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BY JAMES LOCKHART. /■'■ ■■' / /•■/ ^' *' Methods of resolving equations by approximation ought to be considered as of the highest utility, and as being absolutely necessary to the completion of the doctrine of the resolution of algebraic equations, which is the most important branch of the science of Algebra." Maseref. LONDON: PRINTED FOR THE AUTHOR, By L. Harrison Sg J. C. Leigh, 373, Strand; AND PUBLISHED BY R. ACKERMANN, AT TH£ REPOSITORY OF ARTS, 101, STRAND. 1813. 7i^ * THIS WORK IS RESPECTFULLY DEDICATED (by permission) TO THE REV. ISAAC MILNER, D.D. PRESIDENT OF QUEEN'S COLLEGE, AND LUCASIAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE, BY HIS OBEDIENT AND HUMBLE SERVANT, JAMES LOCK HART. London, May l, 7 83 3. QA2.) INTRODUCTION. All the known methods of approximating towards the roots of equations of the third and superior orders, are attended with great labour and uncertainty. Previously to their appli- cation to a given equation, one or two figures of the root must be found by trial, or an assumption made, to serve as a basis for subsequent results. If two correct figures, found by trial, are employed as a first near value of the root, the subsequent result is generally in excess, and frequently seve- ral steps of the same method of approximation are requisite to correct it: if the root x of the equation_j;^— -6£^j[ be ag-_ proximated to by Raphsori's method, and the figures 2.5 taken as a first near value, such excess and slowness of ap- proximation will be found to take place. If an assumption of figures is made, great judgment, founded on long expe- rience, is necessary, otherwise the approximation will be very tedious, and oftentimes not more than two correct figures will be obtained after three applications of any of the methods.. Results obtained by JPLo Lagny's method are, in general, the j most correct, but the operation is tedious, and the approxi-^ I mation cannot be absolutely confided in. The convergency I iiecessary to the resolution of equations of one order is fre- quently inapplicable to others, and even in equations of the same order the same convergency is insufficient. Hence ori- .ginate the unequal errors in excess and defect which persons skilled in approximation continually experience. Nor is this the only obstacle to a speedy solution ; for, when two figures of the root have been assumed or found by trial, and three additional and supposed correct figures have been obtained by a first step of approximation, there is no mode of proving the work but by raising the figures to the given power, and making the necessary additions or subtractions ; and even then doubt will frequently remain. The most expeditious : -i-o (••<•«,•' O^ O f VI and judicious approximator can seldom obtain nine correct figures of the root in less than two hours, and he will in many cases be employed three or four hours about a single equation. If he resorts to the method of series, the length of the operation is intolerable. The object of this work is, the resolution of cubic equa- tions only, including all those belonging to the irreducible case, and rather more than one fourth of those, the roots of which are accurately discoverable by Tartaglia's rule. The method which it contains is, however, more peculiarly appli- cable to the resolution of those equations to which that rule cannot be accommodated. It commences with giving a first near value of the root without assumption or trial, and pos- sesses a due convergency ; the results are always accurate, and the operation is so simple, that a boy capable of extract- ing the square root can perform it with ease. Sometimes all the necessary figures are found by one step of the approxima- tion, and in no case whatever will it require more than half an hour to obtain eight figures of the root of the given equa- tion. Before the author ventured to publish his method, he com- /municated it to some gentlemen profound in mathematical sci- ence, who gave him encouragement to hope, that it would meet the public favour ; and should that hope be realized, he will endeavour to make the same principle suit equations of the fifth order. He has not given all the demonstrations which might have been expected : they would have increased the work to an inconvenient size ; or, by excluding many exam- ples, rendered it unfit for that purpose which he trusts will be its destiny, namely, that of giving the student a taste for approximation. The astronomical purposes to which the method may be adapted, need not be dilated on. The greater part of the examples have been selected in order to resist the method as much as possible. Some of them include all that is difficult in the irreducible case ; and the author assures his readers, that he is not acquainted with a single defect that attaches to this method. METHOD OF APPROXIMATION. 1 HE iiTeducible case is sufficiently comprehended under the two equations .r^ —bx=c and bx — x^ =c, being greater than 6.75 in each equation. c^ The former equation has only one root ; the latter equation has two roots, the sum of which is equal to the root of the former equation. A general and finite expression for the values of the roots, is not known, and all that we can do, in innumerable cases, is to find ap- proximating values. The following rules are necessary to a right understanding of the method contained in this work, and serve to facilitate the requisite operations. AUXILIARY RULES. RULE I. To find the difference of two cubes, their cube roots being given. Make their difference equal to three times the difference of their cube roots multiplied into the square of tlie arithmetical mean of their cube roots, and twice the cube of half the difference of their cube roots : for. :3-3,3=3x-3yx(^) 4-2 {^) or, x^—y^=--x^i+y -\ — 4 4 d being equal to x — j/. 8 Ex. The cube roots 476 and 472 being given, to find the difference of their cubes. 476+472 = 898704 3 I_2 4 2696112 V = 16 2696128 is the difference of the cubes. This rule is applicable when ten or eleven figures have been found by the method, should there be a doubt respecting the accuracy of the last figure. The approximator will then be relieved from the necessity of cubing two high numbers. Cubing need not be resorted to at that state of the approximation, because a more simple method of proving the work is introduced ; but it may be preferred from habit. By means of this rule the differences of very high powers of num- bers may be obtained with comparatively little trouble. Ejc. Let x^ —y^ be required. This rule may be of some use in the method of increments ; and it is obvious, that the difference of any two cubes may be geometrically found by means thereof. RULE II. Three numbers in arithmetical progression being given, to deter- mine whether their product is connected or not with the irreducible case, the middle number being considered as the root of the equa- tion x^ —bx=c. The progression may be represented thus : X — V ^ XXX x-\-'^b=x^ — bx, which may or may not be connected with the case in question. The rule in respect of numbers is this : — Multiply the middle num- ber by l->/. 75 =.13397459-}- : if the product is greater than the first \ t 9 lenn, the product of the three terms is connected uith the irreduci- ble case, otherwise it is not: for ax 1 — '^.75=^: — '^.75.r ; and if this quantity is equal to the first term, the product of the three terms a ^ 7 / ' X ^ becomes a^ —.75x3 = -— , or a— '^.75xx a x.v + v .75x=-— : but this is 4 4 the limit of the two cases, and the rule is therefore correct, Ex. Let 1, 10, 19 be the three terms. 1 — v. 75x10 is greater than the first term ; therefore the product of the three terms is connected with the irreducible case. The resulting equation is x^ — 81x = 190=:their product : for 10-v/81xl0xl0+^/81 = 190, and x-^Slx X X x~[-^^^l=x^ -Six. Let three other terms be 2, 10, 18. 