RARE set eal’ : PSMA NEY Sane ny aagtt cut Serene Ay 4 iv i STUART Taek SUE TRRRIAS ey : Nee ea ehegic Serer trea $2 bt mem 2. ont ws Je 3 q aby Waa tT Meat eG hike Ni % Le Tt Sait 44 % Ace Aw i #934. Ut ARE ng CRT A A ate 8 py taut wed AMy, A ail res: 4be! Ni x WG t Peed is met “ ay at aa ae { ‘ ‘ G i : VON (44EP vagioe it of MSR x % 4 ¢ 2 44 a x cs Sok el ead RES? ASAP TAS a he aa 44 Fight ca Se aaa SEPA ae auras Ba fine Ee NES be CTE) taht aie Gy as a2? TARY PRL ALT 4Att 4 he * a ee ee ro a set , Piel tie = wey enene Se or sealer The ert te Soe als gr SS Ae a (ae Eat, <= riod (SoS ay Mw i rare Os a reererentgeen aes sgaers atone - -e r rn — _ - =a ~ ~ 4 ~ ~ arin pavwer a aes re Cees ty — pte etnlety bombs mat THE UNIVERSITY OF ILLINOIS LIBRARY 513 Se CERO Owes B9Sdueee Vue THE ART AND SCIENCE OF Peed: ARLTHME TIC. hie % Ril Rye aah) ay rat} a a uy yay ia ' ~ f ” ch , o 1.4 a; f Oa \ i 7 ba a ‘ { ‘ ,* ‘ f ; . A + : ’ . i ' , y . ‘ é ‘ * L ; J > A : 1 ’ gl « . nat . 1 y : CT hl dy NA eae 4 Peube 4 ee tm : ; 1 Gh 7h 4 Vt Ue ‘ DUAL ARITHMETIC: A NEW ART. PART THE SECOND. THE DESCENDING BRANCH OF THE “AR7” AND THE “SCIENCE” OF DUAL ARITHMETIC. BY OLIVER BYRNE, FORMERLY PROFESSOR OF MATHEMATICS, COLLEGE FOR CIVIL ENGINEERS. Author of the ‘‘ Young Dual Arithmetician,” and Inventor of the Art and Science of Dual Arithmetic; wnd the Calculus of Form, a New Mathematical Science. LONDON : BELL AND DALDY, 186, FLEET STREET. 1867. [All Rights reserved. sy ‘ ~~ ) . ‘ ¢ wa ‘ A “T * 7 ‘ é \ é é ‘ . A 1 > * ” ‘ A =" ree 7 a - “eo . | , ir t, . - i i - +1 j Fi 4 } ie TA. £2) ee Pe 9 3) 2% / iF / a4 : a i 7 * ; ; . * : a J ra ty She ’ as LONDON : PRINTED BY R. CLAY, SON, AND TAYLOR, BREAD STREET HILL. e f ? 4 ss * « i “ ACE wCy! its 4 ee bos RS ge De OP ee wn 7 : : ata : y ie edeitiny aN DEE ‘ DA iae #7 [ é =e | ia LIN E. GARDINER FISHBOURNE, RN. CB. ETC, ETC, ETC. BY THE AUTHOR. PREFACE, ALTHOUGH a department of greater importance than Dual Arithmetic has not been added to Mathematical Science, I will not occupy the time of the reader in reviewing the trains of thought that led me*to its discovery, or in offering weighty reasons for the methods that I have invented to develop its art and science. I will merely give a concise account of the nature of the five books on the subject 1 have now published, and briefly describe their usefulness. I. In my first work, “ Dual Arithmetic, a New Art,” the ascending branch only is discussed; the descending branch, and the application of both branches combined, designedly received but little attention, to avoid complicating a subject so entirely new. And, although rendered prolix to prevent obscurity, it will be found to effect thoroughly the objects for which it was written; namely, to exhibit the great flexibility, range, and power of dual developments even when subjected Vill PREFACE, to clumsy and restricted treatment, and to show that Dual Arithmetic harmonized not only with Common Arithmetic, but also with Algebra, and the higher branches of Mathe- matics. I. In the present work I have developed the descending branch of the system, connected it with the ascending, and treated Dual Arithmetic as a science. Thus, I have completed the art and science of a calculus of the concrete values of quantities known or unknown, and shown for the first time how all mathematical functions, direct and inverse, can be submitted to the operations of Dual Arithmetic without the aid of Tables. Among the several operations and their converse that can be done with the greatest ease by direct processes of Dual Arith- metic, without the aid of Tables, in an endless variety of ways, I shall only mention here, the Involution, and the Evolution of numbers for any root or power; the direct calculation of the logarithm of any number whatever to any base; and the general methods of determining numerical roots of all orders of equations and also of exponential and transcendental equa- tions, whether the bases be known or unknown. iD “The Young Dual Arithmetician”’ is a work designed to quality young Students to read and understand the larger works, to render the practical calculator independent of tables PREFACE. 1x of common logarithms, and to demonstrate that, if tables be ~ preferred, those of dual logarithms are incomparably the best. Any schoolboy may construct an extensive table of dual logarithms in an incredibly short space of time, and afterwards test its accuracy at any point, if he only understands Common Addition and Subtraction. IV. “The Dual Doctrine of Angular Magnitudes and Functions, and its Application to Plane and Spherical Trigonometry.” In this work, trigonometry is treated in an original and philosophical manner by demonstrating the transcendental for- mule of trigonometry without the aid of impossible quantities. V. “ Tables of Ascending and Descending Dual Numbers, Dual Logarithms, and their corresponding Natural Numbers ; and of Angular Magnitudes.” When operations are performed with dual numbers in their lowest terms, and tables are used, it is not necessary that such tables should range beyond the natural numbers from ‘414213561 to 1°, and from 1° to "70710678 (see the present work, p. 12). But this volume of Tables exceeds these limits, and -is more comprehensive, and more easily used », than any hitherto calculated. These tables are equal in power to Babbage’s and Callet’s combined, and take up less than one-eighth part of their space. Dual Tables ranging b D4 PREFACE. from 1° to 2°99161136, ascending, and from 1° to ‘299161136, descending, have the greatest power in economizing the time and labour of the calculator. I intend now to turn my attention to developing another new mathematical science which I have discovered, and styled the “ Calculus of Form.” It establishes modern analysis on a purely mathematical basis, and rejects the reasoning of the Differential and other methods now current. OLIVER BYRNE. CONTENTS. INTRODUCTION. Page General form of ascending dualnumber. . . . ..... . . Hi Ultimate values in the eight RR Pe mete i Dee IGE EL Et tye me 8 Extension of the notation . . Movealaeh! Lath soba FE ale The comma (, ’) and the period (° a aia) kG es Mi Me a PUPeIUPIUMMIMOCETIGDIS 6 6k oe re als vii, Vili eememmmeemmending branch .. - .., +. hs 8 es) ee we) XK, 1, XU CHAPTER I. THE GENERAL NOTATION APPLIED TO PARTICULAR NUMERICAL EXAMPLES. SHORT METHODS FOR CONVERTING A NATURAL NUMBER TO A DUAL NUMBER; AND A DUAL NUMBER TO A DUAL LOGARITHM; AND VICE VERSA. Articles (1) to (15). Articles , . Page (1)(2) Definitions . . : I (2)(3) Bases of the Reading beanie iacapsh of the fein i. eich both branches combined °. yi (4) The small figures attached | | | ae (5) How the dual logarithms of numbers are indicated. . . . . 4 (6) The arrow and comma of. the descending branch, digits and their position . . Fate) 2) Ri Reet) (6) What a cipher vioedelbes as a adic digit (7) Dual logarithms of the descending branch, notation exchiplifiba 6 (8) How to find any two of the three corresponding numbers (Natural number), (Dual number), (Dual logarithm), by an aN SRS SS ee ee Ae a ok ea XU Articles (9) (10) (11) (12) (13) (14) (15) RESULTS OBTAINED BY DUAL DEVELOPMENTS ; CONVENTIONALLY EXPRESSED IN ALGEBRAIC LANGUAGE. QUOTIENTS. POWERS AND ROOTS. (16) (17) (18) (19) CONTENTS. Page How any dual logarithm may be reduced to a dual number whose first digit does not exceed 3, or ’3; and succeeding digits not to exceed 5, or’5 . Every second digit may be made a cipher Important relations in converting natural nun Boe rs anh numbers . General tabulated form Limits for dual logarithmic tables ve Tables of dual logarithms superior to common opener in both accuracy and precision . ; ; ‘ When dual logarithms are employed, no alison Bs ag be made when arithmetical complements are employed To reduce a dual number of the ascending branch to a dual logarithm. RULE ; : : To reduce a dual logarithm of the seoennine Diaibe és a nat number, RULE . To reduce a dual number of the dasconcitia erin toi a aunt logarithm. RULE : To reduce a dual logarithm of Te deseidee br aie +o} a dina! number. RuLE . CHAPTER ILI. Articles (16) to (55). Recapitulation of Conventional arrangements and Notation . The use of the comma (,) in designating ascending and de- scending dual logarithms. To find the arithmetical comple- ment of a dual logarithm . . Position of the decimal point Aelia : Practical application of the decimal point determined . In the seventh line from top of page + 9°014911, should i ‘QOI4QII Practical examples . ; Bait Numbers that may be omitted in nears oe 19 21. OF SIMPLE OPERATIONS PRODUCTS. 23 24. 25 26 27 28 29 Articles (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) CONTENTS. Tables of dual logarithms shown to be superior to tables of common logarithms Reduction of ordinary formule Aerts patows The use of subsidiary angles avoided . Examples : Examples on the raratin for ‘the Faken 10 Ae oy AMS : questions of interest and annuities Formule and examples in interest Principal, time, and interest being given, to find the renee The amount, rate, and time given, to find the principal The amount, principal, and time given, to find the interest . The rate, time, and principal given, to find the amount The amount, principal, and rate given, to find the time that money will double itself at compound interest The rate, amount, and principal given, to find the time : To find the amount when the principal is increased by the interest every year . General formule . Examples Transformations and Be ietions : t Preliminary reductions required in ARS the chat of equations gS Important fen Glusumter, 5 Property when the first three dual digits are zeros m P To find R” when R¢ is given int a Questions relating to interest and annuities pentane’ Roots of equations calculated by direct methods A convenient dual digit found by a method that eaartie common division Particular equation of the wth vate ee, of iy Aa Eliade The same root under different ascending dual forms The last step independently proves the preliminary colonia Examples introduced for the sake of uniformity . Operations indicated by the signs { and — . ERG WHA 25, toy tas 3,3 + « in conjunction with + The sign of dual subtraction (—), ascending branch The dual sign of plus or minus (+) for the ascending br as Different forms of the same development . Coincidence of corresponding values . xiii Page 31 32 33 35 36 37 38 39 39 40 40 AI 4I 42 43 45 47 47 48 XivV CONTENTS. Articles Page (43) Direct methods of reduction in particular cases . . . . . . 61 (44) To find the dual number and the dual logarithm of a natural number of the form 1’000w,u,u%,pg. RULE. mer Oe (45) When each of the dual digits are less than io. . SAS GF (46) To find the dual number and dual logarithm answering to a natural number of the form 1'oow,u,rspg. RuLE. Examples 63 Examples, observations, contractions, &c. . . . . . 64, 65, 66 (47) Syntheses of practical developments, functions, and their inverse operations and theif revetse. .. 2. +) 22 Hyperbolic system of logarithms . . . 23 ey i AS (48) The dual system of logarithms furnishes all the advantage of both hyperbolic and common logarithms, without retaining any of their defects . . . I (49) Circumstances under which the saloalet of diffarenioas aid the dual calculus coincide . . . . . , eee ; “Fo (50) The operative numbers and consecutive “dua numbers ; failure of the calculus of differences . . . au? Sapo ye (51) A combination of particular factors that may ratalens when made to assume the form of an ascending dual number . . 73 (52) . Examples of an overrated method applied to calculate dual loparithns 2) 2 8S ee, (53) Counterfeit factors .. .0.9.- . 0.5.9.5. .9,> . 7 (54) Operations and their reverse . . . » Me (55) Original form of Rule, Article (12), page (r cal 2 Cee CHAPTER ITI. (56) Ascending dual developments applied to determine the values of unknown quantities under a variety of dualforms . . . 77 (57) Examples of investigation. . . . ree General expressions applied to partiouiie cases .:\.. 7 ae ee Examples of simple equations involving large numbers . . . 81 Details, notation illustrated . . . .. eae (58) Quadratic equations, notation of ascending Beatle illustrated . 83 No practical inconvenience can arise at any time in making ~,, a unit greater or less than the dual ai HY belonging to the ath position . . . yer = Examples, large Bostioiontif éhoidd of tittita’:. (2 er CONTENTS. Articles (59) Cubic equations . (60) Equations of the fourth oes, (61) Equation of the fifth degree . Calculations and general reasoning (62) Exponential equations . stats (63) General solution of the ante w= G+. (64) General solution of the equation a = N . CHAPTER IV. (65) Special treatment of the descending branch of dual arithmetic 102 (66) Details of two methods of reduction . PR (67) How dual logarithms in the first position are found . Examples in reduction iG . 105 positive, those of the ascending are to be considered negative, and vice versd . Elements of descending e Aheh , : : aa? SUL pa tie he LORS (68) When dual logarithms of dieeilice Peles are considered 2i07 To construct a table of dual logarithms, dual Aditi. a their corresponding natural numbers of the descending branch by . 109 common subtraction Extension of descending dual tables . 110 to Operations with the descending branch of dual sega ts inde- pendent of the ascending branch . Elementary examples | + — ascending signs { + — descending signs \ Tabulated form of the operative numbers . Dual developments by the application of both fale oA - General definitions . Elementary examples . mid i's Reductions effected in a great many ways | + —. +. ascending Dual signs < } both branches combined > how employed . { + — + descending Extended developments : . Tabulated form of consecutive bases o Pats digits : Reciprocal of all dual numbers easily found . 108 113 oi ea 4 De 115 to 118 . 119 . 120 - 422 » Bihar EE 123 to 126 . 127 3-F28 129 to 148 . 149 . 150 XVI CONTENTS. CHAPTER V. Page Solution of important problems designed as models and examples of — concise methods of operating, and succinct processes of investi- gation . : .. a ee Recapitulation of the coher soamnte of both branches . 2 eee The logarithm of 10 and 2 being forgotten, it is required to produce them by an easy and direct operation . te iin ee ee To find the logarithms of the bases by direct operate -. 1§8%0:160 The shortest processes to reduce natural numbers to the simplest dualnumbers .. . MMPI mS" Useful practical eco . 164 Reciprocal operations ; the great value of : . 166 Quadratic equation solved with great ease, both piandige ae employed. ... . 6 oh in To find the natural sine and if sine of Io” be an independent calcula- tion from knowing the natural sine of 30° . . . . . 4172t0174 When s represents the sine of the are a to radius 1, then 7s — 568° + 112s° — 648’ = sine of (7a); if 7a be equal to 180°, 360°, 540°, &c. the equation becomes (2787)? — 7 (27s)? + 14 (27s?) — 7 = 0; find the three values of (28). ...... « = - «ean nnes Find the first twelve and the number of figures in the continued product 1.2.3.4.5.6.. Alt eins . 2 ps Find the first eight arid the rane of hee in the continued product 1.2.3.4..... 365 : +) 183 To find the first seventeen stage ea the ante of Ae in fs continued product 1.2.3.4...... 1875 : . 184 The continued product 366 X 367 X 368..... 1875 -/ 185 Equations partly exponential and partly integral . : as oso Rationale of the method of calculation . . . Bla he to 189 Results easily found that defied the combined skill of mathematicians before the introduction of the dual calculus . . Lge Given a* + b¢ = c* to find z - 193 To find the first eight figures of the sontnna rigduee of od numbers » 203 The area of a curve whose equation is y = a e-** between given limite Aye ee MP Se Equations to the catenary .. .. 8). 3.0) ) VND THE ART AND SCIENCE OF DUAL ARITHMETIC. INTRODUCTION. ON THE ASCENDING AND DESCENDING BRANCHES OF DUAL ARITHMETIC WITH EXTENSION OF THE NOTATION, In the work entitled ‘“ Dual Arithmetic, a New Art,” and in its reissue with an® analysis, we showed -that any number whatever, whether great or small, might be reduced to the form [tos fires 2 TOMY 06, 4'8,5 Uys: Ke: where this notation is used for the continued product 2" x 10” x (1°1)" x (1°01)“ x (1'001)"s x &e. Thus | ) ) 2710° |, 3,1,4,1,2,1,1,3, = 2” x 10° x (1'1)° (1'01)’ (1'001)* (10001)' (1'00001)” (1°000001)* (1*000000T)’ (100000001) = 2’ x 10° x 1°34985881 = 5399°43524. The transformation of any common number into a dual number, and’ the converse operation were fully shown in that work. B lv INTRODUCTION. It was also shown how any of the digits of any dual number might be transformed into zero, the remaining digits being altered in value. When the first seven digits were so transformed, the eighth remaining digit was called the ultimate value of the dual number in the eighth position, and was shown to possess all the properties of a common logarithm of eight places of decimals. And then was written 8 2° 1073) sue The method of calculating these ultimate values for every dual digit, as well as for the common numbers 2 and 10, was shown. By this means, every arithmetical operation requiring the use of logarithms was performed without the use of tables, and by methods involving only the simplest processes of arithmetic. Arithmetical solutions of many problems and their converse were obtained by using the dual arithmetic in its simplest form of development, which have defied the skill of previous investi- gators, with all the aids of the highest forms of calculus. It was also shown how the dual system of calculation blended with the operations of common arithmetic without interfering with the generality of either. This was done under great disadvantages, as it was inex- pedient at first to introduce into the art more than one of its branches. The student who understands what has been written and pub- lished on this subject is now in a position to enter into the more extended development of the subject. DUAL NUMBER, ASCENDING BRANCH. Vv Since any number may be represented in the form Nea 2" 10 | 0., U., Uy, Uys Uy KO we may omit the bases 2 and 10, with as much advantage in perspicuity as we omitted the bases I°I, 101, 1°001, &c. and. write the above expression in the form N=” , U, U,, U,, Ke. By using digits to the left of the arrow, the powers of 10 may be dispensed with altogether. Where EAI WO, Us thay Ure Wy OCCe is a notation for a continued product of the form N = (1 + 1000)”s (I + 100)”2 (I + 10)” (1 + 1)” (1 + *1)® (1 + ‘o1)”2 (I + ‘OO1)"s (I + *O001)™* Ke, Now ; I w 3 (I + 1000)"s = rome + cot =n(10')"* Yang I Ww 2 (1 + 100)” = 1022 1 + To se, (10°) Vw.) I Ww 1 (1 + 10)" = ro" + a = (10) Yup Hence, N = Way Wy Wy YY Uy Uy Uy Uy KC. = 10% +2, +3, I” (uv. + w,), (Up + w,), (U, + W,), u, Ke. = (iy + wy + Bw) Im (4, + w,), (u, + w,), (wW, + w,), U, Ke. This last expression shows how powers of Io in a dual number may be replaced by digits on the left of the arrow, such digits representing powers of the bases 11, IOI, 1001, &c. increasing from left to nght. vi INTRODUCTION. And conversely how digits on the left of the arrow may be transferred to the right. Thus, rae 34,2. | 5,6,7,8,9,= "J (5-+2),(4+6),(7+3),8,9,= "J 7,10,10,8,9, Again, ) pak 35,68, = 43, 7 (8-—3),(6—-— 4) 6,8, = aay ie 0168, or, 11> 35,6,8,=3,0,2, |° (3—2),(5—0),(6—3),8=3,0,.2, |° 2,5,3,8, In writing the power of 2, which is always at the middle of the arrow, care must be taken not to confound it with the figure at the top of the arrow, to the right, designating the position of the first dual digit following it. The comma (,) is employed in the operations of dual arith- metic, while the period (*) is retained to separate whole numbers from decimal fractions ; this part of the general notation should be remembered. It will be found that the comma accompanying a dual digit, or a dual logarithm, will be a sufficient distinguishing characteristic without employing the strong black figure, as above, and in the Work previously published. Thus, Pia «§ eld owe Bee ‘VY 5,6,7,8 is a short expression for (10)’ (2)? J) 0,0,0,5,6,7,8, And ny 1;2/3,.4 for. (10)° (2)° |. 0,.0;naaeee In the Work feat referred to, we. showed stingy any: dual number might be transformed into another, any number of Y aa NUMBER, ASCENDING BRANCH. Vil oc digits, Pistia, dies sty fat nent iy ZeX0, the next remaining digit being increased in value. The. extent to which this reduction was necessary to be carried in practice was shown to depend upon the accuracy of the arith- metical result required to be obtained. Thus for results true to four places of figures, it was shown that dual numbers of only five digits were required, and that when four of these were reduced to zero, the fifth gave all the properties of a common logarithm of five places of decimals, &c. For results true to seven places of figures, eight dual digits are required, and when the first seven of these are reduced to zero, the eighth is called a dual logarithm. See ‘Dual Arithmetic, a New Art,” pp. 27, 28. ~ It was shown also that there were several values which give correct results in these fifth, sixth, seventh and eighth positions, but a particular set of values were shown which were termed ultimate values. For calculating these ultimate values, as well as for an account of their properties, we must refer to “ Dual Arithmetic, a New Art,” pp. 212—214. In the generalization of the form of the dual number ascending branch Where Pigee su), , Uy KC. is written under the form N= Wy, Wy, Wy 1” Uy Uy Us Uy Ke. We must remember that m represents a positive whole number. Or N = (1 + 1000)"s (1 + 100)2 (1 + 10)” (1 + 1)" (1 4 ‘1)% (I + ‘or)”2 (1 + ‘001)" (1 + 0001)" &e. Vili INTRODUCTION. It is evident by inspecting the form of this continued product, that by means of the powers of these bases written to the right and left of the arrow, including that of the base (1 + 1) or 2 on the arrow, that any number from + infinity to I, can be ex- pressed to any degree of accuracy. The bases being neglected in the representation of the quantities just as powers of 10 are neglected in ordinary arithmetic, and the bases in logarithmic arithmetic. Quantities less than 1 are represented under the form N=”J"u,, U,, Us, u,, &e. by making m negative, but in that case we do not transfer m to dual digits on the left of the arrow. A more complete method of representing numbers less than 1 will be shown when we discuss the notation of the descending branch of dual arithmetic. DUAL LOGARITHMS, ASCENDING BRANCH. 8 U=y) 4, Here this notation signifies that w is the ultimate value of the common number U in the eighth position, or we may say that u is the dual logarithm of the common number U. As results true to seven places of figures are those most commonly used in arithmetical operations, when we speak of a dual logarithm, without specifying position, we regard it as of the eighth position. As the use of the logarithm of a number is most frequent in symbolical operations, instead of writing Dual log. U = u, when U = J, uw we write J,(U) = u, DUAL LOGARITHM, ASCENDING BRANCH. 1X Thus since 8 2 = | 69314718, J,(2) = 69314718, Or dual log. of 2 = 69314718, a whole number. Hence it will be seen that by attaching a comma to the sign J, we indicate an operation the exact converse of that represented 8 by the sign |. It is often necessary, when using a dual number not reduced to its ultimate position (but which can always be so reduced) to indicate that its logarithm is to be taken. Thus supposing we have to indicate the logarithm of the number represented by the dual number t 7,2,6,0,7,8,2,6, Using the analogous notation to that above, we should write it }, 7,2,6,0,7,8,2,6, But V 7,2,6,0,7,8,2,6, = }, 603 14718, 1), 7,2,6,0,7,8,2,6, = 69314718 Hence on the whole, if a a Ved Nee Ne RS } 603 14718, Then Wi2) = 09314718 = |, 7,.2,6,0,7,8,2,6, And taking away the commas attached to the arrows which indicate logarithms, we have Z =1177,2,0,0,7;8,2,0 This gives all the notation we require at present for logarithmic operations by the ascending branch. x INTRODUCTION. OTT AND EXPLANATION OF THE DESCENDING BRANCH OF DUAL ARITHMETIC. Me Any number N may be written as a continued product of a form sald ad i pa ehate | 10” x (I — ‘I)": (I — ‘O1)”2 (1 — ‘oo1)’s (1 — *0001)"* &e. or’ sto (9) (-99)”* (-999)"s (79999)"+ ce. In analogy with the notation used in the ascending branch of dual arithmetic, this continued product may be written thus ~ ? ? ? ? i) ? m Y Uv, Vs V, v; %V where any of the digits v, v,, v,, &c. as well as m may be positive or negative. Negative digits are only used when the descending branch is not combined with the ascending. As in the ascending branch the power of 10, m, may be taken off the arrow and digits placed to the right when m isa + whole number. Thus 0/00, 0, Pt, tb, Oe. represents the continued product (9) C99)"* ('999)"* 9999)"* (9)" (99)"* (999)"* but the bases 9, 99, 999, or the digits ¢ tay fy | &c. are seldom employed exéept in’ analytical inquiries.: = For the most part this descending branch is only used — iv combination with the ppm one. When so used, positive digits are only employed, and then the descending branch gives-a method of pines all-numbers j Ss Lan pay i Fi BASES OF DUAL NUMBERS, DESCENDING BRANCH. Xi between 1 and minus infinity into dual numbers. When this combination is used, only + digits of both branches are required. Thus any positive whole number between o and + infinity, may be represented under the form ’ ? 0, 0, } Wy Wy Wy Y™ Uy Uy Uy Uy KC. which represents the continued product (-9)"* (-99)"2 ("999)”* (1 + 1000)" (I + 100)”2 (I + 10)”: (I +1) (I +1)” (1 +01)“ (1 + 001)” &e. This gives the power of representing any number, however small or however great, by the combination of the two branches, using only + digits. For analytical purposes, it is necessary to extend the bases of the descending branch, so as to express zero and quantities which are negative of any magnitude whatever in that base. The scheme of the bases of the descending branch may be written under this form. — ©,....