IMAGE EVALUATION TEST TARGET (MT-3) /. ^ /^ fife U/^' Photographic Sciences Corporation ^ «• ^v ^ ^v 4^^ O^ '^^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 L (meaning "CON- TINUED"), or the symbol Y (meaning "END"), whichever applies. Un dee symboles suivents appara?tra sur la dernidre image de cheque microfiche, selon le cas: le symbole — ^ signifie "A SUIVRE", le symbole V signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as rrany frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent Atre filmds A des taux de reduction diffirents. Lorsqiie le document est trop grand pour Atre reproduit en un seu' clich6, *l est filmA i partir de I'anglb supArieur gauche, de gauche i droite. et de haut en has, en prenant le nombre d'images n^cessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 rr £RR^1T^, ir? ooi if^v'tXi'd u, iXv^'-.-'}^^ Page 17, line 13, BD for B " 21, line 22, f/>^ for (r). " 24, last formula, (s-/>J (s--c) for (^^--<^^ (^^--«/ " ^;^, line 23, an!:;/es (or angle. " 55» " 9> 10, Each denominator is square root of 2. " 57j " 29, feet fc" niL/ies. *• 60, •' ■;, cylinder for prism. " ^5) " i9> omit reference. ANSWERS. — ■s IV. 8, Cfor/y-lT, 1553^00- 1521. / V. r», 5 ^or 3 ; 16, •6qii for 09911 ; *43, (5) a for <:. VI. % ,8ir5; £:^78-65. VII. rj'^, 4i°i7';»^7, Lon. in .^2°55-9' W.; *i*^, 44i;6°27.5 W; 4t), 47-28. ^ VIII. !.1fr, 811 -875 ; IX. », 169.6; 15, 62-83. X. 8, lo.i m; 1.6 dec; 5), c? for ^ ; 10, 9:16. XI. '49, IS for 13. 1883. P] r Pre^ ELEMENTARY 4 Practical Mathematics FOR HIGH SCHOOLS, INCLUDING E RUDIMENTS OF TRIGONOMETRY, NAVIGATION, MENSURATION, AND DYNAMICS, 7ITH THE USE OF LOGARITHMS, BY FRANK H. EATON, A.B. (Harv.), '^CHBK OF MATHEMATICS IN THE NOVA SCOTIA NORMAL SCHOOU Prescnbed by the Council of Public Instruction for use in the Public Schools in Nova Scotia. TRURO, N. S. ! D. H. SMITH&CO 1883. <^^^ y c I ^ Eiitt red according to Act of rurlinnipnt of Canada, in the year 1883, by D. H. SMITH & CO. Ill the Office of the Minister of Agr.'oulture at Ottawa. 6'.^> PREFACE. ^ In ^he compilation of this manual, the author has had in view the requirements of students who have been well taught in the rudiments of Algebra and Geometry, and has aimed to give it such a character that its use as a text-book will conduce to the development of mathe- matical power, as well as to the acquisition of valuable practical i:n<>wledgj. It i hoped that, while pains has been taken to guard aginst the enervatirg efieocs of wire-drawn explanations :nd awev'ib..^uH jUustratioin, the text has been made suiiicieotl}' precis, and ^nieili(.ible,and the questions asui exercises there^u bafficiont^y typical and suggestive to render a complete ccmprehtnsion of the principles and methods discussed, e -fily attainable by all for whom :t is designed. The purely geometrical demonstrations of Chapters IX. and X. are rendered necessary by the fact, that editions of Euclid still in use do not contain an adequate treatment of the geometry of solids. W lile most of the exercises are original, not a few have been selected from various works without specific acknowledgment. The author is under obligation to several friends for material aid in preparing the tables and in correcting the proof, but more especially to Alexander Mackay, Esq., of the Halifax High School, for a critical and suggestive revision of the manuscript. HRIDEI.EKRO, Ortohtr, 1S8S. INTKOUUCTION. Pure Mathematics inv(!Htij:jatos tht» relations of numbers ftnd of 8pac(; magnitudes. Applied Mathematics shows how the principles of Pure Mathematics may be utilized for practical purposes. The ultimate aim of this book is to render some of the elementary principles of Pure Mathematics available in the computation of dist>\nces, directions, areas, volumes, and capacities, and of the theoretical relations and effects of forces in their simpler combinations. But, since the method of Logarithms and the principles of Trigono- metry are of special utility in many of such computa- tions, the earlier chapters will be devoted to such a con- sideration of these subjects as is absolutely essential to the general aim. Furthermore, since an accurate representation of dis- tances, directions, and outlines by diagrams is frequently necessary, it will be appropriate to describe briefly, in a preliminary chapter, the construction and use of simple implements by means of which lines and angles can be plotted and measured. ) CONTENTS. V. CHAP. I.-^IMPLE MATHEMATICAL INSTRUMENTS. Diagonal Scale, ..... | Protractor, ••-... j Scale of Chorda, - • • . . 3 EXBROiaiM, • • • . . 9 10 10 10 CHAP II-TRKiOyOMETRICAL DEFINITIONS. Trigonometry. Rectilineal Figures Composed of Right Triwiglea, Similar Kight Triangles, Trigonometrical Ratios, Sine and Cosine, Tangent and Cotangent, Secant and Cosecant, - Illustration, Functions of Obtuse Angles, EXKKCtSlkS, 4 6 6 7 9 10 11 12 12 12 12 12 18 18 13 II 14 14 CHAK III.-ABSOLUTE VaLUES OP TRIGONOMETRICAL FIATIOS. fiimits of Trigonometrical Ratios, - . . 13 Functional Values for Angles of 46', 30\ 60", and 18°, 14 Tables of Natural Functions, • • • - 1ft How to Use the Tables, • • • • 16 EXBRCISSS, • • • . . 16 16 17 16 18 CHAP. IV. -SOLUTION OF PLANE TRIANGLEa Definition, Four General Cases, - Solution of Right Triangles, Law of Sines, - Ambiguous Case, Two Sides aad Contained Angle, Another Formula, When the Three Sides are Known, KXERCISKS, 17 18 19 20 21 22 23 24 20 20 20 21 21 22 23 23 2ft CONTENTS, CHAP. V.-.LO<»ARITHMS. I>«f)nitioii, ..... Uiffuront. Ix>{{arithini of th« SAnte Nrmh^r. • Common or Dvimry Synti-ni. Pro|M)rtioR of Coiumoii Lo^ttritlims, • f^ )garithm> from the Tahlua, • Algebraio Priituiiiles, .... \A)iitkrit\iin» of I'rottuotb ainl Quotionti, Tables of fiOgarithniH of Augiilnr FuDotioiu, KXKIU'IHRM, .... AST. »Aua 85 27 n 87 27 88 88 8tt 80 89 ao 89 31 30 88 30 31 cuAiv VI. rKKuns and distancrs computkd by THK AID OH" TttlGONOMETKY. |)«'tiuitionit, ..... Vertical DiatAitccH, .... Horizontal DiatanceM, .... Aiipliohtioti of the (ieomotry of Cin-Ics, Ratio of Diamuter am! Circumference, KXKRCMKM, .... CHAI'. VlI.