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 DEC 7 
 
 Form L-9-5m-5.'24

 
 TRIGONOMETRY 
 
 AND DOUBLE ALGEBRA 
 
 AUGUSTUS DE MORGAN 
 
 OF TRINITY COLLEGE, CAMBRIDGE 
 
 SECRETABY OF THE ROYAL ASTBONOMICAL SOCIETY 
 
 FELLOW OP THE CAMBRIDGE PHILOSOPHICAL SOCIETY 
 
 AND PEOFESSOB OF MATHEMATICS IN UNIVERSITY COLLEGE LONDON. 
 
 La seule maniere de bien trailer les elemens d'une science exacte et rigoureuse, 
 c'est d'y mettre toute la rigueur et 1' exactitude possible. D'ALEJHBERT. 
 
 Tant que 1'algebre et la geometrie ont et6 separees, leur progres ont ete lents 
 et leurs usages bornes; mais lorsque ces deux sciences se sont reunies, elles se sont 
 prgtees des forces mutuelles, et ont marche ensemble d' un pas rapide veis la per- 
 fection. LAGRANOE. 
 
 LONDON: 
 PRINTED FOR TAYLOR, WALTON, AND MABERLY, 
 
 BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE, 
 
 UPPER GOWER STREET AND IVY LANE, PATERNOSTER ROW. 
 1849.
 
 CAMBRIDGE: 
 
 PRINTED BY METCALFE AND PALMER. TRINITY STREET. 

 
 Engineering & 
 Mathe 
 Scierces 
 Library 
 
 Q /\ 
 
 PREFACE. 
 
 \ 
 
 M 
 
 <N 
 
 THE work before the reader is entirely new, not 
 
 V:> 
 
 being in any sense a second edition of that which 
 I published on the same subject in 1837. 
 ^ It consists of two books. In the first, I have 
 
 endeavoured to give the student who has a competent 
 knowledge of arithmetic and algebra as much for 
 instance as is contained in my works on those sub- 
 jects, to which reference is made in various places 
 ^ a view of trigonometry, as a branch of algebra 
 and a constituent part of the foundation of the higher 
 mathematics. In the second, I have given an ele- 
 mentary view of algebra in its purely symbolic 
 character, with the application of that geometrical 
 basis of significance which affords explanation of 
 every symbol. 
 
 The term double algebra has not yet obtained cur- 
 rency, though that of triple algebra has, of late years, 
 been much employed. It means algebra in which
 
 each symbol stands for an object of thought having 
 two distinct and independent qualities : just as the 
 symbol of a straight line, to be perfect, must desig- 
 nate both the length and direction of the line. I 
 have not, after much thought, and some discussion, 
 been able to fix on a better name of sufficient brevity. 
 If, by the application of a somewhat startling adjec- 
 tive to the word algebra, any of those who are still 
 bewildered by an art in which impossible quantities, 
 or quantities which are not quantities, are made 
 objects of reasoning, should become aware that by 
 slow degrees, and the union of many heads, the art 
 has become a science, and the impossibilities possible, 
 they, at least, will have no objection to the phrase. 
 
 A. DE MORGAN. 
 
 University College, London, 
 Feb. 10, 1849.
 
 LIST 
 
 OF SOME WRITINGS ON THE SUBJECT OF ALGEBRA, 
 
 In which the peculiar Symbols of Algebra are discussed. 
 
 London, 1685, folio. JOHN WALLIS. A Treatise of Algebra, both 
 historical and practical. Reprinted in Latin, with additions, in 
 the second volume of Wall-is' a Works, Lond. 1693, folio. 
 
 Naples, 1687, folio. GILES FRANCIS DE GOTTIGNIES. Logistica 
 Universalis. 
 
 London, 1758, 4to. FRANCIS MASERES. A Dissertation on the use 
 of the Negative Sign in Algebra. 
 
 London, 1796, 8vo. WILLIAM FREXD. The Principles of Algebra.* 
 
 Cambridge, 1803, 4to. ROBERT WOODHOUSE. The Principles of 
 Analytical Calculation. 
 
 Philosophical Transactions for 1806. M. L'ABBE BTJEE. Memoire 
 sur les Quantites Imaginaires (Read June 20, 1805). See also 
 the review of this in Vol. xn. of the Edinburgh Review, April 
 ' - 
 
 In the list which follows the preface, the ].riiit-v .has 
 omitted a note of interrogation which foBowed the v?bj 
 
 Play fair. 
 
 Quantites Negatives, et des Quantites Pretendues Imaginairus. 
 
 Dedie aux amis de T evidence. 
 Cambridge, 1828, 8vo. JOHN WARREN. A Treatise on the Geome- 
 
 trical Representation of the Square Roots of Negative Quantities. 
 Philosophical Transactions for 1829. JOHN THOMAS GRAVES. 'An 
 
 attempt to rectify the inaccuracy of some logarithmic formula;.' 
 
 (Read December 18, 1828.) 
 Philosophical Transactions for 1829. JOHN WARREN. 'Considera- 
 
 tion of the objections raised against the geometrical repre- 
 
 sentation of the square roots of negative quantities. (Read 
 
 February 19, 1829.) The same volume contains JOHN WARREN. 
 
 1 On the geometrical representation of the powers of quantities, 
 
 whose indices involve the square roots of negative quantities.' 
 
 (Read June 4, 1829.) 
 
 Cambridge, 1830, 8vo. GEORGE PEACOCK. A Treatise on Algebra. 
 Cambridge, 1837, 8vo. Anonymous [OSBORNE REYNOLDS]. Stric- 
 
 tures on certain parts of 'Peacock's Algebra,' by a Graduate. 
 
 * An opponent not only of imaginary but of negative quantities. Perhaps this 
 work suggested M. Buee's memoir. 1 have a letter in my possession from M. Bute 
 to Mr. Frend, dated June 21, 1801, by which it appears that the former was desired 
 by a gentleman in whose house he was living (as tutor, perhaps) to write a private 
 reply to Mr. Trend's objections This letter evidently contains the germs of the 
 views which he afterwards published. See the Annual Report of the Royal Astrono- 
 mical Society for 1842. According to Dr. Peacock, M. Bfcee is the first formal niain- 
 tainer of the geometrical signification of V 1.
 
 vi LIST OF SOME WRITINGS ON ALGEBRA. 
 
 Philosophical Transactions for 1831. DAVIES GILBERT. ' On the 
 nature of negative and of imaginary quantities.' (Read Novem- 
 ber 18, 1830.) 
 
 London, 1834, 8vo. Report of the Third Meeting of the British 
 Association for the Advancement of Science. This volume con- 
 tains George Peacock ' Report on certain branches of analysis,' 
 a most valuable historical discussion on, among other things, 
 the advance of algebra. I cite from it the following works, 
 which I have either not seen, or cannot immediately obtain. 
 Paris, 1806, ARGAND, Essai sur la maniere de representer la 
 Quantites Imaginaires dans les constructions geometriques. Also 
 papers or observations by FRANCOIS, ARGAND, SERVOIS, GER- 
 GONNE, in the Annales des Mathematiques for 1813 (and I sup- 
 pose the following year). Also a paper on the arithmetic of 
 impossible quantities, by PLAYFAIR* in the Philosophical Trans- 
 actions for 1778 ; with a Reply, by WOODHOUSE, in the same 
 work for 1802, entitled ' On the necessary truth of certain 
 conclusions obtained by aid of imaginary expressions.' 
 
 London, 1836, 8vo. Anonymous [GEORGE PEACOCK]. A Syllabus 
 of a Course of Lectures upon Trigonometry, and the Application 
 of Algebra to Geometry. 
 
 London, 1837, 8vo. A. DE MORGAN. Elements of Algebra. 2nd 
 edition. 
 
 London, 1837, 8vo. A. DE MORGAN. Elements of Trigonometry 
 and Trigonometrical Analysis, preliminary to the Differential 
 Calculus, .... 
 
 Edinburgh Philosophical Transactions, Vol. XIV. Part 1. D[UNCAN] 
 F[ORBES] GREGORY. ' On the real nature of Symbolical Algebra. 
 (Read May 7, 1838). 
 
 Ladies' Diary. London, 1839, 8vo. (small). THOMAS WHITE. 'On 
 
 the algebraical expansion of quantity, .... and on the symbol 
 
 V-l> which is usually considered* to denote impossible or 
 
 imaginary quantity,' (at page 59). 
 Cambridge Philosophical Transactions, Vol. VII. Part 2. A. DE 
 
 MORGAN. ' On the Foundation of Algebra. (Read Dec. 9, 
 
 1839). 
 Cambridge Philosophical Transactions, Vol. VII. Part 3. A. DE 
 
 MORGAN. ' On the Foundation of Algebra, No. II. (Read 
 
 Nov. 29, 1841). 
 
 Paris, 1841, 8vo. M. F. VALLES. Etudes Philoso2>hiques sur la AY 
 
 du calcul. Premiere Partie. No more yet published. 
 Cambridge Philosophical Transactions, Vol. VIII. Part 2. A. DK 
 
 MORGAN. ' On the Foundation of Algebra, No. III. (Read 
 
 Nov. 27, 1843). 
 Cambridge, 1842 & 1845, Svo. GEORGE PEACOCK. A Treatise on Algebra. 
 
 Vol. I. Arithmetical Algebra. Vol. II. Symbolical Algebra 
 
 and its applications to the geometry of position. 
 London, 1843, 12mo. MARTIN OHM [translated by ALEXANDER 
 
 JOHN ELLIS]. The Spirit of Mathematical Analysis, and its 
 
 relation to a logical system. 
 
 * The author supposes it to be indeterminate, because it can be expanded by 
 help of a divergent series. The paper is marked 'received April 1816.'
 
 TABLE OF CONTENTS. 
 
 THE REFERENCES ARE TO THE PAGES OF THE WORK. 
 
 BOOK L TRIGONOMETRY. 
 
 CHAPTER I. 
 
 Preliminary Notions. 
 
 Definition of trigonometry, 1 ; undulating magnitude, 1 ; periodic 
 magnitude,. 2 ; suggested by angular magnitude, 2 ; gradual mea- 
 surement of angle, 3 ; factors of 360, 3 ; circumference of circle, 4 ; 
 IT, 4 ; multiplication and division by IT, 5 ; arc -f radius, 5 ; arcual 
 measurement of angle, 5, 6 ; gradual and arcual comparisons, 6 ; 
 gradual measurement of arc, 6. 
 
 CHAPTER II. 
 
 On the Trigonometrical Functions, and on Formula of One Angle. 
 
 Axes, origin, projections, co-ordinates, abscissa, ordinate, 7 ; 
 r, 0, x, y, 7 ; sign of r, x, y, 8 ; four quarters and their signs, 8 ; 
 6 and Innr + 0, 9 ; sine, cosine, tangent, cotangent, secant, cosecant, 
 versed sine, coversed sine, 9 ; complement, supplement, opponent, 
 completion, 10 ; trigonometrical functions as abstract numbers and 
 multipliers, 10; curve of sines, &c., 11; fundamental equations, 
 11, 12; limits of value, 12; signs, 12, 13; negative sign of r, 13; 
 initial and terminal values, 13; cosine even, sine odd, 13; tangent 
 odd, 14 ; %nnr + and its rules, 14, 15 ; double value of functions, 
 16; 15, 18, 30, 45, 60, 72, 75, 16, 17; sin 6 -f 0, (1 - cos 6) ^ 0,' 
 tan0 -^ 6, 17, 18; and 1 -^0 2 , 18; older system of definitions, 
 18, 19 ; arese of circle and sector, 20.
 
 viii CONTENTS. 
 
 CHAPTER III. 
 
 Formulas which involve two or more Angles. 
 
 Extended notion of projection, 21 ; distinction of AB and BA, 
 and consequences, 21 ; similar distinction as to angles, 21 ; signs 
 of projections, 22 ; general investigation of cos(< + 0) and sin (< + 0), 
 23 ; connexion of the formulae, 24 ; cases of limited demonstration, 
 25, 26 ; collection of formulae, 26, 27 ; remarks on the formulae, 
 28, 29 ; cosine and sine of the sum of any number of angles, 29, 30 ; 
 coswO and sinwO, 30, 31 ; trisection of an angle, 31 ; series for cos 9 
 and sinO, 32, 33; ditto for tan 6, 34; algebraic definition of trigo- 
 nometry, 34 ; cos" 6 and sin" 6, 34, 35, 36. 
 
 CHAPTER IV. 
 
 On the Inverse Trigonometrical Functions. 
 
 Functional notation, direct and inverse, 37 ; inverse trigono- 
 metrical functions, 37, 38 ; examples in the use of inverse symbols, 
 38, 39, 40. 
 
 CHAPTER V. 
 
 Introduction of the unexplained Symbol *J \. 
 
 Remarks on the evidence of V - 1 in this chapter, 41 ; connecting 
 formulas of trigonometricals and exponentials, 42 ; De Moivre's 
 theorem, 42 ; multiplicity of directions in 6 4- n, 43, 44 ; roots, 
 particularly of unity, 45, 46; transformations of a + b V 1 and 
 selection of meaning in tan" 1 (b -4- a), 46 ; extension of logarithms 
 with Naperian base, 47 ; extension of the Naperian base, 48 ; 
 isolated case of coincidence of logarithms in different systems, 48 ; 
 the negative quantities which have real logarithms, 49 ; equivalents 
 of De Moivre, 49 ; deduction of ordinary formulae from them, 49, 50 ; 
 reduction of sin m cos"0 to a linear form, 50, 51 ; mode of finding 
 2 n cosn0a: n and 2a n sinn6x", 51, 52; connexion of <(x + y V 1) 
 and <O-yV-l), 52, 53; examples, 53, 54, 55; series for 
 tan" 1 a;, 55 ; calculation of -rr, 55, 56 ; inverse connexion of trigo- 
 nometricals and exponentials, 57 ; use of multiplicity of value 
 of logarithms, 57, 58 ; resolution of sin into factors, 58, 59, 60 ; 
 Wallis's form of ir, 61: deduction of approximate form, for 1.2.3...n, 
 61, 62; factors of cos ; logarithms of sin 0, cos 6, tanfl, 63; Vieta's 
 expression for JTT, 63; Bernoulli's numbers, 64; series for tanx, 
 cot a;, and (t x - I)" 1 , 65. 
 
 CHAPTER VI. 
 
 On the Connexion of Common and Hyperbolic Trigonometry. 
 Hyberbola, 66 ; its areas, 67, 68 ; formation of hyperbolic tri- 
 gonometry, and connexion of its formulae with those of ordinary 
 trigonometry, 69, 70.
 
 CONTENTS. ix 
 
 CHAPTER VII. 
 
 On the Trigonometrical Tables. 
 
 Arrangement and extent of the tables, 71, 72 ; distinction of real 
 and tabular logarithms, 72 ; tables recommended, 73 ; argument, 
 interval, function, difference, interpolation, 73 ; method of inter- 
 polating, 74 ; choice of functions for accuracy, 75 ; tangents of 
 angles near to 90, 75, 76 ; first notions of the construction of tri- 
 gonometrical tables, 76, 77, 78. 
 
 CHAPTER VIII. 
 
 On the Solution of Triangles. 
 
 Meaning of solution, 79 ; formulae for right-angled triangles, 
 79, 80 ; cases of ditto ditto, 80 ; tabulated example, 80 ; formulae 
 for oblique triangles, 81, 82, 83; cases of ditto ditto, 84, 85; mode 
 of entrance of double solution, 85 ; tabulated example, 86, 87 ; 
 mention of occasional rules, 87 ; reduction of triangular formulae 
 to identities, 88. 
 
 BOOK II. DOUBLE ALGEBRA. 
 
 CHAPTER I. 
 
 Description of a Symbolic Calculus. 
 
 Object, 89 ; peculiar symbols, meanings, rules of operation, 89 ; 
 possible deficiencies of either, 90, 91 ; complete absence of either, 
 91, 92; symbolic calculus, what, 92; recovery of meaning, significant 
 calculus, 93 ; slight example, with illustration of defects, 93, 94 ; 
 possibility of more than one mode of restoration to significance, 94 ; 
 step from specific to universal arithmetic, and thence to ordinary- 
 algebra, 95, 96 ; necessity for other than numerical distinction, and 
 mode in which distinction is suggested by algebra, 95, 96 ; single 
 algebra, phrase whence derived, 96, 97 ; twofold use of its signs, 
 directive and conjunctive, 97 ; remarks on the progress of algebra, 
 98, 99, 100. 
 
 CHAPTER II. 
 
 On Symbolic Algebra. 
 
 Abandonment of meaning, 101 ; collection of the symbolic laws 
 of algebra, 101, 102, 103 ; instances of symbolic deduction, 104, 105 ; 
 reservation of small italic letters to signify combinations of unit 
 symbols, 105.
 
 x CONTENTS. 
 
 CHAPTER III. 
 
 On Areas and Solids, 
 Meanings of the fundamental symbols under which, algebra be- 
 comes a legitimate mode of establishing the second book of Euclid, 
 106, 107 ; comparison of an inexplicable symbol of this algebra 
 with one of ordinary algebra, 107, 108 ; this system not altogether 
 new, 108. 
 
 CHAPTER IV. 
 
 Preliminary Remarks on Double Algebra. 
 
 Suggestion on V 1 which gave rise to it, 109 ; the old algebra 
 to be incorporated, 109 ; examination of magnitude of one dimension, 
 time, gain and loss, 109, 110 ; a wider basis of significance in length 
 affected by direction, 110, 111 ; geometrical introduction extension, 
 not restriction, 111, 112 ; mode of making the application to problems 
 of one dimension, 112, 113, 114; separation of subject-matter and 
 operative direction in arithmetical addition and multiplication, 
 -15, 116. 
 
 CHAPTER V. 
 Signification of Symbols in Double Algebra. 
 
 Reason of the term double algebra, drawn from the meaning 
 of an isolated symbol, 117; meaning of =, 117; origin, 117; unit- 
 line, 118 ; axes of length and direction, 118 ; meaning of A + B, 
 A B, B, 118 ; coincidences of common and extended addition, 
 &c., 118; addition really junction and its resiilt joint effect, 118; 
 meaning of A x B and A 4- B, 119 ; symbolic representation of 
 double-meaning symbol, (a, a), 119 ; coincidences of common and 
 extended multiplication, &c., 120 ; roots and powers, without refer- 
 ence to exponents, 120 ; developed expression of an algebraical 
 theorem, 121 ; construction of all double symbols by single ones 
 and V 1, 122; re-introduction of trigonometry, 122; demonstration 
 of the symbolic rules, 123, 124, 125 ; deduction of fundamental 
 trigonometrical formula?, 126 ; proof of the validity of that deduction, 
 and of its coincidence with ordinary proofs, 127, 128. 
 
 CHAPTER VI. 
 
 On the Exponential Symbol. 
 
 Assumption of arithmetical logarithms, definition of logometer, 
 129 ; choice of logometer, 129 ; multiple values of logometers, 130 ; 
 definition of A B , 130; limitation of , 130; logometric equations, 
 131 ; proof of symbolic rules, and limitations, 131 ; proof of 
 e Ov-i _ COS _j_ ginfl . ^ _l f 182 ; connexion of e and the arcual unit, 
 132, 133, 134 ; transformation of It s , 134 ; example of reduction 
 to significance, 135; fallacy exposed, 136; formulae which supply 
 those of p. 131 when the limitations are removed, 136, 137.
 
 CONTENTS. xi 
 
 CHAPTER VII. 
 
 Miscellaneous Remarks and Applications. 
 
 Extension of logarithms, 138 ; forms of < (a b V - 1), 138, 139 ; 
 extension of trigonometrical terms, 139, 140; discontinuity of pas- 
 sage from + to + in single algebra, 140 ; interpretation of a problem 
 impossible in single algebra, 140, 141 ; Cauchy's theorem on the 
 limits of imaginary roots of equations, 141, 142, 143, 144 ; Paradox 
 of single algebra which disappears in double algebra, 144, 145 ; 
 change of geometrical representation in passing from real to ima- 
 ginary, 145, 146. 
 
 CHAPTER VIH. 
 
 On the Roots of Unity. 
 
 Power of considering ( 1)" as either quantitative or directive, 
 147, 148 ; properties of the roots of + 1, 148, 149, 150, 151, 152, 153, 
 154 ; solution of x n = 1 when n is prime and a;"" 1 = 1 has been 
 solved, 155, 156, 157 ; Gauss's accession to Euclid's geometry, 157 ; 
 properties of the roots of 1, 158 ; formation of recurring expressions, 
 158, 159 ; Thomas Simpson's method of section of series, 159, 160. 
 
 CHAPTER IX. 
 
 Scalar View of Algebraical Symbols. 
 
 Law of ascent of algebraical operations, 161 ; illustrative notation, 
 162; scalar function, 162; ascent of algebraical operations, 162; 
 imperfection of ordinary notation for scalar representation, 163 ; 
 other law which this notation follows, 164 ; general law of scalar 
 ascent, 164 ; limitation of the scalar function, 164 ; inverse operation, 
 165 ; fundamental basis of algebra, 166 ; scalar notation in extension 
 of that of ordinary algebra, 166, 167. 
 
 ERRATUM. 
 
 XX 
 
 Page 138, after insert 
 A.x 
 
 I
 
 BOOK I. 
 
 TRIGONOMETRY. 
 
 CHAPTER I. 
 
 PRELIMINARY NOTIONS. 
 
 IT is proved in the sixth book of Euclid that when sides and 
 diagonals are given, in number enough to determine a rectilinear 
 figure, angles depend solely upon the proportions of sides, and 
 proportions of sides solely upon angles. If two angles of a triangle 
 be given, all the ratios of sides are given : and, If the ratio of 
 each of two sides to a third be given, all the angles are given. 
 There is then a close connexion between angles, and ratios of 
 lines : the branch of mathematics in which this connexion is 
 examined, suitable modes of expression invented, and results ob- 
 tained and applied, is called TRIGONOMETRY, taking its name 
 from one of its earliest applications, the measurement of triangles. 
 
 Trigonometry contains the science of continually undulating 
 magnitude; meaning magnitude which becomes alternately greater 
 and less, without any termination to succession of increase and 
 decrease. A function of x is continually undulating, when, as x 
 increases continuously, say from to oo , (fox never becomes per- 
 manently increasing nor permanently diminishing, nor permanently 
 approaching to a fixed limit. Ordinary algebra has no such func- 
 tions in its finite forms; and though it has them in its infinite 
 series, yet it cannot easily recognize and establish the undulating 
 property. Trigonometry is the branch of algebra in which undu- 
 lating functions are considered. All trigonometrical functions 
 are not undulating : but it may be stated that in common algebra 
 nothing but infinite series undulate: in trigonometry nothing but 
 infinite series do not undulate. 
 
 Trigonometry is a branch of algebra: nevertheless, it is usually 
 founded on geometrical considerations. This is not an absolute 
 
 B
 
 J ELEMENTARY NOTIONS. 
 
 necessity : but any other foundation would make it much more 
 difficult for the beginner to understand. It will become evident 
 that another mode of establishing the algebra of undulating 
 quantities might have been chosen, in which no geometrical 
 notion need have been even alluded to. 
 
 Of all undulating magnitudes, the most simple is the periodic, 
 which exhibits a perpetual recurrence of the same cycles of 
 alteration. If, however a function may change while x changes 
 from to a, the same changes take place while x changes from 
 11 to 2o, from 2a to '3a, and so on, that function is periodic. 
 The general property of such a function is expressed by the 
 equation (x + ) = 0:r, true for all values of x, and for one 
 value of a, or for its multiples. For (x f a + a) = (x 4- a) = <px ; 
 or (x + 2a) = $x. Similarly (x + 3a) = 0a: ; and so on. 
 
 Consideration of angular magnitude must suggest periodic 
 functions. Let a straight line, fixed at one extremity, revolve 
 about that extremity. The total angle described may go on 
 increasing ad infinitum: the angle itself is not a periodic mag- 
 nitude, though beginners are apt to think so. But the direction 
 indicated is periodic, though not a magnitude. 
 
 There is no direction indicated during the second revolution, 
 which was not previously indicated during the first. Let a wheel 
 turn on an horizontal axle, by a handle at the end of a spoke : 
 the angle turned through by the spoke goes on increasing as 
 long as the wheel turns one way; but the height of the handle 
 above the ground is a periodic magnitude, which goes through 
 the same cycle of changes in each and every revolution. Certain 
 periodic functions, suggested by the revolution of a straight line 
 about a point, form the trigonometrical alphabet, as we shall see. 
 
 If we had now to invent a mode of measuring angles, the 
 most convenient first method would be to adopt the whole revo- 
 lution* as a standard unit; thus the angle '467 would signify 
 
 * Observe this consequence of the periodic character of direction, 
 that the angle has a unit expressible in words without reference to 
 other magnitude exhibited. ' The angle through which a line revolves 
 in regaining the direction with which it first started,' is a perfect 
 description of a definite amount of angular magnitude. But no 
 number of volumes could describe an English foot, if drawing, and 
 reference to length supposed capable of access, were both prohibited.
 
 ELEMENTARY NOTIONS. 3 
 
 467-1000 ths of a revolution. This plan has not been adopted: 
 the usual method is to divide the whole revolution into 360 equal 
 parts, each of -which is called a degree. Each degree is divided 
 into 60 minutes, each minute into 60 seconds; formerly each 
 second was divided into 60 thirds, each third into 60 fourths, 
 and so on. This sexagesimal mode of division was once applied 
 to all kinds of magnitude: thus the sixtieth part of any length, 
 time, weight, area, &c. was called its minute, the sixtieth part 
 of the minute its second, and so on. Nothing of this method 
 remains to us, except in the divisions of a degree of angle, 
 a degree of arc (the 360th part of the whole circumference of 
 a circle), and the hour of time. Thus it would have been said 
 that the circumference of a circle is very near to 3 8' 29" 44'" 
 diameters, meaning 3 + / + g-g^ + FrtoTO of a diameter. 
 
 Degrees, minutes, seconds, &c. are represented by ' " '" &c. 
 But it must be noticed that thirds, fourths, &c. are wholly 
 obsolete, decimal fractions of the second being preferred. Thus 
 18 47' 23" -1774 indicates the following fraction of a whole 
 revolution, 
 
 18 47 23 1774 1 
 
 _L 
 
 360 60 x 360 60 x 60 x 360 10000 60 x 60 x 360 * 
 
 In this mode of measurement it is worth while to remember ; 
 the right angle and its multiples, 90, 180, 270, 360; the half 
 of a right angle and its multiples, 45, 90, 135, 180, 225, 
 270, 315, 360; and the third of a right angle and its multiples, 
 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360. 
 Also the thirds of the revolution, 120, 240, 360; and its fifths, 
 72, 144, 216, 288, 360. And 360 should be well known a> 
 2 3 x3*x5, from which its separation into pairs of factors, 2.180, 
 3.120, 4.90, 5.72, 6.60, 8.45, 9.40, 10.36, 12.30, 15.24, 18.20, 
 will be easily gathered. 
 
 The above method of measurement may be called gradual 
 (pronounced yrade-ual). But it is not the only method in use. 
 There is another, which 1 shall call the arcual* method. To 
 explain this method, it must first be shewn that circumferences 
 
 * I have been in the habit of styling this the theoretical method, 
 as being used in the theory of the subject : but I shall now adopt 
 the term used in the text.
 
 4 ELEMENTARY NOTIONS 
 
 of circles are to one another as their diameters. Let it be 
 granted that the circumference of a circle is greater than that 
 of any inscribed polygon, and less than that of any circum- 
 scribed polygon. 
 
 Draw the circle whose radius is OA. Let BOA be the 2n th 
 part of a revolution: so that 
 2 x BD and 2w x CA are the 
 circumferences of the inscribed 
 and circumscribed regular poly- 
 gons of n sides. These circum- 
 ferences are as BD to CA, or 
 as OB to OC. Consequently, 
 if the angle BOA be made 
 
 small enough, or n great enough, the inscribed and circumscribed 
 regular circumferences may be made as nearly equal as we please ; 
 and either, therefore, as near as we please to the circumference 
 of the circle, which lies between them in magnitude. Now take 
 another figure like the preceding, but constructed on a different 
 radius OB', and with all its letters accented. We know then 
 that the two inscribed regular circumferences of n sides are to 
 one another as OB' to OB', and also the two circumscribed 
 circumferences. Let P and P', C and C', Q and Q' be the 
 circumferences of the inscribed polygons, the circles, and the 
 circumscribed polygons. Then the order of magnitude is always 
 P, C, Q and P', C 1 , Q', and the ratios P : P and Q : Q' are always 
 equal and constant (each being the ratio of the radii) while 
 P and Q, and also P' and Q', can be made as nearly equal as 
 we please. Hence it follows that C : C' is the same ratio as 
 P:P and Q : Q'. Let P and Q be C-M and C+X; and 
 let P and OJ be C" - M' and C" + N'. Then M, N, M', N', 
 may each be made as small as we please; and C-M: C' - M' 
 being always one ratio (that of the radii), the limiting ratio C : C' 
 can be no other (Algebra, p. 157). The same follows from the 
 same use of the ratio (7-f JV: C" + N'. 
 
 The circumference being C, and the radius R, it follows that 
 the fraction CR is the same for all circles. It is always 
 denoted by 27r; that is, TT is always made to represent the fraction 
 which expresses the ratio of the circumference to the diameter. 
 An investigation of the value of TT, such as we can hereafter make,
 
 ELEMENTARY NOTIONS. a 
 
 but of which at present we must assume the result, shews that it 
 is nearly ^, very much nearer to f||, and expressed, as far as 
 twenty places of decimals will do it, by 3-14159265358979323846. 
 Its reciprocal is, to the same extent, -31830988618379067153. 
 I leave the student to demonstrate the following rules, the con- 
 venience of which is the formation of results by successive cor- 
 rections, so that the point at which it is desirable to stop is 
 pointed out by the value of the corrections. 
 
 To multiply by TT, first take the multiplicand 3 times and 
 one-seventh of a time, deduct its 800 th part, the 100 th part of 
 the last, and 2 millionths of the multiplicand. Then add the 
 hundred-millionth of the multiplicand, and 7 3 per cent, of that 
 hundred-millionth. The result is as correct as if thirteen figures 
 had been used in the ordinary multiplication. 
 
 To divide by TT, take seven 22 ndi > of the dividend, one 8000 th , 
 3 millionths, and 7 hundred-millionths ; then deduct 2 thousand- 
 millionths, and add the thousandth of the last. The result is 
 as correct as if thirteen decimals had been used in the ordinary 
 division. 
 
 Let there be an angle of which the arc is s to the radius r, 
 and s' to the radius r', the circumferences being c and c'. Then 
 the angle is to four right angles (Euc. vi. 33) as s to c, and 
 
 
 as s' to c'. Hence s : 2irr : : s' : ITTT' , whence - = . Or, to a 
 
 r r' 
 
 given subtending angle, arcs are to one another as their radii. 
 Let there be another angle, having the arcs S and S' to the 
 
 s S 
 
 radii r and r'. Then the angles are as s and S, or as - and - 
 
 r r ' 
 
 or as - and . That is, any two angles being made central 
 
 angles in any two circles, the fractions obtained from r- are 
 
 rad. 
 
 proportional to the two angles. For instance, the angle which 
 has an arc 6 to the radius 17 is to the angle which has an arc 
 11 to the radius 8 as ^ to -g 1 . 
 
 From this theorem is derived the arcual mode of measuring 
 angles. Let the arcual angular unit be that angle which subtends 
 an arc equal to the radius, and let all other angles be measured 
 by the numbers of arcual units, or the fractions of an arcual 
 unit, which they contain. Then we shall have the following 
 
 n3
 
 6 ELEMENTARY NOTIONS. 
 
 theorem : The number of arcual units in any angle is the quotient 
 of any arc which that angle subtends, divided by the radius. 
 For if 9 be the number of arcual units in the angle which 
 subtends s to the radius r, we have (Euc. VI. 33), 
 
 o 
 
 : 1 : : s : r, or 6 = - . 
 
 fll'C 
 
 When we write the equation angle = -j- , we understand by 
 
 ' angle' an abbreviation of ' number of arcual units contained 
 in the angle'. 
 
 The number of arcual units in four right angles is circum- 
 ference -f radius, or 2?r ; in two right angles, TT ; in one right 
 angle, \TT. Since 180 degrees make TT arcual units, the arcual 
 
 180 
 unit is - - degrees, or 57-295779513; it is also 3437''74677 ; 
 
 7T 
 
 and 206264"-806. It may be remembered, within the hundredth 
 part of a second, as 57 degrees and three tenths, all but one- 
 fourth of a minute and one-fifth of a second. This is 57 17' 44"-8. 
 
 The degree, minute, and second, are severally the fractions 
 01745329, -0002908882, and -000004848137, of an arcual unit. 
 The arcual unit being our usual reference, the degree may ge- 
 nerally be considered as a small angle. Most of the theorems 
 which I assert to be approximately true for small angles, are 
 nearly true for an angle as small as a degree. 
 
 The student must remember not to confound 2?r with 360, 
 nor TT with 180, as is sometimes done, even by writers. That 
 2?r= 360 is true in a certain sense ; and so is 20 = 1, for 20 shillings 
 are one pound. 
 
 When a circle is divided into 360 equal arcs, each is called 
 a degree of arc; and the degree of arc is divided sexagesimally. 
 The radius is 57 47' 44''-8 of arc. On a great circle of the earth 
 (the equator for instance, or a meridian), the second of arc is 
 about 100 feet.
 
 CHAPTER IT. 
 
 ON THE TRIGONOMETRICAL FUNCTIONS, AND ON FORMULAE OF 
 
 ONE ANGLE. 
 
 4* 
 
 LET two straight lines be drawn at right angles to one another ; 
 let them be called axes, and their point of intersection, O, the 
 origin. Let any line, OP, be drawn from the origin; and let 
 
 \ 
 
 N A 
 
 PN, PM be drawn perpendicular to the axes. In the rectangle 
 MOPN, ON and OM are called projections of OP upon the axes. 
 The projections of OP are also called coordinates of the point P : 
 and the coordinates are distinguished by the names abscissa and 
 ordinate. Usually, a projection and a parallel to the other pro- 
 jection are employed, as ON, NP : and then the projection is 
 generally named the abscissa of P, and the parallel to the other 
 projection, the ordinate of P. And generally the abscissa is taken 
 upon the axis drawn horizontal in the page, and the ordinate 
 parallel to the vertical axis. The letter x usually designates an 
 abscissa, y an ordinate ; and the axes are called the axes of x 
 and of y. 
 
 A line terminating at 0, and indefinitely extended, revolves 
 about O, setting out from one side of the axis of x, OA. "When 
 it has described an angle 6, which may be of any magnitude, 
 a distance r is taken off. This distance is always considered as
 
 8 ON THE TRIGONOMETKICAL FUNCTIONS 
 
 positive, when r is taken off on the revolving line : and as nega- 
 tive, if taken off on the opposite side. Thus, the acute angle 
 POA being 30, we may refer OP to a line which makes 30 
 with OA, and then we say that OP is positive. But when we 
 say that OP is part of a line which makes 210 with OA, we call 
 it negative. 
 
 Again, one. particular direction of revolution is considered as 
 positive, the other as negative. If the arrows designate the posi- 
 tive revolution, then OP, being positive, makes an angle with OA, 
 which may be called +30 or -330; but if OP be negative, it 
 makes an angle + 210 or - 150. 
 
 On the axes, each species of coordinate or projection has its 
 proper algebraical sign. The starting-line of revolution is always 
 taken as the positive side of the axis of x ; and the result of -f 90 
 of revolution as that of the axis of y. Thus ON is positive, ON' 
 is negative ;* OH is positive, OM' is negative. 
 
 The axes divide the plane into four quarters: and as a line, 
 revolving positively, passes from to 90, from 90 to 180, from 
 180 to 270, and from 270 to 360, it is said to be in the first, 
 second, third, and fourth quarters of space. But these might 
 equally well be designated as the ++, +-, --, and -+ quarters 
 of space. 
 
 In this system, -I- f , + -, , - +, the first of each pair gives 
 the succession -f + ; and these are the signs of the y projections 
 of lines in the four quarters : the projection on the axis of y of 
 a line in the first quarter of space, is + ; in the second, -f ; in the 
 third, -; in the fourth, -. The second of each pair gives the 
 
 succession + h ; and these are the signs of the x projections 
 
 of lines in the four quarters. The algebraical combination of each 
 of the pairs gives the succession + - + - ; and these are the signs 
 
 * When the revolving line comes into the position ON\ is it 
 negative? I answer, no: OA", as a projection, is considered as part 
 of a line which makes an angle with, the starting-line ; and, on 
 a line so described, is negative. But OJV , as a position of the line of 
 revolution, is part of a line which makes 180 with the starting-iine ; 
 and thus considered, it is positive. The same considerations apply 
 to the other axis. A line may be considered as making with itself 
 an angle or an angle ISO": whatever signs its parts have in the 
 first case, the}- have the opposite ones in the second.
 
 AND ON FORMULAE OF ONE ANGLE. 
 
 9 
 
 of the arithmetical products or quotients derived from the two 
 projections of a line in each of the four quarters. 
 
 Everything that takes place in the first revolution is repeated 
 in the second; and is repeated in an inverted order in the first 
 negative revolution. In all that depends upon the direction in 
 which an amount of revolution terminates, an addition or sub- 
 traction of a whole revolution makes no difference whatever. But 
 in all that depends upon the actual magnitude of the angle 
 revolved through, an alteration by a whole revolution makes an 
 effective difference. Measuring arcually, Itmr + 9 may most often 
 be confounded with 9 when m is any integer, positive or negative ; 
 but not always. 
 
 The primary trigonometrical functions of an angle are the ratios 
 of the projections to the revolving line, and to one another, direct 
 and inverse : these ratios are independent of the length of the 
 revolving line. Let x, y, r be the values, with their proper signs, 
 of the abscissa, ordinate, and radius, or base, perpendicular, and 
 
 hypothenuse. The six ratios - , - , - , - , - , - take each a 
 
 r r x y x y 
 
 name, the etymology of which cannot be explained till we come 
 to exhibit the older definitions : at present they must stand for 
 arbitrary sounds. Let 9 be the angle by revolving through which 
 r has gained its position. 
 
 is called the abbreviated into 
 
 base 
 
 r 
 
 y. 
 
 r 
 
 rad. 
 ordinate 
 
 hyp- 
 
 perpend. 
 
 sine of 9 
 
 sin 
 
 rad. 
 
 hyp. 
 
 y. 
 
 X 
 
 * 
 
 y 
 
 ordinate 
 
 perpend. 
 
 tangent of Q 
 cotangent of 9 
 
 tan 9 
 
 C0t0 
 
 abscissa 
 abscissa 
 
 base 
 base 
 
 ordinate 
 
 perpend. 
 
 r 
 x 
 r 
 
 y 
 
 rad. 
 
 hyp. 
 
 secant of 9 
 cosecant of 9 
 
 sec# 
 
 COS6C0 
 
 abscissa 
 rad. 
 
 base 
 hyp. 
 
 ordinate 
 
 perpend. 
 
 1 - cos 9 
 1 - sinO 
 
 versed sine of 9 
 coversed sine of 9 
 
 versfl 
 covers 9
 
 10 ON THE TRIGONOMETRICAL FUNCTIONS 
 
 This table must be thoroughly learned. The terms base, per- 
 pendicular, and hypothenuse, referring to the right-angled triangle 
 in which the projections are sides, does not mean that what Euclid 
 would call an angle of that triangle is always the angle in question. 
 It is so when 6 is less than a right angle, or when the revolving 
 line is in the first quarter. But in the second quarter, 6 is a 
 supplement* of Euclid's angle ; in the third quarter it is an 
 opponent ; in the fourth quarter it is a completion. All this, and 
 many other things of which only hints are given, must be fixed 
 in the mind by attentive consideration of all the phases of the 
 figure of a line projected on the axes: no amount of description 
 will supply the place of such consideration. 
 
 It is important to remember that all the trigonometrical functions 
 are purely abstract numbers. They are not angles, nor lines, any 
 more than they are weights, or sums of money. They represent 
 the fractions which lines are of lines, the ratios of lines to lines. 
 Thus, the cosine of 60 is J : one-half of what ? Answer, one-half 
 of a time : when the revolving line has described 60, the projection 
 on the axis of x is one-half of the revolving line ; the last words 
 in italics contain the assertion that cos 60 = . 
 
 Thus the functions may be advantageously remembered by 
 their effect as multipliers. The cosine and sine may be called 
 projecting factors: multiplication by cos<? turns r into its pro- 
 jection on the axis of x; multiplication by sin# turns r into its 
 projection on the axis of y. The projections of r are rcosd and 
 r s'mO. The tangent and 'cotangent are interchanging factors; 
 multiplication by tan0 converts the projection on x into that on y , 
 multiplication by cot6> converts the projection on y into that on x. 
 
 We may of course take a line which has as many linear units 
 as a certain angle has of angular units, or as a sine or tangent has 
 
 * The term, supplement has long been used to signify the detect 
 from two right angles : thus 6 and TT Q are supplements. By 
 opponents, I mean angles made by opposite straight lines with one 
 straight line, in the same direction of revolution : thus 6 and IT + 6 
 are opponents. By completions, I mean angles which together make 
 up a whole revolution : thus of and 2-n- 6 each is the completion of 
 the other. Finally, the well-known term complement is arbitrarily 
 used to denote the defect from a right angle : thus 6 and \ir are 
 complements.
 
 AND ON FORMULAE OF ONE ANGLE. 11 
 
 of abstract units, and in this sense it may be permitted (to those 
 who can do it without confusion) to talk of a line and angle being 
 equal, or of a line equal to the sine of an angle. The frontispiece 
 has curves constructed in this manner for each of the six principal 
 functions. The origin is O, the axis of x is OA...: the abscissa 
 is the angle, the ordinate on one curve is the sine, &c. The 
 student may, when he has read a little further, detect for himself 
 the curve of sines, of cosines, of tangents, of cotangents, of secants, 
 of cosecants. 
 
 There are eight trigonometrical 'functions, of which two are 
 absolutely defined by formula?; namely, 
 
 vers<3 = 1 - cos#, covers = 1 - sin#. 
 
 Of the remaining six, we may predict that five independent 
 equations exist among them : for one angle and one ratio of sides 
 absolutely determine all the angles (and therefore all the ratios 
 of sides) of a triangle, whensoever that given angle is a right 
 angle or more. There are easily found more than five relations; 
 but not all independent. First, there are the relations which 
 obviously and necessarily follow from the algebraical form of 
 the definitions, independently of the meaning of the symbols. 
 These are 
 
 cos<? x sec# = 1, 
 
 cos# 
 
 smu x cosecu = 1, tan^ = - , cott/ = - . 
 cose/ saw 
 
 tantf x cot0 = 1. 
 
 Of these only four are independent: the third and fourth make 
 the fifth follow. Secondly, there are the relations which follow 
 from the meaning of x, y, and r. The equation y? + y* = ?*, 
 which follows from the application of arithmetic to Euc. I. 47, 
 gives 
 
 cos 2 + sin 2 = 1, 1 + tan 2 = sec 2 fl, 1 + cot 2 = cosecV, 
 of which one only is independent; for cos s # + sin 2 = 1, gives 
 
 1 + f "Ef V = f - Y. or 1 + tan 2 = sec 2 6>, &c. 
 \costfj \cosO) 
 
 The following collection of formulae, either proved above, or
 
 12 ON THE TRIGONOMETRICAL FUNCTIONS 
 easily deduced, should be carefully remembered: 
 
 
 cos o. seuf - i, cu 
 sin0.cosec# = 1, 
 
 1 + tan 8 = sec 2 0, 
 
 , COS0 
 
 cot0 = 
 
 tan#. cot<? = 1, 
 
 1 + cot^ = cosec 8 6 
 
 i sin 
 
 1 
 
 sin0 tan ^ 
 
 Y 
 
 V(l -i-tan^)' 
 
 
 
 CL 
 
 b 
 
 a V 
 
 (d* + 5 8 ) ' A/( 
 
 ..i , 7 a \ ' 
 
 The student should, as an exercise, express each function in 
 terms of all the rest. 
 
 I now proceed to the examination of several material points 
 connected with the several functions. 
 