(I — 1000)"* (I — 100)"* (1 — 10)" (1 — 1)" (1 - *1)" (I — ‘O1)” (I — ‘o01)”’s Ke. to I. The base continually approaching to + 1 but never exceed- ing it. Also, the descending bases may be employed under the form +0 .... (1000 — I)** (100 — 1) (100 — I) (1 — 1)™ (‘I — 1) (or + 1)% (oor — 1) eo ....... — I. The negative extensions of the bases, however, being solely used for analytical investigations, the base (1 — 1) as well as (1 — 10) (I — 100), &. (1 —1) (‘ol — 1) (‘oor — 1) Ke. are not used in the present work. xil INTRODUCTION. Using the processes of operating on a number by ascending dual numbers, ‘4 nie 9 5 = "9 1 I, 99 9929) = Oils 9999 99 00 299 ae 99999999 99.99.9999 9999999 = ‘9 41,1,0,1,0,0,0,1, Therefore we may say that ‘9 |, I,1,0,1,0,0,0,1, = 1, very nearly. Similarly, it may be shown that 99 ¥9,1,0,1,0,0,0,1, = | _— ‘999 J 0,0,1,0,0,1,0,0, = I "9999 J, 0,0,0,1,0,0,0,1, = I Or I } 0,1,0,1,0,0,0,1, = I I { 1,1,0,1,0,0,0,1, = 1 ? ? ’ > I 4 2 I (0.0,1,00,1,00, — oy | 8 10.0.0,1,000,1, = I THE SCIENCE OF DUAL ARITHMETIC, AND THE APPLICATION OF THE ART, INVOLVING BOTH BRANCHES. CHAPTER I. THE GENERAL NOTATION APPLIED TO PARTICULAR NUMERICAL EXAMPLES, WITH SHORT METHODS FOR CONVERTING A NATURAL NUMBER TO A DUAL NUMBER; AND A DUAL NUMBER TO A DUAL LOGARITHM; AND vice versd. 1. Duau Arithmetic is a new art of maneuvring numbers, and also a new science by which the relations of quantities are investigated with ease and accuracy, with or without the use of tables. 2. The term Dual is employed because the art has two branches, the basis of each branch being composed of two parts, and because the digits of a dual number may be subjected to a variety of changes in magnitude and position, while at the same time the dual number remains equal in value to two unchangeable extremes, namely, a natural number, and a logarithm to a known base. tia DUAL ARITHMETIC. BASES OF THE ASCENDING BRANCH. Limit UR) P=, heat he (10001); (1001) 5 (101); (11); (2); (151); (WoT); Limit (1r@01); (OOUI)]. . Lay. I BASES OF THE DESCENDING BRANCH. Limit ; (B) — ©....(—9999"); (— 999"); (— 99°); (— 9); (©) (9); (99); Limit | (3900)) See I Or Limit Limit (C) + & ... (999°); (99°); (9°); (0); (—°9); (— 99); (—-999);.-- —1 3. The sum of the bases (A) and (B) similarly circumstanced assume the values 7 Qine ott i (2%)5 (Og 42") Bae. fies eee 2. Of (A) and (C) + © .«.4(2000°);(200°); (20°); (2°); (2); (02) aia ees Of (B) and (C) GLRG sk Hes (0); (0); (©); (0)s (O)3 . 8 See fe) The descending branch, and the ascending, and both com- bined may be represented respectively by the three following general symbols. | ” en u, v ~P NOTATION OF DUAL NUMBER, ASCENDING BRANCH. 3 4. A small figure placed at p designates the position occupied by a dual digit, and sometimes points out the leading position occupied by the first of more dual digits than one. m expresses I he ses I 7” ”? I Q™ 1) =e 2 tes I n ” an A dual number of positive dual digits has always an exact value in common numbers when no contractions are employed in the reduction. When eight positions to the right and eight to the left of the signs } J, counting from left to right in both cases, are occupied by ciphers or other digits, the sign J, being placed before the eight ascending digits, on the left, and 4 after the eight descending, on the right ; yet with respect to range the dual number is said to be one of eight digits although sixteen positions, and other position between the signs { and J, may be occupied. If one of the signs { or J is omitted, the positions attached to it are supposed to be occupied by ciphers. 5. When the last dual digit, and all that follow, are rejected, and when the last is 5, 6, 7, 8, or 9, the digit preceding may be counted one more, as in decimal arithmetic. 8 2° = |, 7,2,6,0,7,8,2,6, = ¥, 0,0,0,0,0,0,0, 69314718, = |, 69314718. The 8 being omitted, the expression is written 2 = 169314718, 4 DUAL ARITHMETIC. Then 69314718, is termed the dual logarithm of 2° and written V,(2°) = 69314718, 8 10° = } 230258500, (10) = 2302585009, The diameter of the earth through the poles is said to be 41706091'152 feet, the dual logarithm of which is equal to the whole number 1615789463 ; Then J, (41706091°152) = 1615789463, DESCENDING BRANCH. 6. In this branch the arrow points vertically, and the comma is to the left of the digit and above it, while in the ascending branch the arrow points straight down, and the comma is to the right of the digit and below it. ( (‘9) is represented by ’1 > in the Ist position. mt a @ 2 , 5 Ss (9) ” ” 2 \ ” 9 ro a “a2 ? oF (9) ” ” 3 Mh » ” a { &e. &e. Bs (99) 1s represented by ’o’I f or ’1 {in the 2d pos. = 2 oc oO 28 | C99" » » ‘0'2} or 2 ae oo i os ial oie alice 52.9 a we) oe Ei" L &e. &e. ( (999) is represented by ’0’o' I for’ fin the 3d p. OOO as ? ’0’o0’2 for 2 sig CODO saab rents ’0’0’3 for’3 ses tl &e, Ke. Three decimals in the base DUAL LOGARITHM, DESCENDING BRANCH. 5 Hence, in both branches, if there be 7 decimals in any base, its powers, or dual digits, are placed in the rth position. (9)’ (99)? 18 written °3’2 BOLECUO tots tore 7s Sit (-999)" (-999999)" (99999999)" is written ’o '0°3 0 '0'2'0°6 f _ A cipher being in the first and also in the second positions shows that no power of ‘9 or ‘99 is employed ; the same may be said of other positions occupied by ciphers. ’°0’0°3’0’0’2’0’6 may be written *3’0’0’2’0’6 > 3 at *3 “ "9 4p "6 a 3 6 8 (9) 99)" 999)" (9°) (99°)" is written °3°2°5 ff ’1’2 and may be put under the form ~ 4°4°5 f (io!) ="4'4'5 fe The reduction of *3’2’5 } ’1’2 to ’’4’4’5 $ is similar to that established for the ascending branch. For (1°1)’ (101)* (1'001)° (11°) (101)" = 2,1, } 3,2,5,= 5) 44,5, 7, In the descending branch, as in the ascending, a dual number reduced to the eighth position is also called a dual logarithm, and must be considered negative, if the ascending dual logarithm is taken positive, and wice versd. It will be shown hereafter, that tie (een TOG 30052: or f = "1005034 ‘4 ‘OVOL Ue hae heen '0'0'O I f = "10000 t &e. &e. 6 DUAL ARITHMETIC. Then '2°3'4'5 61778 '9 f= 124544 105en as in the ascending branch, the 8 designating the position is omitted in practice. Again, "705432110 = '3'0'! ( O,5,0,0,1,5,6,3, It has been already shown that 8 vi 0,5,0,0,1,5,6,3, = V 4976728, and that "307 A "3 17082008» 8 Then 8 *765A3211 = 31708206 ( 4976728, = ’26731478 8 8 "31708206 4976728, '267 31478 The dual logarithm of the decimal ‘76543211 is '26731478 written J, (76543211) = '26731478 ; These reductions are introduced to exemplify the notation. How to make all such reductions will be shown when the opera- tions of the descending branch are being discussed. 8. How to find any two of the three corresponding numbers— (NATURAL NUMBER); (DUAL NUMBER); (DUAL LOGARITHM) ; by easy and direct processes, the remaining one being given. SMALLEST DIGITS FOR DUAL NUMBERS. 7 _ Any dual logarithm may be compounded of multiples of 69314718(n) and 230258500(m), and a logarithm numerically less than (34657359) half the dual logarithm of 2. — If 230258509 alone is operated with, any logarithm may be compounded of multiples of 230258500, and a logarithm numeri- cally less than (115129255) half the logarithm of ro. arm = 34657359 meee OO214718 Ign = 103972077 an = 138629436 23n = 173286795 3m = 207944154 Ke. 4m I LES 2o2'5 5 230258509 345387764 460517019 575646274 690775528 Ke. If the given logarithm be greater than > by «, but less than n, then a logarithm less than - If the given logarithm be greater than n, but less than 14” Saa)anetos by y, then a logarithm less than . Again, if the ie te be greater than 1}n by 2, but less than 2n, then, 5 42] n 2n — {—. =-— 2 2 ; bd DUAL ARITHMETIC. which is also less than 3 and soon. A similar process of reason- ing may be applied to im; m; 14m; 2m; &e. g. Because v 3,0,0,9,4,1,0,7, — V 34657359, and ee ed RG St ee ee 337034001 { = 34657359 t hence any dual logarithm may be reduced to a dual number whose first digit does not exceed 3, or ’3; and by operating 2 3 with the logarithms of | 1,; | 1,; ) 1,; &. andor a4; ‘1 $4; 71 4; &. in a manner similar to that explained with 2 3 respect to 69314718, and 230258509 (8), succeeding digits after the first may be found so as not to exceed 5, or ’5. 2 3 Dual logarithms of | 1, and *1 {; J 1, and 1 #3 J 1, 2 and ’r 4. may be arranged in the following order: 3 ‘1’ =°10536052 1° ='1005034 “1 0 tesnaone 2 3 2 3 1, ae 9531018, 1 I,= 995033 bt = 99950, A multiple of 10536052 may be involved so that the remainder will not exceed half of 10536052 = 5268026, which contains 1005034 five times but not six; the same may be said of half 1005034 = 502517, &c. and of half 9531018 = 4765509 ; &e. &e. We have now arrived at these important conclusions, namely that with the dual logarithms of 10° and 2: ({J,(10°) and |,(2°)) and their multiples together with a logarithm, numerically, not SMALLEST DIGITS FOR BOTH BRANCHES. 9 greater than 34657359, or °34657359 the dual logarithms of all the natural numbers between +o ando are instantly determined. The corresponding dual number may be put under the form (A), hu A OR BE Ua ee, Se iF f et 00.0. u,v, * 1 U,y Un) Uy Ugy Uns Ugy Uqy Uys (A); in which it is not necessary that either ’v, or u,, should exceed "3 or 3, and at least half these digits may be ciphers. Therefore, to determine in a direct manner the natural number corresponding to a dual logarithm requires but little numerical labour, since (A) may assume the forms (B), (C), (D), &c. Te °v, 0 ”, 9 fo) fe) OD ie U, ak Us, O, Us, Ug) Uy, U,, (B). V op RS SE SE bog Say 6; ie OO 2 vv, 'v, al Cette, t,, 0, 0, O; & * (CO), ; A '0’v,’0’v7, 0000 * ik Uy Oy Us, O, Uy Uyy Uy Uy (D). Ke. Ini reducing a dual logarithm of the form B, C, D, &c. to a natural number, it is of no moment to have each of the last four digits not greater than 5, or ’5 ; then, when the. positions uw, u, u, U, are occupied, v, v, v, v, become ciphers, and vice versd. Let u, wu, u, u, be a natural number composed of the dual digits w,, u,, w,, w,, and let v, v, v, 0, be the natural number com- posed of the dual digits ’v, ’v, ’v, ’v,; then 5 | U, Ug Uz Us = 10000 4, U, U, U, and a7 eet y AE YO neo) 5 a3 vy, v, U,V, { = LOCOOOOCO — 2, Y, % U%, a fe) _ DUAL ARITHMETIC. These numbers being operated on by such dual numbers as ? eee J > 0’v’0’0 aN Me; On thst 5 4 ‘ F mein O50 O°G bh O,P Rha, Unseen ae, pnee i yi oO" a) is U,, O, Ug) O, V &e. the corresponding natural number will be produced. 10. The solution of the converse problem, that is, to find the dual logarithm of any given natural between +o ando requires no additional labour or skill, since any given number operated on by 10° may be found in one or other of the positions N, N, N, N, N, N, and 1-00000000 ; 2'00000000 and *50000000 may be operated upon by a dual number whose first digit is not greater than |, 3, or ’3 } and also assume one of the positions N,N,N, &e. I*00000000 I*O00000000 Ne N* 1°41421356 "70710678 ai ans 2*00000000: *5 0000000 N, N, 3°16227766 °316227766 Reductions may be often simplified by multiplying or dividing numbers found near the positions N, N, N, &c. by 2. A brief inspection and comparison of the numbers exhibited in the subjoined tabulated form will exemplify our first sketch of these important relations. LIMITS OF TABLES OF DUAL NUMBERS. ITOQOO00000 Numbers Saae a yi intervening. 4 a . rai i | 141421356 | Numbers 729 = "3h intervening rel. = 2% divided ‘by 2. 3 eae | 200000000 | Numbers leat SAPD intervening ; =121 = 12, divided by 2. 3r= {3} | 266000000 Numbers Il = 1, intervening = 12h 2, multiplied by 4. eer 3; 345000000 Numbers 729 = 31 intervening =o $i ="2% multiplied by 2. 9 = If | 499999999 | Numbers Eee = eT, intervening =121 = 2, multiplied by 2. Waa Uo i 3 | 707106780 | Numbers } CMe 6 ? intervening. ( S! = 2¢ 9 Fat | 999999999 | 00 Ir Examples. 1°32898724 Dr Pie tas pad Tn p50 ge we \ 3,0,0,5, 28441721, 1670°74 odo sft 0,31, "1798805 5 239;408438 Fp aa ae ie 3 O105 if 2,0,0,0,3,4,2,6, 18010428, 279701465 003 ‘fe I,2,0,5,0,2,5,73 £I277 101, 3605" Po Ti no "2711003706 ls 0,0,0,0,2, "32711615 054549625 Pit STO20'V'9 it 1,0,2,0,3,0,00, 9733918, 819'672683 2001 sf rin 8 SA A pe ia a8 "18885019 12 ; DUAL ARITHMETIC. 11. From what we have stated, it is evident that if tables be employed, but two are required, one of the ascending branch ranging from 1.9,0,0,0,0,0,0,0, to 3,6,0,9,4,1,0,7, and another of the descending branch ranging from Le ee Fs ee BEY) DPR berms Bee Por BOS ibe 0’0'0’0'0'0 # to °33'0°3'4’ V0 1 with natural numbers and dual logarithms to correspond, proper reductions being made involving powers of both 10 and 2. 700 When powers of ro only are involved, no reductions require to be made. ‘T'wo tables, one of the ascending branch whose natural numbers range from I'00000000 to 2°991611 362 ; (I) and another of the descending branch whose natural numbers range from 999999999 to 2991611362; — (I) are then required. With such tables logarithmic operations may be effected' by mere inspection, and natural numbers are prepared for loga- rithmic operations by simply changing the decimal point one, two, three, &c. decimal places to the right or left until the natural numbers are to be found between 100000000... and 2°99I16113612... or between "299161136.... and ‘99999999.... Since, to change the decimal point one, two, &e. places to the right or left being tantamount to multiplying or dividing by 10, 100, &e,; in resulting natural numbers, the position of the decimal point is more readily obtained than if the operation was performed by common logarithms. It must be remembered that dual logarithms are whole numbers, those of the descending branch have a comma to the TABLES OF DUAL LOGARITHMS. 13 left above, and those of the ascending to the right delow ; thus, the dual logarithm of 2: as well as the dual logarithm of 3 is the whole number 69314718 but written 1, (2°) = 69314718, Y, (5) = ‘69314718 sum = 00000000 = J, (1) If the dual logarithms of the ascending branch be considered positive, those of the descending branch must be taken as negative, and vice versd. Although the calculations throughout this work are made without the use of tables, and by processes designedly rendered prolix for the sake of clearness, yet, before entering upon the general discussion of the descending branch, it may be necessary to show, when tables of logarithms are employed, how vastly superior tables of dual logarithms are to those of common loga- rithms in both accuracy and precision. With what numbers must tables of dual logarithms (1) (II) (10) be entered to find the logarithms of 245°072 98°3657 4°846321 2°345678* 1°35672* 33°4455 0012345 ‘4763 "8765432" “OO! zt I0OO° 1000" 9473° 1276° The numbers marked with the * have the decimal point in the required position; to find the dual logarithms of the other numbers the tables (I) (II) must be entered with 245672 (1). 798365 (II). ‘4846321 (II). 12345 (J). 470g (Lh). 334455 = (11). G ay: I" (1). I° (1). 7 (1). ove Cinta Wh 1'276 (1). 2°45672 becomes the first number if the decimal point be 14 DUAL ARITHMETIC. removed two places to the right, marked 2,; 1'2345 becomes the second number when the decimal point is removed three places to the left, marked ’3; and so on. Multiply 90°986868, 19:4334858, and 295429627 continually together by dual logarithms. 2, . ¥,(2°95429627) = 108326047, 1, 4, (1°94334858) = 06441255, 2, (190986868) = 9445506 1653217096, I, ¥,(1) =’230258500 1, (52237607) = °64936713 Result. Product = 52237607 In practice, subtraction is avoided. in all cases, by substituting the arithmetical complements of that class whose sum is numeri- cally least for the logarithms. It requires but a trifling inspection to decide which of the two class of logarithms has the greatest numerical value. Work of the above Example in a practical form. 1891673953 ar. co. 133558745 ar. co. "9445506 230258501 V, (52237607) = °64936713 Product = 522376°07 The result 0064936713 is written ’64936713, since a whole number is not altered in value by prefixing ciphers to the left. Hence when dual logarithms are employed no allowance has USE OF LOGARITHMIC TABLES. 15 to be made on account of having employed arithmetical com- plements, which is one of the advantages of this over other systems. The management of common logarithms is rendered difficult, because the decimal part is always taken positive, while the whole numbers or indices may be either positive or negative. Thus, the common logarithm of ‘00012345 is made up of two parts, — 4, and + ‘ogI49QI11, written 4:09I4QII. Find the value of 580285005 x 82°550825 x (0092730300 by dual logarithms. 3. V,(2°85095) = = 104765225, 2 Vy, (92730306) ar. co. = 12452512 2 a V, (82550825) ar. co. = 180824393 J, (2°18239151) = 78042130, 00218239151 = the required value. 12. To reduce a dual number of the ascending branch to a dual logarithm. RULE. To the dual’ number taken as a common number, add 31018 times the first digit, and 33 times the second digit; then subtract 5 times the first three digits, a cipher being inserted after each, from the sum and the remainder is the dual logarithm. » Demonstration. Let 4, Ups Ugy Usy Ugy Ugy Uy Uy be the dual number to be reduced to. a dual logarithm ; 16 DUAL. ARITHMETIC.. then, “ Dual Arithmetic,.a New Art,” p. 212. y) 1, = 9531018, = IO000000 — 500000 + 31018 yy I, = 995034, = I000000— 5000 + 33 J, 1,= 99950,= 100000 — 50 Vy" 1, = 10000 VT) = 1000 415 = 100 » I, = 10 ) I, = I V, %» = L0000000u, — 500000u, + 31018 u, J) Up». = 1000000%,— 5000%,+4+ 32%. J, %» = 100000u, — 50u, Y, Ua, = TOOOOU, so Pate 1000 u, Vy Uy = 100 u, Wy Up = LOU, > Ug = lu, But ut hu + hu + Ke. = I,u, %, Uy Ke. also 10000000, ++ I1O000000u, + 100000u, + Ke. being tantamount to writing the dual number as a natural one;. and 500000u, + 5000u, + 50%, = (IO0000%, + 1000%, + I04u,)5,. which is the same as to say, five times the first, second, and third digits supposing a cipher placed after each; hence, the truth of the rule is established. When the analysis of the ascending branch of “ Dual Arith- metic, a.New Art,” was being drawn up, the Author first gave: this Rule, with other short methods of reduction, and some peculiar examples to show, among other things, that when the Theorems of Taylor, Maclaurin (or rather of Stirling), Lagrange and Laplace failed to apply,,the dual method was applicable in all cases without fault or failure. DUAL LOGARITHM FROM DUAL NUMBER. 17 Examples. Ex. 1. Reduce |, 3,4,5,6,7,8,9,2, to a dual logarithm. Y 35455 :6,7,859,2, 93054 =3 x 31018 132 =4 X 33 BAOOT Caz &: 152025 = 304050 x $ Dual logarithm = 33140828, Hx. 2. Reduce |, 7,2,6,0,7,8,2,6, to a dual logarithm. 2= | 7 ,2,0,0,7,8,2,0, 27 US0 = 7 x 31018 66 2 x 33 72825018 351030 = 702060 x § 1, (2°) = 690314718, 13. To reduce a dual logarithm of the ascending branch to a dual number. When the given logarithm is greater than |, (2°), or J, (10°), we have shown (8), how it may be compounded of multiples of 69314718, = J, (2°), and 230258509 = |,(10°) and a logarithm numerically not greater than (34657359), half the logarithm of 2-. Let the remainder thus found be of the ascending branch, and if it does not consist of eight places of figures, establish eight places by prefixing ciphers to the left; then apply the following RULE. Add once, twice, three times, &c. 500000 according as the first figure on the left of the sum, becomes respectively 1, 2, 3, &c. 18 DUAL ARITHMETIC, Subtract 31018 times the first figure, which must not alter after the operation, but reappear in the remainder. ‘Then add once, twice, three times, &c. 5000 according as the second figure to the left of the sum becomes respectively, 1, 2, 3, &c.; subtract 33 times the second figure, which must not change in the opera- tion, but reappear in the remaimder. Again, add once, twice, three times, &c. 50, according as the third figure of the sum becomes respectively I, 2, 3, &c. and the dual logarithm is reduced to a dual number of eight digits. This Rule is the converse of the last, (12) and requires no demonstration. Examples. Fa. 1. Reduce the dual logarithm 547842164, to a dual number. If twice 230258500, and once 69314718, be taken from the given logarithms, the remainder will be 18010428, 18010428, 500000 =I x 500000 LoS toda 2e 31018 =I1 x 31018 18479410 40000 = 8 x 5000 . I8519410 264, =—8 x 33 I8519146 25.0 = 5 X50 ¥ 1,8,5,1,9,3,9,6, nae 1,8,5,1,9,3,9,6, = 547842164, DUAL NUMBERS FROM DUAL LOGARITHMS. 19 Ex. 2. Reduce the dual logarithm 211373490, to a dual number. The given logarithm and ’230258509 together gives ’18485019 a logarithm of the descending branch, which may be reduced by Rule (15), which will be found on page 21. Ex. 3. Reduce 69314718, to a dual number. 69314718, 3500000 =7 X 500000 72814718 ie 20007 4-3 1OLS 72597592 10000 = 2 x 5000 72007592 Ot 20 33 72007526 300 =6 x 50 Dual number = J, 7,2,6,0,7,8,2,6, 14. To reduce a dual number of the descending branch to a dual logarithm. RULE. Add to the dual number written as a natural number, five times the first three digits, supposing a cipher placed after each, 36052,.multiplied by the first digit, and 34 multiplied by the second, the sum will be the dual logarithm. 20 DUAL ARITHMETIC, Demonstration. Since ‘QA, I,1}0,1,0,0,0,1, = "1 "99 ¥,0,1,0,1,0,0,0,1, = I 999 1, 0,0,1,0,0,1,0,0, = I 9999 J) 0,0,0,1,0,0,0,1, = I “99999 J, 0,0,0,0,1,0,0,0, = I, &e. — &e. us (9) + J, 1,1,0,1,0,0,0,1, = 0; but , '1,150,1,0,0,0,1, = 105 36053, vy, (9) + 10536052, = 0. Ina similar way it may be shown that } C99) + 1005034, Y (999) + 100050, J, (9999) + 10000, = o &e. &e. O O | The dual logarithm of ‘9 is negative, and represented by the number 10536052 written ’10536052; but ‘9 1s written “14 and \, (9) 18 written °1 In the same way it may be shown that the dual gS of ’o’1 4 is equal ’2005034 written O71} or 71 = 100508" '0’0'1'} or 71’ = "100050 *0’0’0'1 4} or “14 = *10000 4 &e. &e. DUAL LOGARIFHMS FROM DUAL NUMBERS. 2I "yf = 100000002, + 500000¥, + 36052, ? ? v, "= 1000000v,+ 5000%,+ 34%, ? ? v, {= I1Q0000v, + 500, reg TOOOO”; &e. Ke. But 0, h + 0,'f +0," + &e. =v, "0, "0, &e. "hs and 10000000, + 1000000, + 100000%, + &c. being tantamount to writimg the dual number as a natural one; while 5 (100000, + 1000”, + 10%) = 500000”, + 5000v, + 50%, ; which is the same as tying five times the first, second, and third digits supposing a cipher placed after each. Hence the truth of the rule is established. Let it be required to reduce ’6'6’0'6’8’2’0'24 to a dual. logarithm. '6'0'0'6'8’2’0'2 3030000 =5 x 606000. wLOgt> = 0°x 76052 204 =6%X 34 69314718 15. To reduce a dual logarithm of the descending branch to a dual . umber. RULE. Subtract once, twice, three times, &c. 536052, according as the first figure on the left becomes 1, 2, 3, &c. which first figure must not alter but reappear in the remainder. Then subtract, once, twice, three times, &c. 5034, according as the second to 22 DUAL ARITHMETIC. the left of the remainder become respectively 1, 2,3, &c. Again, subtract once, twice, three times, &c. 50, according as the third fizure of the remainder becomes 1, 2, 3, &c. respectively. Thus the dual logarithm is reduced to a dual number of eight descending dual digits. This rule being the converse of the last requires no demonstration. It is not necessary to say more in this place respecting the descending branch as it will be fully discussed hereafter, both independently and in conjunction with the ascending branch. The practical calculator, however, who requires to be indepen- dent of tables and possess means by which the accuracy of his results may be readily tested, can see, we have no doubt, from these preliminary propositions and examples, how dual arithmetic completely and simply furnishes these requirements. At the same time an operator employing tables will easily perceive how incomparably superior tables of dual logarithms are to those of common logarithms. 23 CHAPTER II. RESULTS OBTAINED, BY DUAL DEVELOPMENTS, OF SIMPLE OPERATIONS CONVENTIONALLY EXPRESSED IN ALGEBRAIC LANGUAGE. PRODUCTS, QUOTIENTS, POWERS, AND ROOTS. Recapitulation of Conventional Arrangements and Notation. 16. THE dual logarithm of A is written J, (A). If US hus ty uy s-. =u, then },(U)=u, and {,u,u,u,,....=4%, J (2) = 69314718, J, (10) = 230258500, A comma is placed on the right of positive dual logarithms, and on the left of negative dual logarithms. Thus, 34567844, is a positive dual logarithm, the same as + 34567844 a positive whole number; and ’45678921 is a negative dual logarithm, the same as — 45678921 a negative whole number. A dual logarithm is changed from positive to negative, and from negative to positive, by simply changing the position of the comma from right to left or from left to right, as the case may be. The arithmetical complements of dual logarithms (ar.co.) do not retain the comma. To find the arithmetical complement of a dual logarithm ;—begin at the left, set down minus 1, written T, then take each of the figures from 9 except the last figure on the right, which must be taken from 10. 45665423, dual log. 765432098 dual log. 154334577 ar. co. 123456702 ar. co. 24 DUAL ARITHMETIC. Logarithms of the ascending branch have the comma to the right, while the comma is to the left of logarithms of the descending branch. Examples. Ea. 1. Find the cube root of (1865 °655)(°02691008)(848:21877) (328°7077)(*13.465965 5 )(6296°168) Ans. *53462388 Representing by (A) and (B) the numerator and denominator of the fraction. (A) (B) 3,| (1865655) =62361210, | 3,/¥,( *3287077) =111258635 '2| J,(2"691098) =98994929, |’1|1,(1°34659555)= 29758037, | 3 V,( *84821877)='16461667 | 4,|J,( 6296168) = *46264392 4, 144894481, 6, '127764990 6 127764990, 3)2 272659471, o and ’2 over "460517018 J, (10°). For the cube root) divide by §3) 187857547 59 Re ye a0 62619182 44,0,6,7,2,5,8,8,8, = *53462388 These dual logarithms and the natural number answering to the resulting dual logarithm (62619182), may be independently calculated at once by the methods and rules laid down in the preceding chapter, or by any of those detailed in “ Dual Arith- metic, a New Art.” 17. If tables of dual logarithms be employed, like those described (11) ranging from 1°00000000 to 2’°99161136 and from "299161136 to *999999999, the required numbers are obtained by mere inspection, and with far less inconvenience than with a table of common logarithms, PRACTICAL EXAMPLES. 