LI BR ARY UNIVERSITY OF CALIFORNIA. GIF^T OK Received C/^ct^ ^iJi^iSg/ . j[ ^a-^jj^-^;/^ yy^. ^4^^ \^? Shelf No. . ^4 V ^ C«* ^C> A TREATISE ON" HIGHEK TEIGONOMETRY. «9- A TEEATISE ON HIGHER TRIGONOMETRY, BY THE REV. J. B. LOCK, M.A. SENIOR FELLOW, ASSISTANT TUTOR AND LECTURER IN MATHEMATICS OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE ; LATE (1872 — 1884) ASSISTANT MASTER AT ETON. ^ 0? THE 'u'HIVBR5IT7] Hont^on: MACMILLAN AND CO. 1884 [All Bights resei-ved.] LC>]5 ©amibritigc : PRINTED BY C. J. CLAY, M.A. & SON, AT THE UNIVERSITY PRESS. PREFACE. The present work is intended to complete the sub- ject of Plane Trigonometry as far as it is usually read in Schools and in the junior Classes at the Universities. The introduction of the hyperbolic sine and cosine is an innovation which seems fully justified by their im- portance in other subjects, and by the simplification effected by their use in the statement of many theorems and formulae. I wish to thank the Master of Gonville and Caius College for his valuable advice and as- sistance and for his permission to insert the proof given in Art. 62. The method of proof employed in Art. 47 was first suggested by Professor Adams. I shall be very grateful for any suggestions or cor- rections from teachers or students. J. B. L. Eton, March, 1884. NOTE. Eeferences to the articles in the Higher Trigonometry are given thus [Art. 100] ; references to the Elementary Trigonometry thus [E. 100]. The Articles and Exercises which are marked with a star should be omitted when the subject is read for the first time. The order of the Chapters may in many cases be varied at the discretion of the teacher; in particular the last two Chapters may often be read as an Appendix to the Elementary Trigo- nometry. Those of the examples which are not original, have been selected from the various Examination Papers which have been set at Cambridge in the Tripos and in the different College Examinations during the past forty years. Various Examination Papers are appended for the information of intending Candi- dates. CONTENTS. CHAP. PAGE I. The Exponential and Logarithmic Series . . 1 II. De Moivre's Theorem 12 III. Results op De Moivre's Theorem .... 22 IV. Proofs without the use op \/( - 1) . . . . 38 V. On the Summation op Trigonometrical Series . 62 VI. Resolution of sin d and cos 6 into Factors . . 74 VII. On the use of V(-1) ^^ VnL The Rule of Proportional Differences ... 91 IX. On Errors in Practical Work 105 X. Examples op the Application of Trigonometry to Geometrical Problems ...... 113 XI. On the Use of Subsidiary Angles to facilitate Numerical Calculation 122 General Miscellaneous Examples .... 132 LIST OF EXAMINATION PAPERS. I, For Admission to Sandhurst. Further Examination. Nov. 1882 ..." 150 n. Cambridge Previous Examination. Dec. 1883 . . 151 III. For Admission to Woolwich. Preliminary. June^ 1882 152 IV. For Admission to Woolwich. Preliminary. Dec. 1882 154 Viu CONTENTS. PAGE V. Mathematical Tripos. * The Three Days.' Jan. 1881 156 VI. Mathematical Tripos, Part I. June, 1882 . . 157 VII. Oxford and CAMSBiDbE Schools Examination. Eton, 1882 158 Vni. Oxford and Cambridge Schools Examination. Eton, 1883 159 IX. Christ's Church, Oxford. Entrance Scholarships, 1883 161 X. Christ's College, Cambridge. Entrance Scholar- ship. 1878 162 XI. St John's College, Cambridge. June Exam. 1879 . 163 XII. St John's College, Cambridge. Minor Scholarship, 1881 165 Xin. Clare, Caius, and King's Colleges. June Exam. 1880 166 XIV. Christ's, Emmanuel, and Sidney Sussex Colleges. June Exam. 1882 168 > 'of TiiE ■ -^e^^ CHAPTER I. The Exponential and Logarithmic Series. 1. The series ^+|T'*'li'"[|"*"[|+®*^-^^^^- is of importance. Hence we prove as follows I. Its value is less than 3. For it is less than ,1111 ^ i.e. less than --Hrl-J. i. e. less than 1+1 + 1. II. Since it is less than 3, the series is convergent. III. Its value is 271828182... This may be easily calculated [See Ex. I. (1)]. IV. It is incommensurable. For suppose that it is commensurable; it can then be put into the form — where m and n are integers. In this case n m n L. ,111,1 1 etc. 2 TRIGONOMETRY. Multiply eacli side of this supposed identity by \n then m\n—l=8i whole number + + ~, ^—7 + etc. ' n+1 {n+l){7i+2) 1 1 But ;, + 7 YV? o\ + ®*^- is a proper fraction; for it is greater than ^ ^^^ ^®^^ than T + 7 Tw + 7 Tv6 + etc., i. e. less than - . n+l (n+iy (n+iy n Hence we have to suppose that m\n-\ (a whole number) = a whole number + a proper fraction; which is absurd. Y. Since the numerical value of the series is incom- mensurable, and we know of no surd or other algebraical expression that is equal to it, it is usual to express its numerical value by the letter e. [cf. E. 28.] EXAMPLES. I. (1) Calculate the vailue of e by taking the first 13 terms of the series. (2) Prove that the first 13 terms of the series will give the value of e correct to 9 significant figures. ,«s -r^ .1 j^ 2 4 6 8 ^ (3) Prove that n + [3 + 1-5 + [7 + ®*c- = ^• /^N T> XT. X 1 . 1 + 2 , 1 + 2 + 3 , 1 + 2 + 3 + 4 , , e (4) Prove that _ + -^-+._|^4- g + etc. = ^. (5) Prove that the series l+i-r + rH +i~^ + ^^^' ^^ conver- LL L± U gent for all values of x. (6) Prove that the series x-\x^ + ly7^-\x^ + Qtc. is con- vergent if X is greater than - 1 and is not greater than 1. ON EXPONENTIAL AND LOGARITHMIC SERIES. 3 2. Expansion of e"" in ascending powers of x. Since (l + 1)" = {(l + i)y always ; and since by the binomial theorem And similarly ., .V , , ■(■-3 ■(■-^(■-1) . (1 +-) = 1 + 1 + ' + , , o +etc. \ nj 1.2 1.2.3 Therefore l'*'*-i-T-* TsTj — **( ccfa; — I xix — ]\os — ~) = !+,;+___ + __ + etc. This statement is arithmetically intelligible and true provided both these series are convergent. They are convergent for all values of w greater than 1. . Therefore they are arithmetically intelligible and true however great n may be. And in the limit, when n is in- finitely increased, the above statement becomes [cf. Art. 8] {11 )' a^ x^ , ^ X x' X^ X* This result is called the Exponential Theorem. 1—2 a. 4 TRIGONOMETBY. 3. To expand a^ in ascending powers of x. We have e* = 1 + p.- + t^^ + t- + etc. Li 1^ E . Let a be any number, and let c — logga, so that e' Then a^={er = e- = e''''''\ Therefore ^ - a; log. a x^ {\o£f.af a;^ (log. a)""* «*= 1 + —~- + ^ .%' ' + — So^-^ + etc. li- li LI The Logarithmic Expansion. 4. In the above expansion put 1 + y for ct, and we obtain (IH- yr = 1 + " '"g-g ^ y) ^- "' i'°g'g ^ y)i' + etc. This may be put into a different form thus : ^ ^ = log, (1 + 2/) + y^ {log, (1 + 2/)}' + terms con- taining higher powers of x = log,(l + ?/) + aj.i?, where i2 is a quantity which is not infinite when a; = 0. The limit of the right-hand side when £c = is log, (1+2/). The limit of the left-hand side may be found thus : i-^^-|l+.2/ + -^-^2/-+ g V ^etc-lj a;-l 2 (a7-l)(jc-2) 3 = 2/+-|2-2/ +^ ^^^— ''2/ +etc. "Uv this, when aj = 0, has for its limit THE LOG ABIT HMIC EXPANSION. O This series is convergent when y is equal to or numeri- cally less than 1. Therefore, when y lies between - 1 and +1 or is equal to 1, log,(l+2/) = 2/-i-2/' + l-2/'-J-2/^+ etc. This is the required Logarithmic Expansion. EXAMPLES. II. (1) Calculate the numerical value of twelve terms of the x^ a^ 1 series \+x+,^-\-nz + etc. when a?= - 1, and show that it = - . Prove the following statements : (3) log,2=| + 3L+i+etc. = l-2i3-ji-,-gl^-etc., and calculate the value of log,, 2 to 2 decimal places. (Result -69...) (^) p + ^ + ^ + etc.=2logey-log.(y + l)-log,(y-l). (5) log,i±|=2{y+^y+i3^+etc.}. ^^^ ^{27^1 "^3(2^+1)3+5(2^+1)^ "^^^^-j =log, -^ = loge (1 +y) - log, y. if (7) 21og,y-log,(y+l)-log.(,y-l) ^ = ^ {2^^1 + 3(2^,2 -1)3 + 5(2^2-1)^ + "'""•j • o TRIGONOMETRY. (8) Prove that log.(„+l)-log.(»-l)=2g +^3 + 5^^+ etc.}. (9) Use the series of Ex. (8) to prove that log, 3 = 1-098612. (10) Use the series of Ex. (7) and the result of Ex. (9) to prove that log, 2 = -693147. (11) Use the result of Ex. (9) and the series of Ex. (6) to prove that log, 10 = 2-302585. On the Calculation of the Table op Logarithms. 5. The series for log^ (1 + ?/) is only convergent provided y is not greater than 1 and is greater than — 1 ; also, unless y is small, the series converges very slowly. It is therefore not a convenient series for the purposes of numerical calculation. We proceed to obtain such a series. 6. Since log, (1 + 2/) = y - J^/' + Iv^ - hf + etc. [Art. 4.] therefore log, {l-y) = -y -^y^ - 12/' - i/ - etc. Hence by subtraction where y must not be numerically greater than 1. Let m and n be positive integers, and let m be > w ; then is less than 1. Put for y in the above result. CALCULATION OF THE TABLE OF LOGARITHMS. i Then log,-=2-^ +^ + etc.>. Let m = n+l in tlie above ; then log,(n + l)-log.» = 2{2^j +3^^3 + eta}. This series is rapidly convergent, and we have thus an easy method for obtaining the logarithms to the base e of successive numbers. Logarithms to the base e are called Napierian Logarithms from their inventor. [E. 227.] The logarithms are calculated thus : Since log,(n+l)-log,r.::=2|^ + 3^J^^^, +etc.|, in this formula put 1 for n. Then since log 1 = 0, we can calculate the value of log^ 2. Next put 2 for n in the above formula, and we can calculate the value of loga 3. And so on. The number of terms of the series which it is necessary to include diminishes as n increases. In this way a Table of the ' Napierian ' Logarithms, of all whole numbers up to any desired magnitude may be determined. On the Calculation of Common Logarithms. 7. We know that log^^w = log,7i - log,l 0. [E. 209, 2 1 0. ] Hence, a table of Napierian Logarithms having been con- structed, from it we take log^lO (=2-3025850...) and cal- culate :; ^ = -43429448 : the table of Common Logs of logelO whole numbers is then formed by multiplying each of the corresponding Napierian Logarithms by -43429448.... TRIGONOMETRY EXAMPLES. III. (1) From the preceding data calculate logio2, logio3, logio9. (2) Find log^ 7 and thence calculate logio 7. (3) If /i=logioe, prove that logio(^ + l)-logio^ __ j 1 I 1 _ ~^ hn+l "^ 3 (271+ 1)3 ■'' 5 {in + 1)5 "^ ®*^' 8. In Art. 2 the limit, when n is infinite, of is assumed to be ,— . \t This is clear as long as r is not comparable with n. That it is true for all values of r may be proved by induction, thus : Assume that the above expression = — + i?, where ^ is a quantity whose limit is zero when n is infinite; multiply each side by the factor \x — ) r , and we obtain x.R r+1 r r+1 •1- \r+l 1 n' . 1 r x^ In this, when n= co . R=0, - = Q, and =- 7- is finite n r+ 1 [r however great r may be. This proves the proposition. ON CERTAIN LIMITS. 9 9. To prove that the limit of Icos-j, when n is in- finitely increased is 1. Since cos' - = 1 - sin^ - ; nf.-a l..a l.-a,l rAi.iT = --^\ sin* - + - sm* - + J, sm^ - etc. \ . [Art. 4.] [Art. 4, since sin' - is less than 1.] This series is less than the g. p. . „ a . aOl ' rO. Sin"* - + sm - + sm - + etc., n 71 n . J a Sin - i. e. less than , that is, than tan* - : 1 - sin - n .*. log f cos- j is less than - ^ •( n tan* - K 2 (a** ^j ^ 2 \ ' that is less than -^l'-^ tan* - - — . a The limit of ~ tan* - when w = oo is 1. [E. 290.1 The limit of — when w = oo is 0. n 71 = 00 .•. log (cos - ) = when and therefore the limit of f cos - j is 1. q. e. d. 10 TRIGONOMETRY, / sin ~ \ 10. The limit qfl — — . I when n is infinitely increased is 1. n 1 We have (E. 289) 1, —. — ^r, -^ in ascending order of ^ ' sm^ cos^ ° magnitude, when 6 is less than 90". Therefore also (1)", , a X * n \ 1 . a /' / aV \ sm— / I cos- ) ^ n^ \ n/ are in ascending order of magnitude. Now let n be infinitely increased and then [ — — J lies a sm- n' between 1 and a quantity whose limit, by Art. 9, is 1. q. e. d. *EXAMPLES. IV. (1) Prove that the Hmit of (cos- j when n is infinitely in- ised is 1. (2) Prove that the limit of ( — ^ J when 6 is infinitely di- ■'2 tt2 creased is 1. (2) Pro minished is 1 (3) Prove that the limit of ( cos - j is e 2 -vvrhen n is infi- nitely increased. (4) Prove that the limit of ( cos - J is zero when n is infi- nitely increased. (5) Prove that the limit of (cos 0)^"" wher6 m is an integer, _a2 and 6 is infinitely diminished is zero, e 2, or 1 according as m is greater, equal to, or less than 2. MISCELLANEOUS EXAMPLES. V. H *MISCELLANEOUS EXAMPLES. V. (1) Since a^={l + (a- 1)}*, prove by expanding the right- hand side that a*= 1 ^- Aio: + ^2^2 + j^^ + etc., where A^ = {a-l)-j^{a-lf + ^ia-lf. (2) Since a^+^=a'=xa^, expand a^+^ and a^ by the theorem of Ex. 1 and by equating coefl&cients of ^, prove that A^ + 2 J jy + 3^3y2 _,_ etc. = Aia''. Expand a^, and by equating the coefficients of the various powers of y find A^, A^, etc. in terms of A^. Result. a^= 1 + A^x + -, - + - .- - etc. 1 (3) Show that aAi in the last example is e. Hence by Ex. 1 prove that log„a=(a-l)-^(a-l)2 + J(a-l)3_etc. (4) Prove that ill log^ ?i = m {(n"* - 1) - -i (71"* - 1 }2 + ^ (n*" - 1 )3 - etc.}. \^ Hence, having given that 102^-1 = -000000000536112, prove that log^ 10 = 2-30258. (5) Prove that log (?i + o?) - log ?i = ;i | - - ^^ + ^^3 - etc.| . Hence if n be a number greater than 10000 and d a number less than 1, prove that .— 2^^ ~ — ^ — = -^ to a sufficient ap- ^ log(7i + a )-log7i d' ^ proximation for all practical purposes. (6) Prove that . /sin^X . e , , e ^ log \-Q ) =log cos - + log cos Yi + log cos 23 + etc. (7) Prove that log sin 2a + log cot a = cos 2a - J cos^ 2a + J cos^ 2a - etc. ( 12) CHAPTER II. De Moivre's Theorem. 11. Def. J-^ is a symbolical expression, whose square is - 1 , which is capable of obeying the ordinary laws of Algebra. Since >/ — 1 obeys the laws of Algebra J -a^ = J -I X a^ = aj - 1. The student must observe that such an equation as A +B J — 1 =a + b J —I can only be true when A^a and We shall often use the letter i as an abbreviation for 12. De Moivre's Theorem. Whatever he the value ofn positive or negative, integral or fractional, cosna + J —I sinna is one of the values of (cos a + ^ — 1 sin a)". I. When w is a positive integer. Consider the product (cos a + ^ - 1 sin a) X (cos /3 + J -Isin /3). DE MOIVBE'S THEOBEM. 13 It is equal to cos a . cos )8 - sin a . sin /3 + ^ — 1 (cos a . sin y8 + sin a . cos ^). That is to cos (a + /3) + ^ - 1 sin (a + P). Similarly the product {cos (a +/3) + J- i sin (a + P)} x {cos y + ^ - 1 sin y} is equal to cos (a + /? + y) + ^ - 1 sin (a + )8 + y). Proceeding in this way we obtain that the product of any number n of factors, each of the form cos a + ^ - 1 sin a is equal to cos(a+j8+y+ ...71 terms) + ^-1 sin(a+/3 + y + ... w terms). In this result let ^ = y = etc. = a, and we have that (cos a + ^ - 1 sin a)" = cos Tia + ^ — 1 sin na. Thus, when n is a positive integer, De Moivre's Theorem is true. II. When 01 is a negative integer. Let 9i = - m. Then ni is a positive integer. And (cos a + V - 1 sin a)" = (cos a + ^ - 1 sin a)"*" = - i== -= 1= [By I.] (cos a + ^ - 1 sm a)"* cos ma + ^ — 1 sin ma 1 cos wa — ^— 1 sinwa = ^ X ^. cos ma + jj — 1 sm ma cos ma - ^ -1 sin ma _ cos ma ~ J —1 sin ma cos^ 7/ia + sin* ma Therefore (cos a + ^ - 1 sin a)" - cos ona - ^ — 1 sin ma = cos (— m) a + y - 1 sin (- m)a = cos na + J -1 sin na. Thus De Moivre's Theorem is true when n is a negative integer. 14 TRIGONOMETBY, III. When n is a fraction, positive or negative. Let 7i = - , where p and q are integers. Now (cos 13 + J -I sin py = C0& q/3 + J - I sin qp. [By I. and II.] Therefore taking the q*^ root of both sides cos )8 + ^ — 1 sin y8 is one of the values of (cos qjS + J — I sin q^y , or, writing a for q/3, cos - + ^ - 1 sin - is one of the values of (cos a + ^ - 1 sin a)' . Therefore (cos - + y- 1 sin - ) , that is cos — + j'- 1 sin — [By I.' p is one of the values of (cos a 4-;^ - 1 sin a)*. Thus the theorem is completely established. EXAMPLES. VI. (1) If ^ stand for cos 2a +^ sin 2a, and B, C, D for similar expressions in terms of /3, y, b, prove that uib + CD = 2cos(a + ^-y-S){cos(a+^ + y + S) + ^sin(a + ^ + •y + 5)}. (2) With the notation of Ex. 1, prove that 1 _ sin(a + i3 + y+&)-^cos(a + ^ + y + 5) AB-(JJJ~ 2sin(a+i3-y-fi) (3) With the same notation, prove that {A-B){C- D) =^ - 4 sin (a-iS) sin (y-5) (cos (a + ^+y + 5) - i sin (a+^+y + S)}. EXAMPLES. VI. 15 (4) With the same notation prove that 1 cos (a+/3+Y+^) -^sin (g+^ +y + d) (A+B)iC+I))~ 4 cos (a -^) cos (y^) • (5) Prove that cos (a + jS + y...) + «^* sin (a + i3 + y+ ...) :cos a.cos/3. cosy ... {(l + i'tan a)(l + ztan/3)(l + itan y) }. //.\ -n .IX sm(a + i3 + y+... ) (6) Prove that ^ ^^-^^ ^ = Si-So + s.-s.4- etc. ^ cosa. cos /3. cosy ... ^ ^ s 4 where s^ stands for the sum of tan a + tan /9 4- tan y-»- etc., Sg stands for the sum of the products of these tangents three at a time, and so on. (7) Prove that tan(a + ^ + y + ...)= V^'"^^'"!^''' where s„ ^ ^ \ r- / / i_^2 + «4-etc. ^' 8.2,... are defined in Ex. 6. (8) Write down the last term of the numerator of the frac- tion in Ex. 7, (i) when n is even, (ii) when n is odd. 13. It is known from the Theory of Equations that there are q different values of jc, and no more, which satisfy the equation a;' = a, where a is real or of the form A + J{-1)B. We can prove that we may obtain q different values by de Moivre's theorem, and no more. 14. The expression cos ^ + ^ — 1 sin ^ is unaltered if for we put {6 + 2r7r), where r is an integer. 0+2r7r f — - . ^4-2r7r Hence cos + j - 1 sm , which is one of the values of ji^ {cos (e + 2r7r) + 7 - 1 sin (^ + 2r7r)}% is one of the values of 1 ^ ^^'^ ^^^^"^^^^^^^^ {cos e + J~^ sin ey. 7 ' v ^^ ™^' ^ |tJHI7ER3lTy 16 TRIGONOMETRY. 15. Bt/ giving to r the values 0, 1, 2 ... q- 1 we obtain q different values of cos — f- J— 1 sin ~ — ; and what- ever integral value we give to r, we cannot obtain more than q different values. re- Take a circle, centre and radius OR. Let ROP^ be the angle - . Divide the whole circumference of the circle q starting from P^, into q equal arcs, P^P^, P^P^^ ^2^3' ^*^- Then each of the angles PfiP^, PfiP^, ^P^z^ •"•• etc. is equal to — ; and in describing the angle ( - + — J, the volving line, starting from OR, turns first into the position OPq and then on through r of the angles PfiP^ , PfiP^, etc. Hence, whatever integral value r may have, OP must stop in one of the q positions OP^, OP^, OP^, etc. and it can stop in no other position. mt £ XI, • ^ + 2^7r , / — . . e + 2r7r Therefore the expression cos + ^ — 1 sm cannot have more than q difierent values. Also no two of these q positions are equi-sinal and at the same time equi-cosinal. EXAMPLES, VII. 17 Therefore this expression has q different values. Also by giving to r the values 0, 1, 2, ... (5'- 1), in suc- cession, OP will be made to stop in each of the q possible positions in turn. Therefore by giving to r the values 0, 1, 2... (5-- 1) in succession, we obtain tlie q different values of the above expression, q. e. d. 1 6. An expression of the form A+ J -I B, where A and B are arithmetical quantities, can always be put into the form r {cos a + a/ - 1 sin a}. Let A = r cos a, B = r sin a. Then A^ + B^ = r^ (cos^a + sin^'a) = r^ and — = = tan a ; ^ ^ reosa ' whence a and r can always be found. [E. 116.] It will be convenient to take r positive : then we must take a in that quadrant which makes cos a the same sign as A. [Cf. E. 148, 149.] Example 1. Express 1 + ^J -I in the form r {cos a + sf-^^ sin a). Here r8ina = l and rcosa=l, .-. r- = 2, tana = l. 1+ ^/^^=V2'{co8450 + V'^8in450}. Example 2. Express ( - a) in the form r {cos a + i sin a). Here rcosa=-a, r8ina=0, .*. r^=a'^, a = (2n + l)7r. .', — a=a{cos(2/i+l) ir+isin(2n + l)7r}, where n is an integer. EXAMPLES. VII. (1) Express 1-^^, V3 + \/^, l+^/sV^ each in the form r (cos a + J^ sin a). (2) Find all the values of (i) (4/^2 + 4 ^2^^)^ (ii) (4V3 + 4V~1)^ (iii) (n/s + V^)^ (3) Find all the values of (i) 1^ (ii) 32^, (iii) 27^. L. 2 18 TBIGONOMETRY. 17. If we express any arithmetical quantity a in the form of a De Moivre's expression we obtain a (cos 2r7r + ^ - 1 sin 2r7r), i. e. the product of a by the De Moivre's expression for unity. Therefore the n nth. roots of any arithmetical quantity a are found by multiplying the arithmetical nth. root of a by each of the n nth roots of unity in succession. The nth roots of unity are therefore important, and are discussed in the following examples. Example 1. Solve the equation x"^- 1 = 0. In other words, Jind all the values o/jyi, or, find the factors o/x"- 1. Since cos2r7r + ,s/-lsin2r7r = l. It follows that a:"=cos2r7r+\/-lsin2r7r, where r is an integer, and therefore 2r7r / — - . 2r7r a; = cos h A^-l sm , w ' n This result is best discussed by means of a figure. _ -P, ,, P. n odd I. Let n be a whole number. 27r Let the angle ROPj^ = —. On the circumference of the circle centre and radius OR, measure off arcs P-^P^, -Pg^s' ^*<^- ^^^h equal to RP^. Then since n . R0Pi = 2ir, n of these arcs will occupy the whole circumference, and OP^ wUl coincide with OR. Also, if r be a DE MOIVBE'S THEOREM. 19 2rir whole number, in describing the angle — the revolving line, starting from OjB, must stop in one of the positions OP^^, OP.2, etc. and in no other. No two of these positions are both equi-sinal and equi-cosinal. mi .1 • 2r7r /— — . 2r7r Thus the expression cos hij -Isin has n different values, and no more ; and these values can be found by giving r in succession the values 0, 1, 2...n- 1. When r =0, a; = 1 : when n is odd this is the only arithmetical value ; when 71 is even, there are two arithmetical values; for let n=2m, then when r=m, x= -1. In any case, the angles ROPi and ROP^-i are equi-cosinal, and sin R OP^ = - sin R 0P^_-^ . The same thing is true of ROP^ , JBOP„_j , and of ROP^y ROP^-^, and so on. „ 27r /— - . 2ir , 2-ir /^ . 2v Hence x - cos w - 1 sm — , and x - cos v J -Ism— , n ^ n n ^ n are factors of x" - 1. Their product is / 2ir\' . » 2t . „ _ 2t , I X - cos — I + sm^ — , I.e. a?-2x cos h 1. \ n J n n Hence we obtain that m being a whole number a;^"» -l = (x2 -1)^x2 -2a; cos l^ + l") U2-2a;cos ^^ + 1 V.-m quad- ratic factors, a^J"H-i - 1 = (x - 1) f a;2 - 2x cos 7,-^ + 1 Vx'! - 2a: cos -^ + 1 V . . m quadratic factors. [Note. Let a = cos 1- ^A^ sin — ; XI 2r7r / — - . 2rTr then cos h J -1 sm — =a^, n n Therefore the roots of the equation x" - 1=0 are 1, a, a^, a^... a""^.] P ^ II. When w is a fraction in its lowest terms = - . Then x' - 1 = 0, or xP-l9 = 0, or xp-1=0. This is the same as the case already discussed. 2—2 20 TRIGONOMETRY. in. When n is incommensurable (e.g. .^/2). Then as before 2nr I — - . 2rir fl;=eos vJ -1 sm — , n ^ n 2rT In this case, r being an integer and n incommensurable, can n never be an exact multiple of 2ir. The angles will therefore not recur geometrically and the equation will have one arithmetical root, viz. 1, and an unlimited number of symbolical roots. EXAMPLES. VIII. (1) Find the roots of the equation ^^-1 = 0. (2) Find the quadratic factors of ^ - 1. (3) Write down the quadratic factors of a^—1. (4) Solve the equation ^^ — 1 = 0. (5) Give the general quadratic factor of ^^o _ q^2o^ Find all the values of v^l. (6) Example 2. Here To find the Quadratic factors o/x" + l=0. 7r + 2r7r . . x + 2r7r X = cos 1- 1 sin ■ , n even n odd 2t In the figure ROPq=- , PoOPi = and n angles each equal to PqOPi make up 2ir; OR bisects P^OP^-i. Also ROPq and i20P„_i are equi-cosinal, while sin i?OPo= —sin i?OP„_i, the same relation holds good for any two angles equi-distant from OR, DE MOIVRKS THEOREM. 21 f Tr + 2rir . . •7r + 2r7r\ , / 7r + 2r7r . . 7r + 2r7r\ . •. [ X- cos ^ sin ) , and x - cos 1- 1 sm ) V n n J \ n n J are factors of a;™+l. Therefore their product viz. ix^- 2x cos "^ ^+ 1 j is the form of the general quadratic factor of a;"+ 1. When n is even and = 2 w there are m such factors. When n is odd and =2w + l there are w such factors; the remaining factor is a; + 1, as is clear from the figure. EXAMPLES. IX. (1) Find the roots of the equation x^-\-\=0, and write down the quadratic factors of ^■* + l. (2) Write down the quadratic factors of .r^ + 1. (3) Write down the general quadratic factor of ^r^^ + 1 =0. (4) Find all the values of v'^. (5) Find the factors ^r^^ + l. (6) Find a general expression for all the values of ^ -\. * MISCELLANEOUS EXAMPLES. X. (1) Prove that sin (ai + Ga + oj . . . 71 terms) =5iC„_i - SgC^-a + Sj^^n-s - etc., where SjC^-r stands for the sum of the products of the sines taken r together each multiplied by the product of the remaining n-r cosines. (2) With the notation of Ex. 1, prove that cos (ai + 02 + cg . . . w terms) = c„ - Cn-^Si + Cn-iS^ - etc. (3) Write down the expansion of sin(a + /3+y + S + e) and of cos(a+i3 + y + 5 + e)- 22 TRIGONOMETRY. (4) Prove that in tiie series of expressions formed by giving to rm ( cos 1- 1 sm J the values 0, 1, 2, Z ...{q-1) in succession, the product of any two equidistant from the beginning and the end is constant. 15 (5) One value of (x/s + sj^lf is - 2^ {J^ + 1). (6) From the identity {x -h){x-c) (x- c) {x - a) (x -a){x-b)_ (^^K^^c) "^ {b-c){h-a) "^ 'lc-a){c-h) ~ ' deduce by assuming ^=cos2^ + zsin 2^, and corresponding as- i.- f I. A xv i. sin (<9 - jS) sin (<9 - y) . „ ^^ v, sumptions for a. o and c that —. — 7 ^^~—'. — ) ~ sin 2 (^ - a) -f ^ ' sm (a - /3) sm (a - y) ^ ^ two similar expressions =0. (7) Prove that the n nth roots of unity form a series in g. p. CHAPTEE III. Results of De Moivre's Theorem. 18. We proceed to deduce many important results from De Moivre's Theorem. We shall generally in this chapter write iiov J -I. RESULTS OF DE MOIVRE'S THEOREM. 23 19. By Art. 12, when n is an integer we have cos 7i6 + i sin n6 = (cos ^ + i sin Oy ; Expand the right-hand side of this identity by the binomial theorem, remembering that i^ = —l and that i'* = + 1. Equate the real part of the result to cos nO. This gives us cos nO = cos"^ - '^^^^'^ cos"-^^ . sin'^ + — 5i ^-^,- — '-^ ^ cos *$ . sin*^ - etc. II Equate the imaginary part to i sin nO, This gives us sin nO^n cos-'O . sin ^ - n{n-^l)(n-2) ^^^„_,^ ^ ^^,^ [£ + __v ^^ ^ ^-^ ' cos" ^0 . sin*^ - etc. [^ 20. In the above n is a positive integer, and the last terms in the series for cos nO and for sin nO will be different according as n is even or odd. EXAMPLES. XI. Prove the following statements : (1) sin 4^=4 cos^^. sin ^-4 cos ^. sin^^. (2) cos 40 = cos* ^ - 6 cos2 $ . sin^B + sin* 6. (3) The last term in the expansion of cos 10^ is - sin^^^. (4) The last term in the expansion of sin 12$ is -12cos^.sinii^. (5) When 71 is even the last term in the expansion of cos n9 is (-l)^sin"^. (6) When n is odd the last term in the expansion of cos nd is n-l (-1)3 ncosd.sin«-i^. 24 TBIGONOMETRY, Exponential Values of Sine and Cosine. 21. By De Moivre's Theorem, when n is any commen- surable number, and x any angle, (cos nx + i sin nx) is a value of (cos x + i sin x)". For X put the unit of angular measurement ; then (cos 71 + i sin n) is a value of (cos 1 + i sin 1)". Let k stand for (cos 1 +i sin 1), then (cos n + i sin n) is a value of ^", where h is independent of 7^. Whatever other values (cos 1 + i sin 1 )" may have, in what follows we shall only use the value (cos n + i sin n). 22. This important result is a symbolical statement of the fact that expressions of the form cosn + ismn are combined "by the laws of indices. 23. Let the unit of angle be a radian. [E. 59.] Then since cos,6 + i sin 6 = k^y [Art. 21.] and consequently cos ^ — 'i sin ^ = ^" ^, where k is independent of 6, .-. 2isiRe = k^-k-^, = 2 1 6 logfi +j^e' (log^ky + etc. I . [Art. 3.] Hence i ~^~ = log^^ "^ T^ ^^ i^^S^W + ®*^- ^log,k + 6\B; where E is finite for all values of (since sin $ is always less than 0, and .*. log^^b is finite). Let 6 be infinitely diminished. Then, since is the circular measure of the angle, the limit of —^ is 1. [E. 290.] Also the limit of the right-hand side is logjc. EXPONENTIAL VALUES OF SINE AND COSINE. 25 Hence i = logjc^ or, k = e . Therefore, when 6 is the circular measure of the angle, cos^ + 7^sin^ = e^^^. 24. Since cos ^ + z sin ^ = e* and cos^ — isin^ = e~* ; . ♦. 2 cos (9 = e*^ + e~^^ ; and 21 sin 6 - e'^ - e~'\ J0 , „—i9 J9 —19 TT e 4- e . e —e Hence and 2i are exponential values of the cosine and sine respectively, when the angle is expressed in circular measure. These results may be applied to prove any general formula in Elementary Trigonometry. Example. Prove -^^^- -**- = tan a. 1 + cos 2a 2i sin 2a _ e^fa - g-2i« _ (e»« + «-*«) («*« _ g-**) 2 + 2 cos 2a ~ 2Te^^*+e-^ ~ (eta + g^taji gja_g-ta 2iBina .^ = itana. q.e.d. gta^g-ia 2 COS EXAMPLES. XII. Use the exponential values of the sine and cosine to prove the following: (1) cos2a + sin2a=l. (2) cos 2a = cos^ a - sin^ a. (3) sin ^ = - sin (-B). (4) cos ^ = cos ( - 0). (5) cos (a + ^) . cos (a - i3) = cos^ a - siu2 ^ = cos^/S - sin2 a. (6) cos3^=4cos3^-3cos^. (7) sin 3^=3 sin ^-4 sin'^. (8) 2cosna.cosa = cos(?i + l)a + cos(n-l)a. (9) 2 sin 7ia{-l) 22 sin2 a = 2 sin (71 + 2) a — 4 sinna + 2 sin (n- 2) a. (10) 2 cos Tia (-1) 22 sin2 a = 2 cos (?i + 2) a - 4 cos na + 2 cos (n - 2) a. 26 TRIGONOMETRY. 