1 — V .75 X 10 is less than the first term ; therefore the product of the three terms is not connected with the irreducible case. The resulting equation is x^ — 64.i=360=tlieir product: for 10- v/64x 10 X 10+^^64 = 360, and X - ^^64 x x x a -f ^^64 =x' - Qix. We may pass rapidly over series of this kind, and determine, almost by inspection, whether they are connected or not with the subject before us. The application of this rule will be shewn hereafter. RULE Hi. To determine whether the equations x^ —bx=c, and bx — x^ =c, are connected with the irreducible case or not. Find the value of -— If — is greater than 6.75, the equations are connected with the case ; otherwise not. We know, that if, in the application of Tar-- taglia's rule to these equations, — cannot be subtracted from — ? the equations are irreducible, and that when these quantities are x^ t-qual to each otlier, .r^ -- hx = — : but tl^is equation is the boun- C 10 dary of the case which is reducible, and of that which is not reduci- b^ c^ b^ ble. If, then, —- = —-5 it results that ^- = 6.75; and consequently ' ' 27 4 ^3 c^ n J that the equation is irreducible if -~is greater than 6.75. Ex. Let.r3-6.76.r=6.75, Let lx-x^=l, Leti^=100x=100; these equations are in the irreducible case : but the equations x' -6.75.r=6.75, X 3_6.ri=6 bx — x'^=-b, are not in the irreducible case. RULE IV. To find the values of the roots of the equation bx — x^=c, the root of the equation .r^ — bx=c being known. Having found the value of x, solve the quadratic equation xt — t^=x^~b, and the two roots of that equation will respectively ' be the roots of the equation bx — x^ —c. This quadratic may be converted into the following convenient expression, to give the values of the roots, the sign + belonging to | the greater root. x_^^^ib^ 2 - 2 One of the roots is always greater than the other, while the equa- tion belongs to the irreducible case; but they are equal Vv^heu b^ =6.75. Henceforward the greater root will be called t, and the lesser root v. e Ex. Let8Ir-x3=190. The root of the equation r^ — 81x=190, is 10. a^^-81 = 19 10^-^ = 19 f=5-i-v^6 z; = 5-'V^6 11 Ex. Let63x-j:2=162. The root of the opposite equation is 9. 9 +^252-243 t and r= 2-_ ^ hence ^ = 6, v = S. RULE y. To find the value of the root of the equation x^—hx=c, when either of the roots of the equation bx — x^ =c is known. Put cZ=the square of the root known, e = b~d, then .; = : Ex. Let 81t- -1^=190. e=50-10^/6 5+v'64-'^^231-30v'6 2 Ex. Let 7x-a^=6. v = l d=l e = Q x=-^—=S PaiLE VL :10 To find a near value of the root of the equation x^ —bx^c, without assumption or trial, and at the same time to discover close limits thereof. In the best treatise on this case that I have read, the limits are thus defined : — " x is always greater than v^6, but less than -/^ ;" and these C 2 12 limits are considered as very narrow by the learned author. But much narrower limits than these may alwg,ys be readily discovei'ed. The method of obtaining them will, in the first place, be exhibited by some examples. Ex. Let ^3- 100x1=1. ' This exam.ple is favourable to the method ; but many others might have been selected, wherein hundreds of correct figures would have been obtained. Take the value of — , which in this case is 1000000, and it? square root is 1000. Take this square root as the numerator of a fraction, and the square root +1 as the denominator. Take the denominator as the numerator of a new fraction, and this numerator -j-l as the denominator. The two fractions are -f?^ and r%%i. Squaring the numerator of the second or right-hand fractioUj gives 1002001 subtracting the numerator 1001 leaves 1001000 b^ but this last number being greater than the value of -;- = 1000000, the two fractions suit the intended purpose. It results, that x is less than "^f — 1001/» 1000 ' 1 • , //100-2^.. that X is o-reater than v ] "" ^ 1001 / „ . , . 10016 J3ut in the given equation, — =100.1 1000 and ?^^= 100. 0999: 1001 hence x is less than v^l00.1 = 10.004998 X is greater than ^^100.0999= 10.004993 : consequently the figures 10.00499 are correct for a first near value of X, and the proposed narrow limits have been found. Ex. Letx^-3x=l. This example is unfavourable to the method, because the value ot — is a low number. c. 13 e = 27 the square root is 5-|-. The integer only is taken, and the fractions derived, as in the former 5 6 example ; they are — and — : the numerator of the last being squared, and the numerator subtracted from the square as before, there remains 30, which is greater than — =27; therefore these fractions suit the c intended purpose. Using the same process with the first fraction, there remains 20, which is less than —=27 ; and it is absolutely requisite that the two remainders should be respectively below and above the value of — . In this example, x is less than '^(— )j and greater than ^ {-t\ \ therefore x is less than 'V^3.6 = 1.897 + x'\^ greater than 'v 3.5 = 1. 870-|-. The two figures 1.8 must, therefore, be correct for a first near value of X ; but the proximity of the limits is not sufficient : by adding .6 to the numerator and denominator, respectively, of the first fraction, . , 5.6 jt becomes — . 6.6 Then 5.6^=31.36 subtract 5.6 h^ remainder =25.76, which is less than the value of -;- ; therefore x is less than ^^f-— -) = v^3.535714 + = 1.880 + , and the ^5.6 ' near valije of, and the proposed narrow limits for, the root .r have been found. Ex. Letx3-2x=l. This example is as unfavourable as possible to the method, and is selected on that account. c"- ■=8 the square root is 2 + 2 3 the fractions are -7- and -— 3 4 but it is impossible that the former should have place in this me- thod, because the quantity x^ — hx cannot rise so high in our case as. 14 J 3 -1 5 .2 — , nor can hx sink to ^- — . The fraction — must, therefore, be re- 4' 4343' jected ; and the two fractions - - and -r- must be employed, and thej^ 4 5 have the requisite properties : for 3^-3 = 6 4''-4 = 12 and the value of ,=8, lies between the remainders. It may be observed, that when the first fraction is rejected, the succeeding fractions will answer the purpose ; and it will not be ne- cessary to square either of the numerators, if we are satisfied with the limits obtainable by them. In the present example, x is less than "^ i — ^ = ^^f --j=1.63-{-, and greater than v^( -'') = v^f-- )=1.5811-H. These limits are not satisfactory, and since the first limit falls with- out our case, because x cannot rise to '^(-7-)} li^'t may be infinitely near to it, the quantity .2 is added to the numerator and denominator, 3 3 '^ respectively, of the fraction—, which becomes -^, and 3.2^ - 3.2 = 7.0i. The root x is therefore less than v^( — ^ =Vf ?:i^ ='v/2.625 = 1.620. ^3.2 I ^3.2^ In the same manner, the second fraction — is reduced to -'- ; and 3.5^ 5 4.5 3.5=8.75, which is greater than—. The root a: is therefore greater '''" ')=v2.571i-f = 1.60+. The figures ■8.4, . // 9 than V ^— --j=v'2.571i-f = 1.60+. The figures 1.6 are correct for a first near value of x ; and the very close limits v^( — ] and '^ [—. ) are obtained. There are two other necessary remarks, which will be made the subjects of the next rule. RULE VII. To determine when tl;e first fraction should be rejected, and to ascertain when the value of the root may be accurately found, by- means of one of the fractions employed, 2 Should the first fraction be -s^, always reject it, and employ tlie frac- 3 4 tions -J- and -r- The general rule will be elucidated by an example. 15 Ex. Let x^ -28.1=28. 5 6 The fractions which first appear, are-^- and — ^ 6 ^ . Attention must first be given to the second fraction, — ; for, when the second fraction answers, we may be certain that the first is correct. 6^ — 6=30, which is prreater than the value of — -, and the two fractions are to be used ; but a nicety is about to be pointed out, not mentioned before, lest it might embarrass the subject. Ex. Let.r3-30x=30. 5 6 The fractions which first appear, are — and -^ 6^-6=— = 39 The two fractions are correct, and are to be used, for the reason given in the next example. Ex, Letx3-30.7x = 30.7. 5 6 The fractions which first appear, are — and - - . . , l^' . 6* — 6=30, which is less than the value of — ; and it would seem, from what has been said, that the first fraction ought to be rejected, nevertheless these fractions are correct : for the general rule is this, that, representing the second fraction by -^-, if the value of g'^ —g is nearly equal to the value of — , but yet a little below it, then the value of g*— g4-"T~ must be found; and if such value is still below hi the value of -Y", then the first fraction must be rejected; but if such b^ value is greater than — , the second fraction is correct, c 6 In the present example, 6^—6+ =30.8+, which is greater than b ' the value of — - ; consequently the vakie of the root x is somevvhat, c though very little, greater than ^{-^)' 16 Ex. Let a3 -30.9^=30.9. ^ ^ The fractions which first appear, are -q- and -=- ; but the first miist be rejected, for the reason given, in the last example, respecting tlie second fraction; that is, because 36 — 6+ — is not so great as 30.9, which is the value of — in this example. '' 6 7 The fractions to be used, are -— and — - ' 7 8 The value of the root x is somewhat, though very little, less thart When the proper fractions have been found, the value of the root x is always less than the square root of the following quantity, viz. ^ ir—], — denoting the first fraction ; and it is always greater than * ^ J the square root of the following quantity, viz. '^^ — V -|- denoting | the second fraction. But this must be understood to relate to those equations only where the root cannot be accurately discovered by means of either of the fractions. We now come to the method of finding the value of the root x accurately, by means of one or other of the fractions. Ex. Let a3 -90.1 = 100. the fractions are — - and — ; 9 h^ but 9^-9+— =72.9= — 10 c- Whenever this agreement occurs, the root can always be found accurately. . ,. , //10x90x ^^ In this example, x=^ ( — - — 1=10, The value of the root might also be found, by multiplying the abso- lute term 100 by the denominator 10, and extracting the cube root; but it M^ould be attended with more labour. Ex, Leta3_72.9a:=72.9. 