 1. Limits of value. No cosine nor sine can fall without the 
 interval -1...+ 1: for neither x nor y can numerically exceed r. 
 For the same reason, no secant nor cosecant can fall within the 
 interval -1...+ 1. But a tangent or cotangent may have any 
 value, positive or negative. Versed and coversed sines always 
 lie in the interval 0...2. 
 
 3C I/ 
 
 2. Siqns. Let r be taken positively. Then - and - have 
 
 r r 
 
 the signs of x and y. Hence cos# (and its reciprocal sec#) have 
 their signs remembered by the succession + + : or the cosine 
 has the sign + in the first quarter, - in the second, &c. But 
 sin0 (and its reciprocal cosec<?) have their signs remembered by 
 the succession + -f . And tan (and its reciprocal cotfl) have 
 their signs remembered by the succession f - + - (page 8). 
 
 The pairs of signs by which the quarters are distinguished, 
 state the signs of the sine and cosine, while the sign compounded 
 of the other two, states that of the tangent. Tims, in the 
 fourth, or - + quarter, the sine is negative, the cosine positive, 
 and the tangent negative. 
 
 It is also worth while to remember, that while all the three 
 pairs are positive in the first quarter, each of the other quarters 
 has only one positive pair belonging to it. 
 
 In the first quarter, all are positive. 
 
 In the second .... only the the sine and cosecant. 
 
 In the third .... only the tangent and cotangent. 
 
 In the fourth .... only the cosine and secant.
 
 AND ON FORMULAE OF ONE ANGLE. 
 
 13 
 
 The whole system remains consistent with itself if negative 
 values of r be introduced, under the definitions in page 8. Take 
 the figure in page 7 as an instance, and say that OP is negative. 
 Then, ON and NP being positive, the sine and cosine are 
 negative. And so they ought to be : for OP, being negative, 
 must be considered as on a line in the third quarter, and d as 
 between two and three right angles. 
 
 Versed sines and coversed sines are always positive. 
 
 3. Initial or terminal values. These are the values when 
 the revolving line begins or ends a quarter of space, and is on 
 one of the axes. In every such case one of the projections 
 vanishes, and the other is of the same length (but not always 
 of the same sign) as the revolving line itself. At = 0, and 
 = TT, y vanishes; at 9 = \TT, and 6 = %TT, x vanishes. At 6 = IK, 
 the values at 9 = are repeated. Examination will give the 
 following table, which should be remembered; partly by the 
 help of the connecting equations. 
 
 Arcual Anyle 
 
 
 
 ATT 
 
 7T 
 
 fir 
 
 27T 
 
 Cosine 
 
 1 
 
 
 
 -1 
 
 
 
 1 
 
 Sine 
 
 
 
 1 
 
 
 
 -I 
 
 
 
 Tangent 
 
 
 
 oo 
 
 
 
 oo 
 
 
 
 Cotangent 
 
 00 
 
 
 
 GO 
 
 
 
 oo 
 
 Secant 
 
 1 
 
 00 
 
 -1 
 
 oo 
 
 1 
 
 Cosecant 
 
 00 
 
 1 
 
 CO 
 
 -1 
 
 so 
 
 Versed sine 
 
 
 
 1 
 
 2 
 
 1 
 
 
 
 Coversed sine 
 
 1 
 
 
 
 1 
 
 2 
 
 1 
 
 Gradual Anyle 90 180 270 j 360 
 It is hardly necessary to say that all the trigonometrical func- 
 tions are periodic, and that 2/r is in every case the angular 
 extent of one period, or of a number of periods. In every case 
 Ft) = F(0 + 2?;-), F representing a primary trigonometrical function , 
 The period of the sine and cosine is 2?r ; of the tangent, ir. I now 
 proceed to examine circumstances connected with this periodic 
 character. 
 
 The cosine is what is called an even function of 6; that is, 
 it does not change at all when 6 is changed into - 0: or cos(- 6) 
 = cos0. But the sine is what is called an odd Junction ; that i>, 
 
 C
 
 14 ON THE TRIGONOMETRICAL FUNCTIONS 
 
 it changes sign when is changed into - ; or sin (- 0) = - sin 0. 
 Let two equal lines revolve, one positively and one negatively : 
 it is clear from the elements of geometry, that whatever equal 
 angles they may have described, the projections on x are the 
 same, identically, and the projections on y differ in sign only. 
 Hence, x f r is the same for both ; and y 4- r is not, but the 
 difference is in sign only. 
 
 The tangent is an odd function ; for tan(- 6) = sin(- 6) 4- cos (- 6) 
 ~ - sm6 -f cos# = - tan0. The cotangent is also an odd function. 
 The secant and versed sine are even functions; the cosecant 
 is odd ; the coversed sine is neither The terms even and odd, as 
 applied to functions in general, are suggested by the properties 
 of the even and odd powers. 
 
 If, and the length of the revolving line being given, we 
 form a new angle thus, one or more right angles 0, it will 
 readily be seen that the right-angled triangle made by the re- 
 volving line is in all cases the same in form and magnitude. 
 But two variations of position occur; sometimes the projections 
 differ in sign from those of the original triangle ; sometimes 
 they change name, the line which was x becoming y, and vice 
 versa. An examination of all the cases will present the following 
 table : 
 
 Angle. Absc. Ordin. Conclusions. 
 
 6 x y 
 
 \TT~Q y x cos(|7r-e) = sin0, sinQ7r-0) = cos0, tan(j7r-0) = cote, 
 
 ITT + O -y x COS^TT -f 6) =-sin0, sin( J?r+0) = COS0, tan(i-7r + 0)=-cot0, 
 
 7T-6 -x y cos(?r-e) =-cos0, sin(?r- 0) =sin0, tan(7r-0) =-tanf>, 
 
 TT + 6 -x -ycos(7r + 0) =-cos0, sin(7r+0) =-sin0, tan (TT~ 0) = tantf, 
 
 '-TT-e -y -x cos(fTT-e) =-sine, sin(2-7r-e)=-cose, tan(-}7r-0) = cote, 
 
 \ir-\ti y -x cos (|TT + 0) = sin0, sm(f/r-i 6)=-costf, tan(f7r-fe)=-cotf, 
 
 ITT - x -y cos(2;r-0)=cos0, sin('27r-6) =- sin0, tan('>7r-0)='- tan*'. 
 
 These transformations, it must be observed, apply to all values 
 of 0. For instance, let 6 lie between Zv and 2?r; then \TT - 
 lies between - TT and - - -. Draw the figure accordingly, and 
 it will appear that the x of either is the y of the other, loth in 
 sign and magnitude. The formulae therefore are universally true :
 
 AND ON FORMULAE OF ONE ANGLE. 15 
 
 but they may be best remembered by the supposition that 9 is 
 a small angle, so that \ir-Q is in the first right angle, \TT + Q 
 
 in the second, as also TT - 9, and so on. All the cases may 
 now be contained in the following rule. According as the num- 
 ber of right angles is even or odd, let the function remain, or 
 let it be changed into its co-function (sine and cosine, &c. are 
 co-functions). Then prefix the sign which the given function 
 has when 6 is less than a right angle ; and lastly, write for 
 the angle. For example, let it be required to simplify tan(f ?r + 6). 
 There is an odd number of right angles; -|TT + 9 is in the fourth 
 right angle, when 6 <|TT: in the fourth right angle the tangent 
 is negative: accordingly, tan(f TT + 0) = - cotd. But in trans- 
 forming cot(7r - 6), we see an even number of right angles, and 
 an angle in the second right angle; accordingly, cot(7r-0) = -cot#. 
 
 The following cases are so important that they should be 
 remembered apart : 
 
 The functions of complements are co-functions, 
 
 sm(\7r - 0) = cos0, cos (TT - 6) = sin 0, 
 Supplements have the same sine, sin(7r - 6) = smO. 
 Opponents have the same tangent, tan(w + 6) = tan0. 
 Completions have the same cosine, cos(27r - 0) = cos0, 
 COS(|TT + 6) = - sin0, COS(TT - 0) = - COS0, 
 sin(|7r + <?) = cos0, tan(?r - 0) = - tan#. 
 
 All the angles which have the same sine as 6 are included in 
 the formulae 2>mr + 0, and (2? + 1) TT - #: all which have the
 
 16 ON THE TRIGONOMETRICAL FUNCTIONS 
 
 same cosine as 9 in the formulae Irmr -t- 6 and 2imr -6: all which 
 have the same tangent as 6 in imr + 0: m being any integer, posi- 
 tive or negative. 
 
 To one sine there is but one cosecant; to one cosine there 
 is but one secant; to one tangent there is but one cotangent; 
 and vice versa. But in every other case a function has two 
 functions of every other kind attached to it, with opposite signs. 
 This appears, firstly, from what precedes : any sine, for instance, 
 belongs to two angles, supplements, which have cosines, &c. 
 opposite in sign. Supplements, opponents, and completions have 
 their functions of equal value, and opposite signs, except in the 
 three cases noted above. Thus, 
 
 sin(7r + 0) = - sin6>, tan (277- - 0) = - ta.nO, &c. 
 
 Secondly, from the equations in page 12, in which it appears 
 that every determination of one function in terms of one other 
 function requires an extraction of the square root, except when 
 the functions are reciprocals. Thus, 
 
 cosfl = + V(l - sin ! 0), tan0 = + </(!- cos 2 0) f cosfl, &c. 
 
 When one function is found, all the others can be found. 
 And by very ordinary geometry we can contrive to express the 
 functions of 15, 18, 30, 45, 60, 72, 75. Of these I shall 
 deduce some, and arrange the whole in a table, of which I 
 leave the student to fill up the demonstration. 
 
 (45). This angle has equal projections, or tan 45 = 1. 
 
 (30, 60). If a, a, c be the sides of an isosceles triangle, 
 and 20 its vertical angle, then c = 2asinO. Let the triangle be 
 equilateral ; then a - 2a sin 30 ; or sin 30 = |. And this is cos 60. 
 
 (18, 72). By Euc. IV. 10, it appears that an isosceles triangle 
 having 36 for its vertical angle has for its base the greater 
 segment of the side, as determined in II. 11. If then a be the 
 side, and c the base, we have a(a - c) = c 2 , or 2c = (V5-l)o. 
 Hence, sin 18 (or cos 72) = J (V5 - 1). 
 
 (15, 75). Take a right-angled triangle having an angle of 30, 
 an hypothenuse 2, and therefore 1 for the side opposite and <J3 
 for the side adjacent to that angle. Bisect the angle of 30; the 
 bisecting line divides 1 into segments which are as ^3 to 2 : the 
 smaller segment is then V3 f (2 + V3), or y3(2 - ^3), or 2^3 - 3.
 
 AND ON FORMULAE OF ONE ANGLE. 
 
 17 
 
 The bisecting line is therefore 
 
 V(3 + 12 + 9 - 12-/3), or yi2.y(2 - y3) 
 
 o /Q ys-i 
 
 MO . / X v 
 
 
 y2 
 
 Hence, sin 15 = 
 
 y3(2 - ys) ys - 1 ye - y2 
 
 
 V3(ye-y2) 2y2 4 
 
 
 
 sine 
 
 cosine 
 
 tangent 
 
 cotangent 
 
 
 
 A*" 
 
 15 
 
 ye-y2 
 
 ye + \/2 
 
 2 ys 
 
 2 + ys 
 
 75 
 
 i 
 12 
 
 4 
 
 4 
 
 AT 
 
 18 
 
 ys-i 
 
 y(io+2ys) 
 
 y5~ i 
 
 y(io+2V5) 
 
 72 
 
 f 
 
 4 
 
 4 
 
 V(10 + 2V5) 
 
 ys- i 
 
 6T 
 
 30 
 
 2 
 
 iys 
 
 i^3 
 
 ys 
 
 60 
 
 JL 
 3 
 
 ATT 
 
 45 
 
 IV 2 
 
 ^y2 
 
 1 
 
 i 
 
 45 
 
 1 
 
 
 
 cosine 
 
 sine 
 
 cotangent 
 
 tangent 
 
 
 
 Let Q be arcually measured, and let it be the 2w th part of 
 four right angles. Describe a circle with the revolving line r for 
 its radius, and inscribe a regular polygon of n sides. One of 
 these sides is 2rsin0, and the whole circumference of the polygon 
 is 2wsin0; that of the circle is 2irr, or 2n0,r. The ratio of 
 
 these circumferences is ; and this ratio, the larger n is made, 
 
 
 or the smaller 0, the more near it is to unity (page 4). That is, 
 when diminishes without limit, the fraction sin# -f approaches 
 without limit to unity. The approach is distinctly seen, even 
 when the angle is far from very small, to ordinary notions : 5, 
 in arcual units, is '0872665, and its sine is -0871557 : these two 
 differ by less than the 800th part of either. 
 
 Again, 
 
 tan# 
 
 sin0 l-cos0 
 
 e cos0 ' e 
 
 
 
 /sin 0v 
 ~ 
 
 the second of which is easily got from 1 - cos z # = sin*0. It follows 
 that the limit of tan 0-^-0, as diminishes without limit, is 1 x 1 
 or 1 ; while that of (1 - cos0) -h is x 1 or 0. As diminishes, 
 then, sinO and tan0 approach to 0, but 1 - cos0 diminishes much 
 more rapidly than 0. Any number being named, however great, 
 contains 1 - cos# more than that number of times, before 
 becomes nothing. When is -0872665 (5 of gradual measure), 
 contains 1 - cos<? more than 20 times. 
 
 C3
 
 18 ON THE TRIGONOMETRICAL FUNCTIONS 
 
 sin ad sinaO /30 a 
 
 It is important to observe that . nn , or . . . --, has 
 
 smpd ad smpO p 
 
 a . 
 
 - for its limit, when diminishes without limit. Also that n sin - 
 
 has the limit 0, when n increases without limit. 
 
 We have seen that tan# and sectf both become infinite when 
 = TT. Now 
 
 - cos<9 , , , ,, 
 
 sec0-tan0 = -- ^- = - . (=0, when = 
 cosO 
 
 consequently, as 6 approaches -|-7r, the difference of tan0 and 
 secO diminishes without limit. Shew in a similar way that 
 cosec0 - cot# diminishes without limit, with 9. 
 
 , , , , - . 
 
 Again, l-cos#=- - 2 (~ i i near ty> when 6 is small). 
 
 Hence, when 6 is small, cos# = 1 - \6* nearly. This, and sin0 = 0, 
 are equations which are near enough to truth for most purposes 
 of calculation, when is small. 
 
 I now give an account of the method of defining the tri- 
 gonometrical terms which was,* until very lately, universal. 
 
 A given straight line, called the radius, revolves from a starting - 
 line OA, as in our definitions ; but it must be of the same length 
 for all angles, which need not be the case in ours. The arc de- 
 scribed by the revolving extremity generally (though not always) 
 takes the place of the angle, f Then BM was called the sine of the 
 arc AB (sinus, bosom, the literal translation of an Arabic word: 
 if BAB ' represent a bow (arcus), half of the string BB" comes 
 against the breast of the archer). And OM is the cosine of AB : 
 this word is an abbreviation of sine of the complement, or com- 
 plemented sine ; it was long before OM was considered as any- 
 thing but the sine of another arc, BA. And AM (once the 
 sagitta, as occupying the place of the arrow) was the versed sine 
 (or turned sine) of the arc AB. And A'L should have been 
 called the coversed-sine, as being the versed sine of the comple- 
 ment : but this term is only a recent invention for the completion 
 
 * But not from all time ; for Rheticus, who gave the first 
 complete trigonometrical table, and invented the secant and cosecant 
 to complete it, used the method of ratios. 
 
 t By constant attention to the arc of a circle, some writers 
 have become unable to think of angle as a magnitude.
 
 AND ON FORMULAE OF ONE ANGLE. 19 
 
 of the system. Draw a tangent at A, and at A \ and produce 
 
 V' A' 
 
 OB to meet them in T and V. Then A T was called the tangent 
 (as being drawn on the tangent) of AS; and A'V, the tangent 
 of the complement, was called the cotangent of AS. Lastly, 
 OT was called the secant of AS, as being on a line which cuts 
 the circle ; and O V the cosecant. 
 
 All these definitions are thus connected with ours : the old 
 linear function, divided by the radius, in every case gives the 
 modern numerical function. 
 
 Denote the linear function by the word commencing with a 
 capital letter; and let OS = r, L SO A = 6. Then we have 
 
 MB 
 OS 
 
 Sin A B 
 
 AT Tan AS 
 
 cos<? = 
 
 OM Cos AS 
 ~OB~ ~~r ; 
 
 OT Sec AS 
 
 cot = 
 
 COSCC0 = 
 
 OM AV Cot AS 
 ~MB~ ~OA~ ~~r ; 
 
 OS OV CosecAB 
 
 vers = 1- 
 
 MS OA' 
 
 _ 
 
 Covers0 = l - 
 
 OM 
 ~OB 
 
 MB 
 'OB 
 
 AM Vers AH 
 ~OB~ ~7 5 
 
 AL 
 ~OB 
 
 Covers AS 
 
 Speaking but of ultimate calculation, the old system is iden- 
 tical with the new one, if we only make r= 1, or take the linear 
 unit for a radius. But there always remains this essential dis- 
 tinction, that the function of the old system is always a line, 
 that of the new one a number. In the old system the sine of 
 the arc of 30 is half a radius, whether that radius be used as
 
 20 ON THE TRIGONOMETRICAL FUNCTIONS, &c. 
 
 the measuring unit* or not. In the modern system, the sine of 
 the angle of 30 is the fraction which half a radius is of the 
 whole radius. 
 
 If we substitute in our formulae the equivalents of the old 
 system, we have such equations as 
 
 Sin AB Cosec AB 
 
 r 
 Cos*AB 
 
 = 1, or Sin AB . Cosec AB = r 2 , 
 
 = 1, or Cos* AB + Sin* AB = r\ &c. 
 
 5 - -r 
 
 r 2 r* 
 
 If it be occasionally desirable to refer to the old system, it 
 may be done without confusion by speaking of the sines, &c. 
 of arcs or the linear sines, &c. of angles. 
 
 The area of the circle is thus found. Inscribing the polygon 
 
 of n sides, and 9 being the 2n th part of a revolution, or - , we 
 have for each of the n triangles, the area |r cos# . 2r smO, and 
 
 7T 7T 
 
 for the whole polygon nr* cos# sin#, or r 2 cos - . n sin - . When 
 
 n n 
 
 n increases without limit, the limit of this is wr 8 , which is the 
 area of the whole circle. The sector which has the angle 0, 
 
 Or* 
 being to the whole circle as Q to 2?r, is . 
 
 * Trigonometry might be defined as that part of the application 
 of Algebra to Geometry which is independent of linear measure; 
 since ratios are independent of the units in which their terms are 
 arithmetically expressed. One disadvantage of the old system is, 
 that it keeps this independence of linear measure out of view.
 
 21 
 
 CHAPTER IIT. 
 
 FORMULAE WHICH INVOLVE TWO OR MORE ANGLES. 
 
 IT will be desirable to gain a more extended notion of the 
 projections of a line. If any line, AS, be taken as belonging 
 to an indefinite line on which sign is recognized, a careful dis- 
 tinction must be drawn between AB and SA : one is positive 
 and the other is negative. Thus, p. 7, NO is not x, but ON, 
 which is there positive, while NO is negative. If we attend 
 to this, we shall find that, however A, B, C, may be distributed 
 on a straight line, 
 
 A C= AB BC= AB-CB = BC- BA; AB + BC+ CA=0; 
 
 Thus AC =4-2, AB = + 1, C=-5, and + 2 = + 7 + (- 5). 
 
 Next, the angle made by P with Q is to be carefully dis- 
 tinguished from the angle made by Q with P. If one be the 
 other is - 0, or, at our pleasure, 2?r - 6. To gain fixed ideas, 
 let us suppose that in the angle made by P with Q, denoted 
 by P A Q, we proceed from the positive direction of Q as a 
 starting-line, and thence by revolution, positive or negative ac- 
 cording as we want a positive or negative angle, to the positive 
 direction of P. But in Q A P, we proceed from the positive di- 
 rection of P to that of Q. Ascertain from this that 
 
 P A Q + Q A P is ITT, 0, or - 2?r, 
 
 according as we 6 take the positive angles in both cases, or one 
 positive and one negative, or both negative. 
 
 When neither end of a line is at the origin, the projections 
 are determined by drawing perpendiculars from both ends of
 
 22 
 
 FORMULAE WHICH INVOLVE 
 
 the line. Thus AA' has NN' and MM' for projections: but 
 
 N 
 
 I - 
 
 A A has N'N and M'M. 
 
 The projections of the line r, making the angle 6 with the 
 axis of x, are always r cos0 and r sin#. Take the preceding 
 figure, and first let AA be -f. If be the angle it makes with 
 the axis of x, that angle belongs to the -f- - quarter : r sin# is, 
 as to sign, + x -f, and is positive; and so is MM'. But r cos0 
 is + x -, or negative ; and so is NN'. But if A A' be negative, 
 the angle (the opposite side being now used, .as in p. 8) is 
 of the - -f quarter; and r sin<? is - x -, or +, as before; while 
 r cos0 is - x +, or -, as before. 
 
 In the language of Euclid, equal and parallel lines have equal 
 projections. But we must say, equal and parallel lines, estimated 
 in tJie same directions, have equal projections. Thus AA and SB 
 have equal projections ; and so have A A and B'B : but A A and 
 BB' have only projections equal in length, and differing in sign. 
 
 If any points, as A, B, C, D, be taken, the projection of AD 
 is the algebraical sum of those of AB, BC, and CD. 
 
 These projections, taking the axis of x, are PS, PQ, QR, RS 
 and, by what precedes, 
 PS=PR+RS=PQ+QRiRS. 
 
 The only question now is this, 
 do PS, PQ, QR, RS, always 
 represent the projections, in 
 what manner soever the lines 
 AD, &c. take signs ? And the 
 answer is in the affirmative,
 
 TWO OR MORE ANGLES. 
 
 23 
 
 since we have seen that any variation of sign is accompanied 
 by a compensating variation in the mode of estimating the 
 angle; so that the projection remains unaltered both in sign and 
 magnitude, so long as the line remains unaltered in direction 
 and magnitude. 
 
 Now let the axes revolve through the angle 0, giving a pair 
 
 of secondary axes ; and let a revolving line, starting from the 
 positive side of the secondary axis x, revolve through a further 
 angle 6; having thus revolved through 0-i-O from the original 
 starting-line. In the diagram, is about 2k right angles, and 
 is nearly three right angles more ; or, if you please, a little 
 more than a right angle negatively. The projections of OP on 
 the primary axes are r cos (0 + 0) and r sin (0 + 9} ; on the se- 
 condary axes, r cos<? and rsin#; and these last projections, ON, 
 and NP or OH, make angles with the primary axis of x which, 
 estimated by our rules, are and + %TT ; for the revolving axis 
 of y is always a right angle in advance of the axis of x. If then 
 we project the secondary projections on the primary axes, we have 
 Projections of ON are rcos#.cos0 and Vcos0.sin0, 
 
 Projections of NP are r sinfl.cos (0 + ITT) and r sin 6. sin (0+2^). 
 
 Looking at the projections on the primary axis of x, we have 
 Projection of OP = Projection of ON + Projection of NP, 
 rcos(0 -l- 0) = rcos0cos0 + 7-sin0.cos(0 + -i?r), 
 
 cos(0 + 0) = cos0cos0 + sin#{cos(0f |-7r)or-sin0}, 
 
 cos(0 t 6) = cos0 cos# - sin
 
 24 FORMULAE WHICH INVOLVE 
 
 Looking at the projections on the primary axis of y, we have 
 Projection of OP = Projection of ON + Projection of NP, 
 r sin(0 f 0) = r cos& sin0 + r s'mO sin(0 -f \ 77), 
 
 sin(0 + 0) = cos sin + sin {sin (0 + |TT) or cos0}, 
 
 sin(0 + #) = sin0cos0 + cos0sin#. 
 
 These formulae being universally true, we might write - 
 instead of 0, and then we have 
 
 cos (0 - 0) = cos0 cos (- 0) - sin0 sin (- 0), 
 
 = cos0cos# + sin0sin#, 
 sin(0 - 6) = sin0cos(- 0) + cos0sin(- 0), 
 
 - sin cos 9 - cos sin 0. 
 
 This foundation of all the ulterior part of trigonometry, may 
 be stated thus, 
 
 cos(0 6) = COS0COS0 + sin0sin0, 
 sin(0 + 0) = sin0cos# cos0 sin0. 
 
 The formulae are not independent: but any one really con- 
 tains all. This has partially appeared. To shew it completely, 
 observe that the operations connected with projection on the 
 axis of y are precisely the same as those connected with the 
 axis of x. If we adopt the axis of y as a starting-line, and 
 preserve the positive direction of revolution unaltered, we may 
 reckon angles from the axis of y, and use cosines in determining 
 the projections, provided that every line which makes an angle 
 /< with the axis of x, be considered as making /< - i TT, or /( + | ir, 
 whichever we please, with the axis of y. If then we want to 
 apply the formula 
 
 cos(0 -f 0) = cos cos 6 - sin sin 0, 
 
 to angles measured from y, we must alter 0, which is measured 
 from x, into - i 77. This gives 
 
 or sin(0 -t 6) = sin0cos# + cos0 sin0. 
 
 The demonstration above given is universal : but it can ( nly 
 be convincing to those who enable themselves to understand, 
 :n the most general s-ense, the preliminary theorems. Any want
 
 TWO OR MORE ANGLES. 
 
 25 
 
 of such mastery over the universal character of theorems in 
 projection will follow the student through all his course, par- 
 ticularly in the higher geometry and in mechanics. To break 
 the difficulty, it may be worth while to examine demonstrations 
 of particular cases, so as to show what manner of arithmetically 
 separate operations are algebraically presented in one by the 
 preceding process. 
 
 [Within these brackets, lines are not affected by any but 
 specified sign : thus A B when negative is written - AS : and 
 no distinction is made between AS and SA. 
 
 Let AOP = 0, POQ = 0, both taken in the positive direction 
 of revolution. In the second diagram is nearly two, and 
 nearly three, right angles. Project OQ on OA and OP, &c. into 
 OM, MQ and ON, NQ; project ON, NQ into OS, SN, and 
 NR, RQ,. In the first diagram, in which and 6 have positive 
 sines and cosines (and RQN^tfi), we have 
 OQ cos(0 + 0) = OM = OS - SM = OS - UN, 
 
 = ONcos(fr-QNsm<p=OQ cos# cos0 -OQ sin# sin0, 
 cos(0 + 0) = cos0 cos6> - sin0 sin6>. 
 Also OQ sin(0 + 0) = QM = RM + QR = NS + QR, 
 
 = O^ r sin0+Q^Vcos0=OQ cos0 sin0+OQ sin# cos0, 
 sin(0 -I- 6) - sin0 cos0 + cos0 sin#. 
 
 In the second diagram, RQN is not 0, but TT - 0. And first, 
 OQcos(0 + 0-27r)= OM= OS+ SM = OS + JtN 
 
 = Q cos QONco sNOS+ OQ sin QONsmNQR; 
 cos(0 + 6 - 27r) = cos QOA r cos NOS + sin 
 
 D
 
 26 FORMULA WHICH INVOLVE 
 
 cos Q ON = cos (d - TT) = cos (TT - 0) = - cos 6 
 sin QON= sin (0 - TT) = - sin (TT - 0) = - sin0; 
 cos NOS = cos (TT - 0) = - eos0 ; sin NQR = sin (TT - 0) = sin 0. 
 Whence cos (0 + - 2?r) = (- cos#) (- cos0) + (- sin#) (sin0) ; 
 
 or cos (0 + #) = cos0 cos0 - sin0 sin0. 
 Again, 
 
 OQ sin (040-27r)= QM = QR - RM= QR - NS 
 = QN cos NQR - NO sin NOS 
 = OQ sin QONcos NQR-OQ cos QO^.sin NOS, 
 sin (0 + - 2?r) = sin Q O^V cos NQ R - cos Q ON . sin NOS 
 = (- sin 0) cos (TT - 0) - (- cos 6) sin (TT - 0), 
 or sin (0 + 0) = sin 6 cos + cos sin 0. 
 The student should repeat the same process on various cases.] 
 
 Observe that a complete proof of the cases of cos (0 -f 9} and 
 sin (0 + 6) is also one of cos (0 - 6) and sin (0 - 0), independently 
 of the substitution of - for 6. For cos (0 - 0) is cos (0 + 2?r - 0) 
 or cos0 cos (277- - 0)-sin0 sin (27T-6) or cos cos + sin sin 0. 
 And similarly for sin (0 - 0). 
 
 From the table in p. 17, verify the first row by aid of the 
 third and fourth : find the sines and cosines of 3, 12, 27, 33, 
 48, 63. From these the sines and cosines of all the multiples 
 of 3 may be easily expressed. 
 
 The only form of the preceding theorems which occurs among 
 the fundamental equations is 
 
 cos (0 - 6) = cosO . cos 6? + sin6> . sin$ or 1 = cos"9 + sin*#. 
 A large collection of formulae may be deduced, as follows : 
 
 1. cos (0 + 6) = cos0 cos0 - sin0 sin0. 
 
 2. cos (0 - 0) = cos0 cos# + sin0 sin0. 
 
 3. sin (0 -f 6) = sin0 cosO + cos0 sinO. 
 
 4. sin (0 - 6) = sin0 co&O - cos0 sin0. 
 
 5. cos (0 - 0) -f cos (0 + 0) = 2 cos0 cos0. 
 
 6. cos (0 - 0) - cos (0 + 6) = 2 sin0 sin0. 
 
 7. sin (0 + 6) + sin (0 - 6) = 2 sin0 cos0. 
 
 8. sin (0 + 0) - sin (0 - 0) = 2 cos0 sin6>.
 
 TWO OR MORE ANGLES. 27 
 
 For write , and for 6 write ^-75 ; 
 
 2t 2t 
 
 6+0 0-0 
 
 9. cost/ + cos0 = 2 cos J! - . cos s-g . 
 2 2 
 
 , . + .0-0 
 
 10. cos0 - co30 = 2 sin - . sin *- - . 
 2 2 
 
 11. sin0 + sin^ = 2 sin 
 
 12. sin0 - sin0 = 2 cos 
 
 13 si "0 ~ sin ^ tan |(0 - 0) sin0 + sin6> + 
 
 sin0 + sine tan (0 + 0) ' cos 0+ cos " :an ~T~ 
 
 14. ton0 + tan* = 5?-l^, tan - 
 ' 
 
 cos0cose' 
 - tan0 
 
 cos0cose' 
 
 si 
 sin 
 
 divide numerator and denominator by cos0 cos0, and 
 
 15. tan (0 0) = sin (^ A g ) = sin0 cos0 cos0 sin0 ^ 
 cos (0 0) ~ cos0 cos0 + sm0 sin0 ' 
 
 which also follow immediately from 14. 
 
 16. sin 20 = 2 sin0 cos 0, sin6> = 2 sin - cos - . 
 
 2 2 
 
 17. cos 20 = cos 2 - sin ! = 2 cos 2 -1 = 1-2 sin*0. 
 
 18. cos z = i + f cos 20, sin 2 e = i _ i cos 20. 
 
 19. l+cos0=2cos 2 ^, l-co S = 2sin s -, 1 " cosg a tan' - 
 
 214- cos0 2 ' 
 
 2 tan0 
 
 20. tan 20 = - 5- . 
 
 1 - tan 2 
 
 91 l-sin0 ITT 0\ 
 
 31. :: - 77. = tan 8 1 --- I. 
 1 + sm0 ^4 2/ 
 
 22. 
 
 4/ tan0
 
 FORMULAS WHICH INVOLVE 
 
 The following remarks may be made on these formulae. 
 5, 6, 7, 8. Remember these formulae thus : 
 product of cosines = half cosine of difference + half cosine of sum, 
 product of sines = half cosine of difference - half cosine of sum, 
 sin greater x cos less = half sine of sum + half sine of difference, 
 sin less x cos greater = half sine of sum - half sine of difference. 
 
 The universal formulae are here expressed (the two last, at 
 least) with some arithmetical limitation; by which the one most 
 convenient for arithmetical operation may be selected. Thus at 
 once we learn to write down 
 
 sin5 cos 18 = sin 23 -f sin 13, sin50 cos4 =sin54 + ism46 . 
 
 We have thus convenient substitutes for multiplication of sines 
 and cosines by one another; of which much use was made before 
 the invention of logarithms : We can also resolve any product 
 of sines and cosines. Thus 
 
 cos a sin b sin c = cos a { cos (b - c) - \ cos (b + c)} 
 = {cos(6 - c - a) + cos(6 - c + a)} - $ {cos (b + c - a) + cos (b + c + a)} 
 = J {cos (6 - c - a) + cos (b - c -f a) - cos (b -f c - a) - cos (6 + c 4- a)}. 
 Or thus : cos a sin b sin c = {\ sin (b - a) + J sin (b + a)} sin c 
 = i (2 cos (b-a-c)-% cos (6-a+c)} + { cos (6+a-c) - | cos (6+a-f c)}, 
 the same as before. 
 
 9, 10, 11, 12. Remember these formulae thus: 
 Sum of sines = twice sine of half sum x cosine of half difference. 
 Difference of sines=twice cosine of half sumxsine of half direct* diff. 
 Sum of cosines = twice cosine of half sum x cosine of half diff. 
 Difference of cosines=twice sine of half sum x sine of half inverted diff. 
 
 Most write the formula 10 as 
 
 . + . 0-0 
 cos0 - cos0 = - 2 sin *-= sin ^-r . 
 & 2> 
 
 But whichever way it is written, no one will ever be expert in 
 the use of trigonometrical formulae until cos (a - b) and cos (6 - a) 
 prevent the instantaneous notion of perfect identity of value and 
 sign : while sin (a - b) and sin (b - a) equally suggest sameness 
 of value with difference of sign. Again, it is frequently desirable, 
 
 * Direct, read in the order of reference ; inverted, read in the 
 contrary order. When </>, 0, are mentioned in that order, <f> 6 
 is the direct difference, 6 tf> the inverted difference.
 
 TWO OR MORE ANGLES. 29 
 
 after observing the effect of an interchange upon one side of 
 an equation, to verify the sameness of the effect on the other 
 side. Thus in 9, interchange of and produce no alteration 
 in the first side: how is it seen that no alteration is produced 
 on the second side ? By remembering that cos |- (9 - 0) and 
 cos i (0 - 9) are the same. Again, interchange of and changes 
 the sign of the first side of 10; and of the second also, since 
 sin (9 - 0) and sin f (0 - 0) have different signs. A person tho- 
 roughly practised in these considerations remembers the general 
 character of the formulae 9-12, and makes the details correct 
 by the habit of satisfying the above conditions. 
 
 15. Two angles differ by a right angle ; how are their tangents 
 related ? If = + TT, tan0 = - cot0, or 1 + tan0 tan# = 0. This 
 result, which is often wanted, is best remembered by the de- 
 nominator in 15 : if tan (0 - 9) be infinite, we must have 
 
 1 + tan0 tan0 = 0. 
 Prove the following formula : 
 
 tan(0| YT | )x tan0 + tan VT + tang- tan0. tan yr. 
 
 1 -tan0 tan ty -tan YT tan# - tan# tan0 ' 
 
 from which it follows that the sum of the tangents of the three 
 angles of a triangle is equal to their product. Also the following : 
 If j be the sum of the tangents of a set of angles, t v t 3 , &c. 
 the sums of the products of every two, every three, &c.; then 
 the tangent of the sum of those angles is ^ - t 3 + t b - ... divided 
 by 1 -t t + t t - ... . This may best be proved by showing that if 
 it be true for any number of angles, it remains true when one 
 more angle is introduced. 
 
 If there be any number of angles, and if S a be the product 
 of all their cosines, and S, t the sum of all the products which 
 have for factors the sines of n of them and the cosines of all 
 the rest ; then the sine of the sum of those angles is S l -S 3 -\-S.,-... 
 and the cosine of the sum is S - S 3 +S^~ ... . 
 
 Suppose this proposition true for any one number of angles, 
 and" let S a , S v &c. have the above meaning. Introduce one more 
 angle, having a and b for its cosine and sine, and let T be 
 now the product of all the cosines, and T n the sum of the products 
 in which n are sines and the rest cosines. Now it is clear that 
 
 D3
 
 30 FORMULAE WHICH INVOLVE 
 
 T is S a. Next, 7\ consists, first, of all the terms which compose 
 S , each multiplied by b, and of those of S r each multiplied by a ; 
 whence 2\ = S b + S^. And T 2 has all the terms in S l each 
 multiplied by b, and all those in S t each multiplied by a ; whence 
 T z = Sfi + S a a. And thus we show that T m = S,,^ + S m a. But if 
 there be k angles in the first set, T^ +l is SJ), and S^ does not 
 exist. But the law of connexion T i+l - S t b + oS^a still exists if 
 we suppose S^ = 0. 
 
 Now if the cosine and sine of the sum of the k angles be 
 S -S t + S t - ... and S l -S 3 + S b - ..., then, after introduction of the 
 new angle, the cosine and sine of the sum of the k + 1 angles are 
 
 (S -S 2 + S t - ...)a- (S, -S 3 S b - ...) 6 or T - T 8 + T 4 - ..., 
 (S,- S, + S s -...)a + (S -#.+ $-) & or Z\ - T 3 + T t - .... 
 If then the theorem be true for k angles, it is true for k- 1 . 
 But it is true for two angles ; for, and 6 being those angles, 
 S is cos0 cos0, and S l is sin0 cos0 + cos0 sintf, and S t is 
 sin0 sinO, S 3 is 0, S t is 0, &c. And cos (0 + 0) is S - S a + S t - ..., 
 while sin (0 + 0) is S l - S 3 + S b - ... . Hence the theorem is true 
 for three angles, hence for four, &c. The beginner had better 
 proceed in one or two cases thus : 
 
 cos (0 + I/' + 0) 
 
 = cos (0 + i/<-) cos# - sin (0 + ^) sin# 
 
 = (cos0 cosi^-sin0 sim/^) cos#-(sin0 cos^ + cos0 sini/r) sin^ 
 =cos0 COST/*- cosO- (sin0 sini/r cos^4sin0 sin# cosY^sin^ sinO cos0) 
 -V% + (*4-0)-(flr.-0)+.... 
 
 If there be n angles, the number of products having m srnes 
 is the number of distinct ways in which we can select m out 
 of the angles, or the number of combinations of m out of : 
 denote this by m n ; accordingly 
 
 - 1 n - 2 - + ! 
 
 m n stands tor n ... . 
 
 2 3 m 
 
 If all the angles be equal, and each of them be 0, each term of S m 
 is c""'"8 m , where c means cos# and s means sin#. Accordingly, 
 S m becomes m,,c"""'s m , and we now have 
 
 cos nO = c" - 2, t c n ^s s + 4 fl c*V - 6,,c"V + ... , 
 
 sin n0 = l n c"-'s - 3 n c"- 3 s 3 + 5 n c n ''^ - 7 n c'^s 7 + ... .
 
 TWO OR MORE ANGLES. 31 
 
 Now the development of (c + s)" is c" + l n c n "' s + 2,,c n ~V + ... ; 
 whence the following theorem : Develope (c + s)" by the binomial 
 theorem, and put together the odd terms, 1st, 3rd, 5th, &c., and 
 the even terms, 2nd, 4th, 6th, &c.; change the alternate signs 
 in each lot, and the results are cos nO and sin nO. Thus we 
 may at once write down 
 
 cos 20 = c* - s", sin 20 = 2cs, 
 
 cos 30 = c 3 - 3cs a , sin 30 = 3c s s - s 3 , 
 
 cos 46 = c 4 - 6cV -f s 4 , sin 40 = 4c 3 s - 4cs 3 . 
 
 The beginner should form some of these successively ; thus 
 sin (30) = sin (16 + 6) = sin 20 . c + cos 20 . s 
 
 = 2cs . c -f (c 2 - s 8 ) s = 3c 2 s - s" 
 sin (40) = sin (30 + 0) = sin 30 . c + cos 30 . s 
 
 = (3c ! s - s 3 ) c + (c 3 - 3cs 8 ) s 
 = 4c s s - 4cs 3 , and so on. 
 
 The question of finding the sine or cosine of the n^ part of 
 an angle is now reduced to that of solving an equation of the th 
 degree. For example, given the sine of an angle, b, it is required 
 to find the sine of its third part. Here 
 
 6=3(1 - x*) x - z? = 3x - 4x s , 
 
 x being the sine of the third part. Hence x is to be found from 
 4a; 3 - 3x + 6 = 0. 
 
 For example, if the angle be 30, we have to solve 
 
 8x 3 - 6x + 1 = 0, 
 
 which, by Horner's method, has '173648177867 for one of its roots, 
 approximately; and this root is sin 10. 
 
 There are three roots to this equation, all real : but three 
 distinct problems are attempted, all soluble. For what we really 
 ask, in the equation, is the sine of the third part of the angle 
 whose sine is |. This last angle may be either 30, 360 + 30, 
 
 2 x 360 + 30, 3 x 360 + 30, 4 x 360 + 30, &c., or 180 - 30, 
 
 3 x 180 - 30, 5 x 180 - 30, &c. Look among the thirds of all 
 these angles, and we find three angles having distinct sines, 10, 
 130, 250; or 10, 50, 250. And the three values of x are the 
 sines of these three angles.
 
 32 FORMULAE WHICH INVOLVE 
 
 From the preceding theorem we can, and with tolerable ease, 
 exhibit the algebraical series which cos0 and sin are equivalent 
 
 (x\ n 
 cos -j 
 
 neither diminishes nor increases without limit when n increases 
 without limit. Of this, a priori, we must be uncertain, for as 
 
 X 
 
 n increases, cos - increases towards unity, while the increase 
 n 
 
 of the exponent has a diminishing effect. Between the increase 
 
 (y. \ 2" 
 cos j , 
 
 / x\ n 
 
 for instance, is greater or less than I cos -I . But, taking n 
 
 \ n l 
 
 X 
 
 so great to begin with as that - shall be between - |TT and 
 
 + ^TT, we easily see by our formulaj that the duplication of n 
 effects an increase. For 
 
 xxx 
 1 -l- cos - cos - + cos - 
 , x n n n x 
 
 cos* - = > > cos - ; 
 
 2n 2 2 n 
 
 I x^" I *\" 
 
 or cos > cos - . 
 
 \ 2n/ \ ml 
 
 ( x 2 / x 4 
 cos- j , (cos- ) , 
 
 &c. is a succession of increasing terms, of which no one exceeds 
 
 unity: for (cos -) cannot exceed unity, unless cos ^ could be 
 \ / n 
 
 greater than unity. Accordingly, the preceding terms severally 
 approach to some limit: let it be L. 
 
 Now take the term which may represent any one of the 
 terms already found in cosnd and sinnO; namely, 
 
 n , . factors)... (cos 0)"- m (sin0) m . 
 
 2 o m 
 
 Let nO = z, a fixed angle : but nevertheless n may be as great 
 as we please, provided 6 be taken = 2 -f n. And as n increases 
 without limit, diminishes without limit. Now take the term 
 preceding, divide it by (cos#)", and at the same time multiply 
 and divide it by 0, m times. It then becomes 
 
 g - 9 nO-0 n9- (m + 1)0 J_ (sin^)" 1 
 ~~ ~ '" " "" ' '
 
 TWO OR MORE ANGLES. 33 
 
 6 z-20 z - (m + 1) 9 /tan6>\ m 
 
 ' 3 ...... 
 
 When Q diminishes without limit, this, for every specific 
 
 22 2 z"* 
 
 value of m, approaches without limit to a. - .-... . l m , or ^ - . 
 
 2 o >n 2,o.*.ni 
 
 Next, after dividing both equations in page 30 by c" or (cos -] , 
 
 perform the preceding compensatory operations on the several 
 terms, and equate the limits of the sides of the equations 
 (Algebra, page 157). We have then, 
 
 cosz z 8 z 4 sinz z 3 z 5 
 
 L 2 2.3.4 L 2.3 2.3.4.5 
 
 These series will be found to be convergent (Algebra, page 186); 
 and these equations themselves determine L. For if we make 
 z = 0, the first gives cosO = L, or L = 1 : if we divide both sides 
 of the second by z, and diminish z without limit, remembering 
 that sinz f z has the limit 1, we also find L = 1. 
 