25. The results 2 cos ^ = e^' + e-^\ 2i sia = e'' - e'^' may be used to simplify expressions containing ^ — 1 . Example 1. Reduce cos (a + i/3) to the form A + iB. 2 cos (a + ijS) = e''*-^ + e-**+^ = e-^ . e*'* + e^ . e-'** = e~^ (cos a + 1 sin a) + e^ (cos a - i sin a) = cos a (e^ + e""^) - i sin a (e^ - e~^). This is in the required form. Example 2. Express log (a + ib) in the form A + iB. Let a+i6=r (cos a + i sin a). Then (Art. 16), r^=a^+b% t&na=^^. Thus, log (a + i6) == log {r (cos a + i sin a)} = log r + log 6*"* =log r + ta= J log (a2 + 62J ^^ tan-i - . This is in the required form. Example 3. Reduce (a + ib)"+^^ to the form A+iB. Let a + i& = r (cos 7 + i sin 7). Then r2 = a2 + &2^ tan 7 =-. And (a + 1&)«+^'^ = r^+^P . e^v (*+^'^) =r^.r'^P.e^y''.e-^y = r« . e-^y . e^^ ^°Sr, g/ya j-por r=elog»*] — yttg-^y I cos (/3 log r + a7) + i sin (j3 log r + a7) } . This is in the required form. EXAMPLES. XIII. 27 EXAMPLES. XIII. Prove the following statements : (1 ) cos (a + i^) + i sin (a + ^/3) = e~^ (cos a + i sin a). (2) 2 sin (a + i^) = {eP + e-P ) sin a + 1 (e^ - e'P ) cos a. (3) cos (a + 1^) - 2 sin (a + 1/3) = eP (cos a - z sin a). (4) 4 cos (a + i^) . cos (a - i^) = e^^ + e"^^ + 2 cos 2a. (5) 4 sin (a + 1/3) . cos (a -i^) = 2 sin 2a + 1 (e^^ - e'^^). 2 (e^ + e-^ ) cos a + 2i {eP - e'P ) sin a (6) sec (a +1/3): e2^ + 2cos2a + e-2^ /^\ + / . -ON 2sin2a + i(e2^-e-2^) (7) tan(a+e^)= ^,^^^^^;^^^^_,/ . (8) (V^)^'-^=ri^. (9) Express a^^^ in the form ^ + iB. (10) Express (a + iby in the form A + iB. (11) log^*=2aan->*. (12) log?!^4^=2itan-i|cotx5^^i:rl. ^ ' °sin(^-ty) ( e^+e-'J /, ^x 1 cos (x -iy) ^ , , , r, e* - e~^^ (13) log — )-— /( = 2i tan-i -^tan^-— — „^ . (14) logsin2(^+zy) =log (e2y - 2 cos 2a? + e-2y) - 2i tan-i -[cot x ^-^ . (15) logcos2(a + i/3) =log(«2^^2cos2a + e'^^)-2itan-ijtana^— ^l. (16) a^/ * = a^°'^^" {cos (sin 45^ log a) + i sin (sin 450 log a)}. (17) Express a>^* in the form A + iB. (18) Express {a + ib + c' )*+'*^ in the form A + iB. 28 TRIGONOMETRY. 26. Since COS ^ + i sin ^ = e**^ = 1 + i^ - ^ - 7:5- + n + etc., [2 |_3 |4 we obtain by equating the real and imaginary parts sm^ = ^-7^+--- etc. |3 1^ These results are very important. In the next chapter will be found a proof independent of v - 1 and a collection of examples. e*^ IS A Periodic Function. 27. cos 6 and sin 0, and therefore also cos 6 + i sin ^, repeat their values every time 6 is increased by 27r. There- fore e*^ also repeats its values every time is increased by 2'jr. When a function of 6 repeats every possible value in exactly the same order each time 6 is increased by a certain value X, it is said to be periodic, and X is called its period. If we are given a particular value of such a function of 0, we can find an unlimited number of values for (each of the form a + nX, where t* is a whole number,) for each of which this function will have that given value. Also, as 6 changes from to 27r, none of the values of e** are repeated. In other words, there are no two values of 6 in the same period for which e^^ has equal values. Example. Given that tan^ = a, and that a is one angle whose tangent is a, then we know that ^ = a + mr, where w is a whole number. 29 28. Hence, if e^'^ = e% we know that 6 and a differ by some multiple of 27r, i.e. that ^ = a + 2mr, where 7^ is a whole number, and the value of n cannot be decided without some further datum. Example. Since 2i sin ^ = e^^ - e~^, and sin tt = 0, therefore e^^ - e~^'^ = or e*'^ - e~*\ This means that ir and - ir are two values of 6 for which the periodic function e^^ has the same value. And since the period is 1x, ir and - IT should differ by 2mr. In this case n is clearly 1. 29. The same thing may be stated thus : since cos a + i sin a = cos (a + 2r7r) + i sin (a + 2r7r), Therefore 62^^*^ _ i ^^s is also e^ddent since cos 2r7r + i sin 2r7r =1). Hence unity has one real logarithm, viz. 0, and also an unlimited number of symbolical logarithms each equal to 2ir7r, where r is some integer. 30. Again, a = axl=ax e^'"'*'^ = elog'«+2«>T . Hence every real quantity a has one arithmetical logarithm, and also an unlimited number of ssrmbolical logarithms, which differ by 2ir7r, where r is an integer. These symbolical logarithms do not interfere in any way with the theory of arithmetical logs explained in Chapter 1. Example. Prove that the equation sin 6 = has no symbolical roots. Suppose that sin(a+^~lj8) = then gm-^_g- + x'" If = (a;" + X-") + n (oj""^ + x-'-'') + ^ii^J) («;''-* + x-'"''') + etc. = 2cosw^ + w.2cos(7i-2)^ + ^?^-^^^^ Also (2i sin ^)" = («-«;-')". First let w be even. Then the expansion of {x - a;"*)" is JC- + «-" - 71 ix--' + x-'-'') + —^ («^""* + x~""'') - etc. .\(2isin^)"=2cosw0-w2cos(7i-2)^ + ^^^^^^ — etc. Next let n he odd. Then the expansion of (x-x'^ is a;" _ a;-» _ n (X-' - x-^"-'^) + ^^^-^^^ (a^""' - ^"^""'') - etc. . •. (2i sin ^)'' = 2i sin w^ - w . 2i sin (w - 2)^ + !L(!^zi) 2isin (n- 4)^ -etc. I? L. 3 34 TRIGONOMETRY. Whence dividing by i and putting i^ = - 1 we have n-l (-1)2 2"^in"^-2sin•/^(9-w.2sin(7^-2)^-etc. It must be noticed that when the last term is indepen- dent of 6 J the factor 2 is omitted. EXAMPLES. XVI. Prove that (1) 128cos8^=cos8^ + 8cos6^ + 28cos 4^ + 56 cos 2^ + 35. (2) 64 cos'' ^=cos 7^+7 cos 5^ + 21 cos 3^+35 cos ^. (3) 64 sin7 ^=sin 7^-7 sin 5^ + 21 sin 3^ - 35 sin 6. (4) 512 sinio ^ = cos 10^ - 10 cos 8^ + 45 cos 6^- 120 cos 4^ +210 cos 2(9 -126. 38. To resolve x^" — 2x" cos n^ + 1 into factors, when n is a whole number. Since x^" - 2cc" cos n^ + 1 = (aj" - cos nOf + sin^ nO = {(«" - cos nB) + i sin nO] {(as" - cos nO) - i sin nO} = {x" - (cos nO-i sin nO)} {x" - (cos n6 + i sin nO)] = {x" - (cos e-i sin ^)"} {«" - (cos ^ + i sin ^)''}, therefore x — (cos ^ - * sin 6) and a; — (cos + i sin ^) are factors of x^" — 2x" cos nO + 1. P. FACTORS OF x'^^-2x^co8ne + h 35 Therefore also their product, i.e. cc^ — 2ajcos ^+ 1, is a factor of a;^" - 202" cos nO + 1. And since cos n$ is unaltered if for $ we write 6 + it follows that x^ — 2a3 cos ( ^ + — J + 1 is also a factor. In the above figure let ROP^^O^ and let the whole circumference, starting from P^, be divided into n equal arcs P^Pj , PjP^ . . . P„_i P^. Hence, whatever be the integral value of r, the angle + is represented by one of the angles ROP^, POP^y etc. Hence in general there are n different values and no more of cos ( ^ + j . [The exceptions are (i) when one of the points P^ coincides with R, (ii) when R bisects one of the arcs P^Py_i; i.e. (i) when nd = 2nc, (ii) when nd={2r+ 1) t, and in these cases x^** - 2x^ cos nd + 1 reduces (i) to the form («« - 1)^=0, (ii) to the form (a;" + 1)3= ; the factors of these forms have been discussed on pp. 18, 19.] And the n different values are found by giving to r the values 0, 1, 2 ... (71- 1) in succession. Hence the n quadratic factors of a^" - 2x'* cos nO+l are (a' - 2ic cos ^ + 1) ja;' - 2a; cos ^^ + ?^) + l| X ... X |a;»-2a;cos(^ + ^ 27r) + l|. EXAMPLES. XVII. Solve the following equations : (1) ;r8-2^*cos60<' + l=0. (2) ^I0-2^cosl00+ 1=0. (3) a:i2-2x6cos§7r + l=0. (4) x^^+^3x^+l=0. (5) Write down the factors of ^ - 2^" ^" cos a + f\ 3-2 36 TRIGONOMETRY, ^MISCELLANEOUS EXAMPLES. XVIII. (1) Prove that if cos 6 and sin 6 be defined by the equations 2 cos B = a^ + oT^, 2i sin Q—ct- oT^, then sin 6 and cos 6 satisfy the fundamental conditions cos2^ + sin2^=l, cos^ = cos(- ^), sin ^= -sin (-^). (2) Prove that if a degree is the unit of angular measure- ment 2cos^=^+^~* where k^^=e^^. (3) Assuming De Moivre's theorem, prove that *sin^'<'-(l-cosx<') 1 7 . ^ n 7N9 . i. ^ ^^=log^+ - (log ^)2 + etc., where k = cos 1* + 1 sin 1^ = e^^^ . (4) Prove that if two right angles be taken as the unit of angle, the exponential values of cos ^ and sin ^ are ^ (e^^^+e~^^^) and -Ji(«'''^-e-*^''). (5) Assuming that e^*''*=l where r is an integer, prove that e*^ is a periodic function of 0. (6) Assuming that e^^ is a periodic function of ^, and that the period is 27r, prove that e^'^=l where r is an integer. (7) Prove that {ib + c^)* = r« (cos ad + i sin a6) where o -. 7o «7 • T X /I ft + sinlogc j-2 = i + 52 + 26smlogc, tan^= z — ^- , cos log c (8) If log (1 + cos 2^ + 1 sin 2^) = ^ + iB, then ^ = log 2 + log (cos 6), (9) Prove that e* - 2 cos ^+e~* =4 sin \{6 + ix) sin \{6 - ix). (10) If cos-i (a + 2/3) = J. + 1^, prove that a2 ^' , ^ «' , ^^ 1 cos'M sin2J. ^' (eP +e-^)2 (^^ _ e-^)2 MISCELLANEOUS EXAMPLES. XVIII. ^^ (11) Prove that log, ( - 1) =i (2n + 1) or. (12) Prove that log {x + ly) = ^ log {x^ + ■if) + i tan~i ^ . Hence prove that , X sin 6 -/i^-^A -3?^-«/, tan~^ — 7= ^ sin ^ - — sm 2^ + — sin 3^ + l+a;cos6^ 2 3 (13) If (a) is such a function of a that 0(a)x0(^) = (^(a+i3) for all values of a and /S, prove that <^(a) = {0(l)}<» for all rational values of a. Show that cos a-\-i sin a is a form of (^ (a) which satisfies the preceding equation, and deduce De Moivre's theorem. (14) Prove that tan-i (cos a + 1 sin a) = {n + J) tt + 1 log (tan \ a). (15) Prove that 1=4 tan-i \ - tan-i 7^+ tan-i J^, and apply the result to find the value of tt to 5 places of decimals. (16) Find the number of radians in the least angle whose tangent is ^^j ; also the number of degrees in the least angle whose tangent is 10. (17) Prove that the general value of e'* is cos (1 + 2r7r) 6-\-i sin (1 + 2r7r) 6 where r is an integer. (18) Defining cmO as the real part and *sin^ as the imaginary part of e**, prove cos^ = cos(-^), sin^= -sin(-^), sin(^ + 0) = sin 6 cos + cos 6 sin 0. ( 38 ) CHAPTER IV. Proofs without the use of J -^. 39. In this Chapter we shall give proofs of most of the preceding results by methods which do not involve the use of ^—1. The student must not on this account suppose that the validity of results obtained by the aid of ^ - 1 is doubtful. We shall make some remarks on this point later on. 40. To prove, when n is a positive integer, -\ — ^ cos ^a. sma n{n-l)(n-2){n~3) n(n-l) „_„ . 2 cos na = cos a ^ — ^r—^ cos a. sin a 1 . Z cos* *a. sin*a-etc., 1.2.3.4 and 7i(n-l)(n-2) „_3 sm na=n cos a . sm a ^ — - — ^-^^ cos a . sin a -¥— 1 » o \ X -COS ^a.sm^a-etc. 1.2.3.4.5 PROOFS WITHOUT THE USE OF */^. 39 These formulse may be proved by induction, thus : Assuming that the above statements are true for a certain value o£ n, we can prove that they must also be true when ?i + 1 is written for n. Since cos (n + l)a = cos na . cos a - sin na . sin a [E. 154.] ; in this, substitute for cos na and sin na those values given above, and we obtain ix «+i (n(n-l) ) -_, . , cos [n + l) a = cos "^ a — ■( y n + ^ f cos a . sm*a ( n(n-l){n-2){n-d) ■*■! 1.2.3.4 n(n-l)(n-2)) ,_3 . 4 + — ^^-:j — ^^ ^ V cos" ' . sin* a - etc. The coefficient of cos""''a. sin'^'a is ^;-y f7^(n-l)...(n-r4■l)(7^-r) n(n-l)...(n-r + l) \ ^"•^^ \ (r+J \r J ^ (_ 1)'-^ f (n-.l)7^(7^-l)...(n-r-fl) ) ^ (n 4-1^ ft Therefore cos (n + 1 ) a = cos"+' a - ^^-^j — -^ cos" " * a . sin^ a 1 . ^ (n+l)n(n-l)(n-2) n-a • * + ^ \ \ „ ^: ^cos" 'asin*a-etc. 1.2.0.4 A similar result will hold good for sin (91+ 1) a. Thus, if the formulae are true when n is any whole number, they are true when 7i + 1 is substituted for n ', But they are true when w = 1 and when ti = 2 ; Therefore they are true when n is any whole number. 40 TRIGONOMETRY. ^2 f\4. A6 41. To prove cos^= 1 -j- + rj- -^+ etc., and smO = 6--r^ + TT-T^ + etc. II 15 ll In the formula n{n-l) „_2 . 2 cos 7ia = cos a \ — ^ cos a . sm a + ^ / \, ^ V ^ cos" 'a . sm^a + etc. write ^ for na, and let w be increased without limit while 6 remains unchanged. Then since a=-, a must be diminished n without limit. We may write the above in the following form, ^ '"''-i";;yf-"' Krm'-"- COS—) is 1, since r is not greater than n ; [Art. 9.] the limit of -^ ^ -^ ^^ -^ IS ,— [Art. 81 : \r [r L J and the limit o£ I — r— J is 1, since r is not greater than n. ^ n [Art 10.] Therefore, by proceeding to the limit, we obtain . . 0' B' e' , cos EXPANSION OF sin a IN POWERS OF a. 41 Similarly, the expansiou for sin na may be written ${e-a)(e- 2a) f^^^ ey-^ /sin a^ 1.2.3 / ^\"-" /sin a\ ^ By proceeding to the limit, we obtain as before sin ^ = ^ - pr- + 1-^ - etc. 3 6 7 42. In the result sin a = a - r^- + ,-^ — ,— + etc. the series [3 [5 [7 is convergent for all values of a. [For the ratio of any term to the preceding is —, — -^. ; and what- n (71 + 1) ever be the value of a, by taking n large enough this fraction can be made less than some quantity which is itself less than unity.] In the proof of Art. 41 no limit was put upon the value of the angle. Therefore the result is arithmetically intelli- gible and true for all values of a, 3 5 7 Therefore the series a - ,^ + -p- - ,^- + etc., which is equal [£ 12 IZ to sin a for all values of a, must be periodic. [Art. 27.] 43. A series in ascending powers of a quantity (a) is chiefly useful when a is small, for the smaller the quantity a, the greater is the relative importance of the earlier terms of the series. Also, the sine (and the cosine) of an angle of any magnitude may be expressed in terms of the sine or cosine of some angle less than ^tt. Hence, the above series are never used in numerical calculations except for values of a less than ^tt. 42 TRIGONOMETRY, 44. "We have = ("-i)^(2"F)^(g"3l)"''*°- Each of the above brackets is positive (provided a^ is not greater than 6 and therefore, a fortiori, if a is less than 1). Therefore sin a is less than a and greater than a — ^a. Again (sin a — a) is negative and = — ^a^ + (a positive quantity), sin a - (a -|^a') is positive and = yj^j^a* — (a positive quantity). Therefore the difference between a and sin a is less than •g^a^; the difference between sin a and a — Ja^ is less than Example. If a = YV (^^ ^ radian), the difference between a and sin a is less than -J^.IO'^, i.e. less than a six-hundredth part of a. The difference between sin a and a - ^a^ is less than xiir x 10~^ which is less than a millionth part of a. 45. The following results may be proved in a similar manner. The difference between 1 and cos a is less than Ja*, cosa and (1 — Ja*) irr**« szii^ nice Example. Find the limiting value of ^ when a is infinitely ^ •' 1- cos na •' ^ diminished. For sin wia, write ma - E^a^ ; for cos wa, write 1 - J n^a^ + R^aK We know that Ri is less than ^m^ and that R^ is less than ^^V^** Then sin^mg _ {m a-R^ a^)^ _ {m—R^a^Y ^ 1 - cos na ~ in?a? - iJgtt^ ~ Jw^ - i^ga^ ' 2771^ hence, when a is infinitely diminished the required limit is — ^ . EXAMPLES. 43 EXAMPLES. XIX. (1) Prove that when a is not large, the difference between (a-^a^ + xl^a^) and sin a is less than -51^ aJ, (2) Prove that when a is not large, the difference between (1 - ;! a^H- 2V a*) ^^^ cos a is less than jh^ a^. IS'' (3) Prove that sin — = -099833. (4) Prove that the value of sin l^' coincides with that of the circular measure of 1* at least as far as five places of decimals. (5) Solve the equation sin (^77 + ^) = '71, neglecting 6^ and higher powers of 0. (6) Given that sin 1' = "0002909, calculate approximately the value of TT. (7) Given ^ = igif , prove $=4^ 24' nearly. o /ox r^- J J.^ 1 - sitl^Tl^-sin^m^ T, /I A (8) Fmd the value of — 7- -z — when ^=0. ^ ' 1 - cos^^ /«x -r. 1 i. sin2 \/m?i^ - sin 7w^ , sin n^ r, a r. (9) Evaluate -rz ^r-^. ' — ^r— when ^=0. ^ ' (l-cosm^)(l-cos?i^) (10) Find the limit of ^~*{^ + ^-^| ^j when 6 is infinitely diminished. (11) Prove that (eight times the chord of half a small circular arc minus the chord of the whole arc) divided by three, is equal to the length of the arc, nearly. (12) Prove by induction „tan«-?iMMtan'd+ tann^=- , n{n-l) . „ . , 1 ^ ^ tan2^+. 7^2 /j3 (13) sin(^+A) = sin ^+7icos^-Txsin^--^ cos^ + etc. If. 2. 44 TRIGONOMETRY. Expansion of (2 cos a)" and of (2 sin a)". 4G. The following notation will be found convenient. Def. cosh X stands for ^ '^J and sinh a; for ^ ~" ^ 2 — -— 2 • cosh X and sinh x are abbreviations respectively for the words hyperbolic cosine of x and hyperbolic sine of x. We shall use the notation cosh^cc for (cosh a;)^, etc. EXAMPLES. XX. Prove the following statements : [Compare Examples XII.] (1) cosh^ a; - sinh^ ^-=1. (2) cosh 2x — cosh^ x + sinh^ ^ = 2 cosh^^ -1 = 1 + 2 sinh^ x, (3) cosh 3^ = 4 cosh^ x-^ cosh x. (4) cosh x — cosh {-x). (5) sinh 3^= 3 sinh x + 4, sinh^ .r. (6) sinh ^= - sinh ( - x). (7) cosh {x +y) . cosh [x-y) = cosh^ x + sinh^ ?/ = cosh^ y + sinh^ ^. (8) 2 cosh 71^ . cosh ^=cosh (^ + 1) ^ + cosh (n - 1) x. (9) 2 sinh nx . 2^ sinh^ x=2 sinh (ti + 2) ^ - 4 sinh ti^ + 2 sinh (w - 2) x. (10) 2 cosh nx . 2^ sinh^ :r = 2 cosh (?i + 2) ^ - 4 cosh nx + 2 cosh (?i - 2) X. (11) cosh ?^^ - cos wa=2 cosh (n - 1) x {cosh .r - cos a} + 2cosa{cosh {n-l)x- cos (^ - 1) a} - {cosh (n-2)x- cos (^i - 2) a}. 47. ^o prove, when n is a positive integer, that 2° cos °a caw he expressed in terms of cos na, cos {n — 2) a, etc. ; -) + etc. = 2 cosh Jiic + 91 . 2 cosh {n —2)x nin—Vj^ . , .. + , ^ ' 2 cosh {n - 4) a; + etc. Therefore also by Art. 47 9"r>nftV=2cos7ia+7i. 2cos(?i-2)a+— ^ — ^'2cos(?i-4)a + etc. :i"cos"a: 46 TRIGONOMETRY. As in Art. 37, wlien n is even, the last term does not contain cosh x, and in this term the factor 2 is to be omitted. n-l *49. To prove f when n is odd, that (-1) ^ 2" sin'' a can he expressed in terms of sin na, sin (n - 2) a, etc. ; that 2° sinh nx can be expressed in terms of sinh nx, siiih (n — 2)x, etc. ; and that the two expressions are the same inform. We have, when ti is a positive integer > 2, 2 sin na (- 1) 2^ sin^ a = 2 sin (w + 2) a - 4 sin wa + 2 sin {n - 2) a. I. 2 sinh nx . 2^ sinh^ x = 2 sinh {n + 2)x- 4: sinh nx + 2 sinh (ti — 2) a?. II. We have also (- 1) 2^ sin^ a = 2 sin 3a - 4 sin a. We proceed as in Art. 47. Multiply each side of this result by — 2^ sin^ a, and we obtain by the aid of I. (- ly 2" sin' a= 2 sin 5a - 10 sin 3a + 20 sin a. Multiplying again by — 2'sin^a, we could obtain by the aid of I. an expression in the required form for (— 1)^ 2^ sin'^ a. By continuing this process we could obtain an expression n-l in the required form for (— 1) ^ 2" sin" a, where n is any odd positive integer. Again, since 2^ sinh^ x^2 sinh 3a; - 4 sinh a;, by making use of II. in the same manner we could obtain an expression in the required form for 2" sinh" x. And since the process on the right hand is the same in each case, the resulting expressions are the same in form. EXPANSION OF (2 sin a)« 47 *50. We have 2" sinh"a; = {e' - e"')" [n odd] = e- _ e-"' _ n (e^""'^ - e"^""*^) + etc. = 2 sinh waj-Ti 2 sinh (?z - 2) a; + ^\^~ ^ 2 sinh (n - 4) ic - etc. Therefore it follows by Art. 49, that when n is odd, n-l (- 1) ' 2" sin"a = 2 sin Tia- ti . 2 sin {n-2)a + \ ^ 2 sm (w - 4) a - etc. *51. To prove that, when n i« even, (- 1)'' 2" sin" a = 2 cos wa - w . 2 cos (tz - 2) a 7i(n- 1) „ , ,. + — ^^ — ^ 2 cos (/I — 4) a — etc. We have, when ti is a positive integer > 2, 2 cos Tia . (-1) 2^sin'a = 2 cos(/i + 2)a -4cos na + 2 cos(7i-2) a 2 cosh nx . 2^^ sinh" x = 2 cosh (w + 2) ic - 4 cosh nx + 2 cosh (w - 2) x. Following the argument of the last article, we have since (-1)2' sin^' a = 2 cos 2a - 2, and 2» sinh' x = 2 cosh 2x - 2. And since 2" sinh" a; = (e'' - e~')" [n even\ = 2 cosh wa;-w2cosh(7i-2)a; + — ^ — ^ 2 cosh (w-4) ic- etc. n Therefore when w is even (— 1)^ 2" sin" a = 2 cos wa - 7i2 cos (n — 2) a + \ ^ 2 cos (n — 4) a - etc. In the last term the factor 2 must be omitted. [Cf. Art. 37.] 48 TRIGONOMETRY. ^EXAMPLES. XXI. (1) Prove that 8 cos* ^=cos 46+4 cos 2$+3. (2) - 64 sin'^^ = sin V^ - V sin 5^+ 21 sin 3^ - 35 sin 0. (3) 128 sinS ^= cos 8<9 - 8 cos 63 + 28 cos 43 - 56 cos 23 + 35. (4) Write down the last term in the expansion in multiples of cos 3, of (i) 22^C0S2**^. (ii) 22^+1 C0S2^+1^. (iii) 2^+2gij^4^4.2 ^^ (5) Any general formula expressed in cosines is also true in hyperbolic cosines. (6) Any general formula expressed in cosines or in squares of sines will be true in hyperbolic cosines and sines if we write - sinh^ 3 for sin^ 3. 62. To prove, when n is a 'positive integer, that cos na can he expressed in powers of cos a ; that cosh nx can be ex- pressed in powers of cosh x ; a7id that the two expressions are the same inform. We have, when n is &, positive integer 2 cos (n + 1) a = 4 cos na . eos a — 2 cos (n — 1) a. (I.) 2 cosh {n+l)x = 4 cosh nx . cosh x-2 cosh {n-l)x. (II.) . In I. put n=l, and we obtain 2 cos 2a = 4 CDs'* a — 2. Next put n^2, and using this last result, we have 2 cos 3a = 8 cos^ a — 6 cos a. Put w = 3, then using the last two results we have 2 cos 4a = 1 6 cos'^ a - 1 6 cos^ a + 2. Next put n = 4, then by the aid of the last two results, we can obtain an expression for 2 cos 5 a in powers of cos a ; and so on. EXPANSION OF cos nd. 49 By proceeding in this way we could obtain an expression in the required form for 2 cos na when n is any positive Again, by making use of (11.) in the same manner we could obtain an expression in the required form for 2 cosh nx. Also, since the process is the same i7i each case, the two resulting expressions are the same in form. Example. Trove that cosh nx - cos na is divisible by cosh x-cosa wlien n is a positive integer. From the above we have cosh nx = An cosh" x + A^_n cosh""^ x + etc. cos 7m = ^„ cos"' a + ^„_o cos"~2 ^ 4. etc. the coefficients in the two expressions being the same. Hence by subtraction cosh nx - cos na = A^ (cosh" x - cos" a) + ^„_2 (cosh""^ x - cos"~2 a) + ... and each term in this expression is divisible by cosh a; -cos a; [for ?/"-z" is divisible by y—z when n is a positive integer] therefore also cosh nx - cos 71a is divisible by cosh x — cos a. EXAMPLES. XXII. (1) Prove that 2cos6^=64cos<'^-96 cos*^ + SGcos^^- 2 also that e^ + e-^ = (e* + e-*)^ - 6 (e' + e"*)* + 9 (e* + e-*)2 - 2. (2) Divide cosh 6^ - cos 6^ by cosh .r - cos B. (3) Prove that x** + ^"~" - 2 cos na is divisible by x+:v~^ - 2 cos a when ?i is a positive integer. L. 4 50 TRIGONOMETRY, ^53. We can find the law of the coefficients in the expansion for 2 cosh 7ix, as follows : since (1 - e^z) (1 - e-'^z) - 1 -;2 (e" + e"") +z' = l -;s2 cosh x + z', .'. log (1 - e'z) + log (1 - e'^z) = log {1 - ;2 (2 cosh x-z)}; . •. e^'z + ie''z' + |e'V + etc. + e-^'z + Je"' V + ^ e'^V + etc. = z{2 cosh x — z) + \z^ (2 cosh x- z)* + \z^ (2 cosh x — ay + etc. In this identity the coefficients of »" on each side must be equal. On the left-hand side the coefficient of z"" is - (e"' + e""'), 2 . . 1 that is - cosh nx. The coefficient of z"" in - 2;" (2 cosh i« - zf is -^2" cosh" a;. The coefficient of ;s" in 3-;s""' (2cosh a; -zY'^ n ^-i ig (^ — 1) 2""^ cosh"~^cc, and so on. Thus we get n- 1 ^ - cosh 7i£c -- cosh" X ^ hi - 1) 2""' cosh"-^a; n n n— i^ w— 2 1 . J 1 (-3)(»-4)(«-5)2„.,^^^j^..,;^^^^_ 7i-3 1.2.3 Therefore also 2 cos wa = 2" cos" a ~ ?* . 2""^ cos""'' a + „ . M 2- CO."- « - « . ^'^ifc^ 2»- cos- a + etc. 1.2 i . z. o *54. This result may be transformed into a more sym- metrical form- as follows. The general term is ^_^^,^^(»-r-l)^«-2r+l) ^2 cose)---. EXPANSION OF cos 710. 51 I. Let n he even; let 2m stand for n. Then n — 2r is even ; let 2p stand for n — 2r. The general term may be written / ^ sm-p n(m + p-l){m+p-2)...(2p+l) ^,^ ^^^,^ ^ ^ ^ \m—p [where p is to have all integral values from to m] nlm + p — l ^ ' \m-p \zp ^ (_ 1)-^ n{m+p-^\){m^p-2).,.(m-p + \) ^^ ^^^^ ^ .=(-ir-x n(2m + 2jp-2)(2m + 2p-4)...(2m)...(2m-2jP + 2) ^ ^ ■ \2p -cos (9 [for there are 2p — \ terms in the series (m +jo- 1), (w + j9- 2) ...(m-^ + 1)] . (- i)^-pK-(2;.-2n{.^(2,-4y}...W , ^^^,^^ Hence putting for p the values 0, 1, 2 ... we have, 2 cos w^ II. Let 91 be odd, then by putting 2m + 1 for ti, and making a similar transformation, we shall obtain n-l 2cosw^ = (-l)' X 2]wcos^ — .^ _ ^^ cos"^+ \ ^ »^^ ■ , ^ cos°g-etc.L C 1.2.0 1.2.3.4.5 j 4—2 52 . TBIGONOMETRY, *55. The following is an illustration of an important method. Suppose that we have a general theorem such as cos n6^A^ + A^ cos + A^ cos^ ^ + ^3 cos^ + etc (I), which is true for all values of 0. For 6 put 6 + h and we have cos 7i6 . cos nh — sin nO . sin nh = A^ + A^ (cos ^ . cos h — sin ^ . sin h) + A^ (cos . cos A - sin ^ . sin hy + A^ (cos ^ . cos h — sin ^ . sin hy + etc. For cos h we may write 1 - Hh^ and for sin h we may- write h — R'lc', where R and R' are both finite when A = ; hence we obtain [Art. 44.] cos nB — n'h^R cos qi9 — nh sin nO + n^h^R' sin n$ ~A^ + A^cos6 + A^ cos^ + etc. - A {^^ sin ^ + 2^2 sin ^ . cos ^ + 3^3 sin (9 . cos' ^ + 4^^ sin ^ . 008=^ ^ + ...} + terms containing higher powers of A (II). This result is true for all values of h, and remembering I. we see that it is divisible by h. Dividing by h we get a result which is true for all values of h, and is therefore true in the limit when h = 0. Proceeding to the limit we obtain + n sin nO = A, sin + 2A„ sin 6 cos 9 + 3A^ sin 6 cos' + etc. *56. The student who is familiar with the methods of the Diflferential Calculus will observe that the above result may be obtained by difierentiating each side of the equa- tion (I). EXPANSION OF cos n9 AND OF sinnd. 53 *57. Applying this result to the series of Art. 54 we have, when n is even 2sinw^-.(-l)' 2 sin ^ X \ncose--^-^cos^e^ 1.2.3.4.5 ' '''' ^ - '''j ^ when n is odd w-l 2 sin nO = {-!)' 2 sin 6 x |l - 4-^-^ cos»^ + ^______i^__^cos^^.etc.|. '"'58. Hence we have the series: I. (?i even), (- 1)' cos n6 = l- ;px cos'^ + — ^^— — ^cos*^- II. ^^ (-l)"^^'sinw^=sin^|7icos^-^^^^^^cos'^+| III. (?iodd), (-lf^cosw^ = wcos^-*^^^2--p<^os'^+... IV. „ (- 1)* sinw^ = sin ^ 1 - .^-^^- cos=^ + ... I . ■^59. In each of the above formulae put - - ^ for 9, then I. (neven), cosng = l -^-^sin'^+ -^- ^ sin'^-... II. „ sin 7id = cos In sm 6 ^ — ^ ' sm ^ + . . . ^ III. (7z odd), sin 71$ = nsmO — r~o~Q ^^^^ ^ + . . • lY. „ cos?i^ = cos^p --y—^ sin'^ + ...[•. 54 TRIGONOMETRY. *60. In the following example an independent proof is given of the result of Art. 54. Example. To expand cos nd in ascending powers of sin 6. From Art. 19, we have when n is even cosnd=l + A2Qin^d + A^sin^e + etG I. The constant term is 1, because when ^=0, cosn^ = l, and sin^=0. For 6 write d + h and we have cos nd cos nh + sin nd sin nh = 1 + Jig (cos ^ cos /i + sin ^ sin /i)2 + ^^(cos^cos /i + sin^sin 7t)*+ ... For cos nh, write l-^n^h^ + etc, for sin nh, write nh-^n^h'^ + etc., and substitute similar expressions for cos h and sin h. In the result we may equate the coefficients of h^. On the left-hand side the coefficient of h^ is - ^n^ cos^ nd. In the term A^r (sin d + h cos d - ^h^ sin d - etc.)^'' the coefficient of h^ is ^r j^^^Y^^^ sin2'-2 ^ cos2 ^ - r sin2»- ^1 . Hence -^n2 cos 71^=^2 {cos2^-sin2^}+u44{2.3sin2^cos2^-2sin4^}+... + ^ar I ^^f^~— sinsi-a d cos^ d-r sin^'* ^ | + . . . In this writing 1 - sin^ d for cos^ d we obtain as the coefficient of sin^^^ \2r{2r-l) I ( (2r+2)_(2r+l) This then gives another form for the expansion of cos nd in ascend- ing powers of sin^ d. The coefficients of these two series are the same. Hence 2r(2r-l) , I , , (2r + 2)(2r+l) 2r+2 -^nU^^ = -A2r T o +^ +^: 1.2 ' ') ' '^+' 1.2 0^ ^2r+3--7o;rTTwo;:Xo\^2r' n'^-{2r)^ (2r+l)(2r-i-2) EXPANSION OF co&nd. f>u Patting for r the values 1, 2, 3 . . . in succession we obtain 71^-22 _ n^(wg-22) ^^-" 3.4 '^2-1.2.8.4' and so on. Thus cos nx etc. n factors. DE MOIVRE'S PROPERTY OF THE CIRCLE. 59 64. This result may be interpreted geometrically. Let OR be the initial line ; with centre and radius equal to x describe a circle; let ROP^ be the angle a. Divide the whole circumference, starting from P^,, into n equal parts, P,P,, F^P,, ... F,_,P,. Let OQ he equal to a, so that Q is any point in OP or in OP produced, then QP/ =0P^' + OQ' - OP,. OQ. 2 cos POP,= x'+ a' -ax. 2 C08 a QP^'=OP,'+OQ'- OP fiQ. 2 cos POP^ = x' + 0* -ax.2cos( a + — ) and so on ("?) Hence the result of Art. 56 may be written 0/>;»- op;. 0Q''2cos nROP,+ OQ^'^QP;. QP^\ QP^. . . QP^_ /. This result is De Moivre's property of the circle. 65. Some particular cases of the above should be noticed. When Q coincides with i?, a = x, and the above becomes OP" .2sm{in, POP,) = PP, . PP, .PP.... PP„,, . . .(I). 60 TRIGONOMETRY. Again if E coincides with one of the points P, then a is a multiple of — , and na is a multiple of 27r and we have (OE" - OQ-y = QP,' . QP^' . QP^\ . . (?i^„_, ; .-. OR''-OQ''=^QP,.QP^,QP^...QP„_, (II). Now if the arcs P^P^, P^P^,-.- are bisected in points p^, Pn Ps '•• Pn--[ respectively, we have by what has just been proved 07?- - 0^-= ^P„. Qp, .QP, . Qp^,..QP^_, . Qp^_^. Therefore, by division OR" + OQ" = Qp,.Qp^.Qp,...Qp„_, (III). The student should notice carefully that in (I) Q lies somewhere on the circumference; in (II) OQ or OQ pro- duced, passes through one of the points P^P^...; in (III) OQ or OQ produced, passes through the middle point of one of the arcs p^p^j VxV^'t ^^^' (II) and (III) are Cote's properties of the circle. MISCELLANEOUS EXAMPLES, 61 * MISCELLANEOUS EXAMPLES. XXV. (1) Prove Euler's Formula, viz. sind e e 6 J . n -— r- =cos - . COS t;^ . COS ^ ... ad int. and deduce from it that sin 6 is less than $ - ^6^. (2) AB is the diameter of a circle and Qq any point on the circumference; Q^, Q2, Q^,-" are the points of bisection of the arcs AQqj AQi, AQ^y-" prove that BQ,,BQ,,BQ,...BQ^^OA'^.