-i>'AVIOATION. Klciiieiitary Conoeptioua, Auxiliary CoiiceptionM, Kepresentationa by Mercator'c Chart, KxpressioQ for Diifeionce of Longitut'e, Expression for Meridional Parts, Trigonometrical Relations of Departure, Difference of Latitude, Diatance, and Course, Trigonometrical Relations Demonstrated, Trigonometrical Relations of Difference of Longitude, Nioridional Difference of Latitude, and CourHe, RoMolution of a Traverse, Variation of the Compass, Deviation, lipdway, CJorrectiou of the Log, KXBKCIMKS, CHAP. VIIL-COMPUTATION OF PLANE AREAS. Area of Parallelogram derived from Base and Altitude, 51 50 Area of Triangle derived from Base and Altitude, • 62 50 Area of Triangle derived frum its Three Sides, - 63 51 33 33 34 38 35 34 .W 35 37 35 37 38 39 • 39 .19 m 40 41 m 41 41 • 42 42 m 4^ 42 • 44 4» '♦ 45 44 46 45 47 45 48 45 49 40 50 46 47 OONTKMTH. ▼H Atm fA Triangle derircd from Sitiss aad I^iaa of Iiuorib«(l ('iroU, - • ■ • . Arsftof OiroU denved from Uadtoa ar Ctmamfsi^noo, Aw* of CirouUr Sector derivud from it« Arc, AmA of Qu»4lril«tei«l darirod from DiaguoAli and iiiei? Triclin»ticD, . • ■ Area of QuAdriUtond derived from its Hide* and Inciinatiuu of Diagonal*, • . . Area of QtiadrilaterKi Imicriiitible in a Cii-olt, Aroa« of SimM.ir Kiguros, .... EXKKCMNM, • . *ai S4 SA M »7 68 fi9 60 rAoa fit n fit R9t U 54 S5 16 CHAP. IX.-COMI»aTATlON OK POLYUKDhAL AND CUllVBD AKKAS. I'clyhodron, .... PrUm, ..... liateral Surface of Prinni, (cylindrical Surface. C'ylind r, liateral Surface of Cylinder, Pyramid, - , . , Frustum of Pyramid, • Lateral Surface jf Pyramid an.l i^Vutttum, Cone, ..... Lateral Surface of Right Cone, Conic Frustum, and Surface ui Frustum of Right Cone, • • • . Spherical Surface, EXSRCISEH, ..... 05 81 4= -cos (ISC-il). From the above considerations it appears that tlie sine of an angle ia equal to the sine of its mpplement, while the cosim of an angle is equal to the cosine of its mppU- ment with a contrary sign* EXERCISES. 1. Verify the following identities (Art. 11); sin A cos A coiA:^ 1 =tan A ; 1 sin A tau^; — cosec A. cos A sin A 1 cos A ~ cot A ; - sec -4 ; tb! /l^tf.^^^V^u'JSht best not toj|ter fully into t'hg discussion of #r:«„;r::::?"*";^"L:"!:"!!-!*'if'"^' ^:!r^***^** eutireiy that relating to functions of angles greater than 180°. 0,r^^^ c TRIGONOMBTRICAL DEFINITIONS. 15 le r n 2. ExpreM in words the roUtions indicftUd in Ex. 1. 3. 'n. Express all the other functions of an angle in terms of its sine. Apply Euc. I. 47. K. Establish the following identities : — tan A n/6« sin »il = 1 - cos 'il ; sec *il = 1 + tan *A. i 7. Construct the two angles whose common sine is { ; the angle whose cosine is 2 ; the angle whose tangent is |. 8. Find tan A and sec A when sin ^ =f. 9. Given coseo .4=1, find cot A and cos i4. ^^0. When cim A = \, what are sin A and cosec A T ^^^*^^11. Express all the functions of an angle in terms of the sine of its complement. 12. With what limitations is the following statement true?— Tho functions of au angle are equal to those, respectively, of its supplement. CHAPTER III. ABSOLUTS VALUES OF TRIGONOMETRICAL RATIOS. 8KCTI0N I. Numerical Valtiea 0/ Functioru. ABC ^l^ of f ri^ooometrical Batios.-I„ the triangle .hn.?/5®\?' ^ ^^^ '"PP"'^'' <="?'''''« of revolution about A; the nearer it approaches coincidence with AB «. the .mailer the angle A. the les., CB becomes, and the nearer AB approaches to equality with AG; while the tZTn?*^^>T»'^'""'"l°'^^^^''^<=°"'^«' the nearer the ength of AB approaches zero, and that of GB to equality with the length of AG It follows then that, a, e^e angle A varies from 0' to 90'. its sine increases from " to i, and its cosine decreases from 1 to Similar processes of reasoning will demonstrate the hmtting values of the other functions (see Ex 1-6) lA-*' ^^?*'T' ^*'"''*" ^°' *°«^'^= °^«°. 30°, 60°, and "T r processes by which functional value. may be determined for most angles between 0° and 9o'» cannot well be explained in this book, yet these values for a few angles are readily obtainable, thus — ABSOLUTE VALUES OF TRKiONOMSTRICAL RATIOS. 17 AngUmf m*.— In the triMigle ABC {Bjt. 4), if ArzW AV\ and AC^BVsf^, But tm il = ^^ ' -'- '" * 2 BO* - = 4\/2, and Angles of 30' and fiO".— If ^(7Z> (fig. 5) be equiUt«ra], uid C^ per pendiottl«r to the boie, then AGB^SO', AB~^^, and BC* = ^ AC* .'. sin 30* = 4, and ain 60'=^. AngU$ of 18* '.-Let ABE (fig. S) be iioaceles, and t ABC- ^i? ^ 18*. IfBZ)« = i5i4i4Z)then BD*-BA {BA-BD) = BA^-BDBA . BD'BD , . BD _ -l+^/8 .. ... . , ' ^ST — 9 * positive root only being taken. Butein ^-BC=^^= _^(Euc. IV. 10) A Jo "AH .'. tin 18'=^^^. iVo^e.-- Having found values for sin 45°, sin 30°, sin IV, and sin 18°, the values of the other > functions may be deduced from them (Chap. II., Ex. 6) ^ SECTION J I. Trigonometrical Tables. 15. Tables of Natural Functions.— The functional values for consecutive angles from 0" to 90° may be com- puted and tabulated for reference. It is, however, unnecessary to extend the Tables be- yond 45°, since the sine, tangent, secant, cosine, cotangent, and cosecant of angles from 0° to 45" are respectively the cosine, cotangent, cosecant, sine, tangent, and secant of angles from 45° to 90° (Art. 11, Note 2). Note 1. —Since the functions of supplementary angles are numericatlv equal (Chap. II., Ex. 12), the same Table will furnish functional values for angles from 90° to 180°, due ragard being had to the appropriate algebraical signs. '^ * This demonstration involves the solution of a quadratic equation, and may be omitted, if necessary. n 18 PRACTICAL MATHEMATICS. JS^oit 2.