8 .9 Jj 3 The fractions are -r- and — j and -- = 72,9 9 10 ' (^ 17 a'=v/(l^^)=v/81=9 or, x=v 10x72.9 Ex. Let x^ - 17i =V6|Ij. = 72.9 The fractions are as before. 9 ' ^9 -^m=^m Let.r3-.7r=6. Ex c^ 9.527 The fractions are '-^„d^-' 4,5 5.3 3.5^-3.5+— =9.527 4.5 //4.5x7x / x=^( ) =V9=3. ^ 3.5 ' Ex. Let o:^ -9.527x = 9.527. The fractions are — -> and -z^_, and -^=9.527 3.5 , 4.5 , b' 4.5 "^^ 575' ^"^ ? =./(ii4^)=v/l2.25 = 3.5, Ex. Let 3^3 _40.875j;=4.2.875. The fractions are -— and — - 7 8 7 72-7-1 = 42.875 ..=./(f)='v^(2i?)=7, D 1& Ex. for the equation is equivalent to the equation 1^ — 81=8, Ex. Let x^ - (.25-.5 + -^Ja-=.083. ^ 1.5' x=.^. This example is not in the irreducible case. The only integral numbers that admit of such development, are 3, 2, 3, 4, 5, 6, and 8 ; tnat is, if we make x"^ —x-] -, equal to any other integral number, the value of x cannot be found ; and it is this inability which forms one of the chief obstacles to the reduction of our case. It is obvious, that an equation of the form x^ —bx=b, is equivalent :)-* x^ to the equation x^ -= ; ^ x^-^2x^-\-x^ {or • — -— =xK .r^+2x+l The principle on which the foregoing rule is founded, may be thus explained. Let ther^i be given for solution, the equation x' —bx=c, b^ b^ and let this be converted into another, of the form y^ y=z — ; the e-" c^ b' / /" x^ y. ^ bx value of y will be Vf c^ )= — ; and the value of x will be '^i- ], or x = --/-. Substituting the quantity n^ — n-\ for — , y ' b & 1 J ^^_l_j ^z> the second equation becomes n the value of y, the root of this equation, is n ; but the square root of the whole quantity, n'^ — n-\ , cannot be so great as w, because the value of the root ?/ must be greater than the square root of the coefficient. If we tate z as the square root of the coefficient, and as the numerator of a fraction, and add unity to z as the denominator, z z we get the fraction : but z^—zA r> is less in value than z-\-l z + 1 19 n^~7t-] ; and consequentlyj/ is less than vf ^ ' " " ' ,^4-1 j z n-i-lx n^ ~?i4- \ , «+l J, the quantity n ^ . ;/ w-f-1 X w^ — w+— --■ being equal to n^ . Again, if we take another quantity, zo greater than r, and so to that the value of w'^ — zoA. -, shall be o-reater than tlie value of a; + l' ^ 7 . w 1 .,, , ^ r\ .//'zo-\-lX7i^ — n-\ \ 7i^ — n-\ ; then j/ will be greater than V r S/-fl J, for this last quantity will be less than ii. In order to discover the limits of the root x, by means of those obtained for i/, if we put d=^, then x='/[^^4^) =v^(-^±iil^'), for d y . — =— ^ — ; and it results, that the limits for the value of x are obtaina- .y J/ + 1 ble by the application of the same quantities that are used to discover those for y ; because the quantity — y has the same ratio to y^, as bx has to JT^. A more detailed investigation would lead us into a field i too extensive for the purposes of this work. RULE VIII. To discover the roots of several equations, the value of tlie root jp of one equation being known ; the value of — - being the samein all the equations. Multiply the coefficients of the given equations by the square of I the root of the equation, the value of whose root is known, and divide the products by the coefficient of the same equation. The square roots of the quotients will be the roots required. Ex. Let the equation, the root of which is known, be X' -90^=100 x= 10 -=..0, D 2 Let the following eq^uations be solved by the rule. x" —x= — ,-— 90^90 100 i)V9i 1 00^^27 x'—Sx=- ,^ 90V90 .r3-72.9a:=72.9 The value of — , is 72.9 in each equation. 10, The value of x in the first equation, J* "^(-r) 30, of X in the second equation, is '^(-^) 9 729, of X in the third equation, is "^ ( ) =9. Hence it appears,, that the ratio of bx to r^, depends on the magni- i b^ . b^ . . ' tude of —-; for, as the mao-nitude of — increases, the ratio of bx to C^' ' ~ " O" - ~ ^2 x^ increases accordingly. This rule is applicable to all those equations of the form x^ —bx=Cf which can be solved by Tartaglia's rule, and also to equations of the form x^ ■^bx=c. RULE IX. To generate an equation in the irreducible case with any coeffi- cient, and with any root that is consistent with the magnitude of the coefficient. Divide the coefficient into any two parts, and make the absolute ; term equal to the product derived from the multiplication of the square root of one of the parts by the other part. Such equation will be in the irreducible case, except the square root of one-third part of the coefficient is taken ; for then the roots t and v will be equal, and the equation will be solvable by Tartaglia's rule. Ex. Let 17 be the coefficient selected. Let it be divided into the pans 4 and 13, and let 4 be the part, the square root of wliich is to be multiplied by the other part. The equa- tion becomes x^ — 17x = 26. *>1 j«W XL ^^+^ ^+^xi'^ -H^u ^= t=^U-l, 1=2. Let the coefficient be divided as before, and let 13 be the pari, tlic square root of wliich is to be mukiplied by the other part. The equa- tion becomes x^ — 17.r=:'^208. and, in general, if d is substituted for the part, the root of which is to be taken, and e for the other part, then x=- This property, independently of the great use it is of in the gene- ration of equations with the coefficient 3, frequently enables us to find the roots of equations very difficult to be solved otherwise. Ex. Required the roots of the equations .r^ — 2-2x=^1575. 'V^1575 = 15V^7 15+7 = 22 therefore i= — r^^^^ =V1.75+v 16.75 >> ;=v^l6.75-^^1.75, v=^^l. The reader will observe, that the roots t and v are given with posi- tive signs, because they are the affirmative roots of the opposite equation bx — i'=c; and this must be continually understood as we proceed. Having found the value of x, the values of t and v are apparent, for v ^ is equal either to t or to v ; and since t is always greater tlian X --, no mistake can occur. If t='^^d, then x—t = v; if v=^d, then x—v = t. Consequently, when x is discoverable, its value may always be found under two forms. Ex. Let x^- 71=6 d=\, e = 6 22 ^l+^^T+il .= 2 =3 ^"4+^/4 + 12 a= -^ =3. There is, however, no rule whereby we can find e universally ; such a rule would perfect the solution of the case. It is necessary to remark here, that x and v are at their greatest magnitudes, and t at its least magnitude, when the absolute term is - /- • , • 1 ./i'^^>\ ./( Jf \ 25 mnnitely near in value to ^ ( -— - ] = v f __ j x — , There is another very easy method of making an equation in the irreducible case, of the form .r^ —bx = i\ Take any two numbers fof t and Vf t being the greater number, and their sum for x : then will be the coefficient, 2 and xtv will be the absolute term. Ex. Let t= 6 t^= 36 v= 4 v^= 16 xtv==2i0 :r=10 a:* = 100 2)152(76 the equation becomes x^ -76.1=240 or, 7Qx-x^=2iO. The rule for the absolute term is well known ; but the expression for the coefficient is, perhaps, new. RULE X. To decompose any number or quantity into three squares, in an infinite number of ways, and in such manner that one of the roots shall be equal to the sum of the other two roots. Take any number or quantity as the coefficient of the equation x^—bx=c. Equations having such number or quantity as a coeffi- cient, may be infinite in number, for the absolute term may be infi- nitely varied ; but in every equation of the form x^ —bx~c, belong- b^ ing to our case, as well as in that where the value of - - is exactly 6.75,a'+^*+i>*=2&. 23 These squares cannot be discovered universally, yet they may be discovered in an infinite number of ways. The means of effecting it may be elucidated by an example. Ex. Let it be required to find 3 squares equal to the number 20. Take 10 for the coefficient of the equation x^ —bx=c. Take any number greater than v 6="^ 10, but not greater than '^ (— ] = n (-7-]; for the value of the root i. Let such value be 3.5, then z-' — 10j = 7.S75 x=3.