GIFT OF 
 Dr. Horace Ivle 
 
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KOBOSON'S MATHEMATICAL SERIES. 
 
 ELEMENTS 
 
 PLANE AND SPHERICAL 
 
 TRiaONOMETEY, 
 
 WITH 
 
 NUMEROUS PRACTICAL PROBLEMS. 
 
 BY ; ; 
 
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 AXJTHOB OF A FULL COtn^'OP ilLflf^^llAlCil.' , . , 
 
 IVISON, BLAKEMAN, TAYLOR & CO., 
 
 PUBLISHERS, 
 NEW YORK AND CHICAGO. 
 
 1880. 
 
i( 
 
 R-OEiisrsoisr's 
 
 SERIES OF MATHEMATICS. 
 
 The most Complete, most Practical, and most Scientific Seeies of 
 Mathematical Text-Books ever issued in this country. 
 
 Mohinson's Progressive Table Book, 
 
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 Itobinson's Progressive Practical Arithmetic, 
 
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 EDUCATION Oe^' 
 
 Enteeed, according to Act of Congress, in the year 1867, by 
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 District of New York. 
 
 Jb' 
 
NOTICE. 
 
 Upon the suggestion of many- Teachers, the Publishers have thought 
 best to bind in a separate volume this Treatise on Plane and Spherical 
 Trigonometry ; continuing, however, as heretofore, to bind it up with 
 Robinson's New Geometry, in one volume. 
 
 In this form it will bo more convenient, and less expensive for those 
 Teachers and Students who do not wish to take up the Geometry in con- 
 nection with it, or who desire to use this Treatise on Trigonometry and 
 some other author on Geometry. 
 
 ,a- -! 
 
 924227 
 
TKIGONOM-ETEY 
 
 PAKT I. 
 
 PLANE TRIGONOMETRY. 
 
 SECTION I. 
 
 ELEMENTARY PRINCIPLES. 
 
 Trigonometry, in its literal and restricted sense, has 
 for its object the measurement of triangles. When it 
 treats of plane triangles it is called Plane Trigonometry . 
 In a more enlarged sense, trigonometry is the science 
 which investigates the relations of all possible arcs of the 
 circumference of a circle to certain straight lines, termed 
 trigonometrical lines or circular functions^ connected with 
 and dependent on such arcs, and the relations of these 
 trigonometrical lines to each other. 
 
 The measure of an angle is the arc of a circle inter- 
 cepted between the two lines which form the angle — the 
 center of the arc always being at the point where the 
 '. two lines meet. 
 
 :• -..The arc is measured by degrees^ minutes, and seconds; 
 •• lli^re being 360 degrees to the whole circle, 60 minutes 
 \\\ fal'/one degree, and 60 seconds in one minute. Degrees, 
 /..' nlinutes, and seconds, are designated by °, ', "; thus, 
 %%\ 27° 14' 21", is read 27 degrees 14 minutes 21 seconds. 
 iV;. .'The circumferences of all circles contain the same 
 •'•*•• number of degrees, but the greater the radius the greater 
 \-"l * (244) 
 
SECTION I. - 245 
 
 is tlie absolute length of a degree. The circumference of 
 a carriage wheel, the circumference of the earth, or the 
 still greater and indefinite circumference of the heavens, 
 has the same number of degrees ; yet the same number 
 of degrees in each and every circumference is the meas- 
 ure of precisely the same angle. 
 
 DEFINITIONS. 
 
 1. The Complement of an arc is 90° minus the arc. 
 
 2. The Supplement of an arc is 180° minus the arc. 
 
 3. The Sine of an angle, or of an arc, is a line drawn 
 from one end of an arc, perpendicular to a diameter 
 drawn through the other end. Thus, BF is the sine of 
 the arc AB^ and also of the arc BDE, BK is the sine 
 of the arc BB. 
 
 H 
 
 4. The Cosine of an arc is the per- j) ^^ 
 
 pendicular distance from the center of 
 the circle to the sine of the arc ; or, it is 
 the same in magnitude as the sine of 
 the complement of the arc. Thus, QF 
 is the cosine of the arc AB\ but 0[F= 
 KB^ which is the sine of BB, ^ 
 
 5. The Tangent of an arc is a line touching the circle 
 in one extremity of the arc, and continued from thence, to 
 meet a line drawn through the center and the other ex- 
 tremity. Thus, AH is the tangent to the arc AB^ and 
 BL is the tangent of the arc BB, 
 
 6. The Cotangent of an arc is the tangent of the com- 
 plement of the arc. Thus, DX, which is the tangent of 
 the arc BB^ is the cotangent of the arc AB. 
 
 Kemark. — The co is but a contraction of the word complement. 
 
 7. The Secant of an arc is a line drawn from the center 
 of the circle to the extremity of the tangent. Thus, CH 
 is the secant of the arc AB^ or of its supplement BBE, 
 
 8. The Cosecant of an arc is the secant of the comple- 
 ment. Thus, CZ, the secant of J?i>, is the cosecant oiAB. 
 
 21* 
 
 J 
 
246 PLANE TRIGONOMETRY. 
 
 9. The Versed Sine of an arc is the distance from tho 
 extremity of the arc to the foot of the sine. Thus, A^ 
 s the versed sine of the arc AB^ and DK is the versed 
 line of the arc BB, 
 
 For the sake of brevity, these technical terms are con- 
 tracted thus : for sine AB^ we write nn, AB ; for cosine 
 4 B, T7e write coz, AB; for tangent AB, we write ta7u 
 AB, etc. 
 
 From the preceding definitions we deduce the follow 
 mg obvious consequences : 
 
 1st. That when the arc AB becomes insensibly small, 
 or zero, its sine, tangent, and versed sine are also 
 nothing, and its secant and cosine are each equal to 
 radius. 
 
 2d. The sine and versed sine of a quadrant are each 
 equal to the radius ; its cosine is zero, and its secant and 
 tangent are infinite. 
 
 3d. The chord of an arc is twice the sine of one half 
 the arc. Thus, the chord, BG, is double the sine, BF. 
 
 4th. The versed sine is equal to the difi*erence between 
 the radius and the cosine. 
 
 5th. The sine and cosine of any arc form the two sides 
 of a right-angled triangle, which has a radius for its 
 hypotenuse. Thus, OF and FB are the two sides of the 
 right-angled triangle, CFB. 
 
 Also, the radius and tangent always form the two 
 sides of a right-angled triangle, which has the secant of 
 the arc for its hypotenuse. This we observe from the 
 right-angled triangle, OAIT, 
 
 To express these relations analytically, we write 
 
 sin.' -f- COS.' = i? (1) 
 
 R^ -I- tan." = sec.' (2) 
 
 From the two equiangular triangles CFB, QAH^ we 
 
 Jiave 
 
 CF ', FB ^ CA I AH. 
 
SECTION I. 247 
 
 That id, 
 
 COS. : sin. = B : tan. ; whence, tan. «= — _ I (3) 
 
 cos 
 
 Also, OF , CB = OA : CM. 
 
 That is, 
 
 COS. : i2 = jR : sec. ; whence, cos. sec. = B^, (4) 
 The tivo equiangular triangles, QAH and ODL, give 
 
 CA I AH =^ DL : DC. 
 
 That is, 
 
 R : tan. = cot. : B\ whence, tan. cot. =■ iT. (5) 
 Also, CF : FB =^ BL : BQ. 
 
 That is, 
 COS. : sin. = cot. : i2; whence, cos. B = sin. cot. (6) 
 From equations (4) and (5), we have 
 
 COS. sec. = tan. cot. (7) 
 
 Or, COS. : tan. = cot. : sec. 
 
 We also have ver. sin. = R — cos. (8) 
 
 The ratios between the various trigonometrical lines 
 are always the same for arcs of the same number of 
 degrees, whatever be the length of the radius ; and we 
 may, therefore, assume radius of any length to suit our 
 convenience. The preceding equations will be more con- 
 cise, and more readily applied, by making the radius 
 equal unity. This supposition being made, we have, for 
 equations 1 to 6, inclusive, 
 
 sin.' + cos.' =1. ( I ) 
 
 1 + tan.' = sec.» (2) 
 
 sin. ,„, 1 ... 
 
 tan. = (3) COS. = — (4) 
 
 COS. sec. 
 
 tan. = (5) COS. = sin. cot. (6) 
 
 cot. 
 
 Let the circumference, AEBH^ be divided into fouj 
 equal parts by the diameters, AB and EH, the one hori- 
 
248 
 
 PLANE TRIGONOMETRY. 
 
 zontal and the other vert- 
 ical. These equal parts 
 are called quadrants, and 
 they may be distinguished 
 as the first, second, tldrd, 
 and fourth quadrants. 
 
 The center of the circle 
 is taken as the origin of 
 distances, or the zero point, 
 and the different directions 
 in which distances are esti- 
 mated from this point are indicated by the signs + and 
 — . 1^ those from Q to the right be marked +, those 
 from Q to the left must be marked — ; and if distances 
 from Q upwards be considered plus, those from C down- 
 wards must be considered minus. 
 
 If one extremity of a varying arc be constantly at A, 
 and the other extremity fall successively in each of the 
 several quadrants, we may readily determine, by the 
 above rule, the algebraic signs of the sines and cosines 
 of all arcs from 0° to 360°. Now^, since all other trigo- 
 nometrical lines can be- expressed in terms of the sine 
 and cosine, it follows that the algebraic signs of all the 
 circular functions result from those of the sine and 
 cosine. 
 
 We shall thus find for arcs terminating in the 
 
 sin. COS. tan. cot. sec. oosec rera. 
 
 1st quadrant, + -|- -f -f- + -f -f 
 
 2d " +___ — — ^ 4 
 
 3d" __4.4.__ + 
 
 4th " — + — — + — -f. 
 
 PROPOSITION 1. 
 
 The chord of 60° and the tangent of 45° are each equal to 
 radius ; the sine of 80°, the versed sine of 60°, and the co* 
 fiiie of 60° are each equal to one half the radius* 
 
SECTION 1 
 
 248 
 
 With (7 as a center, and CA as a 
 ladius, describe the arc ABF, and 
 from A lay off the arcs AD = 45°, 
 AB = 60°, and AE = 90° ; then 
 is EB = 30°. 
 
 1st. The side of a regular in- 
 scribed hexagon is the radius of 
 tLe circle, (Prob. 28, B. lY), and as the arc subtended 
 by each side of the hexagon contains 60°, we have the 
 chord of 60° equal to the radius. 
 
 2d. The triangle CAR is right-angled at J., and the 
 angle (7 is equal to 45°, being measured by the arc AD\ 
 bence the angle at H is also equal to 45°, and the trian- 
 gle is isosceles. Therefore AH — CA — radius of the 
 circle. 
 
