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Ges pe ae ans ig Toa ON Nit Le eRe eae ye or Sh ih coca aaaia s aS Petar this book on or before the Latest Date stamped below. University of Illinois Library as, S4 te? APR 19 1988 npn 24 Wt ies DA lo L161—H41 A TREATISE ON ALGEBRA. Vor sLis ON SYMBOLICAL ALGEBRA, AND ITS APPLICATIONS TO THE GEOMETRY OF POSITION. BY GEORGE PEACOCK, D.D. V.P.R.S. F.G.S. F.R.A.S. DEAN OF ELY, FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, &e. MEMBER OF THE AMERICAN PHILOSOPHICAL SOCIETY OF PHILADELPHIA, PRESIDENT OF THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, LOWNDES’S PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF CAMBRIDGE, AND LATE FELLOW AND TUTOR OF TRINITY COLLEGE. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS; FOR J. & J. J. DEIGHTON. AND SOLD BY F. & J. RIVINGTON, AND WHITTAKER & CO., LONDON. M.DCCC.XLV. // LA 4 rs f ra gay 4 eS ae 15 aa er t sh ; MAU Bik Digitized by the Internet Archive in 2022 with funding from ig University of Illinois Urbana- phan ie ; ‘ weds aut oe mY wen aid > SUP Pes earn ht | a ‘//archive. ora/Celalls eat seo aa 2p tha . ; ¥ ‘ , pein a a ate Fi Soaks i P ah ase WAY RE RAT ICS Ro. SEP aR YEN PREFACE. I nAvE endeavoured, in the present volume, to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are con- sidered; and this difficulty has not been a little increased, im the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained. It is true that the same views of the relations of the principles of Arithmetical and Symbolical Algebra formed the basis of my first publication on Algebra in 1830: but not only was the nature of the dependance of Symbolical upon Arithmetical Algebra very imperfectly developed in that work, but no sufficient attempt was made to reduce its principles and their applications to a complete and regular system, all whose parts were connected with each other: they have conse- quently been sometimes controverted upon grounds more or less erroneous; and notwithstandmg a very general acknow- ledgment of their theoretical authority, they have hitherto exercised very little influence upon the views of elementary writers on Algebra. It may likewise be very reasonably contended that the reduction of such principles, as those which I have ventured > ROIDDD: 9 ete ep ep PS 49 Boy 1V PREFACE. to put forward, to an elementary form, in which they may be fully understood by an ordinary student, is the only prac- tical and decisive test, I will not say of their correctness, but of their value: for we are very apt to conclude that the most difficult theories and researches which have become familiar to us from long study and contemplation, may be made equally clear and intelligible to others as well as to ourselves: and though I will not say that I feel perfectly secure that I may not have been, in some degree, under the influence of this very ‘common source of self-deception and error, yet I have adopted the only course which was open to me, in order to bring this question to an issue, by embodying my own views in an elementary work, and by suppressing as much as possible any original or other re- searches, which might be considered likely to interfere with its complete and systematic developement. It is from the relations of space that Symbolical Algebra derives its largest range of interpretations, as well as the chief sources of its power of dealing with those branches of science and natural philosophy which are essentially connected with. them: it is for this reason that I have endeavoured to as- sociate Algebra with Geometry thoughout the whole course of its developement, beginning with the geometrical interpre- tation of the signs + and — when considered with reference _ to each other, and advancing to that of the various other signs which are symbolized by the roots of 1: we are thus enabled, in the present volume, to bring the Geometry of Position, embracing the whole theory of lines considered both in relative position and magnitude and the properties of rectilineal figures, under the dominion of Algebra: in a sub- sequent volume this application will be further extended to the Geometry of Situation, (where lines are considered in Ee PREFACE, ¥. their absolute as well as in their relative position with re- spect to each other), and also to the theory of curves. The theory of the roots of 1 is so important, not merely with reference to the signs of affection which they symbolize, but likewise in the exposition of the general theory of equa- tions and in all the higher branches of Symbolical Algebra, that I have thought it expedient to give it with unusual fulness and detail: such roots may be considered as forming the connecting link between Arithmetical and Symbolical Algebra, without whose aid the two sciences could be very imperfectly separated from each other. I have not entered further into the general theory of equations than was necessary to enable me to exhibit the theory of their general solution, as far it can be carried by existing methods, reserving the more complete exposition of their properties and of the methods employed for their numerical solution, to a subsequent volume. The plan which I have adopted necessarily brings Tri- gonometry, or to speak more properly, Goniometry, within the compass of the present volume, not merely as forming the most essential element in the application of Algebra to the Geometry of Position, but as intimately connected with | many important analytical theories. Pe) ay aT, (AS i 0 tas iy Wilby? igi AGG ae 44 r + iA Sih + aes a wa a ¥. : y Mia ee ry, as a av’ 4 ‘> z eeu AR a Wile . } ~ ¢ : \ a i a :. =.) ie - i g : SIRS MOREE os asi , aT ar. poll pint CONTENTS. CHAPTER XI. PAGK Own the operations of addition and subtraction in Symbolical Al- NN RORIUM WUD, BS ioe, ends, « kaa SRE Olga DOME 1 CHAPTER XII. On the operations of multiplication and division in Symbolical (te Sy, ae re Ete Bee elec tt AS ETS rec Pere ee Ck eee eS 17 CHAPTER XIII. On the determination of the highest common divisors, and the lowest common multiples of two or more algebraical expres- ANNs te a chin a Ca dh ce deat ees Beak caeiiy Lagat tai tee dndtcyiad 36 CHAPTER XIV. On the reduction of algebraical expressions to their most simple MLVeMG 4OrMs® 5-510. bts aan 4 ft Se Cen Cee ay ee oe 52 CHAPTER XV. Formal statement of the principle of the permanence of equiva- METIS OUE: eae ete Ree tee ete ess cies eassyereecstiescysacsnsccss 59 RENE RCIEMITII COS 24.2.9, 1.9, as sales Wi) as eb See' *) q Qa -6 q ‘(2a —0 we 8 bottom - 24/005 ( 5 ) 2/Leos ( 3 . 289 3 bottom _ log (s26fSnt) log (1- Fe ON) a a 344 4 top byes, 2 cos Ge ; 2 3 357 11 _ bottom = z at It is feared that very few of the errors have been noticed, no sufficient tin having been devoted to the revision of the work. CHAPTER XI. ON THE OPERATIONS OF ADDITION AND SUBTRACTION IN SYMBOLICAL ALGEBRA. 543. Tuer symbols in Arithmetical Algebra represent num- Distinction bers, whether abstract or concrete, whole or fractional, and the yen ies operations to which they are subject are assumed to be identical and Arith- in meaning and extent with the operations of the same name rennin in common Arithmetic: the only distinction between the two sciences consisting in the substitution of general symbols for digital numbers. Thus, if a be added to 4, as in the expression a+, it is assumed that a and 6 are either abstract numbers or concrete numbers of the same kind: if 6 be subtracted from a, as in the expression a—0, it is assumed that a is greater than 6b, which implies likewise that they are numbers of the same kind: if a be multiplied by 6, as in the expression ab (Art. 34), or if a be divided by 6, as in the expression 5 (Art. 71), it is assumed that 6 is an abstract number. In all these cases, the operation required to be performed, whether it be addition or subtraction, multiplication or division, is clearly defined and understood, and the result which is obtained, is a necessary consequence of the definition: the same observation applies to all the results of _ Arithmetical Algebra. 544. But the symbols, which are thus employed, do not The as- : ; : sumption convey, either to the eye or to the mind, in the same manner as (FyP | digital numbers and geometrical lines, the limitations of value to bees ; values 0 which they are subject in Arithmetical Algebra: for they are the symbols | A i employed | equally competent to represent quantities of all kinds, and of all oP eye he relations of magnitude. But if we venture to ascribe to them necessary a perfect generality of value, (upon which a conventional limi- pire tation was imposed in Arithmetical Algebra), it will be found Pendentuse . . ‘ of the signs to involve, as an immediate and necessary consequence, the + and—. Vor. II. A Positive and nega- tive quan- tities, Assump- tions made in symbo- lical addi- tion and subtrac- tion. 2 recognition of the use of symbols preceded by the signs + and | —, without any direct reference to their connection with other symbols. Thus, in the expression a— 6, if we are authorized to assume ~ a to be either greater or less than 6, we may replace a by the : equivalent expression 6 +c in one case, and by 6 —c in the other: : in the first case, we get a—b=b+c—b=b—b+c (Art. 22), =0+c (Art.16) =+c=c: and in the second a—~b=b—c—b =b-—b—c=0-c=-c. The first result is recognized in Arith- — metical Algebra (Art. 23): but there is no result in Arithmetical — Algebra which corresponds to the second: inasmuch as it is assumed — that no operation can be performed and therefore no result can be _ obtained, when a is less than 6, in the expression a — 6 (Art. 13).* eines 545. Symbols, preceded by the signs + or —, without any connection with other symbols, are called positive and negative | (Art. 32) symbols, or positive and negative quantities: such sym- ’ bols are also said to be affected with the signs + and —. Positive | symbols and the numbers which they represent, form the subjects — of the operations both of Arithmetical and Symbolical Algebra: but negative symbols, whatever be the nature of the quantities — which the unaffected symbols represent, belong exclusively to the province of Symbolical Algebra. | 546. The following are the assumptions, upon which the rules of operation in Symbolical Addition and Subtraction are— founded. , Ist. Symbols, which are general in form, are equally gene- | ral in representation and value. ; 2nd. The rules of the operations of addition and subtraction in Arithmetical Algebra, when applied to symbols which are general in form though restricted in value, are applied, without alteration, in Symbolical Algebra, where the symbols are gene- ral in their value as well as in their form. Y, It will follow from this second assumption, as will be after- wards more fully shewn, that all the results of the operations of addition and subtraction in Arithmetical Algebra, will be results likewise of Symbolical Algebra, but not conversely. If we assume symbols to be capable of all values, from zero upwards, we may likewise include zero in their number: upon this assumption, the ex- pressions a-+ b and a—6 will become 0+ and 0—b, or +b and —b, or b and — b respectively, when a becomes equal to sero: this is another mode of deriving the conclusion in the text. 3 547. Proceeding upon the assumptions made in the last Article, the rule for Symbolical Addition may be stated as fol- lows: “Write all the addends or summands (Art. 24, Ex. 1.) in the same line, preceded by their proper signs, colleeting like terms (Art. 28) into one (Art. 29): and arrange the terms of the result or sum in any order, whether alphabetical or not, which may be considered most symmetrical or most convenient.” It will be understood that negative (Art. 545) as well as positive symbols or expressions may be the subjects of this operation, and it is therefore not necessary, as in Arithmetical Algebra, that the first term of the final result should be positive (Art. 22). 548. The following are examples of Symbolical Addition. (1) Add together 3a and 5a. 8a+5a=8a (Art. 29). (2) Add together 3a and — 5a. 8a—5a=—2a (Art. 31). This is exclusively a result of Symbolical Algebra. (3) Add together —3a and 5a. —3a4+5a=24. This result, which is obtained by the Rule, is equivalent to that which would arise from the subtraction of 3a from 5a: or, in other words, the addition of — 3a to 5a in Symbolical Algebra, is equivalent to the subtraction of 3a from 5a in Arithmetical Algebra. (4) Add together — 3a and — 5a. —38a-—5a=-—8a (Art. 31). This is exclusively a result. of Symbolical Algebra: in con- trasting it however with Ex. 1, it merely differs from it in the use of the sign — throughout, instead of the sign +. (5) 8a (0) Paes (7) —abc —5a — 2 12abc +7a —72" 13abe —4a — 42° —20abe a — 92° 4abe Rule for symbolical addition. Negative ter may occu- py the first placein the results of Symbolical Algebra. Examples. Rule for Subtraction in Symbo- lical Alge- bra. 4 In these examples, the coefficients of the like terms, w which have the same sign, are added together, and to the difference of the sums, preceded by the sign of the greater, is subjoined the: symbolical part of the several like terms: it is the rule given in Art. 31, applied without any reference to the signs of the first term. (8) 3a—4b (9) —Ta?+6xry— Ty’ —7a+8b 82°—-4ay-— y? a—3b —3x°— xy+10y" —3a+ b —20°+ xy+ 2y?’ ee In these examples, the sets of like terms are severally come bined into one (Art. 31), and arranged, in the result, in alpha- betical order, no regard being paid to the placing a positive term, when any exists, in the first place. ) (10) —3a-—46+ 5c — a+2b—3d 36-—4¢+ 6e 7c —8d—9e —4a+ 648c—lld—3e ———— The several sets of like terms are collected together out of the several addends. : (11) 72°— 4axr+ a? —102°—llazr— 4a’ — 32° +13a2+ 10a? — 622-— 2ar+ 7° Fa bh a a ne The alphabetical order of the symbols is, in this case, re- versed. It should be kept in mind in this and in all other cases that the arrangement of the terms in the final result, does not affect its value or signification, but is merely adopted as an aid to the eye or to the memory, or with reference to peculiar circum- stances connected with some one or more of the symbols involved: see the Examples in Art. 33. 549. The rule for subtraction in Symbolical Algebra , derived, by virtue of the assumptions in Art. 546, from ug corresponding rule in Arithmetical Algebra: it may be stated as follows. 5 “To the minuend or minuends, add (Art. 547) the several terms of the subtrahend or subtrahends with their signs changed from + to — and from — to +.” The following are examples of Symbolical subtraction. Examples. (1) From a subtract — 6. The result is a+: or the symbolical difference of a and —b is equivalent to the swum of a and 6. (2) Ta (3) -Ta (4) 7a (5) - 7a 3a — 3a — 3a 3a 4a —4a 10a —10a In these examples, the minuend and subtrahend are written underneath each other, as in common Arithmetic: the results are severally the same as in the following examples of addition. (Art. 547) 7a — 7a 7a —Ta —3a 3a 3a —3a (6) a+5 (7) a-b (8) a+b (9) -a-b 26 —2b Qa These examples are respectively equivalent to (6) a+b-—(a—)), or a+b-a+b=2b. (7) a-—b—(a+b), or a—b-—a-—b=- 2b, (8) a+b-—(-—a+b), or a+b+a—b=2a, (9) —a—b-—(a—b), or —-a—b—a+b=-2a. The terms of the several subtrahends are included between brackets, and, when the brackets are removed, all their signs are changed (Art. 24). | (10) a°+3a?x+3ax?+a° e—Sa 4+ 3ar—x 6a2x+22° (11) 3a— 46+ 7e-— gd 2b—10c— 6d+14e 3$a— 664+17c-— 3d—l14e Aa FT Rn I TRS SIRT ER tn BT oer 6 (12) From $r—7y subtract 7+ 2y and —72+4y. Sa—Ty-(a@+2y)—(-Ta+4y) =3a—Ty—x—-2y+7a—4y=92—13y- The several subtrahends included between brackets and pre- ceded by the sign —, are written in the same line with the! minuend; the brackets are subsequently removed and the signs of the several terms which they include changed, in conformity | with the Rule. (138) 2° +2ry—y°—{a'+ay—-y' + (Qry—2°—y")} =2°4+2ry—y?—a°-ay+y?—(2ry- 2 —y’) == ry—Qay+a?+y" = xv? — ry ty’. In this case, we first remove the exterior brackets and reduce | to their most simple equivalent form the terms which are ex- ternal to those which remain: we then remove the remaining brackets and arrange the terms, when reduced, in alphabeticah order (Arts..20 and 21.) (14) a—[a+b—{a+b+c-—(at+b+c+d)}] =a—a—b+{a+b+ce—-(at+b+c+d)}} = —b+a+b+c—(a+b+c+d) = a+c—a-b-—c-—d = —b-d, or —(6+d). In this case, we have a triple set of brackets which are suc’ cessively removed, and the like terms, which they involve, are | obliterated or reduced into one. See the Examples in Art. 33. eee 550. In the exposition and exemplification of the preceding | usedin rules, we have felt it to be unnecessary to repeat definitions. and | Arithmeti- : 3 ‘ : : acl assumptions which are common to Symbolical and Arithmetical Apes Algebra: such are the ordinary uses and meanings of the signs. In Arith- +and -—, of coefficients (Art. 25), of like and unlike terms (Art. 28), NOTE of indices and powers (Arts. 38 and 39), and the methods of the defini- : . : tions deter- denoting the ordinary operations (Art. 9). pane a 551. The use however, of the same terms in these two sciences | Tules 0 ° e e ° operation: Will by no means imply that they possess the same meaning in all’ ee their applications. In Arithmetic and Arithmetical Algebra, ad- | brathe dition and subtraction are defined or understood in their ordinary | 1 f . . BBerition sense, and the rules of operation are deduced from the defini- | determine tions: in Symbolical Algebra, we adopt the rules of operation | 7 | . . . . . which are thence derived, extending their application to all the mean- values of the symbols and adopting also as the subject matter of pati: : our operations or of our reasonings, whatever quantities or forms of eae | : . - : : and of their symbolical expression may result from this extension: but, inas- results. ‘much as in many cases, the operations required to be performed are impossible, and their results inexplicable, in their ordinary sense, it follows that the meaning of the operations performed, as well as of the results obtained under such circumstances, must be derived from the assumed rules, and not from their defini- tions or assumed meanings, as in Arithmetical Algebra. We will endeavour to illustrate this important and funda- mental distinction between these two sciences in the case of the two operations which form the subject of this Chapter. 552. The rule of subtraction, derived rigorously from the Considera- definition or assumed meaning of that operation in Arithmetical 947.0! tree Algebra, directs us to change the signs of the terms of the Cases of, ; : subtraction subtrahend and to write them, when so changed, in the same in the tran- line with the terms of the minuend, incorporating, by a proper pera ue rule, like terms into one. In the application of this rule, three cal to Sym- cases will present themselves, which it may be proper to con- Aleeper sider separately. Ist. Where the subtrahend is obviously less than the minuend. When the If the minuend be a+c and the subtrahend be a, then a+c pa —a=+c=c. If the minuend be 3a+76 and the subtrahend be the minu- 2a + 6b, then 3a + 76 — (2a + 66) = 3a+76 -2a-—6b=a+ b. seer, In these examples, the results follow necessarily from the definition of subtraction in its ordinary usage, merely supposing that the symbols represent magnitudes of the same kind. 2nd. Where the subtrahend may or may not be less than when the the minuend, according to the relation of the values of the subtrahend , be | symbols involved. iad ihe a If the minuend be 3a+46 and the subtrahend 2a+ 58, then eran 38a+4b—-(2a+5b)=38a+4b-2a-—5b=a—b. necessarily If a be greater than 4, this is a result of Arithmetical Al- Eee gebra, but it ceases to be so when a is less than 6: as long therefore as the relation of the values of a and } remains un- determined, it is uncertain whether it is a result of Arithmeti- cal or of Symbolical Algebra: it is one of an infinite number of cases in which these two sciences may be said to inoscu- late with each other. . If the minuend be 3a—46 and the subtrahend be 2a—6, then 3a—46-(2a—b)=3a—4b—2a+b=a—3b6. Unless a be greater than 36 this is an example of Symbolical Algebra only. When the 3rd. Where the subtrahend is obviously greater than the subtrahend minuend. ls greater ’ than ane If the minuend be a and the subtrahend a+c, then minuend, Examples. a—(a+c)=a-—a-c=-c. If the minuend be 2a + 36 and the subtrahend 3a + 46: then 2a+3b- (3a+ 4b) =2a+3b-—3a-—4b=-—a-—b=~-(a+b). If the minuend be 2a+86+4c and the subtrahend be 4a+7b64+10c: then 2a+3b+4c-(4a+76 + 10c) = 2a+3b+4c-—4a—7b-10c =—2a—4b-6c=-(2a+4b+ 6c). In these examples, the results are obtained, by the appli« cation of the rule for the removal of the brackets when pre- ceded by the sign minus and for the incorporation of like terms, under circumstances which are not recognized in Arith- metical Algebra, inasmuch as they are not necessary conse- quences of the definition of the operation of subtraction, though: they are necessary results of the unlimited application of the rule for performing it. In Symbo- 553. Again, in Arithmetical Algebra, it has been shewn heal Alse- to be indifferent in what order, terms connected by the signs eeeece in plus and minus, succeed each other, so long as a positive term what order ; J a Oe occupies the first place (Art. 22) and the several operations in- Bree dicated are possible*. Thus a+6 is equivalent to b+a: a—b +c is equivalent to a+c-—b or c+a—b or c—b+a*. The same rule is transferred to Symbolical Algebra, without any restriction with respect to the values or signs of the symbols involved, in virtue of the assumptions made in Art. 546. Thus * Art. 21 and note. Thus if b was greater than a, but less than a+c, the operation, or rather succession of operations, indicated in the expression a—b+e would be impossible, but would cease to be so in the expression a+c—O6: itis however convenient, even in Arithmetical Algebra, to consider an expression as representing any value which an interchange of its terms would render it capable of expressing: in virtue of this convention, we might consider —b +a, as equivalent to a—b, and recognise its use in Arithmetical Algebra, if a was greater than b. & 9 a—b is equivalent, in this latter science, to — 6+ a, for all values of a and 6: —3a+ 5a is equivalent to 5a—S3a: a—b+e ‘is equivalent to a+c—b or c+a—b or c—hb+a or —b+ar+c or —b+c+a, and similarly in all other cases. . 554. Inasmuch as the results of symbolical addition and The mean- ‘ ' : ing of those subtraction are obtained from an assumed rule of operation, and yosutic af 10t from the definition of the operation itself, it will follow ae ‘ . ¢ : algebra, chat their meaning, when capable of being interpreted, must which are iti 7 ° not com- ye dependent upon the conditions which they are required to mon to _ satisfy: but as the rules for performing these operations and Sen: . Pe . . Calaigebra, he results obtained are or may be made* identical in those must be ag ‘wo sciences in all cases which are equally within their pros.coraimed by interpre- rince, it is allowable to assume that the operations and their tation. esults, within those limits, possess precisely the same meaning: t is only when the results of these rules are not common to Arithmetical Algebra, that it will be found necessary to resort o an interpretation of their meaning, upon principles which ve shall proceed to establish and to exemplify in the case of ‘he operations which are under consideration. 555. The addition of a symbol preceded by a negative Conditions ‘ign is equivalent to the subtraction of the same symbol pre- eke k ‘eded by a positive sign and conversely. of positive y P 8 y and nega- Thus a+(— 6) =a-—-b= a—(+5): ah a—(—b6)=a+b=a+(+)b)t. _ It appears, therefore, that in the case of negative symbols, ne operation of addition is no longer associated with the fun- amental idea of increase, nor that of subtraction with that of ecrease: and thus a change of sign from plus to minus, in te symbol operated upon, is equivalent to a change of opera- ton from addition to subtraction and conversely. 556. The signs plus and minus, when prefixed to symbols Thesigns enoting quantities of the same kind, cannot denote modifica- eee pendently, _* Thus if a be greater than b, the symbolical result —b +a is convertible into feek bebe, ie -b, which is, under this form, a result of arithmetical algebra. vertible af- T These formule express the rule for the concurrence of like and unlike signs fections of symbolical addition and subtraction, whichis as follows: “when two unlike Magnitude sms come together, they are replaced by the single sign — ; and when two like eg sns come together, they are replaced by the single sign +.” Vou. II. B 10 tions of magnitnde*, but only such affections or qualities . the magnitudes represented, as are convertible by the operatior of addition and subtraction: it is on this account that —a c¢ admit of no interpretation, as compared with a or +4, when denotes an abstract number, to which no qualities are attributed io ppeted 557. Quantities and their symbols are said to be real ¢ sible basa possible, when they can be shewn to correspond to real ¢ a possible existences: in all other cases, they are said to be wa real, impossible or imaginary. It will follow, therefore, that whe positive symbols represent real quantities, the same symbo with a negative sign will be said to be impossible or imag nary, whenever they are not capable of an interpretation, whic is consistent with the conditions they are required to satisf, It remains to shew that there exist large classes of magnitud which possess qualities which can be correctly symbolized k the signs + and —, and that consequently the terms negative an impossible+ are not coextensive in their application. Interpreta- 558. Our first example of the existence of qualities tion of the ) : 4 ; E ‘ signs + and Magnitudes which can be thus symbolized will be in expressir Ea ay the opposite directions of lines in geometry, and which wi symbols re- be found to constitute one of the most extensive applicatiol es ii of Symbolical Algebra: the discussion of the following proble geometry. will form the most simple introduction to this most importa’ theory. Problem. ‘¢ A traveller moves southwards for a miles, and then retur northwards for 6 miles: what is his final distance from the poi of departure ?” Its solution = Let A be the point of by the prin- : ciplesof departure and AB the dis- Pale. tance, expressed in mag- z bra. nitude by the symbol a, to 2 which he travels southwards: let BC be the distance, expressi in magnitude by the symbol 6, through which he returns nort * For if +a and —b continue to denote magnitudes of the same kind, they be replaced by the ordinary symbols of Arithmetical Algebra, such as ¢ and when c+d=a+(—b)=a—b is always greater than c—d or a—(—b) or a4 results which are contradictory to each other. + So numerous are the cases in which negative quantities admit of a consist: interpretation, that the term impossible has never been applied to them: it | been uniformly applied however to a second class of symbolical quantities, thot not, as will be hereafter shewn, with perfect propriety. 11 wards: his final distance AC, when he stops, from the point of departure, is the excess of AB above AC, and is correctly ‘epresented by a—6: or if we suppose c to represent the distance \AC, then we have a-—b=c. As long as AB is greater than BC, or a is greater than b, the raveller continues on the same side of the point of departure 4, md the solution of the problem is strictly within the limits of Arithmetical Algebra: but if we suppose the traveller to return arther northwards than he went, in the first instance, south- wards, or AB to be less went han BC, then his distance © A E AC to the north of the point of departure A, is not capable f being represented in Arithmetical Algebra by a—b, since a s less than 6, and the operation thus indicated is impossible: yut inasmuch as, in this case, AC the final distance of the tra- eller to the north of A is the excess of BC above AB, or of ? above a, it will be correctly represented in Arithmetical Alge- ora by 6—a: and if b=a+c, we shall have b-—a=c. _ There are therefore two distinct cases of this problem, when Two dis- . > . . . ti t solved by the principles of Arithmetical Algebra, according as eet Ge the traveller stops on the soulh or on the north of the point blem when #€ departure: and it will be observed that the solution obtained poate n each case expresses the absolute magnitude of the final dis- eet ance only, and not its quality or affection, whether south or cal Alge- north. br In the geometrical solution of the problem, the distances are Its geome- *xpressed and exhibited to the eye, both in quality and mag- Ree solu- aitude: and there is no such interruption of continuity in passing rom south to north of the point of departure, or, in other words, hrough the zero point, as occurs in the solution of the problem _ y the principles of Arithmetical Algebra. Let us next consider the solution of this problem by the prin- jt; solution ‘iples of Symbolical Algebra. Bre te pice Denoting AB by a and BC by 6, when AB is greater than Symbolica BC, the distance AC from the point of departure is denoted gre yy a—b: and inasmuch as the operation denoted by — is assumed 0 be possible for all relations of value of a and 6 (Art. 546), we may suppose 6 equal to a+c as well as to a—c, or to be treater as well as less than a: in one case we get, when 6=a-—c, The signs + and — used as signs of affection of symbols denoting lines, sym- bolize op- posite directions, and oppo- site direc- tions only. . 12 a—b=a—(a—c)=+c=c, and in the other, when 6=a+¢6 a—b=a—(a+c)=~-c. Assuming the first result + c or c, which is common to Arith- metical Algebra, to represent a distance to the south of the point of departure, the second or —c, which belongs to Symbolical: Algebra alone, will represent a corresponding distance to the north of the point of departure: for the expression a—6 be- comes +c or —e, according as 6 is replaced by a —c, or by a+c, or according as the traveller is on the south or the north of the point of departure, and his actual distance considered with respect to magnitude only, is, in both cases, correctly expressed by c: it follows, therefore, that the signs + and — applied to the distance c, under such circumstances, when con- sidered with reference to each other, will symbolize its qualitie whether south or north: thus if +e denotes a distance to thi south, — c will denote an equal distance to the north and con- versely; such an interpretation of the meaning of these signs when thus applied, is consistent with the assumption that the operation denoted by — in the expression a—b is equally pos- sible for all values of the symbols: and it also enables us to include under one formula, and therefore under one solution, the two cases of the problem which are required to be sepa- rately considered in Arithmetical Algebra*. : + 559. We are enabled to conclude, from the discussion of the preceding problem, that the signs + and —, when thus used independently, symbolize relative qualities of the specific mag- nitude which the symbol c expresses, where the nature of the relation between them is determined by the conditions which they are required to satisfy: thus in the case which we have been considering, it is opposition of direction, and opposition 9 direction alone which these signs can correctly express: for if we suppose AB (a) to be the distance A to which the traveller proceeds south- wards, and BC or 6 the distance to which he proceeds not northwards, but in any other direction, such as “north north east, then his final distance AC from the point departure will not be expressed by a-6, whatever be the re- lation of the values of a and 6: in other words, if we replace b by a—c or by a+ce, the result +c or ¢ in one case and —c im * See appendix. ———v cS et et 13 the other, will not express the final distances of the traveller either in magnitude or direction, and consequently will admit of no interpretation which is relative to the problem proposed *. 560. The same problem may be variously modified in form, The same without altering the essential conditions upon which its solution ae by the principles of Symbolical Algebra depends: thus instead variously i i i - , modified in of estimating distances northwards and southwards of a point form and of departure, they may be estimated, with reference to the pi ee observer's position, as lines to the right or left, up and donn, clusion. fo and from, or as reckoned upon any straight line intermediate in position to these three fundamental directions t: and in all these cases the quantity and direction of the movement fo or from the same point, or the magnitude and direction of the geome- trical lines which express them, will be correctly symbolized by the signs + and —: the steps of the processes by which these conclusions are deduced, are absolutely identical with those which have been followed in the discussion of the problem in Art. 558. 561. Equal lines which have the same relative though dif- Equal lines ferent absolute positions, will be expressed by the same symbol Which have , , the same with the same sign: for relative but if 4B and CD be taken “ ri 0 . ioe equal to each other upon the same line, and be estimated, with positions reference to their mode of description, in the same direction, aN by they are identical in magnitude and direction, and are there- pect fore expressed by the same symbol which symbolizes the first, with the ad by the same sign which symbolizes the second. Again, °°” if two equal lines 4B and CD be drawn per- pendicular to the same line or axis XY, and their directions estimated from it, AB and CD a|___8 being identical in magnitude and direction, will de expressed by the same symbol with the same sign: and, more generally, equal parallel lines ¢ lrawn or estimated in the same direction, whe- ther with reference to a line or plane, will be ikewise represented by the same symbol with 'Y X * For the angle between the directions of the movements, a relation different tom that of mere opposition is involved, as well as the magnitudes of a and b, in he determination of this distance and its direction. t If we conceive three planes to pass through the given points at right angles to vach other, two of which also pass through the eye of the observer, the three funda- nental directions in question will be estimated upon their common intersections. 14 the same sign: for if AB be identical in magnitude and ir direction with CD, and if EF be identical in magnitude and direction with CD, it will follow that AB and EF, which may represent any two D equal and parallel lines E 1 drawn towards the same parts, are identical in magnitude and direction, and therefore represented by the same symbol with the same sign. A B Equal pa- 562. It will follow, therefore, that if AB and CD be tw 2 arate equal parallel lines drawn £ in opposite jn opposite directions, A I directions will be ex- they will be expressed by Pee, the same symbol with dif- symbol ferent signs, one plus and the other minus: for if BA be pro: with dif- duced to E, making AE equal to AB or CD: then CD ane Ad pana oe AE are identical in magnitude and direction, and therefor other—. represented by the same symbol with the same sign: if A therefore be expressed by a, AE and therefore CD will bi expressed by —a and conversely. Dv C Eve eune 563. The same system of symbolization will equally ex express press opposite directions of movement upon curved or upol BAtoas oi straight lines: for, if in the problem proposed in Art. 558 w line straight suppose the traveller to move from A to B upon the curved or zigzag line ACB through a distance Re ge equal to a and then return upon , the same path from B to C through : any / : a distance equal to 6, his final VAN ANY : distance from A, reckoned upon the same path, will be expresset by a—b: and as the conditions of the problem and of Symbo lical Algebra will admit of every possible relation of value of and b, we may equally replace b by a—c or by a+c: and th results + c or c in one case and —c in the other, will express th final distance of the traveller from A: every step of the rea soning by which this conclusion is deduced being the same in the problem referred to. | 15 '. 564. Whatever conclusions have been deduced in the pre- The asin ceding articles respecting the symbolization of the affections pa he i i i ‘ i / _ tion is ap- and magnitudes of lines, will be equally applicable to those mag vHeable to nitudes whose affections and magnitudes can be symbolized or magnitudes expressed by the lines themselves. Thus forces in opposite Mabe directions, such as a force which pushes and a force which Pressed in . ° quantity pulls, or a force which attracts and a force which repels, may and affec- be expressed by lines in opposite directions whose lengths are Hee proportional to the magnitudes of the forces: and it conse- quently follows that the same forces can be expressed by sym- bols affected by the signs + and—. Again, time past and time to come and whatever magnitudes are capable of continuous and indefinite extension, and therefore of a geometrical representation, are likewise susceptible of affections which can be similarly sym- bolized. 565. We shall conclude our observations upon this subject pe cae with the discussion of one more example of a problem of very perty xtensive application. erper , A merchant possesses a pounds and owes 0 pounds: his sub- stance is therefore a— 6, when a is greater than 0. But since a and 6 may possess every relation of value, we may replace 6 by a—c or by a+c, according as a is greater or less than 6: in the first case we get a—b=a-—(a-—c)=c: and in the second, a—b=a—(a+c)=-—c: if c therefore express his substance or property, when solvent, —c will express the amount of his debts when znsolvent: and if from the use of + and — as signs of affection or quality in this case, we pass to their use as signs of operation, then inas- much as (Art. 555) a+(—c)=a-c and a—(—c)=a+e, it will follow, that the addition of a debt (—c) is equivalent to the subtraction of property c of an equal amount, and the subtrac- tion of a debt (—c) is equivalent to the addition of property (c) ‘of an equal amount: and it consequently appears that the sub- traction of a debt, in the language of Symbolical Algebra, is not its obliteration or removal, but the change of its affection or 16 character, from money or property owed to money or property possessed. ’ The preceding examples of the interpretation of the meaning of negative quantities, and of the operations to which they are subjected, will be sufficient to shew the student that the province of Symbolical Algebra is not unreal and imaginary, but that it comprehends the representation of large classes of real exist- ences, including some of the most important of those which are the objects of mathematical and philosophical reasoning. eM Eg a ta CHAPTER XII. ON THE OPERATIONS OF MULTIPLICATION AND DIVISION IN SYMBOLICAL ALGEBRA. 566. Tur fundamental assumptions which were made in Funda- Art. 546, with respect to Symbolical addition and subtraction, Teaseata are equally applicable, and for the same reasons, to Symbolical roped Hil ve multiplication and division: they are as follows: plication Ist. Symbols which are general in form, are equally general ae in representation and value. 2nd. The rules of the operations of multiplication and divi- sion in Arithmetical Algebra, when applied to symbols which are general in form though restricted in value, are applied without alteration to the operations bearing the same names in Symbolical Algebra, when the symbols are general in their value as well as in their form. It will follow from the second assumption that all the re- sults of the operations of multiplication and division in Arith- metical Algebra, will be results likewise of Symbolical Algebra, but not conversely. 567. The same three Cases of the operation of multiplication ore ee present themselves in Symbolical and in Arithmetical Algebra: cal multi- they are as follows: ae gt Ist. When the multiplicand and multiplier are mononomials. 2nd. When the multiplicand is a polynomial and the mul- tiplier a mononomial. 3rd. When both the multiplicand and multiplier are polyno- Mials. 568. In Arithmetical Algebra, the rule for the concurrence The rule for of like and unlike signs (Art. 57,) is required in the 2nd and Oe On 3rd Cases only: but in Symbolical Algebra, the occurrence of and unlike : : ; signs re- xymbols or single terms affected with the signs + and — used amined in independently (Art. 544), renders its application necessary in all all the three cases, the three Cases under consideration. Its dedue- ; : tion from 569. In order to shew that the Rule of signs is a necessary the as- sumptions consequence of the assumptions made in Art. 566, we shall con- ;) a\y s6¢. Vou. II. C The Rule. Rule for symbolical multiplica- tion for Case 1. 18 sider the product of a—6 and c—d as determined by the prin- ciples of Arithmetical Algebra (Art. 56), which is (a—6)(c—d)=ab—ad—be+bd. (1). Assuming, therefore, the permanence of this result, or in other words, the equivalence of the two members of which it is composed, for all values of the symbols, we may suppose two of their number to become successively equal to zero: thus, if we suppose 6 =0 and d=0, the product (1) in question becomes Ist. (a—0)(c—0)=ac—ax0-—0xc+0x0, or a te ah, obliterating the terms which involve zero. If we suppose 6=0 and c=0, we get, and. (a—0)(0—d)=ax0—ad—0x0+0 xd, or a x— =—ad. If we suppose a=0 and d=0, we get, 3rd. (0-6) (c—0)=0xc-0x0-—be+6 x0, or —6 xc=—be. If we suppose a=0 and c=0, we get, 4th. (0—b)(0—d)=0x0-0xd-bx0+5bd, or —6 x—d=bd. It follows therefore generally, as a necessary consequence of the assumptions (Art. 566), which form the foundation of the results of multiplication in Symbolical Algebra, that “ when two like signs, whether + and + or — and —, concur in multiplication, they are replaced in the product by the single sign+: and that mhen two unlike signs similarly concur, whether + and —, or — and +, they are replaced in the product by the single sign an 570. We now proceed to exemplify, in their order, the three different Cases of Symbolical multiplication. Case 1. When the multiplicand and multiplier are mono- nomials. Ruxe. “In finding the mononomial product we must de- termine first, its sign; secondly, its coefficient; and lastly, its literal part.” “Its sign is found by the rule for the concurrence of like and unlike signs which is deduced in the last article: when this sign is +, it is commonly suppressed.” “Its coefficient is the product of the coefficients of the seven factors (Art. 37): if the coefficient be 1, (Art. 30), it is generall suppressed, as not necessary to be exhibited.” “Its literal part is found by writing the several letters and their powers in immediate succession after each other, incor porating powers of the same letter into one by the rule givel in Art. 41.” 19 It is proved in Arithmetical Algebra, (Art. 37), and therefore assumed in Symbolical Algebra, (Art. 566), that it is indifferent in what order the several component factors of a product succeed each other: it is on this account that it is usual to follow: the alphabetical order, whenever the peculiar circumstances of the question under consideration do not render a departure from it convenient. 571. The following are examples: Examples GQ) axb=ab or +ax+b=+ab. mi pe (2) -—ax-—b=ab or —ax—b=+ab. (3) ax—b=~—ab or +ax—b=~-ab. (4) —axb=—ab or —ax+b=—ab. These four examples merely express the rule for the con- urrence of like and unlike signs. (Art. 569.) (5) 5ax7b=35ab. (6) -—Tax—9b=63ab. (7) 172 x —19y=— 323 cry. —32 4 —3ary leas bps y uppressing the number 4, which is common to the numerator nd denominator. (Art. 76). (9) llabe ze ls atbict _—1480°b*c® — — 18a btc® 12 4.4 528 48 _” uppressing the common factor 11. (10) Ilex —12yx—-13z=17162yz. “ad ee 3 2,2 Myc. 5 O89 | Jaya" says Sc PS MS Huests 5 S- 972. Case 2. When the multiplicand is a polynomial and 1e multiplier a mononomial. Rute. “Multiply the single term of the multiplier into Rule for very successive term of the polynomial, and arrange the pro- nai! ucts of the several terms in the result, preceded by their proper gns, in any order which may appear most symmetrical or ‘ost convenient.” “If the sign of the mononomial multiplier be negative, the gns of all the terms of the multiplicand will be changed: in very other respect the rule agrees with that which is given in rithmetical Algebra (Art. 50). Examples. Rule for Case 3. Examples. 26) 573. The following are examples: (1) a(a—b)=a’—ab. (2) -a’(a’—ab+ 6?) =—a*+a°b—a’b’. (3) wy(-aty)=—aytay’. (4) re 4\ 3 o 9 4 10 3 574. Case 3. When both the multiplicand and the multi plier are polynomials. Rute. “ Multiply successively every term of one factor int every term of the other, add the several partial products togethei and arrange the terms of the result in any order which may b considered most convenient, without regard to the sign of th first term.” “If there be three factors, multiply the third into the produc of the two first : and similarly for any number of them” (Art. 61 575. The following are examples: (1) 2r—a xr—b rag —ba+ab x’—(a+b)x+ab (2) 3a—5b —7Ta+4b — 21a*+ 35ab +12ab— 206? — 21a?+47ab ~— 206? (3) («+ a) (a*- aa + 0°) = 2+ a’. (4) (a 0?+2bc~c*) (—a°+b?+2bc4 c*)=~a'+ 2a7b?+ 2a — b*+ 267 c?— c+. (5) (#°+ aa +b) (#*- ax +c) = a'—(a2=b ~c) «?= (ab—ac) + be. | 5 2 2 3 a (6) (Ger+ 3az- 22) (oat ax—£) = Sats 14% _ 1010 5 eT ie = +--+. 6 6 eee 21 | The examples of multiplication which are given in Arithme- eal Algebra (Arts. 37, 51 and 69, and also Arts. 478 and 479) re examples likewise of Symbolical Multiplication, the processes jllowed being the same. 576. It is usual to arrange the terms of the product according Letter or _ » the powers of some one letter, which may be called the /etter oyna r symbol of reference: thus x is the symbol of reference in Exam- tles 1, 3, 5, 6, and a in Example 2: the same letter would be the ymbol of reference in Example 4, if the result was reduced to the quivalent form — a*+ 2 (b?+ c*) a?— 44+ 207c?— c*. The same result may be equally arranged under the equivalent orms — b*+ 2 (a’?+c’) b’—a*+ 2a’c?~c* nd — c+ 2(a?+ b*) c?-—a*+2a°b?—b*, shere } is the symbol of reference in one case, and c in the ther. 577. The processes of division in Arithmetical and Sym- Division in olical Algebra merely differ in the additional rule which is \i7020™ equisite for determining the sign of the quotient or of its first erm, when a negative sign affects one or both of the first terms or the only terms when both of them are mononomials) of the lividend and divisor. There are three Cases of the operation of Symbolical Division Three jorresponding to the three cases of the operation of multipli- the oes lation (Art. 567): we shall consider them in their order. eae ey 1V1S10nN. 578. Case 1. When both the dividend and divisor are Rule for , division aononomials. when the Rute. “The sign of the quotient is positive or negative, aE a, ‘ecording as the signs of the dividend and divisor are the same are mono- r different.” pte “The coefficient of the quotient is found by dividing the coefficient of the dividend by that of the divisor.” “The literal part of the quotient is found by obliterating the ymbols or their powers, which are common to the dividend nd divisor: and retaining those which are not thus suppressed.” Art. 78). 29 Examples. 979. The following are examples: a (wigan a= a9 ror <= =a. — aL (2) ~ax+—-x=a: or, ae aa (3) ar+—ax2=-a: or, ~=-a. (4) -axr+x=—a: or, —" =~. These four examples express the rule of signs: it is im mediately derived from the rule for the concurrence of like am unlike signs in Multiplication (Art. 569), by observing that thi product of the divisor and quotient is equal to the dividem (Arts. 70 and 72). (5) Qa’x®+12a*x*: or, _38x3a'axe 32. Ax 8a?a?xa 4a” suppressing the factor 3a?zx*, which is common to the divi dend and divisor. (6) —18abcde+—12bde, or —18abcde 3ac —12bde ~ 2” suppressing the factor 6bde, which is common both to the dividend and divisor. Other examples are given in Art. 77. When the Case 2. When the dividend is a polynomial and the diviso dividend is armononaminl ob eet ] d . . ° . Re divicor RuxE. “Divide successively by the rule in Case 1 (Art nomial, 78), every term of the dividend by the divisor, and the several results connected with their proper signs form the quotient.” (Art. 81). Examples. 580. The following are examples: (1) Divide -ar—dx by —2, —~ax—bex ———— =a+ hb, pad FY iz 23 (2) Divide aa*- a’x’+ a’x*— atx by —aua, 4 2,,3 3 4,2 4 ar*—a’a*+a>x?—ater > age a a =—a+axr*—a'?r+ a? . —ax = — (a°— ax’+ axr—a?)=a’— a’ x + ar’— 2’, ese several forms being equivalent to each other. (3) Divide —12acfg+ 4af*g—3fe*h by — 40°)? fe, ~ 1acfg+4af?g—3fath 3c _ f , 3gh — 4a°b? fg ab? ab? 44076?” Other examples are given in Art. 82. 581. Case 3. When the divisor is a polynomial. The dividend may be either a polynomial or a mononomial : pene the it in the latter case, as will be shewn hereafter, there can be pale ; _ finite quotient. mial. Rue. “ Arrange the divisor and dividend according to the wers of some one symbol or letter of reference, (Art. 576), or -much as possible in the same order of succession, whether dhabetical or otherwise, and place them in one line as in long vision of numbers in arithmetic: find the quantity or expression uich multiplied into the first term of the divisor will produce 2 first term of the dividend: this is the first term of the ‘otient : multiply this term into all the terms of the divisor and btract the resulting product from the dividend: consider the mainder, if any, as a new dividend, and proceed as before, atinuing the process until no remainder exists, or until the acess becomes obviously interminable.” This is the same rule which is followed in Arithmetical Alge- 2, (Art. 84), and the examples which are given in illustration it (Art 86), are examples also of Symbolical Division. 582. The following are additional examples: Examples. (1) Divide — 22°+ 32°y-62y?+9y’, by 27+ 3y’. x+3y?) —2a°+32°y—-6xy’?+Qy*? (-24+3y 24 (2) Divide «*—2aa’+ (a*—96°)a°+ 184 bax —Qa*b’, by 2*—(a- 36) «- 3ab. x’—(a—3b)x-3ab) x'-2ax*+(a’?—9b*)2*+18ab>x—Qa*°b (a’—(a+3b)a+ 3a a*—(a—3b)a°- 3abx’ —(a+3b) x*°+(a?+3ab—9b’)a°+18ab*x —(a+3b)x°+(a’—9b’)a°+3ab(a+3b)x +38aba°—(3a°b—-Qab’)z—9a°b? +8aba°—(3a°b—Qab*)a—Qa°b? (3) Divide a by 1-2, (Art. 432). 1-2) a (@+ar+ax’+&e. aur +ax +axr—ax’ + a2’ +axr’—ax® +ax° The dividend and the successive remainders are mononomia and the successive subtrahends are binomials: it follows ther fore, that the remainders can never disappear, however far th operation is continued, and the series which forms the quotient therefore interminable. (Art. 87). (4) Divide x+a by «+0. a—b) (a—6b)b (a—b)B a+b) «+a CL Sali a or arene — &e. xr+b +(a—b) (q—t) +450" _(a—4)6 _(a-6)b (a-6)6 OE 25 The terms of the divisor and dividend being arranged with ‘eference to the letter x, we consider a—6, which is the first ‘emainder, as forming a single term only. ' If we reverse the order of the terms in the dividend and livisor, the process will stand as follows: : b+x) a+x (ow eae per ee Ax a+— b 2% (1 fy 22 E5 9s . Eos b b _(a—b)a_ (a—6b) 2’ b 6? (a—b) x? (a—b) 2’ Acie ah * (a—b) 2° ine We have indicated the successive reductions by which the “ : Cx : vst remainder is found: 2x and = are like terms (Art. 28), hose coefficients are 1 and ; respectively, (Art. 30): we then a b , ‘ b : ibtract the fraction + from 1 or from its equivalent B having , ‘ b— ie same denominator with 7 (Art. 128), which gives us i 1 : a—b , ; * its equivalent wae) for the coefficient of x in the re- ainder. a— Xx All the terms of this series, after the first, are negative. 6) F421 48), Hd) (aH) ge v x x 1 Vor. II. D Origin of intermina- ble quo- tients. Intermina- ble quo- tients in Arithmetic. 26 at+x a (a+b)x (a+b) 2’ (7) ee ee , a-2% a (a+b)x (a+6)2’ pts bo eT ee r+5 i baht phen Ve: O) oy 4 ae erie 10+2 Oo ee ee 10 LG oe ee are (10) ee 5 + BS ai t &e = r ; nt : ; A 7 and diver- atroduce negative as well as postive terms 1n the quotient, by gent series lowing the subtraction of partial products which are greater may origi- __-e sina 28 nate in the as: well as less than the successive remainders: thus if we divide same ex- A : : pression by & + @ by xr+a, we get the series a change in oa 20° 2a4 the ar- ; eee rangement B-at Te ae + tte Die of its terms. which is convergent or divergent according as x is greater or les than a: but if we reverse the order of the symbols, dividing a’? +a? by a+a, we get the series 9x27 22° 22% a—x+—-——, +7 - &e... a a a which is convergent or divergent according as a is greater or les: than «: it follows, therefore, that though the fraction re | ahy often fy MOATERe. i ! . is identical in value with TRE must adopt, in Arithmetica) Algebra, in dividing its numerator by the denominator, that forn, of the fraction which secures the arrangement of its terms in thi order of their magnitude: the results of the operation, wheneve| this arrangement is departed from, belong exclusively to Sym, bolical Algebra. Conse- 588. If we multiply any number of the terms (x) of th’ quences of : F obliterat- ‘ : “+a sua , : | a cor. quotient corresponding to - , by the divisor a —a, it will | AE: : a : and jj tain cases, 2a" Ls na panel found to differ in defect from the dividend 2? + a’ by x —1 we suppose the quotient series to be indefinitely continued or to be indefinitely great, this difference will become zero (0) ¢ infinity (co), according as a is less or greater than x, or in othe words, according as the series is a result both of Arithmetic? and Symbolical Algebra, or of the latter science only: and if were at liberty equally to neglect the symbols 0 and @ in th reverse process of passing from a series to the expression | which it originates, we might regard the series | 2 aed, r+at+—+— + &e. - x as equally significant for all relations of the values of the symbol; but under such circumstances, the processes which we must ft low for the purpose of effecting this transition, must not be d pendent upon the arithmetical aggregation of any number of tl terms of the series, but upon methods which are altogether ind. pendent of the relative or absolute value of the symbols involve — —- 29 | 589. When symbols are incorporated by the operation of Interpreta- ultiplication, the sign of the product is determined by the Rule sesueeal ‘Signs (Art. 569): or in other words, the signs of the factors products. »termine the sign of the product. When, therefore, the factors ve assigned, the sign of the product is no longer arbitrary, and 3 relative interpretation, when affected by different signs, must 2 dependent upon the interpretation of the factors themselves: fore entering, however, upon this inquiry, it will be necessary » determine the meaning of such products in Arithmetical Al- ebra, when their factors are not abstract numbers. 590. When one of the two symbols in the product ab is Me one cay : the t 1 abstract number, it simply means that one factor is to be feats aie ultiplied by the other in its ordinary arithmetical sense, the On oy aits of the product being identical with those of the concrete ber. ‘ctor : thus, if a denote a line of 6 inches in length, and 6 the ‘umber 7, then a6 will denote a line of 6 x 7, or 42 inches in ngth: and inasmuch as the factors of the product ab are com- ‘utable with each other, without altering its value, we are at verty to suppose the abstract factor to be in all cases the multi- dier and the concrete factor the multiplicand*. / 591. When both the factors are concrete magnitudes, whe- When both | . re, d ; thesymbols ier the same or different, their interpretation, when possible, are con- ust be made in conformity with the following principles. crete. Ist. The product must be identical with the arithmetical pro- Principles uct, when the factors are replaced by numbers. of ne For if not, the operations in Arithmetical Algebra would not 2 identical in meaning with the corresponding operations in ithmetic. 2nd. The product is always the same in meaning and magni- ide, when the factors incorporated are so. For otherwise, the ctors would not determine the product. | 592. One of the most important cases which we have to When the . . : : oe two sym- msider, is that in which the symbols incorporated represent Lola euiee eometrical straight lines: it will be found that their product sent : straight ; lines. * Upon the same principle, if the concrete factor be affected with the sign — , e product will be affected with the same, and the negative units of one will be entical in meaning with the negative units of the other: thus if —a denote a dis- ‘nce of 3 miles to the south of a given point, and b the number 5, their pro- net — ab or — 15 will denote a distance of 15 miles to the south. | Their pro- duct is the rectangular area which they con- tain. 30 will correctly represent the area of the rectangle which thos lines contain. For, let us suppose the lines AB (a) and AD (6), which con tain the rectangle ABCD, to be in the proportion of the numbers 5 and 4, and also to be represented by them: if we i aed ba _|@ divide AB into 5 equal parts in the | points a, 6, c, d, and AD into 4 equal a parts in the points a, £, y, each of | | these equal parts will be a line repre- “ sented by 1. Through the several poe points of division, let lines be drawn parallel to AB and to AD respectively, dividing the whole re angular area into 5 x 4 or 20 equal squares, each of which i constructed upon one of the several linear units into which th sides are divided. Assuming therefore the product ab to denote the rectangl ABCD, it will satisfy the first condition (Art. 591), because i we replace a and 6 by the numbers 5 and 4, then ab is re placed by their product or 20, the units in the factors bein; equal lines, and those in the product being the squares con structed upon them. Again, all rectangles are equal which are contained by th same sides, and all products are equal which arise from the sam factors: it follows, therefore, that in conformity with the secon; condition (Art. 591), the factors a and 6 determine the product al as well as the rectangle of which it is the symbolical expression’ S— | | | * A parallelogram, not rectangular, would satisfy the first condition, but m the second: for if the sides AB (5) and AD (4) of the oblique parallelogram ———— ieee be divided into linear units, and J i" lines parallel to the other sides be drawn through the points of division, it will be divided into a number of equal rhombs, p which is equal to the numerical product | of the numbers of linear units in the two adjacent sides: but inasmuch as parallel- ograms contained by the same sides, but , making different angles with each other, have different areas, it follows that the products ab might correspond to different values when a and 6 were the same or in other words, the factors would not determine the product: no such amb guity exists with respect to the rectangle, where the angle included by | containing sides is determined by the definition of the figure. a b c d B 31 593. Having shewn that, when a and 6 are geometrical Meaning of 4 ‘ . , 2 the product ‘aight lines, their product ab will express the area of the qi, whena ctangle which they contain, it remains to consider its mean- ante g, when one or both its factors become negative as well as lines ad- Lie: For thi ee mitting of sitive. Kor this purpose, 1 we , D O. theienn voduce the adjacent sides BA +and—. d DA of the rectangle ABCD 6 and d respectively, making » ee 6=AB and Ad= AD, and com- ete the rectangles ADcb, ABEd, d E bed, we shall find, by the prin- ° ples of interpretation established in the last chapter (Art. 558), at if AB=a, then 4b=—a, and if AD=6b, then Ad=—): id of the four adjacent rectangles which are thus formed, d4BbC_D contained by +a and +6, and expressed by +ab or ab it. 592): ADcb is contained by —a and + 6, and is therefore ‘pressed by their product — ab: ABEd is contained by + a and 6, and is expressed by their product — ab: and lastly, Abed contained by — a and — 6, and is therefore expressed by + a6 + ab: for the expressions + ab and — ab represent equal mag- ‘tudes (Art. 558) with different signs, and those signs are de- rmined by the rules of Symbolical Algebra: whilst the inter- etation given to them corresponds equally to all values of e symbols involved, whether positive or negative. 594. There are two rectangles ABCD and Abed, which Ambiguity mrespond to the same product ab; and two others AbcD and pst ae “Hed BEd, which equally correspond to the same product — ab: ae e two first arise from the product of +a and +6 and of —a id — b: the two last from the product of —a and + 6 and of +a id—6. In passing therefore from the factors to the product, e product itself and its interpretation are fully determined ; but the inverse process, when we pass from the product to its mponent factors, there are two different pairs of factors which {ually correspond to it, and there is no reason whatever for e selection of one pair of them in preference to the other: ‘in other words, the determination of the product is ambiguous. hus the rectangles ABEd and ADcb, which are both of them ‘pressed by — ab, are equally related to the rectangles ABCD id Abed, which are both of them expressed by + ab or ad. 32 The pro- 595. If we extend our inquiry to the determination of tl ane a meaning of the product abe of three factors a, 6, c, which sey, presenting Yally represent geometrical straight lines, we Been tie shall find that it may correctly represent the eae volume or solid content of the rectangular & which they Parallelopipedon, of which a, 6 and c are contain. —_ three adjacent edges: for if AB (a), AC (6), AD (c), the three adjacent edges of such a parallelopipedon, (any one of which is per- pendicular to the other two, or to the plane which passes through them,) are expressible by integral numbers, and are severally di- vided into equal linear units, (such as Ab, or Ac, or Ad), ar if planes parallel to the bounding planes of the parallelopiped were made to pass through the several points of division, tl solid ABFDGCE would be divided into cubes equal to ea other (and to becgdf), constructed upon a linear unit, the nun ber of which will be equal to the product of the numbers whic express the number of linear units in each edge respectivel the requisite arithmetical condition is therefore satisfied, it beir merely necessary to keep in mind that the units in the produ are equal cubes, whilst those in the factors are equal lines. This solid is of the same magnitude whatever be the ord in which the symbols or the edges which they represent, a taken: and the product abc denotes a rectangular and not ¢ oblique parallelopipedon, inasmuch as the latter solid involv the values of the angles which the edges make with each othe and which its definition does not determine. D Signsofthe 596. There are only two signs of the product of three fa product of : : : Ain three fac- tors, though they may arise from eight different combinations tors, whicl u may He 1c” the signs of the component factors; they are as follows: positive or negative. (1) +ax+bx+c=abe. (2) +ax-bx-c=abe. (3) -ax+bx-c=abe. (4) -ax—-—bx+c=abe. (5) +ax+bx-—c=—abe. (6) +ax—bx+c=—abce. (7) -ax+6bx+c=-— abe. (8) -ax-bx-c=-—abe. 33 | These eight products would correspond to eight different, Their inter- ‘hough equal and similar rectangular paral- aon slopipedons, having a common angle (4), nd constructed upon edges which form everally one of each of the pairs of lines presented by +a(AB) and —a(Abd), b(AC) and —6(Ac), +c (AD) and —c(Ad). hose pairs of solids, which touch by a ‘ommon plane, and which have therefore ‘wo edges in common, have different signs: ose pairs of solids which have one edge nly in common, have the same sign: whilst those pairs which ave only one point in common, and all whose three edges have i different signs, have also different signs. f the meaning of the product of four or more symbols repre- an enting lines, or of two or more symbols representing areas, or bol. f any other combinations of symbols representing lines, areas, or olids, which exceed three dimensions, inasmuch as there is no ‘rototype in Geometry with which such products can be com- red: in other words, the existence of such products is possible 1 symbols only. 5 7. We should fail in attempting to give the interpretation Products of | 598. We have assigned an interpretation to the products Geometry b and abc, when the symbols, which they involve, represent may ee eometrical lines, in conformity with the general principle which as a sym- onnects Symbolical with Arithmetical algebra, and which assumes bora frat when the symbols are replaced by numbers, such products jegenerate into ordinary arithmetical products: if we may sup- ose, therefore, lines to represent numbers, (and there is no relation f magnitude which they may not represent,) they may equally 2present any concrete magnitudes whatsoever, of which these ‘umbers are the representatives: it will follow, therefore, that if , 6 and c are represented by lines, the rectangle contained by 1e lines a and 4, and the rectangular parallelopipedon con- ‘ructed upon the lines a, 6 and c, may represent any specific jagnitudes, which ab and abc may represent, when a, 6 and c re replaced by numbers. We shall thus give to Geometry the haracter of a symbolical science. ey oL. II. E 34 Space de- 599. Thus, if v represents the uniform velocity of a body’ ee motion, and ¢ the time during which it is continued, the produe eee vt will represent the space over which the body has move sented bya in the time ¢: and if we assume one line to represent v, anc wigidey another line to represent ¢, the rectangle contained by then would represent the space described equally with the sym bolical or numerical product vt. Linear re- 600. When, however, lines are assumed to represent quan pre tities like velocity (v) and time (¢), which are different in thei Ae are nature, and therefore admit of no comparison with each other i perfectly respect of magnitude, the first assumption of them must be per arbitrary. fectly arbitrary: thus, if v denoting a certain velocity, be repre sented by an assumed line, »’ denoting any other velocity, woul be represented by another line bearing the same proportion t the former, that v’ bears to v: and in a similar manner, one lin may represent a time ¢, and another line any other time /’, if the bear to each other the proportion of ¢ to /’: but the magnitudes and ¢ admit of no comparison with each other, and therefore t line which represents an assigned magnitude of one of ther can bear no determinate relation to the line which represents a) assigned magnitude of the other: in other words, the lines, whic severally represent their units, may be assumed at pleasure. Numerical 601. The same remark applies, and for the same reason, feat arbi to the representation of essentially different quantities by mea ooo of numbers, the values of their primary units being perfectl arbitrary: thus the unit of time may be a second, a minut an hour, &c. whilst the unit of space or velocity, (for one : the measure of the other) may be a foot, a yard, a mile, &¢ thus, if the units be assumed to denote severally a second time and a foot in space, we may speak of a velocity denote by 1, 2, 3, 10 or 20, being such as would cause a body to moy uniformly over 1, 2, 3, 10 or 20 feet of space in one secon twice those spaces in 2 seconds, three times those spaces i 3 seconds, and therefore through a space which would be denote by vt, if the body moved with a velocity equal to v, during an number of seconds denoted by ¢: and if we now pass from aritl metic to geometry, we may assume a line to represent a secor of time, whilst an equal or any other line represents a foot 3 space or a velocity of one foot: such primary units being on 35 ssumed, all other values of those magnitudes will be represented y lines bearing a proper relation to them. , 602. The meaning of algebraical products, when the factors Algebraical fe any assigned quantities, being once determined, we expe- WoUents. ence no difficulty in interpreting the meaning of algebraical uotients, when the dividend and divisor are assigned both in spresentation and value: the general principle of such inter- retations being, that “the operation of Division is in all cases re inverse of that of Multiplication:” in other words, the quo- ent or result of the division must be such a quantity, that, when wultiplied into the divisor, it will produce the dividend: we vill mention a few cases. 603. If the dividend and divisor be both of them abstract Examples ambers, the quotient is either an abstract number or a nu- poral Be serical fraction (Art. 92). tion. _ If the dividend be concrete and the divisor an abstract umber or numerical fraction, the quotient is a concrete quantity £ the same kind with the dividend (Art. 590). If the dividend and divisor be concrete quantities of the ume kind, the quotient is an abstract number or a numeri- If the dividend be an area and the divisor a line, the quotient a line which contains, with the divisor, a rectangular area qual to the dividend, (Art. 592). ' If the dividend be a solid and the divisor a line, the quotient the rectangular base of an equal rectangular parallelopipedon, ‘f which the divisor is the third edge (Art. 595). ' If the dividend be a space described and the divisor the niform velocity with which it is described, the quotient is the ‘me of describing it (Art. 599). / It is not necessary, however, to multiply examples of such iterpretations of the meaning of quotients, when the principle ‘hich connects them with their corresponding products admits ‘f such easy and immediate application. CHAPTER XIII. ; ON THE DETERMINATION OF THE HIGHEST COMMON DIVISOR{ AND THE LOWEST COMMON MULTIPLES OF TWO OR MORE ALGEBRAICAL EXPRESSIONS. | | | et 604. Tue determination of the highest common divisor meaning of and the lowest common multiples of two or more algebraica eo expressions will be required for the reduction of fractions t divisors their most simple equivalent forms, in the same manner tha oe * the processes for finding the greatest common measure and th OF aleeb least common multiple of two or more numbers are involve ical expres- in the corresponding reductions of numerical fractions (Arts. 9) “ana and 116): we use the terms highest and lowest, with respect t the dimensions of the symbol of reference (Art. 576.) accordin; to whose powers the terms of those expressions, whose commoi divisors or multiples are required, are arranged: the term greatest and least ceasing to be applicable, in the case of ex pressions whose symbols are indeterminate in value. s ‘Two alge- 605. We shall begin by considering the process for di braical eX- be th di . f t ] b . 1 . pressions covering the common divisor of two algebraical expression ele only, and we shall arrange them as the numerator and denc nighes common Minator of a fraction, which it is proposed to reduce to its mos Heine simple form: for we have already shewn (Arts. 75 and 578, may be as- that any factor common to the numerator and denominator ¢ sumed t ‘ F ; . , oth ite . Sea ae a fraction may be obliterated without altering its signification ¢ Pete: value. There are several steps in this process, which it will b duced to its Convenient to notice in their order. most sim- pao. 606. In the first place, there may exist a common numeric: pee divisor of the coefficients of all the terms of the numerator an factors. | denominator, which may be discovered by inspection, or by th common arithmetical rule (Art. 98.) for that purpose: thu 3 is a divisor of every term of the numerator and denominat of the fraction Qx+15y Examples. A oes Nh ng 120°+21y?’ i 37 hich is immediately reducible, therefore, to the more simple juivalent form | Ba+ 5y 4at+ Ty? In a similar manner, the fraction 791a°+ 452ab + 10176? : 14694 + 12436 reducible to the more simple form Ta’+4ab + 96° 13a+116 > 1e common divisor being 113. 607. In the second place, there may exist a simple symbol, Detection of monono- mial alge- * the numerator and denominator may be divided without intro- braical ucing fractions. Such divisors may generally be discovered . ae y the mere inspection of the several terms in which they present 1emselves. Thus a is obviously a divisor of every term of the numerator Examples. id denominator of the fraction a—ax * powers or products of symbols, by which the several terms a+ ax’ hich is therefore reducible to the more simple equivalent form a—-w@& a+ az Again, xy is a divisor of every term of the fraction Baty — 4a°y? + 52x? y? Cary + 7x’ y? hich becomes, when reduced, 3x°—-4xy 4+ 5y* Ox+ Ty ; irselyes to such as can be obliterated by division without intro- meas : ucing fractional terms: without such a restriction, there would One aa eno limit to the number of such divisors, inasmuch as every as factors. -gebraical expression can be divided by any mononomial, with- at producing an indefinite quotient. Thus, we may divide a*- ax | | 608. In the research of mononomial divisors, we confine What class } Detection of com- pound algebraical factors. Rule. 38 : x and a?+awx severally by a’, and their quotients 1 he and 14 will form the fraction a“ Fils a L 1+-— a which is equivalent to a’—az a-—7 but we should abstain from calling a? a common factor of tl —=-—— a ; a’+ax a+z 2 ; a’ —ax | numerator and denominator of 5 ae because, when applic a+ax as a divisor, it introduces the fractional term —, where no su a | fractional term previously existed. 