o -< r- m tig UC-NRLF $B 543 MbD 00 00 fu , T^.. )rtk. \ EXTRACTION REAL ROOTS NUMERAL EQUATIONS DENOMINATIONS. BY WILLIAM HOYLE. IHO LONDON: PRINTED FOR ALL THE BOOKSELLERS, and MAY BB HAD OF W. DEAN, MANCHESTER, AND J. WBSTALL; ROCHDALB. 18^6. CAJORl INTRODUCTION. X HE following small treatise being chiefly intended fw: those who have already made some advance in the science of algebra, it will only be necessary to inform the general reader, that the extraction of the roots of an algebraic equation, and particularly the solution of the irreducible case in cubic equations, has been assiduously sought after during the last four centuries, by the most eminent mathe- maticians of Europe ; amongst whom the following names {with regard to the subject of the present work) stand pre- eminent, viz. Scipio Ferreus, Nicholas Tartalea, Hierony- mus Cardan, Lewis Ferrari, Raphael Bombelli, Vieta, Albert Girard, Harriot, Oughtred, Descartes, Sir Isaac Newton, Maclaurin, John and James Bernoulli, Fontaine, Euler, Waring, Simpson, Legendre, and Lagrange, To the more advanced mathematician, who is already acquainted with the different methods employed by the above mentioned eminent persons, it will be necessary to give a demonstration of the system by which all the real roots (either positive or negative) of a numeral equation, con- taining only one unknown quantity, may be extracted. iv INTRODUCTION. In the following examples a number, as near the real root as possible, is assumed, which is placed in the quotient or root ; the said number is then involved to within one power of the given equation, and the different powers multi- plied by their proper coefficients, the products are then col- lected into one sum, due regard being had to the signs + or — , which sum being placed before the numeral (to which the equation is equal), may be supposed to act as a divisor in common arithmetic, the numeral as the dividend, and the root which has been involved as the quotient ; multiply the divisor by the quotient and it will be the same as though the root had been involved to the proper powers of the given equation, as may be seen by the following ; viz. (i?* — ^-i^'-J- ix^-\-5x'\- 1 ) X A'^i-t'^ — 'Zx"^ +4^-3 -|-5j?^ -^x; the product of the divisor and quotient is now subtracted from the dividend, the remainder will be the difference between an equation of the same order and coefficients, as the given equation, with the assumed number for its root; to the remainder bring down as many ciphers as there are units in the highest power of the given equation; another figure must be placed in the root, and managed according to the following formula. The first figure in the root having had its powers extract- ed, may now be considered as known to be a part of the real ^'oot, and therefore may be represented by a, the second fi- gure, which is not yet known to be a part of the real root, by .r ; the first figure standing to the left may be consider- ed as a tens figure and the second a unit; from whence the present figures in the root may be represented by a'\'X; we 4nust new involve this binomial to the highest power of tiic INTRODUCTION. given equation ; thus, let the given equation be of the fourth order, or