 3d. The triangle ABQ is isosceles, and Bn is a per- 
 pendicular from the vertex upon the base ; hence An = 
 nC = Bm. But Bm is the sine of the arc BE, On is the 
 cosine of the arc AB, and An is the versed sine of the 
 same arc, and each is equal to one half the radius. 
 
 Ilence the proposition ; the chord of 60°, etc. 
 
 PROPOSITION II. 
 
 Given, the sine and the cosine of ttoo arcs, to find the sine 
 and the cosine of the sum and of the difference of the same 
 arcs expressed by the sines and cosines of the separate arcs. 
 
 Let (r be the center of the 
 circle, CD the greater arc, 
 and DF the less, and denote 
 these arcs by a and h re- 
 spectively. 
 
 Draw the radius GD', make 
 the arc DE equal to the arc 
 BF, and draw the chord EF. 
 From i" and ^, the extremi- 
 ties, and J, the middle point 
 
 G »r 
 
250 * PLANE TRIGONOMETRY. 
 
 of the chord, let fall the perpendiculars FM, EP, and 
 IN, ou the radius GO. Also draw DO, the sine of the 
 arc OB, and let fall the perpendiculars III on FM, and 
 ^iTon IK 
 
 Now, by the definition of sines and cosines, BO =» 
 sin. a; 00 = cos.a; FI =^ sin.6; (7/= cos.6. We are 
 to liud 
 
 FM = sin. (a + 5) ; GM = cos. (« + &); - 
 EP = sin. {a — b)', GP = cos. {a — h). 
 
 Because IN is parallel to DO, the two A*8, GBO^ 
 GIN, are equiangular and similar. Also, the A FHI is 
 similar to the A GIN-, for the angles, FIG and HIN, 
 are right angles ; from these two equals, taking away tKe 
 common angle HIL, we have the angle Fill— the angle 
 GIN The angles at H and iVare right angles; there- 
 fore, the A's Fill, GIN, and GBO, are equiangular 
 and similar; and the side HI is homologous to IN 
 and BO. 
 
 Again, as FI = IE, and IK is parallel to FM, 
 
 FH = IK, and HI = ^^. 
 Bj similar triangles we have 
 
 GB : BO = GI : IN 
 That is, R : sin.a = co3.5 : IN-, or, IN^ sin.ygsj^ ( 1 , 
 
 Also, GB '. GO ^ FT \ FH. 
 
 That is, i2 : cos.a = « ^m.h : HF; or, JPir= cos^n^^ ^2) 
 
 Also, GB : GO = GI : GN 
 
 That is, H : cos.a = cos. J : GN; or, GN=^ ^y^-^^^^^ ( 3 j 
 
 Also, GB : DO = FI : IH. 
 
 That is, i? : sin.ci = sin.6 : IH; or, 7ir= ^l^hl^'l'j:!!. (4) 
 
 By adding the first and second of these equations, we 
 bave 
 
 IN -f FH = F3I - sin. (a + 6). 
 
SECTION I. 251 
 
 a,, , . . / , 7\ sin.a C0S.6 + cos.a Bin.ft 
 That IS, sin. (a -{- b) =^ ^ . 
 
 By subtracting the second from the first, since 
 
 IN— FJI= ZZV— iir= UP, we have 
 
 , ,. sin. a C0S.5 — cos.a sin. J 
 8in.(a — J)= ^ 
 
 J3y subtracting the fourth from the third, we have 
 
 GN — IH = GM = COS. (a 4- b) for the first member. 
 
 TT / . IN cos.a C0S.6 — sin. a sin. 5 /.» 
 Hence, cos. (a + 6) = . (5) 
 
 Mb 
 
 13y adding the third and fourth, we have 
 
 GN-^- in= GN+NF= (7P = co8.(a— 6). 
 
 rr , TV COS. « COS. 5 4- sin. a sin. i ,ct\ 
 
 Hence, cos.(a — o) — ~ . ( o ) 
 
 Collecting these four expressions, and considering the 
 r» lius unity, we have C'^^ ' 
 
 sin. (a + J) = sin. a cos.ft + cos.a sin. J ( 7 ) 
 
 sin. (a — 6)= sin. a cos.6 — cos.a sin. 6 (8) 
 
 cos.(a -\-h) — cos.a cos.^ — sin.a sin.i ( 9 ) 
 
 .cos.(a — ^) = cos.a cos.6 + sin.a sin.5 (10) 
 
 Formuloe {A) accomplish the objects of the proposi 
 tion, and from these equations many useful and import 
 ant deductions can be made. The following are the 
 most essential : 
 
 By a Iding ( " ) to ( 8 ), we have ( n ) ; subtracting ( 8 ) 
 fi\>m ( 7 ) gives ( 12 ). Also, ( 9 ) added to ( 10 ) gives ( 13 ) , 
 ( 9 ) taken from ( 10 ) gives ( 14 ). 
 
 sin. (a -\-h)-\- sin. (a — h) = 2sin.a cos.6 ( H ) 
 
 %\\\.(a-\-h) — sin. (a — 6) = 2cos.a sin. 6 (12) 
 
 cos.(a + i) + cos.(a — h) = 2cos.a cos.6 ( 13 ) 
 
 cos.(a — h) — cos.(a + b) = 2sin.a sin.6 (14) 
 
 If we put a-hb = A, and a — b = B, then ( H ) becomei. 
 ( 15 ), ( 12 ) becomes ( 16 ), ( 13 ) becomes ( 17 ), and ( 14 ) be 
 comes (18). 
 
 w 
 
 (B) 
 
252 
 
 PLANF TRIGONOMETRY. 
 
 (0) 
 
 rA-}'B\ /A— By 
 
 Bm.A + sin.^ = 2sin. (^-) cos. (^-^) ( ^^1 
 Bin.^-sm.5=2cos.(^±-?)sin.(^i:^) (16) 
 
 A ^ A ' 
 
 'A^B\ /A — By 
 
 oos.A + COS. J5 = 2cos. (^ -) cos. (^-) ( ^^ ) 
 A-hB^ 
 
 C08.J5 — cos -4 
 
 2 
 ^ — ^> 
 
 ^u^)""(^> ' 
 
 18) 
 
 sin. 
 
 If we divide ( 15 ) by ( -6 ), (observing that — '- = tan., 
 
 COS. 
 
 cos 1 
 
 ttnd -r-^ = cot. = as we learn by equations ( 6 ) and 
 
 sin. tan. 
 
 ( 5 ), we shall have 
 
 Ai-B 
 
 A—Bs 
 
 A+By 
 
 mn.A + sin 
 
 .B ''•'•(-^) "'°^-(-2-) ta°-(^-) 
 
 sin.^ — sin.i? 
 
 Whence, 
 
 A—B^ 
 
 A—B^ 
 
 
 (19) 
 
 A-VBs 
 
 A-By 
 
 sin.JL+sin.^ : sin.J. — sin.5 = tan.( \ : tan. ( ) 
 
 V- f vV ^ ^ 2 ^ 
 
 That is : The sum of the sines' (ff cnty two arcs is to the dif- 
 ference of the same sines, as the tangent of one half the sum 
 of the same arcs is to the tangent of one half their difference. 
 
 By operating in the same way with the different equa- 
 tions in formulae ((7), we find, 
 
 ^sm.A + sin.B /A-\-B\ 
 
 1^) 
 
 sin.^l H- sin.jB /A — B\ 
 
 cos.^ — cos^Z ^ ^^** V 2~~) 
 sin.A — sin.^ ^A — B\ 
 
 co8,A Tco^ == *^^- \ 2~/ 
 sin.J.— .sin..^ _ /^ + A 
 
 cos.^— cos.^ "" ^^^- \ 2~/ 
 
 cos.A + cos.^ ^^^' \~^ / 
 
 co&.B — COS. J. 
 
 tan. 
 
 /A^B 
 
 (20) 
 (21) 
 (22) 
 (23) 
 
 (24) 
 
SECTION I. 253 
 
 These equations are all true, whatever be the value 
 of the arcs designated by A and B ; we may, therefore, 
 assign any possible value to either of them, and if in 
 equations (20), (21), and (24), we make ^= 0, we shall 
 have, 
 
 Bm.A J. A 1 ,ncx 
 
 = tan. — = —I ( 25 ) 
 
 i -f cos.^ 2 cot^A 
 
 Bm,A .A 1 f^^. 
 
 = cot. -^ = — T ( 26 ) 
 
 (g 1-— cos.^ 2 tan.J^ 
 
 1 4- cos.^ cot.^^ 1 
 
 (27) 
 
 1— cos.Jl tan.J^ tan'l^ 
 J£ we now turn back to formulae (A), and divide equa- 
 tion (7) by (9), and (8) by (10), observing at the same 
 
 time that — '- = tan., we shall have, 
 cos. ' 
 
 +« / , i\ sin.a cos. J 4- cos.a sin. 6 
 tan. (a -f 6) = 
 
 tan. (a — 5) = 
 
 cos.a C0S.6 — sin. a sin.b 
 sin. a cos. 5 — cos.a sin. 5 
 
 cos.a C0S.6 + sin. a sin. 6 
 
 By dividing the numerators and denominators of the 
 
 second members of these equations by (cos.a cos.i), we 
 
 find, 
 
 sin. a C0S.5 cos.a sin. 6 
 
 , ,. cos.a C0S.6 cos.a C0S.6 tan.a-ftan.ft 
 
 tan.(a4-5)= , -. -. — ==- — (28) 
 
 cos.a COS. sm.a sin.6 1 — tan. a tan. o 
 
 cos.a COS. 6 cos.a COS. 5 
 sin. a COS. 5 cos.a sin.6 
 
 , ,, cos.a cos.6 cos.a COS.6 tan. a — tan. 5 
 
 tau.(a— 5)= -. — . . ,==-— ^ -— - (29) 
 
 ^ ' cos.a C0S.6 sin. a sin. D l+tan.a tan.6 
 
 cos.a COS. 6 cos.a C0S.6 
 If in equation (H), formulae (B), we make a = 6, we 
 ihall have, 
 
 sin. 2a = 2sin.a cos.a (30) 
 
 Making the same hypothesis in equation (13), gives, 
 co8.2a + 1= 2co8'.a (31) 
 22 
 
254 PLANE TRIGONOMETRY. 
 
 The same hypothesis reduces equation (14) to 
 
 l_cos.2a = 2siIl^a (32) 
 
 The same hypothesis reduces equation ( 28 ) to 
 
 , o 2tan.a ,oq\ 
 
 tan.2a = - — — -i- (3^) 
 
 1 — tsijr.a 
 
 If we substitute a for 2a in (31) and (32)^ we shall hav6 
 1 + cos.a = 2cos.'ia. (34) 
 and 1 — cos.a == 2sin.'Ja. (35) ^ 
 
 PROPOSITION III. 
 
