7/>. 
 
 7-T 
 
 
 BY THE SAME AUTHOR. 
 
 In royal l2mo. price Is. 6d. 
 
 FIRST NOTIONS OF LOGIC 
 
 (preparatory to the study ot geometry). 
 
 The Author of this work on Arithmetic drew up the 
 above-mentioned treatise, from having observed that most 
 students who begin geometry are extremely deficient in 
 the sort of knowledge which it contains, to their great 
 hinderance in all branches of science, even in learning 
 the principles of arithmetic demonstratively. He recom- 
 mends that youths who are trained in arithmetic from his 
 work should learn the elements of logic at the same 
 time, as a help to the precise conception of the connexion 
 of words with each other, and as a preparation for the 
 study of Geometry. 
 
 PAULINE FORE MOFFITT 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA 
 GENERAL LIHRART, BERKELEY 
 
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m 
 
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 Digitized by tine Internet Archive 
 
 in 2008 witii funding from 
 
 IVIicrosoft Corporation 
 
 http://wwwlifcliive.org/details/elementsoftrigonOOdemoricli 
 
 * m 
 
ELEMENTS 
 
 OP 
 
 TRIGONOMETRY, 
 
 AND 
 
 TRIGONOMETRICAL ANALYSIS, 
 
 PRELIMINARY TO THE DIFFERENTIAL CALCULUS: 
 
 FIT FOR THOSE WHO HAVE STUDIED THE PRINCIPLES OF ARITHMETIC 
 AND ALGEBRA, AND SIX BOOKS OF EUCLID. 
 
 BY 
 
 AUGUSTUS DE MORGAN, 
 
 OF TRINITY COLLEGE, CAMBRIDGE, 
 
 SECRETARY OF THE ROYAL ASTRONOMICAL SOCIETY, 
 
 AND PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. 
 
 Tant que I'algebre et la geometrie ont ete separees, leur progres ont 6t€ 
 lents et leurs usages bornes ; mais lorsque ces deux sciences se sont reunies, 
 elles se sont pretees des forces mutuelles, et ont marche ensemble d'un pas 
 rapide vers la perfection.— Lagrange. 
 
 LONDON: 
 
 PRINTED FOR TAYLOR AND WALTON, 
 
 BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON, 
 30 UPPER GOWER STREET. 
 
 M.DCCC.XXXVII. 
 
iDaaTioi Lisa. 
 
 LONDON 1 
 
 FRINTSD BV JAMES MOVES, CASTLE STREET, 
 LBIC&STBR SgUARE. 
 
 (£,JU>c^JU'Cr>i 
 
 

 PREFACE. 
 
 The three great branches of elementary mathematics, mean- 
 ing all that should -precede the study of the Differential 
 Calculus, are Arithmetic, Algebra, and Geometry. Each 
 pair of these gives rise to a new inquiry, namely, the con- 
 nexion which exists between the two. Thus, it is practi- 
 cable to consider separately, 1. The application of arith- 
 metic to algebra, and of algebra to arithmetic. 2. That of 
 arithmetic to geometry, and of geometry to arithmetic. 
 3. That of algebra to geometry, and of geometry to algebra. 
 
 This will, at first sight, appear something like an undue* 
 quantity of distinction : nevertheless, with the exception only 
 of the two-fold comparison of arithmetic with geometry, 
 there is in it no degree of separation which cannot be fully 
 justified,, either as matter of necessity or convenience. 
 
 The application of arithmetic to algebra is made in the 
 very formation of the latter science, though based upon 
 considerations neither known nor admitted in the former. 
 But, nevertheless, those considerations are derived from 
 arithmetic in this sense, — that the want of them is suggested 
 by the imperfections of the latter, and the method of arriving 
 at them by the manner in which those imperfections appear. 
 
 416 
 
IV PREFACE. 
 
 The ideas to which arithmetic cannot fail to bring us, are 
 more than its language has power to express ; and the 
 passage from that science to algebra consists in the method- 
 ical arrangement of the ideas to be expressed, and the 
 invention of a language proper for the purpose. The appli- 
 cation of algebra to arithmetic is a department of a more 
 special character. It is of little consequence whether it be 
 made a separate study, or not ; the indispensable branches 
 of it appear in their proper places, and nothing more is 
 necessary than to point out the connexion. The higher parts 
 of what is called the theory of numbers have offered, as yet, 
 singularly little aid in the application of mathematics to the 
 sciences of matter, and may, therefore, be omitted entirely 
 by the student whose wishes on this subject are bounded by 
 the possession of an instrument for physical inquiry. Neither 
 will the more exclusively mathematical student find any 
 thing in that theory for which he should make special pre- 
 paration in his elementary reading : I have, therefore, omitted 
 'it altogether in my Algebra. 
 
 The application of arithmetic to geometry must be made 
 as soon as the study of proportion begins. But, viewed by 
 the side of arithmetic, geometry becomes the science of 
 continuous magnitude in general : that is to say, the con- 
 siderations on which it is necessary to dwell are such as 
 apply equally to all magnitudes, as well as to spaces or 
 lengths. In the accompanying* treatise, On the Connexion 
 of Number and Magnitude ^ I have endeavoured, at least, 
 
 • I have placed this treatise at the end of the Trigonometry ; but 
 it should be understood as intended to be read first, or, at least, that the 
 two should he read together. 
 
PKEFACE. V 
 
 to place the real difficulties of the subject before the higher 
 class of students ; guaranteeing nothing more than this, that 
 a larger proportion of readers will understand the tract in 
 question than would, by themselves, be able to master the 
 Fifth Book of Euclid. The extension of the arithmetical 
 notion of ratio, (shewn to be necessary, as well as furnished, 
 by the consideration of magnitude in general, but principally 
 of space magnitudes), constitutes the primary portion of 
 the application of geometry to arithmetic. 
 
 The application of algebra to geometry is divisible into 
 two distinct subjects. 1. The science usually called by that 
 name, but which may be styled the theory of curves and 
 surfaces. I may say of this part of the study, that though, 
 on various accounts, it is very desirable that it should be 
 made a separate branch, still it is not indispensable. A 
 student of more than average intelligence might pick up, as 
 he went along, enough of this part of mathematics to enable 
 him to pursue his career : no new principles are insisted on, 
 and the independent value of the subject mostly lies in the 
 very extensive field of practice which it opens in the ele- 
 ments of geometry and the operations of algebra. 2. Trigo- 
 nometry : the subject of the present treatise ; on which I 
 proceed to speak more at length. 
 
 The notion which is now studied under the name of 
 Trigonometry, is that of magnitude in a state of alternating 
 increase and decrease, or periodic magnitude. The term 
 itself merely implies the measurement of triangles, as geo- 
 metry does that of the earth ; and it is still convenient to 
 refer the measurement of triangles (and other figures) to 
 trigonometry, but only as a minute and isolated application. 
 
VI PREFACE. 
 
 Taking the primary idea of quantity alternately in«reasing 
 and decreasing, it is obviously of l\indamental importance 
 to detect a proper method of measurement. The circle 
 presents itself for the purpose in the following way. Con- 
 ceiving a periodic change of magnitude to run through iu 
 whole cycle in a given time, let a point revolve uniformly 
 round a circle in the same time, starting from the end of a 
 fixed diameter. The height of the point above the diameter 
 is a periodic magnitude, which goes through all its changes 
 in the same time as the given magnitude : and it is, in 
 fact, one of the great objects of trigonometry to express 
 periodic variation whose law is known in any way, by 
 means of the simple species of variation just described. 
 
 Upon further examining the question of periodic varia- 
 tion, we discover in geometry, and in geometry only, a 
 species of magnitude which is of necessity periodic, and is 
 utterly exclusive of indefinite increase : namely, direction. 
 In speaking of the direction of a line as a magnitude, we 
 mean to imply that all direction is relative, inasmuch as 
 we only judge the direction of one straight line by com- 
 paring it with another. No straight line can increase its 
 diflference of direction from that of another indefinitely : 
 after a certain quantity of change, coincidence is reproduced. 
 The connexion of direction with length is found to lead to 
 an extension of the algebra of positive and negative quan- 
 tities, which gives the same power of interpreting a-\-b^ — 1 
 relatively to a, as exists already in the case of -}- a or —a 
 relatively to a. This is an application of geometry to 
 algebra ; and, though there does exist a point of view in 
 which geometry may appear not absolutely indispensable. 
 
PREFACE. Vn 
 
 and which I have described in Chapter IV., yet it is pretty 
 clear that abstraction must advance considerably before it 
 will be safe to abandon reference to the science of space in 
 presenting algebra complete to the beginner. The progress 
 of algebra, in this respect, is very curious. In its infancy, 
 geometrical interpretation was rendered necessary by want 
 of power in its symbolic language; was abandoned as the 
 latter grew to maturity; and is finally had recourse to again, 
 because symbols are now sufficient to express relations of 
 magnitude which do not yet exist except in regard to space. 
 
 This modern application of geometry to algebra is traced 
 in Professor Peacock's Report on Analysis to the British As- 
 sociation, printed in the second volume of their Transactions 
 (a.d. 1834). The account there given of this particular 
 point, as well as the rest of the article, should be read by 
 the student of the higher mathematics with great attention, 
 as being equal, in the elementary point of view, and superior 
 in the historical, to any thing which has yet appeared on 
 the subject. The Report cited has rendered any reference 
 to authorities unnecessary. 
 
 I have not inserted any thing on the solution of spherical 
 triangles ; the subject being one which, though of primary 
 use in Astronomy, is but little connected with the funda- 
 mental part of Trigonometry. I should have added a 
 chapter upon the subject, had I not already published a 
 treatise in the Library of Useful Knowledge, to which I am 
 thereby enabled to refer the reader. 
 
 A. DE MORGAN. 
 
 University/ CoUegCy London^ 
 March 1, 1837. 
 
ERRATA IN " ELEMENTS OF TRIGONOMETRY." 
 
 Page 7, Corollaries 1 and 2, for the circles C and C, read the circle C; 
 for P r^arf P'. 
 
 20, line 8, /r^r - + - read - x -. 
 
 20, Une 8, for — + — reaA — x — . 
 
 26, line 2, /or A"0 C', read M"0 C". 
 
 26, line 14, /or C"N, read C"N'. 
 
 31, line 1 1, /or O, read 0. 
 
 42, line 12, strike out sines of the. 
 
 81, line 6, /or read . 
 
 2 2^/3I 
 
 81 , line 8 from the end, for %" + s^w * a/^ read i' Xi^^"^ ^^-^ 
 
 122, line 20, for y— read y~*^. 
 
 ERRATA IN " CONNEXION OF NUMBER AND MAGNITUDE." 
 
 Preface, page iv. line 17, for that, read than. 
 
 Page 5, lines 7, 8, and 9 from the end, transpose P and X ; and Q and Y. 
 
 9, line 9, /or B, read A. 
 
 10, line 19,/or IF =/M, read l¥ =fL. 
 
 22, line 4 from the end ,for processes erabling, read process resembling. 
 
 27, line 16, for -, read -,. 
 
 34, line 13, for are, read is. 
 
 35, line 3, for less, read greater. 
 
 46, line 13, for may be named, read is named. 
 
 46, lastHne,/or t;(B + K), read j<;(B + K). 
 
 47, line 1,/or i;(Q + Z), read w;(Q + Z). 
 
 47, line 2, /or vQ, read wQ, and Z may be as small as we please. 
 
 53, last line, /or B + C + F, read B + D + F. 
 
 • 58, line 5 from the end, /or B, read D. 
 
 • 61, lines 2, 4, 6, 7, for mP, read nP. 
 
 — — 66^ line 14, /or in which, read in which we see. 
 
 74, line 10, for V : U, read V,: L'. 
 
 76, line 17, for X„, read X . 
 
 78, line 10,/or for, read from. 
 
 . 79, line 18, /or a^v^, read a*i;*. 
 
 80, line 16, /or 10, read lOV. 
 
 80, line 20, /or VfeX, read Vfc\ 
 
ELEMENTS 
 
 OF 
 
 TRIGONOMETRY. 
 
 CHAPTER I. 
 
 DEFINITIONS AND FUNDAMENTAL FORMULA OF TRIGO- 
 NOMETRY IN THE CASE OF ONE ANGLE. 
 
 (1.) Trigonometry originally meant simply the measurement of 
 triangles. It now means measurement generally by means of the 
 properties of triangles, in all cases in which the connexion between 
 sides and angles is concerned, not sides only, nor angles only : 
 together with all consequences of such measurements as are useful 
 in the higher parts of mathematics. If algebraical symbols and 
 operations be adopted, as is now universally the case, it is a branch 
 of the application of algebra to geometry. 
 
 (2.) If a line U, taken at pleasure, be called the linear unit, or 
 1 of length, then 2 U is called 2, ^ U is called ^, and so on. And 
 any line incommensurable with U is denoted by a general symbol 
 such as cr, where, if the line specified were commensurable with U, 
 a would be a number or fraction, and aU would represent the line. 
 Let aU still represent the line, where, in the theory, a is a symbol 
 for the ratio of the line to U ; and, in practice, a line as near as we 
 
 please to the incommensurable line is taken, namely, - U, and - 
 
 n n 
 
 is substituted in results instead of a. I here suppose the student 
 
 to have read the preliminary treatise : if not, he must be content with 
 
 the application of the following proposition. 
 
 Two lines being given, A and B, either two whole numbers m 
 
 and n can be found, so that B= -A; or so that -A shall be as 
 near to B as we please. 
 
 The same considerations apply to any other magnitudes; to 
 angles for instance. 
 
Z DEFINITIONS AND 
 
 (3.) Before proceeding to ascertain how a line may depend upon 
 an angle, or an angle upon a line, it may be useful to shew that we 
 are, by means of geometry, able to determine lines from lines, without 
 consideration of angles, and angles from angles, without consideration 
 of lines. The complete determination of some angles of a figure by 
 means of the rest, whenever that is possible, is contained in Prop. 32 
 of the first book of Euclid. If a„ Oj .... On be the n angles of a 
 rectilinear figure which has no re-entering angles (pointing inwards), 
 and if ?r be the angle made by a line and its continuation, or twice a 
 right angle, then 
 
 </! + fif2 + fls + + 6r„ = (n — 2) T 
 
 In a triangle «i + cfg + CTs = -r 
 
 Hence all the angles of a triangle are known when two are known. 
 
 (4.) The determination of lines by means of lines depends 
 mostly upon the two following propositions ; 
 
 I. 47. If A, B, and C, be the sides of a triangle, and if A and B 
 contain a right angle, then the squares on A and B make together an 
 area equal to the square on C. 
 
 Let A, B, and C, contain respectively o, ft, and c, of any linear 
 unit. Then (Arithmetic, §234.), if a be a whole number or fraction, 
 the square on the unit (which call T) is contained aa times in the 
 square on A, or the square on A is oaT. Similarly, the square on B 
 is 66T, that on C is ccT; whence 
 
 tfaT + ^^>T = ccT 
 
 or T taken aa-\-bb times is T taken cc times. Therefore, 
 
 aa -j-bb = cc 
 
 [When * a, h, and f, are either of them incommensurable, this 
 equation no longer exists arithmetically. We shall give the strict 
 developement of this proposition in such a case. Firstly, the ratio 
 of the square on A to the square on U (the linear unit) is that com- 
 pounded of A : U and A : U. Let A be to U as U to X ; then 
 (VL prop. 16.), the rectangle whose sides are A and X is equal in area 
 
 • The parts in brackets are for the student who has read the 
 Introductory Treatise in such a manner as to l)elieve he understands it. 
 
 The references are to the articles of my treatise on Arithmetic, the 
 pages of my treatise on Algebra, and to the books of Euclid. 
 
1 
 
 
 1 
 
 
 aa-\-bb ; 
 
 : 1 
 
 cc : 
 
 1 
 
 aa-\-bb : 
 
 : cc 
 
 FUNDAMENTAL FORMULA OF TRIGONOMETRY. O 
 
 to the square on U. Therefore, the squares on A and U are two 
 rectangles with a common altitude A, and bases A and X ; con- 
 sequently (VI. 1), 
 
 Square on A : Square on U : : A : X 
 
 But A : X is compounded of A : U and U : X, that is, of A : U and 
 A : U. Now, let the ratio of A : U be represented by a : 1, and the 
 compound ratio hy aa : 1. Then we have, proceeding in a similar 
 way with the other sides, 
 
 Sq. on A : sq. on U :: aa 
 
 Sq. on B : sq. on U :: 6 6 
 
 Sq. on A + sq. on B : sq. on U : 
 But, Sq. on C : sq. on U :: 
 
 Therefore, Sq. on A + sq. on B : : sq. on C : 
 
 But, Sq. on A + sq. on B = sq. on C, therefore, aa -\-bb=cc 
 
 or let the ratio of A : U have any symbol a : 1, &c., and let aa : 1 
 represent the ratio compounded of a : 1 and a : 1, and \eim-\-n : 1 
 mean the ratio of the sura of two magnitudes to U which are severally 
 to U as m : 1 and w : 1, agreeably to the definitions in the preliminary 
 Treatise, and we have what should in strictness be written 
 
 aa + hh : 1 : : cc : 1 
 
 "When A, B, and C, are commensurable with U, then a, b, c, 
 mean whole numbers or fractions, and the preceding is reduced to 
 aa-\-bb = cc as before. Ifa, 6, and c, be whole numbers or frac- 
 tions, such that all, 6U, cU, are very near to A, B, C, then we have 
 aa + 66 = cc very nearly; but if a, b, and c, be really symbols of 
 incommensurable ratios, or rather o:l, 6:1, c:l, then the pre- 
 ceding proposition can only be interpreted with reference to mag- 
 nitude ; namely, as aaV -\-bb\J =cc\J where aa U : U is the ratio 
 compounded of «U : U and all : U, or A : U and A : U.] 
 
 (5.) VI. 4. If A, B, C, and P, Q, R, be the sides of equi- 
 angular triangles j namely, the angles opposite to A and P equal, &c. 
 then, 
 
 A:B::P:Q, B:C::Q:R, C:A::R:P 
 
 If the ratios of the sides of any triangle be known, the angles are 
 geometrically known ; that is, we can construct the angles, but cannot 
 yet find their ratios to one another. 
 
DEFINITIONS AND 
 
 If the sides of the first triangle be aU, 6U, cU, and those of the 
 second jsV, yV, rV, we have (a, 6, and c, being numbers or 
 fractions) 
 
 a : h:\ p : q aq = ph 
 
 h \ c\\ q '. r hr = qc 
 
 c : ay. r : p cp = ra 
 
 [If these represent incommensurable ratios, let the student treat 
 these equations in the same manner as aa-\-bb = cc preceding]. 
 
 (6.) I now proceed to a case in which lines are numerically de- 
 termined by lines. Let there be a triangle ABC, in which, U being 
 
 an arbitrary linear unit, we have 
 
 AB = cU 
 BC=:aU 
 CA = bV 
 
 Let AK = xV, KB = yU, 
 
 Then, in the first case, x -\- V = C 
 second . . x — y = c 
 
 required CK, the perpendicular on cU ; and 
 AK and KB, the segments* of AB. 
 
 KC =pU 
 
 In both 
 
 a;2+^2 = l^ 
 
 First Case. 
 
 a;2-?/" 
 
 cix — y) = 6^— a® 
 cix -i-y) = c^ 
 
 2cx = b^+c^—a^ 
 
 Second Case. 
 c{x -\-y) = b"—a" 
 
 ci^x — y) = c^ 
 
 2cx = J2-|.c2— a^ 
 
 • If K bo a point in A B, or A B produced, A K and KB are called 
 segments of AB, in all cases. If K lie between A and B, AK + KB = 
 ABj if K lie beyond B, AK-KB = ABj if K lie beyond A. 
 BK-KA=AB. 
 
'^<t-^^ 
 
 -C^^t-c 
 
 a ' ' 
 
 t^-^a 
 
 FUNDAMENTAL FORMULAE OF TRIGONOMETRY. 
 
 First Case. 
 
 62 + C2- 
 
 a; = 
 
 Second Case. 
 
 62 + c2— «2 
 
 2c 
 
 X = 
 
 2c 
 
 
 y = 
 
 c^ + a' 
 
 2cy = Ir—ar — c^ 
 
 2c 
 
 y = 
 
 2c 
 
 But both cases might have been contained in one, by adopting the 
 conventions of algebra, instead of keeping to arithmetic ; for, if we 
 suppose the second case to be made from the first by moving B to 
 the left, we see that BK will be measured in a direction contrary to 
 that which it had at first. If, then, the formulae of the first case had 
 been used to determine BK in the second case, we should have been 
 warned to measure B K in a direction contrary to that assumed, by its 
 negative sign. For instance, let c = 2, a=:3, 6 = 4; then we have, 
 taking the formula of the first case. 
 
 y = 
 
 4 + 9 — 16 
 
 That is, BK is -U in magnitude, not on the same side of K as in 
 
 the case from whence the formula was arithmetically derived, but on 
 the contrary side. {Algebra, chapters I. and II.) 
 To fiyd p, we observe that 
 
 P' = « - / = -^2 
 
 __ {2ac — (c2 + a2-.i=)} {lac-^-c" -\-a' — b'^} 
 
 __ __ 
 
 {b^—{a-cy] {{a+cy-b^} 
 4c^ 
 
 Let a + b +€ = 2? 
 
 h 4- c — G = 2(5— a) 
 c -^a—b = 2{S'-'b) 
 a -\- b — c = 2(s--c) 
 
 (^a ^ b -{-c) {a -{-c — b){c -\- b — a) (b -^ a — c) 
 4 c" 
 
 then 
 
 c^p^ = 4:s{s^a){s^bXs—c) 
 
 cp = 2'>/s{s—a)(S'^b){s—c) 
 
 As this result will appear in the sequel, we have chosen it as our 
 instance ; and by proceeding with the two propositions in question, 
 it is thus possible to determine lines in terms of lines, without the 
 
 b2 
 
 Q- " 
 
b DEFINITIONS AND 
 
 necessity of employing angles as means of expression. I now pass 
 on to the consideration of an angle connected with a line. 
 
 (7.) From the Sixth Book of Euclid it appears, that though 
 given lines will determine angles (as, for instance, the three sides of a 
 triangle being given, the angles can be constructed, both internal 
 and external), yet that it is not necessary to give the lengths ; for 
 any other lengths which have the same relative magnitudes will 
 give the same angles. Indeed, the Sixth Book of Euclid is, in great 
 part, an inductive proof of the following proposition. If the absolute 
 lengths of all the lines in a figure be altered in the same ratio, the 
 angles of the figure are not altered : hence, angles depend upon the 
 ratios of lines. The line with which an angle is most connected is 
 the arc of a circle, and it will be necessary to know something more 
 of this figure than can be directly found from the elements. We 
 shall indicate the principal steps necessary, which may be readily 
 filled up by a student who understands the Sixth Book. 
 
 (8.) Definition. A bounded figure is called convex, when 
 no straight line whatsoever can meet its boundary in more than two 
 points, unless it be itself part of the boundary. 
 
 (9.) Theorem. If one convex rectilinear figure be entirely con- 
 tained within another, the boundary of the contained figure must be 
 less than that of the containing. 
 
 D 
 
 Let ABCDEF and GHIKL be the containing and contained 
 figures, then, 1. MN is less than MABN, or MNCDEFM is 
 less than ABCDEF A. 2. Similarly, LODCNL is less than 
 MNCDEFM; 3. KPCNLK is less than LODCNL; 
 4. KIQLK is less than KPCNLK; 5. KIHGLK is less than 
 KIQLK. Whence the proposition. 
 
 (10.) Postulate. Let it be granted that this theorem is also 
 true of convex curvilinear or mixtilinear figures. 
 
FUNDAMENTAL FORMULAE OF TRIGONOMETRY. 7 
 
 (11.) Theouem. If the homologous sides of two similar recti- 
 linear figures may be made as nearly equal as we please, the figures 
 themselves may be made as nearly equal as we please, both in length 
 of boundary and area. We leave the demonstration to the student, as 
 it is a very simple consequence from the Sixth Book. 
 
 (12.) Theorem. For every rectilinear figure which can be de- 
 scribed i7i any one circle, a similar figure can be described about any 
 other circle. We shall merely indicate the construction, and leave 
 the student to finish the demonstration. Let C be a circle, and P 
 any inscribed polygon ; let C be another circle ; in it inscribe P' 
 similar to P. Bisect every side of P', and draw radii through the 
 points of bisection. From the extremity of each such radius draw a 
 tangent ; then will the polygon P", formed by all the tangents, be 
 similar to P' and to P. 
 
 Corollaiy 1. The circle^j-G- and C are contained, both as to length 
 of boundary and area, between P and P" 
 
 Corollary 2. By making the radii sufficiently near to equality, P 
 and P" may be made (11.), both in boundary length, and area, as near 
 as we please to each other : and if we are at liberty to give P as 
 many sides as we please, and as small (10), the same follows of the 
 circles C and C themselves. 
 
 Corollary 3. And all these results are equally true of sectors of 
 circles which contain the same angle. 
 
 (13.) Theorem. If in a circle a polygon be described, of which 
 no one side exceeds Z in length ; then, if Z may be made as small as 
 we please, the polygon and circle may be made as nearly equal as we 
 please, both in boundary length, and area. 
 
 For, under these circumstances, as may be easily shewn, the in- 
 scribed and similarly superscribed polygons may be made as nearly 
 equal as we please in both respects; and (10.) the circle lies between 
 them in both. 
 
 Corollary. The same is true of any sector of a circle. 
 
 (14) Theorem. The circumferences of circles (or of similar 
 sectors) are as their radii, and their areas as the squares on the radii. 
 
 Let the radii be R and R', and describe in the circles two similar 
 polygons, having all their sides severally less than Z. Let P and P' 
 be the whole boundary lengths of the polygons, and let C and C be 
 the circumferences of the circles. But the boundary lengths are pro- 
 portional to the lengths of two homologous sides, which are to each 
 
8 DEFINITIONS AND 
 
 Other as the radii, and (12.) by making Z sufficiently small, these 
 same lengths may be C — K and C — K', where K and K' are as 
 small as we please. Thence, we have (however small K and K' 
 may be), 
 
 C-K : C'-K'::R : R' 
 
 If possible, let C be to C not as R to R', but, firstly, in a greater 
 ratio. Then 
 
 C is to C more than C — K is to C — K' ; 
 or wiC exceeds nC where tnC — niK does not exceed nC — nK'. 
 
 Let mC=.fiC'-\-'V ; then nC'+V — mK does not exceed nC — nR', 
 and, still more, does not exceed nC 
 
 Consequently, »/ K is at least equal to V ; but V is determined with- 
 out reference to K from C and C, and K may be as small as we 
 please. Therefore, it may be so taken that wK shall be less than V. 
 But if the supposition under trial be true, mK must at least be equal 
 to V ; therefore, that supposition is not true, or C is not to C more 
 than R to R'. If possible, let it be less ; then we have, by similar 
 reasoning, 
 
 mC is less than 7^C'' where mC — niK is not less than nC' — nK'. 
 Let 7;?C=nC' — V; then nC— V — wK is not less than nC — uK', 
 still more then nC — V is not less than nC — nK'; 
 
 or, nK' is at least equal to V. This, as before, shews that the sup- 
 position cannot be true. Whence the first part of the theorem 
 follows; and the same may be similarly proved of the sectore. 
 
 Let A and A' be the areas of the circles, and Q and Q' the squares 
 on the radii; whence it follows, that A — L and A' — L' being the 
 areas of the similar polygons, (12.) L and L' may be made as small 
 as we please. But (VI. prop. 19.) we must have, 
 
 A-L : A'-L' :: Q : Q' 
 
 and, precisely as in the former case, we find that A is to A' neither 
 niore nor less than Q is to Q'. Hence, A is to A' as Q to Q'. 
 
 Now, let the student shew, that the area of a regular circumscribed 
 polygon of n sides is JnrsT (see next article), sU being one side: 
 and thence, that cU and aT being the circumference and area, we 
 must have a = i^ c r. 
 
FUNDAMENTAL FORMULJE OF TRIGONOMETRY. 
 
 9 
 
 (15.) Algebraically, let rV and r'U be the two radii, cV and 
 c'U the circumferences ; then we have. 
 
 / , c c 
 
 c : c :: r : r or - = - 
 
 r r 
 
 If T be the square on U, the areas aT and a'T are as rrT and 
 
 rVT, or, 
 
 / , , a a' 
 
 a : a :: rr : rr ^ = — ^ 
 
 (16.) The ratios represented by - and —^ are incommensurable ; 
 
 but we have very nearly 
 
 c f^ 355 
 
 355 
 113 
 
 Let the ratio, which is incommensurable, but very near that of 
 355 : 113, be represented by sr ; 1 ; or, in the more common way of 
 
 c 
 speaking, which uses incommensurables as commensurables, let -, 
 
 which is the same for all circles, be called 2?r. Then we have 
 c = 2'7rr a = crr^ 
 
 T = nearly — -r = nearly 3*14159 
 To prove the preceding, we have not yet the means. Let us agree, 
 however, to denote -, whatever it may be, by 2 -r. 
 
 [But for those students who dislike to leave any thing behind 
 which is afterwards to be proved, we shall establish the following 
 proposition, which assigns the approximate value of ^. 
 
 Let BC, DE, be the sides of regular polygons of n sides, in- 
 scribed and circumscribed about the circle where radius is O A. 
 
10 
 
 DEFINITIONS AND 
 
 Draw a tangent at B^ and complete the figure as shewn. Then will 
 BA, AC, be sides of the polygon of 2n sides inscribed, and BK, 
 KA, are halves of sides of the circumscribed polygon of 2/i sides. 
 Let Ip and Ep mean the areas of the interior and exterior polygons 
 o( p sides. Thence, we have, 
 
 I. 
 
 L2n 
 
 I. 
 
 E„ 
 
 triangle OBC 
 
 OBM 
 
 OM 
 
 triangle O B A 
 
 OB 
 OM 
 
 2 X triangle OB A 
 
 OBA 
 
 OA 
 triangle ODA 
 OD 
 OA 
 
 Therefore, 
 
 I„ : l2n '' ^2n • En 
 
 If, then, T be the square on the linear unit U, and if I„ = t'n T, 
 &c. we have 
 
 = '^Len 
 
 Again (VI. 3.), 
 AK : KD 
 AK : AD 
 
 And 
 
 whence 
 
 E2„ : En 
 
 OD : : MO 
 MO+OB 
 MO + OA 
 I„ H-I 
 
 OB 
 
 : AO 
 : MO 
 : MO 
 
 :In 
 
 : 2 X triangle AOK : triangle AOD 
 
 2n 
 
 2AK 
 
 2In : I„ +l2„ 
 
 9, 
 
 AD 
 
 €2n = 
 
 In en 
 
 In -\- ^31 
 
 which formulae, namely, 
 
 hn = "^in^n 
 
 ^2n = 
 
 'i/an^" 
 
 »n -t- *a » 
 
 enable us to pass in numbers approximately from the areas of any 
 inscribed or circumscribed polygon, to those of double the number of 
 sides. 
 
 Let the linear unit be the radius itself, and T, therefore, the square 
 on the radius. We have, then, on inspection (or IV. 6, 7.) 
 
 I, = 2T 
 
 E4 = 4T 
 
FUNDAMENTAL. FORMULAE OF TRIGONOMETRY. 
 
 11 
 
 = 2 
 
 = x/8 = 2n/2 
 
 = 4\/2-n/2 
 
 = 4 
 
 e^ = 
 
 16 
 
 2+V8 
 
 =8(\/2-l) 
 
 
 \/2) 
 
 2\/2+4\/2— n/2 
 
 &c. 
 
 These expressions shew do very easy law; but if we begin with 
 approximate values of the second pair, namely, 
 
 = 2 
 
 = 4 
 
 is = 2-8284271 
 e, = 3-3] 37085 
 
 and proceed with these approximations, it will be found, by a calcu- 
 lation which may appear laborious (but, as we have said, this digres- 
 sion is for students whose industry exceeds their disposition to believe 
 without proof), that we have 
 
 Approximate fraction 
 
 Number of sides 
 in the polygon. 
 
 4 
 
 of T contained in 
 the area of the in- 
 scribed polygon. 
 
 .... 2-0000000 
 
 Ditto, Ditto, of the 
 circumscribed polygon. 
 
 4-0000000 
 
 8 
 
 2-8284271 
 
 3-3137085 
 
 16 
 
 3-0614674 .... 
 
 3-1825979 
 
 22 
 
 64 
 
 .... 3-1214451 .... 
 3-1365485 
 
 3-1517249 
 
 3-1441184 
 
 128 
 
 31403311 
 
 3-1422236 
 
 256 
 
 512 
 
 .... 3-1412772 .... 
 3-1415138 
 
 3-1417504 
 
 3-1416321 
 
 1024 
 
 3-1415729 
 
 3-1416025 
 
 2048 
 
 4096 
 
 8192 
 
 3-1415877 
 
 .... 3-1415914 .... 
 ... 3-1415923 
 
 3-1415951 
 
 3-1415933 
 
 3-1415928 
 
 16384 . . . . 
 
 . . . 3-1415925 . 
 
 3-1415927 
 
 32768 
 
 .... 3-1415926 
 
 3-1415926 
 
 Now, the area of the circle itself always lies between those of the 
 inscribed and circumscribed polygon; and the polygons of 32768 
 sides, inscribed and circumscribed, do not differ, (slight errors of 
 approximation excepted, which may affect the last place), by the ten- 
 millionth part of a square unit. We may say, then, that the area of 
 
12 DEFINITIONS AND 
 
 355 
 the circle is approximately 3'14159T, or — — T very nearly; that is, 
 
 «r = 3*14159 very nearly.] 
 
 (17.) Lemma. If the same magnitude be m units of one kind, 
 and m' units of another kind, or if mil =m'U', then m : m' :: U' : U, 
 and if U' = /cU, then m=zm'k. 
 
 (18.) Let there be two different angles, BOA, B'OA', and 
 describe different circles AB, A'B'. Let the given quantities be 
 
 the radius OA, and the arc AB; the radius OA' and the arc A'B'. 
 Required the ratio of the angles. 
 
 (VL 33.) Angle BOA : Angle BO A' :: BA : VA 
 (14.) OA : OA' :: VA: BA' 
 
 Therefore, BA : B'A', the ratio of the arcs, is that compounded of 
 the ratios of the angles and the radii. Let U be the linear unit, 
 e the angular unit; and let 
 
 BA = sU, B'A' = s'U, OA = rU, OA'=r'U, 
 
 Angle BOA = d 0, Angle B'OA' = ^'0 
 
 Then r : r' 0' :: s : s' 
 
 (/ r 8' 
 
 (19.) We have already seen that there has happened a case 
 {Algebra, p. 226.) in which one system of suppositions is most con- 
 venient in analysis, while another is so in practice ; namely, in the 
 
 choice of a base for logarithms, where 2-7182818 is the base 
 
 of analysis, and 10 that used in applying logarithms to computations. 
 Just so, in the present instance, there is an angular unit which it is 
 convenient to adopt in investigations, while another unit is universally 
 supposed in practical applications. And the neglect of distinction 
 between these two units is the stumbling-block of the beginner. 
 
FUNDAMENTAL FORMULA OF TRIGONOMETRY. 13 
 
 though the necessity for the distinction is too great to allow us for a 
 moment to think of abandoning one or the other unit. 
 
 (20.) 1. The analytical or theoretical unit (there is no distinct 
 term for it in general use) is the angle which has an arc equal to the 
 radius. If, in a solid circular plate, we stretch a thread equal to the 
 radius from point to point of the edge, the thread is then the side of 
 a regular hexagon (IV. 15 ) and subtends two-thirds of a right angle. 
 If, then, we bend the thread over the edge, it will (no stretching 
 being supposed) subtend at the centre somewhat less than tioo-thirds 
 of a right angle, which is the first rough notion of the analytical unit. 
 
 (21.) In the process of (18.), let s=r, and let be the ana- 
 lytical unit ; then 9 is also the same, or = 1, and we have 
 
 ov to determine the number of analytical units in any angle, divide 
 the number of linear units in the arc by that in the radius. 
 
 Hence we can easily ascertain how many analytical units there 
 are in one, two, &c. right angles. The radius being rU, the whole 
 circumference is 2<rrU, and its fourth (or the arc of a right angle) 
 
 is -rU, the number of units in which is -r. This divided by the 
 number of units in the radius, or r, gives -. 
 
 The right angle in analytical units is -, approximately, 1-5707963 
 
 Two right angles are t 3-1415926 
 
 3 
 Three right angles --r 
 
 Four right angles 2 -tt 
 
 (22.) 2. The practical method of measuring an angle is well 
 known to be as follows. Let the 90th part of a right angle be called 
 a degree ; the sixtieth part of a degree, a minute; the sixtieth part of 
 a minute, a second. Let these be denoted by 1°, V, \", which are 
 not symbols of number, but of magnitude. They are angles. We 
 shall always denote the theoretical unit by e. 
 
 1°= 60.1' = 3600.1" 
 
 A right angle = 90.1° = 5400.1' = 324000.1" = 1'5707963 nearly. 
 
 c 
 
14 DEFINITIONS AND 
 
 Hence P = -01745329 nearly = 206264-8 .1" nearly 
 r =-000290888 .... =3437-747.1' .... 
 1" = -0000048481 . . . . = 57-29578.1° .... 
 
 (23.) It is usual to divide the circumference of a circle into 
 360 equal parts, and to call each part a degree ; the sixtieth part of 
 a degree a minute, &c. To avoid confusion, I shall call these linear 
 degrees, minutes, &,c. Thus, every circle has its own linear degree; 
 and the greater the circle, the greater the linear degree. On a circle 
 of the earth, the linear degree is 69 miles and a half (a little less) 
 in length. The student will remember that a linear degree and a 
 degree are two things as different as a line and an angle. 
 
 (24.) It is very common among writers on this subject to con- 
 found ^ and 180° or 180.1°, | and 90° or 90.1°. Thus we see 
 sometimes such equations as 
 
 ^ = 180° I = 90° 
 
 which are equations of as little title to existence as 
 
 1760 = 1 or 20 =4 
 
 instead of 1760 yards = 1 mile, or 20 shillings = 4 crowns. 
 
 The angle, which in theoretical units is tt, in degrees is 180. But 
 it does not, therefore, follow that tt = 180, any more than that 
 12=4, because that length which in feet is 12, in yards is 4. 
 
 (25.) We now proceed to a more general use of the word 
 measurej which frequently occurs in practice. One quantity is said 
 to measure another, even when the two are of different kinds, if any 
 change whatever made in the one is accompanied by a proportional 
 change in the other, so that if A of the one give B of the second, 
 in A of the one always gives mB of the second, whether m be 
 whole or fractional, or the representative of an incommensurable 
 numerical symbol. 
 
 Thus, angles are measured by arcs of given circles ; for on the 
 same circle any alteration of either arc or subtended angle produces 
 a proportionate alteration in the other. Between given parallels, 
 the areas of rectangles are measured by their bases. But a square 
 is not measured by its side ; for if the side be doubled, for instance, 
 the square is not doubled, but quadrupled. 
 
FUNDAMENTAL FORMUL-E OF TRIGONOMETRY. 15 
 
 (26.) One magnitude or ratio is determined by another, when, 
 the first being given, the second is given ; or at least when, the first 
 being given, the second cannot be any thing we please, but must have 
 one or other of a certain finite number of values. Thus, one angle of 
 a triangle being given (VI. 6.), the ratio of the containing sides de- 
 termines the other two angles, or rather, determines one of them ; for, 
 one angle being given, the sum of the other two is given : the ratio 
 of the containing sides, with the relation just mentioned, determines 
 both the remaining angles. 
 
 (27.) If one angle be given (VI. 7.), the ratio of two sides, 
 which do not contain that angle, absolutely determines the other 
 angles, if the given angle be a right angle or more; but if the given 
 angle be less than a right angle, the ratio of the two sides (which do 
 not contain it) determines only two values of each angle, one of which 
 it must be. We shall afterwards have to return to this point. 
 
 (28.) An angle, in Euclid, is only one of the angles or openings 
 made by two straight lines, namely, that opening which is less than 
 the opening made by a line and its continuation, or less than two 
 right angles. But any two lines which terminate in their point of 
 meeting, make two angles, one less and one greater than two right 
 angles, the sum of both being four right angles. 
 
 (29.) The foundations of trigonometrical notation are as follows : 
 Let a straight line OB, setting out from the position OA, revolve 
 round the point O. Let A A'' and A' A"' be at right angles, and let 
 lines measured from O towards A, or from O towards A', be positive, 
 while lines measured from O towards A", or from O towards A"', are 
 negative. Let positive angles be described when the revolution makes 
 
16 DEi*INlTIONS AND 
 
 OB proceed from OA to OA', and let negative angles be described 
 vrhen the revolution makes OB proceed from OA to OA'". And let 
 
 ratios be positive when both their terms have the same sign, and ne- 
 gative when both their terms have different signs. 
 
 Let the Euclidean angle AOB be 9Q, where 9 is the analytical 
 unit, so that 6 is the algebraical symbol for the angle expressed in 
 analytical units ; then the whole revolution, or 4 right angles, being 
 2 7r9, the angle of two revolutions, or eight right angles, being 4 7r9, 
 and so on, the line OB is said to make, with OA, an angle 
 
 6&, or {2'7r-^&)0, or (4-^ + ^)0, &c. 
 
 according as we consider OB to be in its first, second, third, &g. 
 revolution. Or the method by which we take into account the ne- 
 cessity of considering the following proposition, " If a straight line 
 revolve round a point continually, it will, in every succeeding revo- 
 lution, pass again through all the positions which it had in the first," 
 is by making the following assertion : The angles 0, 2 tp -f ^> 4 tt + 0, 
 &c., and, generally, 2w7r -|- 0, w being a whole number, are the angles 
 made by O B and O A in the first, second, third, &c., and, generally, 
 in the (n-f- l)th revolution. The angles which it will be necessary to 
 consider as denoting distinct positions, are those which are less than 
 four right angles or 2 7r9, if we only consider positive angles; or 
 those which are less than two right angles in magnitude, if we con- 
 sider positive and negative angles; that is, which lie between -{-ttQ 
 and — 7r9. We shall, for the present, confine ourselves to these 
 angles. 
 
 (30.) We now proceed to define what we shall call the primary 
 trigonometrical functions of an angle, the names of which are the 
 sine, cosine, tangent, cotangent, secant, cosecant, versed sine, and co- 
 versed sine of the angle. And, firstly, whereas in old books on trigo- 
 nometry these functions are lines, in every modern system they are, 
 or should be, defined to be numbers, or, in the widest sense, 
 ratios. Secondly, the term sine is probably from an Arabic word, 
 but the meaning is not well known ; tangent and secant are of obvious 
 Latin derivation ; but the definition we shall give has nothing to do 
 with their etymology. 
 
 Consider cosine as the abbreviation of " sine of the complement." 
 
 cotangent " tangent of the complement." 
 
 cosecant " secant of the complement." 
 
FUNDAMENTAL FORMULAE OF TRIGONOMETRY. 17 
 
 where by the complement of an angle is meant the algebraical excess 
 of a right angle over the angle in question. Thus, 9 9 being an angle 
 
 -0—09 or {^-—0^ 9 is the complement, and -— its algebraical 
 
 representation- If the angle be greater than a right angle, the com- 
 plement is negative. Similarly, if Al° be the angle, A being a 
 number or fraction, (90 — A) 1° is the complement. 
 
 First, consider an angle AOB less than a right angle. From any 
 point B draw a perpendicular B M ; then the ratio of B M to B O is 
 
 TJ TV T 
 
 the sine of the angle AOB, which we may denote by -—tt, meaning 
 
 that if BM and BO be expressed in linear units, then the number 
 of units in BM, divided by that in BO, gives the number which is 
 called the sine of AOB. 
 
 The cosine, or sine of the complement, is thus deduced : Let the 
 triangles OLK and 0MB be equal in all respects, namely, KLto 
 OM, &c.; then AOB and AOK make up a right angle, or AOK is 
 
 the complement of AOB. Its sine is, therefore, tt^? or jy^, which 
 
 is the cosine of AO B. 
 
 By the tangent of AO B is meant 
 
 BM 
 MO 
 
 KL 
 
 The cotangent of AOB is, therefore, , or 
 
 By the secant of AO B is meant 
 KO 
 
 BO 
 OM 
 
 The cosecant of AOB is, therefore, -qT-^ ^^ 
 
 MO 
 MB 
 
 BO 
 BM 
 
 By the versed sine of AOB is meant — rr-r , or 1 — cosine of AOB 
 
 The coversed sine of AOB is, then, 
 
 c 2 
 
 KO — OL 
 
 KO 
 
 , or 1 — sine of A O B 
 
18 DEFINITIONS AND 
 
 (31.) When the angle is between one and two right angles, as 
 A OB', then let the same definitions be adopted, as follows (p. 15.) : 
 
 sine. 
 B'M' 
 BO 
 
 cosine. 
 OM' 
 OB' 
 
 tangent. 
 B'M' 
 OM' 
 
 cotangent. 
 OM' 
 B'M' 
 
 secant. 
 BO 
 OM' 
 
 cosecant. 
 BO 
 B'M 
 
 which is pos. 
 
 neg. 
 
 neg. 
 
 neg. 
 
 neg. 
 
 pos. 
 
 Remember that OB and OB', &c. have no sign. Lines only are 
 considered as having signs + and — , which must be in one of two 
 opposite directions. 
 
 When the angle is between two and three right angles, as A OB" 
 (positively measured), then we have for the 
 
 sine. cosine. tangent, cotangent, secant. cosecant. 
 B"M" OM" B"M" OM" B"0 B'O 
 
 B"0 OL' OM" B"M" OM" B"M 
 
 neg. neg. pos. pos. neg. neg. 
 
 When the angle is between three and four right angles, as A OB"' 
 (positively measured), then we have for the 
 
 sine. cosine. tangent, cotangent, secant, cosecant. 
 
 B'" M"^ OM"^ B'"M"' OM" B'^"0 B"'0 
 
 B"'0 OB^ OM'" B"M'" OM'" B'"M 
 
 neg. pos. neg. neg. pos. neg. 
 
 The student should verify each of these assertions, which we shall 
 proceed to systematise. 
 
 (32.) When an angle is less than a right angle, say it is in the 
 first right angle ; when between one and two right angles, say it is 
 in the second right angle, &c. Now, remember the following table : 
 
 sine and cosecant + + — — 
 cosine and secant + — — -j- 
 
 tangent and cotangent + — -|- — 
 
 which will be found, on examination, to contain the results of the 
 preceding articles thus. I wish to know whether the sine, in the 
 third right angle, be positive or negative ; repeat 
 
 sine and cosecant, plus, plus, minus f minus, 
 the third of which is negative ^ the answer required. Examine the 
 preceding results, and see that this table contains them all. 
 
 (33.) Now, from the definitions, make it appear that the second 
 column of assertions below is a translation of the first. 
 
FUNDAMENTAL FORMULiE OF TRIGONOMETRY. 
 
 ig 
 
 No sine or cosine can exceed 
 unity. 
 
 No secant or cosecant can 
 be less than unity. 
 
 A tangent or cotangent may 
 have any value whatsoever. 
 
 The side of a right angled 
 triangle cannot exceed the hy- 
 pothenuse. 
 
 Tha hypothenuse of a right 
 angled triangle cannot be less 
 than a side. 
 
 Two lines, of any ratio what- 
 soever, may be the sides of a 
 right angled triangle. 
 
 All this is relative to numerical magnitudes, independently of 
 sign. 
 
 CoroUary. Versed sines and coversed sines are always positive, 
 and never exceed 2. 
 
 (34.) Now make the following abbreviations. 
 
 Let 9 or A. 1° be the angle. 
 
 Let sine of 9 be written 
 
 cosine 
 
 tangent 
 
 cotangent 
 
 secant 
 
 cosecant cosec 9 
 
 versed sine 
 
 coversed sine covers 9 coversed sine 9 
 
 Similarly, let sine of A.l° be written sinA, &c. Observe, that 
 if 09 and A.l° be the same angle, expressed in the two different 
 units, then sin = sin A, cos 0= cos A, kc; for it is obvious, that 
 the angle itself has the same sine, &c. in whatever units it may be 
 expressed. 
 
 (35.) Fundamental properties implied in the definitions. 
 
 sin 9 and read 
 
 sine 9 
 
 cos e 
 
 cosine 9 
 
 tan 
 
 tangent 
 
 cot 
 
 cotangent 9 
 
 sec 
 
 secant 9 
 
 )sec 
 
 cosecant 9 
 
 t'ers 
 
 versed sine 9 
 
 
 b' 
 
 
 Af 
 
 
 B 
 
 
 y 
 
 
 \^ 
 
 ^ 
 
 ^ 
 
 5 
 
 
 A"Mr 
 
 
 -X ^^( 
 
 >\ 
 
 X 
 
 M 
 
 A 
 
 .-'J 
 
 : 
 
 \" 
 
 ^ 
 
 M" 
 
 I 
 
 7 
 
 
20 DEFINITIONS AND 
 
 Let U be the linear unit, and let OM and OM' contain x linear 
 units, the latter being marked — j, on account of the contrary direc- 
 tion of OM'. And, first, let AOB (in first right angle) = ee; 
 
 Then sin d x cosec ^ = " x - = 1 
 
 r y 
 
 cos ^ X sec ^ = - X - = 1 
 r X 
 
 tan ^ X cot ^ = ^ X - = 1 
 X y 
 
 Let AOB' (in second right angle) =:^e; then, in the same way, 
 
 sin d X cosec ^ = - H = 1 cos ^ x sec ^ = -^^^ ■\ ^— = 1 &c. 
 
 r y r — x 
 
 so that these formulae are universal, and we have the following con- 
 sequences of definition. 
 
 The sine and cosecant are reciprocals. 
 The cosine and secant are reciprocals. 
 The tangent and cotangent are reciprocals. 
 
 We have also 
 
 y V-i-r sin 9 
 
 tan^ = ^ = ^-^— =. 
 
 X x-f-r COS0 
 
 . X x-^r cos 
 
 y y-T-r sin 
 
 (36.) Consequences of Euclid 1. 47. We have the equations (4.) 
 
 The first of which (and the rest maybe treated in the same way) 
 may be written in three different forms, namely, 
 
 (^y + {^^^ = 1 or cos 2^ + sin2^ = 1 
 
 1 + Q = (^-X or 1 4- tan «^ = sec«^ 
 
 1 + (-) = (-) or 1 + cot » ^ = cosec » Q 
 
 Hence sin»^ = -^ = ^^r"'^\. = 
 
 tan"^ 
 
 x'+y"* l^-(j/-^Jf)' l-hian'0 
 
 cos a ~"~ — ^ — 
 
 j'+y "" l + (3/-?-x)' ■" l+tan'e 
 
FUNDAMENTAL FORMULA OF TRIGONOMETRY. 
 
 21 
 
 tan 6 
 
 Deduce also sin^ == 
 
 1 
 
 N/r+lan^ 
 cot^ 
 
 To these we may add the equations of definition 
 
 vers ^ = 1 — cos ^, covers ^ = 1 — sin ^, 
 
 (37.) The student should now, as an exercise, express each of 
 the primary functions in terms of every other, as in the following 
 table, where the meaning of t in each column stands at the head, 
 and the value of the expressions in each horizontal line at the 
 beginning. 
 
 COS0 = 
 
 tan0 = 
 
 cot0 = 
 
 cosec 0= 
 
 sin ^ 
 t 
 
 cos^ 
 
 tan^ 
 t 
 
 s/\-.f -7== 
 
 cot 
 
 1 
 
 n/1-^2 
 / 
 
 t 
 
 1 
 
 1 
 t 
 
 t 
 
 t 
 t 
 
 1 
 ~t 
 
 1 
 
 1 t 
 
 1 
 
 ^ 7 
 
 1 
 
 7 ^ 
 
 sec ^ 
 1 
 
 7 
 
 cosec ^ 
 1_ 
 t 
 
 n/^ 
 
 1 
 t 
 
 ^'+'' 7m 
 
 t 
 t 
 
 Thus, if we would know from the preceding the value of the cosecant 
 in terms of the secant only, we find 
 
 sec 9 
 
 cosec d = 
 
 Vsec^e— 1 
 
 (38.) The next question is to determine what we may call the 
 /rans«7ion values of the primary functions, namely, those which they 
 should have when the angle is nothing, or one, two, or three right 
 angles exactly. For, in these cases, a look at the figure of (28.) 
 will shew that the right angled triangle, which is the basis of the 
 definitions, disappears altogether, so that, by any definition yet 
 existing, there are no sines, cosines, &c. But, agreeably to the 
 
22 
 
 DEFINITIONS AND 
 
 conventions of algebra, we shall use the following extensions as 
 abbreviations {Algebra, p. 156.) 
 
 If, when X approaches without limit to a, X diminish without 
 limit, let it be said that when a:=a,. X = 0: if, in such case, X 
 approach without limit to A, let it be said that when j: =a, X = A : 
 and if X increase without limit, let it be said that when T = a 
 X = a, or is infinite. 
 
 (39.) According to these definitions, and observing what species 
 of variation of magnitude each of the functions undergoes, we have 
 the following table : 
 
 1 right angle, 2 right angles, 
 
 r 
 
 3 right angles, 
 3^ 
 
 
 
 or 
 
 2 
 
 e 
 
 or 90.1° 
 
 or 7r 
 
 e or 180.1° 
 
 or 
 
 2 
 
 or 270.1° 
 
 sine 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 — 1 
 
 cosine 
 
 1 
 
 
 
 
 
 
 
 — 1 
 
 
 
 
 
 tangent 
 
 
 
 
 
 
 a 
 
 
 
 
 
 
 oc 
 
 cotangent 
 
 a 
 
 
 
 
 
 
 
 (X 
 
 
 
 
 
 secant 
 cosecant 
 
 1 
 
 
 
 
 a 
 
 1 
 
 
 — 1 
 
 
 
 — 1 
 
 versed sine 
 
 
 
 
 
 
 1 
 
 
 2 
 
 
 
 1 
 
 coversed sine 
 
 1 
 
 
 
 
 
 
 
 1 
 
 
 
 2 
 
 The first three lines are the most important. The student may 
 shew that as the angle approaches either of the four intermediate 
 values, its sine, &c. approach the value in the table. 
 
 (40.) But the following method will be more clear. We shall 
 determine linear measures of the sine, cosine, &c. in the same manner 
 as the arc of a circle is the measure of an angle. Let there be any 
 linear unit U, and with centre O, and radius OA = U describe a 
 circle, and let the angle OAB, or 09, be that in question. The rest 
 
 BM 
 
 of the figure will need no description. Now, because sin B = rr— - 
 
 when BM and BO are expressed in units (let them be x\J and U) 
 we have sin0= x, or sin0.U = BM. Consequently, BM and the 
 sine of© change in the same proportions; and BM may stand for 
 tl>e sine of (as it might represent a sum of money, an area, or any 
 other magnitude) being a linear measure of it, at the rate, to use a 
 common phrase, of AG to a unit. Or let OA be one measure y then 
 
FUNDAMENTAL FORMULA OF TRIGONOMETRY. 
 
 23 
 
 the fraction of a measure in BM is the sine of the angle A OB, and 
 the fraction of a measure contained in the arc AB, is the fraction of 
 an analytical unit contained in the angle AOB. In precisely a 
 similar manner, OM is a linear measure of the cosine. But when 
 we come to consider the tangent, we see that BM is not a measure 
 of it : for, if the angle increase, the change of the tangent is the 
 
 combined effect upon the ratio of a simultaneous increase of BM, 
 and decrease of MO. But both are represented in the increase of 
 the line AT; for, by similar triangles, BM : MO : : AT : AO, of 
 which AO remains the same, and AT changes in the same ratio as 
 the tangent. Hence, AT is a linear measure of the tangent of AOB. 
 Similarly, since O M : MB :: A'V : A'O, or AO, A'Vis a linear 
 measure of the cotangent; and, since BO : OM :: TO : OA, 
 then TO is a linear measure of the secant. Similarly, OB : BM : : 
 VO : OA', or VO is a linear measure of the cosecant. Again AM, 
 the linear measure of 1 — cos 9 and A'L, that of 1 — sin 6, are those 
 of vers 9 and covers 9. Consequently, understanding by the linear 
 unit U the radius OA, and also that the question " How many 
 times ?" also implies " What fraction of a time?" and " What times 
 and fraction of a time?" we have the following list of synonymous 
 questions, in which the blanks may be filled up out of the same 
 horizontal line. 
 
 What is the ( ) of How many linear units 
 
 the angle AOB? are contained in ( ) ? 
 
 sine BM 
 
 cosine O M 
 
 tangent AT 
 
24 DEFINITIONS AND 
 
 What is the ( ) of How many linear units 
 
 the angle AO B ? are contained in ( ) ? 
 
 cotangent , A' V 
 
 secant OT 
 
 cosecant O V 
 
 versed sine AM 
 
 coversed sine A' L 
 
 number of analytical units the arc A B. 
 
 The student may now readily construct the table in (39.) by 
 observing the final states of the linear measures. Thus, as the angle 
 diminishes without limit, the line A'V increases without limit, or 
 (38.) the cotangent of is infinite ; and so on. 
 
 (41.) (Now, according to the old system of trigonometry, not 
 yet exploded for the beginner, though every person must practically 
 get rid of it before he can advance far in analysis, the line BM is the 
 sine, and not of the angle, but of the arc AB. In this system, there 
 is an infinite number of sines to the same angle, corresponding to 
 all the arcs which that angle can subtend ; and, consequently, it is 
 always with reference to the radius supposed that all formulae must 
 be constructed. For example, it is not true that 
 
 sin^^ + cos^^ = 1 unless when the radius is 1 
 but sin'0 + cos'-'0 = r^ where r is the number of linear units in the 
 radius. This embarrassing consideration is always avoided in practice 
 by making the radius the linear unit, and then substituting for the 
 lines called sines, &c. their numerical proportions to the radius : 
 which amounts in fact to the more modern method). 
 
 (42.) We are now to consider, in connection with 9, the angles 
 which exceed or fall short of any whole number of right angles by 0, 
 or which are contained in the following series. 
 
 ^-T ^-^ ^-f ^ ^+i ^ + T&C. 
 
 -T-^ -c-^ -l-d -^d -^+1 -^+cr&c. 
 
 all contained in the form d: w ~ it ^, where m is a whole number. 
 
 But, in the first place, we must observe that any addition or 
 subtraction of four right angles, or multiples of four right angles, 
 produces no change in the position of OB (29.) being, in fact, 
 
FUNDAMENTAL FORMULA OF TRIGONOMETRY. 
 
 25 
 
 equivalent to supposing complete revolutions, one or more, to have 
 taken place, leaving OB the same in position as before. The sines, 
 cosines, &c. depending entirely upon the position of OB, and in no 
 way upon the number of revolutions supposed in attaining that posi- 
 tion, we must have (2 sr being the numerical symbol of four right angles) 
 
 sin & == sin(2'7r + ^) = sin (4-^ + ^) = sin (G'tt + 6) &c. 
 
 = sin(^ — 2^) = sin(^ — 4';r) = sin(^ — Ocr) ^c. 
 
 and generally, F representing the operation by which we pass from 
 an angle to any primary function {Algebra, p. 203.) we must have 
 
 F(^) = 'F{d + 2m'?r) 
 
 where m is any whole number, positive or negative. 
 
 Hence, we may reduce the list in the last page : for, we find 
 
 that F ( ^— is the same as F ^2'^ + d— ^ or F (^ + ^V 
 
 We shall limit ourselves first, to the consideration of the following, 
 
 3^ 
 2 
 
 2cr- 
 
 omitting 2<r4-^, because its primary functions are those of^. And, 
 first, let ^ be less than a right angle, so that m- -\-6 must fall in the 
 (m-f l)th right angle, and m- — 6 in the mih. We have called 
 
 - — i the complement of/; it is also usual to call t — 6 the sup- 
 plement of ^; being, when i is less than two right angles, the adjacent 
 angle of Euclid. Now, draw the following figure, making ^e a small 
 angle for convenience. 
 
