actual mul- 
 tiplication in a variety of particular cases, B method which wo shall here pursue, 
 and give a rigorous demonstration of the proposition in a subsequent article of 
 this treatise. 
 
 a; representing the quantity under tho first radical by a 8 , and that under tho second 
 by — /i-, tho expression takes tho form a+/?\/ — 1 ; hence 
 
 v/<i+V— l=a+/V— l- 
 
 a E. D 
 
BINOMIAL THEOREM. 
 
 101 
 
 Let us form the successive powers of x-\-a by actual multiplication. 
 z -\-a 
 r -{-a 
 r 2 -f- x a 
 
 -f- x a-\-a 2 
 
 r 2 -f-2x a-\-a 2 2d power. 
 
 r -f- a 
 r*+2x'-a+ x~a? 
 
 -{- x 2 a+ 2x0?+ a? 
 
 ./'-(- ;;./-ci+ ;:.v(i : -\-,r- 3d power 
 
 x -}- a 
 
 x 4 +3x i a+ 3.r-a--i- ia 3 
 
 -f- x 3 g+ 3x 2 q 2 4. 3xa 3 -fa 4 
 
 x 4 +4x 3 a-f- 6x 2 a 2 + 4xa 3 -|-a 4 4th power. 
 
 x + a 
 
 x-"*+4.r 4 a+ Cx^-j- 4x 2 a 3 -f ^ 
 
 + x 4 a+ 4x 3 a 2 -|- 6x 2 a 3 + 4xa 4 +a 5 
 i^+5.r»a+10r ! a 2 +10a- 2 a 3 4- 5xa 4 +a 5 5th power 
 
 J + «» 
 
 x« + 5x*a + lOx'a 2 -f- 1 OxW -f 5x 2 a 4 + x~aJ 
 + r^a^- 5r*a 2 4-10x 3 a 3 +10.r 2 a 4 - r - 5xa 5 +a 6 
 
 c +6x 5 a + 15x 4 a 2 +20x 3 a 3 -j-15x 2 a 4 + 6xa 5 +a 6 6th powei 
 
 x -\- a 
 
 x 7 +6x 6 a+15x 5 a 2 +20x 4 a 3 4-15x 3 a 4 4- 6x 2 a 5 -|- xa 6 
 
 -f- .t^a-f- 6.r 5 a 2 4-15.r 4 a 3 4-20.r 3 a 4 4-15.r 2 a 5 4-6.ra 6 4-a 7 
 x 7 + 7x G a+ 21x 5 a 2 + 35x 4 a 3 + 35x ; a 4 -f- 21x 2 a 5 + 7xa 6 -fa 7 .... 7th power. 
 
 In order that these results may be more clearly exhibited to the eye, we 
 shall arrange them in a table. 
 
 TAJSLE OF THE POWERS OF X-\-d. 
 
 (x+a) 
 
 (x + ay 
 
 x 2 +2xa + a 2 
 
 x-f-a 
 
 (x+a)= 
 
 x 3 -|-3x 2 a + 3xa 2 +a 3 
 
 (x+a) 4 
 
 r*-\- 4x 3 a+ 6x 2 a 2 + 4x« 3 + a 4 
 
 (x+ay 
 
 x 5 +5x 4 a+10x 3 a 2 +10x 2 a 3 + 5xa 4 -fa 5 
 
 (x+a)< 
 
 xfi+6x r, a+15.r 4 a 2 4-20.r 3 a 3 +15x 2 a 4 + 6xa 5 + a 6 
 
 (*+«)' 
 
 x" + 7x p a+21x s a 2 +35x 4 tf 3 4-35x' 5 a 4 +21x 2 a 5 + 7xa 6 + a 7 
 
 (x+a)< 
 
 r 8 + 8x 7 a+ 28.r«a 2 4- 56.r 5 a 3 + 70x*a 4 + 5Gx 3 a 5 + 28x 2 a 6 + 8xa 7 + a 8 . 
 
 In the above table, the quantities in the left-hand column are called the ex- 
 vressions for a binomial raised to thejirst, second, third, &c, pouter • the cor- 
 
102 ALGEBRA. 
 
 responding quantities in tho right-hand column are called the expansions, or 
 developments, of those in the left. 
 
 10G. The developments of the successive powers of x — a are prec 
 tho same with those of x-\-<t, with this difference, that the signs of the terms 
 are alternately -4- and — ; thus, 
 
 (r— a) 6 =x*— Sr'a-J-lOrV — WxW+'xca*— a\ 
 
 and so for all the others. 
 
 107. On considering the above table, we shall perceive that, 
 
 I. In each case the first term of the expansion is the first term of tho bi- 
 nomial raised to the given power, and the last term of the expansion is the 
 second term of the binomial raised to the given power. Thus, in the expan- 
 sion of (x-\-ay the first term is ar 1 , and the last term is a*, and so for all the 
 other expansions. 
 
 II. The quantity a does not enter into the first term of tho expansion, but 
 appears in the second term with the exponent unity. The powers of .r de- 
 crease by unity, and the powers of a increase by unity in each successive 
 term. Thus, in the expansion of (.'"+")'' wo have x 6 , a*a. arte 3 , arte 3 , x*a 1 , 
 xa b , a 6 . ., 
 
 III. Tho coefficient of the first term is unity, and the coefficient of the 
 second term is, in every case, the exponent of the power to which the binomial 
 is to be raised. Thus, the coefficient of the second term of (x-^-a)* is 2, of 
 (x+cf) 6 is G, of (x+a) 7 is 7. 
 
 IV. The coefficient of any term after the second may be found by multiply 
 mg the coefficient of the preceding term by the index of .r in that term, and 
 dividing by the number of terms preceding the required term. Thus, in the 
 expansion of (x-\-a) 4 the coefficient of the second term is 4 ; this multiplied 
 by 3, the index of .r in that term, gives 12, which, when divided by "2, the num- 
 ber of terms preceding the third term, gives G, the coefficient of the third term. 
 Again, 6, the coefficient of tho third term multiplied by 2, the exponent of x 
 in that term, gives 12, which, when divided by 3, the number of terms pre- 
 ceding the fourth term, gives I. the coefficient of the fourth term. So, also, 
 35, the coefficient of the fifth term in the expansion of (.r-4-(r) 7 , when multi- 
 plied by 3, the index of X in that term, pives 105, which, when divided by 5, 
 tho number of terms preceding tho sixth, gives 21, the coefficient of that 
 term. 
 
 By attending to the above observations wo can always raise a binomial of 
 the form (r-fa) to any required power, without the process of actual multi- 
 plication. 
 
 EXAMPLK I. 
 
 Raise x-\-a to tho 9th power. 
 
 The first term is 2*0°, 
 
 The second term is . • 9x*a l ; 
 
 9X8 
 The third term is ... . — — x 7 a 9 = 3Gx 7 a* ; 
 
 The fourth term w ... — — x c a :{ = 8ir*<P, 
 
 J 
 
BINOMIAL THEOREM. 103 
 
 The fifth term is — : — x 5 a*=\26x 6 a* ; 
 
 4 
 
 m, • , • 126X5 
 
 The sixth term is — - — r*a 5 =126.r 4 a 5 ; 
 
 o 
 
 ™, , 126X4 
 
 The seventh term is — - — x 3 a G = 8i3?a°; 
 
 o 
 
 The eighth term is — ^— x 2 a 7 = 36x 2 a T ; 
 
 84X3 
 7 
 
 36X2 
 
 The ninth term is — - — x 1 a 8 = 9x 1 a 9 ; 
 
 8 
 
 9X1 
 The tenth tenn is — - — x°a 9 = x°a». 
 
 Hence, 
 
 (x+a) 9 = x 9 +9x 3 a+36x 7 a 2 + 84x 5 a 3 + 126x 5 a 4 +126x 1 a 5 +84x 3 a 6 + 36xW 
 + 9xa 8 +a 9 . 
 
 EXAMPLE II. 
 
 In liko manner, 
 
 (x— a) w =x w — 10x 9 a+45x 8 a 2 — 120x 7 a 3 +210x 6 a*— 252x 5 a s +210x , a 6 — 120 
 a?a? + 45x 2 a 8 — 1 Oxa 9 + a 10 . 
 
 08. The labor of determining the coefficients may be much abridged by 
 attending to the following additional considerations : 
 
 V. The number of terms in the expanded binomial is always greater by 
 unity than the index of the binomial. Thus, the number of terms in (x+a) 4 
 is 4 + 1, or5; in (x+a) 10 is 10 + 1, or 11. 
 
 VI. Hence, when the exponent is an even number, tho number of terms in 
 the expansion will be odd, and it will be observed, on examining the examples 
 already given, that after Ave pass the middle term the coefficients are repeated 
 in a reverse order ; thus, 
 
 The coefficients of (x+a) 4 are 1, 4, 6, 4, 1. 
 
 The coefficients of (x+a) 6 are 1, 6, 15, 20, 15, 6, 1. 
 
 The coefficients of (x+a) 3 are 1, 8, 28, 56, 70, 56, 28, 8, 1. 
 
 VII. When the exponent is an odd number, the number of terms in the 
 expansion will be even, and there will be two middle terms, or two contiguous 
 terms, each of which is equallj* distant from the corresponding extremities of 
 the series ; in this case the coefficient of the two middle terms is the same, 
 and then the coefficients of the preceding terms are reproduced in a reverse 
 order; thus, 
 
 The coefficients of (x+a) 3 are 1, 3, 3, 1. 
 
 The coefficients of (x+a) 5 are 1, 5, 10, 10, 5, 1. 
 
 The coefficients of (x+a) 7 are 1, 7, 21, 35, 35, 21, 7, 1. 
 
 The coefficients of (x+a) 9 are 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. 
 
 109. If the terms of the given binomial be affected with coefficients or ex- 
 ponents, they must be raised to the required powers, according to the princi- 
 ples already established for the involution of monomials ; thus. 
 
104 ALGEBRA. 
 
 EXAMPLE III.' 
 
 Raise (2r , -f-5a 2 ) to the 4th power. 
 
 The first terra will be (2r>)* =l€x" ; 
 
 The second terra will he .... 4(2.r 1 ) 3 X (5a 2 ) =4x8x5iV, 
 
 4X3 , 
 Tie third term will be -^— x (^x (5a : )-=Gx4 x25xV», 
 
 6X2 
 The fourth term will be .... -^-(2x 3 ) 1 X (5a 2 ) 3 =1 X2X ISor'a 9 , 
 
 4 
 The fifth terra will be j^x 3 ) x (5a"-)< = G25a 8 ; 
 
 .-. (2r 3 +5a 2 ) 4 =16x 12 +160x 9 a 2 + 600x^<+1000x 3 a 6 4-625a 8 . 
 
 EXAMPLE IV.* 
 
 In like manner, 
 ( a 3 + 3ab )» = (a 3 ) 9 + 9 (a 3 ) 8 X ( Sab ) + 36 (a 3 ) 7 X ( 3a& ) 2 + 84 (a 3 ) 6 X ( 3a5 )» 
 + 126 (a 3 ) 5 X (3aZ>)' + 126(a 3 )<X (3a&) 5 + 84(a 3 ) 3 X (3a6f 
 + 36(a 3 ) 2 X (3a6) 7 + 9a 3 X (3a6) 8 + (3ai) 9 
 =a 27 -f 27o M 6 + 324a 23 Zr -f 2268a 21 6 3 + 10206a 19 b* + 30618a 17 o* 
 + 61236a 15 i 6 +78732a 13 i 7 +5,9049a u t 3 +19683a 9 i 9 . 
 110. We shall now proceed to exhibit the binomial theorem in a general 
 form. Let it be required to raise any binomial (r+a) to the power represent- 
 ed by the general algebraic symbol n. Then, by the preceding principles, we 
 shall have 
 
 The first term x D ; 
 
 The second term nx^a ; 
 
 n(n — 1) 
 The third term ' x°--a- ; 
 
 n{n — l){n—2) 
 The fourth term Too x n_3 a 5 ; 
 
 n(n — l)(n — 2) In— 3) 
 The fifth term -5 \\~ A l x°-*a*, 
 
 &c &c. 
 
 The last term a". 
 
 The whole number of terms will bo n-\-\, and the coefficients be repeated 
 
 /n + l\ in \ 
 
 a reverse order after the I — - — 1"', or (^+1 )"' terra, according as n is odd 
 
 or even ; moreover, the terms will all have the sign +• '' 'I"' quantity to be 
 expanded be of tho form x-f-a, and they will have the sign -\- and — alter- 
 nately, if the quantity be of the form x — a. Hence, generally, 
 
 n(n — 1) n(n—l)(n—2) 
 (i-fa)"=.r n +7ix n - 1 a+^p r - V~ 2 a 2 +-^ — -'-^ '-x n ~ <r+ 
 
 n(n— l)(n— 2) w(/j — l) 
 
 + ; .,;> Va"- 3 + \ V ,r -•-|_„.m"- 1 + a- 
 
 n(n—l) 
 (r-rt)"=r"-)i.i»- 1 a-| — — x n <r ±a°. 
 
 i • - 
 
 • The 1>' it method of pi I unplei is to r.iise the fourth and 
 
 ninth powers, and then, in thi thua obtained, to snbatitv lory, and! r 
 
 s in the Brat, and ■i i for >/. and Sai for t In die leoond. 
 
 ui 
 
BINOMIAL THEOREM. 105 
 
 In this last case £n be an even number, the last term, being one of the odd 
 terms, will have tho sign -f- ; and if n be an odd number, the last term, being 
 one of the even terms, will have the sign — . 
 
 Both forms may be included in one by employing the double sign. 
 
 (4«)"=.t»±B^"- 1< w(7i-l) x „_ 2a w(»-l)(n-2) jn _ 3a3 & ^ 
 
 L.2 1.2.3 
 
 If we make x and a each equal to 1, (x-\-a) n becomes (l-f-l) n , or 2 n , and the second mem- 
 ber reduces to its coefficients ; hence the sum of the coefficients in the binomial formula is 
 equal to the « ,b power of 2. 
 
 EXAMPLE V. 
 
 To exemplify the application of the theorem in this form, let it be required 
 to raise x-\-a to the power 5. 
 
 Here we havo w=5, n — 1=4, n — 2=3, &c. 
 Hence, 
 
 x n is 2 s = x 5 
 
 nx n ~ l a is 5x*a = 5x*a 
 
 n(n—l) 5.4 
 
 • ' x"-^ 2 is — r¥ ■=10x 3 a 2 
 
 n(n—l)(n—2) 5.4.3 
 
 ' -x n ~ 3 a 3 is x 2 a 3 = 10x 2 a' 
 
 n(n—l)(n—2)(n—3) 5.4.3.2 
 
 1 2.3.4 lx *~ 4ai is 172-374^ = ^ 
 
 n(n — l)(n— 2)(n — 3)(n— 4) 5.4.3.2.1 
 
 -J: i± 11 '-± '- x °a 5 s x°< 
 
 1.2.3.4.5 9 1.2.3.4.5^ 
 
 (x+a) 5 =x 5 +5x 4 a+10r 5 a 2 +10x 3 a 3 +5.ra 4 +a 5 
 
 EXAMPLE VI. 
 
 Raise 5c 2 — 2i/z to the 4th power 
 
 Here, 
 
 .•.x a becomes (5c 2 ) 4 = 625c 8 
 
 nx a ~ l a becomes 4(5c-) 3 X (2yz) = l000c*yz 
 
 .r.=5c 2 
 a=2y: 
 «=4 
 
 n(n— 1) 4.3, 
 
 ~ 2a * becomes p^c 2 ) 2 X (~1/ z f= 600c*yh 3 
 
 11 ~~> I o~~ 3;n "' 3a3 • • -becomes T 1 ^(5c 2 ) 1 X {^y~f= IGOc-yV 
 
 n ' n - 1 f 2 -f''~ r ' ) x--%'becomesi||J ( 5^rx(% : )'= 16^ 
 
 .-. (5c 2 — 2yzy=625c s — 1000c s yz + 600cy-z~— lGOc-ifz^+lGy'z*. 
 
 111. We have sometimes occasion to employ a particular term in the ex 
 pansion of a binomial, while the remainder of the series does not enter into our 
 calculations. Our labor will, in a case like this, be much abridged, if we can 
 at once detennine the term sought, without reference either to those which 
 precede, or to those which follow it. This object will be attained by finding 
 what is called the general term of the series. 
 
 If we examine the general formula, we shall soon perceive that a certain 
 relation subsists between the coefficients and exponents of each term in the 
 expanded binomial, and the place of the term in the series ; thus, 
 
106 ALGEBRA. 
 
 The first terra is 
 ar n , ■which may be put under the form .r n ~ 1+1 ; 
 
 The second term i9 
 nx^-ta =n.r n - 2 + l a 2_1 ; 
 
 The third term is 
 
 n(n — 1) n(n— 3 + 2) 
 
 x n " 2 a 2 =— ^—^x^-Wo*- 1 • 
 
 1.2 — 1.(3—1) - a ' 
 
 The fourth term is 
 
 n(w — 1)(»— 2) n(n — l)(n— 4 + 2) 
 
 — (^ — — .r n - 3 a 3 =— -E_J,*-M+i fl 4 i • 
 
 1.2.3 1.2.(4 — 1) X a ' 
 
 The fifth term is 
 n(n— l)(n — 2)(n— 3) n(n— 1)(»— 2)(»— 5+2) 
 
 — — — -X"~*a* =— '— Vi-W/t5-1 . 
 
 1.2.3.4 * — 1.2.3.(5 — 1) a ' 
 
 The sixth term is 
 
 n(n — l)(n — 2)(n— 3)(n— 4) n(n— l)(n— 2)(n — 3)(n— 6+2) 
 
 ; — r— - — ; — z n_J a ;, = ; ! — x a ~*+ l cP~ • 
 
 1.2.3.4.5 1.2.3.4.(6 — 1) 
 
 Observing the connection between the numerical quantities, it is manifest, 
 that if we designate the place of any term by the general symbol p, the /»* 
 term is 
 
 n(n — l){n— 2)(n— 3) ( n _„ + o\ 
 
 ' nWd (/!) jn - p+laP -'- 
 
 This is called the general term, because by giving to_p the values 1, 2, 3, 4, 
 we can obtain in succession the different terms of the series for (r+a)\ 
 
 EXAMPLE VII. 
 
 Required the 7 th term of the expansion of (.r+a) 12 . 
 Here n = 12 > .-. n— -p+2=7, n — jp+l=6 
 
 p= 7 I ^ — 1=6. 
 
 Substituting these values in the general expression, we find that the term 
 sought is 
 
 12.11.10.9.8.7 
 
 ■= — = — - . - ^ a 6 ,* or 924x e a e . 
 1 . 2 . 3 .4.5.0 
 
 EXAMPLE VIII. 
 
 Required the 5 th term of (2c 4 — 4/t 5 ) 9 . 
 
 Here n = d, p=5, x=2r«, a = -\lr\ 
 
 .-.n— p+2 = 6, ?* — ^> + l=5, p — 1 = 4 ; 
 
 ■ 9.8.7.6, , , 
 
 .-. the 5"' term js /1 (~ t -')"' X (4A 6 ) 4 , or 126 X 32 X 256c 2 % s0 . 
 
 Since tho second term of the proposed binomial has tho sign — , all the 
 even terms of tho expansion will have the sign — , and all the odd terms the 
 sign + ; therefore the 5 th term is 
 
 +1032192c 9 °A 90 . 
 
 i \ vmi'i.i: i\. 
 
 Required the middle term of the expansion of (x — a) w . 
 
 Since the <\|)unent is 18, tin- whole Dumber of terms will bo I?, and heui 
 
 Tli>- operation hero to bo performed is beat effected l>_s canceling 1 1 it- tutors. 
 
HIGHER ROOTS OF NUMBERS. 107 
 
 the middle term will be the 10 th ; and since it is an even term, it will have th* 
 sign — ; hence it will be 
 
 18.17.16.15.14.13.12.11.10 
 - 1.2.3.4.5.6.7.8.9 ^ "-^OAA 
 
 EXAMPLE X. 
 
 Required the third and the last terms of the expansion of [-x-\-2yj 7 
 
 21 
 Ans. — x'nf and 128y 
 
 TO EXTRACT THE nfi> ROOT OF A NUMBER. 
 112. The n th power of 10 is 1 with n ciphers, and the n th power of any 
 number below 10 must be less, and can, therefore, bo composed of no more 
 than n figures. The n th power of 100 is 1 with 2n ciphers, and the n lh power 
 of any number between 10 and 100 can not, therefore, contain more than 2n 
 figures, nor less than n. For a like reason, the n th power of three figures can 
 not contain more than ?,n, nor less than 2n. That of four figures can not con- 
 tain more than 4ra, nor less than 3n, &c. The n th root of a number being re- 
 quired, it is evident from the above that there will be as many figures in the 
 root as there are periods of n figures in the given number, counting from right 
 to left, and one more if any figures remain on the left. The root may be 
 divided into units and lens, and the n lh power of it, or the given number, will 
 be equal, according to the Binomial Theorem, to the n ih power of the tens plus 
 n times the n — 1 power of the tens into the units plus a number of other terms 
 which need not be considered. Tens have one cipher on the right, and hence 
 the »"' power of tens has n ciphers on the right ; the n right-hand significant 
 figures, therefore, make no part of the n th power of the tens ; to find the tens 
 of the root, then, the n a ' root of those figures which remain, after rejecting n on 
 the right, must be sought by an independent operation ; but if there are more 
 than n of these remaining figures, the tens of the root are expressed by a 
 number containing more than one figure, which number may be separated into 
 its units and tens, the n lb power of the tens of which does not contain the n 
 significant figures on the right of that number upon which the independent 
 operation is now performing, and in consequence these n figures arc also re- 
 jected. After rejecting periods of n figures successively, beginning on the 
 right until there remains but one period and part or the whole of another 
 period on the left, let these be considered an independent number, its root will 
 contain two figures, tens and units; the n tii root of the tens is to bo sought in 
 what is left after rejecting the right-hand period ; the n — 1 power of the 
 tens has n — 1 ciphers on the right; so, also, has any multiple of this, and, 
 therefore, n times the n — 1 power of the tens into the units; which last 
 quantity, therefore, is not to be sought in the n — 1 right-hand significant 
 figures ; after subtracting the n ,h power of the tens just found, only one figure 
 of the next period, therefore, is to be placed on the right of the remainder, 
 which is then divided by n times the n — 1 power of the tens; the quotient 
 will not bo exactly the units, for the dividend contains also a part of the other 
 terms of the power of the binomial which were not considered : this quotient 
 may be greater than the units of the root, but never can be less ; it must be 
 diminished till the n lh power of the two figures fouu. is equal to or less than 
 
108 ALGEBRA. 
 
 tlio independent number under consideration. Annex now to this independent 
 number the next period on the right of it, and consider what is thus obtained 
 as a new independent number; the two figures of the root already found will 
 be the tens of the root of the new Dumber; bringing down our figure of the 
 right-hand period of it to the remainder after subtracting the n th power of the 
 two figures of the root just found from the first independent number, and 
 dividing by n times the n — 1 power of the tens, now composed of two figures, 
 a third figure of the root is obtained ; proceeding in this manner, the entire root 
 of the given number will at length be extracted.* 
 
 EXAMPLES. 
 
 (1) V504321, 2366=8,921. (3) ^ 233416517309451. 
 
 (2) VH64532, 07234. (4) !^282429536481. « 
 
 113. By employing the binomial theorem, wo can raise any polynomial to 
 any power, without the process of actual multiplication. 
 
 For example, let it be required to raise x-\-a-\-b to the power 4. 
 Put 
 
 <*+*> =y; 
 
 Then, 
 
 (r+a + by = (z+yy, 
 
 ==.r 4 +4.r 3 ?/-|-6.r 2 j/ 2 -j-4;r7/ 3 - r -7/ 4 , or putting for y its value, 
 =j-»+4.r'(« + fc)4- Gj*(a + b)'+ 4x(a + b) ; + {a+by. 
 Expanding (a-\-b)-, (a-\-by, (a-f-i)S by the binomial theorem, and per- 
 forming the multiplications indicated, we shall arrive at the expansion of 
 (r+a + by. 
 
 It is manifest that we may apply a similar process to any polynomial. 
 The following is a demonstration of a general formula for the 
 
 POWER OF A POLYNOMIAL. 
 
 In the expression 
 
 (a-f-ft+c+rf....)" 
 
 make x=zb-\-c-\-d . . . the above power will be equal to (a-|-.r) m , and by tn*» 
 binomial theorem the term which contains a n in the development of this may 
 be written 
 
 1.2.3.4 tnx a n .r m -° 
 
 1.2.3...nXl.2.3...(/«— n)*t ^ 
 
 Making y=c-\-d... wo have a ,m_ "=(/»-j-y) m " ; , and developing this last powei 
 the term containing />"' may bo put under the form 
 
 * If there be decimal) in the given number, ciphers mual be annexed, if necessary, to 
 make exact periods ofd< id a principle similar to dial explained in (Art. 90). 
 
 It" the index of the rimt to be extracted I"' composed of factors, it can be extracted by 
 means of the successive roots, the degrees of which an i ■ ■•<! by these factors. For 
 
 if the m "y^(/"'"P be required, m have i/a 0U| P=a"P, T/a°P=aP, and y aP=a. 
 
 The I bo extract roots of numbers of a de ree higher than the square is by racaua 
 
 of lo rarithms. 
 
 t This may be obtained from the Ordinary form of the genera] term of the hinoinial 
 formula 
 
 vt{m— 1) (m — ;i-f-l>" n r m - n 
 
 1.2.3....n' ' 
 
 bv inultii'lyin • both numerator ami denominator by 1 . „• . :i ... (m — n). 
 
HIGHER ROOTS OF POLYNOMIALS. 109 
 
 1.2.3.4 {m — n) X b'"y m - n ~ a/ 
 
 1.2.3...«'Xl.2.3 {m—n—n')' 
 
 It is evident that if this quantity be put in the place of a: m ~ n in the ex- 
 pression [a], the result will represent the assemblage of the terms which 
 contain a n b n/ in the power of the given polynomial. This result, after can 
 celing common factors, will bo 
 
 1.2.3.4....mx a n b a ' y m - n -'" 
 
 1.2.3...»Xl-2.3...n'Xl.2.3...(m- n — n')' *■ * 
 Making z=zd-\- ... wo shall have y m - a - a '==(c-\-z) m - n - D ', and the term con- 
 taining c L " will be 
 
 1.2.3.. .(m — n—n') X c n "z m - n - n '-"" 
 1 . 2 . 3 . . . . n" X 1 • 2 . 3 . . . . (m — n — n' — n") ' 
 substituting this expression for y m - a ~ a/ in [b], we have 
 
 1 ■ 2 . 3 . . ■ m X a°b a 'c a "z m - a - n '- r "' 
 
 1.2.3... «X 1-2.3... «' XI -2.3... n" XI -2.3... (m— n— n' — n")' 
 It is evident now, without carrying the reasoning farther, that if V be the 
 general term of the development of 
 
 ( a _|_6_j_c-f-i...) m , 
 this term may be represented thus, 
 
 1.2.3.4 mXa n b n 'c n " ... 
 
 = 1.2.3...wX 1 -2.3... n'X 1.2.3... »"X •• 
 n, n', n" . . . being any positive whole numbers at pleasure, subjected only to 
 the condition that their sum shall be equal to m. So that all the terms of the re- 
 quired development may bo obtained by giving in this formula to n, n', n" . . . 
 all the entire positive values which satisfy the condition 
 
 n-\-n'-\-n" . . . ,=m. 
 When one of these numbers is made zero, V takes an illusory form. If, for 
 example, n =0, the series 1 . 2 . 3 . . . n placed in the denominator is nonsensical, 
 because factors increasing from one will never present us with a factor zero. 
 To. relieve this difficulty, let us recur to the general term [a] in the development 
 
 x m 
 of (a-\-x) m , and observe that the hypothesis n=0 reduces it to - — — — -. 
 
 But the hypothesis «=0 ought to give in this development the term which 
 does not contain a, and this term is x m . Then, in order that this term shall be 
 deduced from the formula [a], it is sufficient to consider the series 1 .2.3...n 
 as equivalent to 1 in this particular case of n=0. The same observation 
 should be extended to the other series of factors contained in the denominator 
 of V, and then V will give, without any exception, all the terms of the power 
 of the polynomial a-{-b-\-c-\-, &c. 
 
 TO EXTRACT THE mP> ROOT OF A POLYNOMIAL. 
 
 The problem is, having given a polynomial, P, which is the in'' 1 power oj 
 another polynomial, p, to find the latter. 
 
 Let us consider the two polynomials as arranged according to the decreas- 
 ing exponents of some letter, x, and call a,b,c, the unknown terms of the 
 
 root p. They must be such that, in raising a-{-b-\-c. . . to the power m, we 
 obtain all the terms which compose P. But if we imagine that we havo 
 formed this power by successive multi plications, it is clear that, in the result, 
 
NO ALGEBRA. 
 
 the term in which x has the highest exponent is the m* power of a ; then wt 
 II know the first U rm of the root sought, p, by extracting the m 1 * root of th» 
 first term of the given nial, P. 
 
 The first term of the rool being found, it will be easy to obtain the second; 
 but I prefer to show at once how, when we know several successive terms of 
 the root setting out from the first, we can determine the term which comes 
 immediately after. 
 
 Let u represent the .sum of tho known terms, and v that of the unknown, 
 then P = (u-}-r)'", or, developing, 
 
 P =u m -\-mu m - 1 v+ku n '-*u' i +k'u m - : <v 3 -\-, &c. 
 
 I have not exhibited the composition of the coefficients k, k' .., this not 
 being necessary, as will appear. From this equality, by subtracting u m from 
 both the equals, we obtain 
 
 ~P—u m =mu m - 1 v-\-Jcu m ->- + /.- V ,: HP+, Arc. 
 
 The first of these equals, P— u m , is a quantity which we can calculate by 
 forming the m th power of the known quantity u, and subtracting it from tho 
 polynomial P. The second is a. sum of products, by means of which we can 
 easily assign the composition of the first term of the remainder P — ?<"', and, 
 consequently, discover the first term of tho unknown part, v. 
 
 First, if we develop « m_1 , it is clear, by the rules of multiplication alone, 
 that the first term of the development, that is, the one which contains r, with 
 the highest exponent, will be a" 1-1 ; then, if we cull/ the fust term of r, the 
 first term of the product mu m ~ l v will be ma m ~f. By a similar course of rea- 
 soning, we perceive that the first terms in the developments of the other prod- 
 ucts will bo respectively ka m --f 2 , k'a m - 3 f 3 , These terms, abstraction 
 
 being made of the coefficients,which have no influence upon the degree of x, 
 can be deduced from the term ma m " l f, by suppressing in it one , or more fac- 
 tors equal to a, and replacing them by as many factors equal to f. But/being 
 of a degree inferior to a with respect to x, these changes can give only terms 
 of a degree inferior to ma m ~f.* Then, after having subtracted from the given 
 polynomial P the 7/i lh power of the part u of the root alrea !y found, the first 
 
 term of the remainder is equal to the product of m times . - power m 1 of 
 
 the first term a of the root by tho first of those terms which remain still to be 
 found. Therefore, dividing the first term of the remainder by m times the 
 power m — 1 of the first term of the root, the quotient will be a new term of 
 this root. This conclusion furnishes the means of discovering successively all 
 the terms of the root as soon as the fust is known. To Jiavc the second term, 
 b, subtract from tin P the m A power of t term of the 
 
 root, then divide the first term <f tlie remainder by ma™ -1 ; to have the thira 
 term, c, of the root, subtract from P the m ; < r of a-f-h. then divide the first 
 item of this remainder by ma'" ', ami so on. 
 
 If in any part of the process, the remainder being arranged according to the 
 powers of x, its first term is not divisible by m times the m — 1 power of | 
 iir.>t term of the root, tin- given polj Qomial will no! have at, exact root of the 
 degree m. , 
 
 Wo may arrange according to the ascending powers of a letter, r, as was 
 
 ' Thai, for example, if a contain a£, and/ conb ■! m=I,0, then a" ; /wi!l r<.. 
 
 J \\ ill <-< nil :i : ti .,' '', Iltnl Sm 
 
FRACTIONAL POWERS OF BINOMIALS. Ill 
 
 remarked at (Art. 80, III.), when treating of the square root, and the above 
 observation will undergo the same modification as there stated. 
 
 It would be superfluous to speak of the case where the letter of arrangement, 
 x, enters, with the same exponent, into several terms. The method of proceed- 
 ing in such a case has been already sufficiently indicated in previous articles. 
 
 EXAMPLES. 
 
 (1) Extract the 5th root of 32.r 5 — 80.r»+80.r 3 — 40.r 3 +10.r— 1. 
 
 (2) Extract the 6th root of 729— 291G.r 2 +48G0.r 4 — 4320x 6 -|-2160.t 8 — 576x"» 
 
 -f-64x 12 . Ans. 3— 2Z 2 . 
 
 tf) Extract the fifth root of ar 20 +15.r- ,6 -5.r- 14 +90.r- 12 — 60x- 10 4-282r-» 
 
 — 252.r- 6 + 505.r- 4 — 496.r- 2 + 495 — 4G5r 2 
 -f 275a.- 4 — 80x 6 -f- 15Z 8 — x 10 . 
 
 Ans. £- 4 -|-3— x 2 . 
 114. In the observations made upon the expansion of (x-\-a) n , we have sup- 
 posed n to be a positive integer. The binomial theorem, however, is applica- 
 ble, whatever may be the nature of the quantity n, •whether it be positive or 
 negative, integral or fractional.* When n is a positive integer, the series con 
 sists of n-\-l terms ; in every other case the series never terminates, and the 
 development of (x-\-a) a constitutes what is called an infinite series. 
 
 Before proceeding to consider this extension of the theorem, we may re- 
 mark, that in all our reasonings with regard to a quantity such as (x-\-a)'\ we 
 may confine our attention to the more simple form (l-fa) n , to which the 
 former may always be reduced. For, 
 
 (x+a) =s(l+j) 
 
 ... (x+ay=\x(l+l)Y 
 
 =z n (l-f--) , or x n (l+w) n , if we put -=u 
 
 i a n{n— 1) a? n(n — l)(n — 2) a 3 
 =x^l+n.-+- 1 - ir .- ; + n¥7 3 .- 3 + 
 
 n (n-l)(n-2)(n-3) a 4 ) 
 1.2.3.4 T 4 +' *°" S f 
 
 Suppose n=-, where r and s are any whole numbers whatever, 
 
 I r 
 
 Then (x-\-a.y becomes (x-\-a) s , and substituting - for n in the series, 
 
 - -/ a \- * 
 
 (x+a)*=:3?\l+-y 
 
 a s\s~ ) a" s\s )\s ~) a 3 
 
 r 
 s 
 
 =*•(! + -•-+ 1>2 - a ,-r L2 .3 •& 
 
 r 
 
 (^XHG- 3 )*. 
 
 + T2T3T4 ?+' &0 - 
 
 * A perfectly rigorous demonstration of the binomial theorem for any exponent what- 
 ever, integral or fractional, positive or negative, will be found towards the close of this 
 treatise. 
 
 t This expansicu may be obtained by substituting, in the general form (Art. 110), 1 foi 
 
 <, and — for a. 
 
 x 
 
112 ALGEBRA. 
 
 Or, reduced, 
 
 ! r a r(r — s) a" r(r — s)(rv-2s) a* 
 W =r ^ + «*x+l.ii.s* •P+ 1.2.3.S 3 x» 
 
 r(r— *)(r-2s)(r-3*) B» ? 
 
 + 1.2.3.4.* •r 4+ ' S 
 
 The binomial theorem, under this form, is extensively employed in analysis 
 for developing algebraic expressions in series. 
 
 EXAMPLE I. 
 
 Expand y/x-\-a in a series. 
 
 V*fa=(*W 
 
 =#+;)*• 
 
 Here r=l, s=2. 
 
 j.j.l) 1 a 2\2 V a 2 2\2 vfe V ^ 
 ( 1 + 2'i"'" 1.2 V+ 1.2.3 V 
 
 i(HG-»)(H «. i 
 
 + 1.2.3.4 'a- 4 "*" ) 
 
 ( 11113 
 
 A 5 la 2 X ~~2 a 2 2 X ~~2 X— 2 a 5 
 ~ X ' I 1_ *"2 ' x+ 1 . 2 ' •r 2 "' 1.2.3 ' *• 
 
 113 5 *j 
 
 4- . — . — 4- ( 
 
 ^ 1.2.3.4 x*^ ) 
 
 _ ±< 1 a 1 a 2 1,3 a* 1.3.5 
 
 — *" } 1 + 2"x — 1.2.4 '^"^1.2. 3.8' ? _ 1.2.3.4. Hi 
 
 a* } 
 
 «+ 1 
 
 which last may be derived at once from [a], and put under the form 
 
 l( la la- 1.3 a 3 1.3.5 a* 
 
 X ~ I + 2 " x~J7i ' x~-"^2 .4.6" Z 3 " - 2 . 4 . G . 8 ' x* 
 
 1.3.5.7 a* i 
 
 + 2.4.6.8.10 " ?~ * &C J 
 
 wherb the law of the series is evident. 
 
 EXAMPLE II. 
 
 Expand -/a 2 — tz 2 c- in a series. 
 
 i 
 
 y/ir— a-c- = {a-— a-c : )- 
 
 —ail—e") 1 Here, r=l, 6=2, -=— c* 
 
 x 
 
 $ i r>(A— sG-OG- 2 ) 
 =« Jl_-. ,,+_ _ . c , — I72T3— • <* 
 
 12 3' "' • • • • \ 
 
 < 1 1 1.'. 1.3.5 / 
 
 =" J 1 -,'-,..' '-■■.,iTaX6l e, -' lVr 'i 
 
FRACTIONAL POWERS OP BTNOMIALS. 113 
 
 EXAMPLE III. 
 
 Expand — in a series. 
 
 m 
 
 =mb (l + -j = Herer=l, S =-2,-= rs . 
 
 = - \ If! ~~g("~ 2~0 * 
 
 b I X — 2 ' b* "•" 1.2 " & 4 
 
 "aV - ^ -1 / \~2~ 2 ) £» 
 
 + "1.2.3 ' 6 8 
 
 -K-1- 1 ) (-4- g ) (-I- 3 ) n\ ( 
 
 + 17273^ • y m 
 
 m ) 1 — ~^ X ~^ ff ~2 X ~2 X ~~2 
 
 = 6'( :1— 2'F+ TT2 "6*"*"" 1.2.3 
 3 5 
 
 2 X 2 X 2 X 2 cl6 
 
 15+ . &c. 
 
 ' 6 G ~ 1.2.3.4 ' i 8 
 
 wi c 1 c 4 ^jj c» 1. 3.5 c>~ ? 
 
 = 6"| 1_ 2T 2 + 2T4'i*~27T76'"P + ' C S 
 
 which last expression might be derived immediately from formula [a]. The 
 same remark will apply in the following examples. 
 
 EXAMPLE IV. 
 
 n 
 
 Expand ~ — in a series. 
 1 y/b^—c^e 3 
 
 n 1 
 
 :n(i 2 — cV) 2 
 
 c 2 e» 
 
 =nb (l— jr) 2 Here r= -l,s=2, -= — 
 
 nc 1 c 2 e 2 "^V - ^" 1 / / c 2 e 2 y 
 
 = 6 \ 1 + 2 * ~6^ + 1.2 • ' \~b*~) 
 
 " 2 \ 2 " l / \ ' 2~ 2 ) /_£l£lV 
 
 1.2.3 
 
 -§(-4- i )(-5- a )(-5- 3 W : > 
 
 c 2 e 2 1.3 c 4 e 4 1.3.5 c 6 e 6 
 
 _n( 1 c 8 e 3 1 .3 c*e* 
 
 2.4.6 * 6 6 
 1 . 3 . 5 . 7 c«e 8 ) 
 
 + 2.4.6.8''F+' &C ' * 'I 
 
 II 
 
114 ALGEBRA. 
 
 EXAMPLE V. 
 
 Expand —===. in a series. 
 r V(w 3 +" 5 ) 3 
 
 p + q 3 
 
 ■■(p+q)(m 3 +ns) * 
 
 a 
 
 =7n~*(p + q)(l+—) Here,r=— 3, «=4, 
 
 a n" 
 
 x ?ft 3 
 3 
 
 3/3 \ 
 
 i£±ll< _3 n* ~4V~4~V /_n_»y 
 
 : i»* j * — 4 ' m 3 + 1.2 ' \m 3 ) 
 
 4\ 4 /\ 4 / /n\ 3 > 
 
 + 1.2.3 * W) S 
 
 (.P + g) , 3 n s 3.7 ^io 3.7.11 
 
 m? } * ~" 4 ' wi 3 "*" 1 . 2 . 4 2 m^" - T" 
 
 2 . 3 . 4 3 
 
 n 18 3.7.11.15 n-° > 
 
 7n9"^1.2.3.4.4 4 m u_ ' ' \ 
 
 EXAMPLE VI. 
 
 1 1 C 2x 3x* 4I 3 . > 
 
 EXAMPLE VII. 
 
 , „ n £ # < -, 3 a* 3 x* 5 x 5 7 a* - > 
 \ ) \ 2 2 c* 2 5 c* 2 7 c 6 2 9 c 8 S 
 
 EXAMPLE VIII. 
 
 —-^ J_(, , 3 i 3.13 x» 3.13.23 £ 
 
 a + 10 1 'l.2.aV 10 3 1 . 2 . 3 . a 1 
 ,3.13.23.33 x* , . ) 
 
 T 10* 1.2. 3. 4. a* 1 ( 
 
 EXAMPLE IX. 
 
 1 
 
 ^—*r T '-3i 1+ io 
 
 x , 6x 3 6.11.x 3 , 6.11.16.x* . 
 i=l : -+ — t <^c. 
 
 (1+x)* 5 5.10 5.10.15 5.10.15.20 
 
 EXAMPLE X. 
 
 7 2618 x 30 
 
 The eleventh terra of the series for (a 3 — .rV is — — . — -. 
 
 ' 4782969 a 33 
 
 115. The binomial theorem is also employed to determine approximate 
 values of t ho roots of numbers. 
 In the formula 
 
 (x+«)"=x n (14-n \-— '- — - - £ «-H )• 
 
 v ~ ' v ~ x 1 . 2 «• 1.2.3 x 3 ' 
 
APPROXIMATE ROOTS OF NUMBERS. 
 
 115 
 
 Let us put «=-, the expression becomes (x+a)r or %/ z+a, and we have 
 
 S/x-\-a= */x(l+-. 
 
 ^ * v ^ r x ' i m2 
 
 1 a r\r / a? r\r / \r / a J 
 
 1.2.3 
 
 x 3 
 
 + 
 
 r/ . 1 a 1 r— 1 a* 1 r— 1 2r— 1 a 3 
 
 r x 
 
 2r 
 
 r yr 
 
 3r x 3 
 
 •) 
 
 If we wished to form a new term, it would manifestly bo obtained by mul- 
 
 3r 1 _ a 
 
 tiplying the fourth by 
 
 and — , then changing the sign, and so on for the 
 
 rest, the terms after the first being alternately positive and negative. 
 
 This being premised, let it be required to extract the cube root of 31. The 
 greatest cube contained in 31 is 27 ; in the above formula let us make r=3 
 r=27, a =4, and we shall then have 
 
 V31= 3/27 + 4 
 
 1 4 1 
 
 -3(l + 3 • ^-3 
 
 4 16 
 
 T 27 2187 ^ 
 
 1 
 
 3 ' 729 
 
 320 
 
 16 1 
 + 
 
 3 
 — , &c. 
 
 1 
 
 3 
 
 5 
 9 
 
 64 
 
 19683 
 
 -, &c.) 
 
 531441 
 The succeeding term will be found by multiplying 
 
 320 
 
 by 
 
 3r— 1 a 
 
 531441 
 
 2 4 2560 
 
 -.— , and then changing the sign, which will give us — , 
 
 In like manner, we shall find the next term by multiplying 
 
 4r 
 
 -, OF 
 
 X 
 
 2560 
 
 43046721 
 
 by 
 
 4r— 1 a . .„ , „ , 2560 11 4 
 
 -57" -x' lt Wdl ' therefore » be ^^x-x-:^ 
 
 112640 
 
 r, and so on 
 
 15~27~ 17433922005' 
 for any number of terms. 
 
 Let us, however, confine our attention to the first five terms of the series, 
 and reduce them to decimals ; we shall have, for the sum of the additive terms, 
 
 f 3=3.00000 ^ 
 
 { 
 
 - = 0.14815 
 
 320 
 
 = 0.00060 
 
 ^ 531441 
 
 And for the sum of the subtractive tenns, 
 
 16 
 
 \ =3.14875. 
 
 2187" 
 2560 
 
 -0 00731 
 ■ 3.00006 
 
 :—0.00737. 
 
 43046721" 
 Hence 
 
 >/3T=3. 14138 
 
 a result which we shall proceed to show is within 0.00001 of the trum. 
 
 116. When the expression for a number is expanded in a series of terms. 
 the numerical values of which go on decreasing continually, we easily perceiv** 
 
116 ALGEBRA. 
 
 that the greater the number of terms which we take, the more nearh/ shall wo 
 approach to the real value of the proposed expression. Such a series ia < . 
 converging. If we suppose the terms of the series alternately positive and 
 negative, we can, upon stopping at any particular term, determine preo 
 the degree of approximation at which we have arrived. 
 
 Let there be a series a — b-\-c — d-\-e — f-\-g — h-\-k — l-\-m comj» 
 
 of an indefinite number of terms, in which we suppose that the quantities a, 
 b, c, d go on diminishing in succession, and let us designate by N the number 
 represented by this series, we shall prove that the numerical value of N lies 
 between any two consecutive sums of any number of the terms of the above series. 
 
 For let us take any two consecutive sums, 
 
 a — b-\-c — d-\-e—f and a — b-\-c — d-\-e—f-{-g. 
 
 Upon considering the first of these, we perceive that the terms which fol- 
 low —/are +(& — ^)4"(^ — OH ; but smce tne series is a decreasing 
 
 one, the positive differences g — h, k — I, &c, are all positive numbers; hence 
 it follows that, in order to obtain the complete value of N, we must add to the 
 turn a — b-\-c — J-|-e — /some positive number. Hence 
 
 a — b + c — d+c — /<N. 
 
 With regard to the second sum, the terms which follow +g are — (h — k), 
 
 — (I — m), ; but the partial differences, h — k, I — m, <5cc, are positive; 
 
 hence — (/( — /.), — {I — m) , are all negative, and, t he re fore, in order to 
 
 obtain the complete value of N, we must subtract some positive number from 
 the sum a — b+c — d+e—f+g. Hence 
 
 a—b+c— d+e—f+g>N, 
 and it has been shown that 
 
 \ a — b + c—d+e—f <N; 
 
 therefore N lies between these two sums. 
 
 From this it follows that, since g is the numerical value of the difference 
 of these two sums, the error committed ivhcn we assume a certain number of 
 terms a — b-\-c — d-\-e—f for the value of N is numerically less than the term 
 which immediately follows that at which we stopped. 
 
 In the preceding example, all the terms after the first being alternately posi- 
 tive and negative, we may conclude that the numerical value of the first five 
 terms 
 
 4 16 320 \'">60 
 
 3 + 27 _ 2187 "^"531441 ~43046721 
 differs from tho true value of V^l u y a quantity less than the value of the 
 
 112640 
 
 sixth term, which was found to bo equal to ; but this fraction is 
 
 17433922005 
 
 by mere inspection less than , therefore, when we assume that 
 
 V31=3. 14138, the result is within 0.00001 of the truth. 
 
 117. From what has been said above it will be seen that, in order to obtain 
 an approximate value of tho n"' root of any number N by the method of series, 
 we may make use of the following 
 
 rum:. 
 
 Resolve the given number N into two parts of the form p n -f- q. whrrc p" is the 
 
 i 
 highest u"' power contained in N, and in t)ir development of (x-f-a)" make 
 
DEGREE OF APPROXIMATION OF SERIES. 117 
 
 x=p n , a=q. The number of terms to be taken in the resulting series wil. 
 
 depend upon the degree of accuracy required, and can be determined by the 
 
 principle just explained. Convert all the terms of which account is taken into 
 
 decimals, and then effect the reduction between the additive and subtractive 
 
 terms. 
 
 q 
 This method can not be employed with advantage except when — is a small 
 
 fraction ; for unless this be the case, the terms of the series will not diminish 
 
 with sufficient rapidity, and it will be necessary to take account of a great 
 
 number of terms in order to arrive at a near approximation. 
 
 It may happen thatp" is <^q ; we must then modify the above process, for 
 
 p a a a . . 
 
 then — or — is greater than unity, and therefore all the powers of - will m 
 
 crease in numerical value as the degree of the power increases. 
 
 Suppose, for example, that the cube root of 56 is sought, 27 being the 
 greatest cube contained in 56, wo shall have 
 
 a 29 
 
 x=27, a = 29 and .-. -=-, 
 
 nnd the terms of the series will go on increasing instead of diminishing (we do 
 not speak of the coefficients, which are fractions differing but little from unity). 
 
 8 1 „ . 
 
 But we may resolve 56 into 64 — 8, or 4 3 — 8 ; but — , or -, is a small fraction. 
 
 b4 8 
 
 On the other hand, if we substitute — a for a in the expression for V r + a > 
 
 we have 
 
 r 1 a 1 Ti — 1 a 2 1 n— 1 In— 1 a 3 
 
 8/ ' x — 6£:=£n(l . . 
 
 n x n 2n x' 2 n 2n 3n x 3 
 If we put x=64, a=8, we shall obtain a series of terms which will de- 
 crease with great rapidity. 
 
 Here all the terms, with the exception of the first, are negative, and we can 
 not apply to this scries the criterion established in (Art. 116) for fixing the de- 
 gree of approximation. But we shall approach very nearly to the required 
 degree of approximation if we take into account such a number of terms that 
 the first which we neglect shall be less, by one tenth, for example, than the 
 decimal place to which we wish to limit the approximation. 
 The student may take the following examples as exercises : 
 
 (1) V39 =V32 +7 =2.0607 true to 0.0001. 
 
 (2) 3/65 = V64 +1 =4.02073 . . . true to 0.00001. 
 
 (3) {/260= V 256 + 4 =4.01553 ... true to 0.00001. 
 
 (4) yil)8 = V128— 20=1.95204 . . . true to 0.00001. 
 
 118. Roots of imaginary expressions of the form a±6 -J — 1 are extracted 
 
 by putting the expression under the form (a ±6 V — l) n , and developing by the 
 binomial theorem ; a series of terms will thus be obtained, which may be put 
 under the form A+B yf — 1, A representing the algebraic sum of the rational 
 terms, and B the algebraic sum of the coefficients of V — 1- Algebra fur- 
 nishes no other general method for this transformation, but when n is a power 
 of 2, it can be effected without the aid of series 
 
lid ALGEBRA. 
 
 Let us consider, first, the two radicals yJa-\-b y/ — 1 and \! a — by/ -1. 
 Placing 
 
 [1] y]a+by/^-L+ya — by/~]=.r 
 
 [2] y]a+byf— I— Va— 6/3i=y, 
 
 and squaring both, there results 
 
 2a+2-/a- ! -H 2 =x s 
 2a— 2y/a^fb 1 =y 2 . 
 
 Whatever may be the sign of a, the value of # x : is positive, but that of y* is 
 negative. From these equalities we derive 
 
 [3] .T=V2a+2 y/at+V, y=\y] —Qa+2y/a*+bV yf^l. 
 But the equalities [11 and [2] givo 
 
 Then, finally, putting for a: and ?/ the values [3], we shall have 
 [4] Ja+by/~l= lj2a + 2y/^f& 
 
 + ±J-2a + 2 Va 2 +i 2 V^ 
 
 [5] Ja-b y/~=l= ^2a + 2V^4^ 
 
 --l-2a + 2y/a' 1 +b"' /^T. 
 Now, if we consider the radical expressions 
 
 \a±b^~l,y]a±:by/~—i, y a±b y/~—i, &c, 
 we observe that the extraction of a single root which is some power of two, 
 can be replaced by successive extractions of the square root; consequently, 
 the repetition of the formulas [4] and [5] will always reduce the above ex- 
 pressions to expressions of the form AJLB y/ — 1. 
 
 Remark. — In each of these formulas tho first member, by reason of th 
 radicals which it contains, may have four different values, and the same 
 true of the second member. In both, tho lour values of the first member arc 
 the same, and this is tho case evidently with tho second member; so that 
 tho two formulas make really but one. They present no difference except 
 when wo use them simultaneously in the same algebraical calculation, because 
 then we ought to regard tho terms into which y/ — 1 enters us affected with 
 contrary signs. But then it is necessary to remark besides, that, by tho very 
 manner in which wo have arrived at these formulas, yf a--\-l> : in them re pre 
 
 sents the product of yJa-{-b y/ — 1 \!a — b yf — I ; consequently, the del 
 inutions of these two radicals ought always t<> be snppo 1 I OCMfted in Ml 
 a manner that their product should have the rigs which is given to y/a : -\-u~ 
 in the second member. Without attention to this tbo formulas might lead to 
 false results. 
 
RATIOS AND PROPORTION. 119 
 
 Another remark of importance may be added here. 
 
 The methods of proceeding in certain operations upon imaginary exiressions, 
 exhibited at (Art. 6G), were suited to the restrictions which in ordinary cases 
 would be understood as pertaining to the radical sign. If, however, this sign 
 have its most general signification, it must be used in its ambiguous sense, 
 that is, as having JL before it. Then \/ — flX V — a would have a more ex- 
 tended sense than simply the square of ■/ — a. It would have, in fact, four 
 values, 
 -j_ y/ — a x + V — a, — V —a x + V — a, + V — «X- V ~ a> 
 
 — V — a x — V— «> 
 
 or 
 
 —a, +c, +a, —a 
 
 These four, in fact, amount to but two, -\-a and — a, which are the values 
 obtained by the ordinary rale of multiplication, -/ — ax V — a= • v /a2 =i a - 
 
 If the quantities under the radical are different, the reasoning will be a little 
 varied. Let the product be required of 
 
 The first of these factors •/ — a may be put under the form a' V — 1, and 
 the second under the form b' V — 1. The product will then be expressed by 
 
 a'b' t/~— 1 X V— 1- 
 But after what has just been said, if there be no restriction in the meaning 
 of the sign -/ , we have -/ — lxV — l = rtl. Hence 
 
 a'6'/^lX V~^-L = ±a'b'. 
 But since the square of a'b' is a'-b' 2 , or ab, we have a'b'= y/ab, and, there 
 
 fore, 
 
 V — aX V — 6=i V a b, 
 the result winch we should obtain by the ordinary rule for the multiplies 
 tion of radicals. We thus perceive that this rule gives us the true product 
 in its most general form wh,en there is no restriction in the sense of the radi- 
 cal sign. 
 
 RATIOS AND PROPORTION. 
 
 119. Numbers may be compared in two ways. 
 
 When it is required to determine by how much one number is greater or 
 less than another, the answer to this question consists in stating the difference 
 between these two numbers. This difference is called the Arithmetical Ratio 
 of the two numbers. Thus, the arithmetical ratio of 9 to 7 is 9— 7, or 2, and 
 if a, b designate two numbers, their arithmetical ratio is represented by a — b. 
 
 When it is required to determine how many times one number contains, of 
 is contained in another, the answer to this question consists in stating the 
 quotient which arises from dividing one of these numbers by the other. This 
 quotient is called the Geometrical Ratio of the two numbers. The term 
 Ratio, when used without any qualification, is always understood to signify a 
 geometrical ratio, and we shall, at present, confine our attention to ratios of 
 this description. 
 
120 ALGEBRA. 
 
 120. By the ratio of two numbers, then, we mean the quotient which arises 
 
 from dividing one of these numbers by the other. Thus, the ratio of 12 to 4 
 
 12 5 1 
 
 is represented by — or 3, the ratio of 5 to 2 is - or 2.5, the ratio of 1 to 3 is - 
 
 or .333 . . . We here perceive that the value of a ratio can not always be ex- 
 pressed exactly, except in the form of a vulgar fraction, but that, by taking a 
 sufficient number of terms of the decimal, we can approach as nearly as we 
 please to the true value. 
 
 121. If a, b designate two numbers, the ratio of a to b is the quotient 
 
 arising from dividing a by b, and will be represented by writing them a : b, or r. 
 
 122. A ratio being thus expressed, the first term, or a, is called the ante- 
 cedent of the ratio; the last term, or b, is called the consequent of the ratio. 
 
 123. It appears, therefore, that, iu arithmetic and algebra, the theory of 
 ratios becomes identified with the theory of fractions, and a ratio may be de- 
 fined as a fraction whose numerator is the antecedent, and whose denominator 
 is the consequent of the ratio. 
 
 124. When the antecedent of a ratio is greater than the consequent, the 
 ratio is called a ratio of greater inequality ; when the antecedent is less than 
 the consequent, it is called a ratio of less inequality ; and when the antecedent 
 
 12 
 
 and consequent are equal, it is called a ratio of equality. Thus, — is a ratio 
 
 12 3 
 
 of greater inequality, —rj is a ratio of less inequality, - or 1 is a ratio of 
 
 equality. It is manifest that a ratio of equality may always be represented by 
 unity. 
 
 125. When the antecedents of two or more ratios are multiplied together 
 to form a new antecedent, and their consequents multiplied together to form 
 a new consequent, the several ratios are said to be compounded, and the re- 
 sulting ratio is called the sum of the compounding ratios. Thus, the ratio t 
 
 c 
 is compounded with the ratio -, by multiplying the antecedents a, c for a new 
 
 antecedent, and the consequents b, d for a new consequent, and the resulting 
 
 ac a c 
 
 ratio t-? is called the sum of the ratios r and -.. 
 bd b d 
 
 m p r t 
 In like manner, the ratios -, -, -, — are compounded by multiplying all 
 
 the antecedents together for a new antecedent, and all the consequents for a 
 
 mprt 
 new consequent, and the resulting ratio, — — , is called the sum of the ratios 
 1 ° nqsw 
 
 mprt 
 
 n' 7' s' w' 
 
 12G. When a ratio is compounded with itself tho resulting ratio is called the 
 
 d indicate ratio, or double ratio of the primitive. Thus, if we compound the 
 
 a a fl ' i a 
 
 ratio j with j, tho resulting ratio, vs, is called tho duplicate ratio of -r- 
 
 a 3 a 
 
 Similarly tj is called tho triplicate ratio, or triple ratio of r- 
 
RATIOS AND PROPORTION. 121 
 
 a n . a 
 
 And, generally, j~ is called the sum of the n ratios y- 
 
 1 
 
 a 2 
 \ccording to the same principle, the ratio — is called the subduplicale ratio, 
 
 b 2 
 
 X 1 1 
 
 f a a 2 . a 2 a 2 a 
 
 -■' half ratio of y-; for the duplicate ratio of— is — X - f=r» 
 
 b b 2 b 2 b 2 
 
 i 
 a 3 
 So, also, the ratio — j is called the sublriplicate ratio, or one third of the ratio 
 
 b? 
 
 I I J. A 
 
 e a -r, . , . n a 3 a* a J a 3 a 
 
 ot r« r or the triple ratio of — is — X — X "T=T* 
 
 b 3 V s h* b 3 
 
 i 
 
 a" a 
 
 And, in general, — is called one n th of the ratio r ; for n times the ratio 
 
 b° 
 liii , 
 
 a a . a n a n a» a 
 
 I's-X-X-X-- ton terms =y-. 
 
 b a b" o" 6" 
 
 3 
 
 Note. — The ratio — g is called the sesquiplicate ratio of y-, for it is com- 
 
 I 1 
 
 a 2 a a 3 
 psunded of the simple and subduplicato ratio ; thus, — Xr=— • 
 
 b 2 b b 2 
 
 127. If the terms of a ratio be both multiplied, or both divided, by the same 
 
 quantity, the value of the ratio remains unchanged. 
 
 a 
 The ratio of a to 6 is represented by the fraction 7 ; and since the value of 
 
 a fraction is not changed, if we multiply, or divide, both numerator and de- 
 nominatoi by the same quantity, the truth of the proposition is evident. Thus, 
 
 a 
 
 a_ma__n or a:o=roa:m&=-:-. 
 
 b~mb~b n n 
 
 n 
 
 128. Ratios are compared with each other by reducing the fractions, by 
 which they are represented, to a common denominator. 
 
 [f wo wish to ascertain whether the ratio of 2 to 7 is greater or less than 
 
 2 3 
 
 that of 3 to 8, since these ratios are represented by the fractions - and -, 
 
 / 8 
 
 which are equivalent to — and — ; and since the latter of these is greater than 
 
 the former, it appears that the ratio of 2 to 7 is less than the ratio of 3 to 8. 
 
 129. A ratio of greater inequality is diminished, and a ratio of a less inequal- 
 ity is increased, by adding the same quantity to both terms. 
 
122 ALGEBRA. 
 
 Let j represent nny ratio, and let x be added to each of its terms. The 
 
 two ratios will then bo 
 
 a a-\-x 
 
 V b+x' 
 
 which, reduced to a common denominator, become 
 
 ab-\-ax ab-ifbx 
 
 b(b+xy b{b+x)' 
 
 a 
 
 If a>6, i. e., if t be a ratio of greater inequality, then 
 
 ab-\-ax ab-\-bx 
 b{b+xyb{b+x) ; 
 
 md ••. r is diminished by the addition of the same quantity to each of its term*. 
 
 a 
 
 Again, if a<b, i. c, if t be a ratio of less inequality, then 
 
 ab-\-ax ab-\-bx 
 b(b+x)<b{b+x)' 
 
 and ••. j is increased by the addition of the same quantity to each of its terms. 
 
 130. If there be any number of ratios in which the consequent of the first ratio 
 is the antecedent of the second, and the consequent of the second the antecedent 
 of the third, and so on, the sum of any number of said ratios is the ratio of the 
 first antecedent to the last consequent. 
 Let the proposed ratios be 
 
 a b c d e 
 F"? 3' ?/•"""• 
 Then, by (Art. 125), their sum is 
 
 a b c d e 
 
 b x ! x d x l x f"~' 
 or 
 
 abede 
 
 bedef----' 
 
 a 
 i. e.,j. 
 
 131. Proportion is an equality of ratios. 
 
 Thus, if a, b, c, d be four quantities, such that a, when divided by b, gives the 
 same quotient as c when divided by d, then a, b, c, d are said to be in propor- 
 tion, or to bo proportionals ; the numbers 20, 5, 36, 9 are proportionals, for 
 
 20 . 36 
 
 -=4,and-=4. 
 
 When four quantities aro proportionals, it is usually enunciated by saying 
 that the first is to the second as the third is to the fourth. Thus, if a, 6, c, d uro 
 proportionals, wo say that a is to b as c is to d, and this is expressed lyr wri- 
 ting them 
 
 a:b::c:d, or a:b=c:d, 
 or as fractions, 
 
 a c 
 ~b~d' 
 
RATIOS AND PROPORTION. 123 
 
 The first or second form of notation is usually employed in geometry, the 
 last in analytical investigations. The signs : : and = have precisely the same 
 meaning. The sign : is the sign of division. 
 
 a c 
 
 132. The expression a : b : : c : d, or 7 =-7, is called a proportion, and a, b, c, d 
 
 are severally called the terms of the proportion. The first and last are called 
 the extreme terms, the second and third the mean terms. The first term is 
 called the first antecedent, the second term t\\a first consequent, the third terra 
 the second antecedent, and the fourth term the second consequent. 
 
 133. When the second and third terms of a proportion are identical, the 
 quantity which forms these terms is called a mean proportional between the 
 other two ; thus, if we have three quantities a, b, c, such that 
 
 a 1 a b 
 
 a:b::b:c, or r==-, 
 
 b c 
 then b is said to be a mean proportional to a and c, and c is called a third pro 
 porlional to a and b. 
 
 If, in a series of proportional magnitudes, each consequent be identical with 
 the next antecedent, these quantities are said to be in continued proportion ; 
 thus, if we have a series of quantities, a, b, c, d, e,f, g, h, such that 
 
 a : b : : b : c : : c : d : : d : e : : e :f: :f: g : : g : h , 
 or 
 
 a b c d e f g 
 b c d e / g A' 
 then the quantities a, b, c, d, e,f, g, h are in continued proportion. 
 A continued pi'oportion is called a progression. 
 
 The following are the most important propositions connected with the sub- 
 ject of proportion. 
 
 I. If four quantities be proportionals, tJie product of the extreme terms vnll be 
 equal to the product of the mean terms. 
 
 Let 
 
 a:b::c:d, 
 
 or 
 
 a_c 
 
 b~d' 
 
 Multiplying these equals by bd, the expression becomes 
 
 ad=bc. 
 
 II. Conversely, If the product of any two quantities be equal to the product 
 oj any other two, these four quantities will constitute a proportion, the terms of 
 one of Hie products being the means, and the terms of the other the extremes. 
 
 Let 
 
 ad=zbc. 
 Dividing these equals by bd, the expression becomes 
 
 a c c a 
 b = d ,0r d == b ; 
 i. e., a : b : : c : d, or c : d : : a : b. 
 In the first, a and are the extremes, and b and c the means ; in the second, 
 b and c are the extremes, and a and d the means. 
 
 III. If three quantities be in continued proportion, the product of the extreme 
 tcrn\s is equal to the square of the mean. 
 
124 ALGEBRA. 
 
 This follows immediately from I. ; for let a, b, c be three quantities in con- 
 tinued proportion, then 
 
 7 7 a b 
 
 a:b::b:c, or v=- 
 b c 
 
 .-. acz=b X b by I. 
 
 = b*, or b= y/ac. 
 
 IV. Conversely, If the product of any two quantities be equal to the square 
 of a third, the last quantity will be a mean proportional between the otfier two 
 
 Thus, if ac=b-, b is a mean proportional between a and c ; for, since 
 
 ac=b 2 , 
 dividing these equals by be, 
 
 -=-, or a:b::b:c. 
 b c 
 
 V. Quantities which have the same ratio to the same quantity are equal to 
 one another, and those to which the same quantity has the same ratio are equal 
 to one another. 
 
 First, let a and b have the same ratio to the same quantity c, then a = b. 
 Since 
 
 a:c::b:c, 
 or 
 
 a b 
 c = c ; 
 multiply these equals by c .•. a=b. 
 
 Again, let c have the same ratio to each of the quantities a and b, then a=J> 
 Since 
 
 c:a::c:b, 
 or 
 
 c c 
 a~V 
 dividing these equals by c, 
 
 11 
 a b 
 .-. a=b. 
 
 VI. Ratios that are equal to the same are equal to one another. 
 Let a:b::x:y 
 
 c Then a:b::c:d. 
 And c:d::x:y 
 
 This is an axiom. 
 
 VII. If four quantities be j>roportionals, they will be proportionals edso alter 
 nando, that is, the first will have the same ratio to the third that the setend hat 
 to the fourth. 
 
 Let a:b::c:d, then, also, a:c::b:d. 
 
 a c 
 Since t=;7. divide each of these equals by c, and multiply each by b 
 
 Then - =-j»" i. e., a:c::b:</. 
 
 c d 
 
 VIIT. If four quantities be proportionals, they will be proportionals also 
 
 invertrmlo, that is, the second ivill have to tin- first the same ratio that the 
 
 fuuitli hat to thf third. 
 
RATIOS AND PItOPOilTION. X25 
 
 Let a:b::c:d, then, also, b:a::d:c. 
 
 a c 
 Since v=j divide unity by each of these equals. 
 
 We have 
 
 1 1 
 
 ©~Q' 
 
 b d 
 
 — =— ; 1. e., b:a::a:c. 
 a c 
 
 IX. If four quantities be proportionals, they will be proportionals also com- 
 ponendo, that is, the first, together with the second, will have to the second the 
 same ratio that the third, together with the fourth, has to the fourth. 
 
 Let a:b::c:d, then, also, a-\-b:b::c-\- d:d. 
 
 or 
 
 Since T = 7' ac ^ 1 to eaca °f these equals, then 
 
 a c 
 
 b + ^d 
 
 a-\-b c-\-d 
 
 a c 
 
 1 + 1=^+1, 
 
 d 
 
 ; i. e., a-\-b:b::c-\-d:d. 
 
 X. If four quantities be proportionals, they will be 'proportionals also divi- 
 dendo, that is, the difference of the first and second will have to the second the 
 same ratio that the difference of the third and fourth has to the fourth. 
 
 Let a:b::c:d, then, also, a — b:b::c — d:d. 
 
 a c 
 Since T—~r subtract unity from each of these equals, then 
 
 a c 
 
 b- l =d~ 1 ' 
 
 or 
 
 a — b c—d . 
 
 —t — = — j— ; l. e., a — b:b::c — d:d. 
 b a 
 
 XI. If four quantities be proportionals, they will be proportionals also con- 
 vertendo, that is, the first will have to the difference of tSie first and second the 
 same ratio that the third has to the difference of the third and fourth. 
 
 Let a : o: :c:d, then, also, a: a — b : :c:c — d. 
 
 a c b (I ' 
 
 Since r= ji then, by prop. VIII., -=7 ; aud hence, subtracting iheso equal 
 
 Quantities from unity, 
 
 b__ d 
 
 a c' 
 
 or 
 
 a — b c — (I 
 ~a~ = '~c~' , 
 
 * 
 
 ar 
 
 a c 
 
 r = ; ; i. e.. a: a — b::c:c — d. 
 
 a — c — d 
 
126 ALGEBRA. 
 
 XII. If four quantities be proportionals, the sum of the first and second wiu 
 have to their difference the same ratio that the swn of the third and fourth hoi 
 to their difference. 
 
 Let a:b::c:d, then, also, a-\-b:a — b::c-\-d'.c — d. 
 
 Since T == ~r we Uave » 
 
 by prop. IX., —=-T; 
 
 a — b c — d 
 and, by prop. X., — -g- =— -j-; 
 
 dividing these equals by each other, 
 a-\-b c-\-d 
 
 :-d 
 
 or 
 
 a-\-b c-\-d 
 
 7= -,; i. e., a-\-b:a — b::c4-d:c — d. 
 
 a — o c — d ' ' 
 
 XI I T. If there be any number of quantities more than two, and as many 
 others, which, taken two and two in order, are proportionals (ex aequali), the 
 first will have to the last of the first rank the same ratio Uiat the first of the 
 second rank has to the last. 
 
 Let 
 and 
 Let 
 
 a, b, c, d .... be any number of quantities, 
 e,f,g,h .... as many others. 
 
 a:b ::e :f 
 
 
 b:c ::f:g { { 
 
 ► Then, also, a:d::e:h. 
 
 c :d::g:h. 
 
 
 For, since 
 
 
 
 a e 
 b = f 
 
 IJL 
 
 c g 
 
 
 c g 
 
 d~h' 
 
 multiplying the first column together, and also the second, 
 
 
 abc efg 
 
 bcl^fglC 
 
 <>r 
 
 
 a e 
 
 77 = r 
 
 ; i. e., a:d::e:h. 
 
 \ 1 V. Jf (here be any number of quantities more than two, and as many 
 nth' r», which, taken tivo and tivo in a cross order, arc proportionals (ox tr-quali 
 perturbata), the first will have to the last of the first rank the same tatio that the 
 first <>f the second rank has to the last. 
 
RATIOS AND PROPORTION. 
 
 127 
 
 Let 
 and 
 
 Let 
 
 For, since 
 
 a, b, c, d . . . . be any number of quantities, 
 e,fg,h....&s many others. 
 
 a : b ::g:h i 
 
 b : c ::f :g> Then, also, a : d : : e : h. 
 
 c :d::e :f ) 
 
 a 
 b z 
 
 g 
 ~h 
 
 b 
 
 c~ 
 
 _f 
 
 g 
 
 c 
 
 e 
 
 d = 
 
 7' 
 
 abc 
 bed' 
 
 
 or 
 
 a e ■ i i 
 
 -j=T ; i. e., aid'.'.e'.h. 
 
 XV. If four quantities be proportionals, any powers or roots of these quan 
 tilies will also be proportionals. 
 
 Let a : b : : c : d ; then, also, a n :b n ::c a : d a . 
 
 Since 
 
 a c r. , , / a \ n / c \" 
 
 t=-j, raising each of these equals to the nth power, It) =1 -7I, 
 
 or 
 
 a n c n 
 
 ^=j n ; i. e., a n :b n ::c a :d a , 
 
 , where n may be either integral or fractional.* 
 
 XVI. If there be any number of proportional quantities, the first will have to 
 the second the same ratio that the sum of all the antecedents has to the sum of 
 all the consequents. 
 
 Let ", b, c, d, e,f, g, h be any number of proportional quantities, such that 
 
 a : b : : c : d : : e :f: : g : h. 
 
 Then 
 Since 
 
 we bave 
 
 and 
 
 or 
 
 a : b : : a-{-c-±-e-{-g: b-\-d-{-f-{-h. 
 
 a c e g 
 l = d = f~h' 
 
 ab =ba 
 
 ad=bc 
 
 af=be 
 
 ah = bg, 
 
 a(b+d+f+h) = b\a+c+e+g) 
 
 a a-^-c-\-e-\-g 
 
 '■ 1—b + d+f+h 
 
 a:b::a-\-e-\-e-\-g:b-\-d-\-f-\-h. 
 
 • The ratio of the resulting proportion is tnc » u uower o' Jie ratio of the jjiven proportion 
 
128 ALGEBRA. 
 
 XVII. If three quantities be in continued proportion, the first wiU have to 
 the third the duplicate ratio of that which it has to the second. 
 
 Let a :b::b:c, then a :c::a 2 : 6 s . 
 
 Since 
 
 a b a 
 
 j=~, multiply each of these equals by j-; then 
 
 a a b a a- a . 
 b X b=-c X b' or ¥=c ; i-e.,a:c::a»:*». 
 
 XVIII. If four quantities be in continued proportion, the first will have to 
 the fourth the triplicate ratio of that which it has to the second. 
 
 Let a, b, c, d be four quantities in continued proportion, so that 
 
 a:b::b:c::c:d ; then, also, a : d : : a 3 : b 3 . 
 Since 
 
 a b c 
 
 
 
 
 b~c— d ,V> 
 
 e i 
 
 lave 
 
 » 
 
 
 
 
 
 a b 
 
 
 
 
 
 
 
 
 b = c 
 
 
 
 
 
 
 
 
 a c 
 
 
 
 
 
 
 
 
 b~d 
 
 
 
 
 
 
 
 
 a a 
 
 
 
 
 
 
 
 
 b = b' 
 
 
 
 
 
 Multiplying 
 
 these 
 
 equals 
 
 together, 
 a 3 bca 
 b i= 7d~V 
 
 
 
 
 
 or 
 
 
 
 
 
 
 
 
 
 
 
 a 3 a 
 l~ 3= d ' U 6 " 
 
 , a 
 
 :d: 
 
 :a 3 
 
 :6 s , 
 
 XIX. If two proportions be multiplied together, term by term, the products 
 will form a proportion. 
 
 Let a: b :: c :d, 
 
 and e:f::g:h; 
 
 then ae:bf::cg:dh, 
 
 a c c g 
 
 for T = ~n a,u ' 7=rj 
 
 hence, multiplying equals, 
 
 b-d' a "\f 
 
 ae eg 
 
 —=— or ae:bf: iCg :dh.* 
 
 Tho compatibility of any change in the on|er of the terms of a proportion 
 may be tested by forming the product of the extremes and means in both the 
 original and changed proportion, when, if they Agree, the change is correct 
 Thus. a:b::c:d may be written d:b::c:>i, for we hare ad=zbc in both. 
 
 I \ IMPLES in PROPORTION. 
 
 (1) The mercurial barometer stands a) a height of 30 inches, and the specific 
 gravity of quicksilver is ].'?'•, ; ! . How high would a water barometer stand .' 
 
 \xa. ."•". feel 1 1 1 inches. 
 
 i i The weights Of a lever have the same ratio as the lengths of the oppo 
 site arms. The ratio of the weights is .">, and the longer arm 10 indies 
 What is tin- length of the Bhorter arm .' a\ns. 2 inches. 
 
 * The ratio of the resulting proportion la ti I 1 1 • ■ - ratios of the two fivi d i , 
 
 portions. 
 
EQUATIONS 129 
 
 (3) The weights of a lever are 6 and 8 pounds, and the length of the .shorter 
 arm 18 inches. What is that of the longer ? Ans. 24 inches. 
 
 (4) At the end of an arm of a lever 5 inches long, what weight can be sup- 
 ported by 2\ pounds acting at the end of an arm 4| inches long? 
 
 Ans. 2 4 8 5 pounds. 
 
 (5) Triangles are to each other as the products of their bases by their alti- 
 tudes. The bases of two triangles are to each other as 17 and 18, and their 
 altitudes as 21 and 23. What is the ratio of the triangles themselves 1 
 
 Ans. 119:138. 
 
 (6) The force of gravitation is inversely as the square of the distance. At 
 the distance 1 from the centre of tho earth this force is expressed by the 
 number 32.16. By what is it expressed at the distance 60 ? 
 
 Ans. 0.0089. 
 
 (7) Tho motion of a planet about the sun for a short space is proportional 
 to unity divided by the duplicate of the distance. If the motion be represented 
 by v when the distance is r, by what will it be expressed when the distance is r 1 1 
 
 Ans.— 
 
 'j 
 
 (8) The times of revolution of the planets about the sun are in the sesquipli- 
 cate ratio of their mean distances. The mean distance of the earth from the 
 sun being expressed by 1, that of Jupiter will be 5.202776 ; the time of revolu- 
 tion of the earth is 365.2563835 days. What is the time of revolution of 
 Jupiter ? Ans. 4332.5848212 days. 
 
 EQUATIONS. 
 
 PRELIMINARY REMARKS. 
 
 134. An equation, in the most general acceptation of the term, is composed 
 of two algebraic expressions which are equal to each other, connected by the 
 sign of equality. 
 
 Thus, ax=b, cx*-\-dx=e, cx 3 -{-gx' : =hx^-k, mx i +nx 3 -{-px' i +qx-\-r=0,aTe 
 
 equations. 
 
 The two quantities separated by the sign = are called the members of the 
 equation, the quantity to the left of the sign = is called the first member, the 
 quantity to the right the second member. The quantities separated by the 
 signs 4- and — are called the terms of the equation. 
 
 135. Equations are usually composed of certain quantities which are known 
 and given, and others which are unknown. The known quantities are in 
 general represented either by numbers, or by the first letters in the alphabet, 
 a. b, c, &c. ; the unknown quantities by the last letters, s, t, x, y, z, &c. 
 
 136. Equations are of different kinds. 
 
 1°. An equation may be such that one of the members is a repetition of the 
 other; as, 2x— 5=2r— 5. 
 
 2°. One member may be merely the result of certain operations indicated 
 in tho other member; as, 5a:- r -16=10x— 5 — (5a:— 21), (x+y){x— ?/)=x 3 — y\ 
 r 5 — i/ 3 
 
 — 2-=X* + Xy + y* 
 
ALGEBRA. 
 
 . All the quantities in each member may be known and given; as, 2.3 = 10 
 + 15, a-{-b=c— d, in which, il" we substitute fur a, b, c, d tho known quan- 
 tities which they Represent, the equality subsisting between the two members 
 
 will bo sell-evident. 
 
 In each of the above cases the equation is called ntical equation. 
 
 4 . Finally, the equation may contain both known and unknown quantities, 
 and bo such that the eufhality subsisting between the two members can not be 
 made manifest, until we substitute for the unknown quantity or quantities cer- 
 tain other numbers, the value of which depends upon the known numbers 
 which enter into the equation. The discovery of these unknown numbers 
 constitutes what is called tho solution of the equation. 
 
 When found and put in the place of the letters which represent them, 
 if they make the equality of the two members evident, the equation is said to 
 be verified, or satisfied. 
 
 The word equation, when used without any qualification, is always under- 
 stood to signify an equation of this last species ; and these alone are the objects 
 of our present investigations. 
 
 ar-|-4=:7 is an equation properly so called, for it contains an unknown 
 quantity x, combined with other quantities which are known and given, and 
 the equality subsisting between the two members of tho equation can not be 
 made manifest until we find a value for x, such that, when added to 4, the 
 result will be equal to 7. This condition will be satisfied if we make x=3 ; 
 and this value of x being determined, the equation is solved. 
 
 The value of tho unknown quantity thus discovered is called the root of the 
 equation, being the radix out of which the equal 'um is formed; the term root 
 hero has a different sense from thai in which we have hitherto used it, viz., 
 that of the base of a power. 
 
 137. Equations are divided into degrees according to the highest power of 
 the unknown quantity which they contain. Those which involve the simple 
 or first power only of tho unknown quantity are called simple equations, or 
 equations of the first degree; those into which the square of the unknown 
 quantity enters are called quadratic equations, or equations of the second de- 
 gree: so wo have cubic equations, or equations of (he third degree ; biquad- 
 ratic equations, or equations of the fourth degrt t ; equations of the fifth, sixth, 
 
 .... 7i th degree. Thus, 
 
 ax -\-b =cx-\-d is a simple equation. 
 
 4i* — 2x =5 — a: 2 is a quadratic equation. 
 
 x 3 -\-px 2 =z'2q is a cubic equation. 
 
 x n -{-j)x n ~ 1 -\-qx n ~ 2 -{-, &c, =r, is an equation of the n"' degree. 
 
 138. Numerical equations are those which contain numbers only, in addition 
 to tin- unknown quantities. Thus, .r -f , r )./- = r..r-f-17 anil ■lx=7y are numer- 
 ical equations. 
 
 I. Lateral equations are those in which tin- known quantities are repre 
 sented by letters only, or by both letters and cumbers. Thus, x -\-j>.i : -\-<i.r=.r 
 r* — Xj)x f y-\-!jqx^if : -\-rxfz=z!') are literal equations. 
 
 140. Lot us now pass on to consider the solution of equations, it being under- 
 stood that /" sol/oe an < quation is t<> find the oalui of the unknown quantity, or 
 to find a number which, when substituted for the unknown <juantity in the equa- 
 tion, renders On first member identical u-itii the second. 
 
SIMPLE EQUATIONS. 131 
 
 The difficulty of solving equations depends upon the degree of the equations 
 and the number of unknown quantities. We first consider the most simple 
 case. 
 
 ON THE SOLUTION OP SIMPLE EQUATIONS CONTAINING ONE UN- 
 KNOWN QUANTITY. 
 
 141. The various operations which we perform upon equations, in order to 
 arrive at the value of the unknown quantities, are founded upon the following 
 axioms : 
 
 //' to two equal quantities the same quantity be added, the sums will be equal. 
 
 If from two equal quantities the same quantity be subtracted, the remainders 
 will be equal. 
 
 If two equal quantities be multiplied by the same quantity, the products will 
 be equal. 
 
 If two equal quantities be divided by the same quantity, the quotients will be 
 equal. 
 
 These axioms, when applied to the two equal quantities which constitute 
 the two members of every equation, will enable us to deduce from them new 
 equations, which are all satisfied by the same value of the unknown quantity, 
 and which will lead us to discover that value. 
 
 142. The unknown quantity may bo combined with the known quantities in 
 the given equation by the operations of addition, subtraction, multiplication 
 and division. We shall consider these different cases in succession. 
 
 T. Let it be required to solve the equation 
 
 x-\-a=b. 
 If, from the two equal quantities x-\-a and b, we subtract the same quantity 
 a, the remainders will be equal, and we shall have 
 
 x-\-a — a=b — a, 
 or 
 
 x=b — a, the value of x required. 
 
 So, also, in the equation 
 
 x+6=24. 
 
 Subtracting 6 from each of the equal quantities x-\- 6 and 24, the result is 
 
 x=24 — G 
 :=18, the value of x required. 
 
 II. Let the equation be 
 
 x — a=b. 
 If, to the two equal quantities .r — a and b, the same quantity a be aided, 
 the sums will be equal ; then we have 
 
 x — a-{-a=b-\-a, 
 or 
 
 x=b-\-a, the value of x required. 
 
 So, also, in the equation 
 
 a:— 6=24. 
 Adding 6 to each of these equal quantities, the result is 
 
 x—24+6 
 =30, the value of x required. ^ 
 
 It follows from (I.) and (II.) that 
 
132 ALGEBRA. 
 
 We may transpose any term of an equation from one member to the other ft) 
 changing the sign of that term.* 
 
 We may change the signs of every term in each member of the equation with 
 out altering the value of the expression.] 
 
 If the same quantity appear in each member of the equation affected with th 
 tame sign, it may be suppressed. 
 
 Ill Let the equation be 
 
 ax=b. 
 
 Dividing each of these equals by a, the result is 
 
 b 
 r=-, the value of x required. 
 
 So, also, in the equation 
 
 6z=24. 
 
 Dividing each of these equals by 6, the result is 
 
 x=4, the value of a: required. 
 
 From this it follows that, 
 
 When one member of an equation contains the unknoum quantity atone, 
 affected with a coefficient, and the other member contains known quantities only, 
 the value of the unknown quantity is found by dividing each member of the 
 equation by the coefficient of the unknown quantity 
 
 TV. Let the equation be 
 
 X - = b. 
 a 
 
 Multiplying each of these equals by a, the result is 
 
 x=ab, the value of x required. 
 
 So, also, in the equation 
 
 x 
 
 Multiplying each of these equals by 6, the result is 
 
 x=144. 
 
 From this it follows that, 
 
 When one member of the equation contains the unknown quantity alone, du 
 tided by a known quantity, and the other member contains known quantities 
 only, the value of the unknown quantity is found by multiplying each member 
 of the equation by the quantity which is the divisor of the unknown quantity. 
 
 V. Let the equation bo 
 
 ax dx m 
 
 b c n ' 
 
 In order to solve this equation, we must clear it of fractions ; to effect this, 
 reduce the fractions to equivalent ones, having a common denominator (Art. 
 41), the equation becomes 
 
 aenr been bdnx ban 
 hi it l>< n be n ben ' 
 Multiply these equal quantities by t ho same quantity ben, or, which u 
 
 • If we transpose a plus term, it subtracts this term from both members ; and if we 
 transpose a minus term, it tddi this term to both. 
 
 t This is, in fact, tho sum.' thing as transposing ovory term in each member of the eque 
 tion, or multiplying tliruuyhout by — 1. 
 
SIMPLE EQUATIONS. 133 
 
 aently the same thing, suppress the denominator ben in each of the fractions, 
 the result is 
 
 aenx — bcen=bdnx — bcm, an equation clear of fractions. 
 
 So, also, in the equation 
 
 2x 3 x 
 
 T-4 = U + 5- 
 Reducing the fractions to a common denominator 
 
 iOx 45 660 12x 
 60~ — 60 == "60~"^"60'' 
 
 Multiplying both members of the equation by 60, the result is 
 40a: — 45=660-j-12.r, an equation clear of fractions 
 If the denominators have common factors, we can simplify the above opera- 
 tion by reducing them to their least common denominator, which is done (see 
 Art. 44) by finding the least common multiple of the denominators. Thus, in 
 the equation 
 
 5x 4x _7 13 r 
 
 The least common multiple of the numbers 12, 3, 8, 6 is 24, which is, there 
 fore, the least common denominator of the above fractions, and the equation 
 will become 
 
 IOx 32.r 312 21 52x 
 *24 "24 ~~ '~2A~2~i~~2A' 
 Multiplying both members of the equation by 24, the result is 
 
 10.r — 32x — 312=21 — 52x, an equation clear of fractions. 
 Hence it appears that, 
 
 In order to clear an equation of fractions, reduce the fractions to a common 
 denominator, and then multiply each term by this common denominator. In the 
 fractional terms the common denominator will be simply suppressed. 
 143. From what lias been said above, we deduce the following general 
 
 RULE FOR THE SOLUTION OF A SIMPLE EQUATION CONTAINING ONE UNKNOWB 
 
 QUANTITY. 
 
 1°. Clear the equation of fractions, and perform in both members all the alge- 
 braic operations indicated. 
 
 2°. Transpose all the terms containing the unknown quantity to one member 
 of the equation, and all the terms containing known quantities only to Hie other 
 \ member, and reduce each member to its most simple form. 
 
 3°. We thus obtain an equation, one member of which contains the unknown 
 quantity alone, affected with a coefficient, and the other member contains known 
 quantities only ; the value of the unknown quantity will be found by dividing the 
 member composed of the known quantities by the coefficient of the unknown 
 quantity. 
 
 The terms containing the unknown quantity are usually collected in theirs* 
 member of the equation, though they may often be more conveniently col- 
 lected in the second ; the second being afterward written as the first member, 
 and the first as the second. 
 
 Sometimes an equation presents itself as one of a degree higher than the 
 first, but both members are divisible by such a power of the unknown quan- 
 tity as to reduce the equation to" one of the first degree. 
 
134 • :.;. 
 
 In other cases, clearing an equation of fractions reduces it, by the cane, 
 of those terms which contain the higher powers of the unknown quantity, to 
 the first degree. 
 
 A proportion containing an unknown quantity in any of its terms can be 
 thrown into the form of an equation by multiplying the extremes, and also th*» 
 means, and setting the two products thus formed equal to each other. 
 
 j \ AMPLE I. 
 Given, 19x+13 =59— 4x. v 
 
 Transposing. 19x+ 4x=59 — 13. 
 
 Reducing, 23x=46. 
 
 Dividing by 23, x=2. 
 
 Verification. — Substitute 2 for x in the given equation, it becomes 
 19x2 + 13=59—4x2, or 
 
 38+13=59— 8, an identity. 
 Let this process be repeated in some of the following examples. 
 
 EXAMPLE II. 
 _. XX XX 
 
 Gl ven, -_ i + 10 = 3-2+11. 
 
 Reducing to least common denominator 12, 
 
 2x 3x Ax 6x 
 
 12-12 + 10 = 12-12+ 1L 
 Multiplying both members by 12, 
 
 2x— 3X+120 = Ax— 6x+132. 
 Transposing, 2x — 3x— 4x + Gx=132 — 120. 
 Reducing x = 12. 
 
 EXAMPLE III. 
 
 5x+3 Ax— 10 
 
 G>ven, "^+ 7 = -To~ +1<) - 
 
 Reducing to least common denominator 20, 
 25x+15 8.r— 20 
 
 ~ir-+ 7 = ^o-+ 10 - 
 
 Multiplying both members by 20, 
 
 25x+-15-|-140= Sx— 20 + 200. 
 Transposing, 25x— 8x=200— 20 — 15 — 140. 
 
 Reducing, 17x= 25. 
 
 25 
 
 Dividing by 17, x= — . 
 
 1 \ \MII.I IV. 
 
 2x— 5 Tx+10 12x — 10 
 
 Given, — ; — — — - — =16 - . 
 
 4 3 5 
 
 Reducing to common denominator, 
 
 30x— 75 140x+200 144x— 120 
 
 U0 — GO = 1G — qq • 
 Multiplying both members by 60^ 
 
 — 75— 140x— 200 =960— 144x+] 
 Transposing, 30x— 140x+144x=960+ ?.'> '+200+ 12a 
 
SIMPLE EQUATIONS. 135 
 
 iveducing, 34£=1355. 
 
 _ 1355 
 
 Dividing by 34, x= . 
 
 Tt is unnecessary to write the common denominator. 
 
 EXAMPLE V. 
 
 12— 4.r 2aM-5 7.r+G0 
 
 Given, - I5 f- = 3+-^~50. 
 
 deducing to least common denominator, 10, and neglecting it, we have' 
 
 12— 4.r—4.r— 10 =30+ 35.T+300— 500. 
 Transposing, — 4.t— 4.r— 35.r=30-}-300 — 12+10—500. 
 Reducing, — 43.r= — 172. 
 
 Changing the signs of both members,* 
 
 43.r= 172. 
 Dividing by 43, x= 4. 
 
 EXAMPLE VI. 
 
 Given, ax-\-b =.cx-\-d. 
 
 Transposing, ax — cx= d — b. 
 
 Simplifying, (a — c)x= d—b. 
 
 d — b 
 Dividing by (a — C), x*= . 
 
 EXAMPLE VII. 
 
 ax CX gX 
 
 Reducing to a common denominator, 
 
 adhx bchx . bdgx 
 
 i^+m+ e =f x +-bdk+ m ' 
 
 Multiplying by bdh, 
 
 adhx-\-bchx-\- bdeh=bdfhx-{- bdgx-\- bdhm. 
 Transposing, adhx-\-bchx — bdfhx—bdgx= bdhm — bdeh. 
 Simplifying, (adh-\-bch — bdfh—bdg)x=bdhm — bdeh. 
 
 bdhm — bdeh 
 Dividing by coefficient of *, X =adk+bch-bdfh-bdg 
 
 bdh(m — e) 
 adh-\- bch — bdfh — bdg' 
 
 EXAMPLE VIII. 
 
 x dx 
 
 Given, — 1— +3aZ>=0. 
 
 a c 
 
 Reducing to common denominator and neglecting it, 
 
 ex — ac — adx-\-3a-bc=0. 
 
 Transposing and simplifying, (c — ad)x=ac — 3a : bc. 
 
 ac(l — Sab) 
 Dividing by coefficient of a:, x=- ——5 — . 
 
 Verification. 
 ac(l—3ab) 
 
 c — ad acdll — 3ab) 
 
 1— — 7^ 7 r J -+3ab = Q; 
 
 a c(c — ad) ' 
 
 * Or dividing both members by — 43, gives x=4. 
 
13G 
 or 
 
 or 
 
 ALGEBRA. 
 
 c(l— 3ab) ad(l — 3ab) 
 
 ■ 1— L 4-3ab=0', 
 
 c — at 
 
 ■ad 
 
 c—3abc—c-\-ad — ad+3a"bd-\-3abc — 3a"-bd=:Q. 
 
 Given, 
 Transposing, 
 
 Given, 
 
 Clearing of fractions 
 
 EXAMPLE IX. 
 
 x+18=3x— 5. 
 
 18+5 =3x— x 
 
 23=2x 
 
 23 
 x ll 1 . 
 
 EXAMPLE X. 
 
 a b d 
 
 x c' e 
 ace=bex-\-cdx 
 ace=(be-\-cd)x 
 ace 
 
 X = 
 
 be-\-cd' 
 
 EXAMPLE XI. 
 
 Given, 
 Dividing by x, 
 
 3x 2 — 10x=8x-|-x 2 . 
 3x— 10 =8 -fx 
 x=9. 
 
 Given, 
 
 Dividing by x m_1 , 
 
 Given, 
 
 EXAMPLE XII. 
 
 x m =ax m ~ l . 
 x=a. 
 
 EXAMPLE XIII. 
 
 ax m — a' 
 
 a 
 
 ■=a- 
 
 r m— i 
 
 x m X" 
 
 Multiplying by x m , ax m — a':=ax m — a"x. 
 Cancebng ax m in both members, 
 
 ■a' = —a 'x .-. x=— . 
 a" 
 
 Given, 
 
 EXAMPLE xiv. 
 ad 
 
 a:bx::c:d ••• bcxz=ad .-. x= 
 
 be 
 
 144. In addition to tho axioms in (Art. 141) we may snbjox the fallowing: 
 
 If two equal quantities be raised to the same power, the results will be equal. 
 
 If (lie same root of two equal quantities be extracted, the results will be equal. 
 
 Hence any equation may be cleared of a single radical quantity by trans- 
 posing all llie other tonus to the opposite* aide, and then raisin:; each member 
 to the power denoted l>\ the index of the radical. If there be more than on 
 radical, tho operation must bo repeated. Thus : 
 
SIMPLE EQUATIONS. 137 
 
 EXAMPLE XV. 
 
 Given, -\/3.f-|- 7 = 10. 
 
 Squaring each member of the equation, 
 
 3x+7 = 100. 
 Transposing, 3x=l()0 — 7. 
 
 Reducing, and dividing by 3, x=31. 
 
 EXAMPLE XVI. 
 
 Given, ■/4a:+2= y/4x-\-5. 
 
 Squaring both sides of the equation, 
 
 4^+2=4x4-10 \/4x"+25. 
 Reducing, — 10</lx=23. 
 
 Squaring both sides, 400x=529. 
 
 _529 
 
 EXAMPLE XVII. 
 
 •/ar+28 t/x+38 
 
 Given, 
 
 Vi-f4 V- r + (j 
 
 Clearing the equation of fractions, 
 
 x+28 V^+6 -v/^+168=a:+38 /ar+4 V^+152. 
 Transposing and reducing, 16=8 -\/x. 
 
 Dividing both members by 8, 2= -J x. 
 
 Squaring both members, 4= x. 
 
 EXAMPLE XVIII. 
 
 Given, tya+x = 2 ^x 3 4-5ax+6 2 . 
 
 Raising both members to the m th power, 
 
 a+x = V^+5a.T+6 2 . 
 
 Squaring both members, a 2 -\-2ax-{-x~=x' 2 -\-5ax-\-b 2 . 
 
 Transposing and reducing, — Zax =b- — a 2 . 
 
 Changing the signs, 3ax =a 2 — b 2 . 
 
 a 2 —b 2 
 Dividing by 3a, x = — - — . 
 
 Given, 
 
 EXAMPLE XIX. 
 
 yx—a* 2 7x— a 
 
 -tyx—a b 
 
 Since tyx is the square of "tyx, and a- is the square of a, wo can perform 
 
 the division indicated in the first fraction, ;md have for a quotient 
 
 2 v/x—a 
 2 yx+a= — £— , 
 
 .:(b-iyy.v=-(b + l)a, 
 (b + l)a 
 
 . 2m/ ? V ' > 
 
 " v x — (6-1)' 
 ( (b+l)a y° 
 
 •'•• r -V b-i ) 
 
 (20) Given 4x+36=5x+34. " Ans. x=2. 
 
 (21) Given 4x— 12+3x+l=2x+4. Ana. x=3 
 
138 ALGEBRA. 
 
 (22) Given .'.(/ + .. — 56+2=76— a+c+6, Ans. z=l2& — ia + r+4. 
 
 (23) Given 13J — — 2x— 8|. Ans x=9. 
 
 (24) Given 12} + 3x — r;_ — ='- 5J. Ans.x=l: 
 
 a: a- i 
 
 (25) Given -+—=-+7. Ansx=l2. 
 
 (26) Given -+r. + - = 13. Ans. x = 12. 
 
 x x 
 
 (27) Given x+- — - = 4x — 17. Ans. x=ti. 
 
 x+4 
 
 (28) Given 5 r^-=i— 3. Ans. x=7. 
 
 3x— 5 2x— 4 
 
 (29) Given *+ — — = 12—— - — . . Ans. x=5. 
 
 x+1 x+3 x+4 
 
 (30) Given -^— +— 7— =— 1-+16. Ans. x = 41. 
 ' 3 4 5 
 
 5x 4x 
 
 (31) Given 5x— — +12=— + Ans. 1=12. 
 
 _ x 4x 41x 
 
 (32) Given 7x+13? — -=— — 8?+ — . =9. 
 
 2 5 o 
 
 (33) Given 6x— 7] — J-X+.10 — 5x— 2 4 '=0. Ans. x=0, or 8$. 
 
 (34) Given 4(5x+7— -?-) = = (3.r+9 — 4). Ans. x=— 1§. 
 
 x+Jx+'x 20x— 25 
 
 (35) Given -^-=— — Ans. x=2,V 
 
 ■7* £ OP] rp 
 
 (36) Given '—— + Gx=—- r —. Ans. x=9. 
 
 11 x 19 x 
 
 (37) Given x+ — - — = — - — . Ans. x=5. 
 
 2x+G lis— 37 
 
 (38) Given 3x+— ~ =5 + — . Ans. x=7. 
 
 n,_4 18— i 
 
 (39) Given— r 2= - + .r. Ans. X=4. 
 
 :;,-_ll ;,,• — :> 97—' 
 
 (40) Given 21+ =— — +— ,— . Ans. r=9 
 
 x— 1 5x+ 14 1 
 
 (41) Given 3x ; 1= — ;. Ans. r=7 
 
 x ' 4 12 
 
 x l <">3 /• 4 + /' 
 
 (42) Given -y- + :i -^— =7-— J- \ns. x=8 
 
 7.,+ r, 16+4i ::< + 
 
 (13) (liven — - — — - — +G= r— . \ns. x = ] 
 
 ' :; 5 ' 2 
 
 3 X +4 7,. — :: ,,_n; 
 
 (II) (liven ; — = — ; . Ans. ,r=2 
 
 ■ > 2 4 
 
 17—31 l;+2 7/+M 
 
 (45) Given — — ^— =5 — Gx+ ina. .; = . 
 
 —3 20- 1 Gx— 8 kr — 1 
 
 (46) Given X r- +4 = — — — <. r=6 
 
 .) 2 / 5 
 
SIMPLE EQUATIONS. 13S 
 
 4i — 21 57 — 3x 5x— 9G 
 v 4?) Given ^-^ + 31+ — — =241-— liar. Ans. x=21 
 
 fa+18 , R 11— 3a: 13— a 21 -2x 
 
 (48) Given -g 4*-- ^-=5x-48— ^— y^-. 
 
 Ans. x=10. 
 
 (49) Given 5.r+ :iL ^— =— 77— + •">• Ans. x=4. 
 
 &.T a 7 a ex . ad 
 
 (50) Given __-— -^ Ans. x=-^. 
 
 5x— 1 3x— 2 llx— 3 13x— 15 8x— 2 
 
 (51) Given 23 + ^-+ — —^~= 3 ~ 7— 
 
 Ans. x=9. 
 
 1 3x— 13 12+7x 9+5x llx — 17 
 
 (52) Given 4x+-_~ __ =7 *_33— 55- — -g— 
 
 Ans. x=15. 
 
 ace (a + Z>) 2 x , , . a 3 e(c— d) 
 
 (53) Given -7— - — ! — — bx=ae — 3bx. Ans. x= , 2 ■ ^w - 
 
 a+3x 7a — 5x 9x x 5x 
 
 (54) Given ______ +3 - T =^ + 65. 
 
 39a& — 14a 3 
 Ans. x== 
 
 ■27ab — 96 + 12" 
 6x (36c+ad)x 5a& _(36e-^aa> 5a(2ft— a) 
 
 (55) Given o^~ 2a6(« + Z>) ~~3c37z- 2a6(a-6) "~ a 3 -& 2 ' 
 
 5a(2& — a) 
 
 Ans. x= — -j — • 
 
 3c — a 
 
 d — c 
 
 (56) Given ax+c=bx+d. Ans. x=^— ^. 
 
 4a' — 3ab 
 
 (57) Given 2ax— &x+2a&=4a 3 — ab— 3ax. Ans. x= 5a _i • 
 
 7a& — 3a 2 
 
 (58) Given (3a— x)(a— 6) + 2ax=4fc(a+x). Ans. g= ■ ft . 
 
 11, 6c 
 
 (59) Given -ax+-6x=c. Ans.s= 3 . 2& . 
 
 x dx . ac(l— 3afc) 
 
 (60) Given — 1 + 3aZ> = 0. Ans. x= — - _^ . 
 
 a2 X aic — ac 2 +6cd — '•'</ 
 
 (61) Given ^— + c?c=6x—ac. Ans. x= j2_j c _ a 2 
 
 ax mx , hcn-\-bdn 
 
 (62) Given T -c=-+a\ Ans. *=-—£_, 
 
 «r ex 8a6 3 +4&3— 12a 2 6 
 
 (63) Given __+ 4 6=^^. Ans. *= 3aB + ||6 _ flC+6c 
 
 Six x—b bx—a* x 4a 9 (a s +a6— 6 s ) 
 
 (64) Given — _— =- i -^_-. Ans. *= 3a 3_ 6a * 6+ai3+66s 
 
 (a+5)(x— &) 4a&— 5 s a'' — 6x 
 
 (65) Given "^ ^ 3^-^p— 2x+~ T - 
 
 a^+S ^fe +4a 3 & 3 — 6a 6 3 +26 < 
 
 Ans. x_ i(4a- + 2a6-26-) 
 
140 ALGEBRA. 
 
 ax b ex px q rx kb-\-kq 
 
 (66) Given — 4- — 4-^-=— — — — -r. Ans. x= — ■; ; — : : 
 
 1 ; m^mT k m m k ka — kp + mc + rm 
 
 W+q) 
 
 k(a—p)-\-m{c-{-r)' 
 
 x ax bx ex mx 
 167) Given - + - + - + -~+ p . 
 
 12 P k 
 Ans. x=: 
 
 ■V2(l—m) + 2{3a-\--2b) + 3t 
 
 ' cq — rb 
 
 (68) Given r{ax+b—c)=c{px+q — r). Ans. x= ra __ . 
 
 x4-px—qx mx—n n ( f ]—p) 
 
 (69) Given —^- - = . Ans. x= \ . 
 
 v ' p — q m m 
 
 (70) Given (£m+i>)($a?-3r)Mfm+2p)(fx-7r). 
 
 r(952m + 4928p) 
 
 9m+208/» ' 
 
 Ans. x=- 
 
 m*x h? frnrx—h-n—Bntgx 
 
 (71) Given — — 4-5rcx= 
 
 x ' n g ' bng 
 
 4n/t 2 
 Ans. x= 
 
 5m s g— 4m*+33fty 
 
 13(5ax— 22 3 6) 24(3ax— 20£Z>) 
 
 (72) Given ^ , = 7c * k ' • 
 
 (2041c— 4406?A-)6 
 (455c — 648&)a 
 
 13m — 7x 4m — x m4-p 
 
 (73) Given ; + = kx. 
 
 v ' m-\-p m — p in — p 
 
 limp — 16m- +p* 
 6ji — 8m-\-k(m : — p 9 )' 
 
 3a6c (2a + b)b-x a-b 2 bx 
 
 (74) Given — ri+ / , L + / ■ M3 =to+7- 
 
 3a i bc(a-\-by--\-a s b i 
 Ans. r=- 
 
 ■(3ac+i)(a+6) 3 — (2a + 6)(a+i)6« 
 
 a 2 — Six 6&x— 5a 2 £x+4a 
 
 (75) Given ax— — <z& 2 =6x-| . 
 
 1 a ' 2a 4 
 
 ■\ab'— 10a 
 
 ADS - a '= 4«-3fc * 
 </-6 
 
 (76) Given rtx c 4-tx=c.f 2 -f-rfx. Ans. = 
 
 </ — c 
 
 D4-B 
 
 (77) Given Ax m 4-Bx m - 1 = Cx m — Dx m -'. Ans. x= _ . 
 
 14a 3 — 2a 2 6x 21a 3 4-5a 2 6x 
 
 (78) Given — 31^= -^ \-20c\ 
 
 v ' 17mx 33mx ' 
 
 105a 3 
 
 Ans. X=: 
 
 '151a-7> + 28611c*m 
 
 4m(K 2 — 5x 2 ) r»n{«* — 2r) 2K 9 
 
 (79) Given — =7mp+- — . Ans. xsr^g— p. 
 
 24x* 5X 8 
 
 (80) Given 3-^=^^. Ans.x=3]>. 
 
181 
 <82 
 
 (83 
 (84 
 (85 
 
 (86 
 
 (87 
 (88 
 (89 
 (90 
 (91 
 (92 
 
 (93 
 (94 
 
 (95 
 
 (96 
 
 (97 
 
 (98 
 
 (99 
 
 SIMPLE EQUATIONS. 141 
 
 _ ax a mx" bm — ap 
 
 Given j— — = — ; . Ans. x= . 
 
 b-\-cx 2 ) ~\~ ( l x a< l — cm 
 
 x 
 Given 12 — x:-::4:l. Ans. x=4. 
 
 5x+4 18— x 
 Given — ^ — : — - — ::7:4. Ans. x=2. 
 
 2 ' 4 
 
 Given 2© :1:: 1:3.1416. Ans. 0=0.1591. 
 
 b 7ad > 
 
 Given a: t::-: 7c. Ans. t 
 
 c 
 
 b * 
 
 c 
 
 Given r : 1 : : c : 3.1416. Ans. r =— rr=. 
 
 0.141b 
 
 Given -v/4x+16 = 12. Ans. x=32. 
 
 Given V2x+3+4=7. Ans. x=12. 
 
 Given -/l2+x=24- y/x. Ans. x=4. 
 
 Given -^+40=10 — y/x. Ans. x=9. 
 
 Given y/x — 16=8 — y/x. Ans. x=25. 
 
 Given V^— 24 = y/x— 2. Ans. x=49. 
 
 _ _ 25a 
 
 Given y/x — a= yfx— \yf a. Ans. x=tt^-. 
 
 - 9 
 
 Given y/bx ■ v / ^+~= y/bx-\-2. Ans. x=— . 
 
 ■1 
 
 _ {b—a\ 
 
 Given y/4a-\-x=2y/b-\-x — y/x. Ans. x= , . 
 
 (6— of 
 
 Given x-}-a + y/2ax-\-x~b. Ans. x== — —? — 
 
 x — ax y/x 
 
 j. 
 
 Given — =-= . Ans. x=; 
 
 y/x x 1 ~ a 
 
 V^+28 V^+38 
 
 Given — z= = — = • Ans. x=4 
 
 "v/x-f-4 y/x-\-Q 
 
 .,. y/x4-2a y/x+4a 1 ab \* 
 
 Given — — r= — — . Ans. x=( j) . 
 
 y/x+b y/x+Zb \a — bj 
 
 3x— 1 V3x— 1 
 
 (100) Given —== = 1+ o • Ans. x=3 
 
 y3x-{-l 
 
 ax—b* y/ax—b 1/. , c 3 \« 
 
 (101) Given =H--^7 ■ An8> ;r =a( 6 +^i ) 
 
 1 t 2 — 4a* 
 
 (102) Given x= a^+xy/b^+x^—a. Ans. x= — ^— 
 
 25 
 
 (103) Given y/b+x4- y/x=—z==., Ans. x=4 
 
 V5+x 
 
 <104) Given a/x+ y/x—Jx— y/x=-yj =• Ans. ^=r^. 
 
 1 1 /r~ n~ 9 
 
 (105) Given -+-=V^+V^+^- 
 
 Ans. x=2a. 
 
1 I I ALGEBRA. 
 
 46 
 
 (106) Given i/lQx+3=7. Ans. x=—. 
 
 (107) Given y/.c — 3:7=10 — V* Ans. x=81. 
 
 — 9 V^-i' — 3 
 
 (108) Given —= — 1 = ^—- . \ ns. x=5. 
 
 V5x+3 
 
 (109) Given h ^/a.v—b=k Vcx+dx—f. Ans. x= — — — J — 
 
 an? — (c-\-d)l<? 
 
 \fa-\-x-\- y/ a — .r 2ay/m 
 
 (110) (liven — = = -Jm. Ans. x=— . 
 
 V«+.r— V«— x 1 + m 
 
 (111) Given V« 2 +c=y ,?**», • Ans. x= s - - 9. 
 
 ' • ' Vrf(.r-|-9) ayu--|-c 
 
 „ mi , nwj (nr-X-mq)(nr — mq\ 
 
 112) (iiven — V/;-x-+fy 2 +- L - =/r. Ans. x= ^ J' K ^. 
 
 ' n J ' * ' w 2>nnpr 
 
 When an equation can never he verified, whatever value we put in the 
 ■ of the unknown quantity, it is said to be impossible ; and when an equa- 
 tion is always verified, whatever value be put for the unknown quantity, it is 
 6aid to be indeterminate. 
 
 CASES OF IMPOSSIBILITY AND [NDEffERMINATION IX EQUATIONS 
 
 OF THE FIRST DEGREE. 
 
 I. Problem. — To find a number such that the third of it, augmented by 75, 
 and five twelfths of it, diminished by 35, shall make three quarters of it, added 
 to 49. 
 
 The equation is 
 
 x 5x 3x 
 
 - + 75 + --35 =1 +49, [1] 
 
 x 5x 3x 
 
 ••• 3+r 2 -T= 9 
 
 .-. 4x-\-5x— 9x=108 
 
 .-. = 108. 
 
 An absurdity. There is, therefore, no value of x which can satisfy the 
 
 equation [1]. 
 
 The impossibility may be rendered evident in the equation [1] itself by re 
 
 ducing the similar terms in the firsl member; thus, 
 
 3x 3x 
 
 T+ 40= T +49. 
 
 It is evident that the two members will always differ by 9, whatever bo the 
 value of x. 
 
 II. Problem. — To find a number such that, adding together the half of it in 
 creased by 10, two thirds of it increased by "J'', and live sixths of it diminish 
 ed by 34, the sum shall lie equal to twice the Bxceas of this number over 5. 
 
 r+io 2(x-f20) r>(./— ::t) 
 
 ... 3x+30 + 4x+80 +. r »x — 170=12x— 60 
 ... 3,-^. ir-f ;,.,— 1 ■.'./•=: 170 — 30 —80 —60 
 I*. <., = 0. 
 The unknown x is, therefore, altogether indeterminate; that is to say, it 
 way be taken equal to 2 or ■''. or any number whatever. 
 
SIMPLE EQUATIONS. 143 
 
 ON THE SOLUTION OF SIMPLE EQUATIONS, CONTAINING TWO OR 
 MORE UNKNOWN QUANTITIES. 
 
 145. A single equation, containing two unknown quantities, admits of an hi' 
 finite number of solutions; for if we assign any arbitrary value to one of Che 
 unknown quantities, the equation will determine the corresponding value of 
 the other unknown quantity. Thus, in the equation y=x-\-lQ, each value 
 which we may assign to a; will, when augmented by 10, furnish a correspond- 
 ing value of y. Thus, if x=2, J' = l~ ; if x=3, y=13, and so on. An equation 
 of this nature is called an indeterminate equation, and since the value of y de- 
 pends upon that of x, y is said to be n function of x. 
 
 In general, every quantity, whose value depends upon one or more quantili/s, 
 is said to be a function of these quantiti :. 
 
 Thus, in the equation y=ax-\-b, we say that y is a function of x, and that 
 y is expressed in terms of x, and the known qualities a, b. 
 
 If, however, we have two equations between two unknown quantities, and 
 if these equations hold good together, then it will be seen presently that wo 
 can combine them in such a manner as to obtain determinate values for each 
 of the unknown quantities ; that is to say, each of the unknown quantities will 
 have but a single value, which will satisfy the equations. The equations in 
 this case are called determinate. 
 
 In general, in order that questions may admit of determinate solutions, we 
 must have as many separate equations as there are unknown quantities ; a 
 group of equations of this nature is called a system of simultaneous equations. 
 
 If the number of equations exceed the number of unknown quantities, un- 
 less the equations in excess conform to the values of the unknown quantities 
 determined by the others, the equations are said to be incompatible. Thus, 
 if we have x-\-y = l0 and x — y = G, the only values of x and y which will satisfy 
 both these equations are 8 for x, and 2 for y. Now, if we were to add an 
 other equation to these, it must conform to these values, and could not be 
 written in any form at pleasure. Thus, wo might for a third equation say 
 zy =16 ; but we could not write .r?/=100, for this third equation would be in- 
 compatible with the other two.* 
 
 * Equations may be incompatible when the number does not exceed the number of un- 
 knowns, as the following problem will show : 
 
 A sportsman was asked how many birds he had taken. He replied, if 5 be added to the 
 third of those I took last year, it will make the half of the number taken this year. But if 
 from three times this last half 5 be taken, you will have precisely the number taken last 
 year. How many did he take in each year ? 
 
 Let x= the number this year, and y= the number last year. 
 
 x ii . 3x 
 -=-+5, y— 5. 
 
 2 3 ~ ' J 2 
 
 Substituting in the first the value of y in the second, 
 
 x x 5 , 
 2 2 3 ~ 
 .-. 3.1- — 3j=30 — 10 
 =20; 
 an absurd equality, whence we conclude that there exist no values of .r and y which satisii 
 the two equations. 
 
 This is because the conditions of the problem are inconsistent with each- other. When, 
 however, the two equations are derived from the same problem, and its conditions <u"o not 
 rontradictory, values for x and y wiV always be found to satisfy them. 
 
144 ALGEBRA. 
 
 146. In order to solve a system of two simple equations c '■■> '• ing two un- 
 known quantities, we must endeavor to deduce from them a single equation 
 containing only one unknown quantity; we must, therefore, make one of the 
 unknown quantities disappear, or, as it is termed, we must eliminate it. The 
 equation thus obtained, containing one unknown quantity only, will give the 
 value of the unknown quantity which it involves, and, substituting the value of 
 this unknown quantity in either of the equations containing the two unknown 
 quantities, we shall arrive at the value of the other unknown quantity. 
 
 The process which most naturally suggests itself for the elimination of one 
 of the unknown quantities, is to derive from one of the two equation? an ex- 
 pression for that unknown quantity in terms of the other unknown quantity, 
 and then substitute this expression in the other equation. We shall see thai 
 the elimination may be effected by different methods, which are more or less 
 simple according to the nature of the question proposed. 
 
 example I. 
 Let it bo proposed to solve the system of equations 
 
 y— *= c (i)? 
 
 y+x=\2 (2) $ 
 
 147. First Method. — From equation (1) we find the value of y in terms 
 of x, which gives i/=x-f-6 ; substituting the expression x-\-G for y in equation 
 (2), it becomes x+G-f-x=12, from which we find the determinate value x=3 ; 
 since we have already seen that y=x-\-G, we find also the determinate value 
 y=s3+6 or 9. 
 
 Thus it appears, that although each of the above equations, considered sep- 
 arately, admits of an infinite number of solutions, yet the system of equations 
 admits only one common solution, x=3, y=9. 
 
 148. Second Method. — Derive from each equation an expression for y in 
 terms of x, we shall then have 
 
 2/= a: -f-6 
 y=12— x. 
 
 These two values of y must be equal to one another, and, by comparing 
 them, we shall obtain an equation involving only one unknown quantity, viz., 
 
 x+6=12 — x. 
 Whence 
 
 x=3. 
 Substituting the value of x in the expression y=x-\-G, we find y=9. 
 The substitution of 3, the value of x, in the second expression, 2/ = 12 — x, 
 leads necessarily to the samo value of y ; thus, 12 — 3=9, for we derived the 
 value of x from the equation x-f-G = 12 — X. 
 
 149. Third Method. — Since the coefficients of y aro equal in the two 
 equations, it is manifest that wo may eliminate y by subtracting Hie tico equa- 
 tions from each other, which gives 
 
 (y+.v)-(y—v) = ii-6. 
 Whence 
 
 2x=6 
 *ss3. 
 Having thus obtained tho value of r, we may deduce that of y by making 
 r=3 in either of tho proposed equations; wo can, however, determine the 
 
SIMPLE EQUATIONS. 145 
 
 •alue of y directly, by observing that, since the coefficients of x in the proposed 
 equations are equal, and have opposite signs, we may eliminate a: by adding 
 the two equations together, which gives 
 
 Whence 
 
 7/ = 9. 
 
 If we examine tho three above methods, wo shall perceive that they con- 
 sist in expressing that the unknown quantities have the same values in both 
 equations. 
 
 These methods have derived their names from the processes employed to 
 effect the elimination of the unknown quantities. 
 
 The first is called the method of elimination by substitution. 
 
 The second is called the method of elimination by comparison. 
 
 The third is called the method of elimination by addition and subtraction. 
 
 Tho rule for the first is to find the value of one of the unknown quantities in 
 one of the equations, and substitute it in the other equation. 
 
 For the second, is to find the value of the same unknown quantity in each of 
 the two given equations, and set these values equal. 
 
 And for tho third, is to make the coefficient of the unknown quantity to be 
 eliminated the same in the two equations, and add or subtract as the case may 
 require. Add, if the signs of the equal terms are different, and if they are 
 alike, subtract. 
 
 By either of these rules a single equation, containing but one unknown quan 
 tity, is obtained. 
 
 EXAMPLE II. 
 
 Take the equations 
 
 2x+3y=13 i (1) > 
 
 5.r-j-42/=22 (2) S 
 
 1°. Eliminating by substitution. 
 From equation (1), we find 
 
 13— 2x 
 
 Substituting the value of y in terms of x in equation (2), it becomes 
 
 13 — 2x 
 5z+4x — 3 — =22; 
 
 *n equation containing x alone, which, when solved, gives 
 
 x=2. 
 This value of x, substituted in either of the equations (1) or (2), gives 
 
 y=3. 
 
 2°. Eliminating by comparison 
 
 13— 2x 
 From equation (1) 2/r= — . 
 
 22— 5x 
 From equation (2) y= — - — . 
 
 13— 2x 22— 5.T 
 Equating these values of y, — - — = — -z — ; an equation containing xonry 
 
 K 
 
146 ALGEBRA. 
 
 Whence 
 
 -O 
 
 Substituting this value for x in either of the preceding exj ressions lor y 
 we find 
 
 2/ = 3- 
 3°. Eliminating by subtraction. 
 
 In order to eliminate y, we perceive that if we could deduce from the pro- 
 posed equations two other equations in x and y, in which the coefficients of y 
 should be equal, the elimination of y would be effected by subtracting one of 
 these new equations from the other. 
 
 It is easily seen that we shall obtain two equations of the form required if 
 we multiply all the terms of each equation by the coefficient of y in the other. 
 Multiplying, therefore, all the terms of equation (1) by 4, and all the terms of 
 equation (2) by 3, they becomo 
 
 8ar+12y=52 
 15*4-12^=66. 
 Subtracting the former of these equations from the latter, we find 
 
 7x=14. 
 Whence 
 
 z=2. 
 In like manner, in order to eliminate x, multiply the first of the proposed 
 equations by 5, and the second by 2, they will then become 
 
 10x+15?/=G5 
 10*4- 8y=44. 
 Subtracting the latter of these two equations from the former, 
 
 7y=2l. 
 
 Whence 
 
 y=3. 
 
 In order to solve a system of three simple equations between three unknown 
 quantities, we must first eliminate one of the unknown quantities by one of the 
 methods explained above ; this will lead to a system of two equations, con- 
 taining only two unknown quantities ; the value of these two unknown quan 
 tides may be found by any of tho methods described in the last article, and 
 substituting tho value of these two unknown quantities in any one of the original 
 equations, wo shall arrive at an equation which will determine the value of the 
 third unknown quantity. 
 
 EXAMPLE III. 
 
 Take the system of equations 
 
 3.r4-2y+ Z=16 (\)\ 
 
 2z4-2y+2z=18 (2) C 
 
 2x+2y+ r=14 (3) ) 
 
 1°. Eliminating by substitution. 
 From equation (1), we find 
 
 r = lG — 3.r— 2y (1). 
 
 Substituting this value of; in equations (2) and (3), thoy become 
 2x+ 2y+2(16— 3x— 2y)=18 . . . ('») ? 
 2ar+2y4- (16— to— 2y)=14 . . . (6) S 
 these last two equations contain x and y only, and. if treated according to any 
 of tho ubove methods, will give us 
 
 x=2, v=3. 
 
SIMPLE EQUATIONS. 147 
 
 Substituting these values of x and y in any one of the equations ( ), (2), (3), 
 4), wo find 
 
 2°. Eliminating by comparison. 
 
 In order to eliminate z? derive from each of the three proposed equations a 
 value of z in terms of 2 and y ; we then have 
 
 2 = 1G — 3.c— 2y 
 2= 9 — x — y 
 2=14 — 2.r — 2y ; 
 equating the first of these values of z with the second and with the third In 
 succession, we arrive at a system of two equations : 
 
 16— 3x— 2y= 9 — x— yt 
 16 — 3x—2y = li—2x—2y \ 
 containing x and y only ; these equations give 
 
 x=2, 2/=3; 
 these values of x and y, when substituted in any of the three expressions for 
 r, give 
 
 2 = 4. 
 
 3°. Eliminating by subtraction. 
 
 In order to eliminate z between equations (1) and (2), 
 
 3x+2y-\- z _i6 
 2x+2y+2z = 18; 
 we perceive that, in order to reduce these equations to two others in which 
 the coefficients of z shall be the same, it will be sufficient to divide the two 
 members of the second equation by 2, for we thus have 
 
 Subtracting this from the first equation, 
 
 3.r+2?/+2 = 16, 
 we find an equation between two unknown quantities, 
 
 2x+y=7 (a). 
 
 Tn order to eliminate z between equations (1) and (3), 
 
 3a;+2?/+2=16 
 2x+2y+z = U. 
 Subtract the latter from the former, which gives 
 
 the substitution of this value of x in equation (a) gives 
 
 y=3, 
 and the substitution of these values of x and y in any of the proposed equa 
 tions gives 
 
 2 = 4. 
 
 The particular form of the proposed equations enables us to simplify the 
 above calculation ; for if we subtract equation (3) from equations (1) and (2) 
 in succession, we have 
 
 (3x+2y+ 2)— (2x+2y+z)=16— 14, whence xz=2 
 (2x+2y+2z) — (2x+2i/+2) = 18— 14, whence 2=4 ; 
 and substituting these values of .r and z in any of the proposed equations, we 
 find 
 
 y=3. 
 
U8 ALGEBRA. 
 
 In order to solve a system of four equal ions between four unknown qua»ti 
 we reduce this case to the last by eliminating one of the unknown quantities. 
 We thus arrive at ;i Bystem of three equations between three unknown quan- 
 tities, from which the value of these three unknown quantities may be found. 
 Substituting these values in any one of the equations which involve the othe; 
 unknown quantity, we deduce iVom it the value of that unknown quantity. 
 
 EXAMPLE IV. 
 Take the system of equatii 
 
 *■'■+.'/ + = + «=" (I)) 
 
 x+y+z- t= 4 (-2) 
 
 .(•-p-// — Z + LV = 11 (3) 
 
 ■'—'/+-+ (4) J 
 
 The first equation give 
 
 t=U—x—y—~ (5). 
 
 Substituting this exp n i'w t in the three other equations, we find 
 
 x+ V+ z=; 9 (G) 
 
 x+ y+3z=n (7) 
 
 *+%+ 2=12 (8). 
 
 In order to solve these three equations betwt ;, we find from the 
 
 first 
 
 r = 9_.r-v/ (9) ; 
 
 and substituting this value of; in the two ther equations, they become 
 
 *+2/=-5 (10) 
 
 2/=3 (11) 
 
 Whence x=2 (12). 
 
 Substituting the values of x and y in equation (6), we find 
 
 z=4 (13). 
 
 Substituting these values of a:, y, z in any of the first five equations, wo find 
 
 t—b. 
 We can arrive at the same result more simply by subtracting equation (1) 
 from the three following in succession ; we shall thus find 
 
 2i=14 — 4, 2z— *=14— 11, -J//— 2<r=14— 18; 
 the first of these three new equations gives i=5 ; this value of/, substituted 
 in the two other equations, gives z = 4, y = 3 ; and substituting these values of 
 y, z, t in any one of the original equations, wo End ..= •.'. 
 
 By following a process of reasoning analogous to the above, wo shall be able 
 to resolve a system of any number of equations of the first degree, provided 
 there be as many equations as unknown quant i' 
 
 It frequently happens that each of the proposed equations do not involve ah 
 the unknown quantities. In this case, a little dexterity will enable us to effect 
 the elimination very quickly. 
 
 KXAMPLK V. 
 
 Take tho system of equations 
 
 '/— :: - y+2: = 13 (1) 
 
 41 — Jr=30 (-'I 
 
 47/4-~': = 14 (3) 
 
 5i/ +3* =32 (4) 
 
 Upon examining theso equations, wo perceive thai the elimination of z be 
 
SIMPLE EQUATIONS. 149 
 
 tweeu equations (I) and (3) will give an equation in x and y, and "that the 
 elimination of t between equations (2) and (1) will give a second equation in 
 r and y. These two unknown quantities may thus be easily determined : 
 The elimination of z between (1) and (3) gives .... 7y — 2x= 1 
 The elimination of t between (2) and (4) gives .... 20?/ -f 6.r=: 
 Multiply the first of these equations by 3, and then add 
 
 them, wo have 41i/=41 
 
 Whence y= 1 
 
 Substituting the value of y in 7y—2x=zl, we have . . . x= 3 
 
 Substitute this value of x in (2), we have At — 6=30 
 
 Whence '=9 
 
 Finally, the substitution of the value of y in (3) gives . . ~= 5 
 
 The following general rule may be given for a system of any number of 
 equations : 
 
 Eliminate one of the unknown quantities by combining the first equation 
 with each of the others, or by combining them all in any way in separate 
 pairs. The number of equations and of unknown quantities is thus made one 
 less. Proceed with these in the same way till there is but one equation and 
 one unknown quantity. Having found the value of this, substitute it in a pre- 
 ceding equation containing but two unknown quantities, which will then have 
 but one, whose value may be found. Substitute the values of the two un- 
 known quantities thus found in an equation immediately preceding, containing 
 only three, and so on, till all the values of the unknown quantities are obtained. 
 
 We have seen in the method of elimination by subtraction that, in order to 
 render the coefficients of the unknown quantity the same in both equations, 
 we must multiply each of the equations by the coefficient of the unknown 
 quantity, which it is required to eliminate, in the other. If the coefficients of 
 the unknown quantity have a common factor, this operation may be simplified; 
 thus 
 
 EXAMPLE VI. 
 
 'T'ake the system of equations 
 
 12.r+32?/=340 (1) > 
 
 8x+ 24?/ =2.34 (2) S 
 
 In order to render the coefficients of y equal, observe that 32 and 24 have a 
 common factor, 8 ; it will suffice then to multiply equation (1) by 3, and equa- 
 tion (2) by 4 ; they then become 
 
 36.?;-4-9G.y=1020 
 32.1- -f96?/=l 016. 
 Subtracting the latter from the former, 
 
 4.r=4 
 .r=l. 
 Again, in order to elimiuate x, since 12 and 8 have a corirnon factor, 4, it 
 will suffice to multiply equation (1) by 2, and equation (2) by 3 ; we then have 
 
 24ar+64y=680 
 24x+72y=762. 
 Subtracting the former of these two equations from the latter, we have 
 
 82/=82 
 2/=10{. 
 
150 ALGEBRA 
 
 (7) Given x+y = 15 (1) 
 
 x-y= 7 (2) 
 
 Ans. x=ll, 7/= 4. 
 
 (8) Given x+y = 10 (1) 
 
 2x—3y = 5 (2) 
 
 Ans. x=7. y = 3. 
 
 (9) Given 2x+3y= 13 (1) 
 
 5x+4y=22 (2) 
 
 Ans. x =2, 1/ = 3 
 
 (10) Given x=4y (1) 
 
 2x-f 3i/=44 (2) 
 
 Ans. x=lG, 2/ = 4. 
 
 (11) Given 2x+3?/ = 70 (1) 
 
 4x+5y=l30 (2) 
 
 Ans. a-=20, y = li) 
 
 (12) Given 3x—5t/=13 (1) 
 
 2.r+72/ = 81 (2) 
 
 \ns. .r=16, i/^7. 
 
 (13) Given ll.r-f 3?/ = 100 (1) 
 
 4x—7y= 4 (2) 
 
 Ans. .r=8, y= 1 
 
 (14) Given |+|=7 (1) 
 
 2 ' 3 
 
 x y 
 
 -4-- 
 3^2 
 
 x y 
 
 Ans. :r=6. J/= 12. 
 
 (15) Given '^+72/=99 (1) 
 
 f+7s=51 (2) 
 
 Ans. arr=7, ?/= 1 4 
 
 7u 
 
 (16) Given 3<+ — =22 (1) 
 
 °t 
 11m— ^=20 (2) 
 
 . / = •">. l/ = 2. 
 
 (17) Given.r-f l:?/::5:3 (1) 
 
 7_L;r 5_ 7/ 42 2.r — 1 
 4 2 -12 4 ^ 
 
 Ans. x=4, ?/ = 3 
 ja 9s 
 
 g"+Io' 
 
 2r 4s 
 (18) Given -+— =G4 (1) 
 
 -4-— =77 (0) 
 
 : i in v ' 
 
 Ans. r=60, s = 30 
 
 (19) (liven 5p+fcr=131J (1) 
 
 13/)— ff=142j (2) 
 
 .p=16J$$, r = : 
 
SIMPLE EQUATIONS. 161 
 
 (20) Given 6?*— 14^=5^+119? (1) 
 
 7^+140=2^ (2) 
 
 Ans. *=— 24.07, V=— 14.24 
 
 (21) Given 9x=4.r' (1) 
 
 .r+.r'=2G (2) 
 
 Ans. x = 8, a:'=18. 
 
 (22) G^T^T = 8i + Z (1) 
 
 21z 1 + 28z 2 =334 (2) 
 
 Ans. z 1 =61 T , T S 9, z 2 = — 33i|f, 
 
 V + 3 n ti\ 
 
 (23) Given 2a:— ^-=8 (1) 
 
 *_*=!_*»-*£ » 
 
 Ans. ar=5, y=5. 
 
 ?/ -8 3.r+4y+ 3 2.r+7- y 
 
 (24) Given 5+— g- = -Jq J5 ■ • • • W 
 
 7.r+6 9y+5.r- 8_ .r+y 
 11 ~ 12 4 l ' 
 
 Ans. .r=7, y=9. 
 
 ,25) Given (x+5)(7/+7) = (x+l)(y-9) + 112. ... (1) 
 
 2:r+10=3y+l (2) 
 
 Ans. x=3, y=5. 
 
 o-j. •;/ 3v 1 
 (26) Given— —4+--+.r=S—j+Y2 U) 
 
 2_-+2=|— 2z+6 (2) 
 
 Ans. x=2, y=7 
 
 x _2 10— .r y— 10 
 (27) Given — g —^—=—j— (1) 
 
 2.V+4 2.r+y g+13 
 
 3 "~ 8 "~ 4 V ~ ; 
 
 Ans. x=7, y=10 
 
 2y 8.r— 2 4+?y a;— y . 
 
 z:3y::4:7 (2) 
 
 Ans. £=12, y=7. 
 
 4y 
 15x4- — 
 3y— 2+.r n , 3 
 (29) Given x ^-j^ =1 + — 33— (!) 
 
 3.r+2y y-5 llx+152 3y+l 
 
 6 "" 4 " 12 2 * * ' K ~> 
 
 Ans. x=8, y=9 
 
 25+51/ 7x— 6 , A 3x— 1 0+7y 
 (30) Given 1 + — p^ 3— =10 jg- - • • U) 
 
 .(2) 
 
 Ans. .r=3, y=7 
 
 __ :D ,.____ :: l:8 (~) 
 
152 
 
 ALGEBRA. 
 
 Ax by 9 
 (31) Given* — + 4=- — 1 
 ' x 2 y" y 
 
 5 4_7 3 
 
 x'y x'2 
 
 (1) 
 (2) 
 
 \32) (Ihi'i) 5x+7#=43 . 
 llx+<ty=G9 . 
 
 (33) Given 8x— 21y= 33 
 6x+35?/=177 
 
 Ans. x=4, ?/=2 
 
 ."CO J 
 
 • (2) S 
 
 Ans. x=3, y=4 
 
 .(2)5 
 
 Ans . .r = 1 J , y =3. 
 
 2x v 3?/ 1 
 
 (34) Given T -4+^+x=8-^+- 
 
 y x 1 
 
 i-2+ 2 = 6- 2 *+ 6 - 
 
 (1) 
 
 
 (35) Given x- 
 
 3x+5.?/ 
 
 fl7=5y-| 
 
 4x+7 
 
 • (2) 
 
 Ans. x=2, ?/=7 
 
 • (1)1 
 
 (2) 
 
 17 ' * l 3 
 
 22— Gy 5x— 7_x-fl 8?/+5 
 
 3 IT~~ 6 ~ ~T8~ 
 
 Ans. .r=8, y=2. 
 
 36) Given ax+ by=c '. . . (1) ) 
 
 fx+gy=h . . (2) $ 
 
 — fr/i a/t — c/" 
 
 Ans. x==- ' 
 
 (37) Given x-\-y=s . 
 x — y—d 
 
 'ag—bf y ag—bf 
 
 (1) 
 (2) 
 
 s-\-d s — d 
 
 Ans. x=— — , y=—z~ 
 
 (38) Given x-\-y=s (1) 
 
 bx=ay (2) 
 
 as bs 
 
 Ans. x= — — , 2/=- 
 
 ■ a +b ,:f —a+b 
 
 (39) Given ax+by=c (1) 
 
 mx — ny=d (2) 
 
 nc-\-lnI vtc — ad 
 
 Ans. x= j — T , w= - r . 
 
 na-\-mb J na-\-mb 
 
 (40) Given 7ax=4& (1) 
 
 2cx + 3dy=4c (2) 
 
 46 28ac— 8bc 
 
 Ans. x=— , y= — - — -. — . 
 la J 2lad 
 
 (41) Given bcxsacy— 26 (1) 
 
 a(J — V) 2b 3 
 
 be 
 
 a4-2b 
 
 Ans. x=i-, v= • 
 
 ic • c 
 
 * These should not bo cleared of fractions, bat tbo unknown fractions bo climi 
 
 Bated by making Uicia Blike, and subtracting. 
 
y— -r-=& W 
 
 SIMPLE EQUATIONS. 153 
 
 a b , 1X 
 
 :42) Given — — =— — - 11) 
 
 ax+2by=d . . . . : (2) 
 
 f Z_Ga 2 +26 2 3a 2 +</-6 * 
 Ans. z= ^ , 2/= T7£- 
 
 (43) (iiven x— •— 7— = c V 1 ) 
 
 a — x 
 
 6~ 
 
 a_a6+6 2 c+6d g+<z&— 6c+6 a rt 
 Ans. xz= ^p , y—~ ¥+ ! ""• 
 
 a-4-46 2a— 36 ,-v 
 
 (44) Given— L . — =t I 1 ) 
 
 v ' ?ft+a: 3m, — 2/ 
 
 5ax—2byz=c ( 2 ) 
 
 #(c 3 — 6 3 ) 26 3 , m 
 
 (45) Given ^+^-^- T =c 3 .r (1) 
 
 b(cx+2)=cy i (2) 
 
 Ans. a;=^. 2/ — — ~ 
 
 336 
 
 (46) Given 17a;— m+(6+10/)2/=/ 3;c ^ 
 
 4.^+52/=^^ ( 2 ) 
 
 a b /n 
 
 (47) Given -+-=?» I 1 ) 
 
 2/ 
 - C +^=n (2) 
 
 6c — ad be — ad 
 
 Ans ' x =nb~^d: 2/= ^=^* 
 
 ,48) Given x -\-y =s (!) 
 
 x^—y^—d ( 2 ) 
 
 s*+d s 2 — </ 
 Ans. .r=-^-, ys=-jg-. 
 