609. Lastly, there may exist compound or polynomial e pressions, which are divisors of the numerator and denominat of a fraction, by the discovery of which their dimensions m: be reduced, and the form of the fraction generally simplifie thus, «+a will be found to be a divisor of the numerator ai denominator of the fraction | x+ a’ 2*— a?’ which is reducible therefore to the form xv?—ax+a? Le. tae 24D xa | which is more simple than the former with respect to the dime sions, though not with respect to the number, of its terms: b in this, and in all similar cases, there is no obvious charact presented to the eye, by which we can immediately discoy such compound divisors when they exist, and we must follo therefore, for this purpose, a process which is similar to th: for finding the greatest common divisor of two numbers (A. 98). It is as follows: « Arrange the numerator and denominator, or the e: pressions whose common divisor is required to be found, 2 cording to the powers of some one symbol of reference (A 576), and divide one of them by the other, making t | 39 pression the divisor which does not involve the highest power the symbol of reference: continue the division until the highest j wer of that symbol in the remainder is less than in the divisor : iike this remainder the new divisor, and the last divisor the jw dividend, and continue the same process until the re- under disappears, when the last divisor is the compound (mmon divisor required: but if the first remainder from which {2 symbol of reference disappears is not identically equal to iro, then there exists no divisor in which that symbol is in- \lved.” ' 610. The application of this rule will very generally intro- Intrusion ce coefficients of the symbol of reference, which are foreign of extra- neous mul- ites most simple form of the compound divisor which is sought ue of , and which a subsequent modification (Art. 611) of the Socata jocess will enable us to exclude: the following is an example mariage (the occurrence of such a coefficient. Let it be required to the Rulein yluce the fraction Art. 609. i w+ 40°+ 5442 ' v+5n+4 Re { its most simple form. w+5a+4) a+ 40°+52+2 (a@-1 | e+ 5xe + Aer —2#7+ £+2 —a°— 52-4 62+6) a +5x+4 (E+5 r+ 2 +4r+ 4 4x+4 | The last divisor 62 +6 divides the numerator and denomi- t-or of the fraction, giving finite or terminable quotients, but \th terms under a fractional form: but if we had struck out the fitor 6, which is common to both its terms, we should have Gained the form x +1 of the common divisor sought for, which | | | Modifi- cation of the Rule by which they may generally be ex- cluded. Lemmas on which the preced- cible from the following Lemmas. ing Rules are founded. 40 would reduce the original fraction to its most simple equivaler form, and which is : v?+324+2 r+ 611. The intrusion of extraneous divisors in this and simil cases may generally be avoided by the following modificati of the general Rule in Art. 609, the proof of which rests of very simple principles. “Multiply or divide the dividend and the successive diviso and dividends by any number, symbol or expression, which hj no common divisor with the original dividend or divisor, so ; to avoid, in every case, the introduction of fractional terms in the quotients, partial products or remainders: the last divisi found by the Rule in Art. 609, thus modified, will be general the most simple common divisor in which the symbol of r ference is involved. 612. The preceding Rules (Arts. 609 and 611), are dedi Lemma 1. If d divide A, it will divide Aa, provided a do not present itself under a fractional form. For if 4 = 2d, Aa =axd: and inasmuch as a does not prese itself under a fractional form, there is no term in its denominat which can obliterate d or a factor of d. Lemma 2. If d divide A and B, it will divide Aa+B provided that a@ and 6 do not present themselves under a fre tional form. For if d=ad and B=yd, then Aa=aaed and Bb=by and therefore da+ Bh=axrd+byd=(ax+by)d: and there no term in the denominators of a and 6 which can obliteré d or a factor of d. Lemma 3. The highest common divisor of 4 and B is t highest common divisor of Aa and Bb, if a and 6 have no co mon divisor, provided a and 6 do not present themselves unc a fractional form. ? | For if d=ad and B=yd, then 2 and y have no com divisor: in a similar manner, we have da=aad and Bb=bi when ax and by have no common divisor, and where neitl a nor 6 can obliterate d or a factor of d. ‘ Lemma 4, The highest common divisor of da and Bb is | highest common divisor of 4A and B, if a and 6 have no comm divisor, and do not present themselves under a fractional form. t 41 For if da=axd and Bb=byd, then A=ad and B=yd: 1 since av and dy have no common divisor, it follows that which is the highest common divisor of Aa and Bd, is also : highest common divisor of A and B*. It is assumed that a and B, and } and A have no common ‘asure. 613. Two algebraical expressions, like two numbers, may be Extended d to be prime to each other (Art. 107), if they have no common paachast 7 sasure or divisor : a single algebraical expression may be said ewise to be absolutely prime, if it be not resolvible into rational ‘tors. 614. The following is the general form of the process for Form and ding the highest common divisor of two algebraical expres- Sono tha ms; its proof will readily follow from the preceding Lemmas. PT°°€ss: ' It is assumed that a, 4, c, d... are prime to each other and A and B (Art. 613), and that they do not present themselves ider a fractional form. We shall prove, in the first place, that every measure of A d B is a measure of D. * Tt is assumed as a principle (which is proved in arithmetic and arithmetical ‘ebra,) that a factor of a product can only be introduced through multiplication that factor or by another which involves it: and that, when once introduced, it aonly disappear through division: thus, if a= cv andb =dy, then ub =cdry, in other words, c, d, x and y or all the separate factors of a and 6 are found in > product of a and b: and again, if a=cwx and p=“, then ab=cd, where the ‘tor x, which existed in a, has disappeared through division, inasmuch as it esented itself also in the denominator of b. ' Vor. II. FP F 42 For if any expression represented by 2, measures A and F measures Aa and pJB, and therefore da—pB or Ce (Art. 6) and if x measures Cc, it also measures C, for c is prime to 4 ; B, and therefore to x, which is a measure of A and B: andi measures C and B, it measures Bb and q C, and therefore Bb=| or Dd: and if measures Dd, it also measures D, for d is pr} to A and B, and therefore to x, which is a measure of 4 and In the second place, it may be shewn that D is a meas; of A and B. For D is a measure of gC and Dd, and therefore of qC +| or Bb: and if D measures Bb, it measures B, for 6 is prime A and B, and therefore to D: and if D measures pB and Cej measures p B+ Cc or Aa: and if D measures Aa, it measures| for a is prime to A and therefore to D. | It follows, therefore, that every measure of A and B meast} D, and that D measures A and B; and consequently D is| highest common divisor of A and B*. When the 615. If the expressions, whose highest common divisoi expressions whose ; : al highest be convenient to multiply all of them by a factor whichi divisor js 2 common multiple (the lowest when discoverable) of tli renee several denominators: the factor, thus introduced, whether s} fractional ple or compound, may be obliterated, when convenient or nei terms. required, involve terms under fractional forms, it will gener} sary, at the conclusion of the process. Other modifications of the preceding Rules, which are so» times useful as tending to simplify the process, will be notik amongst the following Examples. | os il le = to its lowest terms. x Examples, 616. To reduce the fraction pe x°—a") x —a® (2 a — ara (Dividing by a’) ax —a® x—a) r= a (a+a v*— ax axa? ax—a’* * The process assumes that a, b, c, d... are so chosen, that the quoti ), Geass May not take a fractional form: otherwise it would not necessarily fo} that D would measure p B and qC, when it measures B and C. 612. Ja} 43 _ highest common divisor, therefore, of x*—a* and z?— a? —a: and the reduced fraction is v+acr+a’ ‘ zta 3 - &—30xr +70 : (2) To reduce the fraction ~~, ohgbaad to its lowest terms. “—3x—-—70 @—3x2—-70) x°—39x +70 («+3 2— 32°—70e 32° + 3la+ 70 327— Qx-210 viding by 40) 40x + 280 z+7) 2*— 342-70 (4-10 “+ Tx —10x«2-— 70 —~10x2—70 In this case 40x+ 280 is the first remainder in which the hest power of x is less than the highest power of « in the ‘sor: its factor 40 is prime (Art. 613.) both to 2?— 32-70 . 2-—39x+70, and is therefore struck out: the reduced 2? -—7x+10 sion is ——-—_-——. x—10 (3) To reduce the fraction a — 32° 427+ 32-2 Aw’? —Q92°+247+3 ts lowest terms. * Other examples are (1) A $2 4 — 7 2 a+ ar-a*+a a—ar+a by 3 (2) 30037 «= ————_ : the common divisor is a?+ ax + a? a“—@ t—d@ at—a*t g3—aa?+ a? 2x — a3 a? + a3 x?—ax+a? : the common divisor is x + a. & 44: 413-9 7242 x+3 ) —3 23-2243 x—2 4 444—12074-42°+121—8 (x 474— 92°4+277+ Sax — 32°420°+ Ga- 8 -~ 4 Mere ee a a 120° 822362482 (3 1223-27 2? + 671+ 9 19a°—42x +23) 4 v9 124-2 143 19 762°—-17142+38 1+57 (42 761°—16822+92x —- 34a°—542457 — 19 57 12+-102642—1083 (3 57a2— 1262+ 69 (Dividing by 1152) 1152a2—1152 1927-19 x —23 1423 23 1423 The reduced fraction is therefore xv—Q2°-2+2 Aa’?—5a—8 There are two divisions necessary in each of the two | stages of this operation, before we reach a remainder in wh the highest power of (x), the symbol of reference, is less in the divisor: in the first of these stages the multipliers 4 —4*, and in the second, the multipliers 19 and —19 are quired, in order to prevent the introduction of fractional quoti| (Art. 611): these two pairs of multiplications might be seve replaced by one, if we had multiplied by 4° or 16 in one Q and by 19° or 361 in the other: the process would then stan follows: * We multiply by —4 instead of 4, in order to make the first term of the dend positive ; but this is not essential. ae 45 197242243) at—3a%+ 0° +3r—2 16 162*— 482° + 1627+ 48.1 — 32 (4r-8 1624— 3609+ 827+ 122 —12a3+ 827+362 —32 —1203+272°-— 6x — 9 1922+ 422— 23 (423-9194 2748 361 1444 23—3249 22+ 722+ 1083 C 76x43 1444 a? —3192 2? + 1748 x —572?—1026x +1083 —57x2— 126x + 69 —1152x +1152 It is obvious that the first form of the preceding process is re simple than the second, in consequence of its involving dler coefficients*. * The determination of a series of remainders in conformity with this rule, is e the foundation of Sturm’s process for finding the number and limits of the real, 3 of numerical equations: the process, however, becomes extremely laborious, 1 for equations of the 4th or 5th degrees, in consequence of the rapidity with *h the numerical coefficients increase. ‘Other examples of the same class are (1) a—2a2-1524+36 2?+a¢—12, 322—47—-15 32+5 ~ common divisor is « —3. 2) a4— 3a%— 1829+ 320496 2°+ x? — 147 — 24 a: 4e3—Q7?—367r4+32 427+ 72-8 common divisor is x — 4. ’3) —a?—Qae+45 _2?+20—10_ i 3229-22-21 or +7 ‘common divisor is a— 3. ‘4) 1523 + 3522+ 324+7 a Sattl . Q7 «4+ 632°— 1227-282 9a°- 4a common divisor is 3x + 7. In this example, the denominator must be multiplied by 5, which will give 135 the coefficient of the first term, which is the least common multiple of 27 and 15, (5) r+ 227+9 _@ 2422 +3 — . 729—1la?+1l5c+9 Txe+3 common divisor is 2— 22+ 3. (6) 1222+ 552 + 63 ACTER. 63 2? — 36.2? — 3432+ 196 = 91a? 6la+28° common divisor is 3x + 7: the first multiplier is either 4 or 16, (4) To reduce wep a Be teh ie saad pwarr 2a aes: to its lowest terms. If the numerator and denominator be multiplied by 18 (A 615), the fraction becomes 182°— 1lla’?x— 2a? 182°—6a2—12a° 182*— 6axr—12a") 1823—1la*x-— 2a° (w 182°-— 6a2°-12a°x 6az’?+ ax-— 2a? 3 18az°+3a°x—-— 6a? (a 18axz*— 6a°x — 12a? (Dividing by 3a’) 9a? a+ 6a’* | 3x+2a) 18a°- 6ax—-12a° (62-6 1827+ 12ax —18axr— 12a? —~18ar—12a* 62°—4axr-—a? 6 (#—a) rator and denominator by 6, it becomes The fraction reduced is : or, dividing its num (5) To reduce the fraction a’? +b? +0? +2ab+2ac+%be a’— 6*-—c*®—2be to its lowest terms. 4] Let a be the symbol of reference: a’—b'—2bc—c*) a®+2(b +c) a+b74+2be+c? 1 a’ —b°-2bce-—c? ividing by 2(+<)} 2(b+c)a+2(b?+2bc+c’) a+bh+c) a—b’-2bc-c? fa—(b+c) a’+(b+c)a —(b+c)a—b’—2be—c’ —(b+c)a—b?—2be-c? at+b+e ie reduced fraction is F a6 Let 6 be the symbol of reference : —~b°-—2cbh+a’—c*) 0°? +2(a+c)b+a°+2ac+c (-1 b?+2cb—a’+ec° jividing by 2a) 2ab+2a°+2ac | | b+a+c, or a+b+0¢, tich will be found to be the common divisor as before: the m of the process will be precisely similar if c be made the mbol of reference*. actad+be+bd > °. (6) To reduce PEGE T TREE SUN to its lowest terms. ' Make a the symbol of reference: (c+d)at+bc+bd} (e+f)a+ber+bf { * Other examples of the same class are | aca?+(ad+bc)r+bd ecxt+d. (1) a? zg? — $7 + ~ ax=.b common divisor is ax + b. (2) common divisor is ax + by —cz. a+ (1 +a)ar+a7 _ a+z. (3) a ee at — 4% a? — & a®x?— Qacarz — b?y*+ c?2? ar—by—cz | a?a? + 2abuy +b2y2—c222 axrt+by tes" : common divisor is a?+ 2, | 2 . 48 It will be found by trial, that c+d, or the coefficient divides the numerator but not the denominator: if we st it out of the divisor, we get a+b} (e+fjat+be+bf fe+f (e+f)a+b(e+f) ab+ad+bct+bd c+d ae+aft+be+bf e+f Therefore (7) To reduce the fraction Aa®ca—haida+24a*bca—-240bdx+36 ab’ cx—36 ab’da Tabca—7abda'+7acx'—7 ac da’—21 b*d2°+21b* c2a°+21 bc’ a*—-21 to its lowest terms. In the first place, we discover by inspection, that 4az factor of the numerator, and 7.° a factor of the denomin oleae : AaAx 4a. me, dividing the fraction by ro or 7 > in order to simplify remaining part of the operation, and arranging the result acc ing to powers of a, we get (c—d)a’+6(be—bd)a+9 (bc —b*d) (bc—bd+c’—cd)a— 3(b°d—b’c— be*+ cbd) ; In the next place, we find by trial that c —d, the coefiicie a’, is a common divisor of the numerator and denominator: fraction reduced, by dividing by it, becomes a’+6ab+ 9b? (b+c)a+3(b'+be)” Arranging the process in the usual form, we get (b+c)a+3(b'+bc) ) a+ 16ab6+90° ( It will be found that 6+ c, the coefficient of a in divisor, is a factor of the divisor but not of the divid dividing the divisor by it, we get a+ 3b) a’#+6ab+90° (a+ 3b a’+3ab 3ab +96? 3ab4+ 96? 49 ht follows, therefore, that a+6ab+90? _ a +3b (6+c)a+3 (67+ bc) b+e the original fraction, in its lowest terms, replacing the 5] 4a. or => 18 [a Aa (a+ 3b) Ta’ (b+c) ” highest common divisor being a(c—d)(a+3b), or aca—adx+Sbcex—3bdz*. 617. The preceding process will always succeed in detecting The com- mon factors ‘common factors of two algebraical expressions, when exhibited of two alge- ler a rational form, whenever they exist. braical ex- pressions For in the first place, when the same power only of each under a rational abol presents itself (Ex. 6, Art. 616), the common factor, If fm ane -, will be found amongst the successively reduced coefiicients page be oun . such symbols. In the second place, when different powers of the symbol reference present themselves, the common factors, if any, ich are independent of that symbol, are factors of its co- cients, and may be found by the same general rulet. * The following are other examples of the same class: (1) 6ry + 8r+ 9y + 12 _ Sy +4, 10xry—8x2+1l5y—12 5y—4 Q) vyz+ Sry +Qeetyst+6r4+3yt2+6 _ rz+3xr+24+3 . ayz+2ry +2uz+2yz+4e+4y+42+8 az+2r+22+4 2 primary coefficients of x in the numerator and denominator are yst+dy+22+6 and yz+2y+22+4: _ secondary coefficients of y in the primary coefficients of x are +3 and 2, which are not identical: but the secondary coefficients of z derived from same primary coefficients of « are y+2 and y+2, which are identical, 4 form the common divisor required. The coefficients yz+3y +22 +6, 2+, -2 are the successively reduced coefficients of x in the numerator, amongst ich the common divisor, if any, which is independent of x, must be found. + Thus the terms of the fraction (a?— b?) a2— (2a?— ab — b*) ba +a(a— b) b? (a — 6)?x?— (2a?— 3ab + b*) x + a?b?— ab? :arranged according to the primary symbol of reference x: we first find a — b, ich is the common measure of the coefficients a?— b? and (a — b)? of the first ms of the numerator and denominator, and which is found to be a common asure of all their other terms: we subsequently find, by.the general rule, »~—b ‘Vor. II. G Rule for finding the highest common divisor of three or more alge- braical ex- pressions. Extended use of the term mul- tiple. The lowest common multiple. 50 When these factors, if any, are struck out, we finally deter mine, by the same rule, the common factors which involve th symbol of reference, and thus obtain the most simple equivaler form which the primitive fraction admits. 618. If the highest common factor of three algebraical ex pressions A, B and C is required, we find X the highest commo divisor of A and B; and then the highest common divisor of X an C is the highest common divisor of A, B and C: the proof is th same, mutatis mutandis, as of the Rule for finding the greates common measure of three numbers (Art. 106); and_ similarl: for any number of such expressions. 619. An algebraical expression, which is divisible by an other, may be said to be a multiple of it, and the lowest or mos simple common multiple of two or more such expressions i required in the reduction of two or more algebraical fraction to their most simple equivalent forms, in the same manner a the least common multiple of two or more numbers is require in the corresponding reductions of numerical fractions: for thi purpose, we divide their product by their highest common diyi sor* (Art. 116): and when there are three or more of such ex to be a common measure of the numerator and denominator of the reduce: fraction (a+b) 2?— (Qa+b) ba + ab? (a—b) «?—(2a—b) bx +ab?’ leading to the most simple form of the equivalent fraction, which is (a+b)x—ab (a—b)x—ab~ * If x be the highest common divisor of A and B, andif A=axand B=b2 : B 7 where a and b are prime (Art. 613) to each other, then aS = Ab= Ba, is thi lowest common multiple (M) of A and B: for if not, let m be a multiple of A andB whose dimensions are lower than those of M, and let M@=my=Ab=Ba: it follows b a : ; Abie therefore, that m=Ax—= Bx 7 or in other words, that y is a common divisor 0 a and 6, which is contrary to the hypothesis. It is the fundamental principle of the theory of the measures and multiple of numbers that divisors can only be introduced by multiplication, and obliteratec by division: in other words, that ab can have no divisor, which is not a product 0 divisors of a and b, where 1 and a, 1 and b are included amongst them: an this principle is so essentially involved in our primary conception of number, tha little can be gained by any attempt to establish it by a formal demonstration, a in Art. 110, oe Se 51 essions, whose lowest common multiple is required, we apply Hs same rules, mutatis mutandis, as in common arithmetic, (Arts. 620. ‘The following are examples: (1) The lowest or the most simple common multiple of Examples, f-a? and x*-a* is (x? — a*) (a* — a’) t—U =(e+a)(@°- @)=a'+ax*-aa-a. (2) The lowest common multiple of a°—32?+ 7-21 and 1 og 49 is x — 49) (x?~ 3a°+ Tx-21 ee late Dakin ey Ta =F) _ (at — 49) (e- 9) =«°—382'— 49x + 147. (3) The lowest common multiple of w—a’, «’—(a+b)x+ab and 2-6? of (a° + b) x? + a°b?. This may be found, mutatis mutandis, by the rule in Art. 126, afollows: z—a) «—a’, «x —(a+b)r+ab, 2°-B z—b)x+a, « —b > 2-6? x +4, 1 > a+. ‘The product of «—a, 7-6, x+a,2+6is the lowest common b Itiple required. Reduction of single fractions. Reduction of two or more frac- tions, con- nected by the signs + and— to their most simple equivalent form. Rule. Examples. se CHAPTER” :AIN'. ON THE REDUCTION OF ALGEBRAICAL EXPRESSIONS TO THE MOST SIMPLE EQUIVALENT FORMS. 621. Wuewn an algebraical expression presents itself ull a fractional form, whose numerator and denominator admit of common divisor, their dimensions may be lowered and the fo of the fraction generally simplified without altering its signific tion or value, by the process which is given in the last Chapter 622. When two or more algebraical expressions, one more of which are under a fractional form, are connected the signs + and —, they may be reduced to their lowest comm denominator, and subsequently added or subtracted or reduc to their most simple equivalent form, by the same rule, muta mutandis, which is given for the addition and subtraction numerical fractions in Art. 124: it is as follows. Rute. “Find the wowEst common multiple (Art. 619) the denominators of all the fractions, for the new common de minator. Find the successive quotients which arise from dividing 1 LowxEST common multiple by the several denominators of the fn tions and multiply them successively into the numerators of several fractions, thus forming the successive numerators of equivalent fractions mith a common denominator ; connect the 1 numerators together with their proper signs, and beneath the res write the common denominator, reducing the fraction, which ti form, to its lowest terms.” * | 623. The following are examples. a—b : E | to its most sim . ath (1) To reduce the Sayan PENCIL oy equivalent form. The lowest common multiple of the denominators is th product a’— 6°: (a? — 6°) 5 ape x (a+b) =(a+b) (a+b) =a°+2ab+6', a- Saas x (a—b) =(a—b) (a —b) =a’-2ab+ 6° Their sum = 2a’ + 2b? * See Appendix. 53 The final equivalent fraction is therefore 2a°+2b? 2(a?+b") ‘eo ah ee ch admits of no further reduction. x+6 X+2 ecole ii. Sea 9 St WS Se it eo aa an (2) To reduce the expression st simple equivalent form. The lowest common multiple of the denominators (D) is x? +727 — Ox — 63. et) =( +3) (e+ 6) = 2° + 90418, x? —9 Their sum = 4. EE -_ (+2) =—(a+7)(4+2)=- 2°-Qae—14 The final equivalent fraction is therefore 4 ich admits of no further reduction. xr 3 (3) To reduce the expression Bias (RL UE, to its »st simple equivalent form. D=(1+2)(1-2*)=1+a-a3- 2". Dxez ’ =a(14+204+2e+2°)=r+20°+ 207+ 2%, —2 Dx —-2# ‘ 7 Bee at esee) = - fH PH Dx — x fee +2) Ss — 2—x2', Their sum =¢t+ x — x* The final equivalent fraction is therefore xvt+a?— a x(1+a2-— 2°) ee Serr ee ee ee ee hich admits of no further reduction. Mer iue OY ach ars! ¢5 (e+1)P 2(@+1)? 4(at+1) 4(z+3) 3 most simple equivalent form. (4) To reduce 54 D=4(x +1) (a + 8). rir or Sen oo 91g) ae nip =~ (8+ Y (7+ 3)= = 7 ee i@en > S@+0 ers) see 2oe 49984155 say 75H = 52° 152-18 4x? + 40 The final equivalent fraction is therefore Ac? + 4 FF “ge et A A(ex+1)?(e+3) (a@+1) («+ 3)’ which admits of no further reduction. ‘ In the preceding reductions we have adhered strictly to t general rule, though the same result can frequently be obtain by shorter and more expeditious processes, which it is not 1 cessary to notice: they can only be safely employed by a stud who has already become familiar with algebraical operatio and whose memory is stored with an habitual knowledge a great number of their more simple and elementary results * The following are examples of the same class: (1) a4+-b a—b 4ab u—-b a+b a?—b2° 1 ] oy (2) art 4 pay 1] = 4? ] ] 1 LD es eres es - 115 2 BOF ] » 1 (Vetieigerte eT 2° 5) a (ad—be)x a+be ( ¢ c(c+dx) ct+dzr* bh ad — be athe oy) a SNS aa mb ae 3 ee baie ul SP, 3(l+a2) 3(1—a+a?)” 14.3" é 1 1 1 1 (8) 4(14+ 2) t4(] =a) dao CL pe") icant * 55 24. The rule for the multiplication and division of fractions Multipli- ven in Art. 143: and it will be seen, by a reference to the rapes ‘les which precede it, that it is derived by a species of an- fractions. ition, from the principles which are made the foundation of dolical Algebra: those principles, as applicable to the cases r consideration, may be restated as follows. 25. What is the product of ei Pr ee The pro- b d duct of two fractions. mc oye upposing 7 to be the multiplier, we may assume that amongst successive values of c there is one which is equal to md, : ! c md e m is a whole number, making — equal to —— or m: d d r such circumstances, therefore, the product of and S18 a xomes the product of 3 would be the result if the general form of te product of and m, which is —— " (Art. 130): d 7 Was ae and no other form of this product will satisfy equired condition: and inasmuch as it is assumed that the of this product, whatever it may be, is independent of the fic values of the symbols involved ray 546), it follows that which is the form of the product hes 3 and 5 in one case, be its form likewise in all others. : : oe c ; 26. What is the quotient of © divided by =? Quotient of b d one frac- tion divided . + c md another. upposing the divisor 5, as before, to become 7 OF ms by d* e m is a whole number, it will follow that the quotient of ; . : . a will become, under such circumstances, identical with 1 1 1 1 S(a—1)” 4(c@—3) * B(a—5) ~ 28 —9a? + Qn 15" 1 1 1 a~—b (a—b)(x+b) (a—6)(x+a) (w+ap” (+a (ae+b)’ a? b? ce (b-«)(c—a)(x+a) * (a—6)(0—0)(a +b) * (ac) (b—e)(a Fe) 4? ~ (a +u)(a+b)(a+e)' 56 the quotient of i by m, which is Ss (Art. 181): such w be the result, if the form of the quotient of i by , was and no other form of this quotient will satisfy the requ condition: and inasmuch as it is assumed that the for this quotient, whatever it may be, 1s independent of the cific values of the symbols involved, it follows that a W is the form of the quotient of ; divided by 5 in one must be its form likewise in all others. Examples. 627. The following are examples of the multiplication division of fractions. 4 a = I PEM Gna antl ce GD eS ta tan wo ae x? —9 ea (x? —9)(#—1) _ tora One 9 _ a — 408 ota e+ (@+4)(e+8) 2 +7412 z+e when reduced to its lowest terms. a’ + 6? a—b Itipl d : (2) Multiply “api Od Sia aie Sooo A wo abotar oD ane oF CMG I | wep aye) Mae yk I 263 a? +2ab +b?’ when reduced to its lowest terms. — 4 xr+2 Divid : A ie LV eae ace Yordl w—4 2+2 _@ —4)(@+1)_ #+a°—-4x—4 ee 42 gel (a+2)(2+2) 424+ 2rt+4 vt when reduced to its lowest terms. PPO; a—b (4) Divide aavans by Tee a+b? a-b at tae a+ab+ab?+h? ati a—b? a+b ~ (a?— 6°) (a— a—b) a’—ab—al?+6 a’—2al when reduced to its lowest terms. Division by 1 * equiva. 628. The Tre of the division of a by ; is al eae mul. tiplication the product of a and 7 ; 38 _ (Art. 144.): it follows, the by a 57 we change the operation of division into multiplication by ting the divisor, and the operation of multiplication into ion by inverting the multiplier. f 29. By thus changing the character of the operations per- Great ‘ ‘ ° . Tada varieties of ed by inverting the symbols or expressions Involved 1M equivalent ki, or, in other words, by transferring them from the nume- pene: » to the denominator of a fraction or conversely, we shall of their only be enabled to produce an almost endless variety of mt Le valent forms, but likewise, in many cases, to reduce them, tion. 1 given, to others which are more simple and convenient: following are examples of such conversions : (4) d sufficient to guide him in the interpretation of their (ning and use in similar cases. j ~ £ Tee lie (5) MiGs. ), xv — j ° it {9 a i | l—-vr T+2r ‘x 1 2 ae ithe numerator —— + —"— = auld ; and the denominator | 1+ l—«# l1-@ | 2 ee ‘je l+e 1--* VOL. Ti; H : their quotient is therefore 1. | 58 1 ail +2 aloe 1 6 ee — — * (6) 1 1 x Lo Ie following the same process of reduction as in Ex. 5. — 1 ra 1 14+2-1 4 =e for 1 = ———— 2 7) spades, x 1l+2 l+a 1+ l+¢e (5) | athe “Fane l+a 1 i and the fraction becomes, therefore ———=—« = — | x x ' =) | 1 l+¢2 1 x i! (8) em try 1 1+2’ 1+- 1 ea ie x 1 l+e and, therefore, = ™ 1492” by multiplying the numerator and denominator by 1 + z. ee ee ee (9) : s ; =2?—7xr+12: for Ce i eae 1 138 1 2-4 2-3 #-7e24+12 . 2?-7e+12° 1 x®— 492 — 120 (10) Sims 1 Ty ,, e—40 0 & for multiplying the numerator and denominator by 2+ : , we =o 1- ie See flake Caatth naar we ae if bea 1+2¢+2 (12) ore @+3a°4+2254 ot & l+a 14404327 +208 1+2r4+2 CHAPTER XV. MAL STATEMENT OF THE PRINCIPLE OF THE PERMANENCE ) OF EQUIVALENT FORMS. 630. In the exposition of the fundamental operations of aie bi, ition, subtraction, multiplication and division in Symbolical ciple fol- ‘ebra, we have adopted the corresponding rules of operation piesa Arithmetical Algebra, extending them to all values of the or determi- bols involved, as well as to those additional derivative forms* Becvalaha ich are the necessary results of that extension: and we have forms. sequently endeavoured to give to those extended operations | to their results, such an interpretation as was consistent it the conditions which they were required to satisfy. In further developement of this science we shall continue to / . suided by the same principle, making the results of defined ations, or the rules for forming them, the basis of the cor- ‘onding operations and results in Symbolical Algebra, and ‘of the interpretation of the meaning which must be given hem, whenever such interpretation is practicable. 31. This principle, which is thus made the foundation Its formal je operations and results of Symbolical Algebra, has been *‘@tement d “The principle of the permanence of equivalent formst”, may be stated as follows: ‘Whatever algebraical forms are equivalent, when the symbols general in form but specific in value, will be equivalent likewise 1 the symbols are general in value as well as in form.” t will follow from this principle, that all the results of and con- ametical Algebra will be results likewise of Symbolical *“t""°* bra: and the discovery of equivalent forms in the former \ce, possessing the requisite conditions, will be not only | discovery in the latter, but the only authority for their lence: for there are no definitions of the operations in Sym- Jal Algebra, by which such equivalent forms can be de- ‘ined t. ‘Such are +a and —a, and other forms which are thus derived, and which ot recognized in Arithmetical Algebra. ‘Treatise on Algebra, p. 105. Cambridge, 1839. ‘See Appendix. | | - 60 nnapyeng of 632. The term operation is used in the most comprehens operation. Sense, as including every process by which we pass from equivalent form to another: and its interpretation, in Symbol; Algebra, as we have already seen, will more or less change y every change of the circumstances of its application: the in pretations which we have given of the operations of addition ; subtraction, multiplication and division, in the preceding C ters, furnish examples of such changes. = *. eee . ~ ae ee ee See = ee ee - — CHAPTER XVI. | ) THE THEORY OF INDICES. ' 633. Tue continued product of a number or symbol a into Deans of indices lf, repeated as a factor n times, is expressed by a” (Arts. 38 and powers ed 39) and the expression itself is called a power (the n>) of a, eee ed n its exponent or index: and it is easily shewn to be a neces- Algebra, sy consequence of this definition, that the product of two powers faba ¢ the same symbol is also a power of that symbol, whose index mae they iequal to the sum of the indices, of the component factors: that i if a" and a* be two powers of a, then (Art. 44) | Ge ea | It will follow from this conclusion and the known relations the operations of multiplication and division, that, if m be m | a an jeater than 7, ikea a"-": for the product of the divisor a" and {e quotient a"-", or a® x a™~"=a"*"""=a™; we conclude there- ‘re that a”-" is the quotient required, inasmuch as, when mul- ‘slied into the divisor, it produces the dividend. | 634. The definition of a power, in Arithmetical Algebra, The index plies that its index is a whole number: and if this condition of peed : not fulfilled, the definition has no meaning, and therefore metical » conclusions are deducible from it: the principles, however, aye tie ‘Symbolical Algebra, will enable us, not merely to recognise paler e existence of such powers, but likewise to give, in many stances, a consistent interpretation of their meaning. | 635. Observing that the indices m and x in the expressions The equa- hich constitute the equation ponte 633 gene- a" x a" =a™*", ralized by ) - the prin- ough specific in value, are general in form, we are authorized yuan of e perma- » conclude by “the principle of the permanence of equivalent nence of ams” (Art. 631), that in Symbolical Algebra, the same expres- Pa car ons continue to be equivalent to each other for all values of those idices: or in other words, that iil an x.a = a whatever be the values of m and 2. 62 Principle This equation, which forms one of the most important pi Diaries. positions in Symbolical Algebra, is sometimes called “the pr ciple of indices.” We shall proceed to notice some of the numer¢ conclusions which are deducible from it, beginning with examp of the interpretation of powers whose indices are fractional negative numbers. . | Interpreta- 636. What is the meaning of a3? 1 tion of a2. The product a? x a2 =a2*3 = q'— a, (Art. 42) by the “pri ciple of indices” (Art.635): and it likewise appears that,/ax,/a= where ,/a denotes the square root of a (Art. 223): we conclu¢ therefore, that a2 is identical in meaning with ,/a, inasmu as when multiplied into itself, it produces the same result*, Meaning of 637. What is the meaning of as? 1 | vi The product of ai x aixataaititi_ Land the produ of /a x X/a x 2/a =a, where /a is the cube root of a: it folloy therefore, that asis identical in meaning with the cube root of 1 Meaning of | 638. What is the meaning of ax, where n is a whole nur! 1 : ry ber? an, . 1 e e . . . ) If am be successively multiplied into itself, and repeated ; a factor n times, the index of its product will be the sum ; the indices of the component factors, and therefore equal | ; 1 . , .| n times oF 1, and the product itself will be equal to ¢ the x‘ root of a or ,*/a possesses the same property, and ¥ , 1 conclude, therefore, that an =a, Meaning of 639, What is the meaning of a3? 4 ab and | ; * Those quantities which produce the same result, when employed in tl same operations, are considered as identical: this principle, though correct : Arithmetical Algebra, will require some modification in Symbolical Algebra) thus wx a=a?=—ax~—a: and we should not be justified, in concluding fro: thence, that a=—a. We shall have occasion to notice this ambiguity more : length, when we come to the consideration of multiple values and of the Too of 1 and —1. | 4 | 63 The continued. product x4 7 = a = fat x Jat x e/a’ x L's Wiis conclude, therefore, that a3 is identical with a/a‘, or with fifth root of the fourth power of a. pare generally, if m and n be Moe numbers, it ey be i 4 4 4 4 5 *X@5XaA5Xa5xa5=a m™ power of a. 640. The preceding examples will be sufficient to shew ; powers with fractional indices, though not deducible from definitions of Arithmetical Algebra, will correctly express intities which are recognized in that science; we shall pro- te further to shew that powers with negative indices will ‘wise admit of an equally simple and consistent interpretation. Tf m and n be whole numbers, where m is greater than n, we a” ie shewn (Art. 633), that a” x, = =a"-": and it follows, ° n a the « general principle of indices” (Art. 633), that a" x a7" ‘-" We conclude, therefore, for the particular case under sideration, that _we infer, by “the principle of the permanence of equivalent as,” that this proposition is true whatever be the value of n. It follows, therefore, that cial pi, a a” a pee Jee OF A. a pee a sn SET C8 a” a an at) 1 me 1 res 1 a a?" a eet Gas = —— « a’ a? an And, conversely, a= - a : a* a?” 1 1 1 ] iz 1 ass >. 5 a?= i ee ery . 1 ans ans hes at rir | po I C=: a= CER tee. ) a~s im ia. len in the same manner that a> =/a", or the nt root of Negative indices. Proof that ] we Examples. Proof that a? Proof that (a™) %— mn, Examples of the re- duction of expres- sions in which in- dices occur. 64 641. Amongst other important consequences of the P. ceding proposition, it will follow that a’=1. For a m ' 4 La alt; : a -*='9"s and singe “a1, it appears that a°= whatever be the value of a. | 642. Ifn be a whole number, it may be easily shewn tha) (a™)"= a™", | For, in this case, (a”)" denotes the continued product; a” into itself, where a” is repeated as a factor n times, and | index of the power which is equivalent to this product is eq to the sum of the indices of the component factors (Art. 6% which is mn or » times the index of one of them: and si\ nm and m are general in form, though x is specific in value will follow from “the principle of the permanence of equival| forms” that the proposition is true for all values of m and x wl; soever. | It follows, therefore, that (a?)3 = ai=a=,/a’*. ) m on ~ 6 m | (a™)s =a = J/a™. (a2)3 =a6= Ja. c/a. 643. The preceding properties of powers and their ind: will enable us not merely to vary to an almost endless exh the equivalent forms of expressions in which they occur, 1 likewise to reduce them to the most simple forms which t; are capable of receiving: the following are examples. | 1 6= (a~3)-s=a ‘ 1 (i) ea: so attegnts—. | ut NS Beg I (2) a? xa’=a? ®=a’*, a aS Md Abels te , (3) — =a" S=a*=aa*xXa *. me, 1 Dae a én a’ (4) a’xa *=a% f=ah=_., at * The square root of a? may be —a as well as +a. (6) gail 1 xly | (8) a") *t 3 q—-1x-2x-3 _, 9-6 i 1 20 5 4 | @ (abc 8) tg 2687 o—a 0 bc | ; , lucing the indices to a common denominator) GPs Ve iat 40 va (ra)* 40) fab? J(ab') Jab ) J(ab%)|? = jab? (ab)! (ab')* (ab) LOSE 14 eee 2 73 1 Be a8 = 100° x a* b* x a®b§ x ath4{> = {aebe{s=aqrbo, an) {tors HA ta, )} P fe* art =|b3(a +2)! =b2(a+2)'=,/b (a +2). fd | In many of the preceding examples, the indication of roots s ffected by signs as well as by indices; but, in the process reduction it is generally convenient and sometimes necessary oeplace the signs of the different roots by the correspond- n\ indices: it would in fact conduce very greatly to the uni- oaity and clearness of algebraical notation, if the use of radical iis was altogether abandoned. $44. The following are miscellaneous Examples of the re- Other ex- amples of ition of expressions involving radical quantities or indices MP Ceo other and equivalent forms: | 1 | tn) SOT Es) (1) Se = dia +2). | ae For the product of ,/(1 — x) and ,/(1 + 2) is /(1 ~ 2°). You, II. I | 66 xr 1 — 2°) + Sn 7a a, aeOK 1-2 wel Ce We multiply the numerator and denominator by ,/(1 — a? _N@*- 2") (a+ 2) + a”) | | : | (5) r/(a* + 2”) + ,/(a* — 2”) ae Cys) are We multiply the numerator and denominator by J(a’ + 2’). (4) Mokwetreh a2 — 63 This is involved in the proposition that a?— 6’= (a + 6) (a (Art. 66.) a-— b Ribg rk NE hae | (= a3 4+ ash e+ b 4 (Art. 86. Ex. 1 bes } | (6) Jae + ——— at 2 ENE | 4 a— Jaz Ja-Ja" For ana 5 aueCeW a8) a Jax—ax Wie - paethaa , and, therefor — Jax = oe _ aaz— ax+ax | a sax aja} ars aya a— fax a— Jax jae Je j CHAPTER XVII. 0 THE EXTRACTION OF SQUARE ROOTS IN SYMBOLICAL ALGE- BRA: ORIGIN OF AMBIGUOUS ROOTS, AND OF THE SIGN yee. 645. Ir will follow, from the Rule of Signs, (Art. 569.) There are 4 ‘ f always two tit, in Symbolical Algebra, there are always two roots, differ- roots, with ir from each other in their sign only, which correspond to the saa sie square: thus a’ may equally arise from the product a x a which pro- | 2 . duce the al -ax-a: (a +6)? may equally arise from the product came (+b)x(a+6), and —(a+6)x—(a+6): (a—6)’ may equally Square. ase from the product (a—6)x(a—b) and (6—a) x (b—a),* al similarly for all other squares. It follows, therefore, that i passing from the square to the square root, we shall always fd two roots, which only differ from each other in their sign ; It it is the positive square root alone which is recognized in Arithmeti- vithmetical Algebra, and which may therefore be called the cat sigan - root. cthmetical root. +) 646. We have already had occasion to notice these am- Occurrence lzuous square roots in Arithmetical Algebra (Art. 383) in paler «ducing the square roots of squares, such as «’—2ax+a° and Toots in (-2ax+.?, which are identical in their arithmetical value, aUATe ‘ough different in the arrangement of their terms. If the a lation of the values of the symbols 2 and a be known, the ale for the extraction of the square root of these expressions, hich is given in Arithmetical Algebra, would require the terms ‘the square, and therefore of the root to be arranged in the ‘der of their magnitude, and consequently no ambiguity could cist with respect to the arithmetical root, which would be «—a 2 was greater than a, and a—za if x was less than a: but if the ‘lation of those values be unknown, as where x is an unknown amber to be determined from the solution of the equation which ads to the formation of the square, it is uncertain or ambi- uous, whether the root be x—a or a—2, until that relation is * For—(a—b)=b-—a, and therefore — (a—b) x —(a—b) = (b—a@) x (b—a). The sym- bolical roots de- rived by the same process as in Arith- metical Algebra. Use of the double sign +, Rule for extracting the square root, Examples of ter- minable square roots, 68 assumed or determined*. In this case, however, the ambi of the roots originates in the ambiguity of the problem) posed, and not in the independent use of the signs in Symbj Algebra. 647. In extracting the square root, we follow, as in all I operations, the same process both in Symbolical and in Aj metical Algebra, assuming the proper relation of the sym; and the negative is at once found from the positive roo merely changing its sign. It is not unusual, likewise, t¢¢ note the double root by prefixing the double sign + to it:| +a means equally +a or —a, one or both: +(a—6) ni equally a—6 and 6—a: and similarly in other cases. 648. The following is the Rulet for extracting the sik root in Symbolical Algebra: Arrange the terms of the square according to some symb) reference (Art. 576); obliterate the first term of the square, make its square root the primary term of the root to be Sou divide the first remaining term of the square by double then mary term of the root, making the quotient the second term ct root: add this second term, with its proper sign, to doublitj primary, to form the divisor: multiply the last term of the'a into the divisor, and subtract their product from the remall of the square: if there be any remainder, repeat the same pres considering the terms already found in the root as constitutinl single primary term: and so on continually until there is nit mainder, or until the process becomes obviously interminable. second root is found by changing the sign of all the terms gf jirst. 649. The following are Examples in which the process minates. * We do not assume, in Arithmetical Algebra, that r—a or a—«, are ecal the roots of 22—2ax+a?” when the relation of values of x and a is unknowr1b that «—a is always the root of «?—2axr+a?, and a—z the root of a2—2ar4 and it is only when the relation of values of » and a is unknown, that thi nothing to guide us in the selection of one of those forms of the square inp ference to the other. t The Rule for extracting the square root in Arithmetical Algebra ham been formally stated apart from the corresponding Rule in Arithmetic (Arte 2h, | 69 ) To extract the square root of a’— 2ab +b’. r a®—2ab4b* (a—b: the second root is H2a-6) —2ab+6° —(a—b) or b—-a. | l e obliterate a? and make its square root a, the primary (of the root: we double a (2a), and we divide — 2ab by it: uotient — 4 is the second term of the root: we add (Art. 547) 9 2a, making the divisor 2a— 6: and we subtract the pro- ((2a—b) x—6 from the first remainder — 2a + 6’, and there 1 second remainder. y To find the square root of a*— 2a°a+ 3a°2*—2ax*+ x*. l a’ — 2a? x + 3a°x* —2ax’+ x4, 2a°—ax) —2a°x + 3a°x?—2ax°+ x* (a?--axr+.2°: the won et 8 re second root is 2 3 Se) i ea (—Qax+a°) . + 2a°2*?—2ax*® + x* + Qa°x*—Qax® + x* | we divide the first term 2a’x’ of the second re- ider by the first term 2a? of 2a°—2a, and the quotient ‘is the third term of the root: we add +2° to 2a°—2a« ‘rm the second and final divisor. 3) To extract the square root of 4a* + 9b? + 16c?—12ab + 16ac— 24be. Making a the symbol of reference, this expression becomes | 4a? — (126 — 16c) a +9b°- 2460+ 16" (2a-(3b—4¢), (3b —4c)} —(126-16c)a+9b’— 246c + 16° Examples of inter- minable square roots. 70 We divide —(126 —16c) a, which is the first term of the fj and only remainder, by double the primary term of the roo} 4a: we thus get —(3b—4c), which forms the second term the root*. } | The second root is —-2a+3b— 4c. 650. In the following Examples the process leads to, indefinite series. | (1) To extract the square root of a’+ 2”. a+ a? (a+ iaH re - 2a 8a® 16a‘ pm ness pias 2a 4a? ete =) ad asd Aa? x x 8 eee ee 4a? 8a‘ 64a° 5 ve n x fs x ® Ope a oe ee eee naa Moa: 8a* 64a° 6 8 x x x’ av? J Inasmuch as the number of terms in the subtrahend always greater than in the remainder, the process can née terminate. It follows, therefore, that > Cer eee x 5 x® J(a?+2°)=(a? + 2°)? =+ ‘2 hearin Mam Posen &e.} ; If we reverse the order of the terms in the square, we su find, by a similar process, 4 4 6 8 2 <= (ge 2\2 2. a ed mee: J (x? + a*) = (a? + a’) ets at igi Ode * Other examples are a) V/ (8-245) = (F-2). a Ox* 7a? 10la? 147 1 (8) AY ear tee) 32° 7a oe ey pat & 5 +5) it The only arrangement of the terms in the square which wat SA ld be recognized in Arithmetical Algebra, is that in which gent series. r follow the order of their magnitude, Art. 646: thus, if a «greater than x, it is the first (1) of these series only, which onvergent: if a be less than a, it is the second (2): if this rer be reversed, the same series are divergent, and no ap- Iithation is made to the value of the roots by the aggre- 1 19 and, therefore, x — ao + 2? and «=10 or —9. ocess will stand as follows : B-pe-g {2-3 Bite 2 tere } Qa i prt | 4 | | 2 2 | follows, therefore, that (>-5) differs from a?—pa—q by -E 4, and, 0 quently, if «?—pa-—gq=0, which the conditions of the equation require, we / 2 2 b (et) Be | and, therefore, »—Z a y/ (7 +4) , | P pe . p= ib — ; or x 2 Vv (E+a) th the following Chapter, examples will be given of biquadratic and higher Yiions of an even order, which admit of reduction or solution, through the ‘Cary processes for extracting roots. | Tn equa- tion (7), both roots are possible or both im- possible : in equation (8) both roots are negative or both impossible. values be recognized as admitting of interpretation, the solio is also ambiguous like the one preceding. 60 In the equation a+ 4e—21=0, we find (# + 2)?= 214+ 4=25, and, therefore, x + 2=+ 5, and «=$ or —7. It is the positive root alone which is recognized in metical Algebra. 660. Again, it appears from the formule (11) and 1 of solution of the equations | x? —px+q=0 (7) xv’+pxt+gq=0 (8), i that, when p and q are arithmetical quantities, and E gil than q, there are two positive roots of the first equation’ 2 and two negative roots of the second (8): for wi Cae always less than f and therefore the signs of the two | are determined by the sign of E. Thus in the equation x’ — 7x+12=0, | ve Py 2 ( we find ( -3) <4 -12= f6 1 and, therefore, x — Le se 3° and « =4 or 3. In other words, the solution is ambiguous. In the equation v?+122+35=0, we find (# + 6)’ = 36 — 35 = 1, and, therefore, x +6 =+1, and « =— 7 or — 5. This solution is arithmetically impossible: but if neg, | 81 ; 2 If, in the same formule (11) and (12), 4 is less than q, then g is negative, and the roots of equations (7) and (8) will move the square root of a negative quantity and therefore hsign /—1. (Art. 652.) Thus in the equation xv’— 8x+25=0, we find (2 — 4)’= 16-25 =— 9, and, therefore, x-4=+ 3 fie and «= 4+3,/—1, or bead: od i Again, in the equation 97 | x aks aati ee 0, i 3\7. 95 OT 4, we find (# +5) = Siri age 9 9 . 2 and, therefore, x + 5a Ses 3.2 ;—~ 5 ah? Reet and 2 =—>+—,/—1, o1 S oua cr) The unarithmetical or impossible roots which present them- siyes in equations (7) and (8), necessarily assume the form 1 b,/—1 and a—b,/—1, where the term involving the sign .-1 is affected with the sign + in one of them, and with the sa — in the other. 661. The roots, which are thus determined, bear a very Composi- siple relation to the given quantities of the equations to which pony Ly belong; their symbolical sum being always equal to the of the se- ficient of the second term with its sign changed, and their eatin piduct to the last term: thus, if a ae 6 be the roots of the Lape, elation x«’—pax+q=0, we find the roots. Po cn erica Ble Eo} fe E-] -8 (En) Vo. Tk L | 82 A change of sign in p and q, one or both, will not aj this conclusion, it being assumed that a and 6 may be eit positive, negative, one or both, or both of them imaginary,| Resolution 662. The same assumptions being made, it may be furti f - ° Q Sratic ti. Shewn that z—a and «— are the simple factors of the trinor nomial into its simple factors. 2 el For r-a=a—b— [(—4), n? r-b=2-h+,/(E-4), and, therefore, 2 2 (x — a) (x—b) =( -£) - (F-q) =2°— px +0 - + ah =2°—px+q. It will follow, therefore, that «°-— px+q will become e¢ to zero, when one of its factors is equal to zero, or when @| or «=, and in no other case: in other words, a and 6} the only values of x which verify the equation x --pxr+gq=0*. 663. The propositions contained in the two last arti will be found to be included in some general propositions wl! apply to the coefficients of equations of all dimensions, w they assume the ordinary form. Examples 664. The following examples, of the solution of quadz! f the solu- : ; tee Saude equations, and of problems which lead to them, may be dratic sidered as supplementary to those which are given in Arts. ; equations. and 410, and 412f. | _ ™ It may be otherwise shewn, that, if a@ be a value of x, which m a7—pa+q=0, then r—a isa factor of the trinomial x?— px+q for all vi of x: for by the assumption it appears that a?— pa + q=0, and consequently | x?— px+q =a*—patgq —(a?—patq) =a*?—a®—p(xr—a): and, therefore, spite =r+a-—-p=x2—(p—a): it follows, therefore, that a—a is a factor of a2—pa+q: the other fact a—(p—a). | + In Example 1, Art. 389, the second or negative root is—3, and in Examp} it is —14: the two roots in Example 5 are both negative, one being — #,| the other —4: the solutions in Examples 2 and 4 are ambiguous, and both! 83 [ a Examples. 1) ec xr+2 r+] aN Jlearing the equation of fractions, Art. 368, we get ) 102° + 302+ 20—$2°—3x=1027+ 202. Sollecting like terms into one (Art. 372), and transposing known terms to one side of the equation, and the unknown he other, we find 3x?— 7x = 20. Pividing both sides by 3 (Art. 370), which is the coefficient Se ee 20° 5 a Sompleting the square (Arts. 384 and 657), the equation mes (« ny = i ital ent OA 36 Rs 736 Extracting the square root, we get ERY | a ry therefore a = 4 a x=4, or > There is only one arithmetical root: but the negative root fully satisfies, as may be found by trial, the symbolical llitions of the equation. 2 » gh ated x _ 29 2) Let aaa! atta The several processes of reduction and solution follow the e order as in the last example. 102° + 490 — 1402 + 102? = 203.2 — 292’, 49 x? — 3432 =— 490, x’?—7x=-—10, ( - 2) = 8-10-29, 2 4 yp Millegsy z Seria, Or, 2 are determined by the processes of Arithmetical Algebra. In Example 5, 110, the second or negative root is —9: in Example 6, itis — 15: in Exam- » Art. 412, it ig — 15. 84 This is an example of an equation, whose roots are ari metically ambiguous. 32 +25 ‘ ! Ofie acgroneggt rere eG 63.2 + 525 + 62° + 502 — 28 — 840 — 82? = 588 + 562, 927+ 272=—- 91, po Reha Obs 2 2 £=— if, Or ——- This equation admits of no arithmetical root, but its sy bolical conditions are satisfied by both the negative roots . w—4e¢ 32°4+11 Seale Sees 130 4, Ax? —T6x + 3927+ 143 + 416 = 1562, 43a°-— 1722 =— 559, 559 *_ 47 eee eds x — 4x 43 > v’—42+4=-9, z—-2=+3,/-1, L=2Z*B gar There is no arithmetical or negative root which will sati the equation: the two roots are said to be imaginary*. +8= 32a, P ~ : se id 665. We shall endeavour to connect the solution of aati following problems producing quadratic equations, with andinter- illustration of some of the more common principles of in ata ie of preting the results of symbolical operations. sults. ‘ ; . : ‘ * In the examples of quadratic equations which are given in the Note, 389, the Symbolical, as distinguished from the Arithmetical, roots, are in Ex ple 1, —10: in Example 2, —7: in Example 4, =o in Example 5, -—1 Example 6, — 10: in Example 8, -3: in Example 11, 2 + wy aoe ee or 2 85 1) To divide the number 10 into two parts, whose pro- shall be equal to 24. “et x be one of the two parts, and therefore 10 — x the other. Cheir product =x (10-—.2)=24: or 10x— a°= 24, and, therefore, x? - 104 =— 24. Jompleting the square, we get (x — 5)’= 25 —24=1, and, therefore, x-— 5==+1, and wz = 6, or 4. The two parts into which 10 is divided are 6 and 4, and e is no ambiguity in the solution of the problem: but the 1e of 2 is ambiguous, inasmuch as, conformably to the as- \ptions which are made, it may equally represent the greater he lesser of the two parts into which 10 is divided. If the product had been 25 instead of 24, we should have x’ —10x2=- 25, (@ — 5 =25 —25=0, r—5 =, = os In this case 10 would have been divided into two equal ‘ts*, and there would have been no ambiguity either in the ition of the problem or in the value of a. | If the product was required to be 26 instead of 24, the equa- 1 would have been x*— 102 =— 26, (a — 5)?= 25 — 26 =— e-5=+,/=1, v=5+,/-1. In this case, the solution of the problem is impossible in the ise in which it was proposed, and we transfer the same epithet ‘its symbolical roots: but those roots will be found to satisfy symbolical conditions of the problem; for their sum =(5+,/-1)+(5-./-1)=10, | E (* The product of the parts into which a number is divided, will be the greatest mn those parts are equal: for if 2n be the number, and n + a one of those parts, , therefore, n — x the other, their product is (n + x) (n — a), or n®— #*, which le greatest possible when «= 0. | 86 and their product oa =(5+,/—1) x (5—,/—1) =25 +1=26*. | (2) The sum of a decreasing arithmetical series is 7) it first term 21, and the common difference 3: to find the mb of its terms. i" The formula in Art. 422, gives us i 75=|42-3(n~1)| 5, | | where », or the unknown number, is the number of ters the series. i We thence get 4 15n —n’= 50, n*—15n =— 50, 4 ye? 1D D | am —-—-=2+-—, 2 2 ' { R= 10 Or If we take 5, the less of these two values, we get the wvies of five terms | 21, 18, 15, 12, 9, which satisfies the proposed conditions. r If we take 10, or the greater of the two values of x to exes the number of terms, the series will be found to be 21; 18, Ls 12, 9, 6, a 0, — 3, — 6, which likewise answers the conditions of the problem, inasrich as the symbolical sum of the 5 last terms 6, 3, 0, —3,= a equal to zero: but the arithmetical solution of the problen aS proposed, is not ambiguous, inasmuch as it is not possibl te form 10 consecutive terms of the decreasing series 21, 18 15,... without introducing, and therefore recognizing, negi re terms. * Another equation of the same class is given in Example 7, Art. 41¢ the student is particularly recommended to study the problems leading to quaat equations, which are given in that and the following articles, with especial refem to the more enlarged views of the interpretation of symbolical results whic at given in this Chapter. | : 87 Che arithmetical conditions, however, of the equation 75={42-3(n—1)}} =, ) hich the problem leads, are equally satisfied by »=5 and 10: for the first gives n the second, 75 = (42 - 27) x 5, ch are severally equal to each other. 3) Given one line, to find another such, that the rectangle A Soran trical pro- ‘ch they form, shall be equal to the square of their difference. Jem stated Let a be the given line, and «x the line to be determined: generally. 11 by the conditions of the problem, we form the equations ax =(«—- a)’ or (a- 2)’ Ci: ‘ch equally lead to the equation near =— a’. Completing the square, , Sarin Ga’ et 9 CER fa mena, 4 A? ONE 2 Ps 3a a,/5 = Pry gu? t ~ ~~ 1+,/5 dake, | geen ON, ‘Both these roots are positive, since ,/5 = 2.236 ..., and, there- p 1,0 4 , > '2.618a, or .382a, nearly. In the problem proposed, it is uncertain whether the line Its solution includes je determined is less or greater than the given line: and the two cases, tbolical solution, being coextensive with the enunciation of , problem, includes both cases. ‘The first case is equivalent to the following problem. Its first case stated and solved. we 5 qe 5 le = 1.618 and Sethe 618 ... 3 consequently Its second case stated and solved. called x the whole line produced, the resulting equation wo 88 “To divide a given line into two such parts that the recta contained by the Sako and one of the parts shall be equa the square of the other.” If we call x the greater of the two parts, and, theref a—zx the other, the equation will become a(a—2)=2° (2), the symbolical roots of which are acho a, and = es a, one being positive and the other negative: the solution is the. fore not ambiguous, there being only one positive root. ut if x be taken to represent the less of the two parts into whichpe line is divided, the equation becomes axz=(a—.2)’, | the solution of which, as we have seen above, is ambigu this ambiguity, however, is removed, by the necessity of ane : the least of the values of x or .382a, as the only value whh is compatible with the conditions of the problem. | The second case corresponds to the following problem. | “To find a point ina given line produced, such that ie rectangle formed by the given line and the whole line produc, shall be equal to the square of the part of it produced.” If' we call « the part produced, and therefore a + a the wile line produced, the conditions of the problem lead to the equatn a(a+2)=2° (3), i the two roots of which are BEE) a, and en a, differ g ~ merely in sign from those of equation (2): it is the arithmetal root alone which corresponds to the problem: but if we Id Fl have been ax=(«—-a)’, | “1 the solution of which is ambiguous, though the problem is 1h, the less of the two roots being excluded by the conditions wh it imposes. ! It thus appears that the same problem may lead to ¢ equation whose solution is unambiguous, and to another wh is ambiguous: but in the latter case, the conditions of. the p 89 lem will immediately lead to the exclusion of one of the two ots, and therefore to the removal of the ambiguity. _ Whatever geometrical problems can be shewn to depend im- “ediately or ultimately upon the division of a line into extreme jd mean ratio, will lead likewise to some one of the equations hich we have just been considering*. (4) To find a point in a given chord of a circle produced re Lye om whence the tangent drawn to the circle shall be equal blem. ye a given line. F Let AB (2c) be the given chord, hd CD (l) the given line: assume P be the point in the chord BA pro- iced, from which the tangent PT : A: awn to the circle is equal to CD: rea d let PA the part of the chord P=; E "er and which is required to determined, be denoted by z. Then, by the property of the circle ituclid, Book 111. Prop. 36.) we get ioe ii Py” - Gistabae |), | yreplacing PA, PB and PT, by a, x +2c and l, a(a+2c)=L*, or 2° + 8en = i (1). , Completing the square, we get (e+cpac+Pl, | : and #+c==,/(c?+1*), or x=,/(c?+1*)—c and — ,/(c? +l’) —c. It is the first of these values only which belongs to Arith- i tical Algebra, and which corresponds to the problem proposed. i if we had denoted PB and not PA, by x, we should ve found the equation Zo Oo =.L (2); t: positive root of which, or c+,/(c’+J’) gives the value «PB required, and is identical in magnitude with the negative i of the first equation (1). ' In the algebraical solution of this and other problems, we The analy- sin by assuming the solution of the problem to be solved me ese | Such is the problem for constructing a triangle where each angle at the bis double of the angle at the vertex, (Euclid, Book rv. Prop. x.) upon which tlinseription of a regular pentagon in a circle depends, as well as others which et be proposed. Vou. II. M q and we subsequently determine the quantity whose valu required, by following out the necessary consequences of } conditions which it must satisfy until we arrive at one wl: can be made the basis of an equation: thus, in the first solu) of the problem under consideration, we assume P to be the pr in the chord BA (2c) produced, from which the tangent . drawn to the circle, is equal in length to the given line CDi and we denote PA, the assumed distance of P from A, byx the conditions of the problem, which are involved in the genval property of the tangent of a circle, shew that x must ly such a value that . 90 O(P +26) c= be: pos and it is found that the only arithmetical value of x, whch thesis. will satisfy this equation, is ,/(c?+1*) -—c. The preceding process of investigation may be called hi analysis of the problem proposed, and immediately leadito its synthesis or construction, which is as follows. } Bisect the chord BA in E: draw BF perpendicular to |b and equal to CD or I: join EF, which is equal to | J EB + BF’) = (P+ P\=c+a=EP: it follows, therefore, that a circle described from the centre E, 1th the radius EF, will cut BA produced in the point P required Geome- The corresponding process in Geometry, by which, assunig peat the problem to be solved and the unknown line or other qu synthesis. tity to be determined, we are enabled to discover, froma examination of the conditions which it must satisfy or the ‘0 perties it must possess, the direct means of determining its ve is likewise called its analysis: whilst the inverse process,)j which, when those consequences are traced out or discoveid, we are enabled to construct or solve the problem, is callecit synthesis. Thus, in the problem under consideration, we bil by assuming the point P to be that from which the tangent/4 drawn to the circle is equal to CD: we further know, i" the property of the circle, that | PAm eas we likewise know, if we bisect AB, that PA x PB+ AE) equal to PE* (Euclid, Book 11. Prop. 6), and, therefore t PT? + AE? or to CD? + AE’, (since PT’ = CD”), or to the hy thenuse (EF) of a right-angled triangle EBF, of which | and BF (which is equal to CD), are the sides: if, theretie CD and AE be given, the distance EP of the required Pr from the given point E is also given. - — 7 91 | This is the geometrical analysis of the problem: its synthesis, ‘hich is the ordinary form of exposition adopted in Geometry, ‘rects us to bisect BA in E: to draw BF perpendicular to ‘Band equal to CD: to join EF: and in BA produced to take 'P equal to EF: and we finally assert that the tangent PT en to the circle from the point P thus determined, is equal to | e given line CD. We then subjoin the proof, by which the ‘rrectness of this construction, or synthesis, is established. The processes of exposition in Geometry are generally syn- fae, ‘etical, whilst those of discovery in that science, and of dis- Algebraare overy and of exposition in Algebra, are uniformly analytical: tae hehe is this fundamental distinction in the ordinary mode of chibiting or deducing the conclusions in Geometry and Igebra, which has led to the very general application of the 1m analytical to algebraical processes, as distinguished from sose which are geometrical, though the processes of discovery | the two sciences are essentially the same. \ 666. Prostem. Given four points in the same straight nee ie, to find a fifth, such that the rectangle under its distances blem. ‘om the first and second may bear a given ratio to the rect- igle under its distances from the third and fourth. | Let A, B, C and D be the given points, and , » Gg p p fee be the point required: lett 4D=a, BD=6; |—'—'—'— D=c and DP =a, the lines being estimated from 4 towards P Art. 558); it will follow, therefore, that CP=r+c, BP=a2+b id dP =x +a: and the conditions of the problem give us iPAxPB (x+a)(r+6b (PCxPD ~~ («+o)z - gees (1), ‘here é is a given number, and which leads, when reduced, | the equation we (1+e)c|x=ab (2): | or v-(47=*_¢) _ab , one (3). The solution of this equation gives us a+b Sc ee Dine & a AaTkD Be) +34/\(—-2) +22 | ): ie If e be positive and if both a and 6 have the same BIO OF Discussion » drawn in the same direction, the two roots are of the form a ofits solu- id — (Art. 659), and shew that there are two points on dif- rent sides of the point D, whose values those roots deter- ine, and which equally satisfy the conditions of the problem. | Limiting values. 92 If e diminishes, the value of a or of the positive root \. creases and becomes indefinitely great when e is indefinis small: the negative root, under the same circumstances, becor ; i DN ultimately equal to arep ea If, in equation (1), we replace 1 +e by 1—e, the values « will be expressed by which are both of them negative, shewing that there are points on the left of the point D, which answer the conditii of the problem: but if we reverse the directions, and, therefe: the signs of a, b, c and a, the two roots, under the same cumstances, will be both of them positive t. * For the equation (2), if e become evanescent, degenerates into the sime equation —(a+b—c)«=ab, which gives the value of x in the text: the sm : : a+b—e take the negative sign of the square Toot, and make ¢ = ada. c, we get | : “4ab 3 4ab extracting the square root of 1 + oa ‘by the process given in Art. 650, Ex., we get R lI tle + | SI Bian ~ re) “——— I] Nien + l Nile Qab 2a2b2 c=3t—it (14-2 aa + &e. et? = (a+b—c—ec)?x (“*7-*_e); e and if e becomes indefinitely small, then et becomes a + b — c, and e* ¢? is infinite great: it follows therefore, that, under such circumstances > ab ab valent Tol aac ae al et a+b—e all the other terms of the series becoming indefinitely small: the other root the equation (2), as we have already shewn, is indefinitely great. } The following are particular examples of this problem. (1) Let a=8, 6=6, and c =3, and let CE BON sta ba (v+3)x 93 ‘he problem, whose solution we have been considering, was i oe sim to the ancient geometers under the name of the Deter- Determi- te Section, and is said to have branched out, with its kindred eae Sec- <‘ems, into eighty-seven propositions*: it would be difficult lect an example better calculated to exhibit the superior ty and comprehensiveness of symbolical processes. ‘he magnitudes, which are the subjects of our reasoning -eometry, are either given and exhibited to the eye, or He tae evequired to be determined and exhibited: and the conclu- Souctas 9) which we draw concerning them, are in the first instance, aeeceet maed to the specific magnitudes under consideration: it is ized. »by a deductive process that we are enabled to make our musions respecting them general, by shewing that the form ‘/e demonstration would remain unchanged, when applied to h’ magnitudes of the same kind, which present precisely the conditions. hus, if two triangles, which are given or exhibited, have ides about two equal angles, equal to each other, they are én to have their remaining side, angles, and also their areas y to each other: and the demonstration, and therefore the ‘e values of « are 12(DP), or —4(DP’). we change the signs of a, b, c, and therefore the directions of the lines 2present, we merely change the signs, and therefore the directions of the Of 2. 4 3) Let a= 4, b=3, and c=1, and et EF ES > only value of x is — 2: or P is the middle point be- 4 pg p c p Band C I—I—|— II Let a= 4, b=3, c=1, and let EAI OR (x+1)a 2 values of « are — 10.772 and — 2.228 nearly. we change the signs of a, b and c, or the directions in which they are ‘ed, we shall change the signs of the values of 2, and therefore the directions th they are reckoned. 't formed one of the lost treatises of the celebrated Apollonius of Perga, | Pappus has noticed in his Proemium to the 7th book of his Collections, tich Schooten and Robert Simson have succeeded generally in restoring : ct is stated by him in the following terms. “To cut an indefinite straight a point, so, that of the lines intercepted between that and other given points ‘he Square of one, or the rectangle under two, of them, may bear a given ither to the rectangle formed by a given line and another of these inter- lines, or to the rectangle formed by two of them.” The sepa- rate cases of Geome- try are compre- hended under the same for- mula in Algebra. 94 conclusion, being obviously independent of the specific for values of the triangles, or of their sides and angles, is eqy applicable to all triangles whatsoever, and is therefore geir; But if there is a disruption of the continuity of the visiblejor ditions of the proposition proposed, though the truth whi expresses may be general, it will be necessary to establis! th corresponding cases by a distinct demonstration: thus, | } asserted that different parallelograms upon the same basein visible conditions will present three distinct cases for consi; tion; first, when the sides opposite to the common base hay their points in common: secondly, when the sides opposi the common base have one point only in common: and thik when the sides opposite to the common base have no poi, common: and inasmuch as the terms of the demonstrationiy be found to involve the notice or consideration of the J tive position of the sides opposite to the common base jit respect to each other, the precise form of the demonstrio which is applicable to one of these cases, will not be ek cable, without alteration, to the other: but if the demo tion of this or any other proposition was conducted thrig the medium of general symbols, the conclusion obtaine) well as the reasoning by which it was obtained, would be im to be equally general with the symbolical language in Vic it was expressed. It is this necessity of considering all the separate cas | the same proposition or problem in Geometry, and the in sibility of comprehending the extreme or limiting, simultanels with the ordinary, values of the magnitudes which they inv which commonly renders the reasonings and processes of metry more operose and less effective than those whicla conducted by the general symbols of Algebra: and it itl permanence of the equivalence of the forms and _ conelto expressed by general symbols, for all values of those i bols, whether limiting or otherwise, which are once shev be equivalent, when they are general in their form, even thi they may be specific in their value (Art. 631), which en| us to include, under one general formula or conclusion, positions or cases of propositions which must be separately! distinctly considered in Geometry. | The discussion of a general formula, or expression, 95 aig in the solution of a problem or otherwise, consists in y ving the separate cases which it comprehends, corresponding »2e limiting or other values of its symbols, and assigning to _ of them their correct interpretation: it requires us to pass a,eview, the several propositions which, in the geometrical rynthetical exposition of the problem, would require a sepa- ; and successive consideration. The geometrical problems th we have considered in this Chapter, present examples of u discussions and analyses, and they will be found generally ) mstitute the deductive processes which are required in the pications of Algebra to Geometry and Natural Philosophy : | not easy, therefore, to overrate their importance, or to yess too strongly upon the mind of a student, the neces- i of acquiring such clear and accurate notions of the prin- 's of interpretation as may guide him securely in the earch and establishment of his conclusions. CHAPTER XxX. ON THE SOLUTION OF EQUATIONS OF HIGHER ORDERS at THE SECOND, WHICH ARE RESOLVIBLE INTO SIMPLE} QUADRATIC FACTORS. | i Classifica- 667. Equations, as we have already seen (Art. 377) tr a classified according to the highest power of the unknown sy10 which they involve, when cleared from fractional and raica expressions: and if we further suppose the significant term'to be transferred to one side, the general forms of simple, quadric cubic and biquadratic equations would be as follows. | (1) «-p=0, | (2) 2°—pr+q=0, | (3) 2° —pa’?+qr—r=0, | (4) a pa®+qa’?-ra+s=0: | and similarly for equations of the fifth, sixth and higher orde pay ile 668. Any equation may be solved, by processes which Ive Balped cali already been considered, if it be resolvible into simple or qid- which is atic factors: for those values of the unknown symbol (x), wch resolvible into simple severally reduce those factors, and therefore their product pred zero, and no others, will satisfy the conditions of the equatn: factors. and it is merely necessary to solve the several equations wich arise from successively equating those factors to zero, in oe to discover all the roots of the equation ™*. | It is not our intention, in the present Chapter, to atte pt to prove the necessary existence of such simple or quadrit factors in all equations reduced to the ordinary form, or i possibility of discovering them, whenever they exist: butve * Thus if u= PQR, and if w=0, then P=0, or Q=0, or R=0: but u cand become zero, if all its factors P, Q, R retain values different from zero h if uw=a?—7Tae+12=(x—3)(x—4): then w=0, when 2—3=0, or r—4=(OF when x=3, or r=4: but there is no other value of x which reduces u to 0 Again, let w=a?—132°+ 47x—35=(«—1)(r—5) (vx—7): then w=0 «—1=0, or c—5=0, or x—7=0: or when x=1, or r=5, or r=7: but ey value of « different from these, will make u the product of three significant facts and, therefore, not =0. | 97 ll merely notice the existence of large classes of equations the superior orders, which admit of such resolution into f ors, and consequently of complete solution. | /669. To solve the equation The cube roots of 1. v--1=0. ‘Since 2°— 1 =(x—1)(#?+2+1), for all values of x (Art. 69, E 5), it follows that | | x—1=0, en x—1=0, or when 2?+2+1=0. The first equation, e-1=0, gives 2=1, which is the arith- nical root (Art. 645,) of the ps imitive equation. The second equation, or 2°+x2+1=0, gives Mente ee 9 > | otwo unarithmetical and imaginary roots, which also aay the p nitive equation*. Inasmuch as the equation 2*—1=0 gives «*=1, and there- fo x= /1, it follows that there are three cube roots of 1, which a mae 1+ /3,/— =p ia “E ES 1 Me ean tie i \ For (mie V8 =1)\3_-148y8/—14943V/3x/—1_8_, Their pro- ae 2 7 8 — Oi at | pertiess ‘he square of one of these imaginary roots is equal to the other: for NB gS i ged eC 2 F 4 iis 2 ‘ a (clewv3V -1)?