 In any right-angled plane triangle^ we may have the fol- 
 lowing proportions : 
 
 1st. The hypotenuse is to either side, as the radius is to the 
 sine of the angle opposite to that side,^ 
 
 2d. One side is to the other side, as the radius is to the tan- 
 gent of the angle adjacent to the first side, 
 
 3d. One side is to the hypotenuse, as the radius is to the 
 eecant of the angle adjacent to that side. 
 
 Let CAB represent any right- 
 angled triangle, right-angled at 
 A. 
 
 (Here, and in all cases hereafter, we shall represent the angles of a 
 triangle by the large letters A, B, C, and the sides opposite to them, 
 by the small letters a, 6, c.) 
 
 From either acute angle, as C, take any distance, as 
 CB, greater or less than CB, and describe the arc BE. 
 This arc measures the angle 0. From B, draw BF par- 
 allel to BA ; and from JE, draw EGi, also parallel to BA 
 or BF, 
 
 By the definitions of sines, tangents, secants, etc, BF 
 is the sine of the angle 0; EGi is the tangent, CO the 
 secant, and CF the cosme. 
 
SECTION I. 
 
 255 
 
 Now, by proportional triangles we have, 
 
 CB : BA= CD: DF or, a : c = R : sm.O 
 CA:AB=CU: EG or, b : c 
 CAiCB ^ QE : CG 
 
 Hence the proposition. 
 
 ScnoLiuM. — If the hypotenuse of a triangle is made radius, one sidfl 
 is the sine of the angle opposite to it, and the other side is the cosine 
 of the same angle. This is obvious from the triangle CDF» 
 
 R :tan.(7 
 o\\ h : a = R : sec. (7 
 
 PROPOSITION IV. 
 
 In any triangle, the sines of the angles are to one another 
 08 the sides opposite to them. 
 
 Let ABC be any tri- 
 angle. From the points 
 A and B, as centers, 
 with any radius, de- 
 Hcribe the arcs meas- 
 uring these angles, and 
 draw pa, CD, and mn, 
 perpendicular to AB. 
 
 Then, pa = sin.^, and mn = sin.^. 
 
 By the similar A's, Apa and ACD, we have, 
 R : Bin. A = biCB; or, R{CI)) = b sin.A 
 
 By the similar a's, Bmn and BCD, we have, 
 
 R : sm.B = a: CD; or, R(CD) = a sm.B (2) 
 
 By equating the second members of equations ( 1 ) 
 and (2) 
 
 b sin, A = a sin. 5. 
 
 Uence, sin.^ : sin.^ = a : b 
 
 Or, a : b = sin.A : sin.^. 
 
 ScnoLiUM 1. — When either angle is 90°, its sine is radius. 
 
 Scholium 2. — When CB is less than AC, and the angle B, acute, 
 the triangle is represented by ACB. When the angle B becomes B^, 
 it is obtuse, and the triangle is A CB^ ; but the proportion is equally 
 
 (1) 
 
256 PLANE TRIGONOMETRY. 
 
 true with either triangle ; for the angle CB^D = CBAy and the 8iu< 
 of CB^D is the same as the sine of AB^C. In practice we can deter- 
 mine which of these triangles is proposed, by the side AB being 
 greater or less than AC', or, by the angle at the vertex C being large, 
 as A CB, or small, as A CB^. 
 
 In the solitary case in which AC, CB, and the angle A, are given, 
 and CB less than AC, we can determine both of the A's A(^B and 
 A CB'' ; and then we surely have the right one. 
 
 PROPOSITION V. 
 
 If from any angle of a triangle, a perpendicular he let fall 
 on the opposite side, or base, the tangents of the segments of 
 the angle are to each other as the segments of the base. 
 
 Let J.^ (7 be the triangle. Let fall 
 the perpendicular CB, on the side 
 AB, 
 
 Take any radius, as Cn, and de- 
 scribe the arc which measures the A. QV^^^^^B 
 angle C, From n, draw qnp parallel to AB. Then it is 
 obvious that np is the tangent of the angle BOB, and nq 
 is the tangent of the angle AQD. 
 
 Now, by reason of the parallels AB and qp, we have, 
 qn : np = AB : BB 
 
 That is, tan.^OZ) : \s.xi,BQB = AB : BB. 
 
 PROPOSITION VI. 
 
 If a perpendicular be let fall from any angle of a triangh 
 to its opposite side or base, this base is to the sum of the other 
 two sides, as the difference of the sides is to the difference of 
 the segments of the base, 
 
 (See figure to Proposition 5.) 
 
 Let AB be the base, and from 0, as a center, with the 
 shorter side as radius, describe the circle, cuttiiig AB in 
 Q, and AOm F-, produce AQ to E. 
 
SECTION I. 257 
 
 It is obvious that AF is the sum of the sides AO and 
 CB, and AF is their difference. 
 
 -Also, AD is one segment of the base made by the per- 
 pendicular, and BD = BCr is the other; therefore, the 
 difference of the segments is AG. 
 
 Aa A is a point without a circle, by Cor. Th. 18, B. 
 in, we have 
 
 AE X AF = AB X AG 
 
 Hence, AB : AF = AF : AG. 
 
 PROPOSITION VII. 
 
 TJie sum of any two sides of a triangle is to their difference^ 
 as the tangent of one half the sum of the angles opposite to 
 these sides, is to the tangent of one half their difference. 
 
 Let ABQ be any plane triangle. 
 Then, by Proposition 4, we have, 
 
 BQ: AQ^ sin.J. : Qin.B. 
 
 Hence, A B 
 
 «C + ^(7:^C— ^(7=sin.^-fsiii.^:sin.^— sin.^(Tli.9,B.II> 
 But, 
 
 tan. ( — i — j : tan. ( — - — \ = sm.A 4- sin.^ : sin.-i 
 
 — sin.^, (eq. (19), Trig.) 
 
 Comparing the two latter proportions, (Th. &, B. U), 
 we have, 
 
 ^C'+ ^(7:5C— ^C=~tan. (^4-^^ :tan. (^1:=1^) 
 Hence the proposition. 
 
 PROPOSITION VIII. 
 
 Given, the three sides of any plane triangle, to find some 
 relation which they must hear to the sines and cosines of thi 
 respective angles. 
 
 22* R 
 
258 
 
 PLANE TRIGONOMETRY. 
 
 Let ABC be 
 the triangle, and 
 let the perpen- 
 dic alar fall either 
 upon, or without 
 
 thebase, as shown c _ _ 
 
 m the ngures. 
 By recurring to Th. 41, B. I, we shall find 
 
 c p 
 
 CD = 
 
 c* 
 
 2a 
 
 (1) 
 
 Kow, by Proposition 3, we have 
 
 B : COS. = b : CD. 
 
 Therefore, 
 
 nj) _ ^ COS. 
 
 R 
 
 (2) 
 
 Equating these two values of CD, and reducing, we 
 Have 
 
 COS. (7 = — ^ 
 
 2ah 
 
 Arn) 
 
 In this expression we observe, that the part <?, whose 
 square is found in the numerator with the minus sign, is 
 Uie side opposite to the angle ; and that the denominator 
 is twice the rectangle of the sides adjacent to the angle. 
 From these observations we at once draw the following 
 expressions for the cosine A, and cosine B : 
 
 COS. A — 
 
 R(h' + c' — a' 
 
 cos 
 
 2ba 
 
 in) 
 
 (i>) 
 
 :ac 
 
 As these expressions are not convenient for logarith- 
 mic computation, we modify them as follows: 
 If we put 2a = A, in equation ( 31 ), we have 
 
 COS. A + 1 = 2cos.' J J.. 
 
 In the preceding expression, (w), if we consider radius 
 ttxiity, and add 1 to both members, we shall have 
 
COS. J. + 1 = 1 H- 
 Therefore, 2cos.'J^ = 
 
 SECTION I. 259 
 
 J' -f (;> — a« 
 
 2hc 
 26c 4- J' + <?' — «• 
 
 2bc 
 
 ^ (h + g)' — g' 
 26(? 
 CoDsidering 6 + c as one quantity, and observing that 
 (5 + c)' — a' is the difference of two squares^ we have 
 (t-|.c)«— a»=(5+c-|-a) {l-irc—a) ; but (?»+c— a)=Z>4-c+a— 2a. 
 Hence, 2cos.'JA = ( & + <^ + a) ( ^^ + <^ + ^ - 2 a). 
 
 n « ij ^ 2 ) V 2— "";. 
 
 Or, cos.'J^ = J 
 
 0(7 
 
 By putting = «, and extracting square root, 
 
 the final result for radius unity is 
 
 COS. U = sj'-^^. 
 
 ^ DC 
 
 For any other radius we must write 
 
 C08.W = J I^'' (>-"). 
 
 ^ DC 
 
 By inference, cos. IB = v/^'^ (^ ~ ^^. 
 
 AlBO, COS. i(7 = Ay^MfZI). 
 
 In every triangle, the sura of the three angles is equal 
 to 180° ; and if one of the angles is small, the other 
 two must be comparatively large ; if two of them are 
 email, the third one must be large. The greater angle 
 is always opposite the greater side; hence, by merely 
 inspecting the given sides, any person can decide at 
 once which is the greater angle; and of the three pre- 
 
260 PLANE TRIGONOMETRY. 
 
 ceding equations, tliat one should be taken which apphca 
 to the greater angle, whether that he the particular 
 angle required or not ; because the equations bring out 
 the cosines to the angles ; and the cosines to very small 
 arcs vary so slowly, that it may be impossible to decide, 
 with sufficient numerical accuracy, to what particular 
 arc the cosine belongs. For instance, the cosine 0.999999, 
 carried to the table, applies to several arcs; and, of 
 course, we should not know which one to take ; but this 
 difficulty does not exist when the angle is large ; there- 
 fore, compute the largest angle first, and then compute 
 the other angles by Proposition 4. 
 
 But we can deduce an expression for the sine of any 
 of the angles, as well as the cosine. It is done as fol- 
 lows: 
 
 EQUATIONS FOR THE SINES OF THE ANGLES. 
 
 Besuming equation ( wi ), and considering radius unity, 
 we have 
 
 a» + h' — (?» 
 
 COS. a = —^^3—. 
 