 (49) Givena.-+2/:a::a;— 2/:6 (1) 
 
 z 2 _2, 2 = C < 2 > 
 
 a-4-6 Jc " — 6 /c 
 
 Ans. x=- ¥ - y J~ b , y=-2-yJZb' 
 
 (50) Given .r+ y^-f 7y=a (1) 
 
 x+ ■ y /x*—yz=zb (8) 
 
 a 3 +6 2 a6(a— 6) 
 
 Ans - - r =5^+6j' ^ = a+6 
 
 151) Given x 2 +a:2/=« (*) 
 
 2/ 2 4-.ri/=:6 ( 2 ) 
 
 a 6 
 
 Ans. x= — , y= — ■ 
 
 ■y/a + 6 " V a-\-b 
 
154 
 
 ALGEBRA. 
 
 (52) Given 2:r+3y+4z = 16 
 
 3x+2y— 5z= 8 
 5x—6y+3z= 5 
 
 (53) Given 5x—Gj-\-iz = 15. 
 
 7x+4y— 3z=19, 
 2ar+ 2/ + Gz=46, 
 
 (1) 
 (2) 
 (3) 
 
 Ans. x=3; 2/ = 2; r = l. 
 
 (1) 
 (3) 
 
 (54) Given* -+- =a 
 x y 
 
 X ' z 
 
 1 1 
 
 -+-=c 
 y ' z 
 
 Ans. z=3; 2/=4 ! ~ = t>. 
 . ..(1) 
 
 Ans. xz=- 
 
 (2) ■ 
 
 (3) 
 
 o 
 
 ; 2/= 
 
 rt-|-6 — c a — b-\-c b-i-c — a 
 
 (55) Given x+y=36 ; 1+2=49; 7/-fz=53. 
 
 Ans. z=16; y=20; z=33 
 
 (56) Given r+ttf+z =30; »+«?— z=18; f— ?r+: = 14. 
 
 Ans. r=16; ?r=8; z=6 
 
 (57) Given u+Jt) =164; t>+fw=82; 7/ + ]w = 136. 
 
 Ans. u=128; t>=72; 10=40. 
 
 (58) Given ax-\-by=c; my-\-nz=j) ; fx-\-gz=zq. 
 
 Ans. x= 
 
 bnq-\-cgm — b<zp 
 agm-\-bfn ' 
 agp-\-cfn — anq 
 -*~ agm-\-bfn ' 
 •\-hfp — cfm 
 
 agm + bfn 
 
 (C9) Given 3(ax+by)=z ; 5y=7(x+3a) ; llarssfz-f 121. 
 
 o + 189a& 
 
 Ans. x = 
 
 440— 45a — 636' 
 6776+1848a — 189a" 
 y_ 440— 45a— 636 ' 
 _14520«4-5544<zt + 203286 
 440— 45a— 63i ' 
 
 7 5 9 11 13 15 
 (60) Given —=-; j-j^S 7=yZl3' 
 
 Ans. r=— 10-;:; y=-34 ■.. : : = -32. 
 
 a_|_Z» /; — c />-}-'' r — </ i/-\-J: lc — h 
 '61) Giveu — : — = ; ; r-. — = ; -r-: — =7 . 
 
 ' a-\-x b — y b-\-y r — z </-4-; l;—x 
 
 * Do net clear 1 is, but mako --, . 4.C., the unknown quantities. 
 
SIMPLE EQUATIONS. 
 
 155 
 
 (62) Given 2.i- 
 
 3y 1 1 
 
 7:c—5z=y -\-x - 
 x v z 
 3+1 +4 = 58 
 
 • 86 
 
 (1) 
 
 (2) 
 
 (3) 
 
 Ans. £=48; v=54 ; 2=64. 
 
 (63) Given 6x— 4y+5z=2fi 
 
 Ax-\-2>y — 7z = l\ . 
 12x—Gy — 32=3 j . 
 
 (1) 
 (2) 
 (3) 
 
 164) Given 18x—7y—5z = 11 . 
 4y-fc+a§*=108 
 
 (65) Given y+~+ 5 
 
 Ans. x=l; y = \; z=J. 
 
 ... (1) 
 . . . . (2) 
 , . . . (3) 
 
 Ans. .r=12; i/=25; z=6. 
 
 3 o 
 
 x—1 y—2 2 + 3 
 ~ 4 5~ : 10 
 
 fy— 5 
 
 -l 3 — - 
 " ■» 12 
 
 (66) Given ^ + | + ~ 58 
 
 5.r y z 
 7 + 6 + 3 
 
 76 
 
 X 32 u 
 
 2+-8-+5= 79 
 
 (1) 
 
 (2) 
 
 (3) 
 =5 
 
 (1) 
 (2) 
 
 (3) 
 
 y = 7; is. 
 
 (4) 
 
 :30; 2=168; w=50. 
 
 ^ + z + M =248 
 
 Ans. .r==12 ; 2/= 
 
 (67) Given 7x— 2z + 3u=17 (1) ' 
 
 4y—2z+ l = \\ (2) 
 
 5y— 3a:— 2«ss 8 (3) 
 
 4y—3u+2t— 9 (4) 
 
 3z + 8w = 33 (5) 
 
 Ans. £=2; ys=4; z = 3; u=3 
 
 ( = 1. 
 
 Elimination may be effected in a general form, and particular cases be re- 
 solved by substitution in this form. 
 
 We shall illustrate this with a system of three equations. 
 
 Given ax -\-by + cz +^ =°» 
 
 a'x + b'y +c'z +k' =0, 
 a"x+b"y +c"z+k"=0. 
 
 Eliminating among these three equations by any of the foregoing methods 
 v?e find 
 
 (b"c' —b'c")k + (be" —b"c)lc' + (b'c —bc')k" 
 
 .Ts= 
 
 (a'b"—a"b')c+{a"b—ab")c' + (ab' —a'b)c'" 
 
15G ALGEBRA. 
 
 (a'c"— a"c')k + {a"c—ac")k' + (ac' —a'c)k" 
 J The same denominator as in the value of x ' 
 
 (a"b'—a'b")k-{- {ab"—a"b)k' -f {a'b — ab') k" 
 ^ The same denominator as in the value of x 
 To apply this general form to a particular case, take (Example 53) above. 
 
 = ( 1X— 3— 4X6)(— 15) + (— OXC— 1X4)(— 10j + [-»X4— (— 6X- J 3)](— 46)_lg57 
 C (7X1— 2X4)4+(2X— 6— 5X1)(— 3)+(5X4— 7X— I — 419 ' 
 
 ( 42+C)( — 15) + (8-30)( — 19) + ( — 15— 28)(— 4G) _1C7C 
 y ~ 419 ~ -lliJ ~ 4 ' 
 
 (l)(-15) + ( + 17)(-19) + (-C2)(-46) 2514 
 Z — 419 — 419 
 
 Changing the signs of k, k', k", in order that they may be positive in tne 
 second member of the three proposed equations, and performing the multipli- 
 cations indicated in the general values of x, y, and z, they may be written as 
 follows : 
 
 kb'c" —kc'b" +ck'b" — bk'c" + bc'k" — cb'k" 
 
 X= ab'c" —ac'b" + ca'b" —ba'c" + bc'a" — cb'a'" 
 
 ak'c" —ac'k" +ca'k" —ka'c"+kc'a" —ck'a" 
 
 " The same denominator as that of x ' 
 
 ab'k" — ak'b" -f ka'b" — ba'k" -f bk'a"—kb'a" 
 
 The same denominator as before 
 
 By observing carefully the composition of the formulas for two and three 
 equations, we may discover general rules by means of which wo can calcu- 
 late the formulas suitable for any number of equations. 
 
 First Rule. — To find the common denominator in the values of all the 
 unknown quantities. With the two letters a and b form the arrangements 
 ab and ba, then interpose the sign — between them, thus : 
 
 ab — ba. 
 
 If there are but two equations to resolve, placo an accent on the 2° letter 
 of each term, and the result, ab' — ba', will be the common denominator of 
 the values of x aud y. 
 
 If there are three equations, pass the letter c through all the places in each 
 term of the expression ab — ba, taking caro to alternate the signs ; ab will thus 
 givo abc — acl-\-cab ; also, — ba will give — bac-\-bca — cba, and tho whole 
 
 abc — acb-\-cab — bac-\-bca — cba ; 
 then place one accent on the 2° letter of each term, and two on the 3 ', and the 
 resulting expression will be tho common denominator of the values of.r, y, and z. 
 
 If there are four equations, take the letter </, which is the coefficient of the 
 fourth unknown u, and pass it through all the plans in each term of tho sexi- 
 nomial above formed, taking caro to alternate the si^ns of the terms furnished 
 by each of them, beginning with -f- for thoso which result from a term pro- 
 ceded by the sign -f-, and with — for those resulting from a term affected 
 with tho sign — ; finally, place ono accent mi the 2° letter, two on the 3", and 
 three on the 4". Tho resulting polynomial is the common denominator of the 
 four unknown quantities x, y. z, u. 
 
 ab'cf'd' ' — b'd"d"+ad'h"t?"— da'b"&" 
 
 —ac'b"</'" + ,!y,/ , h"— l „lr'ir+ l /„'c"b'" 
 ±ca , l>">i—ca'd"b"' + cd'a"b' — b'" 
 
SIMPLE EQUATIONS. 157 
 
 — ba'c"d"' + ba'd"c'" — bd'a"c'"-\- db'c c'" 
 + bc'a"d"'—bc'd"a"'+bd'c"a'"—db'c"a" 
 —cb'a"d'" -\-cb'd"a'" —cd'b"a"' +dc'b" a' ' . 
 
 if there lie a greater Dumber of equations, proceed in the same manner. 
 
 Second Rule. — The numerators may be derived from the common de 
 nominator. For this purpose, it is only necessaiy to replace, without touch 
 ing the accents, tho letter which serves for coefficient of, the unknown quanta 
 ty we wish to find, by the letter k, which represents tho known term in the 
 second member. Thus : change a into k, to have the numerator of x ; b into 
 k, to have that of y ; and so on. 
 
 There remains still a method of elimination to bo mentioned, which alone 
 is applicable to equations of higher degrees, as well as to those of the first. It 
 u called the method of tho common divisor. It consists, where two equations 
 are given, in dividing one by the other (after transferring all the terras to the 
 first member in both), that divisor by the remainder, and so on till tho letter 
 jf arrangement, which must be one of the unknown quantities, is exhausted 
 from the remainders. The last remainder containing but the other unknown 
 quantity, being put equal to zero, will present an equation from which the first 
 unknown quantity is eliminated. 
 
 If there bo three or more equations, eliminate one of tho unknown quanti- 
 ties in this way between the first and second, then between the first and third, 
 and so on. 
 
 Tho reason which may be given for this rule here, though a better one win 
 be furnished hereafter, is, that the dividend being zero and the divisor zero, 
 the remainder must be zero. 
 
 Let us apply this method to Example (8) above. The two g'ven equations are 
 
 X+ y— 10 = 
 2x—3y— 5=0. 
 
 Elimination, 
 
 2x—3y — 5 
 2x4-2^—20 
 
 z+y-io 
 
 -57/+15 -4-5. 
 - t/4- 3 =0 .-. y=3. 
 Substituting this value in x-\-y — 10 = 0, we obtain z=7. 
 
 EXAMPLE II. 
 
 Given.r- 3 4-3 i y.i-4-3/.r— 98=0 (1) 
 
 3?-\-4yx —2y~ —10=0. 
 Elimination, 
 
 x*-\-2yxi-\- 2y 2 x— 98 
 x*-\-iyx*— Zyix— 10.r 
 
 x--{-4yx — 2^2 — io 
 
 x—y 
 
 — yx--{- 5y*x-\- lOx — 98 
 
 — yx- — 4//2.r-j- 2,7/ 3 -f-10# 
 x*-\- Ayx— Zy? — 10 
 
 9//2+10 
 
 9y*x-\- 10 x— 2?/ 3 — Wy— 98 or 
 {9y* -j-10) x— 2?/3— 10?/— 98 
 
 (9^4-10):^-f(36y3-J_40y)j>_i8y«— 110^2— 100|a;4-]ty34-25$r+49 
 
 (9yg+10)a^— ( 2y3-f.iQy 4.98)3; 
 
 (WyZ+SQy -{-98);e— 18y*— HOyS— 100---2 
 (19yS-j-25y 4-49);r— 9yi— 55^3— 50 
 
 . 9yH- 10 
 
 (9y s +10)(19y3_L.o 5 ^ +49)2:— 81y6—585y*— 1000^— 500 
 (9^2+10) (19 ;/*-f25y -|-49).r — 38?/"— 2 40y*— 1060?/ 3 — 250?/?- .2P40?/— 4 801! 
 
 — 43^i— 345^-«4-1960^3— 750j/--j-2!)40?/4-4e0? 
 
153 ALGEBRA. 
 
 This last remainder, put equal to zero, will make an equation from whiih x 
 is eliminated, and which contains only y. It is called the final equation. 
 
 ON THE SOLUTION OF PROBLEMS WHICH PRODUCE SIMPLE 
 
 EQUATIONS. 
 
 150. Every problem which can be solved by Algebra includes in its enun- 
 ciation a certain number of conditions of such a kind that, in taking at pleasure 
 values for the unknown quantities, it is always easy to see whether or not they 
 will verify these conditions. In the greater part of questions in Algebra, these 
 verifications consist in this, that, after having effected certain operations upon 
 the values of the known and unknown quantities, we ought to arrive at equali- 
 ties. This being understood, if the unknown quantities bo represented by 
 letters, algebraic expressions may be formed in which shall be indicated, by 
 means of signs, all tfee calculations necessary to be made, as well upon the un- 
 known numbers as upon the known, to find the quantities which ought to be 
 equal. Consequently, joining these expressions by the sign of equality, we 
 shall havo one or more equations, which will be satisfied when the true val- 
 ues of the unknown quantities are substituted in the place of the letters which 
 represent them. 
 
 Reciprocally, when all the conditions of the problem are expressed in the 
 equations, the values of the unknown quantities which satisfy these equations 
 must certainly satisfy the enunciation of the problem. 
 
 It is impossible to give a general rule which will enable us to translate eve- 
 ry problem into algebraic language ; this is an art which can bo acquired by 
 reflection and practice alone. Two rules which may be of some service are 
 the following : 1. Indicate upon the unknown quantities represented by letters, 
 and upon the known quantities represented either by letters or numbers, the same 
 operations as uvuld be necessary to verify them if they xcere known. 2. Form 
 two different expressions of the same quantity, and set them equal. We shall 
 give a few examples, which will serve to initiate the 6tudent, and the rest 
 must be left to his own ingenuity. 
 
 PROBLEM 1. 
 
 To find two numbers such that their sum shall be 40, and their difference 
 16. 
 Let x denote the least of the two numbers required, 
 Then will .r-f-16= the greater, 
 
 And x-\-x -f-16 = 40 by the question ; 
 
 That is, &r= 40— 16=24 \ 
 
 Or x=— = 12 = less number, 
 
 And x-f-16 = 12-j-16=28= greater number required. 
 
 What number is that, whose '. pari exceeds its I part by 161 
 
 Let x= number requii 
 
 Then will its \ part be \x, and its J par! 
 
 And, therefore, ' — | ■— n; by the question, 
 Or, clearing of fractions, l.r — 3x=rl92 : 
 Hence X=192, the number required. 
 
SIMPLE EQUATIONS. 159 
 
 PROBLEM 3. 
 
 Divide c£l000 among A, B, and C, so that A shall ha~e <£72 more than B. 
 and C <£100 more than A. 
 
 Let * x= B's share of the given sum, 
 
 Then will ar+ 72 = A -' s share, 
 
 And ar+172= C's share, 
 
 And the sum of all their shares, :r+:r+72+.r+172, 
 
 Or 3x+ 244 = 1000 by the question ; 
 
 That is, 3.t= 1000 — 244 = 756, 
 
 756 
 Or =—r=de252= B's share ; 
 
 Hence x+ 72=252+ 72=,€324= A's share, 
 
 And .r+172 = 252+172=c£42t= C's share; 
 
 B's share o€252 
 
 A's share 324 
 
 C's share 424 
 
 Sum of all . . <£1000, the proof. 
 
 problem 4. 
 
 Out of a cask of wine, which had leaked away -*, 21 gallons were drawn, 
 and then, being gauged, it appeared to be half full: how much did it hold ? 
 Let it bo supposed to have held x gallons, 
 Then it would have leaked *x gallons ; 
 Consequently, there had been taken away 21 + ]£ gallons. 
 But 2 l-f-?a;=ir by the question, * 
 
 Or 12G + 2.r=3.r; 
 
 Hence 3.r— 2.r=126, 
 Or £=126= number of gallons required. 
 
 problem 5. 
 
 A hare, pursued by a greyhound, is 60 of her own leaps in advance of tH3 
 dog. She makes 9 leaps during the time that the greyhound makes only 6: 
 but 3 leaps of the greyhound are equivalent to 7 leaps of the hare. How 
 many leaps must the greyhound make before he overtakes the hare ? 
 
 It is manifest, from the enunciation of the problem, that the space which 
 must be traversed by the greyhound is composed of the 60 leaps which the 
 hare is in advance, together with the space which the hare passes over from 
 the time that the greyhound starts in pursuit until he overtakes her. 
 
 Let x= the whole number of leaps made by the greyhound. Since the 
 
 hare makes 9 leaps during the time that the greyhound makes 6, it follows 
 
 9 3 
 that the hare will make - or - leaps during the time that the greyhound 
 
 3.r 
 
 makes 1, and she will consequently make — leaps during the time that the 
 
 greyhound makes x leaps. 
 
 We might here suppose that, in order to obtain the equation required, it 
 
 3a: 
 
 would be sufficient to put x equal to 60+—; in doing this, however, we 
 
 should commit a manifest mistake, for the leaps of the greyhound are greater 
 
160 ALGEBRA. 
 
 than the leaps of the hare, and we should thus be equating two heterogeneous 
 numbers; that is to say, numbers related to a different unit. In order to re- 
 move this difficulty, we must express the leaps of the hare in terms of the 
 leaps of the greyhound, or the contrary. 
 
 According to the conditions of the problem, 3 leaps of the greyhound are 
 
 7 
 equal to 7 leaps of the hare ; hence 1 leap of the greyhound is equal to - 
 
 ?.r 
 leaps of the hare, and, consequently, x leaps of the greyhound are equal to — 
 
 leaps of the hare ; hence wo have at length the equation 
 
 7x 3x 
 
 Clearing of fractions, 14a:r=360-|-9:r 
 
 x— 7-2. 
 Hence the greyhound will make 72 leaps before he reaches the hare, and m 
 
 that time the hare will make 72 X^ or 108 leaps. 
 
 PROBLEM 6. 
 
 Find a number such, that when it is divided by 3 and by 4, and the quo 
 tients afterward added, the sum is 63. 
 
 Let x be the number ; then, by tho conditions of the problem, we have 
 
 x x 
 3+4= C3; 
 
 I Hearing of fractions, 7x= 63X12 
 
 .r=108. 
 If we wished to find a number such that, when divided by 5 and by 6, the 
 sum of the quotients is 22, we must again translate the problem into algebraic 
 language, and then solve the equation ; in this case we have 
 
 x x 
 
 5 + 6 ' 
 
 Clearing of fractions 1 l.r =22 X 30 
 
 r=60. 
 If, however, wo desire to solve both these problems at once, and all others 
 of tho same class, which differ from tho above in tho numerical values only, 
 
 we must substitute for thoso particular numbers the symbols a, b, c, , 
 
 which may represent any numbers whatever, ami then solve the following 
 question. 
 
 Find a number such that, when it is divided by a and by b, and the quo- 
 tients afterward added, tho sum is p. We have 
 
 x x 
 
 -+T =i>i 
 
 a ' b 1 
 
 (ii-\-b).v= ahp 
 ahp 
 
 151. This expression is not, strictly speaking, the value of tho unknown 
 quantity in OUT problems, but it presents to OUT view the calculations which 
 an r< Min -Hi- for the solution of them all. \x I MB of this nature is call- 
 
SIMPLE EQUATIONS. 161 
 
 ed a formula. This formula points out to us that the unknown quantity is ob- 
 tained by multiplying together the three numbers involved in the question, 
 and then dividing their product, abp, by a-\-b, the sum of the two divisors ; or 
 we should rather say, that our formula is a concise method of enunciating tho 
 above rule.* Algebra, then, may be considered as a language whose object 
 «s to express various processes of reasoning, as also the results or conclusions 
 to which they lead. 
 
 Such is the advantage of the above formula, that, by aid of it, the most ig 
 norant arithmetician could solve either of the proposed problems as readily as 
 the most expert algebraist. The former, however, could only arrive at the 
 result by a blind reliance on the rule which the formula expresses ; but differ- 
 ent kinds of problems require different formulae, and the algebraist alone pos 
 sesses the secret by which they can be discovered. 
 
 PROBLEM 7. 
 
 A laborer engaged to servo 40 days upon these conditions : that for every 
 day he worked he was to receive 80 cents, but for every day he was idle he 
 was to forfeit 32 cents. Now at the end of the time he was entitled to re- 
 ceive $15.20. It is required to find how many days he worked and how 
 many he was idle. 
 
 Let x be the number of days he worked ; 
 
 Then will 40 — x be the number of days he was idle ; 
 
 Also .rx 80 = 80.r= the sum earned, 
 
 And (40— x) X 32=1280— 32.r= sum forfeited ; 
 
 Hence 80.r— (1280— 32.r) = 1520 by the question ; 
 
 That is, 80x— 1280 + 32.r=1520, 
 
 Or 112.r=1520 + 1280=2800 ; 
 
 2800 
 Hence .r=Tpr^-=25= number of days he worked, 
 
 And 40 — .r=40 — 25 = 15= number of days ho was idle. 
 We may generalize the above problem in the following marner : 
 Let n= tho whole number of days for which he is hired, 
 
 a= the wages for each day of work, 
 b = the forfeit for each day of idleness, 
 c= the sum which he receives at theftnd of n days, 
 x= the number of days of work ; 
 Then n — .t= the number of days of idleness, 
 
 ax= the sum due to him for the days of work, 
 b(n — .r)= tho sum he forfeits for the days of idleness. 
 We thus find for the equation of the problem, 
 
 ax — b (?i — x) = c ; 
 Whence ax — bn -\-bx— c 
 
 (a-\-b)x= c-^-bn 
 
 c-\-bn 
 x= — — r-. the number of days of work, 
 a+b J 
 
 * Let the student try this rule upon a variety of numbers ; he will see that the generi 
 brmula embraces as many particular examples as he chooses to imagine. 
 
 L 
 
162 ALGEBRA. 
 
 e-\-bn 
 
 And .•. n — x= n 
 
 a+b 
 an-\-bn — c — ha 
 
 a + b 
 an — c 
 
 , . llio number of dt ,-s of idleness. 
 a-\-b 
 
 By substituting in these general expressions, for the number of days of 
 
 work and number of days of idleness, the particular numerical values of the 
 
 letters, the samo result will be obtained as before. 
 
 problem 8. 
 
 A can perform a piece of work in G days, B can perform the same worK in 
 8 days : in what timo will they finish it if both work together ? 
 Let .r= the time required. 
 
 Since A can perform the whole work in G days, - will denote the quantity 
 
 x 
 he can perform in 1 day, and therefore - the quantity ho can perform in .r 
 
 days ; for the same reason, - will bo the quantity which B can perform in x 
 
 days ; and we shall thus have 
 
 x x 
 
 6+8 = 1 t 
 14.r=48 
 x=3!) days. 
 Let us generalize the above problem. 
 
 A can perform a piece of work in a days, B in b days, C in c days, D in d 
 days : in what time will they perform it if they all work together ? 
 Let .r= the time; 
 
 Then, since A can perform the whole work in a days, - will denote the 
 
 x 
 
 quautity he can perform in 1 day, and, consequently, - will be the quantity he 
 
 x x x 
 can perform in x days; for the samo reason, -r, -, —. will be the quantities 
 
 which B, C, D can perform respectively in x days ; we thus have 
 ^+i+^ =(wll0leW01 ' k )' 
 
 =i; 
 
 abed 
 abc-\-abd-\-acd-\-bcd" 
 What is the rulo expressed by this formula ? 
 
 * Let tbo student translate the formula for the number of days of idleness, and that for 
 the number of days of work, into a rule. 
 
 V P 
 
 t We might represent the piece of work by p ; then , and - would express the quantities 
 
 which A and B can perfoua in ono day, and tlio equation would be 
 
 i, divided throughout bpp, gives the equation in the text WLenthc valao of aquaa 
 titv in immaterial, as in this case, it is best represented by 1. 
 
SIMPLE EQUATIONS. 163 
 
 PROBLEM 9. 
 
 A courier, who traveled at the rate of 31 ^ miles in 5 hours, was dispatched 
 from a certain city ; 8 hours after his departure, another courier was sent to 
 overtake him. The second courier traveled at the rate of 22^ miles in 3 hours. 
 In what time did he overtake the first, and at what distance from the place of 
 departure ? 
 
 Let x= number of horns that the second courier travels. 
 
 Then, since the first courier travels at the rate of 31 J- miles in 5 hours, that 
 
 pi) />Q 
 
 is, r-7. miles in 1 hour, ho will travel — x miles in x hours, and, since he start 
 ed 8 hours before the second courier, the whole distance traveled by him will 
 be (S+x)-. 
 
 Again, since the second courier travels at the rate of 22^ miles in 3 hours 
 
 45 45 . . 
 
 that is, — miles in one hour, ho will hence travel — x miles in x hours. 
 
 The couriers are supposed to be together at the end of the time x, and 
 therefore the distanco traveled by each must be the same ; hence 
 
 45 63 
 
 -* = (8 + *)- 
 
 450.r=(8+.r)378; 
 
 .-. 72.r=3024 
 
 a: =42. 
 
 Hence the second courier will overtake the first in 42 hours, and the whole 
 
 45 
 distance traveled by each is — X 42=315 miles. 
 
 To generalize the above, 
 
 A B C 
 
 ! 1 I 
 
 Let a courier, who travels at the rate of m miles in t hours, be dispatched 
 from B in the direction C ; and n hours after his departure, let a second 
 courier, who travels at the rate of m' miles in V hours, be sent from A, which 
 is distant a. miles from B, in order to overtake the first. In what timo will he 
 come up with him, and what will be the whole distance traveled by each ? 
 
 Let x— number of hours that the second courier travels. 
 
 Then, since the first courier travels at the rate of m miles in t hours, that is, 
 
 7th 7Tb 
 
 — miles in 1 hour, he will travel —x miles in .r horns, and, since he started n 
 t c 
 
 hours before the second courier, the whole distance traveled by him will be 
 
 — m 
 (n+x)-. 
 
 Again, since the second courier travels at the rate of m' miles in V hours, 
 
 7th 771 ' 
 
 that is, — miles in 1 hour, he will travel — x miles in x hours ; but since he 
 
 started from A, which is distant d miles from B, the whole distance traveled 
 
 m' 
 by the second courier, or —x, will be greater than the wt.ole distance traveled 
 
 by the first courier, by this quantity d ; hence 
 
 \ 
 
164 ALGEBRA. 
 
 m' m 
 
 — x—d = {n + x)- 
 
 t' * » >t 
 
 (in' m\ mn 
 
 (mn+fo£)* / 
 
 m7 — mt' 
 
 1 he wnuie distance traveled by first courier, =— . { ■ 
 
 m't — mt' 
 
 The whole distance traveled by second courier, =— 
 
 f m't — mf 
 
 PROBLEM 10. 
 
 A father, who has three children, bequeaths his property by will in the fol- 
 lowing manner : To the eldest son he ' sum, a, together with the /;'" part 
 of what remains ; to the second he leaves a sum, 2a, together with the n' 
 of what remains after the portion of the eldest and 2a have been subtracted ; 
 to the third ho leaves a sum, 3a, together with the n th part of what remains 
 after the portions of the two other sons and 3a have been subtracted. The 
 property is found to be entiroly disposed of by this arrangement. Required 
 the amount of the property. 
 
 Let x= the property of the father. 
 
 If we can, by means of this quantity, find algebraic expressions for the por- 
 tions of the three sons, we must subtract their sums from the whole pro) 
 x, and, putting this remainder =0, we shall determine the equation of the 
 problem. 
 
 Let us endeavor to discover these three portions. 
 
 Since x represents the whole property of the father, x — a is the remainder 
 after subtracting a ; hence, 
 
 x — a 
 Portion of eldest sou, =a-\- 
 
 n 
 
 an-\-x— 
 
 n 
 
 an-\-x — a 
 x — 2a — 
 
 (1) 
 
 71 
 
 Portion of second son, =2a-f- 
 
 =2a4 
 
 n 
 
 nx — 'Ian — r-\-(i 
 
 •!uir-\-nx — 3an — r+a 
 
 x— 3a 
 
 aii-\-.r — a '-nx — 3an — x-\-a 
 
 (2) 
 
 Portion of third BOO, = 3</-^ 
 
 = 3a-l 
 
 n 
 
 3gn'+n«r— 6an*— 2nx+4an+z— a 
 According to the conditions of the problem, the pro] itirerjrdisp 
 
SIMPLE EQUATIONS. 165 
 
 rf. Hence, when the sum of the three portions is subtracted from a, the dif- 
 ference must be equal to zero ; this gives us the equation 
 
 an-\-x — a larP-^-nx — 3<zm — x-\-a "iaip-^rflx — Qan2 — 2na;-\-ian-\-x — a 
 
 clearing the equation of fractions, and reducing, 
 
 ,—6ari i —3ri 2 x-\-10an' i -{-3nx—5an—x-\-a=0 
 ... { n *—3n*+3n— l)x=6a?i 3 — 10are 2 -|-5an — a 
 
 6an?— 10a7i 2 +5aw— a (6w 3 — 10n--\-5n — l)a 
 X ~ ? i 3 — 3?i-+3/i— 1 = {n— 1)» ' 
 
 reflecting upon the conditions of the problem, we may obtain an equation 
 much more simple than the preceding. It is stated that the portion of tho 
 third son is 3a, together with tho » th of what remains, and that the property 
 is thus entirely disposed of ; in other words, the portion of the third son is 3a, 
 and the remainder just mentioned is nothing. 
 
 We found the expression for that remainder* to be 
 rPx — 6an 2 — 2nx-\-4an-\-x — a 
 
 Equating this quantity to zero, we have 
 
 .•. n 2 x — 6an" — 2nx-{-Aan-\-x — a = 
 (n° — 2n-\-l)x=6ari 2 — Aan-\-a 
 Gan- — 4ara-f-a 
 
 » a — 2n -j- 1 
 (6n-— 4n + l)a 
 
 .r= 
 
 - (n-1)" * 
 
 This result is, moreover, more simple than the former. We can easily prove 
 that the two expressions are numerically identical, for, applying to the two 
 polynomials (G« 3 — 10/t- + 5>i — l)a, and (n 3 — 3n-+3>i-\-l), the process for find- 
 ing the greatest common measure, we shall find that these two expressions 
 have a common factor n — 1 ; dividing, therefore, both terms of the first result 
 by this common factor, we arrive at the second. 
 
 The above problem will point out to the student the importance of examin- 
 ing with great attention the enunciation of any proposed question, in order to 
 discover those circumstances which may tend to facilitate the solution ; he will 
 otherwise run the risk of arriving at results more complicated than the nature 
 of the case demands. 
 
 The above problem admits of a solution less direct, but more simple and 
 elegant than those already given. It is founded on the observation that, after 
 having subtracted 3a from the former portions, nothing ought to remain. 
 
 Let us represent by r,, r 2 , r 3 the three remainders mentioned in the enun 
 ciation ; the algebraic expressions for the three portions must be 
 
 r, r„ r 3 
 
 "+* 2a+ t' 3a + ~n 
 
 1". By the conditions of the problem, we have r 3 =0. 
 
 Hence the third portion is 3a. 
 
 * Next above (3). 
 
166 ALGEBRA. 
 
 r, 
 
 2°. Tho remainder, after the second son has received Ba-f — , may be rep- 
 
 n 
 
 r„ (n — l)r 
 resented by r„ =, or - 
 
 n n 
 
 But this is the portion of the third son ; hence we have 
 
 (n — l)r„ 
 
 • — =3a 
 
 n 
 
 San 
 
 n — 1 
 
 tt . „ , , San 3a 
 
 Hence tho portion of the second son is 2a + r— n = 2a-f- r» or » 
 
 1 ' n — 1 ' n — 1 
 
 reducing, 
 
 2an-{-a 
 
 ~n~=T' 
 
 T 
 
 3°. The remainder, after the eldest son has received a-\-—i may be repre- 
 
 eented by r, , or — -. 
 
 J n n 
 
 But this remainder forms the portion of the other two sons ; hence we have 
 
 (n — l)r, 2an-{-a 
 
 n n — 1 
 
 ban* — 2an 
 
 3a 
 
 ■'■ r '~ {n-iy 
 
 ,, , » , ban 2 — 2an ban — 2a 
 
 Hence the portion of the eldest son is a+— ; -rz— -±-n=a+-, ttt, 
 
 1 (n — l) 2 ' (n — 1)' 
 
 or, reducing, 
 
 an^-^San — a 
 
 1 n-— 2m + 1 
 
 Hence the whole property is 
 
 2ara+a a« 2 +3an — a 
 3a 4- — 4- — ; 
 
 reducing tho whole to a common denominator, 
 
 3a{n -—2n + l) + {2,i n + ■/)(/< — l)4-an 2 4-3ara— Jt , 
 
 LI 
 
 performing the operations indicated, and reducing, 
 
 (6/i 8 — 4 «4-l)a 
 n»— 2«4-l ' 
 the result obtained above. 
 
 This solution is more complete than the former, fur we obtain at the same 
 time the property of the father and the expressions for the portions of his 
 three sons. 
 
 We shall now solve one or two problems in which it is either I y oi 
 
 convenient to employ more than one unknown quantity. 
 
 problem 1 1. 
 
 Required two numbers whose sum is 70 and whose difference is 16 
 Let x and y l><; the two numbers. 
 Then, by the conditions of the problem, 
 
SIMPLE EQUATIONS. 167 
 
 .r+2/=70 (1) 
 
 x— y = l6 (2), 
 
 which aro tho two equations required for its solution. 
 Adding the two equations, 
 
 2or=86 
 x=43. 
 Subtracting tho second from tho first, 
 
 2y=5i 
 y=27. 
 Hence 43 and 27 are the two numbers. 
 
 problem 12. 
 on has two kinds of gold coin, 7 of the larger, together with 12 of the 
 smaller, make 238 shillings ; and 12 of the larger, together with 7 of the smaller, 
 make 358 shillings. Required the value of each kind of coin. 
 
 Let x be tho value of the larger coin expressed in shillings, y that of the 
 smaller. 
 
 Then, by the conditions of the problem, 
 
 7a?-|r-12i/=288 (1) 
 
 And 
 
 12;r+ 7#=358 (2). 
 
 Multiplying equation (1) by 7. and equation (2) by 12, 
 and subtracting tho former product from the latter, . . 95.r=2280 
 
 .-. x= 24. 
 Substituting this value of x in equation (1), it becomes 168 + 12?/= 283 
 
 .-. y= 10. 
 The larger of the two coins is worth 24 shillings, the smaller 10 shillings. 
 
 PROBLEM 13. 
 
 An individual possesses a capital of §30,000, for which he receives interest 
 at a certain rate ; he owes, however, §20,000, for which he pays interest at a 
 certain rate. The interest he receives exceeds that which he pays by §800. 
 Another individual possesses a capital of §35,000, for which he receives inter- 
 est at tho second of the above rates ; ho owes, however, §24,000, for which 
 he pays interest at the first of the above rates. The interest which he re- 
 ceives exceeds that which he pays by §310. Required the two rates of in- 
 terest. 
 
 Let x and y denote the two rates of interest for §100. 
 
 In order to find the interest of §30,000 at tho rate .t, we have the pro 
 
 portion, 
 
 30,000a: 
 100: 30,000 ::x: =300x. 
 
 In like manner, to find the interest of §20,000 at the rate of y, 
 
 20,000'/ 
 100 : 20,000 :: y :— — -^=200?/. 
 
 But, by the enunciation of tho problem, the difference of these two sums is 
 $800 ; hence we shall have, for the first equation, 
 
 300.r— 200j/=800 (1). 
 
 Translating, in like manner, the second condition of the problem into alge- 
 braic language, we arrive at the second equation, 
 
168 ALGEBRA. 
 
 350y— 240.r=310 (2) 
 
 The two members of the first equation are divisible by 100, and those of the 
 second by 10 ; they may therefore be replaced by the following : 
 
 3x— 2y= 8 (3) 
 
 35y— 24ar=31 (4) 
 
 In oi'Jer to eliminate x, multiply equation (3) by 8, and then add equation 
 (4) ; hence 
 
 19y=95 
 
 ■•• y= s. 
 
 Substituting this value of y in equation (3), we have 
 
 3.r— 10=8 
 .-. x—6. 
 Then the first rate of interest is 6 per cent., and the second 5 per cent. 
 
 problem 14. 
 
 An artisan has three ingots composed of different metals melted together. 
 A pound of the first contains 7 oz. of silver, 3 oz. of copper, and 6 oz. of tin. 
 A pound of the second contains 12 oz. of silver, 3 oz. of copper, and 1 oz. of 
 tin. A pound of the third contains 4 oz. of silver, 7 oz. of copper, and 5 oz. 
 of tin. How much of each of these three ingots must he take in order to 
 form a fourth, each pound of which shall contain 8 oz. of silver, 3J oz. of cop- 
 per, and 4 j oz. of tin ? 
 
 Let x, y, and z be the number of ounces which he must take in each of the 
 ingots respectively, in order to form a pound of the ingot required. 
 
 Since, in the first ingot, there are 7 oz. of silver in a pound of 1G oz., it fol- 
 
 7 
 lows that in 1 oz. of the ingot there are — oz. of silver, and, consequently, in i 
 
 7x 
 oz. of the ingot there must be — oz. of silver. In like manner, we shall find 
 
 lo 
 
 , 12?/ 4z 
 
 that — '-, — represent the number of ounces of silver taken in the second and 
 10 lb 
 
 third ingots in order to form the fourth ; but, by the conditions of the prob 
 
 lem, the fourth ingot is to contain 8 oz. of silver ; wo shall thus have 
 
 7x 12?/ 42 
 
 ig+Tu-+h]= s W 
 
 And reasoning precisely in the samo manner fur the copper and tin, we find 
 3a: 3?/ 7r_15 
 
 16+ 16 + 16~~T ™ 
 
 6x y 5z 17 
 
 16+ 16 + 16 = T ( 3 ) 
 
 which are the three equations required for the solution of tho problem 
 Clearing them of fractions, they become 
 
 7s+12y+4z=128 (1) 
 
 3x+ 3y+7z= 60 (5) 
 
 6ar-f. i/+5:= 68 (6) 
 
 In these three equations the coefficients ofy are most simple ; it will, ther» 
 fore, be convenient to eliminate this unknown quantity first. 
 
SIMPLE EQUATIONS. 1G!) 
 
 Multiply equation (5) by 4, and subtract equa- 
 tion (4) from the product, wo have 5;r-f-24z=112 . . (7) 
 
 Multiply equation (6) by 3, and subtract equa- 
 tion (5) from the product, we have 15.r-|- 8z = 144 . . (8) 
 
 Multiply equation (8) by 3, and subtract equa- 
 tion (7) from the product, we have 40.r=320 
 
 .-. x= 8 
 
 Substitute this value of x in equation (8) ; it be- 
 comes 120-f- 8z=144 
 
 .-. z= 3 
 
 Substitute these values of x and z in equation 
 (6) ; it becomes 48+3/+15 = 68 
 
 ••• V= 5 
 
 Hence, in order to form a pouud of the fourth ingot, he must take 8 ounces 
 of the first, 5 ounces of the second, and 3 ounces of the third. 
 
 problem 15. 
 
 There are threo workmen, A, B, C. A and B together can perform a cer- 
 tain piece of labor in a days ; A and C together in b days ; and B and C to- 
 gether in c days. In what time could each, singly, execute it, and in what 
 time could they finish it if all worked together ? 
 Let x== time in which A alone could complete it. 
 y= time in which B alone could complete it. 
 z = time in which C alone could complete it. 
 Since A and B together can execute the whole in a days, the quantity 
 
 which they perform in one day is - ; and since A alone could do the whole 
 
 in x days, the quantity he could perform in one day is - ; for the same rea 
 
 son, the quantity which B could perform in one day is - ; the sum of what 
 
 D 
 they could do singly must bo equal to the quantity they can do together 
 
 hence / 
 
 111 
 
 -+-=- 1) 
 
 x ' y a v ' 
 
 In like manner, we shall have 
 1 ( 11 
 X z b 
 
 -K=s (2) 
 
 111 
 
 -+-=- 3) 
 
 y ' z c v ' 
 
 Subtract equation (3) from (1), 
 
 1111 
 
 — =— (4) 
 
 x z a c v ' 
 
 Add equations (2) and (4), 
 
 2_1 1 1 
 x a b c ' 
 2a> 
 
 tzc-j- be — ab' 
 
170 ALGEBRA. 
 
 In like manner, 
 
 2a be 
 J ab-\-bc — ac 
 
 2abc 
 ■ ab-\-ac — be' 
 
 Let t be tho time in which they could finish it if all worked together; then, 
 by Prob. 8, 
 
 /l 1 1\ 
 
 2abc 
 ab-^-ac-\-bc 
 
 (16) What two numbers are those whose difference is 7 and sum 33 ? 
 
 Ans. 13 and 20 
 (1?) To divide the number 75 into two such parts that three times the 
 greater may exceed 7 times the less by 15. 
 
 Ans. 54 and 21. 
 
 (18) In a mixture of wine and cidor, | of the whole plus 25 gallons was 
 wine, and A part minus 5 gallons was cider; how many gallons were there of 
 each ? 
 
 Ans. 85 of wine, and 35 of cider. 
 
 (19) A bill of $34 was paid in half dollars and dimes, and the number of 
 pieces of both sorts that were used was just 100 ; how many were there of 
 each 1 
 
 Ans. 60 half dollars and 40 dimes. 
 
 (20) Two travelers set out at tho same time from New York and Albany, 
 whose distance is 150 miles ; one of them goes 8 miles a day, and the other 7 ; 
 in what timo will they meet ? 
 
 Ans. In 10 days. 
 
 (21) At a certain election 375 persons voted, and the candidate chosen had 
 a majority of 91 ; how many voted for ei 
 
 Ans. 233 for one, and 142 for the other. 
 
 (22) What number is that from which, if 5 be subtracted, - of the remain- 
 der will be 40 ? 
 
 Ans. 65. 
 
 (23) A post is I in the mud, A in tho water, and 10 feet above the water-. 
 what is its whole length ? 
 
 Ai.s. 24 IV 
 
 There is a fish whoso tail weighs 9 pounds, his head weighs as much 
 as his tail and half his body, and his body weighs as much as his head and his 
 Gail; what is the whole weight of the fish .' 
 
 (25) After | way '. and ' of my money. ! had 66 guineas left in my 
 
 pane; what was in it at first? 
 
 Ans. 120 guineas. 
 
SIMPLE EQUATIONS. 171 
 
 (26) A's age is double of B's, and B's is triple of C's, and the sum of all 
 their ages is 140; what is the age of each ? 
 
 Ans. A's =84, B's =42, and C's =14. 
 
 (27) Two persons, A and B, lay out equal sums of money in trade ; A 
 gains $630, and B loses $435, and A's money is now double of B's ; what did 
 each lay out ? 
 
 • Ans. $1500. 
 
 (28) A person bought a chaise, horse, and harness, for 8450; the horse 
 came to twice the price of tho harness, and the chaise to twice the price of 
 the horse and harness ; what did he givo for each ? 
 
 Ans. $100 for the horse, $50 for tho harness, and $300 for the chaise. 
 
 (29) Two persons, A and B, have both tho samo income : A saves J of his 
 yearly, but B, by spending $250 per annum more than A, at the end of 4 
 years finds himself $500 in debt ; what is their income? 
 
 Ans. $625. 
 
 (30) A person has two horses, and a saddle worth $250 ; now, if the sad- 
 dle bo put on the back of the first horse, it will make his value double that of 
 the second ; but if it be put on the back of the second, it Avill make his value 
 triple that of the first ; what is the value of each horse ? 
 
 Ans. One $150, and the other $200. 
 
 (31) To divide the number 36 into three such parts that 4 of the first, i of 
 tho second, and i of the third may be all equal to each other ? 
 
 Ans. The parts are 8, 12, and 16. 
 
 (32) A footman agreed to serve his master for c£8 a year and a livery, but 
 was turned away at the end of 7 months, and received only c -£2 13s. 4.d. and 
 his livery ; what was its value 1 
 
 Ans. c£4 16s. 
 
 (33) A person w r as desirous of giving 3d. a piece to some beggars, but found 
 that ho had not money enough in his pocket by 8 d. ; he therefore gave them 
 each 2d., and had then 3d. remaining ; required tho number of beggars ? 
 
 Ans. 11. 
 
 (34) A person in play lost | of his money, and then won 3s. ; after which, 
 he lost i of what he then had, and then won 2s. ; lastly, he lost j of what he 
 then had; and this done, found he had but 12s. remaining; what had he at 
 first? 
 
 Ans. 20s. 
 
 (35) To divide the number 90 into 4 such parts that if the first be increased 
 by 2, the second diminished by 2, the thud multiplied by 2, and the fourth 
 divided by 2, the sum, difference, product, and quotient shall be all equal to 
 each other ? 
 
 Ans. The parts are 18, 22, 10, and 40 respectively. 
 
 (36) The hour and minute hand of a cle xactly together at 12 o'clock : 
 when are they next together ? 
 
 Ans. 1 hour 5/,- minutes. 
 
 (37) There is an island 73 miles in circumference, and three footmen all 
 start together to travel the same way about it : A goes 5 miles a day, B 8, and 
 C 10; when will they all come together again ? 
 
 Ans. 73 days. 
 
172 ALGEBRA. 
 
 (38) How much foreign brandy at 85. per gallon, and domestic spirits at 35. 
 per gallon, must be mixed together, so that, in selling the compound at 9s. per 
 gallon, the distiller may clear 30 per cent. ? 
 
 Ans. 51 gallons of brandy, and 14 of spirits. 
 
 (39) A man and his wife usually drauk out a cask of beer in 12 days ; but 
 when the man was from home, it lasted the woman 30 days ; how many daj b 
 would the man alone be in drinking it? 
 
 Ans. 20 days. 
 
 (40) If A and B together can perform a piece of work in 8 days ; A and C 
 together in 9 days ; and B and C in 10 days : how many days will it take 
 each person to perform the same work alone ? 
 
 Ans. A 14fj clays, B 17; : ;. and C 21 
 
 (41) A book is printed in such a manner that each page contains a certain 
 number of lines, and each line a certain number of letters. If each pai:e were 
 required to contain 3 lines more, and each line 4 letters more, the numb 
 letters in a page would be greater by 22 1 than before ; but if each page were 
 required to contain 2 lines less, and each line 3 letters less, the number of let- 
 ters in a page would bo less by 145 than before. Required the number of 
 lines in each page, and tho number of letters in each line. 
 
 Ans. 29 lines, 32 letters. 
 
 (42) Hiero, king of Syracuse, had given a goldsmith 10 pounds of gold with 
 which to make a crown. The work being done, the crown was 
 
 weigh 10 pounds; but the king, suspecting that the workman had alio] 
 with silver, consulted Archimedes. The latter, knowing that 
 water 52 thousandths of its weight, and silver 99 thousandths, ascertained the 
 weight of tho crown, plunged in water, to be 9 pounds 6 ounces. This dis- 
 covered tho fraud. Required the quantity of each metal in the crown. 
 
 Ans. 7 pounds 12}§ ounces of gold, 2 pounds 3^ ounces of silver. 
 
 (43) To divide a number a into two parts which shall have to each other 
 the ratio of m to n. 
 
 ma na 
 
 Ans. : — , — ; — . 
 
 m-\-n m-\-n 
 
 (44) To divide a number a into three parts which shall be to each ether 
 
 as m:n:p. 
 
 ma 7i a pa 
 
 Ans. 
 
 7ll-\-n-\-J> , in -\-n -\-j>' m -\- n -\-p 
 
 (45) A banker has two kinds of change ; there must be a pieces of the first 
 
 to make a crown, and h pieces of the second to make the same : now a per- 
 son wishes to have c pieces for a crown. How many pieces of each kind must 
 
 the banker give him? 
 
 a{b — c) /'(c — a) „ . , 
 
 Ans. -r - of the firsl kind. — ; of the second. 
 
 h — a b — a 
 
 (46) An innkeeper makes this bargain with a Bportsman: every day that 
 the latter brings a certain quantity of game he is to receive a sum a, but every 
 day that lie fails to bring it he is to pay a sum 6. After a number n of 
 days it may happen that neither owes the other, or that the first ewes the 
 stM 1, 01 that ti id owes the firsl a Bum c. Required if formula which 
 
SIMPLE EQUATIONS. 17.? 
 
 snail express in all three cases the number of days that the sportsman brought 
 the game. 
 
 Ans. x= — — 7-. 
 a-\-u 
 
 In the first case c=0, in the second case we must take the positive sign, in 
 
 the third case tho negative sign. 
 
 (47) If one of two numbers be multiplied by m, and the other by n, the sum 
 of the products is p ! but if the m * st be multiplied by m', and the second by n', 
 the sum of the products is p'. Required the two numbers. 
 
 n'p — np' mp' — m'p 
 
 Ans. : t; • ; 7~« 
 
 mm! — m'n mn' — m'n 
 
 (48) An ingot of metal which weighs n pounds loses p pounds when weigh 
 ed in water. This ingot is itself composed of two other metals, which we 
 may call M and M' ; now n pounds of M loses q pounds when weighed in 
 water, and n pounds of M' loses r pounds when weighed in water. How 
 much of each metal does the original ingot contain ? 
 
 Ans . !^J-P) pounds f M , n{p ~ q) pounds of M'. 
 
 r — q r — q * 
 
 REMARKS UPON EQUATIONS OF THE FIRST DEGREE. 
 152. Algebraic formulae can offer no distinct ideas to the mind unless they 
 represent a succession of numerical operations which can be actually perform- 
 ed. Thus, the quantity b — a, when considered by itself alone, can only sig- 
 nify an absurdity when a>6. It will be proper for us, therefore, to review 
 the preceding calculations, since they sometimes present this difficulty. 
 
 Every equation of the first degree may be reduced to ono which has all it* 
 signs positive, such as 
 
 az-\-b=cx-\-d (1)* 
 
 Subtracting c.r+6 from each member, we then have 
 
 ax — ex=d — b. 
 Whence 
 
 a:= (2) 
 
 a— c ' 
 
 This being premised, three different cases present themselves ; 
 
 1°. c£>6 and a>c. 
 
 2°. One of these conditions only may hold good. 
 
 3°. b>d and c>a. 
 
 In the first case the value of x in equation (2) resolves the problem without 
 giving rise to any embarrassment ; in the second and third cases it does not, at 
 first, appear what signification we ought to attach to the value of x ; and it is 
 this that we propose to examine. 
 
 In the second case one of the subtractions, d — b, a — c, is impossible ; for 
 example, let b>d and a~>c; it is manifest that the proposed equation (1) is 
 absurd, since the two terms ax and b of the first member are respectively 
 greater than the two terms ex and d of the second. Hence, when we en- 
 counter a difficulty of this nature, we may be assured that the proposed prob- 
 
 * We can always change the negative terms of an equation into positive ones by trans- 
 posing them from the member in which they arc found to the other member. 
 
171 ALGEBRA. 
 
 lorn is absurd, since the equation is merely a faithful expression of its condi- 
 tions in algebraic language. 
 
 In the third case we suppose b^>d and r^>a; here both subtractions are 
 impossible ; but let us observe that, in order to solve equation (1), we subtract- 
 ed from each member the quantity cx-\-b, an operation manifestly impossible, 
 since each member <^cx-\-b. This calculation being erroneous, let us sub 
 tract ax-\-d from each member ; we then have 
 
 b — d=cx — ax. 
 
 Whence 
 
 b-d 
 x= (3) 
 
 c — a v ' 
 
 This value of x, when compared with equation (2), differs from it in this 
 only, that the signs of both terms of the fraction have been changed, and the 
 solution is no longer obscure. "We perceive that, when we meet with this 
 third case, it points out to us that, instead of transposing all the terms involv- 
 ing the unknown quantity to tho first member of the equation, we ought to 
 place them in the second ; and that it is unnecessary, iu order to correct this 
 error, to recommence the calculation ; it is sufficient to change the signs of 
 both numerator and denominator. 
 
 When the equation is absurd, as in the second case, we may nevertheless 
 
 make use of tho negative solution obtained in this case ; for if we substitute 
 
 — x for -\-x, tho proposed equation becomes 
 
 — ax-\- b s= — cx-\- d. 
 
 b—d 
 
 Whence ,r= , 
 
 a — c 
 
 a value equal to that in (2), but positive. If, then, we modify the question in 
 
 such a manner as to agree with this new equation, this second problem, whicl 
 
 will bear a marked resemblance to tho first, will no longer be absurd, and. 
 
 with the exception of the sign, will have the same solution. 
 
 Let us take, for example, the following problem : 
 
 A father, aged 42 years, has a son aged 12 ; in how many years will the age 
 of the son be one fourth of that of the father? 
 
 » Let .r= the number of years required. 
 
 42+3 
 Then — j- =12+*; 
 
 ... x— o. 
 
 Thus the problem is absurd. But if wo substitute — x for -\-r, the equa- 
 tion becomes 
 
 — x 
 = 12 r 
 
 4 
 and tho conditions corresponding to this equation change the problem to the 
 following : 
 
 A father, aged 42 years, has a son aged 12 ; how many years have elapsed 
 sinrc the ago of the son was one fourth of that of the father !* 
 Here -r— 
 
 " As a problem is translated into algebraic ! by means of an equation, so as 
 
 equation maybe translated back into a problem, provided the general uaturoof the probl*' ■ 
 be kivowu. 
 
SIMPLE EQUATIONS. 175 
 
 Take another example. 
 
 What number of dollars is that, the sum of the third and fifth parts of which, 
 diminished by 7, is equal to the original number ? 
 
 x x 
 Here -+--7=.r. 
 
 Whence x=— 15. 
 
 The problem is absurd • but, substituting — x for +.r, 
 
 x x 
 -3-5- 7 = -* ; 
 or 
 
 x x 
 
 3+5 X7 =*' 
 
 which gives 
 
 a:=15; 
 
 and the problem should read, What number of dollars is that, the third and fif tl» ' 
 parts of which, when increased by 7, give the original number ? 
 
 153. With regard to the interpretation of negative results in the solution 
 of problems, then, wo may, from what is seen above, establish the following 
 general principle : 
 
 When we find a negative value for the unknown quantity in problems of the 
 first degree, it points out an absurdity in the conditions of the problem pro- 
 posed ; provided the equation be a faithful representation of the problem, and 
 of the true meaning of all the conditions. 
 
 The value so obtained, neglecting its sign, may be considered as the answer 
 to a problem which differs from the one proposed in this only, that certain quan- 
 tities which were additive in the first have become subtractive in the second, and 
 reciprocally. 
 
 154. The equation (2) presents still two varieties. If a=c, we have 
 
 d-b 
 *=— ; 
 
 in this case the original equation becomes 
 
 ax-\-b = ax-{-d, 
 whence b=d ; if, therefore, b be not equal to d, the problem would seem ab- 
 surd.* 
 
 d—b . , m 
 
 But the expression — - — , or, m general, — , where m may be any quantity, 
 
 m 
 represents a number infinitely great. For, if we take a fraction — , the small- 
 
 er we make n, the greater will the number represented by — beccme; thus, 
 
 n 
 
 m 
 n 
 
 for n=-> -rr-z, Tz-rzi the results are 2, 100, 1000 times m. The limit is in- 
 
 in- 
 
 finity, wliich corresponds to n=0. Or, we may say, to prove — infinite, that 
 
 * The absurdity is removed by considering that finite quantities have no effect when 
 added to infinite ones ; that, in comparison with infinities, finite quantities are all equal to 
 one another, and all equal to zero 
 
176 ALGKBKA. 
 
 a finite quantity evidently contains an infinite number of zeros. The symbol 
 for the value of x in this case is 
 
 By clearing the expression 77=00 of fractions, wo have ??i=0Xoo> from 
 
 TO 
 
 which it appears that the product of zero by infinity is finite. So, also, — =0, 
 
 or the quotient of a finite quantity by infinity, is zero. 
 
 155. If, in equation (2), a=c, and b=d, we have 
 
 
 
 in this case the original equation becomes 
 
 ax-\-b=.ax-^-b. 
 Here the two members of the equation are equal, whatever may be the value 
 of./-, which is altogether arbitrary, and may have any value at pleasure. We 
 perceive, then, that a problem is indeterminate, and is susceptible of an in- 
 finite number of sohdions, when the value of the unknoicn quantity appears 
 
 under the form -. 
 
 
 It is, however, highly important to observe, that the expression - does not 
 
 always indicate that the problem is indeterminate, but merely the existence of 
 
 a factor common to both terms of the fraction, which factor becomes under 
 
 a particular hypothesis. 
 
 Suppose, for example, that the solution of a problem is exhibited under the 
 
 a 5 — b 3 
 forra x=-^-^. 
 
 If, in this formula, we make a = &, then r=-. 
 
 771 
 
 * This infinite value of expressions liko — may be sometimes positive, sometimes nega 
 tive, and sometimes indifferently positive or negative. 
 1°. Let there be the formula x-=- -, in which 7n and 11 are two invariable numbers. 
 
 which we suppose positive, and different from zero, while - can have all possible values. 
 
 in 
 Making z=n, we have .t=--. But as the denominator, («— .ays positive, what- 
 
 ever 1 z may bo, the infinity here should be regarded as designating the positive infinity. 
 
 —771 
 
 S 3 . By analogous reasoning, we see that if we have the formula x=— and *=« 
 
 should have the negative r= — <». 
 
 3 J . Let there be the formula x= . Tl till j=— , but hen 
 
 n — z ' 
 
 the infinity will have an ambiguous sign. Sup; and cause - to incr 
 
 the formula will give increasing values, which will be all 1 On the contrary, takin 
 
 z>n, then diminishing Z till it becomes equal to n, the formula gives increasing \; 
 
 '.'. Inch are negative. Therefore, the hypothesis z=n ought to be considered as causir 
 
 formula to take two infinite values, the one positive and the other negative. This i 
 
 cat :d by writing x=~L<». Tho 00 is here tho transition value between -f- and — . Zen 
 
 1 a 1 ransition value between -\- and — . For, let x=n — X : if r<», and s incre: 
 
 2>n, tho value of x in changing from -f- to — 1 rough 0. Quantities in eh. 
 
 #ign must always pass through or 00 . '1 . pass through or 00 with 
 
 out changing si-n, as in *=(n — s)", and ; r . 
 
PIMPLE EQUATIONS. 17? 
 
 But we must re nark, that a? — b 3 may be put under the form (a — b) 
 
 (a?-{-ab-{-b"), and that a"—b 2 is equivalent to (a — b) (a-\-b) ; hence the 
 
 above value of x will be 
 
 ( fl — 6 )(a«+a&+6 8 ) 
 
 X ~ (£"— 6)(a + 6) ' 
 Now if, before making the hypothesis a = 6, wo suppress the common fac- 
 tor a — b, the value of x becomes 
 
 a*+ab + b* 
 x = a~+b ' 
 an expression wh/ch, under the hypothesis that « = &, is reduced to 
 
 3a 2 3a 
 a?s= 2a a= "2* 
 Take, as a second example, the expression 
 
 a t — b* (a + b){a—b) 
 
 x= 
 
 (a — 6)3 — (a — b){a— 6)' 
 
 
 making # = &, the value of x becomes £==-, in consequence of the existence 
 
 of the common factor a— b ; but if, in the first instance, we suppress the com- 
 mon factor a — J, the valuo of .r becomes 
 
 a+b 
 X —~^—b ; 
 sn expression which, under Vhe hypothesis that a=b, is reduced to 
 
 2a 
 z=-=co. 
 
 From this it appears that the symbol - in algebra sometimes indicates die 
 
 existence of a factor common to the two terms of the fraction which is reduced to 
 that form. Hence, before we can pronounce with certainty upon the true 
 value of such a fraction, we must ascertain whether its terms involve a com- 
 mon factor. If none such be found to exist, then we conclude that the equa- 
 tion in question is really indeterminate. If a common factor be found to exist, 
 we must suppress it, and then make anew the particular hypothesis. This 
 will now give us the true value of the fraction, which may present itself under 
 
 A A 
 one of the three forms ^, — , -. 
 
 In the first case, the equation is determinate ; in the second, it is impossible 
 
 in finite numbers ; in the third, it is indeterminate. 
 
 
 There are other forms of indetermination besides - ; for, whatever be the 
 
 values of P and Q, wo have 
 
 1_ 
 
 P 2._§ 
 
 Q- x Q~r 
 
 P 
 P 
 
 The first of these equivalents of j=r, where P and Q both equal zero, be- 
 
 00 
 
 comes OX 00 ) and the second becomes — , which symbols must, therefore, be 
 
 OD 
 
 
 considered as having the same meaning with -. 
 
 M 
 
i;- ALGEBJ 
 
 i.N OF FORMULAS FURNISHED BT Ti. BRAL EQUATIONS OF TB* 
 
 FIRST DECREE, WITB TWO OR MORE I I N QJ0ANTIT1 
 
 When the common denominator of the general values of die unknown quan 
 tities reduces to zero, it is not readily seen how the given equations are to be 
 verified. We shall examine here the particular cases of this kind which may 
 occur. 
 
 Resume the two equations, 
 
 a x+b y=k [1] 
 
 :+b'yz=k' [2] 
 
 from which we derive the formulas 
 
 Jcb' — bk' ale'— Tea' 
 X= ab' — ba" V= ab' — ba'' 
 First particular Case. — Suppose the denominators to be z the uu 
 
 merators not : then we have 
 
 : — bk' aJc'—l 
 ab' — ba 1 = 0, x=z , y= . 
 
 The values of x and y are then infinite ; that is to say, in order to satisfy the 
 two given equations, they must surpass every assignable magnitude. 
 
 From the equality ab' — ba'=0, we derive a'=-j-, and, consequently, the 
 
 equation [2], by putting in it this value, becomes 
 
 ab' 
 
 -j-X-\-b'y = lc\ .-. b'(ax-\-by) = l 
 
 , The first member is the first member of [1] multiplied by 1/ ; the same re- 
 lation must subsist between the second members, in order that the value of a 
 and y may verify at the same time equations [1] and [2]. Heuce bk'=kb', 
 or, kb' — bk'=0 ; i. c, the numerator of x would be equal to zero, wh'u 
 contrary to hypothesis.* 
 
 In this way the impossibility of finding values of x and y, which satisfy at 
 the same time the two given equations, is made apparent; but this im; 
 bilily is siill better characterized by the infinil . which, at the same time 
 
 that they indicate the impossibility, show besides that it arises from the fact 
 thai the values of the unknown quantities are t->o great to bo as 
 
 [f we suppose ab' — ba' to bo at first a tall quantity, tho values of x 
 
 and y will be very great, but they will alway ' the equati ins until the 
 
 instant ab' — ba' reduces to zero, when, if ct in a direct mannei 
 
 the verification of the equations, it is 
 ignable magnitude.! 
 
 Second particular Case. — Suppose the denominator to he zero at the same 
 time as one of the numerators; for example, thai w< 
 
 ab'—ba'=0 t W — /■/.■■ = 0. 
 
 I maintain that the other numerator will be also equal to zero; for th» 
 two equ Jities aboi e give 
 
 * The note to art i anomaly. The tantitiea kV and W arc i 
 shen compared with ii finity. 
 
 t Considered in relation i<< the question, lie conditions of which nr. I j tho 
 
 problem, infinil may 1><- sometimes :i Irae solot ition. She applies 
 
 ii,„, of try furnishes nam th ra may 
 
 b, cited th 'i \\ bare a i angle ii onknown, and we find for its tangent an infinite vata 
 in clear, tli.n. thai thl an |le mnat be right 
 
SIMPLE EQUATIONS. 179 
 
 al' _kV 
 
 a '— T' b ' 
 
 and, consequently, the other numerator becomes 
 
 akb' alcb' 
 ak' — ka'= — ; — — — ; — =0 
 b b 
 
 Jf at first we had supposed this numerator equal to zero., we could have 
 
 proved in a similar manner that of a: to be so also. 
 
 The present hypothesis then gives 
 
 
 
 Of themselves these symbols indicate indetermination ; I shall prove, by going 
 back to the equations, that they ought, in fact, to be indeterminate. 
 
 For this purpose, substitute in equation [2] the values of a' and k', found 
 ■ibove, and it becomes 
 
 aV , W V. , _ x b' 
 
 -yZ+i'7/ = — , .:^{ax+b7j)=jk. 
 
 Thus we see that it can be formed by multiplying the two members of equa- 
 
 b' 
 tion [1] by -r- ; then all values of x and y which satisfy one of the two equations 
 
 will also satisfy the other. But if we give to x values at pleasure in equation [1], 
 an, by resolving it afterward, find corresponding values of y ; aud as these 
 same values satisfy the second equation, we conclude that the proposed equa 
 lions admit an infinite number of solutions. 
 
 Let it, however, be observed, that the indetermination in this case does not 
 permit us to take whatever value of y, and, at the same time, of x, we please, 
 because tho above explication shows that, when one of these unknown quan- 
 tities is assumed, the value of the other is determined. 
 
 The case before us comprehends that in which k=0, k' =0, ab'—ba'=0, 
 
 because then x and y become -. If we return to the equations proposed, they 
 
 reduce to these, 
 
 ax-\-by=0, a f x-{-b'y=0. 
 
 They give respectively 
 
 a a' 
 
 y=— b x,y = - v x. 
 
 a a' 
 But upon the hypothesis of ab' — ba'=0, we derive t=j-,; then the two 
 
 values of y are equal, whatever be that of x, and there is veritable indeter- 
 mination. 
 
 Yet it is to be observed, that, if we take the relation of y to x, this relation 
 is determinate, because we have 
 
 y a a' 
 
 x = ~b = ~b'' 
 
 If tho condition 7=77 had not existed, the two values of y above could wot 
 
 have been equal, except we suppose .r=0 ; y would have been then zero, and 
 the relation of x and y no longer determinate, but indeterminate. 
 
 A similar discussion to the above might be given to a system of three or more 
 equations, with as many unknown quantities. It would, however, be more 
 rlilficult to investigate the cases of impossibility and iudetermination, and it is 
 
180 ALGEBRA 
 
 not worth while to delay upon them. We shall content ourselves with setting 
 down here some observations intended to caution the student against certaiD 
 hasty conclusions to which he might naturally be led. 
 
 We have seen, in the case of two equations with two unknown quantities 
 that x and y become infinite and indeterminate simultaneously. 
 
 The first error which might be committed would be that of supposing from 
 analogy that, in the case of several equations, the unknown quantities would 
 all become infinite or indeterminate together. Suppose, for example, then* 
 are under consideration the three equations 
 
 ax -\-by -\-cz =/c, 
 a'x -j-o'y +c'z =ld, 
 a"x+b"y+c"z=k". 
 The common denominator of the values of r, y, z, is 
 
 R=ab'c" — ac'b"-\-ca'b" — ba'c"+bc'a"— cb'a' 
 and it may be written in three ways : 
 
 R=a{b'c"—c'b") +a'{cb" —bc") + a"{bc'—cb'y 
 R=b{c'a"—a'c")+b'(ac" — ca") + b"(ca'—ac'y 
 R=c(a'b" — b'a") + c'(ba"—ab") + c"(ab'—li') 
 Place 
 
 b'c"=c'b", cb" = bc". 
 
 From these equations we deduce bc'=cb', and, consequently, R becomes 
 
 eero. Then the numerator of .r, which is formed from R by changing a, a'. 
 
 •a" into k, k', k", becomes zero also. But as the numerator of y is formed by 
 
 placing k, k', k" in R instead of b, 6', b", there is no reason why this numerator 
 
 should become zero, unless we make some new hypothesis. The same may 
 
 be said of that of z. Thus the value of .r can take the indeterminate form -, 
 
 0' 
 
 where the values of y and z are infinite. 
 
 But with regard to this indeterminate form, another error still is to be 
 
 avoided, because it may bo that the indetermination is only apparent (see 
 
 Art. 153). In order to judge better of it, we shall have regard only to th*» 
 
 single relation 
 
 c'b" 
 b'c"=c'b", .:c"=-rr. 
 
 Substituting this valuo of c" in the general value of r, it will be seen that 
 be' — cb' becomes a common factor of both numerator and denominator. But 
 by hypothesis this factor is zero: it i i ace, t! en. which produces the 
 
 appearance of indetermination. Suppressing it, we havt the true value of .r, 
 which appears no longer indeterminate some new hypothesis be joined 
 
 to those already made.* 
 
 * An important observation should bo made bel the subject of indetermi. 
 
 nation. 
 
 When the two terms of n fraction decrease so as to 1 than any assignable 
 
 quantity, if the suppositions which cause one of I entirely 
 
 independent of those which cause the other (•> i of these t 
 
 at zero ii 
 
 fraction, maybe c<]ual to ajiy quantity w 
 
 arrhe when the two terms shall 1 I "f ih.ir • ' will expros» 
 
 aplete Indetermination. Bnl it may happen that tl • rms of the fraction are 
 
 naeted together ii inch a way. that to a < 
 
SIMPLE EQUATIONS. 181 
 
 i56. We shall conclude this discussion with the following problem, which 
 will serve as an illustration of the various singularities which may present 
 themselves in the solution of a simple equation. 
 
 problem. 
 Two couriers set off at the same time 
 
 from two points, A and B, in the same — -pr, \ 4 7^ — 
 
 • i ,• j ii i- ^ A t$ U 
 
 straight line, and travel in the same di- 
 rection, A C The courier who sets out from A travels m miles an hour, the 
 courier who sets out from B travels n miles an hour ; the distance from A to 
 B is a miles. A.t what distance from the points A and B will the couriers be 
 together? 
 
 Let C be the point where they are together, and let x and y denote the dis- 
 tances A C and B C, expressed in miles. 
 
 We have manifestly for the first equation 
 
 x—y=a . (1) 
 
 Since m and n denote the number of miles traveled by each in an hour, that 
 
 s. the respective velocities of the two couriers, it follows that the time re- 
 
 x y 
 mired to traverse the two spaces, x and y, must be designated by — , - ; these 
 
 J & J m n 
 
 two periods, moreover, are equal ; hence we have for our second equation 
 .t y 
 m~n ' * ' 
 
 The values of x and y, derived from equations (1) and (2), are 
 
 am an 
 
 x= , y= . 
 
 1°. So long as we suppose m^>n, or m — n positive, the problem will be 
 solved without embarrassment. For, in that case, we suppose the courier who 
 starts from A to travel faster than the courier who starts from B ; he must, 
 therefore, overtake him eventually, and a point C can always be found where 
 they will be together. 
 
 2°. Let us now suppose m<«, or m — n negative, the values of a: and y are 
 both negative, and we have 
 
 am an 
 
 .r= , y= — . 
 
 The solution, therefore, in this case, points out that some absurdity must exist 
 in the conditions of the problem. In fact, if we suppose m<^n, we suppose 
 tliat tho courier who sets out from A travels slower than the courier who seta 
 out from B ; hence the distance between them augments every instant, and it 
 is impossible that the couriers can ever be together if they travel in the di- 
 rection A C. Let us now substitute — x for -f-.r, and — y for -{-y, in equa- 
 tions (1) and (2) ; when modified in this manner, they become 
 
 a very .small value of the other ; and that, when they converge toward zero, their relation 
 converges toward a determinate limit, which it does not attain till the moment that the 
 
 
 two terms vanish, and the fraction presents itself under the form -.* A particular exam- 
 ple of this last case is the vanishing of a common factor of the numerator and denominator 
 Th» same remark is applicable to the svmbol — . 
 
 00 
 
 * Ti.. principle is fully exemplified in the differential calculun 
 
i£2 ALGEBRA.. 
 
 y—x = a \ 
 
 - = -( 
 m 11 ) 
 
 equations which, when resolved, give 
 
 din an 
 
 r= , y= . 
 
 in which llie values of x and y are. positive. 
 
 These values of x and y give the solution, not of the proposed problem 
 which is absurd under the supposition that ?«<«, but of the following, which 
 is llie translation of the changed equati< 
 
 Two couriers set out at iho same time from tho points A and B, and travel 
 in t B C, tVc. (tho rest as before) ; the values of .r and y mark the 
 
 distances A C, B C, of the point C, where the couriers are together, from 
 the points of departure A and B. 
 
 From this problem, as well as that of the father and son above, may be de- 
 duced the following rule, when the value of the unknown quantity is found to 
 be negative : 
 
 Change (he sign of the unknown quantity in the first equation, or the one 
 deri 'iately from the probh 'nation, translated into 
 
 common lung" . Ill furnish the problem which will give a positive solut 
 
 If the j he at first enunciated in a general manner, then negative 
 
 values of the unknown quantity may he regarded as furnishing a true s 
 but are to be interpreted in a contrary sense. Thus, if positive values repre- 
 sent distance to the right, negative will rejjrcscnt distance to the I si- 
 tive express elistance upward, negative distance downward; if (It e for 
 die* future, the latter must indicate time ])ast ; if the one gain, the o' 
 loss ; if the one a rate of increase, the other a rate of decrease, eye* 
 
 3°. Let us next suppose m=u ; the values of x and y in this case becom« 
 
 am an 
 
 or 
 
 r=oo , 2/ = °° ; 
 
 that is to say, x ami y each represent infinity. In fact, if we suppose m = n, 
 we suppose the coi i sets out from A to travel exactly at the same rate 
 
 as the courier who sets out from I! ; consequently, the original distance, a, by 
 which they are separated will always remain the same, and if tho couriers 
 travel fori vi r, they can never be together.! 
 
 * Applications of this use of positive and e quantities constantly occur in trigtt- 
 
 uomi i 
 
 . n-=J. Ti> im- 
 ir t" the principle stated in We a id a 
 littii All zen ed with finite quantities, 
 but i d with oi I t as x, though x be : 
 but Zx-\-a=x-\-a=a, if£=0. I i the li ■ mpared with 
 
 1 : in the . and .r, a 
 
 with i qual. 
 
 •i, x-\-a t as "■. when oa 
 
 il with infinil other, and all 
 
 ed v, iili . "t when simp 
 
 nnothi !•. h 
 
SIMPLE EGIUATICLVS. 183 
 
 4°. Let us suppose m=n, and also a = ; the values of x and y iu this case 
 hecome 
 
 
 
 that is to say, the problem i -tin ate, and admits of an infinite number 
 
 of solutions. In fact, if wo suppose « = 0, we suppose that the couriers start 
 from the same point, and if we at the same time suppose m=n, or that they 
 travel equally fast, it is manifest that they must always be together, and conse- 
 quently every point in the lino A C satisfies the conditions of the problem. 
 
 •")'. Finally, if we suppose a = 0, and m not =n, the values of X and y in 
 this case becomo 
 
 x=0, t/ = 0. 
 
 In fact, if wo suppose the couriers lo set out .from tlio same point, and 
 to travel with different velocities, it is manifest that the point of departure is 
 the only point in which they can be together. 
 
 ADDITIONAL PROBLEMS. 
 
 (1) The rent of an estate is greater than it was last year by 8 per cent, ot 
 tlio rent of that year ; this year's rent is 1890. What was last year's ? 
 
 Ans. 1750. 
 
 (2) A company of 90 persons consists of men, women, and children ; th* 
 men are 4 in number more than the women, and the children 10 more than 
 the men and women together. How many of each ? 
 
 Ans. 22 men, 18 women, and 50 children. 
 
 (3) From the first of two mortars in a battery 36 shells are thrown before 
 the second is ready for firing. Shells arc then thrown from both in the pro- 
 portion of 8 from the first to 7 of the second, the second mortar requiring a* 
 much powder for 3 charges as the first does for 4. It is required to deter- 
 mine after how many discharges of the second mortar the quantity of powder 
 consumed by it is equal to the quantity consumed by the first. 
 
 Ans. 189 discharges of the second mortar. 
 
 (4) The fore wheels of a carriage are 5} feet and the hind wheels 7 -J feet 
 in circumference ; the difference of the number of revolutions of the wheels 
 is 2000. What is the length of the journey ? 
 
 Ans. 39900 feet, or 7$ miles. 
 
 (5) Three brothers, A, B, and C, buy a house for t £2000 ; C can pay the 
 whole price if B give him half his money ; B can pay the whole price if A 
 give him one third of his money; A can pay the whole price if C give him 
 one fourth of his money. How much has each ? 
 
 Ans. A c£l680, B <£1440, C ,£1280. 
 
 (6) The passengers of a ship were ]• Germans, I French, £ English, I 
 
 but the objection to the doctrine of tlio special and immediate superintendence of Provi- 
 lenc Fairs of men, thnt it implies an incredible decree of condescension in an in- 
 
 finite being', finds in the principle above stated a satisfactory refutation. As compared 
 with infinitj , the 6 ] irtion of matter is equal to the greatest, and it is therefore nc 
 
 more an act of condescension on the part of God to charge himself with the care of an in- 
 dividual than of a nation — with the revolutions of a satellite than with the movements of 
 a system 
 
184 ALGEBRA. 
 
 Dutch, and the residue, amounting to 31, Americans. How many were 
 
 there in the whole ? 
 
 Ans. 120. 
 
 (7) Suppose the sound of a bell to be heard at the distance of 1142 feet in 
 a second in a still atmosphere, and that a wind is blowing sufficient to occa- 
 sion a delay of ' in time. In how many seconds will the sound reach a dis- 
 tance of G000 feet ? 
 
 Ans. 6.304. 
 
 (8) Quicksilver expands, for each degree of the centigrade thermometer, 
 
 s .'.,, of its volume. According to this, how high would the barometer stand 
 
 when the temperature is 0°, if, when tho temperature is 21°, it stands at a 
 
 height of 27 inches . Q ! lines? 
 
 Ans. 27 in. 7 T y 3 8 T lines. 
 
 (9) What degree of heat in a centigrade, thermometer would be required 
 
 to cause the barometer to rise to 26 inches 8 lines, if 0° raised it to 26 inches 
 
 4 lines ? 
 
 Ans. 70ft. 
 
 (10) A piece of silver, the specific gravity of which is 10J, weighs 84 oz. 
 
 How much weight will it lose in water ? 
 
 Ans. 8 oz. 
 
 (11) In a mass of zinc and copper, weighing 100 pounds, 8 parts are of the 
 former and 3 of tho latter. How much zinc must be added, that the propor- 
 tions may be as 14 :5 ? 
 
 Ans. 3 
 
 (12) At the extremities of two arms of a balanced lever, whose lengths are 
 16 and 21 feet, two weights are suspended, which together amount to 65| 
 pounds. How much is suspended at each arm ? 
 
 Ans. 37^ and 28, , 
 
 (13) The rango of temperature of a thermometer during the year was 
 44 T 3 ff °. Tho ratio of the degrees at which it stood at the extreme points 
 above and below zero was 7:4. "What were the points? 
 
 Ans. 28f T ' - above, 16 3 8 5 below. 
 
 (14) In 4000 pounds of gunpowder there are 3210 less of sulphur than of 
 charcoal and saltpetre, 2760 less of charcoal than of sulphur and saltpetre. 
 How much of each of those ? 
 
 . Sulphur 380, charcoal 620, saltpetre 3000. 
 
 (15) It is required to divide the number 99 into five such parts that the first 
 may exceed tho second by .">, be less than the third by 10, greater than the 
 fourth by 9, and less than the fifth by I 
 
 \i. . The i arl > are 17, 1 '. 27, 8. and 33. 
 
 (if,) \ ;:;,d ]! began trade with eq In the first year A tripled 
 
 his stock, and bad B27 to Bpare; B doubled his, e I bad £153 to Bpare 
 
 Now the amount of both their gains was five times the stock of either. What 
 
 was that stock ! 
 
 Ans. .£90. 
 
 (17) What two numbers are as 2 to 3 ; to each of which, if 4 be added, tho 
 
 Bums will be as .") lo 7 ? 
 
 Ans. 16 and 2 I. 
 
 (181 Four places are situated in the order of the letters A B. ('. D. The 
 
SIMPLE EQUATIONS. 185 
 
 distance from A to D is 34 miles. The distance from A to B is to the dis- 
 tance from C to 1) as 2 is to 3 ; and one fourth of the distance from A to B, 
 added to half the distance from C to D, is three times the distance from B to 
 C. What are the respective distances '? 
 
 Ans. AB = 12, BC=4, CD = 18. 
 
 ( I .') A field of wheat and oats, which contained 20 acres, was put out to a 
 laborer to reap for 6 guineas (of 21s. each), the wheat at 7 shillings an acre 
 and the oats at 5 shillings. The laborer, falling ill, reaped only the wheat. 
 How much money ought ho to receive, according to the bargain .' 
 
 Ans. <£4 lis. 
 
 
 
 
 ad only half his army 
 -}-G00 being wounded, 
 aither slain, taken pris- 
 isist 1 
 
 Ans. 24000. 
 
 . party of soldiers, who 
 3k from him j of what 
 took ^ of what now re- 
 leep left. How many 
 
 Ans. 103. 
 ; the expense of d€50 a 
 by i of what remained 
 k- was doubled. What 
 
 
 Ans. 740. 
 ligits, the sum of these 
 are transposed. What 
 
 Ans. 23. 
 mgers. Seven outsides 
 The fare of the whole 
 le coach took up 3 more 
 ce of which the faro of 
 to 15. Required the 
 
 
 s, 18 and 10 shillings 
 
 
 V-.J, jli.o »«»« u , c ^~~.. ...~ .- J* what times will they be 
 
 together during the next 12 hours ? 
 
 Ans. "V, minutes past 1, 10}y«n'mutes past 2, and so on, in each successive 
 hour 5, 5 r later. 
 
 ) A person sets out from a certain place, and goes at the rate of 11 miles 
 in ") hours ; and 8 hours after another person sets out from the same place, 
 and aoes after him at the rate of 13 miles in 3 horns. How far must the lat- 
 ter travel to overtake the former ? 
 
 Ans. 35J miles. 
 
 (27) A i-eservoir which is full of water may bo emptied at two cocks. One 
 is opened and J of the water runs out ; another is opened, and the two run- 
 
184 ALGEBRA. 
 
 Dutch, and the residue, amounting to 31, Americans. How many were 
 
 there in the whole ? 
 
 Ans. 120. 
 
 (7) Suppose the sound of a bell to bo heard at the distance of 11 12 feet in 
 a second in a still atmosphere, and that a wind is blowing sufficient to occa- 
 sion a delay of ' in lime. In how many seconds will the sound reach a dis- 
 'ance of G000 feet ? 
 
 Ans. G.304. 
 
 (8) Quicksi pands, for each degree of the centigrade thcrmom 
 - ' ., of its volume. A 
 
 when the temperature 
 height of 27 inches 
 
 (9) What degi 
 
 to cause the baromctei 
 4 lines ? 
 
 (10) A piece of silvc 
 How much weight wil 
 
 (11) In a mass of zit 
 former and 3 of the lat 
 tions may be as 14:5? 
 
 (12) At the extremit 
 16 and 21 feet, two w 
 pounds. How much i^ 
 
 (13) The range of - 
 44jV>. The ratio of i 
 above and below zero w 
 
 (11) In 1000 pounds 
 charcoal and saltpetre, 
 How much of each of i 
 
 (15) It is required to 
 may exceed the -eecoiu 
 fourth by 9, and less than the nttn By TV. 
 
 \n i. The parts are 17, 14, 27, 8, and 33. 
 
 (In) A and B began trade with equal In tin- first year A tripled 
 
 hie Stock, and had 627 to spare; B doubled his. and had (£153 to spare 
 
 Now the. amount of both their gains was five times the Bl her. What 
 
 was that stock ! 
 
 Ans.. I 
 
 (1?) What two numbers are as 2 to 3; to each of which, if 4 be added, the 
 
 ■urns will be as 5 to 7 ? 
 
 Ans. in and 
 (18^ four places are situated in the order of the letters \ B. C, l>. The 
 
 
SIMPLE EaUATIO 185 
 
 distance from A to D is 31 miles. The distance from A to B is to the dis- 
 tance from C to D as 2 is to 3 ; and one fourth of the distance from A to B, 
 added to half the distance from C to D, is three times the distance from B to 
 C. What are the respective distances ? 
 
 Ans. AB = 12, BC = 4, CD = 18. 
 ( 19) A field of wheat and oats, which contained 20 acres, was put out to a 
 laborer to reap for 6 guineas (of 21s. each), the wheat at 7 shillings an acre 
 and the oats at 5 shillings. The laborer, falling ill, reaped only the wheat. 
 How much money ought he to receive, according to the bargain ? 
 
 Ans. c£4 lis. 
 
 (20) A general having lost a battle, found that he had only half his army 
 -j-3600 men left, fit for action, one eighth of his men -j-GOO being wounded, 
 and the rest, which were one fifth of the whole army, either slain, taken pris- 
 oners, or missing. Of how many men did his army consist ? 
 
 Ans. 24000. 
 
 (21) A shepherd in time of war was plundered by a party of soldiers, who 
 took { of his flock and ] of a sheep ; another party took from him J- of what 
 lie had left, and \ of a sheep more ; then a third party took | of what now re- 
 mained, and \ a sheep. After which ho had but 25 sheep left. How many 
 had he at first ? 
 
 Ans. 103. 
 
 (22) A trader maintained himself for three years at the expense of <£50 a 
 year, and in each of those years augmented his stock by A of what remained 
 unexpended. At the end of 3 years his original stock- was doubled. What 
 was that stock 1 
 
 Ans. 740. 
 
 (23) There is a certain number consisting of two digits, the sum of these 
 digits is 5, and if 9 be added to the number, the digits are transposed. What 
 is the cumber ? 
 
 Ans. 23. 
 
 (24) A coach has 4 more outside than inside passengers. Seven outsides 
 could travel at 2s. less expense than 4 insides. The fare of the whole 
 amounted to <£0 ; but at the end of half the journey the coach took up 3 more 
 oulside and one more inside passenger, in consequence of which the fare: of 
 the whole became increased in the proportion of 19 to 15. Required the 
 number of passengers, and the fare of each kind. 
 
 Ans. 5 inside, 9 outside ; fares, 18 and 10 shillings. 
 
 (25) The hands of a clock are together at 12 : at what times will they be 
 together during the next 12 hours ? 
 
 Ans. 5, 5 ,- minutes past 1, lOAy-wiuutes past 2, and so on, in each successive 
 hou r 5 r 5 T later. 
 
 I) A person sets out from a certain place, and goes at the rate of 11 miles 
 in 5 hours; and 8 hours after another person sets out from the same place, 
 and goes after him at the rate of 13 miles in 3 hours. How far must the lat- 
 ter travel to overtake the former ? 
 
 Ans. 35? miles. 
 
 (27) A reservoir which is full of water may be emptied at two cocks. One 
 is opened and ]- of the water runs out ; another is opened, and the two run- 
 
186 ALGEBRA. 
 
 ning together, empty the vessel in \ c.f an hour more than \vn< require 1 fbi 
 the first cock alone to empty the fourth part. It' the two cocks had been 
 opened at the commencement, the reservoir would have been emptied in \ of 
 an hour sooner. How lung would it have tal first cock, rimning'alone, 
 
 ti) empty the reservoir 1 
 
 Ans. 4 hours. 
 
 INDETERMINATE ANALYSTS OF THE FIRST DEGREE. 
 
 1 57. If there be proposed for solution one. equation of the first d 
 taining two unknown quantities, any value at pleasure may be given to oi 
 the unknown quantities, and tho equation will make known a corresponding 
 value for the other; from which it appears that the equation adm 
 infinite number of solutions. The number of solutions will, however, not bo 
 so unlimited, if it be required that the values of x ami y shall be whole num- 
 ber.- : and still less so, if they must be both entire and positive. 
 
 Let there be the equation 
 
 ax-\-by=c, 
 a, b, c being any whole numbers whatever, either positive or ne: 
 all the factors common to these three numbers could be su] 
 this to have been done. 
 
 And first, let it be observed, that if there should remain now a commin lac- 
 tor in a and b, the equation could not admit of a solution in whole numl 
 for whatever values might be substituted for x and //, the first mi would 
 
 be divisible by this common factor of a and Z>, while the second m ivould 
 
 not, and the equality would therefore be impossible: a and b must, therefore 
 be supposed prime to each other. 
 
 . Take, for example, the equation 
 
 24jr+65//=243 " . . . (1) 
 
 in which the coefficients 24 and 65 are prime to each other. 
 
 Resolving it, with respect to x, 
 
 243—1 3 — 17 v 
 
 rr= - = 10 °?/4- 
 
 24 W ~^+ 24 ' 
 
 In order that x and y may both bo whole numbers, and. at the - 
 
 satisfy tho given equation, it is necessaiy that — — — should be a : 
 
 nun.! 
 Representing this by /, we have 
 
 3 — 17// 
 
 and 
 
 = « (*) 
 
 .,—10 — 2// + / (3) 
 
 i of the giveYi equation in whole numbers then reduces itself to 
 i f the equation (2). 
 We resolved the given equation with re peel to the unknown quantity which 
 had tli*' least co ; doing the 
 
 :; — 
 
 3F- — «+ ir - ; 
 
 nnd procee ing as bei 
 
JNj .'IK ANALYSIS OF THE EE. 
 
 3— 7< 
 
 =*' (4) 
 
 17 V ; 
 
 y=—t + L' (5) 
 
 The solni ion of (2) in whole numbers depends on that of (I, . re- 
 
 o*ct to /, gives 
 
 • 3 — 17 1' 3 — 31' 
 
 «— j ■-=<•+— 
 
 '-=¥='• • (<9 
 
 t=— 2i'+r (7) 
 
 the same way, 
 
 3—7/" r 
 
 /' 1 Of" — 
 
 t- 3 _i a 3 
 ■o=«'" (8) 
 
 f=i— sr— r" (9) 
 
 Equation (8) gives 
 
 /" — ?,','" (10) 
 
 The solutions of the given equation in whole numbers are therefore obtained 
 by giving to the indeterminate quantity V" all possible values in whole mini 
 bers, positive or negative; and for each of these values of I'", the i 
 (10), (9), (7), (5), and (3), determine successively the values of the indeter 
 minate quantities t", (', t, and of the unknown quantities y and x. The equa- 
 tion is therefore resolved in the [uired. 
 
 Formulas may be obtained which give immediately the values of .r and y in 
 terms of V". For, substitutim he value 31'" of t" in (9), we find*'=l — 1'"' \ 
 substituting this value of t' and that of t" in (7), we find t= — 2 + 17t'' ; sub- 
 stituting this last value and that of V in equation (5), we find y=3— 2it'", and 
 from (3), x=2 + 6ol'". 
 
 These last two expressions give all the entire solutions of the proposed 
 equations by attributing successively to t'" all possible values in entire num- 
 bers, positive or negative. 
 
 159. The same process with the general form 
 
 ax-\-by=c 
 would run thus, 
 
 c — ax 
 
 y=-v- w 
 
 Dividing a by /;, aud calling q tho quotient, r the remainder, 
 
 c — (bq-\-r)r c — rx 
 
 y = p =-?*+-£-, 
 
 mak e 
 
 c—rx c—bt 
 
 — =<• ■■■ *=>— < 2 > 
 
 railing «j ' the quotient of b by r, and r' the remainder, 
 
 c—r't 
 x —— q >t+—-—, 
 
 =«',.-. <=--7-. • ' • (3) 
 
 r r 
 
198 ALGEBRA. 
 
 And calling q" the quotient of r by r', and r" the remainder, 
 
 c—r"t' 
 t=-q"t'+-^—, 
 
 make 
 
 c—r"V 
 
 (-1) 
 
 r 
 
 and so on. The process is now evident, and it will be perceived Hint the co- 
 efficierjt8 r, r', ?", which enter into the equations (2), (3), (4), are the suc- 
 cessive r -; which would be obtained in operating as if to find the com 
 ition divisor of a and b. We must at length arrive at a remainder 1, be. 
 a Hi id b are supposed prime to each other. 
 
 For the sake of being more definite, let r" be supposed to be this remaindei 
 then equation (4) gives 
 
 t'=— r'l" + c (5) 
 
 By means of equations (2), (3), (4), and (5), the values of y, x, I, and V idh 
 be written as follows : 
 
 y=— (] z +t 
 xz=—q't -\-t' 
 t = — q "t' + l" 
 t' = —r'l" +c. 
 T of equations shows that any entire value bein°; assumed for t". 
 
 Lng value of V substituted in that of t, tho values of t, tf in tlial of r, and 
 the \ allies of x, t in that of?/, the proposed equation is resolved in whole nui 
 
 Tho success of tho method is founded on the pro .lion 
 
 which division effects upon the coefficients of the indeter 
 reason, however, why the constant term, found in the successh dons, 
 
 should not also be divided. In this way tho calculation will it nailer 
 
 numbers, an advantage which is not to be neglected. 
 For example, take tho equation 
 
 3x— 8?/ = 4 3. 
 ^s the multiplier of x is less than that of?/, resolve the equation .Ter- 
 
 ence to x, 
 
 8y+43 
 s=— — . 
 
 Dividing 8 by 3, the quotient is 2, and the remainder 2 ; and dividing 43 ov 
 .", the quotient is 14, remainder 1; then 
 
 2//+1 
 ar=22/ + 1 '* + ''-J- = '.'/ + 1 1 + ' 
 
 2t/+l=3/ 
 
 3l — \ t — \ 
 
 y=-^-=t+—=t+t' 
 
 l — l=2t' 
 
 in which last equality V may receive all possible entire values. By means oi 
 this value may be found 
 
 v =t+t'=2/'+i+r=r>t'+\ 
 
 i = 2y+14+«=2(3f , +l)+14+2f+l=8«'+17. 
 
 to V the values 0, 1, 2, 3,... we lind 
 
INDETERMINATE ANALYSIS OF THE FIRST DEGREE. 133 
 
 y= 1, 4, 7, 10,... 
 * .r=17, 25, 33, 41,... 
 V may also receive the negative valu 
 
 1 o T 
 
 161. In the above example, the values of?/ and x form two arithmetical pro- 
 gressions, the first of which has the common difference 3, the coefficient of x 
 in the proposed equation ; and the second the common difference 8, the co- 
 efficient of y taken with the contrary sign. This proposition may be sei i to 
 be goneral by elfecting the successive substitutions in the general solution. 
 but the following demonstration is preferable. 
 
 It appears, from the general investigation already made, that the equation 
 
 az~\-by=c (1) 
 
 admits of an infinite number of solutions in whole numbers, whatever may be 
 the signs of a and b, provided they are prime to each other. Suppose one of 
 these solutions to be 
 
 x=A, y=B. 
 These values must satisfy the given equation (1), thus, 
 
 «A-foB=c. 
 Subtracting this equality from (1), we have 
 
 a(x— A)-f% — B)=0 
 _ a(A — x) 
 
 The values of x are to be whole numbers, and such that y shall also be a 
 whole number. Then the product a(A — x) must be divisible by b ; but a is 
 prime with b, (A — x) is, therefore, a multiplier of b (see Art. 84, Note), hence 
 we may write 
 
 A— x=bt; 
 t being somo whole number. From whence 
 
 .r=A — bt, y=B-\-at. 
 
 These formulas exhibit the law of the values to be obtained for x and y, 
 when there are given to t all entire values successively. If t be taken equal 
 to 0, 1, 2, 3, . ... there results 
 
 x=\, A— b, A— 26, A— 36, &c. 
 y=B, B-f a, B + 2a, B + 3a, &c. 
 
 In general, when t increases by unity, y increases by a, and x by --6. 
 The solutions in whole numbers, then, of the equation ax+by=c, are the cor 
 responding terms of trco progressions by differences. In the progression be- 
 longing to each of the indeterminates, x and y, the common difference is equal to 
 the coefficient of the other indeterminate. But it is necessary to be careful to 
 take one of the coefficients with the same sign that it has in the equation, and 
 (he other with the contrary sign. 
 
 It is immaterial which of the coefficients is taken with the contrary sign, 
 because in the formulas which express x and y the signs of bt and — at may 
 be changed, since t can receive all possible values, positive and negative. 
 
 162. In the general equation, if c=0, so that 
 
 ax-\-by = 0, 
 as one solution is eviden.y x=Q, y=0, the general formulas become 
 
 x—bt, y= — at. 
 
I'M ALQE] 
 
 iG3. Again, suppose c to be a multiple of a or b. Let c=bd, then 
 
 -j-/w/ = /;cZ. 
 
 One solution is evidently r=0, // = </ ; hence the general values are 
 
 x—bt, y=d — at. 
 Example, 5.r — 7//=21. 
 
 The evident solution is x=0, y=—'3, and the general values 
 
 x=7t,y=—3-\-r,/. 
 
 164. We shall point out two simplifications which may sometimes be made 
 uj the calculations. An example will explain them. 
 
 80ar-yl77/=39. 
 lolving it with respect to y, 
 
 80a:—: 
 
 If 80 be divided by 17, 80 = 17 X 4 + 12 ; but as the remainder, 12, exceeds 
 
 half the divisor, 17, we observe that we may write 
 
 80 = 17 X (4 + 1) + 12 — 17 = 17X5— 5; 
 
 that is, augmenting the quotient by unity, we have a negative remainder less 
 
 than half the divisor, which causes a more rapid reduction in the numbe 
 
 The 39, divided by 17, leaves a remainder -\-o, which it is unnecessary to 
 
 change. "\\ e have then 
 
 (17 X -3 — 5)x— 17X2 — 5 5.r-f5 
 
 y= rj =5x-2— — . 
 
 But another simplification now ] itself, from the fact that 5 is a factoi 
 
 of 5.r-[-5, and this numerator may be written 5(x-J-l). In order to rendei 
 5(x-\-l) divisible by 17, it is 011I3- necessary to take x-\-l, any multiple what- 
 ever of 17. Whence the auxiliary equation 
 
 .r -4-1 = 17/; 
 .-. .(-=17/ — l.-y = S0t— 7. 
 
 RESOLUTION OF THE EQUATION C.C-\-l,y = C IN NUMBERS EOTII ENTIRE AND 
 
 POSH !VK. 
 
 165. We begin as if the values of x and y were required to be entire only, 
 and thus derive, as before, e ons of the form 
 
 x=\— bt, y=B + at. 
 But now, instead of attributing to / all possible values in whole numbers, we 
 choose only those which will render x and y positive. Hence there result for 
 
 t certain limitations Which are always easy tO determine. 
 
 First, let us consider the case where a and b have the same sign in the 
 
 equation 
 
 ax-\-by=c (1) 
 
 Suppose a and b positive; for if they were both negative, they might be 
 
 rendered positive by changing all the signs of tin' equation. We must also 
 
 ppose c to be positive, otherwise the equation would be impossible in p©si- 
 
 whole numbers. 
 Writo the general values of.c and y under the following form: 
 
 tel (£_i) i3 ^«(«_z2). 
 
 Then we perceive that, to render x positive, it i 
 
 A .-I'- 
 ll! take t<C , , and likewise, in order that 7 may be positive, to tako /> . 
 
V 
 
 INDETERMINATE ANALYSIS OF THE FIRST DEGEE i91 
 
 The signs > and < do not exclude equality ; that is to say, if the first limit 
 were a number n, we might make i = /i. The corresponding value of x would 
 be r=0. 
 
 166. Since t must be an entire number between two limits, it follows that 
 the number of solutions of the equation is also limited. 
 
 And this is evident from the equation itself; for a and b being positive if 
 ilute for x and y positive numbers, the two terms ax-\-by will be al- 
 i positive ; and as their sum has to remain constantly equal to c, it is im- 
 possible that either of these terms should increase indefinitely. 
 
 It may happen that there is no whole number between the limits assigned 
 above for t ; then we conclude that the equation is impossible. Such a case 
 would happen if the limits should be embraced between two consecutive whole 
 numbers like these, <>4.' t and £<4f ; or, again, if they were contradictory, 
 as, for example, <>4g and £<3f. 
 
 1G7. In the second place, consider the case in which a and b are of contrary 
 signs. Suppose the equation in question to be 
 
 ax — by=c (2) 
 
 in which a and b represent two positive numbers. Then the general v. 
 of x and y are of the form 
 
 x=A-{-bt, y = B-\-at. 
 
 But we can write them 
 
 x=b{t-~-),y=a{t-=^). 
 
 And wo perceive at once that to have x and y positive, we must have, at tho 
 same time, 
 
 that is to say, we may attribute to t all entire values above the greatest of 
 these limits without excluding equality, if this limit is an entire number. 
 
 By this wo perceive that the equation ax — b>j=c admits always of an infinite 
 number of solutions, while the equation ax-\-byz=c admits of but a limited 
 number, and even may not have any. 
 
 Let us apply what precedes to some problems. 
 
 168. Problem I. — A company of men and women expend at a feast 1000 
 francs. The men pay each 19 francs, and the women 11 francs. How - 
 
 n and hoiv many ivomen arc there? 
 
 Let x represent the number of men and y the number of women. "Wo 
 have to resolve in entire numbers the equation 
 
 19.r+lly=1000 (3) 
 
 In making the calculation, as in (160), and profiting by the simplifications in- 
 dicated by (Art. 164), we have successively, 
 
 1000 — 19.r • :•>'•— 1 
 
 y= = 91-2.r+- ir -=91--r+; 
 
 3.r — \=\\t 
 11*4-1 \—l 
 
 l—t=3t' 
 t=l—3t'. 
 Arrived at this point, we return to x and y, and they become 
 
1 92 ALGEBRA. 
 
 x=it+t'=m— 3i')+t'=i— a f 
 
 2/ = 'Jl— 2.r+/ = 91 — 2(4 — 11/') + (1 — 3f') = 8-i- r 19t'. 
 Thus, llie general formulas which express .r iiuJ ?/ in terms off are 
 
 x=4 — 11/', ?/ = 84 + 19/'. 
 In order that a: may be positive, it is necessary and sufficient that we ham 
 11/' <4, or £'<C f Y ; and in order that y should be also positive, it is necessary 
 and sufficient that we have 19f > — 84, or /'> — 4,^. Then wo must take /'. 
 one of the series of values, 
 
 t' = 0, —1, —2, —3, —4. 
 
 To these values correspond 
 
 a=4, 15, 26, 37, 48 
 ?/=84, 65, 46, 27, 8. 
 The number of solutions is limited, as we ought to expect, since, in the 
 equation (3), the terms containing x and y are of the same sign. 
 There are five solutions in all, to wit ; 
 
 1st solution, 4 men and 84 women. 
 2d solution, 15 men and 65 women. 
 3d solution, 26 men and 46 women. 
 4th solution, 37 men and 27 women. 
 5th solution, 48 men and 8 women. 
 
 Remark. — From what has been said at (161), it is sufficient to procure a 
 single solution of the equation (3) to form immediately the general values of x 
 and y. Thus, after having found above t=l— 3/', we make t' = ; and if we 
 calculate the corresponding values t = l, x=4, i/ = 84, it is evident that the 
 values x=4, y=84, ought to form one solution of the equation ; then we i 
 place immediately x=4 — 11/', ?/ = 84 + 19/'. 
 
 169. Problem II. — With two measuring rods of different lengths, the one 5 
 feet, and the other 7, it is required to make, by placing them the one after tht 
 other, a length of '23 feet. 
 
 This problem requires tho solution in whole numbers of the equation 
 
 5x+7t/=23. 
 
 Wo derive from it successively 
 
 23—71/ 2+2y 
 
 x=—^-=5-y--^=o-y-2t 
 
 1 + 2/ =5t 
 y=5t-l 
 x=zC> —71 
 
 In order that y may bo positive, we must make />' ; and that x may be 
 positive, /<','. As no whole number falls between ', and f, wo conclude that 
 the problem is impossible. 
 
 Ki makk. — The equation would have had an infinite number of solutions if 
 negative values had been admitted. For example, if /=0, we hue .r=6, 
 i/r= — 1. This solution indicates thai byplacii one of the rods, that of 5 feet, 
 
 6 limes in succession, and placing afterward the rO 1 of 7 l> cut off 
 
 iis length from the end of the distance thus ol tained, the remainder would be 
 the required length, 23 feet 
 
 . PaoBLl M [II. I /rrson pi '.iircs an*/ sheep. 
 
 ha-'- cost him 8 shilling*, and each thru' 27. Jh found Uiat he had paid Jet 
 
INDETERMINATE ANALYSIS OF THE FIRST DEGREE. 193 
 
 the hares 97 shillings more than for the sheep. How many hares did he pur 
 chase, and how many sheep ? 
 
 8x—27y=97 
 27w+97 3y+l 
 
 3y+l=8t 
 
 8t—l t+l 
 
 y=- ir =3t--j r =3t-l' 
 
 <+l=3f 
 t=3t' — l. 
 By making ('=0, we have t= — 1, y = — 3, 2=2. And the general values 
 are 
 
 T=27f+2, yz=Sl' — 3. 
 The values of x and y having to be positive, these formulas show that t 
 ought also to be positive, and large enough to cause 8£'>3, or i'> \. We may 
 then give to V all the values $'— 1, 2, 3, &c, to infinity ; and we form, conse- 
 quently, the table, 
 
 t' = 1, 2, 3, 4, &c. 
 
 :r=29, 56, 83, 110, &c. 
 
 y= 5, 13, 21, 29, &c. 
 
 The problem admits of an infinite number of solutions ; and the answer is, 
 
 that there are 29 hares and 5 sheep, or 56 hares and 13 sheep, or 83 hares 
 
 and 21 sheep, &c. 
 
 171. Problem IV. — To find a number such that, in dividing it by 11, there 
 remains 3, and dividing it by 17, there remains 10. 
 Let the number be represented by N, then 
 
 N = ll.r+3 and N = 17i/+10 
 
 .-. ll.r+ 3 = 17v/+10 (6) 
 
 Proceeding as before, 
 
 17.V+7 . fy-l-7 
 
 x=- jr -=y+— 1 -=y+t 
 
 6y + 7 = Ut 
 Ut — 7 t+l 
 
 y = -^ = 2l-l-^ = 2t-l-f 
 
 t-\-l=6t' 
 t—6t' — l. 
 The hypothesis ('=0 gives t= — 1, ?/= —3, x= —4 ; and then we conclude 
 immediately that 
 
 x=17l'— 4, y=lll'— 3. 
 
 We can not take V negative, nor even t'=0, because x and y would become 
 negative; but we may take t'r=l, 2, 3, &c., to infinity. 
 
 If we wish formulas in which Ave can give to the indeterminate all entire 
 positive values setting out from zero, all that is necessaiy is to chauge t into 
 1+0, 6 being the new indeterminate. Then we have 
 
 i=13+ 170, y=8+U0. 
 By means of these values, we find 
 
 N=llx+ 3 = 11(13 + 170)+ 3 = 146+1870 
 N = 17?/+10 = 17( 8+110) + lO = 146+1870. 
 These two expressions are equal, and they should be, since equation (6) has 
 
 N 
 
J 94 ALGEBRA. 
 
 been formed by equating the values of N. We perceive that there 19 an in- 
 finity of numbers which fulfill the two conditions enunciated, and that they are 
 all represented by the formula 
 
 N = 146+1870, 
 in which 6 is an indeterminate, which may receive all positive values beginning 
 with zero. 
 
 It is easy to 6how that this number N satisfies the enunciation ; that is U 
 say, that if we divide it by 11, the remainder will be 3, and if by 17, the re- 
 mainder will be 10 ; for wo have 
 
 N 3 N 10 
 
 n =170+13 + n , and - = n0 +8 +-. 
 
 172. Problem V. — To find a number such Qiat, dividing it by 11, there 
 remains 3 ; dividing by 17, there remains 10 ; and dividing it by 37, there re- 
 mains 13. 
 
 In the preceding problem we have found the numbers which fulfill the 
 
 first two conditions. Putting x for 0, which we may do, since 6 can be any 
 
 positive whole number, this formula becomes 
 
 N=146+187.r (8) 
 
 But in order that the number N may fulfill the third condition, we must 
 
 have N=:37y-J[-13. Then we have the equation 
 
 37y+13 = 146-r-187;r. 
 
 Then 
 
 1872T+133 2i+22 
 
 V= ^ =5z+3+— ^-=5*+3+2i 
 
 x+ll=37f 
 xz=37t— 11. 
 
 In order that x may be positive, we must give to t only positive values above 
 zero. But in making t=\-\-6, wo can attribute to 8 all the entire positive 
 values beginning by zero. By this change x becomes 
 
 x=26 + 370. 
 And by substituting this value in formula (8), we obtain 
 
 N=5008+ 69190. 
 Such is the general formula of the numbers which satisfy the three condi 
 tions enunciated. 
 
 173. The determination of tho limits led to the necessity of finding (165) 
 the values of tho final indeterminate t, which render positive expressions of 
 the form A-\-bt, or, in other terms, which are such as to make 
 
 A+6t>0. 
 Transposing tho term A, 
 
 &*>— A. 
 If b is positive, dividing by b, 
 
 But if b is negativo, tho division by b changes the signs of the inequality 
 and tho two members are unequal in the contrary sense ; i. e., 
 
 «4 
 
 Suppose, more generally, that we have ilu> inequality 
 
 at+b>ct+d. 
 
INDETERMINATE ANALYSIS OP THE FIRST DEGREE. 195 
 
 By the transposition of the terras, 
 
 (a — c)t^>d — b. 
 
 Then, according as a — c is a positive or negative quantity, we derive 
 
 d-b d-b 
 £> , or /< . 
 
 a — c a — c 
 
 This process is called resolution of inequalities. The whole subject cf in 
 equalities will bo found treated in a subsequent article. 
 
 174. Resolution in whole numbers of several equations of thk 
 first degree, when the number of equations is less than that 
 of the unknown quantities. 
 
 Let there be for resolution the equations 
 
 2x+Uy— 7z=341 (1) 
 
 10.r+ 4?/ + 9c = 473 (2) 
 
 If we multiply the first equation by 5, and afterward subtract the second, 
 we shall have 
 
 66y — 44z=1232. 
 Or, dividing by 22, 
 
 Zy— 2z=56 (3) 
 
 But the entire values of y and z, which suit the proposed equations, ought 
 also to satisfy this ; consequently, applying to it the method already known, 
 we have 
 
 y=2t, z~Zt— 28. 
 If we had but equation (3), we should have its solutions in whole numbers, 
 by giving to t all the whole-number values possible. But this equation takes 
 the place of only one of the proposed, so that it is necessary that the values 
 of y and z should be such that, in adding to them certain values of x, which 
 must also be entire, one of these proposed equations shall be verified. For 
 this reason we substitute the preceding values of y and z in equation (1), and 
 seek for the entire values of x and I, which belong to the resulting equation 
 The substitution gives 
 
 2.r+ 7f=145; 
 and from this we obtain, designating by V any whole number whatever, 
 
 .T=69 + 7i', t = l—2t'. 
 Then place the value t=\ — 2t' in those of y and z, and you find the un- 
 known quantities x, y, z expressed in terms of t\ to wit : 
 x=69+7t', y=2— W, z = — 25— 61'. 
 These formulas make known all the entire values which satisfy the equa- 
 tions proposed. 
 
 If it be desired besides that these values should be positive, t must be so 
 chosen that 
 
 69+7*'>0, whence f>— 9«; 
 2— 4i'>0, whence f< A; 
 — 25 — 6r>0, whence «'<— 4£. 
 From this we find the only values which can be attributed to V are t'z=z — 5, 
 — 6, — 7, — 8, — 9. By substituting these numbers, we 6hall have five solu- 
 tions in positive wholo numbers : 
 
 z=34, 27, 20, 13, 6 
 i/=22, 26, 30, 34, 38 
 z= 5, 11, 17, 23, 29. 
 
19b ALGEBRA. 
 
 175 The p g example shows sufficiently the method to be pun ! 
 
 in resolving equations of the first degree in positive whole numbers, when t lie 
 number of unknown quantities e that of tho equations. But, to leave 
 
 nothing to be desired, 1 shall indicate the method to be pursued in the case 
 of three equations. 
 
 Let there be, then, ' in the unknowns x, y, z, u three equations of the 
 
 1st degree, which I will name collectively the equations [A]. 
 
 By tho elimination of x we shall find between y, z, and u two equations of 
 the 1st degree : I shall name them [B]. 
 
 By the elimination of y we shall deduce from these last an equation of the 
 1st degree between ; and it." 1 shall name it [C]. 
 
 From tho equation [C] wo derive z and u expressed in function of an aux 
 diary indeterminate t. 
 
 These values being substituted in oue of the equations [B], we derive from 
 it an equation between y and t, and from this tho values of y and ( in function 
 of a now indeterminate «' ; consequently, we can also express z and u in terms 
 
 ofr. 
 
 Finally, these values of y, z, u being carried into one of the equations [A], 
 there will result an equation between x and V, which will enable us to find x 
 and I', and, consequently, y, z, and it, in function of a new indeterminate /". 
 
 When the equation is to be resolved in whole numbers of any sign what- 
 ever, we may attribute to tho final indeterminate L" all possible valui 
 whole numbers. But when the solutions are to be restricted to such as are 
 at the same time entire and positive, there will exist for L" limitations which it 
 will be always easy to assign. 
 
 17G. When we have two more unknowns than equations, or several more, 
 the ^determination is still greater ; but the condition of having values which 
 shall be at the same time entire and positive, may limit considerably the num- 
 ber of solutions. Wo shall confine ourselves to two examples, which will suf- 
 fice to show how the method explained above should be modified in such c 
 
 Given to resolve in positive whole numbers the equation 
 
 10x+9?/+7r=5S (4) 
 
 As the unknown z has the smallest coefficient, I derive 
 
 58— 9?/— 10.r 
 
 ;= 1 — ; 
 
 and, effecting the division as far as possible, 
 
 2— Oy— 3.r 
 z=8-y-z+ •- . 
 
 Tho numerator 2 — 2y — 3x must bo a whole numb ; theio 
 
 fore I place 
 
 ? — 2 / — ■:>r = 7l : 
 o_;}. r _7, x+t 
 
 ■■■li= r 2 =l—r-r,/_ -J-; 
 
 and, x-^-t being obliged to bo a whole number divisible by 2, I place, also, 
 
 z+t=2? .-. r=— t+2tf ; 
 and, going hack to y and c, we expr< unknowns in function of t and t 
 
 We have thus the three formulas 
 
 *=— t+2f, y=sl— 2f— 3C, :=7+l/-f/' .... (5) 
 
INDETERMINATE ANALYSIS OV THE FIRST DEGREE. 19? 
 
 In oiiler to have the entire and positive solutions of the proposed equation 
 (4), wo must give to t and t' all the entire values, which satisfy simultaneously 
 the three conditions 
 
 — l+2t'>0, l—2t— 3f >0, 7 + 4H-«'>0 .... (6) 
 From hence result limitations for t and V, which will be discovered by em 
 ploying for these inequalities operations altogether analogous to those of elimi- 
 nation. For greater neatness, suppose the signs > exclude equality ; that is 
 to say, that hone of the three unknowns, x, y, and z, can be zero. 
 First, if we multiply the 1st by 3 and the 2d by 2, they become 
 _3«+6«'>0, 2 — it— 6£'>0 ; 
 adding, t' disappears, and we have 
 
 2— 7«>0 .'.i.<|. 
 A similar elimination between the second inequality and the third gives 
 
 22 + 10«>0.-. <>— 2i. 
 We see that the indeterminate t is embraced between the limits — 2] anu 
 f ; then we shoidd take only 
 
 t=-2, -1,0. 
 Let us consider each of these values successively. 
 1°. If we make t= — 2 in the three inequalities (6), they become 
 2+2«'>0, 5— 3f>0, — l+f>0; 
 .-.«'> -1,«'<1|, f>l. 
 As there is no whole number between 1 and 1|, it follows that the value 
 t= — 2, which furnishes these limits for t', ought to be rejected. 
 2°. If we make t= — 1, the three inequalities (6) become 
 l + 2«'>0, 3— 3«'>0, 3 + *'>0 ; 
 
 .-.f>— |. r<+i.tf>— 3. 
 
 Between — \ and -f-1 there is no other entire number except ; then we 
 can take t= — 1 and t' — O. 
 
 3°. If we make <=0, the inequalities become 
 
 2i'>0, 1— 3f>0, 7-K>0; 
 .:(.'> 0, V<l,V>-7. 
 Between and 1 there is no whole number ; consequently, the value i=0 
 ought also to be rejected. 
 
 The only values of t and V to which positive values in whole numbers of x, 
 y, and z correspond are, then, t-= — 1 and £'=0. By substituting them in 
 the formulas (5), we obtain 
 
 x=l, 7/ =3, 2=3, 
 and this solution is the only one admissible. 
 
 177. For a second example, I propose the two equations 
 
 6.r+ 77/+3r+2*< = 100 
 
 24x+12y4-7z+3tt=200. 
 
 Eliminating v, we have 
 
 3O.r+3?/+5r = 100. 
 
 As in this equation the terms 30.r and 100 are divisible by 5, it will be best 
 
 to take the value of z : this is 
 
 3?/ 
 z=20 — Gx — -~. 
 5 
 
 From which we see that y ought to be a multiple of 5 ; consequently, we have 
 
 y=5t 
 
 z=20— 6x— 3«; 
 
198 
 
 ALGEB1IA. 
 
 then, by substituting these values in the first of the two proposed equation* 
 it becomes 
 
 G.C+35J+G0 — 18x— 9t+2u = 100 ; 
 or, rather, 
 
 — 12x+26J+2u=40; 
 .-. u=20 +6z— 13 . 
 The three unknowns, y, z, u, are thus found expressed in functions oi 2, 
 and of the indeterminate auxiliary t. 
 
 In order to resolve the two proposed equations in positive numbers, it is evi- 
 dently necessary to take x and t positive, since x is one of the primitive un- 
 knowns, and since y=bt. But it is necessary to satisfy also the inequalities 
 20 — Gx— 3<>0, 20-4-fcr— 13*>0. 
 In adding them, x disappears, and there remains 
 
 40— 16O>0 .•. *<2|; 
 then the values which wo ought to give to t are 1=0, 1, 2. 
 With the value t = we should have 
 
 y=0, z=20 — 6x, u=204-G.r; 
 and we see that we can mako x=0, 1, 2, 3. From whence result for the 
 proposed equations 
 
 'x= 
 
 ' x= 1 
 
 r x= 2 
 
 r x= 3 
 
 v= j 
 
 r=20 
 u=20 
 
 •?/= 
 z=14 
 k U= 
 
 y= 
 z= 8 
 
 w=32 
 
 v= ° 
 
 ^«=38. 
 
 With the value t=l we should have 
 
 y=5, r = 17— G.r, t/=7 + 6x; 
 and the only admissible values of x are x=0, 1, 2. Thence result the throe 
 solutions 
 
 x= 2 
 
 y= 5 
 
 z= 5 
 
 a = 19. 
 
 z=ll 
 
 « = 13 
 
 Finally, with the value ( = 2 we should have 
 
 2/=10, z=14— 6x, u = — G + Gx. 
 The only admissible values of .r are x=l, 2; and from thence result the 
 two further solutions 
 
 ( x= 1 (x= 2 
 
 i/ = 10 I i/ = 10 
 
 z = 8 ] z = 2 
 
 u = ( u= ti. 
 
 In all, nine solutions. There would be but three if those w hided in 
 
 which one of the unknowns is zcm. 
 
 1 \ \ M 1 I 
 
 1°. Two countrymen have together 100 eggs. The one says to the other, 
 If I count ti iv eggs by eight 1, there is a surplus of 7. The second answers, 
 If I count mine by tens, 1 find the same surplus of T. How many eggs had 
 each .' 
 
 Ans. Number of eggs of the first, =63 or 23; of the second, =37 or 77. 
 
 To find three whole numbers BUCU that, if we multiply tile lir-t by 3, 
 tlie second by . r ), and the third by 7. the sum of the products shall be 
 
QUADRATIC EQUATIONS. 199 
 
 and such, moreover, that if the first be multiplied by 9, the second by 25, and 
 the third by 49, the sum of the products shall be 2920. 
 
 Ans. First number, =15 or 50. 
 Second number, =82 or 40. 
 Third number, =15 or 30. 
 3°. A person purchased 100 animals at 100 dollars; sheep at 3J- dollars a 
 piece ; calves at 1 J dollars ; and pigs at J- a dollar. How many animals had he 
 
 of each kind ? 
 
 Ans. Sheep, 5, 10, 15. 
 Calves, 42, 21, 6. 
 Pigs, 53, 66, 79. 
 
 4°. In a foundry two kinds of cannon are cast ; each cannon of the first sort 
 weighs 1600 lbs., and each of the second 2500 lbs. ; and yet for the second 
 there are used 100 lbs. of metal less than for the first. How many cannons 
 are there of each kind ? 
 
 Ans. Of the first, 11,36...; of the second, 7, 23 ... . 
 Or, of the first, ll + 25£ ; of the second, 7 + 16f. 
 5°. A farmer purchased 100 head of cattle for 4000 francs, to wit: oxen at 
 400 francs apiece, cows at 200, calves at 80, and sheep at 20. How many had 
 he of each? 
 
 Ans. In excluding the solutions which contain a zero the problem admits of 
 the ten following : 
 
 Oxen, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4. 
 Cows, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2. 
 Calves, 24, 21, 18, 15, 12, 9, 6, 3, 5, 2. 
 Sheep, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92. 
 
 QUADRATIC EQUATIONS. 
 
 178. Quadratic equations, or equations of the second degree, are divided 
 into two classes. 
 
 I. Equations which involve the square only of the unknown quantity. 
 These are termed incomplete or pure quadratics. Of this description are the 
 equations 
 
 rf 
 
 ax*=b; 3a*+ 12=150-0?; 3— Y2+ 3a;3 =2i + 2:c2 +~2T ; 
 they are sometimes called quadratic equations of two terms, because, by trans- 
 position and reduction, they can always be exhibited under the general form 
 
 ax"=b. 
 Thus the third of the equations given above, 
 
 x" 5 n n _ 7 259 
 
 ■3- i2+ 3a? — M+^+'aT 
 
 when cleared effractions, becomes 
 
 8x 3 — 10 + 72.i-=7+48.r 2 +259, 
 or, transposing and reducing, 
 
 32.t 2 =276, 
 which is of the form 
 
 ax-=b. 
 
200 ALGEP. 
 
 II. Equations which involve both the square and the simple power f the 
 
 unknown quantity. These are termed adjected or complete quadratics. Of 
 
 this description are the equations 
 
 x 3 2.r 273 
 
 ax"-+bx=c; z=-10.r=7; — -- + - = 8-j-x"- + — ; 
 
 they are sometimes called quadratic equations of three terms, because, by 
 transposition and reduction, they can always be exhibited under the general 
 form 
 
 ax"-\-bx:=c. 
 Thus, the third of the equations given above, 
 
 5.r 2 x 3 2j o 273 
 
 ~6~"~2 + 4 = 8 — "3 _ *°"+T2~' 
 when cleared of fractions, becomes 
 
 10a?— G.r+9=9G — Sx— 12X-+273, 
 or, transposing and reducing, 
 
 22x 2 +2x=360, 
 which is of the form 
 
 ax--\-bxz=c. 
 
 SOLUTION OF PURE QUADRATICS CONTAINING ONE UNKNOWN QUANTITT 
 
 179. The solution of the equation 
 
 ax- = b 
 presents no difficulty. Dividing each member by a, it becomes 
 
 b 
 
 whence 
 
 x~=-, 
 a 
 
 -±£ 
 
 b 
 
 If- be a particular number, either integral or fractional, we can extract its 
 
 square root, either exactly or approximately, by the rules of arithmetic. If 
 
 b 
 
 - be an algebraic expression, we must apply to it the roles established for the 
 
 extraction of the square root of al quantil 
 
 It is to be remarked, that sinco the square both of -\-m and — m is -j-m 9 . 
 
 ova 
 
 eo, in like manner, both (+./-) ™v\ (— /-) is + -. Hence the ab 
 
 equation is susceptiblo of two solutions, or has two roots : that i<. there are 
 two quantities which, when substituted for x in the origu a] equation, will ren- 
 der the two members identical ; these are 
 
 lb (b 
 
 .r= -I- /- and .r= — /- ; 
 
 for, substitute each of these values in the original equat'on ax' , =b, it become* 
 
 and 
 
 a X\—Jj ■**• or «X-=6i i- e., I> = 
 
QUADHATIC EQUATIONS, oqi 
 
 Henco it appears that in pure quadratics the two values of the unknown 
 quantity are equal with contrary signs.* 
 
 EXAMI>LE I. 
 
 Find the values of a - which satisfy the equation 
 
 4x-— 7 = 3x- 2 +9- 
 Transposing and reducing, .t- = 16 
 
 .-. x r=± -/16 
 = ±4; 
 henco the two values of x are + 4 and — 4, and either of these, if substituted 
 for x in the original equation, will render the two members identical. 
 
 EXAMPLE II. 
 
 x* 5x" 7 299 
 
 3~ 3 +T2 = 24 - ' r2+ "2T• 
 Clearing of fractions, 8x-— 72+10.r 2 = 7— 24.t 2 -}-299 
 Transposing and reducing, 42r 2 =378 
 
 378 
 
 
 
 
 
 i* 
 
 "42 
 = 9 
 
 
 
 
 
 ••. X 
 
 = ±3, 
 
 and the two 
 
 values of 
 
 x are 
 
 +: 
 
 5 and 
 
 —3. 
 
 
 
 
 
 EXAMPLE III. 
 
 
 
 
 
 3a 2 
 
 X 2 
 
 x ■ 
 
 =5 
 
 5 
 
 = 3 
 
 3 
 Since 15 is not a perfect square, we can only approximate to the two va] 
 of a:. We find the approximate values to be 
 
 ar=1.290994, or —1.290994. 
 
 EXAMPLE IV. 
 
 X 
 
 ^/r i -\-x"-—x 
 
 -.m. 
 
 Clearing of fractions, x =??i V r s +a? — mx, 
 
 .-. (m-\-l)x =wi A //--+.r-. 
 Squaring, (m' i -\-2m-\-l)x"=m-(r--{-x-), 
 
 .-. (2m + l)x° = m"r 2 
 
 mr 
 
 ' -/2m+l" 
 
 * One might suppose that in extracting the square root of both members of 
 tion as afl=b, the double sign should be prefixed to x, the root of x", also. But it is to 
 be observed, that it is the value of -\-x that is required. Besides, suppose we were to write 
 -\z x= ^^.V b : combining these signs in all possible ways, there result the four equations, 
 
 ■4-x=-f-v'£> -f-.r= — t/1, — 2-— -f--i/"'- — .r= — ^b, 
 the last two of which may be deduced from the first two by changing the signs of the two 
 members ; the equation J-.r^-J-y'i expresses nothing more, therefore than the equation 
 ar=-J-l/&. We might always omit -J^, si aplied before i/ 
 
202 ALGEBRA. 
 
 i:xAJ:ri.i: v. 
 
 m-\-x-\- \/-J',i r-\-.i : 
 
 = 11. 
 
 m + r — V ~>nx + x- 
 Render the denominator rational by multiplying both terms c the fraction 
 by the numerator, the equation then becomes 
 
 (?«-f-- r + y/2mx-\-3?)* 
 
 m 2 
 Extracting the root, 
 
 ra-}-x-f- ->/2mx-\-x-=±m^n. 
 Transposing, t/ 2/nx-\-x-= ^m <J n — (m+ x). 
 
 Squariag, 2mx-{-x-=m"n^2m V«(»i+x)-|-(»i-}-;r) 8 . 
 
 transposing and reducing, 
 
 ±2m. i/ n(?n-\-x)=m 2 (l-}-n), 
 
 m(l+ | 
 .•. m-f-x s=- 
 
 n) 
 'n 
 
 m(l-\-n) 
 x = —m 
 
 = rLm 
 
 2-y/n ' 
 
 (6) ll(x 2 — 4)=5(x*+2). Ans. x=±3. 
 
 ^_j_7 x 7 7 
 
 '7) ta — = . , ; — =^=0. Ans. x=4-9. 
 
 x 3 — 7x x--f-7.r x- — 73 
 
 m-4- -\/ m z — x 2 x 
 
 (8) =-. Ans. x=± y/2mn— »•. 
 
 (9) V^-^- V>-+* : =P Ang- x=± ; ,«*< ? ,-.g)W(;, + 0» 
 
 •v/p+x-r- -v/v — x /x 
 
 ^ 10 ) V* V? = ' AnS - *=± 2 Vj>q-f- 
 
 180. In the same manner wo may solve all equations whatsoever, of any 
 degree, which involve only one power of the unknown quantity ; that is, all 
 equations which are included under the general form 
 
 ox = b, 
 or equations of two terms. 
 For, dividing each member of the equal nos 
 
 b 
 
 x"=-. 
 a 
 
 Extracting the ri k root on both sides, , 
 
 If n be an even number, then the radical must :ted with tho double 
 
 nign ±, for, in that case, both (+M~) M>d ( — "/-) wiL equally pro 
 
 duce - 
 
 a 
 
QUADRATIC EQUATIONS. 203 
 
 EXAMPLE XI. 
 
 5&— 57=2^+135 
 
 3x 6 =192 
 
 x"=64 
 
 Here +2 and —2 are two of the roots of the above equatior 
 
 EXAMPLE XII. 
 
 p ' x q 
 
 px^Jx 
 
 (p+ x )Vl>+ x =— —• 
 
 2 3 p 
 
 Or, (p+x) L '=x 2 .-. 
 
 Squaring, (^+x) 3 =a- 3 . - 
 
 9 
 
 /- 
 
 f/ 1 
 
 Extracting the cube root, p-\-x=x* 1L 
 
 V 
 
 
 EXAMPLE XIII. 
 
 T 7 - p -i r : -i a /TV — 
 
 Ans. x — '- -•■>*-■» 
 
 — Xq = -x 
 q S 
 
 EXAMPLE Xi v . 
 
 G4?/ 5 — 48?/ 4 +127/ 2 — 1 =-G 1. 
 Extracting the cube root, we have 
 
 = I — |p'— • 
 
 w 
 
 /5 V^ 
 
 42/' 
 
 EXAMPLE XV. 
 .T 3 — 1/ s = 117 (1) 
 
 x -y = 3 (2) 
 
 Cubing the latter equation, 
 
 r 3 — 3x-?/+3x^/-— t/ 3 = o 7 , 
 out x 3 — T/ 3 = 117. 
 
 .-. by subtraction, 3x 2 y—3xy' 2 — 90, 
 
 and ' xy(x—y) = 30; 
 
 dividing by (2), we have .-. xy = 10. 
 
 Now from (2) x 3 — 2xy+y*— 9, 
 
 and 4.ry = 40. 
 
 .-. by addition, x 3 +2xy+y'*= 49, 
 
 and k+J=±7, 
 
 but (2) x— y = 3. 
 
 By addition, 2x = 10, or — 4, 
 
 .-. x = 5, or — 2, 
 and by subtraction 2y = 4, or —10, 
 
 .-. y = 2, or — 5. 
 
201 ALGEBRA. 
 
 (16) J=2a*+26, Ana.x=±i/Ti, 
 
 (17) x- : (18—*)*:: 25 : 16. Ans. r=10 or 90. 
 
 x 14 x * 
 
 ( 18 ) HZ^ : — T" ::1 Ans.x=8or5G. 
 
 75Qr-7) 48(z-4) 
 
 ^ iJ i ' x _ 4 ~= r _ 7 • Ans. x=19 or J 
 
 (20) . •_ /= 10, iy— ^3—15. Ans. .r=±8, y = 
 
 (•-'I) (*— y)r=91, (x— y)»=49. . x=±13, t/=±6. 
 
 (22, (*-y£=24, (*-y)£=6. 
 
 z =12, or i. 
 
 (23) x : y = 13 , xy-= 36. Ans. x=4, y=3. 
 
 (24) £ry= a/^+2/ 2 +2-+2/, a«+y«=:(:r+ .'/)*- i 
 
 A: 3. T=6, y= 
 
 '- ■/'' ?/.r 2 4-?/-.r 
 
 (25) ; ' r --.r +i /, 2_££_ =4 . Al)S . r= 2j y== , 
 
 (26) i a +y°=a, x a -y a = b. 
 
 (2; ,r' + l0.r 3 — I0x- + 5x— 1=32. Ans. r=3. 
 
 (28) -2*«+l= Ans. <=±V6- 
 
 ' - Vy=3, V-r+ Vy=7. Ans. 1=625, y=16. 
 
 (30) .!-•-_ y'=369, .r- — y*=9. Ans. z= | -,. v=±4. 
 
 (31) .r 5 — t/ 3 =56, x—y=—. Ans. .r=4 or —2, y=2 or — 1. 
 
 !) .r-.y + T/^HG, xy?+y=14. Ans. .r=5 or 2-/|, y=4 or 10. 
 
 V*+ \fy=6, x+y=72. Ans. x=64 or 8. y=8 or 6 
 
 34) | =20, ar|+ yy=6. Ans. .r=±8 or i -/8, .7=32 or 1 
 (35) .r 4 +2.r 2 ?/-+y<=129G — 4xy(x 2 +xy+ij : ), x—y= 
 
 Ans. .r=5 or — 1, ;/ = l or — 5 
 181. We have seen that an equation of the form ax" = b 1 roots, or 
 
 that there are two quantities which, when substituted for x in the original 
 equation, will render the two members identical. In like maimer, we 
 find that every equation which involves x in the third power has three roots ; 
 an equation which contains x* h 'oots ; and it is a general proposition 
 
 in the theory of equations that an equation has as many roots as it has di- 
 mensions. 
 
 :. The above method of solving the equatii zb will give us only 
 
 one of the n roots of the equation if n be an odd number, and two roots if n be 
 an even number. Such a solution must, therefore, bo c 1 imperfect, 
 
 iliiiI we must ha arse to different pp to obtain the remaining 
 
 roots. This, however, is a subject which we in one for the present 
 
 SOLUTION OF COMPLETE QUADRATICS. COITTAINII ' \KNn\VN iM\n 
 
 183. [n order to Bolve the general equal 
 
 \-h.r=c, 
 
 let us begin by dividing both members bj . the coefficient of .< ' : the o-jua 
 
 u<m then becomes 
 
 c 
 
 -(--!■=-, 
 
QUADRATIC EQUATIONS. 205 
 
 or, 
 
 x"-\-px=q, 
 
 putting, for the sake of simplicity, 
 
 6 c 
 a 1 a l 
 This form of the quadratic equation may be produced by multiplying to- 
 gether two simple equations. Suppose 
 
 x— a=0, x— /3 =0; 
 ... (a-a){x-(i) = 0, 
 which is satisfied by making x = a, or x=3. 
 
 Multiplying the two factors (x—a) and (i— /3), the equation becomes 
 
 x*—{a+(3)x+a8=0 (1) 
 
 Substituting first a, and then (3, for x, this may be written either 
 
 a 2 — (a+/J)a + a/3 = 0, 
 
 or 
 
 (3- — {a+(3)(3+af3=0, 
 
 which are identical. 
 
 Putting in equation (1) above, p in place of — (<z-f-/?), and —q in place ot 
 a.8, it assumes the form 
 
 a- 2 -|-px — q=0. 
 
 But 
 
 _p 2 = a-+2a,3+/3 2 
 
 — 47= 4a/? < 
 
 By subtraction, • „ , . s — •-, i en ~i ««' 
 
 .-. a — 8 = V7 ;2 + 4 7- 
 
 q+]3=— p. 
 
 By addition and subtraction, a= — 2P+3 v'i'' 2 + 4 5 
 
 (3=-lp-iVp-+*q- 
 
 As a and /3 are the values of .r, and differ only in the sign of the radical part, 
 both may bo written together thus : ' 
 
 Hence tho following rule for resolving a complete or adfected quadratic 
 equation. 
 
 Reduce the given equation to the form x 3 +px — q=0 by clearing of frac- 
 tions, transposing all the terms to the first member, and dividing throughout by 
 the coefficient of the square of the unknown quantity. The equation being thus 
 prepared, the value of the unknown quantity will be equal to ± the coefficient of 
 its first power with the sign changed, ±? the square root of the square of this 
 coefficient — 4 times the knoivn terms of the equation. 
 
 Tho expression x= — ^pzLlVp"-\--lq may, by passing the | under the 
 radical, be written x= — l,p^z V {%?)*-{-& which, translated into a rule, is 
 often the more convenient form. 
 
 EXAMPLES. 
 
 (1) z 2 — yx+2=0. 
 By the rule, 
 
 11 , //11\ 3 11 , /121 11 , , /49 11 , 7 
 
 *=6-±K%) - 4x2 =¥ ± n/-9-- 8 =ir ± Wir=-G ±*« 
 2 
 
 .'. x='3 or -, 
 according as we use the upper or lower sign. 
 
206 ALGEB 
 
 (2) 3r— I s =2 ; changing nil the si 
 X s — 3;r= — 2,0 »— 3x+2=0. 
 
 By tho rule, 
 
 3 
 
 X=5±J-/9— 2X4=s2 or 1. 
 
 Hither of thcso values of x will satisfy the given equaticn. First substi- 
 tuting 2, wo have 
 
 3x2—4=2; 
 and substituting 1, we have 
 
 3X1 — 1 = 2. 
 
 (3) x-+6z= 16. 
 By tho form, 
 
 x=—]p± V{hp)- +q 
 
 x= — 3± a/^+16=2 or — 8. 
 
 (4) x- — 10.r =— 21 
 
 X—5± V25— 21 
 x=7 or 3. 
 
 (5) acx 2 -f tax — adx — bd=0. 
 Dividing by ac, 
 
 }l> <\ bd 
 x 2 -f - — -).r= — . 
 1 \a c/ ac 
 
 .-.by the rule, 
 
 ^-3W£¥+?-- 
 
 .-. x=-, or . 
 
 c a 
 
 (6) a*+6x=27. Ans. x=3, or —9. 
 
 (7) x"-— 7.r+3| =0 Ans. x=6$, or 1. 
 
 10.r 
 
 (8) a?+— =19. ins. x=3, or — 6J. 
 
