QA 
 
 218) 
 
 v^aspc. 
 
IN MEMOmAM. 
 George Davidson 
 
 University of California 
 
lA 
 
 
 DIRECT AND GENERAL METHOD" 
 
 FINDING THE APPROXIMATE ¥ALUEg 
 
 THE ItEAL KOOTS OF NOMERICAL EQUATIONS 
 
 ANY DEGREE OF ACCURACY, 
 
 By d. WJNICHOUSON, A. M., 
 
 I*rcsi(ii;m ;uiii I'rofcssor nt M;ilhem;itics in the Louisiana State University and Agricultural and Mecl-.anical College, 
 
 And author of Nicholson's Mathematical Series. 
 Baton Rouge, La. 
 
 NKW (JULliANS: 
 V. V. UAXSELL & BHO., PUBLISHERS. 
 
 ISftl. 
 
 |)j^|^ 
 
 i 
 
DIRECT AND GENERAL METHOD 
 
 OF 
 
 Finding the Approximate Values 
 
 OF 
 
 THE REAL ROOTS OF NUMERICAL EOOATIONS 
 
 ANY DEGREE OF ACCURACY. 
 
 BY J. W. NICHOLSON, A. M., 
 
 Presultnt and I'rofessor of MaUiematics in the Louisiana State University and Agricultural and Mechanical College, Baton Rouge, i. 
 
 And author of Nicliolson's Mathematical Series. 
 
 NEW ORLEANS; 
 K. F. UANSELL & BRO, PUBLISHERS. 
 
 1891. 
 
I'tUNTED BV 
 
 1/. Graham & Son 
 new okleans. 
 
 Coi'YKiGHT i8gi, iiv J. W". Nicholson. 
 AU rights rrstrvfd. 
 
 
 
INTRODUCTION, 
 
 In this small treatise I have undertaken the difficult task of producing a direct and absolute 
 method of finding the approximate values of all the real roots of numerical equations. The prac- 
 tical utility of the method, whether great or small, has been with me a matter of secondary 
 consideration. 
 
 The most celebrated methods yet discovered of approximating the roots of equations are 
 those of Lagrange, Newton and Horner; but all these a.v& tentative, since they depend primarily 
 on ^/-/'a/ substitutions of numbers for the unknown quantity. It is believed that this want of com- 
 pleteness is for the first time theoretically and practically overcome by the method now proposed. 
 Indeed, judging from the general solutions of cubic and biquadratic equations, if it were possible 
 to express the roots of higher equations in terms of their coefficients by means of radicals, the 
 reduction of the expressions to their simplest forms would be more laborious and less direct than 
 thj process of finding the roots as herein presented, especially when all the roots are real. 
 
 This method may be termed a generalization of that of Newton, of which a competent judge 
 says, "It is comparatively easy of application and rapidly attains its object." However, the gen- 
 eralization, if such it may be called, was effected by the employment of principles and methods 
 which appear to have formed no part of Newton's conception or deduction ; and I trust the follow- 
 ing statement in reference to the matter may not be inappropriate: 
 
 Several years since I was extracting the square and cube roots of numbers by the principles 
 of inequalities (see Nicholson's Elementary Algebra, pp. 212-217), and the idea occurred to me 
 that the same principles might be advantageously applied to the solution of numerical equations. 
 That idea, though vague in its inception, is the germ of the present production. 
 
 In the development of this idea, I disco veered that " successive substitutions " tend to a defi- 
 nite limit in the intervals where the given function is an increasing function of x. Largely by the 
 application of this princijsle, without reference to or even a thought of Newton's method, I obtained 
 the following general formula: 
 
 ^t(^x) 
 
 (j)i — s)fix) — xf'{x) J ^' 
 
 The demonstration of this formula is not essential to our present purpose, but in reference 
 to it I desire to state : 
 
 (1) It, together with the elementary formulas on which it is based, affords the means of 
 finding, without trial substitutions, the approximate values of the real roots of numerical equations. 
 
 -^ -»^ -^iJ^UM^^ /"A O 
 
Introduction. 
 
 (2) It assumes n different forms or values as « is made equal to 1, 2, 3, «, in succes- 
 sion. Each of these forms is peculiarly adapted to finding the corresponding root oi /\.v) = 0, 
 when that root is positive. 
 
 (3) When « = M it reduces to 
 
 ^^'^ = '-r&> ' ; ^'^ 
 
 which is the formula of Sir Isaac Newton. 
 
 Before noticing specially the form which ^ (.r) assumes when « := n, I had prepared a paper 
 on the former and its application to the solution of numerical equations. But, having subse- 
 quently observed that Newton's formula was one of the n forms of ^(.c), the idea then occurred 
 to me that this formula might be made the means of accomplishing all I desired, by employing it 
 according to the principles and methods which led to the discovery of the general formula (1). 
 
 Therefore, since F(_x) is of a simpler form than ^ (a;), although not always so direct and 
 rapidly converging, I abandoned my original treatise and prepared the present pamphlet. 
 
 An impoi'tant feature of this method is its applicability also to equations involving imaginary 
 roots. By it we determine " the number and place of the real roots" by finding the roots them- 
 selves. 
 
 In point of practical utility it furnishes very simple and economic solutions of cubic and bi- 
 quadratic equations. The solutions of higher equations are tedious and laborious by any method ; 
 yet, in many instances, the work is simplified and abridged by the present process. 
 