26 DEFINITIONS AND 
 
 () is at the centre (not marked), let the angles AOB, AOC", 
 A'OB', A'OC, A"OB", A'OC, A'"OB'", A'OC" be all equal to 
 each other, and to 6 e. Let the triangles MOB, MO C", N O C, 
 NOB'", N'OB', N'OC", M'OC, M'OB" be all made equal to 
 each other in every respect; namely, ON to MB, OM to NC, 
 &c. &c. Then we have (all angles being measured positively) 
 
 ZAOB = ^0 zAOB' = (^+^)0 
 
 ZAOB" = (^ + ^)0 zAOB' = (3^ + ^)0 
 
 ZAOC = (f-^)0 /.AOC = (^-^)0 
 
 Z.AOC" = (32-^)0 Z.AOC'" = (2^-^)0 
 
 From hence we can immediately find any primary function in terms 
 of a primary function of 0e, as follows. Suppose it required to find 
 
 cotf 3- — 6\ : we have immediately 
 
 I Kr\rMi^ — ^'^ — BM(with contrary sign) ___ , BM 
 cot(^AUO ) — ^77p^ — o M (with contrary sign) ~ "^ OE 
 
 or cotf 3- — &\ = tan ^ 
 
 (43.) Now, in this investigation, there are 42 cases, but they all 
 fall under the following rules for expressing a function of w - ±: ^ 
 by means of a function of 6. Let F fm - + 6\ be required. 
 
 1. //m he odd, change F into its co-function ; namely, sine into 
 cosine, cosine into sine ;* tangent into cotangent, cotangent into 
 tangent, &c. lifmbe everij let F remain. 
 
 2. Look at the scale of signs (32.) of F, namely, 
 
 for sine and cosecant + -| 
 
 cosine and secant -\ 1- 
 
 tangent and cotangent -f 1 
 
 and, observing in which right angle w - ± ^ falls, prefix the 
 
 • According to our definitions (30.) the cccosine means the cosine 
 of the complement, or the sine. 
 
FUNDAMENTAL FORMULA OF TRIGONOMETRY. 27 
 
 sign which answers to the number of that right angle in the 
 scale. 
 
 For instance, for cot ( 3 - — ^ j , the number of right angles is 
 
 oddf or we write tan for cot : and 3 6 is in the third right angle ; 
 
 f co^ j the proper sign is 4" 
 
 cot (3 ^ — ^) = tan ^ 
 
 Let the student go through the cases /rom the fgure, and satisfy 
 himself that they agree with this rule. 
 
 (44.) The following are some results. The first set is in the 
 definitions. 
 
 sinf- — ^) = cos^ 
 cosf - — ^) = sin 6 
 
 tan (2 — dj = cot ^ 
 
 sin('r + ^)= — sin^ 
 ^ cos (t + ^) = — cos d 
 - tan {t + &)= tan 6 
 
 sinf- + ^ j = cos^ 
 cosf- -h dj = — sin^ 
 tan(^ +N =— cot^ 
 sinr — — ^j = — cosd 
 
 [josf — = — sin^ 
 
 :anf-^ — ^1 = cot^ 
 
 sin('3' — d) = sin^ 
 cos (cr — &)is= — cos ^ 
 tan(-y— d) = — tan^ 
 
 ^"vT" + ^) = ~ cos^ 
 50sf-^ -{■ 6J =■ sin^ 
 tan(5^ + ^) = — cot^ 
 
 sin (2 -TT -7- ^) = —sin ^ sin (— ^) = — sin^ 
 
 cos(2';r— ^)= cos^ cos(— ^)= cos^ 
 
 tan (2':r — ^) = — tan ^ tan (— ^) = — tan d 
 
 The last set is deduced from that immediately preceding, by sub- 
 tracting four right angles (42.). They may be deduced immediately, 
 by observing that MOC" = — ^e (29.)- 
 
 To obtain versed and coversed sines, remember that 
 
 vers ^ = 1 — cos ^, covers & = 1 — sin ^ 
 
 Thus, covers (3^— ^) = 1 — sin(— — ^) = 1 + cos^ 
 
 (45.) When ^e is greater than a right angle, the results are the 
 
28 DEFINITIONS AND 
 
 same as if it were less than a right angle. An easier demonstration 
 will afterwards apply ; in the meantime, suppose ^ = t -{- ^', and we 
 
 want, for instance, cot f 3 - — ^ J . We have then, 
 
 But tan^' = tan(<7r + &') = tan^; 
 
 therefore, cot f 3- — ^ J = tan ^ 
 
 the same as before. Let the student work a number of instances 
 of this kind. 
 
 By means of the last set of the preceding formulae, we can easily 
 
 ascertain Y\6 — m^j. Suppose tan(^— t) is required; then we 
 
 have 
 
 tan (^ — -tt) = — tan('7r— ^) = — (— tan^) = tan^ 
 
 (46.) We shall now proceed to some theorems connected with 
 the limits of the ratios of trigonometrical functions, {Alg. p. 162.). 
 
 B 
 
 The ratio of an angle (in analytical units) to its sine, approximates 
 without limit to unity, when the angle is diminished without limit. 
 Let AOB, AOD, be equal angles; then BM = MD, arcAB = 
 
 AB 
 arcAD ; and, AOB being ^e; or, — — being ^, we have 
 
 . . AB BM .^ T^,- 
 
 6 : sind :; -7-7^- : -j-rr- :: AB : BM 
 AO AO 
 
 :: 2AB : 2BM :: arcBD : chordBD 
 
 Let the chord BD be the side of an inscribed polygon of n sides; 
 then the greater n is taken, tlie less does the whole boundary of the 
 polygon (which is n X chordBD in length) differ from the circum- 
 ference of the circle (which is n X arcBD). Let n x arcBD = 
 n X chord B D -f Z ; then can the length Z be made as small as we 
 please, by taking n sufficiently great. 
 
 But chord BD = — -j—arcBD, and substitution gives 
 
FUNDAMENTAL FORMULJE OF TRIGONOMETRY. 29 
 W X arc B D — /2 X —J- X arc B D = Z, 
 or ^1 — -^y-) X 71 X arc B D = Z 
 that is, (1 — -) X circumference = Z 
 
 But as n increases without limit, the angle BOD, and, therefore, 
 BOM, diminishes without limit; and since, in such a case, Z also 
 
 diminishes without limit, the fraction 1 — diminishes without 
 
 limit, the circumference being always the same. Hence, — - — ap- 
 proaches without limit to unity. Let the student try to demonstrate 
 this in the manner of (4.), supposing ^ and sin^ to be incommen- 
 surable. 
 
 The angle of 5 degrees, which is not for any practical purpose a 
 small angle, and which, in analytical units, is -0872665, has, for its 
 sine, -0871557 ; which gives, 
 
 sin (-0872665) _ '0871557 _ 872 
 
 -0872665 "^ -0872665 ~ 873 ^^^^ "^^'" ^' 
 
 or, when AOB is the eighteenth part of a right angle, if the arc BA 
 were divided into 800 equal parts, BM would be more than 799 of 
 these parts. 
 
 (47.) As the angle diminishes without limit, the cosine approaches 
 without limit to unity ; and 1 — cos^ diminishes without limit, as also 
 does sin^. It will be necessary to examine the ratio of 1 — cos^ to 
 sin^ under this change. 
 
 1 — cos^ (1 — cos^) (1 -f-cos^) sin'*^ sin^ 
 
 sin^ siii^ (1+cos^) sin^(l -|-cos^) 1 4-cos^ 
 
 of which the numerator diminishes without limit, while the deno- 
 minator increases with the limit 1 -|- 1 or 2. The fraction, therefore, 
 
 diminishes without limit, or ■. — ;; — diminishes without limit. 
 
 sni^ 
 
 When ^ is 5 degrees, its sine and cosine are -0871557 and 
 
 •9961947 very nearly. Whence, 
 
 1 _^cos£ __ -0038053 4_ 
 
 ~ sm^ ~ -0871557 '^ 87 
 
 or, for this angle, 1 — cos^ is less than the twentieth part of sin^. 
 
 2 D 
 
30 DEFINITIONS AND 
 
 If we take 5 minuteg instead of 5 degrees (5 minutes in ana- 
 lytical units being -0014544, its sine the same to seven places of 
 decima's, and its cosine -9999989), we find, that if i were divided 
 into 14000 equal parts, the sine would not be less by so much as one 
 of those parts, and that 1 — cos^ is not the 1300th part of sin ^. 
 
 (48.) Again, since — : — = -, the limit is unity when 6 di- 
 
 ' ° sm^ cos^ "^ 
 
 ... . , ... , . tan^ tan ^ sin ^ .... - 
 
 minishes without limit ; and since — - — = -r — 7 . — 7—, the limit of 
 
 this, in the same circumstance, is 1 x 1, or 1. And, because 
 
 1 — cos^ 1 — cos^ ^ sin^ 
 
 6 sin^ 6 
 
 1 **^ cos 6 
 therefore also ; diminishes without limit at the same time 
 
 
 
 as 6. 
 
 (49.) We shall now propose, as a problem, the solution of the 
 equation 
 
 sin a; = siny 
 
 or rather, to find certain solutions ; for we have no means as yet of 
 ascertaining that any given number of solutions is the total number. 
 Looking among the results of (44.), we find the following solutions, 
 premising, first, that x =y is one solution : an angle has but one 
 sine. 
 
 a: = y±2cr ar = y±4'7r a: = y±6cr, &c. 
 
 X = 'TC-^y X =s {^—y)-±i2'7r = {\ir—y or ^'J^^y 
 
 X = {ir—y)-±.A<7r = ^T—y or — S-r— y, &c. 
 
 We now propose tanmj; = cotn_y. Since cotnj/ = tan( - — nxf\ 
 
 we have, firstly, mx=. nyy orar=-( ny). We have, also, 
 
 mx ^ - — ny±.2'!e two: = - — wy ± 4^, &c. 
 
 And since tanx ::= tan(j:ih«')> the following are also solutions : 
 
 mx =■ —ny±.<K or ^— wy±3flr, &c. 
 
 (50.) The following propositions will be readily proved, espe- 
 cially from the figure in (40.). In the same right angle there are no 
 two sines, or cosines^ of tangents, &c., which are equal to each other. 
 
FUNDAMENTAL FORMULJE OF TRIGONOMETRY. 31 
 
 And of angles which do not exceed four right angles, there are two 
 to every sine (or cosecant), and x being one, ^r — jt is the other; 
 two to every cosine (or secant), and x being one, 2* — j" is the other; 
 two to every tangent (or cotangent), and x being the lesser, ^-{-x is 
 the greater. 
 
 Now, X being less than two right angles, so is -r — x ; but -r -{-x 
 and 2?r — x are greater than two right angles. Consequently, where 
 there is question of the angles of a triangle, the cosine of an angle 
 (or secant), or the tangent (or cotangent), being given, the angle is 
 absolutely determined ; for there is but one angle which is contained 
 within the limits of the angles of a triangle (O and jt), to which such 
 cosine, &c, can belong. But, when the sine of an angle is given, or 
 found, as that by which the angle of a triangle is to be determined, 
 there maybe two angles within the limits of the problem; for if j: 
 be one answer, jt — j: is another. 
 
32 FORMULiE CONNECTED WITH 
 
 CHAPTER II. 
 
 FORMULJE CONNECTED WITH TWO OR MORE ANGLES. 
 
 (.51.) The doctrine of ratios, as in Euclid, presents the notions which 
 answer to multiplication, division, raising of powers, extraction of 
 roots ; every fundamental operation, in fact, except onl^ addition and 
 subtraction. The truth is, that it considers two ratios just as two 
 lines are treated in the first book ; that is, as subject to the relations 
 of greater, equal, and less, but without any classification or comparison 
 of the various modes of greater and less. From the definitions, it 
 appears that the ratio of X + Y to Z is more than the ratio of X to Z ; 
 and here the Fifth Book stops . But ratio is a magnitude ; we apply 
 the words greater, &c. to two ratios. It is true that the definition of 
 ratio looks more like that of a criterion than of a magnitude; but, 
 as we have seen, the word angle is in the same predicament :. 
 the definition of opening, or inclination, is only a rough primary 
 conception, while the useful definition of an angle is the criterion 
 which determines the greater or less, or the equality, of two angles. 
 The rough conception of ratio is relative magnitude ; that notion, by 
 which a spectator who knew nothing about numbers, would decide 
 whether the picture of a known object was in or out of proportion ; 
 that notion, by aid of which savages, who have as little idea of 
 numbers as it is possible for a human being to have, comprehend a 
 map as soon as it is shewn to them, and point out the various sites 
 which they know, as soon as they know whereabouts in the map 
 they are for the time, and the direction of north or south upon it. 
 With this notion comes the following: That the relative magnitudes 
 of X and Y to Z, make up the relative magnitude of X + Y to Z : 
 whence, we subsequently come to the general definition of addition 
 of ratios; namely, that to add the ratios of A to B and C to D, re- 
 duce both to other ratios having the same consequent, say X to Z, 
 and Y to Z; then the sum of the preceding ratios is that of X -j-Y 
 loZ. 
 
 We have introduced these considerations again, in order to point 
 
TWO OR MORE ANGLES. 33 
 
 out the difference between geometry and algebra, which the following 
 question, being the fundamental proposition of the present chapter, 
 will exemplify. Given the primary functions of two angles, of which 
 the sum is less than a right angle, required the primary functions of 
 their sum. The geometrical solution is as follows : 
 
 The first angle being less tlian a right angle, take any straight 
 
 line OM, and erect MN, so that the ratio of NM to MO shall be 
 the tangent of the first angle ; then will N O M be the angle in 
 question. Then draw N P perpendicular to O N of such length that 
 the ratio of PN to NO shall be the given tangent of the second angle; 
 whence PON is the second angle, and MOP is the sum of the 
 angles. Draw P Q perpendicular to M; then k the ratio of P Q tQ 
 PO the sine of the sum required, &c. This geometrical construction 
 is a complete solution within the meaning of the terms geometrical 
 solutiorif with regard to which it is matter of definition that lengths 
 are determined, found, or given, when the extreme points are given. 
 But it is not an algebraical solution, of which it is a condition that 
 no magnitude is given, determined, or found, unless its ratio to 
 some given magnitude of the same kind be given, &c. The geome- 
 trical solution is the more easy, because it assumes the harder point, 
 and requires only determination of position ; the algebraical solution^ 
 which requires ratios, carries the geometrical solution further, and 
 demands consequences with which the geometrical solution, by an 
 express definition of exclusion, has nothing to do. And the student 
 will do well to remember this when he comes to read controversy 
 about the relative value of algebraical and geometrical solutions. 
 
 The algebraical solution is as follows, without symbolic language. 
 Draw NR parallel to OM. Then, since PQ is made up of PR 
 and NM, the ratio PQ: PO is the sum of PR : POandNM : PO. 
 But PR : PO is compounded of PR : PNand PN : PO, or by 
 
34 FORMULA CONNECTED WITH 
 
 similar triangles, of OM : ON and PN : PO, and being com- 
 pounded of given ratio3, may be expressed by whatever symbol we 
 adopt to signify composition. Similarly, NM : PO is compounded 
 of NM : NO and NO : PO, two given ratios. Under the common 
 meaning of terms in algebra, we may, if all the pairs be commen- 
 surable, and if O P stand for the number of linear units in O P, 
 &c. we proceed thus : let N O M = ^ e, M O P = ^ e, then MOP = 
 (^ + ^) ©) and we have 
 
 . , , .X _ PQ _ PR , NM __ PR PN NM NO 
 sm(^ -VQ) - pQ-pQ + PO -pN*P(J"^NO'*PO 
 
 MO PN NM NO 
 
 = NO'PO + NO*PO = '°'^'^"^+''"^'°'^ 
 
 If the ratios be incommensurable, we must either, 1 . Imagine 
 commensurables very nearly equal to MO, &c. to be substituted, and 
 the real meaning of the equation will then be (meaning by (a) a very 
 near approximation to a), (cos ^ (sin ^) + (sin S) (cos ^) is very near 
 to sin((p + ^); or, 2. adopt the more general ideas of ratio in the 
 preliminary treatise, and interpret the symbols of operation accord- 
 ingly. Leaving the student to take which course he can, we now 
 proceed, having obtained in every sense of the terms 
 
 sin (^ + ^) = sin ^ cos ^ + cos <p sin & 
 
 similarly cos (^ 4* ^) = cos 9 cos^ — sin <p sin ^ as follows: 
 
 . OQ OM — NR OM ON NR NP . 
 cos(^ + ^) - OP = OP = ONOP •"NPNO*'^- 
 
 Now, construct a figure in which ^ e and 6 e are each less than 
 a right angle, but their sum greater ; shew that the process for the 
 sine remains exactly the same, and that in that for the cosine of the 
 sum, which is negative, OM — RN also becomes negative; whence 
 we still have cos (^ -f (f) with its sign = (O M — R N) -J- O P, 
 and the two formulae are precisely as before. Shew also that by 
 aid of 
 
 cos^^ + sin'^ = 1 cos«^+ sin^*^ = 1 
 
 the sum of the squares of the preceding developements is = 1. 
 
 (52.) Since these formulae are universally true, independently of 
 all values of the angles, within a right angle (as far as we know yet) 
 they will remain true if instead of ^, we write ^— ^. Do this, which 
 gives 
 
TWO OR MORE ANGLES. 3§ 
 
 sin <p = sin(^— &) cos^ +cos(^ — &) * sin ^ 
 cos <p = cos (^ — Si) cos & — sin (^ — &) sin ^ 
 
 Multiply the first by cos ^, and subtract the second multiplied by 
 sin^, remembering that cos^^ + sin '^ = 1, which gives, transposing 
 the sides, 
 
 sin {<p — ^) = sin <p cos & — cos <p sin 6 
 
 cos {<p — &) = cos <p cos ^ + sin f sin & 
 
 Which proves that the two first formulae are true when one of the 
 angles is negative. In sin (^p -{-6) write — 6 for ^, and we have 
 
 sin(^ — ^) = sin^ cos( — &) +cos^ sin( — &) 
 or (44.) = sin 9 cos ^ — cos <p sin 6, as just proved. 
 
 Endeavour to deduce these propositions from an adaptation of the 
 construction in (51.), and verify the value of the sum of the squares 
 as before. 
 
 (53.) We now shew that these formulae remain true whatever may be 
 the magnitude of the angles. We shall take a case, and recommend 
 the student to acquire dexterity in the management of the formulae, 
 by trying various others. Let us suppose our first angle to be in the 
 third right angle, and our second in the fourth, so that the sum 
 must be in the sixth at least, or in the seventh. Let ?r + ^, and 
 
 — - -f- ^ be the analytical units in the angles, and 6 and (p must 
 
 therefore be severally less than a right angle. Then the sum is 
 
 — 4-^ + ^, or2^ + - + ^ + (p, and, therefore, its sine is (42.) 
 *> 2 
 
 ^'"U 
 
 + ^ + ^ j or cos {& -\- <p) or cos d ' cos (p — sin ^ * sin ^ . . (A) 
 
 But sin (tt + 0) = — sin 9 or sin 9 = — sin (7r + 9) 
 
 cos (tt + 0) = — cos . . cos = — cos (tt + 9) 
 
 ■ f^-^ ^ \ ■ /Stt , \ 
 
 sml— -+0l = — COS0 .. cos^ = — sin(— -+01 
 
 cos^Y+0)= sin^ .. sin0= cos (^y + 0^ 
 and, by substitution in (A), we have 
 sin(7r + + 3^ + 0) = — cos(7r+6l) X — sin(y +0) — 
 
 (— sinTT+e) X cos(^y + 0) 
 
36 FORMULiE CONNECTED WITH 
 
 in(7r + + 3^ + 0) =cos(7r + d)sin(y +^) + sin (^r + 0) cos (y + 0) 
 
 let TT -f = 0', •— + = ^', and we have 
 
 sin {& + ^') = cos 0' sin ^' + sin 0' cos 0', the same as before. 
 
 We have then as general formulae, true for all angles, positive and 
 negative, 
 
 sin (0 + 0) = sin cos -f cos sin sin(0 — B) = sin0cos0 — cos0sin9 
 cos(0 -|- ^) = CO80 COS0 — sin0 sin0 cos(0 — 9) = cos^ cos0 + sin0 sin0 
 (54.) We can now, with very slight labour, acquire a large number 
 of very useful formulae so quickly, that previous description will be 
 unnecessary. 
 
 sin (0 ~f- 9) + sin (0 — 9) =■ 2 sin <p cos 9 
 sin(0 + 0) — sin(0 — 9) = 2cos0sin0 
 cos (0 + 0) + cos (0 — 9)=. 2COS0COS0 
 cos (0 + 0) — cos(0 — 9) := — 2 sin 0. sin 
 
 Since these are always true, we may for and write 3 (0 + ^) ^"^ 
 i (0 — 0). Do this, which gives 
 
 sin + sin0 = 2 sin ^ (0 -f 0) cos i (0 — 0) 
 
 sin — sin0 = 2 cos ^ (0 + 0) sin i (0 — 0) 
 
 cos 4- COS0 = 2 cos \{'P -\- 9) cos i (0 — 0) 
 
 cos — cos ^ — 2 sin i (0 + 0) sin ^ (0 — 0) 
 
 sin — sin __ tan K0 — 0) sin + sin __ . w , «v 
 
 «in0+sin0 tan i (0 + 0) cos0 + cos0 ^K9-t») 
 
 (55.) Now, from 2 = + 0, and from sin (0 4- 0) &c. deduce 
 
 
 sin 2 = 2 sin cos sin = 2 sin - cos - 
 
 2 2 
 
 
 
 cos 2 = cos''0 — sin''0 cos0 = cos' sin'- 
 
 2 2 
 
 = 1 — 2sin«0 = i_2sin=? 
 
 2 
 
 = 2cos"0 — 1 = 2 COS* 1 
 
 2 
 
 Q 
 
 1 + COS 2 = 2cos'0 1 -f COS0 = 2cos'- 
 
 Q 
 
 1 — cos 20 = 2sin'0 I— cos0 = 2siu»- 
 
 2 
 
TWO OR MORE ANGLES. 37 
 
 *^^"^ - 1+COS20 *^" 2 - 1+COS0 
 
 "^^^ ^ - 1-COS2 *^" U 2J - 1 + sine 
 
 (56.) tanU + 0) = ^t±ll = sin ^ cose + cos ^ sin 
 ^ '^ \r I y COS (0 + 0) COS COS — Sin (p sin 
 
 divide both terms of the last fraction by cos <p . cos 9 
 
 , , .. tan 4- tan 
 
 tan (0 + ^) = :; ~- -^ ; 
 
 ^^ ^ 1— tan^tanO 
 
 , . . tan — tan 
 
 similarly, tan (a — Q) = — — -f \ 
 
 •" \r J l + tan^tane 
 
 ^ ^ 2 tan ^ . 2 tan i 
 
 tan 2^ = :: rr tan^ = 
 
 1— tan«0 1 — tan2i0 
 
 sin (0 + 0) ^ sin(0 — 0) 
 
 tan ^ + tan ffi = — ^r-^-^ tan ^ — tan ® = — \ ^ 
 
 ~ cos cos ^ cos 0. cos 
 
 (57.) The formulae in (54.) reduce multiplication to addition in a 
 way which may remind us of logarithms, and we shall see more of the 
 same sort of analogy before we have finished. They give, 
 
 sin0cos0 = ^sin(0 4-0) + |sin(0 — 0) 
 cos sin = i sin (0 + 0) — ^sin(0 — 0) 
 
 COS0COS0 = -i-cos(0 + 0) +^cos(0 — 0) 
 sin sin = |cos(0 — 0) — ^ cos (0 + 0) 
 
 Let it be required to reduce the product cos w cos n cos/> ; 
 cosm cosn C0SJ9 = ^cos(?w -f w) .cos/? + Jcos(w — w) cosj!) = 
 J cos (/n -f- w + jo) + i cos (w + n — /;) + ^ cos ( jo + w — w) + 
 \ cos(/) -\- n — w), by applying the same formulae twice. 
 
 (58.) We may apply this method to ascertain the nth power of a 
 sine or cosine in terms of the sines and cosines of the multiples of 
 the angle, as follows; by i^bb^ 
 
 cos''0 = ^ + ^ cos 2 cos30 = i cos + I cos 2 cos 
 
 = i cos + i (2 cos 3 + ^ cos 0) (20 + = 30, 20 — = 0) 
 = icos0 + icos3 or 4cos30 __ 3 cos + cos 3 
 
 Multiply by 2cos0 (we thus avoid fractions) 
 
 8cos^0 = 6cos20 + 2cos30cos0 
 
 = 3 + 3cos20+cos40+cos20 = 3 +4cos20 + cos40 
 
 £ 
 
38 FORMULiE CONNECTED WITH 
 
 Proceeding in this way we get the following set of equations : 
 COS0 = cos 9 
 
 2 cos'0 = cos2 0-f 1 
 
 4 cos*0 = cos30 4-3cos9 
 
 8cos*0 = cos40-|-4cos20 + 3 
 lecos'e = cos50-t-5cos30 + lOcos0 
 32 cos«0 = cos6 0-i-6cos40 + 15cos2 + lO 
 64c0b'd = cos70-f rcos50 + 21cos30-f-35cos0 
 128 cos"© = cos80 + 8cos60 + 28cos40 + 56cos20 + 35 
 
 &c. &c. &c. &c. &c. 
 
 Again, 2sin'0 = 1 — cos20, 4sin=»0 = 2sine — 2cos20.sine 
 
 (54.) (2cos20. sine = sin30— sine) = 2 sin — sin 30 -f sine 
 
 = 3sine — sin3e 
 8sin^e = 6sin'-'e — 2sin3e.sine 
 
 = 3 — 3cos2e — (cos2e — cos4e) = 3 — 4cos2e + cos4e 
 
 Proceeding in this way, we get equations which may be thus most 
 systematically arranged : 
 
 sin e = sin 9 
 
 — 2sin'e = cos2e — 1 
 
 — 4sin3e = sin 3 e — 3 sine 
 8sin*e = cos4e — 4cos2e + 3 
 
 16sin«e = sin 5 e — 5 sin 3e -f 10 sine 
 
 — 32sin«e = cos6e — 6cos4e + 15cos2e— 10 
 
 — eisin'e = sinze — 7sin5e -j- 21 sin 30 — 35 sine 
 128sin''e = cos8e--8cos6e + 28cos4e — 56cos2e + 35 
 
 &c. &c. &c. &c. &c. 
 
 Between these two sets there are strong resemblances and strong 
 differences. It appears that the cosine is a much more simple 
 function, in its relations with other cosines, than is the sine in relation 
 to other sines. The alternation of positive and negative signs, in 
 pairs, here occurs for the first time. We shall now shew how to 
 form th€ inverse expressions, namely, cosn9, &c. in terms of powers 
 of cose, &c. 
 
 (59.) By (55.) we have 
 cos 2 e = cos'e — sin'e sin 2 e = 2 sin e . cos 9 
 
 Let the sine and cosine of be denoted by s and c. 
 
TWO OR MORE ANGLES. 39 
 
 Then cos2 = c^ — s^ sin20 = 2cs 
 
 cosSe = cos(2e + 0) = cos2e.c — sin2 0.s = (c*— s^jc — 2cs.s = c^— 3cs' 
 
 sin30 = sin(20-f 0) = sin20.c -f cos2e.s = 2cs.c -hCc*— s')s = 3c=s— s^ 
 
 COS40 = COS30.C — sin30.s = c*— Gc's'^ + s* 
 
 sin40 = sin30.c + COS30.S = 4c^s — 4cs' 
 
 and thus we get the following equations : 
 * 
 
 cos© .= c sin0 = s 
 
 cos2© = c* — s* sin20 = 2cs 
 
 COS3 = c^— 3cs'* sin30 = Sc^s — s» 
 
 COS4 = c* — 6c^s'-|-s* sin40 = 4c^s — 4cs' 
 
 COS50 = c'— 10c's'+ 5cs* sin50 = 5c*s— lOc^s^^ -|- s* 
 
 and so on ; the law of which will be hereafter investigated. 
 
 (60.) It is easily proved that 
 
 (cos0 + sin0)2 = 1 + sin 20 (cosO— sin©)* = 1 — sin20 
 
 cos^ = ± J >/l4-sin20 ± J \/l--sin20 
 
 sin^ = ± i v/l + sin 20 =F J \/l— sin20 
 
 in which the ambiguity of signs will be afterwards discussed. 
 
 Also cos^ = ^^(1-1-00820) sin^ = n/4(1— cos20) 
 
 (61.) Multiply together the fifth and sixth in (54.), and 
 obtain 
 
 sin(^ H- d) sin(9 — ^) = sin'^ — sin^^ 
 
 (62.) We shall now proceed to some cases, in which the sines, 
 &c. may be exhibited numerically. But, first, by means of this 
 theorem, namely, that (w2-|-n')U, 2mn\J, and {m^ — w^)U, are the 
 sides of a right angled triangle, U being any linear unit, we can at 
 pleasure find the means of verifying the preceding formulae in the 
 most exact manner. 
 
 ^ .« . . 2mn , . vi^ — 11? ^ 2 run 
 
 For, if sm^ = —5— — 5, then cos^ = — j- — - tan^ = 
 
 4 3 4 24 
 
 Let m = 2, w = l ; then sin^ = -, cos^ = -, tan^= -, sin2^=— , 
 
 7 44 117 
 
 cos2^ = , sin3^=-—-; , cos 3^ = — - — ^5 &c. In such a case as 
 
 iO 1 iO 1 ZO 
 
40 FORMULA CONNECTED WITH 
 
 this we do not know the angle in question ; but we will shew that 
 this rude method, though the labour would be very considerable, is, 
 in theory, an unfailing means of finding the angle to a given sine or 
 cosine, as nearly as we please. Suppose an angle to have a sine and 
 cosine, both positive ; that is, to be less than a right angle, or in the 
 first right angle. By finding sin^, sin 2^, &c., and cos^, cos 2^, &c., 
 we are able to find tan^, tan 2^, &c. Now, since the angle in question 
 is less than a rigiit angle, there will be multiples of it in every right 
 angle (one at least) ; that is, no right angle can be left out, or there 
 must be values of n^ lying between m and m + 1 right angles. Con- 
 sequently, since the tangent in the several right angles is alternately 
 positive and negative, we shall always be warned of the value of the 
 multiple angles passing a whole number of right angles, by a change 
 of sign. The first negative sign will indicate that the multiple has 
 become greater than a right angle; the next change, namely, from 
 negative to positive, that the multiple now exceeds two right angles ; 
 and, generally, the m\\i change of sign shews that the multiple in which 
 it appears lies between m and m-\-\ right angles. If, then, we wish 
 to know the angle which belongs to the given tangent within, say one 
 rth part of a right angle, we proceed step by step, and find within 
 what right angles v6 lies, by noting the number of changes of sign in 
 the series tan ^, tan 2 ^, tan 3^ tan t; 6. Let it be between m and 
 
 /« -|- 1 right angles ; then 6 lies between — and of a right 
 
 angle, or is known within one vth part of a right angle. 
 
 The preceding process would be too long and laborious for prac- 
 tical purposes ; but it shews us, theoretically, that the determination 
 of the angle which has a given primary functiony to any degree of near- 
 ness^ is within the means of common algebra. 
 
 (63.) Coming now to the determination of some primary func- 
 tions, we shall express the angles both in analytical and practical 
 units. By (34.), all that we have proved of the primary functions of 
 angles represented in the former way, is true of the latter; except 
 only the theorems in (46, &c.), where tlie angle enters directly with 
 its primary functions. For instance, though 10" is a very small 
 angle, it is obviously neither proved in (64.), nor true, that 
 sin 10" = 10 nearly. If we now rejjresent n degrees by n°, n mi- 
 nutes by n', &c., we have the following equations : 
 
TWO OR MORE ANGLES. 
 
 41 
 
 ^0 = 180°, f = 90^ ^0 = 45°, ^0 = 30°, f = 60°, 
 ^0 = 15°, ^0 = 135°, ^0 = 120°, &c. &c. 
 
 1. -e or 45°. The sine and cosine are evidently equal. In 
 4 _ 
 
 (30.) if OM = MB = U, then OB =: -^211, and 
 
 IT .IT 1 1 .- . T _7r 
 
 COST = sin- = —y=r- = - s/2 ^^"4 ~ ^°^4 ~ ^ 
 
 2. ^0 or 30°. A right-angled triangle, having an angle of 30°, 
 6 
 is the half of an equilateral triangle. The side opposite to 30° is 
 half the hypothenuse ; hence 
 
 . TT 1 
 
 ^^^6 = 2 "^^6 
 
 5 = yi__l = lV3 tan^=;^cot^=V; 
 
 3. -e or 60°, is the complement of ^0 or 30°. Therefore, 
 
 ''''1 = 1 ^i"r-=^N/3 tan| = v3 ^«^I = :;7f 
 
 4. HLq or 15°, is the half of J or 30°. Hence, by the for- 
 1 ^ o 
 
 mula in (60.) 
 
 cos— is either --y/ 1 + -^- + -^^ 1 — - 
 
 or the same, made negative ; but it must be positive, whence we have 
 
 TT v/3 + 1 .... . TT s /3-~l 
 
 5. —0 or 7°. 30' is the half of —0 or 15°. And by (60.) 
 24 12 
 
 -^ = -y(l+5^/6-lv2) + -y(l-i^/6 + iv2) 
 
 -^ = Sv/(l + lv6-iv/2)-lv/(l-1^6 + l./2) 
 
 In this way we may successively find the sine and cosine of 
 3° 45', 1° 52' 30", 56' 15", 28' 7"-5, 14' 3"-75, 7' l"-875, 3' 30"'9375, 
 1' 45"-46875, 52"-734375, the latter angle being -8789063 of a 
 
 E 2 
 
42 FORMULA CONNECTED WITH 
 
 minute. But since the sines of small angles are nea»ly in the 
 ratio of the angles themselves, and since the ratio of magnitudes 
 are the same in whatever units they may be expressed, we have 
 very nearly, 
 
 As -8789063 of 1' : 1' : : sine of the first : sine of 1' 
 
 the fourth term of which can be found from the preceding three. 
 This method of finding the sine of one minute supposes that we do 
 not know how to express the angle of 1' in analytical units; if, how- 
 ever, we assume the results of (22.), we see that the sine and angle 
 of 1' are nearly equal, when the angle is expressed in terms of 9, 
 and, therefore, that sine 1' = -000290888, nearly. But nearly is a 
 vague term ; we must endeavour to find how nearly the sines of the 
 preceding are equal. If we look at the figure in (40), and the 
 postulate in (10.), we see that AT -f-TB is greater than the arc A B, 
 or that 
 
 AT + TB AT + OT — OA . , ArcAB 
 
 -QA- °' OA '^ S''^"*'^ '^"" "UA- 
 
 that is tan + sec — 1 is greater than 9 
 
 But sec 9 is greater than 1 ; therefore, tan 9 is greater than 9. 
 Consequently, we find that 9 — sind is less than tan — sin 0, or 
 
 Q 
 
 than sin 9 {\ — cos 9) — cos 9, or than 2 sin 9 sin'^- -;- cos0 ; or 
 
 2 sin H I sin _ 
 
 6 — sin ^ is less than 
 
 2sin0 Tsin- j 
 
 cosO 
 
 If, then, we increase the numerator of the preceding, and diminish 
 the denominator, we in both ways increase the fraction ; consequently, 
 
 26lf-V 2s\u9(smtY 
 v2/ IS greater than \ 2/ 
 
 Q06^9 COS 9 
 
 or 0— sin© is less tiian \ —5^; diminish the denominator still 
 
 further by substituting 9 for sin0, and we have finally 
 
 9 — sin0 is less than J — - 
 
 Or -000290888 — sin*000290888 is less than a fraction very near to 
 irSS^ °^ -00000000001 very nearly. 
 
TWO OR MORE ANGLES. 43 
 
 Hence, to ten places of decimals, the angle and sine of one minute 
 are the same things : that is, we may assume sin 1' = -000290888. 
 
 Now, COS0 =1 — 2sin2- ^ 1 — 2 (-) very nearly, or 1 — ^ 0'. 
 
 We have, therefore, cos 1' = 1 — J (^0002909)2 very nearly = 
 •999999958 very nearly. Knowing thus the sine and cosine of one 
 minute, we might calculate those of 2', 3' ... . l", 1^ T, . . . . up to 
 89° 59', 90", and by dividing sines by cosines, we might find the 
 tangents. After which, by taking reciprocals, we might find the 
 cotangents, &c. To put this method in practice would increase all 
 necessary difficulties some hundreds of times; but here, as in (62.), 
 we are not pointing out how a table of sines, &c. should be formed, 
 but merely shewing how it may be done, that the student may not go 
 to the tables, as to results of the possibility of arriving at which he has 
 no comprehension whatever. We can, however, make him see that 
 even the labour of this process may be materially lessened. 
 
 1. No cosines need be calculated, nor cotangents, nor cosecants ; 
 for in sin 1' we have cosine 89° 59' ; in sin 2' we have cos 89° 58' &c. ; 
 so that a complete table of sines for all minutes of the right angle 
 is also a table of cosines. For a similar reason, a table of tangents 
 is also one of cotangents, &g. 
 
 2. Half of the secants and cosecants may be formed by simple 
 addition, when the rest are known, by the formulee 
 
 *ce = i(,ania + ooll9) secO = ^^ (tan(^-|) + cot(?-^)) 
 
 which we leave to the student to prove. 
 
 (64.) Suppose, however, that our table is calculated for every mmute 
 of the right angle, it remains to see how the truth of the calculated 
 results may be verified. It must be observed, that in all mathe- 
 matical tables, the danger of an error of printing is greater than that 
 of an error of calculation. An error of the former kind is one to 
 which all places of figures are equally subject; one of the latter is 
 only to be feared in the last figures. The method of verifying a 
 doubtful figure is to calculate the function in question by means of 
 any other functions, using one of the formulae already obtained. 
 Thus, if there be a doubt about sin 16°, as given in the tables, we may 
 remember that it ought to be the same as 2 sin 8° cos 8°, and we may, 
 therefore, double the product of sin 8° and cos 8°, and compare it 
 with what is given for sin 16°. But, as multiplication and division 
 
44 FORMULAE CONNECTED WITH 
 
 are tedious operations, compared with addition and subtraction, we 
 should, for our present purpose, prefer such formulae as contain only 
 the latter pair of operations. And such formulae have received the 
 name o^ formula of verification, meaning, that they are peculiarly 
 applicable for that purpose. We shall now deduce a few of the 
 kind. Let A be an angle measured in degrees, &c. 
 
 (54.) (63.) sin(30° + A) 4-sin(30*'-- A) = 2 sin 30°. cos A = cos A 
 Again, the sine of — e, or 18<», may be tlius expressed. If 
 
 50 = — , we have 
 2 
 
 cos30 = sin20, or (59.) cos^Q — 3cos0 sin^e = 2cos0.sin0 
 divide by cos0, and substitute 1 — sin'-^O for cos^^, 
 
 1 — 4sin'0 := 2sin0; or, taking the positive value of sin d 
 
 sin — = — from which we find 
 
 10 4 
 
 Hence we find (54.) 
 
 n/5 — 1 A 
 
 sin(18° + A) + sin(18°- A) = 
 
 2 
 
 cos (36° + A) + COS (36° - A) = ^^^^ cos A 
 
 Subtract the first from the second, which gives 
 
 cos(36°+A) + cos(36°— A) = cosA + sin(18« + A) + sin(18 — A) 
 
 (65.) Before beginning to use the tables, the student should have 
 a good notion of the changes which take place in the magnitudes of 
 the several functions through the first right angle. And he should 
 also take some method of readily remembering the changes of relative 
 magnitude which take place through the four right angles. The best 
 method of doing it is by remembering the general character of the 
 forms of certain curves, which we shall presently proceed to explain, 
 first premising a more simple illustration of the method. Suppose 
 we wished to take a view of the prices of corn (the average per 
 quarter) in different succeeding years. 
 
Ia^« 15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 J 
 
 
 
 \l 
 
 v' 
 
 
 ^^^^.^^^ ♦ 
 
 /\ 
 
 /" 
 
 (J 
 
 A 
 
 
 \^ 
 
 \y 
 
 
 
 
 5r 
 
 A 
 
 
 
 If 
 
 / \ 
 
 / \ 
 
 
 
 
 
 1 
 
 1 1 
 
 Zondtn. Taylor IrWaOmi, t^ptv (towpr Strut. 
 
TWO OR MORE ANGLES. 
 
 45 
 
 Take two scales perpendicular to each other, as in the figure : let 
 AB represent a year of time, AC a pound sterling of money. If we 
 
 A B P Q 
 
 ■i 1- 
 
 begin, say at the year 1790, let B be marked 1790, P 1791, Q 1792, 
 &c. and at each point erect a perpendicular the length of which, AG 
 being twenty shillings, shall represent the price of the quarter of corn 
 for the year. We have thus a better idea of the magnitude of the 
 changes than we could get by looking at a table of prices. 
 
 Now, take a similar scale for angles and their primary functions. 
 Let angles be measured on OA, at the rate of a right angle, or 90° 
 toOA; let numbers be measured on the perpendiculars to OA at 
 the rate of O B to a unit. The curves are so drawn that if any angle 
 be laid down on OA (that is, if the proper line be measured from O, 
 which is to OA as the angle in degrees, &c. is to 90°), then the six 
 curves will cut off from the perpendicular six lines which repre- 
 sent (if OB represent 1) the sine, cosine, &c. of the angle. It is 
 now for the student to find out which is the curve of sines, which 
 that of cosines, &c., to examine them attentively, until he perceives 
 the truth of all the theorems in (32.) (39.) &c. and to remember the 
 forms of the curves in such manner that the mere words sine, cosine, 
 &c. shall call up the ideas of the variations of magnitude which are 
 peculiar to the function in question. 
 
 For instance, it is long before it is as familiar to a beginner as 
 the word cosine, that the cosine of is 1 ; the notion always being, 
 that the cosine of nothing is nothing. A recollection of the manner 
 in which the curve of cosines begins with the angle, would com- 
 pletely remove the liability to this mistake. 
 
 It would be one of the most improving exercises which the 
 student could impose upon himself, to draw a considerable number of 
 
46 FOBMUL-E CONNECTED WITH 
 
 »uch curves, provided he can obtain paper ruled * horizontaiiy and 
 vertically at small intervals. It will be quite sufficient to divide a 
 right angle into six equal parts, or to take six intervals of length for 
 the right angle. Four intervals of perpendicular length should re- 
 present the unit of the functions in question, by which means that 
 same unit will very nearly represent the analj/tical unit 9 on the line 
 on which angles are measured. Suppose, for instance, the paper is 
 ruled at intervals of a tenth of an inch. Take three inches for a 
 right angle, and two inches for a unit of sine, &c. Suppose the curve 
 required to be that which cuts off sin20 — sin0: then, taking two 
 places of decimals (which will here be sufficient) and remembering 
 that 20 perpendicular subdivisions on the paper count as 1, or each 
 subdivision as -05, divide the two places by 5, and the result is 
 the number of subdivisions. Also, five subdivisions on the line OA 
 represent 15 degrees, or each subdivision represents 3 degrees. The 
 rest of the process is as follows : 
 
 Angle A sin A sin 2 A sin 2 A — sin A Subdivisions Subdivisions 
 
 for the angle, for sin 2 A — 
 
 
 
 •00 
 
 •00 
 
 •00 
 
 15° 
 
 •26 
 
 •50 
 
 •24 
 
 30° 
 
 •50 
 
 •87 
 
 •37 
 
 45° 
 
 •71 
 
 100 
 
 •29 
 
 60° 
 
 •87 
 
 •87 
 
 •00 
 
 75° 
 
 •97 
 
 •50 
 
 — •47 
 
 90° 
 
 1-00 
 
 •00 
 
 — 1^00 
 
 105° 
 
 •97 
 
 — •50 
 
 — 1-47 
 
 120° 
 
 •87 
 
 — •87 
 
 — 1^74 
 
 135° 
 
 •71 
 
 — 1-00 
 
 —1-71 
 
 150° 
 
 •50 
 
 — •87 
 
 — 1-37 
 
 165° 
 
 •26 
 
 — •50 
 
 —•76 
 
 180° 
 
 •00 
 
 •00 
 
 •00 
 
 
 sinA. 
 
 
 
 
 
 5 
 
 4! 
 
 10 
 
 n 
 
 15 
 
 5t 
 
 20 
 
 
 
 25 
 
 -9f 
 
 30 
 
 — •20 
 
 35 
 
 — 291 
 
 40 
 
 -34J 
 
 45 
 
 -341 
 
 50 
 
 -271 
 
 55 
 
 -151 
 
 60 
 
 — 
 
 • The common ruling machine used by stationers will rule paper 
 very well to tenths of inches, with each inch-line broader than the rest. 
 Some years ago, I caused a quantity of paper to be so ruled, which is 
 still on sale with the publisher of this work. In looking at ruled paper, 
 the eye is too accurate a judge : when parallel lines are ruled close 
 together, a very trifling defect is perfectly visible. 
 
TWO OR MORE ANGLES. 47 
 
 The student should now lay down and continue this curve, and 
 others of the same kind. 
 
 {66.) Let A be a small angle, of about the value of T. It has 
 been shewn that for more than eight places of decimals, cos A = I, 
 sin A = h, and hence tan A = h. We have then, with quite sufficient 
 exactness for the tables which it is necessary to use (which never 
 exceed seven places of decimals). 
 
 sin(0 + //) = sin 9 cosA + cos0 sin A = sin0+cos0.A 
 cos {9 + h) = cos 9 cos h — sin sin h = cos 9 — sin9 .h 
 
 sin A h very 
 
 (56.) tan (0 4- A) — tan0 = 
 
 cos (9 + h) cos 9 cos^9 nearly 
 
 tan(0 + A) = tan0 H — . h 
 
 We shall leave the following to the student. 
 
 1 
 
 cot{9-\-h) = cote 
 
 sin^9 
 
 ,« .^ ^ . sind , 
 
 sec(9-\-h) = sec0H -.h 
 
 cos-'9 
 
 cosec (9 + fi) = cosec 9 r-r-r . h 
 
 ^ ' sm^9 
 
 vers(0 + ^) = vers0 + sin0 .A 
 
 covers {9-\-K) = covers 9 — cos 9 . h 
 
 That is, F 9 representing any primary function of 0, we have 
 for sin. tan. sec. or vers. F(^ + A) = F^H-MA 
 for COS. cot. cosec. or covers. F(^ + /«) = F^ — M/i 
 
 whefe M is not a function of A, but of 9 only. (Remember that 
 functions beginning with co. are all decreasing when the angle in- 
 creases, in the first right angle). Let h! = -000290888 the minute 
 expressed in analytical units ; and let h be any angle less than h! : 
 then we have 
 
 F(d + h') — Fd = Mh' ) for functions which increase 
 F(^+A) — F^ = Mh ) with the angle 
 
 whence F(^ + A) = F^ + ^,{F(^ + A') - F^} 
 
 But F{9-{-h') — F0 is the increment of the function, when the angle 
 receives an increment of one minute in value ; it is, therefore. 
 
48 FORMULAE CONNECTED WITH 
 
 immediately found from the tables, in which the values of F^O-j-h') 
 
 and ¥9 are those which follow each other, if the tables be to every 
 
 minute, as is usual. Let this increment be denoted by Dif. for 
 
 differencey as in the tables, where ti)e subtraction is made in a 
 
 separate column. And let h contain s seconds and decimals of 
 
 h s 
 
 seconds. Then will -, = — and we have (if A be the angle 9 in 
 
 degrees and minutes. 
 
 F(A + 5) = FA + ^x Dif. 
 
 and in the same way, if F0 be a function which decreases when 
 increases, we have 
 
 F{d+h) or F(A + 5) = FA-^ X Dif. 
 
 where Dif. now stands for FA — F(A + 1') and is taken from the 
 tables. 
 
 The preceding is, in a more exact form, the representation of a 
 notion which may be more easily given. If we ask what is that 
 function of A which increases uniformly when h increases uniformly 
 (the more easy phrase is, which grows at the same rate as long as 
 h grows at the same rate) the answer is, that the function can only be 
 P-f- MA where P and M are independent of A. In this function, if 
 for h we write successively 
 
 h, h-\-t, h-\-2t, h-^Sty &c. 
 
 the values of the function are 
 
 (P + M/i), (P-hMh) + Mt, (P + MA)-f2M^, &c. 
 
 and, similarly, the function P — MA is of the only form which de- 
 creases uniformly when A increases uniformly. If, then, there be a 
 function of A which does not increase uniformly when A undergoes 
 considerable changes of value, but which increases very nearly uni- 
 formly when A is small, and undergoes small changes; that is, is very 
 nearly equal to some form of P 4- MA; the consequence is, that we 
 may for very small values of A, and very small changes, treat llie 
 function as if it were one which increased uniformly. 
 
 To illustrate this, I take out of the table the cosine and sine of 
 6% 6° 1', 6° 2', and 6° 3', as follows : 
 
TWO OR MORE ANGLES, 
 
 49 
 
 Angle 
 
 Sine 
 
 Dif. 
 
 Cosine 
 
 Dif. 
 
 6°0' 
 
 •1045285 
 
 •0002893 
 
 •9945219 
 
 •0000305 
 
 6°!' 
 
 •1048178 
 
 •0002892 
 
 •9944914 
 
 •0000305 
 
 6° 2' 
 
 •1051070 
 
 •0002893 
 
 •9944609 
 
 •0000306 
 
 6° 3' 
 
 •1053963 
 
 
 •9944303 
 
 
 Hence, at and near 6° 1', the sine is a function which increases 
 uniformly at the rate of -0002893 per minute of angle, and the co- 
 sine diminishes at the rate of -0000305 per minute. This uniformity 
 of increase or decrease, which obtains when the angle changes through 
 successive minutes, will, d fortiori, still remain when the angle 
 changes from second to second in the interval between two minutes. 
 That is, we must, to find the sine of 60° 1' 37"^5, add to sin 6° V such 
 a part of -0002893 as 37^ is of 60, and to find cos6° 1' 37"-5 we 
 must subtract from cos 6° 1' such part of -0000305 as 37^ is of 60. 
 The process may be performed either by common multiplication or 
 division, or in the manner of the rule called practice in commercial 
 arithmetic, as follows : 
 
 2893 
 37 
 
 20251 
 8679 
 
 107041 
 1447 
 
 6,o) 10848,8 
 
 sin 6" V 
 
 = ^1048178 
 
 proportional part ^^Q^Q^gQg 
 
 for 37"^5 
 
 sin6°l'37"-5 = -1049986 
 
 for 60 2893 
 
 for 30 i 1447 
 for 6 \ 289 
 for 1 I 48 
 for '5 ^ 24 
 
 for 37-5 1808 
 
 1808 
 
 305 
 37 
 
 2135 
 915 
 
 11285 
 153 
 
 6,o) 1143,8 
 
 cos 6° 1' = 
 
 proportional part 
 
 for 37^5 ~ 
 
 cos6°l'37"-5 = 
 
 •9944914 
 — •0000191 
 
 •9944723 
 
 for 60 305 
 
 for 30 ^ 153 
 
 for 6 ^ 31 
 
 for 1 i 5 
 
 for -5 ^ 3 
 
 for 37-5 192 
 
 191 
 
50 FORMUL-aS CONNECTED WITH 
 
 Nothing will secure accuracy to a single unit in the last place of 
 tables, which are, therefore, always carried a unit further than would 
 otherwise be requisite. 
 
 The inverse process to the preceding follows immediately from 
 it. Given FA, F(A. +s), and F(A + 1'), required s. Let P be the 
 given value of F (A + «) > then we have 
 
 p = F A -I- ^ Dif. Dif. = F(A + 1') - FA 
 
 (P — F A) 60 
 ' = Dif. 
 
 But, if FA be a decreasing function, we have 
 
 P = FA-±Dif. .= (IA=P)«2 
 
 60 Dif. 
 
 Thus, suppose it is required to find the angle which has •1052111 
 for its sine, and also that which has '9945000 for its cosine. 
 
 P -1052111 P -9945000 
 
 sin 6° 2', or FA -1051070 cos 6°, or FA -9945219 
 
 P — FA 
 
 1041 
 
 FA- 
 
 -P 
 
 219 
 
 60 
 
 
 
 60 
 
 2893)62460(21-6 
 
 305)13140(43-1 
 
 5786 = s 
 
 
 
 1220 
 
 4600 
 
 940 
 
 2893 
 
 
 
 915 
 
 17070 250 
 
 Angle required 6** 2' 21 "-6 Angle required 6° 2' 43"- 1 
 
 (67.) We must proceed exactly in the same manner with the loga- 
 rithms of the primary functions ; and the law of the increase or decrease 
 may be exhibited as follows. If we have (making ^ = -43429 . .) 
 
 F(A-f s) = FA + ^-Dif. this gives logF(A-f5) = log(FA+~-Dif.) 
 
 = logF A H- log(l + ^ ^0 = logFA-f ± ^-^ nearly. 
 (See Algebra, pp. 226 and 237.) But by making s = 60", or 1', we 6nd 
 
 iogF(A + 1') = logFA + '^ or (^ = iogF(A + 1') - logFA 
 
TWO OR MORE ANGLES, 51 
 
 therefore /A. Dif. -7- FA is what is found in the column of differences 
 of the logarithms, and may be taken from the tables. We have then 
 
 logF(A + S) = logF A + ^ X Dif. of log 
 
 or we may remember that the reasoning in page 48 applies to any 
 functions which increase continuously, and, therefore, to the loga- 
 rithms of the primary functions, as well as to the functions themselves. 
 (68.) It depends, then, upon the amount of the difference between 
 F(A+l') and F(A) whether we can pretend very nearly to find the 
 angle which belongs to any intermediate between them. Look, for 
 instance, at the beginning of the table of logarithmic cosines, which 
 by the arrangement of the tables is the end of the table of logarithmic 
 sines. We have for instance, 
 
 log. cos 0° 6' = -9999993 r^r^r^c 
 
 4.cosO°r= -9999991 Dif- = -0000002 
 
 We cannot, out of this 2 of difference, make different cosines for 
 every second between 6' and 7'. The log. cosine here is increasing 
 so slowly, that many successive increments of 1" will not make it 
 shew a difference of a unit in seven places of decimals. We come to 
 2° 41' before an angle increased by 1" has its log. cosine increased 
 by '0000001. And if log. cosines are to shew tenths of seconds, 
 that is, if the log. cosine is to increase so rapidly that 0"*1 added to 
 the angle shall make a difference in seven places of decimals, the 
 angle must be upwards of 25°. But when we come to tangents of 
 angles very near 90°, we find that the preceding method fails, because 
 the increases of the log. tangent for successive increases of the angle 
 are far fjom uniform. Consequently, when the angle to be found is 
 small, avoid expressing it hy means of its cosine, if possible ; when it is 
 nearly a right angle, avoid its sine and tangent, if possible. In the 
 case of a tangent which is very great, denoting an angle near 90% 
 proceed as follows. Let tan A = a, a being a considerable number, 
 so that the angle is nearly a right angle. Remember that {56.) 
 
 tan(A-45°)=^""t~'- =^^ 
 ^ ^ tanA-fl a+1 
 
 Find, not A, but A — 45°, from this formula, and the difficulty will 
 disappear; for near 45° the increase of the tangent is very nearly 
 uniform, and also that of its logarithm. For instance, I wish to 
 
52 TORMVLM CONNECTED WITH 
 
 know, with great exactness, the angle whose tangent is 3000. On 
 looking at the tables I see that it is between 89° 58' and 89° 59', but 
 on examining the increase of the tangent, I see as follows : 
 
 tan 89° 57' 1145-9 ^,. ,^, ^^ 
 
 „ , Dif. = 573-0 7 not nearly 
 
 tan 89° 58' 1718-9 f- ^ 
 
 tan 89° 59' 3437-7 Dif. = 1718-81 equal 
 
 But, the angle wanted diminished by 45° has for its tangent, 
 
 2999 
 -3^ = -9993336 
 
 tan 44° 58' -9988371 
 
 
 4965 
 
 
 60 
 
 
 Dif. = 5813)297900(51-3 
 
 29065 
 
 
 7250 
 
 
 5813 
 
 A— 45°= 44° 58' 51"- 3 
 
 14370 
 
 A = 89° 58' 51"-3 
 
 (69.) Suppose cos A = a, a being very near unity, or the angle 
 very small. We have then 
 
 sm'lA = i(l-a) sinlA=yiHf 
 
 which may easily be calculated by logarithms ; and from i A, A can 
 be found. Similarly, if we have sin A = a, where a is very near 
 unity, we have 
 
 1-a = l-cos(90°-A) = 2sin«(45°-lA) 
 
 sin (45°- 1 a) =\/^ 
 
 and from 45° — i A, A can be found. 
 
 (70.) We have seen that the cosine approaches very near to unity 
 when the angle is small ; so near that 1 — cos 9 is a small quantity by 
 the side of 9 itself, when B is small (48.). But our future purposes 
 will require a theorem which we shall introduce here, to give a further 
 notion of the rate at which cosO approaches unity. If we take an 
 
 Q Q 
 
 angle 0, less than -, and form the series 0, -, -, &c. we have a 
 2 2 3 
 
TWO OR MORE ANGLES. 53 
 
 series the terms of which diminish without limit. If we then 
 take 
 
 COSO, cos-, cos-, COS-, &c. 
 
 we have a series of terms approximating without limit to unity. Let 
 US now take 
 
 COS0, lcos-1 v^^^qJ V^°^I/ 
 
 We know that the powers of a fraction less than unity decrease, and 
 without limit as the exponents increase {Algebra, p. 159), that is, if 
 the fraction operated upon remain the same. But here we have 
 in passing from 
 
 Q 
 
 cos 9 to cos -, increase 
 n 
 
 from cos- to icos-j decrease 
 
 'n V 72/ 
 
 the question is, as n grows greater and greater, which will pre- 
 
 cos-1, by the de- 
 crease which takes place in raising the power, tend to a limit less 
 than unity, or may it be brought as near to unity as we please. 
 To try this, wiite 
 
 cos-j in the form M — 2sin2 — J 
 
 and remember that if 2 f sin— -j =2/a( — J then /a continually 
 
 approaches to unity as n is increased. Substitute, which gives 
 {Algebra, p. 209, and p. 218, for a similar process) 
 
 \ nJ \ 2 rrJ 2 /r 2 4 n^ 
 
 every term of which, except the first, diminishes without limit when 
 n increases without limit ; for Q remains the same, fi approaches to 
 unity, and n has increase without limit, causing the same in all the 
 denominators. Hence 
 
 cos- ) has the limit 1, the same as that of cos- 
 nJ n 
 
 F 2 
 
(71.) We shall now examine the limit of (cos — f- Asin- j under 
 
 \ n nj 
 
 54 FORMULAE CONNECTED WITH 
 
 ; o! I cos — f- A sin 
 \ n 
 
 like circumstances. It is evident that 
 
 f ^ . 7 • 0\" f 0\" f^ . 7 0\" 
 
 I COS — h ksin-j = Icos- ) (I + Atan-1 
 
 for (p + q)'= /'"(l+f)" and i^^ = tan 
 
 in which product the first factor, as just shewn, has unity for its 
 
 limit, and we must examine that of the second factor. If we make 
 
 
 
 tan- = yo.- then (48.), as n increases without limit, the limit of ^ 
 
 is unity. Substitute and develope by the binomial theorem, which 
 gives (as in AlgebrUf p. 218), 
 
 1__L 1-il-i 
 
 = l + f^.kd+-^f.^k'6'+ -^^f,^.k'^+ .... 
 
 Take tlie limit of both sides (limit of ^ = 1) 
 
 Limit of (l + *tan?)"= 1 + ftO + i^' + |f + .... = e*' 
 
 or Limit of (cos- + A sin-) = g*^ 
 
 \ n n/ 
 
 cos- + Asin-J = g*^ 
 
 1. is never absolutely true; 2. is very nearly true, if n be great; 
 3. can be brought as near to truth as we please, by making n suffi- 
 ciently great. Extract the arithmetical wth root {Algebra, p. 110) of 
 both sides, and we have, as nearly as we please 
 
 cos- -h/dsin- = g " I „r Jcosw + Asinw = g 
 
 if u may be as small as we please. 
 