—ln the Tablet arrtnf^d for this book (Tabb I.) the valaee nvtn are for angles at intervala of 10', and are, in general, earned ta four place* of decimala only; the nae of nuch Tables giving reaultfl of luthoittnt accuracy fur Otdinary computations. In Tables to be used in very precise calculations, however, the iutcrvak must be smaller, and tlie number of decimal places greater. Notti 3.— Secants and cosecants are to be derived, when required, from the tabulated cosines ard smes (Chap. II.). 16. How to Use the Tables— Proportional Parts.— The method of finding from the I'ables the function of an angle lying between two succensive tabulated angular ▼alues, called the Method of Proportional Parts, is based upon the assumption that through the given tabular iii- terval the angle and its function vary proportionally. Illvstration. — Required from the Tables sin 43° 3'. Sin 43'i: -6820; sin 43* 10'= 6841 ; Difference of sine for this intervals '0021 .', sin 43* 3'= sin {43" + ^ tabular interval) = *in 43*+^, funotionAl difference = 6820 + A x 002 1 = •682a Again. — Required ccs 25° 37'. Cos 26' 37' = cos (26' 30' + /« tabular interval) = cos 25» 30' - ,V (cos 25° 30' - cos 25' 40*) = •9026 -1^x0013 =-9017. Note 1. — The error incurred by the method of proportional parti di< minishes, of course, with the interval for which the functions are tabulated. Note 2. — The process of obtaining the angle corresponding to a given functional value which lies between two tabulated functions demands no special elucidation, being simpl}' the reverse of that just illustrated. EXERCISES. 1. What is meant by saying that cos 0*= 1? 2. Why are the sine and the cosine invariably proper fractions? 3. Give the limiting values of each of the functions of angle* from 0" to 90°. 4. Of what angles is 1 the sine, the tangent, the cosine, the secant, respectively? 5. What angle has i tor ^ its sine? 2 for its secant! V3 for its tang.^nt? (J/tf-- AHSOr.UTK VALUB8 OK TKiaUNOIlKTHK AL KATI08. 19 «. Oive th« limits of variation of the tangent and of the M«uat of anglua in the aecond quadrant. (JlAkJL'^' **»"** ^ ^^'o oot ^ = VJ ^hen aeo A^ 1- 2 VI. • 8. Find all the functional valaaa of 46* without oiing the Table*. aX9. Show that ooi 60* = 4, and deduce the remaining functional valoea. to. Oirea tan -^ =^ , find A, and de A Inf />! — « /..,_ ^i_^ i. :ii ... - ■ •. •--*• •-• yji '^i .-V, vein that given. .ib^vw iae vrau «• tOJ^UTON or PLANS THUNGLKS. SI \ V 2. -The matbod of io\ving right triiu (or BOA -B0£J) = G-1i (C+^) = i (G-A); ^^^ Ev^=tan FEG (E\ic. III. .•:i. anil Art, 9):-t£" i tn^ a\ -„''— DF ^^=tan EGD (Euc. I. 29)=tan i {G-A), SOLUTION OF PLANE TRIANOLEa 23 •■•^I'^t"^ M§r7)' *''*^' ^^ "=^»"*y of triangles (Euc. VL 4). FC _AF_c + a DE AE c-a • c + ct_ tan ^ (G +A) ' ' c-a tan ^ {C-A)' Sote 1.— Observe that C+ ^ = 180" - B; hence three of the quantities in the equation just proven are known, if c, a, and B constitute the data; the fourth, tan i (C-A), and hence i (0-A), can be found. Xote2.-1iiC+A) + UC-A) = G; and ii (G+A)-^ (G-A)zzA. Note 3. — To fiad the remaining side, apply the principle of Art. 20. tVMgr.— Solve Exercises 18-20. 23. Another Formula. — A formula which is sometimes preferable to that of Art. 22 in solving case (c) is thus demonstrated: — In the triangle ABC (fig. 10), either acute or obtuse, draw AL perpendicular to 3G. A A First. -AB^ = BG^ + CA^-2BGGD (Euc. II. 13); but C2)=i4C.co3 (Art. 8), .*. c'=a''-i ')«-2a5.co8 G. Second.- AB^=BC^ + CA^ + 2BG-GD (Euc. 11. 12); but C'Z)=^C.eo8 ACD= -AG.cosACB (Art. 12), .*. c» = a'» + 6'-2a6.cos G. Note 1. — By the aid of this formula, c can be found if a, b, and G are known ; the remaining angles may be found by Art. 20. Note 2, — It is obvious that the formula of this article is equally ser- viceable wherx the three sides are given, but in the next article a formula more commonly used for this case will be investigated. iiy^flr.— Solve Exercises 21 and 22. 24. wnen the Three Sides are Knowii. — Describe a circle (fig. 11) with AG, any part of a side of the angle i u PRACTICAL MATHEMATICS. A, as radius; draw CB perpendicular to the other side, which must be produced to meet the circuinference in E; join EC. Fig. 11. Tan ^ =tu> ^( Euc. III. 20) = ?,9 (Art. 9) = ^^ = ^^ 2 ' EB^ '"^ EA+AB CA+AB B\\tBC-AC%\nA,a.ndiABz=AC.coiA, A AG. %in A sin A tan .-=.-— ^ 2 AC+AC.cobA 1 + cos^' sin' A 1 - cos« A tan'»"-= 2 (1+C08J)« (1 + cos^) From Art. 23, cos ^ = *' "^ *^' " **' -i^,(Chap.II.,Ex.6) = i^ 1- coa A oobA' tan«4= 2 1- 2bc 26c ' 26c-6«-c« + a» I . ft' + c'-g" 26c+6» + c'»-a" , 2fc Factoring (Todhunter's Algebra, Art. 87), tan* 4 ~ {a + c-b){a + b-c) 2 (a + 6 + c) {b + c-a)' . Let 2a=a + * + c; then a + c-6 = 2(«-6); o-r 6-c=2(« - c)j and 6+c-o=2 (8-o). .•.taa«^=*J-«zl)iiZ^ .-. t»n:l- 2 ^ /(^-6)>-( \/ «(«-«) A^ote. -In applying this formula in case {d), it is advisable generally of sines. 11 (ifrou. SOLUTION OF PLANE TRIANGLES. 25 il. EXERCISES. 1. What is mnant by solving a triangle? Explain the exception in Art. 17. 2. Why is there not an additional case (Art. 18) when the data are two angles and the side adjacent to both? J 3. Find ell the unknown parts of the isosceles right triangle whose (^^^^^^ hypotenuse is 6 s/2. Sug. —By Geom. A - 45*, and o = c = 6. . 4. The perpendicular sides of a right triangle are respectively 6 and OUn^^jQjg . what are the other parts? .Va^.— a = Vi08 = 6 V3, . •. C- 30°, etc. Afru^ 5. In aright triangle a =400, -4 = 53°; find the other parts without ^^^ the Tables; sin 63°= 8. (ijff^^JL 6- Given sin ^ = cos A, a=14, and A + 0=8; find all the unknown '''^^iparts. J , , , ^ 7. If sec A = — ^ find the ratios a : 6 : c, when B= 90". Cl' ^ cos C^ 8. Solve triangles, in each of which ^=90°, from the following addi- tional data, using Tables when necessary:— a=30, 6=40. JAy^-^"*— ^=40°, a=62. d''^' diff«'-«nces. data are a, b, and c. ^^- ^^ ^® modified to find ^ when the 2a Write the equivalents of tan ^ and tan ^ r«,n«nf;- i • x 2 o» respectively, m terms of o, b, and c. " £ 24. Apply both methods 'in finding the angles when- (1) a=5, 6=6, c = 7. (2) a=525, 6=450, c = 300. (3) o=6=45, c=10. (4) a=3, =4, c=5. ii CHAPTER V. LOGARITHMS. Section 1.