 The examples, which are quite numerous for so small a treatise, have been selected with a 
 view of presenting all the conditions and cases that can possibly arise, and to illustrate every part 
 of the subject. I might have chosen examples better adapted to a more favorable display of the 
 method, but preferred to restrict the selection to such as usually appear in our best text-books. 
 
 With the hope that all efforts to extend and perfect science by the removal of limitations, in- 
 dependent of the intrinsic worth of their attainment, are duly appreciated, and with the belief that 
 the present treatise will be awarded that meed of praise commensurate with its merits, the author 
 freely commits it to a generous, intelligent and discriminating public. 
 
NOTATION AND PRINCIPLES. 
 
 NOTATION. 
 
 1, For brevity and uniformity we shall employ the following notation: 
 
 ( I ) An expression like (i>^>6ora<;y<fo signifies that the value of y is between that 
 of a and /y, or that y may have any value in the interval from a to I). 
 
 (2) An expression like r \ F {oc) \ a signifies: 
 
 1°. r is to be substituted for x in Fix), and the result F (r), ( := r, , say), substituted for 
 x'ln F {x), and that result F i^i), ( = '"; , say), substituted for x in F {x), and so on. 
 
 2°. Each new result will be nearer the value of a than the preceding one, so that there is 
 no limit to the accuracy which may be obtained in thus finding the approximate value of a. That 
 is, a is the limit of r [ F (x) | a. 
 
 Hence, when r and F^x) are known, r | F (■») | d is, theoretically, the value of a, and 
 ■practically a formula by which that value may be determined to any degree of accuracy. When 
 we wish to express it in the former sense, we write simply 
 
 r\F{x)^a (3) 
 
 It is our purpose to show how such values of every real root ot any numerical equation may 
 be directly determined. 
 
 (3) The general equation of the n th degree will be denoted by f (,x) = 0, and the succes- 
 sive derivatives of f(x) by/' {x), f" (a;), /'" (x), etc. 
 
 (4) The roots of f^x) = 0, in descending order of magnitude, will be denoted by a, , a, , 
 a,, , etc. ; those of f (a?) = by aj , a'^, aj , etc. ; those of f" (a?) = by a" , aj' , aj , etc. ; and 
 so on. Thus, a^ is read : the third toot of the fourth derivative. 
 
 (5) Imaginary roots will be denoted by i; thus, ffli and a^ = i, is read: a, and «j are 
 imaginary. 
 
 (6) The superior and inferior limits of the roots oi f (x) = will be denoted by /, and L; 
 those oi f' (a;) = 0, by l\ and ij ; of/" {x) = 0, by I" and Vi ; and so on. These limits are deter- 
 mined by the ordinary algebraic methods. When the roots are all positive we shall make ^ = — 
 the coefficient of a;""', and l.^ = — the quotient of the coefficient of x° divided by that of x. 
 
DIRECT AfETriOb OF FINDING THE APPROXIMATE VaLu£s 
 
 (7) The functions F {x), Fi (x), F., (■*■)> etc., will always have the following values: 
 Fix)= X —fj^. F,ix)=.x~C^; F. (.r) = x—^1^. etc (4) 
 
 PRINCIPLES. 
 
 2. If either of the roots of /"(.») = 0, as a, , be substituted for x in F (•<?) the result will be 
 that root. That is, F (a,) = a.. Similarly, Fi («!) = a\; and so on. 
 
 3. If F {x) is an increasing function of X when a < x < h, then P (rt) < P (^)- 
 
 4. F (.)?) is an increasing function of x at all points where its derivative is positive. That 
 is, when F' (x) > 0. 
 
 5. F (x) is an increasing function of x in the intervals where f{x) and f" (.r) have the same 
 sign. For, taking the derivative of the value of F {x) we have: 
 
 F'(x)==.f{x)f"ix)^[rix)y (5) 
 
 Hence, F' {x) > when /(a;)/" (.«) > 0. 
 
 6. Evidently / (a?) /" (x) > 0, (1) when «, < .r < Z, , (2) when ^ < .f < «„ , and (T.) 
 when «,]>»> ay_i or a"_i > ;» > «, . Hence, F (a;) is an increasing function of x between the 
 same limits, or in the same intervals. 
 
 7. Since li ])> «! , and F (*) ^n increasing function of x in the interval from <i, to J, , F (/,) 
 > F (tti) ; but jf' (ttj) = rt, ; hence F (/,) > ffj . 
 
 Again, since l.^ <^ a„ we may show in a similar manner that F (Jj) <C «» • 
 
 DETERMINATION OF THE SUPERIOR AND INFERIOR ROOTS 
 
 OF AN EQUATIOiN. 
 
 8. The superior root of f (.r) = /s ?i | Jp^ (x). 
 
 For, substituting l^ for *• in F (x), remembering that/ (J^) > 0, and /' (/j) > 0, we have 
 
 By Art. 7, F (/J > «i ; hence ?j > F (/j) > ffj. 
 
OF THE REM, ROOTS OF XU.MERICAI, EQUATIONS. 
 
 That is, if Zj be substituted for x in F (.r) the result will be in the interval between ij and 
 a^, and if that result be substituted for x in F (x) it will be still more nearly equal to flj ; and 
 SO on. 
 