 Our results then are (z being an angle in arcual units), 
 
 z 8 z 4 z 6 z 8 
 
 cosz = 1 - 
 
 2 2.3.4 2.3.4.5.6 2.3.4.5.6.7.8 
 
 z* 
 
 sm z = z - - -f 
 
 2.3 2.3.4.5 2.3.4.5.6.7 2.3.4.5.6.7.8.9 
 in which we see verification of the preceding assertions that 
 cosz is an even function, and sinz an odd one that sin z = z 
 and cosz = 1 - z a , nearly, when z is small. 
 
 The readiest mode of calculation from these series is by 
 throwing them into the forms 
 
 z 2 
 
 r, 3 2 r, z 2 r, ^ r, z * r, 
 
 smz = 2 {l - - {l - - {l - ^ {l - - {l - ...... , 
 
 where { indicates that the preceding multiplier is a factor of 
 all that follows. 
 
 Thus, the calculation of cosl (or 5717'44"'8 in gradual units) 
 is obtained to twelve decimal places (see the property of alter- 
 nating series, Algebra, page 184) from
 
 34 FORMULAE WHICH INVOLVE 
 
 Turn 1 -f- 13.14 into a decimal fraction of 13 places, subtract 
 it from unity, divide by 11.12, subtract from unity, &c., keeping 
 13 places throughout: the final result may be depended on to 
 12 places. 
 
 Those who have mastery enough over algebraical division 
 to divide the series for the sine by that for the cosine, will find 
 
 z 3 2z 5 11 z 1 G2z 9 
 
 tans = z + 1- -f H + .... 
 
 3 15 315 2835 
 
 the law of the terms of which is too complicated for the beginner. 
 Every one, however, should verify on the series, cos s z + sin*z = 1, 
 cos's - sin*z = cos2z, 2 sinz cosz = sin2z. 
 
 I said (page 2) that we should soon make it very evident 
 that a purely algebraical basis might have been made for tri- 
 gonometry. If we had chosen to call the preceding functions 
 of z, namely 
 
 z* z 3 z 3 
 
 1 -2 + "" Z -2^ + "" Z+ 3 + -' 
 
 by the names of cosine, sine, and tangent of z, (and their reciprocals 
 secant, cosecant, and cotangent), we might have investigated the 
 properties of these series, and we should at last have arrived 
 at all our preceding formula? of connexion; but with much 
 more difficulty. 
 
 I now go to the converse problem, in which it is required to 
 express cos"0 and sin"0 by means of sines or cosines of 6, W, 30, &c. 
 
 First, let there be n angles a, b, c, d, &c., and proceed as in 
 page 28 with cos a cos b cose. Thus we have 
 
 cosa cos6 = |- cos(a - 6) + -| cos(a -f- b) 
 
 cosa cosft cose = \ cos (a - b - c) + J cos (a - 6 + e) -f cos (a + b - c) 
 -f J cos (a + b + e), &c. 
 
 The final divisor will be 2"~ 1 , the final number of cosines 2""'; and, 
 looking at the manner in which the angles enter, we shall see the 
 cosine of every choice out of + a b c d + ... In every term 
 change the sign of every letter, which will not alter the value of 
 any one cosine, add the results together and divide by 2, which 
 will leave the whole unaltered, and we shall then have 2" for 
 a divisor, 2" for the number of terms, and every variety of 
 abcd ... among the angles. If we now make a, b, c, &c.
 
 TWO OR MORE ANGLES. 35 
 
 all equal to one another and to 6, we shall be able to subdivide 
 all the choices furnished in 4 ... (n terms) into the 
 following. One case of nO, taking all +, and one case of - nd, 
 taking all - : ! cases of (n - 2) 6, with one only taken negative 
 (giving (n - 1) 6 - 6), and as many with - (n - 2) 0, taking one 
 only positive : 2,, cases of (n - 4) 0, taking two only -, and as many 
 of - (n - 4) 0, taking two only 4 ; and so on. But at the last step 
 there will be a separation between the cases of n even and n odd. If n 
 be even, say = 2k, there will be at last k lt cases of {(n - k) - k} 0, 
 or 00, taking k - and k + ; the case of k taken 4 and k taken - 
 not being distinct from the former. But if n be odd, say = 2k + 1, 
 then there are k n cases of {(n - k) - k} or 0, in which k are taken - ; 
 and as many of - 6, in which k are taken 4. Accordingly, cos a 
 cos b cos c... being now cos"0, we have 
 
 2"cos"0 = [cos04 cos(-0) 4 ! {cos(n- 2)04 cos - (-2)0) 
 4 2 n {cos (n - 4)0+ cos - (n - 4)0} 4 3 n {cos(n-6) 04cos-(n-6)0} 
 4 ... ending with k, t cos 09 if n = 2k, and with 
 
 n (cos04 cos -0) if n = 2k+l~\. 
 
 Collecting these, by help of cos (- a) = cos a, we have, for a final 
 
 form, 
 cosn0 = -^-A cosnO 4 n cos(w-2)0 + n - cos (w- 4)0 4 ... I 
 
 on the condition that cos 00, when it occurs, is only to take half 
 the coefficient indicated by the general law. 
 The beginner may proceed thus, 
 
 cos*0 = 4 \ cos 20 cos 3 = cos0 4 i(cos0 4 cos 30) 
 
 = i (cos 30 4 3cos0) 
 cos 4 = (cos 30 cos0 4 3 cos s 0) = | (cos 20 4 cos 404 3 4 3 cos 20) 
 
 = i (cos 40 4 4 cos 20 4 f cos 00), &c. 
 Now let be changed into ^7r-0, or cos"0 into sin"0. If we 
 
 examine cosfm --A\, we begin by rejecting all the fours out 
 \ 2 / 
 
 of m, as indicative of complete revolutions; and the final form 
 of this term depends on the remainder. Call 4k an even even- 
 number, as it is the 2& th even number, 4k 4 2 an odd even-number, 
 being the (2k 4 l) th ; 4&4 1 an odd odd-number, is it the (2k + l) th 
 odd number; and 4k 4 3 an even odd-number, being the
 
 36 FORMULAE WHICH INVOLVE, &c. 
 
 (2k + 2) th . Then, for even even-numbers, the above is + cos^t ; 
 for odd odd-numbers, + sin A ; for odd even-numbers, - cos A ; 
 for even odd-numbers, - sin A. And m - 2 is of the class of 
 even numbers, or of odd numbers, of which m is not, &c. We 
 have then the four following formulae: 
 
 n even even sin"# = j < cosnO - n cos(n- 2)0 + n cos(w-4)#~... > 
 
 1 f n 1 1 
 
 w odd odd sin"<? = -< smnd - n sin(n-2)6 + n - sm(n-4)6~... > 
 
 n odd even sin"# = - -1 cosnd - n cos(n-2)0 + n cos(n~4)6>-... I 
 
 2 \_ 2 J 
 
 even odd sin"0 = - - t < sin ??0 - n sin(n-2)0 + n sin(-4)0-... I 
 
 Of these the beginner should construct instances, as before. 
 He may also try to prove the following theorems : 
 
 S i n2 0- 2 = tan (45 +0)- tan (45 -0) 
 ~ tan (45 + 6) + tan (45 - 6) ' 
 
 tan<9 = cot0- 2cot2(?, 
 
 1-cos^ 6 1 + costf e 
 
 = tan - ; = cot - . 
 
 2 sin^ 2
 
 CHAPTER IV. 
 
 ON THE INVEKSE TRIGONOMETRICAL FUNCTIONS. 
 
 WE may now consider cosO, &c. as functions of 6, and accord- 
 ingly, itself as a function of cos#, or of sin#, &c. If x = cos#, 
 then 6 may be described as ' an angle whose cosine is x.' The 
 continental writers denote this by angle (cosine = x), but in our 
 country it is universally described by a symbol derived from a 
 functional analogy. If 0a: denote a function of x, then (0a?) 
 is denoted by 8 a:, 0(0 2 #) by 3 ar, and so on. On this notation 
 0~'a; should denote the function on which performance of gives x, 
 so that (0~'#) = x, and <p^x should denote that function which 
 gives (f?(<p*x) = x. Accordingly, cos a; being considered as a func- 
 tion of x, and the abbreviated word cos as a functional symbol, 
 cos -1 o: should denote the function which satisfies cos (cos"'a;) = x. 
 Hence, cos" 1 ^ must stand for the angle (meaning any angle) whose 
 cosine is x. Similarly, suT'a;, tan~'#, &c. stand severally for any 
 angle whose sine, tangent, &c. is x. 
 
 The objection to this analogy is, that we do not pursue it. 
 We do not employ sin ! a: for the sine* of the sine of x, but for 
 sin a; x sin a: or (sina-) 2 . The answer is, that we ought to follow 
 the analogy, and that we certainly should, if questions in which 
 the sine of the sine, or the sine of the sine of the sine, were 
 so frequently employed as to require abbreviation. And when 
 such questions actually occur, then sin 3 a: should stand for 
 sin {sin (sin a:)}, and sin a: x sin a; x sin a; should be denoted by 
 (sina:) 3 ; a form which some writers prefer, as it is. But as sines 
 of sines, &c. very rarely occur, it is not necessary to disturb 
 established notation. 
 
 As above defined, 0# and (f)~ l x are what are called inverse 
 functions. But when we talk of the inverse function of 0ar, it 
 
 * The student may ask, How can any thing but an angle have 
 a sine ? I answer, that 6 is not an angle, but the number of arcual 
 units in an angle. Every number has a sine. 
 
 E 
 
 5' :
 
 38 ON THE INVERSE 
 
 is as when we talk of the square root of x, knowing that there 
 are two. Generally speaking, inverse functions have more values 
 than one ; in the case before us, an infinite number. For, let 
 cost's be 6, that is, let 6 be an angle which has x for its cosine ; 
 then so has 2nnr 0, m being any integer, positive or negative. 
 
 Hence it always arises that though 00" 1 * = x, 0~ 1 0o; is not 
 always x, but only has x for one of its values. Thus (^xf is x, 
 but \/(x*) is either x or - x, at pleasure, or else one or the other, 
 as dictated by the particular problem in hand. Similarly 
 
 cos" 1 cos = 2mir 0, tan"' tan# = imr + 0, 
 
 sin" 1 sin = 2mir + or (2m + 1) TT - 0, cot" 1 cot# = rmr + 6, 
 
 cos" 1 sin# = Znnr ( ^ - 0) , tan" 1 cot0 = m-n- + ^ - 6, &c. 
 
 \2 / 2 
 
 This chapter is wholly on expression, and is intended to enable 
 the student to understand the theorems hitherto demonstrated, 
 when expressed in inverse language. All that is wanted, then, 
 is a set of examples for consideration. I shall give two cases 
 with full explanation, and then write down others to be considered 
 
 by the student. 
 
 cos (2 sin" 1 *) = 1 - 2x*. 
 
 This is nothing more than cos 20 = 1 - 2 sin*0. Let x be the sine 
 of 0, that is, let be an angle whose sine is x, and substitute. 
 The formula is to be understood as ' the cosine of double any 
 angle whose sine is x is 1 - 2#V 
 
 tan" 1 * + tan"V = tan" 1 [- ) . 
 \1 -xyl 
 
 In all the formula? which have inverse functions for their terms, 
 we have choice on one side and not on the other. In the formula 
 4^ x 9* = 36^ we are not at liberty to say that any square root 
 of 4 multiplied by any square root of 9 is any square root of 36 : 
 but any square root of 4 multiplied by any square root of 9 is out' 
 of the square roots of 36. And by the above we mean that uinj 
 angle whose tangent is x augmented by any angle whose tangent 
 is y, is one of the angles whose tangent is (x + y) (1 - xy). 
 It is proved thus : 
 
 tan0 + tan0 \ 
 
 
 = tan 
 
 r a> 
 
 l-tan0.tan0 \1 - tan0. 
 
 . / 
 1 
 \1
 
 TRIGONOMETRICAL FUNCTIONS. 39 
 
 or -f is one of the angles, &c. Let tan0 = x, tand = y, or let 
 = tan" 1 *, = tarT'y, and substitute. 
 
 The student may now employ himself on the following: 
 
 smcos" 1 x = V(l-^ 8 )) tansec" 1 *^: VO^-l)* sin (2 sin" 1 *) = 2*i/(l-**), 
 sin (3 sin' 1 *) = 3* - 4x*, sin (4 sin" 1 *) = (4x - 8* 3 ) A/(! - aJ*) 
 
 <2x 3x x 3 
 
 tan (2 tan" 1 *) = - - s , tan (3 tan" 1 *) = -^ a j 
 1 * 1 o* 
 
 taking acute angles, 
 
 = tan" 1 1 + tan" 1 ^ = 4 tan" l - tan' 1 
 
 - tan' 1 & + tan' 1 
 sin" 1 * + sin'ty = sin" 1 (*V(1 - y 
 
 cos" 1 * + cos" 1 ^ = cos" 1 {V(l - * 2 - y* + a^y 2 ) - zy}, 
 
 /** + 1 
 
 cos tan" 1 sm cot" 1 * = \ / , cos . sin '. cos . sin" 1 * = *, 
 \ x* + 2 
 
 V3 - 4* z 
 - 3- 5 
 1 -** 
 
 (6* 2 - 2) V(* 6 - 15* 4 + 15* 8 - 1) 
 
 sm 2 cos ! tan 3 cot '* = 5 - J ^ - - - - - ' . 
 
 x* (x* - 3)* 
 
 This is a chapter on language; and some of the preceding 
 examples are merely hard phrases to be construed from trigono- 
 metry into algebra. But such transformations have an important 
 use in calculation. If we wanted to calculate the value of the 
 last-named function of * when * = 5'1761328, and had such 
 trigonometrical tables as those of Hutton, hereafter described, 
 it would be the easiest plan, beyond comparison^ to proceed by 
 the first side. That is, we should find by the table the angle 
 whose cotangent is 5*1761328, treble it, find the tangent of the 
 trebled angle from the table, pass to the table of cosines with 
 that tangent, find the angle to it, double that angle, and take 
 the sine of the last. Thus sin cos" 1 * is, with tables, easier than 
 V(l - 3?), and sin 2 sin" 1 * easier than 2*V(1 - #*) 
 
 The following are a few instances of reduction to mixed 
 trigonometrical forms : 
 V(o* + & 2 - 25 cos<7) = Vl( t by - 2ab (1 + cosC)} 
 
 2v/ai.cosi<?\ 
 '
 
 40 ON THE INVERSE TRIGONOMETRICAL 'FUNCTIONS. 
 V(rt* + &* - 2ab cosC) = V{(a - W + 2ab C 1 ~ cosC)} 
 
 v/(a* + i 8 ) = a sec tan" 1 - , </( a * - b*) = a cos sin" 1 - , 
 
 (I [I 
 
 - 6 V(&* ~ 4ae) b . .2 Vac 6 . , . . 2 v 7 ac 
 
 ' = cos s i sin ' - , or Mir* sin ' - . 
 
 2a a b a b
 
 CHAPTER V. 
 
 INTRODUCTION OF THE UNEXPLAINED SYMBOL ^-1. 
 
 IF we look at the series for sin0 and cos#, of the form obtained 
 in p. 33, we see that each term is one of those in e 9 (Alyebra. 
 p. 225). We easily deduce 
 
 eos^Adn^l + W-f-ftg+^H.*^-.... 
 
 If there existed such a quantity k, as would give k' = - 1, k* = - k, 
 k* - 1, k* = k, &c., then cosfl + k sin0 would be 6**. Such a quan- 
 tity there is not in Algebra, as hitherto considered : for k 1 - - 1 
 is absurd. If, under pretence of satisfying this equation, we invent 
 k = *J-\, and proceed to use it according to the laws which 
 demonstrably govern our intelligible symbols of positive and ne- 
 gative quantity, we adopt the process of all the algebraists, with 
 a fair statement of what we are doing. A use, which ought to 
 have been called experimental, of the symbol -J-l, under the name 
 of an imf>ossible quantity, shewed that, come how it might, the 
 intelligible results (when such things occurred) of the experiment 
 were always true, and otherwise demonstrable. I am now going 
 to try some of these experiments : the student may rest assured 
 that the new results of this chapter will, in the second book, 
 be rendered demonstrative, upon a system which clearly defines 
 -V/-1 ; or he may doubt it : but he must not think they are 
 demonstrated here, though they will have strong moral* evidence 
 in their favour. By giving precedence to the use of *J-1, under 
 the above stipulation, the student will gain the advantage of 
 familiarity with the language of double algebra, before he ap- 
 proaches the difficulties. 
 
 * It is almost impossible to discredit Woodhouse's remark ; 
 " "Whether I have found a logic, by the rules of which operations 
 with, imaginary quantities are conducted, is not now the question : 
 but surely this is evident, that since they lead to right conclusions, 
 they must have a logic." 
 
 E3
 
 42 INTRODUCTION OF 
 
 Say that we suppose, from the above, 
 
 The processes of algebra constantly lead to this result, and 
 refuse every other; I mean those in which V~l is assumed to 
 be something which, though unintelligible, is governed by the 
 laws of algebra a fellow-subject of the other symbols, with a mask 
 over his features. For instance, common multiplication will give 
 (cos# + sin 0. -/-I) (cos0 + sin0 . V~l) = cos (0 + 6) + sin (040). V~l- 
 Let fe denote cosfl 4- sin6 . V-l ; then fO x/0 =/(0 + 0), and 
 (Algebra, p. 204) fd must be E 9 , where E is independent of 6. 
 Accordingly, 
 
 E 9 -l _ cosfl-1 sinO i 
 
 ~e~ ~e~ T ' 
 
 Diminish 6 without limit, and (p. 17, and Algebra, p. 266) 
 
 log .8=0 + V-l. or E=d~ l , JS 9 =^' 1 ' 1 . 
 If v ~* = cos 6 + sin 0.V~1 universally, then 
 
 e v " = cosO - sin#. V~l whence 
 6 0v-i + e -ev-i e 07 " 1 - e" 6lv " 1 
 
 --- 
 
 Had these forms been intelligible, they would have been the 
 proper algebraical definitions of the cosine and sine of 0; and 
 trigonometry would have been pure algebra in the ancient sense, 
 and a very easy part of it. For assuming tan0 to be sin04-cos0, 
 and sec<?, cosec#, cot#, to be reciprocals of the other three, all 
 the formulse of trigonometry would have been proved by simple 
 algebraical operation. For example, 
 
 cos0 cosfl = i (e^- 1 + e-*"- 1 ) (e ^ + e^' 1 ) 
 
 _ i (g^+flw- 1 + 6 -(^+e)v-i + e (^-OK-i + e -(0-0)v-i\ 
 
 = \ (cos (0 + 6) + cos (0 - 0)}. 
 
 Since e ''' 1 has this property, that a change of into nO raises 
 it to the w th power, we must have . 
 
 (cos0 + sin0 . V~l) n = cos n & + sm n V~l- 
 
 This is called De Moivre's Theorem. The student, instead of 
 referring to it, must take pains to associate cos0 4- sinfl . v/-l 
 with the notion of a quantity which is squared, cubed, &c., by 
 introducing a double, treble, &c. angle. And in like manner 
 he must associate the notion of reciprocals with cos0 + sin0. -J-\
 
 THE UNEXPLAINED SYMBOL V-l. 43 
 
 and cosQ-smO . ^-1, without being obliged to bring them back 
 to e^' 1 and e" ev " 1 . The following equation will assist: 
 (cos0 + sin0 . V- 1) (cos# - sintf .-j-\) = cos ! + sin*0 = 1. 
 
 If n be integer, the first side has one value only, and also 
 the second. But if n be fractional, as f- , the first side has 
 10 distinct values ; the second apparently only one. This intro- 
 duces us to a new consideration of the highest importance. 
 
 We have been using an angle in two different ways: first, 
 absolutely, as a magnitude, in the same manner as any other 
 kind of magnitude; secondly, as generated by a straight line 
 revolving from a given starting-line, and indicating the direction 
 which the revolving line points out when it has revolved through 
 the angle. As magnitudes, and 6 + lir arcual units of angle 
 differ as much as d and 6 + 2?r feet, or gallons, or hours : as 
 indicators of direction, they yield no difference at all they in- 
 dicate the same direction. 
 
 If we begin with as indicating a direction (for which 
 6 + Znnr would have done as well), and if FO be the solution 
 of a problem in which 6 is a given quantity, that problem is 
 equally solved by F(9 J r2mw), ?n being any integer positive or 
 negative. So many different values as we can give to F(d + 2rmr), 
 so many different solutions : but if FO be another angle, used 
 as an indicator of direction, then so many different values as 
 we can find for F(0i ^trnr), no two of which differ by a posi- 
 tive or negative multiple of 2?r, so many distinct answers are 
 indicated. And all that we say of directions applies to the 
 trigonometrical functions, which take value only from the direc- 
 tion of the revolving line, and not at all from the number of 
 revolutions by which it has been attained. 
 
 If Q be an angle which indicates a direction, nO can only 
 indicate one direction, when n is integer. For, using 6 + 2ni7r 
 for 6, nO becomes nO + Intmr, and 2nm is an even integer. But 
 
 if n be a commensurable arithmetical fraction, say - in its lowest 
 
 9 
 terms, then n9 indicates q distinct directions, no more and no 
 
 fewer. For n (0 + 2t7r) or - + TT indicates the same di- 
 q q 
 
 rection for any two values of m, m' and m", in which - 
 
 9 9
 
 44 INTRODUCTION OF 
 
 is an even integer, or (m' - m") - an integer, positive or negative ; 
 
 and for no others. Now since q and p are prime to each other, 
 this can only be when q divides m' - m", positively or negatively, 
 or when m' and m" differ by a multiple of q. 
 
 If then we take the following values of m, namely, 
 0, 1, 2, 3, ... (q-1), 
 
 we get all that give really different directions ; for every other 
 number, positive or negative, differs from one or another of these 
 by a multiple of q. All the distinct directions , then, are indi- 
 cated by one or another of the following angles, 
 
 Now if p and q be prime to one another (Arithmetic, Ap- 
 pendix, p. 195), and if we divide p, 2p, ... (q - 1)^ severally by q, 
 the remainders (which are all we need look to, since every unit 
 in a quotient is 2rr in the angle), whatever order they may 
 occur in, are 1, 2, 3, ... (<?-i), each occurring once somewhere. 
 Consequently, changing the order, we may say that all the di- 
 rections which n (0 + 2)mr) can indicate, are those indicated by 
 
 p p1 2 
 
 which may be expressed thus : If q be the lowest denominator 
 of n, all the directions indicated by nO may be derived from any 
 one of them, by successive advances of the q ih part of a revo- 
 lution each. 
 
 In the last equation, written thus, 
 
 - p p 
 
 (cos0 + sintf . V-l) 9 = COS- + sin - . </-!, 
 q q 
 
 we see q different results on the second side ; which are the 
 q ambiguities of value we are taught by common algebra to 
 give to the first side. 
 
 If cos a + sin a. *J-\ be one of the values of (cos0 + sin0. */-!)", 
 then cos a -sin a. ^-1 is, by similar reasoning, one of the values 
 of (cos0-sin0.V-l) w . If sin0 = 0, that is, if cos0 be either 
 + 1 or -1, then cos0 i sinfl. y'-l is the same as cbs0-sin0. </-!,
 
 THE UNEXPLAINED SYMBOL -/-I. 45 
 
 or both cosa + sina.y-1 and cos a- sin a. y-1 are values of both. 
 This will enable us to arrange our sets with better perception 
 of their connexion. 
 
 I shall now exhibit all the 12th roots of + 1, and all the 
 12th roots of - 1. For the first, 6 = 0, for the second = TT, (which 
 gives ^ TT for a commencement), and on these angles we must 
 make advances of one-twelfth of a revolution at each step, stopping 
 when we have gained 12 distinct values for each. Gradual mea- 
 surement will here be most convenient. 
 
 The twelve twelfth roots of+ 1. 
 
 cosO sinO. y-1 gives only one root, + 1 itself 
 
 cos30 sin30. V-l ...... ^3 ^-1 
 
 cos60 sin60. V-l ...... i iv'3-V-l 
 
 cos90 sin 90. V- 1 ...... V~l 
 
 cosl20sml20.y-l ...... - \ iAV~l 
 
 cos 150 sinl50.V-l ...... - 1V3 W" 1 
 
 cos!80 4 sinl80.V-l ...... only one root, - 1. 
 
 The twelve twelfth roots of - 1. 
 
 cosl5 sinl5. V-l gives ^^ ^^ .y-1 
 
 cos45 sin45. y-1 ...... 1V2 JV2.V-1 
 
 V6 - V2 , V 6 + V2 ; 
 cos7o sm7o. V-l ...... - - - j-* V-1 
 
 cosl05 sinl05.V-l ...... - ^ 6 " V 2 V 6 
 
 cos 135 + sin 135. V-l ...... - ^2 + 
 
 cosl65 sinl65.V-l .. _ V 6 + V 2 V 6 " 
 
 .. 
 4 4 
 
 We have now found the twelfth roots of any quantity, positive 
 or negative. If a be any twelfth root of +- 1, a ^m, m being 
 positive, is a twelfth root of m. For its twelfth power is a lz .m, 
 or m. Similarly, if /3 be any twelfth root of - 1, ffim is a 
 twelfth root of - m. 
 
 We may extend this further, as follows. Two given quan-
 
 46 INTRODUCTION OF 
 
 titles a, and b, cannot be cosine and sine to the same angle, 
 unless a* + 6* = 1 : but they may be proportional to the sine 
 
 and cosine of the same angle. For if a 2 + b* = in 1 , then and 
 
 m m 
 
 are cosine and sine to the same angle, and the tangent of that 
 angle is . Hence we have a transformation of great importance, 
 
 (b b \ tan"'-v-i 
 
 costart" 1 - + sintan" 1 -V-l I = V( a8 +^ ! ) 6 ' 
 a a / 
 
 But this point is to be remembered: tan" 1 a; has two values 
 which indicate different directions; and those values are oppo- 
 nents ; Q being one, + TT is the other. Now 6 and 6 + TT have 
 contrary sines and cosines ; sin (6 + TT) = - sin0, cos (0 + ir) = - cosO. 
 If we set out with a given sign, say the positive one, for ^(a* + 6*), 
 and take the wrong angle, we shall end with - a - b^f-l, instead 
 of + a + 6-/-1 ' we ma y set it right either by altering the angle, 
 or using the other square root of a* + 6*. But as the positive 
 
 root is generally used, the proper value of tan" 1 - may be re- 
 membered as the angle whose cosine has the same sign as a. 
 The following would be the most convenient arrangement. One 
 
 angle which has - for tangent has its sine of the same sign as a, 
 
 and its cosine of the same sign as b; the other has the sine 
 of a different sign from a, and the cosine of a different sign 
 
 from b. Let the first be denoted by tan" 1 - , and the second 
 
 a 
 
 by tan" 1 . Thus we have 
 - a 
 
 b -b 
 
 Show now that one twelfth root of a + 6-/-1 is 
 
 'vYCa* + &') (cos (& tan" 1 |) -f sin (& tan" 1 - W~ 
 
 and that all the twelfth roots may be found, either by successively 
 increasing the angle by twelfths of a revolution, or by multiplying 
 the above by all the twelfth roots of T 1.
 
 THE UNEXPLAINED SYMBOL ^-1. 47 
 
 Returning to our original notation, we have 
 
 x + y^-1 = r(cos0 + sin 0.^-1) = r^' 1 -, 
 x - T/y'-l = r(cos0 - sinftV-1) = rr fllM . 
 
 The fundamental equation 6 v " = cos0 + sin0.</-l, gives the 
 following results ; e 2 '""^ 1 = 1, and e { * m "*"'- 1 = -l. If a be a positive 
 quantity, we have a = e logo = e 1 g a+2 "" r ' v - 1 . j^ then &n y gy^^ z 
 be a logarithm of a, which satisfies a = e 3 , we have such right 
 as we can take in this chapter to say that the ordinary arithmetical 
 logarithm (which we shall still denote by log a) is only one 
 of a class, all contained in log a + Ztmr-J-l, in which m may 
 be any positive or negative integer. Let Aa denote any logarithm 
 of a; then we have \o = logo -t 2//wr<\/-l : and the usual form 
 log a is one case of Xa. Here a was positive. 
 
 Now -a = (-l).a = e ( * n " 1} " f - l .<?* a = e io * a **+ i w- 1 ., whence 
 we may say that negative quantities have logarithms, and that 
 
 X(- a) = log a + (2m + \)TT^-\ ; 
 
 but still, as before, there is no arithmetical logarithm to a negative 
 quantity ; for, m being integer, 2m + 1 cannot vanish. 
 
 Let the student now show that, in this extension, any logarithm 
 of a, added to any logarithm of b, gives one of the logarithms 
 of ab, &c. All our ordinary logarithmic relations remain true 
 in this sense. Since 
 
 x + y^-i = r ^- 1 = re ^- mv ^- 1 = eiog-c^-)/-^ 
 we have all the system of logarithms exhibited in 
 X (x -f y-J-l) = logr + (6 + 2tmr)^-l, 
 
 = $\og(x* + f) + (tan- 1 y - + Zmir] -J-l. 
 \ x / 
 
 Here 2nnr is not necessary, unless we restrict tan" 1 - to be 
 
 x 
 
 HI I 
 
 in the first revolution; otherwise, tan" -(remembering the dis- 
 
 x \ 
 
 tinction between it and tan" 1 ) expresses every case by itself. 
 
 A still further extension of the notion of a logarithm may 
 now be made. The base e is e 1 | --"" IV - 1 . jf x, a represent the
 
 INTRODUCTION OF 
 
 most extensive logarithm of a, which we can get from any 
 
 such form of e, it must be obtained (a being positive) thus: 
 /""* -r- 
 
 { ) = } ( J ' = 6 j 
 
 ,. log a + 2w7rv / -l lo 
 
 or La = i.o^. /!i > -(-) = 
 
 m and w being any positive or negative integers. Proceed in 
 the same way with x + y-J-l, and we have 
 
 It thus appears that there is, to the base e, an infinite number 
 of systems of logarithms, corresponding to the values of m, and 
 an infinite number of logarithms in each system, corresponding 
 to the value of n. 
 
 Two logarithms of one quantity, taken out of different systems, 
 cannot generally be found equal. If, m and m' being two integers, 
 we form the equation {let p = 6 + 27m, p 1 = 6 + 27m'} 
 
 log r + p i/- 1 _ log r 4 p'-J- 1 
 1 + 2TO7T V- 1 ~ 1 + 2mVV- 1 ' 
 
 clear it of fractions, and equate the possible and impossible terms, 
 we get pm' = p'm and 2mV log r + ja = 2m7r log r -f p'. Substitute 
 in the second the value of p' from the first, and we get 
 
 (27TTO log r - p) (in 1 - m) = ; 
 
 either then m' = m, and the systems are the same, or 27rm log r = p 
 and 27rm' log r = p'. In these cases a logarithm of x + y V~ 1 or 
 r , in each system is log r, the arithmetical logarithm of r. 
 But, m and TO' being the indices of the bases, and n and ' those 
 of the particular logarithms to those bases, this requires that r 
 and 6 should be determined by 
 
 + 27rn 6 + 27m' 6 + 27m 
 
 w' - n rim - nm' 
 
 or log r = -- , e = ZTT , -- ; 
 m - m m - m 
 
 so that, for two given systems, and two given values of n in those 
 systems, there is one expression, and one only, which has the same 
 logarithms in both.
 
 THE UNEXPLAINED SYMBOL V - 1 49 
 
 In ordinary algebra it is said that negative quantities have none 
 but impossible logarithms. And this in the face of the result that, 
 Ve being a, ^ is either + a or - a, so that - a has ^ for a logarithm. 
 We can now show how these isolated cases of negative quantities 
 with real logarithms arise. 
 
 Let us solve the general question What expressions have real 
 logarithms, what are they, and in what systems? The follow- 
 ing equation is produced by multiplying both terms of the fraction 
 by 1 - 2w7r -/- 1) 
 
 log r + (6 + Imr) ^1- 1 
 
 1 + 2nnr V- 1 
 
 log r -f 2m7r(0 + 2mr') + 2mr - 2mir log r 
 
 " "^ 1 ~A a a V~ ! 
 
 L(x + 
 
 This is a real quantity only when 2nnr log r = + Imr, in which 
 case L (x + yj- 1) = log r (1 + 4mV) -f (1 + 4wV) = log r. If = TT, 
 in which case y = and 2 is negative (# = re^" 1 = - r) we have 
 
 2n+l 
 
 log (- r) = log r, whenever 2m?r log r = (2n + l)7rorr = e 8W This 
 is precisely the case we might have anticipated : for 8 " +1 has two 
 real (2m) th roots, one negative. But it appears that instead of 
 the system being that of the base e, the base is g U2 "" rv -\ The 
 complete illustration of this difficulty may be gathered from the 
 second book. 
 
 Returning now to the fundamental equations, let z stand for 
 e^' 1 or cos0 + sin0 . V~ ! We have then 
 
 2 = cos0 + sin0. \]- 1, a" = cosnO + sinn<?. J- 1, 
 z" 1 = cos0 - sin0 . J- 1, z' n = cosnO - sin nO . *J- 1, 
 
 2 cos0 = z + z" 1 , 2 cos0 = z n + z'", 
 
 2 V- 1 sinfl = z - z-, 2V- 1 . sinwfl = z" - z n . 
 
 These equations may almost be said to contain trigonometry. 
 Completely established, they would furnish proof of all we have 
 done : the deduction from them of our previous results must 
 be inductive proof that, somehow or other, the use of -/- 1 does 
 lead to true results. 
 
 Required sin 3 in terms of functions of multiples of 6 : 
 
 = - (sin 30 - 3 sin#}, as in page 35.
 
 50 INTRODUCTION OF 
 
 Required cos 49 in powers of sin 6 and cos#; 
 
 z 4 + 3^ (c + sV-l) 4 + (c- 
 cos40 = - = S - * - ' 
 
 - c* - 6cV + s 4 , as in page 31. 
 
 The student must take notice of the manner in which the action 
 of -/- 1 supplies the place of the rule in page 31. 
 
 "We may now extend some of our rules. Required sin"'0 cos"^ 
 in functions of multiples of 6. That is, we have to find 
 (z - z l ) m (z + g-') n 
 
 2'"'(V-ir 
 
 In the numerator, the descent of each developed factor is by two 
 dimensions in each term ; for 
 
 (z - z-') m is z m - l m z m - l z- 1 + 2 m z m -V* - ...... 
 
 Now if we multiply Ka + La*' l b + Jf*'V + .... + Paft*' 1 -f Qb k by 
 n - b or by a + 6, we have as results, 
 
 The shortest way of doing this is by writing down the coefficients 
 K, L, &c. in a row, and under them -ZT+0, -E + JT, &c. In this 
 manner we may rapidly make the multiplications, and in either 
 of two mutually verificatory ways : by coefficients from (z - z" 1 )" 1 , 
 and successive multiplications by z 4 z" 1 , or by coefficients from 
 (z + z" 1 )" and successive multiplications by z - a" 1 . As to the rest, 
 the final divisor will be 2"'' 1 , for a 2 (and >J-l,tf there) will be 
 taken up in the reconversion of z"' z"" into cosine or sine. And 
 if m be even-even, (7-1)'" is 1, and the result is in cosines; 
 if odd-odd, it is V~ 1> anc ^ the result is in sines ; if odd-even, 
 it is - 1, and the result is, we may say, in negative cosines, the 
 sign of each term being changed ; if even-odd, it is - <J- 1, and 
 the result is in negative sines. And the conclusion begins with 
 (w + n)0 and this angle diminishes by 29 at each step. But 
 if z occur, there is no term distinct having z", so that the 2 just 
 mentioned is not taken up in forming cos 09, and must therefore 
 be used in denoting the coefficient ; or only half the coefficient in 
 the result must be used. As an instance, I shall take sin 5 # cos 6 0, 
 in which, if we work both ways for verification, we shall pick up 
 during the process all that is wanting for finding sin'"0 cos'0,
 
 THE UNEXPLAINED SYMBOL </-!. 
 
 51 
 
 for any value of m not exceeding 5, and sin 5 cos"0, for any value 
 of n not exceeding 6. The conclusion which each step prepares 
 us for is written in abbreviation at the commencement. 
 
 c 6 
 
 1 + 
 
 6 + 
 
 15 + 
 
 20 + 
 
 15 + 
 
 6 + 
 
 1 
 
 
 
 
 s c 6 
 
 1 + 
 
 5 + 
 
 9 + 
 
 5- 
 
 5- 
 
 9- 
 
 5- 
 
 1 
 
 
 
 sV 
 
 1 + 
 
 4 + 
 
 4- 
 
 4- 
 
 10- 
 
 4 + 
 
 4 + 
 
 4 + 
 
 1 
 
 
 sV 
 
 1 + 
 
 3 + 
 
 0- 
 
 8- 
 
 6 + 
 
 6 + 
 
 8 + 
 
 0- 
 
 3 
 
 -1 
 
 sV 
 
 1 + 
 
 2- 
 
 3- 
 
 8 + 
 
 2 + 
 
 12 + 
 
 2- 
 
 8- 
 
 3 
 
 + 2 + 1 
 
 S 5 C 6 
 
 1 + 
 
 1- 
 
 5- 
 
 5 + 
 
 10 + 
 
 10- 
 
 10- 
 
 10 + 
 
 5 
 
 +5-1-1 
 
 S S 
 
 1- 
 
 5 + 
 
 10- 
 
 10 + 
 
 5- 
 
 1 
 
 
 
 
 
 8 s C 
 
 1- 
 
 4 + 
 
 5 + 
 
 0- 
 
 5 + 
 
 4- 
 
 1 
 
 
 
 
 SV 
 
 1 - 
 
 3 + 
 
 1 + 
 
 5- 
 
 5- 
 
 1 + 
 
 3- 
 
 1 
 
 
 
 8 5 C 3 
 
 1- 
 
 2- 
 
 2 + 
 
 6 + 
 
 0- 
 
 6 + 
 
 2 + 
 
 2- 
 
 1 
 
 
 s'c* 
 
 1- 
 
 1 - 
 
 4 + 
 
 4 + 
 
 6- 
 
 6- 
 
 4 + 
 
 4 + 
 
 1 
 
 - 1 
 
 8 s C 5 
 
 1+0- 
 
 5 + 
 
 + 
 
 10 + 
 
 0- 
 
 10 + 
 
 + 
 
 5 
 
 + 0-1 
 
 8V 
 
 1 + 
 
 1 - 
 
 5- 
 
 5 + 
 
 10 + 
 
 10- 
 
 10- 
 
 10 + 
 
 5 
 
 +5-1-1 
 
 Attending to the rest of the process, we have 
 
 2*cos*0 = cos60 + 6cos40 + 15cos20+10 
 
 2sin0cos 6 = sin 70 +5 sin 50+ 9sin30 + 5sin0 
 
 2 7 sin*0cos s 0=-cos80-4cos60- 4cos40 + 4cos20 + 5 
 
 2" sin 3 6 cos 6 = - sin 90 - 3 sin 19 + 8 sin 30 + 6 sin 
 
 2 9 sin*0cos 6 = cosl00 + 2cos80- 3cos60-8cos40 + 2cos20 + 6 
 
 2 ro sin 5 0cos'0 = sinll0 + sin90- 5 sin 19 -5 sin 50 + 10 sin 30+10 sintf 
 
 2 4 sin*0 = sin 50 -5sinS6>+10sin0 
 
 2 s sin 5 cos 6 = sin 60 - 4 sin 40+5 sin20 
 
 2 4 sin s 0cos 2 = sin70 -3 sin 50+ sin 30 + 5 sin 
 
 2 7 sin 5 cos 3 = sin 80 - 2 sin 60 - 2 sin 40 + 6 sin 20 
 
 2 s sin 5 cos 4 = sin 90 - sin 70 - 4 sin 50 + 4 sin 30 + 6 sin 
 
 2 9 sin 5 cos 5 = sin 100 - 5 sin 60 + 10 sin 20 
 
 2 w sin 5 0cos s = sinll0+ sin 90 -5 sin 70 -5 sin 50+ 10 sin30+ 10 sin & 
 
 Let a 9 + !# + a^+ ... be a converging series, if extending 
 ad infinitum (which it need not here do), and let it be the 
 development of a known function of x, <px. It is required to find 
 + a^x cos0 + a,? cos20 + ...... 
 
 Multiplied by 2, with z" + z~" written for 2 cos nO in the several
 
 52 INTRODUCTION OF 
 
 cases, it obviously becomes <p(xz) + <p(xz~*). It is then 
 
 H00rs)+0(*z-')}, 
 
 which can be reduced to a real form. Similarly 
 a sin 00 + ctjX sin + a^y? sin 20 + ... is {0 (xz) - (xz' 1 )} -f- 2 ^- 1 . 
 
 Before proceeding to some examples, it will be worth while 
 (seeing before us a field of such extent as the applying the 
 summation of any algebraical series to the summation of one 
 in which the terms are severally multiplied by the sine or cosine 
 of the multiple of an angle), to consider the transformation of 
 (xz) and <fi(xz' 1 ). 
 
 Let there be a function of ^a which, if *Ja had only one value, 
 would itself have only one value. This restriction of value may 
 be, if we please, conventional ; for instance, sin~'5 . ^a is such 
 a function, if we suppose ourselves restricted to one value of 
 shT'i. If, then, F(*Ja) can be thrown into the form P + Q . Ja, 
 where P and Q are wholly unaffected by the change of -/ a into 
 - V F(- V) must be P - Q.^a. 
 
 Now it is a proposition to be carefully remembered, that any 
 function of x + y^/-l, x' + y'^-l, &c., can always be reduced to 
 the form P+ Q <J-1, in which P and Q are wholly independent 
 of V~l> or are rea l quantities. In the second book I shall 
 show this independently of all particular cases : at present we 
 must be content with induction. The proposition is clear enough 
 of sums, differences, and products, however varied ; and also when 
 division enters, if we look at its reduction to multiplication by 
 
 x-y^-l x y 
 
 ' 
 
 As to powers, we can thus reduce the form (x + 
 for in this we see (re*'' I ) p+ *' M > or 
 
 e" lo e '-* cos (q log r + p0) + e p lo * r - q9 sin (q log ; + pO) . V~l ; 
 
 for e'^ ' we have e* cosy + z sin y.-J-l ; for log (x + y <J-\) 
 we have 
 
 If we extend our notion of cosine and sine, taking the ex- 
 ponential forms in p. 42 as definitions, we have
 
 THE UNEXPLAINED SYMBOL V-l. 
 sin (x+y V-l) = ~ {e"- 1 -* - f ^ 
 
 l {e 
 
 ^ . 
 sin a; + - cos x . y-1 
 
 2 2 
 
 We may reduce cos (# + y -/-I) and tan (a: + y V~l) ^ n like manner. 
 If, with alike extension, we take siiT'^ + y V~l)i we may make 
 the reductions as follows : 
 
 -0^/-l = log(cos0-sm0.^-l), = V-1. log (cos0- sin0. V-l), 
 
 sin' 1 a; = V-l . log (V(l -*)-* V~l}. 
 sin' 1 (a: + y V~l) = V~* lo g (V(l - ** + y* - 2*y V~l) - * V~l + y}. 
 
 the second side of which, by preceding processes, can be reduced 
 as required. 
 
 If, in every case, <p(x + y V~l) can be reduced to P + QV~1> 
 in which P and Q are real, then, by our first remark, ff)(x-y V~l) 
 can be reduced to P - Q V~l> whence 
 
 |{0O+y V-l)+0(*-yV-l)} and JV-1 {0(*+y V-l)-0(*-W-l)} 
 are real, being P and - Q. 
 
 "We have now to consider <fi(xz) and 0(s" 1 ), or 
 
 (a; cos0 + x sin6> . V~l) an d (* cos^ - x 
 A few principal examples will here be sufficient. 
 Let <j>x = (1 + x} n , 
 
 the w' h power of which is 
 
 rt sxin f . a; sin . . , * sin Q - T 
 
 (l+2a; cos 0+ 2 )3" <i cos n tan J - - + sm n tan' 1 4 . V~l \ > 
 
 I l+a;cos0~ l+a:cos0 v 
 
 (a;z) + (tfz" 1 ) ,, a; sine 
 
 - ' = (1 4- 2x cos + a;*)*" cos n tan ' ; - - , 
 2 1 -)- x cos 
 
 d> (xz) - <& (xz~ l ) ., _ _ ,.j n . a; sin 
 
 yv Q ,, ' = (1 + 2a; cos -f a; 2 ) 5 " sin n tan' 1 - 
 2 V-l 1 + x 
 
 . 
 