_14+2v3J—1-3_-1+v3/—1 a 2 i 4 a. weak f, therefore, a represents one of these cube roots, a? will represent the other, ui'the three cube roots of 1 may be represented by 1, a, a’: or they may be ented by a, a?, a®, since a?=1. 7 a : ] igain, since a x a?= a*?=1, it follows that a?= FL and a= — and conse- q 1 juitly the three cube roots of 1 may be represented by 1, a and rt where « siie of the imaginary cube roots of 1. ery integral power of a cube root of 1 is a cube root of 1. or every integral power of 1 is 1: and since all numbers are expressible ae formula 3n, 3n +1, 3n + 2(Art. 526, Note), where n is a whole number, lows that ) a” = a®*, or a5*+1, or Freda la where a@°%= 1% 99*t1— 9°*xa=Ilxa=a; 4 and a3"+2— @3"y @?=1] x a?=a?. Tou. II. N The cube roots of —l. The sixth roots of 1. 98 o The properties of the cube and higher roots of 1 are 4 nected with many important theories, and will be more ticularly considered in a subsequent Chapter, (xxi). | 670. To solve the equation e+] = Since #°+1=(#+1)(2?—2+ re for all values of a, it fol W that z*+1=0, when ++1=0, or when 2#*?—27+1=0. The first equation, +1=0, gives r=—1. The second equation, «?—x2+1=0, gives BSW SA = 2 : Inasmuch as the equation, «°+ 1 =0, gives 2° =— 1, and tli fore x= f=] — 1, it follows that the three cube roots of — 1, are | Ly 3,fSt es 1-/3f/=1, Q 2 3 671. The roots of the equation x®—1=0, are included amongst those of x*—1=0 and 2°+1=0. For 2°—1 = (#°—1)(#*+1) for all values of x, and the 0 of x«°—1=0 (Art. 668,) are therefore those of : The same expression «°—1 may be likewise resolved 1 the three quadratic factors z®—1=0 and 2#°+1=0. v’—1, #7+2+1 and 2-241: and the roots of the equations x’—-1=0, 2°+2+1=0 and x«?-27+1=0, are also the roots of the equation «°—1=0. * If a be one of these imaginary roots, the three roots may be exp’ 1 by ~ 1, « and a, or by —1, a and oypt by a, a? and a®: for a&=—1 67. Biquadratic equations, which present themselves under Solution of i biquadratic h form i eile 4 2 . r+ qz°+s=0 1 wanting the ° ‘ : ( ) second and 1 immediately resolvible into the quadratic factors fourth ; ’ terms - the enera wets /(L£-s) and +f, /(t- Ny forte Q 4 2 4 Art. 667. ai consequently admit of solution by the processes given for dratic equations: for if we make a?=u, the equation (1) u’+qut+s=O0, wose factors are (Art. 662.) q (ee ) q_ (e- wtds /(E s) and w+¢ NAG S$); wich become the factors above mentioned, when wu is replaced ‘The four roots of the proposed biquadratic equation are, ‘refore, stat /E-9} StH}, 673. The roots of the equation considered in the last Article, Th same . . . . ex 1- ny be exhibited under other and equivalent forms, which are hited $1etimes susceptible of more complete arithmetical reduction. 0 Oe For if we divide the original equation a+gx’?+s=0 (1) tw, we get Ss 2 L+q+— 5=0 q+3=9, 1 therefore s\? a BY nue ~—)\)=c4 +2 J/s+ G=2,/sta° +=, =2,/s-—q; 2 100 and therefore K Js +a esq) (2). If we multiply both sides of this equation by x, we get x'+ /s=,/(2,/s—q) 2, a= /(2,/s—q) x=—,/Js. Solving this pair of quadratic equations in the ordin manner, we get and therefore and therefore tris AC ark Aare) where the different combinations of the signs + and —1° furnish the four different roots of the primitive equation (1 * If we compare the equivalent expressions for x which are obtained in, 672 and 673, we find V ica (ea) oa (So) Vl a and, consequently, ara) Vaca) “ qd “ may be considered as expressing the square roots of ne a 4 result which may be easily verified: and if we replace — - by a, < —s by b therefore s by a?— b, we shall find eae (24D) any (=a which assumes the form ./a + 4/8 in those cases in which a?—b is a ¢ plete square. Thus (1) /(5+2/6) =/2+ 3. (2) s/(17 — 1124/2) =3 -2./2. (BY lini (S A i) a 1 (430k) 2a) Dae (6) WV (542) =14 5. J2 (6)en/CAT FABER 178 Kae 101 4. Thus, if the equation be Examples. v+7x74+4=0, t m/f B)a- JED /(-1-9 g=0 and s=1, we get eae Na) Ni ge /2 - the equation be z*— 20x + 64 =0, t sae J(5+4)#J/(5—4) =+3+1=4 or — 4, 2 or —2. 5. If we extract the square root of the first member ie of dratic biquadratic equation arranged as in Art. 667, (where all parece nificant terms are tr ansposed to one side,) and if we arrive bese emainder, which, with its sign changed, is either a complete into factors 2 or independent of the unknown symbol, the biquadratic pomp ion may, in all cases, be resolved into two quadratic factors, pane e square $ roots determined by the solution of two quadratic equa- root of their first member as w if we express by X the first member of the equation, arranged in A 'y w the terms which are obtained in the root before the “™ °°” ss of extracting its square root terminates, then X— wu? is 2mainder: and if X — uv? =— v?, ave X=u?—v’=(ut+v)(u—v), his is =0, when «+v=0, or when u—v=0 (Art. 668): ots of the equations «~+v=0 and u—v=0 are therefore the of the equation X=0. 102 : y If v* involve x and be not a complete square, then though a factor of X*, does not exhibit « under a rational but if v? be independent of x, then u+v and u—v wille x under rational forms, whether v’ be a complete square or } Examples. 676. The following are examples. (1) Let «t+ ~~ 240-256 =0: — 23 += _ 240 — 256 ‘e+ fee +=} 32" Ox? 4 Dare" f Ox”. * The condition, upon which the success of this process of resolution d may be easily discovered by extracting the square root of t+ pas qu?+ra+s, (which is the general form of the first member of a biquadratic equation), asfio — Lv 2 ct + par*+qa*+rx+s jee -3 (=~ .) 22 Stree ; elt Berea 2 Qa2+ px —3 ae )}- The last remainder but one, with its sign changed, or (7 -«) Fe) |! , OPE i will be a complete square, if (@ = a) pany - The last remainder will be independent of zx, if 2 one r=0, This last condition will always be satisfied, if p=0 and r=0, an Examples in Art. 674. The preceding method will succeed in detecting w and v, wheney t are of the form a?+ ax and ba-+c: or of the form a2?+ax+b and e: necessarily when they are of the form «2+ax+d and bx+c, inasmuc will generally require the solution of a cubic equation to determine the valu 103 ays + 24a” +256 is a complete square, since i a x 256 = 144 =1 x (24)?*: root is + 16. quadratic equations, into which the primitive equation , are ote on 32 aah ae eo a4 +16 = 0, 3 Pads +b a4 2-2 _16 = z’—16=0: our roots of the biquadratic are _ 8+ ,/= 247 x , 4 and —4*. St eae et I et the equation be q x —2a°+327-22-3=0. a — 24° + 32°—-2a—-3 (a#-a#+1 Q2u°— a2) — 2x + x | | I Qu°-Qr+1) +22°-Qx-3 Qx2°—Qxr+1 — 4 following are examples of the same class. (1) fan re el, I I a —— ctors are #422 49 and «2—9: and the roots are 3, —3, and _ 132 /—155 — Uy ; 8 ctors are «+ rie 1 and ae 1, and the roots are : ayo 8 and —}. 104 i We thus get u*=(2*—a#+1)? and v’=4, and therefore, utv=2°—-x+3=0, u—v=2°—-x-1=0. 1+,/-11 Ds 7/5 The four roots are esate and Lev e) (3) Let the equation be x*— 2Qax°+ (a?— 26°) 2? + 2ab?x — a°b?=0. a*— 2ax°+ (a?- 26") x’ + 2ab? a —a°b? (a —axr—5 22°—-ax) —Qax*+ a?x’? Oy? 2 hte 6°): — 20° x? + 2ab?x — a?b? — 26°27? + 2ab?x + 4 — a?h?— fy Therefore «’= («?— ax +b?) and v?=a*l? + 6‘, and the) of the component factors x’ —axz—b'+b,/(a*+b*) and «—ax—b?—b,/(a?+B%, when respectively equated to zero, are expressed by the forrl Arena etal * This equation results from the solution of the following problem: ‘tov mine a right-angled triangle, the sum of whose sides containing the rightt is a, and the perpendicular from the light angle upon the hypothenuse! If x be taken to represent one of the sides, and therefore a —# the oth, well-known properties of the right-angled triangle will readily furnish the eqt aan = 4/(a? — ax + 242), or ASR (1): This equation is one factor of the rationalized equation {r?— aa + b V(a?— 2ax + 22°)t {e?— ax —bV(a?—Qan + 227) = 1*— 2aa>+ (a? — 2b?) 224+ 2ab? 2 — a2b?=0. | The two roots $ + 4/ {tb da/(a?+b*)} belong to the first of equations, and express the two sides of the triangle which are required | determined: the other two roots a2 ae VAL Gt b+ byV(a? + b7)t belong to the second equation, and are totally foreign to the problem propose This problem is given in the Arithmetica Universalis of Newton (Pro! Cap. II. Sect. 1v.) a juvenile work, but which is every where pregnant with i of the extraordinary genius of its author: the whole section on the Reduc! Geometrical Questions to Equations will richly repay the most careful pert! the student. 105 ‘4) Let the equation be a? —Tx+,/(x°-— 72+ 18) = 24. —[f we transpose 24 and multiply together the two factors ; a? — Ta — 244 ,/(x*— 72 + 18) and x#°— 7x —24—,/(#°— 72+ 18), which the radical expression ,/(«*—7«x+18) has different siis, their product (a? — 7 x — 24)’ (a°— 7a + 18), ‘sational and becomes zero when either of its factors is zero. (a®— 7x —24)°—2°+ 7x-18 (a -Tx- 24-3 »(a°— 7a — 24) — 3) — a? +7a+ 2444 | _ 169 4 "Therefore (2?— Tax — 24 - y=? : and -Te-24—-faae. If we solve this pair of equations, we get the respective pairs + ./173 ovalues 9 and - 2, and Ulan SEARS The two roots 9 and -2 belong to the proposed equation : 7+,/173 2 a? — 7x —-24-,/(a?- 7x + 18) =0, ich is introduced for the purpose of rationalizing the proposed iation, and are altogether foreign to it: the first pair of roots y be considered as the proper roots of the equation, and the sond pair as roots of solution merely *. two others belong to the factor * Equations, such as that in the text, involving square roots of the unknown s}bol or of expressions involving it, may be generally rationalized, by trans- ng all their terms to one side, and by multiplying together the several factors ch arise from changing the signs of the quadratic surds: but it must be kept in d, that there are generally, though not always, roots of the rationalized, which not roots of the proposed equation, and that there are many equations, in- ‘ing such radical expressions, which admit of no solution whatsoever. Vou. Ae oO If 106 Where the 677. When equations appear in, or are easily reducible j ripe ant form where the unknown symbol presents itself exclusivel: power only of an ex- If the proposed equation had been x—,/2 x=4, the rationalizing factors 7 pression in- be <= volving the c—4—)/2r and r—44/22, ae and the resulting equation 22—10x+16=0, whose roots are 8 and 2: occur. it is the first root only which belongs to the proposed equation x — ,/Qq the other root 2 belongs to the factor eam a r— 44/22 =0, | which is introduced in the process of rationalization, and is foreign therefo the original. equation. The adventitious factors required in the rationalization of the équation | | 3 /(112 ~ 8z)— 19 —./(32+7) =0 (1), 34/(112 — 82) 19 + /(32 +7) =0 (2), — 34/(112 — 8x) — 19 —/(32 +7) =0 (3), are — 3/(112—82) — 19+ /(3x4+ 7) =0 (4): . we first form the two factors, which are the products of (1) and (2) and 013) and (4) respectively, and which are 1362 — 752 — 114,/112—82 =0 (5), 1362 — 757 + 114/112 — 82 =0 (6), and the final rationalized equation is their product or 11148 44388 a = 7 625 "+ “625 = 9 (7), | 7398 t 6 and ——. whose roots are 6 an 695 The first of these roots is the proper root, and corresponds to the prim equation (1), or to the rationalizing factor (5): the second is a root of solu and corresponds to the factor (6), or to the subordinate factor (4) : it ma’ further observed, that the subordinate factors (2) and (3) are satisfied by ne of these roots, and that they are equations which admit of no solution whatsoeve In a similar manner, the equation { V(22+7) +/(82—18)=/(724+1), (1), when rationalized, becomes ) 272 162 Pp a -—_— = we _—=0, (2), , the first is the proper root of the proposed equa (1), and the second a root of solution only: also the proper root of the equa. 8 | 7 77 (=-5) — Vv (5 + 45) _ qZV(102 + 56) =0, 14568980 28746409 ° and in both these cases there are irratic factors of the rationalized equation which no value of « can satisfy. of whose roots 9 and -9 is 20, and its root of solution It will be found that 4 and — = are proper roots of the equation +2 /(c7+2+5)—14=0: but there are no values of x which will satisfy its rationalizing factor a—2/r27+274+5)—-14=0, 107 sressions which possess the relation of the square and simple er only, we may replace this expression by a new symbol, proceed to solve the equation with respect to it. Thus, if the equation be 6x — 2+ 3,/(a*—6"+16)=12 (1), make ,/(x’—62+16)=u, or «?—~6x+16=u?, and therefore -a@=106-u*: we thus get the quadratic equation ; v 16 —u?+ 3u=12 (2); re the values of w are 4 and — 1: but inasmuch as the form fhe proposed equation (1) excludes a negative value of u, y confine our attention to the arithmetical root of equation 2*: we thus get u=/x*—6x +16 = 4, | or x*—62+16=16, and «=0 or 6, 2h are the proper roots of the equation. The equation \(w + 3)? + (w+ 8)? —7 (e+ 82 =72r+ 711 sasily reducible to the form {(@ + 3)? + (w+ 3)P— 7 {(e + 8)? + (w+ 3) = 690, Ich becomes, if we make w= (a + 3)? + («+ 3) | u’ — Tu = 690. The values of uw are 30 and —23: those of x are 2, —9, si _— ° which are the four roots of the proposed biqua- ric equation t. n the equation x - 5 gate a+ex+5 f(a?+e+5) 252 i we take the negative value of wu or —1, the resulting roots 3+,/ — 6, ) spond to the equation | 6x —2°—3,/1?— 62+ 16=12, tare merely roots of solution of the original equation. The biquadratic equation, if reduced to the ordinary form, is a*+ 1423+ 6622— 119 x — 630 = 0, 11 may be solved by the extraction of its square root. 108 we multiply both sides by a, and thus get ee ¥, 5a 71105 e+at+5 f(@+ar+5) 25- if we make oe —u, the roots of the resulting equatic 116 2 “+ = — rosie 4: 29 : : . are = and aie of which the second must be rejected. The roots of the equation Yea hae Ld ig flat e455) > oo 20 : . are 4 and in of which the second must be rejected, b merely a root of solution: there is, therefore, only one pri root of the proposed equation. Let 2? + 625 = 891. Make u=«*, and we get u’>+6u= 891, 8 5 “u=2° =27, or — 33, ert (1) x 8, or (— 1)3 133%; L= (1) x 3°, or (— 1)! : (33)8 = (1)8 x 248, or (— 1)8 .,/39135393 t- Thé num- 678. The number of proper solutions of an equation is| ee BPR ~ affected by a common denominator of the index of the unkn Ren a symbol, or its powers: for if the highest power of this syr| ee Hien was 2”, the process of solution would give us the values of | minator of and the number of those values would not be increased by! the index of 1 J ea transition from the values of x™ to those of x: thus the equé symbol. x°+62°=891 _12,/3 ja * There are three cube roots of 1, which are 1, = 15/5 669), and three cube roots of —1, which are — 1, aE | ( Art. 1 there are, therefore, six values of #5 in this equation. + For (1)8= (4)? andi(- 18s (—1): 109 | have the same number of proper solutions with the on x + 62° = 891, he solutions themselves only differ in the roots of one ion being the fifth powers of those of the other. _ however, we should suppose the equation 28 +623 = 891 ~mit equally all the forms which are proper to the different nolical values of x3 (which will be found hereafter to be 5 amber,) we should have 5 equations and 30 roots, of which dy are the proper roots of the proposed equation, the rest i roots of solution. | will appear, likewise, that the reduction of the indices of einknown symbol, when one or more of them are frac- , to a common denominator, will convert roots of solution Koroper roots, unless the change from one form to the other «companied by a limitation of the roots to be extracted, “msequence of such a change, to their arithmetical values 1! thus the proper root of the equation a+ /x=6 (1) 4and 9 is the root of solution: but 4 and 9 are equally proper of the equation 22+ 72=6 (2), ls we are equally restricted to the arithmetical value of |. the two equations (1) and (2). 1 a similar manner, there are only three proper roots of equation R516 = 0.5» (1); here are six proper roots of the equation a’®— 6x 3 22—622—-16=0 (2), if include the roots of the former equation (1). The bino- mial theorem when the index is a whole number. The same series is equivalent when the index is perfectly general in value as well as in form. The pro- duct of the series for +2)" and (l+a)” is the series for (l+a)°t™, CHAR TE Rx THE BINOMIAL THEOREM AND ITS APPLICATIONS. 679. In Chap. vir. (Art. 486...), we have proved, ‘x the index 2 is a whole number, that i 1.2.6 2 (1+a)'=1+ne+n(n—1)—— +n(n—1)(n-2) + and it will be seen, from an examination of this series, a) x’ x a” Toe ED ie? eer eee are n, n(m—1), n(n—1)(n-2),..., n(m—1)... (m—7r4) x, being, for the (1+7)™ term, the continued product of thid scending series of natural numbers from n to n—r+1. | 680. This series for (1 +2)" is perfectly general in its i though n is specific in its value, and it will continue therot by ‘the principle of the permanence of equivalent forms” i 631) to be equivalent to (1+.)", when x is general in I as well as in form: and it will consequently admit, in vi of this equivalence, of being immediately translated intdh whole series of propositions respecting indices and their je pretation, which are given in Chapter xv1*. 681. Thus “the general principle of indices” (Art. shews that (l+a)*\(1+2)"=(14+2)" for all values of 7 and n’, and consequently the product oth series for (1+.2)” or 2 3 H L+netn(n—1) 75 +n(n-1)(n~-2) ~ + &e, * See Appendix. 11] 1 f the series for (1 +)”, or 4 BH (n—1) = — + n(n? 1) fn Beye Bate, —* 1.2 (ors le equivalent to the series for (1 + at”, or 2 1+(n+n')x+(n+n’)(n+n’—1) — | : +(n+n’)(n+n’—1) (n+ n’—2) ae: 37 ke, | the same circumstances. 1 : . Again, tl nes for 1+.x)~” (Art. 640,) The series 2. Again, the series fo (say or (1 + 2)-” ( >) a lie found by replacing n by —n in the series for (1 +a): i tar™ us get x 1 As: @ product —x(--—1) may be replaced by n(m+ 1): the wet —n(—n-1)(—n-&) by —n(n+1)(n+ 2), and simi- yor the subsequent terms. 2 jayt=l-natn(n+ 1) —n(n+1)(n+2) + &e: | 3. The series for (1—.)" is deducible, in virtue of the The series principle (Art. 631), from that of (1+.)", by changing jae igns of those terms which involve the odd powers of x: {is get, as in Art. 491, ; fs n° m_1_ Bay Ye relay oe oe? | oe MY i) 1—nrin(n Lie n(n —1)(n preety ‘a similar manner, the series for (1 - x)-” will become a oe ——"T)) sore x x? 78724 b) (7+ 2) 2———— a | Pe ( )( ) Lire. otek z 1+nx+n(n+1) ‘all the signs and terms are positive *, Since 4 + Au=A(1+u), and therefore (A+ Au)"= A” (1 +4)’, Series for 1 ius oe, =l+e4r7?4 234 —4 oes)? = 14804 8294 4224 ...... PP eaeee” 43.4, 545 —7)-3 — Ngee Se (l—2)-8=14 394 io ft 1.2030 t (A+Au)”. Examples. | Series for (1+.2)3. it follows that u u (A+ Au)'=A ji+nutn(n—1 75+ ("— 1-2) oe 112 32 T 2 (1) Thus, if A +Aunaisat=a'(1 +3), we have nm x f therefore (a? + 2°)"=a° 1} +n. +n(n—1)-s~Gaal x +n(n—1)(n-2)-—g-ga8* Lot a" x 1. a Qn—4 ——* sn (n—1)(n=2) multiplying a" into every term of the series. (yee. A+ Au=a—az=a'(1-"), we get =a*"+ na" 2? +n(n—1) "ae: A=@, u=—-—, WH, U=-G ’ ne 2” @’ and therefore 3 AS MN Asn im x n4 a uz ae hb ae (a’—ax)’*=a"" }1 n—+n(n ie n(n—1) (m 2) al ; 2 3 —a?"—na”—'x+n(n—l Jaen (n—1) (n—2)a°"-8 = 685. Since (1 +)? means the square root of 1 + # (Aip it follows that the corresponding series {replacing by 3 series for (1+ .2)"}; int a 1+$.2+3(4-1)75+4@G-D@G-2 rege expresses the square root of 1+. If in this series we replace the factors - 1 3 1 su Vie hanhs Was oh on aT Ry ees ¥ 35 irae) Pee we get we ig? lak Sa 113 Series for : 1 x*\} lf the series for (a*+ x”)? =a (1 + ~:) be required, we get (424 ,2)3, 2 4 6 8 | “ae ry x a 52 ome) = a0) + — ee ( ) ( 2a° 8a* 16a 128a? ae a =a ‘ re 52° = —— — —— 4 2a 8a°® 16a® 128a’ na similar manner, we shall find Series for 1 2 2\5 A ay a2? a? x 5 x® (a? — x?)2. ED) 5 Se ee 2a 8a° 16a° 128a’ “hese results are identical with those given in Art. 650, 1 and 2, and the student will not fail to observe the readi- with which the terms of the series are formed and the law ‘neir formation ascertained, compared with the ordinary pro- s of extracting the square root. 86. Since (1 +x) means the cube root of 1 + x (Art. 637), The series llows that the corresponding series {replacing n by 3 in es i «series for (1 + x)"t, (l+a)3. ) 1 i 8 1/1 1 ee mot —|——] oi is a A) ee ae ih oe 4G ade 1) Wrcoaa* jesses the same cube root of 1+ 2. "his series becomes, by replacing 1 2 1 Foy 8 any, ——., —— 2 by a~, == ees a 3 igs Praia Sates DES a l Lxi2 2" 1.2.5 x 1299548 Fi 5B — ——— . —_ + —___ - > +e 3 3701.2 3 1.2.3 3 1.2.3.4 ve 2 52° . 102° Sr ca ho at Oe se 3G ce Ste O4S 87. More generally, if we replace x by ; in the series for ee series [| Au)" (Art. 684), and replace the factors pe i ntl (A+Au)o. »... of the numerators of the successive coefficients by | , 7 hh shall find | qi Bae yh PP 2D) es | (A+ Avtar 142 ws uLithe | ASPEN P29) 0 Bees ) q 1.2.3 om II. P a 114 3 | Examples. Thus, in the expansion or developement of (a* + ax) ©, whe . 7 LW “==, p=3, p—q=—7T, p—29571%, p—3q=- 27, . we get +5 VA Arf e Oo: ,3-7-17 +1 n+1 ’ n+l we get r =the a ds _———— : in cases, where * For the complete coefficient of the rt term is and of the (1 +7) term is , and the —1 passes from positive to mn 117 3 and =a , then r is the next whole number, which 2 45 : eater than 38” and therefore the series becomes convergent its second term: if »=—3 and r=—, we find , 11 | Nat Be 99 Fr j r= 7 ee sey . ' 12 | : : 13 convergency begins from the 23rd term: if n Tea and , we have a point of convergency when r= 83, and, sub- o~! 0 F) 4 of the series. | : ie ; 1 ontly a point of divergency, when r=17: if n=—> and | , we find r=5, or the divergency begins with the 6th (98. It appears, therefore, that the series for (1 +.)" and a) are or become, convergent, whenever 2 is less than 1, henever the binomials, whose powers are developed, are metical, both in their arrangement and value: but that « cease to be so whenever « is greater than 1, or whenever cease to be arithmetical either in their value or their ligement: for in this case 1—.z is negative, and therefore ithmetical in its value: whilst under the same circumstances, ymomial 1+.2, though arithmetical in its value, is not so in ‘rangement, inasmuch as the greater term succeeds the less ‘n inverse and interminable operation (Art. 386)*: but it € is greater than 1, we may obtain positive values of r from both these tlz, which shews that a point of convergency is in such a case followed ‘point of divergency. _It is in the inverse operations of Arithmetic and Arithmetical Algebra, « Division and Evolution, that we meet with interminable results, and ‘ich the arrangement of the terms of the expressions, which are the subjects operations, in the order of their magnitude, is absolutely necessary to enable Lie to, when we cannot accurately obtain, the true result which £ ght for: such an arrangement, which is emphatically called arithmetical, ‘J absolutely necessary, though generally convenient, in operations, whether | or inverse, which lead to a terminable result. Thus —————— a The series for (1 + a)” and (1l—2)” are conver- gent or divergent according as 1+ and 1— «are arithmeti- cal or not, both in their ar- rangement and value. 118 should be observed, that in the latter case, the terms resulting series, either are, or become, alternately positi negative, confirming the remark which has elsewhere bee (Arts. 517 and 650), that the expression in which it orig though arithmetical in its value, is not so in the chara the operation to which it is subjected. 694. If the series which arises from the developem (1+ 2)" be arithmetical, or convergent, we shall be enabl only to approximate to its true value or sum (s) by the. gation of its terms (provided we include the term from the convergency begins), but likewise to assign the lin the excess or defect of the aggregate thus formed fro true sum which is required. Thus if o be the aggregate or sum of r terms of the and 7’, with its proper sign, its (1 +r)" term, then the tru s will differ from « by a quantity less than n+1 r % Thus (1+ 2)? =1+2x 42% and (r+ 1)?=22+22+4+1, and the resi 1+2xe+4+ a? and 2?+22+4+1 are identical in their arithmetical value, though not in their arrangement, 1 x be less or greater than 1: but (1 + «)-?=1—29 + 3r2— 4a3+... and aT ah © apey ee (2+ 1)=35 1-43 -SGt 05 x and if « be less than 1, it is the first, and if x be greater than 1, it is the sec these series which is convergent: or, in other words, the convergency 0 ‘ gency of the resulting series depends upon the arithmetical or unarith arrangement of the terms of the binomial. * P Ti ee 7 n+1., ; For if p=1— a, it will follow, sincel — increases as 7 im that the sum of the geometrical series T— T px + Tp? x? — Tp? x3 + 9 pals. (Art. 432), will be less than the sum o’ of the corresponding term px 1+ r—T(1-***)e47 (1-***) (1-F42) ae... r r l+r of the series for (1+ 2)"; but the sum of the geometrical series T—Tx+ Tr?— Ta? +..., which is pres will be greater than o’ since p is necessarily less than 1, wl greater than 7 : it follows, therefore, inasmuch as o’ is intermediate in value b T1y n Example, suppose it was required to determine within Example. mits of error the square root of 5 would be A Si by gation of 5 terms of the series for 2(1 + 1)3, or LA 1 1 5 7 all gc I gc eI Bi E a Ao dnd! aocats Bat Ae? d find o = 2.236026 nearly, T= .0000534 nearly, = .0000321 nearly, ch ae rene 2.2360687 nearly, erefore s differs from 2.2360687 by a quantity less than a T : T . .’ that o + iene will be less and o + eee will be greater, ++o' by a quantity less than aes L+a (l+px)(1+a) — S14 (1- 24h) Base) CHAPTER XXIII. ON THE USE OF THE SQUARE AND HIGHER ROOTS OF SIGNS GF AFFECTION. The signs 695. Tue signs of affection, which we have hitherto of affection hitherto. are +, —, ,/—1 and —,/—1; the two first of which ei aeace: may be conveniently replaced by +1 and —1, if we co nized, are them, like the two last, as factors of the symbols which pot affect*. Thus +a=+1x a and -—a=— 1 xa, in the samem pre roots that a PDI =/-1 xa and — oie =— RE x a (Art. Of these signs, the two first + 1 and —1, or 1 and —1, a two square roots of 1: whilst the two last, /—1 and — are the two square roots of — 1: the entire series of then —1, +,/—1 and —,/—1, may be easily shewn to be th biquadratic roots of 1 ft. | Their use 696. The preceding signs, or their equivalents, aré palin the ficient, as we have shewn in the preceding Chapters, to e} Riess every symbolical consequence which arises from extendin) of Arith- rules for Addition, Subtraction, Multiplication, Division teas the Extraction of the square root, which are proved in . metical Algebra, to all values whatsoever of the symbols to they are applied, in conformity with the general principle permanence of equivalent forms (Art. 631): no further sig affection would be required, if the processes of Arithn; and Symbolical Algebra, were confined to the several opeli above enumerated. ihe 697. But the processes of Evolution in Arithmeti) ae of the Arithmetical Algebra, are not confined to the extraction of |! higher and biquadratic roots, and it will be found necessary to f orders of the roots of duce new signs, in order to give the same extension to thet Ber ia. for the extraction of cubic and higher roots, and whict Ae those we have already considered, are capable, as we shal} ‘volution, : c ceed to shew, of being correctly expressed or symbolized multiple symbolical values of the cubic and higher roots of * See Appendix. + For a*t—1=(«?—1)(22+1)=0: the roots of the equation «2—1: 1 and —1: those of «7+1=0, are foal and oan Fe 121 aoe er y ee ae F In the case 8. Thus a’=1 xa’, and theretore Ja@= lx a: and, af the Sete uch as we have shewn (Art. 669), that the three cube roots of 1. of 1 are > mers) 1. 1 Se) follow that there are also three cube roots of a’, which are mrp S f=), —1—/8i/—1 Mccwnn Co) ah fe es «st of which alone is arithmetical. 9. Ina similar manner, we have — a>=—1 x a’, and there- Of the cube — 3) ‘ roots of — 1. J=a'= ce 1 x a: and inasmuch as we have shewn that yree cube roots of — 1 (Art. 670), are Le Sif 1 by he /B Sl 2 : 2 5] | follow that there are also three cube roots of —a%, which — a, ; Wesel tees eb ge Rane Gi. Foxy aight ae a of which are arithmetical. 0. It may be easily shewn that the three cube roots of The als d the three cube roots of —1, form the six senary roots pad for if x =/1, we have x°=1, and therefore are the senary roots of 1. 2®—1=0 Cho dnasmuch as 2&— 1=(2*°-—1)(2*+1), bd Por (stsveN= y=) ae 2 i Sab e for pee) = 1 (Art. 669, Note) : 3 ’ —ti (=!-weW/=1,): (PEED. : Eee). - | f If. Q Use of the nth roots of l and — 1. 122 7 it follows that the equation (1) is equally satisfied by the roots of «*—1=0, which are the three cube roots of 1, an the three roots of «'+1=0, which are the three cube ro —1. (Art. 671). | 701. More generally, since a"=1xa", and —a"=(-1)a’, it follows that a= <1 x a and 4 Ea Seal x a, and the n roots of a” and — a" (which will be shewn her to be severally » in number), will be correctly expressi symbolized by multiplying a into the several symbolical yj of ./1 in one case, and of ] 2 3 2) ~ } For if we divide 2*»— 1 by x—1, the complete quotient is : heals Ot sitet: OTT Te r+1; {nce all its terms are positive, there is no positive or arithmetical value ivhich can make the sum of its terms equal to zero. } | — 124 ae and if a represent one of the unarithmetical roots, the { roots will be represented by a, a’, a® (Art. 669, Note) *. The biqua- 706. The roots of the equation x«*—1=0, which are! dratic roots of 1. biquadratic roots of 1, are 1, —1, Rak and SN Ss (Art. } Note). The two first are also the square roots of 1: the twa are the square roots of —1, or of that square root of 1, W is not arithmetical: and if a represent ,/—1 or —,/—1, four roots will be expressed by a, a’, a® and a’. The 9 707. The roots of the equation #*—1=0, which are] na . Riles quinary roots of 1, are eck gchar (£2). /( ae : 4 8 : 4. 8a) or 1, .309+951,/—1, —.809+587,/—1 nearly. T bay pro- It appears that the pairs of imaginary roots are sevel perties. j Ss : | reducible to the form a+b ra —1, and that in both cases Jat +b) =1t: | SOrebyac., and 1, one of the imaginary or unarithmetical roots beirt a ; reciprocal of the other. t For 5—1=(«—1)(a*+ 27+ 0?+2+1)=0, where x—1=0 or at+a?4+27+24+1=0. In order to solve the equation a*+ 23+ a2+a2+1=0, we divide it L which gives Gel w+ae+1l4+—4+—=0, o (a 1 1 or a®+-—s+24+-+1=0; fy a ive ] or (0+ 2 4 (: + ;) =e Aik Olle the solution of this quadratic equation gives the values of x in the text. + For if «w=.309 and 6 =.951, we find /(a? + b?) = a/ 999882 = .9999...... = 1 nearly : info Me i or ee 4 and b= (24%) we get 024 Fa), SEN 125 will be shewn, in a subsequent Chapter, to be a property ion to the imaginary roots of 1, whatever be their order*. ‘we represent one of the quinary roots of 1, which is dif- : from 1, by a, their whole series will be expressed by Gee an at: df the same series be continued indefinitely, the same suc- sin of values will be reproduced periodically t. his is a general property of those imaginary roots of 1, which e enominated by prime numbers, and will be demonstrated nally in the following proposition. 8. If there exists a root{ a of the equation 2"— 1, which General a Eg : . property of ferent from 1, when » is a prime number, then its » roots thantlueeare il be expressed by the terms of the series of 1 when x is a prime Cie aa. 2. ae (tae number. 1 the first place, no power of a, whose index is less than n, e equal to 1. milarly, if a = — .809 and b = .587, we get /(a2 + b?) = x/ .999639 = .9999 = 1 nearly: ; 5+1 5— 3 5 —/d i =O and b= a/ ( BO), we get a? +o? tv 4 ~ aie Vania VEN d Thus the pair of the imaginary roots of a#—1=0 is —3}+ 3 i) a=— 1 and p= 2, we therefore get \/(a? + b?) =./(4 + 3) =1. Ry if a = 309 + .951 TER be one of the unarithmetical roots of c°>—1=0, a? = — 809 + .587,/ — 1, a® = — 809 — .587,/ — 1, at = 309 — .951 Reale a=, are also the quinary roots of 1: it will be found likewise, if the series be atued, that a&®’=—a'xa=lxa=a, } t!, the same series of values is reproduced, and so on for ever. : It will be demonstrated, in a subsequent Chapter, that there exists in all © ne, and therefore n — 1, roots of the equation «”— 1 = 0, which are different ich other and from 1: in the absence of such a demonstration, the existence sh roots in the equations 2?—1=0,«4—1=0 and 2°—1=0, where they ea determined, affords a reasonable presumption that they exist likewise other cases. } The indefi- nite series formed by the suc- cessive powers of an imagi- nary root of 2 —l1=0 is periodic. 126 For, if not, let a?=1, where p is less than nm: then \ a? —1=0 and a"—1=0, for the same value of a, it fo that they must have a common factor, which becomes zen that value: but if x be a prime number, a— 1 is the only con; factor of a’—1 and a"—1, and which can only become when a=1 (Art. 703), which is contrary to the hypothes: follows, therefore, that a" is the lowest power of a, whir { ‘ ° { equal to 1. In the second place, the successive powers of a, whos dices are less than m, are different from each other. For, if not, let a?=a‘, where q is less than p, and bo them less than n: dividing both sides by a’, we get a?-: which is impossible, since the index p—q is less than p,u therefore less than x. | In the third place, the first n terms of the series (1 roots of the equation #?—1=0. For.a"= 1, since a is assumed to be a root of the equatioj (a*)* = (a")*= (1)"=1, (a°)* = (a")' = (1)°= 1, (a?)"= (a)? = (1)? =1, (a")"=(1)"=1, it follows that a’, a®.....a", satisfy the requisite condi of the equation equally with a: and since they are all of § different from each other, and ” in number, the eq x"—1=0 can have no other roots. I Again, since 709. If the series 2 3 n Qa, a, Mig «ee A be continued beyond a", forming the terms n+1 n+2 n+3 5) ’ a a Ce. were the same values will recur in the same order. For a’*t!=a"xa=1lxa=a, n+2 a = emi 1 Xia a, a®t =a x a®=1 x a= 2%, “re eee eee ee eee eee eeaeeeeoe It appears, therefore, that the series n n+1 ’ oO Qs ier ene. sie ae oe Saag Sa ‘ is periodic, the same terms recurring, in the same order, | every 2" term. 127 10. The root (a) of the equation x"—1, whose successive Base ‘+ pes eriod an yrs form the complete period huh 2 n number. AG chiara be called its dase: and it is obvious that there are as many as there are roots of the equation different from 1. ; will be farther shewn, in the articles which follow, that if a com;osite number, no root of the equation «"—1=0 will g'ss a similar property; but that in all such cases, a base ‘} complete period of x terms may be formed, which is de- sent upon those roots of 1 different from 1, which are, vally denominated by the prime factors of 7. 1]. The roots of the equation The roots ‘ of the equa- ae tT Sy (1) tion vI—1=0 i p and q are different prime numbers, are the several pro- are the pro- f s which can be formed of the roots of the subordinate que ie it tions z?—1=0, and 2’—1=0. subordinate \ equations or, let a and £ be two roots of #?—1=0, and 2?—~1=02—1=0 s:ctively, which are different from 1, and let pai Oy REE EC eh PAP B, B...8' (8), -)eir corresponding periods; then the several terms of their “uct, which are n in number, form all the roots of the ition 2?7—1=0. ‘or if a’B* be - one of those terms, we have (0" Bt = (arty (ry = 1": therefore a root of the equation «?!—1=0. i all the terms of this product are different from each for, if not, let a” 3° =a"’ G’’, and therefore a’-” = Bat ia is impossible, since 1 is the only root which is common se equations xz?—1=0, and w!—1=0. .; terms of this product, therefore, which are all of them ent from each other and pq in number, form all the roots € equation x?!— 1 = 0. / For a? = (a?)’= 1, since a? = 1: and B??=(f%)= 1, since pt=1. For r—r’ is less than p. and s‘—s is less than q: and if s’—s be negative, ; | pt | Bt=* = a7 = Bra? = B1-&-*’), which is a root of x7— 1 =0, and different ) l, since q — (s—s’) is less than q- 128 ean 712. The product af is the base of a complete peri uct af is : : ; the base of pg terms of the roots of the equation a?’—1=0. ree In the first place every term of the period pris. a8, (af), (@B) --- («BY «- is equal to some one term of the product of the subord, periods corresponding to «?—1=0, and «'—1=0. For (af) =a"f", and a’ is always a root of a” —1=0, B" a root of a1—1=0, whatever be the value of r: and tt fore af" is, by the last Article, always a root of a1 13) Again, all the terms of this period are different from x other. | For, if not, let (af) =(af), where r is greater tha we thus get a’*=/'-"= 6'-°- (Art. 711, note), whichis possible, unless r—s=mp, and tg—r—s=nq, or mp=n’q, where n’=t-+ and since p is prime to q, the least values of m and x’ in| equation are g and p (Arts. 110 and 116): it follows, th fore, that r—s is not less than pq, and consequently the ti (2B) and (af) which are equal to each other are beyon¢ limits of the period. The terms, therefore, of this period, which are pq in nu form all the roots of the equation 2?’— 1 =0. Therootsof 713. Thus the roots of the equation 2°—1=0, are f a®—1=0. two roots of x7—1=0: they are therefore -14+ /3/—-1 -1-,/3/-1 1-/3/—1 1+ JB) i a es a TE the terms of the period whose base (a /) is ep chase a Se ee oe 714. Ifp and q be equal fy each other, as in the equatio| x? —1=0, the series of roots in the subordinate equations z?—1=0 and z'-1=0 become identical with each other, and no new values or 129 formed therefore by their multiplication with each other : under such circumstances, the roots of the equation el, be found to be the p™ roots of those of the equation z?—1=0. For if we make a’?=u, we get x?” —u?=1, and therefore "lu, where the values of w are the roots of the equation 1=0. “8 Again, if we suppose « to be one of the roots of The period x’ 1=0, which is different from 1, and ( one of the pe roots ba rat ofc, we shall find that the p® roots of the equation 2? —1=0 m’ be represented by the terms of the period whose base is «/. For the terms of this period, which are p’? in number, are alidifferent from each other. For if not, let (@f)’ =(af3)’, and therefore a’~*= (*-" where id s are less than p: and since a’~*= (6°, it follows that Be D&-9—1: but since r—s cannot exceed p—1, it follows . (p+1)(r—5) cannot exceed (p+1)(p—1), or p*- 1, and th: the terms of the period af, (af), ...(a)?* form all the s of the equation 2” —1=0. 716. Thus the roots of the ghia 2°—1=0 are the cube The roots of ros of the roots of the equation x*—1=0: they are therefore * 7a tl cube roots of STE OR EN En ea 2 ar as & Slay hore ° ‘The method of extracting the cube and higher roots of ex- Bevel end a Sell td , will be ssions, such as lained in a subsequent Chapter. 717. lf p, q, r be three prime factors of 7, which are dif- The roots mt from each other, and if a, 6, y be severally roots of the hiseaa ations z?—-1=0, #!—-1=0, and w—1=0, which are dif- pep = 0, mt from 1, then the period of n terms, whose base is aBy, and | form all the roots of the se goE 2"—1=0. aber In a similar manner, if x = p’ and if a be a root of «?-1=0, ch is different from 1, and if ( is 2/a, and if y is 2/8, we Wee. II. f General It appears, therefore, that the roots of the equation x” — conclusion. +1) form, in all cases, periods of n terms, which are the cessive powers of a common base: and it is this remarkable itp perty, which will be shewn in a subsequent Chapter, to om the principal basis of their interpretation. Bae Spas 718. The roots of the equation «”*+1=0, which aret ' n roots of —1, are included amongst the roots of the equi r"?—-1=0*. Their If a be one of the roots of «"+1=0, which is different on pepouss) Te ots thedd powers of the base a of the period of 2n terms | Gy Agha pee Cee ee will express the whole series of its roots. | For a*=~ 1, (a*)" = (a")°=(—1)?=-1: | (5) =(@")=(-1)'=-1, | and so on for other terms of the series: it follows thereforeha a, a*,a’,... are roots of 2"+1=0. Again, all the terms, and therefore the odd terms of the puiot d,0", a®. Nee are different from each other: and since they are n in nune they form all the roots of the equation 2"+1=0. The roots of 719. Thus the roots of the equation x‘+1=0 are inchile vc + 1 = 0, ——| oe 0, amongst those of the equation #®—1=0: and if 1+J/@ f 130 shall find that the period of p*® terms, whose base is afy, express all the roots of the equation 2?°—1-= 0. | It is not necessary to give the demonstration of these x positions, nor to shew in what manner they may be extede to the exhibition of the roots, periods, and their bases oth equation «"—1=0, when 2» is any composite number whatscye ve! one of the roots of x*+1=0, all its roots are expressed the period | tg ¢ ae (! of 1)" eae) For «9*—] = (a*— 1) (2% +1). { | Isl We ych are equivalent to oat /=1 -1+J=1 -1-J/-1 1-J=1 ty 1 nhs sleiors eg, SLE ey Liepan aea ')The series formed by the even powers of the same base (Art. fe } ss ee 1) eS ; which are Cpl - 1, =—/ wi and 1, are the ‘es of a equation a*—1=0. (Art. 695). n i 720. We have shewn (Arts. 708 and 710), that when the The order mex n is a prime number, the same period of n terms may ofeuncey ue from (n—1) different bases, which are the (n—1) first apace erio0d 1S “eas of the period, the last or n™ term being 1: but it will ple by lilwise be found that the terms of all these periods follow a a change } va Ss : of its base. ferent order: thus, if in the period ‘| MG a. . aco aunt (1), wecesponding to «7—1=0, we replace the base a by a*, we itl get the period Heed, d, a, as. G5, 1 (2), wose terms are the 2nd, 4th, 6th, 1st, 3rd, 5th and 7th terms the period from which it is derived: if in the same period 1 we replace a by a’, we shall get the period YY Paths Cr en Op Osi d (3), wise terms are the 3rd, 6th, 2nd, 5th, Ist, 4th and 7th of h same period: and it may be observed generally, that if in teries of repeating periods of m terms, we replace the first yy the r'™ term, then if x be prime to r, we shall reproduce 1 eriod involving the same terms, with that from which it is loved, but disposed in a different order, being such as would a2 from taking every 7 term of the periodic series*. _. For if one term (the pt») of the series of rth terms coincides with a final et a” ot 1 of a (the qt") period, in the periodic series a, a?...a", a, a®... a”... it n nm ne iust have pr = qn or Lee =>», and since n is prime to 7, the fraction — cannot q Le deduced to lower terms: and since p and q are the least possible numbers, vl h will answer the required conditions, it will follow that p=n and q=r: or n/her words, it is the nt term of the new period which coincides with the final e1 of the rth period of the periodic series. gain, all the terms of the period thus produced are different from each other : 01 f possible, let a'’= a‘", and therefore a“-%" =1: but since s and t are both of 132 Properties 721. If the » terms, therefore, of a period be arrank ee circularly, and if we mark in succession every r“ term, (r i arranged prime to »), we shall pass round the circle r times, hay eircularly. we return to the first term: thus, if »=7 and r= 2, a) we arrange the terms of the period | 5 + 5 6 a as Ori rend ed round the circumference of a circle, and mark every second term, begin- ning from 1 the succession or new period of marked terms will be a’, at, a’, a; Care (2); where every term of the first period is included once, and once only, and where 2 circuits are made before we return to 1: again, if we make r = 5, we shall form the period G80 00 wed wen (3); where 5 circuits are made before we return to 1: and simill in other cases: and it is obvious that the same periods wilb found, and in the same order, if we replace a, in every ‘¢ cessive term of the period (1), by a? in one case, and by ai the other, depressing the indices in every case to the resii which arises from dividing them by 7. Cyclical 722. If we restrict our attention to the imaginary root arrange- “ “ A . ment of the *"- 1=0, where: n is a prime number, or, in other wordst roots of ~~ the roots of the equation a . at Fae aT pa ev as (y= | at—2 1s. a prime number. we shall find that they are capable of an arrangement, by wh: the terms of the period which they form, may always ri in the same order, upon the replacement of one base by anotlt this may be effected by arranging the terms of the period n—1 CALs eh ac ae: them less than n, it follows that s—tis less than n: but we have shewn all that the least power of a’, which is equal to 1, is a’. | It may be observed, that the proposition is true of all powers of a base wi indices are prime to n, whether n be a prime number or not. 133 h an order that their indices may be the residuals of the sive powers of a primitive root of n (Art. 531), which ehend every number from 1 to m—1 inclusive: thus, if 7, of which 3 is a primitive root, the residuals of 2 3 e oo, Bt 8! Se S20. 4, 5,1? the terms of the period Gum, 2 10%,n0°, ia? (Lh); 71 248; : ~ ; = 0, be distributed in the order of these ponding to als, as follows, meat. a°,*a°, a’, a (2), ien arranged circularly, it will ind that the succession of its will remain unaltered, what- e the base or term of the period rich the assumed base or a is led: thus, if @ be replaced by > get the period \2 6 4 5 3 ra) 3@,%4,@,a,a (3), terms follow the same circu- ‘hese residuals are successively formed by multiplying the preceding resi- the primitive root, or3: thus, 3 being the first residual, the second is 2, s the remainder from dividing 3 x 3 by 7: the third is 3 x 2 or 6: the fourth :temainder from dividing 3 x 6 by 7: the fifth is 5, the remainder from '3x4by7: the sixth is 1, the remainder from dividing 3 x 5 by 7: they ds recur in the same order for ever. If we take the second primitive root ,/uch is 5, we shall get the series of residuals 5, 4, 6, 2, 3, 1, the five first ‘one series being the five first terms of the other series in a reverse order. = 18, there are 4 primitive roots corresponding, which give the following of residuals : 2, 4, ’ ’ ’ ’ ’ , > ? ’ aa 74 dst iGs ee ll, Oro; Lomi Le Peat) Opop lin kane O; a, Os e4ee 2) Pend 0 Mylue uno, 2, le," 7, 3, O,1 4, lle}, BAA feds. 01) hae 25.9, 8.10, 65): sriods form pairs, in which the first 11 terms follow severally an inverse lier: it may be observed, that the terms of the second pair are the 5th terms of the first. student will find the theory of such primitive roots discussed at consider- gth in Art. 531, and those which precede and follow it. Cyclical periods : their great import- ance. 134 lar arrangement with those of the period (2), each term) . . 2 one place in advance: if we replace a by a* or a’, wg adawet Tene sae eg rn ees ath, 8a (4), where each term is two places in advance, when compare . . 3 the same period (2): if we replace a by a or a®, we; period | | where each term is three places in advance: and similar 9 will be observed to follow, whatever be the term in the sep which a is replaced. 723. Such periods of the imaginary roots of 1, oil respond to prime values of the index, may be properlyi cyclical periods, inasmuch as the various derivative perlogal result from changes of their bases, will perpetually fol\ the same order round the circumference of a circle, whie regard is paid to the initial term: they will be found hea to be connected with the most important analytical the and therefore deserve the most careful attention of the studit a, eke a, a’, ai, a® ; (5), * They form the basis of Gauss’ well-known researches respecting - metrical division of the circle, and of Lagrange’s theory of the solutio equation a”— 1 forall values of n. See the Disquisitiones Arithmetice 0} Sectio 7m, and the Résolution des Equations Numériques of Lagrange, Noixi : | | : | | : H CHAPTER XXV. } 4E GENERAL PRINCIPLES OF THE INTERPRETATION OF HE SIGNS OF AFFECTION WHICH ARE SYMBOLIZED BY HE ROOTS OF 1. 4. We have explained in a former Chapter (xxri1.) the miso! ruction and use of the roots of 1, as the recipients of the of 1 as the nof affection, which the application of the general principle Stage g permanence of equivalent forms” (Art. 631) renders ne- ee sy in algebraical operations, and more particularly in those jj.) at iiogd Elution: we have shewn that 1 x r and —1 xr may be conve- bt. ny used as equivalent to +7 and ~r respectively, where 1 111, which are also the square roots of 1, may be considered } recipients of the signs + and —: the extraction of the square tof expressions, such as r* and —r°, or of their equivalents jand —1.x~7°*, leads to results which are correctly sym- te by /1xr and ,/—1xr: the consideration of higher conducts us in a similar manner to expressions such as ! 1 1 and \/—1xr, or to their equivalents (1)* x r and(—1)*r, , ne ay ° e the signs (1)" and (—1)* are used to designate such cms or qualities of their common subject r, as can be vy to be consistent with their symbolical properties. 3 We have subsequently investigated (Chapter XXIV.) Recapitu- ore important of these properties, with a view to the dis- ene ey of the conditions which must be made the basis of their bolical pro- pert ies. > az we have shewn, when x is a prime number, » lt) has necessarily x symbolical equivalents or values*, ved it has one such value which is different from 1: and lve further demonstrated that these equivalent values or ‘nay be considered as the successive powers of a common che term value, when thus used, has no reference to magnitude, but to «al form only, or to the quality of magnitude which its symbolical con- naire competent to express: all the values of éLyer are equal in mag- K but different in affection or quality. 136 base, which is one of them, forming periods of terms, } recur in the same order for ever. oe of 726. If r, therefore, be a specific magnitude to whi) e condl- 1 . . ; . . . . . ' ee rete sign (1)* is attached, or into which it is multiplied, fa retation : 2. : ps Pian ap- the expression (1) r; and if @ be a base of (1)» (Art) plied to, : : ’ . 4 specific or any one of its roots which is different from 1, then (1)# magni- ; Padaa: equally express any one term of the period ON Tee Teall, atte sy Otee (1), The suc- 727. It will be observed that the successive terms | terme ofa Period bear the same symbolical relation to each other, the} period pos- a'r adr atr ar | sess the — , see same sym- arr a tar a aha eae ae being identical with each other, and with a: and it will ; each other: therefore, that whatever be the affection or quality whic! capable of symbolizing, when applied to the specific mag By it must piel eo the affection which conmaal we reach its last term a"7, which is identical with r. and must We may conclude, therefore, that the quantities desi| {barat by the terms of the period quantities GT. aT. tee ree of the same 3 ‘ kind and of must be of the same kind*, and of the same magnituaae the same inasmuch as the successive terms of this period are symbi magnitude. __ eo : different from each other, they are also different in tht fections: but if we enter upon a second period, the sameé will re-appear, and in the same order: and so on from 2 to period, however far the series of them may be extem Their geo- 728. The relative position of equal lines in Geometry (eileen ing equal angles with each other, and which are the qu tion. * Quantities may be of the same kind though different in quality or at thus all straight lines are quantities of the same kind, though they may relative or in absolute position. t For, if not, let the magnitude of the quantity expressed by ar, ec without regard to its affection, ber(1+c¢): then since a*r bears the same? to ar that ar bears to r, it will follow that the magnitude expressed by a7? r(1+c)®: similarly, the magnitude of a’r will be r(1+c)*, that } r(1+c)*... and that of a®r,...r(1+c)": but a*r=r, and therefore r=r or in other words, c = 0. 137 > division of 4, 8, 12 right angles by the number of terms period, will be found to present an exact correspondence e symbolical conditions considered in the last Article, and enable us to give to them a consistent and complete inter- tion. et a circle be described with a radius AB(r), and let its mference be supposed to vided into » equal parts in eee 8... B, es - the radii AB, , AB,, ABs A» AB,_,, and AB being 1 to the several points of on: then, if 4B be repre- vd in magnitude and _ posi- nby r, 4B, may likewise be sented in magnitude and on with respect to 4B in » B, by @’r, AB, by a'r. CAB, my) a"'r, ahead a is a base of the period 1, a, n—1 B n—2 7* or, since AB, AB,, AB,, AB,, &c. are equal in magnitude ddentical in position when considered with respect to each Art. 726, it will follow, that if 4B, can be represented in itude and position with respect to 4B by a x AB, AB, may jually represented in magnitude and position with respect uBpby ax AB,, AB, by ax AB,,...AB,_, by «x AB, 2; gain, if we make AB =r, we get ! AB, =a x AB=ar, Weta h AD eae 7, AB,=ax AB, =<'r Mies =a SAD = aT. | AB <=ax AB,_‘=a"rorr, will follow, therefore, that the successive lines AB,, AB,, AB,_,, AB, which are assumed to be severally repre- iF with respect to each other by mead, 2 xdByax AB,...ax AB,_2; ax AB, 1; |e represented, under the same circumstances, with respect : primitive line 4B or r, by the several terms of the period 27, a@r...a*—'r, a®r, the line represented by the last term ‘on. a1. s ee + Sms of the period (Art. 721), being AB,,, which coincides witli Choice of the base a, which cor- responds to the least angle of transfer. 138 ar, coinciding with the primitive line or 7, in exact confoj with the symbolical condition, which gives a"=1, and and so on, for ever. It appears therefore, that if a, which is an imaginary; 1 . . of (1)*, be multiplied into 7, which represents a given line, the product of « and r or ar, may represent a line of} length, making an angle with the primitive line r, why ~th part of 4, or of some multiple of 4 right angles. Again, if we should make a’ the base of the period in of a, the terms of the new period which arises, or OE, OT 6 Oe TARA A Pa ae . would severally represent the magnitude and position ¢ lines AB,, AB,...AB,_,, AB,, AB, AB, 3 AB, wie line makes, with that which precedes it in the series, an such as B,AB, B,AB,..., which is the double of B,AF which renders it necessary therefore to pass twice roun’ circumference, or through 8 right angles, before we rett! the primitive line AB: and if we should make a’ the bi the period instead of a, the successive lines which its } would represent would be 4AB,, AB,,, AB;,..., the final ! and which we reach for the first time, after passing rou circumference r times, or through 47 right angles: and i} be further observed, that the primitive line 4B or 7, i first of the x radii AB,, AB, ... AB, which, in this ( of circulation, we reach for the second time. 729. We have assumed, in the preceding investigatii to be the appropriate sign of affection, which when mult) into AB or r, shall denote the magnitude and position o} with respect to AB, and therefore «? to be the correspol sign for AB,, a’ for AB,, and a"~' for AB,_,: but inasmuct may express any one of the (n—1) roots of (1)*, whic different from 1 (Art. 708), there is apparently no reaso one of them should enter into the expression which desi the relative position of AB, with respect to 4B correspdl to the least angle of transfer, in preference to any oth’ will be found, however, when we are enabled, in the pr 1359 r enquiries, to assign explicitly the symbolical forms of | (Chap. xxxt.) for specific values of x, that every term in eries will possess a determinate use and interpretation, as e partially seen likewise in the particular examples which : in the absence, however, of such determinations, we issume, as we are at liberty to do, « to express that term » series of roots which corresponds to the least of the ; of position or transfer, or to B,AB, after which assump- he preceding theory will at once assign to all the other of the period their determinate signification. (0. The interpretation, which in the preceding Articles we The pre- applied to the radii of the same circle, which make deter- eee os) : tions are e angles (that is, —th part of 4 right angles, or of some equally n applicable le of 4 right angles) with a primitive radius, are equally Saat ened sable to any lines which are equal to them, and which beens jally make the same angles with the primitive line: for we deterniis . . : nate angles elsewhere shewn (Art. 561), that equal lines which have wel ra jame relative, but different absolute, positions in space, are other. ssed by the same symbol with the same sign, whether —; and the same reasoning, which was applied to those tive signs, will be found to be equally applicable to the , general signs which we are now considering. 31. Again, such interpretations may be indifferently con- A: may »d either with reference to an operation, or to the result of dere atte eration: onsider the multiplicati as) CPt as eration : that is, we may cons the multiplication of a (a pega ees Pld ‘ ration. wh : results of of (1)") into 7, as a specific operation, ose symbolical apevatianal : is ar, and whose geometrical result is the transfer of ne denoted by r (AB) through a determinate angle BAB, : 2 may simply consider it as the result of an operation, or ; simply representing the line AB,, making a determinate BAB, with the primitive line AB. The operation itself is ansfer of the line 4B through the determinate angle BAB,: esult of the operation is the line AB, having a determinate on with respect to AB, considered without reference to manner in which it attained it: it is under this second that we may consider the lines denoted by a"r and by r as jutely identical with each other, insomuch as we suppress all ence to the operation or process of transfer by which one js brought into absolute coincidence with the other. 140 We shall now proceed to apply the preceding theor. some examples, in which the symbolical forms of the root 1 are explicitly determined. | oe 732. The square roots of 1 are 1 and—1: and a=—1 | preceding 704), is the corresponding base of the period —1, (—1 theoryto. 4 4y 1 the geome- , ee : trical inter- In this case, if r represent AB in magnitude and pos pretation of : the square @7” or —7r, will represent AB, roots of 1. which makes an angle equal to { nN two right angles (3 of 4 right an- ‘R ame gles) with AB, a result which Ns indicates, that in passing from the first to the second position, 4B is transferred through 2 } angles (Art. 728): the second or last term a*r of the pi is r or AB; and inasmuch as AB, is transferred through 2} angles in passing from the position AB, to AB, correspor| to the multiplication of ar by a, it coincides in position value, as well as in its symbolical expression, with the prini line AB or 7. It may be remarked that the general theory i cides thus far (and they proceed in common no further) | the interpretations given in Art. 728, and those which fo) Itsapplica- 733. The biquadratic roots of 1, or the values of (1); poner 1, -1, Si and — yeu (Art. 706) : if we make JI (a) i a a 714), the base, the period will be JH1, 1; J“ If we denote AB in magnitude and position by 7, al AB,, AB,, AB,, AB make right angles with each other, then it would appear by the principles of the proposition in Art. 728, that AB, would be re- presented in magnitude and position with respect to AB or r by ar or r Jeg) AR, by o’r or —17, AB, by a®r or > YA, and AB by a*r or r where the primitive line AB is re- produced both symbolically and geometrically *. * If we had made —,/—1 the base, then ar or Ps Pa would he noted AB, or the line AB transferred from AB through 3 right angles direction b, b,, by, b3: also, in that case, a?r or —r, would have denoted .} the line ABz transferred through 3 right angles in the direction b;, ), ) ay or raf — 1 would have denoted AB, , and a‘r would have denoted At 141 vus appears, that if a line, in a given position, be denoted Geome- i i . : Oe hee ee *1] trical inter- qual lines making one or three right angles with it, will Brean cll bolically represented by CaN es and —r rf —T respect- the sign : conclusion of fundamental importance in the theory of N cae erpretation of the results of Algebra when applied to fake ines. ry. re suppose the effect of the multiplication of ,/— 1 into r, g the line AB, to be its transfer from AB to AB, through it angle BAB, in the direction 6b,, the effect of the mul- ion by att = 1, will be its transfer from AB to AB, 1 the three right angles BAB,, B,AB,, B,AB,, in the m6, b,, b,, b,: if we suppose the directions of the move- f transfer to be reversed, the signs corresponding to the agles will be changed from + to — and conversely, the generated being subject to a similar change, Art. 563. The cubic roots of 1, or the values of (1)3 (Art. 705) Applica- tion to the a. Gas cube roots ; Se aN ee I Sh of 1. ‘ 2 2 } ba; el we make pica a) cl the base «, the period will be Mee it /3/-1 —1-J3f/-1 1 2 by] 9 LS de as we draw AB, AB,, AB, dividing the circle into three parts, or making angles | B,AB,, B,AB with each vhich are severally equal | of 4 right angles, or to F hen if r denote AB in % ude and position, t ets), by mote the magnitude and a of AB, with respect to be +d | aie tee) r 142 the magnitude and ae of AB, with respect to Al) esl n,| vga the base, Ligh bee 2 > if we had made — instead of would denote AB, thus passing twice round the circle | we reach AB. Sum and If we produce BA to D, and join B, B,, which bisects] ; difference } of two lines “, we shall find i ee en J3 sidere ly id * with refer- BE = 2° AB ence to position as —/3 well as and EB, = mek: iAB, magnitude, . since EB, and EB, being , |/ considered merely with re- ‘ ference to each other, have opposite signs, one + and the other —: it will follow, therefore, that if EB, be further considered with re- | ference to AB or r, it will be represented both in mag) oa and position by nid r,/—1 (Art. 733), and EB, by — oe r | it will follow, therefore, that | _ 3 FB AB, =— + ea 1=AE+ EB, te We AB, == N38, f= dE EB,, or in other words, that the symbolical sum of the twe AE and EB, of the right-angled triangle AEB, is the} thenuse AB,, the three sides being considered in magnitu; also in position with respect to the primitive line AB: | a similar manner, the symbolical sum of AE and EB,, whe : ° . oe : \ sidered both in magnitude and position with respect to ABs “ For since the angle B, AD is 60° or 4rd of a right angle, the triangl is equilateral, and the perpendicular B,E bisects the base: it follows that AB?= AE?+4 EB,’, and therefore EB,?= AB,?— AE?= AB?— } AB: and therefore EB, = AB. 143 genuse AB, of the right-angled triangle dE B,, where EB, nm in an opposite direction to EB,, and therefore affected a opposite sign*: this is a case of a general proposition, will be established in a subsequent Chapter (xxx1), by it will appear that if @ and 6,/—1 represent two sides d BC (where the position of BC is dto AB) of a right-angled triangle nae Pata their symbolical sum or a+b A ert G é » the hypothenuse AC: and simi- 4 Vel B yf a and —b,/—1 or AB and BC’ ie sides of a right-angled triangle 5 ( their symbolical sum or a—bJ—1 » symbolical difference of AB and BC), will be AC’. ‘ain, it will be found that 4B,=(-5 + wor Ja) (-;- Lor, /=)=-r=AD, 4B, ~ AB,=(-5 43% r/=1) - (-5-% r J=1) = /3r,/—-1=2EB, = B,B,: is. words, the symbolical sum of the two sides 4B, and »f the rhombus 46, DB, is the diagonal AD which they e, and their symbolical difference AB,— AB, is the second g}al B,B, which is at right angles to the former: the same ‘ition will be shewn hereafter to be true with respect to les and diagonals of any rhombus whatsoever. will be found however to be impossible to demonstrate elly these and other important propositions, or to make xtended applications of Symbolical Algebra to Geometry, hit the aid of a knowledge of the theory of angles and Ba cures, and of the various periodical ratios which con- i the science of Trigonometry, or more properly Goniometry, hich we shall proceed to consider at length in the Chapters ‘ immediately follow. he lines EB, and EB,, being drawn in opposite directions, are affected posite signs, one + and the other —: the application of these signs is inde- ¢ of the sign ,/—1, which farther indicates that EB, is perpendicular ‘with reference to which it is estimated. CHAPTER: XXVI. ON THE REPRESENTATION AND MEASURES OF ANGLE The 735. Tue theory of angles, their measures, and the eee odical ratios which determine them, constitutes the scie try, pees Goniometry, whilst the specific application of some of its 1 n ry, ane : , Batliboln to the determination of the sides, angles, and areas of tr gonometry- would be properly termed T'rigonometry, and to reet figures in general Polygonometry: but it has arisen fro associations connected with the progress of our knowlec these sciences, that the least general of these denominatio anticipated and superseded the adoption of the others, an usual to include, under the name of Trigonometry, the s of Goniometry in its most extended applications. Angles in 736. Angles are considered in Geometry, as absolute Saath nitudes only, without any reference to their mode of gener with ee they may possess every magnitude between zero and. two ence to * ° ° a ° e their mag- @gles, which are their limiting values: for lines which ¢ nitudeonly, mg geometrical angle with each other, and which, in conf and not with re- with other views of the generation of angles, make angle: spect t . their mode ach other equal to zero, or two right angles, or any mi Oe eee of 2 right angles, are either parallel to each other, or i same straight line: and such lines are not distinguished each other in Geometry, as being drawn in the same opposite directions*. Angles may 737. As lines, however, may be conceived to be gen be con- . : + : : Brivedio be Dy tue motion of a point (Art. 558), so likewise angles Gpemied be conceived to be generated by the motion or revoluti y the re- volution of : a line * In applying the principles of Symbolical Algebra to the represent round a lines, we have been enabled (Arts. 561 and 562) to form two classes of | Pei: lines (including those which are in the same straight line), according as t. given posi- which form them are drawn in the same or in opposite directions, or acco! tion. they form angles with each other, which are an even or an odd multiple right angles, zero being included amongst the former : this classification of | lines is not recognized in Geometry, where angles cease to exist at the ¢ limits of zero or two right angles. 7 145 2 round a fixed point: when viewed with reference to a mode of generation, and not with respect to their mag- es merely, angles will be found to be not only capable ilefinite increase, but likewise of affections which may be olized by the ordinary signs of algebra. hus, if a radius AP revolve from the primitive position » the position AP, it may ‘id to pass over or gene- Cc he angle BAP, whilst ps em aa oint P passes over or 7 ates the arc BP: if the | P nent be continued from : \ 1 AQ, the are BP will be Hole sed by PQ, and the an- they sub- AP by QAP: and it / Bel dilow from a well-known sition*, that the angles and BAQ will bear to ‘other the same propor- vith the arcs BP and BQ, by which they are subtended > circumference of the same circle. Relation between 8. If the subtending are be the quadrant BC, the cor- Degrees iding angle BAC is a right angle: and if we suppose the pee uant BC to be divided into 90 equal parts, the right angle ve divided by the radii which pass through these points JO equal angles, each of which is called a degree: and :are subtending a degree be divided into 60 equal parts, The sexa- of them will correspond to, or subtend, an angle which ccauiils , i part of a degree, which is called a minute: if the arc |; called a second: and we may proceed similarly to other ir units in the sexagesimal scalet, as far as we choose to Pil them. ‘Muelid, Book vr. Prop. 26. he sexagesimal division of the circle has prevailed since the time of €7or the astronomers who preceded him, (see the article ‘‘ Arithmetic”’ in the ypedia Metropolitana, p. 401), and is so intimately associated with our tif thinking and speaking on such subjects, as well as with our astronomical \ i 108 T instruments, | | 146 Anglesthus 739. It thus appears that the angle generated will in Paeble ot in the same proportion with the corresponding are wh indefinite described or passed over, by the extremity P of the rev mereas’- radius AP: and consequently if one of them admit of inc increase, so likewise will the other: thus, if P advane (Fig. Art. 737), the extremity of the quadrant BC, thea generated is the right angle BAC or 90°: if P move on to P, in the second quadrant of the circle, the correspi angle BAP, is greater than one right angle and less tha bearing to the right angle BAC the same proportion tlt arc BCP, bears to the quadrantal are BC: the revolvinga AP will reach the position 46, after describing two right BAC and CA), and it is in this sense that the lines AB al which are drawn in opposite directions, are said to mak each other an angle equal to two right angles*: if the mo AP be continued, it will reach the position AP, after des} two right angles together with the angle bAP,, or a equal to 180°+5AP.+: when it reaches c, at the extrer the third quadrant, the generating radius has described or} over three right angles, or 270°: if the movement be i instruments, tables, and records, as to make its abandonment somewhiil Advantages Venient and embarrassing : but the superior brevity and uniformity of pres of the cen- computation adapted to the decimal scale is tending rapidly to replace tesimal gesimal by the decimal, or rather by the centesimal, division of the deg) Decca 13° .27’. 42” in the sexagesimal scale is equivalent to 13°.4633 nearl) decimal or to 13 degrees 46 (centesimal) minutes and 33 (centesimals nearly. | TheFrench ‘The French, simultaneously with the establishment of their Systeme ¢ division of décimale, proposed to divide the quadrant into 100 degrees, the centesim| the qua- into 100 minutes, and the centesimal minute into 100 seconds, and soj NE this division was adopted in the Mécanique Céleste of Laplace and «de temporary scientific works. The change however from the nonagesim|| Causes of centesimal degree, was attended with no advantages sufficient to compe) its failure. the great sacrifices of tables and records which its adoption rendered ¥ and its use was speedily abandoned, even in France. If the proposed chi been limited to the centesimal division of the nonagesimal degree, it cou) have failed, when the authority of the great men who proposed it is cal to have been readily and universally adopted. * At this point the geometrical angle contained by AB and the revolv} ceases to exist, inasmuch as the lines which contain it are in the samé§ line: no regard is paid in Geometry to the directions of lines when cis per se. ; + The corresponding geometrical angle, which is formed by AB a BAP, or 360° — (180° + b APz) = 180° — b AP: this is sometimes called th ment of BAP.» to 360°. —_ il 147 ‘ued to P;, in the fourth quadrant, the corresponding angle ited is 270°+cAP;, and when P reaches B, or the point rvhich it started, after describing the entire circumference, ‘wresponding angle generated is 4 right angles, or 360°. ¥ proceed onward for a second revolution, we must add fr 4 right angles to the angles generated at the correspond- Joints i in the first revolution: and similarly for every addi- i revolution of the revolving or generating radius, adding, e (n+1)" revolution, x 360°, or 4% right angles to the wilical or numerical value of the angle in the first. | "0, The angles which are thus generated, and which are Goniome- ae, like other magnitudes, of indefinite increase, may be vee ten wl goniometrical, as distinguished from geometrical, angles: angles. sf the geometrical angle BAP, which AP makes with AB, be eA, the series of goniometrical angles A, 360° + A, 2 x 360°+.A, 10°+ A, ...n x 360+ A, will severally designate the same iim of AP with respect to 4B, implying the additional con- ¢ that this position had been attained in the 1st, 2nd, 3rd, )\.