 Hence (.3), J^ | JF (x) = a^ (6) 
 
 9. The inferior root off{x) = is l^ \ F (x). 
 
 For, substituting l^ for x in F (x), rememberino; that when » is even f (^2) > ^'^^ 
 /' Oi) <^ 0' ^"^' vice versa when n is odd, we have l^ < -f' (^2)- 
 By Art. 7, ^ (Z^) < ff,„ ; hence, Z^ < J?' (Zj) < «,. 
 Therefore, as before, v.-e have (3) : 
 
 h I F(x)=a (7) 
 
 exa.n«f»i-e:s. 
 
 1. .7;' — 6 a? 4- 1 — 0. 
 
 Here, ?, = 6, /, = i, and F (x) = (^illf +1>. Hence (6), (7), 
 
 J (iX/ 0) 
 
 i\ F(x) \ a, = .171 + 
 .17 I " =.171572 =a. 
 
 & \ F (x)\ ai = 5.83 + 
 5.83 I " = 5.828427 = a, 
 
 These answers are correct to six decimal places. 
 
 2. aP ~ G x'' + 9 x — 2 — 0. 
 
 Here, /, = 6, i, = f = .2 + , F^x) =J(J^IZ^J^+J1. Hence (G), (7), 
 
 6 \Fix) Ia,= 4.8 + 
 
 5 I " =4.2 + 
 
 4 I " = 3.7 + 
 
 .3.7 I " =3.731' + 
 
 3.73 I " — 3.73205 = a, 
 
 Tliese results are correct as far as extended. 
 
 .2 I JP (X) 1 «3 = .264 + 
 .26 I «' = .267 + 
 
 .27 I " = .26794 =r a. 
 
 Note.— In finding fli we may use ^ (x), see ( 1 ), instead of F {x) by making n :^Z and 
 s = 1 ; and we shall find that the approximation is more rapid. 
 
 3. x' 
 
 7 j; + 7 = 0. 
 
 Here, '1 = 3, /, 
 
 I and F (.r) = ' 
 
 2 x" ■ 
 
 '6 X- — 7 
 
DIRECT METHOD OF FINDING THE APPROXIMATE VALUES 
 
 3 I F (.*•) 1 fl, = 2.3 — 4 i F (x) I a., = — 3.1 
 
 2 I " =: 1.8 —3 I " =— 3.05 
 
 1.8 I " =1.7 — 3.05 I " " = — 3.0189 =ff, 
 
 1.7 I " = 1.692 
 
 1.69 I " = 1.6920 = «, 
 
 These results are correct as far as carried. It will be observed that the approximation in 
 finding a, is comparatively slow, this is owing to the fact that n. is nearly equal to a, , as we shall 
 subsequently show. See ex. 8. If ^ (x), (1), is used in place of F (a;) the approximation will 
 be more rapid. 
 
 ^ 3 ( (x^ — 4) .r^ 4- 1) 
 4 ( (a-2_6)a; -f 3) 
 
 4. J7* — 12 x= + 12 a; — 3 = 0. 
 Here, h = 4, t, = — 4 and F {x) 
 
 i: \ F {x) \ a^ =3.3 + 
 3 I " = 2.87 + 
 
 2.9 I " = 2.859 + 
 
 2.85 I " = 2.8580 = a, 
 
 5. if' — 10 a^ + 6 a; + 1 = 0. 
 
 Here, ?, = 4, /, = — 4 and F (x) = i-"^^''''' ~ ^Zli. 
 
 . ■' b x^ (^x' _ 6) -{- 6 
 
 -i\F ix) I «, 
 — 3.91 I " 
 
 — 3.91 + 
 : — 3.9073 = a. 
 
 4 I F (.r) I a, = 3 + 
 3 I " 3.056 
 
 3.05 I " 3.0530 = aj. 
 
 — 4 
 
 1 F (.r) 
 
 ". 
 
 = 
 
 -- 3.4 
 
 — 3.4 
 
 1 " 
 
 
 = 
 
 — 3.06 
 
 - 3.00 
 
 1 " 
 
 
 - — 
 
 — 3.0653 
 
 10. Generally, when the successive values do not tend d'rectly towards a definite limit, the 
 root sought is imaginary. It sometimes happens that the successive values jump from small to 
 large numbers, or vice versa, and then tend towards a definite limit. In such cases we are to infer 
 that the root sought is imaginary, and the result is another real root of the given equation. 
 
 «. Find the inferior root of x^ — 6 a'^ + 10 a? — 8 = 0, 
 
 (2 X — 0) x^ -\- 8 
 
 Here, I, = .8 and F (*) 
 
 (3 X — 12) j; + 10 
 
 The successive values of .8 | JF (-i') | a~ are + 2.2, 
 imaginary. 
 
 1.3, — .2, etc. Hence, flj is 
 
6P tHE KKAL toots OF NUMERICAL EQUATIONS. , 9 
 
 DETERMINATION OF ALL THE REAL ROOTS. 
 
 PRINCIPLES. 
 
 11. The ,9 th root oi f (x) = and the (« — l)th root of /" (x) = are both between 
 the « th. and the (« — 1) th roots of /' (.r) = 0, all the roots being real. That is, a'. < a, < 
 «;_! , and «; < <■_, < ai_i . 
 
 12. In the interval from a', to a,, f (,x) and f' (a;) always have contrary siorns, and in the 
 interval from a, to a',_, they always have the same sign. Therefore, remembering that 2*' (.«■) = 
 X — (/(•*) ) -^ (/' (i'?) ), we have: 
 
 (1) Whena?_, <a., a';_, <_ F (a^_,). 
 