 (9) »=3+i2" =10, or 2. 
 
 (10) x 2 — Gx+8 = 80. Ans. x=12, or —6. 
 
 (11) x°— 10x+17 = l . r=8, or 2. 
 
 (12) x*— x— 40=170. Ans. x=15, or —11. 
 
 (13) 3x 2 — 9x — 4 = 80 \> . =7, or —4. 
 
 (14) 7x 2 — 21.r+13 = 293. Ans. x=8, or -5. 
 
 X* 4x 57 
 
 (15) -+7 19=15*. Ans. x=9, or — — . 
 
 o O o 
 
 2x q x 9 
 
 (1G) — + 3J- -+8. Ans. x=3, or — -. 
 
 (17) x-j- I-) = 13. Ans. x= i, or —2. 
 
 3G — u 
 
 (18) 4u— =46. Ans. u=12, or —J. 
 
 5— i 9— 3p 
 
 (19) 1G-- - Tr = - + 3 P . Ans. .. = •;. . 
 
 ~ P 
 
 16+3 1G— G9 
 
 < 20 > HT+ *df= 6 *" An.. y = 5, or -. 
 
QUADRATIC EQUATIONS. 207 
 
 □ _1_7 + 4D 
 
 (21) 14+4 □ —77^7=3 D + — g — Ans. D =9, or 28. 
 
 Ans. A =2, or 
 
 3 
 
 (23) ^-f -?=—;—. Ans. <=2,or|f. 
 
 7A 3 2A 11A + 18 
 (22) — 3-= 33 • 
 
 < + 22 4 9< 
 
 1 < 
 
 <t> 0+1 13 t 
 
 < 24 ) ^i+^T^T Ans. 0=2, or -3. 
 
 (^) xT60=3-xb- An3 - X = 14 ' " - 10 ' 
 
 8?j ^0 
 
 (26) — — - — 6=—. Ans. v=\0, or — §. 
 ' tf+2 3u 
 
 48 1C5 . ' . _ 
 
 < 27 ) ^=7+To- 5 - Ans. »=5 3 , or 5. 
 
 (28) x 3 — 8x=14. Ans. x=9.4772+, or —1.4772+. 
 
 (29) 3x 3 +x=7. Ans. x=1.3699+, or — 1.7032+. 
 
 (30) 6x— 30 = 3x 3 . Ans. x=l + 3 V — 1- 
 
 (31) (x— V 142.334) (.r+ -/ 142.334) =27.22x. 
 
 Ans. a:=13.61± V 327.566. 
 
 (32) 23 : (140+x) = (240+x) : 1041. Ans. x=— 27.4 or —352.6. 
 
 , V34 
 
 (33) (x+6):(3x+12) = (3x— 12): (a:— 6). Ans. ar=±— £-. 
 
 (34) 21:: 3 — 1617z + 20748=0. Ans. z=60.73, or 16.27. 
 
 (35) 3.5g a — 11.75g— 41.25=0. Ans. g=5.5, or —2.14 . 
 
 1+ V1008 
 
 (36) (3x+l)(4x— 2) = (13x+7)(5x— 3). Ans. x= — 
 
 x 7 
 
 (37) -7-77;— ?=0. Ans. .r=14, or —10. 
 
 ' x+60 ox— o 
 
 w+4 7—w 4w4-7 
 
 (38) — ^—— -_=— ^—— 1. Ans. w=21, or 5. 
 
 v ' 3 w — 3 9 • 
 
 15— p 12—3o 23»+60 
 
 (39) —~- Tp -^r=1]> *J—> Ans. i ;=3, or ftf. 
 
 x+11 9 + 4.r 
 
 (40) — + , =7. Ans. x=3, or —A. 
 
 2 ? +9 4o— 3 3(7—16 
 
 ( 41 ) ^9 t -+ 4 i+3 = 3 + ^8- AnS ' ? = 6 ' ° r *' 
 
 2.r — 1 8— x 2 x ,. 
 
 (42K = -4 — . Ans. x=2, or — V. 
 
 **** 3— x 2x— 2 T 2 3 
 
 3 6 11 » o "R 
 
 (43) 4- = — . Ans. x=3; or ff. 
 
 y > 6x— x-^x 3 +2x 5x lI 
 
 4x 3 +7x 5x— x 3 4X 3 a o 9 t 
 
 (441 ! 1 _ — . Ans. x=3, or — |i. 
 
 K ' 19 ^ 3+x 9 "^ 
 
 r»+2x 3 +8 . . ., 
 
 (45) —f. ^=x 3 +x+8. Ans. x=4, or — y 
 
 ' x ; +x — 6 
 
 ac . c± y/<9— \ac 
 
 (46) cx-— b = (a + b)x*. Ans. s= 2(q+&) 
 
 (47) x 3 +tx+cx+at— ac+6c— a 3 =0. Ans. x=a — 6, or — a— c. 
 
ops ALGEBRA. 
 
 (48) 2(o-c)y/2- r -a»=(6-c)«+< 
 
 Ans. ?/ = ~ • 
 
 184. If t=a in the general form (.r— a){x— i)=0, it assumes the partic 
 ular form (x—a)-=x"—2ax+a-=0. 
 
 If tho two values of .r be -\-a and —a, the form (.r— fl)(.r+a)=.r- — a- = l). 
 
 185. Recollecting that the value of the unknown quantity is called the root of 
 the equation, it is seen that every equation of the second degree has two roots, 
 and, by the general form (I), x' : — {a-{-b)x-\-ab=0, that their sum is equal to 
 the coefficient of the second term with the contrary sign, and that their prod- 
 uct is equal to the absolute term or known quantity, when transposed to th< 
 first member. Thus, in Example 4, above, the sum of the two roots 3 and 
 _9 j s _(j ? a nd tho product —27. The same may be seen in other exam- 
 ples. 
 
 The general form ax"-\-bx=c is capable of producing all the particular 
 forms by the supposition of particular values for the coefficients. Thus, if 
 6=0, it assumes the form of pure equations. If c = 0, it may be written 
 
 x(ax-[-b)=0, 
 
 b 
 which we perceive may be verified by making x=0, or ax-{-b=0 .-. x=— -. 
 
 b 
 The roots are, therefore, in this case, and — -. Whenever an equation is 
 
 divisible throughout by the unknown quantity, one of its roots is zero. 
 
 When wo know that the two roots of the equation of the second degree are 
 real, the above relations make known at once the nature of these roots ; for 
 example, admitting that those of the equation x- — 2x— 7 = are real, we 
 conclude immediately that they are of different signs, because their product 
 is equal to the absolute term —7, and, moreover, that the greater is positive 
 because their sum is +2, the coefficient of x taken with the contrary sign. 
 
 18G. Another mode of solution may be derived as follows : 
 
 If wo can, by any transformation, render the first member of the, equation 
 x?_j_p;r = <7 the perfect square of a binomial, a simple extraction of the square 
 root will reduce tho equation in question to a simple equation. 
 
 But (.r-H j;) 3 is x--\-jrx-{- ' 
 
 In order, therofore, that tho fust member may be transformed to a perfect 
 square, wo must add to it the square of './> ; that is, the square of half the co- 
 efficient of the second term, or simph \ ; it thus becomes 
 
 x"-+ ] u+ l r 
 
 which is the square of x-f _,. Bui sine,' \v I to the left-hand 
 
 member of the equation, in order thai tin- equality between the two members 
 may not be '1 troyed wo must add tin- same quantity to the right-hand mem- 
 ber also ; the equation thus transformed will be 
 
QUADRATIC EQUATIONS. 20'J 
 
 or 
 
 p Ip 2 
 
 Extracting the root, x-\- - = ± -y ^-+ ?• 
 
 Transposing, x=—-±yl-—-\-q 
 
 *=- fWf 
 
 — JP ± \Ap 2 + 4g 
 - 2 
 
 the same form for the value of x as we obtained by the first method. 
 
 If- \~f~ 
 
 We affix the sign ± to ■\f-r-\-Qt because the square both of -\-yJ-r-\-q- 
 
 and also of — \/~r + ?> * s +(x"^^)' an( ^ evei y quadratic equation must 
 
 therefore, have two roots. 
 
 From what has just been said, we deduce the following general 
 
 RULE FOR THE SOLUTION OF A COMPLETE QUADRATIC EQUATION. 
 
 1. Transpose all the known quantities, when necessary, to one side of the 
 equation, arrange all the terms involving the unknown quantity on the other 
 tide, and reduce the equation to the form ax 2 -f-bx=c. 
 
 2. Divide each side of the equation by the coefficient ofx". 
 
 3. Add to each side of the equation the square of half the coefficient of the 
 simple power ofx. 
 
 That member of the equation which involves the unknown quantity will 
 thus bo rendered a perfect square, and, extracting the root on both sides, the 
 equation will be reduced to one of the first degree, which may be solved in 
 the usual manner. 
 
 EXAMPLE I. 
 
 12x— 210=205— 3.r 2 +5. 
 Transposing and reducing, 
 
 3.r 2 +12r=420. 
 Dividing by the coefficient of a; 2 , 
 
 2; 2 +4a;=140. 
 Completing the square by adding to each side the square of half the coefficient 
 of the second term 
 
 z 2 +4:r+4 = 140 + 4, 
 o 
 
 (,r+2) 2 =144. 
 
 Extracting the root, x-f-2=± -\/l44 
 
 ' =±12 
 .-. x=— 2 ±12. 
 Hence 
 
 <x=— 2-f 12 = 10 
 (x=— 2 — 12 = — 14. 
 Either of these two numbers, when substituted for .r in the original equation, 
 will render the two members identical. 
 
 O 
 
210 ALGEBRA. 
 
 EXAMPLK II. 
 
 2a: 2 +34 = 20x+2. 
 Transposing and reducing, 
 
 2a: 2 — 20x=— 32. 
 Dividing by 2, a: 2 — 10z= — 16. 
 
 Completing the square, 
 
 a?— 10x+ 25=25— 16, 
 or (x— 5) 2 =9. 
 
 Extracting the root, x — 5= i V 9 - 
 
 r=5±3. 
 
 Hence 
 
 <a:=5+3=! 
 (x=5 — 3=', 
 
 EXAMPLE III. 
 
 3a: 2 — 2a = 65. 
 
 2 65 
 Dividing by 3, x-—-x=z—. 
 
 Completing the square, 
 
 3 + \3/ ~~ 3 + \3/ ' 
 
 or 
 
 (*-$=- 
 
 96 
 
 Hence 
 
 -±¥ 
 
 1 14 
 XBB 3 ± T 
 
 1+14 
 x=-^-=5 
 
 1—14 1 
 
 X =-3~ = - 4 3- 
 
 EXAMPLE IV. 
 
 rS+x— 2 = 0. 
 Transposing, z 2 +x = 2. 
 
 The coefficient of x in this case is 1 ; .-. in order to complete the square. w*» 
 
 /iy 1 
 
 must add to each side I -I , or -. 
 
 •'• *M-*+J= 
 
 = 24— 
 ~t 4 
 
 W- 
 
 9 
 = 4 
 
 1 
 
 X +2 = 
 
 ■±l 
 
 .'. x=l, and ars 
 
 o 
 
QUADRATIC EQUATIONS. 
 
 211 
 
 EXAMPLE V. 
 
 6x— 30=3x 2 . 
 Transposing, — 3x' 2 +6x=30. 
 
 Changing the sign on both sides, - 
 
 3x 3 — Gx= — 30. 
 Dividing by 3, x 2 — 2x= — 10. 
 
 Completing the square, x 2 — 2x-|-l = l — 10, 
 or 
 
 (x— 1)"= — 9. 
 
 Hence 
 
 x— 1 = ± V — 9. 
 
 (x=l + V— 9 
 ?x=l— •/— 9' 
 In the above example, the values of x contain imaginary quantities, and th« 
 roots of the equation are, therefore, said to be impossible. 
 
 EXAMPLE VI. 
 
 5 1 3 , 2 273 
 
 -^--x+- = 8- 3 -,-^+— . 
 
 Clearing of fractions, 
 
 10x 2 — 6x4-9=96— 8x— 12x 2 -f 273. 
 Transposing and reducing, 
 
 22x 2 +2x=360. 
 Dividing both members by 22, 
 
 2 360 
 22' T:= "22 
 
 /1\ 2 
 Adding I — ) to both members, 
 
 i2 
 
 * 2 +™*=l^-- 
 
 a:2+ 22 :r + \22/ == 22 _+ \22/ 
 
 Extracting the root, 
 
 Hence 
 
 ar +22 =± V22"+W) 
 
 /7921 
 = V(22)» 
 
 ~ ± 22' 
 
 1 89 
 x=— — 4- — =4 
 
 22^22 
 
 1 89_ 
 X 22~"22~~ 
 
 45 
 "11* 
 
 Transposing, 
 
 ax 2 - 
 
 EXAMPLE VII. 
 
 ac 
 
 'a + b~ 
 
 :CX — bx*. 
 
 (a-\-b)x~ — cx= 
 
 ac 
 
 a+b' 
 
n2 ALGEBRA. 
 
 c ac 
 
 Dividing by a+b, x °—-^+b ' X=z {a + bf 
 
 Completing the square, 
 
 c c» , cc c 2 
 
 x V~^p ' X +4(a+t) a -(a+6)^4(a + 6^' 
 
 or 
 
 r c ) 9 c"4-4ac 
 
 l X "2(a + 6)S ~4(a+6)»" 
 
 Extracting the root, 
 
 a: — 
 
 c •v/c-4-4gc 
 
 2(a + 6) = ± 2(« + 6)' 
 
 ••• x = 2(a + 6) ' 
 The two values of x here are 
 
 C _L. Vc 2 +4ac _c— jv/c^4-4ac 
 *= 2(a+6) ' X - 2(a+6) 
 
 EXAMPLE VIII, 
 
 m 2 x 2 
 a 2 +& 2 -2&x+x 2 =-^-. 
 
 Transposing, (n 9 — m^x 2 — 2*m 2 x= _n 2 (a 2 -j-i 2 ). 
 Dividing by the coefficient of x 2 , 
 
 2bn*x 9 <*"+&' 
 
 x °" — n 2 — m 2— — W '»»— m 9 " 
 
 Completing the square, 
 
 2fcn 2 x / bn 2 \ 2 _ / 6n 9 \ 2 _ 7T-(a 2 +fr 3 ) 
 ** — ^^Z^>+ Wll^ 2 / "~\n 9 — »»V » 9 — to 9 
 or 
 
 — \ m-{a : 4-L-)—n : a- [ 
 
 Extracting the root, 
 
 I __^L_ = ±-^— ■ v '>n 9 la*+& 9 )-n 9 a 9 
 
 n 2 — m* 
 x 
 
 n 2 — m- ( * 
 
 The two values of x are 
 
 x=^^ | bn+ /m 9 (a 9 +&»)-»"a« \ 
 
 n S I 
 
 x = - „ J l,n— % /m : (a--! r b-) — ri 2 a i t 
 
 n- — mr t ' 
 
 .g\ x 2 4-4x=21. Ans. x=3, x= — 7. 
 
 (10) x 2 — 9z+4j=0. Ans. x=8j, x= 7y 
 
 (11) 622x— 15x 2 =6384. A.M. x=22|» *=18 . 
 
 (12) a-r 9 — 7x4-34=0. Ans. x: 
 
 7+- v /_ln 7— V— 1039 
 
 z 
 
 16 ' 16 
 
aUADltATIC EQUATIONS. 213 
 
 — 1+V^33 _l_ s /"l33 
 
 (13) 3a. a +x=ll. Ans. x= - , x= - 
 
 x 4x 2 
 
 (14) -_ 4_ i»+23?— —=45 — 3j*+4:r. 
 
 o O 
 
 6x 2 — 40 3r— 10 23 
 
 (15) 3x — — — - — — =2. Ans. :r=— , x=4. 
 
 v ' 2x — 1 9 — 2x 2 
 
 90 90 27 5 
 
 (16) ——-—-r———7=0. Ans. x=4, z= — -. 
 v ' x x+1 x+2 3 
 
 3a 2 a: 6a 2 +a&~2Z> 2 Z> 2 .r 2a— b 3a+26 
 
 (17) ate 2 + = - . Ans. x= , x= - 7 . 
 
 v ' ' c c 2 c ac oc 
 
 ■\frnn -\/mn 
 
 (18) m.f 2 — 2mx-Jn=?ix' i — mn. Ans. £=— -. — ; — j-, x=—: j-. 
 
 v ' y m-\- yn ym — yn 
 
 (19) 4o% 2 +4a 2 c 2 ar+4aii 2 x— 9c^ 2 x 2 +(ac 2 +ic? 2 ) 2 =:0. 
 
 ac*+bd* ac^+bd 1 
 
 Ans. x= — - — , ■ , , x=- 
 
 2a + 3rtVc' 2a—Zdy'c 
 
 5a + 10a& 2 (by'a + b (l + 26 s W^c\ ci 
 
 < 20 ) 9T 2 ^3^^-(^-r L -+ 3-a 2 )*+^( a +»> c = - 
 
 _(3— a»)Va+6 _3b*cdy'c 
 AnS ' X_ afc(l+26 2 ) ' X= 6a ' 
 
 187. The above rule will enable us to solve, not only quadratic equations, 
 but all equations which can be reduced to the form 
 
 x- a -\-px n =q; 
 that is, all equations which contain only two powei's of the unknown quantify, 
 and in which one of these powers is double of the other. 
 
 For if, in the above equation, we assume y=x", then y^^zx* 1 . and it be- 
 comes 
 
 y*+py = q. 
 Solving this according to the rule, 
 
 — pAi V^' 2 +4? 
 
 y= 5 • 
 
 Putting for y its value, 
 
 x n—— ? ^ ^ + 4? 
 
 2 
 Extracting the nth root on both sides, 
 
 V o 
 
 EXAMPLE I. 
 
 a- 4 — 25x 2 =— 144. 
 Assume x' i =y, the above becomes 
 
 2/2_25?/=— 144. 
 Whence 2/=16, 2/=9. 
 
 But since x"=y .•• x= ± Vy » 
 
 .-. a-= ± y/16, x= ± V9- 
 Thus the four values of x are +4, — 4, +3, — 3. 
 
214 ALGEBRA. 
 
 EXAMPLE II. 
 
 x*—7x*=8. 
 Assume x-=y, y" 1 — 7y=.8. 
 
 Whence y = 8, y= — 1 
 
 And since x 2 =y •'• *= i V y- 
 
 Whence the four roots of the equation are i -/8, i •/ — 1, t. f e last two 
 of which are impossible roots. 
 
 EXAMPLE III. 
 
 Let x 6 — 2r»=48. 
 
 Assume z-^zzzy, the above becomes 
 
 i/ 2 — 2jr=48. 
 Whence 2/=8, or — 6. 
 
 But since x s =y .•. x= tyy. 
 
 Hence two of the roots of tho above equation are -4- 1/8 and — 1/6; tti»- 
 remaining four roots can not be determined by this process. 
 
 EXAMPLE IV. 
 
 Let 2x— 7 • v / i-=99, 
 
 or 2x— 7x*=99. 
 
 This equation manifestly belongs to this class, for the exponent of .r in the 
 first term is 1, and in the second term half as great, or .',. 
 
 In this case assume ■y/x=y, the equation becomes 
 
 2y 2 —7y=99. 
 
 Whence 2/=9, y = — tt. 
 
 But since -\/x—y .-. x=y 2 
 
 121 
 
 • '. 2'z=bl, Xzzz — - — • 
 
 To account for the two values of x in this equation, it must be observed that 
 one belongs to -\- -\/x, the other to — -y/t' 
 
 This will appear clearly in the following example. 
 
 example v. 
 
 ax=b-\- -v/cx (1) 
 
 Solving this equation in the same manner as the preceding, we shall find 
 2ab-\-c-\- ■v / 4atc-|-c- 2<7?> + c — <J\abc-\-c' i 
 
 T= 2^ ,X= 2aT~ ' 
 
 If wo substituto these two values of x in the original equation, we shall find 
 that the first only will verify it ; the second belongs to the equation 
 
 ax=b — -\/cx (2) 
 
 These two equations, multiplied together, produce the complete quadratic 
 equation 
 
 <»V- (2«/> + r).r+6«=0, 
 whoso roots are tho two valnes of x given above. 
 
 Tho explication of this matter Is, that \/x is always supposed to have the 
 double sign i, and therefore the general form expressed !>y equation (1) in- 
 volves covertly that expressed by equation (2). It i :iy, therefore, it- 
 
aUADRATIC EQUATIONS. 215 
 
 examples of this kind, to tiy the answers obtained, by substituting diem, m 
 order to see which belongs to the given form. 
 
 188. Many other equations of degrees higher than the second may be solved 
 by completing the square ; although, it must be remarked, wo can seldom ob- 
 tain all the roots in this manner. The transformations to which wo subject 
 equations of this nature, in order that tho rule may become applicable, depend 
 upon various algebraic artifices, for which no general rule can be given. The 
 following examples will serve to give tho student some idea of the course he 
 must pursue ; a little practice will soon render him dextrous in the employ 
 ment of such devices. 
 
 EXAMPLE VI. 
 
 Let Vz+12+\/i-+12 = 6 * 
 
 Assume x-j- 12=?/, the equation then becomes 
 
 y*+y*=6, . 
 
 which evidently belongs to the same class as the previous examples ; completing 
 
 the square, we shall have 
 
 i 
 2/ T =2, or —3. 
 
 Raising both sides of the equation to the power of 4, 
 
 y = 16, or 81 
 
 ••. x, or y — 12= 4, or 69. 
 
 EXAMPLE VII. 
 
 Let 2a; 2 4--v / 2.r 2 +l = ll. 
 
 Add 1 to each member of tho equation, it becomes 
 
 W+l-^ V2x 3 +l=12. 
 Assume 2.t 3 +l=2/> thon 
 
 y+y*=\2. 
 
 Completing the square, and solving, we find 
 
 i 
 
 2/2, or -v/ 2i-2 + 1=3 ' and — 4 
 2a: 2 4-l=9> and 16 
 
 , 15 
 r 2 =4, and — . 
 
 /l5 /l5 
 
 Hence x =+2, —2, +\/~> — VIP 
 
 It may be remarked, that it is in general unnecessary to substitute y, which 
 has been done in the above examples for the sake of perspicuity alone. 
 
 EXAMPLE VIII. 
 
 / 8\ 2 8 
 
 Let (x+-) + x=42--. 
 
 Transposing V+x) + r+i) =42, 
 
 g 
 Considering x+~. as one quantity, and completing the square, 
 
 / 8\ 3 / 8\ 1 169 
 
216 ALGEBRA 
 
 8 1,3 
 
 ••• x +i=-2 ± y 
 
 = G, and — " 
 
 Hence we have the two equations 
 
 x" — Gxr= — 8 
 
 x 2 +7x= — 8. 
 
 Solving the first in the usual manner, we find 
 
 x=4, and 2, 
 
 and by the second, we have 
 
 — 7+-/17 , — 7 — Vl7 
 x= , and - , 
 
 which are the four roots of the proposed equation. If we had r^'uc.ed Una 
 equation by performing the operations indicated, instead of employing the 
 above artifice, it would have become 
 
 z'+x 3 — 26x 2 + 8x+64 =0, 
 a complete equation of the fourth degree. 
 
 The roots of equations of the fourth degree, reducible to the second as abore, 
 
 present themselves ordinarily under the form "\JaJc Vb, and frequently of 
 ford an application of the process exhibited at (Art. 104). 
 
 (9) x<+4x 2 =12. Ans. ar=±-/2, or ± V~6. 
 
 (10) x°— 8x 3 — 513=0. Ans. r==3, or — ^"19. 
 
 (11) x*— 13x 2 +36=0. Ans. t=±2, .r=±3. 
 
 (12) (x*-2) 2 =-(x 2 -fl2). Ans. x=±2,x=±- } . 
 
 (13) (a*— l)(x 2 — 2) + (x*— 3)(x 2 — 4)=x*+5. Ans s=±l, x=±3. 
 
 /m± ■/'»-+ 4/A L 
 
 (14) x-°— 7nx n =p. Ans. x={ ^ J ]". 
 
 jte+2 4-Vx 
 
 • 15) — =s = — . Ans. .t=4 * 
 
 4-f- V^ V^ 
 
 (16) V^# == £ = Vf. An8 r= / -ft=fci/4*+4a»+ty 
 a+ a/i V-r \ 2(a+l) / * 
 
 (17) Vx 3 — 2 -v/x— -r=0. Ans. x=4. 
 
 (18) a/^+ -/^=6 v^ Ans. x=2. 
 
 (19)5=22^+^. Ans. x=49. 
 
 3a/^ 
 i o 
 
 5 1 
 
 (20) — — — =0. Ans. x=25 or 49. 
 
 v ' x — 5 20 
 
 (21) x 7 +x- J =756. Ans. x=243, or (— 28) 
 
 1 3 
 
 (22) X s — x 2 =56. Ans. x=4, or (— 7) '. 
 
 i i 
 * In this and sorao of the following examples another value, *=— . is also found, bat if 
 
 •rill not satisfy thu equation, and is, therefore, to b< p. -14.] 
 
QUADRATIC EQUATIONS. $217 
 
 6 
 5 5 / — Q7\ 7 
 
 (23) 3x^+^=3104. Ans. x=G4, or I — -L\ . 
 
 (24) aJ+lJ=c. Ans. *=(* ^^-by 
 
 8 3 
 
 4 /ir 3 " / 74\ *~ 
 
 (25) 3r 3 — — =—592. Ans. x=8, or (_ — ) 
 
 2 \ 5 / 
 
 n 2 
 
 (26) x n — 2ax* = b. Ans. x=(a± y/a*+b) a 
 a:i-f-4i \/7c _A [rr\/x-\-x) 2x* 
 5\/x — x 3 — \/x (5\/x — x)(3 — \/x) 
 
 x x b 
 
 ■\Zx-\-\/'a — x \/x — \/a — x \/x 
 
 r — a; 3 — \/x (5\/x — x)(3 — \/x) 
 
 m * , + _ * =A. Ans . .J&IZ+* 
 
 2 
 
 ,„„. x+\/x 2 — 9 , % 8-4-1/— 11 
 
 (29) ^^ =(x— 2)2. Ans. *=5, or 3, or ■ 
 
 a;— -v/x2— 9 
 
 j 
 
 (30) a;+5=-/a;+5+6. Ans. *==4. 
 
 (31) a;+16— 7-/-c+16=10— 4-/a;+16. Ans. x=9. 
 
 (32) i/x+12+i^o;+12=6. Ans. a;=4. 
 
 (33) x°— 2a:-f6- l /a:2— 2x-f 5=11 . Ans. as=l, or lj-2i/l5. 
 
 (34) 2r2+3a:— 5-v/2^+3a;+9-f-3=0. Ans. a:=3, or . 
 
 (35) [(a:— 2)2— a;]2— (a;— 2)2=88— (a;— 2). Ans. x=6, oi —1, or 5 ± 3 V— 3 
 
 (36) (ar+6)2-f2x*(a;+6)=138+a;*. Ans. x=4 
 
 o 
 
 (37) x— l=2+-^r. Ans. x=4. 
 
 f33) art— 2a^+o;=132. Ans. a;=4, or —3, or ^^ — — 
 
 4 1 9-J--/48I 
 
 (39) 9x-f--/l6a: 3 -|-36.c ! =15a3 — 4 Ans. .r=-, or , or x= 
 
 3' 3 50 
 
 t* n \ 12+8.r J . „ — 34-V— 7 
 (40) x= — ^ . Ans. a;=9, or ±-*- . 
 
 X — 5 
 
 , . 49a;2 , 48 , 6 8 —3-1-1/93 
 
 41 — — U-_49=9+-. Ans. jc=2, or , or ±V - . 
 
 y ' 4 r xi ~x 7 7 
 
 (42) ■— 1 — ^—I7x=8. Ans. x=4-2, or —8, or — -. 
 
 2 4 - 1 - ' ' 2 
 
 («) (-DM*-^!- ' *■— i-v^ 
 
 (44) ar<— (24c+4n2) x s-j-J2 t .2 = o. Ans. a;=-J-\/^+ 2 «'i 2 av'^c+c3. 
 
 (45) a;2-a;+5v/lai=5aT+6=52±-. Ans. JC= 5 =fc^ 1329 > ^ 5C=3j ^ _J 
 
 (46) ^ =-. Ans. ar= 5 (±l/-7-3). 
 
 x— i/x' 2 — ai « 
 
 8' 
 
 Note. — In some of the above examples we have given answers which will not satisfy 
 the equation unless the double sign be understood before the radical. In some cases this 
 Bign is understood, in others not; but whether it is or not will always be known from the 
 probloia from which the equation is derived. 
 
218 ALGEBRA. 
 
 ON THE SOLUTION OF QUADRATIC EQUATIONS CONTAINING TWO 
 
 KNOWN aUANTITE 
 189. An equation containing two unknown quantities is said to be of the 
 second degree when it involves terms in which the sum of the exponents of the 
 unknown quantities is equal to 2. but never exceeds 2. Thus, 
 
 3x-_ 4.r+i/ 2 — xy— 5^+0=0, 7xy—ix-\-y=0, 
 are equations of the second degree. 
 
 It follows from this that eveiy equation of the second degree containing 
 two unknown quantities is of the form 
 
 ay' 2 -{-bxy-\-cx--l r dy-{-ex-\-f=0, 
 where a, b, c, represent known quantities, either numerical or alge- 
 braical; i. c., the equation contains the second power of each of the unknown 
 quantities, the first power of each, and the product of the two. Not that 
 every equation of the second degree contains all these, but when any one of 
 them is wanting the coefficient of that term, in the general form, is said to be 
 zero. *. 
 
 Let it bo required to determine the values of x and y, which satisfy the 
 equations. 
 
 arf+bxy+cx'+dy+ex+f =0 (1) ) 
 
 ary*+b'xy+c?x*+d'y+e'x+f'==0 (2) S 
 
 Arranging these two equations according to the powers of y, they become 
 
 atf+{b x+d)y+(cx"-+ex+f)=0 
 
 ay-+(b'x+d')y + (c'x°-+c'x+f') = 
 
 Put bx+d=h; cx-+ex+f —k 
 
 b'x+d'=h'; c'x"-+e'x+f'=k f . 
 
 .:ay"+hy + k=0 (3) 
 
 ay+h'y+k' = (4) 
 
 Multiply (3) and (4) by a' and a respectively, and also by fc and k ; then 
 
 aa'y"--\-a'hy+a'k=0 (5) 
 
 aa'y"-\-ah'y+ak'=0 (6) 
 
 aky+hk'y+kk'=0 (?) 
 
 a'ky--\-h'ky+kk' = (8) 
 
 Subtracting (6) from (5), and also (7) from (8), we have 
 
 {a'h—ah')y-\-a'k—ak' = (9) 
 
 (a'k—ak')y-\-h'k—hk'=0 (10) 
 
 Multiplying (9) by h'k—hk', and (10) by a'k—ak', wo have 
 (a'h—ah'){h'k-hk')y+(a'k—ak'){h'k-hk') = . . (11) 
 {a'k-akjy+{a'k-ak')(h'k-hk') = . . (12) 
 
 A (a'h—ah'){h'k-hk') = {a'k-aky (13) 
 
 Substituting the values of//, h', k, l J in equation (13), we have 
 
 = I (o-c— a<0i3+( o '«— fl O*-H , l/*— «/' 1 2 
 Hence, by multiplying and expanding, tho final equation in x is of the fourth 
 degree, which will, in general, bo tho degreo of the equation obtained by 
 eliminating between the two equations of the second degreo; but the general 
 form includes a variety of equations, according to the values of the coefficient 
 ,r. 6, c, &c.; when d, e,f, d', • '../" are each =0, the solution may be obtain- 
 ed by quadratics, the resulting equation in x beii 
 
 {(a'6_oo')z+a'd— ad'\ . {(fc'c— b*)x— {dd— cd')}s=(afc— tuff* 
 
QUADRATIC EQUATIONS. 
 
 219 
 
 Although tho principles already established will not enable us to solve equa- 
 tions of this description generally, yet thore are many particular cases in 
 which they may bo reduced either to pure or adfected quadratics, and the 
 roots determined in the ordinary manner. 
 
 EXAMPLE I. 
 
 * 
 
 Required the values of x and y, which satisfy tho equations, 
 
 S *+y=p 
 
 ( xy = cf 
 
 3?-\-2xy-\-y*ss:p* 
 
 4xy =4<7 2 . . . , 
 
 Squaring (1), 
 Multiply (2) by 4, 
 
 Subtract (4) from (3), x 2 — 2xy-{-y 2 —i) 2 — 4q°, 
 
 or 
 
 Extract the root, 
 But by (1), 
 Add (1) to (5), 
 Subtract (5) from (1), 
 
 {x—yf=p 2 —4q 2 . 
 x— y=± Vp 3 - 
 
 x +y=p- 
 
 ■ 4<f 
 
 2x= J p± Vp" — 4-q"~ 
 
 Hence the corresponding values of x and y will bo 
 
 x=- 
 
 P+ /? 3 -4<? 2 
 
 y— 
 
 P — Vp~ — 4<? 2 
 
 x= 
 
 P—Vp 2 —4q"' 
 
 • and 
 
 T- 
 
 p-\- Vp' 2 — 4? 2 
 
 (1)? 
 (2)S 
 
 (3) 
 (4) 
 
 (5) 
 
 EXAMPLE II. 
 
 Square (1), 
 But by (2), 
 Subtracting, 
 
 c x -\-y :=a . 
 
 \ X 2 +7/ 2 =& 2 
 
 x 3 -f 2xj/ + 9/ 2 =a 2 . 
 x" -\-y"'=b 2 . 
 
 2xy =a 2 — Z> 2 
 Subtract (3) from (2), x 2 — 2xy+y"=2b"~— a", 
 or (x— y)-=2¥— a 2 . * 
 
 Extracting the root, x— 1/=± -\/2b 2 — a-. 
 
 But by (1), x+y=a, 
 
 adding and subtracting 
 
 2x=a± \/2i 2 — a- 
 
 2y=a^f ■^2b 2 —a i . 
 Hence the corresponding values of x and y will be 
 
 a+ V2b 2 — a 2 ') 
 x= - ^ x 
 
 and 
 
 a— V26 2 — a'- 
 
 y=- 
 
 a— J2b 2 —a 2 
 
 y=- 
 
 «+ j2b 2 — a 2 
 
 1 
 J 
 
 
 (3) 
 
 EXAMPLE III. 
 
 Cube (1), 
 But by (2), 
 Subtracting, 
 or 
 
 <x + y =m. . . . 
 ( x 3 -\-y 3 =?i 3 • • • 
 x3+ 3x 2 y+ 3xt/ 2 + y 3 = ?n 3 . 
 
 X 3 -4-?/ 3 =tt 3 . 
 
 3x 2 y-\- 3xy 2 —tri 3 — n 3 , 
 3xy{x-\-y)=?n 3 — n 3 . 
 
 an 
 
 (2)V 
 
220 
 
 ALGEBRA 
 
 Substitute for (x-\-y) its value derived from (]), 
 
 3xy . m = /ir — n 3 
 
 xy. 
 
 4xy-. 
 
 3m 
 
 4(w 3 — n ) 
 
 3?7i 
 
 
 Squaring (1), 
 But by (.3) 
 
 Subtracting, 
 
 or 
 
 But by (1), 
 
 x 2 -\-2xy-\-y 2 =mr-. 
 
 •1/7/ = 
 
 4(m 3 — n 3 ) 
 
 3 m 
 
 x 2 — 2xy + y 2 = m 2 
 
 4(m 3 — n 3 ) 
 
 3m 
 
 (x-y)°- 
 
 An 3 — m z 
 3m 
 
 -.x—y =±yj~ 
 x+y —m 
 
 ■711° 
 
 3m 
 
 •. 2x=m±J— 
 
 3m 
 
 — ni- 
 
 ne' 
 m 
 
 Hence the two corresponding values oTx and y aro 
 
 _m Un 3 — mA m hn 3 - 
 
 i — ; f and , f 
 
 m Un 3 —m 3 m An 3 —m 3 
 
 y -2-\J-T2m-) ^2 + V-T^-J 
 
 EXAMPLE IV. 
 
 $**+**y*+y*=a ( 
 
 1 3 3 
 
 C. r 3^_ x L. 2/ 2_|_ 3/ 3 = 7 J ^ 
 
 square (1), tf+xtyS+tf+'Hx 11 . X*y*+2x*y5+2y1 . A"=a» 
 But by (2), a^+gy-fy 8 -_/,. 
 
 Subtracting, 
 
 1 
 
 2W 
 
 ,.*,,* 
 
 nr 
 
 2x- . xV+2a:V+2y T xV=«'— *, 
 
 5 3 3 3 3 3 
 
 r yV+iy+y t )=(i , -i 
 
 But by (1), 
 And by (3), 
 
 Adding, 
 
 or 
 
 .•. 2.r' y* . a 
 2 •'< 3 
 
 2a 
 
 (3) 
 
 2a 
 
 3 3 :t 
 
 <l' — 6 
 
 '~~2u~' 
 
 j a 3a 9 + 6 
 
 (x T + 1/ 4 ) 5 = — 
 
 2a 
 
 •'•'+.'/' 
 
 f =±v^ 
 
 («> 
 
QUADRATIC EQUATIONS. 
 
 221 
 
 Again, from (1), 
 And from (3), 
 
 Subtracting, 
 
 01 
 
 But by (4), 
 
 3 3 3 3 
 
 x^-\-x*y 7C -\-y-=a. 
 
 J 2a 
 
 * J ' On 
 
 3 3 
 
 .•. x*—y* 
 
 3 3 
 
 x*+y* 
 
 . fib—a? 
 
 ~ ± V~2a~ 
 
 , /3a 2 — b 
 
 '. adding and subtracting, x f =A:U— % ±\/ 
 
 36— a 2 
 
 2a 
 
 3 /3a 2 — 6 /36-a* 
 
 15) 
 
 Hence the corresponding values of a: and y are 
 
 
 J- -/3a 2 — 6+ ■/36— a s ) s 
 
 V8a 
 
 x= 
 
 ± -v/3a 2 — 6— -v/3i— a 2 > if 
 
 -I 
 
 and 
 
 -/8a 
 
 -J- -/3a 2 — /,> — V36— a 2 if ^ ± -/3a 2 — 6+ -/3&— a 2 ^ f 
 
 -/8a ) 4 -/8a 
 
 The following require the completion of the square : 
 
 example v. 
 
 C x+2/+2- 2 +7/ 2 =a (1) * 
 
 I x-y+x*-y*=b (2) S 
 
 Add (1) and (2), 2x 2 4-2.r=a+Z> (3) 
 
 Subtract (2) from (1), 2tf-\-2y=a — b (4) 
 
 Equations (3) and (4) are common adfected quadratics; solving these in the 
 usual manner, we find 
 
 — 1± -v/l + 2fl + 2// 
 
 x=- 
 
 — 1± yfl-\-2a— 26 
 
 EXAMPLE VI. 
 
 $ x +y = G (l) > 
 
 I x"+i/ 4 =272 (2) S 
 
 Raise (1) to the 4th power. 
 
 x* + 4.r 3 7/ -f Qxhf + 4xf + y l = 1 296. 
 But from (2), x 4 -\-y 4 = 272, 
 
 Subtracting 4X 3 ?/ + 6x 2 y 2 -f- 4x?/ 3 =1024, 
 
 or 2x?/(2x 2 + 3x?/4- 2t/ 2 ) = 1024 (3) 
 
 But by (1), 2x!/(2x 2 +4x2/4-2?/ 2 ) = 144x i ?/ (4) 
 
 Subtracting (3) from (4), ■2.ry : = liAxy— 1024. 
 
222 
 
 ALGEBRA. 
 
 Transposing and dividing by 2, 
 
 x-y- — 72x1/ =—512. 
 
 Completing tl e square, x"-y' i — 72x_y-f 1296=1296— 512, 
 
 or 
 
 First, let us suppose xy = 8. 
 
 By(i), 
 
 And 
 
 Subtracting, 
 
 (xy— 36)-= 784^ 
 
 .-. xy — 3C) =± v/784 
 
 xy =36 ±28 
 
 =64, and 8. 
 
 X'+2xy+f- = 36, 
 
 4xy =32. 
 x 3 — 2xy-\-y- = 4 
 
 •. x— ?/ =i2, 
 x+y =6. 
 
 and 
 
 But 
 
 .-. adding and subtracting, 
 
 x=4 I 
 
 y=2 $ """ } !/ = 4 
 Secondly, et us take the other value of xy, or 64. 
 By (1), x"-+2xy+y-= 36, 
 
 4xy = 256. 
 Subtracting, x- — 2xy + y- = — 220, 
 
 But 
 
 .•. adding and subtracting, 
 
 .-. x— ?/ =± V— 220. 
 x+y =6. 
 
 x=- 
 
 6+ V— 220" 
 
 y=- 
 
 6— •/—220 
 
 6— V— 220" 
 
 and 
 
 y=- 
 
 6+ V— 220 
 
 ' 
 
 Hence, in the above equations, two of the roots of x and j are pos* -le, and 
 two impossible. 
 
 (7)* 2x + Sy =118 (1) > 
 
 5*2— 7i/-=4333 (2) \ 
 
 x=35 ) x=— 229A> 
 
 Ans. ,_ > and .'. > 
 
 3/= 16 S 7/= 192/, S 
 
 (8) 8x+23y = 2.t 3 4.2y 8 (1) i 
 
 341/4- 6.r°— 52/ 2 =13x^/+24 (2) \ 
 
 — 181) 
 
 Ans. 
 
 x=3 
 
 7/ = 2 
 
 133 
 34 
 
 .'/ = • 
 
 
 X: 
 
 55jpj/H14 
 
 ~~ 26 
 
 y=- 
 
 — 9±3-y/in4 
 
 
 (9) (x-y)(.i*-if) = a (1) , 
 
 CM-yK^+y 9 )^ (2) s 
 
 Ans. x= 
 
 V 26— a ± Va \' 26— a ?V'i 
 
 -2/= 
 
 2V26-a 
 
 2V26— a 
 
 • The following examples, though a valuable . are likely to detain the stu-lont 
 
 io"", and -.._■ be omitted. 
 
QUADRATIC EQUATIONS. 
 
 223 
 
 (10) xyz 
 
 x+y 
 xyz 
 
 =a 
 
 2/+ 2 
 xyz 
 
 = 6 
 
 x-{-z 
 
 (11) x+y=a, 
 x*+y* = b. 
 •-. — 96— x*y*, 
 
 
 — 
 
 
 
 (1) 
 (2) 
 
 (3) 
 2abc(ab-\-bc — ac) 
 
 ^_v 
 
 I 2abc(a 
 "\J (ab-\-ac — bc)(bc-\-ac — aby 
 j 2abc(bc-\-ac — ab) 
 V= ^y]{ab+ac— hc)(,ib -{-be— ocj' 
 I 2abc(ab-\-ac — be) 
 ~ \(ab-{-bc — ac)(bc-\-ac— ab)' 
 a Lb — a 3 a lib— a 3 
 
 Ans. x=4, or 2, or 3i \/21» 
 , or 3 ^p -/21. 
 
 - -/a 2 "— c 2n )°, 
 
 
 
 
 
 - V« 2n — c 20 )"' 
 
 or — 3± -/3. 
 
 or — 3^= -v/3- 
 
 '5, ^=—3-1-1/7, 
 ^=4, or 1. 
 
 or 
 
 —13-j- -/— 39 
 
 or 
 
 -13^1/— 39 
 
 ; also, .r=5, or -, 
 : 5 
 
 Iso, g=3, or — 15. 
 
 Ans. x=9, or 
 
 3' =4 ' or 7 
 
 Ans. x~5, or 
 
 25 
 
 r 
 
 17 
 
 10' 
 
 y=3,or — . 
 
 Ans. £=G or 
 
 9 
 
 y=12 or — 9. 
 
 ^ 
 ^ 
 
 -13-J-i/— 47 
 
 -1-rt-l/— 11 
 
 — 47 l^Sy^c 
 
 -, or 
 
 or 
 
 i±V-n 
 
 
222 ALGEBRA. 
 
 Transposing and dividing by 2, 
 
 xhf—l =—512. 
 
 Completing tl e square, afy 8 — 72iy+ 1296=1296—512, 
 
 or (ry— 36) : = 784._ 
 
 .-. xy — 36 =± a/ 784 
 ry =36 ±28 
 
 =61, and 8. 
 
 First, let us suppose xy = 8. 
 
 By (1), x*+2xy+f-=36, 
 
 And 4xy =32. 
 
 Subtracting, x*—2xy-\-y" = 4 
 
 •. a:— y =±2, 
 
 But .r+2/ =6* 
 .-. adding and subtracting, 
 
 Secondly, et u 
 
 By (i). 
 
 Subtracting, 
 
 But 
 
 .-. adding and subt 
 
 y 
 
 Hence, in the at 
 two impossible. 
 
 (7)* 2x +Sy = 
 5x 2 —7y"= 
 
 (8) 8x-\-23y 
 34i/4- Gx*- 
 
 (9) (.r-7y)(.r°_ : . 
 (x+y)(z»+j 
 
 x =i I and $ a = 2 } 
 
 • The follow i 
 Irm -. 
 
QUADRATIC EQUATIONS. 
 
 223 
 
 (10) xyz 
 
 x+y 
 
 xyz 
 
 =a 
 
 y+ z 
 
 x-\-z 
 
 =b 
 
 
 (1) 
 (?) 
 
 (3) 
 
 111) x+y =a, 
 x 3 -\-y 3 = b. 
 
 (12) 4xy = 9G — x-y** 
 
 x+y =6. 
 
 (13) .T n +7/"=2a n , 
 
 xy 
 
 
 (14) x 2 +x+y = 18 — y\ 
 
 xy=6. 
 
 (15) x»-f 2^+7/-+2x=120— 2^ 
 
 (16) afi+ft—x—i?=78, 
 
 X V +aH-y=39. 
 
 (17) aPy*—7xy*— 9-43=71;-, 
 
 xy— y= l ~- 
 (is) »— *Vxy+y— yx-\-Vy=o, 
 
 V x +Vy= 5 - 
 
 , , x* , 4a; 85 
 a;— 2/ =2. 
 
 i 2abc(ab-\-bc — ac) 
 
 \{ab-\-ac — bc)(bc-\-ac — ab)' 
 
 j 2abc(bc-\-ac — ab) 
 
 ■ ,== -y (ab-{-ac — bc)(ab-fbc—acy 
 
 J 2abc(ab-\-ac — be) 
 
 = \ \ab-{-bc—ac)(bc-{-ac—ab)' 
 
 a Ub—a? a lib— a 3 
 
 Ans. a.ji^-jg-, ^-^-^ 
 
 Ans. x=4, or 2, or 3± \/21, 
 2/=2, or 4, or 3=p ■/21. 
 
 Ans. .r=(a n I t V« 2 " — c 2n )' a , 
 c* c 2 
 
 * (« n rt V« 2n — c 20 )"' 
 Ans. .r=3, or 2, or — 3± V3, 
 i/=2, or 3, or — 3=f V3- 
 
 Ans. a;= — JFFi/ 5, #= — 3-J-i/ 5 > 
 also, a:=6, or 9, y=4, or 1. 
 
 -13-tv/— 3 » 
 
 20 - /-: K/-T — = 2 . 
 
 Vj;-J-y V 3a: 
 
 (21) *<— 2« , ty- r -y 2 =49 
 
 **. -2j ■ - y - -j- y 4 — x-+i/-=i20. 
 
 Ans. a;=9, or 3, or 
 
 Ans. x- 
 
 y=3, or 9, or 
 —19 
 
 — 13=PV— 39 
 
 1 
 
 -; also, x^=5, or -, 
 
 17^6^—2 5 
 
 y= — GjL-v/ — 2; also, #=3, or — 15. 
 
 25 
 
 Ans. a:=9, or — , 
 4 
 
 y— 4 > or 7- * 
 
 17 
 Ans. 1=15, or — , 
 
 —3 
 y=3, or — . 
 
 Ans. x=6 or 
 
 10 
 
 9 
 
 #=12 or — 9. 
 Ans. .r=-|-3, or J^lA' or ir\/— — ^r 
 
 3-J-i/ — 47 
 
 /l.'>4-3i/5 1 /" 
 
 w=y^-±J=HJg^ 
 
 l4--j/— 47 l4-3i/5 
 2, or — 1, or -Si- , or ■ „ -, 
 
 i±!A=n 
 
 or 
 
224 ALGEBRA. 
 
 (22) xi/-\-xy"-=12, 
 
 x-\-xyt=18. 
 
 (23) x— x*=3— y, 
 4— x =y—y. 
 
 Ans. 
 
 x = 
 
 "-, 
 
 or 16, 
 
 
 :■'- 
 
 =■-', 
 
 1 
 
 Ans 
 
 . xz 
 
 =4, 
 
 1 
 
 
 9= 
 
 = 1. 
 
 9 
 
 or -. 
 
 4 
 
 1 —974-^/6045 
 (24) (x2+%=x y +12t>, Ans. *=5, or -■ or 3=J: , 
 
 (x*-\-\)y=xiy* — 744. y=* or 150, or 
 
 58 
 
 r-v/0045 
 
 (25) x -\-y 4-V^+y=12, Ans. x=5, or 4, 
 a<s-|-y :) =189. y=4, or 5. 
 
 (26) x^y^+x— y=132, Ans. x=ll, or — 1, or 61±i/— 3716, 
 ( I 2-f- 3 ,:)( cc _ 3 ,) = 1220. y=l, or ~ n . or — 6lT" y/— 371C. 
 
 >;S") x^ = 2y?, Aqs. x=14| 3 , or 8, 
 
 8x*— y*=14. y=ys"','". or 4. 
 
 ^28) x*-\- y%=3x (see note, page 217), Ans. x=4, or 1, 
 
 x l ~-\-y^—x. y=8. 
 
 (29) ;c-fa;i= : ^ip^+4. Ans. *=4, or 1, 
 
 y+«y=y 3 + 4 y- 5'= 1 ' or — 2 - 
 
 (30) 2x+y=26— 7v^H^+4, Ans. x=2, or —10, 
 2x — \/y 15 2.r-f-i/y 
 
 y=256. or 256]*. 
 
 131) 8^/x— 9x^=9^— 16xy, Ans. x=t, 
 
 5x=4+25y:. y=±? 
 
 o 
 
 (3f ) -Cx— y^=6_yM, Ans. x=4, or 16, 
 
 ar» 12 x 
 
 <B3) /5i/x+5-i/^+-/y=10— \/*. Ans. x=9, or 4. 
 
 •V/x5+-v/y6=275. ^=4, or 9. 
 
 PROBLEMS PRODUCING PURE EQUATIONS. 
 
 (1) What two numbers are those whose sum is to th r as 10 to 7 
 and whose sum, multiplied by the less, produces 270 ? 
 
 Ans. ±21 and ±9 
 
 (2) There are two numbers in the proportion of 4 to 5, and the difference 
 of whose squares is 81. What are tho numbers? 
 
 Ans. ±12 and ±15. 
 
 (3) A detachment from an army was marching in regular column, with 5 
 men nunc in depth than in front; but upon the enemy coming in Bight, the 
 front was increased by 845 men, and by this movement the detachment was 
 drawn up in five lines. Required the number of men ? 
 
 . 4550. 
 
 (4) Two workmen, \ and B, wen • 1 to work for a certain number 
 of day ■ sit different rates. At the end of the time, A, who hud been idle 4 of 
 
QUADRATIC EQUATIONS. 225 
 
 those day;, had 75 shillings to receive ; but B, who had been idle 7 of those 
 days, received only 48 shillings. Now, had B been idle only 4 days and A 7, 
 they would have received exactly alike. For how many days were they en- 
 gaged, how many did each work, and what had each per day ? 
 
 Ans. A worked 15 and B 12 days. 
 
 A received 5 and B 4 shillings per day. 
 
 (5) A vintner draws a certain quantity of wine out of a full vessel that holds 
 256 gallons, and then filling the vessel with water, draws off the same quantity 
 of liquid as before, and so on for four draughts, when there were only 81 
 gallons of pure wine left. How much wine did he draw each time ? 
 
 Ans. 64, 48, 36, and 27 gallons* 
 
 PROBLEMS WHICH PRODUCE AD'FECTED OR COMPLETE QUADRATIC 
 
 EQUATIONS. 
 
 ' PROBLEM 1. 
 
 190. To find -.umber such that twice its square, augmented by three 
 
 times the numbeV, is equal to 65. 
 
 Let x be the number required, we have for the equation of the problem, 
 
 2z 2 +3x=65. 
 
 3 /65 9~ 3 23 
 
 Solving the equation, x= — --j-^— +— = — - J-—. 
 
 13 
 Hence x=5 ; x= — — . 
 
 The first of these two values satisfies the conditions of the problem, as stated 
 in the enunciation ; for, in fact, 
 
 2(5) 2 +3X 5=2x25+15 
 =65. 
 
 In order to interpret the meaning of the second value, let us observe, that 
 if we substitute — x for -f-x in the equation 2x 2 -f-3x=65, the coefficient of 3x 
 alone will change its sign, for ( — x) 2 =(+x) 2 =x 2 . Hence the value of x will 
 no longer be 
 
 ■6* 
 
 3 , 23 
 
 3 23 
 
 but will become x= + - ± — . 
 
 1 4 4 
 
 13 
 Hence x=— • ; x= — 5, 
 
 where the values of x differ from those already found in sign alone. 
 
 13 
 Hence we may conclude that the negative solution — — , considered with- 
 out reference to its sign, is the solution of the following problem : 
 
 To find a number such that twice its square, diminished by three times the 
 number, is equal to 65. 
 In fact, we have 
 
 /13\* 13_169 39 
 
 2 V2/ ~ 3X "2 _=_ 2""""2" 
 = 65. 
 P 
 
226 ALGEBRA. 
 
 PROBLEM 2. 
 
 A tailor bought a certain number of yards of cloth for j 2 pounds. If he had 
 paid the same sum for 3 yards less of the same cloth, then the cloth would 
 have cost 4 shillings a yard more. Required the number of yards purchased. 
 
 Let x be the number of yards purchased. 
 
 240 
 Then is the price of one yard, expressed in shillings. 
 
 If he had paid the same sum. for 3 yards less, in that case the price of eacti 
 
 240 
 would be represented by -. 
 
 X^— o 
 
 iJut by the conditions of the problem, this last price is greater than the 
 former by 4 shillings ; hence the equation of the problem will be 
 
 240 240 
 
 x-3 - x + 4 ' 
 or x 2 — 3x=180. 
 
 Whence a:= -_L_^_ + i 8 n =5 _. fc ._ 
 
 .•. x=15 ; x= — 12. 
 
 The value of x=15 satisfies the conditions of the problem, for 
 
 240 240 
 
 -=lG;-=20, 
 
 the price of each yard in the first case being 1G shillings, and in the last case 
 20, which exceeds the former by 4 shillings. 
 
 With regard to the second solution, we can form a new enunciation to which 
 it will correspond. Resuming the original equation, and changing x into — x, 
 it becomes 
 
 240 240 
 
 i=— :+4i 
 
 — x — 3 — x 
 or 
 
 240 240 
 
 •4, 
 
 x+3 x 
 
 an equation which may be considered as the algebraic representation of the 
 following problem : 
 
 A tailor bought a certain number of yards of cloth for 12 pounds. If he had 
 paid the same sum for 3 yards inore, then the clot li would have cost 4 shillings 
 a yard less. Required the number of yards purchased. 
 The above equation when reduced becomes 
 
 x*+3x=180, 
 instead of x' — 3x=180, as in the former case; solving tho above, we find 
 
 x=rl2; x= — 15. 
 The two preceding problems illustrate tho pr aiciplo explained with regard 
 to problems of the first degree. 
 
 PROBLEM 3. 
 
 A merchant purchased two bills; one for 88770, payable in 9 months, the 
 other for $7-lH-t, puyublo in 8 months. Pot the firsl lie paid (1200 more 
 than for tho second. Required the rate of interest allowed. 
 
aUADRATIC EQUATIONS. 227 
 
 Let x represent the interest of $100 for 1 month. 
 
 Then 12x, 9x, 8x severally represent the interest of $100 for 1 year, 9 
 months, 8 months. 
 
 And 100 + 9x, 100+8r represent what a capital of $100 will become at 
 the end of 9 and of 8 months respectively. 
 
 Hence, in order to determine the actual value of the two bills, we have the 
 following proportions : 
 
 8776X100 
 100 + 9j:100:;8776:' 100+9;c 
 
 7488X100 
 100+8ar;100;:7488: 100 , 8x • 
 
 The fourth terms of the above proportions express the sum paid by the 
 merchant for each of the bills. 
 
 Hence, by the conditions of the problem, 
 
 877600 74880 
 100+9x*~100 + 8x — 12 ' 
 or, dividing each member by 400, 
 
 2194 1872 _ 
 
 100 + 9x- — 100 + 8x — 
 Clearing of fractions and reducing, 
 
 216a: 2 +4396x=2200. 
 
 Whence 
 
 2198 /2200 / 2198 \ a 
 x —~ 216. V 216 "H2I6/ 
 
 — 2198J; V 5306404 
 2l6 
 
 — 2198± -v/5306404 
 
 • 12x - 
 
 • l4X — 18 
 
 — 2198±2303.5 
 
 18 
 
 .-. 12x=5.86 ; and 12x= — 250.08 
 
 The positive solution, 12x=5.86 , represents the required rate of in- 
 terest per cent, per annum. 
 
 "With regard to the negative solution, it can only be considered as connected 
 with the other by the same equation of the second degree. If we resume 
 the original equation, and substitute — x for +.r, we shall find great difficulty 
 in reconciling this new equation with an enunciation analogous to that of the 
 proposed problem. 
 
 problem 4. 
 
 A man purchased a horse, which he afterward sold to disadvantage for 24 
 pounds. His loss per cent, by this bargain, upon the original price of the 
 horse, is expressed by the number of pounds which he paid for the horse 
 Required the original price. 
 
 Let x be the number of pounds which he paid for the horse. 
 
 Then x — 24 will represent his loss ; 
 But, by the conditions of the problem, his loss per cent, is represented by the 
 number of units in x ; 
 
228 ALGEBRA. 
 
 x 
 His loss per cent, on one pound is — -. 
 
 x* 
 .«. his loss per cent, on X pounds must be — , or x times as greut. 
 
 T'his gives the equation, 
 
 x 3 
 
 = r 24 
 
 100 
 
 x=50± Vl00=50±10. 
 
 Hence x=60 ; x=40. 
 
 Both these solutions equally fulfill the conditions of the problem. 
 
 Let us suppose, in the first place, that he paid 60 pounds for the horse, since 
 
 be sold it for 24, his loss was 36. On the other hand, by the enunciation, his 
 
 60 60X60 
 
 'oss was 60 per cent, on the original price; i. c, r-— of 60, or =36 j 
 
 thus 60 satisfies the conditions. 
 
 In the second place, let us suppose that he paid 40 pounds ; his loss in this 
 
 case was 16. On the other hand, his loss ought to be 40 per cent, on the 
 
 40 40X40 
 
 .riginal price ; i. c, r-nr of 40, or =16 ; thus 40 also satisfies the con 
 
 ditions. 
 
 GENERAL DISCUSSION OF THE EQUATION OF THE SECOND DEGREE 
 
 191. The general form of the equation, the coefficients being considered ir- 
 pendently of their signs, is 
 
 x*-\-px-\-q~0. 
 
 P % 
 I., II. Let q be positive and < — , 
 
 !P Ip % 
 I. Ifp be positive, x= — -±W— — q, and both values are negative.* 
 
 P fp* 
 II. Ifp be negative, x=-J--±W— — q, and both values are positive. 
 
 P* 
 TU., IV. Let q be positive and > — , 
 
 ( P fW ' 
 
 III. If j? be positive, z= — -±y — — q, 
 
 ns — 
 
 p IP' 
 
 IV. U] } benegative, x=+— ±y — — 7, 
 
 and both values are imagi- 
 nary, f 
 
 • In this and all the following values of x, calling the term - before the radical the ra- 
 tional part, and - /— -^q the radical part, we perceive that, when q is positive, the radical 
 
 part is greater than the rational, since .. J— alone equals ; , the rational part 1 and tli 
 
 of the wholo expression is that of the radical part ; but when q is negative, the radical 
 part is less than the rational, and the sign of the wholo expression is that of the rational 
 part. 
 
 t In this case, if wo examino the general equation, we shnll Bad that the conditions ora 
 abaurd ; for, transposing q, and completing the square, wo havo 
 
QUADRATIC EQUATIONS. 
 
 ml 
 
 229 
 
 V., VI. Let q be negative and < — , 
 
 P Fp 1, 
 
 V. Up be positive, x= — -±w — - + <7» 
 
 VI. If_p be negative, .r= + -±- > y— + ?, 
 VII., VIII. Let q be negative and >-r, 
 
 w In 5 
 
 VII. Up be positive, z=— ^iy^+tf' 
 
 VIII. Ifp be negative, ar=+!±^/j+?, 
 
 and one value is positive 
 the other negative. 
 
 IX., X. Let q=—, and be positive. 
 
 IX. If p be positive, x= — 
 
 X. If p be negative, x=-\- 
 XI., XII. Let g=0, 
 
 • and the two values are equal. 
 
 P , P 
 
 XI. If jp be positive, .r=— -±^, one value =— p, the other =0. 
 
 2" 1 "2 
 
 P,P 
 2 ± 2 
 
 XII. Up be negative, a:=+-±^, one value =+p, the other =0. 
 
 XIII. Let q be negative. 
 
 {XIII. p=0, x=± V?! the two values are equal with opposite signa 
 
 XIV. Let q be positive, 
 
 {XIV. p=0, x=Az V — q, both values are imaginary. 
 
 XV. Let q = 0, 
 
 {XV. j>=0, then x=0, or both values are equal to 0. 
 
 
 P 1 
 
 but since — — q is, by hypothesis, a negative quantity, we may represent it by — to, where 
 m is some positive quantity ; then 
 
 x3 rt px-\-—= — to 
 
 (*±?)+»=o; 
 
 that is, the sum of two quantities, each of which is essentially positive, is equal to 0, • 
 manifest absurdity. Solving the equation, 
 
 -P 
 
 and die symbol \/ — to, which denotes absurdity, serves to distinguish this case. Hence, 
 when the roots' are imaginary, the problem to which the equation corresponds is absurd. 
 
 We still say, however, that the equation has two roots ; for, subjecting these values of 
 x to the same calculations as if they were real, that is, substituting them for x in the pro> 
 posed equations, we shall find that they render the two members identical 
 
J30 ALGEBRA. 
 
 XVI. One case, attended with remarkable circumstances, stl remains to be 
 examined. Let us take the equation 
 
 a3?-\-bx — c=0. 
 
 Whence x= . 
 
 2a 
 
 Let us suppose that, in accordance with a particular hypothesis made on the 
 
 given quantities in the equation, we have a = 0; the expression for x then 
 
 becomes 
 
 f 
 
 _&±6 
 x= — ; whence 
 
 X =0 
 
 
 
 —2b 
 
 
 
 The second of the above values is under the form of infinity, and may be con 
 
 sidered as an answer, if the problem proposed be such as to admit of infinite 
 
 solutions. 
 
 
 We must endeavor to interpret the meaning of the first, -. 
 
 In the first place, if we return to the equation ax"-\-bx — c=0, we perceive 
 
 that the hypothesis a = reduces it to bx=c, whence we derive x=-r, a. finite 
 
 and determinate expression, which must be considered as representing the true 
 
 value of - in the case before us. 
 
 That no doubt may remain on this subject, let us assume the equation 
 
 ax" 1 -\-bx — c=0, 
 
 and put a.'=-, the expression will then become 
 
 a b 
 
 -+ -— c=0. 
 
 y^ y 
 
 Whence cy" 1 — by — a=0. 
 
 Let a = 0, this last equation will become 
 
 cy"—by=0, 
 
 from which we have the two values y=0, y=- ; substituting these values in 
 jr=-, we deduce 
 
 y 
 
 1 c 
 
 1°. x=- ; 2°. x=t-* 
 
 " To show more distinctly how the indeterminate form arises, let us resume the general 
 value of one of the roots. 
 
 — b+\/b°~{-iac 
 2a 
 If a were a factor of both the numerator and denominator, it might bo supprossed, and 
 then a, being put equal to zero, would give the true value of x. We can not, indeed, 
 ■how the existence of this factor in the two terms of the fraction as it stands ; but if wa 
 
 multiply both numerator and denominator by — b — \/b--\-\ac, it becomes 
 
 _ (~ Z>-fVfr»+4ac)(— b— y/b*+i<t, ) 
 —2a{b+-i/b:+4ac) 
 
QUADRATIC EQ.UATI0N6. 231 
 
 — 26 
 "With respect to the value x= , it is only to be observed that the 
 
 divisor zero, having to be regarded as the limit of decreasing magnitudes, either 
 positive or negative, it follows that the infinite value ought to have the am- 
 biguous sign i. 
 Thus the values of r, to recapitulate, become 
 
 c , 
 
 X=y, X=±CO. 
 
 It is remarkable that, for this particular case, we have three values of a. 
 while in the general case there are but two. 
 
 To comprehend how these values truly belong to the equation ax 2 -\- bx 
 — c=0, put it under the form 
 
 — bx-\-c 
 
 ^ =a. 
 
 x 2 
 
 — bx-\-c 
 When a = 0, the question is to find values which will render — zero 
 
 c 
 We see that x=r will do it; and as the same expression can be written under 
 
 b c 
 the form — - +"^ we perceive that it becomes zero also, from the value9 
 
 X=±aQ.* 
 
 XVII. Let us cousider the still more particular case still, where we have, 
 
 
 
 at the same time, a=0, 6 = 0. Then the two general values of x become -. 
 
 We have seen above that the first may be changed into 
 
 2c 
 
 6+ A /6'-+4ac' 
 Transforming the second in a similar manner, it becomes 
 
 ( — 6 — V6 3 +4ac)( — 6+ V* s +4ac) —2c 
 
 X= 
 
 2a(— 6+ V6 2 +4ac). —6+ *Jb 2 +4ac 
 
 In which, making a=0, 6 = 0, the values of x, thus transformed, both give 
 r=oo ; and here, also, the infinity ought to be taken with the sign i. 
 
 If we suppose a = 0, 6=0, <?=0, the proposed equation will become alto- 
 gether indeterminate. 
 
 The numerator, being the product of the sum and difference of two quantities, is equal 
 to the difference of their squares, to wit : b 2 — (6*-j-4ac)= — 4ac. We see, therefore, that 
 2a is a common factor to the numerator and denominator of the last expression. Suppress- 
 ing it, we have 
 
 2c 
 
 b-\-\Zb*-{-4ac 
 in which, if we make #=0, it gives «=;. 
 
 * In the analytic theory of curves these values answer to the intersections of the axis 
 of abscissas with the curve of the 3° order, the equation of which is yx-\-bx-\-c=0. If this 
 curve be constructed, it will be found to cut the axis of abscissas first at a finite distance 
 from the origin, and besides has this axis for an asymptote both on the side of the positive 
 and negative abscissas, which amounts to saying that it cuts it at infinity in either di- 
 rection. 
 
232 ALGEBRA. 
 
 192. Let us uow proceed to illustrate the principles established in this gen 
 eral discussion, by applying them to different problems. 
 
 PROBLEM 5. 
 
 To find in a line, A B, which joins two lights of different intensities, a point 
 which is illuminated equally by each. 
 
 P 3 A Pi B 1\. 
 
 (It is a principle in Optics that the intensities of the same light at different 
 distances are inversely as the squares of the distances.) 
 
 Let a be the distance A B between the two lights. 
 
 Let b be the intensity of the light A at the distance of one foot from A. 
 
 Let c be the intensity of the light B at the distance of one foot from B. 
 
 Let P , be the point required. 
 
 LetAP,=z; .-. BP,=a- x. 
 
 By the optical principle above enunciated, since the intensity of A at the 
 
 distance of 1 foot is b, its intensity at the distance of 2, 3, 4, feet must be 
 
 b b b b 
 
 -,-,—; hence the intensity of A at the distance of x feet must be —. In the 
 4 9 lb J x* 
 
 c 
 same manner, the intensity of B at the distance a — x must bo -. ; but 
 
 according to the conditions of the question, these two intensities are equal ; 
 hence we have for the equation of the problem 
 
 b_ _c_ 
 
 x*~(a^xy-' 
 Solving this equation, and reducing the result to its most simple form, 
 
 a y/b 
 
 X ~ Vb±Vc 
 We shall now proceed to discuss these two values : 
 
 a V 'c 
 
 a \/b 
 
 1° .1 
 
 a\Jb 
 2° x = - 
 
 Vb-Vc) 
 
 a — xz 
 
 ^ 6 +/ c whence VH-/< 
 
 { a - r =Vi 
 
 — a y/c 
 b^Wc 
 
 I. Let 6>c. 
 
 ay/b y/b 
 
 The first value of x, . , — , — 7-, is positive, and less than a, for — 77—; 
 
 J yb-\-yc ' v'»+v c 
 
 is a proper fraction; hence this value gives for the point equally illuminated a 
 
 point Pj, situated between the points A and B. Wo perceive, moreover, that 
 
 the point P, is nearer to B than to A; for, since l^>r, we have 
 
 Vl>+ V>>> V'+ Vc, or 2 Vb> V^+ V<-. and .-. j* >g, 
 
 (iy/b a 
 
 and. consequently, . . , >;,- 1 his is manifestly the result at which we 
 
 ought to arrive, for we hero suppose the intensity of A to be greater than that 
 
 ofB. 
 
 . a y/c a 
 
 The corresponding value of a — x, . . . — j-, is positive, and less than -. 
 
 " V b 
 Die second value of r, —jt r-. is positive, and greater than a, for 
 
QUADRATIC EQUATIONS. 233 
 
 Vb> Vb- Vc, ••• Vb _ Vc >h and ... -±—- >a . 
 
 This second value gives a point P 2 , situated in the production of A B, and to 
 
 the right of the two lights. In fact, we suppose that the two lights give forth 
 
 rays in all directions ; there may, therefore, be a point in the production of A B 
 
 equally illuminated by each, but this point must be situated in the production 
 
 of A B to the right, in order that it may bo nearer to the less powerful of the 
 
 two lights. 
 
 It is easy to perceive why the two values thus obtained are connected by 
 
 the same equation. If, instead of assuming A P t for the unknown quantity x, 
 
 b c 
 we take A P 2 , then B P 2 =a- — a, thus we have the equation -£=-. rj ; but 
 
 since (x — a) 3 is identical with (a — x) 2 , the new equation is the samo as that 
 already established, and which, consequently, ought to give AP, as well as 
 AP,. 
 
 The second value of a — x, —rr j~, is negative, as it ought to be, being 
 
 estimated in a contrary direction from the first, on the general principle already 
 established, that quantities estimated in a contrary sense should be represented 
 
 with contrary signs ; but changing the signs of the equation a — x= -jr — -7-, 
 
 y b — v c 
 
 a V ' c 
 we find x — a= ., . , and this value of x — a represents the absolute 
 
 length of B P 3 . 
 
 II. Let b <c. 
 
 The first value of x, ., — y- is positive, and less than -, for -\/b-\- V c 
 
 > *"+ ^ - *»+ *>■/* - -jwtA - iwjsl- 
 
 a y c d 
 
 The corresponding value of a — .r, ., . , is positive, and greater than - 
 
 Hence the point P l is situated between the points A and B, and is nearer 
 to A than to B. This is manifestly the true result, for the present hypothesis 
 supposes that the intensity of B is greater than the intensity of A. 
 
 ay/b — a-y/b . 
 The second value of x, —j-r j-, or —. jr, is essentially negative. In 
 
 order to interpret the signification of this result, let us resume the original 
 
 , , b c 
 
 equation, and substitute — x for +x, it thus becomes —=7 — ; — ^. But since 
 1 ' x- (a-j-.r)- 
 
 (a — x) expresses in the first instance the distance of B from the point required, 
 a-\-r ought still to express the same distance, and, therefore, the point re- 
 quired must be situated to the left of A, in P 3 , for example. In fact, since 
 the intensity of the light B is, under the present hypothesis, greater than the 
 intensity of A, the point required must be nearer to A than to B. 
 
 — a V ' c a -\/c 
 The corresponding value of a — .r, — ry j-, or . ., , is positive, and 
 
 the reason of this is, that x being negat ve, a — .t expresses, in reality, an 
 arithmetical sum. 
 
234 ALGEBRA. 
 
 III. Let b=c. 
 
 a 
 The first two values of x and of a—x are reduced to -, fvhich gives the 
 
 bisection of A B for the point equally illuminated by each light, a result which 
 is manifestly true, upon the supposition that the intensity of the two lights is 
 the same. 
 
 a \/b 
 The other two values are reduced to — — , that is, they become infinite, 
 
 that is to sav, the second point equally illuminated is situated at a distance 
 from tho points A and B greater than any which can be assigned. This re- 
 sult perfectly corresponds with the present hypothesis; for if we suppose 
 the difference b—c, without vanishing altogether, to be exceedingly small, the 
 second point equally illuminated, exists, but at a great distance from the two 
 
 a Vb 
 lights , this is indicated by the expression . . _ —r-, the denominator of which 
 
 is exceedingly small in comparison with the numerator if we suppose b very 
 nearly equal to c. In the extreme case, when b=c, or V^ — -/c=0, the 
 point required no longer exists, or is situated at an infinite distance. 
 
 IV. Let b=c and a=0. 
 The first system of values of x and a—x iu this case become 0, and the 
 
 second system -. This last result is here the symbol of indetermination ; for 
 
 if we recur to the equation of the problem 
 
 b__ c 
 x*~ (a— x) 4 ' 
 
 or 
 
 (b—c)z~—2abx=—a% 
 
 it becomes, under the present hypothesis, 
 
 O.z 2 — 0..r=0, 
 an equation which can be satisfied by the substitution of any number whatever 
 for x. In fact, since the two lights are supposed to be equal in intensity, and 
 to be placed at the same point, fliey must illuminate every point in the line 
 A B equally. 
 
 The solution 0, given by the first system, is one of those solutions, infinite 
 in number, of which the problem in this case is susceptible. 
 
 V. Let a=0, 6 not being ==r. 
 Each of the two systems in this case is reduced to 0, which proves that in 
 this case there is only one point equally illuminated, viz., the point in which 
 the two U glits are placed. 
 
 Tho above discussion affords an example of the precision with which algebra 
 answers to all tho circumstances included in the enunciation of a problem. 
 
 ■We shall conclude this subject by solving one <>r two problems which re 
 quire the introduction of more than one unknown quantity. 
 
 mom, km 6. 
 
 To find two numbers such that, when multiplied by the numbers a and b 
 respectively, the sum of the products may be equal to 2&, and the product of 
 the two numbers equal top. 
 
aUADRATIC EQUATIONS. 035 
 
 Let x and y bo the two numbers sought, the equal ons of the problem will 
 be 
 
 ax-\-by = 2s (1) 
 
 X1 J= P (2) 
 
 From(l) 
 
 2s — ax 
 
 Substituting this value in (2) and reducing, we have 
 
 ax 2 — 2sx-{-bp=z0. 
 Whence 
 
 And ••. 
 
 s 1 , 
 
 The problem is, we perceive, susceptible of two direct solutions, for s is 
 manifestly > \/s' 2 — a?bp ; but in order that these solutions may be real wo 
 must have s 2 >, or =a 2 bp. 
 
 Let a = b = l ; in this case the values of x and y are reduced to 
 
 £=s± Vs"— p, 2/=sT V$~—p- 
 
 Here we perceive that the two values of y are equal to those of x taken in 
 an inverse order ; that is to say, if s-f- V ' s" — p represent the value of x, then 
 * — Vs' 2 — P will represent the corresponding value of y, and reciprocally. 
 
 We explain this circumstance by observing that, in this particular case, the 
 equations of the problem are reduced to x-\-y=2s, xy=p, and the question 
 then becomes, Required two numbers whose sum is 2s, and whose product is 
 p, or, in other words, To divide a number 2s into two parts, such Uiat their 
 product may be equal to p. 
 
 PROBLEM 7. 
 
 To find four numbers in proportion, the sum of the extremes being 2s, the 
 sum of the means 2s', and the sum of the squares of the four terms 4c 2 . 
 
 Let a, x, y, z represent the four terms of the proportion ; by the conditions 
 of the question, and the fundamental property of proportions, we shall have as 
 the equations of the problem 
 
 a+z=2s (1) 
 
 x-\-y = 2s' (2) 
 
 xy=a* (3) 
 
 c?-{-a?4-y , 4-z 2 =4c 3 (4) 
 
 Squaring (1) and (2) and adding the results, 
 
 a 2 + .r 2 + 7/ + z 2 -f 2az + 2xy = 4 (s 2 + s' 2 ) . 
 But by (4), a 2 -j-;r 2 +?/ 2 +z 3 =4c-. 
 
 Subtracting, 2az + 2xy = 4 (s 2 + s' 2 — c 9 ) . 
 
 .-. by (3), 4az=4(s-+s' i —c°-)=4ty . (5) 
 
 Squaring (1), a 2 +2«z + c-=4s 2 . 
 
 But by (5), Aaz =4(s 2 +s' 2 — <:'). 
 
 Subtracting, a*—2az-{-z r =^A{c° r —s'-). 
 
 Extracting the root, a— z= ± 2 y/c 2 — *' s . 
 
 But by (1), a+z = 2s. 
 
236 ALGEBRA. 
 
 .•. adding and subtracting, a=s^z \A C — s' J 
 
 Precisely in the same manner we shall find 
 
 X = S';±: -y/c"- 
 
 y=s'=f y/<?—#. 
 
 The four numbers will therefore be 
 
 a=s+ Vc 2 — s' 2 , x=s'+ j<? 
 
 zz=s — V ' c" — s' s , y=s' — -/c 9 — s 5 . 
 These four numbers constitute a proportion, for we have 
 
 az = (s + y fc"- — s" 2 ) {s — Vc-— s"-)= s 2 — c 2 +s' 
 
 :H/ = (s' + V^-^jfs'- -v/c 2 — S 2 )=5' 2 — C' + S' 
 
 9 
 
 (8) What two numbers are those whose sum is 20, and their product 36 ? 
 
 Ans. 2 and 18. 
 
 (9) To divide the number 60 into two such parts that their product may 
 be to the sum of their squares in the ratio of 2 to 5. 
 
 Ans. 20 and 40. 
 
 (10) The difference of two numbers is 3, and the difference of their cubes 
 is 117. What are those numbers ? 
 
 Ans. 2 and 5. 
 
 (11) A company at a tavern had d£8 15s. to pay for their reckoning; but, 
 before the bill was settled, two of them left the room, and then those who re- 
 mained had 105. apiece more to pay than before. How many were there in 
 
 company ? 
 
 Ans. 7. 
 
 (12) A grazier bought as many sheep as cost him <£60, and after reserving 
 15 out of the number, he sold the remainder for <£54, and gained 2s. a head by 
 them. How many sheep did he buy ? 
 
 Ans. 75. 
 
 (13) There are two numbers whose difference is 15, and half their product 
 is equal to the cube of the lesser number. What are those numbers ? 
 
 Ans. 3 and 18 
 
 (14) A person bought cloth for <£33 15s., which he sold again at £2 B». per 
 piece, and gained by the bargain as much as one piece cost him. Required the 
 number of pieces. 
 
 \ us. 15. 
 
 (15) What number is that, which when divided by the product of its two 
 digits, the quotient is 3; and if 18 more be added to it, the digits will bo 
 transposed ? 
 
 Ans. 24. 
 
 (1G) What two numbers are those whose sum, multiplied by the greater, 
 is equal to 77, and whose difference, multiplied by the lesser, is equal to 121 
 
 \ ms. i and 7. 
 
 (17) To find a number such that, if you subtract it from 10, and multiply the 
 remainder by the number itself, the product shall be 21. 
 
 Ans. 7, or 3» 
 
QUADRATIC EQUATIONS. 237 
 
 (18/ To divide 100 into two such parts that the sum of their square roots 
 may be 14. 
 
 Ans. 64 and 36. 
 
 (19) It is required to divide the number 24 into two such parts that their 
 product may be equal to 35 times their difference. 
 
 Ans. 10 and 14. 
 
 (20) Tbe sum of two numbers is 8, and the sum of their cubes is 152. 
 "What are the numbers ? 
 
 Ans. 3 and 5. 
 
 (21) The sum of two numbers is 7, and the sum of their 4th powers is 
 641 . What are the numbers ? 
 
 Ans. 2 and 5. 
 
 (22) The sum of two numbers is 6, and the sum of their 5th powers is 
 1056. What are the numbers ? 
 
 Ans. 2 and 4. 
 
 (23) Two partners, A and B, gained c£l40 by trade; A's money was ? 
 months in trade, and his gain was c€60 less than his stock; and B's money, 
 which was ^£50 more than A's, was in trade 5 months. What was A's stock ? 
 
 Ans. d£l00. 
 
 (24) To find two numbers such that the difference of their squares may 
 be equal to a given number, g 2 ; and when the two numbers are multiplied by 
 the numbers a and b respectively, the difference of the products may be equal 
 to a given number, s 2 . 
 
 as 2 ± b Vs 4 — (a 3 — 6-)o» 
 
 Ans - ^ — n 
 
 a- — b- 
 
 Js 2 ±a-v / s 4 — (a 3 — b-)q i 
 a 2 — ft 2 
 
 (25) There are two square buildings that are paved with stones a foot 
 square each. The side of one building exceeds that of the other by 12 feet, 
 and both their pavements taken together contain 2120 stones. What are the 
 lengths of them separately ? 
 
 Ans. 26 and 38 feet. 
 
 (26) A and B set out from two towns, which were at the distance of 247 
 miles, and traveled the direct road till they met. A went 9 miles a day, and 
 the number of days at the end of which they met was greater by 3 than the 
 number of miles which B went in a day. How many miles did each g6 ? 
 
 Ans. A went 117 and B 130 miles. 
 
 (27) The joint stock of two partners was $2080 ; A's money was in trade 9 
 months, and B's 6 months ; when they shared stock and gain, A received 
 $1140 and B $1260. What was each man's stock ? 
 
 Ans. $960 and $1120. 
 
 (28) A square court-yard has a rectangular gravel walk round it. The side 
 of the court wants 2 yards of being 6 times the breadth of the gravel walk, 
 and the number of square yards in the walk exceeds the number of yards in 
 the periphery of the court by 164. Required the area of the court. 
 
 Ans. 256. 
 
 (29) During the time that the shadow on a sun-dial, which shows trae 
 time, moves from 1 o'clock to 5, a clock, which is too fast a certain number f 
 
238 ALGEBRA. 
 
 hours and minutes, strikes a number of strokes equal to that number of 'ooura 
 and minutes ; and it is observed that the number of minutes is less by 41 than 
 the square of the number which the clock strikes at the last time of striking 
 The clock does not strike twelve during the time. How much is it too fast? 
 
 Ans. 3 hours and 23 minutes. 
 
 (30) A and B engage to reap a field for d£4 10s. ; and as A alone could reap 
 it in 9 days, the}' promised to complete it in 5 days. They fouud, however, 
 that they were obliged to call in C, an inferior workman, to assist them for the 
 last two days, in consequence of which B received 3s. del. less than he other- 
 wise would have done. In what time could B or C alone reap the field ? 
 
 Ans. B could reap it in 15 days, C in 18. 
 
 (31) The fore wheel of a carriage makes 6 revolutions more than the hind 
 wheel in going 120 yards ; but if the periphery of each wheel be increased 1 
 yard, it will make only 4 revolutions more than the hind wheel in the same 
 space. Required the circumference of each. Ans. 4 and 5. 
 
 (32) The intensity of two lights, A and B, is as 7 : 17, and their distance 
 apart 132 feet. "Whereabouts between is the point of equal illumination? 
 
 Ans. 51.595 feet from A. 
 
 (33) The loudness of a church bell is three times that of another. Now, 
 supposing the strength of sound to be inversely as the square of the distance, 
 at what place between the two will the bells be equally well heard ? 
 
 Ans. .3662 of distance between the bells from the second. 
 
 (34) Supposing the mass of the earth to be 1 and that of the moon 0.017, 
 their distance 240 thousand miles, and the force of attraction equal to the mass 
 divided by the square of the distance ; at what point between will a body be 
 held in suspense, attracted toward neither? 
 
 Ans. 27G82.8 miles from the moon. 
 
 (35) The hold of a vessel partly full of water (which is uniformly increased 
 by a leak) is furnished with two pumps, worked by A and B, of whom A takes 
 three strokes to two of B's; but four of B's throw out as much water as five 
 of A's. Now B works for the time in which A alone would have emptied the 
 hold; A then pumps out the remainder, and the hold is cleared in 13 hours 
 and 20 minutes. Had they worked together, the hold would have been emp- 
 tied in 3 hours and 45 minutes, and A would have pumped out 100 gallons 
 more than he did. Required the quantity of water in the hold at Bret, and 
 the hourly influx of the leak. 
 
 Ans. 1200 gallons in the hold, 120 gallons of leakage per hour. 
 
 (36) To divide two numbers, a and b, each into two parts, such that the prod- 
 uct of one part of a by one part of b may be equal to a given cumber, /', and the 
 product of the remaining parts of a and b equal to another given number, j '. 
 
 Ai s ^ gb-(p'-p)± V \ab-(p'-p)\*-4abp 
 
 26 
 
 ab + {p'— p)^ V \ a b — {p'—p)\*—4abp 
 + 26 
 
 _ gb—{p'—p)Az V\ab — (p'—p)\ 1 —4 abp 
 y 2a 
 
 a h+(p— p)zf yf \ gb — {p'—p)\*—4 „b r 
 2a 
 
QUADRATIC EQUATIONS. 239 
 
 (37) To find a number such that its square may be to the product of the 
 differences of that number, and two other given numbers, a and b, in the 
 given ratio, p : q. 
 
 {a-\-b)p± v'(« — l') 2 j) 2 -\-'\abpq 
 
 Ans. — - . 
 
 2{p — q) 
 
 (38) There is a number consisting of two digits, which, when divided by 
 the sum of its digits, gives a quotient greater by 2 than the first digit ; but if 
 the digits be inverted, and the resulting number be divided by a number greater 
 by unity than the sum of the digits, the quotient shall be greater by 2 than the 
 former quotient. What is tho number? 
 
 Ans. 24. 
 
 (39) A regiment of foot receives orders to send 216 men on garrison duty, 
 each company sending the same number of men ; but before the detachment 
 marched, three of the companies were sent on another service, and it was then 
 found that each company that remained would have to send 12 men additional 
 in order to make up the complement, 21G. How many companies were in the 
 regiment, and what number of men did each of the remaining companies send 
 on garrison duty ? 
 
 Ans. There were 9 companies, and each of the remaining 6 sent 3G men. 
 
 DECOMPOSITION OF THE TRINOMIAL X 2 -\-pX — q INTO TWO FACTORS OF THE 
 
 FIRST DEGREE. 
 
 193. If we add to this trinomial, in order to complete the square of the first 
 two terms, the term ^p 2 , and afterward subtract the same, so as not to change 
 the quantity, it becomes 
 
 2-2 +px -\- 1 p- — I p- — q, 
 which may be written thus : 
 
 (*+iiO S -Ui> 8 +?) (2) 
 
 But the difference of the squares of two quantities being equal to the prod 
 uct of their sum and difference, the expression (2) is equal to the following • 
 
 (*+&+ VW+q)(x+hp- V^+q) • • • (3) 
 We perceive from this expression that the two factors of the first degree, 
 
 which compose the trinomial of the second degree, are x minus each of the 
 
 roots of the equation of the second degree, formed by putting this trinomial 
 
 equal to zero. 
 
 Moreover, by equating (3) to zero, we perceive that the only way of satis 
 
 fying the resulting equation is by making one or other of the factors of the 
 
 first degree, of which it is composed, equal to zero. 
 The first, 
 
 X +U J + V!^+?=o, gives x=— \p— Vlp 2 +q; 
 
 and the second, 
 
 x +h 7 — VkP 2 +1= ^ § ives x =—h>+ V\p 2 +q- 
 Hence there are but two values of x which will satisfy the general equatioa 
 
 x 2 -\-px — 9=0. 
 
 EXAMPLES. 
 
 1°. Decompose the trinomial x 2 — 7z-}-10 into two factors of the first de- 
 gree. 
 
240 ALGEBRA. 
 
 From the equation x* — 7x+10=0 we find the roots x=5 and 1=2. 
 Henco 
 
 a?— 7x+10 = (x— 5)(x— 2). 
 2°. 3x°— 5x— 2. 
 
 Equating this trinomial to zero, after dividing by 3, wo obtain the equation 
 x 2 — §x — §=0, the roots of which being x=2 and x= — |, we have 
 3a«— 5x— 2=3(x— 2)(x+|)=(x— 2)(3x+l). 
 
 3°. .7-+5T+3. Ans. (*+$—£ V^)(*+5+Wl3). 
 
 4°. 4a*— 4ar+l. Ans.~(2x— 1)-.« 
 
 5°. x-— 5x+7. Ans. (x— f)*+|. 
 
 194. To complete the analysis of the 2° degree, it would be necessaiy to 
 consider the case where the unknown quantities exceed the equations in num- 
 ber. The moro simple is that when there is but one equation and two un- 
 known quantities. If it be resolved with respect to one of the unknown quan- 
 tities, y, for example, an expression is found generally containing x under a 
 radical ; so that, by giving to x any rational values whatever, irrational values 
 would be found for y. It might be proposed to find rational values for x, for 
 which the corresponding one of y should be rational also. But the difficulty 
 of this problem, unless it be restricted to some very simple cases, is beyond 
 mere elements. We add one or two here. For further information upon 
 the subject, the student is referred to the Theory of Numbers, by Legendre, 
 a separate and veiy elegant treatise, in one quarto volume. 
 
 INDETERMINATE ANALYSIS OF TIIE SECOND DEGREE. 
 
 Resolution in whole numbers of an equation of Oie second degree, with two 
 unknown quantities, which contains but the first poicer of one of the unknowns. 
 
 195. The questions of indeterminate analysis, which depend upon equations 
 of a degree superior to the first, go beyond the limits which we have imposed 
 on ourselves in the present work ; but when an equation of the second degree 
 contains the second power of but one of the unknown quantities, the solutions 
 of this equation in whole numbers may be regarded as a question of indeter- 
 minate analysis of the first degree. 
 
 Equations of the second degree in two unknown quantities, which do not 
 
 contain the second power of one of these, aro represented by the equation 
 
 mxy+nx*+px+qy=:r (1) 
 
 Resolving this equation with respect to y, we find 
 
 — nx- — px-\-r 
 
 mx-\-q 
 
 We deduce from it, by performing the division, 
 
 n nq — mp m*r-\-mpq — nq* 
 
 J m ' ra 2 "*" m 2 (jnx-{- q) ' 
 
 which gives 
 
 N 
 m-y=— mnx+nq— mp + ^^-j (3) 
 
 outting to abridge m^r+mpq — rt^ssN. 
 
 y= — ^ttt, — ( 2 ) 
 
 N_ 
 should bo a wholo number; wo must, therefore, calculate all the divisors of 
 
 In order that X and y should bo wholo numbers, it is necessary that — - 
 
 * This presents a case of what aro oaUed kjuuI roots 
 
For the first equation 
 
 INDETERMINATE ANALYSIS OF THE SECOND DEGREE. 241 
 
 the Dumber N, and put mx-\-q equal to each of these divisors successively, 
 taken with the sign -f- an d with the sign — . If the equations thus obtained 
 furnish for x a certain number of entire values, these values are to be substi- 
 tuted in equation (3); and it is necessary, moreover, in order that y may be o 
 whole number, that the second member which becomes a known quantity 
 should je divisible by m". 
 
 It is evident that the number of entire solutions will be very limited, and 
 that there may not be even one. 
 
 If this method be applied to each of the following equations, 
 
 2xy— 3r 2 + y=l 
 5xy=2x +3?/+ 18 
 xy-\- x°~=2x-\-3y-\-29, 
 considering only the positive solutions, we find 
 
 ( .r=0, y = l 
 ' \ .r=3, i/=4. 
 ( x=l, y=\0 
 For the second equation ..._.< x=3, y=2 
 
 ( x=7, ?/ = l. 
 
 c x=4, y=21 
 For the third equation < ,- „ 
 
 If' the remainder, after the hvision of —nx-—px-\-r by mx-\-q, should be 
 zero, equation (1) would be of the form (mx-\-q)(ax+by-{-c) = ; ai;d we 
 should have all the solutions of this equation by resolving separately the two 
 equations mx-\-q=0, ax-\-by-\-c = 0. 
 
 The method which has just been explained is applicable only in case in is 
 not zero. 
 
 Let ??i = ; equation (1) give,s 
 nx*+px—r 
 
 y=—- q — (4 > 
 
 Suppose that one value of x=a (a being a whole number) gives an entire 
 value for y. If we place x=a-\-qt, t being any entire number whatever, we 
 find 
 
 y= — — - I ^ (2nat-\-nqP +pt); 
 
 by hypothesis, na?-{-pa—r is divisible by q ; the value of y, corresponding to 
 1=0+ qt, will be then a whole number. As this conclusion is true, what- 
 ever be the sign of t, it follows that, if the equation admits of entire solutions, 
 thoy will be found to be such as answer to a value of x between and q. 
 Consequently, to obtain all the solutions in whole numbers, it will be suffi- 
 cient to substitute for x in the equation the numbers 0, 1, 2, 3, . . . q— 1, 
 and each solution in whole numbers corresponding to one of these numbers 
 will furnish an infinite number of others. 
 
 Equation (4), in which the object is to find values of x which render the 
 polynomial ?ix--\-px — r a multiple of the given number q, M. Gauss calls con- 
 gruence of the second degree ; so, also, the equation ax-{-by=c, in which we 
 eeek to render ax — c a multiple of b, is a congruence of the firs'- degree. 
 
 Further matter on the subject of indeterminate analysis will be given in con- 
 nection with the theory of numbers, for which see a subsequent part of the 
 work. 
 
 Q 
 
242 ALGEBRA. 
 
 MAXIMA AND MINIMA. 
 
 196. When a quantity which is capable of changing its valut attains such a 
 value that, after having been increasing, it begins to decrease, cr, Inning been 
 decreasing, it begins to increase, in the first case it is called a maximum, and in 
 the second a minimum. The same quantity may have several maximum or 
 
 minimum values. 
 
 EXAMPLE. 
 
 To find what value of x will render the fraction a maximum or 
 
 *2x '2 
 
 minimum. 
 
 Equating the given function of x to 2, we have 
 
 t' r _i_O 
 
 ~-±±Z=z .-. x=z+l± V~ : -l- 
 
 "We perceive at once that by making r == -j- 1 we have x=2, and that the 
 values of 2, a little less than 1, render x imaginary ; hence the given expression 
 has a minimum value 1 corresponding to x=2. 
 
 In a similar manner, making z= — 1, we have .7=0; and a negative value 
 of 2, a little smaller than 1, would render x imaginary. But in algebra, nega- 
 tive quantities, which, without, regard to the sign, go on increasing, ought to be 
 regarded, when the sign is prefixed, as decreasing; we may, therefore, 
 that a value of 2, a little greater than — 1, renders x imaginary, then z = — 1 is 
 a maximum corresponding to x=0. 
 
 As the subject of maxima and minima is generally treated by the aid of the 
 differential calculus, we shall not dwell further upon it here, though it furnishes 
 one of the applications of equations of the second degree. 
 
 THE MODULUS OF IMAGINARY QUANTITIES. 
 
 197. We have seen (191) in the equation of the second degree 
 
 x*+px+q=0, 
 
 that when q is positive, and greater than "-— , the roots are imaginary. Replace 
 
 \p by — a, to avoid fractions ; and to express that 7>-r. put g=a 5 4-&' ; ; the 
 equation will become 
 
 and, by the formula for the solution of equations of the second degree, 
 
 x=aAz V— &*> 
 or 
 
 x=a±bV~^l (I) 
 
 The absolute value of the square root of the positive quantity a : -{-b : is call- 
 ed the modulus of the imaginary expression (1). For example, the modulus 
 of 3— 4 V — 1 would be V^ + IG, or 5. 
 
 Two quantities, such as a-\-bi/ — 1 and a — b \/ — 1, which differ from one 
 another only in the sign of the imaginary part, are called I mjugatU of each 
 
 other. Two conjugate quantities bars then ihe same modi 'us. 
 
 If we make //=<), the expression a-{-h y/ — 1 reduces to '/. Thus, the 
 
 formula x =a-\-b y/ — 1 may represent all quantities real or imaginary, fl rep- 
 resenting the algebraic sum of the real quantities, and b that of the roellicients 
 
THE MODULUS OF IMAGINARY CJANTITIES. L'43 
 
 of *J — l in the imaginary terms. When the quantity is real, it has for con- 
 jugate an equal quantity, and the modulus is nothing else than the quantity 
 itself, abstraction being made of the sign. 
 
 Now I shall proceed to establish two propositions relating to moduli, which 
 may be often useful. 
 
 Proposition I. — The sum and difference of any two quantities whatever 
 have a modulus comprehended hetween the sum and the difference of their 
 moduli. 
 
 Let there be two expressions a-\-bj — 1, a'-\-h' V — 1. Calling r and r' 
 their moduli, we have r 2 =a 2 -|-Z; 2 , r' 2 =a" 2 -\-b" 2 . Naming R the modulus of 
 their sum, we have evidently 
 
 R>=(«+a'y-\-(b+hy 
 
 — a'iJ r a'-+lr-{-b' 2 +2(aa'+bb') 
 
 But multiplying r 2 by r'\ we have 
 
 r 2 r' 2 =a 2 a' 2 + lrb' 2 +a*b' 2 +a' 2 b' i 
 = (aa' + bb')-+(ab'—ba'y; 
 then the numerical value of aa'-\-bb' is less than, or at most equal to, rr'. Con 
 sequently, it is clear that R 2 is comprehended between the two quantities 
 r i-^-r'--\-2rr' and r°--\-r'~ — 2rr', or, what is the same thing, between (r-\-r') 9 
 and (r — r') 2 . Then the modulus R is comprehended between the sum and 
 the difference of the moduli r and r'. 
 
 The demonstration is precisely the same where, instead of the sum of the 
 imaginary expressions, we consider their difference. 
 
 Proposition II. — The product of two quantities has for modulus the product 
 of the moduli of these quantities. 
 
 In fact, multiplication gives 
 
 [a + b v r ^l)(a' + ?/ J~—i) = aa'—bb'-\-{ab'-l r ba') /— i ; 
 and if wo take the modulus of this product, we find, conformably to the enun- 
 ciation, 
 
 ■J{aa' — bb'y+{ab'-\-ba') i =y/a i a'-+b-b' 2 -\-a 2 b' i -{-b 2 a' i 
 
 = V(a>+b 2 )(a'* + b' 2 ). 
 
 Corollary. — Then the product of any number of factors whatever must 
 nave for modulus the product q of the moduli of all the factors. Then the 
 *> lh power of an imaginary expression has for modulus the n th power of the 
 modulus of that expression. 
 
 The aoove nomenclature and propositions are from Cauchy, who exhibits in 
 a remarknble manner the efficiency of imaginary expressions as instruments in 
 the investigation of the properties of real quantities. The following is a 
 specimen : 
 
 If two numbers, of which each is the sum of two squares, be multiplied to- 
 gether, the product must also be the sum of two squares. 
 
 Let the two numbers be 
 
 a 2 +6 2 anda' 2 -|-Z>' 2 . 
 The first of these may be considered as the product of the factors 
 
 fl-f-6 / — 1 and a — b y/ — I, 
 and the second as the product of the factors, 
 
 a'+b' -/^T and a'—b' /^l ; 
 
241 ALGEBRA. 
 
 so that the product of the proposed numbers will be the product of the fout 
 
 fuctors 
 
 a + b V^-T, a — b ^"^1, a'+b' yf — 1, a' — b' y/ — 1. 
 Actually multiplying the first and third, and then the second and fourth, we 
 have the following pair of conjugate expressions, viz., 
 
 (aa' — bb') + (ab'+ba') yf^l % { a a' — bb') — (ab' +ba') y/^-i, 
 of which the product is 
 
 {aa' — bb'Y+{ab' +ba'y, 
 which is, therefore, the product of the original numbers, and proves that that 
 product must, like each of the proposed factors, be the .sum of two squares. 
 
 If we interchange the numbers a and b, or the numbers a', 6', the terms of 
 the product just deduced will be different; thus, putting a' for b', and b' for 
 a', which produces no essential change in the proposed numbers, we have 
 (a- + b : ){a'- 2 +b' i ) = {aa' — bb')-+(ab'+ba')- = {ab' — ba'y- + (aa' + bb')-. 
 Consequently there are two ways of expressing by the sum of two squares 
 the products of two numbers, each of which is itself the sum of two squares ; 
 thus, 
 
 (5 2 +2-)(3 3 +2-) = ll-+16- = 4-+19 3 
 (2 s - r .l 2 )(3 2 4-2-)= 4--f 7 : = 1-+ 8 3 
 &c, &c. 
 
 METHOD PROPOSED BY MOUREY FOR AVOIDING IMAGINARY QUANTITIES.* 
 
 198. Objections have been made to results obtained by the calculus of imag- 
 inary expressions. The rules observed in the calculus, it is said, have only 
 been demonstrated for real magnitudes; it is by mere analogy that they are ex- 
 tended to the case of imaginary quantities ; we may, therefore, raise reasonable 
 doubts as to the exactitude of the results thus deduced. 
 
 M. Mourey, who has been much occupied with these difficulties, has sought 
 to free analysis from them entirely, in a work published in 1828, entitled the 
 True Theory of Negative Quantities and of the so-called Imaginary Quanti- 
 ties. Without entering into long details, we shall endeavor here to give an 
 idea of the methods proposed by this author. 
 
 Let us resume the expression a-]-b. -y/ — 1< and give it, at first, the form 
 
 b 
 
 ^^{t^+t^^- 1 ] 
 
 If wo take the sum of the squares of tin 1 fractions, which are between the 
 brackets, we find that this sum is equal to 1 ; and from thence we conclude that 
 these two fractions can be regarded as being the .sine and eosim of a same 
 angle a. Designato also the modulus V"' +' by A ; the imaginary expres- 
 sion can be put under the form A(cos a-|- ■/ — 1 sin a). Considering that 
 this expression contains really but two quantities, the modulus A and the 
 angle «, M. Mourey proposes to regard the modulus A a; expressing the 
 
 length of a right line O A. a sing 
 
 the angle LOX ( which this line D 
 with a fixed axis OX. In other w. 
 
 the modulus A represents a lme of a cer- 
 tain length, which at first lay upon the 
 axis < » Vaud which, by making a move- 
 
 • To luidersturul this, a knowledge of the first principles of Trigonometry is accessary 
 
MOUREY'S METHOD FOR AVOIDING IMAGINARY QUANTITIES. 245 
 
 merit round the origin O upward, has departed from this axis by an angle a. 
 M. Mourey gives the name verser to this angle, or, rather, to the arc which 
 measures it; and then, instead of the imaginary expression, he writes simply 
 Aa, a uotation very suitable to recall at the same time the modulus A and the 
 verser a. He proposes even to give the name route, or way, to the length O A, 
 placed in its true position with regard to OX, so that A verser a, orAa,ia the 
 route from O toward A. 
 
 As a line can make around the origin O as many revolutions as we please, 
 and that, also, as well by commencing its rotation below as well as above O X, 
 it follows that the verser may pass through all states of magnitude, and be aa 
 well negative as positive. It will bo positive when the movement of the line 
 shall have commenced above ; it will be negative when the movement com- 
 menced below. From this it follows that the same route can be represented 
 with a verser which is positive, or one which is negative, provided that the 
 sum of tho versers, abstraction being made of the signs, is 360°. 
 
 From the preceding conventions it results that a way can be represented by 
 giving to tho length A an infinity of different versers. Suppose, to fix the 
 ideas, that O A should be a determinate way, and that then the verser A OX 
 should be an acute angle a ; it is evident that the position of O A will undergo 
 no change if we add or subtract from a any number whatever of entire cir- 
 cumferences. Thus is established this important remark, that if we desig- 
 nate by 2tt an entire circumference, or 360°, and by n any whole number 
 whatever, positive or negative, the expression A27r«-j-° will represent tho 
 same route as Aa ; this is expressed by the equality 
 
 When we give to A a verser equal to zero, the length A lies upon the lino 
 OX. When the verser is equal to n or 180°, this length is found in the op 
 posite direction, O X' ; then it is nothing else than the negative quantity — A. 
 Thus we ought to regard as altogether equivalent the two expressions — A 
 and Air. 
 
 After these preliminaries, M. Mourey establishes the rules of algebraic 
 calculus ; then he passes to equations, and reconstructs algebra thus' entirely. 
 I shall not follow this author in all his details ; I shall confine myself to the 
 developments necessary to explain here what sense the new algebra attaches 
 to the old imaginary expression -/ — A 2 . I shall seek, first, the rule to be 
 followed in the multiplication of any two quantities whatever, Aa and B/?. 
 Here the two factors are the magnitudes A and B, measured upon two lines 
 
 O A and O B. which make, with a fixed axis 
 OX, angles A OX, BOX, represented by tho 
 'A' versers a and (8. It is necessary, then, first 
 of all, to give to the definition of multiplica- 
 tion the extension suitable to render it appli- 
 cable to the case in question. But, consider- 
 ing that, the multiplier B/3 indicates a line B, 
 which departs from the fixed line O X by an 
 angle equal to /?, M. Mourey regards multi- 
 plication sis having for its object to take at 
 first the length A in its actual direction as many times as there are units in B, 
 and to turn the new line O A' around the point O, to depart from this direc- 
 
246 ALGEBRA. 
 
 Hon by an angle enual to ft, and to give it the position O . From this it fol- 
 lows that, in designating by AB the product of the two nugnitudes, obstrac- 
 tion being made of all idea of position, the product sougl t will bo (AB)a-|-' 
 Thus we have 
 
 A« X B/3=(AB)rt-f/3; 
 that is to say, wo multiply the moduli according to the ordinary rules of a 
 metic, and tale the sum of the verscrs. 
 
 If the two versers are equal to it or 130°, we shall have \- X B-=(AB).'-. 
 But A^ and Bw are nothing else than — A and — B, and (AB)2t is the same 
 tiling as -|-AB ; then — Ax — B = -|-AB. This is the known rule, — by — 
 gives -f- . 
 
 According to this rule, the square of Aa will be (A-)2a ; that is to say, 
 take the square of the modulus and double the verser. Then, reciprocally, the 
 square root is obtained by extracting the square root of the modulus without re- 
 garding the verser ; then take half the verser. 
 
 Let us come now to the interpretation of the imaginary expression ■/ — A-\ 
 For this purpose, let us observe, first, that it is equivalent to y/ (A: z )2n~-\-- ; 
 then extracting the square root, 
 
 V — A- = An --f- ' T - 
 If n is even, the verser nT-k-'." places the length A in the same position as 
 \n ; that is to say, in the position OP, perpendicular ; < > \. 
 If n is uneven, the verser nrr-\-\- will place the length A in 
 a position O P', perpendicular to O X, but bebw. Thus, in 
 
 O X the system of M. Mourey, the expression -/ — A* offers no 
 longer to the mind any idea of impossibility. It represeuta 
 P' two routes, OP and OP', equal and opposite, both perpen- 
 
 dicular to the fixed axis O X. 
 
 PERMUTATIONS AND COMBINATIONS. 
 
 199. The Permutations of any number of quantities are tin- changes which 
 these quantities may undergo with respect to their order. 
 
 Thus, if we take tlie quantities </, /', c; then </!>>•, acb, bar, bra. cab, ct,a 
 are the permutations of these three quantities taken all together; ab, ac, ba, 
 be, ca, cb nre the permutations of those quantities taken two and two; a, b, i 
 are the permutations of these quantities taken singly, or <>n, and one, &cc. 
 
 The problem which we propose to resolve is, . 
 
 200. To find, the number of the permutations of n quantities, taken p ani/ p 
 together. 
 
 Let a, 6, c, d, k, bo the n quanti' es. 
 
 The number of the permutations of these /; quantities taken singly, or one 
 and one, is manifestly n. 
 
 The Dumber of the permutations <>t these n quantities, taken two and two 
 
 together, will he n(n — 1). For, since there are // quantities, 
 
 a, /', <.</, k. 
 
 If we remove a there will remain (n — 1) quantities, 
 
 b, c, d, k. 
 
PERMUTATIONS AND COMBINATIONS. 247 
 
 Writing a before each of these (n — 1) quantities, we shall have 
 
 ab, ac, ad, ak ; 
 
 that is, (n — 1) permutations of the n quantities taken two and two, in which a 
 stands first. Reasoning in the same manner for b, we shall have (n— 1) per- 
 mutations of the n quantities taken two and two, in which b stands first, and 
 so on for each of the n quantities in succession; hence the whole number of 
 permutations will be 
 
 n(n — 1). 
 
 The number of the permutations of n quantities, taken three and three to- 
 gether, is n(n — \)(n — 2). For since there are n quantities, if we remove a 
 there wil. remain (?» — 1) quantities; but, by the last case, writing (n — 1) for 
 n, the number of the permutations of (/< — 1) quantities, taken two and two, is 
 (n — l)\n — 2); writing a before each of these (n — l)(n — 2) permutations, 
 we shall have (« — l)(n — 2) permutations of the n quantities, taken three and 
 three, in which a stands first. Reasoning in the same manner for b, we shall 
 have (n — l)(n — 2); permutations of the n quantities, taken three and three, in 
 which b stands fivst, and so on for each of the n quantities in succession ; hence 
 the whole number of permutations will be 
 
 n{n — l){n— 2). 
 
 In likq, manner, wo can prove that the number of permutations of n quan- 
 tities, ta.,en four and. four, will be 
 
 n( n — l){n— 2)(n— 3). 
 
 Upon examining the above results, we readily perceive that a certain rela- 
 tion exists between the numerical part of tho expressions and the class of per- 
 mutations to which they correspond. 
 
 Thus the number of permutations of n quantities, taken two and two, is 
 
 n(n — I), which may be written under the form n(n — 2-j-l). 
 Taken three and three, it is 
 
 n(7i — l)(n — 2), which may be written under the form n(n — l)(ra — 3-f-l). 
 Taken four and four, it is 
 n(n — l)(ra — 2)(n — 3), which may be written under the form n(n — \){n — 2) 
 
 (n-4 + 1). 
 Hence, from analogy, we may conclude that the number of permutations 
 of n things, taken p and p together, will be 
 
 n(n — 1)(» — 2)(n — 3) . (,i_p_{-l). 
 
 In order to demonstrate this, we shall employ the same species of proof 
 already exemplified in (Arts. 23 and 78), and show that, if the above law be 
 assumed to hold good for any one class of permutations, it must necessarily 
 hold good for the class next superior. 
 
 Let us suppose, then, that the expression for the number of the permuta- 
 tions of n quantities, taken {p — 1) and (p — 1) together, is 
 
 n (n — l){n—2){n—3) . . . \ n — (p — l) + l } ... (A) 
 It is required to prove that the expression for the number of the permuta- 
 tions of n quantities, taken p and p together, will be 
 
 n(n— l)(n— 2)[n — 3) (n— p+1). 
 
 Remove a, one of the n quantities a, b, c, d k, then, by the ex- 
 pression (A), writing (n — 1) for n, the number of the permutations of the 
 (n_l) quantities b, c, d k, taken (p — 1) and (^ — 1), will be 
 
248 ALGEBRA. 
 
 (»— 1)(»— 2)(n— 3) [(*— ^-(j>— l)+lj, 
 
 or 
 
 (n—l)(w— 2)(n-3) (m— J>+1). 
 
 Writing a before each of these (n — l)(n — 2)(n — 3) ("—/> + !) 
 
 permutations, we shall have (n — l)(n — '-')(«— 3) (n—p+l) per- 
 mutations of the n quantities, in which a stands first. Reasoning in the same 
 manner for b, we shall have (n — l)(n — 2)(n— 3) (n — ]>-\-l) per- 
 mutations of the n quantities, in which b stands first; and so on for each of 'he 
 n quantities in succession ; hence the whole number of permutations will bo 
 
 »(»-l)(n-2)(»-3) (n-p+l) (1) 
 
 Hence it appears that, if the above law of formation hold good for any one 
 class of permutations, it must hold good for the class next superior ; hut it has 
 been proved to hold good when p=2, or for the permutations of ;i quantities 
 taken two and two ; hence it must hold good when p = 3, or for the permuta- 
 tion of n quantities taken three and three ; .-. it must hold good when /> = 4, 
 and so on. The law is, therefore, general. 
 
 EXAMPLE. 
 
 Required the number of the permutations of the eight letters a, b, c, d, e, 
 f, g, ft, taken 5 and 5 together. 
 
 Here n=8, p=o, n — p-\-l=A ; hence the above formula 
 
 n{n — l)(n — 2) .... (n— p + l) = 6x 7 X G X5 X 4=6720, 
 the number required. 
 
 201. In formula (1) let p=n, it will then become 
 
 n(n— 1)(»— 2) 2.1, 
 
 or 
 
 1.2.3 (n — 1)» (2) 
 
 which expresses the number of the permutations of n quantities taken all 
 together.* 
 
 EXAMPLE. 
 
 Required tho number of the permutations of the eight letters a, b, c, </, c, 
 
 fi £i >>■ 
 
 Here n=8; hence the above formula (2) in this case becomes 
 
 1.2.3.4.5.6.7.8=40320, 
 
 the number required. 
 
 202. The number of the permutations of n quantities, supposing them all 
 different from each other, we have found to be 
 
 1.2.3 (n— l)n. 
 
 But if tho same quantity be repeated a certain number of times, then it is 
 
 manifest that a certain Dumber of the above permutations will become identical 
 
 Thus, it" one of the quantities be repeated a times, the number of identical 
 
 permutations will be represented by 1.2.3 <i ; and hence, in order ti> 
 
 * Many writ- the term perm toth the quan- 
 
 tities iiiv taken nil together, nml iriye the title " : tmentt or variations u<: i 
 
 of the n quantities when taken two mid two, three and ■ 
 
 traduction of these nilditional designations appears unnecessary ; bat, in a wrord 
 
 perm ■ . we must always I to mean those rc\<< v (or- 
 
 muln (8), unless the contrary !»■ specified, 
 
PERMUTATIONS AND COMBINATIONS 24 
 
 obtain the number of permutations different from each other, we must divide 
 
 (2) by 1.2.3 a, and it will then become 
 
 1^2.3 (n — l)n 
 
 1.2.3...^... a ' 
 
 If one .if the quantities be repeated a times, and another of the quantities 
 
 be repeated /3 times, then wo must divide by 1.2 aXl.2 /? ; 
 
 and, in general, if among the n quantities there be a of one kind, (1 of another 
 kind, y of another kind, and so on, the expression for the number of the per- 
 mutations different from each other of these n quantities will be 
 
 1.2.3 n 
 
 1.2 oXl-2 /3xl.2 y, &c. * ' 
 
 KXAMPLK I. 
 
 Required the numbers of the permutations of the letters in the word algebra. 
 Here n=7, and the letter a is repeated twice; hence formula (3) becomes 
 
 1.2. 3. 4. 5. G. 7 
 
 — — • = 2520, the number required. 
 
 1.2 • 
 
 EXAMPLE II. 
 
 Required the number of the permutations of the letters in the word 
 caifacarataddarada. 
 
 Hero ra=18, a is repeated eight times, c twice, d thrice, r twice ; hence the 
 number sought will be 
 
 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18 
 
 1.2. 3. 4. 5. 6. 7. 8X1-2X1. 2. 3x1-2 
 
 =6616209G00. 
 
 EXAMPLE III. 
 
 Required the number of the permutations of the product a* b y c% written at 
 full length. 
 
 Here n=x-\-y-{-z, the letter a is repeated x times, the letter b, y times, 
 and the letter c, z time^ ; the expression sought will, therefore, be 
 
 1-2-3 CH-y+z) 
 
 1.2. 3 x X 1 • 2 . 3 y X 1 • 2 . 3 z ' 
 
 203. The Combinations* of any number of quantities signify the different 
 collections which may ba formed of these quantities, without regard to the 
 order in which they are arranged in each collection. Each combination must, 
 therefore, have one letter different from any other of the combinations. 
 
 Thus the quantities a, b, c, when taken all together, will form only one 
 combination, aba ; but will form six different permutations, abc, acb, bac, bca, 
 cab^, cba ; taken two and two, they will form the three combinations etb, ac, be, 
 and the six permutations ab, ba, ac, ca, be, cb. 
 
 The problem which we propose to resolve is, 
 
 To find tlic number of the combinations of n quantities, taken p and p to- 
 gether. 
 
 Each of these combinations of p quantities being separately permutatod, will 
 furnish 1.2. 3... p permutations, which, multiplied by the whole number of 
 combinations, will give the whole number of permutations of n quantities, taken 
 
 * Where numerical or literal factors are combined, the term combination may be con- 
 sidered as signifying the same as product. 
 
250 ALGKBRA. 
 
 p and p. Therefore tho latter, namely, the whol„ numbe- of permutations, 
 
 or ii(i) — lj(n — 2) (n — P+ 1 )- divided by the number rf permutations of 
 
 atch combination, or 1 .2 .:;... p. will give the number of combinations of n 
 quantities, taken p and p. Denoting it by C, we have 
 
 n(/i-l)(»-2) (»-p+l ) 
 
 c ~ r^3 • • {i—i)p "• ■ (l) 
 
 20 J. There is ;i Bpecies of notation employed to denote permutations and 
 combinations, which is sometimes used with advantage from its com 
 The number of the permutations of n quantiti . a p and p, 
 
 are repre ented by ( n ^'j) 
 
 The number of the permutations of n quantities, taken all togeUier, 
 
 are represented by ("i'") 
 
 The number of the combinations of n quantities, taken p and p, 
 
 are represented by ("'/") 
 
 and so on. It is manifest that the above proposition may be expressed accord- 
 ing to this notation by 
 
 InPp) 
 
 (nCp)= K —J-!.. 
 
 M. Cauchy employs the notation (m)„ to express the number of com 
 tions of m letters, taken n at a time. The German notation ame is 
 
 When the scries of natural numbers, or the letters of tho alphabet up to 
 any required number, are to be permuted or combined, an abb: nota- 
 
 tion has been employed as follows: 
 
 P(l, 2, 3) stands for 123, 132, 213, 231, 312, 321. 
 
 P(1..4) stands for 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, 43. 
 
 3 
 
 C(a...e) stands for abc, abd, abe, acd, ace, ade, bid, bec, bde, 
 If one or more of the numbers or letters may be repeated, this can also be 
 expressed in the notation. Thus, 
 P(l, 1, 2) = 112, 121, 211. 
 
 P(l, 1, 2, 3) = 11, 12, 13, 21, 23, 31, 32. 
 
 C(l, 1, 2, 2, 3) = 112. 113, 122, 12 
 
 If all the letters, numbers, or single things may be repeated an equal num- 
 ber of times, this can be expressed with the aid of an exponent; thus, 
 
 C(l, 2, 3)\ P(0, 1,-Jr. C(1..7)». 
 205. If n single things be arranged in combinations of k, or ofn — k=r, \}\» 
 number of combinations in either case will be die same, i. >-., 
 
 f fc = w («-l)...(n-fr+l) _/,^ w (/>-l).. .(n -r+1) 
 n 1.2.3...* n L.2.3..\r 
 
 for every new combination of k letters must leave a new one. ofr letters* 
 By a similar reasoning, if n be divided into three parts, the fust A-, tin- second 
 
 r. and the third 8, it may be shown that 
 
 exc =cxc =rx( k ' , &c 
 
 n n— t ii n k ii i 
 
 20fi Cases may occur in which not all p ssible combinations, but only such 
 
PERMUTATIONS AND COMBINATIONS. 251 
 
 as fulfill certain conditions, are required. Mnny such may be imagined. For 
 instance, where the numbers to be combined increase by a common difference. 
 or by a common ratio, as 1357, 2468. or 124, or 248. The most useful case 
 is where the number in each combination must amount to the same sum. The 
 method of proceeding in this case is to fill up all the places except the last with 
 the lowest numbers, the last place being occupied by the supplementary num- 
 ber necessary to produce the given sum ; then diminishing the last number 
 and increasing one of the preceding by the same amount, taking care not to 
 allow a lower ever to follow a higher number. We give examples of such 
 
 k 
 
 combinations, the general formula for which is r C(l n). 
 
 (1) "°C(1...7)=127, 13G, 145, 2:)r>. 
 
 (2) "C(1...8)=1238, 1247, 1256, 1346, 2345. 
 
 (3) &C(0..5)«=0005, 0014, 0023, 0113, 0122, 1112. 
 
 (4) «>C(3....)rc=33338, 33347, 33356, 33446, 33455, 34445, 44444. 
 
 It is easy to be perceived that in two cases this kind of combination is im- 
 possible. 1°. When the highest form does not amount to the required sum; 
 and, 2°. When the lowest form exceeds it, as in 
 
 i"C(123)«, or ioC(4...)n. 
 
 207. Similar conditions may be imposed upon permutations. In order that 
 the permutations of a given series of numbers, taken a certain Dumber at a 
 time, should amount always to a given sum, the same rule will apply, with this 
 difference, that lower numbers may follow higher; in other words, the com- 
 binations formed by the previous rule may each be permuted. 
 
 The following examples will render this more intelligible : 
 
 (1) oP(1..8) = 18, 27, 36,45, 54, 63, 72, 81. 
 
 (2) tP(1...) = 124, 142, 214, 241, 412, 421. 
 
 (3) 6p(l...)n = 1113, 1122, 1131, 1212,1221, 1311, 2112, 2121,2211,3111. 
 
 (4) iP(0..)n=013, 022, 031, 103, 112. 121, 130, 202, 211, 220, 301, ",10. 
 
 Under this head, also, two contradictory cases occur: 1°. When the high- 
 est form amounts to too little ; and, 2°. When the lowest form amounts ft too 
 much. As, for instance, in 
 
 9p/1..4)n, or''P(5...)«. 
 
 208. The applications of the theory of permutations and combinations are 
 numerous. One of the most useful is the determination of the coefficients of 
 a series of the form 
 
 a + bx+ c.r 2 + rf^-f ex*-f- . . . + kx" . . .,* 
 
 especially the coefficients of the binomial formula, the method of determining 
 which, by the theory of permutations and combinations, will be given here- 
 after. 
 
 Another extensive application of the theory of permutations and combina- 
 
 * These coefficients are supposed to depend upon some given law. A common case is 
 when the number of {actors combined in each coefficient is indicated by the exponent of 
 the letter of arrangement, x. 
 
252 ALGEBRA. 
 
 tions is to be found in geometric relations, such as where the comlinations of a 
 certain number of points, lines, angles, cVe., from among a given number of 
 these, are required. 
 
 Not less useful is this theory in natural science : as in crystalography. when 
 tin- manifold tonus of crystals are required; in chemistry, when the various 
 combinations of chemical elements; and in music, of consonant tones, 6zc. 
 
 Jim perhaps its most important use is in the doctrine of chances, or, as it is 
 mathematically named, the 
 
 CALCULUS OF PROBABILITY 
 
 The nut linos of this extensive subject we shall here briefly indicate, referring 
 the studenl for further information to the admirable treatises of La Place 
 and Lacroix, and to the practical work ofDe Morgan. 
 
 I. Lei there be among m possible cases g, which, as, fulfilling certain requi- 
 sitions, are considered as favorable, {m — g) = u unfavorable. Then the 
 
 of the favorable to all possible cases is called the mathematical jnobalii '<! / for 
 the occurrence of a favorable case. The ratio of the unfavorable to all possi- 
 ble cases is the mathematical improbability of the occurrence. If the first be 
 expn s e 1 by u; the second by », then 
 
 e; u 
 7/=— and v=— (I.) 
 
 The probability is, therefore, the less, the smaller the number of the fa- 
 vorable in comparison with that of all possible cases, and vice versa. Should 
 all possible cases be favorable, then 10=1, which is, therefore, the expre 
 for certainty. Thus the mathematical probability and improbability of a pic- 
 tured card, of which there are 12, being drawn from 52, are expressed by 
 
 _12_3 _40_10 
 ,r== 52 = 13' ' y= 52~13 ; 
 that of drawing one card from 52, 
 
 52 
 t* = - = l. 
 
 II. Let there be among m possible cases g favorable, of different (first, sec- 
 ond, third, cVc.) kinds, expressed by g l% g 2 , g 3 , &c, the partial probabilities 
 by to,, to 3 , to 3 , &c. ; then 
 
 U- = W l +lV._,-\-H',-\-,\c,= (II.) 
 
 that is, the probability of one of several different kinds is equal to the sum of 
 
 their partial probabilities. Thus, for the probability of one of tin- six faces of 
 
 a die, marked 1, 2, or 3, being thrown, we have 
 
 1 1 1 
 
 «,>=-, «,,=-,«,,=-; 
 
 l l l :; i 
 ••• w =6+6+6=6=2' 
 
 III. Let the occurrence be favorable only on the supposition that two or 
 more of the single /avordble cases concur, then the formula for the compound 
 probability is 
 
 g, XiS,Xr •• .... 
 
 ' - J 7/1 , X m X "l :i • • 
 
 in which /«,, /(<„, 7/i ;i , (Vc, express the poss'blo cases of the part nl occurn n- 
 
CALCULUS OF Pfl,OB ABILITIES. 253 
 
 ces ; that is, the probability of the compounc occurrence is equal to the piod- 
 ucts of the partial probabilities. For as each of them, may concur with each 
 of the m„ cases, there will be m,X'« 2 possible cases, which, by the super- 
 vening of m 3 new cases, increase to m,X»ijX?w 3 , and so on. The same 
 reasoning applies to the favorable cases g x , g 2 , g 3 , &c, from whence, by the 
 principles already established, results formula (III.). Let it be required, for 
 example, to draw out of a vase which contains the numbers 1, 2, 3, 4, 5, and 6, 
 first 1* then either 2 or 3, and, finally, 4, 5, or 6, in three drawings ; the prob- 
 ability is expressed by 
 
 1 2 3 1 
 ™_-x 5X4=05. 
 
 If the partial occurrences are equal (that is, repetitions of the same), then 
 
 «>== I — j . Thus, if with each of three dice, G shall be thrown, 
 
 W= Q =2l6- 
 
 IV. Should there be m possible cases, of which .g are favorable and u un- 
 favorable, and of these k-\-r tire to occur, so that k of the favorable, with r of the 
 unfavorable, must come in juxtaposition, then the expression for the probabili- 
 ty of the occurrence of eveiy such order is 
 
 /jA/g__l\ /g— fc+l\ / u \( u—\ \ I u— r+1 \ 
 w — W \^I-[)-\m — k+l) X \m^k) WTT-^Tr'Am— k— r+1/ ( IV ^ 
 
 This depends on (III.), each of the factors in the above value of w ex 
 pressing the partial probability of the single occurrence of a 1st, 2d, . ...kth 
 favorable case, also of a 1st, 2d, ....rth unfavorable case, and the product 
 expressing the probability of these occurring in a certain order. 
 
 EXAMPLE. 
 
 If from 20 tickets, 8 of which are prizes and 12 blanks, 6 are to be drawn, 
 then, in favor of the requisition that exactly two prizes shall be first drawn, or 
 shall occupy any given place in the order, 
 
 77 
 
 W 
 
 ~ \20/ U9/ X \18/ U7/ \16/ \15/ — J 
 
 "3230 
 
 V. Should there be required in the supposition of the last case no particu- 
 lar order for the single cases which occur, the expression becomes 
 
 k + I \m/'"\i7i—k+l/'\m—k/'"\7n—k—r+ir ' ' ' v ' ; 
 Thus it will be found that, if from 30 appointed numbers out of 90, 5 of the 
 whole 90 are to be drawn, so that just 3 of the 30 shall be among those drawn, 
 it being immaterial at which three of the five drawings, the expression for the 
 probability in this case is 
 
 / 5.4.3 \ /30\/29\/28\ /60\ /59\ 20650 
 ~ \l.2.3/ " \907 \89/ \88/ ' \87/ \867 ""126291' 
 
 VI. Should the number of possible cases continue to remain the same, 
 while the other circumstances are as in (V.), the formula would be 
 
 C (g-Y.W. (VI.) 
 
 K+r W/ \Mi/ v ' 
 
251 ALGEBRA 
 
 EXAMPLE. 
 
 The probability of throwing the same lace three times in 7 casts of a die, 
 or one cast of 7 dice, would be expressed l>y 
 
 7.6.5 /l\ :i (5\*_ 21875 
 1.2.3\0V '\6/ : >J36' 
 
 VII. Let the probability be required that of two different occurrences the 
 first, or, if this does not, the second, shall happen ; if the single probability of 
 the lir-~l happening be expressed by w, the probability of its failing wilf be ex- 
 pressed by 1— iv ; this must be combined with the probability of the second 
 happening, according to (III.), giving 
 
 (1 — «'i)w 2 
 for the probability of the second happening, if the first fails : then the com- 
 pound probability required is expressed (II.) by 
 
 W=:W l -\-W a (l—W 1 )=W l +W a —W l .W 2 . 
 
 EXAMPLE. 
 
 Required the probability of throwing with two dice, at the first cast 8, and, 
 if this does not happen, 9 at the second cast. 
 
 w=—* 4-- (l -—)—— J_i- -—™ 
 
 VIII. Above we have considered the absolute probability of the happening 
 of an event ; the relative probability of the happening of two events is ex- 
 pressed by the formula 
 
 — i — ' or — r — • 
 
 W l -\-W 2 M,-|-«!, 
 
 EXAMPLE. 
 
 The relative probability of throwing with two dice rather 7 than 10, is ex- 
 
 . . «>i 6 2 
 
 pressed by 
 
 Wi+w* 6+3~3' 
 IX. When money depends on the happening of an c. . it, the product of 
 the sum risked, multiplied by the expression for the probability of the event 
 on which it depends, is called the mathematical expectation. If there be 
 among m^nii cases, nii favorable for one pan v. and >n : for tho other, the 
 sum risked by the first a„ and by the second ,r, then for tho mathematical 
 expectation of each we have 
 
 Therefore, when r [= r : , it is necessary that -/, : a : =u\ : w : . This principle 
 is important in the subject of annuities and life insurance. For its nppl eation, 
 and that of all tho foregoing theory to which, see De Morgan on Probabilities. 
 
 r\ vm PLES. 
 
 (1) How many binary combinations of oxygen, hydrogen, nitrogen, carbon, 
 
 sulphur, and phosphorus? I low many ternary combinations of the same.' 
 
 (2) llow many combinations of 5 colors among those of the prism, via., red, 
 
 orange, yellow, green, blue, indigo, and violet .' 
 
 19 and 9 can each be thrown with two dice bol La one way, it nn<l 3 each in two 
 ways, to and i in throe ways, r> and 'J in bar ways, G and h in live ways, 7 in six ways. 
 
MKTIIOD OF UNDETERMINED COEFFICIENTS. 255 
 
 (3) What is the probability of throwing with three dice two equal num- 
 bers 1 wilii five dice, three equal ? 
 
 (4) What of throwing with two dice the faces 2, 4, and 6 ? 
 
 (5) What the probability lhat a dollar tossed twice will fall head up onco ? 
 
 (6) Of which is the probability greater, the drawing at three trials from 
 52 cards three cards of different colors, of which there are four, or three face 
 cards, of which there are 12 ? 
 
 (7) What of drawing out of a vase containing 5 white, 6 red, and 7 black 
 balls, in two drawings, 2 red, or else a white and a black ball ? 
 
 (8) What of drawing out of the same vase, in three drawings, 3 of differ- 
 ent colors, or else 2 black and 1 white ? 
 
 (9) What of throwing with four dice 15, or with three dice 12 ? 
 
 METHOD OF UNDETERMINED COEFFICIENTS. 
 209. The method of undetermined coefficients is a method for the expan- 
 sion or development of algebraic functions into infinite series, arranged accord- 
 ing to the ascending powers of one of the quantities considered as a variable.* 
 The principle employed in this method may be stated in the following 
 
 THEOREM. 
 
 If A.ra-fB.r/i-f-C.ry+, &c, =A'.r"' + B'.ni'+C '.t*'+, &c. (1), for all values 
 of x, then must the exponents of a: in the two members be the same, and the co- 
 efficients of the same powers of .r the same. For, dividing (1) by x", we have 
 A + B-rP— » + C.r>'-« + , &c., r=A'.ra-a4-B'.r/3'-^+C'.r>'-«-|-, &c. (2) 
 
 Since x may have any value, make it zero; the first member thus reduces 
 to A, while tl« second becomes zero, unless we suppose a equai to some one 
 
 of the exponents a', /?', y', Suppose it to be a'. Then we have « = a\ 
 
 and .-. A = A'. Suppressing the equal terms A and A'x*'—<* from the two 
 members of (2), and dividing it by xP— ■<, it becomes 
 
 B-|-C.r>'-^+, &c, =r.'.r,.;-,3 + C.r>'-.3+, &C 
 
 Making, again, ,r=0, the first member reduces to B, and the second to zero, 
 which is absurd, unless we make ,6' equal to some one of the exponents of x, 
 say /?', in tne second member, and then 13 =B'. Proceeding in this way, the 
 exponents of .r, and the coefficients of the same powers of x in the one mem- 
 ber, may be proved equal to those in the other. 
 
 The above theorem may be expressed in a modified form ; thus, if all the 
 terms of (1) be transposed to the first member, it becomes, collecting the equal 
 powers of x, a and a', /? and j3', &c, 
 
 (A— A').r« + (B — B').r£+(C — C').rr+, &c, =0; 
 from which, since A = A', B=B', &c, we perceive that when a function of 
 x is equal to zero for all values of .r, the coefficients of the different powers of 
 x are equal to zero separately. 
 
 EXAMPLES. 
 
 (1) Expand the fraction — ; — : into an infinite series. 
 
 v ' * 1 — 2.f+.r a 
 
 Assume 1 _„ ; =A4-B.r-f-C.r- + P.i.-"-f-E.^+ . . . . , 
 
 * A variable quantity is one which is either entirely indeterminate, so that it may have 
 any value at pleasure, or one which varies in conformity with certain conditions imposed. 
 
ALGEBRA. 
 
 in which some of the coeffic • B, C, &<■-, a sei >, and thus certain 
 
 powers of x be wanting; then, mull y ' — ;,J r' • w ' : hwi 
 
 i=A+ Bx-f. Cj + I-* -h i. H 
 
 — .-A/ — >B ' —2D — .... 
 + A/ + J^. ; + Cz*4-.... 
 Hence, by the preceding theorem, we I 
 
 A = l .-. A= ... =1 
 
 B— 2A=0 B=;\ =2 
 
 C— 2B+A=sO C=2B— A=3 
 
 D— 2C-fBs=0 Ds=2C—B=4 
 
 E— 2D+C=0 E=2D— C=5 
 
 &c. &c. 
 
 Therefore x _ .-, x ■ ,., = 1 + 2x + 3x'- + 4 /■ -f ', /." + W -f 
 
 Tlie equality of a function to a series is hypothetical ; and after A, B, C 
 have been found, the result must be carefully examined. If we put the func- 
 tion ; =A-r-Bz-j-, &c, it gives the absurdity — 1=0. We mu 
 
 Az- l + Bx°+Cr+D.r ! -|-, &c. The method of indeterminate coeffi- 
 
 3x— X s 
 
 cients is to be avoided where other methods will apply. 
 
 (2) Extract the square root of 1-j-x. * 
 
 Assume V l+x=A + Bx -4-C./. : +Dx' ) -f- ..., and square both sides; 
 .•.l+x=A 2 +ABx+AC< +AI> +AEx»+.. 
 + ABx+B ; x- 4-BCx»+BDx«+.. 
 
 + ACx- + BCx-+C : x< +.. 
 
 4-ADx' ; +BDr*4-.. 
 
 + AEx«+.. 
 
 Hence, equating the coefficients of the like powers of x, we have 
 
 A =1 .-.A=2 1 
 
 1 1 1 
 
 2AB=1 B= -^= -= § 
 
 2AC + B—0 C = -^-=- — =-- 
 
 \'.r i i 
 
 2AD4-2BC=0 D=— r-= — = — 
 
 A J - 16 
 
 2BD+I 1 <; 1 l > 5 
 
 2AE + 2BD + C-0 E = — £- = -, j - + - \ =--, 
 
 Ate. 
 
 Therefore \Zl + .< — i(l + -A-r— Jj--+ ,' - +....)■ 
 
 3x— ."> 
 (3) Decompose -= — — — t—tt into two I simple binomial d«*- 
 
 x ' ' x a — lox-|-10 
 
 nominators. 
 
 By quadratics we find X* — 13x+40 = (r — 5)(x— ace we may assume 
 
 3x— 5 A _B A(x— 8)+B - (A+B)x— 8A— 5B 
 
 x*_i;jx+40 = x— 5+x— 8 — (x— 5)(x- x*— 13x-f40 ' 
 
 .-. ::,— .-) = (A + B)r — 
 and by thn principle of undetermined coefficients we have 
 
 A + B = 3, and 8A+5B = 
 

 
 
 : 
 
 - 
 
 ' 
 
 I- 
 
 
 
 
 — 
 
 
 •"I - — • " 
 
 1 
 
 
 . 
 
 5 ;"_ 
 
 
 
 
 . 
 
 
 
258 ALGJSBHA 
 
 a — bx 
 (8) Exrand — - — - to four terms. 
 a-\-cx 
 
 Ans. l-(b + c)l+c{b+c£-c"ib+ r £ + . 
 
 x-}-2 
 (9) Resolve — ; ■ into partial fractions. 
 
 a J_ , _3_ 2 
 
 Ans '2(x+l) + 2|x-l)~x 
 
 (10) Resolve -r- . ,,., , — . into partial fractious. 
 
 ' .?.' J (1 — x)-(l-f-r) 
 
 112 1 
 
 X* X* X '2(1— x)* ' 4(1— .r, 4^1+x^ 
 
 (11) Expand x > +2ax+ai to four terms 
 
 2a 3a* 4a 3 
 Ans. 1— -| — r— 3-+ 
 
 XX" X s ' 
 
 (12) Resolve — — - into partial fractions. 
 
 Ads. 
 