— Nature and Properties of Logarithms. 26. Definition.— Assuming a'^b,x is the exponent of that power of a which equals b, and as such receives the name of the logarithm ofh to the base a; more briefly »=loga 6; or loga b=:x. Note.— A series of values for x corresponding to consecutive values of 6 constitutes a system of logarithms, or table of logarithms. 26. Different Logarithms of the same Number- Since a may have any value whatever in the expression log. b = x,it would appear that any number may form the base of a system of logarithms (see Note 5); and that the logarithm oi- a given number will vary with the base. Illustration.— log^ 64 = 6 ; log4 64 = 3; log8 64 = 2; log4=-6; log4=-3; logs 6^ =-2. Note 1. — The logarithm of a number which is an exact power of the base, is a positive integer, while that of its reciprocal is an equal nega- tive integer. Note 2. — If the number lies between two exact powers of the base or their reciprocals, the logarithm lies between two consecutive integers, either positive or negative. Illustration.-S* 7 68 7 8» .*. log- 68 lies between 2 and 3: 1111 also — 7 A 7 A- . •. logg 8« 68 8» 58 ^ --S between - 2 and - 3 68 1 that is, logg 68 =2 -fa fraction, and log^ - = -3 + a fraction. Note 3.— The integral part of a logarithm is called the characteristic, and the fractional part, which can be only approximately found, and lience is conveniently expressed as an interminate decimal, is called the viantisBU. * Algeljra. For the method of finding approximate mantissas, see Advanced 28 PRACTICAL MATHEMATICa over *»^e; ^ogf,^ = - 8 + a fraction might have been ezprcMed log, J- = - 2 - a fraction, but the former is the mode adopted. ^ logS;^^* " *''^'"' ***** * ^°°* '«'•'" *^« base of a .y.tem of SKUTION II. Common Logarithms. 27. Common or Denary System— The base of ihv coiumon system of logarithms is 10, which is usually omitted m expressing the logarithms of numbers. Thus by tog 100 = 2 is meant logu> 100 = 2. 18 2^m2q'lftls^'!fn^*' • *^^ ^'»Pi?"an System, so called from its inventor la ^ 718^81828, and is generally denoted bv e Thin «v.flL Ik i! .iniK)rUnt theoretically.^^ foJno furthVcUide^rtSoi^uSs^b^^^^ 28. Properties of Common Logarithjis.— The follow- ing properties of logarithms in the common system are due to the decimal nature of the Arabic notation of numbers : — (1.) Every integral or mixed number whose integral part consists of n digits, lies between lO""' and 10", and hence the characteristic of its loga- rithm is w - 1. ^ (2.) Every remove of the decimal point to the right or to the left, increases or diminishes the log- arithmic characteristic of the number by a unit, since such r move either multiplies or divides the number by 10. (3.) In a decimal fraction, the position of the ifirst significant digit, in reference to the decimal point, determines the characteristic of its logarithm. iZ^us^ra^io/i.— If the mantissa of log 4356 = -639 then log 4356 = 3-639 , . - ' ' log 435-6 = 2-639 log 43-56 = 1-639 log 4-356 = lo^ '4356 = r+ -639 = 1-639 log -04356 = 2+ -639 = 2~639, etc. LOGARITHMS. S9 29. Logarithma from the Tables.— r/t/Tt! DigltH.-Axi a table of logarithms constructed tor numbers from 1 to 1000 (Table II.), the mantissa for any number consisting of three digits may be found in the column numbered above by its right hand digit, and ttorizontally oppotdte the first two figures of the number as found in the column marked iV above. Illustration.— l^og 467 = 2-6698. Four or more Digits. — When the antilogarithm, that is, the number whose logarithm is sought, consists of more than three digits (e.g., 4676), its logarithm is found from the table by the method of proportional parts illustrated in Art. 16. Illustration. — Required log 4676. Log 46-70 = log 46*7 = 1 6693 ; similarly log 46«C = log 46-3 = 1-6702. Hence the logarithmic difference corresponding to the difference of yu ( = tiftt) between these two numbers is •0009. Assuming then, that the difference betv/een the logarithms of two numbers, differing from each oth jr by only a small fraction of either, is proportional to the dif- erence between the numbers : — Log 46-76 = log 46-70+T% X XKWO = 1 6693+ 0005 = 16698. Note. — The method of finding the antilogarithm, being tlie reverse of that just illustrated . needs no explanation. a- SECTION III, Utilitif of Logarithms. 30. Algebraic Principles- — Assume a* ' = c; from the laws of exponents, — log„ be =-x + p (Art. 25) h, and he .«+» a, . . c h ■= a « loga 6" fix v^ = a" X '. log„7f- = -. '.rf SO PRACTICilL MATHEMATICS. I 31. tofirarithms of Products and Quotients—The re- siUts ot Art. liO are embodied in the following atatements:- (1.) 7//« logarithm of a product is the mm of die logarithms of its factors; and as a result, the. logarithm of a power of a number is the logarithm of the number multivlied by the exponent of the power. (2.) The logarithm of a quotient is the difference between the logarithm of the dividend and thit of the divisor; hence, also, the logarithm of a root of a number is the hgaHthm of the number divided by the index of the root JVb;« l.—Thus, by the appropriate use of locarithms thb reanltn nf .£^^^ 2.— Instead o( subtracting a logarithm, its co-loffarithm. i f th. of «?f °?!?K»"'!"n,'m>y be reaJily written down by Bubtraotinir osoli -SuflT.— Solve Exercises 1'12. SECTION IV. Logarithmic Functicma. 