AI AT 1IIEM A T I C A KE Y. NEW COMBINATIONS IN RISPFCT rO THE in0oi l lyeorenm ani rotaritmnt ANI) A NEW DISCOVERY OF ONE GENERAL ROOT THEOREM FOR THE SOLUTION OF EQUATIONS OF ALL DEGREES: THE EQUATION, X-A, OR ANY SIMILAR ONE NOT EXCEPTED. BY JOSEPH B. MOTT. DESIGNED FOR SUCH AS HAVE FIRST STUDIED SOME SIMPLE WORK ON ALGEBRA, AND DESIRE TO HAVE A MORE PERFECT KNOWLEDGE OF THAT USEFUL BRANCH OF MATHEMATICS. DETROIT: PRINTED FOR THE AUTHOR BY ROBERT F. JOHNSTONE, OFFICE OF THE MICHIGAN FARMIER. 1855. Entered according to Act of Congress in the year 1855, by JOSEPHI B. MOTT, in the Clerk's Office of the United States District Court for the District of Michigan. PREFACE. THE discovery of a general theorem which will develop the roots of equations of all degrees, has seemed to the author to be sufficient apology for offering the subsequent pages to the world. They might have been much extended: many new facts might have been presented, and various examples introduced, which would have greatly enhanced their utility. Indeed the more simple parts of algebra should be so connected with the results here given, as to form one complete chain of reasoning from beginning to end; but the author's health and personal circumstances would not permit him to devote a sufficient portion of time to the accomplishment of such an object. And as an undue extension of plan is often a cause of failure in the undertaking to carry it out, whereby many useful discoveries may be long hindered from seeing the light, I have thought it best to present these researches as they are, without any modification or enlargement, and run the risk as to what I may be able to do hereafter. Few know or can realize the difficulties that must be encountered by one who undertakes to get up anything original: first, the great variety of experiments that must be made, take up a great deal of time, often bewilder the mind, and frequently amount to nothing but to show what cannot be done in that way; then again the mental faculties are apt to be so intensely occupied, that the loss of health is a common result. Those who spend a large portion of their time in this way, have just so much less left to devote to the acquirement of property, or the means of living, for the want of a sufficiency of which they often fail of accomplishing their object at last; or if a partial success attends their efforts, yet what is brought to view is new and unpopular, and perhaps but iv PREFACE. little noticed during the life of the discoverer. On the other hand, however, inventions and improvements follow each other so rapidly at the present day, and so many things have been brought to light that were scarcely dreamed of a short time ago, that public expectation seems more than formerly to be on the lookout for new discoveries and developments. Perhaps, then, among the denizens of this busy and curious world, some may be found to take an interest in the examination of the following pages. The method of developing the coefficients in the binomial theorem, and finding the logarithmic series, etc., I think will be acknowledged as more simple than any hitherto in use; and the application of the theorems given to the finding of converging series, as often as desired, will serve more fully to exhibit their facility and usefulness. As it respects the general root theorem, I claim both the discovery and invention of it as my own. If any other person has made a similar discovery, it must be a late occurrence, and is wholly unknown to me. DAVIES and other late authors say expressly that no direct formula has appeared for the solution of equations of a degree higher than the fourth, except when the roots are part or all of them rational; but it will be seen in the following pages that equations of all degrees, even many cases of infinity, have a direct solution by one general rule, that is less tedious when applied to numerical equations than any rule of approximation, as well as more perfect. One great obstacle to the successful prosecution of algebraical researches heretofore has been the tedious methods of resolving the higher equations: the design of this work is to obviate that difficulty, and bring all equations nearly on a level. The reason why mathematicians have not long ago discovered a general theorem for this purpose, is because they have generally commenced their investigations byv placing tile PREFACE. V terms of the highest powers of the unknown quantity first at the left hand; in which position, no direct formula could ever possibly be discovered for cases of infinity. Indeed all that has ever been done in this way consists merely in proposing a new formula (if not more than one) for every different degree of equation, until arriving at those of the fourth degree, where all perfect rules have hitherto suddenly stopped; and even the formulas which have been used for cubic and biquadratic equations are so tedious in most cases, when numerically applied, that the less perfect rules of approximation are generally preferred. It is easy for a mere tyro in algebra to propose equations that will puzzle an adept to solve by former rules, but will not be so easy after these researches shall become known. There are various forms to which equations may be reduced before applying the root theorem, that will much simplify the result in many cases: these forms are not all presented, but only those which are of the highest importance in respect to solution, though many others might have been added with propriety. My main object has been to present something new and useful: how far I have succeeded, I leave for the competent to judge. The methods given for the resolution of quadratics in the popular works on algebra are not to be overlooked, although not introduced here; but all the methods for solving the higher equations may be dispensed with beside the root theorem, except cases in which some of the roots are rational, or where the equation is solvable by the binomial or' logarithmic theorem. The particular utility of' the root theorem consists in the determination of the irrational roots of degrees higher than the second. JOSEPH B. AMOTT. 1DOWAGIAC, CASS Co. MICH. AIarch, 1855. CONTENTS. Page. Preface - - iii Introductory Remarks - - - 1 Introduction to logarithms 2 Expansion of binomials 6 Recapitulation of formula - - -10 Examples in development - 12 Computation of napierian and common logarithms - 14 The general root theorem - 20 Solution of equations of all degrees - 24 Equations of the second degree -- 25 Equations of the third degree -- - 33 Equations of the fourth degree - 37 Equations of the fifth degree - 41 Equations of the sixth and seventh degrees - 42 Question in Annuities -- 43 Equations of infinite degree - - - - - - - - 44 Exponential equations -47 N. B. It is possible that some slight errors have been overlooked in the writing or printing of the results of examples given in the solution of equations, as many of them were noted down barely by inspection; but the theorems have been guarded in this particular, and are sufficient to rectify all mistakes, should there be any. INTRODUCTORY REMARKS. IF the binomial theorem be well understood, even in its most simple form as first presented in the following paragraphs, it will, I trust, by the method here proposed, be found a comparatively easy task to accomplish what has not hitherto been effiected with equal brevity and perspicuity, if at all. I am aware that in some developments I have presented things in a manner different from the customary proceeding of mathematical writers, and that my reasons for so doing may not always appear at first sight; but the propriety of my assumptions will, generally become manifest in the results. A disposition to experiment in substitution is probably the main reason for the course I have frequently pursued. In my opinion, the methods of algebra may yet be made to cover the whole ground of the calculus, and even to go beyond it. If it be proposed to give a plain illustration of the principle of maximum and minimum, or of the distinguishing properties of a spiral, etc., I should prefer a resort to algebra rather than to the differential and integral calculus; though it is true I have not had time to examine all the mysterious developments that are reported in the higher mathematics, so as to see whether they can be referred to algebraical methods or not. Nevertheless I have not a doubt but that the blind methods of reasoning (so at least they appear to many minds) introduced in the calculus might be wholly dispensed with. As some may wish to study these few pages who have not yet paid any attention to the nature and use of logarithms, I present the following as an [ MATI. KEY.] 1 MATHEMATICAL KEY. INTRODUCTION TO LOGARITHMS. ART. 1. First take the equation b-= N; in which, if we consider b as the base of a system of logarithms, then x is the logarithm of the number N; that is, log N - x, where log stands for logarithm. Also if 10 be the base in the equations 10'1=10, 10 =100, 103- 1000, &c., then log 10 = 1, log 100- 2, log 1000= O3, &c. In like manner, in the equations bx= N, bx'= N', bx" = N", &c..........