-5 ^ = 1.7.54-^^.8125 r = 1.75-v^ftlOa j^ = 12.25 '^= 3.S7c :^= 3.875- 3.5-^.8120 t^= 3.875+3.5v^.8125 20 If any other numbers within the prescribed limits are used for the value of X, the squares mav be discovered bv means of the forefoino- Tule. RULE XL An equation of the form x^-\-dx=c, being given, to find an equa- tion of the form bx — i-=c, having the same root and absolute term. Find the root of the former equation bv the known rule. Then 2j^+<^=Z/. Ex. Let x5+ 4j=240 j=6 Let j^+41a=240 x=4 72+ 4 = 76 32+44 = 76, the resulting equation is 76x — x^ =240, ^ = 6, -c =4, and the absolute term is unaltered. From this connection between the two forms, innumerable and .therwise most difficult equations of the form bx — x^=c, may be olved ; and by the converse of the rule, very complicated expres- ions may be brought to a simple form. 24 Ex. + 9 V45+V_^ J V45+^__3^^^_^_^^ 4 27 2 27 for, let ar-x^=3 ^ = 1.5-^^1.25. To convert this equation into one of the form x'^ -\-dx'=c, put h-'iv'^—d^ the resulting equation is .r2+^45 + l XT=:3, the root of which is also 1.5 — v 1.25, and equal to the value of the preceding complicated expression. The equation Lx — x^=c has always a corresponding equation, ^3 _j_^^;^p^ with the lesser root r, and same absolute term: but it is not always so with t, or the greater root ; for the corresponding equa- tion having the same root and absolute term, may be of the forms J 3 j^dx—c, x^ =c, or x^ —dx=c; but always reducible in this last case. Ex, Let 103r-x2 = 198 r=2, ^=9 ^-2r^=95 5-2r = -59, the corresponding equations are a-3+95x=198, x=2 a3 -59a =198, a =9. Again, let Sx — x' =8, the corresponding equation is .r-'=8i hence, it results, that r, the lesser root of the equation hx — x^=zc _ 3/{^ /(6.7^c--lf-b^—l>tr-v^-l(-i2br-*—iii^ __ ^^[ /f6.7 ^c^-\-b^—b bh-^-\-l'2bv^- -8t' _ ^ 2 RULE XIT. To generate equations of the forms x^ —Lx=c, hx — x^ =c, with any coefficients, so that the roots t and v of equations of the latter form may have a constant ratio to each other, and to the root x of equations of the first form, and so that the values of — may be the same in all c^ the generated equations. 25 Take any whole number, surd, or fraction, for the coefRcient of an equation ; divide it into any two parts, d and e, by Rule IX. Make the absolute term equal to ^de ; then whatever ratio t and v have to each other, and respectively to r, in sucii equation, the same ratios will exist in all other equations of the same form where the coeffi- cients shall be divided in like manner, and the absolute terms made accordingly ; and the values of — will be the same in all the equa- i-i tions. Ex. Let the coefficient be 28, divide it into the parts 4 and 24, the equation becomes x'^ — 28x=4B. 28^ 48 2 = 9.527 r ^4+^4 + 96 X= =C) i) t = i, t7 = 2. Again, let the coefficient be 63. Take — thereof, as in the last example, for J, the parts are 9 and 54, the equation becomes x^ — 63.r=162. V9+V/9+216 „ x= ! = 9 2 The ratios are the same in each equation, and the values of — are alike. If the coefficient is divided into two equal parts, the value of b^ — will be 8. c* RULE XIII. To discover the roots of the equations Z£)i-^\^bzc''-^2bzc=c, z'-{-S^^bz^+2bz=c, ihe roots of the equations x^ —bx=c, bx — x^=c, being known. VvLtw='^b-v,^b-\-x,^^b-t, t^x-^b. E 26 The former equation has always three real and affirmative roots, when thus related to the irreducible case. The latter equation has but one such root. Ex. Let it be required to find the three roots of the equation the roots of the equations x^ — 9x=8, 9x — x^ =8, being known. ar=. 5 4-^^8.25, t ="^8.25 -.5, r = l : then k;=3.5 +'^8.25, 3.5-^^8.25, 2. It has been shewn in Rule II. that the quantity .r^ —bx is equal to the product of thi'ee numbers in arithmetical progression, their com- j mon difference being ^b. The three numbers in this case are, .5+^/8.25-3, .5+^^8.25, 3.5+^/8.25. The last or greatest of these numbers, that is, x-{-'^ b, is always one of the three roots of tlie equation w^ —S^brc^-\-2bu=c. Ex. \ Let it be required to find the root of the equation z3+3As"+18s = 8, the root of the equation x^—Qx — S being Icnown, and which is .5+^8.25: then 2=.r-v 6=^^8.25-2.5; luid consequently s is equal to the first or least number in the arithmetical progression before mentioned. Of these four roots, there are two which more peculiarly appertain to our case, and, by their complicated functions, stand in the way of the reduction of our px-oblem. The two roots are x — '^b and "^ b — t, or the least value of re. The sum of these roots is equal to v, or the least root of the equation hx — x^=c. RULE XIV. To discover the roots of the equation be~ ~e^ =c''-, the roots of the equation bx — x^ =c being known. Put e=Z'-P e=b — v'^ The equation belongs to the irreducible case, except when — is equal to 6.75 ; and it cannot have any real and affirmative roots, unless — is equal to or greater than 6.75, 6 27 Ex. Let the roots of the equation 28e^ —e^ =2304 = 48^ be required, and let the roots of the equation 28.r — .r^ =48 be known. t = i,v=2: then e = 28- 16=12, e=28-4 = 24. When the roots t and v are equal, then the roots e, e, are equal. It is by no means improper to say, that equations have civen roots ; for, in constructions of cubic equations, tlie roots are distinct even when they are equal. RULE XV. To ascertain, very frequently, by inspection, that the roots of the equations ,r^ ~bx = c, bx~x^ =c, may be accurately found. Compare the magnitudes of b and c. If you find c= b-l; then x=.5+^b-.75 :=v^, + l; thenfe -^— J- 2 + 2 and, in general, if it can be perceived that = 26-8 c = 3Z»-27 c=4/>-64 c=nb — n^ the roots may be found by Rule IX. ; that is, d and e may be dis- covered. Ex. Let x^ -71x=\SQ c = Sb~S^ d=9j e=62 ^=^^?±2^±iH^=I.5+v'61.25 2 <=v/64.25-1.5, 1=3. Another method of frequently discovering the roots, may be added. In the equation .r^ —bx=c, a» -j-l=v/(6+2 X x^ -i-cx-l-l) 216 71 remainder of —-=5. 6 28 The root of a proposed equation may not be discoverable by means of this formula ; yet by making use of it in respect of another equa- tion having the value of — , the same as in the proposed equation, the root may sometimes be found. Let the equation to be solved be Here the application of the formula will not succeed ; therefore make an equation having the absolute term equal to 2x, and likewise having the value of -r equal to 9.527. The equation agreeing with these conditions, is y^— 7?/ = 6. y* + l=v/9j/^+6?/+l = ay-fl : hence y=3. By "Rule VIII. theroot^oftheproposedequationisv( : ) This method will succeed in innumerable cases, all different from the present ; but it would lead us from our subject to dilate on it. RULE XVI. To find the cube root of a cube number, vvhen computed tables at hand are not sufficiently extensive. Divide the number by 216, and if there is a remainder, divide it by 6, and reserve the remainder of this last division. Find the cube number next below the first quotient in the tables. Six times the cube root of that cube number, together with the reserved remainder, will be the required cube root. Ex, Let the cube root of 7610498063 be required. :35233787, with remainder 71 The cube number next below 35233787, is 34965783, the cube root of which is 327. ' 6 X 327 + 5 = 1967, the required cube root. There are many other ways of effecting the same object, but this J is ver}^ expeditious, and, perhaps, more so than any other; besides, the same method with other divisors applies to all pure powers of numbers. 29 RULE XVII. To find expressions, similar to that promulgated by Cardan, for a first near value of the root of the equation x^ —bx=c, by means of the fractions made use of in Rule VII. Represent the greater of the fractions by -y- : then the first near value of X will be .,v,,M'+v^:l^^\,v,,^U-.^ ^h h b' 0-3 Wh T b^ ¥ If g* —g-\-y- is equal to — , the expression will be correct for the ft c value of the root x ; if it is only a little more than the value of b^ — , the approximation will be considerable ; but if it is much greater than such value, the approximation will diminish accordingly : but this may be avoided, because g^ ~'g+4~ "^^.y always be taken nearly equal in value to — -. In all cases, except when the expression is correct, the value found will exceed the real value of .r. Ex. Let a:3- 90a = 100. ^, ^ . 8,9 The tractions are 77 and — •Vio ' 10 729 V]o ' 10 729 Hence x=10, and the expression is correct. Ex. Let x3 -210.93749.r =210.93747. The fractions are rr+r^ 15 16 ^=.5/l6xV^ -+ V hi 16 3375+-^^1^^^It-1i6 3375 b^ Whoever will take the trouble to discover the value of the root x by means of this expression, will obtain many correct figures : but the use of such expressions is not recommended ; for the value of 30 the root may be more commodiously and accurately obtained by means of the fractions, agreeably to the means prescribed in Rule VII. The equation x^ ~bx = c cannot be solved generally by expressions similar to these ; the principle from which they are deduced, is as adverse as possible to the reduction of our case. It may be observed, however, that the near value of x obtained by the expression given in this rule, will never appear under an imagi- nary form, because the quadratic binomial will always be possible. RULE XVIII. To find a fust near value of the root of the equation x^ — bx=-b. The method of approximation about to be introduced, depends on the finding a first near value of the root of this equation : two means of effecting it, without trial or conjecture, will be given, and the I'eader will be enabled to select the most appropriate, when he has gone through the examples. If b denotes a number of considerable magnitude, solve the equa- tion X' —bx = b — i, the root of which can always be found, and you will have many correct figures for the root of the equation x^ —bx==bf Ex. Let y^ - 300000^ = 300000. Find the root of the equation :•" - 300000j='299999. 