 I cos &i 4- /f.sinr.1 = ) 
 n n ( ^^ 
 
 if n may be as great as we please. J 
 
 Observe that this is independent of the value of k. In the last form 
 the result is easy to establish, for when ^ is small coSft» + /csin« is 
 nearly 1 +/c«, which {A/gebruy p. 187) is by much the greater part 
 of the developenient of s **. This process, therefore, is one more 
 experience of the confidence to be placed in the developenients of 
 (1 +.i)" ^"^ ^* {Algebra, c. xi, xii.); and also suggests the propriety 
 
TWO OR MORE ANGLES. 55 
 
 of examining further the form cos0 -|- /csinO. Multiply two such 
 forms together, and we have 
 (cose + A;sin0)(cos0'+/csine') = cos0 cosO' + A:=^sin0sin0' -|-/csin(0 + 9') 
 = cos e cos e' — sin sin 0' + (1 + k^) sin sin 0' + k sin (0 + 0') 
 = co3(0 + 0') +ksm{e-\-9') + (1 +/c2)sin0sin0' 
 or, if this function of 0, cos0 + /csin0 be denoted by y 0, we have 
 
 /0X/0' =/(0 + 0') + (l+/c')sin0.sin0' .... (1) 
 Multiply again hy f9" or cos9" -{- ksin9", which gives 
 /0 x/0' Xf9" =/(0 + 0' + 0")+(l +/c2){sin0"sin(0-f0') +sin0.sin0'/0''} 
 since by (1) /(0 + 0')/0" = /(0 + 0' + 0") + (1 + k^) sin(0 + 0') sin0" 
 
 By proceeding in this way we see that if we multiply together y0, 
 J'9', f9", .... we have an equation of the following form : 
 
 f9.f9'.f9" .... =/(0 + 0' -1- 0" + . . . . ) + (1 + /c^) V 
 
 where V is a function of the angles, and of k. This factor 1 + /c^, 
 might be made to simplify the expression materially, if there were 
 such a value of k as that 1 + /c' should be = 0, but there is evidently 
 no such algebraical value, positive or negative, for k^ is always po- 
 sitive, and 1 + k^ greater than 1 . We shall hereafter see the conse- 
 quences of this hint, but we shall leave this formula for the present. 
 As far as we have yet proceeded, every thing seems to render it most 
 likely that, if any function of sines and cosines be identical with a 
 .function of common algebra, it is of the form cos0 -l-/csin0, which, 
 
 though not found to be such, is very nearly represented by s , 
 when is small. To try this supposition, let us (as an experiment) 
 
 make cos ^ + ^ sin ^ = g for all values of 0, 
 
 merely to see whether the consequences coincide with those already 
 obtained or not. Then, if this equation be universally true, we have, 
 writing — for (44.), 
 
 COS0 — ycsin0 = g~^^ = -;^.^. Let S = j? 
 
 Then 
 
 2cos^ = x-\-- 21i^\nQ =z X 
 
 .1- X 
 
 Again, 
 
 cosnd + ksinnd = g^'*^ = (g*^)" =x' 
 
 
 cosn&^k»mnd = i'^'-^ = (g"*^)" = ~- 
 
66 FORMULA CONNECTED WITH 
 
 Whence 2cos7i^ = a;" + - 2ks\i\nO = ^— > 
 
 (72.) Let us try some properties of the sine and cosine witli 
 these supposed values. 
 
 cos 2 = 2 cos'* 0—1 or 2 cos 2 = (2 cos ey — 2. 
 
 If our preceding equations be correct, we should have, 
 
 x^ -] — „ = (.r H- -) — 2; but this is true, therefore 
 this case does not contradict our assumptions. 
 Again sin 20 = 2 sin cos or 2 /c sin 20 = (2 fc sin 0X2 cos 0) 
 
 But x^ 2 .= (^ )\^~^J ^'"^ ^*^^* therefore, is 
 
 no contradiction. 
 
 Again, cos20 = 1— 2sin20 2k'^cos29 = 2/c2— (2;£sin0)« 
 
 But 2h^cos2Q = k''x^-\-—^ 
 
 X* 
 
 2k^-i2ksm()f = 2A^-(a;-i)' = 21i^ -^2-x^ -^y, 
 
 to equate these two is therefore to make /c'-f 1=0, which cannot be. 
 We may next prove that all the equations contained in 
 
 2cosw^ = a;" -I- —■ 2Asinn^ = a:" 
 
 x^ a» 
 
 for all whole values of n from upwards, miist be true if the two 
 first are true; namely, 
 
 w = 2 = a;°H — 5 2AsinO = x° 5 which are true 
 
 X X 
 
 1 1 
 
 71 = 1 2cos(? =z X -\- - 2hsm& =. X (A) 
 
 X X ^ ' 
 
 For it is readily shewn that the truth of any two of these equations 
 involves the truth of the next, as follows. Let a and a H- 1, any 
 two successive values of n, give true results, that is, assume 
 
 2 cos a ^ = a; -'- -r 2 /2 sin a ^ = a: 
 
 2cos(a + l)^ = x^^ ■\-\^^ 27dsin(a + l)^ = ./"^^-L^ 
 
 -, (a + 2)0 + fl0 . , ,. . (rt+2)0 — a0 
 Now, ^^ — ! — ^— ! = (a 4- 1 ) ^ ^ — ■ — '- = d 
 
TWO OR MORE ANGLES. 57 
 
 (54.) cos(a + 2)0 +cosa0 = 2cos(a + l)0.cose 
 sin {a -\- 2) 9 + sin a e z=. 2 sin {a -\- 1)9 cos 9 
 Therefore 2cos(a + 2)^ = 2cos(« + 1)^ .2cos^ — 2cosa^ 
 
 __ «+2 1 
 
 And 2Asin(a + 2)^ = 2A.sin(a + 1 )^.2cos^ — 2^sina^ 
 
 a+2 1 
 
 From these it appears that the third follows from the two first; 
 the fourth, from the second and third ; the fifth, from the third and 
 fourth, with the second, &c. 
 
 We now try the equation 
 
 (cos^ + sin^)2 = 1 +sin2^ 
 or (^.2cos^ + 2^sin^)2 = 4^^+ 2A.2Asin2^ 
 
 (/- + '+-■- iy = 4^^ + 2^ (.^^ 1-0 
 But {{k + l)a: + ~y = (k + lfx'' + 2ik^-l)+^-!^ 
 
 which becomes identical with 4k^ + 2k(x^ ^j only on the im- 
 possible supposition of 1 -}- /c'^ = 0. 
 We shall try one more case. 
 
 Since .t" = cos n + ^ sin n 9, and x = cos + /c sin 9, we have, 
 cos w -f- ^ sin w = (cos 9 -f- /c sin 9)" = (c + ft s)« 
 or cos 2 H- ft sin 2 = c2 -f- 2 /c c s + ft'^s^ = (c^-f 2 ft c s — s^) 
 cos 30 + ftsin30 = c^^- 3ftc=*s + 3ft2cs'^+ ft^s* 
 = (c^ + 3 ft c^ s — 3 c s'^ — ft s3) 
 COS40 + ftsin4 = c^ + 4ftc3s +6ft'c*s=' + 4ft''cs3-|-ft«s* 
 = (c^+4ftc3s — ec^s^ — 4ftcs3 4-s*) 
 
 The values in parentheses are those already found in (59.), 
 simply multiplying the sines by ft, and forming cos n -f ft sin n 0, 
 
58 FORMULiE CONNECTED WITH 
 
 case after case, altering only the order of the terras, so as to put them 
 under those which they resemble in the results of our present method. 
 We see, then, that the results of our present method coincide with 
 those of (59.), by writing —1 for /:», -—k for /r*, 1 for k\ &c., 
 which are all algebraical consequences of the single assumption, 
 A:' = — 1, which gives k^ = —k, k^ = —k^ = 1, /c* = k, &c. 
 And from all that has preceded, we deduce the following remark, 
 which we have as much reason to suppose generally true, as instances 
 can give. 
 
 (73.) The functions a: + - and x have properties corresponding 
 
 in all respects to the trigonometrical functions 2cos0 and 2fcsind; 
 and such that, under the following limitations, the properties of the first 
 may be deduced from those of the second. If an equation which is 
 true of the first functions, undergo substitution of the second for the 
 first, then, if the result do not contain k at all, it is absolutely true of 
 the second ; but if it contain powers of k it is never true of the 
 second. Nevertheless, it becomes true of the second, if for the set of 
 even powers of /c, namely, /c', k*, k^, &c. we substitute — 1, -f-1, 
 — 1, &c. and for the set of odd powers of k, namely, k, /c^, /c^, /c', &c. 
 we substitute A;, — k, -\-k, — /c, &c. ; in which case, the part inde- 
 pendent of k on one side is equal to the part independent of k on the 
 other, and the coefficient of k on one side equal to the coeflficient of k 
 on the other. And the representative of x is cos9 -j-k sin 0, and 
 
 also e'^^ 
 
 Nevertheless, 2cos9 z=z x -\-- is an impossible equation, except 
 
 only when j: = 1, cosO = 1. For, whereas 2 cosO is never greater 
 
 than 2, x 4- - is never less than 2. For, if x + - were less than 2, 
 
 X X 
 
 jt" -+- 1 would be less than 2x, or x^ — 2x + 1, a square, would be 
 negative. And, in fact, if we solve 
 
 ;c + - = 2cos^ we find x = cos^i k/ — 1 sin^ 
 
 X ^ 
 
 X =: 2/isin^ gives X = Asin^± VA^sin'^ + 1 
 
 X 
 
 which agrees in form with the preceding only when /c*" = — 1. 
 
 We have thus laid the foundation of the application of a more 
 
TWO OR MORE ANGLES. oy 
 
 abstruse analysis to the primary functions of an angle. We shall 
 first consider the application of our formula to the solution of 
 triangles, as it is called, that is, the determination of the remaining 
 parts of a triangle, when enough are given to distinguish it from 
 all others. 
 
60 ON THE SOLUTION OP TRIANGLES. 
 
 CHAPTER III. 
 
 ON THE SOLUTION OF TRIANGLES. 
 
 (74.) Let oU, 6U, cU, be the sides of a triangle, U being any given 
 linear unit, and let A, B, C, be the opposite angles, expressed in 
 degrees, minutes, and seconds. When one of the angles is a right 
 angle, it is evident that the tables of sines, cosines, &c. are nothing but 
 registers of the proportions of the sides of such a triangle. Knowing, 
 therefore, any one side, and an angle, we look to the table for the pro- 
 portion of the other side to it, preferring, of course, the logarithm of 
 the proportion for convenience of calculation. 
 
 (75.) The best tables in common use are those of Hutton, which 
 maybe procured of any bookseller. The arrangement by which the 
 sine of 18°, for instance, is prevented from being again printed as the 
 cosine of 72°, will be better understood by consulting the table (and 
 remarking the description of the functions at the top and bottom of 
 the page, and the reckoning in minutes downwards on the left hand, 
 and upwards on the right) than by any explanation. 
 
 But the following point requires some notice. In every mathe- 
 matical table which contains both positive and negative quantities, 
 there is such a liability to error in taking out the signs, that it is 
 most useful, and almost necessary, to form the table in such a way 
 that all shall have the same sign. Suppose, for example, that the 
 following table was in frequent use. 
 
 + 6, +4, -3, +2, -10, -1, +8, -11. 
 
 Now 12 being greater than any one of these, add 12 to each, 
 which converts the table into 
 
 4-18, +16, +9, -1-14, +2, -t-11, +20, +1. 
 
 Every result in this table is to be added ; but 12 is to be sub- 
 tracted whenever the table is used. There is always both an addition 
 and a subtraction ; the sign of the table will not be liable to be read 
 wrong, and the correction of the table is uniform — always a sub- 
 traction. 
 
ON THE SOLUTION OF TRIANGLES. 
 
 61 
 
 The trigonometrical tables consist of — sines and cosines with 
 logarithms always negative — tangents and cotangents with the same 
 sometimes positive and sometimes negative — and secants and cose- 
 cants with logarithms always positive. The plan which is followed 
 is to add 10 to every logarithm in the table, without exception : 
 
 so that, 
 
 True log. FA = Tabular log. FA — 10 
 
 Tabular \og. FA = True log. FA + 10 
 
 For instance, sin 30o = a log. sin 30° = — -3010300. 
 
 In the tables we have, 10 — -3010300 or 9-6989700. 
 
 (76.) The formulae for the solution of right-angled triangles, are 
 as follows : 
 
 Let C be right angle, cV the hypothenuse ; then we have, 
 
 - = sinA = cosB 
 = tanA = cotB 
 
 C2 = tt2 _|. J2 
 
 a = csinA = ccosB 
 a = JtanA = Z>cotB 
 
 b = ^{c^a , c + a) 
 But the following formulae should be remembered in words. 
 side = hypothenuse into sine of opposite angle 
 side =z 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 
 (77.) The following are the cases which may occur, and the loga- 
 rithmic equations for the solution (L sin, &c. mean tabular log. sin, 
 &c. ; side, or angle, in Italics, means given side). 
 
 1. Given the hypothenuse and a side ; required the rest. 
 
 log. remaining side = J (log. h^p. -|- side -f- log. hj/p. — side) 
 L. sin. angle opp. side = 10 -}- log. side — log. hj/p. 
 angle opp. other side = 90° — angle opp. side 
 
62 ON THE SOLUTION OP TRIANGLES. 
 
 2. Given the hypothenuse and an angle ; required the rest. 
 
 log. side opp. angle = log. hj/p. -\- L. sin. angle — 10 
 log. side adj. angle = log. hyp. + L cos. angle — 10 
 other angle = 90° — angle 
 
 3. Given a side and an angle; required the rest. 
 
 Other angle = 90° — angle 
 
 log. hyp. = 10 + log. side — L sin /. opp. side 
 log. other side = log. side -f L tan /. adj. side — 10 
 
 4. Given the two sides; required the rest. 
 
 L tan. AN ANGLE ^ 10 + log. ITS opp. side — log. other side 
 Other angle = 90° — the angle found 
 
 log.hyp. = 10 + log.^SZD£ — L. sin. irS opp. angle. 
 
 (78.) There are two cases in which the sine and tangent of an 
 angle are severally to be found ; from two equations of these forms 
 
 sin A := - tan A = r 
 
 c b 
 
 If in the first case - be very near unity (68.), use the equation 
 o 2/'90°— A\ c — a . (..o A\ I c — a 
 
 If, in the second case, b be very small in comparison of a, use 
 
 tan A — 1 a — b ,. aho^ « — b 
 
 r— — : = — r-r or tan(A — 45°) = --r-; 
 
 tanA + 1 « + /> ^ ^ a-\-b 
 
 When a side (6) is given and a very small adjacent angle (A), 
 the hypothenuse may be determined by its excess above the given 
 side (which is small) as follows : 
 
 J. ^ J. 7.(1 — cosA) oi. • a-'^ 1 
 
 c — 6^ = — = 0-— » — = ZosiTv' —- very nearly. 
 
 cos A cos A 2 "' •' 
 
 (79.) Given the hypothenuse and the sum of the two sides ; 
 required the rest. 
 
 (a 4- J)2_c2 = 2ah = 2c2sinA.cosA == c«sin2A 
 
 L.sin2A = 10 + log(« -i- 6h-c) +log(a + ^ — c) — 2logc 
 
 Given the excess of the hypothenuse over a side, and the difference 
 of the angles ; required the parts of the triangle. 
 
ON THE SOLUTION OF TRIANGLES. 63 
 
 Let c-b = h, A-B = M, A + B = 90°, 
 
 . 90°+ M -D 90— -M , A.cosA „ 
 
 A = — - — , B = — — , b = — — ^ &c. 
 
 (80.) From the following table, in which all tlie parts of a right- 
 angled triangle are given, any data may be chosen, and the preceding 
 formulae verified. 
 
 c = 128-4327 logc = 2-1086756 
 5= 66-1364 log5= 1-8204405 
 a = 110-0951 log« = 2-0417681 
 
 A = 59° 0' 2r''25 B = 30° 59' 38"-75 
 
 L. sin A = LcosB = 9-9330925 
 Loos A = LsinB = 9-7117649 
 LtanA = LcotB = 10-2213276 
 
 (81.) We shall now proceed to the cases of oblique-angled triangles. 
 The three angles are connected by any of the following relations. 
 
 C 
 
 A -I- B + C = 180° A + B = 180°~-C - ^ = ^^"^ ~ ? "^^■ 
 
 sin(A + B) = sinC, cos(A + B) = — cosC, tan(A + B) = — tanC &c. 
 sini(A +B) =cos^C, cos^A + B) = sin ^C, tani(A + B) = cot ^C &c. 
 Again, sin''(A + B) = sin^A .cos'*B + cos^'A.sin^B + 2sinAcosAsinBcosB 
 for cos'^A and cos'B write 1 — sin^A and 1 — sin^B, which gives 
 sin'(A + B) = sin^A ^sin^B + 2sinA.sinB(cosAcosB — sinA.sinB) 
 
 But, sin(A + B) = sinC cos(A + B) = — cosC 
 
 or sin'C = sin'^A + sin^B — 2 sin A sin B cos C 
 
 Similarly, sin'B = sin'^C + sin^A — 2 sin C sin A cos B 
 sin'^ A = sin^ B + sin^C — 2 sin B sin C cos A 
 
 A • X ^ A , T^\ . ri tan A + tan B 
 
 Agam, tan(AH-B) = — tanC = . : — 4—;^ — r> 
 
 " ' \ 1 y 1 — tan A. tan B 
 
 or tan A + tan B + tan C = tan A . tan B . tan C 
 
 (82.) Let pU, gU, and rU, be the three perpendiculars let fall from 
 the vertices of the triangle upon the sides a U, 6U, and cU. And 
 let A', B', and C be the exterior angles of the triangle adjacent to 
 A, B, and C. Then, if A be an obtuse angle, there are right-angled 
 triangles, having for hypothenuses fcU and cU, and for sides opposite 
 
64 ON THE SOLUTION OF TRIANGLES. 
 
 to A', r U and ^ U. If A be a right angle we have 6 = r, and c ■=. q. 
 If A be an acute angle, we have right-angled triangles having for 
 hypothenuses h U and cU, and for sides opposite to A, rU and q\3 . 
 And the same of B and C. Again, A + A' = 1 80°, B + B' = 180°, 
 C -t- C = 180°, and we have 
 
 /? = 6 sin C or h sin C = 6 sin C, in both cases 
 j) = c sin B or c sin B' = c sin B, in both cases 
 
 That is, 6 sin C = c sin B or - = -r— p; 
 
 c sinC 
 
 „. ., , c sinC , a sin A 
 
 Similarly, - = - — - and - = —. — - 
 
 '' « sinA sin B 
 
 That is, any two sides are proportional to the sines of the opposite 
 angles. This is the formula upon which all others relative to 
 triangles will be made to depend. 
 
 (83.) Divide both sides of the value of sin^C in (81.) by sin'C ; 
 substitute the ratios of the sides for those of the sines of angles, and 
 we have 
 
 1 = -„ 4- -„ — 2 - . - . cos O 
 c^ c* c c 
 
 or c^ = a^ + ^2 _ 2ab cosC, cosC = ^A^ 
 
 2ao 
 
 Similarly, L^ z= c^ + a^—2cacos'B, cosB = ^'"^^'"^^ 
 
 a^= b^ + c^-2bc cos A, cos A = ^'"t^""' 
 
 26c 
 
 These formulae may be readily deduced from the triangle itself, 
 it being obvious that/) = 6sinC, and also that 
 
 c2 = (isinC)2 + (« - Z»cosC)2 
 
 (84.) We now proceed to put the preceding expressions in a 
 form convenient for logarithmic computations : 
 
 (a + 6)«--c«— 2a6 (a + 6)«— c» , 
 
 COSO = — I ^ r — 7 1 
 
 2ab 2ab 
 
 Also cosC = ^^ ^ . = -^^ — r-n: H 1 
 
 2ab 2ab 
 
 1 + eosC = C+t+^Xa+t-c) 1 _ ,„3C = (e+a-yc+t-g) 
 2ab ^ab 
 
ON THE SOLUTION OF TRIANGLES. 65 
 
 Let a-hh + c = 2s then a + h — c = 2(5 — c) 
 
 h^c — a = 2(5— «), c + a — b = 2(5— J) 
 
 Substitute these, and also the values of 1 -|-cosC, &c. which gives 
 
 2 ab 2 ab 
 
 Similarly cos^- = -^ -^ sin'*- = ^ — 
 
 '' 2 ac 2 ac 
 
 A s(s — a) . „A (s — b){s — c) 
 
 2 s(s — «) 2 s{s — b) 2 s(s— c) 
 
 . . 2V . -p 2V . ^ 2V 
 
 sin A = -7— sinrJ = — sinO = — r 
 
 DC ac ab 
 
 where V = ^{5(5 — a) (5 — &) (s — c)} 
 
 A A 
 
 This is derived from the preceding, by aid of sin A = 2 sin — cos — . 
 
 sin A sin B sin C 2V 
 
 a b c ab c 
 
 ^fecsinA = ^casinB = -^rt6sinC = V 
 
 (85.) To reduce c' = a^ + t^ — 2fl6cosC to a form adapted for 
 logarithmic computation, proceed as follows : 
 
 1. c^ = (fl + /;)2_2a^'(l+cosC) 
 
 = ^a + bf-4abcos^^^ = (« + 6)^fl_ _i^^cos=^} 
 
 Now, (a -\-by — 4nb being (« — by is positive ; therefore (o -f-^)' 
 
 is greater than Aab and 
 
 4ab , .„ Aab ,C . , 
 
 and still more -7 — r-rz-^cos^-r is less than 1. 
 
 (a + by {a + by 2 
 
 Compute the positive square root of the last expression, and find 
 in the tables the angle of which it is the sine ; or find K fiom 
 
 . ^, 2 \/aT7 cos A C 
 
 sinK = -r 
 
 a ■}- b 
 
 Then c^ = (o + &)2 jl — sin^MJ or c = (« + />) cos K 
 2. c2 = (a-by + 2ab {l-cosC) 
 
 G 2 
 
66 
 
 ON THE SOLUTION OF TRIANGLES. 
 
 Compute K' from tanK' ^ Wa6sin^C 
 
 a — 6 
 
 Then 
 
 c2 = (« — i)2|l-ftan«K'| C = 
 
 a — b 
 cosK' 
 
 (86.) Lastly, ^=^ gives (54.) and (81.) 
 
 a — b __ sinA — sinB __ tanJ^A — B) __ tan j (A—B 
 a-\-b ~ sinA + sinB ~ tan ^A + B) ~ cot i C 
 
 or tanA(A— B)= ^^cotiC 
 
 ^ ^ a-\-b 
 
 (87.) The preceding formulae are sufficient for our general purpose. 
 We shall now proceed to different cases. 
 
 The student may verify the methods as they are produced upon 
 the sides, &c. of the following triangle : 
 
 a = 15-236 loga = 11828710 s — a = 3-098 log(s — a) = 0-4910814 
 b = 12-414 log* = 1-0939117 s — b = 5-920 log(s— 6) = 0-7723217 
 c = 9018 lege = 0-9551102 s — c = 9-316 log(s — c) = 0-9692295 
 
 s = 18-334 logs = 1-2632572 V = 55-96866 logV = 1-7479449 
 
 a-^-b = 27-650 log(a-f-fe) = 1-4416951 a — b = 2-822 \og(a — b) = 0-4505570 
 b-\-c =21-432 log(6+c) = 1-3310627 6 — c = 3-396 log(6— c) =0-5309677 
 «-}-c = 24-254 log (a+c) = 1-3847834 a — c = 6-218 log(a — c) = 0-7936507 
 
 A 
 
 = 89 9 
 
 B 
 
 = 54 33 
 
 C 
 
 = 36 17 
 
 iA 
 
 = 44 34 
 
 iB 
 
 = 27 16 
 
 iC 
 
 = 18 8 
 
 KA-B) 
 
 = 17 17 
 
 KB-C) 
 
 = 98 
 
 i(A~C) 
 
 = 26 26 
 
 Ka 
 
 = 44 41 
 
 IH 
 
 = 59 12 
 
 Ke 
 
 = 70 57 
 
 K'a 
 
 = 77 7 
 
 K'6 
 
 = 59 56 
 
 K'c 
 
 = 71 45 
 
 23-54 
 25-12 
 11-48 
 41-77 
 42-56 
 35 74 
 59-21 
 682 
 603 
 30-6 
 50-4 
 53-6 
 15-5 
 28-9 
 50-9 
 
 L.sin. 
 
 9-9999530 
 9-9109937 
 9-7721922 
 9-8462647 
 9-6611649 
 9 4933102 
 9-4722989 
 9-2007551 
 9-6485379 
 9-8471366 
 9-9340361 
 9-9755783 
 
 L. COS. 
 
 8-1679268 
 9-7633479 
 9 9063714 
 9-8526583 
 9-9487989 
 9-9778520 
 9-9798951 
 9-9944564 
 9-9520365 
 9-8518083 
 9-7091283 
 9-5134140 
 9-3408970 
 9-6997390 
 9-4954464 
 
 L.tan. 
 
 11-8320262 
 10-1476458 
 98658208 
 9-9936064 
 9-7123661 
 9-5154583 
 9-4924039 
 9-2062988 
 9-6965016 
 
 
 
 10-6408380 
 10-2375348 
 10-4821746 
 
 where, by K^ is meant the angle K of (85.), as obtained when a is to 
 
ON THE SOLUTION OF TRIANGLES. 67 
 
 be found, or from a^ = b'^-\-c^-—2bccosA. In the following table, 
 pVj gV, and r\J, are, as before, the perpendiculars on a, b, and c, 
 and a^U means the segment of aV adjacent to 6U, &c. 
 
 p = 7-347 logp = 0-8661039 
 
 g = 9-017 log^ = 0-9550632 
 
 r =12-413 logr =10938647 
 
 a^ = 10-006 logfl^ = 1-0002831 0^ = 5*230 log a^ = 0-7184581 
 
 ^>„= 12-231 log6^ = 1-0892424 b^= -133 log6^ = 9-1230370 
 
 c^ = 8-835 log ^^ = 0-9462189 c^=z -183 logc^ = 9-2618385 
 
 In all the following article, a is the greatest side, b the mean, and 
 c the least. Consequently, A is the greatest angle, &c. { } means 
 that the quantity enclosed is the one found by the process ; all the 
 others in it being given or previously found. 
 
 (88.) First Case. Given the three sides, required the angles. 
 First method. (6.) a^-^ttc = h^— C^ or («6-- «c)« = (^ + c){b — c) 
 log {a^, — flcl = ^og(^ + c) + log(5 — c) — loga 
 Hence, Oj — a^ is found : let it be h. 
 
 {a,}^^{a + h) {«,} = i(a-^) 
 log.cos{C} = log«6 — log 6 log.cos(B) = logflf^ — logC 
 {A} = 180°- (B + C) 
 
 This method is the shortest when all the angles are wanted, but 
 should not be used (68.) when one of the angles is very small ; or 
 one of the sides very small in proportion to the rest. When one 
 angle only is wanted, use the 
 
 Second method. To find A, use (84.) one of these, 
 
 L.sin{^A} = 10 -f Klogs — 6 + logs— c — log& — logc) 
 
 L.cos{^A} = 10 + ^ (logs + logs — a — log6 — logc) 
 
 L.tan{iA} = 10 + |(logs — 6 + logs — c — logs — log(s — o)) 
 
 {A} = 2 X ^A 
 
 (89.) Second Case. Given two sides (a and 6, a the greater), 
 and the included angle C ; required the rest. 
 
68 - ON THE SOLUTION OF TRIANGLES. 
 
 First method. When both the other angles and the remaining 
 side are required, use 
 
 L.tan {^aITb} = log(fl— 6) + L.cot iC — log(a -f ^.)* 
 {^r+B} - 90°- iC 
 {A} = KA + B) + KA-B) {B} = KA+B)-KA-B) 
 log {c} = L. sin C -f- log a — LsinA f 
 
 Second method. When the side only is required (85,), use either 
 of the following : 
 
 L.sin{K} = i(Ioga+-logi) + log2 + LcosiC — log(fl4-6)) 
 log {c} = log (a -\-b) -\- L.cos K — 10 J 
 
 L.tan{K'} = ^^oga + log6) + log2 + L.sin J C — log(a — 6)) 
 log{c} = log(tt — 6) + 10 — L.cosK'j 
 
 Third method. When the given angle is very nearly 1 80°, let it 
 be 180° — C/, where C, is small ; we have then 
 
 c' = a2+ ft''— 2a6cos(180°— C^) = a^-\- b^ -^ 2 ab cosC, 
 = (« + 6)2-4a 6 single, = (^ + 6)' 1 1 - -i^^ sin» i C J 
 By the binomial theorem \/l — x = 1 — ^x nearly, j: being small 
 
 ^ = (« + '') (' - (IS?- ^'"^* ^') = « + * - S^'"'iC' 
 
 very nearly. But sinC^ = 2 sin ^C^. cos ^C,, or cos^C^ being 
 
 very nearly 1, we have 
 
 sinjC^ = j^sinC/, sin' ^ C, = Jsin^'C^, very nearly: 
 
 , , , a 6 sin' C. , 
 
 C = a H- 6 — J 3jT-^ very nearly 
 
 log [k} = 2 L sin C, + loga + log6 — log (a -f 6) — 20 
 c = a -{- b — ^k 
 
 Fourth method. When b is very small compared with a, and the 
 small angle B is wanted, we have 
 
 a sin A sin(B+C) ^ , .„•/-- 
 
 - = -r—r: = — ■ ' — - = cos C 4- cot B .sin C 
 b sm B sm B 
 
 • 10 is not added here to give the tabular logarithm, because L. cot 
 is already too great by 10. 
 
 f The excess of the one tabular log. compensates that of the other. 
 
ON THE SOLUTION OF TRIANGLES. 
 
 69 
 
 COtB = 
 
 a — h cos C 
 
 tanB = 
 
 6 sin C b . r^ 
 
 = -smO 
 
 h sin C a — b cos C 
 
 very nearly ; hence b is readily found, very nearly. 
 
 (90.) Third Case. Given two sides and an angle not included^ 
 required the rest. 
 
 It can be shewn immediately that this is a problem of the second 
 degree, admitting sometimes of two solutions. Let a and 6 be the 
 given sides, B the given angle j we have then 
 
 ^2 = a!^ ^ c^ ^ 2fl5ccosB 
 
 or {c} = acosB ± \/^>2_ a^sin^B 
 
 . (i-^) csinB . C A ) asinB 
 
 s.n|C| = -^ s,n|A| = —^ 
 
 1. There is no such triangle at all when b is less than asinB. 
 
 2. Only positive values of c must be taken, for a negative value 
 of C would give sinC negative, or C greater than two right angles, 
 which is impossible in a triangle. The roots of the equation are 
 both positive {Algebra^ p. 139.) when a? — b^ is positive, or a greater 
 than b ; in this case there are two triangles satisfying the conditions 
 in question. When a = 6 one triangle disappears, for then one 
 value of c is 0; when a is less than b there is only one solution. 
 
 The geometrical construction will also shew this. Lay down 
 the angle B equal to the given angle, and take PQ == aU, With 
 
 centre P and radius PY = JU describe a circle, then, if &U be less 
 than PR or asinB.U, there is no such triangle ; if 6U = asinB.U, 
 there is a right-angled triangle Q PR, under the given conditions; 
 if, PY being greater than PR, it be still less than PQ, there are two 
 triangles, QPX, and QPY^ satisfying the conditions. But if PY 
 
70 ON THE SOLUTION OF TRIANGLES. 
 
 be greater than PQ there is only one such triangle, QPV, for the 
 triangle QPZ obviously has not the given angle, but its supplement. 
 For logarithmic solution, proceed as follows : 
 
 L.sin{A} zr loga + I-'-sinB — log^ 
 
 Here (50.) are two angles which satisfy the conditions; one less 
 than a right angle, which call A^, the other A^^, or 1 80° — A^, greater. 
 
 Or it may happen that a — - — may be greater than unity, in which 
 
 case L. sinA will be found greater than 10, or log. sin A positive, 
 and there is no such angle. In the case where this does not happen, 
 we proceed as follows : 
 
 Either {C} = 180°-A-B or 180- A, -B 
 
 Or C= 180- A- B C,= A-B 
 
 if A^ be greater than B ; for otherwise, C^^ is negative, and is here 
 inadmissible. Consequently, the two values of c being c, and c,^, 
 we have, 
 
 __ &sinC, 6sin(A^H-B) ___ ^sin(A^^B ) 
 
 ^' sinB ^ sinB ^" ~ sinB 
 
 log{c/} = log^ + LsinC, — LsinB 
 
 log|c,,} = log6 -f LsinC,^— LsinB (if Q, be positive) 
 
 (91.) Fourth Case. Given a side and two angles; (aU the 
 side), required the rest. 
 
 {Third angle} = 180°— (sum of given angles) 
 
 \o%b = loga H- LsinB — LsinA 
 
 logc = loga + LsinC — LsinA 
 
71 
 
 CHAPTER IV. 
 
 ON THE EXTENSION OF THE MEANINGS OF SYMBOLS, AND ON 
 THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
 
 (92.) We have as yet explicitly used only two different kinds of 
 symbols: those o( quantity, specific and general (arithmetical and alge- 
 braical), and those o^ operation; specific, as -\ , &c. and general, 
 
 J\ 0, &c. {Algebra, p. 203). But we are not therefore bound never 
 to use any other symbols; the only laws by which our right to 
 such aids is limited, are the following. 
 
 1. Neither the symbols themselves, nor any expressions in which 
 they are used, must have different meanings of any such kind, such 
 that the consequences of one meaning may be confounded with, and 
 used for, the consequences of another. 
 
 2. The consequences of all assumptions must follow logically 
 from the assumptions themselves. 
 
 It therefore becomes of interest to consider in what other possible 
 ways we might use symbols. And first it must strike us that all yet 
 employed may be styled under one name, more general than either 
 operation or quantity. They are, in fact, symbols of discriminutidri 
 or distinction. Thus, in a, b, and c, in which the symbolic difference 
 is only difference of shape, that circumstance is made the distinction 
 between difference of numerical magnitude. In + and — the same 
 distinction, namely, of form, is made that of the direction given, as to 
 which of two fundamental operations is to be performed. In ub 
 and - we see that difference of position, with the employment of a 
 new, and not altogether necessary, symbol, is the distinction which 
 implies difference of operations. 
 
 (93.) Let us now look at the extension of arithmetic into algebra, 
 not confining ourselves to the notion of operation or quantity, but 
 generalising our idea of symbols to that of mere discrimination; the 
 object being to consider, whether, in this point of view, extension is 
 possible, preliminary to the further question of whether it is advisable. 
 
72 EXTENSION OF THE MEANING OP SYMBOLS, 
 
 Looking narrowly at the steps by which we ascended {Algebra, p. 57. 
 and 110.), we see that, so far as the discriminative quality of the sym- 
 bols is concerned, we found the following consequence of our funda- 
 mental and arithmetical definitions of -|- — &c. namely, that old 
 rules were sufficient to express new distinctions consistently with the 
 old distinctions ; in such sort that, whenever the new distinctions dis- 
 appeared, the result was a legitimate consequence of the older notions, 
 such as would have been obtained if the more circuitous method had 
 been adopted, to avoid which the new distinctions were introduced. 
 For example, when we came to the expressions a -}-(-[- 6) and 
 aH-( — &), which have no meaning under the original meaning of 
 4- and — , we had extended our ideas to the following question. 
 The distinction between a magnitude of any one kind and its dia- 
 metrically opposite, being denoted by *6 and %b, if there occur a 
 case of a problem in which *b requires to be added to a, what will 
 the similar problem require, in which 1[6 is used where *6 was used 
 in the other. And we found, 1. That where *b requires addition, 
 IF 6 requires subtraction. 2. That -f-a and — a would themselves 
 consistently express *a and ^a, provided that the rules appertaining 
 to the original meaning of + and — , and no others, should be 
 also applied to them in their new and distinctive capacity. And 
 we also found that the disappearance of -}- and — , in their new 
 character, was accompanied by the disappearance of all traces of the 
 new distinctions ; or that all theorems which in any case took the old 
 forms only, were true under the old meanings. The old algebra 
 (general arithmetic) was, in every sense, part of the new one. 
 
 (94.) Now, this we shall lay down as the restriction under which 
 any further extension is to be made ; namely, that all theorems at 
 present existing, are to be theorems which are true, whenever they 
 are the consequences of any further extension ; and true in the sense 
 in which they exist at present. We might, propose infinite numbers 
 of changes of meaning, under which some theorems would remain true, 
 but in which others would not be so. For instance, if x placed 
 between two quantities, were made to signify that their sines (not the 
 numbers themselves) should be multiplied together, then (61.) 
 
 axa — bxb = {a -\- b) X {a — b) 
 
 would be true; but ax 2b = bX2a would not be true. 
 
 (95.) There might arise cases in which the answers of problems 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 73 
 
 could not be expressed without more power of symbols than is pos- 
 sessed at present. For instance, will any one undertake to say, that of 
 all possible problems, there is no one of which the answer is as follows: 
 Divide 6754321 by 12, in the common way, with this exception, that 
 wl^never the remainder is an even number, it is to stand, but when- 
 ever it is an odd number, it is to be increased by 1, if the preceding 
 remainder were even, and diminished by 1 if odd. The answer to 
 
 this question would be, 
 
 12) 6754321 
 570367^ 
 
 but no symbols, at present possessed, would describe the operation. 
 Again, if we ask what is that expression which, when x is positive, 
 is JT^, but when x is negative, is x^l This cannot be expressed by 
 present symbols ; and the same may be said of many imaginable 
 results. So much for the possibility of further extension, or a new 
 symbol of distinction; the next question must be, is it wanted, and 
 can it be made ? 
 
 (96 ) When the earlier algebraists first began to occupy themselves 
 with questions expressed in general terms, the difficulties of subtrac- 
 tion 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. Igno- 
 rance, the necessary predecessor 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 negationist 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 vitium paralklorum, 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 
 
74 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 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 differ from those previously applied to the old ones. 
 A symbol, the result of operations upon symbols, either meant quan- 
 tity, or nothing at all ; but in the latter case it was conceived to be a 
 certain new kind of quantity, and admitted as a subject of operations, 
 though not one of distinct conception. Thus, 1 — 2, and n — (a -|- 6), 
 appeared under the name of negative quantities, or quantities less than 
 nothing. These phrases, incongruous as they always were, main- 
 tained their ground, because they always produced true results, when- 
 ever they produced any result at all which was intelligible : that is, 
 the quantity less than nothing, in defiance of the common notion that 
 all conceivable 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. Tliis ought to 
 have been the most startling part of the whole process. That contra- 
 dictions 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 theorems 
 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. 
 
 (97.) The complete explanation of the embarrassing circum- 
 stances is comparatively modern ; the latter arise from a very simple 
 logical misconception, the assumption of the truth of a converse, 
 namely, that if B follow from A, B follows from nothing else but A ; 
 or if A always yield B, B when it appears, must have been produced 
 from A. We can imagine a, 6, c, &c. -f-, — , X , &c., defined in many 
 different ways, so that certain of the theorems of algebra should 
 severally be true under more than one set of meanings ; and we have 
 shewn an instance in page 72. We can further imagine it possible 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 75 
 
 that one set of meanings should be so connected with another, that all 
 theorems which are true in the first, should also be true in the second, 
 and that, beside this, there should be other classes of theorems which 
 are true in the second, but not true in the first. This is as possible as 
 that one figure should be entirely contained in another, without filling 
 it. We might thus conceive a succession of extensions of definitions, 
 giving a series of sciences, each containing the whole of its predeces- 
 sors, and more. Whence it is clear, that if the methods of operation, 
 under any science, wandered beyond its limits, without that corre- 
 sponding extension of definitions being made which converted the logic 
 of one science into that of the other, the consequence would be, the 
 appearance of some of the symbolic means of expressing truths in the 
 wider science, without the key to their interpretation. This happened 
 when we first came to the negative symbol in algebra {Algebra, p. 12), 
 and we were under the necessity of adopting more extensive defini- 
 tions. The proof of undue extension in the operations again occurred 
 (p. 110.), where \/ — 1 first appeared ; but it was not then necessary 
 to follow up the extensions necessary for its elucidation. 
 
 This matter is one of difficulty to a beginner, unused to the idea 
 of a finished language having the meanings of all its terms extended 
 so that the old meanings are only part of the new ones. But, in 
 reality, he has gone through the process, by insensible steps, in his 
 childhood.* Let him compare the first impression he was made to 
 receive by the words " I see," with the sense he puts upon the phrase 
 when, if he understand the preceding, he says he sees my meaning. 
 That which I here call the extended use of the term, he will call the 
 allegorical or metaphorical mode, that is, if we translate these Greek 
 terms, the other-speaking, or transferred mode of expression. But 
 in what way is the speech changed ? By using the word to " see," 
 not only as denoting perception by the eyes, but perception by the 
 understanding in any way whatever. If we heard any one speak, we 
 might still see his meaning. 
 
 (98.) Let us consider the most general meaning of any funda- 
 mental equation of algebra : for instance, a -^-6= b -\-a. We restrict 
 
 * The similarity of the views here given, with some in the review of 
 Mr. Peacock's " Algebra," in the ninth volume of the " Journal of Edu- 
 cation," makes it necessary for the author to state that he was also the 
 writer of those articles. 
 
76 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 ourselves to tliis only, that b and «, and +> mean the same thing in 
 both places, that = denotes, gives the same result as, or is the same 
 in effect as, and that the order of the expression is from left to right, 
 that is, on the first side we first examine a, and b on the second side. 
 Then a + 6 merely implying that something, b, is done in some 
 manner, +> after something else, «, has been done ; the above 
 shews no more than that it is indifferent whether a is done first or 
 b, or that the result is the same in both cases. Consistently with 
 a -{- b =z b -^ a, many meanings might be shewn to be impossible, 
 and many possible. Neither need a and 6 stand in any way for 
 quantities : for instance, the preceding would be true if a stood for a 
 line not considered as a length, b for another line, + for the formation 
 of a rectangle out of a and b, and = for equality of space enclosed. 
 The preceding relation may therefore be a truth under an infinite 
 number of different meanings. Let us now take another form, 
 ab := ba, which admits of exactly as many meanings as the first, 
 and denotes merely indifference of order. Suppose we pick out two 
 such meanings at pleasure, and assign them, only requiring that 
 a and b shall mean the same in both relations. We then have defi- 
 nitions for tf, by + and the meaning of juxtaposition, and have ex- 
 plained a-\- b = b -{- a, and ab = ba, from which it follows that if 
 +, as defined, will intelligibly apply to ab, we have ub-\-b=.b-\-ab, 
 &c. But if we would have our new algebra identical wiih the old 
 one in forms, we must choose such meanings for the symbols in the 
 two relations as will also make 
 
 a{a + b) = aa -\- ah represent a new truth. 
 
 This is a restriction upon all the possible allowances of meaning 
 which might be made : for it does not follow that every meaning 
 which makes a -\- b =. b ■\- a, and ab =^ ba, true, also makes the 
 last true. And other relations might be introduced which would still 
 more restrict the meanings, and so on, until every fundamental 
 relation necessary to algebra had been considered. If we could 
 really collect all the possible meanings of each separate relation, and 
 find the method of ascertaining which must be struck out for each 
 and every new combination which the mechanism of algebra intro- 
 duces, we should, if we could classify the remainder uniting those 
 which are particular cases of a general meaning under their general 
 head, be left with an algebra in the widest possible sense of the word. 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 77 
 
 We do not want to change operations; but we want to find all the 
 definitions under which those operations loill demonstratively enable us 
 to pass from one truth to another. 
 
 (99.) Two explanations may be given of the manner in which n/— 1 
 may rationally be used ; the first purely symbolical, that is, employing 
 */ — 1 as a symbol, the meaning of which is given for convenience 
 only ; the second derived from geometry, and an extension of the 
 method by which lines measured in opposite directions are repre- 
 sented by letters with different signs. The first is wholly algebraical ; 
 the second (for a reason which will afterwards appear) is an appli- 
 cation of geometry to algebra. 
 
 Let /c be a symbol which does not stand for quantity, but for a 
 distinction, in whatever way it may be required to distinguish ; that 
 is, ka and a both stand for the same magnitude, but the first has the 
 mark of either being used in a certain way, or appropriated for 
 certain purposes, or liable to be rejected under certain circumstances, 
 or whatever other distinction k may indicate. When two separated 
 terms are multiplied together, ^s ka and kb, let the product be 
 written k^abf in which /c^ merely implies the presence, in a product, 
 of two terms which had the mark of distinction. Similarly, /c^ a c is 
 the distinction of a product made up of three terms which had the 
 distinction, and so on. Let k{a + b) be the distinction between 
 a -\-b and ka + kb, and so on. We are at liberty to assign to k any 
 discriminative power we please. Let it be as follows : /c is to be a 
 distinction which ceases altogether in terms marked with /c*, A^, /c", 
 &c. or A;'*", and with /c-*, k-^y /c-i^^&c. or k'^^, and which is 
 preserved in k% /c^, /c'^, &c. /(;-*, /c-7, k- ", &c. or in /c^n+i vvhere 
 n is any whole number positive or negative. And let /c^, /c^, /c'°, &c. 
 /c-2j fe-^, /c~^o, &c. be distinctions amounting to a change of sign 
 in the terms denoted by them ; so that if for /c^a we write — a, the 
 object of the distinction is fulfilled, and the term need no longer be 
 distinguished. Let k^, W, /c", &c. or k-'^, A— ^, k-^, Sec. imply both 
 a change of sign and also the continuance of the distinction denoted 
 by k; so that k^a means — ka. We have then defined every thing 
 except k itself, by the following identities ; 
 
 A*"a means a, h^^'^^a means koy 
 A'*"+^a means — cr, 7i'*"+^« means —ka 
 
 And thus every algebraical expression, when its distinctions are all 
 
 n 2 
 
78 ON THE EXTENSION OF THE MEANING OP SYMBOLS, 
 
 marked, is reducible to the form P + ^ Q where P and Q are alge- 
 braical. Thus, 
 ao + aik + a2?i^+ .. means flo— «2 + 04+ •• +A-(ai— fls + ^s— &c.) 
 Now k itself is to have meaning as follows : in the equation 
 V + kQ = Fh- AQ' 
 
 let the presence of /c indicate that equality will still remain as well 
 between the parts independent of k, as between those affected by A:. 
 If we had merely P + Q = F + Q' it would not at all follow that 
 P = Q and P' = Q' : but when we mean to make this additional 
 supposition, let us signify the same by the presence of k. 
 
 Now, it will be obvious upon looking from (70.) to (73.) that 
 we have here only made a notation to express distinctions which 
 have been actually arrived at by process of reasoning. We found 
 a method of embodying all the results previously obtained, of this 
 
 kind : 2cos0 cannot be .r +-; but if we work in any manner with 
 
 X -)-- and X , and produce an equation, then that same equation 
 
 will correspond to one or two true equations, if we work in precisely 
 the same manner with 2cos0 and 2 /c sin 0, and then let k have its 
 discriminative powers. And we shall then find that the result is 
 the same as if 
 
 2cosw^ had taken the place of j;" + -„ and 2 fe sin w^ of x** . 
 
 '^ a" x" 
 
 When none but even numbers of ks occur, it is obvious that 
 the result can be only one equation of the form P = P'; but 
 when odd numbers of /cs also occur, the result will be of the 
 form P-{-/cQ = P'-f-feQ', giving two equations, P = P', and 
 Q = Q'. 
 
 (100.) Next, observe that the algebraical symbol \/ — 1, which 
 is certainly no quantity, positive or negative, and therefore not to 
 be reasoned upon as a quantity, yet has this property, that if those 
 rules be applied which would have been applied had it been a quantity, 
 the results will be expressive of the distinction denoted by k. For in 
 that case we have 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 79 
 
 Again, 
 
 (Viri)-iis-V=i; (v~i)-2i3_i. (^irT)-3ig^— 1, 
 
 &c. from which the coincidence is apparent. Therefore, by making 
 \/^ the symbol of the distinction meant by /c, all the remaining 
 distinctions will be drawn by applying the common rules of algebra 
 to V — 1 as if it were a quantity. 
 
 When in (71.) we began to compare cos0 + /csin with s*^, we 
 
 Q'i 
 
 were also, in fact, comparing it with 1 -\- k9 -\- k^ — + > or ('^ 
 
 we apply the meaning of k) with 
 
 Now, if the method of proceeding be valid, which extends the 
 distinctive property of k to the developement of s ^, then 
 
 cosd + ksm$ = ] — ^+ .... + A;(^— ^ + ....) 
 
 or cos^= l--2+iX4 -2.3.4.5.6 + •- 
 . . _ . 03 0^ 07 
 
 sm^ - 0-— + 2.3.4.5'- 2.3.4.5.6.7 "^ "" 
 
 (101.) We shall now go through a strict deduction of these equa- 
 tions, which will shew that what we have done would, had we seen 
 how, itself have been one. It is evident that 
 
 ^ , , . „ ( which amounts to 
 
 cos0 4-A;sm0 = cos0+/csm0 ^ ^ • ^ 
 
 (^ COS0 = COS0, sm0 = sm0 
 
 Square both sides, which gives 
 
 (cos0 + k sin 0)- = cos=0 + A; . 2sin . cos + k^sin^O 
 
 = cos2 — sin20H-/c.2sin0.cos0 = cos2 + /c sin2 
 
 Generally, if cos n + /c. sin n = (cos + /c sin 0)" 
 
 X (cos + /c sin 0) and cosn0.cos0 'i ^r cosw0.sin0 
 
 . 75 • . • n\+^\ . • . , = (cos0+/csin0)«+i 
 + /c'smn0.sin0J (-|-smw0.cos0 
 
 or cos(n + l)0 + k .sin(n + l)0 = (cos0-f-/csin0)'»+i 
 
 So that this relation, if true for one whole value df n, is true for the 
 
80 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 next. But it is true for n = 1 and n = 2, therefore it is true for 
 all. This is called De Moivre*s Theorem. 
 
 Develope the second side, substitute the meanings of A;', k\ &c., 
 and form the two resulting equations, which will be found to be 
 
 2 2 3 4 
 
 _ , n — 1 n — 2 w T , , n — 1 n — 2 n — 3 n — 4 __, , 
 
 2 i3 ji 3 ^ o 
 
 where c means cosO, and s, sin0. Divide both sides by c", and put 
 
 C 72—1 
 
 t (tan0) for -; at the same time divide and multiply n, n——, &c. 
 by a power of n of the same number of factors, which gives 
 
 which is true for all whole values of n, and all values of 9. Now, 
 consider all those vi'hole values of n, and values of 9, which make 
 n0 = JT, a given angle : whence {Algebra, p. 157.) the limits of the 
 two sides of each of the preceding, made by increasing 7i without 
 limit, will be equal. We proceed to find these limits. We have 
 
 C" = (cos-j the limit of which (70.) is 1 ; -, -, -, &c. the 
 
 limits of which are severally 0; wt or wtan- or a: (tan r ) 
 
 the limit of which (48.) is x x 1, or x; and n0, which is x 
 throughout. 
 
 Take the limits of both sides, which give 
 
 x^ X* "^ 
 
 cosa; = 1 — - + — — - — &c. , , 
 
 2 2.3.4 I the theorem m 
 
 Hence we have 
 cos 
 
 if we abbreviate the preceding series into the formula of which it 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 81 
 
 would be the algebraical developement, if k were a quantity. We 
 shall now adopt the symbol v — 1 for k, which gives (for all values 
 of0) _ 
 
 cose + sinOVITi = g^^-i 
 
 For 9 write— 0(44.) cos0 — sinOV^^ = S~^^-'^ 
 
 g^-v/^-L g"^\/^ g^A^^_ g-^v^"^ 
 
 cos^ == sin d = 
 
 that is, if we develope the preceding exponential expressions, paying 
 attention to thedifFerenceofdevelopement denoted by \/ — 1, (n/ — l)^ 
 &c. we shall arrive at series which we have shewn to be the deve- 
 lopements of sinO and COS0. 
 
 (102.) Some marks of distinction are definition, and all the rest are 
 consequences. Thus e*^ has k a symbol of distinction ; but it does 
 not mean /c.e^, but cos0 + ksinG. And there are some results which 
 when enunciated as if V — 1 were a quantity, are yet more incon- 
 ceivable than less than nothing, in an arithmetical sense (^Algebra, 
 p. 62.) For instance, calling kQ the logarithm of s , we have, 
 making B •=. 1mr,n being a whole number (cos0 = 1, sinO = 0) 
 
 thus 1 has, under our present extension, an infinite number of loga- 
 rithms, corresponding to all whole values of n, positive and negative, 
 namely, 
 
 ... -2^^^, -ctn/^, 0, ^VITl, 2-^^^, . 
 among which the arithmetical logarithm is found, namely, 0. Simi- 
 larly we may deduce 
 
 g(2« + i)crVi:i = -1 log(-l) = (2w + l)':r\/iri 
 
 Let X be the arithmetical logarithm of 3/, then we have 
 y _. g* «_ g* X 1 = S*+ g2"^>^^i = gx+2ntrAA:i 
 
 or all the values of x-\-1n'r'^ — 1 are also logarithms of 3^. The 
 theorem log«-t-logt = log a 6 now exists in this form : if any loga- 
 rithm of a be added to any logarithm of h, the sum is one of the 
 logarithms of a 6. Thus, if log. « stand for the arithmetical logarithm 
 of cr, and Xa for its general logarithm, we have 
 
 Xa + >.b = loga H-2<T?iN/ — 1 -flog 6 + 2'3'wV — 1 
 = log(«6) -f-2'T(w-f 7/')^ — 1 = y^ab 
 
82 ON THE EXTENSION OF THE MEANING OP SYMBOLS, 
 
 (103.) We can now give the forms of all the different roots of a 
 number, which was done to the fourth degree in p. 11 3 of the Algebra, 
 
 If we take the equation 
 
 1 = g2"*^-' = cos2wcr + sin2w-rV:ri 
 
 xivi 2^?rV^ 2n7r , . 2n7r / ; 
 
 we have (I)"* = g "* = cos^p + sm— - V — 1 
 
 from which it might appear that there are as many with roots of unity 
 as there are whole values, positive and negative, of n. But it jvill be 
 found that these values recur as we give to n different values in suc- 
 cession ; as follows : 
 
 First root n = First value of (l)*" is cos 4" sin 0. >/ — 1 or 1 
 
 2 TT 2 TT — — 
 
 Second root w = 1 Second cos H sin ^ — 1 
 
 m m 
 
 Third root w = 2 Third cos \- sin v^ — 1 
 
 m m 
 
 mth root n = m— 1 mlh cos -^^ ^ -f- sm -: ^ V — 1 
 
 w m 
 
 The (m4-l)th value is cos h sm . v — i or cos 4- 
 
 ^ ' mm ^ 
 
 sin V~l, the same as the first. The (m + 2)th value is cos ^^ + 
 
 . 2(m + lV / / , 2 7r\ , . / ,2 7r\ ^ 
 
 sm-i i-v_l or cos I 27rH ) -f sm I 27r H J V— l, 
 
 m ^ 7/1 y \ m / ^ 
 
 the same as the second, and so on. To find the twelfth roots, for 
 
 instance, draw a circle, with the centre O. Divide its circumference 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 83 
 
 into 12 equal parts, beginning at A, any given point. Then every 
 point of subdivision shews a root as follows. If the radius be the 
 linear unit, and if PM and MO represent the fractions of a linear 
 unit which are in those lines; then, 
 
 one value of(l)'r2 jg QM + PM. VlTr 
 
 (104.) It nnay be very soon shewn, that if c -j-s v — 1 be a root 
 of unity, c — s>/ — 1 is also a root. Firstly, because if £* -v^^ be 
 a root, or if 
 
 gnjTA/^ = 1 we have (g"-»^^^)"^ = 1 = (s"*'^^^ 
 
 or, if cos :r + sin j; . V — 1 be a root, cos x — sin x n/ — 1 is another. 
 Secondly, in the preceding list of roots we see from 
 
 2(w— l)fl- 27r , . , ^. / 27r\ . . / 2^\ , r 
 
 or 2 -^r that the last root is cos ( ) + sin I ) v — 1 
 
 2v . 2 7r / r , ... 2 tt , . 2 ^r / ; 
 
 or cos sin — v — 1 the second being cos — + sin — v — 1 
 
 mm m m 
 
 The same will also appear from consideration of the figure. 
 
 (105.) B: 
 — 1, we find 
 
 (105.) By proceeding in the same manner with g<2n + i)!rv' i ^^ 
 
 / 1 sm (2 n + 1) TT , . (2 W + 1 ) TT / T" 
 
 (-_1)"» = cos^ ' — ^- h sm^^ — ^ V — 1 
 
 ^ '^ m m 
 
 and, as before, it may be proved, that there are m roots, and no more. 
 But the roots of — 1 may be more easily set forth to the eye, by means 
 of the roots of + 1, as follows. Every root of — 1 is twice as high a 
 root of + 1 ; for if or'rt^: — 1, then a^2m = i. Consequently, if we 
 take all the twenty-fourth roots of unity, or divide the circle in last 
 article into 24 parts, all those 24th roots of -|- 1 which are not also 
 12th roots, are 12th roots of — 1. 
 
 (106.) Let all roots of unity of the same order be called corre- 
 sponding roots : thus there are m corresponding roots of -f 1 of the 
 wth order. Then, all powers of a root are corresponding roots. For if 
 fi»* z= i, then jit2m or (/x2)»» = l, &:c. And this holds equally of 
 all negative whole powers. But it does not therefore follow that, 
 among the powers of any one root, will be found all the other corre- 
 sponding roots. 
 
84 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 Let us denote i"" ^^ by [j] for the present. Then we have 
 
 \nx-\ = M", [x + 2;,] = \x\, [2;,] = 1 
 
 where n and p are whole numbers, positive or negative. The mth 
 roots of unity are 
 
 '-[»]■ [a'Ca-m-'-D-^'i 
 
 Let us consider the case of m = 8. Then the powers of [O] are 
 severally = [O], and never produce more than one root. The 
 powers of [2-H8] are as follows: 
 
 [a.G].B].[a--'.[T]' [€■&]■[¥]-.- 
 
 • or all the roots are produced in order. But the powers of - are 
 
 [a-G]=-^'ra'B]=^'[f]=[g- 
 
 and there is a continual recurrence of half the roots only. The 
 powers of p are 
 
 D].B]-B] -G]. [¥]-'■ 
 
 [?] - m m - [g. ei - m. 
 
 ra - '• 
 
 followed by recurrence. 
 
 Here again are all the roots. Those roots, whose powers give all the 
 roots of their kind, are called primitive. It is enough for our present 
 purpose to know that there is one primitive root of any order. 
 