^^, (3) Find the Umit of (cos ef""^ ^ when 6=0. (4) Find the limit of logt^ne ^i" ^ when ^ = 0. (5) Of what order is the error when =— ^ is substituted ^ ^ 2 + cosd for^? (6) Prove that 2 cos n0 — 2 cos n6 = 2" (cos <^ - cos 3) x -[cos 0- cos ( — + ^)f jcos^-cos ( ^ + d)[-. . _j /tan2^j-tanh2^\ _j /tan ^ - tanh .r\ ^'^ ^^'^ Vtan2^-tanh2a;;^^'' VtTn ^ + tanh .rj = tan~^ (cot ^ coth x) . . , sinh^ , ., cosh^ where tanh ;r = — r— and coth x= ^—r — . cosh X sinh a: (8) Expand cos*" ^ + sin*"^ in a series of cosines of multiples of e, (9) Prove that TT 27r Stt 47r 5ir Gtt Ttt 1 ""' 15 ""^ T5 ""' 15 ""' 15 '"' 15 "'^ 15 ""^ Is = 27 • (10) Form the equation whose roots are tan^^, tau^f^, tan^^f , tm^^, Uv?^^. Jb2 TRIGONOMETRY. CHAPTER Y. On the Summation of Trigonometrical Series. 66. There are two methods peculiarly applicable to the Summation of Trigonometrical Series. FIRST METHOD. 67. Sometimes each term of a series may be transformed into the difference of two quantities. Example 1. To sum the series sina + sin(a + 5) + sin(a + 25)+ ... +sin{a + (w-l);5}. We have 2sina . sin^S=:cos (a-|5)-cos (a + J5), 2sin(a+5) . sin |5 = cos (a + |5) -cos (d + |5), 2sin(a + 25) . sin |5=:cos (a + |5) -cos (a+ |5), 2sm {a+(n-l)5} . sin|5 = cos ( a + — ~ 5 1 -cos( a + — - — 5 Therefore, if >S^ stands for the sum of n terms, we obtain by addition 25„ . sin|5=:COs(a-4S)-cos (a + — o~~^) = 2 sin {a+|(«-l) 3} . sin |n5. rr,, » n sin {a + i (n-1) 5} . siniwS Therefore /S„= ^ . IJ ^— . EXA.MPLE 2. To sum the series cosa + cos(a + 5)+cos(a + 25)+ ... cos {a + {n-l)d}. We have 2cosa . sin^5 = sin(a + |5) -sin(ai -S). Hence, proceeding as in Example 1, we obtain _cos {a + l (n-1) 5} . sin^wS SUMMATION OF SEBIES. 63 The results of these two examples are often useful. The student is advised to become familiar with them in words. The sum of n terms of a series of sines (or cosines) of angles in a. p. is equal to the sine (or cosine) of half the sum of the first and last angle, multiplied hy the sine of n times half the difference, divided hy the sine of half the difference. Examples. To prove that if n=:2ir, then sin a + sin (a + ^) + sin (a + 20) ... +sin {a + (/i-l)0}=O for all values of a. In the result of Example 1, sin ^nS occurs in the numerator, and sin^n5 = sin|n0 = sin7r = O, and the denominator sin|0 is not = 0. Therefore the sum of the series = 0. Similarly COSa + cos(a + 0) + cos(a + 20)+ ... +cos {a + (n-l)0}=O. 68. The results of Example 3 may be stated geometri- cally : Let OJi be the initial line and HOP^ any angle, then if the whole circumference of a circle centre and radius OB, be divided into n equal parts -Po A» ^i -^a' ®*'^* Then the sum of the sines (or of the cosines) of all the angles ROP^, 7?0Pj...i?0P„_jiszerot. t This is an expression of the fact that the centre of gravity of equal particles placed, at the points Po-Pi--. is at the centre of the circle. 64 TRIGONOMETRY. Example 4. To sum the series sin'" a + sin™ (a + 6) + sin"* (a + 25)+ ... +sin"» {a+ (n-1) S}. This may be done by the aid of Arts. 37, 47. Thus, if m be even 2m sinw a = ( - 1) 2 {cosma-wcos(?n-2) a + etc.} and the required sum may be obtained from the known sum of the series { cos ma + cos m (a + 5) + cos w (a + 25) + etc. } + {cos (m - 2) a + cos [m - 2) (a + 5) + cos (m - 2) (a + 25) + etc. }. Similarly we may find the sum of the series cos™ a + cos™ (a + 5) + cos"* [a + 25) + etc. to n terms. EXAMPLES. XXVI. Sum the following series to ?i terms. (1) sin a + sin 2a + sin 3a + (2) cos a + cos 3a + cos 5a + (3) sin a + sin 4a + sin 7a +- (4) sina. cosa + sin2a. cos2a + sin3a. cos3a+.. (5) cos^ a + cos^ 2a + cos'^ 3a + (6) sin^ a + sin^ 2a + sin^ 3a + (7) cos4a + cos*2a + cos*3a+ (8) sin 2a . cos a + sin 3a . cos 2a + sin 4a . cos 3a + (9) sin a . sin 2a + sin 2a . sin 3a + sin 3a . sin 4a + . . (10) cos^ a + cos3 {a + d) + cos^ (a + 2S) + (11) sin4a + sin4(a + S) + sin*(a + 2S) + EXAMPLES XXVI. 65 . (12) Solve the equation sin 6 + sin 26 + sin 3^ + etc. to n terms = cosd + cos 2^ + cos 3^ + etc. to n terms. (13) Write down the value of series (10) and (11) when n3=27r. (14) Prove that sina + sin3a + sin 5o+ ...towterms cos a + cos 3a + cos 5a + ...ton terms = tan no* (15) Prove that sina + sin( a + 5) +sin(a + 2d)+ ... to (2n-l) terms sin a + sin (a + 25) + sin (a + 45) to 71 terms is independent of a, (16) Deduce from Ex. (1) the sum of the series 1+2 + 3+ +n. (17) Deduce jfrom Ex. (6) the sum of the series 13 + 23 + 33+ etc. + n3. (18) Deduce from Ex. (9) the sum of the series 1.2 + 2.3 + 3.4 + etc. + n (n + 1 ). (19) Sum the series sin o - sin (a + d) + sin (a + 25) - etc. to n terms. (20) Sum the series cos a - cos (a + S) + cos (a + 25) - etc. to n terms. (21) Prove that the series sin"* a + sin"* (a + ^) + etc. to n terms, where n(^=27r, is independent of a, provided m is less than n. (22) Prove that the series cos"* a + cos"* (a + <^) + etc. to n terms, where ncj) = 27r, is independent of a if m is less than «. L. 5 66 TRIGONOMETRY. Example 5. To sum cosec 6 + cosec 26 + cosec id + ... to n terms. We have cosec d = cot^9- cot 0, cosec 2^ = cot ^ - cot 26, cosec 2''-i^ = cot2»-2^-cot2"^. Therefore, as in Art. 67, iS„=cot|^-cot2"-i^. Example 6. To sum tan 6 + ^ tan ^6 + ^ tan ^6 + etc. f o n i^rwis. We have tan ^ = cot ^ - 2 cot 26, ^ tan ^ ^ = i cot ^ ^ - cot ^, ^tan J^ = ;|cot J^- Jcot^^, etc. Therefore S„ = ^ J^i cot A _ 2 cot 26. 69. If the result of summation of such series is given, it is often easy from that result to discover the required transformation. For example. The result of the summation in Example 1 has sin |5 in the denominator. This suggests that sin a . sin 1 5 may be transformed into two quantities which are of course ^cos (a-|5) — |cos(a + ^5). Again, in Examples (5) and (6) the required transformation will be at once seen if we put n = 1 in the answer. The student however is advised only to resort to this method of solution as a last resource. EXAMPLES. XXVII. Sum the following series to n terms. (1) sec 6 . sec 2d + sec 20 . sec 3^ + sec 2B . sec 46 + (2) cosec 6 . cosec 26 + cosec 26 . cosec 36 + cosec 36 . cosec 46+ .... (3) cosec 6 . sec 26 - sec 26 . cosec 3^ + cosec 36 . sec 46 - etc. (4) — + — + . ^ + etc. ^ ' cosa + cos3a cosa + cos5a cos a + cos /a ,^. sin a sin 2a sin 3a . (5) rr~+ J-+ ^+etC. ^ ' cos a + cos 2a cos a + cos 4a cos a + cos 6a EXAMPLES XXriI. 67 (6) __cog_'. cos 2a ^ cos 3, ^^^^_ ^ ' COS a - cos 2a cos a - cos 4a cos a - cos ba (7) sin 2^ . siii2 ^ + i gi^ 4^ . sin^ 2^ + i sin 8^ . sin2 4^ + (8) sin2^.cos2^-Jsin4^.cos2 2^ + isin8^.cos2 4^- A 3 6 6 (9) sin^.cos2--2sin-cos2- +4 sin-. cos2 --....., (10) sec a . sin 2a . sec 3a + sec 3a . sin 4a . sec 5a + etc. ,,,. sin 2a sin 4a , , ' sin a . sin 3a sm 3a . sin ha (12) *^"' 1^172 +**""'rr2r3+*^"'rTl:i+^'"- (1^) t^-^^'iTrria:^ +*"""' 1+1:3^^ +**""rT^^+- /, .X X ^ 3a2 , - 5a2 • (14) tan-i^-p3^-2^+tan-i^-^2273^, + etc (15) tan ^ + 2 tan 2^ + 4 tan 4^ + etc (16) tan a + cot a + tan 2a + cot 2a + tan 4a + cot 4a + etc. ( 1 7) sin ^ sin 3^ + sin 2^ sin 6^ + sin 22 6 sin (22 , 3^) + etc. 6 '\6 6 35 (18) sin5.sin35 + sin -sin'- + sin^ .sin ^2 + (19) cot 6 cosec 5 + 2 cot 16 cosec 25 + 22 cot 22 6 cosec 22 5 + ... 6 6 16 6 (20) cot 6 cosec 5 + | cot - cosec - + 02 ^^* 92 ^^^^^ 9^ + • - (21) cosec a + cosec ^ + cosec 2 + 2 22 (22) - sec a + g2 sec asec 2a + 53 ^^^ " ^^° ^^ ^^^ 2^ a + etc. (23) Deduce from Ex. 2 the sum of the series -t-q + 2~3 "^ 3~4 to n terms. (24) Deduce from Ex. 17 the sum of 1.3+2M.3 + 2M.3 + ... to n terms. 5—2 68 TRIGONOMETRY. SECOND METHOD. 70. When the sum of a series of the form A^ + A^x+ A ^x^ + A^x^+ etc. is known, we can obtain the sums of two series of the forms A^ + A^x cos 6 + A^af cos 26 + A^ cos ZB + etc., and A^x sm£ + A^ sin 2^ + A^ sin Z6 + etc. Let C stand for the sum of the first series, and aS^ for the sum of the second series, then, C+ iS=A^->tA^x (cos ^ + ^ sin B) + A^(qo% 29 + i sin 2^) + etc. = A^ + A^xe'^ + A^ {xe'y+A,(xe'y+ etc. The sum of the last series is known by hypothesis. The result of the summation must then be expressed in the form A + iB; whence we have C=A and S=B. [Art. 11.] Example 1. Sum the series 1 + a; cos ^ + a;2 cos 26 + etc. + x^-'^ cos (n - 1) 9. Let (7„ he the sum of this series, and let S„_i = as sin ^ + a;2 sin 2^ + . . . + a;"- 1 sin (w - 1) ^. Then C^ + iS^_:,:^l+xe^^ + xh^'^ + a^e^^+ ... a;«-ie^"-l^**. 1 - x^e^^ This is a series in o. p. .*. its sum = . I'xe^^ Multiply the numerator and the denominator of this result each by 1 - xe~^^^ and we have ""^ "-V" l-xCe^^+c-^'^j + aj^ ~ l-2xcose + x2 _-, , _ l-a;cos0-a;*'cosw^ + a;'»+icos(n-l)^ n, ^ ii-, Therefore C„ = :; — ^ ;; ^ ^ • i^^^- 11 •] " l-2a;co8 + a;2 •■ -* X sin ^ - a;~ sin n^ + a^'^+i sin (n - 1) ^ ^'^o ^»-= i-2xeosg + »' • ., ., SUMMATION OF SERIES, 69 Example 2. Sum tJie infinite series X- x' sin a + x sin {a + S) + -. sin (a + 2/S) + - sin (a +3/3) + ... If. li Let S stand for the sam of the series, and let C=cosa + xcos{a + /3) + r7rCOs(a + 2^)+ ... Then C + tS=te**+xe'V^+^ e^c^2^+ ...=c**^{e^^' } [Art. 3.] 11 -_ gta^ (cos ^ + 1 sin p) = (cosa + isina)e*^^^{cos(j;sin/S) 4-tsin (xsinjS)J. Therefore S = e^ *'^'"' { sin a cos (x sin /S) + cos o sin (z sin j8) J EXAMPLES. XXVm. Sum the following series to infinity. (1) sina-l-^sin2a+a;2sin3a + ... (2) cosa + :rcos(a4-i3)+^2cos(a + 2/3) + ... (3) sin o + cos a sin (a + /3) + cos^ a sin (a + 2^3) + ... (4) cos a + sin a cos (a + j3) + sin^ a cos (a + 2^3) + .. . (5) sin a + T-^ sin 2a + y^ sin 3a + . .. Li H (6) ^sina + 7^ sin2a + r5 sin3a + ... _. _ cosa . , cos* a __ cos' a _. , (7) 1--,^ cosi3+ - cos2/3--|y-cos3i3+... (8) sina-T^ cosasin (a+j3) + y^ cos^a sin (a+2/3) - ... 11 ■ ii 70 . TRIGONOMETRY, (9) sin a - 1 sin 2a + ^ sin 3a-... (10) . C08a + ^co82a + ^cos3a-f... (11) sina.cos/S-^sin^a. cos2/3+^sin^a . cos3/3-... (12) cos^.cos<^+Jcos2^.cos2(^+Jcos3^.cos3^ + ... Sum the series 13, 14, 17 to 20, to n terms. (13) cosa + ^cos(a+i3)4-^^cos(a + 2/3) + ... (14) x&ma-x^ sin (a +i3) + ^ sin (a + 2^) - ... /^-\ 1 n(n-\) _ n(n-l)(n-'2) _ . (lo) l+iicosa+ .q ^ cos2a + — r— ^cos3a + ... to (n + 1) terms. (16) sin a + w.^: sin {a+ ^) + -^ — - x" sin (a + 2^) + . .. to (w + 1) terms. (17) 1 + cos a . cos /3 + cos^ a . cos 2/3 + cos^ a . cos 3^+ ... (18) sin a + sin a . sin (a + /S) + sin^a . sin (a + 2^) + . .. (19) sina + 2sin2a + 3sin3a + ... (20) l2cosa+Q2cos2a + 32cos3a + ... *Siim the following series to infinity. (21) l+^cososin^ + ^^^cos2asin2^ (22) cosa-Jcos3a + ^cos5a-... (23) 1 + TV ecosa cos (sin a) + r^ e^ cosa cos (2 sin o) -f ... , (24) e * cos y — \ e "^ cos 3y + \ e~^ cos 5y - . . . (25) e* sin a? - 1 e^r gfu 2x+ ^ e^"" sin 3j: - ... tlXPANSION IN SERIES, 71 71. The expression £c^ - 2a; cos ^ + 1 is the product of the two factors (I —xe^^){l —xe-^^), and therefore an ex- pression having x" — 2x cos ^ + 1 for its denominator may often be expanded in ascending powers of x by finding its equivalent partial fractions. _ _ ^ , 2 cos a — 2x cos (a - fl) . , . Example L Expand — ^ — ^ ^ tn ascending powers JL — ^X COS p T X o/x. 2 COS a -2x COS (a -/3) e^'* + g-»'*-z(e*'*"*^ + c-^+^) l-2a;cos/3 + x2 {l-xe^^)(l-xe-*P) [Art. 23.] l-xe"^ 1-xe-i^ = e^{l + xe^^ + x''e^'^+...)+e-^{l + xe-^^+x''e-^'P+...} = 2cosa + a;2co8(a+p)+x2 2cos2(a + /9) + .., [Art. 23.] Example II. Expand - — in ascending powers of x. ^ l-2xcosa + x2 ^^ •' This expression may be written (1 - ax^)-^ (1 - 3ce~**)~^ = l + x'^ + x^ + ... + {e''^ + ^~^){x + a^ + 3^+...] + {e^^+e-^'''){x^ + x^+x^ +...}+ etc. = r-^^{l + a2cosa + a;22eos2a + j;32cos3a+...} ^^ /> O** Writing ^ for x we have, if a > 6 (a3-2a&C03a + 62)-i=-^-- jl + 2^cosa + 2^^cos2a + 72' TRIGONOMETRY. Example III. lii any triangle c" = a' - 2ab cos C + b^; let a 6e'>b, then c^ = h^ (^"^*0 (^-^"'^) ' ^^'^' ^^•■' =21oga--(giC^e-iC)_|^^(giC+g-i(7^_etc. [Art. 4.] logc=loga--cos(7-4-2Cos 2C-^^cos3C-etc. [Art. 23.] This series may sometimes be made useful when - is small. 72. The following example is important. Given sin ^=x sin (0 + a), expand 6 in a series of ascending ^powers ofx. 8mceame=x8m{d+a),.'.e'^-e-'^=x{e'^+'''-e-'^-h, [Art. 23.] e^^^-l = x{e'''e^^^-e-% l-xe"" log 6^*^= log (1 - ;re - *«) - log (1 - are*«). 2ie + 2irT=x{e'''-e-'^'') + ix^ (g2^«_e-2»'«)4-etc. [Art. 4.] e + rir = xsma + ^x^8m2a + ^x^Bm3a+... [Art. 23, 28.] If in the above x = -l, then sin^= -sin(^ + a), so that we may put - 2d for a. Hence we obtain when is less than ^tt ^ = sin 2^ - J sin 4^ + ^ sin 6^ - etc. EXPANSION IN SERIES, 73, EXAMPLES. XXIX. (1) Expand ; ^ in a series of ascending powers ^ ' ^ l-2aCOS<^ + a2 of a ; and prove that if jo„_i, p^, p„+i be the coefficients of three consecutive terms 2jo„ cos(f)=p„-i+p„+i. Expand the following expressions in ascending powers of x, . . sin a .^. 1 + ^ cos ^ /..N -I /, r. OS /i-\ sina-^sin(a- fi) (4) log(l-2^cosa + *'). (5) -^-^-—^, (6) e^cona cos (x sin a). (7) caa; cos bx. ■ (8) eaa; cos 6;r + e*^ cos a^. (9) e^; cos a gin (:r sin a). (10) eajcos^cos(a + ^sini3). (11) ea^cos^ sin(a + ^sin/3). (12) In any triangle sin -4 =t- sin {A + (?), hence prove that a, a^ ^ = r^ sin C + gTa sin 2C+ etc. (13) If tan 0=?i tan 3, find a series for in terms of 0, (14) Prove that -^-^^ |(TT^^ + (lT^^^4 ' """"^ '''" pand sec"*^ in cosines of multiples of 0. (15) Prove that cos nacos^a + 1 sin tia cos^a = — — rr rr and ^ ' (1 - i tan a)** expand cos na cos" a in ascending powers of tan a. (16) Sum to infinity the series (i) 4 + 9 cos ^ + 21 cos 2^ + 51 cos 3^ + etc. (ii) 1 + 3^ sin ^ + ll^F^sin 20 + 43x^^n 30+.., 74 TRIGONOMETRY. CHAPTER YI. Resolution of sin^ and cos^ into Factors. 73. Topro.eAr.e^6{lJl){l-^:^){l-i:-)... integer By Arts. 38, 52 or 62 we have when ti is a positive a;'" - 2aj" cos 2na + 1 =^ (a;''-2£ccos 2a+ l)]a;^-2ajcos f 2a + — ) + 1 i ... ^factors in this result let a; = 1, and let 2n(j> = Tr, then 2 (1- cos 2na) = 2" (1 - cos 2a) {1 - cos (2a + 4<^)}... {l-cos(2a + 7i^4<^)}. Now 1 — cos 2wa = 2 sin^ no. ; hence taking the square root ± 2 bin 7ia = 2" sin a . sin (a + 2<^) . sin (a + 4<^) . . . sin (a + 2ncl> - 20). But sin (a + 2n<}> — 2(f>) = sin (a + tt — 2<^) = sin (2^ — a), hence, when n is odd, we have ± 2 sin na = 2" sin a sin {2 + a) sin {2cf> - a) . sin (4<^ + a) sin(4<^-a) ... sin {{n — 1) cji + a} sin{(w-l) <^ — a}. But sin (20 + a) sin (20 - a) = sin* 20 - sin' a. Hence ± 2 sin na = 2" sin a (sin^ 20 - sin* a) (sin* 40 - sin' a) . . . BE SOLUTION OF SIN d INTO FACTORS. 75 Next, divide both sides by sin a, and let a be diminished without limit, and we obtain 2n = 2" sin' 2<^. sin' 4<^ sin' 6<^ ... sin' (n - 1) <}>. Divide the first of these last two results by the second, /i sin' a \ /- sin* a \ thus ife sin na = 71 sin a ( 1 - . o » . 1 1 1 — . a ^ > ) ••♦ V sm' 2/ Write 6 for na, and let 7i be increased while a is diminished without limit, remaining unchanged : then since sin* a w ^, T .^ £ sm'a . &" the limit of . g IS -j; sin' 2^ . 8 7r» •• sin'2 sin — n and the limit of n sin a = that of 7i - , i.e. = 6; n proceeding to the limit we obtain .sin. = .(l-5)(l-,Q(l-3^,)... Now, when 6 lies between and tt, sin 6 is positive and every factor on the right-hand side is positive ; when $ lies between tt and 27r, sin is negative and one factor only on the right-hand side is negative ; and so on. Therefore the upper sign must be taken in the above result instead of the ambiguity ± : and the proposition is established. *74. We can prove that the upper sign must be taken in each of the foregoing identities as follows : — In the figure, let BOP^ = a; produce Pfi to Q, and divide the semicircumference PQ into n equal parts PqP^, P^ Pg, etc. Then since w . 2^ = -r, each of the angles PfiP^ , PfiP^... is equal to 2<^ and ROP^ = a+2, ROP^ = a -f 4<^, etc. Now consider the first ambiguity on page 74. 76 TRIGONOMETRY, I. Let a be less than x (Fig. I.)- Now, since sin ROP^_^ is negative when P„_j is below ROL, the product of sines on the right-hand side will be positive or negative according as the number of the points P^_^ Rn-2'" which are below ROL, is even or odd. Let r be that number. Then a, which =ROPq = LOQ, is equal to (r.2 + X), where \ is less than 2<^ [in Fig. I. r = 3 and X = Z0P„_3]; li^nce na = n {r2(l> + \) = rTT + TiX, where wX is less than tt. Therefore sin na is positive or negative according as r is even or oc^c?, that is, according as sin a . sin (a + 2<^) sin (a + 4<^) ... is positive or negative. II. Let a lie between tt and 27r (Fig. XL). Then P„ is below ROL^ , and if there are also r of the points F^,P^... he- low ROL^ , (r+1) of the factors sina, sin (a + 2<^). . . are 7iegative. And in this case a = 27r - PfiR = 27r - (r . 2<^ + X) [in Fig. II. r = 3 and PfiR = X], . *. na = 2?Z7r - ^tt — wX, where nX is less than TT. Hence sin ?ia = — sin (rTr + X), and therefore is negative or positive according as r is even or oc?d III. Let a be greater than Stt; and let a=2m7r+a' where a is less than 27r. And the proposition, being true for a! by J. and II., must also be true for a. RESOLUTION OF COS d INTO FACTORS. 77 75. In the identity 2ainna= 2" sin a . sin (a + 2<^) sin (a + 4<^) . . . sin {a + {n - 1) 2<^} write a + ^ for a, then na becomes na + ny i.e. na + Jtt, and we have 2 cos 7ia = 2" sin (a + <^) sin (a + 3<^) sin (a + 5) ... sin{a + (271- 1)<^}. From this we can deduce as in Article 73 that, when n is even, 2 cos ria = 2" (sin' <^ - sin' a) (sin' 3<^ — sin' a) . . . {sin' {n—\) — sin' a}. Whence, writing 6 for na as before, we obtain 2'^'\ /, 2'6\ /, 2'^'^ cos -(■-?-)0-S)0-l?.)' [The ambiguity in the sign may be removed by the method either of Art. 73 or of Art. 74.] 76. Many particular identities may be obtained from the results proved in Arts. 73, 74. For example, in the identity 2 cos na = 2" sin (a + <^) sin (a + 3^). ..sin (a + 2n - ), put a = 0, and we have 1 = 2''"*sin^.sin3<^...sin(2n— 1) , where 2n0 = 7r. Again, in the identity 2 sin ria = 2" sin a sin (a + 2) sin (a + 4<^) . . . sin (a + 2?i<^ — 20), let a be diminished without limit, and we have 2/1 = 2" sin 2 sin 4<^. . .sin {2n - 2<^), where 2/1^ = ir. 78 TRIGONOMETRY, 77. The two results and cos e={l-^){l-^)(l- gyj ... could be proved very shortly as follows, if the following pro- position were true. — If a function 0/6 (such as sin 0) vanish for any value a of 0, then $ — a is a factor of that func- tion of 0. This proposition is known to be true of any rational function of ^, but it is not always true for infinite ^3 series like ^ - ,-5 + etc. ; _i 11. [e.g. e G^ vanishes when ^=0, but the series 1 - z^ + lo^ ~ ^*^'' ^^ not divisible by ^. De Morgan's Differential Calculus, p. 176, Lond. 1842.] Assuming the above proposition, it is clear that sin is divisible by ^, ^ ± tt, ^ ± 27r, ^ ± Stt, ... therefore sin ^=^K^-S(i-2Q(^-£0- Also sin does not vanish for any other real values of 0, nor for any imaginary values of 6 [Example, page 29], Hence A does not contain 6. And hence by diminishing indefinitely we obtain that the value of -4 is 1. Similarly, by assuming a corresponding proposition for COS Of we have, since cos 6 vanishes when ^ = ± - , or * -^ , or ±-2-,... »-=-('-?^)('-l5)(-a- and as before A does not contain 6 ; hence putting ^ = we get 4 = 1. FACTORS OF COSH x AND SINH x. 79 *78. Tojlnd the Factors of e'* + e""' - 2 cos 20 ; and of sink a, cosh a. Bj Art. 63, x"" - 2x"f cos 2a + y^" has n factors of the form , _ 2r7r + 2(9 a X — zxy cos + y . Let w be odd. The last factor is _ _ 2mr-2ir + 2e , ic" - zxy cos + 3^, 2_ _ 2^ and this is equal to x^ — 2xy cos + y^. The last factor but one is 2 _ 2n7r-47r + 2^ „ ic - 2£cy cos + y % and this is equal to a;* - 2xy cos + ?/'. And so on. 71/ Hence a;'" - 2a;V cos 2^ + y"* / , - 2l9 A / , - 27r±2^ A = ( a; - zxy cos ~ + y j x ( ^ - 2a;y cos — — + 2/ ) ^ ...(x^- 2xy cos + 2/M . .. 2n factors Where cc* — 2a;v cos — + v' stands for the two factors n ^ / , _ 27r + 2(9 A / « o 27r-2^ A [x- Ixy cos + 2/ ) { ^ ^^ cos -— + 2/1. Now write 1 + - for », and 1 — for y. The general form of factors on the right hand is (■*3'-=0-.?)-'=^"*('-3' 80 . TRIGONOMETRY. thati. 2(1.3-2(1-^,) that IS, 4 sm^ — -^1 5 cot > . n I, n n ) In the resulting equation put « - 0, and we have 4 sm p = 4 sm — 4 sin - — - 4 sin . . . n n ^ Using this result to simplify the right-hand side, we have = 4sin''^( 1 +-^cot''-)... for 0^ ; also they are true when in the resulting identities for ^, we write — for <^. L. 6 B2 TRIGONOMETRY, In the resulting identities we may write a^ for <^, then By the above artilice the results of Art. 77 and of Arts. 73, 74 may be deduced the one from the other without the introduction of J{- 1). 81. Many results may be obtained from the identities >-^,f— (>-5)(>-£.)('^a^^- Example 1. Prove that ^ + ^ + —^+ ...=-^ . From first of the above identities we have Expanding each of these logarithms by Art. 4, we have In this identity we may equate the coefficients of the various powers of 6^. Hence 111^ 7r2 SUMMATION OF SERIES. 83 « m x/. 1 2^ 2^ 20 •Example 2. To prove cot 0=- - ^^^-^, - ^i^^^^-y, - g^^^r^^a " - By Art. 73, log sin = log + log (l - ^A + log f 1 - ^^ j + . . . I. The required result may be obtained by writing + ^ for in this iden- tity, expanding each term in ascending powers of d, and then equating the coefficient of h on each side. Now log sin {d + h)=log sin d + hcotd-^ h^ cosec* - etc. as in Art. 101 log(^ + /i) = logj-0(l + ^)j=log^ + ^ + i^ + ... [Art. 4.] log jl- ^-^/ i=log jl -^^, - ^-,^, - ^4 = log (^1 -^.j - ,^^,2^.-^^ (,.^._ gy -etc. [Art. 4.] Hence, making these substitutions in logsin(0 + A) = log(0 + /i) + log|l-^^±^'|+etc., and equating the coefficients of h, we obtain the required result. EXAMPLES. XXX. Prove the following statements : /,N 1 1 1 1 19 (2) Y^ + Y^ + ^, + -li + '-=^^*' (3) li + ^, + -li + Y, + --=^W. (4) The sum of the products of the squares of the reciprocals of every pair of positive integers is jJ^tt*. (5) ^^22_ _42 ^ ^^ 2 1.3*3. 5*5. 7""' 6—2 84 TBIGONOMETRY, (6) When n is even r\ cri^ • "^ • Stt .71-1 , (i) 2 2 sm-- .sm^-...sm-^^ — 7r=l. ^ ^ 2?i 2n 2?i /••\ o*^- TT 277 w-4 w-2 , (ii) 2^ cos -.cos — ...cos—r — TT. cos-— — 7r=V"'^. ^ ' Sin a \ a/ \ TT-aJ \ n + aj \ ^-ji - a) ^ ^ sm a \ aj\ Tr-aJ\ ir + aj\ ^it - aj ^^ cos a V 7r-2ayV 7r + 2ayV 37r-2ay (10) cos(a4-^) ^/ _^g_\/^^\A__2^ ■ ^ '' cos a V 7r-2aA 7r + 2aA 37r-2a cos ^ + COS g COS \{a- 6) cos ^ (g + ^) ^ ^ 1+cosa "" cos^^cos^^ ^vh^re l-(^, stands for |l-^^}|l_^j. cos^-cosg^/ _^^r ^M/i ^1 sin ^ + sing • '^ . sing =(^-3 o-.y (i-.y (^-4-a) 0-i^j . EXAMPLES, 85 (14) -±i^;:^^=(,-i)(i-JJ)(i+-Pi(i-^_^\... ^ ' sina \ a/V 7r-aJ\ 7r + a/\ 'zn-{-aJ (15) From the result of Ex. (11) deduce the factors of cosh X + cos a, (IB) From the result of Ex. (12) deduce the factors of cosh X - cos a. (17) From the result of Ex. (7) prove that I 2a 2a 2a Cota = a n'-a' 2^n^-a^ aSi^-a* (18) Un^=-^--|^ + 3-f^-3^4-etc. 1 1 2a 2a 2a ^^ sma~a"^7r^-a2 2 V - a^ "^ 3V - a^ •" (20) —^-—J- ^'^ 1 ^ 4 cos a 71^ - 22a 3^^ _ 2V 5V - 22a2 ' (2 1) Since e^ + er"^' - 2 cos 2a = 2 cos 2ix - 2 cos 2a = 4 sin (a + 1^) sin (a - tr), deduce from the factors of sin 6 those of cosh 2x - cos 2a. rrx TTX ^ ^ x^+r^^x^ + 2^^j:^ + b-^^^~4x^^ _'^ ^^•^^ 12+^2 + 22 + ^2 + 32 + ^2 + etC.-2^ • ^irx_^-TX-2:^2- <24) (1 + ^4-^, + ^+...) (1 1 -1 \ 2 4 + 1^ "*" 4 + 32 "^ 4 + 52"^ "7 ^ 8 * (20) cosec-^= -^^•"^(;,:23^2 + ^ ^2-V.^)2 3V^^ ^2(32;r2-^7-^+-- 86 TRIGONOMETBY, ^CHAPTER Yll. On the use of J{-1). 82. We propose now to make a few remarks on the use of ^(-1). ^(-l) has been defined [Art. 11] as a symbolical ex- pression whose square is (— 1), which is subject to all the laws of Algebra. Thus the factor J{- 1) must not be considered a qua7i' tity but a symbol of operation. In the same way the factor ( - 1) is to be considered not a quan- tity but a symbol of operation [cf. E. Chapter VIII.], Now the laws of Symbolical Algebra are identical with those of Arithmetical Algebra. Consequently any general result in Arithmetical Algebra must be considered true for all values of the symbols involved, provided those values are subject to the laws of Algebra. Conversely, when any result in Symbolical Algebra is capable of an Arithmetical Interpretation, we ought to ob- tain a result which is Arithmetically true. ON THE USE OF v^(-l). 87 This important principle is called the Principle of the Permanence of Equivalent Forms. Example 1. In Euler's proof of the Binomial Theorem for any value of the index, having defined / (w) we say that since fim)xf(n)=f{m + n) when m and n are positive integers, therefore /(m) x/(n)=/(m + n) when m and n have any values, provided those values obey the laws of multiplication. Example 2. In Art. 3 we have the following general theorem : " "-^"^ 1 ■*■ 1.2 "^ 1.2.3 ^••• which is proved for all arithmetical values of x and a. Now since i, that is \/(- 1)> is a symbol which obeys all the laws of Algebra, we may put cos o + i sin a for a in the formula ; as we have done in Art. 23. 83. We have in the proposition of Art. 23 a means of testing the truth of this principle. For, from the result of Art. 23 we obtain many important results. Now we shall find, that we are unable by any legitimate process of Algebra to get any result from that of Art. 23 whicli can be proved false by some other means ; in fact every result obtained from Art. 23 which can be tested by some other process will be found to be true. Example. Take the results of Art. 26 and 32; in Art. 41 an independent proof is given of the result of Art. 2G. Gregory's series may be proved by the method of Indeterminate Coefficients ; and the student will find that Gregory's series and all other series of a similar kind, whose Trigonometrical Proofs depend on the use of /vy(-l) and therefore on the principle of the Permanence of Equiva- lent Forms, can be verified by the Differential Calculus. 88 TRIGONOMETRY. 84. To tliose who feel a diflficulty in accepting proofs depending on the use of J[— 1), we recommend a careful comparison of Art. 47 with Art. 37, and of Art. 40 with Art. 19. The proofs in each case are virtually the same, but in the one case each step is capable of an arithmetical interpretation, in the other the work is abbreviated by the aid of the symbol J(— 1). 85. A complete Geometrical interpretation has been given to the symbol ij{- 1) of which the following is a short sketch. The student is referred to Professor De Morgan's Trigonometry and Double Algebra for further information on the whole subject. Take any origin and any initial line OE. Let the letters a, b, etc. be the measures of the lengths of lines expressed in some fixed unit. Then we make the following Convention. + « is to represent a line of length a drawn parallel to the direction from to R. J(- 1 ) a, or la, is to represent a line of length a drawn parallel to the direction from to U at right angles to OR. Then v/(- 1) {V(~ 1) ^'} ^^^^^ represent a line drawn at right angles to J{- \)a oi length a. That is - a will re- ])resent a line drawn parallel to the direction from to L (opposite to OR). This is in accordance with the con- vention already laid down in elementary Algebra [E. 122]. 86. In accordance with the above Convention a + J(- 1 ) h will indicate the ' sum' of two lines, one a units long, drawn parallel to OR, and the other b units long, drawn parallel to OB. GEOMETRICAL INTERPRETATION. 89 U Let 0'N= + a and J^P = J{- 1) 6. Then we may consider O'P equivalent to a + J{— V)h. For O'N + NP means, ' go from 0' to N and from F to P-; O'P means 'go from 0' to /*.' [E. Chapter VIII.] Now OT - J O'N' + PiV^» = V^^^TP. Thus the magnitude of O'P is what is called in Algebra the modulus of the expression a + J{- 1)6. 87. Again, let a + 16 = r (cos ^ + 1 sin 6), then r = Ja' + h^ = 0'P, tan^---tanxVO'P. Thus (cos ^ + 1 sin ^) r (where r is the number of units of length in O'P and 6 = N0'P) represents O'P in direction and magnitude. Hence cos ^ + t sin ^ is a symbol of operation, and in- terpreted geometrically means 'turn the line operated on through an angle ^.' 90 TRIGONOMETRY. Hence we have immediately De Moivre's Theorem. For (cos P + L sin jB) (cos a + 1 sin a) r means first turn the line of length r through an angle a and then on through an angle p. This is clearly equivalent to {cos (a + /?) + t sin (a + j8)} r. EXAMPLES. XXXI. (1) Prove that the two expressions r (cos a + i sin a) + / (cos [3 + 1 sin ^) and P (cos ^ + 1 sin 6) fo r^ f / r>\ J J. /I ^ cos a + r' cos 3 where p^=r^ + r^ - 2rr cos (a - B) and tan ^ = — ^ —^r. ^ ^ ' r sm a + r sm/3 are equivalent algebraically and geometrically. (2) Prove that the factor e^^^'^ where r is a whole number, is a factor which does not alter the quantity multiplied. RULE OF PEOPORTIONAL DIFFERENCES. 