 + x cos 
 
 and these are the expressions for 
 
 cos0 + a; 2 cos 20+ ..., nx sin0 + w a;* sin 20 + ... 
 2t 2 
 
 F3
 
 54 INTRODUCTION OF 
 
 The verification of such results will be useful practice. For 
 instance, it is asserted above that 
 
 (1 + 2* cos + x 8 ) cos 2 tan" 1 , x S1 " 9 = 1 + 2* cos + x* cos 20. 
 1 + x cos 9 
 
 Now cos 2 tan" 1 a =cos* tan" 1 a- sin* tan" 1 a = (1-tan 8 a) + (l + tan* a). 
 The first side of the above is then 
 
 Let n = - 1, and change x into - #. Show that cos (- tan" 1 - o) 
 is 1 : -i/(l + s )j and that sin {- tan" 1 (- a)} is a + V(l + a *)> an l 
 then show that the above expressions give 
 
 1 - x cos 9 
 
 1 - 2x cos + x* 
 
 = 1 + a; cos + z 8 cos 20 + x s cos 30 + ... , 
 
 - = x sin -f x 3 sin 20 + a? cos 30 + ... . 
 
 1 - 2x cos 9 + # 8 
 
 Verify these by the whole method, (fix being 1 + (1 - x). Also 
 show the following, 
 
 , Cos6 x* cos 20 y? cos 30 
 
 e* C(lStf cos (a; sin 0) = 1 + x cos + - - + - + ..., 
 
 Z Z.O 
 
 ^c cosff / /i\ ^ s i n 20 
 
 e* C(IS(/ sin (a; sm 0) = * sin -f - - + 
 
 1 
 
 Let 0o; = log (1 -f x). Then 
 log (1 + x cos + x sin . V~l) 
 
 -l og (1 .2, cos ) + tan- 
 
 
 
 |log(l+2a: cos 0+a; z ) = ^ cos0- cos 20+ cos 30 -- cos 40+..., 
 
 2i o 4 
 
 # sin a; 2 . . , x 3 . . x* . 
 
 tan" 1 - = x sm -- sin 20+ sm 30 -- sin 40+... . 
 1 + x cos 2 4 
 
 If a; be >- 1 and <+ 1, both these series are convergent, and 
 there is no ambiguity in the first: but there is in the second. 
 The second series, when convergent, has one definite value : 
 which is it of all the values which the first side may bear? 
 It must be the angle which lies between - \TT and + \TT : for 
 when x passes from negative to positive through 0, the series 
 does the same. 
 
 When x is greater than unity, these series become divergent,
 
 THE UNEXPLAINED SYMBOL V-l. 65 
 
 and the student should avoid founding results upon divergent 
 series, as the question of their legitimacy is disputed upon 
 grounds to which no answer commanding anything like general 
 assent has yet been given. But they may be used as means of 
 discovery, provided that their results be verified by other means 
 before they are considered as established. 
 
 If x = 1, we have 
 
 cos 29 cos 39 cos 49 
 
 /. 6\ 
 log (2 cos -) = 
 
 O Q A 
 
 sin 29 sin 39 sin 49 
 
 and 6 must, in the second, lie between - ir and + TT. These series 
 belong to a peculiar class ; they are convergent, but their cou- 
 vergency is not easily established. Their extreme cases often 
 present some algebraical peculiarity. If we divide both sides 
 of the second by 9 and diminish 9 without limit, we have 
 J = l-l + l-l-... (Algebra, p. 197). This is not the place, nor 
 even the work, in which to discuss the peculiar character of 
 these series. 
 
 Let 9 = ir. The first of the equations becomes 
 
 Mog (1 + z 2 ) = l-r 2 - iz 4 + . . . , 
 as well known. But the second becomes 
 
 2C Off Ou 
 
 tan" 1 x - x - r 4 ~ ? + 
 o 5 / 
 
 a remarkable series, both for its simplicity, and for the use 
 to which it has been put. It is convergent when x > - 1, and 
 not > + 1; and thus may be made effective when tan" 1 a; > - -JTT, 
 and not > $TT. When x = 1, tan" 1 * = JTT, we have 
 
 TT 111 
 
 4 =1 -3 + 5-7 + ...... ' 
 
 being the first calculable form in which TT has been directly 
 presented. But this series, though convergent, is very slowly 
 so, (Algebra, page 184) and would require that we should cal- 
 culate 500 terms before we could be sure of three decimal places. 
 The following is more convergent, derived from ^ 
 
 111111
 
 56 INTRODUCTION OF 
 
 But it is best to resolve TT or some known fraction of it, into 
 two or more angles whose tangents are known. Thus, 
 
 tan-'i + tan- 1 ^ = $TT, 
 (page 39), gives 
 
 v \ !_!/!_ 1\ 1/1. i\_ 
 4 ~ 2 + 3 3 \2 3 + 3 3 J + 5 \2 5 + 3V 
 
 which may be easily calculated, as follows. "Write p and q 
 
 for and $ (divide by 4 and 9 at every step) : 
 
 p = -50000000000 q = -33333333333 
 
 p 3 = -12500000000 q 3 = -03703703704 
 
 p* = -03125000000 q* = -00411522634 
 
 p 1 = -00781250000 q 7 = -00045724737 
 
 p 9 = -00195312500 q g = -00005080526 
 
 p 11 = -00048828125 q n = -00000564503 
 
 p 13 = -00012207031 q }3 = -00000062723 
 
 p" = -00003051758 q n = -00000006969 
 
 p" = -00000762940 q" = -00000000774 
 
 p M = -00000190735 q" = -00000000086 
 
 p n = -00000047684 q" = -00000000009 
 p K = -00000011921 
 p* = -00000002980 
 p" = -00000000745 
 p* 3 = -00000000186 
 p 31 = -00000000047 
 p 33 = -00000000012 
 p* = -00000000003 
 Now let (p n + q") 4- be denoted by r n . 
 
 r t = -83333333333 
 
 r, = -05401234568 
 
 r b = -00707304527 
 
 r, = -00118139248 
 
 r, = -00022265892 
 
 r u = -00004490239 
 
 r ls = -00000943827 
 
 r 15 = -00000203915 
 
 r 17 = -00000044924 
 
 r, 9 = -00000010043 
 
 r xl = -00000002271 
 
 r^ = -00000000518 
 
 r S5 = -000000001 19 
 
 r n = -00000000028 
 
 r w = -00000000006 
 
 r sl = -00000000002 
 
 84063894899 
 
 05524078661 
 
 
 84063894899 
 
 
 iar= -78539816338 
 
 
 4 
 
 TT = 3-14159265352 
 which is correct, with the exception of the last place.
 
 THE UNEXPLAINED SYMBOL V-l. 57 
 
 The series for tan~ l x may be easily deduced from one among 
 a number of forms which may best be considered together : as 
 follows. 
 
 We have seen a remarkable connexion between exponential* 
 forms on the one hand, and trigonometrical forms on the other. 
 Every trigonometrical function has an imaginary exponential one 
 for its equivalent, and every exponential function an imaginary 
 trigonometrical one. Many imaginary forms of one kind are real 
 ones of the other; and the following is such recapitulation and 
 addition as will put all the most useful transformations together. 
 
 sin 9 = 
 
 2 V-l 
 
 COS0 = 
 
 ton f) _ 
 
 - 
 
 _ _ _ _ _ _ _ 
 
 V-l 1 + e-*"- 1 V-l e^' 1 + 1 ' e^TT 
 
 = Iog(cos0 - sin6>.V-l).V-l = - Iog(cos0 + sin& 
 sin' 1 a; = log{V(l - **) -x^-\}^-\ = - log(V(l - x*) + x* 
 cos-'x = log [x - V( 2 - 1 )}. V- 1 = - log {x + V(^ 2 - 1 )} V- 1 
 
 log {x + V(*' + 1)1 = Sin -^^- } , log {x + V(* 2 - 1)} = cos-'^V 
 
 V-i 
 
 l _ 1 + tan0.V-l 
 
 - sin 0. V-l ~ l-tan0.V-l 
 
 1 1 + X\/-\ 1 X 
 
 e* = cos (zV-1) - sin (ar^-l )V-1, log a; = 2^-1 tan' 1 . V 
 
 \l + a? 
 
 Many of these transformations are hardly ever used in operation : 
 but unless the student has them before his mind, he will be 
 often at a loss to see the connexion of results which stand in 
 the closest relation. 
 
 The multiplicity of value of logo; (or rather \x, which might 
 have been used throughout, as in page 47) is closely connected 
 with that of sin"' x, &c. But the connexion was not very soon 
 
 * Exponential; for the logarithm is only the inverse function 
 of the exponential one.
 
 58 INTRODUCTION OF 
 
 noticed: and the following mode of investigating the series for 
 tan" 1 a; was consequently faulty. 
 
 Take the logarithm of both sides of e 2 ^ 1 = \ + tan -l 
 
 1 - 
 
 a result evidently absurd, for while the first side increases from 
 to oo , the second side goes through recurring periods. For 
 instance, taking periods of convergency, while passes from 
 2?r - ^TT to 2?r + -J-7T, the series repeats itself for the period 
 during which 9 passes from - ^TT to JTT. The error lies here, 
 2{tan0.i/-l + ...} is not any logarithm we please of e 2 ""'" 1 , but 
 some one logarithm; some one case of 20<J-1 + 2i7T/-l. When 
 we say A = B, therefore LA = LB (page 48) we are correct 
 only (except in the case of one isolated relation between the 
 real and imaginary parts of A and B) on the supposition that 
 we take the same system, and pair the proper logarithms of 
 A and B in that system. And the equation A = B only gives 
 (any given logarithim of A) = (the proper logarithm of B in the 
 same system). If we do not know the proper logarithm of B, 
 we must take the general case, and let the conditions of the 
 problem determine its specific meaning. Accordingly, instead 
 of 20V-1, we must write 26V- 1 4- 2rmr\/-\, and thus we have 
 
 which can be made true; and m must be such integer, positive 
 or negative, as will make + rmr fall between - TT and + \ TT. 
 
 The following investigation requires one theorem from the 
 theory of equations : and the rest of this chapter, generally 
 supposes a student who has read more than is supposed in 
 what precedes. 
 
 Multiply together the two factors x - e^' 1 and x - e"^" 1 ; 
 we have x* - 2cos#.a: + 1. Or thus, x - cosO - sin0.V-l and 
 x - cos0 + sin0.V-l, give the product (x - cos#)* + sin*0, or 
 x* - 2 cos 9.x + 1. Now observe that 
 
 x n + n = 2 cos20, or z - 2* n cos20 + 1 = 0, 
 *c 
 
 is satisfied by 
 
 26 
 X + x' 1 = 2 cos(20 v n), or x* - 2 cos .x + 1 = 0, 
 
 H
 
 THE UNEXPLAINED SYMBOL V-l. 59 
 
 or the roots of the second equation are among the roots of the 
 first. If 26 be changed into 20 + 2^, no change is made in the 
 
 /20 27T\ 
 
 first equation, and the second becomes a? - 2 cos + ) ,x + 1 = 0. 
 
 \n n I 
 
 the roots of which are therefore among the roots of the first. 
 Now two equations x* - ax + 1 = 0, x* - bx + 1 = 0, cannot have 
 a root in common ; for then it would be determined by ax = bx, 
 or x = would give the root : and there can be no such root. 
 If therefore we go on, until we have obtained n equations of 
 the second degree, we have got 2n distinct roots for the first 
 of all. Consequently, by the theory of equations, we have, for 
 all values of x, 
 
 - 2 cos20.x n + I 
 
 f (20 (n-l)27r\ -\ 
 
 ...... <x*- 2 cos + - - -}jf+l>. 
 
 I \n n ) J 
 
 When x = l, z*-2 cos0.a:+ 1 is 2 (1 -cos0) or 4 sin 2 ^-, so that 
 
 
 
 * e A *l e *\ *l 2 ""\ *l (n-l)ir\ 
 
 in t ^=48in*-.4sin'(- + - .4sm a - + ]...4sm* - + - - ' }. 
 n \n n] \n n ) \n n / 
 
 Extract the square root of both sides, and divide by 2 ; 
 
 . . 6 . (0 7T\ . (0 27T\ . (6 (n-l)7T\ 
 
 smG= 2" ' sin - sm - + - sm - + ... sin - + i - ' . 
 n \n n) \n n/ \n n / 
 
 On the second side there are n sines, or, exclusive of the first, 
 n -I sines. If n - 1 be an even number we may pair these, 
 the first and last, the second and last but one, &c. But if 
 n - 1 be odd, this pairing will leave one in the middle, and 
 n - 1 being odd, the middle number is %n, whence the middle 
 
 factor is sin ( - + ) or cos - , which, observe, approaches unity 
 \n 2n/ n 
 
 as 9 is diminished without limit. Moreover, the last one is 
 
 f I" 6 \\ (" e \ 
 
 sin < TT - -- - 1 > or sm --- : 
 
 I \n n/j \n n) 
 
 the last but one is, similarly, sin ( --- }, &c. Hence the pair- 
 
 \ n n/ 
 ing just alluded to gives
 
 60 INTRODUCTION OF 
 
 . 6 . ITT 6\ . ITT 0\ . [lir Q\ . llir 0\ 
 
 sm0=2' M sm- sin - + - sin sm + sin - - - ... 
 
 n \n nj \n n/ \ n n/ \ n n/ 
 
 a 
 n factors in all, the last, a single one, being cos - , if n - 1 be 
 
 odd. Divide both sides by sin (0 4- n}, and diminish 6 without 
 limit. The limit of the first side is then that of 
 
 sin#-sin- or x (sin#4-0)4- (sin- 4- ] ornxl4-l, 
 n \ n n/ 
 
 TT . , ITT . 3?r . 
 
 n = 2 ' sin 8 - . sm 2 . sin 2 one for each pair, 
 
 n n n 
 
 and cos (0 4- n), if there, has the limit unity. Now observe that 
 
 ,. a -I n . a-b a+b 
 
 sm (a i 5) sm (a - o) = 2 sin - cos - . 2 sin cos 
 
 22 22 
 
 = (sin a + sin It) (sin a - sin 6) = sin 2 a - sin 2 b. 
 
 Substitute from this theorem, divide sin0 above by n, and 
 we have, dividing both sides by 6, and transferring n, 
 
 n 
 
 with the factor cos (0 4- n) at the end if n - 1 be odd. This 
 second side is always sin<? 4- 0, however great n may be. If 
 we increase n without limit, 
 
 3 
 
 sin Sin i *> 
 
 n \ nl , ., ,. ., 0* 
 
 or ' has the limit -^- 2 , 
 
 sin 2 sin 8 ( ATT . - 
 n \ n 
 
 (see p. 18). If then we increase n without limit, we have 
 
 This is a remarkable converging product from which sin# 
 might be calculated. Whatever may be, the successive factors 
 approach to unity, and therefore produce less and less effect. 
 A large number of factors will give a close approximation. Let 
 = kw, and we have the convenient form
 
 THE UNEXPLAINED SYMBOL V-l. 
 
 77-3153563 a- _ 4 16 36 64 100 
 
 ~2'4'16'36'64'" * 2 ~ 3 ' 15' 35 ' 63 ' "99 " 
 This was once suggested as a mode of approximating to the 
 value of TT ; it. proceeds too slowly for that purpose, but it answers 
 another. If we take n factors, we see in the numerator the 
 square of the product of the first n even numbers, but not the 
 corresponding square of the product of odd numbers in the de- 
 nominator. One odd number more is repeated once : thus in 
 the denominator, taking 3 factors, we have 3.15.35 or 1 8 .3 2 .5 2 .7. 
 Accordingly, the larger n is made, the more nearly is this 
 equation true, 
 
 /2.4.6... 2n \ 2 ,.7r 2.4... In 
 
 (l.8.6...2n-ir (2n + 1) 2 i 1.3...2n-l= V(M7r) ' 
 
 for (w-f|)7r and mr have nearly the same square roots, if n be 
 very great. This, in common language, is true if n be infinite : 
 I mean that it may be made as nearly true as we please, if 
 n be large enough. 
 
 If this last equation were absolutely true, this next one would 
 follow, as I shall show, 
 
 e 
 
 But as the premise only approaches to truth as n increases, 
 so it is also with the conclusion. Assume 1.2.3... n = n"<fon, or 
 let 0n be 1.2.3... nf ". 
 
 We have then 1.2.3.4 ... 2 - 1 . 2n = (2n) 2n (2n), 
 2 . 4 ............ 2n = 2V0n. 
 
 Dividing, 1.3.5 ..... 2n - 1 = (2)" . 
 
 <f>n 
 
 2.4.6... 2w ((fmf 
 
 Dmdmg again, ^^^^ = = V(njr) . 
 
 0(2) (0n) a 
 
 Hence we get vi^o = ^ or /(2n) = w > 
 
 i 
 
 /?i standing for <jm -f V(2w). Let (/)" be i^, 
 
 then / = (Yrw)" and (f 2)* = (fn) s " or ^ 2 = y- 
 Consequently, Y'" is either a constant, or if a function of , 
 
 G
 
 62 INTRODUCTION OF 
 
 does not change when n is changed into In. Such functions 
 
 (lo2T M\ 
 %TT - I . or any function of it 
 log 2/ 
 
 except an inverse trigonometrical one. If, however, n be very 
 great, log n increases very little when n is increased by a few 
 units. For and about one large value of n, tyn is then nearly 
 a constant; and, assuming it constant, we shall be able to show 
 that the way of determining that constant gives the same thing, 
 whatever the value of n may be. But this assumption of ^n 
 = constant, say c, renders the proof imperfect, and a more per- 
 fect one is beyond those who may be expected to read this 
 work. Take tyn = c, then 
 
 fn = c", $n = V( 27 ) c "> and i- 2 - 3 = V( 27r ) <>"" 
 For n write n + 1, and divide by the former result, which gives 
 
 //. 1\ /. l\ n 
 = \l 1 + - .c. 1 + - . 
 V \ / \ y 
 
 .. . 
 ^(wn) c n 
 
 If be very great, this would give 1 = 1 x c x e very nearly 
 (Algebra, p. 225), or c = e" ; so that 
 
 1.2.3 ... n = ^(l-n-n) . e"" . n" nearly, as asserted. 
 
 This formula succeeds very well, on trial, and the first side 
 is found greater than the second in about the proportion of 
 12w + 1 to 12;?. 
 
 Returning now to the form sin&/7- = &7r (1 - 2 )..., for k write 
 2k, and we have 
 
 . , 
 
 But sin for = for 
 
 /. 4&*\ /, 
 
 ( T) I 
 
 \1 JX 
 
 _)(!-_)... 
 
 a corresponding expression for the cosine. From these factorial 
 expressions it is in our power to find series for the logarithms of 
 trigonometrical functions. Let S n represent the series 
 
 r" + 2-"4-3-" + 4-" + ... ad inf. 
 It is easily seen that we have 
 
 S n = I- 4 3-' + 5^ + ... + 2^ {1 + 2-" + 3-" + ..} 
 or 1-" + 3^ + 5^ + 7^ + ... = (1 - 2-") S n .
 
 THE UNEXPLAINED SYMBOL V -1. 63 
 
 Values of S, t to a sufficient extent may be found in my Dif- 
 ferential Calculus, p. 554. Now 
 
 / 7c z \ 
 log sin JCTT = log k-n- + log (1 - k*) + log f 1 - j + ... 
 
 Expand the several logarithms, after the first, and we have 
 
 k* k 6 k s 
 log sin k^ = log kir - SJc 2 - S t -- - S 6 - S 8 - . . . , 
 
 which is convergent when k<\, and very convergent when k < -J, 
 and, for purposes of calculation, it need never be greater. Apply 
 the same method to the expression for cos kir, and we have 
 
 log cos k^ = - (2* - 1) SJP - (2 4 - 1) 4 -y- (2 s - 1) flUr - ..., 
 
 2t o 
 
 which is convergent when k is less than . 
 
 From these series we get 
 logtanA;7r = logA7r+(2 8 -2)^ 2 T(2 4 -2)^| 4 +(2 6 -2) 6 | 6 + ... 
 
 It will be observed, that &,-!, S^-l, &c. diminish very rapidly. 
 We take advantage of this by throwing the series into the form 
 
 log sin k-n- = 
 
 "We have seen that trigonometrical language affords a brief 
 mode of expressing, in language derived from obvious geome- 
 trical ideas, complicated algebraical relations. The following is 
 a striking instance: 
 
 # . # xl n xl n x I x . x 
 
 sin a; = 2 cos - sin- =2 cos - (2 cos -(2 cos -...1 2 cos sin 
 
 sin x - 2" sin - cos - cos cos cos . . . cos . 
 
 Increase n without limit, and 
 
 sin a; x x x x x 
 
 = cos - cos - cos - cos cos ... ad inf. 
 
 x i 4 o lu >- 
 
 Let x = %7r, and show that this is then only an abbreviated form 
 of the following, 
 2 = V2 V(2 + V2) 
 v~ 2 2
 
 64 INTRODUCTION OF 
 
 The student who is acquainted with the theory of equations 
 may be enabled to express the logarithmic series in another form. 
 The rest of this chapter is briefly given, and may be looked on 
 as a succession of exercises. It appears from 
 
 that sin ATT is formed from its radical factors after the manner of 
 an algebraical expression, so that 1 - 4- . . . = may be con- 
 sidered as an equation of infinite dimension whose roots are 
 + TT, - TT, + 27T, - 2?r, &c. Write k for k 2 , and we have in 
 
 A k * k * 
 ~ 2^3 + 2.3.4.5 ~ 2.3.4.5.6.7 + 
 
 an equation whose roots are Tr 2 , (27r) 2 , (3/T-) 2 , ... Hence we easily 
 get the following theorems : 
 
 ,&c. 
 
 2 2.3' m 2 .w 2 2.3.4.5' * P.m*.n* 2.3.4.5.6.7 
 
 where in 2(1 -f P.m*.n s ), we understand that there is a term of 
 the series for every possible combination of a product of three 
 different integers. And by the known theorem for the reciprocals 
 of powers of roots of an equation, we have, V n standing for 
 
 Vl ~23 = ' F2 ~2i + 2. 3.4.5 = ' ^ ~ 2^3 + 27^5 ~ 2T77 = ' 
 and so on. Calculation of a few of these results will give 
 
 21 2 3 1 2 5 1 
 
 V = - - V = V = 
 
 1 1.2'6' 2.3.4 '30' 2.3.4.5.6*42' 
 
 and so on. Now V n is S^TT^" (page 62), and the fractions 
 
 belong to a set which are called Bernoulli's numbers, and are 
 
 denoted by B v JB 3 , b , &c., so that 
 
 *"~ 2 1.2.3... 2 "-' 
 n_i_i P-i R-i - i n-ooj. R -i 
 
 JJ 3~3Oi x *5~41!' -"7~"30> * J ~66> x 'll~273O) -"l3~~6 
 
 all which will be found by continuing the above process. These 
 numbers appear to follow no law, which exhibited as rational 
 fractions ; but when exhibited under a law, as in 
 1.2.3.. .2n fill
 
 THE UNEXPLAINED SYMBOL V-l. 60 
 
 it would be thought very unlikely that they should be rational 
 fractions. 
 
 Substitution, and writing x for JCTT, now gives 
 
 2B l 2 3 a: 4 21B x" 
 
 logsma; = losx- - x 
 
 logcosa; = 
 
 - 
 
 1.2 1.2.3.4 2 1.2.3.4.5.6 3 
 
 (2 4 ~ 1)21B S x 4 (2 6 -l)2 5 J9 5 8 
 - - --- - - - - -- * - ' - - 
 
 1.2 1.2.3.4 2 1.2.3.4.5.6 3 
 
 2 
 
 Next, we have 
 
 log cos (a; -f A) - log cos re = log (cos A - tana;, sin h) 
 
 = log cos h + log(l - tana; . tanA), 
 log sin (a; -f- A) - log sin a; = log (cos A 4- cot a: . sin h) 
 
 = log cosh + log(l + cota: . tank). 
 
 From the series it is obvious that log cos A -f h diminishes without 
 limit with h. Also it is easily deduced from page 17, that 
 
 log(l + P tan A) f h, or P(tanA + A) - (P 2 tan% v 2A) + . .. 
 has P for its limit when 7 is diminished without limit. Hence, 
 dividing the preceding equations by h, the limits are - tana: and 
 cota:. Perform the same process on each term of the series for 
 log cos a: and logsina;: that is, change x into x + k; subtract the 
 term unaltered, and divide by h, retaining only the limit; and 
 thus deduce the equations 
 
 cota: = 
 
 x 1.2 1.2.3.4 1.2.3.4.5.6 
 1 a; a; 3 2a: 5 x 1 2x 9 
 
 a: 3 45 945 4725 93555 
 
 ^ I 3 
 
 " 
 
 1.2 1,2.3.4 1.2.3.4.5.6 
 
 17a: 7 62* 9 
 
 3 15 315 2835 
 
 p*V-l _(_ p-*V-l 
 
 Take 
 
 and thence, writing -x\j-\ for a:, show that 
 
 - 1 x 2 2 2.3.4 2.3.4.5.6 
 
 63
 
 CHAPTER VI. 
 
 ON THE CONNEXION OF COMMON AND HYPERBOLIC 
 TRIGONOMETRY. 
 
 THE system of trigonometry, from the moment that -J- 1 is 
 introduced, always presents an incomplete and one-sided appear- 
 ance, unless the student have in his mind for comparison (though 
 it is rarely or never wanted for what is called use),, another system 
 in which the there-called sines and cosines are real algebraical 
 quantities. This other system will serve to explain the connexion 
 between logarithmic and trigonometrical functions. 
 
 In the ordinary system, a given revolving line, of a unit length, 
 has one extremity in a circle ; and on that circle every radius has 
 its projections connected by the equation x* -f y 1 = 1. Suppose we 
 take all possible points so placed that the projections of their 
 values of r are connected by the equation x*-y*= 1. Those points 
 are all that are in a curve of the following form, called the 
 equilateral hyperbola, (a curve corresponding, among hyperbolas, 
 to the circle among ellipses ; in fact the circle ought to be called 
 
 the equilateral ellipse). The two lines towards which the branches 
 of the curve approach without end, but which they never meet
 
 COMMON AND HYPERBOLIC TRIGONOMETRY. 67 
 
 (called asymptotes), are at right angles to each other, and midway 
 between the principal axes. From any point P draw PK per- 
 pendicular to an asymptote. Let OK = v, KP = w. Then it is 
 easily seen that x = v cos 45 -f w cos 45 = f V2 ( + w), and that 
 y = v sin 45 - w sin 45 - \ \J1 (v - w) : whence 
 
 i- (v + w)* - \(v - wf = 1 or Ivw = 1. 
 
 Now take one of the asymptotes and the curve that falls above 
 it, and take two portions of the area standing on bases which are 
 to one another as their initial distances from the centre ; that is, 
 let OK KL :: OK 1 K'L'. Divide each of the bases KL, K'L' 
 into n equal parts, and draw perpendiculars and inscribed rect- 
 angles in the manner shewn in the figure. 
 
 Let OK=v, KP = w, KL = t; each subdivision is - , the 
 
 n 
 
 ?n th subdivision ends at v + from the centre, so that the altitude 
 n 
 
 of the ??* th rectangle is 1 -^ 2 (v + ) and the area of the m th 
 
 \ n r 
 rectangle is 
 
 But in the second area - is the same as in the first : there- 
 
 
 
 fore the m th rectangle of the first is equal to that of the second ; 
 and the sum of all the rectangles of the first is equal to the 
 sum of all the rectangles of the second. Now the area KPQL 
 is composed of the rectangles, and of curvilinear triangles : 
 these last, put together, fall short of a rectangle having the 
 subdivision of KL for its base, and the fixed excess of KP 
 over LQ for its altitude. Therefore, as the subdivision diminishes 
 without limit, the sum of the curvilinear triangles diminishes 
 without limit: that is, the curvilinear area is the limit of the
 
 68 ON THE CONNEXION OF COMMON 
 
 sum of the rectangles. And as the limits of equal quantities 
 are equal, the curvilinear areas KPQL and K'P'OJL are equal. 
 The area KPQL, then, depends only on the ratio of OL and OK. 
 
 Next, OK being v, let the area APDK (A being the vertex 
 in the first figure) be A, and let v = (A). Put on an area 
 QLEF equal to ADKP (or A) and let ADLQ be B; so 
 that ADEF is A + B. Then, ADKP and QZ.EF being equal, 
 we have OD : OK :: OL : OK 
 
 Let OZ) = n, and this is m : <pA : : 0J? : (A + 5), or 
 
 = ^ . ^ , whence (Algebra p. 204), 04 = me* ; 
 m ? n 
 
 or w = me 4 , whence c is to be determined. Also, OA = 1, and 
 /_AOD = 45, whence m = |V2. Hence ^=(logw- log|V2)-^ log c. 
 To determine c, observe that if we increase v by h, the increase 
 of the area consists of a rectangle and a curvilinear triangle ; 
 and, h diminishing without limit, the ratio of the curvilinear 
 triangle to the rectangle diminishes without limit. So that the 
 ratio (increase of curvilinear area -f- inscribed rectangle) has the 
 limit unity. Now the increase of the area is 
 
 log(o+A)-logV2 _ log v - log h/2 or J_ bw / h\ t 
 logc logc logc \ v/ 
 
 and the area of the rectangle is h ; 
 
 2 (v + h) 
 
 2(v + h) 1 (h 1 h 1 \ 
 
 the ratio therefore is *-= .= - + ..., 
 
 ft log c \v 2 u 2 / 
 
 and the limit of this is 2 -f log c, which must be 1 ; therefore 
 logc = 2, Accordingly A = %\og(-j2.v). The logarithms used 
 are, as is always supposed when nothing to the contrary is 
 mentioned, the Naperian, or hyperbolic logarithms : and they 
 got the second name from their connexion with the hyperbola, 
 the fact that all other systems are equally connected with the 
 hyperbola not being seen when the name was given. 
 
 We can now find the curvilinear area APN. The area 
 DKPNA is made up of the rectilinear areas DKPA and APN, 
 and is therefore DK (DA + KP) + ^AN.PN; 
 
 or i (v - i\/2) ( v'2 4- tc) + l-(x - l)y 
 
 -] V2 (v - w) - J + \xy - \ y = \ xy :
 
 AND HYPERBOLIC TRIGONOMETRY. 69 
 
 since 2vw = 1, and \ \ ; 2 (v - to) = y. That is, the rectilinear areas 
 ONP, DKPNA are equal; take from both the curvilinear area 
 APN, and there remains the sector APO, and the area DKPA 
 (Hog (V2.v)}, equal. Call the former S, and we have 
 
 S = I- log (V2.o) = log (x + y) = i log (z + V(X - 1)}- 
 Accordingly we have x + y = e'"' 5 , x-y - e'~ s 
 
 2 2 
 
 If we now turn back to the circle, and if S be the area of 
 the sector whose angle is 0, we have, the radius being 1, 
 
 S = - , or = 2S. But now, (r being =1), x = cos0, y = sin#, 
 
 m 
 
 and we have 
 
 ,-S-J-l , _-2,$V-l JJ.SV-1 _-2,$V-l 
 
 4-6 66 
 
 2 ' y= ~2V : 1 
 
 If, in the hyperbola, we choose to call the numbers representing 
 x and y the hyperbolic cosine and sine of the number of square 
 units in twice the sectorial area; we have, 2$ being (which 
 is not now derived from an angle), and the difference of system 
 being marked by capital letters in the words sine and cosine, 
 
 a a ti O Q O 
 
 c" 4. c~" c" p~" c" p~" 
 
 Cos6>=-^ , Sin6> = - , - " 
 
 From this it may be deduced that in order to convert a formula 
 of circular trigonometry into one of hyperbolic trigonometry, 
 when no inverse functions enter, we have but to change cos# 
 into Cos#, and sin0 into V-l Sin0. The following are a few 
 of the results : 
 
 Cos 2 6> - Sin 8 = 1 Cos (0 9) = Cos0 Cos0 + Sin0 Sin0 
 
 Cos 2 + Sin 2 ^ = Cos 20, Sin (0 0) = Sin0 CosO Cos0 Sin6 
 
 Cos"0 = ~ (Cosn(9 + nCos(n - 2) Q + n ^- Cos( - 4)0 + ..A 
 2 (_ 2 J 
 
 Si n "0 = -^ {CosnO + nCos(w - 2)9 + ...... } (n even) 
 
 Z 
 
 Sin"0=-^ {Sin0 4- M Sin (w- 2)0 i- ...... } (n odd)
 
 70 COMMON AND HYPERBOLIC TRIGONOMETRY. 
 
 This is sufficient to illustrate the analogy which exists between 
 the two systems. The advanced student may investigate the 
 connexion of the conjugate hyperbola with the trigonometry 
 in which the fundamental equation is Sin*0 - Cos 2 6> = 1. 
 
 If we now take five independent equations from page 11, say 
 
 tan#=- , tan#cot0 = l, cos0sec<?=l, sin0 cosecfl = 1, 
 
 COS0 
 
 cos 2 + sin 8 = 1 ; 
 
 it is plain that the first four may be considered as equations 
 of definition or introduction for tan#, cot6>, sec#, cosec6>; and 
 that, speaking of its operations merely, trigonometry is the treat- 
 ment of the equation # 2 + /* = 1. Now as this equation might 
 be supposed to arise from many different sources, it may be 
 worth while to inquire how much of what precedes is due to 
 this form, and how much to the application of this form to the 
 circle, or to angular revolution. 
 
 If we take the two following equations, 
 
 2 2V-1 
 
 we are not bound to either, by assuming y? -f y* = 1 : but if 
 we take one, we must accept the other, as will appear on trial. 
 And then we shall find that all the direct formulae of trigonometry 
 follow, as soon as we require that x and y shall take the names 
 of sin and cos : the inverse forms depend in some measure 
 on the meaning of a. Let a take the form v ~ , and we then 
 regain the application of angular revolution.
 
 CHAPTER VII. 
 
 ON THE TRIGONOMETRICAL TABLES. 
 
 THE usual trigonometrical tables are given in conjunction with 
 tables of logarithms ; and they more frequently give logarithms 
 only than cosines, &c. themselves. When logarithms were in- 
 vented, they were called artificial numbers ; and the originals, 
 for which logarithms were computed, were accordingly called 
 natural numbers. Thus, in speaking of a table of sines, to ex- 
 press that it is not the logarithms of sines which are given, but 
 sines themselves, that table would be called a table of natural 
 sines ; and the logarithms of these would be called, not logarithms 
 of sines, but logarithmic* sines. 
 
 All trigonometrical tables with which the student is likely to 
 meet, natural or logarithmic, are constructed as follows : 
 
 1. They include only the first right angle, or from to 90. 
 If cos 96 be wanted, - sin 6 must be found ; or sin 6 in the 
 table must have its sign changed. If cos 96 be wanted in mul- 
 tiplication, &c., the logarithm of sin 6 must be used, and the 
 effect of the negative sign must be properly attended to in tlie 
 
 final result. 
 
 2. The arrangement is always what may be called semi-qua- 
 drantal: the table goes only as far as 45, and that for the 
 remaining half of the right angle is seen by turning the table 
 upside down, or reading from the bottom of the page instead 
 of the top. There is an imitation of this in the arrangement 
 in page 17, in which y'3, which is both cos 30 and sin 60, 
 is read as the former by the top and the right, as the latter by 
 the bottom and the left. Open the table so as to get tangent 
 
 * This leads to confusion in the minds of students, who learn 
 some notion of mysterious identity between a number or fraction 
 and its logarithm ; and write down '30103 = 2. The phrase is as in- 
 correct as royal country would be for king of the country, or con- 
 stabulary pariah for constable of the parish.
 
 72 ON THE TRIGONOMETRICAL TABLES. 
 
 of 37 15', and there will be seen, reading from the top and down- 
 wards, tangent of 37 15'; but reading from the bottom and 
 upwards, cotangent of 52 45'. It would perhaps have been better 
 if the sines had run on to 90, and then all the cosines W 7 ould have 
 been in reverse readings : but the present mode is too firmly estab- 
 lished to be shaken. 
 
 In consulting the table inversely, for example, in searching 
 for the angle which has 9-61723 for the logarithm of its sine, 
 the student must not distinguish sine from cosine, nor tangent 
 from cotangent, but must consider sines and cosines as one table, 
 tangents and cotangents as one table, and must cast an eye on 
 both, and get to 9-61723 as fast as he can. For want of this 
 caution, some beginners will turn over page after page, until 
 they come to 45, and then back again, perhaps to the very page 
 that was first opened. 
 
 3. The trigonometrical tables in use were constructed on the 
 system described in page 18, the radius being 10' or ten thou- 
 sand millions. Hence the logarithm of the radius was 10, and 
 that of most sines used 9 and a fraction, 8, 7, and even 6 occurring 
 towards the beginning or end of some tables. This has never 
 been altered; and the consequence is that every logarithm in 
 the tables is too great by ten for us. For that which we call sin 9, 
 is sin 6 10 10 of the tables. Hence, in all cases, 
 
 Real Logarithm = Tabular Logarithm - 10. 
 
 Thus, where the tables say 9-61628, we must* take out T.61628, 
 or 9-61628 - 10: where they give 12-61628, we must take 2-61628. 
 Some tables only increase by 10 where the characteristics are 
 negative ; and give 9 for 1, but do not alter 0, 1, 2, &c. 
 
 When the process is inverse, the logarithm should be made 
 tabular before entering the table with it. Thus, for finding the 
 
 angle whose sine is 2-41729, we should enter the table with 
 8-41729. 
 
 * Many calculators prefer to consider the actual tables as if 
 formed upon the fiction of always avoiding negative characteristics 
 by increasing each of them by 10 ; and actually use the tables, 
 making corrections in the results. For myself, I feel assured that 
 the student should be taught by real logarithms, and left to find his 
 own way to the other practice, which I much doubt his doing.
 
 ON THE TRIGONOMETRICAL TABLES. 73 
 
 The tables* adopt various numbers of decimal places ; usually 
 five, six, or seven. Five-figure tables are exact enough for or- 
 dinary use : they may be considered as calculated to give results 
 within the 10 th part of a minute, or 6". Those for whom five- 
 figure tables are not sufficient, should use seven-figure tables : 
 the six-figure tables are best for those who have much to do 
 for which five figures is hardly correct enough. 
 
 In every table we use the words argument, interval, function, 
 and difference. The argument is a technical term for that with 
 which we enter a table, and opposite to which we expect to find 
 the value of a function of that argument. Thus, in one table 
 angles and the logarithms of their sines are paired : if we have 
 a specimen of either, and want to find the corresponding one of 
 the other kind, that with which we enter is the argument, and 
 the other its function. The interval of the tables is the difference 
 between the successive values of the principal argument, which 
 values are always equidistant. Usually, one minute is the tabular 
 interval of angle ; that is, the tables furnish trigonometrical 
 functions (or their logarithms) for 0, 0!', 02'...1, 11'...2...90: 
 which I should describe as being of the class (!') 45; the 
 table being really in two halves, one of which is only the reverse 
 reading of the other. But there are tables of the following de- 
 scriptions : (10") 45, (1") 45, (1") 3 (!') 45, &c. 
 
 The differences of a table are the successive differences of the 
 functions belonging to the equidistant arguments. Thus, if oppo- 
 site to 0, 6 + h, 9 + 2h, 6 + 3h, &c. we have p, q, r, s, &c., the 
 differences are q - p, r - q, s - r, &c., and q -p is technically 
 called the difference of p, r - q the difference of q, &c. 
 
 The use of these differences lies in what is technically called 
 interpolation, which is the mode of solving this question : Given 
 the tabular function for Q, d + h, 6 + 2 h, &c., required the proper 
 function for 6 + a given fraction of h. If the several differences 
 be equal, or very nearly equal, as q-p = r-q-s-r, &c. ex- 
 actly or nearly, the differences only are wanted, and the differences 
 of the differences, &c. may be neglected. In this case we may 
 
 * Of ordinary tables, Hutton's (which have gone through many 
 editions) are the best of seven figures ; Farley's, of six figures ; 
 Lalande's (reprinted by Taylor and Walton), of five figures. 
 
 H
 
 74 ON THE TRIGONOMETRICAL TABLES. 
 
 be said to use interpolation of the jirst order ; and this is all that 
 will be wanted here. 
 
 The success of interpolation of the first order depends upon 
 the following theorem : If (fix be a function of x, and k a small 
 quantity, then, for every function of x, (x + k) is very nearly 
 equal to fix + <p/x . k, where 0'a; is another function of x, depend- 
 ing upon fix for its form and character. I leave the student to 
 establish the following theorems, all nearly true when k is small, 
 and the angles (or k at least) are in arcual units : 
 
 sin (x + k) = sin x + cos x . k, 
 
 tan (x -t- k) = tan x + 
 
 k 
 
 sin x 
 
 (1 * OAH ! , 
 
 x + k) = sec a; + . A, 
 
 cos (a; + k) = cos x - sin 
 
 cot(ar + k) = cotaj- 
 
 k 
 
 cosec(a; -t- k) = cosec x - 
 
 cos a: 
 
 But the second and third are not approximately true when cos a: 
 is small, nor the fifth and sixth when sin a: is small. 
 
 K k be a minute or a fraction of a minute, the angle in arcual 
 units is sufficiently* expressed by k sin 1': and then we have 
 sin (x -t- k) = sin x + cos x . sin 1'. k, 6cc. 
 
 The mode of interpolation is the same as to all tables. Say 
 that (x + k) = (fix + fix . k, very nearly : let h be the tabular 
 interval, and let it be required to find (x -f /tA), /t being a 
 given fraction. We have then (a; + A) = 0a: -f fix . ^, or the 
 tabular difference is fix. A, very nearly. But fi(x+^ih) = fixifi'x.fj.h, 
 or /t x tab. diff. is to be annexed, with its proper sign, to 0a:. Were 
 it not for calling attention to this theorem, which is often wanted, 
 the interpolation might be more simply explained. Take the 
 log. sine of 3 18' and 3 19'; we find S'76015 and 8-76234, giving 
 tab. diff. = -00219: the table calls it 219, implying that thi 
 number and its results are to be applied in the last places. And 
 this difference, or one very near it, runs on. We may then oon- 
 
 Since small angles, arcually expressed, are very nearly equal 
 to their sines, k x (arcual expression of one minute) and k sin 1', 
 are very nearly the same, if k minutes be a small angle. In astro- 
 nomical books, n sin 1', n sin 1", &c. are very common substitutes 
 for n n" &c. It may be worth while just to notice that, as to 
 trigonometrical functions, it matters nothing how the angles are 
 expressed: sinl (arcual unit) and sin 57 17' 44" -8. . are the same.
 
 OX THE TRIGONOMETRICAL TABLES. 75 
 
 sider the last places as augmenting at the rate of 219 per minute. 
 Accordingly, for half a minute we say 120, for 7-10 ths of a minute 
 153, and so on. And it is very important to notice that sines, 
 tangents, secants, have positive tabular differences, while co-sines, 
 co-tangents, co-secants, have negative ones. If we want to find 
 log cos 81 13'-6, we must, for the augment of 0'-6 to the angle. 
 subtract 6-tenths of the tabular difference. Neglect of this caution 
 causes more error to beginners than anything else in their use 
 of the tables. 
 
 The state of the tabular difference shows what degree of 
 accuracy the tables are prepared to yield. In a table of five 
 places, the smallest change which the table can indicate is a 
 unit in the fifth place of decimals. Now at sin 3 18', there are 
 219 such units in the tabular difference; and each one belongs 
 to the 219 th part of a minute, about the fourth of a second. 
 When the answer is about 3 18', to be determined by its sine, 
 the problem may be solved by five-figure tables within one 219 th 
 of a minute. Accordingly, when there is choice, it is best to 
 go to those parts of the table at which the differences are largest ; 
 shun cosines for small angles, and sines for angles near to a 
 right angle. Again, at the beginning of the sines, the end of 
 the cosines, and both the beginning and end of the tangents, the 
 differences change very rapidly, and the differences of the dif- 
 ferences become of importance. The use of the ordinary table is 
 generally avoided in these cases, and in the following manner. 
 