 (107.) The roots of +\/— 1 and — s/'^ may be now obtained. 
 For we have 
 
 * =V— l=g ,g =_V— l = g 
 
 which may, as before, be shewn to have m values only. 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 85 
 
 To take an instance for verification : the cube roots of H-n/ — 1 
 are 
 
 1 . I / — r 5 , . 5 / 7- 9 , . 9 / T- 
 
 cos - T + sin - -TT V — 1 cos-cr + sin- T V — I cos-'rr + sm- t v — 1 
 6 6 6 6 6 6 
 
 or lV3 + iv3i _V3 + lV~i -V— I 
 (Vs -f ^ V~T)' = 1(3 V3 + 9V31-3V3- ^31) = v/ZT 
 
 (108.) Having shewn that the rules which would apply to V — 1 
 tf it could be considered as a quantity, will of themselves make all 
 the necessary distinctions between the formulae of common algebra, 
 and the shape in which they become formulae of trigonometry, we shall 
 now proceed to the second method of explanation, or rather to the 
 method of application, which shews the geometrical meaning of the 
 symbols in question. As a starting point, we return again to the 
 method of explanation of the negative sign. In looking at a + ( — b) 
 we found that the sign + no longer preserved the meaning of 
 arithmetical addition, while the quantity operated on, — b, was no 
 longer simply a number, but a number with a sign of direction. 
 We might apply the preceding method of explanation to the passage 
 from arithmetic to algebra. In this case ka would signify a dis- 
 tinction of this kind ; terms having /c^, k*, &c. are all to be of one 
 kind, unmarked, while terms having /c, k^, k^, &c. are all to be of 
 another kind, marked with the distinction k. If we adopted this 
 signification we should soon find that all theorems which are true 
 when the distinction k means nothing, and may be entirely abolished, 
 are also true when the distinction k means simply change of sign, 
 if it be arithmcticallj/ allowable. And it would also be found that 
 were it allowable to consider — 1 as a quantity, the rules which 
 would apply to this latter symbol are precisely those by which the 
 necessary distinctions would be drawn in the course of the process, 
 without any particular attention. We might thus dispense with 
 subtraction at the outset, and establish all theorems in which arith- 
 metically additive terms only occur. And subtraction might be 
 introduced in time by means of a distinctive symbol. We cannot 
 make this an illustration for a beginner, because to place it on the 
 same footing as the subject of (73.), we must require him to imagine 
 
 I 
 
86 ON THE EXTENSION OF THE MEANING OP SYMBOLS, 
 
 himself divested of some of his most simple notions, and thereby 
 reduced to learn things which now are axioms, by a long process of 
 reasoning. lie must conceive himself unable to form a distinct 
 notion of subtraction, other than the inverse of addition ; that is, 
 though the notion of taking away that which he himself has just 
 added may be simple, yet the idea of instituting a subtraction inde- 
 pendently of any previously expressed addition, must be one to be 
 learnt with some difficulty. If this were the case, we could make 
 the use of a distinctive symbol facilitate the acquirement of the 
 general operation of subtraction ; as it is, the last-mentioned process 
 is one of which we have a clear idea, independently of addition. 
 
 (109.) We found that the interpretation of a negative quantity was 
 a magnitude taken in precisely the opposite sense and meaning to that 
 which we imagined, when we applied arithmetical process to the 
 determination of that magnitude. And the affairs of life contain so 
 many such interpretations that the extension looks natural; so natural 
 indeed, that many have drawn a great distinction between the nega- 
 tive quantity, and the square root of the negative quantity. Of this 
 the application of the terra impossible to the latter symbol only, is 
 a sufficient instance: both are impossible, according to arithmetical 
 notions, but the latter only has received the name. There is hardly 
 a phenomenon in nature, or a relation of life, which does not admit 
 of the modes of neutrality, excess on one side, or on the other. Where- 
 ever there are two opposites, with a state between them which does 
 not belong to either, we have the means of illustrating the terms, 
 positive^ nothings and negative. Without specifying what we are 
 speaking of, it would at once be granted that wliere the establishment 
 of one state of things would cause increase, that of the opposite state 
 would cause diminution, and that of the neutral state neither increase 
 nor diminution. If we consider time as divided into beforey after, and 
 at; pecuniary relations divided into those of debtor, creditor, and 
 neither debtor nor creditor ; a balloon with weights, as either rising, 
 sinking, and neither rising nor sinking ; electricity as producing 
 attraction, repulsion, or neither attraction nor repulsion, &c. &c. we 
 have the same common modes of existence: and in all the cases we 
 have mentioned, no others whatsoever. It is in considering space only 
 that we have these modes, and others, as follows : 
 
 (1 1 0.) As long as we consider ourselves at liberty to change the 
 position of a point in a given straight line, and in that straight line 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 87 
 
 only, we have the three modes already considered, and no others. 
 The point P may lie on one side or tlie other of a standard point O, 
 given to measure from, or it may coincide with the point O. Let our 
 
 R 
 
 notion of space be contained in length, and we pass with perfect con- 
 tinuity from P to Q, by continually diminishing OP, until P coin- 
 cides with O, and then continuing the motion of P on the left. We 
 have here perfect analogy with all the other kinds of quantity which 
 we can conceive. Let O represent the commencement of the Christian 
 era; take an inch to a year, and we have the means of making a table 
 of all conceivable modes of time, and by laying down points corre- 
 sponding to different events, we might reduce chronology to a science 
 of feet and inches. B^^ifwe allow all space to enter the question, 
 we may bring OP into the position OQ without diminishing its 
 length, by turning it round through two right angles. And if we now 
 consider the first passage from P to Q, relatively to the new notion 
 we have introduced, we see direct discontinuity. The line OP always 
 continues making an angle nothing* with its first position, until P 
 coincides with O, when O P is not a line, and the idea of opening, 
 actual or possible, ceases altogether : immediately afterwards, O P, 
 now become O Q, makes two right angles, or half a revolution, with 
 its first position. But it might have made this at two steps of a right 
 angle each, or at four of half a right angle each, &c. &c. Conse- 
 quently, in geometry, we can pass from positive to negative by an 
 infinite number of gradations, but which, as yet, we have no means 
 of noting, though the conception is clearly attained. 
 
 Now let us try to extend our notion of chronology in the same 
 manner. Let O denote the commencement of the Christian era ; let 
 years A.C be measured to the left in inches, and A.D to the right. 
 
 * The student must carefully distinguish between no opening where 
 opening might have been, and no opening because opening is inconceivable. 
 And this dislinciion must be made throughout the mathematics. Our 
 language wants a phrase to distinguish between the nonexistence which 
 arises from incoherence of ideas, and that which is not, but might have 
 been, the case. 
 
88 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 We can easily say what point of time P represents, or where P should 
 be in order tliat it might represent any given event. But what is the 
 answer to the question. What point of time does R denote ? This 
 question is wholly unanswerable. All the time we can form a notion 
 of, is already expressed on one side or other of O; there remains no 
 idea of time answering to the point R. We have no idea of time 
 except quantity of duration : we have two ideas of a straight line, 
 quantity of lengthy and direction. 
 
 (HI .) We see then, that in preparing an algebra for geometry, we 
 are making one which will be more than we can apply to any thing 
 else. But we shall carry on our geometrical algebra, and then shew 
 that, by a process of pure reasoning, which contains no assailable 
 point,* we can use this method in all other sciences. 
 
 Let all lines be considered as having direction as well as magni- 
 tude, and both essential to their definition ; so that two lines shall not 
 be called equal, unless they be in fact equal and parallel. Lines of 
 equal length in different directions are to have distinct symbols, and 
 to be different things. Let there be only one direction of revolution 
 (for the pres3nt), namely, fiom OA through OA', a direction, OA, 
 
 having been chosen which is to be permanently that of positive quan- 
 tity, or that in which a line is to be expressed by its simple symbol 
 «U, where U is the linear unit, and a the number of linear units, or 
 (if the student think he comprehends the phrase) the symbol for the 
 ratio of the line to the unit. Consequently, a line measured in the 
 direction O A" has such a symbol as — aU, supposed to be already 
 understood. Call OA .... the line of arithmetic , which is part 
 
 • When I SRy this, I mean that all the objections which have been 
 made relatively to negative and impossible quantities in the usual sense, 
 right or wrong, have nothing to do with the reasoning which will he 
 employed. 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 89 
 
 of . . . . A"0 A . . . . that of the algebra of positive and negative quan- 
 tities. In this line the meaning of + and — is fixed by all that has 
 preceded in ordinary algebra ; but as relates to lines not coinciding 
 with O A or OA", nothing must be conceived to have been yet laid 
 down. The meaning of + and — remains to be determined, subject 
 only to the law of extension, that all the limited meaning is to be con- 
 tained in the extended meaning. The question to be determined is, 
 what is the proper representation of the line O P, rU in length, and 
 9 in direction, meaning that it makes an angle 9 with O A. 
 
 Firstly, suppose 9 to be commensurable with two right angles, or 
 
 that 9e = — ttG (m and n whole numbers) whence 2n9 = m,2'rr. 
 n ^ 
 
 Let k^r be the distinctive symbol which marks that rU makes an 
 angle 9 with the arithmetical line; where k not being a symbol of 
 quantity, neither is 9 that of an exponent, in the algebraical sense. 
 Then it is evident that, by turning OP until it has revolved through 2 » 
 times the angle 9, we bring O P into the position of a line making m 
 sets of four right angles with O A, that is, we bring it into the direc- 
 tion O A for the mih time. And because coincidence is restored after 
 every revolution, we must have 
 
 ^^+2»n5r^U signifying the same length and direction as ^ rU 
 
 ^''"^U u 
 
 or ;fe^ + '"'^rU eguals k^rV ^'^" U = U 
 
 for we have said, let lines be equal when they are the same in length 
 and direction. 
 
 If then we want such a symbol as is capable of expressing the 
 necessary distinctions, and which shall be pointed out by the rules of 
 ordinary algebra, we must so assign k as that k .k .k . . . . (2w) = 
 ^2'T /c2*. . . ^im) = 1, which can be done by making k a 2wth root 
 of 1, in the manner already laid down, where s/ — 1 is treated* as 
 a quantity. If we ask which root of 1 is to be adopted, the answer 
 
 * Let the student carefully note the diflference between treating 
 V — 1 as a quantity, when it has been proved that certain purposes of 
 distinction are answered by so doing, and considering it as a quantity. 
 In operations, this difference amounts to nothing, which is precisely the 
 reason for which we adopt the method ; but in theory, the first method 
 gives good reasoning ; the second gives only a part of it, logical deduction. 
 
 J 2 
 
90 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 evidently is, that root which, when raised to the 2 nth power, gives 
 Ihe unit at the end of its mih revolution, or exhibits it in the form 
 cos 2nnr -{■ sin 2 w t . \/ — 1 . That is, we must let 
 
 ft be signified by cos — J- sm ■ v — 1 
 
 •' 2n ' 2n 
 
 (Be careful to remember that sinxN/ — 1 means, not sin (jr V — l) 
 but (sin x) . n/ — l). By De Moivre's Theorem, 
 
 (cos— H sm ) = cos2w7r + sm2mir V — 1 
 
 \ 2n ' 2ti / 
 
 a form of unity. 
 
 But — — = - TT = 0: hence k is cos0 + sin0 s/ — 1 
 
 Secondly, let and tt be incommensurable; the same notation 
 must still be preserved, extending to the symbol of quantity 9 all 
 those considerations which have been heretofore introduced, in respect 
 to the connexion of commensurables and incommensurables. The 
 general result then is, 
 
 r(cos9 + sinOv^ — l) U signifies a line rU inclined at an 
 an?le 9 to the arithmetical line. 
 
 (112.) The meaning of the sign -f before a term distinguished by 
 V — 1, is to be determined from the preceding. If OM = jU 
 (= is here correctly applied), and if MP equals in length 3/ U (the 
 limited definition of =), we find that the line is 
 
 X +y V — 1 ; 
 
 consequently, x -\- 1/ s/ — 1, or O M + M P s/ — 1, is the symbol for 
 O P ; whence we see that our extension of meaning is as follows : 
 whereas, in the arithmetical line, O M + MP (MP being carried 
 forward on that line) would have been OP on that line, then 
 OM + MPn/ — 1, when MP is distinguished as being at right 
 angles to OM, still means OP. That s/ — 1 is the distinction of 
 perpendicular it I/, as — 1 is of contrariety of direction, appears as 
 
 follows : the unit perpendicular to OA is U f cos - -fsin- n/^ j 
 
 or U\/ — 1; but more evidently from the consideration that if AU 
 jlenote perpendicularity, k (/cU) must denote motion through a right 
 angle more, or k^U must denote — 1 . U. 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 91 
 
 (113.) Our extension is now complete : it is yet for us to see what 
 extension the preceding makes in the remaining notions of algebra. 
 1. What is k r + k r' . Let x and x' be values of OM, and 
 
 3/\/ — 1, and y's/ — 1 those of MP for the lines just stated. Then 
 we have 
 
 k r + k r' = X + x' + (i/ + y') ^ — I 
 
 Now, the general form x -^y v/ — 1 may be converted as follows: 
 
 X -\. y s/ — \ == s/ x^ + y^ (cos0 + sine V— 1) 
 where tan0 = -. This follows from the treatment of s/'ZI\ as a 
 
 X 
 
 quantity ; for if we assume 
 rcos^ = X rs\x\& = y; then r = Va;^ +2/^ ^^"^ = -• 
 
 X + y ^ — \ = rcos^ 4- rsin^ V — 1 = s/ x^-\-y^(cosd + sin^ V^ — l) 
 
 Applying this to x -{- x' -\- (7/ -\- 7/) V— 1, and makmg also 
 x' = r'cosO' y = /sin©', we find 
 
 V'(a; + a;')^+(y + y'Y /cos^H- simpV — l), tan ^ = ^LlLJL 
 
 X -\- X 
 
 A/^2^2/2 + a/2 + ?/'2 4-2 {xx' -^-yy') (cos^ + sin^V — l), or 
 k^r + k^'r' = Vr2 + r'^ + 2rr cos (0 — 0') (cos^ + sin^ V^I) 
 
 
 3 
 
 i^^^^.'^ — // 
 
 Jlx 
 
 / \ 
 
 ^^ 
 
 
 Af f ^^.^^ 
 X 
 
 ]N 
 
 V / -k -i 
 
 L 
 
92 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 = OQ(cosQOA + sinQOAN/^) since tanQOA= ~ = 
 
 A yJ 
 
 , \, ' or the sura of O P and O P 'is OQ, the diagonal of the com- 
 pleted parallelogram, which ends at O. The difference of the two 
 lines will be found in the same way to be 
 
 ^— .r' + (y— y)V — 1 = N/r* +r'2— 2rr'cos(0— eO(cos^'+sin^'^ — 1) 
 (where tan ^' =^-^- = tan F PA,) 
 
 = PP'(cosFPA, 4- sinP'PAy^ 
 
 or the 'difference k r — k r' is PP', the other diagonal. Let OP 
 signify the line OP in length and direction. The way of settling the 
 distinction between OP — OP' and OP' — OP, is as follows. 
 The opposite of any line is found by merely changing the sign 
 of r, if we allow 6 to remain the same. For the opposite of 
 rcos0 4" rsin^N/ — 1 is rcos{9 + tt) + rsin(0 + 7r)N/ — 1 = 
 ( — r) (cose + sin0N/ — T) : that is, two opposite lines may be 
 expressed, either by changing the sign of the symbol of length, or 
 adding two right angles to that of direction. We see then that the 
 opposite of r(cos0 + sin^s/ — 1) may either be defined as r at 
 the angle -f "", or as — r at the angle 9. Now, to determine the 
 proper interpretation of O P — O P', since we are to have all alge- 
 braical formulae remain true, let us write it thus: O P + ( — OP'), 
 and add OP and — TJF, or OP and OT; the result is OV'. 
 Similarly (JP' — OP = oF -H (—OP) = OF -|- UW= OV. 
 In truth, when we came to the first square root, namely, that which 
 gave OQ, we should have ascertained that the sum was OQ and not 
 OX. This must be, for the extended definitions are entirely to con- 
 tain the limited ones. Let OP and OP' revolve towards the arith- 
 metical line, and it is finally OQ, which becomes their arithmetical 
 sum, and not OX. 
 
 (114.) The terms greater and less cannot have meaning as applied 
 to lines defined in length and direction. We may have lines greater 
 in length, or greater in direction, than others ; but OP is not greater 
 or less than OP'. Watching this more narrowly, we see that we have 
 defined equal in a sense which only applies to lines in the same 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 93 
 
 direction; this limitation, for such it is, requires a corresponding 
 limitation of the relative terms, greater and less. 
 (115.) As to multiplication, we see that 
 
 r(cos0 4- sine \/ — l) X / (cos0' + sine V — l) = 
 
 rr'(cos(e + e') + sin(e + 0') \/^^) 
 
 or, in multiplication of r and r', the result belongs to a line r7''U, 
 
 with a direction, the sum of the directions of r and r'. Similarly, 
 
 r T" 
 
 -, belongs to a line -,U, at an angle 9 — &, and so on. We see then 
 
 r r 
 
 that the angles of direction have the properties of logarithms rela- 
 tively to the lines, which is also plainly shewn by (or, if not, is a 
 confirmation of) 
 
 ^(cose + sine VlTl) = rg 
 
 Taking this convenient abbreviation, we have a set of equations, 
 
 as follows: 
 
 6^ZT , /^m , (^+^)yzT 
 
 rg X r g = rrg 
 
 c 
 
 m *" ^A , 
 
 (116.) If we look at our first method of introducing \/ — 1, (99.) 
 we see that the one just explained agrees with it, except in one impor- 
 tant particular. That first method did not bind us to any meaning of 
 the sign -f- and — before k, but left us, as we then might suppose, to 
 imagine that P -|-/cQ must be the same as P -|- Q, with a distinctive 
 mark on Q, reserving it for future operations. But I did not notice 
 at the time, wishing to avoid any generality which was not required 
 by the subject, that the definition we assigned to k entirely destroyed 
 all specific meaning for the sign -j- before k, though it continued 
 that meaning in relations among the terms independent of k, or 
 among those affected with it. For if we agree to mean by P-|-/cQ 
 ?= P' -f- /c Q' an equivalent to laying down both that P = P' and 
 Q = Q',the preceding relation remains equally true, whatever may be 
 the meaning of + before /c; since identical operations performed upon 
 equal quantities must give the same result. In applying the result 
 to geometry, we find ourselves led to a more special meaning of k, 
 for /cQ means the line Q perpendicular to the arithmetical line. And 
 
94 ON THE EXTENSION OF THE MEANING OF SYMBOLS, 
 
 with it we find a special meaning of + between a term without k 
 and one with kj namely, the hypothenuse of a right-angled triangle. 
 But still the meaning of k remains in force, for, in the geome- 
 trical application, P = P' and Q = Q'. By expressly defining two 
 lines as equal which have both equal lengths and directions, we 
 require also equal projections on both OA and OA', as is readily 
 proved from the accompanying figure. For, if MN and PQ have 
 
 6 
 
 H 
 D 
 
 JS. 
 
 a! 
 
 
 
 
 N 
 
 
 M^ 
 
 ^ 
 
 
 
 —.^C\ 
 
 
 -V 
 
 t 
 
 1 
 
 ^ 
 
 
 
 
 
 X. 
 
 K 
 
 X V 
 
 
 
 
 
 
 
 the same length and direction, then KV = LX, and GH == DE. 
 But we have 
 
 Sin = kv + ghv/3T 
 
 PQ = LX 4-DEN/:rr 
 
 (117.) We have thus converted every theorem of algebra into one 
 of geometry, not belonging to a class very useful at present, but, geo- 
 metrically considered, of great complexity. And we have thus the 
 satisfaction of putting upon every theorem of algebra a meaning which 
 is as intelligible as an arithmetical operation, when the latter is as 
 complicated ; and this on the supposition that every letter is a (till 
 now) impossible binomial of the form j:-fyv — 1. Thus, in 
 a + ft-j-c = c-\-h-\-a = 6 -|-a-f c, &c. we see the following theorem. 
 If there be any number of straight lines meeting in a point, and if the 
 conterminous diagonal of the parallelogram formed by any two be 
 made one side of a new parallelogram, and a third line another; and 
 if the conterminous diagonal of the last be similarly combined with a 
 fourth line, and so on till all the lines are exhausted ; then shall the last 
 found diagonal be the same in magnitude and position, in what order 
 soever the several lines were introduced. But theorems which are 
 
AND ON SQUARE ROOTS OF NEGATIVE QUANTITIES. 95 
 
 algebraically very simple lead to problems of great complexity. To 
 shew this we shall enunciate the geometrical theorem which answers to 
 
 {a + bf = a^+2ah + h'' 
 
 as an exercise for the student in the meaning of the extension. 
 
 If there be two finite straight lines making angles with a third finite 
 line on the same side, and all meeting in one point, and if we take 
 the third proportional to the third line and each of the two first, and 
 incline such third proportionals to the third line at angles twice as 
 great as the first and second line ; and if we also take the double of 
 a fourth proportional to the third line and the two first and incline 
 it to the third at an angle as great as the angles of the first and second 
 together; and if we then take the conterminous diagonal of the 
 parallelogram whose sides are the two third proportionals, and also 
 the diagonal of the parallelogram which has the last- mentioned 
 diagonal, and the double of the fourth proportional for its sides : then 
 shall this last-mentioned diagonal be in length a third proportional to 
 the third line and the diagonal of the parallelogram on the first two, 
 and shall make, with the third line, an angle double of the angle 
 made by the diagonal of the first two with the third. 
 
 The student should construct some problems of this kind, both as 
 an exercise in the meaning of the extensions, and the use of the 
 instruments. 
 
 (118.) I now pass to a question of much greater importance, 
 namely, the use of the preceding extensions in reasoning. As a certain 
 species of fallacy is called reasoning in a circle, I think the method 
 I am going to describe might be called reasoning in a triangle; for, 
 when there is an obstacle in one side, we pass from one end of it to 
 the other over the other two, which we shall shew are free. To return 
 to our illustration ; suppose a question involving times before and 
 after a certain epoch to be given, and suppose that ordinary alge- 
 braical reasoning, though it produce an answer positive or negative, 
 does it by means of square roots of negative quantities fallaciously 
 entering in the process. I say fallaciously (110.), because we cannot 
 extend to our notions of duration those relations which answer to 
 various directions of lines, except only, that if time before may be 
 represented by a line north, time after may be represented by a line 
 south. But a line east has no correlative in our notions of time. As 
 long as we keep the notion of time, we must, so to speak, keep in 
 
96 ON THE EXTENSION OP SYMBOLS, ETC. 
 
 the algebraical line (111 ) of positive and negative quantity. But 
 let us proceed as follows. Whatever ratios exist between the given 
 times are made to exist between the geometrical lines which represent 
 them in .... A"OA . . . . ; and if there be an answer to the question 
 in timCi there is an answer in the line A"OA to a given geometrical 
 problem, corresponding to the given question. And, conversely, if 
 there be in the line A"OA an answer to the geometrical question, 
 namely, a line whose length has all the required ratios to the given 
 lines, there must be a time whose ratio has all the required ratios to the 
 given times ; for ratios are the same things whether they be of portions 
 of duration or of length. We cannot reason on the problem throughout 
 when the concrete magnitudes are times : for, by our supposition, 
 modes of duration of which we cannot conceive the existence are 
 introduced. But we can reason on the geometrical problem, because 
 geometry can put an intelligible construction upon correlative modes 
 which exist among lines in different directions. But the geometrical 
 result when obtained, gives an answer to the problem upon the rela- 
 tions of time; not depending upon the methods which gave the 
 geometrical answer in any way, but upon a circumstance altogether 
 different ; namely, that relations among lines, positive or negative, 
 however obtained, have their correlative relations among times. That 
 is to say, we may depend upon the results of general algebra, even 
 when the concrete magnitudes under discussion are such as do not 
 admit of the geometrical extension. But, if the geometrical problem 
 give an answer which amounts to supposing a line OP not in the 
 line of algebraical positives or negatives, then we know that the 
 corresponding problems relative to such concretes as do not admit of 
 the extension, are strictly impossible, and (as yet, at least) incon- 
 ceivable in any sense whatsoever. The same reasoning applies if the 
 given problem be upon abstract numbers, with this additional limit- 
 ation, that the question is impossible unless the answer of the cor- 
 responding geometrical problem lie upon the arithmetical line O A . . . 
 We shall not in future consider it necessary to make any dis- 
 tinction between s/ — 1 and other symbols. Those equations in 
 which it occurs are rationally true in the extended sense of the sym- 
 bols, and those in which it does not occur are true in the ordinary 
 algebraical sense; for, as seen, the extended and ordinary meanings 
 coincide when the symbol >/ — 1 is neither expressed nor implied. 
 
ON THE DISTINCTION OF PERIODIC QUANTITIES, ETC. 97 
 
 CHAPTER V. 
 
 ON THE DISTINCTION OF PERIODIC QUANTITIES, AND 
 PRELIMINARY CONSIDERATIONS ON THE INVERSION OF 
 PERIODIC FUNCTIONS. 
 
 (119.) We have, in the notion of continual revolution, an idea of 
 quantity, tlie whole effect of which is represented by the angle which 
 the revolving line makes with the line of commencement ; and in 
 which, so far as position is concerned, or any ratios which depend upon 
 position, it is indifferent whether the place which the line occupies 
 be in the first, second, or any other revolution. If we were to say, 
 let the angle 27r + be the same angle as 0, we should then call an 
 angle a periodic quantity ; but this notion would not be very accurate. 
 Yet sin0, cos0, &c. are really periodic functions of 0, for as 9 increases 
 they do not increase or decrease without end, but circulate in value 
 through various changes, in such manner that what value soever any 
 one has when = ct, it has the same for a-{-2ir, a + 47r, &c. It 
 is usual to distinguish the primary functions of angles from magni- 
 tudes which depend on lines, or other continually increasing quan- 
 tities, by the name o{ periodic quantities. 
 
 (120.) The question of finding the double, treble, &c. of a quantity 
 of revolution is not affected by the multiplicity of values which answer 
 to that quantity, considered as the determiner of angular position. 
 Thus w0, 7n(27r+0), w(47r + 0), &c. are mQ, 2m7r-\-mQ, 4w7r+w0, 
 which all give the revolving line the same position as mO, when m is 
 a whole number. But when w is a fraction, the case is altered. 
 Suppose we ask, how many different quantities of revolution are 
 there, each of which repeated 10 times, will leave the revolving line 
 at an angle 9 with the primary line OA. This position may be 
 obtained by the revolutions 0, + 2 tt, + 4 tt, &c. consequently, 
 the tenth part, or the solution necessary, is contained in the formula 
 
 or . T H , the values of which are 
 
 10 10 ^ 10' 
 
98 ON THE DISTINCTIONS OF PERIODIC QUANTITIES, 
 
 To' 5 +To' T +To •••• "P'° T +T5» 
 
 after which there is only recurrence; for — -- + 77^ '* 2 ir + — -. If 
 
 then, we were to set out with a problem involving the given angle 0, 
 and the answer were, 
 
 Q 
 
 the value required is cos— - 
 
 there would, in fact, be ten answers ; for 9 might at the outset have 
 been called 27r + 9,OT 47r + 0, &c. ; and though these angles have 
 equal cosines, their tenth parts have not. 
 
 Thus, in the formula sin = 2sin J .cos J 0, we see that 
 
 sin0 is either 2sini0.cos^0 or 2sin(7r-+- ^^)cos(?r-f J0) 
 
 the latter arising from writing 2 tt + ^ for 0. And in 
 
 cosi0 + \/^sini0 = ±\/cos0 4- V— isin0 
 
 the second side has two values ; and so has the first side, its second 
 value appearing by writing 2 tt + for 0. 
 
 (121.) The actually periodic character of the series in (101.) is a 
 point of interest, which it may be advisable to verify. An angle is ex- 
 pressed in analytical units by an absolute number, and every number 
 belongs to some quantity of revolution, 1 belonging to that which 
 makes the arc and radius equal. And the remark in (63.) must be 
 particularly attended to. No result yet obtained, which involves 
 angles themselves, is true of degrees, minutes, and seconds, but only 
 of analytical units. If we take a high number for 0, the formation 
 of a few terms will make the series appear very divergent; but we 
 see that convergence must come, for, in the sine, we have 
 
 0^ 
 (n + l)th term = nth term x r—rr — rr^ 
 ^ ' 2n(2n4-l) 
 
 And in the cosine 
 
 0' 
 
 in -}- l)th term = nth term x ; r — 
 
 (2n — 1)2 n 
 
 So that, however great may be, terms must arrive at which 
 2n(2n + 1) and (2n — l)2n bear as great a ratio as we please to 0*, 
 or the (n-|- l)th terms may be made parts as small as we please of 
 the nth. And the terms being alternatively positive and negative, 
 the magnitude of the latter kind will compensate that of the former, 
 
AND ON THE INVERSION OF PERIODIC FUNCTIONS. 99 
 
 leaving a difference always less than unity, as should be the case 
 from (36.). To shew this, we shall take some terms of the sine and 
 cosine of the angle 10, something greater than 37r. We begin with 
 10, and the process is continual multiplication by 10, and division by 
 successive numbers. Let (ti) mean lO** divided by 1.2.3 . . . . w, 
 then we have 
 
 sinlO = (1) - (3) + (5) - (7) + .... 
 
 coslO== 1 - (2) + (4) - (6) + .... 
 
 To pass from (n) to (n + 1) we must multiply by 10 (move the 
 decimal point one place to the right) and divide by n + 1 . To get a 
 sine and cosine from this (at the beginning) very diverging series, to 
 a single place of decimals, will require about 29 places of decimals 
 to be considered in the first term. Let us begin with 
 
 (1) = 10-0000 29 ciphers 
 
 (2) = 50-000 
 
 (3) = 166-66 &c. 
 
 Proceeding in this way, and keeping two decimals from each term, 
 we find as follows : 
 
 (1) = 1000 
 
 (11) = 2505-21 
 
 (21) = 
 
 19-57 
 
 (2) = 50-00 
 
 (12) = 2087-68 
 
 (22) = 
 
 890 
 
 (3) = 166-67 
 
 (13) = 1605-90 
 
 (23) = 
 
 3-87 
 
 (4) = 416-67 
 
 (14) = 114707 
 
 (24) = 
 
 1-61 
 
 (5) = 833.33 
 
 (15) = 764-72 
 
 (25) = 
 
 -64 
 
 (6) = 1388-89 
 
 (16) = 477-95 
 
 (26) = 
 
 -25 
 
 (7) =1984-13 
 
 (17) = 281-15 
 
 (27) = 
 
 •09 
 
 (8) = 2480-16 
 
 (18) = 156-19 
 
 (28) = 
 
 -03 
 
 (9) = 2755-73 
 
 (19) = 82-21 
 
 (29) = 
 
 -01 
 
 (10) = 2755-73 
 
 (20) = 41-10 
 
 
 
 Hence, (1) + (5) + (9) + .... up to (29) 
 
 = 5506-33 
 
 = A 
 
 (3) + (7) + (11)+ .... up to (27) 
 
 = 5506-90 
 
 = B 
 
 sin 10 = A — B nearly = — -57 
 
 1 + (4) 4- (8) + up to (28) = 5506-20 = C 
 
 (2) + (6) + (10) + up to (26) = 5507-03 = D 
 
 cos 10 = C — D nearly = —-83 
 
 10 =: 3 TT + -58 nearly, or coslO =: — cos-58 sinlO = — sin-58 
 .58 9 = -58 X 57-31° = 33-23 nearly in degrees. 
 
100 ON THE DISTINCTIONS OP PERIODIC QUANTITIES, 
 
 And from the tables sin55°i = -54, cos55°J = -84, whence the 
 first decimal is right, as we proposed. We have thus shewn how to 
 verify the actual coincidence of the series with the sine and cosine 
 of the tables. 
 
 (122.) When an operation is performed successively upon a quan- 
 tity and its results, if we denote the operation by^, the results of the 
 repeated operations may be denoted by ff^ fff, &c., which may 
 be abbreviated into f^y f^^ &c. So that suppose x the quantity 
 operated upon, and 2x4-1 the operation to be performed (that 
 is, let f represent a direction to double, and then add one), we have 
 
 fx = 2x + \, /2a; = 2(2a:+l)+ 1 = 4x4-3, 
 
 px = 2{2(2a; +1)4-1} + ! = 8a; + 7, &c. 
 
 The convenience of this notation consists in the analogy which 
 exists between the indices of/ and algebraical exponents. Thus 
 we have 
 
 f^px =f*x = 2[2{2(2x+ 1) + 1} + l] + 1; 
 
 the only difference being, that in f^f^ we consider the preceding as 
 having all between {} first finished, which total operation is then 
 repeated ; and iuf^x we consider all the operations as separate. But 
 both results, when developed, are, in fact, ffffx. To preserve the 
 same equation, we must let/°j: signify x, in order that 
 
 ff^x may be/^+^ar, orf^x, or fx ; 
 
 and the meaning of f~^x may thus be given. Let it be such that 
 the equation /'""'""ar = /'"(/"j^) still exists, and we must then 
 have 
 
 ff-'x=p-'x=rx = x; 
 
 or f~^x means the inverse of Xy the function whose effect is reversed 
 or undone hy f; so that if/~ be first performed upon x, and then 
 /, the latter restores x again. Thus, ii fx be x^jj'^x is s/x, 
 because /^'"^x or (>/x)* is x. 
 
 Since /'""'""x is either /"/"x, orff"'xy to preserve this equa- 
 tion, we should stipulate that/~yx should be the same as/Z'-^x, 
 or that we should only allow functions satisfying this condition to be 
 called /~*x. Thus in x' and Vx, we see that we have 
 
 (Vi)^ = X, and, changing order of operation, ^ x^ = X. 
 
AND ON THE INVERSION OF PERIODIC FUNCTIONS. 101 
 
 But if we take — n/jt iox f^ x, fx being x^^ we find 
 (— Vj:/ = Xf — n/^2 _ ^x, f'^fXj no; being = X. 
 
 To elucidate this point, we must observe that, in algebra, direct 
 operations, which produce but one value out of one value, are gene- 
 rally accompanied by inverses which produce more values than one ; 
 addition and multiplication only excepted. What do we mean by a 
 square root? The expression, which squared, gives the original. 
 This, with regard to x^, answers both to x and — x. But if we had 
 been considering x^ as an operation to be repeated, giving {x^y, &c. 
 and had from thence, in the preceding manner, deduced an idea of 
 an inverse operation, we should have found that if 
 
 fx = x^, and iiff''^X axidf^^fx, are both to be = x, 
 
 we can only admit/- ijr = s/xy and noty-^o: = — Vx. But because 
 it has been customary in algebra to admit all functions to the name 
 of i/iverses of y, which satisfy J[y~^ a:* = x, without inquiring whether 
 they satisfy y'^yj;" = or, we must here adopt the name, but with a 
 distinction of notation. Let those forms be called convertible inverse 
 functions of/, and be denoted by/~^, which satisfy hoihff'^x = x 
 and f^fx = X. And let those forms be called inconvertible inverse 
 functions of/, and be denoted hyf_-^x, which satisfy oi\\yff_^x=. x, 
 and notf_^fx = x. Thus, when /a; = x"^, f~^x = "^ ^j f-\^ = 
 — ^/ X' But it is right to warn the student that, in other works, 
 /"^ and/_j are not distinguished. 
 
 (123.) Let us now turn to q.q%x. What is its inverse function? 
 Let cos J? be denoted by/jr; then 
 
 g^ V^ _ coso? -h V 1 — cos=a; V — 1 =/a:+Vl — {fxy- V — i, 
 
 (as in my Algebra, p. 123, I always mean by v j; the positive square 
 root, and by x^ the general form, either positive or negative). This is 
 true only for angles less than tt (of all in the first revolution) ; for if 
 x be greater than tt, we must have 
 
 z""^^ = cosj: + sino; >/ — 1, 
 
 and the second term (44.) is negative ; 
 
 k2 
 
102 ON THE DISTINCTION OF PERIODIC QUANTITIES, 
 
 whence g' ^^ = /a: + V]-(/a;)2 s/ ZTf; 
 
 (j; between and tt, 27r and Stt, &c.) 
 
 (jT between ir and 27r, 3 tt and 4 7r, &c.) 
 both included in g» a/=i = /a: + (l — (/a;)^)^ V^T, 
 (for all values of x.) 
 Now, if for X we write a; + 2n7r, /x remains as before, and so 
 does g' : we have then every possible form of the preceding 
 equation in 
 
 g.V-i+2««-y--^y(^^27icr) + {l-(/^+2^'}V3l; ...(A) 
 
 in which there is identity of value in both sides of the equation for 
 all values of n. But before we proceed a step further, we have an 
 important remark to make, the neglect of which for a long time 
 embarassed this subject. 
 
 (124.) Identity of form means positive and absolute identity in 
 every respect. We see it in 
 
 X =^ X, X + a = X + a, cosa; = cosa:. 
 
 Identity of value includes all other cases in which the sign = 
 applies. We see it in 
 
 X = x+a—a, (b — ay = {a — by, cos(a:+2T) = cosa:. 
 
 All operations, general or limited, performed upon identities of 
 form, produce the same results. We see this in 
 
 ^x = '^ x as well as x^ =. x . 
 This proposition is only worth stating as a contradictory of the next. 
 
 Limited operations performed upon identities of value with differ- 
 ences of form, do not necessarily produce the same results. Thus, 
 
 (J— a)2 = (a— ?>)2 does not give ^(^b-^ay = \^(a — by, 
 or b — a z= a — b, 
 but it does give \{b — ay\^ = \{a — by\^, 
 that is, every value of the first side is a value of the second side, but not 
 
AND ON THE INVERSION OF PERIODIC FUNCTIONS. 103 
 
 at our pleasure. It iss/ x on the first side, which is equal to the 
 — '^ X of the second, and vice versa: if we attempt to displace this 
 arrangement, and make v jt of the first side equal to v j; of the 
 second, we have no longer an identity ^ but an equation of condition 
 {Algebra, p. ix.) J for then b — a = a — 6, or a = 6. 
 
 (125.) Conversely, when we find that an operation performed upon 
 both sides of an identity of value gives different results, it will be most 
 judicious, before proceeding further, to examine first the operation in 
 question, to see whether it be not a limited case of a more general 
 operation. And we see that an algebraical equation, of which the 
 sides admit of different values, such as arises from performing a 
 general operation upon a general form, may be considered as an 
 ambiguity of this form, 
 
 Ai, or Ag, or A3 ... . = Bi, or Bg, or B3 . . . . 
 
 in which, though we know that Aj has its equal on the other side, 
 we have no right to say that it is Bj, for it may be any other. And 
 the same of Ag, &c. And all we know as yet, and indeed all we 
 shall find, will shew us that, in operations conducted in all their 
 generality, there will be complete identity of this kind: every value 
 of the first side will have its value on the other. We shall not have 
 Aj and A3 both equal to Bj, and Bj left without a value. 
 
 As long as our inversions only involved the production of two 
 values, as in the square root, or of three, as in the cube root, there 
 was little need to embarrass the subject by general ideas upon inver- 
 sion : each case was the best index to its own peculiarities. But 
 now, when the simplest cases we have to consider give infinite num- 
 bers of inversions (seeing, for example, that though an angle has but 
 one cosine, a cosine has an infinite number of angles), we must 
 watch our processes very narrowly. By trusting to limited notions 
 of inversion, many errors have been introduced, some of which we 
 shall point out in the sequel. 
 
 Returning to equation A, we must first find the arithmetical 
 logarithm of the second side. We shall need some considerations, 
 which the student will read with the more attention, when he knows 
 that he is thus in reality entering upon a most essential part of his 
 future studies, namely, the Differential and Integral Calculus. 
 
104 NOTIONS OP THE DIFFERENTIAL 
 
 CHAPTER VI. 
 
 NOTIONS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS, 
 
 SUCH AS WILL BE REQUISITE IN THE SUCCEEDING 
 
 CHAPTERS. 
 
 (126.) Let any function of Xyfx^ have a certain value given to jt, 
 namely, o. Then let x have the value a -j-Zt given to it, and let the 
 difference of the resulting values of the function be taken, namely, 
 /(a + h) — fa. Instances : 
 
 ga+h — gd, sin(a + ^) — sina, (a -f- A)^ — a2. 
 
 This difference, if the function be continuous (^/geira, p.l02), is one 
 which diminishes without limit at the same time as h. When the 
 
 fraction -^ ^ {Algebra, p. 156) has a finite limit, let it be 
 
 called the derived function of fx, for the value x =■ a. Or, in 
 
 f(x + li) fx 
 
 general, let the limit of — j- — —, made by diminishing h 
 
 without limit, be styled the derived function of fx, if it be a rational 
 and continuous function of x. Let it be denoted by D, that is. 
 
 Limit (if there be one) of /(^ + ^)-/-^ ^ D^^,, 
 For example {Algebra, p. 225), 
 g^ + '»-g^ gft_l /. h h^ 
 
 = ,-^ = ,(1^1^^^^....) 
 
 h ^ h " \ • 2 ' 2.3 
 
 Therefore (^%e6ra, p. 157.) Dg"" = g' 
 
 sin Car 4- A) — sin x cos h — 1 , sin A j. /,^^«,v 
 
 ^ T . ^ s= sma: T -\- cosar —r- and by (46,47.) 
 
 h n n 
 
 Dsino: = sinar X H- cosa: X 1 = cosx 
 
 cos(j:+A) — cosj: ^^„ ^ <^°^ ^ — ^ ^;^ ^ sin A / . - .. x 
 
 — ^ J - / = cos a: 1 sma; — r- (46, 47.) 
 
 h n n 
 
 Dcosx = cosx X — sino; x 1 = — sino: 
 
AND INTEGRAL CALCULUS, ETC. 105 
 
 (127.) Now precisely the same results are obtained if we use 
 
 h — k 
 and diminish h and k without limit. For instance, 
 sin(j;4-A) — sin(j; + /c) . cos^ — cos/c , sin A — sin A; 
 
 Lk — - = '""" ■ h-k + '°^^ k-k 
 
 = - siuxsmi{h + k) -jXL-— ^ +cosa;.cosJ (A + k) x\h^j,^ 
 
 the limit of which, diminishing h and k without limit, is as already 
 obtained, 
 = sinjr X X 1 4- cosj:' x 1 X 1, or cos ^, that is, DsinA\ 
 Let the student try the other instances in a similar manner. 
 
 (128.) Theorem. Required the relation which must exist between 
 two functions which have the same derived function. 
 
 Let the derived function in question be P, and let Q and R be 
 two functions, of botk of which the derived function is P. Let 
 R = Q + T, T being the difference of Q and R ; and when x is 
 made x -j- h, let R', Q', and T' be the values of the three, whence, 
 since R = Q + T is supposed to be an identical equation, we 
 must have it true for all forms of x, and therefore we must have 
 R' = Q' + T' ; whence 
 
 R^ — R __ Q^ — Q r~T 
 
 h h h 
 
 Take the limits of both sides {Algebra^ p. 156), diminishing h without 
 limit, and we have 
 
 DR= DQ + DT, or DT = 0, 
 
 because DR = DQ = P. Consequently T must be a function, 
 whose derived function is nothing for all values of x. Now let 
 T(x) denote this function, where we make T the symbol of the 
 functional operation, and express the quantity x, because we wish to 
 express different values of it. Then let us suppose 
 
106 NOTIONS OP THE DIFFERENTIAL 
 
 T(a; + 3A)-T(a: + 2A) = V3 
 
 T(a: + w-U)-T(a; + /i-2A) = V„_i 
 
 And by addition, T(a7 + TiA) — Ta; = Vj + Vg + + Vn. 
 
 Now, let us diminish h without limit, increasing n at the same 
 
 time in such manner that nh shall be always equal to a quantity 
 
 Jixed in valuCf but what we please (keep this in mind), and called 
 
 y. (See the process in (101.)). By the nature of this derived 
 
 V V 
 function, ~, -r . . . • must all diminish without limit with h, for that 
 ti h 
 
 is implied in the limit of these fractions (which are values of the 
 derived function) being always = 0. Consequently, each of V,, Vj, 
 &c. can be made as small a part as we please of hy and Vi + Vj-j- . . 
 . . + Vn as small a part as we please of n^, which is always y, or 
 the sum preceding can be made as small a part as we please of a 
 finite quantity, i e. diminishes without limit. Hence, T{x ■\-y) — T(y) 
 a quantity independent of A, diminishes without limit with A, which 
 is absurd ; for, not containing h at all, or any quantity whose value 
 depends on A, it is the same whatever h may be. Whence does this 
 absurd conclusion arise? Not from Jthe reasoning, but from the 
 initial supposition, namely, T (x -|- A) — T(x), &c. are quantities 
 depending on h. If they depend on h at all, all the preceding rea- 
 soning follows, and the inadmissible conclusion. The truth must be, 
 then, that T(x-\-h) — T (x) is not a function of/* at all. Neither then, 
 does {T(a: +A) — T(a:)} + T(x) contain A, for we have not added a 
 function of h. That is,T(x + h) is not a function of h; that is, T(j:) is 
 not a function of x. For, if it were, h would be found in T(xH-A). 
 Therefore T(x), supposed to be a function of x, and, therefore so 
 expressed, turns out to be a quantity independent of x. Such a 
 quantity is called a constant with respect to .r. Hence the conclusion 
 is, that if Q and R have the same derived function P, then Q can 
 only differ from R by a constant quantity. 
 
 To verify the positive part of the conclusion, suppose/x = Fx +C, 
 
AND INTEGRAL CALCULUS, ETC. 107 
 
 C not changing when x changes. Then fix + A) = F (j: ■+" A) + C, 
 and 
 
 h '^ h 
 
 consequently the limits (A diminishing) are the same, or D/x = DFx 
 
 (129.) Theorem. If 0a; be the derived function of^o;, then tl 
 function off^^x, any inverse ofya: satisfying^/*_ja: = x, is 
 
 (129.) Theorem. If 0a; be the derived function of^o;, then the derived 
 
 1 
 
 For /-i(^ + ^)— /-i^ _ /.i(^ + A)-'/-iJ^ _ /_i(a:4-A)— /-i^^ 
 = 1 divided by -^'^-i^^ + ^^-'^^-i^ 
 
 Let/lio: = y, yiiC'^ + 'O = y + ^» ^hen /c and h diminish 
 without limit together. Substitution gives 
 
 /-i(^+A)-/,i^ ^ ^ ^.^id^j ^y /(3^-ffe)-/3^ 
 
 Let A diminish without limit, in which case k does the same, and 
 we have 
 
 D/_ia: = 1 divided by 'Dfy=.~= ~^^ 
 
 For instance, let fx = o:^, f_^x^= — s/'^ 
 
 Then D/a; = 2a; = ©^ — = ~ 
 
 ^ ^ (px 2x 
 
 ^"^ ^ (p{—'>/x) ~~ —2s/x "" 2>/^ 
 
 Let/o; = sin 07, then we make /_j or = an angle whose sine is x. 
 
 Dtx = cosa; = ©a; — = 
 
 '^ ^ (px cos J 
 
 •^ ^ \ sme = X / 
 
 1 1 
 
 0(ang. wh. sine is x) cos (angle, &c.) 
 1 1 
 
 cos (angle whose cosine is (1 — x^)^ ) (1 — .r')^ 
 The ambiguity of the sign is a subject to be hereafter considered; 
 when actual application takes place, it depends upon the following 
 proposition. 
 
108 NOTIONS OF THE DIFFERENTIAL 
 
 (130.) D/x is positive for all values of x, at which, when x in- 
 creases or dirainishes,yjr does the same; and negative when/j dimi- 
 nishes by increase of jr, or increases by diminution of x. We must 
 suppose in this proposition, that the increments or decrements may be 
 as small as we please; and the general algebraical meaning of increase 
 and decrease is contemplated. Its converse is also true. 
 
 Firstly, let f{x -j- h) be greater than fxy for any value of A, 
 however small ; that is {Algebra, p. 63.) let f{x + h) — fx be 
 positive. Then, h being positive, we have (J'x -}- h — fx) -J- A is 
 always positive, and its limit D/ir must be positive. Conversely, let 
 T>fx be positive, then there are values of h which are so small, that 
 f{x -\- h) — fx must be positive. For if we make 
 
 fj£+^^ = -Dfx + U, 
 
 then H must be an expression which diminishes without limit with h. 
 The latter can then be taken so small, that Dfx -f H shall have the 
 sign of D/'r, that is, shall be positive; that is, h being positive, 
 h{'Dfx-\-}i) shall be positive, or f(x-\~h) greater ihsiu fx. 
 
 The other cases of the direct and converse proposition can easily 
 be established in the same way. 
 
 (131.) Let X increase from — a to -\-a, where a may be as great as 
 we please, so that we thus suppose a variation of j: between any limits 
 of magnitude which it may be necessary to consider. We abbreviate 
 all this into : Let x increase from — oc to -j- oc : then the preceding 
 shews, that whenever {x increasing) Dfx changes from -\- to — , the 
 function ceases increasing, and begins decreasing, and vice versa. 
 
 Apply this theorem to Dsinx = cost, and shew that the sign 
 of the cosine corresponds to the current method of variation of the 
 sine in the manner described in the theorem. 
 
 The following theorems may be established by the direct process 
 in (126.). Symbols not containing x, are supposed independent of a-. 
 
 If (px = afx, D^a: = a Dfx 
 
 If ipx = afiX + bf^x — cf^x, T>(px = a Df^x + h Df^x — cI>f^ ■< 
 
 Bx"*" = limit o(^d±!!l^ZZlJl = limit (wx^-^ + m ^^^ x'^-^h +••.•) 
 
 {Algebra^ Yi. 217), or Dx"* = mx"*~^. 
 
AND INTEGRAL CALCULUS, ETC. 109 
 
 Dr = limit of 7 , or limit of 1. But 1 being in- 
 variable, as h diminishes without limit, it remains the same, or 
 Do: = 1. Similarly, T)ax := a, 'D{ax-\-b) = a. 
 
 D(ao + cLi^ + «2^^ + ••••) = «i + 2aoX + Sa^x^ +.... 
 
 This being universally true, we may write a^ for 2a..^, a^ for Za^, 
 &c. which gives 
 
 (T* x^ \ 
 
 Go +«l^ + ^2 2 +"33 4- ... j = «i+«2^ + «3^~ + &^C. 
 
 (132.) We have hardly thought it necessary to state that if two 
 functions be always the same in value, their derived functions must 
 be the same in value. But (124.) we must not take a limited form 
 of one, and equate its derived function to that of a limited form of the 
 other, without first ascertaining that our limited forms are really 
 equals. 
 
 The derived function of a function of a function of x is thus 
 found. 
 
 Let us suppose <p{fx). Let/(jr + h) =fx + H ; then we have 
 
 D^fx = limit of »./V + ;)-»/^ = limit of tif^}=M^ . H 
 ^•^ fi II h 
 
 = limit of '-^ ^ ^^^^— X limit of'— ^^ — -—^ — '-L- 
 
 11 n 
 
 Now h and H diminish without limit together, and the first limit 
 would be D^i", but that/j; is in the place of .r. It is, therefore, 
 what D^or becomes when^jr is written for x. The second limit is 
 T>fx ; whence 
 
 -p. /. TV /with .r afterwardsN t\ /• 
 
 J)^fx = T><px{ .hanged into/J >< ^^/^ 
 
 Thus Dcosma; = —sina;(jr changed into wj,r) X Dmx 
 (131.) = — smmx X m 
 Dsin(^^) == coso^' (change 2> into r') X T>x^ 
 =z cosa;^ X bx^: and so on. 
 
 (133.) Dlog.ris limit of ^ (log.r+A —log^r) ; or of ^ log (1 + -); 
 
110 NOTIONS OP THE DIFFERENTIAL CALCULUS, ETC. 
 
 or 
 that is, 
 
 Dloga: = - Dlog/x = — x T>fx 
 
 We shall now be able to proceed with the expression of the 
 inverse trigonometrical functions, and with the extension of what has 
 been already done with the direct functions. 
 
CONTINUATION OF CONNEXION OF FUNCTIONS. Ill 
 
 CHAPTER VII. 
 
 CONTINUATION OP THE CONNEXION OF DIRECT AND INVERSE 
 TRIGONOMETRICAL FUNCTIONS. 
 
 (134.) By sin~^x, cos""^x, tan'^j, we mean in analogy with (122.) 
 ihe angles which have x for their sine, or cosine, or tangent, when we 
 have both sin(sin'"^j) = x, and sin~\sinjr) = x. That is, of all 
 the possible angles contained in such a formula as db 2m7r, and 
 having the same sine, there is one specific angle which is denoted by 
 sin~^j:, answering to the convertible inverse f^x. But all other 
 angles contained in the same formula will be denoted by sin^jx, 
 answering to the inconvertible inverses denoted generally hyf^^x. 
 Whatever a sine, or a tangent, may be in value, there is an angle 
 
 lying between and -\ — , to which that sine or tangent belongs. 
 
 2 ji 
 
 Not so with the cosines, all of which are positive between those 
 limits. But all cosines are found to belong to angles between and 
 TT. Let the fundamental angles to which any given primary func- 
 tion is attached, be chosen between those limits. For instance, by 
 
 sin~^( — 4), is meant—-; but we have 
 
 sin_i( — i) = — ^±2^*^, or ^— (— ^±2?WtJ 
 
 6 
 that is ^ ± {2m + 1)'T 
 
 We find, in fact, the following general equations: 
 sin^iX ^ sin^'^x ±.2m<:r or ± (2m + 1)<t— sin'^o; 
 cos^iX ^ cos'^x ± 2m<7r or ± 2m'7r — cos~^x 
 tan^jo; = tan"^a; ± 2m'T or tan'^a; ± (2m -f !)'»' 
 
 These are, in fact, but a combination of the propositions that there 
 are in the first revolution two distinct angles, having the same sine, 
 
112 CONNEXION OF DIRECT AND 
 
 cosine, or tangent, and that any number of revolutions added or 
 subtracted, does not alter the sine, cosine, or tangent. 
 (135.) We have shewn (129.) that 
 
 Dsin-^ar = (1 -a,-2)-i, 
 
 but which sign we should take has not been determined. If we 
 consider that while x increases from — 1 to -f- 1, the angle increases 
 
 from — -to + -, we see (130.) that the positive sign must be 
 chosen, or that we have 
 
 Dsin"*^a; = the positive value of (1 — X-)"^. 
 Expand this by the binomial theorem, which gives 
 
 Dsin-.^ = l+la-^ + Lix* + i^^+ 
 
 a series which (in common with all others of the kind derived from 
 the binomial theorem, Algebra^ p. 210) is convergent when x^ < 1, 
 or when x lies between -|- 1 and — 1, that is, in every case we 
 propose to apply it to. And by (131.) we have 
 
 •n • -1 T\( . 1 x' , 1 .3x5 1.3.5x' V 
 
 \^23 ^2.4 5 ^2.4.6 7 ^ / 
 
 But (128.) two expressions which have equal derived functions, can 
 only differ by a quantity independent of x, whence we have 
 
 • 1 i-. . , 1 x^ 1.3 jr« 
 sm-i:. = C+^+^3 +-,3 + 
 
 where C is as yet undetermined. We find it thus: since it is 
 independent of x, whatever value it has for one value of x, it has the 
 same for all. Let then x = (read carefully Algebra, p. 189.) The 
 
 only angle between — - and -f- -, which has for sine, is 0, 
 
 and the preceding then becomes = C -f- 0> or C = 0. Tliat is, 
 
 . , .1 x^ 1.3 j» . 1.3.5 x' , 
 
 sm-^X = X+-- +--- +2—67 + .-•• 
 
 Now, the only angle between and tt which has x for cosine, is 
 — sin ~^x, so that we have 
 
 , TT la' 1 . 3 X* 
 
 cos-^X= 2-^-23 -2745 -•••• 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 113 
 
 which is the same result as we should obtain by going through the 
 same process from the beginning, as we now proceed to do. 
 
 (136.) Returning to (129.), \etfx = cos a-, then Bfx or ^.r = 
 — sin x ; therefore, 
 
 Dcos "^x = — 
 
 sin (cos -1 j:") {[--x-f 
 
 where the sign is undetermined ; but as x diminishes from -f 1 to 
 — 1, while cos~^j: increases from to tt (the limits (134.) between 
 which it is contained), we must take a negative sign for Dcos-^x, or 
 a positive sign for (1 — j:^)*. We have, therefore, by the same steps 
 as before, 
 
 Dcos-ia:= -(l +lx^ +hl,x'+ ....) 
 
 1 13 
 
 or cos -^x = C — 1 — - o;^ — r-^T a;* — &c. 
 
 2 2.4 
 
 and C may be determined, as before, by making x = 0, which gives 
 cos^^O = C. But between the limits of definition, - is the only 
 angle of which the cosine is 0, and we have, therefore, 
 
 cos~^a: = - — X X'^ —x"^ — .... as before. 
 
 2 2 2.4 
 
 (137.) But we must now avoid all appearance of an unusual 
 degree of caution without any particular reason shewn, by further 
 examination of the proposition in (128.), namely, that if D ^ a- =: 
 Dfxj then <p x —fx must be a constant independent of x. That 
 demonstration, conducted by reducing ^.r — j'x = Tx, a real function 
 of X, to an absurdity, supposed throughout that D T jr presented, be- 
 tween T X and T (x -f- y\ no peculiarities which would remove it out 
 of ordinary rules, x and y being specific quantities : and also that 
 D (^ JT — /x), or D ,r — D/jt, fulfilled the same condition. But 
 there may happen cases in which ^(^x •\-K) — ^ j:, or f{x -f K) — fx, 
 do not diminish without limit when h diminishes without limit, or at 
 least do not exhibit a form which will entitle us to draw that conclusion 
 
 vvitliout further examination. If, for instance, ^x = -, and x = 0, 
 
 j: -h A = A, we can apply no conclusion of ordinary algebra to v , 
 
 when h diminishes without limit. We might investigate a large 
 
 L 2 
 
114 CONNEXION OF DIRECT AND 
 
 number of cases, but we shall confine ourselves to one, namely, that 
 in which either of the derived functions becomes infinite. And first, 
 having shewn that so long as there are derived functions D ^ j 
 and DyXf within the sense of the definition, we must arrive at 
 T{x-\-i/) = Tjt, or Tx is independent of x; we may assume ^(jr) 
 =fx + (the same constant from x = a to x = b) provided that 
 between these limits D^.r and T>fx are finite; if, therefore, a 
 change take place in the value of the constant, it must be at the 
 point which our reasoning does not include. For example, let 
 Dfx be finite and intelligible from x = a to x = bj and also from 
 X = b -^k to X = c: but between x = b and x = b-\-k let there 
 be a value which makes DJ'x = oc . VVe know, then, that if 
 l><px = DfXf 
 
 r (C,to be determi ned,but 
 from X =i a to X = b, <pa: =fx -f-< of one value between 
 
 [ those values of x). 
 from X = b + ktoX =i= c, (px =Jx + (C, ditto ditto). 
 
 Are we at liberty to say that C, must be C ? Certainly not : for 
 if we attempt to reason by the process in (128.), from x = a to j = c, 
 we include the case about which we can draw no conclusion, where 
 D/jT = a, and where it is by no means evident that Vj+Vg^ .... 
 4-Vn (in that article) will diminish without limit. We cannot shew, 
 nor is it always true, that when T)fx is infinite, a change takes place 
 in the constant; but we have shewn the converse, namely, that if a 
 change do take place in the constant, it must be when Dfx under- 
 goes some remarkable change of algebraical condition, either passing 
 through infinity, becoming impossible, or the like. Let us suppose 
 ourselves at a point at which the change of value takes place, and let 
 it be when x =i t. Then, when x is t — h, the function in question is 
 represented by f(Jt — A) + C, but when x — t-\-h,hy f{t + A) -f C, . 
 
 Hence, calling ^ x the general form of the function, 
 
 (p{t-\-h)-(p{t-h) =/(^+/0-/(^-/O + Ci-C. 
 
 Now, in cases where D<px and Dfx are finite for x =z t, 
 ,(, + A) -,(,_,) ,„,. /(^ + *)-/(^-A) ^ ,,,„ (,„.^_,^<,.,.. 
 
 nishingA without limit, give the derived functions of 0j and/x; and 
 from the preceding it appears, that the difference of those derived 
 functions will be the limit of C — C) -r- A, or infinite. But the 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 115 
 
 difference of two finite quantities cannot be infinite ; whence we may 
 conclude, that D^x and D/x are not both finite, and as they are 
 always equal, both will increase without limit together. 
 