-(1) logN = x, log N'= x', logN" = x", &c. Now if we take the product of the equations 10: = 10, 10 = 100, 103 = 1000, we have 10 X102 X 103 = 10 X 100 X 1000 = 1000000 = 106 = 101+2+3 but according to our former position, in the equation 10 = 1000000, log 1000000 = 6 = 1+2+3 3 = log 10 + log 100 + log 1000. Again, if we take product of equations (1), then bx b' x b" = Nx N'x N", or x+x'+x" = NXN'XN"; therefore log (N X N' X N") = x+x'+x" = log N + log N'+ log N". Hence we come to the conclusion that the logarithm of the product of any number of factors is equal to the sum of the logarithms of the factors.................. (THEOREM 1.) Now if we take the logarithms of 2 and 3 as hereafter computed, we have log2 =.301030 nearly, and log 3-.477121 +-; then by the above theorem, log 4 = log (2 X 2) = log 2 + log 2 = 2 log 2 = 2 X.301030 =.602060. Also by former considerations, log 10 = 1 - log (2X5) = log 2 + log 5 = 1, or log 5 = I - log 2; that is, log 5 = 1-.301030 =.698970. INTRODUCTION TO LOGARITHMS. 3 log 6 = log(2X3) = log2+log3 =.301030+.477121=.778151. log8 = log (2 X 2 X 2) = 3 log2 = 3X.301030 =.903090. log 9 = log (3 X 3) = 2 log 3 = 2 X.477121 =.954242. log 12=log (3 X 4) = log 3 + log 4 =.477121+.602060=-1.0791.81. We also have in the equation p ab, logp _ log (ab) - log a + log b..-.....(1) Or by division, P - b: then log b - log P-; a a but from equation (1), log b - logp - log a; therefore, log 1 logp - log a: that is, the logarithm of a fraction is equal to the logarithm of the numerator, minus the logarithm of the denominator. (THEOREM 2.) Or, for a more general theorem for fractions, let us resume the equation log - log p- log a, and make p- qrs and a def: then log Qf = log qrs - log def; but by theorem 1, def log qrs = log q + log r + log s, and log def = log d+ log e+ logf; therefore, log rs logq logr +f log s - logd - loge -logf: def that is, the logarithm of a fraction composed of factors is equal to the sum of the logarithms of the factors constituting the numerator, diminished by the sum of the logarithms of the factors constituting the denominator..-.- (THEOREM 2m.) Therefore by the two last theorems, etc., we have 1. log 5 - log Li- log 10 - log 2 = 1-.301030 =-.698970. 2. log -- log 2- log 3 -.301030-.477121 -.176091. 3. log X5 log 3X 4 log 5 - 6 log 2 (as 2x4x8 = 2X2x22X2X 2), =.477121 +.698970 + 1 —6x.301030 =.369911 = log 152. 4 MATHEMATICAL KEY. 4. In the equations xy - a and -x_ b, we have y log x + logy = log a and logx - log y = log b, or l6g x = J(log a + log b) and log y = (log a - log b). 5. log1 - log - logn - log n 0: log'- = 0 - 3 log 3 (since 27 = 3X3X3), or log-1 -= - 1.431363. 6. log 310 -- log3 -1 -1.477121. 7. log 2x - 2 log 2 - 2 log 3 - - 1.556302 2X3x6 = log-3 6. Thirdly, as log (aaaa) -- log a4 - 4 log a, therefore log a"- n log a; r r and ifn -, then log a loga: m m that is, the logarithm of any power or root of a number is equal to the logarithm of the number multiplied by the exponent. ----------- ------------------- (THPOREM 3.) 1. log 81 - log 34 - 4 log 3 - 4X.477121 = 1.908484. 2. log 1024 - iog 2'0 - 10 log 2 - 10X.301030-3.010300. 3. log-r/2 - log 2 -= A log 2 - ~X.301030-.150515. 4. log /3 -log 3 _ log 3 7= X.477121 -.159040. 5. logV5 log 51, log 5 - oX.698970 -.069897. 2 6. If 2 -= 100, then x log 2 - log 100l - 2, or x- = g 2 7. If 3X+y- 9 and 5x- -- 6, then log 9 2and xy log 6 log 3 log 5 or x A(2~ 778151) and y ( (2 77SI51) or -- ~2(2 q-698970 and y 698970 8. If a = b, then x loga = log b, or x log log a' y loga 9. If cZ= a, then Y log c log a, or z z log c INTRODUCTION TO LOGARITHMS. that is, the logarithm of any power or root of a number? divided by the logarithm of the number. is equal to the exponent. ------------------ ------------ (THEOREM 4.) 10. If xY a and bxc -d, then ylog x loga and logb + clog x — logd, or 1 1 1 logx - loga and log x - (log d - log b) -log a; y C ii c log a therefore, Y log d log b 11. Given wxy -a, wxz -b, wyz - c and xyz- d, to find the logarithms of w, x, y and z. ANS. log w = ~(log a+log b+log c+log d)-log d, log x = log w+log d —log c, log y = log w+log d —log b, and log z = log w+log d-log a. 12. Given xyz= 4, = _S and x'y2z"- - 2, to find the logarithms of x, y and z. 15 ANS. log x 15log2, or x= 211; log 2 5 211 logy —-— flog2, or y=; 32 log z = 32 log 2, or z = 211o Before much of the utility of logarithms can appear, we must have a table of them, in which the logarithm of any given number, and conversely the number answering to any given logarithm, can be readily found to a considerable degree of exactness by inspection: the method of computing such a table will be presented hereafter. Having such table, let it be required to find how long it would take any sum s of money to double itself at compound interest, at a given rate r. By the arithmetical rule, the interest of the First year = rs, and the amount = s+rs = s(lq-r); Second year = rs(l+r), amount = s(l+r) +rs(l-r) = s(l+r)(l+r) =s(1+r); Third year = rs(l+r), amount = s(l+r)2+rs(1+r)2= s(l+r)(l+r)=s(1+r)3; Fourth year = rs(1+r-)3, amount = s(l+r)3+rs(1+r)3= s(l+r)(1+r)3=s(l+r)4, &c. &c. 6 MATHEMATICAL KEY. Therefore if A -amount, s = sum, r- rate and t - time, the general equation A - s(lH-r)t will express the amount of any sum of money at a given rate per cent compound interest, for any length of time. If the sum be at interest just long enough to double itself, we should have A - 2 s, or 2 s - s(l+r)t, or (l+r)t = 2 then, by theorem 3, t log(l+r) = log 2, or t log2 log(l+i)' If the sum remain at interest until it trebles itself, or until it becomes ten time its first value, then log3 log10 _ 1 log(l+r) log(l+r) log(l+r)Hence any sum, at 7 per cent compound interest, will double in the time denoted by the equation log2.301030 Yrs. mos. das. log2. 3010.2447 years-10 2 28nearly. log 1.07.029384 At 6 per cent, t log_ 2 301030 11.8956 years11 10 23 nearly. log 1.06.025306 One arithmetician gives 10 years 87 days as near the time at 7 per cent. Others give 11 years 10 months 22 days for 6 per cent: 23 days come nearer to the truth than 22; 226, nearer than either. EXPANSION OF BINOMIALS. AnT. 2. Let it be required to expand the binomial (l+r)8. First let us find the 2d, 3d, 4th, &c. powers of (1+r). Thus 1+r and 1+2r+r2 into l+r into l+r gives 1+2r+r2 = (l+r)2, gives 1+3r+3r2+r3 = (l+r)3. And in like manner we may find (1+r)4= 1+4r+6r2+4rq3+r4,........................(1) (1+?r)5= 1+5r+ lOr+ 10-r3+5r4+r5,..................(2) (1 +r)6= 1+6r+ 15r+ 20r3+ lr4+6r5+r6, (3) (1 +r)7= 1 + 7r+ 21r+ 35r3+35r4+ 21r5+ 7r6+r7,......(4) (l+r)8= 1+ 8r+28r 2+56r3+70r4+56r5+28r6 +8r7+r8, -- (5) &c. &c. EXPANSION OF BINOMIALS. 7 ART. 3. We now propose to find the general law that governs the coefficients and exponents in equations (1), (2), (3), etc. By inspection, it is seen that the exponents commence in the second term with unity, and increase in each succeeding one by the same to the last term. The terms without the coefficients are 1, r, r2, r3, etc.; in which we see that each term (after the first) is a multiple of the preceding term, and has one more factor. Now let us make a similar supposition in regard to the coefficients, by assuming A - coefficient of r, A B - coefficient of r2, A B- C of r', A B C'D of r4, &c.*. Then we have (l+r)'= l+Ar+A'Br2+AB Cr3+AB C'Dr4+&c...(S) Now if we make s = 8, and compare equations (5) and (S), (a! they become equal) we have A=s=-8, A'B=28, AB.C=56, AB'C'D —=70, A-B'C'D'E=56, &c. 28 28 7 A-1 s —1 Hence B- A A 8 2; 56 56 6 A-2 s-2 A- 28 3 3 3 3 70 70 5 A —3 s-3 A -BC 56 4 4 4 &c. &c. If we substitute these values of A, B, C, etc. in equation (S), we get (-r) s —1 s-2 s-1 s-2 s-3 (1+r)'= i+sr-s.- r2+s.. -r3+S. 2 2 3 2 3 -' +&c................................. (k) when s- 8. To see whether this series will remain the same when s has a different value, let s = 6, and compare equations (3) and (S): then we have A=6=s, A'B=15, A'B'C=20, A'B'C'D=15, &c. " When dots are used for x to denote multiplication, they will (except when placed between fractions, or between a fraction and a whole number) be elevated, in order not to confound the decimal point. $8 MATHEMATICAL KEY. 15 15 5 A-1 s —1 Consequently B - 2 2A 6 2 2 2; 20 0 4 A-2 s-2 A-B 15 3 3 3 D 15 15 3 A-3 s-3 A-B-C 20 4 4 Substituting these values of A, B, C, etc. in equation (S), we get (l-r) — l-sr-s -lr-2 s —1 s-2 s s — s2 s-3 4 1+7r)-= i+sr+s.