(n+1)" revolution of the generating radius AP. Jmay be observed that goniometrical angles, which are less In what 7 80°, or two right angles, coincide both in magnitude and Bites sy t: position which they express, with the corresponding wrical angles: but that goniometrical angles which are greater 180°, or two right angles, are different in magnitude, but ident, in the position which they express, with the corre- ing geometrical angles, the geometrical angle being formed ¥ or below the line BAb, according as the goniometrical Iicontains an even or an odd multiple of 180°*. iL. We recognize no distinction in Geometry, where the eles gtudes of angles alone are considered, between those which pone) opposite ermed upon different sides of the primitive line AB, or ran “generally between those which are generated by move- tion have ) of revolution in opposite directions: but in Symbolical eas era, where some of the affections of magnitudes, as well i magnitudes themselves, are capable of being considered, “if the goniometrical angle be expressed by 2n x 1809+ A, where A is less 1) right angles, the geometrical angle corresponding is A, and is above the b: if the goniometrical angle be expressed by (2n + 1) 180° + A, the cor- 1 ng geometrical angle is 180° —~ A, and is formed below the line BAb. 148 we shall find that such angles, when expressed symboli will differ in their signs, one series being positive and the { negative, or conversely. For we have shewn in Art. 563, that if an are BP dese| by the movement of a point from B to P, be represented by a or +a, an equal are Bp, P described by a corresponding | movement from B to p, in an opposite direction, would be correctly symbolized by — P| —a: and the same reasoning which is there applied, would equally shew, that if an angle BAP, generated by a radius ? revolving from AB to AP be represented by A or + A, an equal angle BAp generated by the radius revolving i! opposite direction, or from AB to Ap, would be correctly : p sented by — A, and conversely: and generally, whenever g are generated by movements of revolution which are in op) directions, they will be correctly expressed by symbols ? senting their absolute magnitudes, with different signs, and the other —*. } * The process of reasoning applicable to this case is as follows. Let i B represent the magnitudes of the angles BAP and PAQ as used in Arithmetical Algebra, and therefore re A—B will express their difference, which is the angle BAQ: but, in Sym- bolical Algebra, A and B, which are ’ general in form, are general likewise in value, and therefore we may sup- pose B to be greater as well as less 1 ee eee than A. Let us suppose B= A+ C, where C represents the magnitude of the angle BAq or the excess of the angle PAq above BAP: we thus get ? BAq=A-—-B=A-(A+C)=-C: é; or in other words, an angle formed on the opposite side of AB to that on which the angle + A is formed and which is generated by a movement in a)j site direction, is represented by a symbol expressing its absolute magnitu}, with a different sign. 149 2. In considering, therefore, the different modes in which The ni . . : . o4° > eneral @X- erating or revolving radius may reach a given position with areata of t to a primitive line, we must have regard not merely ihe acca t2 quantity, but to the direction of the movement of revo- means of | goniometri- cal angles. it by which we pass from one to the other. hus, if AB be the primitive position of the generating radius q in Art. 741), we may arrive at the position 4P, making _ igle A, less than 180°, with AB, after describing the series | mometrical angles, A, 360°+ A, 720°+ A, ...m x 360°+ A, dceaching AP once, twice, thrice,... (1+) times: but if e verse the movement, generating negative angles, the cor- sjnding series of goniometrical angles will be —360°+ 4, 7/°+ A, —1080°+ A... —n x 360°+ A: and similarly if — 4 be emgle which Ap makes with AB, and therefore on the site side to that on which the angle 4 is formed, then the r sponding series of goniometrical angles formed by direct c:eversed movements will be |) 860°— A, 720°— A, 1080°— A, ... (1+) x 360°— A, fend — 4, — 360°— A, — 720°— A, ... —n x 360°— A: 4, therefore, as regards the position of one line with respect other, which makes with the primitive line an angle + A, hy. A is less than 180°, it will be equally expressed by all ezoniometrical angles which are included in the formula n 360°+ A in one case, and +n x 360°- A in the other. 43. And conversely, we may pass from any goniometrical Transition g: to the corresponding mie , is the proper and determinate measure of the corresponding ane The ratio of the cir- cumference to the radius ex- pressed by Q7r. The mea- sure of 90° is ex- pressed by Tv z° Unity is the mea- sure of an angle of 579.17’ .45", and -0017453 is the mea- sure of an angle of 1°, Angles and their mea- sures are very often designated by the same symbols and the same deno- minations. 152 Thus, the ratio of the whole circumference to the dis, though they are not commensurable, has been approxi, determined by various methods, some of which will b sidered hereafter. This ratio, which very frequently p; itself in analytical formule, is approximately expressed ; number 3.14159, and is also generally represented by the ih a: the ratio of the circumference to the radius, which) , double of the ratio of the circumference to the diameter, is) fore expressed by 27*. The ratio of a quadrant to the radius, which is the ma of a right angle or 90°, is one-fourth part of the ratio — circumference to the radius, and is therefore expressed I If it be required to determine the angle whose measuji or, in other words, the arc whose length is equal to the ¢ it will be found to be 57°.2958 or 57°.17'.45” nearly:f measure of an angle of one degree be required, it will bel to be as an approximi) . ° . . 1 | of a, which differs in excess from its true value by less than 5; th pag diameter: it is this value, which is generally used by workmen, in their es of circular work. Peter Metius, by a similar process, found the remarka: ao which is correct to 5 places of decimals, and which may be very easily bered, by observing that if we write each of the three first odd digits twici cession, as in the number 113355, the three last digits form its numer)r the three first its denominator. Later researches have assigned its value | to 208 places of decimals, a prodigious approximation, which is effected y cesses which are extremely simple and expeditious, and well calculated how much the most complicated calculations may be shortened by a ji selection of formula. See Phil. Trans. 1841, p. 281. + Aight angle is capable of being determined geometrically, and | fore an invariable standard of angular magnitude, to which all other aré referrible numerically, and some also geometrically : if we should call a rig @ unity, and its centesimal subdivisions grades, minutes, seconds, we shou | the French division of the quadrant, or of the right angle, with its scale’ increased one hundred-fold. There is no geometrical mode of determining an absolute unit of a 0 magnitude. ‘Thus a linear unit must be sought for in some assumed end 153 tly and easily into each other, yet in the case of certain tical ratios, which will be considered in the next Chapter, are equally determined by them, it is usual to represent by common symbols, and to call them by common denomi- gs thus, = is considered equally as the representative of ~ t angle, and its measure: and similarly 0 or @ (which are ls very generally employed for such purposes) and any symbols are applied indifferently to designate both angles jr measures. 9. Upon the same principle, if @ is used to denote either pais ie zle or the measure of an angle, C nOune Aaa , angle and is used to denote its comple- Women its comple« \ ment. or the measure of its comple- | es to 90°: or, in other words, if 6 sd B \" the angle or the measure of aie : « p hgle, BAP, then ao is used note the angle as well as the re of the angle, PAC. 0. In speaking of geometrical angles and their comple- pytended we assume them both to be less than 90°: but if 0 ee »s any goniometrical angle whatsoever, or its measure, comple- er greater or less than 90°, we still continue to apply ps T ° me term complement to 5-4, though it may no longer a s its ordinary geometrical meaning. ‘rus if 6=120° or BAg, then 5 9 =— 30", or BAp (Art. therefore, in conformity with the principles of Symbolical ra, that the meaning of the term complement, when no longer the English standard yard (which no longer exists), or by reference to variable standard in nature, such as the length of a pendulum vibrating in vacuo in a given latitude at a given height above the sea, or to the tl f a quadrant, or other definite part of the earth’s meridian, such as the éf France: the same remark applies to the units of time, force, and all r/aysico-mathematical units. n U -_. and if 6=—30°, or BAp, then = 6= 120°, or BAq: it fol- 154 geometrical, is interpreted altogether with reference to th ditions which its symbolical representative is required to 4 Cay : i Neual 751. In a similar manner, if 6 denote the angle Bu) mode of de- ; ‘ oa Oa Boring the its measure, then z—@ is used to denote a | supplement its supplement 180°—9(PA)d), or its mea- ofan angle. fk sure. f In speaking of geometrical angles and _ their supplements, we suppose them both ? | to be less than 180°: but 7-9 continues to be called this plement of 0, when 0 is any goniometrical angle whatsoer Various 752. The angle + 0, or its measure, corresponds to thisa meal geometrical angle with the measure + 2na +0, where n_ Pane ne te whole number: in other words, if we merely regard the pit geometrical of one line with respect to another, we may add to, or sitt ena. from, its measure, any multiple of 27, or of the meas 4 right angles: this conclusion follows immediately fro 4 742. | In a similar manner, = £0 corresponds to the same {oI +4n+4+1 trical angle with +2na7+<—5 “+00 ee ee ee 0: «= responds to the same eel cial angle with +2na +76, 38 (+2n+1)7+0@: and = + @ corresponds to the same geontr : 3 +4n+3 angle with + 2na + = +0 or CAVE 5 LG) | These equivalent measures, corresponding to the san § metrical angle, lead to the most important consequences will be more particularly considered in the subsequent oF : Great im- 753. The propositions in this and the following (4 t f he Aces will be found to constitute the grammar, as it were, of t? Potioo cn guage of Trigonometry, and will be made the foundatic | Trigono- very extensive analytical theory, admitting of the mostval oe. and important applications to every branch of mathematil physical science. It is for this reason that the relations an sl of affection of goniometrica] angles, and their measures, the periodical ratios (Chap. xxv1.) which determine them, i? 155 portant as some of them may at first sight appear, to consequences of very great interest and value*: not therefore be too carefully studied and remembered. whole theory of music is dependent upon the properties and conversions mall number of numerical ratios: yet how vast and complicated is rructure which is raised upon them! It should be the first lesson t, in every branch of science, not to form his own estimate of the of elementary views and propositions, which are very frequently r uninteresting, and such as cannot be thoroughly mastered and re- ‘ed without a great sacrifice of time and labour. a inci in cai . | ——————— SSS a ——— ~ ete ene aS << > el ge Ratios which de- termine, but do not measure, angles. Definition of the sine and cosine of an angle. CHAPTER XXVIII. ON THE THEORY OF THE SINES AND COSINES OF ANGI), 754. Aw angle, such as BAP, is determined by its a which is a (Art. 746): but there are other ratios which equally determine an angle, though they do not measure it, possessing properties of very great importance, which 4 we shall now proceed to consider. If from any point whatsoever P, in one of the lines taining the angle MAP or 0, we draw a perpendicular PM upon the other, we shal] form a right-angled triangle MAP, of which one side PM is oppo- site, and the other adjacent, to the Dp Zz angle @: then the ratio as is called 4 i mnie 4 the sive, and the ratio 7p called the cosine of the PAM or 0: and inasmuch as these ratios remain unaltered ever be the distance of the point P from A*, they are s determine the angle, inasmuch as a definite value of the s cosine determines a definite value of the corresponding angl * For if we take any other point whatsoever p in AP or in AP produc draw pm perpendicular to AM or to AM pro- 1 7 i i duced, we shall find, from a well-known pro- ae ; PM | perty of similar triangles, that ca TP? and iy A AM : hae and therefore the ratios called sine and cosine are always the same for the A + same angle: if therefore the angle be given, they will possess a determinate value. + It was formerly the practice to define the sine and cosine as li not as ratios: thus if we take an arc BP subtend- P ing an angle BAP (@) in a circle whose radius is AB, and if from the extremity P of the are we draw PM a perpendicular upon the radius AB passing through its beginning, then PM was defined 9 to be the sine of the angle BAP, and AM its cosine: it followed from this definition, that the sine 157 55. If we form the right angle MAm, drawing Pm per- The ae of zicular to Am, then the angles PAM ,, fhe eas >; is common to both of them. ‘he same remark applies to the angles BAQ and BAg, vhh are greater than right angles. ‘hese equations, combined with the fundamental equation sin’?6 + cos’*0=1, " pltely determine the sine and cosine of an angle and their elions to each other. 63. The sine of an angle is identical with the sine of its The rela- ujlement (Art. 751), both in magnitude and sign: but the ony nT: ose of an angle is equal to the cosine of its supplement in cosines of ; ? ry) ? angles and naiitude, but differs from it in sign. their sup- ‘or, if the angle BAQ (Fig. in Art. 760), be the supplement Plements. PM Qm f of ip angle BAP, their sines —,; TB and AB are equal in mag- litle, since PM = Qm; and they have the same sign, since PM m(Qm are parallel and estimated in the same direction with re- AM ; to the line BA‘: but the cosine of the angle BAP or AB sual , mu to the cosine of its supplement BAQ, wth i is ae but differs from it in sign, since AM and Am 4 lrawn in opposite directions from A. “he symbolical enunciation of this proposition is - gin (w — 0) = sin 0 Chat); cos (7 — #) =— cos 0 (12). on. IT, x 7 162 The rela- 764. It follows, from the proposition in the last Nic tions of the sines and that cosines of men fs = and sin G + 0) = cos 0 (13), 0. vie . cos ( + 0) =— sin 0 (14). For a 0 is the supplement of 4 — 6, and therefore sin (5 a a) = sin ¢ me 0) =cos0” (Art. "75595 Cos G + 0) = — COs (J _ ) =—sin@ (Art. 755). The same conclusion may be deduced geometrically fro th figure given in Art. 760. | abs hei 765. Again, since the sines and cosines of all goniom ead ¥ angles, which differ only by multiples of 27 or 360°, are iditi¢ sash g, With each other (Art. 761), it follows that sin (7 + 0) =—sin (7 — 0) =— sin 8 (15);59 cos Nate Toh (1 - haat pil (16) 8 =— cos 0. : Hl Oe simi- The same principle will enable us to convert oat ar rela= 5 ° . . . ° ay tide: expressions of a similar kind into others which are equille to them. Thus, sin (= - ) =— cos 0 (17). For sin (“F-0) = sin (0-2 n)=sin- (5 +0) At =— sin G + a) =—cos@. (Art. 764). fe l9 Similarly, we find al Aly Pk | sin (e + a) =— cos 0 (18), db 30 s| cos ar 0) =— sin 0 (19), | cos Gy + 0) = sing (20). 55 i 163 milarly, if the goniometrical angle, whose sine or cosine is 4 ed, exceeds 27 or 560°, it may be immediately reduced «> of the preceding cases by the subtraction or addition of e -eatest multiple of 27, which it contains. Thus sin (27 + 6) = sin 0, | | sin (= +0) =sin (5 +8) = cos 0, sin (3m + @) =sin (7 + 0) =- sin 8, sin (2 + 8) = sin (= ) =—cos 8, cos (27 + 0) = cos 0, | | cos (5 +0) = cos (Z +0) =—sin a, cos (37 + 0) = cos (7 + 0) =— cos 8, | cos (= + a) = cos (= 4 0) = sin 0. 6. The general methods which are adopted for deter- Angles ng and registering in tables the numerical values of the whe and cosines of every angle, expressed in degrees and admit of a simple nu- nes, between 0 and 90°, or in other words, of forming a merical ex- of sines and cosines, will form the subject of a subsequent P's!" er (xx1x): but there are some angles, which are aliquot tof 4 right angle, whose sines and cosines admit of a very ne and immediate numerical expression, by the aid of their o-trical properties, and which it will be found to be very el: for the student to remember: the following are examples. The sine a i € sine and cosine of 45° are equal to each other and to SE ae | or sin 45° = cos (90° — 45°) (Art. 755), =cos 45: and since | sin’ 45° + cos’ 45°= 1 (Art. 758), of 45° ory. 1 et 2sin?45°=1, and therefore sin 45°= v2 = cos 45°: the ~ : ve sign is taken, since 45° is less than — = (Art. 760). he same conclusion follows immediately aa reference to 2 ‘igure in Art. 760. | | } | | 164 The sines . 0 : ake /3 ae and cosines 767. The sine of 60° or the cosine of 30° is “yt ant 0 and : , 30°, orof cosine of 60° or the sine of 30° is 2. 5 and 5 For if, upon the base AB we describe the equilateral tri DAB, the point D will be found upon the cir- cumference of the circle and the angle BAD © will be one third of two right angles or 60°: and if DE be drawn perpendicular to AB, it will bisect it in EF: it follows therefore that cos BAD = cos 60° = a =i =sin 30°: and inasmuch as (Art. 758), cos* 60° + sin? 60° = 1, we get, by replacing cos’ 60° by 1, 4+ sin’ 60°= 1, or sin’ 60°= 3, J3 or sin 60° = “go = Cos 30°, the positive sign being taken, since 60° is less than : : The sine 768. The sine of 18° is Mien » and the cosine of and cosine 4 . BE A9h 1610 412 a5) 10° 4 For if CB be the side of a decagon inscribed in a cir length is the greater of the two portions into which the radius AB is divided, when cut in extreme and mean ratio (Euclid, Book 1v, Prop. 10): it follows therefore that cB oe 1 . AB (Art. 665, Ex. 3), 7 0 and the angle CAB which it subtends is aut or 30. If we draw AD perpendicular to CB, it bisects the! _ 165 and also the chord BC: the angle BAD is therefore 18°, Jie semichord BD = sh nh x AB: and since the sine of BD J/5-1 BAD= iin aka? ee flows that sin yer 1 and cos 18°= ,/(1 — sin’ 18°) = STE) (Art. 758). ‘9. It will readily follow from the three last Articles, and : which precede ae that f) sin 135° = ace 9. Sin 225°=sin 315°=— a ‘ 8, Cos135°=-— aE = cos 225° a Cos 315°= a Be Sin 120°= t. 6. Sin 210° = sin 330°=— 7. Cos120°= cos 210=—N3 ; 8. Cos 330°= we : 9. Sin 150°= "2 : 10. Sin 240° = sin 300°=— oe ’ / 11. Cos120°= cos 240°=- 12. Cos300°=1 18. Sin 1099-N 10+ 2/5) | A | 14 Sin 208°= sin 342° =— ee : 15. Cos108°= cos 252° =— wee | 16. Cos 342°= (10 + 2,/5) ) ae : he numerical values of the sines and cosines of any multiple dsubmultiple of an angle, whose sine and cosine is given, Joe determined by the aid of the following proposition and 7 . | 166 those which follow it, which are of fundamental impor} in the theory of the sines and cosines of angles. Given oe 770. (Proposition.) Given the sines and cosines ofy sines an | cosines of angles, to find the sine and cosine of their sum and difference two angles , Perna the © Let the angles whose sines and cosines are given, be 6 ar ¢ sineand and let it be required to find the sines and cosines of 0 +O cosine of ; their sum 0-06’. eo Let the angle BAC = 6, CAD =’, and therefore BAD = = ) : and let CB be drawn perpendicular to AB, D Ac CD to wD, DE to AB, and CF to DE. Then, from Art. 757, we get AC=ADcos@’ and CD=AD sin @&’: and also BC = AC sin 0 and AB= AC cos 0: and if, in the expressions fer BC and AB, ‘| we replace AC by AD cos 6’, we shall get EB BC= AD sin @ cos #’ and AB= AD cos 6 cos @. Again, CF=CDsinCDF=CDsin0 and DF=CDs since the angle CDF is equal to BAC or @: and if, in the en sions for CF and DF, we replace CD by AD sin 6’, we shalle CF =BE=ADsin0sin@’ and DF = AD cos8@ sin®, | and therefore | DE = BC+ DF = AD sin8™ cos &’+ AD cos 8 sin 6’, and AE=AB- CF=AD cos 0 cos &’— AD sin @ sin &, We thus find sin C2 Oniapate tnt sata (a), cos (8 +0’) = AD = °° 0 cos 0’ — sin @ sin 0’ (b). Again, if we rate BAC=0, CAD=@, “and theri BAD=0-—6'; and if we draw CB per- Cc | pendicular to AB, CD to AC, DE to AB produced, and DF to CB, we shall find, as before, ii SS ae AC=ADcos@’ and CD=ADsin@’: and also BC=ACsin@ and AB= AC cos 0: F B ’ i 167 df, in the expressions for BC and AB, we replace AC by as #, we shall get | BC=ADsin 6 cos 0’ and AB= AD cos 0 cos @’. lso CF = CD cos DCF = CD cos 6 and DF = CD sin 0, since eagle DCF is equal to the angle BAC or @: and if, in the pssions for CF and DF, we replace CD by AD sin 6’, we al get CF = AD cos @ sin & and DF = AD sin @ sin 0’, Shetefore | DE=BC-CF=ADsin 60 cos &’— AD cos 9 sin 0”, i AE=AB+ DF= AD cos 0 cos 0’+ AD sin 8 sin 6’. e hus find . ep Tt, , Rs ie (0-6’) = AD 7 iP 8 cos 6’ — cos @ sin 6 (c), d Ne AE a. , : : ’ Men) = 4p = O89 0080 + sin 8 sin 6 (d). 1. Of the four formule (a), (b), (c) and (d), which are Three of i in the preceding Article, the three last may be easily eae the ‘d from the first (a) or from last Article : may be de- | sin (0 + 0’) = sin @ cos 0’ + cos @ sin 6’* (a), thease > aid of propositions previously established in this Chapter, Hiled we assume this formula to be true for all values of 0 i / *\Che investigation of this formula, which is given in Art. 770, is incom- anless it is made to comprehend all values of 0, 6’ and @ + 6’, which are the limits of geometrical angles, to which the definition of sine and cosine m: for it is only when definitions cease to be applicable, that ‘the principle armanence of equivalent forms” comes into operation: this might be effected ically, though not without some difficulty, by a proper adjustment of the ™to all the different cases which this proposition comprehends; a very hi xamination, however, of the forms which the two members of the equa- |) assume for any values whatsoever of @ and 6’, will shew that the a/1 is correct in all cases. I's, if we replace 6 by 3+ 8, and assume the formula (a) to be correet case, we get } Tv ° | vt (F +040") = sin (5+ a) cos 6 + cos (F +0) sin 0 ; | and 168 Thus, if in (a) we replace 0’ by — 0, we shall find sin — 0’=— sin @’ and cos— @’=cos @, and replacing sin é £04 a’) by cos (0 + 6’), sin (5 + 0) by cos @ and cos ts F 0) by — sin, (Art. 764), we obtain cos (8 + 6’) = cos @ cos 6’ — sin 6 sin 0’ (b), an equation which has been shewn to be correct when 0, 0’ and @ | | less than = 5 : the formula (a) assumed, therefore, is correct in this casts it leads to a correct ee Again, if we replace 0 by > st 6, 0’ by 5+ 0’, and assume the forma to be correct for this case nn we shall get : ere ba Eid es | sin (7 + 64 0’) =sin € +6] cos (5 +0') + 0s (5 +0) cos (5+ and replacing sin (a + 6 + 6’) by — sin (0 + 6’) (Art. 765), sin (F fe e) 4) cos (x + 8’) by —sin 6’, cos (F + a) by —sin @, and sin ‘6 + e’) bro (Art. 764), we obtain — sin (0 + 6’) =— cos @ sin 0’— sin @ cos 6’ : | or changing all the signs, sin (0 + 6’) = sin 8 cos 6’ + cos 0 sin 8’ (a), a correct equation. If, in the same formula (a) we further replace 0 by 7+ ang neo 6’, and assume its correctness for this case also, we shall get a tS +04 a’) Bein rer 8) Con & + a’) f cos (ayaa 64 ais rf if we now replace sin (F +60+ a’) by — cos (0+ 0’) (Art. 765), sin’ by —sin@, cos(7+@) by —cos@, sin (= + a’) by cos 6’, and cos by —sin 6’, we shall obtain ; — cos (0 + 6’) = sin @ sin 6’ — cos 0 cos 0’, or, changing all the signs, cos (0 + 0’) = cos 6 cos 0’ — sin 0 sin 6’ (b), a correct equation. ; It is obvious that we may apply the same process to the verificatio fundamental formula sin (@ + 6’) =sin @ cos 0’ + cos @ sin 6’, for all vali | 0’, and 6+ 0’ Marae including those in which @ or 6’ become ‘@ = or to any multiple of = z? which would require a distinct geometriciil tigation. 169 derefore | sin (0 — 0’) = sin 0 cos & — sin 6” cos 0 (db). gain, if in (6) we replace @ by 57% we get | sin (= - @— ®’) = sin (Z- 8) cos 6’ — cos (@- a) sin 6’: : if we further replace sin ( 9 @’) by cos (0+ 0’), sin ( - a) by cos@ and cos (Z _ a) by sin 9, | fe aall get | _ cos (0 + 0’) = cos 6 cos &’ — sin @ sin 6” (c). _ 3 in this formula (c) we replace 6” by — 0’, we get : | cos (0 — 6”) = cos @ cos 6’ + sin @ sin 8” (d), be ‘ving that cos—6’ is replaced by cos’, and sin- 6 by 3 0. 72. Inasmuch as Other deri- : : vative for- : sin (6 + 6’) = sin 8 cos 6’ + cos @ sin & (a), mule. : sin (0 + 6’) = sin 6 cos 6’— cos 8 sin & (b), re eadily obtain by the addition of (a) to (6), sin (6 + 6’) + sin (0 — 6’) = 2sin 8 cos (e), n by the subtraction of (+) from (a), sin (0 + 6’) — sin (0 — 6’) = 2 cos 8 sin 0 (a): And again, since | cos (8 + 6’) = cos 8 cos 6’ — sin @ sin 0 (a); cos (8 — 6’) = cos 0 cos & + sin @ sin & (d), eind, by the addition of (d) t : ie : ‘i ( ) vs (c), Formule cos (0 + 6’) + cos (0— 6’) =2 cos A cos’ (g), for express- ‘ ing the sum mi by the subtraction of (d) from (c), or oe rs . . P ence oi the cos (0 + 6’) — cos (6 — 6’) =—2 sin @ sin 6 (h). Sane chee | sines of two ‘ P angles in 73. Inasmuch as it may be readily shewn that varia ee the | product of g-(*2° 4 0-0 eld the sines or P) 9 ; cosines of } tthe yar their aa ie sum an a a —) — 2) =s-—d, semi-dif- 2 2 ference. 170 we get sin @ + sin 6’ = sin (s + d) + sin (s — d) = 2 sins cos d, sin 6 — sin 6’ = sin (s + d) — sin (s —d) =2 cos s sind, cos 0 + cos 6’ = cos (s + d) + cos (s— d) =2 cos cos d, cos 8 —- cos 6’ = cos (s + d) ~— cos (s—d) =—2 sins sind: and if we replace s by Lee and d by ss we find ) cos (“") (2), “4 I. 0 Ma Q’ cos 0+ cos 0’ = 2 cos (“ aa: cos ( ) (2), é ; ‘ 0 sin 6 + sin 6’ = 2 sin (= sin 6 — sin 0’ = 2 cos( 2 2 6+0\ . 0-0 3 ) sin ( 5 ) (m), and also of the cosines of two angles, in terms of the produc ¢ the sines and cosines of half their sum and half their differ cos 0~ cos =~ 2 sin ( equivalent to them, and more particularly in their adaptio to logarithmic computation. 774. If we resume the formule sin (0 + 6’) + sin (8 — 6’) = 2 sin @ cos 6’ (e), | cos (6 + 0’) + cos (8 — 6’) =2 cos 0 cos 0 (zg), | and if we transpose sin (9 — 6’) in one case, and cos (0 — 6’) ir other, from the left-hand side of the equation to the right, will assume the form sin (0 + 6’) = 2 sin @ cos 6’ — sin (0 — 6”) (n), cos (9 + 8”) = 2 cos 6 cos 6’ — cos (8 — 6’) (0). } If we further replace 6 by (n— 1) and 0 by @, were 0+0’=ng, and 0-6’ =(n- 2), and the formule (n) ando) become sin np = 2 cos @ sin (n — 1) p—sin(n—2) (p), Cosnp = 2 Cos p cos (n—1) —cos(n—2) (q). 171 hese formule are useful, as expressing the sines and cosines iltiples of an angle in terms of sines and cosines of inferior uples of the same angle: they are very frequently referred 5. If in the formule given in the last Article, we replace To express’ the sine k 2, we get and sone sin 2 =2 sin ¢ cos ¢ — sinO palaces ; terms of the =2 sin ¢ Cos d: sine or co- sine of the rno=0: angle. / cos 2 = 2 cos ¢ cos d — cos 0 =2cos*?—-1=1-2sin’¢: posO=1. follows, therefore, that if the sine and cosine of an angle yen, we can find by these formule the sine of double the g: and also, if either the sine or cosine of an angle be given, :m find the cosine of double the angle. hus, the sine of 30° is 3, and its cosine ee therefore ! sin 60° = 2 sin 30 cos 30 . nano f8 : : — I ; : 10+2 he sine of 18° is se , and its cosine RO there- r | sin 36°= 2 sin 18° cos 18° _N5-1 J10+2/5 J5—J5 aE: 4 TIEN” milarly, from the second formula, we get cos 36°= 2 cos? 18°-. 1 = 1 - 2 sin? 18° | 4 it +1 | 4 F Given the sine or CO= | sine of an (6. The same formule will likewise enable us to express angle, to find the sine . ine or cosine of half an angle in terms of the sine or and esana 52 of the angle itself. of half the angle. My ie. 172 Thus, let it be required to express the values of sin Ot; s cos 5 in terms of sin @. | Inasmuch as ernie kin iter a 2 ea oe and ee 9 1=sin’ > + cos’ 5 (Art. 758) we get, by adding 3 (a8 n4+0 0 0 1 + sin 8 = sin’= +2 sin 5 cos > + Cos’ 5; and, by subtracting : eu ee ) 1—sin 0 = sin’ 5 —2 sin 5 cos 5 + Cos" 5 - 200 0 If sin 5 be greater than COS 5 the terms of the co squares on the right-hand side of these equations are arr in the order of their magnitude (Arts. 587 and 650), ar) extraction of their square roots will give us | JQ + sin 6) = sin 5 ~ cos 5 : mee 6 /(1 — sin 8) = sin 5 — cos 5 ; and therefore, by adding and subtracting, we get sin 5 =} J(1+sin €) +4 Y(1-sin 0) (1), | cos 2 =4,/(1+sin6)-—4,/(1 —sin 0) (2); ah 0 | But, if sin 5 be less than cos ay. the order of the ters the squares Rel ar LU 6 6 ; : i] | sin’ 5 +2 sin 5 cos5 + Cos’ 5 » and sin*S —2 sin 5 cos 5 +60 must be reversed: the extraction of their roots, when form is thus modified, will in that case give us 173 Ce eid J/( + sin 0) = cos 5 + sin 5 : Oi. eau J(1 — sin 8) = cos 5 — sin 5; ; : : : ; erefore cos =1,/(1+sin 0) +4,/(1 — sin 8) (3); sin S=h/(1+sin0)—}Y(1-sind) (4), ) be between the limits 0 and 3 = and 27, we must ts expressions (3) and (4): but if 6 be between the limits aw we must use the expressions (1) and (2): it should kit in mind, however, that for greater values of 8, which a & correspond to a negative geometrical angle, these ex- ns must change their’ sign. (See Art. 781). ua ; ; 1 Gis. un, let it be required to express sin 5 and cos = in terms } ip S ce 3 ~w| & | 0 , | cos 9 = 2 cos'>-1=1~2 sin’ nf ridily obtain the equations 6 2 cos* = = | + cos 8, ~ ee 2 sin’ > = 1 — cos 8, > ~ erefore | 0 1 + cos @ i cos = ,/( - ) (5), i m0 1— cos 0 | sin 5 = Wh C= ) (6). , { be between the limits z and 37, 5 and 77, 97 and 117, it I, found that cos = 7, ( — -): and if @ be between Examples. To deter- mine a series of equisinal angles. 174 | pike | the limits 27 and 47, 6a and 872, 107 and 127, thers Ve) 777. The following are examples. (1) Given the sine of 30°, which is 3 (Art. 767), tif the sine and cosine of 15°. sin 15°= 3,/(1+ sin 30°) -23,/(1 - sin 30°) cos 15°= 3,/(1 + sin 30°) + 3,/(1 — sin 30°) 341 =3/3+3/3= ee = .96609. (2) Given the cosine of 18°= nh e isl °) = .897, to filt sine and cosine of 9°. sin oo= JS) = J(.0515) = .2269, 1 ;18° cos =, /(A*SI) - M978) =.9729. It is obvious that if the sine or cosine of any angle @ be ¥ we can apply the same method to find the sine and cosine |; whatever be the value of x. 778. To find the series of values of 6 which have ai sine*. | In the first place, all angles are equisinal or have thea sine, whose measures differ from each other by multiples (Art. 761). : In the second place, if @ be an angle whose sine is | required magnitude, its supplement 7-6 is also equisina} it. (Art. 763). * Or, in other words, to find all the values of @, which satisfy the sin 6 = a, where a is any positive or negative number less than 1. a 175 -jwill follow, therefore, that all the angles in the two series 0, Qn+ 8, 4nr+06,... Qna+,... ip ah 3m — 0, 5a—-0,... (2x +1) 7-9, at ued both ways, are equisinal with @: or in other words | sin 0 = sin (2mm + 0) = sin {(2n + 1) — 6} ein is zero or any whole number, whether positive or nega- e. i). To find the series of values of @ which have a given To deter- . mine a — a series of I the first place, all angles are equicosinal or have the same edicosinal : : angles. it whose measures differ from each other by multiples of .| Art. 761). I the second place, if @ be an angle whose cosine is of the ued magnitude, then —@ is equicosinal with it. (Art. 762). Iwill follow, therefore, that all the angles in the two series 6, Q7 +8, 4nr+6,... Qnr+0,... — 0, Qa — 8, 4n—O,... Qn7r—O,... tiaed both ways, are equicosinal with @: or, in other words, cos 4 = cos (27 + 8), el is Zero or any positive or negative whole number. '. The propositions in the two last Articles deserve the Great ue areful consideration of the student, as furnishing the eee f the explanation of the multiple values not merely of of the sines . ‘ ; : Y 4 and cosines i¢al and equicosinal angles, but likewise of the periods of equisinal n, by the sines and cosines of their submultiples, which pean it 1 '2 found to be intimately connected with some of the ie ang . e . oO eirsubd- tnportant theories in analysis. multiples. 1s, if the angle be given, there is only one value of its id cosine: there is also only one value of the sine or cosine given multiple or submultiple of it, which form likewise nae and determinate angle. liowever the sine or cosine of an angle be given, and the ef the angle be required to be determined from the value Sine or cosine, then not only may the angle corresponding _ one or more of the series of equisinal or equicosinal The periods of the sines of the sub- multiples o equisinal angles. . . . 1 6: but the sines of a series of angles, which are — part of f = m 176 angles, but the sines and cosines of their submultiples y periodic. | 781. Thus the angles in the series 27+ are equisina th in the series 2n7+0, form periodical series, the sines of tl m terms being different from each other, and afterwan curring periodically in the same order. Thus, the equisinal series being 6, 2r+0, 47 +8,... 2(m-1)7+9, Q2m7r+8,.. the series of their submultiples by m, will be @ Qn+0 40+6 2 (m—1) +0 ? 5) m m m m and the sines of the m first terms are different from each : Oo Paubils as but the (m+1)" term is 27 +— which is equisinal with a which is equisinal with © : 2 (m+2)" term is 27+ and so on successively: it follows therefore that the se sines of the series of submultiple angles is periodic, each consisting of m terms. Again, if we take the second series of equisinal ang] cluded in the formula (2n+1) 7-0, which is a—0, 37-0, 5x-O,... (2m - 1) 7-89, (2m+1) 2- the corresponding series of their submultiples by m will a —-0 Sr —80 5ar—0 (2m—1) 7-0 T - > 3’ axe 9 9n+— m m m m n and the sines of the m first terms are different from each that of the (m+1)" term being equisinal with the first, (m+ 2) with the second, and so on for ever. The sines fore of the second series of submultiple angles or measi angles is also periodic, each period consisting of m term It follows therefore that there are, in the two perio submultiple angles, less than 360°, whose sines may be d from each other, corresponding to the same sine, or tot of any term of the two series of equisinal angles: bi number, when m is odd, may be reduced one half, by the i ‘ J 177 -sines of the terms of one period with those of the other, arms reckoned from the beginning of one period being lly the supplements of the terms of the other reckoned the end, and therefore the second period may be altogether at of consideration: and in the case in which m is even, ‘st half of the terms of each period will bear to those of scond the relation of ¢ and 7+, whose sines are equal, ‘ith different signs. th ‘2. If the cosines of the angles which are the a part The periods m of the co- ‘e equicosinal angles included in the formula 2n7+0 che ce 779) be required, we form the double series ples of equicosinal 8, Qr=+O, 4re0,... 2(m—1)x+O, 2mr+,... angles, a. < ie cosines of the first m terms of the series of submultiples 2m 0 4a 8 2(m~-1)7 , 8 al » mo > m: im? * m m? ae m” ; js: but whether m be odd or even, we find cos =Ccos—— , Hes oe UE SUE Be - |, cos (++ -) =COos pen a mM m m m m m be different from each other, comprising 2m different thus reducing the number of different values of the cosine 2m to m: and further when m is even, the first > terms the beginning of the period are the supplements of the terms reckoned from the (3 + 1)" term to the end, and ‘several cosines therefore differ from each other in their only. 3. The following are examples: Examples. ) Given sin 0, to find sins. le series of equisinal angles are OG; Qi + 8, 4a + 0, ., se nx—0,3r-0, 5c—0,...... fi Li. Z. / will express these four values, two of them being equ 178 The corresponding series of semi-angles are 6 0 2? Tits 20 +55 oat eee an 8 3x 0 5r 0 6 Tig Ritg TE te gil g AAS . of which sin d =— sin (= “a >) and ~~ sin (3 5) i (=. s) = 608 5 CY ae eae CCH ee The expression, given in Art. 776, if we assign the db sign to the square roots, or , 1+ sin 0 fi 1—sin a2 ere x} oe} ein’ and the others to + sin ¢ 5) or mpi 26 22 25 (2) Given cos 6, to find cos : ; The double series of equicosinal angles is =O" 2Qor' 0, Aare Oe ee The corresponding series of semi-angles is - The period of cosines is Saha cos ( en | Q” oe 5) ; 0 6 6 6 of which cos ater cos — 5 and cos (r+ 5) =- 6085, 1 The expression, given in Art. 776, 0 ‘ 1 + cos 6 —-=+ See cos 5 ie Vi ( : ) will express both these values, if we assign to it its double sil (8}) Given ein sound sin 5. | —— le : eee ahr oO . [2a Cic equation, whose roots are sin =, sin (G s 3 , and —sin ary ard 179 ‘he two series of equisinal angles are 0, 2r+0, 4r+8, Ow+9,...... w—O0, 37-0, 57-0, Tr—8,...... ‘he corresponding series of ternary submultiples of these ng’s are ' 6 Qn 0 Aa ‘ zi | 3? 3 3? 3 t= 5 beh rs erate s | Metin Ser, 0 § OF. .6 | re 3 o? eT 3 3° 6 esis es ; ( 5) os 5) sin{-—-—), i ee tle ag & 3 3 8 | 0 Br Pla eG an? @ ‘i —j lement of —-— —, -—-+-=of =--= u since 3 is the supplement o Rok ais CP Ge 3° and n 0 Shor : a> — of Sd - (after rejecting 2), it follows that the values vo 3 fie sines of the first period are identical with those of the cad: and there are consequently only three such values, th are different from each other *. 4) Given cos @, to find cos 2 ; ‘he double series of equicosinal angles is £0, 2r+0, 47+0, O69 +0, 8r+8....... “he corresponding double series of quaternary submulti- le of the angles of this series is 0 a 9 iG 3 O ; ro “i. ESR IAIA OT A amet data | It appears by the formula in Art. 774, that | | 030 4) ote a ae o me sin 8 = 2cos 5 sin sin 5 . ri tet « ips O , ey BP F. Bo, a | = 4cos* = sin 5 — sin 3 (replacing sin by 2 cos] sin 3) mews. G0 =4(1—sin’ 5) Ba hs 10 tes =3sin | — 4sin 3? 0 3 a} @ is the least angle whose sine is of the given magnitude, Caution re- specting the appli- cation of the princi- ple of equi- valent forms. 180 The double period of cosines is , als a 9 AG 3m 9 4 cos PER bate: OH) Paras A Py eat We 4)? which are reducible to 0 ar 6 r 6 6 . ia COR A? Cos m4, » COS a4 » COS a 3 | which alone are different from each other, and of whicl | fourth and third only differ from the first and second in j¢ sign *, | 784. It should be observed, that the definitions of thet f and cosine apply to all geometrical angles, and therefo ( those which are negative as well as positive, extending hy beyond the limits of Arithmetical Algebra: and whateyi bi the propositions, which come within the range of these lef nitions, they must be established by means of them, and n by the generalizations of Symbolical Algebra: and though it | ) the aid of Symbolical Algebra alone that we are enabli 4 form and express negative lines, negative arcs, and_neitiy angles, yet as soon as we have given to them a consistent te pretation, they come within the proper province of Geontr and are subject to the same definitions and propositions, as (jp to the geometrical quantities which they represent in afin as well as in magnitude: it is for this reason that in deor strating the formule of Goniometry, we must conside'tli negative values of the quantities involved as equally poibl with those which are positive, and as equally within the ng principle of the “permanence of equivalent forms,” but byrei sonings founded upon the definitions of the sine and cin and the geometrical or other properties of the magnitud). which they apply: it is only when the definitions, whic at the foundation of our reasonings cease to be applicable,thi we are at liberty to resort to the generalization of Symbic Algebra. | ( . * The-corresponding biquadratic equation is cos 6 = & cos4 ‘ — 8 cos® ' | CHAPTER XXVIII. | INCHE TANGENTS, COTANGENTS, SECANTS, COSECANTS, AND VERSED SINES OF ANGLES. 7). THE ratio of the sine to the cosine of an angle is called Definition of tangent ce and the reciprocal ratio of the cosine to the sine and See angle is called its cotangent* gent. sin 0. Zs if 0 be the angle, then the tangent of @ or tan 0 = ark : cos 0 ee cotangent of 0 or cot 0 = —- sin U ! . | 7. The tangent and cotangent of an angle determine the They de- g] equally with the sine and cosine: nate the BAC -be any angle, and if any SS n whatsoever P be taken in AC, A LY PM be drawn perpendicular to pe a the ratio wai is the tangent, Lhe reciprocal ratio ae is the 7 ns PM went of the angle BAC: and if this ratio be given, it eines the species of the triangle PAM, and therefore the lat A, and conversely. Euclid, Book v1, Prop. 5. } /. The following properties of the tangent and cotangent Properties a . . > 73 “1s of the tan- ‘vable immediately from their definitions. seatand 1 cotangent. cot 0 f } ' he tangent and cotangent were formerly defined as lines, and not as s hus, if BP be the are of a circle sub- it the angle 0 at the centre, and if BT of Tee Coe drawn from B and from the extremity Bs " quadrant, tangents to the circle meet- Ea A produced in T' and t respectively, then ier w called the tangent, and Ct the cotangent i¢igle BAP or 0, ie fe MB 182 s@ and cot 6= -, and the first ratio i} sin ae . sin 9 For tan @ = cos 0 reciprocal of the second,’ and conversely. (2) Tan—0=—tan 6, and cot —@=— cot 0. @ —sm?é FOr tan —@= Lb neh et SF @, and cos—@ cosd =a | 0 bot ie a =-— cot 0. sin — 0 8G 3) Tan(— —6)=cot 0, and conversely. 2 y Tr . (? 0) te (= cos @ For tan (z-6) - een tae ee COREE (4) Tan (1 —@)=— tan @ and cot (7 — 6) =— cot 6. APTN U pees hg sin@ For tan (x > 0) = cos (7—0) pee a — tan ps and cos (7 — @) _— cos 8 ot Cast) sin (7 — 6) sin @ =— cot @. By a similar process it may be shewn that ! (5) Tan (g + 0) =~—cot @ and cot @ + 0) =— tan 0. (6) Tan (= ~ 0) = cot 6 and cot (= = 0) = tan @. (7) Tan (> = 8) =—cot9 and cot (a He 8) =~—tané.| (8) Tan(r+0)=tan0 and cot (a + @) = cot 0. Inasmuch as the sines and cosines of all angles which) by multiples of 27 or 360°, are identical, the same obsery, as is obvious from their definitions, will extend to their tal and cotangents: we thus get . | (9) Tan 0=tan(2n7 + @) =tan (a + 6) = tan (22+ 1): and cot 6 = cot (2nm + 0) = cot (m + 9) = cot {(2n + 1) r +01, where 2 is any whole number, whether positive or negativ! ‘ ie 183 : : 1 / 70) = d 20) 2-5 ) ,/(1 + tan’ 6) and ,/(1 + cot? @) ais | | E sin’ ae cos? @ + sin? @ 1 6=1+—| = —_,,_ —__ => — 3 _ {me 1 + tan PRY: cos? oe and there t + tan’ 0) Seeea and again, ; ant cos’ _ sin’ @ @+ as 0 1 eat Pent Sep sin? 0 sin® 0 ~ sin? 0? 1 | One. daerefore ,/(1 + cot? 0) = ey peiprocally, we find cos 0 = _ ee and sin @= sank : ~ ./(1 + tan? 0) ~ /(1 + tan? 0) ° 9 6 tO rises (1) Band +co sin 20 in 6 0 sin? g 1 ee Srey core nce ree eT | cos@ sin @ cos @ sin @ 4sin 20 - | 2 | . ~ sin 20 (2) Tan 0 —~cot d= 2 cot 20. sin@ cos@ sin® g— cos efi) cos 260 o an 6—~cot 6 = = f é. ls sal cos@ sin@ cos@sind 4 sin 2 a Sots 8. The tangent and cotangent may have every possible Changes of between zero and infinity, whether positive or negative. ind for values of @ between 0 and = sign and value of the . tangent and sin cotangent. the tangent increases ihe increase of @: for values of 0 which are greater than d less than 7, the value of the tangent is negative, and. i shes from infinity to zero, which it reaches when @= 7; ' t will be seen in Art. 793, that _— is called the: secant of 8, and : scant of 6, | He following form, tan @ + cot @ = 2cosec 20, ) ] sin 86> 2 { teevy = 2 cosec 26 (Art. 793), and the proposition in the text may be put 184 for values of @ which are greater than a and less tha ; the tangent is posilive, and increases from zero to infinit) ; for values of @ between = and 2a, the tangent is 4 2 4 and decreases from infinity to zero. The cotangent being the reciprocal of the tangent, itiig are the same for the same angle, but its values follow a rex f ) ors It may be observed that the tangent and cotangent : words, at the points where @ is 0, or 5 this subject, which is not without importance, will be | considered in a subsequent Chapter*. Given the 789. Proposition. Given the tangents of two angs, tangents of : : two angles find the tangent of their sum and difference. to find the Let 0 and 6’ be the angles, whose tangents are given, id te tof ; theirsum it be required to express the tangent of 0+ 0. and differ- ber Tan (0 + 0’) = cos (0+ 0’) cos 0 cos 0+ sin@ sin ” (dividing the numerator and denominator by cos 6 cos 6’), | sin @ | sin 0" 6’ . a cosd cos cos 0’ _ tan 6+ tan 6” sin Qo sin 0’ 1 tan @ tan 0’” - / cos ~ cos 6 In a similar manner it may be shewn that cot 0 cot / +1 Diickh),) > ae 5 ee pote ) cot ’= cot é ~ Shed 790. If, in the expression for tan (0 + 6’) we replace — tann@in and 0 by (n—1) 4, we shall get terms of | tan 0 and AEN Le tan (x —1)0+ tan 0 4 tan (n—1)0. : ~ 1—tan 6 tan(m—1)0° | * The signs of geometrical angles change in passing through zero 1 t right angles: the signs of the sines and cosines change in passing thre only. é order, one being infinite when the other is zero, and one ite ° . . . e@ : | ing when the other is diminishing. i! ‘a their signs when they pass through infinity and zero, or, mt ~,ora multiple f | sin (0+ 6’) _ sin @ cos 6’ cos 0 sin 6” | | = ' Ls rt) 185 9]. If n=2, the formula in the last Article gives us mye! sion tor 2 tan 0 tan 20 in tan 20= rence Gk terms of 1 — tan tan 6, and ) . . . . its con- f we solve this equation with respect to tan 0, we get a ae tan 6=- cot 2@=+,/(1 + cot’ 26). f the value of 20 is required to be determined from the a2of tan20, then the angle may be any term in either of two series, (Art. 787, No. 9), 20, Sr 420, 40+90,.....- ar+20, 37+20, 574+20,...... ‘he corresponding series of semiangles are Oa Oe 2at0, BO, bac 2 Par ant s a 2 > rte +8, the tangents of all the angles in the first series are identical wi. tan @, and in the second series with —cot@: and it will be ound that tan 6 =— cot 20 + ,/(1 + cot? 20)* | — cot @=— cot 26—,/(1 + cot? 20). pe The following formule are not unfrequently useful, mare very easily proved: in (6 + 6’) 9 ,_ sin ( (1) tan 0+ tané Seah cow! Me ,__ sin (0— 6’) (2) tand—tan&’ = ia ah | Since ./(1 + cot? 26@) = cosec 26 (Art. 795), it follows that ) tan 8 = cosec 20 — cot 20, | cot 6 = cosec 26 + cot 20. in the formula in Art. 790, we make n= 3, we get Bet tan 20 + tan 0 _ 3tané — tan? 0 | 1 — tan 0 tan 20 1 — 3 tan? . e] cing tan20 by cant “ | he three roots of this equation are found by a process similar to that given in . ext, and are tan 0, tan(60°+ 0), tan(120° + 0), or ) ve fant tan 6 — 4/3 ; —/3tan0’ 14/3 tand” or in 60°= 4/3 and tan ait = A/a. 7 on IT. AA | ; tan 0, Defimtion of the se- cant and cosecant. Their changes of sign. , cos (6 — 6’) (3) tan0+ cot @’= prey ener (4) tan @—cot @’= cca il| 2 ") ; cos @ sin @ tan@+tan 6’ sin (0 + 6’) tan @— tan U’ sin 6 + sin 6’ cos 6+ cos 0’ ~ 186 ~ sin (0-0’)° G2 n 2 sin 6 —sin 6’ cos0 +cos 6’ on 38). sin 0 +sin 0’ _ ae an (“S) -) sind—sin&’ | /0—0\" tan e ) (8) 793. The reciprocal of the cosine of an angle is calle it secant, and the reciprocal of the sine of an angle is calle it cosecant. If @ be an angle, then the secant of 6 or sec 0 = —i S 1 the cosecant of @ or cosec 6 = ——. sin 0 If BAC (Fig. in Art. 786) be an angle, and if from point P in one of the lines containing it we draw a peel dicular PM upon the other, then the ratio & is the sia is its cosecant*. ge PL of BAC or 0, and the ratio PM being equally convenient to use the actual reciprocals of thé and cosine in most of the expressions in which they would ce wise occur. | of the cosine and sine respectively. Likewise the secant of it * The secant and cosecant were formerly defined as lines, and not as i0 thus, if BP (Figure in Note, Art. 785), be the are of a circle subtendi angle 6 at the centre, and if BT and Ct be respectively drawn from B and fr extremity C, of the quadrant BPC, tangents to the circle, then AT was ile the secant, and At the cosecant of the angle BAP or 0. i s = 0 Y i ‘ : ° | 187 es from 1 when 6@ = 0, to infinity when 6 = 5 : it then changes ign and decreases from infinity to —1 when 0=7: from T ee : : 3 tod= - it increases from — 1 to infinity: and from @ = oe =2n, it decreases from infinity to 1: the cosecant of 6 is ty when 0=0, and 1 when 6 = =: it becomes infinity again ee Sais we 10=7, where its sign changes: when 0 = 3» ,1¢ is —1, and | 9=2rm, it is infinite and negative. 9. The relations of the tangent and cotangent with the t and cosecant are expressed by the equations (Art. 787, 10), J/(1 + tan? @) = sec 8, J/(1 + cot? 6) = cosec 0. ‘6. The versed sine of an angle is the abbreviated expres- 0] The rela- tions of the tangent and cotangent with the secant and cosecant. The versed < Ace . sine of an ‘or 1—cos@: it is written vers 6. ’ BAP be an angle, and if PM be drawn perpendicular 13 (Fig. in Note, Art. 7 85), then the ratio ae is the versed f the angle BAP*. never changes its sign, and the limits of its values are 0 hl} 8=0, and 2 when @=7: it increases whilst 0 increases » g } F % ree 0 to x, and it diminishes whilst 0 increases from o to 2r. de use of this term is now almost entirely abandoned. “The versed sine was formerly defined as the line BM, intercepted wa the beginning B of the arc and the perpendicular PM drawn through ‘emity: it was sometimes called the sagitta of an arc, for if the arc PB ubled, and PM produced to make a complete chord, then BM would be ition of the arrow upon the stretched bow: it is chiefly with a view to make ‘ings of the older mathematicians intelligible, that it is expedient to refer to nnd definitions which have now fallen into disuse. angle. CHAPTER XXIX. ON THE CONSTRUCTION OF A CANON OF SINES AND COSINI TANGENTS AND COTANGENTS, SECANTS AND COSECANTS, What is 797. Wes have hitherto considered sines and cosines, tang Peta or and cotangents as possessing determinate values for determi sines,co- angles, without attempting to assign them, except in the Scot angles of 45°, 30° and 18°, and their successive binary 1 tiples and mabintcrples (Arts. 766, 709, 777): in the pr Chapter we shall proceed to shew in what manner their n rical values may be determined generally for every minute degree of the quadrant, with a view to the construction Table or Canon, in which those successive values may be 1 tered: for in the ordinary applications of Trigonometry, sine or cosine, corresponding to a given angle, and conver the angle corresponding to a given sine or cosine, are not fi by the actual calculation of their values, but always by refer to such a Table. To find the 798. As the basis of our enquiries, we shall begin witl sine and ‘ calculation of the numerical values of the sine and cosine of 1 ee If, in the formula (Art. 776), rad, é : sin5=3V( + sin @)—3,/(1-sin@), we replace @ successively by 30° {whose sine is 3, (Art7 80° 30° 30° 0 i “a asgme ashe and so on as far as a we shall find, PaeShty ; ; ; Peavsaih + 3)-4,/(1 —4) =.258819 = 8, sin = 1/0 +5)-3/U — 5) =.1305262 = 52, eoceandtseaene tae Fd Bee vti eevee esesoene COs'e €8 9 ¢ COMA . 80° sin Sir = 4 (1 + S10) — $V C1 — S10) = Su = 000255028. 189 80° 30x60x1’ 15x15x1’_ 225 oe 256 x 1’, and is there- e first of the successive binary submultiples of 30° which than 1’: and inasmuch as the sines of very small angles e and diminish very nearly in the same proportion with gles themselves*, it follows that 2 5 / 25 z 2 556 * 1 = 95g Sim’ = 000255625, sin ierefore 9 | ad sin 1’ = S5e x ,000255625 = .0002908882 nearly. e corresponding value of the cosine of 1’ derived from (uation cos 1’=,/(1— sin’ 1’) us cos 1’ =.9999999577.- J. The knowledge of the sine and cosine of 1’ of a degree, SNe orm the basis of our calculation of the sines and cosines of cosine of I’, f 7 to construct sles differing from each other by 1’, between an angle of 1’ 2 anon of m sines and jy this purpose we make use of the formule, (Art. 774), Cruel sin (n + 1) 0 =2 cos @ sin nO — sin (n— 1) 4, cos (n + 1) 8 =2 cos 6 cos nO — cos(n — 1) 8. *The truth of this proposition may be inferred from the equation sin 20 sin 20 =2cos@sin 9 or te | 2 = 2cos 0, eliwhen @ is very small, cos@ is very nearly equal to 1, and therefore the >| almost exactly doubled when the angle is doubled, and conversely: 3 s 03 Sr = 9999999674, which differs from 1 by .0900000326 only: again, Wecur to the calculation in the text, we shall find sin Sy = .0010224959, 0 sin an = .0005112482, —- rs = .0002556254, gil ie a it will be seen, for very small angles, how nearly the sines are d, when the angles are so. 190 If we replace @ by 1’, and substitute successively the n, numbers 1, 2, 3, 4, &c. for , we shall get sin 2’= 2 cos 1’ sin 1’ = 0005817764, sin 3’= 2 cos 1’ sin 2’— sin 1’ = .0008726645, sin 4’ = 2 cos 1’ sin 3’— sin 2’ = .0011635526, sin 5’= 2 cos 1’ sin 4’ — sin 3’ = .0014544406, cos 2’ = 2 cos 1’ cos 1’ — cos 0 = .9999998308, cos 3’ = 2 cos 1’ cos 2’ — cos 1’ = .9999996192, cos 4’ = 2 cos 1’ cos 3’ — cos 2’ = .9999993231, cos 5’ = 2 cos 1’ cos 4’ — cos 3’ = .9999989423, Not neces- In this manner we might proceed as far as the sine and ¢ ary‘? Pt” of 45°, in proceeding beyond which we shall find the same vali yond 45°. sines and cosines which have been previously determined: for} much as sin (45° + 0) = cos (45° — 0) and cos (45° + 6) = sin (45! it will follow that the sines of the angles in the series asce' from 45° will be severally equal to the cosines of the comple of angles in the descending series which have been al} determined, and conversely : it thus becomes necessary to con; the canon of sines and cosines as far as 45°, and no further. Beyond 800. But it will be observed that the processes of ¢ Bes of iation, which are founded upon the formule in the last Ar calculation involve the formation of a very laborious product, which: may be greatly sim- expedient to supersede, when practicable, by others vi plifed. require the more simple and expeditious operations of ad& and subtraction only: one of the most convenient of the the formula sin (30° + 6) = cos 6 — sin (30° — 6)*, by which the sines of the angles from 30° to 60° may be} culated by the mere subtraction of the sines of (30°—6) (¥ 6 may have every value from 1’ to 30°), from the cosine | and since | sin (30°+ @) = cos (60° — @), z the determination of the series of sines from 30° to 60° give us the corresponding series of cosines in an inverse ordé * For sin (30° + @) + sin (30° — @) = 2 sin 30° cos 0 = cos @, since 2 sin 30" i 191 In the formule which we have given, we have cal- Process of : . verifying the values of the sines and cosines, at least as far as tha coracte dependently of each other, from the ascertained values 23s of the sine and cosine of 1’, and consequently we may use the tained = of the sine and cosine of any assigned angle 0, to fest curacy of the calculation, by substituting them in the nm cos’@+sin’@=1: for if this equation be not satisfied, ues of one or both of them are necessarily incorrect: thus ‘ake the values of the sine and cosine of 5’, which are in Art. 799, we shall find cos’ 5’ = .99999788460111 8729209, sin? 5’ = .00000211 559745892836, cos® 5’ + sin’ 5’ = .99999999999857765765, ujver which differs from 1 by a quantity less than -0000000000015, isepancy which is referrible to the influence of terms in (culated values of cos 5’ and sin 5’, which are necessarily 1, as being beyond the 10" place of decimals, to which ristered values are limited. The methods of calculating the successive terms of a Process of i of sines and cosines, are methods of continuation, where Y™ic@ton ; a . ’ by the in- ar committed in the determination of any one of them terposition Lees . F me oes of stops or smitted to all those which succeed it: and it is in order of values it the continued propagation of errors, as well as to verify cleus mrectness of the calculations, at different points of their methods. 8, when no such errors exist, that it is usual to interpose, 3, the values of any such terms in the series as can be tned by independent methods. Such are the sines and hn} of 45°, 30° and 18°, and their binary submultiples, or H sum and difference of any angles in the series thus 1¢ *., ¢ A canon of tangents may be formed from a canon of Formation ad cosines, by dividing the sines by the cosines: and a °% Mgents mf cotangents may be similarly formed by dividing the and cotan- by the sines., If however we have found the tangents ne e thus get 2 (18° — 15°) = sin 3°= sin 18° cos 15° — cos 18° sin 15° = .0523360, ———— 192 between 1’ and 45°, we may form the tangents between -# 90°, or the cotangents between 1’ and 4.5’, by means of the fy tan (45° + 0) = 2 tan 20 + tan (45° — 6)*, which involves the most simple operations only. | Thus if we replace @ successively by 1’, 2’, 3’..., we g tan 45°. 1’ = 2 tan 2’ + tan 44°, 59’, tan 45°. 2’= 2 tan 4’ + tan 44°. 58’, tan 45°. 3’= 2 tan 6’ + tan 44°. 57’, Acanonof 804, A canon of secants and cosecants may be formedn secants and cosecants, diately from a canon of cosines and sines, the secant bes reciprocal of the cosine, and the cosecant the reciproca f sine: or much more rapidly, from a canon of tangei cotangents by means of the formule 0 0 cosec 6 = 4 (tan 5 + cot 5) tT secO=2 3 {tan (45-4 5) + cot (450 + =) : Formuleof 805. Formule of verification are equations between t) alas? and cosines, tangents and cotangents of different anglesw if satisfied by their values, as given in the canon, or not verify their correctness, or the contrary. Such are the equations: | (1) Sin @=sin (60° + 0) — sin (60°— @). I" (2) Sin @ = cos (30°— 8) — cos (30° + 4). | (3) Cos @= cos (60° + @) + cos (60° — 6). (4) Cos @=sin (30° + 6) + sin (30° — 6). (5) Tan @=cot 6-2 cot 20. iy 459+ tan 0 tan 45°— 18 — tan 45” tan 0” 1 + tan 45'M for tan 45°= 1 ( ) “ For tan (45°+ @) — tan (45° — @) = _ 1+ tan? 1 —tan@ ~]=—tané@ 1+ tané 4 tan 0 2 tan 0 =Totanr9 = tan 26 {for tan20 = Taal (Art. 791)}, t For, by Note, Art. 791, we find tan 0 = cosec 20 — cot 24, cot 6 = cosec 28 + cot28, and therefore, by adding, we get tan 0 + cot @ = 2 cosec 20, 198 ) Sin 6+ sin (72° + 4) — sin (72°— 6) = sin(36°+ 6) — sin (36 — 8). 7) Sin (90°— @) + sin (18° + 4) + sin (18°— 8) = sin (54° + 0) + sin (54°— 0). 8) Cos 0 + cos (72° + 0) + cos (72° 0) = cos (36° + 0) + cos (36°— 6), §@ Cos (90° 8) + cos (18°~ 8) — cos (18° — 0) = 0s (54° — 8) — cos (54° + 6) *. ormule, however, are better suited to verify the cor- canons of sines and cosines, tangents and cotangents, med and calculated, than to aid us in providing against on and transmission of errors in the progress of their equations are easily verified by developing the sines and cosines ns or differences of the angles which they involve, and substituting lical values of the sines and cosines of 30° and 18°, or their multiples. CHAPTER XXX. ON DEMOIVRE’S FORMULA, AND THE EXPRESSION, BY ME¥ rt, OF THE ROOTS OF 1 AND OF axb,/—1. Denne reis 806. Ie we multiply together two such expressions § ormuia. cos @ + asin@ and cos 4 + a sin 8, we shall find (cos p + asin @) (cos 0 + a sin @) ~ cos @ cos 6 + a® cos p cos 0 + a (cos } sin 0 + sin @ cos If we replace a by /—1, and therefore a’ by — 1, thi tion becomes (cos f + ,/—1 sin @) (cos 8 + ,/=1 sin 8) = cos # cos 6 -- sin: +/—1 (cos ¢ sin 0 + sin ¢ cos 4) = cos (p + 8) + /— 1 sin( (Art. 770.) If we make ~=9, we get (cos 0 + ./ — 1 sin 6)’= cos 20 + /—1sin2@ If we make ¢=20, we get (cos 20 + ./ = 1 sin 26) (sin 6 + ,/ —1 sin 8) = (cos 8 + RES = cos 30+,/—1sin36. | If we make ?=30, we get (cos 30 + ,/ — 1 sin 36) (cos 0 + Ce sin 0) = (cos 6 + JER cos 40+,/—1sin 40. The law of formation of this formula being thus iil we may assume (cos 6 + el sin 0)"~' = cos (n—1)0+ er sin (n- ' and making ¢=(n—1)8, and therefore ~ + 0=n8, vi {cos (n—1) 0+ J 1sin (x - 1) 6} (cos 0 + / a sin) =cosn0+,/—1sinn6 =(cos 0 + ,/ —1sin 0)*-? (cos@ *W| = (cos @ + J —1sin 6)". 195 thus appears, that if the formula os (n — 1)0+,/-1 sin (n — 1) 0=(cos0+/—1 sin 0)"~? 1e, then the formula | cos n0+,/—1sinné = (cos 0 + f= 1 sing) essarily true (Art. 447): and inasmuch as it has been shewn true when vn is 2, 3, 4, it is necessarily true when x is .. and so on, for any whole number whatsoever. a similar manner it may be shewn that cos n0 — eat sin n0 = (cos 0 — Pleat sin 6)". gain, if, in the formule just established, we replace @ by - }may be any angle, great or small, positive or negative*), et 0 nO mae 450 a es n foe! fai sin? —(con +1 sin =) » m an mn mM . Oe og ee RN Pon : cos6+,/—1sin@ = cos — + ,/—1 sin — : (cherefore cen. an 0 Bn ere | (cos 0+ ,/- 1 sin 0)” =cos—+,/—1sin—, j J {lows that n@ ote ps ay cos MO + =i sin = (cos +,/—1sin@)*} =(cos6+,/—1 sin 6)". also, since sos 0+ ,/— 1 sin 0) (cos 6 — ,/— 1 sin 8) = cos? 0 + sin’ 0 = 1, lows that | cos 6 +,/—1 sin @ = (cos @—,/—1 sin 0)~’, ‘therefore (cos@ + ,/-1sin 0)” =(cos 0 — FET tity) Se : For if in the formula cosn8 +/—1 eae = (cosd- Hale sin 0)” ye place 8 by — 9, we get | cos n0 —af —1 sin n@ = (cos0—v/—1 sin 0)", | . : t : (clusion which may be otherwise shewn to be true. ) . Exponen- tial expres- sion for the sine and cosine of 0, 196 But SREY os 1 site con (0 Heel sin 6)” m m =cos(0-,/—1sin@) ™: ; nO — .n@ and, inasmuch as cos — + Ry =~ 1 sin 2 becomes m ” n LG cos —~,/—Isin—, m m ; | mf, On when “ is replaced by — >> it follows that — . no ween oa sey fa 1 sin — =(cos0+,/— 1 sin 8) *% m wn consequently n@ . no ates ae s—=,/—1sin—~ =(cos@ —lsin@) ~, ae m m ( as J in 6) It thus appears that the formula cosn0+=,/—1 sin 0 = (cos 0 + /—1sin oy" * is true for all values of the index. . It is known, from the of its discoverer, as Demoivre’s formula, and constitute of the most important propositions in the whole range of an ~ 807. If, in virtue of the preceding proposition, we s suppose a? to be equal to cos 0+,/—1sin 0, and therefore a”? = cosn@+,/—1sinn0, we might treat a®® and cosn6+,/—1sinné as possessing common properties, and as immediately convei with each other: we should thus find cos 0+,/—1sin@ =a! cos0—,/—] sin?=a-®, * The demonstration in the text is dependent upon the properties of i by the aid of which we are enabled to shew that cosn0 = /—1sinnO and (cos@ + a/ —1 sin 8)" are in every respect convertible into each other: and as the general pro of indices are referrible for their authority to the ‘‘principle of the perm: of equivalent forms,” (Art. 631), so likewise must the formula under co ation be ultimately referrible to the same principle for its establishment the properties of indices being once admitted as algebraical truths, we refer t¢ as furnishing the immediate authority for other truths deducible by me them, and not to the fundamental principles upon which they rest in cor 197 ) 0 a? + aqa-9 | cos 0 = — eats f) ~¢@ : a’ —a sin @ = —— | ‘gain, since | cosn0+,/—1 sin nd = a"®, | | cosnO—,/—1sinnO=a-”®, ould find, in like manner, are + q-nd cos n@ = ease ei ; qr? — q@-7e sin 20 = A ie s, the fundamental equations for those quantities, i value of a: for ) , ao+a-%\? /q9— a-%\? / costO+ sinto=(“ ) +( g « 2,/-1 26 =o 9 ae Q~+2+a-°9 g20_o + q-20 = A a. S A =n Art. 759, $0 a-9+ @ fost = 3 = Coa } | -0__ 40 . a sem fo? . , | sin— 0 = —__- =~ sin, Art. 762, | a OS herefore, adding and subtracting, and also dividing their vy 2 and their difference by 2,/—1, we should find &3. The exponential expressions for the cosine and sine of They azle 0, which are given in the last Article, will be found to S2usty the fundamen- whatever tal equa- tions for the sine and cosine for all values of a. It should be observed that the definition of the sine The value ine of an angle is dependent upon the goniometrical angle of a de- measure no further than it serves to determine the geo-~ pendent upon the i angle, and therefore will remain the same whether the *™«4 or of the arc to the diameter, or any other ratio w measure of if the ordi- hatever nary mea- increases or diminishes in the same proportion with it; SY b¢ ‘is for this reason, that if the value of 0, for a given geo- alan il angle, be indeterminate, the value of a will be indeter- VT f: y | of the goniometrical angle be the ratio of the are to the an angle: taken, it may be re- There are nm different values of @ which give the same value of cosn 0+ V—lsinn6, but dif- ferent values of cos 6 + /— sin 0. 198 minate likewise: but if we agree to assume, as the meas': angles, the ratio of the are to the radius, as adopted in Artj the analytical value of a (for in no case does it possess an i metical value) will be shewn in a subsequent Chapter | determinate and equal to e¥~!, where e=2.7182818 is the of Napierian logarithms: replacing a by this value, we shal} eoV=14 e-ON=1 . eo N-1_ e-ON=1 cos 0 = ——— — —,, 810 = oa 9 3 expressions, which comprehend all the conditions which wa imposed upon the sine and cosine of an angle and its met and from which we may very readily deduce all the fo) which are given in Chapter xxvn, independently of theis metrical definition. 810. If in the equation cos n@ + eal sin 20 = (cos 0 + JeSi sin 0)" the values of n@ are to be determined from the values cosine and sine, then there are » different values of 0 whies different values of cos 6 + ea sin 0, and which equally satisfy the equation. For, inasmuch as the terms of the two series nO, 24+n0, 4r+n0,... 2(n—1)r+n0, 2naw+ no, n—nd, 3rn—nO, 52r—N8,... (Qn —1) a—n0, (Qn+1)7- and no others, are equisinal, (Art. 780), and those of tl series | nO, Qr+n0, 4r+n0,... 2(n—1) a+ né, onmet —~n0, 2r—n0, 47—n0,... 2(n-1)47—n, ona) and no others, are equicosinal, (Art. 781), it follows tt terms of the series cos nse 1.sin n@, when both cos7@ and sinn@ are given.* * If cosn@ =a and sinn@ =), then a?+?=1: if one be given lf the other is determined in magnitude, though not in sign. i; 199 tis the period of n terms Z2(n=I1 2(n Ey ae é, 6, —+9, —+ Oe 7 7 30 others, which give different values of cos @ and sin @, one oth of them, and therefore different values of cos 0+,/—1 sin 0, which give the same values to both the terms of the ex- sion cosn+,/—1 sinnd: form therefore the values of 0, and the only values of 0, ich satisfy the equation (cos 0 + ,/— 1 sin 6)"= cos 28 + J/-1 sin nd. Any one therefore of the » following expressions cos 0 + a sin 0, 9 a 19 cos(*7 +0) +. f= 1 sin (—" +0), SE paieim € ole @ C16 (m6) 6.0/0, 6.00.0 00's as {oie e060, 0'0 6 e910, 0:42 08) °o =. — ne n) cos {2 =D" gh s JT sin @C— 7 sa}, wen raised to the n power, will give the same value of the sression cosn0+,/—1 sinné ten nO is to be determined from the value of its sine and 1 {an gh, (rising Oo" 2(n—r)t | vol 200 and the preceding series of » different values may be red: to equivalent forms, which involve no angle exceeding 18 Determina- = 81]. Let it be required to express the x roots of 1: (|. tion of the nrootsof 1, Other words, to solve the equation x"—1=0. If, in the equation | cos(=27 +0) + J — i sin(*7™ 40) b = cos 20 +, we make @=0, and therefore cos n8=1 and sinn@= 0, we ge 2ra ert a ier hoci COS r= += S10 oe I 7 nv and therefore TT A poe (Lap in which r may be made 0, 1, 2,...(n—1) successively, gi nm results and no more, which are di ifferent from each other: are the n roots of 1, or the x roots of the equation ze"—1=0. 2rT a i cos —— he 1 sin n The cube 812. Thus, let it be required to express the cube roots ol oo In this case, » = 3 and the roots are expressed by (1) §cos 04/20) sin 00) =a! ar an 2 pide (2) cos cho sin = cos 120°+ ,/—1 sin 120°= =1+ Ja Aa (3) COs yf I sin = cos ey oa isin = cos 120°- ,/—1 sin 120°= -1-/3/9) These results agree with those given in Art. 669. The biqua- 813. Let it be required to express the biquadratic roots dl ee aot In this case » =4, and the roots are expressed by _ (1), Coso0+,/—1sin0=1. (2) Cosa" +/=1sin "= 04/51 sors 201 Cos < + = sin =” =~} 60 — . On Qu —— Qa — —+,/—1sin—= ——,/—1 sin —=-—,/- Cos mn ei 4 cos 4 1 sin m / i; lese results agree with those given in Art. 695. 8. Let it be required to express the quinary roots of 1. — The qui- [this case n= 5, and the several roots are expressed by of a 23 0+,/—1sin0=1. 2 2 : — ee +N 1 sin = cos 72°+,/—1 sin 72°=.309 + qe 1x 951. ) 4a _ Aw ° 8 +/—I sin = C08 144°+,/—1 sin 144° ; =— cos 36°+,/—1 sin 36°=~.809+,/—1 x 587. ) UR J —1sin === cos 79° f=] 1 sin 72°=.309—,/—1 1x .951. ) . —. sod, ) se [1 sin =— cos 36° /=1 sin 36°=-.809 —/=1 x.587. ) 1 sse results agree with those given in Art. 707. | 8|. Let it be required to express the septenary roots of 1. The septe- I this case n=7, and the roots are of ae : , ) 0+,/-1sin0= 1} ' 2 x Sbetne ; s+ 4S sin = cos 51°. 26’ + wi 1 sin 51°. 26/ = 623 + ,/—1 x .782. 4 28 Hat so +f 1 sin 2 =~ cos + f= 1 sin 7 =— cos 77°.9/+ ,/—1 sin 77°.9’ =—.229+,/—1 x .975. 6 see s+ ] sin" =— cos” +,/—1 sin | =— cos 25°. 43’+ ,/— 1 sin 25°. 43’ =— .893 + /—1 x .450. A 87 B50) 8a 7 ties T Ss —4+ a) ai ee COS — — eg ee a 7 BON Tg =—.893 —,/—1 x 450. o/ II. Goa 202 (6) Cos 7 —/=1 sin =~ cos <7 ~ /=1 sin 3a =— .222 —,/—1 4a (7) Cost /=1 sin = Semin he bitle ime 623—,/-1 3 if] The preceding Examples will be sufficient to shew ly rapidly the roots of 1, of any order, may be found by mex of a Canon of sines and cosines, in which their successive valb are registered. | To express’ 816. Let it be required to express the m roots of —1: the n roots . : ore in other words, to solve the equation x*+1=0. If, in the equation 2 ony 2 n ae jeos( 7 +0) +,/=i sin(= +0) =cosn0+,/—1 sin . we make n@=7, and therefore cos n@ =—1, and sin x0 = 0, we} WTS ge ee n {eos C7 EU 5 Ji sin =) 5 Sale bi and therefore 1 eee a arp ye Sas gay —“S in which r may be 0, 1, 2... (%—1) successively, giving sults and no more: these are the n roots of —1, or the MX of the equation z*+1=0. The nroots 817. The x roots of —1 are the 2nd, 4th, 6th and 2n" te of —lare of the period of 2 roots of 1, arranged in their order, or e included ; amongst the alternate roots of the equation 2 2 A eee. For the 2nd, 4th, 6th, ... 2n roots of x*"—1 are Qa V2 or O67 . On ey —. ——+,/-—1sin—.,... ‘a Se Loe Sd a See Q2(n—-1)r7 ,— . 2(n—-1)z C08 Se ea) 1 pin Seeks ? 203 i w ae te ™ 33 ee eo 9 cos +/—I1sin—, cos =~ +/ Tain —, ... aie. AL ph Oe Um i 7 i are also the 2 roots of — 1. a Let it be required to express the complete values of To express ; : : the com- (- a)", considered as derived respectively from a and —a, plete values -é values of n. of a” and : (a). 4sume p to represent the arithmetical value of a® or (— a)’, li¢ is independent of any sign of affection, and is the same rore for both, > then find a" =(1)" p = (cos Quran +f — 1 sin2nzrz) p (:7)"= (—1)"p = {cos (2r+1)na+,/—1 sin (2r+1) nz} p, | e} r is any number in the series 0, 1, 2, 3. Im be any whole number, a” = p In be an even whole number, (—a)"=p: for in that case, .) nm is an even multiple of z. n be an odd whole number, (—a)"=—p: for in that case, .) nm is an odd multiple of 7. n be a rational fraction, having a denominator p: then lues of r may be limited to the terms of the series 4 : We ae 2,-..p—l1: j { these limits, whether the series be continued backwards vards, the same values of a” or (— a)” recur. n be an irrational number (Art. 245), every different value 1 the ascending series, will give a different value of (a)" (,2)": and such values, therefore, never recur. . Let it be required to express the complete values of To express ae aaa the com- ) (a+b nf (a 1)" and (a—b,/- Le eae aes 0 si red as derived respectively from (atbV—1)” y» fer in af- J(a+ 6") Me + 6°) fection only |e case, and by the sign cos @—,/--i sin @ in the other: these amet musions will be found to be of fundamental importance in enterpretation of those signs, when applied to lines in geo- ely, and which will be given in the following Chapter. 21. Let it be required to express the complete values of — To express the values | (a+6/=1)4(4—b,/-1),, aa f of ence of _ (a+b f=1¥—-G—b f=1y. (a+ba/—1) dopting the expressions for (a+6 at) =i iand(a—b of aye (a-bJ-1)" hh are given in Art. 819, we find Phi sty (ab (ay =p {cos n (2r7+0)+,/—1 sinn (2r7 + 0)} + p {cos n(2ra+6)—,/—1 sinn (2rm + 0)} =2p cosn(2ra +98), similarly (a+b,/—1)'-(a-b f—1)"=2p,/-1 sin(Qr7+ 0), | a b he cos @ = —_.—_ and sin@ = ——,—x,» and where p is J(a’ + 6?) J (a? + 6) érithmetical value of (a? + 2)? et ete Example. 206 822. Thus, if n=1, we find (a+6,/—1)8+ (a—b,/-1)3=2p eos (=) where p is the arithmetical value of (a°+6°)', and rj term in the series 0, 1, 2, 3 3 ee Qrr+0 The different values of 2 p Cos are 2 (a? + b*) cos » —2,/(a’+ b*) cos ar ue | | | and | — 2,'/(a’+b?) cos am Thus, if it was required to express the different values « te EDI + e-/E-De 3 2 Lute greater than =; we should find, making O- Jt | where ed uss (G+2 4 oy 9 and oe San CO8 2 Eichler tia Q7 é. ue 2 PP td L=2p cos J hoo: ff = #52 p cos 8 4 cos =", 6 fa = 2p COS --2, [L057 : If more terms in the series are taken, the same value recur in the same order. * This formula will be shewn, in a subsequent Chapter, to express th roots of the equation e®—gqr+r=0, Mea he 7? f and it will be observed that if 57 be less than 7? there is no term in the f which involves 4) —1: or, in other words, b = 0. WAP TER: XXX I. | [E REPRESENTATION OF STRAIGHT LINES BOTH IN POSITION [1D MAGNITUDE, AND THE APPLICATION OF ALGEBRA TO THE ‘TEORY OF RECTILINEAR FIGURES. (3. THE investigations, in the last Chapter, have enabled Restate- it merely to prove the necessary existence of roots of the mS Cu tg u ion tained in , s the last a —1=0, Chapter : 2 = : respecting i. are different from 1, {a proposition which we had previously the roots ued (Art. 708)}, but likewise to determine, in all cases, their ° n ete analytical values: we have there shewn that those roots 2 ;pressed by the x values (and there are no more, Art. 811) t: formula 2rT a aT, cos—— +,/—] SI, n n tponding to the successive values of r between 0 and n—1 live. {4. In the application of the successive terms of the period The base a correspond- ) ? ing to the . ; least angle nd by the x roots of 1, as signs of affection to denote the of transfer, “(sive positions of the n radii which divide the circumference ae tie! ‘4 aircle into n equal parts (Art. 728 and 729) it was shewn 2ate. tf a was the appropriate sign used to denote the least of ‘fecessive angles of transfer, then the other terms of the > in their order, would correspond to the other angles of ner in their order: but even when the roots of 1 were itly given, as in the case of its cubic, biquadratic and n y roots, we were unable to connect a specific root with I, ific angle of transfer, inasmuch as there existed no mani- t mbolical connection between this angle and the analytical Nof the root which exclusively corresponded to it: this 7 ainty however will no longer be found to prevail in the 2 —1 LP AN eh eR oe rl h i SS ee a ee eae ee 208 analytical form of the roots of 1, which Demoivre’s fo; enables us to assign to them: for, if we make 20 — . 24 =e OS a hf a Pe n n the base of the period 2 n—) ly Qa yee. a ey we shall find — . 2n\? 4 — . 4 at= (cos +f sin") = 008 fel sin n n n n Qa — 2Q7\3 Or — . 67 ae STAT a = — —1 sin— a (cos es af 1 sin “) sitel eal BS C0 6 bo 6 6 eC Oe © 6 60 6 6 Ue © 0 Os 6 6 & 6 66 6 @.6 00 © © 6) 0 >) see 4a On : 20 where the successive angles of transfer nn follow the order of the successive powers of a: but if wh assumed Avg — . 4a a s— =f es as the base of the period, we should have found 8a — 8a 2 J a’ = cos — +,/—1 sin — n J n? 1 mnths Le w= cos" +,/=1 sin—, coeoeceveoree roe eee ere eee weer ew eee eeeeee where the order of the powers of the base of the period de follow the order of the angles, and where we must pa round the circumference of the circle before we return | 209 ‘ve line: in a similar manner, if the base of the period 2en assumed to be 2r7 eee? dF a COS tal 1 an; n n sould have returned to the primitive line after 7 transits n the circumference, and not before. It appears, therefore, 1e base a, which corresponds to the least angle of transfer le ‘minate, and is in all cases that root of 1, which is expressed t formula 20 sr ee | cos —+,/—1 sin—, | n n | Qa. F eh s le: — is the angle of transfer, and n is the denomination || 2 tl root. £5. It follows, therefore, in conformity with the conclu- niestablished in Art. 728, that if AB or p be the primitive endif AB, AB,, AB,,...A4B,_, be drawn from the centre tlicircle to a series of points dividing the circumference into | . 