 (2) When <_,>«., «:■_!> J' (C_0- 
 
 13. Since F («) is an increasing function of x in the interval from a"_, to a,, Art. C, we 
 have : 
 
 ( I ) When <_, < a. , F «_,) < F (fl.) or < a, , Art. 2. 
 
 (2 ) When a;'_i > «. , F (ai'_i) > JF («.) or > a, , Art. 2. 
 
 In either case then, see Art. 12, ^ «'_,) is in the iriterval from <_, too,. Similarly, 
 r being any number between ffi"_, and a,, F (>') is between r and a, . Hence, 
 
 a':^, \ F {X) = a. (9) 
 
 14. Now let us find a','_i. Since /"^(o;) sustains the same relation to/'"(j?) that the latter 
 does to f {x), we have : 
 
 ar_2 I F, (,<^) == a';_, (10) 
 
 Similarly, aJL, | F^ix)=a':^, (H) 
 
 etc., etc., etc., etc. 
 
 16. It will be seen that the solution of f (x) = depends on the solution off" (a;) = 0; 
 the latter on that of /" (as) = ; and so on, as follows: 
 
 I. We have (6) (7), I, | F {x) = a, , L \ F (x) = a„ (12) 
 
 II. By (12), A- I F, {X) = < , l\ i F^ (a?) = aS_i. 
 
10 DIRECT METHOD OF KlNOiNG THE APPROXIMATE VALUlJS 
 
 Now ill (9) make 8 = 2, and s = n — 1, and we lia\c: 
 
 «!■ \ Fix)=a,_, «;;_, I F (.r) =«„_,. 
 
 Therefore 
 
 l\' \ F^ (x) = a1 ; a'.; \ F (x) = n, \ ^ ^^^ 
 
 '2 i F., (.v) = o;;_ 2 ; u-'_^^ \ F ix) = fl„_ . j 
 
 III. In a similar manner we obtain 
 
 17 I F, (X) = aV ; a\' \ F^ (a?) = «■■ \ a^ \ F (.r) = a, > _ 
 
 i'J I l*'^(a?)=C-4; C-, I -P2 (•T')=«;;-3; C-s I F{x)^a„^S 
 
 The process may evidently be carried on indefinitely. 
 
 16. A Descending Solution of an equation consists in finding the roots successively, begin- 
 ning with the largest; and an Ascending Solution consists in finding the roots similarly, beginning 
 with the smallest. 
 
 WHEN THERE ARE IMAGINARY ROOTS. 
 
 17. In a descending solution if all the roots greater than a, have been found to be real, 
 and a"_i \ F (^x) = a, proves to be imaginary, we infer that «, + i is also, and hence proceed to find 
 the second root below, viz: a".^i \ F {x) == «,_^2 . 
 
 In an ascending solution under similar circumstances we seek the second root above, viz : 
 o?_3 I F{,x) = a...^. 
 
 18. Under the same conditions, if a" and a"^■^ are also imaginary, a"_i | F (:») will ^^ pass 
 over" the two imaginary roots a, and a, + , and produce the root «. 1-2 . 
 
 For under the conditions stated f {_x) f" (a?) > in the entire interval from a;'_, to «,+ 3 . 
 Hence, 
 
 <-i \F{x)=a.^^ ■ (15) 
 
 In general, if all the roots from a, to a. + 3,-1, and from «;' to a" + 2<-i' inclusive, are 
 imaginary, we have: 
 
 al'_, I F{x)=a.+ ^, (IC) 
 
 Thus, making « = 1, ( = 1, and a;'_, = /, , we have 
 
 l,\ Fix)=a, (H) 
 
 And, when < = 2, k \ F ^x) = a, (1«) 
 
6v THE RKAL ROOTS OK NUMEiUCAL EQUATIONS. 11 
 
 19. Again, in an ascending solution under the conditions stated in Art. 17, if «"_j and a". 3 
 are also imaginary, «f,_i | -F (x) will pass over the imaginary roots a, and «,_i and produce the 
 root a,_2- 
 
 For under these conditions, f (,x) f" (j") > in the intei-val from a"~i to a,.,. Hence, 
 
 C_i I l^(,c) ^a._2- (19) 
 
 In general, if all the roots from rt, to «.,_3,_ j , and from a"^^ to al'_2,_3, inclusive, are 
 imaginary, we have: 
 
 ai-.i I F (,ij) = «._,. (20) 
 
 Thus, making s = 1 , ( = 1 and a"_ ^ == /^ , we have 
 
 h I J' («)=:«„_, .....(21) 
 
 If « = 6, t = 1, a\ I ^ (.r) = fl„ (22) 
 
 20. The preceding principles apply also' to /" (a;) = and /"' (.r) = 0, to /" (.r) = and 
 f" (,r) = 0, and so on. Therefore we are enabled in any given equation to determine all the 
 real roots. The following illustrations will render this fact very clear. 
 
 21. Let/(.r) = be an equation of the fourth degree. 
 
 We need not consider the case in which the roots oi f" (iC) = 0, viz: fflj' , a'^ , are real, for 
 in that event the application of formulas (13) is obviously direct. Let us suppose, then, that rt^' 
 and aj = i. 
 
 If the roots oi f (x) =; are real, the roots of /" {x) = are also ; hence, if the latter are 
 imaginary then two or more of the former are also. 
 