 1 1 1 
 
 4(x—l)~4(x+l) — 2(^+1)' 
 
 LOGARITHMS. 
 
 210. Logarithms are artificial numbers adapted to natural numbers, in 
 jrder to facilitate numerical calculations ; and we shall now proceed to explaiu 
 the theory of these numbers, and illustrate the principles upon which their 
 properties depend. 
 
 Definition. — In a system of logarithms, all numbers are consl i the 
 
 powers of some one number, arbitrarily assumed, which is called the base of 
 Oie system, and the exponent of that power of the base which is equal to any 
 given number is called the Logarithm of that number. 
 • Thus, if a be the base of a system of logarithms, N any number, and x such 
 that 
 
 Naca x , 
 then X is called the logarithm of N, in \\ e Bystem whose base is a. 
 
 The base of the common system of logarithms (called, from their inventor, 
 "Briggs's Logarithms") is the number 10. Hence, since 
 
 (10)°= 1, is the logarithm of 1 in this system, 
 (!())'= 10, 1 is the logarithm of 10 in this system, 
 (10)-= 100, 2 is the logarithm of 100 in this system, 
 (10) :, = 1000, .". is the logarithm of 1000 in this system, 
 
 (10) 4 ±10000, I is the logarithm of 10000 in tins system, 
 flee. = dec. iVc 
 
 211. In crder to havo the numbers corresponding to the logarithms L, j or 
 0.5, I or 0.»5, &c, it is necessary to extract the Bquare, 4th, and so on, root 
 of 10, or tt ttttct the square root successively, as exhibited in die following 
 tuble : 
 
LOGARITHMS 
 
 259 
 
 Nnm.-er of times that llie 
 square root ia extracted 
 successively. 
 
 Exponent* 
 Numbers. or 
 
 Logarithms. 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 
 10,000 0000 
 3,162 2777 
 1,778 2794 
 1,333 5214 
 1,154 7819 
 1,074 6078 
 1,036 6329 
 
 1,000 0000 
 0,500 0000 
 0,250 0000 
 0,125 0000 • 
 0,062 5000 
 0,031 2500 
 H.Ol.-, U250 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1,018 1517 
 1,009 0350 
 1,004 5073 
 1,002 2511 
 1,001 1249 
 1,000 5623 
 
 0,007 8125 
 0,003 9062 
 0,001 9531 
 0,000 9765 
 0,000 4882 
 0,000 2441 
 
 13 
 
 14 
 15 
 16 
 
 17 
 
 18 
 
 1,000 2811 
 1,000 1405 
 1,000 0702 
 1,000 0351 
 1.000 0175 
 1,000 0087 
 
 0,000 1220 
 0,000 0610 
 0,000 0305 
 0,000 0152 
 0,000 0076 
 0,000 0038 
 
 19 
 20 
 21 
 22 
 23 
 24 
 
 1,000 0043 
 1,000 0021 
 1,000 0010 
 1,000 0005 
 ■1,000 0002 
 1,000 0001 
 
 0,000 0019 
 0,000 0009 
 0,000 0004 
 0,000 0002 
 0,000 0001 
 0.000 oooo 
 
 By means of the above table, to calculate the logarithm of any number (A) 
 between 1 and 10 accurately to 5 places of decimals, take out from the second 
 column the nearest number to A, but less, and divide A by this. Take out, 
 again, the next logs number than the quotient B, as a divisor for B, and so on 
 until the last quotient contains only millionths ; the logarithm sought is the 
 sum of all the exponents or logarithms in the third column corresponding to 
 the divisors used from the second. For, calling these exponents a, (3, y, 6 
 we have 
 
 A =B; i c c_ D D 
 
 10 a 10" 10 y 10 d 
 
 .-. A=10 a B = 10 a X 10^C = 10 a X 10^ X 10 y D=10 a . 10^. 10 y . 10* 5 . . . 
 
 ...A = lO a +0+ } + 5 -. 
 
 Any exponent beyond 6 being added to the others would not affect the 
 millionth place, or fifth decimal. Q. E. D. 
 
 Now, inasmuch as all numbers lying between the 1st, 2d, 3d, &c, powers 
 of 10 must have broken numbers for logarithms, these numbers will be of the 
 
 . k a k_ 
 
 form 10 m r=10 .10 m ; hence the calculation of their logarithms will in every 
 case depend on the calculation of a fractional logarithm such as has been just 
 exhibited. 
 
 A table of logarithms is a table containing all numbers from 1 up to 10000 
 or 100000, or some high number, with their corresponding logarithms. 
 
 These tables are made with certain abbreviations and conveniences, which 
 we shall presently explain. 
 
 From the scheme of numbers in (210) it appears, that in the common sys- 
 tem the logarithm of every number between 1 and 10 is some number between 
 
260 ALGEBRA. 
 
 and 1, i. e., is a fraction. The logarithm of every .number between 10 and 
 100 is some number between 1 and 2, i. c., is 1 plus a fraction. The logarithm 
 of every number between 100 and 1000 is some number betwoen 2 and 3, i- < • , 
 is 2 plus a fraction, and so on. The whole Dumber, or integral part of the 
 logarithm, is called the index, or, more commonly, the characteristic. 
 
 212. In the common tables of logarithms the fractional part alone of tht> 
 logarithm is registered, and from what has been said above, the rule usually 
 given for finding the characteristic, or index, will be readily understood, vi/. : 
 The index of the logarithm of any number greater Otan unity is equal to ont 
 less than tfie number of integral figures in Oie given number; for if the num 
 ber be between 10 and 100, it will contain two integral figures ; if between 100 
 and 1000, it will contain three, and so on. Thus, in searching for the logarithm 
 of such a number as 2970, we find in the tables opposite to 2970 the number 
 4727564; but since 2970 is a number between 1000 and 10000, its logarithm 
 must be some number between 3 and 4, i. e., must be 3 plus a fraction ; the 
 fractional part is the number 47275G4, which we have found in the tables ; 
 prefixing to this the index 3, and interposing a decimal point, we have 3.47275G4, 
 the logarithm of 2970. 
 
 We must not, however, suppose that the number 3.4727564 is the exact 
 logarithm of 2970, or that 
 
 2970 = (10) 3 - , " 75 <* 
 
 accurately. The above is only an approximate value of the logarithm of 2970 , 
 we can obtain the exact logarithms of veiy few numbers; but, taking a sufficient 
 number of decimals, we can approach as nearly as we please to the true 
 logarithms. 
 
 213. It has been shown that in Briggs's system the logarithm of 1 is ; con- 
 sequently, if we wish to extend the application of logarithms to fractions, we 
 must establish a convention by which the logarithms of numbers less than 1 
 may be represented by numbers less than zero, i. <?., by neg /mbers. 
 
 Extending, therefore, the above principles to negative exponents, since 
 
 or (10) _1 =0.1, — 1 is the logarithm of .1 in this system, 
 
 10 
 
 1 
 100 
 
 1 
 
 1000 
 
 1 
 
 or (10)~ 5 =0.01, — 2 is the logarithm of .01 in this system, 
 or (10) _3 =0.001, —3 is the logarithm of .001 in this system, 
 or (10) _4 =0.0001, —4 is the logarithm of .0001 in this system, 
 
 10000 
 
 &c &c. 
 
 It appears, then, from this convention, that the logarithm of every number 
 between 1 and .1 is some number between and — 1 ; the logarithm of every 
 number between .1 and .01 is some number between — l and — 2; the 
 logarithm of every number between .01 and .001 is some number between 
 — '.' and — .'!, and so on. 
 
 From this will bo understood tho rulo given in books of tallies for finding 
 the characteristic, or index, of tho logarithm of n decimal fraction, viz. : The in- 
 form decimal fraction is a negative number, equal to unity, added to tltf 
 number of zeros immediately following the decimal point. Thus, in searching 
 for a logarithm of tin- number such as .00462, we find in the tables opposite to 
 
LOGARITHM*. 26 J 
 
 462 the number 6646420; but since .00462 is a number between .001 and .01 
 its logarithm must be some number between — 3 and — 2, i. c, must be — 3 
 ^lus a fraction; the fractional part is the number 6646420, which we have 
 found in the tables; therefore — 3-f-'6G40420 is the logarithm of .00462. It 
 is customary to write the sign — over the characteristic to show that it affecta 
 that alone, and not the decimal part of the logarithm, which is positive ; thus, 
 3.6646420. 
 
 GENERAL PROPERTIES OF LOGARITHMS. 
 
 214. Let N and N' be any two numbers, x and x' their respective logarithms* 
 a the base of the system. Then, by definition, 
 
 N =a* (1) 
 
 N'=a"' (2) 
 
 I. Multiply equations (1) and (2) together, 
 
 = a x + x ' 
 .-. by definition, x-\-x' is the logarithm of NN' ; that is to say, 
 The logarithm of the product of two or more factors is equal to the sum of the 
 logarithms of those factors. 
 
 II. Divide equation (1) by (2). 
 
 N a* 
 N' = a T ' 
 
 -x> 
 
 =a x 
 .-. by definition, x — x' is the logarithm of ^r, ; that is to say, 
 
 The logarithm of a fraction, it of the quotient of two numbers, is equal to the 
 logarithm of the numerator minus the logarithm of the denominator. 
 
 III. Raise both members of equation (1) to the nth power. 
 
 N n =a"". 
 .-. by definition, nx is the logarithm of N n ; that is to say, 
 The logarithm of any power of a given number is equal to the logarithm 
 of the number multiplied by the exponent of the power. 
 
 IV. Extract the n til root of both members of equation (1). 
 
 1 X 
 
 N 7l =a 7) . 
 
 x I 
 
 .-. by definition, - is the logarithm of N" ; that is to say, 
 
 The logarithm of any root of a given number is equal to the logarithm of the 
 number divided by the index of the root. 
 
 Combining the last two cases, we shall find 
 
 m n\x 
 
 N~°=ef", 
 
 mx ™ 
 
 whence — - is the logarithm of N". 
 
 n ° 
 
 It is of the highest importance to the student to make himself familiar with 
 the application of the above principles to algebraic calculations. The following 
 examples will afford a useful exercise : 
 
 (1) Log. {a, b, :,d )= log. a-\- log. b+ log. c+ log. d.... 
 
 (2) Log. (^) = log. a+ log. 5+ log. c— log. d— log. e. 
 
26-2 ALGEBRA. 
 
 (3) Log. (a' n b"cv )=m log. a-f n log. b+p lug. c... 
 
 (4) Log. \-^r) =m log. a + n log. £>— J>. log. c. 
 
 (5) Log. (a 2 — :r 2 )=log (a+ar)x(a— x)= log. («/ + .r) + lo g- («*-*)• 
 1 . 1 
 
 (6) Log. V"-'— i'- = o lo S- ("+0 + .J ^g- ("—■'•)• 
 
 1 3 15 
 
 (7) Log. (a 8 V<^)= lo g- fl3 +7 lo g- " 3 = 3 lo S- a + 4 lo §- a = 7 lo S- a - 
 
 m m 
 
 (8) Log. V(a»-ar>)»=- log. (a-*)+- log. (d»+ax+i») 
 
 =— {log. (a—a:)4- log. (a+a:+z)+ log. (a+z—z) J 
 
 where z-=ax. 
 
 (9) Log. V« Q +^-=oi lo g- ( a + x + : )+ lo §- ( a + x — =)}» where ; 2 = 2a.-> 
 
 (10) Log- ^!{_ x) !=2l l0 S- («-*)— 8 '°g- («+*)!• 
 
 TABLES OF LOGARITHMS. 
 
 m he principal French tables are those of 31. Callet, an American edition ot 
 which has been made by the late Mr. Hasler. The first of these table 9 
 marked Chiliade I., occupying only five pages, contains the series of numbers 
 from 1 up to 1200, with their logarithms Expressed to eight places of decimals, 
 the numbers Being in the column marked N, and their logarithms in the column 
 marked Log.* The second table, which is of far greater bulk, exhibits the 
 logarithms of all entire numbers from 1020 up to 10800. The numbers are in 
 the column entitled N, and their logarithms in the following column, marked 0. 
 The characteristics of the logarithms are not written in the tables, since they 
 may be known without, being always one less than the Dumber of dibits of 
 which the number to which the logarithm belongs is composed. The logarithms 
 of numbers containing one figure more than those in the column N, are found 
 by means of the columns marked at top 1, 2, 3, ... 0. Thus, to find the 
 logarithm of 27796, seek in the column N the number 2779; run along the 
 horizontal line which contains this number to the column marked i> ; yon find 
 there the last four figures of the logarithm sought : the first three figures of it 
 are found in the column marked 0, t>> the left of the period, on the Bame 
 horizontal line, or a little above. You obtain thus, after prefixing the proper 
 characteristic, 
 
 log. 27796=1.11.;" 
 
 It will be seen, by inspecting the tables, that the diiVerences of the consecu- 
 tive logarithms is constantly the same for b considerable number of them, and 
 
 as the diH'eivntes of the consecutive numbers is also constant, it follows that 
 
 * This table also miliums an arrangi intent tor reducing uiiiiutfs and aeoooda to aacoada 
 without the trouble of multiplying by 60. Tims, on the fourth page, we find i-' in tin 
 of tlio columns marked I"--, and against 30, in the Brat column marked ". we Bad 740, 
 which is the Dumber of seconds, in i~' 80". By this arrangement we find readily tho 
 I tlim of the seconds in aj n Dumber of minutei and seconds, which i* 
 
 nt in astronomical calculations. It is evident that theae numbers might be J 
 
 as degrees and minutes, or hours and minutea, as well as annates and seconds 
 
LOGARITHMS. 263 
 
 the differences of the logarithms are proportional to the differences of the 
 numbers. Suppose, then, that the logarithm of 14518469 were required. 
 
 From the tables we find, as before, neglecting for the present the charac- 
 teristic (see a page of the tables of Callet at the end of this volume), 
 
 log. 14518 = 1619008. 
 
 This is also the logarithm of 14518000, which differs from the logarithm of 
 the next number 14519, or 14519000, viz., 1619367 by 299, while the num- 
 bers themselves differ by 1000. But the number 14518000 differs from the 
 given number 14518469 by 469, the last three figures not yet used ; hence 
 the proportion 
 
 Dif. Nos. Dif.Logs. DiCNos. Dif.ofLogs. 
 
 1000 : 299 :: 469 : x=141, 
 
 which result, added to 1619068, gives 7.1619209 for the logarithm required, 7 
 
 being the proper characteristic for the logarithm of a number consisting of 
 
 eight figures. 
 
 299 
 The proportion is solved by multiplying the difference 469 by ttjtjt:, or by 
 
 2 9 9 
 
 — 4- -J- . Now, by inspecting the last column of the page, this differ 
 
 10^100^1000 » J f s it.' 
 
 ence, 299, will be found ready calculated, and its product as nearly as it can be 
 
 12 3 
 expressed injiwo or three figures by — , — , — , &c, or .1, .2, .3, &c, the 
 
 multiplier being in the left hand and the product in the right hand of the two 
 small columns of figures under the difference, 299. These multipliers may be 
 regarded as hundredths or thousandths, only giving the products their proper 
 place. With this explanation, the following calculation will be understood : 
 
 Log. 14518 1619068 
 
 0.4 120 
 
 0.06 18* 
 
 0.009 3 
 
 Log. 14518469 7.1619209 
 
 215. To find the number corresponding to a given logarithm, say 1619209, 
 look in the column marked for the nearest less logarithm, and take the cor- 
 responding number, which is 1451. Run the eye along the horizontal line till 
 the number most nearly approaching 9209, forming the last four figures of the 
 given logarithm, is found. This is 9068, which is found in column 8. Sub- 
 tract this from 9209, and the difference is 141. Find in the right hand of the 
 two columns of small figures marked dif. et p., or simply dif., at the top of the 
 page, the nearest less number than 141 ; this is 120, which answers to 4 in 
 the left hand. The difference between 120 and 141 is 21. Multiply 21 by 
 10, and seek, as before, in the small column, the number nearest 210 ; this is 
 209, which answers to 7. The calculation is below. 
 Log. .r=1619209 
 
 For 1619068 14518 
 
 First remainder, 141 04 
 
 Second remainder, 21 007 
 
 a-= 1451347. fc 
 The numbers 4 and 7 thus found may be simply annexed to 14518. 
 
 * The number in the table is 179 ; but, us the 9 is rejected, the 7 is increased by 1, sinew 
 179 is nearer 180 than 170. 
 
264 ALGEBRA. 
 
 If the characteristic of the logarithm had been 
 
 6, the number would have been 1451847 ; 
 5, the number would have been 145184.7 ; 
 4, the number would have been 14518.47 ; 
 1, the number would have been 1 1.51847; 
 
 0, the number would have been 1.451847; 
 
 1, the number would have been .1451847 ; 
 
 2, the number would have been .01451617. 
 
 This table contains in tho first three columns an arrangement for reducing 
 any number of degrees, minutes, and seconds, or hours, minutes, and seconds, 
 to seconds, which is particularly useful in astronomical calculations, where the 
 logarithm of the number of seconds in a given number of degrees, minutes, and 
 seconds is frequently required. 
 
 EXAMPLE I. 
 
 Reduce 0° or b 24' 57" to seconds. In the table (see last page), at the 
 head of the first column, find 0°, and immediately under it 24' ; descending 
 this column to 55", near the bottom, and opposite 57", which is understood to 
 be two numbers below, is found 1407, the number of seconds required. 
 
 If the degrees or hours exceed 3, the proceeding is different. 
 
 EXAMPLE II. 
 
 To reduce 4° or 4 1 ' 2' 39" to seconds. Find 4° 0' at the head of the 
 second column, and below, in this same column, 2' 30", to which corresponds, 
 in the third column, 1455. Thus, 4° 2' 30" = 14550" .-.4° 2' 39" = 14559" 
 
 EXAMPLES OF THE APPLICATION OF LOGARITHMS. 
 
 (1) To find the value to within 0.01 of the expression 
 
 _ 7340X3549 
 r= G81.8x 593.1' 
 By the properties of logarithms, 
 
 log. x= log. 7340-f log. 3549— log. 681.8— log. 593.1. 
 The following is the calculation : 
 
 log. 7340=3.8656961 
 
 log. 3549=3.5501060 
 
 sum =7.4158021 
 
 log. 681.8=2.8336570 
 
 log. 593.1 =2.7 731279 
 
 sum = .j.b'Uu'7649 
 
 First sum, =7.4158021 
 Second sum, =5.60678 49 
 Dift*. or log. .r= 1.8090172 
 
 216. The arithmetical complement of a logarithm is what remains after the 
 logarithm is subtracted from 10. Thus, the arithmetical complement of the 
 logarithm 2.7190826 is 10— 2.7190826=7.280917 1. which ifl obtained by be- 
 ginning on the right and subtracting each figure (carrying 1 to all except the 
 Bret) from 10, or beginning on the left and subtracting each figure of tho 
 logarithm from 9, except the last, which is subtracted from LO. 
 
 217. Thojaperation of subtraction of logarithms can be replaced by addition, 
 if wo use the arithmetic complement ; for n'. to a given logarithm, log. <;, wo 
 add the arithmetical complement of another logarithm, such as 10— lug. 6 
 we have 
 
LOGARITHMS. 265 
 
 log. a -f- 10 — log. b, 
 from which, rejecting 10, the result is 
 
 log. a— log., 6, 
 the same as would be obtained by simply subtracting the second logarithm 
 from the first. 
 
 We have then the following rule for operating with arithmetical comple- 
 ments : Add the arithmetical complements of the logarithms of the divisors and 
 the logarithms of the multipliers of a formula together, rejecting 10 from the 
 sum for every arithmetical complement employed. 
 
 The above example would be wrought by this rule as follows : 
 
 log. 7340 = 3.8656961 
 
 log. 3549=3.5501060 
 
 ar. comp. log. 681.8=7.1663430 
 
 ar. comp. log. 593.1=7. 2268721 
 
 sum rejecting 20=T78090172=log. x, .-. a:=64.42. 
 
 We thus obtain the same result as by the other method. The number cor- 
 responding need be taken from the tables only to four figures, because, the 
 characteristic being 1, the entire part of the number will contain but two 
 places, which will leave two places for the decimal part, as required, since the 
 value of a: was to be obtained to within 0.01. 
 
 (2) To find the value within 0.00001 of the quotient. 
 
 (•v/146298) 4 
 x=- 
 
 (V988789) 5 
 By the rules, 
 
 log. ar= flog. 146298— flog. 988789, 
 and the calculation will be as follows : 
 
 flog. 
 
 146298. 
 
 1 log- 
 
 988789. 
 
 log. 14629 
 
 
 0.1652146 
 
 log. 98878 
 
 . . . 0.9950997 
 
 for 0.8 . 
 
 
 238 
 5.1652384 
 
 for 0.9 . , 
 
 ... 40 
 
 
 
 
 log. 146298 . 
 
 
 
 
 
 20.6609536 
 
 
 . . . 29.9755185 
 
 
 
 4.1321907 
 
 
 . . . 4.9959197 
 
 
 
 |log. 146298 = 4.1321907 
 
 
 
 ar. 
 
 comp. flog. 988789=5.0040803 
 sum —10, or log. x=1.1362710 
 
 
 
 
 .-. x=0 
 
 .13686. 
 
 
 /13 
 (3) Required V/~ by means of logarithms. 
 
 13 log. 1.1139434 
 27 log. 1.4313638 
 
 11)1.6825796 
 
 — 
 
 V/— =.9357149 log. 1.9711436 
 
 The division by 11 is performed by adding — 10 to the negati'e part of the 
 logarithm and +10 to the positive. 
 
 The logarithm to be divided is viewed as if written thus : 
 
 — 114-10.66:25796. 
 
266 ALGEBRA. 
 
 EXERCISES IK LOGARITHMS. 
 
 (4) Calculate the logarithm of 8 from the table on page 259. 
 
 (5) Also of 7, 70, 700, 7000, 70000. 
 
 (6) Also of 356, 35G00, 3560000 
 
 (7) From the tables find the logarithms of 314, 3.721, 41.2. 
 
 (8) Also of 7315, 8416, 91.75, 34760, 1708000. 
 
 (9) Find the numbers the logarithms of which are 0.13130, 4.5651 ! 
 
 (10) Also those the logarithms of which are 3.6520528, 7.4891114. 
 
 (11) Those the logarithms of which are 4.49010, 0.GG200, 5.72403. 
 
 (12) Find by proportional parts the logarithms of 314761, 440736, 37U2C-40U, 
 2111768. 
 
 (13) Also of 22.3345, 137.2014, 46.27835. 
 
 (14) Of .75, .341, .7391, .0347, .000536, .0000083. 
 
 5 3 6 £ _7_ 
 
 (15) Of -, g, —, i3> 4Q- 
 
 (16) Find the logarithm of the product of 9.734 and 5.639. 
 
 (17) Also of 35.98 X 7.433 X 6.543 X 29.78. 
 
 (18) Also of 22.74 X 31.201 X 0.0067 X 0.9298. 
 
 (19) Divide 3758000 by 4986 by means of logarithms. 
 
 (20) Also 16.87:0.07658 and 1.687:7658. 
 
 (21) Also 14.307:30415, 761.23:0.01871, 3.16:0.942. 
 
 7 125 31 734 1 
 
 (22) Find the logarithm of—, — , — , — , — . 
 
 (23) Find the power (5486) 1 by means of logarithms. 
 
 (24) Also the powers (37.49) 9 , (106.4) 5 , (0.032) 7 , (7.0034) 8 . 
 
 /i\!b /sy /iy /3\« /i27\ 13 
 
 (25) Also y , {,) , y , (^ , [ m ) . 
 
 I 1\6 / 1\8 / 20\ 9 / 1 \ 3 
 
 (26) Also (3+3) , (4--) , (7+-J , (lOO-—; . 
 
 (27) Find the cube root by logarithms of 1728000. 
 
 (28) Also V34-782, ^23990, VC2S.73. 
 
 11337 /9466 /120300 Kt 
 
 (29) Also r^— , »]— , u/-^, V0.1563, ^0.0082. 
 
 (30) Also y?; 1 -!^, ^45390000, V800.9. 
 
 (31) Also V(1347) 8 , 7(70.44)", V(8.664)». 
 
 J/1722\ 5 //0.006\ ;i j/72.93\ 7 
 
 < 32 ) Also vU • V Wi ' vfeos/ ■ 
 
 (33) Find by means of logarithms, using the arithmetical complement, the 
 
 27630X2678X5428 
 valueof 36940 X 5302X7013" 
 
 207.3 X 50.66 X 38.09 X 2713 X 0-098 
 
 (34) Also of 344 x o.763 x 0.4 X 6984 X7034.JJ ' 
 
 If -85762X0.00853 
 
 (35) Also of y 7.58913X86.24 ' 
 
 GAUSS LOGARITHM-*. 
 
 218. The common logarithms, or logarithms of Briggs, are applicabls only to 
 the operations of multiplication, division, formation ofpowera, or extraction of 
 roots, and do not apply when the required operation is that of addition or nib- 
 
LOGARITHMS. liC7 
 
 traction, indicated in formulas by the quantities to be operated upon being con- 
 nected by the signs -\- and — . 
 
 A system of logarithms has, however, been invented by Gauss,* designed 
 exclusively for sums and differences. The arrangement of these tables, which 
 contain three columns, marked A, B, C, is founded upon tho following simple 
 considerations. 
 
 We have for tho form of a sum p-{-q, and of a difference p — q, tho follow- 
 ing identities : 
 
 *+»(*£*) « 
 
 p- ( i=p-\~r q ) ( 2 ) 
 
 ••• log- (p+q)= iog..p+ log. (^) (3) 
 
 and log. (p—q)= \og.p— log. (j^r) 
 
 The logarithms of the sum p-\-q and the difference^? — q appear, therefore, 
 in these formulas, equal to the sum or difference of two logarithms, the first 
 of which is to bo considered as directly given, but the second of which must 
 be found by the Gauss tables. They contain, 
 
 T. In the column A logarithms of numbers of the form (-1, increasing from 
 0.000 to 5.000. 
 
 II. In column B logarithms of numbei's of the form ( J, decreasing 
 
 from 0.30103 to 0.00000. 
 
 p-\-q 
 
 III. In column C logarithms of numbers of the form , increasing from 
 
 0.30103 to 5.00000. 
 
 Now, therefore, inasmuch as log. (~)= log. p — log. q, by the tables of 
 
 common logarithms, the first thing to bo done is to take the difference of the 
 common logarithms of p and q, enter with this column A in the Gauss loga- 
 rithms, and take out the corresponding number from column B. The addition 
 of this number to logarithm p will give, according to (3), the logarithm sought 
 ofp+q. 
 
 In order to find the logarithm of the difference^ — q, by means of the loga- 
 rithms of p and q, two cases must be considered : 
 
 /; 
 1°. Where -<2 .•. log. p — log. ^<0. 30103, it is only necessaiy to enter 
 
 with this difference column B, and to subtract tho adjoining logarithm of 
 column C from logarithm p. For, corresponding to the logarithms of numbers 
 
 of the form (— 1 in B, C contains the logarithms of those of the form ( j. 
 
 - P 
 2°. If — >2 .-. log. p — log. 5>0. 30103, and, therefore, is contained in 
 
 the column C ; subtract the corresponding logarithm in column B from loga- 
 * They arc found in the latest edition of the tables of Vega, and those edited by Kohler. 
 
268 ALGEBRA. 
 
 P 
 rithm p ; because, if the numbers in C are considered =— , the con esponding 
 
 P 
 
 numbers in B are 
 
 p — q 
 
 The existence of the foregoing relations between B and C is easily per- 
 ceived if we substitute in II. and III. the value p — 7 for p, and afterward q 
 for p — q. 
 
 EXAMPLES. 
 
 (1) Let log. p=8.24502 and log. £ = 2.74194, to find log. (p + q). We 
 enter column A with the log. p — log. 5 = 0.50308, and the corresponding log. 
 in column B = 0. 11861, .-. 
 
 log._p+B=3.24502-L.0.11801 = 3.36363= log. 2310. 
 
 (2) From log. p=3. 32675 and log. 7=2.09482, to determine log. (p—q)- 
 Find by means of proportional parts for the value of log. p — log. q in column 
 B the corresponding log. in C =0.38325; consequently. 
 
 log. p_C=3.32675— 0.38325=2.94350= log. 878. 
 
 (3) From log. p=2. 64207 and log. 7 = 1.87640 the log. of (p—q) is found 
 by subtracting from the nearest value of log. p — log. 7=0.76567, in column 
 C, the corresponding log. from B = 0. 08171. Thus, 
 
 log. j>— B=2.64207— 0.08171=2.56036= log. 363.4. 
 The Gauss logarithms would be applicable in the solution of the exponentials 
 on page 269. 
 
 (4) Find by the Gauss logarithms the log. of 3/2004- ^100- 
 
 (5) Also the log. of [(0.7345) 3 -f (0.2349) 3 ]. 
 
 (6) Also the log. of the difference ( V36— ^27). 
 
 (7) Also of {(1.237)"— (0.9864) 15 }. 
 
 219. Let us resume the equation 
 
 N=a*. « 
 
 1°. If rt>l, making a:=0, we have N=l ; the hypothesis x=\ gives 
 N=a. As x increases from up to 1, and from 1 up to infinity, N will in- 
 crease from 1 up to a, and from a up to infinity ; so that .r being supposed to 
 pass through all intermediate values, according to the law of continuity, N in- 
 creases also, but with much greater rapidity. If we attribute negative values 
 
 to x, we have N=a _x , or N =— . Here, as x increases, N diminishes, so 
 
 that x being supposed to increase negatively, N will decrease from 1 toward 
 0, tlie hypothesis z=co gives N=0 ; i. c, the logarithm of zero is an infinite 
 negative quantity. 
 
 2°. If a<l, put a — j, where Z>>1, and we shall then have N=i-, or 
 
 N=l', according as we attribute positive or negative values to x. We hero 
 arrive at tho same conclusion as in the former case, with this difference, that 
 when x is positive N <0 . and when x is negat ive N ~> 1 . 
 
 3°. If a = l, then N = 1, whatever may be the value of.r. 
 
 From this it appears that, 
 
 I. //; every system of logarithms the logarithm of I is 0, and the logarithm 
 oftlte base is 1. 
 
LOGARITHMS. 269 
 
 II. If the base be >1, the logarithms of numbers >1 are positriee, and the 
 logarithms of numbers <1 are negative. The contrary takes place if the base 
 be <1. 
 
 III. The base being fixed, any number has only one real logarithm ; but the 
 same number has manifestly a different logarithm for each value of the base, so 
 that every number has an infinite number of real logarithms. Thus, since 
 9 2 =81 and 3 4 =81, 2 and 4 are the logarithms of the same number 81, accord- 
 ing as the base is 9 or 3. 
 
 IV. Negative numbers have no real logarithms ; for, attributing to x all 
 values from — co up to -\-cd , we find that the corresponding values ofN are 
 positive numbers only, from up to -f-ao . 
 
 520. In order to solve the equation 
 
 c=a x , 
 where c and a are given, and where x is unknown, wo equate the logarithms 
 of the two members, which gives us 
 
 log. c=x log. a. 
 Whence 
 
 x= l ° g ' C 
 " log. a 
 
 To determine the value of x in the equation 
 
 ' Aa*+Btt*- b +Ca I - c + =P, 
 
 we "have * 
 
 ^ A +l +£ + >= P ' 
 
 or 
 
 Q«* =P, 
 
 substituting Q for the term in the' parenthesis. 
 
 log. P- log. Q 
 
 
 log. a 
 
 If we have an equation a L =zb, where z depends upon an unknown quantity, 
 x, and we have 
 
 z=A.r u 4-Bx n - 1 4- 
 
 Since z= r— — =K some known number, the problem depends upon the solu- 
 tion of the equation of the n th degree 
 
 KrrA^ + B^^-f 
 
 For example, let 
 
 Hence 
 
 '© 
 
 :2— 5*+4 
 
 = 9. 
 
 /2\ 9 
 
 (. r *_5:r+4) log. [-) — log. - 
 
 ... x"-— 5x+4 =—2;* 
 
 an equation of the second degree, from which we find z=2, x=3. 
 
 (3\« 9 3 9 
 
 o) == 7 •'• 2 '°a- 7> r — 1°S- "J an( l log 
 
 3 , 2 
 
 2~ ° 3" 
 
27U ALGEBRA. 
 
 To find the value of X from tlio equation > 
 
 a 
 
 b" » ==c mjyi-p 
 
 1 aking the logarithms of each member, 
 
 ("— -) Io g- b=mx\og. c+(x- p) log./, 
 or 
 
 (ro log. c+ log./).i- 2 — (n log. b+p log. f)x+a log. 6=0, 
 a quadratic equation, from which the value of x may be determined. 
 In like manner, from the equation 
 
 c mx =a& nI_1 , 
 we find 
 
 log. a — log. b 
 ~m log. c — n log. b' 
 Equations of this nature are called Exponential /.' 
 To resolve the exponential equation 
 
 /117\» 8493 
 V337/ = 73 
 By the rule, 
 
 x(log. 117— log. 337)= log. 8493— log. 73 
 log. 8493— log. 73 
 ' X= ~ log. 337— log. llf = 
 Calculation, • 
 
 8493 log. 3.9290G11 337 log. 2.5276299 
 
 73 log. 1.8633229 I 117 log. 2.0681859 
 
 diff. 2.0657382 . . . . log. 0.3150752 
 
 diff. .0.4594440 log. 1.6 622326 
 
 x=— 4.49616 log.=diff. 0.6528426 
 This example admits the use of the Gauss logarithms. 
 
 Let 10< = — 100 .'.x log. 10= log. ( — 100) ; log. ( — 100) hero must be re- 
 garded, like an imaginary quantity, as a symbol of absurdity. It is evident that 
 there is no power of 10 equal to — 100. 
 
 221. Let N and N + l be two consecutive numbers, the difference of their 
 logarithms, taken in any system, will be 
 
 log. (N + l)- log. N= log. (^p)= log- (l+^)« 
 
 a quantity which approaches to the logarithm of 1, or zero, in proportion as 
 
 ■r} decreases, that is, as N increases. Hence it appears that 
 
 The ill (Terence of the logarithms of two consecutive numbers is less in propor- 
 tion as the numbers themselves are greater. 
 Let a x = N and A V = N : then we have 
 
 .r= log. N to the base a, or x= log. „N* 
 ?/= log. N i" the base l>. <>r //= log. i,N. 
 Hence log. a N= log. ,&-"=</ log. J> ( Kv\. 21 1. 111.); 
 
 .-. r=y log. J>, 
 
 * Understanding by the not A the logarithm of N in the lyatem whoae baaa 
 
 bo. 
 
LOGARITHMS. 271 
 
 and 
 
 y= , r • x ; (A) 
 
 J log. J) 
 
 anil by means of this equation we can puss from one system of logs, tc another, 
 
 by multiplying x, the log. of any number in the system whose base is a, by the 
 
 reciprocal of log. b in the same system ; and thus we shall obtain the log. of 
 
 the same number in the system whose base is b. 
 
 The factor : -, is constant for all numbers, and is called the Modulus, 
 
 log. a « 
 
 that is to say, if Ave divide the logs, of the same number c, taken in two sys- 
 tems, the quotient will be invariable for these systems, whatever may be the 
 value of c, and will bo the modulus, the constant multiplier which reduces the 
 first system of logs, to the second.* 
 
 If we find it inconvenient to make use of a log. calculated to the base 10, we 
 can in this manner, by aid of a set of tables calculated to the base 10, discover 
 the logarithm of the given number in any required system. 
 
 For example, let it be required, by aid of Briggs's tables, to find the log. of 
 
 2 . 5 
 
 - in a system whose base is -. 
 
 Let x be the log. sought, then by (A) 
 
 2 
 Jbg.g 
 
 ' 5 
 
 & 7 
 
 log. 2 — log. 3 
 
 log. 5 — log. 7' 
 
 Taking these logs, in Briggs's system, and reducing, we find 
 
 —0.17609125 
 
 1 ' = — 0.14612804 
 
 o 5 
 
 = 1.2050476= log. ^ to base -. 
 
 2 3 
 
 Similarly, the log. of -, in the system whose base is 5, is 
 
 log. 2— log. 3 
 X ~log. 3— log. 2~~ ' 
 which is manifestly the true result ; for in this case the general equation 
 
 N =a* becomes -= (- ) = ( - ) , and x is evidently = — 1. 
 
 In a system whose base is a, we have 
 
 log. n , 
 
 for, by the definition of a logarithm in the equation n=a x , x is the log. n. 
 Tn like manner, 
 
 n , =a log. («") = /log.« 
 
 ,* The term Modulus, of a system of logarithms, is generally understood to be the num 
 ber by which it is necessary to multiply Napierian logarithms of numbers, in order to ob- 
 tain the logarithms of the system in question. The peculiar character of Napierian loga- 
 rithms will be presently explained. 
 
272 ALGEBRA. 
 
 \MTLES FOR EXERCISE. 
 
 (1) Given 2 2x +2 x =12 to find the value of x. 
 
 (2) Given x-\-y = a, and m (>:_y '=« to find x and y. 
 
 (3) Given m x n x =a, and hx=ky to find x and 3/. 
 
 answers. 
 
 (1) x=l-5849G2, or x=log. ( — 4)-j- log. 2. 
 
 (2) x= .', j a+ log. n4- log. »n } and y= !,\a— log. n -^- log. ?n } . 
 
 (3) x= log. a-^-(log. ?«+ 1°S- 7! ) ar, d .V=7j°g' ce-t- (log. m + l°g- ra )- 
 
 THE EXPONENTIAL THEOREM. 
 
 222. It is required to expand a* in a series ascending by the powers of x. 
 
 Since a = 1 -f- a — 1 , therefore a x = \l-\-(a — 1 ) } x ; and by the binomial theorem 
 
 we have 
 
 t , 1 x(x— 1) x(x— l)(x— 2), 
 
 {l + (a-l)^ = l + x(a-l)+-4^(a-ir-+-^- T ^ L (a-1) 3 +-- 
 
 = l+{(a-l)-l(a-iy-+} s (a-iy-\(a-iy+....\x+Bx* 
 
 + C.C 3 ... 
 
 where B, C denote the coefficients of x 2 , x 3 ; and if we put 
 
 A=(a-I)-\(a-iy-+l(a-iy+l(a-iy+ 
 
 Thena I = l + Ax+Bx 2 +Cx 3 - r -Dx<+Ex''+ 
 
 For x write x-\-h ; then we have 
 
 a I + h = l + A(x+/j) + B(x+/()-+C(r- r -/;y 5 -f 
 
 = l-fAx-f- Bx°-+ Cx 3 + Dr» + 
 
 4- Ah 4- 2Bxh+3Cx-h+4'Dx 3 h + 
 
 -f Bh-+3Cxh~ + GDx-h- + 
 
 + C/i 3 +4D.* + 
 
 + m* + 
 
 Buta J + h =a I xa*=(l+Ax+Bx 2 +Cx 3 +....)(l + A/i + B/i 2 +C/i 3 + 
 = 1 4-Ax+Bx 2 -fCx 3 +D.X 1 +.... 
 
 + A/i+A 2 x/i + ABx 2 /i+ ACrVi + . . . . 
 
 + }ik- Jf-XB.rh- -{-])■ rlr + . . . . 
 
 + l.7r 4-ACx/t 3 -r-.... 
 
 4-D/t* + .... 
 
 Now these two expansions must be identical ; and we must, therefore, have 
 
 tne coefficients of like powers of x and // equal; hence 
 
 A* 
 2B=A 2 ••• B = — 
 
 AB \ 
 
 3C = AB C=— = ;- 
 
 AC v 
 4D = AC D=-=— 
 
 &c. flee. &c. &c. 
 
 \ \ A*X* 
 Hence a x = l + Ax+— "+ \ ....;; + \ ij^H 
 
 which is the exponential thcoit m ; when' 
 
 ^=(a-l)-J(a-l)«+S(a-l) -i(a-l)H- 
 
LOGARITHMS. 273 
 
 Let e be the value of a, which renders A = l, then 
 
 («_i)-4(e-i)»+K«-iy-K«-i)«+...=»i 
 
 X 2 3? X* 
 
 Now, since this equation is true for every value of x, let x=l ; then 
 e — 1 + 1 + 1 . 2 + 1 . 2 . 3 + 1 .2.3.4+ 
 
 ="l+l+«l)+*^)+*G^) + 
 
 =2-718281828459 
 
 223. We add another method of calculating the logarithm of any given 
 number. 
 
 Let N be any given number whose logarithm is .r, in a system whose base 
 is a ; then 
 
 a* = N and a"=N x . 
 
 Hence, by the exponential theorem, we have from the last equation 
 
 1 + A.rz+A^+--- = 1 + A,z+A 1 ^+....; 
 
 and equating the coefficients of z, we get Ax=A 1 ; hence 
 
 A, (N-l)-i(N-l) 2 +!(N-l) 3 -.... 
 X ~ A ~(a -l)-i(a -iy+h(a -l) 3 - . ... ; 
 because A =(a — 1) — \{a — l) 9 +,i( a — l) 3 — ... in the expansion of a n 
 and A t = (N— 1)— |(N — 1) 3 +^(N — l) 3 in the expansion ofW 
 
 224. To find the logarithm of a number in a converging series. 
 We have seen that if a x =N, thon 
 
 ( N-l) -J(N-l)'+ ^(N-l )'-j (N-l)«+... 
 *-( a _l)_i( tl -!)»+*(« -lf-l(a -1)*+... 
 Now the reciprocal of the denominator is the modulus of the system ;*aad, 
 representing the modulus by M, we have 
 
 x= log. N = M{(N-l)-KN-l) 2 +J(N-l) 3 -i(N-l)*+...} 
 Put N = l + n ; then N — 1=», and we have 
 
 log. (l4-n)=M( + n— in«+in 3 — Jn«+J» s — ...) . . .[Al 
 
 .Similarly, log. (1— »)=M(-n- > s — |» s — in*— $n* ) 
 
 ... bg. (1+n)— log. (1— tt)=2M(n+j7i 3 +$n B -Kn 7 +...) 
 
 or lo S- S =2M(n+-J» 3 +Jn«+^n'+ ...) 
 
 • If, in the expression for a* deduced in (Art. 222), we make z=-r t we obtain 
 
 1 it i 
 
 which is tho value of e, given at the end of the same art. : 
 .-. a A =£ .". <z=:e a /. A log;, e= log. a .: —= 
 
 A log. a log. a' 
 
 if t be the base cf the system of logarithms expressed by log. ♦ Therefore — = ia 
 
 oy a previous definition (Art. 221), the modulus for passing from the system whose base ia 
 to that whose base is a. If log. a refers to the base a, - becomes oqaal to log. c 
 
 A 
 
 s 
 
274 ALGEBRA. 
 
 1 I'-f -• 2P l + n I'-j-l 
 
 Put * = 2P+7 ; then 1+n== 2P+I' 1- " = 2P+T' and I^7t =_ p -: 
 
 consequently, 
 
 log. (P + i)- log. P=2M \ -^ +5 ^-_ +5 pJ_+... I 
 
 .••log.(P+l)=log.P + 2M { ^+3(2^+5(2^+... | 
 
 Hence, if log. P be known, the log. of the next greater Dumber can be found 
 by this rapidly converging series. 
 
 By substituting the -cries of Dataral numbers for N n this formula, the cor- 
 responding values of x will be their logarithms. 
 
 224. To find the Napierian logarithms of numbers. 
 In the preceding series, which we have deduced for log. (P + l). Wl " '""1 & 
 number M, called the modulus of the system ; and we must assign some value 
 to this number before we can compute the value of the series. Now. as the 
 value of M is arbitrary, we may follow the steps of the celebrated Lord 
 Napier, the inventor of logarithms, and assign to M the simplest possible 
 value. This value will therefore be unity, and we have 
 
 log. (P + l)=log. P + 2 \ .Tp^ + 3( . 2 p 1 +1) 3+^p^ T y,+ ••• \ 
 Expounding P successively by 1, 2, 3, 4, &c., we find 
 
 log. 3= log. 2 + 2Q+ 3 -^+^+^- 7+ ...) =1-0080123 
 log. 4=2 log. 2 =1-3862944 
 
 log. 5= log. 4+2(^+3^+—^+ -) =1^4379 
 log. 6= log. 2+ log. 3 =1.7917595 
 
 ,og. 7= log. 6+2(^+3-^+^,+ ) =1-0459101 
 
 log. 8= log. 2+ log. 4, or 3 log. 2 =2-0794415 
 
 log. 9=2 log. 3 =2-1972246 
 
 log. 10= log. 2+ log. 5 =2*3025851 
 
 In this manner the Napierian logarithms of all numbers may be computed. 
 
 225. T<> Ibid the common logarithms of nunr 
 The base of the Napierian system is £=2-718281828..., and the base of tbe 
 common system is 6 = 10, the base of our common system of arithmetic 5 then 
 we have 6 = 10, and a=e=2»718281828..., and consequently, if N denote any 
 number, vre shall have 
 
 log. 10 N= j— — Jq • log. £ N ; that is, 
 com. log. N= 2 . 30g5851 ^ Nap. log. N=-43429448X Nap. I 
 
 " To liml the value of trio Napierian base, observe that, since com. log. N='43499448X 
 Nap log. N., if wo make in this expression N=c, the Napier! 
 
 com. log. (='43499448. 
 From a table of common logs., therefore, we find the number corresponding to the log* 
 
 • 
 
LOGARITHMS. 275 
 
 and the modulus of the common system is, therefore, 
 
 1 
 
 M= r-rr: =-43429448 .-. 2M = -86858896 
 
 2-3025851 
 
 Hence, to construct a table of common logarithms, we have 
 Jog. (P + l)= log. P + -86858896 \ $^±%^tf+ 5{2P \ 1)6 + ■ ■ \ 
 Expounding P successively by 1, 2, 3, &c, we get 
 
 log. 2 = -8685889gQ+^+-^+...) 
 
 = •86858896 X -3465736 = -3010300 
 
 log. 3= log. 2 + . 86858896 Q + _L + l + ...) . . = -4771213 
 
 log. 4=2 log. 2 = -0020600 
 
 log. 5= log. 1 jJ °= log. 10— log. 2 = 1— log. 2 . . = -6989700 
 log. 6= log. 2+ log 3 = -7781513 
 
 log. 7= log. G + -86858896 (-+^+t^33+...)= -8450980 
 
 log. 8= log. 2 3 =3 log. 2 = -9030900 
 
 log. 9= log. 3 2 =2 log. 3 = -9542426 
 
 log. 10= =1-0000000 
 
 &c. &c. 
 
 1-4-77. 
 
 226. Since log. — |— = 2M(n+lri 3 +|n 5 +> 7 + ...) 
 
 14-71 P— 1 
 
 let = P ; then l+n=P(l — n), or w = p 
 
 cp_i i /P— 1\ 3 1 /P — IV" I 
 
 ••^• p = 2M Spqp+rlp+l)+5-lp+l)+'--i 
 
 and thus we have a series for computing the logs, of all numbers, without 
 knowing the log. of the previous number. 
 
 EXAMPLES. 
 
 (1) Given the log. of 2 = 0-3010300, to find the logs, of 25 and -0125. 
 
 100 10 3 
 Here 25=—=—; therefore log. 25=2 log. 10—2 log. 2=1-3979406. - 
 
 125 1 1 
 
 A S ain ' ' 0125 = 10000=80 = 10^23 
 
 .-. log. -0125= log. 1— log. 10 — 3 log. 2=— 1 — 3 log. 2=2-0969100 
 
 (2) Calculate the common logarithm of 17. 
 
 Ans. 1.2304489. 
 
 (3) Given the logs, of 2 and 3 to find the logarithm of 22-5. 
 
 Ans. 1 + 2 log. 3—2 log. 2. 
 
 (4) Having given the logs, of 3 and -21, to find the logarithm of 83349. 
 
 Ans. 6 + 2 log. 3+3 log. -21. 
 
 rithra -43429448, which is 2-7182818, the Napierian base. This also furnishes us with an- 
 other definition of the modulus of the common (or any other) system of logarithms ; it is the 
 common (or, &c.) logarithm of the Napierian base. See further note at the end of Progres- 
 sions. 
 
876 ALGEBRA. 
 
 PROGRESSIONS. 
 
 ARITHMETICAL PROGRESSION. 
 
 227. When a series of quantities continually increase or decrease by the 
 addition or subtraction of the same quantity, the quantities are said to be 'id 
 Arithmetical Progression. A more appropriate name is Progression by Dif- 
 fer t.j-.es. 
 
 Thus the numbers 1, 3, 5, 7, which differ from each other by the ad- 
 dition of 2 to each successive term, form what is called an increasing arith 
 metical progression, or progression by differences, and the numbers 100, 97, 
 
 94, 91, which differ from each other by the subtraction of 3 from each 
 
 successive term, form what is called a decreasing progression by differences. 
 
 Generally, if a be the first term of an arithmetical progression, and 6 the 
 common difference, the successive terms of the series will be 
 
 a, a±<5, a ±2(5, a±3<5, 
 
 in which the positive or negative sign will be employed, according as the series 
 is an increasing or decreasing progression. 
 
 Since the coefficient of (5 in the second term is 1, in the third term 2, in the 
 fourth term 3, and so on, in the » tt term it will be n— 1, and the n th term of 
 the series will be of the form 
 
 a±(n-l)«J (1) 
 
 In what follows we shall consider the progression as an increasing one, since 
 all the results which we obtain can bo immediately applied to a decreasing 
 geries by changing the sign of 6. 
 
 228. To find the sum ofn terms of a series in arithmetical progression. 
 
 Let a= first term. 
 1= last term. 
 «5= common difference. 
 n= number of terms. 
 S= sum of tho series. 
 
 rhen S=a+(a+<5) + (a+2<5)+ +!. 
 
 Write the same series in a reverse order, and we have 
 S= I +(Z_«5)+(!-2d)+ + a 
 
 Adding,2S = (fl-H) + (a + + ( a +0 +......+(«+*) 
 
 =n{a-\-l), since the series consists ofn terms. 
 
 s= n -<^ (2) 
 
 • > 
 
 Or, since l=a+{n — 1)6 (Art. 227), 
 
 s jna+««-iy (3) 
 
 Hence, if any three of the five quantities a, I, (\ n, S be given, the remain- 
 ing two may be found by eliminating between equations (1) and (2). 
 
 It is manifest from tho above process that 
 
 The sum of any two terms which arc equally distant from the extreme b 
 w equal to the sum of the extreme terms, and if the number of terms in the series 
 be uneven, the middle term will be equal to one half the sum of the extreme terms, 
 or of any two terms equally distant from the extreme term.';. 
 
PROGRESSIONS. 277 
 
 EXAMPLE I. 
 
 Required the sum of 60 terms of an arithmetical series, whose first term is 
 b and common difference 10. 
 Here a=5, (5=10, n =60 
 
 .-. l=a-\-{n — 1)<5=5+59X 10=595 
 
 (5 + 595) X 60 
 •'• b ~ 2 
 
 = 600x30 = 18000= sum required. 
 
 EXAMPLE II. 
 
 A body descends in vacuo through a space of 16^ feet during the Qrst 
 second of its fall, but in each succeeding second 32^ feet more than in the one 
 immediately preceding. If a body fall during the space of 20 seconds, how 
 many feet will it fall in the last second, and how many in the whole time 7 
 
 193 386 
 
 Here a= ~L2' "lsF' n=20 
 
 193 386 
 
 7527 
 
 :627{ feet 
 
 12 
 (193+7527) X 20 
 
 S = 
 
 = 6^33} feet. 
 
 EXAMPLE III. 
 
 To insert m arithmetical means between a and b. 
 
 Here we are required to form an arithmetical series of which the first and 
 last terms, a and b, are given, and the number of terms =»i + 2; in older, 
 then, to determine the series, we must find the common difference. 
 Eliminating S by equations (1) and (2), we have 
 
 2a+(n— l)(5=J+a 
 / — a 
 
 But here l=b, a=a, n=m+2 
 
 •. the required series will be 
 
 «+H5)+K^)+ +K^)+(°+^-) 
 
 or 
 
 b-i-ma 2i+(wi — \)a mb-\-a 
 
 a+ -^+r + m +i + + ^+r +J - 
 
 (4) Required the sum of the odd numbers 1, 3, 5, 7, 9, &c, continued to 
 101 terms ? 
 
 Ans. 10201. 
 
 (5) How many strokes do the clocks of Venice, which go on to 24 o'clock, 
 strike in the compass of a day? 
 
 Ans. 300 
 
278 ALGEBRA. 
 
 (G) The first term of a decreasing arithmetical series is 10, the common 
 difference i, and the number of terms 21; required the sum of the series. 
 
 Ans. 140. 
 
 (7) One hundred stones being placed on tho ground in a straight line, ;it 
 the distance of 2 yards from each other; how far will a person travel uno 
 shall bring them one by one to a basket which is placed 2 yards from the Bret 
 stone ? 
 
 Ans. 11 mik'S and 840 yards. 
 
 The relations (1) and (2), in which five quantities, a, 6, n, I, S, enter, will 
 
 serve to determine any two of these when the other three are given. Thus 
 
 thay furnish the solution of as many distinct problems as there are waj 
 
 taking two quantities from among five ; and, consequently, the number of 
 
 5-4 
 problems will be —or 10. In order that they may be possible, it is necessary 
 
 that the value of n should be not only real, but entire and positive. Without 
 entering into the details of the calculation, we place below the solutions of 
 these ten problems. 
 
 I. Given a, 6, n. ( , . _ . , 
 
 Required I, S.V =«+<— W S-M8.+(— 1 M 
 
 III. Given a, n, I. ( I — a 
 
 a <<$== -, S=',n{a+l). 
 
 Required 6, S. ( n — l ' 
 
 IV. Given 6, n, S. < 2S— n(n — 1)6 2S + n{n — 1)6 
 Required a, I. I 2n 2n 
 
 V. Given a, n, S. < _^_ j 2 ( S ~ an ) 
 
 Required 6, I. \ ~~ n ~~ ' ~" n(n — 1) ' 
 
 VI. Given I, n, S. < _^_, 2(nl — S) 
 
 Required a, 6. ( n " n{n — l) ' 
 
 VII. Given a, 6, I A ?—g (l + a){l—a+6) 
 
 Required n, S. ( 6 ' 2<5 
 
 VIII. Given a, I. S. < _2S^ (l+a) ( l— a) 
 
 Required n, <J. < n ~ a + V = " 2S — (/+a) ' 
 
 IX. Given a, <5, S. 
 
 d—la-lz V(J— 2«/) : -f 8dS 
 
 " 2d 
 
 Required /, n. ^ =a+( „_ 1)(5 . 
 
 X. Given £, (5, S. 
 
 <5+2Z± V( (, 4- -0-' — 8dS 
 
 Required a, n. f . , .. . . 
 ^ v. a = £ — (n — l)o. 
 
 GEOMETRICAL PROGRESSION. 
 
 229. A series of quantities, in which each is derived from that which im- 
 mediately precedes it, by multiplication by a constant quantity, is called a 
 Geometrical Progression, or Progression by Quotients. 
 
 Thus, the numbers 2, 4, 8, 16, 32, in which each is derived from the 
 
 preceding by multiplying it by 2, form what is called an increasing geometrical 
 
I 
 
 • ■ 
 
 I 
 J 
 
 .' 
 
 
 : - ■ 
 
 mtmxpty uom »iues ot the equation by p, 
 
 Sp= ap-\-ap°-\-ap*-\- -j-ap<>-i_j_ap\ 
 
 Subtract the first from the second, 
 
 S(p— l)=ap n — a 
 
 ••• S= U ' (1) 
 
 p — 1 v ' 
 
 Or, since 
 
 l=ap n ~ 1 
 L, pi — a 
 
 s =^r < 2 > 
 
 If the series be a decreasing one, and consequently p fractional, it will be 
 convenient to change the signs of both numerator and denominator in the above 
 expressions, which then become 
 
 S _a(l-P n ) 
 1—P 
 a — pi 
 l— p 
 
 231. If two progressions have different first terms, but the same ratio, the 
 ratio of the sums of the two is equal to the ratio of their first terms. For 
 
 (a + ap+ap°-+ap*+, &c.) : (J+Sp+^+ft/^-f , &c.) 
 =a(l+p+ P '+ ps+, &c.):6(l+P+ p-+ P 3 +, &c.)=a:6 
 
 232. It appears that if any three of the five quantities, a, I, p, n, S, bo 
 given, the remaining two may be found by eliminating between equations (1) 
 and (2). It must be remarked, however, that when it is required to find pfrom 
 a, n, S given, or from n, I, S given, we shall obtain p in an equation of the n 01 
 degree, a general solution of which can not be given. If n be required, it will be 
 convenient to apply logarithms, as the equation to be resolved will be an expo 
 nential. 
 
M 
 
 2: 
 
 : 
 I 
 
 th 
 
 
 si 
 
 
 8t 
 
 
 ' 
 
 
 K 
 
 
 t\ 
 
 
 tn 
 
 
 P ; 
 
 
 tl 
 
 
 e 
 
 
 tl 
 
 
 ' 
 
 
 Required a, b. ( 
 
 III. Given a, n, I. < I —a i_f-j_n 
 Required 6,S.l n — V - ^ t ; 
 
 IV. Given 6, n, S. < 8S— n(n-l)d ^ 2S+n(n-l)d 
 Required a, I. i °~ 2n 2n 
 
 V. Given a, n, S. 5 . 2S 2(S-an) 
 
 Required 6, I. ( n 11(71 — 1) 
 
 VI. Given I, n, S. < fl _^_; j_ 2(»*— S) 
 Required a, 6.1 ~ n ' n(n — 1) 
 
 VII. Given a, (5, I. < E— a (J + a)(i — a + J) 
 
 Required n, S. i "~ «J + ' 2<» 
 
 VIII. Given a, J. S. < _2S (E+a)(i-a) 
 
 Required n, J. i n_ a + Z' 2S— (J+a) " 
 
 IX. Given a, <*, S. )n=- — r^j 
 
 Required i, n. ^ _ a+ ( n _ 1)( i. 
 
 X. Given I, «J, S. > »= r^j 
 
 Required a, n. ( a = i_ (n _ l yi 
 
 GEOMETRICAL PROGRESSION. 
 
 229. A series of quantities, in which each is derived from that which im- 
 mediately precedes it, by multiplication by a constant quantity, is called « 
 Qeometrica.1 Progression, or Pre ■> by Quotients. 
 
 Tims, the numbers 2, 4, 8, 16, 32 in which each is derived from the 
 
 pm-rdim; bv multiplying it by 2, form what lb called an increasing geom 
 
PROGRESSIONS. 279 
 
 progression ; and the numbers 243, 81, 27, 9, 3, ... in which each is derived 
 from the preceding by multiplying it by the number -, form what is called a 
 
 decreasing geometrical progression. 
 
 The common multiplier in a geometrical progression is called the common 
 ratio. 
 
 Generally, if a be the first term and p the common ratio, the successive 
 terms of the series will be of the form 
 
 a, ap, op 2 , ap 2 
 
 The exponent of p in the second term is 1, in the third term is 2, in the 
 fourth term 3, and so on ; hence the n a term of a series will be of the form, 
 
 ap" -1 . 
 
 230. To find the sum ofn terms of a series in geometrical progression. 
 
 Let a== first term, 
 1= last term, 
 p= common ratio, 
 n= number of terms, 
 S = sum of the series. 
 Then 
 
 S =a4-ap+ap 2 +ap 3 4- + crp n - 1 . 
 
 Multiply both sides of the equation by p, 
 
 Sp= ap-\-ap"-{-ap 3 -\- ^_ap"-i_j_ap°. 
 
 Subtract the first from the second, 
 
 S(p—l)=ap"—a 
 a(p n — l) 
 
 Or, since 
 
 l=a P n ~ l 
 
 *£r « 
 
 If the series be a decreasing one, and consequently p fractional, it will be 
 convenient to change the signs of botli numerator and denominator in the above 
 expressions, which then become 
 
 S __ «(l-A?) 
 
 l— P 
 
 1— p 
 
 231. If two progressions have different first terms, but the same ratio, the 
 ratio of the sums of the two is equal to the ratio of their first terms. For 
 
 (a+ap+ap' i +ap*+, &c.) : (b + b P + bp* + bp 3 +, Sec.) 
 = a(l+p+ p*+ p3+,&c.):i(l+P+ p 2 + p 3 +, &zc.)—a:b 
 
 232. It appears that if any three of the five quantities, a, I, p, n, S, b» 
 given, the remaining two may be found by eliminating between equations (1) 
 and (2). It must be remarked, however, that when it is required to find pfrom 
 a, n, S given, or from n, I, S given, we shall obtain p in an equation of the 71 th 
 degree, a general solution of which can not be given. If n be required, it will be 
 convenient to apply logarithms, as the equation to be resolved will be an expo 
 nential. 
 
5280 ALGEBRA. 
 
 EXAMPLE I. 
 
 Requirea the sum of 10 terms of the series 1, 2, 4, 8, . 
 Here a = l, p=2, n=zlO 
 
 fl(p"-l) 
 p — 1 
 
 = OI0 J 
 
 = 1023. 
 EXAMPLE II. 
 
 Required the sum of 10 terms of the series 1, -, -, — , 
 
 3' 9' 27' 
 
 Here assl, p=y n=\0 
 
 ...S: 
 
 2 
 : 3' 
 
 q( i— p") 
 1-p 
 
 l- 
 
 ■©' 
 
 174075 
 : 59049 ' 
 
 EXAMPLE III. 
 
 To insert m geometric means between a and b. • 
 
 Here we are required to form a geometric series, of which the first and last 
 terms, a and o, are given, and tho number of terms =?n-\-2 ; in order, then* 
 to determine the series, we must find the common ratio. 
 
 Eliminating S by equations (1) and (2), 
 
 ap n ~—a=pl — a 
 
 But here 
 
 l = b, 71=771-4-2 
 
 Henco the series required will be 
 
 lb 16* lh m ~ l lb m #">+» 
 
 or 
 
 CL+ ™+y a m b-\. ™+fa m - l b- + ...-{- m +^/a' 1 b m - , -\- ^ab^+b, 
 or 
 
 m 1 m— 1 1 8 m— 1 t m 
 
 a -\. a™+ l b™+ i + a™+ 1 b™+ 1 + . . . + a n, + ' 6" , "+ 7 + a*"+i £"+» -f b. 
 
 233. To ^"fi ^c sum of an infinite series decreasing in geometrical jrro- 
 gression. 
 
 Wo have already found that tho sum of i i terms of a decreasing geometric* 
 series is 
 
1-P* 
 
 which may dc put. under the form 
 
 PROGRESSIONS. 281 
 
 a—ap n 
 
 S = r ^ 
 
 '1-P l-p- r - 
 
 Since p is a fraction, p n is less than unity, and the greater the number n, the 
 
 smaller will be the quantity p n ; if, therefore, we take a veiy great number of 
 
 ap a 
 terms of a decreasing series, the quantity p n , and, consequently, the term , 
 
 a 
 will be very small in comparison with ; and if we take n greater than any 
 
 assignable number, or make n = co, then p 1 " 1 will be smaller than any assignable 
 number, and therefore may be considered =0, and the second term in the 
 above expression will vanish. 
 
 Hence we may conclude that the sum of an infinite series, decreasing in 
 geometrical progression, is * 
 
 1— P 
 
 a 
 Strictly speaking, is the limit to which the sum of any number of 
 
 terms approaches, and the above expression will approach more or less nearly 
 to perfect accuracy, according as the number of terms is greater or smaller 
 Thus, let it be required to find the sum of the infinite series 
 
 1 
 Here a = l, p=^, n=ao 
 
 1-P 
 1 
 
 1 
 X ~3 
 
 3 
 
 ~~2' 
 
 3 
 
 The error which we should commit in taking - for the sum of the first n 
 
 terms of the above series is determined by the quantity 
 
 ap n _3/l\ n 
 l^p = 2\3/ ' 
 ™ .. 3/l\s 1 1 
 
 Thus, if »=5, tl on - 2 {~) =27F=162 ; 
 
 n = G,then-y =— -— . 
 
 3 
 Hence, if we take - as the sum of 5 terms of the above series, the amount 
 
 would be toe great by y^-. 
 
282 ALGEBRA. 
 
 3 1 
 
 If we take - as the sum of G terms, the amount w.ll be too great by —— . 
 
 ~ 4 DO 
 
 and so on.* 
 
 * I. The theory of progressions involves that of logarithms. Let there be two progres- 
 sions, the one geometric, beginning with 1, the other arithmetical, beginning with 0. 
 
 -ffl:2:4:8:16:32:64:128, &c. 
 -H). 3. 6. 9. 12. 15. 18. 21, &.C., 
 which exhibit a notation sometimes employed. 
 
 If we compare these with each other, we perceive that, multiplying together any two 
 terms of the first, and adding the corresponding terms of the second, we obtain two I 
 sponding terms, again, of these same progressions. Thus, 4X1G=64, 6-}-12=13 ; ai. 
 perceive that 18 corresponds to 04. Thus a multiplication is effected by addition. Tiiid 
 simple observation is, no doubt, very ancient ; but it was the genius of Napier, a Scottish 
 baronet, which derived from it the theory of logarithms, one of the most useful of modern dis- 
 coveries. It was published in 1G44, under the title of Mirifici Logarithmorum Dejcriptio. 
 
 Logarithms, then, according to Napier, were regarded as a series of numbers in arith- 
 metical progression, while the numbers themselves corresponding, formed a geometrical 
 progression. I proceed to explain his method of constructing them. 
 
 In order that the geometrical progression fluiull embrace all numbers greater than 1, it 
 is necessary to conceive it formed of tonus which increase in an insensible mam 
 out lVom 1 ; and, to have their logarithms, it is necessary to conceive the arithmetical pro 
 gression as composed of terms which vary by insensible decrees, setting out from zero. 
 
 At their origin, the simultaneous increments which the terms 1 and receive are inap- 
 preciably small ; but, however small they may be, we may conceive that there is a certain 
 relation established between them, which is entirely arbitrary. Thus, when these incre- 
 ments begin to arise, we can suppose that that of the logarithm is double, triple, &c., of 
 that of the number 1. This relation is called the modulus of the logarithms, which i 
 nate by M. 
 
 Suppose, now, that to the terni 1 of the geometric progression an increment u, 1 i ry 
 small, but yet appreciable in numbers, is given. The corresponding increment of the tenii 
 zero of the arithmetical progression will be very nearly equal to Mu) ; and we can take for 
 the two progressions these : 
 
 -ff-1 : 1-f-w: (l+w)2: (l-fw) 3 : (l+o)4:&c. 
 -^-0. Mu. 2Mu . 3Mu . 4Mo .tic. 
 
 We have said that the relation or modulus M can be taken at pleasure ; consequently 
 according to the values attributed to it, will be obtained different systems of logarithms. 
 The logarithms which Napier published were derived from the progressions 
 
 4f 1 : 1+u : (1+up : (1+^) 3 : &c. 
 -i-0. u. 2w . 3cj .&.C., 
 which supports M=l. 
 
 This avoids the multiplications by M. The logarithms of numbers in Napier's table 
 serve to find those of any other system, by simply multiplying each by the modulus of that 
 system. 
 
 The terms of these two series vary slowly, so that, in prolonging both as far as we please, 
 we are sure of finding in the first, terms equal numbers 2, 3, &c., or so nea» 
 
 them that the difference may be neglected. The corresponding tonus of the second may 
 tfieu be taken for the logarithms of these numbers, those written in the tables. 
 
 By this we perceive that these logarithms are not exactly those of the numbers beside 
 which they are written. But there is another cause of inaccuracy, viz., that <j represents 
 only approximately the increment, which the logarithm takes when cj is that taken by 1 
 The . i is, however, the greater the ezactni 
 
 1 1 Let it be proposed to determine the error produced by assuming that the differen t 
 the numbers is proportional to the difference of their logarithms, when the number of phvu'i 
 in the numbers is 5, and their difference not greater than 1. 
 
 If in the series [A], Art. 221, we make «=-, we have 
 
 'C-?)=»+"-"= M i;-, 1 .+,!-s+'*°i' 
 
PROGRESSIONS. 283 
 
 As in arithmetical progressions, all the questions which can be p-oposed foi 
 
 •olution in geometric progressions reduce to 10, the solutions of which are de 
 
 duced from 
 
 l=ap n ~ l (1) 
 
 _ pi — a 
 
 S=- (2) 
 
 p — 1 v ' 
 
 from which it appears generally that as the number x increases, the difference of the loga- 
 rithms of x and \-\-x diminishes. Also, since - is 
 
 x 
 
 diminished by more than it is increased, we have 
 
 rithms of x and \-\-x diminishes. Also, since - is greater than the whole series, - being 
 
 x x 
 
 M 
 Z(l-j-x) — Ix-^—. 
 x 
 
 If the base be 10, we have seen that M=0.4342...<-. Hence, in this case, 
 
 l(l+x)-lx<±. 
 
 • 
 
 If x consist of five places, its least value is 10000. Therefore the greatest value of 
 
 l(l4-x) — Ix is less than =0.00005. 
 
 20000 
 
 Hence we may infer that the logarithms of every two consecutive whole numbers con- 
 sisting of five places must agree in the first four decimal places at least. 
 Now let 
 
 A = lll4-x)—lx=l 1 -^. 
 x 
 
 A / = Z (2+x) _ Z(1+a;)=Z ?±?. 
 i-\-x 
 
 A-A'=z!±f-Z 2 +* 
 
 But by [A], Art. 224. 
 
 l+x 
 
 (1+s)' / 1 \ 
 
 x{2+x) \ nr x{2+x)t 
 
 Z Kl2^,)= M ^ 
 
 1 
 
 x(2+x)!~ " dx(2-f<r) &F>(2+a:) a ' 3a^(2-f-a;)3 
 
 1 
 
 A-A'<- 
 
 "2x(2+2;) 
 If x consist of five places, its least value is 10000, and, therefore, the greatest valae of 
 
 A — A' is less than = , which, when reduced to a decimal, has no 
 
 20000 X 10002 200040000 
 
 significant figure within the first eight places. Hence, in tables which extend only to 
 
 seven places, we may assume that A — A'=0, or A=A'. 
 
 Thus we infer that, under the circumstances which have been supposed, the logarithms 
 
 of numbers in arithmetical progression will themselves be in arithmetical progression 
 
 P 
 
 Let now n and w-f-1 be two consecutive whole numbers, and n-f-— an intermediate frac- 
 
 1 
 tion. These may be looked upon as three terms of an arithmetical progression, whose first 
 
 1 p 
 
 term is n, whose common difference is -, whose (p-\-l) til term is n-\ — , and whose fe-f-l}* 
 
 term is n-\-\. By what has been already shown, the logarithms of the several terms of 
 this scries will also be in arithmetical progression. 
 
 Let <5 be their common difference. The (p+l) th term of this series will be 
 
 ln-\-p8, 
 which will be the logarithm of the (^-f-l) th term of the former series ; 
 
 .-. ln+pS=l '«+-) [Bl 
 
284 ALGEBRA. 
 
 r F]>*<i«k solutions are contained in the following table : 
 
 I. <?«ven a, p, n. ( _ n _, pi— a a(p n — 1) 
 
 Required I, S. 2 l ~ ap " ' ,, — 1 ~~ p — 1 ' 
 
 Given J, p, n. < I /(p"_i) 
 
 Required a, S. I a ~^ S= p — i( p _ l) 
 
 n — 1/7" ■>— I; 
 
 III. Given a, w, Z. < n _,/7 V*"— V" n 
 Required p, S. J p = V a' S= ""^7— ""-1/a ' 
 
 IV. Given p, n, S. < S(p— 1) Sp n - J (p — 1) 
 
 Required a, /. C " p n — 1 ' "I p" — l ' 
 
 V. Given a, n, S. S S 
 
 Required p, I. fTM^ . + !=-, l=ap^. 
 
 VI. Given £, n, S. 
 Required p, a. 
 
 / 
 
 
 VII. Given a,p,l.<pl—a \og.l— log. a 
 
 Required n, S. t - p — 1' ~ ' log. p 
 
 VIII. Given a, I, S. < S— a _ log. I— log. a 
 
 Required p,n.( S — V ~ ~*~ log. p 
 
 IX. Given a, p, S. < a-f-S(p— 1) log. Z— log. a 
 
 liequired I, n. ( p ■ log. p 
 
 X. Given Z, p, S. ( , • log. I — los:. a 
 
 t, . , <a=lp— S(p— 1), n = l-\ — ^—r -2— . 
 
 Required a, rc. c v /» ~r j g > ^ 
 
 HARMONICAL PROGRESSION. 
 
 234. A series of quantities is called a harmonical progression when, if aut 
 three consecutive terms be taken, the first is to the third as the difference of 
 the first and second to the difference of the second and third. 
 
 Thus, if a, b, c, d bo a series of quantities in harmonical progression, 
 
 we shall have 
 
 a:c::a — b:b — c; b:d::b — c:c — d, cce. 
 
 The reciprocals of a scries of terms in harmonical progression are in arith- 
 metical progression. 
 
 Let a, b, c, d, c,f be a series in harmonical progression. 
 
 Then, by definition, 
 
 Also, the last term of the latter series, which will be 
 
 ln-\-qd, 
 will be the logarithm of the last term of the former series ; 
 
 .-. l{n+l)—ln-\- q6, .: I{n+i)—ln=q6. 
 Hut by [B], lL+£\-4n 
 
 \ ql /(/j-j-1) — hi q 
 
 But, also, 
 
 (»+l ) — n 7 
 Hence Vac difTeroncci of the logarithms arc as the diflVroncos of the numbem 
 

 
 
 
 - / 
 
 a 
 
 
 285 
 
 
 
 
 
 
 V 
 
 
 • / 
 
 
 (a 
 
 
 
 
 ca. 
 
 in 
 
 
 
 <y 
 - 
 
 
 
 
 
 v> 
 
 ■ : 
 
 
 Z' / 
 
 
 
 - 
 
 i«a 
 
 Cr CL 
 
 
 ' 
 
 
 eu 
 
 SIMPLE INTEREST. 
 
 Problem I. — To find the interest of a sum pfor t years at the rate r. 
 
 Since the interest of one dollar for one year is r, the interest of p dollars for 
 one year must be p times as much, or pr ; and for t years t times as much as 
 for one year; consequently, 
 
 i=ptr .... (1) 
 
?84 
 1 
 
 \ 
 V 
 
 I 
 
 23' 
 three 
 the fi 
 
 Th 
 we si 
 
 Th 
 meticc 
 
 Le 
 Then 
 
 Alsc 
 will be 
 Hut by [B], 
 
 Bat, also, 
 
 iL+'A-i,, 
 
 («+i ) — » 7 
 
 Hence t\ic difference! of the logarithms are as the differences of the numbem 
 
INTEREST AND ANNUITIES. 095 
 
 a c::a — b:b — c ; b:d::l — c:c — d; c:e::c — d:d — e, &c. 
 .•• ab — ac=ac — be, be — bd=bd — dc, cd — ce=ce — cd, &c. 
 ab ac ac be 1-. hd bd dc cd ce ce ea 
 ' abc abc abc abc 1 bed bed bed bed' cde cde cde cie 
 or 
 
 1 11 1 1 11 1 1 11 1 
 c b b a' d c c V e d d c' 
 
 from which it appears that tho quantities -, j, -, -?, -, &c, are in arithtreVica. 
 
 CI U C Co C 
 
 progression. 
 
 To insert m harmonic means between a and b. 
 
 Since tho reciprocals of quantities in harmonical progression are in aritn 
 
 1 , 1 
 metical progression, let us insert m arithmetic means between - and r 
 
 Generally, in arithmetical progression, 
 
 l=a+(n — 1)6 
 
 I — a 
 
 .'. (5= -. 
 
 n — 1 
 
 r , • , 1 1 , o— b 
 
 In this case, I =7, a=-, n=m4-2, and .-. ()■=-. — , ,, , . 
 a (m-j-l)ao 
 
 The arithmetic series will be 
 
 1 a+mb 2a + {m—l)b (vi — l)a+2b ma+b 1 
 
 a r '(m+l)a6"' (m-\-l)ab + (m+l)aZ» "^(m+l)a6"^"6* 
 
 Therefore the harmonical series will be 
 
 (m-\-l)ab (m-\-l)ab (7?i-\-l)ab (m-\-l)ab 
 
 °^ a+mb +2a+(wi— 1)6"^" "*"(m— l)a+26"*" ma+b + 
 
 INTEREST AND ANNUITIES. 
 
 235. The solution of all questions connected with interest and annuitf-M 
 may be greatly facilitated by the employment of the algebraical formulas. 
 In treating of this subject we may employ the following notation : 
 Letp dollars denote the principal, 
 r the interest of $1 for one year. 
 t the interest of p dollars for t years. 
 s the amount of p dollars for t years at the rate of interest denoted 
 
 by r. 
 t the number of years thatp is put out at interest. 
 
 SIMPLE INTEREST. 
 
 . Problem I. — To find the interest of a sum pfor t years at the rate r. 
 
 Since the interest of one dollar for one year is r, the interest of p dollars for 
 one year must be p times as much, or pr ; and for t years t times as much as 
 for one year; consequently, 
 
 i=ptr .... (1) 
 
286 ALGEBRA. 
 
 Tuoklkm II. — To find the amount of a sum p laid out for t years at simple 
 in '■■ rest at (he rate r. 
 
 The amount must evidently be equal to the principal, together with the in- 
 terest upon that principal for the given time. 
 
 Hence s=p+ptr 
 
 =p(l + tr) (2) 
 
 EXAMPLE I. 
 
 Required the interest of $873.75 for 2\ years at 4 J- per cent, per annum. 
 It will be convenient to reduce broken periods of time to decimals of a year. 
 By the formula (1) wo have 
 
 i=ptr. 
 In tho example before us, 
 
 p =$873.75 
 
 r = $.0475* 
 
 I =21 years =2.5 years. 
 
 .-. z"=873.75x 2.5 X -0475 dollars. 
 =$103.7578125. 
 The amount of tho above sum at tho end of the given time will be 
 
 s=p-\-ptr 
 = $873.75 + 5103.757. 
 
 PRESENT VALUE AND DISCOUNT AT SIMPLE INTEREST. 
 
 The present value of any sum s due t years hence is the principal which in 
 the time t toill amount to s. 
 
 The discount upon any sum due t years hence is the difference between that 
 sum and its present value. 
 
 Problem III. — To find the present value of s dollars due t years hence, 
 simple interest being calculated at the rate r. 
 
 By formula (2) we find the amount of a sump at the ond of t years to be 
 
 s=j)-\-plr. 
 Consequently, p will represent the present value of the sum s due t years 
 hence, and we shall have 
 
 ^njh-r < 3 > 
 
 for the expression required. 
 
 * r is the interest of Si for one year. To find tho value of r when interest is calculated 
 
 at the rate of $4j or $4.75 per cent, per annum, we have the following proportion : 
 
 $100 :Sl:: $4.73 :r 
 
 • 4.7."i 
 
 .\r=$ =$0.0475. 
 
 100 
 
 In like manner, 
 
 When the rate of interest per cent, is $7, then r=$0.07. 
 
 When the rate of interest per cent, is 6, then r= 0.06. 
 
 When the rate of interest per cent is 5, then r= 0.05. 
 
 When lli.' rate of interest per cent, is 4], then r= 0.0475. 
 
 When the rate of interest per cent, is 4*, then r= 0.045. 
 
 When the rate of interest per cent, is 4|, then r== 0.0 
 
 When the rate of interest per cent, is 4. then r— o.oi. 
 When the rate of interest per cent, is '1\, thenr= 0.0375. 
 dec. 4.c. 
 
INTEREST AND ANNUITIES. 287 
 
 P.wquired tho present vame of 100 dollars, payable in 10 years, at 7 per cent, 
 per annum. 
 
 In this example s = 100 
 
 tz= 10 
 r = .07 
 
 100 
 ••• *= 1 + 10X.07 =S58 - 82 - 
 
 Problem IV. — To find the discount on s dollars due t years hence, at the 
 rate r, simple interest. 
 
 Since the discount on s is the difference between s and its present value, we 
 shall have 
 
 s 
 
 d=s 
 
 1 + tr 
 str 
 
 (4) 
 
 ~~ l + /r 
 
 EXAMPLE. 
 
 Required the discount on $100, due 3 months hence, interest being calcu- 
 lated nt the rate of 5 per cent, per annum. 
 Here s =$100 
 
 £=3 months = .25 years. 
 r— =8.05. 
 
 100X.25X-05 
 '" ~ l-f-.25X.05 
 
 1.25 
 
 — 1.0125 
 =$1,235 dis. 
 
 ANNUITIES AT SIMPLE INTEREST. 
 
 Problem V. — To find the amount which must he paid at the end oft years, 
 for the enjoyment of an annuity a, simple interest being allowed at the rate r. 
 At the end of the first year the annuity a will be due ; at the end of the 
 second year a second payment a will become due, together with ar the in- 
 terest for one year upon the first payment ; at the end of the third year a 
 third payment a becomes due, together witu 2ar the interest for one year 
 upon the former two payments, and so on ; the sum of all these will be the 
 amount required. 
 Thus: 
 
 At the end of the first year, the sum due is a. 
 At the end of the second year the sum due is a-{-ar. 
 At the end of tho third year, the sum cue i? a4-2ar. 
 At the end of the fourth year, the sum due is a-\-3ar 
 
 &c. &c. &c. 
 
 At the end of the t th year, the sum due is #+(f — l)ar 
 
 Hence, adding these all together for the whole amount, 
 
 5 = to-L.ar(l + 2+3+. . . . (<— 1))- 
 Or, taking the expression for the sum of the arithmetical senes, 1-J-2-J-3 
 
 4- («-l) 
 
 t(t—l) 
 s=ta+ra. ± g (5) 
 
288 ALGEBRA. 
 
 Problem VI. — To find the present value of an annuity npayablefor t years 
 simple interest being allowed at the rate r. 
 
 It is manifest that the present value of the annuity must be a sum such that, 
 if put out at interest for t years at the rate r, its amount at the end of that 
 period will be the same with the amount of the annuity. 
 
 Hence, if we call this present value p, we shall have, by Problems I. and V., 
 p-^-ptr= amount of annuity. 
 t{t—l) 
 
 t(t—l) 
 ta-\-ra.- 
 
 p= ' 1.2 
 
 1 + tr 
 _ta 2+(<— l)r 
 ~~2 ' ~~ : l + tr ' 
 
 (6)' 
 
 COMPOUND INTEREST. 
 
 Problem VII. — To find Oie amount of a sum p laid out for t years, com- 
 pound interest being allowed at the rate r. 
 
 At the end of the first year the amount will be, by Problem II., 
 
 p+pr, orp{l-{-r). 
 
 Since compound interest is allowed, this sum ^(l+ r ) now becomes the 
 principal, and hence, at the end of the second year, the amount will be 
 p(l-\-r), together with the interest onp(l-\-r) for ene year; that is, it will be 
 
 P(l+r)+pr£+r)> orjp(l+r)«. 
 
 The sum p(l+r) 5 must now be considered as the principal, and hence the 
 whole amount, at the end of the third year, will be 
 
 p(l +r y+pr(l + r )-, or^l + r) 3 . 
 And, in like manner, at the end of the t ih year, wo shall have 
 
 s=p(l + ry (7) 
 
 Any three of the four quantities, 5, p, r, t, being given, the fourth may al- 
 ways be found from the above equation. 
 
 EXAMPLE I. 
 
 Find the amount of $15.50 for 9 years, compound interest being allowed 
 at the rate of 3,V per cent, per annum, the interest payable at the end of 
 nacli year. 
 
 By equation (7), 
 
 s=p(l + ry 
 .-. log. s= log. p+t log. (1-f-r). 
 But J9=S15.50 
 
 t = 9 years 
 r = S.035 
 .-. log ^ = 1.1903317 
 Hog. (l+r) = 0. 1 344627 
 
 .-.log. 5 = 1.3247911 = log. of 21.12181 
 .-. 8=321.12481. 
 
 * It is unnecessary to give any examples under this rule, as the purchase of annuities 
 nt simple interest can never be ol 
 
INTEREST AND ANNUITIES. 289 
 
 EXAMPLE II. 
 
 Find the amount of .£182 12s. 6cl. for 18 years, 6 months, and . days, at 
 the rate of 3£ per cent, per annum, compound interest, the interest being 
 payable at the end of each year. 
 
 In this case, it will bo convenient, first, to find the amount at compound in- 
 terest of the above sum for 18 years, and then calculate the interest on the 
 result for the remaining period. 
 By formula (7), 
 
 s= P (l + ry 
 log. s=log. p+t log. (1+r) 
 
 Here p=<£l82. 12s. 6d.= £182. 625 
 r= =£.035 
 
 t= =18 years 
 
 .-. log. p =2.2615602 
 Hog. (1+r) = 0.2689254 
 
 .-. log. s = 2.5304856= log. of 339.224. 
 Again, to find the interest on this sum for the short period, we have 
 
 i=st'r 
 .'. log. i= log. s-\- log. i'-|- log r. 
 Here 5=66339.224 
 r=c£.035 
 
 t'=6 months, 10 days— .527402 years 
 .-. log. s = 2.5304856 
 log. r=2.5440680 
 log. i' =1.7221401 
 
 .-. log. si'r=0.7966937= log. of 6.2617200 
 
 .-. srr=£6.26172. 
 
 J'he whole amount required will, therefore, be 
 
 s+s V r=c£339.224+c£6.26172 
 
 =£345 95. 8±d. 
 I 2 
 
 EXAMPLE III. 
 
 Required the compound interest upon $410 for 2| years at 4| per cent, per 
 
 annum, the interest being payable half yearly. 
 
 In this case the time t must be calculated in half years ; and, since we have 
 
 r 
 supposed r to be the interest of $1 for one year, we must substitute -, which 
 
 will be the interest of $1 for half a year ; the formula (7) will thus become 
 
 S =i , ( 1 +l) 31 
 
 .-. log. s= \og.p+ 21 log. (l + o) ■ 
 
 Here p = $410 
 
 r=$.045 
 
 21 =5 half years 
 
 .-.log.;; = 2.6127839 
 5 log. 1.0225 = 0.0483165 
 
 log.5=2.66110()4 = log. of 456.2471 
 .•• s= $458.2471. 
 T 
 
£90 ALGEBRA. 
 
 The inte » ~*iust t>cs the difference between this amount ar \ the original 
 principal ; 
 
 .'.i=s — p 
 
 = $458.247— $410 
 =§48.247. 
 
 EXAMPLE IV. 
 
 $400 was put out at compound interest, and at the end of 9 years amounted 
 to $569,333 ; required the rate of interest per cent. 
 Here s, p, t are given, and r is sought. 
 From formula 
 
 we have log. (1 + ?■)=-( log. s — log.^>). 
 
 Here s = 8569.3333 
 
 ;?=$400 
 
 <=9 years 
 
 .-. log. s = 2. 7553666 
 
 log.jp=2.6020600 
 
 .-.log.s— log.p= .1533066 
 
 .1533066 
 log.(l+r)= - 
 
 = .0170340 
 =log. of 1.04 
 .•. r= .04= 4 per cent. 
 
 example v. 
 
 in what time will a sum of money double itself, allowing 4 per cent, com- 
 i mnd interest ? 
 
 Here s, p, r are given, and t is sought. 
 From the formula (7) we have 
 
 s=p(l+r)\ 
 But here s=2p 
 
 .•.2j?=jp(l+r) 1 
 .-.2=(l+r) t 
 log. 2 
 ' = Iog.(l + r) 
 .3010300 
 
 "".0170333 
 = 17.673 years 
 = 17 years, 8 months, 2 days. 
 In like manner, if it bo required to find in what tune a sum will triple itself 
 •t the same rate, we have 
 
 log. 3 
 t = 
 
 log. 1.04 
 .4771213 
 
 .0170333 
 =28.011 years 
 =28 years, o months, 3 days. 
 
INTEREST AND ANNUITIES. 291 
 
 PRESENT VALUE AND DISCOUNT AT COMPOUND INTEREST. 
 
 If we call p tlio present value of a sum s due t years hence, and d its dis- 
 • count, reasoning precisely in the same manner as in the case of simple inter- 
 est, we shall find 
 
 *=(IT^ (8) 
 
 M 1 -^) < 9 > 
 
 ANNUITIES AT COMPOUND INTEREST. 
 
 Problem VIII. — To find the amount of an annuity a continued for t yeais, 
 compound interest being allowed at the rate r. 
 
 At the end of the first year the annuity a will become due ; at the end of 
 the second year a second payment a will become due, together with the in 
 terest of the first payment a for one year, that is, ar ; the whole sum upon 
 which interest must now be computed is thus, 2a-{-ar. 
 
 At the end of the third year a further payment a becomes due, together with 
 the interest on 2a-{-ar, i. e., 2ar-\-ar^; the whole sum upon which interest 
 must now be computed is 3a-\-3ar-\-ar 2 . The result will appear evident 
 when exhibited under the following form : 
 Whole amount at the end of first year, =a. 
 Whole amount at the end of second year, =za-\-a-\-ar 
 
 =a + a(l+r). 
 Whole amount at the end of third year, =a-\-a-\-a(l-{- r)-\-ar -\-ar{l-\-r) 
 
 = a + a(l + r) + a(l+ry: 
 Whole amount at the end of fourth year, =a-\-a -f- a(l-\-r) -j- a(l-|-r) 8 + o.r 
 
 +ar(l+r) + ar(l + r) 2 . 
 = a+a(l+r)+a(l+r)-+a(l+r) 3 
 5cc. &c. &c. 
 
 Whole amount at the end of t h year, r=a4-o(l + r)+a(l + r) 3 +a(l-f -) :i 
 
 + a(l+r)<-i. 
 
 Hence the whole amount is, in terms of the sum of a geometric progression 
 
 5 = ajl + (l + r) + (l+r)'+ + (1 + ^1 
 
 S-^i — (10) 
 
 Problem IX. — To find the present value of an annuity a payable for X 
 years, compound interest being allowed at the rate r. 
 
 It is manifest that the present value of this annuity must be a sum such, 
 that if put out at interest for t years at the rate r, its amount at the end of that 
 period will be the same as the amount of the annuity. 
 
 Hence, if we call this present value p, we shall have, by Prcbs. VII. and 
 VIII., 
 
 p(l-^-r) l = amount of annuity 
 
 (i+rr-i 
 
 =a. 
 
 r 
 
 1 r{l + ry 
 a (l-|_ r)«— 1 
 
 P= r( i + r) . - a 
 
 (H) 
 
Now 
 
 Also, 
 
 292 ALGEBRA. 
 
 EXAMPLE. 
 
 What is the present value of an annuity of $500, to last for 40 years, cow 
 pound interest being allowed at the rate of 2\ per cent, per annum. 
 By formula (11), 
 
 _a (1 + r) 1 — 1 
 i ? = ;- (1+r) 1 " 
 
 Here 
 
 a =$500 
 r =$.025 
 t =40 years; 
 
 .-. (l+r) t =(1.025)*'. 
 
 log. (1.025)«=40 log. 1.025 
 = 40 X .0107239 
 = .4289560 
 = log. 2.685072 
 .-. (1.025) 40 =2.685072=(l+r) t . 
 
 a 500 
 -r=^5 = 2000 ° 
 
 1.685072 
 ••^= 20000X 2^85072 
 =20000 X-62757... 
 = 12551.40 dollars. > 
 
 REVERSION OF ANNUITIES. 
 
 Problem X. — To find the present value (P) of an annuity a which is to com 
 mence after T years, and to continue for t years. 
 
 The present value required is manifestly the present value of a for T-f f 
 
 years, minus the present value of a for T years. 
 
 a (l + r) T + l — 1 
 By Problem IX., the present value of a for T+* years =- . - . T+t 
 
 „ , m a (!+ r ) T — I 
 
 By Problem IX., the present value of a for T years =- . . 77 — 
 P=". \ (l+r)-T-(l+r)-<T + t>£ (12 ) 
 
 PURCHASE of estates. 
 
 Problem XL — To find the present value p of an estate, or perpetuity, whose 
 annual rental is a, compound interest being calctdated at the rate r. 
 
 The present value of an annuity a, to continuo for t years, by Prob. IX., is 
 
 p=\\l-{l + r)-<\; 
 
 but if tho annuity last forever, as in tho caso of an estate, then t= t\ and 
 
 •. - — ; — - = _=0 ; hence, in tho present case, 
 
 (I-)-?")' OD 
 
 P=°Z (W) 
 
INTEREST AND ANNUITIES. 293 
 
 EXAMPLE. 
 
 What is the value of an estate whose rental is $1000, allowing the pur- 
 chaser 5 per cent, for his money ? 
 Here 
 
 a=$1000 
 r = $.05 
 
 1000 
 ••• P =~^b 
 
 ==20000, or 20 years' purchase. 
 
 REVERSION OF PERPETUITIES. 
 
 Problem XII. — To find the present value of an estate, or perpetuity, whost 
 annual rental is a dollars, to a person to whom it will revert after T years, 
 compound interest being allowed at the rate r. 
 
 By Problem X., the present value of an annuity, to commence after T years, 
 and to continue for t years, is 
 
 p=£j(l+r)-*-(l+r)r(*wj 
 
 In the present case, <=co , and .-. (l+r) ~ (T+t >=0 ; hence we shall have 
 
 P=7-(i+7jx (14 > 
 
 EXAMPLES FOR PRACTICE. 
 
 (1) Find the interest of 3555 for 2^ years at 4£ per cent, simple interest. 
 
 Ans. $ 65.906. 
 
 (2) In what time will the interest of $1 amount to 75 cents, allowing 4| per 
 cent, simple interest ? 
 
 Ans. 16 years, 8 months. 
 
 (3) What is the amount of $120. 50 for 2| years at 4£ per cent, simple in- 
 terest ? 
 
 Ans. $134,809. 
 
 (4) The interest of d£25 for 3| years, at simple interest, was found to be 
 d€3 185. 9d. ; required the rate per cent, per annum. 
 
 Ans. 4|. 
 
 (5) Find the discount on d£100 due at the end of 3 months, interest being 
 calculated at the rate of 5 per cent, per annum. 
 
 Ans. c£l 4s. 8\d 
 
 (6) What is the present value of the compound interest. of c£l00 to be re- 
 ceived five years hence at 5 per cent, per annum ? 
 
 Ans. <£78 7*. 0\d. 
 
 (7) What is the amount of <£721 for 21 years at 4 per cent, per annum 
 
 compound interest? 
 
 Ans. .£1642 195. $\d 
 
 (8) The rate of interest being 5 per cent., in what number of years, at com- 
 pound interest, will $1 amount to $100 ? 
 
 Ans. 94 years, 141.4 days. 
 
 (9) Find the present value of <^430, due nine months hence, discount being 
 
 allowed at 4i per cent, per annum. 
 
 Ans. d£415 19s. 2±d. 
 
294 ALGEBRA 
 
 (10) Find the amount of 81000 for 1 year at 5 per cent, per annum, coni- 
 pound interest, the interest being payable daily. 
 
 Ans. $1051.288 nearly 
 
 (11) What sum ought to be given for the lease of an estate for 20 years, of 
 
 the clear annual rental of £100, in order that the purchaser may make 8 per 
 
 cent, of his money ? 
 
 Ans. .£981 16s. 3^(1 
 
 (12) Find the present value of £20, to be paid at the end of every five years, 
 
 forever, interest being calculated at 5 per cent. 
 
 Ans. £72 7s. 911. 
 
 (13) What is the present value of an annuity of £20, to contiuue forever, 
 and to commence after two years, interest being calculated at 5 per cent. ? 
 
 Ans. ,£362 16s. 2}d. 
 
 (14) The present value of a freehold estate of £100 per annum, subject to 
 the payment of a certain sum (A) at the end of every two years, is £1000, 
 allowing 5 per cent, compound interest. Find the sum (A). 
 
 Ans. A =£102 10s. 
 
 (15) What is the present value of an annuity of £79 4s., to commence 7 
 
 years hence aud continue forever, interest being calculated at the rate of 4), 
 
 per cent. ? 
 
 Ans. £1293 5s. U\d. 
 
 INTERPOLATION. 
 
 236. This name is applied to the process of finding intermediate numbers 
 between those given in tables. 
 
 Tables are generally calculated from an algebraic formula in which there 
 are two variable quantities, the one of which is called a. function of the other, 
 the latter being usually called the argument of the function. 
 
 Thus, logarithms are functions of the numbers to which they belong, the 
 numbers being the arguments. Several formulas expressing the relation be- 
 tween a number and its logarithm have been seen by the student, and will 
 serve to exemplify the formulas in general of which we are now speaking. 
 
 The substitution of successive numbers for the argument, the calculating of 
 the corresponding values of the function, and writing the residts in a table, is 
 called tabulating the formula. 
 
 If the formulas which have been derived under our articles upon interest 
 and annuities should be tabidated, they would furnish what are called interest 
 tables. 
 
 The function frequently depends upon two arguments, as in the formula 
 for simple interest, 
 
 i=ptr (1) 
 
 Here the function is i, the interest, and the arguments are, p the principal, and 
 r the rate. This requires a tablo of double entry, the usual form of which is 
 a table in several columns occupying the whole width of the page, the ai JU- 
 ments being placed, the successive values of the one in a horizontal line at ihe 
 heads of the columns, and of the Other in a vertical line at the side of the | 
 the corresponding values of the function being placed in the column wader ana 
 uf its arguments, and on the horizontal lino of the other. The formula (1) 
 
INTERPOLATION. 
 
 295 
 
 above may employ a table of triple entry, the three arguments being the prin- 
 cipal, the rate, and the time. Such a table is formed by giving a whole page 
 to the argument of rate, the side and top being occupied by the arguments of 
 principal and time. 
 
 Where the differences of the functions are proportional to the differences 
 of their arguments, then the interpolation is made by simply solving a pro- 
 portion, the first two terms of which are the difference of the tabulated func- 
 tions and the difference of their arguments; the third term being the differ- 
 ence between one of the tabulated arguments and that whose function is to 
 be interpolated ; the fourth, or unknown, term of this proportion will be the 
 interpolated function required. This is called the method by first differences, 
 and has been exemplified in taking out logarithms of large numbers not found 
 exactly in the tables. 
 
 When the differences of the functions are not nearly proportional to the 
 differences of the arguments, as in the case of the logarithms of small numbers, 
 the method of interpolation above described would not be sufficiently accurate. 
 The nature of the variation of the function, as the argument varies in value, is 
 made sensible by taking the difference between each two of three consecutive 
 functions in the table, and comparing the difference between the first and sec- 
 ond with the difference between the second and third. If these differences 
 are the same, we have seen, in the note to (Art. 233), that the method of first 
 differences already explained applies ; but if they are not, their difference, 
 which is called a second difference, will, by its magnitude, indicate the degree 
 of inaccuracy of the method of first differences. This exposition will serve to 
 exhibit, in a general way, the nature and office of second differences. We 
 proceed to give a more analytic development of the use of second, third, &c. 
 differences, the latter holding ths same relation to the second differences that 
 these do to the first. 
 
 Let/and/'4" ( 'i represent two consecutive functions in the table, Jj being 
 their first difference. The next consecutive function, if the first differences 
 were constant, would be expressed byf-{-2S l ; but as they are supposed not 
 to be, it must be expressed by the formf-\-2S l -\-d 2 , c5 2 being the second dif- 
 ference, or difference between the two first differences, (5, and ^,-j-^,. The 
 scheme below will show the form of the successive functions : 
 
 Function* 
 / 
 
 /+4<J, + 6<J 2 +4<J 3 -H 4 
 
 and so on ; from which we perceive that the coefficients are the same as m the 
 expansion of a binomial, that of the second term being the number of the con- 
 secutive functym after the first function. Denoting this number by n, we 
 have for the general form of the nth function after the first, 
 
 -l)(n-2), 
 
 1st Differences. 
 
 2d Differences. 
 
 3d Differ- 
 ences. 
 
 4th Dif 
 Terence* 
 
 t5,-|-f5 2 
 
 <5 2 
 
 <5 2 + <5 3 
 d a +2<J 3 +3 4 
 
 <>3 + <'4 
 
 ^ 
 
 n(n — ') n(n- 
 
 -*3+ ••• + *. 
 
 [C] 
 
 1.2 " a ' 1.2.3 
 
 Suppose, now, that, a value of the function intermediate between the first and 
 
 second of the series in the table bo required, n here, instead of be/ng an entire 
 
 number, is a fraction. If the value of the function be required, corresponding 
 
m 
 
 ALGEBRA. 
 
 to a value of the argument midway between its consecutive values in the tabic 
 n becomes equal to -. If the arguments of the tables differ by 24 hours, and 
 
 3 1 
 
 the function be required for 3 hours, n becomes equal to — , or -. If the tabu 
 
 lar arguments differ by 1 hour, or 60 minutes, and the function be required for 
 
 15 1 
 an argument 15 minutes beyond an even hour, n=—=-. 
 
 EXAMPLE. 
 
 Given the logs, of 15, 16, 17, 18, 19, to find that of 17.25. 
 
 Arg or No. 
 
 Func. or Log. 
 
 1st Difs. 5 1. 
 
 2d Difs. S». 
 
 3d Dife, r':i. 
 
 ,54. 
 
 15 
 16 
 
 17 
 18 
 lit 
 
 1.17609126 
 1.20411998 
 1.23044892 
 1.25527251 
 1.27875360 
 
 2802872 
 2632894 
 2482359 
 2348109 
 
 — 169978 
 
 — 150535 
 
 — 134250 
 
 + 19443 
 + 16285 
 
 — 3158 
 
 The numbers in the third column are obtained by taking the differences of 
 
 the consecutive numbers in the second. The numbers in the fourth column 
 
 from the second in the same way. 
 
 9 9 
 
 As 2.25 is - the interval between 15 and 18, we make n=-, and have for 
 4 4 
 
 formula (C), taking <J 1= 2802872, <l= — 69978, (5 3 =19443, <5 4 =— 3158. 
 
 The result would be nearly the same by neglecting <5 4 and using the mean of the two 
 third differences* 
 
 
 f= 
 
 1.176126 
 
 
 n6 l = 9 -6- 
 
 306462 
 
 
 "(n — 1 )* _ 45 ,< _ 
 1.2 2 32 s 
 
 — 239031 
 
 
 n(n — l)(n—2) A __ 45 A _ 
 1.2.3 3 — 384 3 
 
 2278 
 
 n(n- 
 
 -l)(n-2)(n-3) A 135 A _ 
 1.2.3.4 * 6144" * 
 
 69 
 
 Value of func. required, viz., log. 17.25 = 1.23678904 
 The formula for interpolation may be derived very elegantly by the method 
 of indeterminate coefficients. Thus, let y represent the value of the interpo- 
 lated function to be found, A the argument in the table, m the number of parts 
 (4 tl ' 8 in the example above) between A and the consecutive argument of the 
 table, and n the whole number of parts (4 in the above example) between 
 these consecutive arguments. It is evident that y, depending on A and m. 
 may be expressed in terms of these. Assume, therefore, 
 
 ?/ = A4-Bm4-Cm ! 4-Dm+, &c., 
 in which B, C, D, &c, are undetermined coefficients, whose values are to be 
 found. 
 
 Now let m have successive values, represented by 0, n, 2n, 3w, &c, then 
 the corresponding values of y will be 
 
 * As means are much used in calculations with tables, it may be well to advertise. th« 
 student that a mean of three numbers is obtained by adding diem together and dividing by 
 3 ; of five numbers, by adding thorn together and dividing the sum by 5, an* «o on. 
 
INTERPOLATION. 297 
 
 A (1) 
 
 A + Bn4-Cn 2 +Dn 3 +,&c (2) 
 
 A+B.2n+C(2ra) 2 +D(2rc) 3 +,&c (3) 
 
 A+B.3tt + C(3w) 2 +D(3n) 3 +,&c (4) 
 
 &c. 
 Subtracting successively (1) from (2), (2) from (3), &c, and representing 
 the remainders by P', Q', R/, &c, and dividing by n, we have 
 
 P' 
 
 — r=B + C.n+D>i 2 +,&c (5) 
 
 Q' 
 
 — =B-f C.3/i + D7» 2 +,&c (6) 
 
 IV 
 
 R' 
 
 — =B + C.5»+Dl9?i 2 +,&c (7) 
 
 &c. &c. 
 
 Again, subtracting successively (5) from (6), (C) from (7), &c, and repre- 
 senting the remainders by P", Q", &c, and dividing by 2n, we get 
 
 ^=C+D.3»+,&c (8) 
 
 ^=C + D.G«+,& C (9) 
 
 &c. &c. 
 
 Next, subtracting (8) from (9), &c, and representing the remainder* Uy P"\ 
 &c, and dividing by 3n, we have 
 
 ^= D + ' &C < 10 > 
 
 0"— P" _ R'— Q' „ Q'— P' 
 But T'"=^- ; alsoQ" = — and P"=- ; 
 
 2n n n 
 
 i 
 
 (R'-Q')-(Q'-P') 
 
 "~ 2ra 2 
 
 Putting d 3 for the numerator of this fraction, we have by (10), 
 
 n— — —- ^2. 
 
 3/i 6/i 3 ' 
 Substituting this value of D in (8), and transposing, there results 
 
 r— — ^l 
 
 2w — 2ra 2 ' 
 
 Q'— P' 
 But P"=— , and putting d 2 for Q'— P', we obtain 
 
 2n 2 2ra 2 * 
 Again, substituting these values of D and C in (5), and transposing, we have 
 
 „ P' do d 3 <*3 
 
 n 2n r 2n 6«.' 
 
 or, putting d, for P', and simplifying, 
 
 d, do do 
 
 n 2n 3n 
 
 Finally, substituting these values of the coefficients B, C, D ... in the as 
 
 sumed equation, we obtain 
 
 to 1 m (m \ 1 ?«/m 2 3m „ \ , 
 
 w=A+-d, + - -(-_i) ( 5„+--(— 4-2)d 3 + , &c, 
 
 * ' n ' 2 n\n I - ' 6 n \ n 2 n ' / a ' 
 
298 
 
 ALGEBRA. 
 
 as the formula for interpolation, which coincides with the one obtained before 
 ^i» ''si ^3 •• being the first, second, and third differences of the functions, a, is 
 evident from the manner in which tiny have been assumed above. 
 
 Let us apply it to a table in the Nautical Almanac, which gives the moon's 
 latitude at noon and midnight for every day in the year. 
 
 EXAMPLE. 
 
 Let it be required to find the moon's latitude for August 4, 1842, at 16" 18" 
 mean time at Greenwich, that is, at 4.3 hours after midnight. 
 
 Moon's Latitude. 
 
 Si- 
 
 Oo* 
 
 Mean Secocd Difference. 
 
 o ' " 
 
 Aug. 4. Noon, +0 45 48.1 
 Midni-rht, +0 5 54.6 
 
 Aug. 5. Noon,* —0 34 33.1 
 Midnight, —1 14 49.4 
 
 — 39 53.5 
 —40 27.7 
 —40 16.3 
 
 + 34.2 
 — 11.4 
 
 + 11.4 
 
 Now, to apply the formula, we have 
 
 A=0° 5' 54".6, (5, = — 40' 27".7, or —40.463 minutes; 
 
 m 4.3 m 
 
 -=—=0.358, -<5. = — 14' 29".16; 
 12 n l 
 
 n 
 
 m mlm \ . 
 o\ = + ll".4, — 1 = — 0.642, i -(- l)<J.r=— 1".31. 
 
 1 n - n \n I - 
 
 Therefore, y = — 0° 8' 35".87, which, without the sign — , is the moon's 
 correct latitude south at the time for which it was required. 
 
 Second differences will ordinarily insure sufficient accuracy. Third and 
 fourth differences are rarely used. 
 
 INEQUATIONS. 
 
 237. In discussing algebraical problems, it is frequently necessary to intro- 
 duce inequations, that is, expressions connected by the sign >. Generally 
 speaking, the principles already detailed for the transformation of equations 
 are applicable to inequations also. There are, however, some important ex- 
 ceptions which it is necessary to notice, in order thai the student may guard 
 against falling into error in employing the sign of inequality. These excep- 
 tions will be readily understood by considering the different transformations in 
 succession. 
 
 I. If we add the same quantity to, or subtract it from, the two members of any 
 inequation, the resulting inequation will always hold good, in the same sense 
 as the original inequation ; that is, if 
 
 a~>b, then a+a'>t + a', and a — n'>b— a'. 
 
 Thus, if 
 
 8>3, wo have still 8+5>.'+'>, and 8— 5>3— 5. 
 So, also, if 
 
 — 3<— 2, wo havo still — 3 + 6 <— 2+6. and — 3— 6<— 2 — fc.f 
 
 ,* Tin' i> n'l latitude is marked + when north, — when south. 
 
 t Tin- negative qt iter numerical « al te ia ata :.. - lered less titan the 
 
 negative quantity ol' 1'> S numerical \ 
 
INEQUATIONS. 209 
 
 The truth of tliis proposition is evident from what has been said with refer- 
 ence to equations. 
 
 This principle enables us, as in equations, to transpose any term from one 
 member of an inequation to the other by changing its sign. 
 Thus, from the inequation 
 
 a°~+ b°~>3b-—2a 2 , 
 we deduce 
 
 a 2 +2a 2 >3& 2 — 6 2 , 
 or 
 
 3a 2 >26 2 . 
 
 II. If we add together the corresponding members of two or more inequations 
 which hold good in the same sense, the resulting inequation will always hold 
 good in the same sense as the original individual inequations ; that is, if 
 
 a>&, c>d, e>/, 
 then 
 
 a + c+e>b+d+f. 
 
 III. But if we subtract the corresponding members of two or more inequations 
 which hold good in the same sense, the resulting inequation will not always 
 hold good in the same sense as the original inequations. 
 
 Take the inequations 4<7, 2<3, we have still 4— 2<7 — 3, or 2<4. 
 
 But take 9<10 and 6<8, the result is 9— 6> (not <) 10—8, or 3>2. 
 
 We must, therefore, avoid as much as possible making use of a transforma- 
 tion of this nature, unless we can assure ourselves of the sense in which the 
 resulting inequality will subsist. 
 
 IV. If we multiply or divide the two members of an inequation by a positive, 
 quantity, the resulting inequation will hold good in the same sense as the original 
 ineauation. Thus, if 
 
 a b 
 a<,b, then ?na<^mb, — <- 
 
 -a^> — b, then — na^> — nb, 
 
 m m 
 a o 
 n^ n 
 
 This principle will enable us to clear an inequation of fractions. 
 
 Thus, if we have 
 
 a-—b"- c^—d- 
 
 > 
 
 2d <■ 3a ' 
 
 multiplying both members by Gad, it becomes 
 
 3a{a?—¥)>2d(c' 2 —d i ). 
 But, 
 
 V. If we multiply or divide the two members of an inequation by a negative 
 quantity, the resulting inequation will hold in a sense opposite to that of the 
 original inequation. 
 
 Thus, if we take the inequation 8>7, multiplying both members by — 3, 
 we have the opposite inequation, — 24 < — 21. 
 
 8 7 8 7 
 
 Similarly, 8>7, but ^^<-^$ or —3 < — 3" 
 
 VI. We can not change the signs of both members of an inequation unless we 
 reverse the sense of the inequation, for this transformation is manifestly the same 
 thing as multiplying both members by — 1. 
 
300 ALGEBRA. 
 
 VII. If both members of an inequation be positive numbers, we can raise them 
 to any power without altering lite sense of the inequation ; that is, if 
 
 a>l, then a n >&\ 
 Thus, from 5>3 wo have (5) 2 >(3) 2 , or 25>9. 
 
 So, also, from (a+ty^c, we have (a-|~^) : >c s . 
 But, 
 
 VIII. If both members of an inequation be not positive numbers, we can not 
 determine, a priori, the sense in which the resulting inequation will hold good, 
 unless the power to which they are raised be of an uneven degree. 
 
 Thus, — 2<3 gives (— 2) 2 < (3) : , or 4<9; 
 
 But, _3>_5 gives (—3) 2 <( — 5) 2 , or 9<25; 
 
 Again, — 3>— 5 gives (— 3) 3 >( — 5) 3 , or — 27>— 125, 
 In like manner, 
 
 I X. We can extract any root of both members of an inequation wiOioul alter 
 ing the sense of the inequation ; that is, if 
 
 a>6, then V*a> Vb. 
 If the root be of an even degree, both members of the inecuation must 
 necessarily be positive, otherwise wo should be obliged to introdu :e imaginary 
 quantities, which can not be compared with each other. 
 
 EXAMPLES IN INEQUATIONS. 
 
 (1) The double of a number, diminished by G, is greater than 24 ; and triple 
 the number, diminished by G, is less than double the number increased by 10. 
 Required a number which will fulfill the conditions. 
 
 Let x represent a number fulfilling the conditions of the question ; then, in 
 the language of inequations, we havo 
 
 2x— 6>24, and 3.r— G<2x+10. • 
 From the former of these inequations we have 
 
 2x>30, or .r>15; 
 and from the latter we get 
 
 3x— 2x<10 + G, or a:<lG; 
 therefore 15 and 16 are the limits, and any number betwcoa these limits will 
 satisfy the conditions of the question. Thus, if we take the number 15-9, we 
 nave 
 
 15-9X2 — G>21 by 1-8, 
 while 15-9X3 — 6<15-9X2+10 by 0-1. 
 
 5 4 
 (2) 3x-2>^-- 
 
 .-. 30r— 20>25.r— 8 
 30z— 25x>20 — 8 
 6x>12 
 12 
 X >~5- 
 
 3) 43— 5.r<10— 8x. 
 
 7 5 
 (4) 6-4*<8-2.r. 
 
 Ans. r< — 11. 
 
 A 
 
 An-, r- -. 
 
INEQUATIONS. 30J 
 
 12 
 In tlie second example, — , or 2|, is an inferior limit of the values of r. 
 
 82 
 In the t third, — 11, and, in the fourth, — , or 9^, are superior limits of the 
 
 value of x. If the second and fourth of the above inequalities must be verified 
 simultaneously by the values of x, these values must be comprised between 
 2§ and 9£. If the third and fourth, it is sufficient that it be less than — 11. 
 Finally, there is no value which will verify at the same time the 2° and 3°. 
 
 (5) 3x— 2;y>5, 5z+3i/>lG; 
 
 < 5+2y ^ 16-3y 
 .-. ar> — — and a:> — . 
 
 We can attribute to y any value whatever, and for each arbitrary value ol 
 y we can give to x all the values greater than the greatest of the two quan- 
 tities 
 
 5+2?/ 16—3i/ 
 — 3~~ ' ~~ 5~ " 
 We determine, also, from the proposed inequalities, 
 
 ^3x— 5 ^ 16— bx 
 
 K-g-' y>— — • 
 
 In order that these last two may be fulfilled, 
 
 Zx—h 16— 6a; 
 
 47 
 Thus x can receive only values superior to — , or 2 T 9 g, and for each value 
 
 of x there should be admitted for y but values comprised between the two 
 limits above. 
 
 (6) ar 3 4-4x>12 
 .-. x 2 +4ar+4>16 
 
 z+2>±4 
 
 r>2, or —2. 
 The inferior limit of x is -}-2. 
 
 (7) z 2 +7a;<30. 
 
 Ans. x<C3 or — 10. 
 
 The superior limit of a: is — 10. 
 
 P 
 
 (8) Reduce £ n >£ n to its most simple form. 
 
 Ans. x^>p-\-1 
 
 j.n-r+1 
 
 (9) Reduce a: n >_p — to its most simple form. 
 
 Ans. r>l-f- \Jp. 
 
302 ALGEBRA. 
 
 GENERAL THEORY OF EQUATIONS. 
 
 THE NATURE AND COMPOSITION OF EQUATIONS 
 
 238. The valuable improvements recently made in the process for the de- 
 termination of the roots of equations of all degrees, render it indispensably 
 necessary to present to the student a view of the present statg of this interest- 
 ing department of analytical investigation. The beautiful theorem of M. Sturm 
 for the complete separation of the real and imaginary roots, and for discover- 
 ing their initial figures, combined with the admirable method of continuous 
 approximation as improved by Horner, has given a fresh impulse to this branch 
 of scientific research, entirely changed the state of the subject, and completed 
 the theory and numerical solution of equations of all degrees. 
 
 We recapitulate here two or three 
 
 DEFINITIONS. 
 
 1. An equation is an algebraical expression of equality between two quan- 
 tities. 
 
 2. A root of an equation is that number, or quantity, which, when substi- 
 tuted for the unknown quantity in the equation, verifies that equation. 
 
 3. A function of a quantity is any expression involving that quantity ; thus. 
 
 ax 1 + b 
 ax--\-b, flr'-f-cx-f-rf, , , a 1 are all functions of x ; and also ax 2 — by 9 , 
 
 . 2x-f3y 
 
 V4x— 5?/, 3x— 2f 2/ 2 +2/*+ a;S! +a 2 +&+2, are all functions of x and y. 
 
 These functions are usually written/(x), and/(x, y). 
 
 4. To express that two members of an equation are identical or true for 
 every value of x, the sign az is sometimes used. 
 
 PROPOSITION I. 
 
 Any function ofx, of the form 
 
 x n -\-j)x a ~ l -f qx n ^-\- rx"- 3 -f- 
 
 when divided by x— a, will leave a remainder, which is the same function of a 
 that the given polynomial is ofx. 
 
 Let f(x)=x n +px a - l + qx"-"-+ ; and, dividing/(x) by x— a, let Q de- 
 note the quotient thus obtained, and R the remainder which does Dot involve 
 x ; hence, by the nature of division, we have 
 
 /(*):rQ(z-a)+R. 
 
 Now this equation must be true for every value ofx, because its truth de- 
 ponds upon a principle of division which is independent of the particular values 
 of the letters; hence, if x=a, we have 
 
 */(a) = + R; 
 and, therefore, the remainder R is the same function of a that the proposed 
 polynomial is of x. 
 
 EXAMPLES. 
 
 (l) What is the remainder of x- — Gx+7. divided byx—2, without actually 
 performing the operation ? 
 
 * The rtndenl will recollect tliatyfa) stands for x n +px— l +, $p., ami feat, therefore, 
 
 I lor^-f/,," X -\-qn n "-f. fee. 
 
GENERAL THEORY OF EQUATIONS. J3 
 
 (2) What is the remainder of x 3 — 6x"-\-8x— 19, divided by r+3 1 
 
 (3) What is the remainder of x 4 -\-6x*-{-7x' 2 -\-5x— 4, divideo by a:— 5 1 
 
 (4) What is the remainder of z*-\-px 2 -\-qx-\-r, divided by £ — al 
 
 ANSWERS. 
 
 (1) R=2 2 — 6x2+7 = — 1. 
 
 (2) R=(— 3) 3 — 6( — 3) 2 +8(— 3) — 19 =— 124. 
 
 (3) 1571. 
 
 (4) a?-\-pa' i -{-qa-\-r. 
 
 PROPOSITION II. 
 
 if a is the root of the equation, 
 
 x a + A^"-^ A 2 x n - 2 + An-^-r- A„_ 1 j4-A n =0, 
 
 the first member of the equation is divisible by x — a. 
 
 If the division be perfoi - rned, the remainder, according to the preceding 
 proposition, must be of the form 
 
 a n +Aia n_1 -|- A 2 a n_2 {• A n _ 2 a 2 + A n _ia-j-A n ; 
 
 i. c., the same function of a that tho first member of the proposed equation is 
 of x ; and, therefore, since a is a root of the equation, the remainder vanishes, 
 and the polynomial, or first member of the equation, is divisible exactly by 
 x — a. 
 
 Conversely, if the first member of an equation f(x)=0 be divisible by x — a, 
 then a is a root of the equation. 
 
 For, by the foregoing demonstration, the final remainder is f(a) ; but since 
 f(x), or the first member of the equation, is divisible by x — a, the remainder 
 must vanish; hence /(a) =0 ; and therefore, a being substituted for x in the 
 equation /(.r) = 0, verifies the equation, and, consequently, a is a root of the 
 equation. 
 
 PROPOSITION III. 
 
 239. The proposition that every equation has a root, has in most treatises 
 on Algebra been taken for granted. It has, however, of late years been 
 thought to require a demonstration, and we add one which is as brief and clear 
 as any of the best modifications of that by Cauchy. 
 
 As it will prove a little tedious, the student may, if he please to admit the 
 proposition, pass on to Prop. IV. 
 
 It will be necessary to premise a few lemmas relating to the properties of 
 moduli, some of which have been already demonstrated (Art. 197), but we re- 
 peat them here for convenience of reference. 
 
 Lemma I. — The sum or difference of any two quantities whatever has a 
 modulus comprehended between the sum and difference of the moduli of the 
 two quantities. 
 
 Lemma II. — The modulus of a product of two factors is equal to the product 
 of their moduli. 
 
 Corollary. — Hence the product of the moduli of any number of factors is 
 the modulus of their product, and the modulus of the ft ,h power of a quantity 
 is the n" 1 power of its modulus. 
 
 Lemma III. — In order that a quantity of the form a+b\/ — 1 may be zero, 
 it is necessary, and it is sufficient, that its modulus should be zero ; for a and 
 b being real quantities, let 
 
304 ALGEMIA. 
 
 a + by/ — 1=0. 
 
 As the real part a can not destroy the imaginary part b y/ — 1, we must 
 have separately <z = and 6 = .-. \/a--\-b-=Q. 
 
 Lemma IV — Let there be a polynomial of the form 
 
 X=x m — 2->x m ~ l — qx m ~ 2 u, 
 
 in which the coefficients of all the terms after the first are essentially nega- 
 tive. A value of x can always be found sufficiently great to render the first 
 term x m greater than all the others together, and, consequently, the expression 
 X essentially positive, and as great as we please. 
 
 For we can write X thus, / 
 
 \ x x s x m J 
 
 in which, if x be supposed to increase indefinitely, the negative terms in 
 the parenthesis will decrease indefinitely. As soon as x has attained a value 
 "K sufficiently great to make these negative terms together equal to 1, the 
 value of the expression X will go on increasing indefinitely, and be always 
 positive. 
 
 If A be taken negatively instead of positively, X will still be positive, provided 
 m be even ; but if m be odd, then, when —A is put for x, the leading term will 
 be negative, and, consequently, X negative. 
 
 Corollary. — If the first tevmp of a series p-\- qx-\- rx-+, &c, be constant, 
 x may be taken a sufficiently small fraction to make the sign of the whole de 
 pend on that of the first term.* 
 
 * From the above it may be shown, that in every equation of an odd degree two values 
 can always be found, which, when separately substituted for the unknown quantity, will 
 furnish two results with opposite signs, and that in every equation of an even degree 
 two such values can also be assigned, whenever the final term or absolute number is 
 negative; for, in this case, the substitution of zero for x will give a negative result, viz., 
 the absolute number itself, and the substitution of -f-A or —1 will give a positive result. 
 
 From these inferences it may be proved, without difficulty, that every equation of an 
 odd degree, without exception, has a real root, and every equation of an even degree, pro- 
 vided its final term be negative, has two real roots, the one positive, the other negative. 
 This conclusion might be deduced immediately from what has just been established, if it 
 be conceded that every polynomial/(x), which gives results of opposite signs when twe 
 values a, b are successively given to x, passes from/(c) to/(i) continuously through all in- 
 termediate values, as x passes continuously from a to b. But this is a principle that re- 
 quires demonstration. We proceed to establish it with the necessary rigor. 
 
 PROPOSITION. 
 
 If in the polynomial 
 
 /(x)=:r n +A n _ ia : n - 1 .... +Ajfi+A l as+TX 
 
 x be supposed to vary continuously from z=a to x=b, then the function f(x) will varv 
 continuously from /(a) to f(b). 
 
 DEMONSTRATION. 
 
 Let </' be any value intermediate between a and b. Substitute af-\-h for x in the pofo 
 
 nomial. ami it will become 
 
 f(a'+h)={a'+hy+A a _ l (a'+h)*- 1 .... A,K+ A)-+A, (a'+AJ+N; 
 
 that is, actually developing, in the second member, by the binomial theorem, and ami . 
 the r.sulis according to tho ascending powers of A, 
 
GENERAL THEORY OF EQUATION'S 
 
 305 
 
 1'RELIMINARY DEMONSTRATION. 
 
 240. Each of the equations 
 
 has a root of the form a-\-l> V — 1- This is true of the equation .r ,r ' — -j-li 
 whether m be even or odd, sinco x=l always satisfies it. It is also true of 
 the equation x m = — 1 when m is odd, for then .r= — 1 satisfies it. 
 
 When m is even, it must either bo some power of 2, or else some power 
 of 2 multiplied by an odd number ; if it be a power of 2, then the value of x 
 will be obtained after the extraction of the square root repeated as many times 
 in succession as there ore units in the said power. Now the square root of 
 the form a-\-b •/ — 1 is always of the same form (Art. 118). Hence, when 
 m is a power of 2, each of the equations x m = — 1, x m =± -\f — 1 has a root 
 of the form announced. When m is a power of 2 multiplied by an odd num- 
 ber, then, if we extract the root of this odd degree first, there will remain to 
 be extracted only a succession of square roots. 
 
 We have, therefore, merely to show that, when m is odd, a root of rl- V — 1 
 is of the predicted form. 
 
 Now the o<M powers, 1, 3, 5, &c, of -f V — 1> are (Art. 66) 
 
 + V— if.— V— i. + V— !••• 
 
 and the same powers of — V — 1 are 
 
 — V — 1. +V— h — V— 1 
 
 <»nsequently, when m is odd, a root of i V — 1 is either -{- y/ — 1 or 
 - -/ — 1. Hence the predicted form occurs, whether m be odd or even. 
 It follows from this proposition that, whatever positive whole number m 
 
 maybe, ( — l) m and ( yf — l) m will always bo of the form a-\-b sj — 1; or, 
 
 more generally, ( — 1)'" and ( -/ — l) ,n will always be of this form, n and m be 
 ing any integers positive or negative (Cor. to Lemma II.). 
 
 THEOREM. 
 
 241. Every algebraical equation, of whatever degree, has a root of the form 
 
 +A n _ 1 a' n - , -f(«-l)A„_ 1 «'"^ 
 
 -\-n{n — \)af 
 
 +(»-l)(»-2)A n _ 1 «' 
 
 / 
 
 /n— 3 
 
 -{-Ata'i 
 +A,a' 
 +N 
 which mny be written 
 
 ?-+fc° 
 
 ..h n . 
 
 Now, by what has been above shown, a value so small may bt given to h that the sum 
 of the terms al"ter/(a') shall be less than any assignable quantity, however small. Hence, 
 whatever intermediate value a' between a and b be fixed upon for .r iny(.r), in proceeding 
 to a neighboring value, by the addition to a' of a quantity h ever so minute, we obtain fot 
 f{af-\-h) a like minute increase of the preceding value f \af). In other words, in proceed- 
 ing continuously from a to b in our substitutions for x, the results of those substitutions 
 must be, in like r anner. continuous, or all connected together without any unoccupied in- 
 terval. 
 
 u 
 
306 A.LGEBRA. 
 
 a-\-b^/ — 1. whether the coefficients of the equation be = 1 1 1 real, oi any of 
 them imaginary and of the same form. 
 
 Let/(r) — r"-|- \ ,• '+...A : ,r 1 +A,j--fA,/-(-.\ = il (1) 
 
 represent nny equation the coefficients of which are either real or imagiuary. 
 
 It" in this equation we substitute /,-^-,/y/ — i for r, p and q being real, the 
 first member will furnish a result of the form l' + <i V — '• P ; " 1 '' v i being 
 real (Lemma II.). Should i>-\-qy/ — 1 be ;i root of tin 1 equation, this r< 
 must be zero ; or, which is the same thing, the modulus of 1' + ^ \/ — L v ' z ' 
 V^'+Q 1 ' must l>e zero (Lemma 111.). Ami we have now to prove that 
 values of/' and q always exist that will fulfil] this latter condition. 
 
 In order to this, it will be sufficient to show that whatever value of 
 y/ |» -_|_( £-, greater than zero, arises from any proposed values ofjJ aid q, 
 other values of ]> and 7 necessarily exist, for which -v/P'+'i still 
 
 smaller, so that the smallest value of which ^ps-j-Q 9 is capable must be zero, 
 and the particular expression p-f-0 V — 1, whence this value has arisen, must 
 be a root of the equation. 
 
 For the purpose of examining the effect upon any function, /(x), of eha 
 introduced into the value of x, the development exhibited at Art. 239, Note, is 
 very convenient. By changing x into .r-\-h, the altered value of the function is 
 thus expressed by 
 
 /(r+70=/(.r)+/ 1 (x)/ i +/,(x)^+/ ;! (.r) 1 -^...^ (2) 
 
 where /(.r) is the original polynomial, and J\(x), /-(')• &C., contain none but 
 integral and positive powers of x (Art. 239, Note). 
 
 The first of these functions, /(x), becomes P-f-Q yj — 1 w\\ex\ p-\-q^ — 1 
 is substituted for x; the other functions may some of them vanish for the 
 same substitution, for aught we know to the contrary : but till the terms 
 f(.v) can not vanish ; the last Ji n , which does not contain x, must necessarily 
 remain. 
 
 Without assuming any hypothesis as to what terms of /(r-|-/() vanish for 
 the value x=p-\-q^ — 1, which causes the first of those terms, /(.r), to be 
 come P + Q V — li lot °8 represent by h m the least power of h for which the 
 coefficient does not vanish when j>-\-q-\/ — 1 is put for X. This coefficient 
 will be of the form li-j-S-y/ — 1, in which R and S can not both be zero. 
 
 When p-{-q \/ — 1 is put for X, we have represented f(.r) by 1' + ^-/ — 1- 
 In like manner, when p^-q^/ — 1 — |— A is put for X, we may represont the 
 function by P'+Q'V — 1- The development (2) will then be 
 
 P' + QV-l=(P + Qv'-l) + (K + >V-l)><" , + I'-nns 
 /<"'+'. /;'"+-', .... //". 
 
 Now // is quite arbitrary; we may give to it any sign and any value we 
 
 please, provided only it come under the general form <t-\-l>-\/ — 1. Leaving 
 
 the absolute valm still arbitrary, we may therefore replace it by either -|-fc 
 
 I 
 
 or — /,-, or dj ( — 1 )'"/,•; and thus render//" 1 either positive or negative, which' 
 
 I 
 ever we please, whatever be tin- value of m ; and we have seen that ( — i)™ 
 comes within the stipulated form ( \rt. '.MO)- llence we may write the fore- 
 
 going development thus, the Bign ofjfc™ being under our own control: 
 
GENERAL THEORY OK EQ.UAT10NS. 307 
 
 P +QV-l = (P+Q^-i) + (R+S^-l)t"+ terms in 
 /■•'+ 1 , /■"'+-, .... k". 
 
 But in any equation of this kind the real terms in one member are together 
 equal to those in the other, and the imaginary terms in one to the imaginary 
 terms in the other. Consequently, 
 
 P'=P-f 1U"'+ the real terms in k m +\ k m + 2 , . . . . k" : 
 Q' = Q4-S/ 1 :'"-f- real tonus involving powers above k'". 
 Hence the square of the modulus of P'+Q'V — 1 is 
 P' s +Q /s =P s +Q s +2(PR+QS)jfc m + real terms in k'"+\ k'"+\ . . k*\ 
 
 Now Jc may be taken so small that the sum of all the terms after P 2 -f-Q a 
 may take the same sign as 2(PR-}-QS)/»-"' by (239), which sign we can always 
 render negative whatever PB--J-QS may be, because, as observed above, k m 
 may be made either positive or negative, as we please. 
 
 Hence we can always render 
 
 P /3 +Q /a <P a +Q s , or ^Fi+Q*< VP 2 +Q 2 - 
 
 In other words, whatever values of p and q, in the expression p-\-q yf — 1, 
 cause the modulus -y/P^+Q 2 fo exceed zero, other values exist for which the 
 modulus will become smaller; and, consequently, one case at least must exist 
 for which the modulus, and, consequently, the expression P-f-Q-y/ — 1, must 
 become zero. 
 
 This conclusion presumes, however, that PR-f-QS is not zero. If such 
 jhould be tho case, then our having chosen the form of h, so as to secure a com- 
 mand over the sign of '2(PIl-f-QS), will have been unnecessary. The form 
 must then be so chosen that a command may be secured over the sign of the 
 first term after 2(PR+QS)& m , in the above series, for P' 2 -f Q' 2 , which does 
 not vanish, when the preceding conclusion will follow. 
 
 242. The values of a and b in the expression a-\-b y/ — 1, which, when put 
 for x in/(.r), cause that polynomial to vanish, can never be infinite. 
 We may write /(.r) as follows, viz., 
 
 n v „( -, i A n — i , A n _ 2 N\ 
 
 /(.r)=.r^l + — +— + ...^5 
 
 or, putting P + Q-y 7 — 1 for what f(x) becomes, when p-\-qy/ — 1 is substi- 
 tuted for .r, we have 
 
 P+Qa/^1= 
 
 / A n _! A n _ 2 N \ 
 
 Now the modulus of a quotient is the quotient of the modulus of the divi- 
 dend by the modulus of the divisor (Lemma II.). In each of the dividends 
 A n _i, A n _ 2 , &c, above, the modulus is finite by hypothesis. Hence, if either 
 p or q be infinite, and, consequently, the modulus of every denominator or 
 divisor also infinite, the modulus of each quotient must be zero. Hence, in 
 this case, each of the above fractions must itself be zero (Lemma III.), and 
 therefore the modulus of the entire quantity within the parenthesis simply 1 ; 
 and the modulus of a product is the product of the moduli of the factors, so 
 that the modulus of the preceding product, viz., -/P' + Q"' i s t ne modulus of 
 (P~\-qV — !)"• But the n th power of j>-{-qV — 1 has for modulus the n ,h 
 oovyer of the .modulus o(p-\-q V — 1, that is, the n xb power of Vl }2 -{-<f (Lemma 
 
.JU8 ALCiKHRA. 
 
 IL, Cor.), which is infinite; consequently, VP'+^i" must be infinite. But 
 when p + 7 V — 1 is a root of the equation f(x) = 0, \/P"+Q 2 is zero. Hence, 
 in this case, neither p nor q can he infinite. 
 
 '243. An objection may be brought against the preceding reasoning that 
 ought not to be concealed. It may be denied that the modulus of the product 
 above referred to is simply the modulus of (p-\-q V — l) n in tho case of j? or q 
 infinite; for il may be maintained that although in this case all the quantities 
 within the parenthesis after the 1 become zero, yet tho combination of these 
 with {p>-\-<] V — 1)". which involves infinite quantities, may produce quantities 
 also infinite ; and thus the modulus of the product may differ from tho modu- 
 lus of (p+<7 v — l) n by a quantity infinitely great. It is not to be denied that 
 there is weight in this objection. But it is not difficult to see that although 
 the true modulus may thus differ from tho modulus of {p-\-q-J — l) n by an 
 infinite quantity, yet the modulus of (p+tfV — l) n , involving higher powers 
 than enter into the pait neglected, is infinitely greater than that part. This 
 parti therefore, is justly regarded as nothing in comparison to the part pre- 
 served, the former standing in relation to the latter as a finite quantity to in- 
 finity. 
 
 But the proposition may be established somewhat differently, as follows: 
 
 Substituting {]i-\-qV — *) f° r x m f( x )i wo have 
 
 r+QV^i= 
 
 (l>+? V-l) n + A„_ 1 (jJ + g\/-l) r '- 1 +.-A 1 ( i ?+?-/-l) + N. 
 
 Call the aggregate of all these terms after the first P' + Q' -/ — 1 ; then it 
 js plain that tho modulus of the first term, that is, ( yfp*-\-q*) n , must infinitely 
 exceed the modulus VP' 2 +Q' 2 °f the remaining terms whenever p or q is 
 infinite, because in this latter modulus so high a power of the infinite quantity 
 p or q can not enter as enters into the former. Now the modulu s of t he 
 whole expression, that is, of the sum of (p-\-q \f — l) n and P'+Q'V — 1- is 
 not less than the difference of the moduli of these quantities themselves 
 (Lemma I.), which difference is infinite. Nonce, as before, "/PM-Q 1 must 
 be infinite whcn_p or q is infinite. 
 
 PROPOSITION IV. 
 
 244. Every equation containing hut one unknown quantity has as many roots 
 as there are units in the highest power of the unknown quantity. 
 
 Let/(.r) = be an equation of tho n ,h degree; then if a x be a root of this 
 equation, we have, by Proposition II. , 
 
 (.r-^)/,(.r)=:/(.r) = (1) 
 
 where/, (.1) represents the quotient arising from the division of /(.»•) byx — a,, 
 nnd will be a polynomial, arranged according to the powers of x, ono degree 
 lower than the given polynomial f(x). The equation (1) may be satisfied by 
 making either x — a , =0, x=a , . or by making/, (z)=0. Bui j\{.r)=0 must 
 have a root, as a a (see Prop. III., large edition) .'./^(x) must bo divisible by 
 r— a 3 , .•./,(.,) = (./■-,/, )/,(./■). 
 
 Substituting this value of/,(x) in (1), it becomes 
 
 Proceeding r. this manner, it' s 31 (/ ,, CL t , a, are roots of the BUCCi 
 
« GENERAL THEORY OF EQUATIONS. 
 
 factors / 2 (x)=0,/ 3 (x) = /„(.?•) = 0, the degree of the quotient reducing 
 
 l>y one each time, the equation will assume the form 
 
 (x—a l ){x—a 2 )(x—a 3 ) (x— ge„) = 0; 
 
 and, consequently, there are as many roots as factors, that is, as units in the 
 highest power of .r, the unknown quantity; for the last equation will be veri- 
 fied by any one of the n conditions, 
 
 x=a lt x=a. 2 , x=a 3 , x=a A , .... x=a a ; 
 
 and since the equation, being of the n a degree, contains n of these factors of 
 the 1st degree, (x — </,), &c, there are n roots. 
 
 Corollary 1. When one root of an equation is known, the depressed equa- 
 tion containing the remaining roots is readily found by synthetic division. 
 
 Corollary 2. The number of factors of the 2° degree in an equation is n(n — 1 ) 
 4jl . 2; of the 3°, n{n — l){n— 2)+l .2.3, and so on (see Art. 2031. 
 
 EXAMPLES. 
 
 (1) One root of the equation x 1 — 25x 3 +60x— 36=0 is 3 ; find the equation 
 •ontaining the remaining roots. 
 