 (138.) The preceding results (135.) may be immediately made to 
 confirm this conjectural reasoning, for it is hardly more : we see that 
 
 from X =■ to X = H — , we have 
 
 . -1 1 x» 1.3 j;5 , 
 
 s,n x = x+~- + _- + .... 
 
 which, when x = 1, gives a slowly converging series for — . But x 
 never increases beyond 1, therefore this series can never represent 
 2 7r + sin~ x, which is the first value of sin __^x greater than -, be- 
 longing to the supposition on which this series was deduced (135.), 
 namely, that X and sin -l.r increase together. In fact, if we suppose 
 X to vary from — 1 to +1, sin-i x then increasing^ we shall find that 
 
 l)^to(4m + l)| 
 
 SO that the general value of sin_ij; (when x and sin^ix increase 
 together) is as follows. That value which lies between 4 m — 1 and 
 Am -\-\ right angles, is 
 
 ^ \ x^ 1 . 3 x^ . 
 
 2m, + ^- + -3 +— -+ .... 
 
 for all values of w, positive and negative. 
 
 Similarly, the value of cos_i.r, which lies between 4w and 
 4»j + 2 right angles (which includes the cases where cos-^x decreases 
 as X increases, as in (136.)) is 
 
 sin'^X 
 
 varies 
 
 from 
 
 ""2 
 
 2-T+ sin-^o; 
 
 varies 
 
 from 
 
 37r 
 2 
 
 2m'7: -\- sin"^a; 
 
 varies 
 
 from 
 
 {Am 
 
 (4-+l)i 
 
 _ 1 _ Li? ff 
 
 '^ 2 3 2.4 5" 
 
 And at the points where the constant changes x is +1, or — 1, and 
 Dsin_jX, and Dcos.jX (129.) become infinite. In a note at the 
 end of the work we shall collect the various cases, including the 
 change in the form of the series itself, arising from the change of 
 sign in the differential coefficients. 
 
116 CONNEXION OF DIRECT AND 
 
 (139.) We have now to consider tanx and tan_|X. The deve- 
 lopement of the former in a series of powers of x is of little conse- 
 quence as a result ; but the methods by which it will here be obtained 
 are important in other and more useful deductions. 
 
 Theorem. A function of x, which is unaltered by changing the 
 sign of x, or which satisfies the equation <px = (p{ — x), cannot be 
 expanded in a series of even and odd powers of x, but can only be 
 expanded in even powers : and a function which is changed in sign 
 only, and not in numerical value, by changing the sign of x, or 
 which satisfies the equation ^( — x) = — ^(-r), can only be expanded 
 in odd powers of x. 
 
 Let <px = ao-{- aiX + ttQ x^ -^-a^ x^ -{■ a^ X* -{- 
 
 Then <p{ — x) = Oq — a^^ X + a^ x^ — a^ x^ + a^ X* — 
 
 If ^(x) = ^( — x), addition and division by 2 gives 
 
 <px = Oo -\- Oo x^ + a^ x"^ -\- 
 
 If (p{ — x) = — ^x, subtraction and division by 2 gives 
 <px = OiX + a^ x^ + 05 x^ -\- 
 
 In the first case, then,^!^; + a'sX^-\- • • = or (Algebra, p. 188) 
 
 flj = O3 = &c. 
 In the second case, ^^ -f. ctg x^ -f- . . = or (Algebra, p. 188) 
 
 (140.) The derived function of tanx is , or 1 4- tan^'x; 
 
 cos'^x 
 for (56.) 
 
 tan (x + ^) — tan x sin(x4-A — x) 1 sin A 
 
 h A.cos(x-|- A)cosx cos(x4-/0cosx * h 
 
 the limit of which gives Dtanx = — — = 1 -f tan^'x. 
 
 cos'x 
 
 Again, tanx, when the sign of x is changed, becomes — tanx, or 
 tan( — x) = — tanx; whence (139.) the series for tanx must consist 
 of odd powers only. We have, therefore, to investigate what series 
 of odd powers, P, will give DP = 1 + P'. 
 
 Let p -- ^^^ _{_ Q^ ^3 ^ a^x'^ i- Oj x"^ + oox^ -j- 
 
 Then, DP = a^ + 3asX^-h5a,x^ + 7aja^-\-9ao3.''+ ••.. 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 117 
 
 P2 = a'.x'^ -\-1a^a^x^ ^-la^a^oi^ -^la^a-ix^ -^ .... 
 
 4- .... 
 
 Make DP equivalent with 1 +P^ which gives 
 
 1 2 
 
 fli = 1, 3fl3 = n\ or ^3 = -, 5^5 = 2flfi«3 or g- = ^5' 
 
 ^ r» 2 4 1 17 17 
 
 7a7 = 2aifl5 + «3 = 13 + 9 = sTTs ^'^ ^^ = 0^5' 
 
 n o . o 34 . 4 62 62 
 
 H = ^a^a, + 2«3«5 = 3^7^ + iT5 = ^tTTT^' ^9 = 3:7:91:^ 
 .v-hence, a" + ^ :i;3 + A^5 ^ ^ ^.7 ^ _|_ ^9 + .... 
 
 is the only series of odd powers which has a property such as the 
 tangent must have ; namely, to be a function of x, whose derived 
 function is 1 + the square of the function. But tanjr, if developable 
 at all in powers of .r, must be developable in odd powers only (139.) ; 
 consequently, the preceding series must be the tangent of j:, if there 
 be any series whatever of whole powers of x which is = tan x. But, 
 
 by common division, we may see that sinr, or x — — + .... 
 divided by cosx, or 1 — - -|- .... does give an equivalent series of 
 
 i 
 
 powers of -r; consequently, we must have 
 
 . ^ <^ . 2 5 17 - , 62 Q , 
 
 The actual division of the sine by the cosine will verify this, as 
 follows : 
 
 1-- + -^- 
 2 ^ 2.3.4 
 
 )^'-o + iixb-- ••(^' + ^^' + ■^^' + •• 
 
 ■^■"" 2 + 2X4 "■ 
 
 1 ^ 4 , , 
 
 - X^ X ■¥ .... 
 
 3 2.3.4.5 ^ 
 
 3>r3~ — x'^ .... 
 
 2.3.5 
 
118 CONNEXION OF DIRECT AND 
 
 The preceding series must cease to be convergent when x = -, if 
 
 not before ; and must, therefore, be regarded {Algebra, chap, ix.) as 
 an algebraical equivalent of the tangent. 
 
 (141.) We shall now proceed to the determination of tan_jj. 
 
 Let us first take that value which lies between and -f -, (134.) 
 
 which we have agreed to call tan-^x. 
 
 From (101.) tana: = -7=^ — — = — ^ = -7= — -, — 
 
 ^ V— 1 g'A/^n^g-'v^TT V— 1 garVTT^^ 
 
 the latter fraction being made by multiplying numerator and deno- 
 minator of the preceding by g' *^^. From this we find 
 
 g2.rv/=T = ~L' ^""^ and (Algebra, p. 226, formula 3.) 
 
 1 — V— 1 tanj: 
 
 logg2xv/rr^2(N/in.tanar + J(^/^tana:)' + J(^^tanj)*-f ... 
 
 = 2 v^--l(tana;— J tan^o; + J tan^a; — &c.) 
 
 Before proceeding further, annex lirms/ — 1 to the second side 
 (102.), bficause (124.) we have no right to conclude that any one 
 logarithm of the first side is equal to any one logarithm of the other. 
 And, to make the question clear of all difficulties, except one, take a 
 specific value for the tangent of x, say -1. We want to determine 
 the angle or angles which have -1 for their tangent. Calculate 
 
 T -001 , -00001 - 1 „ •. 
 
 •1 — -f- — - — — &c. and call it v. 
 
 3 o 
 
 Then, taking the general logarithms of both sides, we have 
 
 2^>/1T + 2^wn/^^ = 2z;N/3T + 2^m\/~l 
 
 or X -V U'K •=. V ■\- mx, and X = ?; + (?W — w)t, 
 
 which is the same as X ■= V + prr, 
 
 where p is any whole number, positive or negative. By proceeding 
 similarly with the general series, we obtain 
 
 X = (tana: — J tan ^o: + I tan^o; — &c.) -f jwt, 
 
 a result of a true form; for (44.) whatever angle v has a given 
 tangent, v+/)7r lias the same. The only question is, with a given 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 119 
 
 tangent, to which of the angles does the series itself belong ? To in- 
 vestigate this, we shall produce the series itself by aid of the last 
 chapter. Firstly, we must investigate Dtan_jr (129.) 
 
 Here fx = tana;, D/x = <px = \ + tan^or, 
 
 -Pj^ _ 1 _ 1 1__ 
 
 ^J~\^ — ^{f_^x) ~ l+(tan.tan_ixy 1 +a:» 
 
 or Dtan .a; = -7—2 = I — x^ ~\- x'^ — x^ + 
 
 {Algebrttf p. 16), from which, by reasoning similar to that in (135.) 
 
 tan ,x = X — - + 7 — + C. 
 
 -i 3 5 
 
 Now, if we look at (128.), we see that the only supposition on 
 which the reasoning there given can fail, is where x itself is infinite ; 
 
 for o never becomes infinite for possible values of x. Conse- 
 
 1 + A'' 
 
 quently, between tan_iX = — -, and tan_i.r = +- there is no 
 change of value of the constant. Let Cj be the value of the constant 
 which belongs to the interval between — - and + - (in which tan.jX 
 is tan_2 j:) ; then 
 
 tan-' X = ^—3 +5 — +Ci; 
 
 but when j: = 0, tan-^x = 0, and therefore, as in (135.), C, = 0, 
 or we have 
 
 x^ 
 
 tan-ia; = a;-- +5-^ + 
 
 for X write tanj:, and (134.) we have 
 
 X = tan a; — 5 tan^a; + 3 tan^a; — &c. 
 that is, the value of the series is the angle which has the tangent 
 specified in it, and which lies between — - and ■\ — . 
 
 i At 
 
 This series is convergent only when tanj; is less than 1. But we 
 may immediately produce a series which shall always be convergent, 
 as follows. From (135.) 
 
 • -1 1 jr3 1.3 /r^ 
 
 sm ia; = a;+--+--- + 
 
 for x write s n.r, which gives 
 
120 CONNEXION OF DIRECT AND 
 
 ,11.3 , 1.3 1 . . . 
 X = sino: + - - sin'^ + F4 5 ^'"^ "*" 
 
 for sinjT write tanx -f- \/(l + tan'j-), and then write tan~^x for x, 
 
 which gives 
 
 . _i _ 1 fi 1 1 1 1.3 1 1 7 
 
 which will be found {Algebra, p. 182) to be always convergent. 
 
 (142.) As an exercise, deduce from x = tan j: — ^ tan'x + &c. 
 by taking derived functions of both siles, the following series: 
 
 cos^ar = 1 — tan^j: + tan^o; — -land verify thera 
 
 sin^jr = tan^o: — tan* a; + tan ^a; — • • . . 3 otherwise. 
 
 (143.) A series of periodic quantities, such as 
 
 sin^ + sin2^ + sin 3^ + sin4^ + adinf. 
 
 can neither be called convergent nor divergent. If for we take any 
 value, these sines will be a succession of positive and negative quan- 
 tities, in parcels, no one of which exceeds unity. If, for example, we 
 
 TT 
 
 had = -, the series would be 
 
 G V2 + 1 + J V2 + - 1^2 - 1 ^ J V2 - 0) 
 
 + a second parcel of the same form ; 
 
 + a third ; -1- ad inf. 
 
 to which we may give the form 
 
 (1 + V2) {] - 1 + 1 ~ 1 + adinf) 
 
 The meaning of this series has been already discussed in Algebra, 
 p. 197. The question is, are we now to say that the preceding series 
 is one-half of 1 + v 2. In the chapter cited, we found the series 
 1 — 1 +1 — 1 + .... to be a (then) unintelligible limiting form 
 of 1 — X •\- x"^ — x^ -\- .... which was always arithmetically equal 
 
 to — ; — , as long as x was less than 1 ; whence we assumed i = 
 
 1 -j- X 
 
 1 — 1+1 — > and found that the one side was an algebraical 
 
 equivalent of the other. Let us then examine the series 
 
 a;sin^ + a;«sin2(J +Ji3sin3d+ (ar<l), 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 121 
 
 which we may now call convergent, as its terms are severally less than 
 those of the series x -\- x^ -{• x^ -^ . . . . , which is convergent. We 
 may sum this series in various ways, of which we shall choose two. 
 
 1. sin(n + I) & + s\n{n -^ \)d = 2smnd.cos& gives 
 
 w = 1, ar sin 2 ^ 4- = 2 a: sin ^ . cos ^ 
 
 n = 2, a;2sin3^ + aj^sind = 2a:2sin2^ . cos^ 
 n = 3, a7^sin4^ + a;^sin2^ = 2a;^sin3^ . cos^ &c. 
 
 If, then, we call the sum of the whole series S, and sum these 
 equations ad infinitumj the sum of the first column, x sin20 + j;^ 
 sin 3 H- .... is (S — x sm&) ~^ x ', that of the second column is 
 xS; that of the third column is 2Scos0. Consequently, 
 
 S — jTsine c? oc A c X sinQ 
 f- a: S = 2dcos^, or o =: 
 
 x^^ 2xQ0^9 + 1* 
 
 2. Let the sines be replaced by their exponential values, which 
 gives 
 
 1 xy 1 xy-'^ 1 xy — xy'^ 
 
 Sn/I^ \—xy 2V^ \—xy-^ 2>/^ 1— (>+3/~^)'^ + - 
 X y — .y-i xiinQ 
 
 \^(^yj^y-\)x'\-x^ 2n/— 1 1— 2cose.^+j^'* 
 
 the same as before. Now, if in this result we make a = 1, we 
 find it become 
 
 sinO 2 sin ^0. cos A . 
 
 2 
 
 Q 
 
 that is, Jcot- = sin^ + sin2^ + sin3d + .... 
 
 on which we must remark, that arithmetical equality has ceased, 
 and that the sign = can only mean {Algebra, chap, ix.) that one side 
 may be substituted for the other without producing discordances in 
 any arithmetical consequence of the substitution. For instance, we 
 shall consider the following result, which those who have any idea of 
 
122 CONNEXION OF DIRECT AND 
 
 arithmetical identity between the two will be surprised at. The 
 smaller we make 0, the greater is the value of the series. On which 
 remark, that this is not a series of terms with fixed signs, but one in 
 which the signs are positive and negative in parcels, the number of 
 terms in each parcel depending upon the value of 9. First, let 9 be 
 diminished until the first three angles lie in the first two right angles 
 the signs then are, 
 
 + + + 
 
 let further diminution take place until six angles are less than w; 
 the signs are then, 
 
 + + + + + + 
 
 and so on. Now, one very simple case of the series is that in whic li 
 is a measure of tt, say n0 = ir. We have then sin(n0 + 9) = 
 — s'm9, sin(n0 + 2 0) = — sin 2 0, &c. ; so that, if we put a for 
 sin -f sin 2 + • • . • + sin w 0, we have a — a + « — a-f- •••• or 
 ^a for the series. It ^remains then, only to find an expression for a; 
 now we have 
 
 2V — 1 U— 3/ 1 — ^-1 3 
 
 - 2>/i::i 2-(3^+j/-i) 
 
 sin0+sinn0 — sin (w + 1)0 
 
 "~ 2 (1 — COS0) 
 
 which, if n0 = TT, sinn© = 0, sin (n + 1)9 = — sin 6, gives 
 ; and - is the sum of the series already found. 
 
 2(1— cose)' 2 
 
 ffi 
 (144.) If = - TT, m and n being whole numbers, and if F© 
 
 be a primary function of 9, or any function whatsoever of primary 
 functions which is really periodic, the series F0 + F20 -f- .... can 
 always be reduced to the form a — a -{-a — a-\- ....; and the whole 
 question of assigning an algebraical equivalent to the series, \n finite 
 terms, is, therefore, made to depend upon the considerations deve- 
 loped in the ninth chapter of my Algebra. And, when 9 is not com- 
 mensurable with iTy we know that 9 may be found to lie between 
 
 _i'^ and TT where n is greater than any number named: so 
 
 n n 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 123 
 
 that the question in any other case is resolved into the general 
 question discussed in the treatise on Number and Magnitude. 
 
 (145.) When both the series and its algebraical invelopement * 
 are periodic in value, there is generally no difficulty with regard to 
 the constants which may enter. But it may happen tliat one side 
 increases without limit while the other side is periodic, in which 
 case a discontinuous constant must enter. 
 
 For instance, let the series be 
 
 sin^ + -sin2^ + -sin3^ + 
 
 2 3 
 
 which differs from that in (143.) by having terms which decrease 
 without limit; but which are positive and negative in parcels deter- 
 mined by the value of 0. This series is, 
 
 ^,[y-y-' + Kf-y-') + l{f-r^)+ •■••} 
 
 Now, (Algebra, f. 226) y + -^rf +5^' + •• = — log(l — y) 
 
 r'+i3'-^+5r'+-- = -iog(i-i) 
 
 whence the series = — j=- ■} log f 1 J — log(l — ?/) i 
 
 2v/_i "3/(1— i/) 2n/— 1 V y^ 2n/— 1 
 
 Because i = g-^^~i — 1 = g-('' + '^)^^^; and, as in (102.), 
 
 y y » \ J. 
 
 the general logarithm gives ) 
 
 2 2n/— 1 2 
 
 It remains to determine the value of n, corresponding to the 
 condition that 9 lies between certain limits, which limits are also to 
 
 be found. Let = -; the series becomes 1 — , "I" i — • • • • or 
 (141.) tan"'^!, that is -, whence we find 
 
 * I use this word as opposed to developement ; thus ^ is the inve- 
 lopement of 1—1 + 1—1 + 
 
124 CONNEXION OF DIRECT AND 
 
 itn = 7, or w = 1 when ^ = r 
 
 4 4 I 
 
 Similarly, -tt/i— -= — - orw = when ^ = — - 
 
 Now, the addition of 2 tt to the first side of the equation creates 
 no difference ; neither must it do so on the second side. To satisfy 
 this condition, consistently with those just deduced, and also with 
 another which appears immediately, namely, that a change of sign 
 in Q changes the sign of the first side, and, therefore, of the second, 
 
 let w, be the value of n between = and © = -, and Wj between 
 
 i 
 
 © = and = . We have then 
 
 i 
 
 ^7lj_ - ^_ = — \^^nc^ j or wi = — wg-f 1 
 
 the only value of which is n^ = 0, w, = 1, since the only change 
 consistent with periodicity is the addition of w, whenever B has re- 
 ceived an increase of 2 7r : and since n is 1 when = -, and when 
 = — -, and the change takes place when Q changes from negative 
 
 to positive, it must be as just stated. Such an expression as a 
 
 can only thus be made identical with a periodic series ; for, when 
 
 Q 
 
 becomes 0-f-27r, then the preceding becomes a — it — -, in which, 
 
 i 
 
 if the same succession of values is to recur while the angle changes 
 from + 27r to + 4 tt, a must be increased by tt, so that a — ir 
 may be then what a was. The result is that 
 
 from & ■=. — 2 T to d == the series is 
 
 ^ = to ^ = 2^ <T 
 
 ^ = 2'7r to ^ = 4^ 2t 
 
 and so on. 
 
 fl46.) Let us consider the series 
 
 asin^J + - sin2d + -sin3^ + 
 
 2 o 
 
 0+^ 
 
 2 
 
 lying w 
 between 2 
 
 and 
 
 ""2 
 
 + 7r 
 2 
 
 TT 
 
 2 
 
 •• 
 
 2 
 
 + ir 
 2 
 
 TT 
 
 2 
 
 • • 
 
 ■~2 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 125 
 
 1 l_oy-l 
 
 log 
 
 which is reduced to the preceding by a = 1 . Substitute for sin 6, 
 as in (145.), and the preceding becomes 
 
 _^{,„g(l_p_log(l-«y)} o. ^^_.,„,,_^^^ 
 
 But y = cos^ -{- \/ — 1 sin ^ - = cos ^ — n/ — 1 sin 
 
 . . 1 , 1 — acos0 + asin0\/— 1 
 
 whence the series is — 7=- log . ^ / 
 
 2v— 1 1 — tfcosfl — asin0v--l 
 
 Let 1 — acos0 ^ rcos^, asinO = rsin^, whence <p = 
 
 tan .(—^^ V The value of the series becomes 
 
 * \1 — acos9/ 
 
 / — 'J —\ 
 1 cos^+sin^v— 1 1 £_ 
 
 1^~\ cos^ — sin^V^ ~ 2n/^ ^^g~^^^ 
 
 asin^ + - sm2^ + = wr + tan M ;; -) 
 
 where tan_j is changed into tan-i, as the difference between the two 
 is expressed in the remaining term. In the form here given, n will 
 have only one value ; for tan-^x is not a continually increasing term, 
 
 being limited to that angle which lies between and + -. And, 
 
 when a is very small, the preceding series is very small, which cannot 
 be true of the second side, unless n = 0. Consequently, we have 
 
 tan M -) = asin^ + - sin2^ + 
 
 \1 — acosOJ 2 
 
 This brings us to consider the difference between tan-i tanO 
 
 and 0, which are equals only when 9 lies between — - and -|- ^ : 
 
 but, if 9 be greater than -, it is not tan -1 tan which is = 0, but 
 
 one of the values of tan_itan0. Consequently, as 9 increases in 
 successive revolutions, the expression tan-^tan© is periodic, varying 
 
 continually between - and — -. And similar considerations may 
 
 be applied to sin-isin0, which has the same limits, and to cos-^cosd 
 which has the limits and «-. We shall immediately see the use of 
 
 M 2 
 
126 CONNEXION OF DIRECT AND 
 
 this when we proceed to the series of last article, considered as a 
 particular case of the present one. Let a = 1, then we have 
 asmO 2sinfcos| g . q. 
 
 l-acos0 = 2sin«^ = ^^^ 2 - ^"'^ \2 " 2) 
 
 2 
 
 or tan-UanfJ — -) = sind + -sin2d + -sin3^ -f 
 
 n2 2/ 2 o 
 
 if we write the first side - — r» we are in error, for the two are not 
 
 the same, except when - — - lies between — - and -f -, or when 
 
 lies between and 2*. Such a transition would require a con- 
 tinual correction, amounting, in fact, to the process of the last article : 
 
 but tan- 1 tan J" is, by definition, the angle which lies between — - 
 
 and + -, and has x for its tangent. 
 
 The student should now prove the following theorems, illustrative 
 of the limitation of F-i, as distinguished from F_i . 
 
 sin-icosar = - — x from x = to a: = t 
 cos-isino: = - — a; from x = — - toa;= +:: 
 
 cot-Man a; = - — X from x = Otoa;=+T 
 
 (147.) We now proceed to some adaptations of the theorems in 
 chap. ii. to the notation of inverse functions. 
 
 sin(sin"-ia:-l-sin-i3^) =sinsin-ix.cossin-^3/4-cossin-^j'.sinsin-i^ 
 
 = j.\/ 1 _y -\-y^ 1 — x^ 
 
 cos(sin-icr-f-sin-i3^) = \/l — x"^. n/i — y — xy 
 
 sin(sin-ij; + cos-'y)^.rj/ -|- >/l — x^ N^l — y 
 
 sin(2sin-ijr) = 2x\/l— .r^ cos(2 sin-^a') = 1 — 2x' 
 
 sin(2cos-ij:) = 2x'v^l--x'» cos(2cos-ix) = 2j:'— 1 
 
 Is it true that sin (2 sin-^j:) = sin (2 cos-^j)? if not, why? 
 Investigate the consequences of the ambiguity of sign implied in 
 V , and also ascertain within what limits the following is true, 
 and when and how it must be corrected. 
 
 sin-ij: -f- sin-i_y = sin-^ (xVl — y* •{■ yv 1 — j') 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 
 
 127 
 
 Next, tan(tan— ijT + tan— ^v) = ^, tan (2 tan-i j:) = 
 
 1 — xi/ i —' X- 
 
 tan— ij; + tan-i^ = tan— ^f '^ "r 3^ \ 
 
 This latter equation is true whenever the sum of tan-^j; and tan—ij/ 
 lies between — - and - : but if it do not, say, for instance, it lies 
 
 2 'i 
 
 between - and *, then either ?r must be subtracted from the first 
 
 side, or tan__i must be used on the second ; and so on. In the same 
 manner deduce (with similar limitation), 
 
 tan~i:r + tan-iy + tan~i^ = tan-i I , ^ ^— ) 
 
 '^ \1 — xy — 1/z — zx/ 
 
 (148.) Let — —-^ =1 or V = - — ; — which gives 
 ^^^,1 — xy *^ 1 -\- X ° 
 
 , . , /'I — A cr 
 
 tan-* X + tan-i I :; — ; — 1 = - 
 W -^ xJ 4 
 
 r 1 1,1 , 1 . 1 1 5^ 
 
 Let a" = -, y = -, and then tan— i — f- tan—i- = -, 
 2-^3 2 3 4' 
 
 which gives an easy method of computing the value of ir. For (141.) 
 
 1 1 _ 1 1 
 
 2 3 2^ • 5 2^ 7 2' 
 1 1 
 
 3 3 33 ■ 5 3^ 7 37 
 
 _,1._1_1111_11 
 
 2 "■ 2 3 2^ "^ 5 2^ 7 2' "^ 
 
 ,1 1 11.11 11, 
 
 tan-i- = ---_+-----, + 
 
 4 — 2^3 3 \23 ^ 3V ^ 5 \2^ ^ 3V '^^■ 
 Write p and q for - and - (divide by 4 and 9 at every step). 
 
 p 
 
 :s; 
 
 •50000000000 
 
 q 
 
 = 
 
 •33333333333 
 
 f 
 
 = 
 
 •12500000000 
 
 93 
 
 = 
 
 •03703703704 
 
 f 
 
 zr: 
 
 •03125000000 
 
 9* 
 
 = 
 
 •00411522634 
 
 f 
 
 :r:z. 
 
 •00781250000 
 
 9' 
 
 ^ 
 
 •00045724737 
 
 f 
 
 = 
 
 •00195312500 
 
 f 
 
 ^ 
 
 •00005080526 
 
 i>" 
 
 = 
 
 •00048828125 
 
 qn 
 
 = 
 
 •00000564503 
 
 ^,3 
 
 = 
 
 •00012207031 
 
 y.3 
 
 ^ 
 
 •00000062723 
 
 ^,5 
 
 = 
 
 •00003051758 
 
 15 
 
 = 
 
 •00000006969 
 
 ^.7 
 
 = 
 
 •00000762940 
 
 ^•7 
 
 = 
 
 •00000000774 
 
 p.. 
 
 = 
 
 00000190735 
 
 9'' 
 
 = 
 
 •00000000086 
 
 f^ 
 
 = 
 
 •00000047684 
 
 q^y 
 
 = 
 
 •00000000009 
 
 /3 
 
 = 
 
 •00000011921 
 
 
 
 
 f 
 
 = 
 
 00000002980 
 
 
 
 
 f 
 
 = 
 
 •00000000745 
 
 
 
 
 p- 
 
 = 
 
 •00000000186 
 
 
 
 
 P'' 
 
 = 
 
 00000000047 
 
 
 
 
 C 
 
 = 
 
 •00000000012 
 •nnnnnoonnnri 
 
 
 
 
128 
 
 CONNEXION OF DIRECT AND 
 
 Now let (/>" + y") -r- w be denoted by r^. 
 r, = -83333333333 
 r. = -00707304527 
 
 r« = -00000000119 
 
 •84063894899 
 
 ^3 
 
 = 
 
 •05401234568 
 
 ^1 
 
 =: 
 
 •00118139248 
 
 r,, 
 
 = 
 
 •00004490239 
 
 '•is 
 
 = 
 
 •00000203915 
 
 r,9 
 
 = 
 
 •00000010043 
 
 r^ 
 
 = 
 
 -00000000518 
 
 '-27 
 
 = 
 
 •00000000028 
 
 ''31 
 
 = 
 
 •00000000002 
 
 
 •05524078561 
 
 
 - 
 
 •84063894899 
 
 I'T : 
 
 •78539816338 
 
 
 
 4 
 
 
 __ r 
 
 l-l415Q9fi.';rH52 
 
 which is correct, with the exception of the last place. 
 
 (149.) sin-ij: = cos-^n/i— x' = tan-i 
 
 -, &c. Form 
 
 Vl-x«' 
 
 all similar equations, and explain under what limitations they are 
 true. 
 
 It must be observed that every such equation becomes true (134.) 
 when sin_i, &c. are substituted for sin-i, &c. But it does not follow 
 that the same value of m in the general equation 
 
 sin.jo; = '\im<7t + sin-^a* = (27» + 1)^ — sin"^a: 
 
 must be applied on both sides of the equation. 
 
 (150.) The angle sin-io: is made, by convention^ 3. periodic angle. 
 It changes through 0, -, 0, — -, 0, &c. while x chaTiges through 0, 1, 
 0, — 1,0, &c. 
 
 If X be greater than 1, the angle sin-^j: becomes impossible. 
 If in the expressions for sinj;- and cosi", we write xn/ — 1 for .r, 
 we have 
 
 sin(x>/^) 
 
 Jin(a:\/ — l) = - , — 
 :os(a;N/ — l) = - — 
 
 =- or — 
 
 2 n/=1 
 
 * - = cos(a:'\/ — l)- 
 
 2 2 
 
 The right hand sides of the equations are no longer periodic; and in 
 
INVERSE TRIGONOMETRICAL FUNCTIONS. 129 
 
 the same way as all functions of sines, cosines, &c. may be expressed 
 by exponential functions of a: and \/ — 1, so all exponential functions 
 may be expressed by forms of sines and cosines of xy/ — 1. 
 
 (151.) We may thus always give to periodic series their correct 
 periodic values, either in terms of the primary functions, sines, &c. 
 which are periodic, or by the introduction o( periodic angles, sin-^sin0, 
 cos-^cosO, &c. Having illustrated this point, on which freedom 
 from error mainly depends, I shall proceed in the next chapter to 
 some applications of trigonometry, which will give a first view of the 
 manner in which it is used in astronomy and other branches of 
 physics* 
 
130 APPLICATIONS OF THE PRECEDING CUAPTERS. 
 
 CHAPTER VIII. 
 
 APPLICATIONS OF THE PRECEDING CHAPTERS. 
 
 (152.) TuE following question occurs several times in astronomy: — 
 If tan = wtan0, required the developement of ^ in a series of terras 
 which shall be functions of m and 9 : the supposition being that m 
 is nearly unity, so that one value of <p is nearly 0, or = nearly 
 is the solution to which better approximation is required. Let g'^''^ 
 be called »»; then we have 
 
 Now, 
 
 ^~hi _ 1 ^--"'rri __ (i-m)x+i + 
 
 l + iZI^ X l + w^Jli (lH-wi)x + l-m 
 
 -2<p (1 — w)»3^^4-(l + w/) OJ^'^+l ^ l~m 
 
 whence r ^ = 57- = — 5:5 where c = r^ — 
 
 or log»3-2^ = log(l +C»j2^)-log(l +C7,-20-log7j2^ 
 
 But r'^ = g"^^^^^ log^'^ = -2^N/^^+2a/iVirr,&c. 
 
 whence —^^VlTl +2crw V^ = —26^^^—2'xnW^ 
 
 Divide both sides by — 2'>/ — 1, and since n-\-n' is simply any whole 
 number positive or negative, which call r, 
 
 2 c' 2 c' 
 
 ^ — r-r = ^ — 2csin2^ + -— sin4^ sin6^ + &c. 
 
 2 3 
 
 Since is to be the value nearest to 9, we must have r = 0, or 
 
APPLICATIONS OF THE PRECEDING CHAPTERS. 131 
 
 ^ = ^ — 2csin2^ + -^ sin4^ ^ sin6^ + &c. 
 
 (153.) Let it now be proposed to expand the first side of the fol- 
 lowing equation in a series of the form of the second. 
 
 = «o + «i cos^ + fl2Cos2^ + «3COs3^ + 
 
 1 -\- e cos 9 
 
 Assume x^ + f = I 2x1/ = e 
 
 1 +ecos^ = x^ + 2xij X j(g^^~i -f s-^^~)+?/2 
 = {x + ye^^'){x + i/B'^^') 
 
 Let ^ =: z s^^^ = w. Then, 
 
 X 
 
 1 1 
 
 l-}-ecos0 \-\-zu} 1 +2w~i 
 = (1— Zw + zV— ...) (1— Zw-i+;s2w~2— ...) 
 
 = -J— 2 |l— 2zcos^ + 2<22cos2^ — 2;S^cos3^-h....} 
 
 Now, 2a; = N/l+e+ n/1 — e 2y = \/l +e— V 1_« 
 
 2r = -^ = ^.=^= 7== = {Algebra, p. 119.) 
 
 a:2(l_^2) _ ^2_2/2 = VI— e2 
 — i_ = -^.jl-2(l-^^^cos^ + 2(l-^^^ycos2^-..| 
 
 If we take the other sign for v 1 — e^ throughout, we have a 
 developement which is an algebraical equivalent of the first side, but 
 which is divergent when the one just found is convergent. It appears 
 also that this form of developement is not arithmetical, unless e be 
 less than 1. We shall now develope the series in powers of e, so as 
 to include every term up to the third power of e. Firstly we have 
 (rejecting every term above the third power) 
 
132 APPLICATIONS OF THE PRECEDING CHAPTERS. 
 
 — ^- = \e -\- Je^+ .... whence the series is 
 
 (l4-ic2) (l-2(Je + ^e^)cosd+2(Je+|e3)2cos2^ 
 
 -2(Je + Je3)''cos3^ + ••••) 
 Develope the second factor, rejecting all powers of e above the 
 third, which gives 
 
 (l + J«2)(l_l£±i!cos^ + |cos2(J-|%os3^) 
 Multiply, rejecting e*, &c. which gives 
 
 l + je2_li±li'cosd4-|'cos2^- ^cos3^ 
 Multiply both sides by 1 — e^ with similar rejection, which gives 
 = 1 — : — cos ^ + - cos 2 ^ -— - cos3 d 
 
 l-i-ecos0 2 4 "2 4 
 
 (154.) We might also obtain the preceding result as follows 
 (Afgebra, p. 161). 
 
 = (1—6^) (l--ecos^ + e2 cos2^— e3 cos3^), rejecting e*, &c. 
 
 !l +ecos0 
 
 = l—e^— (e— e3)cos^ + e2(J + ^cos2d) — e^dcosd + JcosS^) 
 which, reduced, gives the same result as before. 
 
 (155.) Theorem. While 9 varies from to 2 7r, or from a to 
 a + 27r, every expression of the form asin0H-6sin20+ . . . . or 
 fltan + 6 tan 2 + .... &c. has as many positive as negative values ; 
 that is, for every positive value which the series has for one value of 0, 
 there is a negative value which it has for another, numerically equal 
 to the positive value. 
 
 This theorem depends upon another, namely, that in the same 
 revolution there is always a second angle ff to any given angle 0, 
 such that 
 
 sin ^' = — sin ^, sin 2 ^' = — sin 2^, sin 3 ^' = —sin 3^.... adinf. 
 
 tan^' = — tan(), tan2<)' = —tan 2^, tan3<)' = —tan 3^ adinf. 
 
 The angle in question is 2 tt — 0, as appears from (44). It is, 
 therefore, evident that F0 being either series mentioned, F(2 7r — Q) 
 
APPLICATIONS OF THE PRECEDING CHAPTERS. 133 
 
 = — ¥9. The cosine has not a similar property, but is contained 
 with the others in the following general theorem, of which the one 
 just given is a more simple equivalent as regards the sine and 
 tangent. 
 
 Theorem. If 9 be an aliquot part of 27r and FO = sm9 or 
 cos 9, &c. 
 
 F^ + F(2^) + F(3^)+ +F(2^ or nd) = 
 
 This is already proved for the sine and tangent, since the series may 
 be made from the beginning and end, into one of terms such as 
 F j; _|_ F (^2 TT — x), which are severally =0; and the middle term, 
 if there be one, is F(7r) = 0, and the last term F(2 7r) = 0. For 
 the cosine, if there be a middle term, F(7r), that is, if n be even, 
 all the series preceding Fir may be arranged in terms of the 
 form F^ + F(7r— :r), each of which =0; and all from F(7r + 0) 
 to F(2 7r — 0), the last but one, may be arranged in terms of the 
 form F(7r + J^) + F(2 7r — x), which are severally = 0. And the 
 remaining terms, Ftt and F2 7r, give F7r + F27r = 0. When there 
 is not a middle term Ftt, or when n is odd, we must make use of a 
 general demonstration derived from the method in (143), by whicli it 
 it is proved that 
 
 cos0 + cosn0 — 1 — cos(n+l)0 
 COS0-1-COS20 + COS30 + 4-cosne= ■ —r ^ — !— ^ 
 
 the second side of which becomes when n9 = 2 7r. 
 Similarly, in (143), we have the series 
 
 • n ■ ■ «o , • o/. ■ . • n sin + sin w — sin(w + 1)0 
 sm + sm 2 + sm 3 + + sm w0 = ' — -— -^ — ! — ^ 
 
 *^ (1—^ COS ij J 
 
 which becomes = when n0 = 2 7r. And the same, in holh cases, 
 
 if n9 = 2nnr. 
 
 Hence it appears, that if we have such an expression as 
 
 a sin -f- 6 sin 2 + .... , and if we substitute for successively 
 
 2 rr 47r 
 
 .... up to 27r, the sum of each of the sets of terms corre- 
 
 n n r J 
 
 sponding to asin0, to i sin 2 0, &c., is = 0, whence the following 
 
 result. If we take angles uniformly distributed in value between 
 
 and 2 tt, the sum of the negative values of a sin + Z* sin 2 -f- ... . 
 
 will be equal to the sum of the positive values ; and the same of 
 
 fl cos -{- 6 cos 2 0+....; more briefly expressed thus : the mean 
 
134 APPLICATIONS OF THE PRECEDING CHAPTERS. 
 
 value of either of the preceding expressions is nothing. If, tlien, 
 we have an expression such as «„ + «, sin + "a ^'" 2 + . . . . 
 the sum of n values, the angles of which are uniformly distributed 
 through four right angles, will be na^ + 0; that is, Oq, or a^ is the 
 mean value of the series. Hence an infinite number of ways of de- 
 termining a magnitude which oscillates on one side and the other of 
 a given magnitude : such as is almost every magnitude which is 
 considered in astronomy. 
 
 (156.) We shall now inquire into the method of forming the 
 
 product of two series such as «<, + a, cos 9 -\- a^ cos 2 -\- 
 
 and b^-\-b^cos9 -\-b^cos2 9 + Let 2 <j) (n) be the abbre- 
 viation of the series whose terms are the values of ^n for all whole 
 numbers from to oc. Then the preceding series are denoted by 
 2a„cosnO and 2 6„cosn0; their sum is 2 (a„ + 6«) cos n 0, and its 
 mean value flfo + ^0 ? ^^^^•'" difference 2(fl„ — b„)cosn9, and its mean 
 value Uq — b^. Their product is 2 r/„ 6« cos w cos w 0, for every pos- 
 sible simultaneous pair of values of m and n, both being positive 
 whole numbers. It is evident, moreover, that 2 {p„ -f g,,) = "Spn + 2^,. 
 If, then, we give the general term 
 
 Onhmcosn & cos m$ its value lanbm^os(rn + n)& + ^anhr^coB{m-^n)& 
 
 we have 
 
 2rfncosn0x'2bmcosm& = J 2a„5,„cos(m + n)^-t- J 2a„ft„cos(7w — w)^ 
 
 To find the co-eflBcient of a given cosine, cosp9, in the pro- 
 duct, we must inquire in how many ways m-\-n may be made to 
 give Pf and w — n either p or — j9, for cos( — p9) ^ cosp9. Let 
 us, then, first ask for the mean value of the product, or the term 
 independent of 0. The only way in which m -\-n gives nothing (m and n 
 always positive whole numbers, included) is when m = n = 0, 
 which contributes ^Oo^o to the mean value in question. But m — n 
 = whenever m = n, which contributes haQb^•^la^b^-\-^a^b^-{^ , . . . 
 ad inf. ; so that the mean value of the product is flo''o+i'^i''i + i^2^3 
 
 -\- The student might perhaps think that the mean value 
 
 ought to have been a^^b^^ and may reason thus : If there be a magni- 
 tude which, one time with another, is w^, and a second which is /»„, 
 the product is as often above as below aj)^^ and the latter should 
 therefore be the mean value. But this is not true, as the most simple 
 instance will shew. The average of 7 and 3 is 5 ; that of 4 and 12 
 
APPLICATIONS OF THE PRECEDING CHAPTERS. 135 
 
 is 8 ; the product of the averages is 40 ; the average of the products 
 is 34^. In the series deduced in (153), the mean value of the square 
 of 1 -f-(l + e cos 0) is, by the preceding rule, oo + 2 ^'i + ^ ^'2 + • • • • 
 
 1 , 1 4^2 4^4 
 
 1 — e^ -^ 1 — e^ ^ 1 — e^ 
 
 (157.) I now ask for the term of the product which contains 
 cos 50. The number of ways in which in-\-n can be made = 5 is 
 finite, namely, 
 
 71 = W2 = 5; 71=1 7?2=4; 7i=2 /w = 3; 
 
 71=3 7W = 2; 71 = 4 771 = 1; 7i = 5 7W = 
 
 which contributes 
 
 to the co-efficient. Now all the ways of making m — w = 5 or — 5, 
 are infinite in number, giving 
 
 772 = 5, W = 0, or 772 = 0, 72 = 5; 772=6, 72 = 1, 
 or 772 = 1, 72 = 6 ; 772 = 7, 72 = 2, Or 772 = 2, 72 = 7, &C. 
 
 which contribute 
 
 K«fo^5+«5*o)+ K«1^6 + fl^6W + 2(«2^7+«7^!2) + 
 
 The co-efficient is therefore 
 
 {ao^5 + «1^4+«2^3+«3^2 + «4^1+«5^0 "| 
 
 -f a5&0+«6^1+«7^2+^8^3 + «9&4+ J 
 
 Similarly, the co-efficient of cos w0 is 
 
 the 
 
 half of < , , , , 
 
 The student should now actually multiply some terms of 
 2«„cosn0 and sfe^coswO, and thus produce the result here con- 
 densed, in a more expanded form. 
 
136 APPLICATIONS OP THE PRECEDING CHAPTERS. 
 
 (158.) The product 2a„sinn0 x 26„sin w0 must be developed 
 in a series of cosines : shew how the several terms may be de- 
 duced, and compare them vrith those of tlie last. The product 
 ^OnSinnOx'sbmCOstne must be expanded in a series both of sines 
 and cosines : proceed in the same way. 
 
ADDITIONS TO PRECEDING CHAPTERS. 137 
 
 CHAPTER IX. 
 
 MISCELLANEOUS ADDITIONS TO THE PRECEDING CHAPTERS. 
 
 In this chapter I propose to touch slightly on several points which 
 it is desirable the student should consider, but not necessary, so far 
 as Trigonometry is concerned, that he should enter to any great 
 depth. 
 
 (159.) Problem. Required a solution of any equation of the 
 second degree which has possible roots, by help of the trigonometrical 
 tables. 
 
 Let the equation be reduced to the form x^-\- 2ax -{• b = 0, or 
 x^-\- 2 ax — 6 = 0, a being either positive or negative, and b bein;;; 
 positive. Under one or other of these forms every equation can be 
 reduced. 
 
 1- x'^ + 2ax + b=^0 gives a; = — fl5± t^«"— /^ 
 
 Assume b = a^ sin^Q, or find 9 from sin0 = \/b-~a ; which can be 
 done, for, the roots being possible, a is greater than ^ b. 
 
 X = —a± Va^— a'^^.m^Q = — « (1 q= cos ^) 
 
 9 G 
 
 and — 2asin^- and — 2«cos^- are the roots required. 
 
 2. x^+2ax—b = gives x = ~-a± Va" + h 
 
 Assume b = a^tan'e, or find 9 from tan0 = s/'b-^a^ which can be 
 done, as the tangent of an angle may have any value. 
 
 .V COS0H-1 ./7 cos 0-1-1 
 
 q= sec^) = — « '=— = —Vb — ■.■^^- 
 
 ^ cos 9 sm0 
 
 — 9 —9 
 
 and v^^.tan- and — V^cot- are the roots. 
 ■i 2 
 
 N.B. If « be negative, as in j;'— 2.r — 2 = 0, it will be more 
 convenient to solve x^-\- 2.r— -2 = 0, and to change the sign of t!ie 
 roots of the latter. {Algebra, Chapter V.) 
 
 .\ 2 
 
138 
 
 MISCELLANEOUS ADDITIONS TO 
 
 Examples (with obvious roots for verification.) 
 
 .r«4-3xH-2 
 
 = 
 
 x' 
 
 ^+^- 
 
 12 
 
 = 
 
 « = l-5 b 
 
 = 2 
 
 a 
 
 = ^5 
 
 b 
 
 = 12 
 
 log 6 
 
 2)-30l0300 
 
 
 logb 
 
 
 2)1^0791812 
 
 
 •1505150 
 
 •5395906 
 
 logo 
 
 •1760913 
 
 
 logo 
 
 
 1-6989700 
 
 
 r9744237 
 
 0-8406206 
 
 
 10 
 
 
 
 
 10 
 
 Lsin70°31'43"^5 99744237 
 
 L.tan81°47'12"-5 10-8406206 
 
 logsia35°15'51"^8 9-7614394—10 logtan40°53' 36"^3 9-9375310—10 
 2 log >/& -5395906 
 
 log a 
 
 19-5228788-20 
 •1760913 
 •3010300 
 
 log 1-000000 ,20 0000001— 20 
 
 log 3-000000 10-4771216—10 
 
 logcos35°15'51"-8 9'9119544— 10 log cot 40° 53' 36"- 3 0-0624690 
 
 log\/6 
 
 •5395906 
 
 19-8239088-20 
 -1760913 
 -3010300 
 
 log 4-000000 0-6020596 
 Roots, 3 and — 4. 
 
 log 2-000000 20-3010301—20 
 Roots, — 1 and — 2. 
 
 (160.) On the preceding there is a remark to be made, in con- 
 tinuation of (120). Having found one root of the first equation 
 to be — 2asin'' ^0, we perceive that at the point where 6 was first 
 introduced, in the expression sin0, we might, whatever 9 may be, 
 have substituted tt — 6. This shews us that in the result, since 
 — 2 rt sin' i is a root, — 2 a sin' i (tt — 0), or — 2acos'i0 
 is another root; that is, the other root, since there can be but two. 
 "We shall illustrate this subject by giving a method of solving an 
 equation of the third degree. Let the equation be 
 
THE PRECEDING CHAPTERS. 139 
 
 Assume x •=.y — /c, which gives 
 
 Let li^ — l=p, 2k^ — 3kl + m = q, whence q = Spy —y'^ 
 
 But (59.) writing 1 — s^ for c^, we have sin 30 = 3sin0 — 4sin30; 
 if, then, we assume Spy = 3Asin9, and y ^^^ 4 Asin^^, we have 
 g = A sin 3 0. But the first two equations give A = 2\/j9^, whence 
 the equation can be solved if 
 
 sin3^ = y -j-j- for l^en y = 2Vp.sin9 
 
 This requires that 9^ should not exceed 4 p^, in which case the first 
 equation is rational. But 3 ih 2 7r has the same sine as 3 0, whence, 
 by the same process, we discover that the three following expressions 
 are values of 3/, which make q = 3pi/ — y. 
 
 2V^^sin(^-^) 2V^sin^, 2 t/psin (^ + ?^) , . . . (A) 
 
 (161 ) If for g3^^/:n" ^yg ^rite z, we find 
 
 sin3^=-^-(^--|= JJL 
 
 Z = 
 
 4p^ 
 
 2\/ 
 
 S( 
 changed. Let us assume, then. 
 
 1 
 If for z we take either of these values, - is the other with its sign 
 
 Extract the cube rooi of both sides, and we have 
 
 2^p^Z- or 2'/;^/-^^ = v/l/4^3__^2^ V-— 2 
 
 2V^^-3 or e' S~^^^ = v/v'4p3__52__v^ZY 
 
 7/ = 2p SUi^ =- -y- — .(g —2 ) 
 
 V — I ' 
 
 = P7:f^(V 1/47)^^2+ VZp-N/v'4y>3_ rf^ I^^O 
 which, since V — _ '/ — n/ — 1, and — V a = v ^a, is 
 
140 MISCELLANEOUS ADDITIONS TO 
 
 = ^-l + yf^+V-l--/?^'.... (B) 
 This is the formula known by the name of Cardan, for the solution 
 ofy — 2p^-\-q =0. It may be verified as follows :— Assume 
 J/ := M + V, which gives 
 
 u^ -\- v^ + 3uv{u -h v) — 3p(u -\-v) -^q = 
 
 (u6+<7M3 + p3 _ 0) 
 
 Let uv = p, then U^+v^-hq = 0, or ^ . "^ , ^, A 
 
 (v^ -^ q v^ -i- p^ = O3 
 
 whence w' and «' are the two roots of the last equation, treated as of 
 the second degree. The expression (B) for 5/ is then easily deduced. 
 
 The utility of this result (which is of great historical importance) 
 is not considerable, and I shall give the following only as exercises 
 for the student. 
 
 1. The expressions (A) and (B) cannot both have possible forms. 
 
 2. If 1, k, and k^ (106) be the cube roots of unity, and if A and B 
 be the arithmetical cube roots of7<^andiv'; then m + v, considered 
 alone, may have the following forms 
 
 A + B, A + B^, A + BA2, B + AA, B+AAS 
 BA+AA2, AA + B^S 
 
 none of which (from uv = p) are admissible, except 
 A-fB, A/^^-B;t^ AA2+BA, 
 
 which are the three roots of the equation, or values ofy. 
 
 3. The expressions (A) are possible when all the roots are pos- 
 sible ; and (B) when one only is possible. 
 
 (162.) The equation jr^"--2cos . a'"+l, is solved by every 
 
 value of A' which solves a:' — 2 cos ( j .a-f-1 =0: for (72) 
 
 2 cos ® = a; -f - gives 2 cos w a = a,'* H 
 
 Shew from this that the roots of the given equation are found by 
 giving successive whole values to fn in the formula 
 
 /0 + 2w7r\ , . /0 + 2OT7r\ ./ r 
 
 and shew, as in (103), that there cannot be more than ;i roots. 
 
THE PRECEDING CHAPTERS. 141 
 
 (163.) Any quantity of the form a + 6 V^ can be reduced to 
 the form r(cos + sin On/^). The conditions evidently are, that 
 tan 9 = b-^Of and r = \^a^-\-b^' Hence it is easy to shew, that 
 all functions of a + 6>/IIi, &c. may be reduced to the same form : 
 thus (115) we have 
 
 (« + Jl/Zri)(a' + 6V3l)=rr'{cos(^ + + sin(^ + 0^— ^} 
 {a-hbV^T = r"{cosW^+sinW^l/'^n'| 
 
 We may express a-\-b xZ—T in the form rt^^-^, which is the most 
 compendious method of expressing any algebraical quantity what- 
 soever, and allows of r being supposed positive. A negative quantity 
 is expressed by making any odd multiple of tt (102); a positive 
 quantity by making any even multiple of tt : a quantity of the 
 
 form +6v — 1 or — 6v — 1, by making of the form {^m-\-\')-^ 
 
 or (4w-|-3)-, and one of the form a + 6>/ — 1 by a value of 0, 
 
 which is no whole multiple whatsoever of a right angle. 
 
 (164.) Among possible expressions, those of the form Acos(fl0+«) 
 or A sin (a +a), are such that the sum or difference of any of them 
 is always reducible to the same form, whatever the values of A and a 
 may be, provided only that a always remains the same. The two 
 forms are not essentially different, for 
 
 Asin(«^-fa) is Acos^«^ + «— ^j 
 
 To prove the theorem, observe that 
 
 A cos (a ^ + a) + A' cos (a ^ + a) '— (Acoscc + A'cosa)cosa^ 
 
 ■— (A sin a + A' sin «') sin a & 
 Assume 
 
 A cos a -{- A' cos a' = L cos X A sin a -f- A' sin cl =- \x sin >- 
 
 or L = V'A2 + A'2 + 2AA'cos(a-a'). 
 
 A sin a + A' sin a' 
 tan A = -r — -TT 1 
 
 AcOSa-|- A COSa 
 
 whence Acos («^ + a) + A'cos(a^ + a') = Lcos(«^ + ?.) 
 (165.) I shall conclude this chapter with some examples of a 
 
14^ MISCELLANEOUS ADDITIONS TO 
 
 process knowD by the name of successive substitution. Let us sup- 
 pose any function of j:, ipx, and that, beginning with a value of x, 
 say a, we form tpa = b, ^6 = c, <pc = e, &c. Or, agreeably to the 
 notation of (122), suppose that we form <pa, tp^a, <p^a, &c. Then the 
 limit towards which we approach, if we approach any limit at all, 
 must be a solution of <})X = x. 
 
 For instance, take the function l-\-^x: begin with x = 0, and 
 successive substitution then gives 
 
 1, 1+^, l + J(lJ)or li, l+|ai)or li, &c. &c. 
 the limit is 2, which is the solution of l-{-^x = x. 
 
 To prove this generally, suppose it found that we can bring the 
 results of successive substitutions as near as we please by carrying 
 the process far enough. Let k and k-\-z he the results of two 
 successive substitutions: then, by hypothesis, k -\- z = <pk. The 
 smaller z is, the more nearly does k solve the equation 0j = t. 
 Let / be the limit, and let /c = / + ^> where ^ may be as small as 
 we please. Then ^/ -f ^ + 2" = ^(/ + 3), which being true for values 
 of I and z as small as we please, gives {Algebra, page 157) (pi ^= I. 
 
 (166.) As an example, let the function be 20 + sin jr, where 20 
 means 20 degrees, and sinx is to be reckoned as a fraction of a 
 degree. We begin with x = 0, which gives 20° ; then 20"+ sin 20° 
 is 20°-342, or 20° 20'^, and 20°+ sin (20° 20^) is 20''-347; which 
 is a near approximation to the solution of jt = 20 + sin x where 
 1 means one degree. 
 
 (167.) Successive substitutions, finite in number, will sometimes 
 give theorems by which the logarithmic tables may be examined in 
 many parts at once. For instance, 
 
 sin a; = 2 sin - cos - or 2 sin - = sin j; . sec - 
 
 Z il i « 
 
 4 sin - = 2 sin - sec - = sin x sec '- sec - 
 4 2 4 2 4 
 
 cs . X c\ ' X X !*! XXX 
 
 CJ sm - = Z sm - sec - sec - = sm x sec - sec - sec - 
 8 2 4 8 2 4 8 
 
 and so on ; whence it follows that 
 
 2" sin -- = sin a; . sec - sec - sec — - 
 
 2" 2 4 2" 
 
 XXX X • . om • •*■ 
 
 or cos - . cos - . cos - cos — = sm a; -f- Z sm — 
 
THE PRECEDING CHAPTERS. 
 
 143 
 
 If n increase without limit, the limit of the last divider is x; 
 
 whence 
 
 X X X X J • /• u *u 1 ••* sina- 
 
 cos - cos — cos — cos — . ... ad mf. has the limit 
 
 2 4 8 16 -^ X 
 
 This affords an easy method of verifying the value of tt, as follows: 
 assume J? = -, we have then 
 
 
 TT 
 COS - . 
 
 TT 
 20S - . COS 
 
 8 
 
 TT 
 
 16 
 
 __ 2 
 
 rr 
 
 log cos 45° 
 
 0' 
 
 0" 
 
 
 9-8494850 — 10 
 
 22 
 
 30 
 
 
 
 
 9-9656153 — 10 
 
 
 
 11 
 
 15 
 
 
 
 
 9-9915739 — 10 
 
 
 
 5 
 
 37 
 
 30 
 
 
 9 9979037 — 10 
 
 
 
 2 
 
 48 
 
 45 
 
 
 9-9994766 — 10 
 
 
 
 . 1 
 
 24 
 
 22-5 
 
 
 9-9998692 — 10 
 
 
 
 
 
 42 
 
 11-3 
 
 
 9 9999673 — 10 
 
 
 
 
 
 21 
 
 5-7 
 
 
 9-9999918-10 
 
 
 
 
 
 10 
 
 32-9 
 
 
 9-9999980 — 10 
 
 
 
 . 
 
 5 
 
 16-5 
 
 
 9-9999995 — 10 
 
 
 
 
 
 2 
 
 38-3 
 
 
 9-9999999 — 10 
 
 
 109-8038802—110 
 
 
 141593 
 
 
 •3010300 
 
 log 3 
 
 •4971498 
 
 I should, however, recommend the student not to proceed further 
 in the subject of developement of trigonometrical forms, until he 
 is able to apply the Differential Calculus, without which the theory 
 of series will always be very incomplete. 
 
 THE END. 
 
 LONDON: 
 raiNTEI) BY J. MOVES, CASTLE STREET, LEICESTER SQUARE. 
 
ADDENDUM 
 
 TO THE 
 
 TREATISE ON TRIGONOMETRY. 
 
 In page 42, line 18, there is an obvious mistake in the reasoning 
 contained in the words, " But sec0 is greater than 1 ; therefore tan 9 
 is greater than 0." To set this right, let the student omit that 
 sentence and the preceding, and supply their places by the follow- 
 ing proof, that tan 9 is always greater than 0, whenever the latter i.i 
 less than a right angle, 
 
 1 . Let = n ^, where w is a whole number, whence 0, 2 <}>, 
 'i(p .... ntp, are severally less than a right angle. Then 
 
 „ 2tan0 . , 
 
 tan 20 = ^ — or tan 20 is greater than 2tan0 
 
 ^ 1 — tan^ v 6 r 
 
 tan2 0-|-tan0 
 
 tan 3 = ^ ^-- > tan 2 + tan > 3 tan 
 
 ^ 1 — tan20.tan0 ^ y t v ^ y 
 
 and so on : observing that all the denominators must be positive 
 (since the fractions themselves and their numerators are positive) 
 and less than unity. Proceeding in this way, we shew that tan n 
 is greater than n tan : whence 
 
 „ 9 tanO tan0 
 
 tan e > - tan 0, or -— - > ---^. 
 
 (7 
 
 2. If then we can shew that, by taking n sufficiently great, or 
 sufficiently small, tan -r- is greater than unity, it follows that 
 tan 0-7-0 is also greater than unity. Let a polygon of k sides be 
 circumscribed about a circle whose radius is r ; then each side of 
 
 the polygon is 2rtan-, and the vv^hole periphery is 2/crtan-, 
 
 K K 
 
 which is greater than the circumference of the circle or 2 tt r. 
 Hence, 
 
 tanr is greater than 7- (k being a whole number), 
 
 3. Now, — may be made as near as we please to 0, and either 
 greater or less, by properly assuming n and k (whole numbers). 
 But, by combining the results of (1) and (2), 
 
 tan — IS greater than -7— 
 
 or tan (0 ±: a) ± a 
 
 where » may be as small as we please. Hence it follows that tan 
 is greater than 0. 
 
THE 
 
 CONNEXION 
 
 OF 
 
 NUMBER AND MAGNITUDE 
 
 AN ATTEMPT TO EXPLAIN 
 
 THE FIFTH BOOK OF EUCLID. 
 
 BY 
 
 AUGUSTUS DE MORGAN, 
 
 OF TRINITY COLLEGE, CAMBRIDGE. 
 
 La seule maniere de bicn trailer les eldmens d'une science exacte et rigoureuse> c'est d'y 
 mettre toute la rigueur et I'exactitude possible. — D'Alembbrt. 
 
 LONDON; 
 
 PRINTED FOR TAYLOR AND WALTON, 
 
 BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON, 
 30 UPPER GOWER STREET. 
 
 M.DCCC. XXXVI. 
 
LONDON: 
 
 J. MOYKS, CA8TLB 8TRKKT, LKICESTKR hQlAIlK. 
 
PREFACE. 
 