 (1) Suppose a, and a^ = i, then, Art. 18, ^, | F (.») will pass over n, and a.^ and give a^ , 
 and Ij \ F (x) will give fl^ . 
 
 (2) Suppose a^ and a, = i, then i, | F (x) will give a^ , and l^ | F (.») will give a^ . 
 
 (3) Suppose tts and a, = i, then, Art. 19, I2 \ F («) will pass over a^ and a, and give a^ , 
 and li 1 F (x) will give a, . 
 
 (4) Suppose «! , fflj ) «3 1 <»* — '1 t'le" 'i I ^ (,•"') ''*"'• 'a i -*^ C*^') ^''^ '^'"^'^ S'^^ *'■ 
 
 22. Again, let/'(.r) ^ be an equation of the fifth degree. 
 
 Suppose a" and a'!, = i, then two or more of a, , a^ , «, , a, are imaginary. Now this case 
 is entirely similar to the one which we have just considered ; and the same will be the case if we 
 suppose «.V and a" = 1. 
 
12 
 
 DIRECT METHOD OK KlNDlNCi THE AH'KOXIMATE VAl.UkS 
 
 23. Therefore, formul:is (12), (13), (14), etc., are general; that is, the real roots of 
 f" (.t) =; 0, together with /, | F (J^) and l^ \ F^-v), are sufficient to determine all the real roots 
 of/(.x) = 0; similarly, the real roots of /"(,*•) r= 0, together with ll' \ F^ (ar) and Z'^ JFj (.t), are 
 sufficient to determine all the real roots of /"(a;) = 0; and so on. 
 
 The process is fully illustrated in the solution of the following examples, in the selection of 
 which an attempt has been made to anticipate all the conditions and cases that can possibly arise. 
 
 24. When there are imaginary roots, and especially when the given equation is incomplete, 
 it sometimes happens that two or more of the results are equal to each other. Now in such cases, 
 if we know there are no equal roots, which can always be determined and generally by inspection, 
 we are to infer that the results which differ from each other are the real roots, and the only real 
 roots of the given equation. See Examples 15 and 22. 
 
 25. In finding the roots of /'" (a;) = 0, f" (a;) = 0, etc., no great accuracy is required. 
 One place of decimals, and often the integral part of the root, will suffice, unless the roots of 
 / (a;) = are very nearly equal to each other. 
 
 ClTBir EQUATIONS. 
 
 26. Let/'(.r) r= be an equation of the third degree. 
 By formulas (12) and (13), we have 
 
 .(23) 
 
 Note. — Evidently nj' 
 
 -| of the coefficient of J'-*. 
 
 EXAMPLES. 
 
 7. a^ — 9 .^2 + 22 a; — 11 = 0. 
 
 Here, /, = 9, a'{ = 3, /, = .5, and F (r) = (2 a^ — 9)3^^-1- 11^ 
 
 ^ (3 a- — 18) a; -f- 22 
 
 Therefore, 9 | F (a;) = «! , 3 | ^(a:) = w.^ , .5 | F (.r) = a^ . 
 
 9\F{x)\a, = l + 
 7 I " =6 + 
 
 6 I " =hA + 
 
 3 I F (.r) I ttj = 3.2 
 
 3.2 ! 
 
 3.201G =rt., 
 
 .5.4 I J?' (a-) I rt, = 5.17 + 
 5.1 I " =5.129 + 
 
 5.13 I " =5.1284 = a, 
 
 .5 I F (a-) 1 a, = .65 + 
 .fi5 I " = .669 + 
 
 .66 I " = .6699 c=a„ 
 
 These answers are correct as far as carried. 
 
OF rilK REAL ROOTS OF NUMKKICAI, EQUATIONS. 
 
 13 
 
 8. Find the intermediate root of example 3. 
 
 2 J'' — 7 
 
 Cl' = U, :iii(l F (.<■) = 
 
 6 X- 
 
 1 I " = 1.25 
 
 1.2 I " = 1.32 
 
 9. .r' — 6 x'^ + 15 ,r — 10 = 0. 
 
 I, = 6, a'i = 2, L = .6, and F{x)=^ ^^ j; - 6) x« + 10 
 ' ' ' ^ ' (3a;— 12)x'+15 
 
 The successive values of 6 ] J^ (.i) are 4 -f , 3 — ,2 — , .6, .9, 
 wc infer that flj and a, == i, and a~^=-\-l. See Art. 10. 
 
 1.3 \ F {x) \ a^ ^ 1.35 
 1.36 I " = 1.3568 = rt^, 
 
 10. J-'^ — 15 a; + 21 ^0. 
 
 i, = 5, «;• z= 0, /, -^ _ 5, J' (aO 
 
 2 .r^ — 21 
 
 3 j;2 — 15' 
 
 5 ( F {X) == 2.67247 = «, 
 I FXx) = 1.76915 =ffj 
 b \ F{x)=,~ 4.44162 = «, 
 
 1. From which 
 
 .(21) 
 
 EQUATIONS OF THE FOURTH DEGREE. 
 
 27. Let/"(j;) = be vn equation of the fourth degree, then we have (12), (13), 
 
 h \Fix) =a, -) 
 
 li' I F^ix) :=.«/■; «!■ I Fix)=a, ! 
 
 I'i I F^ (.»•) = a.\ ; «'^ I 1*^ (.r) = «, I 
 
 h I 1^(0 =«. 3 
 
 Norn. — We m.^y evidently use «, and «, for I'i and /'/, , rcspc-ctivelv. It is best to tind ((',' and 
 aj by liic ordinary method. 
 