 1 _|_o —25 -f GO — 30 (3 
 
 3 + 9 —48+36 
 1 +3—16+12. 
 Henco . r ' , + 3.r : — 1 Or +12 = 
 
 is the equation containing the remaining roots. 
 
 (2) Two roots of the equation x' — 12x 3 +48x 2 — 68x+15=0 are 3 and 5 ; 
 find the quadratic containing the remaining roots. 
 
 1 —12 +48—68+15 (3 
 
 3 —27 + 03 — 15 
 
 1 _ «j +21— 5 (5 
 
 5 —20— 5 
 
 1—4+1 
 ... x-— 4x+l=0 
 •s the equation containing the two remaining roots. 
 
 (3) One root of the cubic equation x 3 — 6x 2 +llx — 6=0 is 1; find the 
 ■juadratic containing the other roots. Ans. x 2 — 5.r+G=0. 
 
 (4) Two roots of the biquadratic equation Ax 4 — 14a? — 5x 2 + 31x+6 = are 
 2 and 3; find the reduced equation. Ans. 4x- + Gx+l=0. 
 
 (5) One root of the cubic equation x?-\-Zx' i — 16x+12=0 is 1 ; find the re- 
 maining roots. Ans. 2 and — G. 
 
 (6) Two roots of the biquadratic equation x* — Gx :, +24x — 16=0 are 2 and 
 — 2 ; find the other two roots. Ans. 3+ ■s/b. 
 
 proposition v. 
 
 245. To form the equation whose roots are a,, a 2 , a 3 , a 4 , a n . 
 
 The polynomial, /(x), which constitutes the first member of the equation 
 required, being equal to the continued product of x— a l , x—a.j, x—a 3 , . . 
 r — a„, by the last proposition, we have 
 
 * There can be no other factor of the form [x — a) which will divide /x. for, if there 
 vrere, it must divide some one of the factors (x — a,), (x— a 2 ), Sec. (See note, p. 83.) 
 
510 
 
 ALOE Bit A. 
 
 (x — a^x— a a )(x — a 3 ) (x— a u )=0; 
 
 and by performing the multiplication here indicated, we have, whex 
 
 n=2, x 8 — ai 
 
 — a. 
 
 n=_3, r 3 — d) 
 — a, 
 
 x -\-a l a 2 =Q 
 
 x — a l a : a :i =z0 
 
 x-j-aiU ; (7 3 « 4 =0, and so on. 
 
 .r--\-a l a z 
 
 — a 3 -\-a 2 a 3 
 « = 4, .r' — «! r 3 +aia2 z 2 — CiOs^s 
 — a 2 -f-«i«3 — aia. : a 4 
 
 — a 4 +flia 4 — a."/'-, 
 
 + "■'■'1 
 
 By continuing the multiplication to the last, the equation will be fonnil 
 whose roots are those proposed ; and from what has been done we learn that 
 
 (1) The coefficient of the second term in the resulting polynomial will be 
 the sum of all the roots with their signs changed. 
 
 (2) The coefficient of the third terra will be the sum of the products of 
 every two roots with their signs changed. 
 
 (3) The coefficient of the fourth term will be the sum of the products of 
 every three roots with their signs changed. 
 
 (4) The coefficient of the fifth term will be the sum of the products of 
 every four roots with their signs changed, and so on ; the last or absolitt' 
 term being the product of all the roots with their signs changed.* 
 
 * I. The generality of this law may be proved as follows : Let us suppose it to hold 
 good for the product of n binomial factors, we shall prove that it will for the product of 
 n-\-\ of these. Let 
 
 x n —A l x"- 1 +A i x n --~, &c, 4-A„ 
 
 represent the product of n binomial factors, in which A, represents the sum o 1 -f-fl. ; +<i 1 
 -f-, &.C., -\-a n of the n second terms of the binomials, A,, the sum of their products two and 
 two. A 3 the sum of their products three and three, and so on, and A n the product of all the 
 n second terms o i a <2 a 3 , &c, a n . 
 
 Introduce now a new factor (x — a n+1 ). Performing the multiplication of the above poly- 
 nomial by this new factor, 
 
 x n — A^-'+Aji"-' 2 — , &c, ± A n 
 
 x — a 
 
 n + l 
 
 -"+'- 
 
 -A 1 x n +A i x n - 1 — , ice, ±A a x 
 - qn+1 .T"+A 1 a n+ ,x"- 1 -, &c. -T^ 
 
 +« 
 
 r" + '-A, 
 
 'n+l 
 
 *°+A« 
 
 + A l r7 n+l 
 
 —A. 
 
 -, &c, T-V'„ +1 
 
 Here the coefficient of the second t is composed of A , the sum of all tho 
 
 — "n+l ' 
 
 ■econd terms of the n binomials (x — O,), (.r — n.X fca, and <r n + I , the second term of the 
 
 ( n -4\-l)"' binomial, and is, therefore, eqntQ to "he sum of the second terma of the h+1 Mno> 
 
 +A 
 minis. Tin- in, ■nil-lent ot the third term is composed of A., the ram of the prod 
 
 ! ^"n+l 
 
 nets of the w second terms two and two, and Vn+i' tnc suiu °f •'' '"' ' ,,| '" 1S . °acb 
 
GENERAL THEORY OE EQUATIONS. 311 
 
 Corollary 1. — If the coefficient of the second term in any equation be 0, 
 that is, if the second term be absent, the sum of the positive roots is equal to 
 the sum of the negative roots. 
 
 Corollary 2. — If the signs of the terms of the equation be all positive, the 
 roots will be ull negative, and if the signs be alternately positive and negative, 
 the roots will be all positive. 
 
 Corollary 3. — Every root of an equation is a divisor of the last or absolute 
 term. 
 
 +A 2 
 multiplied by the new second term a n _f., ; hence will be the sum of the products 
 
 af the w-j-1 second terms two .and two- 
 
 The last term A„« Ir+1 is the product of A n , which is the product of all the n second terms 
 multiplied by the new second term «, l+1 , so that A n a n , l is the product of all the w-f-1 sec- 
 ond terms. 
 
 We have thus proved that if the law for the formation of the coefficients above stated 
 hold good for a certain number of binomial factors ti, it will hold good for one more, or /i-f-1. 
 We have seen, by experiment, that it holds good for four, it therefore holds good for five i 
 if for five, it must for six, and so on ml infinitum. 
 
 II. One might imagine, at first view, that the above relations would make known the 
 roots. They give at once equations into which these roots enter, and which are equal in 
 number to the coefficients of the equation (excepting the coefficient of the first term, which 
 is unity). The number of these coefficients is equal to the number of the roots of the equa- 
 tion. Unfortunately, when we seek to resolve these secondary equations, we are led to the 
 very equation proposed, so that no progress is made. 
 
 For simplicity, I will take the equation of the 3° degree. 
 
 *3+rV;-fQ,E+R=0 (1) 
 
 Designating the three roots by a, b, c, we have, to determine the roots, the three re- 
 lations 
 
 P=— a~b— c 
 
 d=<ib-\-ac-\-bc (2) 
 
 R= — abc 
 
 To deduce from them an equation which contains but the unknown a, the most simple 
 mode of proceeding is, to multiply the 1° by a 2 , the 2° by a,' nd add tliem to the 3°. 
 There results 
 
 Pa»4-Q.ra+R=— n 3 — a?b— <&c 
 
 -\-a"b-\-a"c-\-abc 
 y — abc. 
 
 Reducing, and transposing the term — a 3 , we have 
 
 rt 3 -f-Pa2+a«+R=0. 
 
 The unknown quantities b and c are thus eliminated, but the equation resulting is of th* 
 same degree with the proposed. From the symmetrical form of the relations (2) we per 
 ceive that the elimination of a and b, or a and c, would have been attended with similai 
 consequences. 
 
 III. To find the sum of the squares of the roots of any equation. 
 
 -A^a+b+c.+l; 
 
 .'. A 1 «=a2+Z>'2+ C 2 . . . -\-l2-\-<2{ab+ac-{-bc-{- ....) 
 = sum of the squares -|-2Aj ; 
 
 .•. sum of squares ^A, 2 — 2A ,. 
 To find the sum of the reciprocals of the roots. 
 
 (— I)"" 1 A n _ 1 =/>r- . . . l+ac . . . l+ab . . 1+ . . 
 ;— l) n A n =abc...l; 
 
 a^b^c W A„ 
 
312 ALGEBRA. 
 
 Corollary 4. — In any equation, when the roots are all real, and Uie last or 
 absolute term very small compared with the coefficients of the other terms, 
 then will the roots of such an equation be also very small. 
 
 EXAMPLES. 
 
 (1) Form the equation whose roots are 2, 3, 5, and — G 
 
 Here we have simply to perform the multiplication indicated in the equa- 
 tion 
 
 (. r _2)(x-3)(x-5)(x+6)=0 , 
 and this is best done by detached coefficients in the following manner : 
 
 1— 2 (—3 
 — 3+ G 
 1—5+6 (—5 
 _ 5 + 25— 30 
 1 — 10 + 31—30 (G 
 G — 60+18G— 180 
 
 * 1 — 4 —29 + 156 —1 80 
 ... x 4 —4^—29x 2 +15G.r— 1-^0 = is the equation sought. 
 
 (2) Form the equation whose roots are 1, 2, and — 3. 
 
 (3) Form the equation whose roots are 3, — 4, 2+ \/3, and 2 — y/3. 
 
 (4) Form the equation whoso roots are 3+ y/5, 3 — -\/5, and — G. 
 
 ANSWERS. 
 
 (2) x 3 — 7x+6=0. 
 
 (3) x 4 — 3.V 3 — 15x*+49x— 12 = 0. 
 
 (4) r 3 — 32.r+24 = 0. 
 
 PROPOSITION VI. 
 
 246. No equation whose coefficients are all integers, and that of the Itighesl 
 power of the unknown quantity unity, can have a fractional root. 
 
 If possible, let the equation 
 
 .r n + A n ^' 1 x"- 1 + ... +A 3 ^+Ao.r + A l .r+N = n, 
 whose coefficients arc all integral, havo a fractional root, expressed in its low- 
 est terms by j. If we substitute this for x, and multiply the resulting equation 
 bv b"~ l , we shall have 
 
 X + A n _ 1 a n -'+ (-A 3 a 3 6 n - 3 +AaZ. n »+No" ' = 0. 
 
 I o 
 
 In this polynomial, every term after the firsl is integral : hence the first term 
 must be integral also. But j being a fraction in its lowest terms, y must also 
 
 be a fraction in its lowest terms, and can DOl be an integral (See Note to 
 
 Art. 84.) Therefore the jn-oposed equation can not have a fractional root. 
 
 PROPOSITION VII. 
 
 247. If the signs of the alternate terms in an <• [nation he changed, the signs 
 of all die roots will In cluing, </. 
 
 Let r*+AV 1 + A.; + V t X+A =0 • ■ • • (1) 
 
 be u: equation ; then, changing the signs of the alternate terms, we have 
 
 . r n_\ iX .n-l_|_ A , c n • _ ^A, ,,. | £.„ — <) 0- . (2) 
 
 or _,■" + . V ' — A ,r" : + "i \, ,x±A n =0 ... (3) 
 
GENERAL THEORY OP EQUATIONS. 313 
 
 But equations (2) and (3) are identical, for the sum of the positive terms in 
 each is equal to the sum of the negative terms, and therefore they are identi- 
 cal. Now if a be a root of equation (1), and if a be substituted for x in equa- 
 tion (1) and —a in equation (2), if n be an even number, or in equation (3) 
 if n be an odd number, the results will be the very same ; and since the for- 
 mer is verified by such substitution, a being a root, the latter, viz., equation 
 (2) or (3), as the case may be, is also verified, and thereforo —a is a root of 
 the identical equations (2) and (3). 
 
 Corollary.— -If the signs of all the terms are changed, the signs of the roots 
 remain unchanged. 
 
 EXAMPLES. 
 
 (1) The roots of the equation r> — 6x 3 +llx— 6=0 are 1, 2, 3. What are 
 the roots of the equation r 5 -f-6x 2 +ll:r-f-6 = ? 
 
 Ans. —1, —2, —3. 
 
 (2) The roots of the equation .r 1 — 6x a + 2Ax— 16=0 are 2, —2, 3± y/5. 
 Express the equation whose roots are 2, —2, —3+ y/5, and —3— y/5. 
 
 Ans. tf+Gx*— 24x— 16=0. 
 
 PROPOSITION VIII. 
 
 248. Surds and impossible roots enter equations by pairs. 
 
 Letx n +A 1 x"- 1 + A,.r n - 2 -l A„^ 1 r+A^=0' be an equation having a root 
 
 of the form a-\-b V — 1 > then will a — by/ —I be also a root of the equation ; 
 for, let a-\-b y/ — 1 be substituted for x in the equation, and wo have 
 
 (rt + ftV^Tr+Ai^ + fc V Z: l) n_1 + ••••A, 1 _ 1 (a-f&-/ :: ^l) + A n=0- 
 Now, by expanding the several terms of this equation, we shall have a series 
 of monomials, all of which will be real except the odd powers of 6 V — 1, 
 which will be imaginary. Let P represent the sum of the real and Q yf — 1 
 the sum of the imaginary terms of the expanded equation ; then 
 
 P+QV^T=o, 
 
 an equation which can exist only when P = and Q=0, for the imaginary 
 quantities can not cancel the real ones, but the real must cancel one another, 
 and the imaginary one another separately. 
 
 Again, let a — b \/ — 1 be substituted for x in the proposed equation; then 
 the only difference in the expanded result will be in the signs of the odd powers 
 of by/ — 1, and the collected monomials, by the previous notation, will assume 
 
 the form P— QV — 1 but we have seen that P=0 and Q = ; 
 
 ... P_QV=1=0, 
 
 and hence a — by/ — \ also verifies the equation, and is therefore a root. 
 
 Such roots are called conjugate. 
 
 In a similar manner, it is proved that if a-\- y/b be one root of an equation, 
 a — y/b will also be a root of that equation. 
 
 Corollary 1. — An equation which has impossible roots is divisible by 
 
 ja-— (a + b V^T) \ \x — {a — b y/ —1)\, or x 3 — 2ax-\-a*+b*, 
 
 and, therefore, every equation may be resolved info rational factors, simple or 
 q ladratic. 
 
 ( 'orollary 2. — All the roots of an equation of an even degree may be impos- 
 
314 ALGEBRA. 
 
 sible, but if they are not all impossible, the equation must have at least two 
 real roots. 
 
 Corollary 3. — The producbof every pair of impossible roots being of the 
 form a"-\-b' 2 is positive; and, therefore, the absolute term of an equation 
 whose roots are all impossible must be positive. 
 
 Corollary 4. — Every equation of an odd degree has at least one real root, 
 and if there be but one, that root must necessarily have a contrary sign to 
 that of the last term. 
 
 Corollary 5. — Every equation of an even degree whose last term is nega- 
 tive has at least two real roots, and if there be but two, the one is positive 
 and the other negative. 
 
 PROPOSITION IX. 
 
 249. The m roals of the equation X=0, or 
 
 a - m _l_p i .n,-i_^Q r n,- 2 _|_ ) &c> _ [ A -j 
 
 jnust he of the form a-\-b V — 1, of which form we have already shown (Art. 
 241) that it must have one. 
 
 For, let a-f-i-y/ — 1 bo the root whose existence is demonstrated. We 
 know (Prop. II.) that the polynomial x'"-\-. See., is divisible by .r — (a -|- V — 1): 
 but when we effect this division, the quantities a-\-b -/ — 1, P, (I- &c., can 
 combine only by addition, by subtraction, and by multiplication ; then the co- 
 efficients of the quotient x m_1 -+-, cVc, will still bo of the form fl+i-/ — 1. 
 Consequently, the equation x m ~ 1 -^-, &c, will also have al leasl one root of the 
 form a'+fe' V —1 ; dividing x'"-'-^, &c, by x—(a'-\-b' V — 1), the coefficient* 
 of the quotient x m-2 4-i &C., will be still of the same form. Continuing to 
 reason thus, it is evident that the primitive polynomial X will be divided into 
 m factors of the form X— {a-\-b \/ — 1), and, consequently, the roots of the 
 equation will all be of the form a-\-b V — 1. 
 
 PROPOSITION X. 
 
 250. The roots of the two conjugate equations, 
 
 Y+z/=T=o (1) 
 
 Y — Z V— 1=0 (2) 
 
 will be conjugates of each other. 
 
 Letx=a4-6-v/ — 1 be a root of equation ( I ), and Y'-J-Z' ■/ — 1 the quotient 
 of its first member, by x — a — b V — 1, we have the identity 
 
 (Y'+Z'v^K*— a— &/^i)=Y+Zv f "^I (3) 
 
 Effecting the multiplication in the 1° member, we find 
 
 (r — ,/)Y' + oZ'+[(r— «)Z' — M"]-/3l 
 ( lhanging now in tho two factors Z' into — /', and b into — b, we see that 
 in the producl the part which does not contain V — 1 remains the same, and 
 thai thai which does contain \/— ! only changes its sign; by virtue of (3), 
 therefore, w e have 
 
 (V_ZV^l)(,r- a-\-by/^l) = Y-Z -/^T .•••(!) 
 Prom whence we conclude that a—b^ — 1 is a root of (2); that is, all the ro 
 qf (•>) are obtained b\ changing in those "!' (l) the sign of ■/— 1. The real 
 Mots, according to this, musl bo the same in the two equatio 
 
GENERAL THEORY OF EQUATIONS. 31b 
 
 We may now consider the following beautiful proposition as demonstrated 
 from the foregoing. 
 
 PROPOSITION XI. 
 
 An algebraic equation which has real coefficients is always composed of as 
 many real factors of the 1° degree as it has real roots, and of as many real 
 factors of the 2° degree as it lias pairs ofHmaginary roots. 
 
 DEPRESSION OR ELEVATION OP ROOTS OF EQUATIONS. 
 
 PROPOSITION. 
 
 251. To transform an equation into another whose roots shall he the roots oj 
 the proposed equation increased or diminished by any given quantity. 
 
 Let a.T n + A 1 .r n - 1 4-A 2 a: n - 2 -j- A n _!.r4- A„=0, be an equation, and let it 
 
 be required to transform it into an equation whose roots shall be the roots of 
 this equation diminished by r. 
 
 This transformation might lie effected by substituting y-\-r for x in the pro- 
 posed equation, and the resulting equation in y would be that required; but 
 this operation is generally veiy tedious, and we must therefore have recoi 
 to some more simple mode of forming the transformed equation. If we write 
 y-\-r for x in the proposed equation, it will obviously be an equation of the 
 very same dimensions, and its form will evidently be 
 
 «r+ B i2/ n_1 + B ^ n_2 + B I1 _ 1 7/4-B n =0 (1)* 
 
 in which Bi, B 3 , &c, will be polynomials involving r. But y=x—r, and there- 
 fore (1) becomes 
 
 a(z—r)-+B 1 (x— r)»-»+ B n _ 1 (a:-r)+B n =0 . . (2) 
 
 which, when developed, must lie identical with the proposed equation; for, 
 since y-\-r was substituted for x in the proposed, and then x — r for _y in (1) 
 the transformed equation, we must necessarily have reverted to the original 
 equation ; hence we have 
 a(x—ry+'Q x (x— r) n ~ ] + . .B n _,(.r— r)-f B n =a.r n + A,a: n - 1 + .. A^x+A,,. 
 
 * It will be of the same form with the development in the note to (Art. 239). We give 
 it again below, arranged according to the powers of r instead of y. After substituting y-\-f 
 for r, we write the development of each term of the proposed equation in a horizontal line ; 
 the first horizontal line is the development of a.r n , the second of A^x*— 1 , and so on. 
 
 n I n-l , an { n — 1 )„n-S 2 , 
 
 ay*+any r -\ ^— y r +... 
 
 ..„_!,., ,, n -2 , A](«— l)(? t— 2) n _ 3 „ 
 
 4-Aj^ -fA^H— \)y H Y7z y *" + •■• 
 
 I A "-2 , A , ^ n-3 , Ac(M— 2H«— 3) „_, „ 
 
 . +A;# +A 2 (w— <2)y r-\ — y r + ... 
 
 +A„. 
 In which the first column is of the same form as the proposed equation ; the second 
 column, or coefficient of r, is derived from the first by multiplying the coefficient of each 
 term by its exponent, and diminishing the exponent l>y unity; the third column, or coclfi 
 
 cient of , is derived from the second in a similar manner, and so on. 
 
 1.9 
 
 If we designate byf[x) the first member of the given equation, and byf'(x) the first de- 
 rived function, by/"(.c) the second derived, and so on, we shall have 
 
 fix) f'"(x) 
 
316 ALGEBRA. 
 
 Now, if we divide the first member by X — r, every term will evidently be divis- 
 ible, except the last, B n , which will be the remainder, and the quotient will be 
 
 a(z— rJ-'+B^ar— r)-*+ B n _ : (.r-r)+B„_ i; 
 
 and since tne second member is identical with the fust, the very same quotient 
 and remainder would arise by dividing this second member also by x — r ; 
 hence it appears that if the first member of the original equation be divided by 
 x — r, the remainder will be the last or absolute term of the sought transformed 
 equation. 
 
 Again, if we divide the quotient thus obtained, viz., 
 
 t/ (. r _ r )..-i + B I (.r-r)»~- + .... B n _ 2 (r- r) + B„_, 
 by x — r, the remainder will obviously be B„_i, the coefficient of the term last 
 but one in the transformed equation; and thus, by successive divisions of the 
 polynomial in the first member of the proposed equation by X — r, we shall ob- 
 tain the whole of the coefficients of the required equation. 
 
 RULE. 
 
 Let the polynomial in the first member ofHhe proposed equation be a func- 
 tion of r, and r the quantity by which the roots of the equation are to be di- 
 minished or increased ; then divide the proposed polynomial by x — r, or .r-f-'" 
 according as the roots of tho proposed equation are to be diminished or in- 
 creased, and the. quotient thus obtained by the same divisor, giving a second 
 quotient, which divide by the same divisor, and so on till the division termi- 
 nates : then will the coefficients of the transformed equation, beginning with 
 the highest power of the unknown quantity, be the coefficient of the highest 
 jkower of the unknown quantity in the proposed equation, and the re- 
 
 mainders arising from the successive divisions taken in a reverse order, the 
 first remainder being the last or absolute term in the required transformed 
 equation. 
 
 Pjpte. — When there is an absent term in tho equation, its place must be 
 supplied with a cipher. 
 
 EXAMPLES. 
 
 (1) Transform the equation 5-r 4 — 12r'-4-3.r 2 -|-4.r — 5=0 into another whose 
 roots shall be less than those of the proposed equation by 2. 
 
 X— 2) 5r> — 12.r 3 +3.r 2 -f4.r— 5 (5.T 3 — 2x*— x+2 
 5-r 4 — Id-' 1 
 
 — 2r 3 +3.r 2 
 
 
 
 
 — 2r 3 +4.r 2 
 
 
 
 
 — .r--j-4.r 
 
 
 — a- 2 +2.r 
 
 
 
 
 2x— 5 
 
 
 2.1—4 
 
 First 
 
 remain 
 
 
 — 1. 
 
 der. 
 
 r— 2) 5-r 3 — 2r- — r+2 
 
 (.', 4_e.r-fl5 
 
 
 5a?— lOr* 
 
 
 
 
 8x Q — X 
 
 
 
 
 8.r 2 — 1 1: -• 
 
 
 
 
 1 5x 
 
 + 2 
 
 
 1 5x 
 
 —30 
 
 
 
 Second remainder. 
 
GENERAL THEORY OF EQUATIONS. 317 
 
 x— 2) 5x 2 +8:r+15 (5.r+18 
 5x 2 — 10.r 
 
 18*+ 15 
 18x— 36 
 
 51. Third remainder 
 x—2) 5x+18 (5 
 5x— 10 
 
 28. Fourth remainder. 
 Therefore the transformed equation is 
 
 53/ 4 +28?/ 3 +5l7/ e +327/ — 1=0. 
 This laborious operation can bo avoided by Horner s Synthetic Method of 
 division, and its great superiority over the usual method will be at once ap- 
 parent by comparing the subsequent elegant process with the work above. 
 Taking the same example, and writing the modified or changed term of the 
 divisor x — 2 on the right hand instead of the left, the whole of the work will 
 be thus arranged : 
 
 5 — 12 + 3 +4—5 (2 
 10 — 4 — 2 4 
 
 ~H ~—[.-. B i= — 1 
 30 
 
 ■&.' 
 
 2 
 
 — 1 
 
 10 
 
 1G 
 
 8 
 
 15 
 
 10 
 
 36 
 
 18 
 
 51 
 
 10 
 
 
 32 .-. B 3 =32 
 . .*. B 3 =51 
 
 28 .-. B 1= =28 
 
 ... <jyi^.23y*-{-!>iy' 2 -\- 32;/ — 1=0 is the required equation, as before. 
 (2) Transform the equation 5y 4 +28 i )/ , +51i/ 2 +322/ — 1=0 into anothei 
 having its roots greater by 2 than those of the proposed equation. 
 
 5+28 + 51 _|_ 32 _i (_2 
 
 — 10 
 
 — 36 - 
 
 -30 
 2 
 
 — 4 
 
 18 
 
 15 
 
 —5 
 
 — 10 
 
 — 16 
 
 2 
 
 4 
 
 
 8 
 
 — 1 
 
 
 — 10 
 
 4 
 
 
 
 — 2 
 
 3 
 
 
 —10 
 
 
 
 
 — 12 
 ... 5x 4 — 12a 3 +3:i?+4a: — 5=0 is the sought equation, which, from the trans- 
 formations we have made, must be the original equation in Example 1. 
 
 (3) Find the equation whose roots are less by 1-7 than those of the equation 
 
 z 8 — 2z s +3z— 4=0. 
 1—2 +3 —4 (1 
 
 1 
 
 — 1 
 
 —1 
 
 o 
 
 1 
 
 
 
 ~~ 
 
 2 
 
 1 
 
 
 1 
 
 
3 13 ALGEBRA. 
 
 Now we know the equation whose roots are less by 1 than thjse of the 
 iriven equation* it is 3?-\-3?-\-2x — 2=0 ; and by a similar process for -7, re- 
 m. -inhering the localities of the decimals, we have the required equation ; 
 
 tlaiii : 
 
 1 + 1 +2 —2 (-7 
 .7 1-19 2-233 
 •233 
 
 1-7 
 
 3-19 
 
 7 
 
 1-68 
 
 2-4 
 
 4-87 
 
 7 
 
 
 3-1 
 
 .-. 3/ 3 +3-l2/ 2 +4-873/+-233 = is the required equation. 
 
 This latter operation can be continued from the former without arranging 
 the coefficients anew in a horizontal line, recourse being had to this second 
 operation merely to show the several steps in tho transformation, and to point 
 out the equations at each step of the successive diminutions of the roots. 
 Combining these two operations, then, we have the subsequent arrange- 
 ment. 
 
 1—2 
 
 + 3 
 
 —4 (1-7 
 
 1 
 — 1 
 
 — 1 
 
 ~~ 2 
 
 2 
 ^2 
 
 1 
 
 ~0 
 
 
 2 
 
 2-233 
 •233 
 
 1 
 
 1-19 
 
 
 1-7 
 
 3-19 
 
 
 •7 
 
 1-68 
 
 
 2-4 
 
 4-87 
 
 
 •7 
 
 
 
 3-1 
 
 
 
 1—2 
 
 1-7 
 
 + 3 
 — -51 
 
 —4 (1-7 
 4-233 
 
 — -3 
 
 2-49 
 
 •233 
 
 1-7 
 
 2-38 
 
 
 1-4 
 
 4-87 
 
 
 1-7 
 
 
 
 3-1 
 
 We have then the same resulting equation as before, and in the latter of 
 these we have used 1-7 at once. It is always better, however, to reduce 
 continuously as in the former, to avoid mistakes incident to the multiplier 1-7. 
 
 (1) Find the equation whose roots shall be loss by 1 than those of the 
 equation 
 
 ;r>— 7.r +7 = 0. 
 
 (5) Find the equation whoso roots shall he less by 3 than the roots of the 
 equation 
 
 x* — 3.r» — 1 -,.r-+4'.t.r— 12 = 0. 
 mid transform the resulting equation into another whose roots shall bo groater 
 by I. 
 
GENERAL THEORY OF EQUATIONS. 319 
 
 (6) Give the equation whose roots shall be less by 10 than the roots of the 
 
 equation 
 
 r«_j_ o r i_|_ 3. C 2_|_ 4x— 12340 =0. 
 
 (7) Give the equation whoso roots shall be less by 2 than those of the 
 equation 
 
 x* -J- 2x a — 6x 2 — 1 0x+ 8 = . 
 
 (8) Give the equation whose roots shall each be less by | than the roots oi 
 the equation 
 
 2x*— G.r , +4.r 2 — 2.r+ 1=0. 
 
 ANSWKRS. 
 
 (4) 2/3 -|_ 3jy 2 — 4t/ -j- 1 = whence x=y-\- 1 
 
 (5) y»_|_ 9y-i_|_i2j/ 3 — 14t/ = whence x=7/4- 3 
 
 and z 4 —7z 3 +66z — 72=0 whence x=z— 1 
 
 (G) i/<4-427/ 3 4-6G37/ 3 +4GG4 - y = whence x=y + 10 
 
 (7) 7/ 5 4-10/y'4-42?/ :, 4-86?/ 2 4-70^/4-12 = whence x =y+ 2 
 
 (8) 2y 4 — Sy*— 2?/ 2 — ^+8 = ° whence x=y-\- i 
 
 PROPOSITION 
 
 252. If the real roots of an equation, taken in the order of their magnitudes, be 
 
 ^11 ^2' ^3' ^41 ^5^ • 
 
 where a,?'s the greatest, a 2 tfie we.r<, and so on ; then if a series of numbers, 
 
 ft,, fto, ft-j, ft 4 , ft s , 
 
 sn which b, is greater than a,, b 2 a number between a, a«(/ a 2 , b 3 a number 
 between a 3 ««<Z a 3 , and so on, be substituted for x in the proposed equation, 
 results ivill be aMernatcly positive and negative. 
 The polynomial in the first member of the proposed equation is the product 
 u!' the simple factors 
 
 {x— a t )(x — a a )(x — a ;i )(x— a 4 ) 
 
 and quadratic factors, involving the imaginary roots; but the quadratic factors 
 have always a positive value for every real value of x (Art. 248, Cor. 3) ; there- 
 fore wo may omit these positive factors ; and substituting for x the proposed 
 series of values, ft,, b 2 , ft 3 , &c, we have these results: 
 
 (ft,— a,)(ft,— a,.)(ft,— a 3 )(b l — a A ) .... = + . + • + ■+ = + 
 
 (h.,-a l )(b 2 -a 2 ){b 3 -a 3 )(b a -a A ).... =_. + . + .+ =- 
 
 {b :t —n t )(h,—a.,){b.,—a 3 ){b 3 —a A ) ....=—. — .4-.+ = + 
 
 V>4—<h)(l>i—<i:){b A —a 3 ){b A -a A ) ....= — . .-f- = — 
 
 &c. &c. 6cc. 
 
 Corollary 1. — If two numbers be successively substituted for X in any equa 
 Hon, ami give results with different signs, then between these numbers there 
 must be one, three, five, or some odd number of roots. 
 
 Corollary 2. — If the results of the substitution in corollary 1 are affected 
 with like signs, then between these numbers there must be two, four, or some 
 even number of roots, or no root between these numbers. 
 
 ( 'orollary 3. — If any quantity q, and every quantity greater than q, renders 
 the result positive, then q is greater than the greatest root of the equation. 
 
 Corollary 4. — Hence, if the signs of the alternate terms be changed, and if 
 p, and every quantity greater than p, renders the result positive, then — p is 
 less than the least root. 
 
320 
 
 ALGEBHA 
 
 i \ ample. 
 Find the initial figure in one of the roots of the equation 
 
 X 3 — i. ,-•_(;.,• -|-8 = 0. 
 Here one value of x does not differ greatly from unity, for the value of the 
 given polynomial, when x=l, is — 1, and when r=-9, it is found thus; 
 
 1_4_G 4-8 (-9 
 •9—2-79—7911 
 
 —31— 8-79+ -089. 
 The value, therefore, when .r=-9 is (Art. 251) -089. Hence the former 
 alue being negative, and the latter positive, the initial figure of one root is -9. 
 
 PROPOSITION. 
 
 253. Given an equation of the n"' degree to determine another of the (n — 
 lefrrcc, such that the real roots of the former shall separate those of the latte 
 
 Let a., a 
 
 a-,, a 
 
 ■21 "3' 
 
 |1 
 
 a a be the roots taken in order of the equation 
 
 x n 4- A , .r"- ] 4- A 2 x n - 2 4 A n _!.r 4- A n = ; 
 
 then diminishing the roots of this equation by r (Art. 251), we have the U 
 lowing process, viz. : 
 
 14-A,4- A 2 4- A n _ 2 4- A n _,4- A n (r 
 
 r 
 r 
 
 rB, 
 rC, 
 
 rB n _ 3 rB n 
 
 B„_ 3 
 rC„_ 3 
 
 C n _2 
 
 B n _ l 
 
 rCn-? 
 
 C„_! 
 
 rB n _ x 
 "b7~ 
 
 Whence 
 
 C n _ 1 =A 1 _,+ r B _ 2 -f r C„_ 3 • 
 
 =A n _!4- r(A n _ 2 4- r B n _ 3 )4- r (A n _, +r B n _ 3 4-rC n _ 3 ) 
 =A n _!4-2r A„_s+2r« B n _ 3 + r" C B _ 3 
 
 = A„_i + 2r A n _ 2 4-2r-(A„_ 3 + r B n _ 4 )4-r-(A n _ 3 4-rB„_,4-rC _^) 
 =A»_i+2r A n _ 2 4-3r 2 A n _ 3 4-3r> B n _, 4-r 3 C n _, 
 
 =A n _ 1 4-2r A n _ 2 4-3r 2 A n _ 3 4- (n— l)i*-*Ai+ftr 
 
 or 
 
 ( 
 
 C n _ 1 =nr"-'4-(»— l)A 1 r"--4-(/i — 2)A. : r"- 3 4- . . . 2A n _ S r+A B - I 
 Again, the roots of the transformed equation will evidently be 
 fli — r, a 2 —r, a 3 —r, a^—r, .... a n — r, 
 and as we have found the coefficient, C„-n of the last term but one in th 
 transformed equation, by one process, we shall now find the same coefficien 
 Co-d by another process (Prop. V., p. 3(fe) ; il is the product of every (n — 1) 
 roots of the equation (1) with their signs changed; hence wo have 
 
 C n _ l = (r—a l ){r—a : ){r — a i ) to (n— 1) factors ' 
 
 + (r— ai)(r— th)[r— a<) to (n— 1) factors 
 
 4-(r— ai)(r— ".)('•— oi) to (n— 1) factors 
 
 4-(r — a : ){r — flj)(r— a*) to (n— 1) (actors 
 
 Now these two expressions which we have obtained for C n _i are equal to 
 
GENERAL THEORY OF EQUATIONS. 321 
 
 
 
 
 
 
 
 
 
 ' 
 
 N 
 
 
 
 ,< 
 
 
 
 
 
 
 . 
 
320 
 
 ALGKBHA 
 
 
 
 5- " 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
GENERAL THEORY OP EQUATIONS. 321 
 
 one another, and, therefore, whatever changes arise by substitution in the 
 one, the same changes will be produced, by a like substitution, in the other; 
 nence, substituting a u a 2 , a 3 , &c, successively for r in the second member of 
 equation (2), we have these results : 
 
 («i — a 2 )(ai— fi 3 )(ai— a 4 ) = + . + .+ = + 
 
 (a 2 — ai)(«2— a 3 )(a 2 — a 4 ) =—. + .-{- = — 
 
 (a z —ai){a 3 —a 2 ){a 3 —a 4 ) = _._._|_ =4. 
 
 &c. &c. &c. 
 
 But when a series of quantities, a u a 2 , a s , a t , &c, are substituted for the 
 unknown quantity in any equation, and give results which are alternately -\- 
 and — , then, by Art. 252, these quantities, taken in order, are situated in the 
 successive intervals of the real roots of the proposed equation; hence, making 
 C„_ir=0, and changing r into .r, wo have from equation (1) 
 
 nx n - l -\-(n — l)A l x n - n '-\-{n—2)A i x"- 3 -\ 2A n _ 2 .r+A n _ 1 = ... (3) 
 
 an equation whose roots, therefore, separate those of the original equation 
 x»4.A 1 a^- 1 +A s a; ,l - 3 + . . .. A n _,.r-f A n =0, 
 
 and the manner of deriving it from the proposed equation is to multiply each 
 term of the proposed equation by the exponent of x, and to diminish the ex- 
 ponent one. It is identical with the second column of the development in 
 the note to Article 251. It is known by the name of the derived equa- 
 tion. 
 
 Let a u a 2 , a 3 , a 4 , &c, be the roots of the proposed equation, and b t , b 2 , b 3 , 
 &c, those of the derived equation (3), ranged in the order of magnitude ; then 
 the roots of both the given, and the derived equation will be represented in 
 order of magnitude by tho following arrangement, viz. : 
 
 a x , 6 M a 2 , b 2 , a 3 , 63, a 4 , 6 4 , cr 5 , b 5 , &c. . . 
 
 Corollary 1. — If a 2 =a u then r — a^ will be found as a factor in each of the 
 groups of factors in equation (2), which has been shown to be the separating 
 equation (3), and, therefore, the separating equation and the original equation 
 will obviously have a common measure of the form .r — «[. 
 
 Corollary 2. — If a 3 =a 2 =a 1 , then (r — «i)(r — a^) will occur as a common 
 factor in each group of factors in (2) ; that is, the separating equation (3) is divis- 
 ible by (x — a x )~ ; and, therefore, the proposed equation and the separating equa- 
 tion have a common measure of the form (x — ai)". 
 
 Corollary*?,. — If the proposed equation have also a 4 =a 5 , then it will have a 
 common measure with the separating equation of the form (x — a^ (r — a 4 ), 
 and so on. 
 
 Scholium. — When, therefore, we wish to ascertain whether a proposed 
 equation has equal roots, we must first find the separating equation, and then find 
 the greatest common measure of the polynomials constituting the first mem- 
 bers of these two equatious. If the greatest common measure be of the form 
 (x— ai y (x—a 2 y{x—a 3 ) r 
 
 then the proposed equation will have (p-f-1) roots =a l , (<7+l) roots =a 2 
 (r-j-1) roots =a 3 , &c. The equation may then be depressed to another of 
 lower dimensions, by dividing it by the difference between .r and the repeated 
 root raised to a power of the degree expressed by the number of times it ia 
 repeated. 
 
 X 
 
322 ALGEBRA- 
 
 IC IMPIiES. 
 
 Fiud the equal roots of tho equation 
 
 .r_f-5.i*_j_Gx 5 — 6x*— 15s»— 3.c : +8x+4 = (1) 
 
 The derived polynomial is 
 
 71*4-30.^ -r-30.r<— 24.r ! — 4o.f 3 — 6.r-f 8 (S 
 
 and the common divisor of (1) and (2) 
 
 r , +3.r 3 +2-— 3.r — 2 (3, 
 
 The values of x, found by putting this equal to zero, would be the repeated 
 roots of the proposed equation. This itself will be found to have equal roots, 
 for its derived is 
 
 4x 3 +9£ 3 +2z— 3, 
 and their common divisor 
 
 r+1. 
 Hence, by the rule, 
 
 (z+1) 9 (4) 
 
 is a factor of (3), and 
 
 ,(.t+l)' 
 u factor of the proposed. 
 
 Dividing (3) by (4), the quotient is 
 
 .r- + .r— 2, 
 which, put equal to zero, gives 
 
 x=l, or — 2. 
 Hence (3) may be put under the form 
 
 (s+1) 9 (s-1) (*+2), 
 
 and by the rule in tho above scholium the given equation may be put under 
 the form 
 
 (. r+ l)3 (*_!); (a: + 2)», 
 so that in the proposed equation there are three roots equal to — 1, two to 
 -f-1, and two to — 2. 
 
 (2) x 3 — 3a-x— 2a 3 =0. 
 By the process above it may be transformed into 
 
 {x+a)* (x-2a)=0, 
 so that the three roots are two equal to — a, and the third 2a. 
 
 (3) x 8 — 12x 7 +53.r 6 — 92r' i — 9.r«+212.i' 3 — 153x s — 108.r -f-108 = 
 decomposes into 
 
 (.r— 1) (x— 2)- (.r+1)- (.i — 3) 3 =0. 
 254. The most satisfactory and unfailing criterion for the determination of 
 the number of imaginary roots in any equation is famished by tho admirable 
 theorem of Sturm, which gives the precise number of real roots, and, conse- 
 quently, the exact number of imaginary ones, since both the real ami imagi- 
 nary roots are together equal to tho number denoted by the degree of the 
 proposed equation. 
 
 PROPOSITION. 
 
 To find the numher of real and imaginary roots in any proposed equation. 
 
 Tho acknowledged difficulty which has hitherto been experienced in the 
 important problem of tho separation of the real and imaginary roots of any 
 proposed equation is now completely removed by the recent valuable re- 
 searches of the celebrated Sturm; and we shall now give the demonstration 
 of the theorem by which this desirable object has been so fully accomj 
 
THEOREM OF STURM. 323 
 
 ed. nearly as given by the author himself, deeming it far more* satisfactory thun 
 any other version which we have seen. 
 
 THEOREM OK STURM. 
 
 I. Let N.r'"+P.i m - 1 + Q.r m -=+ +Tar-j-U=0 
 
 be a numerical equation of any degree whatever, of which it is proposed to 
 determine all the real roots. 
 
 \V r e begin by performing upon this equation the operation which serves to de- 
 ne whether or not it has equal roots (Art. 253, Sch.), in a manner which 
 we proceed to point out. If V designate the entire function N.c m -f-P* m-1 + » 
 &c, and Vi its derived function (which is formed by multiplying each term 
 of V by the exponent of ;r in this term, and diminishing that exponent by uni- 
 ty), wo must seek for the greatest common divisor of the two polynomials V 
 and Vj. Divide, at first, V by V,, and when a remainder is obtained of a 
 degree inferior to that of the divisor Vj, change the signs of all the terms of 
 this remainder (the signs -}- into — and — into -f-). Designate by V 3 what 
 this remainder becomes after the change of signs. Divide in the same man- 
 ner V, by Vn, and, after having changed the signs of the remainder, it becomes 
 a new polynomial V 3 , of a degree inferior to that of V 2 . The division of 
 y \ ;, conducts, in the same manner, to a function /.,, which will be the 
 remainder resulting from this division after having changed the signs. This 
 series of divisions is to be continued, taking care to change the signs of the 
 terms of each remainder. This change of signs, which would be useless if 
 our object was to find the greatest common divisor of the polynomials V and 
 V,, is necessary in the theory about to be explained. As the degrees of the 
 successive remainders go on diminishing, we arrive finally either at a numeri- 
 cal remainder independent of x, and differing from zero, or at a remainder a 
 function of x, which exactly divides the preceding remainder. 
 
 We shall examine these two cases separately. 
 
 II. Suppose, in the first place, that, after a certain number of divisions, we 
 arrive at a numerical remainder, which may be represented by V r . 
 
 [n this case wo know that the equation V = has no equal roots, since the 
 polynomials V and V, have no common divisor function of x. Representing by 
 
 Q,, Q 2 Qr-i, the quotients given by the successive divisions, which leave 
 
 for remainders — V s , — V g — V r , we have this series of equalities : 
 
 V =V,Q l -V. 2 
 
 V I = V : ()..-V 3 
 V : =V 3 Q 3 -V 4 (1) 
 
 • * • 
 
 V r l,=V r ^ I Q r _ 1 -V r . 
 
 Thus much being premised, the consideration of this system of functions 
 
 V, Vn V 3 V r furnishes a sure and easy means of knowing how many real 
 
 oots the equation V=0 has comprehended between two numbers A and B of 
 any magnitude or signs whatever, B being greater than A. The following is 
 the rule which attains this object : 
 
 Substitute in place of x the number A in all the functions V, V,, V 3 
 
 Vr_!, V r , then write in order, in one line, the signs of the results, and count 
 the number of variations which are found in this succession of signs. Write, 
 in the same manner, the succession of signs which these same functions take 
 by the substitution of the other number B, and ccmnt the number of variation* 
 
324 ALGEBRA. 
 
 which are found tn this second succession. The number of varia ons which it 
 has less than the first will be the number of real roots of the equation V = 
 comprehended between the numbers A and B. If the second succession has a.« 
 many variations as the first, the equation V=0 has no real root between A 
 and B. 
 
 III. We shall demonstrate this theorem by examining how the number of 
 variations formed by the signs of the functions V, V,, V ; . . . V n for any one 
 value whatever of x, can change, when x passes through different states of 
 magnitude. 
 
 Whatever may be the signs of these functions for one determinate value of 
 c, when x increases'by insensible degrees to beyond this .value, there can take 
 place no change of signs in this succession of signs, unless one of the functions. 
 V, V, ..., changes sign, and, consequently (155, note 3°), becomes zero 
 There are then two cases to examine, according as the function which van- 
 ishes is the first, V, or some one of the other functions, V,, V 2 . . . V r _,, in- 
 termediate between V and V r : the last, V r , can not change sign, since it is a 
 number positive or negative. 
 
 IV. Let us see first what alteration the succession of signs experiences when 
 r, in increasing in a continuous manner, attains and passes by a value which 
 destroys the first function V. Designate this value by c. The function V, 
 derived from V, can not be zero at the same time with V for x=c, because 
 by the hypothesis the equation V=0 has not equal roots. We see, besides, 
 by the equations (1), without falling back upon the theory of equal roots, that 
 if the functions V and Vi were zero for x=c, all the other functions, V a , V 3 
 . . ., and, finally, V r , would be zero at the same time ; but, on the contrary. V r 
 is by hypothesis a number different from zero. V, has then for x=c a value 
 dilferent from zero, positive or negative. 
 
 Let us consider values of x very little different from c. If in designating by 
 u a positive quantity as small as we please, we make by turns x=c — u and 
 z=c-\-u, the function V, will have for these two values of x the same sign 
 that it has for .r=c ; because we can take u sufficiently small, to insure that V , 
 shall have for these two values of x the same sign that it has for x=c ; since 
 we can take u so small that V t will not vanish, and not change sign, while x 
 increases from the value c — u to c-\-u.* 
 
 We must now determine the sign of V for u -=c-\-u. Designate for a mo- 
 ment V by f(v), V, by /'(a"), and the other derived functions of V by/"(.r), 
 f'"(x) . . . • ,/"'(')' which are not to bo confounded with V a , V 3 , &C., these 
 latter not being derived functions. When we make x=c-\-u, V becomes 
 f(c-\-u), and we have (see note to Prop. III., Art. 239, or Art. 251) 
 
 /(c+tt)===/(c)+/'(c)u+Y^ 8 +n^« 3 +.&c.; 
 r^ne^, observing that/(r) is zero, and that /'(c) is not, 
 
 f(c+u)=u[f'(c)+ f ^»+{-^>S+-]- 
 We see from tl's expression of/(c-|-u), that in attributing to u very small 
 
 " The deft cat; point on whit li the theorem hinges is the one stnted here. Let it he dis- 
 tinctly seen that sHic* V, can not be zero at the 6amc time with V when .r — <-, therefore, 
 however little c maj dt£er from a value which reduce* V, to zero, u may be taken smaller 
 than this difference 
 
THEOREM OP STURM. 325 
 
 positive values, f{e-\-u) will have the same sign as/'. \* and, consequently, 
 J (c-f-w) will have also the same sign &sf'(c-\-u), since ''(c-j-w) has the same 
 sign as /'(c). Thus, V has the same sign as V, for x=c-\-u. 
 By changing u into — u in the preceding formula, we have 
 
 /( c _ M )- = _ M [/'( c) _^i ) t4+) & c .] 
 
 A.nd we perceive, in the same manner, that f(c — u) has a sign contrary to 
 that of /'(c), from whence it follows that for x=c — u the sign of V is contrary 
 to that of V , . 
 
 Then, if the sign of /'(c) or of V t , for x=c, is -f-t ^ ie s 'g n °f V W 'U be 4" 
 for x=c- T -u, and — for x=c — u. If, on the contrary, the sign of V! is - 
 for ar=c, that of V will be — for x=c-\-u, and -|- for x=c — u. Besides, V, 
 has for x=c-\-u and for x=c — u the same sign as it has for x=zc. 
 These results are indicated in the following table : 
 
 V V, V V, 
 
 cx=c—u, (-, -| , 
 
 For < x=c, +, or else — , 
 {x—c-\-u, + +, • 
 
 Thus, when the function V vanishes, the sign of V forms with the sign ol 
 V , a variation, before x attains the value c, which reduces V to zero, and this 
 variation is changed into a permanence after x passes this value. 
 
 As to the other functions, V 2 , V 3 , &c, each will have, as V,, either for 
 .r=c-|-w or for x=c — u, the same sign that it has for x-=c, that is, if none of 
 them vanish for x=c at the same time with V. 
 
 The succession of the signs of the functions V, V 1( V 2 ... V„ loses then a 
 variation, when x, going on increasing, passes over a value c, which reduces 
 the first function V to zero without destroying any of the other functions, V,, 
 V 21 &c. It is necessary now to examine what happens when one of these 
 functions vanishes. 
 
 V. Let there be a function, V„, intermediate between V and V r , which is 
 lestroyed when x becomes equal to b. This value of x can not reduce to zero 
 cither the function V„_i, which precedes immediately V n , or the function 
 \'„+i, which follows V n . Indeed, we have between the three functions V n _!, 
 \ r „, V n+1 , the following equation, which is one of the equations (1). 
 
 V n _ 1 = V n Q, 1 -V n+I . 
 
 It proves that if the two consecutive functions, V n _ H V n , were zero for the 
 same value of x, V„ +1 would be zero at the same time ; and as we have also 
 
 we should have, again, V 11+ n=0, and so on, so that we should have finally 
 V r =0, which is contrary to the hypothesis. 
 
 The two functions, V„_, and V n+l , have then for x=b values different 
 from zero : moreover, these values are of contrary signs, because the same 
 equation, 
 
 V a _ 1= V n Q n -V n+1 , 
 
 gives V„_i = — Vn+n when we have V n =0. 
 
 * This depends upon a principle demonstrated at Art. 239, Cor., that if a function of a be 
 arranged according to the ascending powers of v, u may be taken so small that the sign 
 of the whole function shall depend upon that of its first term. 
 
326 ILttKBRA. 
 
 This being established, substitute in place of two numbers, b—u and / 
 very little different From b; the two functions, S , nod V n+ i. will have ton 
 these two values of x the same Bigns as they have for x=zb, sine an al- 
 
 ways take a sufficiently small, to insure thai neither V„ ; nor V ,, +1 shall chi 
 sign when r enlarges in the interval from b — u to A-}-//. Whatevel may he 
 the sign ot V„ for xssi — u, as it is placed in the iion of Bigns between 
 
 those of V n _[ and V r n+1 , which are contrary, the Bigns of these three consecu- 
 tive functions, V„ ,. Y„. V , +! , for /='* — u, will form always either a perma- 
 nence followed hy a variation, or a variation followed by a permanence, as i* 
 seen in the following scheme 
 
 V^ V„ V n+1 V , v„ v n+l 
 
 For X=b — U -f- rb —, or else, — i +• 
 
 Similarly, die signs of the three functions, V n _i, V„, V, l+ |, for .r = b-{-v, 
 whatever may be that of V„, will form one variation, and will form but one. 
 
 Besides, each of the other functions will have the same sign for X=b — 
 and .r = l)-\-u, provided no one of them is found to be zero for x = b at the 
 same time as V n . 
 
 Consequently, the succession of the signs of all the functions, V, Vj ... V r , 
 for X=b-\-U, will contain precisely BS many variations as the succession of 
 their signs for x=b — U. Thus, the number of variations in the succession of 
 signs is not changed when any intermediate function whatever passes through 
 zero. 
 
 Oue arrives evidently at the same conclusion, if many intermediate functions, 
 not consecutive, vanish for the same value of X. But if this value should de- 
 stroy also the first function, V, the change of sign of this one would then make 
 one variation disappear at the left of the succession of signs, as has been shown 
 in IV. 
 
 VI. It is then demonstrated that each time that the variable x, in increasing 
 by insensible degrees, attains and passes a value which renders V equal to 
 zero, the series of the signs of the functions V, V,, V a ... V, loses a varia 
 tion formed on its left by the siy;ns of V and V t , which is replaced by a per- 
 manence, while the changes of signs of the intermediate functions. V,, V 
 — V r _,, can never either augment or diminish the number of variations which 
 existed already. Consequently, if we take any number whatever. A, positive 
 Or negative, and any other number whatever. B, greater than A, and if we 
 make X increase from A to P>. as many values of X as are comprised between A 
 and B, which render V equal to zero, so many variations will the succession 
 of signs of the functions V, V, ... V r for .''=1! contain less than the suc- 
 cession of their Bigns for ..'= A. This Was the theorem to be demonstrated. 
 
 Remark. — In the successive divisions which Berve to form the functions V . 
 V3, &c, we can, before taking a polynomial for a dividend or divisor, multiply 
 
 or divide it by any positive number at pleasure. The functions V, V,, V . 
 .... V rl obtained by this operation, will differ only by positive numerical fac- 
 tors from those which we have previously considered, and which appear in 
 equations (l), so that they will have respectively the same Bigns as these for 
 
 each value of X. , 
 
 With this modification we can. when the coefficients of the equation V=fi 
 are whole numbers, form polynomials V . V . &c., the coefficients of w 
 shall be also entire, lint it is necessary to take good care thai the num 
 factors thus introduced oi suppressed be all positive. 
 
THEOREM OF STURM. 327 
 
 VII. This theorem gives the means of knowing the whole number of real 
 roots of the equation V=0. 
 
 In fact, an entire polynomial function of x being given, we can always as- 
 sign to x such a positive value as that for this and eveiy greater value the 
 polynomial will have constantly the sign of its first term (see Art. 239). It is 
 the same with all negative values of x below a certain limit. All the real roots 
 of the equation V = being comprised between — co and -\-<x>, it will be suffi- 
 cient, in order to know their number, to substitute — co and -|-co instead of A 
 and B, iu the functions V, V,, V 2 ... V r , and to note' the two successions of 
 signs for — co and -j-oo. When we make x= + co, each function is of the 
 same sign as its first term. For x= — co, each function of an even degree, in- 
 cluding V r , has the same sign that it has for .r = -f- co ; but each function of an un- 
 even degree takes for x= — co a contrary sign to that which it has for x = + co 
 The excess of the number of variations formed by the signs of the functions V, 
 V x ... V r , for x= — co, over the number of variations for .r=-|-co, will express 
 the whole number of real roots of the equation V=0.* 
 
 To determine the initial figures of the roots, we may substitute the sue 
 cessive numbers of the series 
 
 0, —1, —2, —3, —4, 
 
 till we have as many variations as — co produced; and if we substitute the 
 numbers of the series 
 
 * One might be curious to know how the succession of signs of the functions V, V If V 2 
 
 . .V r must undergo change so as that a variation is lost every time that V vanishes. 
 
 We have seen (IV.) that if c is a root of the equation V=0, the two functions V and 
 Vi must have contrary signs for .r=c — ?/, and the same sign for x=c-\-u. So that if we 
 designate by d the root of the equation V=0, which is next greater than c, so that be- 
 tween c and d there is no other root, V, will have for x=zd — u a sign contrary to that of 
 V. But V has constantly the same sign for all values of x comprised between c and d ; 
 and as V! has the same sign as V for x=c-\-v, and a contrary sign to that of V for x=d 
 — u, we see that Vj has two values with contrary signs for x=c-\-u and for x=d — u ; 
 then, while x increases from c-\-u to d — u, Y Y must change sign once, or an uneven num- 
 ber of times (I., or Prop, of Ait. 252, Cor. 1). 
 
 Let y he the only value of x, or the least value of x between c and d, for which V, 
 changes sign. V and V 2 will have for .T=y — u the same common sign that they have for 
 a;=c-f-M. For a?=}'-f-w V will have this same sign ; but V[ will have the contrary sign. 
 V, will have a sign contrary to that of V for the three values for y — v, y, and y-f-w (V.). If, 
 for example, V is positive for x=c-\-u, we have the following table : 
 
 v V, v s 
 
 For x=y — u + -\ 
 
 x—y + — 
 x=y+u -j 
 
 Thus, before x attained the value c, which destroys V, the signs of V and V! formed a 
 variation which is changed into a permanence after x has overpassed this value c ; this 
 permanence subsists until Vi changes sign, then it is anew replaced by a variation after 
 the change of sign of Vi ; but, at the same time, there is a variation fonned by the 
 of V] and Vj which changes into a permanence, so that the number of variations in the 
 total succession of signs is neither increased nor diminished. 
 
 If Vi changes sign a second time for a new value of x comprehended between c and d , 
 the variation which the signs of V and V form before x attains this value will he b 
 replaced by a permanence ; and still, on account of V;, the number of variations will re- 
 main the same in the succession of signs. As Vi can thus change sign only an uneven 
 number of times, after its last change the signs of V and Vi will form a variation which 
 
 will in itil attains the valne d, which destroys V. We have not to consider here 
 
 u, e V va..i - is without changing sign. 
 
328 ALGEBRA 
 
 0, 1, 2, 3, 4, 
 
 till wo arrive at a cumber which pro luces as vaaaj variations as -|-ao then 
 the numbers thus obtained will be die limits of the roots of the equation, and 
 the situation of the roots will be indicated by the signs arising fron rhe sub- 
 stitution of the intermediate numbers. 
 
 We shall now apply the theorem to a few 
 
 EXAMPLES. 
 
 (1) Find the number and situation of the roots of the equation 
 
 x 3 — 4a: 2 — Gx+8=0.* 
 Here we have V = a? — 4a? — Gx-f-8 
 
 V 1= 3.r 2 — 8.r— 6; 
 then, multiplying the polynomial V by 3, in order to avoid fractions, 
 3x 2 _8z— 6) 3.1^— 12x 2 — 18x+24 {x — 1 
 3.r»_ 8.r-— 6x 
 
 — 4x 2 — 12x-|-24, multiply by J; 
 or — 3.C-— 9x-f 18 
 
 — 3.r 2 + 8.r-r- 6 
 
 — 17x+12 .-. V 2 =17x— 12 
 3j2_ 8r _ G 
 
 17 
 
 17x— 12)51x-— 13Gz— 102 (3x 
 51x 2 — • 36* 
 
 — lOOx— 102. 
 It is now unnecessary to continue the division further, since it is very ob- 
 vious that the sign of the remainder, which is independent of x, is — ; and, 
 therefore, the series of functions are 
 
 V = I s — 4x" — Gx+8 
 V 1= 31 s — 8x —6 
 V B =17z— 12 
 
 V:; = + - 
 
 Put -\- co and — oo for x in the leading terms of these functions, and tha 
 signs of the results are 
 
 * The process applied to the general cubic equation a^-f-tf-r-'-f/u-f-c^O, gives the fol- 
 lowing functions, viz. : 
 
 With the second A 
 
 V = .??+ ax°~\-bx-\-c 
 
 V,=3^-i-2r7j: +b 
 
 V a =2(a 2 — 3%-}-rt£— 9c | 
 
 V 3 =— 4a3c+a 2 i=— \8abc— ilr>— 
 
 0) 
 
 Without ; m, or a—0. 
 
 V = tf+bx+c 
 
 V,=3.r*-f/> 
 
 Y. i =—^bx—2c 
 
 V a = — 16 ! — -J7r- 
 
 (2) 
 
 These functions in (1) and | J) « ill frequently he {bond useful in the application of Sturm's 
 theorem to equations of the third de , i the ions in any particular ex- 
 
 ample may be found by substitution only. In order that all the roots of the equation 
 o?-\-bx-\-c=0 may be real) the first terms ei' the {auctions must he positive ;t hence — 26* 
 and — 46 3 — 27c 2 must be positive : and as — 27c a is always negative, b must he negative, 
 in order that — 4i 3 and — 2b may he positive; therefore, when all the r.*>t-; are real, 4b 1 
 
 C'V I V 
 
 must be greater than 27c", or l-l greater than I ( ) • V\ hen, therefore, £ is negative anil 
 
 (')Hr 
 
 3' 
 all the roots are real, a criterion which has been long known, and as simple as 
 
 -..in be given. 
 
 T Si 
 
THEOREM OF STUIlil. 
 
 329 
 
 For 2 - =-j-oo, + + + + no variation, 
 
 ■v= — », 1 1- three variations, 
 
 .-. 3 — = 3, the number of real rbots in the proposed cubic equatun. 
 
 Next, to find the situation of the roots we must employ narrower limits 
 
 than -(-co and -co. Commencing at zero, let us extend the limits both ways. 
 
 WiVjVs Var. VViViV., Var. 
 
 2 
 
 For.r = signs -\ 1- 
 
 2 
 
 For x= s'iejds -| \- 
 
 .r=l.... -- + + 
 
 1 
 
 x=-l.... + + - + 
 
 2=2. .. + + 
 
 1 
 
 x=-2.... -+- + 
 
 ,r=3.. . + + 
 
 ] 
 
 
 x=4 . . . h + + 
 
 1 
 
 
 x=5.... + + + + 
 
 
 
 
 We perceive, then, by the columns of variations, that the roots are between 
 and 1, 4 and 5, — 1 and — 2 ; hence the initial figures of the roots are — 1, 
 0, and 4 ; and, in order to narrow still further the limits of the root between 
 and 1, wo shall resume the substitutions for x in the series of functions as 
 before. But as the substitution of 1 for x, in the function V, gives a value 
 nearly zero, wo shall commence with 1, and descend in the scale of tenths, 
 until we arrive at the first decimal figure of the root. 
 
 Let x= 1 signs 1- -f- one variation, 
 
 .r=-9 . . . . -| }- -4- two variations ; 
 
 hence the initial figures are — 1, -9, and 4. 
 
 (2) Find the number and situation of the real roots of the equation 
 
 ^_|_.r 3 — x 2 — 2.r+4 = 0. 
 Here the several functions are 
 
 V = r»+ a?— x"— 2:c+4 
 V,= 4x s -\-3x°~—2x —2 
 V. 2 = x*+2x —6 
 V 3 =- x +1 
 V, = + . 
 
 Let x= -f- qd, signs of leading terms + + H H lrsvo variations 
 
 .r= — co -| f- -{- -|- two variations ; 
 
 and all the roots of the equation are imaginary. 
 
 When, in seeking for the greatest common divisor of V and V\, we arrive 
 at a polynomial V n (for example, at that of the second degree), which, put 
 equal to zero, will only give imaginary values of x, it is not necessary to cany 
 the divisions further, because this polynomial V n will be constantly of the snma 
 sign as its first term for all real values of x ; for if it gave a plus sign for one 
 value, and a minus for another, there must be a real root between.* 
 
 (3) Required the number and situation of the real roots of the equation 
 
 2x-'— llx 2 +8.r— 16=0. 
 The first three functions are 
 
 V= 2x 4 — ll.r 2 +8.r— 16 
 V,= 4.i- — llx+4 
 V. = ll.r 2 — 12x +32; 
 
 * This consideration is :»f importance, as the calculations for letemiining the functions 
 v*. ; , V;j are long, especially toward the last, on account of the magnitude of their numerical 
 coefficients. 
 
33C ALGEBRA. 
 
 and the roots of the quadratic 111 8 — 12x+32=0 are imaginary, for 11x32 
 X4 ia greater than 12 s ; hence V must preserve the Bame sign f r ever) 
 value of .r, and the subsequent i'uixt i<ms can not change the Dumber of varia- 
 tions, for a variation is only losl by the change of the sign of V. Honce, 
 
 For rr=-f assigns + + + "" variation, 
 
 ,r = — x . . . -| \- two variations; 
 
 and the proposed equation has two real mots, the one positive and the othei 
 negative, since the last term is negative. (Prop. VIII., Cor. 5, p. 014.) 
 
 When ./■ = () signs — + + x= sigus — + -f- 
 1=1.... r- xsr— 1 h + 
 
 
 o 
 
 -+ + *=-2.... + 
 
 i=3 1- + + x= — 3 ^ H. 
 
 Hence the initial figures of the real roots are 2 and —0. 
 
 When two roots are nearly equal to each oOier. 
 y\) Find the roots of the equation 
 
 a-s+llz 2 — 102.r-f 181=0. 
 The functions are 
 
 V= r»+ll.r ! — 102x+181 
 V,= 3x-+22.r — 102 
 V. = 122.r —393 
 V 3 =+; 
 and the signs of the leading terms are all + ; hence the substitution of — >- 
 and +ao must give three real roots. 
 
 To discover the situation of the roots, we make the substitutions 
 
 .r = which gives -| \- two variations, 
 
 a=l -\ 1- 
 
 •r=2 + • + 
 
 ar=3 -4. 1- two variations, 
 
 x=l -| — J — | — j- no variation ; 
 
 he ce the two positive roots are between 3 and 4, and we must, therefore, 
 form the several functions into others, in which X shall .)0 diminished by 
 ?,. This is effected by Art. 251, p. 315 ; and wo get 
 
 V = 7/ r, 4-20y-— 9y-f 1 
 
 V'x= rw/- + -l(i_y — 9 
 
 V'„ = 122>/ —27 
 
 v :i =+: 
 
 Make the following substitutions in these functions, viz. : 
 
 y= signs -| \- two variations, 
 
 y = -l ...+__ + 
 
 ■?/=-2 . . . -1 \- two variations, 
 
 ?/=-3 . . • + + + + »" variation : 
 bence the two positive roots arc between 3*2 and ■"■■"•. und we must, again. 
 transform the Inst functions into others, in which y shall bo diminish ed by '- 
 Rffecting this transformation, we have 
 
 V" = 2 -1-J0C.: — — : -4--003 
 V",= V.: '+ 11 -J: —-88 
 \ — ;'■<■ 
 
 V" 3 = + . 
 
THEOREM OF STURM. 33 
 
 Let z= Q then signs are -| (- two variations, 
 
 ; = .01 -\ (- two variations, 
 
 z = -Q2 [-one variation, 
 
 z = -03 -p- -J* -(- + no variation ; 
 
 hence we have 3-21 and 3-22 for the positive roots, and the sum of tht roots 
 is —11 ; therefore, —11 —3-21 — 3-22= — 17-4 is the negative root 
 
 When the equation has equal roots. 
 
 255. When the equation has equal roots, one of the divisors will divide the 
 preceding without a remainder, and the process will thus terminate without a 
 remainder, independent of x. In this case, the last divisor is a common meas- 
 ure of V and V,; and it has been shown (Art. 253, Scholium, p. 321) that if 
 (a: — a{){x — a.,)" be the greatest common measure of V and Vi, then V is di- 
 visible by (x — «i)-(r — a 2 ) 8 , and the depressed equation furnishes the distinct 
 and separate roots of the equation, for Sturm's theorem takes no notice of 
 the repetition of a root. The several functions may be divided by the great- 
 est common measure so found, and t the depressed functions employed for the 
 determination of the distinct roots ; but it is obvious that the original functions 
 will furnish the separate roots just as well as the depressed ones, for the for- 
 mer differ only from the latter in being multiplied by a common factor (29) ; and 
 whether the sign of this factor bo -|- or — > the number of variations of 
 must obviously remain unchanged, since multiplying or dividing by a positive 
 quantity does not affect the signs of the functions ; and if the factor or divisoi 
 be negative, all the signs of the functions will be changed, and the number of 
 variations of sign will remain precisely as before. 
 
 Find the number and situati in of the real roots of the equation 
 a" — 7.r' -4- 1 C.r 1 -4- .r- — lG.r+4 = 0. 
 
 By the usual process, wo find 
 
 V = x 5 — 7.r' -4- 1 3s 3 + x- — 1 6.r + 4 
 
 V,= 5.1-*— 28.r ! +39x 2 -4-2.r — 16 
 
 Vn = ll.r"— 48.1'-— 51.r +2 
 
 V 3 = 3a: 2 — 8a: +4 
 
 V 4 = x— 2 
 
 V 6 =0. 
 
 Hence x — 2 is a common measure of V and V\ ; and if 
 
 .r= — co the signs are 1 1 four variations, 
 
 .r= — 2 1 1 four variations, 
 
 •>'=-! 0+- + - 
 
 X— -1 (- -\ three variations, 
 
 x= 1 \- -| two variations, 
 
 x= 2 
 
 x= 3 |--f"~f" one variation, 
 
 x= 4 -j — J — [ — j — f- no variation. 
 
 Therefore we infer that there are four distinct and separate roots ; one is — 1, 
 for V vanishes for this value of x ; another between and 1 ; a third is 2, and 
 a fourth is between 3 and 4. The common measure .r — 2 indicates that the 
 polynomial V is divisible by (.r — 2) 2 ; and hence there are two roots equal to 
 2 (Art. 2"53, Cor. 1) 
 
M2 ALGEBRA- 
 
 ll may happen that one of the functions, V,, V a ... V r _i, should be found 
 zero either for X=A or u = B. In this case it is sufficient to count the varia- 
 tions which are found in the succession of signs of the functions V, V,, V 3 
 ... V r , omitting tlie function which is zero. This results from the demonstra- 
 tion in Ail. 254, V, for the case where an intermediate function vanish) , 
 
 When the number of the auxiliary functions, V ,. \ '_.. ecc, is equal to the 
 ee of the equation, as is ordinarily the ca b, in consequence <>f each re- 
 mainder in Booking for the common divisor being one degree less than the pre 
 ceding, the number of imaginary roots in the equation may be found by the fol- 
 lowing rule : Tlie equation V = will have as many pairs of im ■ roots 
 as there are variations of sign in the succession of the signs of the first terms of 
 the functions V,, V 2 , &c, to the sign of the constant V m ita 
 
 This follows from tlie fact that two eon. ecutive functions, V„_i, Y„. are 
 the one of an even, the other of an odd degree. Then, if the two timet ions 
 have the same sign for :r = -|-cc, they will have contrary for .r= — c, and 
 
 versa. So thai if we write the succession of signs of V, V,, V, V . for 
 
 r = — co and for x=-j-cc, each variation in the one succession will correspond 
 to a pennant nee in the other. Tims, the number of permanences for .>•= — co 
 is equal to tlie number of variations for .r=- r -cc. 
 
 But for .r=-f-co the number of variations will be that of the first terms of 
 the functions V, V, . . . V„„ which denote by i. Then there will be i per- 
 manences for .r= — co and m — i variations. The excess of the number of 
 variations m — i for x= — co over the number i for x=-f-co, is m — 2t, which 
 is therefore tlie number of real roots of the equation, and therefore -J/ the 
 number of imaginary roots, the whole number of roots being m. 
 
 horner's method of resolving numerical equations or all okokrs. 
 
 256; The method of approximating to the roots of numerical equations of 
 all orders, discovered by W. G. Horner, Esq., of Bath, England, is a pi 
 of very remarkable simplicity and elegance, consisting simply in a Bucce 
 of transformations of one equation to another, each transformed equation 
 
 9 having its roots less or greater than those of the preceding by the cor- 
 responding figure in tlie root of the proposed equation. We have shown bow 
 to discover the initial figures of the roots by the theorem of Stobm : and by 
 making tho penultimate coefficient in each transformation available as a trial 
 or of the absolute term, we arc enabled to discover the succeeding figure 
 of the root; and thus proceeding from one transformation to another, we 
 enabled to evolve, one by one, the figures of the mot of the given equation, 
 and push it to any degree of accuracy required. 
 
 GENERAL RL'I.l I, 
 
 1. Find tho number and situation of the roots by Snirm's theorem, and let 
 the root required to be found l>e positive. 
 •_'. Transform the equation into another whose roots shall be less than those 
 
 of the proposed equation by the initial figure of the root. 
 
 .;. I divide the absolute term of the transformed equation by the trial divisor, 
 
 or penultimate coefficient, and the n. \t figure of the root will be obtained, by 
 which diminish the root of the transformed equation as before, and proceed in 
 this manner till tho root bo found to the required accUTI 
 
NUMERICAL SOLUTION OF ALGEBRAIC EQUATIONS. 333 
 
 Note 1. — When a negativo root is to be found, change the signs of the alter- 
 nate terms of the equation, and proceed as for a positive root. 
 
 Note 2. — When three or four decimal places in the root are obtained, the 
 operation may be contracted, and much labor saved, as will be seen in tho 
 following examples : 
 
 EXAMPLES. 
 
 (1) Find all the roots of the cubic equation 
 
 3?—1tx-\- 7=0. 
 By Sturm's theorem, the several functions are (Note, p. 328), 
 
 V = a*— 7z+7 
 
 V, = 3x~ — 7 
 V i= 2x —3 
 
 Hence, for x=-\-fx> the signs are + + + + no variation, 
 
 x= — oo 1 \- three variations ; 
 
 therefore the equation has three real roots. 
 
 To determine the initial figures of these roots, we have 
 
 for x=0 signs -| (- for :r = signs -| 1- 
 
 x=l . . . -\ 1- *=— 1 • • • -\ h 
 
 x=2...+ + + + x=-2 . . . + + - + 
 
 x=-3 . . . + + - + 
 x=-4 . . . - + - + 
 nence there are two roots between 1 and 2, and one between — 3 and — 4 
 
 But in order to ascertain the first figures in the decimal parts of the two 
 roots situated between 1 and 2, we shall transform tho preceding functions into 
 others, in which the value of a: is diminished by unity. Thus, for the fui ction 
 V we have this operation : 
 
 1_|_0 —7 +7 (1 
 1 1 —6 
 1 ^6 +T 
 
 1 2 
 
 2 ^4 
 1 
 
 ~3 
 And transforming the others in the same way, we obtain the functions 
 V =f+3y*— 42/+1; V\=3y*+6y— 4; V' 3 =2y— 1; V'.« = + . 
 
 Let 2/=* 1 t ^ ien tf 10 SI 8 ns nre ~\ 1~ two variations, 
 
 y = -2 -\ 1- do. 
 
 y=-o + + do. 
 
 y= -4 1- one variation, 
 
 2/ = -5 T+ do - 
 
 2/-6 - + + + do. 
 
 y = -7 + + + "f* no variation. 
 
 Therefore, tho initial figures of the three roots are 1-3, 1-6, and — 3. 
 
 The rest of the process, with a repetition of the above, is exhibited and 
 afterward explained below. 
 
334 
 
 
 ALGEBRA 
 
 
 
 1 + 
 
 1 
 
 1 
 1 
 
 — 7 
 1 
 
 — 6 
 o 
 
 
 + 7 (1 -35689580 7 
 —6 
 
 •1... 
 
 —903 
 
 o 
 
 1 
 
 — *4.. 
 99 
 
 *97 . . . 
 
 — 86625 
 
 •33 
 3 
 
 — 301 
 108 
 
 — *1 9 3 
 
 1975 
 
 •10375 . . 
 
 —9048984 
 
 36 
 3 
 
 •1326016 
 
 — 1181430 
 
 *39 5 
 5 
 
 — 17325 
 2000 
 
 36 
 
 141586 
 — 132923 
 
 40 
 5 
 
 —•15 3 25 
 
 243 
 
 8663 
 —7382 
 
 *40 56 
 6 
 
 — 1508164 
 24372 
 
 1281 
 
 — 1181 
 
 40 62 
 
 6 
 
 — *1 48379 
 3 25 
 
 g 
 
 4 
 
 8 
 4 
 
 100 
 
 —89 
 
 *j40|68 
 
 — 148053 
 325 
 
 11 
 — 10 
 
 
 — 147 7218 
 3|6 
 
 
 1 
 
 
 — 1 4 7 G 9 
 3 
 
 •> 
 6 
 
 
 — 1|4|7|6|5 
 The process here is similar to that on p. 318. Tht< numbers marked with 
 stars are the coefficients of the equation having the reduced roots. Thus, *3, 
 *4, and *1 are the coefficients of the equation whose roots are 1 less than 
 those of the proposed equation. The right-hand 3 of *3C he ."> tenths add- 
 ed in the next step of the process, which has for its object to reduce the roots 
 by -3. The coefficients of the resulting equation are *39, — *193, and *97. 
 Now, instead of going on in this manner to obtain the following figures, 568, 
 dec, of the root, lite method of proceeding changes : the 193, which is t lie 
 penultimate coefficient, becomes a trial divisor, by which dividing the absolute 
 term 97, which is .097, the divisor being 1-93, the quotient is 5, the next fig- 
 ure of the root, which is .05. This 5 is annexed to the '•".!>. and we proceed 
 as before; that is, multiply the *395 in the first column by this 5, producing 
 •1975 in the second column, and by addition, 1*7325, and so on. To show that 
 the quotient figure 5 is obtained by means of the trial divisor, observe that the 
 1-7325 is nearly equal to the U-:i3 above, and that the -086625 in the third 
 column, which is the product of 1*7325 by the -05, is nearly equal to tile *-097 
 above; hence the quotient of •• 097 by 1*93 is nearly this same '05. 
 
 The further W6 DTOCeed, the inure accurate this process becomes, for the 
 
 first figure of each Dumber in the first column being units, this, multiplied by 
 the decimal figure found in the root, which is thousandths, tens of thousandths, 
 
 and so on, that is. soon a very small fraction. gives thousandths, tens of thou- 
 sandths, and so on, or a \eiy small fraction, for the product : atnl. the first 
 
NUMERICAL SOLUTION OF ALGEBRAIC EQUATIONS. 
 
 335 
 
 liixire in the numbers of the second column being also units, these numbers 
 tire not much affected by tho addition of the above-named products.* 
 
 When the number of decimal places in the numbers of the third column 
 hecomes equal to tho number of decimal places required in the root, il will 
 not be necessary to obtain any more in the third column ; and as each new 
 decimal figure in the root, multiplied by the number in the second column, 
 would make one more place in the third, it will be necessary to cut off one 
 figure in the second column, and, for a similar reason, two figures in the first 
 column. As soon as the figures are all cut off in the first column, the process 
 becomes simply one of division, the divisor and dividend rapidly diminishing. 
 
 Wo have thus found one root x=l-356895867 , and the coefficients 
 
 of the successive transformed equations arc indicated by the asterisks in each 
 column. To find another, we have the following : 
 
 1 + —7 +7(1-692021471 
 
 1 1 -G 
 
 1 
 1 
 
 2 
 
 1 
 
 -G 
 2 
 
 •4 . . 
 216 
 
 36 
 6 
 
 42 
 6 
 
 48 9 
 9 
 
 49 8 
 9 
 
 50 72 
 2 
 
 50 74 
 2 
 
 1... 
 
 ■1104 
 
 • 104... 
 100809 
 
 ■ 18 4 
 252 
 
 68 . . 
 44 01 
 
 112 1 
 4482 
 
 15 6 8 3.. 
 
 10 144 
 
 1578444 
 10148 
 
 15885 
 
 1 
 
 -3191... 
 3156888 
 
 —34112 
 
 31774 
 
 —2338 
 1589 
 
 —749 
 635 
 
 — 114 
 111 
 
 |50|76 1|5|8|8|7| 
 
 Another root is .r=l-692021471 . . . 
 
 For the negative root, change the signs of the second and fourth terms. 
 
 * To show tliis in a more general way, let 
 
 aa; n +Bx n - I +B.z n -- .... +B n _,.r+B n =0 
 
 oe one of the depressed equations which is to furnish the next decimal place of the root of 
 ,he proposed equation; the value of a; in this depressed equation will of course be a very 
 jraftU fraction; hence the higher powers of it may, without much error, be neglected. The 
 ,1. •pressed equation thus reduces to 
 
 B^.t+B^O. 
 
 Hence the value of x, without regard to its sign, is 
 
 x- 
 
 "B n 
 
 nearly ; that is, it may be obtained by dividing the ultimate by the penultimate coefficient 
 
336 
 
 ALGEBRA. 
 
 -0 
 3 
 
 3 
 
 3 
 
 ~6 
 3 
 
 7 
 9 
 
 90 4 
 
 4 
 
 90 8 
 
 4 
 
 91 28 
 
 8 
 
 9136 
 
 8 
 
 .|91|44 
 
 2 
 
 18 
 
 20 ... . 
 3616 
 
 2,4 
 
 203616 
 3632 
 
 2 7 2 4 8 
 730 
 
 20797824 
 73088 
 
 2087091 
 823 
 
 2 
 
 
 
 2087914 
 823 
 
 2 
 
 
 
 208873 
 
 7 
 9 
 
 
 208874 
 
 6 
 9 
 
 — ? (3-048917339* 
 
 + 6 
 
 — 1 
 
 814464 
 
 — 185536... 
 166382592 
 
 — 19153408 
 18791228 
 
 —362180 
 
 208875 
 
 — 153305 
 
 146212 
 
 — 7093 
 6266 
 
 — 827 
 626 
 
 — 201 
 188 
 
 — 13 
 L2 
 
 2|0|8|8|715 1 
 
 Hence the three roots of the proposed cubic equation are 
 
 x= 1-356895867 
 
 x= 1-692021471 
 
 x=— 3-048917339 
 
 (2) Find the roots of the equation .r 3 -f ll.r 3 — 102x+ 181 = 0. 
 We ltave already found the roots to be nearly 3-21, 3*22, and —17. 
 1 b 4, p. 330.) 
 
 +11 
 
 3 
 
 —102 
 42 
 
 page. 
 
 + 181 (3-2131277.0 
 — 180 
 
 14 
 3 
 
 — 60 
 51 
 
 1... 
 
 — 992 
 
 1 7 
 3 
 
 — 9 . . 
 
 4 04 
 
 8... 
 — 6739 
 
 2 02 
 o 
 
 — 496 
 
 408 
 
 1261... 
 
 — 1217403 
 
 ■jiil 
 2 
 
 — 88. . 
 2061 
 
 43597 
 --34183 
 
 2 06 1 
 1 
 
 6739 
 2 6 '.' 
 
 9414 
 
 — <;787 
 
 2 06 2 
 
 — 4677 . . 
 Carried to next 
 
 2627 
 
 (See 
 
NUMERICAL SOLUTION OF ALGEBRAIC EQUATIONS. 
 
 a37 
 
 2 Oi 
 
 2 06 33 
 3 
 
 2 06 36 
 3 
 
 4677 . . 
 61899 
 
 •2106139 
 
 — 405801 
 61908 
 
 — 343893 
 2064 
 
 — 341829 
 206 4 
 
 — 3397 6 
 . 4 1 
 
 
 — 3393 
 4 
 
 5 
 
 1 
 
 2627 
 —2372 
 
 255 
 — 237 
 
 18 
 —16 
 
 3|3|8|9 
 
 In a similar manner, the two remaining roots will be found to be 
 
 and 
 
 .r=3-22952121 
 
 x= — 17-44264896. 
 
 (3) Given a- 4 -|-:r' , -{-t 2 +3:r — 100=0, to find the number and situation of 
 the real roots. 
 Here we have 
 
 V r- .r'+ x 3 -f- a:2_j_3 x _ 100 
 V 1 ^=4x 5 +3x^2x -\-3 
 V 2 =— 5.r 2 — 34a:+1603 
 V 3 =— 1132.r+6059 
 V, = — 
 
 Iiet a'= — co then signs are -| 1 three variations, 
 
 .r=-4- c o +H one variation ; 
 
 hence two roots are real and two imaginary ; and the real roots must have 
 contrary signs, for the last term of the equation is negative. To find th 3 si^ 
 nation of the roots 
 
 in V ViVsVsV, 
 
 Let x=0 signs \--\--\ 
 
 x=l. . ._+++_ 
 x=2. . ._+++- 
 x=3. . .+ + + + - 
 
 in V V,V 9 V,V< 
 
 Also, x= signs h4H 
 
 .r=— 1 • • • — + + — 
 
 x=-2. . . + + - 
 
 T= _3. . . + + - 
 
 x=-4. • .+- + + _ 
 In this example the function V x vanishes for .r= — 1, and for the o^.... 
 value of x the functions V and V 2 have contrary signs, agreeably to V., p 
 325, and writing -4- or — for gives the same number of variations. The 
 initial figures of the root are, therefore, 2 and — 3. 
 
 Y 
 
338 
 
 ALGEBRA. 
 
 To find the negative root, we have the following operatiou 
 
 11 4 
 4 
 
 11 8 
 4 
 
 12 2 
 
 4 
 
 ~ n]~63 
 
 3 
 
 12 66 
 
 3 
 
 12 69 
 
 3_ 
 
 12 723 
 
 3 
 
 " 12 726 
 
 3 
 
 ~~ 12 72!» 
 
 3 
 
 |-121732 
 
 I— 1 
 
 3 
 
 2 
 
 + 1 
 
 6 
 
 7 
 
 3 
 5 
 
 15 
 22 
 
 3 
 
 8 
 
 24 
 ~46 . . 
 
 3 
 
 4 56 
 
 50 56 
 
 
 4 72 
 
 
 55 28 
 
 -! 88 
 
 
 60 16 . . 
 
 37 89 
 
 
 60 53 89 
 
 37 98 
 
 
 60 91 87 
 
 38 07 
 
 
 61 29 91 
 
 3 81 69 
 
 61 33 75 69 
 
 3 8178 
 
 61 37 57 47 
 
 3 81 87 
 
 61 41 39 
 
 34 
 
 63 
 
 6 
 
 61 42 02 
 
 9 
 
 63 
 
 6 
 
 61 42 66 
 
 5 
 
 63 
 
 6 
 
 |61|43|30 
 
 21 
 
 1 8 
 
 66 
 
 84 . . . 
 
 2 0224 
 
 10 4 2 2 4 
 
 2 2 112 
 
 12 6 3 3 6 
 
 1816167 
 
 128152167 
 
 18 2 7 5 6 1 
 
 12 9 979728. . . 
 
 184012707 
 
 13016 3 740707 
 184127241 
 
 034786794 
 3 7 10 14 
 
 1 3037857809 
 3071332 
 
 13 4 9 2 9 14 
 43003 
 
 1304135 9 17 
 4 3 3 
 
 13 0417892 
 430 
 
 13 418322 
 
 430 
 
 13041875 
 4 
 
 ■ 
 
 1 
 
 3 ii 
 
 i 
 
 1 
 
 .- 
 
 - u 
 I 4 
 
 
 — 100 (3-433577863365M 
 
 —46. .. 
 416696 
 — 43104... 
 384456501 
 
 390491222121 
 
 ~; 
 
 651892890 16 
 
 — 10154478833 
 
 9128951421 
 
 — 1025527 
 
 91292 
 
 — 112599158 
 
 101335040 
 
 14118 
 
 .130 
 — l: 
 391 
 
 -47732 
 391 
 
 — - 
 
 —781 
 
 — 129 
 
 117 
 
 — 12 
 II 
 
 1 
 
 For the positive root wo have a similar operation, 
 
 1 +1 +1 +3 —100 (2-80285121815- .': 
 hut this wo shall leave for the student to perform, and the two roots will b« 
 found to be 
 
 x= 2-8028512181582 . . . 
 
 x=— 3-4335778633659 . . . 
 
 (1) Find tho roots of the equation a- 5 +2.( •' + ."..< "'+ 1 r'-f">.r— 20=0 
 Here we have V = a*+ 2r , -f- 3r>+ I r- + 5.r — 20 
 V,=5.r , 4- 8a?4- 9z"+ar+5 
 \ — — 7./-' — 2 1 ./-• — 1 2 x + 255 
 V 3 = — 13r+14 
 V 4 = — 
 
 For r = — j. we have signs (- + H ,NV " varialione ; 
 
 tzzz-^-cc -|"H um; variation. 
 
NUMERICAL SOLUTION OF ALGEBRAIC EQUATIONS. 
 
 3W 
 
 Hence the difference of variations of sign indicates the existence of one real 
 -ind four imaginary roots, the real root being situated between 1 and 2. 
 
 1 + 2 
 
 4- 3 
 
 + 4 
 
 
 + 5 
 
 
 
 
 — 20 (1-125790.. 
 
 1 
 
 3 
 6 
 
 6 
 
 
 10 
 
 
 15 
 
 3 
 
 1 
 
 15 
 
 
 1 
 
 4 
 
 1 
 20 
 
 20 
 35 . . 
 
 387171 
 
 4 
 
 1 
 
 — 112829 
 
 1 
 
 5 
 
 15 
 
 37171 
 
 87005 
 
 5 
 
 15 
 
 o • • • 
 
 38 7 1 7 1 
 
 — ■25824 
 
 1 
 
 6 
 21.. 
 
 2171 
 3 7171 
 
 3 9 4 14 
 
 22285 
 
 6 
 
 42 6 5 8 5 
 
 —3539 
 
 1 
 
 71 
 
 22 43 
 3 94 14 
 
 844 
 
 
 3136 
 
 71 
 
 2171 
 
 43 5 2 
 
 5 
 
 —403 
 
 1 
 
 72 
 2 243 
 
 23 1G 
 4 17130 
 
 8534 
 
 403 
 
 72 
 
 44 3 5 
 
 ii 
 
 
 
 1 
 
 73 
 2316 
 
 4 7 
 
 21 
 
 •"> 
 
 
 
 73 
 
 4 22 
 
 
 
 44 5 7 
 
 1 
 
 
 1 
 
 74 
 |..2|390 
 
 4 
 
 7 
 
 21 
 
 5 
 
 
 
 74 
 
 4 26 
 
 7 
 
 44 7 
 
 8 
 
 
 1 
 
 
 4 
 
 7 
 
 
 2 
 
 
 
 
 I- -75 
 
 .4|31 
 
 4418 
 
 Hence the real root is nearly 1-125790 ; and by using another period of ciphers 
 we should have the root correct to ten places of decimals, with very little ad- 
 ditional labor. 
 
 ADDITIONAL EXAMPLES FOR PRACTICE. 
 
 (1) Find all the roots of die equation r 3 — 3.r— 1=0. 
 
 (2) Find all the roots of the equation x 1 — 22a: — 24=0. 
 
 (3) Find the roots of the equation a^-f-.r-— 500 = 0. 
 
 (4) Find the roots of the equation 2?-\-a?-\-x — 100 = 0. 
 
 (5) Find the roots of the equation 2r ! -f-3.r 2 — 4.r — 10 = 0. 
 (G) Find the roots of the equation r'— 12.i--j-12.r— 3=0. 
 
 (7) Find the roots of the equation r 1 — 8.r , - T -14.x' 2 +4.r— 8 = 0. 
 
 (8) Find the roots of the equation x* — x*-\-2x*-\-x — 4 = 0. 
 
 (9) Find the roots of the equation .r — 10.r ? 4-6.r- r -l=0. 
 
 (10) Find the roots of the equation .r'4-3.r , 4-2r''— 3x 2 — 2.r — 2 = 
 
 (11) Find all the roots of the equation 
 
 a<U 4r r ' — 3x 4 — 16.r"+llr--|-l 2.r— 9 = 0. 
 
 ANSWERS. 
 
 (x= + 1-879385242 
 
 (1) \ x=— 1-532088886 
 |^.r= — -347296355 
 
 (x= + 5-162277660166 
 
 (2) \ .r= — 1-162277660166 
 [,r=— 4 
 
 (3) a-=7-61727975593e 
 
 (4) x=4-264429973156 
 
 (5) x=l-6248190834°4 
 
 (6) 
 
 (7) 
 
 f.r= + 2-858083308178 
 ar=4- -606018306959 
 x=+ -443276939592 
 x= — 3-907378554730 
 
 (x=-f 5-236067977500 
 J x=+ -763932022500 
 ) x= -j- 2-732050807569 
 [x=— -732050807569 
 $x= + l-146994592039 
 ).r= — 1-090593586698 
 
:nO ALGEBUA. 
 
 r a:=— 3-06531579] 
 x= — -691576280490080 
 
 (9) \ .( = — -i: 
 
 x=4- •879508708414-li;n 
 x= + 3-0530581<- 
 
 (10) ssrl-0591090034618 
 
 ( x=—l ; x=— 3; .t = 1 
 
 (11) <J ar=— 3? x=l 
 
 x=l 
 
 i 
 
 257. The theorem of Sturm gives a simple means of establishing the cod 
 
 mis of the reality nl' the roots. As the real roots are comprised between 
 two limits, — L' and + L, the one negative and th<- other positive, which n 
 he chosen as large as we please, the question reduce- to seeking the condition;' 
 necessary, in order that from x = — L' to x=-{-h the series V, V , Y . \c, 
 should lose a number of variations equal to the degree of the equation. 
 
 Supposing this degree to be to, it must then lose m variations. But in order 
 that it may have m variations, it is necessary that it should have at least rn-^-l 
 1 as it can not have more, we are sure that the quantities V, Y,, V ; , 
 &c, exist to the number m-j-1, and that they are respectively of the degree 
 m, m — 1, to — '2, &c. The last, which does not contain x, will then be repre- 
 sented by V m . 
 
 When in the polynomial functions of x we substitute very large numbers, 
 positive or negative, for X, we know t lint the results are of the same sign as if 
 each polynomial were reduced to its first term : therefore, in the present in 
 vestigation, we need occupy ourselves only with the first term. Let us take 
 the equation V = under the ordinary form 
 
 x m -\-px m - 1 -\-qx m -*+, &c., =0. 
 
 The first term of Y is .r m ; that of the derived polynomial, Y 1( will be mx m ~ l . 
 With regard to those of tho polynomials V ; , Y 3 , &c, they are functions com- 
 posed of the coefficients^, q, &c, determined by tho successive divisions in 
 accordance with the rule. Let us represent these functions by < r a , < ! . . . . G m 
 and write in order the m-f-l quantities, 
 
 .r m , mr™- 1 , G. 2 x m - 2 , G 3 x m ~ 3 . . . G m . 
 The question will be reduced to finding the conditions which will cause the 
 loss of to variations from this series when we pass from .)•=: — L' to .?•=-)- L. 
 In order that this may bo the case, it must have m variations upon tho substi- 
 tution of — L\ and m permanences upon the substitution of +L. But in this 
 series the powers of .r go on decreasing by unity ; consequently, if it has noth- 
 ing but permanences when .r = -j-L, it will have nothing but variations when 
 x= — L'. Thus, the conditions are reduced simply to such as require this 
 series to have only positive coefficients, that is to say. to the following, 
 
 G s >0, G 3 >0 .... G m >0. 
 
 These conditions will never be greater in number than m — 1, but they may 
 be less in Dumber, inasmuch as soino of the above inequalities may involve the 
 ere. 
 
 KXAMI : 
 
 .'".-. Find the conditions necessary for the reality of the roofs of tl 
 
 .- -\-,jr-\-r=Q. 
 I [ere we have //<=:;, and the conditions an- only two in number, G 9 >0 and 
 
 <;,>o. 
 To find <! : and : ; . we calculate V. ami V i ■. sive divisions, as Pol. 
 
 low 
 
RULE OF DES CART: 341 
 
 First Division. ^vision. 
 
 •<.-'+ qx + r 
 ■)i -\-?,ijx-\-^r 
 
 W+q 3a*+ q 
 
 — 2q X — -:.r 
 
 1-j, ■ 4- !,/• _ 6qx+ \)r 
 
 3.r 3 + qx 1-V +1- 
 
 2qx + 3r — ltiqrx + 
 
 . . V r , = — 2or— 3/-. — lfyrr — 2 
 
 .-. V 4 = — 4<y 3 — 27r 2 . 
 Consequently, the inequalities G,.>0, G 3 >0, become 
 
 _-.\ 7 >0, _4g 3 — 27r 3 >0; 
 observing, however, that the fust inequality is embraced in the second, since 
 r* is always positive; and changing the signs of the second, •we have for the 
 sole condition of the roots of an equation of the third degree, being real, 
 
 -h/'-f -27/--<0. 
 
 We have now given so much of the general properties of equations of all 
 ees, and such moles of proceeding, as will insure their numerical solution 
 iu a manner the most certain and infallible, and ordinarily the best. 
 
 There are, however, many transformations of equations, which, by i 
 their degree, or by giving them a particular form, serve to facilitate their 
 tion- in certain cases. There are also many general principles applicable to 
 the resolution of equations of the higher orders by the methods in use previ 
 ous to the discovery of Sturm, which, with these methods themselves, it is de- 
 sirable to know for many purposes in the application of algebraic analysis to 
 the higher launches of both pure and mixed mathematics, for ulterior improve-; 
 menls in the general theory of equations itself, and even for use in the .- 
 tion of equations, in some cases, to which they are more conveniently adapted 
 than the method of Sturm. A treatise on algebra could scarcely be regard: 
 as complete without some notice of these. We shall therefore give as extei 
 sive an exhibition of them as can in any way be useful in an elementary 
 like the present, commencing with the well known 
 
 RULE OF DES CARTES. 
 •2o*J. An equation can not have a greater number of positive roots than there 
 arc variations of sign in the successive terms from -4- to — , or from — to -{-. 
 nor ran it have a greater number of negative roots than there are permanent, • . 
 or successive repetitions of the same sign in the successive U rms. 
 
 Let an equatiou have the following signs in the successive terms, viz.-*: 
 
 + -+ + + + ->or+ + _ + + + . 
 
 Nov. if we introduce another positive root, we must multiply the equation by 
 r — a, and the signs in the partial and final products will be 
 
 -H — + + + + -++ +-+ 
 
 4- — 1 — ±± + ±± — h +-±± + -+±±- 
 
 where the ambiguous sign Az indicates that the sign may be -f- or — accord 
 »ng to the relative magnitudes of the terms with contrary signs in the partial 
 products, and where it will be observed the permanences in the proposed 
 
3 12 AL<. 
 
 equation are changed into signs of ambiguity; hence the permanences, tui^ 
 the ambiguous sign as yon will, are not increased in the final product by Hie in- 
 troduction of the positive root -\-a ; but the number of signs is increi 
 one, and, therefore, the Dumber of variations must be increased by one. He 
 it is obvious that the introduction of every positive root also introduces • 
 additional variation of sign, and, therefore, the whole number of positive 1 
 can not exceed the number of variations of signs in the successive terms of the 
 proposed equation. 
 
 Again, by changing the signs of the alternate terms, the roots will be changed 
 from positive to negative, and vice versa (see Prop. VII.). Moreover, by tin- 
 change the permanences in the proposed equation will be replaced by varia- 
 tions in the changed equation, and the variations in the former by perm 
 in the latter; and since the changed equation can not have a greater number 
 of positive roots than there are variations of sign, the proposed equation can 
 not have a greater number of negative roots than there are permanences ol 
 sign. 
 
 Let v be the number of variations, c' tin' number of variations of the trans- 
 formed equation obtained by changing x into — ar. The number of real ro 
 of the equation can not surpass r-\-r'. Then, if this sum is less than the de 
 gree m, the equation will have imaginary roots. 
 
 The sum v-\-v' is never gi than the degree, and when it is less the 
 
 difference is an even number. (See Art. 248.) 
 
 EXAMINES. 
 
 (1) The equation .z- 6 +3.i- 5 — 41r 1 — 87r , +400.r 2 -(-44-U-— 720=0 has six real 
 roots. How many are positive ? • 
 
 (2) The equation .r 4 — 3.v> — 15.r--f 49.r — 12 = has four real roots. How 
 many of these are negative ? 
 
 2G0. We give next the repetition of a principle already presented, but which 
 may be derived as a direct consequence of the theorem of Sturm. 
 
 THEOREM OP ROLLE. 
 
 Let F(.r) = be an equation which has no equal roots, F '(.'') its derived 
 polynomial. We have seen that as .r increases, the series of Sturm loses a 
 variation every time that x passes over a root of the equation F(.r)=0. and 
 that it can not lose one in any other way. Moreover, we have seen that this 
 variation is lost at the commencement of the series of functions, in conse- 
 quence of F(.r) changing sign, while F'(ar) does nut ; so that F(.r) is always 
 of a sign contrary to that of F'(ar) for a value of x a little less than the root. 
 and always of the same sign for a value a little greater. 
 
 Thus, when we ascend from a root r to a root /■', which is immediately 
 above r, F(j must be of the same sign as F'(') for a value of x a little greater 
 than r, and of a sign contrary to F '(.'") for a value of a; a little less than ;'. Hut 
 in the interval F(.r) docs not clue >; then F'(ar) must change sign at 
 
 least once ; therefore the equation F'(x)=0 has at least one root between •• 
 and r'. 
 
 I , I a, b, c, d . . . g bo the real idols of F(x)=0, arranged in order of iim 
 
 tude, beginning wid the largest ; and let a,, o,, c, .. . g - , be the real roots of 
 
 K7 ;) — (>. disposed in tho.-anie manner. We have jn-t seen that tie 
 
 are comprised, some between /; and h. some between '' and c, dec. ; but as tha 
 
THEOREM OF 110LLE 343 
 
 degree of V'(x), and, consequently, the number of its roots, is one less than 
 the degree and number of roots of F(.r) = 0, it follows that the equation 
 F(.r) = can have but one root above a,, but one between a, and 6, . . ., and, 
 finally, but one below gv This property, which has been long known, and of 
 which we have given an independent demonstration at (Art. 253), is identical 
 With the theorem of Rolle. 
 
 2G1. The considerations which lead to the theorem of Itolle furnish also 
 the means of determining whether the m roots of the equation F(.r)=0 art- 
 real and unequal. 
 
 Since d\ is between a and b, b v between b and c, &c, it is easy to see (Art. 
 , 252) that if we substitute successively a l5 b L , &c, in place of x in F(.r), the 
 results will bo alternately negative and positive ; so that 
 
 For F(o,), F(M, F(c,), &c, 
 
 we have .... — , -f-, — , &c. 
 But we may apply to the function F'(.r) and its derived function F"(ar) all 
 that has been said in the preceding article of F(.r) and F'(.r) ; then, 
 
 For .... F"(a,), F"(6i). F "( r ')' &c ' 
 we have. . +> — , -{-, &c. 
 
 Thou the products F(a 1 )xF"(a J ), F^JxF"^,), &c., of which there are 
 m — i, will be all negative. 
 
 But if we make F(r) xF"(x)=y, and eliminate ^as at p. 157) x between 
 the two equations, 
 
 F'(.*)=0, F(x)xF"(x)=y (2) 
 
 the m — 1 roots of the final equation in y will be precisely tho products above; 
 but since all these products are negative, the equation in y will have only 
 negative roots, and, consequently, all its terms will have tho sign -f-- Thus, 
 when the equation F(.r)=0 has none but real and unequal roots, the theorem 
 of Rolle shows that the roots of F'(.i)=r:0 must be real and unequal also; and 
 from what has just been said above, it appears that besides this, the signs are 
 all plus in the equation in y, resulting from the elimination of x between the 
 equations (2). 
 
 262. Conversely, these conditions being fulfilled, we can demonstrate that 
 all the roots of F(.r) = will be real and unequal. And first, the m — 1 roots 
 of F'(-r)=0 being real, from what has just been said, those of F"(.r)=0 must 
 he real, and the m — 1 values of y, or F(.r) xF"(.r) real also; and the roots 
 of F'(.r)=0 being by hypothesis unequal, the theorem of Rolle proves that the 
 quantities F"(«i), F"(&i), £cc, have their signs alternately -f- and — . Again, 
 since the equation in y has its signs all +' we conclude that it has no positive 
 roots ; and since all its roots are real, they can only be negative ; then the 
 m — l products 
 
 V(a i )xF"(a 1 ),F(b l )xF"(b 1 ),6cc, 
 are negative. But the second factors have their signs alternate'}- -(- and — 
 then the quantities F(a 1 ), F(5,), &c, must have their signs alternately — and 
 ■•\-. Then there exists above «i a root of the equation F(.r) = 0, another be-, 
 tween a x and b u another betw T een b x and Cj, &c, therefore tho m roots of this 
 equation are real and unequal. 
 
 The conditions drawn from the equation in y may be regarded as actually 
 known, because this equation is obtained by simple elimination. As to the 
 
J44 ALGEBRA. 
 
 other condition which requires that the roots of F'(.r)=0 be real, let it be ob 
 served that this equation is of the degree m — 1, and, applying to it the same 
 reasoning as to F(x)=0, we reduce the question to determining the reality of 
 the roots of F"(x)=0, which is only of the degree m — 2. Continuing thus, 
 we descend to an equation of the second degree, the derived function of which 
 jeing of the tirst degree, can not have an imaginary root. Then fho only con 
 dition to fulfill will be that the equation y, which is also of the first degree 
 have its two terms of the same sign. 
 
 Remark. — By recurring to the reasoning which led to the use of the equa 
 tion y = F(r)xV"(.i), it is easily perceived that this may be replaced by 
 M x F(r) X F"(x), M being any positive quantity whatever. We can then in- 
 troduce or suppress in the polynomials F(x), F'(x), F"(.r), &c, such positive 
 factors as may be judged suitable to simplify the calculation. 
 
 263. The equation in y, resulting from the elimination of x in the equations (2», 
 being of tho degree m — 1, will have m — 1 coefficients, thus presenting m — 1 
 conditions to be fulfilled; the second equation in y, obtained by eliminating x 
 between the two, F"(a:) = 0, y = F'(.r) X F'"(.r), will be of the degree m — 2, 
 and present m — 2 conditions to be fulfilled, and so on, till we arrive at an equa- 
 tion of tho first degree in y, which will give but a single condition ; then, 
 taking all the conditions in au inverse order, their number will be express 
 ed (Art. 228) by . 
 
 m(m — 1) 
 1+2+3 |-ro— 1=-^— — -. 
 
 264. For an application of the above, let us take the general equation of tho 
 second degree, 
 
 x 9 +px+f=0. 
 
 Here we have F(x)=x*+px+<jr, F'(x)=2x-\-p, F"(.r) = 2, and we per 
 ceive at once that F'(x) has no imaginary root, since it is of the tirst degree. 
 
 In order to have the equation in //. the two equations between which we 
 must eliminate x are 
 
 2x+p = 0, y = {.i-+ F .u+q) X 2. 
 
 The elimination gives 
 
 y+ 
 
 2(\p>-q)=0. 
 
 Then, in order that the terms of this equation may have the same sign, we 
 
 1 
 must have -p' 1 — <7>0; and this is the only condition necessary to insure the 
 
 reality of tho roots of the equation of the second degree. It accords with 
 what we have seen at (Art. 191). 
 
 265. Let us consider next tho general equation of the third degree. The 
 second term, it will be seen hereafter, may be made to disappear without 
 changing the number of the real roots; we may therefore take it under tho 
 form 
 
 x 3 +9X+r=0. 
 
 In this case F(x)=x»+gx+r, F'(x)=3x«+?, F"(x)=6x. It is oeceei 
 
 [fisl. thai the derived (•(iiiation. :: ./-'+ q — (I, should have only real and unequal 
 roots; and for this the condition is evidently '/<!). 
 
METHOD OF FOURIER. 345 
 
 Secondly, it is necessary to eliminate x between the two equations 
 3.r 2 +?=0 (1) 
 
 7/ = (.r 3 +«7.r+r)x6.r, 
 
 or 
 
 The first gives 
 and (2) becomes 
 
 y — G.r i -{-r, IJ .r--\-Gr:r (2) 
 
 *= -3? ■•■*= j 
 
 y=—-q*+6rx 
 
 3y+4q* 
 
 Substituting this in (1), we have, after reducing, 
 
 2/ 9 +3<? 3 ?+9?(4$ 3 +27r*):=0. 
 
 In order that the three terms of this equation may have the same sign, it is 
 necessary, and it is sufficient, that the known term should be positive. We 
 have already seen that g must be negative, but q" in the second term is posi 
 tive ; then the new condition is 4q :i -{-27r-<^0. Finally, as this new condition 
 can lie fulfilled only when q is negative, it is the only one necessary, in order 
 that the roots of the equation of the third degree should be real and unequal. 
 
 FOURIER'S METHOD OF SEPARATING THE ROOTS. 
 
 266. We shall now give another method of separating the roots, proposed 
 by Fourier, which has the recommendation that the auxiliary functions em- 
 ployed in it are/(.r) and its successive derived functions, which can be form- 
 ed by inspection ;* so that the method can be applied nearly with equal ease 
 to an equation of any degree ; in particular, the intervals in which no real root 
 can be situated are, by Fourier's method, immediately assigned. The objec- 
 tion to this method is, that by its immediate application we only find a limit 
 which the number of real roots in a given interval can not exceed, and not the 
 absolute number; and that the subsidiary propositions by which this defect is 
 supplied are not of the same simple character as the origin 1 *! theorem. The 
 enunciation and proof are as follows. 
 
 THEOREM. 
 
 The number of real roots o/"f(x)=0 which lie between two timbers a and b, 
 can not exceed the difference between the number of variations *f signs in the 
 residts of the substitutions of a and b for x, in the se~>es formed &*. i'(x) and its 
 derived functions : viz.,/(z), f'(x),f"(x), . ..f"(.r). 
 
 If none of the equations 
 
 f(x)=0,f'(x)x=0, &c, 
 have a root between a and b, it is manifest that the substitution of a and b, and 
 of any intermediate quantity, in/(.r), f'(x), &c, will always produce exactly 
 
 * Tin- in. tliod of Sturm employs only the given and first derived function f{x) pw) f'(x), 
 which are the same as V and \' u the other functions in his method, viz., Y .. Y .. \r. be- 
 ,i obtained by the method of the common divisor, which, in pra tedious forfixuo 
 
 lions of the higher degrees, especiallyif they have large coefficients. For t'sim- 
 
 plifying these Laborious opera'aons, see Young's Tin ory and Solution oft] Equations 
 
346 ALGEBRA. 
 
 the same series of signs; but if any of these equations have roots between a 
 and 6, then changes in the will occur in substituting gradually 
 
 ascending quantities from a to b ; our object is to show that by such substitu- 
 tions the number of variations of signs can never increase, and that one varia- 
 tion will be lost every time the substituted quantity passes through a real root 
 /(r) = 0; this we shall do by examining separately each of the cases in 
 which the series of signs can be affected ; namely, 1, when /('") Blone 
 vanishes; 2, when some, derived function, f'"{x), alone vanishes; 3 and 4, 
 when some group of derived functions, of which /'( ') either is nut or is a 
 part, alone vanishes; and lastly, when several or all of these cases of vanish- 
 ing happen at the same time. 
 
 First, suppose that x=c (c being some quantity between a and b) makes 
 J'(.r) vanish, without making any of the derived functions vanish; then the 
 result of substituting c-\- h for x in/(.r) and f'{x) is (supposing h so small that 
 the signs of the whole of the two series which express f(c-\-h) and f'(e-\-h) 
 depend upon those of their first terms, and writing down only the first terms) 
 
 /i. /'(c) and /'(c), 
 
 which have different or the same signs according as h is — or -f- ; therefore, 
 in passing from c — h to c-\-h through a root of the equation, a variation of 
 signs is lost, but none gained. • 
 
 Secondly, suppose that x=c makes one of the derived functions, / In (j), 
 vanish, without making any other of the derived functions, or/(.r), vanish ; then 
 the result of substituting c-j-/i for x in the three consecutive functions 
 
 /-'(.r), /'"(.r), /"+.(,-), 
 (these being the only terms which it is necessary to examine)* is 
 
 /-*(<:), h.f°+i(c),f°+i{c). 
 
 If, then, the first and third terms have the same sign, there will be two varia- 
 tions when h is negative, and two permanences when It is positive; if the 
 extreme terms have contrary signs, there will bo one variation, and one only 
 whether h be negative or positive; therefore, in | from c — h to c-\-h 
 
 i a value which makes one of the derived functions vanish, either two 
 variations or none will be lost, but none ( 
 
 Thirdly, suppose that z=c I r consecutive di rived functions vanish, 
 
 without making any other derived function, or/(.c), vanish; then the result of 
 the substitution of c-\-h for x in the sei . 
 
 /— r (.r), /'«-'+'(.r), .../•»-«(.«■), /-(*), /"' +1 (<). 
 (these being the only terms necessary to be examined) is 
 
 / ,n - r (<0> |77" ,+1 (<-). • • •• |T/ ,n+ '('-)- y/"' +1 (<-). / m+1 (0. 
 
 where \r denotes 1.2.3..../'. 
 
 If, then, the extremes of this series have the same Blgn, there will be r oi 
 r-|-l changes (according as r is even or odd) when /; is ni and no 
 
 when h is positive; if the extreme terms have contrary signs, t' 
 
 * " 
 
 * It i- ; attei d to the other functions of tb of derived fanotions, be- 
 
 . - : i . pposed so small tint net one i I hea bj the substitution of any 
 
 ; i bet • i c — k ami c-\ h, end tJ 
 --i-h. 
 
METHOD OF FOURIER. 34? 
 
 will be r or r-j-1 variations (according as r is odd or even) when h is negative, 
 and one change when h is positive; therefore, in passing from c — h to c-j-A 
 through a value which makes r consecutive derived functions vanish, r or rJtl 
 changes are lost (according as r-is even or odd) but none ever gained. 
 
 Fourthly, suppose the vanishing group to consist of f(x) and the first r — 1 
 derived functions (which corresponds to r roots =c in J'(.r)=0) ;* then the re 
 suit of the substitution of c + /t for x mf(x),f'(x), . . . /^(x), f*{x), is 
 
 j7/ r ( c )'f^/ r ('')---Y/ r ( c ')'/ r ( c )' 
 
 in which there are r variations when It is negative, and none when h is posl 
 five ; therefore, in passing through a root which occurs r times in the equation, 
 r changes are lost, but none gained. 
 
 Lastly, suppose the substitution of x=c to produce several, or all of the 
 above cases at the same time ; then, because the conclusions respecting the 
 effect of the passage through c upon the series of signs in ono part of the 
 series of derived functions are not at all influenced by what happens, in con- 
 sequence of the same passage, at another distinct part of the series, by what 
 has been proved, several variations will be lost, but none ever gained. 
 
 Since then, in substituting gradually ascending values from a to b, variations 
 of signs are generally lost for every passage through a quantity which makes 
 one or more of the derived functions vanish, and invariably one for every pass- 
 age through a root of f(x)=0, but none under any circumstances gained, it 
 follows that tho numler of roots of/(;r) = 0, which lie between a and b, can 
 not be greater than the excess of the number of variations given by x=a, above 
 that given by x=&. 
 
 267. Hence, if the li.nits, a and b, be — co and -j- 00 ' or any two numbers 
 the first of which gives only variations, and the second only permanences ; and 
 if, in the series formed by/(.r) and its derived functions, 
 
 c be substituted for x and be then made to assume all values between these 
 limits, the series of signs of the results will have the following properties ; 
 there will at first be n variations of sign, and at last no variation, but n per- 
 manences; these variations disappear gradually as c increases, and when once 
 lost, can never bo recovered ; one variation disappears every time c passes 
 through a real unequal root of/(.r)=0; r variations disappear every time c 
 passes through a root which occurs r times in/(.r)=0 ; either two or none of 
 the variations disappear every time one only of the derived functions vanishes, 
 without f{x) vanishing at the same time; an even number p of variations dis- 
 appears every time an even group of p functions (not including the first f{x)) 
 vanishes; and an even number q^zl of variations disappears every time an 
 odd group of q functions (not including the first/(.r)) vanishes. Also, if a value 
 causes /(.r) and the first r — 1 derived functions to vanish, and an even group 
 of p functions in one part of the series, and an odd group of q functions in an- 
 other part, to vanish at the same time, the number of variations lost in pass 
 ing through that value will be r-\-p-\-q±l. 
 
 268. Hence, if f(x) = Q have all its roots real, no value of x can make any 
 of the derived functions vanish, and thereby exterminate variations of signs 
 
 * See (Art. 253, Schol). 
 
34ri ALGEBRA. 
 
 without .it the same time making /(x) vanish; for if it could, since tliose vari- 
 ations can never be testored, and since b variation must disappear for ev< 
 passage through a real root, the total number of variations lost would surpass 
 /'. tin- degree of the equation, which is absurd, since there are but n derived 
 functions in all. Whenever, therefore, variations disappear between vnluesof 
 X which do not include a root of/(x)=0, there is. correspond Dg to thai oc- 
 currence, an equal number of imaginary roots of /'(.;) = 0. He ce, if .r=c 
 produces a zero between two similar signs, or if it produces an e\ m i hi i er 
 p of consecutive zeros either between similar or contrary signs, tl U be 
 
 respectively two, or p, imaginary roots corresponding; or if it produces an 
 odd number q of consecutive zeros, there will be q^zl imaginary i 
 sponding, according as they stand between similar or contrary signs ; <■. of 
 course, not being a root of /(.r) = (l. 
 
 Observation. — Since the derivatives which follow any one f T (.c) ma 
 supposed to arise originally from it, it is manifest that the same conciue • - 
 respecting the roots of/ r (.r) = may be drawn from observing the part of the 
 series of derived functions 
 
 f'(x),f'+>(x),...f°(z) 
 
 as were drawn respecting the root of/(x) = 0from the whole series. 
 
 269. Des Cartes's rule of signs is included in Fourier's theorem as a par 
 ticular case. 
 
 For when, in the series formed by /(.<■) and its derived functions, we put 
 •T= — cc, there are?i variations ; and when we put .r=0, the signs of the series 
 of functions become the same as those of the coefficients of the proposed equa- 
 tion 
 
 P^jPa-U •••Pi, 1. 
 
 Let the number of variations in this series of coefficients =/,-, and there!" 
 the number of permanences (supposing the equation complete) =n — k ; if 
 we make ./ = + ao, the signs of the functions are all positive, and the number 
 of variations =0. Hence, between x= — oo and ./=(), the number of varia- 
 tions lost is 7i— k; therefore in a complete equation there can not be more 
 than n—k negative roots, i. e., than the number of permao i the set 
 
 of coefficients ; also, between x = and x= , . the number of variations lost is 
 Tc, whether the equation be complete or incomplete: hence in any equation 
 there can not be more positive roots than/.-, i.e., than the number of variation 
 in the series of coefficients, which is Des Cartes's rule ofsiens. 
 
 270. Fourier's theorem may also be presented under the following form : 
 If an equation have m real roots between a and }>, then the equation whose 
 
 roots are those of the proposed, each diminished by a, has at least m more 
 
 variations of signs than the equation whose roots are those of the proposed, each 
 diminished by b. 
 The transformed equations would be 
 
 '+a)=0,/(y+o)=0; 
 and if these were arranged according to ascending powers of y, the coefficients 
 
 would be the values assumed by /'(r), /"'(,), ,\ ,-.. when a and b are respectively 
 
 written for x. Therefore, whatever number of variations of signs is lost in tho 
 
 series /'(,•),/'(,/■), cVc, in passing from ,i to /-. the same is lost in passing from 
 one transformed equation to the other; but the series for a has at least m 
 
METHOD OF FOUIUER. 349 
 
 more variations than that for b ; therefore /(?/ + a) = has at least m more 
 variations than j\i/-\-b) = 0. 
 
 271. To apply this method to find the intervals in which the roots of 
 f(x) = are to be sought, we must substitute successively for .r, in the series 
 formed by/(>) and its derived functions, the numbers 
 
 _„ ..._10, —1,0, 1, 10,..., +/3 (1), 
 
 I a and -\-(i being the least negative and least positive number, which lmvo 
 respectively only variations and permanences), and observe the number of 
 variations of sign in each result. 
 
 Let h and k be the numbers of variations of sign when any two consecutive 
 terras in series (1), a and b, are respectively written for x ; therefore h—k is 
 the number of real roots that may lie between a and b : if this equals zero, 
 f(x)=0 has no real root between a and 6, and the interval is excluded; if 
 h—k = l, or any odd number, there is at least one real root between a and b ; 
 \l'h—k = 2, or any even number, there may be two, or some even number, or 
 none ; the latter case will happen when, as explained above (Art. 268), some 
 number between a and b makes two or some even number of variations vanish, 
 without satisfying/^) =0. Similarly, we must examine all the other partial 
 intervals ; and when two or more roots are indicated as lying in any interval, 
 their nature must be determined by a succeeding proposition. 
 
 The two former of the following examples are extracted from Fourier's 
 work. 
 
 EXAMPLE I. 
 
 / (x)= .c 6 — 3x*— 24.x- 3 -f 95.r 2 —46.r— 101 = 
 
 /' (,r) = 5x 4 — 12.T 3 — 72x 3 +190x — 46 
 
 f"{x) = 20.r ; — 36x 2 — i n.r +190 
 
 f'"(x)= 60.i : — 72.C —144 
 
 f* (x) = 120.f —72 
 
 P (x) = 120. 
 
 Hence we have the following series of signs resulting from the substitutions 
 of — 10, — 1, 0, &c., for x, in the series of quantities 
 
 j j, J„ jn, 
 
 (-10) - + - + 
 
 (-1) + - + - 
 (0) - - + 
 
 (i) - + +- - 
 
 (10) + + + + 
 
 Hence all the roots lie between —10 and -\-10, because five variations have 
 disappeared; one root lies in each of the intervals — 10 to — 1, and — 1 to 0, 
 because in each of them a single variation is lost ; no root lies between and 1, 
 because no variation is lost between those limits; and three roots may be sought 
 between 1 and 10 (because three variations have disappeared), one of which is 
 certainly real ; it is doubtful whether the other two are real or imaginary. 
 
 Observation.— -When any value c of x makes one of the derived func- 
 tions,/'"^), vanish, we may substitute c^zh instead of c, h being indefinitely 
 small; then all the other functions will have the same sign as when x=c, and 
 the sign of /'"(c i/') will depend upon that < f ±// m+1 (e); i. e., it will be the 
 
 r 
 
 r 
 
 — 
 
 + 
 
 — 
 
 + 
 
 — 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
350 ALGEBRA. 
 
 same or contrary to that of the follo\. Viitive,/ m + 1 (c), according as h is 
 
 positive or negative, or according us we substitute a quantity a little less or a 
 little greater than the value which makes f m (x) vanish. The use of this re- 
 mark will be seen in the following example. 
 
 EXAMPLE II. 
 
 / (r)= 3*— 4a?— 3a:+23=0 
 
 /' (*)= 
 
 4.r 
 
 l — 12.r ; 
 
 ! — 3 
 
 
 
 f"U) = 
 
 L2a? 
 
 '—24a: 
 
 
 
 
 /'"(*)= 
 
 24x 
 
 —24 
 
 
 
 
 r (x)= 
 
 24. 
 
 
 
 
 
 / 
 
 
 /' 
 
 /" 
 
 /"' 
 
 r 
 
 xz=o + 
 
 
 — 
 
 
 
 — 
 
 + 
 
 x=0j-h, + 
 
 
 — 
 
 ± 
 
 — 
 
 + 
 
 x=l + 
 
 
 — 
 
 — 
 
 
 
 + 
 
 x=l=fh, + 
 
 
 — 
 
 — 
 
 =F 
 
 + 
 
 1=10 + 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 Every value less than gives results alternately -\- and — , therefore there 
 i? do real negative root; for .r=0, we have a result zero placed between two 
 similar Bigns, and therefore corresponding to it there is a pair of imaginary 
 roots. There is no root between and 1, but there may be two roots be- 
 tween 1 and 10. 
 
 EXAMPLE III. 
 
 f(x)=x 6 — 6.r , +40r«+G0.r ! — x— 1=0. 
 
 Here there is no root <C — 1; there is one, and there may be three, be- 
 tween — 1 and 0; there is one root between and 1, and there may be two 
 roots between 2 and 3. 
 
 272. The above process will determine the intervals in which the roots are 
 to be sought, but not always their nature ; when an even number of roots is 
 indicated, they may all turn out to be impossible. The series of magnitudes 
 between — co and -f-oo, to bo substituted for x in the d< rived functions, has 
 divided into intervals of two sorts, each contained by a I limits, a 
 
 and b. The first sort of interval is one within which no root is comprehended, 
 i. c, the limits of which give the same number of variations of Bigns in the 
 of derived functions. The second sort is one within which roots may 
 lie, i. e., where the number of variations resulting from the substitution of o 
 is bss than the number resulting from the substitution of a, in the s< ries of 
 derived functions. This second sort of interval has two subdivisions, viz., 
 cases where the indicated roots do really exist, and others where they are 
 imaginary. When we have ascertained that a certain number of roots may 
 tie between a and b, we may substitute c (a quantity between it and l>) in the 
 Beries of dorived functions, and if any variations disappear, our interval is broken 
 into two others; if no variations disappear, we may increase or diminish r, and 
 
 make a second substitution, and it may still happen that no variation is lost, and 
 
 80 on continually; and we may be left, after all, in a stqjo of uncertainty, 
 whether the separation of the routs is impossible because they are imaginary. 
 
 or only retarded because their difference is extremely small. This uncer 
 tainty is relieved by taking the interval so small as to be sure to include the 
 
 real roots, if they exist. 
 
TRANSFORMATION OF EQUATIONS. 351 
 
 One method of arriving at the proper interval is by means of the so-called 
 equation of the squares of the differences of the roots of the given equation, 
 which we shall hereafter have occasion to deduce. This process is tedious in 
 practice; and as our object in unfolding the method of Fourier was to pursue 
 it only so far as it threw light upon the general theory of equations, we shall 
 here leave it. 
 
 We should now introduce the theorem of Budan, but it requires a trans- 
 formation which we have not yet exhibited, and we therefore take this op 
 portunity to complete a subject, one proposition of which (Art. 251) we liave 
 already had occasion to anticipate. 
 
 TRANSFORMATION OF EQUATIONS. 
 
 PROPOSITION I. 
 
 273. To transform an equation into another whose second term shall be removed 
 
 Let the proposed equation be 
 
 ar n +Ai2 n - 1 4-A B a: n -*+ A n _ 1 .r+A n =0 ; 
 
 and by Art. 245 we know that the sum of the roots of this equation is — A! , 
 therefore, the sum of all the roots must be increased by A! in order that the 
 transformed equation may want its second term ; but there are n roots, and 
 
 Ai 
 hence each root-must be increased by — , and then the changed equation will 
 
 have its second term absent. If the sign of the second term of the proposed 
 
 equation be negative, then the sum of all the roots is -f-Ai ; and in this case 
 
 Ai 
 we must evidently diminish each root by — , and the changed equation will 
 
 7b 
 
 then have its second terra removed. Hence this 
 
 KULE. 
 
 Find the quotient of the coefficient of the second term of the equation 
 divided by the highest power of the unknown quantity, and decrease or in- 
 crease the roots of the equation by this quotient, according as the sign of the 
 second term is negativo or positive. 
 
 EXAMPLES. 
 
 (1) Transform the equation .t 3 — 6.r 2 -|-8.r — 2=0 into another whose second 
 lerm shall be absent. 
 
 Here A[ = — 6, and n=3 ; ••• we must diminish each root by £ or 2 
 
 1 _6 +8 —2 (2 
 2—8 
 ^4 "^2 
 2 —4 
 
 —2 —4 
 o 
 
 
 •'• V 3 — 4y — 2=0 is the changed equation. 
 And since the roots are diminished, we must have the relation x=.y-^-2. 
 
 (2) Transform the equation X* — 16a? — 6.r-f-15=0 into another whose 
 second term shall be removed. 
 
 (3) Transform the equation x?+15x*-\-123?— 20.r 2 +14.r— 25 = into an- 
 ithi-r whose seond term shall be absent. 
 
ots% ALGEBRA. 
 
 (4) Change the equation r--\-ax-l r b = into another deficient of the second 
 
 term. 
 
 (o) tb. [uation ./ ■■■■ + . /r + &.r+c = into another wanting the 
 
 • I term. 
 
 A NSW] 
 
 ( >) ,,'■ —gey 8 — 518i/— 777=0. 
 
 (3) ■ _;• y ^412^—757^4-401=0. 
 
 (4) :--" T + l, = ». 
 
 (, i- \ ab 
 
 (5 ) ^(__ 6 )z+_- y+C =0. 
 
 PROPOSITION II. 
 
 2?4. To transform an equation into another whose roots shall be the reetpro 
 tals of the roots of the proposed, equation. 
 
 Let «.i n + A 1 x"- 1 + A;.r' 1 --+ A B _iX+A B =0 be the proposed equa- 
 tion, and put V=-', then x=- and by writing - for x in the proposed equa- 
 
 1 J x y y 
 
 tion, multiplying by y n , and reversing the order of the terms, we have the 
 equation 
 
 A„2/ n +A, , ! r' + A„_ ; jr : + A.y- + A,</ + «=0. 
 
 whose roots are the reciprocals of the roots of the proposed equation. 
 
 The transformation is then effected by simply changing the order of the co- 
 efficients of the given equation. 
 
 Corollary 1. — Hence an equation may be transformed into another whose 
 roots shall be greater or less than the reciprocals of the roots of the proposed 
 equation, simply by reversing the order of the coefficients, and then proceed- 
 ing as in the Proposition to Art. 2^3. 
 
 Corollary 2. — If the coefficients of the proposed equation be the same. 
 whether taken in reverse or direct order, then it is evident that the trans- 
 formed equation will be the same, as the original one ; and, therefore, the roots 
 of such equations must be of the form 
 
 »"i»-; ^r; r*-? r ->»7- ^ c - 
 
 ~i "a ^3 ' i 
 
 Corollary 3. — If the coefficients of an equation of an odd degree be the 
 same whether taken in direct or inverse order, but have contrary signs, then, 
 also, the roots of the transformed equation will be the same as the roots of the 
 proposed equation; for, ch is of all the terms, the original and 
 
 transformed equations will be identical, roots remain unchanged when 
 
 the signs of all the terms are changed. \nd this will likewise be the case in 
 un equation of an even degree, provided only the middle term bo absent, in 
 order thai the transformed equation, with all its signs changed, may be identical 
 with the original equation. 
 
 Equations whose coefficients are the same when taken either in dire. 
 reverse order, are, therefore, called recurring equations, or, from the form of 
 the root 3, r» ciprocal equ 
 
 Corollary 4. — It' the sign of the last term of a recurring equation of an odd 
 legr, ,. l u . _J_, one of the roots of such equation will be — l : and if the 
 
TRANSFORMATION OP EQUATIONS. 353 
 
 of the last term be — , one root will be -j-1. For the proposed equation and 
 the reciprocal have one root, the same in each, and 1 is the only quantity 
 whoso reciprocal is the same quantity ; hence, since each of tho other root* 
 has the same sign as its reciprocal, tho product of each root and its reciprocal 
 must be positive ; and, therefore, the last term of tho equation, being the 
 product of all tho roots with their signs changed, must have a contrary sign to 
 that of the root unity. 
 
 Hence a recurring equation of an odd degreo may always bo depressed to 
 an equation of the next lower degreo by dividing it by .r-j-1, or x — 1, accord 
 ing as the sign of the last term is -{- or — 
 
 Corollary 5. — A recurring equation of an oven degree may always bo de- 
 pressed to another of half tho dimensions. For let the equation be 
 £ 2n +A 1 .T 9n - 1 + A 2 z- n - 2 + A 2 .r 3 +A 1 x+1=0; 
 
 dividing by .r n , and placing the first and last, the second and last but one, &c. 
 in inxtaposition, we have 
 
 *"'+?+Ai(*" 1 4~t) + A n _x(x+^)+A n =0 [2] 
 
 1 
 Assume y=x-{-~, then we have 
 
 1 1 
 
 x +x=y •'• x +x —y 
 
 .T-+-+2 x*+-=y*-2 
 
 (*+l) 4 =- r, +^+ 4 (- T2 +^)+ 6 ^ , +^=y 4 -4(y-2)-6 
 
 &c. &c. &c. =y 4 — 4?/ 2 -f-2; 
 
 substituting these values of 
 
 *+-. *?+& • • • * n +^ in [2] 
 
 the resulting equation is of the form 
 
 2/ »+B 1 i/»- 1 +B 2 2/ n - 2 + B n _ l2 /+B n =0; 
 
 and tho original equation is reduced to an equation of half the dimensions. 
 
 EXAMPLES. 
 
 (1) Transform the equation x 3 — 7ar-f-7=0 into another whose roots shall 
 be less than the reciprocals of those of tho given equation by unity. 
 
 7 —7 +0 +1 (1 
 7 _0 
 
 "~ o ~~ o ~T 
 
 7 7 
 
 7 
 
 7 
 
 14 
 
 ., 7z 3 +14; 2 - r -7r4-l=0 is the equation sought, where ;- r -l=-, or x== 
 
 x 1 2+1 
 
 Z 
 
354 ALGEBRA. 
 
 (2) Find the roots of the recurring equation 
 
 x 6_Gx^+5r 5 +5x J — 6x+l=0. 
 By Cor. 4, this equation has one root x= — 1, and the depressed e^ uation i» 
 
 a*— 7a«+ 1 ±l- 2 — 7.r 4. 1 = 0. 
 Divide by x 2 , and arrange the terms as in Cor. 5 ; then 
 
 ^+7 i - 7 ( r +x) + 12=() - • - (A) 
 
 1 ut x+-=r ; then x 2 +-=: 2 — 2 ; hence, by substitution, (A) becomes 
 x •*, 
 
 2 2_O_ 7r _|_lO_0 ; 
 
 or z 2 — 72 + 10=0; 
 
 and, resolving the quadratic, we get 
 
 7 , /49 
 
 2 =2±Vt- 10 
 
 7 + 3 
 =~2~~ 
 =5, or 2=2. 
 
 Hence x+-=5, and x+-=2, and the resolution of these two quadiatus 
 x x 
 
 gives 
 
 x=i(5i V21) a°<i *=+*• or +1, 
 
 and the five roots are 
 
 -1, +1, +1, -^ , and - ; 
 
 where >JZd* Jg-Va^+yg. 25-2j, = -A =) whkh is the. 
 2 2 5+V21 2(5+ V21) 5+^21 
 
 5+ V21 
 reciprocal of the root - . 
 
 (3) Give the equation whose roots are the reciprocals of the roots of the 
 equation 
 
 tf— 3.x 6 — 2r , +3x 3 +12x 2 +10x— 8=0. 
 
 (4) Find the roots of the recurring equation 
 
 5T / 5_ 4 yi_^3, / 3_3^_j_4, / _5_o. 
 
 (5) Find the roots of the recurring equation 
 
 ;r s+ r i + . r 3_}_ T :_4_. r _|_ 1 = 0. 
 
 ANSWERS. 
 
 (3) 8x°— lOx 8 — 12x« — 3x 3 +2x 2 +3x— 1 = 0. 
 
 l_i_ V— 3 1— V— "5 -3 + 4V-1 -3-4-/-1 
 
 (4) i, — r 2 — . — — . 5 . ™* 5 • 
 
 /_1+V"^3 /_1_V^3 /-1+ -/^3 
 (5) -1, V -^ ■ V o • -V 2 ' 
 
 -\ o ' 
 
 and 
 
TRANSFORMATION OF EQUATIONS. 355 
 
 PROPOSITION III. 
 
 275. To transform an equation into another whose roots shall be any pro- 
 posed multiple or submultiple of the roots of the given equation. 
 
 Let x n -\- A^-' + AaX" -2 -! A n _,.r4-A n =0 be any equation ; then putting 
 
 y=mx, we have x=— , and by substituting this value of x in the given equa- 
 tion, and multiplying each term by ?n a , we have 
 
 y"-\-mA i y a - l -\-m 2 A. i y u -' 2 -\ m l, - 1 A n _ 1 2/+m n A n =0 ; 
 
 an equation whose roots are m times those of the proposed equation. Hence 
 we have simply to multiply the second term of the given equation by m, the 
 third by m?, the fourth by m 3 , and so on, and the transformation is effected. 
 
 Corollary 1. — If the coefficient of the first term be m, then, suppressing m 
 in the first term, making no change in the second, multiplying the third by m, 
 the fourth by m?, and so on, the resulting equation will have its roots m times 
 those of the given equation. 
 
 Corollary 2. — Hence, if an equation have fractional coefficients, it may be 
 changed into another having integral coefficients, by transforming the given 
 equation into another whose roots shall be those of the proposed equation 
 multiplied by the product of the denominators of the fractions. 
 
 Corollary 3. — If the coefficients of the second, third, fourth, 6cc, terms of 
 an equation be divisible by m, ?« 2 , m 3 , and so on, respectively, then m is a com- 
 mon measure of the roots of the equation. 
 
 EXAMPLES. 
 
 (1) Transform the equation 2.T 3 — 4x~-\-7x — 3 = into another whose roots 
 mall be throe times those of the proposed equation. 
 
 (2) Transform the equation 4x* — 3x 3 — 12x 2 -}-5x — 1=0 into another whose 
 roots shall bo four times those of the given equation. 
 
 1 1 
 
 (3) Transform the equation aP+^a? — -x-f-2 = into another whose roots 
 
 shall be 12 times those of the given equation. 
 
 ANSWERS. 
 
 (1) 2x ! — 12x 2 +63x— 81 = 0. 
 
 (2) x*— 3X 3 — 48x 2 -f-80x— 64=0. 
 
 (3) r 3 4-4x 2 — 36x+3456 = 0. 
 
 PROPOSITION IV. 
 
 276. To transform an equation into another whose roots shall be the square* 
 of the roots of the proposed equation. 
 
 Let x n 4-A 1 x n_I + A 2 .T n_2 -f- -f- A n-iX-f-A n =0 be any equation ; then 
 
 r n — AiX n-1 - r -A 2 a: n_2 — i A n _iX=pA„ = is the equation whose roots ar« 
 
 the roots of the former, with contrary signs (Prop. VII., Art. 247). 
 
 Let «i, a 2 , a 3 , &c, be the roots of the former equation, anc — a,, — a 2 , — a 3 , 
 5cc, those of the latter ; then we have 
 
 (x n + A 2 x n ^-| ) + (A^-i-f A 3 x n ~ 3 -f- . .) = {x—a l ) (x— a : )(x— a 3 ) .... 
 
 (x n + A 2 x- B + . . . .)_(A 1 x"- 1 + A 3 x'- 3 + . . .) = (x+a 1 )(- c +<z*)(j:+«.'«) .. 
 
 Hence, by multiplying these two equations, we have 
 x n +A sl x»- 2 4-...) 2 — (A 1 z^- I - r .A 3 x»- 3 + ...) 2 =(x 3 — ai 2 )fx»— a^fx 3 — a, 9 ) .. 
 