-r-S.-. r3+s.-..-. r 2 2 3 2 3 4 + &c...... - -.................................. (K) when s — 6, the same as when s 8-, and a like result will be obtained with any value that may be given to s. We may therefore regard theorem (K) as a general rule for finding any power of (l+r): indeed it will be found the key theorem to the more intricate and comprehensive ones. 1. Let it be required to expand (1+r)'~ by this theorem. We have 10-i 10-i 10-2 (l+r)= l+lo+10r1. r2+10 0. — 3 —+&c. 2 2 3 or (1+r)1~= 1+ 10r+45r;+ 120r3+210r4+ 252r5+ 210r6 — 120r7 + 45r8+ l Or9 + r10. By inspecting the last example and equations (1), (2), (3), etc., it becomes manifest that the number of terms in the righthand member will always be one more than the number of units in the exponent s. A more scientific way of arriving at the same conclusion s —1 s-2 would be to inspect the factors s,, etc. of theorem (K), in which it is seen that these factors continue to diminish until reaching s — 0, where the series must end. The term containing the last mentioned factor is the one numbering s-]-2, which proves the observation correct. 2. Given (a+b)", or its equal a"(l+-)), to expand. a EXPANSION OF- BINOMIALS. 9 Here, in theorem (K), let r - 6 and s = n: then b b n-1 b2 n-1 n-2 b1 a a 2 an 2 *3 a' n-1 n-2 n-3 b4 2 3 4 a. or (a-+b)" — a'Fna-1lb+n.- - a -2b2-+n. l.n —a n-:b: 2 2 3 n-i n-2 n-3 +n. n1. — 9-~3-. a" n4b4.+L &c. 2 3 4 3. Given al(1 Z )n to expand by theorem (K). z m s —1 s-2 Let r - and s ~ then the factors s, -, &c. a n'2 3 m m —n m-2n m-3n will become-, I, -3 —, &c. Therefore, n 2n 3n 4n (a+z)a m z+m m-nz m n-n m —2nz &c n a n 2n a2 n 2n 3n a Now if m - 1, this becomes 1L! z__ I1 1-n zlzl 1 1-n 1-2n- z )3 (a+z)n= a(i+ &C.- -----. na n 2n a n onn 3 a and if moreover a = 1, we get 1 Il 1-n I-n 1-2n 3 (l+z)v= 1+l +-. ~- + 1n n 2n n 2n 3n n 2n 3n 4n ART. 3. For mnore convenient reference, let us recapitulate: [r MATHI K EY2 L i4 s-i 2 s-i s-2- s-i- s-2 s-3 (t+r)' 1 + sr + - s. r S S +S. r+&c.- s.-. ~- + s&. r......... 2 2 3 2 3 4 a +na + n — b+ n- -2a n -I n-2 n-3 b + (A) 2 2 3 2 3 4 1l$z~_ 1 1[_ 1-n _ 1. n 1 -1 1-21 1 1-n i-2n 1-3n4 (); - +'Z+ z-2- 3' I:''C ~.~.-' ~n au 2?n n 2n 3n n 2n 3n 4n /lzl / -nz1 1 1 —n 1-2n I -n z3 1 1- 1-2n 1-3n4 (a+z) nan 1+ —+. _ +.. (C) ~ X a n 2na2 3n a3 n 2n 3n 4n a4 m' [_m z m m —n mz rn- m-n m-2n 2m -n m-2 -3nz4 (a+z)-: [~ - +-.~ -+.. - +. &c. (D) Although theorem (K) is the key to all the others, and might be used in all cases, each one will be found more appropriate to some particular examples. O EXPANSION OF BINOMIALS. 11 If we examine theorem (B), or any other one having a fractional exponent, it may be seen that the series will not i 1-n 1-2n end. The reason of this is that the factors -, n n 2n 3n etc., though produced ever so far, will never come to a term whose value is zero, except when n - 1, which would make - 1, and thereby render the exponent integral contrary to the position taken. The same observation may be made in regard to formula (D), whenever the fraction - can not be reduced to a whole number. Also in either of the formulas, when the exponent is supposed to be negative, the series will be infinite, and every other term will be negative. At other times, the terms will be all negative or all positive, according to the particular values given to the terms of the binomial and to the exponent. The nature of binomials, and also of the more complicated developments, is such that a correct knowledge of the properties of positive and negative quantities is of the utmost importance when we engage in the substitution of numbers for letters. Most algebraical authors enter into a separate examination of binomials with fractional and negative exponents, without deducing them from integrals by substitution as heretofore presented; but the same authors would consider it perfectly compatible to transform the equation x2+ ax- b into one whose roots should be reciprocals of those of the primitive, by making x —; or into various other forms, by making y n rn-n x -, x f n+y, X -y, - x —, x mn, &c. &c. Z -+n With equal consistency, then, may fractional and negative exponents be made to depend on integral and positive ones. If any one wishes to determine the series that the binomial (1+r)n will produce without the aid of theorem (K), first find by the common rules the square root to three or four terms of (].+r), and elevate it to the third power, or take the cube 12 MATHEMATICAL KEY. 3 root of the same binomial and square it; that is, find (l+r)2, 2 s (l+r): then assume (1-[r)-= 1+Ar+A.Br2+A.B.Cr3+&c. and make - equal to 3- in the former example, and to 2- in the n latter, and compare the coefficients in each development with the corresponding ones found by the first operation. Any other fractional power of the same binomial, whose development to three or four terms by the common methods would not be too cumbersome, might be treated in the same manner, to familiarize the student with the formation of the coefficients. To study the properties of binomials having negative exponents, without resorting to theorem (K), we may first find a few terms of the series of such binomials as the following, 1 1 1 (1+r)' (+-)'2 (1+9r)' &c. by the common methods, and then compare as before with the assumption (1 +r)-S — I+Ar+A'Br-+A'B'Cr3+&c. EXAMPLES REQUIRING THE THEOREMS (K), (A), (B), (C) AND (D). 1. Expand (1+2c)Y by theorem (K). ANS. (1+2C)' = 1+6c+12c2+8c3. 2. Expand (2+3y)4 by theorem (A). ANS. (2+3y)4 =16.+96y.+216y2+216y3.+81y4. 3. By theorem (B), 1 3 3.5 3'5.7 (1+-) lA —-.. —..- -&c.& 1C2. 24 2'46 2A468 24-6'8'10 4. By theorem (C), 196 4 1 1 3 3_5 ( oo00 100 22492 2 4 6 8.494 5. By theorem (D), i34(1+2 2 2 22 4 2'4'7 (8+1)3- 4( 1+3.83.6- 2 3*6983 3-6.9.12.84 ) f81. In examples 3 and 4, it is seen that the square root of 2 is expressed by series having very different terms. EXPANSION OF BINOMIALS. 13 The series derived from (1+1)1 is the one commonly chosen to exhibit the square root of 2: it is, however, of little worth for computing the same; but the series of Ex. 4 converges very rapidly. To obtain /2 = (1.96+. 04)%, first find by trial that the square root of 2 is near 1.4: then make 1.42 or 1.96 the first term of the binomial, and the second will be 2-1.96.04. A still more rapidly converging series can be found, by first taking a nearer value of /2. 5. Required the cube root of 2. First let us try 2 3- (1+1)3: then by theorem (B), (1+1) t 2 2'5 2.5'8 \i(1-1) 3 1 i-{ 3-..6+3. -36.9. 12 +&c.- nearly. 43 64 Now to get a more converging series, let 337 be made the first term of the binomial, when the second will be 2- —' A; that is, ( _o —?) lY2. Then by theor. (C), a —27 or a~= 3, z- -7 z= -35-, =n- 3; and therefore a 8(64 10\?j15 2.52 - 2.5.532 -- ~ ~- 3_ 3. — _- __- &C. f (27 27 ~( 3f32 36322 3s6-9-323 - 1.259921. To find a series that will converge still faster, make the first term of the binomial equal to 1. 263 — 2.000376: we shall then have -. 000376 for the second term; and by the theorem, 376 2'3762 2'5'3763 f/2 — 1 26(21 3 &C. 3'2000376 3 3762000376 2000376 6. Required the fifth root of 10. By a few trials, 1.6 is found to be a near value of the root required: then we have 1.65- 10.48576 for the first term, and -.48576 for the second; that is, 1 1 10s- (10.48576-. 48576)5. Then by theorem (C), 105-1 6(1- 48576 4.485762 4'9'485763 5(1048576 5.10.10485762 1 0 15-10485763 - 1.58489319246. 14: MATHEMATICAL KEY. 7. Develop (16+1) by theorem (D). We have a-16, z —1, m 3, nX4, anz8, -f-l-, -=4 ea n -n 9l m-n _ — 2, &c. Hence 2n 3n s 3 3 3'5 3'5'9 (16-{1)~- S' 1- 4 t6- 4S8.16 + — 4812-1-,-3 4.8.12.6,1616 3 -174. ART. 4. The use of the binomial theorem in the raising of powers and extracting of roots has now become apparent. We next propose to apply it to the determination of unknown exponents, and to the development of series for the computation of logarithms. Ex. The equation BxR 1+a - N being given, to develop in certain cases. From this equation, we have B - (1+-a)x. To get an easy development in terms of a, assume (1+a)x — (1 —nx)X - B: then N - Bx - ((l+nx)'X)x- (I+nx)7l- l+a, or 1+nx - (I+a)n; whence x -(i+a)"- -, N (1+nx)?, B — (l+nx)'-. n X Developing the value of x by theorem (K), and those of N and B by theorem (B), we have n — a2 n-1 n-2 as n-1 n-2 n-3 a4 x=a+ —. —- 3 -- -' -&c.; 1 2 1 2 3 1 3 4+ x n 11-2n 1-n 1 —2n 1-3n 1 i2 P12 3 42 3 4 1 14 B -1+1(1-n x)l 2+(1-nx)(2nx)l 3 +(1-nx)(i-2nx)(1-3nx)4 + &c. As these equations are true for all values whatever of n, they will be so when n — 0; on which supposition they will reduce to their simplest possible forms, namely: DEVELOPMENT OF EXPONENTIALS, 15 a2 a3 a'4 a5 x -a — --' + I- (k) 2 3 43 5 + 2 + + 2 + &c. = 1+a, ". ^i(N 1 1 12 B 1 + + 1- + 1 4 &c.- 2.718+. 