 EXAMPLES. 
 
 11. Find the second and third roots of example 4. 
 
 «;' ::^ 1.4 , iC == — 1.4, F (X) = 
 
 1.4 1 F {X) = .OOfiOlS ^ ,ij I 
 
 3 ( (x^ - 4) x^ 4- 1) 
 
 4 ( (.r'^— 6) a; + 3) 
 
 1.4 I Fix) == .443270 = a, 
 
14 
 
 DIRECT METHOD OF FINDING THE APPIIOXIMATE VALUE!> 
 
 12. x* — ir' — Sx + 21 = 0. 
 
 7, = 4, /, = - 3, 
 flV = 2, «'i = 0, 
 
 i \ F (X) =3.6796 =a, 
 2 i " = 2.2674 = («, 
 
 13. x' — 12 jr' + 39 J;-^ — 48 X' + 10 = 0. 
 Here, /, = 12, I., = .2, «i' = i.b, a'' = 1.4. 
 
 12 I F (x) = 7.741 =«, I 
 
 4.5 I " =i =a^ I 
 
 14. X* — a n^ -\- 32 .«2 — 77 *■ + 2.5 = 0. 
 
 ^ '^ (4 a; — 12)i'2 — 3 
 
 I F (.») = J = rt, 
 
 3 1 " = i = a. 
 
 1.4 I J^ (j;) = i = a, 
 
 .2 I " = .258 + = a, 
 
 Here, /j = 5, l^ =; .3, a" = i, a'j =»'. Hence, Art. 23, the only possible real roots are 
 5 I *'(«) and .3 j F {v). 
 
 6 I F (x) = 2.6180 = Og I 
 
 15. a;* — 28 x — 4.5 = 0. 
 7, = 4, ^2 = - 2, «V = "i' = 0, F (X) = 
 
 .3 I JP (a;) = .3819 = a, 
 
 3 X* + 45 
 
 4 07' — 28 
 4 I JP (a;) = 3.4494 ; | ^ (x) — — 1.4494 ; ^ 2 | ^ (a;) = — 1.4494. 
 
 Now, as there are no equal roots, Art. 24, the only real roots are 3.4494 and — 1.4494, 
 
 16. x*+ 2 X + 3.75 = 0. 
 
 7,=0, 72=-2, «i'=«'i=0, F(x) 
 
 3x* — 3:lo 
 
 ix^ + 2~' 
 
 I F (.r) = * and — 2\F (x) = i. 
 .*. «] , a.,, ti^ and a, are imaofinary. 
 17. x' — 19 x^ — 23 .r — 7 = 0. 
 / - 5 7 - -4 Fix)- <l^^^^^±-l. 
 
 fl5' = 1.7, «v= — 1.7 
 
 5 I F (.«■) = 4.8977 = «, — 1.7 j ^ (.!■) = — .7124 = rt, 
 1.7 I ^ (a) = — .5522 = rt^ ^ 4 ] JP (.e) = — 3.6331 == «, 
 
6f the real roots of numerical equations. 
 
 15 
 
 .(25) 
 
 EQUATIONS OF THE FIFTH DECREE. 
 
 28. Let/ (a?) = be an equation o! the fifth degree, then by (12), (13), (14), we have 
 
 h\ F(x) = «, " 
 
 i;- I F^ ix) ^ rt'/ ; rtS' I F (x) = a^ 
 
 flr I F^ (.»•)=«'•. ; «2 I Fix) = a, 
 
 /■■ I F^ (.0 =«" ; "s I Fix)=a, 
 
 h I F (.r) =(h 3 
 
 Note. — In place of l\ and V^ we may write a^ and a^ respectively, and in place of a'l , — J 
 of the coefficient of .»* . 
 
 18. Find rtj , rt.3 Bnd a, in example 5. 
 
 4 it^ r^^ 5^ 1 
 
 «!■ := 1.7, «■■ =0, «;,' = — 1.7, and Fix) = 5 ^2 (^.2 _ 6) + 6 * 
 
 1.7 \ F ix) = .87950870 • = a^ . \ F (.»') ::= — .17567479 = a,. 
 _ 1.7 I j^ (a;) =: — .69157628. 
 
 19. x'—ax'+ 25 Cf^—iOx^ + .34 .x; — 12 = 0. 
 
 (2 a! — 4.8) .c'+ 4 
 '' = «' '-^ = -3' «-^ = 1-6. ^3 (^) - ^3¥^l ,.6)a;+7.o - 
 
 We shall use a' and ffj instead of i" and l'^ . 
 
 8 I JF (.r) = 3 = «,, 3 j 1^2 (.r) = 2.2 ; 2.-2 \ F ix) = 2 = a^ . 
 1|F, (*o=i. 
 .3 I Fix) = l=a,. 
 
 Hence, the three real roots are 3, 2, 1. See Art. 23. 
 
 20. X'— 11 X* + 49 x' — 109 x' + 120 * — 50 = 0. 
 
 11 I Fix) = 1. 
 This is evidently the least root; hence all the others are imaginary. See (18). 
 
 21. 9 a;' — 3 .X* — 26 .1;' — 1 = 0. 
 