01 CHAPTER Till. The Rule of Proportional Differences, otherwise called The Theory of Proportional Parts. 88. The logarithms in tliis Chapter are Common Lo- garithms. In the Elementary Trigonometry a Rule called the Rule of Proportional Differences was given, and it was shown that, assuming the Rule to be practically true, we are enabled to use Tables of a more moderate size than would otherwise be necessary. [Cf. E. 218—222.] The Rule is as follows. The differences between three numbers are proportional to the corresponding differences between the logarithms of those numbers, provided the differences between the numbers are small compared with the numbers. 92 TRIGONOMETRY. In this Chapter we shall prove that this Rule of Pro- portional Diflferences is practically true as applied to the Table of Logarithms of Numbers, and that it is also prac- tically true in general as applied to Tables of Trigonometrical Katios and their Logarithms. 89. The Kule as applied to the Tables of Logarithms may be stated thus : Let n be any number greater than 10000 and less than 100000 ; let d be any number not greater than unity; then as far as seven places of decimals the following proportion is true : log (n-rd)- log n _d log (n + I) - log 71 1* 90. To prove the Rule for the Table of common lo- garithms. We have log (?^ + cZ) = log w (1 + - j = log n + log f 1 + - J (d Id' Id' \ r A ^ ^ 1 -log^ + /^|--2^+3-3-....j. [Art. 4.] Let n be not less than 10000 and d not greater than 1; also /x the modulus [Art. 7] is -4342945... . Hence fL is less than -5, - is not greater than '0001. Therefore ~ -^ i^ not greater than ^(-0001)', i.e. not greater than -0000000025; 73 ^ -^ is much less than this. 6 n Hence at least as far as seven decimal places log{n + d)-logn = ^ . PBOPORTIONAL DIFFERENCES. ^3 Similarly log (')i+l)-\ogn n Therefore jos(« + ^) -l"g« , ^ log {n+l) -log 71 1 which proves the Rule. 91. To prove the Rule for the Table of natural sines. That is, To pi^ove iliat if a he any angle, and 8, 8' angles not greater than 1', then sin (a 4- 8) — sin a 8 sin (a + 8') — sin a 8' as far as seven decimal places. We have sin (a + 8) = sin a cos 8 + cos a sin 8 = sina{l-J8'+...} + cosa{8-J8'+...} [Art. 41.] = sill a + 8 cos a — J 8^ sin a — ^ 8' cos a + ... . 8 is here the circular measure of an angle not greater than 1', .-. 8 is not greater than -0003 [E. Ex. x. 17] ; .*. |8^ is not greater than -00000005 and sin a is not greater than 1. Hence, as far as seven places of decimals in the value of the sines, sin (a 4- 8) — sin a = 8 cos a. Similarly sin (a + 8') — sin a = 8' cos a. Therefore «i"it8'(cot^-J8-. ..)'+... = logsin^ + /A8cot^-J/A8'{l+cot'^}+... = log sin 6 + ^h cot 6- J/m8' cosec' 6+ .... Hence, omitting higher powers of 8 than 8', we have L sin (^ + 8) - Z sin ^ - />i8 cot 6 - i/x8' cosec' 0. T ^ L. I 98 TRIGONOMETRY. If the Tables are calculated for every ten seconds, S is not greater than -00005, and therefore, unless cot 6 is small or cosec'' 6 large, we have Z sin (^ + 8) - Z sin ^ - /xS cot as far as seven places of decimals, which proves the Rule to be generally true. 102. When 6 is small, cosec 6 is large. Suppose that the Tables give the Zsin of every 10". Then 8 is not greater than the circular measure of 10", which is '0000484... and /A is not greater than J. Hence | fiW cosec^ 6 is not greater than — =-^0 — • ^^ order that this may not affect the seventh decimal place 6 cosec^ B must not be greater than 10^, that is 6 must not be less than about 5*^. Also when 6 is small, cot^ is large. Hence when the angles are small, the differences of consecutive L sines are irregular and they are not insensible ; so that the rule of Proportional Parts does not apply to the L sine when the angle is less than 5". 103. When 6 is nearly a right angle cot 6 is small and cosec 6 approaches unity. Hence when the angles are nearly right angles, the dif- ferences of consecutive L sines are irregular but they are at the same time insensible. 104. The case of the Table of L cosines. Similar conclusions concerning log cos 6 may be inferred from the formula log cos (^ + 8) = log cos + fjiS tan^ — J8^ sec^ 6+ The differences in this case will be irregular and large when is nearly a right angle, and irregular and insensible when 6 is nearly zero. This is also clear because the sine of an angle is the cosine of its complement. FAILURE OF THE RULE. 99 105. We find then that the Rule of proportional differ- ences cannot be applied to interpolate between the L sines of angles which differ by 10", when the angle is less than 5°. Three methods have been proposed to replace the Rule. I. The simplest plan is to have Tables giving the L sines for each second, for the first few degrees of the quadrant. II. In the following method we require a Table of the same size as that in method I., but it is a Table in which the differences are insensible. Accordingly we can with this table calculate the log sine of an angle which lies between two consecutive seconds. The method is as follows : Let be the circular measure of n seconds. Then when 6 is small 6 = n .sinl" very nearly. Hence sin^ Sinn" • /. , , . -,„ log -^ = log ^^^^y , = logsin n -logn- log sin 1"; T ' //I sin 6 J. . , ,, , .'. Lsmn = log — ^ + Lsm I + log n sin 6 log n + (log —^ + Z sin V'\ Hence, if a table is constructed giving the values of sin^ (log— ^ +Lsnil ] for every second, for the first few degrees of the quadrant, we can, when the angle is known, find the value of sin^ r • 1" log ~^— + // sin 1 from this table, while the value of log n can be found from the ordinary Table of the logs of numbers. And hence L sin n" can be found. 7—2 100 TRIGONOMETRY. Moreover -—^1 —^^^ + higher powers of 6, and 6 is by hypothesis the circular measure of an angle less than 5", so that —^ = 1-^6'^ approximately. Hence the differences in log — ^ when 6 is small, will be insensible {i. e. will not affect the seventh place of decimals in the result). [For, log '^+-^^-log^=log{l- 1(^+5)^} -log (l-i^^) the largest term in which, is -^dd, i.e. the product of two small quantities.] Therefore we shall not introduce any sensible error in a result obtained from the formula L sin n" = log n + (log — ^ + L sin 1" j if we take the nearest value of log —^ — hZsinl" in the Table. The ordinary table gives the value of log n. Hence we can find Z sinn" even when n is not a whole number. III. Maskelsme's Method. [This method is used in the absence of the special Tables required in I. and II.] When is small, we have sin ^=^(1-1^^); cos^-1-1^^ . •. — ^- = 1-— =(1-1 O^y approximately, u b = (cos 6)^, neglecting higher powers of B than 0^. Hence log sin ^ == log ^ + J log cos 6. Now, when is small the differences of log cos are in- sensible (Art. 104), and if 6 be given we can therefore find log sin at once. MASKELYNE'S METHOD. 101 If we are given log sin ^, we must first find from the table the nearest value of 0, and thence find a value of log cos 6 which will not sensibly difier from the exact value, and then we get log 6 = log sin ^ - 1^ log cos 0, and we thus get a second approximation for the value of 0. With the table of L tangents we proceed thus : log tan 6 = log sin 6 — log cos 6 = log ^ — f log cos approximately from above, and this result may be used in a similar manner. Example 1. Find L sin 1030'27"'2. Let X seconds = d radians. Then irx = 180 x 60 x 60 ; .-. log ^ = log a; + 6-6855749. Here 1030'27"-2 = 5427-2 seconds; .-. a; =5427-2. .♦. 10 + log^= log 5427-2 + 4-6855749; .-. L&md = 3-7345758 + 46855749 - i (-0001504) = 8-4201006. Example 2. Find when L sin ^=8-1021832. From the Tables by the ordinary Kule we find ^ = 43'30". Hence if x be the number of seconds in 10 + log^=loga: + 4-6855749 = L8in^ + i(Lsec^-10); .-. log x= 8-102832 + 5-3144251 + i (-0000348) = 3-4166257= log 2609-88; .-. 5 = 43'29"-88. The student should notice the equation 10 + log = log X + 4-6855749. 102 TRIGONOMETBY. EXAMPLES. XXXII. (1) Find the following Tabular Logs. (i) Xsinl044'36''-8. (ii) Zsin39'8"-4. (iii) Z tan l044'36"-8. (2) Find the angle d from the following equations : (i) Z sin ^=8-4832462. (ii) Z sin ^ = 8-2089620. (iii) Z tan ^ = 8-4834473. (3) Prove that, if n be the number of seconds in an angle $ Ztan^=log7i + 4-6855749 + |(Zsec^-10). 106. In practical work it is always advisable to avoid as much as possible that part of a Table in which the differ- ences are insensible. For example, a slight error in the calculation of the sine of an angle nearly 90" would entail a large error in the derived magnitude of the angle. This point is of such great practical importance that we have treated it at some length in the next chapter. 107. The preceding articles afford examples of an im- portant general principle which is of great use in higher mathematics. If a continuous function of a variable x increases as x approaches a certain value a, and begins to diminish directly X has passed the value a, then the ratio of the differences of the function to the corresponding small differences in the variable x will diminish and approach to zero as a limit when X approaches a. Thus, sin ^ is a continuous function of 6 which increases as approaches Jtt, and it begins to diminish directly 6 has passed through the value Jtt ; hence, as is proved in Art. 92 the ratio of sin (0 + S) - sin ^ to S tends to become insensible as 6 approaches Jtt. MASKELYNE'S METHOD. 103 108. To sum up the Kesults of this Chapter. The Rule of Proportional Differences may be used with- out sensible error in the following cases : I. For a Table of Common Logarithms giving the logs of all numbers from 10000 to 100000. II. For a Table of Trigonometrical Ratios calculated for intervals of one minute from 0** to 90". Except in the case of the tangent and secant of angles greater than 45", the cotangent and cosecant of angles less than 45". III. For a Table of the Tabular Logarithms of Trigono- metrical Ratios calculated for intervals of 10" from 0° to 90". Except in the case of the L sines and L cosecs of angles less than 5", the L cosine and L secants of angles greater than 85", the L tans and L cotans of angles less than 5" and greater than 85". 109. The results of this Chapter may also be obtained without actual reference to the expansions of sin 6 and cos 6 in terms of 0, by the aid of the fact that the difference be- tween sin 8 and 8 is less than \W] so that when 8 is less than 1 degi-ee, sin 8 and 8 differ by less than -0000008. The method of procedure is suggested in the following examples. 104 TRIGONOMETBY. EXAMPLES. XXXm. [In these examples 8 is the circular measure of any angle less than 1'.] (1) Prove that sin (B + b)- sin ^=sin d cos ^ (1 - tan tan ^S) : Hence prove that as far as seven places of decimals sin {B + d)-smd=d cos 6. (2) Prove that cos {d-8)~ cos ^ = sin S sin ^ (1 - cot 6 tan |8) : Hence prove that as far as seven places of decimals cos {6 -d)- cos 0=8 sin 6. (3) Prove that tan ($ + 8)- tan ^=tan S sec^^ ( - — r — th. — s ) : ^ ' \1 - tan ^ tansy Hence obtain the results of Arts. 95, 96. (4) Prove that cot {d — 8)- cot B==h cosec^ approidmately. (5) Prove that ,^ ^, . tan S sin ^ (1+ tan is cot ^) sec (0 + 8)- sec0= ^■.,, . — ^r — ^:^ - ^ ' cos^ ^ (1 - tan tan 8) hence prove, except when is small or nearly equal to ^, that sec (^ + S) - sec ^ = S sin sec^ 0. (6) Prove as in example (5) that cosec {0 - 8) -- cosec 0=8 cos cosec^ 0. ERRORS IN PRACTICAL WORK. 105 CHAPTER IX. On Errors in Practical Work. 110. We have already [E. 217, 227] called the student's attention to the approximate nature of all observed measure- ments. Example. Let the student take any well-defined length, say of 6 or 7 inches, and attempt to ascertain its measure, say to the hundredth part of an inch ; and let him repeat the process at another time with different instruments. He will find that unless he makes his measurements with the utmost care, and unless his instruments are very accurately constructed, his two results will in all probability be different. Such an observation as the above even when made with the greatest care can only be taken as correct to three significant figures. If the measurement has to be made correct to a thousandth part of an inch or to any higher degree of accuracy, the student will easilj' understand that it will be necessary to employ specially constructed instruments. The ordinary diagonal scale or vernier cannot be read with accuracy to the thousandth part of an inch. 106 TRIGONOMETBY. 111. The student must carefully distinguish between mistakes and errors. By taking sufficient trouble a calcu- lation can always be made to attain any required degree of accuracy; so that in what follows we are not concerned with mistakes or inaccuracies in calculation at all. 112. An error may be defined as follows. Suppose an observation made and the result known to be accurate as far as a certain number of significant figures, according to the degree of approximation thought necessary or possible, under the circumstances. The measure taken may possibly give the magnitude of the quantity with absolute accuracy, we cannot say whether it does or not. What we do know is that the difference between the actual magnitude of the quantity and the as- sumed magnitude is less than a certain quantity. This quantity is the possible error; and it should be so small that it is either considered of no importance, or is beyond the limit of observation in the circumstances of the case. 113. It is clearly not necessary to carry our calcu- lations to any higher degree of approximation than that represented by the assumed measure. 114. In the practical application of Trigonometry to En- gineering and Land Surveying we are concerned with two different kinds of measurements. (1) The measurement of lines. (2) The measurement of angles. The measurement of a line of any length with anything like the accuracy of five or six significant figures is a very difficult and tedious opera- tion, and is but rarely performed. We know that by the methods of Trigonometry the known length of one line may be made the basis of the calculation of the lengths of all •other lines in the survey of a country. ERRORS IN PRACTICAL WORK. 107 115. The importance of an error in linear measure- ment is generally measured by the ratio of the error to the estimated length of the distance under consideration. Example. The problem of calculating the distance of the Sun's centre from that of the earth is beset with such great practical difficulties, that astronomers are only able to say that it is about 92 millions of miles. If we knew the distance to icithin a hundred thousand miles, that is, to within about a thousandth part of the distance we should consider the distance to be known with wonderful accuracy. In a distance of this magnitude an error of a few thousands of miles is of no importance. The importance of an error in angular measurement depends in general simply on the magnitude of the error. 116. If the measure of any length is known accurately to seven figures it is practically exact. In other words it is known to within the limits of observation. Example. A base line on Salisbury Plain measured with extreme care for the purposes of the Ordnance Survey in England is about 36578 feet in length, and the error is considered to be certainly less than 2 or 3 inches. That is, the error is less than a hundred thousandth part of the whole, and the measurement has been made correct to six significant figures. The greatest accuracy possible in the measurement of angles is attained when the error is known to be not much greater than the tenth part of a second. The tenth part of a second is about the two millionth part of a radian. This degree of accuracy is only attainable under special condi- tions and with the largest and best instruments. 117. It sometimes happens in the course of a calcula- tion that an error rises in importance in consequence of its being multiplied by a very large number. 108 TRIG ONOME TR Y. We may illustrate this by an example. The height ^ of a tower is ascertained by measuring a horizontal line a from its base and observing the angle of elevation of the top of the tower from the end of that line. Then we obtain h~a tan 0. Now supposing that we are liable to an error not greater than 8 in the observed magnitude of we require to know how this will affect the accuracy of the calculated height h. We know that the error in 6 does not exceed 8. Hence we know that the consequent error in h cannot exeeed k where h-vh^a tan {0 + 8). Hence h = a {tan (^ + 8) - tan &\ - ah sec^ 6 neglecting squares and higher powers of 8. [Art. 95.] Hence the ratio of the error k to the calculated height h is 8 28 sin ^ cos ^ sin 2^* 118. The above result is very instructive. Suppose the measurements are made with the greatest possible care so that 8 is beyond the limits of observation and may be neglected. Then we see that in general the importance of the possible * error' in the calculated height, i.e. 28cosec2^ is, in general, comparable with 8, and is therefore very small. Also this error is least important when cosec 20 is least, i. e. when is ^tt. ERRORS IN PRACTICAL WORK, 109 There are two cases however when the error may become of sufficient importance to render the result practically in- accurate. I. when is small, II. when is nearly ^tt. In the first case the error itself is not large but it is large compared with the height to he measured. In the second case the error itself is very large, and although the height to be measured is large compared with the base a, the importance of the error is also large com- pared with 8. It is a difficulty of this latter kind which renders the estimated distance of the sun from the earth so untrustworthy. 119. We have seen that the ratio of the difference in the sine of an angle to the difference in the angle, is small when the angle is nearly Jtt. That is to say, to a small error in the sine would correspond a large error in the angle. Now, if the sine of an angle has been calculated from observations, and it is found that the value of the sine is nearly unity, we could not without risk of a large error use the value of the angle obtained from the Tables. For, our observations are known to he liahle to errors (whose magnitude depends on the instruments used, etc.), and therefore the calculated value of the sine under consideration is liable to an error of the same kind. Consequently the calculated value of the angle would be liable to a much larger error. And this larger error would possibly affect all results in which the magnitude of the angle was used. Accordingly in practical work an observer would when possible arrange his measurements so as to avoid such a difficulty — in the working out of a problem — as the necessity for obtaining from the value of its sine, the magnitude of an angle nearly equal to a right angle. 110 TRIGONOMETRY. 120. The metliod of Art. 117, which may be applied generally, is of very great importance in practical work; for an observer can often in this way discover beforehand whether any proposed arrangement of his measurements is defective and likely to give unreliable results. Thusj if the measure of a distance or of any trigono- metrical function of an angle be found by means of observed angles and distances, the result is expressed by some formula containing the Trigonometrical functions of the observed angles. If a small error 8 be known or suspected in an observed angle $, we can find the consequent error in the calculated distance by expanding this formula by the methods of the last chapter in ascending powers of S. Then, 8 being so small as to be detected with difficulty, 8^ and higher powers of 8 must be quite beyond the limits of observation. We can in this way estimate the importance of a small error in observation. Example. A vertical pole a feet high stands on the top of a cliff, and from a point on the shore the angles of elevation a and /3 of the top and bottom of this pole are observed. The height of the cliff h is sin 3 cos a r-n, /r.^ -i given by ^ "^ ^ sin (a - ^S) ' ^ ''^''"* ^^^'^ Now suppose an error 5 to have occurred in the observed measure- ment of the angle a, required the consequent error h' in the calcu- lated height of the cliff. TTT u I, , 1./ sin/3cos(a + S) We have h+h'=a . , ^ — r-^ sm (a - /3 + 5) _ a sin /3 { cos a - 5 sin a - etc. } " sin (a - /3) + 5cos (a - /3) - etc. a sin S cos a {1-5 tan a} , ,. .-„ , ERRORS IN PRACTICAL WORK. HI , , ^ a sin 8 cos a , , , ^, , h' = d — — /- — -r-{cot (a-/3)-tana}. sin (o - /3) ^ t-' ' Thus the ratio of the error to the estunated height is 5 {cot (a -]8)- tan a}. EXAMPLES. XXXIV. (1) A triangle is solved from the given parts A, b, c; if there is a small error 8 in the angle A prove that the consequent error in the calculated area of the triangle B is approximately ^Sbc cos A . (2) A triangle is solved from the given parts A,bjC; if there is a small error 8 radians in A, prove that the consequent error in ^ is - 8 sin B cos Ccosec^ radians. (3) If the sides of a triangle be measured and a small error c' exist in the measured value of c, prove that the consequent error in the diameter of the circumscribing circle is c' cos A cos B sin A sin B sin C (4) The height and distance of an inaccessible object are found by observing the angles of elevation a and /3 at two points A and 5 in a horizontal line through the base of the object, the distance between A and B being known ; if the same error be made in each in consequence of an imperfect observation of the horizontal, shov that the ratio of the error in the calculated height of the object to the calculated distance is tan(a + /3) : 1. (5) The area of a quadrilateral AOBQ right-angled at A and 5 is to be determined from observations of the angle AOB, and the length (p and q) of OA and OB. Prove that the area is ^ {2ab - (a2 + 62) cos 6} cosec $, and that if a small error 8 be made in the observation of the angle AOB the consequent error in the area is U.AB^.cosec^AOB. 112 TRIGONOMETRY. (6) If the angles of a triangle, as computed from slightly erroneous measurements of the length of its sides, be J., B, (7, prove that approximately, a /3 y being the errors of the lengths, the consequent errors in the cotangents of the angles are propor- tional to j3cosC + ycos^-(i, ycos^ + a cos (7-/3, acosi5+^cos^ -y divided resiDectively by sin A, sin B, sin C. (7) It is observed that the altitude of the top of a moun- tain at each of the points AB and C where ABC is a horizontal triangle is a. Shew that the height of the mountain is ^ a tana cosec J.. If there be a small error n" in the altitude at C the true height is very nearly , a tan siuul ai cos (7 sinn") . \ sin A sin B ' sin 2a j ' (8) If in a triangle ABC the observed lengths of a, b, c are 5, 4, 6 and there is known to be a small error in the measurement of c, determine which angle can be found from the formula **°2-\/l sis-a) / with the greatest accuracy. [Result. A.] CHAPTER X. Examples op the Application of Trigonometry to Geometrical Problems. 121. In this chapter we shall use the following notation : i), Ej F are the feet of the perpendiculars drawn from the angular points A, B, C of the triangle ABC to the opposite sides. AD, BE, CF intersect in a point P which is called the orthocentre of the triangle ABC. DEF is called the pedal triangle of the triangle ABC. A'B'C are the middle points of the sides BC, CA, AB. AA', BB\ CC intei-sect in a point G, which is called the centre of gravity of the triangle ABC. I, /j, /g, /g are the centres of the inscribed and escribed circles of the triangle ABC ; r, r^, r^, r^ are their radii. [E. 276, 278.] is the centre of the circumscribing circle and R its radius. The circumscribing circle of the triangle BEF passes through A'B'C and through the middle points of each of the lines PA, PB, PC. It is called the nine-points circle. We shall denote its centre by N. [Proofs of the propositions referred to above maj' be found in the appendix to Todhnnter's Euclid.] L. 8 114 TRIGONOMETRY. Example 1. To prove that P, N, G and are in one straight line; and that PG = 2GO = 4NG, i.e. that N is the point of bisection, and G a point of trisection of I'D. N is the centre of a circle passing through D and A'. Therefore N lies in the line bisecting DA' at right angles. This line produced bisects OP. Again, the nine-points circle passes through E and B', therefore its centre N lies on the line bisecting EB' at right angles. This line produced also bisects OP. Therefore N is the middle point of OP. , . ,„ AE AE c cos A Again AP= =.-.- = -r—^ = . ,, ; cos PAE sin C sm C AP = 2RcosA. But 0^' = EcosjBO^'=Ecos^; AP=20A'. Hence if AA' cut PO in G, AG : GA'=PA : 0A'=2 : 1 ; .'. AG = 2GA' or G is the centre of Gravity. Also PG :GO=PAiOA' = 2: 1. q.e.d. GEOMETRICAL PROBLEMS. 115 122. It is often convenient, in attempting the solution of a geometrical problem, to express the lengths of lines involved each in terms of some common unit. When the problem is one concerning a triangle, the Kadius of the Circum- scribing circle may be employed as the unit. Its con- venience is shewn by the symmetry of the following re- sults : — EXAMPLES. XXXV. Prove the following statements : (1) a=2RsiaA, 6=2Esini?, c=2RamC. (2) s=R{aiD. A+sm B + aiaC)=4Rcoa^A . cos^B . cos^C. (3) r=4R sin ^A . sin ^B . sin ^C. (4) 7*1 = 4R sin ^A . cos ^B . cos ^C. (5) ^i) = 2/2 sin ^. sin a (6) PD=2Rco3B .cosC, (7) ^P=2i2cos^. (8) OA'=RcosA. (9) S=^ 2i22 sin A.ainB. sin C. (10) The radius of the nine-points circle = hR. (11) The sides of the triangle DBF are i2sin2^, i2sin2i?, R Bin 20. (12) The area oiDEF=\m sin 2A . sin 2i5 . sin 2C. (13) i?i>=^/j;(2cos5sinC + sin^). (14) dN=]^RcoB{B-C). (15) The distances of the centres of the escribed circles from that of the inscribed circles are 4i2sin^^ 4R&m\By 4i2sin^(7. (16) AE'=R{mnB + smC-B,mA\ where E' is the x)oint in which the inscribed circle touches AC. 8—2 116 TRIGONOMETRY. Example. To prove that the nine-points circle touches the inscribed circle. Draw ID' perpendicular to B€ and NH perpendicular to ID'. (I). Again and Then (see Figure on page 117) Nd=i{PD + OA') = ^ {2R cos B cosC + RcoaA) = ^Rco8{B-C); HI=D'I-dN=r-iRcos{B-C) Bd = i {BD + BA') = l R {2 cos B sin C + sin A), BD'-S -h = R (sin A-8inB + sin C). [E. 280.] IIN=^ R {2 cos B sin C-sin^ + 2sinB- 2sin(7) = R {sinB-sinC-isin{B-C)} (U). Hence IN'^ = r^ + (i R)^ - Rr cos {B-C) + R^{ (sin B - sin C)- - (sin B - sin C) sin {B - C) J . The last bracket is equal to R^ [ninB - sin C) { 2cosi (JS + C) sini (^ - C) - 2 sin|(5 - (7)cos J {B - C) J =r JJ2 4 cos i (B + (7) sin2 ^ (B -Cf) {cos ^{B + C)- cos 1{B-C)} :r. - 8^2 sinS ^{B-G)sin^A .sin^B . sin -| C [xxxv. (3) .] r= -2i2rsin2J(B-C)= -i?r {l-cos(5- C)} ; JA"^^ = ?-2 + (i i2)2 - i2r =: (r - |i2)2. That is, the distance between the centres of the circles equals the difference of their radii, q.e.d. GEOMETRICAL PROBLEMS. 117 EXAMPLES. XXXVL Prove the following statements: (1) The radii of the circles eircumscribing AEF, BFD, CDE are respectively R cos A, R cos B, R cos C, (2) AI=4.R^m\B.BmW. (3) AN^ .-= J^ (3 + 2 cos 2.1 - 2 cos 22? - 2 cos 2C). (4) 0P = B^-2Rr. (5) 0I^^=R^^-2Br^, (6) OP^=R^ (1 - 8cos.i. cos5. cos C) = 9Rr--a'^-b^-c^. (7) /P2=4^2(8sin2^yl.sin*'*i5.sin«^(7-cos J[. cos^.cosC). (8) The area of the triangle /0P= - 2i22 sin ^{B-C). sin H^'- ^1) • sin ^{A- B). (9) T^N^={\R + r,Y. 118 TRIGONOMETRY. 123. To prove that the orthocentre is the centre of the circle inscribed in the triangle J)EF. The circle of which PC is diameter passes through U and I). Therefore the angle UDC ^ EPC = the complement of FCU = A. Similarly FDB = FPB=^EPC = A. Therefore PDF = W-A= PDF. Therefore P is the centre of the circle inscribed in DEF. Similarly A, B, C are the centres of the escribed circles of the triangle BFF. EXAMPLES. XXXVII. Prove the following statements : (1) ABC is the pedal triangle of the triangle l^^Iz' (2) The radius of the circle circumscribing Ij2h i^ 2^. (3) DEF is the pedal triangle of the triangles APB^ BPC, CPA. (4) The radius of the circumscribing circle of the triangle APB=R. (5) The circle circumscribing DEF touches the circle in- scribed in ABP. GEOMETRICAL PROBLEMS. 