1 1P2 P92 3 1-2-3-4 Hence the equation BX 1+a becomes a2 a3 a4 (2.718+) 2 3 4 +., which comprehends the napierian system of logarithms; that is, if 2.718+ be the base, then a2 +a3 414 a5 I (1+a) = a + - - - + - &c., 2 3 4 5 where I denotes napierian logarithm; and hereafter L will be used to denote the common logarithm, when placed before a quantity, instead of log; that is, the expression 1 (l+a) should be read: napierian logarithm of 1+a, and L (1+a).-: common logarithm of 1+a, or so understood. a2 a3 a4 a5 Resuming l (1-a) - a --- --- -- & (1) 2 3 4 5 to find a more converging series, let a- - b, and we have,2 ba b4 b5 1 (l- - - b.......&c....(2) 2 3 4 5 Subtracting equation (2) from (1), we get a2_b2 a-+b3 a4-b4 a5+;c (1+a) -l(1 —b)- (a+b) —b + 3 4 + &.; 2 3 4 5 or, if we make b a, and consider l(l+a) -1 ( 1 —b) -ll1a * The number answering to any napierian logarithm may be determined from equation (N), and a slight modification will give a formula for determining the number of any common logarithm. ]16 MATHEMATICAL KEY. then 1(1+a) + a3 a5 I lq-a] -3 5This series might be converted into various other forms, but the present is about as simple as any, if we combine with 1+a n-Wi it the equation n -, which, reduced, gives a1i~~~~+-a n1 or if we wish to compute the logarithm of a fraction, let m 1-1+a m -n -- —, whence we find a n 1-a m -n 1. To compute the napierian logarithm of 2. Make 2 — a, whence a - ~: then by theorem (1), 1-a we find 12 = 2(+ ( + - + 7-+ &c.); 3.33 5.36 7.37 and by employing eight or nine terms of this series, we get 12 -.69314718. 2. For the computation of the napierian logarithm of 4, we have 14=-1 (2X2) =212 - 1.3S629436. 3. To find the napierian logarithm of 5, since 15 - 1(4X-) 14 + 1 -, it is only necessary to compute 1 I: then make i +a, whence a- -, and by theorem (1), 1-a 1 — 2 (+ 3- 3+ + + &c.) 3.93 5.95 7.97 - 0.22314355, and 14 -1.38629436; 14 + 1 - 15 -- 1.60943791. 4. To find 110, we have 12 + 15 - 110 - 2.30258509. To compute the napierian logarithm of 10 by a somewhat different method (although the foregoing is nearly the same as that adopted by most authors), we may take the equality (~)3 __-. then 2 _6- 45+6-3- _ 14 5(t-x4) _53 _ 53 (or 1l2 - 31 +-l- J(4t~~~~r ~'2r NAPIERIAN AND COMMON LOGARITHMS. 17 also 14 - 212 = 61- + 211 ~I; but 1 z415 -14, or 15 — 14 +V1 i =(61 + 211 5 71 i + 21~ =a- 15; and as 110 - 12 + 15, therefore 110 -- (7 13 + 21 J-25) + ( 1 a + I2) 10 1- + 31-L-a5; but by formula (1), 1 1 1 1Az2(~+ + -+ +&c.), 4 2 (+ 3- 93 + 5-95+ 797 + 33 35 37 and 1 ~ - --- ---- + &c.): hence, 3-2533 + 5-2535 72537 1 1 33 35 110 20- i -0 + 5+ &C.) + 6(&+3 — +33 +. 11 - 0(9 3 93 59 ) 325 F3 5253 & - 2.3025850929940456S401799145468S4. Let m - I - 0.43429448190325182765112S9189166 - modulus: 110 2 m- 0. 868588963S06503655302257S378332. AnT. 5. From the foregoing data respecting napierian logarithms, we may proceed to compute common logarithms. For this purpose, let us first find the value of the unknown exponent x in the equation bx - l+a. Taking the napierian logarithm of both members, we have (a2 a3 a4 x b - (1-+a) -a - a -- -- + &C., 2 3 4 a2 a3 a a5 or x (a - + -- - &c ); lb 2 3 T 5 that is, if b be the base of a system of logarithms, this value of x will be the logarithm of 1+a in that system: therefore if b - 10 (the base of the common system), then a2 a3,4 a5 I: (a -- -} &C); 110 2 3 4 5 or, in other words, the value of x in the equation 10x- l-+a is the common logarithm of l-+a; that is, 1 a2 a3 a4 x - L(l+a) - - -- 2T + &c.).....(1) The factor.43429448190325182765112S9189166 -- m 110 (as before found) is called the modulus of the common system of logarithms. [ MATI. KEY.] 3 18 MATHEMATICAL KEY. Now if we substitute a = y and 1-i = m in equation (1), we have L (1+t-y) -- rn(y -- + + &c. (L) 2 3 4 " To find a more converging series, let y -- z: then we have L (1 —z) - m (-Z - 3 4 &c.). 2 3 4 5 Subtracting the last equation from equation (L), and Y2 ~2 y3SZz y4-Z4 L (1+y) - L (1 —z)= m (y+z - 2 3 4 or if we make z - y, and recollect that L (1+-y) - L (1 —z) =L(1+Y), the last equation becomes L(l-jY) = 2m(y+ + y+ 7 &c.) --—.(L') Ex. 1. To compute the common logarithm of 2. Let 2 - whencey -- therefore, 1 —y 2+1' L2 = 2m(*+ 33+ + &c.)......... 3-33 5-35 7.37 This is the series commonly used for computing L 2; but to find a more converging one, we may first consider 23 8 = 10-2 10(1 —io-) = O1X-8:- = 10X 5; and then L2=A L 23:-SL(10X)) =: (1+L): then make Al+ly 4-5 = i-, whence y — 5; and by formula (L'), L A -- 2 m( + 39+ 5-9 + - + &c.):. 3.93 5.95 7.97 hence because L 2 = - (1 + L q), L2 = [1 —2m(+- -+ 5 —-+ 7+ &c.)]. (2) 3.93 5.95 7.97 Or to find a still more converging series for L 2, we might consider 210 = 1024 = 1000X-2!- 1000Xz~-5- then L2= L 21~ =- L (100OXL) =1 (3+ L ), = 0j o3 L 1o/, COMPUTATION OF LOGARITHMS. 19 as L 1000 = 3. Now employ formula (L'), and make I an ~+Y 128 —125 -2 5 and - 253: consequently 1 —y 128+125 33 35 37 L m + = n + + + 3.2533 5.2535 7.2537 and therefore 33 35 37 L2 = [3 +- 2m(q,- 3. - + - &c. (3) 332533+ 5253~ 7253 )] () The sum of eight or nine terms of this series gives L 2 =.30102999566398119521373889472449, to 32 places. Series (1) will require twice as many terms as (2), and four times as many as (3), to determine to the same accuracy; but all will give the same sum.3010, etc. Ex. 2. To compute L 3, we might deliberate thus: L3= L(2X) = L2+L; L but when high accuracy is required, it is better to take L 3 = 4 L 34 ~ L 81 -= L (80X -o) = ( L 80 + L -8-) = 9 (1 + 3 L 2 +L-L), as L 80 = L(10X23) = 1 + 3 L 2: then by employing theorem (L'), we have -8 - 1'l whence S1 —SO Y = 8 1+ 61; and therefore 81+80 1 1 1 =0 9 m6(1 511- 7161 -- &c.): hence 1 1 1 313 51615 57161 7161) L 3 - k [1 + 3 L 2 + 2 m (- &-C-g.16ia-[-5-Ti~-~)] =.47712125471966243729502790325511. Ex. 3. To compute L 7, we might first make L7 =L(6GX -) L 6 + L = L 2 + L 3 + L; but it will be better to take 74 = - L 2401 - L (2400X 2 -o1-) L (L24oo + L -), where L 2400 = L (100X23X3) = 2 + 3 L 2 + L 3; 20 MATHEMATICAL KEY. and by theorem (L'), L 240o = 2 m ( —4 — 8 340154sO +-741 &c.) ~ hence 3 480 548015 748 4L7 = L4L + 2m (4 si'l13o+5:0i~+ ) The next prime numbers are 11, 13, 17, 19, 23, etc., the logarithms of which may be computed in a similar way to those of 2, 3 and 7; and the logarithms of all other numbers are easily determined from those of the primes. DISCOVERY OF A GENERAL ROOT THEOREM FOR THE SOLUTION OF EQUATIONS OF ALL DEGREES. ART. 6. As an introduction to this theorem, let us refer to Art. 4, equations (k) and (k'), which are x2 X3 XI a=x+ —2 + + - + &c... (k') 2 2-3 2-3+4 a2 a3 a4 and x a — -- -- _....- t_ &c. (k) x a 2 3 4 Both of these equations are comprehended by the equations 1 1 1 B = (l+a)Y and (1+a)si - (l+nx)%X' that is, when n = 0. Let us prove this position by substituting the value of x as represented by equation (k), for x in equation (k'). a2 as a4 x2 a2 a3 1 la4 a- -- &c 2 2 2 24 __ a3 a4 2-3 6 4 x4 _ a4 23-4 24 &c. &c.; which, by summing up, gives X2 X3 X4 X +r + _- +X- &c. = a -1- Q-j)a_ + (-+-)a2 + )a_ 2 213 22346 +- ( 1 ~ra+ )a4- &c. = a;.........(s) THE GENERAL ROOT THEOREM. 21 because ~-, = 0, -j+6 — - = 0, - = — 0 —1 0, &c., whence the equation reduces to a=a, and proves our position correct. Now take the equation a = -+ bX2+cx3 +dx4 +&c., - (G) and make x = a+Aa2+Ba3+-Ca4+-&c., and then substitute this value of x in equation (G), similarly as was before done with the value of x in equation (k'): the substitution made, we have x = a + Aa2 + Bas + Ca4 + &c. bx2 = ba2 + 2bAa3 + (bA2+2bB)a4 + &c. cx3 = cas + 3cAa4 + &c. dx4 - da4 + &c. &c. &c.; and by summing up, x+bx2+cx3+dx4+&c. = a = a+- (A+b)a2+ (B+2bA+c)as + (C+6A +2bB+23cA+d)a4+&c. Now the coefficients of a2, as, a4, etc. must in each case be equal to zero, for the same reason as in the equation a =a + 2 + )a3+ &c. Hence A+-b 0, or A= -b; B+2bA+c 0, or B = -2bA-c _ 2b2 —c; C+bA2q-2bB+3cA+d = 0, or C — bA2-2B —3cA-d -b32 —b(2b2-c)+ 3bc-d = -5b3+5bc-d. By extending the substitution as far as the term containing gX7 in the one and Fa7 in the other member of the foregoing series of equalities, the values of D, E and F may likewise be found; that is, in the equation x = a+- Aa2+ Ba3- Ca4+- Da5+- Ea6+ Fa7+ &., we have A - -b, B = 2b2 —c, C = -5b3+5bc —d, D 144h4-21bc2+6bd+3c2-e, E = -42b5+8463c-28b2d-28bc2+7be+7cd-f, F - 13266-3 3064c —120b3d+-1l80b2c"-36b6e-72bcd+8bf+Sce - 12c3 -- 4d2 -g &c. &c. 22 MATHEMATICAL KEY. Therefore the true value of x in the equation a - x + bX2 - CX3 + dx- +&c.,....(G) by restoring values of A, B, C, etc., stands a= - ba2 + (2b -c)a3 - (5b3-5bc+d)a4 + (14b4-21bc~ -6bd-+ 3c2-e)a5 - (42b5- 84b"c+28b2d+28bc2 — be-7cd.