 /j = -f 3, ai' = 1, fl'^ = 0, n^ = — .8 and '3 = — 2, 
 
16 
 
 DIRECT RfETHOI) OF FINDING fHE APPROXIMATfe VAi,(jES 
 
 1 I F (_x) =i = a^ 
 I F (r) ^i = a^ 
 
 22. a-'^ — 14 a? — 3 — 0. 
 
 3 I J' (a;) = 4- 1.879385242 = a, 
 — .8 I F {x)—— .347296355 = a^ 
 — 2 I ^ (a;) = — 1.53208888(; ^ a. 
 
 Here, i, = 2, i^ = — 2, a? = a" = aj = 0, JP (a?) = g-^J^u* 
 
 Now, Art. 24, as tliere are no equal roots the only possible real roots are 2 | F (•<), 
 — 2\F (x), i F (*■)• 
 
 2 I J'(a?) = 1.9+, \ F(,x) = - .21- , — 2 j JF (x) == — 1.8 — . 
 
 EQUATIONS OF THE SIXTH DEGREE. 
 
 29. Let/(.») =0 be an equation of the sixth degree, then by (12), (13), (14), we have 
 
 h\ F(x) =a, 
 
 l'{ I F^ (0?) =. al ; a\ \ F(x) = a^ 
 aY I F^ ix) = a'i ■ a\^ \ F (x) = a^ 
 
 a'; I F^ (X) ^ a'i ; a'i \ F (a) = a, 
 ll I F. ix) = «V ; «V I -F (X) = a, 
 ?2 I -P (•») = «0 
 
 •(26) 
 
 EXAMPLES. 
 
 88. .■»« — 8 ,r< + 15 a'i' + 2 a; — 1 = . 
 
 Here, /i = 4, l^ = — 4, «;'= 1.68, 
 a^ = .59 ; aV =^ — .59 ; a\'=z~ 1.68 
 
 4 I J' (a;) =2.128419 = Oj 
 
 1.7 \ F(x)= 1.879385 = a^ 
 
 .6 I ^(a;) = .201639 = as 
 
 Fix) = 
 
 5 a^ — 24 a;' + 15 a^' f 1 
 
 6 x" — 32 a;'' 4- 30 x + 2 " 
 
 — .6 I J' (a;) z= — .347296 = «< 
 
 — 1.7 I J' (a;) = — 1.532088 = «j 
 
 — 4 \ F(x)= — 2.330059 = a, 
 
 24. a^ _ 14 ar' 4- 70 a;* — 164 x^ + 203 x'— 126 a; + 22 = 0. 
 
 ?, = 14, Ji' = fli, flr= 3.2, «!/= 1.4 ; l^^a,, l, = -17 + (.2, say). 
 
 Fffr} — 22—3;" (203 — a; (328 — x (210 — a; (56 — 5 a;)))) 
 ' 126 — a; (406 — a; (492 —a; (280 —x (70 — 6 a;))))* 
 
 F, ix) 
 
 203 — x^ (420 — a; (280 — 45 x) ) 
 492 — X (840 — ar (420 — 60 x)) ' 
 
OF THE REAL ROOTS OF NUMERICAL EQUATIONS. 
 
 17 
 
 14 i F ix) = 6.236068 = d, 
 
 6 I F., {X) = 4.8 + ; 4.8 , F (.») = 3.732050 = a, 
 3.2 ! F^ (.%•) = 2.4 + ; 2.4 | F (,t) = 1.763932 = n, 
 
 1.4 j F2 (^) == t = «• = «4 
 
 .26 Fiix) =zi =i. = (t, 
 
 .2 I F ix) = .267949 = nr, 
 
 25. 2 .r' — 25 ,tr + 13 .1; — 3 = 0. 
 
 Here, /i^= 3, l, = — 2, two of the roots of/" (_x) = are 1.3 + and 0, and the other two 
 are imaginary. Hence, the only possible real roots are (Art. 23) : 
 
 3 I F{x) = 2.2469 
 1.3 I F (.r) = .5549 
 
 I F(x) = .2675 
 — 2 I F ix) = — .8019 
 
 EqUATIONS OF THE SEVENTH DEGREE. 
 
 30. Let/"(.i') = be an equation .of the seventh degree, then by the preceding formulas 
 
 the roots are 
 
 h I Fix) =a. 
 
 V; I JPs (x) = «;' 
 
 /r i jp', {x) = flr 
 
 «r' I F, (a;) = a'^ 
 V; I F, (a;) = a'; 
 /'^ I Fi {x) = rti' 
 h\ F ix) 
 
 F(.x) 
 
 «r I F^ (X) = «i' ; 
 
 «2 I F._ ix) 
 < I F2 ix) 
 a'i \ F ix) . 
 
 a 
 
 fts ; a; 
 : ay ; a] 
 
 ^ , F ix) = «5 
 J?' (a;) = a, 
 F ix) = a, 
 
 = fffi 
 
 -= "7 J 
 
 .(27) 
 
 exa.ivif>i>e;s. 
 
 26. a;'— 6 ic« + 23 x*— 49 ar! + 225 x^ — 120 ,r — 200 =0. 
 
 In this solution we shall begin at the top and bottom and work to the center. Hence, we 
 make «, = 6 , l^— ~4, ?i' = «, , I'^ — a^ , f," = «," , l'^ = a'^ , «[■ = « r= .8 , say. 
 