356 ALGEBRA. 
 
 Or a*— (Ai 9 — 2A,)a : + (A : ; — J A ; A.+2A ,)./-"-•'— . . . 6cc s=(a*— a,«j 
 
 (a? — «/)('-' — Oa 9 ) by actually squaring and arranging according to tho 
 
 powers of x. Now, for ' : write y, and wo have 
 
 y°-(\, -- .' \ : )^-' + (A, : -2A 1 A 3 +'JA,).y^-.. &c, =0/-Gr)(7/- . 
 (y—a 3 -) . . . 
 •. 2/ n — (A, 2 — 2A 2 )7/ n_1 + (A ?— >A, \ ,+2A 4 )y - 9 — =0 is an equation 
 
 whose roots are the squares of the roots of the given equation. 
 
 I KAM? 
 
 (1) Transform the equation x 3 +3x : — ftr — 8 = into another whose- roots 
 are the squares of those of the proposed equation. 
 
 Here x 3 — 6x= — 3x 2 -|-8 by transposition, and by squaring we have 
 a* _ 1 2x* -f- 36I 9 = Ox 1 — 4 8.r- + 64 
 .-. x 6 — 21x 4 +84x 2 — 64=0, 
 or 
 
 y»— 21y»+84y— 64=0 
 is the required equation. 
 
 The roots of the given equation are — 1, — 4, 2; and those of the trans- 
 formed equation are 1, 4, 16. 
 
 (2) a*+a*+3a*+16ar+15=0. 
 The transformed equation is 
 
 x 6 +2x , +33x 3 +23x 2 +166x— 225=0, 
 winch has (Art. 259) only one positive root, and therefore the proposed h«* 
 only one real root. 
 
 (3) Transform the equation x 3 — x 3 — 7x-|-15 = 0. 
 
 4) Transform the equation x 4 — 6x 3 +5x 2 +2x — 10=0. 
 
 (5) Transform the equation x 4 — ix 3 — 8x-|-32=0. 
 
 (6) Transform the equation x 4 — 3X 3 — 15x 2 +49x — 12=0. 
 
 ANSWERS. 
 
 (3) y*— 15^+79^—225=0. 
 
 (4) 3/« — 2Gy»+29y 3 — 104y+ 100=0. 
 
 (5) y* — 16f — 64y+ 1024 = 0. 
 
 (6) y*— 39y 3 +495y 9 — 2041y+144==0. 
 
 PROPOSITION V. 
 
 277. To transform an equation into another wanting any given term. 
 
 By recurring to the transformed equation in Art. 251, note, in which the 
 roots of the proposed &ro increased or diminished by a quantity represented 
 by r, it will be seen that in ordor to know what value r must have to make the 
 coefficient of any power of x disappear, it is only necessary to place the column 
 of quantities by which that power is multiplied equal to zero, and the result- 
 ing equation, when resolved, will furnish the proper values of r. This equa- 
 tion will be of tho 1° degree when it is required that the second term shall dis 
 appear, it will bo of the 2° degree when the third is to disappear, and so on. 
 The last term can be made to disappear only by means of an equation of the 
 same degreo as tho proposed. 
 
 By removing the second term from a quadratic equation, we shall ne imma- 
 • ! ately conducted to the well-known formula for its solution. Thus, the equa- 
 tion being 
 
TRANSFORMATION OF EQUATIONS. ^f/7 
 
 .T*-fAx-j-N=0, 
 tne transformed in x'-\-r will be 
 
 + A + Ar[=0; 
 +N S 
 unci, that its second term may vanish, we must have 
 
 2r+A=0 .-. r= — |A, 
 which condition reduces the transformed to 
 
 .r'*— jA 2 +N=0 
 
 .-. x=.r'+r=— iA-J; V]A'-— N; 
 which is the common formula for tho solution of a quadratic equation. 
 
 PROPOSITION VI. 
 
 278. Tb transform an equation into one ivhose roots are the squares of tne 
 i/ijfcrcnccs of the roots of the proposed equation. 
 
 [f in the given equation, /(.r)=0, we make x=a l -\-y, a x being one of the 
 rpots, y will be the difference between a x and every other root. If we make 
 r=a i -\-y, y will be the difference between a« and every other root, and so on. 
 
 But since a^ a 2 , &c, are rooto of/(x) = 0, they must satisfy it ; hence 
 f( ai ) = 0,f(a,) = 0, &c (1) 
 
 If wo eliminate a x or a 2 i &c, between either of these equations (1) and the 
 corresponding ones, f{a x -\-y) = 0, f(a. t +y)=:0, &c, tho final equation in y 
 will bo in each case the same, and is therefore the equation whose roots are 
 the differences of tho roots of the proposed equation. It is evidently the same 
 thing to eliminate between /(.r) nm\f(x-\-y). 
 
 The form of tho equation f{r-\-y) is (Art. 251), 
 
 f{x)+Mx)y+r%f+ r- 
 
 The first term is identical with the proposed equation, and vanishes, and the 
 whole is divisible by y ; we thus deduce 
 
 m+ f £%y+ • • • r- 1 ( 2 ) 
 
 The equation (2) is of the ra — 1 degree, and by elimination with the pro- 
 posed equation of the degree m will produce a final equation of the degree 
 m(m — 1), as will be hereafter shown. It is evident, indeed, that the roots 
 being of the form ax— a 8 , a 2 —a u ai—a 3 , a 3 —a u a. 2 —a n , &c, will be equal in 
 number to the permutations of m letters, two and two, which is m{m— 1) 
 (Art. 200). The factors m and m — 1 will the one be even and the other odd, 
 and the product m(m — 1) must therefore necessarily be even ; moreover, since 
 if one root, ai—a : , be represented by /?, another, a 2 — rti, will be represented 
 by — /?, and the equation (2) will be composed of factors of the form (y — 
 (y-\-j3)=y' i — /?-; and hence will contain only even powers of y. It maj 
 therefore be written under the form 
 
 y*»+pyim-*+qy*m-4^. y & Ci) _|_^ — . . . . (3) . 
 
 and if we make y"---~, we have 
 
 2 m +_p: m - 1 +ry; m -'"+, ecc. +< = (4) 
 
358 ALGEBRA. 
 
 as the equation whose roots are the squares o* the differences of the roots of 
 die proposed equation. 
 
 I. As an application of the foregoing princi] fes, let us find the equation of 
 the squares of the differences for the equation of the third degree. In tin- 
 first place, 1 snail make the general remark, that equations (3) and (4) ought 
 not to change when we augment, or when we diminish, by the same quantity 
 all the roots of equation (1). Consequently, if the second term of a giveD 
 equation be not wanting, we can cause it to disappear (Art. 273), and then 
 find the equation of the differences for the transformed equation ; we shall 
 thus find the same equation as if we had not made the second term vanish, since 
 the differences of the roots will be the same as before, while the calculations 
 will be less complicated. This being premised, I will suppose that the equa- 
 tion of the third degree wants its second term, and has the form 
 
 r>+qx+r = [AJ 
 
 Designate the given equation by/(.r)=0, and the derived polynomials of 
 
 f(x) byj\(x),fi(x)tf 3 (.v) ; the rule for finding the equation of the squares 
 
 of the differences is to eliminate between the two equations 
 
 /(.r) = 0,/ 1 (.r) + ^(,)-/ + T ^/;(.r)/4- . . .' =0 [BJ 
 
 But in the case before us we have 
 
 f(x)=i*+qx+T, f l {x)=3afl+q, f;(x) = 6x, f 3 (x)=6. 
 Substituting, therefore, these values in equations [B], we shall readily perceive 
 that the elimination of x ought to be performed between equation [A] and the 
 fo Flowing equation, 
 
 3x*+q+3xy + y*=0 [C] 
 
 We shall, therefore, arrange this equation with reference to x, and then elimi- 
 nate x by proceeding as if we had to find the greatest common divisor of equa- 
 tions "A] and [C]. 
 
 First Division. 
 
 x^-\- qv -4- r 
 3x34- 3qx +3r 
 
 :^+3v.r-r-7/*+ 7 
 
 x — y 
 + 3r» + 3>/3*+( ? /°+ q)x 
 
 —3yx 2 —(y*—2q)x+3r 
 
 — 3}/.r : — :;//•'.;■ — ,, — . 
 
 Second Division. 
 
 +q)x+f+qy+9r 
 
 y.r + ::(■■/ + ;;i-3r) 
 
 3.r-+ 3?.r+ tf+q 
 
 6(?/ 2 +7)^4- ,; (.'/-' + V).V+- , (.'/ : + V) : 
 + 6(?/ , +7)^+3(?/ 3 +7.V +3r).r 
 tyf+W -3r)x+2(y'+7) 9 
 6W+q){tf+qy -3r)x+4(3f+?) 8 
 gjy+jT)(g+gy —3r)x+Z(y*+qy+3r){ y*+qy—3r) 
 Hr + 7y- :; (r +'l!t + '>i-)(!i :t +<!!'-*r). 
 In the laal division we have multiplied twice by y : -\-<j in order to render th* 
 divisions possible, but if we take v'-f v=°. the divisor reduces to Sr, a quau 
 tity in general differing from 0. 
 
BUDAN'S CRITERION. 359 
 
 Making the last remainder equal to zero, and performing the operations in- 
 dicated, the equation of the differences is 
 
 i/-l r 6rpf+Dq"y~ + 4qi+27r i = ; 
 taking y 2 =z, the equation of the squares of the differences becomes 
 
 za+6gz 8 +92 3 z-f4g 3 +27r 8 =:0. 
 For the equation I s — 7.r+7:=0, we have q== — 7, r=-J-7; and hence the 
 equation in z becomes 
 
 z 3 — 42z 2 +441z— 49=0. 
 
 BUDAN'S CRITERION 
 For determining the number of imaginary roots in any equation. 
 
 280. If the real positive roots of an equation, taken in the order of their 
 magnitudes, be a u a 2 , a s , a. t a n , where a x is the smallest, and if we dimin- 
 ish the roots of the equation by a number h greater than a u but less than a 2 , 
 then the roots will be a x — h, a 2 — h, « 3 — li, ...a a — h, and the first of these 
 will now be negative. Rut tho number of positive roots is exactly equal to 
 the number of variations of sign in the terms of the equation when the roots 
 are all real ; and as we havo changed one positive root into a negative one, 
 the transformed equation must have one variation less than the proposed 
 equation. 
 
 Again, by reducing all the roots by Jc, a number greater than a 2 , but less 
 than a 3 , we shall have two negative roots, «i — Jc, a 2 — Jc, in the transformed 
 equation, and, therefore, we shall havo two variations of sign less than in the 
 proposed equation, for two positive roots have been reduced so as to become 
 negative ones. Hence it is obvious, that if we reduce the roots by a number 
 greater than a n , all the positive roots will become negative, and the transform- 
 ed equation, having all its roots negative, will have the signs of all its terms 
 positive (Art. 259),' and all the variations will havo entirely disappeared. 
 
 We see, then, that if the roots of an equation be reduced until the signs of 
 all the terms of the transformed equation be -j-» we have employed a greater 
 number than the greatest positive root of that equation ; and, therefore, its 
 reciprocal must be less than the smallest real root of the reciprocal equation. 
 Now, if we take the reciprocal equation, and reduce its roots by the reciprocal 
 of the former number, we should have as many positive roots left in this trans- 
 formed reciprocal equation as there were positive roots in the proposed equa- 
 tion, unless the equation has imaginary roots ; hence the number of variations 
 lost in the former case should bo exactly equal to the number left in the latter, 
 when the roots are all real ; and, consequently, if this condition be not fulfill- 
 ed, the difference of these numbers indicates the number of imaginary roots. 
 To explain this reasoning more cloarly, we shall suppose that an equation has 
 threb positive roots ; as, for instance, '1, 2-5, and 3. Now if the roots of the 
 proposed equation be reduced by 4, a number greater than 3, the greatest 
 positive root, tho three positive roots in the original equation will evidently be 
 changed into three negative ones in the transformed one, and hence three va- 
 riations must be lost. Again, the equation whose roots are the reciprocals of 
 the proposed equation must have three positive roots, 1, |, and \ ; and it is 
 evident that if we reduce the roots of the reciprocal equation by j, the recip- 
 rocal of the former reducing number 4, we shall not change the character of 
 Khe three positive roots, because j is less than tho least of them, and 1 — i 
 
360 ALGEBRA. 
 
 § — h 3 — l are a ^ positive ; hence the three variations introduced by the 
 three positive roots must still be found in the transformed reciprocal equation, 
 and, therefore, three variations aro left in the latter transformation, indicating 
 no imaginary roots. The theorem may, therefore, be stated thus : 
 
 If, in transforming an equation by any number r, there be n variations lost, 
 and if, in transforming the reciprocal equation by \ (the reciprocal of r), there 
 be m variations left, then there will be at least n — m imaginary roots in the 
 interval 0, r. 
 
 For there aro as many positive roots in the interval 0, r of the direct equa- 
 tion as there are between l - and - of the reciprocal equation ; hence, if n, tho 
 number of variations lost in the transformation of the direct equation by r, be 
 greater than m, the number of variations left in the transformation of the re- 
 ciprocal equation by -, there will bo a contradiction with respect to the charac- 
 ter of a number of the roots, equal to tho difference n — m. Hence these* 
 roots are imaginary. 
 
 EXAMPLE. 
 
 Find the number of imaginary roots of the equation 
 
 x*—x*-\-2x*+x — 4 = 0. 
 
 Direct. 
 -1 +2 +1 
 10 2 
 
 2 3 
 
 -4(1 
 3 
 
 — 1 
 
 Reciprocal. 
 
 -4 + 1 + 2-1+1 (1 
 
 _ 4 _ 3 _ 1 _o 
 
 _ 3 _ 1 _ 2 — 1 
 
 1 1 3 
 13 6 
 
 
 — 4 — 7 — 8 
 
 — 7 — 8 —10 
 
 1 2 
 
 2 5 
 
 
 — 4 —11 
 —11 —19 
 
 1 
 
 
 — 4 
 
 3 
 
 
 — 15 
 
 Here two variations are lost in the transformation of the direct equation, 
 and no variations aro left in the transformation of the reciprocal equation ; 
 therefore this equation has at least two imaginary roots ; and it has only two, 
 for the sign of the absolute term is negative, implying the existence of two 
 real roots, tho one positivo and the other negative. (See Art. 248, Pr. VIII., 
 Cor. 5.) 
 
 EXAMPLE. 
 
 To find, the number and situation of tho real roots of tho equation x*+ar* 
 +z 2 +3x— 100=0 by Budan's method. 
 
 If the roots of this equation be all real, tho permanences and variation indi- 
 cate throo negati>o roots and one positive root. 
 
 (1) To find the positivo root. 
 
 1 + 1 + 1+ 3 — 100(2 1 + 1+ 1+ 3 — 100(3 
 
 3 + 7 + 17— 6G 
 
 .1 + 13+1-J+ 26 
 
 In the transformation by 2, one variation is left, and. in transforming I 
 there is no variation left ; therefore the positive root is between 2 and 3 
 
 (2) For tlif negative roots. 
 
THE LIMITS OF THE ROOTS OP EQUATIONS. 361 
 
 Reciprocal Equation. 
 — 100— 3+ 1— 1+ 1(1 
 — 103—102 — 103 — 102 
 
 Direct Equation. 
 1 — 1 + 1 — 3 — 100 (1 
 0+1 — 2 — 102 
 1 + 2+0 
 
 2+4 signs all — 
 
 3 
 
 Hero two variations are lost in iho direct transformation, and no variations 
 ire left in the reciprocal transformation ; therefore the two roots in the inter- 
 nal and — 1 are imaginary. 
 
 1 — 1 + 1— 3 — 100 (3 
 
 o_|_ 7 _|_i8_ 4G 
 
 1_1+ l_ 3 — 100 (4 
 3+13*+49 + 96 
 
 Hence the negative root is obviously situated between — 3 and — 4. 
 
 DEGUA'S CRITERION. 
 
 281. In any equation, if we have a cipher-coefficient, or term wanting, and 
 
 if the cipher-coefficient bo situated between two tornis having the same sign, 
 
 there will be two imaginary roots in that equation. 
 
 Let the order of the signs be 
 
 + + -0- + . 
 
 and for writing + or — we have either 
 
 + + - + - + ,or+ + + 
 
 In the former of these we find two permanences and five variations, and in 
 the latter we have four permanences and only three variations ; hence, if the 
 roots are all real, wo must, in the former case, havo five positive and two neg- 
 ative roots, and in the latter, three positive and your negative roots (Art. 259) ; 
 hence wo have two roots, both positive and negative, at the same time, and, 
 therefore, these two roots can not be real roots. These two roots, which in- 
 volve the absurdity of being both positive and negative at the same time, must, 
 therefore, be imaginary roots. 
 
 In nearly the same manner it may be shown that 
 
 (1) If between terms having like signs, 2n or 2n — 1 cipher-coefficients in- 
 tervene, there will be 2n imaginary roots indicated thereby. 
 
 (2) If between terms having different signs, 2/1+1 or 2?i cipher-coefficients 
 intervene, there will be 2«. imaginary roots indicated thereby. 
 
 EXAMPLE. 
 
 The equation x* — r 5 +6:r 3 +24 = has two imaginary roots, for the absent 
 term is preceded and succeeded by terms having like signs ; and the equation 
 r' + l, having the coefficients 1 + + + 1, has also two imaginary roots 
 
 EXAMPLES FOR PRACTICE. 
 
 (1) How many imaginary roots are in the equation 
 
 xs+r 5 — 2x- 3 +2.r— 1 = ? 
 
 (2) Has the equation x* — 2z 2 +6x+10 = any imaginary roots ? 
 
 THE LIMITS OP THE ROOTS OP EQUATIONS 
 282. The limits of any group of roots of an equation are two quantities be- 
 tween which the whole group lies; thus, +co and are limits of the positive 
 roots of every equation, and aud — oo of the negative roots. But in practice 
 We are required to assign much closer limits than these, usually the two con- 
 
362 AlaEBUA. 
 
 gecutive whole numbers between which each root lies, so that the inferior 
 lirnit is the integral part of the included root. This may be effected without 
 knowing any of the roots of the equation, as will be seen in the following prop- 
 ositions. The roots spoken of hi this section are the real roots. 
 
 SUPERIOR AND INFERIOR LIMITS OF THE ROOTS. 
 
 283. The greatest negative coefficient increased by unity is a superior limit 
 of the positive roots of an equation. 
 
 Let — p bo the greatest negative coefficient; then any value of x which 
 
 makes 
 
 x n — p(x a - l -\-x B -*-\- \-x 2 -{-x-\-l) positive, 
 
 x n l 
 
 or x-^^x^+x—H yxz+x+l)^?—- — ,* 
 
 will, a fortiori, make 
 
 x"+p]x a - l +p 2 x"-*+ |_p n _ lX +p , 
 
 otf(x) positive, because in the latter, all the terms after x n will not generally 
 be negative, and of the negative terms not one is greater than the correspond- 
 ing term in the former expression. 
 
 x n — 1 
 Now the inequality x n >_/; is satisfied, if 
 
 x n = or >x " , or x— 1= or >_», or x= or >^;-(-l. 
 
 Since, therefore, p + 1 and every greater number, when substituted for x. 
 will make/(x) positive, the numerical value of the greatest negative coefficient 
 Increased by unity is a superior limit of the positive roots.f 
 
 284. In any equation, if £> r x n-r be the first term which is negative, and — p 
 the greatest negative coefficient, 1-f- \/p is a superior limit of the positive 
 roots. 
 
 Any value of x which makes 
 
 x">p(x n - r +x n — '+ . . . +x+l)> / x _~ , 
 will of course make x n - r -_p 1 x n_1 - r -p 2 x n_2 -l- . . . positive. 
 
 T n_ r+ 1 _ j 
 
 Now the inequality_x n ^>p — , is satisfied if 
 
 ^-P'xTrp 0T3? ~ 1 ( x — 1 )>Pi or if (*— lr-'fa— 1)= or >p, or (x— 1) = 
 
 or >_/?, or x= or >1+ Vp. 
 
 Since, therefore, 1-f- \J p and every greater number gives a positive result, 
 1 -|- V p is a superior limit. 
 
 This method may bo employed when the first term is followed by one or 
 more positive terms. 
 
 EZAMFl 
 
 x*+lla«— 25x— 61i»3. 
 
 Here r=3, and a limit of the positive routs is 
 
 1+ Vo'l, or 5, taking the next higher integer. 
 
 285. If each negative coefficient, taken positively, be divided by the sum of 
 
 * Sec (Art. 23). t This is oommonly known as Maclaurin's limit. 
 
SUPERIOR AND INFERIOR LIMITS OF THE ROOTS. 3G3 
 
 all the positive coefficients which precede it, the greatest of the fractions thus 
 forned, increased by unity, is a superior limit of the positive roots. 
 Let the equation be 
 
 x n +j) l x n - l -i r p 2 x a - 2 +{—2h)x n - 3 -\- ■ • • 
 I ... + (_p r ).r"-'+...+p a =0; 
 
 then, since (Art. 23), 
 
 p m x™=jJ m {x-l){z«>- 1 +x m -*+ ... +x+l)+ Pm , 
 if we transform every positive term by this formula, and leave tho negative 
 tevms in their original form, we shall have 
 
 = (.r-l)x"- 1 + (.r-l).r"--+(.r-l)x n - 3 +...+.r-l + l 
 
 +i>,(x— l)i^+pi(x— l)x»- 3 + . . . +pi(x—l)+pi 
 
 +jp 3 (x— iy**-*+ • • • +M x — 1 )+p* 
 
 — l) 3 X n ~ 3 
 
 + 
 
 Now if such a value bo assigned to x that every term is positive, that value 
 will be the superior limit required ; in the terms where no negative coefficient 
 enters, it is sufficient to have x> 1 ; in the other terms, each of which in- 
 volves a negative coefficient, we must have 
 
 (l+JPi+JPh)(*— 1)>1* 
 
 (i+Pi+.P2+---+iv-i)(*— i)>Pm &c, 
 
 or 
 
 *>T+^+ 1 ' X> l+p,+ P X-+Pr-^ ^ 
 
 If, then, x be taken equal to the greatest of these fractions increased by 
 unity, this value, and every greater value, will make f(x) positive, and there- 
 fore will be a superior limit of the positive roots. This method gives a limit 
 easily calculated, and generally not far- from the truth.* 
 
 EXAMPLES. 
 
 (1) 4X 5 — 8x 4 +23x 3 +105x 2 — 80x+3 = 0. 
 
 8 80 8^ 80 , _ 8 
 
 The fractions are - and 4 ■ 23 ■ 105 » and i>i3o > therefore -+1 = 3 is a 
 
 superior limit. 
 
 (o) 4.r 7 — 6.r6— 7x 5 +8.r 4 +7.r 3 — 23x 2 — 22x— 5=0 ; 
 
 here 3 is a superior limit. 
 
 Observation.— ;The form of the equation will often suggest artifices, by 
 means of which closer limits may be determined than by any of the preceding 
 methods ; thus, writing the equation of Example 1 under the form 
 
 4ar*(x — 2) + 23x 3 +105x(x— — ) + 3=0, 
 
 we see that x= or >2 gives a positive result, therefore 2 is a superior limit. 
 Similarly, by writing the example of Art. 284 under the form 
 
 z(is_25) + ll 
 
 (* 2 -n)= ' 
 
 we see that 3 is a superior limit. 
 
 We have seen (Art. 248) that an equation of an even number of dimension!* 
 with its last term positive may have no real root ; but we shall now show that 
 
 * This is the method of Bret. 
 
SJ4 ALGEBRA. 
 
 m any equation whatever, if the absolute term be small compared wtu the 
 >ther terms, there will be at least one real root also very small. 
 286. In the equation 
 
 Po^ n +i»^ n ~' + ' &c -> +r—r=0, 
 rvhere r is essentially positive, and which may represent any equation what- 
 ever, if r<C „/, , , where v is numerically the greatest coeflicient, then there 
 
 4(1 +p) 
 
 is a real positive root, <2r. 
 
 By dividing by the coefficient of x, and changing the signs of all the terms, 
 and of all the roots, if necessary, eveiy equation may be reduced to the form 
 
 — r+x+, &c., -fp 1 .r— 1 4-p o .r n =0 .... (1) 
 where r is essentially positive ; let p bo numerically tho greatest coefficient, 
 then any value of x<C\ which makes 
 
 -r+x>p{a*+&+, &c, +*")> \_ x , 
 
 will make the first member of (1) positive ; and this condition is fulfilled by 
 
 «.r 3 
 -r+x= or >— , 
 
 because 1>1 — r n_1 , or 
 
 (l+p)-v°—(I + r)x+r=0, 
 or 
 
 2(l+iv).r=(l + r)-V(l + r)— 4r(l+p); 
 if, then, 4r(l-\-p)<C.l, the radical will have a real value >r, and there will be 
 
 for x a real value less than Q , which makes the first member of (1 ) posi- 
 
 tive, while .r=0 makes it negative ; therefore, in any equation reduced to the 
 above form, if r •<[ .,,. . , there is a real small positive. root, <C '-'''■ 
 
 EXAMPLE. 
 
 r»_|_l&r 3 — 2l2- 2 — 12x+l = 
 
 has a real root between and -. 
 
 u 
 
 287. To find an inferior limit of the positive roots, we must transform tho 
 equation into one whose roots are the reciprocals of the roots of the former ; 
 and tho reciprocal of tho superior limit of the roots of tlie" transformed equa- 
 tion, found by the preceding methods, will be the quantity required. 
 
 Hence, if j> r denote tho greatest coefficient of a contrary sign to the last 
 
 m 
 
 term, p„, an inferior limit of the positive roots is ; . For the transformed 
 
 equation will be (Art. 271) 
 
 Pn-l Pi 1 
 
 y n + — y a ~ l -\ \—y r + ...+—= o, 
 
 Vr Vi 
 
 of which — is tho greatest negative coefficient ; therefore 4- 1 is a supenoi 
 
 P a 
 limit of its roots; and, consequently, — — — an interior limit of the positive root* 
 
 o the proposed equation. 
 
A13W TON'S METHOD OF FINDING THE LIMITS OF THE ROOTS. 365 
 
 I 
 
 EXAMPLE. 
 
 jJ_42x 2 +441x— 49=0. 
 
 49 1 
 
 Here p Q =49,j? r = 141, .-. f , . . , or — is an inferior limit of the positive 
 
 roots. By putting x=-, we may discover a limit Cjoser to tho roots ; for the 
 
 *j 
 
 transformed equation is 
 
 6 1 6/ 1\ 
 
 y*- 9 y* + -y-- = 0,oi-if(y-9) + -[y--)=0, 
 
 which evidently has 9 for the superior limit of its positive roots, and, theie 
 fore, the proposed has - for its inferior limit. 
 
 288. To find superior and inferior limits of the negative roots, we must 
 transform the equation into one whose roots are those of the former with con- 
 trary signs (Art. 247) ; and if a, /3 bo limits, found as above, of the positive 
 roots of this equation, then — a and — /? will be limits of the negative roots of 
 the proposed equation. 
 
 EXAMPLE. 
 
 a«_ 7z+7=Q; 
 
 putting .r= — y, we get y" — 7y — 7=0, of which 1+ y/7 or 4 is a superior 
 
 limit. 
 
 1 113 
 
 Also, putting y=-, we get z»+z 2 — -=0, or z 3 — — +z 3 — — = 0, of which 
 
 - is a superior limit ; therefore the negative root of the proposed lies between 
 o 
 
 —4 and —3. 
 
 newton's method of finding limits of the roots. 
 
 289. The limits, however, deduced by any of the preceding methods sel- 
 dom approach very near to the roots ; the tentative method, depending upon 
 the following proposition, will furnish us with limits which lie much nearer to 
 them. 
 
 Every number which, written for x, makes/(x) and all its derived functions 
 positive, is a superior limit of the positive roots. 
 
 For, if we diminish the roots a, b, c, &c, of/(x) = by h, that is (Art. 251V 
 substitute y-\-h for x, the result isf(y-\-h)=0, or 
 
 m+fv>)i +/' / ( / o 1 ^7+ ••• +/ n -'(/o^+2/ n =o- 
 
 Now, if we give such a value to h that all the coefficients of this equation 
 are positive, then every value of y is negative ; that is, all the quantities, a — h, 
 ft — h t c—h, &c, are negative, and therefore h is greater than the greatest of 
 the quantities a, b, c, &c, or is a superior limit of the roots of the proposed 
 equation. Similarly, h will be an inferior limit to al the roots, if the coefficients 
 be alternately positive and negative. 
 
 EXAMPLE. 
 
 To find a superior limit of the roots of 
 
 z 3 — oj?+7x— 1 = 0. 
 
3G6 ALGEBRA. 
 
 The transformed equation, putting y-\-h for x, is 
 
 (h 3 —5h*+7h — l) + {3lr—U)h + 7)ij+(6h-..10)^+y=Q; 
 
 in which, if 3 be put for h, all the coefficients are positive; therefore 3 is a su- 
 perior limit of the positive roots. 
 
 Observation. — This method of finding a superior limit of the roots by de- 
 termining by trial what value of x will make/(.r) and all its derived functions 
 positive, was proposed by Newton. 
 
 waring's or lagrange's method of separating the roots. 
 
 290. If a series of quantities be substituted for x in/(.r), then between every 
 two which give results with different signs au odd number of roots of/"(.r) = 
 is situated; and between eveiy two which give results with the same sign an 
 even number is situated, or noue at all; but we can not assure ourselves that 
 in the former case the number does not exceed unity, or that in the latter it 
 is zero, ami that, consequently, the number and situation of all the real roots 
 is ascertained, unless the difference between the quantities successively sub- 
 stituted be less than the least difference between the roots of the proposed 
 equation ; since, if it were greater, it is evident that more than one root might 
 be intercepted by two of the quantities giving results with different signs, and 
 that two roots instead of none might be intercepted by two of the quantities 
 giving results with the same sign, and in both cases roots would pass undis- 
 covered. We must, therefore, first find a limit less than the least difference 
 of the roots ; this may be done by transforming the equation into one whose 
 roots are the squares of the differences of the roots of the proposed equation. 
 Then, if we find a limit k less than the least positive root of the transformed 
 equation, y 7 k will be less than the least difference of the roots of the proposed 
 equation; and if we substitute successively for x the numbers s, s — V k, 
 s — 2 -//>"> &c- ( s being a superior limit of the roots of the proposed), till we 
 come to a superior limit of the negative roots, we are sure that no two real 
 roots lying between the numbers substituted have escaped -. ami that every 
 change of 6igns in the results of the substitutions indicates only one real root. 
 Hence the number of real roots will be known (for it will exactly equal the 
 number of changes), as well as the interval in which each of them is contained. 
 
 Observation. — This method of determining the number and situation of 
 the real roots of au equation was first proposed by Waring : ii is, however, of 
 no practical use for equations of a degree exceeding the fourth, on account of 
 the great labor of forming the equation of differences for equations of a higher 
 order. 
 
 I \ LHPLE. 
 
 x i — 7x+7=0. 
 
 The numbers 1 and 2 give each a positive result, but yet two roots lie be- 
 tween them. The equation whose roots are the squares of the differences is 
 (Art. 279) i/ 3 — 42?/ a -f-44l2/ — 49 = 0. an inferior limit of the positive roots of 
 
 which is - (Art. 287); therefore, - is loss than the least difference of the 
 9 o 
 
 5 4 
 roots of I s — 7.r-f-7 = (>, ami, substituting 2, -, .,, the results are -|-, — • -+■ 
 
METHOD OF DIVISORS. 3C7 
 
 5 5 i 4 
 
 hence one value of x lies between 2 and -, and one between - and - •, and, 
 
 O O u 
 
 similarly, we find the negative root, which necessarily exists, to lie between — 3 
 and — 3x. 
 
 METHOD OF DIVISORS. 
 
 291. The commensurable roots of/(x) = 0, which are necessarily whole 
 numbers, may bo always found by the following process, called the method of 
 divisors, proposed by Newton. 
 
 Suppose a to be an integral root; then, substituting a for .r, and reversing 
 the order of th,e terms, we have 
 
 Pn+i 7 n-ia+p n -2a 3 + . . . -f^a"- 1 +0"= ; 
 
 ••• ^+P"-> +JW* -\ — +^i« n - 2 +« n_1 = o- 
 
 Hence, — is an integer which we may denote by qi ; substituting and di- 
 viding again by a, we get 
 
 ? ' + a Pn ~ 1 +Pn- 2 + • • • +lha"- 3 +a»-* = 0. 
 
 Similarly, "~ 1 is an integer =q 2 suppose ; and proceeding in this man- 
 
 ner, we shall at last arrive at 
 
 a ' 
 
 Hence, that a may be a root of the equation, the last term, p a , must be di- 
 visible by it, so must the sum of the quotient and next coefficient, qx-\-p D -i ; 
 and continuing the uniform operation, the sum of each coefficient and the pre- 
 ceding quotient must be divisible by a, the final result being always —1. 
 
 If, therefore, we take the quotients of the division of the last term by each 
 of the divisors of the last term which are comprised within the limits of the 
 roots, and add these quotients to the coefficient of the last term but one ; di- 
 vide these sums, some of which may be equal to zero, by the respective 
 divisors, add the new quotients which are integers or zero (neglecting the 
 others) to the next coefficient and divide by the respective divisors, and so on 
 through all the coefficients (dropping every divisor as soon as it gives a frac- 
 tional quotient), those divisors of the last term which give —1 for a final re- 
 sult are the integral roots of the equation ; and we shall thus obtain all the in- 
 tegral roots, unless the equation have equal roots, the test of which will be that 
 some of the roots already found satisfy /'(x) = 0, and the number of times that 
 any one is repeated will be expressed by the degree of derivation of the first 
 of the derived functions which that root does not reduce to zero, when written 
 in it for x (Art. 253). It is best to ascertain by direct substitution whether 
 -\-l and — 1 are roots, and so to exclude them from the divisors to be tried. 
 
 EXAMPLE I. 
 
 a«4-3a*— 8a:+10=0. 
 
 Q 
 
 Here the roots lie between -+1 and —11 (Arts. 285, 288), and the divi- 
 sors of the last terra are ± {2, 5, 10 }, 
 
3(J8 ALGEBRA. 
 
 .: a = 2 — 2 — 5 —10 
 
 g 1== 5 — 5 — 2 — 1 
 
 ?l + (_8)= —3 —13 —10 — 9 
 
 ']:= 2 
 
 ?s= — !• 
 
 Therefore — 5, being the only one of the divisors which leads to a last quo- 
 tient — 1, is the only commensurable root, and it is not repeated, since it does 
 not satisfy the equation /'(.r) = 3.r 2 - r - 6x — 8=0. 
 
 EXAMPLE II. 
 
 zs_5xi+:r>-f i6x 2 — 20x+16 = 0. 
 Here limits of the roots are 6 and —4 ; and the commensurable roots are 
 
 EXAMPLE III. 
 
 a -i_|_5 x -3_o x 2_6x- r -20=0; x= — 2, or —5. 
 
 292. The number of divisors to be tried may be lessened by observing, that 
 
 if the roots of /(.r) = were diminished by any whole number, m, the last 
 
 term of the transformed equation, f{y-\-m) = 0, would bef{m) ; if, therefore, 
 
 a were an integral value of x, a—m would bo an integral value of y, and would 
 
 be, therefore, a divisor of f(m). Hence, any divisor, a, of the last term of 
 
 f(m) 
 f(x) is to be rejected which does not satisfy the condition -— — = an integer, 
 
 when for m any integer, such as ±1, ±10, &c, is substituted. 
 
 EXAMPLE I. 
 
 r 5 — 5.r 2 — 181+72 = 0. 
 Changing the signs of the alternate terms, we have 
 
 ar»+5a«— 18r— 72=0, or a*— 78+6r(x— r-) =0-, 
 
 therefore- the roots lie between 19 and — 5. 
 
 But /(l)=50,/(-l)=S4,/(-3)=54; 
 
 and the only admissible divisors of 72, which, when diminished by 1, divide 
 50, are 
 
 G, 3, 2, -4 ; 
 
 also, all these divisors, when increased by 1, divide 84; but only 6, 5,-4 
 when increased by 3, divide 54 ; 
 
 .-. G, 3, —4, 
 
 are the only divisors which need to be tried ; and they will all bo found to be 
 roots. 
 
 EXAMPLE II. 
 
 :j3_ 6i»+169x— (42) 8 =0. .r=9. 
 
 293. If a proposed equation have fractional coefficients, or if its first term 
 be affected with a coefficient, since (275, Cor. 2) it can be transformed into an- 
 other equation with first term unity and every coefficient a whole number 
 this method will onablo us to find the commensurable roots of every equation 
 undor a rational form. If the coefficients be whole numbers and the first term 
 be^v", and wo only wish to find the roots which are integers, no transfcrmtr 
 
NEWTON'S METHOD OF APPROXIMATIC N. 369 
 
 turn will bo necessary, only every divisor of the last term which is a root will 
 lead to a result — p instead of —1. 
 
 EXAMPLE. 
 
 6x*— 253?-\-26x-+4x— 8 = 0. 
 
 It is the same as 
 
 {x— 2) 2 (3z— 2)(&c+ 1) =0. 
 
 newton's method of approximation. 
 
 294. When we know an approximate value of a root, we may easily obtain 
 other values of it, more and more exact, by a method invented by Newton, 
 which rapidly attains its object. We shall give this method, first in the form 
 in which it was proposed by its author, and afterward with the conditions 
 which Fourier has shown to be necessary for its complete success. 
 
 Let/(z) = be an equation having a root c between a and b, the difference 
 of theso limits, b — a, being a small fraction whose square may be neglected in 
 the process of approximation. 
 
 Let Ci, a quantity between a and b, be assumed as the first approximation 
 to c, then c=c l -\-h, where h is very small ; 
 
 .-. 7(^4-70=0, 
 
 01 
 
 f(c i )+f'(c i )h+f"(c l )^+ . . . +h»=0. 
 
 Now, since h is very small, /j 2 , /t 3 , &c, are very small compared with h ; 
 also, none of the quantities/"(ci),/'"(ci), &c, can become very great, since they 
 result from substituting a finite value in integral functions of x; therefore, pro- 
 vided f'(ci) be not very small (that is, provided /'(.r)=0 have no root nearly 
 equal to Ci or to c, and, consequently, f{x) = no other root nearly equal to c 
 besides the one we are approximating to), all the terms in the series after the 
 first two may bo neglected in comparison with them ; and we have, to deter- 
 mine h, the resulting approximate value of h, the equation 
 
 /(ci)+Vte)=<>; 
 
 • i ~ net)- «/(*)>—/ 
 
 and the second approximation is 
 
 _L.7 U {X) 1 
 
 Similarly, starting from c a instead of e u the third approximate value will be 
 
 $ A*) I 
 
 C3-C- I fi{x) $ _ ; 
 
 and so on; and if we can be certain that each new value is nearer to the truth 
 than the preceding, there is no limit to the accuracy which may be obtained. 
 
 EXAMPLE I. 
 
 x 3 — 2x— 5=0. 
 Here one root lies between 2 and 3, and the equation can have only on» 
 
 * This notation signifies, that after the division indicated is performed, the particular 
 vmlue ci, is substituted for x. 
 
 A A 
 
370 ALGEBBA. 
 
 positive root; also, upon narrowing the limits, we fiud that i=2 gives a nega- 
 tive, and ar=2-2 a positive result; therefore, 2-1 differs from the root bv a 
 quantity less than 0-1, and wo may assume c t =2'l. Hence 
 
 (x 3 — 2j— 5\ 
 
 0-061 
 11-23' 
 
 or 
 
 Similarly, 
 
 c 3 =2-l— 0-0054=2-0940. 
 c 3 = 2- 094 55 14 9. 
 
 EXAMPLE II. 
 
 X s — 7x— 7 = 0. 
 
 There is only one positive root lying between 3 and 3-1, and it equal* 
 3-016917339. 
 
 Observation. — To guard against over correction, that is, against appl 
 such a correction to an approximate value as shall make the new value differ 
 more from the root by excess than the original approximate value did by de- 
 fect, or vice versa, we must be certain that each new value is nearer to rue 
 truth than the preceding; this gives rise to the following conditions, first no 
 ticed by Fourier. 
 
 295. For the complete success of Newton's method of approximation, the 
 following conditions are necessary. 
 
 1. The limits between which the required root is known to lie must be so 
 close that no other root of/(x)=0, and no root of/ (i)=0, or/"(.r) = 0, lies 
 between them. 
 
 2. The approximation must be begun and continued from that limit which 
 makes f{x) and/"(r) have the same sign. 
 
 Let c be a root of/(.r)=0 which lies between a and 6, a<6, c x the first ap- 
 proximate value, and h the whole correction, so that c=Ci -f-/t ; then 
 
 /(c l +A)=0,or/(c 1 )+V''( i )=0. 
 
 A being some quantity between c, and c (Art. 239, Note). 
 
 Therefore, supposing A=c,, which amounts to neglecting all powers of h 
 above the first, and requires that/(.r) = have no root besides c in that interval, 
 and calling the resulting approximate value of h, h u we have 
 
 /(c,)+V(ci)=0. 
 
 Now the true value is c=c l -\-h ; 
 
 The first approximate value is c t with error // ; 
 
 The second approximate value is c i =Ci-^-h l with error h—h u which (ueg 
 lecting signs) must be less than h, 
 
 i. e., h* — (h — hi) 9 must be positive, or 2/j/;, — h l ' 2 = -{-, 
 
 or ^-' = +' or 7W- i=+: 
 
 which condition (since A is an indeterminate quantity between c, and r, or be- 
 tween a and b) can not in all cases be secured unless /'(.r) be incapable of 
 changing its sign between a and b, i. C, unless /'(x)r=0 have no root between 
 a and b. 
 
 Moreover, we must have j77^\>^ or >1, the latter insuring the former. 
 
 Now, if/"(x) preserve an invariable aign between a and />, i. c, \(f"(r)=Q 
 
NEWTON'S METHOD OF APPROXIMATION*. 371 
 
 have no root in that interval, then f'(r) will increase or diminish contirually 
 from a to b ; therefore Ci must be taken equal to that limit which gives /'(<) 
 its greatest numerical value without regard to sign. 
 
 First, let/'(.r),/"(.r), have the same sign from a to b ; thcn/'(.r) increases 
 continually in that interval; therefore we must have Cj=6, or we must begin 
 from the greater limit. But/(6) has the same sign as f(c-{-h)=f(c)-\-hf'{c) 
 =hf'(c), or as /'(c) ; therefore we must have c x equal to that limit which makes 
 f(x) andf'(x) have the same sign. 
 
 Secondly, ]et f'(x), f"(x), have contrary signs from a to b ; then/'(.r) di- 
 minishes continually in that interval ; therefore we must have Ci=a, or we 
 must begin from the lesser limit. But f(a) has the same sign as f(c — h) 
 =f(c) — hf'(c) = — hf'(c), or as —/'(c); therefore, in this case, equally as in 
 the former, we must have c x equal to that limit which makes f(x) and/"(.r) 
 have the same sign. 
 
 Theso conditions being fulfilled, we have 
 
 ''<"'> .l=+,or* -■ 
 
 c—c 2 
 
 or 
 
 C 2 — Ci 
 
 therefore <■, lies between c and Ci ; hence, the new limit, c 2 fulfills the requi- 
 site conditions, and wo may with certainty from it continue the approxima- 
 tion. 
 
 296. To estimate the rapidity of the approximation, we have 
 
 error in first approximate value c u ssk, 
 error in second approximate value c 2 , =h — h { ; 
 But f(ci)+hf>( Cl ) + lh*f"(n)=0, 
 
 /(ci)+/->/'(<->)=0; 
 
 .-. (h-i h )f'{ Cl )+yii*fy)=o, 
 
 " /'(ci) 
 Let the greatest value which f"(x) can assume between a and b (whicn 
 will be either/"(a) or f "(b), if /'"(.r) = have no root in the interval) be di- 
 vided by the least value of 2/'(.r) in that interval which will be either 2f'(a) or 
 2/'(6), and let the quotient be denoted by C ; then, neglecting signs, 
 
 hence, if the first error h in ^ be a smad decimal, the error h — hi with which 
 c 2 is affected (since C will not, except in particular cases, be very large) will 
 be very small compared with h ; and if tho quantity C be less than unity, the 
 number of exact decimals in the result will be doubled by each successive 
 operation. The quantity C, when thus computed for a given interval, pre- 
 serves the same value throughout the operations which it may be necessary to 
 make in order to approximate to the value of the root lying in that interval ; 
 and as we thus know a limit to the difference between the approximate value 
 already found and the true value, we may always avoid calculating decimals 
 which are inexact, and only obtain those which are necessarily correct. 
 
 EXAMPLE. 
 
 6X 3 — 141x+263=0. 
 This equation has two positive roots, one between 2-7 and 2-8, and the 
 
M2 ALGEBRA. 
 
 other between 2-8 and 2-9. Now f'(x)=18x a — 1-11 = has a root 
 
 -£ 
 
 = 2-798, betwoen 2-7 and 2-8, therefore these limits are not sufficiently close ; 
 but this root is greater than 2-79 ; also, 2-7 and 2'79, substituted in/(x), give 
 results with different signs; and 2-7, substituted in f(x) and f"(x), gives re- 
 sults with the samo sign ; therefore, c,=2-7. 
 
 With regard to the other interval, 2-8, 2-9,/'(.r) = 0./"(.r)=0 have no roots 
 between these limits, and 2-9 makes /(x) and f"(x) have the same sign; 
 therefore, c,:=2-9; and starting from these values, we are certain in each 
 case to get a value nearer to the truth. 
 
 f"{x) 
 
 Again, the greatest value which ^, can assume in the interval 2-7, 
 
 2-79, is nearly equal to 10 ; hence, if hi, h 2 , be consecutive errors, we have 
 
 The samo formula will be found to be time for consecutive errors in the in 
 terva] 2 8, 2-9. 
 
 IiAGRANGE's METHOD OF APPROXIMATION BY CONTINUED FRACTIONS. 
 
 297. To approximate to the roots of an equation by the method of continued 
 fractions. 
 
 Let the equation f(x) = have only one root between the integers a and 
 
 a-\-l ;* then, writing a-\-- for x, the first transformed equation will be 
 
 /(«)+/'(^+/»n^.+ -+^=o (i); 
 
 and, since only one value of- lies between and 1, y has only one value greater 
 
 than 1 ; if, therefore, we substitute successively 2, 3, 4, &c, for y, stopping 
 at the first which gives a positive result, the integer preceding that is the in 
 
 tegral part of the value of y. Let this be b, and in (1) write b-\-- for y ; then 
 
 the second transformed equation will have only one root greater than unity, 
 the integral part of which, as before, will be the whole number next less than 
 tho one in the series 2, 3, 4, cVc, which first gives a positive result when 
 written for z ; let this bo c, and in the second transformed equation write 
 
 c-|-~ for z, then the third transformed equation will have only one root greater 
 
 than unity, the integral part of which may bo found as before, and so on. 
 We thus obtain successively the terms of a continued fraction 
 
 c +d' &c - 
 which expresses the required valuo of x. The method of reducmg such a 
 tion, callod a continued fraction, will be hereafter given. 
 
 * Tho roots of the equation may be made to differ by at least unity, if wo fnJ by meani 
 of the equation of tho squares of die differences the 1 it limit to tho differences of tho 
 roots of the proposed equation, and then find n brand quation whose roots shall bo 
 
 that multiple <>f those of the proposed, which is expressed by the denominator of tho least 
 limit of tho differences. 
 
LAGRANGE'S METHOD OP APPROXIMATION 373 
 
 If any of the numbers b, c, d, &c, is an exact root of the 101 responding 
 transformed equation, the process terminates, and we find the exact value of a 
 Also, if one of the transformed equations be identical with a preceding one, 
 the continued fraction expressing the root is periodical; for, after that, the 
 same quotients will recur in the same order ; in this case a finite value, in the 
 form of a surd, may be obtained for the root (see Continued Fractions) by solv- 
 ing a quadratic whose coefficients are rational, both of whose roots will be roots 
 of the proposed, sinco the coefficients of the latter are supposed rational ; con- 
 sequently, the first member of this quadratic will be a factor of the first mem- 
 ber of the proposed equation, which may, therefore, be depressed two di- 
 mensions. 
 
 EXAMPLE. 
 
 To find the positive root of x z — 2.r — 5=0 under the form of a continueu 
 fraction. 
 
 Comparing this with x 3 — qx-\-r=0, we find that 
 
 r 2 q s 25 8 
 
 -——=——— is a positive quantity ; 
 
 therefore (Art. 258) the equation lias two impossible roots ; and since its last 
 term is negative, its third root is positive. Substituting 2 and ?, the results are 
 
 — 1 and +16; therefore the root lies between 2 and 3. Assume x=24--i 
 
 y 
 
 and the transformed equation is 
 
 2/ 3_107 / "-_67/-l=0, 
 
 in which 10 and 11 being substituted, give — 61, +54. Assume i/ — 10+-i 
 
 and we obtain 
 
 Gl: 3 — 94--— 20z — 1 = 0, 
 whose root lies between 1 and 2. Proceeding in this manner, we find 
 
 1111 
 
 • r=2 +Io-+i+T+2--- 
 
 the value of the root in a continued fraction ; the method of reducing which 
 to a common fraction will be hereafter given. 
 
 This method may be combined with Sturm's theorem. 
 
 Here finishes our recapitulation of the older methods. What follows be- 
 longs to the present more improved state of algebraic science.* 
 
 * We shall here point out a method of finding; the equal roots of an equation, which 
 avoids the laborious process of seeking the common divisor, and which may be employed 
 when any other than Sturm's process for discovering the roots of an equation is used. 
 
 1. If an equation whose coefficients are commensurable have a pair of equal roots and no 
 greater number, these roots must be commensurable ; for the common measure of the first 
 member of this equation, and the function derived from it, will be a binomial expression of 
 the first degree with finite coefficients, and which, when equated to zero, will furnish one 
 of the equal roots ; these roots, therefore, must be commensurable, that is, either integers 
 or fractious. 
 
 2. If the leading coefficient in the supposed equation be unity, and the others integral, 
 the equal roots mist be integral, because no fractional root can exist under these condi- 
 tions (Art. 246). , 
 
 3. If an equation with commensurable coeffici ;nts have three equal roots, and no more, 
 these also must be commensurable ; for, in this case, the common measure will be of the 
 second degree, and, when equated to zero, will give two of the equal rorts : these mots, ! s 
 just remarked, must be coninici>surable ; hence all the three roots must be commensurable 
 
374 ALGEBRA. 
 
 BINOMIAL EQUATIONS. 
 298. Binomial equations are those which can be reduced to the t rra 
 
 x m =A ori" 1 — A = (1) 
 
 A being any known quantity whatsoever. 
 
 An I, as before, if the leading coefficient be unity, and the others integral, the equal roott 
 will be integral. 
 
 4. By the same reasoning, if an equation with commensurable coefficients have m equal 
 roots, and no other groups of equal roots, these m roots must be commensurable ; and they 
 will be integral if the leading coefficient be unity and the other oo< integers. 
 
 5. When the leading coefficient is unity, and the other coefficients whole numbers, and 
 /« equal integral roots enter, we may infer, from the formation of the coefficients (245), that 
 
 absolute number, and the coefficient of the immediately preceding term, that is, the 
 licient of x, will admit of a common measure involving m — 1 of these roots ; that the 
 coefficients of x and x" will have a common measure involving m — 2 of them ; and so od 
 till we come to the coefficients of a**-* and a"*-', which will have a common measure in- 
 volving the multiple root once. For, if the depressed equation containing only the unequal 
 roots be considered, it will involve none but integral coefficients, since its last term is form- 
 ed from the penult coefficient of the proposed divided by one root ; so that if the equal roots 
 be now introduced, they can combine with none but integral factors. Hence, if the root occur 
 twice, it will be found among the integral factors of the common measure of the coefficients 
 A n (the final coefficient) and A„-i ; if it occur three times, it will be found among the fac- 
 tors of the common measure of An, An-i, and A n -e, and so on. And, therefore, by trying 
 several factors of the common measure in question, by actually substituting them for x in the 
 proposed equation, when from any circumstance multiple roots are suspected to exist, we 
 may remove all doubt on the subject. In analyzing an equation, the doubts that may arise 
 as to the entrance of equal roots arc confined to certain defiuite intervals, or within deter- 
 minate numerical limits ; so that, of the factors adverted to above, only those falling within 
 these limits need be regarded. 
 
 And further, if the repeated root occur but twice, the square of it must be a factor of a 
 or A n ; if it occur three times, the cube of it must be a factor of A n , and the square of it a 
 factor of An-i ; if it occur four times, the fourth power of it must be a factor of An, the cube 
 of it a factor of A„_i, and the square of it a factor of An-:, and so on. And thus, of the 
 factors of A„ to be tested, those only need be used whose powers also are factors, entering, 
 as here described, according to the multiplicity of the roots. 
 
 6. These inferences may be easily generalized : they apply, whatever be the integral 
 value of the leading coefficient, and whether the repeated root be integral or fractional. 
 
 Thus, let the repeated root be x=j, a and b having no common factor ; then, if the root en- 
 ter m times, the original polynomial will be divisible by (bx—a) m , giving a quotient in- 
 volving the n roots, and into which none but intei ral coefficients enter (953). Let 
 us now return to the original polynomial by multiplying this quotient by bx—a m tin 
 the first multiplication by bx—a will evidently uive a product, into the lirst term of which 
 b must enter as a factor, ami into tho last of which a must enter ; the next multiplication 
 must, therefore, give a product, into the Brat term of which b- must enter, into the second 
 b, into the last ./-, and into the last but one a ; the third multiplication, therefore, must 
 give a product whose first three tonus involvi & last tbn 
 a< re these last in reverse order, and so on. 11. nee the coefficients Ai, A.. A . Ac, 
 will be divisible by b m , b m ~ l , b a ', fco., respectively, down to b; and the ooefflcii nta A . 
 An-i, An a. Ac, by a m , a 1 "-', a™-'-', Ac., down to a. In other words, the coefficients, I 
 in order, reckoning from the beginning, will be divisible by the corresponding decreasing 
 powers of the denominator of the repeated root; and the coefficients, reckoning from the 
 will be divisible by the like powers of the tor. 
 
 7. The inferences* still have place 1 , whatever be tin of the multiple factor • 
 ;„ ■ the proposed polj normal, so Ion ■ as this factor, as weB as tb i J porynonrid, I 
 none bat integral coefficients. This is plain, fro,,, the reasoning in the pr< 
 
 which remains tl.e : ame. BS respects the entrance of the factors b, fl, Whether the 
 multiplier bo bx—a or b.i ,n -f- • • • • +"• 
 
BINOMIAL EQ.UATION1-. 375 
 
 Wo perceive immediately that the m roots of this equation are different 
 from one another ; for the first member x m — A has no common factor with its 
 derived function ma:™ -1 , and hence the proposed equation (Art. 253, Schol.) 
 can not have equal roots. The roots, if we raise them to the power m, ought 
 each to produce A, since they are tho same as tho values embraced in the ex- 
 pression x= t/A. We know, then, that this radical has m different values ; 
 but we shall recur to this subject again, and more at length. 
 
 299. When m is any composite number, tho solution of equation (1) re- 
 duces itself to the solution of several binomial equations, the degrees of which 
 are the factors of m. 
 
 Suppose m=2)qr, instead of the equation 2rPV=0, we can take the equations 
 
 xp=x', z"i=.r", x'"=.A, 
 
 in which x', x" are new unknowns. 
 
 It is evident that, after we have solved the equation x'"=A, the preceding 
 equation .r"'=.x" will make known the values of x', and that then the equa- 
 tion x?=x' will give all tho roots of the proposed equation. This agrees with 
 the formula demonstrated in the theory of radicals (Art. 63), viz., 
 
 \/\A^ 
 
 ' VA=TA. 
 
 300. Designate by a a quantity whose m th power is A, and take x=ay. 
 The equation z m = A becomes a m y m =.a m ; dividing by a m , 
 
 hence y = ™/l 5 and, consequently, x=a,yi. 
 
 We conclude, therefore, that the roots of the equation x m =A can be ob 
 tainod by multiplying one of them by the roots of the equation i/ m =l ; or in 
 general, that the different m ,h roots of a quantity can be obtained by multiply- 
 ing on© of them by the m" 1 roots of unity. 
 
 301. Let us consider more particularly the case in which A is a real quan- 
 tity ; and, to distinguish the hypothesis of A being positive or negative, writo 
 the binomial equation in this form : 
 
 2 m =±A (2) 
 
 These conclusions will greatly simplify the research after equal roots, and will either 
 enable us wholly to dispense with the laborious process for the common measure, or will 
 at least, render the more tedious steps of it unnecessary 
 
 EXAMPLES. 
 2.1^—12^+19x2—6x4-9=0 (1) 
 
 a;7+5i-fi4-6x5— 6.r*— 15a-3— 3x2-f 83+4=0 . . . (2) 
 
 The first of these can have no fractional repeated roots, because the leading coefficient 
 2 has in factor a perfect power; the equal roots, if any, must, therefore, be integral. 
 Unity, which always has claims to be tried, does not succeed ; and from the factors of 9 
 and G, it is plain that +3 and — 3 arc the only other numbers to be tested ; and as no 
 higher power of 3 than the square enters 9, we infer that more than two equal roots can 
 not have place in the equation. By testing 3, we find this to be one of a pair of equal 
 roots. Equal quadratic factors could not possibly enter the equation, since, as the first co- 
 efficient shows, the polynomial is not a complete square. In the second of the above equa- 
 tions no fractional roots can enter. Applying, therefore, +1 and — 1, we discover that 
 -(-1 is twice a root, and — 1 three times. The remaining equal roots — 2 and — 2 are 
 found from the resulting quadratic obtained by suppressing from the given equation the 
 five factors of the first degree. 
 
376 ALGKIJRA. 
 
 Wo can determine, at least by approximation, ft positive quantity a such 
 that wo have a'"=A. Take, again, xssay, equation (2) will become 
 
 This is tlio equation to which I BhaU confine myself exclusively. 
 
 302. 'Tin- following remarks may be made with regard t<> this equation: 
 
 1. Winn m is an odd number, and tin- equation is y m = l or ?/ m — 1=0, it 
 evidently lias the root <y=l : and it has no oilier real root, for every other 
 positive value of y will give y' n ^>l or tf m <l, and a negative valuo will render 
 y m negative. To obtain the equation on which the ?« — 1 imaginary roots de 
 pend, we shall divide y m — 1 by y — L, and thus obtain the equation 
 
 which belongs to the class of equations called reciprocal. 
 
 2. When /// is an odd number, and the equation is y' n = — 1, it has evi 
 dently lor a root y = — 1. By a reasoning analogous to the preceding, 
 it may be proved that; the other roots arc imaginary; and wo obtain the 
 equation on which they depend by dividing y' n -\-l=0 by y-\-\. 13ut to 
 obtain all the roots of tlio equation y m = — 1, it is well to remark that this 
 equation can be derived from y m ss — 1 by changing y into — y. It will suffice, 
 then, to take all tlio roots of i/'" = l with contrary signs. 
 
 3. Supposo m is an ovon number, and let ?/i = 2rt, the equation y" n — 1, 01 
 y" n — 1 = 0, has fur its roots y = -\-l ami y= — 1. Tho other loots are imagin- 
 ary, and the equation which contains them can bo obtained by dividing '/ — 1 
 =0 by (?/ — l)(?/-f-l), or y" — 1 ; but it will bo well to observe that y" n — ) 
 = (?/" — l)(. 7 / n +l)» nnil that, consequently, the equation if — 1 = can bo re- 
 placed by two others moro simple, 
 
 iy"— 1=0, y+i=o. 
 
 4. Finally, when tho equation is y-" = — 1, or y""-\-l=0, we know that tlio 
 even powors of real quantities will always give positive results ; wo hence 
 conclude that all tlio roots aro imaginary. Taking y- = z, tho equation reduces 
 to the degree n, and becomes simply ;"= — 1. 
 
 303. I now proceed to determine the solutions of the equations y m — 1 = 0, 
 y m -\-l=z0, in some particular cases. 
 
 Letm=2; tho equations to be resolved are 
 
 ;// — 1=0, whence 7/=il ; 
 
 i/* -)-l=0, whence y=i \/ — 1. 
 Letm=3; to resolve the equation y — 1=0, observe that it has for a root 
 V = l; we divide it. by y — 1, and it becomes 
 
 y+y+ 1 ~Q> wh '" y= 5 • 
 
 Hence, the three roots are 
 
 _l+V— '* -I--/— •: 
 
 ?/ = !-?/ = 77 .2/=— , -• 
 
 If we take the equation if -\- I = I), W6 shall oh BrVe that its rooti are tho 
 
 samn, except as regards Bign, with those of y — i=u; consequently, they 
 
 will bo 
 
 1_V^3 1-f ,/_:! ■ 
 
 y=-hy= 5 ,3/ = 
 
BINOMIAL EQUATIONS. 377 
 
 Lot m=4 ; tho equation y i — 1=0 may bo decomposed into two others, 
 y* — 1=0, 7/ 2 4-l=0; and from these equations we dorivo tho four roots 
 
 ?/±l,?/± V^l- 
 The equation y f -\-l will be resolved differently; by adding 2if to both 
 membors of tho equation, wo can write it thus: 
 
 (^+1)8=2^; 
 wo can then decomposo it into two others, 
 
 2/ 2 + 1 —y V^, tf+ 1 = — v \/2 ! 
 
 and, finally, from those wo dorivo tho four valuos of y, 
 
 y = ±y/2±yV=<l,y = -lV2±lV^2. 
 "Wo could havo treated tho equation 2/ 1 -j- l== ^ n "reciprocal equation. 
 We might havo observed, also, th;il; it; gives y"=zh V — *» nn( t tnat i taking 
 successively +-/ — 1, — a/ — 1, wo havo 
 
 1= ±V + l/^li V= i V - V^-L- 
 
 V- 
 
 We havo then only to rcduco theso values to tho form a-4-/3-\/ — 1 by tho 
 process in Art. 104. 
 
 By raising tho equation ;y m =pl=0 successively to tho 10° degreo, we 
 shall find that its resolution depends on that of tho preceding cases, or on the 
 resolution of reciprocal equations, which reduce it to a degreo less than tho 5°. 
 
 Lot us examino, first, tho odd degrees. If wo havo the equation ?/"• — 1=0, 
 having observed that.it has tho root y = l, wo divide it by y — 1 ; it then bo- 
 comes 
 
 a reciprocal equation, which we shall reduce to tho 2° degreo. To do this, 
 wo first write it undor tho form 
 
 (^) + (!/ + J) + l=0. 
 
 Then take t/+-=z, which gives y--\-—=z~— 2; and, consequently, the 
 
 y y 
 
 equation in y will bo changed to tho following : 
 
 — 1±a/5 
 2 2 -}-z — 1=0, whence z = . 
 
 Theso valuos being known, thoso of y will bo by tho relation y-\-—=z, for 
 this relation gives 
 
 z 4- Vz 2 — 4 
 
 y= § ; 
 
 mid Ave have only to substitute instead of z successively each of its two values, 
 in order to find the four imaginary values of y. Wo havo then the fivq values 
 
 of y, 
 
 y=l, 
 
 -1+V5 V10 + 2V5 — 
 
 y=-- 4 ± r -V-i, 
 
 -1- -/5 . V10-2 V5 — - 
 y= -. ± 7l V-l. 
 
378 ALGEBRA. 
 
 The equation if — 1=0 will lead to the ematicn ;'-(-: : — "2: — 1=0, and 
 the equation if — 1 = to the equation ; 4 4-; 3 — 3; : — 2z-f-l=0. 
 
 The equations i/ 5 -|-l=0, i/ 7 -|-l = 0, t/°+ 1=0 have, except as regards the 
 signs, the same roots as if their second terms had been — 1. 
 
 Let us examine the even degrees. The equations f — 1 = 0, if — 1 = 0, 
 y 10 — 1=0 do not offer any difficulty, because each of them can be decom 
 posed into two others whose roots are known. 
 
 Taking -4-1 instead of — 1, the analogous equations are 
 
 2/6+1 = 0, whence y=\ V — li 
 2/ 8 -l~ 1 =0, whence 2/=\/ V — 1, 
 
 , y°+ 1 = 0, whence y=yj fy —1. 
 
 But we know the values of fy — 1, \f — 1, %J — \ ; we have, then, only to 
 extract the square roots by the processes in Art. 104. But it will be simpler 
 to treat these equations as reciprocal ; for tho transformed equations in z, on 
 which they depend, have roots which aro real, and are veiy easy to resolve. 
 
 We add some propositions upon binomial equations, preparatory to giving a 
 general method for solving those of all degrees. 
 
 PROPOSITION' I. 
 
 304. If a be one of the imaginary roots of tho equation .r n — 1=0, then any 
 power of a will be also a root. 
 
 For, since a is one root of the equation .r n — 1=0, therefore a n = l, and, con- 
 sequently, 
 
 a"-" = l, a 3n =l, a 4n = l, &c, also a-" = l, cr^ssl, a~ Sn = l, &C, 
 the values « 
 
 a, a 2 , a 3 . . . ., a -1 , a~°, a~ 3 , . . . ., 
 thus satisfying the conditions of the equation, are roots of it. 
 
 Corollary 1. — It hence appears that tho roots of tho equation i n — 1=0 may 
 be represented under an infinite variety of forms, each term in the following 
 series being a root, viz. : 
 
 a- 3 , a-=, a~\ 1, a, a 5 , a 3 , a"-', a n , c"+ l , a 8n , a= n +', 
 
 in which series, however, there can not be more than n quantities essentially 
 different, otherwise tho equation would have more than « roots. 
 
 PROPOSITION II. 
 
 305. If a be one of tho imaginary roots of tho equation x n -f-l=0, then any 
 odd power of o will be also a root. 
 
 For, since a is ono root of the equation .<' = — 1. therefore a n = — 1 ; and. 
 
 since every odd power, whether positive or negative, of — 1 is also — 1. 
 
 therefore, 
 
 n 3 "=— 1, a'">= — 1, «•"=_ 1, &C, 
 
 also 
 
 a~ 3n = — 1, a-' r "'=— 1, a- 7n =— 1, \'c. ; 
 
 so that tho quantities 
 
 a, a 3 , a-' , «-', a~*, <i '• , 
 
 arc runts of the equation. These roots, therefore, assume an infinite variety 
 of forms, although there can nol be more than n essentially different. 
 
BINOMIAL EQUATIONS. 379 
 
 PROPOSITION III. 
 
 300. To determino the roots of the equation x n — 1 = 0, wh&n n is tho square 
 •f a prime number p. 
 
 Put xp=z, then xp — 2=0, and z p — 1 = 0, and let the roots of this last equa- 
 tion be 1, (3, /3 2 , /? 3 , .... /3p _1 ; then, by substitution, 
 
 'af— 1 =0, 
 
 xP - Z =\ X? -P=°> 
 
 ] 2-P — /? 2 = 0, 
 
 &c. &c. 
 Hence the pp values of x, in these p equations, will evidently be all difTerent, 
 and will be the roots of the equation x pp — 1 = 0. 
 
 To determine these roots, it will bo sufficient to advert to Art. 300, which 
 proves that the roots of xp — (3=0 are equal to the roots of x? — 1=0 multi- 
 plied by y[i ; and, in a similar manner, the roots of xp — |3 2 =0 are equal to the 
 roots of xp — 1 = 0, multiplied by tyj3 2 , &c. ; therefore, we immediately con- 
 clude that tho roots of 
 
 xp-1 =0arel, /?, (3*, (3 3 (3^ ) 
 
 xp _/? =0 ycpypipyii /3P- 1 vp [ - u n 10 " ™°™ ot 
 
 tp -13* = V t 3\ [3 y s 3\ (3' Vi 32 • • • • Z 3 "" 1 VP > * 
 
 &c. &c. &c. 
 
 For example, let it bo required to find the roots of x 9 — 1 = 0. 
 The roots of x 3 — 1=0 are 
 
 -1+ V-3 -l-V-3 
 7 -2 ' 2 
 
 hence the roots of x' J — 1=0 are 
 
 . -l+V-3 -l-V-3 
 
 l+y_3 -l+V -3, , -l+V-3 
 v « > o V o i 
 
 _l_V-3 3 -1+^-3 3/ _l-V 
 
 v — v — . v- 
 
 -1+ V-3 -1--/-3 - l-y-3 -l-V-3 
 o V o ' o V o 
 
 iv <6 <6 iv 
 
 The foregoing propositions have been devoted chiefly to an examination of 
 the properties and relations of these roots, and not to the actual exhibition of 
 their values, although, as in the proposition above, one or two examples of the 
 solution have been given to illustrate the reasoning. To obtain the imaginary 
 roots, however, in their simplest form, that is, in the form a+6 V — 1> an d 
 for all values of the exponent, requires the aid of a theorem, borrowed from 
 the science of Trigonometry. 
 
 307. The theorem to which we refer is the well-known formula of De 
 Moivre, given in most books on Analytical Trigonometry. 
 
 (cos a+ sin a . V — l) n = cos na+ sin na . V — 1 ; 
 which, if the arc 2lcn (tt being a semi-circumference, and k any integer) be 
 substituted for na, becomes 
 
380 ALGEBRA. 
 
 2&t , 2kn . , . 
 
 (cos ± sin . V — 1)"= cos 2/cTTztz sin 2/c~ . y — 1 ; 
 
 v n n y > 
 
 that is, since 
 
 cos 2ktt=1, and sin 2/c~=0, 
 
 nJcn 2/,'t 
 
 (cos - — ± sin . V — 1)" = 1 ; 
 
 so that the expression 
 
 2kn . licit 
 
 cos i sin . y/ — 1, 
 
 n n 
 
 couiprehends in it all the n roots of unity, or all the particular values c£ x 
 which satisfy the equation x n — 1 = 0. 
 
 Although, in this general expression, the value of k is quite arbitrary, yet 
 assume it what we will, the expression can never furnish more than n differ- 
 ent values. These different values will arise from the several substitutions of 
 
 up to the number — - — , inclusively, if n is odd, and up to -, if n is even ; and 
 
 for substitutions beyond these limits the preceding results will recur. To 
 prove this, let us actually substitute as proposed ; we shall thus have the fol 
 lowing series of results, viz. : 
 
 for k = .... .r= cos i sin . ■/ — 1=1 
 
 2k . 2w . 
 
 k = l .... x= cos — ± sin — . V — 1 
 
 n n 
 
 4t 4~ 
 
 A:=2 x= cos — ± sin — . V — 1 
 
 n n 
 
 Qk . 6k 
 
 £=3 .... .t= cos — rt sin — . -J — I 
 
 n n 
 
 „_1 (n-l)T , . (n-l) ff — 
 fc = — -— . . xss cos ± sm . v — 1- 
 
 2 n n 
 
 Each of these expressions, except the first, involves two distinct values : g 
 
 that, omitting the value given by k =0, there are n — 1 values, and, consequent 
 
 ly, altogether, there are n values ; and that they are all different is plain, be 
 
 cause the area 
 
 2k 4k 6k (n — 1)t 
 
 u, , , , . • • • ~~ , 
 
 7i n n n 
 
 being all different, and loss than it, have all different i Tin arcs which 
 
 would arise from continuing the substitutions aro 
 
 («4-1)tt (;i+3)x (n+3 )ir 
 n ' n ' n 
 or, which aro the same, 
 
 -, &c. ; 
 
 ( W -1)t ( w -3) y ( w -5)x 
 
 r— , 2tt , 2ir , cVc, 
 
 7i 71 n 
 
 and the sines and cosines of theso are respectively tin- same us the sines ind 
 of 1 1 io arcs 
 
BINOMIAL EQUATIONS. 381 
 
 (n — l)n (n — 3)?r (n — b)n 
 
 n n n 
 
 which have already occurred.* 
 
 w 
 If n is an even number, the final substitution for k must be - instead of 
 
 * i 
 
 ■ , as above ; and, therefore, the final pair of conjugate values for x will be 
 
 x= cos ffi sin 7r . -/ — 1 = — 1, 
 which values of x differ from all the other values, because in them no arc oc- 
 curs so great as n. 
 
 The arcs which would arise from continuing the substitutions beyond 
 
 j n 
 kz=i- are 
 
 (n-\-2)n (n+4)?r (n+G)ir 
 
 — , &c. ; 
 
 n n n 
 
 or, which are the same, 
 
 (n — 2W (n — 4)tt (n — 6)rr 
 2tz—- '-, 2n— -, 2?r— -, &c, 
 
 74 74 71 
 
 and the sines and cosines of these are respectively the same as the sines an<. 
 cosines of the arcs 
 
 (n — 2)k (n — 4)xr (n — 6)k 
 
 , , , (.VC.j 
 
 n n , n 
 
 which have already occurred.* 
 
 It is easy to see that in every pair of conjugate roots, each is the reciprocal 
 of the other. In fact, whatever be k, 
 
 2kn 2Z.-7T ; , 2&7T 27ctt . 
 
 (cos 4- sm .v — 1) (cos — sin . v — 1) = 
 
 v n ' 7i ' v 7i n ' 
 
 2kn 2kir 
 cos 2 4- sin 2 =1, 
 
 J 71 ' 71 
 
 which shows that the two factors in the first member are of the form a, -. 
 
 a 
 
 We have proved (Art. 304) that every power of an imaginary root of the 
 
 binomial equation is also a root ; but, unless n be a prime number, wo could 
 
 not infer that all tho roots would ever be produced by involving any one of 
 
 them. Such would not, indeed, be the case. There is always, however, 
 
 one among the imaginary roots of which the involution will furnish all the 
 
 others; it is the first imaginary root, or that due to the substitution k=l, in 
 
 the foregoing series of values ; for, by De Moivre's formula, the powers of this 
 
 produce all the others, thus : 
 
 2ir 2t 47T 47r 
 
 (cos — -j- sin — . -J — 1;-= cos — 4- sin — -' — 1 
 7i ' n ' n ' n 
 
 6tt 6t 
 
 (cos — + sin — . V — iy= cos — + sin — . V — 1 
 
 v 7i ' n ' n ' n 
 
 2tt . 2tt , !!zi n— 1 n— 1 
 
 (cos — 4- sm — . V — 1) - = cos 71-4- sin . v — 1. 
 
 * Tlie signs of the sines will, however, be different; but the only effect of this difference 
 J evidently to furnish each pair of conjugate roots in an inverse order. 
 
o82 ALGEBRA. 
 
 These powers of the first imaginary root, which we may cal a, th.is fur- 
 nish one half of the entire number of imaginary roots, and the reciprocals of 
 these being the other half, all of them are determined from the first ; the 
 imaginary roots are, therefore, 
 
 n— 1 
 
 o, 
 
 a 2 , a 3 , 
 
 1 
 
 1 1 
 
 1 
 
 a 
 
 a»' a 3 ' 
 
 When n is even, the last power will be 
 
 Mn Mrc 
 
 (cos — 4- 6in — . \/ — lY J =cos tt+ sin n . J — 1 ; 
 and the imaginary roots are, therefore, 
 
 n 
 
 a, a s , a 3 , .... a* 
 111 1 
 
 a' ^ *5 1 
 
 o? 
 
 308. By the general formula (Art. 307), we are enabled to determine all tLo 
 
 roots of the equation 
 
 x n + 1=0 ; 
 
 for, since 
 
 cos-(2&+l)7r=— 1, and sin (2&4-l)7r=0, 
 
 that formula gives 
 
 2/fc+l , 2fc+l . 
 
 (cos tt± sin n . \/ — l) n = 
 
 n n ' 
 
 cos (2A:+l)7r± sin (2fc-fl)jr -V— las— 1; 
 hence the n values of x are all comprised in the general expression 
 
 2&+1 ' . 2&-fl 
 
 x= cos ttJL sin 7r . -/ — 1 ; 
 
 ft 71 
 
 which, by putting for k the values 0, 1, 2, 3, &c, in succession, furnishes the 
 following series of separate values, viz. : 
 
 n . 7T 
 
 for&=0 .... .t= cos-i sin - . J — 1 
 
 n n 
 
 3ir" . 3rr . 
 
 «=1 .... t= cos — ± sin — . / — 1 
 
 n n 
 
 5t . 5t 
 
 &=2 .... x= cos — i sin — . -J — 1 
 
 n n v 
 
 , «— 1 . • ,- — 
 
 Jc= — - — . . x= cos n± sin t . -/ — 1 = — 1 ; 
 
 or, when n is even, 
 
 71 — 2 / ~\ ir 
 
 jfcsss — ; — . . . .T= cos 1 7r — -J - sin (rr — - . -J — 1). 
 
 Now that the foregoing system of n roots aro all different is obvious, siiK* 
 
 7T 3k 5tt tjt tt 
 
 - , — , — .... — , or n— -, 
 n n n n n 
 
 are all different arcs, of which the greatest does not exceed a serai-circum 
 
BINOMIAL EQUATIONS. 383 
 
 ference. If the preceding series be extended, it will be easy to prove, after 
 
 what has been done in Art. 307, that the values formerly obtained will recur. 
 
 As in the former case of the general problem, so here, each root may be 
 
 derived from the first pair of the series ; thus, denoting the first root, cos 
 
 7T W 1 
 
 -rt sin - . -/ — 1, by a or -, according as the upper or lower sign is taken, 
 
 TV Tt Q> 
 
 wo evidently have, for the preceding series, the following equivalent expres 
 sions, viz. : 
 
 a, a 3 , a 5 , .... a" -\ 
 
 111 1 > when n is odd, 
 
 a' a?' tf a" ) 
 
 and 
 
 a, a 3 , a 5 , a" -1 \ 
 
 111 1 ( when n is even, 
 
 a' a 3 ' ~afi a^ ) 
 
 For further researches on tho theory of binomial equations, the student may 
 consult Lagrange's Traite de la Resolution des Equations Numeriques, Note 
 14 ; Legendre's Theorio des Nombres, Part V. ; the Disquisitiones Arith- 
 meticae of Gauss ; Barlow's Theory of Numbers ; and Ivory's article on Equa 
 tions, in the Encyclopaedia Britannica. 
 
 309. We have already frequently had occasion to notice multiple values of 
 radicals, without fixing the precise number which might exist, except for rad 
 icals of the second degree. It is time to introduce the following proposition : 
 
 Every radical has as many values as there are units in its index, and has 
 no more ; in other words, every quantity has as many roots of a given degree 
 as there are units in the index of that degree. 
 
 If the given radical be represented by the general form \/A, this radical 
 designates evidently all the quantities, real or imaginary, which, raised to the 
 power m, reproduce A ; consequently they are merely the values of x in the 
 equation ar m =A. But we know, from the general theory of equations, that 
 every equation of tho m tb degree has m values of the unknoAvn quantity, which 
 will each satisfy it ; hence the proposition is proved. 
 
 This will serve to explain some paradoxes. Let there be the expression 
 May/ — 1. By reducing the second radical to the index 4, it becomes 
 y( — Yf, and the given expression reduces to \Ja,9. result which might be 
 supposed absurd, because, a being positive, the result represents a real quan- 
 tity, while the proposed expi'ession appears to be imaginary. 
 
 There is here a confusion of ideas. If in the expression tya V — 1 tn o 
 radical is an arithmetical determination, it is true that this expression is 
 imaginary ; but if V a be taken in all its generality, and we represent it by a' 
 multiplied by the four roots of unity, or 
 
 a', —a', a' yf — I, —a' y — 1, 
 
 we perceive that some of these values of V«, multiplied by *J — 1, cause this 
 imaginary factor to disappear, and the proposed expression becomes real. 
 
 I shall terminate this article by the explanation of a paradox which presents 
 itself in the employment of fractional exponents. Let there be the expres- 
 
 2 3 L 
 
 sion a f . If the fraction £ be simplified, the expression a* becomes a-. Then, 
 
 in repassing to tho radicals, we have t/a-= ^a. This equality, however is 
 
384 
 
 ALGEBRA. 
 
 not wholly true, because the first member has four values, and the second 
 but two. 
 
 The difficulty may be presented in a general manner by placing 
 
 mp m 
 
 and in concluding from thenco that 
 
 To discover the cause of this error, we must remember that the fractional 
 exponent is but a convention, by means of which we express in another way 
 that the root of a certain power is to be extracted, and, therefore, this expo- 
 nent must not be regarded in the light of an ordinary fraction. 
 
 TILE DETERMINATION OF THE IMAGINARY ROOTS OF EQUATIONS. 
 
 310. In what relates to the limits of roots at Art. 283 and following, real roots 
 only were in view. We shall show here how the limits may be obtained 
 for the moduli of all roots, whether real or imaginary. Let us consider the 
 equation 
 
 a: ra +Px m - 1 + Qx ro - 3 ...=0 (1) 
 
 in which P, Q . . . may be real or imaginary. In order that a value of x may 
 be a root, it is necessary that, after having substituted it in the result, the 
 modulus should be zero. 
 
 Call v the modulus of x, and p, q, . . . those of the coefficients P, Q Ac- 
 cording to Art. 239, those of the terms of the equation will be v m , pv m ~\ 
 
 qv m ~\ . . ., and that of the part Pa^+Q^-H can not surpass the sum 
 
 „ r m-i_i_o V m-2 t . t Then, if we choose for v a value 1 such that we have 
 
 v ™—p V ™-i—qv m -^ =0, or >0 . . . . (2) 
 
 we are sure, by virtue of the article just cited, that the modulus of the first 
 member of the equation (1) will not be less than the above difference ; and that 
 from this point the modulus will not be zero, or, what is the same thing, the 
 value substituted in place of x will not be a root of the equation. Every value 
 of v above "K will render this difference greater ; then 1 is a superior bmit of 
 
 the moduli. 
 
 The quantity "k will be always easy to determine, because it will be sufficient 
 to substitute in the difference (2) in place of v, increasing positive values until 
 this difference becomes positive. If tho coefficients P, Q . . . are real, the 
 moduli p,q,--- will be these coefficients themselves, but taken positively ; and 
 if we designate tho greatest of these values by N, we can take at once for die 
 superior limit A=N-f-L 
 
 To have an inferior limit, wo make ar=-, determine in the transformed in y 
 
 the superior limit of the moduli of the roots, and finally divide unity by this 
 limit. 
 
 311. It has already been proved that imagina ry ro ots always enter into 
 equations in conjugate pairs of the form a±i3i/—l. And this previous 
 knowledge of tho form which ovory root musi take suggests a method for the 
 actual determination of the proper numerical values for a ami >' in any proposed 
 case. The method is as follows : 
 
 Let .r" + A -,.r" l H Aj + N=0 
 
IMAGINARY ROOTS OP EQUATIONS. 38* 
 
 be an equation containing imaginary roots; then, by substituting n-f-JV — * 
 for x, we have 
 
 (a+3V-l)" + An- 1 («+;W-ir 1 +..A(a+/3V-l) + N = 0; 
 or, by developing the terms by the binomial theorem, and collecting the real 
 and imaginary quantities separately, we have the form 
 
 M + NV-1=0, 
 an equation which can not exist except under the conditions 
 M = 0, N = (1) 
 
 From these^wo equations, therefore, in which M, N contain only the quan- 
 tities a, /•, combined with the given coefficients, all the systems of values of a 
 and j3 may bo determined ; and these, substituted in the expression a-\-fi -\/ — I, 
 will make known all the imaginary roots of the proposed equation ; those that 
 are real corresponding to fS=0. 
 
 It is obvious from the theory of elimination as developed at page 1-37, and 
 from the method of numerical solution explained in Art. 255, that the labor of 
 deducing from this pair of equations the final equation involving only one of the 
 unknowns a, /?, and of afterward solving the equation for that unknown, will 
 in general be veiy laborious for equations above the third degree. Lagrange, 
 by combining with the principle of this solution the method of the squares of 
 the differences explained at Art. 278, avoids both the elimination and subse- 
 quent solution here spoken of. It is easy to see how this may be brought 
 about if we have any independent means of determining one of the unknowns 
 0: for the adoption of these means would enable us to dispense with the elimi- 
 nation ; and as the substitution of the valuo of (3 in both of the equations (1) 
 would convert those equations into two simultaneous equations involving but 
 one unknown quantity, their first members would necessarily have a common 
 factor of the first degree in a, which, equated to zero, would furnish for a the 
 proper value to accompany (3 ; and thus, instead of solving the final equation 
 referred to, we should only have to find the common measure between the 
 two polynomials M, N containing the unknown quantity a. 
 
 Now corresponding to every pair of imaginary roots a-\-(3 -<J — 1, a — {3 \/ — 1, 
 there necessarily exists, in the equation of the squares of the differences, a 
 real negative root — 4/3*; so that if all the negative roots of the latter equation 
 be found, the quantity — 4/3 3 must appear among them ; from which the value 
 of /3 would bo immediately obtained, and thence, by aid of the common meas- 
 ure as just explained, tho corresponding value of a. 
 
 But the equation of the squares of tho differences may have a greater num- 
 ber of negative roots than there are pairs of imaginary roots in the proposed : 
 which, however, can not happen except two non-conjugate imaginary roots have 
 equal real parts, or except a real root be equal to the real part of an imaginary * 
 root. Lagrange discusses these peculiarities, and establishes the exactness 
 and generality of tho principle in question, as follows : 
 
 When the real parts, a, y, &c, of the imaginaries 
 
 n-r-jS -v/ — 1' a — PV— 1 
 
 &c. 6cc. 
 
 are unequal, as well when compared with one another as when compared with 
 the real roots a, b, c, &c, it is evident that the equation of the squares of tho 
 
 Bb 
 
386 ALGEBRA. 
 
 differences can not have any other negative roots than th:6e furnished by the 
 several pairs of conjugate imaginary roots, and which are . 
 
 _40», — 4cP, &c. 
 All the other roots, not arising from the differences furnished by the real 
 roots, a, b, c, &c, will evidently be imaginary ; those between the real and 
 imaginary roots supplying the forms 
 
 (a-b + pV-l)\ {a—h—pV—iy 
 &c. &c. 
 
 and those between the non-conjugate roots tho forms 
 
 \(a- 7 ) + ^-6)V-l L \"-, \(a- } )-(3-6)V-l ' 
 U*-7) + iP+8)V-l\*, \(a-y)-(3+6)V-l\* 
 so that in this case every negative root in the auxiliary equation will indicate a 
 pair of imaginary roots in the proposed, and will, moreover, supply tho value 
 of the imaginary part. But if it happen that among tho quantities a, y, occ, 
 thero be found any equal among themselves, or equal to any of the quantities 
 a, b, r, &c, then the auxiliary equation will necessarily have negative roots, 
 corresponding to which there can be no imaginary pair in the proposed equa- 
 tion. 
 
 For let a=a, then the two imaginary roots (a— a-\-(iy/ — l) 3 , (a — a — /3 
 -/ — 1) : will become — /J 2 and — /?», and, consequently, real and negative ; so 
 that if tho proposed equation contain only two imaginary roots, a-\-(3 y/ — 1 and 
 a — j3 \/ — 1, then, in the case of a=a, the equation of the squares of the differ- 
 ences will contain, besides the real negative root — 4/3°, the two — j9», — CP, 
 both negative and equal. 
 
 We thus see that when the equation of the squares of the differences has 
 three negative roots, of which two are equal to one another, the proposed may 
 have either three pairs of imaginary roots, or but a single pair. 
 
 If the proposed contains four imaginary roots, a+/3-\/ — 1, a — 3 -y/ — 1, 
 y+^V — 1, y — (5\/ — 1, then tho equation of the squares of the differences 
 must contain tho two negative roots — 4 -'- and — 4cP; if a=<7, it must also 
 contain the two equal negative roots — /J 2 , — /J 2 ; and if, moreover, y=b, it 
 must contain, in addition to these, the negative pair — 6 2 , — <5-; and lastly, if 
 assy, the four imaginary roots 
 
 j(a-y) + (/?-H)V-lp, f(a->)-(^+<5)V-lj* 
 will be converted into the two negath o pairs 
 
 -(/i—5) 2 , -(/?-d)»; -(0+«f)», -(,-3+<5)-\ 
 Hence we may deduce tho following conclusions, \i/.. : 
 (1) When all the real negative roots of tho equation of tho squares of the 
 differences are unequal, then tho proposed will necessarily have so many pairs 
 of imaginary roots. 
 
 If in this case wo call any one of these negative roots — w, wo ahaD have 
 
 ■\f w 
 fl=— — ; and if this value be substituted fur >' in the two equations (1). and the 
 
 operation for tin- common measure of their 6rsl members be carried on till we 
 krnve at a remainder of tho first degree in a, tho proper value of a will be ob 
 
IMAGINARY ROOTS OF EQUATIONS. 3£7 
 
 ♦ained by equating this remainder to zero. Thus, each negative root, — u\ 
 will furnish two conjugate imaginary roots, a-\-(3 V — 1, and o — (3 y/ — 1. 
 
 (2) If'among the negative roots of the equation of the squares of the differences 
 equal roots arc found, then each unequal root, if any such occur, will, as in 
 the preceding case, always furnish a pair of imaginary roots. Each pair of 
 equal roots may, however, give either two pairs of imaginary roots or no im- 
 aginary roots, so that two equal roots will give either four imaginary rpots or 
 none ; three equal roots will give either six imaginary roots or two ; four equal 
 roots will give either eight imaginary roots, or four, or none ; and so on. 
 
 Suppose two of the negative roots, — w, — w, are equal ; then putting, as 
 
 ■\/ w 
 above, /?=——, we shall substitute this value of /3 in the two polynomials (1), 
 
 and shall carry on the process for the common measure between these poly 
 nomials till we arrive at a remainder of the second degree in a ; since the poly 
 nomials must have a common divisor of the second degree in a, seeing that the 
 equations (1) must have two roots in common, on account of the double value 
 of/?. 
 
 Equating, then, this quadratic remainder to zero, we shall be furnished witn 
 two values for a : these may be either both real or both imaginary. In the 
 former case call the two values a' and a" ; we shall then have the four imagin- 
 ary roots 
 
 a'+pV—l, a'~ PV — li o"+j3V— 1, a"— /?•/— 1. 
 
 In the second case, the values of a being imaginaiy, contrary to the condi- 
 tions by which the fundamental equations (1) are governed, we infer that to 
 the equal negative roots — w, — w, there can not correspond any imaginary 
 roots in the proposed equation. 
 
 If the equation of tho squares of the differences have three equal negative 
 
 roots, — w, — w, — w, then, putting, as before, /?=——, we should operate on 
 
 the polynomials (1), for the common measure, till we reach a remainder of 
 the third dogreo in a; this remainder, equated to zero, will furnish three values 
 of a, which will either be all real, or one real and two imaginary. In the first 
 case six imaginary roots will be implied : in the second only two ; the imagin- 
 ary .values of a being always rejected, as not coming within the conditions im- 
 plied in (1). 
 
 It follows from the above, and from what has been established in Art. 259, 
 that there are at least as many variations of sign in the equation of the squares 
 of differences as there are combinations of two real roots in the proposed 
 equation. Also, it must have at least as many permanences of sign as there 
 are pairs of conjugate imaginary roots in the proposed equation ; or, in other 
 words, it can not have a less number of permanences of sign than half the num- 
 ber of imaginaiy roots in the proposed equation. 
 
 Hence we may infer, that if the equation of the squares of the differences 
 have its terms alternately positive and negative, there can be no imaginary 
 root in the proposed equation. 
 
 The foregoing principles are theoretically correct ; but the practical appli- 
 cation of them, beyond equations of the third and fourth degrees, is too labo- 
 rious for them to become available in actual computation. We give the follow 
 ing illustration of them from Lagrange- 
 
388 ALGEBRA. 
 
 312. To determine the imaginary roots of the equation 
 
 r< — 2.r— 5=0. 
 Computing the equation of the squares of the differences from Hie genera. 
 formula for the third degree at Art. 279, viz. 
 
 2»-J-6pz 9 +9p B z+4p»+27g*=0, 
 in which p = — 2 and q = — 5, we have 
 
 z»— 12z»+36z+643=0. 
 In order to determine the negative roots of this equation, change the alternate 
 signs, or put ; = — M>, and then change all the signs, converting the equation 
 into 
 
 vfi+l2il?+3&W— 643=0, 
 and seek the positive root, which is found by trial to lie between 5 and 6. 
 Adopting Lagrange's development. Art. 297, this root proves to be 
 
 *>= 5 +6 + i 1 
 
 6 + , &c, 
 
 from which we get the converging fractions (see Continued Fractions) 
 
 31 160 991 
 
 5 ' ~6~' "3P 192' ' 
 
 ■\/w 
 Knowing thus an approximate value of ic, wo know /3= — — -. 
 
 In order now to get the equations (1), p. 385, substitute a-\-{3^ — 1 for xin 
 the proposed equation, and form two equations, one with the real terms of 
 the result, the other with the imaginary terms ; we shall thus have the equa- 
 tions (1) referred to, viz., 
 
 o 3 — (3^+2)0—5=0 
 3a2_/y>_2 = 0, 
 in which (i is known. 
 
 Seeking now the greatest common measure of the first members of these 
 equations, stopping the operation at the remainder of the first degree in a, and 
 equating that remainder to zero, we have 
 
 15 
 C =-8^H' 
 and thus both a and /? are determined in approximate numbers. 
 
 313. There is another method of proceeding for the determination of im- 
 aginary roots, somewhat different from the preceding, being independent of 
 the equation of the squares of tho differences. It is suggosted from the fol- 
 lowing considerations : 
 
 Since tho quadratic, involving a pair of imaginary conjugate roots, is always 
 nf the form 
 
 — 2ax+a 2 4-/^ = 0, 
 every equation into which such roots enter must always be accurately divisible 
 liy a quadratic divisor of this form; that is, the proper values of a and |9 are 
 s:icli that the remainder of the firsl degree in r, resulting from the dh 
 must 'oo zero. This furnishes a condition from which thoso proper values of 
 o ami /? may be determined ; the condition, namely, thai the remainder spoken 
 of, A.r — B, must be equal to zero, independent of particular values of .r; and 
 thii implies the twofold condition 
 
IMAGINARY ROOTS OF EQU VTIONS. 389 
 
 A = 0, B = 0, 
 from which a and (i, of which A and B are functions, may bo determined. 
 As an example, let the equation proposed bo 
 
 a;4_j_ 4 x s_j_ e 2 .2_|_ 4^4. 5 _ o. 
 
 Dividing the first member by 
 
 tf—Zax+cP+p, 
 
 we have for quotient 
 
 i-s +. (4 4. 2a)x-\- 6 + 8a + 3a' — (P, 
 and for the remainder of the first degree in x 
 
 (4+12a+12a s 4-4o 3 — 4a/3»— 4j3*)x— 
 
 (o«+^)(6 + 8o+3oS— j3 2 ) + 5, 
 which, hoing equal to zero whatever be the value of x, furnishes the two equa- 
 tions 
 
 4+12a-fl2a B +4o3— 4flj8»— 4j9 s =0 
 
 (o'+j8s)(6+8o+3o=— /P) + 5=0. 
 
 From the first of these we get 
 
 /?*=(! + ar- 
 and this, substituted in the second, gives 
 
 4a-<-f] 6a" -{-24a 2 -f 16a=0, 
 two roots of which are and —2 ; the other two are imaginary, and must, 
 consequently, he rejected as contrary to tho hypothesis as to the form of th* 
 indoterminate quadratic divisor. 
 
 The two real values of a, substituted in the expression above for /3 2 , give 
 fora= 0, (T- = V .-. /3=+l 
 
 a=-2, /?*=(-l)'- .-. /3=-l 
 
 and, consequently, the component factors of the original quadratic divisor, viz.. 
 the factors 
 
 x—a—Pi/—l, x— a+/3-/^-li 
 furnish these two pairs of imaginary roots, viz., 
 
 x= V— i, •?=— V— i) 
 
 and 
 
 x=z— 2— -/ — 1, x=— 2+ V — 1. 
 This method, like that before given, is impracticable beyond very narrow 
 limits, because of the high degree to which the final equation in a usually 
 rises. And it is further to be observed of both, and, indeed, of all methods 
 for determining imaginary roots by aid of the real roots of certain numerical 
 equations, that whenever, as is usual, these real roots are obtained only ap- 
 proximately, our results may, under peculiar circumstances, be erroneous. 
 For instance, in the, two methods just explained we have two equations, 
 /(a)=0, F(/3)=0, where the coefficients of a in the first are functions of ; . 
 and the coefficients of (3 in the second functions of a ; hence, whichever of 
 these symbols be computed approximately, in order to furnish determinate 
 values for th# coefficients of the other, these coefficients must vary slightly 
 from the true coefficients ; and v consequently, under this slight variation of the 
 coefficients, real roots may become converted into imaginary, and imaginary 
 into real. 
 
390 ALGEBRA. 
 
 The terms imaginary and impossible have been thought objectionable when 
 applied to the roots of equations, inasmuch as definite algebraic expressions 
 are always possible for these roots. 
 
 \ specimen of a strictly impossible equation would be the following: 
 
 2z— 5+ -^—7=0, 
 
 when plus before the sign y/ implies the positive root ^x 1 — 7. No ex- 
 pression, either real or imaginary, can satisfy the condition or represent a root 
 of this irrational equation. 
 
 The terms imaginary and impossible, when used, should be understood 
 rather as applying to the solutions of the problem from which the equation is 
 derived than to the expressions for the roots. The number of solutions which 
 the problem admits will ordinarily be expressed by the degree of the equa- 
 tion, but certain suppositions affecting the values or signs of the coefficients 
 may cause some of these solutions to become absurd or impossible, and these 
 will be indicated by the form a-\-b V — 1 for the roots,in which b is not zero. 
 
 THEORY OF VANISHING FRACTIONS. 
 314. From the principles established in (Art. 253), we readily derive the 
 following consequences, viz. : 
 Since 
 
 f(x) = {x—a l )(x—a 2 )(x—a 3 ){x—a,) 
 
 and 
 
 /,(x) = (.i-— a ){x— a. 2 ){x— a 3 ) -\-{r—a i )(r — a 2 ){x—a i ) . . . +, <fcc, 
 
 it follows that 
 
 m± . . . . _i_ + ^_ + _i_+-j_ ....<„ 
 
 f(x) x— a 4 ' x— a 3 ' x— a 2 ' x — a x ' 
 
 In like manner, for any other equation F(x) = 0, we have 
 
 F(x) x—b^x—b^x—b^x—bi . v ' 
 
 Suppose the two equations 
 
 /(x)=0, F(x) = 0, 
 have a root in common, viz., a^b^ then, dividing (1) by (2), we have 
 
 1 _J_ _1_ 1 
 
 /,(.;•) F(x) x— ^~*~.t— g^~x— a^x— a l 
 
 F,(x) ' f(x)~ 1 , 1 
 
 -+7-T+ 
 
 x — fc 4 ' x — /> ; ,'x — 1>>~ x — bx 
 Hence, multiplying numerator and denominator of the second member by 
 r— a u and then substituting for x its value x=rc/,. we have 
 
 /i(«i) F(».) 
 
 /.(a.) /(«.), 
 
 ' ' Fi(a,)~F( (ll ) ' 
 
 from which wo learn, that if any two equations have a common root a, and 
 
 their derived equations be taken, the ratio <>!' tin original polynomials, when '.' 
 
 us put for r, will l>o equal to the ratio of the derived polynomials when ,i it put 
 
 for t. 
 
 This property furnishes us with a ready method of determining the value 
 
THEORY OF VANISHING FRACTIONS. 391 
 
 fix) 
 
 of a fraction, such as „. v , when both numerator and denom nator vanish for 
 F(.r) 
 
 a particular value of x, as, for instance, for x=a. For we shall merely have 
 
 to replace the polynomials in numerator and denominator by their derived 
 
 polynomials, and then make the substitution of a for x. If, however, the 
 
 terms of the new fraction should also vanish for this value of x, we must treat 
 
 it as we did the original, and so on, till we arrive at a fraction of which the 
 
 terms do not vanish for the proposed value of .r. The following examples wfl 1 
 
 nufficiently illustrate this method : 
 
 (1) Required the value of 
 
 x 2 — a* 
 x — a ' 
 vhen x=a. 
 
 M a ) 2 « 
 Here „ . , =-— =2a, the required value 
 F x {a) 1 7 
 
 (2) Required the value of 
 
 nx^ 1 — (n-\-l)x a +l 
 
 (T~rf '* 
 
 when .t=1. 
 
 fi(x) n(n-\-l)x n — n(n+l)x n ~ l 
 
 ¥Jx) = -2(1 —a:) • 
 
 This still becomes - for x=l, 
 
 fi.(x) n*(n+l)x*- 1 — n(»+l)(n— l)s—« 
 F^j- 2 
 
 / 2 (1) n(n+l) 
 
 the value sought. 
 
 F.(l)' 
 
 o 
 
 (3) Required the value of 
 
 when .r=l. 
 
 (4) Required the value of 
 
 1 — x a 
 
 Ml) -n 
 F^l)--!-"' 
 
 b(a — \fax) 
 
 a — x ' 
 for x=a. 
 
 We may here put \/x=y, and thus change the fraction into 
 
 Ha— a 2 y) 
 a—y* 
 
 My) -bJ f^ah h- 
 
 WhA = ~^r •'• — ~=o' the value re q mred - 
 
 * 
 
 This is the expression for the sum of n terms of the series 
 
392 ALGEBKA. 
 
 (5) Required the value of 
 
 m m 
 
 f(y) (a+g)°— (g+y)~ 
 
 F(y)~" x-y 
 
 when x--=y. 
 
 Put a-\-y=zz n , then the fraction is chauged into 
 
 in 
 
 (a-f-.r) n — : m 
 x— z"-\-a 
 
 m 
 
 /,(;) — mz m ~ l m z m m {a-\-y)" 
 Fi(z) — — n:"- 1 ~n ' z a ~ n ' a-\-y ' 
 and, therefore, the value, when x=y, is 
 
 m 
 
 m (a-\-x) a 
 n ' a-\-x 
 
 ELIMINATION. 
 
 RESOLUTION OF EQUATIONS CONT A I NINO TWO OR MORE UNKNOWN 
 QUANTITIES OP ANY DEGREE WHATEVER. 
 
 315. We have already indicated, at p. 157, the possibility of eliminating one 
 of two unknown quantities from two equations by the method of the common 
 divisor. The general theory of equations which has since been unfolded will 
 afford the means of giving a more full development to this subject. 
 
 The two given equations may be thus expressed : 
 
 .F(.r, y)=0,f(x,y)=0 (1) 
 
 They are said to be compatible if they have common values of x and y. This 
 is the case with two equations derived from the same problem, the conditions 
 of which, for the determination of the required quantities, are expressed by 
 the two given equations. 
 
 Suppose now that one of the common values of y were known, and substi- 
 tuted for y in the two equations (1), the first members of both would become 
 functions of X, and known quantities ; the common value of .r, corresponding to 
 this value of y, must have tho property of every root of an equation pointed 
 out at Prop. II. of Art. 238 ; that is to say, if a denote this value of x, each 
 of the equations (1) must be divisible by (x — o) ; in other words, they must 
 have a common divisor containing x. If, therefore, without knowing and sub- 
 stituting the value of?/, wo proceed with the two given equations (1), accord- 
 ing to the method for finding tho greatest common divisor, until wo arrive at a 
 divisor of the first degreo with respect to r, and to a remainder independent 
 of .r, or containing only y, as this remainder would have been zero if I lie value 
 of y had occupied its place during ilio process, the value of y ought to !>,• such 
 as to reduce this remainder to zero. The values of y which will do this are 
 found by putting this last remainder equal to zero, and thus forming what 
 called the final equation in y only. The values of y which satisfy the final 
 
 equation are the only compatible values of this unknown in the two given equa- 
 tions (l). The corresponding values of.;- are found by substituting these 
 vulucs uf ?/ successively in the last divisor, which will ordinarily he of the flat 
 
ELIMINATION. 393 
 
 degree with respect to x, and setting this equal to zero ; each value of y gives, 
 ! >y means of this divisor, the corresponding value of x, which, substituted with 
 .1 in the given equations, will satisfy them. Should this divisor reduce to zero 
 by the substitution of the value of y, we must go back to the previous one of 
 the second degree, which, put equal to zero, will furnish two values of x for 
 each of y ; if this reduco to 0, we must go to that of the 3° degree, and so on. 
 
 316. This conclusion may be arrived at in another manner. Denoting by 
 A=0, for simplicity, the first of the two given equations F(.r, y)=0, and by 
 13 = the second /(.r, y) = 0, by Q the quotient of A by B, and by R the re- 
 mainder, we have 
 
 A=BQ+R (2) 
 
 It follows from this equality that all the values of the unknewn quantities x 
 and y, which give A=0 and B = 0, must also give R=0, since the quotient 
 Q can not become infinite for finite values of x and y, the given equations be- 
 ing supposed to be entire functions, or capable of being rendered such with 
 respect to x and y. (See Art. 275, Cor. 2.) 
 
 For the same reason, all the values which will give B = and R=0, will 
 also give A=0. The system of equations A=0, B = may, therefore, be 
 replaced by the more simple system B=0, R=0. 
 
 If now B be divided by R, and a new remainder, R', be reached, it may be 
 shown in a similar manner that the system B=0, R=0 can bo replaced by 
 the system R = 0, R' = 0, R' being of a lower degree with respect to x than 
 R, and so on, till we arrive at a remainder independent of .r. Let R" be this 
 remainder. Then the original equations are replaced by the system R'=0, 
 R"=0, in which R"=0 is the final equation in y only, and R' generally of 
 the 1° degree with respect to .r. 
 
 317. The same conclusion could not have been arrived at had y been sup- 
 posed to enter into any of the denominators in the above process. Suppose, 
 for instance, that Q in equation (2) contained denominators functions of y, 
 then Q, might possibly become infinite by the values of y reducing these de- 
 nominators to zero, and BQ thus might bo finite (see Art. 156,. 3°), though B 
 were zero. 
 
 318. If, in order to prevent the occurrence of y in the denominator of the 
 quotient when affecting the division of A by B, it had been necessary to mul- 
 tiply the polynomial A by some function of y, foreign roots might thus be in- 
 troduced, not belonging to the proposed equation. For, call c this function, 
 and represent by Q still the quotient obtained after this preparation, and by R 
 the remainder, we shall have 
 
 cA=BQ+R. 
 
 This equality proves that the solutions of the equations B = 0, R=0 are the 
 same as those of the equations cA=0, B=0. But this last system divides 
 itself into two others, A = 0, B = 0, and c=0, B = 0, consequently the equa- 
 tions B=0, R = will admit all the solutions of the proposed equations ; but 
 they will admit, also, all those of the equations c=0, B=0, which can not be- 
 long to the equation A = 0. The same may be shown for any foreign factor 
 Decessary to bo introduced to effect any subsequent division. 
 
 On the other hand, factors are sometimes suppressed for convenience in the 
 process for finding the common divisor. If these factors were such as would 
 reduce to zero on attributing to y its proper values, the process ous;ht to ter 
 
394 ALGEBRA. 
 
 rmnato, since the whole remainder becomes zero with one of its factors, and 
 the preceding divisor would be a common measure of the two polynomials; 
 and yet these values of y which produce this common measure would not 
 have been presented by tho final equation arrived at had the factor in question 
 been suppressed without notice. 
 
 From tho foregoing considerations we see that, to obtain tho values of y 
 which belong to the proposed equations, we must equate to zero the remain- 
 der which is independent of x, as also each of the factors in y which have 
 been suppressed in tho course of the operation, and resolve each equation 
 separately ; secondly, that among the values thus obtained there maybe some 
 which, on trial in the proposed equations, prove extraneous, and which must, 
 therefore, bo rejected. 
 
 319. Simplifications may sometimes be employed, the nature of which is 
 explained conveniently by the aid of symbols, as follows : Let the polynomials 
 A and B, the first members of the given equations, be put under the form 
 
 A=dd'd"uu'u", B=dd'd"vv'v", 
 In which d represents a common divisor of A and B, containing x only ; d' 
 another, containing y only ; and d" a third, containing both x and y. The 
 other factors, w, u', u", v, v\ v", have a similar meaning, except that they are 
 not common to the two polynomials A and B. The proposed equations may 
 be satisfied by placing d=0 ; this equation contains only x, and, when re- 
 solved, furnishes a limited number of values of this unknown quantity, to 
 which may be joined any value whatever of y, and tho given equations A=0 
 and B = will be satisfied. Again, d' = Q will satisfy them, which gives simi 
 larly limited values for y, unlimited for x. Finally, suppose d"=0 ; as d" 
 contains both x and y, an arbitrary value may be given to one of the unknown 
 quantities, and this equation will make known a corresponding one for the 
 other. 
 
 The other modes of satisfying tho given equations consist in equating to 
 zero simultaneously one of the factors u, u', u" of the first, and one v, v\ or 
 v", of tho otheY. But v and u can not be simultaneously equal to zero, since 
 they each contain only .r, and are supposed to havo no common divisor, d having 
 been understood to comprise all the common factors depending on x alone. 
 For a similar reason, u' and v' functions of y alone can not at the same timo be 
 equal to zero. But u" and v", being put equal to zero, are to be proceeded 
 with by the method of tho common divisor, as already explained, and will fur- 
 nish a limited number of values for y, and corresponding values limited also 
 for x. 
 
 320. Should the remainder, in seeking for a common divisor, not contain y, 
 but only known quantities, it could not be put equal to zero. In this case tho 
 given equations would be incompatible. 
 
 EXAMPLES. 
 
 (1) Let the equations bo 
 
 (_2x*+2)y 8 +(x«— 2a?— 2x»-f te-f-l^+fx 6 — 2i 8 +x)y=s0, 
 
 ( _.r-J- l)j/4- ( —.«.-' + ./■) y '+ (.r 3 — x 2 )y-f (x*— *»)y«=0. 
 There arc numerous simplifications of these, for they can be decomposed into 
 factors like the following: 
 
 7/(.r-l )(,-+?/) X (X+ 1 H I '-•-V/-1) = 0, 
 
 y(x-l)(x+y)Xy(*«-3 1=0. 
 
ELIMINATION. J95 
 
 Equating to zero first the common factors, e;ich in its turn, we obtain 
 ' < y=0, < y indet, $ y indet., 
 
 I x indet., \ x=l, \ x = — y ; 
 
 next equating to zero the other factors, we have four systems of equations, viz. 
 
 First system ) ^~ S whence \ ^ = , 
 
 J lx-\-l=U $ I x= — 1. 
 
 Second system < -{~ n , „ > whence <^ , ) 2/ = , 
 J ^ x- — 2y — 1=0 $ } x=l, \x— — 1. 
 
 Third system j * "f = Jj | whence \ ^-l j ^ 
 
 Fourth system 5 ^"^ = ° l whence \V = 1 + V2 _ ( y=l+ V2 _ 
 J a*— 2y-,l=0 S \ x=±(l + V2) J *s=±(l— \/2) 
 
 In the first three systems, all the solutions, except .r= — 1, y= — 1, have 
 already been found; in the fourth, those in which we have x= — y are also 
 already known ; hence, in reality, wo have only determined three new solu- 
 tions, viz., 
 
 C7/=_l cyssi+va cy=i-Vj 
 
 \x=-\, \x=l+y/2, lx=l-V*- 
 (2) To resolve the two equations 
 
 x 3 — 3yx 2 -|-(3y 2 — 3y+l)x— y 3 -f-y 2 — 2y = 0, 
 &— 2yx +y»_ y=0. 
 These equations can not be decomposed into factors ; hence wo pass imme- 
 diately to successive d 'visions. This remark will apply also to equations 3 and 4. 
 
 First Division. 
 
 x 3 -3ya?+{3y»-y4-l)x-y8+y 3 -2y 
 
 -i-x'— 2y — x-{ —7/ + / ) /• 
 
 x 3 — 2yx-\-y- — y 
 
 x—y 
 
 - y* 9 +(2y s +l)a:-y 3 -T-y s --2y 
 
 — yx i -\-2y-x — y*-{- y" 
 
 x— 2y 
 
 Second Division, 
 x- — 2yx-\-y- — y\x — 2y 
 -4-x 2 — 2yx \x 
 
 ¥^- 
 
 Hence, the final equations are x — 2y=0, y" — y=0. "We deduce from 
 these 
 
 and as we have neither introduced nor suppressed any factor, these • wo solu- 
 tions are those of the proposed equations themselves. 
 (3) To resolve the two equations, 
 
 (y— l)a«+*ac— 5y+3=0, 
 
 yx°+9x — l0y=0. 
 
 First Division. 
 
 (y—l) x 9 -f 2x — 5j/+3 
 (y— lJya^+Syx— 5y=+3y 
 
 yx*-\-9x—l0y 
 
 y-i 
 
 -f(y — l)yx" — (— 9y-f- 9)x — 10y"-+10y 
 (_7y+9)*+ 5y»- 7y. 
 
30C ALGEBRA. 
 
 As we have multiplied by y, it is necessary to resolve the equations y=0, 
 yx -' + 9x — lOy = 0, which give x=0, y=0, and to examine whether these 
 values make the dividend equal to zero. As this is not the case, it follows that 
 they form a foreign solution, which it will be necessary to suppress. 
 
 Second Division. 
 
 {-7y+9)*+fy—7y 
 
 yx + (_ 5 i/' + 7»r-63y + 81) 
 
 yx 2 + 9x— lOy 
 (-7y + 9)yx"-+(-G3y+81)x+70if-00y 
 
 ( ~ 7y + 9)yx' - ( - 57/3+ 7y y. x 
 
 (_5i/> + 7y- — 6:Jy + 81).<-+70y — 90y 
 
 (—5,f+7f— 63y+81)(— 7y+9)z-490^+1200?/ 2 _ 81Qy 
 ( -5 ^+7^— 63.y+81)(—7y+9)3.-—2 57y 5 +70 y— 364^+8463/-— 5G7,y 
 
 25?/ s — 70?/ 4 — 126.y 3 +414?y 3 — 243w. 
 
 The equations which it is neccssaiy to resolve are 
 
 (_73 / +9).r+5y 2 — 7?/ = 0, 
 
 25y 5 — 70y 4 — 120y :! 4-414?/ 2 — 243y = 0. 
 
 The second gives the results, which may be readily verified, 
 
 — 3rb3V"I6 
 2/=0, yrsl, y=3, y= ^ . 
 
 By substituting these values in the first of the given equations, we obtain for 
 x the corresponding values x=0, 1=1, ar=2, rr= — 5+ -v/10. 
 
 In the second division we have been compelled to multiply by — 7y-\-9, but 
 no foreign solution has been introduced. 
 
 We have, then, only to suppress, in the five solutions above, that which has 
 been introduced by the first division. There remain, then, for the given equi 
 tions the four following solutions : 
 
 ( _3_l_3-v/l0 r —3— 3-/10 
 
 ix—l, lx—2 t ( x= _ 5 _ A /io, lx=— 5+-/10. 
 
 (4) Let the equations be 
 
 z 2 +(8 i /-13).r-f7/-73/+12=0, 
 a*_(4y+ l).r+2/ 3 +5?y=0. 
 
 .Firs£ Division. 
 z* + (dy-l3)x+if—7y + U h*-(iy + l)x+y°-+5y 
 +x 2 -(4y+ l)x+y'+5y | l 
 (12y— 12).r— 12y-f-12 
 This remainder can be decomposed into tho factors 12(y — l)(x — 1); the 
 calculations will be simplified, and wo shall have those two systems of equa- 
 tions : 
 
 y — 1 = (r— 1=0 
 
 i s -(4y+l)z+y 9 +5y=0, i i»-(4y+l)x- 
 Each of these can bo at once resolved, and we find 
 
 \ 
 
 Cy = l Cy=l <7 / = o S.'/=— 1 
 (x=2,(x=2, ).r=l. <> tsl, 
 (5) r , - r -2y.r-+2y(?/— 2)r4-v : — 4 = 0. 
 
 z"+2yx+2y»— 5y+2=0. fys=2 JyssS 
 
Ans. 
 
 METHOD OF LAIJATIE 397 
 
 (6) z*— 3yx*+3x 2 +3y 2 x— 6yx— x— y+3t/ 2 +2/~ 3 =°. 
 r 3 -}- 3?/:r 2 — 3.c 2 4-3?/ 2 i-— 6yx— X+y 3 — 3?/ 2 — i/+3 = 0. 
 
 I 1 irst system < ^ < ^ < • / 
 
 ^ <a-=0, 4 x=2, lx=— 2. 
 
 Second system \ * ) ^ <| % \ y — ~ 
 
 ( x=l, ( x= — 1, I x=l, I x= — 1. 
 
 Third system \V = 3 SV= — 1 
 (x=0,(x=0. 
 
 (7) z*+yx*-(f~+l)x+y-if=0, 
 a.-3_2/x2_(7 / 2+G?/+9).r+2/ 3 +G^ 2 +93/=:0. 
 
 The first division gives the remainder 
 
 2/^+(37+4).r-(2/ 3 +37 / 2 +47 / ). 
 
 To be able to perform the second division, we multiply the dividend by y , 
 in the same way we prepare the first remainder to be divided. We thus ar- 
 rive at a remainder of the first degree in x, which can be put under the form 
 
 32(y*+3y+2)(x-y). 
 
 Dividing, then, the remainder of the second degree by x — y, we obtain the 
 quotient 
 
 yx+if+3y + 4 = 0, 
 and there is no remainder. 
 
 From these calculations we conclude that the first members of the proposed 
 equations are divisible by x — y, so that they can be verified by all the solutions 
 of the indeterminate equation x — ?/=0. The other solutions are furnished 
 by the system of two equations, 
 
 if+3y + 2=0, yx+7f+3y+4=0; 
 hence we obtain tl e solutions 
 
 y=-l,x= + 2; y=-2, r = + l. 
 
 METHOD OF LABATIE. 
 
 321. Having thus stated the principles on which the ordinary method of 
 elimination depends, we shall now proceed to show how this method has 
 lately been perfected by Labatie and Sarrus. By the aid of the theory which 
 they have introduced, we shall be able to perform the required eliminations 
 without introducing any foreign solutions. 
 
 Suppose that A and B represent the quotients which we obtain by dividing 
 the first members of the given equations by all of their factors which depend 
 only on y. 
 
 Let c be the factor by which it is necessary to multiply A, in order that we 
 may be able to divide it by B ; represent by q the quotient that we obtain in 
 this division, and by Rr the remainder, r designating those factors of this re- 
 mainder that are not dependent on x. Let c t be- the factor by which we 
 must multiply B to render it divisible by R ; represent by q x the quotient, and 
 by "RiTi the quotient that we obtain in this second division, ?-j designating the 
 product of those factors of this remainder which do not depend on x, and so 
 on. Finally, suppose, for the sake of simplicity, that at the fourth division 
 we obtain a remainder independent of x, and designate this remainder by r 3 . 
 We have the equalities 
 
"J93 ALGEBRA, 
 
 c A =B q + Rr 
 
 CiB = R q i -\-\\ l r i /jx 
 
 c_.Il = K ; y i-f-RsTj 
 (_c ;i R 1 = i;.7 .+r 3 
 
 Let J be the greatest common divisor of c and r, rf_ the greatest common 
 
 CC_ CC1C9 CC^r.j 
 
 divisor of — and r,, </> th:it of -ry and r 2 , d 3 that of , ,", - and r 3 . We shall 
 
 proceed to prove that we can obtain all the solutions of the system A = 0, 
 B = 0, without any foreign solution, by resolving the following systems : 
 
 ( r i r { i r. ( r n 
 
 \d~ \d x ~ j d z j d 3 (2) 
 
 (b=o, (r=o, (r i= o, (r,=o 
 
 To establish this proposition, we shall first prove that the solutions of the 
 systems (2) all agree with those of the system A = 0, B=0; we shall after- 
 ward show that the solutions of the system A = 0, B = 0, are all comprised in 
 those of the systems (2). 
 
 [a] Dividing by d the two members of the first equation of system (1), it 
 becomes 
 
 3 A 4 B +5 R < 3 > 
 
 ■4 is entire, for c andr, by hypothesis, are divisible by d ; hence, qB is divisible 
 
 by d ; but B, by hypothesis, is prime with respect to d ; therefore, d divides q. 
 
 Equation (3) shows that the values of x and y, which satisfy the equations 
 
 r c c r 
 
 B = 0, "j = 0, destroy also jA.; but -3 and - are prime with respect to each 
 
 other. Consequently, 1°, all the solutions of the system B = 0, -j = 0, agree tcith 
 
 those of the system A=0, B=0. 
 
 r_ 
 [h] To obtain a relation between A, R, and -y, we multiply equation (3) by 
 
 c., and in the resulting equations place, instead of c.B, its value as found in the 
 second member of the second equation of system (1) ; wo thus obtain 
 
 -d A =\— -d~) R +d riRl - 
 
 The quantity - — -, — - is entire, because r and q are divisible by J : more- 
 
 over, tliis quantity is divisible by d v ; for d v divides -j and r„ and it is prime 
 
 with respect to R. Dividing the two members of the above equation by d u 
 
 q C\T-\-qq\ 
 
 and taking, to abridge, *5=M, — -p — = Mi, it becomes 
 
 ^-A = M 1 R+MR,y (4) 
 
 dd\ di 
 
 To obtain a relation between B, R, and y, wo first multiply the second 
 
 c fcC] cq x c <v, 
 
 equation of system (1) by -., which gives -yB*=-yl\4--fRi>V bince -r and 
 
METHOD OF LABAT1E. 399 
 
 r x are, by hypothesis, divisible by d u it follows that d x divides also -jR ; but 
 
 d x is prime with respect to R ; hence, d x divides —r. Div iding all the terms of 
 
 c cq\ 
 
 the equation by d u and taking, to abridge, -y=N, -jy =N;, it becomes 
 
 ^B=N,R+NR,^ (5) 
 
 Equations (4) and (5) prove that all the values of x and y, which reduce the 
 
 r x CC\ cci cci r { 
 
 polynomials R and -r to zero, destroy also -jr A and -3-7-B i DUt ~tt and -r 
 
 are prime with respect to each other; consequently, 2°, all the solutions of 
 the system R=0, -j- = 0, agree with those of the given system, A = 0, B = 0. 
 
 [c] We obtain a relation between A, R n and -j-, by multiplying equation (4) 
 
 by c 2 , and placing, instead of c 2 R, its value found in the second member of tho 
 third equation of system (1) ; we thus find 
 
 By hypothesis, d 3 divides the first member of this equation, it also divides r 3 ; 
 it ought, then, to divide RJ Mj q. 2 -\- Mc^-r-j; but R t and d 2 are prime with re- 
 spect to each other ; d % then divides the term by which R t in the above equa- 
 tion is multiplied. Designating the quotient by M 2 the equation becomes 
 
 3H A -"*+**a < 6 > 
 
 Multiplying equation (5) by c 2 , and then placing, instead of c 2 R, its value 
 found in the second member of the third equation of system (1), it becomes 
 
 ^B = R I (N 1 , : +No^)+N 1 R,r, 
 
 We can demonstrate as before that the multiplier of Ri is divisible by d 3 , 
 and, representing the quotient by N2, we find 
 
 • SI B = N - R '+ N ' E ^ • • • • • (7 > 
 
 Equations (6) and (7) show that all the values of x and y, which reduce the 
 
 r 2 
 polynomials R t and -f to zero, destroy also the first members of these two 
 
 «2 
 
 cciCj , r 2 ... , , 
 
 equations ; but , , ; and -j are prime with respect to each other ; conse- 
 
 quently, 3°, all the solutions of the system R!=0, ■y-=0, s "^ ^ l0se of the pro 
 
 posed system, A=0, B=0. 
 
 r 3 
 
 [d] The equation which gives a relation between A, R 2 , and ~r, can be ob 
 
 tnined by multiplying equation (6) by c 3 , and placing, instead of c 3 Rj, its value 
 as i^iven in the second member of the fourth equation of system (1) ; we thus 
 Snd 
 
400 ALGEBRA. 
 
 ^A = R ; (.r 7;;+( ,M^)+M,r, 
 Dividing the two members of this equation by da, and designating by M3 the quo 
 tient obtained by dividing the entire polynomial M 2 </ 3 +c 3 Mi-t- oy d^. theie 
 results 
 
 To obtain a relation between B, R : , and y, wo multiply equation (7) !•• 
 
 and put in iho place of c 3 Ki the second member of the fourth equation of the 
 system (1), which gives 
 
 cCxCjPa 
 
 gB=R 2 (N, 73 + C3 N^)+N 2 r 3 . 
 
 dd-ji a 
 Dividing both members by dz, and designating by N 3 the quotient obtained 
 
 by dividing the entire polynomial N^a-J-^Ni-T- by d s , it becomes 
 
 Jg|B=N,K, + N>2 CD 
 
 Equations (8) and (9) show that all the values of x and y, which reduce the 
 
 r 3 
 polynomials R 2 and -j- to zero, destroy also the first members of those equa- 
 ls 
 
 tions ; but , , , , and — are prune with respect to each other ; consequent 
 
 r 3 
 ly, 4°, all the solutions of the system R 2 =0, ^-=0 concur with those of the 
 
 •proposed system, A = 0, B=0. 
 
 (II.) It remains still to be proved that any system whatsoever ofwalucs 
 which satisfy the equations A = 0, B = 0, is a part of the systems of values 
 which furnish equations ( - 2). • 
 
 To form the equations which demonstrate this second part of the theorem, 
 
 c q 
 
 let us first place in equation (3) N instead of -3, and M instead of -% ; it will 
 
 become, transposing the term MB, 
 
 NA — MB = R^ (10) % 
 
 Eliminate now R between equations (4) and (5). Wo can oflfect this elim- 
 ination by subtracting one of these equations from the other, alter wo have 
 multiplied tho first by Ni, the second by M . remembering the values previ 
 ously given to Ni and M L ; but the calculations will be simpler if we multiply 
 equation (4) by B and equation (5) by A. Subtracting the two rosidting equa- 
 tions the one from tho other, we find 
 
 (M t B— N, ^)R+(MB— NA)R,~0. 
 
 Placing instead of MB — NA its \a!ue previously determined, — R-j. and 
 suppressing tho factor R,. this equation becomes 
 
 N,A — M,l! = — R,-r/ .... (11) 
 
METHOD OF LABATIE. 401 
 
 Finally, we eliminate R x between equations (G) and (7). To do this, mul 
 tiply equation (6) by B and equation (7) by A ; then subtract the one of the 
 resulting equations from the other, wo thus obtain 
 
 (M3— NsAJRi+CMjB— NiA.)Rsft=0, 
 
 Placing in this equation, instead of MiB— N X A, its value, determined in (11 j, 
 Rj-tt, and suppressing the factor R H it becomes 
 
 ■ Hi_I(fl-fcS£ (12) 
 
 In the same manner we obtain the equation 
 
 Equation 13 shows that eveiy system of values of x and y which gives 
 A=0, B=0, ought also to satisfy the equation 
 
 ~L r -i. r ± r jL—c\ 
 d d\d>ids 
 
 an equation which requires that one of its factors equal zero, whence it fol 
 lows that the equations 
 
 r r x r 2 r 3 
 
 3=°- ir"' 5= ' 3-3=°' 
 
 give all the correct values of y. 
 
 This being established, let x=a, y=z(i be a system of correct values of the 
 
 equations A = 0, B=0. 
 
 r 
 Jf the value y=P is a root of the equation -^=0, it is clear that the system 
 
 r 
 r=a, 2/=/3 will be a solution of the system B = 0, -i=0. 
 
 r 
 If the value y=P does not verify the equation -y=0, and if it is a root of 
 
 the equation -f=0, we perceive, by equation (10), that the system x=a, 
 
 dl 
 
 y=fi will give R=0 ; consequently, it will be a solution of the system R=0 
 
 r Ti 
 
 Tf the value y=fi verifies neither the equation -7=0 nor the equation ^-=0, 
 
 and is a root of the equation -f=0, we see, by equation (11), that the system 
 
 d% 
 
 r=a, y=P will give Ri=0 ; consequently, it will be a solution of the system 
 
 R 1= =0, 5=0. 
 
 r Ti 
 
 If the value y=(3 does not verify any one of the equations ^=0, ^"=°> 
 
 y = 0, and is a root of the equation -r=0, we see by equation (12) that the 
 aystem x=a, y=@, will give R 3 =0 ; conseruently, it will be a solution of the 
 
 ystem R ; =0, "T=0. 
 
 C c 
 
402 ALGEBRA. 
 
 Hence, all the systems of values which satisfy tiie equations A = 0, B^eO, 
 form part of the values which furnisli equations ("-')• 
 
 The equation ~j-~r --J • _ r = 0, which gives all the correct values oi y, it 
 called the final equation in y. 
 
 REMARKS ON THE PRECEDING METHOD. 
 
 y 
 
 It may chance that in one of the equations of system (2), for example, y 
 
 «i 
 
 =0, R = 0, a value of y, derived from the first equation, destroys some of the 
 coefficients of the powers of r in the second equation, after the highest power 
 of x ; in this caso wo only obtain a Dumber of #Iues of x inferior to the de- 
 gree of the equation 11=0; and if the substitution of the value of 
 destroy all the multipliers of tho powers of x in R, the equation R=0 v 
 not give any value of x. In fact, it can bo proved, by a method similar to that 
 which wo have employed with reference to the general equation of the second 
 degree (Art. 191), that if in an equation of the form Sx"+Hx B '-f-K 
 -f- . . . =0, we suppose that tho quantities which enter into the coefficients 
 S, H, K, occ, are of such a nature that wo have S = 0, H=0, &c., the equation 
 lias infinite roots equal in number to the consecutive coefficients which are re- 
 duced to zero. But it should bo remarked that the theory by which we have 
 proved that the solutions of systems (2) are the same with those of the system 
 A = 0, B = 0, only applies to solutions expressed by finite values of x and >/. 
 
 To prove that the solutions of systems (2), in which the value of X is in- 
 finity, also suit the proposed equations A=0, B=0, suppose that y = . 
 
 fying the equation — = 0, causes one or more of the multipliers of the higher 
 powers of x in R to vanish. If, in tho two members of tho equality (4) we 
 
 T 
 
 make y=[3, the term MR,j will bo reduced to zero, and the degree of the 
 
 term M,R will be lowered with respect to x one or more units. 
 
 Again, we can not suppose thai the terms of 3I t R which are reduce*] to 
 
 zero, havo been destroyed, until we have made y=/3 in tho terms of MR,-4, 
 
 because the degrees of A, B, R, Ri, &c., are decreasing, and we see without 
 difficulty, from the relations which exist between M, Mj, M ; , &*"., that the 
 degrees of these quantities with respeel to x go on increasing. It will be 
 
 necessary, then, in order thai y may have the vain.' \ thai the de> \ 
 
 with respect to x be lowered as many units a> the degree of R is lowered. 
 We can prove, in tho same manner, thai the value y = 3 ought also to cause 
 one or more of tho coefficients of tho higher powers t f x in B to vanish. The 
 equations A = 0, B=0 will give then for '/= ; one or more infinite values i 
 
 As to the reciprocal proposition, thai the solutions of the equations V=0. 
 B=0, in which z is infinite, oughl to be found amon.; the solutions oi systems 
 (2), it is not the fact, as will bo seen in the Becond example following. 
 
 i \ vmi'i.i: |. 
 
 (y-i)*»+y(y+i)*M-(3y , +2'-2)x+2y= 
 
 (y-l)x»+y(y+l)x + 3y«-l=G. 
 
ELIMINATION. 41 3 
 
 The first division gives at once the remainder (y — l)x-\-2y ; taking this re- 
 mainder for a divisor, we obtain, without any preparation, the remainder y" — 1. 
 We shall obtain then all tho solutions of the proposed system by resolving the 
 equations 
 
 3/3—1=0, (y— l)x+2y=0. 
 
 The first equation gives j=il. For the value y= — l we find x= — 1, 
 and this system will satisfy tho proposed equations. For the value y=-\-l 
 wo find a.-=oo. This system, also, will satisfy the proposed equations; for 
 dividing each of these equations by tho highest power of X, and taking x=oo, 
 tho two equations will be reduced to y — 1 = 0. 
 
 EXAMPLE II. 
 
 (y-l)x"-+yx+y°--2y=0, 
 (y-l) X Jpy = Q. 
 
 The division gives the remainder y°~ — 2y — 0; the solutions, therefore, of the 
 proposed equations depend on the system 
 
 y*-2y=0, (y-l)x+y = 0. 
 These equations give the two systems 
 
 y=0, x=0; y = 2, x=—2. 
 But the proposed equations possess, besides, another solution, y=l, Xs=<x>, 
 since the value y=l causes the multiplier of tho highest power of x in each 
 of these equatious to vanish. 
 
 322. The following method of elimination avoids the introduction of foreign 
 roots, and enables us to determine tho degree of the final equation : 
 
 Let equation A or .r m -L.Px m_1 -l-Qc m_2 +T.r-f- V be supposed equal to 
 
 {x— a)(x m - 1 + Ax m - 2 +B.t m - 3 +, &c.) . . . . C; 
 and equation B or ^-fP'x^ + Q'.x"- 2 . . . +T'x+ V to 
 
 (x— a)(x n - I + A'.r n - 3 +B'a.- n - 3 +, &c.) . . . . D; 
 also, let equation A be multiplied by x a ~ l -{- A'.r D_2 +B'x n - 3 , &c, and equation 
 B be multiplied by x m - 1 -f-A.r m_2 -L-B.r m_3 , &c, it is evident that the products 
 must be equal ; therefore, 
 
 (z m + P.-c m - 1 + Q.r m - 2 + , &c. ) (.r"- 1 + A'.r"- 2 + B'x"~ s + , &c. ) = (.i n + P 'af 1 -f 
 Q' x n-s^., &c.)(x m - 1 +Ax m - 2 +Bx m - 3 4-, &c.) E. 
 
 Performing the multiplications and making equal to each other, the coeffi- 
 cients of the same powers of x (Art. 209), m-\-n— 1 equations are obtained 
 
 between the indeterminate quantities A, B, C, . . . . A', B', C, Now, 
 
 the number of indeterminate quantities in equation C is m — 1, and in equation 
 D, n — 1 ; therefore, the number in equation E is m-\-?i — 2. Of the m-\-n — 1 
 equations m-\-n — 2 suffice to determine A, B, C, . . .A', B', C, . . . . ; and one 
 
 equation remains between P, Q, R P', Q', R' . . . ., which it is necessary 
 
 to satisfy in such a manner that the equations C, D may have a common di- 
 visor, x — a; this equation of condition is the final equation required. 
 
 Since none of the indeterminate quantities A, B, C . . . A', B', C .... is 
 multiplied by itself, the equations by means of which those quantities are de 
 termined are of the first degree. 
 
 The final equation being resolved, and the values of y successively substituted 
 in A. B, C, . . . . A', B', C, . . ., the results are obtained from the division of the 
 polynomials C, D by the common divisor x — a. 
 
ALGEBRA. 
 
 If the equations A. B are incomplete, the two products E can not be com- 
 plete polynomials of the degree rre-J-n— 1; but the terms which are deficient 
 io one are found in the. other. For, taking the least favorable case, viz.. 
 x m +P=0; x n +P' = 0; 
 the identity which results from the equality of the two products is 
 
 (x m +P)(x n - 1 +A'x D -*+. &c.) = (x n -fP')(x m - 1 4-Ax ra --+, &c.j 
 
 * 
 
 EXAMPLE. 
 
 Let x 2 +Px+Q=0; 
 
 X 2 +P'x+Q' = 0. 
 Denoting by x— a the factor which is to bo rendered common to these equ& 
 . I ions by the suitable determination of y, the first equation may be considerea 
 the product of x— a by a factor, x+ A, of (tie first degree ; and the second the 
 product of x— a by a factor, x+A', also of the first degree. 
 
 ... a -* + p z+Q = (x-a)(x+A), 
 i«+P'x+Q' = (x— a)(x+A'), 
 and (.r= + P.r+Q)(x+A') = (x 2 +P'x+Q')(x+A), 
 
 a?+P 
 
 + A' 
 
 x 2 -|-Q' x+AQ'. 
 + AP' 
 
 x 2_j_Q x+QA'rzz.r'+P' 
 
 + PA' +A 
 
 Making the coefficients of the same powers of x equal to each other, 
 
 P + A'=P' + A or A-A'=P-P' (1) 
 
 Q+PA' = Q' + AP'orAP'-PA'=Q-Q' (2) 
 
 QA' = AQ' orAQ' — QA' = (3) 
 
 By mean of these three equations of the first degree the two indeterminate 
 quantities A, A' can bo eliminated, and a single equation obtained in terms of 
 the quantities P, Q, P', Q'- 
 
 For, if from equation (1), multiplied by P, or AP — PA' = (P — P')P, equa 
 tion (2) be subtracted, or AP' — PA' = Q — Q', the remainder is 
 
 AP-AP' = (P-P')P-(Q-Q')- 
 (P-P')P-(Q-Q') 
 Whence A = p p, • 
 
 q. ., , a, (P-p , )P , -(Q-Q / ) 
 
 Similarly, A = p_p, • 
 
 If these values of A, A' are substituted in equation (3), 
 
 (P-P')P-(Q-Q') v0 , (P-P-)P'-(Q- Q') 
 p3p7 XH— P_P' xv£_u, 
 
 or (P_P')PQ'-(Q-Q')Q'-(P-P')Q p '+(Q-Q)Q= - 
 (P_P')(PQ'_QP')+(Q-Q')(Q-Q')=o, 
 
 (P_P')(PQ'-QP') + (Q-QT = 0. 
 
 The quantities P, P', Q, Q', containing only y and known quantities, this is 
 the final equation in y. 
 
 It has been already noticed that, if this equation is identical, the prop 
 equations have at least ono common factor of the form x— a. whatever be the 
 value of y ; and that, if it contains only known quantities, theso equations are 
 contradictory. 
 
 When tho final equation has the proper form, the factor r— a is obtained by 
 dividing tho first of the proposed equations by x-\-A ; thus. 
 
THE DEGREE OF THE FINAL EQUATION. 4G5 
 
 x+A) x-+Px+Q (.r+P-A 
 
 X 2 + A.T 
 
 (P-A)a:+Q 
 
 (P— A).r+(P — A)A 
 
 Q— (P-A)A. 
 
 The quotient is x-{-P — A, and the remainder is considered equal to zero, 
 because it is reduced to zero by the substitution, for y, of a value deduced 
 from the final equation. 
 
 Making the quotient .r-j-P — A equal to zero, the value of x is x=A — P, 
 and by substituting the value of A, 
 
 (P-P')P-(Q-Q') 
 
 or 
 
 P — P' 
 
 Q-Q' 
 
 -P, 
 
 This example is given as an illustration of the general method. From its 
 particular form it admits of resolution by another and a much shorter process. 
 For if from x 2 +Px+Q=0 
 
 x 2 -|-P'- r +Q'=0 is subtracted, 
 the remainder is 
 
 (P_P')x+Q_Q'=0; 
 
 . . .(, p p 7 . 
 