2 = .5+v^299999.25, by Rule XV. ; or, c = 548.2218 + . All these figures are correct for a first near value of the root ?/, and the same number of correct figures will always be obtained when b denotes a number as gi-eat as in the present example ; but to be cer- tain of the correctness of these figures, let a near value of the root^ be found by means of the fractions, rry f ,• 548.2 548.3 1 he tractions are — — and 549.2 549.3 for 548.2^-548.2 = 299975.24 548.3^-548.3=300084.89 j/ is less than ^^( ^^^•-^•'""""" )=v^300547.24553082 + =548.221 80 + ^ \ 548. i2 / ' ' .y is greater than ^^(^i^^^P^^)=A/300547.14572312+ =548.22180 + therefore these seven figures and the cypher are assuredly correct for an approximation to the value of i/, and they agree with those obr 31 tained for the value of z to seven places of figures; but those ob- tained for the value of 2 were found with much less labour. If the value of b is greater than 30, we shall get two figures of the root by solving the equation x^ — bx=:b—\ ; and as the numerical value of /; increases, so will a greater approximation take place. But a certain and most speedy method of finding a first near value of the root of the equation x^ ~bx = b, is as follows : To the root of b add .4 for such near value ; and that such near Value is correct, may be thus shewn r Let it be supposed that the value of the root x is "^ b -\- .5 : then a- =(5»^^/»H-1.5Z; + .75V'a + .1'25 subtract bx =b'^b-\- .5b leaves /* + .75v^6 + .12o but this remainder is greater than b, and consequently the value of x cannot be so great as ^ b-\-.5. And that the value of x cannot be so little as ^' b-\-.i, while the equation belongs to our case, may be thus shewn : The equation which limits this case, is x^ — 6.75a =6.75, where x=S ; but ^^6.75 = 2.59807 + adding .4 gives 2.99807 + which is not sufficient for the value of the root .r ; and since the root a, in the equation x^ ~bx=:h, must always be greater than 3, it fol- lows, from the nature of cube numbers, that the value of such root must be greater than '^£ + .4; but it is not a consequence that we should universally derive two correct figures of the root by this means :- the knowledge, however, of these limits is sufficient to enable us to continue the approximation with absolute certainty. There is another means of finding a near value of the root x, de- pending on trial or conjecture ; but the conjecture, or trial, lies within very narrow limits. In the equation bx~x^ =by the lesser root v cannot be so little as 1, nor can it be so great as 1.5, while the equation is in the irreducible case; for it is only equal to 1.5 in the equation 6.75^ — x^=6.75, which is the limiting equation, where the roots t and v are equ;d to each other. The equation 6.75 — zxr — T^ =6.75 — 3 is impossible, s being an in- finitely small number; but the equation 6.75+2 xx-x^' =6.75+= i^ 3i the commeiicemehtof the irreducible case, becaiise the absolute term x^ is less than — . The roots t and v are now of unequal magnitudes ; 4 t is greater than 1.5, and v is less than 1.5, continually approaching Unity, but yet not capable of arriving at it. The root t may be of any magnitude whatever, because there is no limit to the possible magnitude of b. Since the root v cannot be so little as 1, nor so great as 1.5, while the equation is in the irreducible case, we may avail ourselves of the knowledge of these limits, and by the aid of Rule IX. find a first near value of the root .r, which may be exemplified thus : Ex. Let x3 -80.1 = 80. Divide the coefficient into two parts, whereof one part shall be greater than unity, but less than 1.5, and in such manner, that the product of the square root of the lesser part and the greater part, .shall be nearly equal to the absolute term 80. Take 1.014 as the square root: then d= 1.028196 € =78.971804 80 Vf/e= 80.077409256. Make an auxiliary equation z^ — 802 = 80.077409256 , , I.OI4+V316.9OI2I6 then 2= -!-— =9.40785+. The four figures 9.407 are correct for a first near value of the root of the equation a:^ -80a: = 80; but the next figure 8 is in excess, the nearer value of the root x being 9.4077. As this method of obtaining a near value of the root x depends ori trial or conjecture, it must not be considered as forming an essential part of the method introduced in this work. RULE XIX. To find the value of the root of the equation x^-bx=^c, when the root is a whole number. This may be always easily effected. It has been shewn in Rule IL that the quantity x^ — bx, and consequently its equal c, is equal to the product of three numbers in arithmetical progression, their common 33 ditFerence being '^h, and the middle number being equal to x ; and we are enabled by Rule III. to determine whether such progression is connected with the irreducible case or not. Therefore, when an equation of this form is given for solution, and it is required to ascertain whether a: is a whole number or not ; then, if h should not be a natural square number, take the square root of the next rational square, either above or below the value of 6, and set it down as the middle number of the progression : find tlic pro- duct of the three terms, and if such product is near in value to the absolute term, add to it the product of h and such middle number, if the higher square root has been taken ; or subtract the product of h and such middle number, if the lower square root has been taken : should the result be equal to the absolute term, the middle number will be the value of x. The following examples will elucidate the rule. £.r. Leta3-80.r=200. Taking the square root of 81, the series becomes 1, 10, 19 : the product of the three terms is 190 add the middle term 10 the sum is 200 and x=10. Let the same equation be tried with the lower square i>4. The series becomes 1, 9, 17 : the product of the three terms is 153 ; but as this product is consi- derably beneath the value of the absolute term, each term in the series must be increased by unity, whereby a new series results, namely, 2, 10, 18, the product of which is 360, which is greater than the absolute term 200. Thesethree last terms are connected with the equation .i^ — Glx^rSGO, the root of which is 10 j and if from this product 360 we subtract 162 = 160 there remains 200 and consequently 10 is the root of the equation :r2 -80r=:200, 34 Ex. Let x'-990a:= 5614. The coefficient is not a square number ; taking the nearest rational square root, 31=v 961, as the common difference of the three terms, the serieses are, 1, 32, 63 2, 33, 61 3, 31, 65 4, 35, 60 and selecting that series, the product of which is nearest in value to 5644, namely, 3x34x65 = 6630 and subtracting 29 x 34= 986 there remains 5641 hence r = 3l. Ex. Let x^ - 134689^ = 435297600. The coefficient is a square number, and when that is the case, unity must be added to the square root, for the middle term of the progression. The first series is 1, 368, 735 : but as the product of the three terms is much beneath the value of the absolute term, it is evident that serieses containing much higher numbers must be made. Let such serieses be 100, 467, 834 200, 567, 934 300, 667, 1031 400, 767, 1134 500, 867, 1234 L ^he root is a whole number, it must lie between the numbers 767 and 867, because the product of the series, whereof tlie former is the middle term, is less than the absolute term ; and the product of the series, vv^hereof the latter number is the middle term, is greater than the absolute term. Confining the investigation to these limits, the series 450, 817, 1184 is obtained, the product of which is equal to the absolute term, and consequently a:=817. The equation does not belong to the irreducible case. 35 Ex. Let x5-60j: = 6800. The serieses in succession are, 1, 9, 17 5, 13, 21 10, 18, 26 12, 20, 28 the product of the last is 6720 : but the series is connected with the equation x^ — 64t = 6720 adding 4r=4x 20= 80 - the sum is 6800 hence a =20. The equation is not in the irreducible case. It is obvious, that the rule extends to equations of the form a:^ —bx=c, where the coefficients and absolute terms are surds : one example of this kind will be given. Ex. Let x^ - (ll+v'sOja— '^180+^/l50. The series becomes \/54.\/6_\/(ll-j-\/30), 'V^5+v^6, v/5+^^6+v^(l 1+^^30); the product of the three terms is equal to the absolute term : hence a:=v 5+'^6. F 2 36 EXEMPLIFICATION OF THE METHOD. It has been shewn in Eule VI I. that an equation of the form x^ —l}x = c, may be converted into an equation of the form ?