 Subtracting each member of this equation from unity, 
 gives 
 
 l_eos.a=l-(fl±|^). (1) 
 
 Make 2a = (7, in equation (32) j then a = ^0, 
 and 1 — COS. (7 = 2sin.4(7. (2) 
 
 Kv^uating the second members of (1) and (2), 
 2ab — a' — 6» -f c» 
 
 2sin.»J(7 = 
 
 2ab 
 
 _ e'-{ a- bf 
 2ab 
 
 (c -{- b — a){c -\- a — b) 
 * 2^ 
 
SECTION I. 261 
 
 /c + b — a\ /c + a — h\ 
 
 Or, Bin.4(7 = iZIUlL^ZAZZ. 
 
 ab 
 
 BuL—^~= — ^ a, and--- = — ^ 1 
 
 Put ^ = «, as before; then, 
 
 ^ ab 
 
 By taking equation {p ), and proceeding in the same 
 manner, we have 
 
 ac 
 
 From (n), sin.iA = \/EEK5jE^. 
 ^ cb 
 
 The preceding results are for radius unity ; for any 
 other radius, we must multiply by the number of unita 
 in such radius. For the radius of the tables we write 
 R; and if we put it under the radical sign, we must 
 write R^; hence, for the sines corresponding with our 
 loejarithmic table, we must write the equations thus, 
 
 8in.jA= ^MESEEZ). 
 
 ^ be 
 
 ^ ac 
 
 ^ ab 
 
 A large angle should not be determined by these 
 equations, for the same reason that a small angle should 
 not be determined from an equation expressing the 
 cosine. • 
 
 In practice, the equations for cosine are more gener- 
 ally used, because more easily applied. 
 
262 PLANE TRIGONOMETRY. 
 
 The formulae which we have thus analytically devel- 
 oped, express nearly all the important relations between 
 the sines, cosines, and tangents of arcs or angles; and 
 we have also demonstrated all the theorems required for 
 the determination of the unknown parts of any plane 
 triangle, three of the parts of which are given, one at 
 least being a side. 
 
 Such relations might be indefinitely multiplied, but 
 those already established are »^ufficient for most practical 
 purposes, and when others are required, no difficulty 
 will be found in deducing them from these. 
 
 The following geometrical demonstrations of many of 
 the preceding relations, are oflered, in the belief that 
 they will prove useful disciplinary exercises to the stu- 
 dent. > 
 1st. Let the arc AD =A ; then I)G-= sin. A ; CGr=zcoa,A , 
 i>J=sin.iJ.; J.i)=2sin.JJL; CI=co3.iA; 
 CI=I)0; and i>^=2i>0=2cos.JX 
 The angle, DBA, is measured by 
 one half the arc AD ; that is, by J J.. 
 Also, ADa = DBA = iA, 
 Now, in the triangle, BDCr, we have 
 
 sm.DBa : i>(7=sin.90° : BD. 
 That is, sin.J^ : sin.J.=l : 2cos.JJ.. 
 Or, sin.J.=2sin.JJ. cos.}^; 
 
 which corresponds to equation ( 30 ). 
 
 In the same triangle, 
 Bin.9d° . BD==am.BDa : BG; and sm.BDa^coa.DBG 
 That is, 1 : 2cos.JJ.=co8.JJ. : l+cos.J.. 
 Or, 2co8.' J^=l-f-cos.J[, same as equation (34). 
 
 In the triangle, DGA, we have, 
 
 Bin.90° j ^i> = 8in.G^i)^ : a A, 
 That is, 1 : 2sin. J^ = sin. J J. : 1 — cos.J.. 
 Or, 2sin.'' ^A = 1 —cos. J., same as equation ( 36 X 
 
SECTION I, 
 
 268 
 
 By similar triangles, we have, 
 
 BAiAD^ABiAa,- 
 
 That is, 2 : 2sin.J^ = 2sin.Jj4 : versed sin. J.. 
 
 Or, versed sin. JL = 2sin.' ^A, 
 
 2d. From C as the center, with CA as the radiu^ 
 describe a circle. Take any arc, 
 ABj and call it A ; and AD a less 
 arc, and call it ^ ; then JBI) is the 
 difference of the two arcs, and must 
 be designated by (A — B) ; arc AG 
 K= arc AB ; therefore, 
 
 £iroI)a=-A + B; Ua==sm.A; 
 
 £n = 8m.B; 6^n=:sin.J. + sin.^; 
 Bn = sin.^ — sin.^. 
 Fm = ml) = CH= cos.B ; mn — cos. J. ; 
 therefore, Fm + mn = cos.^ + cos.^ = Fn ; 
 mD — mn = cos.^ — cos.^ = nD ; 
 
 and i)6^ = 2sin.(^-±^). 
 
 Because, NF-^AB) AB-^ NF=-A-^ B\ 
 
 therefore, 180° — (^ + j5) = arc FB^ 
 
 or. 
 
 90°-(^4^)=iarcZfi. 
 
 But the chord, FB, is twice the sine of J arc FB ; 
 
 that 
 
 A + B. 
 
 is, FB = 2sin. (90° - 1 ) = 2cos. {~^\ 
 The L_w(ri) = [_BFB, because both are measured 
 
 by one half of the arc BB; that is, by ( — - — V and the 
 
 two triangles, GnB and FnB, are similar. 
 The angle, GFn, is measured by ( — - — \ 
 In the triangle, FBG, Fn is drawn from an angle per 
 
264 PLANE TRIGONOMETRY. 
 
 pendicular to the opposite side ; therefore, by Propositioi* 
 5, we have, 
 
 an : nB^tsin.aFn : tsin.BFn, 
 
 A-{-B^ 
 
 That is, 8iu.J.+sin.^ : sin. J. — sin.J5=tan.( — — — ") : 
 
 tan. 
 
 {^^-j^y This is equation ( 19 ). ^J^' 
 
 In the triangle, GnD, we have, 
 
 sin. 90° : I)Gr = &m.nI)G : Crn; sm.nBG-^cos.nCrl), 
 
 That is, 1 : 2sin. (f^) = cos. (^^) : sin. JL+sin.^. 
 
 Or, sin.^ + sin. J5 = 2sin. (^y^) cos. {^-^)> 
 
 the same as equation (15). 
 3d. In the triangle, FnB, we have, 
 
 sin.90 : FB = sin. BFn : Bn. 
 
 That is, 1 : 2cos.(^±^) = ^m.{^?-) : sin.^^sin.i^. 
 
 Or, sin.^-sin.^ = 2cos. {^~) sin. (^~^), 
 
 the same as equation (16). 
 4th. In the triangle, FBn, we have, 
 
 sin.90 : FB = cos.BFn : Fn. 
 
 That is, 1 : 2cos. {^~^) = cos.(^^i:p?) : cos.^+cos.J9. 
 
 Or, cos.^ + cos.^ = 2cos. (f^^) cos. (^ ~^), the 
 «ame as equation ( 17 ). 
 
 5th. In the triangle, GnB, we have, 
 
 sin.90° : ai) = ain.nai) : nJ), 
 
 That is, 1 : 2sin. (-^— ) = sin. (-^- ) : C08.5— cos.^ 
 *.he same as equation ( 18 ). 
 
 6th. In the triangle, FGrn, we have, 
 
 sin. GrFn : G-n = cos. GFn : Fn. 
 
SECTION I. 265 
 
 That is, sin. ^ : 8in.J.+sin.^ = cos. —^~— : cos.A-^ 
 oos.B, 
 
 Or, (sin.^ + sin.j^) cos. { — - — ) = (co8.-4. + coe.J?) sin. 
 A-{B 
 
 r-¥)- 
 
 . A + B 
 
 sin. 
 
 cos.^ — 2 — 
 
 same as equation (20). 
 
 7tli. In the triangle, FnB, we have, 
 
 Fn : nB : : 1 : tau.BFn, 
 
 Thatis, C08.J5 + C0S.J. : sin.JL— sin.jB :: 1 : tan.J(A — B)» 
 
 ^ &m.A — sin.5 , /A — B\ ,, 
 
 Or, — i - = tan. ( — - — ), the same 
 
 ' cos. J- -f COS..B \ 2 r 
 
 as equation (22). 
 
 8th. In the triangle, GnB, we have, 
 
 On : nD :: 1 : tan.wG^i). 
 That is, 
 
 — o — )» 
 
 cos. B — COS. A , /A — B\ 
 
 or, — i J— -r. = tan. ( — - — ). 
 
 ' sin. A + sin. B \ 2 / 
 
 NATURAL SINES, COSINES, ETC. 
 
 When the radius of the circle is taken as the unit of 
 measure, the numerical values of the trigonometrical 
 lines belonging to the different arcs of the quadrant, bo- 
 come natural sines, cosines, etc. They are then, in fact, 
 but numbers expressing the number of times that these 
 line^ contain the radius of the circle in which they are 
 taken. The tables usually contain only the sines and 
 cosines, because tliese are generally sufficient for practi- 
 §3 
 
266 PLANE TRIGONOMEIR Y. 
 
 cal pui-poses, and the others, when required, arc rnailily 
 expressed in terms of them. 
 
 We proceed to explain a method for computing a table 
 of natural sines and cosines. 
 
 It was shown, in Book Y, that the linear value of thci 
 arc 180°, in a circle whose radius is unity, is 
 
 3.141592653. 
 
 This divided by 180 x 60, the number of minutes id 
 180°, will give the length of one minute of arc, which ia 
 
 .00029088820867. 
 
 But there can be no sensible difference between the 
 length of the arc V and its sine ; and, within narrow 
 limits, that sine will increase directly with the arc. 
 
 Hence, 
 
 sin. 
 
 1' = 
 
 .0002908882. 
 
 
 sin. 
 
 2' = 
 
 .0005817764. 
 
 
 sin. 
 
 3' = 
 
 .0008726646. 
 
 
 sin. 
 
 4' = 
 
 .0011635528. 
 
 
 sin. 
 
 5' = 
 
 .0014544410. 
 
 
 sin. 
 
 6' = 
 
 .0017453292. 
 
 
 sin. 
 
 7' = 
 
 .0020362175. 
 
 
 sin. 
 
 8' = 
 
 .0023271057. 
 
 
 sin. 
 
 9' = 
 
 .0026179939. 
 
 
 sin. 
 
 10' = 
 
 .0029088821. 
 
 Beyond this, the error which would arise from taking 
 the arc for its sine, upon which the above proceeds, 
 would affect the final decimal figures; and we must, 
 therefore, continue the computation of the series by 
 other processes. To find the values of the cosines of 
 arcs, from 1' to 10', we have 
 
 cos. = ^1 — sin.^ = 1 — J sin.*, nearly. 
 
 That is, when the sines are very small fractions, an ih 
 the case for all arcs below 10', we can find the cosine htf 
 subtracting one half of the square of the sine from unity. 
 
SECTION I. 267 
 
 WlieDce, COS. 1' = .9999999577. 
 
 COS. 2' = .9999998308. 
 
 COS. 3' = .9999993204. 
 
 COS. 4' = .99999932304. 
 
 COS. 5' •= .99999894290. 
 
 COS. 6' = .99999847753. 
 
 COS. 7' = .99999792735. 
 