2 ; q! parts, and making angles equal to = with each other, n he several radii thus drawn will be represented both in gyude and in position, in their order, by Pa). p, (n) p \¢0s Ai o ek sin =U"), | at if the formation of these analytical values, according ch same law, be continued, the same series of values will be riuced in the same order: for 2na7 — . 2nn c¢C —— + =i sin=27) = p, n n i2(n+1 — . — , Be icin 228) (oos2 + Jinn), b Resumed considera- tion of the interpreta- tions given in Art. 728. = 210 p (cos * AE" 5 J=7 sin? C*7)") eeoeeresocer eee ee eee eee eee eee eee eee ee eee e eeu eetseeeeeseeeee eee and, consequently, the roots of 1, connected as signs of aft with a symbol p which denotes a line, will represent thes ; line both in magnitude and position in n different posp : 2 ; ” | making angles equal to — with each other, and zn no pp the successive lines, in their order, being severally repre by the successive values of Qrar — . 2r7r COS af ee eee where r has successively every value in the series 0,1; 2, .. (n= 1). The same 826. And, generally, if p be the length of a line, ni interpreta- ; eo ° | tions gene- an angle 8, with the primitive line, then lized. : ne (cos0+,/—1sin®) p will, in conformity with the preceding theory, express 19 of r and » are whole numbers, then cos 0+ ,/—1sin@ is one of the » roots of 1, and therefore (cos =" hy | sin =“) p> or the equivalent expression (cos @+,/—1sin®@) p will represent the magnitude and position of a line, eqt magnitude to p, which makes an angle an or 6 with th mitive line: and if no finite integral values of r and a found, which make ahd absolutely equal to 6, yet we can ; determine, by the theory of converging fractions or oth 211 | ; 2 ; |values of them as will make the value of — approximate * near as we chase: it will follow, therefore, under such castances, that ( 2r1 aa ee A | cos —— + =isin=*) p n n »present a line in magnitude and position (Art. 820) as near mainse to that which is assumed to be expressed by (cos0+,/—1 sin 8) p: the approximation be indefinitely continued, we may con- le'the line which is represented by one of these formula as m rically coincident (Art. 169) with that which is assumed b represented by the other. “ius, if @=13°.14’, and if we make =rT 6, we shall Example. die series of fractions 1 pn 7 49 397 27’ 109’ 136’ 1333’ 10800 ging to the value of : » the last of them being equal to it: thie, the second fraction gives arr an = 13°. 12’.66, which le than @: the third gives sal = a = 13°. 14.1, which ‘ 2rar 987 ater th ath — = —— = 13°. 13’. giater than @: the fourth gives n= 139g 7 13° 13’. 998, ic differs from 0 by less than a th part of a minute: the last, ic gives the accurate value of 6, is the line expressed in gtude and position by 794 1 ae (cos 9500 + ~ 15 70800) 0 sign of affection is the 397th term of the period formed t: 10800 values of the 10800th root of 1. ‘ Inasmuch, therefore, as a line p, making an angle 6 Examples . primitive line AB, is expressed, both in magnitude and ofthe inter- s 1 j pretations itn with respect to it, by \ cos 0 + ° —1 sin®@ (cos 6+,/—1sin 0) p> for given values of @. { | | as. 212 it follows, that if @= 90°, or if AC be perpendicular to A B! AC is expressed by p Rael (Art. 733): if 0 be 120°, as in the position AC’, then AC” is ex- pressed by 1 Sigh 3 Ga ) p (Art. 734): if @ = 180°, as in the position 4b, opposite to AB, then Ab is expressed by (— 1) p or —p (Arts. 732 and} if @=240° as in the position AC’, then AC” is exp by (' id ve —1) p (Art. 733) : 2 2 ~p,J/-1 (Art. 733) where Ac, being opposite to AG; t tinguished, in its representation, from it by the additional st the preceding results are in accordance with those es y in Chap. xxv, and are sufficient to shew the ease and rei with which this principle may be applied. If two sides 828. Again, since ea ke angled tri- —., f Lee et iatee p (cos 6+ ,/—1 sin @)=pcos0+ p sind ,/—1,. aves it will appear that if, upon the primitive line tude and = 4B, we take AD= pcos @ and DC equal in position by ‘ : ' aand | magnitude to psin@ and perpendicular to AB, ba/— I, its then the hypothenuse AC of the triangle 4DC otne- 5 . Prati beds equal to p, and makes an angle 0 with AB, were and is therefore represented in magnitude and Beeb ay I. in position by pcos 0+ sin 0 1 And if we further replace p cos @ by a and p sin @ by | a and b,/—1 express the two sides AD and DC of thei angled triangle ADC both in magnitude and position (Art and a+b,/—1 likewise expresses the magnitude and pos¢ its hypothenuse: and in a similar manner, if we replace 0 and therefore sin 0 by — sin 0, and if, upon the primitive li we take AD = p cos 8 and express Dec (equal and opposite) by — psin 0, then the hypothenuse Ac of the right-angled (a ‘a 2 fad 13 will be correctly expressed both in magnitude and posi- \'Y | p(cos—0+,/—1 sin — 0) or p(cos 0—,/—1 sin 8): cis equal in magnitude to p, and makes an angle — @ with Sanitive line: in this case, therefore, a and —b,/— ae will 3s the magnitudes and positions of the two sides of the angled triangle ADc, and a— b,/—1 will express the mag- 2and position of its hypothenuse: it follows therefore, ge- y, that the symbolical sum of two lines AD and DC which presented in magnitude and position by a and iy win ual to the hypothenuse of the right-angled triangle which form, whose magnitude is expressed by ,/(a? + 6*) and which 3 an angle with the primitive line whose cosine is and sine +B) b Ge J (a +B?) Iso that the symbolical difference of two lines AD and DC, @-b,/-1 1, will be the hypothenuse of the right-angled triangle od by AD and Dc, where De is equal and opposite to (and where Ac is equal in magnitude to ,/(a’+ 6) and s an angle with the primitive line or a@ whose cosine is ae d whose sine is f +8) and who CE BF)” 29. It appears, from the last Article, that if dC and Ac (p) C ual to each other, and make the angles «d —@ respectively with the primitive AB, they are severally represented -agnitude and position by p (cos 0 + ,/— 1 sin @) p (cos 0 = ae sin 6). f we add and subtract these expressions for AC and Ac, we for their symbolical sum, 2p cos 0, and for their symbolical rence 2p sin @,/—1: but if the rhombus ACBc be completed, to diagonals 4B and Cc be drawn, then we find | AB=2AD=2AC cos 08 and Cce=2CD=2AC sin 0: The sum or difference of two equal lines considered in position as well as magnitude are the diagonals of the rhombus which they form. 214 and inasmuch as Ce is perpendicular to the primitive linc{ it is expressed both in magnitude and position by 2p sin @ aH of the diagonals of a rhombus, therefore, the one is the symilic sum of the two sides which include it, and the other is hg symbolical difference. | The sum of 830. More generally, if AC(p) and A c(p’) be two two adja- cent sides making angles CAB(0) and cA B(6’) with keene the primitive line AB respectively, then considered they are represented in magnitude and in position Ade as wellas position by magnitude, a ig the 7 p (cos 0 + i 1 sin 8) diagonal which they and their die p’(cos 6’ + ,/—1 sin 6’) ference is 4 the other respectively: and if we add and subtract these exprei ol diagonal. we shall find for their symbolical sum i p cos 8 + p’ cos & +(p sin 0 + p’sin 0’), /—1, and for their symbolical difference p cos 8 — p’ cos 6’ + (p sin 0 — p’sin 6’),/—1: | if we now complete the parallelogram ACDc contained bAl and Ac, and draw CE, ce and Dd perpendicular to the prini line AB, and DN and cn perpendicular to CE or CE proce either way, we shall find | AE = AC cos 6=p cos 0, and 4de=DN= Ed=p' cos 0, and therefore | Ad=AE+ Ae=p cos 0+ p'cos@. Again, CE = AC sin@=psin0, and ce=CN= Ac sin 0’ = p’ sin t and therefore NE = Dd=p sin 0 + p’sin &. It follows, therefore, that Ad and Dd are represente i magnitude and in position by p cos@ + p’cos 6’, and (p sin @ + p’sin 6’) ,/—1 thenuse AD of the right-angled triangle which they form. ® 215 | a we similarly find | — Ee=—cn=p cos 0 —p’ cos 0’: Cn = p sin 0 — p’ sin 0’: d nce cn, which is parallel to the primitive line 4B, is repre- nt| in magnitude and in position by p cos 0 — p’ cos 6’ and Cn, hi, is perpendicular to AB, is represented in magnitude and sin by (p sin 0 — p’ sin 0’) els it follows that Cc, which is e pothenuse of the right-angled triangle which they form, and hi: is that diagonal of the parallelogram contained by AC and ce hich is not included by them, is the symbolical difference 4) and Ac. il. Inasmuch as the diagonal AD is expressed in position Expres- d lagnitude by sions for the diagonal, p cos 0 + p’cos 6’ + (p sin 6 + p’ sin 6’) ,/—1, Beale position. dnasmuch, as if d was its length and ¢ the angle which it 4 with the primitive line, it would equally be expressed by d(cos¢+,/—1 sin 9), fcows that (Art. 828) d=,/{(p cos 8 + p’ cos 6’)? + (p sin 8 + p’ sin 8’)’} | = Jip" +p? + 2 pp’ cos (0 — 6’)}, nicos ¢ =? 208° +P cos 6 , te corresponding expressions for the second diagonal Cc ( and the angle ¢’ which it makes with the primitive line, aul be d= Jip p*— 2p cos (0+ 89) _ pcos 8 — p’ cos 0 Cos a _ and cos ¢’ = 7 { i2. The interpretations which form the subjects of the pre- Applica- dg articles, enable us to represent lines generally in position A erpretia ll as in magnitude, and thus to bring their properties under oe to ve ory O e!ominion of Algebra: it may tend to illustrate the theory of Se ciices cli interpretations, as well as the relations of Algebra to Geo- f8¥¢s- et’, if we proceed to apply them to some of the more common o)rties of geometrical figures which are immediately deducible o1 them. li Symbolical representa- tion of the three sides of a tri« angle. The sum of the three sides of a triangle, taken in order, is Zero. Funda- mental equations in Trigono- metry pro- perly so called. 216 833. Let it be required to express symbolically the} sides AB, BC, and CA of a triangle, taken in order. Let the sides of the triangle ABC be expressed in mag by a, 6, c, and the angles opposite to them Cc by A, B, and C: and let c be the primitive z line: then BC or a, which makes an angle | x — B with c, will be expressed by A Dem a {cos ( — B) + ,/—1 sin(a - B)}, or a(—cos B+,/—1 sin B)*; and AC or 6, which makes an angle with c equal to 27 or 7+ A, will be expressed by b(—cos A —,/-—1 sin A). 834, The sum of the three sides of a triangle taken ind is equal to zero. For, if CD be drawn perpendicular to AB, we shaifi asinB=CD=bsin A: and AB~AD*DB=b co. 440d and it follows therefore that the sum of the three sides aq triangle ABC, taken in order, is equal to c+a(—cosB+,/-1 1 sin B)+b(—cos A- ,/-1 i ia =c—a cosB—b cosA+,/—1 (a sin B—b sin A) 17 =(Ocp The same conclusion is expressed by saying that they bolical sum of two sides AC and CB of a triangle ACB, tan AB, and not in the reversed direction BA. It will be shewn, in a subsequent Chapter, that : relations of the sides and angles of triangles, upon whiclth * For the exterior angle at B is 7—B, and at C is r—C: and the angle of transfer, in passing from the position AB to CA, is 27 —-B—C ah for A+B+C=rn. + The same conclusion follows from the proposition proved in Art. 829: for if we complete the parallelogram ACBD, then the symbolical sum of AC and AD is AB: and since AD is equal to CB and estimated in the same direction with it, it is symbolically identical with CB: it fol- lows therefore that the symbolical sum of AC and CB is AB. 217 on or determination, from the requisite data, will depend, nowhich constitute the proper science of Trigonometry, are ec'c cible from the three equations | c=acosB+bhcosA | asin B=6 sinA | BRE Cs 35. It will follow, as an immediate consequence of the The propo- rcasition in the last Article, that the sum of the sides of any 4° "834, ineal figure, taken in order, when estimated both in position extended to n(magnitude, is equal to zero: for if ABCDE be any recti- apo" nil figure, then the sum of the consecutive sides 4B and BC figure. je line formed by joining AC: the of AC and CD, or of AB, BC, and is the line formed by joining AD: Fr 14um of AD and DE, or of AB, BC, ¢ and DE is the last side of the figure d~ ah VA and, finally, the sum of EA and AE VAN B ‘AB, BC, CD, DE, and EA is zero, got +4 aauch as AE and EA have opposite 4 gi: the same reasoning will apply, whatever be the number f des. 36. It follows, therefore, that if we draw Ac, Ad, Ae, and General yarallel to BC, CD, DE and in the direction of EA pro- pea a ud respectively, and if 0, 6’, 6”, 0” be the goniometrical nj2s which Ac, Ad, Ae, and Aa make with the primitive n AB, and if we represent the lengths of the sides 4B, BC, di DE and EA by a, b, c, d, and e, then the symbolical ui of the sides will be -a+b6(cos6+,/—1 sin 0) +c (cos & + ,/—1 sin 0’) }+d(cos 0” +,/—1 sin 0”) +e (cos 0’ + ,/—1 sin 0”) =0, wh becomes, by collecting together those terms which are ‘ted, and those which are not affected, by the sign a1; a+6cos0+c cos 6’+d cos 6” + € cos 0” +(bsin6+c sin 0’+d sin 0” +e sin 0”) ,/—1 =0, n quation which is equivalent to the two equations * |For if Be Bini 1 = 0; then A=0 and B=0: for we get A=—B,/ - 1, n¢herefore A? = (-1)?. B?. (—1) = — B?, a condition which no arithmetical als of A and B or of —A or —B, which are different from sero, can ati y. ou. II. EE ‘i a+b6cos8+c cos&’+d cos 0” + e cos 0’”=0 (a), 6 sind +c sin 6’ +d sin 0” +e sin 0” =0 (b) 218 Modifica- 837. If A, B, C, D, E be the interior, and A’, B’, C’, 1 CaN the corresponding exterior angles of the figure, and therg 1 - tionsof (Euclid, Book 1. Prop. 32, Cor. 1 and 2), figure re- | eis A'+ B’+ C'+ D'+ E'=2r7, | aed A+B+C+D+E =32 ©) we get ee eer | 0 = B’+C'=2nr-(B+0), 0" = B+ C'+ D'=3827-(B+C+D), { 0” — B+ C'+ D+ E’=40-(B+C+D+B); | if we now replace 6, 6’, 0”, 6” in equations (a) and (0 these values, they will assume the form a—bcos B+c cos (B+C)—d cos(B+C+D) + e cos (B+C+D+ : 6 sin B—c sin(B+C)+d sin(B+C+D) —e sin(B+C+D+!1 More generally, if the rectilineal figure have m sides g,.--d,,, and if its interior angles be A, A,, A,,.. (where A is the angle at the point from which a is recke then the equations (a), (6), and (c) will become * | a-a, cos A,+a,cos(A,+A,)—...+(-1)""a,_, cos(A,+A,+...4,4)= a, sin A,—a, sin(A,+ A,)+...+(— 1)” a,_, sin (A,+Agt...Ana)= At Ass Az tivate- pee estes A, = (n—2) 7 | Again, since | A,+A,+... A,,=(n—-2) r- A, | A,+ Ag+... A,2=(n—-2)7-(A+A,_,), | A,+ Ag+... A,_3=(n—2) +—(A+A4,_,+ A,_3); | eooeevrerer eee eee eee eee eee eee eee eee seweeeee eee aeaeeee * We write the last term +(—1)"-! an_, cos (A, + Agt ... An—y) i to intimate that it is preceded by a positive or a negative sign, accor n is odd or even: for in the first case (—1)"-1=+1, and in the (-1)""!=-1. 219 quations (a) and (6) may be put under the form 41, cos A, + a, COS (A, + A,)—...+(—1)""’4,_3 cos (4 + aot) +(-—1)""'a,_, cos A =0 (a), | sin A, — a, sin (A, + 4,) +... +(—1)*~" a,-, sin (A + 4,-1) + (—1)*a,_, sin A =0 (b). ‘he equations (a) and (b) may be called equations of figure, ha” eauch as they are not satisfied unless the lines, from whose ey yolical sum these equations are derived, either form a figure, re capable of forming one, when arranged in consecutive ree : if their sum is not equal to zero, but to a+ # J/—1, then 3,/—1 willexpress the line, in position and magnitude, which fie the figure. 0 98. Thus, if a be the side AB of a regular hexagon Expression {)]DEF, then its several sides are re- pak r ented by a, a (cos 60° + al — 1 sin60°), regular a, —a(cos 60° + ,/—1 sin 60°), hexagon. a (cos 120°+ ,/ — 1 sin 120°), | i the three last sides are parallel to the three first, but esti- d in opposite directions. The sum of AB and BC or — a (cos 120°+ ,/ — 1 sin 120°), a are , AC =a+a(cos60°+ fat sin 60°) =“ xe whose length is yh (fe +t). a,/3, and which makes agle with AB whose cosine is se and sine 4, and is therefore 0 Che sum of AB, BC and CD or =a+a(cos60°+ ,/—1 sin 60°) + a (cos 120° + ,/— 1 sin 120°) =a+a,/3,/—-1, th is a line whose length is 2a, making an angle with AB 220 whose cosine is 3 and sine , and therefore of 60°: thi is a diameter of the circumscribing circle. | The sum of AB, BC, CD and DE or AE =a + a(cos 60°+ ,/—1 sin 60°) +a(cos 1200+ ,/—1 sin 12 a7 /3 pos | which is a line whose length is a ,/3, and at right angles to J The sum of 4B, BC, CD, DE and EF or AF =a+ta (cos 60° + ,/—1 sin 60°) + a (cos 120° + /—1 sim 12) — a —a (cos 60° + ,/—1 sin 60°) = a (cos 120°+ ,/—1 sin 12 In a similar manner, it may be shewn that the diagoni( is the sum of CD and DE, or it is the difference of then of AB, BC, CD, and DE, and of AB and BC, or it ik difference of AE and AC: in whatever manner it may be ded it will be found to be represented by _3a 4/3 a nay 9 ~ which is equivalent to a,/3 (cos 120°+ ,/ — 1 sin 120°), which is a line whose length is a,/3, and which makes an ig of 120° with the primitive line AB. | It is obvious that the same principles may be appli. express, both in magnitude and in direction, the diagoné any rectilineal figure whatsoever. The ortho- 839. In the preceding Articles we have considered lin ee p g pretcons their relative, and not in their absolute, positions, no dis of lines ; being made in the representation of lines which are eq upon rect- ° * ° ales parallel to each other and also estimated in the same direct wade but if the primitive line be extended both ways, and an line be drawn at right angles to it through its beginnin origin, then such lines are called rectangular axes, and Ui gs § ? may always be expressed in magnitude and in relative posi tl 221 heir orthographical projections upon such azes are given: sf AP be a line drawn from ‘gin or intersection of the axes, Pz and Py be drawn perpen- to AX and AY respectively, n) x and Ay are the orthographica] ycions of AP: and if QR be a line aind parallel to AP and estimated i same direction with it, and if d Rr, Qq’ and Rr’ be drawn perpendicular to AX and / spectively, then gr and q’r’ are its orthographical pro- ). The orthographical projections of a line which is repre- 4 line is ¢ in magnitude and direction by rieeineade Se Us and relative a(cos@+,/—1 sin 6) position when its gos 0 upon the primitive line, and a sin @ upon the line which SN ere evendicular to it: and conversely, if a and £ be the ortho- jections are cal projections of a line upon the primitive line and upon °°" as perpendicular to it, then the line itself will be ex- ssl in magnitude and position by a+fB/f-1 b the equivalent expression J/(@’ + 8”) (cos +,/—1 sin 9), a . _ where cos ¢ = Wa B) and sin ¢ = Tae *}8*) being the length of the line, and @ the angle which it K@ with the primitive line. Again, the perpendiculars let fall from a point P upon Relations 0 axes AX and AY at right angles to each other, are also Lee fe. tits co-ordinates, and they are likewise equal to the ortho- ordinates pical projections of the line AP, which is drawn from the pi sen, i A to P, to which they are respectively parallel: namely, P™ections. -ordinate Px is equal to Ay and the co-ordinate Py to id if we take the co-ordinates of the points Q and R, at the fities of the line QR which is equal and parallel to, and ically identical with, AP, then the difference qr of the nates which are parallel to AX will be equal to the ee lS ae, '- . Ie 9 “cod?! eee Op eee eee The theory of co-ordi- nates is the basis of the application of Algebra to Geome- try, where absolute as well as relative position is considered. Conditions of the movement of a point which per- petually describes the same closed figure. In a case of a point de- scribing a regular pentagon, whether geometrical or stellated, 222 orthographical projection of QR upon AX, and the dir qr’ of the co-ordinates which are parallel to AY will bye to the orthographical projection of QR upon AY. The theory of co-ordinates will be found to form thi} it is a theory full of important consequences and whi ) require very extensive developement, and we shall thief reserve the further consideration of it to a subsequent this work. 842. If we should conceive a point to be moved spaces or lines represented in magnitude by Os Gs, Ogee Gets and in directions making the goniometrical angles Osn0 705,20 Uotys with the primitive line, then the point would return to thor of the movement, and the lines described would form il figure, when the equations in Art. 837, were verified: feif symbolical sums of the orthographical projections upon e rectangular axes are equal to zero, the line joining theptl and the final point would be equal to zero likewise: and iso further follow, that if the sum of the angles of transfer i A'+ A’, + A’n4+..- Aas including the angle of transition from the n™ to the | line or from a,_, to a was 27 or any multiple of 27, an if conditions of the movement remained unaltered, the poi continue to circulate in the same figure for ever: we wil vour to illustrate our meaning, and to demonstrate the col to which it leads, in the case of the movement of a poinwh describes a regular pentagon. i i 843. Thus, if the successive spaces or lines descrieé equal to each other (a), and if, at the end of each line, the direction of the movement is instantaneously changed through an angle E of transfer of 72°, then the point will return into itself after describing five such lines: and if the movement be further continued in con- A formity with the same conditions, the point | will for ever circulate in the same figure: but if, all tl ot | 223 tions remaining, we suppose the angle of transfer, at the each line to be 144° instead of 72° the point would still y into itself after describing five syut the lines would intersect her, and the figure would be d: thus, if AB was the primi- ie, the order of description of —-< es would be AB, BC, CD, Bad EA, the angles at 4, B, | E, being severally 36° or » soplements of 144°: again, if » igle of transfer at the end of ‘ne was 3 x 72° or 216°, the ne stellated pentagonal figure ul be described, but in a re- ‘s¢ position: and if the angle of ns was 4 x 72° or 288°, the or- at pentagon of geometry would dizribed, but on a different side th primitive line 4B, to that on ic the pentagon corresponding inngle of transfer of 72° was de- ibl: but if the angle of transfer a When the ~ ‘ 0°, the point would continue to movement eed p d . h di . continues ve orward in the same direction ; in astraight lijche same straight line for ever. line. If we should define a regular pentagon to be a figure Theory of i equal sides and five equal angles, without any assumed ex- Periodical movements si, of re-entrant angles or intersecting sides, it is obvious that corre- ‘slated figures referred to in the last Article would equally ‘Paorent Ee w the conditions of such a definition with the regular and pe a sss, of angies. yy, tely bounded pentagon of Geometry: the essential distinc- ; 1 | tween them presents itself in the equation of angles, which 01 nonly assumed in Geometry to be unique for all figures of Sae number of sides: thus the equations of angles for the ett angles of transfer which are considered in the last Article 4, B, C, D, E be the interior angles, / G) 4+B+C+D+E=37, (2) 4+ B+C+D+E=7, (3) 44+B+C+D+E=- ~«~as™ 224 (4) 4+B+C+D+E=—3r, (5) A4+B+C+D+E=—5n. Inthecase Thus, in the figures corresponding to the equations (1) al ee ~ the several interior angles are equal to each other, but w nt ferent signs, and the figures are formed on different sidesf primitive line: the same remark applies to the stellated which correspond to the equations (2) and (3): in Gehe these distinctions of position, as above and below the pri line, are not recognized, and our attention therefore j { science is necessarily confined to one figure of each &ld fifth equation intimates, that as each interior angle is = as the sum of each exterior and interior angle is 7, each¢ ar angle is 27, and consequently no figure is formed. Tn the case 845. The angle of transfer in a regular hexagon is = | 6 of hexago- 3 eee if we should suppose the sum of the exterior angles doub ; angle of transfer would be doubled or become 120°, in white the corresponding hexagon would degenerate into an equat or regular triangle and the describing point would pasity over each of its sides in six changes of movement: if we supa the angle of transfer tripled, the figure corresponding wed would again become an equilateral triangle in a reversed I to the former: if sextupled, the point would continue tf forward in the same straight line, in the direction of its tive motion: if still higher multiples of the angle of were taken, the same series of movements would be'e} duced, and in the same order. Inthecase The series of movements corresponding to an ail ee On ' aia octagonal transfer of a2 and of its successive multiples, would bei move- ments. buted into successive periods of seven figures or lines, oW the Ist and 6th would be the regular heptagons of Gum in reversed positions, the 2nd and 5th, 3rd and 4th sik figures in reversed positions, and the 7th a continued ai 7 225 » straight line: in the series of movements corresponding | 20 Ms ‘ : _ angle of transfer of roa and of its successive multiples, the stind 7th would be the regular octagons of Geometry in od positions, the 2nd and 6th squares in reversed posi- io, the 3rd and 5th octangular stellated figures, the 4th a Beto and the 8th an indefinite, straight line. It is not eissary to extend these observations further, as may be easily lo , to the series of movements corresponding to other angles ansfer and their suceessive multiples.* bac) |The general theory of such periodic movements requires that the sums of he thographical projections of the m first lines described should be equal to Qa f ; : : er( where —» or any of its successive multiples, is the least angle of transfer : r iwe express this angle of transfer by 0, then the equations of the figures esibed or of the movement of a point, which, after describing n equal spaces, etu's into itself, are p{1+cos@+ cos 20 4......... cos (n—1)0}=0, (s) p {sin@ + sin 20 + ......... sin (n — 1) 6} =0. (s') 6, ,-6 ) we replace cos6 by a — (Art, 807), the first series (s) becomes } Regn? a2, «20 oP ccd | p( + 5 + 5 Ne Seer 5 Bein ta? + ......... at16) 0 yan 40-20 4 ieee a eles } e(e ge £1 ae byt ane 1 ) (Art. 429). ‘we divide the numerator and denominator of the first and second of the | 6 6 Bhs of which this expression is composed by a? anda 2 respectively, we get ) Cae % o (ar—H9_g—(n-88 4 g?_a ? Coe: So] | Say, eS PE aed a( aes? yl hi becomes, if we replace a!”~¥)9 — a-™-#9 hy 2,/—1sin(n—4)0 and rf 8 5 “2 by 2,/ =T sin 5 4 | ' Ww: . no (n—1)0 ) sin (n — 3) 8 +sin 5 p sin —- cos 5 —— he t= ' ~ (Art. 773). sin = sin = 2 2 . Ina ou. IT, Kr _ ae = 226 Numberof 846. In a figure of » sides, there are 2n elements, and) e d es = . af) | Tee” three necessary equations amongst them, one of which | La equation of angles: there are, therefore, 2n—3 elements Wi which are . assumable are indeterminate and assumable at pleasure, though not ) | in a figure oe . % . : ' | Bees. lutely unlimited in themselves,* which are n sides and (7- | | In a similar manner, if s’ be the sum of the series p (sin 0 + sin 20 + ........ sin n@), we shall find PME O Loe ©) ate ee | p sin sin ( 5 )e | fae , 2 9 . ; If we make i= —, as in the case of the regular n sided pélyh : CS) p SIR 7 COS os Rei eae Bk al fig MAE ich tha n 3 : nm— 1 p sina sin WT , n er = 0. 29 ar i" sin — n Geometry, we find Le Qua 7 Again, if we make 0= , where m is less than n, we get e n-1 p sin mar COS m1 Fic =), . mmr sin —— n : . {n-1 p sin mar sin ma n s’ — eee Ce QO. Mm TT sin — n If we make 0=27, we find p sinnz cos(n—1)7 0 $in 7 0; , _ psinnm sin(n—1)mr_ 0 " sin 7 0 Under this form, these expressions are indeterminate: but it will apar methods which will be investigated in a subsequent Chapter, that the» le s, under such circumstances, is np, and that of s’ is 0: it will follow t 4 that the movement of the point, when @=27, is one of continual pros in the direction of its first motion. * It appears from the first equation of figure (a) Art. 836, a — a, Cos Ay + dg cos (A; + Ag) — -»» +(—1) 97} Gn—; CoS (41+ Ast | i 227 2s, or (n—1) sides and (n—2) angles, or (”— 2) sides and n 1) angles: in order to determine a figure, therefore, there n(n —1)(n—2) 1.2.3 ‘three classes of data, one class admitting of n(n—1) is eo binations (Art. 453), the second of 7. combinations on the third of the same number. When the independent da are fewer than (2n — 3) in number, the figure is not de- terined : if they exceed (2n—3) in number, they may furnish relts which are inconsistent with each other. (Arts. 402 ar 403). : e In the case 847. Thus, the equations of quadrilateral figures are ae aus Piss lat — a, cos A, + a, cos (A, + Ay) — a; COS (A, + A,+ A;) =0 (1), pee generally. a, sin A, —a, sin (A,+ 4,) +a, sin(4,+A,+4;,)=90 (2), A+4A,+A,+A,=2t (3). If we combine equation (3) with (1) and (2) (Art. 836), we get a—a, cos A, + a, cos (A, + A,) — a, cos A =0 (4), a, sin A, — a, sin (A, + A,) — a; sin A = 0 (5). Of the 7 quantities a, a,, a2, as, A, A,, A,, if five of them b assigned within the requisite limits of value, the remaining t» may be determined from the equations (4) and (5). /848. Parallelograms are defined to be quadrilateral figures *, Eng a viich have their opposite sides parallel: and it will follow as oe oa anecessary consequence of this definition and from the pro- jties of parallel lines, that the sum of every pair of conse- «ive angles in the figure will be equal to two right angles, 21 therefore that if the value of one of its angles be known | « assumed, the rest are determined: for if A+A,=A,+ A,=A,+A,=A,+A=7, \ find A,=7--A, Iris wee , that is of ac to ab, and, therefore, of ac to cd: in the manner, the ratio of the complement of AC and CD to CD, is of AD to CD, and, therefore, of 4D to DE is the same at of the ratio of the complement of ac and cd to cd, that ‘ad to cd, and, therefore, of ad to de: lastly, the ratio e oO of AD and DE to DE, that is of AE to DE >same as that of the complement of ad and de to de, that ae to de: it follows, therefore, that the three triangles i 232 ABC, ACD, ADE, and abc, acd, ade, similarly formed i figure, are similar to each other: and the same conclusic y equally follow, whatever be the number of the sides | figure : it follows, therefore, that all rectilinear figures are , to each other, in which all the triangles similarly formed j are similar to each other. Figures 852. In the geometrical theory of rectilinear figures, ‘h with re-en- . : ; : § trant angles A represents an interior, and 4’ the corresponding aa may ca’ 6we assume A and 4d’, in the equation under the general pe and (ies | | equation of nl figure ana to be both of them positive, and also less than 7: but angles. : ‘ may be conceived to be formed possessing re-entrant angi, ( we have already seen in the case of stellated figures), whe o of these angles may exceed 180°, and where this equation m be satisfied unless the other be negative, and conversely : tis, the movement of transfer, in passing from the line BA 1d be from right to left in one case (Fig. 1), and from ‘4 | right in another (Fig. 2), the external angles (x AC) ger ati will have different signs: and if these movements of ii Fig. 1. Fig. 2. Fig. 3. Fig. | ) | c if | ; A f {iA B B c c I be continued through more than 180°, the interior anglBd will be negative in the first case (Fig. 3), and greate 4 right angles in the second (Fig. 4): for if A’=x+¢ A+A'’=n, gives A=—; and if A’=—ar—®@, then A+A’=7, gives A=2r+¢. | All these relations of exterior and interior angles are xe Fig. 1. Fig. 2. Fig. 3. Fig. 4. t ct c D ey Dp A D : : D = ry | A | : Ke B EB | | ; plified in the formation of the equilateral pentagonal 233 ¢ (3) and (4), of which the first is the regular pentagon etry: the second has the re-entrant and interior angle sater than 180°: and therefore A’ negative: the third angle at A negative, and the fourth the angle at 4 » and greater than 4 right angles. rectilinear figures, therefore, though not geometrical, he equations of figure and of angles, and they may be ned, like geometrical figures, by the aid of those equa- from the requisite data. 7 CHAPTER XXXII. ON LOGARITHMS AND LOGARITHMIC TABLES AND THEIR be Definition 853. THe index or exponent 2, in the equation ofa logarithm. a*=n | is called the logarithm of n to the base a: this definition in Vi the primary notion of a logarithm, in which the term orig) q which is mentioned in Art. 2753. It will be shewn in a subsequent Chapter, that if a and numbers {where the term number is used in its largest (Art. 169 and 416)} greater than 1, there is always an ith metical value of x which satisfies the equation ane or, in other words, there is always an arithmetical logarith ¢ n to the base a. | 7 Sera iy 854. , which we have noticed above (Art. 860), that the loga- ns, in that system, of all numbers, whether whole or deci- expressed by the formule Nx 10" and a or in other wds, of all numbers which are expressed by the same suc- eion of significant digits, may be found from one opening Thus | log 96498 = 4.9845228, log 96498 x 10 = log 964980 = 5.9845228, log 96498 x 100 = log 9649800 = 6.9845228, log eo = log 9649.8 = 3.9845228, ) log ‘ee = log 964.98 = 2.9845228, | og ae = log 96.498 = 1.9845228, | log pee = log 9.6498 = .9845228, log ee = log .96498 = 1.9845228, ; log (ae = log .096498 = 2.9845228, 96498 log —~——*—_ = log .0096498 = 3.9845228. °S yo000000 ~ 08 009989 ? For log N x 10” = log N + log 10" =n + log N, log = log N — log 10" = tog N—n: he logarithm of 10", if 10 be the base, is 7. Tables of logarithms give man- tisse only. Incon- veniences of tables of logarithms to a base which is different from 10. 240 In the three last cases the sign — is placed above, an before, the characteristic, which is alone affected by mt decimal part of the logarithm, or mantissa,* remaining pos but if we subtract the mantissa from 1, forming what is « its arithmetical complement, and diminish the negative chariéte istic by 1, we shall obtain the correct negative logarithm ¢ sponding: we thus find log .96498 =— .0154772, log .096498 = — 1.0154772, log .0096498 = — 2.0154772. | 866. The tables, therefore, of a system of logarithms, wos base is coincident with the base of our scale of arith notation, give the mantisse only of logarithms in one colin with the significant digits, in their proper order, of the or responding numbers in another, inasmuch as the charactewti may be always supplied from the number of integral ples or in case there are none, from the position of the decimal yin with respect to the first significant digit: thus, the mantis: 0 the logarithm of 53399 is .7275331, which alone is reco et in the tables, the complete logarithm 4.7275331 being at | supplied by prefixing a characteristic which is less by 1 ‘an the number of integral places in the proposed number: anii a similar manner, we find, from the same mantissa, the comp}te logarithm of .00053399, which is 4.7275331, where the nee characteristic 4 exceeds by 1 the number of zeros, which m mediately follow the decimal point. I 17 867. If, however, the base of the system was not coincic with the radix or base of our system of arithmetical notatn the logarithms of all numbers included in the formule 10* eo and which are therefore expressed by the same suc sion of significant digits, would not be known from one operig of the tables: thus if the base was the number 2.71828928 & and * This term was introduced by Euler, and may be conveniently use t0 designate the decimal part of a logarithm, in the absence of any simple dig nation which can be supplied by our own language. | * 241 , and if we assume the prefix log to designate Napierian not tabular logarithms*, we should find - log 964 = 6.8710911, log 9640 = log 964+ log 10 = 6§6.8710911 + 2.3025851 = 9.1736762, log 96400 = log 964 + log 100 6.8710911 + 4.6051702 11.4762613, | log 96.4 = log 964 —log 10 | = §.8710911 ~ 2.3025851 = 4.5685060. (It thus appears that we cannot pass from the Napierian arithm of N to that of N x 10” ening of the tables for the purpose of finding the Napierian \arithm of 10". (868. The following is a specimen of the form of our ordinary oF Are tle of the logarithms of numbers. gs ‘ s. 1 2 3 4 5 6 | 98 7372 | 7536 | 7700 | 7864 | 8027 | 8191 | 8355 | 8518 | 8682 | 8846 : ie aie A Be SRA 8 Oe EE ate Eee 9 | 147 ef ef | | | | 424 06 0809 | 0972 | 1136 1300 | 1463 | 1627 | 1790 | 1954 | 2117 * Some modern writers are accustomed to subscribe the base to the prefix log, order to designate the particular system of logarithms which is employed: is log,, means tabular logarithms: log. means Napierian logarithms, whose base >= 2.7182818, and similarly in other cases: such refinements of notation, how- , are rarely necessary, as the circumstances under which they occur will 1erally indicate the nature of the logarithms, whether tabular or Napierian, “ich are used. | Vo. II. Hu . | | tae 9009 | 9173 | 9336 | 9500 | 9664 | 9827 | 9991 | 0154 | 0318 | 0482 i T 242 In the first column are placed, underneath the letter Nt} first four digits of the number, the fifth being written in the m line with it at the head of the successive columns. Inlh column headed 0, are written the mantisse of the logari'n of 26500, 26510, 26520,...... , the three first digits being D pressed as long as they remain the same with those in} first line: in the column headed 1, are written the four hg digits of the mantisse of the logarithms of 26511, 2621 20531, ......: in the column headed 2, the four last digi: of the mantissa of the logarithms of 26512, 26522, 26532, im and so on for the remaining columns headed by the rem nine digits, in their order. It will be observed that the mantissa of the logarithn } 26546 is 4239991 and that of the logarithm of 26547 is 4.240 4 this change of the third digit of the mantissa from $ to 4, is ili cated, in the place where it first occurs, by writing the fourist digits thus 0154. The loga- 869. If the mantisse of the logarithms of the succesye rithms of : : ; ye large num- Humbers, which are given in the Extract from the tabled bersin- __ the last Article, be subtracted from each other, their differcee crease very nearlyin will appear to be between .0000164 and -0000163, and wilbe rages found, by actual reference of the Tables, to continue nea with the same for the whole series of numbers between 26500 and 268): numbers , ee themselves. We are authorized to conclude, therefore, that the mantissa large numbers will increase, within small limits, very nearlyin the same proportion with the numbers themselves, and th consequently, if the mantissa of the logarithms of two a large numbers, and therefore their difference, be given, te mantisse of the logarithms of all intermediate numbers maye found approximately, if not accurately, by a simple proporin and conversely*: this property of logarithms is very importit, as furnishing us with very simple and expeditious methods) greatly extending the range of the tables. $ fo find the = 870. Thus, if it was required to find the logarithm oa ] tl seal ofa num. Number of more than 5 digits (and therefore beyond the rae ber of more of the ordinary tables), we should proceed as follows. than five places. We should find from the tables, the logarithm of the numlr * This proposition will be easily shewn to be an immediate consequence! the logarithmic series, which will be given in Chapter xxxvy. 243 fy ed by the first five digits ; and to this we should add, from th column of proportional parts headed Prop., the number | , . or 1 ~esponding to the 6th digit and irk: of the. number cor- ‘@onding to the 7th digit, by advancing the number given the column one place to the right: their sum would be the irithm required to seven places of figures: and if the number lecimal places of the logarithms recorded in the Tables was giater than what is commonly given, the same method might be ended to find the logarithms of numbers of more than seven pies*. | Thus, if it was required to find the logarithm of 265.4678, we Example. fil from the tables x 265.46 = 2.4239991 yp. part of logarithm corresponding to digit 7 n the 6th place 114 Sailarly for digit 8 in the 7th place 13 ‘le logarithm of 265.4678 = 24240118 ‘en. Again, if it was required to find the number corre- To find the mding to a non-tabulated logarithm, we should proceed as eater lows: ing toa ‘ \ : logarithm We find the difference between the mantissa of the given notin the ‘arithm and the next inferior mantissa in the tables and we “les. ite down the corresponding number : in the column of propor- nal parts, we find the number which is equal to, or next below, » difference thus found and the digit opposite to it is the 6th rit of the number sought for: if there be a remainder, we sub- na zero to it, forming a new difference (removed one place the right) and the digit in the column of proportional parts lich is opposite to the number equal to or next below, it, 1s {2 7th digit of the number required ; and similarly, if more rits than 7 are required to be determined. . For if D be the difference in the mantisse corresponding to a unit of the F 2 and — will be the differences, in conformity with the »perty assumed in the text, corresponding to the two next inferior units in the ular number D b ‘ n and 7th places respectively, and therefore a and will be these dif- if a and b be the digits in those places: the first is found in the column proportional parts opposite the digit a: the second, removed one place to 2 right, is found in the same column opposite the digit 6. Example. Tables of logarithmic sines, cosines, tangents, Ec. The reason why the logarithms of the nu- merical values of the sines, cosines, &e. are in- creased by 10. 244. | Thus, let it be required to find the number, whose loge is 1.233678, and whose precise value is not found in the abl, The given logarithm is 1. 233678 The next less tabular-logarithm of 17.122 is 1.2335545 . aan Their difference is 133 The table of prop. parts gives the digit 5 for 127 Their difference (6) followed by zero gives The same table gives 2 for 51 5 Their difference (9) followed by zero gives i, The same table gives 3 for 76 } The same table gives 5 for 127 7 872. Inasmuch as tbe sines, cosines, tangents, ie of angles enter into formule which are the subjects of calcd equally with other symbols possessing assigned numerical vais, a register of the logarithms of their successive values becons equally necessary with that of the logarithms of the seriesif natural numbers. Tables or canons’ of natural sines, (Ch XXVIII.) cosines, tangents, cotangents, secants, cosecants, ll contain their successive values for every minute (and in soe tables for every ten seconds) of all angles between 1’ and and consequently between 1’ and 90°, if taken in an inverse ord, when the sine is replaced by the cosine, the tangent by cotangent, and the secant by the cosecant: whilst tables of ¢ logarithms of sines, cosines, tangents and cotangents, secas and cosecants, will contain the logarithms of their natural valis increased by the number 10, arranged in precisely the same ord, each page of the natural values being opposite to a page of th corresponding logarithmic values. Their difference 14 followed by zero gives 0 The number required is therefore 17.1225235 nearly. = 873. A very little consideration will shew the great co venience, for the purposes of calculation, of increasing the log: ithms of the goniometrical quantities, as recorded in the tabl by the number 10: for the natural values of the sines and cosi are included between 0 and 1 and the characteristics of the logarithms are therefore necessarily negative: thus | 245 lin 1/ = .0002909: its unaltered logarithm is 4.4637261, j ¥. ME OL ZSO24 Soc ccc sce erences scessesee 20418553, 1.50" = .7660444 ; Res Clie ad Le ed eee 0 (Oe 1.9999836, 98 30° = 8660254 ; GE ER tid ba A ETE 58 85° = .0871557; Seem ee oo) 20 en 0402000. | *such logarithms, therefore, were registered in tables, their 247 _f the six elements which present themselves in every tri- Two only f the th , namely, the three sides and the three angles, there are rales Ee nvolved in the first of the preceding equations, four in the ee d, and three in the third. The last equation, which is the specific qu ion of angles, determines the third angle when two of them ena eone iven: it follows, therefore, that two only of the three angles termina- ve considered as dependant upon the specific* conditions of an he riangle. 76. The solution of the second of the preceding equations, Of the four a involves two sides and the sines of their opposite angles, ee es us at once to express one of these four elements in pie and their op- ; of the other three. Boats angles, any /e thus find . eee b sin A a sin B determine — : 2 (1. ees ET the fourth. sin B sin A ’ aut , 4, sin A = 5 sin B, sin B=-— sin ZA. a _ ikewise, if in equation (1), ~ e-acos B—b cos A=0, | asin B ye ‘eplace b by ie get asin B c=acos B+ — cos A in A _ 4(sin A cos B+sin B cos A) ] sin A _asin(A+B) asinC- inten sin ofi3 Sviatsinl 4+ Dy equation (3), we find } sin (4 + B) =sin(r— C)=sinC. 7]. The conclusions in the last Article may be exhibited Thesides of ! atr > nir the form ae eae “ sind a_sind sin B. ars ~snB’ ¢c sinC’ ¢ sinC’ the oppo- site angles. _| The specific conditions are those which are not common to _all triangles, it seuliar to that which is under consideration: the three sides and two , when given, constitute specific conditions, provided the sum of no two ae gm the third and the two angles together do not exceed 180°: if three be given, they are not the angles of a triangle, unless their sum be 180° : is two of them only, whose sum is less than 180°, be given, there is always [ of which they may form two of the angles. i- 7 248 and it follows, therefore, that the sides of a triangle ary portional to the sines of the opposite angles. Expression 878. The fundamental equations in Art. 875, will ahd for one side ‘ ‘ ‘ . : ; ofatriangle furnish us with an expression for one side of a triangle in eats of of the two other sides and of the cosine of the angle eet hee they include, which is of fundamental importance in the sen an e cosine of of Trigonometry, properly so called. aoe For from equation (1), we get | ee c’=(a cos B+ 6b cos A)’ =a? cos’ B + 6* cos’? A + 2ab cos A Gk From equation (2), we similarly get P 0=(a sin B- 6 sin A)’= a’ sin’ B + 6° sin? A — 2ab sind siB. Consequently, by adding, we find c’ = a’ (cos’ B + sin’ B) + 6? (cos? A + sin? A) + 2ab (cos A cos B - sin A sin B) = a’ + 6° + 2ab cos(A+ B), =a’ + b’—2ab cosC: (a' for cos’ B + sin’ B= 1, cos? A + sin? A = 1, cos A cos B — sin A sin B = cos (A + B) = cos (x — C) = =o; In a similar manner, we shall find [ a’ = b? + c?—2bc cosA (6 b?=a’+c?—2ac cosB (c I Expression 879. If we severally solve the equations (a), (6), ar aS of With respect to cos C, cos A, and cos B, we shall find an angle ofa triangle os gil at + ahi 7 in terms of ais 2ab (a, its sides. 5° +c’? — a? cos A = e| 2bc ( aso —b* ; cos B= Buc , which are expressions for the cosines of the several angles) . triangle in terms of its sides. | Expression for the sine i 4 oe Bek 880. Corresponding and more symmetrical expression: 1t terms O ° ° . its sides. | be found for the sines of the angles of a triangle in terms | 249 s, and which present themselves likewise in a form which is er suited, as will afterwards appear, to logarithmic computa- : we proceed to obtain them, as follows: ‘From equation (a), Art. 878, we get 2ab cos C =a? + b?-— ¢*. 2ab (1 + cos C) =a? + 2ab+b?—¢? =(a+by—c =(a+b+c)(a+b-—c). Art. 66. (g). Again, if from 2a we subtract 2ab cosC on one side, and a b’—c* on the other, we get 2ab (1 —cos C)=c?—a?+2ab—0b? = c’— (a?— 2ab + 6’) =c’—(a—b)y? =(a+c—b)(b+c-a), (h). {f we multiply the two sides of the equations (g) and (*) roectively together, we find | la*B* (1 - cos’ C)=(a+b+c)(b+c—a)(a+c—b)(a+b- C). 4 Replacing 1 — cos’ C by sin’ C, dividing both sides by 4076’, ‘| extracting the square roots, we get CK Ni(a+b+e)(b+e-a)(a+e—b)(a+b-c)} 2ab atb+e ma If we make s = , we find a+b6+c=Qs. 6+c-a=b+c+a—2a = 25~—2a =2(s—a), a+c—~b=a+c+b—2b =2s—2b =2(s-5), a@+b-c=a+bh+c—2c=2s—2c¢=2(s-—c), therefore sin C = 2 /{s es) (s—c)} (i). In a similar manner, we find as 2 /{s (s —a)(s—5)(s—c)} sin A = wis ( sen Fs (A), amy levies Xe edt gb (I). Vor. II. It 250 The ex- 881. It will be observed, that the several expressionsf r . e * . Bata 4 sin A, sin B, and sin C, have the same numerator | sin B, and sin C, have Q Aire (s — a) (s a b) (s i C)ts the same | Sei eraaae which likewise expresses twice the area of the triangle. «which Is Petpithe For the area of the triangle ABC is equal to one-half of; ‘a of the rectangle contained by its base AB and | riangle. ‘ . . the perpendicular CD let fall upon it c from the opposite angle C, and it is | therefore expressed (Art. 592) by 3AB y w | PHOTONS YSU arta) eda ery OE ELE | i 2 A oD - since CD=a sin B=b sin A. © | If in Reuss we replace sind by the expression give : the last Article, we get the area of the triangle (4) =,/{s ( ! (s—6)(s—c)}, or it is equal to the square root of the cont” product of the semi-sum of the sides and of its several exese above the three sides of the triangle. | The pre- 882. If it be granted that the value of an angle may be of a a determined and calculated from the value of its sine, cosine,an | are ade- gent, &c. (and the limits of the values of the angles of triavle Bete foe will be found to remove all ambiguity when the value oth a oop angle is not to be determined from the value of the sine* Trigono- expressions given in the preceding Articles will enable 11 ee calculate the sides, angles, and areas of triangles wheneverth data are adequate to determine them. | We shall now proceed to consider the different cases of alt which are sufficient to determine a triangle, and to exemlif their logarithmic computation. Bes pie 883. Case 1. Given a, A and B, to find C, 6 and c. side an ‘ x y two angles, § The third angle C is found from the equation to find the remaining A+B+Ce=n, parts. which gives C = 7 —(A + B). * For the sine of an angle is identical in magnitude and sign with thsi of its supplement; but the cosine, tangent, and cotangent of an angle are2vé rally equal to the cosine, tangent and cotangent of its supplement in magridi but different from it in sign. . 251 [he equations in Art. 876, give us _4 sin B asin C sin A * ~ sind The same equations, adapted to logarithmic computation, «ome | log 6 = log a + log sin B — log sin A, i] log c = log a + log sin C — log sin A. | Let a = 749.6, A = 37°14’, and B= 67°27’: to find 6, c Example. w C. _ In the first place, to find C, we have | Pa ga a, A= 37.14! B= 67.27' i} A+B 104.41' C= 75°.19' f | ‘To find b, we have 1 log 749.6 = 2.8748296 log sin 67°.27’ = 9.9654582 12.8402878 log sin 379.14’ = 9.7818002 tt 7 . | log 1144.16 = 3.0584876 or b = 1144.16. ee use of the arithmetical complement of log sin 37°.14’, ch is found by subtracting it from 10, will enable us some- wit to simplify this process, | log 749.6 = 2.8748296 | log sin 67°.27' = 9.9654582 i arith. comp. log sin 37°.14°= .2181998 log 1144.16 = 3.0584876 We reject 10 from the final characteristic, as forming simply | tabulated augment of the logarithm of sin 67°.27’. To find c, we have | log 749.6 = 2.8748296 log sin 75°.19! = 9.9855798 arith. comp. log sin 67°.27’= .2181998 log 1197.42 = 3.0786092 or c = 1197.42. SE 252 Given two 884. Case 2. Given a, 6 and 4: to find B, C ande. sides and an angle The equations in Art. 876 give us Opposite to : one of “tity spa them, a OOM EY SRET, : When 4 and B are known, we find C and ¢ as in the first ea, when the Jess than 90°, unless the connexion of the data with other mn to the less Thus if 6 be less than a, or if the angle whose vah case, therefore, there is no ambiguity *. in which the angle opposite to the greater side in one of ‘en Thus if A, which is the given angle, be it will cut AB produced in two points, B ; ——ae | but the two triangles formed have the same data, which are / which becomes, when adapted to logarithmic computation, log sin B = log sin A + log b — log a. The solu- If the value of B is to be determined from that of sin Bti Hear uncertain whether it is B or — B, or whether it is great z gl : : Mn at Aileet arcing perties of the triangle remove the ambiguity. } of the two , 4 ; i rp sides, sought for is opposite to the less of the two given sides, ie B is less than A and therefore necessarily less than 90°: irhi But if 6 be greater than a, or if the angle, whose val) i sought for, is opposite to the greater of the two given sles then there are two triangles, possessing precisely the same it is supplemental, and therefore equisinal to the corresporm angle in the other; and in this case the solution, therefoi' ; necessarily ambiguous. opposite to a, the less of the two given sides a and 0b, then if we describe an arc of a circle with centre C and radius CB, and B’, forming two triangles CAB and CAB’, whose angles B and B’, opposite to the common sidell or 6, are supplemental, and therefore equisinal to each ot: CA, and the angle A in one triangle, and CB, = CB, CAn the angle A in the other. * For the greater side of a triangle is opposite to the greater anglell conversely, a conclusion which is proved in Geometry (Euclid, Book 1. Prop! and may be easily deduced from the fundamental equations in Art. 875: a be greater than 6, then sin A is greater than sin B: if A be greater thar! then B is less than 90° and therefore less than A, inasmuch as the st the angles A and B is less than 180°: but if A be less than 90°, then Bu be less than 90°, and also less than A: for if B be greater than 90°, itsi plement z — B must be less than 90°, and therefore less than A, since sin (a 5 is less than sin A: but ifw—B be less than A, then = is less than Al which is impossible: it follows therefore, that B is necessarily less than .! that the greater side is opposite to the greater angle. 253 b. ambiguity, therefore, which we have shewn to exist in ,¢e under consideration is not a consequence of the imper- ti of our formula, but is essentially dependent upon the con- io of the problem proposed. “e sides of a triangle are 17.09 and 93.451 and the angle Example. pote to the greater of them is 93°.16’: to find the angle pcte to the less. Je logarithmic formula is | log sin B = log sin A + log 6 - log a. log sin 93°.16 = log sin 86°.44’ = 9.9992938 : . | log 17.09 = 1.2327421 | 11.2320359 log 93.451 1.9705840 log sin 10°.31’ 9.2614519* the conditions of the problem had been reversed, and if the le 17.09, 93.451, and the angle 10°.31’ opposite to 17.09 had eisiven, to find the angle opposite to the greater side 93.451, e eration would have stood as follows: | log sin 10°.31’ = 9.2614519 log 93.451 1.9705840 11.2320359 log 17.09 1.2327 420 log sin 86°.44’ or log sin 93°.16’ = 9.9992939 ‘olem. 5. Casz 3. Given a, 6 and C: to find c, A and B. Given two 1e equation (a) in Art. 878, gives us are a c' = a’? + 6? — 2ab cos C, aed since — 1.9705840 is equivalent to 2.0294160, (Art. 865) we may put the ‘e in the text under the following form log sin 93°, 16’ = 9.9992938 log 17.09 = 1.2327421 arith. comp. log 93.451 = 8.0294160 log sin 10°, 31’ = 9.2614519 ig 10 from the characteristic. the ohne 2 The expression a? + 4—2ab cosc, whose terms are con ‘“‘adapted by the signs + and —, and whose values cannot be asteil ease without the performance of arithmetical operations of consic+ Ee difficulty and labour, is said to be not adapted to logaih computation: and formule generally are said to be adayd not adapted to logarithmic computation, when they consist ao ducts, quotients, powers or roots of easily calculated ter which do not require a mixed application of loparici numerical computation in order to determine their valuc Its vague- ness. Solution of the pro- posed prob- lem without the use of subsidiary angles. 254 where c is expressed in terms of the given quantities a, cos C. | When c is found, A, and therefore B, is determined) Case 2. | The phrase, however, and its usage, is somewhat vag’ indefinite, inasmuch as it does not determine absolutely tl ditions of greatest convenience in the selection or preparalt formule for the purposes of computation: in other wos formula which is not adapted to logarithmic computation, a0 ing to the technical usage of the term, may admit, in many; of more rapid computation, by mixed or even by merely f metical means, than one which is so: the selection theref e one or the other, when more than one method are with reach, must be determined by the judgment and experienc sometimes by the taste of the computer. The introduction of subsidiary angles will enable us tot the expression | or any other, whose terms are connected by the signs +i to logarithmic computation, and we shall reserve for the folly Chapter, which will be devoted to their theory and us further consideration of the various methods which are em] for this purpose; in the mean time the following metho enable us to solve the problem proposed. a’ + b?—2ab cos C Since > = sin 4 (Art. 877), we have (Art. 792, No. 8) ib eaten a—b snA-—snB Ata Rye tan ( ) _—_—_—_—_ 255 is known, since C is given: it follows there- on (5) -Gadons A-B yn, and therefore is known: consequently A+B A-B A+B A-B A= 9 as 2 and B= 9 - 7) own, when c may be found by the methods given in Art. t the two sides of the triangle be 27.04 and 74.67, and Example. : angle included between them be 117°.20’; and let it be ed to find the remaining side and angles. this case a ='74.67, 5 = 27.04, aerefore a—b=47.63 and a+6= 101.71, A+ B=r- C=180°— 117°.20’ = 62°.40’, aerefore A+B _ sya, tan (- . =) = log tan (===) + log (a — 6) — log (a + b), log tan 31°.20’= 9.7844784 log 47.63 = 1.6778806 11.4623590 | log 101.71 ~ = 2.0073637 | log tan 15°.55'= 9.4549953 | A+B. a Shy ier 4 nee 5 = 15°.55 256 Therefore A = 47°.15’, and B= 15°.25’. Also g se e and therefore | nA | log c = log 74.67 + log sin 117°.20’ — log sin 479.15! log 74.67 = 1.8731462 log sin 117°.20’ = log sin 62°.40’= 9.9485852 11.8217314 log sin 47°.15’ = 9.8658868 log 90.333 = 1.9558446 We have thus determined the remaining side and anp the triangle. Given three 886. Case 4. Given a, 6 and c to find A, B and ( sides. also the area of the triangle. The cosines (Art. 879) of the three angles of the ei expressed by be ee cos 4 =—_____ , 2be 2 3: 4 a act —b cos B = —_____ ,, Qac 2 2 2 Oo paar cos C =—_____ , 2ah The ex- which are not in a form adapted to logarithmic computi fiche inasmuch as the separate terms of which the numeratc ee ns composed cannot be ascertained without a logarithmie, e€ angies. Roe not Somewhat laborious arithmetical, computation. ambiguous, are not It should be observed however, that the three angi eatin determined by these formule without ambiguity: for thes ey hag of the angle will be positive or negative according as it is § greater than 90°. If we make k = jis (s— a) (s — b) (s pas EE, where s = 257 j1 arithmetical process, we shall find (Art. 880) é k k é Qk 7 edad and-sin -C=—,, c ab v ' bec’ : 2 sin B = a zh are in a form adapted to logarithmic computation. Ve thus get log sin A = log k + 10 + log 2 — (log b + log c) = 3 {log s + log (s — a) + log (s — 6) + log (s — c)! +10 + log 2 — (log 6 + log c). Che log sin A, which is here sought for, is the tabulated loga- 2k be Ve have already shewn (Art. 881), that & is the area of the i gle, expressed 1 in units which are the squares described upon (inear units in terms of which the sides are expressed, whether (2s, feet, yards or miles. an n, exceeding by 10 the proper logarithm of t should be kept in mind that the angle 4 determined by iformula is ambiguous ; triangle can be greater than 90°, and the greatest angle yposite to the greatest side, this ambiguity is confined to angle alone ; and this angle will be greater or less than (+c?- a 2be etive or positive: or in other words, according as 6? +c? ss or greater than a’. ¥ according as the expression for its cosine is ‘Again, if the three angles 4, Band C be determined from oreceding formule, and if, assuming them to be acute, their }or d+ B+C=7, then their values are correctly determined: f not, this equation will be satisfied, (assuming that the arith- bah and logarithmic processes are correctly performed), by ig that value of the equisinal (Art. 776) angle opposite the test side, which is greater than 90°. et the three sides of the triangle be 107.9, 193.4 and 217.12 3: and let it be required to find the three angles of the ligle and its area. ‘on IT. Kk but inasmuch as only one angle of where the factors are formed therefore by a very easy and The ex- pressions for the sines of the angles, though am- biguous, are adapted to loga- rithmic computa- tion. Example. bo cr oe) aejrorng b = 193.4 Rael 7512 2) 518.42 s =259.21: logs = 2.4136518 a= 107.9 $—a@=151.31: log (s—a) = 2.1798676 sraepy.21 6 = 193.4 s—b= 65.81: log (s— 6) = 1.81882919 $= 2)021 C= 217.12 | s—e= 42.09: log (s—c) = 1.6241789 | 2) 8.0359902 | log k = log 10423.06 = 4.0179951 | 10 + log 2 = 10.3010300 14.3190251 log b = log 193. 4 = 2.2844565 | log c = log 217.12 = 2.3366998 = 4.6211563 log b + logc log sin 29°. 46’ = log sin A = 9.6958686 Again, | 10 + log 2 + log k = 14.3190251 log a = log 107. 9 = 2.033021 4 log c = log 217.12 = 2.3366998 log a + log ¢ = 4.3697212 log sin 62°. 51’ = log sin B= 9.9493039 259 ina angles A and B being found, the angle C is known: t may be found as follows: 10 + log 2 + log k = 14.3190251 log a = log 107.9 = 2.0330214 log 6 = log 193.4 = 2.2864565 4.3194779 log sin 87°. 23’ = log sin C= 9.9995472 Therefore A = 29°.46' B = 62°.51' Cee B70 25° 180° Also = 10423.06 square yards = 2.1535 acres, which is the ait of the triangle. ) eee cations of the preceding formule might be proposed, wceh, in some cases, would enable us to obtain the required nalts, by shorter and more expeditious processes than those w ch are given above*: but it is not our object, in this Chapter, tive a complete treatise of practical Trigonometry, but merely ‘xplain generally the method by which we adapt our formule, me of their most useful applications, to arithmetical and loga- mic computation. ; | Thus, it will readily follow, from the investigation in Art. 880, that ee? —C¢ ens Oo cae ye ab Pere Sone S See) US vee) 2 ab ; therefore ( C_ s—a)(s—b) tan S Re eee! GD ST Te f therefore we make Ray/{E-VG—Oe—e) | 2 nmetrical expression with respect to a, b, ¢, (and equal to the radius of the cle inscribed in the triangle) we get ta Ms - tan = Ss t = 3 a a —— 7) tT ae | 7 a ai oD 2 s—a’ ey essions which are not only more easily calculated than the expressions in the te but are also free from ambiguity. See ‘‘ Geometrical Problems and Analytical F\oule, with their application to Geodetical Problems ;” p. 21, a very ingenious 41 original work by the late Professor Wallace, of Edinburgh. \ CHAPTER XXXIV. ON THE USE AND APPLICATION OF SUBSIDIARY ANGLES hatte 887. A suBsrpI4RY angle is one whose sine, cosine, ) cubadiagy Bent, &c. does not exist in the primitive formula, but wh angle. is introduced for the purpose of modifying its form, or of ti litating its computation, by means of logarithms or otherwise. veal inthe A subsidiary angle is generally necessary in the copia adaptat : TAs Rpiaria of expressions consisting of two or more terms connected 4) Diatee the signs + and —, and which cannot otherwise be adaptecy computa- logarithmic computation, as will more fully appear in many) tion which h i hich foll are not the examples which follow. otherwise adaptible. > ; F Tithe ace 888. Let the expression to be adapted to logarithmic ¢) ofa+b. putation be a+6, where a and 6 are positive quantities, wh without the aid of logarithms. In the first place ( =) a+b=a\(1+-—): a es b a | if we make tan’ @ = Pe where 6 is the subsidiary angle, we sl find a a+b=a(1 + tan? 6) =a sec? § = —_-: ( ) cos? @ and therefore log (a + 6) = log a+ 20 — 2 log cos 6 = log a + 2 arith. com. log cos 6. Example. Thus, as an example, let it be required to find the yat of the expression * Therefore log tan = 10 + 3 (logb — loga). 261 397 cos 14° + 410.7 sin 67°. 10’ log 397 = 2.5987905 log cos 14° = 9.9869041 12.5856946 log 6 (rejecting 10) = 2,5856946 log 410.7 = 2.6135247 log sin 67°. 10’ = 99645602 = 12.5780849 log a (rejecting 10) = 2.5780849 logb-loga= 2) .0076097 log tan 45°. 15’ = 10.0038048 arith. com. log cos 45°. 15’ = .1524183 2 2 arith. com. log cos 45°. 15’ = .3048366 _loga = 2.5780849 log 770.56 = log (a + b) = 2.8829215 9. Let the expression be a~6, where both a and 6 are In the case sive, and a greater than 6. ofa— b. | the first place, we find a-b=a(1-7), a b . »7 is less than 1: we may assume, therefore, sin? @ =— , a 1 gives log sin 0 = 10 — 3 (log a — log 6): us get a—b=a(l1-sin’ 0) =a cos’8, herefore log (a — 6) = log a + 2 log cos 0 — 20, i 262 So b In a similar manner, if we make — = cos? 0, we get a a—b=a(1-— cos’ 6) =a sin’ 0, and therefore log (a — 6) = log a + 2 log sin 6 — 20*. —b In the a 890. Let the expression be of the form ae where a «+b" and 6 are positive and a greater than b. If we make : = tan 0, we get b pees a—6 a Lie tan 0 — aaa os 0). a+b AS 1+tand aT ag ) ees Thus the expression (Art. 886), : A) = 5h tm (4) i, ( 2 aaa 2 * ‘Thus, if two sides a and b and the included angle C of a triangle be we find a_sinA_sinfr—(B+C)}_ sin(B+C) basin Ba sin B sin B and therefore, dividing by sin C, _@ __ sin(B+C) —————~ =cotB tC ‘ bsinC sin B sin C pidaiodary 2 and consequently j pee See C cot ain C cot 7 , bh cos C Sarre a ). beosC If'we aks a cos? 0, we get ’ a a sin? 0 . = i= 0) eee cot B re c ¢ cos? 0) = bsinC’ { ay: : , acosC In a similar manner, we shall find, making cos?@ oN ye: b sin? 6’ Sg ri sin C This is a very convenient and expeditious method of solving a triangle two sides and the included angle are given: see Prof. Wallace’s Geom Theorems and Analytical Formule, &c. Edinburgh, 1839. p. 42. 263 tan ()- = tan (45° — @) tan (é : =) > Let the expression be ,/(a’— 6’), where a is greater In the case 0 4 V(@? — ke 7 = 008 0, and we get J(@’ — 6’) =a,/(1 — cos’ 0) =a sin 0. us the expression in Art. 885 or c=,/(a* + b°— 2ab cos C) =,/{a?+2ab+b?—2ab (1+ cosC)} : 2a6(1 + cos C) (a+b), /{1- oe a cay ama Ware 4.ab cos? — = b eto) (a+ SCE =(a +b) ,/(1— cos’ 0) = (a + 6) sin 8, 2 Jab cos — cos 0 aaa, ‘this case | C g cos 0 = log cos > + log 2 + 4 (log a + log 6) — log (a + b) log c = log (a + b) + log sin 0 — 10. 2. Let the expression be ,/(a’+ 6°). Th the cane f Ma2+ b?). we make tan all we get Raa” se J(a’ + 6’) =a,/(1 + tan’ 0) =a sec 0 a cos 0° 264 Thus the expression in Art. 885, or | c=, /(a+5°— 248 cos C) | =,/(a?—2ab +b? + 2ab -2ab cos C) gee ry /\r+ , 2ab0 ae 2 SO! 4ab sin? = | {ney ee aa | | = (a — 5) ,/(1 + tan’ 6) | Ree) | | cos@ ? 2,/ab. sine a—b Inthecase 893. Let the expressions be of the form | . of a £4/(a2+b2). a+ ,/(a’— 6°), or a+,/(a’+ 6’). If, in the first of these expressions, we make a = sin 0, a we get a+,/(a@-b’)=a(l1 + cos 0) =2a cos" 3 ’ ° id 0 a—,/(a’*—b’)=a(1— cos 0) =2a sin’ >t. * The same expression is also adapted to logarithmic computation by rk a tan 0 = which gives C (a —b cos = c= (a—b) cos > sec 0 = : + The roots of the equation v—pr+q=0 Pp p? are 9 = Ata oe a) 265 ff, in the second, we make w = tan 0, a w get | a+,/(a’+ 6) = 5 (cot 0 + cosec 6) (Art. 791) 6 =b cot P) > a—,/(a’ + 6’) = b (cot 6 — cosec 0) (Art. 791) Oy =—6 tans. 394. Let the expression consist of a series of terms connected In the case ° of a series Ww 1 the sign 5 such as of terms a+b+ct+d+ &c. ect In the first place, sign +. syvay b a+b=asec’9, if tan’0=~—. Again, replacing a sec’@ by a’, we find a+b+c=a'+c=d' sec’ 0’ =a sec’ @ sec’ 0’, , Cc Cc Cc a a+b = asec’é Similarly, replacing a sec? sec?’ by a”, we find a+b+c+d=a"+d=a" sec’ 0” =a sec*@ sec’ 0’ sec’ 0”, ar if we make 2 : ad = sin 6, P iiwo roots will be expressed by 0 0 1 poe m2 — p cos" 5 and p sin 5° 2 tis assumed that q is less than a a7 The roots of the equation «?—px—q=0 p p° are ba/ (i+) ; and if we make aM = tan 0, |two roots will be expressed by 0 8 V4 cot 5 and — /q tan 5- Vou. II. Li a 266 7 d d : d if: tan® 0" 2 4) ee eee a’ at+b+e_ asec?0@ sec’ 0’ and so on, whatever be the number of terms of the series, When the 895. Let the expression consist of series of converging tet series 1S . ° convergent Connected alternately with the signs + and —, such as and the terms con- a-—b+c—-d+... nected with the signs In the first place + and -—. Bag i e a—b=acos*?0=a’, if 7 sin’ 6. Again a’ a cos’ @ a—b+c=ad +¢=—y = — ar ; cos?@’—s_ cos’ & ” c if == etAnlu. a a-b If we include a fourth term, we get a cos’ 8 cos’ 0” =a a—b+c—d=a"- d=a" cos’ 0” =——_,___ =a", cos’ 6 . ya a | if — = ——__ = sin’ 0”. a’ a—b+¢ For five terms, we get mt 2 29” gO BCE dh eal Cote cos’ 0 cos’ 0’ cos? 6 and similarly for any number of terms. In the case 896. Let the expression be a continued product, such ot a Con- / a am (a +b) (a’ +b’) (a +b")... factors of the form or (a _ b) (a’ = b’) (a” a b’’). a+b or a—b. In the first case we make b b’ 6” tan’ =~ : tan’ ’=—,, tan’ 0" = 7 5-- : and we get q i K q Ca OC q cos? @ cos? 6’ cos? 0”... © (a + b) (a’+ B’) (a + 6”) ... 267 n the second, we make 4 6 ’ b’ : i sin’@=-, sin’ 0 = 55 sin? 0” = —, ,... a a a we get (a — b) (a’—b’) (a” — 6”) =aa' a” cos’ 6 cos’ 8 cos’ 0”. 197. There are many applications of analysis, particularly in Conver- onomy, in which it will be found to be extremely useful to one Bh ert expressions, such as mule by 3 means of asin A+b cos A, subsidiary angles. acos A +b sin A, equivalent expressions of the form asin(d+@), or acos(A+9), neans of the subsidiary angle 0. Thus, if we make a= ,/(a’+ 6"), = = cos @ and we sin 0, we a asin A+6 cos A=a sin @ sin Aa sin 0 cos A =a sin (A + 0) =,/(a’ + 6’) sin (A = 9), a cos A+b sin A =a cos 0 cos A +a sin @ sin A =a cos (A = 8) = ,/(a’ + 6’) cos (A = 8). An expression of the form a sin? A — 6, re 6 is less than a sin? 4, becomes, by making 6 =a sin’ @, a sin? A —a sin? @=c sin(4 + 8) sin (A — 9), odification of its form which is not unfrequently used. CHAPTER XXXV. ON EXPONENTIAL AND LOGARITHMIC SERIES. Develope- 898. THE exponential expression a‘, the series into wit ment of a*. i¢ may be developed, and the various logarithmic and ot} series which may be deduced from it, enter very extensiy) into analytical enquiries, and deserve the most careful exami. tion of the student. We shall begin with the developement) * into an equivalent series. If, in a*, we replace a by 1 +(a—1), making a®*={1+(a-1)!}’, we shall find, by the binomial theorem, (Art. 680) a=1+2(a—1)+0(e—1) So Lie te 1) (a— a) a 48 if we actually multiply the factors of the exponential coefficie; (Art: 688), and collect together the terms which severally inital the same powers of «2, denoting their successive coefficients } k, A,, A;, As, &c., we shall get xe a =l+ka2+A,-~ gt As syretgtee (TE ein ar cual where k =(a- 3 i a series which possesses, as we shall afterwards shew (A 906), some important properties; it remains to determine coefficients A,, A, .-. : * For the last terms of these several products are —x, 1x22, —1x2x3a, 1x2x%3x 47, 269 in equation (1), we replace x by y, we get 2 a@=1+hy+ A, 9+ 4, — a ste (2). we multiply together the two sides of the equations (1) mxa@—-at-=l+ka+ A eae hak seh +hy+k olf: (3), y” Lye pe | ee | 7, pe eS eo aioe y° $A, 4+ 1 RR: it if, in equation (1), we replace « by x+y, we find . (ety , (2@+y)’ a’ gett katy) +, ae a as ui becomes, by expanding the successive powers of «+ ' it involves, a a oe 1+hker+ A, — A wey if “wi ica hit ou gyn xy +k +24,, +3A, Se rks A), y ay 1s often peti) 2 ry 'e thus obtain two forms of the developement of a**’, which n'aly become identical in form as well as in value, when the Jctive terms of both series which involve x and y similarly entical with each: thus if we compare together the terms 2 two series which involve y only, without its powers, Q2Ax 2 kx a or A,=k’, 2 2 kA, xy _ 3Asa y 1 Paws PRS ey BaD vy = Or A.= kA, = kh Conditions of identity of equiva- lent series. Is the ex- istence of the series for a* ne- cessary or not? ee 270 if we replace A,, A,, A,,...by their values, we get a od : ots Le) Lee: a series in which & and wx are similarly involved. a=1+k + ects e sf 3 is one of great importance, and capable of very extensive jp cation: it is presumed that the form of the series wh equivalent to a* is independent of the specific value of aa therefore the same when wz is replaced by y, or by 4 or by any symbol or combination of symbols whatsoell if a’*’ be equivalent to the product of a® and a’ f, values of x and y, then likewise the series for a®*’ 4 equivalent to the product of the series for a® and a’ und, same circumstances: and this equivalence of the results fi are obtained implies that they are identical in all those x in which 2 and y, one or both, are similarly involved: thus enabled to obtain a series of equations, expressint conditions of identity, by which the form, or analytical ‘ld of the successive coefficients are determined. 900. It may be further observed that the existence equivalent series for a*, or of a series which shall posseit same analytical properties with a’, is a necessary conseq? of the binomial series in its general form, and involves’ principle of the permanence of equivalent forms” no fith than it is involved in the binomial theorem: but whe binomial or any other theorem is once established, whatey the principle upon which it rests, it becomes one of the kd and acknowledged results of Symbolical Algebra, and mi employed in the deduction or establishment of other conelhid equally with the results of the definitions of Arithmetical Als it is thus that the bases of Symbolical Algebra are perpe'l enlarged, and the great principles which present themsel3 the first and most elementary of the generalizations wh! requires, are speedily replaced by other and successive lir3 the long chain of consequences which are found in the prd of our enquiries: such results, therefore, though their exis per se may not be necessary, yet become necessary results } considered with reference to each other and to the proposit upon which, they are finally dependent. | 27] =1+1+——~ + ——_—__- ae . (2 Eee 1. Resuming the consideration of the equation The nume- E rical value | he? x? kh? x a ) ob ke + — +——_ +... a ofa®. ) Pee Bee re” ee, (4), ) 1 » aall find, by making =F; | , ; 1 1 ridly converging series, from which its numerical value may leulated to any required degree of accuracy: the aggre- 1 of 14 of its terms gives us 1 a* = 2.7182818, i is correct as far as the last figure: but no finite decimal yer can express its accurate value *. . is usual, in all. cases, to denote this number 2.7182818 ie symbol e; it is the base of Napierian logarithms (Art. which are exclusively used in analytical formule. 