/' ry= — , where x=~, or v k/j_ — _ ]. It has also been shewn, that \n an c~ b ^ y ' equation of the form i/= ~b}/^=b, such equation being in the irreduci- I ble case, the root ?/ cannot be so little as 3 ; consequently the quantity 1 , ,. , 1 cannot be so little as -—. y + i ■ 4 If, then, an equation of the form x^—bx = c, is given for solution, b^ b^ and it is transformed into an equation of the form j/^ — -^-?/=---, or c^-^ c^ substituting d for — , into an equation of the form y^ ~dy = d, the fol- i; lowing equahon results ; namely, The method is founded on this equation. When d is represented by a low number, then the equation 7/^ —dy^=d is solved with the greatest difficulty, because the quantity 1 . is immediately brought into action : whereas, when ^7+.4 = 34- Here one figure only will be taken, for the sake of an observation about to be made at the next step. Then, 7.1111 .75 6.3611 .2499= — —.0001, and this subtraction is made, because thevalueof ^ 1.1 6.6110, the root is 2.57 ^^1 cannot rise to — adding .5 gives 3.07 as a second value of y. . Again, 6.36111111 .24570024=^^ 4.07 "6.60681135, the root is 2.5703 adding .5 gives 3.0703 as a third value of y. Here it was necessary to take a fifth figure, because a cypher ap- peared; the work could not have been continued without a new figure. Again, 6.3611111111 .2456821364= — ^ 4.0703 6.6067932475, the root is 2.57036 adding .5 gives 3.07036 as a fourth value of y. Only one figure has been taken, in order to adhere to the plan on !,vhich the preceding accurate results have been found. 54 Again, 6.36111111111111 .24567851492251: 4.07036 6.60678962603362, the root is 2 5703675 adding .5 gives 3.0703676 as a fifth value of y. Presuming these figures to be correct, I take j:='v/(ri:l^]=:\/2L21110241001761 = 4.6055512;and these figures y are all correct, for ar=l+v 13. In the second original equation, w=.'^ {J \. 55 Ex. XIV. L€t.r3-9ar=20. This equation may be solved by Rule XV.; fox- x^ -{-1=^ [6+2X x^ +cx-\-l) x^-\-l=^[lQ0x^-{-20x+l) a?^ + l = 10x4-l j;=10. It may be also solved accurately by means of the fractions, and by the method prescribed in Rule XIX., but it is now to be done by approximation. — = 2352.98, the cotemporary equation isj/^ — 2352.98?/=2352,98 v'2352.98+.4=48.9+. Then, 2352.98 .75 2352.23 .020020=- 49.9 2352.250020, the root is 48.500 adding .5 49 as a second value of y Again, 2352.23 2352.25, the root is 48.5 adding .5 gives 49 as a third value of y. And here we must necessarily stop, because no new divisor can be obtained. It follows, that 49 is the exact value of y; and, taking ^=-r= TTir^l^j t^e accurate value of x is discovered, o 98 50 Ex. XV. let ^3- 401=24 23_.l2=.003 — = 111.1, in all these equations^ the cotemporary equation is 3/^ — 111.1^=111.1 v^lll. 11 +.4 = 10.9 as a first value of y. Then, 111.111111 .75 110.361111 1 .084033 = 11.9 110.445144, the root is 10.509 adding .5 gives 11.009 as a second value of y. Two figures were taken at the preceding step, because the work tould not have been continued without the accession of the last figure. Again, 110.36111111 .08327088=— i 12.009 110.44438199, the root is 10.5092 adding .5 gives 11.0092 as a third value of y. Again, 110.361111111111 .083269493388 = ? — 12.0092 110.444380604499, the root is 10.509252 adding .5 gives 11.009252 as a fourth value of y, a:='v/(?d:iili5)=\/43.63330769429203 = 6.6055512, which 1^ y correct in every figure. a;=|-=3.3027756 w z=^= .33027756 57 Ex. XVI. Letx3-ar=10. This equation was proposed foi" solution by Cardan to Tartaglia. c" =7.29 tli€ Cotemporary equation is y^ — 7.-29// = 7.'29 '^7.29+. -1 = 3.1 as a first value of i/. Then, 7.29 .75 6.54 J .243902 = 4.1 6.783902, the root of which is 2.604 adding .5 gives 3.104 as a second value of y. Again, 6.54 .2436647173 = -!- 4.104 6.7836647173, the root is 2.60454 adding .5 gives 3.10454 as a third value of i/. Again, 6.54 .24363266041992 = 4.10454 6.78363266041992, the root is 2.6045407 addinij .5 gives 3.1045407 as a fourth value of ?/. These figures are all correct, and taking x='^(-^^^!^^ ] =3.4494897; y these figures are also correct. If the operation had been continued without a new divisor, one if not two more correct figures would have been obtained, for it has been tried. The roots of both these equations may be correctly found ; and as they are connected vvith the ancient history of our case, it will be shewn how they may be discovered. I 58 4 Take, by Rule IX., -^of the coefficient of each equation for i, and the remaining —ths for e. Then, c?=4, e=5 for the case jt, £?=3.24, e=4.05 for the case y, a:=l+V6, y=.9+v^4.86, and there are two other real expressions for the values of these roots. /=2, i;='^6— 1 for the case x. There is no other equation of the form >r^ —6j:=i+l, 5 being a number consisting of integers onlj^, that can be accurately solved^ the equation being in the irreducible case. ) I : I ) Let.r'-.8x=.02 3 jyj _\/ 1280a' =1 59 Ex. XVII. — = 1280 the cotemporary equation is y^ — 1280j/=1280 v^l280+.4 = 36.1. Then, 1280 .75 1279.25 .0269= ^ .. 37.1 1279.2769, the root of which is 35.76 adding .5 gives 36.26 as a second vahie of i/. Again, 1279.25 . .02683843 = 37.26 1279.27683843, the root is 35.7669 adding .5 gives 36.2669 as a third value of y. Again, 1279.25 .02683346^475 = - * 37.2669 1279.276833463475, the root is 35.766979 adding .5 gives 36.266979 as a fourth value of y. Presuming these figures to be correct, I take x=v^i ~)~ v^.8220586335575400 = . 90667449, which value is correct in every figure; for a:^ -.8x = .019999997+, using the discovered value. The value of the root w may be discovered to as many figures, by taking w equal to \^(. yTl X^12 80 J . Neither of these three equations has accurately discoverable roots. Letx'-16.25r=l. 60 Ex. XV III, c* =4291.015625 the cotemporary equation is y' -4291.015625y=4291.015625 ^/4291.01 +.4 = 65.9 as a first value of y. Then, 4291.0156 .75 4290.2656 .0149=- ^ 66.9 4290.2805, the root of which is 65.5002 adding .5 gives 66.0002 as a second value of t/. All these new places were necessarily taken, on account of the appearance of two cyphers ; the work could not have heen continued without a new final figure. Again, 4290.265625 .014925328581 = ! — 67.0002 4290.280550328581, the root is 65.500233 adding .5 gives 66.000233 as a third value of y: , ,. , . . ^ . 65+v^4489.0625 and this value is correct in every figure, tor ^= 2 y-v/^y+lxl6-25j _2+\/4.o5=4.06155281+, of which every figure is correct. In this example the coefficient is divided into sixty-five equal parts, whereof sixty-four are taken for d; and the remaining part, namely, .25, is taken for e, by Rule IX. Let x^ - 16.5:r=2. 61 Ex. XIX. —=1123.03125 the cotemporary equation is y^ - 1123.03125^=1123,03125 V'l 123.03 + .1 = 33.9 as a first value of y. Then, 1123.03125 .75 1122.28125 .02865 ^ 34.9 1122.30990, the root of which is 33.5008 adding .5 gives 34.0008 as a second value of y. Three places of decimals were taken at this last step, for the rea- son given in the last example. Again, 1122.28125 .028570775525= ^ 35.0008 1122.309820775525, the root of which is 33.500892 adding .5 gives 34.000892 as a third value of y. These figures are all correct, for the exact value of y is 33+^1225.125 —^ — =34.000892+. ^_^/3r+l2<26^\_2^^^ 5^^^2132034+, which figures are also ^ y > correct. In this example the coefficient is divided into thirty-three equal parts, whereof thirty-two parts are taken for c?, and the remaining part for e, by Rule IX. Let j^-280x=792. c^ m Ex. XX. =34.996il299357208+ the cotemporary equation is ?/3 - 34.99641299357208y =34.99641299357208 ^^34.99+. 4 = &.3 as a first value of y. Then, 34.996412 .75 34.246412 .136986 = - 34.383398, the root of which is 5.863 adding .5 gives 6.363 as a second value of y Again, 34.2464129935 .1358142063= ^ 7.363 34.3822271998, the root is 5.86363 adding .5 gives 6.36363 as a third value of y. Again, 34.24641299357208 .13580258649606=- * 7.36363 34.38221558006814, the root is 5.8636350 adding .5 gives 6,3636350 as a fourth value of y, Prestiming the work to be correct, I take jr='/(-^±liil5?)=V324.000009428573= 18.000000+, which is cor- rect ; for x is exactly 18. If the work had been continued, more cyphers would have been obtained. The student is requested to give particular attention to this example, and to take the value of — to a greater extent, in oi'det tp continue the operation. 63 Ex. XXI. Letx5-9i=8. —=11.390625 the cotemporary equation is t/^ — 11. 390625j/=l 1.390625 V1I.39 + . 4 = 3.7 as a first value of 1/. Then, 11.390625 .75 10.640625 .212765= A, 4.7 10.853390, the root of which is 3.291 adding .5 Again, 10.640625 .2085940758: gives 3.794 as a second value of 1/. 4.794 10.8492190758, the root is 3.29381 adding .5 gives 3.79381 as a third value of ?