32. Tables of Logarithms of Angular Functions— bmce the values of circular functions usually involve four or more places of decimals, it is best in operations involving these values as factors (Chapter IV.) to employ logarithms. The logarithms of the natural functions of angles from 0' to 90° have been tabulated, therefore constituting a^afe^e of Logarithmic Functions (Table III )' ''^^ntZt'^^r-'' ^ *'^ -^ ^' '^^ tabAe^tX^o^^ Note 2. — Thn iiifit.hnd nt fii.^'^/* *i._ i -a.^. • . and that of l^nain, ^^^^o^^^^SuSn^^i^^SS ^^.^S^i LOGAUlTHMa 81 being limiUr to the methods already doiioribod for ocing the other tables, needi no farther explanation. Note 3, — Instead of Bubtraotins the lop;arithm of a fanotion we may add the logarithm of its ruciprocid ( ^ rt. 31 (2), and Chap. IL Ex. 1). £X KRC«.S]SS« 1 . Define logarithm. 2. What is the logarithm of »oe base in any system t 3. Of what number is the logarithm the rame in all systems ? What is its logarithm ? 4. Of what numbers U 6 the logarithm to the bases 2, 3, 4, 6 re«neo< tirely ? Sug. 2» = 32.-. logj32=6. 6. What are the respective bases of systemji in which log 2*25 = 2, log 64 = -1, log ^ = 2, logi = -2, log 6 = 2, log 2 = 2. log 2 =3? 1\-1. 64 log^64 = -l. 6. Name three numburs whose logarithms to the base a are integral numbers ; three numbers whose logarithms are mixed numbers greater than 4; three numbers whose logarithms are proper fractions. 7. Distinguish between manttsaa and characteristic. Is it necessary, ^ or only convenient, to consider the mantissa as always positive ? Why ? *^™ Illustrate. S. What are the logarithms of 16 to the bases 5, j, ^ . respectively ? Generalize. 2 4 10 Sug. (ly^= 16, etc. 9. Show that in the common system the number of units in the ohar- acteristio of the logarithm of a number is always one leas than the number of digits in the number. Illustrate. 10. What are the logarithms of '1, "Ol, "OOl respectively, to the base 10? 11. Show that in the common system the characteristic of the loga- rithm of a decimal fraction % U be - {n+ 1) if, in the fraction, n ciphers precede the first significant digit. 12. Given log 2 = -30103, and log 3 = '47712, find without tables the logarithms of the following numbers :— 8, \'Q, '27, .5^, -081, 5, 6, U 310 •06. ^ 36 -064. (i;:)*. *f PRACTICAL MATHKMATICa. Sug,. rx)g i^ = log J2 . log (I6x i) = log 2* +log * = WIOS x 4 ^ (-1)= -20412. Log 6 = log ^^, etc. 18. What obvious advantages has the common system OTor all othera f 14. Find from Table 11. the logarithms of 7606, 35-49, •000«542: also the antdoganthms of 3-4567, 1-0999. ' ^^mx'i^m^fJei^'^ *"**'*'* (3+ -4667) = mntUog 3 x aatUog -4567 15. Perform by means of logarithms the operations indicated i— 4569 X Wir,, 00.36 x 689-6 : 76-08-7-69, -9897 -^-9011 J V6869*, 864* -706*. oJ QA'^'"'\ ^^u".J*^^". "^- **»« logarithmic sines of angles denoted by -» .W, and 78 25 ; also the logarithmic uines of thoir complements and their supplements. 17. Find log tan 13° 10', and log cot 48 50'. 18. Find log sec 65°, log cosec 67^ 40', log tan 25° 38' 30 '. >Sug. Sec 66° 1 cos 55^ • ■ ^^^ '^'^ ^'^" ^ ^°« ^ "^^^ "^^ ''^•'''°' •^^ ^® *" result (Art. 32, Note I). 19. Test the correctness of the logarithmic functions of the animlar magnitudes 36°, 65°, 75°, as given in Table III, by the aid of Tables I. and 11. • "^n ^tl,"^® nurnKv-s in Table III. are but the logarithms of values given in Table I. with 10 added. * 20. Of what angles ia 1-9235 the actual logarithmic sine, cosine, tangent, and cotangent respectively ? 21. Why, in the tables of logarithmic functions, is a single difference column sufficient for tangent and cotangent, while sines and cosines require one for each ? 22. Why is it impossible to obtain from the Tables accurately the natural or logarithmic sines of very small angles ? Is there the same diHiculty m regard to the cosines of small angles? 23. Solve by the aid of logarithms the triangles in which— (1) 5 = 90°, i4 = 47° 30', and a = 5689. (2) B = 90% a = 466, and 6 = 784 -5. (3) a --- 100, 6 =600, c =781. (4) a =. 2-56, 6 = 74 78, ^ = 35° 30'. (6) A = 5^° 25' 30 ", B = 27° 28' 17", c (6) a = 630-6, b = 4283, c = 396-5. 236. CHAPTER VI. HEionra and distances computed by the aid op TKIOONOMETRY. 33. Dehaitions— Tho application of the principles explained in the preceding chapters in the computa- tion of distances, heights, and directions, presupposes the ntcossary data to be furnished by actual measurc- niunt. The distance between two remote objects is said to mibtend an angle at the observer's eye. This angle is a vertical or a horizontal angle, according as it is subtended by the vertical or by the horizontal distance between the objects. The acute angle included between the oblique line connecting two positions of ^ ^ different altitudes, and the horizontal through either of them, is called the angle of elevation of the higher, and the angle of depression of the lower, thus : — DAB (fig. 12) is the angle of elevation of ^ aa seen from A; and CBA (= DAB) is the angle of depression of .^ as seen from B. 34. Vertical Distances- — If the elevation of a distant point above an observer's level is required, any horizontal base line may be measured; li'^ewise, the angled made with the base by lines drawn from its extremities to the foot of the vertical through the distant point ; and lastly, the angle of elevation of that point froir one extremity of the base. 12. I f 34 PRACnOAL MATUEMATICB. one TbMr^If Z) (Bg. IS) is the dcTaUd point, and O the obMrrer"! /) poflition, the neceM'^ry (fig. 14) be the two points, and XY the measured tnanglesPAr and OXY. resDeciivtlv. f^an ha co^^"*^-* -_3^r^.,„ PQ in the triangle fXQ. ' '" ' ~" '"'*'"•"-'' ="-» "ajwi/, BEIOHTH ANli DI8TA3fCS& 85 Kote. Th« problan ia dmpU if on« of the pointt, P, b tcoowibU from X, •ud the other, Q, viaibU from both I* md X (fig, U). 