119 MISCELLANEOUS EXAMPLES. XXXVm. Prove the following statements : (1) If a new triangle is formed by joining the centres of the three escribed circles of a triangle ABC the distances of the centres of its escribed circles from the centre of its inscribed circle are SR sin i{B + C\ 8Rsm^{C+A), SRsin^iA + B). (2) The areas of the triangles IJ^Izi ^ih^i ^z^hi ^h^2 ^^® ^^ one another inversely as the ratio of r : r^ : rg : r^. (3) The radii of the escribed circles are the roots of the equation {x^ + s-) {a;-r) = 4Rx^. (4) PAf PBj PC are the roots of the equation ^ - 2 (R + r) x^+ (/'2 - 47^2+0 ^- 2^ (s^ - (r+2i2)2} = 0. (5) If Pi, p-ij Ps) — i^i are the perpendiculars from ABC £im\ P on the sides of the triangle DEF^ then p^, jOgj Pz^ Pi ^® *^^^ roots of the equation a:* - 2R^+ {^^ - 2Rr' - r'^\ x"- - ^^ r'2 = 0, where / is the radius of the circle inscribed in the triangle BEF. (6) The area of the triangle formed by joining the points of contact of the inscribed circle is 2i?2 sin A . sin 5 . sin C . sin ^A . sin ^B . sin ^C. (7) If the points of contact of each of the four circles touch- ing the three sides of a triangle be joined, and the area of the triangle thus formed from the inscribed circle be subtracted from the sum of the areas of those formed from the escribed circles, the remainder will be double the area of the original triangle. (8) If i?i, R2, R3 are the radii of the circles BGC, CGA, AGB, then a2(52-c2) , 62(c2-a2) c2(a2_62) i2i2 ' u^z ' /i^z = 0. 120 TRIG 0N02IE TR Y. (9) If ^ stand for AI, y for i?7, z for (77, then «%'* + hY + c^^* + (a + ?> + c)2 ^2^V _ 2 {6 Vy%2 + c2a%2^-2 + aWxY']- (10) If lines join the points of contact of each escribed circle of a triangle ABC with the produced sides and these lines form a new triangle, then the lines joining the corresponding vertices of the triangle are perpendicular to the sides of the former triangle and are equal to the radii of the escribed circles. (11) Given the circumscribed and inscribed circles of a tri- angle, prove that the centres of the escribed circles lie on a fixed circle. (12) The sum of the reciprocals of the perpendiculars of a triangle is equal to the sum of the reciprocals of the radii of the escribed circles. (13) If from a point P perpendiculars PZ, PJ/, PN are drawn to the side of the triangle xiBC^ prove that twice the area of the triangle LMN= [B? - [POf] sin A sin B sin C. (14) The centres of the escribed circles must lie without the circumscribing circle, and cannot be equidistant from it unless the triangle is equilateral. (15) rUI^ . 11^ . II^^IA'^ . ZS2 . I0\ (16) The area of the triangle whose angular points are the points of contact of the inscribed circle is to the area of the tri- angle ABC as r : 2R. (17) If JDEF ixre the points of contact of the inscribed circle with the sides of the triangle ABC, then if AI)% BE\ CF^ are in A. P., a, b, c are in h. p. (18) From DBF perpendiculars are drawn to the adjacent sides of the triangle ABC ; prove that the feet of these six per- pendiculars lie on a circle whose radius is B (cos2 A cos^ B cos^ C+sin^ A sin^ B sin- (7)* GEOMETRICAL PROBLEMS. 121 (19) If on one side BC of a triangle ABC a triangle A'BC is described without it such that the angles BA'C, CBA', A'CB are equal to a^y respectively, then AA"^ sin a = sin ^ sin y {a?- cot a + ^^ cot j3 + c ^ cot y + 4A). (20) If a triangle be cut out in paper and doubled over so that the crease passes through the centre of the circumscribing circle and one of the angles A^ the area of the doubled portion is ^/>2 sin2 (7 cos (7cosec i^C- B) sec {C- A), C being greater than B. (21) If 10 = IX, prove that one angle of the triangle ABC is 600. (22) If two of the angular points and the radius of the cir- cumscribing circle of a triangle are given, the loci of the centre of the nine-point circle and of the ortho-centre are circles. (23) Prove that a triangle can be constructed whose sides are a cos yl, 6 cos i?, c cos C and that its area is 2 A . cos A . cos B . cos C. (24) If i^i, ^2 J ^3 ^^® ^^^ Yi\.d.n of the circumscribed circles of BIC, CIA, AIB, prove that B^^ . R./ . R^'=R^ . AI. BI . CI. (25) If the two straight lines which bisect the angles A and C of a triangle ABC, meet the circumference of the circumscribing circle in R and S, then RS is divided by CB, BA into three parts which are in the ratio sin^i^l : 2sin^^ . sin^i? . sin^C : sin^lC. (26) If a point be taken in an equilateral triangle such that its distances from the angular points are proportional to the sides of a triangle ABC, the angles between these distances will be ^ + A,lir+B,^7r + C. /o-N J. T^T . 2(smj5-sm(7) (2 i ) tan IOIi=± -^^ -. — p-^ . ^2 cos A - I CHAPTER XI. On the Use op Subsidiary Angles to facilitate Numerical Calculation. 124. In the Elementary Trigonometry j Art. 185, we have shewn how the Tables may be made use of in the solution of Simple Trigonometrical Equations. It is usual to shew how the Tables may be made use of to facilitate the calculation of the roots of quadratic and cubic equations. The solution of such equations is however rarely required in practical work, so that the method is not of much practical importance. 125. To obtain the num,erical values of the roots of a quadratic ecLuation. I. Let the equation be x^ - 2px + q ~0, where p and ^ are positive. Solving, we obtain NUMERICAL EQUATIONS. 123 First, let q be less than p^; then we can find from the Tables an angle a such that sin^a = ^. Whence we obtain x = p{l ±cosa}. Secondly, let q be greater than p^ ; then we can find from the Tables an angle a such that sec^a = ^ , then x=p{l^^{-l) tana}. II. Let the equation be x' + 2px +q = 0. Then the roots of this equation are equal to those in Case I. with the signs changed. III. Let the equation be oc^ — 2p.r — q = 0. Solving, we obtain We can find from the Tables an angle a such that q tan a = -^ , P and then x =p{l ± seca}. IV. The roots of x^+ 2px -q = are equal to those of Case III. with the sign changed. Example. Calculate the value of the roots of the equation a;2 - 3 -^eSl*- 7-6842 = 0. 3-4651 L //, 4x7-6842\| Solving, x=-^- |li ^{^1 + -^^^^)^ . 124 TRIGONOMETRY, „ ^ 30-7368 Hence tana= ; (3-4651)2' .-. L tan a=: log 30-7368 - 2 log 3-4651 + 10 = 10-3057240; .-. a = 630 40' 55"; .-. a; = 1-73255 {1± 2-255356}. 126. The student will observe that this method is the same as that of adapting the expression jt? |l ± ^/(l + "af) to Logarithmic calculation by means of the Trigonometrical Tables. EXAMPLES. XXXIX. Solve the equations : (i) ^2 + 3.416^^ _ 8-794=0. (ii) ij;2_ 7.941^4.2-7001 = 0. 127. To obtain the numerical value of the roots of a cubic ecLuation. Let the equation be y? + 3^03'' + Zqx + r = 0. Write y — j) for a?, and the equation becomes 2/'-3(/-?)y + (2/-3^^ + r) = 0. Therefore any cubic equation can be transformed into another in which the second term is wanting. CUBIC EQUATION. 125 Hence we may take as our standard equation x^ - 3ax +6 = 0. To solve this writs wy for x, thus n^l/ — Sany + 6 = 0, ,3 But, if a be any angle, we have (E. 167) cos'a - J cos a - J cos 3a = 0. Hence, if we find a such that 46 cos 3a = ^ , while n = 2Ja, then cos a is one of the roots of the equation. Also since cos(2?i7r± 3a) = cos3a, the other two values of 2/ are cos(f7r + a) and cos(j7r - a). But X = ny. Therefore the required roots are 2(a)* cosa, 2(a)^ cos(§7r + a), 2(a)^ cos(f7r-a). 3a can be found provided ^ba~* is less than unity, i. e. provided 6* is less than ia'*. EXAMPLES. XL. (1) Solve the following equations : (i) ^-s.a^+i^^o. (ii) .r3-|a:-^=0. (iii) ;p3_ 3^2 + 3 = 0. (iv) :j73 + 12x2 + 42^' +44 = 0. ( v) x^~ 3V(3)x2 - 3a? + V(3) = 0. 126 TRIGONOMETRY. (2) Solve the equations : (i) ^3 _ 439^ _ 101 = 0. (ii) ^3 _ 17651^ _ 371462 = 0. (iii) 1001.^?3 _ 18472^ - 7941 = 0. (3) Adapt the following expressions to logarithmic com- putation : (i) a±b. (ii) a cos ^±5 sin ^. (iii) sin J. + sin 5 + sin C - sin {A+B+ C). (iv) 1 + cos {26 - 2md) - cos (26 - 2a) - cos {2md - 2a). (v) a cos A + b cos B + c cos C, where A, B, C are the angles of a triangle. (4) If ke^^ be one of the roots of the equation aP + 2qx+r=0 prove that Sq= -k^ {1 + 2 cos 26) and r = 2P cos 6. 128. We shall conclude this chapter with some ex- amples of Elimination. Example I. Eliminate 6 from the equations a cos 6 + b 8110.0=6, a'cosd-i-b'Bine=c'. Solving these equations, we obtain „ h'c -he' . „ c'a -cd cos 6 = —r, jr. Bin 6 = ah' - a'b aV - a'b ' But cos2^ + sin2e=l. Therefore {b'c - bc'f + {c'a - cdf = {ab' - db)\ This is the required result of ehmination. ELIMINATION. 127 Example II. Prove that the result of eliminating 6 from the equations X cos^-?/sin^=2acos2^, X ^ind + y cos d = 2a sin 26 ^ is {x + y)^-\-{x-yY = 2a^, Solving these equations for x and y, we obtain X = a cos 2d cos 6 + 2a sin 2^ sin ^ = a cos3 ^ + 3a sin^ 6 cos ^, y = 2a sin 2^ cos ^ - a cos 26 sin ^ = 3a cos^ ^ sin ^ + a sin^ 6 ; .-. x + y = a (cos 6 + sin ^)3, x-y=a (cos - sin ^)3 ; .-. (a; + y)^=a^{l + 2cos^sin^}, (x - y)^=o^ {1-2 cos e sin 6}, and the result follows immediately. EXAMPLES. XLI. ( 1 ) Giveu that ^, cos ^ = ^. cos ^ + "^ cos 0, Md * - ^ '- prove that sin(d + (^) sin(^-0) sin 25' sin 5 62 sin (^ a^ ' (2) Eliminate 5 from the equations x=2a cos 6 cos 25 - a cos 5, y = 26 cos 5 sin 25-6 sin 6, 128 TRIG ONOME TR Y. (3) Eliminate a and /3 from the equations j; = {a sin'' a + b cos^ a) cos^ /3 + c sin^ ^, 7/ = acos^a + bsm^a, z = {b - a) ain a coa a cob ^. (4) Eliminate ^ from the equations a; + a = a(2 cos 6 - cos 2^), y = a (2 sin ^ - sin 2^). (5) Eliminate 6 from the equations bcos,^0+asm^$=0, & cos^ (^ + 0) + a sin^ (^ + 0) = 0. (6) Eliminate 6 from the equations ^ = a (cos 6 + cos 2^), y = 6(sin^ + sin2^). (7) Eliminate 6 from the equations (a + 6) tan (^ - ^) =. (a - 6) tan ((9 + 0), a cos 2(^ + 6 cos 2^ = c. (8) Eliminate d and ^ from the equations ax by „ ,„ cos ^ sm ^ ax by ^ cos SHI ^ 6-cji = ^7r, (9) Eliminate ^ from the equations cos^ 6 sin"^ ^ cos (a - 3^) sin (a - 3^) (10) Eliminate a from the equations sin d cos 6 1 ^2 - 1 2^ sin 2a 1+2^ cos 2a 4-/^=^ ' shewing that ^ = tan ^ (tt + 2^). ELIMINATION. 129 MISCELLANEOUS EXAMPLES. XLU. (!) If acos(3+(t)) + hco8{d-cl)) + c = 0, a cos ((fi + yJA) + b cos { differ by an even multiple of tt. (10) If iA+B + C) = . and if cos 2X = g-^[*^|- , then tan A + tan B + tan (7= + 1. (1 1) If cos a = cos j3 cos (^ = cos /3' cos 0', and sin a = 2 sin ^ sin ^ , then tan ^ = tan ^ . tan ^ . 2 2 2 /I o\ T^ A ^ si^ a sin ^ , sin a sin (12) If tan d) = — ^ , then tan 6 = ^ ^- . ^ ' ^ cos ^- cos a cos ^ + cos a ELIMINATION. 131 (13) If j3 and y be two values of 6 which satisfy -cos^ + ^sm 6=-, a c R-Uy 8 + y — y then ' acos^-— -^ = 6sin~-^ = ccos^-— ^. Z Ji ^ ( 14) Given a^ cos a cos /3 + a (sin a + sin /3) + 1 = 0, a2cosacosy + a(sin a4-sin'y) + l = 0, prove that a^ cos /3 cos y+a (sin j3 + sin y) + 1 = 0, and that cos a + cos j3 + cos y = cos (a+^ + y), /3 and y being unequal and less than tt. (15) If ^1 and 02 ^^^ *wo values of 6 which satisfy cos ^ cos (^ sin ^ sin ^_ J- T " 7} "T" • o ' — '-'i cos^ a sm-^ a ' shew that B^ and ^2 if substituted for B and in the equation will satisfy it. (16) Solve the equations cos (^ + a) = sin (^ sin /3, cos (0 +^) =sin 6 sin a, and shew that if ^^ and )02 ^ *^® *^'o values of y X /^ , ^ N sin 2/3 tan (i + 60) = -.- - V, „^ o- . (17) If cos(a + ^) + mcos^=?i, n^ cannot be greater than 1 + 2m cos a + m^. (18) Eliminate 6 and ^ from the equations ^cos^ ysind_ a; cos (/> y sin ^ _ a ha 6 ' proving that -^ + '^^ = a" 6" 1 - cos 2a 9—2 GENERAL MISCELLANEOUS EXAMPLES. XLHI N,B. — For convenience in printing, some writers use n ! to denote 1.2.3...n. (1) A person walks from one end J. of a wall a certain distance a towards the West, and observes that the other end B then bears E.S.E. He afterwards walks from the end B a distance (^/2 + l) a towards the South, and finds that the end A bears N. W. Shew that the wall makes an angle cot~i2 with the East. (2) A man on the top of a hill observes the angles of depression a, 8, y of three consecutive milestones on a straight horizontal road running directly towards him; prove that the height of the hill is ^^'K cot^a-2cl% + cot2y ) ^^^^^• (3) sin2 03+y) + sin2(7 + a) + sin2(a + ^) = 4 sin a sin ^ sin y cos(a+^+'y) + 4 cos a cos/3 cos y sin(a+j84-y). (4) If 2a + 2^+2y=?i7r, then sin 2 (^+y) + sin 2 (y + a) + sin2 (a+iS) n-l = 2(-l)^ {1— (—1)"} cos a cos ^ cosy + 2(-lf {l + (-l)"}sinasin/3siny. GENERAL MISCELLANEOUS EXAMPLES. XLIIL 133 logJog^JV logjlogjiV log Jog, 6 logjlogja (5) ^iog^b Vlogja s/log^b Vic (7) If ^+i? + (7=7r, and sin3 6 = sin (^ - 0) sin (i?- ^) sin (C- $), then will cot ^ = cot ^ + cot ^ + cot C. (8) Eliminate ^ from the equations (a + b) (^ +^) = cos ^ (1 + 2 sin2 ^), (a -6) (^-y)=sin^(l + 2cos2^). (9) If cos {d — cf}) is a mean proportional between cos(d + (fi) and sin (^ + 0), then cosec 2^ + sec 20 = cosec 20 + sec 2 •^ ' ^ sm (y + /3)cos2a and cos ^ = (a + /3)cos^y sin (y + a) of these fractions is e sin (j3+y) sin (y + a) sin (a+^) cos O + y) cos (y + a) cos (a+/3) + sin2(a + i3+y)' (20) If \/2cos.4=cosi? + cos3^, ^2 sin A=smB- sin^ J5, then ± sin (^ - J.) = cos 2^= J. GENER4L MISCELLANEOUS EXAMPLES. XLIII. 135 (21) If 4 cos {x — y) cos (y - 2) cos (2 ~ ^) = 1 , prove that 1 + 12 cos 2 (^-y) cos 2 (y—^) cos 2 (2-0;) = 4 cos 3 (^—y) cos 3 {)/—z)cosZ{z-x). (22) If sin O+y) -^ sin (a + S) = sin (y+a)-^ sin + S) = sin (a + i3) - y?: sin (y + S), where a, ^, y are unequal and each less than tt, then will P=l, and each member of the equations = 0. /23) If cos(/3 + a) + cos(a + y) ^ COS Q + y ) + c os (a + y ) COs(^-a) + COs(a— y) C0s(y-^) + C0S (a — y)' then sin j8 , cos 3 COS y - cos a sin a - sin y sin (a - y) ' (24) If X- cos a cos /3 + .^ (sin a + sin /3) + 1 = 0, and ^2(.os/3cosy+^(sin/3+siny) + l=0, prove that x"^ cos y cos a + .r (sin y + sin a) + 1 = 0. (25) If ^+y cosa + 2sina=cos(/3-y), ^+ycos3 + 2sin/3=cos(y-a), ^ +y cos y + 2; sin y = cos (a - /3), prove that ^ = 4 cos ^ (a - /3) cos |(/3 - y) cos ^ (y - a). (26) If a cosa+ 6 cosj3 + ccosy=0, a sina+ 6 sin)3 + csiny=0, aseca+ 5 sec/3 + csecy = 0, then a4 + 54 + c4 - ^b\'' - ^c^a? - ^aW = 0. (27) If acos^+6sin^ + c = 0, a cos + 6 sin + c = 0, cV2=d=a±6, prove that either ^ or ^ must be of the form \nn^-\'K. 136 TRIGONOMETRY, (28) If m is a positive integer, then cos (m + 1) ^=cosm^i2cos^-- — — X. ^ ^ 1 2cos^- 2cos<9- cos^J * where 2 cos 6 is repeated m times. (29) If 2s=;r+y + ^, prove (i) tan {s-x)->r tan {s-y) + tan {s — z) — tan s _ 4 sin ^ sin y sin z 1 - cos^;^; — cos^j/ — cos% + 2 cos X COS y cos z ' (ii) tan~i {s — x)+ tan~i {s—y) + tan~^ {s — z) — tan~^ s = tan-i \Qxyz (^2 4.^2 + ^2 + 4) _ 4 (^2^2 + 2V + ^2y2) • no^ (W sin (^-^) sin (^-y) sin (^-y) sin (^- a) ^ '' ^^ sin (a - ^) sin (a - y) "^ sin O - y) sin (/3 - a) (ii) sin (^-a)sin(<9-^) sin (y - a) sin (y — /3) ~ ' sin(^-a) sin(^-^) sin (a - ^) sin (a — y) sin (j3 — y) sin (/3 — a) sin(^— y) sin (y — a) sin (y — /3) = 0. (31) If a\ b^j (? are in A. p., then tan J., tan B^ tan C are in H. p. /oo\ Ti?*aii^ tan 5 tanC ,, , 1 r x- • (32) It — = , prove that each fraction is X y z equal to . , . „ . ^/l 1 1 IN sin ^ . sm ^ . sm C7 - ^ 1 ; — -- . \x y z x+y + zj (33) OP + OI^^-hOI^^+OI^^=l2Rl (34) If a, (3, y are the radii of the circles OAB, OBC, OCA, a h , G abc then a + ^ + y^W GENERAL MISCELLANEOUS EXAMPLES. XLIIL 137 (35) The sides of a triangle are prove that the angles are in a.p., the common difference being (36) AR {r.f^ + r^r^ + r^r^) = {r^+r^) (r^ + r^ (^i + r,). (37) A = s2 2I cos 6 cos cos \/r, where cos 2^=tan J5 + tan^(7, cos 20 = tan ^(7+ tan ^^, cos 2>/a = tan \A + tan \B. (38) 4(J:^'24.^ij'2 + c'C"2) = 3(a2 + 62 + c2). 16 (^i?'2 , (7(7'2 + CC"2 . ^^'2 + AA'^ . BB'^) = 9 (62^2 + ^ZaS + ^252). \Q{AA'^ + BB'^ + GC'^)=^^ {a^+h^+<^). [Art. 121.] (39) If a, Z), c, G? be the lengths of the sides of a quadrilateral such that one circle can be described about it and another inscribed in it, then the radius of the latter circle is 2 sjjahcd) a+h+c+d' (40) Given that .a7=y cos Z+scos F", y^z cos X+x cos Z^ and that A'+ J''+^is an odd multiple of n, then z = x cos Y+y cos JT, and cosZ= / ~^ . 2yz (41) If ^ + i?+C=1800and y sin C— z sin B _ z sin J. - ^ sin C x—ycosC—zcoaB y—zcosA — xcoaC* X, y, z being real, then . — -r = ~~i^ = -. — 7,. ''^' ^ ' sm^ smB smC 138 TRIGONOMETRY, (42) In any triangle 722(^4 + 54 + c* - 262^2 cos 2 A - 2c'^a^ cos 2B - 2a?-lf- cos 2C) = 2a262c2. (43) The radii of the escribed circles of a triangle are the roots of the equation /S'^H 5 (2 - S2) ^2 + 52^_^ = ^2^^ where 22 is the sum of the squares of the sides. (44) 12E>S'=a3cos (i? - (7) + 6^ cos {C-A)^(? cos {A-B), (45) If be a point within a triangle, such that A 0, BO^ CO are inversely proportional to the sides jBC, CA^ AB; R, R^, R^, R^ are the radii of the circles described about ABC, BOC, CO A, A OB; then (46) ABC is a triangle in which the sides AB, AC are equal. Circles are described with centres A, B, C touching each other externally. Prove that the distance between the centres of the circles that can be drawn touching these three circles is (1-cos^) (l-2cosi?) 46 4 — 5 cos B (47) Perpendiculars OB, OB, OF to the sides of a triangle when produced meet the circumscribing circle in F, Q, R; prove that 4r {FD + QE+RF) = 2bc + 2ca + 2ab -a^-b^- c\ (48) If pi, P25 Ps ^^^ "t^® distances of any point in the plane of an equilateral triangle, whose side is a, from the angular points, then pipi+p,W+PiW - Pi' - P2' - P3'+«' {pi'+p,'+P3')=ci\ (49) From the angular points of a triangle ABC are drawn perpendiculars to the opposite sides and also lines bisecting the angles : if ^ be the angle between the two lines drawn from A, and (f), ylr be corresponding angles at B, C, prove that 1 + cos ^ + cos 1 and not greater than |. (68) sin A + sin i? + sin C is never less than sin 2 A + sin 2B + sin 2C,i{ A+B + C= 180^ (69) U cos a - —^ —-i ..\ \ 2 cos a- 2 COS a- / = 2cos(a + i3)-- 2cos(a + /3)- 2cos(a+)8)- * (70) Prove by induction that sin (a+^+y + ...n angles) = (^S^iC^.i) - {S^C,,.^) + {8,0^,-,) - etc. cos (a + ^ + 7 + . . . n angles) = ((7„) - {S^C^_^ + (>^4(7„_4) - etc. where {SrC^-r) stands for the sum of the products of the sines sin a, sin/3, sin y... taken r together, each multiplied by the product of the {n - r) remaining cosines. Shew that De Moivre's Theorem is equivalent to these two theorems. 142 , . TRIGONOMETRY. (71) Prove that if n is an odd integer the two series 1.2 "^ 1.2.3.4 •••' _ n(n-l)(n-2) n (n-l) (n-2) (n-S) {n-4) _ ^ 1.2.3 "^ 1.2.3.4.5 "•••' are numerically equal, and if 7i is an even integer one of the two series is zero. (72) Prove that -^ = -i- + -1- + ^ 1+^ 1+^ 1-ajC 1-^a;' where - 1, a, j8 are the values of (-1)^, and deduce by writing :i7 = cos 2^ + i sin 26 that 3 tan 3^ = tan ^ - cot (^ + Itt) - cot (^ - Itt). (73) If sin log (a + ib) = a + i^, then log sin (a' + ib') = a+ i^' ; when a'=logV(a2 + &2)^ 6' = tan-i-, a'=logV(a2+^2), /3'=tan-i^. a (74) By writing in the identity 1 1 1 {x-a){a;-b) {a-b){x-a) {a-b){x-bY cos 2^ + x/( - 1 ) sin 26 for ^ and similar quantities in terms of a and ^ for a and b, prove that cos (2^ + a + iS) sin (a - /3) = cos (2a + ^ + /3) sin (^ -- ^) -cos (2^+ ^ + a) sin (^). (75) Prove that cos (.r + iy) = cos .r cosh y-i sin ^ sinh y, sin (ar+ ^y) = sin .r cosh y + 1 cos :r sinh y. GENERAL MISCELLANEOUS EXAMPLES. XLIIL 143 (76) If 2^' = 4 cos a cosh b, 2y =4 sin a sinh h, then sec (a + 16) + sec (a - 15) = ^2-^:^2 > sec (a + 16) - sec (a - 16) = ^^-^^ . (77) One of the values of sin~i (cos 6+i sin 0) is cos-i(Vsin 6) + i log {>/sin ^ + /v/(l + sin 6)} when ^ is between and ^tt. (78) Reduce tan-i (cos ^ + 1 sin ^) to the form A + iB, and hence prove that cos ^ - ^ cos 3^+4- cos 5^ - ... = ±| tt ; the upper or lower sign being taken according as is positive or negative. (79) If CD, 0)2 are the imaginary cube roots of - 1, -^cosa + wcos fa + g j+co^cos (" + ^ )f jcos^ + o>cos ^/3 + gJ+a)2cos f/3+y U = ||cos(a + /3) + cocosCa + /3 + |Va>2cosL + ^ + ^')|, and deduce the value of jcosa + wcos (« + o)+<"^^^^(" + 'T)r» when w is a positive integer. (80) Prove that the real part of (1 + i tan ff)'* is / cos (log cos 6). (81) (a + 46)*+*^ will be wholly real or wholly imaginary according as ^/3 log (a^ + 52^+0 tan-^ - is an even or odd multiple of ^TT. (82) Prove that all solutions of the equation sinh:r=sinha are included in the expression x=in7r + {—l)"a where n is any integer, positive or negative. //■ OP THK ftJNIVERSITr] 144 TRIGONOMETRY. (83) Prove the following rule for finding the length of a small circular arc; to 256 times the chord of one-fourth the arc add the chord of half the arc; subtract 40 times the chord of half the arc and divide the remainder by 45. (84) If two sides a, b and the included angle (7 of a triangle are given, and a small error d exists in C, the corresponding error in R is ^da cot A cos B cosec O. (85) If the unit of measurement be a right angle, find the limit of ;i5 as is indef. diminished. (86) The limit when n is indefinitely increased of I cos - + sm — ) IS eStt. \ n n J (87) If ^ = III , then ^= 5^ nearly. o ob4 (88) If ^= Jtt nearly and ti is > 1, prove that , . ,,i n-l + (n + l)sin^ - (sm ey = . ., . ; ,; . , nearly. (89) If cos-i — -x- cos~i — , — =7, and 6 and x are both ' a + b + x a + x '' small compared with a, then a sin^v x= 8 (90) If in the equation tan^ = (^-^)''^^">y- cot oi + cot 02 cot 03 + cot 04 the angles oj, og, 03, 04 are all nearly equal, then one value of 6 is very nearly I (oj -f 03 + 03 + 04). than - (91) ^ differs from tan d by less than J tan^ ^, ^ being less TT 4* GENERAL MISCELLANEOUS EXAMPLES. XLIIL 145 tan2 e - sinh2 a cotli2 a sin2 ^ - 1 (92) taii2 6 - sinh2 ^ coth^ ^ sin'^ ^ - 1 ' (93) sin(a- 2ni3) + sin{a-2(7t- l)i3} + siii{a~(2n - 2)^} + ... + sin (a + 271^3} = sin a cosec ^ sin (2/1 + 1) /3. (94) tan a tan (a + ^) + tan (a + /3) tan (a + 2^) + . .. -.N ^w / ON tan (a + 71^)- tan a + tan {a + {n - 1) /3} tan (a + ^jS) = ^ ^JJ (95) sec a sec (a + /3) + sec (a + /3) sec (a + 2/3) + sec (a + 2/3) sec (a + 3^) + ... to 71 terms = cosec /3 {tan (a + ti^) — tan a}. (96) Sum the series tan-i T. — - — 7 + tan-i , -— + tan-^ 1 + 3.4 1 + 8.9 1 + 15.16 (97) tan-i I + tan-i | + tan-i -^ + tan-i ^ + ...ion terms =tan-i— 7 _. 91 + 2 (98) Siun to n terms the series 1 . tan ^ . sec 2^ + sec ^. tan 2^sec3^+sec2^ tan 3^. sec 4^+.... (99) (1 +sec 2^) (1 + sec 4(9) (1 + sec 8^) ... (1 +sec 2"^) _ tan 2"^ ~ tand * (100) (2cos<9-l)(2cos2^-l)...(2cos2"-i^-l)=^^^|~^j^^ . (101) Sum the series sin^ 3 sin 3^ . 32sin32^ . , + ^^ — - + ;r^j^ — r^ ... to 71 tcrms. 2 cos ^ + 1 2 cos 3^+ 1 2 cos32^ + 1 L. 10 146 TRIGONOMETEY. (102) The n^^ convergent of 1 1 1 etc., is 2tana+ 2tana+ 2tana + (tan a + sec a)" - (tan a — sec a)" (tana + seca)" + i-(tana-seca)" + ^' (103) The sum of n terms of the series cos a cos 2a ^ .„ 1 + + — ;7— + ... =0, if na=7r. cos a cos- a ' (104) Sum the series log (1-2 cos $) + log (1-2 cos 20) + log (1-2 cos 22 0) + etc. to n terms. 3sin^-sin3^ . 3sin3.r-sin32.r 3sin32^-sin3^;r (105) cos 3^ 3 cos 3^0; 32cos33^ to n terms = | ( tan^j . (106) The roots of the equation are sin^^Tr, sin^^Tr, sin^^. (107) Solve the equation x^ + ^2 _ 2^ _ i = o. [Result. 2 cos f tt, 2 cos f tt, 2 cos ^tt.] (108) (^-cosfTr) (^-2cosf7r) (^-2cos-|7r) (^-2cos|7r) =a^ + 2x^-a;'^-2a; + l. (109) The sum to n terms of the series 12 cos 2a + 22 cos 4a + 32 cos 6a + ...to n terms _n^ siYi.{2n + l) a n cos 2na sin27ia.cosa ~ 2 sin a 2 sin2a ^ sin^a (110) Sum the series ^ , ^sin2^ , (92 sin 3^ , ^^__^^^^, a' f ^"^1 sin2^ + 1.2 sin^^ + 1.2,3 sin^l?"*"- GENERAL MISCELLANEOUS EXAMPLES. XLIIh U7 (111) Sum to n terms (i) 1 + e sin a + e^ sin 2a, ^ A 6 (ii) tan - sec ^ + tan ^ sec - + etc. (112) Sum to infinity (i) ?isma+ ^^ ^ ^ sm2a+ ^ ^ ^ ^ ' sm3a+... (113) Prove that 1-A2 = 1 + 2A cos ^ + 2^2 cos 2x + etc. (1 - A)2 cos'-* ^^ + ( 1 + A)2 sin2 ^^ (115) l + ^^ + ^, + ...=i{e' + 2e-i'coai{^>j3)}. (116) The roots of the equation a;" sin ?ia — nx""'^ sin (^ia + /S) + — ^-j — - ^~^ sin (Ma + 2^) — . . . = are given by ^=sin(a + ^ — ^<^) cosec(^ — ^'0) where k has all integral values from to n- 1 and 71^= it. (117) Find the general value of 6 which satisfies the equation (cos 6 + i sin 6) (cos '2.6 + i sin 2,6) ... (cos n6 + i sin n6) = 1. (118) When n is even and if 710= TT, tan a tan (a +0) tan (a + 2(^) . . . tan {a + (/I- 1) <^} = ( - 1)^. (119) When 7w is odd tanm<^ = tan<^cot [^ +^J tan ^<^ ^^ ... 10—2 148 TBIGONOMETRY. (120) tan ^ + tan f^^ + d\+ tan (^ + o) where fS^=^ the Sum of the m"' powers of the root of equation ^2-1 = 0. (121) If^=^, 2" cos B cos 2$ cos 22 ^...cos 2"-^^ = l. (122) If a,b,c,... are the roots of the equation ^" -Pi^r'^"! + ^2^"""^ -jOg^"'"^ + etc., then tan-la + tan-^S + tan-ic + . . . = tan-i ^i~-?^3+i^5- •- • ^ (123) Prove that .'^ + 6-^^2(1 + 2'^) {1 + (§)2}{1 + (|)2}... (124) The sum of the products of the reciprocals of the fourth powers ot every positive integer is . /TORN + y 2 2 2 2 2 2 (125) tan ^ = + + 2 TT-y TT + y 37r-y 37r + y Stt-?/ 57r+y' ,,„„, ,y 2 2 2 2 2 (126) cot^ + 2 y 27r-y 27r+y 4n--y 47r+y' (127) Prove that the coefficients of 6^ and 6^ in the expression H'-m'-i-yM'-'M'-S'H vanish ; explaining d priori why they do so. (128) Having given the formula . /, 22^2N / 22^2\ deduce the expression for sin 6 in factors. GENERAL MISCELLANEOUS EXAMPLES. XLIIL U9 (129) The coefficient of ^" in the expansion of (i+.)(i+|)(i+|,) ,. IS (2n + l)!* (130) By putting a^ia for 6 in the expression of sin^ in factors, prove that tan~i -^ + tan~i ^— -; + tan"-' .— -^ + . . . ad inf. = ;^ 77 - tan~i {tanh a cot a) + titt. (131) If a series of points are distributed symmetrically round the circumference of a circle, the sum of the squares of their distances from a point on the circumference is twice that from the centre. (132) If A^, A2, A2, ... A2n + i are angular points of a regular polygon inscribed in a circle and any point in the circumference between A^ and -42n + ij then the sum of the lengths OA^ + OAs + OA,+ ... + OA^,^^ = the sum OA2 + OA^-\-OA6+... + OA2n. (133) If from a point P straight lines PB^, PB2,...PB„ be drawn to the middle points of the sides of a closed polygon AiA2...A„, and if the angles PB^A^, PB^A^, ... PB„A^ be denoted by a^, a2^...a,^ respectively, and the triangles PA^A^j PA2A2J ... PA^A^ by Ai, A2, ... A,„ prove that Aj cot ai + Ag cot 03+ .. . + A„ cot a„=0. EXAMINATION PAPERS. I. Sandhukst — FuETHEE. Nov. 1882. 1. Name and define the trigonometrical ratios. Prove that sec^ A + cosec^ A = sec^ A cosec^ A . 2 If the cosecant of an angle between 90^ and 180^ is —j= ,. what is the V3 secant? And if the cosine of an angle between 540° and 630*' is - \, what is the cosecant? 2. Prove the following identities : — i. (sin 2^)2=2 co8M(l- cos 2^). ii. 2 cosec 4^ + 2 cot 44 = cot ^ - tan A . iii. 2 tan-i | + tan-^ ^ + 2 tan"! i = j • 3. In a plane triangle ABC prove that — i. tan^ tan J5 tan (7=tan4 + tanJ5 + tanC. ii. asin4 + & sinJ5 + c sin (7=2 (acos4+^cosB + 7COS (7), where dbc are the sides and a/Sy the perpendiculars let fall on them from the opposite angles respectively. 4. Prove that the area of a triangle = ^a2sin2i? + ^62sin24; and if JJ, r are the radii of the circumscribing and inscribed circles ahc Iir= 4(a + 6 + c) EXAMINATION PAPERS. 151 5. Given log li = -0791812 and log 2f =-3802112, find the value of ^{S'6)^x ^^\~>^8^, the mantissae for 46929 and 46930 being 6714413 and 6714506. In a triangle ABC, 6 = 14, c = ll, ^ = 600; find the other angles, having given L tan IP 44' 29" =9 -31774. 6. A measured line is drawn from a point on a horizontal plane in a direction at right angles to the line joining that point to the base of a tower standing on the plane. The angles of elevation of the tower from the two ends of the measured line are 30" and 18*^. Find the height of the tower in terms of I, the length of the measured line. II. Cambridge Previous Examination. Dec. 1883. • 1. Define the cosecant and tangent of an angle. Shew that cosec-^ =1+ cot' A. Find all the trigonometrical ratios of 30^'. 2. A man wishes to measure the distance between two points A and B between which lies an obstacle. He therefore walks from A to C in a direction at right angles to AB a distance of 50 yards. He now finds that he can walk directly from C to I? and that CB makes an angle of 60*^ with AC. Find the distance from A to B. 3. Prove that sin {90<* + ^) = cos ^. 4. Find a formula for all angles having the same tangent as a. Solve completely the equation tan^ = 1. 5. Shew that sin [x + y)=8'mx cos y + cos x sin y. Prove that sin (a - j3) cos 2j3 + cos (a - /3) sin 2j3 = sin (j3 - a) cos 2a + cos (|8 - a) sin 2a. If sin 05 = f, cosy = y'V, find sin (a; + y). 6. Prove that ,,, . 1 . , /l-cos^ (1) smi^=±^ 2 • 152 TRIGONOMETRY. (2) cos3^ = 4cos3^-3cos^. (3) cos 4:A = cos^ A + sin^ A-Q sin^ A cos^ A . Determine the sign of the radical in (1) when A lies between 360° and 720". 7. If a, b, c be the sides of a triangle ABC, shew that a^ = h^ + c^-2hc cos A. If ABC be an equilateral triangle each of whose sides is eight inches, and in BC a. point P be taken three inches from B, shew that ^P is seven inches. 8. In any triangle ABC shew that sin-- /tSzS tan^- /^lIMEI ''''2-\/ Vc ' *^^2-\/ s{s-a) ' Find all the angles of a triangle whose sides are 13 ft., 14 ft., and 15 ft. in length, having given log 2= -30103, log 3 = -4771213, log 7 = -8450980 and Ltan 260 33' = 9 -6986847, tabular difference for l' = 3159, itan 29<> 44'= 9-7567587, tabular difference for l' = 2933, L tan 330 4r =^9-8237981, tabular difference for l' = 2738. 