/. The three new figures are correct. The final figure at the first step was in excess. Again, 10.640625 .20860234343872=- ^ 4.79381 10.84922734343872, the root is 3.2938165 adding .5 gives 3.7938165 as a fourth value of y. ^ . , , ^ r 1 cy 8x3.793^165 Presuming that these figures are correct, 1 take x=—= y =3.3722813+, which figures are all accurate ; for x = .5-'-^8.25, or, , J . V^ [8.5 -V 8.25 + Wl0.5 + ;i^^b«.-J5 by a second expression, x= ! 1 — ! ■ It is of considerahle use to know, that there are always two expres- sions equivalent to x, whereof one is much less complex than the other, and consequently more easily brought into figures. The singularity noticed in Ex. III. occurs in this present example; that is, the decimals of a* and x are the same. 64 Let x^ — Tx=l, Ex, XXII. ^=343 the cotemporary equation is r/^ — 343y=:343 v^343+.4 = 18.9. Then, 343 .75 342.25 .0502 = 19.9 342.3002, the root of which is 18.501 adding .5 gives 19.001 as a second value of y. The appearance of a cypher made it necessary to take the last figure. Again, 342.25 .04999750=—-— 20.001 342.29999750, the root is 18.5013 adding .5 gives 19.0013 as a third value of y. Again, 342.25 .049996750211 20.00J3 342.299996750211, the root is 18.501351 adding .5 gives 19.001351 as a fourth value of y. Presuming the work to be correct, I take x=^^^—. j? or a:=^ = 2.7144787, which is true in every figure; for j^-7.i = 1.- 7 .9999993+, using the discovered value. 65 Ex. XXIII. Let x^ - 77'284x = 8013128. This equation was formerly solved by the learned Mr. Raphson, and is on that account peculiarly worthy of the attention of the reader. —=7.18893596637029 + the cotemporary equation is ?/^-7.18893596637029z/=7. 18893596637029 'V^7.18 + .4 = 3-|- as a first value oft/. Then, 7.18 .75 6.43 .24=A_.oi 4 6.67, the root of which is 2.58 adding .5 gives 3.08 as a second value of ^. Again, 6.438935 .245098 = — 4.08 6.684033, the root is 2.585 adding .5 gives 3.085 as a third value of //, Again, 6.4389359663 .2447980416=-^ 4.085 6.6837340079, the root is 2.58529 adding .5 gives 3.08529 as a fourth value of y. Again, 6.43893596637029 .24478066428576= 1 — 4.08529 6.68371663065605, the root is 2.5852885 adding .5 gives 3.0852885 as a fifth value of ij. K 66 Presuming the work is correct, I take a:='/(?^^iIZ??f) = '/l02333.1971820463 = 319.89560, and these y ' figures and the cypher are all accurately true ; for 319.89560^ =3-2735938.782247 + subtracting 77284x1 = 24722811.5504 there remains 8013127.231847 The values of the roots t and v will now be given by means of the discovered value of x ; and for this purpose Rule IV. will be em- ployed, and the equation xt-t^=x''-h solved. ^and _319•8956O+^^[iO2333.1948098b- 10019(5.77959744 o = 159.9478+^(534.10382548 using the sign +, ^ = 183.058486 , r= 136.837114 using the sign 319.89560 See Part II. of Trend's Principles of Algebra, pages 44, 45 ; edit. 1796. 67 Ex. XXIV. Let j^^^- 1000.1 — 174. See Frend's Algebra, Part II. page 42, where this equation is quoted from Raphson. — = 33029.4622803540 c* the coteniporary equation is ?/2 - 33029.4622803540// = 33029. 4622803540 ^^33029.46 + .4 = 182.1 as a first value of y. Then, 33029.462280 .75 33028.712280 .005461- ^ 183.1 181.738 33028.717741, the root is adding .5 as a second value of gives 182.238 V' Again, 33028.7122803540 .0054573832-. ^ 183.238 33028.7177377372, the root is 181.73804 adding .5 gives 182.23804 as a third value of i/. Presuming the preceding work to be correct, I take -,,_v/( : taking the discovered value of y in the second equation given for solution in this present example, then 2.7647^=7.64356609 subtract 2.7647 4.87886609 y adding , = .7 34374 gives 5.61324 + . 77 EQUATIONS In the Irreducible Case, njlth correct approximating Answers, for the Stude/tt''s Practice. X' - 3x=l Answer a = 1.879385241 x^ - 4t=I 1=2.1149075 x' - 5j=1 .r=2..3300587 x'i - Qx=l .7=2.528917957 x^ - 7x=il x=2.7144787 x^ - 8x=l j:=2.8889694 x^ - 9x = l 2=3.0540842 r3-10a = l a:=3.2111393 x3-llx=l 1 = 3.36117758 x'-Ux=l T=3.5050397 x3-13j=1 1=3.6434143 a^-14j:=l 1 = 3.7768729 a^-15j = l 1=3.9058959 .r^- 16.1 = 1 1=4.0308912 x'-l'!x=l x= 4. 1522085 a- -18.1 = 1 1=4.2701503 x^-19.r = l 1 = 4.3849801 .t3-20x=1 .r= 4. 4969293 X'- x= .273 1 = 1.1156603957 x^ - 2t= .573 1=1.540143877 x^ - 3x=1.125 1=1.895643923 1 J - 4x=2.625 r=2.270690632 a 3 - 5x = 4.125 1=2.570027472 Find the roots t and v of the opposite equations, Z>x — i'^ —c. .and. = ^-^-V^ ft6-3i- '•" 2 the sign + belonging to t. 7S REMARKS ON THE PRECEDING DIAGRAM. (-CONSTRUCTIONS aftord little or no assistance to approximations, but they frequently exhibit the nature of the difficulties which prevent a perfect solution. The accompanying diagram is an accurate repre- sentation of the irreducible case, as connected with the equations x^ —hx=-Cy hx — x^ =c. It may be familiarly explained thus : Let the line A B, revolving about the fixed point A, be supposed to generate the semicircle BED; let the line B F be supposed to accompany A B in its progress, and to continue in the same straight J line with it ; but B F, exterior to the circumference, previously to its motion, must be supposed, at the instant of motion, to insinuate some ; infinitely small portion of its length within" the semicircle, and to I receive there some magnitude or increment in a direct line towafds the fixed point A. Let such motion be supposed to continue, and such insinuation and accession of magnitude constantly take place, so as to generate the curve B G A by means thereof, in the same ^ 79 time that the semi-circumference B E D is generated. Let the ex- terior ])oint F be supposed to generate the curve F I H in the same time. Under these circumstances, let the progress of the motion be sup- posed to stop at A M L K ; then will the line K M, bounded by the exterior and interior curves, repi'esent the root x of the equation x^ —bx=c=chord L B ; and the lines K L, L M, meeting at the semi- circumference at the point L, will respectively represent the two roots if and v of the equation bx — x^ =c = choYd L B : and a similar representation will take place at whatever situation the pi'ogress may be supposed to stop, the absolute term, equal to the chord at such situation, increasing or decreasing accordingly. Again, let the progress be supposed to arrive at A D H ; there the two roots t and v are equal, being each equal to radius, the line A H=:x being equal to their sum ; and that situation is precisely the boundar3' of the irreducible case : but we know that the boundary is not included in the case. Whatever numerical magnitude may be assigned to tlie line B F='^b, the diagram can undergo no alteration ; for the ratios which it comprehends, are constant under all possible change of numerical magnitude that may be assigned to the line B F. The line B F is the parent, if it may be so termed, of this case, and it represents the equations x^ —bx=0, or bt — t^ =0 ; so that both X and t are equal to the line B F=v Z». The length of the line B F is that of the chord of 120° of a circle, of which A B is radius. The irreducible case is now about to commence, for at the instant A B F moves forward under the circumstances before mentioned, the root V comes into existence. It has been usual with writers on trigonometry to consider radius as unit}-, and a happy coincidence attended the assumption ; namely, that at the first boundary alluded to, the value of the root x and of the absolute term were each equal to 2, or to the length of the diame- ter. The same assumption may be now used. If, then, radius is called unity, B F is equal to "^3, and the equations, represented by the line B F, are x^^ox—.0,St—t^=0; so that x and t being each equal to '^3, are equal to each other. At the instant of motion the irre- ducible case begins; for .r has received an increase of magnitude; t, has lost part of its magnitude, being no longer equal to ^3 ; v is 80 generated, and is accurately equal to the sum of the increase of x, and of the decrease of /. If the increase of x is called z, and if the decrease of t is called w, then V is equal to z and re, and the following equations result : z'-\-S'^bz^- ■i-2bz=c. Here occurs the first difficult}- of our case ; for, in order to obtain the numerical value of v, without which neither x nor t can be accu- rately found, the roots of both these equations must be obtained : but this cannot be generally effected ; for by exterminating the second terms, the equation x^ —bx=c is constantly produced. Returning to the equations x^ — 3.i = 0, 3^ — ^^=0, let the increase of .r be supposed equal to .1, then the following equation arises : x^-3r=.03^^3+.601 where j='^ 3-f .1 : but in the equation 3^'--j;2==.03v 8-f .601, the values of t and r are,. X ^^{i b-'^x'' T- T the sign -f belonging to t. Hence the increase of x is considerably less than