86. Application of the Geometry of Oirclei— There are many trigonometrical probloma whose solution in- volvijM a recognition of the properties of circlea Thus, it is frequently necessary to find the distance of a station of observation from three points whose dis- tances from each other, and the horizontal angles at the observer's position, are known ; as, for instance, refiuired the distance of a ship at sea from each of three heatilands whose distances from each other are laid down upon a chart, and whose directions from the ship are observed. In fig. 15, AB, BC, CA denote the known di«tancea; the anfflea CAD and DC A are constructed eciual, respectively, to the angles under which the distances d.'tioted by iiC and AB i^re observed, a circle is 'escribed about ADO, BD is produced to meet the circumference in A', and A E and CE are joined. E denotes «he o'tserver's position, iiince the angles A 4'Z) and ACD are F'«- l^* ciiual, as also &rifCED muI CAD (Euc. III. 21) Then AD ot the triangle ADC n BA C of the triangle A BC ; whence in the triangle for^j; BE, a triangle A CE. triangle A in the triangle BAD ; the next step . „ .^ for AE, BE, and the angle BAE ; lastly, Ci- i» obtainea by solving the */» 37. Ratio of rotameter and Circnmference of a Circle.— Many compui^auions of lengths, both those which involve the trigonometrical methods and those which do not, require that either the circumference or the diameter of a circle shall be expressed in terms of the other, ^o do this exactly can be shown to be impossible, that is, the two quantities are incommensurable, but in various ways n-c xai^iu uui»wc\/il ineili may Do luUnCi 10 WUfilil U'fiy required degree of approximation. 96 PKACTICAL MATHEMATICS. One of the siinple.st of these methods is *o compute by trigonometry, if necessary, the perimeter of a regular inscribed polygon in terms of the radius ; then, consider- ing this polygon as the tirso of a series of regular inscribed polygons, each of which has twice as many sides as that which precedes it in the series, to compute the values of the successive perimeters by the data furnished at each step. And since the circumference is the limit of such inscribed perimeters, it is evident that the successive values, thus found, more and more nearly express the ratio of the circumference to the radius. In this way the following results have been obtained, regarding the radius as 1 : — No. of Sides in In- scribed Polygon. Perimeter. 6 6- 12 6-211658 24 6-2()o257 48 6 -278700 98 6-282066 192 6-282905 384 6-28.3115 768 6-283163 I sue 6-283181 3072 6 283183 A comparison of the last two results shows that the ratio sought has been obtained true to five decimal places; that is, — Circum. = diam. x 3-14159+ {see Ex. 20.) This rallo is generally denoted by ir ; hence the formula Circum. =27rr. Note 1.— From this formula it is easy to find the length of a circular arc approximately, in terms of the radius, having given either the number of degrees m the arc, or the length of the chord subtending it. Note 2.— For many purposes it is sufficient to regard ir as equal to ot. HEIGHTS AND DISTAKCES. 37 EXERCISES. 1. A horizontal angle is measured by an arc of the horizor., while a vertical angle is measured by an arc of a celestial meridian. Explain this statement Sug.— An observer's celestial meridian is in the same plane as his terrestrial meridian. 2. Distinguish between antfle of elevation and angle oj depression. 3. Devise some simple means of measurine horizontal and vertical angles. 4. The jhadow cast by a tree is 40 feet long, and that of an upright stiiik 3 feet long is 2 feet 6 inches; how tall is the tree? what is the altitude of the sun? and what will be the length of the shadow when tlie altitude of the sun is only 10" 40' ? ^ 5. From the foot of a tree standing on a horizontal plane, a straight line is measured to a distance of 155 feet, and the angle of elevation of its toj) found to be 34°, Required the height of the tree. 6. The angle of elevation of the top of a hill from the foot of its slope is 45" 50'; while, at a point 500 feet horizontally distant from the foot, the angle of elevation is 24° 30'. What is the vertical height of the hill? 7. A tower stands on a hill, the angle of elevation of which from one point of observation is 40° 30', that of the tower being 59° 50' ; from a second point of observation 275 feet horizontally more remote, the angle of elevation of the top of the tower is 26° 25'. Required the height of the tower. 8. Fifty feet from the foot of a tower, situated on the summit of an incline, the angle subtended by its height was 52° ; seventy-five feet further down the incline the angle subtended was 31° 40'. Required the height of the tower. 6'ug.—(l) Sin 20° 20' : sin 31° 40' : : 75 : ?; (2) apply Art. 22. 9. From the foot of a mountain the ground slopes away at an angle of I(F with the horizon. At a certain position on the slope the vertical angle subtended by the mountain is 50° 30'; 200 yard? rurther up the (slope the subtended angle is 55°. What is the vertica I distance of the mountain top above the first station ? 10. From the north end of a chuich 60 feet long and 40 feet high, a tower rises to the height of 55 feet from the ground. How tar must an observer, whose eye is 5 feet 6 inches from the ground, stand from the basement on the south end, in order that he may just see the top of the tower ? 11. A lighthouse is standing on the summit of a precipitous cliff; the keeper sees a distant ship in line with a buoy which is moored i of a mile from the foot of the cliff, the angle of depression of the ship and of the buoy being 20° 30', and 50° 30', respectively. How far away is the ship? 12. From a point beyond the end of a mountain spur, both extremities of a proposed tunnel through it are visible, subtending an angle of 69° .W; the distance to one point of emerireiice is l.\ miles, to the othf^r- l n-Ue,' How long will the tunnel hi ? " ' 88 PRACTICAL MATHEMATICS. 13. An observer on the hank of a river wishinff to juinonf.;« uu distance from an objec# on the opposite hZ] sfght tL^bS^d also «^oJl kT ^"^ ^r "'^" H^ ^^^ "^«'- 75 yards further dowCfindSitte 2?aii K.