25 __ The number of places of figures the decimal point has to be removed to the right or left, being noted, (C), the dual logarithm of any number, as 1*865655 may be employed to represent the logarithm of 001865655 01865655 "1865655 1865655 18°65655 186°5655 1865°655 &e. (C) In practice it is not necessary to set down, as at (A), (B), (C), the number of places which the decimal point is removed to the right or left to produce the number to be operated with. Indeed, the final number (’2) found by taking the amount of B (6,) from the amount of A (4,) may be instantly counted before commencing to operate. Referring to (A), (3,) with the comma to the right is considered positive, and signifies that the period is to be removed three places to the right to bring 1°865655 to 1865°655. Again ’2 with the comma to the left, is considered negative, and indicates that the period removed two places to the left will bring 2'691098 to ‘02691098 ; and so on. If the value of (1865 °65 5 )(2691'098)(848'21877) ef (328°7077) (1°3465965 5)(629°6168) had to be found, then (A) and (B) would become (A) (B) 3> 3» 7) 3, 6 3, co For the cube root divide by 3) ay 9, I, and O over. 26 DUAL ARITHMETIC. In the first case ’2 does not contain 3; then ’2 is over and twice the dual logarithm of 10 is incorporated under a negative form with the amount 272659471,; in the latter case, the natural number corresponding to the dual logarithm answering to the amount 272659471, has to be ane by 10; or the natural number answering to 42400962, must be mraltiphed by 10. For, 272659471, 230258509; Vy (10) |, (1'52807622) = 42400962, 152°807622 is the result in the latter case. It is almost unnecessary to remark, that, instead of adding and subtracting as above, the resulting logarithm may be found by addition. 18. Before taking the arithmetical complements, the commas of the logarithms of the dividing factors have to be changed from right to left, or from left to mght, as the case may be. Then, the arithmetical complements of the logarithms with the comma to the left, if that class be Zeast in amount, have to be taken. On the contrary, the arithmetical complements of the other class, if that class be less in amount, are to be taken, (12). yy OO5055) Tia eee 137638781 ar. co. W220 1008) i ee. fue IOIOO5O7I1 ar. co. ytd SO4G21677)". 2. ae "16461667 MEL SEO TOT) iio cee eke 1888741365 ar. co. Divnor (134639085) Bae! ey 29758937 Wit ZO0TOS) oe. a 153735008 ar. co. ah Cise) ston "460517018 3 ) 187857547 J, (°53462388)...... "62619182 It is evident that no allowance has to be made on account of having to employ arithmetical complements, which is one of the ——— = PRACTICAL EXAMPLES. 27 many advantages of this over any other system. The manage- ment of common logarithms is rendered difficult because the decimal part is always taken as positive, and is the only part given in tables, while the whole number or indices may be either positive or negative; for example, the common logarithm of .00012345 is made up of two parts, — 4 and + 9’014911, written 4OQI491T. Hix, 2. Find the cube root of BS TS SS by the ascending branch involving powers of 10° and 2° and also by the shorter method by addition and powers of only 10. Since | Bae X% 799% 355 _ 345 x 703 X 355°. 8°40 x 366x887 840 x 366 x 887°’ then (3°45), = 123837420, (8°40) = 212823170, (7°63), = 203208780, 1,(3°66) = 129746310, V(3°55), = 126694750, 1,(8°87) = 218267470, 453740950, 560836950, These dual logarithms may be found at once by the methods given in the previous chapter, and in “ Dual Arithmetic, a new Art,” page 213. These methods, however, will be considerably simplified in the course of the present work. + 453740950, =,(3°45) + 1.(7°63) + 10(3°55) — 560836950, = Y,(8'40) + ¥,(3°66) + J,(8°87) For the cube root SES divide by 13) _ 107096000, — 35698667, = log of required cube root. Vv (2) 69314718, 33616051, log of 1°39956362 the half of which is ‘69978181 the cube root required. The nearest number corresponding. to the dual log ~ 35698667, written 28 DUAL ARITHMETIC. °35698667 may be found by the methods indicated above. If required, the number could as easily be obtained to eight, nine, &c. places of decimals. The result — 2365333, being negative, the addition of log 2 renders it positive, then 1°39956362 has to be divided by 2. Second and shorter method. 2 13(345) = 7106421089 Si3> 1763) = 727049728 uns V.C355) = 7103563757 2, = ¢ (840) 182564661 ar. co. 4 3, I | 16366) 1899487802 ar. co. S 3, 6 \J,(887) 188008963 ar. co. diff. o 3) ’107096000 V,(69978181) = “35698667 = "34070375 It requires more skilled labour to obtain similar results by common logarithms, for common logs are made up of a combina- tion of whole numbers and decimals ; besides, as before observed, the common log of a decimal fraction is part positive and part negative, while dual logarithms are always whole numbers, either positive or negative. (0376 Xx 198 V66°8 x °947 Ex. 3. Find the value of t to nine places of decimals. 66°8 x ‘947 = 6°68 x 9°47 "0376 x 19°38 = °376 x 1°98 = (70) 3°76 x 1°98 J,(1°98) 68309680, 1,(6°68) = 189911790, 1,(3°76) 132441890, V,(9°47) = 224812880, + 200751570, Square root 2) 414724670, y,(10) — 230258500, 207362335, For cube root 3 ) — 29506939, — 9835646, PRACTICAL EXAMPLES. 29 — 9835646, — 207362335, — 217197081, 9) _ 7) — 434395962, — 62056566, V, (2) = + 69314718, 7258152, = ¥, (1'07528041) 2) 1'07528041 " *§37640205 required root. Shorter method involving \,(10.) but not ,,(2.) (A) (B) I, ¥,(1'98) 131690320 ar. co. 2, J,(.668) = ’40346718 I (376) = 97816619 © 1,(947) = 75445629 o 3) 29506939 2) 2, 2) °45792347 9835646 oe 22896174 177103826 ar. co. 177103826 230258509 ‘217197981 2 7) 434395962 | 62056566 = q,(’537640205) 19. The small numbers under (A) and (B) may be omitted in practice, and also ar. co. to designate the arithmetical comple- ment, which is sufficiently indicated by being always preceded by minus one (1). Ex. 4. Muliioly 548648128 by 386°344448 and take the square root of the product ; and also the fifth root. 548648128 = 27. 10°. (1'37162032) BOO'S4AAAS 210s (193172224) 30 DUAL ARITHMETIC. (1°37162032) = — 31599277, 1,(1°93172224) = 65841199, 5 (10) = 1151292545, 3( 2) = 207944154, 2) 1456677175, 728338588, V(10") = 690775527, 37563061, log of 145590923. 1455°90923 is the required square root. Again, | 5 ) 1450677175, 291335435, V,(10) = 230258509, 61076926, = (1°84184672) 18°'4184672 is the required fifth root. Practical Method. 4, ¥,(548648128) 139970204 3; 1,(-386344448) 104897408 2)7, 3, and I, over |,,(10.) = 230258500, 2) 75120I2a V,(1'45590923) = 37563061, 1455°90923 = square root. 139970208 5)7> 104897408 I, and 2, over J,(10°) = 460517018, 5)305 384630 J,(1'84184672) = 61076926, 18'4184672 = fifth root. PRACTICAL EXAMPLES. 31 20. It may be contended by some that such results as we have obtained might be more conveniently and expeditiously found by a common table of logarithms. T’o which we reply that without the use of tables of dual logarithms our methods might require more labour, yet their results may be depended upon and tested for their accuracy up to the last figure. It must be remembered that the effective use of a table of common loga- rithms is acquired only from considerable practice, and even when this skill is attained, we have no means of testing our results. All that we can be sure about, supposing the tables to be correct, is that the first five or six digits of the products, powers or roots of numbers, obtained from logarithmic tables to seven places of decimals can be depended on, though we, have no independent methods of testing the results. In every respect tables of dual logarithms such as described (1. and IT.) in section (11) page 12, are incomparably superior to any tables of logarithms that have hitherto been calculated. Kix. 5. Find the ? root of *832516529. "832516529 = ip (1°04004560) = 1% 1, 0,4,0,0,4,0,0,3, Y 2,4,0,0,4,0,0,3,= 3984135 V8) = = +207944154 211928289 230258509 a y,(10) — 18330220 LES 5)— 54990660 — 10998132 1(2) + 69314718 58316586 = |, 6,1,1,3,5,4,9,5, = 1'79170165 2)1'79170165 ‘895850825 the required root. G 32 DUAL ARITHMETIC. Shorter Method. V.(832516529 = 718330220 Oa 5)'54990660 V,(895850825 = ’10998132 Before the introduction of Dual Arithmetic, the calculator would find it difficult to solve such a question as the last without tables. By means of a table of common logarithms, he might worm out *89585, and no more. But whether this result was right or wrong, he could not decide with any certainty, unless he constructed a whole table of logarithms. In many cases tables of common logarithms are found to be very inconvenient. Thus, if it were required to find the value of (07) by logarithms, the operation would be as follows, common log of ('07)...... — 2 + *8540980 07 — ‘I4 + ‘059156860 = — ‘080843 IA’ a result which is negative, and therefore cannot be found in a table of common logarithms. But, — 08084314 = — I(1 — °08084314) = — 1 + ‘QIQI15686, written T'91915686 REDUCTION OF ORDINARY FORMULA FREQUENTLY EMPLOYED. 21, Diet 32ers, 8.5 8.’ on be the dual number corre- _ sponding to the common number N, and n the dual logarithm of af 2,5 th Uae Ge pasts A) 2,5: 2,, welch wee = 2, SUBSIDIARY ANGLES SUPERSEDED. 33 Then § N = 2? 107} u,, tay Way ee oe = 2P 10!) a,; To avoid the use of the bases 2 and 10 in the expression 2” 10% we may write it under the form p‘ g, the inverted comma between the p and q indicating that the base 2 and 10 are sup- pressed. Thus, N=p‘q)u...=poqyn In applying the same notation to any other common num- ber M, then, M=rsl)u,...=rsym JV) =7)(2)+sJ),(10) +m; +, VN? + M? = {2p 2g J20, + 27 28 },2m,}? Of m and n let m be the greater, Then, (V+ MY = ropnf PEF 4 Vom 2r2s 2n =rsin {2(p—r) 2(¢—s) + }2(m—n),}3 {2(p — r) 2(q — 8) + ¥2(m—7n),} may always be reduced to the form 2 a‘ 26 J, 2¢, and J, 2(m — n) is less than 2, since neither m nor n can ever be equal to 2. The square root of 2 a‘ 2b), 2¢, = a OC, (N?+ M*)! =(@+4+r)‘(0 +s) )6=(a+7r) O+s) (+0), Ex. 6. Find the value of V(635°297388)” + (253692174) Here M = 635:297388 = 2” 107 (1°58824347) = 2‘ 2 |, 46262870, sree i, N = 253692174 = 2" 10° (1°26846087) = 1° 3 23780432, = pan, 34 DUAL ARITHMETIC, The calculations with 2 and 10 are extremely simple. — V2 (m — n), = ) 44964876, = 1°56776145 Hii S$ i-o, 7 eee 2 + 156776145 = “ (1 + 062710458) = oe ), 6082269, and corresponds to the expression 2 a‘ 26 26, Hence, /(635°297388)" + (253692174) = 2°10° |, 23780432, = ) 3041135 = 2 x 10° |, 26821567, = 2615'2587 And in this way the value of any expression of the form VM? + N? can be obtained, whatever numbers M and N may be. It may here be noted that such an expression as this cannot be solved by common logarithms, except by adapting the expression VM? + N? to logarithmic computation by means of the intro- duction of a subsidiary angle—a method which requires the use of logarithmic tables of Trigonometrical functions. The value of (N¢ + M‘)i, may be found in a similar manner. Thus to find the value of v/ (4567°8346)* — (4321-695)! to nine places of figures. (4567°8346)'={2710°(1'14195865)}*=(2" 10°, 13274495,}8 = 24107 |, 8849663 (4321'695)#=(2"10'(1'08042375)}' ={2710°. 7735337}! = 2410") 5156891 3692772 SUBSIDIARY ANGLES SUPERSEDED. 35 4 of 5156891 = 1718964 and J, 3692772, = 1:03761800; then {(4567°8346)8 — (4321°695)3}# = 2! 108), 1718964,{ 103761800 — 1}8 103761800 at i} = 2. (188090000) } os fe A} 63175040} Ube 21058 2¢ | 1718964, |, 21058347, = |, 766838758, = 2°15 305,460, the required root. ‘I'he general reasoning employed in the last example may be applied to all examples of this class. The more complex methods of reduction are designedly employed. To find the value of VM?—N?, which cannot be found by ordinary logarithms without the use of subsidiary angles by the dual method. : Ex. 7. To find the value of V(15676:06501)° — (12649'47259)* to nine places of figures. Since, (A + B) (A — B) = A’?- B’ Put M=A+B,andN=A-B; According to the notation before employed, a aes oT TL Poy ate G++ 8 mMt—-n Re MN oat M = 28325'5376 = 14 | 34803152, N = 3026°59242 = 1‘2 |, 41420021, 2°7 | 76232173, 1S |, 38116087, = 2" 107 J, 38116087, = 9259'04208. 30 DUAL ARITHMETIC. EXAMPLES ON THE NOTATION FOR THE BASES 2 AND 10, APPLIED TO QUESTIONS OF INTEREST AND ANNUITIES. 22. The notation previously employed will be rendered more complete, in many cases, by putting one letter instead of two to the left of the sign J, to express the combined powers of 2 and 10. It will be found that this contraction will neither render expressions obscure, nor curtail their generality. N = 2? 102 J u,, Ug) Ugy 0 = PI YD, was expressed under the form N=piq)u ..--=pavs which by the method above indicated will become N=nJu.....=n 3 where the italic (n) represents the dual logarithm of the dual number : SF TR A ek OF ay U,, 26,, tt, sae : and the roman (n) represents pg or 2” Tot Log of N, = J,(N) = J.(m) + , {2? Tot}? = 2”? 10% = vp‘ vg = UN, » being a whole number or a fraction, positive or negative. Again, let 8 M = 2" 10° ,%,, %, U,)-- + = 2° Or i ee ene = 98.) on, Or M=myu,..... =m ym, NOTATION FOR POWERS OF 2 x 10. 37 Then MxN=(min)Jut+u)..... = (m+n) )(m + n), so that m + n indicates (p + r)‘ (¢ + 8), or 2? +" Iotts, And Bea) 466, —»).22 2m Y@- 9) Then m — n indicates (7 — p)‘ (s — q) or 2"—? 10°? Hence, it must be observed, that in the addition and sub- traction of the small roman letters to the left of |, the powers of 2 can only combine with the powers of 2; and powers of 10 with powers of 10. For examples in interest and annuities : Let P denote the principal in pounds. I the rate of interest. . : I z the interest of one pound for a year, = am Y the time in years. M the amount of P at compound interest. R = (1 + 7) the amount of one pound in a year. 23. Then it is shown by elementary writers that, M= PR. 38 DUAL ARITHMETIC. Ex. 8. How much would £327°436 (P), amount to in (Y) 27 years at 3% (1), per cent per annum, compound interest ? Here P = 327°436 tex 27 And I = 33 And P, Y, and I are given to find M. P being a common number can be expressed, as shown above, in the form 2™ 10" |p, orp yp, And R being less than 2, since mn Th ded 35 te eee can be expressed in the form R= fr, R* =f Yr, . M=PR' =p dp, VYn=py(irtp), - R = 1035 = ¥ 3440148, R” = 4.27(3440148), = 192883996, P = 2' 10° (163718000) = p |p, = 2' 10° |. 49297588, M=PR* =2'10°Y,49207588, 192883996, =2'10", 142181584, = 2" 10°, {2(69314718) + 3552148} = 2°10 ) 355148, = £828°927884. EXAMPLES IN INTEREST. 39 Ez. 9. How much money must be placed out at compound interest to amount to £3,000 in 20 years, at 5 per cent. 2 24. Here M, R, and Y are given to find P. Follett coat yr =my(m — Er); But | M = 3000 = 2" 10°), 40546562, =mJm, Y= 20. R = 105 = |, 4879021 = J7, the ¥ ¥r, = 1 (4879021 x 20), = |,97580420, = 21, 28265702, ee mym, 2’ 10°) 40546562, rat 2 |, 28265702, = 10°], 12280860, = £1130°66794 Ex. 10. At what interest must £422°3575 be placed out to amount to £666°666 in 15 years ? i 25. Here M, P, and Y are given to find I. OA); m@- By (m—p),_ Mm—p,m— en (p)F= a a's = vi M=666'666 = ae = 2°10°|,5 1083464,=mm, P = 422°3575 = 2°10°(1'05589375) = 2°10) 5438761,= pp, M p = 945644703, oa 45644703 R= a 2) = |, 3042980, = 103089748. trast Vestas @ = 03089748 and 7 aeaa I = 3089748, the interest, or rate per cent. H 40 DUAL ARITHMETIC. Ex 11. What will £927 10s. amount to in 18 years, at © per cent. compound interest, payable half-yearly ? 26. Here ; R, Y, P, are given to find M. M= PR* R= 1°03; ¥ =36; P= og R= Yr, = 9 2955883, RY = Vr, = 4 36(2955883), = 24 37097070, P=pJp, = 2° 10°) 14788110, . M=PR* =PJ(Yr +p), = 24 10° § 1885 180, = £2688" 159856. Ex. 12. In how many years will a sum of money, lent at 5 per cent. per annum, compound interest, double itself ? 27. Here M, P, and BR are given to find Y. aye R = 1°05 = |, 48709015, = Jr, Hie 2 : 1 Yr, = | 60314718, and Yr = 69314718 y — 09314718 _ 69314718 00511563 2 = 68802900, = J), (1'98978978) It may also be shown (13), p. 17, that 68802900, = J, 7,2,0,9,5,7,0,8, and | 7,2,0,9,5,7,0,8 = 198978978 31. The next examples, numbered I. II. III. &c. may be con- sidered an amplification, and are selected to illustrate other simple but important dual reductions which have often to be made. INVOLUTION AND EVOLUTION. 47 PRELIMINARY REDUCTIONS OFTEN REQUIRED IN CALCULATING THE ROOTS OF EQUATIONS. 20 I. If }7,0,0,0,0,0,0,0, = R ; then R = J 0,3,3,5,0,9,0,7, Reduction, (12). Why aG 7 x SIOLS 1} 7,0,0,0,0,0,0,0, Subtract 3 8 SY Wp SL ep rae £0,606 771,26, 03335856, OO15 150 = 003030 x 5 03351006 99 = 3 X33 (13) R = 10,3,3,5,0,9,0,7, It is scarcely necessary to add that, Reduction, (12) signifies that the succeeding reduction is made according to the principles explained in article (12); and that .-. (13), denotes therefore by article (13). 20 Il. If R = J7,0,0,0, then R = J,0,3,5, nearly; this result may be found by mere inspection, for taking the dual as a common number, we have 7 — 035 which isa little over 0335 20 ts 355 ues ol £3 5; xis am, Dearly. The succeeding remarks should be particularly observed. 32. tS Dual developments never result in approximate values ; however unlike equal dual forms may appear, every form | I 48 -'.. DUAL: ARITHMETIC. of the required value is true to the designed degree of accuracy, which may be as great as we please. From the flexibility of dual numbers we are not obliged to resort to trial and error, nor are we confined to one set of numbers and particular narrow intervals to exhibit required results. ga 28. ILL. In the above example R= \} 0,3,3)5,0,9,0,7, = 1°033921 16 and 2a? oe ese R= "00 48643 073,5; = 103392116 also In finding J, 0,3,3,5,0,9,0,7, = 1'03392116 in a direct manner without being obliged to retrace our steps, 1703392116 is not found by approximation because ’0’0’1’4’8’9'4°3 y0,3,5, is also found equal to 1°03392116 by a like direct procedure. 20 IV. If |,0,0,9,0,0,0,0,0 = R , then R = J,0,0,0,4,4,9,7,8, Reduction. (12). J, 0,0,9,0,0,0,0,0, Sub. OO00045 20) 00899550 . V 9,0,0,4,4,9,7,8 009 Ne AL , 59 7 00045; which is not much greater than 0044978. 33. When the first three dual digits are zeros, or ,0,0,0..... the remaining five dual digits may be treated, and reduced as if they were common numbers ; for example, if |,0,0,0,9,8,0,2,8, = R*, then R = J,0,0,0,3,2,6,7,6, See Article (9). Reduction. 3 ) $0,0,0,9,8,0,2,8, 1, 0,0,0, 3,2,6,7,6, INVOLUTION AND EVOLUTION. 49 This property may be generally established thus, the dual logarithm of YO, 0, O, t6,, Ups Uy Uy Us written J, 0,0,0, Uy Uay Uy Uz»Uy (See page Q). is equal 10000u, + 1000u, + 10Ou, + Ou, + u, and is the natural number expressed by the dual digits taken as common digits. 3 4 Neer te ==) 2, 3,4,5;0,7,9,9, then BR =f 2,5,0,7,3,0,3,2, Reduction. See Articles (12) (13). 62036 + twice 31018 V 2,3:4)5,6,7,8,9, Subtract 101520 — five times 2,03,04,0 99 + three times 33 22503724 3 ye eg A Ue 33755586 5 7) 168777930 oar Gi.) 4 “f 1O + twice 5..... ZED ii ae 62036 — twice 31018 25049097 + 2500 + five times 5... 25074097 165 — five times 33. 1:2,530;7,930)432) 50 DUAL ARITHMETIC. 34. These reductions are so readily effected, and so fully illus- trated here and elsewhere, that, in future, such trifling calcula- tions, in most instances, will be omitted, and in such cases, for example, we shall say, 20 Hy A ie aly, . LW = §0,3)3)5)0 eee 3 5 Ik Av, = 4h 235455 075050) °. KR =) 25.0.7 2 m » It is a very easy dual operation to find R when B is given, yet a’ practical solution of this simple problem, without the use of tables, defied the combined skill of mathematicians before the Author, Oliver Byrne, discovered and developed dual arithmetic. QUESTIONS RELATING TO INTEREST AND ANNUITIES: CONTINUED. Ex. 16. Suppose an annuity of £50 to amount to £1413°98528 in 20 years, what is the rate per cent. compound interest ? From (5), page 42, or MR-AR=M-A 20 1413°98528R — 50R = 1363'98528; (K). R = 1, will satisfy equation (K), but this value of R is not admissible, since R is always greater than 1. Hence another value of RK must be found, such that 50 times R in the 20th power taken from 1413°98528 will leave a remainder = 1363°98528, (KX). Many dual numbers may be found, and each reducible to R= 1035, which will also satisfy equation (ix); one set of these dual numbers will be presently found, showing that when 20 R = ¥7,2,0,9,5,5,6,3, then R = | 0,3,4,5,5,2,3,6, = 1035 ROOTS OF PARTICULAR EQUATIONS. 51 35. It has been proved in particular cases (27) (I.) (IL) &e. and it will be generally established hereafter, that if oe Yo. . and iia )w... then the ratio of |u.. to Jw.. taken as natural numbers approaches the ratio of n tom. This knowledge would be of little value unless we also had the power, when R is given or assumed, to determine R true to the limit of the designed degree of accuracy, and besides every path pointed out by the ratio of m to m should lead in a direct way, without deviation, to the exact value sought. These demands are fully supplied by dual arithmetic. Nor is this all, for we may take dual digits much greater or much less than any particular digits pointed out by the ratio of n to m, andwyyet obtain, without guessive artifices, a result as near the truth as we please. Since this method gives the same result, as near the truth as we please, by several direct processes, it presents a series of direct operations, and not a succession of approximate trials. A method may be direct, and yet give results that con- tinually approximate to correct results. Returning to equation (IX) which may be put under the form 20 ax—bxw =e In which a-—-b=ce chy nN ae es tah . ; | re The fraction that aus of 6 will point out the first dual digit. Ty ARE pan sae 20. + 20 ae TARO hae : Ze erat 7 20 If R = 7, then R= J§0,3,3,5,0,9,0,7, (34). 4 9,3,355,0,9,0,7, (1) 1 7:0,0,0,0,0,0,0, (1) + 141398528 (2) — 50°0000000_ (2) + 1461°94934- (3) — 974358550 (3) 52 DUAL ARITHMETIC. (2) multiplied by (1) produces (3). This arrangement is maintained throughout. See ‘“ Dual Arithmetic, a New Art,” pp. xxvi. and 173. In finding a second convenient dual digit for a dual value of 20 R , many of the succeeding figures might have been omitted. +73 geoff + 1461°94934 ame tf once — 97'435855 — 24 1364°513485 take 1363985280 from (K). 24) $28 (+ Yon, 3 — 48 36. It may be observed that the method above instituted re- sembles common division, but the quotient figure is a dual digit, and may be in excess or defect, greater or less than 9, without involving error; the case is otherwise with common division. 20 If ry .=0,2, then +7 =410,0,0,0,0,5,0,3) 0am 20 The remainder of the process employed to find R and then (34) R may be arranged in the succeeding order. Mult. by 0,0,0,9,9,5,0,3, y 0;2, + 1461°94934 — 97°4358550 gives 1463'40408 — 993943157 TRI TAN ed Of gee A034 400 = 9989;2 one 99 50m — 26:22 ) 1364°01,03643 take 1363°98/530 _— from (K) —2\506 (+ 0,0,0,9, — 2/358 If 7, = 10,0,0,9, then *, = {,0,0,0,0,4,5,0,0, (34). Mult. by 1}, 0,0,0,0,4, 5,0,0, J 9,0,0,9, + 1463'40468 — 99°3943157 gives + 1463°47055 — 99'4838064 ROOTS OF PARTICULAR EQUATIONS. 53 erro 7 + 1463'470/55 — 99°48 — 99'483)8064 —_—_—_—_— — 26°31 + 1363:986|7436 take + 1363°985|28 from (K) a 2631) — | 4636 (,|,0,0,0,0,5,5,6,3, 1/3155 1481 1316 165 158 7 20 R = b 752;0,9,55550,3, and .*. R=0,3,4,5,5,2,3,6, = 1°03499987 = 1°035 nearly. The rate per cent. may be said to be 34. If our decimals had been carried out sufficiently far, we should have obtained 1°035 instead of 1:03499987. This example is the reverse of example 15, page 44. fix, 17. An annuity of £140 left unpaid for 33 years amounted to £16650'71552, compound interest ; what was the rate per cent, ? According to the formula employed in the last problem 83 16650°71552R — 140R = 1651071552; (K) R = 1 satisfies (K), but this value of R is not admissible, for R is always greater than 1. If the operation of extracting the next root to 1, be commenced with 6, 7, 8, or 9, respectively, 54 DUAL ARITHMETIC. then it will be found that 33 R = Os) 4,6,0,0,3,5,2,5, — 7 Ay 2,9,6,6,5,6,9,4, = 8) 155 50)253300554: => 9 J0,3,5,6,6,4,3;55 = 9325. It is evident that 8 may be substituted for Re since 8 times 140 = 1120, which approaches ‘o7 times 16650. How to find such limits of the values of unknown quantities will be discussed presently. However, in this equation (Kk), the 33rd root of 6 = |,0,5,.... .the 33rd Trooh 0b) 7) —=saeeeee the 33rd root of 8 = J,0,6,... & the 33rd root of O=og Hence, a mistake can scarcely be made, even by those ignorant of the theory of equations, so great is the range of convenient dual forms which the required value may be made to assume. -33))207944154, =dual log. of 8 06301338 Y 0,6,3,3,1,2,9,0 = 33rd root of 8. Multiply by J,0,6,3,3,1,2,9,0 bese + 16650..... — 7400. aa gives sen ke ee — 1120 che Tina ee OL Tid / iio ae ; — 1120 once — I1/20..... — 583) 106/13 ie take 105/10. eae from (K) — 1/03 (+ ¥1, vale 58 If y= 1, then r = }0,0,2,8,8,9,1,9, (34). ROOTS OF PARTICULAR EQUATIONS. 55 Mult. by |, 0,0,2,8,8,9,1,9, Wal; yee ake SOLIS ats Result Jyh) ne — 1232 + 538 35 of Ree oe — 1232 Cienvete beta |, — 604) LOU Gaui: take BOP WOry feo. from (KX) —42 Path Oe ee () O;5; If r, = 0,5, then 7, = |0,0,1,5,0,8,1,3, (34). Mult. by J 0,0,1,5,0,8,1,3, 9,5, + 1778498 . . — 123200 Result + 1781182... — 129484 + 5397 g3s0f + 17811/82... — 1294°8 once — 1294 84 . — 7551) 1651698. .. take 1651071 ... from (Kk) aye. — 6104... (Y, 0,0,8,2, 23 The root being so far determined by contracted operations, let 35 Heo, 1,5,0,2, then R = J.0,6,7,0,5,9,0,8, (34). The succeeding operation by using the coefficients of R and 33 Rin the original equation (p. 53) is independent of those above employed bs show that 8 | 1,5,8,2,.... 1s a convenient dual form of R. What follows, not only determines R to the designed degree of accuracy, but also proves the preliminary calculations. Mult. by J, 0,6,7,9,5,9,0,8, + 16650°71552 + 17816°24553 8 |, 1,5,8,2, 140°000000 — 1305°500527 Result 56 DUAL ARITHMETIC. + 539886 sof + 17816'24/553 — 1305500 once — 1305°50|053 a 765614 “165 10°74|500 take 16510°71\552 from (K) 7656 ) ont 948 () 0,0,0,0,3,8,5,2, at 33 Sis sikh. Se Goal Lp yeepey.s (Oss pe R = |,0,6,7,9,6,0,2,5, = 1.07; and .*.. 7 is the rate per tent. In Examples 16 and 17, dual methods of calculating unknown but well-defined magnitudes are applied without referring to the general dual system of solving equations hereafter discussed. Ex. 18. If the yearly rent of a freehold be £200, what ts tts present value at 54 per cent. compound interest ? From (7), page 43, ye Rear ae = 3636374, the present value. This example is introduced for the sake of uniformity, the required result is found by common division. The dual method is chiefly applied where logarithms have been found peculiarly serviceable, and in cases when neither logarithmic nor common arithmetical operations will apply, as in Examples 16 and 17. EXAMPLES IN INTEREST. 571 Hix. 19. Required the present value of an annuity of £140, which is to continue 33 years at 7 per cent. compound interest. The amount is £16650°71552. See Example 17. (33) PR = M = 16650'71552 Hie 507 =.) 