 This Treatise is intended ultimately to form part of one 
 on Trigonometry. The place in which most students con- 
 sider number and magnitude together for the first time, is 
 in the elements of the latter science, unless they have un- 
 derstood the Fifth Book of Euclid better than is usually the 
 case. Previously, therefore, to commencing Trigonometry, 
 I consider it advisable to enter upon the consideration of 
 proportion in its strict form ; that is, upon the Fifth Book 
 of Euclid. There is no other method with, which I am 
 acquainted which gives any thing like demonstration of the 
 general properties of ratios, though there is a doux oreiller 
 pour reposer une tete bien faite, which many of the con- 
 tinental mathematicians have agreed shall be called demon- 
 stration, and which is beginning to make its way in this 
 country. 
 
 Hitherto, however, it has been customary for mathe- 
 matical students among us to read the Fifth Book of Euclid; 
 frequently without understanding it. The form in which it 
 appears in Simson's edition is certainly unnecessarily long, 
 and the tedious repetition of " A B is the same multiple of 
 C D which E F is of G H," in all the length of words, 
 renders the reasoning not easy to follow. The use of 
 general symbols of concrete magnitude, instead of the straight 
 line of Euclid, and of a general algebraical symbol for whole 
 
IV PREFACE. 
 
 number, seems to me to remove a great part of the difficulty. 
 Throughout this work it must be understood, that a capital 
 letter denotes a magnitude ; not a numerical representation, 
 but the magnitude itself: while a small letter denotes a 
 number, and mostly a whole number. And by the term 
 arithmetical proportion^ when it occurs, is signified, not the 
 common and now useless meaning of the words, but the 
 proportion of two magnitudes which are arithmetically re- 
 lated, or which are commensurable. 
 
 The subject is one of some real difficulty, arising from 
 the limited character of the symbols of arithmetic, con- 
 sidered as representatives of ratios, and the consequent 
 introduction of incommensurable ratios, that is, of ratios 
 which have no arithmetical representation. The whole 
 number of students is divided into two classes : those who 
 do not feel satisfied without rigorous definition and de- 
 duction ; and those who would rather miss both that take a 
 long road, while a shorter one can be cut at no greater ex- 
 pense, than that of declaring that there shall be propositions 
 which arithmetical demonstration declares there are not. 
 This work is intended for the former class. 
 
 AUGUSTUS DE MORGAN. 
 
 London f May 1, 1836. 
 
CONNEXION 
 
 OF 
 
 NTTMRF/R AXrn TU A rtMinrTTTM? 
 
 ERRATA. 
 
 Page 35, line 3, for less read greater. 
 
 47, ... 1, /or t;(Q+Z) read w(Q + Z). 
 
 — ... 2, /or uQ read wQ. 
 
 , .,«v*»v^ ^^iv-ociii, iiaeu eiiuer 10 an inattentive 
 
 reader, or to one whose whole attention is engrossed by the difficulty 
 of comprehending terms which cannot yet have become familiar to 
 him. Before proceeding, therefore, to explain Trigonometry (the 
 measurement of triangles), which, in the widest sense, includes all 
 the applications of algebra to geometry, it will be right to inquire on 
 what sort of demonstration we are to pass from an arithmetical to a 
 geometrical proposition, or vice versa. 
 
 Geometry cannot proceed very far without arithmetic, and the 
 connexion was first made by Euclid in his Fifth Book, which is so 
 difficult a speculation, that it is either omitted, or not understood by 
 those who read it for the first time. And yet this same book, and the 
 logic of Aristotle, are the tvvo most unobjectionable and unassailable 
 treatises which ever were written. 
 
 The reason of the difficulty which is found in the Fifth Book is 
 twofold. Firstly; — It is all reasoning, unhelped by the senses: most 
 of the propositions have no portion of that intrinsic evidence which is 
 seen in " two sides of a triangle are greater than the third ; " but, at 
 the same lime, the propositions of arithmetic which correspond to 
 
 ;fe 
 
IV PREFACE. 
 
 number, seems to me to remove a great part of the difficulty. 
 Throughout this work it must be understood, that a capital 
 letter denotes a magnitude ; not a numerical representation, 
 but the magnitude itself: while a small letter denotes a 
 number, and mostly a whole number. And by the term 
 arithmetical proportion^ when it occurs, is signified, not the 
 common and now useless meaning of the words, but the 
 
 duction ; and those who would rather miss both that take a 
 long road, while a shorter one can be cut at no greater ex- 
 pense, than that of declaring that there shall be propositions 
 which arithmetical demonstration declares there are not. 
 This work is intended for the former class. 
 
 AUGUSTUS DE MORGAN. 
 London f May 1, 1836. 
 
CONNEXION 
 
 OF 
 
 NUMBER AND MAGNITUDE. 
 
 When a student has acquired a moderate knowledge of the operations 
 and principles of algebra, with as many theorems of geometry as are 
 contained in the first four books of Euclid's Elements, it becomes 
 most desirable that he should gain some more exact knowledge of the 
 connexion between the ideas which are the foundation of one and 
 the other science, than would present itself either to an inattentive 
 reader, or to one whose whole attention is engrossed by the difficulty 
 of comprehending terms which cannot yet have become familiar to 
 him. Before proceeding, therefore, to explain Trigonometry (the 
 measurement of triangles), which, in the widest sense, includes all 
 the applications of algebra to geometry, it will be right to inquire on 
 what sort of demonstration we are to pass from an arithmetical to a 
 geometrical proposition, or vice versa. 
 
 Geometry cannot proceed very for without arithmetic, and the 
 connexion was first made by Euclid in his Fifth Book, which is so 
 difficult a speculation, that it is either omitted, or not understood by 
 those who read it for the first time. And yet this same book, and the 
 logic of Aristotle, are the two most unobjectionable and unassailable 
 treatises which ever were written. 
 
 The reason of the difficulty which is found in the Fifth Book is 
 twofold. Firstly; — It is all reasoning, unhelped by the senses: most 
 of the propositions have no portion of that intrinsic evidence which is 
 seen in " two sides of a triangle are greater than the third ;" but, at 
 tjie same time, the propositions of arithmetic which correspond to 
 
 B 
 
^Z CONNEXION OF 
 
 those of the Fifth Book are very evident, and the student is therefore 
 led to escape from the notion of magnitude, and fly to that of number. 
 Secondly; — The non-existence of any very easy notation and system of 
 arithmetic in the time of Euclid, made geometrical considerations re- 
 latively so much more simple, that the form of his book is (to us) 
 unnecessarily remote from all likeness to a treatise connected with 
 numbers. The difference between our day and his lies in this : that 
 in the former the exactness of geometry was gained with some degree 
 of prolixity and (to a beginner) obscurity ; in the latter, the facility 
 of arithmetic is preferred, and perfect demonstration is more or less 
 sacrificed to it. I shall now endeavour to present the Fifth Book of 
 Euclid in a form which will be more easy than the original, to those 
 who have some acquaintance with algebra. 
 
 By number is here meant what is called abstract number, which 
 merely conveys tiie notion of times or repetitions, considered inde- 
 pendently of the things counted or repeated. By magnitude, or 
 quantity, is meant a thing presented to us, not as to its form, if it 
 have form, or as to colour, weight, or any other circumstance, but 
 simply as that which is made up of parts, not differing from the whole 
 in any tiling but in being less ; so that, if we consider separately a 
 part and the whole, we have only two inferences : 
 The part is less than the whole. 
 The whole is greater than the part. 
 
 Every thing we can see or feel presents to us the notion of mag- 
 nitude or quantity. And here we must observe, that we have to pick 
 our words from among those in common use, which never have very 
 precise meanings. For instance, we have magnitude, the nearest 
 English word to which is greatness; and quunlity, for which the 
 word, if it existed, should be so-much-ness. These words are of the 
 same meaning, and the more indefinite we now leave them (except 
 only in assigning that tliey are to be considered as applied to any 
 thing which can be made mo7e or less), the better for our purpose ; 
 since it is the object of this treatise to deduce from that indefinite 
 notion a method of making mathematical comparisons of quantities, 
 by aid of the notion of number. 
 
 Upon two magnitudes, our senses will enable us to draw one or 
 other of the following conclusions : 
 
 1. The first is sensibly greater than the second. 
 
 2. The first is sensibly less than the second. 
 
NUMBER AND MAGNITUDE. d 
 
 3. The first is sensibly equal to the second ; meaning that the dif- 
 ference, if any, is so small that our senses cannot perceive it. This is 
 what is meant by equality of magnitudes in common life. The English 
 foot and the Florence foot are equal for common purposes : they 
 differ by about the twentieth part of an inch, which in a foot is called 
 nothing. 
 
 Perfect equality is a mathematical conception, which never can be 
 absolutely verified in practice ; for so long as the senses cannot per- 
 ceive a certain quantity, be it ever so small, so long it must always be 
 possible that two quantities, which appear equal, may differ by as 
 much as the imperceptible quantity. But we are not reasoning upon 
 what we can carry into effect, but upon the conceptions of our own 
 minds, which are the exact limits we are led to imagine by the rough 
 processes of our hands. The following, then, is the postulate upon 
 which we construct our results : 
 
 Any one magnitude being given f let it be granted that any number 
 of others may be found, each of which is {positively and mathematically) 
 equal to the first. 
 
 Let A represent a magnitude — not as in algebra, the number oP 
 units which it contains, but the magnitude itself — so that if it be, for 
 instance, weight of which we are speaking, A is not a number of 
 pounds, but the weight itself. Let B represent another magnitude of 
 the same kind ; we can then make a third magnitude, either by putting 
 the two magnitudes together, or by taking away from the greater a 
 magnitude equal to the less. Let these be represented by A -f B and 
 A — B, A being supposed the greater. We can also construct other 
 mE^nitudes, by taking a number of magnitudes each equal to A, and 
 putting any number of them together. Thus we have 
 
 A -{- A which abbreviate into 2 A 
 
 A+A + A 3A 
 
 A + A + A-f A 4A 
 
 and so on. We have thus a set of magnitudes, depending upon A, 
 and all known when A is known ; namely, 
 
 A 2A 3A 4A 5A &c. 
 
 which we can carry as far as we please. These (except the first) are 
 distinguished from all other magnitudes by the name oi multiples of A ; 
 And it is evident that they increase continually. Let the preceding be 
 
4 CONNEXION OF 
 
 called the scale of multiples of A. It is clear that the multiples of mul- 
 tiples are multiples ; thus, 7 times 3 A is 21 A, m times wA is (mw)A, 
 where mw is the arithmetical product of the whole numbers m and n. 
 The following propositions may then be proved. 
 
 Prop. I. If A be made up of B and C, then any multiple of A 
 is made up of the same multiples of B and C ; for 2 A must be made 
 up of 
 
 B C B C 
 
 of which B and B make 2B, C and C make 2C ; so that 2 A is made 
 up of 2 B and 2C. Similarly, 3 A is made up of 
 
 B C B C B C 
 
 orof 3B and 3C. 
 
 Corollary. Hence it follows, that if A be less than B by C, any 
 multiple of A is less than the same multiple of B by the same mul- 
 tiple of C. For, since A is less than B by C, A and C together make 
 up B ; therefore, 2 A and 2 C make up 2 B, or 2 A is less than 2B by 
 2 C. The algebraical representations of these theorems are as follows : 
 
 If A = B + C mk=zmB + mG 
 
 If A=B-C w2A = mB-7nC 
 
 m being any of the numbers 2, 3, 4, &c 
 
 Prop. II. However small A may be, or however great B may 
 
 be, the multiples in the scale 
 
 A, 2 A, 3 A, 4 A, 5 A, &c. 
 
 will come in time to exceed B, by continuing the scale sufficiently 
 far : B and A being magnitudes of the same kind. This is a pro- 
 position which must be considered as self-evident : it must be re- 
 membered that B remains the same, while we pass from one multiple 
 of A to the next. Put feet together and we shall come in time to 
 exceed any number of miles, say a thousand. But the best illustration 
 of the reason why we formally put forward so self-evident a pro- 
 position, will be to remark, that it is not every way of adding mag- 
 nitude to magnitude without end, which will enable us to surpass 
 any given magnitude. To a magnitude add its half; to that sum add 
 half of the half; to which add the half of the last: and so on. No 
 continuation of this process, were it performed a hundred million of 
 times, could ever double the first magnitude. 
 
NUMBER AND MAGNITUDE. O 
 
 Prop. III. If A be greater than B, any multiple of A is greater 
 than the same multiple of B. This follows from Prop. I. And if A 
 be less than B, any multiple of A is less than the same multiple of B. 
 This follows from the corollary, Prop. I. And if A be equal to B, 
 any multiple of A is equal to the same multiple of B. This is self- 
 evident. 
 
 Prop. IV. If any multiple of A be greater than (equal to, or less 
 than) the same multiple of B, then A is greater than (equal to, or 
 less than) B. For example, let 4 A be greater than 4B; then A 
 must be greater than B; for, if not, 4 A would be equal to, or less 
 than, 4 B (Prop. III.). 
 
 Prop. V. If from a magnitude the greater part be taken away ; 
 and if from the remainder the greater part of itself be taken away, and 
 so on : the given magnitude may thus be made as small as we please, 
 meaning as small as, or smaller than, any second magnitude we 
 choose to name. 
 
 Let A and Z be the two magnitudes, and let A diminished by 
 more than its half be B, then 2 B is less than A. Let B diminished 
 by more than half be C ; then 2C is less than B, 4C is less than 2 B, 
 and still more less than A. Let C diminished by more than its half 
 be D, then 2D is less than C^8D is less than 4C, and still more 
 than A. This process must end by bringing one of the quantities 
 
 A, B, C, D, &c. below Z in magnitude. For, if not, let A, B, C, &c. 
 always remain greater than Z. Then, since 2B, 4C, 8 D, 16 E, &c. 
 are all less than A (just proved) still more must 2Z, 4Z, 8Z, 16Z, 
 &c. be less than A. But this cannot be ; therefore, one of the set A, 
 
 B, C, &c. must be less than Z. 
 
 [The reductio ad absurdum, as this sort of argument is usually 
 called, is a difficult form of a simple inference. Suppose it proved 
 that whenever P is Q, then X is Y. It follows that whenever X is 
 not Y, P is not Q. It is usually held enough to say, for if P were Q 
 X would be Y. But the form in which Euclid argues, supposes an 
 opponent ; and the whole argument then stands as follows. " When 
 X is Y, you grant that P is Q ; but you grant that P is not Q. I say 
 that X is not Y. If you deny this you must affirm that X is Y, of 
 which you admit it to be a consequence that P is Q. But you grant 
 that. P is not Q ; therefore, you say at one time that P is Q and that 
 P is not Q. Consequently, one or other of your assertions is wrong, 
 
 B 2 
 
6 CONNEXION OF 
 
 either ' P is not Q' or * X is Y.' If the first be right, the second 
 is wrong : that is, ' X is not Y ' is right.'' 
 
 The preceding argument runs as follows ; — when A, B, C, &c. are 
 all greater than Z, then 2 Z, 4 Z, &c. are all less than A : but 2 Z, 
 4 Z, &c. are not all less than A ; therefore. A, B, C, &c. are not all 
 greater than Z]. 
 
 Corollary. The preceding proposition is equally true when, 
 instead of taking more than the half at each step, we take the half 
 itself in some or all of the steps. 
 
 Prop. VI. If there be two magnitudes of the same kind, A and 
 B, and if the scales of multiples be formed 
 
 A, 2 A, 3 A, &c. B, 2B, 3B, &c. 
 
 then one of these two things must be true ; either, there are mul- 
 tiples in the first scale which are equal to multiples in the second 
 scale ; or, there are multiples in the first scale which are as nearly 
 equal as we please to multiples (not the same perhaps) in the second 
 set: that is, we can find one of the first set, say wA, which shall 
 either be equal to another in the second set, say nB, or shall exceed 
 or fall short of it by a quantity less than a given quantity Z, which we 
 may name as small as we please. 
 
 Let us take a multiple out of each set, any we please, say p\ and 
 jB. IfpAand^Bbe equal, the first part of the alternative exists; 
 if not, one must exceed the other. Let pA exceed 5B, say by E; 
 then we have 
 
 ;?A = ^B + E (1) 
 
 Now E is either less than B, or equal to B, or greater than B. If 
 the first, let it remain for the present ; if the second, we have 
 ^A = (9 + 1) B, or the first alternative exists : if the third, then B 
 can be so multiplied as to exceed E. Let (^ + 1) B be the first 
 multiple of B which exceeds E ; that is, let the next below, or t B, 
 be less than E, say by G, then we have 
 
 E=^B+G pA = <7B4-^B + G 
 
 or pA = ((/ + OB + G 
 
 Now G must be less than B ; for E or ^B + G is less than (i + 1) B, 
 or ^B -f B. We have then made this first step (observe that «/ -f ^ is 
 
NUMBER AND MAGNITUDE. 7 
 
 only some multiple of B; call itrB). Either the first alternative 
 exists, or we can find p A and r B, so that 
 
 J5 A = 7- B + G where G is less than B (2) 
 
 Now G can be so multiplied as to exceed B; let yG and (v + 1)G 
 be the multiples of G, between which B lies, so that 
 
 vG is less than B, say vG = B — K 
 (u + 1 ) G is greater than B, say ?;G + G = B + L 
 
 and it follows that K + L=:G, for since (v-{-l)G and vG differ 
 by G, if a magnitude lie between them, their difference must be made 
 up of the excess of that magnitude over the lesser, together with its 
 defect from the greater. Consequently, either K and L are both 
 halves of G, or one of them falls short of the half. Suppose K is less 
 than the half of G : then take both sides of (2), v times, and we have 
 
 vpA = ^JrB + vG 
 or vpA = z;rB + B — K 
 
 or vpA = (vr -{-\)B — K(K less than half G) 
 
 But if L be less than the half of G, take both sides of (2) ?; + 1 times 
 which gives 
 
 {v + l)pA = {v+l)rB +(?; + l)G 
 
 or (?;+l)pA= v + l-rB +B + L 
 
 or v+l pA = (v + l r + 1)B + L (L less than half G) 
 
 If K and L are both halves of G, we may take either. And if (a case 
 not yet included) a multiple of G, vG, be exactly equal to B, we 
 have then 
 
 vpA = vrB -{-vG = {vr -\-l)B • 
 
 which gives the first alternative. Consequently, we either prove the 
 first alternative, or we reduce the equation 
 
 pA = rB + G (G less than B) 
 
 to an equation of the form 
 
 p'A = r'B ± G' JG' not greater than the half of G. 
 
 We may now proceed as before ; but, to exemplify all the cases that 
 may arise, let us take 
 
8 CONNEXION OF 
 
 p'k = r'B-G' 
 
 If i/G' be exactly B, we prove the first alternative, as before ; but if 
 B lie between v'G' and (v'+l) C, let us suppose 
 
 v'G' = B-K'i and K'+ L' = G' 
 
 (l?' + l)G' = B + L' J as before, 
 
 in which one of the two, K' or L', will not be greater than the half of 
 G', so that we obtain by the same process, an equation of the form 
 
 p"A = q'k±G"\ 
 
 G" not greater than 
 the half of G'. 
 
 By proceeding in this way, we prove either, 1. The first alternative 
 of the proposition; or, 2. the possibility of forming a continued set 
 of equations 
 
 ;?A=^B±G, p'A = ^/B±G', p"A = ^'B±G", &c. 
 
 where, in the scale of quantities G, G', G", &c., no one exceeds the 
 half of the preceding. Consequently, we may (unless interrupted by 
 the first alternative) carry on this process until one of the quantities 
 G, G', G" &c. is smaller than Z (Prop. V.) that is, we have either 
 the first or second alternative of the problem. And exactly the same 
 demonstration may be applied to the case, where at the outset 
 j9A = 9B — E. 
 
 This proposition proves nothing of a single magnitude, but it 
 establishes two apparently very distinct relations between magnitudes 
 considered in pairs. There may be cases in which the first alternative 
 is established at last : and there may be cases in which it is never 
 established. We shall first take the case in which the first alternative 
 is established. 
 
 Suppose it ascertained by the preceding process that 
 
 8A = 5B 
 
 Here is an arithmetical equation between the magnitudes : and there- 
 fore any processes of concrete arithmetic will apply. Take the 
 40th part (8x5 = 40) of both sides, 
 
 ,. ^ . 8A 5B A B 
 
 which gives 40 = 40 ""' 5 = 8- 
 
 consequently the fifth part of A is the same as the eighth part of B, 
 or that which is contained 5 times in A is also that which is contained 
 
NUMBER AND MAGNITUDE. 9 
 
 8 times in B. Let this fifth of A or eighth of B be called M; then 
 A = 5M, B = 8M, and A and Bare both multiples ofM. Con- 
 sequently, when the first alternative of Prop. VI. exists, both A and 
 B are multiples of some third magnitude M. The converse is readily 
 proved, namely, that when A and B are both multiples of any third 
 magnitude, the first alternative of Prop. VI. is true. For if A=.rM, 
 B=3/M, we have yA = 3/jrM, xB = xi/Mj or i/A = xB. The 
 term measure is used conversely to multiple, thus : if A be a multiple 
 of M, M is said to be a measure of B. Hence in the case we are 
 now considering, A and B have a common measure, and are said to 
 be commensurable. We have therefore shewn that all commensur- 
 able magnitudes, and commensurable magnitudes only, satisfy this 
 first alternative. 
 
 There remains, then, only the second case to consider, which it is 
 now evident contains those magnitudes (if any such there be) which 
 have no common measure whatsoever. The question therefore is. 
 Are there such things as incommensurable magnitudes? On this 
 point the second alternative shews that our senses cannot judge, for 
 let Z be the least magnitude of the kind in question, which they are 
 capable of perceiving (of course with the best telescopes, or other 
 means of magnifying small quantities which can be obtained) then 
 we know that p A may be made to difier from ^B by less than Z, 
 that is, we may say that all magnitudes are sensible/ commensurable. 
 But it evidently does not follow that all magnitudes are mathema- 
 tically commensurable ; and it has been shewn, by process of de- 
 monstration, that there are * incommensurable quantities in such 
 abundance, that take almost any process of geometry we please, the 
 odds are immense against any two results being commensurable. 
 
 The suspicion that all magnitudes must be commensurable led to 
 the attempt, which lasted for centuries, to find the exact ratio of the 
 circumference of a circle to its diameter. And even now, though the 
 adventure is never tried by those who have knowledge enough to 
 read demonstration of its impossibility, no small number of persons 
 
 • Legendre, and others before him, have shewn that the diameter 
 and circumference of a circle are incommensurable ; and the student will 
 find in my Algebra, p. 98, or in the Lib. Useful Know., treatise on the 
 Study of Mathematics, p. 81, proof that the side and diagonal of a square 
 are incommensurablea. Also in Legendre's Geometry, or Sir D. Brew- 
 ster's Translation. 
 
10 CONNEXION OP 
 
 exercise themselves by endeavouring to make an elementary ac- 
 quaintance with geometry (and sometimes none at all) overcome this 
 difficulty. It is our business here to shew how strict deductions 
 may be made upon quantities which are incommensurable, with the 
 same facility, and in the same manner, as upon commensurables. If 
 we call any length (say that known by the name of a foot), the unit 
 of its kind, and denote it in calculation by 1, we must call twice such 
 a magnitude 2, and so on ; half such a magnitude i, and so on. We 
 may then apply arithmetic, every possible subject of which is con- 
 tained in the following infinitely extended table. 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 2 &c. 
 
 
 
 
 1 
 
 2 
 
 
 
 1 
 
 
 3 
 
 2 
 
 
 
 2 &c. 
 
 
 
 1 
 
 3 
 
 
 2 
 3 
 
 
 1 
 
 4 
 3 
 
 
 5 
 3 
 
 
 2 &c. 
 
 
 
 1 
 
 4 
 
 &c. 
 
 2 
 
 4 
 
 
 3 
 4 
 &c. 
 
 1 
 
 5 
 4 
 &c. 
 
 6 
 4 
 
 
 7 
 4 
 &c. 
 
 2 &c. 
 
 And every length which is commensurable with the foot is in- 
 cluded, in many different forms, in this table. Let F represent the 
 foot, L any other length, let M be their common measure : let 
 
 F=/M, L = ZM, then ZF=/M, or 
 
 L = ^ F = -7. when F is called 1. 
 
 But it is plain that we cannot, by any arithmetic of length founded 
 upon the foot as a unit, draw conclusions as to lengths which are in- 
 commensurable with the foot, though we can perhaps do so for any 
 practical purpose. Let L be a length which is such, and let Z be a 
 length so small as to be immaterial for the purpose in question. Then, 
 we can determine / and^, so that 
 
 /L = Z F ± G (G less than Z) 
 
 SO that, by assuming L = -7.F, we commit an error, in excess or defect, 
 
 less than G, and therefore immaterial. With such a process many 
 minds would rest contented ; but there is a consideration which will 
 
NUMBER AND MAGNITUDE. J 1 
 
 Stand in the way of perfect satisfaction, or at least ought to do so. 
 Granting that in the preceding case the error at the outset is imma- 
 terial, let us suppose the student disposed to substitute for all incom- 
 mensurables, magnitudes very near to them which are commensurables, 
 and thus to continue his career till he comes to the highest branches 
 of applied mathematics. Let us suppose a set of processes, beginning 
 in arithmetic, continued through algebra, the differential calculus, &c., 
 up to a point in optics or astronomy, in a series of results, embracing, 
 we may suppose, ten thousand inferences. If he set out with an 
 erroneous method, what security has he that the error will not be mul- 
 tiplied ten thousand fold at the end, and thus become of perceptible 
 magnitude. If somebody acquainted with the subject have told him 
 that it will not so happen, he might as well skip the intermediate 
 sciences and receive the result he wants to obtain on the authority of 
 that person, as study them in a manner, the correctness or incorrectness 
 of which depends on that person's authority. If he answer that the 
 result, namely, such multiplication of errors, appears extremely im- 
 probable, it may be replied, j^rsZ/j/, that that is more than he can 
 undertake to decide; secondly, that by pursuing his mathematical 
 studies on such a presumption, he makes all the pure sciences present 
 probable results only, not demonstrated results ; more probable, per- 
 haps, than many parts of history, but resting on an impression which 
 raust, in his mind, be the result of testimony. 
 
 It appears, however, that we may expect series of collateral results, 
 the one for commensurables, the other for incommensurables, and 
 presenting great resemblances to each other ; for we may, by any 
 alteration, however minute, convert the latter kind of magnitude into 
 the former. But this we may prevent, by extending our notions of 
 arithmetical operations, or rather by applying to magnitude processes 
 which are usually applied to number only, as follows : 
 
 If we examine the processes of arithmetic, we find, 1st, Addition 
 and substraction, to which abstract number is not necessary, since the 
 concrete magnitudes themselves can be added or subtracted. 2d, Mul- 
 tiplication, the raising of powers and the extraction of roots, in all of 
 which abstract number is essentially supposed to be the subject of 
 operation. 3d, Division, in which it is not necessary to suppose 
 abstract number in finding the whole part of the quotient, but in 
 which we cannot, without reference to numbers, compare the remain- 
 der and divisor, in order to form the finishing fraction of the quotient. 
 
12 CONNEXION OF 
 
 4th, The process of finding the greatest common measure of two quan- 
 tities, in which the remainder is not compared with the divisor, except 
 in a manner which is as applicable to the case of concrete magnitudes 
 as of abstract numbers. To shew this, we shall demonstrate the me- 
 thod of finding the greatest common measure of two magnitudes. 
 
 Let A and B be two magnitudes, which have a common measure 
 M ; let A =a M, B = 6 M. Then, it is clear that 
 
 a:A-F?/B or {xa+yh)M, xA — yB or {xa--i/b)M 
 have the same measure, unless it should happen that in the latter case 
 xa = i/b, in which case xAz=i/B. Let A be the greater of the two, 
 and let A contain B more than jS and less than /3 -f 1 times, so that 
 A=/3B-1-B', when B' is less than B. Then B' being A— /3B, is 
 measured by M. Let B contain B' more than /3' and less than yS' + 1 
 times ; or let B=/3'B'+ B'' where B" is less than B'. Let B' contain 
 B" more than /3" times, &c., or let B'=)3"B" + B'", and so on. And 
 B" or B — iS'B'is measured by M, &c. We have then the following 
 conditions : 
 
 A is a multiple of M 
 
 B M 
 
 A = /3 B -f B' B' < B, but is a multiple of M 
 
 B=/3'B-fB" B" <B' 
 
 B'=/3"B" + B"' B" <B" 
 
 Now, since B B' B" are decreasing quantities, and all multiples 
 of M, they are all to be found in the series, 
 
 M, 2M, 3M, 4M, &c. 
 
 in which continual decrease must bring us at last to nothing, or we 
 must end with an equation of the form.* 
 
 ■Q(n) __ ^(n+1) I3(n+1) _^ Q 
 
 that is, one remainder is a multiple of the next. To lake a case, let 
 the fifth equation finish the process ; so that, in addition to the pre- 
 ceding, we have 
 
 B" = /3"B' -f B'^ 
 
 B'" = /3'^B'^ 
 
 • When a letter denotes an indefinite number of accents, it is dis- 
 tinguished from au exponent by being placed iu brackets, and higher 
 numbers of accents than three are usually denoted by Roman numerals. 
 
NUMBER AND MAGNITUDE. 13 
 
 In the fourth, substitute B" from the fifth, giving 
 
 In the third, substitute B" and B'" as found, giving 
 
 B' = (/3" /3"' /3i^ + /3" + /S^O ^'" 
 
 In the second, substitute B' and B" as found, giving 
 
 B = (/3'/3"/3'"/3i^ + (3' 13" + (3' (3^ + /3'"/3'^ + 1) B^^ 
 
 In the first, substitute B and B' as found, giving 
 
 A = (/3/3'iS"/3"'/3i^ + ^(3^ 13" + /S/S'/S^^ + /3/3"'/3i^ +/3",S"'/3i^ + /3 + /3" + /3'0 B^ 
 
 Consequently, B^" is a common measure of A and B ; but, since M at 
 the outset is any common measure we please, let it be the greatest 
 common measure. Then B^^ must be M, for it is in the series M, 
 2 M, &c. ; and were it any other than M, there would be B'^ a com- 
 mon measurg, greater than the greatest. Hence this process deter- 
 mines the greatest common measure, and also the number of times 
 which each of the two, A and B, contains the greatest common 
 measure. 
 
 It is here most essential to observe, that this whole process is in- 
 dependent of any arithmetic, except pure addition and subtraction, 
 which can be performed on the magnitudes themselves, without any 
 numerical relation whatsoever; the only thing required being the 
 axiom in page 3. We shall actually exemplify this on two right 
 
 lines. 
 
 Ah 
 Bh 
 
 z k 
 
 A = B -h (xij) B = (XT/) + (zk) (x7j) = 2(zk) 
 Therefore B = 3(zk) A = 5(zk) 
 
 In this case, by actual measurement (supposed geometrically exact) 
 B and A are found to be respectively 3 and 5 times zk. 
 
 When the preceding process has an end, we therefore detect the 
 greatest common measure ; and we have shewn, that where there is a 
 common measure, the process must give it, with the converse. Con- 
 sequently, in the case where there is no common measure, this process 
 must go on for ever, and we have an interminable series of equations, 
 A=/SB-HB', B=/3'B'-f-B", B' = /3"B"-{-B'", &c., the conditions of 
 
14 CONNEXION OF 
 
 which are, that B, B', B", B''', &c., are a continually decreasing 
 series, though it does not follow that each one was less than the half 
 of the preceding. We shall now examine the effect of successive sub- 
 stitutions from the beginning, first making the following remark : If 
 there be any two incommensurable quantities A and B, of which A is 
 the greater, then there follows an interminable set of whole numbers, 
 /3, /3', /S", .... which are not subject to any particular law, but can 
 be found when A and B are given ; and an interminable set of quan- 
 tities, A, B, B', B", .... connected with the former by this law, that 
 A contains B between /5 and /3 -f 1 times ; B contains B' between /3' 
 and A'-j-l times, and so on. 
 
 We have B' = A— jSB 
 
 B" = B-/3'B' = B-iS' (A~/3B) 
 or B" = (/3/3'-M)B-/3'A 
 
 B'" = B'-/3" B" = A-/3 B-(/3/3' H- 1)/3"B -|-/3'i8"A 
 = (/5'/3"+ 1) A-(/3^/3"4-/3 + /3") B 
 
 and thus we go on representing the remainders alternately, in the form 
 pA. — qB and qB — pA. We may easily find the law of the co- 
 efficients, as follows : 
 Suppose we come to 
 
 B<"> = ^B--;?A 
 B("+i) = p'A-r/B 
 Then we have B^") = /3("+i) B<"+i) + B<"+2) 
 or B<"+2) __ B(«) __ j3(n+l) £(«+!) 
 
 = ^B-j9A-iS<"+^) (p'A^q'B) 
 = (/S("+^> r/ + ^)B-(/3("+^)p'+p)A 
 or if, continuing the preceding notation, we suppose 
 
 B("+2) = q"B-p"A 
 we have p" = /3<"+^) p -\- p g" = /3<"+^) (/ + q 
 
 so that, if we write the values of B', B", .... with the following no- 
 tation, putting opposite to each the /3{«) which occurs /or the first time 
 in the B(") of the equation ; namely, 
 
 B' =p,A^q,B (3 
 
 B" = q.B^p^A /3' 
 
NUMBER AND MAGNITUDE. 15 
 
 B"' ==j93A-^3B /3" 
 
 B- =^,B~/>,A r 
 
 &c. &c. &c. 
 
 we have the following uniform method of forming pn and gn for dif- 
 ferent values of n in succession. 
 
 />i = 1 qi = ^ 
 
 P2 = /3' 92 = /3'/3 + 1 
 
 Ps = /3>2 4-pi 93 = /5"^2 + 5'i 
 
 &c. &c. &c. &c. 
 
 in which it is plain, from the method of formation, that j9, p^ &c. 
 ^i ^2 &c- 3^6 increasing whole numbers, so that we may continue, 
 supposing B' B" .... never fail, till pn and qn are greater than any 
 number named. And since B' B'' .... are all less than B, and 
 therefore less than A, we have the following succession of results, 
 ad infinitum. 
 
 ^1 A is greater than 5'iB but less than (5'i+l)B 
 
 a?g A is less than q^ B but greater than {q^ — 1 ) B 
 
 ^3 A is greater than g'sB but less than (5'3+l)B 
 &c. &c. &c. 
 
 Hence, it appears that 
 
 A is greater than 
 
 less than 
 
 greater than 
 
 less than — B he. ad inf. 
 
 Now, from this table of relations, we can determine whether any 
 given multiple of A, or A, is greater or less than any given multiple of 
 B, 3/B. To do this we must inquire between what two consecutive 
 multiples of B does jtA lie. 
 
 We now proceed as follows : 
 
 1. We must shew that any fraction, such as 
 
 a4-m .. . ^ a j ^ 
 
 T-^ — lies between -r and — 
 6 + n b n 
 
 2± 
 Pi 
 
 B 
 
 2± 
 
 P2 
 
 B 
 
 1± 
 P3 
 
 B 
 
16 CONNEXION OF 
 
 unless where the two latter are equal, in which case the first also is 
 the same. The preceding must be true if 
 
 a + 171 lies between r(b + n) and - (6 -f n) 
 
 , an . bm , 
 
 or a + and f- m 
 
 n 
 
 , an hm 
 
 or a-f-;w + -7- — mand a +m-{- a 
 
 n 
 
 or a + m -\- n-r -^ n— and a + m + h o-r 
 
 on no 
 
 which is evidently true : for if - be greater than — , the first is greater 
 
 than a -t- m, and the second less ; if ^ be less than — , vice versa. 
 
 n 
 
 2. We now see that 
 
 iL or ^;;^ lies between ^^ and ^-1 
 
 ^» and li 
 f^ Pi 
 
 or 
 
 P. P>"'P,-\-P2 
 
 Jfz ra 
 
 and so on. Consequently, to arrange all the fractions thus con- 
 sidered, in order of magnitude, we must write them thus, 
 
 1l h. 1l .... .. . -^ ^ -^ 
 
 P\ Pa Ps "" Ps Pi Pi 
 
 3 . We can thus bring two fractions as near together as we please : 
 to prove this, take three consecutive fractions 
 
 £m gm+l / <7m+2 ^^ /^ gm+1 + ^m \ 
 
 Pm Pm+1 \pm+2 (i'^"'+^%+i -\- fm J 
 
 which reduced to a common denominator, the first and second, and 
 the second and third, give 
 
 yw pm+l gm+l Pm 
 
 Pm Pm+l Pm+l fm 
 
 and ^^"""^^Vm+l Pm+l + Pm qm+l ^^^ ^^""^^^qm+l pm+\ + Pm+l qn 
 Pm+l Pm+2 Pm+l pm+i 
 
 in which it is clear that the difference of the numerators is the same 
 in each couple, but that if the first numerator be the greater of the 
 
NUMBER AND MAGNITUDE. 1 7 
 
 first couple, the second numerator is that of the second ; a result we 
 might have foreseen, having proved that 
 
 'tt±L lies between ^^^ and ^. 
 
 Hence it follows that the numerator of the difference of any two 
 successive fractions of the set 
 
 Ij. ll _^ 
 Vi P2 ?3 
 
 is the same as that of the difference immediately preceding, that is, 
 the difference of — and ■ has the same numerator as 
 
 Pn pn-l 
 
 the difference of ■ ■ and ^^~ which has the same 
 
 Pn-l Pn-2 
 
 numerator as the difference of — and — ; but 
 
 P2 Pi 
 
 li.__ ^ ^ J_ ^ 
 
 therefore this numerator of the differences is always 1, or 
 ^" and ^^±i. differ by ^ 
 
 Pn Pn+l Pn Pn+1 
 
 Hence the difference may be made as small as we please, or smaller 
 than any fraction — named by us, since pn itself can be made greater 
 than m, much more^w pn+i- 
 
 4. These fractions cannot for ever lie alternately on one side and 
 
 V 
 
 the other of any given fraction -. 
 
 For if this were possible, then, since A lies between 
 
 £^B and 2^^B 
 
 Pn pn+l 
 
 V 
 
 and since by the supposition -B does the same, and since the couple 
 
 just mentioned can be made to differ by as small a fraction of B as 
 we please, then we should have 
 
 A = -B ± K 
 
 J? 
 
 c 2 
 
18 CONNEXION OF 
 
 where K may be made as small as we please. Now this is saying 
 
 V 
 
 that A ^ - B ; for A must either be 
 
 X 
 
 -B or -B ± some definite magnitude; 
 ic X 
 
 but the latter it is not ; for the supposition we are trying leads to 
 
 A = -B + a magnitude as smaH as we please. 
 
 Consequently, our supposition that the series of fractions lie alter- 
 
 nately on one side and the other of a definite fraction -, leads to the 
 
 conclusion that A and B are Commensurable, or the process of finding 
 B' B" .... finishes, as we have shewn. But it does not finish, by 
 hypothesis; therefore the series of fractions cannot lie alternately 
 
 V 
 
 on one side and the other of -. 
 
 X 
 
 We can now shew between what multiples of B xA must lie. It 
 is clear that 
 
 a; A lies between -^^B and f^^llB : 
 
 Tpn /Jn+l 
 
 now it is not possible that any whole number v should always lie 
 between — ^ and ; for if so, then would 
 
 - always lie between — and "^ 
 X pn pn+i 
 
 which has been proved to be impossible. Consequently, 
 
 ■^ ^" "R A '^ ^""♦"^ "R (^^^^^ approach each other 
 Pn pn+i without limit) 
 
 must come at last always to lie between two multiples of B ; and 
 still more must x A, which lies between them. Hence, by proceeding 
 far enough, we can always find between what multiples of B lies xA; 
 and thence whether xA is greater or less than j/B. 
 
 We have thus divided all pairs of magnitudes into two classes, 
 1. CommenmrahleSf in which we can always say that A = -B, 
 
 g and p being whole numbers, and can always tell exactly by what 
 fraction of A or B, j A exceeds or falls short of 3/B. For we have 
 
NUMBER AND MAGNITUDE. 19 
 
 xA^yB=z fx--y^\A=: fx^—y'SB if xA>yB 
 (yB-xA)= (|3/-:r)A= {y-x^-)B if xA<yB 
 
 2. IncommensurableSf in which we can never say A = - B, but 
 
 in which we can assign a series of fractions alternately increasing and 
 decreasing, but making less and less change at every step, 
 
 ll ll ll 
 
 Pi P2 Pa 
 
 and such that A is greater than — B, less than— B, &c. ad infinitum: 
 
 Pi P2 
 
 so that we can always assign 
 
 A = ^B + K 
 
 Pn 
 
 where K is less than any magnitude we name ; and such that we can 
 always tell by them whether xA exceeds or falls short ofi/B, but not 
 exactly how much. 
 
 Let us suppose, as an example, that we have two magnitudes A 
 and B, which tried by the process in page 13, give 
 
 A = B + B', B = B' + B", B' = B" + B", &c. ad inf. 
 
 or suppose /3 = 1, ^' = 1, /3" = 1, &c. ad inf. 
 
 Hence the several values ofp and q are p, = l, />2 = 1> ^3 = 2, &c. 
 as in this table, 
 
 12 3 4 5 6 7 8 9 10 11 12 _ 
 
 _^ - &c 
 
 I? 1 1 2 3 5 8 13 21 34 55 89 144 
 
 9- 1 2 3 5 8 13 21 34 55 89 144 233 &c. 
 
 A>B<2B>|b<|B>2b<Hb &c. 
 
 Hence A lies between B and 2B 
 
 2A 3B .. 4B 
 
 3A 4B .. 5B 
 
 4A 6B .. 7B 
 
 6A 8B .. 9B&C. 
 
 If we wish to know between what multiples of B 100 A lies, 
 
 we find 
 
20 CONNEXION OP 
 
 A> WB < |->; 100A> leillB < 16lli|B 
 
 or 100 A lies between 161 B and 162 B. 
 
 We can thus form what we may call a relative multiple scale made 
 by writing down the multiples of A, and inserting the multiples of 
 B in their proper places; or vice versa. In the instance just given 
 the commencement of this scale is 
 
 B, A, 2B, 3B, 2A, 4B, 3A, 5B, 6B, 4A, 7B, 8B, 5A, 9B, &c. 
 
 which we may continue as far as we please by simple arithmetic. If 
 the magnitudes in question be lines, we may represent this multiple 
 scale as follows : 
 
 01 1 )f-4 HX i )H h-^ 1 Bf 1 
 
 Measuring from O, the crosses mark off multiples of A, and the bars 
 multiples of B. Thus 
 
 01i = B Ol2=2B Ol3=3B&c. 
 Oxi = A 0x2= 2A Ox3 = 3A&c. 
 
 We shall now proceed to some considerations connected with 
 a multiple scale, for the purpose of accustoming the mind of the 
 student to its consideration. We may imagine a scale like the pre- 
 ceding to be equivalent to an infinite number of assertions or nega- 
 tions, each one connected with the interval of magnitude lying between 
 two multiples of B. Thus, the preceding scale contains the following 
 list ad infinitum. 
 
 1 . Between and B lies no multiple of A 
 
 2. Between B and 2B lies A 
 
 3. Between 2B and 3B lies no multiple of A 
 
 4. Between 3B and 4B lies 2 A 
 &c. &c. &c. &c. 
 
 Now, on this we remark, 1st, That the negatives of the above 
 series, though they appear at first to prove nothing, yet in reality 
 have each an infinite number of negative consequences. From the 
 third assertion of the preceding list, namely, neither A, nor 2 A, nor 
 3 A, &c. lies between 2B and 3B, we immediately deduce all the 
 
 2 
 
 following : A does not lie between 2 B and 3 B, nor between - B and 
 
NUMBER AND MAGNITUDE. 21 
 
 3 2 3 2 3 
 
 -B, nor between -B and - B, nor between -B and -B, &c. &c, 
 
 2 ' 3 3 ' 4 4' 
 
 2d, Observe that every affirmative assertion in the above includes a 
 
 certain number of the affirmative ones which precede., and an infinite 
 
 number of parts of the negative ones preceding and following. For 
 
 instance, we find that 100 A lies between 161 B and 162 B, or A lies 
 
 between — B and - — B, that is between B and 2B. Again, 2 A 
 
 322 324 
 
 lies between — B and B, or between 3B and 4 B. Similarly, 
 
 100 100 ' •' 
 
 3A lies between B and B, or between 4B and 5B; and it 
 
 100 100 ' 
 
 might thus seem at first as if every affirmation made all the affirm- 
 ations preceding its necessary consequences. But if we try 90 A by 
 the preceding, we shall find that it lies between 
 
 ^^^^^ B and ii^B or between 144 B and 146 B 
 
 100 100 
 
 so that we can only affirm 90 A to lie either between 144 B .and 
 145 B; or between 145 B and 146 B^ but we 4o not (from thisj know 
 which. But we Can say that 90 A (Joes not lie between 146 B and 
 147 B, or between 147B and 148 B, &c. The points at which any 
 affirmation does not determine those preceding may be thus found. 
 Let kk lie between /B and (^-}-l)B ; or 
 
 A lies between tB and B 
 
 A ml-r, , ^(^+1)t> 
 
 mA -T-B and -A_L_^B 
 
 k k 
 
 If-^ and —-^ — ^ lie between t and ^+1, then mk lies between 
 
 ^B and {t 4-l)B: but if, in going from the first to the second, we pass 
 through a whole number, or if?;?/, divided by k, gives a quotient t 
 and remainder r, and m{l-{-l), divided by /e, gives a quotient t-{-l 
 and remainder r', then we have 
 
 or m =■ k — r -\-r' or r + m = ^ + r' 
 
 and in all cases where r-\-mh greater than k, this condition can be 
 fulfilled. The process may be shortened, by using instead of / the 
 
22 CONNEXION OF 
 
 remainder arising from dividing / by k. Suppose, for instance, it is 
 required to determine what preceding affirmatives are ascertained by 
 the proposition 10 A lies between 33 B and 34 B. We have then 
 ^=33, A: = 10, remainder of /-T-/: = 3. 
 
 m = 2 tz= 6 r = 6 2 A lies between 6B and 7B 
 
 m=3 t^ 9 r' = 9 
 
 w = 4 t = 13 r = 2 4A 13B and 14B 
 
 W2 = 5 ^=16 r = 5 5A 16B and 17B 
 
 m = 6 t = l9 r = 8 
 
 w = 7 jf = 23 r=l 7A 23 B and 24 B 
 
 w = 8 tz=26 r = 4: 
 
 m = 9 t = 29 r = 7 
 
 By proceeding thus, it will appear that there is no perceptible law 
 regulating the places of A, 2 A, .... among B, 2B, &c., derivable 
 from the sole condition of /cA lying between ^B and (/-f-l)B. Never- 
 theless, it is easy to prove, that if all the rest of the relative scale be 
 given from ahd after any given point, that the whole of the preceding 
 part can then be determined. For, suppose /cB to be the commence- 
 ment of the part of the scale given, and let the place of w A be asked 
 for, which precedes A A, the first multiple of A appearing in the 
 scale. Multiply m byg, so that wg shall be greater than A. Then 
 mgA appears in the portion of the scale given, say between wB and 
 (m> + 1)B. Therefore 
 
 mA lies between -B and ^^^ B 
 g g 
 
 and if — and ■ lie between t and ^ + 1 , the question is settled ; 
 
 but this must always be the case, if we include the case where - or 
 
 -» is itself a whole number. 
 
 g 
 From all that precedes, we draw the following conclusions : 
 
 1 . Having given A and B, two incommensurable magnitudes of 
 
 the same species (both lengths, both weights, &c.), we can assign, by 
 
 a processes embling that of finding the greatest common measure in 
 
 arithmetic, the relative scale of multiples of A and B, which points 
 
 out between what two multiples of B any given multiple of A lies, or 
 
 vice versd. 
 
NUMBER AND MAGNITUDE. 23 
 
 2. Any part of the beginning of this scale being deficient, we can 
 construct it by means of the rest. 
 
 3. We can find a magnitude which shall be commensurable with 
 A, differing from B by less than any magnitude we name; and can 
 assign the fraction which it is of A. 
 
 Given the two magnitudes, their relative multiple scale is given ; 
 but when the scale is given, the two magnitudes are not given. For 
 it is easily proved that there is an infinite number of couples of mag- 
 nitudes which have the same scale with any given one. Let the scale 
 
 P P 
 
 of A and B be given ; then will the scale of - A and - B be the same, 
 
 where J!? and q are any whole numbers whatsoever. 
 
 For if ^A lie between ZB and (Z+1)B 
 
 then ^^ A lies between Z^B and (/ + 1)^B 
 
 q q ^ ' q 
 
 or making ^A = A' ^ B = B' 
 
 q q 
 
 A A' lies .between ZB' and (Z+1)B' 
 
 whence the scale of A and B is the same as that of A' and B' for any 
 value of /c. 
 
 What is it, then, which is given when the scale is given ? Not the 
 
 magnitudes themselves ; for if the scale belong to A and B, it also 
 
 p p 
 
 belongs to every one of the infinite cases of -A and - B. The scale, 
 
 q q 
 
 therefore, only defines such a relation between the magnitudes as be- 
 longs to 2 A and 2B, 3A and SB, &c., as well as to A and B. It is 
 usual to call this relation the proportion between the two quantities in 
 common life, and in mathematics their ?'atio ; in Euclid the terra is 
 
 Xoyos, 
 
 Two magnitudes, A and B, are said to have the same ratio as 
 two other magnitudes, P and Q, when the relative scales of the two 
 are the same ; that is, when the multiples of Q are distributed as to 
 magnitude among those of P, in the same way precisely as those of B 
 are distributed among those of A. And P and Q may be two mag- 
 nitudes of one kind, two areas, for instance, while A and B may be of 
 another, two lines, for instance. 
 
 It is easy to shew that this accordance of scales is equivalent to 
 the common idea of proportion, such as it would become if we took 
 
24 CONNEXION OF 
 
 all means of comparison away, except that of multiples. Let us 
 imagine A and B to be two lines in a picture, and P and Q the two 
 corresponding lines in what is meant for an exact copy on a larger 
 scale. Set an artist to determine whether P and Q are in the proper 
 proportion to each other, without any assistance except the means of 
 repeating A, B, P, Q, as many times as he pleases. He will reason 
 as follows : " If Q be ever so little out of proportion to P, though it 
 may not be visible to the eye, yet every multiplication of the two will 
 increase the error, so that at last it will become perceptible. If there 
 be a line 100 A laid down in the first picture, and if it be found to lie 
 between 51 B and 52 B, then should 100 P lie between 51 Q and 
 52 Q. But if Q be a little wrong, then lOOP may not lie between 
 51 Q and 52 Q." 
 
 It only remains to see whether this definition of proportion will 
 include the case of commensurable quantities. These satisfy such an 
 equation as A;A = /B, k and / being two whole numbers, and it is 
 easy to shew that the whole relative scale is divided into an infinite 
 succession of similar portions. Firstly, this one equation determines 
 the whole scale ; for we have 
 
 A = Ib mA='^B 
 
 k k 
 
 or if — lie between ^and^ + l, wiA lies between t B and (^ + 1 )B 
 
 if -— = ^ mA = ^B 
 
 k 
 
 Let us suppose, for instance, A = - B. Then we have 
 
 A lies between B and 2B 
 
 2A 3B and 4B 
 
 3A 5B and 6B 
 
 4 A is equal to 7 B : or the scale is 
 
 B A 2B 3B 2A 4B 5B 3A 6B 
 
 7B 
 
 4A 
 
 From this point the scale begins again in the same order. Thus, 
 the second portion is 
 
 ^^ 8B bA 9B lOB 6A IIB 12B 7A 13B ^^? 
 
 4A 8A 
 
 and so on ad infinitum. The arithmetical definition of A having the 
 
NUMBER AND MAGNITUDE. 25 
 
 same ratio to B which P has to Q, is simply that of A being the 
 same fraction of B which P is of Q : or if 
 
 A=iB P=iQ 
 
 Now, since the scale depends entirely on -, it is the same for 
 
 both ; conversely, if the scale of A and B be the same as that of 
 P and Q, then if/cA = ZB, kF must = /Q. Hence the two defi- 
 nitions are synonymous : if one applies, the other does also. 
 
 When the multiple scale of A and B is the same as that of P and 
 Q, we have recognised the proportionality of A and B to P and Q. 
 But these scales may differ. The question now is, may they differ in 
 all possible ways, or how far will their manner of differing in one 
 part of the scale affect their manner of differing in others ? Am I, to 
 take an instance, at liberty to say, that there may be four magnitudes 
 such that 20 A exceeds 18B, while 20 P falls short of 18Q ; but that, 
 for the same magnitudes, 13 A falls short of 17B, while 13P exceeds 
 17 Q ? Such questions as this we proceed to try. 
 
 When only two things are possible, which cannot co-exist, each 
 is the complete and only contradiction of the other : the assertion of 
 one is si denial of the other, and vice versa. But when three different 
 things are possible, one only of which can be true, the assertion of 
 one contradicts both of the other two ; the denial of one does not 
 establish either of the other two. 
 
 The want of a common term, which may simply mean not less, 
 that is, either equal or greater, without specifying which, and so on, 
 causes some confusion in mathematical language. To remind the 
 student that not less does not mean greater, but either equal or 
 greater, we shall put such words in italics. Thus, not less and less, 
 not greater and greater, are complete contradictions : the denial of 
 one is the assertion of the other. 
 
 If A and B be two magnitudes of one kind, and P and Q two 
 others, of the same or another kind, such that 
 
 mA is /ess than 7^B, mP is not less thdni nQ 
 
 then it is impossible that there should be any multiples such that 
 
 m'A is greater than n B, m'F is not greater than w'Q 
 
 For we find, from the first of each pair, 
 
26 CONNEXION OF 
 
 A is less than — B, A is greater than —, B 
 
 m ' ° rn' 
 
 still more is — B greater than — 7 B or — greater than — r 
 
 But P is not less than — Q, P is not greater than —7 Q 
 
 Now, all the four combinations of this latter assertion contradict 
 
 n . , n' „ „ 
 
 — IS greater than — • ; as follows : 
 
 gives — = — ; 
 mm 
 
 P = i^Q, P = ^,Q, 
 
 m ^' m' ' 
 
 P greater than — Q, P = — ^, Q, gives — less than — •, 
 
 P = — Q, P less than — Q, gives — less than — ; 
 
 m ' m' ' * m m' 
 
 P greater than — Q, P less than — • Q, gives — less than — , 
 m ' m' ' ° wi m 
 
 Hence the two suppositions above cannot be true together : the 
 happening of any one case of either proves every case of the other to 
 be impossible. 
 
 If we range all the possible assertions which can be made, we 
 have as follows : 
 
 mA is greater than wB 
 mA is equal to w B 
 
 mA is less than nB 
 
 m'A is greater than %'B 
 m'A is equal to w'B 
 m'A is less than n'B 
 
 Ai 
 
 P3 
 
 mP 
 
 is greater than 
 
 «Q 
 
 p. 
 
 mV 
 
 is equal to 
 
 «Q 
 
 Pi 
 
 wzP 
 
 is less than 
 
 «Q 
 
 Pt 
 
 m'F 
 
 is greater than 
 
 n'Q 
 
 P2 
 
 mT 
 
 is equal to 
 
 »'Q 
 
 Pi 
 
 mT 
 
 is less than 
 
 n'Q 
 
 Four of these must be true, one out of each triad ; and there are 
 81 ways of taking one of each, so as to put four together. But we 
 shall take the sets A and a together, and find what inference we can 
 draw by taking one out of each. 
 
 A3 a^ proves nothing as to — and —7. It merely says that 
 
 — B and — : B are both exceeded by A, which may be whether 
 m m 
 
 — is greater than, equal to, or less than — ;. The same for P3 y;, 
 
NUMBER AND MAGNITUDE. 27 
 
 A3 a^ proves — -, greater than — ; as does P3 p^ 
 A3 a^ proves —7 greater than — ; as does P3 p^ 
 
 Aq a^ proves —, less than — ; as does Pg p^ 
 
 A2 ^2 proves — -, equal to — ; as does Pg p^ 
 
 A2 «i proves —7 greater than — ; as does Po pi 
 
 Ai a^ proves — ; less than — ; as does P^ p 
 
 Ai ^2 proves — ; less than — ; as does V^ p^ 
 Ai Gi proves nothing ; neither does Pi Pi 
 
 Now, if we put these pairs together, or make pairs of assertions, 
 in the manner already done, we have 81 distinct sets of four asser- 
 tions, divisible into those which may be true together, and those which 
 cannot be true together. An inconsequential supposition, such as 
 Agflg, may co-exist with any of the rest from the other set Pp; but 
 
 those which give -y necessarily greater, equal to, or less than — in 
 
 the set A a, can only co-exist either with the similar ones from the 
 set Pp, or with those which are inconsequential. Thus we have 
 
 A3 ^3 may be true with any marked Pj5 
 
 A3 «2 requires either Pa^^s, 
 
 Aatti 
 
 "2 ^3 
 A2 «2 
 A2 fll 
 
 Ajfl 2 
 
 Ps^a, 
 P3P3. 
 Psi's. 
 P3P3, 
 
 P3P2, 
 
 Pa/'i, 
 
 P3P.. 
 
 Pa/-!, 
 
 P.Ps, 
 
 ViP» 
 
 -p.p.. 
 
 
 PsiJa. 
 
 Pai'i. 
 
 P.P3. 
 
 Pii's. 
 
 P.i's,. 
 
 ^iPz. 
 
 P2P1, or 
 
 P«/>i, •• 
 
 Pi^Jj, •• 
 
 P«Pi. •• 
 
 P1P2. •• 
 
 PiPs, •• 
 
 Pi^Ji 
 Pii'i 
 P.fi 
 Pi Pi 
 P,Pi 
 Pii'i 
 Pii'i 
 
 Ai tti may be true with any marked Pp 
 
 The remaining thirty cannot be true ; but it is unnecessary to specify 
 
28 CONNEXION OF 
 
 them, as a simple induction from the preceding will shew how to 
 classify those which may and cannot be true. Attach an idea of 
 magnitude to the phrases greater y equal, and less ; say that " u4 is 
 greater than B" is higher than " A is equal to B," and this again 
 higher than ** A is less than B." We have marked the highest 
 phrases by the highest numbers. Say that in Agflj, Ag/z,, &c. (calling 
 A and a the antecedent clauses of any four marked A, a, P, p), the 
 antecedents are descending; in Aga^, A^Oj, and A, a,, stationary; 
 and in Aj Oj, Aj a^, &c. ascending. Then all the propositions which 
 imply the co-existence of any two antecedents, and any two conse- 
 quents of the form A a Fp, may be divided into those which may be 
 true, and those which cannot be true, by the two following rules ; 
 Ascending antecedents cannot have descending consequents. 
 Descending antecedents cannot have ascending consequents. 
 
 Precisely the same rules will apply if we take two propositions 
 A P for antecedents, and two others ap for consequents; as we may 
 either deduce in the same manner, or by simple inversion. For if 
 Ao Pp, with any numerals subscribed, do not contradict either of the 
 preceding rules, neither will the corresponding case of AP ap do so, 
 and the contrary. Instances, A, a, P3P3 and A, P3 a^p^ ; A,c, V^p^ 
 and A2P3 fliPa? ^c. 
 
 Let us then take a case of A, B, P, Q, in which we find one 
 ascending assertion relative to mA, nB, wP, nQ, for some partiisular 
 values of m and n ; for instance 
 
 * p JS A is less than 4 B 
 ^ ^ I3P is greater than 4Q 
 
 which, as we have seen, is never contradicted in form by any assertion 
 that can be true of any other multiples. These four quantities are 
 not proportionals: for 3 A being less than 4B, and 3P greater than 
 4Q, P cannot lie in the scale of P and Q in the same place as A in the 
 scale of A and B. But to what more common notion can we assi- 
 milate this sort of relation between A, B, P, and Q, namely, that 
 all true assertions of the form (AP) are either ascending or stationary, 
 and never descending ? Have we any thing corresponding to this in 
 the arithmetic of commensurable quantities ? Let us suppose A and 
 B commensurable, and also P and Q : say that 
 
 A = -B P = -,Q 
 
 V V 
 
NUMBER AND MAGNITUDE. 29 
 
 Then 3-B is less than 4 B ; 3 ^, Q is greater than 4Q ; 
 
 - is less than - ; -, is greater than - ; - is greater than - 
 V 3 V 3 V V 
 
 or A is a less fraction of B than Pis of Q ; which in arithmetic is 
 
 also said thus, A bears a less proportion to B than P does to Q, or 
 
 P bears to Q a greater proportion than A bears to B. Hence we get 
 
 the following definitions, in which we insert the previous definition of 
 
 proportion, and the accordance of the whole will be seen. 
 