* *'"',9^ "•^e of *he nver he obser^-es, simUarly. that the 5Mflr.— Compare Art. 35. ««?«;„ ^^®?°''® ^^^^ ? circular race-course is 1 ^ milo* lone. Three ''^y^^f *^ ** """^ selected, such that the angle under which the aeronS Tr^t'^nAfXtrt ?r '^' ^."^ ^' ''^' 37'. iSid that 8u£tend^' bythe toJces of th«i5nw '"'°"^' " ^*f 30'. Required the shortest dis XZ porttsTthe'feTce?'^ '^"°*'"' '"^ *^^ '^^^^^ '' *^« -*«- «t,fej:^Vov^'''!^^"^ proportional to the circumferential angles they Sug.— Find the angles ^, B, and C 17. Lines joining three objects are respectively 125-6. 130-4 and 112 Sl?ro'f/r'''i' '""^ ^l^ ^:S"*'°^ °"*"^« ^f th« triangle ?hefi^? 1*1 tte last of these lines subtend angles which are respectively 48» 58' ^d objecte. "^""'""^ *^' ^'--"^'^ ^'^ *^^ «*"*^^^ from each ^of the th^e iS'Mgf.— Apply Art. 36. « i^ J« ^®'f^* "^i.^^^/v '5.^^ ^"°^^«' »^^ *^e chord of half the arc w 4 feet 6 mches. Fmd the diameter of the circle. ^ugr. —Construct diagram. From similar triangles — Diameter = (4-6)2 ~ 1-25. ' ^ 19. 'The chord of an arc is 20 feet, and its height, 4 feet. Find the cir^cumference of the circle and the length of the arc subtendeHy tue Au^^'~^^^ piameter=4+I0»^4 by similar triangles: (2) arc • t x diam. ; : angle at centre : 360'. * , \^/ am . r x 20. Find for circle whose radius is 8, the perimeters of a series of in s;,el;et"''^'"''^ "^'" ^'''''"^ '' 6,T2rrd"i? sidS; CHAPTER VII. NAVIGATION. 38. Elementary Oonceptiona. — In addition to the principles of trigonoiiietry, the computation of distances and directions at sea involves the application of mathe- matical principles based on the fact of the earth's virtual sphericity. The fundamental elements of less difficult problems in Navigation are known technically as : — Distance — Expressed in nauticd miles, i.e. miles of 60V 6 ^eet each; the track of the ship being called a Rhumb Line. Course— The angle made by the rhumb line with a meridian, as indicated by the compass. Difference of Latitude and Difference of Longitude.— Respectively the differences between the latitude and longitude left and the latitude and longitude arrived at. Any two of these elements may be the data from which the remaining two are to be found. Let PE and PQ (fig. 16) denote meridians which intercept EQ, a A, Bh, and AB, area re- spectively of the equator, two parallels, and a rnumb line; then EQ denotes difference of l.i^gitude, aB or Ah difference of latitude, and the angle hAB the course. pj»^ jg^ 39. Auxiliajpy Conceptions. — To facilitate computa- tion, certain subsidiary elements are employed : — Meridian Distance. — The number of nautical miles 40 PRACTICAL MATHEMATICS. between two given meridians at a given latitude. It is greatest at the equator, diminishing gradu- ally as the poles are approached (ficr, 16). Departure. — The number ot* miles easting or westing actually made in sailing from one meridian to another. Since the meridian distances at the initial and final latitudes are unequal when the course is obliqr.e, the departure in that case is equal to the mean meridian distance, which is not found, however, on the parallel of the mean lati- tude, but a little nearer the poles. Thus, departui-e may be denoted by dp (fig. 10), more than half- way from ^a to bB. Meridional Difference of Latitude.— A minute of lati- tude, which is everywhere the same, and a minute of longitude at the equator are equal, each being a nautical mile in length. Hence, if two meridians are a mile apart at the equator, because of their convergence the meridian distance diifers more and more from a minute of latitude as the distance from the equator increases. In other words — relatively to the corre- sponding meridian distance, a minute of latitude becomes greater and greater the higher its latitude. From this it follows that, if. to represent two con- secutive meridians, two vertical straight lines be drawn (fig. 17), thus making the meridian distances at all latitudes the same, then the distances between the ^' 2' J' \ ^ I ^ B i ^ 1 ^ C \ b B [..-(?. A i xi.\ji.i.£t\jixua,i. iinca Q Fig. 17. ^ - intersecnng them, designed to represent consecutive minutes of latitude, must be made sue- NAVIGATION. 41 cessively greater antl greater, in order that, at any given latitude, the lat. x\ 42 PRACTICAL MATHEMATICS. Hi 42. E^pre.*^8ion for Meridional Parts— Since the dif- ference of longitude exceeds the meridian distance at any latitude m tne ratio of the secant of that latitude (Art. 41). It follows that. if. as in fi;;. 17. the meridian dis- tances are all made equal to the difference of londtude ea.h exceeds the distance it represents in the ratio of the secant of its latitude, which is therefore the ratio in which eaxjh corresponding minute division of the meridian must be mcrea^sed to preserve the proportion (Art 39) In other words, since the true meridian distance at lat V ha^ been multiplied by sec. 1'. arid the meridian distance at lat. 2 ha^ been multiplied by sec. 2'. etc., hence in representing latitude, Qa - 1' x sec. r, a6 - 1 x sec 2' 6c = 1 X sec. 3', etc. and consequently the distance of the parallel of latitude a:" from the equator is equal to 1' x (sec 1 + sec. 2' + etc. ... + sec. x'), that is -- Mer part, for lat. x" = sec. V + sec. 2' + etc. . . .'+ sec. x\ 4d£n~af ^aft^^rX^^^^ tables of 43. Tngonometrical Relations of Departure, Difference ^ of Latitude, Distance, and Course. —•The lines on a Mercator's Chart which denote respectively a rhumb line and the corresponding meridio- nal difference of latitude and differ- ence of longitude form a right tri- angle, in which the course is denoted by the acute angle adjacent to the meridian line; so, also, may the cor- ^.nof ?• \ responding parts of a similar triangle denote respectively, actual distance, difference of lati- tude, departure, and course (Art. 40). Let ACB (%. 