670557 1, R =433 x 6765871, = } 223273743, M = 10°), 50986811, = 10°), 281245 320, M _ mm, 10°) 281245320, ee = = = 10° 1577, 223273743, ‘© ¥57971577 157971577, = 1°78553179 P = 1785'53179 Ex. 20. In how many years will an annuity of £50 amount to £2000, at 4% per cent. per annum, compound interest ? From (5), page 42, x Re 1 M=Ay—., i 4 Ce nt VY¥r,=R = [ aE aaa , which put = Js, Udoicr, Vey Yr V7.5 5 -'0r Yo= . R = 1:04375, _é eo PEN, er oss7>) 4 a I 2°75 2°75 = 21, 31845377, = | 101160095, = 1s, 58 DUAL ARITHMETIC. and 1°04375 = ) 4282001, = 7, _ 8 — LOLI60095 r 4282001 = 23'6245 years. or Y = 233 years nearly. OPERATIONS INDICATED BY THE SIGNS + AND —. 37. Before proceeding’ further, it is necessary to explain how the dual sign of addition (+), and the dual sign of subtraction (—-), of the ascending branch, differ from the ordinary signs of addition and subtraction, and how these new signs are operated with. rN ME To 7 Then A= 0|0\0) O O oO O 6 0/0 alae 65/1 3/20 3 2/5 7 ay = 39/4 1|6 0/6. o|o. which put = A, 19 7108 480. 319 4)1 7O- 3:94/2. 20 27° 0\2 1% 414\270 A {U,,%» = 414 which put = A | UU Uy = 4146190 4 8/8 2440/7 Lie which put = A, ANS, ots 4°1"A 1B 1490 7 which put = A, DUAL SIGNS + AND —. 59 Then, by employing the dual sign of addition (+), this con- tinuous and well-known process may be indicated as follows :— U, =AtuA = AMT eg ) = AL's A | hie A | U,) Up) a PO, eA tuA = AL; mera, = A | 0,0, 4,5 = Al uA A, A | Uy Uy Uy Uy = A, | 0,0,0,0,t,, = a + UA, Py; A, ; Ke, ke, Ke. | vu, is written |w,, or |?u, |0,0,¢,, is written |u,, or |* 2, | 0,0,0,2,, 18 written | w,, or |*w, &e. &e. 38. The units w,, wv, uv, .... In conjunction with +, may be operated with in any order whatever, provided that all the units are incorporated. A |0,0,z, =A+tuA =A(itu,) which put = B, ; Biv ou, =B iu, =B,+u,B, which put = B,; A|u,,0,u,4, = B,|*u, = B,4B,u, which put = B,; A | u,, u,v, = B,|?u, = B,+u,B, which put = B,; Ke. Ke. Ke. Then A, = Bu. Again, Alu, = Afu,A = A(ifu)=A,; poe == At An eA (It) = A, ; =(AtuA)tu, (Atu, A) =A(itu)tuA(itu,) . = A[(rtu) +, (14 4)] A | Uy UUs) ao A, Pus, 7 A, + uA, on A, ; ={A[(1 $m) $m (1 $m) utA [C1 $%) $m (1 $4) ]}} =A tatu dy dy [re de) $e (dey &e. Ke, 60 DUAL ARITHMETIC. 39. ‘The sign of dual subtraction (—), ascending branch. re. =A|u, =A—u,A = A(1—u,) which put = D, ; A apne : fa, 7A Vly =D, ty=D,—u,D,=D, (1, which put=D, ee =A|%,,U,Uy = D, |’u, = D,—u,D, which put = D,; &e. &e. Develop A |.w,,u,,%,,;%, .... in a form involving both + and —. Bak = A,{u,A,=A,; A, | Us,Uys = A|?u, = A,—wA,= AR Ay | uty, == A, lou, = Ay te A, | UU tyu, = A*lu, = A,—u A, =A; &e. &e. 40. ‘The dual sign of plus or minus for the ascending branch is written +. When the dual digits w,, u,, u,,.... are either plus or minus, one of the many developments of A |w,, u,, Uy... . will be {{(A}u,A) + u,(A+u,A)] $u, [(A+u,A) +4, (AduA)]} + u,{[(Atu,A)+u,(Atu,A)] 44, [(A+u,A) +u,(Atu,A)]} + &e. which is more concisely expressed under the form A{[itujtultu}tu........ } (Z). 41. The development (Z) may be given under as many different forms as there are variations in the permutation of all the dual digits taken together. When eight consecutive ascending positive dual digits are combined, (Z) may be given under 40320 forms, each expressing the same result, for 8x7x6x5 xX A4.x 3 X 2 xl = 40320, ‘In the latter mone when A = 1, (Z) may be given under the form ! + [%, + [~, + [u,+ [", + [~, . [u, + [“.+ [™, (Y). DUAL SIGNS + — +. 61 Suppose each of the dual digits to be less than 1o, and (Y) limited to eight places of decimals, then (Y) becomes (100000, 1,1,U,) + [U, + [u,+ [us + [yy In future each dual digit is supposed to be less than 10 if the contrary be not specified. 42. Coincidence of the corresponding values of the 5th, 6th, 7th, and 8th dual digits ina tabulated form; u,=0 u,=0 u,=0 U,=0. Natural Numbers. Dual Numbers, Dual Logarithms. Particular case 1°00004763 | 0,0,0,0,4,7,6, 3, 4763, General form — 1-00002,11,00,0, | 0,0,0,0,2,,UyM,Ugy UU ps Ugy (1000)w, + (100)%, + (10)u, + u, 18 represented by w,u,u,u,, 100000000 + (1000), + (100)u, + (10)u, + u, is represented by 1OO0OU,U,U,U, No error can be involved through considering 1o00ow,w,2,u, a whole number while being operated upon. 43. To find the natural number corresponding to a dual number of the form | 0,0,0,24,,2/,,W,)U Ug) Let the operative numbers or binomial coefficients . ad = . / ” for the dual digit w, be represented by 1 wu, wu, wu," ... am ie ls 7 a , " for the dual digit ~, be represented by 1 wu, u,' wu," ... &e. &e. Corresponding values of the 4th, 5th, 6th, 7th and 8th dual digits in a comparative tabulated form, when uw, =o U, =O Uy =O. Natural Numbers. | Dual Numbers. Dual Logarithms. ( 100054763 ie) 2 / i U 5 (-47) | 0,0,0,5,4,7,6,3, | j 100054775 | eur | | | + |u,, | | | 0,0,0,04,,2,)2 Uz) Uey | UUM UUs ; 1OOOU,U,U,U,|Wey | | | | } a7 UW, ("20,%,) | | General form | Particular case | 1000u,u,u, p q 62 DUAL ARITHMETIC, In this case (Y) becomes (10000u,u,u,u,) + [u, = 1OO0OU,U,U,W, + U,(1OOOO) + U,("U,U,) + WU, = - 2 , *, 1LOOOOW,06,1,.UzU, + Uy + W,(CU,M,) Hence the natural number is equal to the dual number, taken as a natural number, + the third operative number for w, + the nearest whole number to vu, multiplied by the decimal w,u’. This rule is readily reversed. Hence 44. ‘lo find the dual number and the dual logarithm of a natural number of the form 1:000u,u,u,7 q. RULE. Subtract the third operative number for the digit wu, and the nearest whole number to w, x decimal (-w,1,), the remainder is the required dual number when | is put for 1. ‘The correspond- ing dual logarithm is expressed by the five dual digits thus obtained. Examples. Ex. 1. Find the dual number and dual logarithm corresponding to the natural number 1°00054775. * The third coefficient for 5 is 10; the coefficients or operative numbers for 5 being I 5 10 105 I. 2° nearly. 5(*47) 100054775 12 = 10-2 Dual number = | 0,0,0,5,4,7,6,3, (16) (page 23)|, (100054775) = 54763, = |, 0,0,0,5,4,7,6,3, PRACTICAL SYNTHESES. 63 Hix, 2. Find the dual number and dual logarithm of the natural number 1*00078987. The third coefficient for 7 is 21, the operative numbers being Mee ehe 35. 35) 2b 7 1; 7 (89) = 6 nearly. 100078987 27=21+6 | 0,0,0,7,8,9,6,0, dual number. 78960, dual logarithm, Natural number. Dual number. Dual logarithm. 1:00078987 | 0,0,0,7,8,9,6,0, 78960, 45. Let each of the dual digits be less than 10, (Y) limited to nine places of figures, and uw, = 0, u,=0; then (Y) becomes (10000U,U,U,U, ) + [w, + [w, (X) Fs In accordance with (41), (X) may be put under the form (1o00w,u,u, p q) - [u, + (1000u,u,U,p q) + U, (1O00u,4,"u, ) te us 100°0U, ) ti u," ‘I) = (LO0w,u 4,4, pq) + Wy (LOO) + Uy" (*1) + U, (uyuym,) + Uy’ (OW) u, (ou,) seldom amounts to a unit, and in most cases may be neglected. In reversing the process u,(w,u,"u,) must be put under the form U, (w,0°) + U, (u,U,). 46. To find the dual number and dual logarithm answering to a natural number of the form 1:00u,u,rspq. RULE. From the given number subtract the natural number corre- sponding to |0,0,u,; from the remainder take u,(u,o') and w,(u, decimal w,), and also wu,’ (‘ou,) when it amounts to a whole L 64 DUAL ARITHMETIC. number; the last remainder is of the form u,u.u,q, which may be reduced to a dual number by the Rule (44). put for 1. Examples. | being Ex. 1. Find the dual number and dual logarithm answering to 1007891709. Given number | 0,0,7, 7 (80°) 7 (6'4) 21 (‘08) Rule (oy, 28 + 8 (64) Dual number Ioo I0oO 789 702 179 104 Binnie Gy. —|560 Jooos6515 —| 45 }00086|470 2 000/18 646)8 {ooo BS | 0,0,7,8,6,4,3,5, oe * | 0,0,u,, u,, (U,0’) u, (uu, ) uw, (OU, ) U, + U, (UU, 7X5. .*. (16) Rule (12), |,(1:00789179) =7 86085, Dual number. } 0,0,7,8,6,4,3,5, Natural number. 1':00789179 Dual logarithm, 786085, Ex. 2. What is the dual number and dual logarithm corre- sponding to 1006? 100,600|000 }0,0,5, 100 501 oe J}? 2, | Ed|098/999 . 5 (90) —l450 &, (u,0') | 00/098/549 5 (3°5) — 43 Us (u,.U, ) —I wu,’ (-ou,) | 00 0908 505 PRACTICAL SYNTHESES. 65 Yo this last result we have to apply the Rule given in article (44). Jooo9g8505 36 +9 (84) —44 u, + U, ("u,u,) Dual number | 0,0,5,9,8,4,6,1, Natural number. Dual number. Dual logarithm. 1'006 | 0,0,5,9,8,4,6, 1, 5982211, (12), u,(5°) 25° 59821, Ex. 3. What is the dual number and dual logarithm corre- sponding to the natural number 1°00456081 ? : TOO|45 6/081 }4, 100/400/600 |u,, loolos 5/481 4(50) —|200 «(4 Joolo5 5/281 4(5'2) = 21 u(t) Jooos 5/260 O u, (OU,) oools5 5 26/0 10 + 5 (*52) — 113 we + U, (UU) 5 (40). Dual number | 0,0,4,5,5,2,4,7, Natural number. Dual number. Dual logarithm. 100456081 | 0,0,4,5,552:4,75 455047, (12), w,(5°) 20° 455047, In this example, given previously, page 45, u” (‘ow,) = 0, and u,(uu,u,) may be subtracted all together, without involving error, as the values of uw, u, u, in this case can be anticipated. 66 DUAL ARITHMETIC. The Author of the present Work communicated examples under Rule (46), page 63, when he first made known Rule (12), see page 16. The original form, in which Rule (12) was delivered, was slightly altered in being analyzed, but the altera- tion did not involve error; however, the change renders the reversing of the rule difficult; the reverse rule is given in article (13), page 17. The case is otherwise, from changing the original forms when analyzing the examples under Rule (46), as error may be involved; these discrepancies will be discussed towards the end of this chapter. Ex. 4. Find the dual number and logarithm corresponding to the natural number Ol. : fea nah )9, — 1903608 100109 613.92 9 X 90° —|810 4u,(u,0’) Jo0095582 9(5°5) = 5 2 Uz (w,"%,) 36(‘09) 3 4/(ou,) Joo095529 Then, (40), —41 U, + u,('U,u,) T equired dual number | 0,0,9,9,5,4,8,8, |, (101) = 995033, (16). The natural number 1°01 may be represented by two dual numbers, namely, | 0,1,0,0,0,0,0,0, and | 0,0,9,9,5,4,8,8, the dual digits of each being less than 10; other natural numbers may be similarly expressed. u,(u,u,u,) must be subtracted in two parts, u,(w,0.) and u,(u,w,~,) and not all together. In the ollowing syntheses these matters will be attended to. DUAL AND HYPERBOLIC LOGARITHMS. 67 SYNTHESES OF PARTICULAR DEVELOPMENTS. FUNCTIONS AND THEIR INVERSE. OPERATIONS AND THEIR REVERSE. 47. The remainder of this chapter is devoted to matters of importance, which require particular attention. The dual logarithm of any given number 2, divided by 10° gives the hyperbolic logarithm of n, to eight places of decimals ; that is 1) _ log, n. 10, The young student may be deceived by this coincidence, and imagine that the dual system of logarithms is established by similar processes of reasoning to those used for hyperbolic and common logarithms. That such is not the case may be readily established as follows. Writers on logarithms, with much difficulty and by a series of artifices, show that in the equation r =n; x being the logarithm of any given number 1, to the base 7, that _ (= 1)-4— 1 + $@— 19 -3m—1'+.... Oe ede—1,+3@-1)-tm—1+-... © But the expression (Q) cannot be practically applied except in very rare cases. When the denominator is put = 1, that is, when (r—1)—$(r—17 +47 —19% —F(@r—1)*4+....=1, then w= 2718251025. 2. which, by writers is generally represented by e, and the system is usually termed the hyperbolic system of logarithms. Let (2718281828 ...) =10; then from (Q), w=(10—1)—4(10—1)? + 4(10—1)*~f(Io—I)*....; but to sum this series is practically impossible. 68 DUAL ARITHMETIC. Again, let (2'718281828',..) = 2. Then from (Q), w= (2—1)—4(2—1)? +4(2-—1) —42@-—1) 4+... =1—ds+i4-44+.... 69314718 the hyperbolic log. of 2. ll To find the sum of the series 1—$+3-—-14+..... step by step until we arrive at ‘69314718 is a very tedious process. The hyp. logs. of 11; tor; roo1; &e, are more readily found by (Q), for let | (2°*718201520.., 4) a ae then 2=(r1—1)—-$(1—1) +401-1) —4(r1 — 1)* +... OG AOS ay ali aad =), 10 t(i) +4(96 bs Ge tee ='09531081 the hyperbolic log. of 1°1 the denominator of (Q) being = 1° when r = 2°71828.... Now let (I°00O00000I)* = 2. then 7 = 1:00000001 and n = 2° in this case all the terms of the denominator of (Q) may be neglected except 7” — 1 = ‘00000001 = —~ ite) for $(r — 1)’; $(7 — 1)°; 4(7 — 1); &e. are very small. In this latter case (Q) gives Ow Fits — 3 + ONe *QOOOOOO!I +... DUAL AND HYPERBOLIC LOGARITHMS. 69 48. Hence, the value of x in (2°718281828...)” = 2° found by (Q), and multiplied by 10°, is equal to the value of a in (1°00000001)" = 2° to eight places of decimals, found -also by (Q). The same may be said in applying (Q) to the equations (2°718281828 ...) =1°r and (1°0000000T)” = I°I (2°718281828 ...)) = 101 and (1'QQ00000I) = 1'0F &e. Ke. % and. generally to e =n and (1'00000001)" = 2, but, as before observed, the cases in which @) is practically applicable, are very_few. These remarks apply to developments with hyperbolic loga- rithms given in the Analysis of “ Dual Arithmetic, a New Art,” pp. 39 to 46. Without care, those developments, with hyper- bolic logarithms may give a wrong impression. Although (Q) indicates that 1 (x) ’) = the hyperbolic log. of n, true to eight places of aieiehis. for any given number n, yet none of the processes or devices usually employed to apply (Q), to establish it, or to give it a more practical form, in any way resemble the dual system for finding the logarithm of any given number n, to any base 7, which is by a direct and ex- tremely simple procedure. The young student will avoid being deceived, by carefully comparing (Q) and ‘“ Analysis,” pp. 39 to 46, with the correct dual methods of reduction, Chapters I. and IV. of the present work, and ‘‘ Dual Arithmetic, a New Art, opp. 212°to 214. The dual system of logarithms furnishes all the advantages of both hyperbolic and common logarithms without retaining any of their defects. 7O DUAL ARITHMETIC. CIRCUMSTANCES UNDER WHICH THE CALCULUS OF DIFFERENCES AND THE DUAL CALCULUS COINCIDE, 49. In the accompanying Figure let each of the angles AOB, BOC, COD, &c. be equal 30°; OA=1; OB=]1,; OC=]2,; OD =| 3,; &c. to OM =| 12,; and let a logarithmic spiral pass through the points A, B, C, &. to M. Again, let Ob =’1f; Oc =’2}; Od =’3}; &e. to Om =’121; and let a logarithmic spiral pass through the points A, b,c, d, &c. The radius vector | 2,5, falling half-way between C and D falls beyond the curve, for the length of a radius vector drawn to the curve half-way between C and D, that is forming an angle of 75° with OA, is equal | 2,4,7,8,5,7,2,7, true to eight places of decimals. In the descending branch the radius vector ’4’5 | does not fall on the curve half-way between e and J, or which is the same thing, the radius vector ’4’5| does not make an angle of 135° with OA. "4’5'2’42’75'6{ is the length of the radius vector to the curve in middle between e and f. It is easily observed that the DUAL DIGITS OF THE SAME RANK. 71 calculus of differences will only apply to ten consecutive digits of the same rank when true, and not approximate values, are re- quired. A radius vector whose length is | 2,5, makes an angle of 75° 39° 35'°7 with OA, and not an angle of 75°; and a radius vector of *4’5{ makes an angle of 135° 43’ 58:4’, and not an angle of 135°. 50. The operative numbers will not apply to all natural numbers that correspond to consecutive dual numbers. One or two of the numerous examples that might be selected will illustrate this matter. | 0,9,9, = 1°10356790 | 1,0,0, = I*'10000000 | 1,0,1, = I'101 10000 | 1,99, = 121392468 | 2,0,0, = 1°21000000 | 2,0,1, = I'21121000 &e. &e. One example from the descending branch will be sufficient. 65998506 = '3'9'9 | ‘65610000 = ’4’0’0 f 65544390 = "401 | Ke. &e. Although the operative numbers do not conduct us from the value of | 1,9,9, to the value of | 2,0,0, &c. yet they will apply in passing from | 5,5,0,9, to |5,5,1,0,; |5,5,2,90, &e. | 5,5,0,9, = 1°69418619 169419 otherwise | 5,5,1,9, = 1°69588038 169|418|619 169588 508/256 ae pee 508 | 5555259, — 1'69757626 pO a ak A 169758 1°69 927 383 | 5155359 = 1°69927384 See (37), page 58, &e. Xe, M /2 DUAL ARITHMETIC. A logarithmic spiral may be made to pass through the radi vectors |5,5,0,9,; |5,5,1,90,; |5,5,2,0,; &c. with the angular distances between every consecutive pair equal. The operative numbers, are employed both in the calculus of differences, and in the dual calculus, but under different restric- tions ; for example, 1 5,5,9,7, = 1°70915315 | 5,5,9,8, = 1°70932407 | | 5,5,9,9, = 1770949500 5,6,0,1, = 1°70975978 The calculus of differences will show that | 5,5,9,9,5,4,8,8, = 1°70958882 but | 5 ,6,0,0,0,0,0,0, = 1°70958882 The equality here established is only correct as far as eight places of decimals, for | 5,6,0,0,0,0,0,0, = 1°70958881774441651 exactly. | 5,5,0,0,5,4,8,8, has an exact value also, (32) page 47, the first nine figures of which do not differ a unit from 170958882. The calculus of differences fails to determine the exact value of |5,6, by the consecutive differences above em- ployed. As we proceed, other comparisons and parallel developments will be instituted, and it will be finally demonstrated that the calculus of differences, when properly restricted, becomes a branch of the dual calculus. When the Analysis was being drawn up, the Author of the present Work introduced the calculus of differences to show how the operative numbers might be derived without reference to the binomial theorem, and also to show how to construct a table of ascending dual numbers with their corresponding natural numbers, by common addition, and independent of the operative numbers; the alterations made in analyzing the first communication, rendered the explanations, entered into here, necessary. 73 7 A COMBINATION OF PARTICULAR FACTORS THAT MAY MISLEAD WHEN MADE TO ASSUME THE FORM OF AN ASCENDING DUAL NUMBER. 51. The factors of the imitative arrangement are me 1°8 ZEF Sige fie: Lt wie: I'l Bee Oat By ei BON, 1‘Ol (Ff) eee O00. POGOe ete... 1‘OOI Ke. &e. It is easily shown that 1°8389270996 = (1°7) (1°08) (1'001) (10005) (1°00009) (1°000003) (1'0000006) (1'00000004). The factors 1, ror, roor, &c. are seldom incorporated in such products. 1:3 is put for (1'1)°; 1°9 for (1°1)’; 106 for (1'01)’; 1007 for (1:001)'; &e. These counterfeit factors are rigid and inflexible, and have no branch to imitate descending dual digits. The log of 1:1 being given, the log of 1’9, 1°7, 1°6, &c. cannot be readily found; while in the dual system numbers are expressed by indices, and not by coefficients, and are very flexible, besides, when log 11 is known, log (1°1), log (1°1)', log (1'1)° &c. are easily found. 52. An overrated method is given in the Analysis, pp. 61 to 72, “Dual Arithmetic, a New Art,” to find dual logarithms by limited tables of the logarithms of the factors (I*); this method may be applied to other systems of logarithms, but not without limited tables which the method cannot supply in any case. 74 DUAL ARITHMETIC. However, the logarithms of the factors (IF) can only be inde- pendently calculated by the dual method. For example, the dual or any other logarithm of 1°3389270996 may be found by adding together the logarithms of the factors (1°7), (1:08), (1°001), &c. taken from tables previously prepared. 53. Lhe factors (I) have also been employed to approximate to the roots of particular equations; a root so determined might be put under a form to imitate a dual result, but a slight inspec- tion renders the difference apparent, even to those who merely understand the application of the ascending branch of the dual calculus to find unknown quantities under a variety of dual forms, which subject is fully discussed in the next Chapter. OPERATIONS AND THEIR REVERSE. 54. It is ‘a very important feature of the dual calculus, especially in finding the roots of equations, that inverse dual functions are not only expressed compactly but also readily determined; and in most cases, dual operations are readily reversed. Before the introduction of dual arithmetic but few elementary functions possessed these useful properties. A direct rule should be so framed that the reverse one may be easily deduced when required; these important features have not been dwelt upon, indeed, they have been much neglected. The rule article (12), page 15, is taken from the expression |, UW, UpsUq Ws Uigs UUs Ugy = U,UU,U,U,1k. UW, — 5 (,0U,0U,O) + 31018u, + 33u,; (16). But in reversing the Rule, 5 (w,ow,ow,0) has to be added, and 31018u, and 33u, to be subtracted. Hence the expression for the reverse Rule given in article (13) page 17 must be put under the form Us, Ug Ugg yUipUgyUryUyy — 5 (4400000) + 31018U, —5(wu,000) + 33%, rab (u,0) Since 5 (w,0w,0u,0) = 5 (w,00000) + 5 (w,000) + 5 (u,0). OPERATIONS AND THEIR REVERSE. 75 This modification is necessary, as the values of wu, wu, have often to be anticipated in applying the reverse Rule. For example, let it be required to reduce the dual logarithm 29950000, to a dual number. Dual log 29950000, 5(w,00000) + 1500000 31450000 Z1018u, —-93054 31356946 5 (u,000) +5000 343361946 33U, ie Oyo 31361913 5 (u,0) +150 Dual number | 3 15330.2, 0103; 5 (300000) 3(31018) 5 (1000) see (16). Again, let it be required to reduce | 3,1,3,6,2,0,6,3 to a dual logarithm. Dual number | 3,1,3,6,2,0,6,3, 5(u,0u,ouw,0) —1505150 29856913 u, (31018) +93054 4, (33) +33 Dual logarithm 29950000, 5 (301030) 3 (31018) 1 (33) see (12). The Author regrets having allowed the original form under which he communicated this Rule (12), to be altered when being analyzed, for, as before remarked, page 66, although the change did not involve error, yet it renders the reversing of the rule difficult. 76 DUAL ARITHMETIC. ORIGINAL FORM. Article (12), page (15). 55. Rule:—From the dual number of eight digits taken as a common number, subtract five times the first three digits, supposing a cipher placed after each, add 31018 times the first digit, and 33 times the second, the amount is the dual logarithm. In most cases, this reduction may be effected in one operation. Thus, tid Gutaed £3 +93054 = 3(31018) | 3,1,3,0,2,0,6,3, Dual number. — 150515 * = 5(301030) Dual logarithm 29950000, It is advisable that the student, before proceeding further, should thoroughly understand the criticisms instituted from Article (41) to Article (55). Functions and their inverse, operations and their reverse, will be discussed in a general manner when the ascending and descending branches are combined, for not until then can the great power of the dual calculus be applied. In Chapter IV. ‘the descending branch will be treated of systematically, and in detail. a7 CHAPTER III. ASCENDING DUAL DEVELOPMENTS APPLIED TO DETERMINE THE VALUES OF UNKNOWN QUANTITIES UNDER A VARIETY OF DUAL FORMS. 56. In this Chapter we do not discuss the theory of equations, nor establish any abstract criteria respecting the nature of the roots of equations, but apply a system that will determine the values of unknown quantities under a variety of ascending dual forms, and that too without being obliged to keep within very narrow boundaries. In fact, we first propose to show the power and scope of the machinery to be put in motion, and afterwards to restrict the operations of the whole machine to concise and convenient limits. Examples. Ex. 1. Given 276°593124% = 7634'83528, to find the 7th root and dual logarithm of x. _ 7634°83528 2°10°(1°90870882) _ A 276 593124 _ (2)10°(1°38296562) < (2) (10) | 3,3,0,4,1,8,2,9, 8 (2)(10) | 34310:4594552,0; core | 331792909, |, (©) = 331792909, _— Article (16). = 47398987, = |, 4,9,3,1,9,7,6,8, | 4,9,3,1,9,7,6,8, = 1'60639071, the 7th root of «. 1, (a) 7 78 DUAL ARITHMETIC. INVESTIGATION. 57. Put 1°38296562 = A and 1°90870882 = B; Now s is greater than 1°33, but less than 1°46, hence the first dual digit is |3,. Or, z2 a than °46, which also shows the first dual digit to be | 3,; the three first figures of A and B have only to be inspected to arrive at this result. Should a digit be taken too great, the work mav be continued by making the succeeding digit negative. = is greater than °33, but less B=190870882 A= 113 4 8|2/910/5 6/2 1/4|8)8/9.6.9 1 2 4}1/4|8/8 9 7 [1/3|8]2'9\7 AtuA=A(itu,) =184072725 put=d. Now : is greater than 1°030, but less than 1040, or B—b divided by 0 is greater than ‘030, but less than ‘o40, hence the next dual digit is | 0,3, A(1+u,) = 1814 0]7 2|72|5 . put = bd 5.5/2 2|18/2. 5 SP2iee | 0,3, 184. A{(itu,)+[u} =o +4,) = 189/65 0/313 put=c I|I 3 7\9Q02 2|845 |0,0,6, 4 A{(r+4,) + [u,4 [u} = e(t4+%,) = 190791 064 put = d 7631/6 Ijt_ |0,0,0,4, - 190867 391 put=e NOTATION ILLUSTRATED. 19086/7391 put=e Ge Pee take © AGT math 3 5 1909 = |-%, 1582 1527=|8, ets 38 =| 2, 17 8 17=19, A{(r$,) + [e+ [eet [ut [eu $ [ue $ [e+ Lad = A | U,,UyUyyU,U Ui pV Uy = B Or A | 3,3,6,4,1,8,2,9 = B B A ee | 3,3,6,4,1,8,2,9, b{(14%,)+[uyt----- +[u,} =B; e{(1 +u,) + [mF AER + [u,} -B: AY(1 fu) t[u,$--.-- +[u,} = B; &e. &e. Since, (1t,)+[u,+[u,4 2 Othe = ihu=0 minus +|[u,}+[u,+... (1); 79 In finding uw, we have not to estimate the value of Fut lyt - will not increase the value of u, a unit. N . all we want to know respecting it is that it 80 DUAL ARITHMETIC. Take = I'79..., as another example to illustrate the notation; then’ uw, = |6, for (1°1)’ = eee end (1°1)’ = | 7, = 1'9487171. Hence, between |6,=1°771561 and |7, = 1°9487171 V[u.tlu+... has a range of values betweeno, and *1771561, which is the ;4 of the natural number corresponding to the lesser dual digit. Neglecting {[u,+[u,+...in (1) and putting At+Au,=B ey B- es Then it is clear, if 4 * so ‘79 or ‘87 or any number up to ‘95, u=|6; butif aa °231 and “4641, %, = 12, = °34 or ‘42 or any number between Again, because (14%) + [y+ [My $- 02 = d B 4 b : B :. .Ifu, = 7 minus +[u,t[u,t... (2) u, may be so chosen, that the rejection of +[u,+[u,+ ... will not decrease the value of w, a unit, or render w, negative. If = 1'032, then w,'=| 0,3, and wu, may be taken =| 0,3, for all values between 1°030301 and 1:04060401; hence, between | 0,3, and }o,4,; +[u,+[u,+... has a range of values between o and ‘01030301, which is the 45th part of 1:03030I. Neglecting +[u,+[u,+.. in (2), then b(i+u,} =blub=B $M, ara NOTATION ILLUSTRATED. SI If ade ear O3an thet, = 10,3, if a ='074.., \ u, = |0,7, for 074... is greater than ‘07213535210701 and less than ‘0828567056280801. Ina similar manner wv, may be found from B T + UW, = ae Bb —~e ? or from +U, = reserving for further consideration,;the surplus, represented by +[u,+[u,+. The process being continued, the dual number is found under the simplest form, when made up of ascending dual digits. It may be necessary to! observe, that as many places of decimals are taken as the required dual number is to have digits Hx. 2. Given 7634°83528x% = 276593124, to find a, rts fifth root, and dual logarithm. _ 276593124 _ _(10)"(2'76593124) _ © = 763483528 (2)(10)(1'90870882) _ Ho | 3)8;5,451,957,2, or, &% = 03622777875. |, (4) = 138629436, |, (10) = 2302585009, |, (40) = 368887945, |, (2s) = 7368887945 See Article (17) F 3,8,5,4,1,9,7,2, os 37095040, 5 ) 331792905 dual log of = or |, (a) “66358581 166358581 | = $1 2956137, = $ | 0,2,9,6,6,5,1,9, = *5 1500129 ; the fifth root of « = *515001209. 