2. If, in equation (a), we replace x by 7Z> we get We series or e”. = Fd rv fend + at + + ———-+...: asiee hes eee *The process for this purpose is very simple and expeditious: divide l, and /ecessive quotients which thence arise by the successive natural numbers : jn of the quotients, increased by 2, is the number required: thus, Ce: 16666666 4) .04166666 5) .00833333 6) .00138888 7) .00019841 8) .00002480 9) .00000276 10.) .00000027 11) .00000003 12) .00000000 2.71828180 thas been shewn, in Art. 203, that e is an incommensurable number: it is pable, therefore, of being expressed by any finite decimal. - 272 1 “ but a*=e, and therefore a‘'=e*: we thus get i x e=1+x2+— +—— ~+... This series, which constantly presents itself in ei |) comes convergent from its r™ term, if 7 is the first whole n greater than z. | 1 The Na- 903. Since a‘=e, we get pierian logarithm a=e, of a. and consequently / is the logarithm of a to base e, or, in words, it is the Napierian logarithm of a. ‘ Modulus of 904. Again, since a=e", we find a system of logarithms. a®= &* =n, | " » we shall obtain the correspilli where kx is the Napierian logarithm of n, whilst x logarithm to the base a: if we multiply, therefore, Na logarithms by ; or low logarithms to base a; and the multiplier zis called dulus (u) of that system. The series 905. Again, since k = log a, we get (Art. 890) for log n : y ( ° y in terms 0 a—1 fy aimee 1 Soca n. 1 = (a —1)-—-*—~* + S —S 4A 4+... L og a=(a—1) 5 : n or replacing a by n, " (n—1)? (n—1)? (n-1)* log n = (n — 1) - ET Nem eet Roar &c. ((, a series of great importance in analysis, and which ae the Napierian logarithm of a number in terms of the mal itself. et series 906. This series is divergent when n exceeds 2; but! or log n may be ‘simple modification of its form will enable us to make made con- al] cases, as rapidly convergent as we please. vergent 1n all cases. I 1 i For (n™)"=n and log (n= )" = m log (n™) =log n, and i fore log n =m log (n™)=m{(n™ —1)-4 (n™ —1)?+ L(n™ —1)—9 273 f we suppose 7 to be greater than 1, it is always possible, 1 y suming m sufficiently large, to make n™ exceed 1 by a uitity as small as we choose: thus, if x= 10, we find 102 —1 =2.162277, — (oe ES | — II “778279, 108 —1 = .333521, II 1 10% —1] -154781, . 1 ' 102—1 = .074607... 1 1027 1= .000000000536112, hh, multiplied into 2°*, gives 1 9 (10* — 1) = 2.3025851, rult which expresses the accurate value of the Napierian githm of 10, as far as it goes: for it will be found that ) 1 eecond term of the series (d), which is 2 (102" — 1)’, pos- s3 no significant digit in the first 8 places of decimals. J]. The reciprocal of log 10 or The modu- 1 1 i ee tabu- = 434294481, rithmeet log 10 10 2.3025851 t modulus (Art. 904) of tabular logarithms, or it is the factor hich Napierian logarithms, when calculated by the series e last Article or by other and more expeditious methods, gre to be multiplied, in order to reduce them to the cor- sinding logarithms of the tables. 08. The preceding method of calculating tables of loga- Practical lis is theoretically perfect, but the operations of multipli- antes t 1, division and extraction of roots which it involves, makes ceune f method 0 Re Ss calculating , Briggs, in his Arithmetica Logarithmica, has found logarithms. 1 . 102% 1 = ,000000000000000127819149320035 : sme work contains the tabular logarithms of the natural numbers as far 17 to 61 places of decimals, an unrivalled monument of labour and ingenuity, ‘period when the various expedients which modern analysis supplies for i ing such calculations, were almost entirely unknown. le Wg Mm | a | O74 its application for such purposes extremely tedious and e1 rassing: but in the calculation of the logarithms of sucesj numbers, we shall be enabled to resort to much more exped'o methods, founded upon logarithmic series, or upon the diffeiy of successive logarithms, some of which we shall proce notice. Loga- 909. Since (Art. 905) rithmic series. logn=(n—1)—4(n—1f+3(n-1)-... we shall find, by replacing x by 1 +2, | i «ike Ve Log (1+2). log (1 Tl) at er ieee (e), a series of very simple form, and which is very frequ referred to. | 910. If, in this equation (e), we replace z by - x, we Bt 2 3 4 f Log (1-2). BY NEE SFE 2 CaO APs OB | og (1-2) log (1 — 2) = 5-3-5 CAL 911. If we subtract the series for log (1-.«) from th: log (1 +2), we shall get iq lie SC aarks 1 = —+—+... ei Log 2 912. If in this equation (g), we make ke rs nd i q” 2 fore x=P—1, we get Pad =. — q\? —qQqys lo Pog {P—4, 4 (P=2) +4(2—2) + &e} hy. |) °q \p+q °\p+q/ *\p¥q “ § Rak Sal ST ey | | h ay If we make p=q+1, we get oer = Oar’ ae i fore Clie \ 1 1 1 OR =2 opt +4 @qniy 1 * &el, or 1 \ , log (q+ 1) =logq +2 jg + SOVPS ioe (2 275 ' (his is a series adapted to the calculation of the logarithms f accessive numbers, being rapidly convergent in all cases, and ne particularly so, when the numbers, whose logarithms are o ht for, are large. 13. If p and q are resolvible into factors which involve Expression h first power of a number x only without a coefficient, and pt hie lore “ey seloade | a number fue fraction be of such a form as to decrease rapidly i 3 terms of ee he loga- ith f vin n increases, then we may express the logarithm of the rithms 0 nr test of the factors of p and q, in terms of the logarithms eA fine other factors, and of the series (h), See P g Pat, q,. 4 (P= q ‘+ &eh. log? . Par p+q Thus, let p= a? and g=2*—1=(#+1)(«—1), and therefore eee we find log (= 2 log 2—log (# +1) —log (#~ 1), i therefore | log (x + 1) = 2log x — log (x — 1) ' 1 1 os ee es Se Ree . lene *baaoypt > y which the logarithm of a number is expressed in terms of hlogarithms of the two next inferior numbers, and of a rapidly “verging series. Again, if we assume p=2 (4-7) (@ + 7)? = 2°— 98.2*+ 2401 2’, q = (a — 3) (w+ 3) (a@— 5) (a + 5) (a — 8) (x + 8) = 2° — 98 2* + 2401 x? — 14400, oi Bes 7200 p+q «—98a* +2401 2? — 7200’ | we find 1 therefore log (w + 8) =2log (x +7) + 2log x + 2log (x — 7) — log (# + 5) — log (« + 3) — log (# — 3) — log (a — 5) } fies (—8)—2 | 550; sagt 7200 Tf z=100, the first term of the series, or | 7200 x°— 98 r* + 2401 x? — 7200 +&e.p. a y 276 has no significant digit in the first '7 places of decimals: forhj and higher values of « therefore it may be altogether ; lected, and the calculation of the logarithms of a considel; succession of such numbers may be made dependent uporth logarithms of 8 inferior numbers only. Roa 914. The practical and most expeditious method of garithms of lating the logarithms of successive numbers will be found the numbers, dependent upon the Theory of Differences, which will fornth subject of a subsequent Chapter of this work: we shall niicg it no further in this place, than is requisite to justify the {les which are given in Art. 870, for finding the logarithm’ of numbers which are not in the tables, and conversely. Inasmuch as »+1l=n (1 + -) > we find log (n + 1) =log n + log (1 +5), and therefore i alga + &e.: coca fete 2 ean, i" log (n + 1) —- log n= 4 ; and, if x be large, - may be taken as an approximate vahi oi this difference: thus, if = 10000, aie will have no signifant digit in the first 8 places of decimals. Again, since (n+ a)=n (1 + =) , we find log (n + x) = log n + log (: ey . and therefore ax? x’ Re log we ee log (n + x) — log n " 3 tan | | and if « be small compared with x, then : will be the app) mate value of the difference of log (x +x) andlogn. If thero A=- be the approximate value of the difference log (n +1) — log n, we shall find that «A will express the approximate value 0 difference log (n + x) — log n. | it a, the difference of the logarithms of +a and n will e early proportional to the difference of the numbers n+ jer (Art. 869): and that if the difference of log (+1) and g be given, the difference of log (n+) and logn, when « jail compared with n, may always be found approximately y simple proportion. f the logarithms under consideration be tabular and not igerian, then log (+1) ~log n=» (5-554 &e.), n n! } 2 log (n +2) log n= (=~ 25 +&e.), re p is the modulus of tabular logarithms (Art. 907): if \ .erefore be assumed to be , then the formula log (n+ a)=logn+nA 7 furnish an approximate value of log (n+ x): it will be at m: seen that this proposition is the basis of the rules in i 870, to which we have before referred. | Limits of series. In a geo- metric series. CHAPTER XXXVI. ON THE LIMITS OF THE VALUES OF SERIES PROCEEDIN(A CORDING TO ASCENDING OR DESCENDING POWERS SYMBOL, WHICH IS CAPABLE OF INDEFINITE DIMINt OR INCREASE. 915. Tux following propositions relating to the limi the values of series, proceeding according to ascending old scending powers of a symbol, which is capable of inde} increase or diminution, will be found to be extremely vf in many enquiries, and more particularly in those which x the application of Algebra to the properties of curvilinear fst The complete theory of limits, properly so called, is only parjlly involved in them, and will be more expressly considered) subsequent Chapter of this work. 916. If we make x= ca the sum of the geometric sels: a2t+ar+ax’?+a2°+... is equal to a+06. For the sum of this series, when x is less than 1, is — : é (Art. 432): and, if we make &=7——, we get a _a@(at+s) | a+. has It will follow, therefore, that a value of x, in such a s€és, may always be assigned, which will make its sum (s) differ Dm 6 6 ; : i Seat LE — oa fro. / is less than eT say a (ax NE will make s differ fre by a quantity less than 6. 279 } 1 A limit of a series or expression is a fixed value to yh it approaches nearer than for any assignable difference, yhit a symbol upon which it is dependent is indefinitely ‘y\ished or indefinitely increased, but which it never attains vhit the value of that symbol is different from zero in one sor from infinity in the other. 18. It will follow therefore that the limit of the value of he eries i 31 first term or a: for it has been shewn that a value of « Ataxrt+axr*+ax'+... na be assigned, which will make its sum differ from a by a; tity less than any other that can be assigned: and it never tts to this limit, though it may approach indefinitely near o , whilst « is different from zero. 19. If in the series | a +“ +5 —; g+5 at. ie : ats pr beding according to inverse powers of x, we make x= a “Be sum is equal to a+6. ‘or, if we make a : , this series becomes | ataytayt+ay t+... and therefore r= ato . whie sum, when y= + ei! 18 a+o. t will follow, therefore, that, in such a series, a value of play always be assigned, which will make its sum (s) differ ro, its first term by a quantity less than any which may be rea: for if 6 be the quantity thus assigned, then any value (a+0) acne will make s less than a+ 6. of greater than “ é a _ che limit, therefore, of the value of this series, when 2 is int finitely (see Chap. xxxvil.) great, is its first term or a. a | a 20. The terms of the geometric series Atarx+ara«e+arcn2'+.. («), severally either equal to or greater than the corresponding ; ib of the series ; } ] : (8), A+ AL + Agu? + a,xr°+... Definition of a limit. The limit of a geo- metric series pro- ceeding ac- cording to the ascend- ing powers of a symbol is its first term. The limit of a geo- metric series pro- ceeding according to inverse powers of a symbol. Formation of a geo- metric series which is a supe- rior limit to a given series pro- cording to 280 | | ceeding ac-if 7 be the greatest inverse ratio of any two consecutiv oq powers of the same symbol. efficients. a, a, 4a, t In the first place, let — be greater than —, —, and ally a a, a, ER : : (Sh, ; a sequent ratios of a similar kind: then we have = a | ‘ Pepi | therefore a,=ar: 7 is less than r, and therefore a, is es than a,r, and therefore also less than ar’, since a,=ar: \ less than r, and therefore a, is less than a,r, and therefor es than ar*, since a, is less than ar*: and similarly for all yb sequent Coefficients of the series: it follows, therefore, thath first and second terms of the geometric series (a) are equ te the first and second terms of the series (8), but that allt subsequent terms of the first series are severally greater those corresponding to them in the second. a In the second place, let any other inverse ratio, sue a — » and not the first 42 be the greatest, and therefore eu n—1 ; a to r: we have consequently % less than r, and therefore a, 5 fs . f than ar: = less than 7, and therefore a, less than a,r, al @ fortiori less than ar’, since a, is less than ar: and so on, | we come to a,_,, which is less than ar’-!: the next to =r, and therefore a,=a,_,r, which is less than ar’, ce a, Qn-1 a,-, 18 less than ar"~*: and similarly for all subsequent cifli cients: it will follow, therefore, that the terms of the seriesa), after the first, are severally greater than those correspon nig to them in the series (/). | ; A a If the successive ratio, ae —, —, as far as the nt 1 2 an , be equal to each other, but greater than all those wc n—1 fe a follow them, then if ie the n first terms of the series, 2) and (8) are equal to each other: but all the subsequent ten of the series (a) are severally greater than those corresponiig to them in the series ((). 281 ‘We may assume the coefficients of the series (8) to be posi- ti, and to increase perpetually as we recede from the first ten: but the proposition which we have demonstrated above w. be true a fortiori if the coefficients form a decreasing series, oif one or more of them become zero or negative. ‘It will follow generally, therefore, that if r be the greatest irerse ratio of any two consecutive coefficients, the terms of the gimetric series (a) are severally equal to or greater than the ecesponding terms of the series (@), and that consequently a i. of x may always be determined, which will make the at of the series (@) differ from its first term by a quantity le than any which may be assigned: thus, if rea, ane 8 th-efore Sarat)’ the sum of the series (a) is a+6, and thefore the sum of the series (8) is less than a+8: and the lat of its value (Art. 917) is a, or the first term of the series. Thus, in the series | Examples. yA 1x24+2x3r+3x402+4x 502+... (8), thvalue of r or of the greatest inverse ratio of two consecutive coficients is 3: and the terms of the geometric series | 24+2x324+2x 3?4°+2x 3®2? +... (a), ar severally equal to or greater than those of the series pro- : ped (8): if SCRE the sum of the terms of the series o (4 will differ from its first term 2 by a quantity less than 8. y | \ the series proposed had been a 1+1x22+1x2x3a°+1x2x3x 4a?+... th ratio of the n to the (n—1)™ coefficient is x, which increases infinitely : the series is therefore infinite, if indefinitely con- ir 2d, whatever be the value of x. . } 4 + i /21. Proposition. If there be three quantities whose On the vaes are expressed by series proceeding according to powers pan of ie same symbol, and if, for the same value of that symbol, whose first i terra 1s thi first be necessarily greater than the second and the second unknown, ths the third; then, if the first and third series have the same Pt which a Ey is included fir term or the same limit, the first term or limit of the second in value be- Dom II. Nn 282 | tween two series will be necessarily equal to it: it being assumed thatthe others ° . ; : [. which have NVerse ratio of any two consecutive coefficients of those aie Peeea is always finite. a Thus, if the three series representing the three quantities > A+ A,2+0,0°+a,2°+... (ia b6+6,24+ 6,07 + b,2° +... } where the inverse ratio of any two consecutive coottclal always finite, then values of 2 exist which will make ei arithmetical values differ from their first terms by quanie less than any which may be assigned: let such values olth series be a+8, 6+8,, a+5,: and since it is assumed that! is greater than 6+6,, and 6+ 6, greater than a+ 6,, it will Hi that the values of a—b+6—6, and 6—a+6,-8, are arithmetical and positive: and, if possible, let us supp b=a+d or a—d: in the first case, the preceding expres yn become ( , ‘ ‘ ( A+C,L+ Cyt + C0 +... ( —d+6-—6, and d+6,-6,, and in the second d+é—6, and —d+6,-6,: and inasmuch as these expressions are necessarily positiy it will follow that, in the first case, 5-6, is greater than d,t} greater than d+6, which is contrary to the hypothesis, rai is supposed less than any quantity which may be assigned :ind in the second case, 6,—6, must be greater than d, or 3, gr ter than d+6,, which is also contrary to the hypothesis, sin’? has been supposed less than any quantity which may bias signed: it follows, therefore, that 6 is necessarily equal 1 which is the proposition to be proved. — | This is a proposition of very extensive application, inasrid as it will very frequently happen that an expression, whir i not capable of direct development into a series by the ai ¢ assumed definitions or known theorems, may be shewn, by a considerations, to be included in value between expressions witl admit of direct development, and which have therefore ascer' able limits; and it will follow that, if those limits be the sue the limit of the unknown expression or undeveloped seri / necessarily the same likewise. ya 283 22, The correct notion of a limit is not easily formed, mae ing a in nuch as it is necessarily connected with a conception of a the con- ception of » of existence of magnitude, either in itself, or in Some 2 Timit. quitity upon which it is dependent, which is incapable of arimetical or geometrical representation: and like all our i ete therefore, which ultimately involve considerations of ne) oF infinity (Chap. xxxviut.), it is entirely negative: it is omhis account that it becomes of the utmost importance that wrshould confine our attention exclusively to the definition C 917), which we have given of it, and altogether discon- ne: it, like other definitions, from every consideration which is,ot essentially involved in it. A limit, in conformity with its definition, may be zero, but A Tit n infinity: for though we are incapable of conceiving zero panant her as ome of the successive states of existence of magnitude, we not infi- a capable of conceiving its existence in a state in which it he dirs from zero by a quantity less than any which may be Misied, and therefore when zero becomes, as it were, the fixed litt of the definition: but we are utterly incapable of con- cing the existence of a quantity which is not infinite, but Ww ch at the same time differs from infinity by a quantity less th any which may be assigned: and therefore, under no ‘cxumstances can infinity answer the conditions of a limit, wich the definition assigns to it. } 7 The series 923. Iv has been already shewn (Art. 808) that the vu nee of a in the exponential expressions Ora. a*+a~ a*—a~* and The mea- sure of an angle which is assumed in CHAPTER XXXVII. ON THE SERIES AND EXPONENTIAL EXPRESSIONS FOR TI) SINE AND COSINE OF AN ANGLE. 2 2,/=1 | is indeterminate, as far as it is dependent upon the defingr of the sine and cosine of an angle only: we shall proceecit the Articles which follow, to shew that it ceases to be ind ¢ minate when the measure of angles ceases to be so. . The exponential series deduced in Chapter xxxv give 1 eee gy eee ya Hatin bth ¥ Tele hea sa Bh. So eR i he? x? IP x? kA x me Tes Sees ae where k=loga: we thus find i An be k? x? ke x* COs x = ea pa Benn 2 1 enteoug ead sin # = C~a ki x ki x 1 ai yates tre past If we further replace & by c,/—1, these series become, COS & 1 Godt oe oa ee lee ea, cea? cn? 924. It may be easily shewn that these series will sati) the equation (Art. 758) sin’ + cos*a = 1 j when substituted in it, whatever be the value of c: but if 285 1e, as we have already done (Art. 746), x or the measure Art, 746, St ° . ; . ; determines *, angle, to be the ratio of the are which subtends it to the fe!emm ds of the circle in which it is taken, then it may be demon- of the sym- r:dthat 1 is the only value of c, in the series for sinx and is hag A 9s, which will satisfy the conditions to which it leads. volve. ‘or, if we assume 2 or the measure of the angle BAC to be 1. then since sin z= si reet maecP AB. CP. i—_ = a AB BEC BEC ncinasmuch as ex ca? in=cr - ~~ +-——~——_-... Pr 1.2.3" 1.2.3.4.5 ve ret | sin Cx e cat SRT PE al EEE eS, sin x vive c, which is the limit of _ (for it is the limit (Art. 918) f 1e series which is equivalent to it), is required to be deter- ni d. ‘or this purpose, we observe that the ratio ee or arc BEC here BEC: but CP=AB sina, and the chord BC=2 AB sin | GP : chord BC ** ie since the chord BC is less than ricer than nd ° BG aes *ax 2 cos = sin — \ CP sin 2 ae a? thord BC a: = 2 (ATC 75) = Coss: | Q Serlcp m(again, inasmuch as the tangential line BT is greater than the | GP BT: sin « CP -EC* : mt pe = fe Mali rGEC*, and since tan z FP = AB? We 8% tng O88 = BT 6} d therefore less than baat aR 4 CP ag is less than xe BEG? ™ : | This may be easily shewn from geometrical considerations: if from the xtiaity C of the arc BEC, we draw Ct a tangent meeting BT in t, then we have st, Ct: but Tt is greater than Ct, being opposite to the greater angle TCt: be wre BT is greater than Bt + Ct: again, if from the middle point E of BEC, ve aw the tangent cEb, meeting Ct in c and Bt in b, then we have bt+ct, : | greater | 286 sin It appears, therefore, that = is interposed in value-bere are arranged in the order of their magnitude; and sinc {h first and third have a common limit, which is 1, them c of the second series is identical with it, and therefore to 1. (Art. 919). x e e,e cos 5 and cos x: or, in other words, the quantities c? x? c' x cos—-=1— + ete = Laer er ls rae sin x c8 x? c® x4 eenmeE AG Be Uo. 8 WI. Scio k 6 Bo 479.8 5 eh Aer Cx c' a SS See 2 A Se i eCR Ce greater than bc, and therefore Bt + Ct, and therefore a fortiori BT, greatitha Bb+bc+cC: in asimilar manner, if we bisect the arcs BE and CE, an'to their middle points draw tangents meeting Bb, be and Ce, then the sum’ fl tangents thus formed will be less than Bb + be + Cc, and therefore « jtio than BT’: by continuing this process, we should increase the number and di ni the magnitude of these small tangents, until their sum, which is necessari than BT, shall differ from the are BEC by a quantity or line less thaa that may be assigned: or in other words, the arc BEC is necessarily lesh BT: by asimilar course of reasoning, the arc BEC may be shewn to be @ sarily greater than the chord BC. Again, if we take r for the radius of the circle, the series Yi HH H rtang, 2rtans, 2?rtan 5B ee 2°r tan = will express BT, and the several sums of the successive circumscribing tan 1s in a similar manner the series . a ye . ack Qr sin 5» 2?r sin 53 » ‘eS antl r sin S41 will express the chord BC and the several sums of the successive inscribed chis the ratio of the nth term of the second series to the nt term of the first is x v : lgj cos — 2” +) sin Seal Qn Se —— eee 5 £, L 2" tan — coo: gerd and the limit of this value is 1: this is another mode of arriving at the conclusi' the text. This relation of the chord, are and circumscribing tangent, is one of | damental importance in the application of Algebra to the theory of cus it applies to the arcs of all curves of continuous curvature, as well as to of the circle. x? a* os f= 1-—— —- . Too tease 4 a a® PE Lae woes eo 4 x is the measure of an angle, which is formed by dividing «ec which subtends it at the centre, by the radius of the in which it is taken. e 15. It appears, from the preceding investigation, that ) c=1 and k=c,f—1=,/-1 inasmuch (Art. 903) as k=loga=,/-1, e nd eV=1, at=etV-1 and a-*=e7*N-1, tore at+an? et N=14 e-2V-1 (SUPE OB EE es gg een ee ae a STL 2 2 iis a an e-2V=1 1 sin ¢ = ——— = ——___—_—__ | 2 final ai hus obtain determinate and explicit exponential expressions he sine and cosine, adapted to the commonly assumed mea- of an angle. 26. Inasmuch as cos 2 = ———___——__ 2 MM os Ni eae ~i /- lsin = Ti Cee get, by adding | cosz+,/—1sina=e*V-1, by subtracting | cosx—,/—ising =e-*V-1, )27. It follows, therefore, since (Art. 811) ar ey sD art 1 cos + J—1sin=" f—1=6¢ ened 3 1}R, Determi- nate expo- nential expressions for the sine and cosine. Proof that e*Ni=3 = COs @ AY Pa sina. 288 that the x roots of 1 are expressed by 20 = 4m N= 2(n-1)r oye ayaa = and if the series be continued, the same values recur in thear order. 928. Again, since COs 44/1 sine slesay aie or tere. ale it follows that log J—1=— mA] Conclusions of this kind are symbolical only, and cann'b made the basis of calculation: and it is only in those ¢asi j which the symbols or signs which such expressions invye are capable of being interpreted, that we are enabled to mn ceive the nature of the connection between them and the spf numerical or other value which they may be shewn to denote pe ae 929. Among many other important conclusions which low immediately from the exponential expressions for the ne and cosine, it will be found that the formule of Demoie which are given in Art. 806, are immediately deducible {in them. For, if in cosv+,/—1sina=e*V-1 we replace x successively by @ and by n0, we get cos 6+ ,/—1sin@ =e9V-1, and cos 70 + ,/—1 sinn@ = en? V=1— CREME (cos 6+ ,/—1 sint and if in cos x—,/—1 sinz=e-*V=1 we replace x successively by 0 and by 20, we get 289 cos 0—,/—1 sin 0 =e-9V=1, 30. The substitution of the exponential expressions for Develope- hsine, cosine or tangent of an angle, in the goniometrical natn onule in which they are involved, will enable us to resolve tan B= ha, in many cases, into series, which are not only very remark- ene in their form, but which admit of very useful applications. by means Thus, in a triangle, if a, 6 and the angle C be given, to de- GES ie nentials. sine A or B, we find a@_sind_ sin(B+C) | C cos B sin C Panwe*s sino ot “asin Da therefore sin B 6 sin C cosB a—beosC f e replace the sines and cosines by their equivalent expo- eial expressions, we get eBV-1_ ¢—-BV-1 b (e€ V=1_ e-€V=1) in therefore eBV-1 BO ly ee ea eee SHY To be % —~BV=1 na betN=1 e of we take the logarithms of the two members of this equa- io we get log e28 V-1 = log (a — be-© V=1) — log (a— be€¥=1) = log a + log (1 ih ec V-1) — loga—log (1 ~2 eeV=1) : b 6? bie a oB rt rey hence... 2 3 Bega Oo Seveyai te 3 e~scN=1— 4 a 20° 3a ict m2 cc zr—y c—d * For if yaa? we eet =e g teat and conversely. Art. 296. Oo a 290 or _b ecv—1_ ahaa 3? efON alee mary 2,/=1 2a?" Bolte eee pe Pea ut sin 2C aoe 3C =— sae a8 “er sin Ae oe This series converges rapidly if 6 is small compared with , Develope- 931. Similarly, in the equation ment of the ale cay tan «=m tana, ~™mlan@s which very often occurs in the applications of analysis, wale ex N=T_e-sV=i m(enV=1_ ee a1) i SAE kae hy cite and therefore |. "bael ae te oy oo Val = e2aVa1 i Dice ; e2aV—1 from whence we derive, as in the last Article, w= a+ —— ; singa+3(™— +) sin 4a +86. If we replace m by tan@, this series becomes, since m— ow 7 w tan (0 — 45°), =a + tan (6 — 45°) sin2a + 3 (tan 0 — 45°)’ sin 4a +. Series for 932. Inasmuch as its tangent. cel (e#V-14 e-2V=1) we readily find 1+,/—1tanex | onl tan a and therefore, taking the logarithms " ax, f/—-1=2{/—1tane+4(/—1 tanz)+4(/- “i tan a) or e2tV 1H x=tanz—$tan®r+itan’r—... 291 ries* remarkable for the simplicity of its form, by which ch measure of an angle is expressed in terms of its tangent. {f we make 2 the measure of 45°, and therefore tan 45°= 1, w get F Beas 1.00 Siping A 4 Bihar 4 a'ries which is very slowly convergent. 4 I (f we make x the measure of 30°, and therefore tan zx = 73 : wiget ) T 1 Le bY | 1 1 saa mt? a + ° 6 =a 3°37 er Age ries which converges with great rapidity. 383. Among many other conclusions which are deducible The sines ° P . : small fr the series for the sine and cosine of an angle, it may eae ave eadily shewn that the sines of small angles are very nearly 2° !y pro- portional to pryortional to the angles themselves, a proposition whose truth the angles wihave assumed in a former Article (798): for if @ and 0’ be themselves th measures of two small angles, we find, neglecting those poers of @ which are higher than the third, that 62 sin 0 ‘ 6 a eee Ayr. Shas A Zs sin 0 go" 0 6 7 Itio which becomes very nearly that of Sy when @ and 6’ ivery small, or differ very little from each other. Thus, let ingles measured by 0 and 0 be 1’ and 2’ respectively: inas- mth as an arc, which is equal in length to the radius (Att. ; 1 2 dy, 30 — t Has Fe (4, contains 57.3° = 4438’ nearly, we get 4438 and eT in therefore sin 1’ : =1%x 25 57 = $ x 1.0000000 | = 4 nearly. | This series was discovered by the celebrated James Gregory in 1771: f 15 times the series in which tanz =< we add that in which tana = x : ubtract from their sum the series in which tan av = a? the result will be pression for 7: it was by this method that the remarkable approximation, oned in the Note to Art. 747, was obtained. . - 292 Syn 934. Again, the ordinary formule for the solution of tr; solution of Sometimes give results, which are less accurate than thosevh ae ar are commonly obtained, when they involve the cosines of; volve the which differ little from zero or 180°, or the sines of those; eles of differ little from 90°, inasmuch as, under such circum: ( pee Be their values change slowly for considerable changes in thiya from zero, Of the angle: the series, however, for the sines and cos@ or 180", °F angles, will enable us to deduce formule which are adaje eae i such extreme cases, and where the minutest changes ojya fer little in the quantities sought to be determined will become m from 90°. diately sensible. Thus, in a triangle, where 4 and B are very small andyl C is very nearly 180°, we get (Art. 885), if + — 0 be the mas of .C, e=a’?+ b?+ 2ab cos 8 2 z 6? — 7 ae 7 +2ab 1-3 = (a +b)*— ab6?, | omitting all terms of the series for cos @ beyond the seed being too small to affect the result within the limits f recorded places of figures ; by extracting the square root, y ab? c= (a + b) " —— cath nearly. Thus, if C= 179°, a=20, 6 =16, we find gat ee f 57.3 10 3? ! ~ 81 cant = 35.999424 nearly. Again, if « and § be the measures of A and B*, we ge 8s . a sin d=a—— nearly, 6 3 sinB= Pp —=& Sor sin C=sin(4+B)=a+p-“*8) * When we speak of the sines and cosines of angles, it is indifferent’! the angles are expressed by their measures, or by degrees and minus it should always be kept in mind that the numerical values of the meas angles enter essentially into the series which we are now considering. 293 | ie c sin A e( -%) esi C . =e 2508) pay, asp e add these expressions together, we find +6 a+b=c capB orc ee + + 3cap 1+3ap’ nvenient and very accurate formula for expressing one side of a triangle of ihe two others and of the measures of the adjacent angles, provided les are small. Meaning of 935. Tue terms infinite and indefinite are frequently } the terms infinite and Ndiscriminately by mathematical writers, though, if due re; indefinite, Infinity and zero. CHAPTER XXXVIII. ON THE RELATIONS OF ZERO AND INFINITY, . | e . } rg ‘ ’ e was paid to propriety of language, they should be distingui| from each other: they are negative terms whose meaning 14 be defined by that of the terms finite and definite, which» respectively opposed to them. | A finite number, a finite line, a finite space, a Jinite t) would denote any number, line, space, or time, which is ei¢ assigned or assignable: whilst the term definite could propl be applied to such of those quantities only as were alrél assigned or determined: in other words, the term Jinite is nr comprehensive than definite, being limited only by the poe possessed by the mind of conceiving the relations which magnitudes, to which it is applied, bear to other magnitis of the same kind. Ny | An infinite number, an infinite line, an infinite space, ani Jinite time bear no conceivable or expressible relation to a fil number, a jinite line, a finite space, or a Jinite time: the tix indefinite, properly speaking, when applied to these quantiti would imply nothing more than that they were not determi or not assignable. | 936. Magnitudes may be infinitely small as well as infini' great, and the abstract term infinity should be, properly speakis equally applicable to both, though it is confined, by the usi of language, exclusively to the latter, whilst the term 4 exclusively applied to the former: the general term infinit | superseded by the specific terms immensity and eternity in / case of space and time*. } | * The phrase for ever, though properly expressing infinite duration of ti? is commonly applied to denote infinite repetition as well as infinite time: t the processes which never terminate are said to be continued for ever: 295 37. The symbol « is used to denote magnitudes which Symbols of nfinitely great, in the same manner that the symbol 0 is pm aoe al to denote those which are infinitely small: they are con- el by the equation = co, and —=0, where a may be a though indeterminate magnitude: in the first case, we con- ve onsider 0 as the quotient of a divided by ~; such results 1 be interpreted by considering the dividend as the product ie divisor and quotient: thus, there is no finite number shh, when multiplied into zero or an infinitely small number tional or decimal) will produce a finite product: there is inite line, which multiplied into sero or an infinitely small n will produce a jinite area, and similarly in all other cases. 38. The product of into 0, or of an infinitely great and Different a finitely small number, line, or other magnitude, in the sense atte with we have attached to those symbols and terms, may pro- cen ey m du a finite result, but it does not follow that it must do so: symbol. equation am oo is universally true, when a is finite: but the ai? equation is also true, when a is infinite, and therefore jiced by «: in other words = co: in such a case, the injity denoted by the symbol «, on one side of the equation, is|tid to be infinitely greater than the infinity denoted by the bol on the other: for one is equivalent to ho and the sym q 0 c 0 tir to . , and the ratio of the second to the first, or = x= “= or infinity. 3389. The mind is as incapable of conceiving the relation one ae oilifferent orders of infinities as it is of conceiving infinity infinities Bs « : ° denot sf, and it is only when the relation between them is the fea Phe essary result of symbolical language, and of those general ame im come defi- f) series which are said to be continued in infinitum, are also said to be con- cepts iid for ever: but it should be observed, that the notion of infinity of time stances of ‘sely associated in the mind with all our notions of indefinite repetition. their origin are known. Example. 296 laws of their combination, which the rules of Algebra iy upon them, that they can become the proper object ( 9 reasonings: for the same symbol is used equally to ng all magnitudes which are infinitely great, and the same sph 0 to denote all magnitudes which are infinitely small; the symbols « or 0 were used, in a course of operatils ordinary symbols, when the circumstances of their usage slwe a common origin, and therefore indicated something beyid. mere symbolical identity, we must adopt the results of cer tions upon them, whether finite or not, in the same maniy any other necessary results of Algebra: thus, if the sybo co and o denoted the same infinite magnitude (such as — re . ax@ when x=1), the relation of ax @ and bx, or VW qd bxa a, be equally 7? as if o, in this ratio, had been replaced a ordinary symbol of algebra: but, if the course of our reason 3 should call our attention to ax « and bx w, as simply den g two infinite magnitudes, without any reference to the relion which the circumstances, in which they originated, me them bear to each other, we might properly represent a’ and 6x by the common symbol o, and the relation bet»en axc and 6x would become alt AES indeterminate. , But though the symbol © might not have the same ovit in the expressions ax © and bx «, yet if the precise symbcci conditions of their origin in both cases were known, the i ile termination of their relation to each other might be remo d: thus, if the symbol » in ax originated in the a 2 rere when «=1, and the symbol o in bx « originated inhe ‘ 2 expression when x=1, we should find 1-2? 2 a x CAD. L——7 b Ree ee “WW fbi < 1-2 a) ax0 ] = ~-— a rae (when «= 1) eer 297 inasmuch as it appears that | ax(1—2*) a(1+z2) b x (1-2) iT b ll values of « whatsoever, and therefore when «=1, it will fo w that, under these circumstances axxo 2a bxao 5b’ mcthe relation of the infinite magnitudes represented, in the nu'erator and denominator of this expression, by the common yol ©, or by the common symbol 0, under another and eqwvalent form, is necessarily that of 2 to 1. 40. It may be useful to illustrate, by some other examples, Further ex- thorigin of the indetermination which exists, in certain cases in asa Be exjessions which involve the common symbol 0 or the common origin of different sy bol ©. orders oN [he sum (s) of the series annie | axz+azr*+ax°+... an the value of x, are denoted by the common symbol 0, when z zero: but inasmuch as RY | pT OtGL+ ant... wise value is a when z is zero, it will follow, that if we denoted thirst zero by 0’ and the second zero by 0, we should find 0” 0 = a, orn other words, a definite relation would thus be shewn to t between the zeros denoted in this particular instance by 0’nd 0: but all traces of this relation would be obliterated if ' and 0 were replaced, as is usual, by the common symbol 0 1e common and only representative of zero, and if no reference w made to the circumstances of their origin. Again, the sum (s) of the series | axr’+az*+ax'+... e | issero, when x is zero: if we divide s by 2x, we get lap ; ae 2 3 mate tar +a2r +... | x Wor. II. Pp 298 ‘ . . . ef e . . § f which is zero when « is zero: if we again divide : by a, we bell s 2 = = a te ax + ax + eee x whose value is a, when wz is zero: if we denote the zeros i responding to s and to -, when 2 is zero, by 0” and 0/, a the zero corresponding to x by 0, we shall find o=0, al 4 , 0 . 0 therefore or 0, whilst ance in other words, the relation tween the two first zeros is zero, but between the two last isi: but if we had denoted the three successive zeros 0”, 0’ and 0 the common symbol 0, and had suppressed the considerat of all reference to their origin, their relations to each other wor have been altogether indeterminate. Again, the sum (s) of the series | | a 2 is infinite when 2 = 0: and similarly the sum s’ of the series a a : —=+-+a+ar+ar*+... yee is also infinite, when « = 0: if we denote the first of these infini magnitudes by « and the second by ’, we shall find, sin / Aaa pad hy ull | ote and therefore se iaa $ $= = when x«=0, or, in other words, the ratio of the infinite magnitudes denote, in this particular instance, by ’ and o, is also an infinite ma; nitude: if, however, we had denoted these infinite magnitud without reference to their origin, by the common symbol o, should have lost all traces of the relations which existed amon them. a 941. It appears, therefore, from the preceding consider ofexpres- tions, that expressions, which, under certain circumstances, beco sions which 0 oe Recome 5 Of &» may admit of determinate bald which are differel 299 fm 0 and ow; and that this determination may always be ected when such expressions admit of equivalent forms, which 0 CO : - not become a or —> under the same circumstances. 942. Thus, the obliteration of a common factor, which be- Oblitera- nes 0 or « for specific values of the symbols which it involves, pat oe uy furnish an equivalent form of the original expression, which facta 0 ee) : which be- es not become Dirac? under the same circumstances. lee 0 or La As an example, the fractional expression ——; x*-a 3 0 becomes 0 Examples. ‘hen e=a: but if we obliterate the common factor «—a (which ‘comes 0 when x= a) of its numerator and denominator, the action assumes the equivalent form a+ ax+ x’ Gra ie Pe Sa* 3a : , hich becomes paweg = when x ='a: its value, therefore, is no nger indeterminate. | Again, the expression + az®°—Qa’?2’?+ lla’ a—4a* x'—ax>— 3a’x*+ 5a°x— 2a 0 : ; ecomes 5 when 2 =a: but if we obliterate the common factor »-a) of its numerator and denominator, the original fraction . r+4a : ssumes the equivalent form , which becomes when xr+2a 3 943. But if different powers of a common factor exist in When dif- ‘he numerator and denominator of the original fractional ex- scene Ae yression, the discovery and obliteration of what is common to common | Joth of them by the ordinary rule, will leave a residual power cai be f this factor in one of them, which becomes 0 or « for the spe- baci vific values of the symbols which are under consideration: under tor. uch circumstances, the value of the fraction is either zero or nfinity. Thus, a—2x is found to be a common factor of the nume- vator and denominator of 3a°— 10a? x + 402° + 82° Te wea ] | 7 > 300 which is thus reduced to the equivalent form 3a°— 4an—427 | | a+ Qe : | but inasmuch as a — 2~ still continues a factor of its -— . ‘J a . * . the value of the fraction is zero when x =—: in a similar man rhs) the value of the fraction 8a? — 4ax2—42° } a°—2a°x—4ax*+ 82°’ ? O a ; Lip which becomes i when « = a and which also becomes 7 —_ > when the common factor a—2.2 of its numerator and denominair : ; ; a ; is obliterated, will be found to be o when x= a for a— 22 sll continues to be a factor of its denominator. | When the 944. In many cases, however, the expressions which becoe expressions 0 a which be- — oy —, for specific values of the symbols which they invol: come O CO 0 = i 9 or = are do not present themselves under a rational form, and the ordin: y irrational rule, therefore, for detecting their common factors ceases ; or involve : exponential be applicable. In the absence of the more prompt and cert pee methods of finding their determinate values, under such c- metrical cumstances, whenever they exist, which will be given in a si- Ue sequent Chapter of this work, we may very frequently succel in determining them by the following method. . Let the fraction 5 become or = when 2=a: replacer by a+h, and develope P and Q (by the aid of the binoml theorem or other methods at our command), according to pows of h, whether integral or fractional: divide the series whi result by any powers of k, which are common to all their term, and subsequently making h zero or infinity, we shall find t? determinate value of a? which is sought for. Examples. Thus, the expression J(a— 2) + ,/(ax — x’) J (a* — x”) 0 . becomes 5 when a =a: we replace x by a—h, and it becomes 301 Jh+/(ah—h’) J(2ah — h’) ey yeaa q IN a Sal 2 at h Ig 7 h Oy a 6g ee we al a— Z log a — log «’ shich becomes + when x =a: if we replace « by a- h, we get 2 a’—(a—hy na*-'h—n(n—-1)a"~’ — 2a loga—log(a—h) hile 20° h n—l n—2 ree —n(n—l)a cans in rer i ey Fen oP =na", when h=0 or «=a. Let the expression be @—1-log(1+2) x d | * It may be observed, that the common factor of a series of irrational ex- ptessions of the same order, or which are reducible to the same order, may be ound, by finding the common factor of the rational expressions which are ‘neluded under the same radical sign: the common factor, when subjected to whe common radical sign, which is thus found, is the common factor of the rrational expressions under consideration. Thus a — x is the common factor of a— 2, ax —2* and a®— 2°: and therefore V(a-1) is the common factor of the numerator and denominator of V(a—2) +4/(ax — 2°) Vay’ wy l+/zr which becomes, when reduced, Warn) 302 which becomes 5 when «= 0: if we develope e* and log (1+ we get x? x a of —— + ——____. —- ] — (7 —- —+ —-.... Le ere Cee ( ) which becomes 1, when x= 0. Let the expression be sin 26 cos (n — 1) 8 Se > which becomes 4 when r=27 (Art. 844, Note); if we ma: §=22-—h, this expression becomes n' h® (n 1) h? ~sin nh cos (n= 1) h_(#~7 9,3 + 8)- "7g + 8 — sinh o sive. om n®h* (n — 1)? h? re 1.20907 ee hh? rk prpere Pte oe oe =n, when h =0*. * An expression which assumes, for specific values of its symbols, the fo. 0 0x co, becomes — 9 by inverting the expression which becomes o : thus . Tw is Oxo, when r= 5° but if we replace tanz by — pat Bacon [yee ‘ - - - -~ - a he ee ; ; ae ie fe and its determinate value is sh In a similar manner, an expression such r 1 r—1 log’ : which becomes © —co, when x=1, assumes, by reducing the fractions to a cor mon denominator, the eiuiaiar form clogr—a+]1 (r—1) loge ” -_— —..- 2. : 0 , which becomes 0 when x =1: its determinate value is 4 303 945. In the preceding examples, the values of : antl << Daasein which asidered with reference to the expressions in which they ori-9 ,.2 ;, ate, were determinate, zero or infinity: and if zero and infinity theeiek ee sre considered as comprehended within the range of those pos- indetermi- jle existences which we are capable of contemplating equally mea th those which are the objects of calculation, we might regard em as determinate in all cases*: but in the examples which follow, - : ; or — will be found to be symbols of indeterminate magnitude, '* In the contemplation of continuous magnitude, we are accustomed to con- Jerzero and infinity as equally included in the successive states of its existence, “th those which are finite and determinate: thus, whilst the measure of an angle T “16 . : : : . ysses through all values between oi h and 5+ h, including oe its cosine 1s sid to pass through sero, and its tangent through infinity under the same cir- mstances, and we regard zero as the cosine, and infinity as the tangent of , equally with the cosine or tangent of any other determinate angle. _ It may be stated however as an objection to our considering such values as suc- ssive states of existence of the cosine and tangent, that they merely mark a point of ansition in the condition of those successive magnitudes, from positive and negative, hich may be considered as an interruption of their continuity : but it is easy to pro- yse examples of a transition through zero and infinity, in which no such change of fection or sign takes place: thus if a curve Cp touch the line ACB in C, and if it be mmetrical with respect to it (such as the = p vc of a circle) on each side of the point C: ien if, from successive points between P ad p, we draw successive ordinates or per- = “9 G 7 3 endiculars from points in the curve, such ;PMand pm to AB, they will pass through zero at C, and will be identical in mag- itude and sign at points equidistant from C: in a imilar manner, if the two curves APD and Bpd be ymmetrical with respect to an indefinite line CF, Di | 1d hich they never touch, but which they approach earer than any assignable distance, (or in other ords, if, in conformity with a phrase which will e afterwards explained, the indefinite line CF be common asymptote to these curves), then for 2 qual distances from C, the ordinates PM and ‘m will be identical in magnitude and sign, and vill increase indefinitely as they approach C: in ther words, PM may be said to pass through nfinity, in passing through C, without any change n those conditions of its existence, which the igns of algebra are capable of expressing. iF 304 or the appropriate analytical forms by which the failure of} requisite conditions of determination is indicated. Examples. Thus, in the equation az—b=a'«—b’, the ordinary process of solution gives us b’—b a—a’ a = If we suppose 6’=6, and a’ not equal to a, we get z=0. If we suppose a’=a, and Db’ greater than 4, we get f= 0. If we suppose a’=a and b’=64, we get art In the first case, 0, and in the second, , are the only val: of z, which will satisfy the proposed equation, in conforr distinguish them from ordinary symbols. | In the third case, all values of x will equally satisfy the e. ditions of the equation, which is therefore absolutely inder. expression for z could assume, which would not give a é terminate value, whether finite, infinite, or zero: the interje tation, therefore, which we have given of the expression = this case, is absolutely determined by the circumstances wl! characterize its occurrence. Again, if we solve the simultaneous equations ax+by=c (1), a’x+by=c' (2), ” * The proposed equation, under these circumstances, becomes an ident equation whose symbols, when not actually assigned, may assume every pos value, + For the value of x may be expressed by — equally with > which becomes = when a’=a and b’=b. 305 the ordinary process (Art. 395), we shall find _ Wc—be’ ree Tae? § a’b?’ fetid Vib aon If we suppose a’= ma, b’=mb, c’=mc, we shall get _mbc—mbc ev ~‘mab—mab 0’ _ mac—mac late ~ mab—mab 0° jich shew that x and y are indeterminate as far as these ex- essions are concerned : for the equations (1) and (2), under such cumstances, are not independent equations (Art. 391), the second ing derived from the first, by multiplying each of its terms by 2 same symbol m, and it furnishes, therefore, no new con- ion for the determination of the symbols which it involves. _ The values of 2 and y, in the equation ax+by=c 2 separately, though not simultaneously, capable of all values aatsoever, and are, therefore, when separately considered, abso- tely indeterminate: but if we consider them as connected in 2 proposed equation with each other, the value of one of them termines the other, and one of them only can be considered as solutely indeterminate with reference to the other. ' 946. In performing the operation of division and of the ex- Indetermi- iction of roots (Arts. 586, 650 and 693), when the operations do es ate it terminate, and when the symbols are not arranged, as required Se | Arithmetic and Arithmetical Algebra, in the order of their whose agnitude, even when the expression is arithmetical in its ori- lugar nal value, we obtain diverging series, whose terms are alter- ean itely positive and negative*, and whose values are therefore determinate: for, under such circumstances, no approximation made to a determinate value by the aggregation of any number ‘their terms: and it has been observed that this indetermination -referrible to the neglect of that arrangement of the terms * Mr De Morgan, in a singularly original and learned Memoir in the Trans- tions of the Cambridge Philosophical Society (Vol. VIII. Part 1.) classifies ivergent Series, as alternating or progressing, according as their terms are ternately positive and negative, or have all the same sign. Vor. IT. Qa i The values of cos. and sino. 306 which are the subject of the operations performed, which rules of Arithmetic and of Arithmetical Algebra prescribe :1 such cases also the restitution of the arithmetical order of rangement would convert alternating divergent, and there indeterminate, series, into others which were convergent and der minate, to whose values we can approximate as nearly as) choose, by the aggregation of a sufficient number of their terr It would be premature, in the present stage of our prog's in analysis, to speculate generally upon the origin and charae of alternating and progressing divergent series. Do alternay divergent series originate in expressions, which are always art metical in their value, though they may not be arithmetical the circumstances and process of their developement? Do if gressing divergent series always originate in expressions wit are not arithmetical, either in their origin, in their arrangem or in the process of their developement*? Will divergent sev either of one class or the other, when substituted, in algebrats operations, in place of the expressions in which they originé or in which they are presumed to originate, give correct | equivalent results? | These questions, which are of fundamental importance! many of the most delicate and difficult applications of analy have given occasion to much dispute and controversy, and ly not hitherto been satisfactorily settled. It will be prudent thi fore for a student, whenever he is called upon to consider 2 infinite values, whether of series or of expressions under a Cf nite form, to consider very carefully the circumstances in wh they have their origin, before he draws any conclusion respect} the interpretation which they are capable of receiving. ( 947. The symbol will sometimes present itself in pressions, which have periodical values, when the result 1 2 3 Thus (+a) 1 — 2x7 + 3x? -— 477 + &c. and 1 ——— = 1+2r+32? + 42° +&c.; Ga) +32? + 40° + &c.; and when « is greater than 1, the first series is an alternating, and the sect 1 1 —————ee GN ————-— are der th (laa) Gia) , under these cumstances, equally arithmetical in their value, though not in their arrange? progressing divergent series: but ; 1 J Se of their terms: but can we consider ———,, as also arithmetical in its or (1-7) 307 apparently, if not absolutely, indeterminate, in consequence our being unable to assign the position in the period to hich it corresponds: thus, the values of cos@ and sin@ are riodical, their limits being 1 and — 1, through the intervals of hich they pass, whilst 0 passes through intervals of value equal Qn. What then is the value of cos and sin «*? What is e value of tana or cot«? The answers to these questions are unded upon indirect considerations, to which we should never ur, when other resources are within our reach: they form ints of transition between the demonstrated and acknowledged ths of deductive science and the less definite results of me- physical speculation. * See Mr De Morgan’s Memoir above referred to, Sect. 111: it is usual to sign 0 as their values, inasmuch as 0 is the mean of their successive periodic ues: for it is equally probable that the value in question is @ or — a: and if e value is assumed to be wnique, 0 is the only value which the doctrine of nances, or similar considerations, would assign to it. aS CHAPTER XXXIX. ON THE DECOMPOSITION OF RATIONAL FRACTIONS WITH (C¢ POUND DENOMINATORS INTO PARTIAL FRACTIONS. Rational 948. Ir is usual to distribute arithmetical fractions into t algebraical : : fractions Classes as proper and improper (Art. 95), according as the numera yan is less or greater than the denominator: and the same denomi classes - tions are extended to rational algebraical fractions according as’ er an : . : : J iteieen dimensions of the symbols in their numerators are lower or hig than those in their denominators: thus : is a proper and. 2 . . . e e 1 + i an improper numerical fraction: and similarly, Tt ; Is a pro x Levers 1-2’ and is an zmproper algebraical fraction. Improper 949. Inthe same manner, that improper numerical fractic Peete are reducible, by the division of the numerator by the dei oe to ; minator, to an integer and a proper fraction, so likewise impro; Tationa expression algebraical fractions are reducible by a similar process of divis: Staber to an integral expression and a proper algebraical fraction: fraction. ; 30. d A j this manner, i338 reduced to the mixed number 2 ie? and sit larly + srs is reduced to «?—1+—~— where «’—1 is an integ Tine 14+2° algebraical expression and Tag 2 Proper algebraical fractio and similarly in all similar cases. Example 950. But the proper fraction, which is thus obtained, rate ofan 22¢ther resolvible into other and more simple fractions, when algebraical denominator is resolvible into factors: thus suppose it was fraction. : 7 : quired to resolve the fraction val xv? —Tx+12 | 309 . / : : : , 9 partial and more simple fractions, whose denominators are ‘+. factors «x —3 and x—4 of its denominator x*—7x+12: | this purpose, we should assume r+1 inn 4 i B SoTeHI2:- (2-3 2-4 yere A and B are unknown and are required to be determined. If we add the partial fractions together, we get r+1 _Ax-4A+Br—3B x?—Tx+12— a? — Tr +12 . Ld therefore 2+1=(4+B)2-(44+3B). _Inasmuch as the value of x is indeterminate, this equation n only exist for all values of z, by supposing the corre- nding terms of both its members to be severally identical ith each other; we thus get | A+B=1, 44+3B=-1, ad therefore d=—4, and B=5: we consequently find r+i1 et i 5 e—7e+12 «-3 2-4 951. The principles which are involved in this process, Principles tough simple and obvious, are of considerable importance which are involved in ad of very extensive application. this pro- CeSsse x+i1 eS In the first place, the relation between Be Oe’) is not merely one of equality, but also f identity, where « may have any value whatsoever, provided is simultaneous in both: and we thus get as many conditions, Yr equations, as there are corresponding terms on each side, nd therefore as many equations as there are unknown quan- ities A and B to be determined. Identity and equality. General process for resolving rational fractions -with de- composable denomina- tors. When the factors of the deno- minator are possible and un- equal. 310 More generally, if any two series, proceeding according, the same law, such as atbet+cu?+dx?+...... (ie a+Bat+yartoae+...... (2) be identical as well as equal, the corresponding terms of eih series must be identical also, which will furnish as many eq tions as there are terms in each: for if not, the value of ti be no longer indeterminate, admitting of all values betwin zero and infinity inclusive: we have assumed the truth of ts proposition in the investigation of the series for a* in Art. 898. If the members of the equation zr+1=(4+ B)x-(4A4+ 3B) were merely equal to each other and not identical, the symbs A and B would be indeterminate, and the value of «x (as woud appear by the ordinary solution of the equation) would be :- pressible in terms of them, and therefore be dependent upon the: it is the additional hypothesis of the identity of form as wl as of the equality of value of the two members of this equatil, which furnishes the equations for the determination of A al B, and leaves x indeterminate, as is the case with the symbs involved in any other identical equation. 952. The following method of resolving rational fracti with decomposable denominators, into an equivalent series f partial fractions, is general, and will serve to illustrate the p ceding observations. Let s be a proper and rational fraction, and let- a+ba | a factor of N: if we make N=(a+62)Q, we get Me Mee A eee N (a+b2)Q a+bax Q’ and therefore M P(a+ bz) —~ = A +—-_~—+. Qt a * For if not, by subtracting the equal expressions (1) and (2) from ej other, we get O=a—a+(b—B)a+(c—y)a?+(d—d) 2% 4+ &e., where x can be neither zero nor infinity, unless a — a and the other coefheis are also zero. | 311 If we now suppose M to become m and Q to become 4, b ibn a+bx=0 or r=, we shall find A=—*. L ‘Thus, let it be required to resolve Example. x (a +1) (a + 2) (x + 3) jo partial fractions. Assume ~_—— hess fm, A, As : Fb (a +1) (@+2)(@+3) v+1 2+ Oo eS : thus get of e+1=0 and #=-1: 4\=— : 1 -_= =O and 7~=— : ak 1h Se SE es | (@+2)(@+3) mq (2-1)(8-1) 2" Hl Pa pi 4 | Mi(c+ij@+s) and See teas ECs a x . m 9 9 1 (x+1)(@+2) 2+3=0 and «=—3: Nes (723)@=3) 7 : | Consequently x’ 1 4, ) | @41)@+2) +8) 2@+l) #+2 2(2+8) 953. If (a+b2) be a factor of N, where r is greater than 1, Whar a of the fac- will be found, if we make N=(a+ 62) Q, that a +bx will still iors are P(a +b2) possible Q 2 a factor of Q, and consequently will become 5 and equal. ad not necessarily 0, when a+ 6x=0: under such circumstances emake N=(a+62)' Q, and assume Oh. Ay A, A, se N(G@+bay G@+bayt B+ be 0 > * It will be found that P =° , under the same circumstances: for it may le@teadily shewn that P= and if a+bx=0, then M—AQ=0, other- it follows therefore that a+bx is also a factor of the value of P will be ‘vise P would be infinite: ‘1- AQ, and if this factor be obliterated by division, ound. 312 } and therefore | it nA + A, (a+bx)+... A, (a+bx)' 4 —- S| consequently, if we make a+bx=0, we get Again, transposing 4,, we find Hd Q a both members of which equation are divisible by a+ba: if) make eee we get a+bex — A,=4A,(a+br)+... A, (a+bay+7 (a+bay, = 4a+ Ay(a+b2) +... A (a+ bays 5 (a+ bx) which gives A, =i M’-— A,Q, In a similar manner, by making M”= a+be >» We fir m’” : : : ; A, = ai and the continuation of this process will enable + successively to determine 4,, 4,,... as far as AM Example. Thus, suppose it was required to resolve w+ Qe (x — 2)*(a + 3) into a series of partial fractions. Assume x+Qx Seely cs vi A, B (t#-2)?(@+3)° («#-2)8 (@—2) "2-2*aa3? which gives OQ > eng 741+ As (#—2) + Ay (@ ~2)°+ B re 3 M 2#°+22 (a — 2)° + and therefore, making «—-2=0 or «=2, we get i 4 m A,=—-=—. q af 313 In the second place, we get M 22° 36 18 Breet tere aoe aa) Q ae 5 5 a ae 2-2 (e+3)(@-2) wv+38 Q- 1d therefore, making 2=2, we find _m 58 Pe 26 In the third place, we get M’ 18 58 42 eT ae. pci Sk: Sed L Q 1a +24 + ; ap (tt8) +95 mr r—2 (a + 3)(a- 2) mei eC) nd therefore, making x=2, we find +i! 3 92 egy Oty Lastly, if we make M=2°+22 and Q=(x- 2)’, we find, by naking 1+3=0, or r=- 3, ty m 83 Consequently, Meee 2 ty 4 8S (@- 2) (@+3) 5(w-2)° 25(a@—-2) 125(w@—-2) © 125(@ +3) | 954. If one of the quadratic factors of N be of the form Whensome of the fac- x? —Qart+a’ +p’, tors involve ‘a Val. whose simple factors are e—a—/},/—1, and z—a+f,/—1, we ‘bollect the terms A fos ees ae meister so. A,fa1 into a single term of the form Ax+B a —Qaa + a*+ B®” M_ Az+B As N «w?-Q2art+a?+p? «x-—a; Rr + wae 314 | If we make N=(e2°—2axr+a*+*)Q and replace - a+B,/—1, when M becomes m+m,,/—1 and Q become q+q/ —1, we get m +m, /—1 _ ad amet ps OLD) Weel | gow alien q+q° q+q | =A(a+Bf-1)+B=Aa+B+AB /-1. We thus find _(™@%=—™q) p_ B(mg+maqu)—a(mq = mq) | RO Aq). B(q? +’) Thus, let us suppose f L Ax+B Ce . ES tt: (a—4)(a?+404+5) a?+4274+5 a«—4 N : : where M=.2 and Qe arp aise =a—4, Replacing # by ~2—,/>} we get a=-2, B=—-1, m=-2, m,=-1, g=—6, q,=—1, an therefore Howes and raphe 37 37 If, in the second place, we make M=2, Q= 2° +4245 an x=4, we get " A ’ we therefore find £ &. A i —-4x+5 z*—-112—-20 87(4@—4) 37(a?+42+4+5)° If there be a single factor of N of the form (x?- 2ax+a°+ 6) =u" we assume M VAS Be Be | A,x + B, Hig LN co ieg i aloes Ml ap arent rape * For m-+m V1 _ (m+mV—1)(q-n V—1) qt+uV—-1 (qtuaV—-lN)q-aV—!1) RR BG Fie oC my q)V = 1 an! a or ee at! oe 7 ¢ 315 hich gives : V 7 =A,4+ B,+(A,2+ B,)u—...+(4,¢+ Bu + Pu’ Q > e then determine A, and B, as in the last case, and subsequently ° aking M-(4,2+B,)Q_ u M’, we get 5 =A,2+ B,+(4,0+ But... (A,v+ Bu? + oe : e then determine A, and B, as before, and proceed in the same anner as far as A, and B,. | By the application of this process we shall be enabled to »solve the fraction ‘ ¢ ” (a? + 4)? (a — 5) ito the equivalent series of partial fractions 4(a+5) 25(#+5) je 2D 29 (a? +4)? 207(47+4) 20°(x- 5) © CHAPTER XL. ON THE ASSUMPTION AND DETERMINATION OF SERIES, Threct and 955. THE methods which are employed for the determinatii indirect : ; oe as renee ak of equivalent forms, may be considered as constituting two gr determin- classes, according as they are direct or indirect: direct methc ing equi- Silent are those in which the transition from the primitive to the equit forms. lent form, is effected by means of defined or definable operatio), such as multiplication, division, the raising of powers, and t extraction of roots; whilst indirect methods are generally resort! to, when we are unable to express in words or otherwise ti nature of the operations, which connect the primitive forms wi those which are equivalent to them: but, under such circus stances, the primitive form will commonly furnish the conditie which the derived or secondary form must satisfy, in order tl its equivalence may be determined. Use of in- 956. Though the employment of indeterminate coefficiei pea (Art. 898) whose use, in one important application, we ha indirect exemplified in the last Chapter, properly belongs to the seco! Bored Pe of these methods, yet it is likewise applicable to cases of dird developement, by a simple change, from direct to inverse, | the character of the operations which are required to be pr formed: the following examples will be sufficient to explain ol meaning. Example. The series corresponding to the fraction A+Bzx a+bxex+cz2? may be found by the actual division of the numerator by t denominator: but as it is readily seen that it proceeds by asce ing powers of x, we may begin by assuming A+ Bz aibarogt 40+ Ait + Aya? + A,a*+ &e. 317 ere A,, 4,, Ay, Ag--- are indeterminate* coefficients, whose lues are required to be assigned: for this purpose we multiply th sides of this equation, instead of dividing A+ Ba, by -bat+ ca*, which gives us A+Br=A,a+ A,ax+ A,ax’?+ Ajaa’+ Aar*+... + Ajba + A,ba*? + A,ba*? + Azsbat+... + Aca’? + A,cx’?+ A,ca*+... ‘In order that these results may be identical as well as equal, eg must have the corresponding terms identical with each other: 'e thus get dia A; A,a+A,b=B, A,a+ A,b+ A,c=0, A,a +A,b+ A,c=0T, he successive solution of these equations, in which 4,, 4,, A,.-. are the unknown quantities, will give us A leva B_ Ab | ae eras fee ene eae Ses Me a a a iD 2A a tA bc Ayame te a a a ‘The law of formation of the two last coefficients A, and A, * The term indeterminate, which is generally used in such cases, might be ore properly replaced by undetermined or unknown: there is generally nothing definite in its character when thus applied. + Such series, resulting from the ordinary process of division in algebra, are Scale of metimes called recurring series, and the multipliers employed in the formation telation. the successive coefficients, connected with their proper signs, constitute what salled the scale of relation: thus the scale of relation in the series under con- eration is — > — =, and in the numerical example which follows, itis —2 —3. 318 will extend to all those which follow, and the series may be continued at pleasure. Thus, if 4=1, B=3, a=1, b=2, c=3, we find 4A,=1, A,=1, A,=—5, A,=7, &e., and therefore 1+ 32 TA PEM WE. far +0 — 52+ 72®—llat+ oles Again, assuming oma—esin(o—8) = A, sina+eA,sin (a+b) +A, sin (a +25). and multiplying both sides by 1—2e cosh +e’, we get sina—esin (a—6)= A, sina +e A, sin(a +b) +A, sin (a +26)- —2eA,sinacosb-2e?A,sin(a+b)cos +e°A, sina. Equating the coefficients of corresponding terms, we fi A, sina=sina; and therefore 4,=1: A, sin(a+ 6)—2 sina cos6 =~ sin (a—b), or A, sin(a+6)=2 sina cos 6 —sin (a- b)=sin (a+b); ) and therefore 4A, =1: A, sin(a + 2b)—2 sin (a+ 6) cosh + sina =0, : or A,sin(a+26)=2 sin(a+6) cosb—sina=sin (a+ 2b) and therefore A,=1: and similarly for all subsequent te we thus find ied Rye: 3 . | ) peed eI sin (a+b) +e’ sin(a+26) + a a result which is equally deducible for all values of e. * This series is a recurring series, whose scale of relation is 2e cosh —e® e? sin (a + 2b) = 2ecosb x esin(a +b) — e?. sina, e’sin (a + 3b) = 2ecosb x esin(a + 2b) —e®sin (a + b). If we call s the sum of this series indefinitely continued, we find 5 =sina + esin(a + b) + 2ecosb(s — sina) — es, and therefore s(1- 2ecosb +e?) =sina —esin(a—b), 319 7. The form of the series in the last Example is peculiar, This pro- such as could not have been assumed arbitrarily, or without iunted by » previous piamcUee of the relation existing between the 2” erro- ting expression* and its developement: but in the absence need ach a knowledge, the process under consideration might Vik ann ar likely to lead sometimes to the assumption of series, series. er they possessed a necessary existence in the form as- to them, or not: or, in other words, whether the opera- pric. on one side of the equation are indicated and not ed, would conduct us when they are performed to such ties or not: but a very little consideration will be suffi- t to shew that the process employed for the determination of mdeterminate members of the assumed series, involves the es- lal conditions upon which the equivalence of the resulting essions depends: and that the failure, in the determination hose coefficients that are erroneously or unnecessarily as- ed, or their entire disappearance from the final result, would ish the requisite correction of the first assumption, or, in ir words, would be considered as the proper indication of the existence of the assumed equivalent expression or series, or least of some one or more of its terms, under the form ch was assigned to them. Thus, if we should assume Examples 1 A of series ee A Art Net eae 3 assumed, 1+2 r with super- fluous proceed to the determination of.the indeterminate coefficients, the comparison of the terms of the two identical results de- ed in the ordinary way, we should find 4_,=0; and it would | therefore that the assumption, in this case of a term of a n, epnich has no existence in the equivalent or resulting series, ild lead to no error in the result obtained: again, if we should ime 1-2’ B1+ 2’ — x‘ a sina —esin(a—b ‘: and §= paces? 4 s is the expression which generates the series in the text: it is only when s than 1 that this reasoning can be considered as strictly correct, unless 4 BD icfinitely distant from the beginning, may be equally neglected, whether nfinitely small or infinitely great. * Instead of generating expression, it is more common to use the term gene- i function. terms. — A,+A,a+ A,2?+ A,x°+ A,xt+Azx?+... 2. . and proceed, in a similar manner, to the determination ot} several coefficients of the series, we should find 4,, A,, A,,% all other coefficients of the terms which involve odd powers 4, severally equal to zero, and the correct result 1—22°+ 32*-—52°+8a2°-... would be determined in the same manner as if we had menced with the assumption of the series A, + Agu? + A,at+ Agoet+... in which all superfluous terms were omitted. Considera- 958. Though the correct assumption, therefore, of the Bec of the equivalent series in the first instance, is not absolvl theassump- essential to the correct determination of the series itself, yet iti pea always, more or less, tend to simplify the process for that purgse by lessening the number of quantities to be determined, , by not encumbering the equations, whose solutions are req i with unnecessary symbols: it is for this and other reasons, it becomes important, in all such researches, to avail oursee of any considerations which may serve to guide us in the fin which must be assumed for the series. | The general principle of the method of sndelerminiaa efficients, as we have already seen, (Art. 898) is to dedugi expressions or series, which, from the nature of the projs by which they are obtained, are identical with each otle and this identity will be found generally to exist for all ve whatsoever of the symbols which they involve: but the primiy expression,.and the assumed series which is equivalent to it, ‘3 not always admit of the same range of values with the iden‘a expressions to which the developement of the series may 1 1 thus some of these values may change the form of the prim : expression by making factors or terms of it 0 or «©, whenh assumed series may cease to be equivalent: whilst others 13) make the assumed series divergent, when the arithmeva equality between it and the expression from which it ise rived, will cease to exist, though their algebraical equivalece may, in a certain sense, be still said to continue *. If thex * It would be premature, atthis stage of a student’s progress, to enter (0 the formal discussion of the much disputed question of the possible algebiva 321 yression, whose developement is required, be finite, or zero, when ‘he symbol or combination of symbols, according to which the ‘eries is required to be arranged, is zero, then in neither case can ts negative powers present themselves in one or more terms of ts developement ; and it will be found that its first term is the imite value thus determined in one case, and zero in the other *: mut if, under the same circumstances, the expression becomes nfinite, then there is one term, at least, of the resulting series vhich involves its negative power. Again, if the sign of the expression, in which a series ori- sinates, changes or does not change its sign, for a change of quivalence of a finite expression and a diverging series, of course confining he application of the term equivalence to its use as a factor in multiplication in any other known algebraical operation: but it may be asked, if the pro- {uct of such a series with a given expression, or the result of any other alge- yraical operation performed with it, is a definite symbolical result, when the ‘eries, which is thus employed, though general in its form, is assumed to be onvergent, why should it not necessarily produce the same symbolical result, when, without any change of its form, it is assumed to be divergent? It is in act at least as difficult to reject divergent series, when viewed either as the esults or the subjects of operations, as it is to admit them when viewed as the epresentatives of arithmetical or other magnitudes. _ Thus the expression or generating function (Art. 956) sin a — esin (a — 5) ~1—2ecos b + e? vill generate the series sina +esin(a+b)+e?sin(a+2b)+... vhatever be the value of e, whether less, equal to, or greater than 1; and in he two first cases, it will likewise express the arithmetical value or sum of the eries continued in infinitum; in the third case, when e is greater than 1, it will equally generate the series, though it will not express its arithmetical sum which, under such circumstances, is incapable of arithmetical expression: but t may be considered equally equivalent for some symbolical purposes at least, if 1ot for all, whether it is arithmetically equal to it or not. * Thus, if we should assume the existence of a series for (1+ 2)", pro- seeding according to positive powers of x, its first term would be necessarily 1, masmuch as 1 is the value of (1+a)", when x=0O: in a similar manner, he first term of the series for (a +x)", if proceeding according to positive sowers of x, would be a”: but the first term of the series, for the same expres- a—a : , would be 2a”, inasmuch as a+ jon, proceeding according to powers of x i : =0, or when r=a: in a si- x a —_— Ya" would be the value of (a+ 2x)", when ci a nilar manner, the first terms of series for cos x, proceeding according to powers of « Tv i é = wr of 77% would be 1 in one case, and 0 in the other, inasmuch as cos2x . Tw s 1 or 0, according as x =0 or 5° Vou. II. Ss An equi- valent series should pos- sess as many values, if more than one, as the expression in which it originates, Whatever value a radical or 322 | sign of the symbol or combination of symbols according { which it is arranged, we should at once conclude that i odd powers alone presented themselves in one series, and i even powers in the other: thus in assuming series, proceedi according to powers of # for sinz and cosx, we should on th account, exclude all terms involving the even powers of « fro; one series, and all terms involving its odd powers from the othe 959. A series, in order to be completely equivalent to tl expression from which it is derived, should possess the saw number of values with it, when those values are more the one: but the process of developement will in many cases a ply exclusively, in the first instance, to the deduction of th; form of the series which represents its arithmetical value, an which is wnique, leaving the other values to be supplied by variou methods, some of which will be explained hereafter: thus tk series which the binomial theorem, as commonly applied, giv for (1+ x)t is gy 32° eh Bed ad La 144 #.1.2 8.1.2.8 4.1.2.8.4° which expresses the arithmetical value of the biquadratic ro of 1+ 2, when « is less than 1: its complete developement wou be expressed by | x 32? CER der st Saya a { RMTLABEEY 2, Saye a cee Me ra) LINO ade ye 7 | where 11 is the recipient (Art. 724) of the multiple values of t 1 14(1 + ; ; ; Irn id biquadratic root, which may also be replaced by cos wa RB) a sin — , whose several values are 1, — 1, ra ~1, and — /-1. Similarly, we find A 1 oF le Bere? 1 sa | vlan 2)= (0%) 1 Ties o sidom as aen8 on ° where the value of the series, included between the bracket is unique, but that of (ax), which is multiplied into it, is multiple the successive substitution of the three values, which (az) admi | of, will give the triple series which represents the comple developement. 