 1. Generally speaking, a table of sines with smaller interval 
 is annexed, extending over all or great part of the first degree. 
 And the tangents and sines are very nearly equal : up to half 
 a degree there is no practical separation. 
 
 2. At the end of the tangents, the best way is to use the 
 tangent of the complement, which is very small, and has very 
 nearly the same logarithm as the sine of the complement. For 
 instance, I want to find with accuracy to five figures the tangent 
 of 89 46' -18, using the English reprint of Lalande. This is the 
 complement of 13'-82, the tangent of which cannot be distinguished 
 from its sine. Looking into the second small table at the end, 
 I find 7-60360 and 7-60674 for log sin 13' -8 and log sin 13' -9 
 (tabular interval O'-l). The tabular difference is 314, belonging 
 to O'-l ; for 0'-02 I must take 2-10 th * of this, or 63, which added
 
 76 ON THE TRIGONOMETRICAL TABLES. 
 
 to the last places of 7-60360, give 7*60423; or really, 3-60423. 
 This is the real logarithm of the cotangent : that of the tangent 
 is 2-39577. The tangent then is 248-74. 
 
 I want to find the angle whose tangent is 3174. Its logarithm 
 is 3-50161; that of the tangent (which confound with the sine) 
 of the complement is 4-49839 ; in the tables 6-49839. In the first 
 of the small tables appended to the English reprint of Lalande, 
 6-49175 and 6-49849 (with difference 674) belong to 1'4" and 1'5". 
 But our unattained part* is 664 ; and 674 for the whole second 
 gives 664 for 664 4- 674, or -985 of a second. The complement 
 of the angle required is then 1' 4" -985, or the angle is 
 89 58' 55" -01 5. 
 
 Nothing but actual practice can give expertness in the use 
 of the tables. I should recommend a student to begin by verify- 
 ing some formulae. For instance (page 29) the sum of the 
 tangents of the three angles of a triangle is equal to their product, 
 since tan 180 = 0. Choose three angles which make 180, find 
 the tangents from their logarithms, and add : add the logarithms, 
 and find the natural number to that sum : the two results ought 
 to agree. Choose a, b at pleasure, and calculate (a + 6) f (ab - 1). 
 Find the angles to which these three are tangents: their sum 
 ought to be 180. 
 
 On the construction of trigonometrical tables I shall say no 
 more than to show the student that such a thing is possible 
 without any impracticable amount of calculation. If tables were 
 now to be constructed, methods derived from the calculus of 
 differences, which I cannot here describe, would take the place 
 of those which I mention. But even these last are much more 
 easy than those to which we owe the tables in use. 
 
 If we really wanted to find the sine and cosine of one minute, 
 which, arcually expressed, is -0002908882, we should easily find 
 that O 7 4- 2.3.4.5.6.7 has no significant figure in the first twenty 
 decimal places. If twenty places were enough, the following 
 would be quite sufficient : 
 
 * This is a phrase which I use for the excess of what we want 
 to find over the nearest below it which we can find. Thus if the 
 log. tangent of the angle I want to find be 10-37466, and the nearest 
 underneath it in the tables is 10-37461, the unattained part is 5.
 
 ON THE TRIGONOMETRICAL TABLES. 
 
 Next, we have 
 
 cose + sin0 = Vtcos'e + sin'e + 2 sine cose) = V(l + sin 20), 
 
 cose - sin0 = V(cos*e + sin 2 6> - 2 sine cose) = V(l - sin 20), 
 
 when e < JTT, both square roots must be taken positively; and 
 we have 
 
 cose = | V(l + sin 20) + | vt 1 - sin 26) ; cos 9 = ^ (1 + cos 20), 
 sine = f V(l + sin 2e) - J V(* - sin 25) ; sin = VI C 1 - cos 2^). 
 
 So that, either the sine or cosine of an angle being given. 
 both the sine and cosine of its half can be found by two 
 extractions of the square root. 
 
 Now (page 26) we may assume that we start with the sine 
 and cosine of 3, 6, 9, fully expressed for calculation. Thus 
 we have (proceeding as directed in page 26), 
 
 cos 30 = (V3 + 1) (Vo-1) + (V3 + 1) V(5 + V5) 
 sin 8 V2 8 
 
 Hence the sines and cosines of all the multiples of 3 may 
 be calculated first, as verifications of the process. Having de- 
 termined sin V and cos 1', it is now possible, by the formulae 
 sin (x + 1') = sin x . cos 1' + cos x . sin 1', &c., to calculate the sines 
 and cosines of 2', 3', &c., up to 2700' or 45; which complete* 
 the table of sines and cosines, from which the tangents, &c. 
 may be calculated by division. 
 
 Much shorter methods might be introduced, as before re- 
 marked, from the calculus of differences. But even from common 
 formulae, the above labour might be considerably reduced. I 
 leave the student to prove the following formulae : 
 
 \ *>/ \ 
 
 sin(30+ 0) = cos - sin (30 - 0), cosectf = tan0 + cote, 
 
 tan (45 + 0) = 2 tan 2e + tan (45 - 0), sec = tane + tan ^45 - fY, 
 
 \ 2/ 
 
 cosece = cote i tan - . 
 
 m
 
 78 ON THE TRIGONOMETRICAL TABLES. 
 
 From which we gather that when all sines and cosines up to 
 30 are calculated, the rest can be found, the sines by simple 
 subtraction, the cosines by one multiplication only : that when 
 the tangents are found up to 45, the rest can be found by simple 
 addition : and that all the secants and cosecants can be found by 
 addition only, from the tangents and cotangents. 
 
 The student may also prove the following formula, which is 
 often cited as a mode of verifying the tables, by instances selected 
 at hazard, 
 
 cos (36 + 0) + cos (36 - 6) = cos0 + sin (18 + 0) + sin (18 - 6).
 
 CHAPTER VIII. 
 
 ON THE SOLUTION OF TRIANGLES. 
 
 THIS subject, in which (and in spherical trigonometry) 
 trigonometry was first constituted a distinct branch of mathe- 
 matics, is now of little importance in a general course of mathe- 
 matics. It consists mainly in the finding of convenient formulae 
 for the answer to the different cases of the following question : 
 Given some parts of a triangle, to find the rest. This is called 
 the solution of a triangle. But, in truth, the method given is not 
 a solution of the problem, but a reduction of it to the solution 
 of a right-angled triangle. And the maker of the tables it is who 
 solves the right-angled triangle, rather than the user of them. The 
 former registers, for every acute angle which consists of an exact 
 number of minutes, all the proportions of the sides of a right- 
 angled triangle which has that angle for one of its angles ; and 
 thus gives all the factors necessary to convert any known side 
 into another before unknown. The latter makes use of the 
 register, calls himself the sole solver of the triangle, and learns 
 an inaccurate conception of what he has been doing. 
 
 Let the sides of a triangle contain a, b, c linear units; and 
 let the opposite angles, gradually measured, be A, B, C. And 
 first, let C be a right angle. By the formation of the register 
 just alluded to, we have 
 
 - = sinA = cosl?, a = c sinA = c cosB, 
 c 
 
 T = t&nA = cotJB, a = 6tan-4 = b cot-B, 
 o 
 
 But the following formulae should be remembered in words. 
 side = hypothenuse into sine of opposite angle, 
 side - hypothenuse into cosine of adjacent angle, 
 hypothenuse = side by sine of opposite angle, 
 hypothenuse = side by cosine of adjacent angle, 
 
 side = other side into tangent of opposite angle, 
 side = other side by tangent of adjacent angle.
 
 80 
 
 ON THE SOLUTION OF TRIANGLES. 
 
 The following are the cases which occur, and the formulas 
 of solution : 
 
 Given 
 
 
 
 
 a, 6 
 
 
 b a 
 
 A 90 
 
 B. 
 
 a 
 
 sinB cosB ' 
 
 
 c, b 
 
 a = ^(c-b.ci 6), 
 
 sinE = - , 
 c 
 
 A = 90 - 
 
 B. 
 
 c, A 
 
 b = c cos^i, 
 
 a = c sin^4, 
 
 B = 90 - 
 
 A. 
 
 a, A 
 
 
 a 
 
 7? 90 
 
 A. 
 
 
 sin A ' 
 
 
 b, A 
 
 , . 
 
 b 
 
 B 90 
 
 A. 
 
 
 cos A ' 
 
 
 The following are the parts of one right-angled triangle with 
 logarithms, for exercise in these formulae, previously to taking 
 other examples : 
 
 c = 128-4327, log c = 21086756, 
 b= 66-1364, log 6- 1-8204405, 
 a = 110-0951, logo = 2-0417681, 
 A = 59 0' 21"-25, B = 30 59' 38"'75, 
 log.sin^ = log.cosJ5 = 1-9330925, 
 log.cos^i = log.sin B = 1-7117649, 
 log.tan^i = log.cot B = 0-2213276, 
 log.tan^ = log.cot A = 1-7786724. 
 
 Special cases sometimes occur in which departures from these 
 formulae may be advisable : as, Given b, A, where A is very small. 
 Here c - b cos A is not an advantageous formula (page 75): 
 but if we take 
 
 , , (l-cos.4) . A 
 
 - b = b - -- - = 2o snr , nearly, 
 2 
 
 since cos^4 is very near unity, we get the excess of c over b very 
 accurately. 
 
 We now proceed to triangles in general. Draw a perpendicular 
 from the angular point of C upon c. If this perpendicular fall 
 within the triangle, it is clear from the definition of a sine that 
 it is b sin A, and also a sm. If the perpendicular fall outside 
 the triangle, either A or B should have its external angle sub- 
 stituted for it : but an internal and its external angle are supple-
 
 ON THE SOLUTION OF TKIANGLES. 81 
 
 ments, and have the same sines. Therefore, in all cases, 
 
 ,. a sin^4 a b . . 
 
 = bsmA, or - = , or-; - = - , ora:b::sinA:sinB. 
 b sm smA smls 
 
 Sides, then, are to one another as the sines of their opposite 
 angles. The angles then being given (or rather, two of them 
 being given, and the third found), the proportions of the sides 
 are found, being those of the sines. 
 
 I shall make this the fundamental formula from which all 
 others are deduced, namely 
 
 = b = c (j). 
 
 sin A sin_B sinC 
 
 Show that each of these three, a -sin A, &c., is the diameter of 
 the circle circumscribing the triangle, found in Euc. IV. 5. 
 
 The angles of a triangle being each less than two right 
 angles, opponents and completions (page 10, note) cannot both be 
 angles of any triangle : but supplements can, that is, one may 
 be an angle of one triangle, and the other of another. When, 
 therefore, an angle is determined by its cosine or tangent, there 
 is but one such angle belonging to the solution : but when it 
 is determined by its sine, there are two angles which may belong 
 to the solution; that is, there may be two distinct solutions. 
 
 Now take the expanded form of sin (A + B), square both sides 
 of the equation, and substitute values for cos 2 ^t and cos'jB; this 
 gives 
 sin 2 (^4f B) = sin s ^4(l - sin a .Z?) + ( 1 - sin 2 ^4 ) sin s .B + 2 sin A sin_Z? cos A cosS 
 
 =sinM -f sin 2 !? + 2 sin A sin.B cos(A -f ), 
 if A, B, C be the angles of a triangle, we have 
 
 A + .5 = 180 -C, sin (A + ) = sin C, cos(A + B) =- cosC; 
 whence sin 2 C = &in*A + sin 2 .B - 2 sin A sin B cos C; 
 divide* both sides by sin* C, for sin A -= sin C and sin B + sin C, 
 write a + c, and b c, and then multiply by c 2 . This gives 
 
 c 8 = a 2 + 6 s - lab cosC ..................... (2). 
 
 * This process supplies the want of a theorem with which the 
 student ought to be acquainted in its general form. Prove that 
 if an equation be homogeneous with respect to a set of letters 
 p, q, r, &c., that equation remains true if p, q, r, &c. be erased, 
 and p, q', r, &c. substituted, provided that p' is to p as q' to q, 
 and as r' to r, &c.
 
 82 ON THE SOLUTION OF TRIANGLES. 
 
 Show that this proposition is the arithmetical representative of 
 Euclid II. 12, 13; and that the introduction of the distinction 
 of positive and negative quantity prevents our needing two pro- 
 positions. 
 
 As in page 39, we may express the above thus : 
 
 - Ma-tysectaiT 1 v " ...(3). 
 \ a + b I a-b 
 
 The formula (2) may be proved thus : From the vertex of A 
 draw a perpendicular upon a. In all cases it will be seen that 
 each side of a triangle is the sum of the projections of the other 
 two upon it, provided each projection be called positive or nega- 
 tive, according as the angle of projection is acute or obtuse. Thus 
 
 a = b cos C+ c cos B, b = c cos A + a cos C, c = a cos B + b cos A. 
 Now <? = (c cos Af + (c sin Af 
 
 = (b - a cos C)" + (a sin Cf = V - 2ab cos C -f a*. 
 
 The form cos C = "* + f~ * . . . (4) 
 
 2ab 
 
 is often useful. From it we have 
 
 n (a + J) 2 - c 2 , C (a + b + c) (a + b - c) 
 
 1 -I- cos C= -. , cos = - . 
 
 2ab 2 4a6 
 
 n c*-(a-by t C (b -f c - a) (c + a - b) 
 
 1 - cos C = ^r-j '- sin* = . 
 
 2ab 2 4ab 
 
 Let a + b + c = 2s, then 
 
 a + b- c = 2(s-c), b + c-a = 2(s-a), c + a - b = 2(s - b). 
 By substitution we have 
 co s .g = ^ - n .g _(-)(-) 
 
 2 06 2 06 2 s(s-c) 
 
 s(s -b) 
 
 be 2 be 2 s (s - a) 
 
 ...(5). 
 
 Let p = \l - , which it will presently be 
 
 \ $ 
 
 shown is the radius of the inscribed circle (Euc. IV. 4). 
 Show that 
 
 ten = ^' ^2-% 6' tan 2 = ^ C < 6 >>
 
 ON THE SOLUTION OF TRIANGLES. 83 
 
 cos - gives sm^ = >J.s(s-ct)(s-b)(s-c)...(l), 
 
 and similar forms for sin B and sin C. 
 
 If p , p , p , be the perpendiculars let fall upon a, b, c, 
 from the opposite vertices, we have 
 p =5 sinC=c sinB, p=csmA = asinC,p=asmB = bs'mA...; 
 
 "a * b r e 
 
 and the area of the triangle is expressed by any one of the 
 following seven equivalent forms: 
 
 ap a bpi cp c ab sin C be sin A ca sin B 
 T' T' T' 2 ' ~~2 ' 2 ' 
 
 V{s (s - a) (s - b) (s - c)} or ps (8). 
 
 _. . . b sin . a - b sin A - sin B 
 
 The formula -- - gives r = ---. -. , 
 
 a sin A a + b sin A + sin B 
 
 v. * 1 1 A \ ab C 
 
 or tan (A-B)= - cot- ...(9X 
 + o 2 
 
 There are four circles which touch the three lines of a triangle : 
 one, the inscribed circle of Euclid, touches the three sides; of 
 the others, each touches one side and the other two sides pro- 
 duced. Let p be the radius of the first, and p a , p ti p c , those 
 of the other three. The area of the triangle is $ (pa ( pb + pc) 
 or ps : whence p, now used, is the same as p of the preceding for- 
 mula (6). Again, the area of the triangle is (p a b + p a c - /a a a) or 
 ( - a) p a ; whence 
 
 V-s (s - a) (s - b) (s - c) 
 
 P- = > CCL. 
 
 ' s- a 
 
 1111 
 
 Show that - + + - = - . 
 
 Pa Pb PC P 
 
 Let 4 denote the projection of a on b, with its proper sign, 
 &c. Then 
 
 a t - a cos C, &c., b = a a + c 4 , &c., and we have 
 
 We can now treat all the cases of oblique-angled triangles. 
 Of two given sides or angles, let the greater of the two, when 
 there is one, be denoted by the prior letter of the alphabet
 
 84 ON THE SOLUTION OF TRIANGLES. 
 
 1. Given the three sides to find the angles. 
 If one angle only be wanted, say A, take 
 
 A ls(s- a) A l(s-V) (s-c) 
 
 - ~ - -- 
 
 preferring the first for the greatest angle, the second for the least. 
 If all the three angles be wanted (or even two), take 
 
 P = V{(s - a) (s - b) (s - c) -^ s} 
 
 tan \A -p^r(s-a\ tan \B = p -=- (s - V), tan %C= p -f (s - c), 
 which verify each other, since ^A+^-B + ^C- 90. 
 This method was once much used. Since 
 
 (a - c) (a + c) 
 = -- ' 
 
 from which determine a b and c b , and then use 
 
 2. Given two sides and the included angle (a, b, C), to find 
 the rest. If all be wanted, calculate the angles by means of 
 their half sum and half difference, thus, 
 
 \A 4 *B = 90 - |C, tan (\A - \B} = i tan (90 - JC), 
 
 sin C , sin (7 , . , 
 
 Lastly, c = a - b - - , which ought to agree. 
 sin A sin B 
 
 If only the side c be wanted, take either 
 
 . . 2 -Jab cos i(7 ., 2 '/aft sin iC 
 
 c = (a + 6) cos sm ' = (a - 6) -r cos tan ' ; 
 
 a -(-o a - o 
 
 say, for reference, 
 
 (a + b) cos S c and (a - 6) -- cos T c . 
 Or thus, b a = 6 cos (7, p a = b sin (7, which find : 
 
 c a = a -b a , tan -B = ^ , c - ^=, , 
 c cos 5 
 
 which is a direct reduction of the solution to that of right-angled 
 triangles.
 
 ON THE SOLUTION OF TRIANGLES. 85 
 
 3. Given one side (c) and two angles, required the other 
 sides. Calculate the third angle, and then use 
 
 sin A , sin I? 
 a = c -: ^ , o = c . -^ ; 
 sin C sm C 
 
 but if C be obtuse, use A 4- B instead of C. Or, in any case, 
 A + B may be used, taking the cosine of the excess above 90 
 (which excess is easily found without pen and ink) when A + B 
 is obtuse. 
 
 4. When two sides and an angle not included are given, 
 
 (a, b, B), required the rest. 
 
 a sin B 
 
 First calculate A from sin A =- . 
 
 b 
 
 If a sin B > b, sin A > 1, and there is no solution. If a sin B - b, 
 then sin -4 = 1, and A is a right angle, as is its supplement: 
 there is this one solution only, and c = a cos B, C= 90 - B. 
 But when a sin B < b, sin^i<l, and there are two solutions, 
 one acute and one obtuse. Let these be A and A" ; and let 
 C" and C", c' and c", be the corresponding values of the third 
 angle and side ; then C' = 180 - B - A, C" = 180 - B - A", and 
 
 a sin C" b sin C' a sin C" b sin (''' 
 fj ___ m 
 
 sin A sin B ' sin A sin B 
 
 So far it seems as if we were sure of two solutions, when- 
 ever a sin B < b. But in trigonometry we are often made to 
 observe what meets us so frequently in ordinary Algebra, namely, 
 that in constructing the conditions of a problem, we are com- 
 pelled to take in those of cognate problems. If we have not 
 until now met with such a circumstance in this present chapter, 
 it is because in mere solution of triangles, we have not introduced 
 into this isolated subject our conventions as to the measurement 
 of angles, which would, as in page 10, oblige us to consider 
 completions and opponents of Euclid's angles as among the angles 
 of a triangle, and each cosine and tanyent as belonging to two 
 possible angles of a triangle. But each of two supplements may 
 be the angle of a triangle : and when, in our construction, we 
 have to use the sine of a yiven angle, we conjoin with our 
 problem that one in which the supplement of the first given 
 angle is made the given angle. Let B' and B" be the acute 
 
 I
 
 86 ON THE SOLUTION OF TRIANGLES. 
 
 and obtuse supplements : then all we are entitled to say is that 
 the preceding solutions belong, each of them, to one or the 
 other of the triangles (a, b, B) (a, b, B"). 
 
 First, suppose the given angle to be opposite the lesser of 
 the given sides (b < a). Then B < A, and there cannot be any 
 obtuse value of B: that is, both solutions belong to the acute 
 value of B. Secondly, let the given sides be equal (b = a) : 
 then B = A, and A must be acute, and 180 - B - A", or 
 180 - A - A", is = 0. One of the solutions vanishes into a 
 straight line, and the other is an isosceles triangle. Thirdly, let 
 the given angle be opposite the greater of the given sides (b > a). 
 Then B > A, there cannot be any obtuse value of A, but both 
 values of B may be used, and one solution belongs to each 
 value. The following diagram will explain these cases. 
 
 This double solution is, as might be supposed, indicative 
 of the problem being one of the second degree. We have 
 
 b 2 = c 8 + a* - lac cos B, c = a cos B J(W - a* sin 2 B). 
 
 Here a sin B is p c , and -j(tf - o 2 sin 8 B) is b c with its sign, on 
 the supposition that c is measured positively on the side of the 
 acute value of B ; a cos B is a c ; and the above equation is only 
 c = a c + b c , in which c has its proper sign. The consideration 
 of this problem, and of its connexion with Euclid vi. 7, will 
 be a useful exercise. 
 
 The following table shews all the parts of a triangle and 
 their logarithms for exercise. 
 
 a = 15-236 loga = M828710 s - a = 3-098 log(s - a) = 0-4910814 
 b = 12-414 log& - 1-0939117 s - b = 5-920 log( - b) = 0-7723217 
 c = 9-018 logc = 0-9551102 s-c = 9-316 log( - c) = 0-9692295 
 s = 18-334 logs = 1-2632572 P s = 55-96866 logps = 1-7479449 
 
 a f b = 27-650 log(a + b) = 1-4416951 a - b = 2-822 log(a - b) = 0-4505570 
 6 +c = 21-432 log (b -f c) = 1-3310627 b - c = 3'396'log(6 - c) = 0-5309677 
 a . c - 24-254 log(a -f c) = 1-3847834 a - c = 6-218 log(a - c) = 0-7936507
 
 ON THE SOLUTION OF TRIANGLES. 
 
 87 
 
 8. 
 
 T a 
 
 TO 
 T f 
 
 log. sin. 
 
 log. cos. 
 
 log. tan. 
 
 1-9999530 
 
 2-1679268 
 
 1-8320262 
 
 1-9109937 
 
 T-7633479 
 
 0-1476458 
 
 1-7721922 
 
 1-9063714 
 
 1-8658208 
 
 T-8462647 
 
 1-8526583 
 
 1-9936064 
 
 1-6611649 
 
 1-9487989 
 
 1-7123661 
 
 1-4933102 
 
 1-9778520 
 
 1-5154583 
 
 1-4722989 
 
 1-9798951 
 
 1-4924039 
 
 1-2007551 
 
 1-9944564 
 
 1-2062988 
 
 1-6485379 
 
 1-9520365 
 
 1-6965016 
 
 1-8471366 
 
 1-8518083 
 
 
 1-9340361 
 
 1-7091283 
 
 
 1-9755783 
 
 1-5134140 
 
 
 
 1'3408970 
 
 0-6408380 
 
 
 1-6997390 
 
 0-2375348 
 
 
 1-4954464 
 
 0-4821746 
 
 4. =89 9 23-54 
 B = 54 33 25-12 
 =36 17 11-48 
 A = 44 34 41-77 
 B = 27 16 42-56 
 C = 18 8 35-74 
 
 - B) = 17 17 59-21 
 
 - (7) = 9 8 6-82 
 
 - (7) = 26 26 6-03 
 ? = 44 41 30-6 
 ?* = 59 12 50-4 
 
 = 70 57 53-6 
 
 = 77 7 15-5 
 
 - 59 56 28-9 
 
 = 71 45 50-9 
 p a = 7-347 logp a = 0-8661039 
 p b = 9-017 log^ ft = 0-9550632 
 p c = 12-413 log^ c = 1-0938647 
 
 b a = 10-006 logi a = 1-0002831 c a = 5-230 logc fl = 0-7184581 
 a, = 12-281 Ioga 3 = 1-0892424 c 6 = -133 Iogc 6 = 1-1230370 
 a c = 8-835 log a,. = 0-9462189 b c = -183 log b c = 1-2618385. 
 
 Cases may occur in which the particular values of the data 
 render special methods convenient. For instance, when a, b, C, 
 are given, and C= 180 - (7,, C / being very small, we may 
 proceed as follows: 
 
 c 3 = a 2 + b 2 - 2ab cos (180 - C,) = a* + V + 2ab cos C, 
 
 = (a + by - 4a& sin 2 1(7, = (a + by jl - 4fl ^ sin 2 \C\ . 
 
 By the binomial theorem, *J(\ - x) -\ - -\x nearly, x being small. 
 2ab 
 
 very nearly. But sin (7 = 2 sin C . cos CJ , or cos |-(7 ( being very 
 nearly 1, we have 
 
 sin i C, = i sin (7 , sin 8 C, = sin 2 (7, , very nearly : 
 
 , 1 absin*C. 
 
 c = a + o - r , very nearly. 
 
 2 a + b
 
 88 ON THE SOLUTION OF TRIANGLES. 
 
 If the circumscribed circle be drawn, the angle of the radii 
 drawn to the extremities of a, is the angle at the centre to 
 which A is the angle at the circumference. There is, then, 
 an isosceles triangle, in which r and r include either 2A, or 
 360 - 2 A, the third side being a. Consequently, 
 
 a = 2r sin (A or 180 - A) : 
 
 that is, 2r = a 4- sin A. The three sides of the triangle are then 
 2r sin A, 2r sin B, 2r sin C; 
 
 and all the formulae become trigonometrical identities, if these 
 be substituted for a, b, c. Thus, substitution in the formulae 
 for sin 2 ^C gives us 
 
 . t C _ (sin.5 + sinC*- sin A) (sin (74- sinA - sinJB) 
 ~ 
 
 2~ 4 sin A 
 
 which is always true when A -h B + C = 180. 
 
 Shew that the line drawn from the vertex of A bisecting 
 the side a is *J{\(V + c 8 ) - i 2 } or \ V{6 2 + c z + 26c cos A}. Also 
 that the line bisecting the angle A is 2bc cos \A 4- (6 4- c). Shew 
 that 
 
 16s (s -a)(s- b) (s-c) = 26V + 2c 2 o 8 + 2aW - a 4 - 6 4 - c 4 .
 
 BOOK II. 
 DOUBLE ALGEBRA. 
 
 CHAPTER I. 
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 THE object of this book is the construction of Algebra upon 
 a basis which will enable us to give a meaning to every symbol 
 and combination of symbols before it is used, and consequently 
 to dispense, first, with all unintelligible combination, secondly, 
 with all search after interpretation of combinations subsequently 
 to their first appearance. 
 
 In arithmetic and in ordinary algebra we use symbols of pre- 
 viously assigned meaning, from which meaning, by self-evident 
 notions of number, &c., are derived rules of operation. The 
 student must understand by symbols, the peculiar symbols of 
 arithmetic and algebra : strictly speaking, the written or spoken 
 words by which meaning is conveyed are themselves symbols. 
 And symbols must be explained by other symbols, except when 
 they denote external objects or actions, in which case the symbol 
 may be explained by pointing to the object present or the 
 action taking place. Language itself is a science of symbols 
 (namely, words) having meanings (which are described in the 
 dictionary by words of the same or another language) and rules 
 of combination (laid down in its grammar}. 
 
 No science of symbols can be fully presented to the mind, 
 in such a state as to demand assent or dissent, until its 
 peculiar symbols, their meanings, and the rules of operation, 
 are all stated. In this case we have but to ascertain first, 
 whether the peculiar symbols be distinguishable from each other ; 
 secondly, whether the meanings are capable of being distinctly 
 apprehended, each symbol having either one only,- or an attain- 
 able and intelligible choice; thirdly, whether the given rules 
 
 13
 
 90 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 of operation be necessary consequences of the given meanings 
 as applied to the given symbols. If these inquiries produce 
 as many affirmative answers, the basis of the science is so far 
 unobjectionable; and all intelligible conclusions which are drawn 
 from a correct and intelligible use of the rules of operation, 
 are true. But yet it may be imperfect. 
 
 First, it may be incomplete in its peculiar symbols. There may 
 be a want of symbols which those already in use suggest, but 
 which are not made to appear. This is not the incompleteness to 
 which algebra is most liable : it suffers more from its symbolic 
 combinations growing much faster than the ordinary language in 
 which they are, if possible, to be occasionally expressed. 
 
 Secondly, it may be incomplete in its meanings. For example, 
 it may be capable of applying, with the same symbols, to more 
 subjects than its actual meanings take in. This is one possible 
 incompleteness, of a very obvious character. Another, of a much 
 less obvious character, and which probably nothing but actual 
 experience of it would have suggested, is this : symbols, defined 
 in a manner which makes them separately intelligible, may be 
 unintelligible in combination ; their separate definitions may 
 involve what, in the attempt* to combine them, produces con- 
 
 * The student may be surprised at my saying that we should 
 never have imagined such a result in algebra without actual ex- 
 perience of it : for it may strike Mm immediately that in ordinary 
 language AVG may have not merely unmeaning, but contradictory, 
 combinations. But the answer is that we are so accustomed to 
 contradictory combinations, used in some emphatic sense, that 
 they are recognised idioms : it even happens that they express 
 more and better meaning while they are fresh, and before use 
 makes the contradiction wear off, than afterwards. When General 
 Wolfe first used the expression ' choice of difficulties ', which was 
 contradiction, choice then meaning voluntary election, he made 
 those to whom he wrote see his position with much more effect 
 than could have been produced a second time by the same words. 
 Ordinary language has methods of instantaneously assigning meaning 
 to contradictory phrases : and thus it has stronger analogies with 
 an algebra (if there were such a thing) in which there are pre- 
 organized rules for explaining new contradictory symbols as they 
 arise, than with one in which a single instance of them demands 
 an immediate revision of the whole dictionary.
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 91 
 
 tradiction. The second case may be a consequence of the first, 
 or it may not : contradictory combinations may arise from 
 limitation of meaning, and may cease to be contradictory under 
 extended meanings ; or it may happen, either that no such 
 abolition of contradiction is possible in the case thought of, or 
 else that every extension of meaning which destroys contra- 
 diction in one combination creates it in another. 
 
 Thirdly, it may be incomplete in its rides of operation. This 
 incompleteness may amount either to an absolute privation of 
 results, or only to the imposition of more trouble than, with 
 completeness, would be requisite. Every rule the want of which 
 would be a privation of results, may be called primary : all 
 which might be dispensed with, except for the trouble that 
 the want of them would give, may be treated merely as con- 
 sequences of the primary rules, and called secondary. 
 
 Each of the three great objects of consideration, peculiar 
 symbols, assigned meanings, and rules of operation, may then 
 be defective, independently of the rest. Can we carry the defect 
 so far as to imagine one or more of them to be entirely wanting ? 
 The cases of absolute deficiency, which it may be worth while 
 to notice here, principally to accustom the student to the idea 
 of the separation, are as follows : 
 
 1. Meaning* and rides without peculiar symbols. Unques- 
 tionably algebra might be deprived of its peculiar symbols, 
 ordinary words taking their places. There is no more truth, no 
 more meaning, and no more possibility of drawing consequences in 
 
 (a a - 6 2 ) - (a - b) = a + b, 
 
 than in ' the difference of the products of two numbers, each 
 multiplied by itself, divided by the difference of those numbers, 
 gives the sum for a quotient.' Before the time of Vieta, algebra 
 had always been much retarded by the want of a sufficient use 
 of peculiar symbols. 
 
 2. Peculiar symbols, and meanings, without rules of operation. 
 In this case the only process must be one of unassisted reason, 
 thinking on the objects which the symbols represent; as in 
 geometry, which has its peculiar symbols (as AB, signifying 
 a line joining two points named A and ). But no science
 
 92 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 of calculation* can proceed without rules; and these geometry 
 doesf not possess. 
 
 3. Peculiar symbols, and rules of operation, without assigned 
 meanings. Nothing can be clearer than the possibility of dic- 
 tating the symbols with which to proceed, and the mode of 
 using them, without any information whatever on the meaning 
 of the former, or the purpose of the latter. A corresponding 
 process takes place in every manual art in which an assistant 
 obeys directions, without understanding them. The use of such 
 a process, as an exercise of mind, must depend much (but not 
 altogether) upon the value of the meanings which we suppose 
 are to be ultimately assigned. A person who should learn how 
 to put together a map of Europe dissected before the paper is 
 pasted on, would have symbols, various shaped pieces of wood, 
 and rules of operation, directions to put them together so as 
 to make the edges fit, and the whole form an oblong figure. 
 Let him go on until he can do this with any degree of expert- 
 ness, and he has no consciousness of having learnt anything: 
 but paste on the engraved paper, and he is soon made sensible 
 that he has become master of the forms and relative situations 
 of the European countries and seas. 
 
 As soon as the idea of acquiring symbols and laws of com- 
 bination, without given meaning, has become familiar, the stu- 
 dent has the notion of what I will call a symbolic calculus ; 
 which, with certain symbols and certain laws of combination, 
 is symbolic algebra : an art, not a science ; and an apparently 
 
 * A calculus, or science of calculation, in the modern sense, is 
 one which, has organized processes by which passage is made, 
 or may be made, mechanically, from one result to another. A 
 calculus always contains something which it would be possible 
 to do by machinery. 
 
 t Those who introduce algebraical symbols into elementary geo- 
 metry, destroy the peculiar character of the latter to every student 
 who has any mechanical associations connected with those symbols ; 
 that is, to every student who has previously used them in ordinary 
 algebra. Geometrical reasoning, and arithmetical process, have 
 each its own office : to mix the two in elementary instruction, 
 is injurious to the proper acquisition of both.
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 93 
 
 useless art, except as it may afterwards furnish the grammar 
 of a science. The proficient in a symbolic calculus would na- 
 turally demand a supply of meaning. Suppose him left without 
 the power of obtaining it from without : his teacher is dead, and 
 he must invent meanings for himself. His problem is, Given sym- 
 bols and laws of combination, required meanings for the symbols 
 of which the right to make those combinations shall be a logical 
 consequence. He tries, and succeeds; he invents a set of mean- 
 ings which satisfy the conditions. Has he then supplied what 
 his teacher would have given, if he had lived? In one par- 
 ticular, certainly : he has turned his symbolic calculus into a 
 significant one. But it does not follow that he has done it in 
 the way which his teacher would have taught him, had he 
 lived. It is possible that many* different sets of meanings may, 
 when attached to the symbols, make the rules necessary con- 
 sequences. We may try this in a small way with three symbols, 
 and one rule of connexion. Given symbols M, N, -f, and one 
 sole relation of combination, namely that M + N is the same 
 result (be it of what kind soever) as N + M. Here is a sym- 
 bolic calculus : how can it be made a significant one ? In the 
 following ways, among others. 1. M and N may be magnitudes, 
 + the sign of addition of the second to the first. 2. M and N 
 may be numbers, and + the sign of multiplying the first by the 
 second. 3. M and N may be lines, and + a direction to make 
 a rectangle with the antecedent for a base, and the consequent 
 for an altitude. 4. M and N may be men, and + the assertion 
 that the antecedent is the brother of the consequent. 5. M and N 
 may be nations, and + the sign of the consequent having fought 
 a battle with the antecedent: and so on. 
 
 We may also illustrate the manner in which too limited or 
 too extensive a meaning interferes with the formation of the 
 most complete significant calculus. In (1), limitation to mag- 
 
 * Most inverse questions lead to multiplicity of answers. But 
 the student does not fully expect this when he asks an inverse 
 question, unless he be familiar with the logical character of the 
 predicate of a proposition. A always gives B : what gives B ? 
 answer, A always, and, for aught that appears, many other things.
 
 94 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 nitude is not necessary, unless ratio and number be signified 
 under the term. In (2), if M (only) were allowed to signify 
 number, N -f M would be intelligible, but M + N would be un- 
 intelligible ; an impossible symbol of this calculus. In (3), (M +N) 
 signifying the rectangle, (M ' + N) + P would be unintelligible 
 at first: further examination would show that the explanation 
 is not complete ; and that the proper extension is that M + N + P 
 should signify the formation of the right solid (rectangular paral- 
 lelepiped) with the sides M, N, P. But M + N + P -f Q will 
 be always unintelligible, as space has not four dimensions. In 
 (4), the extension of M and N to signify human beings, would 
 spoil the applicability of the rule, unless the meaning of + were 
 at the same time extended to signify the assertion that the 
 antecedent was brother or sister (as the case might be) of the 
 consequent. 
 
 But when the symbols are many, and laws of combination 
 various, is it to be thought possible such a number of co- 
 incidences should occur, as that the same symbolic combina- 
 tions (unlimited in number) which express truths under one set 
 of meanings, express other truths under another? Could two 
 different languages be contrived, having the same words and 
 grammar, but in which the words have different meanings, in 
 such manner that any sentence which has a true meaning in 
 the first, should also have a true, but a different, meaning in 
 the second? This last question may almost certainly be an- 
 swered in the negative : the thousands of arbitrary terms which 
 a language presents, and the hundreds of grammatical junctions, 
 present a possible variety of combinations of which it would be 
 hopeless to expect an equal number of coincidences of the kind 
 required. But Algebra has few symbols and few combinations, 
 compared with a language : more explanations than one are 
 practicable, and many more than have yet been discovered may 
 exist. And the student, if he should hereafter inquire into the 
 assertions of different writers, who contend for what each of 
 them considers as the explanation of -/-I, will do well to sub- 
 stitute the 1 indefinite article. 
 
 We can now form some idea of the object in view; and we 
 must ask, first, what are the steps through which we have gone,
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 95 
 
 to arrive at algebra as it stands in the mind of the student 
 who commences this book. They are, very briefly, as follows: 
 
 Beginning with specific or particular arithmetic, in which every 
 symbol of number has one meaning, we have invented signs, 
 and investigated rules of operation. An easy ascent is made 
 to general or universal arithmetic, in which general symbols of 
 number are invented, the letters of the alphabet being applied 
 to stand for numbers, each letter having a numerical meaning, 
 kndwn or unknown, on each occasion of its use. And thus, 
 omitting many circumstances which have no particular reference 
 to our present subject, we arrive at a calculus in which the 
 actual performance of computations is deferred until we come 
 to the time when the values of the letters are found or assigned. 
 Accordingly, whereas in particular arithmetic every computation 
 is completed as it arises, or declared impossible, in universal 
 arithmetic we have a calculus of forms of computation, in which 
 each numerical computation is only signified, and not performed; 
 the proviso, if possible, being annexed by a reasoner to every 
 step of every process in which a chance of impossibility occurs. 
 
 Out of a few cases of difficulty, there is selected one, which 
 appears at first sight destined always to make the proviso above 
 mentioned an essential part of most processes of universal arith- 
 metic. It is the impossible subtraction; the constant appearance 
 in problems of a demand to take the greater from the less, to 
 say how many units there are in 6 - 20, for instance. An ex- 
 amination of the circumstances under which such phenomena 
 occur shows, inductively, that their producing cause is always 
 this, that either in the statement of the problem, or in its treat- 
 ment, some one quantity is supposed to be of a kind diametri- 
 cally opposite to that which it ought to be. 
 
 Simple number, the subject of abstract arithmetic, be it par- 
 ticular or universal, fails to show any acknowledgment of a 
 distinction which strikes us in almost every notion of concrete 
 magnitude. Measure 10 feet from a givon point oa a given 
 line: the command is ambiguous until we are told which of 
 two directions to take. A sum of money in the concerns of 
 A and Co. is incapable of being entered in their books until 
 we know whether it be gain or loss. A weight is generally
 
 96 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 of one kind, but not always: the weight of a balloon is a ten- 
 dency in the direction opposite to that of most weights ; or 
 rather, the word weight being by usage not allowed a double 
 signification, we say a balloon has no weight, but something 
 which is the direct opposite of weight. A time, one extreme 
 epoch of which is mentioned, is not sufficiently described until 
 we know whether it is all before, or all after, the epoch. And 
 so on. In every one of these cases, the numerical quantity 
 of a concrete magnitude, described by means of a standard 
 unit, is not a sufficient description; it is necessary to specify 
 to which of two opposite kinds it belongs. This specification 
 must be made by something not numerical: number is wholly 
 inadequate. 
 
 The first suggestion would be, it might be thought, to invent 
 signs of distinction : but universal arithmetic makes a sugges- 
 tion which forces attention, before the necessity for distinction 
 is more than barely perceived. Should we ever suppose that 
 the result of a problem is gain, or distance in one direction, 
 or time after an epoch, &c., when it is in reality, say 4 of loss. 
 or of distance in the opposite direction, or of time before 
 the epoch, &c., the answer always presents itself as 0-4, or 
 m-(m + 4), or as some version of the attempt to take away 
 4 more than there are to be taken away. It is then judged con- 
 venient (that the convenience amounts to a necessity is hardly 
 seen at that period) to make - 4 the symbol of 4 units of a kind 
 directly opposite to those imagined in 4, or + 4. And this 
 is the first of the steps by which universal arithmetic becomes 
 common, or single algebra. See Algebra, pp. 12-19 and 44-66, 
 for more detail. 
 
 This word single, as applied to algebra, is derived from space 
 of one dimension, or length, in which it is always possible to re- 
 present the effect of every intelligible operation of single algebra, 
 and the interpretation of every result which admits of any 
 interpretation at all. When we reckon time, gain and loss, &c., 
 it is always possible to translate our reckoning into terms of 
 length, as follows :
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 97 
 
 Take any point O, in a straight line, which call the zero-point, 
 from which all measurement is to begin. Let OU represent the 
 unit of any particular magnitude, and let magnitudes of one 
 kind, say gains, be measured towards A, and losses towards B. 
 Successive gains and losses may be taken off, and the final balance 
 exhibited, by the compasses. As long as the result is always of 
 one kind, so that an assumption to that effect would never 
 render the processes of pure arithmetic unintelligible, the suc- 
 cessive results always appear on one side of O: but the moment 
 a result of the contrary kind appears, (which, unless the arith- 
 metical computer were aware of it, and had provided accord- 
 ingly, would leave him with an attempt at impossible subtraction 
 on his hands,) it is indicated on the opposite side of O. 
 
 The convention as to the meaning of -j- 1 and - 1, namely, 
 that they shall represent units of diametrically opposite kinds, 
 is a very bold one : not merely because it takes up signs which 
 are originally intended for nothing but addition and subtrac- 
 tion, and fixes another signification on them ; but because it still 
 employs them to connect quantities, and ly a new kind of 
 connexion. The signs in fact are used in two senses, the direc- 
 tive, and the conjunctive. + (- 3) tells us, by virtue of -, what we 
 are talking of, and by virtue of -f ( ) how we are to join what 
 we talk of to the rest. As conjunctive signs, + means junction, 
 or putting on what we speak of; and - means removal. Thus, 
 if + and - in the directive sense indicate gain and loss, the 
 question, What is 
 
 ( _ 3 ) + (+S)-(-7) + (-4)-(+3)? 
 
 is the following : A man loses 3, and gets a gain of 8, with 
 the removal of a loss of 7, the accession of a loss of 4, and 
 the removal of a gain of 3 : what is the united effect of all 
 these actions on his previous property ? The answer is, the 
 accession of a gain of 5, + (+ 5). 
 
 The mere beginner is allowed to slide into single algebra 
 from universal arithmetic in a manner which leads him to under- 
 rate the magnitude of the change. I do not see that it can 
 be otherwise : but, at this period, my reader may be made to 
 observe that the process by which we shall pass from single to 
 double algebra, is the surest and most demonstrative (perhaps 
 
 K
 
 98 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 the only demonstrative) mode of passing from universal arithmetic 
 to single algebra. It is not until he can drop all meanings, 
 collect the laws of combination of the symbols, and so form 
 a purely symbolic calculus, and then proceed to furnish that 
 calculus with extended meanings, that he becomes fully master 
 of the change. But the close resemblances, which make the 
 slide above referred to so easy, might make it doubtful whether 
 he would be fit to take proper note of this case of reduction 
 and restoration* until he has seen a more striking form of the 
 same process, namely, that which is exhibited in the transition 
 from single to double algebra. 
 