276593124 \ 3, 82 DUAL ARITHMETIC. These and similar results may be found by other dual methods of operating, and under a variety of dual forms; how- ever, the importance of the particular treatment here employed, will be presently seen when we come to find the roots of complex equations. Details of the es) of the last Example. 0|8|7|0|8 |8 |2 7\2\6/1 |2|615 5 1712 |6|1 |2 17 1|9|0|8 |7|I 1/9 5 25 2 098 375 2 CDV fe I 821 494 7 Oo" 5 7700 Los o I 27 0.4. I I 7 272) e 5455 217 1016 2689 2489 4049/1 415 - O 3\2 3/9 3/2- 7\1 113 3|8. TiAi2 217). ree: Le 7669 2765 1908 1908) es ae 25404) 2255 763 ——_——- 04 95 (1088 .. 2032 223 203 1495 (10054 .. 1376 119 a | | | 0,0,5, Therefore {u, = 13, because 1°45 is less than 1°4641 = | 4, but greater than 1334 en }.35 Therefore | U, - | 0,8, because 1°088 is less than T0936 . i= $0,9, but greater than 1'082856.. = |0,8, ‘Therefore | 0,0,%4, = | 0,0,5, because 1'0054 is between and |0,0,6, = 10615... l'O510 wa | NOTATION ILLUSTRATED. 83 58. Given az’ + ba =, to find the value of a. Substitute 7 for z, so that it may be possible to represent x under the form PU AGE te «0 « the nearer 7 approaches the value of x, the less will be the dual number | w,,w,,,,... Which has an extensive range of forms and values in each particular case. Criteria to determine the range and convenient limits of r, in equations of all degrees, will be treated of in another place. Let ar + br=c, and suppose the first dual digit to be |u,, then substituting 7 |u,, for x, ar’ |2u,+br|u,=c; . ar(it2u,) + oriitu,) =c; ar’ + brt}2ar’u,tbru, =c; C, + 2ar*u,t bru, =c; oy oe =< od. Shela We ri 1 Leartbr Uy Uz, &e, may be found in a similar manner. Since w,, u,, &c. are whole numbers, positive or negative, +2ar*+br in most cases may be treated as + 2ar° + br. -It may be necessary to state that but little practical incon- venienace can rise at any time in making w,, a unit greater or less than the proper dual digit belonging to the zth position of the required dual number presented under its simplest form ; for if un, be taken too great, then by making the following dual digit negative, the process may be continued without retraction, interruption, or error. The same may be said if w,, be taken too small, with this difference, that the succeeding dual digit will be greater than 9,. 84. DUAL ARITHMETIC. Ex. 3. Given 357°836528a2* — 573'456388x2 = 8107°37676, to find both values of x, and the 7th root and dual logarithm of the lesser value. If we examine (3'°5....)a°—(5°'7...)a=810... (ine given equation divided by 100), it appears that 3|u,3 4|u,; 5 |u, may be substituted for ~; 6|u, also may be substituted for x, but then u, becomes negative, and the operation is not as easily performed as when 5|w,, is put for a Then putting z=5|wu, the given equation becomes 8945'9132 | (2u,), — 2867:28194 | u, = 8107°37676, which has to be compared with the general equation (58) ar’ |2u,—br|u,=c + 178 twice +-/89°. ... “aes — 28 once — 28... (d)r (+ 2a" — br) + 150 +6r... {@) +81... 150) +20... (c—¢,) 15... (aye Substituting r|w,,u,, for « the given equation becomes 10824°5549 | 2u, — 3154'01013 |n, = 8107°37676 + 2164 twice + 10824'5549 — 310 once — 315401013 — 1854) + 967). 2et iv + S8To ce se 43 NOTATION ILLUSTRATED. 85 The next step furnishes the equation 11264°0752 | 2u, — 3217°40573 | u, = 8107°37676 + 22528 twice + 11264... — 3217 once moe 7.n ty Moone 19311) + 8047... NS + 8107... 60 (|0,0,3, = | u, 58 The next step furnishes the equation 11331°8288 | 2u, — 3227°'06760 | u, = 8107°37676 + 22663 twice + 11331°8288 — 3227 once —. 3227:06760 19436 + 8104°76120 take aes + 8107°37676 from 2 61556 19436 (| 0,0,0,1,3,4,5,7, 6719 5831 888 777 112 a 15 tf aS | 1,2,3,1,3,4,5.7) = 5°62915577 Coefficient of second term with its sign changed 573°450388 == = 1°60256526 357°830528 ie From 1°60256526 Take 5°62915577 Also w= — 4'02659051 86 DUAL ARITHMETIC. 5 | 1,2,3,1,3,4,5,7, = | 172778181, the dual logarithm of zx. x may also be found under the form, x = 4 | 3,5,5,8,0,7,7,0, 7) 1172775101, Seventh root | 24682597, = | 2,5,6,4,5,6,9,6, = 1°27995632 59. Given a2’ + ba”? + cu = d, find a. Let x be taken so that the required root may be found under the form r|w,, uv, u,...- As before remarked, 7 has a great range of values, but it is evident that the nearer r is taken to the value of x, the less will be the affixed dual number Tp Rae pra which may be made to assume a variety of forms in each particular case. Substitute 7 |u, for x in the given equation, then ar’ |3u,+ br|2u,+ecr|u,=d, ar (I+3u,) +br(1t2u,) +er(itu) =d, ar’ + br’ + er} 3u,art2u,brtuer =d, d—d, UL = —__ > 1 $¢3art2br ter’ putting d, for ar°+br’+cr. The dual digit u, may be | obtained by employing + 3a7°+2br°+ cr as a division instead of t3ar°{2br°tcr. To find x, put ar*|3u,=a,; br’? |2u,=6,; cr|u, =c,; and substituting r|w,, u,, for x, the given equation becomes a,|3u, + b,|2u, a c, | u, = d, or, a,(t$3u,) + O,(t42u,) + o(14%) =d d—d, Uy =7— 3 434,428, 76, d, being put for a, +6,+c¢,. As in the case of w,, the value of uw, may be found by employing + 3a, + 26, + ¢, as a divisor ROOTS OF EQUATIONS CALCULATED. 87 instead of +3a,+20,+¢,. Again putting a, | 3u, = a,; b,{2u,=6,; ¢,|u,=¢,; then by substituting r|u, uv, uw, for x, the given equation becomes a, | 3Us ag. b, | 2U, +, | U, wa d, U, = eae Gass 2. 7 73a 28,F 0,” By continuing the process and extending our notation d—d, deo “4 =T3a,t20,te,’ my ley Ae Cia 60. In a general equation of the fourth degree, aa’ + ba’ + ca’ + da=e, let two or all the roots be real, then as in the last case. a e—eé, o Tgar*t36r'}2ery}dr’ 2 a a va $44,430, 42444,’ Ca 4, = —_ OTF = T4a, 4 30,4 20,44, &e. &e. 61. If r|u,, wu... be a root of the equation ax + bx + ca? +da*+ex=f, then w,, u,, vu, &c. may be found from oa oe 1 tart 4br't 3cr{2dr'ter O 88 DUAL ARITHMETIC. at hath : $54,445,436,4 20,42,’ om toy, | . +5a,+46,+ 3¢,+2d,+e,’ &e. * &e, Other equations may be treated in the same general manner. Ex. 4. Given 34°56789a”° — 2345°678a* — 123°4567 2° + 456'7891 x" + 56789°12u@ = — 415978976065 to find a value of x, true to nine places of decimals. It requires but little observation to see that a value of x lies between 0 and 100, and on closer inspection it will be found that a value lies between 10 and 30. ‘Then 20 being put for 7, and 7 | «, being substituted for x, vu, may be obtained from 1, 2S OE AE I 1 t5 art 4br*t 3crtadrter’ ar’ =a, br* = b, er = 0) dr =a, er = €@, +110617248' — 375308480°— 98765 3°6 + 182715°64 +1135782'4 + 650 5 times. +1II0...... —A500 “times —875 oe a times. 8 aa ok O 2 2 times’ =F PEE Ie ‘Ty CMR She Beesedye ge — 264...... Cf) tae — AIS it..ee (f/f) from =; 1 ee (+{1,=]u, a,)5u,=a, b\4u,=6, ¢|3u,=¢ dlen=d + 178150185" — 549489146 — 1314566942 4+ 221085'924 €, |, ee + 1249360°64 ROOTS OF EQUATIONS CALCULATED. + 8905 — 21976 ee 4 a + 12 — 1309 ) a, | 5 U, = a, 89 5 times + 1781. 4 times — 5494..... Bewmes | — 12 eves. 2times + 2.eeee Otte es 2 STEP od ea « (j,) take ALSO. o Hanes (f) from AAR s HOO (+ 10,3, =| 4%, —' 397... b, |4uU, = 6, C, | 3U, = ¢, d,|2u, = d, + 206826835° — 619178123° — 1437722°51 + 234687°163 + 103410 — 247668 €, | U, = e + 1287217°52 5 times 4 times 3 times 2 times I time -+- 429 46 128 — 144513 ) | 0,0,2,5,0,0,0,0, square cube 4th 5th + 20082.... — 619Q17.... co iad Ge apes og ny oes + 328. — 41227.... (f,) take —41597.... (f/f) from mae AN a | 0,0,2,5, = + | UqsUys — 289 81 249900, 499800, = | 0,0,5,0,0,0,5,0, 749700, = | 0,0,7,5,0,0,5,0, 999600, = | 0,1,0,0,4,5,6,7, 1254500, = | 0,1,2,5,4,4,6,7, gO DUAL ARITHMETIC. a, | O, I 52,5,4,4,0,7, b, | O, 1,0,0,4,5,6,7, C, | 0,0,7,5,0,0,5,0, + 209427135° — 625398465 — 1448548°10 d, | 0,0,5,0,0,0,5,0, €, | 0,0,2,5 ,0,0,0,0, + 235863°006 + 1290438'27 + 1047135 5 times + 209427135" — 2501593 4 times — 625398465" _ 4345 3times — 144854810 + 471 2times + 235863'006 + 1290 Itime + 1290438'27 — 1457042 ) — 415893577, _— take a — 415978976° from a 85399 (+ | 0,0,0,0,5,8, — 72852 —~ 12547 — 11656 — 89gI If the process be continued another step, the value of x will be found to be 20| 1,3,2,5,5,8,7,6, which value might be found under many forms; for example :— Common number. Dual numbers, Dual log. of x. ( 22 | 0,3,2,5,5,8,7,5) 20 | 1,3,2,5, 5» 8 Bok x = 22°724071 MY = 312345320 724071 Se ao t138, 7 oe 12 | 6,6,6,9,8,4,4,7, &e &e &e. Whence it is evident that « may be found by putting any number from 22 to 12 for 7; 20 is selected, because its square, cube, &c., are easily obtained and operated with. ‘To determine a value of « in such equations as the given ones, by any other known method, would be almost impossible, on account of the laborious calculations and other perplexing circumstances in- volved. ROOTS OF EQUATIONS CALCULATED. gl EXPONENTIAL EQUATIONS. 62. Given a = 8: to find the value of «a. Logarithms of any system being employed, it is well known that xv loga = 8; and it will be presently shown that 2°38842348 (2°38842348) = 8, |, (2°38842348) x 2°38842348 = |, (3°). 8 But 2 = | 7,2,6,0,7,8,2,6, = | 69314718, or |, (2°) = 69314718, since the dual digits reduced to the 8th position is termed the dual logarithm. Dual log of 2, or |,(2°) = 69314718, and because 2° = 8, therefore, 207944154, = 3 |, (2°) = the dual log of 8. § Again, 2°38842348 =2 | 1,9;255 57545533; oe | 87063353, |, (2°38842348) = 87063353, then 37063353, multiplied by 2 | 1,8,2,5,7,4,5,3, or its equal 2°38842348, must give 2°07944154, if 2°38842348 be: the value of x. 92 DUAL ARITHMETIC. Proof. The details of the work. 87063353 Be sda AAs Tolalalalla| a es ee 115 3/2 3}/1 5 513 O13 1 LO 13 im OV"O -O A 207|4091698 211s ee | 20.7 2 O81 tues 21 Tyas 1lo3g1|2 2\I 20792|8657 2 | 1,8,2,5, nt, Rael ag iis 832 1O4 6 |; (8) =20 7 9 4 4 I 5 4; 2 | 18,255 57145535 63. Therefore, the dual log of 2°38842348 multiplied by 2°38842348 gives the dual log of 8, and hence, 2°38842348 is the value of « in the equation z* = 8 true to nine places of figures. It is evident that a formula, to be established presently, which in all cases reverses the above direct process, will give the value of « in the general equation a =a. Since, an in- dependent and direct solution of this equation has defied all attempts of mathematicians by arts previously known, it may ROOTS OF EQUATIONS CALCULATED. 93 be necessary, before delivering the general formula, to give at length the solutions of one or two particular examples, to prevent any misunderstanding. |, (2) = 69314718, |, (3°) = 109861229, |, (4°) = 138629436, |, (5°) = 160943792, |, (6°) = 179175948, |, (77) = 194591016, |, (8°) = 207944154, |, (9°) = 219722458, 10°’) = 230258500, II*) = 239789528, 2|, (2) = 138629436, 3 |, (3°) = 329583687, 41, (4) = 554517744, 5 |, (5°) = 804718960, 6 |, (6+) = 1075055688, 7 |,(7°) = 1362137112, 8 |, (8) = 1663553232, . 9|, (9°) = 1977502122, 10 |, (10°) = 2302585090, II |, (11°) = 2637684808, These numbers may be expressed under the general form n|,(n); but few of them are required. They are used in a manner similar to that in which the squares and cubes of the nine digits are employed in extracting the square and cube Ht will be found convenient to arrange the well-known dual numbers |1, |0,1, |0,0,1, &c. in roots of common numbers. the following order :— rt =| 1, =| 9531018, Pot) = it, = | 995033, rool = |r, = | 99950, 1'0OOI = 4, = . 10000, 100001 = | I, = r 1000, &e. &e &e. or, or, or, or, or, (tI) = 9531018, (a). |, (P01) = 995033, (6). |, (Toor) = 99950, (¢). |, (10001) = 10000, (d). |, (00001) = — 1000,” (@). &e. &e. Given x* = 8, to find the value of «. If z*=N, then = |, (x) = |,(N); In the given example « |, (x) = |, (8) = 207944154,. DUAL ARITHMETIC. 29 6O01g9f2 zt 91/9 16/$ Z|r 06g 6\1z ol6|6/6 | eleleh 2 29=06S666 g £4£06g1z £)0'0|6)6 1 slo0'0.6 6x —_——— Zz 9=££0566 D Qfozg061 z D = QIOIfS6 ‘o'%” gicoL+y CN 4 ‘o'n (Nt Yon (u) ‘fu ZLOG OL O Gz a “Sal ‘Sl6Q9|o1 ate eis "9Q/S giZ Liz Te eG ale jae ae rad oy oa GOO 016 oor ZIV \1 z\Zipl1 “elolelel 9530:2.00.61 O¢.70 909 C1 a C S EAP t 0.9. (2) ‘(u) ‘tu Aq poyuasordos ‘(z) ‘| z ‘€ pue Z WEIAMJOY SI @ Jo on[eA oy} ywY) quoptad st yt ‘ZeQLeS6zF fad en —P ( 6£o9FF Bee Cc € P (Nn) 4 Coe aus Ti Dorsee - 96£90z = Cnn ‘f (tt CAGE 7) ‘6Q1z OI ogtfZ1 -* a Cn) ( saeltadie a ras 7) MIT PR Ob! EFOzOgE I (uy Fu YIM poouswM0d st ssodoid oy orOJaIEY]} = (£)}€ pue ‘ofb6zogt1 = (c) Iz 90UIg 95 ROOTS OF EQUATIONS CALCULATED. ‘en | "oblchgge.z = EGVLS QI [z= a ‘UOISIAIp UouuI0D Aq punoy ore “E°S'H'Z | syLOLp ONT, g eR is ie ii s}Igtp oy} Surpuy ur days yovo ye poydde st }UoUT}eeT] IepIWIs W ‘poUurltustoyop Ajsnotaoid useq oaey ‘p pue "2 9oUIs poure}qo A[Ipvot ST PN) 6 fen =nen | u pue @ Surmouy wor punog st “a sny} 41 sapooord yey wos poureyqo st days eM bqog016L02Z 1|z fj0 6 Fol olF6908402 te et Oe os | S252 Sy Or7 LO govsivit cigizLlzlozg —_—— cl cl Sze CCz ggsi IcOc 'S4'L.1) GoSIe o6zfe — bvogor6Loz (NK) ‘f bvSivb6loz (N) ‘tf ( Lobb ORT O] 6208. 4-—-— OND ‘I DUAL ARITHMETIC. 9 L6z$S6g 6 gq = ££0566 op. - 2Z£10Z2259 - 6 p = QIOILS6 YO gbgo0gSS Crt Zo teo1s onc v0 02 OILIO9 oor’ 109) ¢ tlog'9 fvooz ( 16889892 ZEeizo Sf £61 ‘g'n (w) ‘| («)‘{u ‘cz1z0S$ZZ61 = (6) ‘| 6 “| as) << ele Ogi “01907 ('x)'t “+++ 96Eg0z (N) ee eee 6£920z (XD es Secreta Fee (u) ai rye aug Se0z (nw) 4 (2h 6oe '2 908976 ee ZOISQOE. ae z Cx ‘f ac LOS 1zoSZZ61 OLE ( hoe re ite ZQTZ ‘» zgi6ZZSg = zizoSZZ61 ——— (w) { 2 ‘O1 UBT} S89] Nq 6 ULYY Ta}veIS oq ysnuA x ‘OGOSgSzofz = (01) for pue ‘zzIzOSZZ61 = (6)‘{6 asnvoog] (nN) ‘t = ‘o1P696Eg0z = (C00ZZbEz11) f= (@) ‘fe *. ‘gi h696£ goz {| = ‘1O'g 1c‘ 1'Z'I | OL = OOOLAVETII = 2X ‘NOOLLV ECL = ot uoynnba ay} Ur x fo anpoa ayy puif i synos OOO'LLVETIT 70 payoulnjsa st (9981) quasaud yo yzuna ayy fo uoymndod ayT 9 “*5T 97 SPECIAL SOLUTION OF AN EXPONENTIAL EQUATION. VEOTLits .O.=‘9'1'e'9'c'9'S'0 16 =m if ££6='9°7'9‘F ‘06x (001) Cx" ’a 1££6=‘c'9'E'0 | 6 x (C00!) Bn p §gz£6 =‘9'€'0f6x(o0001) (*N) ‘tf 2 908926 I *p"n oL|z (ND ‘1 £3\69|z 0S|$6/6g 6 9= 06 666 [06296 £€ goz 9 VolgQI|I Bey 12-2002 gvorZ EY SESUE S.0z | A IQiI99 1 V S gIgVo fie g oz SPAS SOz7II£ 902 1|V Og OI\€ Sie eae Slr zcfjgg o'Z oz oii Zot LLY vEOg ee ist ant Oa GBs. ( Z10€ So6zg6£goz ("N) ‘t =f £6 gIP696£g0z (N) ‘f b90z Mele Sle QVzZ ( ggIo£ STELOQUS INDY fe Pl SED "**O96€g0c (N) ‘f L£Q02 "Qh =eiZ 1} 19 (Se ““*911zgoz (°N) {| ‘p Sgzté "**696£g0z2 (N)‘f I1£QO07 98 DUAL ARITHMETIC. 64. General solution of the equation at =N. If C= N|UjUyMy.. = n{it[ut[m%tl%+---F Then n| UU, = n{it[u, +[u,| n{it[u,} {1 +[u,} m{1+[w,} {1 + [a,} | tty, = (14 [ey + [ey + [} nf foe, + Leg} {1+ Loy} n(x ley} (14 Ea} (14 Oy nf + [u, + ey} {14 Ley} Ke. &e. i] I N | Uy Uy = n{It[w, + [w+ [e+ [e,} = nf ad [oy 4 Log Leg} {14 Lm} &e. &e. Multiply L=N | Us,Up)Uyy eves mit t+ [w, + [uy + [+ [w,t..... } by loga= |,(n)+u,atu,b+uct+.... aRO\eE CeCe Ce eee +nu,a,i+t[u,t[u,tlu,t...-. } +nu,b{1+[u,+[u,t[ut..... } +nu,clt+[u,+[,+[ut....- + &e. &e. GENERAL SOLUTION OF EXPONENTIAL EQUATIONS, 99 Multiply ni + [w,} by |,(n)+u,a@ (n \, (n) +nU, a)\ u 7 [w,3 2 {; (N,) ) na being put =a, Multiply om +U,a+U,d) |,(n)+u,a+4,b n{I +[u,+ [w,} j ~ ntti + [w,}{1 7 [w,} (|,(N,) +nu,b {1+ [u,})tr $[~,} put = |,(N,); nb{1+[u,} being put = d, Multiply |, (n) tu,a+u,b + u,¢ by n\ I + [w, + [+ [w,} ar ! + [w+ [w,} { t + [ws ({,(N,) + mmol $ [04 + [iy + [e%93) (1 + [uh = |, (N,) 5 ne{i+[u,+[u,+[u,} being put =<, Multiply |, (2) +ua+U,b+u,c+ ud by n\ t . [w, + [w+ [4 + [w,5 a n{ I + [u, + [w, + [w,} { 1 + [w,} (J, (Ny) + meg{ 1 [ety + [eg + [2%,}) (0 4 fon} = |, ON) 5 nd\1+[u, + [u, +[w,} being put =d, In the same manner the development may be continued. When f,(N); 2|,(~) and a, become known, then uw, may be found. 100 | DUAL ARITHMETIC. If x=n|u, then n{1+[u,} multiplied by |, (n)+«,a = |, (N). m{,(m) {1+ [u,} + mata $ [yt = |, 0) n|,(n) +n], (n) [u, + nu,at...= |, (N) +n|,(n) [u, + nau,+..=|,(N) —x],@) and .*. wu, may be determined from (N) — n], (x) a) (N) — |], (n) a o ae ean: a: \( + Then |,(N); |,(N,); and 6, become known, and uw, may be found, for if «= n|u,,u,, then | (nr) +u,a+ 4,6 multipled by = {,(N,) {1 [ut + mab {t + Le} ; oa i (N,) + |, (N,) [w, a u,b, + Pid |, (N) +4, (N,) [u, si u,b, + he? or Ne (N) pre \ N,) and .*. wu, becomes known from Ly (N) ‘ona it (N,) I Then |,(N); |,(N,); and c, becomes known, and uw, may be found, for if «=n|m,,u,,u, then |,(m) + u,a + u,b + u,¢ multiplied by {1 + [w, + [u, + [u,} = (1, (Ny) + me,){1 4 [x] |; (N Be! " [wv ot a: Use, { I “ [w,} te, |; (N) =LO)+h Dy + med. = 40) ph (N.) Lu, ay C,Us + “pe to BAS |; (N) By ; (N,) and .*. wu, becomes known from |, (N) x3 i; (N,) 1 (N,) i GENERAL SOLUTION OF EXPONENTIAL EQUATIONS. IOl In the same manner w,,w,, &c. may be found, and B= | WU, Uy -- becomes known. Although the value of $n iP (7) [u, 3 U, Ay + at he aes |, (N) chant |; (2) may be accurately found, yet it may be put under the form (Ae) “3 a,) wu, = |,(N) — »], (x) for if u, be taken too great or too small the process may still be continued, as the excess or defect will be corrected by the succeeding steps. The same remark applies to cae (N,) [2 bP, Ut OR rs |, (N) % |, (N,) which is put under the form N 4 2) + o)u, = 1) - 1.) to determine w,; and so on with respect to w,, u,, &e. Consequently the process is continuous without interruption, since « may be represented under a vast number of dual forms, all amounting to the same natural number. Tor example, x was found under the form 2 | 1,8,2,5,7,4,5,3, in the equation 2* = 8; but a might be found under the form ’0’1’3’0’8’2’1’7 1] 2, each of these dual numbers when reduced becomes 2°33842348, TO2 CHAPTER IV. SPECIAL TREATMENT OF THE DESCENDING BkANCH OF DUAL ARITHMETIC. 65. ALTHOUGH we have defined and described the nature of this branch of dual arithmetic, yet, to avoid complication, the special treatment and practical application of this branch of the art have been postponed until now; the method thus pursued will be found to possess many advantages. When both branches are judiciously combined, great power is gained and much labour saved. Operations, that might be cumbersome with either branch, may be rendered simple and concise by combining both. In the descending branch, the bases ‘9; *99; ‘999; Wc. or I—‘I; I—‘O1; 1— ‘001; &c. are employed in a manner similar to that in which the bases I'l; I°OI; I°OO1; &e. or I+‘I; I+ ’O1; 1 + ‘001; &c. are engaged in the ascending branch. See Articles (1) to (7), pp. 1 to 6. (1 — *1)° (1 — ‘o1)* (1 — ‘Oo1)* (1 — *0001)’ = (9)’ (99)" C999)" (9999) = 508715701 and is represented thus, 6,45355,0,0,0,0 | = 508715701 103 66. DETAILS OF TWO METHODS OF REDUCTION. By continually Subtracting. By Binomial Coefficients, or 1°00 00 00000 1000 00000 90 00 00000 © 9 0000000 8I O00 00000 8 I 000000 72 9 000000 7 2900000 65 6 I COCOCO 6 5 6 10000 590490000 5 99 49000 53|I 4.41000 5 314410 5261 26590 5 261266 520865324 5 208653 515656671 5.156567 51 0|5 00104 5 10500 5099 89604 5 09989 509479615 5 09479 50 89/701 36 Operative Numbers. T\olo clo ojolojolo + O}O;O,E0/0/0/010 — 1/5 0.0|0|0|\0/o0 + Git 2 0Olo\ololoio — It|slololololo + 6lo|o0|\0 — 1\0l00 + 531414 1looo. + 2112/5 716410. — 4 | 3\1 818 6l5 . + : 2620 S10 500/1I04 + 3h} 153 1/590 — 2 Hes 2+ | — 5 089/701 3|5 + 5 | 2544815 — Seine MoOe71.5 701 In particular cases, the reduction will be found more convenient by operating with binomial coefficients, when proper contractions are employed; but the re duction by continually subtracting is best suited for forming a table, or to obtain a consecutive series of these num- bers. The following table (A), will be Q 104 DUAL ARITHMETIC. 508 9\70136 found useful in reducing numbers to a 50897 dual form; it is constructed with the 7 | BO 89 Ic 9230 greatest ease, each natural or common 4 50892 number being obtained from the pre- es re Ae ceding one, by subtracting each digit of heer) LSE the line above with 0, suppose after it, pon from the digit to its left. For example, 508817460 take 6561, which imagine to be 65610. 50882 Then 1 from 0 gives Q, carry 1 to 508 7 66578 6 = 7 from I gives 4, carry I to 50877 5 = 6 from 6 gives 0, carry 0 to Bay asi 6 = 6 from 5 gives 9, and 1 from 6 gives 5; thus 59049 is instantly found. TABLE (A). | Common number. | Dual number. | Dual log negative. | Coy ‘9 | t... P| 10536052 | (-9)" ‘81 | PREY) 21072104 | | (9) 729 | 73---1 | 731608156} | | Egy sos Gr | 4... 1 “| "one : ieee CO) 89049 | ’5---f | 752680260] Co)” | 531441 | Oe 63216312 | (9y' | 4782069 | 7... | °737523641 | Oy Prenat ON Sk ee | 84288416 | | | | 387420489 | ’9...] 94824468} mae) 67. The dual logarithms are found by multiplying 10536052 by 1, 2, 3, &e. respectively. ELEMENTS OF THE DESCENDING BRANCH. 105 10000000 [ = 710536052 [ =’1 J 8 ‘oO VOO0OO0OO| = 71005034 (i ='] | ‘OO VOODOO] = 100050 =I 1 ‘WOOO L000] = ~—- 10000 | ='I | &e. Ke. &e. Because (9) | 1,1,0,1,0,0,0,1, = 1° and by (12), | 1,1,0,1,0,0,0,1, = | 10536052, and mie, “10536052 f = | — 10536052, 8 or "105 360052 | = | — 10536052, Dual log of (9), or |, (9) = — 10536052, A dual number reduced to the eight position being termed a a dual logarithm, the 8, marking this position, for the sake of brevity, is omitted in the descending as well as in the ascending branch. — 10536052, = (9)’| = ’10536052 Again, (99) | 0,1,0,1,0,0,0,1, = 1° but |0,1,0,1,0,0,0,1, = | 1005034, '1005034 {| = | — 1005034, or ‘t { = ’1005034 | |, (99) = — 1005034, written 71005034 And again, because (999) | 0,0,1,0,0,1,0,0,= 1° but | 0,0,1,0,0,1,0,0, = | 100050, "100050 | = | — 100050, or. "1 {| =’100050 { . 3 8 &e. = «ec. ¥00 DUAL ARITHMETIC. Then it is readily shown that "508715701 = 67586598 | = "6435 O00 0f '6'4'3'50'0'0'0 | 3216312 six times 536052 20136 four times 5034 150 three times 50 67586598| So that any descending dual number may be reduced to its eight position by adding to the dual number, considered as a common number, 536052 multiplied by the first dual digit, 5034 multiplied by the second, and 50 by the third dual digit. See Rule, Article (14), page 19. Ex. Let it be required to reduce ’6'6'0'6'8'2'03} to tts eight position, or to its dual logarithm. 66068203 3216312 six times 536052 30204 six times 5034 69314719 69314719 | = ’6'6'0'6'8'2’03 | 8 This Rule is easily reversed, and a descending dual logarithm reduced to a descending dual number. See Rule, Article (15), page 21. Examples. 1. Reduce the descending dual logarithm 69314718} to its corresponding dual number. ELEMENTS OF THE DESCENDING BRANCH, 107 69314719 3216312 six times 536052 66098407 30204 six times 5034 ’°6'6'0'6'8'2’0'3 | 69314719 | = ’6'6'0'8'2’0'3 | As the third dual digit is 0, no multiple of 50 has to be subtracted, 2. Reduce the descending dual logarithm ’67586598 | to its corresponding dual number. 67586598 3216312 six times 536052 64370286 20136 four times 5034 64350150 150 three times 50 ’6'4'3'5' 0000 | SESE Ih IP Ee | [ 67586598 | = 64350 0'0'0 68. If dual logarithms of the ascending branch be considered positive, those of the descending branch must be considered negative, and vice versd. Because = (99999999) very nearly. I (1*0000000T) I (Foooo000T)? = (99999999) 108 DUAL ARITHMETIC. But I 8 ene fiooo000l}? hee and (99999999)? = 'p | : Actual division of 1° by 100000001 ; I‘OOOOOOOI ) 1'000000000 ( ‘99999999 QOO000009 999999910 QOO0000009 999999010 QO0000009 999990010 QOO0000009 999900010 QOO0000009 999000010 © QO0000009 QQ0000010 QO0000009 QOO0000 IO QOO0000009 Remainder 1 Hence, the log of *508715701, found by the descending process, is the same as the log of *508715701, with a negative sign, found by the ascending process ; for 2(*508715701) = 1'017431402 = | 0,1,7,3,3,4,4,0, and | 0,1,7,353)45440, a4 | 1728123, > ELEMENTS OF THE DESCENDING BRANCH. 109 But \,(2) + |, (508715701) = |, (1'017431402) |, (508715701) = |,(1°017431402) — |, (2) — |, (2) = — 69314718, il OL7AL.... y= 1728123, "508715701 = | — 67586595, It was before shown that ‘508715701 = 67586508 | results that may be said to coincide. Table (B) is very easily constructed by subtracting as follows ; and adding the constant log ’1005034 |. "99000000 = ’0’! . . | = 71005034 | 990000 ‘98010000 = 02... | = ’2010068 | 980100 107020000 = (03 F.\f'=""3015 102) 970299 OG OOOT = "O04, -, [== 4020136 | 960596 "95099005 = 05... | = 5025170 Ke. Ke. Ke. The numbers corresponding to ’1’0..{ ’2’0..{ ’3’0..] &c. are obtained from Table (A). 110 Common Numbers. "QQO00000 ‘98010000 ‘97029900 "96059601 "95099005 ‘94148015 "932065 35 ‘92274470 ‘91351725 ‘“QOOO00000 °89 100000 *88 209000 87326910 8645 3041 "85589105 84733214 83885882 83047023 82216553 ‘*8 LOOO00O0 *801Q0000 "79388100 78594219 77808277 77030194 76259892 75497293 74742320 DUAL ARITHMETIC. TABLE (B). Descending Dual Numbers. . . . . > ° . . . : = . & . . . . ‘ . . * . 7 . ° . . . so errr > es me rm rrr r--— eesa— seem ae seer eon reer c - e es ed see eeemricrem— err—_ -COoema eee eae See eee eee ee se Se Dual Logarithms ; negative. "1005034 °2010068 °3015102 4020136 "5025170 6030204 7035238 "8040272 "9045 306 "105 36052 "11541086 "12546120 "13551154 "14556188 "15561222 "16566256 ‘17571290 18570324 "19581358 °21072104 22077138 23082172 °24087 206 ’25092240 '26097 274 27102308 28107 342 29112376 Yi TABLE (B)—continued. Common Numbers. yuna ual ogni 739948907 29..f °30117410 "7 2900000 0 F47 31608156 "72171000 BL eat 32613190 71449290 322] 33018224 707 34797 13°31 34623258 "70027449 "314 oc 35628292 "69327175 B35 ey 36033326 68633903 BG ke 37638360 "67947564 OR sal 38643394 67268088 eS ast 39648428 66595407 39+-{ ’4065 3462 "65610000 Oe. iy "42144208 "6495 3900 4094 ‘43149242 64304361 Ae tN '44154276 "63661317 43-04 45159310 "63024704 ‘440.4 °46164344 62394457 455-4 47179378 ‘61770512 Ak SM "48184412 ‘61152807 "47-4 "49189446 60541279 eee 50194480 "59935866 49... 4 "51199514 "59049000 150, 4.34 "52680260 58458510 Saks Beal 5 3685204 57873925 52..f "54690328 57295 186 53 4.4 "55695 362 "56722234 Le a 56700396 “56155012 Bea ett "57705430 55593462 56... | "58710464 55037527 Sa "59715498 54487152 ahh cat 607205 32 "53942280 59... | '61725566 112 Common Numbers. 