 COS. 8' = .9999973035. 
 
 COS. 9' = .9999965730. 
 COS. 10' = .9999957703. 
 
 The natural sines of arcs, differing by 1', from 10 up 
 lo 1°, raaj be computed from those of arcs less i ad 
 10', oy means of equation ( H ), group B, which is 
 
 sin. (a -{■ h) = 23in. a cos. b — sin. {a — b); 
 
 And when a = b, this equation becomes 
 
 sin. 2a = 2sin.a cos. a. Eq. (30). 
 
 To find the sine of 11', we make a = 6', and 6 = 5'; 
 
 then sin. 11' = 2sin. 6' cos. 5'— sin. 1'== .00319976^13 
 
 a = 6 = 6', sin. 12' = 2sin. 6' cos. 6'. 
 a = 7', i = 6', sin. 13' = 2sin. 7' cos. 6' — sin. V. 
 a=b=:7, sin. 14' = 2sin. 7' cos. 7'. 
 a = 8, 6 = 7, sin. 15' = 2sin. 8' cos. 7' — sin. V 
 
 And so on to the 
 
 sin. 30' = 2sin. 15'cos.l5'. 
 sin.l° = sin. 60' = 2sin. 30'cos.30'. 
 sin. 2° = 2sin.l° cos. 1°. 
 sin. 3° = 2sin. 2° cos. 1*" — sin. 1°, etc., etc., etc. 
 
 This process may be continued until we have found 
 ll:c sires and cosinea of all arcs differing by 1', from 
 to 90°, the values of the cosines being deduced success- 
 ively from those of the sines by means of the formula. 
 
 COS. = v/l — sin.'. 
 In this calculation, we began by assuming that, for 
 small arcs, Ihe sines and the arcs were sensibly equal. 
 
268 PLANE TRIGONOMETRY. 
 
 [t must be remembered that this is but an approxiina. 
 tion ; and although the error in the early stages of the 
 process is not sufficient to affect any of the decimal lig- 
 ures which enter the tables, it will finally become so, 
 since it is constantly increased in the operations by 
 which the sines and cosines of the larger arcs are de- 
 duced from those of the smaller. When the error has 
 been thus increased until it reaches the order of the last 
 decimal unit of the table, which assigns our limit of 
 error, we must have the means of detecting and correct 
 ing it. 
 
 Thij^ consists in calculating the sines and cosines of 
 certain arcs by independent processes, and comparing 
 them vTith those found by the above method. 
 
 We have seen, for example, (Prop. 7, B. V), that the 
 chord of 
 
 80^^ 517638090 ; whence, sin. 15° =.258819045. 
 
 15" :r= .2610523842; « " 7° 30' =.130526192. 
 7^30' ^ .1308062583; " " 3° 45' =.0654031291. 
 
 And so on to 
 
 sin. 14' 3" 45'" = .004090604. 
 
 etc. etc. etc. 
 
 The following elegant method of deducing, from the 
 sine of an arc, the sine and cosine of one half the arc, is 
 given, assuming that the student is familiar with the 
 simple algebraic principles upon which it depends. 
 
 Let us take the natural sine of 18°, which is .3090170, 
 
 18° 
 and make x = sine, and y the cosine of 9° = -— -. 
 
 Then, a:* + 2/' = 1; W 
 
 and 2xy = .3090170 (2); Eq. (30) 
 
 Adding, we have 
 
 jr'+ 2xy +/ = 1.3090170; 
 
SECTION 1. 269 
 
 Taking the square root, we have 
 
 a: + 3/ = 1.144123. (3) 
 Subtracting (2) from (1), 
 
 x^ — 2xy + y' = .690983; 
 taking the square root, 
 
 rr — y = —.831254* (4) 
 Adding (3) and (4), 2x = .312869, 
 
 hence, x = 8in.9° = .1564345 
 
 Subtracting (4) from (3), 2^^ = 1.975377 
 hence, y = cos.9° = .9876885 
 
 Now, by making x = the sine of 4° 30', and y = cosine 
 of 4° 30', and as before 
 
 x» + i/» == 1 
 and 2xy = .1564345, 
 
 we obtain the sine and cosine of 4° 30'; and another ope- 
 ration will give the sine and cosine 2° 15', etc., etc. 
 
 We may in this manner compute the sines and cosines 
 of all arcs resulting from the division of 18° by 2, and 
 we may make their values accurate to any assigned deci- 
 mal figure. 
 
 This has been carried far enough to show how a table 
 of natural sines, etc., could be computed ; but in conse- 
 quence of the tedious numerical operations which the 
 process requires, other methods are resorted to in the 
 actual construction of the table. 
 
 The Calculus furnishes formulae giving the values of 
 the sines and cosines of arcs developed into rapidly con- 
 verging series, and from these the sines and cosines of 
 all arcs from 0° to 90°, can be determined with great 
 
 * AVhen an arc is less than 45°, the cosine exceeds the sine; and 
 
 when the arc is between 45° and 90°, the sine exceeds the cosine. 
 
 Hence, when the arc is 9°, y, its cosine, exceeds x, its sine ; and we 
 
 therefore plac »,d tJie minus sign before the second member of Eq. (4). 
 
 23* 
 
270 
 
 PLAJE TRIGONOMETRY. 
 
 accuracy and with comparatively little labor. In the lajjt 
 two columns on each page of Table II, will be found the 
 v^alues thus computed of the sines and cosines of every 
 degree and minute of a quadrant. 
 
 TRTGONOMETRICAL LINES FOR ARCS EXCEEDING 90°. 
 
 From the annexed figure, 
 the construction of which 
 needs no explanation, are 
 deduced by simple inspec- ^ 
 tion the results given in the 
 following 
 
 TABLE. 
 
 90° + a° 
 
 270° — a° 
 
 Bin. = COS. a, COS. = — sm. a 
 
 sin. = — cos. a, cos. = — sin. a 
 
 tan . = — cot. o, cot. = — tan . a 
 
 tan. = cot. a, cot. = tan. a 
 
 sec. = — cosec. a, cosec. = sec. a 
 
 sec. — — cosec. a, cosec. = — sec. a 
 
 180° — o° 
 
 270° 4- a° 
 
 sin. = shi. a, cos. = — cos. a 
 
 sm. = — cos. o, COS. = sm. a 
 
 tan. = — tan. a, cot. = — cot. a 
 
 tan. = — cot. a, cot. = — tan. a 
 
 sec. = — sec. a, cosec. = cosec. a 
 
 sec. = cosec. a, cosec. = — sec. a 
 
 180° -fa° 
 
 360° -a° 
 
 sin.=— sin. a,cos. =— cos. a 
 
 sin.. = — sin. a, cos. = cos. a 
 
 tan.= tan.a,cot. = cot. a 
 
 tan. = — tan. o, cot. = — cot. a 
 
 8ec. =— sec. a, co8ec.=— cosec.a 
 
 sec. = sec. a, cosec. = — cosec. a | 
 
 1 
 
 By means of this table, the values of the tngonoraet- 
 rical lines of any arc between 90° and 360°, can be ex- 
 pressed by those of arcs less than 90°. 
 
 K, for examj)le, the arc is 118°, we have 
 
SECTION 1. 271 
 
 Sin. 118° = sin. (90° + 28°) = cos.28° ; 
 tan. 118° = tau.(90° -f 28°) = — cot.28° ; 
 etc., etc., etc. 
 
 For the arc 230°, we have 
 
 sin. 230° = sin. (270° — 40°) = — cos 40° ; 
 flec.230° = 6ec.(270° — 40°) = — cosec.40° ; 
 etc., etc., etc. 
 
 In many investigations, it becomes necessary to con- 
 M iQv the functions of arcs greater than 360° ; but since 
 the addition of 360° any number of times to the arc a, 
 will give an arc terminating in the extremity of a, it id 
 obvious that the arc resulting from such addition will 
 have the same functions as the arc a. And hence it fol- 
 lows that the functions of arcs, however great, may be 
 ftxpressed in terms of the functions of arcs less than 90°. 
 
272 
 
 PLANE TRIGONOMETRY. 
 
 SECTION II. 
 
 PLANE TRIGONOMETRY, PRACTICALLY APPLIED. 
 
 In the preceding section, the theory of Trigonometry 
 has been quite fully developed, and the student should 
 now be prepared for its various applications, were he 
 acquainted with logarithms. But logarithms are no part 
 of Trigonometry, and serve only to facilitate the numeri- 
 cal operations. Trigonometrical computations can be 
 made without logarithms, and were so made long before 
 the theory of logarithms was understood. 
 
 For this reason, we proceed at once to the solution of 
 the following triangles. 
 
 1. The hypotenuse of a right-angled triangle is 21, 
 and the base is 17 ; required the perpendicular and the 
 acute angles. 
 
 Let CAB be the triangle, in 
 which OB = 21, and CA z= ■ p 
 
 17. With C as a center, and 
 CD = 1 as a radius, describe 
 the arc DE, of which the sine 
 IS DF^ the tangent is EG, and 
 the cosine is CF. 
 
 By similar triangles we have 
 CB : CA '. 
 
 that is, 21 : 17 : 
 
 17 
 21 
 
 Hence, 
 
 COS. C 
 
 I 
 
 
 C . 1 
 
 FE A 
 
 CD : CF', 
 
 
 1 : cos. C. 
 
 
 = .80952+. 
 
 
SECTION II. 27a 
 
 We must now turn to Table II, and find in the last two columns 
 the cosine nearest to .80952, and the corresponding degrees and 
 minutes will be the value of the angle C. 
 
 On page 57, of Tables, near the bottom of the page, and in the 
 column with cosine at the top, we find .80953, which coriesponda 
 to 35*" 56' for the angle C. The angle B is, therefore, 54° 3'. 
 
 'T'ais Table is so arranged, that the sum of the degrees at the top 
 and bottom of the page, added to the sum of the minutes which are 
 found on the same horizontal line in the two side columns of the 
 pagC; is 90° 
 
 Thus, in finding the angle C, the number .80953 was found in 
 the column with cosine at the head. / We therefore took the de« 
 grees from the head of the page, and the minutes were taken 
 from the left hand column, counting downwards. 
 
 For the side ABy we have the proportion 
 
 CF : FB :: CA : AB] 
 
 or, cos. : sin. C i: 17 : AB] 
 
 that is, .80953 : .58708 : : 17 : AB. 
 
 From which we find AB = .58708 x 17 -i- .80953; 
 
 whence, AB = 12.328. 
 
 If we had formed a table of natural tangents, as well as of natii- 
 ral sines, AB could have been found by the following proportion • 
 CE : EG :: CA : AB 
 
 or, 1 : tan. C :: 17 : AB; 
 
 whence, AB = 17 tan. C. 
 