y^-^by^ — cy^ — dyz=zO^ and the fourth power of a +^=a*+4a'.r+6a^:i;^+'^^^''+^^; ^" which we may con- sider a^ to have been already extracted, and as the sum will have to be muhiplied by x at last, we shall have 4a3-f-6a*x +4a2"^+a;', but as a may be considered to be ten times the value of the figure by which it is represented, we have, in the following examples, begun with a?', and retreated one figure tothe left, it being the same way that is practised in common numeral multiplication, in order to save the repetition of ci- phers : now, as i/^ has only 1 for its co-efficient, the sum of 4a3 4-6a^a?4-'la^+^^ ™^y ^^ now left to represent t/*. Then a^\^\^ =a^ -^-^a^ x-^-Zax^ -\-x^ where the same allow- ances for a 3 and a lower power of :r will have to be made as before, whence we shall have 3a^+3a^-f-;r*, which (as y* is multiplied by 4-^) must be multiplied by i, and then 3a2Z>+ ^ahx-^-hx^ will represent +^^^* Then a'{-X{^=.a'^-{-2ax^x^ where, proceeding as be- fore, we have — 2ac — ex to represent — ct/*. Then aJ^x will be merely — d!, to represent — dy^ there- fore we shall have Aa^ '\'Qa^ X'\'\ax^ -{-x^ 2a^h+2ahx+hx'' — 2ae — ex —■d which, added together, will give the second divisor. But in the addition of numerals it will have to be ob- served, that a: is a decimal, and that therefore as many ci- phers must be prefixed to the right of — d as there are units ▼1 INTRODUCTION. within one in the highest power of the given equation ; i& the above case it will be 4 — lz=3, the number of ciphers; and generally let n represent the number of decimals in the root and m the number of units in the highest power of the given equation, then the number of ciphers to be added will be 3/*=m— lX» j/*=m — 2xn y3=rm— SXWj&c. whrch will be readily seen by adding together the root, square, cube, &c. of a decimal, as .2 .02 .22=. 04 .02«=z:.0004 .2»=.008 .023=. 000008 .2*=. 001 6 .02*=. 000000 16 where, to make an equal quantity of decimals in each line, k wrll be necessary to add 3 ciphers to .2 two to .04, 1 to .008, 6 to .02, 4 to .0004, and 2 to .000008. When the second divisor has been multiplied by the last figure in the root and the product subtracted, to the remain- der as many ciphers must be brought down as before, and every thing brought on in the same manner. The two figures in the root will now be known to be a part of the real root, and therefore will be a, the figure next put in the root will be x. After the first two or three divisors have been got it will be easily seen what the next quotient figure will be, as the significant figures, or those to the left in the divisors, will not then vary much. INTRODUCTION. VU This treatise would not have been published in its present form, if the author could have got it inserted in any of the periodical publications. _ThejoIul^^ case in cubic equations was sent to the editor of the Me- chanics' Magazine, London, on the 1st of August ; its receipt was acknowledged on the 13th of August, but, as it has ne- ver yet appeared, what use the editor has made of it is not known. The solution of a cubic equation was sent to the editor of the Kaleidogigij^, Liverpool, on the 12th of Sep- tember, and appeared in it on the 4th of October ; but the editor, in a note, November 1st, declined inserting, for the present, any thing more in mathematics. OldhamyDec. hi, 1825. Ml-.VK SOLUTION EQUATIONS, CONTAINING ONLY ONE UNKNOWN QUANTITY. Example I. Given x^-^-l 5x^+63x^50=0. 