53144100 "52612659 "520865 32 "515656067 "5 1050010 "50539510 50034115 ‘49533774 "49038436 "48548052 ‘47829690 “47351393 46877879 .46409 100 ‘45945009 "45485459 "45030004 ‘44580298 ‘44134495 "43693150 ‘43046721 ‘426016254 ‘421QO091 ‘41708190 "41350508 ‘4093 7003 ‘40527633 ‘40122357 "39721133 "39323922 "38742040 DUAL ARITHMETIC. TAaBLe (B)—continued. ty Ae 1S Os k Oe wOLe Descending Dual Number. * . . . . . . . ° . . . . . . . . . ° . . . . . . . . . Dual Logarithms ; negative, 63216312 "64221346 65226380 66231414 "67236448 68241478 69246512 "70251546 "71256580 "72201614 73752304 74757398 , 75702432 '76767466 77772500 78777534 79782568 "80787602 81792636 '82797670 84288416 °85293450 86298484 87303518 88308552 ’89313586 90318620 '91323654 92328088. 93333722 '94824468 DESCENDING DUAL NUMBERS AND LOGARITHMS. 113 Table (B) is easily extended in a similar manner by con- tinually subtracting and adding 100050 to the logarithms as follows :-— , "99900000 = ’o'o'l .. ] = '100050 | 99900 "99800100 = 'o'o'2.. | = ’200100 | 99800 "99700300 = ’0’0'3.. | = *300150] 99700 "99600600 = "0'0'4. ote | 1 ’400200 QQ60I "99501099 = '0'0'5 .. | = °500250} 99501 "99401598 = '0'0'6... | = ’600300 | 99402 99302196 = 007 ts | eet 700350 | 99302 "99202894 =’9/08.. | = 800400 | 99203 "99103691 = 009... | = *900450] "99000000 = ’0’!’0 .. | = 71005034 | &e. &e. &e. The numbers corresponding to ’o’'1.. {| ’o'2..] ’03..[ &e. are obtained from Table (B), and those corresponding to fee aie *3,.. {) &e. are given im fable (A). Hence, a table of descending dual numbers may be formed with great ease, in a short time, to any required extent. 114 OPERATIONS WITH THE DESCENDING BRANCH OF DUAL ARITH- METIC, INDEPENDENT OF THE ASCENDING BRANCH, When operating with the descending branch alone, the digits, for the sake of convenience, may be placed to the right of the sign {; a dual number so expressed is sufficiently distinguished from one of the ascending branch whose digits are to the right of |; the point of the arrow pointing up in the former case, and down in the latter. Examples. 1. Find the dual logarithms of 2° by the descending process. ,“5 0000000 —s 4 = |’6'6'0°6’8’2’0'3 3216312 six times 536052 30204 six times 5030 | 69314719 Dual log (3), written |, (4) = — 69314719, = 69314719 Dual log of (2°) written |, (2) = + 69314719, which agrees, to a unit in the eight place, with the logarithm of the same number found by the direct process. 50034115 = | 6,6, may be taken from Table (B) or directly calculated. OO. ....1 == "5003/4 To Ueeab 3100 20 — 6 | 7 4 SS: _Aulebeeek HB it VP ca aa, 02 ; 4'0 09 2 | 102 ; 1|00 3 { 2 i 2 °6'6'0'6'8’2’0'3 { above written [’6’6'0'6’8’2’0'3 = §. REDUCTION OF DESCENDING DUAL NUMBERS. T15 2. Find the dual logarithm of 10° by the descending process. ? Lee | 1024 | ’0’2’3’6'1’4’3’2 = 1000 = IO” |’0'2’3’6'1'4°3'2 10068 two times 5034 150 three times 50 2371650} |, (2) = 693147180, 2371050 3 ) 690775530 = |, (10°) ; |, (10) = 230258510 | This result agrees with that found by the direct process to a unit in the ninth place, the logarithm before found being 230258500. The reduction in detail. a ID2 ae O00}. + = (2) 2 le 800|0 . 1024\0. i] hace hee 301087 301 6 {| 1000/61 45/4 6003/7 15 1OOOO|I 43 2 I+ ++ +4 a ag dy ’0'2’3'6'1'4’3'2 | 1024 above written 1024 | 0,2,3,6,1,4,3,2, is equal 1000 = 10’. 116 DUAL ARITHMETIC. 3. Find the dual logarithm of 94701510 by the descending process. 947°01510 = (10°) (947015 10) ‘94701510 = ’0’5’4'1’8'6'5’6 | 25170 five times 5034 200 four times 50 5444026] |, (10°) = + 690775527, |, (94701510) = 5444026 685331501, = |, (9470151) Reducing Process. (94701510 _ | given number. ’o’5 .. f = 950/990 ca) T | from table (B). 3|8 0 3/96 4 Sigua ‘It 9471\9180 Neale 94709708 | 8 | 8]198 diff, bh BPS a 6 | 621 . 5'68 5 | 5/3 6 | 6 0°5’4’1’8'6'5'6 | = ‘94701510, | which is written ’0’5’4’1’8'6'5’6 | for the sake of convenience. REDUCTION OF DESCENDING DUAL NUMBERS. ry? It is easily shown by the ascending system that 947°O15 YBa 2°(10)” | 1753571452355 RAR ’0'5'4'1'8'6'5°6 | (10)" = 2°(10)" | 1,7,35714)2,355 ’0'5’4'1’8'6'5’6 { (10) = 2° | 1,7,3,7,4,2,3,5; ia 21 1,7 1397145253555 . may be represented by F re e(74 Br (7 e442 413 415} 5 0,5,4,1,8,6,5,6, | (10) and is represented thus ro{t tot [SF l4tls $ (8 Fl64 [5 +16} + being the sign used in the ascending branch, and + the sign employed in the descending. Meee 7 4(37--} = 5{rrlotistlat:...} 4. Find the dual number and logarithm of 179°170165 by the descending method. T’o prepare a number to be operated upon by the descending process, it is necessary to reduce the given number to a decimal fraction; the nearer it is brought to *99.... the more readily are its dual representatives found. The necessary preliminary reductions are easily effected by the use of the numbers 10 and 2. 179170165 = (2) (10)" (895850825) "895850825 = ’1'0'5'4'1'8'3'7 | 5 2 O0.5)2 2G 10998139, negative, written 10998139 118 DUAL ARITHMETIC. |, (10°) 4605 17018, 1, (2) 69314718, + 529831736, |, (895850825) = 710998139 518833597, = |, (179170165) Reduction. ‘895850825 | number to be reduced. TO. ites ipleeclin + 4\5 00|000 — 5 | gi000 + se | 9-| 895 5(08 99/1 + 4 { 3|5 820)3 + : go+ 895 867284 heii af 116459 ) 8959 8 | 7\5 00 : 7/167 3 | 3|33 2\69 hs 6/4 8 6|3 In the dual number ’1’0’5’4’1’8’3’7 | one of the digits "4 { is 4 negative, the operating numbers for ’4 in the descending process are +13; +4; +10; + 20; &e. | ELEMENTS OF THE DESCENDING BRANCH. 11g In the ascending branch, the operating numbers for 4, are fee Ay + 10: — 20;) &e:. 10541837} = tit [1+ [o+[5—[4+....} descending branch. )2,3,5,1,={1+[2+[3—[54...} ascending branch. — being the negative sign in the ascending branch. and <— the negative sign employed in the descending branch. |; +; —-5; ascending signs. {; +; —; descending signs. In ascending developments the natural numbers continually increase or ascend, while in descending developments the natural numbers continually decrease or descend; but in both branches the arrows point to the greater number. Bet 7OLOs =) 1'0'5'4 18°37 | (10°) (2) = (107) (2)irt¢ [it fot(s—[4tlr t(8t[3 +17} 8 = | 518833597, and |, (179170165) = + 518833579, or dual log of (179'I 70165) = 518833579, The operating numbers or binomial coefficients for both the ascending and descending processes, and for both positive and negative dual digits may be determined and registered in the following convenient tabulated form. (See page 120.) 8 120 DUAL ARITHMETIC. 792/| 1287 / / / po ie ; ] ae “ha i a 4 Hig. z | 792/| 1716/| 3432/) 6435 VAR / | | / | / / Ve * ELEMENTS OF THE DESCENDING BRANCH. I21 When the perpendicular and horizontal lines of units are set down, the other numbers are found by simply adding diagonally. 2 For a dual digit, as 4; on the fourth diagonal line the operating numbers 1; 4; 6; 4; 1 are found, and on the fourth horizontal or perpendicular line the operating numbers I; 4; 10; 20; 35; &c. are found. These numbers, with the proper signs, are employed in both the ascending and descending operations for the dual digit 4 whether it be positive or negative. The operating numbers for any other dual digit from 1 to 9 are found in the same way in the above table. Ascending branch. For the dual digit 4 When positive +1; +4; +6; +4; +1 When negative +1; —4; + 10; + 20; — &e. _ Descending branch. For the dual digit 4 Raia +i; —4; +6; —4;+4+1 \ When negative + 1; +4; +10; + 20; + &e. 1 | 4, = 1464100000 | 4,= 683013455 656100000 shes _, II | I 1524157503 I22 DUAL DEVELOPMENTS BY THE APPLICATION OF BOTH THE ASCENDING AND DESCENDING BRANCHES OF THE ART COMBINED. d b bd N =%0,’,'v, «. . nfm Ut, Uy 1. 1M, or, . 8 N= 'v’v’o... alma, >< sees Dual log of N = n, is written ],(N) = x, 5 |u, may be written |w,, or |0,0,0,0,u, ee | v | may be written ’v,..[ or ’oov.. | 3 m represents some power of 10 and » some power of 2. It may ,be observed that a dual development has TWO branches, (’v’v’v... |) and (| u,2,u,...); A DOUBLE sign (}) ; TWO ultimate values, the natural number (N) and its logarithm (n,) to a known base; and TWO powers (m and x) of TWO simple numbers (10 and 2); hence the art has a good claim to the title Dual Arithmetic. To find the dual log of 3° and g 8 10 = 9 |.1,1,0,1;0,0,0,1, = 9|, 10536058) |, (10) = 230258500, 105 36052, |, (9) = 219722457, |, (3) = 109861229, ° COMBINATION OF BOTH BRANCHES. 123 1. Find the dual logarithm of 3°1415926535 and give all the figures employed in the operation. 3°1415926535 ~] 9'42477790.. ‘942 47 7796 06..f= 941/48 oF 49 iL 941480 ; 942 421629 ROMmG? 14 3° 4l7 1 21 9° AO 8 814 82 5/64 | slog 1° 3141 59265 ite ee 76’0'0'0'0'0'0 { 3 | 0,0, 1,0,5,9,6,0, 30204 50 6030204 IO5Q10, 105910, negative 5924294 |, (10) = 230258500, |, (3) = 109861229, 120397280, |, (7) aa 114472986, . 124 DUAL ARITHMETIC, 5 3 6 7 w= OLY TLO4 [1+[5+ lo} [6} in which but five digits are employed, one belonging to the descending branch, namely, (6) and four belonging to the ascending branch, (1,5,0,6,). By com- paring developments by each branch separately, and with both branches combined, the advantages to be gained by the combined operation will be readily perceived. Yet, in particular develop- ments, the application of the ascending or descending branch alone will be found most convenient. Descending. IILIIIII = TOOOOOO00| = — 10536052 | 5 |, (VILLILIIL) = + 10536052, Ascending. I'TIILIIII = | 1,1,0,1,0,0,0,1, = | 10536052, Descending. ee We) ’ 1'23450789 =’2’00'00000| = — 21072104 [. 8 |, (1'23456789) = + 21072104, Descending. 137174211 = 3’'0000000 | = — 31608156 | 8 |, (1737174211) = + 31608156, 2. Find the dual logarithm and dual number corresponding to *765432110. COMBINATION OF BOTH BRANCHES. 125 BY BOTH BRANCHES COMBINED. 765432110 given number oolo oolo oo0!o 29/0 316 If 7661 86/326 7 66 3) > 721900 1 316 4|50 7\20 7 186 765420140 T1970 5 monaees cl 1) 4316 6 3827 15; 489 7 Be 1.03 Not w 2) ‘76543211 = 3'0 100000] 0,5,0,0,1,5,0,3, = 26731478 | Reduction to a Dual Logarithm. See (12), (13), (14). Soak het EB Ue h ee Ge + °3’0'1'0’'0'0'0'0 J 0,5,0,0,1,5,6,3, + + 150005 25000 — + 108156 165 + Rly 7 renee 7 7 2 8, 49767 28, 26731478 |, (76543211) = 26731478 126 DUAL ARITHMETIC. Reduction by the Ascending Branch. 7 ) ‘765432110 Number to be reduced 1'093 A744 A = ~ (109347444) 2 }9,= .1093/6852/7 + #2 2187|4 — 12, 3+ 1093 4665 6 sob | ee 788 ¢ 76 6 | 7, is 23 iy 8 bane 8 76543211 = f. | 0,9,0,2,0,7,2,0, = | — 26731476, See Rule, Article (12). 297 + nine times 33 | 0,9,0,2,0,7,2,0, 45000 — subtract. 8936017 (7) = 194591016 20357 03 57 |, (10) = 230258509 — 26731476— or °26731476 COMBINATION OF BOTH BRANCHES. 127. Reduction by the Descending Branch. ‘765 432110 given number. 940 + S12 — Seat lara ees Reade 2\2970)8 a3 + 534 Gih.93 5 » DPS 2'5...| = '770|301 4\6 21 6 | iI 3 765 69 1 67|0 (eat Yee 70543211 = '2’5'6'3'3'9'0'2 | = '26731476 f and |, (76543211) = ’26731476 Reduction by the Rule, page fo. '2'5'6'3'3'9'0'2 | 1025300 5 times 205060 72104 2 times 36052 170 5 times 34 26731476 Reductions may be effected in a great number of ways by both branches of the art, jointly or severally applied. Every method is continuous without interruption, since no incon- venience is experienced by intermediate results becoming too 4 128 DUAL ARITHMETIC. great or too small within proper limits. However, the best method to be employed in each particular case must be left to the skill or design of the operator. 76sqg211 = 7 (14 (942+ [7 $12} {1241s +[6+[3+[stlo4 2} I 2 5 6 7 8 (rts $6540 4 [5 $164 13} In general terms, ‘ODO A} Gs te = ‘ 1+[u,+[v,— [u,—[e,+ [w,} v.’0'v,'0'” [| O, LO, U,, O, % 1 3 bade i = "ert fo, — Lu, — Loy [04 [ee [e} =~ {1 $[o,—[u,} {1 —[o,t [ut lon} {1 + [4} &e. Ke. Ke. From the foregoing practical applications of the dual signs, it is presumed that their functions, or the operations indicated by them, will be understood. | + — 4+. ascending. Dual signs | { both branches combined. f + — +. descending. ¢ A dual number reduced to the eight position, on account of its practical importance, is termed a dual logarithm. In practice it is seldom necessary to use more than eight dual digits, or a dual number reduced to a higher position than the eight, since results are rarely required to be true beyond the seventh decimal place. However, a system may be soon framed to secure any required degree of accuracy; as an example of such extensions, 1, = |Q,95557151907139557 515255904) 717) = | 10,1, = 10,0,9,95514)8,7,3)1,044,5,5200 =| [0,0,1 = |0,0,0,9,9,9,5,4,5,7,8,4,6,0,5,9,6, =I Ps = |0,00,0,9,9,9,0,514514,8:755,76 = | ee = 10,0,0,0,0,9,9,9,9:945:4)51455:8:05 = | |0,0,0,0,0,1, | =|0,0,0,0,0,0,9,9,9,9,9,9,5,4,5,45, = i | 0,0,0,0,0,0,1, = | 0,0,0,0,0,0,0,9,9,9,9,9,9,9, 5,45, = | | 0,0,0,0,0,0,0, I, = | 0,0,0,0,0,0,0,0,9,9,9,9,9,9,9,9,5, = [ EXTENDED DEVELOPMENTS. 129 dual numbers to the seventeenth position are employed in the following developments. 95 31017980432486,016 995033085 316808,288 EARS EET {9999500033330833 99999500003 3,333 Gorda ae 033 9999999500 ‘999999995, The work of one of these reductions will show how the above results are obtained. 5 1000000000/0000 ole (eye) 900000000010 00 s\6a bd ole 8/4 00 10000 9/00 0 3 60/00 8 4.00 See 00 3 2 40 316003 24 8 10000990044461|1972 190008 91 0400 316004 picasa 55 8 3/76 9000899 1/4! 3160 oe 5 00 04.9997 I [130 DUAL ARITHMETIC, 10000 9999954507875 x 45 492125 40 00 4.0 00 M, 5488125 5 0005 00 4 10001 = | I, = | 0,0,0,0,9,9,9,9,5,4,5,4,8,75557,0 (See “Dual Arithmetic, a New Art,” page 42, and “ ‘The Young Dual Arithmetician,” pp. 72 to 80.) 1 2 3 The dual numbers | 1,; | 1,; | 1,; &c. are reduced to the seventeenth position as follows. a7 8 | 0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,= | 999999995, = | I, 1 7 Mult. by 9 | 0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,= | 8999999955, (a) Add 9;9;9,9:9;9)5 455 999999545; 17 7 }0,0,0,0,0,0,0,9,9,9,9,9;9,9,51455= | 9999999500, = | I, alvg T | 0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,= | 9999999500, = | I, 17 Mult. by 9 | 0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0, = | 89999995500, (6) 17 9,0,0,0,0,0,0,0,0,0,= | 8999999955, (a) Add 9;9,9,9;5,4555455, 999954545; 17 6 | 0,0,0,0,0,0,9,9,9,9,9,9, 5,455,455, = | 99999950000, = | I, EXTENDED DEVELOPMENTS, 17 6 | 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,=| 99999950000,=| I, Mult. 17 by 9 1 10,0,0,0,0.9,0,0.0,0,0.0,0,000.0,=| 8999995 50000, 9,0,0,0,0,0,0,0,0,0,0,= 89999995500, (c) (0) 9,0,0,0,0,0,0,0,0,0,= 8999999955, (a) Add 939551455 34955759) = 995454579; 17 | 0,0,0,0,0,9,9,9,9;9,55455,45557,9,=| 999995000033, ay 5 131 | 0,0,0,0, I,0,0,0,0,0,0,0,0,0,0,0,0, =| 999995000033,=| I, Le Mult. by 9 +1 6,0,0,09,00,0000,000000.=| 899995 5000300, 9,0,0,0,0,0,0,0,0,0,0,0,= 8999995 50000, 9,0,0,0,0,0,0,0,0,0,0,= 89999995500, 9,0,0,0,0,0,0,0,0,0, = 899999995 55 Add 5141514585755 57,0, = 545487570, 17 }0,0,0,0,9,9,9,9,5,455,4,8,7,5,7,0,=| 999950003 3331, 17 | 0,0,0, I,0,0,0,0,0,0,0,0,0,0,0,0,0,=] 9999500033331, Mult. 7 by 9 | 0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,= | 89995 500299977, 9,0,0,0,0,0,0,0,0,0,0,0,0,= 8999955000300, 9,0,0,0,0,0,0,0,0,0,0,0,= 899999550000, 5 ,0,0,0,0,0,0,0,0,0,0, = 49999997500, 4,0,0,0,0,0,0,0,0,0, = 3999999980, Add 557,8,4,6,0,5 9,6, = 578460596, 17 | 0,0,0,9,9,9,5,4,557,9:4,0,0,5,9,0,= | 9995003330835 3, (d) (c) (6) (a) 132 DUAL ARITHMETIC. 17 S |0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,=| 9995003330835 3,=| I, Mult. 17 by 9 |0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,=| 8995 50299775180, (J) 9,0,0,0,0,0,0,0,0,0,0,0,0,0,= 89995500299977, (é) 5,0,0,0,0,0,0,0,0,0,0,0,.= 4999975000167, |5, 4,0,0,0,0,0,0,0,0,0,0,0,= 399999800000, | 4, 8,0,0,0,0,0,0,0,0,0,0,= 79999996000, { 8, 7,0;0,0,0,0,0,0,0,0,= 6999999995, | 7; Add 3,1,0,4,4,5,5,2,0,= 310445520, 17 10,0,9,9,5,458,753, I 10,4,4,5,5,2,0,=1 995033085 3 16800, 17 2 JO, 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,=| 995033085 3168090,=| 1, Mult. ey ; by 9 1 1o9000000000000000=| 895 5297767851275, | 9, 5,0,0,0,0,0,0,0,0,0,0,0,0,0,0, = 499750166541767, |5, 7,0,0,0,0,0,0,0,0,0,0,0,0,0, = 69996500233316, [ 7, 5 ,0,0,0,0,0,0,0,0,0,0,0,0, = 499997 5000167, i Si 9,0,0,0,0,0,0,0,0,0,0,0, = 8999995 50000, | 9 7,0,0,0,0,0,0,0,0,0,0, = 69999996500, V7, 3,0,0,0,0,0,0,0,0,0, = 2999999985, | 3 Add 55711,2,5,94757) = 571259477; 17 10,9,5575:9571315979152,5,9,457575=| 953 1017980432487, 17 | 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,=| 953 1017980432486, EXTENDED DEVELOPMENTS. 133 The numbers here registered are true to the last figure ; to secure the designed degree of accuracy, the calculations were made for dual numbers of twenty digits, and when broken off to seventeen, the proper allowances were made. The same degree of accuracy is established in the following tabulated multiples in a similar manner. : Ree > 8 The values of ’1}; ‘O'1f; ’o'o'1|; &c., in the seventeenth position were found as follows. Since ‘9 | 1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0, = I. 17 | 1,= | 9531017980432486,016 | i= 995033085 316808,288 4 } 1,= 9999500033 330,833 J i= 999999995, 16 | 1,= 10, Bem 3252005782030 (5 p= 5208025 7828913150 | '2t= '21072103131565260 {3 fe 208 154007347800 | "4. | = 7421442062631 PLE te {='032 £6309394005780 | Sz '84288412526261040 | 9 | ='94824464092043670 | 105 3605 1505782630 7 | = 73752360960478410 | 134 ao nN DUAL ARITHMETIC. 95 31017980432486, "19062035960864972, | | 28595053941207458 | "38124071921729944, | '47655089902162430, | '57186107882594916, | 66717125863027402, 75248143843459888, | "85779161823892374, 2 1, 1 995033085 316808, | " 199006617063 3616, 1 | 2985099255950425, 17 7 | 3980132341267233, [4975 165426584041, | s970198511900850 '6965231597217658, | 7960264682534466 | "8935207767851275, wt. SJ | 1, 7 999500333083 53, "199900066616707, 1 b | | '299850099925060, +399800133233413, | "499750166541767, 7 599/00 i ' | '609650233158475, | "799600266466827, | "899550299775180, | 9999500033331, | 1999900006662, | 29998500099993, | 3998000133323, | 49997500166654, | 59997000199985, | "6999650023316, | 7999600026647, . 17 | 8999550029997, > ete er EXTENDED DEVELOPMENTS. sd Chics Wao aie ta S) C\ ture 5 11, 17 | 999995000033, 17 | 1999990000067, 17 | 2999985000100, 17 | 3999980000133, 17 | 4999975000167, 17 | 5999970000200, 17 | 6999965000233, ir | 7999960000267, 17 | 8999955000300, 6 | 1, 17 } 99999950000, 17 | 199999900000, 17 } 299999850000, iy | 399999800000, Py. | 499999750000, 17 } 599999700000, leg | 699999650000, iy | 799999600000, 17 , 899999550000, a OV, Ma, > i ta co Xe) I 7 x, 17 | 9999999500, 19999999000, 29999998500, 39999998000, "49999997500, 59999997000, |69999996500, | 79999996000, 17 | 89999995500, 1 1 1 1 | | | 7 7 7 7 11, 17 | 999999995, | 19999999000, | 29999900985, | 3999999980, |'4999999975, 17 | 5999999979, hy | 6999999965, 17 | 7999999960, 17 | 8999999955, 136. DUAL ARITHMETIC. ee Sanne eat OR, we ‘I | . OF, 5g 1 3 '10536051565782630 | I '100050033358353. i I '21072103131565260 i 2 '200100066716706 | 2 ys tO08 54607347200 11113 "30015010007 5059. 3 '42144206263130520 J 4 4002001 33433412 . 4 ; Ramee ts | 5 0250166791765 | 5 163216309304695780 i 6 '600300200150118_ t 6 73752360960478410 J 7 | °700350233508471 | | 7 | '84288412526261040 J 5 | | 0400266866824 | 8 | '94824464092043670 | | 19 — ’900450300225177 | | 9 aes or, FOr 7 | hes or, ‘OOOIf 1005033505350144. 4 I '10000500033390 I - The dual logarithm of N is equal to the dual logarithm of V plus m times the dual logarithm of 10; plus n times the dual logarithm of 2, plus wv, the dual logarithm of the dual number Wet... (4). ese Fs Oo 8 : POE BPE ON SU Cars Let V = ’v,'0,'0, 0,0, 0, 0,0, | and may be written |’v,'v,'0,'0,0, U5 YU, p pee ne De ee pe PS te =’oOo00000'0v{ and may be written {’0’0’'0'0'00'0'v =y{|=’v} the eight being omitted in all cases. The 8 digits may be on the right or left of {| when not connected with independent ascending dual numbers. ISA DUAL ARITHMETIC. In (5) the comma is at the barb of the arrow, and desig- nates that the dual logarithm of the descending dual number ’y,'v,'v,... | is equal to ’v; and (5) also indicates that the dual logarithm of V is equal to ’v which is negative if wu, be con- sidered positive, and on the contrary ’v is positive when w, is taken as negative. Since Mian \| PATNA a | 18 Sa ah z V,'VyV5.V V5 Vg Vz Vg MLN Uy, Woy Way Way Wes Upy Urny Wes ee WN eS EA feats fall ee 0'0'0'0'0'0'0'v _mfn O, ,0,0,0,0,0,0,2%, v min U, These latter expressions are also put under a logarithmic form by simply placing a comma at the barb of the arrow. Thus |, (N) = |, (VU) = |,(V) + |, ©). | "U, Un Vg s+» Mls Wy Ugy Ugg « Il ‘y min U, m|,(10) + n |, (2) + (u— vv), or, n|,(10) +], (2) +’(v —w). Dual numbers composed of ascending and descending dual digits having but one digit in each position, may be placed to the right or left of {, because ascending digits are sufficiently distinguished from descending by the accompanying commas, and the digits of both branches are ranged in order from left to right. : dpe ee Tae | b] >) b y 0'V,'0'0,,0'0'0'U, | Uq,O, UgyO, Uys Wey Ups O, may be written mm Us Vp Way V, Ugs Wey Uny Vg 3 ene Pa Fe Went et a Da : U,/V,000'U, 0,0, m]n O,0,Us) U,)UzyO,0,0, may be written 070 Mb, U., Us VU, U,V min : 1n2y oe be a ee ? GENERAL NOTATION. 155 if the leading digit be an ascending one, | is in most cases placed on the left, but when the leading digit is a descending one, then } is generally put on the right of the compound dual num ber. | °0'v,0'0 0,0, 0, U, | alo" (2)* 1, O, Ug) 0,0,0,0,0, 5) ’ 4 min Uy) Vy Ug Vgy V 59 Vey Urs Ve The dual logarithm of » ee e Y 3 ? ? O'V,'0'0'0, U4 Uz, Vy | UW43 O, Way Uyy is thus indicated a7) Uy Vg b] >] |, 04) Vp Way Uy Ue Ue The dual logarithm of "D,'V, | X (R) x | 0,005, 204) Wy Whey Ury Ups may be put in the form 15 (70,0, Ugy Uigy Uy Ury Lay | R} or written >] , >) |; (R) 7% vY, VU, Uys Usy Wey Ugy gy | As the ordinal arrangements of both ascending and descending digits are from the left to right, the sign | { or | may be placed to the right or left of ascending, descending, or mixed dual numbers when all the positions are occupied without giving the expression an ambiguous meaning. Yet this change of sign (| | {) from left to right, or vice versd, may be employed to designate the reduction of a natural number to a dual number, and the converse operation. Let N be a natural number reduced to a dual number | Wy Uys Uigy Wry Wey Uigy Uy Uy then ING Sere eee Mee a and let | Uy Uyy bay oo os be reduced to the natural number N, then 4 4 4 v -—— ug Wry Uys Uns Uy; Urey Way Ung Ut, mlm == N, 156 DUAL ARITHMETIC. Again, if ’ , ’ ) bf ’ y b b] 2 J J ) bf b 4 U,0,0,U,U,0,0,0,| =M; M={’v/0,0,0,0,0, 0, %, a eee Tak hes. if P = | Wy'0, V4 Up Ups Up Uy Us Deak faa eh r then Un) Up Vz Uy Uns Uz Vy V5} =P; 4 b y d if Q = 70,40, V,U,y 00,0, nm also mw 0, Uy) O,'U, Uy V,0,0, = Q. |, CN) = ma yy Ug Uys «02 = Y, then Uy | = min by, Us) Uy oo os i 8 or | = Ue ae == Ni hence the law is manifest. Since Nie? m)n Uy, N ess 1 1o™ 2” N 9 A ’ |; Feaes, ="v|,u,=(u—v), or ='v— 4) according as v is greater or less than uw. See pp. vill. ix. 2, 3, 4, Il. PROBLEMS. Ex. 1. Suppose the dual logarithms of 10: and 2: to be forgotten, it is required to reproduce them by an easy direct operation. Pat moe and. "10 ¥ sania hich Sa — = 8 = '2'72'3'7' 1171: which put = a. ae y So ae P a 710 frm — = —; = 1024 =’000'1'8'2'1’3 | 0,2,4,0,0,0,0,0, ; which put = 8. CONCISE PROCESSES, PROBLEMS, ETC. 157 Then i iA baile ¥ = 6 Se 5 Aah a a b Oi: eda ta yo }(@)= 1,6) — 34,4) =1@) and L(y) =31,(@)—- |L@ =|, (o) Calculation. | 1,(6) = 2371653, |, (a) = 22314355 then minus 3], (a) = 66943065, |, (2) = 69314718, 3 |, @) = 207944154, minus |,(a@) = 22314355, 230258509, = |, (10). Hx 2. Let it be required to find the dual logarithms of the ~ bases 1°13 1'01; 1001; and Q; ‘99; 999; by simple, direct, and independent operations. In the first example a few additions and siesta: will ee show that 8 = 21237117 | aes O24 = 182.13 { 2,4, but the reduction of these and other dual numbers to dual 108 arithms requires the application of the Rules pp. 15 to 22. 158 For example, 2237 117 | IOO510 See Rule, 72104 page 19. 34 "22314355 DUAL ARITHMETIC. ’0'0’0' 182’ 13 f 0,2,4,0,0,0,0,0, 001020 2389800 66 2389866 1S 3 See Rules, pages 15, 19. 2371653, The Rules by which these simple reductions are made, depend upon the dual logarithms required in the present question. Because (Introduction, xii.), ‘9 | I, 1,0,1,0,0,0,1, = I. ‘99 | O, 1,0, I 30,0,0, I ae | | . ‘(999 | 0,0,1,0,0, I,0,0, = I. F (9) t ih I,1,0,1,0,0,0, 1, = O. |. (99) aE \; O,1,0,1,0,0,0, I= O. : (999) i \; 0,0, 1,0,0,1,0,0, = O. Since the dual log of 1° = o. Again, a few simple additions and subtractions will establish the following equations. PA AT A= Boe Pe 004270112 I'L = | 1,0,0,0,0,0,0,0, | 0,10,0,0,1,0,0,0, LOI = | 0,1,0,0,0,0,0,0, Spe DR Ye be ea AL 0000'475'0'0 | 0,0, 10,0,0,0, 3,3, L‘OOI = | 0,0,1,0,0,0,0,0, ee) ae De Te OOOO 00 45 } 0,0,0,10,0,0,0,0, CONCISE PROCESSES, PROBLEMS, ETC. 159 In the first place we have to find the dual logarithm of (1°0001)”° = J, 0,0,0,10,0,0,0,0, |, (1'0001)" = 10 |, (10001) For ten consecutive digits T‘OOOr = | 0,0,0, 1 ,0,0,0,0,0,0, H ’0'0'0'0'0'0'0'0 45 | 0,0,,0,0, 10,0,0,0,0,0, 45 999955, To eight digits |, (10001) = 9999,°55 or 10000, |, (1'0001)” = 10 |, (1:0001) = 99995, nearly. |, (1001) = |, 0,0,1,0,0,0,0,0, I DES Wey eR Pl DE BS Me 0'0'0'0'0'0'4’5 }, 0,0,0, 10,0,0,0,0, 99995 — 45 = 99950, _ —— (A). Then it is evident that |, (17001) = |, 0,0,1,0,0,0,0,0 = 99950, .. |, (11001) = 10], (1:001) = |, 0,0,10,0,0,0,0,0, = 999500, iM (I ‘0 1) ss i, 0,1,0,0,0,0,0,0, I 2 gee Uy WER POR IE SE I | 0'0'0'0'4'5'00 | 0,0,10,0,0,0, 3,3, 999500, 33) 995033, (B). Z 160 DUAL ARITHMETIC. |, (999) + |, 0,0,1,0,0,1,0,0, = 0, but |, 0,0,1,0,0,1,0,0, = 99950, + 100, = 100050, from (A). |, (999) =— 100050, written ’*100050 And because 700 100000’| = |, (999) = 7100050 (C). ’°00'4'0'0'0'00 | = ’400200 f (99) + ie O,1,0,1,0,0,0, I, =O but from (B) |, 0,1,0,1,0,0,0,1, = 995033, + 10001, = 1005034, | |, (99) = — 1005034, written "1005034 (D). ‘oO V000000’t = |, (99) = 71005034 |, (4:1). 1,0,0,0,0,0.0 o'c'4'2’0'1' 12 f, 0,10,0,0,1,0,0,0 il |, 0,10, = 9950330, (B) x 10. 1000, 995 1330, 7004’f = ’400200 (C) x 4. 9551130, "2OLie 9531018, (E). 7 |, (1'1) = J, 1,0,0,0,0,0,0,0, ="953iguee CONCISE PROCESSES, PROBLEMS, ETC. 161 ; (9) F ie I,1,0,1,0,0,0,1, = O. ~ But from (B) and (EF) |, 1,1,0,1,0,0,0,1 = 9531018, + 995033, + IOOOI, |, (9) = — 10536052, written 710536052 ia000000 | = |. (9) "10536052 (F). Hx. 3. Let tt be required to reduce by the shortest processes, and least possible amount of calculation, the seven natural constant numbers (a), (b), (c), (2), (e), (f), (9), to the most simple dual numbers, and to dual logarithms best adopted to logarithmic com- putation. 