9. A base line 400 feet in length is measured from the foot of a vertical tower and at the end of this line the angular elevation of the top of the tower is observed to be 26*^33' 54"; shew that the height of the tower is very nearly 200 feet. Refer to question 8 for the necessary logarithms. III. Woolwich — Pbeliminaky. June, 1882. 1. Prove that the angle subtended at the centre of a circle by an arc equal in length to its radius is an invariable angle. One angle of a triangle is 45*^, and the circular measure of another is 1|. Find the third, both in degrees, and in circular measure. 2. Define the secant of an angle, and shew how your definition applies to angles between 180° and 270^. If sec ^ = - 2, what two values between 0° and 360" may A have ? EXAMINATION PAPERS. 153 3. Obtain a formula embracing all the angles which have a given tangent. Determine all the values of $ which satisfy the equation : \/3 tan2 d + l = {l + \/S)tSind. 4. Find an expression for tan 3^ in terms of tan A. Shew also that tan SA tan 2A tan A = tan 3^ — tan 2 A - tan A . 5. Prove that sinl80=^^^^^^; and that sin^ 30" = sin 18" sin 5A\ Shew that in any circle the chord of an arc of 108® is equal to the sum of the chords of arcs of 36'* and 60". 6. Demonstrate the identities — ,,. {cosec^ + sec^)2 . ^ cosec^ ^ + sec^ ^ (2) ^m 3^ = 4 sin A sin (60" + A) sin (60" - A). (3) 4 (cot-i 3 + cosec- V5) = t. 7. What are the advantages gained by the use of logarithms cal- culated to the base 10 ? If logio2 = -30103, find the logarithms of 5, j^^, and 4 V^T, to the base 10. 8. Prove that in any triangle — (1) 2bc cos A = b^ + c^~a\ I+cosjA-B^cobC _ a*+6 ^ ^' l + cos{A-C)coaB~a^ + c^^ 9. If r^ be the radius of a circle touching the side a of a triangle and the other two sides produced, shew that — A B C rjcos- =aco8 — cos— . If a be the side of a regular polygon of n sides, and i2, r, the radii respectively of its circumscribed and inscribed circles, prove that JJ + r=4a cot 5-. 154 TRIGONOMETRY. 10. Two sides of a triangle, which are respectively 250 and 200 yards long, contain an angle of 54" 36' 24". Find the two other angles, having given L cot 270 18' = 10 -2872338, diff. for V= -3100 ; L tan 120 3' 50" = 9-3329292 ; log 3 = -4771213. 11. The eye of a soldier in a straight trench of uniform depth is 2 feet above a level plain on which he sees two men standing in the same straight line as the trench ; the parts of their bodies above the level of his eye subtending at it the angles tan~i -00416 and tan~^ -004. On walking 200 ft. towards them in the trench he notices that the height of one exactly hides that of the other ; and, on approaching 596 feet 8 in. closer still he finds that the portion of the height of the nearer above the level Of his eye subtends at it 45". Find the heights of the men. IV. Woolwich — Pkeliminaey. Bee. 1882. 1. Shew how to express in degrees, minutes, and seconds, an angle whose circular measure is known. Find, correct to three places of decimals, the radius of a circle in which an arc 15 inches long subtends at the centre an angle containing W 36' 3-6". (7r = 3-1416.) 2. Define the sine of an angle, and prove that sin ^ = sin (1800 ->4) = sin {-(1800 + ^)}. Write down formulae including all angles which satisfy — (1) 2sin^ = l, (2) 2sin2J = l. 3. Prove that cos [A+B) = cos A cos B - sin ^ sin B, and deduce expressions for cos 2A , cos ^A in terms of cos A. 4. Given cos .4 = -28, determine the value of tan \A, and explain fully the reason of the ambiguity which presents itself in your result. EXAMINATION PAPERS. 155 5. Prove that (1) tan^ + cot5 = \/{sec2^ + cosec''^}. (2) sec^-tan^ = tan(^7r-^^). (3) cos 200 + cos lOQO + cos UO^ = 0. (4) cos-i U + 2 tan-i i ^ gin-i | . 6. State and prove the rules by means of which you can determine by inspection the integral part of the logarithm of any given number. Given log 4-9G = -6954817, log 4-9601 = -6954904, find the logarithms of 496010, -000496, and 49600-25. 7. Shew that in any plane triangle a=6co8(7 + ccosB. If c = \/2, ^ ^- 117^ B = 450, find all the other parts of the triangle. 8. Find the greatest angle of the triangle whose sides are 50, 60, 70 respectively, having given log 6 = -7781513, L cos 39n4' = 9-8890644, diff. l'=1032. 9. Express the area of a triangle in terms of one side and the two angles adjacent to it. Two angles of a triangular field are 22^^ and 45'* respectively, and the length of the side opposite to the latter is a furlong. Shew that the field contains exactly two acres and a half. 10. Find an expression for the diameter of the circle which touches one side of a triangle and the other sides produced. If d^, do, c?3 be the diameters of the three escribed circles of a triangle, shew that fZ^tZg + ^2^3 + ^3^1 = {« + & + c)^. 11. A man standing at a certain station on a straight sea-wall observes that the straight lines drawn from that station to two boats lying at anchor are each inclined at 45^ to the direction of the wall, and when he walks 400 yards along the wall to another station he finds that the former angles of inclination are changed to lo** and 75'' respectively. Find the distance between the boats, and the perpendi- cular distance of each from the sea-wall. 156 TBIGONOMETRY. V. Mathematical Teipos. 'The three days.' Jan. 1881. 1. Explain, and state the several advantages of, the chief systems of angular measurement in use. Prove that the circumferences of circles vary as their radii ; and mention the approximations to their constant ratio which are practi- cally employed. Shew that there are eleven pairs of regular polygons which satisfy the condition that the measure of an angle of one in degrees is equal to the measure of an angle of the other in grades: and find the num- ber of sides in each. 2. Define the sine of an angle; and find the value of the sines of angles of ISS^, 2400, 292io, 4320. Shew that sin^ 100 + cos2 200 - sin lO^ cos 20" = sin^ 10^ + cos^ 400 + sin lO^ cos 400= f . 3. Prove geometrically that sin x + 8my — 2 sin ^{x + y) cos i {x - y). Solve the equation cos X + sin Sx + cos 5x + sin 7x + ... + sin (4w - 1) cc = | (sec x + cosec x) . 4. Find an expression for cos [x-^ + x^ + x^) in terms of sines and cosines of x-^, x^, x.^. State the corresponding theorem for the case of n angles ic^, x.;^, ...£c„. If cos {y-z) + cos {z-x) + cos [x - jf ) = - f , shew that cos3 [x+9) + cos3 {y-\-e) + cos^ {z + d)-S cos [x + 6) cos {y + 6) cos [z + 6) vanishes whatever be the value of 6. 5. Shew how to solve a triangle having given the three sides: proving from the formulae obtained that there cannot be more than one triangle, though there may be none, with the given parts. The perpendiculars from the angular points of an acute-angled triangle ABC on the opposite sides meet in P: and PA, PB, PC are taken for the sides of a new triangle. Find the condition that this should be possible: and if it is, and the angles of the new triangle are a, /3, 7, shew that ^ cos a cosS. cos 7 , . -r, ^ 1-1-- ; -} ^-1 ;^=isec^ sec B sec C. cos A cos B cos C EXAMINATION PAPERS. 157 6. Find the radii of the inscribed, the circumscribed, and the nine-point circles of a given triangle. If be the centre of the first, 0' of the second, and P the centre of perpendiculars, shew that the area of the triangle OO'P is - 2i22 sin 1 {B - C) sin h{C!-A) sin I [A - B), where B is the radius of the circle circumscribing ABC. VI. Mathematical Tripos, Part I. June, 1882. 1. Explain the different methods of measuring angles. Find the number of degrees in each angle of a regular polygon of n sides (1) when it is convex, (2) when its periphery surrounds the in- scribed circle m times. Find correct to -01 of an inch the length of the periphery of a decagon which surrounds an inscribed circle of a foot radius three times. 2. Prove geometrically the formula cos a + cos ^ = 2 cos ^ (a + /3) cos ^ (a - j3). Prove that 2 cos (a-/3) cos (^ + a) cos {6 +/3) + 2 cos (/3 - 7) cos (^ + /3) cos (^ + 7) + cos(7-a)cos(^ + 7)cos(^ + a)-cos2(^ + a)-cos2(^ + /3) -cos2(^ + 7)-l is independent of 0, and exhibit its value as a product of cosines. 3. Prove geometrically the formula , ^, tan a + tan B ^ ^' 1 - tan a tan ^ Prove that if a, /3, 7, 5 be four solutions of the equation tan(^ + ^7r) = 3tan3^, no two of which have equal tangents, then tan a + tan /3 + tan 7 + tan 5 = 0, and tan 2a + tan 2/3 + tan 27 + tan 25 = | . 4. Prove that in general the change in the cosine of an angle is approximately proportional to the change in the angle. 158 TRIGONOMETRY. Prove that if in measuring the three sides of a triangle small errors X, y be made in two of them a, 6, then the error in the angle G will be -('^cotS + ^cot^V and find the errors in the other angles. 5. Prove that in any triangle acos 5 + 6 cos^ = c, and deduce the formula 0^ = 0^ + b'^ - 2ab cos C. Prove that if be the centre of the circumscribing circle of the triangle ABG, the sides of the triangle formed by the centres of the three circles BOG, CO A, AOB will be proportional to sin 2^ : sin 2i? : sin 2C. Find the angles of the new triangle correct to one second when the sides of the triangle ABG are in the ratio 4:5:7. 6, Find the radius of the inscribed circle of a triangle in terms of one side and the angles. Prove that if P be a point from which tangents to the three escribed circles of a triangle ABG are equal, the distance of P from the side BG will be |(& + c) sec lA sin IB sin IG. VII. OXFOBD AND CaMBBIDGE ScHOOLS EXAMINATION. EtOU, 1882. 1. Prove that the cosine and sine of an angle have their signs changed, but their magnitudes unaltered, if the angle be increased by two right angles. Investigate a general formula for all angles whose tangent is equal to tan A. 2. Find the cosine and tangent of 45" and 60^'. Apply Euclid vi. 3 to find tan lb\ 3. Prove that sin {A-B) = sin A cos B — cos A sin P, and that sin 2A cos A sm ^ = 1 + cos 2^ 4. Express cos ^ (/3 + 7 - a) cos J (7 + a - §) cos ^ (a + /3 - 7), as the sum of cosines of separate angles. EXAMINATION PAPERS. 159 5. Express sin hA in terms of sin A ; and prove, d priori, that to any given value of sin A , four values of sin ^A must correspond. Having given sin 18*' = ;i (\/5 - 1), find cos 81<'. 6. Shew how to find the height of an inaccessible object by obser- vations of its angles of elevation, taken at two points on a straight line through its base. I stand on a hill on one side of a lake, and observe the angle of elevation (a) of the summit of a mountain across the lake, and also its angle of depression (/S) as seen by reflection in the lake. If ^ be the known height of the mountain, shew that its distance is 2/ico8a C08/3 sin (a + /i) ' it being given that the ray of light from the top of the mountain makes the same angle with the vertical after reflection from the lake as it did before reflection. 7. Express the sine of half an angle of a triangle in terms of the sides. Prove that, in any triangle, 2 (cos \A- sin ^A )2 cos IB cos ^C = (cos ^C + cos^^ -cos^Z^) (cos^^+cos^jB-cos^C). 8. Find the radius of a circle described about a triangle. If the radius of this circle be equal to the least side of the tri- angle, what is the magnitude of the least angle ? 9. Sum the series — (1) 1 + e~"** cos nx + e'^*^ cos 2nx + ad infinitum. (2) cos(a + j3) + cos(a-f3/3) + + cos (a + 2/1 + 1/3). VIII. Oxford and Cambridge Schools Examination. Eton, 1883. 1. Given 7r = 3-1416, find the number of degrees in the unit of circular measure of angles. 2. If sin e= -l^~, find tan 6, cos 2d. 160 TBIGONOMETRY. 3. Prove that all angles included in the formula 2mr ± a have the same cosine as a. Solve the equation cos 6 + ,J3 sin 6 = 2. 4. Prove the equivalents : (1) sec2 d + cosec2 d = sec^ d cosec^ 6 ; (2) cos e - cos 3^ = (sin 3^ - sin d) tan 29. 5. ABC is a triangle right-angled at A ; BD meets AC in D: find AD in terms of CD and the angles ABC, ABD. 6. Shew that in any triangle (2) tanJ(B-(7) = ^%oti^; (3) (6 -c) cot lA + [c-a) cot ^5 + (a - h) cot iC=0. 7. Find an expression for the radius of the circumscribed circle of any triangle in terms of the sides. The bisector of the angle A meets the side BC in D and the cir- cumscribed circle in E : shew that DE = ^-7^ — ^ . 2 (6 + c) 8. K the ratio of two sides of a triangle is 2-\-J%; and the in- cluded angle is 60*', find the other angles. Q Q 9. Shew that cos - + \/( - 1) sin- is one of the values of {cos0 + V(-l)sin0}«, n being a positive integer. What are the other values? 10. Sum the series : (1) cos^ + cos2^ + cos3^+ to n terms; (2) sin^ + isin2^ + |sin3^+ to infinity. EXAMINATION PAPERS. 161 IX. Christ's Church, Oxford. Entrance Scholarships. 1883. 1. Prove geometrically that d>s'md>d-^; 6 being less than a right angle. 2. Prove the identities /•X ^,, T.X C0t4C0tjB-l X • 1, (1) cot {A+B) = —^^^--—^, geometricaUy. (ii) (cos A + sin ^){cos 2 A + sin 2^) (cos A - sin 3^ ) = cos 2^ cos 4^. (iii) 26 (cos«^ + sinS^ ) = cos 8^ + 28 cos 4.A + 35. (iv) 2 cos (n cos-^as) = (« + Jx^ - 1)** + (x - ,^x'* - 1)». 8. Shew that Having given that log,3 = 1-0986, find the value of logj^ 3. 4. Eliminate a, j3 from the equations x={a sin2 a + b cos^ a) cos^ j3 + c sin^ /3, y = a cos^ a + 6 sin^ a, 2 = (6 - a) sin a cos a cos /3. 5. If circles can be both described about, and inscribed in a quadrilateral, whose sides are a, 6, c, d, and the angle between the diagonals 6, then ac~bd ^ = cos-i ac + 6ci 6. Solve a triangle, having given the base a, altitude h, and the difference of the angles of the base a. Account for the two values obtained for the vertical angle, and shew which of them is possible. 7. Shew that in a plane triangle (i) S(&-c)(s-a)cos^ = 0. (ii) iR^ra + n+Tc-r. (iii) ^ = -^.QinB + ^ ( -^y am2B + l (-^YBmBB + .., "" ' 2 a + b \a + bj * \a + 6/ L. 11 162 TRIGONOMETBY. 8. The triangle A'B'C circumscribes the escribed circles of the plane triangle ABC; shew that . B'C _ C'A' _ A'B' a cos A~ b cos B~ c cos C ' 9. If K be the centre of the nine-point circle of the triangle ABC, then 4:AK^ = R^ + b^ + c^-a^, where R is the radius of the circum- scribing circle. 10. If cos {d + (psj -1) = cos a + /^ - 1 sin a, and a, 6, are real, prove that tan^^-tan^a^sin^^sec'^a, and find a relation between 6 and + ... X. Christ's College, Cambridge. Entrance Scholarship. 1878. 1. Find the general expression for all angles which have a given tangent or cotangent. Solve the equation sec^ ^ + 3 cosec^ ^ = 8. 2. Prove geometrically the formulas : (1) cos {A-B} — cos A cos B -h sin A sin B. (2) sin A + BmB = 2sm h {A + B)cosl {A -B). Shew that cosec^ +cosec (4 + |7r) + cosec(^ + |7r) = 3cosec3^. 3. If 6 be the circular measure of an angle less than a right angle, prove that sin 6 is less than 6, but greater than 6 - ^6^. 4. Prove that if a, /3, y are any three plane angles (cos a + cos /3 + cos 7) { cos 2a + cos 2^3 + cos 27 - cos (/S + 7) - cos (7 + a) - cos (a + j3) } - (sin a + sin /3 + sin 7) { sin 2a + sin 2^ + sin 27 - sin (/3 + 7) - sin(7 + a) - Bin(a + /3)} =cos 3a + cos 3/3 + cos 37-3 cos (a+/3 + 7). EXAMINATION PAPERS. 163 5. Shew that r = iR sin ^A sin iS sin iC, where R is the radius of the circumscribing circle and r of the inscribed circle of the triangle ABC. If A, A' be the areas of the two triangles, in the ambiguom case (given A, a, b), prove that the continued product of the inscribed and escribed radii to the side b is equal to A A'. 6. State Be Moivre's theorem, and prove that there are n values and no more for the expression {cos9+\/{-l)sme]». Write down the fifth roots of ( - 1). 7. Prove that „ I sm^ - \ / sm- — \ sin^ = nsin-| 1- l| 1- I and deduce the expression for sin 6 in factors. Shew that the sum of the series 34 + 54 + 74+ 94+---(j4\^^ 12/* XI. St John's College, Cambridge. June Exam., 1879. 1. Explain the method of measuring angles by degrees, minutes, &c. The numerical measures of the angles ^, i?, C of a triangle when referred to units i", m", n^, respectively, are in arithmetical progres- sion, and when referred to units p^, q^, r^ respectively, they are in geometrical progression. Find A, B, C. 2. Define the sine and cosine of an angle, and prove that sin2^+cos2^ = l. If cos^ ^ + cos j5 = 1 = sin'-* A + sinBj find A and B. 11—2 164 TRIG ONOMETR F. 3, Prove geometrically that (1) Bin2^=2 sin^ cos^, s'm2A (2) tan A l + cos2^ If ^ = 2,rjri, prove that 2" cos cos 2^ cos 22^ cos2"-i^ = l. 4. Prove that cos .4+ cos ^ = 2 cos |(^ +5) cos 1(4-^). Find and from the equations cos a {cosa + cos(a + ^)}=cos/9{cos/3-|-cos(;S+0)}, cos a { sin a + sin (a + ^) } =cos |3 { sin /3 + sin (j3 + 0) } . 5. If 6 be the circular measure of an angle less than a right angle, prove that sin 0, 0, and tan 0, are in ascending order of magnitude. If the unit of measurememt be a right angle, find the limit of tan - sin e^ as is indefinitely diminished. 6. Expand log„ (1 + aj) in a series of powers of x. Prove that 2(coSii+^cos3^ + |cos5^ ^^cos^i^-sin^p+Kcos^i/i-sin^p) + ^{cos^^A-sin^^A) + 7. In any triangle the sides are proportional to the sines of the angles opposite to them. Through the angular point C of a triangle ABC is drawn any line CMN on which are dropped perpendiculars AM, BN. Prove that MN=AM cot B~BN cot A. 8. Express the sine and cosine of half the angle of a triangle in terms of the sides. If ABC, A'B'C be two triangles, such that 1 + cos ^ _ 1 + cos 5 _ 1 + cos C a' ~ h' ~ c' ' prove that tan \A tan \A' = tan ^B tan ^B' = tan |C tan |C". EXAMINATION PAPERS. 165 9. Give the formulae for the solution of a triangle in which one angle and the containing sides are given. If C = 440, a = 43 ft., i; = llft., find ^ and ^. Having given log 2 = -3010300, log 3 = -4771213, L tan 22« = 9-6064096, L tan 340 17' = 9-8336109, L tan 34® 19' = 9 '8338823. 10. Enunciate and prove De Moivre's theorem. If n be equal to 3mrt 1, prove that n ^ (n{n-l)n{n-l){n-2)) + j n(n-l)(n-2)(n-3) ^ n(n-l)(n-2)(n-3)(n-4) j ^, _ ^^ ^ ^ 11. Find the sum of the following series, each to n terms: (1) cosa+ cos3a + cos5a+ (2) tan-. 2 + tan-' j^^+ tan- — «_ + tan- j^j|-^g + 12. Resolve x^"- - 2x^ cos ^ + 1 into factors. If n be an even integer, prove that • o^ o« 9, ■.^" ^ 27r + ^ 4t + 9 {2n-2)7r+^ ^^2 2"-2 ( - 1)* cos - cos cos— 008^ ' 2 n n n n XII. St John's College, Cambridge. Minor Scholarship, 1881, 1. Shew that in the expression for tan — in terms of tan A we should a priori expect a double result. Find tan 112°. 30'. 2. A triangle is such that the product of two sides is equal to the square on half the base : prove that the difference of the sides varies as the distance from the vertex to the middle point of the base. 3. (i) If X, y, z be any angles, prove that &iii^{x-y-z) sin^(y-2) + sin^ (cc + y-z) sin ^ (y + z) = sin ^a: sin y. (ii) Also \i A, B, BhQ the angles of an isosceles triangle, 2 sin2 (4 _ i^) (2 - cos A) = (sin^ ^ + 2 sin^ £) (1 - 8 cos A cos^ B). 166 TRIGONOMETRY. 4. (i) Eliminate 6 from 2cos2^+ Xsec^ = 3| 2sin2^ + /ACOsec^ = 3j' * (ii) If x^ cos a cos /3 + a; (sin a + sin /3) + 1 = 0] and x^ cos /3 cos y + x (sin /3 + sin 7) + 1 = OJ ' prove that x^ cos 7 cos a + x (sin 7 + sin a) + 1 = 0. 5. Prove that the distance between the centre of the inscribed circle and the intersection of perpendiculars from the angular points on the opposite sides of a triangle is 2R {vers^ vers B vers C- cos A cos J5 cos C}^, •where R is the radius of the circumscribed circle. 6. Prove that {cos + \/(-l) sin ^}^ admits of no more than q values. Find the continued product of the 4 values of {cos-^7r + V(- 1) sin^Tr}^. XIII. Clabe, Caius, and King's Colleges. June Exam., 1880. 1. Draw a curve representing the change in sign and magnitude of tan 26 while 6 changes from to tt. Do the same for tan 26-2 tan 6. ii. Prove geometrically /i\ ± ,4 r}\ tan ^ - tan B (1) tan(^-^)=- — -, ^ ' ^ ' l + tan^tan5' (2) cos3^=:4cos3^-3 cos^. If 3(l+tan2^tan2J5) + 8tan^tanJ3 = tanM+tan2 5, A and B differ by some multiple of ^r. 3. If sin 3^ be given, and from this value tan A is to be found, shew d 'priori that six values are to be generally expected. Prove by help of this, or otherwise, that tan2 a { tan2 (iTT - a) + tan2 + a) } + tan2 { 1 TT - a) tan2 (Itt + a) = 6sec23a + 3. EXAMINATION PAPERS. 167 4. If ^ + + ^ = 0, prove that tan i^ tan i^ tan |;/'= - ^ —■ (1). Find cos x from the equation {4cos {x + a)-l] {4cos(a;-a)-l}=5(2cos2a-l) (2). Eliminate a from the equations sin $ _ cos ^ _ 1 ^-1 ~ 2/S siu 2a " 1 + 2^ cos 2a + ^2 ♦ shewing that /S=tan (^Tr + i^) (3). 5. State the principle of proportional parts in the use of tables of functions. What is meant by saying that the differences are (1) in- sensible, (2) irregular? Prove that they are both insensible and irregular in the case of the logarithmic sine when the angle approaches ^tt. Determine a limit to the error which can be made in finding the logarithm of ^+^7^ from seven-figure tables from those of N and N+1, where a lies between and 100 and N consists of 5 digits. 6. Explain fully the method of solving a triangle, given two sides, the included angle and a table of logarithms. ABC, AB'C are two triangles having ABy BC equal respectively to AB\ B'C, and A, C, C are collinear. If the angle BAB' is 1", find correctly to a tenth of a second the angle between BG and B'C, where AB = 2BC and z ^I^C=60o. 7. ABC is a triangle and tangents are drawn to the nine-point and circumscribing circles at the four points where the perpendicular from A on the opposite side BC meets them. Prove that the four tangents form a parallelogram of area _2 cos A cos B cos G ~t&n{B~C)~' 8. Find the limit of the expression (90 - ^) tan 6*^ as 6 approaches 90. 168 TRIGONOMETRY. 9. Given that x'^+—- — 2cos'md for all values from to m, shew that the formula holds when m + 1 is written for w. Deduce the r I value of a;» + — , in the most general form. 10. Prove that sin ^ = ^ - - + — \6 5 and deduce the exponential value of sin d. Shew that sin-i (cosec e) = {2\ + l) ^tt +^{-1) h cot | (Xtt + $), wliere X is any integer positive or negative. 11. Assuming the factorial expressions for sin 6 and cos 0, prove that tan d>6, provided 6 Ue between and ^tt. By means of the result in question 8, or otherwise, prove that the infinite product 4.55.66.77.8 . ,^16 274-476-6T8-8-710 ^^^^^^^to-. 12. Sum the series, cosa+cos(a + /S) + ... to 7i terms (1), sina + 3sin2a + 5sin3a+... to n terms (2). XIV. Christ's, Emmanuel, and Sidney Sussex Colleges. June Examination, 1882. 1. Define the cosine and the tangent of an angle. Trace the changes in sign and magnitude of /^v \/3 + tan 9 as 6 varies from 0« to 360o. 2. Prove geometrically that Bin A-smB^2&ml{A-B) sin I (A + B), EXAMINATION PAPERS. 169 Shew that (1) cos 2a cos- (jS + 7) + cos 2/3 cos^ (7 + a) + cos 27 cos^ (a + /3) = cos 2a cos 2/3 cos 27 + 2 cos (/3 + 7) cos (7 + a) cos (a + /3), sin Ha + /3) sin i (a + 7) sin ^ ( ^ + 7) si n ^ {/3 + o) sinHa-^)sia^(a-7) "*" sin H^ - 7) sin ^ (/S - a) ^ sin i (7 + a) sin i (7 + j3) , . . + • 1/ ( • 7 mCOS7 = cos(a + /3 + 7). smi(7-o)sinH7-/3) v r // 3. Find the limit of — r— when ^ is diminished indefinitely. a In order to ascertain the distance of an inaccessible object P, a person measures a length AB = 20 yards in a convenient direction; at A he observes that the angle PAB = QO^, and at B that the angle PB A = 119^ 20'. Find approximately the distance BP. To what degree of accuracy is your result correct, supposing (1) that there is no error in the measurement of the angles, (2) that there is an error of 1' in the measurement of each angle? 4. In any triangle ABC, shew that a = b cos c + c cos B, and that a^ = h^ + c^- 2bc cos A . If N be the foot of the perpendicular from C on ^Z?, and the circle on CN as diameter cut CA, CB in P and Q respectively, shew that the angle BPN is equal to the angle AQN. 5. Express the area of a triangle in terms of its sides. A straight line ^-B is divided at C into two parts of lengths 2a and 26 respectively. On AC, CB and AB as diameters semicircles are described so as to be on the same side of AB. If O be the centre of the circle which touches each of the three semicircles, shew that its radius _ ah{a + h) "aS + aft + ftS' and that its diameter is equal to the altitude of the triangle A OB. 6. Shew how to find the height and distance of an inaccessible object on a horizontal plane. 170 TRIGONOMETRY. A person wishing to ascertain the height of a tower stations him- self in a horizontal plane through the base at a point at which the elevation at the top is 30"^. On walking a distance a in a certain direction he finds that the elevation of the top is the same as before, and on walking a distance five-thirds of a at right angles to his previous direction, he finds that the elevation of the top is GO*'. Shew that the height of the tower is \/^a or VII*- Explain the two results. 7. In a triangle ABC, I, I' and are the centres of the inscribed circle, the escribed circle opposite A and the circumscribing circle respectively, and R is the radius of the latter circle. Shew that (1) 012 = ^2 (1- 8 sin 1^ sin ^5 sin IC), (2) t^nIOr=J-^^^'^^. ' 2 cos A-\ 8. Explain the meanings of sin-icc and tan~ia;. How many bounding lines are required to construct all the angles included in the formula sin~i a + cos~^6 + tan~i c ? Shew that sin-^ a + cos-i 6 = sin~i {ab + Jl-d?' Jl - b'-). If X7/ + yz + zx = l, prove that one of the values of ''^ {i+x^){i + y^)+''^ (1+2/2) (1+,^) + ^^^ (iT.2r(i+x2)-2'^- 9. Assuming De Moivre's Theorem find the expansions of sin nd and cos nd as homogeneous functions of sin d and cos 6. Find the equation whose roots are tan^ — , tan2 — - , tan2 -- , tan2 — , and tan2 — . Find also the sum of the fourth powers of these tangents. 10. Investigate Gregory's series for the expansion of tan'^a? in powers of x. Expand tan~^ (x + cot a) in powers of x. 11. Prove that C0SW^=2"'-l( COS^-COS-^ ) ( COS^-COStt^ ) ( cosg- ^~ ^^ ). EXAMINATION PAPERS. 171 Shew also that ii p<.n . (2r + l)7r „(2r+l)7r sin ^ — ^—^ cosP ' --^_ (-ir cosnd nr^ ^ ^ (2/- + l)7r cos 6 - cos — -^ — 2n 12. If J, JB, C, D be the angular points of a regular polygon of n sides inscribed in a circle of radius a and centre 0, shew that PA^ . PB^ . PCP' ...= r-"" - 2a"r" cos nd + a-\ where OP=r and the angle AOP = d. Prove also that the sum of the angles that AP, BP, CP, ... make with OP IS tan r" cos nd - a'* ANSWERS TO THE EXAMPLES. II. (3) The student should observe that each of these series is very slowly convergent. III. (2) log, 7 = 1-9479. IV. Examples (1), (2), (3) indicate a method of obtaining the Log- arithmic and Exponential Expansions. (6) This is a form of Euler's Formula sin 6 = 2^ cos TT cos ■^, cos —, .... cos — - sm — 2 2,^ 2^ 2** 2" when Q is indefinitely increased. VI. (8) (i) (-if-'^n-i. (ii) {-lf^"^«. VII. (1) \/2 { cos (- 45") + i sin ( - 45°) } , 2 (cos 30" + i sin 30°), 2 (cos 600 + i sin 600). answehs to the examples. 173 (2) (i) 2(co8l50 + tsinl5<'), 2(008 1350 + 1 sin 135"), 2(cos2550 + tsin255<'). (ii) 2 (cos IQO + i sin 10"), 2 (cos 1300 + i sin ISO"), 2 (cos 250" + i sin 250"). (iii) A^2 (cos 6" + i sin 6"), ^2 (cos 78" + i sin 78"), a5/2 (cos 150" + i sin 150"), ^2(cos294" + isin294«). (3) (i) ±1, ±V{-1). (ii) 2, 2 {cosi(2r7r) + tsini(2r7r)} putting 1, 2, 3, 4 for r successively. (iii) 3, |{-l + iV3}, f{-l-iV3}. vni. (1) l,cosi(2r7r)4-isin4(2r7r), wherer=l, 2, 3, 4. (2) (a;2-l)(x2-V2a; + l)(a:2 + l)(x2+V2« + l). (3) (x - 1) {x^ - 2x cos tV (2nr) + 1} six factors putting 1, 2, 3, 4, 5, 6 for r. (4) {x'^-l)(x^-x+l){:c^ + x + l). (o) a:2-2aa:cosT3o(nr) + a2. (6) cos J {rvr) + i sin \ [rir), r having each integral value from up to 11. TSL (1) ±-l^iV(-l);^, (x2-xV2 + l)(x' + xV2 + l). (2) [x^ - V3x + 1) (x2 + 1) (x2 + V3x + 1). (3) a;2-2xcos(l+2r)9" + l. (4) Solve the equation (x^ - V3^ + 1) (^^ + 1) (^2 + ^3^: + 1) = 0. (5) (« + 1) j a;2 _ 2x cos j\ (tt + 2r7r) + 1 { seven factors in all. (6) cos + 1 sm , where r may have any mtegral value. 174 TRIGONOMETRY. X. (3) With the notation of Ex. (1) sin (a + /3 + 7 + 5 + e) = 51C4 - /S'gCg + ^5 cos{a + ^ + y + 5 + €) = Cs-S^C.^ + S^C^. xin. (9) Put h = in Ex. 3, p. 26. (10) Put a = 0, /3 = 1 in Ex. 3, p. 26. (17) v(- 1)= cos — ^ — + i sin — - — ; making this substitution o o a^* is expressed in the form a**"^ . Then proceed as in Ex. 3, p. 26. (18) c* = e'^°^^^cos(logc) + isin{logc), .-. a + ib + c^ = {a + cos (log c)} + i{b + sin (log c)}. Then proceed as in Ex. 3, p. 26. XIV. (1) The equation e^''^=e'^, does not assert that 6i7r=:0, but that QiTr = + 2irTr. (2) e''(«-'^)^gt(«+'r)^ (Joes not assert that i (a - tt) = i (a + 7r). XVII. (1) (x2 - 2x cos 150 + 1) (ic2 - 2x cos 105<> + 1) {x^- - 2 cos IQo" + 1) (x2 - 2x cos 2850 + 1) = 0. (2) (a;2 - 2x cos 2" + 1) (x^ _ 2x cos 74" + 1) (x^ - 2x cos 146« + 1) X (a;2 - 2a; cos 218o + 1) (x^ - 2x cos 2900 + 1) = 0. (3) x^ - 2x cos ^ [Srir + tt) + 1 = 0, six quadratics. (4) a;2 + 2x cos (r x 72^ + 6^) + 1 = 0, five quadratics. (5) x^ - 2xy cos 1- y^, n factors. ANSWERS TO THE EXAMPLES. 