960. But if a radical or other term, admitting of multip values, presents itself in a generating expression or function, an 323 | : “ans < A : hich it is other ex- ibsequently reappears 1n the equivalent series into whic t is cama eveloped, it should be kept in mind that, whatever be the value admitting a s,s : : : of multiple “hich it is assumed to possess on one side of the sign of equality, paineeoa” |. must continue to retain it on the other: thus, in the series sesses in one term of a series, it =1-J/xe+a-2,fa+a—2 far... are 1 1+ /zx yhichever of its two values, «x, whether + or —1, is assumed possess in ies = ae the same is retained throughout the series: tis only, when such radical or other expressions (such as equi- final and equicosinal angles) present themselves in equations, vhose members are neither identical nor reducible, by the per- ormance of the operations indicated, to identity, and where no ‘onvention expressed or understood limits the multiplicity of heir values, that we admit every combination of such values 1s being equally possible, and as being equally included within he range of the different cases to be considered. The consideration of this subject, which is one of the most mportant in the theory of series, will be resumed in a subse- quent Chapter. 961. When an expression, denoted by a symbol y, is de- Inversion veloped in a series proceeding according to powers of any other ent symbol or quantity x2, which it involves and upon which its value is dependent, we may invert the operation, and express z or the symbol according to which the first series is arranged, by means of another series, proceeding according to powers of the symbol y: thus if y=A,2+ Aja’ + Ana +... then the inverse series may assume the form f= AY + Agy? + azy° +... and the problem proposed for solution is the following: “Given ‘the first series or its successive coefficients A4,, 4,, A,... to find the second series or its successive coefficients @,, d,, d3....” ‘We shall exemplify it in the determination of the series for the measure of an angle in terms of its sine, the series for the sine i an angle in terms of its measure being given. Inversion of the series 324 Since therefore (Art. 924) ' | for the sine ane . ne x a (1), pete 112 380 AB sEaas e its measure. . . ey we may assume, for the inverse series a x= A,sinx+ A, sin?x+ A, sin’x+... (2): | for it is obvious that this series (2) must be confined to odi powers of sin 2, for the same reason that the series (1) for sin, was confined to odd powers of «x: replacing in series (1), « an¢ its powers by the series assumed to express it in (2), we get Din z= A, sine + A; sin? + A, sin'e +... a A sin®x 34,24, sin’ a Th ee 1.2.3 1.2.3 eine Tt Ab inte fi) opis hace 1.8.3 4.578 ra tie ee me EN CLO yO Cite Cho oe eee 4 Consequently, adding together the terms on both sides, we get A? wie 342A 5 \ a sinz= A, sina+(4,— ° ) sinta + Cree + sine Leno 1.2.3 1.2.3.4.5 and equating the coefficients of corresponding terms, we find | ae A,? 1 So Sa a pees 3 d = > A; roe O, and therefore A, igens 3A2A AS 9 . Vs Naa ae Sa a 4 d therefi = ——_~_ ° 7.2.3 °1.2.3.4.5 0 and therefore 4, 1.2.3.4.8" Tyee PASEO LS 2 ihe O10) 8s0s'8 8 ene (S's leRahe s 050 one iefe'a (ole fs\ie lel eka eletetatte or een tet teen and therefore pei ites The law of formation of the coefficients of this series is not made manifest by the number of its terms which have been thus determined, and it appears that the extension of the process to other terms would involve the formation of the successive powers | of A,sina + A, sin? x +A, sin'x+... 325 requiring the aid of a theorem, called the multinomial irem, which we have not hitherto investigated: in the pro- , sof our investigations, however, we shall find other and more seditious methods of effecting this and other developements, yvhich the laborious formation of complicated powers and es will be altogether avoided. 962. It has been shewn (Art. 778) that all measures of angles Thei pee. ruded in the expressions 277+a and (2r+1)7—« are equi- ee: Rae 1, and we may conclude that all such values of x are equally '° for the —. ‘ least of the puded in the equation measures of iG the equisi- : x? - ow? ; nal angles. sin ¢ = 2 ———____ + —_—____—__-..... : Lies ook Lae coe el D ( ) it is the least of these equisinal values of x which is alone sin’ x x 9 sin’ x pe eae ia So Sires: 5 r=sing+ teks vich is necessarily convergent and arithmetical in its value. lis observation will be found hereafter to apply to inverse es and expressions generally, and is connected with important ries. ti CHAPTER XLI. ON THE SOLUTION AND THEORY OF CUBIC EQUATIONS. | Theextrac- 963. We have given, in a former Chapter, rules for 4 tion of sim- extraction of simple and compound cube roots (Art. 242), ple and compound far as those rules could be properly considered as includ SS led within the province of arithmetic: the processes in quest es are equivalent to the arithmetical solution of binomial equation. other cubic equations, in cases where they necessarily posi an arithmetical root, whose determination involves no ani guity: we propose, in the present Chapter, to consider general theory of the solution of cubic equations, and to ext plify the arithmetical or other rules to which it leads. The second 964. A cubic equation, cleared of fractions and radi Eaticlets expressions, by the rules given in Chap. V., may be alw’s ! cubic equa- reduced to the form | | tion may ] always be obliterated. x —ax’+ba—c=0, (1) where a, 6, c are whole or fractional numbers, positive i negative: but this equation may be farther reduced by a vi simple process, to a form in which its second term will ¢- appear: for, if we make 2=y +e, we get 327 ‘Adding together the several terms on each side of the sign { quality, we find 2 3 b iy *4ba-c=y—(S—b)y- (AE \=0: w—ar+be—c=y & b)y 27 cus 0 2 9 7? if we replace Sah d by q, and heat Li by 7, we get 3 ZL 3 Y Fay mn: (2) or y= qy +r*. (3) (965. If, in equation (3), we replace y by —2, we get The signs of the roots 2 = gt—7r; (4) of an equa- tion may be {| it follows, therefore, that if « be an arithmetical root of eactsa station (3), —a is a symbolical root of equation (4), and jiversely: the same process therefore which determines one ‘ration (3) to equation (4) or conversely, the negative roots ‘one equation being the positive roots of the other, and : it is usual therefore, in the solution of cubic and (ier equations (as will be afterwards seen), to consider nega- fe and positive roots as determined by the same arithmetical jocess, whenever such a process can be found: they are also Distinction (led real roots, to distinguish them from those whose ex- ‘ obreatad id imaginary jession involves the sign nba —1, and which are called unreal '°t- Die _ (imaginary roots. \" Thus, the equation is y? + Gy? + 9y+4=0 ‘reduced, by replacing y by «—2, to ce? —32+2=0. ‘The equation x? + Qla? + 1462 + 335 = 0 ‘reduced, by replacing « by y —7, to y>—y—1=0. | The equation 539° Hy | 91 etalon 1d, reduced, by replacing y by x—#, to +5 th = 0 328 In what 966. All roots, however, which admit of interpretation aaa oe equally real and significant, whether positive, negative, or Sears fected with a sign involving ,/—1: but it will be usefu Sat *S the theory of equations, to recognize the preceding distin Q imaginary. Of real and imaginary roots, as characterizing two great claje of roots, which are distinct from each other, both in the fy in which they present themselves in equations, and in } practical arithmetical processes which are employed for ty determination. Solution of 967. If we make p the arithmetical cube root of a (wlre eae a a is a positive number, whole or fractional), there are, as y equations. have seen (Art. 669), three symbolical cube roots of a, wh are -14+,/3/=1 =1~\/3 fH1 & ( 2 )e ; ( ) 2 which are the three roots of the binomial cubic equation 2 —a=0. (1) In a similar manner, it will be found (Art. 670) that } -r (2D, (iH, 3 are the three symbolical cube roots of —a, or the three ros of the binomial cubic equation The root p in equation (1) and —p in equation (2), ; called real roots, in conformity with the conventional langue adopted in the last Article: the other roots, which are affect] with the sign ,/—1, are called umaginary roots. r+a=0. (2) General 968. When g was a negative and r a positive whole nu- ocess for . : 7 | Oe nol ber, we were able to determine a value of x which satis of cubic the equation equations wanting the e=gqrtir 1 second 7 ( ) term. by a direct arithmetical process (Arts. 240 and 241): and it my be shewn generally, if we assume PAG Re Bee 329 x to be equal to the sum of the cube roots of s and 1, it such values, whether arithmetical or symbolical, may be nigned to \/s and \/t, as will satisfy the same equation for -yalues of q and r: for we thus get Hts af = ( G/s+,/t) =3\/st(J/s+th+st+t =3\/st.c+s+t, (2): placing /s+,/t by x: but if the values of # and of 2° : the same in equations (1) and (2), then 3 /st.c+s+t, qzu+r e identical expressions, and therefore | 3/st=q, and s+t=r; q° st= oF? and s+t=r. | It will follow therefore that s and ¢ are the roots of the given aadratic equation We thus get c= lst rt “5+ /G-5) e-/G-E) If the equation proposed for solution had been e 2 —az’?+bx—c=0, a 2a® ab a ve must repl ——b, rb dal glydr and add —, which | us eplace q by 3 % Y ov 3 3? ‘ives * For if s and t be the roots of the equation u2—au+b=0, then a='s+t, and b=st. (Art. 661). Vou. II. T 1 The values of tls and sd t are deter- mined by the solu- tion of an equation of six dimen- sions. 330 \etite ja abc, Geri a TAS LileVadigtin 48 Pini, Catena wile ip i a +e- AE a OM Tay a 6ita8 27° a7 6 27 4] This expression will shew how much the formula of {> solution of a cubic equation is simplified, by the suppressiy of its second term. } 969. The values of s and ¢ are derived, as we have ap from the quadratic equation 3 ui u*—ru+—-=0 Oy and those of ,/s and of °/¢ from the binomial cubic equatio; 5. BO cea eee NAG a7)” and or if we begin by representing °/s by v and ms t by z, we get v+ =r (1); 3vuz=q (2); and if we replace z, in the first equation (1) by a derivl from the second (2), we find a) es eno ph 3 or v'— ry? + Le : 27 an equation whose roots are the three values of JES) and the three values of eV G-P it is in this sense that the solution of the equation 3 rT; x—gxr—r=0 may be said to be dependent upon that of an equation of) dimensions. 331 970. The general expression, however, for 2, which is given The gene- Art. 968 will be found to admit of more values than are com- Maccrdas* atible with the conditions which the sum of the cube roots of s eh ee mits 0 ad ¢ are required to satisfy: for if we represent the arithmetical extraneous be root of s by o, and that of ¢ by 7, and the three cube roots cae Re fiby 1, a, a’ (Art. 669, Noe then co, ac, a’o are the three 2) roots ube roots of s, and 7, at, a’ are the three cube roots of t, and Seana ne of the first triplet of values may be combined with one of equation. ye second in nine different ways, as follows: l. o+T. to ac+a’r. 3. ao + aT. (4. o+arT. aes | 6. a®o+a’r. 7. o+a7t. 8. a®’o+T. Q. aoc+art. But it should be observed that the equation of condition 3 Nie i= q Biicts the selection of the combinations of such cube roots to hose whose product is equal to d, and which are therefore ndependent of « or of those powers of a which are different tom 1: it is obvious that this condition is satisfied by the hree first combinations only, which exclusively represent there- ‘ore the three roots of the proposed cubic equation. 971. The three triplets of combinations of the cube roots The nine of s and t, which are bracketted together above, will severally Se orm the roots of the cubic equations cube roots of s and t x—qxeu—r=O0, are the ~ roots of an a —aqz—r=0, equation of w—a’qru—r=0: nine di- mensions. and the nine values of those combinations will be the roots of the equation \(@°—qe-r)(2°-agqa—r)(x*—a?qa—r)=2°—3ra°+(3r°— q°) #—r°=0. Cases in which there is only one real root which is arithmeti- cal. Where one root only is real, but not arith- metical. Where Es root only real hich: is arithme- tical or not, according as r Is posi- tive or negative. Case in which all the roots are real, and one only, or at most two, are arith- metical. 332 It appears therefore that the expression which the se process has given for «, furnishes the general solution of an equ tion of 9 dimensions, of which the proposed cubic equation ( x*—qur—r=0 | is a factor: and we are not authorized to conclude, from th investigation, that there is any symbolical expression which ¢; be formed, which is capable of expressing simultaneously t], three roots of the cubic equation x —qr—r=0, | and those three roots only. | 972. If r and q in the equation | | | a®—qr—r=0 3 be both positive, and if £ be greater than a then o auf (Art. 968) are arithmetical and real, and the three roots are (1) o+7, (2) Bear. ure INE Vea i fi Ce (3) ett on gsJ5 ~ of which the first alone is arithmetical, the other two bein imaginary. a2 3 If r be negative, q positive, and 3 greater than _ , the o and 7 are negative and real: in this case there is onl one real root, which is not arithmetical: the other roots ar imaginary. If q be negative, then o and 7 are real, but with differe signs: there is, therefore, only one real root, which is arithmetica or not, according as 7 is positive or negative: the other root, are imaginary. 973. If q be positive, and — Ness than f, then the tw roots s and ¢ of the reducing ete equation | 3 w—ru+tLe 27 are imaginary: they have therefore no real and arithmetical cube roots. 333 “re however, in this case, we make c=~— and d= xi (E _ 7) er 3 3 2 27 4 5] shall find a=(c+d,/- 18+ (c—d,f/—1)3: id if we make r cos 0 = cos (27m # 8) = Tay = : =, /¢ 27 » shall find (Art. 822) Qn +0 3 | teen 1 x =2(c’? +d)’ cos gz? 2 (c? + d’)° cos L 2 or 2(c’+d*)’ cos an d ; 1 3\1l .d, inasmuch as (+ @y=(4 ir Ne i, the three values of 27 ‘eS a” =2 Ce each lp a Bs | These three roots of the equation are real, and one at least, it not more than two, of them, are arithmetical: for their sum equal to zero”. | 974. The term irreducible has been applied to the case The irredu- © cubic equations considered in the last Article, where the a payeny ‘ree roots, though all of them are real and one of them, at least, lato lithmetical, are not capable of being determined, as in all called. cher cases, by the extraction of roots, or the other processes f Arithmetic and Arithmetical Algebra: the difficulty which | 9 ler +6 or —6 * For a’ $07 2" =2V/ E§ cosy + cos ( st ) +08 aca be 4g 9 Qa 68 -a/ 2 (cos 5 +205 Lea 5) 0 0 -a/4 (cos 5 e0s 5) =, 334 : was thus presented to the earlier writers on Algebra*, to w the principles of Symbolical Algebra were almost entit| unknown, was insurmountable, and involved the solut; as the preceding investigation shews, of the problem of j secting an angle, which was also beyond the province of pli Geometry: it is not the only case in which the separat) of Arithmetical and Symbolical Algebra, and of plane «\ the higher Geometry in which the different conic sections i other curves appear, will be found to be marked by comin limits. Geome- 975. The principles of interpretation, which we have gin say nia. ia former Chapter (xxx1.), will enable us to represent the ( tion pine portions of which the several roots in the preceding soluth roots of a cubicequa- are Composed as well as the roots themselves. tion in the irreducible Let the angle BAC=90, and let the angle BAP =< = Bi case. Qr—-0 P a the angle BAP’= = BAp', and " Qn+0 the angle BAP” = =BAp”: and join Pp, P’p’ and P’p” cutting BA, and BA produced, in Q, Q’, and Q”: / Pp’ if we assume the radius AB, /1, we shall find that | ) : i | ) 2 (oa! Jia ®) na» /2 (ob! nl J Ecos 5 +a/ 1 sin) and 4 (cos 5 ag sa will be expressed in magnitude and in position by AP and 4, ] and their symbolical sum, or 2 NE: cos & by 24Q: ina simil manner, the values of the other two roots will be the symbolic sums of AP’ and Ap’, and AP” and Ap”, which are respect equivalent to 24Q’ and 24Q”. * An excellent account of the progress of the researches and discover! of ‘Tartalea, Cardan, Ferrari, Bombelli, Vieta, Des Cartes, and other ea algebraists, in the solution of cubic and biquadratic equations, is given by Mo tucla in his Histoire des Mathématiques, Tom. I. Pars m1. Livre unt. | 4 305 \If the angle 0 be between 0 and 90°, or if its cosine be posi- Case in 5 which one | Qr+0 Qr—-0 . or two of 12, then <7 *" and ~—— are both greater than 90°, and their is bs 3 3 roots are {ines are negative: in other words, one root of the correspond- RET b e e,@ . . 1ca . |; equation is positive, and two are negative: but if @ be iz ae : : 0 | ween 90° and 180°, or if its cosine be negative, then "i and 976. Before we proceed to the consideration of numerical Problem, amples of cubic equations, we shall notice the following pro- ee Im as well calculated to illustrate the origin of their multiple of the ater guous solu- ‘utions, and to exhibit the composition of their coefficients. _ tions of “To find three numbers, whose sum shall be equal to a, eee 2 sum of whose products shall be equal to 6 and their con- ued product equal to c.” | Let x, y, = be the three numbers required ; then the con- ons of the problem give us et+y+z=a4, ay +xz+yz=5, LYZ=C. | From the first equation we get | y+s=a-a, id from the second | a(y+2)+y2=), or t(a—x)+y2=5), or yZ=b-—axr+2x’: e third equation gives us axyz=a(b-—axr+2x’)=C¢, or x°—ax*+ herd Soret ) Inasmuch as «x may represent indifferently any one of the ree unknown numbers or quantities which were assumed to e represented by 2, y, 2, which are all similarly involved in original equations assumed, it must equally represent them 336 all; or in other words, x will admit of three values, whi are those of the several unknown symbols*. : If the sum of these three values or a=0, then one them at least is positive, and two negative, or two of thy positive, and one negative, when all the roots are real:) otherwise one of the three values is real, whether positive negative, and the symbolical sum of the two others, which » imaginary, is equal to it with a different sign. Compo- 977. In considering the relations which the coefficients) sition of the : ; coeflicients a cubic equation cae x —ax?+bx—c=0, equation. bear to its roots, it is evident likewise, from the result of i preceding elimination, (and the same may be easily provd from other considerationst), that the coefficient (a) of the seced * The same remark applies to the values of the unknown symbol inn equation, which results by elimination from any system of equations which symmetrical with respect to the several unknown symbols which they invol’: thus, if we eliminate y from the system of symmetrical equations a? + y? = 35, r+y= 5, we get the quadratic equation a? -5r+6=0, where x has two values, which are those of « and y in the proposed syst of equations ; for « and y are obviously interchangeable with each other. But if the system of equations had been x? + y® — 35, G—Y= Fe which are not symmetrical with respect to x and y, the equation resulti ; from the elimination of y would have been | 3 3x Yea od: Pa peek re ey Sel IB Dy a cubic equation, in which x has one real value only, the other two bei imaginary. t For, if @ be a root of the equation a—aa?+br—c=0, then x —a is a factor of it: for if a—aa?+ba—c=0, then vw — aa?+br —c¢ =23— a®—a(72- a*)+b(x—a), which is divisible by r—a: 337 cm is the sum of the roots, the coefficient of the third term ) is the sum of their products two and two, and the last ‘em is their continued product: and it will therefore follow at if the second term disappear, the sum of the roots is neces- Maite (7-£)--= 337311 aR avo7 | a 4 a Se GS) Seer mp2 ay 6 A : e=a/(- 581+ BION pase OK 4, 9 ata we 581 a Srey _-7-,/39 4: Q Consequently a’ =o+7T=-—7; ty Seer) (ont) (S87 ven . g g 9) 3 Pet) (er) LE ey i 2 2 2 2° _ In this example, we have put down the values of o and 7, Tentative mder a finite form, as determined by a tentative process of the finding for ollowing kind. the cube roots of ‘ aber binomial nd since a, B, and y are roots of this equation, it thus appears that x — a, surds, when |-B, and x—y are factors of it, and therefore there can be no more: we such roots De cet gah a finite a3 — ax? + bx—c=(x—a)(x—f)(x#—- 7) form. =8-(a+Ph+y)er+(aptraytBy)r—aBy, vhich proves the proposition in the text. imey or. II. Uv 338 Thus, if it be suspected, from the nature of the case or oth wise, that a numerical expression of the form a+,/b whose cube root is required, (as in the case of s and ¢ int Example just given) is a complete cube of an expression of t form r+y pears where « and y are, one or both of them, quadratic surds, a which are not reducible to a common quadratic surd, then shall also have 3 a Rey J (a Jb) ay Rae . it will follow, therefore, that er Aye Diver n(@?— 6) = “JR , or /{(a’- 6) R}=2°-y?=c, an expression, in which R may be always so assumed, th (a*—6) R may be a perfect cube* and therefore x°— y’ a ratior number. Again, since J {(a + ./b)? R} = a? + Quy +y’, Ni(a — /b)° R} = 2 — 2ay + 9’, Nia + /6)? R} + f(a — /b)? R} = 2a? 4 4 pe and inasmuch as it would follow from the hypothesis which y have made, that x? and y° are whole numbers, the value 2x°+2y" is necessarily equal to the sum of the two nean integral values 1, ’ of J/{(a+,/b)*R} and w/{(a—,/b)? R}, one them being taken in defect and the other in excess: we thus fir we get / t+e > x? + y? = x? — y? = C. * If a®?—b be not a perfect cube, or not resolvible into factors which ; repeated, then the least value of R is (a?—b)?: thus, if w= 1162, a 4/b = 13849244, we get a*—b=1000, R=1 and 2?-—y?=10: if a? - b =54=2 x 27=2x 33, then R=4 and x? — y= 6; but if a? b =58=2x 29, then we must make R = 582 and a? — y? = 68, 339 Adding and subtracting these expressions to and from each her, we get | +142 +/+2c Qe? =- 7 © and oe +i — +i/—2¢ agra tt =e and (tees) lit , which is thus determined, S/R | found to be equal to a+,/b, the problem is solved, and one the roots of the equation is a whole number, or a rational action: if not, there is no such root of the equation, and its proximate value must be determined by the actual extraction the roots involved in it. It appears that unless ,/(:+:/+2c) and ,/(i+’—2c) are even ‘umbers, x? and y® cannot be whole numbers, but this may be voided, by multiplying a+ ,/d, in the first instance, by he Thus, in the example under consideration, we find If, upon trial, the cube of 581 + ,/(337311) 1162 + ,/1349244 Sir YEE Se 2a Hi 93 y) ,__ 581 ~,/(337811) __ 1162 —,/1349244. oe A i 98 : nd therefore a=—1162 and b=,/1349244: consequently | a?— b= 1000 = 10°: ve thus find R=1, and | 8/(a? — 6) = «°— y= 10. Again a +,/b =— 1162 + 1161.569 = — .431, nearly, ind therefore | 3/(a+,/b)?= .7554 nearly: say 1: and a—,/b =— 1162 — 1161.569, = — 2323.569 nearly ; 2/(a —/b)? =175.45 nearly: say 175. We thus get x’ —y’= 10, x+y’ = 88, zv’=49 and r=—7, y°= 39 and y = N39. 340 Therefore M(- 1162 + ,/1349244) =~ 7 + ,/39, which upon trial is found to be true. (2) Let 2°+8r-9=0, _ 81+,/12705 944784 +,./1829520 — OF i —— ’ (Se 6° : , _81-,/12705 _ 944784 — ,/1829520 HE SccciUiotapal ins Te, 3+,/105 7 ae _ 3-,/105_ T= 6 > and therefore =1, gl = lit /— Bi moc =— 35 Nar raid ie sore (3) Let #®9-92+14=0, s =—7+,/22 =— 2.3096, t =—7—,/22 =~ 11.6904, o =— ,/2.3096 = — 1.3208, + =— 3/11.6904 = — 2.2696; w! =o6+7=— 3.5904, ce!” =-7— eAGesia por a 1.7952 + .82166 ,/=1, nm o+T (o— 7) 5 carpi hanks = 1.7952 — .82166 ,/=1. The process followed, in the two last Examples, for exhibitiz the roots under a finite form, is not applicable in this case. (4) Let «°—182?+872-—70=0. If we make «—-6=u or c=u+6, we get the transform! equation ue—21u+20=0, in which = in whic 4” a7 = 100-343 =— 243, all) 341 ae Sie 10 | If we make cos 0 =——=3 = ——; and therefore 7 OS eee 2 10 + log 10 == jh log 7 = .4225490 3 log 7° = 1.2676470 1.2676470 foe cos 57°. 19’. 12” = 9.7823530 id therefore 6 = 0 — 57°. 19’. 12” = 122°. 40’. 48”. 3 3 log 2 = .3010300 log p/ 2 = 5 log 7 = .4225490 To find the value of wu’ =2 Jt Bell -7235790 log cos : = log cos 40°. 53’. 36” = 9.8784812 log wu’ = log 4 = .6020600, omitting 10. SE In —-O To find the value of wu” =2 Jt cos as : -7235790 log cos ere = log cos 79°. 6’. 24” = 9.2764210 log u” = log 1 = .0000000 To find the value of w/” = 2,/2 cos ai 5 3 342 -7235790 -log cos 2 - G =~ log cos 160°.53’.36” = 9.9753910=log cos 19°.6".| —log wu” = log 5 = .6989700 and therefore xu’ =— 5. The corresponding values of x or the roots of the origi equation are 10, 7, 1 (5) Problem. Three consecutive chords, whose lengths 1, 2, and 3 inches respectively, subtend a semicircle: to find diameter of the circle. Let the chords AB=1, BC=2, and CD=3: let the diame AD=x, and draw the diagonals AC C (uw), and DB (v): inasmuch as it ap- pee pears, by a well-known property of &% fe quadrilateral figures, that hs) ACx BD=ADx BC+ABxCD, A we get uv=22+ 3. But, since ABCD is a semicircle, the angles ABD and A@ are right angles, and, therefore, =,/(#?-1) and v=,/(a?-9): we thus get J (2°—-1) /(#?- 9) = 22 + 8, which gives, when rationalized a2*—142°-—122=0, or dividing by a, x®*—14¢—12=0*. * If the three chords had been expressed by a, b, c, the resulting equalt would have been 3_— (a2 +b? +c?) r—abc=0: @ , r2 it is not difficult to prove that in this case 3 is necessarily less than yi ‘Is problem is selected by Newton in his Arithmetica Universalis, Sectio 4ta. C I., and solved in several different ways as an illustration of the various mos in Mahieh the same problem may be reduced to the form of an equation. 343 } 3 2 In this case 7 is greater than ve and the process followed 27 the last example gives us v’ = 4.1133, x” =— .9118, a!” = — 3.2015. | It is the positive root which expresses the unique value of > diameter, which solves the problem: the negative values 2 roots of solution’ only (Art. 676), and have no reference to 2 geometrical conditions of the problem proposed*: the solu- m is not therefore ambiguous. | (6) Problem. From the extremity of a given arc of a ‘cle, to draw a line in such a manner that the portion of it, tercepted between the circle and a diameter produced which isses through the beginning of the arc, may be equal to the dius. ! Let BC be the arc, and let CP be rawn to meet AB produced in P, in such (T» E manner that DP = AB. P B Make A4B=r, AP =z, and the angle py (‘AC =x — 0: then since ERs RES PDSPC, e get _PBxPE (#-r)(et+r) 2#-r Be Dara ae eee ar ut since PC? = AP? + AC’?-2AP x AC cos (x — 8), fe get 2 2\2 ie Meee oho ofa b r rhich gives zt —3r'az?+2r' cos 0x =0, or 2° —3r'2x—2r' cos 0=0, 3 or (=) -3(=)-2c0s 0=0, r r * See Appendix. Difficulty of proving a universal negative in the geo- metrical solution of problems. geometrical reasoning, from the failure of one construction, tlt 344 Consequently s=cos6+,/—1 sin 8, t =cos0—,/—1 sin 6, Xf 6 ra = 2\C08—, a”! Qn —-60 = = 2 Cos 9 ; ut Qr+0 —=2C 3 Thus, if 6=60°, we find / ~ =2cos 20°= .187949, tf “ =2c0s 100°=— .34'740, at cos TAC? - — 1.53209. It is the arithmetical root 1.87949 which answers the ¢) ditions of the problem proposed: the other roots merely indics that if the angle BAD be 100° in one case and 140° in the ottr. a point to the left of the centre 4 and at the distance BAT from it in one case, and 1.53209r in the other, may be foul, whoce distance from D is equal to r: and the angle BAD\s 4¢ EAC. The geometrical problem, whose analytical solution we '\ Just given, is one of the many forms to which the celebraid problem of the trisection of an angle has been reduced, but r which no construction, which can be effected by the rule ed compass only, has been hitherto discovered: and it is presuml, from the failure of these attempts as well as from other c- siderations, that it is a problem whose solution is beyond te powers of plane geometry. It is very difficult, however, if not impossible to shew, fri geometrical considerations only, that the solution of this orf any other problem similarly circumstanced, is geometrically i- possible, inasmuch as there is no limit to the number of cc structions which may be proposed, all of which should be she to fail; and we are not enabled to conclude, by the forma 345 Jl others will be equally inadequate or inapplicable: it is for his reason that Geometers have not hitherto succeeded in as- igning any form of demonstration, which is sufficiently conclusive nd clear to check the attempts of geometrical adventurers, whose mowledge is imperfect or whose understandings are not sound nd well disciplined, to solve impossible problems. It will be found hereafter, however, that problems which dmit of solution by straight lines or circles, or by combinations ff them, will be, in all cases, reducible to simple or quadratic yquations, or to such as are resolvible into them: and we shall ye enabled to conclude therefore, that if the algebraical solution f a problem leads, as in the case just considered, to an irreducible subic or higher equation, there exists no construction, which plane yeometry can supply, which is capable of effecting it. Vor. II. AS CHAPTER XLII. ON THE THEORY AND SOLUTION OF BIQUADRATIC EQUATION General Pec Ra ies 979. Tue general form of a biquadratic equation is biquadratic equation. z*—azx+ba?—cxr+d=0 (1), Its trans- : : formation Where a, 6, c and d are rational numbers, whole or fractior Aid| 2 positive or negative ; and it may, in all cases, be transformed, — Ww Ss . . . . . second the following process, into another biquadratic equation, whe aoe roots differ from those of the given equation by a given nu ber, but which shall want the second term. a a For this purpose, we make «x — gut OF e=u+7: @ therefore 24? wu a’ f=ut+awt+ + aa 16 256 a Sa°u? = SB a®u a aha 4 16. 64 Qbhau ba? + ba? bu? + ‘3 v 4 16 c cu ae — Pi = —_ mee A. +d + d. If we add together the several terms on each side of tl sign =, we get Bs al ar 8) ut (£ ba ) (ee ae ca ); O=u 5 b)u Pe iy ho Thee 256 Te d); and if we replace 8a? Baeiba aes by q, ae eoue Dyer, 347 id $a*’ 6a*- ‘ca 286 16t'a? by Ss, e get | u*—gqu?—ru—s=0. 4 It is obvious that if the values of « can be determined by ie solution of the transformed biquadratic equation, those of erally to them: in the research, therefore, of general methods r the solution of biquadratic equations, we may confine our tention to those which want the second term, though it is at, in all cases, necessary to do so. 980. We have given, in a former Chapter (xx), many ex- Ferrari’s nples of the reduction of biquadratic equations, either wanting soperiees eir alternate terms or possessing peculiar relations amongst biquadratic om - . : ° equations. ‘eir coefficients, to equations of an inferior degree: and we lall now proceed to apply one of the most common and suc- sssful of the expedients which are there exemplified, which msists in adding or subtracting such terms to or from both ir members, as may make them complete squares. : the original equation are immediately found, by adding * Thus the equation a* — x3 — 52° 4+ 122 -6=0 ‘comes, by writing y + } for z, 43, 75 851 Sed au Ois WG, | The equation | y* — 20y8 + 148y? + 464@ + 480 = 0 comes, by putting x +5 for y, xt — 22? 4162 —15=0. The equation ) ; 7 comes, by putting ted for y, 1472? 19lz 3811 — ee —— ee 348 For this purpose, let the proposed biquadratic equation — put under the form ao gn+re+s;3 (1) and inasmuch as the square of w°+u is a*+2ua’+u’, let | add 2ua*+u? to both its members: we thus get (2? +u)y=(Qut+q)at+re+ults (2). In order that the second member of this equation may } also a complete square, it is necessary that we should have 4(2u+q)(w?+s)=7°,* or r Quo + quit 2su+qs=—7, or 2 w+ tuts sus+ t= =0 (2): It appears, therefore, that in order to determine the val: of u, so that, if 2ua+u® be added to both members of equati (1), they will severally become complete squares,’ it is necesséy to solve the cubic equation | ae w+ dal RAI pes Sp ret at oe If we suppose this equation solved, and a real value o determined, we replace it in equation (2), and proceed as folloy Extracting the square root on both sides of the equation, » etus+(/2u+q. a+ /u+s); and we thus obtain the quadratic equations a*— /2u+q. whu— fers s=0 (4), | a+ J/2u+q.c+ut+/w+s=0 (5), | ae ee as we use the upper, or lower sign: the solution | get or « in the proposed biquadratic equationt. * For if (ax +b)? = a2 x? + 2aba +b, then 4a? x b? =(2ab)?. + It was F. Ferrari of Milan, a disciple of Cardan’s, who, about the middl the 16th century, discovered this method of resolving biquadratic equations,? occasion of the proposition of the following problem. 349 ’ | Let the proposed equation be Examples. xz‘—62°-48r-11=0. The reducing cubic equation will be found to be w+ 3u?+1lu—255=0, {: real root of which, determined by the ordinary process, is 5. The quadratic factors of the proposed equation are x°— 4x-1=0, whose roots are 2+,/5 and 2—,/5: id 2442+11=0, whose roots are —2+./—7 and - 2- fear Let the proposed equation be x*— 252° + 60x — 36 =0. ‘The reducing cubic equation is ue+12.5u*+36u=0, hose roots are 0, — 8, and —4.5. If we make u=0, the system of quadratic equations is x?—5x2+6=0, whose roots are 2 and 3: ad 42+ 5x2-—-6=0, whose roots are 1 and —6. For it appears that 25?— 60 + 36 is a complete square, and bquires no addition to its terms to make it so. If we make u=—8, the system of quadratic equations is xz?—3xr+2=0, whose roots are 1 and 2: nd x? +3x2—18=0, whose roots are 3 and — 6. In this case, we add — 1627+ 64 to 25x2°— 60.2 + 36, which ‘roduces the complete square 9x°— 60x + 100. If we make u=— 4.5, the system of quadratic equations ts x? — 42 +3=0, whose roots are 1 and 3: To find three numbers in continued proportion, whose sum is equal to LO, nd the product of the first and second of which is equal to 6.” This problem leads immediately to the biquadratic equation v4 + 627 — 60x + 36=0, ynd to the reducing cubic equation u? — 3u? — 36u — 342 = 0. The pre- ceding process ap- plicable to biquadratic equations, all whose terms are complete. Example. 350 and 1 x*+4x2—12=0, whose roots are 2 and —6. In this case, we add —9x*+ 20.25 to 252?— 60.2 + 36, wh} produces the complete square 16.2*— 602 + 56.25. The roots of this biquadratic equation, as well as those its reducing cubic, are all real, and the different solutions ex respond to the three different ways of resolving it into quadré factors, involving three different combinations of the roots; is only when all the roots of the biquadratic equation are r that all those of the reducing cubic are real also. 981. The process in the last Article is equally applicas to the solution of a biquadratic equation, all whose terms complete: for if such a biquadratic equation be put under fe form a —paer=qetret+s; (1) we shall find 2 2 (2222 + w) = a — part (E.. 2u)a*— pus + u?, which becomes, by replacing x‘—pa* by qa?+re+s, 2 2 (2 - PE +n) =(F4q+2u)at-(pu-r)erutss, both sides of which are complete squares, if 2 s(% 4g +2u) (u?+s)=(pu—r)’, or if 2 = } ws Luts (o+Pr) PoP iol den Be u Mats | Ri trae aa (2) If the value of w, in equation (2) be found, the equation ( is immediately resolvible, as in the former case, into two qua ratic equations. Thus, if it was required to find two numbers whose su was 3, and the sum of whose fifth powers was 33, we shou readily obtain the equation x*— 64° + 182°-274+14=0, which would lead to the system of quadratic equations ®—32+u=+(/2u—9- a+ fut 14), 351 ere w is found from the reducing cubic equation §3u 225 u> —~ 2 bad SE 2 eR wo — Que + ») 8 The real root of this equation is + which gives x equal to He v9, The two numbers required are, therefore, 1 and 2. Itogether independent of any antecedent assumption concern- { . . . > jsients: if we conclude from it, however, as we are authorized |do, that every biquadratic equation zw —ax?+bx2°-cx+d=0 four roots, and if we denote those roots by a, (3, y, d, then ja, «—, x—y, and x—6 may be easily shewn to be severally wors of the equation, and therefore ( — a) (w- f) (w— 7) (w—2) - 2% +(a+B+7+8) a+ (aB+aytad+By+ BS4+y8) a? +(aPy+aPBd+ayo+PBys)x+aByd =2*-ax°+ba’—-cxr+d: j thus find, by equating corresponding terms of these identical alts, that a=at+P+y+6, the sum of the roots: b=aB+ay+ad+PBy+ Bd+ 79, the sum of all the products of the roots taken two and two: | pe aead fo ays ays the sum of all the products of the roots taken three and three: d=aB 6, the continued product of the roots. It will follow therefore that if the coefficient of the second 982. The preceding method of solving biquadratic equations Composi- the existence of their roots, or the composition of their co- equations. The same conclusion otherwise obtained. 352 term of a biquadratic equation is zero, the sum of the re is zero*. 983. The same conclusion would follow as in Art. 976 fr the result of the elimination of three out of the four symb; a, y, 2,v in the following system of symmetrical equations. e+y+s+v=a (1), ry+astxvt+yetyvt+2v=b 2), ryzs+ayv+azvt+yzv=c (3), | ryzv=d (4). | From equation (1), we get Yte2z+rv=a—Z. From equation (2), we get x (y+zto)+yz+yv+sv=5, and therefore ystyvt+ sv=b—-2(a—-2)=b-art+ae’. From equation (3), we get x (yst+yvt+sv)+ysv=c, and therefore yzv=c—a (b—aar+2*)=c—bar+ax*—2’. And from equation (4), we also get d 2 3 yzv=—=c—be+ce — i, and therefore z*—az®+ bx?-cx#+d=0. The values of x in this equation are quadruple, for a uw} equally represent the value of « or y or 2 or v. ¥| ia * The statement of this proposition supposes the terms of the original equa) to be alternately positive and negative: if the form assumed had been = | a*#+par+qar?+ra+s=0, | we should have found p equal the sum of the roots with their signs changed, é | equal to the sum of their products three and three with their signs changed. 14 ae |: 353 If a, 8, y, 6 be the several values of x, y, z, and v, we then ad as in Art. 982 at 8 munky lg 3 6 = a, aB+ay+ad+PBy+Bd+ d=), aBy+aPit+ays+PByo=c, ays =d*, 984. If we resume the consideration of the quadratic factors Mees ot . . ° ° te > s) and (5) in Art. 980, into which the equation a fae of the roots of za? +u)?—(2Qu+q) 2?—-re—v’—s=0 the biquad- ( ) ( q) ' ratic equa- tion. ad u+/w+s=afs, u—/W+s= 748, id therefore by addition ‘id inasmuch, as we may interchange the roots a, 8, y, 6 in this ‘pression at pleasure, the three values of wu are aBp+ys ay+ fo ad+ Py “Ber tae Dae OPER ? one 5) id there are no moret: we thus discover the reason why the ilues of «, upon which the determination of the roots of the ‘quadratic equation, in this method of solution, are dependent, ve expressible by means of a cubic equation. 985. In the preceding process we have decomposed a fyxamina- quadratic equation, under a modified form, into two quadratic ton of the ) conditions * The same method may be easily extended to explain the composition of the efficients of an equation of n dimensions. + For no more combinations can be formed by the interchange of the symbols _B, y, 6, in the semi-sums of these pairs of products. myo. II. Yy Bian: 354 whicha factors*, by whose solution its roots are found: we shall noy Sprain proceed to examine generally into the conditions which a tring biquadratie mial such as | equation | must sa- zit+ar+b (1), | tisfy. ; ; ; must satisfy, in order that it may be a factor of the equation | a‘—qa*-rz—s=0 (2) | under its unaltered form. . i For this purpose, we shall divide (2) by (1) as follows ; | v?+axr+b)a*—qa’—-ra—s(a’?-axr+a°-b-q | xttaror —ax—az*—aber | | te | —ax*—(b+q) #’-re : : (a?—b—q) 2°+(ab-r)a—-s a? —b —q) 2+ (a—ab —aq) 2 +6 (a?—b ae q q q, —(a°—2ab—qa+r)«—b(a?—b-q)-s | But if «7+ axz+6 be one factor of «*—qa°—ra—s, the 2?—ax+a’>—b—q must necessarily be the othert, and therefo the two terms of the remainder | —(a®~2ab-—aq+r)«—b(a*—-b-q)-s must be identically equal to zero: we thus get the two simu taneous equations a®’-~2ab—aq+r=0 (3), i b (a?-b—q)+s=0 (4). Oy From equation (3), we get Tyre ea b=4(a TNS? eee * If 1? + ar + bisa factor of «t— qa2—rx—s, it continues to be a factor? all its values, and therefore when , , a7+axr+b=0, | in which case also rt*—qa*?—ra—s=0. + For no negative power of x can appear in their product, nor, therefore,? the factors themselves. 355 ind therefore (0’) a’-b-q=4(@-9)- 3, 2a The substitution of these values in equation (4) gives us r? | 1 (a’- q)'- 25 +8=0, which reduced, becomes a’ — 2qa‘+(q?+ 4s) a—1r?=0 (5) FJ replacing a* by u ue— 2qu?+ (q?+4s)u—r*?=0 (6), a cubic equation, whose solution gives a value of w, and therefore of a’. A value of a being thus determined, the two quadratic fac- tors of a*—qz?—-rxz—s=0 are 5 r saz+}(at—g+7)=0, Tr ot ar+4(at—q-")=0. We thus obtain a method of determining the quadratic factors of a biquadratic equation, wanting its second term, without the necessity of any previous modification of its form. It was first given by the celebrated Des Cartes in his Geometry*, a work whose appearance formed a remarkable epoch in the history ‘of Algebra and its applications. Thus let the proposed equation be a*—172°—202-6=0t. Examples. ue — 34u’ + 313u — 400 = 0, | | _ The reducing cubic equation is | | : one real root of which is 16, which gives a=4: we thus get * Lib. III. t Ibid. 356 The component quadratic equations are therefore t+4xe2+2=0, x? —42—-3=0, whose roots are —2+,/2, —2- 2, 2+,/7 and 2—,/7. Euler’s so- 986. If we denote the roots of the equation lution of a biquadratic a‘—g2?—-re—s=0 equation and its de- | duction by @ A; 7,6, we may replace 6 by —(a+ +), inasmuch as x fromthat sum is equal to zero: but since a is the coefficient of the secon teste term of one of the component quadratic equations, its valu may be the sum of any two roots a, 2, y and — (a+ +r) i the proposed biquadratic, and therefore equal either a+ 8, a+q B+y, or, to -(a+f), —(a+), —(8+y): the values of u or a are therefore (a+ 8)”, (a++)* and (8++)’, which are only thre in number and are, consequently, the roots of a cubic equation. _ Again, if we suppose ¢, t’, t” to represent the three root: of the reducing cubic equation thus found or of | ue— 2qu*+(q?+45)u—r2=0, then, inasmuch as t=(a+ PB)’, ’=(a+y)%, t’=(8 ++), we get Jt+ Jt + /t’=(a+ B)+(a+y)+(6+y7) =2(a+B+y)=—26, | Nt— J/t'— jt” =a+ B-(a-7)-(8 + y)=-2y, | Jt— Jt — Jt"=at+y7-(a+ B)-(B +4) =-28, | Jt'— Jt— Jl =B+7-(a+8)-(a+y)=~2a. | It follows therefore that we can determine immediately a roots of the proposed biquadratic equation, by means of the roots of its reducing cubic equation u*— 2qu?+(q?+ 4s) u—r?=0, without the formation or solution of its component quadratic equations. | The same conclusion may be otherwise deduced by the fol- lowing process, which was first given by Euler: let | 357 7 assumed to represent a root of the equation xv’ —ga’?—rx—s=0 th); 21 let ¢, ¢’, t” be the three roots of the cubic equation u®’— Pu? + Qu-—-R=0 (2)5 thus get ECAH 2 JP 42 fit" +2 JE) = P+aA( Jit + fet?s+ Jvr) : (2" -5)= E(it' 5 tt 4 it" 42 Ste’ (fit flit ft) Q = Z ae JR a. _Transposing the significant terms to one side, we get E P?— a Fe — /R.x + eee (3): Comparing the terms of the identical equations (1) and (3), find a | Brel 2 Ciel =") TOs or Q=q°+45, } id the reducing cubic equation (2), becomes u®—Qqu’?+(q?+4s)u—r?=0, ich is identical with the reducing cubic equation in Des Cartes’s ution. —=7 Rk. ; It should be observed that ,/tt’t” = g 1s an equation of con- jon, which makes it necessary that the values of ,/t, ,/t’, ,/t” ‘ould be either all of them positive or two of them negative. ) 987. The following are examples of the SHE of this Examples. tthod of solution. | Let the equation proposed for solution be | x — 252° + 60x — 36 =0. The reducing cubic equation is u®— 50u? + '769u — 3600 = 0. The roots of this equation are / 9, 16 and 25. | I . 358 Therefore a=s (tt ttt) =5 +242= 6, E any Pag ae - Sa ee) 3 | =-5- 2435-1. Again, let the proposed equation be a*— 40a + 39 =0. The reducing cubic equation is u>— 156u — 1600 =0, whose roots are 16, — 8+ Get —8-—6 Lote Therefore a4 (4+ Seu OYE] 4 / S=ON a =J{ar(tsey as (1—3 aye st = 114 — (Lega es (ous ee yen: Epa 441 +3\/=1)—(1= 8 ,f=4)| =-25 3.) 1. =4{-4-(14+3,/—-1)+(1-3,/-D}=-2-3/-—h Acommon 988. The different methods of solving biquadratic equatio Pearibles4 which we have given in the preceding Articles, have been fou all methods t9 depend upon the formation of a cubic equation, whose ro Vee were determinate combinations of those of the equation requid equations: t 9 be solved, whether with or without its second term: ¢ inasmuch as there are only three roots of a cubic equati those combinations alone would succeed, which admit of th values and no more: such combinations are | afi+yo ay+ Bo ado+ Py Qe er Fhe foo! (+f) (7¥+8), (a+y7) (B+), (a+8) (8+), (a+B—- 7-6), (a+7-B-S8), (2+8-f-y)’, or 359 4 meee ch ys ty 8 a8-By a+PB—(y+8)’ aty—(B+8)’ a+o—-(P+7) : jich are only three in number, whether a, (3, and 6 be the ‘ots of a biquadratic equation, with or without its second term : ch also are | (a+ fy, (ata, (a +d), ‘the four roots are capable of being represented by a, B, ¥ hid — (a+ +7), as in the case of a biquadratic equation which ants its second term: we may conclude, therefore, as far as ese examples will authorize us in doing so, that the methods ‘solving biquadratic equations (which will be considered under ‘more general point of view in Chap. Xv.) put us in possession . no specific process for the general solution of equations distinct ‘om those of equations of the third degreet. 989. But it may be asked what is meant by the general What is lution of an equation of the third, fourth, or any higher order ? ea Lis the discovery or determination of an expression, involving the mea vefficients of an equation when denoted by general symbols, whose ‘wltiple values shall express all the roots of that equation and no wore: such equation being supposed to possess the most general mm of its order, or such other form as is deducible from it by a veneral process. It is a relaxation of the strictness of this defi- ‘ition of the meaning of the general solution of an equation, if ‘ve allow the multiple values of the expression or formula thus ‘liscovered, to be limited by means of one or more equations of ‘ondition: without such a restriction, the formula of Cardan is iot a general solution of cubic equations, and the formula of ‘olution of a quadratic equation would alone, of known methods, ‘atisfy all the conditions required. |. When we speak of the numerical solution of an equation What is ‘ : 4, meant by of any degree, we mean the discovery of a process which will the nume- mable us, in all cases, to determine its roots to any required rical solu- y tion of legree of accuracy, when its general coefficients are replaced equations. / * These combinations form the basis of a very ingenious solution of a biquad- vatic equation which is given in the Ist volume of the Cambridge Mathematical Journal, a publication which is justly distinguished for the originality and elegance of its contributions to almost every department of analysis. + See Waring, Meditationes Algebraice, p. 189. ‘ 360 by numbers; and it will be seen, in a subsequent volume | this work, how great is the progress which has been made effecting the complete solution of this problem: but it should’ always kept in mind that the general and the numerical solutiv of equations are problems of a different nature, and almost entire: independent of each other: in one case we seek to determine t» roots one by one, by the repetition of a prescribed process: the other, we are required to discover a single general formu whose multiple values will express all the roots, and those ro only: and it should be observed that no such general form has been found, even with the aid of equations of conditi for a general equation beyond the fourth degree. But is the existence of such a formula, for equations of th fifth and higher degrees, possible; and if not, is there any simp mode of demonstrating that the conditions which it is require to satisfy are incompatible with the laws of Algebra? We ha before remarked (Art. 978) upon the difficulty of proving | universally negative proposition, even in cases which involy relations of a very simple kind, and the attempts which hay been made for this purpose by Abel and others are not al culated to lessen the impression which those observations al intended to convey: the clearest understanding gets ewildaiel by the extreme generality and complexity of the relations whic it is necessary to consider in such investigations, and our fin | assent to the conclusion obtained is rather a formal act of ay quiescence in reasonings whose entire force and relevancy we ca_ neither fully appreciate nor easily refute than a spontaneous ac mission of a truth whose evidence is complete and irresistible. It is contrary to probability that a formula should one formed by the fundamental operations of Algebra only, whic should embrace all the coefficients of an equation, and expre: | indifferently all its roots, and no more: for no such formula he hitherto been found, which rigorously fulfils those — beyond equations of the second degree: and the failure of th various attempts which have been made to discover it in the cas — of higher equations is more likely to be owing to the impos I sibility of the problem proposed than to the inadequacy of th methods which have been made use of for that purpose, or t want of skill in applying them, . f CHAPTER XLIII. 1 THE SOLUTION AND REDUCTION OF RECURRING EQUATIONS. 990. In the absence of general methods of solving equations eee and other degrees superior to the fourth, there are many cases where equations }uations present themselves with particular relations of their Which may be depres- efficients, which admit of reduction to binomial and _ other sed to : uations, whose degrees are within the limits of general solu- a a jm: some of these we have considered in a former Chapter sree: ie: and another extensive class are denominated recurring }uations, whose coefficients from the beginning and the end le the same; of this kind are the equations a+ px+ga?+qut+prt+1=0, r+ px +guett+rar+ gz? +px+1=0, e first of which is of odd, and the second of even dimensions. | 991. + 58°u—a=0, ‘n equation of the fifth degree whose roots have therefore, veen determined. If s and ¢ be the arithmetical 5™ roots of = +" th (5- ®*), and ! Tl i : 2 —1 -— <-*) respectively, and if a be a root of sarge 0, then ) of the 25 values of u which are given by the preceding formula, +and t, as and a‘t, a‘s and al, a@’s and a’t, and a’s and a*é are he only combinations which express the values of w in the pro- dosed equation. Again, the formula ‘ 1 lL fa (f a) é if AG 6\) 8 — + — =f -4—-— = a=\8 \2 4 if 2 4 ) will express the roots of the equation u°— 6sut+ 9s°u?-—2s°-a=0: and, in a similar manner, we may proceed to determine gene- ally, the form of the equation whose roots are expressed by the formula a cane ‘both when nv is an odd and an even number. . CHAPTER XLIV. ‘ ON THE SOLUTION OF SIMULTANEOUS EQUATIONS, AND TH THEORY OF ELIMINATION. Dependent 997. ‘THE object proposed in the solution of those equation Poa which have been considered in preceding Chapters of this worl, symbols. has been to express one in terms of all the other symbols ¢ the equation, upon whose values therefore, whether assigne) or assignable, the value of this symbol is dependent, and b. means of which it is determined: and the same necessary de pendence between one symbol and all the others is presume, to exist in all equations whatsoever, whether the law and natur of such dependence is assignable or not: it is for this reaso} that such a symbol is termed the unknown quantity in the pri mitive equation, when all the other symbols are assigned o assignable, and when its determination is the object propose! by the solution of the equation; or it is termed the dependen symbol, when the others are not assigned, but are considere) as perfectly arbitrary and independent of each other: in othe words, the dependent symbol can only become known and de termined, by assigning specific values of the several symbols in terms of which it is expressed. Connection 998. Thus the equation of the cha- racter of ax+by—c=0, dependent ; andinde- when solved with respect to x, gives us Jet symbols “e with the gis Pd. solution of a the equa- i ; tion which Where a is dependent upon y and the symbols a, 6 and ¢, which ae are all of them equally arbitrary and independent, so far as thi conditions of the equation determine them: but if we supposi (as is most commonly the case), that the first letters of th alphabet a, 6, c, denote quantities which are known or deter! minate, then x is the dependent, and y the independent symbol | 2 ere 367 , however, we should solve the primitive equation (1) with spect to y, we should get , hen y will become the dependent, and x the independent symbol : ne character of dependence and independence, therefore, as dis- nguishing one symbol of an equation from the others, which ‘e unknown and indeterminate, is convertible, and is deter- ined by the solution, or presumed solution, of the equation ‘ith respect to one or other of those symbols. Kee 999. The dependence of one symbol upon an independent Meaning of ymbol or symbols, whether accompanied by others which are panera nown and determinate or not, is usually expressed by the term explicit and unction ; and the function (for the term is also used absolutely) oe 4 said to be explicit or implicit, according as the equation which avolves the dependent and independent symbols, is solved with spect to the dependent symbol or not: thus x is an explicit janction of y, in the equation a Sead bPs jut 2 is an ¢mplicit function of y, or y of x, (these relations being onvertible), in the equation ax+by—c=0. 1000. The term function is not only used to denote the de- Presumed endence of one symbol or quantity upon another or others, ‘»ctons. then their dependence is completely exhibited in a symbolical quation, whether implicitly or explicitly, but also when such ependence is presumed to exist, from the nature of the case, nteriorly to the investigation of any equation by which it may e expressed: thus if we should wish to express the dependence f the space described by a body when acted upon by deter- inate forces according to determinate laws, upon the time of ‘s motion, we should say that the space was a function of the ime: and if we should agree to denote the space by s, and : 368 the time by ¢, we should express the same proposition by meat of the equation s =f (t), where the letter f, prefixed to the ind pendent symbol (¢), is used to express the term functzon. The exhi- 1001. When more than one unknown or indeterminate syn! bition of the dependence bol presents itself in an equation, the solution of the equatic vie with respect to any one of them, and therefore the exhibiti ae the of its dependence upon the others must be effected by the gener Bagi methods which have been taught in the preceding Chapter and must be limited by the limitation of those or other method: thus the equation e+yt J/@ty)=12 may be reduced to the equivalent equation Rise Oi Ys where the actual dependence of x upon y is exhibited: in a simil manner, the equation x 2) 2 Ee r+y a0 2 y Jy y leads to the equivalent equation = ey a Wie ie ae and ai — Qa°y? + y*— 2a? a? + 2a°y’ = b*— a’, to the equation ra (atebea yh. Contri- It is obvious, however, that this exhibition of the dependeni fae to the Of One symbol upon one or more other indeterminate symbo pete leaves them necessarily as indeterminate as in the primitiy values. equation: and however important such reductions may ]) for many of the purposes for which such equations may 1} applied, it can only be by the aid of other hypotheses + conditions, that their values can be absolutely determined’ such conditions, however various when considered in connecti« with the problems from which they arise, will generally resol’ themselves into the simultaneous existence of as many ind * Such is the condition, considered in Chap. 1x., which restricts the values the unknown or indeterminate symbols to whole numbers. 369 yendent equations, as there are indeterminate, and in this case, own quantities involved in any one of them. 1002. Thus, if we have two equations involving x and ¥, Process by i a ; a : which such ssessing simultaneous values in both of them, then it is obvious ectiations aat the value of x determined, in terms of y, from the first may be quation, must be the same as that of z determined, in terms sab ne te f y, from the second: if we equate these values, we get an pasrtagul uation involving y only: the solution of this equation gives eliminated. ae absolute value or values of y, and leads us therefore neces- arily to the absolute value or values of x: the following are xamples, and many others have already been considered in (1) 7z- 9Y=7 ; Examples. 32 +10y = 1005" The first equation, solved with respect to z, gives us = {ss oY 3 7 | The second equation, solved likewise with respect to 2, ives us ) 100 — 10 ee 3 The values of « and y in the two equations being assumed to 7+9y 100—10y f 3 21+ 27y = 700 - 70y, 97y = 679, y=7; and therefore nM sieses Bia Bh 10" 10. 7 7 7 zw? Ag 35 SOS eG @) y Yy 9 Vou. IT. 3A i Elimina- tion : ; final equation. 370 If we solve the second equation with respect to 2, we gi ew=2Q+y. If we equate these values of x, we get 1 fer cy sty=2+y: 4 and therefore y =3 or + The corresponding values of x are 5 and r P+2 yt+4 24-5 7 2) Epa gee TO A z+y+z—-12=0 Clearing the first equation of fractions and solving it wi respect to 2, we get < ce “yon From the second equation, we get ‘ r=12-y-%. Equating these values of z, we get 5y —42 + 24 A or 13y+ 42-72 =0. =12-—y-2, We have thus reduced the two primitive equations to 01, involving two unknown, and, in this case, indeterminate q tities: the third unknown quantity has been eliminated fra them, and by the process employed for that purpose we he? lessened the number of the proposed equations by 1. | If we had commenced by eliminating y, instead of 2, fra the primitive equations, we should have obtained the equa i 132+9z-—84=0: obtained the equation A 4a —Qy+24=0. The elimination of symbols or unknown quantities, wheth determinate or indeterminate, from equations in which they 2 7 371 d the processes which are employed for that purpose, which e very various in form, are generally identical with those em- oyed for the solution of such equations, or rather for their duction to a single final equation: if this final equation in- ves one unknown quantity only, its value may be determined ysolutely, by means of it, by the aid of the known methods for lving equations, as far at least as those methods extend: if it volves two unknown quantities, they are both of them indeter- inate, and one of them is independent and arbitrary: if the jal equation involves more than two unknown and indeterminate mbols or quantities, then all but one of them are independent id arbitrary. 1003. In the preceding examples, the process of elimination The num- * one unknown quantity has reduced the number of equations aten Oil yone: and a very little consideration would shew that if was Peer iy BE -e number of the primitive equations involving any number of py the aknown quantities, of which x was one, then the number of in- ce 2pendent equations which would result from the elimination of quan ‘would be x—1: for if each of these equations be solved with ~ bam spect to x, (and we assume the practicability of such solutions) id if we should thus obtain Rs AT Ae. Cis Aga want ds a= A, here 4,, A,, A;,...-..A, are the symbolical values of x sxived from the several equations, then we should find, by juating the first value of x with each of the others, the following »- 1) equations, Rea yf 4 Sol ae yoo, Ape All other similar combinations of the quantities 4,, 4,, A,, .. A, with each other, though equally admissible with the pre- ding, will lead to equations immediately derivable from them, id therefore presenting no new and independent conditions for ae determination of the unknown symbols which they involve : lus, AD a A= (4Aj—4,) — (4Ap-4,) = 0, A, — A, = (A, - A,) — (A, — Az) = 0, 4a - A, =(4,~ 4,)—(4,~ 45) = 0, Ge ee d/4 @ U6 Cs OCR HE OE ROC COR Oe Awe Equations indepen- dent of each other. 379 in other words, the equation corresponding to any such cor. bination will always arise from subtracting from each other sor: two of the (n—1) equations which resulted from the first ser; of those combinations which it was considered most proper al convenient to select. Again, inasmuch as the elimination of one unknown quan from any number of (x) equations, diminishes the number { independent equations by 1, it will follow that the successi: elimination of (z — 1) unknown quantities will diminish the nw ber (x) of equations by (n—1), and will therefore leave a sing final equation remaining: if, therefore, the number of unknoy quantities be the same as the number of equations which invols them, the final equation will involve one unknown quantity on which will admit of determination by any methods which enak us to solve the equation itself: but if the number of unkno quantities exceeds the number of equations by m, the final equ. tion will involve (m+1) unknown quantities, which are therefe indeterminate in common with all the others, and m of them # independent: but if, on the contrary, the number of equati« exceeds the number 2 of unknown quantities by m, all [ae eoeereree nN fluous, if they are not inconsistent with each other. (Arts. 36, 402, 403). : When we speak of equations as independent of each oth, we mean such as severally contain conditions for the determi tion of the unknown quantities which they involve, which # not supplied by the other equations, nor derivable from the: we must exclude therefore all equations which are multiples! any other of the equations proposed, when the multiplier is assigned quantity, and independent of the unknown quantits involved in the equations: for the equation which thence aris can express no new condition for the determination of the qué tities or symbols which are involved in it. Again, we mit exclude equations which are multiples of other equations, wh = ko6ic. ob kbc, + kb, Co — k,05C; : Ay b4C3— Ay b5 Cy — Ayb, C3 + Aab3Cy + A3b,Co— AzboCy" d, by a similar process, we likewise obtain 4a Qa, kc; i a, kc; — a,k,c; + agksc, + ask, Co a askoc1 a, boc; ae Qa, Oaes a a bcs > a,65c, a a;b,c2 Fd a. Se, ; af a, bok; = a,b; Ke oe a,b,k; + a,b;k, + azb, ke Se azb,k,* Ay bees — A,b5C, — yb, C3 + dyb3C, + sb, Cy — a3b2C, | 1014. If the number of equations (n +1) exceed the num- To deter- r of unknown quantities (x) by 1, we may determine their ree of lues by means of x equations, and substituting those values ron the (n+ 1), we obtain an equation of ie rdicont which must number of » satisfied, in order that the equations may be consistent with equations ; exceeds by ch other: thus, if we have the three equations 1 the num- ber of quan- ' b tities to be ax + by =C, (a) eliminated. an+by=c, (6) a2 + by =c": (c) id only two unknown quantities 2 and y to be eliminated, e find from the two first ° _Oce—be' a’'o—ac’ ~ab—ab’?™ abl—ab? hich substituted in the third equation, give us, when multi- lied by ab’—a’b, the equation of condition a’ (b'c — bc’) +b” (a’c —ac’)=c"’(ab’—a’b). (d) * See Art. 404. Appendix, Vol. I. p. 399. , 384 The pro- 1015. If the coefficients, one or more of them, involve o blem of aL elimination Or more unknown symbols, the equation of condition (d) w from equa- be the final equation to which the elimination of x and y, | higher this process, will lead: and it will be found to be very eas, 3 Baal to by a very simple artifice, to extend this method of eliminati| elimination t9 a system of equations in which different powers of one from equa- tions of the more of the symbols which are required to be eliminate, first degree. present themselves *. Thus, let the equations be ax’+br+c=0, (1) a’x° + ba' +c’ =0, (2) where the coefficients a, 6, c &c., one or more of then ij volve y and its powers: and let it be required to eliminate If we multiply these equations (1) and (2) by a, we g the additional equations | ax’? +62°+cx=0, (3) a’ x* + bx? +c 2 =0: (4) and if we consider 2°, 2’, and z as distinct and independe) symbols, like z, y, 2, the number of equations, thus formed, w exceed by 1 the number of symbols to be eliminated. If we eliminate 2z* from equations (3) and (4), we get t equation (a’b — ab’) x’ + (a’c —ac’)x =0, (5) or Ax’?+ Br=0, replacing a’b—ab’ by A and a’c—ac’ by B. If we now eliminate «* from equations (5) and (1) in o case, and from equations (5) and (2) in the other, we get | (Ab-—aB)x+ Ac=0, (6) (Ab’—a’ B)x+ Ac’ =0, (7) which leads to the final equation or determinant (Ab-—aB)c' —(Ab’-a’ B)c=0: or, replacing the values of A and B, (ac’ —a’c)’ + (a’b — ab’) (c'b — cb’) =0. * Cambridge Mathematical Journal, Vol, 11. p. 232 and 276. 385 Thus, if the proposed equations be 2x°—3(y-1)x+y?-3y-6=0, 3.x? — (Sy — 4) x —10y?— 4=0, ie elimination of x, by this process, gives us the final equation | ; ) | 430y' — 252y° — 440 y? + 150y + 112 = 0, . hich is the same result as would be obtained by the method ‘the greatest common divisor. It is this reduction of the general problem of elimination » that of elimination from equations of the first degree only, hich gives additional importance to the latter problem, and ) the research of the most easy and expeditious rules, both ibs the expression of the unknown symbols which they involve, for the formation of the final equation, wnder its most simple orm, which results from their elimination, when the equations sceed by one the number of such symbols: the discussion of se methods and formule to which this inquiry would lead, is ae of no ordinary difficulty and extent, and is such as would ‘idly be consistent either with the object or the limits of iis work: we shall confine ourselves therefore to the single ‘toblem of the result of the elimination of x unknown symbols ‘om x+1 equations of the first degree. 1016. Resuming the expressions for 2, y, 2, which are Enuncia- (ven in Art. 1013, we shall be enabled to enunciate the law of Hoe oF ne ‘rmation of the numerators and denominators of those ex- mation of the expres- ‘fessions, and to ascertain the principle upon which it may be sonahirthe ‘tended so as to express the values of the unknown quantities lamar | four or any greater number of such equations. denved from three In the first place, the numerator of each fraction differs from equations 3denominator in having &, with its subscript numbers, in the Epon! vace of the coefficient (with the same subscript numbers) of the aknown quantity whose value it expresses. In the second place, the number of terms in each numerator ad denominator is 6, or is equal to the number of permutations } ‘the subscript numbers 1, 2, 3, (Art. 444). In the third place, the algebraical sign of any term of the ‘amerator or denominator will be the same with, or different Vou. II. ac ~ have the same sign. Determi- nation of the expres- sions for the unknown quantities in a system of four or of n equa tions. 386 from, the first term in each, according as it involves an odd | an even number of quantities which are different from those | the first: thus, the terms a,b,c, and a,b,c,, which involve t} quantities, 6,, c;, and 6,, c,, which are different from each oth have different signs, whilst the terms a,b,k, and azb,h,, e2) involving three quantities which are different from each oth It will follow as a consequence of the last observation, tl the corresponding terms in the numerator and denominator, | in other words, the terms with the same subscript numbers | the same order, will have the same algebraical signs. We shall now proceed to consider the formation of cor! sponding expressions for the unknown quantities, when the are four or a greater number of such equations. Let e- 5), ThE eh — expressions for the three unknown quantities in three equatic which are given above (Art. 1013); and let the four equati« be be assumed to represent t: ax+by+ces+du—-h=0...... (1) aga + boy +cge+d.u—h,=0...... (2), a,t+b,yt+c,s+d,u—k,=0...... (3), ea Woy rane + 4u —khy=0 22.00 (4). If in the expressions N,, N2, N; we replace k,, hy, hs k,—d,u, k,—d,u, k,—d,u respectively, and if m,, m,, m3 taken to represent the values of N,, N,, Nz, when /; is replac by d,, kz. by d., k, by d;, then we shall find, from the three” a equations (1), (2), (3), N,— 1,u N,— neu N, — gt 9 ess al daa ak Dal ea eee pressing fractions, we get (A,n, + by Nn» + C43 ao d,D) u— a,N, a b, Nz ay C4 N, + k,D= 0, and therefore _ aN, +6,N, + ¢,N3— ky D | 4 uw a4Ny he b, Ns VC, ta d,D : | 387 dif N’, N”, N’” with their proper subscript numbers be suc- ssively taken to represent the values of the numerators of e expressions for any other three amongst the four unknown tantities in the three equations (1), (2), (3), namely, of , z, y, u, when c is replaced by d and d by ec, : x, 2, u, when 6 is replaced by c and ¢ by 8, y, 2, u, when a is replaced by 6 and 5 by.a; dif the corresponding values of and D be denoted by n’, n”, nl”, (oe iby? De en we shall find a, Nj + b, No’ + cs Nz’ — kD’ TE EA Ro, AUB agN,+'b, Ne" + ¢y'N,— & D” ayn; % Oa 2 anf Sdlp”? a, rt ab b, N"! a C; N,/” Ee ky, DY" i. a,n,” a bn” 4 Cy ni, +t: d, rye ~ The number of terms in the numerator and denominator of ch of these expressions is four, or is equal to the number of iknown quantities : and if similar expressions were formed for ‘e unknown quantities and five equations, they would severally mtain five terms in their numerator and denominator; and nilarly for whatever number of equations such expressions ere investigated: for the substitution of the expressions es ma —1) equations but adapted to a system of m equations and unknown quantities, in the n'* or additional equation, will id to an expression for the new unknown quantity, whose imerator and denominator consist of x terms; or if 2, denote e last unknown quantity introduced, and / and m (with their oper subscript numbers) be the coefficients of «,_, and 2,, en we shall find _ 4, P, +1,P,+6,Pet--- b,Peii— hn Q | a ted te Oi LO) — oe OA id it is obvious that corresponding symmetrical expressions for (n—1) unknown quantities derived from : Ln 388 may be obtained for all the other unknown quantities a, X,—-9) «++ 2, in the inverse order of their introduction. Enuncia- It appears therefore that the number of factors in e2 pee law product involved in the numerator and denominator of the of forma- pression for xz, will be 2: for an additional factor is introdud aaa for every additional unknown quantity or additional equatio, for the n f poknown Again, the number of terms, when they are completely « Si aphadl hibited, in the numerator and denominator of the expression 3 ofnequa- x, is 1x2x3x...m, or is equal to the number of perm loa tations of the subscript numbers 1, 2, 3, ..., (Art. 449): 1 the number of terms in the numerator and denominator of { expression for x, is m times the number of them in the expr sion for z,_, in a system of (n—1) equations, nm (m—1) e the number of them in the expression for z,_, in a system f (x—2) equations, and so on, until we descend to the expressi for the unknown quantity in a system of two equations. Again, the number of positive and negative terms in numerator and denominator being the same in the expression 14 the unknown quantity in a system of two and three equatio), will continue the same likewise in the corresponding expressios in a system of x equations: for the new numerators of the expressions are formed by multiplying the series of literal u- merators (which are the same as far as the letters involved a their signs are concerned) of the (—1) first unknown quantit and also their common denominator, into a,, 6,, c,... k,, respe tively, and connecting their results with the sign +: if therefe the number of positive and negative terms be the same in t numerators and denominators of those expressions for (n—'! unknown quantities and (n—1) equations, it must continue t? same therefore when there are x unknown quantities and} equations: and inasmuch as this number was the same, whi there were two equations and two unknown quantities, it mu continue the same therefore whatever be their number: the sar observations apply, with a very trifling modification, to t! number of positive and negative terms in the denominators. _—_——_ Lastly, the same law which was noticed as determining t negative and positive terms in the case of the expressions for f! unknown quantities in three equations, will prevail likewise f 389 ay number of such equations: for whatever condition deter- ‘ines the sign of the separate terms in the numerators and ‘enominators of the expressions for the unknown quantities, Lien there are (n—1) unknown quantities, will determine their igns likewise when there are » unknown quantities: for the vries of terms involved in their numerators and denominators fe multiplied into new factors ] nd therefore the conditions for the determination of the signs of 4e resulting products in each series remain the same as before. 