1359593 (a), density of mercury at the temperature of o° centigrade, according to Regnault. . 17329625 (6), cubic inches of water in an ounce avoirdupois, at the temperature of 62° I’ahrenheit. 2085 3654 (c), feet, the semi-axis minor of the earth, according to Bessel. 3437°7468 (d), length of the radius in minutes. 39°37079 (e), inches in a metre. 61':09908 (f), cubic inches in a litre. 98087952 (g), metres, the accelerating force of gravity. In reducing natural numbers to dual numbers of the simplest form by the shortest and easiest methods of calculation, any. natural numbers, as (a), (b), (c), (d), (e), (f), (g), that may be selected, must occupy one or other of the positions between the limiting natural numbers I, II, IIL, &., and reduced in a similar manner to (a), (0), (c).... See “The Young Dual Arithmetician,” pp. 83 to 98, and the present work, pp. 7 to 12. 162 DUAL ARITHMETIC. (a) 13°59593 | II. 141421356 | | I (2) "86648125 | III. 200000000 | lI (¢) 2 Wena neg | 10426827° (d) x 4 V. 345000000 | (e) x 2 | VI. 500000000 | (f) x2. | VII. 707106780 | (9) = 98087952 = 7874158 = 122°19816 ee | VIIT. 999999999 | = 13750°9872 PM pe 1| 35251,3,5545751, 3 = b — "V4 Ja 2,2 Aes 3 . = At se 6 4 te d seis 25 ARG alz ee ret 4 8 .. € = "2/0175 Fone, . f=" strisgigooeaa, 4 4 eee! foe 2 a 7:9,5,0;8, a CONCISE PROCESSES, PROBLEMS, ETC. 163 Dual logarithms best adopted to logarithmic computation. |,(1°359593) = 30718541,; —_|, (1°7329625) = 54983241, ; |, (270853654) = 73494409,; _—|, (34377468) = ’106751434 ; |, 7874158) =’23899883; — |, (6109908) = ’49267338 ; |, (98087952) = ’1930560. See Article II. pp. 12 to 14, respecting the general tables where these logarithms may be found by inspection. Ea. 4. Reduce such unwieldy dual numbers as (A), (B), (C), to the most convenient forms for reduction to natural numbers. Also find the dual logarithms which determine the corresponding natural numbers through merely inspecting the general tables. IO (A). ’8's'7’6'9'2'3'4| x Sz written ’8’s’7’6’9'2'3'4 af2 Pe (B). Tot * | 7,8,9,4,5,0,3,4, written 71 7,8,9,4,5,6,3,4, (C). '3'7'7'6'5'5'4'3 [ 2° 10* | 9,3,7,8,4,6,6,7, written '37765'5'4'3 4]2 9,3,7,8,4,0,6,7, Reduction of (A). ’8'5'7'6'9' 234 | = '90083170 = ’20768452 — |, (2) 3 8'5'7'6'9'2’3'4 B[z ='2 T]2 3,0,3,8,0,2, = 5°07789524 To find the most convenient log to enter the tables with. ’90083170 — 3 |, (2) + 2|,(10) = 162489695, and 162489695, — |, (10) = 67768814 = |, (50778952) found through mere inspection. 164 DUAL ARITHMETIC. Reduction of (B). |, 7,8,9,4,5,6,3.4, = 6307856, + |, (2) F|,1 7,8,9,4,5,0,3,4, = £|,2 0,6,3,3,7,8,0,8, = 000426044192 Entering the table with ’85321217 the corresponding natural number will be found = ‘426044192. ’85321217 [3 = Z|2 0,6,3,3,7,8,0,8, Reduction of (C). 37765543 4]2 9,3,7,8,4,6,6,7, = ’2’0'1’0'0'0'0'0 {8 0,2,0,0,6,6,6,1, = 66040°77616 The tables must be entered with the dual log ’41489782 to which corresponds the natural number ‘6604077616 ; for °3'770'5'5'43 [,2 9,3,7,8,4,0,6,7, = 41489782 = 50139291, + 2|,(2) — |, (10) and the decimal point has to be moved five places to the right, which brings 6604077616 to 66040°77615. USEFUL PRACTICAL CRITERIA. tas If a student has sufficient skill to solve the preceding four examples without the use of tables or other extraneous aids, it may be fairly presumed that he understands the elements of dual arithmetic. Ex. 5. Hind a convenient dual number to represent when I aye? g = 32°1816762. z |, (9) = 347139724, > CONCISE PROCESSES, PROBLEMS, ETC. 165 4 of 347139724, = 247956946, the reciprocal of which is '247956946 = 17698437 —|,(10), which may readily be put under the simple dual form bMS LSE Oe RES OE Re et } 20011 2)3) 3:21 '0'3'4 ui gi I Ea. 6. Reduce ———————— to a stm le V(2530°92172)" + (635°297388)" . dual number. Put G=— 253002172 “and b= 635207388: then the expression becomes therefore feral EHC G6 -) af (7352973°° ( ’ \2°5 3692174 = |, (06271045) = 46664207 — |, (10) 2 ) = 2 (45366213 + ’93095145) a b 2 ie [ a ( | = |,(1'06271045) = 6082268, -, AG 42 + (.) } FI; @) = 3041134 + 93095145 — 3], (10) ’2’0'2’0'0'0'00 8 | T 0, 5,0,1,1,5,2,9, = "96136279 — 3 |, (10) But J, (38237143) = 96136279 2 Hence ’3’0’2 3{t 5,0,1,1,5,2,9, 18 the simple dual number and 00038237143 is the natural number of this reciprocal. (21), p. 23. 166 DUAL ARITHMETIC. Ex. 7. Find the reciprocals of the dual numbers ’3'0'0'0'0'0'0'0 | ; A NE Ey a eet Nee je) | 0,5,0,4,3,2,1,7,; and ’3’0°1’0'0'0'00 | 0,5,0,4,3,2,1,5 . The ease with which the reciprocals of dual numbers may be found greatly facilitates the work of calculating the roots of equations. 370000000} = "31608156 ; reciprocal 31608156, = |, 3,0,0,3,0,0,0,3, |, 0,5,0,4,3,2,1,7, = 5018382, ; reciprocal 5 "5018382 =’5 { 6,7,8,6, 2 370 1'0:0:000 |, 0,5,0,4,3,2,1,5, II 266898 36 reciprocal 266898 36, I '2”5'6'00'0'0'0 | 0,0,0,0,7,7, 3,8, See Rules, pp. 15 to 22. Ex. 8. Find the roots of the quadratic equation x’ + ax+b=0, and give the results when a = 2108" and b = 3844. Divide by x, then, b £—-a+—-=0 eG Z+- Sa x CONCISE PROCESSES, PROBLEMS, ETC. 167 yb being substituted for « this last equation becomes b b a eee UG yn 7 k a ° _=— ==, h h HA: y SNe, which ca Ge. When 8 numerically less than either + 2 or — 2 the roots are imaginary, since every number, whole or fractional, positive or negative, added to its reciprocal, gives results which cannot be numerically less than + 2° or — 2. In all such cases y may be put =’v.. | ab the reciprocal of Pee? A/0 which is ee |v,... because ei must always be greater, and, at the same time, nearly equal to y. Besides, this substitution does not require the application of other particular tests or criteria. / Then putting 7 = oh and s =”, and supposing 7 to be taken greater or less than F to facilitate the calculation, ’v and v, vary in accordance, but may be found as follows :— LN ed ee r(t¢[e) + (1 ¥¢[v) = trlv+ts[v=A-—(r+3) In order to find a convenient value and position for », this last expression may be put under the form —7rv+sv=A—(rt+s) and may be positive or negative when the process is continued. AA 168 DUAL ARITHMETIC. 2108 Jb = /3844° = 62°; The iGoe ee (A), 7 = 34° and s, reciprocal of 7, = " = '029411765 aa Nich aa Fs) — (02941 ....) ~ —(r—s) — (+ 3397058 ....) then we have _| 0,0,0,9,0,0,0,9 reciprocal of ’0’0’0’9 }. gives ’0'0'0'9 f; Ld > 8 I Again, 0’0’0’9 { 34 + rh | 0,0,0,9,0,0,0,9, or, — 33°96941224 + ‘02943825 = 33°99885049; (7 + 5). A—(r+s) +4 ‘OOI1I4951 —(r-—8) —33.93997399 then we have ’0’0'0’0’3’3’8’7 | reciprocal of | 0,0,0,0,3,3,8,7, gives | 0,0,0,3,3,8,7, y = 34 multiplied by ’o’0’0’9’0’0'0'0 | 0,0,0,0,3,3,8,7, ; 5 *. 34 x 62 = 2108 mult. by ’9 { 3,3,8,7, = 2089'17446 . Hence quadratic equations may be solved with great ease and certainty, without completing the square. xz also = 2108' — 2089'17446 = 18°92554 Kix. 9. Find the roots of the equation x*—ax+b=0 and apply the general reasoning to the particular case. rs a — 1866°538714a@ + 649'539 =O | (=) =b@ -4h. Since y/d is put = x (@ = |,(y) + 21,@. CONCISE PROCESSES, PROBLEMS, ETC. 169 1; (a); |, (186658714) = 62411173, — $]|,(5); —4],(649539) = 21574620, 83985793, = |, (2°31603784) ! : at y += = 2'31603784= 5 (A), Hence the equation has two real roots since (A) is greater than 2° Put r = 2° then reciprocal s = °5 ; A—(r+s) —*184... ik isn te a ee a which designates that ’1 { is a convenient value for ’v,; then we have | 1,1,0,1,0,0,0,1, the reciprocal of ’1 [. Again, "1 {2 +5 | I,1,0,1,0,0,0,1, or, S00 + 555°. - = 2'°355 2.5 (r+). A—(r+s) —(039...) —(r—s) att 2d. ws.) which designates that ’0’3{ is a convenient value for ’v,; then we have | 1,4,0,4,0,0,0,1, reciprocal of 7173 f. Then, 173 [2+°5 | 1,4,0,4,0,0,0,1, or, 1°74653820 + 57256118 = 2°31909...; (r + 8). A—(r+s) _— (00306...) OS Eee eae e730 te) which shows that ’o’o’2} is a convenient value for ’v,; then as in the foregoing, | 0,0,2,0,0,2,0,0, 1s the reciprocal of ’o’o’2 [. 170 DUAL ARITHMETIC. Lastly, Pe ey 0'0'2 | 174653820 + °57256118 | 0,0,2,0,0,2,0,0, or, 1'74304688 + °57370800 = 2731675488 ; (7 + 8). A -(@}+ 8) p< (OOO7T 704) 7m. > Os ey Oe SZ yresie = (1.16933888) gives 00061 Cee | | y = 713261321 {2° and _],(y) = 55502113, But, |, (@) = |, (y) + 41, @) = 55502113, + 21574620 = 33927493, 33927493, = |, (1'40392925) ; x% = 1403°92925 or, x = 1866°58714 — 140392925 = 462°65789. The value of this new method of finding the roots of quad- ratic equations becomes very apparent when the coefficients are large, as in the present example; and when closer limits are taken, the work will be much curtailed.. For instance, if a number a little less than Beg be Jb substituted for y in ¥ ie the result approaches a In the foregoing solution, if 231 — 2 eee put =7, 204 then the value of y would be found under the simple form 0327613754 { (1°8). : : : : a bi The reason is obvious, since the reciprocal of —- — vb is Vb a t/0: Noe a aia Ou 3). eee panties Ah owe 7, 3 a (— -*) + ; 38 nearly at Ci = YE erties a : a but evidently greater than it. Hence ’v| a ~ v) is con- veniently put for 7. CONCISE PROCESSES, PROBLEMS, ETC. E71 Kix. 10. Find the roots of the equation «+ ax—b=0, and apply the general formule to the particular equation xu -+5a2—8I =0. When the signs of the roots are changed, the equation becomes xv —ax—b=0, therefore B= — = a, x Putting yb for x, as in the last example, the equation becomes a kre 7 fp? (A). I y If y be supposed of the form 7|w,.. then we have t) U5. w+. fs = A, r(it[u) —s(1+[u) =A, tr{wu—tifu=A-(r—-s). In order to find a convenient value and positive for « the last equation may be written +7u+su=A-—(r—s) _A-(r-3) hes fee UW When A is less than 1, Le - B approaches the value of ; b 7, but when A is greater than 1, then ah + “ approaches the value of 7. In the present case, a = 2, a proper fraction and 1°3 may be put for 7 as : 4. 172 DUAL ARITHMETIC. $= — = 769230 catia 79923077 ee T+8 2'06923077 b Pee os} |,0,1,2, = 1194933, reciprocal ’1194933 = ’0'I'2 | 0,0,0,1,0,2,0,1, 1°3 | 0,1,2, + (*76923077)’0'1’2 | 0,0,0,1,0,2,0,1, or 1°31561731 — "76009374 = 55553357 Az ae "00002198. — eaten r+s 2°07572105 eet ded Nat Seat bad Jo hs, Np iey I°3 | 0,1,2,0,1,0,5,6, — 1°31564121 a = 9(131564121) = 11°84077085 Ex. 11. Find the natural sine and log sine of 10° by an independent calculation from knowing the natural sine of 30° to be equal 4. It is well known that if a be the sine of an arc 2 to radius 1, when m is odd, then, the sine of mz will be _ mm — I) 3 m (m* — 1) (m* — 9) 4° 2°3 2°3°4°5 m (mm! — 1) (m* — 9) (m*—25) 0, ge 2°3°4'5°6 Ma In the present example m = 3, then m = 3, z= an are of io to radius I, and 2 = the sine of z. I ; z | wu, will be a convenient dual form for the value of w, since CONCISE PROCESSES, PROBLEMS, ETC. 173 x is a fraction, x may be neglected, then = aero, = Hence the given equation may be put under the dual form () l2u+t(3)= 735 @) and the value of « may be found to any required degree of accuracy. Put (=) = and (3) = 8, r(14[2u) + s(t [u)="753 (A) tr[2uts[u=A—(r +58). 2 - In order to find a convenient value of wv, this last expression may be put under the form +2ru—su=A-—(r+s) I A G3) 7 36 ii Coa Tecan aes 72 |,0,4, = 3980132, reciprocal = 73980132 = ’0°4 | 0,0,0,4,0,0,0,4, Again, putting (-) |0,4,4%, for « the given equation becomes (z) ‘eee 4 (:) ‘O4u ! 0,0,0,4,0,0,0,4, = pi Ae ; (A). 2 If (=) | 0,8, = 03007935 be put for 7, 3 4 and (3) "4 { 4,0,0,0,4, = °72073527 for s, 2 ae ae Bo (1 18)es 00081462: 5. pF; pee * A eT 66057757 + | 0,0,1,2,3, 174 DUAL ARITHMETIC. Because |, 0,4,1,2,3,0,0,0,0,0,0, = 41030813597, and the reciprocal ’41030813597 = ’0’'4'0'8'2’9' 4300775 | Then putting (-) | 0,4,1,2,3,u, for x, we have r= (eh | 0,8,2,4,6, = 030153408486 re Bm ey SS | we (§) 0'4’0'8'2'9'4’3'0'0'7’5 | = “719849665962 _A—(r+8) _ °000003074448 _ , 179929 909 - 2r7—s 659542848992 _ 40CT Gee U c= (=) |, 0,4,1,2,3,4,0,0,1,4,8,4, = 17364817766 |, (z) = 551861098745, — |, (10). Therefore the natural sine of 10° = ‘17364817766 and the dual log sine = 551861098745, — |, (10). Ex. 12. When s represents the sine of the arc a to radius 1, | then 78— 56s + 1128° — 648’ = sine of (7a); tf 7a= 180, 7a = 3607, 7a = 540, de., the equation becomes (2's)! — 7 (2's + 14(2%s) —7 <0; find the three values of (23). The given equation may be put under the form 2°s°)8 oh A A leis: ue 2°s*— 1, which when divided by 2’s* becomes v7 eT a , >, putting » = 2s the last equation becomes V7 205 v I 7 en (K). ; ae i= V7 = 37796447 = «. V7 NENG CONCISE PROCESSES, PROBLEMS, ETC. 175 A value of v being something less than 2° put ’w{ (2) for »v, the equation (K) becomes 2a(t+[u) +4(14[2u) = 1. 2at+2a[u+4til[ew=1. 4+2al[ut4[2u=1— (2a + }). or, (— 2a+4)u=1-— (2a + 4) at a4 Lat Be eA) OP Eat —24a+% —'25592894 The process being continued, one value of v is found to be PnP pg? phe = (2) }’0’'2’5’3’ 1,1,5,5, usually written ’0’2’5’3 { (2) | 0,0,0,0,1,1,5,5, 0'2’5’3 | 0,0,0,0,1,1,5,5, = 97492802 = sine of ae Chord or 25s = 1'94985604. Also v may be found = (3) |,0,4,1,7,6,2,8,4, = 156365975 = 2s s = ‘781820988 = sine of xe The given equation may be put under another form, since 2 .2\3 ae (2’s*— 1)? or =(1 — 27s")? f (v')3 2 eee oe Vv v7 v =[—%v Nz ma 176 DUAL ARITHMETIC. I oY, 5 av=1; (K) ; (a) being put for 7 = M7 = 37796447301 If v be put =’uf then (K) becomes |2u,—’uf(a)=1 1f+[2u—a(it[u) =1 +[2u—talu=1—-1+a=a To find a convenient value for w this last equation may be written +2u+au=(2+a)u=a If ’wu] =’1{ the square of the reciprocal will be | 2,2,0,2,0,0,0,2, now equation (K) becomes (1'23456780) | 2u, — ’u, | (34016802) = 1. or, b|2u,-—"u,f(@=1 b(1 + [2u,) —e(t 4 [u,) = 1 +b[2u,—telu,=1—-b+e This last expression may be put under the form (+ 2b+c)u,=1-b+6¢ to find a convenient value for w, _1—b+6€_ ‘10560013 1" 2b+¢ 2'80030380 which indicates that ’o’3 | is a convenient value for ’w,. | 0,6,0,6,0,0,0,6, is the square of the reciprocal of ’o’3[. CONCISE PROCESSES, PROBLEMS, ETC. 9 By continuing the process the first six digits will be found to I sd ty ee | The next step not only verifies the foregoing work, but also determines the next five or six digits following DE ek ig oar i reduced to the twelfth position = TAIS 31540201 the logarithm of the reciprocal squared will be 283663080562, = ’0'0'2’2'6'0'4’4'7'8'7’7 | 3, 1363.17 | (37796447301) = 32798525862 (¢,) '0'0'2’2’6'6'4'4'7'8'77 | 3, = 1327985 33128 (d,) @,) [2% — wt (@,) = 1. b, (1 4 [2u) — (0 F[w) = 1 + 6,[2u oth $6, [w= the ia b, and ultimately, (+ 26,+¢)u=1+e¢,—8, _1+¢,—6, — ‘00000007266 _ I 2 oh yoo +. 2O8A4is.. 15, Tage es v ='1'3'6'3'1'7700000 | 0,0,0,0,0,0,0,2,4, 3,4, = 433883724578 = s = sine of — NK] Ss he It may be shown by plane geometry, that sine (3@) = 3s—4s*; sine (5a)=5s—20s°+16s; sine (7a)=7s=56s°+1128'—64s'; &e. (See Leslie’s Elements of Geometry. Prop. II. p. 356, 1811.) Putting 2 for v’=2°s" and sine (7a) =0, as before, the equation becomes 2— 724+ 142—-7=0 To take away the second term, substitute # + Z for 2, then the equation becomes 3 178 DUAL ARITHMETIC. Let AGFH be a circle, radius OF = OH=a, ABC an inscribed equilateral triangle; take any point E, between C and B, make the are EC = CF = FG, and draw the other lines of the figure. Fig. 2. Now, by Prop. D, Simson’s Euclid, VI. , AE x CB=CAx EB+ EBXAE dividing by CB=CA=AB gives AK =EB+ EC. Consequently, if EC and EB represent two of the roots of a cubic equation, wanting its second term, AE with a contrary sign will represent the third root. (Saunderson’s Algebra.) Suppose the chord EG of the three arcs EC, CF, FG, to be known and put = 2b. It is easily shown that EB is } of the are EBAG. CONCISE PROCESSES, PROBLEMS, ETC. 179 Let a = the chord EC = OF = FG, and y = the chord EF = CG, | | GH = V 4a a The two quadrilateral figures CFGH and EGFC furnish the following equations :— 2a x y= ange — a + eV 4a? — 2 yXxXy=s=uxunt+2bxe 2®—3a@e%+2ab=0 (M). the chord CE =z, will be one of the positive roots of this equation ; and the chord EB of the third part of the are on the other side of EG must be the affirmative root; for if x be put for the chord of one-third of the are EBHG, the equa- tion will be the same. Comparing equations (L) and (M), also, 20°b = 4 ban eee half BC. 27 6 We subjoin the solution of an ingenious geometrical question connected with a class of cubic equations ; it was proposed in “The Lady’s, Farmer’s, and Mathematical Almanack,” Dublin, 1861; and reproposed in the same work, 1862 and 1863, by Mr. Matthew Collins. No solution of this question has been before published as far as the Author of the present work is cognizant, Question. Prove geometrically that the division of a right angle, or the circumference of a circle, into 7 equal parts, can be effected by means of the trisection of another given angle whose tangent is 3/3. 180 “DUAL ARITHMETIC. The following construction may be employed to divide the circumference i a given circle, BT AZ, into seven equal parts by plane geometry, when the are DP or the are PnQD of the circle DQxP is trisected. aa) oN Take OA = AD = DE=1; on OD describe the equilateral triangle OF D and draw FE. Draw DH parallel to OF fae EH will be the radius of the circle abnQ. Take DP = 5 ae Fig. 3. DE, or of the radius OA, then the straight line DG =— DP of OA, OG = ~AF, ON] In the right-angled triangle CDG, CD = HE = a = the radius of the circle abnQ; DG= GP = otf AO; and A F = v3 e CONCISE PROCESSES, PROBLEMS, ETC. - 181 ae The tangent of the angle OD Game = ee == 33: Let Pd be equal one-third of the are PdaD, then if bn be the side of an equilateral triangle inscribed in,the circle, Pn. will be’ one-third of the greater arc PnQD3-Qn, nb, are*the-sides of an inscribed equilateral triangle. Make PL = PQ, and draw LR “perpendicular to OA meeting the semicircle ORA in R, then OR equal to the side of the regular Heptagon BSTV XYZ. It is evident, from what has gone before, that PQ is the negative value of « in the equation pee ose. 3 27 OP =%=2 She OSA) DB But AO OU OR sor fi 6 OL =.0 8 OR = 2s when s = sine of ~. The positive values of a apply to and r ix. 13. Find the first twelve and the number of figures in the continued product 1.2.3.4.5......... 18. It is shown in the Author’s work, “The Calculus of Form, a New Science,” that Py Lh C2). 45 LG) ads G4) be oso: |, (x) = 5|(2n) + (x +5) |(2) — 10%Q; I 2 e * 182 DUAL ARITHMETIC. B,, B,, B,,.... are the well-known numbers Bernoulli, the ; : I 1% I I I first ten being B, = G3 Butigag! Bao Boag B, = G3 PR 691 B =7, B jee, B _ 43867 | _ 1222277 1 2730? 13 "a 510 ? 17 798 ? 19 2310 XZ = 18'0O000COOOOO00000 + B, 1 . em I I “5 ei 0046296296296 — = a5 th iaa fed 4 ; hae I aaa 0000004762990 + = 360 x Tg es ‘0000000004200 — = Rt x se SOg t 1260 teaiee or Mee ‘OOOO0000000009 + = Py ee 78 og" IT = 680 * 78? 17 9953708462503, = 10"Q |, (18) = 289037175789616471, in the 17th position. 184 |, (18) = 53471877521079047 13, $|,(27)= 91893853320467275, 5439081605 4283710988, in the 17th position. 17995 3708462503, in the 13th position. \ 36395445 2080334, = (10”) | 4,8,9,5,3,5,6,0,8,1,1,3,2, 4(10") | 4,8,9,5,3,5,6,0,8,1,1,3,2, — 640237 3705 730000 This result shows that the product consists of 16 places of figures, the first twelve being 640237370573, which is true to the last figure, for by common multiplication L231 hee. 18 = 6402373705728000. The calculation becomes more easy the greater # becomes, as the two following examples will show. ‘This problem of so CONCISE PROCESSES, PROBLEMS, ETC. 183 much importance in the theory of chances, the calculus of finite differences, definite integrals, &c. received great attention from Euler, Legendre, and Laplace, and yet, with extensive tables and tiresome approximations, very few of the first figures of their results could be depended upon. ‘The dual calculus, in this as in numerous other instances, in a direct manner, removes with ease all impediments. fix. 14. Lind the first eight and the number of figures in the continued product 1.2.3.4.5.6.......... 365. In this example x = 365° and He I I I I I ee A. a 360, 365°" 3600 ¢ 48627125’ which will not amount to a unit in the eight decimal place ; Q in this case = a — = + = (365°) — (00022832) (10°)Q = 36499977168, ar. co. 163500022832, \,(3°65) = 12947271676, in the roth position. (3654) |, (3°65) = 4732227797578, in the roth position. = 47322277976, inthe 8th position. |, (365) = |, (3°65) + |, 0") (3654) |, (365) = (3653) |, (3°65) + |, (10™) These simple preliminary arrangements and calculations being made, the required result is almost instantly obtained, and that, too, without the use of tables or other outlandish dodges. (365%) |, (3°65) = 47322277976, (10°) Q 163500022832, ar. co. 2 |, (27) 91893853, |,(10") +92044726,= 10914194661, =|,(10") +2], 2,3,6,8,3,1,7,3, 2 | 2,3,6,8,3,1,7,3, = 2°5 1041286 CC 484 DUAL ARITHMETIC. The first nine figures of the continued product 1.2.3.4.5....... 365. will be 251041286 and the complete product will consist of 779 places of figures, for 10 x 10% = 10 and 2'510... multiplied by 10™ gives 779 places of figures. Ex. 15. Find the first seventeen figures and the number of figures of the continued product 1.2.3.4.5.6.7........... 1875. |, (1875) = 62860865942237405, in the 17th position. (18754) |,(1°875)=117895 55407466625 3078, in the 17th position. 4|,(27) = 9189385 3320467275, in the 17th position. 11798744792798672035 3, = |,(10°") + 95091166601581537 |,(1875) = |, (10°) + |, (1875) (18754) |, (1875) = {, (10°) + (18754) |, (1875); (18754) |, (1°875) =|, (10°) +95091166915815 37, 4 |, (27) + (18755) |, (1875) =|, (1094) + 95091 166691581537, 1875° = 3515625 I . oe " {875 ~ 00053333333333333 = A A I (1875)" (1875) = ‘00000000015170370 = B es I 5 3 = ‘O0000000000000004 = C (1875)* (1875) j (C) and the succeeding terms may be rejected. CONCISE PROCESSES, PROBLEMS, ETC. In this example Ge BAe Re & m 122) ye aks ce i my Tat tag 300 ws = (1875') — (00004444444444444) + *000000000000421 40 107) = 187499995555555597696, = |, (10°) + 69568985840279344, 4 |, (2m) + (@ + 4) |, (2) — 10"Q = |, (10°48) + 2552218085 1302193, = |,(10%*) + 140651435501004477, 17 | 14065 1435501004477, ol 4 | a RNC RO ee PY OY fe Be By oe eee = 408170321 111040128 185 This product will consist of 5325 places of figures, the first seventeen of which will be 4081 70321TIIIO4013.... Ex. 16. Required the first eight figures of, and the number of Figures tn, the continued product 306.367.368.369 Pee. - 1875.) = |, (10°4) + 140651436, Ce ee 365.) = |, (108) + 92044724, |, (10#*) + 48606712, But. 48606712, = |, 5,0,0,5,2,0,7,2 = |, (16259091). en .6 Oh Consequently, the first eight figures of the continued product eo 4. «© 8 ¢ 366.367.308.369 1875, will be 16259091 and plete product will consist of 4547 figures. the com- 186 DUAL ARITHMETIC. Ex. 17. Suppose (a) 371, the number of chances for the happening of an event in a single trial, and 3597 (b), the number of chances for tts failing ; find how many trials must be made to have an even chance that the event will happen once. Let x be the number of trials. Then, according to De Moivre p4hUS ish (a +b" 2 2= 4 Buh}, Cn Pe ees |, (2) _ 69314718, a\ |,(1°10314151) 9816207, |, (1 + 5) «x = 700612 or 7 times nearly. ix. 18. Suppose (a) the number of chances for the happening of an event in a single trial, and (b) the number of chances for its fathing ; find how many trials (r) must be made to have an even chance that the event will happen (7) times at least. First put, @ =\§3) 0: |7000 > sant vee a then = 200 = q; = = ‘005 ; b Qo ae a BR — ‘000025. The chance that the event will happen at least three times in « trials is equal to the first a — 2 terms of the expansion of sot heidi a b i; (5 OS ey sak Bars) and this chance by hypothesis is 4. Hence the last three terms of the expansion will be equal to 4, that is a (a — b* + wabe—-1 4+ <5 ) a?h®*-2 = (a pe CONCISE PROCESSES, PROBLEMS, ETC. 187 Dividing by 6* this equation may be written a\" - Gt ea pee +5) ; (1) Pat At oe os annne Since @ = 5° and 6 = 1000" (1) becomes a4? + (‘O000125)x + 0049875 + = Hang (2). It is easily shown that w has some value between 100° and 1000°. Put 500), for x, then (2) assumes the form 100 Vesaty 500%, +(‘0000125)(500|u,)+'004987 5 = a OTEy ; +3 (3). We may here premise that |’ 1,0,1,0,0,0,1, is the reciprocal of | 1, P 2:2,0,2,050,0;2} 8 2. [°3 3,0,3,0,0,0,3, 7 3 | 3, &e. &e. and vice versd. ‘The same may be said of {o0,1, and |’0’10,1,0,0,0,1,; of |0,0,1, and [’0’0’1001,0,0, &e. |, (1005) = 498755, 498755, x 500 = 249377500, = |, (005 )°°° (3) put under a logarithmic form becomes I ete + 625 | u, + °49875 t a 249377500, t (249377500,) [ 2, ri 3 (10) ne \; U,. = 19118991, } (249377500,) [w, — |, %, Although the reductions here instituted are extremely simple, yet we have been careful to record every step in establishing a convenient form (4) to operate with, because this is the first equation of the sort ever solved by a direct process. 