-f)a6 + (132b6-330b4c+ 120b3d 18Ob2C2 —36be — 72bcd+ S1bf +~8ce-12c+4d-g )a7 - &c. (PR) When the coefficient of X2 is zero, that is, when b - 0, these equations (G) and (R) become a = x + cX3 + dx4+ ex5 + &c..........................(G') x = a - ca3-da4+(3c2-e)a5+ (7cd-f)a6+(8ce- 12c3+4d"-g)a7 + &c........................................... (R') It will hereafter be seen that three or four terms of equation (R) can be made to suffice in all cases, by transforming once or twice; and if only the first term a were used, and the equation transformed several times successively, it would amount to the same thing as Newton's rule of approximation. ART. 7. As a preparatory step to applying theorem (R) to particular cases, let us consider the equation a ='x+ 2+ dx'X3 + e'x4 + &C., whose form is different from that of a = x+- bX2 + C3 3+ dx4+ &c.; but if we divide both members of the former equation by Cb, it becomes a = ~ c2 d' e' b X + bx2 + _ X3 + 4 + &C. b, b, b a c then make a,, b =, etc., and we get a - XC- ca+ - dx4 + &c., which is equation (G), the root of which is expressed by theorem (R). Hence the first thing to be done to an equation, before applying this theorem, is to divide it throughout by the coefficient of x. GENERAL ROOT THEOREM. 23 ART. 8. Again, an equation of' the form a = Xn + bXn+l + C'Xn+2 + d'Xn+3 + &C. must be transformed by substituting x = r+y; and then by dividing both members by the coefficient of y, it will be reduced to the form of equation (G). To exemplify this, take the equation 4 = 2x2+ 5x3, and let x = l+y: then 2x2 = 2 (I+y)2 = 2+ 4y+ 2y2, 5x3 = 5 (I+y)3 = 5+15y+15y2+5y3; and therefore 4 = 7+19y-4-17y2-+5y3, or, by transposition and division by 19, -_ - -1 = Y + i 9y + lt5~. 19 1.9 -- 9 9-,3. This equation now has the first term of the second member such that its coefficient and exponent are both unity; and to this form every equation must be reduced, before the theorem can be applied. ART. 9. It must also be borne in mind, that if we wish to apply theorem (R) to the equation a = y+y2 + y3 +dy4 + &c., or to the equation a = z -+ bz2 + cz3 +- dz4 + &c., then for x in the lefthand member of the equation (R) must be put y or z, or whatever letter expresses the root of the equation that is to be developed. ART. 10. It will also be found, wherever the theorem is applied, that the least root of the equation is always the one that becomes developed; but as it is easy to transform any equation into a shape such that the least root of its derivative shall constitute a portion of the greatest or any other root of the primitive, therefore either root of any equation may be determined if we first suitably transform the given equation, and then develop the least root of this transformed equation by the theorem, and add it to the known assumed part of the root of the given equation. 24, MATHEMATICAL KEY. To illustrate this, take the equation x3 —6x = 2, or — 2 ( = ) = - 163. Now in the present form the theorem will only determine the least root; but if we wish to obtain one of the other roots, we can soon find by trial that one root lies between 2 and 3, perhaps near to - then let x = -4-+y in the given equation, and we have X3 (5-+y)3- 1-2 _ 5+ 15 2 +3 -6x =-6(+y) -30 6 therefore 2 = + 51y + 1i5?2 + y3 or (2')- 51 -- y- 3 232 2451 2-51 + 1 The least root of this last equation, as the theorem would tell, will give the greatest value of x in the primitive equation; that is, if substituted for y in the assumption x = — +y. As a further illustration, take the equation x3- 6x2 + 1x = 6: its three roots are 1, 2 and 3. If we make x = 4+y, the three roots of the transformed equation will be (as y = x —4) 1-4, 2-4 and 3-4, or -3, -2 and -1. Now while the rule determines only the least root of equation X3- 62 + lix = 6, it will determine the least root of the equation whose roots are -3, -2 and -1, and thereby render known the greatest root of the given equation. SOLUTION OF EQUATIONS OF ALL DEGREES, BY TIlE GENERAL APPLICATION OF THE ROOT THEOREM. ART. 11. To have the equation a = + b- + cx3 + d4 + &c..........+.+. (G) and the corresponding theorem z = a + A +a2 Ba3 + Ca4 + &............. () express an equation of the first degree, make SOLUTION OF EQUATIONS. 25 b = 0, c = 0, d - 0, etc.: then A = -b = O, B = 2b2-c = O, C = — 535b3 c-d = 0, etc. Hence, under this supposition, equations (G) and (R) become a = x and r = a. QUADRATICS, OR EQUATIONS OF THE SECOND DEGREE. Ex. 1. Given the equation 5 = 10x+x2, or d= — x- x2, to to find the least value of x. In the theorem x = a + Aa2 + Bas + Ca4 + &c., make a =, b = l-, c = 0, d = 0, &c.: therefore A = -b = — o, 2 2 B = 22-c = - _O 102 102' 5 5 C — 5b3+5bc-d = -l+0 —-0 = 203 ~ ~ 10a 14 42 132 and in like manner D- -, E —=o-5, F =-T6, &C. Hen1 2 5 14 42 132 Hence x + b- -i+ —- +- &c. 10e22 102.23 103.24 104'25 105'26 106,27 -. 477225X-. Ex. 2. Given the equation a = x+bx2, to find the least value of x. In order that the equation a = x+bx2+cx3+dx4+&c. and the corresponding theorem x - a+Aa2+Ba5+Ca4+&c. shall express the equation a = x+bx2 and its least root, make c=0, d=- 0, e=0-, &c.: then A = — b, B 22 —c = 2b; and in like manner, C =- 5b3, D = 14b4, E = -425, F- 132b6, &c. Hence x = a —ba +2b2a3-5b'a4+ 14b4a5- 42b6a6+132b6a-&c... (2) This formula is sufficient for all quadratics, and is more convenient than the general theorem. 3. Given 54 = 365x-x2, or x',+- = x — -5x2, to find the least value of x. In equation (2), let a - W5 and b -- _ —: then 26 MATHEMATICAL KEY. 2 54 254S 542 +-2b2a3 - 2 365 4 X 63&c - 3652 = 365X 365 3653 54 542 2X 54 5 2544 Hence x 3 -- 54+54 -5-54 &c. =.148005220+. 365 3656 3655 365 = Formula (2) is so simple in form, that the terms of the series expressing the least root can be readily written down by inspection in many cases, as in some of the following examples. 4, x +x- IL%~ 1 1 2 5 14 -&c x ---- 1, o ~ q- 10 — 104 -0 i, by making a = 1 — and b = 1. 5. 3x2-2x =, or -- = x-3X2. 3 2'32 5.33 2.162 22.163 23. 16 6. 15x2 —50x - -7, or 50o- x —a3x2' 3.72 2.32.73 5.33.74: = +-5 —-+ + 5 + &c. 10.502 102.50 103504 7. -13 = 739x+123x2, or - 3- -x+L 2X32. =, s3 123'132 2.1232.133 5'1233'134 39 ~-~- -&c. 739 7393 73397 8. 20 = 20y+y2, or 1- =Y+2 —Y2: 2 5 14 y = 1 — ~o + 2 03 + 20 — &C. 9. 100 = 25z+2l9-z2 or 4 = z+61-sz2 2= z+9 64 7z 5Z2 11.42 2.112.43 5.113.44 z -- 4- --- -. +. 6475 64T75 64753 10. 1= X+X2: x = 1-1+2-5+14-42+132 —&c. 11. 100 = Xt-2: x = 100 - 1002 + 2-1003 - 5'1004 + &c. EQUATIONS OF THE SECOND DEGREE. 27 ART. 13. We see that examples 10 and 11 produce not converging series, but diverging ones; and as the results are more indefinite than the equations themselves, such equations should always be transformed once or twice, when a new equation will arise whose least root will be a small fraction. For example, make x = r+y in the equation 100 = -x+2, and let r be a near value of x (which can be easily obtained by trial): the resulting equation, when developed, will form a rapidly converging series; and the nearer r comes to x, the swifter will be the convergence. From this it is evident that an indefinite number of series can be found, that will express the same root of an equation; a fact analogous to what we have seen in the case of logarithms, and of which further instances of confirmation will appear among the following examples. Let us resume the equation 100 = x+x2, in which it is easily found by trial that x is less than 10 and greater than 9. 10~. Then let x = 9+y, and we have X= 9+ y, ~2 = 81+18y+y2, and 100 = 90+-19y+y, or 100-90 lo y+ y2. 19 1 Now by the quadratic formula, 1 102 + 210 5 104 y 19 193 19- 197. and as x = 9+y, therefore 9 +=+1 102 2'103 5'104 - + -9 + 1- — 1 + &c...() 19 195 19, 2~. Or, we may substitute x = 10+y in the given equation 100 = x+x2, and will then obtain — I- = y+_y2; whence,. 102 2'103 5'104 by developing, y = & 21 3 21 &C., 21 215 217 and therefore 102 2'103 52104 21" 21" 21 77 28 MATHEMATICAL KEY. 30. A still more converging series may be found, by observing that 9.5 is a nearer value of x than either of those used in series (1) and (2): then let x = -2-+y,,and the transformed equation will be 420 Y+-2; whence 1 1 2 5 Y = 4'20- + - + &c., and 42.203 4 3.20+ 4 24.204 1 1 2 5 z 9 = 3- +'4-~-5- &c. 3.o 4(3) By using seven terms of series (1) or (2), x = 9.5124922 —; and with only five terms of (3), we find x = 9.5124921973. By taking successively nearer values of x, we may find other series at pleasure, each more convenient than the preceding, but perhaps none more simple than (3); yet if we make x = 9.5125-ky, and then transform, we shall get a series that will converge twice or thrice as fast, but each term will be more complicated with large numbers. 