175 XIX. (5) d is the smaller root of the quadratic e^- -26 + {-0029... )2 J -2 = 0. (7) 2™-. (8) ?^^^. (9) '-^^. (10) ^v. XXI. 2.1(2/1-1)... (>^ + l) ^^ ^^ «(7i-l)... 2.1 ♦ (2n + l)2n(2n-l)_,^2^^^^^ ^ ' n. (n-1) ... 2. 1 (iii) .- (4n+2) (471 + 1) ... (271 + 2) (271 + 1)271(271-1) ... 2.1* XXIV. (1) n even, {(- 1) ^j - cos 77a} 2-^+^ ; n odd, 2-"+^ cos na. n-z} n (2) n even, ; n odd (- 1) 2 __ _ . ^ ' ' ' cosTia (3) n even, (- l)i (1 - cos 71a) 2-"+! ; n odd (- 1) 2 2-"+^ sin na. (4) n even, , ; n odd . „ ' 1 - cos 7ta sin^ na .tan^-^^i^-^tan3^+... (5) (6) (7) tan 77a = —^ l.'^^^^Jtan^^ + etc. by Art. 40 is an equation of the n"' degree in tan 6, of which tan a is one root, and tan ( a+ - j is another. Hence as in Art. 61. (5) The sum, n even, = — „ h n (71 - 1) ; ' '' tan2 7ta ^ ' 77, odd = 77- tan^ 77a + 7i (71 - 1). (6) The sum, 77 even, =-7icot77a; 7i odd, = Titan 71a. (7) The sum = 77 cot ?7a. 176 TRIGONOMETRY. XXV. (3) Let u = (cos d)^^ ^ then log w = cot ^ log (cos 6) the limit of this = when ^ = 0, therefore the limit of u is 1. (4) - 00 . (5) Of the third order. (8) 2-^"+^ jcos ind + ^'^ i^"" ~ ^^ cos 4.{n-\)d 4w(4w-l)(4n-2)(4n-3) ., ^, . , + — ^ '\ '^ -^cos4(n-2) 6 + L . Z , o , ^ (10) By Art. 40, or Ex. 12, p. 43, the equation , ^ n(n~l){n-2) 71 tan d ^— — ^^^-- — tan^^ + . . . tanwa= — has for its roots a, a + - , a -I •> ... n n Put a = 0, ?i = ll, and divide by tan 6, when we have an equation in tan d, viz. 0=1-15 tan2^ + 42 tan^^ - 30 tan^^ + 5 tan^^ - ^V tan^^^, whose roots are ± tan ^\ ir, ± tan y\ tt , ... ± tan ^\ tt, writing x for tan^^, we have the required equation. XXVI. .. sin ^ fw 4- 1) a sin I na ,^, cosna.sinwa (1) : • {a) -. — • ^ ' sm^a sin a sin 1(371 - 1) a sin I na ... sin fw + 1) a sin wa ^ ' sin f a * ^ ' 2 sin a ,„. . f cos(« + l)asin7ia] ,^, ,j . „ ,._ . . _ , (5) \\n+ ^ .' V. (6) Usesin3a = |(3sma-sm3a). (7) Usecos^a = ^cos4a + ^cos2a + |. (8) =|(8in3a + sina) + ^(sin5a + sina) + ^ (sin 7a + sina) + ... ANSWERS TO THE EXAMPLES. 177 (9) =i{cosa-cos3a + cosa-cos5a + cosa-cos 7a+...}. (10) Use cos3a = ;i(cos3a + 3 cos a). (11) Use sin^a = |^sin4a-isiii2a + f . (12) 6= - , or :: {rTT + ^ir) where r is any integer. (13) and f n. (16) in(n + l). (17) In^ (yi + l)^. (18) ^n (n + 1) (n + 2). (19) Write 5 + tt for 5. (20) Write 5 + tt for 5. XXVII. (1) cosec^{tan(n + l)^-tan^}. (2) cosec^{cot^-cot(M+l)^}. (3) sec d {tan (n + 1) (^ - J r) + cot 6}^ this may be proved by putting ^-^Trfor ^inEx. (1). (4) Each term in this series is one-half the corresponding term in Ex. (1). (5) I cosec ^a {sec ^ (2/i + 1) a - sec J a}. (6) I cosec ^a {cosec J a - cosec ^ (2n + 1) a}. (7) isin2^-2^sin2«+^^. (8) isin2^-(-l)«-^i8in2«+i^. (9) isin2^-(-l)«2"-2sin^. (10) The result is similar to that in Ex. (5). (11) ^seca {cosec a- (-1)" cosec (2» + l) a}. (12) tan-i(n + l)-tan-il. (13) tan-i {n + 1) a - tan"! a. (U) tan-i(w + l)^a^-tan-ia2. (15) cot 0-2** cot 2»d. L. 12 178 TRIGONOMETRY. (16) The series reduces to [See Ex. 5, p. 66] 2 (cosec 2a + cosec 4a + cosec 8a + ... to w terms). (17) i {cos 2^ -cos 2"+!^}. (18) | cos ^- cos 4^ (19) |cosec2|-2«-icosec2 2'»-i^. (20) ^cosec^ |^-cosec2^. (21) cot — -cot a. (22) The series reduces to [Ex. 5, p. 66] sin a (cosec 2a + cosec 4a + cosec 8a + ... to n terms). (23) 1-^^. (24) 2-^"-l. xxvm. , sing cos a - a: cos (a - j8) ^ ' l-2a;cosa + a;2* ^ > l-2xcos/3 + x2 * sin a - cos a sin (a - j3) .. cos a - sin a cos (a - /3) ^ ' 1-2 cos a cos /3 + cos'-* a * ' 1 - sin 2a + sin^ a (5) fi^^^** sin (sin a). (6) e^ ^°^ ** sin (x sin a). (7) e-«°s"^°^^cos(cosasin/3). (8) e-<^o^*^°^^sin(a-cosasin^). (9) la. (10) -log (2 sin ^a). (11) |log (l + 2sinacos jS + sin^ a). (12) -Iog(l + 2cos^cos0 + cos2^). cos a - a; cos (a - /3) - aj^ cos (a + w/3) + a;"+i cos {a + (n-l)j8} (13) 1 -2a; cos /3 + 052 (14) a; sin g + x^ sin (g - jg) + ( - a;)»+i sin {a + n^)~{- a;)^+^ sin { a + (n - 1) )S } l + 2a;coSj8 + a;^ (15) 2''cos'^iacos^na. ANSWERS TO THE EXAMPLES. 179 (16) r^ sin (a + n - sin in (t- a) (u) tan«-tan^„. (112) — ^„-*.-X_-'. ANSWERS TO EXAMINATION PAPERS. 5. .469296; 71044'29';48015'31". 6. H^^rrgo^^^tW ' n. 4. d = lnT + iir. 5. ||. 6. -. 8. 530 7' 48", 590 29' 23". m. 1. 1300 _?Z2!, |^_3. 2. 120°, 2400. TT 3. mr + lir, mr + iir. 7. l-log2, 31og2-3, f log2-l. 10. 740 50' 38", 500 32' 53". 11. 5 ft. 4 in., 6^ ft. IV. 7. C=180, 6 = co8ecl80, a = cotl80-l. 8. 78° 27' 27". 11. 156-4, 556-4. V. 3. a; = r7r-|7r, inx = rir + l'n; VI. 2. 2 cos (jS - 7) cos (7 - a) cos (a - /3). a: 2/ cot C « ^ . ^ 4. , ■ ^ - •^—,— , -^ r - „ - - cot C. sin C a sm C a 184 TRIGONOMETRY. VII. 4. |{cosna + /3 + 7) + cosi03-h7-3a) + cosn7 + a-3/3) + cos^{a + j3-37)}. 9. (1) e^~"'''cos(e-"»^sinna;); (2) cos{a + (w + l)j3}smn^cosec/3. Yin. 2. tan^=± 2»m cos(0 + -j7r)=l. , cos 2^ 5. AB {m^ + ri-y CD sin ABD cos ABC 11. sin (^ jy a -^i^D) J5-C=900. 10. (2) i^-i(2r + l)7r. IX. (i) See Ex. XXVIII. (22). (ii) See Art. 32. (iii) See Ex. XXX. (18). XI. 1. ^ + ^ = 'J1^^=(^)\ao + BO + CO = 1SOO. I n m pq \ r J 2. ^ = 2/3 + 2w7r, 0r=2a + 2n7r. 5. ^V^. 11. (2) See XLni. (96). 6. -1. XII. XIII. 12. (2) (2n + 1) sin yia - (27i - 1) sin (n+ 1) a - sin a 9. 10. 2 (1 - cos a) XIV. See answer to Ex. XXV. (10). ^TT + ^TT - {a;-! - ^x-^ (sin a)-^ sin 2a + ^x'^ (sin a)-^ sin 3a - CAMBRIDGE ; PRfMTEVBY C. "tlATriifi. & SOg; »* . USIVBRSITY CKITEBSIXT PRESS. <: K4LlSki^ A Catalogue OF WORKS ON Mathematics, Science, ANO History and Geography pum.isnED CY Macmillan & Co., Bedford Street, Strand, London. CONTENTS. mathematics- Arithmetic . . . .. Algebra . . • • • Euclid and Elementary Geometry Mensuration Higher Mathematics . science- Natural Philosophy Astronomy . . . • Chemistry . . . • Biology . . . • Medicine . . • ■ Anthropology Physical Geography and Geolog Agriculture Political Economy Mental and Moral Philosophy HISTORY AND GEOGRAPHY 3 5 5 6 7 14 19 19 21 25 2S 25 26 27 27 29 AND 30, Bedford Street, Covent Garden, London, W.C, April 1884. MATHEMATICS. (i) Arithmetic, (2) Algebra, (3) Euclid and Ele- mentary Geometry, (4) Mensuration, (5) Higher Mathematics. ARITHMETIC. Aldis.— THE GIANT ARITHMOS. A most Elementary Arith- metic for Children. By Mary Steadman Aldis. With Illustrations. Globe Svo. 2s. 6d. Brook-Smith (J.).— ARITHMETIC IN THEORY AND PRACTICE. Bv J. Brook-Smith, M.A., LL.B., St. John's College, Cambridge ; Barrister-at-Law ; one of the Masters of Cheltenham College. New Edition, revised. Crown Svo. 45. 6d. Candler. — help to arithmetic. Designed for the use of Schools. By H. Candler, M.A., Mathematical Master of Uppingham School. Extra fcap. Svo. 2s. 6d, Dalton.— RULES AND EXAMPLES IN ARITHMETIC. By the Rev. T. Dalton, M.A., Assistant -Master of Eton Collie. New Edition. iSmo. 2s. 6d. [Ansiucrs to the Examples are appended. Pedley.— EXERCISES in arithmetic for the Use of Schools. Containing more than 7,000 original Examples. By S. Pedley, late of Tamworth Grammar School. Crown Svo. 5x. Smith. — Works by the Rev. Barnard Smith, M.A,, late Rector of Glaston, Rutland, and Fellow and Senior Bm-sar of S. Peter's College, Cambridge. ARITHMETIC AND ALGEBRA, in their Principles and Appli- cation ; with numerous systematically arranged Examples taken from the Cambridge Examination Papers, with especial reference to the Ordinary Examination for the B.A. Degree. New Edition, carefully Revised. Crown Svo. \0s. 6d. ARITHMETIC FOR SCHOOLS. New Edition. Crown Svo. 4J. 6d. A KEY TO THE ARITHMETIC FOR SCHOOLS. New Edition. Crown Svo. 8s. 6d. 4 MACMILLAN'S EDUCATIONAL CATALOGUE. Smith. — Works by the Rev. Barnard Smith, M.A. {contimied)-— EXERCISES IN ARITHMETIC. Crown 8vo, limp cloth, 2s. With Answers, 2s. 6(i. Answers separately, 6d. SCHOOL CLASS-BOOK OF ARITHMETIC. i8mo, cloth. 3^. Or sold separately, in Three Parts, is. each. KEYS TO SCHOOL CLASS-BOOK OF ARITHMETIC. Parts L, IL, and III., 2s. 6d. each. SHILLING BOOK OF ARITHMETIC FOR NATIONAL AND ELEMENTARY SCHOOLS. i8mo, clotli. Or sepa- rately, Part I. 2d. ; Part II. ^d. ; Part III. ^d. Answers. 6d. THE SAME, with Answers complete. i8mo, cloth, is. 6d. KEY TO SHILLING BOOK OF ARITHMETIC. i8mo. 4^. (>d. EXAMINATION PAPERS IN ARITHMETIC. i8mo. i^. 6d. The same, with Answers, i8mo, 2s. Answers, 6d. KEY TO EXAMINATION PAPERS IN ARITHMETIC. i8mo. 4J. 6d. THE METRIC SYSTEM OF ARITHMETIC, ITS PRIN- CIPLES AND APPLICATIONS, with numerous Examples, written expressly for Standard V. in National Schools. New Edition. i8mo, cloth, sewed. 3^. A CHART OF THE METRIC SYSTEM, on a Sheet, size 42 in. by 34 in. on Roller, mounted and varnished. New Edition. Price 3^. 6d. Also a Small Chart on a Card, price id. EASY LESSONS IN ARITHMETIC, combining Exercises in Reading, Writing, Spelling, and Dictation. Part I. for Standard I. in National Schools. Crown 8vo. gd. EXAMINATION CARDS IN ARITHMETIC. (Dedicated to Lord Sandon.) With Answers and Hints. Statidards I. and II. in box, is. Standards III., IV., and V., in boxes, IS. each. Standard VI. in Two Parts, in boxes, is. each. A and B papers, of nearly the same difficulty, are given so as to prevent copying, and the colours of the A and B papers differ in each Standard, and from those of every other Standard, so that a master or mistress can see at a glance whether the children have the proper papers. MATHEMATICS. ALGEBRA. Dalton.— RULES and examples in algebra. By the Rev. T. Dalton, M.A., Assistant-Master of Eton College. Part I. New Edition. i8mo. 2x. Part II. i8mo. 2s. (id. Jones and Cheyne. — ALGEBRAICAL EXERCISES. Pro- gressively Arranged. By the Rev. C. A. Jones, M.A., and C, H. Cheyne, M.A., F.R.A.S., Mathematical Masters of West- minster School. New Edition. i8mo. is. 6d. Smith.— ARITHMETIC AND ALGEBRA, in their Principles and Application ; with numerous systematically arranged Examples taken from the Cambridge Examination Papers, with especial reference to the Ordinary Examination for the B.A. Degree. By the Rev. Barnard Smith, M. A., late Rector of Glaston, Rutland, and Fellow and Senior Bursar of St. Peter's College, Cambiidge. New Edition, carefully Revised. Crown Svo. los. 6d. Todhunter. — Works by L ToDHUNTER, M.A., F.R.S., D.Sc, late of St. John's College, Cambridge. " Mr. Todhunter is chiefly known to Students of Mathematics as the author of a scries ot admirable mathematical text-books, which possess the rare qualities of beinj^ dear in style and absolutely free from mistakes, typographical or other." — Saturday Review. ALGEBRA FOR BEGINNERS. With numerous Example.-. New Editioii. iSmo. 2s. 6d. KEY TO ALGEBRA FOR BEGINNERS. Crown Svo. 6j. ed. ALGEBRA. For the Use of Colleges and Schools. New Edition. Crown Svo. 7^. 6d. KEY TO ALGEBRA FOR THE USE OF COLLEGES AND SCHOOLS. Crown Svo. loj. 6d. EUCLID & ELEMENTARY GEOMETRY. Constable. — geometrical exercises for be- GINNERS. By Samuel Constable. Crown Svo. 3^. 6d. CuthbertSOn. — EUCLIDIAN geometry. By Francis Cuthbertson, M.A., LL.D., Head Mathematical Master of the City of London School. Extra fca^i. Svo. 45-. 6d. 8 MACMILLAN'S EDUCATIONAL CATALOGUE. Boole. — Works by G. Boole, D.C.L., Sac: {continued)— A TREATISE ON DIFFERENTIAL EQUATIONS. Sup- plementary Volume. Edited by I. Todhunt£R. Crown 8vo. The CALCULUS of finite differences. Third Edition, revised by J. F. MouLTON. Crown 8vo. ioj. 6d. Cambridge Senate-House Problems and Riders, with Solutions : — 1875— PROBLEMS AND RIDERS. By A. G. Greenhill, M.A. Crown 8vo. 8x. 6d. 1878— SOLUTIONS OF SENATE-HOUSE PROBLEMS. By the Mathematical Moderators and Examiners. Edited by J. W. L. Glaisher, M.A., Fellow of Trinity College, Cambridge. 12s. Cheyne. — an elementary treatise on the plan- etary THEORY. By C. H. H. Cheyne, M.A., F.R.A.S. With a Collection of Problems. Third Edition. Edited by Rev. A. Freeman, M.A., F.R.A.S. Crown 8vo. Ts. 6d. Christie.— A COLLECTION OF ELEMENTARY TEST- QUESTIONS IN PURE AND MIXED MATHEMATICS ; with Answers and Appendices on Synthetic Division, and on the Solution of Numerical Equations by Horner's Method. By James R. Christie, F.R.S., Royal Military Academy, Woolwich. Crown 8vo. 2>s. 6d. Clausius. — MECHANICAL THEORY OF HEAT. By R. Clausius, Translated by Walter R. Browne, M.A., late Fellow of Trinity College, Cambridge. Crown 8vo. los. 6d. Clifford. — THE ELEMENTS OF DYNAMIC. An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies. By W. K. Clifford, F.R.S., late Professor of Applied Mathematics and Mechanics at University College, London. Part I. — KINEMATIC. Crown 8vo. 'js. 6d. Cotterill.— A TREATISE ON APPLIED MECHANICS. By James Cotterill, M.A., F.R.S., Professor of Applied Mechanics at the Royal Naval College, Greenwich. With Illustrations. 8vo. {In the press. Day." — PROPERTIES OF CONIC SECTIONS PROVED GEOMETRICALLY. Part I. THE ELLIPSE. With Prob- lems. By the Rev. H. G. Day, M.A. 8vo. 3^. 6d. Day (R. E.)— ELECTRIC LIGHT ARITHMETIC. By R. E. Day, M.A., Evening? Lecturer in Experimental Physics at King's College, London. Pott 8vo. 2s. MATHEMATICS. Drew. — GEOMETRICAL TREATISE ON CONIC SECTIONS. By W. H. Drew, M.A., Sf. John's College, Cambridge, New Edition, enlarged. Crown 8vo. 5^. SOLUTIONS TO THE PROBLEMS IN DREW'S CONIC SECTIONS. Crown 8vo. 4J. dd. Dyer.— EXERCISES IN ANALYTICAL GEOMETRY. Com- piled and arranged by J. M. Dyer, M.A., Senior Mathematical Master in the Classical Department of Cheltenham College. With Illustrations. Crown 8vo. ^. 6d. Edgar (J. H.) and Pritchard (G. S.).— NOTE-BOOK ON PRACTICAL SOLID OR DESCRIPTIVE GEOMETRY. Containing Problems with help for Solutions. By J. H. Edgar, M.A., Lecturer on Mechanical Drawing at the Royal School of Mines, and G. S. Pritchard. Fourth Edition, revised by Arthur Meeze. Globe 8vo. 4^. 6(/. Ferrers. — Works by the Rev, N. M. Ferrers, M.A., Fellow and Master of Gonville and Cains College, Cambridge. AN ELEMENTARY TREATISE ON TRILINEAR CO- ORDINATES, the Method of Reciprocal Polars, and the Theory of Projectoi-s. New Edition, revised. Crown 8vo. 6j. 6ii. AN ELEMENTARY TREATISE ON SPHERICAL HAR- MONICS, AND SUBJECTS CONNECTED WITH THEM. Crown Svo. 7^. Cd. Frost. — Works by Percival Frost, M. A., D.Sc, formerly Fellow of St. John's College, Cambridge ; Mathematical Lecturer at King's College. AN ELEMENTARY TREATISE ON CURVE TRACING. By Percival Frost, M.A. 8vo. 12s. SOLID GEOMETRY. A New Edition, revised and enlarged, of the Treatise by Frost and Wolsteniiolme. In 2 Vols. Vol. I. Svo. 1 6 J. Hemming. — an elementary treatise on the DIFFERENTIAL AND INTEGRAL CALCULUS, for the Use of Colleges and Schools. By G. W. Hemming, M.A., Fellow of St. John's College, Cambridge. Second Edition, with Corrections and Additions. Svo. gs, Jackson. — GEOMETRICAL CONIC SECTIONS. An Ele- mentary Treatise in which the Conic Sections are defined as the Plane Sections of a Cone, and treated by the Method of Pro- jection. By J. Stuart Jackson, M.A., late Fellow of Gonville and Caius College, Cambridge. Crown Svo. 4^. 6d. lo MACMILLAN'S EDUCATIONAL CATALOGUE. Jellet (John H.). — a treatise on the theory of FRICTION. By John H. Jellet, B.D., Provost of Trinity College, Dublin; President of the Royal Irish Academy. 8vo. Ss. 6^. Johnson. — integral calculus, an Elementary Treatise on the ; Founded on the Method of Rates or Fluxions. By William Woolsey Johnson, Professor of Mathematics at the United States Naval Academy, Anuopolis, Maryland. Demy 8vo. Ss. Kelland and Tait. — INTRODUCTION TO QUATER- NIONS, with numerous examples. By P. Kelland, M.A., F. R.S., and P. G. Tait, M.A,, Professors in the Department of Mathematics in the University of Edinburgh. Second Edition. Crown 8vo. 7^. 6d. Kempe.- OW to draw a straight line : a Lecture on Linkages. By A, B. Kempe. With Illustrations. Crown Svo. IS. 6d. {Nature Series.) Lock. — ELEMENTARY TRIGONOMETRY. By Rev. J. B. Lock, M.A., Senior Fellow, Assistant Tutor and Lecturer in Mathematics, of Gonville and Caius College, Cambridge ; late Assistant- Master at Eton. Globe Svo. /^s. td. HIGHER TRIGONOMETRY. By the same Author. Globe Svo. 3^. 6d. Both Parts complete in One Volume. Globe Svo. ^s. 6d. Lupton. — ELEMENTARY CHEMICAL ARITHMETIC. With i,ioo Problems. By Sydney Lupton, M.A., Assistant-Master in Harrow School. Globe Svo. c^s. Merriman. — elements OF THE METHOD OF LEAST SQUARE, By Mansfield Merriman, Ph.D., Professor of Civil and Mechanical Engineering, Lehigh University, Belhhhem, Penn. Crown Svo= 7^. 6d. Morgan. — a collection of problems and ex- amples in mathematics. With Answers. By H. A. Morgan, M.A., Sadlerian and Mathematical Lecturer of Jesus College, Cambridge. Crown Svo. 6s. 6d. Millar. — elements of descriptive geometry. By J. B. Millar, C.E., Assistant Lecturer in Engineering in Owens College, Manchester. Crown Svo. 6s. Muir. — A TREATISE ON THE THEORY OF DETERMI- NANTS. With graduated sets of Examples, For use in Colleges and Schools. By Thos. Muir, M.A., F.R.S.E., Mathematical Master in the High School of Glasgov^ Crown Svo. 7^. 6d. MATHEMATICS. Parkinson.— AN elementary treatise on me- CHANICS. For the Use of the Junior Classes at the University and the Higher Classes in Schools. By S. Parkinson, D.D., F.R.S., Tutor and Fraelector of St. John's College, Cambridge. With a Collection of Examples. Sixth Edition, revised. Crown 8vo. 9^. 6d, Phear.— ELEMENTARY HYDROSTATICS. With Numerous Examples. By J. B. Phear, M.A,, Fellow and late Assistant Tutor of Clare College, Cambridge. New Edition. Crown 8vo. Pirie.— LESSONS ON RIGID DYNAMICS. By the Rev. G. PiRiE, M.A., late Fellow and Tutor of Queen's College, Cam- bridge ; Professor of Mathematics in the University of Aberdeen. Crown 8vo. 6j-. Price and Johnson DIFFERENTIAL CALCULUS, an Elementary Treatise on the ; Founded on the Method of Rates or Fluxions. By John Minot Price, Professor of Mathematics in the United States Navy, and William Woolsey Johnson, Pro- fessor of Mathematics at the United States Naval Academy. Third Edition, Revised and Corrected. Demy 8vo. l6i-. Abridged Edition, %s. Puckle.— AN ELEMENTARY TREATISE ON CONIC SEC- TIONS AND ALGEBRAIC GEOMETRY. With Numerous Examples and Hints for their Solution ; especially designed for the Use of Beginners. By G. H. Puckle, M.A. New Edition, revised and enlarged. Crown Svo. 1$. 6d^ Rawlinson. — elementary STATICS. By the Rev. George Rawlinson, M.A. Edited by the Rev. Edward Sturges, M.A. Crown Svo. 4j. 6d. Reynolds.— MODERN METHODS IN ELEMENTARY GEOMETRY. By E. M. Reynolds, M.A., Mathematical Master in Clifton College. Crown Svo. zs. 6d. Reuleaux.— THE kinematics of machinery. Out- lines of a Theory of Machines. By Professor F. Reuleaux. Translated and Edited by Professor A, B. W. Kennedy, C.E. With 450 Illustrations. Medium Svo. 2ls. Robinson.— TREATISE ON marine SURVEYING. Pre- pared for the use of younger Naval Officers. With Questions for Examinations and Exercises principally from the Papers of the MACMILLAN'S EDUCATIONAL CATALOGUE. "Robinson— {conitnued) — Royal Naval College. With the results. By Rev. John L. Robinson, Chaplain and Instructor in the Royal Naval College, Greenwich. With Illustrations. Crown 8vo. 'js. 6d. Contents. — Symbols used in Charts and Surveying — The Construction and Use of Scales — Laying off Angles — Fixing Positions by Angles — Charts and Chart- Drawing — Instruments and Observing — Base Lines— Triangulation — Levelling — Tides and Tidal Observations — Soundings — Chronometers — Meridian Distances — Method of Plotting a Survey — Miscellaneous Exercises— Index. JRouth. — Works by Edward John Routh, M.A., F.R.S., D.Sc, late Fellow and Assistant Tutor at St. Peter's College, Cambridge; Examiner in the University of London. A TREATISE ON THE DYNAMICS OF THE SYSTEM OF RIGID BODIES. With numerous Examples. Fourth and enlarged Edition. Two Vols. Vol. I. — Elementary Parts. 8vo. 14^. Vol.11. — The Higher Parts. 8vo. [In t^e press. STABILITY OF A GIVEN STATE OF MOTION, PAR- TICULARLY STEADY MOTION. Adams' Prize Essay for 1877. 8vo. Ss. dd. Smith (C). — CONIC SECTIONS. By Charles Smith, M.A., Fellow and Tutor of Sidney Sus.sex College, Cambridge. Second Edition. Crown 8vo. 7^. 6d. Snowball. — the elements of plane and spheri- cal TRIGONOMETRY ; with the Construction and Use of Tables of Logarithms. By J. C. Snowball, M.A. New Edition. Crown 8vo. 7-^' ^f/. Tait and Steele.— a treatise on dynamics of a PARTICLE. With numerous Examples. By Professor Tait and Mr. Steele. Fourth Edition, revised. Crown 8vo. \2s. Thomson.— A TREATISE on the motion of VORTEX RINGS. An Essay to which the Adams Prize was adjudged in 1882 in the University of Cambr'dge. By J. J. Thomson, Fellow and Assistant Lecturer of Trinity College, Cambridge. With Diagrams. 8vo. 6^. Todhunter. — Works by I. ToDHUNTER, M.A., F.R.S., D.Sc, late of St. John's College, Cambridge. " Mr. Todhunter is chiefly known to students of Mathematics as the author of a series of admirable mathematical text-books, which possess the rare qualities of being clear in style and absolutely free from mistakes, typographical and other." — Saturday Review. TRIGONOMETRY FOR BEGINNERS. With numerous Example^. New Edition. i8mo. is. 6d. KEY TO TRIGONOMETRY FOR BEGINNERS. Crown 8vo. 8j. 6d. MATHEMATICS. 13. Todhunter. — Works by I. Todhunter, M.A., &c. {continued)— MECHANICS FOR BEGINNERS. With numerous Examples. New Edition. i8mo. i^s. 6d. KEY TO MECHANICS FOR BEGINNERS. Crown 8vo, 6s. 6d. AN ELEMENTARY TREATISE ON THE THEORY OF EQUATIONS. New Edition, revised. Crown 8vo. 7^. 6d. PLANE TRIGONOMETRY. For Schools and Colleges. New Edition. Crown Svo. $s. KEY TO PLANE TRIGONOMETRY. Crown Svo. los. 6d. A TREATISE ON SPHERICAL TRIGONOMETRY. New Edition, enlarged. Crown Svo. 4J. 6d. PLANE CO-ORDINATE GEOMETRY, as applied to the Straight Line and the Conic Sections. With numerous Examples. New Edition, revised and enlarged. Crown Svo. 7j. 6d. A TREATISE ON THE DIFFERENTIAL CALCULUS. With numerous Examples. New Edition. Crown Svo. lOs. 6d. A TREATISE ON THE INTEGRAL CALCULUS AND ITS APPLICATIONS. With numerous Examples. New Edition, revised and enlarged. Crown Svo. ioj. 6d. EXAMPLES OF ANALYTICAL GEOMETRY OF THREE DIMENSIONS. New Edition, revised. Crown Svo. 4f. A TREATISE ON ANALYTICAL STATICS. With numerous Examples. New Edition, revised and enlarged. Crown Svo. los. 6d. A HISTORY OF THE MATHEMATICAL THEORY OF PROBABILITY, from the time of Pascal to that of Laplace. Svo. iSy. RESEARCHES IN THE CALCULUS OF VARIATIONS, principally on the Theory of Discontinuous Solutions : an Essay to which the Adams' Prize was awarded in the University of Cam- bridge in 1 87 1. Svo. 6s. A HISTORY OF THE MATHEMATICAL THEORIES OF ATTRACTION, AND THE FIGURE OF THE EARTH, from the time of Newton to that of Laplace. 2 vols. Svo. 24s. AN ELEMENTARY TREATISE ON LAPLACE'S, LAME'S, AND BESSEL'S FUNCTIONS. Crown Svo. 10s. 6d. Wilson (J. M.).— SOLID GEOMETRY AND CONIC SEC- TIONS. With Appendices on Transversals and Harmonic Division. For the Use of Schools. By Rev. J. M. Wilson, M.A. Head Master of CUfton College. New Edition. Extra fcap. Svo. 3J. 6d. 14 MACMILLAN'S EDUCATIONAL CATALOGUE. Wilson.— GRADUATED EXERCISES IN PLANE TRI- GONOMETRY. Compiled and arranged by J. Wilson, M.A., and S. R. Wilson, B.A. Crown 8vo. 4^. dd. "The exercises seem beautifully graduated and adapted to lead a student on most gently and pleasantly." — E. J. Routh, F.R.S., St. Peter's College, Cambridge, (See also Elementary Geometry.^ Wilson (W. P.).— A TREATISE ON DYNAMICS. By W. P. Wilson, M.A., Fellow of St. John's College, Cambridge, and Professor of Mathematics in Queen's College, Belfast, 8vo. Woolwich Mathematical Papers, for Admission into the Royal Military Academy, Woolwich, 1880— 1883 inclusive. Crown 8vo, 3^-. (>d, Wolstenholme.— MATHEMATICAL PROBLEMS, on Sub- jects included in the First and Second Divisions of the Schedule of subjects for the Cambridge Mathematical Tripos Examination. Devised and arranged by Joseph Wolstenholme, D.Sc, late Fellow of Christ's College, sometime Fellow of St. John's College, and Professor of Mathematics in the Royal Indian Engineering College. New Edition, greatly enlarged. 8vo. 18^. EXAMPLES FOR PRACTICE IN THE USE OF SEVEN- FIGURE LOGARITHMS. By the same Author, lln preparation. SCIENCE. (i) Natural Philosophy, (2) Astronomy, (3) Chemistry, (4) Biology, (5) Medicine, (6) Anthro- pology, (7) Physical Geography and Geology, (8) Agriculture, (9) Political Economy, (10) Mental and Moral Philosophy. NATURAL PHILOSOPHY. Airy. — Works by Sir G. B. Airy, K.C.B., formerly Astronomer- Royal : — UNDULATORY THEORY OF OPTICS. Designed for the Use of Students in the University. New Edition. Crown 8vo. 6s. 6d. ON SOUND AND ATMOSPHERIC VIBRATIONS. With the Mathematical Elements of Music. Designed for the Use of Students in the University. Second Edition, revised and enlarged. Crown 8vo. 9^. A TREATISE ON MAGNETISM. Designed for the Use of Students in the University. Crown 8vo. gy. 6d, SCIENCE. 15 Airy (Osmond).— a TREATISE ON GEOMETRICAL OPTICS. Adapted for the Use of the Higher Classes in Schools. By Osmund Airy, B.A., one of the Mathematical Masters in Wellington College. Extra fcap. 8vo. 3^. dd. Alexander (T.).— elementary applied mechanics. Being the simpler and more practical Cases of Stress and Strain wrought out individually from first principles by means of Ele- mentary Mathematics. By T. Alexander, C.E., Professor of Civil Engineering in the Imperial College of Engineering, Tokei, Japan. Crown 8vo. Part I. 4X. 6^. Alexander — Thomson. — ELEMENTARY APPLIED MECHANICS. By Thomas Alexander, C.E., Professor of Engineering in the Imperial College of Engineering, Tokei, Japan ; and Arthur Watson Thomson, C.E., B.Sc, Professor oif Engineering at the Royal College, Cirencester. Part 11. Trans- verse Stress ; upwards of 150 Diagrams, and 200 Examples carefully worked out ; new and complete method for finding, at every point of a beam, the amount of the greatest bending moment and shearing force during the transit of any set of loads fixed relatively to one another — e.g.y the wheels of a locomotive ; continuous beams, &c., &c. Crown 8vo. \os. 6d. Awdry.— EASY LESSONS ON LIGHT. By Mrs. W. Awdry. Illustrated. Extra fcap. 8vo. 2s. 6d. Ball (R. S.). — EXPERIMENTAL MECHANICS. A Course of Lectures delivered at the Royal College of Science for Ireland. By R. S. Ball, M.A., Professor of Applied Mathematics and Mechanics in the Royal College of Science for Ireland. Cheaper Issue. Royal 8vo. ioj. 6d. Chisholm. — the SCIENCE OF WEIGHING AND MEASURING, AND THE STANDARDS OF MEASURE AND WEIGHT. By H. W. Chisholm, Warden of the Standards. With numerous Illustrations. Crown 8vo. ^. 6d. {Nature Series.) Clausius.— MECPIANICAL THEORY OF HEAT. By R. Clausius. Translated by Walter R. Browne, M.A., late Fellow of Trinity College, Cambridge. Crown 8vo. ioj. 6d. Cotterill. — a treatise on applied mechanics. By James Cotterill, M.A., F.R.S., Professor of Applied "Mechanics at the Royal Naval College, Greenwich. With Illus- trations. 8vo. [In the press. Gumming. — an introduction to the theory of electricity. By Linn/eus Gumming, M.A., one of the Masters of Rugby School. With Illustrations. Crown 8vo. 8^. (>d. i6 MACMILLAN'S EDUCATIONAL CATALOGUE. Daniell. — A TEXT-BOOK OF THE PRINCIPLES OF PHYSICS. By Alfred Daniell, M.A., Lecturer on Physics in the School of Medicine, Edinburgh. With Illustrations. Medium 8vo. 2is. Day.— ELECTRIC LIGHT ARITHMETIC. By R. E. Day, M.A., Evening Lecturer in Experimental Physics at King's College, London. Pott 8vo. 2s. Everett.— UNITS and PHYSICAL CONSTANTS. By J. D. Everett, F.R.S., Professor of Natural Philosophy, Queen's College, Belfast. Extra fcap. 8vo. 4^. 6d. Gray — ABSOLUTE MEASUREMENTS IN ELECTRICITY AND MAGNETISM. By Andrew Gray, M.A., F.R.S.E., Chief Assistant to the Professor of Natural History in the Uni- versity of Glasgow. Pott Svo. y. 6(/. Huxley.— INTRODUCTORY PRIMER OF SCIENCE. By T. H. Huxley, P.R.S., Professor of Natural History in the Royal School of Mines, &c. iSmo. Is. Kempe. — how to draw a straight line ; a Lecture on Linkages. By A. B. Kempe. With Illustrations. Crown Svo. IS. 6d. {Nature Series. ) Kennedy.— MECHANICS OF machinery. By A. B. W. Kennedy, M.Inst.C.E., Professor of Engineering and Mechani- cal Technolo^^y in University College, London. With Illus- trations. Crown Svo. [/;? the press. Lang. — EXPERIMENTAL PHYSICS. By P. R. Scott Lang. M. A., Professor of Mathematics in the University of St. Andrews. Crown Svo. [/« preparation. Martineau (Miss C. A.). — EASY LESSONS ON pieat. By Miss C. A. Martineau. Illustrated. Extra fcap. Svo. IS. 6d. Mayer. — SOUND : a Series of Simple, Entertaining, and Inex- pensive Experiments in the Phenomena of Sound, for the U?e of Students of every age. By A. M. Maver, Professor of Physics in the Stevens Institute of Technology, &c. With numerous Illustrations. Crown Svo. is. 6d. {Nature Series.') Mayer and Barnard. — light : a Series of Simple, Enter- taining, and Inexpensive Experiments in the Phenomena of Light, for the Use of Students of every age. By A. M. Mayer and C. Barnard, With numerous Illustrations. Crown Svo. 2s. 6d. {NalKre Series.) SCIENCE. 