 The perpendicular AB may also be found b}' the proportion 
 CD : DF :: CB : AB; 
 
 or, 1 : sin. C :: 21 : AB; 
 
 whence, AB = 21 sin. C = 21 x .58708 = 12.32868. 
 
 2. The two sides of a right-angled triangle are 150 ?nd 
 125 ; required the hypotenuse and the acute angles. 
 
 We may employ the same 
 figure as in the preceding prob- ^ 
 
 lem. 
 
 Then, from the similar trian- 
 gles, CFD and CAB, we get 
 
 CF : FD :: CA : AB; 
 
274 PLANE TRIG ONOMETBY. 
 
 that is, COS. C : sin. C : : 150 : 125 : : 6 : 6, 
 which gives 6 sin. 6^ = 5 cos. 0} 
 
 hence, 36 sin.'C = 25 cos.«C. 
 
 A.dding member to member, 36 cos.^6' = 36 cos.*C. 
 
 we have 36 (sin.^C4- cos.'^C^ = 61 cos.* C. 
 
 Butsin.^C-f cos.'C = 1, (Eq. (1) Trigonometry); 
 whence, 61 cos.^C7 = 36; 
 
 cos.^a == '^ = .5901639344: 
 61 
 
 and COS. C = .76822, nearly. 
 
 To find the angle of which this is the cosine, we turn to page 6(1 
 of tables, and looking in the column having cosine at the head; we 
 Bee that .76822 falls between .76828, which has 48' opposite to it 
 in the left hand column, and .76810, which has 49' opposite to it 
 in the same column. Now, the cosines of arcs less than 90° de- 
 crease when the arcs increase, and the converse ; and while the 
 increase of the arc is confined within the limits of 1', the increase 
 of the arc will be sensibly proportional to the decrease of th^ cosine. 
 
 0.76828 .76828 
 
 Hence, 0.76810 .76822 
 
 18 : 6 : : 60" : sf' 
 
 which gives a;" = 20". 
 
 The angle C is, therefore, equal to 39° 48' 20", and the angle 
 B s 90° — 39° 48' 20" = 50° 11' 40". 
 To find CB, we have 
 
 CF : (W : : CA i CB 
 
 or, COS.C : 1 :: 150 : CB 
 
 that is, .76822 : 1 :: 150 : CB 
 
 150 
 whence, CB = --4>o = 1^5.26—. 
 
 3. The base of a right-angled triangle is 150, and the 
 tii gle opposite the base is 50° 11' 40" ; required the 
 h> potenuse and the perpendicular. 
 
oECTION II. 
 
 275 
 
 Let CAB be the triangle. 
 Then, (Prop. 4, Sec. I), 
 riu.50° 11' 40" ; sin. 90" :: 150 : CB. 
 Whence, 
 
 CB 
 
 150 
 
 :j;j = 195.26, 
 
 .708:^2 
 
 »lie same as in the preceding example. 
 To find ABj we have 
 
 CD : DF :: CB : AB) 
 that is, 1 : sin. (7 or cos. 5 :: 195.26 : AB) 
 
 from which we find 
 
 AB = 195.26 sin. 39° 48' 40"; 
 or, AB =^ 125.0015. 
 
 4. Two sides, the one 30 and the other 35, and the in- 
 cluded angle 20°, of a triangle, are given, to find the 
 other two angles and the third side. 
 
 *Let BA C be the triangle, in which BC 
 = 35, BA = 30, and the angle B = 
 20°. From A^ the extremity of the 
 shorter -side, let fall on BC the perpen- 
 dicular AD, thus dividing the triangle ^ 
 into the two right-angled triangles BAD and CAD. 
 
 Then, from the triangle BAD, we have 
 
 1st, 
 
 or, 
 
 2d. 
 
 BA 
 
 : AD) 
 
 30 
 
 '. AD ^ m sin. 20** 
 
 BA 
 
 :BD; 
 
 30 
 
 : BD = BO cos. 20° 
 
 sin. D : sin. B 
 1 : sin. 20° 
 1 : COS. B 
 or, 1 : COS. 20° 
 
 In the table of natural sines, we find sin. 20° = .34202, and tht 
 .'OS. 20° =x .93969; hence, AD = 30 X .34202 = 10.26060, and 
 fii; = 30 X .93969 = 28.19070, and therefore DC := BC -^ 
 BD = 6.8093. 
 From the triangle CAD, we have 
 
 1st, AC=^ ^AD' + DC^ = v/(10.26)* + (6.8+/ ^ '»*' ^^'^ 
 2d. Ar^ : AD :: Sxn.90° : sin. C: 
 
276 
 
 or, 
 
 whence, 
 and the 
 
 PLANE TRIGONOMETRY. 
 12.367 : 10.26+ :: 1 ; sin. C) 
 
 «^"- ^ = 1W7 = -^^^l^- 
 ande C = 56° 26\ 
 
 If, now, we add angles B and (7, and take the sum from 180* 
 the remainder will be the angle BA G, 
 
 Hence, \__BAC=^ 180° — (56° 26'+ 20°) = 103° 34'. 
 
 5. Two sides, the one 18 and the other 21, and the 
 angle opposite the side 24 equal to 76°, are given, to find 
 the remaining side and the other two angles. 
 Let X denote the angle opposite the side 18. Then, 
 
 24 : 18 :: sin. 76° : sin.ar, (Prop. 4, Trig.). 
 or, 4:3:: sin. 76° : sin. x. 
 
 sin. a; = f sin. 76° = f X .97030 = .72772; 
 
 whence the angle opposite the side 18 is 46° 41' 45". 
 Adding this to the given angle, and taking the sum from 18V®, 
 we get 57° 18' 15" for the third angle. 
 
 To find the remaining side, denoted by y, we have 
 
 sin. 76° : sin. 57° 18' 15" :: 24 : y; 
 
 or, .97030 : .84155 : : 24 : y. 
 
 24 X .84155 
 
 .97030 
 
 = 20.815 = 3d side. 
 
 6. The three sides of a triangle are 18, 24, and 20.816 j 
 required the angles. 
 
 This problem may be solved by Prop. 6, or by Prop. 8, Trigo- 
 nometry. 
 
 First By Prop. 6. 
 
 In the triangle ABCy make CB = 
 24, ^(7= 20.815, and AB = 18. 
 
 Then, 
 
 24 : 38 815 :: 2.815 : CD -^ BD. 
 
 e7>-5i>= 1^-264225^ 
 24 
 
SECTION II. 277 
 
 Put CD + BD ^ OB = 21. 
 
 By addition, we get 2CD ^ 28.5527; 
 
 dividing by 2, and CD = 14.2763+. 
 
 And hence, BD = CB — CD = 24 — 14.2763 = 9.7237. 
 
 In the triangle ADB, we have 
 
 BA : BD :: 1 : cos. 5 
 
 or, 18 : 9.7237 :: 1 : cos. ^ = .54020 
 
 rr ki TT T> ^Q (cos. 57° 18' = .54024) 
 Table 11, Page 53, | ,,, 5,0 .p. ^ .^^ppp } 
 
 diff. = 24 : 60" : : 4 : 10" 
 
 hence, L ^ = ^7*^ 18' 10". 
 
 It will be observed that Examples 5 and 6 refer to the same tri- 
 ungle, and that in Example 5 the angle B was 57° 18' 15". Thia 
 slight discrepancy in the results should be expected, on account of 
 the small number of decimal places used in the computations. 
 
 Second. By Prop. 8. 
 
 Sum of the sides, = 62.815, 
 half sura denoted by s, = 31.4075 
 a = 24 
 
 s — a = 7.4075 
 
 Formula, cos. \ A = \ / —^- ^, radius being unity. 
 
 s{s — d)=z 31.4075 X 7.4075 = 232.65105625 
 he = 20.815 X 18 = 374.67 
 ^^^"""-^ « .62095 very nearly. 
 
 V':62095 = .78800. 
 
 Hence, cos. \A = .78800, and M (Table II, page 59) » SS* 
 Tery nearly ; the angle A is therefore equal to 76°, which agrees 
 with Example 5. 
 
 7. Given, the three sides, 1425, 1338, and 493, of a tri- 
 angle; required, the angle opposite the greater .side, using 
 the formula for the sine of one half an angle. 
 24 
 
278 PLANE TRIGONOMETRY. 
 
 Make a = 1425, h x= 1338, and c = 493 ; then the [_ i ii 
 opposite the side a, and the formula is 
 
 Ann A = (^ - ^) (« -A 
 be 
 
 in which s denotes the half sum of the three sides. 
 
 Then we have s = 1628, s — h = 290, s^c=^ 1135, (s — h) 
 
 is-^c) = 329150, ic = 659634, (^ — ^) (^ — ^ .498988 
 
 be 
 
 H.nce, sin. U = ^.498988 = .70632. 
 
 In the table we find sin. 44° 5Q' 28.5" ~ .70638. 
 Therefore, lA = 44° 56' 28.5", and A = 89° 52' 5l" ; — but 
 little less than a right angle. 
 
 In these seven examples we have shown that it is possi- 
 ble to solve any plane triangle, in which three parts, one 
 at least being a side, are given, without the aid of loga- 
 rithms. But, when great accuracy is required, and the 
 number of decimal places employed is large, the necessary 
 multiplications and divisions, the raising to powers, and 
 the extraction of roots, become very tedious. All of these 
 opera tiors may be performed without impairing the cor- 
 rectness of results, and with a great saving of labor, by 
 means of logarithms ; but, before using them, the student 
 should be made acquainted with their nature and pro- 
 perties. 
 
 LOGARITHMS. ^ 
 
 Logarithms are the exponents of the powers to which 
 fi fixed number, called the base, must be raised, to pro- 
 duce other numbers. 
 
 The exponent of a number is also a number express- 
 ing how many times the first number is taken as a factor. 
 
 Thus, let a denote any number ; then a' indicates that a 
 has been used three times as a factor, a* that it has been 
 aoe.l four times as a factor, and a" that it has been thus 
 •i3cd n times. 
 
OECTION II. 279 
 
 Kow, instead of calling these numbers S, 4, n, 
 
 exponents, we call them the logarithms of the powers a », 
 aS a\ 
 
 To multiply a^ by a', we have simply to ^^Tite a, giving 
 it an exponent equal to 2 + 5 ; thus, a^ x a^ — a\ 
 
 Hence, the sum of the logarithms of any number of factoit 
 is equal to the logarithm of the product. 
 
 To divide a" by a», we have oul}^ to write a, giving it 
 au exponent equal to 12 — 9; thus, a"-r-a' = a'; and, 
 gererally, the quotient arising from the division of a"* by 
 a", is equal to a"'~". 
 
 Hence, the logarithm of a quotient is the logarithm of the 
 dividend diminished hy the logarithm of the divisor. 
 