1X1=1 1 +63 —15 49)50(1.028=1: 49 64 —15 357604)1000000 —15 715208 49 1st divisor. 35425744)284792000 283405952 1386048 2X2= 4 3x10x2= 60 3x10x10=300 202 —15 —3030 30604 + 630000 +660604 —303000 357604 2nd divisor. SOLUTION OF EQUATIONS, 8X8= 64 2048 3X102X8:^ 2448 —15 3x102x102=31212 3145744 +63000000 +66145744 —30720000 —30720 35426744 3rd divisor. Solution 2. or' — 1 5x^ +6Sx=:50 9)50(6.576=^: 54 6X6=36 6 — • 63 —15 —725;— 4000 _ >_ —3625 99 —90 : —90 —49301 )— 375000 — —345107 9 1st divisor. -4577004)— 29893000 —27462024 —2430976 5X5= 25 125 3x6x5— 90 —15 3x6x6=108 • —18750 11725 +18025 +6300 + 18025 — 725 2nd divisor. SOI.UTION OF EQUATIONS. 7X7-= 49 3x65x7= 1365 3x65x65—12675 1307 —15 —1960500 + 1911199 —49301 3rd divisor. 1281199 + 630000 + 1911199 6x6=1 36 3x657x6;:i: 11826 3x657x657r=:1294947 13146 —15 1Q71Q0000 129612996 + 63000000 + 192612996 45177004 4fh divisor + 192612996 When A—6,576. ;i;3zzi+284.37 10709 — 15a;2=:--648.656640 +63:ir-+4 14.288 — 50. 76 +.002430976 the above remainder. Solution 3. x^^\5x^+63x=50. 7xT— 49 7 7)50(7.395—^. +63 —15 49 + 112 —105 189)1000 —105 567 7 1st divisor. 44991)423000 404919 5172175)28081000 25860875 2220125 SOLUTION OF EQUATIONS. 3X3= 9 3X7X3= 63 3X7X7=147 livisor. 143 —15 -2145 15339 + 6300 21639 —21450 189 2nd( 9X9= 81 3X73X9= 1971 3X73X73=15987 1469 —15 —22035 3rd divisor. 1618491 + 630000 2248491 —2203500 44991 5x5= 25 3X739X5= 11085 3x739x739=1638363 14785 —15 —221775 163947175 + 63000000 226947175 —221775000 5172175 4ih divisor. SOLUTION OF EQUATIONS. When x=7,295. a:3=r+404.4031 54875 ~.15:p2z=-^820.290375 +63;r=+465.885 +49.997779875 + the above remainder = .002220125 50.0000000000 .028 The three values of x -^ +6.576 .395 C+1.0! of a: ^+6.5 ^+7.3i , 1 . QQQ 5s6cond term with its "*" * I sign changed. Ex. 11. Given j:^—8.r» + 14;r»+4;p=:8; or to get the negative root a;*+8.r' + I4^2 — 437=118. 7X7X7=343 7x7=49 8 392 7 10063)80000C.732 14 70441 — 732=:.r 9800 29753947)95590000 3920 89261841 343 30855146488)63281590000 14063 61710292976 —4000 10063 1st divisor. 1571297024 6 SOLUTION OF EQUATIONS, 3x3x3— 27 3x3= 9 143 4x7x3x3iz= 252 3x7x3:^ 63 14 6X7X7X3= 882 3x7x7.~147 4x7x7x7=1372 2002 15339 1462747 8 12271200 20020000 122712 33753947 — 4000000 29753947 2nd divisor. 2x2x2== 8 2x2= 4 1462 4x73x2x2= 1168 3x73x2= 438 14 6x73x73x2= 63948 3x73X73=15987 4x73x73x73=1556068 20468 1603084 1562474488 8 12824672000 20468000000 12824672 34855146488 - 4000000000 30855146488 3rd divisor. Solution 2. x^—Sx^ + lix^+ix-^S. 4 2 +8) 8(2.732=ar —8 14 16 —32 28 —10657)— 80000 8 —74599 28 4 +8 Ist divisor. 4 —16837253)— 54010000 —50511759 — 17057525512)-349824 10000 - 34115051024 —867358976 SOLUTION OF EQUATIONS. 7x7x7= 343 4x2x7x7= 392 6x2x2x7=168 4x2x2x2=32 7X7= 49 3x2x7= 42 3x2x2=12 IfifiQ 53063 —8 47 14 658 —133520 + 122863 — 10657 2nd divisor. 4000 65800 53063 + 122863 3x3x3= 27 4x27x3x3=: 972 6x27x27x3= 13122 4x27x27+27=78732 80053947 3x3= 9 3x27x3= 243 3x27x27=2187 221139 —8 —176911200 + 160073947 543 14 7602 3rd divisor —16837253 4000000 76020000 80053947 160073947 8 SOLUTION OF EQUATIONS. 