 OP THE DEGREE OP THE PINAL EQUATION. 
 323. The degree of the final equation can not be depressed by the reduction 
 of each of the coefficients P, Q, R . . . P', Q', R' . . . in the equations 
 
 ( ,n_J_p 7 .m-l_|_Q r m- 2 m m t m _)_Tx+V=0, 
 
 .r"4-P'2 n ~ 1 +Q'-'.' n ~ 2 .... +T'.r+V / = 0, 
 to the term of the highest exponent in y which it contains ; for the degree of 
 each of the equations is not changed by the reduction. Therefore, the reason- 
 ing may bo applied to the equations 
 
 x m -i r ayx m ~ 1 -\-lifx m - 2 -{-(y m - l x-\-vy m =0 .... (1) 
 
 x" -fa'2/:c n - I - r -&y.r n - 2 +?y*- 1 x+v'y n =0 .... (2) 
 
 which are of the same degree respectively as the preceding equations. The 
 latter are reducible to 
 
 /x\ m /x\ m ~ l /.r\ m - 2 x 
 
 \y) + a [y) + h \y) ' ' ' ' + t y+ V =° < 3 > 
 
 \y)+ a iy) + b iy) V ..-+^+«'=0 (*) 
 
 x 
 in which the unknown quantity is -, and a, b, . . .1, v ; a', b', . . . t', v', ar» 
 
 numbers. 
 
 Denoting by a, (3, y . . . the numerictv roots of equation (3) 
 and by a', /3', y' . . .the numerical roots of equation (4) 
 these equations may be decomposed into 
 
 
40G ALGEBRA. 
 
 Whence (a; — ay )(x — &y){x — yy ), cVc. =0 (5) 
 
 (x-ay'){x-!i'y)\ X - 7 'D), Sec. =0 (6) 
 
 Substituting in equation (5) the roots of .r from equation (6), viz., ay 
 B'y, &c., 
 
 (a'y— ay){a'y— 0y){a'y— yy), &c. =0, 
 
 ip'y—*yWy—PyWy—7y)* &c. =o, 
 
 {y'y — ay){y'y—iiy){)'y — }y), &c. =0. 
 
 Or, since the number of factors in equation (5) is m, and the number of 
 roots in equation (G) is n, 
 
 y^(a'-a){a'-l3)(a'- 7 ), &C. =0, 
 y™(0' -a)(/3' -0){0'-y), &c. =0, 
 y-(/_a)(/-j8)(/-7), &c. =0. 
 
 Consequently, there are n equations, each of the degree ?» ; these give all 
 the solutions in y. The product of these roots (or solutions) of y is the final 
 equation, sinco it becomes zero for all the values of y which render its factors 
 zero, and only for these values. Now, this product is evidently of the degree 
 tnn. Consequently, the degree of the final equation (unless roots not belong- 
 ing to the proposed equations are introduced by the process of elimination) 
 can not exceed the product of the degrees of the proposed equations. 
 
 It ought to be observed that the numerical values of the roots of y are 
 changed by this process, but that their number remains undisturbed by it 
 
 IRRATIONAL EQUATIONS. 
 
 32 !. All the direct methods, employed for the solution of equations suppose 
 that the unknown quantities in them aro not affected with any radical sign ; 
 when, therefore, the unknown is found under a radical sign, it will be neces- 
 sary, before applying the process of solution, to employ some preparatory 
 method of rendering the equation rational. Such a method is at once sug- 
 gested by the theory of elimination. For, if we equate each of the irrational 
 terms with an unknown quantity, and remove the radical from each of these 
 new equations by involution, we shall have a series of equations (including the 
 original one, with its irrational terms replaced by the new symbols) without 
 radicals, from which the quantities, temporarily introduced, may Je eliminated, 
 and thence a rational equation obtained, involving only the original unknown 
 quantities. 
 
 The following examples will fully illustrate the mode of proceeding : 
 
 (1) Let the equation be 
 
 .,_ ^/.r— 1+ y7+T=o. 
 
 Put 
 
 y= y/x— 1, z= Vx+l\ 
 
 and we then have the three following rational equations from which wo may 
 eliminate y and r, viz., 
 
 y*=x—l, z»=a:+l, x—y+z=Q. 
 
 From the last equation we gel </=.'■-{-:, and. by substituting mis value in the 
 first, y becomes eliminated, and we have these two equations in z and r, viz., 
 
 — X+l=sO 
 
 4-0,. r + . r . _, + !_(); 
 
EXPONENTIAL EQUATIONS. 407 
 
 and, to eliminate z from these, wo apply the process explained in the preceding 
 articles, and thus get the final equation 
 
 x « _ 3.f5 _(_ 8.x" + .t 3 + 7.r 2 — 7x + 2 = 0. 
 
 (2) Let the equation be 
 
 V4x+7+2Vi'-4 = l- 
 Putting 
 
 7/= V4x+7, z=y/x— 4, 
 wo have the system of equations 
 
 if=Ax+l, z 2 =x— 4, 
 
 EXPONENTIAL EQUATIONS. 
 
 325. An exponential equation is an equation in which the unknown appears 
 in the form of an exponent or index ; thus, the following are exponential equa- 
 tions : 
 
 a* = b, x*=a, a h *=c, x** = a, Sec* 
 
 To resolve the equation 
 
 10 x =:2 
 
 put x=— , then 
 
 10 x/ =2 .-. 10=2 X '. 
 The value of x' lies evidently between 3 and 4 ; place it, therefore, equal 
 to 3 plus an unknown fraction, and wo shall have 
 
 i i 
 
 10=2 3+ *", or 10=2 3 x2 i " 
 
 10 ! (by 
 
 . ox" . 1 — 1 — o 
 
 "8 " W — A 
 
 The value of x" lies evidently between 3 and 4, ••• place 
 
 ' and proceed as before. The value of x is thus obtained in a continued fraction. 
 
 _1_1 _1 1 
 
 which may bo carried to any extent at pleasure, and the value found by the 
 method explained hereafter. (See Continued Fractions.) 
 
 When the equation is of the form a* = b, or a h *=c, the value of x is readily 
 obtained by logarithms, as we have already seen in Art. 220. But if the equa- 
 tion be of the form x x =a, the value of .t may bo obtained by the rule of double 
 position, as in the following 
 
 EXAMPLE. 
 
 Given £ x =100, to find an approximate value of x. 
 
 * Exponential equations, and those in which logarithms of unknown quantities enter, 
 belong to a claw called transcendental. 
 
408 ALGEBRA. 
 
 The value of x is evidently between 3 and 4, since 3 3 =27 and 4*s=25G ; 
 hence, taking the logarithms of both sides of the equation, we have 
 
 a- log. x— log. 100=2.* 
 
 First, let £i = 3-5; then 
 
 3-5 log. 3-5= 1-9042380 
 
 true no. = 2-0000000 
 
 error =—-0957020 
 
 Second, let x 2 = 3-G ; then 
 
 3-G log. 3-6= 2-0026890 
 
 true no.= 2-0000000 
 
 error = + -0020890 
 
 Then, as the difference of the results is to the difference of the assumed 
 numbors, so is the least error to a correction of the assumed number corre- 
 sponding to tho least error ; that is, 
 
 •098451 : -1 : : -002G89 : -00273; 
 hence x=3-6 — -00273=3-59727, nearly. 
 
 Again, by forming the value of x x for £=3-5972, wo find the error to be 
 — ■0000841, and for x=3-5973, the error is + -0000149; 
 
 hence, as -000099 : -0001 : : -0000149 : -0000151 ; 
 therefore, x=3-5973 — -0000151=3-5972849, the value nearly. 
 
 EXAMPLES FOR PRACTICE. 
 
 (1) Find x from the equation z x =5. Ans. 2-129372. 
 
 (2) Solve the equation x* = 123450789. Ans. 8-G400268. 
 
 (3) Find x from the equation x' t =2000. Ans. 4-827822G. 
 
 DEMONSTRATION OF THE BINOMIAL THEOREM. 
 
 CASE I. 
 
 326. If, at Prop. V., Art. 245, we suppose the second terms a u a 2 , a 3 , &c, of 
 
 the binomials to be all positive instead of negative, and all equal to a, then the 
 
 products two and two will all become a 2 ; those three and three, a 3 , and so 
 
 on ; and, by recurring to Art. 203, we perceive that the number of combiua 
 
 tions or products two and two, if we suppose that there are n binomials, will 
 
 , , n{n — 1) , , , , «(« — l)(n— 2) 
 be expressed by — — — , the number three and three by — -— -, and 
 
 so on. Hence, where n is a whole number, 
 
 (x+a) n =x n +nax u - l +-Y : ^a-x a -°--\-, &c, -fa", 
 
 or 
 
 (a+») n =a n +na n - 1 z+Aa n - 2 .t-+Bu I: - : \r 3 +, <5cc (1) 
 
 Dy reversing the order of the terms, and disregarding the particular form of 
 tho coefficients after the second term. 
 
 CASE II. 
 
 If tho exponent bo fractional, wo have 
 
 (</ + .r)» = V («+x)'"= \Z«'"4-'"«'" - l x+\W" •.;-•+, &c. 
 
 * In equations of this kind the following method may be adopted: Let x*=a ; then 
 x log. x= log. a; put log. x=y, and log. a—/i ; then xy=b, and log. x-f- log. y= log b; 
 hence y-\- log. y= log. b. Now y may be found l>y double position, as abovo, and then r 
 
 becomes known. When a is less than unity, put sb=- and rt=- : then We have /< T =v 
 
 h 
 
 .-. y log. })■=. loir. >/, and if log. b=C, and log. V- U : then cy=Z. and log. r-f- Iolt. _v= ' 
 
 or log. c-\-z= log. z. Hence - mvv be found by the preceding method, and then y and x 
 
 become known. 
 
DEMONSTRATION OP THE BINOMIAL THEOREM. 4CJ 
 
 Applying the rule at Art. 113 for extracting the root of a polynomial, the 
 
 m 
 
 first term of the root will be a ° ; the divisor of the second term of the given 
 
 (m\ m _m 
 
 a*) =na a ; and the quotient or second term of the root 
 
 will be -a m ~ ^ m *'x=-a* x. When the two terms of the root thus found 
 n n 
 
 are raised to the ri h power, and subtracted from the given polynomial accord- 
 ing to the rule, the first two terms of the latter will be canceled, and the next 
 
 m— _ ... , ra— 2 
 
 highest power of a to bo divided by the constant divisor na u will bo a 
 multiplied by a?, and the quotient, which is the third term of the root, will 
 
 contain a to the power n— 2 — (ra— J = 2 with a: 2 , and so on, so that the 
 
 root may be written under the form 
 
 m jjj 2—1 —— 2 ™— 3 
 
 a^-l—a." x+A'a n x 2 +B'a n r'+.&c., 
 
 the same form, so far as regards the exponents, as when tho exponent is a 
 whole number. Tho coefficients will be examined for this and the next case 
 together. 
 
 CASE III. 
 
 When the exponent is negative, either entire or fractional, as a consequence 
 of what has just been demonstrated, we have 
 
 1 1 
 
 But if the division bo effected according to the ordinary rules, the quotient 
 will be indefinite, and of the form 
 
 a- m —ma- m - l x-\-k"a- m ~' 1 x 2 -\-, &c. ; 
 then, whatever be the exponent of a binomial, its development, as to the co- 
 efficients of the first two terms and the exponents of all, is of the same form, 
 viz., that indicated by equation (1). 
 
 Now, to examine the coefficients of the other terms, for the sake* of gen- 
 erality, I shall consider two consecutive terms of any rank whatever, and I 
 shall write 
 
 (a + .r) m = a m + ma ,a ~ 1 x \- Ma m_n .T" + Na m - n_1 a; n + 1 + , &c. 
 
 Let us change throughout x into x-\-y ; as the unknown coefficients con- 
 tain neither a nor x, the above expression becomes 
 
 (a-\-x-i r y) m z=a m -\-ma m - l (x-\-y) 
 
 , .+Ma m -"(.r+i/) n +Na m - n - 1 (x-f-2/) n + 1 -f , &c. 
 By changing a into a-\-y, we should have found 
 
 (a+y+rp = («+?/) m +™(0+2/) m_I *- 
 r -M(a+2/) m - n .i n + N(a-f-2/) m - n - 1 x u +i4-,&c. 
 
 In the two proceding equalities the first members are equal, therefore the 
 second members must bo equal also ; and this is the case whatever values x 
 and y may have. Then, if they bo arranged according to tho powers of y, 
 they must be identical. It is true, they contain binomials, but we know the 
 first two terms of each of these binomials, so that we can form the part which, 
 in each second membei-, contains y to the first degree, and that will suffice for 
 our purpose. Designating it by Yy in the one and by Y'y in the other, it 
 is easy to find 
 
410 ALGEBRA. 
 
 Y =ma m ~ l r -Mna m -^- 1 + N(«-- r -l)a m - n - I .r B .... 
 
 Y'=ma m - 1 ... +M(»i — «)a m - n -'.f D +N(m — n — l)a rr --°-V+ 
 
 These two quantities must be equal, whatever be the value of r ,• the co- 
 efficients, therefore, of the same powers of x must be equal. Considering 
 only those which pertain to a m-n ~ 1 x", we havo 
 
 N(n + l) = M(»i— n) .-. N = M( "!~"\ 
 
 We see by this according to what law, in the development (1), any coeffi- 
 cient whatever is formed from the preceding. It is the same that we have 
 found for the case of a positive exponent (Art. 107, IV.) ; and as we have 
 seen that the first two terms are composed in the same manner, whatever 
 be the exponent m, it will bo so also with all the other terms. 
 
 An abbreviate notation, sometimes employed to express the coefficients of 
 the binomial formula, is the initial letter B of the word binomial, with the ex- 
 ponent of the power of the binomial before it, and the order of the coefficient 
 above. Thus, the coefficient of the 1° term, if the exponent bo n, is ex- 
 
 1 2 
 
 pressed by "B;'of the 2°, n B ; of the 3°, n B, &c. ; of the Z.-" 1 term 
 
 r»(»— l)...(n— k+1) >?, 
 
 — — — = by n B, or otherwise simply n k . 
 
 SERIES. 
 RECURRING SERIES. 
 
 mill a ' 
 
 327. To develop the expression } in a series, place 
 
 ^=A+B.r+C^ +) ccc, 
 
 ind proceeding by the method of undetermined coefficients, explained at Art 
 
 209, we find 
 
 a' b b b 
 
 A=- B = — A, C = — B, D = — C, ccc. 
 a a a a 
 
 From which we perceive that each coefficient is obtained by multiplying the 
 
 b 
 preceding by — -. Hero the series is a simple geometrical progression. 
 
 Proceeding in a similar manner with the fraction 
 a'+bx 
 
 we obtain 
 
 a' b' — Ab c b b 
 
 A=-, B= , C = — A- B, D= — B- ■ C, &c. 
 
 a a a a a a 
 
 Hero each coefficient from tho 3° is tho sum of the twc preceding, multi- 
 
 c b 
 
 plied respectively by — - and — -, or each term is the sn n of the twi 
 
 rr : h r 
 
 ceding multiplicil by — — and — — . 
 1 J a a 
 
 Again, in the development of 
 
RECUIUIENG SERIES. 4U 
 
 a'+b'x+c'x' 2 
 
 a-{-bx-[-cx 2 -\-dx i 
 
 each term will be composed of the three preceding, multiplied respectively by 
 
 dx 3 ex 2 bx 
 
 a a a 
 
 finally, it becomes now evident that in general a fraction of the form 
 
 a' + b'x+c'x 2 . . . +/t'x m - 1 
 
 a -\-bx-\-cx~ . . . -\-lc x m 
 
 produces a series, each term of which from the (m-\-l) a is composed of the 
 
 Jc Ji c b 
 m preceding, multiplied respectively by — -x m , — -x m ~ l , . . . — -x 2 , x. 
 
 Sei'ies of this form are called recurrent, and the assemblage of quantities by 
 which it is necessary to multiply several consecutive terms to obtain the fol- 
 lowing term, is called the scale of relation of the terms. 
 
 328. Problem. — A recurring series being given, to return to the generating 
 fraction. 
 
 In this enunciation it is supposed that the recurring series is arranged with 
 respect to an indeterminate x. Let 
 
 S=A+Bx+Cx 2 -f-. . . 
 be such a series, having for a scale of relation [px 3 , qx 2 , rx~\. Since this scale 
 contains three terms, the generating fraction is of the form 
 
 a'+b'x+c'x 2 
 
 a -f- bx -\- ex 2 -f- dx 3 ' 
 
 If this fraction had been given, wo have seen that the scale of relation would 
 
 r d c b ~\ 
 
 bo x 3 , — -x 2 , — —x . But the generating fraction can be written thus, 
 
 a' b' c' 
 
 — +-.T+-Z 2 
 
 a ' a 'a 
 
 bed' 
 14—X+-X-+-X 3 
 ' a a 'a 
 
 and then we perceive that the three terms in x of the denominator can be at 
 once obtained by taking those of the scale of relation with contrary signs. 
 Thus, we can put the generating fraction under the form 
 
 a + j3x+yx°~ 
 1 — rx — qx' 2 — px* 
 and wo shall only have to determine a, /?, y. To do this, place 
 a _i_/j. r J_ yx 2 
 1 — rx — qx 2 — px 3 
 and since, after clearing it of fractions, the equation ought to be identical in 
 form, we derive from it, having regard only to the first three terms, 
 
 a+/?x+7.r 2 =A+B x+C x 8 
 — Ar —Br 
 
 -\q ' 
 
 Consequently, we shall have for the generating fraction 
 
 A + (B — Ar)x+ {c—Br—\q)x i 
 
 S=- 
 
 1 — rx — qx 3 — px 
 
 ■ •■ 
 
412 ALGEBRA. 
 
 For example, let S = l — 2r — .v ; — .">,-{- 4. r 4 — ... he a recurring series, 
 whose scale of relation is [-{-x 3 , +4a?, — 2x], Taking the above formula, we 
 shall have 
 
 A=l, B = — 2, c= — l,p = l, ?=4, r=— 2, 
 %nd we shall find 
 
 1 + 2j— 4x-— a- 3 ' 
 
 329. Problem. — A scries being given, to determine whether it be recurring, 
 
 and, in this case, to return to the generating fraction. 
 
 Let the given series be 
 
 S=A+B:r-fCx 2 -f-L\rH 
 
 a' 
 Let us determine first whether it be equal to a fraction of the form — r-r-. 
 
 1 a-\-bx 
 
 a' 
 and place S= , . From this equation we derive 
 
 1 a-\-bx a b 
 
 S = ~a T ~ = a' Jt ~a' X; 
 the quotient, therefore, of (1), divided by the series, ought to be exact, and of 
 the form p + ax ' Then the generating fraction will be expressed thus : 
 
 P + <l x 
 If the division does not stop at the second term this series will not be recur- 
 ring, or else it will arise from a more complicated fraction. 
 
 a'4-b'x 
 
 Place S= — r-7 — ; ;, we shall have 
 
 a-\-bx-\-cx' 
 
 1 a _|_6.r-f c .r 2 a"x 2 
 
 S = ~a T +b ; x~ =P+qX ^"a^+Vx ; 
 
 that is to say, dividing (1) by the series S, if wo stop the division after wo 
 have obtained as a quotient terms of the form p-\-qx, the series Si.r : , which is 
 the remainder that we then have, and which is always divisible by a-, will bo 
 
 Si a" 
 
 such that, after we have removed this factor, wo must havo 
 
 ^h + qtx; 
 
 S~~a'+b'x' 
 Hence wo derive 
 
 S a'+b'x 
 
 S~ a' 
 
 that is to say, the new division ought to terminate at tho socond term in the 
 quotient ; and then, to find tho generating fraction, wo shall have the two 
 equations 
 
 1 S, S 
 
 g=p + <7.r+-g.r 3 , g-=p 1 + 7 1 r, 
 
 whence 
 
 . , S, ' S-jp.+jiX 
 
 Consequently, tho generating fraction will be 
 
 {]> + <! '■)(!'' + 'h'l+x* 
 
RECURRING SERIES. 413 
 
 Suppose that the quotient of S by Si is not exactly p t -L. 71X ; if the series 
 e recurring, it will bo of an order superior to the second. Let us examine if 
 
 a'+h'x-lr-c'x* 
 
 we can have S = — —, — : — — — r^- 
 a-\- bx-{-cx i -\-dx i 
 
 We derive from this equation 
 
 1. a"+b"x 
 
 S=P+? X + a'+b'x+c'x^ ; 
 
 that is to say, after having obtained the first two terms of the quotient of 1, 
 divided by the series Si, wo shall find for a remainder a series, all of whose 
 terms will contain z 2 ; and if wo designate this remainder by Six 2 , we shall 
 have 
 
 Si a"+b"x 
 
 This equality gives 
 
 ti—a'+b'x+c'x- 
 
 -r=pi + ?1 x+ 
 
 ,-.• 
 
 hence, designating by S 3 .t 2 the series which we find for a remainder after 
 having earned the division of the series S by the series Si to the terms of the 
 quotient pi-\-qiX, we should have 
 
 S x — a"+b"x 
 From this last equality we derive 
 
 ■^-=p.-\-q. 1 x. 
 
 Here the operations stop ; for, returning to the generating fraction, we shall 
 have the equations 
 
 1 S, S S a Si 
 
 S=P:H' r +g-* 2 > ■Q—Pi + qiX+~x' i , ■^-=p. 2 +q 2 x; 
 
 and from these equations we derive 
 
 IS, 1 S, 1 
 
 S: 
 
 . . Si ' S S 2 ' Si pn+qjx 
 
 We have, then, only a fow substitutions to make in order to obtain a frac- 
 tion equal to S. 
 
 Without proceeding further, the reader will doubtless perceive that the 
 successive operations for finding the quotients jp+<p, pi-\-qiX, &c, and for 
 returning to the generating fraction, bear a striking analogy to those which are 
 necessary in reducing an ordinary fraction to a continued fraction, and in re- 
 turning to the ordinary fraction. This observation will take the place of a 
 general rulo. If we arrive at a division which gives an exact quotient ot the 
 form p+5-r, we know that the series is recurring. (See Contin. Fractions.) 
 
 EXAMPLE. 
 
 Suppose we wish to determine Whether the series of numbers 1, 2, 3, &c, 
 be recurring. It is not this numerical series which we must consider, but the 
 equation 
 
 S = l+2a:4-3x 2 4-4r ? + . . . 
 
 We perceive that the operations will be performed as follows : 
 
414 ALGEBRA. 
 
 Division of 1 by S 
 1 n_|_2:r-f.3.r 2 - r -4x 3 -|- 
 
 l + 2j + 3.r 8 +4r'4- |1 — 2 x 
 
 — 2x — 3x- — 4i J — bx* — .... 
 
 — 2x— 4x- — Gx 3 — 8x* — 
 
 x 2 +2x 3 +3x 4 + = Si.r 3 
 
 Division o/S by Si. 
 
 l-4-2x + 3. r =4-4r»+ 
 1 _|_ o. r _j_ 3X2+4X 3 + 
 
 1 + 2 •-|-3x'4-4.r 5 4- 
 1~ 
 
 
 
 l s,.s 
 
 Hence, the series S is recurring, and we have ^ = 1— 2x-|--^£ 2 1 oi = l- 
 
 IS 1 
 Wo derive from this S= 5 ~- = l ; consequently, S=- — : — : 
 
 l_2.r+-rx 3 
 
 (i-xy 
 
 Remark. — In finding a rule to determine whether a series is recurring, we 
 have considered the series as derived from a fraction whose numerator is of a 
 degree inferior to the denominator. But even if this last condition does not 
 have place, it is easy to perceive that the same explications, and, consequently, 
 the same rule, will always subsist. 
 
 329. Problem. — To find the general term of a recurring series. 
 Suppose that the series has for a generating fraction 
 
 a / +6'a+ j-frz" 1 - 1 
 
 = a -\-bx + \-kx' u 
 
 We can write this fraction thus : 
 
 F=(a'jf b'x.. . +h'x m - 1 ){a+bx+ \-kx™)--. 
 
 It is evident, then, that by developing the power — 1, ob^-r.ning the product 
 of the two factors in this equation, and taking in this product the part which 
 contains x to any power whatsoever, we shall have the general term of the re- 
 curring series. But the problem is resolved ordinarily by another process, 
 which I proceed to exhibit. 
 
 Wo divide first all the terms of the fraction F by k, and write it under the 
 form 
 
 U_ a'x m - l +!J'x m -*+ . . . 
 y— a .m_|_ flr n-i_|_^ l .„ a _^_ _ _ _• 
 
 The fraction is supposed in all cases to be reduced to its most simple form, 
 so that U has no common factor with V. 
 
 Wo decompose, then, the denominator into binomial factors, such as X-\-a, 
 whether it be by equating this denominator to zero, or by some other method, 
 and then the fraction is regarded as resulting from tho addition of many others, 
 which have for denominators these different factors. We determine all these 
 partial fractions, anil then form the general term of tho development of each : 
 then, taking the sum of these general terms, we shall have the general term 
 of the recurring series. 
 
 [n this decomposition into partial fractions it is necessary carefully to dis 
 
SUMMATION OF SERIES. 415 
 
 tinguish in V the simple faoiors from those which are raised to powers For 
 each simple factor x-\-a we shall takea fraction of the form 
 
 M 
 x-\-a 
 For each factor, such as (r-f h) n , wo might take one of the form 
 
 Ax"- 1 + B.r'- 2 4-... 
 (x+b) n ' 
 
 but it is more convenient to have only fractions with monomial numerators ; 
 instead, therefore, of a fraction like the preceding, we take n, like the fol- 
 lowing : 
 
 N N, N 2 N n _ x 
 
 (x-\-b) n ~i~ {x+b) n ~ l + {x+b)«- 2 *~ x _^ b > 
 
 M, N, Ni... representing quantities independent of a:. 
 
 Consequently, if we suppose that V=(.T-f-a)(j-'+&) n - ■ ., wo can place 
 U M N N. N n _, 
 
 V~ x+a+(r- r -i)" + (z+i)"- 1 ' ' ' + x+b^ ' 
 and the question will be reduced, for the present, to the determination of the 
 numerators M, N, N lt &c. But these have been determined in Art. 209, (3) 
 
 The preceding decomposition being effected, the determination of the gen 
 oral term of the recurring series does not offer any difficulty. 
 
 Each partial fraction can be put under the form P(p-\-x)~^, designating by 
 H an entire positive number, which can be equal to 1. If we develop thi? 
 power, we readily find that the term affected with x a is 
 
 1 . 2 . 3 ... n rp 
 
 It is the sum of like expressions, all containing x n , and resulting from the 
 different partial fractions which compose the general term required. 
 
 When the denominator of the generating fraction contains imaginary fac- 
 tors, these factors introduce imaginary quantities into the general term. If 
 we suppose, however, that the coefficients of the numerator and denominator 
 of the proposed fraction are all real (and they are always taken so), it is evi- 
 dent, a jniori, that, as we find the development of this fraction by division, the 
 general term can not embrace any imaginary factors ; consequently, wo are 
 sure that all the imaginary quantities which arise from the factors of the de 
 nominator will disappear. 
 
 SUMMATION OF SERIES. 
 
 The summation of series is the finding of a finite expression equal to the 
 proposed series, even when the series is infinite, and in many cases this finite 
 expression is found by subtraction. 
 
 EXAMPLES. 
 
 Ill . 
 
 (1) Required the sum of the series ttj+fTq + o"! - !" • • • • t0 infinity. 
 
 ad infinitum. 
 
 „ 1 1 1 1 1 1 
 
 LetS = l + 2+3+4 + 5+G+ 
 
 111111 ,. , 
 
 .-. S — l=-+-+-+g+-+-+ ad infinitum. 
 
416 ALGEBRA. 
 
 Hence, by subtracting the latter from the former, wo have the required sum 
 11111 
 1^+2^3 + 31+4^5 + 5.0 + 
 
 1 1 1 
 
 (2) Required the sum of the series 7-^+-rr+TT+ to n terms. 
 
 l.o —.4 o.O a 
 
 1111 1 
 
 LetS =I + 2+5+^+ - (a) 
 
 1111111 1 
 
 - S - 1 -2+^H + ^+2=3+4 + 5 + G+ Z+2 <*> 
 
 Subtracting (b) from (a), we have 
 
 11 12 2 2 2 2 
 
 1 + 2 _ ?i+l _ 7i-f2 = r3+2T4+3T5+4l;+ n(w+2) 
 
 J_ 1 J_ J_ _1_ _1$ 1 /1 1 \ > 
 
 •'' 1.3 + 2T4 + 3^ + 4.G + "7i(«+2)~2^ 1 + 2 — Vi+1 + 71+2/ $ 
 
 ~2t 1 ~~7i- r -l + 2 _ n + 2> 
 
 n n 
 
 = 2ft+2+4n+8' 
 
 Wlien n is infinitely great, then wo have 
 
 1111 , . „ . 1/ 1\ 1 1 3 
 
 t-^A-^-t-\-7tt+t^+ ... ad infinitum =-( 1 + -) — — =7. 
 
 1.3~2.4~3.5~4.6 ' 2\ '2/ co oc 4 
 
 1111 
 
 (3) Sum the series 7^—777+^— TT+ ad infinitum. 
 
 Ans. 1 
 4 
 
 (4) Sum the series 7^+0^X5+3^X6+ ad iQfinitura ' 
 
 Ans. jg. 
 
 e f? 17 
 
 (5) Sum the series 7^3+2^4+3X5+ t0 n termS " 
 
 3 2 1 
 
 Ans. 7: — — TT + — TTi- 
 2 71+1 ' 7» + 2 
 
 (6) Sum the series a-f- 2ar4-3ar 2 +4ar 3 -f- . ... to 71 terms. 
 
 f 1 — r° nr a ) 
 
 Ans. a < yz rz — - ? . 
 
 I (1— r) s 1 — r > 
 
 (7) Sum the series l + 3r+5x 2 - r -7.r 3 4-9x 4 .... ad infinitum. 
 
 An9 - $=& 
 
 DIFFERENCE SERIES. 
 330. Let there be the ai-ithmotical progression 
 
 a, a + c5, a + 2t, a + 36 
 
 If wo begin with a now term, b, and add to it successively each term of the 
 above, wo obtain 
 
 b, 6-fa, b+2a+t, b+3a+36, & + 4a+GJ . . ., 
 which is called a difference series of the 2° order, and so on, as in tho follow 
 ing scheme : 
 
 
DIFFERENCE SERIES. 417 
 
 Merles ' 1C teTm - 2 ° *'"'""■ 3 ° '*"" ""' ' er:D ' 
 
 "l a, a+d, a-\-2d, . . . a+(n—l)6. 
 
 H. b, b+a, b+Za+6 . . . b-\-{n— 1)«+ (W ~^ " - rf. 
 
 .„.. ., iw , ( a — S )(it— 1) _ , (»-3)(«- 2) (k— 1) , 
 III. c, c+4 c+2b+a . . . c+(»— 1)H — «+ x 2 ~ 3 * 
 
 tec. &c. 
 
 EXAMPLE. 
 
 I. order, 2, 5, 8, 11, 14 . . 
 
 II. order, 4, 6, 11, 19, 30 . . 
 
 III. order, 5, 9, 15, 2G, 45 . . 
 
 331. From the manner in which these difference series are formed, it is 
 evident that if we subtract from one another the successive terms of any or- 
 der, we obtain the terms of the preceding, and continuing i.i this way till wo 
 subtract the successive terms of the first from one another, we obtain between 
 them the constant difference 6. 
 
 332. If the order of a series be unknown, its order may be found from what 
 has been said above. Thus the series 
 
 5, 9, 15, 26, 45 , 
 taking the difference of the consecutive terms. 
 
 4, 6, 11, 19 
 
 2, 5, 8 
 
 3, o, 3, 
 
 after three subtractions of consecutive terms presents a constant differenco, 
 and is, therefore, a series of the 3° order. 
 
 333. To separate the roots of an equation by means of difference series. 
 The x lh term of a series of the order m would be expressed by 
 
 *+(*—!)/+ x . 2 g+ ' ' • • + 1 . 2 . . . m ' 
 
 which, arranged according to the powers of x, would be of the form 
 
 M;r m 4-A.r m - 1 + B.r m - 3 .... +Gx+K; 
 that is, of the form of the first member of an equation of the m tU degree, X=C. 
 If, now, we give to x the values . . . — 4, — 3, — 2, — 1, — 0, 1, 2, 3, 4, ... . 
 representing the values which the polynomial X assumes by 
 
 X_4, X_3, X_2, X_i, X , Xi, X 2 , &c (1) 
 
 these quantities will form a difference series, since x denotes the order of the 
 term in a series of which X is the general term. There is no objection to x being 
 negative, as a series may be continued below as well as above the first term, 
 observing the same law in a contrary sense. 
 
 Taking a sufficient number of terms of the series (1) to obtain, by subtrac- 
 tion of its successive terms, the series of next lower order, and from this, in 
 the same manner, that of the next lower order still, till we arrive at constant 
 differences, the terms of the series (1) may be extended indefinitely to the 
 right and left by forming them according to (Art. 330), without the trouble ot 
 substituting numerical values for X, and calculating the corresponding values 
 of X. Those values of X which have contrary signs will (Art. 252, Cor. 1) 
 have one or an odd number of roots between them. 
 Take, for example, the equation 
 
 9. r 4_3. c 3_i30x2_l7.r+260=0. 
 D D 
 
418 ALGEBRA. 
 
 Giving x the values — 2, — 1, 0, 1, 2, we have the following results inc'os*u. 
 in the parentheses : 
 
 j£._4 -X._3 X_j A — i Xo A.) A : A3 A_t 
 + 744_ 49(_ 58 -J-159 + 260 +119 —174)— 313+224, 
 
 forming a series of the fourth order. The eriea of the third order is 
 —793 — 9( + 217 +101 —141 —293) — 139 + 537 ; 
 A the second, +784 + 226( — 116 —242 — 152)+154+67l 
 
 of the first, —558 — 342(— 126 + 90) +306+ 522 ; 
 
 equal differences, +216 +216(+216)+216+216. 
 
 By substituting other values, as —3, — 1, —5, — G, and +3, +4. + 5, +6, 
 &c, we may extend the top series to any length. 
 
 To save the time and trouble of substituting consecutive numbers and calcu- 
 lating the result, the method of difference series is employed, thus : 
 
 Substitute a number of consecutive values one more than the degre 
 equation; the smallest numbers, being more easily substituted, are preferred. 
 In the present example, substituting — 2, — 1, 0, 1, 2, we obtain thai pi 
 of the first scries which is of the 3° order, included in brackets ; from 
 by subtracting its consecutive terms, the corresponding portions of the s 
 of the 2° order, and so on ; and, finally, the diiference, 216. Using this dif- 
 ference, we may extend the top series at pleasure, according to the methud 
 in Art. 330. 
 
 The roots of the equation lie between those numbers the substitutions of 
 which produce unlike signs in the result ; thus, in the above there is one root 
 between — 3 and — 4, one between — 1 and — 2, one between 1 and 2, and 
 one between 3 and 4. 
 
 334. There exists between the coefficients of two consecutive powers of 
 r+rt relations from which many useful consequences may be deduced. 
 
 Suppose the m lil power of x+a to be 
 
 x m +Aax m - 1 +Ba 2 x m - 2 +Ca 3 x ,n - 3 +, Sec. 
 Multiplying the polynomial by x+a, there results 
 
 x m+i + A«x m + Ba 2 x m - 1 + Ca 3 x m - 2 + . . . 
 + ax m +Aa x^ + Ba x m - 2 + . . . 
 
 From which we conclude that, to ohtabi the coefficient of any term of the 
 (m + iy* power q/"x + a, it is only necessary to add to the coefficient of the term 
 of the same rank in the m' h power that of the preceding term. 
 
 335. According to this rule, we can form the coefficients of the sucw»«s!vo 
 powers of x+a, as may be seen in the following table : 
 
 1, 1, 1, 1, 1,1, I, 1, 1 . .. 
 
 1, 2, 3, 4, 5, 6, 7, 8 . . . 
 
 1, 3, 6, 10, 15, 21, 23 . . . 
 
 1, 4, 10, 20, 35, E6 . . . 
 
 1, 5, 15, 35, 70 . . . 
 
 1, 6, 21, 56 . . . 
 
 1, 7, 28 . . . 
 
 1, 8 . . . 
 
 1 . . . 
 
 The first vertical column of this tablo is formed of the single number 1. The 
 •econd column is formed of the number 1 written twice. We fbfl)* the third 
 
THE DIFFERENTIAL METHOD OF SUMMING SI IUES. 119 
 
 column by placing at the side of each term in the second column the number 
 obtained by adding it to the term above it; we find thus, for the first term of 
 the third column 1 + or 1> tne second term is 1-f-l or 2, and the third 
 + 1 or 1. The fourth column is deduced from the third in the same manner 
 that that is from the second, and so on. The two terms of the second column 
 may be considered as the coefficients of the first power of :r+«. It results 
 from the above rule that the terms of the third column are the coefficients of 
 the development of (ar-j-a) 3 , those of the fourth column of (■•■'+ '')\ &c. 
 
 This tabic, which may be indefinitely extended, is called the Arithmetical 
 Triangle of Pascal. 
 
 336. It is easy to see from the composition of the arithmetical triangle that 
 the p th term of any horizontal line is the sum of the p first terms of the pre- 
 ceding horizontal line. Because if we consider, for example, the term 56, 
 which is the sixth of the fourth line, this term is formed by adding the two 
 numbers 21 and 35, which are placed at its left in the third and fourth lines; 
 but the second of these two numbers, 35, is the sum of 15 and 20 ; the last 
 number, 20, is the sum of 10 and 10, and the last number, 10, the sum of 6 
 and 4 ; finally, 4 is the sum of the two numbers 3 and 1 ; we have, therefore, 
 56 = 21 + 15 + 10 + 6 + 3 + 1. 
 
 THE DIFFERENTIAL METHOD OF SUMMING SERIES 
 
 337. Let a, 6, c, d, e, . . . . be a series of terms, in which each term is less 
 than the succeeding one ; and, taking the successive differences, we have 
 
 c d e, Sec. 
 
 c — b d — c e — d, &c. 
 
 ■ 2b+a d — 2c+b e —2d+c, Sec. 
 
 d— 3c+36— a e—3d+3c—b, &c. 
 
 e — 4o?+6c — 46+a, Sec. 
 
 Putting dii d», d 3 , d+, for the first terms of the first, second, third 
 
 fourth, .... differences, we have 
 
 b — a =d 1 .-. b=a-{- d x 
 
 c — 2b-\-a =d 2 .'. c=a+2c? 1 + d. t 
 
 d—3c-\-3b—a =d 3 .-. tZ=a + 3d 1 +3rf 2 + d 3 
 e — id-\-6c — 4b-\-a=d 4 •'• c r=« + 4d, + 6c£ 2 +4<f 3 +(: : 
 &c. &c. 
 
 Hence the (n+1)"' term of the proposed series is evidently 
 
 . A . fr- 1 ) ^ , »(n-l)(n-2) , 
 a+nd l +n~Y^-d i + 1.2.3 * 3 + 
 
 and, therefore, the n th term is (by writing n — 1 for n) 
 
 n±t„ iw .. (n- l)(n-2) ( B -i) ( w -2)(n -3) 
 
 338. To find (S) the smn of n terms of a series. 
 Let a, b, c, d, e, Sec. 
 
 and 0, a, a-\-b, a-\-b-\-c, <z + i + c+d, &c., 
 
 be two series, of which the («. + l) th term of the latter is obviously the sum of 
 n terms of the former: but the first terms of the first, second, third, fourth 
 .... differences in the latter, are 
 
 a 
 
 b 
 
 (dx) 
 
 b—a 
 
 (A) 
 
 c 
 
 (d 3 ) 
 
 
 («*0 
 
 
420 ALGEBRA. 
 
 a, b — a=di,* c — 2& + fl = </ ; , d — or.-\-%b — a, =d z , &c ; 
 honce the (tt + l) th term of the latter series, or the sum of n terms if tb 
 former, is, by (1) in the last article, 
 
 ^ ~ 1.2 l ~ 1.2.3 *~ 1.2.3.4 3 ~ ' 
 or 
 
 n{n — 1) n(n — 1)(» — 2) , n(n—l)(n— 2)(n — 3) , 
 
 T 1.2 ^ 1.2.3 -^ 1.2.3.4 "i-r-'-l" 
 
 EXAMPLES. 
 
 U) To what is 1.2+2.3+3.4 + 4.5+ . . . n(n+l) equal? 
 2, 6, 12, 20, 30, is the given series ; 
 
 4, 6, 8, 10, differences of the consecutive terms; 
 2, 2, 2, differences of these again, d 2 ; 
 0, 0. 
 Hence, a=2, di=4, d 2 =2, and d 3 , d 4 , <kc. =0; therefore 
 
 <5 -- ■ W ^ n ~ 1 ) .i . *»(n— 1)(n— 2) 
 S=na+ - d,+ — d 3 ; 
 
 =2n+2»(n— l)+« n (« — l)(n— 2) 
 = I«(n+l)(n + 2). 
 Proceed always in this way till the differences become the same.f 
 
 (2) Find the sum of n terms of the series 1, 2 3 , 3', 4 3 , 5 3 , &c. 
 
 (3) Find the sum of n terms of the series 1, 4, 10, 20, 35, &c. 
 
 (4) To what is 1.2.3+2.3.4 + 3.4.5+ n(»+l)(»+2) equal? ■ 
 
 (5) Sum n terms of the series 1, 3, 5, 7, 9, 11, &c. . . . 
 
 (6) Find the sum of 15 terms of the series 1, 4, 8, 13, 19, &c 
 
 (7) Sum 8 terms of the series 1, 2 4 , 3\ 4 4 , 5 4 , 6 4 , ecc. 
 
 (2) 
 (3) 
 
 ANSWERS. 
 
 n"(n-\-lf 
 4 ' 
 n(n+l)(»+2)(n + 3) 
 
 1.2.3.4 
 (4) >(n + l)(n+2)(n+3). 
 
 (5) n*. 
 
 (6) Jn(n«+6n — 1)=785. 
 
 n 5 n* n 3 n 
 
 (7) T+-+^-rn=8772. 
 
 5 ' 2 t 3 30 
 
 POWERS OF THE TERMS OF PROGRESSIONS 
 339. If all the terms of a geometrical progression 
 
 -ffa : aq : aq z : ag 8 aq n ~ x 
 
 are raised to the same power m, the result is the series 
 a m , a m q m , a m q" m , a™^" 1 a a f^- l \ 
 
 which is a geometrical progression, of which the first term is a m , the ratio q m , 
 and the number of terms n. 
 
 >40. If the terms of 11 p ion by differences, whose first term is a and 
 
 ■'minion difference <*, bo each raised to the »r power, we ha^e 
 
 * Tliis is the <h of the former series, but the dg of the latter. 
 
 t The terms of tlio formula ('J), containing those orders of differences which becomo zero, 
 
 like ''.!, </i. i\i-., iu 1 .-mi]il.' l, will all vanish, ami tin ■ expression for S will be composed 
 imly of the prec Dg I rmfl. 
 
POWERS OF THE TERMS OF PROGRESSIONS. - 421 
 
 a m ±E:a m . 
 
 m(m — 1) 
 (a+ d) m =a m +ma m - 1 6+ — Q m ~ 2 (P+, &c. 
 
 JL • A 
 
 m(m — 1) 
 (a— 2i5) m =a m +wia m - 1 2<5+— j-^— ! -a m --U 2 '-\- , &c. 
 
 ra(m — 1) 
 (a— 3J) m =a ra +ma m - 1 3(J+-Y-2- i a ra - 2 9(5 2 +,&c. 
 
 &c. &c. 
 
 Taking the differences of the consecutive terms, 
 
 m(m — 1) 
 (a+ 6)'"— a m =ma m - l 6-\- V "-^ 8 + , &c. 
 
 (a+2(5) m — (a+ (3) ,n =ma m -M-{- ^~ a m - 3 3<P+, &c. 
 
 (a+3<5) m — (a + 2(5) m =ma m -M4- ? ^Y^^a ra - 2 5(3 2 +,&c. 
 
 These differences being not the same, the same powers of the terms ol »n 
 
 arithmetical progression do not form an arithmetical progression. 
 
 341. To find the sum of the m th powers of an arithmetical progression. Let 
 
 —a .6 .c.d. ..Te.l 
 
 be any arithmetical progression, of which the common difference is <5. Then 
 
 b=q+3, c=b+6, l=Jc+6. 
 
 Raising these equalities to the power m-{-l, 
 
 , . , (m-\-l)m 
 
 b' n + 1 =za m + l + (wi + l)a m <J+ 1 -y L -^ L -a m - 1 6 2 -^, &c. 
 
 X • A 
 
 + 1 ==& ra + 1 +(m+l)i m d+^i-^-V- 1 (5 2 +, &c 
 
 \, 
 
 C 
 
 / m + 1 ==^>+ 1 + (m + l)7c m J+^Y 1 J— &™-M*+, &c. 
 
 Adding all these equalities, suppressing the common terms in the two equa 
 sums, viz., b m + 1 , c m+1 , &c, and transposing a m + 1 , we have 
 
 Jrn+l _ a m+l _ ( m _i. l),5(a m _L- ft- .. . . _j- &»), 
 
 -f^—^— 6*(a m - 1 + b m ~ l . . . . +7c m - 1 ), 
 
 + ,&c. 
 1 o abridge, let 
 
 a +6 +c+d \-Tc +1 rsSn 
 
 a 2 +6* +Ar 3 + p .=: S si 
 
 G «>_L.6m^_ + £ m + Z m =S m . 
 
 Then the last expression becomes 
 
 The value of S m deduced from this is 
 
 l<"+i—.a m+l m m(m—l) 
 
 S m ==^+^ ?I q :i ^-^(S m _ I -^->)--^-' < 5HS ra _ 2 -Z'«- 2 )-, &c. (1) 
 
 The law of the unwritten terms is sufficiently apparent, and the series must 
 evidently end with the term preceding that which contains the fac':or ri — m 
 or 0. 
 
422 ALGEBRA. 
 
 By formula (1) the sum S m can be found, when the sums of the inl >r.ot 
 powers are known; for this purpose, make ?n = 0, the formula gives S ; 
 making m = l, it gives S^ and so on to the sum of the powers required. 
 
 If the progression -4-a.a + <5.a + .2'5 is replaced by -^-1 .2.3 N (or 
 
 the series of natural numbers from 1 to N), i. e., a = l, 6=1, i = N, then for- 
 mula (1) becomes 
 
 N m + l — 1 m „ m(m — l) 
 S '" = N °+-^+r-2 (Sm - 1 - Nm_I) -273 (S^-N™- 2 )-, && (2) 
 
 If m=0, (2) becomes 
 
 N°+' — 1 N — 1 
 S =N°+— — = 1+-^ =N (3) 
 
 N(N+1) 
 Sl ~ 2 (4) 
 
 S 9 =N*+^=i-(S 1 -N)-i(S -No), 
 
 Ifmrrl, 
 Ifm=2, 
 
 3 v ' ' 3 V 
 
 :N* 
 
 N 3 — 1 /N 2 +N \ 1 
 f-— — (-J— Nj-3(N-1), 
 
 XT N 3 1 N* N xr N 1 
 = N2 +3-3-T-2+ N -3+3' 
 _N 3 N« N_2N»+3N S +N 
 — ~3^~Y^ ~6~~ 6 ' 
 
 N( N+1)(2N + 1) 
 
 &2=_ ~6 (5j 
 
 formula (3) expresses the sum of l°+2°+3° to N terms, or of 14-1 
 
 -f-1... to N. 
 
 EXAMPLES. 
 
 (1) If 7n = and N=10, S =N = 10. 
 
 Formula (4) expresses the sum of 1 + 2+3 |-N. 
 
 10(10 + 1) 110 
 
 (2) Ifm = landN=10 S 1= = ,/ =—=55. 
 
 Formula (5) expresses the sum of 1- + 2- + 3- + N 5 . 
 
 10 v 11 v °l 
 
 (3) If m = 2 and N=10, S»= ^-^-=385. 
 
 PILING OF BALLS AND SHELLS. 
 
 342. Balk and shells are usually piled in threo different forms, called trian- 
 gular, square, or rectangular, according as the figure on which the pile rests 
 is triangular, square, or rectangular. 
 
 (1) A triangular pile is formed by continued horizontal courses of balls 
 shells laid one above another, and these courses or rows are usually equilateral 
 triangles whose sides decrease by unity from the bottom to the top row, which 
 is composed simply of one shot. 
 
 Denoting by N the Dumber of balls contained in on of the equilateral 
 
 triangle which forms the base of the triangular pile, it is evident thai tl e num- 
 ber of balls in the base will bo expressed by 1+2+3 . . . + N or S, which 
 by (4) is equal to 
 
 N 3 +N 
 
PILING OF BALLS AND SHELLS. 42„ 
 
 If in tnis expression N is successively replaced by the numbers 1, 2, 3 ... ., 
 the number of balls in the successive layers, beginning at the top, will be ob- 
 tained. These are, 
 
 i»-fi 
 
 in the first, — - — =1 ; 
 
 2 2 +2 
 in the second, — - — =3 ; 
 
 3 2 +3 
 in the third, — - — =G ; 
 
 2 
 
 4 2 +4 
 in the fourth, — - — = 10. 
 
 Whence the sum of the whole number of balls contained in the pile is 
 
 1»+1 2 2 +2 3 2 +3 N 2 +N 
 
 ~ o~~ ^ o + 7T~ • • • + o ' 
 
 iv <w & <w 
 
 which is sometimes used. A better form may be obtained from this by writing 
 it first 
 
 19_|_2 2 +3 3 . . . + N 2 1 + 2 + 3 . . . +N 
 
 or 
 
 or 
 
 2^2' 
 Si+S! 1/2N 3 +3N 2 +N N 2 +N\ N 3 +3N*+2N 
 
 + S 1 _1/2N 3 +3N 2 +N N 2 +N \ ] 
 ~2~ = 2\ ~~ 6 ^~~ 2 / = 
 
 N(N + l)(N + 2) 
 
 6 
 the most convenient expression for the number of balls in a triangular pile 
 
 EXAMPLE. 
 
 How many balls in a triangular pile, the side of whose base contains 35 ? 
 
 35(35+l)(35 + 2) 
 Ans. v ^ j v ^ =7770. 
 6 
 
 (2) A square pile is formed by continued horizontal courses of shot laid one 
 above another, and these courses are squares whose sides decrease l>y unity 
 from the bottom to the top row, which is also composed simply of one shot ; 
 and hence the series of balls composing a square pile is 
 
 N(N+1)(2N+1) 
 1 + 4 + 9 + 1G + 25-1 |-N 2 =S a = ' - , 
 
 where N denotes the number of courses in a pile. 
 
 EXAMPLE. 
 
 If a side of the base of a quadrangular pile contains 35 balls, how many in 
 
 the pile ? 
 
 35X3(5X"1 
 
 Ans. s = 14910. 
 
 o 
 
 (3) A rectangular pile is one in which the layers, except the uppermost, are 
 arranged in rectangles. Representing by m + 1 the number of balls in the 
 top row, the layer below it must contain 2 rows of m+2 balls, the next layer 
 3 rows of ??i + 3 balls, and so on, to the N'\ which contains N rows of ??j + N 
 balls each ; and the number in this pile is 
 
424 ALGEBRA. 
 
 ( m +l)+2( m +2)-j-3(i»+3)+4(m+4)+ .... N(i»+N) 
 =m+2j»+3m+4ro+ .... Nm-4 r l 9 +2 -^9 -f-4 8 + .... N- 
 
 =m (l + 2+34-4+ . . . .N)-f square pile 
 
 N(N+1) 
 
 = . 7n-\- square pile. 
 
 (4) The number of balls in a complete triangular or square pile must evi- 
 dently depend on the number of courses or rows; and the number of balls in 
 a complete rectangular pile depends on the number of courses, and also on the 
 number of shot in the top row, or the amount of shot in the latter pile depends 
 on the length and breadth of the bottom row; for the number of courses is 
 equal to the number of shot in the breadth of the bottom row of the pile. 
 Therefore, the number of shot in a triangular or square pile is a function of N, 
 and the number of shot in a rectangular pile is a function of N and m. 
 
 The expression for a rectangular pile, 
 
 N(N+1) N(N + 1)(2N + 1) 
 __ m+ _ f 
 
 may be written 
 
 N(N + l)(3/n + 2N+l) 1 
 
 6 ; =gN(N+l)[2(m+N)+m+lJ. 
 
 But m + 1 is the number of balls in the top row, N is the number in the smaller 
 side of the base, and ra-j-N the number in the greater side, 2(m-f-N) the 
 
 L • , „ , ., N(N-fl) 
 number in the two parallel greater sides ; moreover, is the number 
 
 of balls in the triangular face of each pile; hence we have also this general 
 -ule for rectangular or square piles. 
 
 RULE. 
 
 Add to the number of balls or shells in the top row the numbers in its two 
 parallels at bottom, and the sum multiplied by one third of the slant end or 
 face gives the number of balls in the pile. 
 
 EXAMPLES. 
 
 (1) How many balls are in a triangular pile of 15 courses ? Ans. 6&0- 
 
 (2) A complete square pile has 14 courses: how many balls are in thf» pile, 
 and how many remain after the removal of 5 courses ? Ans. 609 and »54. 
 
 (3) In an incomplete rectangular pile, the length and breadth at bottom are 
 respectively 4G and 20, and the length and breadth at top are 35 and 9 -. how 
 many balls does it contain ? Ans. 71 
 
 (4) The number of balls in an incomplete square pile is equal to fi times 
 the number removed, and tho number of courses left is equal to tho number 
 of courses taken away : how many balls were in the complete pilo ? 
 
 Ans. 385. 
 (.')) Let h and k denote the length and breadth at top of a recUngubu 
 truncated pile, and N tho number of balls in each of the Blanting edges; tLen. 
 if B bo tho number of balls in the truncated pile, prove that 
 
 N< ) 
 
 B=- \ 2N 9 +3N(A+Jfc)+6Mr— Stt+fc+NJ+l £ . 
 
VARIATION. 425 
 
 VARIATION. 
 
 343. Let a Jenote a constant quantity, or one which does n. t change its 
 talue, and x a variable which is supposed to increase or diminish. 
 
 The product of the quantities a and x being denoted by X, if x is increased 
 
 or diminished, X will be increased or diminished in the same proportion. 
 
 Thus, if x become x', and, consequently, X become X', we shall have 
 
 x:x' ::X:X', 
 
 for 
 
 ax x X 
 ax=X and ax'=X' .-. — ,=— =tt„ or x : x' : : X : X'. 
 
 (IX x Ji. 
 
 Under these circumstances X is said to vary directly as x. 
 
 The symbol of variation is x ; and the expression X varies directly as x, is 
 indicated by the combination of symbols X <x x. 
 
 344. If the product of x and y be constant, and x, y both variable, since 
 
 xyz=x'y' = C, 
 
 1 1 
 
 J J y y< 
 
 In this case as x varies as the reciprocal of y, x is said to vary inversely as y, 
 and the symbolical expression is 
 
 1 
 
 X OC -. 
 
 y 
 
 If xy=X and x'y' = X', then X : X' : : xy : x'y'. 
 
 The variation of X in this case depends on the variation of two quantities 
 V and y, which is expressed thus, 
 
 X oc xy. 
 
 X X' XX' 
 
 345. If xy=X and x'y'*=X', then, *=— and x'=— .-. .r : x? : : — - : — -. 
 
 In this case x is said to vary as X directly, and as y inversely. The symbol is 
 
 X 
 
 x oc — . 
 
 y 
 
 x y y z 
 
 346. Letxx2/,i.e.,.r:.r'::2/:2/'or-='-,andlet?yo:2,i.e.,y y' : z : : z' or - =r- 
 
 • i y y z 
 
 X z 
 
 X' 2 
 
 Gnat is, if one quantity vary as a second and the second as a third, the first 
 varies as the third. 
 
 1 1 
 
 347. In like manner, if .t qc y and y oc -, x oc -. 
 
 — =— or x : x' : : z : z', i. e., xoc z ; 
 
 2 
 
 Again, let xoc y and zoc y .'. xoc z, or x:x'::z:z', or x:z::x':2' 
 
 .-. x±2:z::x'±z' :z', or x^zz:x'±z' 
 But z:z' ::y:y', .-.x^zZix'^zz'-.-.y-.y', l. e., yxx±z. 
 
 Again, since xocy, x:x' ::y : y', and since zccy, z :;'::;/ :?/', .-• xz:.r"2 
 . y- : y' 2 , and tfxz : yftfz' : : y : y', or y a -y/ 1 - » tnat is > ^ Uvo quantities vary 
 respectively as a third, their sum, difference, or pquare root of their product, 
 varies as this third quantity. 
 
 348. If x x y and m be a constant quantity, integer or fractional, since x : y : : 
 
42G ALGEBRA. 
 
 x' : y', .-. x : y : : mx' : 7ny' (Art. 107), i. e., x oc my ; that is, if one quantity vary 
 as another, it varies as any multiple or part of this other. 
 
 When x cc y, and, consequently, x oc my, so that x : x' : : my : my' or x : my 
 ::x':my', then, if x=?ny, x' will be equal to my' in all cases; whence, if .r 
 vary as y, x is equal to y multiplied by some constant quantity. 
 
 349. If X and Y are two corresponding values of x, y, 
 
 X=mY, .-. 7n = y:', 
 
 from which it follows that, when two corresponding values of x, y are known, 
 the constant m may be found. 
 
 350. Let xx y .-. x : x' : :y :y' .•. x m :x' m : :y m :y' m .•. x m a:y m ; 
 
 m being any exponent integer or fractional. Whence, if one quantity vary as 
 another, any power or root of the first quantity will vary as the same power 
 or root of the second quantity. 
 
 351. Let xij, and let t be another quantity, either variable or constant, and 
 of which I, I' are either equal or different values. Then, since 
 
 xazy, x : x' : : y : y\ and t:t'::t:t'; 
 .•. xt : x't' : :yt: y'l', or xtccyt ; 
 x x' y ?/' x y 
 
 Vl'-'-'l'-'l" 01 "^!' 
 
 that is, if one quantity vary as another, and if each of them be multiplied or 
 divided by any quantity, variable or constant, the products or quotients will 
 vary as each other. 
 
 XV x 
 
 Consequently, if xoc y, -a -, or — oc 1. 
 
 x 
 Whence, i( x <x y, - is constant. 
 
 J y 
 
 352. Let xy a X, i. e., xy : x'y' : : X : X ; 
 by alternation, xy : X : : x'y' : X' ; 
 
 X X' X 
 
 and similarly, .roc— ; 
 
 y 
 
 that is, if the product of two quantities vary ns a third quantity, each of the 
 two quantities varies as the third directly, ami as the other inversely. 
 
 353. If X=X'= constant, xy : 1 : : x'y' : 1 ; 
 
 1 1 1 
 
 .-. .r : - : : x' : — , or x x - ,- 
 
 y y y 
 
 that is, if the product of two variable quantities be constant, these quantities 
 vary inversely as each other. 
 
 354. Let a be a constant, and .r, y, z variables, and let 
 
 a : x : : y : z, a : x' : : y' : : . A 
 .-. az=.ry, az' = .i'i/', &c. ; 
 .-. az : (/:' : : xy : .r'//', or :::':: xy : x'y' 
 .-. z x.ry ; 
 
 that is, if four quantities arc always proportional, and one or two of them are 
 constant, tl thers being variable, it can be found bow the latter vary. 
 
 355. Let x, y, z be three quantities, of which, c cy when : is constant, and 
 
SYMMETRICAL FUNCTIONS. 427 
 
 x<xz when y is constant; it is required to determine the variation of X wheu 
 y, z are both variable. 
 
 Suppose, first, that .r is made to vary as y, and that wheu y becomes y', x 
 becomes x'. 
 
 Next, that x' (varied from x by the variation of y) is made further to vary 
 as z, and that when z becomes z', x' becomes x". Then, since 
 
 x : x' : : y : y' , and x' : x" :z:z' 
 .•. xx' : x'x" ::yz: y'z', 
 or x : x" ::yz: y'z' ; ' 
 
 i. e., x<xyz. 
 Therefore, if x vary as y when z is constant, and as z when y is constant, 
 when y, z are both variable, x varies as the product yz. 
 
 Similarly, it can be proved, that if t vary as v, x, y, z separately, the others 
 being constant when v, x, y, z are all variable, t varies as the product vxyz. 
 
 SYMMETRICAL FUNCTIONS OF THE ROOTS OF AN EQUA- 
 TION. 
 
 356. There are certain functions of the roots of an equation which may be 
 expressed, in a general manner, by means of the coefficients of that equation, 
 without the equation itself being resolved. 
 
 These functions, which form a very extensive class, are termed rational 
 and symmetric functions, or simply symmetric functions. 
 
 They are called rational, because the roots do not enter into them under 
 the radical sign, nor with fractional exponents; the roots are combined only 
 by addition, subtraction, multiplication, and division. These functions are 
 called symmetric, because the roots are combined in such a way that any two 
 of them may be interchanged without altering the value of the function. 
 
 For example, the expressions 
 
 ab ac be 
 ac+bc+ab, a°-+b'+c\ — + — + — _3a6c 
 
 are rational and symmetric functions of a, b, c. 
 
 All the coefficients of an equation are symmetric functions of its roots, as 
 may be seen in the expressions for the coefficients in Art. 245 ; for, in these 
 expressions, if a t were written in every place where a 2 occurs, instead of a 2 , 
 and a 2 in every place where <?! occurs, instead of a M or if any other two of 
 the roots were interchanged, the values of the expressions would not be 
 altered. 
 
 Several quantities, a, b, c, &c, being given, if we arrange them two and 
 two, in every possible way, and if in each arrangement, e. g., ab, we give the 
 exponent a to the first factor and the exponent to the second, we have a se- 
 ries of products such as a"bP, whose sum is evidently a symmetric function 
 of the quantities a, b, c, &c. This function is called a double function, be- 
 cause each term contains two of the given quantities ; it is represented, 
 abridged, by S(a a b@), the letter S being here employed to denote the word 
 sum. In like manner, triple, quadruple, &c, symmetric fuuetions are repre- 
 sented by S(a"ZA>), S(a"b^d s ), &c. 
 
 In accordance with this notation, simple symmetric functions, as a a -\-b a 
 
428 ALGEBRA. 
 
 -\-c a -\- , will be represented by S(a"), which, for the sake of abridgu.ent, 
 
 is ordinarily written S a . In like manner, we have 
 
 < l = a +b +c + ... 
 
 &c. cVc. 
 
 The notation of which we have been speaking applies to entire symmetric 
 functions; but when the terms of a symmetric function are fractional, we 
 can, by reducing them to a common denominator, express the function by a 
 single fraction, whose numerator and denominator are integral symmetric 
 functions. Thus : 
 
 ab ac be 
 
 which is a fractional symmetric function of a, b, c, becomes, by reduction, 
 
 a*b* + aV + b*c* — 6a*b*c* 
 
 357. An equation being given, to find the sums Si, S 2 , &c, of the like and 
 entire forcers of its roots. 
 
 Let the equation be X = 0, 
 or x m 4-Px ra - 1 + Q.r m - 3 +R.r m - 3 . . . +Tx+U = .... (1) 
 
 and call the m roots a, b, c, d. 
 
 We can find by Art. 238 the quotients obtained by dividing X by each of its 
 factors, x — a, x — b, x — c, &c. ; and we know (Art. 253) that by adding these 
 m quotients together, the sum must be equal to the derived polynomial X', or 
 
 m . r ™-i+(wi— l)P;r n '- 2 + («i — 2)Q.c m - 3 4-(wi— 3)Rx ra -^. . . -f T. 
 The coefficients, therefore, of the powers of .t, in this sum, must be equal to 
 the coefficients of the same powers of x in the derived polynomial X', each to 
 each. In this manner the required sums can be determined. 
 Let us take, then, the quotient of X divided by .r — a, 
 X 
 
 _.r m-1 -r-« .r m - 2 +a 2 
 
 x (l 
 
 -fP| +Pa 
 
 + Q 
 
 r m-3_j_ a 3 
 
 + Pa- 
 + Qa 
 + 11 
 
 ; m — * . . . -fa" 1-1 
 + Pa m - 9 
 4-Qa™- 3 
 
 + T. 
 In order to have the other quotients, it will bo sufficient simply to substitute 
 for a, in this expression, successively b, c, d, &c. [f we add these quotients, 
 and put Si, S a , S 3 , &c, instead of the sums a-\-b-{- c-{- . . ., a 9 +6*+C*-f" • • 
 a 3 -J- 6 s -J- c 3 -}- . . ., we shall have 
 mx n 
 
 n - l +s, 
 
 .r m --+S, 
 
 X"n-»+S 3 .r" 1 -*.. 
 
 • • -\- ^m— 1 
 
 + mP 
 
 +ps, 
 
 +PS 
 
 + PS m _* 
 
 + '" ( i 
 
 +QS, 
 
 +QS m ^ 
 
 
 
 + ///K 
 
 + RS m _, 
 
 + »iT. 
 Hence, equating the coefficients of corresponding terms :n these denfiral 
 repressions, we get 
 
SYMMETRICAL FUNCTIONS. 42<» 
 
 Si+mP—(m— 1)P, 
 
 S 2 +PS, + mQ=(m— 2)Q, 
 S 3 +PS, + QS 1 +mR=(m— 3)R, 
 
 S^ + PS^+QS^ |-mT=T, 
 
 or, simplifying, 
 
 S 1 + P = 0, 
 
 S 2 +PS 1 + 2Q=0, 
 
 S3+PS*+QSi + 3R = 0, (2) 
 
 S m _i+PS m _ 2 +QS m _ 3 • . • +(m-l)T=0. 
 By means of these equations it will bo easy to calculate successively Si, S 2 , 
 S 3 , &c, and, finally, S m _ 1, i. e., the sums of all the similar powers of the roots 
 whose index is less than the degree of tho equation. In order to determine 
 the sums of tho higher powers, expressed by S, n , S m+ i, S m+2 , &c, Ave substi 
 rute successively a, b, c, . . . in equation (1), and thus obtain 
 
 a m_|_p a m-l_^.Q a m-3 _|_T« + U = 
 
 i">4.Pin,-l^_Q im - 3 +T6+U=0 
 
 &c. 
 Wo multiply these m equalities respectively by a n , b", &c, and then add 
 them ; we thus obtain 
 
 S m +n-f"PS m +n— l + QStn+n— 2 • • • • -f- TS ll+1 -{-US n = 0. 
 
 We can make successively «=0, 1, 2, &c., and thus determine S m , S m+ i, 
 
 S m+2 , ; we find 
 
 S m +PS m _ 1 +QS m _ 3 . . . +TS 1 +US =0 
 S m+1 +PS m +QS m _ 1 ... +TS.,+US, = (3) 
 
 S ra+2 +PS m+ , +QS ra . . . + TS 3 +US a =0 
 In the first of these equations we can put in place of US , mil, for S 
 =a°- r -o°-4-c°-{- . . . =m; we shall thus find that these formulas follow the 
 same law with those in (2). By means of the first of theso we can determine 
 S m , and, passing successively to each of the succeeding formulas, we shall bo 
 able to determine each new sum by means of the sums already calculated. 
 
 It may be well to observe that all the sums, Si, S 2 , S 3 , &c, may be ex- 
 pressed without any denominator in functions of P, Q, R, &c. This results 
 from the fact that the first term in each of the relations (2) and (3) has unity 
 for its coefficient. 
 
 EXAMPLES. 
 
 (1) For a numerical application take the equation x 3 — 7.r 4-7=0. Here 
 P = 0, Q=— 7, R=7. Since P=0, the relation S, + P=0 gives S 1= 0. 
 Tho relations, then, which determine the sums Si, S 2 , . . . S 6 , reduce them- 
 selves to 
 
 S,=0, S. 2 +2Q=0, S 3 +3R = 0, 
 S 4 +QS 2 =0, S 6 +QS 3 +RS s =0, S 6 -fQS 4 +RS 3 =0; 
 and, by substituting the values of Q and R, we readily find 
 
 S,=0, S 2 =14, S 3 =— 21, S 4 =98, S 6 =— 245, S 6 =833. 
 
 (2) Calculate the sums of the similar and entire powers of the rooi-s of th 
 equation x 4 — .r 3 — 19x 2 +49x— 30=0. 
 
 Ans. S, = l, S s =39, S 3 = -89, S 4 =723, S 5 =— 2849, S 6 =16419, &c 
 
430 ALGEBRX. 
 
 (3) t«+rr+«=0 
 
 Ads. S, = 0, S. = 0, S J =—3r, S 4 = — is, S s =0, SgsSf*. 
 
 :358. In the equation S m+n +PS ID+1 ,_^-QS„ H . II _ 9 |-TS t+1 + US n =0 
 
 n can be a negative number, and thus the sums of the negative powers of the 
 
 roots can be determined. But it will be more simple to change x into - in 
 
 the proposed equation, and to find successively, by means of formulas (2) and 
 (3), the sums of the positive powers of the roots of the transformed equation. 
 It is evident that these powers are the negative powers of a, b, c, ... . 
 
 359. To determine double, triple, Sec., functions, represented by S(a a b^), 
 S(a a zA- y ), &c. 
 
 In order to find S(a"£r) we multiply together the two sums 
 
 a a +b a +c a -\ =S«, 
 
 we have 
 
 + «"^ +aV 3 4-iV ? H 
 
 This product contains two series of terms. The first series is the sum of all 
 the powers a-{-/3 of the roots, and may bo expressed by S a _j-/3 ; the second 
 series is the sum of all the products which are formed by multiplying the 
 power a of any root whatsoever by the power ft of any other root, and may 
 be expressed by S(a"bft). We have, then, 
 
 S a+/ ?+S(a"6^) = S a S,;: 
 and from this equation we derive, for double functions, the formula 
 
 S(a a 6^ = S a S / 3-S u+/? . 
 To find the triple function S{a a b^c y ), multiply together the three sums 
 
 <I a + fc a +c a -f-... = S u , 
 
 a«+b S +cV+...±=Sp, 
 a y +b Y +c r + ... = S Y . 
 The product is a symmetric function, which evidently comprises all the 
 terms contained in each of the five forms 
 
 a a +P+r, a a+/ V, a a+ *b , a^'b\ a% li c y ; 
 nence we have 
 
 S aW ,+S(a a+ /V) + S(a n +>^) > 
 
 + S(/+>7/') + S(a a i'V) p^ b >" 
 But the formula for double functions gives 
 
 S(a a+ V)=S a+j8 S y --S B+J 8 + y > 
 
 S(a a+ > b ) = S a+y Sp- S u+/Hr , 
 
 S(/ +) '« a ) = S^ + >S u -S, i+ . +; . 
 
 By substituting these values in the preceding equality, and then deriving 
 
 from this equality the value of S(</'7. V"), we obtain for triplo functions the 
 
 formula 
 
 S(a n ^c y ) = S a S / jS y -S a +^S y -S a 4 ) S, ( _S, H) S iI + -jS, : ;, +> .. 
 
 In tho same manner might the quadruple function S{a"lrc Y cr), or the sum 
 of any succeeding combinations, bo expressed by tho stuns of the powers. 
 
SYMMETRIC FUNCTIONS. 431 
 
 360. Every rational and symmetric algebraic junction of the roots of an 
 equation can be expressed rationally by the coefficients of that equation. 
 
 Since Si, S 3 , S 3 , &c, can bo expressed without denominators (Art. 357) in 
 functions of the coefficients of the proposed equation, and the double, triple, 
 quadruple, &c, functions can be expressed by the sums of the powers, it fol- 
 lows that all these symmetrical functions can be expressed by integnd func- 
 tions of the coefficients. And as every symmetrical polynomial in a, b, c . . . 
 must be composed of the assemblage, by addition or subtraction, of several 
 symmetric functions of the form S{a a b^c Y d & . . .), it follows that the value of 
 every rational symmetric function whatever of the roots of an equation (with- 
 out the roots being known) can bo expressed by the coefficients of the equa- 
 tion. 
 
 USE OF SYMMETRIC FUNCTIONS IN THE TRANSFORMATION OF EQUA 
 
 TIONS. 
 
 361. Symmetric functions present themselves in the transformation of 
 equations, whenever the roots of the transformed equation must be rational 
 functions of the roots of the given equation. 
 
 Let a, b, c ... be the roots of the given equation ; for the sake of definite- 
 ness, I suppose that two of its roots enter into the composition of each root 
 of the transformed equation, and I represent by F(a, b) the rational function 
 which expresses the law of this composition. 
 
 Suppose that, after we have made all these combinations, two and two, of 
 a,b,c... we put successively in F(a, b) instead of a and b, the two roots of 
 each arrangement, it is clear that wo shall thus have all the roots of the trans- 
 foi-med equation, to wit : 
 
 F(a, b), F(a, c), F(6, a), F{b, c) &c 
 
 Consequently, this equation, decomposed into factors, will be 
 [z-F(a, b)] [z—F{a, c)] . . . . =0. 
 
 This product does not vaiy in making between a, b, c . . . . the proposed ex- 
 change ; for, if we make the change, the factors can only place themselves in 
 some other order. We are sure, then, that, after the multiplication, the co- 
 efficients of the different powers of 2 will be symmetric and rational functions 
 of a, b, e . . . 
 
 Thus, by following the method of procedure hitherto explained, we can 
 express these coefficients by means of those of the proposed equation. 
 
 362. But there exists another method, often preferable, of employing sym- 
 metric functions. 
 
 It is founded on the observation that the relations [2] and [3] in Art. 357, 
 existing between the coefficients of an equation and the sums of the similar 
 powers of its roots, can be used to discover the coefficients of the equation 
 when they are unknown, provided wo know these sums as far as that sum of 
 the powers whose order is equal to the number of unknown coefficients,- i. e., 
 to the degree of the equation. 
 
 Hence, to arrive at the transformed equation, we determine, first, of what 
 degree this equation is to be. Wo next find the sums of the first, second, &c., 
 powers of its roots, as far as the sum of the powers whose order is equal to 
 the degree of this transformed equation; then, by means of these sums, we 
 calculate the unknown coefficients. It is clear that these different sums are 
 
432 ALGEBRA. 
 
 symmetric functions of the roots of the proposed equation, and that they can 
 be expressed by the coefficients of this equation. Hence they can readily be 
 determined. 
 
 3G3. As an illustration of the preceding method, I will resume here tin- 
 question of the equation of the squares of the dilferences, already treated of 
 in Art. 278. Symmetric functions give the most simple and elegant solution 
 of which it is susceptible. The question is this : 
 
 To find the equation whose roots are the squares of the differences of the 
 roots of a given equation, 
 
 x ra +Px ra - I + Qx m - I 4-. . . . = [A] 
 
 Represent the transformed equation by 
 
 z n 4-jpz n - 1 +92 I1 -*4-rz ,, - 8 +. . . . + ?r + w = . . . [B] 
 
 The m roots of [A] being a, b, c . . . those of [B] will be 
 
 {a— by, (a—c)°, (a — d)-, . . . (6— c) 3 , . . . {b—dy, {c—dy, . . . &c 
 
 The number of these squares is evidently that of the combinations, two and 
 two, that can be made with the m quantities, a, b, c . . . ; hence the degree of 
 the required transformed equation will be n=\m{m — 1). 
 
 The coefficients p, q, r . . . may easily be found when we know the sums 
 of the similar and entire powers of the roots of equation [B] ; since the sum 
 of the first powers is equal to that of the n lh powers. Let us designate these 
 new sums, then, by f,f 2 ,f 3 , <fcc, and find the general value of/ a , a being any 
 entire and positive number whatsoever. 
 
 The roots of the equation [B] are, as has already been stated, (a — by, cVc- 
 Raising these roots, then, to the power a, we have 
 
 / a =(a— 6) M +(a— cyt+ia— df a [■{b-c)- i +, &c. 
 
 In order to find this sum, consider tho expression 
 
 <p{x) = {x— a y a +(x— b)" a + {x— c) ia + 
 
 which contains the m binomials .r — a, x — b, x — c If we make in this 
 
 expression successively x=a, b, c, . . ., and add the m results, we evidently 
 
 obtain 
 
 2f a =<p(a) + <p(b) + $(c) + . . . 
 
 If we develop the powers which compose tf>(.r), we find 
 
 2a(2a— 1) 
 
 x 2a — 2aax- a ~ l + , t — i a»l Ba -«+ a sa 
 
 6(r)=i 2n(2a — 1) 
 
 +, <fec. 
 or, more simply, by using tho notation S lt S. : . Arc, 
 
 2o(2a— .n 
 
 #(x) = m.i-- , -2aS I i aa - , + — - — — ... + S M . 
 
 Substituting a, b, r, . . . in this expression instead of x, and adding tho re- 
 sults, we obtain 
 
 2n(2a — 1) 
 
 -,;/;, = ,»S.,,--.'..S,S., I „ 1 + \ 2 ; S,S, a _ 2 . • • +>' 
 
 In this second member it will be perceived thai the terms at an equal dis- 
 tance from the extremes are equal; consequently, stopping at the middle term 
 
 of the expression, and taking only the hall' of that term, we have the geMnl 
 vn'-ie of_/;,, to wit, 
 
QUADRATIC FACTORS OF EQUATIONS. 433 
 
 ,/u — mh 2a — 2abiSj a _i-j - — — — S 2 S JU _ 2 . • . , 
 
 5a(2a — 1) ( 
 1.2 
 
 I | 2«(2a -l)(2«-L)...(a+l) c , 
 2 1 . ( > 3 a ba^a- 
 
 As the signs are alternately -{- and — , there will never be any uncertainty 
 as regards this last term. Let us view, then, the operations which must be 
 performed. 
 
 1°. "We calculate the sums Si, S 3 , S 3 .. up to S :a by means of the known 
 relations S, + P=0, S 2 +PS! + 2Q=0, &c. 
 
 2°. In the formula which expresses^ we make successively a=l, 2, 3, 
 ..«, and we thus have, to determine the n Bums f u f 2 ,f 3 , ...f n , 
 /=mS. 2 — S 1 S„/,=mS 4 — 4SiS 3 +3S : S 2 , &c. 
 
 3°. Finally, the relations existing between these n sums and the n coeffi 
 cients^j, q, r, ... will give the values of these coefficients, viz., 
 
 P=~fu q=-h(fi+PAh r=-l(f,+ pf i +gf t ), &c. 
 
 364. A method entirely analogous to that which has been employed in find 
 ing the equation of the squares of the differences can be employed in a great 
 number of cases, and particularly in those where the roots of tho transformed 
 equation are similar, and entire powers of the difference, of the sum, of the 
 product, or of the quotient of any two roots whatsoever of the given equation. 
 
 For example, suppose that each new root is to be the power k of the sum 
 a-\-b of two roots of equation [A]. Taking ns=|m(m — 1), the transformed 
 equation ought to have the form 
 
 2 n_J_p 2 n-l.J_g 2 n-3_|_ [-tZ-\-U = Q ..... [C] 
 
 and if we make 
 
 f a = (a+b)^+(a + cY«+ . . . +(6+c) k «+, &c, 
 the calculation will reduce itself to expressing/ a by a general formula. To 
 do this, we take the function 
 
 ^(x) = (x+a)"«+(.r+&)^+(x+c) ka + ,&c, 
 the development of which is 
 
 kaUca — 1) 
 ^(a:)=ma Jlg +feaSia* n - 1 + ' &&**-*+ . . . + S ka . 
 
 J. • Urn/ 
 
 But if, before the development, we substitute in ^(.r) successively a, b, c, 
 
 . ., instead of x, the sum of the resultants will be equal to ~f a -\- 2 ka S k „; 
 
 hence it is easy to perceive that by making the same substitutions in the 
 
 development, we shall have 
 
 2/a+2 kfl S ta =mS k ,+^aS 1 S ka _, . . . +mS k «. 
 
 Finally, we derive from this equation the required formula, 
 
 kalka — 1) 
 /a=(m-2 ka - 1 )S ka +7caS 1 S k ,_ 1 +-A r -^ ! S,S ku _. 2 +, &c. 
 
 When ka is even, we stop at the term which contains S with two equal in- 
 dices, and we take only the half of it ; but when ka is uneven, we stop at the 
 term in which the two indices are l(ka — 1) and r,(7ia4-l), and we take the 
 entire term. ^ 
 
 QUADRATIC FACTORS OF EQUATIONS. 
 
 365. Every equation of an even degree has at least one real quadratic factor. 
 Let the proposed equation be 
 
 E E 
 
434 ALGEBRA. 
 
 x n -\-p l x a ' 1 +^ J x n_2 + kP°=°» having roots a, b, c, &c, and let ji = 2/j. /» 
 
 being an odd number. Let it be transformed (Art. 3G2) into an equation 
 whose roots are the combinations of every two of its roots, of the form », =a 
 -\-b-\-mab, m being any number; also, let the transformed equation be 
 (£ m (?/) = 0; then its coefficients will be symmetrical functions of a, b 
 and, therefore, rational and known functions of p u p : , ice. ; and its degree wiU 
 
 be .which is odd; therefore, 6 m (y)=z0 will have at least ;one real roou, 
 
 whatever be the value of m. Hence, making m=\, 2, 3, . . . {/*('-." — 1) + 1 \> 
 successively, each of the equations (p { (y) = 0, 6 : (y) = 0, &c, will hire at least 
 one real root; that is, we shall have p(2fi — 1)+1 real values for combinations 
 of two roots of the proposed equation, of the form a-{-b-{-mab ; but there are 
 oi\]y u(2u — 1) such combinations which are differently composed of the roots 
 a, /;, c, cVc. ; therefore, two of these combinations, for which we have obtain- 
 ed real values, must involve the same pair of the quantities n, b. c, &c. : I»*t 
 this pair of roots be a, b, and a, a', the real roots of the corresponding equa- 
 tions 6jy)=0, </> m .(t/) = 0, so that 
 
 a-\-b-{-7nab=a, a-\-b-{-m'ab = a ; 
 therefore, a-\-b and ab are real, and the proposed equation has at least one 
 real quadratic factor, and two roots, either real, or of the form o±^V — *• 
 Hence every equation whose degree is only once divisible by 2 has at least 
 one real quadratic factor. 
 
 We shall now prove that if it be true that every equation has at least one 
 real quadratic factor when its degree is r times divisible by 2, or when n=2 r /i, 
 where n is odd, the same is true when the degree of the equation is r-f-1 
 times divisible by 2. For, let n=2 r+1 fi; then the degree of the transformed 
 equation will be 2 r / u(2 r +V — 1)> which is only r times divisible by 2 ; therefore, 
 by supposition, the transformed equation, 6 m (y) = 0, will have two roots, either 
 real or imaginary. If they are real, then, exactly in the same way as for the 
 preceding case of the index being only once divisible by 2, it may be shown 
 that die proposed equation has at least one real quadratic factor. If they are 
 imaginary, we shall have y=a± l 3-</ — 1, each of which expresses the value 
 of some one of the combinations a -\-b-\- »iul>, a-\-c-\-mac, &c. Suppose, 
 therefore, that we have a-\-b-\-mab=a-\- 3-/ — 1 ; then, as shown above, we 
 can give m such a value m', that $ m .(y) = shall havo a root corresponding to 
 the combination of the same letters, so that a-\-b + m ab=a '-\- ; i' V — 1 ; from 
 which equations we can obtain values of rib and a-\-b under the forms 
 
 a+b = y+t -/^T, 
 
 a6 = ?' + <<V-l, 
 
 ... a*— (y+oV— lJx-f.y'-l-d'V — l is a factor of /(z) ; 
 
 but if any real expression have a factor of the form M-J-N \/ — 1, it most a.»u 
 have one of the form M — N -J — 1 ; 
 
 ... x i — { y — dJ~^-i)r+y — i'j~~^-\ is a factor of/(r) ; 
 if, therefore, these two expressions have no simple factor in common, their 
 product will be a biquadratic factor of /'(.')• 
 
 (x'-K+^+O'r-^)-'. 
 which car always be resolved into two real quadratic factors. (See solution 
 of liiquad atics.) If they have a factor in common, since they may be written 
 
QUADRATIC FACTORS OF EQUATIONS. 435 
 
 x>—>x+y- V-l(dr-<5'), x— yx+y+ yf —l(&x-&% 
 it can only be of the form r — £ ; and the factors themselves become 
 
 [x—K+\<f^i)(x—e), (x— k — ?. /3T)(x— e) ; 
 and, therefore, the proposed equation admits the real quadratic factor 
 
 Hence an equation whose degree =2 r +V* will have a real quadratic factor, 
 provided an equation whose degree =2 r fj. has one ; but we have proved this 
 to be the case when r=l ; therefore it is universally true that every equa- 
 tion of an even degree has at least one real quadratic factor. If now this fac- 
 tor be expelled, the depressed equation will have its coefficients real and its 
 degree even, and will, therefore, as before, have one real quadratic factor. 
 Hence the first member of every equation of an even degree may be resolved 
 into real quadratic factors. 
 
 366. Hence if we divide the first member of any equation 
 x n +p 1 i n - 1 +^ ; r n -=+ . . . +p a = 
 by x 2 -|-a.r-f-6, admitting no terms into the quotient that have x in the de- 
 nominator, we shall at last obtain a remainder of the form A.r-f-B, A and B 
 being rational functions of a and b; and in order that x--j-ax+6 may be a 
 quadratic factor of the proposed equation, it is necessaiy and sufficient that 
 this remainder should equal zero for all values of x, which requires that we 
 separately have A=0, B =0. The different pairs of values, real or imaginary, 
 of a and b which satisfy these equations will give all the quadratic factors of 
 the proposed; and as the number of these factors is \n(n — 1) (Art. 244, Cor. 
 2), the final equation for determining one of the quantities a, b, obtained by 
 eliminating the other between the two preceding equations, will be of the 
 degree ±n{n— 1), which exceeds n, if n>3 ; therefore, the determination of 
 the quadratic factors of an equation will generally present greater difficulties 
 than the solution of the equation. 
 
 As the proposed equation has necessarily \n or \{n— 1) real quadratic fac- 
 tors, according as n is even or odd, there will always exist the same number 
 of pairs of real values of a and b, satisfying the equations A=0, B = ; and 
 if any of these pairs of real values be commensurable, they may be easily 
 found ; and the commensurable quadratic factors being known, the equation 
 may be depressed. 
 
 EXAMPLES. 
 
 (1) To resolve a^— 6i 2 -L.7i£— 3=0 into its factors. Dividing by i 8 -f ax+b, 
 we find a remainder, 
 
 {n + 2ab+6a — a 3 )x— {a-b— o 2 — 66+3) ; 
 therefore, to determine a and b, we have 
 
 n + 2ab + Ga— a 3 =0, 
 a?b — b 2 — 6b + 3 = 0. 
 Solving the former with respect to b, and substituting in the latter, we find 
 
 (a 3 — 4) 3 = n 3 — 64, or a=\'4-L- fyn 2 — 64 ; from whence 6, and the other 
 quadratic factor, 
 
 I s — ax+a" — 6—6, 
 may be determined. 
 
436 ALGEBRA. 
 
 (2) The resolution of r 4 - r -/>r , -|-7 r ' : + r - r 4" 5 into its two quadratic factors 
 - 3 - r - m -r+"> x"-\-m'x-\-n, may be effected by the following formulae : 
 
 in 
 
 =?(?+ Vz),m' = \( 2 >- y/z), 
 
 r — am -\- pin 2 — m 3 r — qm ' + /'<'-' — m '* 
 
 n = —^ , n' = ■- , . , 
 
 p — 2m j 1 — 
 
 wnere z is a root of the equation, 
 
 z3_(3p«_ 8?)z 9 -j-(3p*— 16p 8 j-fl6o s - T -16pr— 64s)z— (8r— 4pj+j>») 8 ~0, 
 
 which has necessarily a real root. 
 
 elimination BY SYMMETRIC FUNCTIONS. 
 
 367. Symmetric functions furnish a method of elimination which has the 
 advantage of making known the degree of the final equation. 
 
 Let the two equations be 
 
 x m_|_p x m-l_J_Q a .m-2_^ Rxm -3 # .._0 (1) 
 
 i»4.P'f-'^QV-2^.R'i»-»... = (2) 
 
 in which P, Q..., P\ Q'. . . are functions of y. If we could resolve (1) witn 
 respect to x, we would derive from it m values, a, b, c..., of x, which would 
 be functions of y ; and, by substituting these values of x in equation (2), we 
 would have, for determining the values of y, in equations free from x, viz., 
 
 a n +P'a n - 1 + Q'a n --+R'a n - 3 ...=0 ^ 
 
 6n4-P'6 - 1 -fQ'6 B - s -fR / 6 ,t - 3 ...=o[ .... (3) 
 
 C "_}-P'c n - l + Q'c n --+R'c n - ;5 ...=:0) 
 &c. ckc. 
 
 But, in general, the resolution of equation (1) is impossible, and the prob- 
 lem is to obtain a final equation which embraces all the values of y without 
 distinction. 
 
 We shall have an equation which will fulfill this condition by multiplying 
 together the m equations (3), for the resulting oquation will be satisfied by 
 each value of y derived from any one of them, and it can not be satisfied in 
 any other way. But the factors of this resultant can only change places, 
 whatever permutations we may make between the quantities a, b, c . . . ; the 
 product, then, will only contain entire and rational symmetric functions of 
 these quantities ; hence we shall bo able to express these factors by means 
 of the coefficients of equation (1), and in this way wo shall have the final equa- 
 tion in y. 
 
 This method of elimination leads, in general, to \ calculations, 
 
 but it has the advantage of giving a final equation containing all the roots that 
 it ought to embrace, without any complication of foreign roots. 
 
 368. This method has also the advantage of leading to a general the. 
 with respect to the degree of the final equati [n the ] re rti( le the 
 first equation is of the d< the second of the . and P, Q..., I", 
 Q'... are any functions whatsoever of y; but, for the theorem in question, 
 
 ■ functions must evidently be polynomes, such that the sum of tl 
 ponents of .r and y Bhall be, at most, equal to m in each term of equation \l), 
 hiuI, at most, equal to n in each term of equation (2). We have, then, to de- 
 termine to what degree y can be raised in the symmetric functions which 
 compose the product of equations (3). 
 
 Each term of this product is the product of m t m respectively from 
 
ELIMINATION BY SYMMETRIC FUNCTION'S. 437 
 
 the m equations (3) ; hence, designating these terms by Ya a , Y'Zr, Y"c y , the 
 term of the product will be YY' Y". . . a a bPc r . . . But f lie product of these m 
 equations being symmetric with respect to the quantities a, l>, c..., all tho 
 terms should have tho same form with the one that we have given above ; 
 consequently, we know that the product embraces all the terms represent- 
 ed by 
 
 YY'Y"...X.S(a"^c> / ...) (4) 
 
 We have now to determine the degree of y in this expression. Observing 
 that the degree of?/ in Y is, at most, equal to n — a, in Y' to n — ,', in V" to 
 n — y, &c, we shall readily see that in YY'Y". .. its degree will be, al most, 
 equal to mn — a — j3 — y On the other hand, if we refer back to the rela- 
 tions (Art. 35G) from which the sums S u S., S 3 , tVc, are derived, we shall 
 see that, P being, at most, of the first degree in y, Q of the second, R of the 
 third, and so on, tho degree of y in these sums can not surpass the subscript 
 number of S ; and, in like manner, if wo refer (Art. 359) to tho formulas 
 which express double, triple, &c, functions, we shall perceive that in 
 S(«"i' <■'.. .) the degree of?/ can not surpass a-f-/3-(- }- . . Hence in expres- 
 sion (4) the degree of y will be, at most, equal to mn. 
 
 The same remark will apply to all tho symmetric functions whoso sum 
 composes the product of the m equations (3) ; therefore, lastly, the final equa- 
 tion can not be of a degree superior to mn. 
 
 The demonstration seems to require that equation (1) contain m. Bet wc 
 can suppose that at first .r m had a coefficient, A, independent of y, and that we 
 have divided the whole equation by A. The final equation ought to subsist, 
 whatever may be the value of A ; we can make A=0, and it is evident that 
 this supposition will not raise the degree of the final equation. Finally, the 
 theorem is to be thus understood : that tho elimination between two general 
 equations, the one of the degree m, the other of tho degree n, ought to give a 
 final equation of the degree mn ; but that, in particular cases, the degree of 
 the final equation can be loss than mn. 
 
 EXAMPLES. 
 
 Tho two eqiiations, x — y m = 0, x"-\-ay n -\-by-\-c = 0, although veiy simple, 
 will give a final equation fully of the degree mn ; for, by substituting in the 
 second the value of x derived from the first, it becomes y mn -\-ay n -{-by-\-c—(). 
 
 On the other hand, in eliminating x between the equations x n — i/ m =0, 
 x"-\-ay"-}-by-\-c=0, wo obtain a final equation of a degree less than mn, viz., 
 y™+ay" + by+c=0. 
 
 369. For extending the theorem to any number whatsoever of equations, 
 we have the general theorem given by Bezout, viz., that If, between equations 
 equal in number to that of the unknowns, ice eliminate all the unknowns, except 
 one, the degree of the final equation will be, at most, equal to the product of the 
 degrees of these equations. 
 
 Before Bezout, the theorem had been known for the case of two equations : 
 and Cramer, in the appendix to his Introduction to the Analysis of Right 
 Lines, has given a very simple demonstration, which, in reality, does not differ 
 from that which we have stated. It has been a desideratum that tho same 
 demonstration should be capable of being applied to all other cases; this has 
 been accomplished by Poisson, in a memoir which appeared in the i leventh 
 volume of the Journal dc VtlcoU Polytt ' '.que. 
 
138 ALUEB11A. 
 
 METHOD OF TSOHIENHAUSEN FOR SOLVING EQUATIONS. 
 
 370. As another application of the theory of elimination, we shall briefly 
 illustrate the principle upon which Tschirnhausen proposed to accomplish th« 
 general solution of equations, but which, as observed at Art. 277, was 
 found to be of but very limited application, not extending beyond equations ol 
 the fourth degree ; and, even within this extent, too laborious for general 
 The principle consists in connecting with the proposed an auxiliary equation 
 of inferior degree with undetermined coefficients, and of as simple a form as 
 possible consistently with the office it is to perform, but involving, besides the 
 unknown quantity x, a second unknown y. The unknown, common to both 
 equations, is then eliminated according to the method at Art. 315, and a Final 
 equation in y thus obtained, of which the coefficients are functions of the un- 
 determined coefficients in the auxiliary equation. The arbitrary quantities, 
 thus entering the coefficients of the final equation in y, are then determined 
 80 as to cause certain of these coefficients to vanish; by which means the 
 equation is ultimately reduced to a prescribed form, supposed to be solvable by 
 known methods. 
 
 371. As an example, let it be required to reduce the cubic equation 
 
 xr i ^-ax' 2 -{-bx-\-c=0 (1) 
 
 to the binomial form 
 
 y*+k=0. 
 Assume an auxiliary equation 
 
 x 2+a'x+b' + y = (2) 
 
 and eliminate x from (1) and (2) in the usual way. The remainder arising 
 from dividing the first member of (1) by the first member of (2) is 
 
 (a' 2 — aa'+b — b' — y)x+(a'— a){b'+y) + c, 
 which, equated to zero, gives 
 
 (a-a')(b'+y)-c . 
 x —a' i —aa'+b—b'—y , 
 
 and this value of x, substituted in the proposed equation, transforms it, after 
 reduction, into the form 
 
 y*+hf-+iy + k=0 (3) 
 
 where 
 
 h— 3b'— aa'+a' 2 — 2b 
 i=3b' 2 —2b'(aa' — a 2 -\-2b)-\-a'-b 
 +(3c— ab)a'+b 9 — 2ae 
 k = b ri —ab' i a' + bb'a" i —ca' 3 +{a-—1b)b'*-) r 
 (3c_a6)a'6 , +aca' 2 +(t a — 2ac)b' — bra' + c*. 
 Hence, in order to reduce (3) to the prescribed form, wo must determine 
 the arbitrary quantities a', b' conformably to the conditions /i = 0, i=0 ; that 
 is, these quantities must satisfy tin- equations 
 
 3B'_ aa'+cP— 26=0 
 
 3&' 9 — 2o'(oa'— a 9 +2&)4-a"o+ 
 
 (3c— a6)a'+o 9 — 2ac=0, 
 
 of which tho first is of the firsl degree with respect to a' and /-', and the otuei 
 
 of tlu> Becond degree, bo thai their values may be determined by a quadratic 
 
 aquation. And these values, or. rather, the expression for them in terms of 
 
METHOD OF LAGRANGE. 439 
 
 the given coefficients, being substituted in the preceding expression for fc, reo 
 der that symbol known ; and thus tho required form 
 
 y 3 +k=0 
 is obtained. 
 
 372. In a similar manner may the general equation of the fourth degree 
 
 x* + ax* + bx* + ex + d = 
 be transformed into one of the form 
 
 y*+hy 2 +k=0, 
 which is virtually a quadratic, by eliminating x from the pair of equations 
 
 a: 4 -f- ^.r 3 + bx" -\- cx-\- d = 0, 
 
 x *+a'x+b' +y=0, 
 which elimination will conduct to a final equation in y of the form 
 
 from which the second and fourth terms will vanish by the equations of con- 
 dition 
 
 the first of which will be of the first degree as regards the arbitrary quantities 
 a', b', and the second of the third ; both quantities are, therefore, determina 
 ble by means of an equation of the third degree, and thence the quantities 
 h, Jc, which are known functions of them. 
 
 All this is very laborious, but it really does effect the object proposed thus 
 far ; that is, it reduces the solution of equations of the third and fourth do 
 grees to those of inferior degrees ; but beyond this point the method fails, as 
 the conditional equations resolve themselves ultimately into a final equation 
 that exceeds in degree that which they are intended to simplify. 
 
 On this subject we may add that Mr. Jerrard has greatly extended the prin- 
 ciple of Tschirnhausen, and has succeeded in reducing the general equation 
 of the fifth degree 
 
 z 5 + A 4 r* + A^ + A,.r 2 -f Ax + N =0 
 to the remarkably simple forms 
 
 x 5 ^-ax i +b=0 
 
 .r 5 -far 3 +6=0 
 
 x 5 ^-ax' i -\-b=0 
 
 x 5 -\-ax +6=0; 
 
 so that tho solution of the general equation of the fifth degree might be con- 
 sidered as accomplished if either of the above forms could be solved in general 
 terms. 
 
 For a very masterly analysis of Mr. Jerrard's researches, the reader is re- 
 ferred to the paper of Sir W. R. Hamilton in the Report of the sixth meet- 
 ing of the British Association. 
 
 METHOD OP LAGRANGE FOR SOLVING EQUATIONS. 
 
 373. A remarkable application of the theory of symmetrical functions is that 
 made by Lagrange to the general solution of equations ; by that means he 
 solves the general equations of the first four degrees by a uniform process, 
 and one which includes all others that have been proposed for that purpose, 
 the common relation of which to one another is thus made apparent. 
 
 It consists in employing an auxiliary equation, called a reducing equation, 
 whose root is of the form 
 
440 ALGEBRA. 
 
 denoting by x_, x : , . . .r„ the n roots of the proposed equation, and by a one of 
 the n' ] ' roots of unity; and the principle on which it is based is as follows- 
 Let y be the unknown quantity in the reducing equation, and let 
 
 y = a l x l J r a,r : -\ + n„x a , 
 
 d, cia, ...do denoting certain constant quantities; then, if n — 1 values of y, 
 and suitable values of the constants o„ a,, . . . a,„ can be found, so that we may 
 have n — 1 simple equations, these, together with the equation 
 
 —lh = i-i+r;+ ••• +X m 
 will enable us to determine the n roots. 
 
 Now, supposing the constants in the value of y to preserve an invariable 
 order, a M a 2 , &c, since the number of ways in which the n roots may be com- 
 bined with them to form the expression aj£j-f-a>r s -f-, cVc, is the Bame as the 
 number of permutations of n things taken all together; therefore, the expres- 
 sion for y \\\]1 have 71(71 — 1) . . . 3.2.1 values, and the equation for determining 
 y will rise to the same number of dimensions, or will be of a degree higher 
 than that of the proposed equation ; hence the method will be of no use, un- 
 less such values can bo assumed for the constants a,, a : , . . . a n as shall make 
 the solution of the equation in y depend upon that of an equation, at most, of 
 n — 1 dimensions. Now this may be done (at least when n does not exceed 
 4) by taking the n lh roots of unity a , a, a 8 , a\ . . . a n_1 for a,, a*. . . . a Q , so that 
 
 y=za°Xi-{-ax.--{- . . . -|- a r -'x r -}- a r .r H 1 -f- j-a a -'.r . 
 
 For, in the first place, with this assumption, the reducing equation will 
 contain only powers of y which are multiples of n ; for, since a n =l, 
 
 a n - r y = a°-*x 1 + a»-<+Kv. z + . . . +x r+ i+ax t+i + j-tf"-*-^, 
 
 or a a ~ T y = a n .r r+1 + a.r r+; + \- a»-Kc r , 
 
 wliich is the same result as if we had interchanged x x and x r+ i, x 2 and x^, 
 &c, so that if y be a root of the reducing equation, a n ~ r y is also a root ; there- 
 fore, the reducing equation, since it remains unaltered when a n -'y is written 
 for y, contains only powers of y which are multiples of n ; if, therefore, we 
 make y n =z, wo shall have a reducing equation in r of only 1.2.3 . . . (n — 1 
 dimensions, whose roots will be the different values of z which result from 
 the permutations of the n — 1 roots .?•.., .r,, . . . r n among themselves. We shall 
 now have, expanding and reducing, 
 
 z=y n = u -\-u l a-{-u 2 a-+ . . . +tt _iO n - 1 , 
 
 in which w . «i, v,, . . . m„_, are determinate functions of the roots, which will 
 be invariable for the simultaneous ehau into ..- r+1 , r s into x r+: , cVrc, 
 
 since z = (a r y) n ; and when their values are known in terms of the coefficient! 
 of the proposed equation, Ave shall immediately know the values of the roots 
 For let z , z\, z,, . . . z n _, be the different values of :, when 1, c, /?, y, . . . A 
 the roots of i/ n — 1=0, aro substituted for a ; then, since y= V :, we have 
 
 Xi+X % + ■ . . +•'•„= V£o 
 *l + «•»••:+ • • • + a " '•'•„= V-l 
 
 therefore, adding, an 1 taking account of the properties of the suiur of the 
 powers of I. a, 8, y, Sic., (Art. .",.'>7, [2]), we gel 
 
METHOD OF LAGRANGE. 4-H 
 
 nXl = V~o+ V?i+ • ,. • + V~i- 
 
 Again, miltiplying the above system of equations respectively by 1, a" -1 
 3"-', . . . A"-i, we get 
 
 nx 2 = Vz +a n - 1 ^z 1 +/3"- 1 V-H +^ _1 Vzn-i, 
 
 and so on for the rest. Hence, since — pi = V z ot and .-. ( — j> 1 ) n =z r=M # 
 
 + M H \-ita— n the problem is reduced to finding the values of «,, w ; , . . . u„_i. 
 
 374. When n is a composite number, the above general method admits of 
 simplifications. For let n have a divisor m, so that n=mp, and let a be a rool 
 of 7/'" — 1 = 0; then, since a m =l, a m +'=rt, a ! "+' = « 2 , &c, fl 2m = l, a^-Hssa 
 -Sec, we have 
 
 y=x i -\-av i -{-a"x.,-{- . . . +a n ^ 1 x„ 
 
 =X 1 +aX s +a*X 3 + h am_1 -Xmi 
 
 where X r =.r r - r -.r m+r -f-.r 2m+r - r - . . . +- r 'i-i«+" an( i consists of ^? roots ; 
 
 .-. :=7/ n = 2i +w 1 a-|-W2 a2 4" • • • + M m-i° m-1 » 
 where w , Wi, &c, are known functions of Xu X 2 , &c. ; and when they ari 
 found in terms of the coefficients of the proposed equation, wo shall be able to 
 determine immediately the values of Xj, X 2 , &c, as before. To deduce tho 
 values of the primitive roots x u x % , x s , . . . .r„, we must regard separiitely those 
 which compose each of the quantities Xj, X 2 , &c., as the roots of an equa- 
 tion of p dimensions. Thus, let the roots whose sum is X! be those of the 
 equation 
 
 xp— XiZP-^ + La*- 2 — Mxp- 3 + . . . =0, 
 where L, M, &c, are unknown ; then the first member of this equation is n 
 divisor of the first member of the proposed, since all its roots belong to tho 
 hitter. Hence, effecting the division and equating to zero the coefficients of 
 rP _1 , a:P~ 2 , &c, in the remainder, we shall have p equations in Xi, L, M, &c, 
 of which the first p — 1 will give the values of L, M, &c, in terms of X, by 
 linear equations. It will then remain to solve the equation so formed of p 
 dimensions. Similarly, substituting the value of X 2 in place of that of X u we 
 shall have an equation giving tho next group of roots .r 2 , .t^j, &c. ; and so on 
 
 EXAMPLE I. 
 
 X s — px~-\-qx — Tz=0. 
 Let the roots be a, b, c, and let 
 
 y=a-\-al^-a-c ; 
 .-. zz=zy*=a?+b*-{-c*+6abc+ 3(a 2 i + o 2 c+c 2 a)a+3(a 2 c+& 2 a-f c 2 Z>)a«, 
 =u a -{-u,a-\-2t i a'. 
 But «i, Ui are roots of the quadratic 
 
 u- — {ni + !<;)" + w 1 w. 2 = 0, 
 and ul-{-u.2=3^(a i b) = 3pq— 9r (Arts. 357, 359), 
 u l u,=0\abcS 3 +Z{aW)+3a°b°c' i \ 
 = 97 3 + 9(pi — 6pq)r+ Sir 2 . 
 Hence «i, u 2 are known, 
 
 and .•• itos^p 3 — ("i + W;), is known. 
 Hence, denoting by z : , z s , the values of z when a and a 2 are respectivef) 
 written for a, we have 
 
ALGEBRA. 
 
 a-\-b-\-c—p 
 a -\-ab-\-a"-c=yT L 
 a-\-a"b-\-ac= ^/z 2 ; 
 from which we obtain the values of a, b, and c, viz., 
 
 EXAMPLE II. 
 
 x* — px? -f- qx* — rx -j- s = . 
 Since 4=2.2, let a be a root of y- — 1=0, so that <z 2 =l ; 
 then 3/=r 1 + ar 2 +T 3 4-aj: 4 =X 1 - r -aX 2 , 
 
 if Xi=:.Ti-^-ar 3 , Xo=.T;-}-:r.j ; 
 
 •. z=y 2 =u -\-au 1 
 
 where « =X 1 4-X^, w 1 =2XiX ! , and u +Ui=z —p' > ' 
 
 Hence u l =2(x l -\-x 3 )(x 2 -\-Xi), by interchanging the roots among themselves, 
 Will admit the two other values 2(.r 1 +x 2 )(.r 3 -f ar 4 ), and 2(.r 1 +x < )(.r. ;: - r -x 3 ), and 
 will, therefore, be a root of an equation of the form 
 
 «J— MttJ+Ntt,— Pr=0; 
 
 the coefficients being symmetrical functions of x u x 2 , x 3 , x 4 , and, consequently, 
 assignable in terms of p, q, r, s. It is easily seen that if we make u l =2q — 2u, 
 we shall have an equation in u whose roots are 
 
 X^+XiXi, .r^ + .Ts-r.,, XiX 4 -L.X 2 X 3 ; 
 
 and the transformed equation is (Art. 362) 
 
 u s —qu' i -\-{pr—As)u — {2r—iq)s—r i =0. 
 Let u' be a root of this equation, then u^=2q — 2w' ; hence, making 
 a=— 1, z, = m — U\=P' 2 — 2ui=])- — Aq-\-Au' ; 
 .-. X l + X,=]y 2 _Xi— X : = <fz~ x \_ 
 
 Hence x l , x 3 may be regarded as roots of a quadratic x 2 — X,x-j-L=0; 
 dividing the proposed by this, and putting the first term of the remainder equal 
 to zero, we find 
 
 x;-j> x;-HX,-r 
 
 L - 2X,-p ; 
 
 therefore, x,, x 3 are known; and x 2 , x 4 will result from tlie same forinuhe 
 by interchanging X, and X 2 , or by changing tlie sign of the radical -y/-i- 
 
 EXAMPLE III. 
 
 x n — 1 
 
 - = 0, n being n prime number. 
 
 If r be one of the roots, and a be a primitive root of the prime number n 
 (that is, a Dumber whose several powers from 1 to n — 1, when divided by n, 
 leave different remainders), it will be proved hereafter that all tlie roots of 
 his equation may be represented by 
 
 r,r>. i to"- 11 . 
 
 L,>r yssr+ora+aVH l. o n - «r» n - , l 
 
METHOD OP LAGRANGE. 413 
 
 a beiiig a root of the equation y n ~ l — 1=0. Therefore, observing that a°- l =. 1 
 and r"=l, 
 
 z=2/ n_1 = M -J-nM,+a 2 M L ,-|- . . . + a "~ 2M u-2, .... (1) 
 u , m,, &c, being rational and integral functions of r which do not change hy 
 the substitution of r", r« 3 , r aS , &c, in the place of r; for these quantities, re- 
 garded as functions of x x , x„, x 3 , &c, do not alter hy the simultaneous changes 
 of x, into x 2 , .To into x 3 , &c, nor by the simultaneous changes of x, into x 3 , 
 ,r 2 into x 4 , &c, to which correspond the changes of r into r", into r>* 2 , &c. 
 
 Now every rational and integral function of r, in which r n = l may be re- 
 duced to the form 
 
 A + Br-fCr 2 -f Dr 5 -}- p-Nr"- 1 , 
 
 the coefficients A, B, C, . . '. N being given quantities independent of r ; or, 
 since in this case the powers r, r 2 , r 3 , . . . r n_1 may be represented, although 
 in a different order, by r, r«, r« 2 , . . . r« n_2 , we may reduce every rational 
 function of r to the form 
 
 A-fBr+Cra-f-Dr« 2 4- ... +Nr«"- 2 . 
 Therefore, if this function is such that it remains unaltered when r is 
 changed into r a , it follows that the new form 
 
 A+Bra-fCra 2 -f Dr^-J + Nr, 
 
 coincides with the preceding; 
 
 .-. B = C, C = D, D=E, &c, N=B, 
 and therefoi*e the function is reduced to the form 
 
 A+B(r+r<- + r«' :! + \-r« n --), or A— B, 
 
 since the sum of the roots — — 1 ; hence each of the quantities w , u lt «. : , 
 &c, will be of the form A — B, and its value will be found by the actual de- 
 velopment of z=?/ n-1 ; so that we have the case where the values of w , u t , n : , 
 &c, are known immediately, without depending upon the solution of any 
 equation. Hence, if we denote by 1, a, /3, y, &c, the n — 1 roots of the equa- 
 tion x n ~ l — 1=0, and by z , z u z 2 , &c., the value of z answering to the substi- 
 tution of these roots in the place of a in equation (1), we shall have, as in the 
 former cases, 
 
 "--^+-^+-^1;+ . . . + n -y"zZi 
 r = ^r~ 
 
 an expression for one of the roots of the equation x n — 1 = ; and the othei 
 roots are r 2 , r 3 , &c. 
 
 Thus, the solution of x n — 1=0 is reduced to that of the inferior equation 
 2/ n_1 — 1 = 0, of which 1, a, /?, y, &c, are the roots ; also, since n — 1 is a com 
 posite number, the determination of «, [i, y, &c., will not require the solution 
 of an equation of a higher degree than the greatest prime number in n — 1 ; 
 that is, the solution of x n — 1=0 (n prime) may be made to depend upon the 
 solution of equations whose degrees do not exceed the greatest prime number, 
 which is a divisor of n — 1. 
 
 EXAMFLK IV. 
 
 x 5 — 1 = 0. • 
 The least prim itive root of 5 is 2 ; for the powers of 2 from 1 to 4, when 
 divided by 5, leat e remainders 2, 4, 3, 1 ; 
 
144 ALGEBRA. 
 
 • 
 .•. 1/ = 7•+a/-- + a 3 /- , + a :, r , ; 
 
 also ^=1, r''=l, and r + r : -\-r*-{-r' = — 1; 
 
 .-. z=y*= — l + 4a+14a- — 16a?. 
 
 But the four roots of y 4 — 1 =0 are 
 
 1, -1, V"l, - V^l ; 
 
 .-. Zb=l, ^ = 25, z 2 = -15 + 20 v — 1, 
 z 3 = — 15 — -20 V— l; 
 
 ••• .r=}{— 1+ V5 + V — 15 + 20-/— 1 + V — 15— 20V3lf. 
 
 375. For the proof that, in the general equation of the »"■ degree, the 
 
 formation of the reducing equation will require the .solution of au equation of 
 
 1 .2.3... n 
 
 1.2.3... (n — 2) dimensions, when n is prime ; and of ■; ■ 
 
 ' ' (to— l)m(1.2.3...j>) m 
 
 dimensions, when n is a composite number, and =zmp, where m is prime ; 
 
 and that, consequently, the method fails when n exceeds 1. the reader is 
 
 referred to Lagrange's Trade de la resolution dcs equations nunuriques, note 
 
 xiii., from which the matter of this section is taken. 
 
 RESOLUTION OF THE GENERAL EQUATIONS OF THE THIRD AND 
 
 FOURTH DEGREES. 
 
 RESOLUTION OF THE EQUATION OF THE THIRD DEJREE. 
 
 376. I shall suppose that wo have made the second term of the equation of 
 the third degree disappear, and, to avoid fractious, I will write this equation 
 under the form 
 
 x 3 +3px+2q=0 (1) 
 
 Among the different modes of resolving it, the most simple consists in form- 
 ing a priori an equation of the third degree, without a second term, which ad- 
 mits of one known root, but expressed with indeterminates, and to make use 
 afterward of these indeterminates to render the equation identical with the 
 proposed equation (1). To establish this identity, it will be net y to 
 
 write two equalities, and for this reason we employ two indeterminates. 
 
 Let there be made x=a-\-b : the cube will be x 3 =a 3 -\-b 3 +3ab(a-{-b) ; 
 then, replacing a-\-b by ar, and transposing, we shall have 
 
 x 3 —3abx—a 3 —b 3 =0 (2) 
 
 an equation which admits the root x=a-\-b, and which it is necessary to ren- 
 der identical with equation (1). Therefore wo place 
 
 ab=— p, a 3 +b"= — 2<y . . . . (3) 
 
 Tho first of these equalities gives a s b 3 =—j>\ Tims we know the sum 
 a 3 +b ? ; and tho product d'b\ Then the values of a 9 and b 3 aro roots of an 
 equation of the second degree, in which the coefficient of tho second term is 
 equal to +27, and tho last term equal to —j> 3 (seo Art. 101) ; so that this 
 equation will be, calling ; the unknown, 
 
 + '-V— q>' = 0. 
 This is called tho redact J equation. 
 
 lis two roots represent the values rf«* and 6»j moreover, we can take' 
 or of them indifferently for the valuo ofo 8 , because this amounts to chang- 
 1 h. and b into a, in tho valuo x=a + b. I will take 
 
\ 
 
 EQUATION OF THE THIRD DEGREE. 445 
 
 
 Each radical of t\ie second degree here has but one value, but each one of 
 the third degree has three. If we could satisfy equation (3) without making 
 any choice between these values, we could also, by the same values, render 
 equation (1) identical with equation (-2) ; and since a-{-b is a root of the sec 
 ond, the first ought to be satisfied by taking 
 
 *=V-?+ Vf+i*,+\l-q- Vq*+i* • • (4) • 
 
 which is the formula of Cardan. 
 
 But an important remark presents itself: it is, that since each radical of the 
 third degree has three values, the above expression must have nine, while 
 the equation (1) ought to have but three roots. It is necessary to explain, 
 then, whence comes this multiplicity of values, and to discern among them 
 which ought to be true roots of the equation (1). 
 
 For this purpose, let us observe that, properly speaking, it is not the reso- 
 lution of equations (3) which has given a and b, but rather the equations 
 a?b 3 =— p\ a s -\-b 3 =—2q ... (5) 
 
 Now if we designate by a and a? the two imaginary cubic roots of unity, 
 which, as we know, are the one the square of the other, it will be readily 
 seen that the equation « :! i n = — p* may result indifferently, from raising to tho 
 cube these following : 
 
 ab= — p, ab= — np, ab = — n-p. 
 
 Hence it follows that the nine values contained in formula (4) ought to give 
 the roots of the three equations, 
 
 a?+3px+2t/=0, s?+ 3opx+2g=0, x 8 +3a'*px+2q=0 .... (6) 
 
 We can, moreover, consider these nine values as the roots of the equation 
 of the 9° degree, which would be obtained by multiplying together the three 
 equations (6). But it will be more simple, and will amount to the same thing, 
 to raise to the cube either one of these equations, after transposing to the 
 second member the term which contains p. In this manner we find at once 
 
 (&+2q) 3 = — 21 fa*. 
 
 As to the roots which belong especially to each of the three equations, what 
 precede^ furnishes the means of distinguishing them ; because, according as 
 the coefficient of x shall be 3p, Zap, or Za-p, it is clear that we ought to add 
 only the values of a and b, for which we have abz= — p, or ab = — ap, 
 or ab = —a' 2 p. 
 
 By this rule it will be easy to form the roots of the proposed equation 
 r 1 -\-3px-\-2q=0, the only one with which we have to do. Designate by A 
 one of the values of the first cubic radical, and by B ono of the values of the 
 second ; tho values of a and b will be 
 
 g = A, aA, cfiA; 6=B, aB, a : B. 
 
 Moreover, suppose, for {hurts admissible, that A and B represent the values, 
 the product of which is — p. From what has just been said we ought to add 
 only the values, the product of which is AB ; then, recollecting that a , =l. 
 we must take 
 
446 ALGEBRA. 
 
 x=A + B, i=oA+a?B, x=a 2 A+aB; 
 and, besides, we know (3U3) that we have 
 
 a = , a> = 
 
 II* we replace A and B by the two cubic radicals, and a and a* by thej 
 va. ues, we shall have 
 
 =V-<7+ Vf+P 3 +\!-q- 
 
 Vt+p 3 , 
 
 g= -i+v-3 ^_ g+ ^ + ^ + - 1 - v -y _ g _ v^+/a 
 
 -l-V- 
 
 -y[--q+ v^+F'+ ~ 1+ /~y -7- Vg*+f»- 
 
 These are the roots of the proposed equation, but we must take care to at- 
 tach to the two cubic radicals the same restricted sense as to A and B, with- 
 out which we should find false roots. 
 
 377. To discuss these valves, it will be more convenient to leave A and B 
 
 substituted for the cubic radicals, and to isolate the one which is multiplied by 
 
 V — 3. By this means we have 
 
 x=A + B, 
 
 A + B A— B _ 
 ar=— f- +— J- V3, 
 
 A+B A— B - 
 
 sss — 2 — ~ T~ V3 - 
 
 I shall suppose, also, as is done ordinarily, that tho coefficients 3p and Mcj 
 represent real quantities. Then equation (1), being of an uneven degree, has 
 always one real root, and it is admissible to suppose that A and B are the 
 values of a and b, which give this root; so that A + B will be a real quantity. 
 This being premised, let us return to the two radicals 
 
 !=V— ?+ Vf+f, b=\] — q— vAyH 
 
 a=y — </+ Vq 2 +p\ 6=V — <7— vr+r' 1 - 
 
 If q"-{-p 3 ^>0, each of them has one real value ; then we can suppose A and 
 B real. Consequently, A + B and A — B will be so also; then the first root 
 x=A+B is real, and the other two are imaginary. 
 
 If <7 2 +j? 3 =0, we have A = B, and then the three roots will be Cas2A, 
 x= — A, x= — A. They are all three real, and the last two are equal with 
 one another. 
 
 Finally, let 7 2 +p 3 <[0, which requires p to bo negative. Then a and b 
 have no longer any real determination, and, consequently, the three values of 
 X are found complicated with imaginary quantities. However, we know that 
 one of them must bo real, and, indeed, it is evident that tho cases in which 
 the three roots of equation (1) are real and unequal can only be found on tho 
 hypothesis in question, thai 7 +/''<j>, as may be seen by referring to the 
 supposition just above of 5 9 +p 3 ">0. It would be wrong, then, to affirm that 
 the values of x are imaginary. I will prove, in fact, that neither of them are 
 so; and as we can always suppose thai A and B are determinations such that 
 the sum A + B represents the real root, the existence of which is demon- 
 strated, u:e whole is reduced to showing that the pari '.(A — B) v — 3, which 
 
EQUATIONS OF THE THIRD DEGREE. 447 
 
 is found in the other two values of x, must be real. By the rules of algebra 
 alone we have (A— B)(A 2 +AB + B 2 )=A 3 — B 3 ; then 
 
 A 3 — B 3 A 3 — B 3 
 
 A_B — A 2 +AB + B i — (A+B) 2 ^AB" 
 
 But, because of the values of a 3 and of i 3 , we have A 3 — B 3 =2 V ( f-\-p 3 \ and, 
 
 by the manner in which A and B have been chosen, we have AB = — p ; 
 
 2 v/ry 2 4-» 3 
 
 then, making A+B=x', there results A — B= — — ; consequently 
 
 x -j-p 
 
 A-B — j-.i{cf-+f) 
 2 V— J- x , i+p 
 
 But by hypothesis we have q"-^-p a <^0 ; then the quantity above is real ; then 
 the three values of x are also. 
 
 It is thus demonstrated that, upon the hypothesis of q 2 -\-]?K.b, the imag- 
 inary quantities which affect the three values of x must destroy one another. 
 It would seem, therefore, that analysis ought to furnish the means of making 
 them disappear, but as yet it has not been found capable of effecting this re- 
 duction. For this reason, the case under examination has been called the ir- 
 reducible case. Whenever the equation falls under this case, the general ex- 
 pressions of the roots will be of no use in calculating their numerical values. 
 and then we can recur to the methods of Arts. 290-297. 
 
 EXAMPLES. 
 
 (1) r 3 — G.r— 9=0. 
 
 9 7 
 
 We have j9 = — 2, q= — - .-. ■\/q--\-p [i =-, which gives 
 
 •W= 
 
 16 
 
 9+ Vq i +P i =y-7T=2, 
 
 B=\J-q- Vq> + f=\ll=l. 
 
 Thus the three roots are 
 x=3, 
 
 *=-!+ V3V-l + J(-l-V3V-l) = '(-3+ V3V-1), 
 
 (2) x 3 — 21x+20 = 0. 
 
 Here p=— 7, 9=10; 
 
 .•.x=\j — 10 + 9 V^+V— 10 — 9-/^3. 
 This example is one of the irreducible case. The general value of x ap- 
 pears in an imaginary form, and yet the roots are real, being the numbers 1, 
 4, and — 5, which, by substitution, will be found to verify the given equation. 
 378. The solution of the irreducible case may be obtained, also, by the help 
 of a table of sines and cosines. We subjoin the method, for the benefit of the 
 student acquainted with trigonometry. 
 
 Solution of the irreducible case by trigonometry. 
 ;os 20=2 cos- 0—1 
 cos 30=2 cos 20 cos — cos 6 
 
44ft ALGEBRA. 
 
 Substituting the first expression in the second, 
 
 cos 3*9=4 cos 3 0—3 cos 0. 
 Whence 
 
 3 ! 
 
 cos 3 0— - cos 6— - cos 30=0 (1) 
 
 4 4 x ' 
 
 In the proposed cubic equation, which we may write under the form 
 
 x a +3px+2q = .- (2) 
 
 x 
 put the unknown r cos for x ; or, which is the same thing, put - for cos 6 
 
 und (1) becomes 
 
 3 1 
 
 x 3 — -r-x— -r 3 cos 30=0. 
 
 4 4 
 
 Comparing this with (2), we have 
 
 1 
 
 -r 3 cos 30=— 2q, 
 
 and 
 
 -r 2 = — 3p .-. r=2 -y/ — p, which is real, p being negative , 
 
 2o q 
 
 .-. cos 30=- ' 
 
 pr j _ps 
 
 Consequently, the trigonometrical solution of the proposed cubic equation, 
 that is, the determination of 0, and thence of r cos 0, depends upon Qie triscc- 
 lion of an arc, or the determination of cos from cos 30. 
 
 The mode of proceeding by aid of trigonometrical tables is obvious ; we are 
 
 to seek in the table of cosines for the angle whose cosine is q-J jj this will 
 
 be the angle 30, and, consequently, one third of it will be ; and the cosine of 
 this, multiplied by r, or 2 -/ — p, will give r cos 0=x for one of the real roots 
 
 of equation (2). As the given cosine, J-v/'^-y belongs equally to three arcs 
 
 viz., 30, 2n-\-3d, and 2tt — 30, by taking the cosine of one third of each of the 
 latter two, wo shall have the values of the remaining roots. Thus all the 
 three roots will be expressed as follows : 
 
 2 V r ^ cos 0, 2 y/— P cos -(2tt+30), 2 V — p cos -(2~— 30). 
 
 Or, using the supplements of the two latter arcs instead of the arcs themselves, 
 and remembering that the cosine of an arc is equal to minus the cosine of its 
 supplement, we have somewhat more simply (he three values of x in the fol- 
 lowing form : 
 
 2 V— p cos0, — 2-v/— p cos (G0°— 0), —2-/—/' co3(6O° + 0). 
 This method, with a single exception, applies to the irreducible case; for. 
 as the trigonometrical! cosino of an arc is always less than unity, except when 
 thai arc is a multiplo of 180°, we must have 
 
 »vi 
 
 ,,.<» .:q : <-p\ 
 
 or f+p*<0. 
 
 When 30 is a multiple of 180°, two roots must be equal. 
 The reducible case may also employ the aid of trigonometry. 
 
EQUATIONS OP THE THIRD DEGREE. 449 
 
 379. If in the expression 
 
 we put cot tpssA-J , it becomes -JU— cot ^± cosec $)L 
 Hence, reducing, the real root of r i -\-qx-{-r=0 is 
 
 which, by putting tan -= tan 3 0, may be further transformed into 
 
 -4 
 
 cot 26. 
 
 q 3 r- 
 Similarly, the real root of x 3 — qx-\-r=0, -^z <j, becomes (by putting cosec 
 
 3 
 
 7-/3V2- ^ 
 
 *=oU ' tan 2= tan 3 0), 
 
 \q/ 
 
 la 
 
 cosec 20. 
 
 -4 
 
 380. The following method of arriving at a now and valuable formula for the 
 solution of cubic equations will be found an excellent exorcise for the student :* 
 Let the given equation be 
 
 &+px+q = (1) 
 
 Placing 
 
 x=m+y (2) 
 
 we obtain 
 
 y 3 -\-3imf-\-(3m?-{-p)y-\-m 3 -{-pm+q=0 (3) 
 
 Taking 
 
 y=\ ( 4 ) 
 
 we obtain 
 
 \l) + 3m (\y + (2m*+py:+7rv>+pm+q=0 ; 
 which gives 
 
 3m 2 +p 3m 1 
 
 1 m 
 Placing 
 
 z 3 + , , , 2 2 4—T-i r- z +-r~, r-=° • • • (5) 
 
 1 m 3 -\-pm-\-q ' m s -{-pm-\-q ' ?Ji 3 -\-pm-\-q v ' 
 
 z-w- 3m2+i? • (6) 
 
 3(?n 3 -\-pm-\-q) ' ' ' \ i 
 
 we find 
 
 3p?n* +9qm— p* — 27qm 3 -{-18p"ni 2 -\-27pqm+27q' 2 -\-2p' i _ 
 
 W *+ 3{m 3 +pm+qy W + 27 (m a +pm+q) 3 = ° * ' ' 7 ) 
 
 * It is the production of an old pupil of the author's, Mr. James S. Woolley, whom ill 
 health, and other discouraging circumstances, have not prevented from making- some im- 
 portant discoveries in alycbra, which it would be premature at present to publish to the 
 world. 
 
 F F 
 
150 ALGEBRA. 
 
 The value of m, which renders the coefficient of w zero, may be foi nd thus 
 
 3pm 8 +9«/m— p»r=0. 
 Then 
 
 3<7 , 3 1 «• 
 m = -^, X ^ ; +I 
 
 The value of w in (7), substituting the value of m, found in (8), is ezpree 
 in the foUowing four equations, (9), (9, a), (9, b), (9, c), the last three being 
 obtained by decomposing (9) into factors. 
 
 1C: 
 
 xv. 
 
 w. 
 
 ( 8i g+ i2 )(-K /J^f) 
 («£-»)(-W±*4) 
 
 (9) 
 
 (9, a) 
 
 (9,6) 
 
 tV-|±7^ 3 +tX\/ 1 ^+ 81 | 
 
 ?i?—_ — £— 
 
 (9,e) 
 
 Substituting in (G) the values of ?ra and w, found in (8) and (9, c), we shall 
 have 
 
 ( 81 I+ 12 )(-IS&?) 
 
 Substituting in (4) the values «f z, given in (10), and decomposing one more 
 nf its terms into factors, we shall have 
 
 ■■Kjs^g_ 
 
 Hence 
 
 .y?&*f' ' /,'-',) v ^rn > vfe"+") ! 
 
 + 
 
 (continuing the numerator) ( 8l !^ 18 ) ( _ f ±\/a7+ ?) ( ia ) 
 
EQUATIONS OF THE THIRD DEGREE. 451 
 
 But the first term in the numerator of (12) may be transformed thus : 
 
 And the last term in the numerator of equation (12) is 
 Therefore the sum of the first and last terms of the numerator of (12) is 
 Therefore, 
 
 p'\27 J - ' 4\ 2~V27 J ' 4/ ' V * ' p* 
 
 54 
 
 7 
 
 54 n g» 
 
 Dividing both numerator and denominator by -p^2\/o7^ ?3 ~^T' we e 
 
 
 I= (_l ±% /^f) + l(_l ± ^^l) i 
 
 The numerator of this value of x is equal to 
 
 The denominator is equal to 
 
 Dividing numerator and denominator by the common factor, we have 
 
 x=- 
 
 (-^V^ 
 
 4V 
 
 27^+4 
 
 This formula may be reduced to that of Cardan by dividing the numerator 
 by the denominator, and observing that 
 
 we thus obtain 
 
 — HW^+9*+ (Hi W^+iT- 
 
 But the first form is preferable, as it gives only the three values which satisly 
 

 452 ALGEBRA. 
 
 equation (1), whereas Cardan's formula gives nine values, six of w aich have 
 to be rejected. 
 
 A partial division gives 
 
 1 
 
 r\ i. 
 
 3\3 
 
 •={-1+ P-4-t) 
 
 K-Wfe+9 
 
 which is an advantageous form, inasmuch as but one third root has to bo ex- 
 tracted, both radicals having the same form. 
 
 A shorter solution of the above might bo given, but we have already extend- 
 ed our article on cubics sufficiently far. 
 
 IRRATIONAL EXPRESSIONS ANALOGOUS TO THOSE OBTAINED IN THE RESO- 
 LUTION OF EQUATIONS OF THE THIRD DEGREE. 
 
 is V A± y/B ; 
 
 381. One of these expressions is"yAi -\/B ; but it frequently happens 
 that A and B are rational numbers, and then it may be possible to reduce 
 these radicals to simpler expressions, in which there are no longer radicals 
 over radicals. This problem has already been resolved for radicals of the 
 second degree, and it is now proposed to resolve it with reference to radicals 
 of the higher degrees. 
 
 VA+VB. 
 
 I shall", commence with the cubic radical y A+ VB. We can not suppose 
 for this root a quantity of the form V a-\- y/b, for we have 
 
 ( V&+ "/*)'=« Va+3«y6-f 3&V"a + 6 Vb 
 = (a+3b) V"a + (3a + J) -Jb, 
 
 a result which contains the radicals V 'a and -Jb. But the preceding calcula- 
 tion shows that we should have a result of the form A-j- \/B, by raising to 
 the third power the expression a-\- y/b and (a-j- V b) yc. I will choose this 
 last expression as the more general ; we shall then have 
 
 -4, 
 
 A+VB = (a+V6)Vc (1) 
 
 Raising both members to the thud power, it becomes A-|- -v/B = c(a 3 +3a6) 
 +c(3a 3 -f &)-/&>* equating tho rational parts together, and the irrational parts 
 by themselves. 
 
 A=c{a*+3ab) (2) 
 
 VB=c{3a°-+ b) yfb (3) 
 
 The problem, then, is, to find for a, b, c rational values which satisfy these 
 two equations. But squaring these equations, and then subtracting the one 
 from the other, we have 
 
 A 2 — B=c 2 (a 6 — 3a<i + 3aW— b*)=c-(a*— o) 3 ; 
 
 , vTa 2 — B)<- 
 
 hence a? — b = . 
 
 c 
 
 Since a and 6 ought to be rational, it will be accessary to take c such that 
 (A 3 — B)c be an entire or fractional cube, which is always possible. Calling 
 the second member of the above equation M. we shall have </-' — />=>!. 
 whence i=a 3 — M. By substituting this value of b in equation (3), it wiD 
 become 
 
EQUATION OF THE THIRD DEGREE. 453 
 
 4ca 3 — 3Mca— A=0 (4) 
 
 This equation must give for a at least a commensurable -ulue, without 
 which the transformation (1) will bo impossible. 
 
 If, instead of VA-f- \/B, wo should have to reduce y] A — y/ti, ir would 
 suffice to change throughout in the preceding method the sign of V b. 
 
 '/ — 
 
 For example, let the expression be \/ 14 i \/200. We shall have A = 14, 
 
 B=200, A 2 — B = — 4; hence (A 2 — B)c= — 4c ; we shall then have the 
 perfect cube —8, by taking c=2. Consequently, M = — 1, b = u-+], and 
 equation (4) becomes 8a 3 4-Ga — 14 = 0. It can be satisfied by the commen- 
 surable value a = l, which gives 0=2. Again, we have already obtained 
 c=2; hence, finally, 
 
 1/ 
 
 14± A /200 = (l± y/2)V>2. 
 
 J ZZ 
 
 Again, leu the expression be \ — II ±2 •/ — 1. We will pass 2 und^r tne 
 
 radical of the second degree ; we shall then have A= — 11, B=— 4, A 2 — B 
 = 125. As 125 is already the cube of 5, it will suffice to make c=l. Con- 
 sequently, we have M=5, &=a'-— 5, and equation (4) becomes 4a 3 — 15a 
 + 11=0. But this equation is satisfied by the value a=l ; hence fc=— 4, 
 and, consequently, 
 
 V 
 
 _ll±2V-l=(l±V-4)Vl. 
 
 382. Let us consider the more general expression yAi -\/B, and take 
 
 J 
 
 'A± VB = (ai Vb)Vc (5) 
 
 The problem, again, is to determine rational numbers for a, 6, c, if it be 
 possible. 
 
 Raising (5) to the power n, and equating separately the rational parts, we 
 
 obtain 
 
 n(n — 1) , n{n— 1)(»— 2)(n— 3) . , , , ,„, 
 
 VB= C [na^+^^=^a»-»6+,&c.]V6 • • • (7) 
 
 We can, as in the ca>e of the cubic radical, square these two equalities, and 
 subtract the one from the other; but the reductions will be immediately per- 
 ceived by observing that we ought to have, at the same time, 
 
 A+ VB=c(«+ Vb)*i A — VB=c{a— V~b) n ; 
 and that, consequently, 
 
 A»— B=d>(a+ V~f>) n (a— • v / "o) l, = c g (a 8 — 6)" ; 
 
 V(A 8 — B)c"-- 
 
 whence a- — b = . 
 
 •c 
 
 We see from this that it will be necessary to take c of such a value that the 
 second member of this last equation shall be rational. Calling this second 
 member M, we shall have a 3 — 6=M, whence &=a*— M; subsrirutin 
 
454 ALGEBRA. 
 
 value of b in (6), the resulting equation in a will have a commensurable root 
 every time that the transformation (5) is possible. 
 
 383. In the resolution of equations of the third degree, what renders the ir- 
 reducible case so remarkable is, that although we are assured that the three 
 roots are real, it is, nevertheless, impossible to make the imaginary quantities 
 disappear otherwise than by means of series. This difficulty is not confined 
 to the equation of the third degree ; it will be encountered equally in the gen- 
 eral formula 
 
 VA+BV— 1 + V^ 
 
 'A-BV-l (8) 
 
 which formula I shall stop to consider for a moment. 
 
 To consider this expression in its most general sense, we ought to comrjine 
 the n determinations of the first part with the n determinations of the second, 
 so that we shall have, in all, n- values. But the expression is rarely taken in 
 so general a sense, and I proceed to define that which we ordinarily attach 
 to it. 
 
 As the two radicals which have the index n represent the roots of the bi 
 nomial equation, their determinations are equal in number to the quantities 
 which have the form f-\-g V —1. Moreover, it is manifest that to each de- 
 termination of the first radical there corresponds one of the second, which 
 only differs by the sign of yf —1. But we suppose that these corresponding 
 values are those which ought to be added in formula (8) ; and, with these re- 
 strictions, the values of x are all real, and only n in number. 
 
 The product of these two radical values, thus taken in a same pair, is real 
 and positive ; but for the product of the two radicals we have, in general, 
 
 Va+bv-ix\/a-bv-i = V 
 
 VA^ + B-\ 
 
 and the radical which expresses this product can only have a single real ana 
 positive value ; hence, if we represent it by K-, we ought to be able to charac- 
 terize the conjugate values, which must be added in formula (8), by the con- 
 dition that their product be equal to K 2 . 
 
 Formula (8) can be regarded as a general expression of the roots of an equa- 
 tion whose degree is marked by the number of values of which the equation 
 is susceptible ; hence, provided that it be taken in its greatest extension, oi 
 with the restriction which wo have just mentioned, the degree of the equa- 
 tion must bo either n" or n. 
 
 This last remark leads us to explain how we form an equation, when we 
 know the expression for its root; that is to say, thai an equation being given, 
 susceptible of taking different values, by reason of the multiple values of the 
 
 radicals which it contains, it is required to find an equation free from radicals 
 which has these values for roots. I will tako, for example, the same expr 
 sion (8). 
 