 When all true assertions on (m A, n B) {m P, n Q) are either 
 ascending or stationary, and never descending, A is said to have to B 
 a less ratio than P to Q; when always stationary, the same ratio; 
 when always descending or stationary, and never ascending, a greater 
 ratio. 
 
 This amounts in fact to the definition given by Euclid, the 
 opening part of whose Fifth Book we shall now make some extracts 
 from, with a few remarks. 
 
 Definition III. Ratio is a certain mutual habitude (^o-^^ics, 
 method of holding or having, mode or kind of existence) of two 
 magnitudes of the same kind, depending upon their quantuplicity 
 (^^yiXixornSf for which there is no English word ; it means relative 
 greatness, and is the substantive which refers to the number of times 
 or parts of times one is in the other). 
 
 In this definition, Euclid gives that sort of inexact notion of ratio 
 which defines it in commensurable quantities, and gives some light 
 as to its general meaning. It stands here like the definition of a 
 straight line, " that which lies evenly between its extreme points" 
 prior to the common notion, " two straight lines cannot enclose 
 space," which is the actual subsequent test of straightness. In most 
 of the editions of Euclid we see " Ratio is a mutual habitude of two 
 magnitudes with respect to quantity,''^ which makes the definition 
 unmeaning. For quantity and magnitude in our language are very 
 nearly, if not quite, synonymous ; or if any distinction can be drawn, 
 it is this: magnitude is the quantity of space in any part of space. 
 But as Euclid is here speaking of magnitude generally (not of sj: ace 
 magnitudes only) the words magnitude and quantity are the same.* 
 
 * Euclid again uses the word ^yiXixoms (Book VI. def. 5) in a 
 manner which settles its meaning conclusively. The more advanced 
 reader may consult Wallis, Opera Mathematica, v. II. p. 665, 
 
 D 2 
 
30 CONNEXION OP 
 
 Definition IV. Magnitudes are said to have a ratio to each 
 other which can, being multiplied, exceed *' one the other." This 
 means that quantities have a ratio when, any multiples of both being 
 taken, the relation of greater or less exists. It is usually rendered 
 " Two magnitudes are said to have a ratio when the lesser can be 
 multiplied so as to exceed the greater." But the above is literally 
 translated, and the sense here given to ratio makes the next definition 
 consistent. It is a way of expressing that the two magnitudes must 
 be of the same kind, which requires that the notion of greater and less 
 should be applicable to them. That this notion should be applicable 
 to the quantities themselves as well as their multiples, being the 
 necessary and sufficient condition of the possibility of the comparison 
 implied in the next definition, is here assumed* as the distinction of 
 quantities which have a ratio. 
 
 Definition V. Magnitudes are said to be in the same ratio 
 the first to the second, and the third to the fourth : when the same 
 multiples of the first and third being taken, and also of the second 
 and fourth, with any multiplication, the first and third (multiples) are 
 greater than the second and fourth together, or equal to them together, 
 or less than them together. 
 
 This amounts to our definition of proportion, namely, that the 
 relative multiple scale of A and B is the same as that of P and Q. 
 For, take the same multiples of A and P, namely, wA and mP, and 
 the same multiples of B and Q, namely, nB and nQ. Then, if the 
 relative multiple scales be the same, let m\ lie between vB and 
 (y + l)B, it follows that mF lies between vQ and (v4-l)Q. If, then, 
 n be less than v, nB is less than vB, and nQ less than vQ. And 
 W2 A being greater than vB must be greater than nB, while, for the 
 same reason, ;nPis simultaneously greater than nQ. In the same 
 way the other parts of the definition V. may be shewn to be included 
 in that of identity of multiple scales. Now, reverse the supposition 
 
 • The common version is several times referred to afterwards, and the 
 definition 4 expressly alluded to, in the editions of Euclid. But it must 
 be remembered that the Greek of Euclid contains no references to pre- 
 ceding propositions, these having been supplied by conmientators. The 
 reader may, if he can, make Aoyov 'ix*'* ^i^f aXXnXa fnyiSri kiytren, 
 a ^vvuTxi 'rokkxTXatna^ofAivtx, aXkrikuv vTifi;^i/v mean, " Magnitudes are 
 said to have a ratio, when the less can be multiplied so as to exceed 
 the greater." 
 
NUMBER AND MAGNITUDE. 31 
 
 and assume Euclid's definition. If, then, mA lie between vB and 
 (y-\-\)B, it follows that mA is greater than vB, whence, by the 
 assumption wP is greater than vQ. Similarly, because m A is less 
 than (v+l)B, mP is (by that definition) less than (t;+l)Q. There- 
 fore, mP lies between vQ and (v+l)Q, or in this instance, or for 
 any one value of m, the scales are accordant, and the same may be 
 proved in any other case. It follows, then, that the two definitions 
 are mutually inclusive of each other. 
 
 The manner in which Euclid arrived at this definition has been 
 matter of inquiry. But any one who will examine the first nine 
 propositions of the tenth * book, will see that he had precisely the 
 same means of arriving at it as we have used. But, besides this, he 
 might have come by the definition from a common notion of practical 
 mensuration, as follows. Suppose two rods given, one of which is 
 the English yard, the other the French metre, but neither of them 
 subdivided. The only indication which looking at them will offer, is 
 that the metre exceeds the yard apparently by about ten per cent. 
 To get a more exact notion, the obvious plan will be to measure some 
 great distance with both. Suppose 100 yards to be taken off with 
 the yard measure, it will be found that that 100 yards contains about 
 91 metres and a half, the half being taken by estimation, and we will 
 suppose the eye could not thus err by a quarter of a metre. Then 
 the yard must be "915 nearly of a metre, and the error upon one yard 
 cannot exceed the hundredth part of the quarter of a metre, or '0025 
 of the metre. But the mathematician, to make this process perfectly 
 correct, will suppose distance ad infinitum, measured from a point 
 both in yards and metres, or in fact will form what we call the relative 
 multiple scale. He then looks along this scale for a point at which 
 a multiple of a yard, and a multiple of a metre end together. If 
 this happen, and it thus appear that m yards is exactly equal to 
 
 n metres, the question is settled, for a yard must be — of a metre. 
 
 But it will immediately suggest itself to a mind which is accustomed 
 not to receive assumptions without inquiry, that it may be no two 
 
 • There are two English editions of the iohole of Euclid, and there 
 may be more : that of John Dee (now old and very scarce) and that of 
 J. Williamson, London, 1788, in two thin quarto volumes. The disser- 
 tations in the latter are a strange mixture of good and bad, but the text 
 is very literally Euclid, in general. 
 
32 CONNEXION OF 
 
 points ever coincide on the multiple scale. But in this case it is 
 very soon proved that mA may be made as nearly equal to nB as 
 
 we please, by properly finding m and n; so that a fraction — may 
 
 be found such that A shall be as nearly — B as we please. Even 
 
 m ^ 
 
 admitting that this would do to assign A in terms of B, it leaves us 
 no method of establishing any definite connexion between A con- 
 sidered as a part of B, and P considered as a part of Q. 
 
 The word part usually means arithmetical part, namely, the 
 
 3 
 result of division into equal parts. Thus - is a part of 1 made by 
 
 dividing 1 inio 7 equal parts, and taking 3 of them. The phrase of 
 
 Euclid in the books on number (VII. to X. both inclusive) is that - is 
 
 7 
 
 3 
 part of 1, - h parts of 1. And it is easily shewn that, in this use of 
 
 the word, every quantity is either ;?ar^ or parts of every other quantity 
 which is commensurable with it. And of two incommensurable quan- 
 tities, neither is part or parts of the other. But in the original sense 
 of the word part, any less is always part of the greater. This notion 
 of incommensurability, the non-existence of the equation w A = nB, 
 for any values of m or n, obliges us to have recourse to a negative 
 definition of proportionality, a term which we proceed to explain. 
 Examine the definition of a square, namely, " a plane foursided figure, 
 with four equal sides and one right angle." It is clear that the ex- 
 amination of a finite number of questions will settle whether or no a 
 figure is a square. Has it four sides? are they in the same plane? 
 are the sides equal ? is one angle a right angle? Proof of the affirm- 
 ative of these four propositions proves the figure to be a square. Now, 
 examine the number of ways in which a figure can be shewn to be 
 not a square. All propositions are either affirmative or negative ; 
 A is B or A is not B. The affirmative can be proved or the negative 
 disproved, with one result only, for both give A is B. But the 
 affirmative can be disproved, or the negative proved, with an infinite 
 number of results; it is done by proving that A is C, or D, or E, &c. 
 &c. ad infinitum. Thus there may be an infinite number of ways of 
 shewing that a figure is not a square, but there is only one way of 
 shewing that it is a square. This we call a positive definition. 
 
 Now examine the definition of parallel lines, " those which are in 
 
NUMBER AND MAGNITUDE. 33 
 
 the same plane, but being produced ever so far do not meet." We 
 are not considering where the lines meet, if they do meet, or dis- 
 tinguishing between lines which meet in one point and in another, but 
 simply dividing all possible pairs of lines into two classes, parallels 
 and intersectors. Now here it is impossible to prove * the affirmative 
 of the proposition, " A and B are parallels," by means of the de- 
 finition only, without proving an infinite number of cases. To see 
 this more clearly, remember that every proposition relative to the 
 intersection or non-intersection of straight lines, is an assertion which 
 either includes or excludes every possible couple of points which can 
 be taken, one on each straight line. " Lines intersect " means there 
 is a couple of such points which coincide. " Lines are parallel '' 
 means that there is no such couple whatsoever, of all the infinite 
 number which can be taken. 
 
 The first proposition in which Euclid proves the existence of 
 parallels (the 27th) does not shew that the lines are parallels, but 
 that the proposition, " the lines are intersectors," is inconsistent with 
 preceding results. The proposition, " A and B are parallels," though 
 it appears affirmative, yet is in Euclid a negative, for his express 
 definition of parallels does not define what they are, but what they are 
 not, " not intersectors." This we call a negative definition. 
 
 Now, to examine further Euclid's definition of equal ratios, we 
 must consider his definition of greater and less ratios. They amount 
 to the following. A is said to have to B a greater ratio than P has 
 to Q where there is, among all possible whole numbers m and w, 
 ani/ one pair which give mA greater than nB, but mP equal to or 
 less than w.Q; or which give wA equal to nB, but mF less than »Q ; 
 which give in fact, in any one case, what we have called a descending 
 assertion. And A is said to have to B a less ratio than P has to Q, 
 when any one pair of whole numbers m and n gives mA less than 
 wB, but mV equal to or greater than nQ, or w A equal to wB, but 
 otP greater than nQ : which give in fact, in any one case, what we 
 have called an ascending assertion. Here, to a mind the least inqui- 
 sitive, appears at once a decided objection. Our notions of the terms 
 
 • The celebrated axiom of Euclid evades this, and in point of fact 
 amounts to another and a positive definition of parallels, the assumption 
 being that the old definition agrees with it. Or rather we should say, 
 that the first twenty-five propositions of the first book establish a part of 
 the connexion of the definitions, and the axiom assumes the rest. 
 
34 CONNEXION OP 
 
 greater and lest will never allow us to suppose that any thing, quan- 
 tity, ratio, or any thing else, can be both greater and less than another 
 quantity, or ratio; and yet, on looking at the definition of Euclid, 
 we see that for any thing which appears to the contrary, one pair of 
 values of m and n may shew that A has a greater ratio to B than P 
 to Q, while another pair may shew that it has a less. The objection 
 is perfectly valid ; the only fault to be found is, that it should not 
 have arisen before, when the definitions of the first book were pro- 
 posed. How is it then known that there can be such a thing as a 
 foursided figure with equal sides and one right angle, or as lines 
 which never meet ? The confusion arises from placing the definitions 
 in the form of assertions, before the possibility of the assertions which 
 they imply are proved. The defect may be remedied (we take the 
 square as an instance) in two ways. 
 
 1. Write all definitions in the following manner. To define a 
 square, for example, " If it be possible to construct a plane figure 
 having four equal sides and one right angle, let that figure be called 
 a square." 
 
 2. Omit the definition of a square, head the 46th proposition of 
 the first book as follows. 
 
 " Theorem. On a given straight line, a four-sided figure can be 
 constructed which shall have all its sides equal to the given straight 
 line, and all its angles right angles." Having demonstrated this, add 
 the following definition : Let the figure so constructed be called 
 a square. 
 
 We have shewn that all sets of four magnitudes, A and B of 
 one kind, P and Q both of the same kind with the first, or both of 
 one other kind, can be divided into three classes. 
 
 1. Those in which simultaneous assertions on mA and n B, and 
 on mP and nQ, are all (for all values of wj and n) either ascending 
 or stationary. 
 
 2. Those in which they are all stationary. 
 
 3. Those in which they are all either descending or stationary. 
 For we have shewn that the only remaining possible case o priori, 
 
 namely, that in which there are both ascending and descending 
 assertions for different values of m and n, is a contradiction amounting 
 in fact to supposing one fraction to be both greater and less than 
 another. And it has been shewn that all the three cases are possible, 
 for commensurable quantities at least. We are now, therefore, in a 
 

 NUMBER AND MAGNITUDE. 35 
 
 condition to say, let A and B in the first case be said to have a 
 less ratio to B than P has to Q ; in the second, the same ratio; in 
 the third, a !•••• ratio. The only question now is, are these definitions f ^-H-^ 
 properly negative or positive. It will immediately appear that, out 
 of the three, the first and third can be directly and affirmatively 
 shewn to be true of particular magnitudes, and that the second cannot. 
 By which is meant, that the comparison of individual multiples may, 
 by a single instance, establish the first or third, but that no com- 
 parison of individual multiples, however extensive, can establish the 
 second. For the second consists in stationary assertions ad infinitum, 
 and the first and third are proved by a single ascending or descending 
 assertion. 
 
 As an instance, suppose 
 
 A = 951 feet B = 497 feet 
 1902 994 
 2853 1491 
 3804 1988 
 4755 2485 
 
 P = 1300 lbs. Q= 679 lbs. 
 2600 1358 
 3900 2037 
 5200 2716 
 6500 3395 
 
 In these first five multiples, there are none but stationary assertions, 
 of twenty five which might be made. Thus 
 
 4755 > 994 1 2853 > 2485 1 951 < 994 j 
 6500 > 1 358 1 3900 > 3395 \ 1300 < 1358 \ ^''' 
 
 but neither of the three definitions is thereby shewn to belong to these 
 four magnitudes. Now, take the first and third 498 times, and the 
 second and fourth 952 times, and we have, going on with the series 
 of multiples, 
 
 473598 473144 647400 646408 
 474549 473641 648700 647087 
 
 and here the process may close, for we have 473598 less than 473641, 
 while 647400 is greater than 647087. Consequently, we have 
 proved, by comparison, that 951 feet has to 497 feet a less ratio than 
 1300 lbs. to 679 lbs. 
 
 But the case in which neither greater nor less ratio exists can never 
 be established by actual comparison of multiples, except only in the 
 case where the pairs of magnitudes are commensurable. For, remark 
 that the mere circumstance of the relative multiple scale of A and B 
 
36 CONNEXION OF 
 
 agreeing with that of P and Q up to any point, is neither proof nor 
 presumption that the two magnitudes given are actually proportional, 
 though, as we shall see, it is certain evidence that they are nearly 
 proportional, if the multiple scales agree for a great number of 
 multiples. Proportion is not established until the similarity of the 
 multiple scales is shewn to continue for ever. Now, though it would 
 not be remarked at first, this insertion of an infinite number of con- 
 ditions to be fulfilled, is tantamount to a negative definition, if we 
 wish to make the definition specifically speak of one absolute cri- 
 terion of disproportion or proportion. Disproportion is where there 
 is an ascending or descending assertion somewhere in the comparison 
 of the multiple scales. Proportion is where there is no descending 
 or ascending assertion. 
 
 In the case of commensurable quantities the definition is positive, 
 because there is then a single stationary assertion, which, being proved, 
 all the rest are shewn to follow. If A and B be commensurable, let 
 wA = wB; then if mP = nQ, there is proportion; if not, there is 
 disproportion. See page 24 for the proof as to the rest of the mul- 
 tiple scales. 
 
 We have said, that, when the multiple scales agree for a long 
 period, there is proportion nearly; and it is proved thus: Suppose 
 that the scale of A and B agrees with that of P and Q, up to 10,000 
 P and 10,000 Q, but that we have disagreement as follows : 9326 A 
 lies between 10,000 B and 10,001 B, whereas 9326 P lies between 
 10,001 Q and 10,002 Q. Or the scales run thus : 
 
 10,000 B 9326 A 10,001 B 10,002 B 
 10,000 Q 10,001 Q 9326 B 10,002 Q 
 
 How much must we alter A to produce absolute proportion ? Not 
 more than would be necessary to make 9326 A greater than 10,002 B, 
 or less than would still keep it less than 10,001 B. That is, we must 
 so alter A as to add somewhere between and 2B to 9326 A, or 
 somewhere between 
 
 »■"* 9^6^ '° ^ 
 
 Consequently, the addition of a small part of B to A would make an 
 accurate proportion. 
 
 We might now proceed to the propositions of the Fifth Book of 
 
NUMBER AND MAGNITUDE. 37 
 
 Euclid ; but there are three difficulties in the way of the student's 
 perfect satisfaction with the definition. 1st, He may have a mys- 
 terious idea of incommensurables. 2d, He may not be satisfied of 
 the necessity of departing from arithmetic. 3d, He may find it diffi- 
 cult to imagine how the existence of proportionals can ever be esta- 
 blished, with, apparently, an infinite number of conditions of definition 
 to satisfy. We suppose that the gravity of tone which elementary 
 writers adopt, is inconsistent with the statement of a beginner's diffi- 
 culties, in the words in which he would express them. We shall 
 remove all necessity for preserving such dignity in a case where it 
 may be inconvenient, by a simple supposition. Let ^ be a beginner 
 in the stricter parts of mathematics ; that is, a person apt to mix pre- 
 viously acquired notions with the meaning he attaches to definitions 
 which are intended to exclude all but the ideas literally conveyed in 
 the words which are used ; much better pleased with the apparent 
 simplicity of an incorrect definition, gained either by omitting what 
 should not be omitted, or by supposing what cannot be supposed, 
 than with the comparatively cumbrous forms which provide for all 
 cases, and distinguish differences which really exist; and, finally, 
 when a doubt exists, rather predisposed against, than in favour of, the 
 necessity of demonstration. Let JB be another person, who has sub- 
 jected his mind to that sort of discipline which has a tendency to 
 remove the propensities aboveraentioned. We can imagine them 
 talking together in this manner : 
 
 A. — I have been trying to understand the meaning of incommen- 
 surable quantities, and cannot at all make out how it can be that one 
 given line may be no fraction whatsoever of another given line, though 
 both remain fixed, and certain lines ever so little greater or less than 
 the first are fractions of the second. 
 
 B. — A little consideration will teach you, that neither in arithmetic 
 nor geometry are we at all concerned with how things can be, but only 
 with whether they are or not. Do you admit it to be demonstrated 
 that the side and diagonal of a square, for instance, are incom- 
 mensurable ? {Algebra, page 98). 
 
 A. — I cannot deny the demonstration, but the result is incompre- 
 hensible. Does it really prove, that if I were to cut the diagonal of a 
 square into ten equal parts, each of these again into ten equal parts, 
 and so on for ever, I should never, by any number of subdivisions, 
 
 £ 
 
38 CONNEXION OP 
 
 succeed in placing a point of subdivision exactly upon the point 
 which cuts off a length equal to the side. 
 
 B. — I take it for granted you have sufficiently comprehended the 
 definitions of geometry, to be aware that a thin rod of black lead, or 
 a canal of ink, are not geometrical lines ; and that the excavations 
 which you perforate by the compasses are not points. 
 
 A. — Certainly ; I now have no difficulty in imagining mere length 
 intersected by partition marks, which are not themselves lengths. 
 
 B. — Then, in the case you proposed, you need not go so far for 
 a difficulty; for your method of subdivision will never succeed in 
 cutting off so simple a fraction as the third part of the diagonal. 
 
 ^._ Why not? 
 
 B. — You see that 9, 99, 999, &c., are all divisible by 3, so that 
 10, 100, 1000, &c., cannot in any case be divisible by 3, but must 
 leave a remainder. Your method of subdivision can never put to- 
 gether any thing but tenths, hundredths, Sec. If possible, suppose 
 one-third to be made up of tenths, a in number, added to hundredths, 
 b in number, added to thousandths, c in number. Then we must have 
 
 3 10 ^ 100 ^ 1000 
 Clear the second side of fractions, and we have 
 
 i^ = ax 100 + ^^x 10 + c 
 
 o 
 
 or is a whole number, which is not true. And the same rea- 
 
 3 
 
 soning might be applied to any other case. 
 
 A. — This is conclusive enough; but it seems to follow that the 
 third part of a line is incommensurable with the whole. 
 
 B. — So it is, as far as the one method of subdividing which you 
 propose is concerned. Let tenths, hundredths, &c., be the measurers^ 
 and one-third and unity are incommensurable. But the word with 
 which we set out implies all the possible subdivisions of halves, thirds, 
 fourths, fifths, &c. &c., to be tried, and all to fail. 
 
 A. — But here is an infinite number of ways of subdividing. Can 
 it be possible that no one of them will give a side of a square, when 
 the di:\gonal is a unit ? 
 
 B. — In the first place, it would be a sufficient answer to this sort 
 of difficulty to say, that, for any thing you know to the contrary', the 
 
NUMBER AND MAGNITUDE. 39 
 
 number of ways in which you may fail is as infinite as the number of 
 ways in which you may try to succeed. In the second place, there 
 is also an infinite number of ways of subdividing, which will not give 
 one-third. Let your first subdivision be into any number of equal 
 parts, except only 3, 6, 9, 12, &c.; and your second subdivision the 
 same, or any other, with the same exceptions, &c. The same rea- 
 soning will prove that you can never get one-third. 
 
 A. — But look at the matter in this way. Suppose the halves, the 
 thirds, the fourths, the fifths, &c. &c. of a diagonal laid down upon it 
 ad infinitumy so that there is no method of subdividing into aliquot 
 parts, how many soever, but what is done and finished. Would not 
 the whole line be then absolutely filled with subdivision points, and 
 would not one of them cut off a line equal to the side of the square. 
 
 B. — You have now changed your use of the word infinite^ and 
 applied it in the sense of infinity attained, not infinity unattainable. 
 As long as you used the word to signify succession, which might be 
 carried as far as you pleased, and of which you were not obliged to 
 make an end, the word was rational enough, though likely to be mis- 
 understood ; but as it is, you may as well suppose you have got be- 
 yond infinite space, at the rate of four miles an hour, and are looking 
 back upon the infinite time which it took you to do it, as imagine 
 that you have subdivided a line ad infinitum. But if the idea of in- 
 finity attained be a definite conception of your mind, you meet the 
 difficulty of incommensurable quantities in another form. The defi- 
 nition of the term incommensurable was shaped in accordance with 
 the exact notion, that, subdivide a line as far as you may, you must 
 stop at some finite subdivision; and incommensurable parts of a 
 whole are those which you never exactly separate arithmetically, stop 
 at what finite subdivision you please. But, if you will contend for 
 infinite subdivision attained, and imagine the line thus filled up by 
 points, then it will be necessary to divide all parts of a whole into two 
 classes, those which are cut off by finite subdivision, and those which 
 are not attainable, except by infinite subdivision ; the former answering 
 to commensurable, the latter to incommensurable, parts. The diffi- 
 culty remains then just as before ; in other words, why should the 
 side of a square be not attainable from its diagonal except by infinite 
 subdivision, when the sides of a rectangle, which are as 3 to 4 (instead 
 of 3 to 3), are attainable by a finite number of subdivisions ? 
 
 In the next place, you have spoken of a line filled up by point?, 
 
40 CONNEXION OP 
 
 the infinitude of the number of points being the compensation for each 
 of the points having no length whatsoever ; at least, it is not easy to 
 see what else you can mean, 
 
 A. — Certainly that is what I mean ; and the common expressions 
 of algebra are in accordance with what I say. For, if I cut aline 
 into n equal parts, it is plain that the sum of the n parts makes up 
 the whole, be the number n great or small. But by making n suffi- 
 ciently great, each of the parts may be made as small as I please ; 
 and, therefore, allowing it to be rational to say that P takes place 
 when n is infinite, in all cases in which we may come as near to P as 
 we please, by making n sufficiently great (which is the expressed 
 meaning of in/in«Ye in algebra), it follows that we may say, that the 
 line is made up of the infinite number of points into which it is cut 
 when divided into an infinite number of equal parts. 
 
 B. — I see every thing but the last consequence. 
 
 A. — Why, surely, the smaller a line grows, the more nearly does 
 it approximate to a point. 
 
 B. — How is that proved ? 
 
 A. — Suppose two points to approach each other, they continually 
 inclose a length which is less and less, and finally vanishes altogether 
 when the two points come to coincide in one point. So that the 
 smaller the straight line is, the more near is it to its final state — a 
 point. 
 
 B. — You have not kept strictly to your own idea (which is a 
 correct one) of the way in which the words nothing and infinite may 
 be legitimately used. You have supposed a line to be entirely made 
 up of points, each of which has no length whatsoever, because you 
 may compose a line of a very large number of very small lines, each 
 of which, you say, is nearly a point. Let us now consider whether 
 your final supposition is one to which we can approach as near as we 
 please by diminution of a length. Any line, however small, can be 
 divided into other lines by an infinite number of different points; for 
 any line, however small, admits of its halves, its thirds, &c. &c. So 
 that there is a theorem which is not lessened in the numbers it speaks 
 of, or altered in force or meaning, in any the smallest degree, by 
 diminishing the line supposed in it ; namely, any line whatsoever 
 admits of as many different points as we please being laid down in it. 
 "NoYff of y OUT Jinal length, or limit of length — the point — this is not 
 true : consequently, you throw away a result at the end, which you 
 
NUMBER AND MAGNITUDE. 
 
 41 
 
 cannot throw away as nearly as you please during the process by 
 which you attain that end ; nor will the denial of it, near the end, be 
 less in the consequence or amount of the error, than if the rejection 
 were made further from the end. Therefore, in asserting that a dimi- 
 nishing straight line approximates to a point, you have abandoned 
 the condition under which you are allowed to speak of nothing or 
 infinite. 
 
 Again, the «th part of a line taken twice is certainly greater than 
 the simple wth part, however great n may be. Now, what do you 
 suppose two points to be, which are laid side by side without any in- 
 terval of length between them ? 
 
 A. — They are, of course, one and the same point. 
 
 B. — But in your infinite subdivision, two nth parts must be greater 
 than one nth part, or two of your points must be greater than one ; 
 but these two points are the same point, which is therefore twice as 
 great as itself. Such are the consequences to which the supposition 
 of a line made up of points will lead. 
 
 A. — I have frequently heard of lines being divided into an Infinite 
 number of equal parts. 
 
 B. — But you never heard those equal parts called points. I 
 can soon shew you that, in the mode of allowing infinity to be 
 spoken of, this fundamental condition is preserved, namely, that no 
 theorem, limitation, number, nor other idea whatsoever, which forms 
 a part of any question, is allowed to be rejected or modified when n is 
 infinite, unless it can be shewn that such rejection or modification 
 may be made with little error when n is great, with less error when n 
 is greater, and so on ; finally, with as small an error as we please, by 
 making n sufficiently great. Now, remark the following truths, and 
 the form of speech which accompanies them, when n is supposed 
 infinite. 
 
 General Theorem. 
 The greater the number of 
 equal parts into which a line is 
 divided, the less line is each of 
 the parts : so that an aliquot 
 part of any line, however great, 
 may be made less than any 
 given line, however small. 
 
 Terminal Theorem. 
 If a straight line be divided 
 into an infinite number of equal 
 parts, each part is an infinitely 
 small line. 
 
 e2 
 
42 
 
 CONNEXION OF 
 
 General Theorem. 
 
 Any line, however rmall, 
 may be cut by as many points 
 as we please. 
 
 No straight line, however 
 small, ceases to be a length ter- 
 minated by points. 
 
 Terminal Theorem. 
 
 An infinitely small line may 
 be cut by as many points as we 
 please. 
 
 An infinitely small straight 
 line is a length terminated by 
 points. 
 
 Now, taking your notion of infinite subdivision attained, it may 
 be shewn that incommensurable parts necessarily follow. For, how- 
 ever far you carry the subdivision, you do not, by means of the sub- 
 division points, lessen the number of points which may be laid down. 
 For each interval defined by the subdivisions contains an infinite 
 number of points. Consequently, if you will suppose the infinite 
 subdivision attained, you cannot do it without supposing an infinite 
 number of points left in the intervals, or an infinite number of in- 
 commensurable quantities. This I intend only to shew that the 
 proof of the existence of incommensurable quantities is, upon your 
 own supposition, somewhat better than that of their non-exigtence. 
 But it would be better to use nothing and infinity as convenient 
 phrases of abbreviation, not as containing definite conceptions which 
 may be employed in demonstration. 
 
 A. — I do not see how your objection applies against nothing; 
 if we cannot attain infinity by continual augmentation, we can cer- 
 tainly attain nothing by continual diminution. 
 
 B. — So it may seem at first, and in truth you are right as to one 
 sort of diminution, that which is implied in the word subtraction. 
 From the place in which there is something take away all there is, 
 and you get nothing by a legitimate process. But subtraction is the 
 only process which leaves nothing ; division, for example, never leaves 
 it. Halve a quantity, take the half of the half, and so on, ad infinitum: 
 you will never reduce the result to nothing. 
 
 A. — But however clearly you may shew that incommensurable 
 quantities actually exist, as a necessary consequence of our de- 
 finitions of length, number, &c., I should feel better satisfied if 
 you could give something like an account of the way in which 
 they arise. 
 
 B. — If you will consider the way in which number and length 
 are conceived, perhaps the difficulty may be somewhat lessened. Let 
 
NUMBER AND MAGNITUDE. 43 
 
 a point set out from another point, and move uniformly along a 
 straight line until the two are a foot distant from each other. It is 
 clear that every possible length between and one foot will have 
 been in existence at some part or other of the motion. Now, suppose 
 a number of points as great as you please, to set off from the first 
 point together; but, instead of moving in the straight line let them 
 
 move off in curves, the first coming to the straight line at - and 1 
 
 12 12 3 
 
 of a foot ; the second at - - and 1 ; the third at - - - and 1 of a 
 3 3 4 4 4 
 
 foot ; and so on, as in this diagram. 
 
 Can you feel sure that these contacts of curves with the line, 
 separated as they must always be from each other by finite intervals, 
 will ever fill up the whole line described by a continuous motion. 
 If not, this figure will always supply presumption in favour of in- 
 commensurable parts, which will of course be increased to certainty 
 by the actual ^roof of their existence. And this should be sufficient 
 to overturn a doubt which after all is derived from confounding the 
 mathematical point with the excavation made by the points of a pair 
 of compasses. The practical commensurability of all parts with the 
 whole is a consequence of there being magnitudes of all sorts below 
 the limits of perception of the senses (see page 3). 
 
 A, — Granting, then, that there are such things as incommensurable 
 quantities, it is admitted, that though A and B are incommensurable, 
 yet A and B + K i«ay be made commensurable, though it be insisted 
 on that K shall be less than any given quantity, say less than the 
 hundred thousand million millionth of the smallest quantity which 
 the senses could perceive, if they were a hundred thousand million 
 of million of times keener than they are at present. Would it not be 
 sufficient, when incommensurable quantities, A and B, occur, to 
 suppose so slight an alteration made in B as is implied in the above, 
 and reason upon A and B + K so obtained, instead of upon A and B. 
 Surely such a change could never produce any error which would be 
 of any consequence ? 
 
 JB. — Of consequence to what? 
 
44 CONNEXION OF 
 
 A. — To any purpose of life for which mathematics can be made 
 useful. 
 
 B. — I am still at a loss. 
 
 J. — What process in astronomy, optics, mechanics, engineering, 
 manufactures, or any other part either of physics or the arts of life, 
 would be vitiated by such an alteration, or its consequences, to any 
 extent which could be perceived, were the error multiplied a million 
 fold? 
 
 B. — None whatever, that I know of. 
 
 A. — What, then, would be the harm of introducing a supposition 
 which would save much trouble, and do no mischief? 
 
 B. — I am not aware that I admitted such a supposition would do 
 no mischief, when I said that it would not sensibly vitiate the appli- 
 cation of mathematics to what are commonly called the arts of life. 
 I see that your idea of mathematics is very much like that which a 
 shoemaker has of his tools. If they make shoes which keep the 
 weather out, and bring customers, he need not wish them to do more, 
 or inquire further into any use, actual or possible, which they may or 
 might have. The end he proposes to himself is answered, when he 
 has sewed the upper leatiier firmly to the sole. But whether his art 
 serves any higher purpose — whether the possibility of obtaining con- 
 veniencies, and avoiding hardships (which it creates in one respect), 
 excites industry and ingenuity, creates property to equalise the 
 fluctuations of harvests and commerce, and prevent the community 
 from undergoing periodical pests and famines — makes men so de- 
 pendent on each other that internal war is next to impossible, and 
 external war a grave and serious consideration, &c. &c., are not 
 matters for the thoughts of a working shoemaker ; nor will similar 
 considerations ever enter the mind of a working mathematician. You 
 have spoken of the purposes of life ; I do not know what the purposes 
 of your life may be, but if among them you count such a discipline 
 of the mind as may always render your perception of the force of an 
 argument properly dependent upon the probability of ihe premises, 
 and the method by which the inferences are drawn, it will be one of 
 your first wishes to propose to yourself, as a standard and a model, 
 some branch of study in which the first are self-evident, or as evident 
 as any thing can be, and the second indisputable and undisputed. 
 For though you may find no other science which will compete with 
 this in accuracy, yet you will be more likely to iufer correctly, when 
 
NUMBER AND MAGNITUDE. 45 
 
 you have seen what you know to be correct inference, than you would 
 have been if you had never, in any case, distinguished between de- 
 monstration of certainties and presumptions from probabilities. And 
 still more, will you be qualified to refute, and refuse admission to, 
 that which takes the form of accuracy without the reality. If the 
 mathematical sciences be good as a weapon, they are a hundred fold 
 better as a shield. I have seen many who were visibly little the 
 better for their mathematical studies in what they advanced; but very 
 few indeed who were not made sensibly more cautious in what they 
 received. 
 
 A. — But is not my notion adopted in practice by a great part of 
 the mathematical world, particularly on the continent. 
 
 B. — It is certainly true, and it is particularly the case with the 
 French, who, though they have done more than any other nation, since 
 the time of Newton, to advance the mathematical sciences, have been 
 by no means anxious to consider them as resting on other evidence 
 than that — not of the senses — but of the limits of the senses. One 
 of their most celebrated elementary writers considers none but arith- 
 metical proportion, and begins his work by shewing either that two 
 straight lines have a common measure, as in page 12, or that the 
 remainder " echappe aux sens par sa petitesse." All his propositions, 
 therefore, in geometry, are either true, or so nearly true, that the 
 difference is imperceptible. The phrase we have quoted is an honest 
 and a valuable admission; it shews you, that in the opinion of one 
 of the most useful and extensive elementary writers that ever lived, 
 arithmetical proportion makes geometry a science of approximate, 
 not absolute, truth. 
 
 A. — I see as much; but cannot the slight shifting of one of the 
 quantities which I proposed be somehow or other corrected, so as to 
 make a strict and useful theory of the proportions of incommensurable 
 quantities ? 
 
 B. — Yes, and in a very simple way; by adopting the definition 
 of Euclid. This may surprise you, but I will soon shew that the 
 most natural correction of your notion leads direct to the definition of 
 Euclid. Let it be granted that A and B being commensurable, and 
 mA=7iB, proportion between A, B, P, and Q means that m P=nQ. 
 Now you want, when A and B are incommensurable, to be allowed to 
 substitute B + K instead of B, where K is excessively small. I 
 suppose you would be perfectly content if it could not be made 
 
46 CONNEXION OF 
 
 visible by any microscope. Now I am of a somewhat more abstract 
 turn, and should not like my geometry to be put in peril by the 
 abolition of the excise on glass ; which it might be by the allowance 
 of experiments for the improvement of that article, which are now 
 effectually prevented. 1 cannot admit B + K, where the magnitude 
 I want to reason upon is B. But as the definition of proportion of 
 incommensurables is not yet settled, let us examine this case : A and 
 B being incommensurable, let P and Q be quantities of such a kind 
 that A and B + K are commensurable, and also P and Q -f Z, and 
 that the four just named are arithmetically proportionals. Let it be 
 possible, these conditions subsisting, to make K and Z as small as 
 we please : not as small as this, that, or the other small quantity, but 
 smaller than any whatsoever which may be named, being still some 
 quantities. You wish to substitute B -j- K and Q + Z for B and Q : 
 I prefer to use the conditions laid down to ascertain how B and Q 
 themselves stand related to A and P. Let us suppose we name two 
 small magnitudes, K' and Z', of the same kind as A and P, or B and 
 Q, or K and Z, which we are at liberty to make as small as we 
 please. We can then find K and Z less than K' and Z', and such 
 that A, B + K, P, Q + Z are proportional. Suppose A and B -f K 
 commensurable, and let 
 
 mA = n(B + K) whence mP = w(Q + Z) 
 
 whence it is easily proved, as in page 24, that the relative scale of 
 multiples of A and B + K is the same as that of P and Q + Z. I 
 say it follows, that the relative scale of A and B is the same as that 
 of P and Q ; for, if not, the two latter scales must differ somewhere. 
 Let it be that vA is greater than m;B, but v P less than wQ. Then, 
 since uA is greater than wB, let K be taken so small (it may be as 
 small as we please) that v A shall also exceed u>(B + K), whence, by 
 the proportion assumed in the hypothesis, v P exceeds w{Q + Z), 
 while, by the hypothesis we are trying, rP is less than wQ. This is 
 a contradiction, for uP cannot exceed wQ-{-iuZ and fall short of w?Q 
 at the same time. In the same way, any other case may be treated ; 
 and it follows that our suppositions, if K may be as small as we 
 please, amount to an hypothesis from which Euclid's definition fol- 
 lows. If, in the above, we suppose vA less than wB, while uP is 
 greater than wQ, we see that vA also falls short of v(B -f K), whence, 
 
NUMBER AND MAGNITUDE. 47 
 
 by the proportion, vP falls short of i(Q + Z), which cannot be if it /iXT 
 exceed tQ. /„'* 
 
 A. — But is not this deduction, namely, Euclid's definition, more 
 cumbrous than the form from which it has just been deduced ? 
 
 JB. — How so? 
 
 A. — Does it not involve an infinite number of considerations, ex- 
 tending the whole length of the multiple scales ? 
 
 B. — And does not your definition do the same thing, unless you 
 stop somewhere with the values of K and Z ? Is it not necessary, if 
 we would not be merely microscopically correct, but absolutely cor- 
 rect, to suppose that K and Z may be diminished and diminished ad 
 infinitum ? And what difference is there, as to the number of con- 
 siderations in question, between two magnitudes which are to diminish 
 without limit, and a set of increasing multiples of two given mag- 
 nitudes ? 
 
 A. — But Euclid's definition seems to wander such a way from the 
 quantities in question, while the other remains close to them, and we 
 never seem to quit them, except for something very near to them. 
 The actual application of the definition I prefer will require nothing 
 but the division of all magnitudes into aliquot parts. 
 
 JB. — Your objection amounts to this ; that you feel the fractions 
 of a quantity to be more closely connected in your mind with the 
 quantity itself than its multiples. This may be the case ; and, if so, it 
 is some reason for preferring the form to which you seem most inclined. 
 But there may be a stronger reason for preferring the other ; and, 
 undoubtedly, as long as difficulties exist, every system of science must 
 be a balance of inconveniences. But Euclid is, of all men who ever 
 wrote, the one who has a reason for the course which he takes, where 
 there are two or more. I suppose you cannot but admit that it is 
 better to found a definition in geometry upon the result of something 
 which can actually be done, by the means of geometry, than upon 
 something wliich can only be conceived or imagined to be done, with 
 what certainty soever ; for instance, you would not wish to be obliged 
 to use other means than the straight line and circle, or to suppose an 
 object gained without using any means at all ? 
 
 A. — Certainly not. That there shall be no assumption of me- 
 chanical power beyond that of drawing a straight line or circle; is the 
 foundation of pure geometry. 
 
 B. — Then the question is settled in favour of Euclid's definition j 
 
48 CONNEXION OF 
 
 for, without either assuming more mechanical means, or making a 
 gratuitous assumption, no angle, nor arc, nor sector of a circle, can be 
 divided into 3, or 6, or 9, &c. parts, unless it be a right angle, or a 
 given half, fourth, eighth, &c. of a right angle. There are some other 
 exceptions ; but, generally, to cut any angle into three equal parts is 
 a geometrical impossibility, and certain algebraical considerations 
 furnish the highest presumption that it will always remain so. 
 
 A. — But this difficulty is still left: how are we ever to shew 
 that there are such things as proportional quantities? 
 
 B. — We can do this so easily, that the greatest stumbling-block 
 of the process lies in its being so easy and perceptible, that a beginner 
 does not very well see where lies the knowledge he has gained, unless 
 he has paid profitable attention to the definition of proportion. From 
 the first book of Euclid it is evident that a rectangle is doubled by 
 doubling the base, trebled by trebling it, and so on ; and also, that of 
 two rectangles between the same parallels, the greater base belongs to 
 the greater, and the lesser base to the lesser. Now, let B and B' 
 represent two bases, and R and R' the rectangles upon them, the 
 altitudes, or distances of the parallels, being the same. If then we 
 take the first base m times, giving m B, the rectangle upon that base 
 is m R : if we take the second base n times, giving n B', the rectangle 
 upon that base is nR', the parallels always remaining the same. 
 Hence it follows, that m B and n B' are bases to the rectangles m R 
 and nR' between the same parallels; accordingly, therefore, as viB 
 is greater than, equal to, or less than n B', so is twR greater than, 
 equal to, or less than n R' : and this being true for all values of in 
 and n, it follows that B has to B' the ratio of R to R', or the bases 
 of rectangles between the same parallels, and the rectangles them- 
 selves are proportionals. 
 
 A. — Am I to understand then that there are difficulties in the 
 way of considering magnitude in general, which are not found in 
 arithmetic, or the science of abstract number? 
 
 B. — Quite the reverse: the difficulty arises from the deficiences 
 of arithmetic itself, and from their being ratios which the ratio of 
 number to number cannot represent. 
 
 A. — But how is that? arithmetic always seemed clear of such 
 difficulties as we have been considering, 
 
 B. — And so would this subject, if the disposition to be satisfied 
 with what is in the book, which is part and parcel of almost every 
 
NUMBER AND MAGNITUDE. 49 
 
 beginner had been permitted to rest quietly upon a theory of com- 
 mensurable ratios. But did you never, in arithmetic, hear of the 
 creation of a nonexistent number or fraction, in spite of there being 
 no such thing, by agreeing that there should be such a thing, and 
 drawing a picture to represent it? 
 
 A. — I do not understand the jest; but I suppose you allude to 
 algebra, and to quantities less than nothing? 
 
 B. — Not at all ; 1 am speaking of pure arithmetic. To me, — 2 
 is a much easier symbol, or picture, than n/2; and even the difficulties 
 of s/ — 2 lie as much in the n/ as in the — . 
 
 A. — But I do not understand what you mean by saying that s/2 
 does not exist; it is the square root of 2, and multiplied by itself it 
 gives 2. You may find it as nearly as you please. 
 
 B. — If it be the object of arithmetic, commonly so called, it is 
 either a whole number or a fraction. Which of these is it ? 
 
 A. — It is a fraction ; 1-4142136, very nearly. 
 
 B. — I did not ask you what it is very nearly, but what it is ? 
 
 A. — It cannot be given exactly, but we all know there is such a 
 thing as the square root of 2. 
 
 B. — If the objects of arithmetic were numbers, fractions, and 
 things, and the latter term had a definition, I might admit what you 
 say. And in concrete arithmetic, where 1 is a things a foot, a 
 pound, or an acre, I admit that there is such a thing as v 2. But 
 that thing is not attainable arithmetically by taking any aliquot part 
 of the thing 1, and repeating it any number of times. In abstract 
 arithmetic the square root of 2 is an impossibility; and having no 
 existence, I do not see how one fraction can be said to be nearer to 
 it than another, except in this sense, that 2 + z may be made to have 
 a square root where z may be less than any fraction we name. The 
 independent existence of v 2 is an algebraical consideration of some 
 difficulty; that is, belongs to the science which has relations of 
 symbols, under prescribed definitions, for its object, without reference 
 to their numerical interpretation. The difficulties of \/2 are precisely 
 those of incommensurable magnitudes ; in fact s/^ is the diagonal 
 of a square whose side is 1. But it is to algebra that difficulties of 
 this kind should be referred. The student, if he use ^2 in pure 
 arithmetic, must expressly understand it as a fraction whose square is 
 nearly 2, and must consider this part of arithmetic (without algebra 
 
50 CONNEXION OF 
 
 as a science of approximation, unless geometry, or some other science 
 of concrete quantity, be supposed to lend its aid. 
 
 A. — But I cannot divest myself of the idea that >/2i V's? and 
 \/6 are really fractions, and that the product of the two first gives 
 the last. I suppose, in some sense or other, you admit this pro- 
 position ? 
 
 £. — Certainly. If \^2-f x and ^3-{-j/ and ^6-\-2 be made 
 to exist, by giving proper values to x, 3/, and z, which may all be as 
 small as I please, and if, moreover, x, 1/, and z be so related that 
 2 = 3 j: + 23/ + xi/f which condition does not interfere with the last ; 
 I can then admit that 
 
 V2-J-X X V3+y = n/6 + z 
 
 But I do not allow myself to suppose that (understanding by multi- 
 plication the taking of one number or fraction as many times or parts 
 of times as there are units or fractions of a unit in another), there can 
 be such a truth as that 
 
 \/ 2 (neither number nor fraction) wM/<i/j/t>d by \/ 3 (do. do.) = \/ 6 (do. do.) 
 
 But this is beyond our subject, except so far as it shews that the 
 difficulty lies more in arithmetical than in geometrical considerations. 
 
 A. — Might we not then dispense with arithmetic altogether, and 
 make a definition corresponding to proportion for geometry? 
 
 B. — Yes; but the difficulty would appear in another shape, of 
 the very same substance. Let four lines be called proportional when, 
 being straightened without alteration of length, if necessary, the 
 rectangle made by the first and fourth is equal to that made by the 
 second and third. Let areas be proportional when, being converted 
 into rectangles with a common altitude, their bases are proportional. 
 Let angles be proportional, when they are angles at the centre of 
 proportional arcs of the same circle. But here would immediately 
 arise this difficulty, — to make a straight line equal to a given arc of 
 a circle ; which is out of the power of the geometry of straight lines 
 and circles. 
 
 A. — Is not the reductio ad absurdum (which is very much used 
 in the establishment of the tneory of proportion) rather a suspected 
 method. I have heard it called indirect demonstration; and it is 
 frequently slated as a defective method, not to be used if it can 
 possibly be avoided. 
 
NUMBER AND MAGNITUDE. 61 
 
 B. — The complaints against this method of demonstration have 
 become much more frequent, if not entirely made their appearance, 
 since the time when logic was a necessary part of a liberal education, 
 as it once was, and as I hope it will be again. I have sometimes 
 wondered whether this argument would have been considered objec- 
 tionable if it had been reduced to the form " A is B, B is C, therefore 
 A is C ;" as follows : " Every contradiction of P is a contradiction 
 of the proposition that the whole is greater than its part ; but every 
 contradiction of this proposition is false : therefore every contradiction 
 of P is false ; or P is true." The reduclio ad absurdum is as conclu- 
 sive, and may be made as intelligible, as any other argument. And 
 if any argument be good in proportion to the effect upon the mind, 
 where is the affirmative proposition, in geometry or not, which the 
 mind seizes as readily as it recoils from an absolute contradiction in 
 terms ? Where is the likeness or resemblance between things which 
 are alike, that is so forcible as the unlikeness or want of resemblance 
 of two ideas which palpably contradict, such as black is white ? 
 
 A, — fs there then no advantage in the direct over the indirect 
 demonstrations ? 
 
 B, — D'Alembert has said that the former are to be prefen-ed 
 " parce qu'elles ^clairent en m^me-temps qu'elles convainquent," which 
 is a good description of the diiBerence. But even this must be taken 
 with some allowance, for there are many indirect demonstrations 
 which are highly instructive. 
 
 Recapitulation. By the ratio of A to B, we mean (without any 
 further specification at present) a relation between the magnitudes of 
 A and B, determined by the manner in which the multiples of A are 
 distributed, if each be written between the nearest multiples of B in 
 magnitude. That is, if B, 2 B, 3B, &c., be formed, and A, 2 A, 
 3 A, &c., and if A lie between B and 2B, 2A between 3B and 4 B, 
 and so on, the relative scale 
 
 B, A, 2B, 3B, 2A, 4B, &c. 
 
 is to be the sole determining element of the ratio, so that there is to 
 be nothing but the order of this scale on which the ratio depends. 
 And if P and Q be two other magnitudes with the same order in 
 their scale, P compared with A, and Q with B, then A and B are to 
 be said to have the same ratio as P and Q. But if any multiple of A 
 
52 CONNEXION OF 
 
 precede among the multiples of B the place which the correspondiDg 
 multiple of P occupies among the multiples of Q, then A is to be 
 s;\id to have to B a less ratio than P has to Q. But if a multiple of 
 A come later in the series of multiples of B than its corresponding 
 multiple of P in the series of multiples of Q, then A is said to have to 
 B a greater ratio than P has to Q. It is plain that the ratio of A to 
 B must be greater than, equal to, or less than that of P to Q, and 
 also, that in saying A is to B as P to Q, we also say that B is to A 
 as Q to P. 
 
 [We must remind the student that we have now nothing to do 
 with the reasons of this definition, or the accordance of its parts with 
 each other, or with any notion of ratio more than is contained in it. 
 We are merely now concerned to know what follows from this 
 defin'tlon. The numbering of the following propositions is that in 
 Euclid.] 
 
 When A has to B the same ratio as P to Q, the four are said to 
 be proportionals, and are written thus : 
 
 A : B:: P : Q 
 
 which is read A is to B as P is to Q. 
 
 IV. If A : B : : P : Q 1 m and n being any 
 
 Then mA I niB y, nV : nQ) whole numbers. 
 
 This we know when we see that any quantities being arranged in 
 order of magnitude, so will be their multiples. If the scales be 
 
 B A 2B 3B 
 
 Q P 2Q 3Q 
 
 the following scales 
 
 wiB mA 2mB 3//iB 
 
 wQ nP 2wQ 3nQ 
 
 will also be arranged according to magnitude. Whence the pro- 
 position. 
 
 VII. IfA, B, C, be three homogeneous magnitudes (all lines, or 
 all weights, &c.) and if A = B ; then 
 
 A : C :: B : C 
 and C : A :: C : B; 
 
 for the scales must evidently be identical. 
 
NUMBER AND MAGNITUDE. 53 
 
 VIII. A + M has a greater ratio to B than A has to B, and B has 
 a less ratio to A+M than B has to A. Let M be muUiplied so many 
 limes that it exceeds B ; say mM = B + K : then 
 
 m 
 
 (A + M) = mA + B + K 
 
 Let OT A lie between tjB and (u + l) B ; then m(A + M) lies between 
 vB + B+Kand (i; + l)B + B4-K, and certainly beyond (u + l)B. 
 Consequently, in the scales of A-j-M and B, and A and B, a multiple 
 of A + M is found to be in a higher place among the multiples of B 
 than the same multiple of A among the multiples of B. Whence, by 
 definition, A+ M has to B a greater ratio than A to B. The second 
 part of the proposition is but another way of stating the first, as 
 appears from definition. Thus we may also say that A has to B a 
 less ratio than A + M has to B. 
 
 IX. If A : C : : B : C then A = B 
 or if C : A::C : B then A = B 
 
 For (viii.), if A be greater than B, A has to C a greater ratio than 
 B to C, which is not true. If A be less than C, A has to C a less 
 ratio than B to C, which is not true : therefore A=B. The same 
 reasoning proves the second case. 
 
 X. If A have to C a greater ratio than B has to C, then A is 
 greater than B. For if A were equal to B, then these ratios would be 
 the same: if A were less than B (viir.), then would A have to C a 
 less ratio than B has to C. Therefore, A is greater than B. Similarly, 
 if A have to C a less ratio than B has to C, A is less than B. And 
 if C have a greater ratio to A than to B, A is less than B ; if C have 
 a less ratio to A than to B, A is greater than B. 
 
 XI. If the ratios of C to D and of E to F, be severally the same 
 as that of A to B, then C has to D the same ratio as E to F. This 
 answers to a case of the general axiom, that two things which are 
 perfectly like to a third in any respect, are perfectly like each other 
 in that respect. The multiples of C are distributed among those of 
 D in the same manner as those of A among those of B, as are those 
 of E among those of F. Therefore, the multiples of C are distributed 
 among those of D as are those of E among those of F. Whence the 
 proposition. 
 
 XII. If A be to B as C to D, and as E to F, then as A is to B 
 so is A+C + E to B+C+F. 
 
 !• 2 
 
54 CONNEXION OF 
 
 For mA lying between wB and (n+l)B, then mC lies between 
 nD and (w4-l)D» and 7«E between wF and (n4-l)F, and, con- 
 sequently, TwA + wC -f wE, or»i(A-f-C+E) between n(B-|-D-|-F) 
 and (n+1) (B-f-D+F). Whence the proposition. 
 
 XIII. If A have to B the same ratio as C has to D, but C to D 
 a greater ratio than E to F, then A has to B a greater ratio than 
 E to F. 
 
 This is one of a class of propositions which come under this 
 general theorem : for any ratio, an equal ratio may be substituted, 
 and all consequences of the first ratio are consequences of the second. 
 This, which seems very evident, may appear so upon mistaken evi- 
 dence. Ratios, as far as we have yet gone, are not quantities, but 
 expressions of that relation between quantities upon which the order 
 of magnitude of their multiples depends. For quantity, we may 
 substitute other quantity equal to the first in magnitude wherever the 
 relation is one which depends only on quantity; we may not sub- 
 stitute a triangle of the same area instead of a square, except there be 
 question of nothing but superficial magnitude, or area. Ratio, again, 
 IS to us at present the order of the multiples, so that if A and B have 
 their multiples arranged among each other in a given order, if P and 
 Q have the same, we may say that whatever is true of the order of 
 multiples of A and B, is also true of the order of P and Q ; whatever 
 connexion the order of multiples of A and B establishes between A 
 and B and other magnitudes, the same connexion exists between P 
 and Q and those other magnitudes, because the accident of A and B, 
 which is the sole connexion between them and the consequence inferred, 
 is also an accident of P and Q. The necessity for going over such 
 considerations, arises from its never being allowed to be taken for 
 granted that a mathematician has studied logic. Hence Euclid* is 
 frequently obliged to reiterate the same assertions in different forms. 
 To take the proof of the present proposition ; to say that C has to D a 
 greater ratio than E to F, is to say that m C can be found greater than 
 
 • Euclid was a contemporary of Aristotle, as is generally supposed, 
 and may, therefore, never have seen the science of the hitter. It is free 
 to us to suppose that if he had, he would have distinguished between a 
 purely logical and a geometrical consequence : tliat is, would not have 
 reiterated the same proposition in different forms; or, if you please, 
 different cases of the same verbal truth as if they were distinct truths : 
 and we will suppose so accordingly. 
 
NUMBER AND MAGNITUDE. 65 
 
 nD, while w E is equal to or less than n F. But to say that A has to 
 B the ratio ofC toD, is to say that whenever wC is greater than nD, 
 mA is greater than nB. Therefore, to say that the ratios of A and B 
 and C and D are the same, but the latter greater than that of E to F, 
 is to say that m A may be greater than nB, while mE is equal to or 
 less than nF ; or that A has to B a greater ratio than E has to F. 
 
 Now, let the student compare this with the following proposition. 
 A and B are greens of exactly the same shade : but B is a darker 
 green than C, therefore, A is a darker green than C. Would it be 
 unnecessary to prove this ? then it is equally unnecessary to prove the 
 preceding. But we will prove this in the same manner as we prove 
 the preceding. Let there be a test of greenness, which decides between 
 two greens (there is a test of comparison of ratios in Euclid), and 
 apply the test to B and C The result is, of course, that B is the 
 darker. But A being by hypothesis exactly the same as B, the testing 
 operation would be self contradictory if it did not exhibit, when 
 applied to A and C, the very same intermediate process by which 
 we were able to compare B and C, with the same result. If the 
 above be unnecessary, then the demonstration of Euclid's proposition 
 is unnecessary. 
 
 The fact is, that there are in geometry two distinct sorts of de- 
 monstration, the first of which is only a portion of the second. The first 
 is the verbal treatment of the terms of an hypothesis, and the deve- 
 lopement of all assertions which are necessarily included in the terms 
 of the proposition, without drawing upon any other axioms or 
 theorems for evidence. It is the purely logical process, by which we 
 make two assertions put together shew their joint meaning, and 
 express what, without deduction, they only imply. Thus, from 
 " Every A is B," and " no B is C," we make it evident thot in these 
 assertions is necessarily contained a third, that " no A is C." Thus 
 it has been shewn that we cannot allow simultaneous existence to the 
 two propositions, " A is to B as C is to D," and " C is to D more 
 than E is to F," without almost expressing, and certainly implying, 
 by the mere meaning of our terms, this third proposition, that " A 
 is to B more than E is to F." 
 
 The second process is that in which the demonstration, besides 
 the purely logical process of extracting implied meanings out of the 
 expressions of the hypothesis, appeals to propositions which are not 
 in the hypothesis, and which, for any thing the hypotheses tell us to 
 
56 CONNEXION OF 
 
 the contrary, may or may not be true. Of course — not logic, but — 
 reason requires that these propositions should have been previously 
 proved, or assumed on their ovs^n evidence expressly. Let us take the 
 following proposition, " The sum of the circles described upon the 
 two sides of a right-angled triangle is equal to the circle described 
 upon the hypothenuse." Now, take every notion implied in this 
 hypothesis, " Let there be a right-angled triangle, and let circles be 
 described on its three sides." The united faculties of man never 
 proved that the sum of the circles on the sides was equal to the 
 circle on the hypothenuse, without assuming with Euclid, to the effect 
 that only one parallel can be drawn through a point to a given right 
 line ; with Archimedes, to the effect that the chord of a curve is shorter 
 than its arc, &c. &c.; and various consequences. But are any of these 
 propositions necessary to our complete definition of a right angle, a 
 triangle, or a circle ? If not, we have a broad and easily recognised 
 distinction between the first and second method of demonstration ; 
 the first, an operation of logic, or deduction from the premises of the 
 hypothesis ; the second, introducing premises from without. 
 
 There are two classes of reasoners whose ideas we recommend 
 the student closely to examine, before he finally decides : 1. Geome- 
 trical writers in general, who pay no attention to the methods which 
 they are using, but let the first book and the fifth book of Euclid 
 contain no difference by which it may be remarked that the processes 
 contained in the two are different acts of rhind. Did they ever 
 think that geometry could be made the engine by which the student 
 could examine certain operations of his own faculties, or did they only 
 imagine that it was a method of making very sure that squares, 
 circles, &c. had such and such properties? 2. The class of metaphy- 
 sical writers, who express themselves to the effect that all mathema- 
 tical propositions are contained in the definitions and axioms, in a 
 sense in which other results of reasoning are not. Put them to the 
 proof of this assertion as to geometry, and then as to arithmetic. 
 