the decrease of /, and V is equal to the sum of such increase and decrease. But there is a limit to the decrease of t and to the magnitude of r, which limit takes place when they become equal to each other ; and beyond this limit their existence is impossible. There is, however, no limit to the mascnitude of x. The equations under consideration, are supposed to have the con- stant coefficient 3, equal to the square of the chord of 120° ; the limit beyond which t and v cannot exist, takes place when the equa- tions become .r^ — 3x=2, 3x — a'^=2, for then t and v are each equal to unity ; H D being equal to t, and A D being equal to v ; and this is the only place in the diagram where they can conjunctively exist without appertaining to the irreducible case. In this state, the whole increase which x has received, is 2 — ^3; the decrease which t has experienced, is '^3—1 ; and v has risen from an infinitely small quantity to 1. The curve F I H has ap- proached the semi-circumference in the ratio of the decrease of / ; the curve B G A has continually approached the fixed point A, and ultimately fallen into it, according to the combined ratios of the de- ct.ei^s^ of t and of the increase of a\ But there is another curve 81 I deserving the attention of the reader, although it is not introduced into the diagram. This curve may be supposed to originate at the point B, to be formed by the continual union of z and w^ and to separate the root v into two parts. These three transcendental curves might be drawn with all possible accuracy, but I see no particular use it would lead to. The diagram itself might have been well spared in this w^ork ; and indeed it has been introduced only for the sake of making the reader familiar with the constitution of the case in question. It may be observed, that when the root t is perpendicular to the diameter, it is equal to the chord of 90°, and the value of — is exactly 13.5 : consequently, one half of the equations of the form x^ —bx=c, while in the irreducible b^ case, have their values of — less than 13.5 ; and these equations are the most difficult of solution : the roots t and v continually approach b^ nearer to each other in value, as the value of — approaches its li- mit 6.75. The difficulty decreases as the equations approach the point B ; for the value of v decreases rapidly, and ultimately vanishes : and yet a great facility of solution does not take place until the absolute term, or chord by which the absolute term is represented, decreases to nearly 20° ; for at the chord of 60°, where the equation becomes b^ a 3 — 3x=l, and where the value of — is 27, the solution is effected with only a little more ease than where the value of — - is only 13. On referring to the diagram, and applying radius from B to the cir- cumference at N, B N is the chord of 60° ; GO is equal to a, N O to /, and G N to r, which ai'e the roots of the equations x^ — 3x=l and 3x — j^=l, or of any other equations of these forms where the values of —^ are equal to 27. Perhaps the reader would wish to know my opinion on the possi- bility, or probability, of a reduction of the case. The impossibility of a reduction has not been proved hitherto, and I think the epithet irreducible improperly applied to the case : it may therefore be possible ; but the probability is not great. There is no Mecsenas to encourage men in such arduous undertakings ; and even some societies, from which we might expect the most brilliant disco- veries to emanate, have, in consequence of incapacity of councils, M 82 or inattention to science, degenerated into emporiums of ease and indolence. I should not notice here, were it not for the public good, that I wrote a letter to the Secretary of the Royal Society of London, offer- ing to communicate the method of approximation which I have given in this book, and requesting to know if such communication would be agreeable. My letter reached the Board, but no answer was re- turned. I would ask the managers of this societ}^, how they can expect that the chief end of their institution should be fulfilled, namely, " To make the way more passable to zchat remains wirevealed,^^ , if they disdain the offerings of thinking men ? It is probably ow- ^ ing to similar neglects, that the Philosophical Transactions have been of little importance for many years past. This concerns the reader, and I entreat him to look into Dr. Button's celebrated Mathematical Dictionary, a work with which he cannot dispense in his library, and under the article " Royal Society," 1 he will find that learned man's opinion on the present state of the society ; and will be convinced, that the preceding remarks have been made only with a view to the public welfare. Now, little book ! go forth into the world without fear or aj)prehen- sion, for thou art engaged in the cause of truth. Present thyself to those whose lamps are always burning. Enter into the study and retired room ; but tarry not among those archives where the works of the renowned Abraham Sharp are mouldering in obscurity. If thou visitest the colleges, perhaps thou mayest be greeted ; but if a more retired life should please thee, thou mayest seek the parterres of Argyle, and perhaps be welcomed by some gardener's boy ; or, perchance, some shepherd may take thee to those lofty fells where thy master made thee, and place thee on the honied ling while his flock is feeding. Be not ambitious, and farewell ! 83 APPENDIX. A METHOD OF AFFORDING SOME FACILITY TO THE RESOLUTION OF QUADRATIC EQUATIONS. 1 HE resolution of quadratics being perfect, all that is attempted here is, to afford some convenience in the operation. By freeing the terms containing the unknown quantities from fractional or high coefficients, such facility will be obtained. This may be done in the following manner : Let ax' -\-bx=c putting ?y = putting z =6rjc y h r +h = ac z ^+s- ac ~b^ and , hz consequently — =x Ex. Let4.r=+3.r= =22 r+^= = 88 z^--\-z = 88 9 8 bz ^ — o x= :2 Equations of the preceding form have onl}' one real and affirmative root; and therefore there can be no other number than 2 that can answer the conditions of the equation 4.r^ 4-3x=2-2. Let ax^ —bx=c putting ?y=:cfa: ^/^ — bi/=ac —y putting z=-^ z'^ — z—T ac yi. V Z^ J .1 ^^ and, consequently, — = .r. 84 Ex. Let 4:r*-3j:=370 y^-3y=1480 1480 ~2 „ Z — ,4, 9 40 hz 2 = — a: = — = 10. 3 Equations of the preceding form have only one real and affirma- tive root; and, consequently, there can be no other number than 10 that can answer the conditions of the equation 4x" — 3j: = 370. Let ax — bx^ = c putting y=bx ai/ — 7/^=bc V putting 2=^- a 2-2^= — - az msequently, -f=x Ex. Let 10j:-2,r' = 12 10.'/ - r = 24 2 - 2" = .24 2=:. 4 X = ='':-2. i But equations of this form, when possible, have always two real and affirmative roots, and their sum is equal to the coefficient of the term which contains the unknown quantity, when the coefficient of the term containing the square of the unknown quantity is unity ; but the coefficient of 2* is 1, therefore another value of z is .6; and, con- sequently, another value of x is ^=3. 85 Ex. Let 40jr- 3x^ = 125 375 15 2 - 1600 64 3 40z , ^=8-' x_3_5 5 40z 200 25 ^-8' ^-3 - 24 3 Since equations of the three forms, having coefficients of any magni- tude, may be thus easily converted into other equations, having unity for the coefficient of each term, the three following known rules for their solution, may be applied, in order to find the value of z, by which means some length of operation may be frequently avoided : When the form is az — z^=c, z= " — When the form is z* +oz=c, '^ (a^-\-ic a -=—r, Y When the form isz^-az=c, a+^[«^+4c ^~ 2 If the reader will solve the following equations in the usual man- ner, and also by the method now given, he will be enabled to judge whether the pi'oposed convenience has been afforded or not. Ex. Let 3x"-f-5x=:182 y y^-f-5j/=546 ^V^25+2l8l_2.5 = oi x=—-=7. 3 Ex. Let =22— 2 3 6 V 1 JL -L's/f I _|_ 5 3 2 1 20 x=2y=7. 86 Ex. Let - ^ =25.2 6 ^10 •^ ^10 6 z^ + s = 420 '=20, .y=j^=2, a=6y=12. £r. 3i Let -^ — [-I- =62.5 o 4 .y'+^=37.5 ■1 z^ +z=()00 z=24, y=f- = 6, a=^:^=10. 4 3 Let -—-6.1 = 12 y- — 6{/ = 51 y=10.74506+, a-=i^ = 2.528+. Ex. 6r^ Let — -5x = 10 ?y^-53/=6 J/=6, ^=B^-=io. , 4fr^ X Let-^ 1- — = 30 y"+^=52.5 ^2 z^ + z=210 ^=14, 5/=|- = 7, j:=^x7 = 4. Er 87 Ex, Let— +-2=10-5 3// 9 28 z=y, i/=hz=2, a=^=3. Let-— — (-1 = 1 . 4y 2 THE END. E R R A T J. Page 24, line 6, read ( ^45 + l) X *• Page 63, line 6 from the bottom, in a few copies only, read ♦' (8.5 — t'b.a5. Printed by L. Harrison & J. C. Leigb, 373, Strand "^ w g i€ RNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY Of THE UNIVERSITY OF Gl y^'SS?^ <5^^ ^ UNIVERSITY OF CALIFORNIA LIBRARY BERKJELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. -^ NIA LI nDec'5£JL,| DEC 4 ^95S CAUF. 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