12. Given x2+6x - 8, to find one value of x. By inspection, one value of x is a little greater than unity: then let x = -1+y; whence a = 1-t-2y+y, 6x = 6+6y, and 8 = 7+8y-y2, or s —7 8 = y —_ y2: then developing, 1 2 5 14 y - + 8- 8 8- + 8- &c., and 1 2 5 14 8+ 8 75 8 89 13. Given 179 = 3x+17x2, to find one value of x. As x is but little greater than 3, let x = 3+y, and the new equation is found to be ~-7 = y + y2; whence 7 173 2'175 5'177 i 3- 1 0 5 —7 1073-2-17 - &c., 05 1053 1055 1057 or l J73 2 175 5 177 1.053 10-55 [Q057 EQUATIONS OF THE SECOND DEGREE. 29 Or if there be put x = - 16 + y, the new equation will be 35 5y? 2 -39 8 4 -Y + 29-2 whence is found 51.3.52 2.512.355 X 6 399 8 4 332'39842 3322.39843 14. Required the square root of 2, or the value of x in the equation x2 - 2. Let x = 1 + y; then x2 1+2y+y2 = 2, or = y2: then as y =+ - - — + - - &c., I 2 5 14 therefore x - 1 + - + + 5 - &c.-/2 Or if x = 1.4 + y, then (1.4+y)2 = 1.96+2.8y+y2 - 2, or 2-1.96 2 -2 7 0 y + y2; whence 2,8 5 252 5.53 Y 7 0 + _0- 3 04 + &c.: hence 14.702 142.703 143 704 I4 + 145 2.52 5.53 X - q- + -- 142.703 143'704 + &c. 2. 15. What is the square root of 3? I1 2 2'22 5'23 Answer: - - 5 — - &c. 7.8 73.8 783 7784 ART. 14. In the preceding examples, we have only attempted to determine one of the roots in each equation, but now propose to find both. For this purpose, let the two roots of any quadratic be denoted by a' and b'; while the same, when unknown, may be indifferently represented by x. Then we shall have x-a' = 0 and x-b' = 0; and also (a-a') (x-b') =,- (a'+b')x + a'b' = 0: therefore the sum of the roots, with the sign changed, will be equal to the coefficient of x, when the coefficient of x2 is unity. ART. 15. Again, a cubic equation has three roots, which may be denoted by a', b' and c': then -a,' = O, x —b' = 0 and x —c' = 0; 30 MATHEMATICAL KEY. and by multiplication, (x-a') (-b')(x-c') = x3- (a' + + c')x + ('b'+ a'c'+ b'c')x-a'b'c' = 0. ART. 16. In a similar manner, if a', b', c' and d' denote the four roots of a biquadratic, we should find X4 _ (a'+ b'+c'+ d')x3 + (a'b'+ a'c'+ a'd'+ b'c'+ b'd'+ c'd')x2 - (a'b'c'+a'b'd''+a'c'd'+b'c'd')x + a'b'c'd' = 0. ART. 17. From the foregoing, it is concluded that any equation of the form x-+A'xn-l'+ B'xn-2+ C'xn-3+ &c. to (last coefficient) xn-n = 0, is such, 1~, that the coefficient of the second term (A'), with its sign changed, is equal to the sum of all the roots. 20 That the coefficient of the third term (B') is equal to the sum of all the products that can be formed by every two of the roots. 3~ That the coefficient of the fourth term (C'), with its sign changed, is equal to the sum of all the products that can be formed of every three of the roots; and so on to the last term. 40 That when the last term is made to represent one member of the equation (as might be done by transposition in the preceding equations), it will be equal to the product of all the roots, with its sign changed for quadratics, biquadratics, and all equations of even degree, but not changed for odd degrees. 50 Therefore, from the first conclusion, it follows that when one root of a quadratic equation is known, the other may be found by subtracting the former from the coefficient of the second term of the equation. Let us resume example 12, which is x2+6x - 8. We found 1 + 2 5 14 - & 8Ti 8 87 89 then, as the sum of the roots is -6, 1 2 5 14 is ---- 7 otr + r83 s5 + + is the other root. EQUATIONS OF THE SECOND DEGREE. 31 Or the equation x2-6x 8 may be transformed by making z= -7-+y, which gives -- = y —y2: whence 1 2 5 - - + 837- &., or 1 2 5 14 x -7 - - -H +- - + - &c. as before; which, subtracted from -6, gives 1 2 5 14 z = 1 +-+ s —- + 8 &c. 83 85 8Z7 S for the other value. Let us refer to example 13, which is 17x2+3x - 179, or x2+ _37-x -_ -7-. One root was found to be 19 35 51.352 2.512.353 3984 33239842 3322.39843 then as -— r = the sum of the two roots, therefore 51'352 2'512'353 l 7 1 6 9 8 -34 +- 332.39842 3328-39-.43 is the other value. 16. What are the roots of the equation x2-7x - -11 Here one value of x being near 2 and the other near 5, we may either put x =-2+y or x = 5+y; but in one case we would obtain the lesser value of x, and in the other the greater. Thus if x 2-]+y, then x2 - 4+4y+y2, -7x - -14 —7y and -11 -10 —3y+y2, or, =- yWy2. Then by developing, 1 2 5 y + 3 3 -+- 37 &c.: therefore 1 2 5 x = 2 + i+ + + - - + &c. - the lesser root; 33 35 37 and as the sum of the roots is 7, we also have 1 2 5 x 5 3 - - - - - &c. the greater root. 33 36 5i 32 MATHEMATICAL KEY. Or if it be desired to obtain the greater value first, then let x = 5+y in the equation x2-7x- - -11, and we have -- - y+y-,y2; whence 1 2 5 y = --- _ -- -&c.; -33 35 37 but as x = 5 + y, consequently 1 2 5 x 5-i-33 3 5q-&c. the greater root, as before. To find a more converging series, let x = (5-* —)+y (5 +y +y, instead of 5+y: this gives - 2761 -1y2; 27'61 and then developing, and introducing x -- + y, we find 12 5 19 1 92 2'193 27'61 27'617 3 27861S 27 27i 61 27613-27'615 &c. the greater, and (as 7 - the sum of the roots), 7 27 + 2719 192 2 19 x -W - -F- --- -+ —-6&c. — the lesser 27-61 27-61 27.61 & root; that is, x -4.61S034+, or x= -2.381966-*. * By taking the equation — 1 = x+x2, and applying the quadratic formula, we get x = —1 —2 —5-14-42 —-&c. =- co for one, - -i —-/(-3) by completing the square, and x = 2+6+14+42+&c. = + co = — +4-/(-3) for the other root. Whenever the roots are imaginary (or impossible), the sum of the series will be infinite. Every equation of an odd degree will have at least one real root, but those of even degree may have all their roots imaginary. A mathematical question, so proposed as to contradict itself, will often produce an equation with imaginary roots, which circumstance will be shown if the sum of the series becomes infinite, whether the equation be quadratic, biquadratic, or of any higher even degree. The theorem will not determine an imaginary root, if there be any other. EQUATIONS OF THE THIRD DEGREE. 33 ART. 18. CUBIC EQUATIONS. 1. Given xa3 -- 3x2-x- 10+o x 1, to develop the least root, without transforming, by theorem (R), which is x = a- +Aa2 + Ba3 +- Ca4 + &c. Here A -,-b A o -, 3 10 B - 2b —c= 2 X 102 - 10- 10, 3 15 C - -b+ 5bec- d — 5 X X o 0 = -o; 456 4074 and in like manner, D - - E --- &c 105 Hence, for the least root, we find 3 8 15 456 4074 x - 10- 10 - - {- - 10 -.103078. 103 10+ 5 107 109 1011 2. Given a - x+bx2+cx-3+-0x4+0x5+&c., to develop x. In the general equation a = x + bx2 + c3x + dx' + ex5 + &c., and the corresponding theorem x = a +- Aa + Ba' + Ca4 + Da5 + &c., let d, e, f, etc. be made zero: then the equations become a = x+bx'+cx'+ 0 and (by restoring values of A, B, C, etc.), x = a - hba + (2b2-c)a3 - (5b3-5bc)a4 + (14b4-21b'c+3c2)a' -(42b5-84b3c+28bc2)a'+ (132b6-330b4c+ 180b'c2- 12c')a7-&C. (3) This formula will answer just as well for all cubics as the general theorem, and is rather more simple, especially after the fourth term, but it becomes still further simplified when b - 0; that is, if, in the equation a- x+bx2+cx3, and the corresponding formula, b = 0, then a = x-+cx3, and x = a - ca3 + 3c2a' - 12c3a7 + &c............. (3') [ MATH. KEY.] 5 34 MATHEMATICAL KEY.. +1, 3. Given 5 - 25x+15x2++x3, or - x+3-2+xI3, to find x. Here, by formula (3), a -5, b -, and c-: 3 17 24 hence x - - 1 + &c. *5 55 56 Or, to obtain a more converging series, let x - -+y; or a still more converging series will be obtained by making --- 3 117 x.18+y. To find the decimal,.18 -= - + - nearly, 53 55 is not difficult; and then transforming from the assumption x-z a - + y, will give 8168 155400 2 10000 * _ m+ + Y-; 30497200 Y 304972 304972 y whence, developing, etc., we obtain 18 168 155400.81682 x_ -l 818 + - + &c. 30497200 100233049723 Two terms of this series give x=.1802678, so far correct. 4. Given x3+6x 2, to develop x. First we have - * - x z -+6jx3; and by formula (3'), we have a =- ~ and c - hence x 3 6 6337 -+ &c..32748+. 5. Given x3-6x = 2, or - x — 6-x3, to develop x. 1 3 12 Ans. xc- 6-6i - - &c. = —.33987+. 6.33 6 A.55 6a.37 In the last two examples, if greater accuracy be required, make x —.33+y in the former, and x -.34+y in the latter: the developments of the transformed equations will then converge with great rapidity. If it be required to determine another root of the equation x3-6x - 2, it may be depressed one degree by dividing it by x+.33987, when the quotient will be a quadratic from which the other two roots may be obtained; but the better way will be to find by trial a near value, and then transform. By trial, we find 2 too small and 3 too large: therefore 2.5 will be EQUATIONS OF THE THIRD DEGREE. 35 near one root. But suppose we first transform by making x-z3+y: whence we obtain -1 y + 32_ I 3 377 and by developing, etc., we find _3 47 x - 3 — ~ — - &c. - 2 6+. - - 732 2 34 Next let x - 2.6+y - " —+y in the equation x3-6x ~ 2, and 1 5'13 52 we shall find - y+ 12+ 3; and developing, 5~-119 119 3.119 1 13 179 x -+ -&c. 2.60168-. 