 When the earlier algebraists first began to occupy themselves 
 with questions expressed in general terms, the difficulties of 
 subtraction soon became obvious, inasmuch as the greater would 
 sometimes demand to be subtracted from the less. The science 
 has been brought to its present state through three distinct steps. 
 The first was tacitly to contend for the principle that human 
 faculties, at the outset of any science, are judges both of the 
 extent to which its results can be carried, and of the form in 
 which they are to be expressed. Ignorance, the necessary pre- 
 decessor of knowledge, was called nature ; and all conceptions 
 which were declared unintelligible by the former, were supposed 
 to have been made impossible by the latter. The first who used 
 algebraical symbols in a general sense, Vieta, concluded that 
 subtraction was a defect, and that expressions containing it 
 should be in every possible manner avoided. Vitium ncyui 
 was his phrase. Nothing could make a more easy pillow for 
 the mind, than the rejection of all which could give any trouble ; 
 but if Euclid had altogether dispensed with the vitiitm paral- 
 
 * Algebra, al jebr e al mokabala, restoration and reduction, got 
 its Arabic name, I have no doubt, from the restoration of the term 
 which completes the square, and reduction of the equation by ex- 
 tracting the square root. The solution of a quadratic equation 
 was the most prominent part of the Arabic algebra. Alter the 
 order of the words, and the phrase may well represent the final 
 mode of establishing algebra ; reduction of universal arithmetic to 
 a symbolic calculus, followed by restoration to significance under 
 extended meanings.
 
 DESCRIPTION OF A SYMBOLIC CALCULUS. 99 
 
 telorum, his geometry would have been confined to twenty-six 
 propositions of the first book. 
 
 The next and second step, though not without considerable 
 fault, yet avoided the error of supposing that the learner was 
 a competent critic. It consisted in treating the results of algebra 
 as necessarily true, and as representing some relation or other, 
 however inconsistent they might be with the suppositions from 
 which they were deduced. So soon as it was shewn that a 
 particular result had no existence as a quantity, it was permitted, 
 by definition, to have an existence of another kind, into which 
 no particular inquiry was made, because the rules under which 
 it was found that the new symbols would give true results, did 
 not difier from those previously applied to the old ones. A 
 symbol, the result of operations upon symbols, either meant 
 quantity, or nothing at all; but in the latter case it was con" 
 ceived to be a certain new kind of quantity, and admitted as 
 a subject of operation, though not one of distinct conception. 
 Thus, 1-2, and a - (a + b), appeared under the name of negative 
 quantities, or quantities less than nothing. These phrases, in- 
 congruous as they always were, maintained their ground, because 
 they always produced a true result, whenever they produced any 
 result at all which was intelligible: that is, the quantity less 
 than nothing, in defiance of the common notion that all con- 
 ceivable quantities are greater than nothing, and the square 
 root of the negative quantity, an absurdity constructed upon an 
 absurdity, always led to truths when they led back to arithmetic at 
 all, or when the inconsistent suppositions destroyed each other. 
 This ought to have been the most startling part of the whole pro- 
 cess. That contradictions might occur, was no wonder ; but that 
 contradictions should uniformly, and without exception, lead to 
 truth in algebra, and in no other species of mental occupation 
 whatsoever, was a circumstance worthy the name of a mystery. 
 
 Nothing could prevail against the practical result that theorem! 
 so produced were true ; and at last, when the interpretation of 
 the abstract negative quantity shewed that a part at least of 
 the difficulty admitted of rational solution, the remaining part, 
 namely that of the square root of a negative quantity, was 
 received, and its results admitted, with increased confidence.
 
 100 DESCRIPTION OF A SYMBOLIC CALCULUS. 
 
 The single algebra, when complete, leads to an unintelligible 
 combination of symbols, V~l : n t more unintelligible than was 
 -1 when it first presented itself; for there are no degrees of 
 absurdity in absolute contradiction of terms. The use of ^/-l, 
 which leads to a variety of truths (page 41), points out that it 
 "must have a logic" (page 41, note). I now proceed (page 92) 
 to collect the symbols and laws of combination of algebra, or 
 to describe Symbolic Algebra.
 
 CHAPTER II. 
 
 ON .SYMBOLIC ALGEBRA. 
 
 IN abandoning the meanings of symbols, we also abandon 
 those of the words which describe them. Thus addition is to 
 be, for the present, a sound void of sense. It is a mode of 
 combination represented by 1 ; when + receives its meaning, so 
 also will the word addition. It is most important that tht 
 student should bear in mind that, with one exception, no word 
 nor sign of arithmetic or algebra has one atom of meaning 
 throughout this chapter, the object of which is symbols, and 
 their laws of combination, giving a symbolic algebra (page 92) 
 which may hereafter become the grammar of a hundred distinct 
 significant algebras. If any one were to assert that -f and - 
 might mean reward and punishment, and A, B, C, &c. might 
 stand for virtues and vices, the reader might believe him, or 
 contradict him, as he pleases but not out of this chapter. 
 
 The one exception above noted, which has some share of 
 meaning, is the sign = placed between two symbols, as in A = B. 
 It indicates that the two symbols have the same resulting mean- 
 ing, by whatever different steps attained. That A and JB, if 
 quantities, are the same amount of quantity ; that if operations, 
 they are of the same effect, &c. 
 
 The following laws are not all unconnected : but the unsym- 
 metrical character of the exponential operation, and the want 
 of the connecting process of + and x, pointed out in the last 
 chapter, renders it necessary to state them separately. 
 
 I. The fundamental symbols of algebra are 
 
 0, I, +, -, x, -f, (), and letters. 
 
 In ( ) there is the best mode of expressing the peculiar case 
 in which the symbol consists in position ; as in A s , in which 
 the distinctive symbolical force of the form lies in writing B 
 over A. 
 
 K3
 
 102 ON SYMBOLIC ALGEBRA. 
 
 II. It is usual to call + and - signs, and them only : but in 
 laying down the laws of symbolic algebra, the close connexion 
 existing between + and - on the one hand, and x and -f on the 
 other, requires that the latter should also be called signs. Let 
 the former be called term-signs, the latter factor-signs. It is to 
 insist on this connexion that I do not (for a while) introduce 
 the more common synonymes for AxB and A^B, namely AB 
 
 ,A 
 and-. 
 
 III. A symbol preceded by + or - is a term; by x or 4- a 
 factor. In A*> A is the base, B the exponent. When an ex- 
 pression consists of terms, let them be called co-terms ; when of 
 factors, co-factors. 
 
 IV. Let and 1 be a co-term and co-factor of every symbol, 
 + and x being the connecting signs of the symbol, but either 4- 
 or -, either x or -f, those of and 1. As seen in 
 
 Thus and 1 are a kind of initial or starting symbols, the first 
 of terms, the second of factors. 
 
 It is seen that + and x, placed before a symbol, do not alter 
 it : x A is A, having reference to 1 understood, as in 1 x A ; 
 and + A is A, having reference to understood, as in Of A. 
 
 V. Co-terms and co-factors which differ only in sign, are 
 equivalent to term and factor 1. 
 
 + A - A = 0, x A v A = l. 
 
 The more usual form of the last is 1 x A -f- A = I. The starting 
 symbol is frequently used in factors, but rarely in terms. The 
 student is well accustomed to + A and - A, in abbreviation of 
 + A and - A : but not to x A and 4- A for 1 x A and 1 -f A. 
 But he must use the latter a little, if he would see the complete 
 analogy of the term and factor signs. 
 
 VI. A symbol is said to be distributive over terms or factors 
 when it is the same thing whether we combine that symbol with 
 each of the terms or factors, or whether we make it apply to
 
 ON SYMBOLIC ALGEBRA. 103 
 
 the compound term or factor. Thus, looking at 
 
 
 
 A BCD and ABCD, 
 we see the * of the first distributed in the second. 
 
 VII. Term-signs are distributive over terms, and factor-signs 
 over factors: as in 
 
 at full length + (0 *A -) = + (0 + ^) + (0 - B), 
 l^-(lxA^B) = l~(lxA)^(l^B). 
 
 VIII. The tenn-signs of factors may belong, each one of them, 
 to any factor of the compound, or to the compound. 
 
 - A x - B = - (- A) x B = - (-) (A x B). 
 
 IX. Like term-signs in combination produce +; unlike, -. 
 Like factor-signs in combination produce x; unlike, -f. As in 
 
 + (-A)=-A, -(-A) = +A, x(-rA)=-rA, +(rA)=xA. 
 
 X. Terms and factors are convertible in order, terms with 
 terms, factors with factors. As in 
 
 XL Factors are distributive over the terms of any cofactor 
 with the sign x. (The corresponding law for -=- factors can be 
 deduced, and is not to be set down as fundamental). As in 
 
 and x (B - C) 4- A = B -f A - C~ A. 
 
 XII. The relations of the starting symbols and 1, a ex- 
 ponents, are A" = 1, A 1 = A. 
 
 XIII. The exponent is distributive over factors with x (the 
 case of v is deducible). As in 
 
 XIV. The operations of x and the exponential operation ( )( \ 
 successively repeated with the same base, are reducible to the 
 lower operations + and x performed with the exponents. As in 
 
 A* x A c = A^ c , ^ B ) c = A** c .
 
 104 ON SYMBOLIC ALGEBRA. 
 
 Any system of symbols which obeys these rules and no others, 
 except they be formed by combinations of these rules and 
 which uses the preceding symbols and no others except they 
 be new symbols invented in abbreviation of combinations of these 
 symbols is symbolic algebra. Ordinary algebra contains all these 
 symbols and all these rules, but its assigned meanings do not 
 make all results significant. I now proceed to combined symbols, 
 and to a sufficient amount of proof by instance, that one who 
 admits these rules admits, as consequences, all the combinations 
 of ordinary algebra. 
 
 Let 1 + 1 be abbreviated into 2 ; 2 + 1 into 3 ; 3 + 1 into 4, 
 and so on. Now introduce the abbreviations of A x B and 
 
 A 4 B, namely, AB and . 
 Jo 
 
 "We have then A + A - 2 A ; for (iv), A + A is 1 x A + lx A 
 or (xi) (1 + 1) .4 or 2.4. Similarly, A -f A + A = 3A. Again, 
 4.447 is f.4: for (x), 1x4x^4-7 is Ix447x.4, or .4x447, 
 or (VII), (VIII), A x (x 4 4 7), or A ; 
 
 (A-B)(C-D) is (xi) (A-B)C-(A-B)D, or, (xi) again, 
 AC-BC-(AD-BD), or (vn), AC-BC-(+AD) -(- BD\ 
 or (ix), AC- BC-AD + BD; 
 
 A AC 
 
 = ^TTT, for x A x C 4 (B x C) is (vn). xAxC+B^C, or 
 
 Jo .oG 
 
 (x), x A -4 B x (74 C, or (v), x A 4 B, or ; 
 
 jQ 
 
 .4x0 = 0; for(v) ^4x0 is A (+ B - B), or (xi) 4- AB - AB, 
 which (v) is 0. 
 
 A 1 
 
 From what precedes = is p This is an instance 
 
 Jj + C Jj C 
 
 2*2 
 
 of the deducible part of (xi) ; it is xA -f (B+ C) = + ( 
 The complete rule XI, in all its parts, fundamental and deducible, 
 is this: A factor may be distributed over the terms of its 
 eofactor, with its factor-sign or the contrary, according as the 
 receiving eofactor is x or v. Thus 
 
 f A -f (B -t- C) is V (A x B + A x C) ; 
 
 ^ , C , AD CB ^ AD + BC 
 
 -g ^ has been shewn to be , or (xi) - .-.
 
 ON SYMBOLIC ALGEBRA. 105 
 
 * is (xiv) ^ B+( - B) , or (ix) ^ B - B , or (v) A\ or (xn) 1. 
 So that A-* = A ; and ^ = ^ B ^- c = ^ B - ; 
 
 A z is ^ 1+1 , or A 1 A\ or ^ ; ^ 3 is AAA, &c. 
 
 ^3 gives (^ty = ^f 1 = ^4 (xn), or A* A* A* = -4 ; 
 
 - ^ x - .B is (VIII) -(-)A x ^, or + yf-B, or ^45; 
 
 ^4 x (BC) is ^4 x (x B x (7), or (vn) ^ x (x 5) x (x C), 
 
 or (IX) AxSxC. 
 
 In this way the student must examine narrowly a large 
 number of fundamental operations, satisfying himself that he 
 could produce them from the rules alone, independently of every 
 notion of meaning. The question is this, Might a machine, 
 which could, when guided, make introductions and alterations 
 by the preceding rules and no others, be made to turn one of 
 the alleged equivalent combinations into the other. 
 
 It will be exceedingly convenient to reserve the small letters 
 a, b, c, &c. most strictly to signify pure combinations of the 
 unit-symbol 1, with any term or factor-signs, as +2, -f, &c.: 
 and to use the capitals A, B, C, &c. for other cases. With the 
 exception of I shall use Greek letters only for angles.
 
 ( 106 ) 
 CHAPTER III. 
 
 ON AREAS AND SOLIDS. 
 
 I MAKE the first example of significant algebra to be an 
 application of symbolic algebra to the geometry of right areas 
 (rectangles) and right solids (rectangular* parallelepipeds), be- 
 cause the application is useful, and abounds in instances of the 
 distinction between symbols which become significant under the 
 meanings given, and those which are not significant. 
 
 However clearly a student may see that the ordinary arith- 
 metical proofs of the propositions in the second book of Euclid 
 are not sound, except for lines which are commensurable with 
 one another, yet, considering that every proposition which can 
 be proved by such arithmetical proof must be true\ (as may be 
 otherwise established) .for all lines whatsoever, it may be sus- 
 pected that the mechanism of the arithmetical proof is really 
 the mechanism of some sound and general proof. And so it 
 turns out, namely, that one of the significant algebras is the 
 method of proof desired. 
 
 f and - are simple addition and subtraction; A, B, &c. are 
 lines, if not otherwise specified, and it is easy to confine them 
 to lines. Again, x in A x B makes the symbol mean the rect- 
 angle under A and B ; the second x in A x B x C makes the 
 symbol mean the right solid under A, B, C. The symbols 
 0, 1, 2, &c. are as in arithmetic : thus 2AB is twice a rect- 
 
 * The length of this phrase is intolerable : and I am in the habit 
 of using the following extension of the word right. As a right 
 line is formed by the simplest and most direct motion of a point, 
 so the term right area might be applied to that formed by the most 
 direct motion of a right line, and right solid to the solid formed by 
 the most direct motion of a right area. Accordingly, the rectangle 
 and the rectangular parallelepiped would be the right area and right 
 solid. 
 
 f The perfect confidence which a mathematician puts in these 
 proofs does not arise, as he knows, from their proving that their 
 conclusions are true, but from their proving that they can (other- 
 wise) be proved to be true.
 
 ON AREAS AND SOLIDS. 107 
 
 angle; *, after a symbol derived from I, meaning common mul- 
 tiplication. Exponents, save only 1, (understood, xn.) 2, and 3, 
 need not appear. Heterogeneous terms are insignificant when 
 put together : thus AS + C, the area of a rectangle added to 
 a length, is unmeaning : as an area, the length is nothing. Again, 
 A -f S is merely the ratio of the two lines ; all the rules become 
 true under this meaning, joined with the others. AB^ C is the 
 other side (C being one) of the rectangle equal to the rect- 
 angle under A and B. And ABC^D is the area of the base 
 (D being the altitude) of a right solid equal to that under 
 A, B, C. And ABC^DE is the altitude of the same, DE 
 being the base. And A* or AA is the square on A; A 3 or 
 AAA is the cube on A. 
 
 It will be veiy easy now to establish that these meanings 
 give truth to all rules which have significance : to see the follow- 
 ing for instance. 
 
 A (B - C} = AB -AC, or, between the same parallels, the 
 rectangle under the difference of two bases is equal to the dif- 
 ference of the rectangles under those bases. 
 
 ABC=ACxB, or the remaining side (C being one) 
 of the rectangle equal to the rectangle under A and B, is equal 
 to the proportion of B, which is expressed by the numerical ratio 
 of A to C. 
 
 As far as -f and - are concerned, this system is that of pure 
 arithmetic. And A*, ABCD (space not having four dimen- 
 sions), are unintelligible. And we have instances of forms which 
 are significant, while equivalent forms are insignificant. Thus 
 ABCD -f E is unintelligible; there is no solid of four dimen- 
 sions. But the equivalent form of symbolic algebra, A^Ex BCD 
 is significant : it is such proportion of the right solid BCD as 
 A is of E. Shall we then say 
 
 A^Bx BCD = ABCD^E? 
 
 Shall we say, in common algebra, 
 
 _ 6 L _?L _e"- 1 +e-'"- 1 
 
 " + 2^4 '" ~ 2 
 
 Both questions are to be answered alike. Those who can, 
 in common algebra, find a square root of - 1 , will be at no loss 
 to find a fourth dimension of space in which ABC may become 
 ABCD: or, if they cannot find it, they have but to imagine
 
 108 ON AREAS AND SOLIDS. 
 
 it, and call it an impossible dimension, subject to all the laws 
 of the three we find possible. And just as V~l> m common 
 algebra, gives all its significant combinations true, so would it 
 be with any number of dimensions of space which the specu- 
 lator might choose to call into impossible existence. 
 
 The rules having been proved true, so far as significant, all 
 results produced by none but significant steps are pure geo- 
 metry. 
 
 Thus (A + B) 2 = A 9 + 2AB + B* is Euclid II. 4: not an arith- 
 metical representation, but the proposition itself. 
 
 A 3 B 3 
 
 And = A 3 + AB + B*, significant when A > B, means 
 _<4 JL> 
 
 that the base of a right solid which equals the difference of 
 two cubes, the difference of their sides being the altitude, is 
 equal to the sum of the squares on the sides and their rect- 
 angle. 
 
 The student must not call this significant phase of algebra 
 modern, though in its separated form it may be so. The su- 
 periority of the Greek geometry over Greek arithmetic, in means 
 of expression and demonstration, caused much of the notion 
 on which the former is founded to find its way into the latter. 
 It is from this mixture that we get the terms square and cube, 
 as applied to a x a and a x a x a (numbers). Vieta, who so 
 materially improved the symbolic language of algebra as to be 
 rightfully considered the founder of its modern form, was so 
 thoroughly possessed with the idea of linear, areal, and solid 
 representation, that he would have written such an equation as 
 XXX + A XX + BX = C, under the idiom 
 
 XXX + AXX + B planum X = Csolidum, 
 
 if he had used exactly our symbols. To have done otherwise, 
 to have allowed B and C to be the same species of magnitude 
 as X and A, would have appeared to him like asserting that 
 two solids and an area could make a line. 
 
 I should recommend the student to consider this algebra 
 well, and, when he meets with any circumstance of ordinary 
 algebra in which significance is difficult to conceive or absolutely 
 unattainable, to try if he can imagine the corresponding case 
 of the subject of this chapter.
 
 ( 109 ) 
 
 CHAPTER IV. 
 
 PRELIMINARY REMARKS ON DOUBLE ALGEBRA. 
 
 IF, taking the rules of symbolic algebra, we were to ask 
 for an assignment of meaning to (-1)^ which would make all 
 those rules true of it, we should naturally be led to select for 
 consideration the rule (XIT.) on which the symbolic character 
 most depends. It is 
 
 (-1)4 (-1)4 = (-1)W = (-l)' = _i, 
 or - 1 = V-l x {V-l x 1}. 
 
 Consequently, V~l must satisfy this condition, that twice 
 successively applied to + 1 by the process of x (whatsoever that 
 be) it has the effect of changing + 1 into - 1. 
 
 There may be many significant algebras in which this is 
 done. But the demand made by common consent is, that 
 our completely significant algebra shall be an extension of the 
 defective system with which we commence : meaning, that so 
 far as that system goes, significantly, it shall be a part of the 
 new system. It would not help us, with reference to the mathe- 
 matics now established, if fifty completely significant systems 
 were produced, unless in one or more of them the same story 
 were told as in the old algebra, so far as this last tells any story 
 at all. We must have, if possible (and I am to show that it. 
 is possible), all that we do understand still understood in the 
 same sense, with such enlargement of meaning as will give 
 significance to symbols which we do not now understand. Ac- 
 cordingly, + 1 and - 1 are still to signify diametrically opposite 
 units. 
 
 Let us then examine one of the usual systems of explanation, 
 in which we have a distinct conception of two diametrically op- 
 posed directions of measurement, and of no more. Let it be time. 
 Can we form any notion of an operation upon time (+ 1 being 
 an hour future measured from a certain aera) which being 
 twice repeated, shall produce an hour past, (-1)? The answer 
 
 L
 
 110 PRELIMINARY REMARKS 
 
 seems obvious : go back an hour, and then go back an hour. 
 But a little consideration will show that this process cannot 
 be represented by V-l. For then </-l.l would mean 1-1, 
 or 0, and ^/-1{^-1.1} would be *j-\ {0}, which (page 104) must 
 be 0. Besides, this operation, go back an hour, is - 1 ; and 
 1-1-1 is - 1, as required. Moreover, since by the laws of 
 algebra V-l cannot be any positive or negative quantity, it 
 would be absurd to say that *J-\ . 1 could be so interpreted 
 as to mean any time future, all which is already taken up by 
 positive quantity, or any time past, all which is taken up by 
 negative quantity. And we have not any other notion of time : 
 there is nothing (except 0, which will not do, as seen) intermediate 
 between time future and time past. We may then safely assert 
 that when + 1 means a unit of future time, - 1 of past time, 
 this algebra, significant as to all positive and negative quantity, 
 must remain insignificant as to \'-l. 
 
 Next try the simple notion of gain and loss. If we could 
 imagine a commercial event, which changed 1 of otherwise 
 certain gain into something of an intermediate character, not 
 truly described either as gain or loss ; but such that, should 
 the event happen again, it would convert the intermediate state 
 into 1 of certain loss we might be prepared to hope for a 
 significant algebra on this basis. I will not say that such a 
 basis of significance is impossible ; but only that it has never 
 been produced, though it has been before those who think on 
 this subject, as a suggestion, for more than forty* years. When 
 any one shall succeed in producing such an intermediate state 
 between gain and loss, then the symbolic algebra will become 
 significant on a system of gain and loss. 
 
 At present, however, take what notion we may, which pre- 
 sents the two diametrically opposite states (page 95), we find 
 ourselves at a loss to make a notion of anything intermediate, 
 except in one case. Length and direction in a plane f offers 
 
 * See Phil. Trans., M. Buee "Memoire sur les Quantites Ima- 
 ginaires," read in 1805 : also the first edition of Dr. Peacock's 
 Algebra, pp. 366, 367. 
 
 f I dismiss, without anything more than such allusion as will 
 prevent my being supposed to denyjthem, all the bases of significance
 
 ON DOUBLE ALGEBRA. Ill 
 
 an immediate solution of the question. We can pass from a 
 line to its opposite, not only along the line, but also by supposing 
 the line to turn round. The condition at the beginning of this 
 chapter is satisfied by supposing V~l x to be a revolution through 
 a right angle : for a repetition of the process turns a line through 
 a second right angle, opposes its direction to that which it had 
 at first, and satisfies the equation - 1 =-\/-l x {V~l x 1}. This 
 observation contains the first thought which led to the inquiry 
 into the question whether a completely significant algebra could 
 be constructed on definitions involving, not only opposite lengths, 
 but lengths in other directions. And hence it is frequently 
 stated that this result is derived from assuming *J-\ as the symbol 
 of perpendicularity. But this statement does not give a fair 
 representation ; that -y'-l represents a unit of length perpen - 
 dicular to that represented by + 1 is a consequence, not an as- 
 sumption: and a consequence of assumptions of a much more 
 simple character. 
 
 In inventing such a system, we obviously found an algebra 
 on a geometrical basis of significance. Why this limitation ? 
 Because, except in geometry, we nowhere find the varieties of 
 distinct conception which will afford meaning to our symbols. 
 As before seen, we are not bound to this system. The moment 
 any one shall afford us a distinct notion of time, or of mercantile 
 result, intermediate between past and future, or between gain 
 and loss, in a manner analogous to that in which a perpendicular 
 is intermediate between the two sides of its correlative per- 
 pendicular, that moment the system of symbolic algebra is as 
 ready to apply itself to time, or to gain and loss, as now to those 
 geometrical ideas on which it will presently be established in 
 significance. 
 
 B OA 
 
 If OA stand for a pound of receipt, and OS for one of 
 
 which may be found in length considered in three dimensions, or 
 on other than plane surfaces, or in lengths which are not recti- 
 linear.. The number of such bases is, I have not a doubt, quite 
 unlimited.
 
 112 PRELIMINARY REMARKS 
 
 expenditure, it would be perfectly easy to keep a cash account 
 with a pair of compasses, on the line - + indefinitely extended. 
 Measure off the amount in hand at the beginning, and set oft 
 all receipts (at OA for 1) towards the right, and all out-goings 
 towards the left, and the last point touched by the compasses 
 would never fail to show the balance in hand. But what money 
 does OC represent? Again, with OA to represent the first 
 year after the Christian sera, and OB the first year before, 
 a pair of compasses -will assign its place to every stated event 
 that ever did happen, or that we can imagine to have hap- 
 pened. But what event happened at OC"...the Christian aBra: 
 and what adverb is proper to take the place of before or after 
 (neither of which will do) in the blank. . . By such considerations 
 we may see that we do not restrict ourselves to geometry, but 
 extend ourselves to it : with ample means of representing all the 
 notions we have, and introducing others for which most notions 
 of magnitude afford us no analogues. And we may see the 
 propriety of extending the meaning of a geometrical term, 
 and calling time, loss and gain, &c., magnitudes of one dimension. 
 But then arises the following question : Granting that we help 
 ourselves in geometry, of what use is this algebra out of 
 geometry, in problems which have data derived from time, 
 or loss and gain, &c. ? To put this question properly, it should 
 be resolved into two, as follows : 
 
 1. Suppose the problem is 'at what time after a certain 
 epoch will an event take place which... [here describe the conditions 
 of the problem]...' Suppose the answer to be, that the event 
 must happen at 4 4- 3^/- 1 hours after the epoch : what does 
 this mean? It means that it is really and truly impossible* 
 that an event should happen, under the prescribed conditions, 
 at any imaginable moment, past or future : and that the assertion 
 that it can happen contains the assertions that what is, is not, 
 that a whole is no greater than its part, &c. 
 
 * The word impossible has been so misused in algebra, in the 
 sense of inexplicable, that the impossible of ordinary life, which 
 "can't be and never, never comes to pass," requires some ad- 
 ditional epithet to express it in an algebraical work.
 
 ON DOUBLE ALGEBRA. 113 
 
 As long as the meanings of symbols* remain unextended, " the 
 essential character of imaginary expressions is to denote im- 
 possibility: and nothing can deprive them of this signification. 
 Nothing like a geometrical construction can be applied to them ; 
 they are indications of the impossibility of any such construction, 
 or of anything that can be exhibited to the senses." 
 
 2. Suppose that to the above problem we obtain an answer 
 that the event takes place, say in 4 hours from the epoch, and 
 that our solution is obtained by aid of V~l> which however 
 disappears in the result. How are we free from the imputation 
 of applying reasoning to contradictory terms, seeing we do 
 not profess that, when time is the basis of significance, ^-l 
 has any meaning. I answer that, if time continue to be our 
 basis of significance, we are unanswerably open to that impu- 
 tation : but that, if we translate the terms of our problem, that 
 is, substitute geometrical ones, and work a geometrical answer, 
 our whole process is intelligible; and so many units of length 
 as our geometrical answer contains, so many units of time does 
 the answer to the original problem contain. Algebra takes 
 cognizance only of units, not of what units they are, whether 
 of length or time, &c. Each of its transformations is made 
 in one way, Avhatever may be the magnitudes from which the 
 units represented by its symbols are derived. A problem given 
 in terms of one magnitude may be solved in terms of another, 
 provided that every condition of the problem be faithfully pre- 
 served. 
 
 * The quotation which follows the words in italics is from the 
 review of M. Buee's memoir on Imaginary Quantities, in vol. xn. 
 (1808) of the Edinburgh Review. The earlier writers on this 
 subject were much given to supposing the explanations of V 1 
 to be absolute, and to be a demonstrable part of ordinary algebra : 
 the extension of meaning, or of the field of significance, was not 
 distinctly announced, and I imagine, indistinctly conceived. Hence, 
 as against M. Buee, there is an amount of propriety in the reviewer's 
 remark. But, nevertheless, it is a striking instance of the con- 
 fusion between ignorance and nature, alluded to in page 98. The- 
 re-viewer ought to have seen that in pure arithmetic every part 
 of his dictum, applies to negative quantities. 
 
 L3
 
 114 PRELIMINARY REMARKS 
 
 This is a point which, to the beginner, may require some 
 illustration. Suppose, then, we have this question : Two youths 
 are aged 6 and 16 years ; when will one be twice as old as 
 the other? Answer, in 4 years. Now propose it thus: Two 
 youths have 6 and 16 apples ; when will one have twice as many 
 apples as the other ? The data of the question are insufficient ; 
 there is no connexion expressed or implied between the number 
 of apples one may get, and that which the other may get. But 
 there was a connexion between their increments of age, implied 
 in the mention of time, and capable of being expressed. I did 
 not say ' Two youths are aged 6 and 16 years, and for each 
 year which one advances in age, the other advances a year also, 
 required, &c.' ; because the words in italics are necessarily due 
 to the mention of age. Now add to the second problem the 
 condition that for each apple which either gets, the second gets 
 one also, and we have the first problem, in which each 1 is 
 derived from a year, faithfully rendered into another in which 
 each 1 is derived from an apple : and the answer is, ' When each 
 has got four apples'. 
 
 It is a true method of finding the half of ten apples, or the 
 half of ten years, to describe an equilateral triangle upon a line 
 of ten inches, to bisect the vertical angle (Euclid I. 11), and to 
 show that each of the segments of the base is 5 inches. The 
 student must take care, in applying a complete significant double 
 algebra to questions of non-geometrical magnitude, that he does 
 not fall into an error analogous to that of supposing the equi- 
 lateral triangle to be described upon the ten years, or the ten 
 apples. 
 
 The separation of essential from non-essential notions is a 
 very important process to all who would think upon mathe- 
 matical subjects. In the first problem we see that the answer 
 is what it is, not because it is time of which we spoke, except 
 so far, that between any two moments, all person's ages have 
 received the same accession. The distance run over by persons 
 in the same carriage would have done as well. The formation 
 of symbolic algebra itself is a separation of the essential con- 
 ditions of operation from the non-essential : the rejection of all 
 meaning over and above the points of meaning on which trans- 
 formations depend.
 
 ON DOUBLE ALGEBRA. 115 
 
 There is another instance of separation of essential notions 
 which it will be necessary to use. In thinking of a process 
 of arithmetic, for instance, there is the subject-matter of the 
 science, and the mode of operation: these two things are dis- 
 tinct, to those who can separate them. But there may be a 
 difficulty in doing this : is it possible, for example, that we 
 could think of addition without thinking of number or magni- 
 tude, or thinking of more ? This point we shall try. The 
 subject-matter of arithmetic is number ; its primary operation 
 is counting or numeration. This counting proceeds from 0, 
 which represents, and must represent, the state of the mind 
 with respect to the number attained, before the counting begins. 
 Memory (and, for high numbers, reductive modes of expression) 
 save us from counting every time we produce number for use. 
 Any one who had forgotten seven must begin as children do, 
 first with none at all, put on one, put on another and say two, 
 &c. until he comes to seven. Now let us suppose that he is 
 to add seven to three, and that he has forgotten both seven and 
 the total. He must proceed first by counting seven, and then 
 by repeating the process of counting seven, with no alteration 
 except substituting 3 in place of 0, to start from. Thus we 
 have 
 
 ln sevTn. ing } > add one > !; add another, 2; 6; add one, 7. 
 
 Ina See e . Ven }' 3 > add one, 4; add another, 5; 9; add one, 10. 
 
 Accordingly, a and b being two integers, the direction given in 
 forming the arithmetical symbol a + b is ' proceed from a, first 
 formed, in the same manner as you proceed from to form &.' 
 Now if a and b stand for numbers, we must of course think 
 of number in doing this. Nevertheless the description of the 
 operation contains no numerical idea, except w r hen the subject- 
 matter is numerical. It is only 'Do with X as you did with Y 
 to make Z,' and every book of art, on any subject whatever, 
 abounds with this species of direction. It is seen in our symbolic 
 algebra : for B is + B ; so that in A + B it is seen that A 
 only takes the place of 0. 
 
 Again, let us think of arithmetical multiplication. Here the 
 separation of notion of operation from notion of subject-matter 
 is even more easily made. "What is 7 times 3 ? It is a number
 
 116 PRELIMINARY REMARKS. 
 
 which has a 3 for every unit which there is in 7. The direction 
 then is, Substitute 3 for 1 in the formation of 7. In place of 
 + 1 + 1 + 1 + 1 + 1 + 1 + 1 write 0+3 + 3 + 3 + 3 + 3 + 3 + 3. 
 Accordingly, a x b is always the result of substituting a for 1 
 in the formation of b, or of proceeding with a as we proceed with 
 unity in forming b. This is seen in the symbols : for B is 1 x B ; 
 and in A x B, A takes the place of 1.
 
 CHAPTER V. 
 
 SIGNIFICATION OF SYMBOLS IN DOUBLE ALGEBRA. 
 
 Tins particular mode of giving significance to symbolic algebra 
 is named from its meanings requiring us to consider space of 
 two dimensions (or area), whereas (page 111) all that ordinary 
 algebra requires can be represented in space of one dimension 
 (or length). If the name be adopted, ordinary algebra must 
 be called single. I first commence with the mere description 
 of the symbols, and then proceed to establish the rules in 
 Chapter II. 
 
 All the symbols which in single algebra denote numbers or 
 magnitudes, in double algebra denote lines, and not merely 
 the lengths of lines, but their directions. Thus two lines of the 
 same length, but in different directions, or two lines in the 
 same direction, but of different lengths, must have different 
 symbols. Accordingly, each symbol is meant to convey a double 
 signification : it describes the length, and direction, of its line. 
 
 Two finite lines have the same direction, when they are 
 parallel, and when they run in the same direction* on these 
 parallels. Thus, A and B being points, AS and BA are not 
 entitled to the same symbol : and if A, B, C, D be the points 
 of a parallelogram in order, AB and DC have the same symbol, 
 but not AB and CD. Thus AB = DC is true : AB = CD is not. 
 
 The symbol has reference to one particular point, arbi- 
 trarily chosen, but steadily kept to, which may be called the 
 on'i/in. By X = 0, we mean that X has no length : it is the 
 equal of a line, so to speak, which begins and ends at the 
 origin. The line 1, is a line arbitrarily chosen as to length 
 and direction, but steadily kept to. When 1 is drawn from 
 the origin, the line in which it is, indefinitely extended both 
 
 * The word direction is used in two different senses. Thus 
 north and south are different directions on a line, and the line of 
 north and south is one direction among lines out of an infinite 
 number.
 
 118 SIGNIFICATION OF SYMBOLS 
 
 ways, is called the unit-line, Afterwards, and particularly with 
 reference to symbols of the form A B , it will recal properties 
 if we designate the unit-line as the axis of length, and the 
 perpendicular to it as the axis of direction. 
 
 Since A and are found from O by progress over certain 
 lengths in certain directions, let us first describe the line we 
 choose to call A, and then, proceeding from its extremity, let 
 B be set off, commencing from the completion of A. Let the 
 third side of the triangle, if we take the 
 B which commences at the completion of 
 A, or the diagonal of the parallelogram, 
 if we take the B which commences from O, 
 be denoted by A + B. Then the operative 
 direction in page 115, is strictly applied to } 
 a different subject-matter. To form A + B, 
 we put A in the place of in + B. 
 And just as in arithmetic 11 + 7 tells us how far we are from 
 when 7 has been counted from and after 11, so here A + B 
 is supposed to indicate how far we are from O, and in what 
 direction, when -t- is joined to A. And since (A+B-B) is 
 to be A + 0, or + ^4, or A, annexing - B must be equiva- 
 lent to going over a line equal and opposite to B. And A- B 
 represents the length from O, and direction attained, by going 
 over, first A, and then an equal and opposite to B. And - B, 
 standing alone, is - B, or a line equal and opposite to B 
 from O itself. 
 
 If A and B be in the same direction, A + B and A- B are 
 as in single algebra: this will appear by following the above 
 rules. And if we take the unit-line, it Avill appear that 1 + 1, 
 or 2, is two units of length in that line ; 2 + 1 three units of 
 length in the same; and so on. All the symbols derived from 1, 
 represented by small letters (p. 105) are lines in the unit-line, 
 continued both ways : this partly appears already, and will be 
 seen further. 
 
 It thus appears, that what we here denominate addition is truly 
 not addition of magnitude to produce magnitude, but junction 
 of effects to produce joint effect. It is the process of the seaman, 
 when he represents himself as having only made ten miles (that 
 is, on his way to port), when perhaps he has gone, on two tacks,
 
 IN DOUBLE ALGEBRA. 119 
 
 24 miles altogether; but his effective progress is only 10 miles. 
 In this sense, describing two sides of a triangle, of 12 miles 
 each, may be of no more useful effect than describing the third 
 side of 10 miles. Nor is there, in one sense, the slightest 
 objection to saying that 12 and 12 make 10. 
 
 Let us now consider by what process 1 (OU) becotnes B. 
 There is a change both in length and direction: the change 
 of length is accomplished by altering OU in the ratio of OU 
 to the length of B (or multiplying O U by the number of linear 
 units in the length of JB). The change of direction is made 
 by turning OU through the angle made by B with OU. Now 
 substitute A in the place of 1 : 
 multiply its length by the num- 
 ber of units in B, and turn it 
 through the angle made by B 
 with OU. This process strictly 
 follows the direction in page 116, 
 and if we agree that the result 
 shall be denoted by A x B, we 
 have the following rule. The u 
 
 length of Ay. B is the arithmetical product of the lengths of 
 A and B, expressed in units ; and the angle of A x B with the 
 unit-line is the sum of the angles of A and B. 
 
 Before going further, the student must observe that we can 
 invent a method of representing the duplicity of our symbols. 
 Let letters placed within parentheses have their meaning in single 
 algebra, and let (a, a) signify a line of units of length inclined 
 at an angle a to the unit-line. Thus 1 is (1, 0), 2 is (2, 0), <tc.; 
 -1 is (1, TT), and (a, a+2wz7/-)=(a, a). Let A=(a, a) and B=(b, /3), 
 then we have 
 
 AxB = (at), a + /3). 
 
 This transformation is very easy: but addition is expressed 
 with more difficulty. We have 
 
 _L T? f /( 2 , w i o j, 1 13 -i a sma & sin/31 
 
 B = < v{ + " i 2a cos(/3 - )}, tan ' 
 
 I acosaicos/3J 
 
 >ince A x B f 
 A -r B as follows : 
 
 a cosa + icos/3/' 
 Since A x B -f B is to be A, we have for the meaning of
 
 120 SIGNIFICATION OF SYMBOLS 
 
 or the division of this algehra consists in dividing the length 
 of the dividend hy the length of the divisor for the number 
 of units of length in the quotient, and subtracting the angle 
 of the dividend from the angle of the divisor for the angle of 
 the quotient. Observe that we need not, unless we please, 
 use any negative number inside the parentheses : thus (- 2, a) 
 is (2, a + TT) and (2, - a) is (2, 2?r - a), or (2, 4?r - a), &c. Per- 
 haps at first it will be best to avoid negative quantities within 
 these parentheses. The following are some examples : 
 
 ,, 
 3 x 4 = (3, 0) x (4, 0) = (12, 0) ; = 
 
 - 3 x 4 = (3, TT) x (4, 0) = (12, TT) = - 12, 
 
 - 3 x - 4 = (3, TT) x (4, TT) = (12, 27r) = (12, 0) = 12. 
 
 Hence it appears that in the unit-line, multiplication and 
 division are precisely those of single algebra. But for all other 
 directions except ( , 0) and ( , TT), lines of the same direction 
 have not products in that or the opposite direction. 
 
 Let AA, AAA, AAAA, &c., without any reference to ex- 
 ponents, be called the second, third, fourth, &c. powers of A. 
 And let >JA, ~J A, V 'A, &c., be lines of which the meaning is 
 defined by ^A x ^A, -J A x '*] A x \J A, &c., being each equal 
 to A, and they may be called the second, third, fourth. &c. 
 roots of A. Then we have immediately 
 
 AA = (aa, 2a), AAA = (aaa, 3), AAAA - (aaaa, 4), &c., 
 
 ^A = fa, ? , &c. 
 
 As explained in pages 43, 44, choice of values immediately com- 
 mences, as soon as we have occasion to take a subdivision of 
 an angle. Thus, since a + 2rmr may take the place of , we 
 may infer, as in the pages cited, that ^A has two directions 
 whose angles differ by IT, half a revolution ; that VA has three 
 directions, indicated by angles differing by a third of a revo- 
 lution; and so on. In fact that 
 
 f /"/ a 2?T\ 
 
 is any one of V , - +m ) , 
 \ n n I
 
 IN DOUBLE ALGEBBA. 121 
 
 where m is any integer. Thus - 1 being (1, TT) we have for 
 V-l either (l, ^ or (l, ^), 
 
 or Ae square roots of unity are units* perpendicular to the unit 
 line. If, to draw a distinction, we denote (1, |TT) by ^/~1, then 
 -V-l will be denoted by (1, f TT). 
 
 As yet, every symbol or combination of .symbols from the 
 unit line, in obeying the laws of double algebra, obeys also 
 those of single algebra; the code of the latter being merely 
 a local chapter in the code of the former. But, for symbols 
 in general, the theorems of algebra are assertions of a much 
 wider kind. When we say in double algebra that 
 
 (7 x 7 - 2 x 2) -f (7 - 2) = 7 + 2, 
 
 we repeat in substance a proposition of arithmetic, the greatest 
 difference being that our additions and subtractions are rather 
 carryings forwards and backwards with the compasses than nu- 
 merical efforts of mind. But in establishing 
 
 (A A - SB) ~(A-S) = A + , 
 
 we shall establish nothing less than the following geometrical 
 theorem. 
 
 If there be two given lines inclined at given angles to a 
 line of standard length and direction, and if to the standard and 
 each of them a third proportional be taken, and placed at an 
 angle with the standard double of that made by the original : 
 and if from the end of the first line so resulting, a line be 
 drawn equal, parallel, and opposite to the second: and if the 
 line joining the common intersection of the standard and given 
 lines with the last found extremity of this last line be called 
 a first result: and if from the extremity of the first given line 
 two lines be drawn equal and parallel to the second line, in 
 the same and opposite directions: and if the lines joining the 
 common intersection before named with the last found extremities 
 be called second and third results : then the second result is a 
 fourth proportional to the third result, the standard, and the 
 
 * Whatever may have been suggested by the considerations in 
 page 109, the reader will see that double algebra is far from being 
 founded on the assumption that V~ 1 denotes perpendicularity. If 
 suggestion be foundation, it is more nearly founded on the separation 
 of operation and quantity in arithmetical addition and multiplication. 
 
 M
 
 122 
 
 SIGNIFICATION OF SYMBOLS 
 
 first result, inclined to the standard at an angle equal to the 
 excess of that of the first result over that of the third result. 
 The student should verify some general theorems of algebra 
 by actual drawing: this would give him practice in the meaning 
 of the terms. 
 
 The unit line, produced both ways, might well be called 
 the line of single algebra ; and the positive side of it the 
 line of pure arithmetic. And it readily follows that all symbols 
 of double algebra are capable of being expressed by symbols of 
 single algebra, combined with V~l : or V-l is the only peculiar 
 symbol of double algebra. 
 
 To show this, first observe that a V~l is (a, 0) x (1, f TT) 
 or (a, |TT), or a units of length perpendicular to the unit line. 
 Let there be a line R, and let it be projected upon the unit line 
 
 IV-l 
 
 and its perpendicular into projections of a and b units of length. 
 The first projection (a, 0) is properly represented by a: but 
 NP is (b, |TT) or b V~l : NM is b; and R is a + 6 V-l, by the 
 definition of +. Thus we have a representation of any line, 
 by means of symbols from the unit line and */-!. 
 