17 Newton. — PRINCIPIA. Edited by Professor Sir W. Thomson and Professor Blackburn e. 4to, doth. $is. 6d. THE FIRST THREE SECTIONS OF NEWTON'S PRIN- CIPIA. With Notes and Ilhistrations. Also a Collection of Problems, principally intended as Examples of Newton's jNIethods. By Percival Frost, M.A. Third Edition. 8vo. 12s. Parkinson. — a TREATISE ON OPTICS. By S. Parkinson, D.D., F.R.S., Tutor and Prtelector of St. John's College, Cam- bridge. New Edition, revised and enlarged. Crown 8vo. los. 6d. Perry. — STEAM. AN elementary TREATISE. By John Perry, C.E., Whitworth Scholar, Fellow of the Chemical Society, Lecturer in Physics at Clifton College. With numerous Woodcuts and Numerical Examples and Exercises. i8mo. 45-. 6d. Ramsay.— EXPERIMENTAL PROOFS OF CHEMICAL THEORY FOR BEGINNERS. By William Ramsay, Ph.D., Profe.-sor cf Chemistry in University College, Bristol. Pott Svo. 2S. 6d. Rayleigh. — THE THEORY OF SOUND. By Lord Rayleigii, M.A., F.R.S., formerly Fellow of Trinity College, Cambridge, 8vo. Vol. I. 12s. ed. Vol. II. I2s. ed. [Vol. III. in the press. Reuleaux. — the kinematics of machinery. Out- lines of a Theory of Machines. By Professor F. Reuleaux. Translated and Edited by Professor A. B. W. Kennedy, C.E. With 450 Illustrations. Medium Svo. 2\s. Shann. — an elementary treatise on heat, in RELATION TO STEAM AND THE STEAM-ENGINE. By G. Shann, M.A. With Illustrations. Crown 8vo.4J'. 6d. SpOttiswOOde.— POLARISATION OF LIGHT. By the late W. Spottiswoode, P.R.S. With many Illustrations. New Edition. Crown Svo. y. 6d. [Nature Series.) Stewart (Balfour). — Works by Balfour Stewart, F.R.S., Professor of Natural Philosophy in the Victoria University the Owens College, Manchester. PRIMER OF PHYSICS. With numerous Illustrations. New- Edition, with Questions. iSmo. is. (Science Primers.) LESSONS IN ELEMENTARY PHYSICS. With numerous Illustrations and Chromolitho of the Spectra of the Sun, Stars, and Nebulae. New Edition. Fcap. Svo. 4^. dd. i8 MACMILLAN'S EDUCATIONAL CATALOGUE. Stewart (Balfour).— Works by {continued)— QUESTIONS ON BALFOUR STEWART'S ELEMENTARY LESSONS IN PHYSICS. By Prof. Thomas H. Core, Owens College, Manchester. Fcap. 8vo. is. Stewart — Gee. — practical physics, elementary LESSONS in. By Professor Balfour Stewart, F.R.S., and W. Haldane Gee. Fcap. 8vo. • Part I. General Physics. \_N'early ready. Part II. Optics, Heat, and Sound. [/;/ preparation. Part III. Electricity and Magnetism. [/« preparation. Stokes.— THE NATURE OF LIGHT. Burnett Lectures, ^y Prof. G. G. Stokes, Sec. R.S., etc. Crown Svo. 2.s, 6d. ON LIGHT. Burnett Lectures. First Course. On the Nature of Light. Delivered in Aberdeen in November 1883. By George Gabriel Stokes, M.A., F.R.S., &c., Fellow of Pem- broke College, and Lucasian Professor of Mathematics in the University of Cambridge. Crown 8vo. 2s. 6d. Stone.— AN ELEMENTARY TREATISE ON SOUND. By W. H. Stone, M.B. With Illustrations. i8mo. 3.^. 6d. Tait. — HEAT. By P. G. Tait, M.A., Sec. R.S.E., Formerly Fellow of St. Peter's College, Cambridge, Professor of Natural Philosophy in the University of Edinburgh. Crown 8vo. 6s. Thompson. — elementary LESSONS in electricity AND magnetism. By Silvanus P. Thompson. Pro- fessor of Experimental Physics in University College, Bristol. With Illustrations. Fcap. Svo. 4-r. 6d. Thomson. — the motion of vortex rings, a TREATISE ON. An Essay to which the Adams Prize was adjudged in 1882 in the University of Cambridge. By J. J. Thomson, Fellow and Assistant-Lecturer of Trinity College, Cambridge. With Diagrams. Svo. 6s. Todhunter.— NATURAL philosophy FOR BEGINNERS. By I. Todhunter, M.A., F.R.S., D.Sc. Part I. The Properties of Solid and Fluid Bodies. iSmo. 3^. 6d. Part II. Sound, Light, and Heat. iSmo. 3-r. 6d. Wright (Lewis). — LIGHT ; A COURSE OF EXPERI- MENTAL OPTICS, CHIEFLY WITH THE LANTERN, By Lewis Wright. With nearly 200 Engravings and Coloured Plates. Crown Svo. 7^. 6d. SCIENCE. 19 ASTRONOMY. Airy.— POPULAR astronomy. with Illusti-ations by Sir G. B, Airy, K.C.B,, formerly Astronomer-Royal. New Edition. i8mo. 4^-. 6d. Forbes.— TRANSIT OF VENUS. By G. Forbes, M.A., Professor of Natural Philosophy in the Andersonian University, Glasgow. Illustrated. Crown 8vo. y. dd. {Nature Series.) Godfray. — Works by Hugh Godfray, M.A., Mathematical Lecturer at Pembroke College, Cambridge. A TREATISE ON ASTRONOMY, for the Use of Colleges and Schools. New Edition. 8vo. lis. 6d. AN ELEMENTARY TREATISE ON THE LUNAR THEORY, with a Brief Sketch of the Problem up to the time of Newton. Second Edition, i*evised. Crown Svo. 5^. 6d. Lockyer. — Works by J. Norman Lockyer, F.R.S. PRIMER OF ASTRONOMY. With numerous Illustrations. iSmo. IS. {Science Privurs.) _ ELEMENTARY LESSONS IN ASTRONOMY. With Coloured Diagram of the Spectra of the Sun, Stars, and Nebulce, and numerous Illustrations. New Edition. Fcap. Svo. <^s. 6d. QUESTIONS ON LOCKYER'S ELEMENTARY LESSONS IN ASTRONOMY. For the Use of Schools. By John Forbes- RolBERTSON, iSmo, cloth limp. is. 6d. THE SPECTROSCOPE AND ITS APPLICATIONS. With Coloured Plate and numerous Illustrations. New Edition. Crown Svo. 3 J. 6d. Newcomb. — popular astronomy. By S. Newcomb, LL.D., Professor U.S. Naval Observatory. With 112 Illustrations and 5 Maps of the Stars. Second Edition, revised. Svo. i2>s. "It is unlike anything else of its kind, and will be of more use in circulating a knowledge of Astronomy than nine-tenths of the books which have appeared on the subject of late years."— Saturday Review. CHEMISTRY. Fleischer.~A SYSTEM OF volumetric ANALYSIS. Translated, with Notes and Additions, from the Second German Edition, by M. M. Pattison Muir, F.R.S.E. With Illustrations. Crown Svo. 7^. 6d. € 2 20 MACMILLAN'S EDUCATIONAL CATALOGUE. Jones. — Works by Francis Jones, F.R.S.E., F.C.S., Chemical Master in the Grammar School, Manchester. THE OWENS COLLEGE JUNIOR COURSE OF PRAC- TICAL CHEMISTRY. With Preface by Professor RoscoE, and Illustrations. New Edition. i8mo. 2s. 6d. QUESTIONS ON CHEMISTRY. A Series of Problems and Exercises in Inorganic and Organic Chemistry. Fcap. 8vo. 3^. Landauer. — blowpipe analysis. By j. landauek. Authorised English Edition by J. Taylor and W. E. Kay, of Owens College, Manchester. Extra fcap. 8vo. 4s. 6d. Lupton. — ELEMENTARY CHEMICAL ARITHMETIC. With 1,100 Problems. By Sydney Lupton, M.A., Assistant-Master at Harrow. Extra fcap. 8vo. 5^-. Muir.— PRACTICAL CHEMISTRY FOR MEDICAL STU- DENTS. Specially arranged for the first M.B. Course. By M. M. Pattison Muir, F.R.S.E. Fcap. 8vo. is. 6d. RoSCOe. — Works by H. E. RoscOE, F.R.S. Professor of Chemistry in the Victoria Ll^niversity the Owens College, Manchester. PRIMER OF CHEMISTRY. With numerous Illustrations. New Edition. W'ith Questions. l8mo. is. {Science Primers). LESSONS IN ELEMENTARY CHEMISTRY, INORGANIC AND ORGANIC. With numerous Illustrations and Chromolitho of the Solar Spectrum, and of the Alicalies and Alkaline Earths. New PMition. Fcap, 8vo. 4^. 6d. A SERIES OF CHEMICAL PROBLEMS, prepared with Special Reference to the foregoing, by T. E. Thorpe, Ph.D., Professor of Chemistry in the Yorkshire College of Science, Leeds, Adapted for the Preparation of Students for the Government, Science, and Society of Arts Examinations. With a Preface by Professor RoscoE, F.R.S. New Edition, with Key. i8mo. 2s. Roscoe and Schorlemmer.— inorganic and or- ganic CHEMISTRY. A Complete Treatise 011 Inorganic and Organic Chemistry. By Professor H. E. RoscoE, F.R.S., and Professor C. ScHORLEMMEK, F.R.S. With numerous Illustrations. Medium 8vo. Vols. I. and IL— INORGANIC CHEMISTRY. Vol. I.— The Non-Metallic Elements. 21s. Vol. II. Part I.— Metals. i8j-. Vol. II. Part IL— Metals. iSj. Vol. III.— ORGANIC CHEMISTRY. Two Parts. THE CHEMISTRY OF THE HYDROCARBONS and their Derivatives, or ORGANIC CHEMISTRY. With numerous Illustrations. Medium 8vo. 21s. each. SCIENCE. Schorlemmer. — a manual of the chemistry of THE CARBON COMPOUNDS, OR ORGANIC CHE- MISTRY. By C. Schorlemmer, F.R.S., Profes.^or of Che- mistry in the Victoria University the Owens College, Manchester. With Illustrations. 8vo. 14^. Thorpe. — a series of chemical problems, prepared Avith Special Reference to Professor Roscoe's Lessons in EleiBcn- tary Chemistry, by T. E. Thorpe, Ph.D., Professor of Chemistry ill the Yorkshire College of Science, Leeds, adapted for the Pre- paration of Students for the Government, Science, and Society of Arts Examination?. With a Pi-eface by Professor RoscOE. New Edition, with Key. i8mo. 2s. Thorpe and Riicker.— a TREATISE ON chemical PHYSICS. By Professor Thorpe, F.R.S., and Professor RiJCKER, of the Yorkshire College of Science. Illustrated. 8vo. [In preparation . Wright.— METALS AND THEIR CHIEF INDUSTRIAL APPLICATIONS. By C. Alder Wright, D.Sc, &c., Lecturer on Chemistry in St. Mary's Hospital Medical School. Extra fcap. 8vo. 3j. 6d, BIOLOGY. Allen. — ON THE COLOUR OF FLOWERS, as Illustrated in the British Flora. By Grant Allen. With Illustrations. Crown 8 vo. 2>^.6d. {Nature Series.) Balfour. —A TREATISE ON COMPARATIVE EMBRY- OLOGY. By F. M. Balfour, M.A., F.R.S., Fellow and Lecturer of Trinity College, Cambridge. With Illustrations. In 2 vols. 8vo. Vol. I. i8j-. Vol. IL 2IX. Bettany.— FIRST LESSONS IN PRACTICAL BOTANY. By G. T. Bettany, M.A., F.L.S., Lecturer in Botany at Guy's Hospital Medical School. i8mo. i.f. Darwin (Charles). — MEMORIAL NOTICES OF CHARLES DARWIN, F.R.S., &c. By Professor Huxley, P. R.S., G.J. Romanes, F.R.S., Archibald Geikie, F.R.S., and W. T. Thiselton Dyer, F.R.S. Reprinted from Nature. With a Portrait, engraved by C. H. JEENS. Crown 8vo. 2s. 6d. {Nature Series.) 22 MACMILLAN'S EDUCATIONAL CATALOGUE. Dyer and Vines. — the structure of plants. By Professor Thiselton Dyer, F.R. S., assisted by Sydney Vines, D.Sc., Fellow and Lecturer of Christ's College, Cauabridge, and F. O. Bower, M.A., Lecturer in the Normal School of Science. With numerous Illustrations. \In preparation. Flower (W. H.) — an INTRODUCTION TO THE OSTE- OLOGY OF THE MAMMALIA. Being the substance of the Course of Lectures delivered at the Royal College of Surgeons of England in 1870. By Professor W. H. Flower, F.R.S., F.R.C.S. With numerous Illustrations. New Edition, enlarged. Crown 8vo. \os. 6d. Foster. — Works by Michael Foster, M.D., F.R.S., Professor of Physiology in the University of Cambridge. PRIMER OF PHYSIOLOGY. With numerous Illustrations. New Edition. l8mo. is. A TEXT-BOOK OF PHYSIOLOGY. With Illustrations. Fourth Edition, revised. Svo. 21s. Foster and Balfour.— the ELEMENTS OF EMBRY- OLOGY. By Michael Foster, M.A., M.D., LL.D., F.R.S., Professor of Physiology in the University of Cambridge, Fellow of Trinity College, Cambridge, and the late Francis M. Balfour, M.A., LL.D., F.R.S. , P^ellow of Trinity College, Cambridge, and Professor of Animal Morphology in the University. Second Edition, revised. Edited by Adam Sedgwick, M.A., Fellov/ and Assistant Lecturer of Trinity College, Cambridge, and Walter Heape, Demonstrator in the Morphological Laboratory of the University of Cambridge. With Illustrations. Crown Svo. los. 6d. Foster and Langley. — a COURSE OF ELEMENTARY PRACTICAL PHYSIOLOGY. By Prof. Michael Foster, M.D., F.R.S., &c., and J. N. Langley, M.A., F.R.S., Fellow of Trinity College, Cambridge. Fifth Edition. Crown Svo. p. 6d. Gamgee.— A text-book of the physiological CHEMISTRY OF THE ANIMAL BODY. Including an Account of the Chemical Changes occurring in Disease. By A. Gamgee, M.D., F.R.S., Professor of Phy?iology in the Victoria University the Owens College, Manchester. 2 Vols. Svo. With Illustrations Vol.1. l2>s. [Vol. II. in the press. Gegenbaur.— ELEMENTS OF COMPARATIVE ANATOMY. By Professor Carl Gegenbaur. A Translation by F. Jeffrey Bell, B.A. Revised with Preface by Professor E. Ray Lan- KESTER, F.R.S. With numerous Illustrations. Svo. 21s. SCIENCE. 23 Gray.— STRUCTURAL BOTANY, OR ORGANOGRAPHY ON THE BASIS OF MORPHOLOGY. To which are added the principles of Taxonomy and Phytography, and a Glossary of Botanical Terms. By Professor AsA Gray, LL. D. 8vo. los. 6d. Hooker. — Works by Sir J. D. Hooker, K.C.S.I., C.B., M.D., F.R.S., D.C.L. PRIMER OF BOTANY. With numerous Illustrations. New Edition. i8mo. is. {Science Primers.) THE STUDENT'S FLORA OF THE BRITISH ISLANDS- New Edition, revised. Globe 8vo. los. 6d. Huxley. — Works by Professor HuxLEY, P.R.S. INTRODUCTORY PRIMER OF SCIENCE. iSmo. is. {Science Primers.) LESSONS IN ELEMENTARY PHYSIOLOGY. With numerous Illustrations. New Edition. Fcap. Svo. 4^. dd. QUESTIONS ON HUXLEY'S PHYSIOLOGY FOR SCHOOLS. By T. Alcock, M.D. i8mo. is. 6d. PRIMER OF ZOOLOGY. i8mo. {Science Primers.) [In preparation. Huxley and Martin. — a COURSE OF practical IN STRUCTION IN ELEMENTARY BIOLOGY. By Professor Huxley, P.R.S. , assisted by H. N. Martin, M.B., D.Sc. New Edition, revised. Ci^own Svo. 6s. LankeSter. — Works by Professor E. RAY Lankester, F.R.S. A TEXT BOOK OF ZOOLOGY. Crown Svo. [/« preparation. DEGENERATION : A CHAPTER IN DARWINISM. Illus- trated. Crown Svo. 2s. dd. {Nature Series.) Lubbock. — Works by Sir John Lubbock, M.P., F.R.S., D.C.L. THE ORIGIN AND METAMORPHOSES OF INSECTS. With numerous Illustrations. New Edition. Crown Svo. 3J. dd. (Nature Series.) ON BRITISH WILD FLOWERS CONSIDERED IN RE- LATION TO INSECTS. With numerous Illustrations. New Edition. Crown Svo. 4^^. dd. {Nature Series). M'Kendrick.— OUTLINES OF PHYSIOLOGY IN ITS RE- LATIONS TO MAN. By J. G. M'Kendrick, M.D., F.R.S.E. With Illustrations. Crown Svo. 12s. dd. Martin and Moale.— ON THE DISSECTION OF VERTE- BRATE ANIMALS. By Professor H. N. Martin and W. A. Moale. Crown Svo. [/« preparation. (See also page 22.) 24 MACMILLAN'S EDUCATIONAL CATALOGUE, Miall. — STUDIES IN COMPARATIVE ANATOMY. No. I. — The Skull of the Crocodile: a Manual for Students, By L. C. MiALL, Professor of Biology in the Yorkshire College and Curator of the Leeds Museuiii, 8vo, 2s. 6d. No, II, — Anatomy of the Indian -Elephant, By L, C. Miall and F, Greenwood. With Illustrations. 8vo, 5^-, Mivart. — Works by St, George Mivart, F.R.S, Lecturer in Comparative Anatomy at St, Mary's Llospital. LESSONS IN ELEMENTARY ANATOMY. With upwards of 400 Illustrations. Fcap. 8vo. 6s. 6d. THE COMMON FROG. With niia^erous Illustrations. Crown Svo. 3J. 6d. {Nature Series. ) MuUer. — THE fertilisation of flowers. By Pro- fessor Hermann Mljller. Translated and Edited by D''Arcy W^ Thompson, B.A., Scholar of Trinity College, Cambridge. With a Preface by Charles Darwin, F.R S. With numerou;; Illustrations. Medium Svo. 2is. Oliver. — Worlician=, London. A TREATISE ON MATERIA MEDICA. 8vo. lln the press. TABLES OF MATERIA MEDICA : A Companion to the Materia Medica Museum. With Illustrations. New Edition Enlarged. 8vo. \os. 6d. Hamilton. — a TEXT-BOOK OF PATHOLOGY. By D. J. Hamilton, Professor of Pathological Anatomy (Sir Erasmus Wilson Chair), University of Aberdeen. 8vo. \^ln preparation. Ziegler-Macalister. — TEXT-BOOK OF PATHOLOGICAL ANATOMY AND PATHOGENESIS. By Prt)fessor Ernst ZiEGLER of Tubingen. Translated and Edited for English Students by Donald Macalister, M.A., M.B., B.Sc.,M.R.C.P., Fellow and Medical Lecturer of St. John's College, Cambridge. With numerous Illustrations. Medium 8vo. Part I. — GENERAL PATHOLOGICAL ANATOMY. 12s. 6d. Part IL— SPECIAL PATHOLOGICAL ANATOMY. Sections I.— VIII. I2J. 6d. [Part III. in preparation. ANTHROPOLOGY. Flower.— FASHION in deformity, as Illustrated in the Customs of Barbarous and Civilised Races. By Professor Flower, F.R.S. , F.R.C.S. With Illustrations. Crown 8vo. 2s. 6d. {Nat2i7-e Seties). Tylor.— ANTHROPOLOGY. An Introduction to the Study of Man and Civilisation. By E. B. Tylor, D.C.L., F.R.S. With numerous Illustrations. Crown 8vo. ^s. 6d. PHYSICAL GEOGRAPHY & GEOLOGY. Blanford.— TPIE rudiments of physical geogra- phy FOR the use of INDIAN SCHOOLS ; with a Glossary of Technical Terms employed. By H. F. Blanford^ F.R.S. New Edition, with Illustrations. Globe 8vo. 2s. 6d. 26 MACMILLAN'S EDUCATIONAL CATALOGUE Geikie.— Works by Archibald GiiiKiE, F.R.S., Director General of the Geological Surveys of the United Kingdom. PRIMER OF PHYSICAL GEOGRAPHY. With numerous Illustrations. New Edition. With Questions. i8mo. is. {Scieitce Primers J) ELEMENTARY LESSONS IN PHYSICAL GEOGRAPHY. With numerous Illustrations. Fcap. 8vo. 4J. ^d. QUESTIONS ON THE SAME. is. 6d. PRIMER OF GEOLOGY. With numerous Illustrations. New Edition. i8mo. is. {Science Primers.) ELEMENTARY LESSONS IN GEOLOGY. With Illustrations. Fcap. 8vo. [In preparation. TEXT-BOOK OF GEOLOGY. With numerous Illustrations. 8vo. 28J. OUTLINES OF FIELD GEOLOGY. With Illustrations. New Edition. Extra fcap. 8vo. 3^. 6d. Huxley. — PPIYSIOGRAPHY. An Introduction to the Study of Nature. By Professor Huxley, P.R.S. With numerous Illustrations, and Coloured Plates. New and Cheaper Edition. Crown 8vo. 6s. AGRICULTURE. Frankland. — agricultural chemical analysis, A Handbook of. By Percy Faraday Frankland, Ph.D., B.Sc, F.C.S.j Associate of the Royal School of Mines, and Demonstrator of Practical and Agricultural Chemistry in the Normal School of Science and Royal School of Mine?, South Kensington Museum. Founded upon Leitfaden fiir die Agricultur- Chemische Ajtalyse, von Dr. F. Krocker. Crown 8vo. ^s. 6d. Tanner.— Works by Henry Tanner, F.C.S., M.R.A.C, Examiner in the Principles of Agriculture under the Government Department of Science ; Director of Education in the Institute of Agriculture, South Kensington, London ; sometime Professor of Agricultural Science, University College, Aberystwith. ELEMENTARY LESSONS IN THE SCIENCE OF AGRI- CULTURAL PRACTICE. Fcap. 8vo. 3^. ed. FIRST PRINCIPLES OF AGRICULTURE. i8mo. is. THE PRINCIPLES OF AGRICULTURE. A Series of Reading Books for use in Elementary Schools. Prepared by PIenry Tanner, F.C.S., M.R.A.C. Extra fcap. 8vo. I. The Alphabet of the Principles of Agriculture. 6d. II. Further Steps in the Principles of Agriculture, is. III. Elementaiy School Readings on the Principles of Agriculture for the third stage, is. SCIENCE. POLITICAL ECONOMY. CoSSa.— GUIDE TO THE STUDY OF POLITICAL ECONOMY. By Dr. LuiGi Cossa, Professor in the University of Pavia. Translated from the Second Italian Edition. With a Preface by W. Stanley Jevons, F.R.S. Crown 8vo. 4^, dd. Fawcett (Mrs.) — Works by MiLLiCENT Garrett Fawcett:— POLITICAL ECONOMY FOR BEGINNERS, WITH QUES- TIONS. Fourth Edition. i8mo. 2s. 6d. TALES IN POLITICAL ECONOMY. Crown 8vo. 35. Fawcett. — a manual of political economy. By Right Hon. Henry Fawcett, M.P., F.R.S. Sixth Edition, revised, with a chapter on " Stale Socialism and the Nationalisation of the Land," and an Index. Crown Svo. 12s. Jevons.— PRIMER of political economy. By W\ Stanley Jevons, LL.D., M.A., F.R.S. New Edition. iSmo. is. {S.icnce Primers.^ Marshall. — THE ECONOMICS OF INDUSTRY. By A. Marshall, M.A., late Principal of University College, Bristol, and Mary P. Marshall, late Lecturer at Newnham Hall, Cam- bridge. Extra fcap. Svo. is. 6d. Sidgwick.— THE PRINCIPLES OF POLITICAL ECONOMY. By Professor HENRY Sidgwick, M.A., Prelector in Moral and Political Philosophy in Trinity College, Cambridge, &c., Author of "The Methods of Ethics." Svo. i6s. Walker. — POLITICAL ECONOMY. By Francis A. Walker, M.A,, Ph.D., Author of "The Wages Question," "Money," " Money in its Relation to Trade," &c. Svo. los. 6d. MENTAL & MORAL PHILOSOPHY. Caird. — moral philosophy. An Elementary Treatise on. By Prof. E. Caird, of Glasgow University. Fcap. Svo. [/« preparation. Calderwood,— HANDBOOK OF MORAL PHILOSOPHY. By the Rev. Henry Calderwood, LL.D., Professor of Moral Philosophy, University of Edinburgh. New Edition. Crown Svo. Clifford.— SEEING AND THINKING. By the late Professor W. K. Clifford, F.R.S. With Diagrams. Crown Svo. 3J. ^d. {Nature Series.) 28 MACMILLAN'S EDUCATIONAL CATALOGUE. Jevons. — Works by the late W. Stanley Jevons, LL.D., M.A., F.R.S. PRIMER OF LOGIC. New Edition. iSmo. i^. {ScicKce Primers. ) ELEMENTARY LESSONS IN LOGIC ; Deductive and Induc- tive, with copious Questions and Examples, and a Vocabulary of Logical Terms. New Edition. Fcap. 8vo. 3^ 6^, THE PRINCIPLES OF SCIENCE. A Treatise on Logic and Scientific Method. New and Revised Edition. Crown 8vo. lis. 6J. STUDIES IN DEDUCTIVE LOGIC. Crown 8vo. 6s. Keynes. — formal LOGIC, Studies and Exercises in. Including a Generalisation of Logical Processes in their application to Complex Inferences. By John Neville Klynes, M.A., late Fellow of Pembroke College, Cambridge. Crown 8vo. los. dd. Robertson. — elementary lessons in psychology. By G. Croom Robertson, Professor of Mental Philosophy, &c., University College, London. \_Iu preparation. Sidgwick.— THE METHODS OF ETHICS. By Professor Henry Sidgwick, M.A,, Prselector in Moral and Pohtxxl Philosophy in Trinity College, Cambridge, &c. Second Edition. 8vo. 14J. HISTORY AND GEOGRAPHY. Arnold. — the Roman system of provincial ad- ministration TO THE accession OF CONSTAN- TINE THE GREAT. By W. T. Arnold, B.A. Crown 8vo. 6j. "Ought to prove a valuable handbook to the student of Roman history." — Guardian. Beesly. — STORIES FROM THE HISTORY OF ROME. By Mrs. Beesly. Fcap. 8vo. 2s. 6d. " The attempt appears to us in every way successful. The stories are interesting in themselves, and are told with perfect simplicity and good feeling." — Daily News. Brook.— FRENCH HISTORY FOR ENGLISH CHILDREN. By Sarah Brook. With Coloured Maps. Crown 8vo. 6s. HISTORY AND GEOGRAPHY. 29 Clarke.— cr,ASS-BOOK of geography. By C. B. Clarke, M.A., F.L.S., F.G.S., F.R.S. New Edition, v.ith Eighteeii Coloured Maps. Fcap. 8vo. 35-. Freeman.~-OLD-ENGLISH history. By Edward A- Freeman, D.C.L., LL.D., late Fellow of Trinity College, Oxford. With Five Coloured Maps. New Edition. Extra fcap. 8vo. 6s. Fyffe.— A SCHOOL HISTORY OF GREECE. By C. A. Fyffe, M.A., Fellow of University College, Oxford. Crown 8vo. [/// preparation. Green. — Works by John Richard Green, M.A., LL.D., late Honorary Fellow of Jesus College, Oxford. SHORT HISTORY OF THE ENGLISH PEOPLE. With Coloured Map?, Genealogical Tables, and Chronological Annals. Crown Svo. 8j. 6d. Ninety-ninth Thousand. " Stands alone as the one general liistorjr of the country, for the sake of which all others, if young and old are wise, will be speedily and surely set aside."— Academy. ANALYSIS OF ENGLISH HISTORY, based on Green's "Short History of the English People." By C. W. A. Tait, M.A., Assistant-Ma ,ter, Clifton College. Crown 8vo. J^s. 6(i. READINGS FROM ENGLISH HISTORY. Selected an I Edited by John Richard Green. Three Parts. Globe 8vo. IS. 6d. each. I. H-^ngist to Cressy. H. Cressy to Cromwell. III. Cromwell to Balaklava. A SHORT GEOGRAPHY OF THE BRITISH ISLANDS. By John Richard Green and Alice Stopford Green. With Maps. Fcap. 8vo. 3^. dd. Grove. — a primer of geography. By Sir George Grove, D.C.L., F.R.G.S. With Illustrations. i8mo. is. {Science Primers. ) Guest. — lectures on the history of ENGLAND. By M. J. Guest. With Maps. Crown 8vo. 6s. " It is not too much to assert that this is one of the very best class books of English History for young students ever published." — Scotsman. Historical Course for Schools — Edited by Edward a. Freeman, D.C.L., late Fellow of Trinity College, Oxford. I.— GENERAL SKETCH OF EUROPEAN HISTORY. By Edward A. Freeman, D.C.L. New Edition, revised and enlarged, with Chronological Table, Maps, and Index. i8mo. 3^. 6d. 30 MACMILLAN'S EDUCATIONAL CATALOGUE. Historical Course for Schools. Continued— IL— HISTORY OF ENGLAND. By Edith Thompson. New Edition, revised and enlarged, with Colou:-ed Maps. i8mo. 2J. 6d. III.— HISTORY OF SCOTLAND. By Margaret Macarthur. New Edition. i8nio. 2.s. IV.— HISTORY OF ITALY. By the Rev. W. Hunt, M.A. New Edition, with Coloured Maps. i8mo. 3^. dd. v.— HISTORY OF GERMANY. By J. Sime, M.A. iSmo. VI.— HISTORY OF AMERICA. By John A. Doyle. With Maps. iSmo. 4r. 6d. VII.— EUROPEAN COLONIES. By E. J. Payne, M.A. With Maps. iSnio. 4^. dd. VIII.— FRANCE. By Charlotte M. Yonge. With Maps. iSmo. 3J. dd. GREECE. By Edavard A. Freeman, D.C.L. \Tn preparation. ROME. By Edward A. Freeman, D.C.L. \In preparation. History Primers— Edited by John Richard Green, M.A., LL.D., Author of "A Short History of the English People." ROME. By the Rev. M. Creighton, M.A., late Fellow and Tutor of Merton College, Oxford. With Eleven Maps. i8mo. \s. "The auth'.r has been curiously successful in telling in an intelligent way the story of Rome from first to last,"— School Board Chronicle. GREECE. By C. A. Fyffe, M.A., Fellow and late Tutor of University College, Oxford. With Five Maps. i8mo. \s. "We give our unqualified praise to this little manual." — Schoolmaster. EUROPEAN HISTORY. By E. A. Freeman, D.C.L., LL.D. With Maps. i8mo. is. "The work is always clear, and forms a luminous key to European history." — School Board Chronicle. GREEK ANTIQUITIES. By the Rev. J. P. Mahaffy, M.A. Illustrated. i8mo. is. " All that is necessary for the scholar to know is told so compactlj' yet so fully, and in a style so interesting, that it is impossible for even the dullest boy to look on this little work in the same light as he regards his other school books."— School- master. CLASSICAL GEOGRAPHY. By H. F. Tozer, M.A. i8mo. i.f. " Another valuable aid to the study of the ancient world. ... It contains an enormous quantity of information packed into a small space, and at the same time communicated in a very readable shape." — John Bull. HISTORY AND GEOGRAPHY. 31 History Primers Continued— GEOGRAPHY. By Sir George Grove, D.C.L. With Maps. i8mo. IS. "A model of what such a work should be. . . . We know of no short treatise better suited to infuse life and spirit into the dull lists of proper names of which our ordinary class-books so often almost exclusively consist." — Times. ROMAN ANTIQUITIES. By Professor Wilkins. Illus- trated. i8mo. is. "A little book that throws a blaze of light on Roman history, and is, moreover intensely interesting."'— School Board Chronicle. FRANCE. By Charlotte M. Yonge. i8mo. is. "May be considered a wonderfully successful piece cf work. ... Its general merit as a vigorous and clear sketch, giving in a small space a vivid idea of the history of France, remains undeniable." — Saturday Review. Hole.— A GENEALOGICAL STEMMA OF THE KINGS OF ENGLAND AND FRANCE. By the Rev. C. Hole. On Sheet, is. Kiepert^A manual of ancient geography. From the German of Dr. H. Kiepert. Crown Svo. ^s. Lethbridge.— A SHORT MANUAL OF THE HISTORY OF INDIA. With an Account of India as it is. The Soil, Climate, and Productions ; the People, their Races, Religions, Public Works, and Industries ; the Civil Services, and System of Administration. By Roper Lethbridge, M.A., CLE., late Scholar of Exeter College, Oxford, formerly Principal of Kishnaghur College, Bengal, Fellow and sometime Examiner of the Calcutta University. With Maps. Crown Svo. 5^. Michelet. — a summary of modern history. Trans- lated from the French of M. Michelet, and continued to the Present Time, by M. C. M. Simpson. Globe Svo. 4s. 6d. Ott^.— SCANDINAVIAN HISTORY. By E. C. Ott6. With Maps. Globe Svo. 6s. Ramsay.— A SCHOOL history' of ROME. By G. G. Ramsay, M.A., Professor of Humanity in the University of Glasgow. With Maps. Crown Svo. [In preparation. Tait. — ANALYSIS OF ENGLISH HISTORY, based on Green's "Short History of the Enghsh People." By C. W. A. Tait, M.A., Assistant-Master, Clifton College. Crown Svo. 3^. 6d. Wheeler.— A SHORT history of india and of the FRONTIER STATES OF AFGHANISTAN, NEPAUL, AND BURMA. By J. Talboys Wheeler. With Maps. Crown Svo. lis. " It is the best book of the kind we have ever seen, and we recommend it to a place in every school library."— Educational Times. 32 MACMILLAN'S EDUCATIONAL CATALOGUE. Yonge (Charlotte M.).— a parallel HISTORY OF FRANCE AND ENGLAND : consisting of Outlines and Dates. By Charlotte M. Yonge, Author of "The Heir of Redclyffe," &c., &c. Oblong 4to. 3?. dd. CAMEOS FROM ENGLISH HISTORY.— FROM ROLLO TO EDWARD II. By the Author of "The Heir of Redclyffe." Exti'a fcap. 8vo. New Edition. 5^-. A SECOND SERIES OF CAMEOS FROM ENGLISH HISTORY. — THE WARS IN FRANCE. New Edition. Extra fcap. 8vo. 5^. A THIRD SERIES OF CAMEOS FROM ENGLISH HISTORY. —THE WARS OF THE ROSES. New Edition. Extra fcap. 8vo. 5J. CAMEOS FROM ENGLISH HISTORY— A FOURTH SERIES. REFORMATION TIMES. Extra fcap. Svo. 5^. CAMEOS FROM ENGLISH HISTORY.— A FIFTH SERIES. ENGLAND AND SPAIN. Extra fcap. Svo, 5^. - EUROPEAN HISTORY. Narrated in a Series of Historical Selections from the Best Authorities. Edited and arranged by E. M. Sewell and C. M. Yonge. Fii-st Series, 1003— 1 154. New Edition. Crown 8vg. ds. Second Series, 1088 — 1228. New Edition. Crown Svo. ds. LONDON; R. CLAY, SONS, AND TAYLOK, PKINTERS. 14 DAY TTQ17 RETURN TO DESK PROM W^^ BORROWED LOAN DEPT. Renewed books are subject to SdUte recaU. ,^v 41 :mB^(i ^^^vsumrm^ — j Lp21A-60m.6,'69 (J9096sin^47«_A General Library UNIVERSITY OF CAUFORNIA UBRARY