5'119 5.1193 3.1195 The terms -3 +51 =- 2.60163, which is as near the true value of x as can be expressed without more decimals. The third root may be found by a series similar to that of the others; but by Art. 17, 10, the sum of all the roots is equal to minus the coefficient of x2 in cubics, which, in the equation 3 —6x - 2 is zero: then if the three roots be denoted by a', b' and c', we have a'+ b'+ c' -O; but we have found a' - -.33987 and b' 2.60168; whence c'= 0 — (a'+b') - -2.60168 +.33987 - — 2.26181. Hence the three roots of the equation x —6x - 2 are 2.60168, -2.26181, and — 33987. 6. Given x3-12x = 15, to find the three roots of the equation. To obtain a converging series for either of the values in the example, we must first transform. As one value is near 4, let x = 4+y: then (4+y)3 — 64 + 4Sy + 12y2 + y3, - 12(4'+y) -48 - 12y; hence 15 - 16 + 36y + 12y2 + y3, or a-3 — y + y2 + 3. 1 1 7 5?= e3-1 7. - -5 &c.; therefore 6 33 6 3362 364 36 xz~ = 4 1 &c. 1 3.9719608 = a' 3.36z ~364 365 36 MATHEMATICAL KEY. We might also find 2 _4 300 512500 x 220 + - 3+ - _220 — +&c. — 2.3954260-= b', by first assuming x- -2- 4+y; and, finally, x _ c -(a=-b) = =-1.5765348. The three last examples are solved in Bonnycastle's Algebra, and some of the following will be found given as examples by different authors. 7. Given x3+10x2+5x - 2600, to find x. First let x- ll+t-y, as one value of x appears to be near 43 1 11: then we find I y 147Y2+ y and 44-1477 4'147y, X1 - 11 1 7- + -&c. = 11. 00679934. 8. Given 5x3+9x2-7x 2200, to find one value of x. Let x = 7+y, and -s- - y 4y2 + — 5 y3 114.932 2122.933 4973820.934 x — 7 q -L - -zr-3 — +-&c. 4 8543 854 854 &C. 7.1073536. 9. Given 5x3 —32 —2x- 1560, to find one value of x. Let x-=7+y, and W69= y + 69-y2 + 96.62 14977.6s x = 7 +- 91 +-69 - &c. -- 7.0086719. 69 13 691r 10. Given 2xS —3x2 3, to find one value of x. Let x = 2+y, and — r —-y= + -2y2 -_22y3: 9 186 x - 2 — -- & c. 123 125 11. What is the cube root of 2, or the value of x in the equation x -- 2? Let x = 1 +y, and x3 1728 + 2+yS, or 10 03 +0"Y + ~C~2 10S 102 _2 7 2_ y + 2_6 _0 -1 0 0 272 432y2+ 3 x _ x2- _72 ~~~62'2722 63.2723 10,4343 434. EQUATIONS OF THE FOURTH DEGREE. 37 A much more convenient series may be found by substituting x — 1.26+y. ART. 19. BIQUADRATIC EQUATIONS. For the purpose of more easy reference, let us again bring to view the general equation a = x + bx2 + cx3 + dx4 + ex +- &c., -------— (G) and the corresponding theorem x a - ba2 + (2b2-c)a3- (5b3-5bc-d)a4 + (1464 —221b2c+6bd+3c'-e)a5 - (42b6 — 84b3c+28b2d+28bc2- 7beh- 7cdf )a6 + (132b6-330b4c+ 120b3d+180b"'c2-36bSe-72bcd+-8bf + 8ce- 12c3+ 4d2-g)a7 _ &c...................... (R) This will be the most convenient formula for all equations higher than the third degree, except when some of the coefficients b, c, d, etc. of equation (G) are equal to zero. 1~ If, in equation (G) and theorem (R), we make b = 0, we have a - c3 + dx4 -cx ede5 + &C.; (G') and for the root of the same, x = a - ca3-da4+(3c2-e)a5+(7cd-f)a6+(8ce-12c3+4d2-g)a7 + &c................................ (R) 2~ If, in equations (G) and (R), b - 0, d -- 0, f= 0, etc., we have a -x + crs + ex5 +- gx7 + &c.;.(G") and for the root of the same, x = a - ca3 + (3c2-e)a5 - (12c3-8ce+g)a7 + &c.... (R") 3~ If, in equations (G) and (R), c = 0, they become a = x + bx2+ dx4 + ex5s +f6 + &c., (G"') and x = a - ba2 + 2bha3 - (5b3+d)a4 + (14-4+6bd —e)a5 - (42b-+28b2d-7be+f)a6 + (132b6+ 120b3d-36b2e+ 8bf+ 4d2-g)aT —&c. (R"') These are all the modifications it is necessary to make, except such as will occur in practice: indeed theorem (R) might be used on all occasions, and the modifications made 38 MATHEMATICAL KEY. at the time of application; but formulas (R'), (R") and (t"'), when required, will mostly dispense with such labor. 1. Given 4-20x+ —1 1x2+-9a-x4, or 11 9 1 X -s 2 -x3 x4 4. 5 4.5 4.5 to find one value of x by theorem (R). From the theorem, we have a, 11. 9 1 b 45, c —45, and d — -: therefore 4.5 i4.5 + 62 +3645 16006 4x3 -42-5 43 4 4.5 + & 1796+ To find a more converging series, first transform by making =5~ +y, or by making x-. 1S+y. 14 2. Given x4 5x+ 9x = 4-, or - = x. -— 4, to 5.9 find one value of x by formula (R'). 14 We have a- 59 c = —, and d - -, e-0, f=O, etc.: 14 5'143 144 x -+ - - + &c. =.329+. 5.9 53.94 54.95 Or by making x -= +y in the given equation, we get 11 7 1 7n2 99 $3 + 2 7o 4 15202 (R) we find and then, by theorem (R), we find 11 117.112 47376.113 ~ + &c 15.202 1552-20- 1532025, and x =.32971055. 3. Given x4 — 1Ox= —1, or -= x- -jox4, to find one value of x by formula (R'). Observe that a = -b-, c - 0, d = -Ao, e = 0, f 0, etc.: then x= 1 4+ then x - -- -C. -.100010004. Io 109 EQUATIONS OF THE FOURTH DEGREE. 39 4. Given x4 —x3+-10x - 1, or' - x —-x3+-fx4, to find one value of x by formula (R.'). A 1 3 7 52 Ans. 0 10 — + 10+10 &c. By tlieoremn (R), the four following examples are quite simple, and easily developed almost by inspection. 5. 1 - X+X2+X3+14 1 1 1 21 X 10 0 - +1-&c x2 - T103 1024 +05 7. ~ —N -x 2+- - x4 1 1 1 2. -- 10 3 1104 105 * 1 (} —i- X+ A- I- & - 10 1+02 + 103 + 104 + 105 + &C. 81. 1 2 z llo I 0 -t- + &C. 9 5 40x+20X2+12 1 o+5X4 x - - 2 - + 482 8 8 + 16 + 10. 7 - 67x-29x2+17x3 —11lx4 7 29.72 543.73 6169.74 67 + 67 +- +.. 67 673 + 675 677 11. Given x4+-x3+4x2-8x = 8000, to find one value of x. Let x = 9+y, and 2 — 4 y — 5 4 y2 6Y+ 3 4 63 Y 4 9 1 544'2712 460164271.90 34663 34665 12. Given x4 —x'2. -3, to find one value of x. Let x - 1+y, and - y-y 2- + -- — Xy3-J-y4 40 AMATHEMATICAL KEY. 9 151- 3029 x — 1 —~ —-11 —~ —115 -- 117 — 1&c, 13. What is the biquadrate root of 2, or the value of x in the equation x4 - 2? Let x 2-+y, and 12 736 86400'7362 1161216000'7363 1 ~ 6 9 1 2 0 -_ 00 _&C.-4 69120 - 691205 14. Given x4-2x - 13, to find one value of x. Let x - 2+y, and 3-0 3, 20 3 1 3y4. 24 912 41220 X — 2 +3 0 + 3 o- 3o +&c 303 305 307 If it be required to determine another root of the equation x4-2x — 13, it is soon found that there is a negative value near -2. Then let x — 2+y, and 7= y 4yY 2+ 30_y_3 -y4 24.72 912.73 41220.74 X 2 o 3 - -+ —t+ -+&c. 3 0 303 + 305 307 To find a more converging series for the last root, make x=: — + 4y. In this way, any real root of any equation, or all of them, may be expressed. 15. Given x4-3x2+75x - 10000, to find one value of x. 597 2 8 1 1 4 Let x=-10+y, and — 9-3 y + 5.803y2 + 8 3 + y 597.902 55221S8903 10 - 890 - 5803 - 52 8035 -&c. - 9.88600+. As the decimal.886 is near to 8, then to find a more converging series, let x - 9-+y — 9+y; or let x - 9.896+y, and three terms of the series constituting the value of y will give y and x to over twenty places: one term alone will give x = 9.88600270094. EQUATIONS OF THE FIFTH DEGREE. 41 ART. 20. EQUATIONS OF THE FIFTH DEGREE. 1. Given -1- =x+x2+x3+x4-+x5, to find one value of x. Employing theorem (R), we have a- -o, b 1, c- =1, d = 1, e = 1, f- O0, g 0-, etc.; and therefore 1 1 1 I 1 6 x + — A- + o — +&c. 1 0 103 104 +15 1+7 2. -— =- X-+X3+.4+aX5. By formula (R'), I 1 2 7 z -- 1~ —lo —-~3 + 1y4 + o o &C., - 1 2 4 103 105 107 4. o = x+x5. By formula (R"), -x - 1-. + &C. 5. Given X5+2x4+3x3+4x2+5X - 16. Let x= 1+y, and = + y2 + y3 1 y4 + 5: then by theorem (R), etc., 1 1 7 11 639 5.7 52.72 54.73 55.74 57.76 6. What is the fifth root of 160000, or the value of x in the equation X5 = 160000? Let x - ll+y, and 1051 - 2jJ13310 2 + 12 4,3 + 1 7-3 205- = y + y 325Y 73205oY 73205Y5: 1 0o 5 1 13310'10512 265734150'10513 Y 7-3 2 05 732053 732055 x 10.985605- - 5T/160000. [ MATH. KEY.] 6 4(2 MATHEMATICAL KEY. ART. 21. EQUATIONS OF THE SIXTH DEGREE. 1. Given 2x-6 x5+3x4 —x3+4x2-19x- -, to find one value of x. First we have 7 3 3 5 2 - x- x -x2 12+ x3- 4 x+ x5 _ 2x6: 3.19 9 2'19 19 219 19 then in theorem (R), let 7 4 3 3 5 2 azz b-, b — c= - de1- e 3-19 19 2 1919 2 19 19 g = 0, h = 0, etc.: therefore 7 4+.72 7.73 S33.74 21157'75 3'19 32193 2'33 195 34'197 43199 & 7 To fin;d a more converging series, make x = *19+Y; or, x - +-y would be more simple, and give a series nearly as convergent. 2. Given 1i- x-x3- 4x -x6, tofind One value of x by formula (R'). 1 1 1 1 xz -- A:~ q - -- +- + — q- &c. _ 2_10 2+105 6.106 2+107 + &C. ART. 22. We might multiply examples of the sixth degree, then of the seventh, and so on; but as the same theorem, or its derivatives, answer for all cases, it is unnecessary. One example, however, of the seventh degree, is added. Find one value of x in the equation x + X2 +XI3 + XI+4-x5 ++ - x X7. By theorem (R), we have 1 1 1 1 1+ 9 + 93 94 95 96 97 98 To find a more converging series, make x —+y or —. 9.9~~~~-1yo -— y QUESTION IN ANNUITIES. 43 ART. 23. The general equation for annuities is A' =p[(l+r)-n1] A' Axr/, or 1+ - r (+r)...