 If it is required to raise a number denoted by a', to the 
 fifth power, we write a, giving it an exponent equal to 
 3x5; thus, {a^)^= a ", and, generally, (a «) "» = a " *". 
 
 Hence, the logarithm of the power of a number is equal to 
 the logarithm of the number multiplied by the expoiient of the 
 power. 
 
 To extract the 5th root of the number a', we wTite a, 
 giving it an exponent equal to |; thus, \/a'=a^, and, 
 generally, to extract any root of a number, we divide the 
 exponent of the number by the index of the root, and the 
 quotient will be the exponent of the required root. 
 
 Hence, the logarithm of a root of a number is equal to the 
 quotient obtained by dividing the logarithm of the number by 
 the index of the root. 
 
 Now, understanding that by means of a table of loga- 
 rithms we may find the numbers answering to given 
 logarithms, with as much facility as we can find the loga- 
 rithms of giv^en numbers, we see from what precedes that 
 multiplications, divisions, raising to powers, and the ex- 
 traction of roots, may be perfonned by logarithms ; and 
 the utility of logarithms, in trigonometrical computations, 
 mainly consists in the simplification and abridgment of 
 these operations by their use. 
 
280 PLANE TRIGONOMETRY. 
 
 The common logarithms are those of which 10 is the 
 base ; that is, they are the exponents of 10. 
 
 Thus, 10' = 10 Hence the logarithm 10 =1. 
 
 10^=100 " " " 100 =2. 
 
 10' =1000 " < " 1000 =3. 
 
 10* = 10000 " " « 10000 = 4. 
 
 etc. etc. etc. etc. etc. 
 
 10 dr 
 
 Since 2Q = 1 = 10'-' = 10", and generally -^^ = a" = 1, 
 
 it follovva that in this, as in all other systems, the loga- 
 rithm ot 1 = 0. 
 
 From what precedes, it is evident that the logarithm of 
 any number between 10 and 100 must be found between 
 
 1 and 2 ; that is, its logarithm is 1 plus a number less 
 than 1; and any number between 100 and 1000, will 
 have for its logarithm 2 plus some number less than 1, 
 and so on. The fractional part of the logarithm of a 
 number is expressed decimally. 
 
 The entire number belonging to a logarithm is called 
 its index. The index is never put in the tables, (except 
 from 1 to 100), and need not be put there, because we 
 always know what it is. It is always one less than the 
 number of digits in the integer. Thus, the number 3754 
 has 3 for the index to its logarithm, because the number 
 consists of 4 digits; that is, the logarithm is 3 and some 
 decimal. 
 
 The number 347.921 has 2 for the* index of its loga- 
 rithm, because the number is between 347 and 348, and 
 
 2 is the index for the logarithms of all numbers over 100, 
 and less than 1000. 
 
 All numbers consisting of the same figures, whether 
 integral, fractional, or mixed, have logarithms consisting 
 of the same decimal part. The logarithms differ only in 
 their indices. 
 24* 
 
SECTION II. 
 
 281 
 
 Thus, 
 
 the number 
 the number 
 the number 
 the number 
 the number 
 
 7956. has 
 795.6 has 
 79.56 has 
 7.956 has 
 
 3.900695 for its log. 
 2.900695 
 1.900695 
 0.900695 
 
 .7956 has —1.900695 
 
 the number .07956 has —2.900695 
 
 From this we perceive that we must take the logarithm 
 01' t of the table for a mixed number or a decimal, the 
 same as if the figures expressed an entire number; and 
 then, to prefix the index, we must consider the value of 
 the number. 
 
 The decimal part of a logarithm is always positive; 
 Dut the index becomes negative when the number is a 
 decimal ; and the smaller the decimal, the greater the 
 negative index. Hence, 
 
 To prefix the index to a decimal, count the decimal 
 point as 1, and every cipher as 1, up to the first significant 
 figure, and this is the negative index. 
 
 For example, find the logarithm of the decimal 
 .0000831. 
 
 ISTum. .0000831; log. —5.919601. 
 
 The point is counted one, and each of the ciphers is 
 counted one; therefore the index is minus five. 
 
 The smaller the decimal, the greater the negative 
 index ; and when the number becomes 0, the logarithm is 
 negatively infinite. 
 
 Hence, the logarithmic sine of 0° is negativehj infimite^ 
 liowever great the radius. 
 
 A number being given, to find its corresponding logarithm. 
 
 The logarithn of any number consisting of four figures, 
 or less, is taken out of the table directly, and without the 
 least difficulty. 
 
 Thus, to find the logarithm of the number 3725, wf 
 24 * 
 
282 PLANE TRlGONOxME TRY 
 
 find 372 at the side of the table, and in the columfl 
 marked 5 at the top, and opposite 372, we find .571126, 
 for the decimal part of the logarithm. 
 
 Hence, the logarithm of 3725 is 3.571126. 
 the logarithm of 37250 is 4.571126. 
 the logarithm of 37.25 is 1.571126, etc. 
 
 Find the logarithm of the number 834785. 
 This number is so large that we cannot find it in the 
 table, but we can find the numbers 8347 and 8348. The 
 logarithms of these numbers are the same as the loga- 
 rithms of the numbers 834700 and 834800, except the 
 indices. 
 
 834700 log. 5.921530 
 834800 log. 5.921582 
 
 DiiFerence, 100 52 
 
 Now, our proposed number, 834785, is between the 
 two assumed numbers ; and, of course, its logarithm lies 
 between the logarithms of the two assumed numbers; 
 and, without further connnent, we may find it by propor- 
 tion thus, 
 
 '100 : 85 = 52 : 44.2 
 
 Or, 1. : .85 = 52 : 44.2 
 
 Hence, for finding from the table the logarithm ot a 
 number consisting of more than four places of figures, 
 we have the following 
 
 RULE. 
 
 Take from the table the log. of the number expressed by the 
 the four superior figures ; this, with the proper index, is the 
 approximate logarithm. Multiply the number expressed by the 
 remaining figures of the number, regarded as a decimal, by 
 the tabular difference, and the product will be the correction 
 to he added to the approximate log. to obtain the true log 
 
SECTION II. 283 
 
 EXAMPLES. 
 
 1. What is the log. of 357.32514? 
 
 The log. of 357.3 is 2.553033 
 
 No. not included, .2514 
 Tabular diflf., 122 
 
 Prod., 30.6708 ; correction, 81 
 
 log. sought, 2.553064 
 
 The log. of 35732.514 is 4.553064 
 
 " .035732514 " —2.553064. 
 
 2. What IS the log. of 7912532 ? 
 
 Approximate log., 6.898286 
 .532 X 55 = correction, 29 
 
 True log. = 6.898315. 
 
 A logarithm being given, to find its corresponding number. 
 
 For example, what number corresponds to the log. 
 6.898315? 
 
 The index 6 shows that the entire part of the number must con« 
 tain seven places of figures. With the decimal part, .898315, of 
 the log., we turn to the table, and find the next less decimal part 
 to be .898286, which corresponds to the superior places, 7912. 
 
 The difiercnce between the given log. and the one next less is 
 29. This we divide by the tabular difference, 55, because we are 
 working the converse of the preceding problem. Thus, 
 
 29 H- 55 = 52727-f . 
 
 Place the quotient to the right of the four figures before found, 
 And we shall have 7912527.27 for the number sought. 
 
 This example W3.S taken from the preceding case, and 
 the number found should have been 7912532 ; and so it 
 would have been, had we used the true difterence, 29.26, 
 in place of 29. 
 
 When the numbers are larere, as in this example, the 
 
284 PLANE TRIGONOMETRY. 
 
 result is liable to a small error, to avoid which the loga- 
 rithms should contain a great number of decimal places; 
 but the logarithms in our table contain a sufficient num- 
 ber of decimal places for most practical purposes. 
 
 Hence, for finding the number corresponding to a ay 
 given logarithm, we have the following 
 
 RULE. 
 
 Look in the table for the decimal part, of the given lego- 
 rithm, and if not found, take the decimal next less, and take 
 out the four corresponding figures. 
 
 Take the difference between the given log. and the next less 
 in the table ; divide that difference by the tabular difference, 
 and write the quotient on the right of the four superior fig^ 
 ures, and the result is the number sought. 
 
 Point off the whole number required by the given index, 
 
 EXAMPLES. 
 
 1. Given, the logarithm 3.743210, to find its corres- 
 ponding number true to three places of decimals. 
 
 Ans. 5536.177. 
 
 2. Given, the logarithm 2.633356, to find its corres- 
 ponding number true to tw^o places of decimals. 
 
 \,_^S^^ ^ns. 429.89. 
 
 3. Given, the logarithm — 3.291746, to find its corres- 
 ponding number. Ans. .0019577. 
 
 4. What number corresponds to the log. 3.233568 ? 
 
 Ans. 1712.25. 
 \ What is the number of which 1.532708 is the log. T 
 
 Ans. 34.0963. 
 6. Find the number whose log. is 1.067889. 
 
 Ans. 11.692. 
 
 EXPLANATION OF TABLE II. 
 
 Table I is merely a table of numbers and their corres- 
 ponding logarithms, and requires no explanation other 
 
SECTION II. 286 
 
 than that which has been given in connection with the 
 suljject of loganthms. 
 
 Table II, with the exception of the last two oohmnib, 
 which contain natural sines and cosines, is a table in 
 which are arranged the logarithms of the numerical 
 valucc of the several trigonometrical lines corresponding 
 to the different angles in a quadrant. The values of 
 these 'ines are computed to the radius 10,000,000,000, 
 and i^eir logarithms are nothing more than the loga- 
 rithms, each increased by 10, of the natural sines, co- 
 sines, and tangents, of the same angles ; because the 
 values of these lines, for arcs of the same number of de- 
 grees tiiken in different circles, are directly proportional 
 to the radii of the circles. 
 
 The natural sines are made to the radius of unity; 
 and, of course, any particular sine is a decimal fraction, 
 expressed by natural numbers. The logarithm of any 
 natural sine, with its index increased by 10, will give 
 the logarithmic sine. Thus, the natural sine of 3° is 
 .052336. 
 
 The logarithm of this decimal is — 2.718800 
 
 To which add 10. 
 
 The logarithmic sine of 3° is, therefore, 8.718800 
 
 In this manner we may ffnd the logarithmic sine of 
 any other arc, when we have the natural sine of the 
 same arc. 
 
 If the natural sines and logarithmic sines were on the 
 eame radius, the logarithm of the natural sine would be 
 the logarithmic sine, at once, without any increase of 
 the index. 
 
 The radius for the logarithmic sines is arbitrarily 
 tiiken so large that the index of its logarithm is 10. It 
 might have been more or less ; but, by common consent, 
 it is settled at this value; so that the sines of the smallest 
 area over used shall not have a negative index. 
 