2x2x2= 8 4x273x2x2= 4368 6x273x273x2= 894348 4x273x273x273=81385668 81475146488 2x2= 4 3x273x2= 1638 3x273x273=223587 22375084 —8 ^179000672000 + 161943146488 5462 14 76468 —17057525512 4th divisor. 4000000000 76468000000 81475146488 161943146488 Solution 3. 343 49 —8 7 14 9800 4000 343 14143 -3920 10223)80000(.763=ar. 71561 —392 13201896)84390000 79211376 13171136187)51786240000 39513408561 12272831439 10223 SOLUTION OF EQUATIONS. 6X6X6= 216 4x7x6x6= 1008 6x7x7x6= 1764 4X7X7X7 = 1372 1558696 6xe=z 36 3x^X6= 126 3X7X7 = 147 146 14 20440000 15996 —8 4000000 1558696 — 127968 25998696 12796800 2nd divisor 13201896 3x3x3= 27 4X76X3X3= 2736 6x76x76x3= 103968 4x76x76x76=1755904 1766328187 3x3= 9 3x76x3= 684 3X76X76=17328 1739649 —8 -13917192 ,1523 14 21322000000 4000000000 1766328187 27088328187 -13917192000 13171136187 3rd divisor. c 10 SOLUTION OF EQUATIONS. 125 Solution 4. X 25 —8 5 14 —200 70 4 125 -200 1st divisor — 1 x^^Sx^ + \ix^+4x=S. —1)8(5.236 -~5 53288)130000 106576 64626747)234240000 193880241 66509117736)403597590000 399054706416 4542883584 2x2x2= 8 4X5X2X2= 80 6x5x5x2= 300 4X5X5X5=500 530808 2x2= 4 3X5X2= 30 3x5x5=75 7804 ^8 —62432 102 14 142800 4000 530808 677608 -624320 2nd divisor 53288 3x3x3= 27 . 4X52X3X3= J872 6x52x52X'^^ ^8672 4x52x52x52=562432 567317947 3x3= 9 3x52x3= 468 3x52x52=8112 815889 —8 -6527112 SOLUTION OF EQUATIONS. u 1043 14 146020000 4000000 567317947 717337947 —652711200 64626747 3rd divisor. 6x6x6= 216 4x523x6x6= 75312 6x523x523x6= 9847044 4 X 523x 523 X 523=572222668 6xQ= 36 3x523x6— 9414 3x523x523=820587 82152876 —8 -657223008 573208125736 10466 14 146524000000 4000000000 573208125736 723732125736 —657223008000 4th divisor 66509117736 Four values of x ~ 1571297024 — 867358976 + 12272831439 + 4542883584 J • QQo ^second term with its I sign changed. - .732 ) +2.732 + .763 +5.236 12 SOLUTION OF EQUATIONS. Ex. 11. Given ^*— 12:r2 + 12^- roots. 8 2 12 — —12 8 — —24 —24 — 1 1 —4 1st divisor. —3=0 to find the four -4) 3(2.858 —8 232)110000 89856 34388125)201440000 171940625 36*491110112)294993750000 291928880896 3064869104 8x8x8= 512 4x2x8x8= 512 6x2x2x8=192 4x2x2x2=132 48 — 12 —576 568.32 12000 —57600 11232 2nd divisor. 5x5x5= 125 4X28X5X5= 2800 6x28x28x5= 23520 4x28x28x28=87808 565 — 12 —6780 90188125 12000000 —67800000 84.388125 3rd divisor. SOLUTION OF KQUATfONS. 13 8X8X8= 512 4X285X8X8= 72960 6 X 285 X 285 X 8== 3898800 4x285x285x285 = 92596500 929871 101 12 12000000000 —68496000000 5108 — 12 —68496 36491110112 4th divisor. 216 Solution 2. ^4_i2^2 + 12^-— 3ii:z0. 12000 5016)3.0000(.606=^, -7200 30096 ^ 216 —72 —1594953384;— 9600000000 5016 1st divisor. —9569720304 6 -12 30279696 6x6x6= 216 4x60x6x6= 8640 6x60x60x6= 129600 4 X 60x 60x 60=864000 1206 12000000000 —12 877046616 —14472000000 —14472 8T7046616 2nd divisor 1594953384 Solution 3, 64 4 —12 —4800 12000 64 c4__i 2:1:24.120;— 3=0. 7264)3.0000(.443=a;. 29056 2217024)9440000 8868096 . 1748236667)5719040000 7264 1st divisor. 52447i0001 474329999 14 SOLUTION OF EQUATIONS. 4x4X4= 64 4X4X4X4= 256 6X4x4x4= 384 4x4x4x4=256 84 —12 1008 297024 —10080000 12000000 2217024 2ik1 divisor. 