(1) r P Let us now assume A' 1413.98, - 50, and n 20: then 1413.98 50 [(iSr) 1] or 285796 - (1r)-1) r r an equation from which it is supposed by some to be practically impossible to obtain r, except by successive approximations. The root theorem will make the approximations at once, or by one series. A' If, in equation (1), we assume - q, it will stand 1+qr = (l+r) n-i n-i n-2 n-i n-2 n-3 = l+nr+n.- r2+1.-..-+n.-..-4+~&C.: 2 2 3 2 4 or if we transpose, and divide by nL. —i r, we get 2q —2n r n —2 n-2 n-3 + n(n-1) 3 3 4 but from the above, q 28.2796 and n = 20: therefore by substituting these values, etc., there will result.043+ - r + 6r2 +- 25.5r3 + &c.; and by the root theorem, r -.043 - 6X.0432 + (2.62-25.5).0433 —&c. -.043 -.011 +.003 -.035 nearly. To find a more convergent series, let A =. 035, and r = A+x in the equation 1+qr- (l+r)n, and we have 1+Aq+qx - (I+A+x)n; or if 1-+-Aq B and I+A =-, then B-+qx-= (+xz) - gn(+~)', or CB + A 1 +) n n n-1 nn-1 n-2 @ m z nx —2+ _- 23- @ 2r&c.; = ~ ~ ~ ~ z 1 } — ~~x — ~~. 44 MATHEMATICAL KEY. or, transposing, and dividing both members by (g, B_-n + n(n-W 2 x 2+ n(nl)(n-2) -- + &c. (2) Lastly, apply theorem (R), by considering a 6 =,=(n-1) = -l)(n-(2)- ) &c. a --- b C n q 2(n,~-' —q)' 6(,~,, —p) Equation (2) may then stand a = x bx2 + cx3 + &c., whose root is x - a - ba2 + (2b2 —c)a3 - &c.; therefore r - A + a- ba2 +- (2b2 -c)a3 - &c..(r) From the above data, the numerical values of a, b, c, etc. may be determined: that of a will be a very small fraction, which will cause series (r) to converge with great rapidity. ART. 24. CASES OF INFINITY; OR SUCH AS WOULD BE COMPREHENDED BY THE EQUATION a = x + bX2 + CX3 + dx4 + &c., when the series does not terminate. 1. Given 1-j- = x+x2+x3+-x4q+&c. without end, to find the true value of x by theorem (R). We have a= - 1-, b = 1, c = 1, d = 1, etc. forever; and therefore 1. 1 1 1 - - — &c.=.09090909, &c.= —. X: 10 102 1Ci1 104 2. -i = — X'2+X3-X4+&C.: 1 1 1 x = — + + + + &c..11111, &c. = 9. 102 1O3 1O4 3. - x% = +3+x5+X7+&C.: 1 2 25 x + - - _+ &c. = /(26)-. 0 0a 107 EQUATIONS OF INFINITE DEGREE. 45 1 1x41 4. a = x+-Xc2+-:-f2''4 x+&c.: x: C -- ~2 + Wa -- a4 + &c. 1 (1+a). 5. a - x —~x2 + XS — X4+ &c.. x = a -~ a2+ -'a3 4 a4 +. & +a). 6. a = x - X3+- x5- x7-+ &c.: 2 17 x- a a- a3+ a5 a7+ &c. 3.5 5.7.9 7. Given the equation 1 1 n-x 3(n-x)3 +(n-)5 7(n_)7 + & to find the value of x. Make = y; which gives x _ - n —1, and n-x y y therefore ny-i- _ y = y3 + 5 _ y7 _ &c., or ny —i y2- y4+ y6 7y8 + &C.; and by transposition, 1 = ny -2 + *4 y6 + -1y &c or 1 y2 1 4 1 1 or -=y- y2+-y y _ya Y -y &c. n n 3n 5n 7n Here, by formula (RP"'), 1I 1 1 1 1 a -, b= —, d eO, f = g=0, h -, etc.; and therefore n nn n n3 n (4 3n2W) n5 12081324 1 1 -— ~ 6 -- -,,~ + - + -Aq-&c.; n5-3n3 + 5n1 n n 3n4 5n2 9W2 n7 + or, by simplifying, 1 1 5 16 302 y= —++ - + + &c. n n3 + 3n + 5n7 45n9 If n = 4.71238898, y =.222548, x = n — =.218979. y 46 MATHEMATICAL KEY. J3 X5 X7 8. Given x - x - 2'3 2'34'5 234'5 6'7 to find x. By formula (R"), 1 1 1 a= 4, c = —, e- g'6 2i3 2345' g 2-34567 1 32 32. 52 32. 52' 72 23-2: 2'345'25 2'3'34-5-67-27 2'345'678'9'29; or, for convenience in summing up the terms, let us consider A = first term (4), B = 2d, C = 3d, etc.: whence 32 52 72 x = + A + B+ - C + D+&c. 2.3.22 4.5.22 6-.722 8 9-22 Five terms of this series will give x =.5236 nearly: eleven terms will give x =.52359S775, 6x = 3.14159265, 3x =.78539814 nearly. The solidity of a cubic inch, and that of a sphere 1 inch in diameter, are to each other as 1..523598775. The diameter of a circle, to its circumference, is as 1.: 3.14159265. The surface of a square inch, and that of a circle 1 inch in diameter, are as...........!. ~.78539814. To make every thing plain, the source from which equation X3.5 X7 ~ -- x — + +- &c. 23 2345 2-3-4'567 has been derived, should be understood; but our present object is to give the solution of equations. 9. Given a x - -*3+P + - P5 7 + &c., to find x. 2 17 Answer: x = a 3 + -a5 + -37a7 + &c. 15 315 Examples 1, 2, 3, 4, 5 and 8, and many other equations of infinite degree, produce recurring series, so that as many terms may become known as desired; but cases of infinity are not easily transformed like equations of finite degrees. EXPONENTIAL EQUATIONS. 47 ART. 25. EXPONENTIAL EQUATIONS. Let there be given the equation (a'x)b1' = c', (e) to find the value of x, when a', b' and c' are known. If we convert the equation into logarithms, it becomes b'x( La'+ Lx) - Lc'. -. - (1) Now assume x = r(1-+y) = r+ry; then b'x = b'r(l+)y and L-= Lr + L (i+y). Restoring these values of b'x and Lx in equation (1), we get b'r(l+y)[ La' + L Lr + L(1+y)] = Lc', L c' or (1+y)[La'+ Lr+- L(l+y)]' L c' or (l+y)( La' -- Lr) + (l+y) L(1-ky) - b-; which, considering (l+y) ( L a' + L r) = L a' + Lr + ( L a' + L r)y, and transposing La' + Lr, gives L c/ (La' -- L r)y + (l+y) L(-+y) b — a- L cr L c'- b'r(La' - Lr) then by formula (L), L (1+y) = m(y y2 t_ y y4 + 1y5 &c.). Also we may find (1+y) L(l+y) = m(y + jy2 - y3 + y4 4- y5 + &c., 2~3 3*4 4-5 and (La' + Lr)y - (1+y) L(l+y) =(m + La' + Lr)y +2 m ~rn ) m m y _ my4_ my5 + &c -6 2 0+&c. = Lc' - b'r(La' + Lr). b'r and, dividing by m + La' + Lr, gives Lc'-b'r(La' +Lr) m _ m 3 b'r(m+La'-+L?-) =) 2(m+La'+ Lr) 6(m+La'+ Lr) + 12(m —a-f- L) 20(m,+ La' - +Tr) -- 5 +_ &c. 12(m+La`+ Lr) 20(m+ La'+ Lr) + 48 MATHEMATICAL KEY. Now let a Lc' - b'r(La' + Lr),:and -- b'r(m+La-+Lr) m+La+ Lr then we have (a& n) a y +?Y2 - y93 +n y4 n y6_+ &c; and by the root theorem, nl 2 ( n n )3 ( 5n2 n y= a -=s- a +- + ~ ++( 7n4+ 7n3 n2 n ) &; or, because x = r+ry, we have for the true value of x, n 2 n2 n n3 5n2 n a + a -- q- a a 2 2 6) 8 12 12 (7V4 77 n2 5 + — 8-+- +~ + 20... 1. Given (6x)x = 96, to find the value of x. Comparing this example with equation (e), we have a' = 6, b' 1, c' = 96; and as x is near to 2, let r - 2 then _ Le'- b'r(La+ Lr) L96 - 2( L6 + L2) b'r(n+ La'+ Lr) 2(m+ L6+L2)' or as L96 = L32 +- L3 = L25+ L3 = 5L2+L3, and L6 = L2+L 3, therefore L96 -2(L6 +L.2) = 5L2 + L,3 — 2(2L2 + L3) =L2- L3, and 2(m+L6+L2) = 2mn+4L2+2L3, or L2-L3 m m a = and n = = 2m+4 L2+2 L3 m-4-L6+L2 m-m+2 L2+L3 Using the logarithms of 2 and 3, and the value of m as determined (Articles 4 & 5) to seven places, gives a =. 0581745 and n =.5739; and then applying formula (x), we find = 2 + 2 (-. 116349 -.000971 -.000035 -.000002) - 1.882643 +. To find a more converging series, make r -1. 88. EXPONENTIAL EQUATIONS. 49 If we ]have a table of logarithms, and only wish to determine x to six or seven decimal places, it is best to use only two terms of formula (x), which will give x = r + ar + &c., and then we must substitute successively two or three times; but in equations (a & n), it will only be necessary to determine a. T'aking this course with the equation (6x)...z.. 96, r. 2, we first find L2 — L3.301 —.477 -, —._.05_+, 2m-+4L2+2L3 2 X.434+4X.301+2 X.477 and thlerefore x,: - r + ar- 2 -.058X2 _ 1. 8S4 nearly. Next let. r 1. SS; then in equations (a & n), which are Lc' ——.b'r(La'-L?) md m a ~ zX~Wm-[Land' +m we get Ir. -f —ln+ Lr) m+La'+ iEr L96 -- l.SS (L6 +- LI.SS) 1. 88 (m + J6 + L l.s88).982271 - 1. SS(.77S151 --.274168) 1.88 (.4343 +.7781 +-..2742).003930.002643....:: -: —-- I, a nd.1. 88 X 1. 4866 1.88.002643:, - - r -I- c. - 1. 88+ 1.8 S X 1.88. (Given --- 100, to find x. Here as x is less than 4, let r -- 3; whence Lc' -'r(La'+Lr ) L 100-3(L1- L3) b'r(m+ La'+ l,r) 3 ( m- +L1 +- Lr) 2 - 3 L 3 2 -- 3 X.477 569 ___ =. —.2~: 3 ( t + L3) 3 (.434 +.477) 2733 then x -- r+&-ar-r c. - 3 + 3 X.2 -- 3.6 nearly. 2 - 3.6 L3.6 Again, let r — 3.6 and 6).3.6( (m + L3.6)' WVe may here consider 32 X 22 L3, ~ (; -- I, 3 _ -J- 2 (L 3 - L 2) --; and therefore [ MATh. KEY.1 7 50 MATHEMATICAL KEY. 2 - 3.6 L3.6 5.6 -7.2(L3 + L2).002689 3.6(m+L 3.6) 3.6(m+2 L3+2 L2-1) 3.6 x.990596.002715.00275 - a, or ar —.002715, and 3.6 x = r+ar = 3.6 —.002715 = 3.597285+. To determine x to a greater degree of accuracy, take the logarithms of 2 and 3 as computed heretofore to 18 places, and we may find 2 (L3- + L2)-1 = L 3.6 - 0.556302500767287265, m 0.434294481903251828, m + L 3.6- 0.990596982670539093, 2 - 3.6 L 3. 6 = 0.002689002762234154, 3.6(m+ L3.6) - 3.566149137613940734, a — 0.000754035419851601, n -- (m + L 3.6) = n = 0.438416923835606983, a2 — 0.000000568569414391, n2- 0.1922093991055, as -— 0.000000000428721477, n3- 0.0842678, a4- 0.000000000000323272. Now by formula (x), a — 0.00075403541985160, - n a2 — 0. 00000012463522682, + (\ + )a3= -0. 00000000007252862, - 8- + -2 12) a4 —0.00000000000005472, fifth term = —0.0000000000000004. Let s = sum of five terms; then s - 0. 00075416012766180, 3.6s = rs =- 0.00271497645958248; or, as x - r+rs, therefore x - 3.5972850235404175, true thus far.