 6 
 5 
 3 
 .8 
 
 — 1.6 
 
 — 2.8 
 
 — 4 
 
 Fix) = 5.2465 = a, 
 
 Fzix)^ 3+; S \ F ix) = 2.8284 =a2 
 
 F^ ix) = 2.4 ; 2.4 I F^ ix) = 1.5 ; 1.5 | F ix) = 1.4231 = a^ 
 Ft ix) == .5 ; .5 I J'a ix) = i a. 
 
 Ft ix)=— .5 ; — .5 I F^ ix) = i a, 
 
 F^ix)^ —1.6; —1.6 \ F ix) =— .6696 = a, 
 
 Fix)= = — 2.8284=0, 
 
18 DIRECT METHOD OF FINDING THE APPROXIMATE VALUES 
 
 It appears that «< and ffj = i , but in reality «4 = — .6696, and a, and a, = i. See (22). 
 
 27. x' — 6 a;» + 17 a; — 29 = 0. 
 
 Here, 1^ — -{- o , /^ == — 4, and /" (.r) = 21 x-' — 60 .t'. Hence the roots of /" (.r) = , 
 are ].9, 0, 0, 0, — 1.9. Therefore, there being no equal roots, the only possible real roots are 
 I F(.r), 1.9 i l-'Ca;), | F (x), — 1.9 | F(w), and — 4 | F (x). 
 
 5 I F ix) = 2.4 + 
 1.9 \ Fix)= i 
 
 I F (X) = i 
 1.9 I F(x) = — 1.9- 
 
 i \ F(x) = — 2.1 
 
 EQUATIONS OF THE Nth DEGREE. 
 
 31. It is evident that the process may be applied to any numerical equation. 
 
 ROOTS OF NUMBERS. 
 
 32. In extracting the roots of numbers by formula (1), the approximation is very rapid. 
 
 EXAMPLES. 
 
 28. Find the cube root of 60. 
 
 Make (a; — 3 )' = 30 ; whence af — 9 x^ + 27 a; — 57 = 0. 
 
 ( (9 a; — 54)a; + 171) a; 
 
 Substituting in (1), making « = 1, ^ (x) -. 
 
 (.«2 _ 27) a; + 114 
 
 6 I ^ (a') I 3 + v^ = 6.107 + 
 6.107 I " =6.1072325050. 
 
 ••• 1/30 =3 3.10723250.50. 
 
 29. Find the square root of 3. 
 
 Make {x — 2)^= 3 ; whence .-r? — 4 .r -j- 1 =: ; and substituting in (1), making .f = 1, 
 
 ~^^ ' (X + 1) (.r - 1) 
 
 4 I ^ (a,') I 2 + ^/J= 3.73 + 
 3.73 I " = 3.7320508 + 
 
 3.7320508 | " =3.732050807568877293527 
 
 .-. ,/y= 1.7320,50807568877293527, 
 
 These results are correct as far as carried, 
 
OF THE RKAI, KOOTS OK NUMEKICAI. KQUATIO^^S. 19 
 
 NOTES. 
 
 1. We have made no attempt to estimate the rapidity of the approximation, because the method of 
 doing so is given in many of our best text-books. See Todhunter's Theory of Equations, page 1^9. Suffice to say 
 that "the number of exact decimal places is nearly doubled at each step." 
 
 2. In all the practical applications of equations a greater degree of accuracy in the roots than three or four 
 places of decimals is seldom, if ever, required. 
 
 3. In order to avoid the possibility of an error in seeking a root, the number to be substituted for x should 
 be a little larger or smaller than the partial result of the last substitution, according as the root sought is less or 
 greater than that result, respectively. This is especially necessary when two roots are very nearly equal 
 to each other. 
 
 CONCLUSION, 
 
 The following is an extract from " The Teaching and History of Mathematics in the United States," page 
 363, by Prof. Florian Cajori, M. S.: 
 
 " In this connection a quotation from Gerhardt's Geschichle der Mutheinatik in DeulscUland (p. 205) is 
 instructive. 
 
 "Savshe: 'The advocates of the combinatorial analysis were of the opinion that with the complete 
 solution of this problem (of reversion of series) was given als > the general solution of equations. But here 
 they overlooked an important point — the convergency or divergency of the series which was obtained for the 
 value of the unknown quantity. Modern analysis justly demanded an investigation of this point, inasmuch as 
 the usefulness of the results i.s completely dependent upon it.' It thus appears that, through the misuse of 
 infinite series, the Germans were temporarily led to believe that they had reached a result which mathematicians 
 had so long but vainly striven to attain, namely, the algebraic solution of equations higher than the fourth 
 degree. It will be observed that their method lacked generality, since it could at best not yield more than one 
 root of an equation. But in the determination of this one root the combinatorial school was deceived. The 
 result was a mere delusion — a mirage produced by the refraction of the rays of reasoning from their true path 
 while passing through the atmosphere of divergent f-eries." 
 
 The "general solution" referred to above is a direct and absolute method of finding the approximate 
 values of the real roots of numerical equations of the higher degrees. Therefore I claim, I trust without immodesty, 
 to have " reached a result which mathematicians have so long but vainly striven to attain." 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 tiMajreODF 
 
 LD 21-100m-ll,'49(B7146sl6)476 
 
GAYLAMOUNT i 
 
 PAMPHLET BINDER 
 
 Mano/ttduftJ iy 
 
 1 6AYLORD BROS. Inc. j 
 
 Syr«cui«, N. Y. 
 
 Skocllon, Calif. 
 
 iVs28798S 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
^^