1017. If it be required to find the final equation, when the umber of equations (n+1) exceeds by 1, the number of (7) ymbols to be eliminated, we find from the z first equations, ne w= Hs vhere N and D are formed by the general law which we have Just investigated: we find, in a similar manner, from the x ‘ast equations, | nN’ t= D’ 3 where N’ and D’ are determined as before: we then equate the values of x, which are thus found, which gives NEN (3, DY or ND/- N’D=0, Which is the final equation required. Thus, if there be three equations, ax +by —c =0, aa +b’y—c' =0, a2 + by -.c” =0, we find from the two first equations, (ab’—a’b) x —(b’c — bc’) =0, and from the first and last, (ab”— ab) « —(6"c— bc”) = 0, Mode of obtaining the final equation when the number of symbols eliminated is less by 1 than the number of equations. 390 which leads to the final equation (ab’— a’b) (b"c — bc”) — (ab"— ab) (b'c—bc)=0; or, dividing by 6, | ab’c” — ab" cl —a'bc’ + a’ Bt a’bc’—a’bc =0. It will be observed that this final equation is found — making the denominator D=0, in the expression for any 0: of the three unknown symbols arising from the solution of thi: equations of the first degree which are given in Art. 1013. | ’ CHAPTER XLV. YON THE ALGEBRAICAL SOLUTION OF BINOMIAL EQUATIONS. 1018. We have shewn in a former Chapter, that the roots The Bon . metrica } . . . *the binomial equation solution of the equa- x"—1=0 tion z*—1=0 e completely expressible by means of the different values of aoe sf. imply its ne formula algebraical cos —— + /— 1 sin U properly so called. there r may be replaced by any term of the series 0, 1, 2. 3, ee hes 1e same roots recurring, in the same order, if additional terms f this series are taken. But this solution, though complete, is not algebraical, in the trict sense which is attached to that term, for reasons which ave been explained before (Art. 990); but it will be found that ae symbolical properties of those roots, and of the cyclical yeriods (Art. 722) which they form, will be sufficient to deter- ‘ine them algebraically or rather arithmetically in all cases. _ For this purpose it will be necessary to investigate some wropositions, which are equally applicable to the roots of bino- ‘nial and other equations, and which will enable us to bring he various methods of solution of cubic and biquadratic equa- ions, which we have already considered, under the operation if a common theory *. vis 2 Uu HOL9. Let p, pis par-+- Pea be the roots of any equation of Formation 2 dimensions, and let 1, a, a’,...a”" be the roots of the binomial of Seat equation in from an x"—1=0 assumed form of its root. >-quation *® The substance of this theory is given by Lagrange, in notes x11. and xtv. to he Resolution des Equations Numeriques. 392 of the same degree: and let it be required to investigate tl conditions requisite for the determination of the roots of tl equation which are expressed atin by the formula | té=p+ap,+a’p.+ ete es Cnr ae In the first place, the number of values of ¢, and therefore tl dimensions of the corresponding reducing equation will be Ni do irs Hh 3 Aen for if we suppose Petr a ere to retain their position, we may arrange the n roots Po Pir Paces Pn-i inlx2x3x...m different ways, each of which will produc a different value of ¢. | But a very little consideration will shew that these 1 x 2 x 3 --. ” roots or values of ¢ may be distributed into 1x2x3x... (m—-1) groups of x roots each, whose n™ powers are identical: for if t=p+ap,+a*ps+... ap, 4, then also at=ap+a°p, + a> pst wos Pais at=a’p + ap, + @ pg tee Pais a"'t=a"'p+p,+ap,+ ve “el eee which are all of them equally included in the number of th 1x2x3x...m combinations which form the roots of the re quired equation: but inasmuch as Polat)" — fo tyes (ae n ys. it follows, that if we make 6 = 1" * This form of the root of the reducing equation, or equation resolvante, ¢ he terms it, was suggested to Lagrange by an examination of the form of th roots of the reducing equations in the solutions of cubic and biquadratic equa tions: and it should be kept in mind that the propositions which follow hav) reference to the\specific form assumed for t only or to such as are reducible t it, and therefore do not close the i inquiry as far as other forms, which may a) assumed for t, are concerned. 393 | th : 5 1 ‘he number of different values of @ is only the — part of the The lowest | n dimension to which it wmber of values of ¢, and consequently the values of @ are jg generally xpressible by means of an equation of reducible. : | 1x2x3x...(n-1) fimensions. We may assume, therefore, é=—?/"= A,+aA, +a°A,+ cee aA) ; vhere A,, A,, A,...A,_,, are combinations, whether symmetrical* w not, of the roots p, pi, po-++ Pr Of the equation to be solved: or it is obvious that 6 can involve no power of @ higher than v7, inasmuch as all powers of a are reducible to some one term of the series 2 n—1 Po Oe) Got te, Oe 1020. If it be assumed hypothetically, that the values of The roots 4,, A,, A,... A, are expressible by known numbers, or in eae verms of the coefficients of the equation to be solved, it may will deter- * : . mine those then be easily shewn that all its roots may be expressed in terms of the . : n : h . proposed of the different values of 2/0, which arise from replacing a by aquiatines the successive roots of r—)=0; For if we denote the terms of the series Lan OO by Let an Tare 43% aris and if we denote the corresponding values of @ in the equation 6=A,+aA,+0°A, +... aA; by 95.915. Og 2 <5 Ona 5 09 we shall find, since 0=2", and therefore _ * A symmetrical combination of the roots of an equation is invariable, inasmuch as it is not changed by a change in the order of the roots: and it may be easily proved that all symmetrical combinations of the roots of an equation are expressible in terms of its coefficients, and are therefore given, when the equation is given. Vou. II. aD 394 /0=t=ptapt+a®prt ... a" pyr, that N90 = p + pit po + e+ Pn-1> VO, = p+ Tipit Ty? pet o-+ 11" pars 0. = p + 1opi t+ 12" pot cee Te Deo s w/0,-5 = P “7 Tr-1P1 a Pel Pe THs ene Bey mn If we add together the terms on both sides of the sign = we get 0/8, + 0/0, + 2/04... 9/8, , =np*, NO + Oi + 82+ «+» Ona ‘ or, Again, since 2/8, = (BS oll ta Ve Pat eeeeeeve Pn-1> ry Un) Ope 13" 2p Pits ipet)- + 1 2-Per ao, 9 _ — rot 0g = 72" p + pit Teapot -. 12" Bais —1 2” i! -1 - Tn=i Meme a tp * P1 7 Pri P2 + eee 1 .-1PENS we get N80 + 71" 8, + 73" A/02 + «2. THT1 R/O, = np, or _ NO + ri? Os + 7e"* / 8a + wee 1D} Nn n In a similar manner, we find a + 7°" 8/0, + 7" Si Heo ee TaN Ona pointer tt, _N 4, + 710/81 rails + ees Tra VOn4 Pe eA For Ll+ry tre + ..... Peay 1 + 7,7 + ro? + 2... ri. = 0; 1 + 7° -b r,° bea pean Toa} => 0, 395 It thus appears that all the x roots of the proposed equation ‘te successively expressible in terms of 0,, 0,, 0)... 6,-, and of le n roots of z*—1=0: ad it is in this sense, and in this sense only, that the general solution of equations may be said to be dependent upon that f binomial equations of the same degree. . 1021. Before we proceed with the developement of other Applica: arts of this theory, we shall proceed to apply it in the first bee istance to the solution of quadratic, and subsequently to those formule to : : ; ; the solution f cubic and biquadratic equations. of a quad- ratic equa- tion. oo Let the quadratic equation be xn*—px+gq=0, ad let its roots be p and p,: the roots of the corresponding inomial equation Dee | te 1 and —1, and therefore a=—1. Let t=pt+ap,, | 6 = t= p’? + p’?+2app'= A, +aA,. But p’+p’* and 2pp’, or A, and A, are symmetrical com- inations of the roots, and are expressible therefore in terms of le coefficients of the proposed equation; we therefore find A, = p’— 29, PES Gp We thus get 05 = p?- 2¢+2q =p’, 0,=p’-2q+2aq=p’—4q, ad therefore _N%+/8 _ p+n/(p?- 49) p Bar Q > _Nb+0/8 p—J/(p?- 49) Pi Q 7 2 , Xpressions which coincide with those given by the ordinary ‘Tocess of solution. (Art. 658.) 396 A modifica- 1022. It may be observed, that inasmuch, as when a= tion of the A form of the WE Have root of the L=pt+ pit pot o-- Pars reducing equation. or equal to the sum of the roots of the equation, which is alga expressed by the coefficient of its second term, with its s changed, it will follow therefore that, if p be this coefficient, shall have O,=(p + pit pet -++ Par) =p", which is therefore always known. Again, inasmuch as Oy Ags Ai Ast see Heads we get Ay = p"— Mik soa A,— cee ACS: and if we substitute this value of A, in 0 = A, + aA, + a’ A, ae os ats Fe it assumes the equivalent form 0 =p"+(a—1) A, +(e?-1) A+... (a"?-1) A; and therefore 0, = p"+(m—1) 4, 4+ (7-1) 4.4... (n""-1) Aw, 0, = p"+ (r,- 1) A, + (ro? — 1) A, + eee Cre a 1) Bass 6,1 = p" + (Tra — 1) Ar + (7-1-1) Ag+ --- (ret - 1) Ana 7 pcos of 1023. The great, and in most cases, insuperable difficu equation. Of the problem consists in the determination of the values A,, Ay... A,1, or the coefficients of a, a?... a*' in them pression for @: thus in the cubic equation x —azx?+be—c=0, if we make t=p+ap,+a*po, we get 6=#=A,+aA,+0?A,, =a°+(a-—1) A, +(a?-1) Ap, where A, = 8 (p* pi + pr’ po + po’p)s A, = 3 (p* po > pi p = po pi) Zz 397 “hich are convertible into each other, if we change p, into p,, id p, into pi: if we form the quadratic equation, therefore, of hich A, and A, are the roots, we get u?— Pu+Q=0, Vhere P and Q are invariable, or in other words, involve the jots of the proposed equation symmetrically. It is always possible, as will be shewn generally hereafter, ) assign symmetrical expressions of the roots of an equation in wms of its coefficients, and the application of this theorem would nable us, in the case under consideration, to assign the values of >and Q in terms of the coefficients a, 6, and c: but if with a jew to the simplification of the very complicated expressions, vhich would thence result, we suppose a=0, and if we replace 1 by -q and c by 1, we shall find P=-9r, T= 0 (97? — q°)*, ind therefore We thus get 0, =(a—1) u, + (a?—1) u%, nd therefore (Art. 1020) — f81 + 0/80 ee 3 ? i a 3/0, + @° 0/0, Pil ai Ful Ae a’? J0+a,/0. pee Sev gae OF These expressions coincide with those which are given in Art. 968. * The complete expressions when the coefficients are P>» 4)", are as follows : =—3pq-—9r, Q=-993 +9 (p? +6pq) r+ 8lr’. 398 An equa- 1024. More generally, if we consider the equation tion of n ; dimensions, = nn fan get co: se BLA: 3 &, when nis a a igh Pov 4 pee eae. where x is a prime number, we find pendent, in " s - its ultimate 0 = p"+(a—1) A, + (a? — 1) ys BETS te (a"! — 1) Ags analysis, 2 ATE ; : eae which is given, as we have shewn in Art. 1019, by an equation equation o pe ee Bae hee (02) 1x2x3x... (n—1) dimensions. dimensions: but if we form the equation u!— Pu?*+ Pu — ... = P,,=0, whose roots are 4,, A.,...A,,, and if q be the number different forms which this equation may assume for differe values of P, P, ... P,_,, then the final equation which aris from their continued product will have invariable coefficient and will have q(m—1) dimensions: and inasmuch as q(m—-1)=1x2x3x ... (n—2) (n—-1), it follows that q=1x2x3x... (n-2): it appears, therefore, that whilst the values of @ are expresse by an equation of 1x2x3x...(n— 1) dimensions, the valu A,, A,...A,4, in terms of which it is expressed, will be di pendent upon an equation of 1x2x3x... (2-2) dimensions, Tn the case Thus, if 2=3, as in the case of cubic equations, the value ape of 0 are given by an equation of 1x2 dimensions, and the « efficients of the equation which- expresses the values of A, an A, will depend upon an equation of the first degree, and the are therefore unique: this coincides with the result given in Ar 1023. In the case If n be 5, or if the proposed equation be of 5 dimensions, th I ee values of @ will depend upon an equation of 1x2x3x4 or 2 5th degree. dimensions, and the coefficients of the equation which expresse the values of 4,, 4,, A,, A, will depend upon an equation o 1x2x3 or 6 dimensions, exceeding by 1 degree the dimension of the proposed equation. . ten the 1025. If the number which expresses the dimensions of the umber which ex- Proposed equation be not prime but composite, then it will bi presses the found that the process under consideration may be very greatly 399 t aplified: for if »=qm, where m is not greater than q, then the dimensions - of the equa- 1es tion is not . LS ava t7e ay prime, but Y composite. the roots of z"*—1=0 ry be distributed into q groups or periods m—1 Pega oe, Fl asd sea y curring after every m term: we thus get EP. A Pye ine tle edie deve GS" ays + Pm +O Pings + eee eee eee a Pem-15 + Pom + & Pom+1 + . seer ee ee P3m-—l> eee eee eee reevr eee eee eee + Pq-iym + &Pq—tymt1 t +++ @™ Pgm-r? 1 if we make ‘9 P+ Pmt Pam tH seceeecenecse Prete sas Pl i Pm+1 ws P2am+1 tel eben chery P q-1) m4+1 ae M¢y ceeoeerere eer eer eer eer eeeseeseeeesee see eee Pm-1 rif P2m-1 ar Psm-1 + eee Pqm-1 = 4km-19 > shall find : b= he PU AY aa ate ee Considering therefore X,, X,, X_...X,,, as the roots of a Ww equation of m dimensions only, we shall find : §=("=A,+a4,+a*?A,+ coer ee eeey Ane. 4 = p"+(a—1) 4, +(a?-1) 4,+... (e™'-1) A, » be the coefficient of the second term of the proposed equa- om with its sign changed: and therefore xX = RLOn Ot wl Ua tes. sin) Oo Com m > x RO ea) Ove Tar in| Og Hee Pe = NOt ° Ka Oo + 11 01 + 2 Oat to + Ft Our m—1 — m Dimensions of the final equation. 400 Having thus determined the values of the groups of t; roots of the proposed equation which are severally denoted } X,, Xi, Xo.-- Xn+, we proceed to apply the same process j the several equations of g dimensions which they form: and} q is still a composite number, we may again proceed to 1 distribute the roots of each primary into secondary groups, ai to repeat the same process for their determination. 1026. The number (1) of values of ¢ in the expression. t=X+aX,+a°X,+ ... aX, , is Lt OS i gee : (195 2-KS se BS Rg)es for the number of permutations of the roots in each group} 1x2x3x...q, and therefore all the permutations which m su groups can form with each other is (1x2x3x...q)": andf the number of values of ¢, when all the terms in the series 2 n-) Ra Fy Ce hyo are different from each other be 1x2x3x...m, and if t recurrence of the same period, after every m term, distribus them into groups admitting of (1x 2x3x...q)" permutatio), which become identical with each other, it will follow that te number N of different values of ¢, under such circumstance will be expressed by dividing 1x2x3x...” by (1x2 xk reid) yas Again, the number N’ of different values of 0 is only te t h = part of the number of different values of ¢ (Art. 1019), an therefore DV aren m Lastly, inasmuch as 0=p"+(a—1) A, +(e®—1)A,+... (e"—1) An, it will follow that, if we form the equation un) — Py Qumn—. ee O, whose roots are A,, 4....A,,,, the number N” of sets of sul coefficients (arising from different values of 4,, dA, ... A, 401 thich enter into them will be (m-—1)N”, which is also equal » N’: we thus get N’ N N’ = Siew m—1 m(m—1) cn ee EE i 1.x 2: 3 Xi. 8 ") ~ m(m—1)(1x2x3x eit number which expresses the dimensions of the final reducing quation upon which the formation of the coefficients P, Q, R, xc., and therefore the solution of the proposed equation, in its iltimate analysis, may be said to depend. 1027. Let us apply the formule in the last Article to the PaaS i s : , ion of the olution of the biquadratic equation preceding theory to be at px 5% qx" — re+s=0 (1). the solution of a biquad- In this case we have m=q=2, and a or the root of nr, beh which is different from 1, is —1: we thus find +f t= X,+aX,, where X,=p+p, and Xo = pit ps: Pee Aa at pee loee Ay _ The triple values of 4,—2X,X, are expressed by the roots of the cubic equation i, w—Puw+Qu—-R=0 (2), ‘where P, Q, R involve the roots of the proposed equation (1) symmetrically, and are therefore expressible in terms of its co- ‘efficients and are consequently known. Inasmuch as A, =2(p + pa) (p. + ps) =2(ppit ppa+ pipe + pops) =2q—2 (ppat pips) =29—2u, )where w’=pp2+ pips, we may transform the equation (2) into u!?— P'u!? + Q’n’ — Rf’ =0, (3), Vow. If. 3K Theory of Euler’s so- lution of a biquadratic equation. 402 when P’=q, Q’=pr-4s, R’= (p*—4q) s+7° are assigned by th process given in Art. 981 *. We thus find 4,=2q—2uw’, and therefore 0,=p’—24, = p?—4q + 4u': if we find a real value of uw’, or of 0,, we get Again, since we may consider as a factor of the proposed equation: and if we proceed to ex. amine the conditions (as in Art. 985) which this factor must satisfy, we shall readily find 5 ae? Aiat OX re 2X,-p ; we thus finally get, as in Art. 1021, ecu. - 2_ Ay (we 28 ip shes fhe and f2= xX, — = ) and similarly for p, and p,. 3 1028. If instead of forming the equation which expresses’ the triple values of 4,, we had begun at once to form the equa- | tion which expresses the triple values of 8, we should have * The processes of solution of biquadratic equations, which we have considered in Chap. xiiz., are equivalent to the corresponding processes for forming the reducing cubic equations, whose roots are such symmetrical combinations of those of the proposed equation as have triple values only: such are (p + ps) (p1 +3). or (pp2+pips), Or (p+p2—p;—p3)?, and others which may be formed: such processes are always greatly simplified, by supposing the coefficient of the second term of the biquadratic equation to be zero: thus the equation, whose roots are (p+pe—p; —p3)®, investigated in the Article which follows, when all the terms of the proposed equation are complete, is u’ — (3p? —8q) u?+(3p4— 16 p2q + 16pr+16q?—64s) u—(p?—4pq+4+8r)?=0, which becomes, when p=0, ue + 8qu?+ (169q?— 64s)—6472=0: it is only in this latter form that the particular process of transformation (if so it may be termed) which is commonly known as Euler’s solution (Art. 986), is | practicable: and it is to this form of the original equation evelusively that Des Cartes’ solution (Art. 985) is capable of application. 403 P=pP + Pot Pit Ps» £=p+pa- pir- Pa» om whence we should obtain g(p+t)=pt+ps, and 3(p—t)=p, + pa: we farther make u+v=pp,, and u—v=p,p;, the two quad- tic factors of the proposed equation would become x —1A(pt+it)r+u+v=0, ?@_1(p—tha+u—v=0, ‘hich multiplied together would give the equation a — pat {2u+i(p?—?*) 2? -(pur+iv)e+u?—v'=0: ‘we equate the terms of this biquadratic equation with those of e equation proposed, we get 2u+h(pt—f)=4, puttv=r, Ur =O = $i If from these three equations, we eliminate wu and v by the ‘dinary methods, and replace ¢’ by 4, we get the cubic equation mught for, which is 6 —(3p?—8q) # + (Sp‘— 16p’q + 16pr + 16q? — 645) 9 —(p®?-4pq+ 8r)’ =0.* If 6@,, 0,, 9, be the three values of @ in this equation, we jet pt JO: + 0+ /6s BS 4 at _ P= N01 + JO — J ad : Pi p+ rJ0,—r/02—/8s le A ie ae p- 91 — /92 + /%5 pa = i hie in aie * Ivory. Article ‘‘ Equations” in the Encyclopedia Britannica. 404 It will be observed that the last term of the equation whie expresses the values of 4 with its sign changed (Art. 1027, No or 0, x 0, x Qs = (p°—4pq + 87) and therefore N91 x n/ 02 x 85 == (p*— 4p + 87). If we suppose p,=0, p»=0, p3=0, and therefore p=p, the J8, x ./02 x ./03= p? = p®, which shews that the upper or positir sign must be taken: if we suppose therefore p?—4pq+8r 1 be a positive quantity, then either all the values of ,/0,, ,/0., Re / must be positive, or they must become negative by pairs, as 3 the form of solution just given: but if p*—4pq+8r be a né gative quantity, then either all the values of ,/0,, ,/0,, and ,/ must be negative or two of them positive, in which case tl solution would assume the following form: — p- JO, — f0.— fs EP coe 4 rae. hy P+ J/81—/02 + /8s 4 > pi= p—JO.+ /0.+ /0 Pa Ge.) Yo SOME Me _ptJ/i+/b2— oven Past 4 Failure of 1029. If we should proceed to apply the same method { 2 aa the expression of the roots of the equation | Pera x’— pat + qa®*—rz’+sx—s=0, aca we should find dimensions. t=pt+ap, +a°po+a°pz+a'p,, where «a, a’, a*, a‘ are the roots of the equation x ag 1 = 0, which are different from 1: if we now make @=?°, we should o | tain an equation in 0, with determinate coefficients of 1 x 2 x 3x dimensions, which is decomposable into 2 x 3 biquadratic equi tions of the form i H'— P+ Q0— RO+S=0, rt 405 yhere each of the coefficients P, Q, R, S admits of 2 x3 (Art. 024) different values for different permutations of the roots: or 1 other words, those coefficients will be dependent upon the jlutions of equations of the sixth degree: and it appears, there- mre, that the solution of an equation of the fifth degree, will ecessarily lead, by the process under consideration, to an equation f the sixth degree. 1030. If we apply the preceding process, however, ve shall find that it will enable us to assign its roots, by ilgebraical processes alone, for all values of n. to the Applica- tion of this olution of the binomial equation theory to . binomial fo es! li OC equations. For this purpose, let us suppose ” to be a prime number, Oa of a cycli- ind let us represent the roots of cal period of the roots ‘where 4,, 4,, A,... A,» are rational expressions involving r, which do not change upon the substitution of ra, ra’, ra’, &e. teat when 7 is a seh a prime x—1 number. oy r, r’, r°...r"*, and let us suppose them to be further dis- ‘mibuted into the cyclical period AR RP a A a een ee by means of any one of the primitive roots (Art. 531) of , such asa: we shall thus find that the substitution of r¢, re’, 74%, &c, or of any one of its terms, in the place of r, will reproduce the same series in the same order, if no regard be paid to the position of the first term, or, in other words, if they be supposed to be arranged in a circular order (Art. 722). : 1031. Assuming therefore « to be a root of Tietemminae tion of the Ce =O, roots of the reducing which is different from 1, we may make and from i ees of t ' t=r+aré+arr?+... a™?ra’, ive cual and therefore moe 6=t71=A4,+a4,+0°A, +... aA, ., 406 for r: for if we replace ¢ by a”*t, at... at, successivel we get 7 a"Vt=a"*r+ratare+ ..... a" ra? at =a* *r a yas yee at ra. OF 8 OO 2185 eee OO ais ee eo OLS 6 CEE eee Bale at=ar+a°ra+a'ra?+ , Ta: and if again in the same expression for ¢, we substitute for the successive terms of the series ra, ra”, ra®.., ra"-* we shall g ratarv’+e@°?ray ... g™® r= at r?+ares@*? rat4 ... gra ak: TU arg ra 2) gare at; it thus appears that the values of @ or of Ej fr-! as (a a aes me (a*t)"" =... es AES: are the same, whatever be the term of the series fd which we substitute for r, and that consequently those value; admit of absolute determination, inasmuch as it thus appears tha’ they involve the roots r, ra, ra?... pa? symmetrically. . 3 falte 3 n—2 PETE To ae, Poe = If we represent therefore the values of A, in the expression - I D= Avia Ave a OE EPs bine by i P+Qr+Qr?+... Q,.r™ (1), # it will remain unaltered, in conformity with the proposition just demonstrated, if we replace r by r¢, when it becomes P+Q,74+Q,r?+...Q or (2), ] and therefore equating corresponding terms of the identical expressions (1) and (2), we get nt Q = Q,; Q, sae Q,, Q, = Q; OK Q,-s a Qo Pi and therefore Av=P+Q(r+re+... ree = P-A, | 407 r+rap yar .., po? ]: a similar manner we find A, <= P, - Q, , Ay = P, zt Q., A,» = P. a2 Q,-2 fue the symbols P, P,... P,., Q,Q,...Q umbers. » denote rational nr— | The values of 6 are therefore = P-Q+ Gla Gy (Pa Qs) tense (Pog — Q,,») 1= P — Q + ay, (P; aa Q:) “t ay” (P, — Q:) “psd a," * (Px cia Q,-2)> a= P—-Q+ oz) (P; i Q:) 7 ay” (P, = Q:) a a Gem? 3 Q,,-2), lee P-Q+a,_. (P,-Q) a" an_2(Ps—Q,) fee Gnas (Pes = Qe); ae mangle ee tt — ppt m2 re =A, Ag=a', ag=a ...a,5=a “. ty | We thus find a BN Uy al Or-re Oat eae Ones a—1 ; SH —2 —1 at —1] | a — Eyer | Opa nf O5F 28 afd 7/0, 2 n—l Tree CSO SCSCHOE SMP SFCSE KR BOTT BR DB OCOHEeHEO RO eC HCC CHE KBPEOSC EC SKS \ipar? "Noo + a1 * "M91 +e “Oa + -+ + me “0,2 n—1 11032. A very little consideration, however, will be sufficient Simplifica- ae ‘ ti f thi ‘shew, that it is not necessary to form the expression for a eyhben We [wer of 0 so high as 6""', when n is a prime number: for, under Apu 5 sah | é e root, th circumstances, we have n—1 =mq, and it is obvious that which is terms of the expression for ¢ may be distributed into groups TORR Of ‘m or q terms as in Art. 1025: thus if a be a root of the bino- mial equa- tion whose ip — Je 0: i 408 index is the which is different from 1, we have least factor 5 “ 2 —l py! of n— 1. t= + @TO4 G7 Fae ee eee is eee a Vas ms + 7a 4 getty oF rae Mae ee arty ete as oe rag-)™ ie qrag-Dmt a a? pag-}) m2 BRP i a” rar"! : and if we assume D. Ee eo a eee og ae PP ae rae, Xo = 74. pant} 4 parmth OR eI pa—mtt Xo TO ae oe Lee 5 an it will obviously appear, from an examination of the terms which these several expressions are composed, that by replaci r by ra, X, becomes X,, X, becomes X;,... X,,. becomes X,, similarly, by replacing 7 by r«’, X, becomes X,, X, becomes ... Xm—. becomes X,: and similar consequences will follow fr replacing r by other terms of the cyclical period of the roo it will follow therefore that $= X,+aX,+0°X,+ ... a” XX, will become, when r is replaced by 74, X, aX, +a? Xie eames ts when r is replaced by 7’, it will become D.C POP Setiot (ee Geet e *, it will become and when r is replaced by rv" Xo Gk Aah ee FT A eee It will follow, therefore, that if Ope (a Leer Ge Lycee Leet ya = Apt ait at Aad nna iol oe A,, A,, Ao.-- An» will be rational expressions involving ) X, ...X,, in such a manner as not to change, when X, is chan; into X,, X, into X,, and so on: and that they are conseque of the form P+ Q(X, + X,+ X,+... X,,) or P+Qs, where Pi Q are determinate numbers, and s is either —1, or as will 409 ° en hereafter, some known rational or irrational number: if we ow find therefore the different values of 0, which are B, = P, + Q,s + (Pi + Qs + ...... + (Pri + Qn), Oi = P, + Q,8 +4; (P, + Q,8) +... +0," (Pr + Qui); 6, = P, + Q, + ae (Pi + Qus) + ..... tae" (Pra + Qn18), 6,1 = P+ Qot+ Gms (P+ Qs) +... Fanti il ge cangs Q-15); e shall find, making °/0,=s, BX. = 5+ 7/0, + 2/8, + — + ™f/On1 (eS ee eer ct ie a5 m Brads th Ppgpaced [UR ae Boge Sid files ate Caen Ped Mee $e m rts Dg stat Ot ag Yet tot YO 3 m ’ ; KX, = SHO + eta Oa + ++ + we Oe w, m Having thus found the values of X,, X,.-: X,,, it remains ) determine the several roots, which are g in number, in the averal groups which they form: for this purpose, we consider tem as forming the roots of an equation of q dimensions, and ssume t =T + ara” 4 azra™ se, ad) rae), * If mg=n-—1, and if q=2, and m="—*, we find n—l 1 X,=rtra* =r+-: (Art. 532) Xjara¢e, ay eg | X3 = pag ea If we make therefore — 2 i pte bisa re my sin dias yt n n ve find 9 2 X,;=2 i Xo = 2 cos oh 5g she ett nN n n Vou. II. 3F Applica- tion of the preceding theory to the solution of the equation x5—]=0, ee 410 where a is a root of the equation -—1=0, a which is different from 1: we then form 0=t'=A,+aA,+ a®A,+... + ial Foon where A,, A,, A, ... A, are reducible to the form P+ Q; where P and Q are severally determinate numbers, and s = X, whose value has been previously found: the several values of ( and consequently those of re”, ra’”.. . ra") are determined, as i the last Article, merely replacing s by X, and m by g. If, however, it should be found that q is still a composit number, the several roots which are comprehended in X,, X, X;, X,, may be further distributed as before, and the proces in the last Article repeated: for it is always desirable to redue as low as possible the index of the power to which ¢ is require to be raised. We shall now proceed to apply the preceding theory to th solution of binomial equations. 1033. Let it be required to solve the equation : a°—-1=0. ‘ The number 2 is a primitive root of 5, and the cyclical perior of the roots of this equation is therefore we Bee andy ac or ME AM BON Ae, In this case, since 4=(n—1)=2x2=qm, if we make a=—] or that root of x’—1=0, which is different from 1, we find — — xt ape Xs; where X,=r+r7* and X,=7r*+ 7°. Consequently 0==A,+aA,, where A, =2X,X,=2(r+r4+r4rt)=- and A,=s*— A, (Art. 1022) =3: therefore fe oa “2G 15: i ere 0 aes oe + 41] We thus get, since /0,=s=—1, /8,+/0- -1+/5 X, = ig ge MDs Xx J8,+a/6, —-1-/f/5 2 = = 2 2 But inasmuch as X=r+r* and X,=7r2 47%, e find, by repeating the same process, =rt+ar', d therefore = t* =r? +754 2a= X,+2a=X,—-2. We thus get, since ,/0’,= Xi, oats 6 ay X J (Xp 2) re ; g i? ae X,+a,/6, _X- J - 2), 2 2 ; id again, by changing X, into X,, and X, into X,, we find WEEN CCD 2 3 vo Xe- (Ki = 2) 2 If we replace X, and X, by their numerical values, we get RAST. ta ai Y eee ars prem Di iN Ga 10—2,/5) RN 2k uae ee ey? TS o ~1-/5+,/(—10+2,/5) Ly RS ae Bee VAP eT ee Bed G10 2/0) - ; These will be found to coincide with the values given in rts. 707 and 814. 412 Solution of 1034. Let it be required to solve the equation the equa- Elen ei eh he Since 3 is a primitive root of 7, the cyclical period of th roots of the equation is fo fast tet, Teta Again, since n—1=6=3x2=qm, if we take a the root of a*— — 0, x—l we get 1 = X, + aXe, where X=rt+rstr, Xoaretr'+r’. Therefore G27 =A, + adj; where A, =2X,X,=2(34+r+rt+r?+r+r't r’) =2(3+s5)=6+2s=4, since s=—1, and therefore A,=s*'-A=1-4=- 3. We thus find @=— 3 + 4a, and therefore 0, =— 7. Consequently x eres chee Ole rat ae jee 2 Se 9 3 eh hee et SER hee 2 Again, since q=3, if we make a a root of the equation z’?—1 =i x—l 413 id consider 7, r’, 7*, which are involved in X,, as the roots ‘a new equation, we shall find t=r+ar+a’r, id therefore @’=t'?= A, +aA,+a" Ag, there by the actual formation of the cube of ’’, we find A,=6+r+7r°+7r°=6 +X, A,=3(r+re+r*)=3X, A,=3 (re +r° 47°) =3Xo, ind therefore | 6’ =6 + X,+ 3aX, + $a’ X. We thus get and therefore at J+ 2/0’, _ 3 X, + 03/0’, +a 3/6", a 3 > X,+ea 0, + a? 0's = DAS Cie Sale se - r? r The values of r°, r°, and r® may be found in a similar manner. 1035. Let it be required to solve the equation Solution an the equa- xa? —1=0. tion p7—1=0. Since 3 is a primitive root of 17, the cyclical period of the roots of is 414 Since n—1=16=8 x 2=qm, we make a the root of z*—] ~0, r—i1 and assume t=X,+aX,, where Xy =r trp rg pl 4 ploy phy pty ps X= Pe rg hg pl g lt gy oT g pl 4 8, Therefore @= A,+aA,, where A, =2X,X, =8s=— 8, and A, =s°— A, =14+8=9. We thus get and therefore = = ~ 1,5615528, SS SS ED *. , 7) 3 2.5615528 * Generally, if n be a prime number of the form 4m+1, and if tk cyclical period formed by the roots of be distributed into two groups X, and Xz of 2m roots each, formed by takin its alternate terms; then if we consider X, and X, as the roots of the quad ratic equation u? + Au+ B=0, we shall find A=(X,+X_,)=—s=—1, B= X,X,=ms=-G—1), eal for, there are (2m)? =4mxm=M@=U (n—1) terms in their product, forming = LAr a : ; —1 {- original cyclical periods, and therefore equal to “ xs= (n—1), the equation consequently becomes wu" 20, and therefore 415 Again, since g=8=4x2=4q'm, we proceed to distribute the is in the primary groups X, and X,, into the secondary groups )Y., Y;, Y, assuming Pal, +a Yo, ere Years? + rio 4 ot Ye ate ade oh pa ao We thus get P=t? = A’,+aA',, therefore g —1l+/n aT RD HAe fhus if n=5, we find —-1+//5 > Re If n=18, we find —1+,/13 meio wate But if n be of the form 4m + 3, then if X, and Xz be assumed as before, 1 be supposed to be the roots of the equation u24+Au+ B=0, 2 shall find f A=-(X,+ X,)=-s=1, B= X,X,= 3 (n+1); there are (2m+1)? or m(n—1)+2m-+1 terms in their product, forming complete cyclical periods of n—1 terms, and (2m-+1) terms which are erally equal to r” or 1; we thus get B=ms+2m+1l=—-m+2m+1 n+l = 1= x m + 4 > equation thus becomes n+1 wu? + ut =, 4 mL EAS se ot whose roots are ae eh aes Thus, if n=7, we find (Art. 1034) ~-14+V-7 i. 5 : f n=11, we find his) _-1l+v-11 U — 2 ° This proposition is given in Gauss’ Disquisitiones Arithmetice, Art. 356. 416 where AG =2Y,Y,=2s=-2, and A',= X- A= Bee, Therefore A ASST ia ag 7 0 Se re ae + 2 c= ar Oana? and also vie ae Re se ACSI = 2.049481. Fyn Ot ew USN SA en De eer In a similar manner we find vee SN ete) Sel eh yyy Y,= se pee EME IRE gh) = — 2.905704. Again, since q/ =4=2x2=4q"m, we proceed to distribute t! & q q p secondary groups Y,, Y., Y2, Y; into ternary groups Z,, Zs, Lu, Zs, 45, Z;, Z,: taking the same value of a, we assume t” = fe + Qi. where if A=r+r’=r+ a 1 Ze ars rts r+ a * We thus get 0) = pit 2 — A” + af"): where AY = 22, 2,= 2 ar +7") = 2 Y,, and A” "~.= Zi? + Lica 4, + VAS * The value of A”, is more readily found from Z,?+4 Z,2 than Y,2= A" 417 Therefore 6”, =4+Y,-2Y,, nd me Y,+/0", aa = 4-14 /17+ J34-2J17)- 4417+ 3/17 - V34—-2,/17 ~2,/34+2,/17), = 1.864944, im mey,—,/60’ eee = 184537. _In a similar manner, we find ] . 15 1 ZA, =7 qe ‘ here the specific radical ,/x would retain throughout its single ithmetical meaning: but if the equation was here. es “e+ a z 2, where x? admits of double and 2? of triple values, and where It is not assumed that the values either of x? or of 2 are simultaneous on both sides of the equations, it would be equally capable of 36 different forms, and if rationalized would become am equation of 6 dimensions, all whose roots would be equally admissible as proper roots of the equation.* 1053. Such a range of uncertainty, ‘however, would not be Such am- ° ° A r : 1guities lowable in series or expressions derived from known operations are not al- in which the specific values of the radical terms involved in the !owable in expressions generating expressions are, in all cases, assumed to be transmitted cae to the results to which they lead: thus if it was required to by Enowi ; operations. * Appendix. Vou. II. Sul Ambigui- ties which arise when an expres- sion and the ordi- nary form of its deve- lopement have not the same number of values. 434 multiply 1+ x? into 1+2%, the specific values of a? and x would be assumed to prevail in the product 1 1 5 l+a°+a°4+2°: and in the developement of a series such as (1 —x*)3 which gives Legale rat eS Hao aes whatever be the specific value of xz? which is assumed in the generating expression, the same is transferred to the series which is generated: in the absence of such an assumed community of values, the results of the operations of Symbolical Algebra would, in such cases, lead to inextricable confusion. ; 1054. In the transition from a generating expression, more ) especially when it is deduced, in the first instance, for values of the symbols involved in it which are general in form but specific in value, and which are subsequently generalized “by the prin- ciple of the permanence of equivalent forms,” or by means of some proposition to which the application of that principle leads, we sometimes adopt a form of the series whose value is unique when that of the expression in which it abet and to which the series i 3 1+: +n (n—1) (n— 2) Stee 4 which originates in ce mee is unique for all values of n: but | inasmuch as the values of (1+.)" are multiple whenever n is. fractional, the series in question, under such circumstances, ceases to be coextensive with, and therefore completely equivalent: to, the expression from which it is derived: if however, we | should call s the arithmetical value of the series | x x? l+nzr+n(n—1 stn —1 pps ta na+n(n—1) (n—1) (n—- ee Bae we should find (1+#)=1(1+2), and (1x) *t1t (et eyes, where 1” is considered as (Art. 724) the recipient of the multiple values of (1 +)": and inasmuch as it appears (Art. 810) that 1"= cos 2urm + wpe 1 sin2urzm, 435 (where r may have any value in the series 0, 1, 2,3...), we may form the equation ) (1+.2)"= (cos 2nrm+,/—1 sin 2urz) s, which is true for all values of n. We thus get (1+.2)?=13.s=(cosrr+J-1 sin Fx} $= 2s, (1 +2)i= 13,5= (cos te ae Sa sin “7” ) $ | apie ——) evar (A) n and similarly in other cases. The neglect of a sufficient attention to this peculiar source of ambiguity has led to many imperfect and erroneous gene- ralizations, more particularly in the case of an extensive class of goniometrical series, some of which we shall now proceed to investigate: it is only within a very recent period that their correct forms were first exhibited by Poinsot. EEO - 1055. Let it be required to find a series for (2 cos 6)” in Series for terms of the cosines or sines of multiples of 0. fe Coe Let p represent the arithmetical value of (2 cos 0)”, and the sines or cosines of therefore mulgoles (2'co3, 0)" = 1% ip; of 0. if 2cos @ be positive, and (2 cos 0)"™= (— 1)"p, if 2cos@ be negative. Again, since 2 cos 0 = Fey ot ee (Art. 926), we get (2 cos by" = 1" (PV 4 PV =m, , =. m(m—1 is 5 — 4" fen 4 meme 7, MM =D) e(™ 4) O+/ 1 ae Wh But 1"=cos2mrmr+,/—1 sin2mrm, em9V-1 = cos m0 + ,/-1 sin mé, em V=1 _ cog (m —2)0+,/—1 sin (m— 2) 8, e™- 49-1 _ cos (m— 4) 0+,/—1 sin (m—4) 8, a 8° 6.5. Ue. 8 @eeoeevevreeeveer?eeeerer eevee eeeseeeeee ene —— 436 Therefore (cos 2mrm+,/—1 sin2mrm) p = (cos 2mrn+,/—1 sin 2mrm) {cos mO +m cos (m— 2) pee cos (m—4)0+...} ty Ee (cos 2mra + fii sin 2mrm) {sin m6 m (m — 1) 5 sin (m—2)0—...}: +m sin(m—2) 0+ but (cos 2mrm +, /—1 sin 2mrm) (cos m0 +,/=1 sin m8) = cos 2mrm cosmé +,/—1 sin2mra cos m0 + /—1 cos 2mrm sin m6 —sin2mrm sin m0 = cos m (2rm + 0)+,/—1 sinm (2ra 4 0). We thus get (cos 2mrn +,/—1 sin2mrm) p, = cos m (2rm + 6) +m cos (m—2) (2Qr7+ 0) ies y cos (m— 4) (2rar+6)+.. +f=1 {sin m (2rma +0) +m sin (m—2) (2ra + 6) BARA sin (m—4) (2ra +6) + ort =¢,+/-1 Sr» where c, represents the series of cosines, and s, the series of sines of the multiples of 0. If we take the second case, in which (2 cos 0)"=(—1)"p, we shall find, in a similar manner {cos (27+ 1) mmr+,/—1 sin (2r +1) ma} p=c,+/—1 5, where c, and s, represent the same series of cosines and sines of multiple angles as in the expression for 1”, merely putting (2r+1)7+0 for 2r7+9, . When m is 1056. It remains to discuss the different forms which these j a whole number. Series will assume for different values of m. Let m be a whole number. 437 In this case, we may make r equal to zero in both members if the equation: or, in other words, we may replace c, by ¢) ind s, by 5; If 2 cos @ be positive and m even or odd p=¢,. If 2cos@ be negative and m even P=Ly- If 2cos@ be negative and m odd p=—-%- In all these cases, the value of p is expressed, therefore, by 1 series of cosines, the corresponding series of sines being equal 0 zero. m(m—1)...(m—p+1) 1.2...p The (p+1)™ term of c)= cos (m—2p)0 =t cos (m— 2p) 0. The (m—p)” term of c,)=t cos (m—2m-+2p) 0 =t cos (m —2p) 0. The terms of the series for c, being, therefore, the same from the beginning to the end, it follows that - C= 2 {cos m0 +m cos (m—2)0+— RiP mr cos (m- A)O+..} 1 terms, . m : m+ sontinued to at 1 terms, when m is even, and to when m is odd: the last term, in the first case, being Lox od saa (m1). of, m xX. 2Hedieg. oS which is a modified form of the half of the middle term of the dinomial (1+ 1)” (Art. 489): and in the second case being L355 x ohm vt. S eis 27 cos @. 1 i ee: ae 1057. Let m be not a whole number. Wi her ee in ope not a whol If 2 cos @ be positive, we find ciate C, a rs P~ cos2mra. sin2mrn’ 438 If 2cos@ be negative, we find C, 5, pcos m (2r + l)w sinm(2r+1)7° In both cases, therefore, p is expressible either by a seri of cosines or of sines, unless cos2mra=O or sin2mrr=0 j one case, and cosm(2r+1)7=0 or sinm(2r+1)=0 in tl other. In examining such cases, we may suppose m a rational fra tion in its lowest terms of the form f, and also that r doe not exceed n—1. If p is odd, » divisible by 4 or pariter par (Art. 516, Note n 3n 2Qprr T 3pqT and r= or 7-, we get “—" = or -_, and therefo 2prar cos = n In these cases p is expressible by a series of sines only, an p=8, or —S3,, If p be of the form 4i+1 or pariler impar, an z z p=—S_ OF $3,, 1f p be of the form 41+3 or impariter wmpar. te Fy Secondly, if r=0, or if p be odd, n even and r== » we fing 2prm _ a sin 0. In these cases, p is expressible by a series of cosines only and p=c, or —«¢,. F Thirdly, if p be odd, m even and of the form 41+2 or im =i 2 pd 3n-—2 p@r+i)x 4 n ariter par, and r= ~~ » we find cos P P “ 4 In these cases, p is expressible by a series of sines only, anc P =— San 2 OF Sgn-25 according as p is pariter or impariter impar. ra ie ae = 9 2 ai we find sin pars iri @ n Lastly, if 2 be odd and r= In these cases, p is expressible by a series of cosines only, and p=c,, or c,_, according as p is even or odd. 2 2 ; ~. 439 1058. Again, let it be required to find a series for (2 sin 6)", ie et a terms of the sines or cosines of multiples of 0. aha Since 2,/-1 sin @= Glee pov! (Art. 926), we get @ /Hisin ayrai™femoV'-1 _ melm-2)0V=1 4 ™ ey elms OV-1_ ; = cos m (2ra+0)—m cos (m—2) (2rr+O)+... ,/=1 {sin m (2rar +0) —m sin (m—2) (2r7+0)+...} eee 21. f c, be taken to represent the series of cosines, and s, the series if sines. If p denote the arithmetical value of (2,/—1 sin 6)”, then, f 2sin @ be positive, we get {cosm (2r+4)r+,/—1 sinm(2r+4) 7} p=c,+5,J—1 md if 2sin@ be negative {cos m (2r+3) r+,/—1 sinm(2r+4)} p=C,+5,)—1. 1059. If m be a whole number and even, then When mis a whole Q”"" (sin 0)" = + {cos m0 —m cos (m—2) 0+... th PaCS ia) Rye Pv2'.. v1 he sign + or — being used according as m is of the form 42 mr 4242. If m be a whole number and odd, then gn! (sin 0)" == {sin m0 —m sin (m—2)0+... to “ ! terms}, the sign + or — being used, according as m is of the form 47+1 or 4724+ 3. 1060. If m be not a whole number and 2 sin@ positive, then When mis not a whole Ce S, number. cosm(2r+23)m sinm(2r+23) 7’ fi 4.40 Hi ° ° . e « . . a or its value is expressible either by a series of cosines or sines' of multiple angles, except under the following circumstances, — If m= e (in its lowest terms), p and n being odd numbers, then | P= $,-; OF — Sn-19 alan uh according as p is of the form 4i+1 or 42438. If p be even and n of the form 47+1, then Pp aly Ol ee Cy_is 4 ay according as p is of the form 47 or 4742. If m be not a whole number and 2 sin @ negative, then rdf eh Ms 5, Z cosm(2r+%)a sinm(2r+4)a its value being expressible, either by a series of sines or of cosines, except under the following circumstances. If m=E (in its lowest terms), p and x being odd numbers, then P= San or = S3n41 9 4 4 | when x is of the form 4n +3, the sign + or — being used, ac-, cording as p is of the form 4i+1 or 42+3 in the first case, or the contrary in the second. ; If p be even and x of the form 43 +3, then 4 4 _ the + or — sign being used, according as p is of the form 47. or 42+2 in the first case, or the contrary in the second. We have been thus minute and critical in the deduction of | all the separate cases which this problem comprehends, not only on account of the intrinsic importance of the problem itself, but likewise as affording a very instructive example of the proper mode of discussing and interpreting a formula when it is expressed in very general terms: for it will very generally be found, that the more comprehensive is the form in which a problem is stated’ 441 d investigated, the more remote and difficult its application vill be to the particular cases which it includes. A converse problem to the one which we have just been Series for onsidering, would require us to assign a series for cos mo cosmy ae nd sinmé@ in terms of the sines and cosines of 6: the inves- ae of Betion however of such series, which branch out into a great sin 0. iy fariety of forms and cases, cannot easily be effected without the’ id of principles and processes which will be given in a sub- equent volume of this work. 1061. The same principles will likewise find their appli- Theory of ation in the general theory of symbolical as distinguished from gears rithmetical logarithms: if we assume p to represent the arith- guished aetical logarithm of a or of its powers, we may extend our beep nquiries to determine the most general symbolical forms of the !0gtithms. dgarithms of (1 xa)", of (-1x a)", or of —1x(1xa)”, or, in ther words, we may suppose a or a” to be affected by any sign, yhich is recognized in Symbolical Algebra, and their logarithms 9 be required, assuming e to be the common base to which they re referred. Since it has been shewn in Art. 927, that The Na- 1"= cos 2mrn+,/—1 sin Qmrm = ermrAV -1 fogarttttihe nd of 1” and uy Patol ee (—1)"=cosm(2r+1) r+ J—1 sinm (2r+1)r=e™@rttV-1 ° ; will follow, in conformity with the definition of Napierian »garithms (Art. 901) that log 1" =2mr,f-1 log (—1)"=m(2r +1) 7-1. The logarithm of 1” is zero, when m=0 or r=0: the loga- ithm of (—1)” can only become zero when m=0 and therefore -1)’=1: the other logarithms are all imaginary, and are un- mited in number: it is the logarithm of 1° or (— 1)? only, which ; essentially zero and which admits of no other value. 1062. If m be a fraction in its lowest terms, with an ever Case in which one value of the logarithm ; . ‘ of 1” coin- qual to —1, corresponding to r=: in this case we find cides with one value ‘ ee — of a loga- log 1 =2.£ na J-1= pr J-1, rithm of Vou. II. 3K i enominator of the form f, there is one value of 1” which is Symbolical logarithms of a”, The sym- bolical logarithms of (—a)”. 442 where p is an odd number, a result which coincides with one of the values of log (—1)", when m=1 and r= Ps ‘ i i 1063. Inasmuch as a ar={) Kft)Sar™ iat, i we find 5 log a” = log 1"+ log a”, i =2mran,/—-1 +p, where p is the arithmetical logarithm of a”. When we seek for the general logarithms of a”, we presum that a is viewed as a symbolical and not as an arithmetical que tity, and we replace it by 1 x a, where 1 is made the recipient o the affections of a, whilst the other factor a, is considered as an arithmetical magnitude merely: this is a distinction of funda- mental importance, both in this and other theories, and which our ordinary notation is not competent to express, a defect which is a constant and very embarrassing source of ambiguity*. i | | | 7! 1064. Again, since (-ayh=(-1)" x a", log (— a)" = log (— 1)" + log a”, =m(2r+1) 7 J—1+p. * Mr D. F. Gregory, in a very able memoir ‘‘ On the Impossible Logarithms of Quantities,” which is given in the first volume of the Cambridge Mathe- matical Journal, has proposed to distinguish these arithmetical and symbolical values, by a and by +a, where the sign + takes the place of 1, in the notation adopted in the text, as the recipient of the affections of a, and where +” is considered as equivalent to 1”: such a use of the sign +, as the subject of symbolical operations, is opposed to the ordinary conventions 0 notation, and is calculated to keep out of sight the peculiar symbolical pros perties of the various roots of 1, by which all our signs of affection are sym bolized: it is certainly no bbyecuiey to its use, as urged by Mr Gregory, thai its tendency is to recall arithmetical ideas. I cannot refer to this memoir of Mr Gregory, the inheritor of the name and honours of a family singularly illustrious in the history of the sciences without expressing the deep sense which I feel, in common with all who kne him, of the loss which the mathematical and philosophical world has sustainet by his premature death: his memoirs were remarkable for the large and origina views which they take of the principles of mathematical reasoning, and gav ample promise of the valuable results which could hardly have failed to have fol lowed from the full developement of his powers. we get ce Ti 4.43 If m=2, we find The loga- ad aye of 4 ] _ a = 2g — ] P (-— a an og (— a)?= (47 +2) r /—14+2 loga Cs. aes age It appears, from the last Article, that identical, log a? =2rn,/—1+2 log a. It follows, therefore, that, though & a)?= a* nd though the logarithms of (— a)? are always included amongst hose of a’, the converse proposition is not true: it consequently ppears that we are not authorized in inferring, as has sometimes en done, the identity of the logarithms of (—a) and of a®, from e identity of the symbolical results to which they lead when the igns of affection or their recipients are suppressed. 1065. Similarly, since The sym- Pouca — 4" = (— ie bg ogarithms . ( Ee nia of — a”. ve find log — a" =log (— 1) + log 1” + log a™ =(2r+1)7,./—14+2mr'r J-1 +p =(2r+2mr' + 1) J—1 +p. This is the most general form, in which the logarithm of The loga- + ; ., rithm of -a”™ can be exhibited, and it remains to consider, whether it nepeevee dmits, in any case, of an arithmetical value. quantity * Mr Gregory, in the memoir to which I have before referred, has con- luded from the equation (—1)?=1, T, as expressed in his notation Coit that —=-+#, ; 2n+1 md he infers from thence that + 2 is the general representation of the ign —: but it might, with equal propriety have been taken, as the general epresentation of the sign +; for the equation +? =+ 3 equally true with the equation ) (-P=+. 4.44 at Bree If we suppose, as in Art. 1062, m, in its lowest terms, to be cases, be pie? E °F of the form fF, and if we make r’=n, and 2r+1=~—p, then that of a i & positive i sgantty. (2r+2mr +1) =0, ji and there would apparently be one value of the logarithm of —a”, which is possible and equal to p or to the arithmetical value of log a”™. : A very little consideration, however, will shew that this case is not admissible*. ‘ 9 For one of the values of 1% j is, as we have already shewn, (art 1062) equal to —1, which is that which corresponds to 7’ =n: and inasmuch as 4 -1=(-1)=(-1)7, it will follow, that, under such circumstances, P ~a"=(-1) (1)"a"=(-1) (1) a" ee =(-1) (1) a =a" a form which is necessarily a (oe a inasmuch as it is assume that (— 1) (1 x a)” is equal to —a” and not a": we may con clude, therefore, generally that there is no possible logarithm of a negative quantity. » Re iS Difficulty 1066. The ambiguity which necessarily results from the pie identical representation of arithmetical and symbolical quantities, poe and the essential difference of the form of their logarithms, makes theory of a vigilant attention to the most delicate shades of difference arched between them absolutely necessary, in order to avoid defective or excessive generalizations: thus some writers have considered the base of the system of logarithms as equally capable of mul- tiple values with the quantities whose logarithms are required, not sufficiently considering that the system of logarithms itself is altered by every change of its base: thus, if we suppose ie eo they replace y by 1x y, and a™ by 1* x a®, assuming a® in 1* x @ to be arithmetical: we thus find : * The existence of a possible logarithm of a negative number was asserted in my first work on Algebra, page 569, and the mistake was pointed out by Mr Gregory, in the memoir to which I have before referred. B log 1 + log y = log 1* + log a’, arm J/-1+ logy = 2(2r'r J/—-1) +a; d therefore a loey tere J-1 14+2ra/f-1 ’ nd if y=1, we get x or the logarithm of 1 Ls QraJj—1 14+2r'x7,/- 1 nd not 2r7,/—1, as determined above, Art. 1061. wv. + APPENDIX. Drthe Page 12, Art. 558. In Arithmetical Algebra, zero and infinity Sats gd te are the extreme limits of the values of the symbols which we the symbols employ, though the circumstances of their usage will very gene- hora rally confine them within a much less extensive range. Arithmeti- Thus, in the expression a—6, a may have every value be- Seb tical tween infinity and 6, whilst 6 may have every value between a Algebra and 0: the expression a—6 itself may have every value between 0 and infinity. | But in Symbolical Algebra, a, 6, and a—b may severally have every value between positive and negative infinity, zero being included in their number: and if, as in the case of lines drawn in opposite directions, the positive and negative values equally admit of interpretation and are consequently equally real, it will follow that their symbolical is coextensive with their geometrical representation. It may be further remarked that Arithmetical and Symbolical Algebra are equally competent to represent continuous magni- tude between the limits of the values of their symbols, or of the expressions which they form. c. Magni- The magnitudes represented by the symbols of Arithmetic tudes repre- and by the expressions, whether fractional or decimal, which - sented by thesym- they form, are essentially discontinuous, being incompetent to bols of } Arithmetic €Xpress the continuous values which are included between their ea successive units: and it is only by the indefinite subdivision of 7 the primary units, whatever they may be, that we can approxi- — ‘ mate, in this science, to the representation of continuous mag- _ nitude. oe On the contrary, the symbols of Arithmetical or Symbolical — Foes Algebra are incompetent to represent, like those of Arithmetic, a bols of | discontinuous magnitudes: for whilst in Arithmetic, magnitudes — Sa pal are represented at their limits only, however numerous they may Symbolical be, in the other science, they are represented between their Algeb cscgiekieey limits only, whatever those limits may be. nuous. a 447 It is, for this reason, that the symbols of Algebra are not competent to express generally the properties of numbers, which are essentially discontinuous in their nature: and it is owing to this cause that the theory of numbers, properly so called, as distinguished from the theory of numerical operations, being not reducible to the general representations of symbolical language, has made no advancement which is comparable to that of the Kindred science of Algebra. Page 52, Art. 622. We have assumed the rules for the ad- dition and subtraction of numerical fractions as the basis of the corresponding rules for those operations in Symbolical Algebra: ‘but a further consideration of the principles upon which those tules are deduced (Art. 127 and 128) would rather lead to the conclusion that the interpretation of the meaning of numerical fractions which is there assumed, is not strictly deducible from the meaning attached to the quotient of the division of one ‘number by another, with which it is required to be coincident. _ Assuming, therefore, that the only property which the defi- nitions of the fundamental operations of arithmetic would assign ‘to numerical fractions is, that their values are not altered when ‘their numerators and denominators are multiplied or divided by the same number, we will consider the case of two numerical fractions 7 and # which are to be added together; their sum ‘will be represented by phage and if we further suppose that ‘amongst the different values which may be given to m and n, ae there are some which are multiples of p, so that - ar r and 4 = ~E =s, then, under such circumstances, we should find Ripa (r+ s) p gee Y P Pee Dis Oh LP a : ‘ = song) eh — my 8 . ‘it appears, therefore, that when m=rp and n=sp, we get P tiaiieD and inasmuch as this formula or expression represents the sum m uf] Mt +71. ee 448 of the fractions = and : when the symbols involved in it are general in their form, but specific in their value, it will continue to represent their sum when the symbols in it are general both i their form and value: in a similar manner, it may be shewn that m nt m—n TRARY EP Considering therefore such fractions, in the first instance, as expressing the quotient of the division of the numerator by the denominator, we should proceed to interpret their meaning, as in the case of the results of all other symbolical operations, with reference to the symbolical conditions which they are required to satisfy; we shall thus be readily enabled to arrive at the in- terpretation of their meaning, which we have made the foundation of the conclusions in the text. The prin- Art. 631. “The principle of the permanence of equivalent Sane the forms” is that which expresses, in the most general terms, the nence of + nature of the connection between Arithmetical and Symbolical | equivalent forms is the Algebra. | aioe: All the conclusions of Arithmetical Algebra are considered | sion of the to be the necessary results of the defined operations of addition, which °" subtraction, multiplication and division, involution and evolution, tua be- when applied to numbers or quantities whose relations are fully | Arithmeti- understood: and such conclusions when represented through the | aes medium of symbols which are general in form but specific in| Algebra. value, or by rules which are general in their form, though ap- | plied to quantities which are specific in their value, are assumed | to be true likewise when the symbols which represent such mag- | nitudes are equally general in their form and representation. In Arithmetical Algebra it is the definition of the operation. | (whether expressed or understood) which determines the result, — and also the rule for obtaining it: in Symbolical Algebra it is | the rule which determines the meaning of the operation, such rule | being determined by the principles of Arithmetical Algebra, | when the symbols, though general in their form, are yet so | specific in their value, as to come under the operation of its | definitions: the rules of operation are the same in Arithmetical _ and Symbolical Algebra, and therefore the results are the same _ as far as they proceed in common: but it is at the point of | transition from Arithmetical to Symbolical Algebra, when the | 449 symbols or the conditions of their usage, cease to be arithmetical, that the meaning of the operations must be determined, not by definition, but interpretation: and such interpretations must vary with every change in the circumstances of their application. The results of Arithmetical Algebra may be said to exist fo ea by necessity, as consequences of the definitions: and those defi- metical nitions, whether expressed or understood, (for they are never nena ‘formally enunciated) may be considered as derived immediately sary, those from the relations of numbers, and as consequently involving a ee © lical as tar nothing which is arbitrary or variable in our conceptions of ** ies oS their nature or essence: they may be said, therefore, to possess mon to _ in an eminent degree, the character of mathematical necessity. PS. The case is very different, however, with the results of Sym- bala bolical, as far as they are not common likewise to Arithmetical, #onl. ‘Algebra: inasmuch as they may be said to exist by convention only, for the rules for forming them are not proved as conse- quences of definitions, but are borrowed or adapted from a kindred science: and it is only when specific values are assigned ‘to the symbols, that their relations or properties can become the ‘subject of our reasonings, with a view to their interpretation in ‘those cases and in those cases only, where the requisite corre- ‘spondence, between the symbols and the quantities which they ‘are assumed to represent, can be shewn to exist. Again, such interpretations must not be confined to the The inter- meaning of the operations performed merely, but must extend ae likewise to the nature of the connection which exists between must be ‘the operation and its result: the sign =, which is universally o aoaen ‘used for this purpose, means arithmetical equality in Arithmetical ee Algebra, when placed between the primitive expression and the ne primi- iresult of the operation which it involves, whether the result, to sion Se ‘which it leads, presents itself under a finite or indefinite form: Emre ‘but in Symbolical Algebra, in cases which are not likewise com- from it. ‘mon to Arithmetical Algebra and in which the operation which produces the result requires interpretation, the sign =, in common ‘with the expressions which it connects, must necessarily be in- ‘cluded in it: its most comprehensive meaning will be that the ‘expression which exists on one side of it is the result of an ‘operation (using this term in its largest sense, Art. 632) which ‘is indicated on the other side of it and not performed: this ‘view of its general meaning will include as a consequence, arith- Wow LI. 3L Expres- sions may be algebra- ically equi- valent which are not equal. Misappli- cation of the term false to diverging series. 450 metical equality or algebraical equivalence, according as either one or the other of them may be shewn to exist. The phrase algebraical equivalence, as distinct from alge- braical identity or arithmetical equality, would be applied in the case of expressions which, though not possessing either of these characters, are capable of reproducing or representing the# symbolical properties of the expressions from which they are’ derived: thus in the case of the series which result from the! developement of binomials such as (1 +2)” and (1 +2)’, by the} general theorem considered in Chap. xx1, we find | Hecate ey} (1), GS = I"{l+me +m(m—1) = 2 (lt a)=1" {1+ n24+n(n—1)——4...} (2), and it appears that for all values of x, m, and n, we have (1+a)"x (1+2)"=(14+ 2)", and likewise that the product of the series (1) and (2), if they are multiplied. together, term by term, according to the ordi-+ nary rule for that purpose, will be the series ; x? eae which arises from replacing m in the series (1), or in the series | (2), by m+n: it thus appears that the products of the binomials | (1+)" and (1+ .2)", and of the series (1) and (2), are equally formed by replacing m or x by m+n, in the binomials and the series corresponding to them, and in this sense, and to this | extent, they are said to be equivalent to each other: but if | m or n be likewise rational numbers and zx less than 1, then | the binomial (1 + x)", and the series (1) corresponding to it, may _ be considered not merely as algebraically equivalent, in the sense | which we have attached to the term, but likewise as arithmetically equal to each other. It has become a common practice with many distinguished analysts to denounce all diverging series as either false or wmsecure, or, in other words, as not capable of replacing the expression in which they originate in any algebraical operation, without leading to erroneous results: it would, however, be more. in accordance with large and comprehensive views of Symbolical - Algebra and its operations, if it should be said that diverging © thts (1+ (m+n) xv + (m+n) (m+n-1) Ld ig tt See 451 ‘series are not arithmetical, and therefore incapable of arithmetical computation by the aggregation of their terms: that inasmuch as they rarely, if ever, originate in expressions which are arith- ‘metical both in their arrangement and value, it would be more ‘correct to term them false under such circumstances, if they gave arithmetical values of expressions which were not themselves arithmetical: and further, that an algebraical equivalence may ‘exist between an expression and its developement, when they are not arithmetically equal: and likewise that whenever general processes of Arithmetical Algebra may enable us to determine the expression which generates a series, when the symbols which it ‘involves, though general in their form, are so modified in value _as to produce a series which is arithmetically equal to it, they will equally enable us to assign it, when those symbols are general both in their form and value. (Art. 958, Note). Page 106, Art. 680. The existence and form of the series for (1 +2)", is proved, when x is a whole number, by considerations derived from the fundamental operations of Arithmetic and Arith- metical Algebra: and the same series, exhibited under an inter- minable form, is extended to all values of the index, by the principle of the permanence of equivalent forms. The primary assumption of indices in Arithmetical Algebra leads directly to the propositions (’+2)"x (1 4i2)= (T+), {(1+ax)"\"=(1+.2)"", where m and n are whole numbers: and we derive from the ‘same assumption of indices, the extension of the same conclu- sions to the series which are deduced from them, when m and nm are whole numbers, or in other words, we are thus enabled to conclude that the product of the series for (1 + x)" into the series for (1+.2)" will be the series for (1+.)""", or, in other words, their product will be that series which arises from putting m-+n in the place of m in one series, or of » in the other: and also that the series for the n‘ power of (1+«)", which is (1+)"", is formed by putting mz in the place of m in the series for (1+.2)”. It is the principle of the permanence of equivalent forms, as we have shewn before (Appendix, p. 450), which enables us to extend these conclusions to all values of m and n. 452 Euler had drawn the same conclusion, nearly in the same manner, in his celebrated proof of the series for (1 +a)" *, though he at the same time denied the universal application of a prin- ciple equivalent to that of the permanence of equivalent forms, which alone could make it valid: he produced, as a striking exception to its truth, the very remarkable series aire SAH Croke rear NdeI le oes, ae oo Le a a l-a 1 —a’ 1-a’ whose sum is m, when m is a whole number, but not so for other values. | A little consideration, however, will be sufficient to shew that the principle of the permanence of equivalent forms is not ap-. plicable to such a case: for if m be a whole number, as in Arithmetical Algebra, the connection between m and its equi- valent series in the identical equation —_ 7” —_ Am 1] m—l1 De m — 7m! _— 7n—2 md eal a™)(1l-a yan a”) (1—a PAs, a th l-a l—a l-a eee 2 is not given, or, in other words, there is no statement or definition | of the operation, by which we pass from m, on one side of the sign =, to a series under the specified form on the other, and there is consequently no basis for the extension of the conclusion to all values of the symbols, either by the principle of the per- manence of equivalent forms or by any other: it is only when the results, which are general in form, but specific in value, are derived by processes which are definable and recognized, that they become the proper subjects for the application of this prin- ciple. Page 120, Art. 695. The fundamental operations of Sym- bolical Algebra conduct us necessarily to expressions such as 1 1 1 +a or a, and —a, (+ a)" or a*, and (— a)", which are easily shewn : to be symbolically equivalent to +1xa or 1xa and — 1 x a, 1 1 1 = fas i ee las Li (+1)"x a" or 1*x a", and (-1)'x a", where a and a” are used as__ in Arithmetical Algebra: it would thus appear that 1 and —]_ may be very conveniently used as the recipients (Art. 724) of — the known and recognized signs of affection of Algebra: for all modes of representation in this science are considered to be * Acta Petropol. for 1774. 453 equivalent to each other, which lead, in conformity with its rules, to identical symbolical forms*. This replacement of the signs of affection in Symbolical Algebra, by the roots of 1, considered as their recipients and epresentatives, will enable us to consider all expressions in Algebra as composed of two factors, one being the appropriate root of 1 or —1, and the other an arithmetical magnitude, or, in other words, such a magnitude as is recognized in Arith- metical Algebra: and what is more important, it will further enable us to determine the proper symbolical forms of such signs of affection, and thus to reduce the expressions, which involve them, to their most simple equivalent forms: it is this circum- stance which gives such peculiar importance to the theory of the roots of 1, and which connects their properties and their symbolical determination so essentially with the progress of Algebra. Page 343, Art. 979. The subject of the roots of solution of equations which are formed in the solution of Geometrical pro- blems has been discussed at some length in Art. 1044, and those which follow it. It is there shewn, that it by no means follows, that negative roots, though interpretation may make them real, will be therefore relative to the problem proposed: and what * Mr Gregory, in an article above referred to, p. 442, distinguished + a ‘and —a from each other, the former as belonging to Symbolical Algebra and ‘considered as opposed to —a, and the other as a simple unaffected symbol in ‘Arithmetical Algebra: such a distinction, however, would rarely enable us to ‘remove ambiguities, if it could be preserved, and it is farther opposed to the ‘very important and fundamental rule, which allows us to. suppress the sign + ‘when it is connected with a symbol which has no other symbol before it: it is contrary to all just views of the relations of Arithmetical and Symbolical ‘Algebra, to retain any distinctions between them which the ordinary rules of ‘operation would lead us to suppress. Mr Gregory further proposes to make the pppmery signs Gy and — omsg kee Be palects of operations, and to use aS instead of fF , and (— ‘ instead ‘of (-— 1: considering the use of 1 and —1, as the recipients of signs of affection, as objectionable, in consequence of their tending to recal arithme- ‘tical notions, when the circumstances, in which such signs originate, present ‘themselves exclusively in Symbolical Algebra, and may, therefore, according to this views, be altogether independent of arithmetic: but there may be a deri- ‘yative though not an immediate dependance of one of these sciences upon the ‘other, and I believe that no views of the nature of Symbolical Algebra can be ‘correct or philosophical which made the selection of its rules of combination _arbitrary and independent of arithmetic. 454 is more, that even positive roots may, in some cases, fail te answer the required conditions. Page 357, Art. 986. The entire consequences deducible from the equation of condition Jttlt” =r are not correctly stated in the text, inasmuch as no notice is taken of the case in which r is negative as well as positive: it should be added, that if r be negative, then all the values of Jt, JU, ,/t” are negative, or two of them are positive: we should thus find, that when 7 is positive, we have a=g(t+J/l+ Jt), P=3(Jt—Jt— ft"), 1=3 (ll -Jt- At, 3 =4(Vt"— Jt): and if 7 be negative, we have a= it— Jt ~ s/t", B=4(Jt+ a)! At") P= (ttt! — It), 6=3(/t'+/t’—,/é). The theory of this solution, as applicable to the complete equation xv — pa —gqa*—ra—s=0 is still more completely developed in Art. 1028. Page 360, Art. 989. The argument of Abel entitled “De- monstration de |’ impossibilité de la resolution algebrique des equations generales qui possent le quatriéme degré,” is given in the first volume of his works* and has been revised and amplified by Sir William Hamilton in the 18th volume of the Transactions of the Royal Irish Academy: and though no sub- ject could pass through the hands of those great analysts without retaining the impression of their extraordinary sagacity and power, yet it may easily be conceived that even they have failed — to make investigations of a character so refined and difficult * Published by Professor Holmboe at Christiania, 1839. 455 intelligible to a reader whose mind has not been rendered long familiar with the highest generalizations of symbolical language. There is only one remark which I would venture to make yn the subject of this demonstration: its authors would appear to have omitted to notice the effect of equations of condition m limiting the multiple values of the final expression for the roots, a consideration which enters essentially into all specula- tions on the general solution of equations. Page 404, Art. 1089. Attempts have been made by Meyer Hirsh of Berlin in the first instance, and subsequently by Professor Badano of Genoa, to discover cyclical periods amongst the roots of this reducing equation and thus to apply the methods, which are shewn in the Articles which follow, to be successful ‘with binomial equations, to the solution of an equation of the fifth degree: the error of the first attempt was subsequently ‘discovered and acknowledged by its author, a mathematician of ‘no ordinary attainments and merit: that of the second, which resents itself in a very plausible form, has been made the sub- ject of a singularly elegant investigation by Sir William Hamilton n the 19th volume of the Transactions of the Royal Irish cademy. Page 365, Art. 996. It is to the form w—5su°+5s°u—a=0, gy ‘that the resolution of a general equation of the fifth degree has ‘been attempted to be reduced by Mr Jerrard, in some very remarkable researches in which he has succeeded in shewing that ‘this great problem may be reduced to the resolution of an equa- tion under the very simple form { ace e involves ,/—1: their failure in advancing beyond this point, has been shewn by Sir William Hamilton in an “ Inquiry ‘into the validity of a method recently proposed by George B. Jerrard, Esq., for transforming and resolving equations of ele- vated degrees,” which appears in the sixth Report of the British Association for the advancement of Science. v+an+e=0, THE END. Return this book on or before the Latest Date Stamped below. A charge is made on all overdue books, University of Illinois Library UNIVERSITY OF ILLINOIS-URBANA 512P31T C002 v002 A TREATISE ON ALGEBRA CAMBRIDGE [ENG.] on 0973