 To abridge, let us make 
 
 A- r BV^T=(/, ^.BV~l=5; 
 the problem reduces itself to eliminating y and i between the three equation! 
 
 y+z=.r, >/ = a, v = h. 
 But hero the elimination can be conducted according to a very an i\> ,>i o- 
 
EQ.UATION OP THE FOURTH DEGREIv 465 
 
 cess, analogous to that which has been employed for reciprocal equations. By 
 the rules of multiplication we have 
 
 (r+~ m )(2/+2)_=r +1 + zm+1 +2/-(r_^_+= m - 1 )- 
 
 But y-\-z=x and yz=%/ab ; hence, making y ab=c, the equation will 
 become 
 
 yn-H -\-z m+1 =x(y m -\-z m ) — c(?/ m ~ I -f-;'" _1 ). 
 
 By means of this formula wo express, in function of x and c, successively all 
 the quantities y"-\-z 2 , y 3 -\-z 3 , &c. When we have arrived at y n -\-z", we re- 
 place y n -\-z n by a-\-b, and then we shall have' the required equation, which 
 will be of the degree n in x. 
 
 This equation contains c ; but we have c= yah— ?JA 2 -\-B 2 ; hence, c is, 
 in general, susceptible of n different values. By putting in the equation each 
 of these n values in its turn, we shall have n equations, and, consequently. 
 nXn, or n 2 values of x. This, in fact, ought to be the case, from what has 
 been said at the close of the preceding article. If we should wish to have a 
 single equation which has all these values for roots, it would be still necessary 
 to eliminate c between the equation of the degree n in x and the equation 
 c n =ab. 
 
 But if in formula (8) we only wish to associate the radical values whose 
 product is real, it is this real valuo solely which we must choose for c, and we 
 shall only have a single equation of the degree n for determining all the values 
 of x. 
 
 RESOLUTION OF THE EQUATION OF THE FOURTH DEGREE. 
 
 384. After having made the second term disappear, the general equation of 
 the 4° degree is 
 
 x 4 +2 }x2 +<l x + r = (!) 
 
 If we make x=a-\-b-\-c, squaring, there results 
 
 ;r 2 =a 2 +& 2 -fc 2 +2(a&-f ac+ic), 
 )r, transposing, 
 
 a -2_( a »+Z. 2 +c 2 )=2(ai+flr+6c) ; 
 raising anew to the square, we have 
 
 ;r*— 2(a 2 +& 2 +c : ):r 2 +(a 2 -}-^+e- 2 ) 2 ^^ 
 then, replacing a-{-b-\-c by x, and transposing, we obtain 
 
 . r 4_2(a 2 4-6 2 +c 2 ).r 2 — 8rtic.r+(a 2 -j-?, 2 -f c 2 ) 2 
 _ 4 (a-b 2 + a 2 c 2 -f b-d 2 ) = . 
 
 This equation is without a second term, and by the manner in which it has 
 been formed, we know that it admits of the root x=a-\-b-\-c. Thus, we re- 
 solve equation (1) in determining a, b, c, by the condition that it shall hi iden- 
 tical with the preceding, which gives 
 
 — 2{a"+b- + c 2 )=;> 
 — 8abc = q 
 (a 2 -f 6 2 +c 2 ) 2 — 4(a 2 6 2 +a 2 c 2 +i-c-) = r. 
 
 These equalities show that, by taking a 2 , 6 2 , c" for unknowns, these three 
 quantities are the roots of an equation of the 3° degree, the coefficients of 
 which are (see Art. 245) 
 
456 ALGEBRA. 
 
 _(a»+6»+c»)=:| 
 
 2 
 
 p 2 — 4r 
 
 • lo 
 
 9* 
 64 
 
 Consequently, this equation of the 3° degree is 
 p ■))"- — 4r q" 
 
 + 2~ + 16 ■- 64 _U W 
 
 Such is the reduced equation upon which the solution of equation (1) depends. 
 Suppose that the three values of z have been determined, which designate 
 by z', z", z'", we shall have 
 
 «=± V*, 6=± V^ 77 , c=± V~'- 
 If the signs be combined in all possible ways, there will result eight values 
 for a-\-b-\-c or x. But as the last term of the reduced equation (2) was 
 
 formed by squaring the equation abc= — -q, it follows that the values contain 
 
 o 
 
 rtot only the roots of the proposed equation, but also those of an equation 
 which would differ from it in the sign of q. 
 
 At the same time it may be perceived that, to have only the roots of the 
 
 proposed, it is nocessaiy to add only the values of a, b, c, for which abc= — -q, 
 
 8 
 
 and the product of which has, consequently, the contrary sign to q. In each 
 
 particular case it will be easy to determine for the radicals three values, A, B, 
 
 C, which shall fulfill this condition ; and afterward, with these values, we 
 
 form the four roots of ihe proposed, to wit, 
 
 .r= + A+B + C, .t= + A— B-C, 
 
 x= — A+B — C, x=— A — B + C. 
 Generally, instead of A, B, C, the three radicals are placed, and the values 
 of x are written Jhus : 
 
 x= + ■/£+ V^- V£5 x= + -/£- V£+ v5f . 
 
 x=- yfz'+ V~"+ Vz'", x=—^z'- Vz"- V-'". 
 But it is necessary to understand that in applying these formulas to particu- 
 lar cases there must bo taken for -\/z', -J z", yf ':'" three determinations, tho 
 product of which shall be of tho 88 n as q. This observation is im 
 
 portant; failing to have regard to it, we might find false ro 
 
 385. The nature of the roots of the reduced equation will make known the 
 nature of tho roots of tho proposed. But tho reduced ha\ last term 
 
 negative, has always one positive mot (see \rt. '.'!-. Prop. VIII., Cor. 4), and 
 the product of tho other two roots should bo positive ; then, if these last are 
 not imaginary, they will be both positive or both Degative. I pass over the 
 case in which q=0, because then the proposed would b - solved by the rules foi 
 the second degree. Consequently, there are three cases only to be examined. * 
 
 1°. Cast where the three roots of the reduced equation arc positive. There 
 
 the four values of .(• are evidently real, and if the radical 
 
 regarded as representing positive determinations, their pro luct will be positive; 
 
 This explains an operation in Art 
 
DIOPHANTINE ANALYSIS. 457 
 
 then tn« preceding formulas will bo specially applicable to the case of <7^>0. 
 For q<^0 it would be necessary to change the sign of one of the radicals. 
 
 2°. Case where the reduced has one root z' positive, and two z", z'" negative. 
 The radical V z ' will be real, but the radicals V ' z" and -<Jz'" will be imagi- 
 nary ; consequently, the four values of X will be imaginary also, unless 2"=;'". 
 When z"=z'",one of the two quantities V~"+ y/z 777 and y/~ — -/p 7 will 
 become zero, and supposing it to be the latter, the values of x will be 6imply 
 
 x= -v/z 7 , x = -/I 77 , x= — •v/I 7 +2 y/U 7 , x= — V~'—2 Vz"- 
 
 The first two are real, since z' is positive, and the other two avo imagi- 
 
 naiy, since z" is negative. Besides, as in the reduction, wo have supposed 
 
 ■\/z"= y/-z'", we ought to have here y/z' \J z" y/z'"=z" y/z' ; so that tins 
 
 product can only have the sign of q by choosing for y/ z' a sign contrary tc 
 
 that of q, since, by hypothesis, z" is negative. 
 
 3°. Case in which the reduced has one root z' positive, and two roots ■/.", z' ' 
 imaginary. The positive root z' being known, we can divide the reduced by 
 x — z', and wo shall have an equation of the second degree, which will give for 
 z" and z'" imaginary values of the form 
 
 z"=f+g V=l, z'"=f-g /="l. 
 Consequently, two of the values of .r will contain the sum 
 
 yjf+g /ZI+V/-« V"^l ; 
 
 and the other two will contain the difference 
 
 Vf+gV-i-\lf-gV-i- 
 
 THE DIOPHANTINE ANALYSIS. 
 
 386. This branch of analysis derives its name from its inventor, Diophan- 
 tus, of Alexandria, in Egypt, who flourished about the year 360, A.D. It 
 relates chiefly to the finding of square and cube numbers. 
 
 The solutions of the questions must frequently be left, notwithstanding the 
 various rules that have been given for this purpose, to the talents and ingenui- 
 ty of the learner, who, in pursuing these inquiries, will soon perceive that 
 nothing less than the most refined algebra, applied with great skill and judg- 
 it, can surmount the various diff which attend them ; and, in this 
 
 respect, no one, perhaps, has ever excelled Diophantus, or discovered a greater 
 knowledge of the extent and resources of the analytic art. 
 
 When we consider his work with attention, we are at a loss which to ad- 
 mire most, his singular sagacity, and the peculiar artifices he employs in form- 
 ing such positions as the nature of the problems requires, or the more than 
 ordinary subtilty of his reasoning upon them. 
 
 Every particular question puts us upon a new way of thinking, and fur- 
 nishes a fresh vein of analytical treasure, which can not but prove highly use- 
 ful to the mind in conducting it through other difficulties of this kind when- 
 ever they occur, and also in enabling it to encounter more readily those that 
 may arise in subjects of a different nature. 
 
 The following directions for resolving questions in the Diophantine analysis 
 
458 ALGEBRA. 
 
 will be found useful ; but no general rule can be given, and, therefore, tne 
 student must often be loft to depend solely upon his own ingenuity and skill. 
 
 \ 
 
 RULE. 
 
 Substitute for the root of the square or cube required, one or more letters, 
 such, that, when they are involved, either the given number or the highest 
 power of the unknown quantity may vanish from the equation : and then, it' 
 the unknown quantity bo of the first degree, the problem will be solved l»y 
 reducing the equation. But if the unknown quantity be still a square or a 
 higher power, some other new letters must be assumed to denote the root, 
 with which proceed as before, and so on till the unknown quantity is but of the 
 Irst degree, and from this all the rest will be determined. 
 
 EXAMPLES.* 
 
 (1) To find two square numbers whose sum is a square. 
 Let x" and y- be the two squares ; let 3z and 4; be the roots. 
 
 Then 25: 2 = D f=n — bz\- = n°~— 10«; + 25z 2 ; 
 
 .-.z=— — ; if n = 10, z=l, then ;X3 = 3 and :X4=4 , 
 
 and the two squares are 9 and 1G, whose sum is 25, a square, if n = 20, :=2 ; 
 and from this we get another value of x and y, and so on. 
 
 (2) To find two square numbers whose difference is a square. 
 Let x- and y- be the two squares. 
 
 Assume a: 2 — y 2 =:{x — ny)~=x" — 2nxy-\-n"y". 
 Then —if=—2nxy+n°tf-, 
 
 or 2nx=(n 7 -\-l).y ; 
 
 n 2 -4-l 
 
 •■• x =-n — -y- 
 
 2/i J 
 Suppose y=2n, then x=n"-\-l. If n=2, y=l, and x=5 ; also J 8 — y 3 
 = 25 — 16=9, a square number. If n=3, y = G. and .r=10; also x* — y 9 
 = 100 — 36=G4, a square number. 
 
 (3) To change the sum of two squares into the sum of two others any num- 
 Der of ways at pleasure; for example, in three different ways. 
 
 Let a" and 6 s be the given squares, and let a — x and ex — b be the roots of 
 the required squares; then, by the question, we get 
 
 a—x\■ z +'cx^l>\ : = a- + L^ 3 ■, 
 by involution, a 2 — 2a.r-|-.r 2 -f.c 2 .r 2 — 2b<\r+lr = u ■ + /.■ ; 
 
 by transposing and dividing, 
 
 _o rt _j_. r _j_ C <! X _ o ( 7,_o, 
 
 26c+2a 
 
 cir c <i x4-x=2bc-l r -2,i and .r= — — — — , 
 
 1~T" 
 
 where c may bo taken at pleasure ; for example, 
 
 c = 2, 3, and I , 
 
 4o-|-2a 66+2a Bo+2a 
 
 then, '= — , — — — , and — — — . 
 
 * Many of these problem! are sell cted from tin- Arithmi deal Questions of Diopbantos, 
 of which six out of tin' ow remain. The best edition is that published at 
 
 Paris, bj /■' " het, in the year 1670, with notes by Fern 
 
 t This sign □ denotes that the number placed equal to ii is a perfect scjuaru 
 
DIOPIIANTINE ANALYSIS. 459 
 
 (4) To divide a number which is the product of the sum of two squares oy 
 the sura of two others, into two squares two different ways. 
 
 Let a"-\-l~ bo tho sum of two squares, and c 2 +rf 2 the sum of two others, 
 whose product (a 2 +& 2 ) . (c' i +d i ) = {ac + bdy+(bc—ad)*={ac—bd)- + (6c 
 •\-ady, as required. 
 
 (5) To find a number, .r, such, that .r+1 and x— 1 shall bo squares. 
 
 Let 
 and 
 
 or 
 
 and 
 
 
 x+l = 
 x — 1 = 
 
 = b* 
 
 
 
 
 .-. 2= 
 
 =a*- 
 
 -b* by 
 
 subtraction ; 
 
 
 .-.2X1 = 
 
 3: 
 
 = (a+b)(a- 
 =2a, and a 
 
 3 
 
 = 2' 
 
 
 a 2 
 
 9 
 - 4 ; 
 
 
 
 
 .-. .r+1 
 
 9 
 
 = 4' 
 
 and x= 
 
 5 
 
 "4' 
 
 x+1 
 
 =f 
 
 
 
 
 x 
 
 =f — \ 
 
 
 
 
 Or thus 
 
 x — 1=2/ 2 — 2= □ — s — ?/|-=6- c — 2s2/+3/ 2 
 
 .-. s 2 — 2sy = — 2 
 
 s 2 +2 
 2st/=s 2 +2, andy=-£- ; 
 
 9 5 
 
 take s = l ••• 2/=i> anc ^ z=^ 3 — 1 = 7 — 1=: I' as before. 
 
 (6) Required to find four square numbers whose sum shall be a Q . 
 
 Let 1, 4, 9, and x 2 be the required squares; then, by the question, we get 
 
 14+x 2 = D =7i— x| 3 =n 2 — 2nx+x 2 , 
 n- — 14 
 and x =—2n- 
 
 5 25 
 
 where n may bo any number at pleasure, if w = 3, .r= — p x 2 =— , or if n=4, 
 
 1 11 225 ~15 
 
 x=-, and the numbers are 1, 4, 9, and — ; then 1 + 4+9+— = —-=— 
 
 as required. 
 
 (7) Divide 2 into three rational squares. 
 Let x, 2.r — 1, and 3x — 1 be the roots of the three squares respectively: 
 
 then x 2 +4x 2 — l.r+l + 9x 2 — 6x+l=2 ; 
 
 by transposing and dividing, 
 
 5 3 8 
 
 x=-, 2r — 1 = 7' 3x — 1= 7 > the roots; 
 
 and the Q 's will be 
 
 25 , 9 , . 64 
 
 * 2 =49' 2a; - 1 l 2 =i9' anl 3^~ li 3 =49' 
 
 „ , , 25 9 64 98 
 the sum of which is - + TJ^+TJj^ — ~ the P roof - 
 
 Or thus : 
 
 Let 1, x 2 , and y- be the squares ; then 
 
46G ALGEB&A. 
 
 1-f r 2 +?/-=2 and ^+i/ : = l, 
 or x 2 = ] — ?/ 2 = D = 1 — ny\ 2 = 1 — ~>nj -J- "' 3 2/* 
 
 2rc 
 
 4 
 where n may bo taken any number greater than 1 ; if n = 2, then y= - and 
 
 16 9 
 
 y 2 =— ; then will x 2 =— , and the sum of these plus 1 is evidently 2. 
 
 (8) Divide - into three rational squares. 
 
 Let x, 2x — -, and 3x — -, be tho roots of the rational squares, and then 
 
 squares are 
 
 1 1 
 
 x 2 , 4x 2 — 2x+-, 9X 2 — 3x+-, 
 
 and x 2 +4x 2 — 2x4-7+9x 2 — 3.r+-=-, 
 
 i 141 '42 
 
 5 25 
 
 and x will be found to be — , from which we get the three squares, viz., ~, 
 
 9 G4 1 
 
 77^-, tt^i a »d their sum is evidently -, as required. 
 19o Ufa 2 
 
 (9) To divide a given square number, 100, into two such parts, that each 
 of them may be a square number. 
 
 Let x 2 be one of the parts, then 100 — x 2 , the other part, will be a square 
 
 number. 
 
 Assume 100— x 2 =(2x— 10) 2 =4x 2 — 10x-flb0. 
 
 .*. x=8, and 2x — 10 = 6 ; honce 64 and 36 are the parts requin J. 
 
 The same problem may be resolved generally in the following manner : 
 
 Let a 2 be the given square, x 2 = one of its parts, and a 2 — x 2 the other. 
 
 Assume a 2 — x 2 = (nx — a) 2 =n 2 x 2 — 2anx-\-cP\ 
 
 Then — x 2 = ;t : x 2 — 2a«x ; 
 
 2na an 2 — a 
 
 -, and nx — a: 
 
 •■-- „* + l'-"" "* "-n=+l ' 
 / 2na y lan 2 —a\ 2 
 
 2-1-1/ \ri--\-] 
 
 are the two squares required ; in which expressions a and n may be any whole 
 numbers whatever, provided n be greater than unity. 
 
 (10) To liud a numboiyr.sucb.that x-j-128 and x+192 shall bo both square 
 numbers. 
 
 Assume x-f- j.28 = : : •*• £= - — 1 '-' 3, which is one condition answered; then 
 2 s — 128+192r=z s +64=E D =a 2 .■.z 2 =a-— 64 ; then wo have only to assume 
 such a value for a as will make <c — 04 a square; but it is plain that if a be 
 taken =10, then a 2 — -64=36= □ , and ; 2 = 36; but this would make the value 
 of x negative, then, in order to find values for : that will make X positive, take 
 a = 17, and the.i «*=289, and ••• 0*— fa 1=225= □ .-. :-' = 225 and ••• s = -22o 
 — 128=97, the value required. 
 
 (11) To divide a given Dumber, 1 3, consisting of two known squares, 9 and 
 4, into two other square numbers.* 
 
 * in the eolation given of the above problem, n awl m may be taken emu] to any nuin 
 
DIOPHANTINE ANALYSIS. 4G1 
 
 Let nx — 3 be the root of the first squaro sought, and mx — 2 the root of 
 the other squaro. 
 
 Then nx—3\ 2 -\-7nx—2\ 2 =13, 
 
 or (n 3 -fm 2 ) . z 2 =(4wi + 6?i) . x; 
 
 6n-\-4m 
 
 x = 
 
 n-^-m 
 
 2 ' 
 
 3n 2 +4?;m— 3m 2 
 
 whence nx — 3= — — = the root of the first squire. 
 
 n--f- m 
 6mn— 2n 2 +2m 2 
 
 and mx — 2= — — - = the root of the second. 
 
 ?i 2 +m 2 
 
 „ , , 3ra 2 +4m«— 3m 2 17 
 It n=2 and m=l, we have — — 5 =— = the rootot one square, 
 
 (§m— 2rc 2 +2m 2 6 
 and — — ^ =7= tne ro °t of the other square. 
 
 (12) Let 14 bo divided into three rational squares. It is well known that 
 the least three squares in whole numbers are 1, 4, and 9, which will answer 
 the question; but to give a general solution, 
 
 Let 1, 3x — 2, 2x — 3, be the roots of the required squares; 
 
 24 
 then l + (3.r— 2) 2 +(2z— 3) 2 =14, or.r=— ; 
 
 24 72 „ 
 
 then — X3 = — , from which subtract 2 ; 
 
 J. O J. O 
 
 /46\ 2 2116 24 48 
 
 then y— J =-^n » T3 X 2= i3' "' om wllicl1 subtract 3 ; 
 
 /9\ 2 81 2116 81 
 
 then {-) =^ .-. 1+ — +_ = 14. 
 
 (13) To find two square numbers whose difference shall be equal to any 
 given number. 
 
 Let x be the root of the lesser square sought ; and let d, the given difference 
 of the squares, bo resolved into any two unequal factors a and b, of which a is 
 the greater. 
 
 Let x-{- b be the root of the greater square ; 
 
 then (x-{-b) a — x n -=d=ab, 
 
 i. c, 2x-\-b=a, 
 
 a — b 
 Whence x=— - — = the root of the lesser □, 
 
 a + b 
 and a:-f-& = — — = the root of the greater. 
 
 Tf d=60, and a X 6=30 X 2, we have 
 
 30—2 30+2 
 
 — — =14, and— J-^=16; 
 
 whence 16 2 and 14 2 are the squares required whose difference =60. 
 
 bers whatever, provided their ratio be not that of 3 : 2. For if n were to to as 3 to 2, the 
 roots of the squares sought would be found the same as the roots of the known squares. 
 Tf it were required to divide a given square, ofi, into twe other squares, 
 
 Since {m*-\-n")-={m°— n"-)--\-{fimn)^, 
 
 .-. ( m 3_J_ ¥ jj)s . x i—{rrfl— iv i)i . ^-(-(2/»kVJ . x -2, 
 
 where m and n may be assumed at pleasure, m being greater than n. 
 
4b'2 ALGEBRA. 
 
 (14) To find two numbers, such, that if either of them be added to the 
 square of the other, the sum .shall be a square number. 
 
 Let x--\-2xy and y bo the required numbers ; 
 then x 2 +2xy-\-if= D =x-l r y\ % ; 
 
 hence it only remains to make 
 
 2/-j-2.n/+j.-'p= U =x i -\-ny\-=x*+2nx 2 y-\-n i y*, 
 
 l^_4.,-3_2«.r !J 
 
 •"• y = w 2 — 4x 2 ' 
 
 3 , 19 
 
 If n=2j, and z=l, then 2/=r^ and £ 2 -|-2.n/=— , which are two numbers 
 
 that will answer the conditions ; for 
 
 Y 
 
 13 
 
 2 19 256 16 
 + I3 = 169 = r3 
 
 19 
 and 13 
 
 ' 3 _400_20| 2 
 
 13 169 13 
 
 Or thus : 
 
 Put - — x and x for the numbers; then - — x-|-x 2 =(- — x) , a square, and 
 
 = D , where .r may be taken 
 
 1 1 .t 1 a: 1 
 
 + X =16-5+ 3: +- T -i6 + o4- 2 =4+.r 
 
 4- r 
 
 at pleasure, provided it be less than —. 
 
 (15) To find two numbers whose sum and difference shall be both square 
 
 numbers. 
 
 Let x and y be the two numbers ; then, by the question, 
 
 x-\-y= o =a 2 and x — y=. d =i s ; 
 
 add both squares, and we get 
 
 2or=a 2 +6 2 ; 
 
 a 2 4-i 3 
 hence x= — - — . 
 
 Again, by subtraction, 
 
 2y=a 2 — 6 2 and y= — - — , 
 
 where a and b may be taken at pleasure, provided a be greater than b ; if 
 
 9 + 1 9 — 1 
 
 a = 3 and b = l, then — - — =5 and — -—=4, whose sum and difference are 
 
 both squares. Or thus : 
 
 Let x and x* — x be the numbers. 
 
 It is evident that their sum is a square ; and, in order to satisfy the other 
 condition in the question, 
 Assume x — n| 2 =x 2 — 2x, the difference of the numbers ; 
 
 whence 
 
 2n— 2' 
 
 .-. X —X= ^ 2n _ 2 | — o ;i _o' 
 
 « 2 i n 2 ) 2 n 2 
 
 Hence the two numbors aro -— iuid < > — -, ia which n inav 
 
 2n — 2 { 2n — 2 ) 2n — 2 
 
 9 
 bo taken at pleasure, provided it be greater than 1. If 7i = 3, J=t, and 
 
 45 
 
 x° x . 
 
 16 
 
DIOPHANTINE ANALYSIS. 
 
 4G3 
 
 ^16) Find two numbers Whose sum is a square, the sum of their squares a 
 square, and either added to the square of the other a square. 
 
 Lot 7 — x and x bo the numbers ; then their sum 7 is a sauare, and - — x 
 4 4 4 
 
 +x 2 
 
 D=--x 
 
 1 x 1 » 
 
 a square, and — — --{-x-|-x 2 = u =--|-x a square ; and, 
 
 in order to satisfy the other condition, we assume 
 
 1 x 1 
 
 nx 1 
 
 n- 
 
 • 1 3 14 4 
 
 which, solved, gives x=77j^, if ra=4, x=— , and 7— x=— , so that — 
 
 3 
 and — - are numbers that answer the condition? as foUows : 
 
 .CO 
 
 3 
 
 28 
 
 + 
 
 28 
 
 25 __5_ 
 
 : ^=28 
 
 and 2l 
 
 '■ 4 9 112 121 111 3 
 
 + 28 — ^p+-fSl2— =] 3 — 28 5 
 
 28 
 
 also, 
 
 4 
 28 
 
 i 3 16 84 100 10 
 
 1 oq msirr;! 7r^~,i oq 
 
 28 28l 2 281 
 
 28 
 
 ;js 
 
 (17) Find two such numbers, that if their product be added to the sum of 
 their squares, the sum shall be a square. 
 
 Let 2x be their sum and 2y be their difference ; then the greater will be 
 
 r-\-y and the less x — y ; hence x" — y" = their product, and 2x 2 -f-2;y 2 = the 
 
 sum of their squares; then, by the question, 3x 2 -{-y-z= □ =nx — y\ 2 and 
 
 2n y 
 
 x= : — ; if n=2 and V=2, . . x=8, which will answer the conditions. 
 
 n 2 — 3 
 
 (18) To find two square numbers, such, that the difference of their cube 
 roots shall be a square number. 
 
 Let x 6 and y 6 be the required numbers. Then X? — y"= O ; consequently, X 
 and y may be any two numbers which are the hypotenuse and one leg of a 
 right-angled triangle, and the least lumbers of this description are 5 and 3, and 
 the numbers themselves 15625 = 125 2 and 729=27 3 . 
 
 (19) Find three numbers, such, that not only the sum of all three of them, 
 but also the sum of every two, shall be a D . 
 
 Put 4x, x 2 — 4x, and 2.r+l for the three numbers ; then it only remains to 
 render 6x-|-l = □ • 
 Assume its root n — 1 ; 
 
 then 
 
 whence 
 
 6x+l=?i — ll 2 =n, 2 — 2»+l; 
 
 X= : 
 
 ■xn 
 
 if ns=12, x=2^, which will answer the conditions of the problem. 
 
 (20) Find two numbers, such, that the sum of their squares and the sum of 
 their cubes shall be both squares. 
 
 Let b be the base, p the perpendicular, and h the hypotenuse of a rational 
 
 right-angled triangle, x any multiplier of b,p, and h ; then (bx)--{- (px) 2 =(hxy, 
 
 but (bx) z -\-(px) 3 = a rational square =r\r 2 ; hence (b 3 -{-p 3 ).x=r i , 01 
 
 r" 
 x= , 3 , ■ ; ; now if r=b 3 -\-p 5 , . . xss&'-j-p 3 , and .-. bx=b(b 3 -\-p 3 ), px=py 
 
464 ALGEBRA 
 
 (bi+f), now let 6 = 3, p = i; then is x= ( Jl, 6x=273, a 
 6=6 and p=8, then x=728, 6r=4368, and pr=5824, and so on 
 
 (21) Find a number to which if 8 be added, the sum shall be a cube. Bud 
 from which if 1 be subtracted, the remainder shall be a cube. 
 
 Let x be the number; 6=2, esl; then r+6" = a cube and x—?= a 
 cube ; 
 
 hence •+6 3 =(6 + ^0 3 = 63 + 3c2a + -£r« 2 +^ 3 : 
 
 Assume x— c* = {a— c)»= a cube =a 3 — 3a 2 c+3ac 2 — c\ and .-. x=a> 
 
 _9 a 2 c _j_3ac 2 ; and, equating both values of x, we get 
 
 3c 4 c 6 
 a 3 — 3a 3 c+3a^=3ac 2 +-^-a 2 +^- 6 a 3 , 
 
 6 3 +c 3 7 3c6 3 
 whence a== 6 6 — c* X3c 6 3 — c 3 ' 
 
 ■ad, putting the right-hand member of this equation into numbers, we get 
 
 3X8 24 
 
 a -8-l~7 '' 
 
 5256 
 hence x — "343"' 
 
 (22) To find three square numbers,such, that the sum of every two of thera 
 «hall be a square number. 
 
 Let x 2 , y\ and z 2 be the numbers sought. 
 
 Then x 2 +z 2 , 2/ 2 +z 2 » and x 2 +t/ 2 are the three numbers ; i. e., 
 
 X 1 yi X 1 y* 
 
 -+1,^+1, and -+^ 
 
 are threo square numbers. 
 
 x m 2 — 1 y n 2 — 1 
 
 Assume 7= -gjp *nd - =— , 
 
 we have 
 
 x 2 m -«_i-2m 2 +l , y 8 » 4 +2n 8 +l 
 
 -:4-l = r - ; ) a nc » ~7+ 1: = n. < 
 
 2 2 ~ 4»i 2 = 2 ' 4» 8 
 
 x ? +i/ J 
 
 which are evidently two squares; and therefore it remains to make — ~r~ a 
 
 square number. 
 
 Now 
 
 x 2 +y 8 _ ( m 2 — 1 \ 2 i w 8 — 1 ) - (w 8 — l) a (n 8 — 1) 3 _ 
 ~~?"~ — {"l^Tf "M 2« J ~ 4m 2 + 4n 2 
 (ot°— l)3.n g +(w 8 — l) 8 .m 3 
 4m 2 7i 3 
 a square number. 
 
 Hence 
 (ro 2 — l)».n 9 +(n«— l) 2 .m 2 , or (>« + l) 2 . (m-l) 2 .7i 8 4-(w+l) 2 . (n — ])'. m»=a 
 a square number. 
 
 Let m-\-\z=.n — 1 .-. 7i = m-j-2. 
 
 Hence (wi+1) 2 . (m — l)-' .(»' -+--)• + '"• ' 
 
DIOPHANTINE ANALYSIS. 165 
 
 or (m — 1) s • (?h + 2) 2 -|-to 2 . (m+3) 2 , 
 
 or 2>n 4 +8m 3 -\-6m 2 — 4m + 4, 
 
 is a square number. 
 
 , 5m 2 
 Let the root of this quantity be assumcd = — — — 7n-\-2. 
 
 /5m 2 \ 2 
 
 Then (— — m+2j =2wi 4 +8m 3 +Gm 2 — 4/re+4 
 
 whence m = — 24, and n = — 22. 
 
 x m 2 — 1 575 y n-— 1 483 
 
 Also, tsss— - = — — , and -=— -z — = — — ; 
 
 z 2m — 48 z 2n —44 
 
 575: , 483« 
 
 henco x= — — , and y= — — — -. 
 
 48 J 44 
 
 To obtain the answer in wftolo numbers, let 2=528 ;* then r= — 6325, and 
 
 y = — 5796. Hence 528, — 5796, — 6325 are the roots of the squares, and 
 
 528 2 , 579b 2 , 6325 2 are the squares required. 
 
 (23) To find three cube numbers, such, that if from every one of them a 
 
 given numoor 1, be subtracted, the sum of the remainders shall be a square. 
 
 Let l-f-.r, 2 — r, and 2 represent the required roots. 
 
 Then, per question, (l-f-.r) 3 — 1 + (2— xf — 1 + 8 — 1= a ; 
 
 or (l+x) 3 4-(2— .r) 3 +8— 3 = a . 
 
 ar 5 +3x- 2 +3.r-j-l + 8 — 12.r+6.r 2 — .x^+S— 3= D ; 
 
 9x 2 —9x+U = a , =(a— 3x) 2 =a 2 — 6ax+9x* ; 
 
 11 — 9a-=a 2 — Gax; 
 
 a 2 — 14 
 and 6ax — 9.r=a 2 — 14 .•. .r= 
 
 6a— 9 
 _17 
 15 T~ X — 15'" "15' 
 
 16-14 2 17 28 
 
 Suppose rt=4 ; tuen x= — — — =Tt> an " l+- r:= T7> anc * ~ — x= 
 
 4913 , /28\ 3 21952 
 
 ••• <*+*> -3375' &- X) -U ="337^ and 8 
 are the numbers. 
 
 (24) It is required to find three integral square numbers, such, that the dif- 
 ference of every two of them shall' be a square number. 
 Let the roots of the required numbers be denoted by 
 
 s-+tf<, s 2 -y 2 , and r 2 +x 2 . 
 Assume r 2 — x-=s--\-y-; 
 
 then r- — x 2 —s-=y-= O 
 
 and y* = r* — 2>--x- — 2 r 2 s- -\- x* -\- 2x"s' -f- s 4 ; 
 
 but ( f 2+a«) a — (s s —y-)- = D 
 
 = (r 2 +x 2 ) 2 — (s 2 — r°~+x 2 +s 2 ) 2 =r i + 2r 2 x 2 +x 4 — s*+2r-s-— 2s"-x 2 — 2s*— *• 
 + 2r 2 x 2 + 2r 2 s 2 — x 4 — 2s 2 x 2 — s 4 = □ 
 =4r 2 x 2 +4?-V— 4s 2 .r 2 — 4s 4 = D 
 = 4 (r 2 .r 2 + rV — s-.r 2 — s 4 ) = D , # 
 
 .-. r 2 x^-{-r 2 s 2 —s"'x 2 —s 4 = a =a 2 , 
 and (r 2 — 5 2 ) • x"- = a 2 — r 2 s 2 -f s 4 , 
 
 fl S_ r V+«< fl3 
 
 and 2?= r : — = — -—s 2 ; 
 
 r - — S^ 7- — S 2 
 
 take r=21 and s = 13, 
 
 * The least coirunon* multiple of the denominators, 49 and 44. 
 
 G G 
 
466 ALGEBRA. 
 
 r- 
 
 ** n - 2 =i4T^l69- 169 
 
 a 
 
 2 
 
 Take a = 340, 
 
 then a^=25G and y"=r 2 — s-— x 2 =441 — 25G — 169 = : 6, 
 
 .-. (/---f.r-)- = (441- r -256) 2 =(697) 2 = one numter, 
 and (r 3 — * 3 ) 2 = (s 2 +2/ 2 ) 2 = (441 — 25G) 2 = (185) 2 = the second number, 
 and (s 2 — ?/ 2 ) 2 =(169 — 1G) 2 = (153)-, which is the other number. 
 
 (25) To find three square numbers such, that their sum, being severally 
 added to their three roots, shall make square numbers. 
 
 Let 2t,*6x, and 9.r denote the three roots; .-. by the question, 
 
 121z 2 +2.r=D, * 
 121x 2 +6.r= d. 
 121z 3 -r-9.r= D- 
 
 Assume x=-^- ; then 121x=y ; and .-. 121.r-=-^-, and 121x s +2.r=^- 
 
 +121' 
 
 Hence, we got 
 
 2/ 2 4-2?y= D, 
 2/ 2 +G 2 /=n, 
 2/ 2 +9t/=D. 
 
 (Z' — \Y , /2 3 — 1\- Z*— 2z«+l 
 
 \ S sume 2 / 2 +23/=^- 5r j ; and .-. if +2^1 = [-^-j +1= ^3- 
 
 2«_2z 3 +4^4-l z 4 +2z 2 +l /2 J +1\ 2 
 + 1= — = — =y-^~J ; and, consequently, 7/-fl 
 
 =--|-l z s +l z 2 — 22+1 (2 — l) s 
 
 =— - — .-. =y=— — — 1= = — ; houce, by substitution in the 
 
 22 ^z ' ' " Lc 
 
 second equation above, wo have 
 
 (z — lY {z — l) a (2 — 1)< (:-l) 2 
 
 Rut 42 2 is a square number ; 
 
 ... (z—l)«+12zx(z— l) a =D 
 = (2-l) 2 x(^-l) 2 +122.(^-l) 2 = (r-l)=x|(2-l) 2 +12r|. 
 But (2 — l) 2 is n, 
 
 ... ( z — 1)»+12*= D =2 2 +102 + l= a . 
 
 4gain, by substitution in the third, wo have 
 
 ^=^4-9 x^=^- D (*-l) y8*X(z-l)» 
 
 4 2 a -r J * o r — u — 4. j "t" 4,3 — Ui 
 
 .. (2-l)'+183X(:-l) : =a,aud.-. (c-l)-\(2-l) 2 -fl8:.(:-l)«=a 
 
 Hence (:— l) 2 X {(2 — 1) 3 +18: j = d, 
 
 and .-. (2 — 1) 2 +182=D =: : +lC: + l ; 
 
 hence (z»+16z+l)— (: : +10r-fl) = G: = ::: X - 1 . 
 
 ,1 3z+2 
 
 the - sum of which factors is — - — =— + 1, the root of the greater Q . 
 
 /3z \« 
 *»+16z+l = ^+l) = T +3: + l, 
 
ana 
 
 DIOPHANTINE ANALYSIS. 467 
 
 Or 2 
 «»+16*=— + 3i, and 42 2 +64z=9,z 2 +122, 
 
 52 
 .-. 4z+64 = 9z+12, 5z=52, and ? =— ; 
 
 o 
 
 /52 \ 2 /47\2 22 
 
 09 2209 
 
 ,5 / \5/ 25 25 
 
 •2/=- 
 
 52 52 52 
 
 ~X"7~ ~X "T" ~""c" 
 
 15 
 
 5 
 
 _2209 y _ 2209 
 
 —"520 a ' ld *-"121' ~" 62920 ; 
 4418 13254 , 19881 
 •' We SC6 that 62920' 62925' and 62920 are the r °° tS ' 
 
 QUESTIONS FOR EXERCISE. 
 
 (1) Required six numbers whose sum and product shall be equal. 
 
 Ans. 1, 2, 3, 4, 5, and -^-. 
 
 (2) Required five square numbers whose sum shall be a square 
 
 Ans. 1, 4, 9, 16, and j. 
 
 (3) Divide the number 3 into four rational squares. 
 
 16 1 9 49 
 
 Ans. — , — , — , and — . 
 
 2o 2o 2o 25 
 
 (4) Divide unity into three rational squares. 
 
 9 4 36 
 
 AnS - 49' 49' aQd 49" 
 
 (5) Find two numbers whose sum is a cube, and difference a square. 
 
 Ans. 1512 and 216. 
 
 (6) Find two numbers whose product plus their sum or difference is eacb 
 i square. 
 
 5 , 5 
 i Ans. — and 4—. 
 
 (7) To find two numbers, such, that when each is multiplied into the cubo 
 of the other, the products will be squares. 
 
 Ans. 2 and 8. 
 
 (8) To find two square numbers whose difference is 40. 
 
 Ans. 49 and 9. 
 
 (9) To find two square numbers, such, that their sum added to their prod- 
 uct may be a square number. 
 
 A X A 4 
 
 AnS - 9 and 9* 
 
 (10) It is required toJind two whole numbers, such, that their difference, 
 the difference of their squares, and the difference of their cubes shall be squares. 
 
 Ans. 10 and 6. 
 
 (11) Find two numbers, such, that the sum of their squares shall be both 
 a square and a cube. 
 
 Ans. 75 and 100. 
 
 (12) Find two numbers whoso sum shall be a cub» but their product and 
 quotient squares. 
 
 Ans. 25 and 100. 
 
46d ALGKBRA. 
 
 (13) It is required to find three integral sqiaro numbers that shall be to 
 arithmetical progression. 
 
 Ans. 1, 25, and 
 
 (14) To find three square integral numbers in har.monical progression. 
 
 Aits. 1225, 49, and 2 >• 
 
 (15) To find three numbers, such, that if to the square of eacli of them the 
 
 sum of the other two be added, the three sums shall be all squat 
 
 8 , 1G 
 Ans. 1, -, andy. 
 
 (1G) It is required to find three whole numbers,8iich, that if to the square of 
 each of them the product of the other two be added, the sums shall be squares. 
 
 Ans. 9, 73, and 3 
 
 (17) It is required to find three whole numbers in geometrical progression, 
 such, that the difference of every two of them shall be a square number. 
 
 Ans. 5G7, 1008, and 17 
 
 (18) It is required to find three integral square numbers, such, that the dif- 
 ference between every two of them and the third shall be a square number. 
 
 Ans. 149 s , 241 s , and 21 
 
 (19) To fiud three square numbers, such, that the sum of their squares 
 shall also be a square number. 
 
 , 144 
 Ans. 9, 16, and -—jr. 
 
 KmtJ 
 
 (20) To find throe biquadrate numbers the sum of which shall be a square. 
 
 Ans. 12*, 15 4 , and 20 4 . 
 For generalization of Diophantine problems in certain cases, see Bonny- 
 castle's Algebra. See,' also, Theory of Numbers. 
 
 THEORY OF NUMBERS. 
 
 i 
 
 387. We have already had occasion to demonstrate some propositions which 
 
 fall under this head, and which would have been reserved for this place had 
 they not been required for the elucidation of previous parts of the work. 
 
 We recur to one or two of these for the purpose of exhibiting some of the 
 other methods by which they may bo established. 
 
 T. To prove that axb = bxa- Suppose d^>b and c their dillerence; 
 
 .-. aXt = (H c)b = l- + cb; 
 i. e., b taken b times and c taken b times, and 
 
 bxa = b(b+c) = b"+bc; 
 ■ ' ., b taken b times and also c times. 
 
 We perceive that the product <ixl> will be the Bame ;ls &X<*> it" the partial 
 product rxb is equal to /'X' - - But, by similar reasoning, the equality of 
 and br will be proved by the equality of two smaller products, <-,/ and de; and 
 continuing thus, we arrive necessarily at the case where the two factors are 
 equal, or at the case where one of them is equal to unity. Tn tho first i 
 the equality is manifest ; in the Becond, it will follow, from the fact that /ixl 
 is // as well as 1 x''- Then the product a X b 18 always equal to the product 
 OXi- 
 
THEORY OF NUMBERS 4(j9 
 
 IT. To demonstrate that NxaXi=Nx«i, I observe, first, that the prod- 
 act ah is nothing else than a+ a + a + > &*•> tuo number of these terms beingo. 
 ThenNxai=Na + Na + ]Na + , <5cc, to £ terms, =Na X b. Q. E. D. 
 
 III. Nab=Nba ; for Na=N+N+N-|- ... to a terms ; then, to multiply 
 N« by b, it is necessary to take each of the terms b times, 
 thus Na6=No+N6-fN& . . . =N6a. Q. E. D. 
 
 Corollary 1. — If all the factors of N be 1, then 1 Xab = l X bu, or ab=ba. 
 according to I. 
 
 Corollary 2. — The above reasoning applies only to entire factors. The prin- 
 ciple is equally true, however, when some of the factors are fractions ; because, 
 if the entire factors, which are combined with the fractional ones, lie written 
 in a fractional form by placing unity under them, all the factors to be multi- 
 plied together will be fractions ; the product of these, we know, is obtained by 
 taking the product of the numerators and denominators separately, which are 
 entire numbers, and therefore the order is immaterial, from what 1ms been 
 proved above. 
 
 Corollary 3. — If the factors be incommensurable, it is to be observed that 
 the product of two incommensurable quantities has no precise meaning. 
 
 P>ut by regarding the incommensurables as limits to which approximating 
 commensurables tend, since the above reasoning applies to the latter, and their 
 order is immaterial, we may infer that the order is immaterial also in a prod- 
 uct of incommensurable factors. 
 
 Corollary 4. — We have seen that, from the above proposition, it follows that 
 the order of factors in a product is immaterial ; hence it follows that if a 
 number, P, contains the factors a, b, c, &c, it is divisible by their product. 
 
 Corollary 5. — If a number, P, is divisible by another, Q=a6c, then is P 
 divisible by each of the factors a, b, c. 
 
 THE FORMS AND RELATIONS OF INTEGRAL NUMBERS, AND OF THEIR 
 SUMS, DIFFERENCES, AND PRODUCTS. 
 
 388. I. The sum or difference of any two even numbers is au even num- 
 ber. For, let A=2n and B=2n' be any two even numbers ; then 
 
 A±B=2»±2»'p2(»±»')= 2n "' 
 
 which, being of the form 2«, is an even number. 
 
 II. The sum or difference of two odd numbers is even, but the sum of three 
 odd numbers is odd. 
 
 Let A=2m-(-1, B=2«'+l, and C=2n"-\-l, be three odd numbers; then 
 A + B = 2m + 2»'+ 2 = 2»", 
 and A+B + C=2»+2w'+2«"+ 3=2»'"-f 1 ; 
 
 the former having the form of an even, and the latter of an odd number. 
 
 In a similar way it may be shown, 
 
 (1) That the sum of any number of even numbers is even. 
 
 '2) That any even number of odd numbers is even, but that any odd num- 
 jer of odd numbers is an odd number. 
 
 (3) "That the sum of an even and odd number is an odd number. 
 
 (4) That the product of any number of factors, one of which is even, will 
 be an even number, but the product of any number of odd nun bers is odd 
 and hence, again, 
 
470 ALGEBRA. 
 
 (5) Every power of an even number is even, and every power of an odd 
 number is an odd number. 
 
 (6) Henco the sum and difference of any power and its root is an even 
 number. 
 
 For the power and root will be either both even or both odd, and the sum 
 or difference in either case is an even number. 
 
 III. If an odd number divide an even number, it will also divide the half 
 of it. 
 
 Let A = 2n, B=2rc'-{-l be any even and odd number, such that B a a 
 divisor of A ; let the division be made, and call the quotient p ; then we have 
 
 , 2n=p(2n'+l); 
 
 consequently (4), p is even, or of the form 2n" ; 
 hence 2n=2»"(2»'-|-l), 
 
 and 2-^+l= n " ; 
 
 that is, n=|A is divisible by B, if A itself be so. 
 
 DEFINITIONS. 
 
 389. (1) A perfect number is that which is equal to the sum of all its ah 
 
 quot parts, or of all its divisors. 
 
 6 6 6 
 Thus, 6=--\---\--, and is, therefore, a perfect number. 
 
 (2) Amicable numbers are those pairs of numbers each of which is equal tu 
 all the aliquot parts of the other. Thus, 284 and 220 are a pair of amicable 
 numbers, for it will be found that all the aliquot parts of 284 are equal to 220, 
 and all the aliquot parts of 220 are equal to 284. 
 
 (3) Figurate numbers are all thoso that fall under the general expression 
 
 w(n+l)(w-f 2)(«-f 3). . . .(n+?n) 
 1.2. 3. 4. ...(?/i+l) ' 
 
 and they are said to be of the 1°, 2°, 3°, &c, order, according as ?n = l 
 2, 3, &c 
 
 (4) Polygonal numbers are the si^is of different and independent arith- 
 metical series, and are termed lineal or natural, triangular, quadrangular 
 or square, pentagonal, &c, according to the series from which they are 
 generated. 
 
 (5) Natural numbers are formed from a series of units ; thus : 
 
 Units, 1, 1, 1, 1, 1, &c. 
 
 Natural numbers, 1, 2, 3, 4, 5, &c. 
 
 (6) Triangular numbers are tho successive sums of an arithmetical series, 
 beginning with unity, the common difference of which is 1 ; thus : 
 
 Arithmetical spi 1, 2, 3, 4, 5, &c 
 
 Triangular numbers, 1, 3, 6, 10, 1">. 6 
 
 (7) Quadrangular or square numbers are the sums of an arithmetical se 
 
 ries, beginning with unity, and the common difference of which is 2 ; thus : 
 
 Arithmetical series, 1, 3, 5, 7, 9, 11, &c. 
 
 Quadrangular or > 
 
 square numbers, 
 
 1. I. 9, 16, 25, 36, &c. 
 
THEORY OF NUMBERS. 471 
 
 (8) Pentagonal numbers are the sums of an arithmetical series, beginning 
 with unity, the common difference of which is 3 ; thus : 
 Arithmetical series, 1, 4, 7, 10, 13, 16, &c. 
 Pentagonal numbers, 1, 5, 12, 22, 35, 51, &c. 
 
 And, universally, the m — gonal series of numbers is formed from the suc- 
 cessive sums of an arithmetical progression, beginning with unity, the com- 
 mon difference of which is m — 2. 
 
 DIVISIBILITY OF NUMBERS. 
 
 390. I. The product of two numbers, a and b, is divisible by every number 
 which exactly divides one of the two facAxyrs a and b. 
 
 For let 6 be a number which divides b, so that b=c6, we have by the fore 
 going ab=acX6- Then ab, divided by 0, gives the exact quotient ac. 
 
 Corollary. — To divide a product of several factors, divide one of the factors 
 and multiply the quotient by the others. 
 
 On this subject we must observe that a number may sometimes divide a 
 product when it will not divide any factor. Thus, 20 divides neither 12 nor 
 15, but does their product, 180. This is because 20 is composed of factors 
 some of which are found in 12 and others in 15. But if the number 20 had 
 no common factor with one of the factors, it must divide the other. (See 
 Art. 84, note.) 
 
 II If there be n numbers, each of them divisible by k, then is their product 
 divisible by k n . 
 
 For a=kq, b=.kq', c=kq". . . .-. abc...=k n .w, 
 
 w being equal to q X q' X q" X 
 
 III. The sum of several numbers, a+b-j-c-j-d. is divisible by a number, k, 
 when the sum of th& remainders obtained by dividing each by k is divisible by 
 this number. 
 
 For a=kq-\-r, b=kq'-\-r', c=kq"-\-r", &c. 
 
 ... a + b + c+ d=k{q+q'+q" + , &c.)-f-r+r / +r"+, &c. 
 Whence it is evident that a-\-b-\-c, <kc, is divisible by k when r-f-r'4-r", 
 &c, is. 
 
 IV. The difference of two numbers, a and b, is divisible by a number, k, 
 when, if each be divided by k, the remainders are equal. 
 
 For a=kq-\-r, and b=kq'-\-r 
 
 .'. a — b=k(q — q'). 
 
 V. Every number consisting of units, tens, hundreds, Sfc, is divisible by a 
 number, k, when the sum of the jiroducts of the number of units, tens, 8fc, by 
 the remainder, after dividing the units, tens, Sfc, each by k, is divisible by this 
 number. 
 
 For, representing by A, B, C, &c, the quotients, and by a, )3, y, &c., the 
 remainders of the units, tens, &c, by k, we have 
 
 10 n =Afc-f a .-. a . 10" =aA.k + aa 
 
 10 n -'=BA"+/3 b . 10f-'= bBk+bp 
 
 10 n - 2 = C&+y c . 10°--*= cCk+cy 
 
 10 3 =Dfc-f <J d . 10* =dDk+dd 
 10 1 =EJfc-j-e e . 10 1 =eEk + ee 
 10° = ... 1 /. 10° = / 
 
472 ALGEB1LA. 
 
 VI. The proauct, P, of several numbers, a, b, c, d, . . . is divisible by a 
 number, k, only when the product of the remainders, after dividing each of the 
 factors by k, is so divisible. 
 
 For, let a=kq-\-a, b = k /-{- I, c=k/'-{-y, &c, 
 
 .-. ab=kz-\-a.(3. 
 abc=kz-\-ap.y, &c. 
 
 VII. The product, P, of several factors, a, b, c, d, . . . is divisible by a prime 
 number, k', only when one of the factors is divisible by Otis prime n umber 
 
 For, let a=k'q-\-a, b=k'q'-\-p, c=zk'q"-\-y, &c, 
 
 .'. V = k'z-\-a.[i.y . . . 
 
 Therefore, if k' divido P, it must divide a, 3, y . . . 
 
 But k' is not fouud among the factors a, (3, y, . . . since, being remainders 
 to the divisor k', they are all less than it. Neither is k' any combination of 
 them, since it is supposed to be a prime number. Hence a, j3, y, . . . and 
 therefore P is divisible by k' only when one of the remainders =0. 
 
 VIII. If the factors, a, b, c, . . . of a product, P, are prime to k, then is the 
 product not divisible by k. 
 
 For, if k bo an absolute prime number, this follows from VII. Again, if 
 
 k be a multiple of a prime number, as p'v; then, if P be divisible by k, \\t 
 
 have 
 
 P a.b.c.... 
 
 ~r= ; =m ••. a.b.c... =mp'v ; 
 
 k p' . v 1 
 
 thereforo a.b.c... must be divisible by p' , which by VII. is impossible. 
 
 391. I. Problem. — To find all the divisors of any number ivkatever. The 
 first thought which presents itself is to try successively as divisors each of the 
 numbers 1, 2, 3, &c., to N. But this groping process may be abridged. Lei 
 D be a divisor of N, and D' the quotient, we have DD' = N, or, under anoth- 
 er form, DD'= -/N X VN ; then, if D_is < y/N, D' will be > -/N. Tb 
 after having found all the divisors < -\/ S, the quotients which shall have been 
 obtained in dividing N by theso divisors will be the divisors > -J V 
 
 For example, lot N=360. The square root of 360 is comprised between 
 18 and 19 ; thus, we divido 3G0 only by the numbers 1, 2, 3 ... 18. [a this 
 manner we find all the divisors of 3G0, to wit : 
 
 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18. 
 360, 180, 120, 90, 72, 60, 45, 40, 36, 30, 24, 20. 
 
 392. II. Problem. — To form a table of prime numbers. When the ;• 
 proceeding produces no divisor, the number ia a prime number. To 
 the long calculations necessary in these cases, tallies have been construt 
 whirl i co itain the prime numbers up to certain limit 
 
 The in" ! simple manner of constructing it is to write i -ion the 
 
 series of an even numbers 3, 5, 7, &C., to such n limit as we seek, and to efl 
 all the multiples of 3, of 5, of 7, evrc. It is evident that the prime numbers 
 are all that remain. At the head of these numbers it must not be forgotten to 
 place 1 and 2. 
 
 Nothing is easier than to know what multiples to efface. Those of 3 are 
 
 * The student is referred to the tables of Burckharlt, ia which the prinii- Mjnben ex 
 tend to .103G000. 
 
THEORY OF NUMBERS. 473 
 
 tound by counting the numbers 3, 5, 7, &c, in threes, setting out from 5 ; 
 those of 5 in counting them in fives, beginning with 7, and so on. ■ 
 
 393. Remark I. — The series of prime numbers is unlimited. For, suppose 
 it to be otherwise, and that n is the greatest : if we form the product 
 P=2.3.5 . . . n, which contains all the prime numbers, then P + l, which 
 >n, must be divisible by some one of these numbers ; but this is impossible, 
 because there will always be the remainder 1. Then it is impossible that the 
 series of prime numbers should be limited. 
 
 II. In comparing all numbers with multiples of the same number, we are 
 led to present them under different forms, of which use is often made. For 
 example, if wo compare them with multiples of 6, they may be represented 
 first, by one of the six formulas, 
 
 6.r, 6x+l, G.r+2, Gx+3, G.r+4, 6x-f 5, 
 in which x is any whole number whatever. 
 
 But if we wish to consider only prime numbers, it is necessary to preserve 
 only the two formulas, 
 
 6z+l and Gx-{-h ; 
 
 because the others give numbers divisible by 2 or by 3. 
 
 We can also, in place of 6.r+5, write 6(.r+l) — 1 or G.v— 1, since x is any 
 entire number whatever. Thus all the prime numbers except 2 and 3, which 
 are divisors of 6, are comprised in the formula 
 
 N = G.r±l. 
 The reasoning woidd be analogous for any other number than 6. 
 
 394. III. Problem. — To decompose a number into prime factors, and to find 
 afterward all its divisors. 
 
 A number N, if it be not a prime number, can be represented by the product 
 of several prime numbers a, b, c, &c, raised each to a certain power, 60 that 
 we can always suppose N=a m 6 n cP . . . This is the decomposition which it is 
 required to effect. 
 
 Take, for example, the number 504. Divide it first by 2 as many times as 
 possible ; we find thus, 
 
 504=252 X 2=126 X 2 X 2 = 63 X 2 X 2 X 2. 
 Then divide 63 as many times as possible by 3, which is the sirallest prime 
 number greater than 2 : 
 
 63 = 21X3=7X3X3. 
 Then we have 
 
 504 = 7X3X3X2X2X2, 
 
 or, rather, under another form, 
 
 504=2 3 X3 : X7. 
 
 The divisions by 3 have led to the quotient 7. If the quotient had not been 
 a prime number, we should have continued the operations by trying success- 
 ively the other prime numbers, 5, 7, &c. 
 
 We can now readily form all the divisors of 504. They are, in fact, the 
 numbers which we obtain in taking all the prime factors one by one, two by 
 
 * Conceive a board pierced with holes in -which the numbers 3, 5, 7, &c., are placed in 
 order. Then, as we arrive, in counting them by threes, fives, lVc, at the multiples to be 
 effaced, suppose these multiples to fall tln-ough the holes, there will remain only prime 
 numbers. Such was the famous sieve of Eratosthenes, of Alexandria, who lived 280 B.C 
 
47 4 ALGEBRA. 
 
 two, Arc. That we may be sure not to omit any : ivisor, we adopt the fol- 
 lowing arrangement : 
 
 1, 
 
 504 2 2, 
 252 2 4, 
 126 2 8, 
 63 3 3, 6, 12, 24, 
 21 3 9, 18, 36, 72, 
 7 7 7, 14, 28, 56, 21, 42, 
 84, 168, 63, 126, 252, 504. 
 The first column on the left contains the given number and the quotient ol 
 the successive divisions. By the side of these numbers, in a second column, 
 are written the prime numbers, which we employ as divisors, and which 
 are the prime factors of the number 504. Finally, we place at the right of 
 this column all the divisors of 504 ; and I now proceed to state how we obtain 
 them. 
 
 At the top of the third column, but on the line above, that which contains 
 504, we write unity, which may be regarded as the first divisor of 504. We 
 multiply this unity by the first number of the second column, and thus obtain 
 the divisor 2, which we write by the side of this first prime number. We 
 next multiply 1 and 2, the divisors already found, by the second number of the 
 second column, and, neglecting the product 1X2, or 2, which has already been 
 found, we obtain the new divisor 4, which is written on a line with the last 
 multiplier. We proceed in the same manner, multiplying the number of the 
 second column on the horizontal line which wo are forming by each of the 
 numbers above it in the third column successively, until we multiply, finally. 
 by the last number of the second column, which gives a last series of divisor-. 
 which series will always be terminated by the given number. 
 
 When we know the prime factors of a number, we can find its divisors by 
 another process. Suppose that a number N, when decomposed into prime 
 factors, gives 
 
 N=a m i"cP . . .; 
 the divisors of N will be represented by the formula a m 'b a 'ci" . . .. in which tin 
 exponents to', n', p* . . . can not surpass to, n, p . . . 
 
 Hence we know that these divisors will bo the different terms which w« 
 obtain in effecting the product 
 
 P = (l + «+«N a m )(l + /,-fZ,:+ . . . i'>)(i-f_ ( . +c -_j_ . . . cp) 
 
 395. Remarks. — Tho multiplication of tho first two polynomes gives a 
 number of terms equal to (m-fi)(«-|_i) ; consequently, thai of the first three 
 polynomes gives a number equal to (m-r"l)(n+l)(jp-f-l), and so on; hence, 
 the number of all the divisors of N is sed by the formula 
 
 (m+l){n+l)(p+l) 
 
 We also see that P is tho sum of all theso divisors. But we know that the 
 
 <;"■+' — 1 
 
 polynomes which composo P are respectively equal (Art 23) to 
 
 </ — 1 
 6"+' — l 
 -f ) _ l » & c. ; hence, tho sum of all the divisors of N can bo expressed by the 
 
 formula 
 
THEORY OF NUMBERS. 475 
 
 a H-l_l fcn+l — 1 C?+ l — 1 
 
 a — 1 o — 1 c — 1 
 
 For example, taking N=504=2 3 X3 2 X?, we shall have to=3, ra=2, 
 psssl. Hence the number of divisors of 504 will be 4x3X2=24, and the 
 sum of all the divisors will be 
 
 2<— 1 3 3 — 1 7 3 — 1 
 
 IT^l X 3^1 X 731 = 15 X 13 X 8 = 156 °- 
 
 396. IV. Problem. — How many times is a prime number, 0, factor in a 
 series of natural numbers, from 1 to n ? or, in other words, what is the highest 
 power of which divides the product 1 . 2 . 3 . . . n ? 
 
 Let n' be the entire part of the quotient of n by 0. In tli9 proposed series 
 of natural numbers we find the n' factors, 0, 20, 30 . . . ., of the product 
 6.26.30. . .n'Q; and it is clear that they are the only numbers of the series 
 which are divisible by 6. This product can be written thus : 
 
 1 . 2 . 3 ... n' X n '. 
 Hence we shall obtain the required power of 6 by multiplying n/ by the high- 
 est power of 0, contained in the product 1 . 2 . 3 . . . n'. 
 
 The same reasoning may bo repeated with reference to this product ; 
 hence, calling n" the entire part of the quotient of n' by 0, we readily perceive 
 that, the highest power of contained in the last of the above products is com- 
 posed of the power n " multiplied by the highest power of which is contain- 
 ed in the series 1.2.3 . . . n". 
 
 In like manner, calling n'" the entire part of the quotient of n" by 6, we are 
 led to seek the highest power of contained in the product 1.2.3 . . . n'". 
 
 We continue this process till we arrive at a quotient <C,0. For the sake of 
 definiteness, suppose that n'" is this quotient ; then we conclude that the 
 highest power of contained in the given product 1 . 2 . 3 . . . n is 6 a,+a "+ n '". 
 
 For example, suppose we wish to l?how what is the highest power of 7 
 which divides the product 1.2.3 . . . 1000. 
 
 We make n=1000, and taking only tltt entire parts of the quotients, we 
 
 shall have 
 
 1000 142 20 
 
 -—- = 142, —=20, — =2. 
 
 7 7 7 
 
 The sum of these quotients being 164, it follows that the required power is 7 16 *. 
 
 397. Corollary. — Let m, n, p, q be entire numbers, such that we have 
 m=n-\-jy-\-q-\- . . . ; the expression 
 
 1.2. 3. 4. to 
 1.2 » X 1 • 2 pXl-2 qX, &c. ^ 
 
 will always represent an entire number. To prove this, let be a prime factor 
 of the denominator ; wo shall have 
 
 to n p q 
 
 Calling these entire quotients to', n', p', q' . . . ., we shall have also 
 
 to'= or >«'+_Z''+9'+i &c. 
 If wo divide again by 0, and call the new entire quotients m", n" . . . ., we 
 shall, in like manner, have 
 
 to"= or >n"+p"+q"+ , &c. 
 We continue this process as long as the quotients are not all less than 6 
 
476 ALGEBRA. 
 
 Then adding, we shall have 
 
 '+„*"+...)= or >(n'+n"+...) + (jp'+p"+...)+(?'+5 r "+-)+. *c. 
 
 l!iii these different sums make known the highest powers of 0, by which we 
 
 divide the products which compose expression (1); hence there is no 
 
 ie factor in the denominator which does not exist of a power at least equal 
 
 in the numerator of the fraction. This expression, therefore, represents an 
 
 entire number. 
 
 393. Perfect numbers are expressed or determined as follows : 
 
 Find 2" — 1, a prime number, then will N=2 B— I (2" — 1) be a perfect number. 
 
 For, from what has been demonstrated in the preceding section, the sum of 
 
 2" l (2° 1) : 1 
 
 all the divisors of this formula will bo represented by — — — x 77^ — rr — r ; 
 
 because 2" — 1 is a prime by hypothesis. But in this expression 1 is inclu 
 as a divisor, which must be excluded in the case of perfect numbers : pxcIi 
 sive of this, therefore, the formula will be 
 On_! ( 2 n_ l)s_l 
 
 X: '. On-2 1 /->n T \ 
 (On ]\ I ~ L \~ *■> 
 
 (2» — l)x(2 n — 1 + 1) — 2 n -'(2 n — 1) = 
 2(2 n — l)2 n+1 — J" '(-'" — l)=2 n - 1 (2" — 1) = N, 
 that is, the sum of all the aliquot parts of N, exclusive of itself, or of 1 as ;i 
 divisor, is equal to N, and is, therefore, by the definition a perfect number. 
 The only perfect numbers known are tho following eight: 
 
 G, 3355033G, 
 28, 8589869056, 
 496, 137438691328, 
 8128, 23058430081399521 ■-■-. 
 
 399. To find a pair of amicable numbers N and M, or such a pair that each 
 shall be respectively equal to all tho divisors of the other. 
 
 Make N=a m i n cP, &C, and M=a f, P v y n ; then, according to the definition and 
 from what has been demonstrated in the last section, we must have 
 a m-H_l &n+i_i CP+ 1 — 1 
 
 a — 1 — 1 c — 1 ' 
 
 ae+i_l pH-i-l y *+i_l 
 
 T=r x i^r x ^=T- =? ' r + N - 
 
 Find, therofore, such a power of 2, as 2 r , that 
 
 3.2 r — 1, 6.2 r — 1, and 18. 2 r — 1 
 may bo all prime numbers ; then will 
 
 N=2 r+1 (iaud M = 2 r +'6c 
 
 be the pair of amicable numbers sought. 
 
 The least three pair of amicable numbers are 
 
 284, 220. 
 
 17296, 18416, 
 
 9363583, 9437056. 
 
 400. We shall hero introduce the student to tho nomenclature and notation 
 of Gauss, given in his Disquisitiones Arithmetics, which is now generally 
 adopted by writers upon the theory of numbers. 
 
CONGRUENCE OF NUMBERS. 477 
 
 CONGRUOUS NUMBERS IN GENERAL. 
 
 101. If a number a divide the difference of the numbers b and c. b and c are 
 said to be congruous with reference to a ; if not, incongruous. The quantity 
 a is called the modulus ; each of the numbers b and c a residue of the other in 
 the first case, a non-residue in the second. 
 
 The numbers may be either positive or negative, but entire. As to the 
 modulus, it ought evidently to be taken without regard to the sign. 
 
 Thus, — and -{-16 are congruous with reference to the modulus 5; — 7 
 is a residue of 15 with reference to the modulus 11, and not a residue with 
 reference to the modulus 3. 
 
 Zero being divisible by all numbers, every number may be regarded as con- 
 gruous with itself with reference to any modulus whatever. 
 
 All the residues of a given number, a, with reference to a given number, m, 
 are comprised in the formula a -{-km, I: being an entire indeterminate num- 
 ber. This is self-evident. 
 
 The congruence of two numbers is expressed by the sign EE, joining to it 
 the modulus, when necessary, in a parenthesis, thus :* 
 
 — 16 = 9(mod. 5), — 7=15(mod. 11). 
 
 402. Theorem. — Let there be m entire successive numbers, a, a-f-1, a-J-2, 
 ...a-f-m — 1, and another, A; one of the former will be congruous with A, 
 with reference to the modulus m, and but one. 
 
 a — A 
 
 For if is entiro, a = A; if it is fractional, let k be the nearest entire 
 
 m 
 
 n a — A- 
 number; above, if be positive; below, if it be negative; A-\-km will 
 
 fall between a and a-\-m,\ and will bo the number sought; but it is evident 
 
 a — A a-{-l — A 
 
 that the quotients , , &c, are comprised between k — 1 and 
 
 1 m m r 
 
 A"-f-l)t therefore one of them only can be entire. 
 
 403. It follows from this that every number will have a residue as well 
 in the series 0, 1, 2...m — 1, as in the series 0, — 1, — 2... — (m — 1\ 
 They are called minima residues ; and it is evident that, unless zero is the 
 residue, there will be two, the one positive and the other negative. If they 
 
 are unequal, the one will be <^-^\ if they are equal, each of them =^-, with- 
 out regard to the sign; from which it follows, that any number whatever has 
 a residue which does not surpass the half of the modulus; this is called the 
 absolute minimum residue. 
 
 For example : — 13 relative to the modulus 5, has for a positive minimum res- 
 idue 2, vWiich is at the same time its absolute minimum, and — 3 for its nega- 
 tive minimum residue; -j-5, with reference to the modulus 7, is itself its 
 
 * The analogy between equality and congruence led Legendre to employ the sign of 
 equality itself. This modification of it has been introduced by Gauss to avoid ambiguity. 
 
 t This maybe seen from the equality — — =A — n, where ?j<w. 
 
 ?n 
 
 i This may be seen by observing that = — - — I — and it is not dill the nunie- 
 
 J * ° tin in in 
 
 ratorof— increases to m that the quotient Jc increases to £-f-l. 
 in 
 
•173 ALGEBRA. s 
 
 ;■■ residue; — 2 is the negativb •ninimum residue, and, at the 
 
 same time, the absolute mini mum. 
 
 404. The following consequences follow from tho above : 
 
 Numbers which are congruous with reference to a composite modulus are so 
 with reference to any of its divisors. 
 
 If several numbers arc congruous with the same number with reference to the 
 same modulus, tkey will be congruous with each other with reference to this 
 modulus. 
 
 The same modulus must be supposed in what follows : 
 
 Congruous numbers havo the same minima residues ; incongruous have 
 different. 
 
 405. If the numbers A, B, C, &c. ; a, b, c, &c, are congruous each to each, 
 i- e., AEEa, BEE&, &c, we shall have 
 
 • A + B + C . . . = « + i + c . . . 
 Tf A~ a, BEE b, we have also A — BEE a — b. 
 
 406. If A =. a, tee have also k\ EE ka. * 
 
 If k is positive, this is but a particular case of the preceding article, in 
 which A = B = C . . . and a = b~c . . . 
 
 If 1c is negative, —Jc will be positive; then — A.-AEE lea .-. kA = ka. 
 
 If A = a, BEE b, then ABEEai; because ABEE AB EE Ab = ba. 
 
 107. If the numbers A, B, C . . . = a, b, c . . ., each to each, then 
 
 ABC . . . =abc . . . 
 for, by the preceding article, ABEEafc; for the same reason, ABC = ab<, 
 and so on. 
 
 By taking all the terms, A, B, C . . . equal, and a, b, c . . . also equal, if 
 A=a, A*EEa*. 
 
 408. Let X be a function of the indeterminate x of the form 
 
 A.t a -fB;r 6 +Cr c +, See., 
 A, B, C . . . being any entire numbers whatever. If we give to x congruous 
 values with reference to a certain modulus, the resulting values for X will be 
 congruous also. 
 
 Let/ and g be congruous values of x ; by the preceding articles, /" I 
 and kf"~kg" ; in the same way we have Bf'' = Bg b , &c. 
 
 This theorem may be easily oxtended to functions of several indetermi 
 nates. 
 
 409. If, then, we substitute in place of r all entire consecutive numl 
 and seek tho minima residues of tho values of X, they will form a series in 
 which, after an interval of m terms (m being the modulus), the same terms 
 will be again presented; that is to say this Belies will be formed of a period 
 of in terms repeated indefinitely. 
 
 Let there be, for example, X=r — -V-f-G, and /// = ">; for x=0» 1, 2, •'?. 
 &c. ; the values of X give for positive minima residues 1, 4, 3, 4, 3, 1. i. &c, 
 or the five, 1, 4, 3, 4, 3, arc repeated indefinitely ; and if wq continue the 
 series in the contrary direction, that is, if we give to x negative values, the 
 same period will reappear in an inverse onler: whence it follows that the 
 series contains no other terms than those which compose the period. 
 
 410. Then, in this example, X can nol become = 0, nor EE*J(nu>d. 5); and 
 still less =0 or —'J'; from which it follows thai the equations .;•'• — 8x4-6s0 
 ami r 3 — 8x-4-4=0 have not entire roots, and, consequently, not rational roots. 
 V. <• Bee, 'u general that when X is of the form 
 
CONGRUENCE OF NUMBERS. 479 
 
 X n +Ax n -» + B.r"- 2 +, &c, + N, 
 
 A., B, C . . . being entire quantities, and n entire; and positive, the equation 
 X=0 (a form to which every algebraic equation may be reduced) will have no 
 rational root, if it happen that, for a certain modulus, the congruence X = be 
 not satisfied. 
 
 411. Many arithmetic theorems may be demonstrated by tin; aid of the 
 foregoing principles, as, for instance, the rule for determining whether a num- 
 ber is divisible by 9, 11, or any other number. 
 
 With reference to the modulus 9, al! the powers of 10 are congruous with 
 unity; then, if the number is of the form ir-\-lQb-\-100c-^-lQQ0d-\-, &c, it 
 will have, with reference to the modulus 9, the same minimum residue ^ 
 a-\-b-\-c-\-, &c. It is clear from this, that if we add the figures of the number 
 without regarding their place value, the sum obtained and the proposed num- 
 ber will have the same minimum residue. It", then, this last is divisible by 9, 
 the sum of tho figures will bo also, and only in this case. It is the same with 
 the divisor 3. 
 
 Many of the properties of prime numbers, the divisibility of products already 
 given, &c, may be demonstrated by the aid of this system, but we shall not 
 repeat them. 
 
 412. The term congruence is analogous to equation, and the determination 
 of such values, for an indeterminate x, as to produce congruence in expression, 
 is called resolving them. There are congruences resolvable and irresolvable. 
 
 Congruences are also divided, like equations, into algebraic and transcend- 
 ental. Those which are algebraic are divided, again, into congruences of the 
 first, second, and higher degrees. There are congruences, also, containing 
 different unknown quantities, of the elimination of which Gauss treats. 
 
 413. The congruence oar-j-oEEc may be solved when its modulus m is 
 prime with a ; thus, let e bo the positive minimum residue of c — b. We find 
 necessarily a value of x<0«, such that the minimum residue of the product 
 ax, with reference to the modulus m, shall be e. Call v this value, and we 
 shall have 
 
 av = e = c — b ; 
 then av-{-b~EEc(mod. m). 
 
 Here v is called the root of the congruence. It is evident that all the num- 
 bers congruous with v, with reference to the modulus of tho congruence, wil 
 also be roots (Art. 408). It is »lso evident that all the roots should be con- 
 gruous with v; in fact, if t be another root, we have av-\-b = at-{-b ; then 
 at = av; and therefore v=Et. We may therefore conclude that the congru- 
 ence xEE t>(mod. ?/i) gives the complete resolution of the congruence ax-\-b EzEc. 
 
 The foregoing exposition will serve to show how the algorithm of (iauss 
 connects itself with the indeterminate analysis, and we shall hero quit the 
 subject. 
 
 414. No algebraical formula can contain prime numbers only. 
 
 Let 2 , + ax -{- TX '-^ s ^ & c -' 
 
 represent any general algebraical formula. It is to be demonstrated that such 
 values. may be given to x, that the formula in qui- 'ion shall not. with that value 
 produce a prime number, whatever values are given to p. q, r, cVc. 
 
 For suppose, in the urst place, that by making x=m, the formula 
 
480 ALGEB11A. 
 
 V =j>-\-qm-\-rm' l -\-sm :i -J r . &c, 
 is a prime number. 
 
 And if now we assume xr=»i-|-^P, we have 
 
 P= P 
 
 <F= 7 m +?'? p 
 
 rx-= r/zr + 'J/vz/ol'-fro-P 2 
 
 ix'= s;/i :, +3s?/t, 2 ^P + :j.s m P +sfP 3 
 
 &c. cVc. 
 
 Or 
 
 p -\- q.r -\-r.v- -\- si a =(p -\- qm -\- riir -\- siir -{- , &c.,) + 
 P w _|_ 2rm0 + 3s»t-<j>) + P s (r^ + 3smf) -f . I ' 
 = P + P(^+2m04-35m*f.) + 
 P-(7-^4-3sm^)+s£ 3 P 3 . 
 But this last quantity is divisible by P ; and, consequently, the equal quantity 
 
 p-\-qx-{-rx'--\-sx*, &c., 
 is also divisible by P, and can not, therefore, be a prime number. 
 
 Hence, then, it appears, that in any algebraical formula such a value may 
 bo given to the indeterminate quantity as will render it divisible by some other 
 number ; and, therefore, no algebraical formula can be found that contains 
 prime numbers only. 
 
 But, although no algebraical formula can be found that contains prime num- 
 bers only, there are several remarkable ones that contain a great many; thus, 
 r 3 +.r-|-41, by making successively ar=0, 1, 2, 3, 4, &c, will give a series 
 41, 43, 47, 53, Gl, 71, &c, the first forty terms of which are prime numbers 
 The above formula is mentioned by Euler in the Memoirs of Berlin (1772, 
 p. 36). 
 
 To the above we may add the following: ,r : -|- r + ] " an ^ 2.r' : + 29 ; the 
 former has 17 of its first terms prime, and the latter 29. 
 
 Fermat asserted that the formula 2 ,n -f-l was always a prime, while m was 
 taken any term in the series 1, 2, 4, 8, 16, &c. ; but Euler found that 
 
 4-1=641 X 6700417 was not a prime. 
 
 ■11">. If a and 6 bo any two numbers prime to each oilier, ami each of the 
 terms of the series 
 
 6, 26, 36, 46, &c, {a — 1)6 
 be divided by a, they will each leave a different remainder. For if any two 
 of these terms, when divided by a, leave the same remainder, let them be rep 
 ented by .r6, yb ; then it is obvious thai .*/' — yb would be divisible by a, or 
 (r — 7)6 would be divisible by a. But this is impossible, because a is prime to 
 6, and x — y is less than a; therefore b{x — >/) is not divisible by a, but it 
 would be so divisible if the terms xb, yb left the same remainder; these do 
 not, therefore, leave the same remainder; consequently, every term of the 
 series 
 
 6, 26, 36, Arc, (a— 1)6, 
 
 divided by a, will leave a different remainder. 
 
 DEDUCTIONS. 
 
 416. Since the remainders arising from the division of each term in the -• 
 
 6, 26, 36, &c, (a— 1)6 
 oy a are different fr)m each other, and <* — 1 in number, and each of thera 
 
CONGRUENCE OF NUMBERS. 481 
 
 necessarily loss than a, it follows that these remainders include all numbers 
 from 1 to a — 1. 
 
 417. Hence again, it appears that some one of the above terms will leave 
 a remainder 1 ; aniHhat, therefore, if b and a be any two numbers prime to 
 each other, a number .t<> may be found that will render hx—1 divisible by 
 a, or the equation 6a: — ay = ~L is always possible if a and b are numbers prime 
 to each other. 
 
 And it is always impossible if rf and b have any common measure, as is evi- 
 dent, because one side of the equation bx — ay = ^ would bo divisible by this 
 common measure, but the other side, J, would not be so; therefore, in this 
 case the equation is impossible. 
 
 418. If a be any prime number, then will the formula 
 
 1.2.3.4.5, &c., (a — 1) + 1 
 be divisible by a ; for it is demonstrated in our preceding second deduction, 
 that if a and b be any two numbers prime to each other, another number x 
 may be found <[«, that renders the product bx — l-^-*a, or, which is the 
 samo thing, &ar=ya+l ; and that there is only one such value of x<C.a, may 
 be shown as follows : 
 
 The foregoing equation gives, by transposition, 
 
 bx — ay = l ; 
 and, if it be possible, let also 
 
 bx' — ay'=l ; 
 tnd make x'=x±m and y'=.y^zn, where m is necessarily less than a, be- 
 cause bother and x' are so by the supposition. 
 Now, by this substitution, we have 
 
 (bx^zbm) — (ay^zan) = l ; 
 DUt /;./• — «]/ = l ; 
 
 therefore ^ : bm = : ^ : an, or bm-^-a ; but this is impossible, since b is prime to 
 « and m<a, as in Art. 415. There can not, therefore, be two values of x less 
 \han a, that render the equation 6.r — oy = l possible. 
 But, in the series of integers 
 
 1 .2.3.4.5 a— 1, 
 
 eveiy term is prime to a except the first, a being itself a prime ; if, therefore, 
 we write successively 6=2, b' = ?>, b"=4, &c, a corresponding term x, in 
 the same series, may be found for each distinct value of 6, that renders the 
 product xbzcay-\-l, x'b'3zay'-\-l, x"b"^zay"-\-l, &c. ; and it is evident that 
 no one of these values of x can be equal either to 1 or a — 1 ; for, in the first 
 case, we should have 1 x6=#2/+l> which is impossible, because b<^a ; and 
 the second would give (a — l)b=ay-\-l, or a(b — y) = b-\-l; that is. b-\-l-±±a, 
 which can only be when b=a — 1, or when b=.c, which case is excepted, be- 
 cause we suppose two different terms of the series. In fact, since (a — l) 2 
 
 ac «?/-{- 1, there can be no other term in the same series that is of this form ; 
 for if x~32(iy'+l, then (a — 1)" — x" would be divisible by a, or (a — l-\-x) 
 
 X ( rt — 1 — ,r)-H-a, which is impossible, since each of these factors is prime to 
 
 * To save the repetition of the words " divisible by, - ' which frequently occur, the sign 
 ^- is used to express them ; and, for the same reason, the symbol n: is introduced, to ex- 
 press the words "of the form of," which are also of frequent occurrence. 
 
 Ha 
 
482 ALGEBRA. 
 
 a, as is evident, because .r<a, and a is a prime number. Hence cur product 
 
 1.2. 3. 4. 5. ...(a— 1) 
 
 becomes 1 .bx.b'x' .b"x" a — 1; 
 
 but each of these products, bx, b'x', b"z", &c., is, as wo have seen, of the 
 form ay-\-l ; therefore their continued product will have the same form, and 
 the whole product, including 1 and a — 1, will be 
 
 a= (ay+1) X {a-\)^a-y+ay+a-l, 
 to which if unity be added, the result will bo evidently divisible by a ; that 
 is, the formula 
 
 1.2.3.4.5 (a — 1) + 
 
 is always divisiblo by a when a is a prime number. 
 
 DEDUCTIONS. 
 
 (1) The product 
 
 1.2.3.4.5 (« — 1) 
 
 is the same as 
 
 l(a-l)2(«-2)3(a-3), &c., (^^f 5 
 
 and this product, as regards remainder, when divided by a, is the same aa 
 
 ±i-.2 2 .3*.4* (^) ; 
 
 the ambiguous sign being -4- when a — 1 is even, and — when a — 1 is odd ; i. e., 
 
 -f- when a is a prime of the form 4n-{-l, and — when a is a prime of the form 
 
 in — 1 ; also, this last product is the same as 
 
 , / a— 1\ 2 
 
 ±(1-2.3.4 —) ■ 
 
 therefore, from what is said abovo relating to the ambiguous sign we shall have 
 
 l(l-2-3.4 ^-Zl)- +1 ^ a 
 
 when azc4n-f-l ; and 
 
 {('■•■■■« *-?)•-}* 
 
 when azcin — 1. 
 
 Hence eveiy prime of the form 4/i-f- 1 is a divisor of the sum of two square! 
 Again, the latter form may be resolved into the two factors 
 
 1.2.3.4... 
 
 ¥)-?• 
 
 which product being divisible by a, it follows thai a is a divisor of one or otho 
 
 of these factors when it is a prime number of the form 4fl — 1. 
 
 (2) From the first product, which we have shown to be divisiblo by 6, v« 
 
 1.2.3.4, &c, («_l) + l 
 =c, an integer, 
 
 we may derive a great many others, as 
 
 1«.2».3.4, &c, (a— 3)(a — 1)+1 
 
 =c, an intejror, 
 a ° 
 
 P. 2*. 3'. 4. 5, <kc, (a—4)(a — 1)+1 
 =r, an integer, 
 
 and so on till wo arrive at the same form as that in the tir>t deduction 
 
PRIMITIVE ROOTS. 483 
 
 PRIMITIVE ROOTS. 
 
 419. Theorem. — If p, a number prime to a, divide the successive powers 1, 
 b, a 2 , a 3 . . . there will be one at least, before arriving at aP, which ivill leave 
 the remainder 1. 
 
 The remainders being each less than p, there can be but p — 1 different 
 ones, and, therefore, in the p first terms of the series 1, a, a~, a 3 . . . a p_1 , 
 there are at least two which will give the same remainder. Representing 
 them by a m , a m ', and their common remainder by r, suppose 
 
 a m = Ep-f r, a' n '=E'p-\-r (1) 
 
 ... a m '— a m = (E'— E)^, or a m (a m '- m — 1) = (E' — E)p; 
 and, as p is prime to a, it must divide a m '- m — 1. Therefore we have unity 
 for remainder in dividing by p the power a m '~ m , which is <a". Q. E. D. 
 
 420. Let a n designate the lowest power other than a , which gives tho re- 
 mainder 1. All the preceding remainders are unequal. For, if for two 
 powers, a m , a mr less than a n , we could have the equalities (1), we might con- 
 clude, as just now, that a""~ m would give the remainder 1. Consequently, 
 a n would not be the lowest power to which this property belonged. 
 
 THEOREM OF FERMAT. 
 
 421. If p be a prime number which will not divide a, the division of&V' 1 by 
 p will give 1 for a remainder. In other words, a p_1 — 1 is exactly divisible 
 hyp. 
 
 It must be carefully observed that p is an absolute prime number, and not 
 simply prime to a. 
 
 Call q, q', q", . . ., and r, r', r", . . . the quotients and remainders of the 
 p — 1 quantities a, 2a, 3a . . . (p — l)a, divided by p. If we multiply these 
 quantities, and suppose E to be an entire number, we have 
 
 a. 2a. 3a {p — l)a=(qp + r){q'p + r'){q"p+r") . . . 
 
 =E+rr'r" . . . 
 
 The first member is equal to 
 
 1.2.3.... (p— l)aP-' 
 and, as the remainders r, r', r" . . . are all different (Ait. 415), the product 
 rr'r" . . . must evidently be that of the whole series of natural numbers, 1, 2, 
 3 , . . (*j — i), from 1 to (p — 1). Hence the above equality becomes • 
 
 1.2.3 {p—l)xaP- 1 =Kp + l .2.3... (p— 1) 
 
 .-. 1.2.3... {p — l)(aP-' — l)=Ep. 
 
 Tho 1° member of this equality is, therefore, divisible by p ; but since p is 
 a prime number, it can not divide any of the factors 1 . 2 . 3 . . . (p— -1) ; it 
 must, therefore, divide a? -1 — 1. Q. E. P. 
 
 Suppose that we take for p only prime numbers ; if we wish that the pow- 
 ers a n , a 1 . . . a p-1 should give for remainders all the numbers inferior to p, it 
 is necessary to choose a, such that aP _1 should be the lowest power above a°, 
 which gives the remainder 1 ; and if, among those which fulfill this condi- 
 tion, we take for a only numbers below p, Ave have -those which Euler calls 
 primitive roots. 
 
 For the best method of calculating them, the student is referred to the 
 article by Mr. Ivory, in the fourth volume of Supplement to Encyclopedia 
 Britaunica. We shall limit ourselves to setting down hero the primitive root* 
 of numbers as far as 37. 
 
484 
 
 ALGEBRA. 
 
 Numbers p. 
 
 Primitive roots of p. 
 
 3 
 
 o 
 
 
 
 
 
 
 
 5 
 
 o 
 
 3 
 
 
 
 
 
 
 7 
 
 3 . 
 
 5 
 
 
 
 
 
 
 11 
 
 2 
 
 6 
 
 . 7 
 
 . 8 
 
 
 
 
 13 
 
 2 
 
 6 
 
 . 7 
 
 . 11 
 
 
 
 
 17 
 
 3 . 
 
 5 
 
 . 6 
 
 . 7 
 
 . 10 
 
 . 11 
 
 . 12 . 14 
 
 19 
 
 2 
 
 3 
 
 . 10 
 
 . 13 
 
 . 14 
 
 . 15 
 
 
 23 
 
 5 . 
 
 7 
 
 . 10 
 
 11 . 
 
 13 
 
 14 
 
 15 . 17 
 
 29 
 
 2 . 
 
 3 
 
 . 8 
 
 10 . 
 
 11 . 
 
 14 . 
 
 15 . 18 
 
 31 
 
 3 . 
 
 11 
 
 12 
 
 13 
 
 17. 
 
 21 . 
 
 22 . 24 
 
 37 
 
 2 
 
 5 
 
 . 13 
 
 15 . 
 
 17. 
 
 18 . 
 
 19 . 20 
 
 20 . 21 
 19 . 21 
 
 26. 27 
 
 22 . 24 . 32 . 35 
 
 THE FORMS OP SQ.UARE NUMBERS. 
 
 422. Every square number is of one of the forms An or 4»-{-l. 
 
 Every number is either even or odd ; that is, every number is of one of the 
 forms 2n or 2n-\-l; and, consequently, every square is of one of the forms 
 
 4» s :n4n, 
 4n 2 +4n+l3z4?i+l. 
 
 DEDUCTIONS. 
 
 (1) Every even square number is divisible by 4. 
 
 (2) Since every odd squaro by the above is of the form 4(n--{-n)-{-l, and 
 since n 2 -4-tt is necessarily oven, it follows that every odd square is of the form 
 8n-f-l; and, consequently, no number of the forms 8n-{-3, Sn-\-5, 8n-\-7 
 can be a square number. 
 
 (3) The sum of two odd squares can not be a squaro ; for 
 
 (8w+l) + (8/i+l)^:4n + 2 ) 
 which is an impossible form. 
 
 423. Every square number is of one of the forms 5re or 5nil. For all 
 numbers, compared by the modulus 5, are of one of the forms 
 
 5ra, 5«il? 5ttrb2 : 
 and all squares, therefore, are of one of the forms 
 
 25n 2 =c5«, 
 
 25ft'±10w4-l3=5«4-l, 
 
 25n 2 +20rt+4 3:5n+4 or on — 1. 
 Therefore all squares are of one of tin; forms 5« or 5?j±1. 
 
 DEDUCTIONS. 
 
 (1) If a square number be divisible by 5, it is also divisible by 25 ; and if a 
 number be divisible by 5 and not by 25, it is not a square. 
 
 (2) No number of the form 5n-\-2 or 5h-|-3 is a square Dumber. 
 
 (3) If the sum of two squares be a squaro, one of the three is divisible by 
 5, and, consequently, also by 25 ; for all the possible combinations of the three 
 forms 57?, 5n-\-\, and on — 1 are as follows : 
 
 (5„ 4-1)4. (5n' + 1) a: on 4- 2, 
 
 (5n — 1) + (5»'— l)=c5n— 2nr5tt4_:;. 
 
 5n 4" ,r,rt ' =c5n, 
 
 5„ 4_(-,„'4_i)-i-- ) „_4_i, 
 
 gfl +(on' — l);n5/i — 1, 
 
 (r„,-|_i)4-(.-,«'_i)^:5«. 
 
FORMS OF SQ.UA11E NUMBERS. 485 
 
 Now, of these six forms, the latter four have one of the squares divisible by 
 5, and, therefore, also by 25. And the first two are eacli impossible forms 
 for square numbers ; that is, neither of thcso two combinations can produce 
 squares ; thorofore, if the sum of two squares be a square, one of the three 
 squares is divisible by 25. 
 
 (4) In a similar way, it may be shown that all square numbers compared by 
 modulus 10 are of one of the forms 
 
 lOn, 10ra + 5, 10/1+1, 10rc+6, 10;i+4, or 10n+9. 
 Therefore, all square numbers terminate with one of the digits 0, 1, 4, 5, 6, 
 or 9 ; and hence, agaiu, no number terminating with 2, 3, 7, or 8 can be a 
 square number. 
 
 (5) By examining, in like manner, the forms of squares to modulus 100, we 
 may deduce the following properties : 
 
 (6) A square number cau not terminate with an odd number of ciphers. 
 
 (7) If a square number terminate with a 4, the last figure but one must be 
 even. 
 
 (8) If a square number terminate with a 5, it must terminate with 25. 
 
 (9) If the last digit of a square be odd, the last digit but one must bo even : 
 and if it terminate with any even digit except 4, the last but one must be odd. 
 
 (10) A square number can not terminate with inoro than three equal digits, 
 unless they are 0's ; nor can it terminate with three, unless they are 4's. 
 
 424. All square numbers are of the same form with regard to any modulus, 
 <z, as the squares 
 
 2 , l 2 , 2 2 , 3 3 , &c. (ia) 3 , a being even ; 
 and as 
 
 s , l 2 , 2 2 , 3 2 , &c. y-^-) , a being odd. 
 
 For eveiy number may be represented by the formula an^r, in which / 
 shall never exceed \a. 
 
 Now (cm±r)"=a 2 n' i A:2arn-^i- 2 , 
 
 where it is obvious that r" and (an^r) 2 will leave the same remainder when 
 divided by a; therefore, (an^r)" 2 and r 2 will be of the same form compared 
 by modulus a ; but r never exceeds \a, therefore all numbers compared bv 
 modulus a are of the same forms as 
 
 s , l 2 , 2 2 , 3 2 , &c, r 2 , 
 or, as the squares, 
 
 2 , l 2 , 2 2 , 3 s , &c, {\af, when a is even, 
 and as 
 
 3 , l 3 , 2 2 , 3 2 , &c, \— — J , when a is odd. 
 
 DEDUCTIONS. 
 
 (1) When a is even, the general formula 
 
 (/ : /r : r t:2anr-j-r 2 
 becomes 4a' 2 n 3 i4a'nr-f-r 2 
 
 3r4a'(a'ra 3 ±«r)-f-r 2 . 
 Therefore, all square numbers are of the same form to modulus 4a as the squares 
 
 2 , l 2 , 2 3 , 3 2 , &c., a" ; 
 and hence we see immediately that all square numbers to modulus 8 must bf> 
 of the same fonns as the squares 
 
486 ALGEBltA. 
 
 : , 1«, 2 s 
 
 that is, they are all of the form 
 
 8ra, 8ra-fl, 8n+4, 
 as we have already demonstrated. 
 
 (2) The following tables exhibit the possible and impossible forms r* square 
 numbers for all moduli from 2 to 10. 
 
 Possible Formula. 
 2n, 2ra+l, 
 3n, 3n + l, 
 An, 4 7i + 1 , 
 5ra, hn^-l, 
 
 6n, 6w+l, 6?i + 3, 6?i+4, 
 7n, 7n+l, 7rc + 2, 7n-j-4, 
 8w, 8«-fl, 8re+4, 
 9n, 9ra+l, 9n + 4, 9/i + 7, 
 
 10«, 10n±l, 10n±4, lOwio 
 
 Impossible Formulce. 
 3«, 
 
 4«, 4« + 3, 
 
 5n, 5n-j-3, 
 
 6n, 6n -f-^i 
 
 7ra, 7n-f5, 7« + G, 
 
 8rc, 8n±3, Sra-f.7, 
 
 9n, 9?i±3, 9«-f 5, 9«+8, 
 
 lOrc, lO/iztS. 
 
 CONTINUED FRACTIONS. 
 425. Tue name continued fraction is given to an expression of the form 
 1 i 1 
 
 + 4 + i i U+i i 
 
 64-- c4-- 
 
 ^8 «' + , &c-i 
 
 l. c, a fraction whose denominator is a whole number and a fraction, and 
 which latter fraction has also for its denominator a whole number plus a frac- 
 tion, and so on. 
 
 An expression whoso numerators and denominators are any quantities what- 
 ever, may have the form of a continued fraction ; but continued fractions, of 
 which the numerators are 1 and the denominators whole positive numbers, are 
 the kind which most usually occur. 
 
 These expressions arise in various ways, and are of great use in finding (lit; 
 approximate values of fractions and ratios that are expressed in large numbers, 
 as well as in the resolution of certain unlimited problems of the first and second 
 degrees: in the latter of which the answer can not be easily obtained in whole 
 numbers by any other method. 
 
 Thus, in order to represent tile irreducible fraction or ratio by a continued 
 
 I) 
 
CONTINUED FRACTIONS. 487 
 
 fraction, let b bo contained in a, p times with a remainder c ; also, let c be con- 
 tained in b, q times with a remainder d, and so on, according to the following 
 scheme : 
 
 b) a (p 
 
 c)_Hq 
 d) c - 
 e) d (s 
 f, &c, 
 and we shall have, by the principles of division, 
 
 a c b d c e 
 
 c d 
 
 p, q, r, &c, are called partial quotients, and p-\-r, q-\-~ » &c, complete 
 
 quotients. 
 
 By taking the reciprocals of the second, third, &c, of the above equations, 
 we have 
 
 £_1 d_l 
 
 .:%=P + l=p + - + d=p+ l - 1 
 
 Whence, by extending the number of terms and generalizing the formula, wo 
 shall have 
 
 a 1 a 1 
 
 7=P+~ . 1 or t=— , 1 « 
 
 b q-\ — 1 b p4-- l 
 
 According as the numerator is greater or less than the denominator ; for in the 
 latter case we should invert the first as well as the second, third, &c, equations. 
 To convert a given irreducible fraction into a continued one, we have the 
 following 
 
 RULE. 
 
 Divide the greater of the two terms of the fraction by the less, and the last 
 divisor continually by the last remainder, till nothing remains, as in finding their 
 greatest common measure ; then the successive quotients thus found will be 
 ,.ho denominators of the several terms of the continued fraction, the numera» 
 tors of which are always 1. 
 
 EXAMPLES. 
 
 2431 
 (1) Reduce . _-, to a continued fraction. 
 * ' 1051 
 
 1051) 2431 (2 
 2102 
 
 329) 1051 (3 
 
 987 
 
 64) 329 (5 
 320 
 
 9) 64 (7 
 63 
 
 1) 9 (9. 
 
488 A! - 
 
 2431 1 
 
 Hence 1051 — + 3 + 1 - 1 
 
 5+- 1 
 7+-. 
 
 (2) 1096_1 
 
 9119 = 8+I 1 
 
 (3) 421_1 
 
 <I72 — -2+- 1 
 
 As the fracticn -, in every case of this kind, is supposed to be irreducible, or in its low- 
 b 
 
 eat terms, it is evident, by following the above process (which is similar to the method 
 used for finding the common measure of true numbers), that we shall necessarily arrive at 
 a remainder equal to 1 ; or otherwise a and b would have a common divisor, which is con 
 trary to the hypothesis. 
 
 Tho continued fraction obtained will consist of a greater or less number of terms, accord- 
 ing as the fraction - is more or less complicated ; but they will always terminate when - 
 
 \s rational. 
 
 426. A continued fraction may be converted into a series of vulgar fractions 
 by finding the successive sums of its several terms, after the manner of redu- 
 cing complex fractions to simple ones, in common arithmetic ; and the result 
 will be more or less accurate, according to the number of terms of the con 
 tinued fraction employed. 
 
 Each of these results is called a convergent, and they are numbered in 
 order. 
 
 Thus, if it were required to reduce the following continued fraction, 
 
 1 
 
 H — 1 
 
 C+-, &c, 
 
 to a series of common vulgar fractions, the operation will stand thus : 
 
 a . 1 ab-\-l , 1 c abc4-a-\-c 
 
 « =I (1), a + l =-±- (2), fl+ - + l =a+ _ = _J!lJ^ 
 
 or 
 
 (ab + l)c+a 1 1 cd+1 
 
 — ^-p— (3),a+j 1 !=*+£+ rf=«+£rf+&+2 
 
 C-f--. ('(/-)- 1 
 
 abcd-\- ab + ad+cd+l [{ab + l)c+a]d-\-tib + l 
 
 (4) 
 
 — bcd+b+d ~ (oc-fl)</+6 
 
 (1), (2), (3), and (4) are called the first, second, third, and fourth convergent. 
 427. By inspecting tho above convergent^, we perceive that each may be 
 formed from tho preceding by the following 
 
 nui.r. § 
 
 Add the product of tho numerator of the convergent already found by tht> 
 denominator of the next term of the continued fraction, to the preceding 
 
CONTINUED FRACTIONS. 489 
 
 numerator, for the next numerator and follow the same process for the de 
 nominator.* 
 
 EXAMPLE I. 
 
 1 
 
 3 +5+i 1 
 
 denominators or quotients 3, 5, 2, 7, arranged in horizontal line , 
 
 3 16 35 261 
 convergents ^ "F> JT' "go"' 
 
 3 16 
 After having formed the convergents - and — , the rule applies. Then mul- 
 
 tiply 16, the second numerator, by 2, the third quotient, and add 3, the pre- 
 ceding numerator, it gives 35 ; and multiplying 5, the second denominator, by 
 the same quotient 2, and adding 1, the preceding denominator, it gives 11 ; 
 and so on. This method may proceed from the commencement, if we write 
 
 - before the first convergence. 
 Thus, 
 
 
 3, 
 
 5, 
 
 
 7, 
 
 1 
 
 3 
 
 16 
 
 35 
 
 261 
 
 
 
 1 
 
 5~ 
 
 11 
 
 ~82~ 
 
 "When the continued fraction is not terminated, the numerators and denom- 
 inators form two series increasing to infinity. 
 
 428. The convergents are alternately less and greater than the value of the 
 continued fraction ; for the first in the general form is equal to a, and as the 
 fractional part which is added is neglected, this is too small. The second 
 
 convergent is a-{-7' *md, sinco b is too small by -, the fraction 7 is too great, 
 and, consequently, the whole convergent ; and so on. 
 
 EXAMPLE 11. 
 
 It is shown in geometry that the ratio of the circumference of a circle to 
 
 31415926535 . 
 its diameter is j „»..». „..»..» , which, by being converted into a continued frac 
 
 tion, and the successive convergents found, will be as follows : 
 
 3 22 333 355 103993 
 1' 7"' 106' 113' 33102' ; t 
 
 * The generality of tbis rule may be proved as follows : 
 
 N N' N" 
 Let — , — , — ■ be three consecutive convergents, m the quotient, of the same rank as 
 
 N" 1 1 
 
 the convergent — -, and - the partial fraction which follows — ; and let N"=N'w-f-N and 
 D" n m ' 
 
 N v 
 D"=D'm-f-D, according to the rule. The convergent which follows — is formed by sub- 
 
 1 N" N'm-f-N 
 
 stituthig m-\ — for ?i in -— . Making this substitution in its equivalent — — — , we have 
 n 1) D/n-j-D 
 
 N' (m+-) +N rtT/ 
 
 (N'm+NJn+N' N"»+N' 
 
 22 
 t Tlie second, — , was the ratio assigned by Archimedes ; the third, which is mucu 
 
 more accurate, that by Metius 
 
490 ALGEBRA. 
 
 and either of these will be the approximate value of the ratio, more and more 
 accurate as we advance. 
 
 429. The difference between two convergents is equal to 1 divided by the 
 product of the denominators of the two convergents. Thus, in the above ex 
 
 ample, the difference between the first and second convevgents is -, hetwenn 
 
 the second and third it is - — — -, or — — , between the third and fourth ■ , 
 
 and as the true value of the continued fraction is somewhere between any 
 two consecutive convergents, we have its value to within less than the fraction 
 
 ?> ^7o' or nn-rc. ' & c# ' according to the convergent which we take. 
 / 74J llU7o 
 
 To prove this in a general way, let 
 
 N N' N" 
 D' IT" D 7 ' 
 
 be three consecutive convergents, and m the quotient, of the same rank as the 
 
 N" 
 convergent t— ; then N" = N'm-|-N ; D"=D'»i-|-D. 
 
 N' N DN'— DN 
 But D-D=— BW- W 
 
 N" N' N'm+N N'_N'D'm+D'N-N'D'm-DN' 
 '* W'~ D 7== D'7ft+D" - D 7=_ D'(D'm + D) 
 
 D'N— DN' D'N— D N 7 
 "~ D'(D'm + D) — D'D~ ' ' ^ 
 
 The numerators of (1) and (2) are the same, with contrary signs ; and, to 
 
 a 
 find its value, we have only to go back to the first two convergents - and 
 
 rtfc + 1 1 
 
 — r — i the difference of which is 7. 
 b 
 
 430. Since the denominators of the convergents increase to infinity if tlie 
 series continue sufficiently far, it is possible to take two consecutive convergents 
 whose difference shall bo less than any assignable number ; wherefore, as two 
 consecutive convergents comprehend between them the value of the continued 
 fraction, it follows that a convergent can be found whoso value shall ciller from 
 that of the fraction by less than any assigned number. 
 
 For example, if the valuo of a continued fraction be required to within 
 
 , the convergents must bo continued till the product of tho denominators 
 
 of the last and last but one is at least 3 000. The last convergent will then 
 have the degree of approximation required. 
 
 N 
 The convergents aro fractions in the lowed terms ; for if a converge 
 
 admits of lower terms, some quantity </ must be a cmiiimii.ii measure of N and 
 D. Whence ( \rt. 29) q must be a measure of the multiples N'D and N I > . 
 and of ( \rt. •-".») DN' — ND or i 1 , which is impossible. 
 
 431. Each convergent is n nearer approximation to the tree value of the 
 
CONTINUED FRACTION'S. 491 
 
 N" N'?/j -4- N 
 tmued fraction than tbnt which precedes. For, let 7777=-^ — r~r: he a cmver- 
 
 D Dm-\-D 
 
 gent in which m is the last quotient employed ; then, if the complete quotient 
 
 M-|---{-) &c. 30 denoted by y, and y bo substituted for m in the expression 
 
 N " 
 of y—, it is evident (employing x to denote the value of the continued fraction; 
 
 that 
 
 N'y+N 
 
 N N' 
 Subtracting each of the convergents =r, r-- from this value of x, 
 
 N'y+N N (DN'— ND')// ±y 
 
 D'y+D D~~ D(D'y + D) ~"D(D'y + D)' 
 
 N'y+N N' ND'— DN' qpl 
 
 •nd 
 
 D'y+D D' — D'(D't/ + D) — D'(D't/+D)* 
 But 2/>l and D'>D .-. D'(D'y+D)>D(D'7/+D) ; 
 
 ?/ ;> ' 
 
 ••D(D'3/ + D)^D'(D'7/+D)- 
 
 N' N 
 
 vVhence ■=?, is a nearer approximation to the value of a; than =r. 
 
 432. Among continued fractions those have been particularly distinguished in 
 which the denominators, after a certain number of changes, are continually 
 •epeated in the same order, in which the continued fraction so formed is said 
 to be periodic, and may then always be considered as the root of a quadratic 
 equation or a surd. 
 
 To prove this, take a continued fraction entirely periodic. 
 
 1 
 xs=- 1 
 p+- 1 
 
 PJr p+, &c. 
 
 Then, since the number of these fractions is unlimited, it follows that the 
 
 sum of all after the first is also .r ,- whence 
 
 1 
 .r= — ; — .■. ar 3 +«.r=l 
 p-\-x ' * 
 
 ••• x=— 5i»±-J Vp*+4\ 
 in which case the above continued fraction serves to determine tr e vahie of 
 v ^p2_|_4 ) since we have, by transposition, 
 
 P 1 
 -4— l 
 
 2^>+- 1 
 
 P+ p+,tcc; 
 and if p in this last expression be put equal to 2, we shall have 
 
 1 
 
 lVp*+*=lp+z=%+: 
 
 V2 = l+- 1 
 
 v ~2_l_- l 
 
 ^2 + - l 
 
 2-L- 
 
 ^2+, &c 
 A continued fraction is also called periodic when the denominators occur 
 periodically in pairs, threes, fours, &e. ; thus, 
 
492 ALGEBRA. 
 
 1 1 
 
 - 1 or- ] 
 
 P+- 1 P+- 1 
 
 9+- 1 ?+- 1 
 
 P +q+,&C. r +p + - 1 I 
 
 *+f. ' 
 
 Again, a fraction may be irregular in some of its first terms, and only become 
 periodic at a certain distance from its commencement. 
 
 In either of these cases, as above, the value of x, the sum of all the terms, 
 may be obtained by the resolution of an equation of the second degree. To 
 prove this in a general manner, let 
 
 a, o, . . . . &c., be the quotients which form the non-periodic part, 
 p, q, . . . . &c, be the quotients which form the periodic part ; 
 
 then x=a-\—r 
 
 ": 1 
 +- 1 
 
 P+ q+, &c.J 
 
 and, representing by y the value of the periodic part, 
 
 1 
 P ~*"q+, &c, 
 
 we have xz=a-\-r and y=p-\-- 
 
 ': 1 ': 1 
 
 + " 4— 
 
 y y 
 
 Consider these continued fractions as terminating with the partial fraction 
 -, and deduce the convergents ; we have (Art. 426) two equations of the fol- 
 
 k/ 
 
 lowing form : 
 
 _Py+P R'y+R 
 
 The value of y, given by the first of these equations, is 
 
 P-Q-r 
 y — Q'.r— P" 
 which substituted in the second, gives, after reduction, 
 P— Qx R'(P — Q,r)- r .R(Q',r— P') 
 
 Q' x _P'-S'(P_Q.r) + S(Q'.x-P')' 
 which is an equation of the second degree in x. 
 By way of illustration, take the following fraction : 
 
 V V 
 
 x=a+- , p (1) or .r— a=- , v (2) 
 
 + «+f+£,to. «+*+. &«. 
 
 p 2a — (7+ ■v /< /"-M/> 
 
 .-. x — a= — ; ; or, resolving the equation, x= — 
 
 q -\-x — a ' 2 
 
 2a 
 If we transpose — or a, and substitute for x — a its value (2), we have 
 
 vV-+4p— q _ P 
 
 ■ 2 ?+?. /' , 
 
 q + . &c.; 
 
 or, making q = 2a, 
 
CONTINUED FRACTIONS. 4<J3 
 
 P 
 
 2a, &c 
 A similar mode of solution may be applied to continued surds or expressions 
 of the form 
 
 V«+V< 
 
 J* 
 
 a + V a i ^ Cc *' 
 
 the value of which, though apparently infinite, is always determinable by a cer- 
 tain equation, and in some cases in a real integral or fractional quantity ; for, 
 putting 
 
 x=\] a-\-~\Ja-\- -/a, & c -» 
 we shall have, by squaring both numbers, 
 
 u*=a-\-yl a+ Va+, &c, 
 the latter part of which is evidently equal to the original surd ; whence 
 
 x 2 =a-\-x, or x- — x=a .-. x=\^z -/{ + «, 
 where, if a=2, the expression becomes 
 
 r 2+V2+ V2+, &c., =2 or —1. 
 433. The process for developing any quantity, x, in a continued fraction, 
 
 1,1 1 
 
 consists in making successively x=a-{-—, x' = b-\-—, x"=zc-\-—, &c, a be- 
 ing the greatest whole number contained in .r, b the greatest whole numher 
 contained in x', c the greatest whole number contained in x", &c. 
 
 The numbers a, b, c, &c., being found, it is evident that if x', x'' &c., are 
 replaced by their values, the required development is obtained, viz. 
 
 1 
 x=a4-T , 1 
 
 d-\-, &c 
 
 EXAMPLE. 
 
 Let it be required to convert -\/19 into a continued fraction. 
 
 V~19=4+i ... ,-'= -JL— _ V^+4 , 
 x V19-4 - 3 J_ 
 
 '^£±1-^1-0 + i. . 3 . V19+2 
 
 rty proceeding in this way we shall obtain the following : 
 
 0.1 — I L o_i_ — . 
 
 x— 3 -+- 2 . n . 
 
 a/19 + 3 1 
 
 r III 1.. ' Q_L 
 
 — 2 — ' x lv ' 
 * Multiplying both numerator and denominator by i/ 19- M- 
 
494 ALGEBRA. 
 
 V19 + 3 1 
 
 riv -L 1_| 
 
 x — c — i_ r,.v 
 
 V19 + 2 1 
 
 — — = 2-| 
 
 3 ^V 
 
 1 
 
 ,V 1 _L 0_L. 
 
 x VI = -^19 + 4=8+ 
 
 X V1I1 
 
 VH3 + 4 1 
 
 Hence V~19 = 4+- 1 
 
 • 1+- 1 
 
 3 + 1 l 
 
 1+- 1 
 
 • VJI being tlio same as r 1 , it is evident that, omitting the 4, the greatest in- 
 tegral part of -y/iy. the quotients 2, 1, 3, 1, 2, 8, already found, wil always 
 return again in the same order to infinity. 
 
 Should it be required to convert the square root of 19 into a series of con- 
 verging fractions without first reducing it to the continued form, they may be 
 obtained, after the method before employed, from the integral parts of the 
 above results only. 
 
 Quotients, 4, 2, 1, 3, 1, 2, 8, 2. 
 1 4 9 13 48 61 170 1421 
 6' 1' 2' 3~' IT' 14' ~39"' 326 ' 
 
 EXAMPLES. 
 
 Ans. Quotients, -, 22, 1, 4, 2. 
 o 
 
 1 22 23 114 251 
 Convergents, -, -, -, — , — . 
 
 Ans. Quotients, -, 7, 1, 2, 4, 5, 1, 2. 
 
 1 7 8 23 100 iZZ 623 1769 
 Convergents, -, -, -, -, — , — , — , — . 
 
 Ans. The quotients are 5, 1, 1, 3, 5, 3, tVc. 
 
 5 6 11 39 206 657 . 
 And the convergents, -, -, j, y, — , —, &C 
 
 Ans. The quotients are 5, 3, 2, 3, 10, &c. 
 
 5 16 37 127 1307 
 And the convergents, y, — , —, — , -gr— , Arc 
 
 Ans. Quotients, 6, 1, 2, 2, 2, 1, 12, iVc. 
 
 6 7 20 47 114 161 2046 
 Convergents, j, j, -, y , — , — , — . 
 
 434. The converse of the proposition stated in Art. 432 is true, viz., that tho 
 root of an equation of the second degree may be expressed in functions of the 
 coefficients of tho equation by continued fractions. 
 
 The general form of tho equation of tho second decree may be wi Ken 
 
 ax- — bx— c=0 (1) 
 
 in which b and c may be essentially negative. This may be put under the 
 form 
 
 (H 
 
 251 
 764* 
 
 (2) 
 
 1769 
 5537' 
 
 (3) 
 
 t 
 
 •/31 
 
 (*) 
 
 V28 
 
 (5) 
 
 V45 
 
CONTINUED FRACTIONS. 4fl5 
 
 , c 
 ax=b-\--. 
 
 c 
 Multiplying the fraction - above and below by a, it becomes 
 
 ac ac 
 
 'ax b 4-_ ac 
 
 ^ 6+,&c. 
 
 1/ ac 
 
 "*" 6 +, &c.J Q. E. D 
 
 If a=l, tliis becomes 
 
 If 6 = 0, 
 
 0+ 6+,&c. 
 
 x 2 =c, 
 
 *=°+lL c - 
 
 U + + ,&c, 
 
 which has no signification. But if we make 
 
 ar 2 =(z — a) 2 , 
 
 a 3 being the greatest square contained in c, we have 
 
 x 2 =z 2 — 2az-}-a 2 =c,* 
 
 ••. z 2 — 2az = c — a 2 ; 
 
 or, putting c — a 2 =7, 
 
 z 2 — 2az = y, 
 
 z -2a =21, 
 
 and - Z =2 ° + L+, &c. 
 
 7 
 But since x=z — a, x=a-\-— y 
 
 ~a-\- — ■ 
 
 ' 2a+, &c. 
 
 To apply this, let the equation be 
 
 a: 2 =8 .-• a=2, 7=4, 
 
 •-=2+1 4 
 
 ^4 + , &c, 
 or 
 
 + 1 + , &c. 
 The above result may be obtained in a more simple manner ; tlnis, put 
 x 2 =c=o»4-/3 .-. re 2 — a 2 =/3 .-. (x—a)(x-\-a)—3 
 
 •'• X = a +^+x= a+ 2-a + l 
 
 T T 2o+, &c, 
 
 which shows that the square root of any number which is the sum of a 
 
 square, and of another number, is a continued fraction. 
 
496 ALGEBRA. 
 
 Thus, if we have .r-=7 .-. o=2, ,?=3, 
 
 r- 3 
 
 a/7=2+- 3 
 
 M-f, &c. 
 
 435. Continued fractions furnish ;i method of resolving in whole numbeis 
 
 the indeterminate equation 
 
 ax-\-by=c (1) 
 
 In this equation a, b, c are whole numbers, and the first two are supposed 
 
 to have no common factor. Let us conceive that we have developed the 
 
 a 
 tion r into a continued fraction, and that we have calculated all the con- 
 o 
 
 vergents ; the last will be equal to this relation itself. Let us subtract from 
 
 a' 
 it the next to the last, which I represent by y r The numerator of the differ- 
 ence will bo ah' — ha', and by the property of Art. 430 we have 
 
 ab' — ba'=±l (2) 
 
 Multiplied by + <-', this equality becomes 
 
 aX±&'c+6x faV=c; 
 then equation (1) is satisfied by taking ./. •= + &'<:, y=^ z a'c. 
 
 This solution being known, we Know (Art. 1G1) that all the others are given 
 Dy the formulas 
 
 x=^zb'c — bt, y=^a'c-\-al, 
 
 t designating any whole number whatever. We take the upper or lower sign 
 
 according as we have -f- °r — in the equality (2), or, what is the same thing, 
 
 a 
 according as the convergent 7 is of an even or uneven rank. 
 
 EXAMPLE. 
 
 Let there be the equation 
 
 261a:— 82y ==117. 
 
 261 
 Tf we reduce -77/ to a continued fraction, we find 
 
 Quotients, 3, 5, 2, 7. 
 
 3 16 35 261 
 Convergents, -, -j, — , — . 
 
 If we take the numerator of tho difference - — — — , and observe that 77-7 
 
 ~ .' 11 82 
 
 is a convergent of an even rank, we have 
 
 261x11—82x35=4-1. 
 
 Then, multiplying by 117, 
 
 261 X 11 X 117 — 82 x .".5 X 117 = 117. 
 The equation, then, is satisfied by making r=ll x 117 = 1287 and y=35 
 Xll7=4096; then, finally, the general values of x and y aro 
 
 x=1287+82«, y==4095+261fc 
 tf we divide 1287 by 82, and 4095 by 261, we find 1287 = 82x15+57 and 
 4095 = 261 xl">-f-180. Then, observing that t is any whole number what- 
 ever, we can write more simply 
 
 r=574-82f, y = 180 + 261/. 
 
RATIONAL SOLUTIONS OF QUADRATIC INDETERMINATES. 497 
 
 430. The following theorem will be found useful in the resolution of inde- 
 terminate equations of the second degree. 
 
 Let p' : — A(/=iD be an indeterminate equation, in which D<C \/A. 1 
 
 assert, that if this equation is resolvable, tho fraction - will be found among 
 
 the (ructions which converge toward -/A. 
 
 From the above equation we derive p — q-\/A = =, and, therefore, 
 
 p+qy/A 
 
 V — rb<* rfcD D« 
 
 -— yf A, which I represent by — rr= — ; then d=- 
 
 q T 7 ( i; + (? VA) p + qjA 
 
 Let — be tho converging fraction which precedes -, and which is of such a 
 
 nature that the sign of 6 will bo tho same with that of D ; it will remain to 
 
 Dq q — 
 
 be proved that we have — < — — , ovD(q-{-q )<^p-i-q ^ A. 
 
 .p-\-qy/A ( l+% 
 
 • _ (5 
 In the second member, instead oi p, I put its value, qy/A^-; tho in- 
 equality to bo proved can then be written thus : 
 
 (7+7o)( VA-D) + (<7-<7o) VA±^>0. 
 
 But this inequality is manifest, sinco we have VA>D, q^>q , and since 
 
 the part (q—q ) VA, which is at least equal to VA, by itself surpasses -, 
 
 •7 
 
 which is less than unity. — , then, will always be found in the fractions v, 
 
 converge toward VA, so that it will only bo necessary to- develop -/A in a 
 continued fraction, and to calculate the converging fractions which result, in 
 order to have all the solutions in entire numbers of the equation 
 
 x 2 — A2/ 2 =iD, 
 D being < -/A. 
 
 METHOD OP RESOLVING IN RATIONAL NUMBERS INDETERMINATE 
 EQUATIONS OF THE SECOND DEGREE. 
 
 437. Let the proposed general equation be 
 
 ar + b.vy+cf+d.v+ey+f—O, 
 in which x and y are the indeterminates, and a, b, c, d, e,f\he given entire 
 numbers, positive or negative. We first derive from this equation the fol- 
 lowing : 
 
 2ax+by+d=V[(1>y+d)*-4a(cy*+ e y+f)]. 
 If we make, to abridge, the radical =t, i 2 — 4ac=A, bd—2ae=g, 
 d* — 4af=h, we shall have the two equations 
 
 2<z.r-f by-\-d=t, 
 Ay*+2gy+h=zt*. 
 If we multiply the last of these equations by A, and make, again, Ay-\-g 
 =i\ g 3 — Ah=T>, we *ha!l have the transformed equation 
 
 v 2 — At?=&. 
 Reciprocally, if wo can find values of v and t which satisfy the equation 
 
 Ii 
 
498 ALGEBRA. 
 
 v 3 — A^=B, we deduce from it the values of the inde terminates x and y in 
 the proposed equation, viz., 
 
 v — g t — by — d 
 
 in which we should observe that both v and t may be taken with either sign, 
 as we may desire. 
 
 If we find the solution of the proposed equation in rational numbers, it will 
 ■uffice to resolve, by means of these numbers, the transformed v z — A< : =B ; 
 but if we wish to resolve the proposed in entire numbers, it will not only be 
 necessary that t and v be entire numbers, but that the values of t and v, sub- 
 stituted in those of x and y, give for these indeterminates entire numbers. At 
 present wo will only occupy ourselves with the resolution in rational numbers. 
 
 438. Every indeterminate equation of the second degree can be reduced, 
 
 as we have just seen, to the form v- — Af : =B; but, whatever may be the 
 
 rational numbers t and v, wo can suppose that they are reduced to a common 
 
 x y 
 
 denominator. Hence, making »==-, t='-, we shall have to resolve the 
 
 ° z z 
 
 equation 
 
 .r 2 — Ai/ 2 =B2 5 , 
 in which now .r, y, z are entire numbers. 
 
 We can suppose that theso three numbers have not a same common divisor, 
 for if they had had one. we could have made it disappear by division. 
 
 In the same manner, we can suppose that the numbers A and B have no 
 square divisors ; for if wo had had, for example, A=A'Z; 2 , B=B'l\ we might 
 have made J:y=y', lz=z', and the equation to be resolved would have become 
 
 xt—A'y'—B'z' 2 , 
 in which A' and B' have no longer a square factor. 
 
 The equation a; 3 — Ay' : =Bz- being thus prepared, we shall observe that any 
 two of the indeterminates x, y, z can not have a common divisor ; for if #-, for 
 example, should divide a: 2 and y", it must necessarily divide also Bz 2 ; but it 
 can not divide ; : , since the three numbers x, y, z have no common divisors ; 
 neither can 2 divide B, sinco B has no squaro factor, x and y, therefore, are 
 prime with respect to each other ; for the same reason, x and z are primes 
 with respect to each other, as well as y and 
 
 I assert, moreover, that A and B can be supposed to be positive ; for we 
 can only have, as regards the signs of the terms of one equation, tho following 
 three suppositions : 
 
 r=_A7-=4-l!: . 
 x*—Ay- = -iV. 
 x 2 +Ay 2 = + B:. 
 
 (I omit the supposition x--\-Ay" = — B:-\ sinco it is evidently impossible.) 
 Of these three combinations the second coincides with tho third hyn simple 
 transposition; but if wo multiply the third by B, and make* B:=;', AB = A\ 
 we shall have 
 
 z'- — A'y-=Bx': 
 The equation to bo resolved, therefore, can always be reduced to the form 
 
 x*— By '= \ 
 in which A and B aro positivo numbers, and do not contain any square factor 
 
RATIONAL SOLUTIONS OF QUADRATIC INDETERMINATES. 499 
 
 439. The method which we shall proceed to follow for the resolution ot 
 this equation is that given by Lagrange, in the Memoires de Berlin, 1767. It 
 consists in producing, by means of transformations, the successive diminution 
 of the coefficients A and B until one of them becomes equal to zero, in which 
 case the solution can be immediately deduced from known formulas. 
 
 The equation thus reduced is of the form x" — y- = Az", or x 2 — By*=z 2 , 
 
 but these two formulas do not differ, and it will suffice to give the solution of 
 
 the first, x- — y 2 =Az". To do this, decompose A into two factors a, ,8 (which 
 
 will always be prime with regard to each other, since A has no square factor), 
 
 and suppose that z also is decomposed into two factors p, q, such that we 
 
 have A=a/3, z=pq, we shall have the equation (x-\-y)(x — y)=a[ip-q-, which 
 
 we can, in general, satisfy by taking x-{-y=ap' 2 , x — y=Pq i ; this supposition 
 
 gives 
 
 au 2 -f-/3<7 2 c/^—dq 2 
 x= , y= g -, z=pq; 
 
 hence the three indeterminates x, y, z will be expressed by means of two 
 
 arbitrary quantities p and q ; if it should happen that the values of x and y 
 
 contain tho fraction A, x, y, z must each be multiplied by two. 
 
 Such is the general solution of the equation x\ — ?/ 2 =Az 2 , a solution which 
 
 v. ill comprise as many particular formulas as there are ways of decomposing 
 
 A into two factors. 
 
 For example, if A=30; there are four ways of decomposing 30 iuto two 
 
 factors, viz., 1.30, 2.15, 3.10, 5.6 ; hence will result these four solutions of the 
 
 equation x* — t/ 2 =30z 2 , 
 
 1°. ,r= p 2 -f 30<7 2 , y== p 2 —30q' 2 , z = 2pq, 
 2°. x=2jf--\-15q~, y = 2p' i —\bq 2 , zz=2pq, 
 3°. x=3p~+10q 2 , y = 3p°- — 10</ 2 , z = 2pq, 
 4°. x=bp 1 -\- 6q 2 , y=5p' 2 — 6q\ z = 2pq. 
 
 440. Let us proceed to the general equation x- — By 2 =Az 2 ; observe that 
 this equation, being the same with x* — Az 2 =By", we can, without diminish- 
 ing the generality o/ the theorem, suppose that the coefficient of the second 
 member is the greater of the two. In case of equality, the reduction that we 
 shall indicate would always be employed. 
 
 Let, then, the proposed equation be .r 2 — Bt/ 2 =Az 2 , in which we suppose, 
 at the same time, A>B, A and B positive, and free from any square factor. 
 
 We have already proved that x and y are primes as regards each other ; ?/ 
 and A, therefore, are equally prime to one another ; for if t/ 2 and A had a 
 common divisor 6, x* also must, necessarily, be divisible by 6, and x and y 
 would not then be primes to one another. 
 
 But since y and A are primes to one another, if we suppose that the 
 proposed equation is resolvable, and that we can, therefore, find determinate 
 values of x and y, .t=M, y=^N, we shall also be able to satisfy the equation 
 of the first degree, 
 
 M=nN— y'A, 
 in which M, N, A will be given numbers prime to one another, and n, y' two 
 indeterminates. 
 
 Hence, in general, without knowing the particular solutions x=M, y=N, 
 we can suppose x=ny — Ay' n and y' being two indeterminates; and, sub- 
 stituting this value of x in th* proposed equation, we shall have, after having 
 divided by A. 
 
500 ALGEBRA. 
 
 /«- — B\ 
 
 \— L —)tf-2™ni'+W i 
 
 But since y and A are prime to one another, this equation can not subsist 
 
 n=— B , ■ 
 
 unless, — - — be an entire number. Let this entire number = A'k 2 , k* being 
 
 the greatest square which can be a divisor of it, we shall have 
 
 \'k-, 
 and the equation to be resolved will become 
 
 A'k'y- — 2nyy'-{- Ay '- = ;-. 
 
 We perceive that if there be any value whatsoever of a which renders n 3 B 
 livisible by A, this value can be augmented or diminished by any multiple of 
 A, without ?r--B ceasing to be divisible by ce, we can suppose that 
 
 its value is comprised between the limits and A, or even between the more 
 extended limit \ — .'A and +' ^ • 
 
 We conclude from this, that in trying successively for n all the entire num 
 bers from — 'A to -\-} ,A, we shall encounter, necessarily, one or more values 
 which will render n" — B divisible by A, provided, howevor, the equation is 
 resolvable ; and in case these values will not render n- — B divisible by A, we 
 can conclude with certainty that the proposed equation is not resolvable. 
 
 441. Suppose, then, that we have found one or more values of ?i which 
 fulfill the required condition, tf will be necessary with each of these vain 
 continue the calculation in the following manner : 
 
 Resume the equation A'k-y : — 2nyy'-\-Ay'*=:z*; if we multiply it by A'k* 
 and if we make, to abridge, 
 
 A'k"y — ny'=x', kz = z', 
 
 tbe transformed will be 
 
 x'x'—By'y'=A'z , z' 
 
 This transformed could be resolved, if we could determine the solution of 
 the proposed equation, since the values of x', y', z' are easily deduced from 
 those of x, y, z ; reciprocally, the proposed will be resolved, if we find the solu 
 tion of ils transformed. For, from the known values of x', y', z', we cac 
 equally deduce those of x, y, z ; and it matters little whether these last value- 
 be under an entire or fractional form, since we have regard only to the resolu 
 tion in rational numbers, and since, after we have found any fractional values 
 of x, y, z, we can reduce them to a common denominator and suppress it. 
 
 Since we can suppose the number n <.' A, it is clear that "Try - or A' will 
 
 be <C}A, and, at the same time, positive; for n can not be < VB, since 
 
 otherwise n" — B would be <B, and could not be divisible by A. The 
 
 osed equation, therefore, will be reduced to ;ion in every respect 
 
 similar, in which the coefficient A', which takes the place of A, is less than 
 
 '. If we have, again, A'>B, wo can, in like manner, from the equa 
 '-• — By'"=zA'z' : , deduce a second transformed, 
 
 -n/'-=A' 
 
 in which A." will be <JA', '""l always positive. To obtain this Becond trans- 
 formed, there will be do new condition to bo full' [ready fa 
 
GAUSS'S METHOD OF SOLVING BINOMIAL EQUATIONS. 501 
 
 ■B 
 
 =Aft 3 , if we make n=/iA'+n', and if we take the indeterminate /* in 
 A 
 
 n' 2 — B 
 such a way that «/<^A', it is easy to see that — -r-, — will be an entire positive 
 
 number less than ] A' ; we have, consequently, 
 
 n' 2 — B = A'A"7c' 3 , 
 A" being less than JA', and not containing any square factor. 
 
 If it should happen that A", again, were greater than B, we should continue 
 this system of transformed equations, in which B is constaut, until we arrive 
 at one of this form 
 
 x 2 — By 2 = Cz\ 
 in Which C will be positive and <B. 
 
 443. But after we have passed into the second member of this equation the 
 term which has the greatest coefficient, which gives 
 
 x 2 — Cz"-=By 2 , 
 we can proceed in a similar maimer to the roduction of the coefficient B by a 
 second system of transformed equations 
 
 x" 2 — Cz' 2 =B'y' 2 . 
 
 x" 2 —Cz" 2 =B"y" 2 , 
 
 &c. &c, 
 
 in which the coefficients B', B", &c, will be positive, and will diminish in at 
 least a quadruple ratio, and thus we shall soon arrive at a transformed 
 
 x 2 — Cz*=~Dy 2 , 
 in which the coefficient D will be less than C. 
 
 But the series of positive arid decreasing numbers A, B, C, D will not go 
 on ad infinitum ; it will terminate necessarily at unity, and when we shall 
 have arrived at this term, the resolution of the last transformed, which is given 
 at once, will make known tho?e of all the preceding equations, and, consequent- 
 ly, that of the proposed. 
 
 GAUSS'S METHOD OF SOLVING BINOMIAL EQUATIONS. 
 
 444. The solution ofx" — 1 = 0, it has been proved (Art. 299), can al- 
 ways be reduced to tho caso where n is prime; ami the case of n a prime 
 number, by a method invented by Gauss, may be made to depend upon the 
 solution of equations whose degrees do not exceed the greatest prime number 
 which is a divisor of n — 1. The leading feature of Gauss's method is to rep- 
 resent the imaginary roots by a series of powers of any one of them, whose 
 indices form a geometrical instead of an arithmetical progression. Thus, if m 
 be a number (and such, called primitive roots of n, can always be found) whose 
 several powers from 1 to n — 1, when divided by n, leave different remainders, 
 and a be any imaginary root, then all tho roots may manifestly be represent- 
 ed by 
 
 o m , a m ", a"' 3 , . . . a" 1 " -1 ; 
 
 or, since m n_1 = t un -4-1, where /t is an integer, by a, a m , c m3 , &c, a™" - *. 
 
 445. The advantage of this mode of representing the roots is, (1) that they 
 can bo distributed into periods, each of which, when continued, will produre 
 the roots of that period i'i the f :ne order; and ( ') tha 
 
ALGEBRA, 
 till 
 
 r of such periods will be equal to the sum of a certain number of periods, 
 tho importance of which properties will be seen in the use made of them. 
 
 (1) Let n — l=rs, r being a prime factor of n — 1, and let m T =.li ; then the 
 roots may be written in vertical columns, each consisting of r terms, as follows : 
 
 a a h a 1 '* . . . a' 1 ' 
 
 a m a mh a mli _ < < a mli 
 
 ~r— 1 r— 1. • 1, 2 r— 1,,b — 1 
 
 a m a m b a m ^ . . . a "> b , 
 
 and if any one of the periods formed by the horizontal rows be continued, tne 
 roots in that period will be produced in the same order; thus, if the first 
 row were continued, the indices would be /i'r=?» r3 =m n_1 =^« + l' 7i ,+1 =ttl ^ *" , " , 
 = (lin^-l)in r =imh-\-h, &c, and the corresponding roots, a, a' 1 , &c. 
 (2) Let any two of the above periods be represented by 
 
 a* +o lb + ^^4., &c, 4-a ah5_1 
 a b_^ a bh_|_ a bh 2 _|_ ? & C-) ^.abh 8 - 1 , 
 
 and let us multiply them together, using each term of the lower line in suc- 
 cession as a multiplier, and starting at that term of tho upper line which stands 
 over it, and producing the upper line so as to supply the terms neglected at 
 the beginning, the result is 
 
 a 3+b _^.a»H-b - r -a ;lh '+ b -f, &c, 4-a ab9-1 + b 
 
 a(a+ b )l> _|_ a (ah+b)h _|_ a (nb 2 +b)h _L_ , &C., + a:^ 8_1 -H>)h 
 ata+bjb 3 _J_ a (ab+b)b 2 4_ a (ab 2 +b)b 2 4., SCC, -f U, ' S_ ' + b ) h * 
 
 a (a+b)b 8 - 1 ^_ a (ab+b)U , - 1 _^ a (aU 3 +b)^- I _J_ i ^CC, -f C ^ -1 +b)!»' , ~ l ; 
 
 and therefore, collecting the vertical columns into periods, we get 
 
 2(a a )S(a b ) = S(^+ b )4.v( a ai,+b)_|_v( ft ,b-+b^ #i- .. , ~ 1 +^), 
 
 or the product of two periods is equal to the sum of 5 periods ; and, conse- 
 quently, the product of any number of periods will be equal to the aggregate 
 of a certain number of periods. 
 
 kxample 1. 
 
 x' — 1=0 ; 6=3.2, .-. r=3, s = 2 ; also, 3, 3 : , 3\ 3', 3\ when divided by 7, 
 leave different remainders, viz., 3, 2, G, 4, 5 ; .-. m = 3, and the roots are 
 
 pi = a -\-a G 
 /' : =a?-\-a* 
 />—.;' 4- a 6 , 
 
 and pi4-p a +jp 3 =— 1- 
 
 AlflC p l p i = a* 4- .r-j-ir' -\-<i- =,]>. + /,. 
 
 ]>:}).< = aP -f Cfi 4- a -|- if- =p 1 -f /)., 
 p l p 3 = a 3 -\.a 4-a''4-„<=/> 4 
 
 ••■lh/>:-\ ]>■!>; + I'll'.— — ■■'• 
 
 nn-l ^i^.;': 1 =/'.- + /'' + / , - = ' :! +/'-'+/'.+/ , =' 1 - 
 
 Therefore the cubic which has /-,. y . ,\ For it- ro >te, is /''+/'■— ~p — l=i» 
 
GAUeS'S METHOD OE SOLVING BINOMIAL EQUATIONS. 503 
 
 EXAMPLK II. 
 
 c 17 — 1 = ; 1G=2.8 ; also, the powers of 3 from to 15, wheD divided by 17 
 leave remainders 
 
 1 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6, 
 .-. pssa +a° 4.a 13 +a :8 -i-a 1 «4-a 8 +a< + a 2 
 9=a3 4-a 1 "+a r '+a 11 4-a 1 «- r .a 7 4-a 12 +a«; 
 
 tuen p-\-q= — 1, and 
 
 p<7 = a»-f a 12 -f a^-f a -f a 2 -f a n -f a 7 -fa 5 
 a 2 -f a" _f a 8 4-o 9 +a -f a u -f a 12 -f a" 
 
 a 8 -fa 7 4- a i5_|. a 2_j_ a 4_j_ a 5 _|_ u^_ a io 
 
 =P+1 +P +P+P+Q +7 +'?=— 4; 
 therefore, p and 7 are roots of z 2 -f z — 4=0. 
 
 Next, the periods p, q may bo resolved respectively into the periods 
 r = a -f a 13 -f a ,6 -f a 4 ? £ = a 3 -f a 5 -f a u -f a 12 > 
 S = a 9 -f a''-f a 8 -f a 2 V w = a 10 -f a n -f a 7 -f a^ > ' 
 .-. r+s—p, 
 and rs = a 10 -f a 5 -f a 8 -f a 13 ' 
 
 a ii_|_a 14 4-a 2 +a lli 
 
 ^ =p+'?= — 1 ; 
 
 a s -fa 3 -f a 15 -f a 
 
 therefore, r, s are roots of z- — pz — 1 = ; and, similarly, t, u are roots of 
 — qz — 1=0. 
 Lastly, the periods r, s, t, u may be resolved respectively into 
 
 ri=a +a IG > s l = a^ -fa 8 > < 1 = a 3 -f a 14 ) tti=a 10 -fa 7 
 r,=a 13 -f a' 1 ) ' s 2 = a 15 -f a 2 S ' ^ = a 5 -f a 12 S ' w 2 = a u + a 6 
 
 then ri-fr 8 =r, 
 
 r 1 r 2 =a 14 -f « H + « 3 + <z f> =<, 
 .•. r,, r 2 are roots of: 2 — rz-f £=0 ; 
 
 Iff 
 
 and n, the greatest root of this equation, =a-f -=2 cos — . 
 
 For further information upon the theory of numbers,the student is referred 
 to the TliCorie des Nombres of Legendro, the Disquisitiones Arithmetics of 
 Gauss, of which there is an excellent French translation (Recherches Arith- 
 metiqucs) by Poullet-Delisle ; to Barlow's Theory of Numbers ; to the article 
 of Ivory in the fourth volume of the supplement of the Encyc. Britan. ; and 
 to the Memoirs by Libri, in tome v., 1838 (JEtrangcres), and by Cauchy, iu 
 tome xvii., 1840, of the MCmoires of the French Academy of Sciences. 
 

 
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