3x3x3= 27 4X44X3X3= 1584 6x44x44x3= 34848 4x44x44x44=340736 883 12 10596 344236667 —10596000000 12000000000 1748236667 3rd divisor. o;^— 12a;2 + I2^— 3=0, or to tret a — root x^ — 12^* — 12i: — 3=0. . ^ihcrefore -^3.907 12 —63 27 3 —21) 3(3.907 < • ^ ^ ^ IS a root. — 36 72249(660000 — 12 650241 +27 — 132231584743)975900000000 —21 1st divisor. 925621093201 50278906799 SOLUTION OF EQUATIONS. 15 9x9x^- '^29 4x:^X9x9= 972 6x3x3x9zzi 486 4x3x3x3=108 69 -12 -828 167049 —82800 —12000 +72249 2nd divisor. 7x7x7= 343 4x390x7x7= 76440 6x390x390x7= 6388200 4 X 390 X 390 x 390=237276000 -93684 237915584743 —93684000000 —12000000000 132231584743 3d divisor. 2.858 +.003064869104 .606 —.000030279696 .443 +.000474329999 -3907 —.050278906799 16 SOLUTION OF EQUATIONS. Ex. IV. Given ^*+6a;^— 10x3— 112x'2_207j:— 110=0. 4^=256 64 16 —175) 110(4.464 = ^; 6 —10 —700 +.384 —160 16676496)81000000 +256 66705984 —160 4 —448 —112 222439918096)1429401600000 _207 1334639508576 448 Istdiv— 175 2310222205747456)9476209142400000 9240888822989824 2353203194101T6 4X4X4X4= 256 4x4x4= 64 5x4x4x4x4= 1280 4x4x4x4= 256 10X4X4X4X4= 2560 6x4x4x4= 384 10X4X4X4X4= 2560 4x4X4X4=256 ' A ^/S » Xf^'JIKJ 297024 17821440 15629056 15629056 33450496 —16774000 4x4= 16 3x4x4= 48 3x4x4 = 48 5296 — 10 2nd divisor 16676496 84 — 112 —9408000 —2070000 —5296000 —52960 — 16774000 SOLUTION OF KQUATIONS. IT 6x6x(3x6= 1296 5X44X6X6X6== 47520 10X44X44X6X6= 696960 10X44X44X44X6=1 51 J 1010 5X44X44X44X44 = 18740480 192586012496 6X6X6= 216 4X44X6X6= 6336 6X44X44X6= 69696 4X44X44X44=340736 347769176 208661505600 192586012496 401247518096 — 178807600000 3rd divisor 222439918096 6x6= 36 886 3x44x6= 792 —112 3x44x44=5808 --.99232 588756 —10 —58875600000 —99232000000 —20700000000 — 178807600000 18 SOLUTION OF EQUATIONS. 4X4X4X4— 256 5X446X4X4X4= 142720 10X446X446X4 X4zz: 31826560 10x416x446x446x4r=i: 3548661440 5 X 446 x446 X 446 X 446=197837875280 1981930598323456 4X4X4= 64 4X446X4X4= 28544 6X446X446X4= 4773984 4X446X446X446=354866144 355343827904 +6 2132062967424000 1981930598323456 4113993565747456 —1803771360000000 4th divisor 2310222205747456 4X4= 16 8924 3X446X4= 5352 ~I12 3X446X446=1596748 999488 59728336 —10 —597283360000000 —999488000000000 — 2O7OCO0OOO0O0OO —1803771360000000 SOLUTION OF EQUATIONS. [9 a;S^(\x^^]0xB—]l2x'-—201s—]l0—0. or to find a -root 6x^—x^ + \0x'^—\l2x^ +207x~l\t^=zO ; on trying this, — 2 will be found to be a root. Dividing the equation by j;+2 we have x^-{-4x^ — \Sx^ — 76x — 55=0; or to find a — root ^''* — 4:r' — JSj7^+'^6«*'' — 55^:iO, whence — 1 is evidently a root; and again dividing this equation by ^-)-^9 ^'^ ^^^^^ x^ -{-Sx^ — 21a; — 55:=:0 : or to find a — root 3x^ — x^'^2\x — 55=0; by inspection — 5 is a root, and again dividing, we have a; ^ — 2x — llnzO, we now have four roots. The original equation will only admit of one more, and by adding the roots together we find it must be — , therefore a;*+2a? — 11 =0. 2 2 44 20 486 4924 200 2000 4 64 686 6924 The five roots r+4.464 C -2.464 4)11(2.464 V^'''^°''r^-'^^^ ^ g^ > IS a root. 64)300 256 ■ 6 686)4400 4U6 6924)28400 27696 704 (n^ >*f iGAVLAMOUNr! ' »*AMPHUT WNOEii Manufactured by ;©AYLORDBROS.I«e.i Syr«co«», N. Y. Stockton, Calif. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 2oMar'50a/ 60V t r ^^'^