188 DUAL ARITHMETIC. | 4 ‘20000000 | u, = 19118991, + (249377500) [vw —|,u, (4). + ‘62500000 | u, + ‘49875000 ‘ ‘20000000 ‘62500000 ‘4987 5000 |, (1°32375000) = 28046962, Now let us suppose w, to be in the first position, then for every unit in it the right-hand member of (4) will be increased by 24937750, and diminished by 9531018, hence the increase for », units will be at least = (24937750 — 9531018) wu = 15400732w. Again, for every unit in uv, the left-hand member of (4) becomes ‘20000000 } ’£1,0,1,0,0,0,1, + "62500000 |, + 1°49875000 = 1'36806818 at least. And = (1°323:75000) | 0,3,3,0,8,1,5,5 = 1°36806818 the logarithm 28046962, is increased at least by the logarithm of | 0,3,3,0,8,1,5,5, = 3293104, for every unit in wu, of the first position. Hence the equation 28046962, + 3293104,u = IQII189QQI1, + 15406732,u will give a convenient value of u, 12113628u = 8927971 gives uw = ‘07 Consequently, | 0,7,7,, may be taken as a convenient value for |u, To avoid being misunderstood, these directions are given in detail, and the results carried far beyond the extent required to find a convenient value for u,; indeed, we might have taken wu, = 41, without inconvenience. CONCISE PROCESSES, PROBLEMS, ETC. 189 {’0°7’27,0,2,0,7, is the reciprocal of | 0,7,2, {’0’70,7,0,0,0,7, is the reciprocal of | 0,7, | 0,7,%,, being put for | w, in (4) the equation becomes |, (18654360 (1 + [u,) + 67008460 (1 + [u,) + “49875} = | (1°005 )500 ¥%7,%3, (2) (5) Lorine = 30142692, } (267366433) [u,—|,(u); (5). ‘18654360 67008460 "49875000 |, ( 135537820 ) = 30408052, For each unit in wu, the right-hand member of (5) will be at least increased by 267366 and diminished by 99950, hence the increase for uv, units will at least be (267366 — 99950) u, = 167416u,. ‘The left-hand member of (5) becomes 18635706 + ‘67075468 + ‘49875 = 135586174 for wu,=1 and (1°35537820) | 3,5,6,7,1, = 1°35586174; consequently, the logarithm of the left-hand member is increased at least-by 35671, for each unit in w,; 30408052, + 35671u, being put = 30142692, + 167416u, _ 265360 _ = ds 7 a" aigeeter 2,01. Pe ee | The reciprocal of | 0,0,2,0,1, is |’0’0’20°12,’0’0 Again, to obtain greater accuracy, put | 0,0,2,0,1,w,, for uw, in (5) and the equation becomes |, {18616911 (1 + [u,) + 67143215 (1 + [w,) + 49875} a | (1°05) 5001 9,7,2,0,1,%0, (2) (5) | 47,2,0, Lt = 30479471, + (267904112) [w,—|,(u,); (6). I9gO DUAL ARITHMETIC, ‘18616011 67143215 49875 |, (135635126) = 30479822, For each unit in u, the right-hand member of (6) will be at least increased by 168, therefore the increase for w, units will be 168u, = (268 — 100)u,. The left-hand member of (6) becomes ‘18616930 + °67143278 + ‘49875 = 1°35635208 for u, = 1, hand \{ 135635 peg) | 6,1, = 1°35635208; consequently, the logarithm of the left-hand member of (6) is increased by 61, for each unit in uw. 30479822, + 61u, = 304790471 + 168u, x = 500 | O57,2,0,1,3,2,9, = 5 37°14748. Ex. 19. Let 19(a) be the number of chances for the happening of an event in a single trial, and 200(b) the number of chances for tts failing ; find how many trials (a) must be made to have an even chance that the event will happen 3(r) times at least. ae . . —,, = 0036125 ; a Ria 2b t b and the general equation becomes (11085) 2 24 + 085 + 0036125 (@ — 1) b= which may be written o4* + 00361250 + 0813875 } - er hur CONCISE PROCESSES, PROBLEMS, ETC. Igi The dual logarithm of 1-085 in the 12th position = 81579986989, |, (17085) = 8157999, The value of x is situated between 30 and 40, since for 40, equation (1) becomes ia | | (1'085)*° 40 is + °0036125(40) + 0813875 } and ne a 3a (1:085)*° Lig + 578+ 325551f and FOOSE The logarithm of the left-hand member is |, (£100355000) = 354374, but the logarithm of the right-hand member is greater = 40 |,(1'085) — |, (2) — |, (10) = 26746721, When a = 30] u, equation (1) becomes 30] u, ea ” +. 0036125 (30 | u, ) ai 0813875 } nd we tebe + 65025 | u, fe 488325 } — (3) Fae’. (2) Neglecting |u, and taking the logarithms of both sides of (2), we find |, (1°338575) = 29160565, and 30 |, (1085) — |, (10) = 244739961, — 230258500, = 14481452, When GAO 354374, islessthan 26746721, pee SC) 29160565, is greater than 14481452 Hence the value of x lies between 30 and 40, limits sufficiently close to find the value of x to any required extent. DD 192 DUAL ARITHMETIC. For each unit in |u, of the first position, the logarithm of the left-hand side of (2) is increased by 3439640, at least, and at the same time, the logarithm of the right-hand number of (2) is increased by (24473996, — 9531018,) = 14942878, at least. 29160565 + 3439640wu being put = 14481452 + 14942878 u will point to a convenient value for | u, _ 14679113, Oh =a eee 11503238, | 1,2,u,, may be put, for in (2) we obtain |, {°17823564(1 + [u,) + "72965203 (1 + [u,) + 488325} = L°O8 ec )3041,2,%s, Doe rees = 32845564,4274625157[u,—|,u»; (3) For each unit in uw, the logarithm of the left member of (3) is increased by 39500, and the right-hand member by 174675 ; hence, putting 33370334, + 59500u, = 32845564, — 174675 u, gives u, = 3'9 xz may be put = 30| I,2,5,9,% the next step gives a 30 | 1,2,3,9,1,0,0,0, = 33°795352 This result, so easily found by the dual calculus, defied the combined skill of Laplace, De Moivre, the Bernoullis, and other writers on the theory of probabilities. The method here insti- tuted will apply to equations of all dimensions equated to exponential equations. Kx, 20. Required the value of x in the equation Te 4 ed ote 9. Ans. x = 3'904136670. CONCISE PROCESSES, PROBLEMS, ETC. 193 The solution of this simple-looking question has heretofore defied the skill of mathematicians. (2) Es (5) ors ( (2) em 25131443 oe 2’4’0'3'9'2’0°3 | , (;) = "11778304 = 7171'2’3'7°1'1'8 t Put z=n\|u The given equation may be put under the form (2’40'3’9'2’0'3 [tt tasar « ae Sb ean ea oh A i Pie atari = 7; C240) = 72) = "4685.2. nearly. Mtl 21 \oenis 3 | = 707. 2... nearly. 1175 too great. ee AG) = 0 Of 364 i nearly. Be 1 2.1) ='4'4) = "630... ... nearly. ‘994 too small. Consequently x is less than 4, but greater than 3; however, we may commence operating with either of these numbers. 11778304 3 25131443 =|, (702331962) = 35334912 3 ee "75394329 = |,('470507545) A TBO: 2" (A), Excess. 194 $59 75u, ='I7 ts 05975 U, uf "1728 = (a) 05975 - | 2, { 2, 7\5|3/914|3|2I9 35334912 '1|5\o sola 7066982 7\5|3|9|4|3 353349 DUAL ARITHMETIC. C75 ---) 44) (47...) = $35 25%, (73/5...) (4 [u,) (70 ..) = +424 50u, 71583 2800)=71's'2'7'1'4’8'7 | 7420331 Oi 227 1275 "42755243 — 70'°7'3'8'4 (470507545) (15'2'7'1'4'8'7 f) = (702331962) (0'7'3'8'4'9 43 t) = "AO1610974 652103212 am ae 94 Then the corresponding results may be thus arranged :— 91227138 = |, (401610974) |, (652103212) = '42755243 1053714186 Iixcess, (B). (o1|2...)(}%,) (402) = 3666241, (42|7...) (¢u,) (652) = 2784044, 645028 u, = ‘0537 3f CONCISE PROCESSES, PROBLEMS, ETC. 195 2 2 | 8, | 8, '91|22|71|38 °42|75|52143 — 7|29/81/71 3142|04|19 25/54/30 11|97/15 Pie 23/04 64 30 i 3 5425 58 = '0'3'5'2’7'2'0°6 | 7558781 = '0'7’5'23'2'9'3 "46 29 78 O1 '98 78 59 19 (401610974) (0'7'5’2’3'2’9°3 |) = 372373023 (652103212) (0'3'5’2’7’2’06 |) = "629406473 The next»convenient dual digit may be found by comparing these results. 98785919 = |, (372373023) |, (629406473) = ’46297801 1001779496 001779... Excess, (C). ('987|8 ....) (4 u,) (372 -. ) = (368... )us, (46219 ...-)G4,)(629..) = (291... )u, 659u,= 001779 ~— (C). U, = 2, 3 3 tg; lide: ~ ’987|859\19 ee 1\975|72 925/96 99 46 Horo 7t 0010762: | 926 42 =’9'2'6'4’2 { 989 835 90 "463 904 43 196 DUAL ARITHMETIC. (372373023) (1'97'6'2'1 |) = 371637659 3 ('629406473) (9'2’6'4’2 |) = 628823629 ; 4 1°000461288 000461288 ixcess, (D). We are now in a position to operate to find the next con- venient dual digit, or w,, ('9898| ...- ) Fm) (3716) = 3678. . u, (4639|....) (4 %,) (6288) ai 2OL7 a ares 6595 u, = 000461288 toes Uu,=7) 4 4 1 7; \ 7, ’9898|3590 46390443 6|9288 32473 21 IO | 69309 ="'6'9'3'0'9 | 32483 = °5'2'4'8'5 | 9905 2899 46422926 (371637659) (6'9'3'0°9 |) = "371380158 (628823629) (’3'2’4'8'3 t) = ‘628619392 4 eRe SNe. ‘000000450 Defect, (P). The remaining digits may be found by common division, by retaining the last divisor (D). (D) 6595 450 (’68 f ee 7 396 54 CONCISE PROCESSES, PROBLEMS, ETC. 197 x =’000000°68 | 3 | 2,8,2,7,0,0,0,0, | 3°94139052 yi 3°94139052 BS 83 94139052 9 3°94139052 Hix, 21. Given 3% + 7* = 16" to find x. Ans. x = ‘57812480. As in Example 1, it is evident that Yak Ne ry = (3) a ie ere |, (3) ='167397643 WR eat and \, (2) = ’82667857 Put L= AU, Uy Uy - - It may be soon confirmed that a may be put = 4; for 2 ) °167397643 oe ’83698822 = |, (43301270) 2 ) 82667857 |,(66143783) = 41333929 109445... 090445... Excess, (A). To find a convenient value for w,, (83...) (4%) (43) = 35694, (41...) (tu,) (66) = 2706u, 6275u, = 09445; = (A). u, may be put = T, 198 DUAL ARITHMETIC. 9 219 an ah sponte ='0'8’3'2’90'4'6 0} 313 a "92068704 i (43301270) (0'8’3’2’9'4’6’0 [) = *39824534 (66143783) ('0'4’1'1’3’2’0'7 |) = 63465535 L'0320)08 0320.08 To find a convenient value of w,, 92068704 = |, (‘398245 34) Nr me LS Se ae Le 0'4'1'1'3'2’07 f Excess, (B). |, (63465535) = 45467322 .. )4u,) (398) = 36616 u, (45...) ({ ) (634) = 285304, 65146u, = ‘0329... Hence wu, may be taken = 5, 2 2 \ 5, \ 5, '92|06|87 \04 '45/40|73|22 4)60)34] 35 2|27|33|00 9/20/69 4|54167 g\21 + - 5 (B). 469 64 30 = '0'4'67'5'9.9'4| 2319290 =0'2'3'0'9'0'7'2 | '96 76 51 34 ‘47 78 66 12 (398245 34) ('0°4’6'7"5'0'9'4 |) = °37997443 (63465535) ('0'2’3'090'7’2 |) = 62010525 1'00007968 "00007968 Excess, (C). CONCISE PROCESSES, PROBLEMS, ETC. 199 The remaining digits may be found by common division. 90765134 = |, (37997443) |, (*62010525) = "47786612 G2. ee te M.-.) (3799...) = 296036 (wu... ~) ya) 0 eee Mee eo) C4778... == 36746 (ws . 41) 66382 (u....) = ‘00007968 6638 ) 0000 7968 4 4 6638 (| 1,2, = | Uy Uy ee 4 4 152; }) 152, °9676|5 134 ea ie ae + - 9677 Wn 1935] 956 M1612 —"000'F1'6'1'2 f 5735 ='000°0°5'7'3’5 { 9677 6746 ‘4779 2347 (37997443) (11612 |) = SPS: (62010525)( ’5’7’3’5 |) = ‘62006968 99999999 Mise 410,50; 1,2, 57 OL 2ZA50 Or, peace tee ae wishes a 1 657812489 Ha. 22. Given (4255805712) + (52918°7469)* = (60000°)* to find x. Ans. «= AOL i223: EE 200 DUAL ARITHMETIC. Let L = PB | Wy Uy Uy -- - 42558°05712 = °709300952; and 2201S ie = ‘881979115 60000" / 60000" Therefore, CV2'0'12'5'7'1 {as UryWay ee ('3'2'9 2SGGe [) By testa, coat — ne & B may be put = 3, for if B be put = 4, the result is ereater than 1. aioe = ih high umreeeb ie sum greater than I. 2('3’2 1\(= Gav enéarlyy eat Ones he I . 36! Wine me \ sum greater than I. 3(3'21) =’96] nearly = 686...” But if 8 be taken = 4 then the sum would be less than I. Hence {3 may be conveniently assumed = 3, yet, 2 or 4 may. be put = 8 and a correct value of x obtained. 8 may be either a multiple or submultiple, great, if w,,w,,... be small, and small if w,w,,... be great. However, by one or two rough trials like those above instituted, a value may be given to 8 that will render its application convenient. (709300952) + (881979115)* =1 I | (32'7'2’8'Q'6'2 {)* =e (Ii 20 12 oe ty ere | | 343475 36 "12558691 3 6} |, (356854885 ) =" 103042608 ’3767607 3 =|, (686080203) ‘686080203 "356854885 1°042935088 : OASIS. fA, Inxcess, (A). CONCISE PROCESSES, PROBLEMS, ETC. 201 (376...) (t[u) (686... (1030...) ([u) (356... 2579 .. (u) 3677 « - (u) 7505. (Uy =n Odeo p | ttt): It is evident that u,=0; then wu, = 6, tor 6256u, = 04/29... (6, B17 Seen 2 2 | 6, | 6, °37|67|60)7 3 10|30!42|60]8 2|26/05|64 (61|82|55|6 5|65|14 1]54/56/4 7154 2|06] 1 6 15 231 78 38 = 023076720] 63 39 196 = 06'3'0'8'8'42 | "3999 3911 "1093 81 804 (686080203) ('0'2’3'0’7'6'2’0 |) = 670360856 (356854885) ('0'6'3’0'8'8'4’2 |) = °334935266 '109381804 = |, (334935206) |, (670360856) = °39993911 1°005296122 005296122 Excess, (B). To find %,. (1093 ..... ) (+ [,) (3349) = 366... u, bac... ) ¢ [u,) (6703) = 268... u, | 634u, = 00529 (B). u, may be put = 8, — 202 DUAL ARITHMETIC. 3 3) {8 | 8, 109}381/804 °399/939|11 875/054 3|199|51) 3/063 | 11/20 6) 2 Peay sie bag ty O7812% = 0087 7.72 3t 321073 = 00320923 '110 259 927 3 "403 149 84 (334935266) (0'0'8'7'7'7'2'3 |) = “332006985 (670360856) ('0'0°3’2’0'9'2’3 |) = 668211852 1'000218837 ‘000218837 '110259927 = |, (332006985 Dia ok Phi e |, (668211852) = ’40314984 Zo find u,, (1102)... .% ) [%) (332) = 365864 u, (HOS e\ece ) ¢ [u,) (668) = 2692044, Ey Ley ee Excess, (C). 635... U, = 000218837 Then wu, may be put = 3, and uw, = 4, for 635 ) Sees (3,4, 1/905 2833 2540 CONCISE PROCESSES, PROBLEMS, ETC 4 4 \ 3.4, \ 3.4, ‘1102|5992|7 "403 14984 3307 8 ice 3 | | 4412| =| 4, pees put 3749 3 = (00037493[ 13708 ='0'010'1'3'7'08 | °1102 97420 (332006985) (0'0'0'3'7'4'93 |) = °331882524 (668211852) ((0'0'0'1’3'7'0'8 {) = ‘668120257 1000002881 luxcess, (D). (11029....) (+ [u,) (33188 ....) = 36608 x, ('4033....)(¢[u,) (66812.... 63553u, = 000002881 (D). 63553 ) 000002/881 ( 4, He 2|5.42 339 318 (5, Dee, O.0. 98 AAG 32ie 2 2 E 123297 Kix. 23. Find the first eight figures of the continued product of the odd numbers 1°3'5°7°9 In the Author's Work on the “Calculus of Form, a New Science,” it is shown that Sy are (2a—1)) = ax], (x) + (xv + $) |, (2) — 10°a. 203 204. DUAL ARITHMETIC. In this example x = 505. |, (505') = 622455842,'9275 |, (2) = 693147 18,056 a |, (a) = 314340200678,3875 35038589977,308 = (5054) |, (2) 34937879065 5,085 5 IO°x% = 50500000000, 29887879065 5,685 5 This result divided by |, (10) = 230258509,2994 gives 1298. in the quotient with 3245585, remainder. 3245585, = | 0,3,2,6,0,5,8,6, = 103298827. Hence, the product will consist of 1299 places of figures, the first eight of which is 10329882. Ex. 24. In the curve whose equation is y = eae jind y T when «= 3. And x when y + ; € being equal to 2°718281828... |, (€) = 100000000, 1, (y) = |, (2) — $1, (a) — 2], (e) |, (2) = 69314718,0559045 33 2 |) (7) = 57236494,292470008 = |, (==) T 12078223,763524525 |, (y) = 12078224, + ’900000000 = ’887921776 887021776 = 33112261, — |, (10°) 33112261, = | 3,45553,9,3,255 = 173925305 y = 00013925305 when w= 3. CONCISE PROCESSES, PROBLEMS, ETC. 205 ; I Again I, (=) = 12078224, — x” (100000000,) re 1842068074, + 12078224, IQOOQOO00000, i 541 APe9 x% = 4°3059799 when y= ‘OOOOOOOO! Ex. 25. ind the area of the curve expressed by 2 f° , ‘ aa | e—*“da and the differences for intervals of « = ‘O01; TW! o mie O2: = 03; «ec. First ordinate, |, (y) = 12078224, — (-01)* (100000000,) 12068224, = |, 1,2,5,4,7,3,9,0, = |, (1°12826630) y = 11282663 when «= ‘OI Second ordinate, |, (y) = 12078224, — (*02)’ (100000000,) = 12038224, = |, 1,2,5,1,7,3,9,0, = |, (1'12792789) y = 112792789 when «= ‘02 Third ordinate, |, (y) = 12078224, — ('03)’ (100000000,) ait 1988224, = 5 1,2,4,6,7,3,4,0, ae he (1'12736404) y = 112736404 when w& = ‘03. Fourth ordinate, 1,(y) 12078224, — ('04)* (100000000,) 11918224, = |, 1,2,3,0,7,2,9,0 = |, (1°12657515) y = 112657515 when w= ‘04. 206 DUAL ARITHMETIC. Fifth ordinate, |, (y) = 12078224, — (05)° (L00000000,) = 11828224, = |, 1,2,3,0,7,2,9,0, = |, (1°12556174) y = 112556174 when # = ‘05. Sixth ordinate, |, (y) = 12078224, — ('06)" (100000000,) = 11718224, = |, I,2,1,9,7,1,9,0, = |, (112432425) y = 112432425 when «x = 06. Seventh ordinate, 1, (y) 12078224, — ('07)” (100000000, ) 11588224, = |, 1,2,0,6,7,1,4,0, = 4,(Pilazsasso) y = 1112286359 when «= ‘07 y = 112118147 when «w= '08 y = 111927617, when x= ‘09. Ke. &e. Two or three hundreds of these ordinates may be calculated ina few hours, and a table of corresponding areas formed for any range, as from 2w2=0 to «=2 or wx=3 &c. When greater accuracy is required the intervals must be made less. 2 : 2 : When a = 0, —~ e—* becomes ae 1'1283792 the ordinate at the origin. i? Hd 11283792 Yy, = 11282663 2) 2°2560455 1'1283228 x ‘O1 = ‘0112832 area between oO and y,. CONCISE PROCESSES, PROBLEMS, ETC. 207 1°1282663 11279279 2) 2'°2561942 1°1280976 ‘O112810 = area between y, and y,. 11279279 1°1273640 2) 2°2552919 1°1276459 ‘0112765 = area between y, and y,. ‘0112697 = area between y, and y,. 1126575 1125562 1'124324 1122864 i218! 1°125562 1°124324 1°122864 VI21181 I'119276 me 2) 2°252137 2)2'249886 2)2°:247188 2)2'244045 2)2'240457 ———ooooe 1'126069 1°124943 1123594 1'122023 1°120229 In the practical application of the theory of probabilities, a table for the values of oe | “ena? dx = A, for intervals of a> @ a each = ‘ol is often employed. To construct such table by the dual calculus, requires but little numerical labour. F F 208 DUAL ARITHMETIC. Specimen. x A Diff. fexere) O0°’0000000 112833 oxo}! O'0112832 T12810 0°02 0°0225642 112765 0°03 00338407 112697 ; 004 OAS TI04 112607 The differences 005 00563711 112494 are calculated as LON 3 00676205 112359 shown above. 0°07 070788564 112202 0°08 0'0900766 112023 0°09 O'1012789 Ex. 26. What is the difference of the areas corresponding to the ordinates 1:53 and 1°54, tin the curve of the previous Example ? |, (y) = 12078224, — (1°53)’ (100000000,) = 8246733, — |, (10) 8246733, = |, 0,8,2,8,6,5,6,0, = |,.(1085004mRm y = "108596315 when aw = 1°53 | and y= de helt: when 2 = 1°54 2) 213909383 106954602 xX ‘OI = ‘0010695. ‘0010695 1s the required difference. n ee € 2x2, when Ex. 27. Required the value of C = ~/ 27 em, m = 14000°; x= 7200; wx, = 6800, and J = 163. LO =h(. _ (163)? (14) 10° 2:(-72) (68) 10° — nl? 2a, ~ 29r (7200') (6800) — Hx. 28. What is the area of the curve y = Ome As? 1, (©) CONCISE PROCESSES, =) 2rxx.) | a hO3) a (72) (1°36) ‘42973856 LOS PROBLEMS, ETC. nl? Z2LX 1429 209 Meee eee, = 3'79867239 n > | (=) = "499878930. = 41287868, — |, (10°) 41287868, = |, 4,3,1,7,8,7,4,7, = |, (151116161). C = OOOI5IIIOIOL. = °499878930 + 379867239 2 : — e~™ from x= 3 or We have before shown how the ordinates may be almost instantly calculated from the formula |, (y) = 12078224, — 10° 2’. — 10° x’. *Q00000000 *Q961000000 "1024000000 "1089000000 ’I 156000000 °T225000000 "1296000000 “1 369000000 ?1444000000 «’ 1521000000 1600000000 "168 1000000 "1764000000 "1849000000 y. ‘00013925 "00007 566 ‘00004030 ‘00002104 ‘00001076 "00000540 ‘00000265 ‘000001 28 ‘0000006 1 "00000028 ‘OOOOO0OI 3 ‘00000006 ‘OOOO00002 ‘OOOOOOO I 210 DUAL ARITHMETIC, Tuomas Srmpson’s RULE. To the sum of the first and last, or extreme ordinates, add 4 times the sum of the 2d, ath, 6th, &c., or even ordinates, and twice the sum of the 3d, 5th, 7th, &c, or odd ordinates, not including the extreme ones; the result, multiplied by 4 the ordinates’ equidistance, will be the area. 7566 4030 41488 four times even ordinates. 2104 1076 10894 twice the odd ordinates. 540 265 13925 first ordinate. 128 61 1 last ordinate. ; 28 13 ——_—. 6 2 00066308 1 ordinates’ equidistance. 10372 5447 4 2 3)'000066308 ———— —-- es 41488 10894 ‘000022103 area between 2 = 3 to x = 4°3. area between t 999977897 “=O and 2 =3 The area (A) of the curve whose equation is y = - -*" may T be represented by either of the following series :— Tree pane Dee A =-—= (2-= = + matt dh ap pease Xfce t.0)3 (1). Convergent. < —... 2 ; (2). Divergent. | The first of these formule may be employed when z is less than 2, and the second by the aid of continued fractions, when xis greater than 2; but to apply either (1) or (2) by common Arithmetic, when a is a compound number, almost amounts to’ an impossibility, except to obtain rough approximations. CONCISE PROCESSES, PROBLEMS, ETC. 211 Hix. 29. What is the area of the curve whose. equation ts 2 y=—— e* from x =2 to x = 3? Nor 2. — 10° 2 y. 20 *400000000 02066985 (1) at ’44.1000000 01371565 (2) 2:2 "484000000 00892216 (3) 2% "5 29000000 00568902 (4) 2°4 '5 75000000 00355515 (5) 2°5 625000000 00217828 (6) 2°6 676000000 00130805 (7) a7 "7 2Q000000 "00076992 (8) 2'°8 "784000000 00044421 (9) 2°9 841000000 00025 121 (10) 30 *QO0000000 ‘0001 3925 (11) 1371565 892216 568902 355515 217828 130805 76992 44421 25121 1412957 2260408 2 4 shill shad — 2825914 9041632 02066985 09041632 02825914 00013925 "13948456 ey 3 ) 013948456 004649485 = area from x = 2° to x = 3. 000022103 = area from 2 = 3° to # = 4'3. 004671588 = area from x = 2° to x — 4'3. Ar. Co. ‘995328412 = area from x =0' to x= 2. 212 DUAL ARITHMETIC, Ex. 30. Suppose 18 boys are born to 17 girls, in which case out of 14000 births the most likely individual case is, that 7200 should be boys, and 6800 girls, what is the probability that the number of boys shall fall between 7200 + 163? In Laplace's formula (Q) 2 = 7200 w#, = 6800 n= 14000 i= 163 and m9 = 3'14159265 ... ke ¢€ = 27182515200 eee n72 ¥ n $id Shanti Ll fo Q 2722, eee nee 27 LH, 2 (Q) 2) In Example 27. + ./ e€ 22, is found to be ‘OOOTS5 1116161, and ste = 379867239, which put = 2’, then 2 = 1'94901832. Then making z the independent variable, ant eee 2 pe Dai J« 222, dl becomes Ye: i € dz. oO Zone a7 The area of the curve between x = 1°94901832, and z = 2°00 may be found as in previous examples. When Zi 72" ordinate = ‘02066985 2 = 1'°94901832 ordinate = 02527625 2) 05098168 04594610 Half diff. 02549084 0459461 x ‘02549084 = ‘00117120 = area. CONCISE PROCESSES, PROBLEMS, ETC. 213 According to Example 29 the area between the ordinates for z = 0 to z = 2’ was found to be = 99532841 OOII7120 ‘99415721 = area between 0 and 194901832 “OOO 5111 994308 32 = the value of (Q). It is therefore 99430832 to 00569168, or 175 to I, that the number of male births shall be within the limits specified. Suppose a perfectly flexible chain of uniform density and thickness to be suspended from two fixed points, A and B, and when in equilibrium to form the curve AOB; this curve RS S 8 a & Ra s3 pay SS & g D CG Fig. 4. ig termed the catenary. The equations to this useful curve being of a mixed and exponential kind, the calculations that have been made respecting it amount to little more than mere guess work. 214 DUAL ARITHMETIC. Let O be the lowest point of the chain AOB; OM=2; MP = y; and the arc OP = s. Again, let » be the length of a portion of chain which is, equal to the tension at O. If we suppose the part OP rigid, after it has assumed the form of equilibrium, it will evidently be supported in the same manner, and the tensions at O and P will be the same as when it was loose. OP is therefore kept at rest by three forces, namely, the tension at O acting in the direction of the tangent OY, the tension at P acting in the direction of PT, the tangent to the © curve at the point P and the weight of the piece of chain OP acting in a vertical direction. Because the three forces just described are respectively parallel to the three sides of the triangle TMP, the forces being in equilibrium will be propor- tional to these sides. Weight of OP : tension atO :: TM : MP. But ds, dy, and da are respectively parallel to PT, PM, and MT; ds, dy, dx form a very small right-angled triangle at the point P. | In every plane curve ds’ = dz’ + dy’; hence in general d: dy’\3 ; —— (1 + oY) .. In the catenary, A EAN dx ds We+8 sds 8S COR = da 8 Vu + s° Taking the integral of this last equation, and observing that s =O when 2 =O; we obtain et+v—rJ/(v +s") or P= a? 4+ 20H...... (a). Having determined v, (a) is the equation of the curve expressed by a and s as variables. CONCISE PROCESSES, PROBLEMS, ETC. 215 d j Lada Again, because oY —~ = —__*__. which being integrated we s Wy 4 20a gives Prt he L+VtNG + 20% eS v Ber Vet + 2020 (8) e& rae 7) v See SAO es O).0 e being the base of the hyperbolic system of logarithms, (8) is the equation to curve between the variable co-ordinates x and y. The equation (@) may be put in the form (y) to render it convenient for dual calculation. earta(t)+ {(2)(J(r43)() 5 om ‘Transposing SES, and squaring both sides of (@) we obtain uy Yaetv. (+r)? wv +20 8 SCE opens ie a ie i em eam w fe —4 a+o=F(erre *)) 2 and because (a2) S=a2'+2ve or S=(@+v)—v vf. ae —— bitty = }). s(e@-e"*)s © which is the equation of the curve between the variables s and y ; v being unknown but not variable in each particular inquiry. Suppose ¢ to be the length of a portion of the chain which is equal to the tension at any point P, then, Pesce Mee dst oo Sas =tde: GG 216 DUAL ARITHMETIC. Since s? = x* + 2va, differentiating gives sds = xdx + vda. ida =2dx+vdeandt=ax2+. Suppose the tension at P to be balanced by means of PD, a portion of the chain passing over a pulley at P and hanging freely, then PD=2+v=O0M+4 0; YD = OC =», which is evidently a constant quantity, although unknown. Hence, if the tension be supposed to be balanced by means of portions of the chain hanging over pulleys at points P, A, &c. the lower ends will be in the same horizontal line DC. Example. In a suspension bridge, let the central span AB, between the piers be 677712 feet, the droop or deflection of the chain OX = 52°02 feet, the weight of the chain 365 tons; find the strains at the highest points A and B, and at the lowest point O. AK 7. XOg, Vihaees 338'56 Then if AX=y=1; OX =a =°15365076; = *153650756. = = ‘07682538; and (2a)! = 55434783. Putting z for “ in equation (vy) it becomes ef = 1 + 153650762 + 55434783 23 (1 + 076825382)8; (1). It will be hereafter shown that which 2 approaches ; OR cai Le ay + a is a rough limit to 6x ; ay bal 30490208 in the present Example. 2 may be put = 3025 |u,=4]|2,u, then 2e=4]| 1 =: ’ CONCISE PROCESSES, PROBLEMS, ETC. Zu According to this design equation (1) becomes 802514 —[1+:046479036|u,+'304891 30|5 (I+'02323968 |w,2]; (2). The dual logarithm of the left-hand member of (2), is 30250000, and the dual logarithm of the right-hand member is 30372259, when u,=o0. Hence it is evident from mere inspection, that it is convenient to suppose uw, a dual digit in the third position. It is further evident that for each unit in w, the dual logarithm of the left-hand member of (2), will be at least increased by 30250, for 302/5 OO|00, 3.02/50, for |0,0,1,. Again, for each unit in |0,0,u, the dual logarithm of the right-hand member of (2), will be at least increased by 15065,. Hence the following equation points out a convenient value for t,, ; 30250000, + 30250, uw, = 30372250, + 15065, u, _ 122250, io 15185, = 8, nearly. Then |0,0,8,,, being substituted for uw, in equation (2), it becomes @ 30492847 4 w.4, = [1 ate 04685350 | w,, + °30611270 | (I + 02342625 | w,,)#]; (3). When uw, = 0, the dual logarithm of the left-hand member of (3) is 30492847, and the dual logarithm of the right-hand member is 30493068, and inspection shows that the next position to be occupied is the fifth. Again, for each unit in u,, the logarithm of the left-hand member of (3) will be increased by 305, for 30492847, 305, for | 0,0,0,0,1. 218 DUAL ARITHMETIC. By substituting | 0,0,0,0,1, for w,, in (3) its loparthes will be at least increased by 225, therefore, putting tHe 30492847, + 305 u, = 30493068, + 225 wu, 5 221 gives uUu= ine — 255 % = *3025 | 0,0,8,0,2,7,0,0, = 4 |'2:0,8,0,2:7,010, = v = reciprocal of 4 | 2,0,8,0,2,7,0,0, = 2 | 5,1,8,0,0,0,5,9, v = 3'2793085 v X 33856 = 1110'263 feet. Tension at O = 1110°263 feet of chain. . Tension at A = 1162°283 feet of chain. Although it is not the design of this work to discuss the limits of the roots of equations, we have on many occasions taken the In the present inquiry we assumed that OQ | a most convenient limits. did approaches 2, which is readily shown. => rr 3y° + : Y UV uv Me \ dx We+te2va V2 2U vl elt ree ip 2 340 I2v from integrating and neglecting all the terms after the second. 2 ay? = 2ux Goes =.) 6v I44v x > may be Again, in operating with the dual calculus 2 2 neglected, then v = St | v4, Uy Uy»... to any required de- gree of accuracy. The dual number | w,,w,,u,,.... x + 3 y 6x makes good all the defects of: R. CLAY, SON, AND TAYLOR, PRINTERS. BREAD STREET HILL, aS Tran bite t > é 3 io Be HERE F Riku by hee fee $tbws - pbbe a atad oo he » abi bibcsebbi- ~atriesel hs 7 Abe PPR) bee Ree UNIVERSITY OF ILLINOIS 513.5B990 DUAL ARITHMETIC LONDON -URBANA C001 vo02 7101871 — Er anne aeinm meet Oa oS mne eo at ee " 7 i ag ig is Sig Fa capers gen ips omet es 8 aoe ~ os *~ Sie wo Sale AF ae 5 Py : : Pas ite b be ki LAr, AN GAGS rghs , Kae : ae A ry ens + hehe % ‘ak F f Py Anes. wi Abt » ) ie ; } re : +. hy 1 i . a e H : ‘ 1 ~y 7 q , tik 7! ’ j Se Rs he eats): + NS See ant SURES AEE) tte Ba De --% ‘ Sy yb : sy Coe ous teed ae bis? wees hs L , ; sceterpapen ‘ +e et i $8 te 3 BED mag et ee REE Le BES ek AED Lee BEET D ; Z PERE BAGS Rie RS BES \ fy 7 ? ” ip wehl> Robie See het 5 Sty 5 ; ‘ae Peet one ey Sar step nhs a ake atSpee neve BRS Deets . Are 4 Joey Beye kbty SP etbinathys ‘ i