 The whole of the process in the fifth book is purely logical, that 
 is, the whole of the results are virtually contained in the definitions, 
 in the manner and sense in which metaphysicians (certain of them) 
 imagine all the results of mathematics to be contained in their 
 definitions and hypotheses. No assumption is made to determine the 
 truth of any consequence of this definition, which takes for granted 
 "more about number or magnitude than is necessary to understand the 
 
KtTMBER AND MAGNITUDE. 57 
 
 definition itself. The latter being once understood, its results are 
 deduced by inspection — of itself only, without the necessity of looking 
 at any thing else. Hence, a great distinction between the fifth and 
 the preceding books presents itself. The first four are a series of 
 propositions, resting on different fundamental assumptions ; that is, 
 about different kinds of magnitudes. The fifth is a definition and its 
 developement; and if the analogy by which names have been given 
 in the preceding books had been attended to, the propositions of that 
 book would have been called corollaries of the definition. 
 
 XIV. If A be to B as C is to D, all four being of the same kind, 
 then if A be greater than C, B is greater than D j if equal, equal, and 
 if less, less. 
 
 A must either be > = or < C. Let A be greater than C ; then 
 mK is greater than mC Let mK lie between wBand(w+l)B; 
 then will mQ lie between wD and ^«-|-l)D. But because A exceeds 
 C, 2A exceeds 2C by twice as much, &c., and wA exceeds mC by 
 m times as much ; or wA may be made to exceed mC by a quantity 
 greater than any one named, say greater than B and D together. 
 Then the order of magnitude of the four multiples mC (w+l)D, «B, 
 »»A must be as written : for (n-f-l)D does not exceed mC by so 
 much as D, and wB does not fall short ofmA by as much as B, 
 while mh. exceeds wC by more than B and D put together. There- 
 fore, nB is greater than (7i4-l)D, and still more than wD. That is, 
 B is greater than D. 
 
 Let A be equal to C. If B exceed D at all, wB may be made to 
 exceed wD by more than D, or twB may be made, from and after 
 some value of w, greater than {m + 1)D. That is, the order of mag- 
 nitude may be made 
 
 mH (m + l)D mB (m + l)B 
 
 Having gone so far on the scales that this order becomes per- 
 manent, go on till a multiple of C (/cC) falls between the two first. 
 Then, by the definition, kK falls between the two last, which is ab- 
 surd ; for, because A=:C, /cA==/cC; therefore, B does not exceed D. 
 In the same way it may be shewn that B does not fall short of D. 
 Therefore, B = D. 
 
 The remaining case (A less than C) may be proved like the first. 
 
 XV. A is to B as /wA is to wjB 
 
58 CONNEXION OF 
 
 The scale of multiples of A and B is nowhere altered in the order of 
 magnitude by multiplying every term by w. If p A lie between ^13 
 and ((7-|-l)B, (/?;«) A which is p(mA) lies between q{mB) and 
 (7 + 1) (mB). 
 
 XVI. If A be to B as C is to D 
 and if all four be of the same kind, 
 
 Then A is to C as B is to D. 
 
 (iv.) mA is to mB as nC is to nD 
 
 (xiv.) If mA be greater than nC, wB is greater than nD, if 
 equal, equal ; if less, less. Therefore, A is to C as B to D. 
 
 X Vir. If A + B be to B as C + D to D, then A is to B as C is 
 to D. If 7«A lie between nB and (n + 1)B, it follows that 7nA-\-mBj 
 or m(A + B) lies between (m + n)B and {m-{-n-^l)B. Then, by 
 the proportion, m(C + D) lies between (m + n)D and (w + n + 1) D, 
 or mC + wD lies between mD + wD and mD + (n + l)D, or mC 
 lies between «D and (n4-l)D. Therefore, the scales of A and B, 
 and of C and D, are the same ; whence the proposition. 
 
 XVIII. If A be to B as C is to D, then A + B is to B as C + D ^ 
 is to D. A proof of exactly the same kind as the last should be given 
 by the student. 
 
 XIX. If A : B : : C : D, C and D being less than A and B, then 
 A : B : : A — C : B — D. For the hypothesis gives A to C as B to 
 D, and A is C + (A — C), and B is D + (B — D), whence, 
 
 C + (A — C) is to C as D + (B — D) is to D 
 (xvir.) A — C is to C as B — D is to D 
 
 (xvi.) A — C is to B — D as C to D, or as A to B 
 
 XX. If A be to B as D to E 
 and B to C as E to F 
 
 fgreater than ") ^greater than 1 
 
 Then A is ] equal to > C, when B is < equal to > F 
 (less ihan } (less than ) 
 
 Let A be greater than C ; then A is to B more than C is to B ; 
 but A is to B as D to E, and C to B as F to E ; therefore, D is to E 
 more than F is to E, or D is greater than F. In a similar way the 
 other cases may be proved. 
 
NUMBER AND MAGNITUDE. 
 
 69 
 
 Hence it follows, that A is to C as D to F. For, 
 
 (vi.) mh. is to 7«B as mD to wE 
 
 ?iB is to nC as nE to wF 
 therefore, mA is > = or <wC when mD is > =: or <nF 
 whence, A is to C as D to F. 
 
 XXI. If of the magnitudes 
 
 ABC , A: B::E:F 
 
 D E F "^'^^^ B:C::D:E 
 Then A > = or < C when D > = or < F 
 
 Let A be greater than C ; then A is to B more than C is to B : as 
 bi^fore E is to F more than E is to D, or D is greater than F. Simi- 
 larly for the other cases. 
 
 XXII. If there be any number of magnitudes, 
 
 A B C D 
 
 P Q R S 
 
 and if any two adjoining be proportional to the two under or above 
 them, then any two whatsoever are proportional to the two under or 
 above them. For, since (xx.) 
 
 A : B : : P : Q-) 
 
 B.C.Q.IIJ ^^^^ C:D::R:SJ 
 
 Therefore, A : D::P : S, 
 &c. 
 
 XXIII. In the hypothesis of (xxi.), by proof as before in (xx.), 
 A is to C as D to F. 
 
 XXIV. If A be to B as C to D ^ then A+E is to B 
 and E be to B as F to D J as C + F to D 
 
 For, A:B::C:D1 , .„ ^ ^ 
 
 > whence, A : E : : C : F 
 and B : E : : D : F J 
 
 (xviJi.) A + E:E::C + F:F 
 
 But, E : B : : F : D 
 
 Therefore, A + E:B;:CH-F:D 
 
 XXV. If A : B : : C : D, all being of the same kind, the sum 
 of the greatest and least is greater than that of the other two. First, 
 which are the greatest and least ? If A be the greatest, then C is 
 greater than D ; and because A : C *. *. B : D, B is greater than D ; 
 
60 CONNEXION OF 
 
 therefore, D is the least. Now, prove tliat if B be the greatest, C is 
 the least; and that, by inverting the proportion, if necessary, it may 
 alv^rays be written with the greatest term first, and the least last. 
 
 When A is the greatest, since A — B .* A : ; C — D : D, A — B 
 is greater than C — D ; therefore, ( A — B) + B + D is greater than 
 (C — D) + B + D, or A -f D is greater than C + B. 
 
 If there be a given ratio, that of A to B, and another magnitude 
 P, there must be a fourth magnitude Q, of the same kind as P, such 
 that A is to B as P to Q, or Q to P as B to A. 
 
 Firstly; Q may certainly be taken so small that (mB being 
 greater than nA) niQ shall be less than nP. Find m and n to 
 satisfy the first conditions, and let K satisfy the second. Then K is 
 to P less than B is to A. Now (wB being less than nA), Q may be 
 taken so that mQ shall be greater than nP. Find m and n to satisfy 
 the first, and let L satisfy the second. Then, L is to P in a greater 
 ratio than A to B. And it is immediately shewn that every magni- 
 tude less than K is to P less than B to A, and every magnitude 
 greater than L is to P more than B to A. Whence, it is between 
 K and L that the fourth proportional Q is found, if any where. 
 There cannot be more than one such value of Q ; for, if there be two 
 different magnitudes V and W, since, then, by taking m sufficiently 
 great, we may make mV and mW differ by more than P, it is impos- 
 sible that both mV and wW can lie between the same consecutive 
 multiples of P, as those of B which contain between them w A. And 
 the above also evidently shews, that if we suppose a magnitude Q, 
 changing its value from K to L, it cannot during its increase become 
 of the same kind as L, namely, more to P than B is to A, and then 
 again become of the same kind as K. For, whatever magnitude has 
 this property of L, every greater one has the same. There is then 
 only one point between K and L at which this change takes place, 
 and we have, therefore, this alternative: Either G (between K 
 and L) is less to P than B is to A, and every magnitude greater than 
 G is more ; or, some magnitude G between K and L is the same to 
 P as B to A, and is the intermediate limit lying above all those 
 which are less to P, and below all those which are more. By dis- 
 proving the first alternative, we prove our proposition. If possible, 
 let G be less to P than A to B, G-f V more, however small V may be. 
 
NUMBER AND MAGNITUDE. 61 
 
 Then may mG be made less than wP (mA being greater than wB), 
 while w(G+V) is greater than mP. For the ascending assertion 
 must be converted at least into a stationary/ one. Let m G fall 
 short of mP by Z ; then V may be taken so small that mV shall 
 not be so great as Z, or wzG-f mV not so great as wG+Z, that is, 
 not so great as mP. But the first clause of the alternative supposes 
 that w(G4-V) must be greater than wP, how small soever V may be; 
 therefore this clause cannot be true, or the second must be true. 
 
 This fourth proportional to A, B, and P, then, must exist; but 
 whether it can be expressed by the notation, or determined by the means 
 of any science, is another question. It can be expressed in arithmetic 
 when A and B are commensurable : it can be found in geometry 
 (by the straight line and circle) when A and B are lines or rectilinear 
 areas. But if they be angles, arcs of circles, solids, &c. it cannot be 
 assigned by the straight line and circle, except in particular cases. 
 
 Let us suppose the ratio of A to B given, that is, not A and B 
 themselves, but only the answer to this question for all values of w, 
 " Between what consecutive multiples of B lies w A ?" Suppose also 
 the ratio of B to C given ; how are we to find the ratio of A to C, or 
 can it be found at all ? that is, is it given or determined by the two 
 preceding ratios. Take any magnitude P, and determine Q so that 
 P is to Q as A to B, and then determine R so that Q is to R as 
 B to C. Then the ratio of P to R (page 59) is that of A to C; not 
 that P is A or R is C (for they may even be magnitudes of different 
 kinds), but P is to R as A is to C. 
 
 The process by which the ratio of A to C is found by means of 
 those of A to B and B to C, is called by Euclid composition of these 
 ratios ; or the ratio of A to C is compounded of the ratios of A to B 
 and B to C. What, then, ought to be meant by the ratio compounded 
 of the ratios of A to B and X to Y. Our guide in the assimilaion 
 of processes, and the extension of names, is always the following 
 axiom. 
 
 Let names be so given, that the substitution of one magnitude 
 for another equal magnitude shall not change the name of the process ; 
 and, generally, that the same operations (in name) performed upon 
 equal magnitudes, shall produce the same result. 
 
 Let X be to Y as B to N, where N is a fourth proportional to be 
 determined. Then the ratio of A to N is that compounded of A to B 
 and B to N, and is what must be meant by that compounded of 
 
62 CONNEXION OF 
 
 A to B, and X to Y. It is proved in Prop. 20, that ratios com- 
 pounded of equal ratios are equal ratios. 
 
 Again, to find the ratio compounded of the ratios of A to B, 
 C to D, and E to F ; let the process by which the ratio of A to D is 
 derived from those of A to B, B to C, and C to D, still be called 
 composition. Then take B to M as C to D, and M to N as E to F : 
 the ratio of A to N is that compounded of the three ratios. 
 
 In the beginning of this work, we deduced the necessity for con- 
 sidering incommensurables in some such manner as that of Euclid, 
 from the notion which, as applied to coramensurables, admits of a 
 definite representation, derived from the idea of proportion. But the 
 method of the fifth book is different. It is there implied, that where- 
 ever two magnitudes exist, their joint existence gives rise to a third 
 magnitude, called their ratio, of which magnitude no conception is 
 given except what is contained in certain directions how to apply the 
 terms equal, greater, and less, to two of the kind. On this the 
 natural question is, what sort of magnitude is this, and how do we 
 know that there is any magnitude whatsoever which admits of this 
 apparently arbitrary exposition of definitions? This question is very 
 much to the point, and the want of an answer at the outset is a main 
 cause of the difficulty of the Fifth Book. The answer implied in the 
 work of Euclid is this : Let us first consider what will follow if there 
 be such things as ratios, or magnitudes to which these definitions of 
 equal, greater, and less apply; we shall then shew (in the Sixth Book) 
 that there are different pairs of magnitudes, of which it maybe said 
 that they have ratios, and we shall never have occasion to inquire 
 what ratio is. 
 
 "We may take a case parallel to the preceding from the First Book. 
 The notion of a straight line suggests nothing but length ; that of two 
 straight lines which meet, suggests a relation, which we may conceive 
 stated in this way. If A, B, C, and D, be straight lines, of which 
 A and B, and C and D, meet ; let A and B be said to make the 
 same angle as C and D, when, if A be applied to C, and B and D 
 fall on the same side, B and D also coincide : but let A be said to 
 make a greater angle with B than C with D, when, in a similar case, 
 B falls outside of C and D, &c. To this it would be answered, that 
 the preceding definitions are a circuitous way of saying that the angle 
 made by two lines is their opening or inclination ; an indefinite term, 
 which, though it distinguishes angle from length, does not serve to 
 
NUMBER AND MAGNITUDE. DO 
 
 compare one angle with another. And just in the same manner, if it 
 were not that the definition is more complicated, and refers to an 
 abstract, not a visible or tangible, conception, it would immediately 
 be seen that i^utlo is illative magnitude, — a term which is sufficient 
 to distinguish the thing in question from absolute magnitude, but 
 which does not give any means of comparing one thing of the kind 
 with another. The immediate deduction of this idea is as follows : 
 If, whenever mA lies between «B and (n -\-\) B, it also happens that 
 »nP lies between wQ and (n -f 1) Q, it follows that A, lying between 
 
 two certain fractions of B, — B, and — '■ — B, then P lies between 
 
 n 
 the same two fractions of ^. Or, if m A = n B, that is, if A= — B, 
 
 m 
 
 then P is the same fraction of Q. Or we may state it thus : if B be 
 
 made unity, for the measurement of A, and Q for the measurement of 
 
 P, then A and P are the same numbers or fractions of their respective 
 
 «nils. 
 
 Euclid has commenced the subject with a rough definition, as we 
 liave seen, p. 29, and the translators have spoiled it, by not distin- 
 guishing between quantity, and relative quantity; that is, by so 
 wording the definition as to say nothing more than that ratio is a 
 relation of magnitudes with respect to magnitude. 
 
 We now come to consider the application of the preceding notions 
 to arithmetic. Let us first separate all that part* of arithmetic which 
 relates to abstract and definite numbers, from the rest, and let us call 
 it primnri/ arithmetic. A little observation will shew that abstract 
 number as distinguished from concrete, is really the same thing as 
 ratio of magnitude to magnitude. What is three^ for example? It is 
 an idea which we obtain equally from looking at 
 
 and 
 
 From putting such concretes together, we bring away a notion of there 
 being the same relative magnitudes existing between the individuals 
 
 • The whole of the First Book of my Treatise on Arithmetic ^ with the 
 exception of § 158, 165-169. 
 
64 CONNEXION OF 
 
 of each pair. In the first, it is repetition^ in the second, it is length, 
 in the third, it is opening, we are reminded of; but in all three, we say 
 the first is three times the second. Now this word times is, in fact, a 
 limitation, which will not do for our present purpose ; it implies that 
 we will have no other ratios except those of line to line in the series 
 
 A h 1 
 
 B h -T 1 
 
 c I- , -1 ! 
 
 D 1 , , , 1 &c. 
 
 made by repetitions only : but there may be ratios which are not those 
 of line to line in any repetition, how far soever carried. 
 
 Here is a point at which we are compelled to pause, to adjust the 
 "Well-known terms of number to the new idea we have put upon them. 
 Abstract numbers are certain ratios; abstract fractions are certain 
 other ratios ; but all possible ratios are not found among numbers and 
 fractions ; whence it arises, that primary arithmetic, though it may be, 
 so Jar as it goes, a theory of ratios, is not a theory of all ratios, nor 
 are its operations such as can be performed upon all ratios. 
 
 That ratios are magnitudes, we must have supposed from the 
 beginning, seeing that they bear the terms equal, greater, and less. 
 But there was still this defect, that our test of A being to B more than 
 C to D, was one which left us with no idea how much more A was to 
 B than C to D; which amounts but to this, that we could not define 
 the ratio of ratios without having first defined ratio. But, in like 
 manner as arithmetic was made the guide to that notion which is 
 properly* called the ratio of incommensurable quantities, so will the 
 ratio of two ratios in arithmetic lead us, after a little consideration, to 
 the meaning of the ratio of ratios of incommensurables. 
 
 When we say two, we refer to the repetitions of the smaller in a 
 ratio of magnitudes, thus visibly related : 
 
 When we say twice two, there is a change of idiom in our language. 
 It might be, instead of twice two is four, two twos a^re four ; that is, 
 where there exists that idea of relative magnitude which we signify by 
 
 • Consistently; so as to couple with operations upou problems of 
 oommensurables those operations which apply to the same problems upon 
 incommensurablest 
 
NUMBER AND MAGNITUDE. 65 
 
 (wo, let the idea o( relation be coupled with the idea of a larger relation, 
 in exactly the same manner as our idea of magnitude, when we look 
 
 at , is increased when we look at ; and we shall then, 
 
 by considering the result as one of relative magnitude, be led to the 
 
 idea of the relation between and . This, 
 
 of course, does not give a better comprehension of twice two is four ; 
 but what it explains is, that we are using the term ratio in a consistent 
 sense, when we say that the ratio of 2 to 1, increased in the ratio of 
 2 to 1, is the same as the ratio of 4 to 1 ; and, generally, that the 
 ratio of w to 1, increased in the ratio of w to 1, is the ratio of tnn to 1. 
 And the notion of relative magnitude contained in the words, ratio of 
 m to p, must be the same as that contained in the words, ratio of mn 
 io pn ; and, conversely, the notion in the latter is that implied in the 
 former. I doubt if any thing that deserves the name of proof can be 
 given of this proposition, which seems to be worthy the name of an 
 axiom. What idea we form of magnitude as portion of magnitude 
 from A and B, the same do we form from 2 A and 2B. Nor can 
 I imagine these propositions extended to fractions in any more funda- 
 
 77Z 7) 711 D 
 
 mental manner than by observing, that as — taken - times is — 
 •' ° n q nq 
 
 times (times mean times, or parts of times, either separately or both 
 
 tn , J) 
 
 together,) a unit, the ratio of — to 1, altered in the ratio of- to 1, 
 
 is the ratio of — to 1 : or that the ratio of m to n. altered in the ratio 
 nq 
 
 ofp to q, is the ratio of mp to nq. These are propositions in which 
 the line between deduction and mere establishment of the synonymous 
 character of terms is very indefinite. I recommend the student (o 
 examine his own idea of what he would have meant by " the pro- 
 portion of 3 to 2 increased in the proportion of 5 to 4, is the pro- 
 portion of 15 to 8." If he be a metaphysician, I refer him to his 
 oracle, on condition only that the response shall not contradict the 
 preceding proposition. 
 
 The multiplication of ;w and n is, then, the alteration of the ratio 
 of m to 1 in the proportion of w to 1 ; and the ratio of magnitudes 
 w A and nA is the same as the ratio of magnitudes mB and jiB, and 
 of w to n. Hence, to alter mA : nA (which is m : n) in the ratio of 
 pB to qB, which is {p : q), is the formation of?/?/) : nq, or mpA to 
 72 J A, or mpB to nqB. Now, this is precisely what Euclid has 
 
 G 2 
 
bo CONNEXION OP 
 
 termed the composition of these ratios ; for, let wjA : nA : : vB : j^B, 
 then V B : j? B compounded with pB : gB, is vB : qlif or v : q. But 
 
 mA : nAi: m : n vB : pB ::v : p 
 
 Therefore m : n is v : p or — : 1 is - : 1 
 ^ n p 
 
 pm pm 4 A 
 
 V =' — V : q is ^-— ' q or pm : nq or pmA : nqA 
 
 or pmB '. nqB. 
 
 Hence, composition is multiplication of terms, when the ratios are 
 those of number to number. Let, then, composition of ratios stand 
 for multiplication of terms, and be considered as the corresponding 
 operation in the case of incommensurable magnitudes. 
 
 Prove from this, that if U : A and U : B be compounded, giving 
 U : C, that when A = aU and B = 6U, we have C=a6U, and that 
 if U : A and B : U be thus compounded, giving U : D, we have 
 
 D = -U, D:U::y:l; in which operations, corresponding to mul- 
 tiplication and division. 
 
 It may be a matter of some curiosity to know whether Euclid carried 
 with him the notion of multiplication of numbers in the composition 
 of ratios. In the Fifth Book, the notion of the numerical magni- 
 tude of a ratio is entirely suppressed, except only in the single word 
 *7iXixorm (see page 29.) Composition* is defined to be the taking 
 an antecedent of one ratio with the consequent of another; and it is 
 not even specified that the intermediate terms are to be the same. 
 But in the Sixth Book we find composition, or collocation of ratios, 
 to mean the multiplication of their quantuplicities (see page 29). 
 
 * IvvSiffis y^oyov iffr) X^-^is vov iiyoufAivou f/ttra tou iftefiivev us tvif vf^of 
 
 ivTO TO ITOfAiVOV' V. Dof. 15. 
 
 Aoyas \k Xoyuiv cvyxilf^eci xiytron oretv at ruv koyu* ^tiXiKOTnrts i^' 
 iavras ^rokkefrXeca'iafS-iTg'at Toiutri rtvei, — VI. Def. 5. 
 
 The second of these definitions has usually been omitted in modern 
 editions. But it is worthy of remark, that, in the first, to compound is 
 evvTt^iff^ut ', in the second, o-vyKutr^on ; and the second is the word after- 
 wards used by Euclid, though in the sense of the first. The reason of the 
 omission appears to have been a disposition on the part of commentators 
 to consider Euclid as a perfect book, and every thing which did not 
 •ccord with their notions of perfection, as the work of unskilful editors or 
 interpolators. 
 
NUMBER AND MAGNITUDE. 67 
 
 The addition and subtraction of ratios can only be primarily con- 
 ceived when the latter terms of the ratios are alike. Thus, 
 
 A 
 
 B 
 
 C D 
 
 we must imagine the idea of relative magnitude given by BC com- 
 pared with A, and by C D compared with A, to be put together, in 
 order to make up the relative magnitude of BD to A. Addition and 
 subtraction are, as to ratios, ideas not so simple as multiplication and 
 division. Shew that the preceding is the only way in which 7n : 1, 
 increased in the ratio of /i : 1, will give ww : 1, consistently with 
 the notion of multiplication of whole numbers being successive 
 additions. 
 
 When ratios have not the same consequents, they must be reduced 
 to the same consequents. Thus, A : B and C : D are added by 
 taking A : B : : P : Z and C : D : : Q : Z, and P + Q : Z is the sum 
 of the ratios. This answers to addition of fractions. 
 
 Let P be the mean proportional between A and B, meaning that 
 A : P as P : B. It may be proved, as in page 60, that there must be 
 such a magnitude as this mean proportional, and we may also prove 
 that we can find A : P as P : Q, and P : Q as Q : B, thus forming 
 two mean proportionals. It is readily proved, that if A==aU and 
 B = 6U, then P=cU where cc = ab. If, then, a6 be a number or 
 fraction which has a square root, P can be found commensurable with 
 A and B; but if a& have no square root, number or fraction, then P 
 is incommensurabla with A and B, but not, therefore, unassignable as 
 a magnitude, though unassignable as a numerical fraction of A or B. 
 Consequently, when we speak of \/2, it must be with reference to 
 magnitude, and we mean \/2 M, an accurate representative (if we 
 choose to define it so) of the mean proportional between M and 2M. 
 Similarly, when there are two mean proportionals, we find P, if 
 A.=.a\J and B = 6U, to be cU where ccc-=ah, and this is incom- 
 mensurable unless a 6 be a cube number or fraction. But we may 
 define \/2M to be the first of two mean proportionals between M 
 and 2M ; and so on. 
 
 Are we, then, to use long processes and comparatively obscure 
 definitions, whenever the ratios of a problem are incommensurables ? 
 By no means ; we proceed to shew that it may always be made pos- 
 
68 CONNEXION OF 
 
 sible to let the processes of arithmetic (or rather of algebra) be used 
 as if the ratios in question were commensurable ; and that we may 
 thus deduce a result which may either be interpreted strictly at the 
 end of the process, or made to give a result as near as we please to 
 the truth in arithmetical terms. Let us suppose this Problem : Two 
 pounds are spent in buying yards of stuff, and as many yards are 
 bought as shillings are given for a yard. Let x be the number of 
 yards, then x yards at x shillings a yard, gives xx shillings; whence 
 xjr= 40, which is arithmetically impossible. Now, turn from num- 
 bers of pounds to quantities of silver, and let S be the silver in a 
 shilling, X that in the price given ; let L be a yard, and Y the length 
 bought. Then it is required that 40 S should be given, and that X 
 should bear the same ratio to S as Y bears to L. Now, if X be given 
 for L, what must be given for Y ? Take P of such relative magni- 
 tude to X, as Y is to L ; that is, let 
 
 L : Y::X : P = 40S 
 But as L : Y : ; S : X Therefore S : X : : X : 40S 
 
 or X must be a mean proportional between S and 40 S. Now, if we 
 make our symbols general, and let x stand for any ratio, numerically 
 possible or not, but proceed as we should do if it were arithmetical, 
 we proceed as in the first case, and finda'=v40, which, being in- 
 terpreted as a magnitude, with reference to its ratio to S, means, when 
 the symbols are geneml, n/40S, the mean proportional between S 
 and 40 S. If we wish for an approximate numerical result, we must 
 suppose 40 + a to be the sum, where 40 -|- a has a square root, and 
 then we have jr = v40+a; and since a may be made as small as 
 we please, we can make this problem as near the given one as we 
 please. 
 
 The following table should be attentively considered. In the first 
 column, an incommensurable ratio x, of X to U, is given, or a func- 
 tion of it and other ratios, under arithmetical symbols ; in the second 
 is the ratio which the function really gives, when the symbols on the 
 first side are extended in meaning. 
 
 a; or o: : 1 the ratio of X to U 
 
 y--2/ : 1 Y .. U 
 
 z.. z : I Z .. V 
 
NUMBER AND MAGNITUDE. 
 
 xyz 
 
 xy\-yz 
 
 1 : : 1 : T or U : X 
 
 as - : 1 
 
 X 
 
 compounded of :r : 1 and j; : 1 or X : U and X : U. Let 
 U : X : : X : P then P : X and X : U compounded give P : U 
 or j:^ : 1 is the ratio which a third proportional to U and X 
 bears to U. 
 {x : 1) (3/ : 1) (^ : 1) ratio compd.of X : U, Y : U, and Z : V, 
 
 Let X : U : : P : Y ; the above is then compounded of 
 P : U and Z : V. Let P : U : : Q : Z. The result is then 
 Q : V or xyz : 1 is Q : V 
 P a fourth prop, to U, X, and Y 
 
 Q U,P, .. Z 
 
 .rj/: 1 is P :U when U :X:: Y:P 
 yz:l is Q:V .... U: Y::Z;Q 
 Take Q : V :: M : U or V : Q :: U : M 
 P H- M : U is the ratio required. 
 
 X- : 1 compd. of X : U and V : Z 
 
 LetX : U :: P : V and P : Z is the ratio required. 
 
 Now, we have assumed the operations of finding^ a fourth pro- 
 portional, a mean proportional, two mean proportionals, &c. Whether 
 these can be done, or whether any or all cannot be done, is a question 
 for every particular application. In arithmetic, we will suppose the 
 data arithmetical; a fourth proportional can always be found. In 
 geometry, a fourth proportional can be found to lines or rectilinear 
 areas; but not to angles, &c. And a mean proportional cannot 
 generally be found in arithmetic, but can be found in geometry, 
 between two straight lines, or two rectilinear areas. But two mean 
 proportionals cannot be found in geometry or in arithmetic. 
 
 It must be remembered, that while we are here speaking of 
 geometry or arithmetic, we are not speaking of every conception we 
 can form of these sciences, but of the subjects as limited by the de- 
 finitions of what it has been agreed shall be called arithmetic and 
 geometry. Elementary arithmetic means the science of numbers and 
 fractions : elementary geometry, the science of space, so far as the 
 same has properties which can be deduced by allowing o^ fixed 
 straight lines and circles. To say that an angle cannot be trisected 
 geometrkally, means, that it cannot be trisected by means of straight 
 
70 Connexion op 
 
 lines and circles as defined. But there is an abundance of curves, 
 the stipulation to draw any one of which would secure the means of 
 trisecting an angle. And, by simply granting that a circle should be 
 allowed to roll along a straight line, and that the curve described by 
 one of its points should be granted, we can either square the circle, or 
 find the ratio of any two arcs. And, just in the same way, if we 
 were to define a journey to be 100 miles or less, it would be perfectly 
 true that we could not make a journey from London to York, but that 
 we could from London to Brighton. 
 
 It is surely time that the verbal distinction between different parts 
 of the same sciences should be done away with. Every conception 
 which can be shewn to be not self contradictory, can be as easily 
 realised by assumption as the drawing of a circle, which is itself a 
 perfect geometrical idea, and can only be roughly represented by 
 mechanical means. Whatever can be distinctly conceived, exists for 
 all mental purposes ; whatever can be approximately found, for all 
 practical uses. 
 
 It may be worth while to make the student remark the close 
 similarity which exists between the process in page 64, and that by 
 which we enlarge our ideas in algebra, from the simple consideration 
 of numerical magnitude to that of positive and negative quantities. 
 In both, we set out with a notation insufficient to express all the 
 results of problems ; in both, this circumstance is marked by the 
 appearance of unexplained results, the examination of which, on 
 wider grounds, shews the necessity for attaching more extensive ideas 
 to symbols ; and in both, the partial view first taken is wholly included 
 in the more general one : while in both, the processes conducted 
 under the wider meanings are precisely the same in form and rules 
 as those which are restricted to the original meanings of the symbols. 
 The principal difference is, that in extending arithmetic to the general 
 science of ratios, we are not engaged in interpreting difficulties 
 arising from contradictions, but from results which are only approxi- 
 mately attainable. But in both the reason is, that we set out with 
 our symbols so constructed, that we cannot undertake a problem 
 without tacitly dictating conditions to the result. In beginning 
 algebra, we make quantities indeterminate in magnitude, with symbols 
 of operation so fixed in meaning, that they cannot be used without 
 an assumption that we know which is the greater and which is the 
 
NUMBER AND MAGNITUDE. 71 
 
 less of two unknown quantities. We have, therefore, to examine tb€ 
 different cases of problems which present different results according 
 as one datum is greater or less than another; and thus we obtain those 
 extensions of meaning which will make the problems and the symbols 
 equally general. In beginning arithmetic, we invent no symbols of 
 ratio, except those which represent the ratios of magnitudes formed by 
 the repetitions of a given magnitude. These we find to be not 
 sufficient to represent all ratios ; though it is shewn that we can make 
 them represent any ratio which magnitudes can have, as nearly as 
 we please. The invention of new symbols of ratio must require 
 the generalisation of operations ; that is, we cannot speak of multi- 
 pUcution or division of ratios generally, while these words have a 
 definition which applies only to ratios of repetitions, or commensurable 
 ratios. 
 
 There is a difference between the impossible of primary arith- 
 metic, and that of geometry. The first is unattainable by a restricted 
 definition, the second by restricting the cases of general definitions 
 which shall be allowed to be used. In arithmetic, we attempt a 
 science of relative magnitudes, by running from the general notion of 
 relative magnitude to the more precise and easy notion of the relative 
 magnitudes of one certain set of magnitudes, A, an arbitrary, A + A, 
 A + A-f A, &c. We are very soon taught that our symbols will 
 not express all ratios, that is, if we have a general notion of ratio to 
 think about : whence our definitions are not sufficiently extensive. 
 But in geometry, having assumed notions and definitions from which 
 we cannot help conceiving an infinite number of different lines and 
 curves, we immediately proceed to cut ourselves off from the use of 
 all except the straight line and circle ; that is, the straight line between 
 or beyond two given points, and the circle which has a given centre 
 and a given radial line. Until these demands or postulates are 
 looked upon as restrictionSj their sense is never understood. (See 
 the Appendix.) 
 
 This difference is, however, not very essential ; since it is much 
 the same whether we define in too limited a manner, or whether we 
 limit ourselves to the use of only a part of a general definition. We 
 shall in the sequel discard the restrictive postulates, and suppose 
 ourselves able to draw any line which we can shew to be made by 
 the motion of a point. 
 
 The method by which Euclid first exhibits four proportional 
 
72 CONNEXION OF 
 
 Straight lines, though elegant and ingenious, has not the advantage 
 of exhibiting the notion of ratio directly applied to two straight lines. 
 The following theorem is directly proved from the first book, and 
 maybe made the guide. If a series of parallels cut off consecutive 
 equal parts from any one line which they cut, they do the same from 
 every other. This premised, suppose any two lines OA, OB, and 
 take a succession of lines equal to OA and OB, drawing through 
 every point a parallel to a given line. Draw any other line, OCD, 
 intersecting all the parallels : from which the preliminary proposition 
 shews, that whatever multiple Oa is of OA, the same is Oc of OC ; 
 and whatever Ob is of O B, the same is O c? of O D. And if O a be 
 greater than, equal to, or less than Ob, Oc is greater than, equal to, 
 or less than, 0</. Hence the definition of equal ratios applies pre- 
 cisely to the lines OA, OB, OC, and OD, which are, therefore, 
 proportionals. This gives the construction of Book VI. Prop. 12, or 
 one analogous to it. 
 
 A B fe « 
 
 The method of finding a mean proportional between two straight 
 lines is given in Prop. 13 ; but as we now wish to make the straight 
 line the foundation of general conceptions of magnitude, we shall 
 pass at once to those considerations which involve any number of 
 mean proportionals. It adds considerably to the interest of this part 
 of the subject, that we are thus brought to the notions on which the 
 first theory of logarithms was founded. 
 
 Let there be any number of lines, V, V„ V^, V^, in con- 
 tinued proportion ; that is, let all the ratios of V to V„ V, to V,, Vj to 
 Vj, &c. be the same. And let V, be greater than V ; in which case 
 Vj is greater than V„ &c. If V, were equal to V, then would V, be 
 equal to Vj, &,c. And, first, we have the following 
 
 Theorem. By however little V, exceeds V, the series V, V„ &c. 
 is a series of magnitudes increasing without limit : so that, however 
 great A may be, a point may be attained from and after which every 
 terra is greater than A : but in all cases whatsoever, \\ may be taken 
 
NUMBER AND MAGNITUDE. 73 
 
 SO near lo V, that the terms of the series V, Vj, &c. between which 
 A lies, shall be as near to A in magnitude as we please. 
 
 Firstly, the series increases without limit. For, since V : Vj : : 
 Vj : Vg, and V and V^ are the greatest and least, we have 
 
 V + V2 is greater than Yi + Vi 
 or Vo — Vi is greater than V^ — V 
 
 Or, Vg exceeds V, by more than V, exceeds V. Similarly, V3 exceeds 
 Vj by more than V^ exceeds Vj ; and so on. But it to V were added 
 continually the same quantity, the result would come in time to 
 exceed any given magnitude ; still more when a greater quantity is 
 added at every step. 
 
 Secondly, since then we come at last to Vn less than A, while 
 Vm+1 exceeds A, it is plain that A will not differ from either by so 
 much as they differ from each other. But because 
 
 V„ : V„^.i : : V : V, 
 we have V„+i-V„ : V„ : : V^-V : V 
 
 If then Vi— V be so small that m (V,— V) shall not exceed V, 
 neither will ?w(V»+]— V„) exceed Vn, and of course not A. Let m 
 be any given number, however great, and let Vj — V be less than the 
 mth part of V ; then will Vn+i— Vn be less than the ?nth part of A; 
 or, by taking m sufficiently great, may be made as small as we please. 
 Whence the second part of the theorem. 
 
 Theorem. In the preceding series, the selection 
 
 V V„ V,„ V3„ &c. 
 
 constitutes a similar series of continued proportionals. For, since 
 any two consecutives in the upper line next given are proportional 
 to those under them in the lower, 
 
 V, v„ y, v„ 
 
 v„ v„+, v„+, v,„ 
 
 we have (xxii.) V : V„ : : V„ : Vgn : and so on. 
 
 If between each of the terms of the series we insert tJie same number 
 of mean proportionals, the series thus formed will have the same pro- 
 perties as the original. Let us say we insert two mean proportionals 
 between each two terms. Then we have 
 
 H 
 
74 CONNEXION OF 
 
 V K K' Vi L L' Vg M M' V3 
 
 Now the only question about the continuance of the same ratio from 
 term to term is in the ratios Vj : L, Vj : M, &c. But I say that since 
 
 V : K : : K : K' : : K' : Vi V : V^ 
 V,:L::L: L ::L' lY, ^""^ V^ : V, 
 
 that V : K : : V, : L. For if not, let these latter ratios differ; say 
 
 V is to K more than V, is to L. Then is K to K' more than L is 
 to L' ; and hence (presently will be shewn) the ratio compounded of 
 
 V to K and K to K', or V : K', is greater than that compounded of 
 V, : L and L : L' or V : U. Similarly, V to K' and K' to V, being 
 more than Vj : L' and L' : V^, we have V : V, is more than V, to V^ 
 which is not true. Therefore V is not to K more than V, to L; a 
 similar process shews that it is not less : consequently, 
 
 V : K : : Vi : L 
 
 or the continuance of the primary ratio is uninterrupted. 
 
 The theorem assumed in the above is thus proved. If A : B more 
 than P : Q we have in A greater than nB, while wj P is less than nQ ; 
 or any other descending assertion. And if B : C more than Q : R, 
 we have xB greater than yC, while j;Q is less thanyR. Or we have 
 
 mxA greater than nx^, nx^ greater than nyC, or mxA greater than nyC 
 mxP less than nxQ, nxQ less than nyB., or mxF less than nyR 
 
 that is, A is to C more than P is to R; which is what we assumed. 
 If then we insert a mean proportional between V and A, giving 
 
 V M A 
 
 if between each we insert a mean proportional, we have 
 
 V M' M M" A 
 
 If we proceed in this way, we shall come at last to a series of the 
 
 form 
 
 V Vi V, V„.,(V„ = A) 
 
 in which no two quantities differ by so much as a given quantity K. 
 We can actually insert one mean proportional between any two 
 quantities ; it is done in geometry between two lines, and (page 60) 
 two magnitudes of any sort may be made (one being given) propor- 
 tional to two lines. Thus, let A, B, C, be continually proportional 
 
NUMBER AND MAGNITUDE. 75 
 
 lines, or let B be a mean proportional between A and C. Then if 
 A and C were taken proportional to (say angles) M and K, it follows 
 that if A : B : : M : L, that M, L, and K are continued proportionals, 
 by a proof of the sort given in the lemma of the last theorem. 
 Granting, then, that every two magnitudes have one mean propor- 
 tional, we may now shew that they have any number of intermediate 
 proportionals ; as follows : 
 
 We set out with 2 quantities, and the first insertion adds 1, the 
 second 2, the third 2% the fourth 2^ and the nth 2"-i. Con- 
 sequently, n complete insertions add 
 
 1+2 + 22+ + 2«-i or 2" - 1 
 
 to the first 2 ; giving 2« + 1 in all. Now, let us suppose that 2« + 1 
 divided by p leaves a quotient q, and a remainder r which is not 
 greater than p. Consequently, we have for the whole number (V and 
 A inclusive) after n insertions, 
 
 V = pq + r which is also p {q + I) — (p -- r) 
 
 and p — r is also not greater than p; and V,n=A when m:=v and 
 is greater or less than A, according as m is greater or less than v. 
 If then out of the series (the proportion being continued up to 
 Vp(«+i)) we select 
 
 V Vg Vgg (Vpq less than A) 
 
 ^ "^9+1 "^2 (g+1) (yp(q+i) greater than A) 
 
 We see V and Vp^, and V and ^piq+i) each with p — 1 mean pro- 
 portionals inserted between them, namely, 
 
 V, V2g ^(P-I)q and Vg^i V2(qj^l) V(p_i)(g+1) 
 
 But from Vpq to Vp(g+i) there are p passages from term to term 
 of the complete series, consequently, since each passage may be made 
 by an augmentation less than K, the difference between the two may 
 be made less than p K, which call Z. Hence we have the following 
 
 Theorem. To find two magnitudes, one greater and the other less 
 than A, but differing from it by less than a given quantity Z, between 
 each of which and V, p — 1 mean proportionals shall exist, obtained 
 by continual insertion of one mean proportional, continue the in- 
 sertion until no two successive terms shall differ by so much as the 
 
76 
 
 CONNEXION OP 
 
 pih part of the quantity Z : then the quantities required and the mean 
 proportionals shall be in the set so found. 
 
 Hence it can be shewn that there are p — 1 magnitudes (whether 
 attainable or not with any given means is not the question) which 
 are mean proportionals between V and A. Let Pp and Qp be mag- 
 nitudes, one greater and one less than A, which have such mean 
 proportionals, namely, let the following be continued proportionals, 
 
 V Pi Pg Pp_i (Pp greater than A) 
 
 V Qi Q, Qp_i (Qp less than A) 
 
 obtained by the preceding method, from which it is apparent that 
 Pj is greater than Q,. Now, exactly as in page 60, if we assume 
 Xj to set out in value = Q,, so that V : Xp more than V : A (Xp 
 
 bring the pth of the set of continued proportionals V, Xj, X^, ) 
 
 and to change through all possible intermediate magnitudes up to 
 Xj=Pi, or V : Xp less than V : A, there is but this alternative; 
 EITHER at some intermediate point V : Xp as V : A, or Xp = A, 
 OR, there is a point at which V : Xn more than V : A, being always 
 less when Xj is greater by any magnitude however small. The latter 
 may be disproved, or the former proved, as in the page cited. 
 
 To resume the original subject. It appears, then, 1st, that if be- 
 tween V and A we continually insert mean proportionals, in such 
 manner that at every step one mean proportional is inserted between 
 every two consecutive results of the preceding step. 2d, If the series 
 be continued beyond A, preserving still the same ratio between the 
 consecutive terms of the continuation which exists between conse- 
 cutive terms lying between V and A ; then will this process leave us 
 at last with a series of consecutive proportionals, having consecutive 
 terms so near together in magnitude, that every magnitude lying be- 
 tween V and any we please to name, shall have a term of the series 
 differing from it by less than Z, however small Z may be. 
 
NUMBER AND MAGNITUDE. 
 
 77 
 
 Let us now make OK and HL perpendicular to any cliosen 
 line OM, and let V be the line OK, A the line HL. Bisect OM in 
 C, and erect CD the mean proportional between OK and HL. 
 Bisect OC and CH, and erect the mean proportionals between OK 
 and CD, and between CD and HL. Continue this process, and we 
 shall thus get an increasing number of points between K and L, 
 which will soon give to the eye the idea of a curve line rising from 
 K to L. When we have thus divided OH into 2« parts, by n in- 
 sertions, giving 2" + 1 lines, we may, by setting off portions equal 
 to those intercepted in OH, continue that line on one side and the 
 other, and thus continue the scale of proportionals and the series of 
 points on one side and on the other of O and H. However far we 
 may go we can never complete this curve ; but if we admit that a 
 curve exists, wherever a series of points can be laid down, as many as 
 we please, and consecutively as near as we please, then we have 
 a right to assume this curve as existing, and, for purposes of rea^ 
 sorting, as constructed. Call this the exponential curve, (exponere, to 
 set forth), which expounds ratios, a phrase to which we shall presently 
 give meaning. That the student may not suppose we are using an 
 old word in a new sense, it is necessary to inform him that this curve, 
 or rather the process which we have illustrated by it, is older than the 
 algebraical symbol a", and that x gets the name of exponent from it. 
 We shall presently see the analogy. 
 
 The exponential curve being given, every line OG has its place 
 MP among the ordinates of the curve, and its abscissa OM, which 
 expounds or sets forth that place. From the nature of the formation, 
 it is evident that a given line has but one exponent, and that the 
 order of magnitude of lines (to the right of O), is also that of their 
 exponents. 
 
78 CONNEXION OF 
 
 And the main property of the curve is this : that a fourth propor- 
 tional to any three lines (V being one), OK, MP, M' P', may be 
 found by adding the exponents OM and OM' (making OM'^, 
 and finding the line M"P" expounded by that sum. To prove this, 
 make n sets of insertions in O H, and suppose M P to lie between 
 Vm and Vm+i, while M'P' lies between Vm' and Vm'+i. Now, ia 
 the series of continued proportionals, 
 
 V V. V, (VA. = A)V,%2 .... 
 
 I s.y that V : V„ : V^ : V„+„, 
 F»' V V. V, V„., V„ 
 
 »m' ^ m'+l ^ m'+2 *m'+m-l ^ m'+m 
 
 we have V : Vj : : V^^ : V^'+i &c. &c. 
 
 whence V : V,„ : V,„, : V^^^, 
 
 Similarly, V : Y^^, ; V^.+i : V^^^^+2 
 
 Now, by a lemma we shall presently shew, since M P lies be- 
 tween Vm and Vm + i, and M'P' lies between Vm' and Vm'+i, the 
 fourth proportional required lies between Vm+m' and Vm+m'+2. Let 
 K be the value of one of the last subdivisions of OH ; then we have 
 supposed OM to lie between wiK and (w + l)K, and OM' between 
 m'K and (m'-|-l)K. The preceding makes it evident that the fourth 
 proportional has an exponent between {m-\-m')K and {?/i-\-m' -\-2)K; 
 while the sum of the exponents OM and OM' also lies between 
 (77i + m')K and {m -{-m'-{-2)K. Since K can be made as small as 
 we please, it must follow that the sum of the exponents is the ex- 
 ponent of the fourth proportional; for two diflerent magnitudes can- 
 not lie between two quantities which can be made as near as we 
 please, as can {m -f- m') K and (w -f m') K + 2 K. If the two approxi- 
 mating magnitudes approach to each other, keeping one of two 
 different magnitudes between them, they must, at last, leave out the 
 other. 
 
 The lemma alluded to is as follows : If 
 
 A: B::C: D 
 and A : B + B' : : C -f C : D + D' 
 
 Then if A, B-^X, C-fY, D + Z, be also proportionals, where X 
 and Y are less than B' and C, then Z must be less than D' ; for A is 
 
NUMBER AND MAGNITUDE. 79 
 
 to B -|- X more than A is to B -|- B' ; or (substituting equal ratios), 
 C + Y is to D + Z more than C + C is to D + D'. Still more is 
 C + C (remember that C is greater than Y) to D + Z more than 
 C + C to D + D'; that is, D + Z is less than D+D', or Z less 
 than D'. 
 
 The following property we leave to the student to deduce from 
 the last. If there be any three lines, Xj X^ Xg, expounding Yj Yj Y3, 
 any lines whatsoever greater than V, then the exponent of the fourth 
 proportional is X^+Xg — X,. 
 
 These are all properties of algebraical exponents, or of logarithms, 
 (^keyuv a^i6[jt,oi, numbers expounding ratios). We shall now make it 
 appear, that the line expounded by x is of the form a^. 
 
 Let the numerical symbol of V or O K be v ; let that of H L or A 
 be a. Then, if arithmetical mean proportions be continually inserted, 
 we have 
 
 V {a v)^ a 
 
 V c^v^ a^v^ a^v^ a 
 
 V cc^i^ c^v^ c^v^ a^v^ a 
 
 or generally, when 2" — 1 (say p — 1) mean proportionals are in- 
 serted between v and a, the mth of these proportionals is 
 
 m . m m 
 
 (2"=^) aP V ~P which is tikp 
 
 if we suppose a=:vk. Now, let us suppose a number 1/ thus ex- 
 pounded by X ; and after n insertions, let this number x lie between 
 ma and (m-i-l)a, a. being the pih part of OH, (let OH be c). We 
 have then 
 
 X lies between m- and (/w + 1)- 
 P ^ ^P 
 
 m . m , c 
 
 or between c — and c — + - 
 
 P P P 
 
 or 
 
 -=«T + ^ (^<£) 
 
 P 
 
 Therefore, — = 
 
 p c c 
 
 c 
 
 Consequently the number expounded by m a, or m-,orx — /3 is 
 V k ' 
 
80 CONNEXION OF 
 
 and since /s diminishes without limit as the insertions continue, the 
 
 X 
 
 number expounded by x is vk^ . That is, if we adopt general nu- 
 merical symbols, let O M = jt, MP =y, and we have 
 
 X 
 
 or, if we let OK represent the linear unit(v = l), and let 0H = 
 OK = l, HL = 10, or /c = 10, we have 
 
 y = 10^ 
 
 or X is the common logarithm of y. 
 
 From the curve we see how it is that magnitudes less than V 
 are expounded by negative quantities, with other well-known pro- 
 perties of logarithms. 
 
 We see then, that the assertion " the common logarithm of 2 is 
 •30103 very nearly," may be thus made; which is perhaps the most 
 distinct view that can be given of a numerical logarithm. If we make 
 10 V the hundred thousandth magnitude in a, series of proportionals, 
 
 ^9 ^ !■> *^2> (* 100,000= '0) Vioo^ooi, &C. 
 
 then will the 30103rd of these proportionals, or V^^^^^ be very nearly 
 equal to 2 V. 
 
 If we chose, we might, granting that the exponential curve can be 
 constructed, make V/cX by definition the line MP; where X stands 
 for O M, and k for the ratio of H L to OK. From this it would 
 readily be deduced, that when k represents a commensurable ratio, 
 
 and X is — linear units where V/c'" has an arithmetical existence, 
 
 P 
 the results of this theory are the same as those of common algebra. 
 And from hence it appears, that the science known by the name of 
 the application of algebra to geometry (of which it is the foundation, 
 that a linear unit being given, every expression of algebra may be 
 considered as a length, or at least the symbol of the ratio of a length 
 to that unit) does, in point of fact, make this additional assumption, 
 while an application of geometry (with this assumption) to algebra, 
 would take away all want of rigorous conception of the meaning 
 of algebraical formula, so far as the meaning of the exponent is 
 concerned. 
 
 The view above given is very nearly that by which logarithms 
 
NUMBER AND MAGNITUDE. 81 
 
 were first calculated, but the method was not so general. The natural 
 logarithms (see my Algebra, p. 226) arose thus. If we suppose a 
 very large number of mean proportionals, then V and Vj will be very 
 nearly equal. Let Vj = V -f X, then if we assume O H, so that X 
 shall expound V + X when X is very small, or more correctly, if we 
 suppose the limit of (V+X) divided by the magnitude expounded by 
 X, as X diminishes without limit, to be unity, we have the first, or 
 Napier's system. 
 
APPENDIX. 
 
 ON THE DEFINITIONS, POSTULATES, AND AXIOMS 
 OF EUCLID. 
 
 I HERE propose to endeavour to make such a subdivision of the 
 definitions, &c. at the beginning of the First Book, as may enable the 
 student to review the reasoning of the whole. 
 
 I shall consider the 10th and 11th axioms as among the postulates, 
 firstly, because some old manuscripts support this change ; secondly, 
 because the older translations (from the Arabic) support it also, and 
 even place the 12th axiom in the same list; thirdly, because it is 
 utterly impossible to place them in Euclid's list of common notions. 
 For he uses no such word as axiom (Greek though it be), but calls 
 " the whole is greater than its part," koiwi Iwoia., that which is in the 
 conceptions of every one. Now, what is the probability that he 
 considered " all right angles are equal," as a truth familiar to the 
 understanding of every beginner in geometry ? His postulates 
 {a,irnfji,a,rtx.^ demands) do, according to the etymology of the word, 
 include those axioms, if not the 12th also. 
 
 I also place out of view the axioms which belong to all kinds of 
 magnitude as much as to space, namely, from the 1st to the 9th in- 
 clusive. There remains then in the shape of limitation, or assumption, 
 six postulates, namely, three which I will call restrictive^ being those 
 commonly called postulates^* and three assumptions, being the 10th, 
 nth, and 12th axioms, so called. 
 
 Some of the definitions contain assumptions of certain conceptions 
 existing to which names are to be given ; namely, those of a point, a 
 
 • 1 have seen the word postulate defined as a self- evident problem ; 
 and axiom as a self-evident theorem. This definition is derived from 
 the character of the postulates and axioms as usually given ; but from no 
 other source. 
 
84 APPENDIX. 
 
 line, the extremities of a line, a straight line, a surface, the extremities 
 of a surface, a plane surface, a plane angle, a plane rectilineal angle. 
 Others assume the possibility of certain relations existing, as will 
 appear from the form in which they are put. I shall now give the 
 definitions, classified with the corresponding postulates, in the manner 
 which appears to me to be most systematic, and placing in [ ] such 
 additions as seem requisite. 
 
 1. A point; an indefinable notion; but two persons, whatever 
 their idea of it may be, can reason together in geometry who deny a 
 point all parts or magnitude. Let it be granted that a point has no 
 parts or magnitude, and that we are concerned with no other property 
 of it, if there be any. 
 
 2. Aline; also indefinable, but those whose ideas of it allow it 
 length, and deny it breadth, can proceed. Let it be granted that all 
 reasoning upon lines is to be founded only upon the assumption that 
 they have length without breadth. [Thickness should have been 
 added, but breadth may mean breadth in any direction.] 
 
 3. The extremities of a line are points. [If this define any term, 
 it must be the term extremities, for the other two have been defined. 
 To me it appears something like a theorem, as follows : That which 
 ends a line cannot have length, for it would be a part of the line ; it 
 cannot have breadth or thickness, which a line has not ; it has there- 
 fore the only qualities of a point on which we reason, or comes within 
 the definition of a point.] 
 
 4. A straight line ; an indefinable notion, except by the rough 
 idea that it does not go on one side or the other of the two points, 
 [which is no definition, because it assumes the thing in question.] 
 Let it be granted, as a common notion, that two straight lines do not 
 enclose a space, or have not two points in common, without having 
 all intermediate points in common. Whatever the idea of a straight 
 line may be, this is the only property which will be appealed to. 
 
 5. Surface ; an indefinable notion ; those whose ideas give it 
 length and breadtli, but deny it thickness, have the means of reasoning 
 upon it in geometry. 
 
 6. The extremities of a surface are lines. (See remarks on 3.) 
 
 7. A plane surface; an obvious notion, roughly defined by lying 
 evenly between bounding straight lines. [This notion, however 
 obvious, does admit of a stricter definition. It is a surface of such 
 kind that any two points in it being joined by a straight line, all 
 
APPENDIX. 85 
 
 intermediate points of the straight line are on the surface. This 
 property is tacitly appealed to throughout.] 
 
 8. A plane angle ; the inclination, or bending towards each other 
 of two lines in a plane. [This definition is superseded by the next ; 
 no angle except one made by straight lines is ever used.] 
 
 9. A rectilineal angle (plane), the inclination of two straight lines. 
 [An obvious notion of opening; it is tacitly assumed that we know 
 how to determine when two angles are equal, or when one of them 
 exceeds the other, as in the fourth proposition.] 
 
 10. Right angles are those made by a straight line, called a 
 perpendicular, which falls on another straight line, making equal 
 angles on both sides. Postulate ; let it be granted that all right angles 
 are equal. [This is far from an obvious postulate ; the reason for it 
 seems to have been as follows : That two straight lines which coincide 
 in two points coincide in every point between them, has been admitted ; 
 it is sufficiently obvious to sense that they coincide beyond or on each 
 side of the two common points ; that is, they coincide altogether, 
 throughout all possible length. This seems an infinite assumption ; 
 and if it be assumed instead that all right angles are equal, it may be 
 proved afterwards that no two straight lines have a common segment ; 
 that is, that two straight lines which coincide for any length, never 
 afterwards separate. But it may be shewn, that the assumption of all 
 right angles being equal, amounts to the same infinity of assumption ; 
 as follows : The right angle is by definition the half of the opening 
 which two straight lines make, when one is the continuation of the 
 other, as AB, BC. To assume that all right angles are equal, is to 
 assume that the doubles 
 
 1 T- -I 
 
 V -T H 
 
 D E F 
 
 of right angles are equal ; that is, that if we lay B on E, with E D 
 coinciding with BA, then EF and BC will coincide. Now it is 
 precisely the same thing to assume, that when AB is made to coincide 
 with DE up to the point E, that the two coincide beyond it. 
 
 I should recommend the student to make to the assumption that 
 two straight lines cannot coincide in two points without coinciding 
 between them, the addition that they also must coincide beyond them. 
 
86 APPENDIX. 
 
 It may then be directly proved that all doubles of right angles are 
 equal, and thence that all right angleis are equal.] 
 
 The definitions 11, 12, 13, 14, need no remark, being purely 
 nominal. 
 
 14. The circle, a plane figure, having all points of its boundary 
 (15. the circumference) equally distant from a given point (16. the 
 centre) within it. [Here is tacitly a postulate, namely, that this point 
 lies within the figure. It is also assumed in the first proposition, that 
 if any point of a circle be within another, the two circles must 
 intersect. There are several assumptions of this kind, which shew 
 that Euclid did not affect that extreme form of accuracy which sub- 
 sequent commentators have attributed to him. The assumption of a 
 circle assumes the existence of an isosceles triangle.] 
 
 17. A diameter of a circle is a line passing through the centre, 
 and terminated both ways by the circumference ; it divides the circle 
 into two equal parts, or (18. semicircles). [Here is a demonstrable 
 theorem positively assumed. The application of one part of the 
 circle to the other (as by revolution of one-half round the diameter) 
 as in the fourth proposition, would prove it.] 
 
 From (19.) to (23.), the definitions are merely nominal. 
 
 24. If there be a triangle having three equal sides, let it be called 
 equilateral. [In this form I give all definitions, the existence of the 
 objects of which is to be established.] 
 
 25. An isosceles triangle is one having two sides equal. 
 
 26. A scalene triangle has the three sides unequal.] This defini- 
 tion is never used.] 
 
 (27.) and (28.) are nominal ; (29.) tacitly refers to the thirty- 
 second proposition ; and from (30.) to (33.), should be written in 
 the manner of (24.) 
 
 (35.) If there be two right lines, which being produced ever so 
 far on the same side never meet, let them be called parallels. And 
 let it be granted, that if two right lines falling upon a third make 
 interior angles together less than two right angles, they are not 
 parallels. [This bone of contention, when reduced to the form in 
 which it is most palpable to the senses, is as follows : Let it be granted 
 that two right lines which meet in a point, are not both parallel to any 
 third line. This assumed, Euclid's axiom follows. For he is able to 
 shew that the one parallel which he afterwards draws, through a point 
 to a given lint, has the property of making the two internal angles 
 
APPENDIX. 87 
 
 equal to right angles : there is but one parallel ; consequently all lines 
 which have not that property are not parallels.] 
 
 It remains to add what I have called the restrictive postulates. 
 I cannot believe that Euclid, who appears to assume veri/ obvious 
 propositions, even when he might prove them, could have intended to 
 require formally the admissions that a straight line may join two 
 points, and may be continued, and that a circle may be drawn with a 
 given centre and radius. If this had been the case, why not assume 
 (Prop. IV.) that two straight lines may be drawn making equal angles 
 with two other straight lines, — a conception more difficult than that 
 a straight line may be drawn. I conceive, therefore, that the meaning 
 of the three assertions commonly called postulates, is as follows : 
 Let it be considered as intended, that no assumption of processes shall 
 be made, except only the drawing of a straight line between two given 
 points, the continuation of any terminated straight line to any in- 
 definite (not given) distance, and the construction of a circle with a 
 given centre and radius. 
 
 THE END. 
 
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