 Let R = (r, />), and let the projecting factors, by which a line 
 at the angle p is converted into its projections, be called cosp 
 for the unit line, and sin/> for the perpendicular. Remember 
 that we here recommence trigonometry ; nothing out of my first 
 book will be used in this second until it has been proved again 
 as a consequence of double algebra. We may consider cosp 
 and sin/j as by definition, the lengths of the projections of (1, />). 
 Accordingly, by similar triangles, a = r cos p, b = r sin p, and 
 R or (r, p) is r cosp + r sin/) . V~l- And ^ eri we nave 
 (1, /)) = cos/) + sin/) . V-l, (r, p)=r (cos/) -f sin/> . V~l)-
 
 IN DOUBLE ALGEBRA. 123 
 
 I shall defer the consideration of the symbol () until it 
 has been established that all the rules in Chapter II., except 
 xil., XIII., xiv., are necessarily true of the above symbols. Re- 
 member that the symbols in parentheses, as (a, ), are strictly 
 those of single algebra, and can even be made those of pure 
 arithmetic : and that those not in parentheses are always symbols 
 of double algebra. Thus, at this moment, I have hardly a right 
 to say 3x4 = 12: but in (3, 0) x (4, 0) = (3 x 4, 0), common 
 arithmetic gives the right to say that 3x4 in the parentheses 
 is 12 : so that 3x4 is (12, 0) or 12 of the double algebra. 
 
 I. All the symbols have been made significant, except the 
 exponential symbol () . The new symbols, </> V, &c., though 
 made significant, must be deferred till we treat of exponents. 
 
 II. III. The student may now freely use AB and -= for 
 A x B and A -f- B. 
 
 IV. In + A we see nothing but A, or rather a case of A, 
 which may have an infinite number of positions, and + A is 
 that one which begins at the origin. In A we only see 
 injunction not to proceed from the second extremity of A in 
 either direction. In common arithmetic, 7, for instance, written 
 alone, might be the last 7 in 18 or any other number : but + 7 
 is the first 7 which is counted from : and 7 is a direction 
 not to count beyond 7, either forwards or backwards. In 1 x A 
 we have A described as the unit altered into the length of A, 
 and made to turn through the angle of A : in A x 1 or A -f 1 
 we see A described with further direction, 1 being (1, 0), not 
 to alter its length, nor its angle. In common arithmetic, 1x7 
 is unity altered into 7 ; and 7xlor7-flis7 unaltered. 
 
 V. The definitions of - and -f were constructed to satisfy 
 + A - A = 0, and x A -^ A = 1. 
 
 VI. VII. Any case of VII. may easily be shown thus. The 
 application of + or - to a compound term is a direction to let 
 the result stand, or to change it into the opposite line. Now 
 if we apply + to each of the simple terms, each of them stands, 
 and therefore their compound stands, which is equivalent to 
 applying the sign + to the compound. But if - be applied to 
 each simple term, or if each be changed into its opposite, it 
 will appear from common geometry that the compound is also
 
 124 SIGNIFICATION OF SYMBOLS 
 
 changed into its opposite; so that the sign - is applied to the 
 compound. 
 
 Again, in 1 -f- (1 x A -f JB) we see that the operations are 
 
 -?, 27r-(a-/3)} 
 
 But in 1 -f (1 x .4) 4- (1 -r B) we have, since 
 
 as before. 
 
 VIII. The application of a term sign affects only the 
 angle : nor even that, unless the sign be - ; in which case a 
 revolution through two right angles is produced. Now whether 
 this alteration be made on a factor or on the whole compound, 
 matters nothing; for whether the factor sign be x or -f , revo- 
 lution through two right angles is of the same effect whichever 
 way it is made. 
 
 IX. The effect of + is merely permanence, that of -, oppo- 
 sition. Thus, - (- A) is + A, for the line equal and opposite 
 to the line equal and opposite to A must be A itself: other 
 cases may be proved with equal ease. 
 
 Again, (f A) or 1 -f (1 4- A), A being (a, a), is (1, 0) f- , -J, 
 or -1 1 -^ - , - (- a) I , or (a, a), or 1 x A. 
 
 X. The first part of this rule, that relating to terms, is 
 obvious : + A -f JB is the diagonal of a certain parallelogram, of 
 which + A and f S are sides ; and 4- B + A is the same diagonal.
 
 IN DOUBLE ALGEBRA. 125 
 
 Hence any two consecutive terms may be made to change places ; 
 for A + B + C - D + E = (A + B) + (+ C- D) + E 
 
 And if in any arrangement any two consecutive symbols may 
 be made to change places, it follows that, by change after 
 change, any one order may be converted into any other. 
 
 As to the factors, it is plain that x and 4- each indicates two 
 distinct operations, either of which is capable of being per- 
 formed without the other. These operations are separately of 
 the convertible character, and their joint result is the same : for 
 instance, 
 
 A x = (a, a) x (b, j8) = (db, a + /3) = (Jo, ft 4- a) = B x A, 
 
 , ., 
 
 XI. It may help us here, and elsewhere, to remark that there 
 is no essential distinction between + and -, or between x and -K 
 Thus A + B is A - (- B), or (a, a) + (b, ft) is (a, a) - (6, TT + ft). 
 And A -f B is A x (1 -f- B). All cases of this rule may then 
 be contained under 
 
 A( + C) = A + AC. 
 
 If any number of lines be multiplied by A, it is obvious 
 that the products make the same angles with one another as 
 the originals, since each angle made with the unit line is in- 
 creased by a. Again, the lengths are all increased in the same 
 proportion, their units being all multiplied by a. If then the 
 sides and diagonal B, C, B 4- (7, be all multiplied by A, we have 
 AB, AC, A(B + C), sides and diagonal of another parallel- 
 ogram. Therefore A (B + C) = AB + A C. 
 
 With the exception of what relates to exponents, we have 
 now a right to affirm that symbolic algebra is truly rendered 
 significant by the preceding definitions ; and that, so far, every 
 identical equation of ordinary algebra is also an identical equa- 
 tion of double algebra. And further, that all ordinary or single 
 algebra is so much of the double algebra as relates to the 
 symbols of lines taken in the unit line or its continuation. These 
 consequences are inevitable, unless it can be shown, first, that 
 some indispensable rule of operation is omitted in, and cannot 
 
 113
 
 126 SIGNIFICATION OF SYMBOLS 
 
 be deduced from, the rules in Chapter II.; and secondly, that 
 such omitted rule, when brought forward, is found not to be 
 a necessary consequence of the definitions in this chapter. 
 
 But inevitable consequences are not always easily credible : 
 particularly when very extensive and easily deduced consequences 
 stand upon a very small basis of definition. And it is not 
 easily credible that the whole of trigonometry should be capable 
 of re -establishment as a consequence of these definitions, after 
 throwing every part of the first book away except the defi- 
 nitions of co&9 and sin#. 
 
 A close examination of all the definitions and of all the 
 demonstrations of the symbolic rules will show that nothing of 
 geometrical theorem is assumed except the doctrines of parallel 
 lines and similar triangles. Nevertheless, what amounts to an 
 arithmetical demonstration of Euclid I. 47, can be immediately 
 produced. 
 
 It is seen that (1, 0) x (1, -6) = (1, 9-6) = 1. But (1, 9) 
 = cos# -t- sin0. V~l and (1, - 9) - cos0 - sinfl.y'-l, and their 
 product is cos<?cos# + sin(?sin#, which is therefore = 1. Accord- 
 ingly r cos6> . r cos9 + r sin0 . r sin# = rr, which is the arithmetical 
 form of I. 47. Now it is undeniable that I. 47 is proved again 
 (without reasoning in a circle) from parallels and similar triangles 
 in VI. 31. There must be then, in our definitions, and in the 
 operations which are performed in 
 
 (cos0 -f sin0 V~l) x (cos0 - s,m9*/-l), 
 
 something which amounts to such a deduction as is made in 
 VI. 31. And this, it may be shown, is the fact. 
 
 Take the wider question following. From (1, 0) x (1, 9) 
 = (1, + 6), we have 
 
 (cos0-f sin0.V~l) (cos0-f sin0.V-l) = cos(0+0) + sin (0 + 0).*/-l 
 (cos0cos6> - sin0 sin0) + (sin0 cos# + cos0 sin6>). -/-I 
 = cos(0 + 9) + sin (0 + 9) . ^-\. 
 
 But a -i- b i/-l = d -f V y-1 gives a = d and b = 6', since equal 
 and parallel lines have equal projections. Hence we have 
 
 cos (0+<?) = COS0COS0 - sin0 sin0, sin(0+0) = sin0 cos# + cos0sin0. 
 Now it can be shown that the steps of the preceding multipli-
 
 IN DOUBLE ALGEBRA. 
 
 127 
 
 cation are, in significance, the steps, not merely of a proof of 
 these theorems, but of one very commonly given. Let OM be 
 the unit line, and OM the unit, <AOM= 9, <BOM=$, <BOC=0, 
 <COM=<fi + 0, and construct the obvious figure. 
 
 Accordingly, 
 
 OP = cos0, OQ = cos0, OR = cos(0 + 0), 
 
 OR = OX + XR = O X + (- R X) = OX + (- FT), 
 
 Now, as to lengths only, 
 
 OB : OT:: OQ : OX or 1 : cos0 :: cos0 : OX= cos0cos0, 
 OB:BQ::CT:TV VT = sin0 sintf, 
 
 OB: BQ:: OT: TX TX = sin0 cos6>, 
 
 O^ :OQ::TC: VC VC = cos0 sin^. 
 
 Therefore, using the geometrical designations as symbols of 
 double algebra, 
 
 OX= (cos6cos0, 0) = (cos0, 0) x (cos0, 0) = OQ x OP, 
 
 - VT= (sin0 sine, TT) = (sin0, ATT) x (sin6, ^TT) = QB x PA, 
 
 XT=(sm<t)COse, \TT} = (sin0, ^TT) x (cos0, 0) = QB x OP, 
 
 FC = (cos sin 0, I-TT) = (cos0, 0) x (sin0, |TT) = OQ x P^4, 
 
 OAxOB = (OP + P^t) (OQ + Q.B) 
 
 = OP.OQ 4- Q5.P^4 + OP.QB + OQ.PA, 
 = OX + (- VT) + XT+ VC, 
 = OR + RC= OC. 
 
 Here, first, we have formed OC, or (1, + 0) from OA x O.B, 
 or (1, 6) x (1, 0) with an account of the actual geometrical trans-
 
 128 SIGNIFICATION OF SYMBOLS. 
 
 formation which goes on at each step : secondly, in so doing we 
 
 have been led to 
 
 OR= OP.OQ+QB.PA, 
 
 or cos(0 + 0) = COS0COS0 + (sin0. >J- 
 = cos cos 6 - sin0 sin#, 
 RC = OP.QB + OQ.PA, 
 
 or sin(0 + 0). <J-\ = (sin0.^-l) cosO -t- 
 
 And by this and similar instances we may satisfy ourselves that 
 the mechanical operations of double algebra are, when the mind 
 takes cognizance of their significance, true proof of their results, 
 just as is the case when they represent no more than arithmetical 
 notions. The great difference is, that in the latter case we are 
 much more familiar with the subject-matter, and more readily 
 learn to make mere operation carry conviction.
 
 ( 129 ) 
 
 CHAPTER VI. 
 
 ON THE EXPONENTIAL SYMBOL. 
 
 IN proceeding to treat of exponents, it is necessary to assume 
 the knowledge of some one system of arithmetical logarithms. 
 We cannot therefore (or certainly not at first) allow the word 
 logarithm to be divested of its meaning, and to pass into double 
 algebra to receive an extended meaning. Now since our system, 
 dealing in lines, gives results by measurement, the word logometer 
 suggests itself as a convenient variation of the word logarithm, 
 Let the logometer of A (denoted by \A) be defined as the 
 result of some convenient operation on A which has the follow- 
 ing property, 
 
 \A -f \S = X (AB). 
 
 An infinite variety of such operations may at once be given. 
 For, since the angle enters in multiplication and division with 
 the properties of a logarithm (as in the angle of the product is 
 the sum of the angles of the factors, &c.), we shall find that, 
 with respect to (r, p), Mlogr + Np, M and N being any fixed 
 symbols whatever, has all the property required. 
 
 If \R = M logr + Np, we have, R being (r', //), 
 
 A. {RR or (rr', p + p')} = M logrr' + N (p + p) 
 
 = Mlogr + Np + Mlogr' + Np 
 - A R -t- \R. 
 
 Now as we wish to preserve the single algebra intact as to 
 all unit-line symbols, we must make M = 1 ; for otherwise \(r, 0) 
 would not be logr. As to N, the most convenient assumption 
 is the form &V-1, which would give \R = logr + kp.^-l. But 
 it does not limit us if we make & = 1 : for neither the base of 
 the logarithms nor the mode of choosing an angular unit is yet 
 settled (in this book), and the power of changing the value of k 
 is supplied by that of changing the unit in which the angle is
 
 130 ON THE EXPONENTIAL SYMBOL. 
 
 expressed. Our definition of \R is now contained in 
 
 \R - logr + p*J-\, 
 
 or the logometer of any line has the logarithm of the length 
 for its projection on the unit line, and the angle (meaning a line 
 of as many linear units as the angle has of angular units) for 
 its projection on the perpendicular. And this is the connexion 
 of the two axes with length and direction from which the terms 
 suggested in page 118 are derived. Thus we have, it appears, 
 a species of logarithm to R on each axis ; or a symbol which 
 is augmented by addition in multiplication, &c. 
 
 In this symbol, \R, occurs, for the first time, a choice of 
 meanings ; and that choice is unlimited. For p we may write 
 p 2m7r without altering the meaning of R : but for each value 
 of m we have a distinct logometer to R. And all the logometers 
 of R are diagonals of rectangles standing on one base, logr, 
 with altitudes p, p + 27T, p + 4?r, &c., p - ZTT, p - 4?r, &c. 
 
 But though every line have an infinite number of logometers, 
 yet every logometer has only one primitive line. For if + &</-! 
 be a logometer, its primitive can have no length except the 
 number whose logarithm is a, and can be in no direction except 
 that indicated by the angle b. Consequently, if two primitives 
 be equal, we can only say that any logometer of the first is one 
 of the logometers of the second : but if two logometers be equal, 
 we can assert that the primitive of the first is equal to the 
 primitive of the second. 
 
 Now take the following as the definition of the symbol A B . 
 Let its logometer be B\A : that is, let it mean the line whose 
 logometer is \A. If we use, for a little while, the inverted 
 letter \, so that \A shall signify the line whose logometer is A, 
 we may state that we use A* as an abbreviation of \(S\A). 
 
 Let e be the base of the arithmetical logarithms used in the 
 unit line: and remember that is strictly to denote a line at 
 an angle to the unit line; it is (e, 0), and (E, 77-), or (f, 4?r) 
 is, for a base of logarithms, or in connexion with logometers, 
 to be distinguished from (E, 0). Also, observe that in like 
 manner as we have abandoned (till we recover it) the mode 
 of measuring angles, so we do not yet say that e is the peculiar 
 base of the Naperian logarithms: let it be any which it can
 
 ON THE EXPONENTIAL SYMBOL. 131 
 
 be, until we show cause for preferring one base to another. 
 We have then 
 
 \R = lo r 4 V-1 = VClog 2 r + A tai 
 
 \J2 = (number whose log. is r cos p, r sin /a), 
 
 X (a 4 &V-1) = ilog (a 8 4 6 2 ) 4 taiP - .J-l, 
 
 a 
 
 \ (a 4 &V~1) = (number whose logarithm is a, ). 
 
 Having shown that the fundamental formulae of trigonometry- 
 are deducible from the double algebra, I now use the first book 
 of this treatise, in every point except specifying the mode of 
 taking an angular unit. 
 
 The following are the proofs that, under the above definition 
 of A*, the laws of symbolic algebra are true. 
 
 XII. In A\ we are to see \(Ox\A), or -y/(0 + OV-l), or 
 (1, 0), or 1. In A", we have \(1 x X^4), or \\A, or A. 
 
 XIII. We prove two symbols identical in meaning, if we 
 prove any one logometer of the first equal to any one of the 
 second. Now, by definition, the logometer of (AB) C is C\(AB) 
 or C(\A 4 XJ?), or C\A 4 C\S. But the logometer of A C B C is 
 \A C 4 \B C , or C\A 4 C\B. Therefore (ABf = A C J3 C , provided 
 that we use the same logometers of A on both sides, and the 
 same of B : that is, the same cases of a 4 2.rmr and ft 4 2m?r 
 on both sides. 
 
 XIV. Since \(A*A C ) is \A B 4 \A C , or B\A 4 C\A, or 
 (B 4 C)\A, which is a logometer of A* +c , it follows that 
 
 A*A C = A B+C , 
 if we use the same logometer of A throughout. 
 
 Again, X {(^ B ) c }= C\A* = CB\A, which is a logometer of A* c . 
 Therefore (A*) c = A B , 
 
 provided the same logometer of A be used on both sides. 
 
 Next it is to be shown that when the exponent is a symbol 
 of the unit line, as in A m , the above definitions of the exponent 
 agree with those of ordinary algebra. This is in fact, contained 
 above : for A 3 is A m , or A 1 A\ or AA ; A 5 is such that
 
 132 ON THE EXPONENTIAL SYMBOL. 
 
 whence A* is V 'A ; and so on. But, once for all, 
 
 A m = -y (m\A) = y (m log a + ma V~l) = (# m > ma), 
 
 from which all the cases can be deduced. 
 
 For A,(e, 0), we have l-fO^-l or 1. Hence for A we have 
 \(A\) or \A. That is, 6 A must be our future way of ex- 
 pressing \A ; and we have, as in ordinary algebra, A = e^ A . 
 Again, (1,0) has for its logometer + 6^-1 ; therefore (1, 0) 
 is fi*'" . But this is cos0 + sin0.^-l : therefore 
 
 Here again occurs the difficulty of page 126. We get this 
 fundamental equation on terms so cheap, that we suspect its goodness. 
 And moreover, it cannot be always true, while e and the angular 
 unit are both unnamed. The second side does not depend for 
 its numerical value upon what number is, but only upon what 
 angle it represents. The first side is dependent for its numerical 
 value upon those of e and 0. If, for instance, we choose to 
 halve the angular unit, so that the angle now containing units 
 contains 20 units, the second side is unaltered. But 
 
 cos 20 + sin 20.^-1 is e""' 1 , 
 
 which is not 6 ^' 1 . Nevertheless it will be easy both to establish 
 that some such equation must exist, and that a connexion exists 
 between the base to be taken for the logarithmic system and 
 the unit of angular measure. 
 
 Having established all the fundamental rules, we may by the 
 process in page 205 of the Algebra, interpreting the symbols as 
 in the double system, show that any function which possesses 
 the property fA xfE =f(A + S) must be of the form C A , where 
 C is independent of A : and cos0 + sin 0.^-1 is such a function 
 of 0. Accordingly we must have 
 
 C 9 = cos0 + sin0y-l. 
 
 This result only differs from the former in that C, which is 
 quite arbitrary, takes the place of e^" 1 , which is wholly un- 
 determined. Return to the first, and observe that it gives 
 
 ^' 1 - cos 1 + sin !.</-!>
 
 ON THE EXPONENTIAL SYMBOL. 133 
 
 which is the relation that must exist between the angular unit 
 and the base of the logarithms. If we were to make no appeal 
 to common algebra, we should proceed with this equation to 
 define the above relation, and in process of time we should 
 arrive at the result that if the method of angular measurement 
 be arcual, the base of the logarithms must be Naperian ; that 
 is, that if angle 1 have an arc equal to the radius, s must be 
 
 1 1 
 
 1 + n-+o+- 
 
 But as it may be unsatisfactory to leave such a point behind 
 us, I will establish it on the following basis: the binomial 
 theorem with a positive integer exponent, and the theorems 
 that log (1 -;- x) 4- x and tan x -f x both have the limit unity 
 when x is diminished without limit; with, of course, the ex- 
 plained symbols of double algebra. And in assuming tan x -=- x 
 to have 1 for its limit, we assume the arcual unit. 
 
 Now 
 
 (page 119) (1,0) + ~ P 
 
 (/ 1\ R / / 1 \ 
 
 1 + ^-) Its logometer is H\[l + X=r-l . 
 -tt I \ R I 
 
 R or (l { -^ -r (r, p) is (- , ~ ^ , say (k, | - 
 
 , -- 
 
 2 / I \l4A;sin/vJ 
 
 X (l + ^] = ilog(l + 2&sin/j f **) 4 tan' 1 7 - 8 /- V-l. 
 \ t ] 1 + A; sm/j 
 
 And, J2 being (cosp + sin/>.\/-l) f k, we have 
 
 m (l + ^) = P + Q V-l, where 
 
 cos/j , 
 P = - log(l 
 
 = - log(l + 2k sin/, + *) + tan - 
 
 2A; A; 1 + Asin/a 
 
 Now let the length of _R increase without limit, or let k 
 diminish without limit. Then we have 
 
 loe(l + 2& sin/> H Z; 2 ) log(l 4 2k sin/; 4 /,-*) 
 
 - ~ = -- (2 
 
 N
 
 134 ON THE EXPONENTIAL SYMBOL. 
 
 and, taking that angle which diminishes without limit with its 
 tangent, we have 
 
 1 _j & cos/> _ t k cos/) . k cos/> cos/> 
 
 A; 1 + k sin/j 1 + k sin/> 1 + & sin/> 1 + & sin/j ' 
 
 and the limit is cosp. Hence P has for its limit 
 
 cos/>.2 sin/) - sin/> cosp, or : 
 Q has \ sin/j.2 sin/> + cosp.cosp, or 1 ; 
 and P + Qy'-l has V~l- 
 
 If then the length of R increase without limit, (1 + V~l - -R) R 
 has for its h'mit \(0+ 1 ^-1) or (1, 1), provided that the logometer 
 used have an angle between - TT and + TT, and that the logarithms 
 used be of the system which gives log (1 + x) -f x the limit unity. 
 Let R = n*J-\, n being integer; then 
 
 (1 \nV-l ff Ixnl^- 1 
 
 1+- or^l+- I has the limit (1, 1) : 
 nj L\ / J 
 
 but, as n increases without limit, (1 + 1 -f- n) n approaches the 
 limit 1 + 1 + i -f ... (Algebra, page 225). Consequently we have 
 
 / 11 V- 1 
 
 (1 + 1 + -+ + ...j = cos 1 + sm 1-V-lj 
 
 or e has the value used for that letter in single algebra. 
 
 We have now a completely significant system of algebra, and 
 the whole contents of Book I. Chapter V. are established by 
 demonstration, if that chapter be now inserted here. 
 
 The symbol \R is the first in which multiplicity of meaning 
 occurs ; a property which it communicates to 22 s . All the mean- 
 ings of this last symbol, the distinction between the cases in 
 which their number is infinite and those in which it is finite, 
 &c., will be best seen by reducing JR S to another form. Let
 
 ON THE EXPONENTIAL SYMBOL. 135 
 
 since re^" 1 is r cos + r sin 0.^- 1 or (r, 6). Here (page 46) 
 
 tan" 1 (b -f a) may be any angle with b ~ a for its tangent, in 
 
 P 
 
 which the cosine and sine have the sign of a and b ; and (a" f b*f 
 is taken with a positive sign. 
 
 First, it appears that when q is not = 0, that is, when the 
 exponent is not a symbol of single algebra, the number of values 
 is absolutely unlimited. But even in this case, when p is a 
 rational fraction, the number of directions is no more than one 
 for each unit of the denominator: and when p is an integer, 
 there is only one direction (pages 43, 44). 
 
 Next, when q = 0, we have 
 
 f ? 61 
 
 (a + b V-l) p = <^ (a* + 6 2 ) 2 , p tan' 1 - j> , 
 
 which has only one length, and as many directions as there are 
 units in the denominator of p. If p be incommensurable, the 
 number of variations of direction is infinite. The case of b = 
 is discussed in pages 45, 46. 
 
 The effect of the term qt/-l in the exponent, is the addition 
 of %q log (a* + 6 2 ) to the angle, and the subtraction of q tan" 1 (b 4- a) 
 from the logarithm of the length. 
 
 The student should exercise himself in the reduction of different 
 forms of J2 S to significance, first, by the complete process, next 
 by the formular result. For instance, ^/-l v . Here >j-\ is 
 any case of (1, 2nnr + ^TT), and its logometers are contained in 
 
 (2m?r + \TT)^-\, or (2rmr + |TT, Imr + \TT). But 
 
 for it is not worth while to distinguish Inir and 2nir + Ik-. 
 n and k being any integers we please. This last is the logometer 
 of the result required; therefore 
 
 rVLl. / 1 A , 1 / 1 . 
 
 Otherwise V- 1 = + 1 V~l = E 
 
 or V-l*" 1 is any power of e"^, whose exponent is in the series 
 
 ...-3, 1, 5, 9, &c. 
 
 The following fallacy has before now been seriously proposed 
 as an argument against the introduction of imaginary quantities
 
 136 ON THE EXPONENTIAL SYMBOL. 
 
 into ordinary algebra. Since 1"=1, I/' 1 = 1 : but 1 = s 2 ^' 1 , 
 therefore ( 27rV ' 1 ) v " = 1 O r ~ 2 = 1, which is absurd. If we try 
 1 ' J ~ , we have XI = 2imr V-l, V-l XI = - 2?7r, which is the logometer 
 of ~ 2 "" r . Accordingly, we admit the equation l v ~ l = ~ 2m7r , 
 meaning that if m be any integer, positive or negative, e 
 is one of the values of 1 . And if m = 0, this is 1. But this 
 last 1 is not 7rV " : the first 1 is (1, 0), the second is (1, 2;r). 
 How these should give different logometrical results, double 
 algebra makes manifest enough. The logometric operation makes 
 differences of form and value both out of differences of form 
 ivithout difference of value. 
 
 In the Rules xm. and xiv. it is demanded that the same 
 logometers of each symbol shall be used throughout; otherwise 
 the relations are not true. Does the neglect of any analogous 
 regulation lead to errors in single algebra ? To try this, let us 
 see if error may be produced. First take A*.A C = A** c , and 
 observe that AP = XQ -I- R means P = Qs B . Use a particular 
 logometer in A*, call it A^4, and another, \A + Zrmr V-l, in 
 A c . The logometer of A*A C , thus taken, is 
 
 Hence A B .A C = A* +c e 2 ""^ 1 , 
 
 in which A* and A E+C are formed from the same logometer. 
 A very simple instance of the truth of this equation will show 
 that beginners may commit a mistake in ordinary algebra. Let 
 j? = C = J, n = 1. Then we have A^A* = Ae^' 1 = - A. But 
 it ought to be +[A. This beginner's mistake is like the following : 
 V4 = + 2, V* = - 2, therefore V 4 V 4 = - 4 > or + 4 = - 4. The 
 two different forms of .4* are formed from different logometers. 
 
 Unity, when exhibited in the form 2 "" v ~ 1 , is formed from the 
 logometer + 2tmr V-l : and a, exhibited as ae" m * " , is formed 
 from the logometer loga + 2imr V-l. If we consider as primary 
 that form of a symbol which takes its angle from the first 
 positive revolution, or from inclusive to ITT exclusive, and if 
 \ (1 -/ denote the primary logometer thence formed, and if \ m A 
 denote the logometer loga -f ( + 2imr) -/-I, and A m B the value
 
 ON THE EXPONENTIAL SYMBOL. 137 
 
 of A* formed from it, we have the following equations: 
 
 ^ B _ ^ B g 2m7T B ^-l A S = A B 2 t m -")' rBV - 1 
 
 j B A B ,< B 2(m+n-p)7TB'/-l / A -n \C ; c p C 2(m+n-*-Z)rCV-l 
 -^m ^n = -^ E ' V-^m-^n^ = ** -l S > 
 
 (A *) c = A BC g{ 2Cm- *) BC + 2 " c }' lV - 1 . 
 
 V tii Jn k 
 
 The following equation, ^~ = TT, very often found in 
 
 y-i 
 
 books of algebra, merely means, when brought to significance 
 by adoption into double algebra, that TT >/-! is one meaning 
 of \ (- 1). 
 
 In former days, it was not uncommon to object to the equa- 
 tion -/-I . *J-\ = - 1, on the ground that it should be V(- 1 x - 1) 
 or A/1 or ! But it was hardly seen that, on this mode of 
 reasoning, </a . <Ja is V 2 or i a > i n all cases. And moreover this 
 last is true, if ^a be indefinite. For then it has two values ; 
 and if in y'a.y'a, we are not bound to use the same value in 
 both the first and second factor, then ^a . <Ja is a, 4 a or - a, 
 according as Ja and ^/a represent the same or different square 
 roots. The two square roots of a are constructed on "different 
 logometers ; one on log a + 2m7r</-l, the other on 
 
 loga + (2w + l)7ry-l.
 
 ( 138 ) 
 
 CHAPTER VII. 
 
 MISCELLANEOUS REMARKS AND APPLICATIONS. 
 
 THE theory of logarithms admits, and even requires, an 
 extension above what has been given to it. The loyometer of 
 the last chapter answers to the ordinary Naperian logarithm 
 of algebra; we are now to examine what answers to the loga- 
 rithm to any base. 
 
 It will, at a future time, when a significant algebra is made 
 the basis of elementary instruction, be a question whether the 
 symbol should not indicate the amount of revolution of a line 
 as well as its length and direction : whether, for instance, (a, a) 
 and (a, a -f 2?r) should not be distinguished by some difference 
 of symbol. But even at present, in all that relates to logo- 
 meters, it will be convenient to adopt this distinction. Ac- 
 cordingly R may denote (r, p), p lying between and 2;r; 
 while R m may denote (r, p + 2i?r). 
 
 Let B or (b, (B) be the base ; it is required to find the logo- 
 meter of X or (x, ) to this base, defined by the equation ,B*B X = ^Y. 
 The logometer of the last chapter has (t, 0) for its base. De- 
 noting A _AT simply by \X, we have \ B X.\B = \X, or 
 
 The extension in page 48, supposes B to be (e, 2m7r), and X 
 to be a symbol of the unit line, or = 2mr or (2 + I)TT. When 
 
 logo; : log 6 :: : /3, 
 we have X B X = logx : log b = log (base b) x. 
 
 That (a + 6 y'-l) must take the form p + q <J-1, a propo- 
 sition collected by a laborious induction in incomplete algebra, 
 is now no more than was, in that algebra, the assertion that 
 a real function of a real quantity is a real quantity. For 
 every combination of symbols can be explained, and everything 
 explicable is a line of definite length and direction, and every
 
 MISCELLANEOUS REMARKS. 139 
 
 such line can be represented by p + q <J-1. Nor is it more 
 difficult to prove that if (f)x be a real, or, as we should now 
 call it, unit line, function of x, (>(x - y V-l) must be p - q\>-l. 
 For the same operations, performed on the same lines, will 
 produce the same resulting lines, by whatever symbols they 
 are denoted. Change the positive and negative directions on 
 the axis of direction, and also the positive and negative direc- 
 tions of revolution. All the symbols of the unit line still repre- 
 sent what they did before; but the lines which were a + b V~l 
 and p -f q */-!, are now a- b 1/-1, and p - q -/-I. Therefore, the 
 same operations on the same lines producing the same result, 
 we have (a - b V~l) = P ~ 1 V~l : hut if the function should 
 contain other double symbols, as a' + b 1 i/-l, &c. and if 
 (a + b V-l, a'+ i'V-1, &c.) =p + q </-!, 
 then (a - 6 y~l, a' - b' -/-I, &c.) =p - q V~l- 
 
 It would seem as if there is still left one source of inexpli- 
 cable result : what is the angle a + /3 *J-\ ? what is the length 
 a + b^/-l? or what is meant by the symbol (aib^-1, a + ^3^/-l)? 
 There is nothing here except such a confusion of symbols as 
 arises in arithmetic when 7 + 4-5, for instance, is by mere 
 inadvertence of operation presented as 7 + (4 - 5). We have 
 
 ( a 
 
 = (V(a 8 + 6 2 ) e-/ 3 , tan' 1 - + a), 
 a 
 
 a line of intelligible length and direction. The suppositions 
 which, assuming (r, p) as the solution of a problem, end with 
 r - a + b^/-l, p = a + fi *J-1, are analogous to those which in ordinary 
 algebra introduce the impossible subtraction into the process of 
 solution, when it is not necessarily produced in the answer. 
 
 If the assumption of a length a, should lead to a + /3 V~l 
 as the requisite angle, it means that the length a will not do, 
 but that ae'^ 3 will do, with the angle a. 
 
 If we extend the definitions of cos# and sin0 so as to derive 
 them from the equations 
 
 _
 
 140 MISCELLANEOUS REMARKS 
 
 we have, when 6 is a unit-line symbol, their meanings unaltered ; 
 and when 6 is not, still an intelligible signification. Thus 
 
 COS($V-1)~ n ' Sin (^ V-l)- o V"l & C - 
 
 Similarly sin" 1 (a; V-l), &c. can be interpreted. 
 
 The notion of continuity generally derived from ordinary 
 algebra is corrected in the double system. If a unit symbol 
 gradually change from positive to negative, passing through 0, 
 there is at the moment of passing through 0, an instantaneous 
 accession of TT to the angle, and ?rV-l to the logometer. Ac- 
 cordingly, the square roots are at once advanced by %TT, the 
 cube roots by -|TT; and so on. But we are apt to think only 
 of length, which, in the case in question, does change continu- 
 ously. The only perfectly continuous way of passing from x = + a 
 to x = - a, is by supposing 6 to change from to TT, or from 
 to - TT, in the formula x = a(cosO 4- sin#V~l) : an d the cor- 
 responding continuous passage from x = a to x = b is obtained 
 by the same change made in 
 
 x = f (a + 6) + | (a - b) cosd -f- 1 (a - b) sin0. V-l- 
 
 In this change all the roots also change continuously. In 
 many parts of the integral calculus, results which are inexpli- 
 cable on the supposition of change of length, are at least in- 
 telligible on the supposition of revolution of length, though the 
 connexion of the two is not yet elucidated. 
 
 When the data of a problem are those of the significant 
 system, any one of the problems which are really impossible 
 while the terms are those of ordinary algebra, becomes possible 
 as soon as the terms are allowed the extension of double algebra. 
 For instance, it is required to divide 2a into two parts with 
 the product b. The parts are a + V( 2 - &) and a - V( 2 - b). If 
 a and b be numbers, the problem is arithmetically soluble if 
 a 8 - b be positive : that is, if a and b be unit-line symbols, the 
 parts required in the problem are also unit-line symbols. But 
 if a* - b be negative, the parts are 
 
 a V(&-a*-).V-l, or JV&, 
 
 tan" 
 
 the product of these is (b, 0) or b, and their sum is a, for
 
 AND APPLICATIONS. 141 
 
 the parts are the sides of an equilateral parallelogram, of which 
 (a, 0) is the diagonal. But the parts are not now entitled- to 
 that name, arithmetically speaking : they are components, but 
 under a law of composition which is not merely addition of 
 magnitude. 
 
 The following theorem, given by M, Cauchy for the deter- 
 mination of the number of imaginary roots of an equation, and 
 for the proof that every equation has as many roots as dimen- 
 sions, can be established with clearness by the use of double 
 algebraical meanings. 
 
 Let x and y be the projections of z in (z, ), or the coordinates 
 of the extremity of Z. Let $Z = AZ n \ SZ n ' l + ... an integral 
 function of Z: A, B, &c. being symbols each of which has only 
 one value, or at least, of which only one value is to be here em- 
 ployed. Write x 4- y^J-\ for Z, and let (x + y^f-V) = p + q*J-l, 
 where p and q are real, or unit-line, functions of x and y and 
 the unit-line symbols of A, B, &c. When x and y are such that 
 (x -)- ?/V~l) = 0, let the point of which they are coordinates 
 be called a radical point, single, double, triple, &c., according 
 as there are one, two, three, &c. roots equal to x + y^-1. Let 
 the extremity of Z traverse any bounded contour whatsoever 
 in the positive direction of revolution. As it traverses, note the 
 
 7) 
 
 changes of sign in - , at which the passage is through 0, neg- 
 lecting the changes at which the fraction passes through oo. 
 Let k be the number of times that there is a change from + to - , 
 and I the number of times that there is a change from - to +. 
 Then | (k - 1) is the number of radical points within the contour, 
 on the supposition that each radical point counts as often as 
 the root it indicates occurs. 
 
 First, if the theorem be true for each of the contours into 
 which the figure of a larger contour is divided, it is true for 
 the whole. For if all these contours be severally described 
 in the positive direction of revolution, each part, except the 
 external boundary, will be described twice, in two opposite 
 directions. Accordingly, every change from + to - or from - 
 to +, on any part which is described twice, is met by another 
 change from - to -f or from + to - when the same line is
 
 142 MISCELLANEOUS REMARKS 
 
 described in the opposite direction: and this for every thing 
 except the external contour. 
 
 If then we form k - I for each contour, and sum the results, 
 it is the same as if we had formed it for the external contour 
 only. 
 
 Suppose the whole contour divided into as many as may be 
 necessary of smaller ones, each of which may be as small as we 
 please. In order that k - I may have any value on a contour, 
 both p and q must vanish on that contour. For if neither 
 vanish, p v q does not change sign at all, and k - I is but 0-0. 
 If q only vanish, p-r q can only change sign in passing through oo ; 
 and such changes are not to be reckoned as part of k or I. If 
 p only vanish, all the changes of sign are made when p 4- q 
 passes through 0, and are all counted : and if all be counted, 
 there must be as many from + to - as from - to 4 ; or k = I, 
 k - I = 0. To give value to k - I there remains only the case 
 in which p and q both vanish. Now if smaller contours be 
 described within those first taken, and smaller within those 
 again, and so on, it must be at last (whatever it may have 
 been at first) that only those contours which have radical 
 points within them, have both p and q vanishing on them. For 
 suppose we take one in which there is no radical point, and 
 subdivide it perpetually, and always find internal subdivisions, 
 on the contours of which p and q change sign. We may pro- 
 ceed in this way until the extreme values of x, throughout and 
 within each contour, differ as little as we please, and also the 
 extreme values of y. That is, the values of x and y for which p 
 vanishes approach as nearly as we please to those for which q 
 vanishes, within the contour which has no radical point within 
 it: or 0(# + ?/v'-l) ma y diminish without limit within that 
 contour; which is absurd.
 
 AND APPLICATIONS. 143 
 
 It is, then, to contours having radical points within them, 
 and no others, that we must look for the possibility of k - I 
 having value. Let the subdivisions be so far increased in number 
 that no one possesses more than one radical point, single or 
 multiple as the case may be. Consider one of them, containing 
 a radical point P, to which s roots belong ; and let If be the 
 symbol of OP. Let Q be a point on the contour, and let OQ 
 be Z, and PQ, It. Then Z= H + It, and we have 
 
 0Z = 0(^4 4 H) = MR + NR' +l + .. 
 
 because 0Z is divisible by (Z - H) s , or (A 4 K) by If. 
 We have then, 
 
 0Z = mr s cos(sp 4 /*) 4 nr a * cos{(s + 1) p 4- K} 4 ... 
 
 4 (nir' sin (sp 4 /<) + nr" fl sin{(s + 1) p + } 4- ...] <*/-!, 
 
 p _m cos (sp + /LI) 4 w cos{(s 4 1) p + "} + ... 
 q m sin (sp 4 ,) 4 r cos{(s 4 1) p 4 "} + ... 
 
 Let the contour be made so small that the sign of this expres- 
 sion is not afiected by r : then it depends at last on that of 
 cot (sp 4- /M). In this, while the extremity of Z traverses the 
 contour, passes through an interval of 2ir, and sp + v through 
 s revolutions. In each of these, the cotangent changes sign 
 from + to -, passing through twice ; but the corresponding 
 change from - to + is made in passing through oo, and must 
 not be regarded. Hence k = 2s, 1 = 0; and * (k - 1) = s, the 
 number of roots which the radical point represents. 
 
 Since, then, the value of ^ (k - I) for the whole contour is 
 made up of the sum of the values for all the contours of the 
 subdivisions ; since no subdivision yields anything except it 
 contain a radical point, when it yields as many units as that 
 point represents roots ; the theorem stated follows at once. 
 
 Now we have 0Z = AZ" + Z"' 1 + , 
 
 whence it follows that, if 0.Z be p + q -/-I, 
 
 p __ az" cos(w + a) + bz"~ l cos{(w - 1) 4 /3} -f ... 
 
 q az" sin(ra -t a) 4- 6z" ' cos{(/t - 1) 4 /S} 4 ... ' 
 
 Let the contour in question be a circle, with the origin for its 
 
 centre, of radius Z so great in length as to contain all the 
 
 radical points ; and further, let z then increase without limit.
 
 144 MISCELLANEOUS REMARKS 
 
 The sign of p ~ q is ultimately always that of cot ( + a), which, 
 as before, is shown to yield 2n changes from + to -, and no 
 others in which p passes through 0, while passes through ITT. 
 Hence k = 2n, 1 = 0, or ^ (k - I) = n : that is, every integral 
 expression of the n th degree has neither more nor less than n 
 roots. 
 
 Algebraical paradoxes disappear under the application of our 
 significant symbols. The equation x^ = c (- xf is satisfied in- 
 dependently of x, by 1 = c (- l)i Change x into - x, which it 
 seems we may do, since the equation is now identical, and we 
 have (- x^ = c (x)^. Multiplication gives x* (- x)& = <?x^ (- a;) 5 , 
 or c 2 = 1. But - c 2 = 1. 
 
 The explanation is as follows. Let (x, ) be the symbol 
 denoted by x in the above equation : if it be real, is a mul- 
 tiple of 77 ; but this matters nothing. Then - x is (x, + TCTT) 
 where k is some odd number. Accordingly, x being a positive 
 arithmetical symbol, we have 
 
 2 I 
 
 ktr\ 
 
 which is satisfied by 1 = c ( 0, ) . When we say, change x 
 \ 2 / 
 
 or (x, ) into - x, we may, if we like, take a different value of k, 
 and then we have, k' being also an odd number, 
 
 f , + (A; + # 
 
 = C V *' 2 
 
 which is satisfied by the same value of c as before. Multiply 
 the equations together, and we have 
 
 ' 2 
 
 and k and &' being odd numbers, k + k' is even, say 2&". Un- 
 
 doubtedly, then, 
 
 one value of x* x one value of (- a;) 
 
 = (? {one value of (- xfy (one value of a$) ; 
 
 but we are not now sure of any common factor by which to 
 divide. And the double division is impossible. Let us make it 
 possible with respect to (- x$, which must be done by taking 
 k and k' both of the form 4m + 1, or both of the form 4m + 3.
 
 AND APPLICATIONS. 145 
 
 We have, then, the same form of (- x)' 1 on both sides. But 
 then k + k' is in either case of the form 4m f '2, ancl therefore' 
 the division gives 
 
 * 
 
 which is always satisfied by c 8 = - 1. 
 
 If the successive changes of sign in x be made by one con- 
 tinuous method, say addition of IT to the angle, then, starting 
 with one particular form of x*>, say a, we pass successively through 
 a, V-1> ~ a > -V~1> a i & c - If by addition of - TT, then we 
 proceed through , - a*J-l, -a, a*J-l, a, &c. And similarly 
 for other roots. The remembrance of the meanings of the 
 symbols will save rules : whatever angle we add to x, we add 
 the wi th part of that angle to its m th root. 
 
 The same difficulty occurs in treating the equations 
 
 (- x) = c<px, i } = c(px, &c., 
 \x I 
 
 all of which may be made to appear to require <? = 1, as above. 
 But the first is satisfied by 
 
 <px = x m , if (- l)'" = c; 
 and the second by 
 
 x = (log*), if (- 1)'" = c. 
 
 And the explanation is of the preceding kind in both cases. 
 
 As long as a and b are unit-line symbols, and the length 
 of b less than that of a, ^(a? - 6 2 ) is a unit-line symbol, as 
 follows. Let OA and OB be a and b (the reader may supply 
 the diagram), and take AC = OB on the limit line. From O 
 draw a tangent to the circle having centre A and radius AC; 
 let the point of contact be P, and take OQ on the positive 
 part of the unit line = OP. Then OQ is + V( 2 - & 2 )- But when 
 b is of greater length than a, V(a 8 - 5 2 ) is the symbol of a line 
 on the axis of direction : O is now within the circle, and if the 
 circle cut the positive axis of direction in _R, OR is + V( fl2 - & 2 )- 
 All this is evident from geometry ; and nothing in the higher 
 parts of modern geometry is more remarkable than the con- 
 stant connexion of the shortest semi-chord passing through a 
 point within a circle with the length of the tangent drawn from 
 a point without it. 
 