19) be the triangle from a Mercator's s NAVIGATION. 48 Chart of which AC and CB and the angle A denote njspectively me,, diff. lat. diff. Ion., and c^mvse; then if AD denotes true aifference of latitude, DE and AE will be the departure and distance respectively; and the solution of the triangle ADE, will give any two of the four elements under consideration, that is, diff. lat, den diM and course, the other two being furnished as data! ' 44. Trigronometrical Relations Demonstrated.— That a plane nght triangle may represent the mutual relations of course, departure, difference of latitude, and distance can be directly proven : — Let AB (fig. 20) be a rhumb line cut by meridians into portions, so small that each may be considered as a straight line; correspond- ing to these minute rhumbs Ac, cd, etc., hB, let hc,ed, etc., jB, at right angles to the' consecutive meridians, denote the minute departures, the aggregate of which con- stitutes the total departure; and Ab, ce, etc., hj, the minute differences of latitude,' whose aggregate makes up the whole difference of latitude; then since the ^-e- ^». triangles are all similar, one angle in each being equal to the course, — Ab:bc:Ac::ce:ed:cd:: etc. : : hj :jB : hB, .'.Ab:hc '.Ac:'.Ab^ce-\-ctc. +hj: ',j + ed+etc. +JB :Ac + cd + etc.+hB. Hence AbibcAc:: diff. lat. : dep. : dist. ^ But the elementary triangle Abe is similar to any plane nght triangle, one of whose acute angles equals the course; hence the relations affirmed in Art. 43 are established. /f«S \Z^}^ T*^°1 °^ computation, thus suggested, is called Plane baiUng, because it involves the properties of a pl^e triangle. rdSL7tL^k^!.±^l7?i!?.^^^^^^^ K^^Jo detennine the Note a.-A table, in which are recorded the departures and differences Fig. 20. 44 PRACl'ICAL MATHEMATICa of Utitude, computed for coniocutive connea And diiUnoes. iu c»Iled » Traver$e Tablt (««« TabU V.). Note 4.— It ii uieful to know the departure, chiefly as an aid in det. r- mining the difference of longitude, as ahown in the next article, ^u^.— Solve Exerciaea 1-23. 45. Trigonometrical Relations of Difference of Longi- tude, Meridional Difference of Latitude, and Course.- At aoiiie latitude intermediate between the initial and ^ final latitudes the meridian distance must equal the departure (Art 39); denote this latitude by z, then, — diff. Ion. =dep. x aec. z (Art. 41) If, then, the parts of the triangle ADE (fig. 21) denote departure, etc. W in Art. 43, and AD be produced till AG =^ AD ^ sec. z, the side (75 will de- note difference of longitude, and At\ meridional difference of latitude, as Fig. 21. in Art. 43. Hence from the similar triangles — Diff. Ion. _ mer. diff. lat. dep. diff, lat. > Diff. Ion. = mer. diff. lat. x dep. diff; lat. = mer. diff. lat. x tan. course. This method of findinc the difference of longitude is called because the relations involved are those of lines on Notel Mercator'a SaiUmj, a Mercator'a chart. Note 2.— If the course is on a parallel, the difference of latitude is nothing, and the meridian distance is equal to the distance sailed Hence (Art. 41) diff. Ion. rrdist. x sec. lat. In such cases the solution is said to be by Parallel Sailing. Note 3.— The methods thus far discussed are sufficient for the solution of all problems relating to simple courses. Note 4.— In low latitudes, and when the course is greater than 45° the intermediate latitude at which the meridian distance is ecual to the departure (Art. 39) may be assumed, without serious error, 'to be the mean of the initial and final latitudes. On this assumption— Diff. Ion. = dep. x sec. mid. lat. Longitude found by this method is said to be found by Middle Latitude oailtng. Sug. —Hoive Exercises 24-37. NAVIOATION. i6 46. Resolution of a Traverse— In all but exceptional voyages a ship's course is a compound one, of which the various simple courses and distances are recorded in the log book. When such a compound course or traverse is to be resolved, it is necessary to ccipute, or to find directly from the Traverse Table, the departures and differences of lacitude for the several courses. The algebraic sums of these will be, respectively, the net departure and differ- ence of latitude for the traverse. The distance in direct course, and the difference of longitude, can then be found as if the course were a simple one (Arts. 43 and 45). N^ote 1. — The reaolution of a traverse is called Traverse Sailing. Note 2. — The first course on a traverse is usually called Departure Course. It is simply the bearing and distance of ;?me headland, light- house, etc., from the ship, observed just before the land is lost sight of. Tliis bearing ust be reversed, of course, in the computation. 47. Variation of the Compass.— As the magnetic and geographical poles are not coincident, it is only in certain longitudes that the needle points directly towards the North. The readings ^^^ the compass in all other longi- tudes must therefore be corrected for vacation, which is either easterly or westerly according as the north end of the needle is deflected towards the east or towards the west. Note. — The amount of variation in any place may be known by con- sulting a chart, or by comparing the direction of the needle with that of the sun at noon. 4S, Deviation. — The influence of the beams and other iron in a ship upon the needle occasions an additional error in the compass readings. The amount of this error, which is called Deviation, varies with the direction in which the ship is heading, being least, generally, when „ ._„,,,.._ I. j._,j TTXVil fiiU XtlCSlTiiCUiV; l.ll'Ct.l.'Liiaill. A record of the errors for all directions of a ship's head, f 46 PRACTICAL MATHEMATICS. Obtained by sxHnging the ship, as it is called, constitutes a demahon table for that ship. "•-"uwa JVo<