 O
 
 146 MISCELLANEOUS REMARKS. 
 
 The student may investigate for himself the difference of 
 meaning of the following theorem, 
 
 (the second term having the sign of b) in the cases in which 
 a~ - V is positive, and those in which it is negative.
 
 ( 147 ) 
 
 CHAPTER VIII. 
 
 ON THE ROOTS OF UNITY. 
 
 THE roots of unity are really, when algebra is made com- 
 plete, of an intermediate character between the quantitative 
 
 symbols A, S, &c., and the directive symbols + and -. They 
 
 
 may be given absolutely to either class of symbols. Thus (1)" 
 
 is [1, ), k being any positive or negative integer: and thus 
 
 V n I 
 
 
 
 we find (p. 132) that (1)" is the unit of length inclined at any 
 
 number of w th parts of a revolution. No question, then, that 
 
 i 
 
 (1)" is a perfect particular case of A'\ But if, considering -f 
 
 and - in their directive character, we had chosen* to designate 
 
 !_ 
 by (+)" the prefixed sign of a change of direction which would 
 
 restore + A back to that form after n performances of its opera- 
 
 1 
 tion : and by (-)" a sign of such change of direction as would 
 
 change + A into - A after n such performances ; we might 
 
 i i 
 
 have established the laws of exponents over (+)" and (-)", and 
 
 \_ i^ 
 
 (i)"A and (-)"A would have had + A and - A for particular 
 
 i \_ 
 
 cases. But the (+)"A and (-)"A of the second view are abso- 
 
 i i 
 
 lutely identical with (+ 1)" x A and (- 1)" x A of the first. 
 
 The present chapter treats these roots of unity in a manner 
 which is by no means uncommon ; and which in itself involves 
 
 * The only point in which I differ from the view taken by my 
 deceased friend, the late D. F. Gregory, one of the most profound 
 thinkers who has ever attended to the subject, lies hi this, that 
 he advocated either the necessity or the unavoidable expediency 
 of the second view ; and I look upon the two as equally sound, 
 and the choice as a question of convenience which is settled 
 merely by usage.
 
 148 ON THE ROOTS OF UNITY. 
 
 a full half-leaning to the purely directive definition. The pro- 
 perties of these roots are established on the definition and 
 nothing else : no knowledge of the algebraical forms is de- 
 manded, or established for use. The properties, for instance, 
 
 of the forms of (I) 5 , their relations to one another, and to the 
 i 
 
 forms of (1)" for values of n other than 3, are quite independent 
 of the fact (unknown, it may perfectly well be) that their common 
 forms are 1 and \ (1 </- 3). Accordingly, the whole of this 
 chapter might be translated into an algebra of directive signs, 
 of which -f and- are mere instances. Thus, that a is a directive 
 sign which repeated n times has the same meaning as -f, might 
 be expressed by " = -t- ; = signifying identity of directive mean- 
 ing. I first consider the roots of + 1. 
 
 LEMMA. If m and n be integers prime to one another, 
 integers can be found, a and b, such that mb - na = 1. Turn 
 m -f n into a continued fraction, and let the approximation pre- 
 ceding the final restoration be a -f b. Then, by the property of 
 the successive approximations, mb - na is either +1 or - 1. 
 
 1. Every m th root is an (?nn) th root, m and n being any 
 integers whatsoever. For if "' = !, then (a m ) n = 1, or o m "=l. 
 The first root 1 T or 1, is a root of every order. 
 
 2. Every power of an m th root is an i th root: for if "* = 1, 
 (a n ) m = (a'")" = 1, so that " is also an m th root. This holds 
 whether n be positive or negative. 
 
 3. If m and n be prime to one another, no w th root (except 1) 
 is an M* root. Find a and b so that mb - na = i 1 ; if then 
 '" = 1, a" = 1, we must have ('") 4- (")" = 1 or a" = 1, or a = 1. 
 
 4. If an mth root be an w th root, it is also a & th root, where 
 k is the greatest common measure of m and n. Let m = km', 
 n - kri, m' and n' being prime to one another. Let m'b-n'a= 1, 
 then mb -na = k. Let be both ??z th and ?z th root : then a mb ' 
 
 = 1, as before, and a~* = 1, or a* = 1. For instance, we see from 
 (1) that the 8* roots are both 32 nd and 40 th roots : we now see 
 that the 8 th roots are the only ones which are both 32 nd and 
 40* roots. 
 
 5. There cannot be more than n th roots. For x n - 1 is 
 (x - a) (x - /3)...a, (3 being all the roots. As soon as n separate
 
 ON THE ROOTS OF UNITY. 149 
 
 roots are discovered, the product becomes of the w th degree, and 
 is then identical with x n - 1. And this product cannot vanish 
 except for x - a, or /3, &c. Nor can any of these roots be equal, 
 since nx tl ~ l has no root except 0. 
 
 6. If n be a prime number, and a be one root (not 1). 
 then 1, o, a 2 , ... a"" 1 are n different roots. For if a - a, k and / 
 being less than Z, we have a k ~ l - 1 ; but n is greater than, and 
 therefore prime to, k - I (being a prime number, and prime to 
 all numbers except its multiples), and an n l}l root cannot be 
 a (k - /)th root. 
 
 7. If m, n, p, &c. be each prime to all the rest, and if 
 a and a' be two different TO* roots, /3 and /3' two w th roots, &rc.. 
 it is impossible that 0/37... = a'(B'~/'... For an instance, take 
 three classes of roots. Then because a and a' are m th roots, 
 and up"/ = a'/3Y> it follows that p'"^ = /j""-/"", and because 
 /3" 4 and /3"" are n' h roots, 7'"" = 7""", therefore 7 - 7' is an (wm) th 
 root. But 7 P = 7* = 1, therefore 7 4- 7' is a jth root, or p and 
 ?H, being prime to each other, 7 -f- 7' is both a _pth an d an (?n) th 
 root, which cannot be. 
 
 8. If n be not a prime number, let P, Q, R, &c. be its 
 prime factors, and let n = P v QiRr. . . Then if a be any P p th 
 root, ft any Q 3th root, 7 any R r *h root, &c., a/37... * s an nth 
 root. And all the n th roots can be thus found, and no more. 
 First, (/37...)" or a"(B n ^"... is Ixlxl..., since n is a multiple 
 of P p , and therefore a" = 1, &c. Therefore 0/37... t's an n'h root. 
 Next, (by 7), no two such products can give the same th root, 
 since P p , Q 9 , &c. are prime to each other. Thirdly, since there 
 are P p P*** 1 roots, Q q Q 7 th roots, &c. the number of combi- 
 nations of one out of each set is P p Q q ... or n. Therefore all 
 the varieties of such products give n different n th roots, or all 
 the n th roots and no more. 
 
 Accordingly the whole question of finding roots has been re- 
 duced to that of prime orders and power-of-prime orders. All 
 the 360 th roots, for instance, are found whence the 2 3 t h 3 3th and 
 5 th roots arc" found. 
 
 9. Every order has some roots which belong to no lower 
 order. If n be a prime number, this is the case with n - 1 ot 
 
 03
 
 150 ON THE ROOTS OF UNITY. 
 
 the roots (all except 1). If n be of the form P p , P being prime, 
 any n lh root of a lower order than n must be (7) of the P p ~ ' 
 order: for, P being prime, P* is the only form of common 
 measure of P p and lower numbers, k not exceeding p - 1. Hence 
 there are P" - P" l or P p ~ l (P - 1) of the P pth roots which 
 are of no lower kind. Next, if n be P p Q q R r ... and if we take 
 cifi-/... where a is one of the P*** 1 roots which are of no lower 
 kind, &c. then apy... is an th root of no lower order. For 
 (a/ty...)'" = 1 must give a m = 1, ft'" = 1, &c. : if a'" be not = 1, 
 a'" = (/3-y. ..)""', and '" being a P^h root, so is (/*/...)""'. But 
 this last is a (Q q R'. ..) th root, and PP and Q q R r ... are prime 
 to each other. Therefore '" = 1, &c. Now since a m = 1, and 
 is a P^ 11 , and no lower root, m has P p among its factors; 
 since {3'" = 1, &c., m has Q q among its factors ; and so on. Hence 
 m cannot be less than n, or P p Q q R r ...; while it is obvious that 
 it may be n. 
 
 Hence the number of n th roots which are of no lower order 
 is P p - l Q q '\.. x (P - 1) (Q - 1)... : that is, (Arithmetic, p. 196) 
 for every number less than n and prime to it, (1 included) 
 there is an n ih root which is no lower root: and all the other 
 M^ roots are lower roots. 
 
 Let those nth roots which are no lower roots, be called prin- 
 cipal* w th roots. Then there are 4 principal 12 th roots: for 
 less than 12, and prime to it, we have 1, 2, 7, 11. 
 
 Grant one principal root, and all above follows immediately. 
 
 For if 1, a, *, a" ' be all different, and if we select a*, 
 
 in which k is prime to n, then 1, a*, a 2 *, ( ' M) * are all dif- 
 ferent, and embrace the whole of the first series in a different 
 order. For the succession 1, k, 2k, ... (n - 1) k with each term 
 divided by n, gives the same remainder in no two cases 
 (Arithmetic, p. 195). But if Ik = I'n + r, aft = a r : and therefore, 
 in the second series, we see nothing but the first series with 
 its terms altered in order. Thus, if a be a principal 12 th root, 
 
 * I would have said primitive nth roots, but Gauss has used 
 this last word in connexion with the subject of roots. Moreover, 
 it is not that these roots are primitive nth roots, so much as that 
 'nth' \s the primitive ordinal of these roots.
 
 ON THE ROOTS OF UNITY. 151 
 
 the principal 12 th roots are a, 5 , a 7 , ", and if we form the 
 powers of these, dividing by 12 whenever it occurs, we have 
 the 12 th roots, all of them, arranged in the following sequences : 
 
 1 a a* 
 
 1 a 5 a" 
 
 1 a 7 a* 
 
 1 a" a" 
 
 10. All the powers of an n th root are cyclical. Thus, if a 
 be a principal root, we have cycle of n; for we have 1, a, ... 
 a"" 1 , a"(=l), a"' 1 (= a), ...... But if be not a principal root, 
 
 the cycle is in number sub-multiple of n. If, for instance, 
 n being 12, a be a sixth ro.ot, we have a 6 = 1, 7 = , a* = a*, &c. 
 The negative powers are only the same cycle repeated back- 
 wards ; thus a~ l = a"" 1 , a' 2 = a"" 2 , c. 
 
 The most convenient way of considering the roots is by 
 arranging them in reciprocal couples, or from the beginning 
 and end of the cycle. Thus, being a principal 12"' root, 
 we distribute the 12 roots into 1 ; and a", or and "' : 
 a" and ', or a 2 and er a ; a 3 and ~ 3 ; 4 and ~ 4 ; 5 and '''; and 
 lastly, a, not a 6 and "*, for * = a" 6 , and each must be - 1. 
 The student must remember not to couple + 1 and - 1. 
 
 The cycles of couples have a reverse order, both in the couples 
 and in their succession. Thus the double cycles of 12 th roots run 
 thus: 1, a and a" 1 , a 2 and a" 2 , a 3 and a" 3 , a 4 and a" 4 , a 5 and " 5 , 
 o 6 or -1, a 7 and a" 1 , which is a" 5 and a b , a 8 and 8 , which is 
 <T 4 and a 4 , &c. 
 
 [Hitherto we have had nothing to distinguish one principal 
 root from another. But when we consider the values of the roots 
 (page 45) we see one pair of roots, both principal, and principal 
 among principals. They are the ones which have the smallest 
 angles in the first revolution, positive and negative : namely, 
 
 '2- . 27T / 27T\ . / 27T\ 
 
 cos + sm . J-l and cos -- + sin -- ) . -/-I, 
 n n \ n ) \ n ) 
 
 -. . 2?:- sin 2w . 
 
 contained in cos + 
 
 n n 
 
 (2w\ 
 1, j short of 
 
 the w*i is an equivalent of (1, 0). But they are distinguished
 
 152 ON THE ROOTS OF UNITY. 
 
 from all other principal roots, in that they, by their powers, 
 furnish the simplest forms of all the other roots, namely, with 
 angles in the first half-revolutions, positive and negative. They 
 ought to be called radical n th roots.] 
 
 11. We show a mode of forming all the 12' h roots whenever 
 we show a mode of proceeding from number to number, in 
 such manner that by casting out 12 whenever it arises, we get 
 
 the results 0, 1, 2, 11, in any order whatever. Thus, 
 
 beginning with any number, and proceeding by additions of 1, 
 or 5, or 7, or 11, we obtain all the succession 0, 1, ... 11, as 
 in (9). Can we now do this by successive multiplications f 
 Trial will give reason to announce, in any case we may take, 
 that, leaving* out (and consequently a or 1), we can always 
 find a multiplier or multipliers which will succeed with a prime 
 number. With 13, for ins'ance, the following multipliers will 
 succeed: 2, 6, 7, 11. Take any number to begin with, say 4; 
 choose 6 as a multiplier ; throw out 13 as it arises, and we 
 shall have the succession 4, 11, 1, 6, 10, 8, 9, 2, 12, 7, 3, 5, 
 (cycle complete) 4, 11, 1, &c. Beginning with 1, as most con- 
 venient, we have for the 13 13t h roots of unity, a, 6 , a 10 , a 8 , 
 a 9 , o ! , a K , a 1 , 3 , a*, ', a 11 ; which, with 1, complete the list. 
 Of this cycle it is immediately seen that if for a we write any 
 other, as a 5 , the cycle is only made to begin in another place, 
 and its successions are uninterrupted. Thus a 5 , ( 5 ) 8 , ( 5 ) 10 , (a 5 ) 8 , 
 &c., are a 5 , a 4 , a", a, &c. That is, we have a method of arrang- 
 ing the roots in recurring cycles such that the substitution of 
 one root for another only disturbs the commencement of the 
 cycle, and not the order in which the roots occur. I return 
 to this subject again. 
 
 12. Every function of the n w th roots, or of any of them, 
 which admits of being expanded in integer powers, positive or 
 negative, of them all, is always reducible to the form 
 
 A a + AJI + A^? + ... + A^a"' 1 , 
 a being a principal root. For when the expansion is made, so 
 
 * That we must leave out is obvious enough, after what we 
 have seen of it as a starting symbol of addition, as opposed to 1, 
 the starting symbol of multiplication.
 
 ON THE ROOTS OF UNITY. 153 
 
 that every term is of the form a p /3 9 7 r ..., a, (3, 7, ... being 
 th roots, substitution of the values of /3, 7, ... in terms of a 
 will give a series of powers of a, which is reduced to the pre- 
 ceding form, since a" = 1, a n+1 = a, &c. Observe, I here speak 
 of the form only: that form may not be fit for calculation, 
 for A , A v &c., or some of them, may be divergent series. 
 
 13. The sum of the w th roots, the sum of the products of 
 every two, of every three, &c., is : but the product of all is 
 - 1 or + 1, according as n is even or odd. This follows from 
 the structure of x n - 1, and the theory of equations. 
 
 14. The sum of the m th powers of the th roots of unity 
 is always 0, except where m is n or a multiple of it, positive 
 
 or negative, and then it is n. For 
 
 1 - a""* 
 
 l m + a m + a*"+... + aW n = \ m . 
 1 -a 
 
 If TO be n, or a positive or negative multiple of it, the first 
 side is obviously 1 -f 1 + ... or n. In every other case, a m is 
 not = 1 and a"'" is = 1 : whence the sum of the terms is 0. 
 Better thus, if it were not that proofs of unexpected simplicity 
 are suspicious. Multiply the sum by a'", it undergoes no altera- 
 tion except transferring 1'" from the beginning to the end. If 
 the sum be S, we have then S=a m S, or S=0, unless a"* = l. 
 
 15. Any symmetrical function of the n w th roots, otherwise 
 real, is real : for every such symmetrical function is a real func- 
 tion of the sum, the products of every two, &c. 
 
 16. If in any function of -JA, n \' B, ^ C, &c., we multiply 
 -\/A separately by every m th root, ^ B by every n th root, &c., 
 and introduce every combination of these values into the function, 
 giving mnp . . . functions in all, and multiply the resulting func- 
 tions together, the product will be a rational function of A, B, C, 
 &c. For example, ^x + -Jy : 1 and - 1 are the square roots 
 of unity, let 1, a, a 2 , be the cube roots. Then I say that 
 
 (V*+Vy) (-V*fvV) (V*^ 7 ?/) (-V^ 8 Vy) (V^^Vy) (-Vx+a*fyy) 
 
 is a rational function of x and y. 
 
 A rational function of A is known by its presenting the 
 same value, if for ^ A be substituted in it a V-4, being any 
 th root of unity.
 
 154 ON THE ROOTS OF UNITY. 
 
 If in the product preceding, which is a symmetrical function 
 
 "/ B 7 "/ 
 
 of V.4, a v ^4, a"' l vA, a being a principal w th root, we 
 
 substitute a k y A for V-4> we have the same function of a* ^/A, 
 4+1 V^, &c....a" +i ~ 1 V^4, or of the same quantities interchanged 
 in order, which, as the function is symmetrical, makes no dif- 
 ference. Hence the product is a real function of A ; and so 
 of the rest. 
 
 The product of the six factors in the example is y~ - x 3 . 
 
 17. If a, ft 7, ... be all or some of the th roots, and if 
 </>(a, ft y, ...) be a function of a, ft y, ... capable of expansion 
 into A + Ap + ... + A^a 1 ' 1 -, then 
 * (a, ft y, . ..) + * ( s , 2 , y 2 , ...)+... + * (a"' 1 , /S"", y- 1 , ...) = nA v 
 
 Returning to the mode of arrangement in (9), we see that 
 if m and m' be two numbers which, divided by p, leave re- 
 mainders r and r / , say qp + r and q'p + r 1 , the remainder of the 
 product mm' is that of rm'. If then we take a, a 8 , ... until 
 we get above p, and then reject the multiple of p, take only 
 the remainder r, go on with ra, ra*, ... until we get above p 
 again, and proceed as before, we really form the remainders of 
 the successive powers of a. Thus, if we want to know the 
 remainders of the powers of 2 divided by 11, we have but to 
 form the series 2, 4, 8, (16, reject 11) 5, 10, (20, reject 11) 9, 
 (18, reject 11) 7, (14, reject 11) 3, 6, (12, reject 11) 1, 2, 4, 8, 
 &c. Now it is proved, in works on the theory of numbers, that 
 if any prime number be taken, n, there are numbers less than n 
 for which the powers, successively divided by n, yield all the 
 n - 1 possible remainders before any recur. That one of these 
 should always exist, is enough for our purpose : but, in truth, so 
 many numbers (1 included) as are less than and prime to n-l, 
 so many numbers less than n are there for which the powers 
 yield all remainders before any recur. Thus, calling such num- 
 bers primitive.* subordinates of n, and examining 19, a prime 
 number, we find that 18 has 6 numbers below it, and prime 
 to it. There are then 6 primitive subordinates of 19, and they 
 
 * Gauss calls them primitive roots of the integer n : but this 
 term would cause confusion, unless the analogies by which it is 
 justified were introduced.
 
 ON THE ROOTS OF UNITY. 155 
 
 are 2, 3, 10, 13, 14, 15. That is, taking 14, 14 1 , 14 2 ... 14 18 , 
 all yield different remainders when divided by 19; so that all 
 the numbers 1, 2, 3, ... 18, are among those remainders. If then 
 a be a principal 19* root of unity, all the 19'h roots, except 1, 
 are contained in the set 
 
 and no one twice. The advantage of this is, that if for a we 
 write another, we only change the commencement of the cycle. 
 Thus, if for a we write a 14 , we only remove the first and second 
 above to the end. This is not the case when we write one for 
 another in the more natural cycle a, a 2 , ... a"" 1 . Remember par- 
 ticularly that the root 1 never enters this series. 
 
 Let all the (n-l) th roots be known, and let w be one of them. 
 
 Let us consider the expression 
 
 2 ., Hi"" 1 
 
 P = a + a w -f a or + ...... + a w , 
 
 m being a primitive subordinate of n. Remember that u>"~ l =l. 
 We see that change of a into a"' is here equivalent to multipli- 
 cation by iv ; change of a into a"' 2 to multiplication by to" ; and 
 so on. So that P"~ l is not aifected by writing any other root 
 for a. Hence if P"" 1 be really constructed by multiplication, it 
 will be found independent of a, or a function of w only; say Q. 
 Hence P = Vfi can be expressed. Let the form of it em- 
 ployed be called 0w. Do this for each root 1, w, w 3 , ...w"'*, 
 iv being a principal (n-l)th root: and let a^ a 2 , a 3 , ... be the 
 successive w th roots a"', ... We have then, taking the obvious 
 equation when 1 is used for w, 
 
 0w = Oj 4- a f 
 0a> 2 = ttj+ fl 2 
 
 00)"- 2 = a l -f f/X 1 ' 2 + 3 w*^ + + a 
 
 From which, by the property in (14), we find 
 
 (n- !)! = -!+ 0f + 0w 2 + + 
 
 (n - 1) a, = - 1 + w"- 2 0w + W 8(n - 2) 0a- 2 + + 
 
 (n - 1) 3 = - 1 + w"" :< 0' 1 ' + t^ (n -^w z + +
 
 156 OX THE ROOTS OF UNITY. 
 
 Whence it appears that when n is a prime number, the n th 
 roots can be expressed in terms of the (n-l)th roots, and are 
 therefore algebraically determinable when the latter are so. 
 
 Writers on this subject give methods of reducing the labour 
 of the preceding : but as my object* is to show the possibility 
 only of finding the w th roots when n is a prime number, and 
 the (n - l) th roots are known, I shall content myself with giving 
 at length the determination of the fifth roots ; 5 being a prime 
 number and the 4 th roots known. One primitive subordinate of 
 5 is 2; and the succession is 2, 4, 3, 1. Hence, a being a fifth 
 root other than 1, and w a fourth root, the fourth power of 
 a z + a 4 w + aW + aw 3 is independent of a. Now, remembering 
 that w 4 = 1, w b = tv, &c., the square of the preceding is 
 
 (a 4 + a 8 + 2) + 2 ( 4 + a) w + (a 3 + a* + 2) w z + 2 (n s + 2 ) w 3 , 
 and the square of this will be found, remembering that 
 
 1 + a + a* -f a 3 + a 4 = 0, 
 
 to be - 1 + 4u) + 14w* - 16aA Let tc v w v . u- 4 , be the fourth 
 roots, and Q lt &c. the values of the preceding. Then we have 
 
 VQj = a* + a 4 a>j + a?u<* + ait>*, 
 
 and similarly for the rest. If w l = V~l a ' 2 = - li w s = ~ V-l, H-\ = 1, 
 we have Q! = - 15 + 20 V-l, Q 2 = 25, Q 3 = - 15 - 20 V-l, Q 4 = 1. 
 We now proceed to discover which of the fourth roots is to 
 be used; nothing being known except that we are to take the 
 same form in all cases. With no restriction, there are 4x4*4x4, 
 or 256 different systems of equations. One form is determined 
 by the question : V^ 4 must be - 1 ; for a* + a 4 4 a 3 + a = - 1. 
 Hence the form of Vl required is that of a principal fourth 
 root moved through an odd number of right angles. Now in 
 the case of a + b V~l, each form of the fourth root has all the 
 properties of a principal form ; for no one of these fourth 
 roots is a square root. And to a + I V~l & n( l a - b ^/-l cor- 
 responding forms are such as p 4- q V~l and p - q V-l, sym- 
 
 * The hint of this limitation of object is taken from the late 
 Mr. Murphy's work on the Theory of Equations : but I have not 
 thought it necessary to enter, even so far as Mr. Murphy has done, 
 into the methods of reduction.
 
 ON THE ROOTS OF UNITY. 157 
 
 metrically disposed with respect to the axis of length. Take 
 such a pair at pleasure, and move them in the same direction 
 through an odd number of right angles, and we have a pair 
 such as - p + q V~l and p + q V~l> which are symmetrically dis- 
 posed with respect to the axis of direction : and such is the 
 pair which must be chosen. 
 
 Now if we extract the fourth roots of - 15 20 V~l hy the 
 formula 
 
 -i - 
 
 we shall find them all contained in 
 
 pq V-!> and (f q V-l) V-l> 
 
 using like signs in the two terms for -15 + 20 V~l> and unlike 
 signs for - 15 - 20 V-L And j = VM 5 + V5), q = Vi( - V5). 
 Choosing a pair symmetrical with respect to the axis of direc- 
 tion, we form the following equations: 
 
 -p + q V~l = a 8 + * V~l - a a - a V"1| 
 
 - V<5 = a* - a 4 + a 3 - a, 
 p + q V-l = a* - a* V~l - a 3 + a V~l, 
 
 - 1 = a* -f a 4 + a 3 -f a. 
 Sum these as they stand, and then sum them after multipli- 
 
 cation by - V-l, -1, V~l> 1 ; - 1, 1, - 1, 1 ; V" 1 . -1 - /-! 1. 
 We thus obtain 
 
 2 = - i (V5 + 1) + I q V-l, a 4 = i (V5 - 1) + i^ V~l- 
 
 3 = -i(V5 + l)-i?v/-l, =i(V5-l)~ W-l, 
 
 which are well-known values of the fifth roots. Changes of sign 
 
 in p, or q, or both, have no other effect except different appor- 
 
 tionment of the above expressions among the roots a, a*, a 3 , a 4 . 
 
 The extraction of the square root of a + 6 V~l i s an operation 
 to which Euclid's geometry is competent; it requires only the 
 bisection of an angle, and the determination of a mean propor- 
 tional, to obtain (V(a 2 + 6 ! ), -J tan" 1 6 -f- a} from {V( 2 + &*)> tan" 1 b -f a}. 
 
 Hence it follows that wherever n is a prime number, and 
 n - 1 is a power of 2, the formation of the nth roots of unity is 
 a geometrical* operation, in the ancient sense. Euclid mastered 
 
 * This is the discovery of Gauss, and is the most remarkable 
 addition to the power of construction which the ancient geometry 
 has received since the time of Euclid. 
 
 P
 
 158 ON THE ROOTS OF UNITY. 
 
 the cases n - 3, n = 5; the next one is n = 17, and the next 
 n = 257. 
 
 The theory of the roots of - 1 is really contained in that of 
 the roots of + 1. Since a: 2 " = 1 is solved both by x n = 1 and 
 x n = - 1, it follows that all the nft 1 roots of - 1 are among the 
 2n th roots of + 1. If a be a principal 2n^ root of + 1, \ve must 
 have a" = - 1, a 3 " = - 1, &c., and the n n^ roots of - 1 are seen 
 in a, a 3 , a 5 , ... a 2 "" 1 . Speaking now of roots of - 1 only, we have 
 the following theorems, answering to some of those in page 148. 
 The student may make a complete list of analogous theorems. 
 Every m th root is an {m(2n + l)}th root. Every odd power of 
 an m th root is an m* root. If m and n be prime to one 
 another, no m th root (except - 1, if both be odd) is an w th root: 
 for a 2wi th root of + 1 would then be a 2n th root, which can only 
 happen as to a square root. 
 
 If a be a principal 2n th root of 1, it is a principal nth root 
 of - 1. For in that case a, a 3 , ... a 2 "" 1 are all different, and 
 only a" is - 1. And there are no other principal n* h roots of 
 -1. 
 
 Let a be a principal M* root of - 1, or 2n th root of + 1. Then 
 a, a 3 , ... a 2 "" 1 are all different ; multiply each by a, and a 8 , a 4 , ... a 2 " 
 are all different. Nor can any one of the first set be the same 
 as any one of the second : for if a u = a 2W , we have a" = a 2n (*-'^ 
 or - 1 = + 1, which is absurd. Therefore there are as many 
 principal w th roots of- 1 as 2w th roots of + 1, and no more. 
 
 The sum of the & th powers of the w th roots of - 1 is always 0, 
 except where k is a multiple of w. The series a* -f a 3 * + . .. -f at 2 "- 1 )* 
 is not altered by multiplying by a- k , except by removing the 
 first term to the end: consequently it is except when a- k - 1, 
 that is, except when k is a multiple of n. If it be an even 
 multiple, the sum is n ; if an odd multiple, it is - n. 
 
 Among the uses which may be made of the roots of unity, 
 the following are remarkable. 
 
 An expression may be formed, which goes through recurring 
 periods of changes while x, of which it is a function, takes suc- 
 cessive integer values. Let n x stand for (* + ft? + }^ n 
 
 a, ft, &c., being all the n th roots of -f 1. Then as x changes 
 through 0, 1,2,..., + 1, &c., S z changes through I, 0,0, ...1,0,... 
 Thus fl^g represents the z fh payment of a rent of a, which
 
 ON THE ROOTS OF UNITY. 159 
 
 is due only every fourth year, the year after next being a year 
 of payment. This is 
 
 - <^ cos(:e - 2) - + cos(2# - 4) - + cos(3x - 6) - + cos(4a; - 8) ^ I , 
 it v, & ^- J 
 
 the coefficient of the imaginary part always vanishing in a sum 
 
 of powers. 
 
 . . a I TTX 37rx\ 
 
 This is - 1 + COSTTX - cos - - cos - - . 
 4 \ 2 2 / 
 
 Again, a/i x + a l n x _ l + a./i j .. z + . . . + a^n^^ represents an ex- 
 pression which takes the cycle of values a , a v ... a, M as x 
 passes through 0, 1, ... n- 1; and repeats the same while x 
 changes through n, n + 1, ... 2n - 1: so that (fax = a r , when r 
 is the remainder in the division of x by n. 
 
 If n x be formed from the w th roots of - 1, the above represents 
 an expression in which the second cycle is formed by changing 
 all the signs, the third by restoring them ; and so on. 
 
 If 0# be A + A^x 4 Aj? + ... a finite or infinite series, the 
 roots of unity enable us* to give a finite form to 
 
 A m + A nMl x + A nMtn x* + ...... 
 
 First, suppose in < n : for x write ax, a being one of the /* th 
 roots of 1, and multiply by """', forming a"" m 0az. Do the same for 
 each root and add the results. The theorem on the sums of 
 the powers of the roots then gives 
 
 ~S,i"' m (hax 
 
 _ A f*.' 11 i A ,>."*"* I ^ /^''" 
 
 - * T -"wj+ji 3 ' T -" L nH--in X 
 
 Divide by x m , and write \j x for x, and the required result is 
 obtained. Ifm>n, say m-pn + k(k<n); find A^x k + A /t+;1 a^ m +... 
 and subtract as many of the first terms as are requisite. 
 
 If the n*h roots of - 1 be used, we may in the same manner 
 find A. m - A nt+n x + ... 
 
 For instance, let it be required to find 
 
 x x* x 3 
 
 + 6?L8!9 + 6.7...12J3 + 6.777.16.17 + 
 
 * In my Differential Calculus, I gave this as (to me) new, 
 expressing a doubt that so apparently obvious a method should 
 never have struck any one. I have since found it given by Thomas 
 Simpson, in the Philosophical Transactions, as read Nov. 16, 1758.
 
 160 ON THE ROOTS OF UNITY. 
 
 Write x* for x, and multiply by a? 1.2.3.4.5, and we have 
 selected terms out of e*, each fourth one being taken, beginning 
 at x h -f- 1.2.3.4.5. Begin at x, and we should obtain the whole 
 series from 
 
 HIV 4 (- D 3 *-* + (V-i) 3 ^' 1 + (- V-i) 3 ^- 1 }, 
 
 Whence 1.2.3.4.5 
 
 is the value of the required series. 
 
 In using this method, it will be best to take a pair of corre- 
 sponding roots, of the form cos6sm6^/-l, and work out, 
 in general terms, that part of the sum of the functions which 
 arises from that pair. Suppose for instance, (f)x = (1 + x) 4 , and 
 that A m x m + A nVrn x* n 4- ... is required, m being < n. If 
 
 a = cos# + sin0 -/-I, 
 we have (ax) = (1 + x cos0 + x sin# . V~l)* 
 
 Multiply this by a n ' m or by a" 1 ", then change the sign of 0, 
 and add, and we have for the part of ~S,a n ~ m (pax which depends 
 on the two roots in cos# sin#.'/-l, 
 
 (1 4 2x cose + s?Y . 2 cos |& tan' 1 ^^ ^ - m0j
 
 ( 161 ) 
 
 CHAPTER IX. 
 
 SCALAR VIEW OP ALGEBRAICAL SYMBOLS. 
 
 IT will be admitted that the view of the extended meanings 
 of A + B and A x B given in pages 118, 119, is a very natural, 
 and even necessary, consequence of the separation of subject 
 matter and operative direction in pages 115, 116 : and that it makes 
 the entrance of the extended subject-matter dictate the mode 
 of assigning significance to A + B and A x B. But the transition 
 to A 3 appears destitute of sufficient obligation to previous sug- 
 gestion. This chapter, which is above most elementary student^, 
 is intended to defend the mode of transition, and to show that 
 the adverse judgment which may be given upon it is partly 
 to be compared to the opinion which a beginner forms upon 
 the law of a series when he has expanded but one or two 
 terms, and which he retracts when he sees those that follow ; 
 and partly due to a failure of consistency in algebraic notation. 
 
 In A 4- B and A x B we see convertible operations, and we 
 take the hint to denote convertible operation by a symbol 
 placed between the subjects: thus AB, A*B, might denote 
 other convertible operations performed with the instruments A, It. 
 Again, we see also a character of ascent, and a connexion : namely, 
 the right of distribution of the higher operation over the ten/in, 
 or separate instruments, of the lower. Let us continue on this 
 hint, and invent an operation which bears to A x B the -saim 
 relation as A x B to A + B ; and another yet again above 
 the last; and so on. Having symbols for the first two, \vt 
 keep them : and provide a notation indicative of the degrn- 
 of ascent, just as we retain the terms square and cube, ami 
 then pass to third power, fourth power, &c. 
 
 Let A"' B, A'"B, A"B, indicate the successive steps of 
 the ascent, ro that our equations of definition are
 
 162 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 
 
 + = 
 
 AxB = BxA, A"'(BxC) = (A'"B)x(A'"C) 1x1 = 1 
 '"C) = (A"B)'"(A"'C) O 3 "'O 3 = Q 
 '"C) = (A t ' B) llf (A* (7) j Q 4 '"Q 4 = Q 
 &c. &c. 
 
 &c. 
 
 Again, in A + B and A x B we have initial symbols and 1 ; 
 of which it is the property, as in + = 0, 1x1 = 1, that when 
 no instrument except the initial symbol is used, nothing but 
 the initial symbol results. The equations of the second and 
 third columns are formed by obvious extension. 
 
 Now let us denominate by the name of scalar function or 
 operation the function which has this property, that its per- 
 formance on the compound is equivalent to the next lower 
 compound of its performance on the separate terms: so that, 
 if X^ liH be the symbol of the scalar function connecting A (n ' l} B 
 and A M B, we have 
 
 It might at first be supposed that we could have different 
 scalar functions at every transition : but a moment's consideration 
 will show that the perfect accordance of the different symbolic 
 relations would enable us to generalize the scalar system, so 
 as to make all its steps alike, if it should so happen that in 
 any one part of the system we found a scalar function of a 
 more general character than had theretofore appeared. The 
 utmost variety that we can admit is, that in one ascent one 
 particular case should be taken, and in another, another, of 
 the most general form which exists. 
 
 Nor is any argument against the above to be derived from 
 the fact of the sequence of operations having a commencement; 
 for there is in truth no commencement. The operation which 
 precedes A l B or A + B is AB, satisfying 
 
 A + (BC) = (At B)(A + C). 
 
 And, understanding any sign + or - in parentheses, in an alge- 
 braical sense, we have 
 
 A g (B <- C) = (A B) <-" (A C), &c. 
 The initial symbols may be represented by Q , Q^, fi^, &c.
 
 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 163 
 
 The system of operations is then interminable in both direc- 
 tions. Let X be the symbol of the scalar function, and let \ 
 represent its inverse function : so that \\A = \\A = A, for one 
 value at least. Suppose, in order to assimilate our system to 
 that of algebra, that whatever forms \A may have, \A has but 
 one form; so that \\.A is A, though \\A may only have A 
 for one of its forms. It would be best, as in algebra, to con- 
 struct a main system upon one form of \A, and to express the 
 results of other forms of \A in terms of that system. In this 
 trunk-system, \ve may consider \ and X as having each only 
 one form; and all the direct operations as having each only 
 one form. It is now clear that if only one of the convertible 
 operations be given, and the scalar function, all are given ; and 
 that we have 
 
 for y (\A m !)) = \\A = A ; and this is A" v \Q nt whence 
 O m = \D. 
 
 The symbols of ordinary algebra, considering the various 
 accidents by which they have attained their positions, have 
 great, but not perfect, consistency. We have A + B, Ax B, 
 and the scalar function log^4 ; the last derived immediately, by 
 Napier, from the connexion of A + B and AB, and not in- 
 directly* from the exponential function, which he knew nothing 
 of. The exponential function is out of the system, strictly speak- 
 ing : for it will be found that the scale of operations having 
 the indices ...... (- ll), (- l), 0, I, n, ill, IV, &c., is, in the ordi- 
 
 nary language, X being used for log., 
 
 ...... X 3 (V 6 " + &6 ), X 2 (e " + e* 6 ), X(e -1- e 6 ), a + b, 
 
 X(Xa + X6) or ab, v*(\*a + X 2 6) or e XaA * or a Xi or 6 Xa 
 X 3 (\ 3 a + \ 3 b) or s ** x \ & c . 
 The initial symbols are X 3 0, X 2 0, XO, 0, 1, e, e , &c. 
 
 * So that by going back to the sources, we find the logarithm 
 first exhibited as the scalar function by Napier, its inventor, and 
 the trigonometrical system first appearing as founded on ratios, in 
 the writings of Rheticus, who first presented it complete.
 
 164 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 
 
 Accordingly, in A E , there is not scalar relation between A 
 and B, but between \A and B. And the preceding is not 
 merely justification, but even proof, of the necessity of making 
 y (\A.\ B) the definition of the next step to AJB, and of making 
 \ (\A) the definition of A B , our symbolic departure from 
 
 The origin of A B is connected AVith a notion which is developed 
 in the following. . Looking at and 1 not as the initial symbols 
 of two distinct operations, but as the initial symbol of ah opera- 
 tion (+) and the scalar step of its subject, let successive operations, 
 and Y-> be defined by the relation that 
 
 TH^ + 1)=0(TM, 3). 
 
 That is, let the several definitions be derived from the solutions of 
 1) = (j)A + B, which gives <pA = AB, 
 1) = .80-4 ..... </>A = B\ 
 l) = B^. &c. 
 
 This last gives a function not capable of finite representation 
 under existing symbols, though we may commence with 
 
 01 = a, 02 = jg* 03 = J5 CBaj , &c. 
 
 It is neither to be expected nor desired that any substitute 
 should be adopted for A B ; but the more the mind accustoms 
 itself to consider this as a function rather of log A than of A, 
 the better. 
 
 Any two convertible functions of x and y, (x, y) and ty (x, y) 
 being given, as two consecutive members of a scale, the follow- 
 ing condition of distribution must be satisfied, 
 
 V' (*. (y =)} = W ( x > y ) V-^ (x, =)}, 
 
 arid the scalar function must be" determined from 
 
 \yr (x, y) = (A*, Ay), 
 A. being a functional symbol. 
 
 Every solution of this system is a chance for the invention 
 of an algebra, in which, (x, y) being denoted by z + y and 
 YT (x, y} by xy, and \ (\x, Ay) by x^ y or y x , all the laws of 
 ordinary algebra remain good. 
 
 In double algebra, the scalar function, in its most general 
 
 form, is 
 
 \R = (m -f n V-l) log r + (/* + "V" 1 ) e >
 
 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 165 
 
 the condition X (r, 0) = log r, which is necessary to the complete 
 and unaltered inclusion of single algebra, gives m = l, n = Q. 
 
 And it will be found on inquiry*, that the adoption of 
 
 log r + (M + v v 1) 
 
 is of no effect whatever, except what would in common algebra 
 be called the choice of e (/wwv ~ lj instead of e"*' 1 for a base of 
 angular exponentials. 
 
 If a moment's hesitation should arise on the retrograde symbols 
 of the scale, the reader may try the equation 
 
 ABC=(A + B)(A + C), 
 
 A 4 log (6 B + 6 C ) = log (6 A+B + 6 Atc ). 
 
 When two successive operations have the required distribu- 
 tive character, that character necessarily attaches to the next 
 one, if formed from the scalar function. Thus, if A '"B be de- 
 fined as \ (\A x \B), we have 
 
 A '" (BxC) = \ {\A x (\B + XC)} = \ (X^t x \JB + \A x \C). 
 But P x Q = \(XP 4- XQ) or \(P -f Q) = \P x \Q ; so that 
 A'"(B xC) = \(\Ax \S) x\(\Ax X(7) = A'"S x A'"C, 
 and so on for the rest. 
 
 When the inverse scalar function is used, the regressive 
 system has the same properties as the progressive one with 
 the direct scalar function: for 
 
 Hitherto I have said nothing of inverse operations. Let 
 A n B be the inverse operation of A"B; so that A"B n B - A. 
 And for A,B and A tl B use A - B and A- B. If any one of 
 the inverse signs follow the rule of signs (p. 103), so does the 
 next. That is, if for instance Q 3l/l (Q, 3lll A) give Q^'A or A, 
 we have Q^(Q^A) = QfA - A. For 
 
 A or X (A^B) = \A , \, 
 = XQ 4/// (XQ 4// M) 
 = \A, by hypothesis, 
 
 * The complete investigation will be found in a paper " On 
 the foundation of Algebra, Part in." in vol. viii. of the Cambridge 
 Philosophical Transactions.
 
 166 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 
 
 or Q t/l ,(Q i/y A) is A. And conversely, if the rule of signs be 
 true for Q 4/K -4 it is true for Q a ,,A. 
 
 An algebra, similar to ours, requires but the following fun- 
 damental basis. 
 
 Two consecutive operations, A + B, A x B, convertible, so 
 that A -f B = B + A and A x B = B x A, and having the second 
 distributive over the first as in (B + C) x A = B x A + Cx A. 
 
 A scalar operation, \A, having the property 
 
 X (A x B) = \A + \B. 
 
 One starting symbol, 0, wholly ineffective in its own operation, 
 so that 
 
 + = 0, + A = A, 
 
 An inverse operation, seen in A - B, so that (A-B) + B = A; 
 and giving - (0 - A) = A. 
 
 Strictly speaking, one operation and its inverse, and the 
 scalar function and its inverse, are sufficient for expression : 
 thus \ (\A + \JB) is sufficient to express Ax B. And hence 
 the whole system of scalar functions and starting symbols may 
 be deduced. But the invention of two operations, followed by 
 that of a scalar function, has been the order of discovery. 
 
 The formation of a symbolic system on the seven operations 
 of addition, subtraction, multiplication, division, involution, evo- 
 lution, and formation of a logarithm, is both redundant and 
 unsymmetrical : but the redundancy is rich in means of ex- 
 pression, and the reduction to symmetry is easy to one practised 
 in the language of algebra as it stands. This last will be best 
 seen by assimilating the notation more closely to that of common 
 algebra. Let 0, /( 0,,, 0,,,, &c., be thus defined: 
 
 0, = 1, 0,, = e, O,,, = e , O v = s , &c., or ;i = \"0. 
 Let n, = E", n /t = s ", &c. or n k = \ k n. 
 
 Let the progressive symbols be -}, +,, +, &c. and x, x,, x,,, 
 &c. thus connected; x is +,, x, is 4-,,, &c. Then, the convertible 
 and distributive properties remaining, we have all theorems of 
 ordinary algebra holding good, when any one suffix is placed 
 below + x and all numerical coefficients. Thus 
 
 (a + b) x,, (a + 6) = a x,, 6 + 2,, x,, a x,, b -f,, b x,, b
 
 SCALAR VIEW OF ALGEBRAICAL SYMBOLS. 167 
 
 means the following in ordinary language. The first side is 
 
 
 
 e 
 
 the second side is 
 
 whence the equation may be easily verified. This chapter may 
 serve to show the necessity of connecting successive operations 
 by the scalar or logarithmic operation, and the ease with which 
 it may be done without any permanent disturbance of established 
 notation. 
 
 THE EXD. 
 
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