286 PLANE TRIGONOMETRY 
 
 In our preceding equations, sin. a, cos. a, etc., rofei 
 to natural sines; and by such equations we determine 
 their values in natural numbers; and these numbers are 
 put in Table II, under the heads of iT. sine and M, cos., 
 as before observed. 
 
 When we have the sine and cosine of an aye, the 
 tangent and cotangent are found by Eq. (3) and (6) ; thus, 
 
 , E sin. ,^. , E COS. 
 
 tan. = ( o ) cot. =- - - — : 
 
 COS. 6.i»:. 
 
 and the secant is found by equation (-^'/j that is, 
 
 E' 
 
 sec. = 
 
 COS. 
 
 For example, the logarithmic sine of 6° is 9.01923^, 
 and its cosine 9.997614. From these it is required to 
 find the logarithmic tangent, cotangent, and secant. 
 
 R sin. 19.019235 
 
 Cos. subtract 9.997614 
 
 Tan. is 
 
 9.021621 
 
 E cos. 
 
 19.997614 
 
 Sin. 
 
 subtract 9.019235 
 
 Cotan. is 
 
 10.978379 
 
 R'is 
 
 20.000000 
 
 Cos. 
 
 subtract 9.997674 
 
 Secant is 10.002326 
 
 The secants and cosecants of arcs are not given in 
 our table, because they are very little used in practice ; 
 and if any particular secant is required, it can be deter- 
 mined by subtracting the cosine from 20; and the cose- 
 cant can be found by subtracting the sine from 20. 
 
 The sine of every degree and minute of the quadrant 
 is given, directly, in the table, commencing at 0°, and 
 extending to 45°, at the head of the table ; and from 45° 
 to 90°, at the bottom of the table, increasing backward. 
 
SECTION II. 2S1 
 
 Tho column having sine at tlie top has cosine at the 
 lK)ttom, and the opposite, because angles read from abo^ e 
 are complementary to those read from below. The differ- 
 ences of consecutive logarithms corresponding to 10" are 
 given for both sine and cosine, but the tangents and cotan- 
 gents have the same column of differences for the reason 
 that log. tan. -flog. cot.=log. R^ and is therefore constant, 
 llence, by just as much as log. tan. increases, log. cot. de- 
 sreases and the converse. 
 
 Ab cosines and cotangents decrease when arcs increase, 
 and increase when arcs decrease, the proportional parts 
 answering to seconds for them must be subtracted. 
 
 Mca?n2?le. Find the sine of 19° 17' 22". 
 
 The sine of IQ** 17', taken directly from the table, is 9.518829 
 
 The difference for 10" is 60.2; for 1" is 6.02; and 
 
 for 6.02 X 22 = 132 
 
 flence, log. sine 19" 17' 22" is 9.518961 
 
 From this it will be perceived that there is no difficulty 
 in obtaining the sine or tangent, cosine or cotangent, of 
 any angle greater than 30'. 
 
 Conversely : Given, the logarithmic sine 9.982412, to 
 find its corresponding arc. The sine next less in the 
 table is 9.982404, which gives the arc 73° 48'. The differ- 
 ence between this and the given sine is 8, and the dif- 
 ference for V is .61 ; therefore, the number of seconds 
 corresponding to 8, must be discovered by dividing 8 by 
 the decimal .01, which gives 13. Hence, the arc sought 
 LS 73° 48' 13". 
 
 These operations are too obvnous to require a rule. 
 SVTien the arc is very small, — and such arcs are sometimes 
 required in Astronomy, — it is necessary to be very accu- 
 rate; fortius reason we omitted the difference for seconds 
 or all arcs under 30'. Assuming that the sines and tan- 
 gents of arcs under 30' vary in the same proportion as 
 tlie arcs themselves, we can find the sine or tangent of 
 any yqij small arc, with ^reat exactness, as follows: 
 
288 PLANE TRIGONOMETRY. 
 
 The sine of 1', as expressed in the table, is 3.463726 
 
 Divide this by 60; that is, subtract logarithm 1.778151 
 
 The logarithmic sine of 1", therefore, is 4.685575 
 
 Now, for the sine of 17", add the logarithm of 17 1.230449. 
 
 Logarithmic sine of 17", is 5.916024 
 
 In the same manner we may find the sine of any other 
 am all arc. 
 For example, find the sine of 14' 21 J"; that is, 861.5". 
 
 The logarithmic sine of 1" is 4.685575 
 
 Add logarithm of 861.5, 2.935254 
 
 Logarithmic sine of 14' 21 r', 7.620829 
 
 Two lines drawn, the one from the surface and the 
 other from the center of the earth, to the center of the 
 sun, make with each other an angle of 8.61". What ia 
 the logarithmic sine of this angle ? 
 
 The log. of the sine 1" is 4.685575 
 
 Log. of 8.61, 0.935003 
 
 Log. sine of sun's horizontal parallax = 5.620578 
 
 GENERAL APPLICATIONS WITH THE USE OF 
 LOGARITHMS. 
 
 L EIGHT-ANGLED TRIGONOMETRY. 
 
 One figure will be sufiicient to represent the triangle 
 lu all of the following examples ; the right ansjle being 
 at B. 
 
 PRACTICAL PROBLEMS. 
 
 1. In a right-angled triangle, ABC, ^ 
 given the base AB, 1214, and the angle 
 4, 51° 40' 30'', to find the other parts. 
 
SECTION II. 289 
 
 To fiud BC. 
 
 Radius, 10.000000 
 
 : tan. ^, 51° 40' 30", 10.102119 
 
 :: AB, 1214, 3.084219 
 
 : BC, 1535.8, 3.186338 
 
 Remark. — AVhen the first term of a logarithmic proportion is rad/UB, 
 fche required logarithm is found by adding the second and third loga- 
 rithms, rejecting 10 in the index, which is dividing by the first term. 
 
 In all cases we add the second and third logarithms together ; which, 
 in logarithms, is multiplying these terms together ; and from that sum 
 we subtract the first logarithm, whatever it may be, which is dividing 
 by the first term. 
 
 To find ^ a 
 
 Sin. (7, or cos. A, 51° 40' 30", 9.792477 
 
 : AB, 1214, 3.084219 
 
 .: Radius, 10.000000 
 
 : AC, 1957.7, 3.291742 
 
 To find this resulting logarithm, we subtracted the first logarithm 
 from the second, conceiving its index to be 13. 
 
 Let ABO represent any plane triangle, right-angled 
 at ^. 
 
 2. Given, AC 73.26, and the angle A, 49° 12' 20"; 
 required the other parts. 
 
 Ans. The angle C, 40° 47' 40" ; BC, 55.46 ; and AB, 47.86. 
 
 3. Given, AB 469.84, and the angle A, 51° 26' 17", to 
 9 ad the other parts. 
 
 Ans. The angle C, 38° 88' 43";' BC, 588.7 ; and ^(7, 752.9. 
 
 4. Given, AB 498, and the angle C, 20° 14' ; required, 
 the remaining parts. 
 
 Ans. The angle A, 69° 46'; BC, 1888 ; and AC, 1425.5. 
 
 5. Let AB = 831, and the angle A = 49° 14' ; what are 
 the other parts? 
 
 Ans. AC, 506.9; BC, 388.9; and the angle C, 40° 46'. 
 
 6. If ^(7=45, and the angle (7=87° 22', what are the 
 remaining parts ? 
 
 Ans. AB, 27.81; 5(7, 35.76; and the angle ^, 52° 38 
 25 T 
 
290 
 
 PLANE TEIGOI^OMETRT. 
 
 7. Given, ^(7=4264.3, and the angle J. = 56° 29' 13", 
 to find the remaining parts. 
 
 Ans.AB, 2354.4 ;5C, 3555.4; and the angle (7, 33° 30'47". 
 
 8. If 4^ = 42.2, and the angle A = 31° 12' 49'', what 
 are the other parts ? 
 
 Ans, A 0, 49.34 ; BO, 25.57 ; and the angle 0, 58° 47' 11". 
 •j. It AB = 8372.1, and BO ^ 694.73, what are the 
 :>ther pai-ts? 
 
 iAC, 8400.9 ; the angle C, 85'' 15' 23" ; and the 
 ^''^^- } angle A, 4^ 44' 37". 
 
 10. If AB be 63.4, and AC he 85.72, what are the 
 other parts ? 
 
 i BO, 57.69 ; the angle C, 47° 41' 56' ; and the 
 ^^'^- ( angle ^, 42° 18' 4". 
 
 11. Given, AQ == 7269, and ^^ = 3162, to find the 
 other parts. 
 
 . f ^(7, 6545 ; the angle 0, 25° 47' 7" ; and the 
 '^^' I angle ^, 64° 12' 53". 
 
 12. Given, AO =^ 4824, and BO r= 2412, to find the 
 other parts. 
 
 . r The angle ^ == 30° 00', the angle = 60° 00', 
 ^^' \ and AB = 4178. 
 
 13. The distance between the earth and sun is 91,500,000 
 miles, and at that distani^.e the semi-diameter of the snn 
 subtends an angle of 16'. What is the diameter of the 
 sun in miles? ' Ans. 887,674. 
 
 In this example, let B be the center of the earth, aS^ that of the 
 Bun, and BB a tangent to the sun's surface. Then the A BBJS 
 is right-angled at B, and BS is the semi-diameter of the sun. The 
 «^lue of 2BS is required. 
 
SECTION II. 291 
 
 14. The equatorial diameter of the earth is 7925 miles, 
 and the distance of the sun 91,500,000 miles. What angle 
 will the semi-diameter of the earth subtend, as seen from 
 the sun ? Ans. 8.94". 
 
 This angle is called, in astronomy, the sun's horizontal parallax. 
 The preceding figure applies to this example, by supposing E to 
 be the center of the sun, S that of the earth, and BS equal to 
 3956 miles. 
 
 15. The mean distance of the moon frv>m the earth is 
 60.3 times 3960 miles, and at this distance the semi- 
 diameter of the moon subtends an angle of 15' 32*'. 
 VVhat is the diameter of the moon in miles ? 
 
 /-v ^ns. 2157.8 miles. 
 
 n. OBLIQUE-ANGLED TRIGONOMETRY. 
 
 PROBLEM I. 
 
 Irt a plane triangle, given a side and the two adjacent 
 angles, to find the other parts. 
 
 In the triangle ABC, let AB = ^ 
 
 376, the angle A = 48° 3', and the 
 angle B = 40° 14', to find the other 
 parts. 
 
 As the sum of the three angles of every ^ ^ 
 
 triangle is always 180^, the third angle, C, must be 180° — 88® 
 17' = 91° 43'. 
 
 To find AC. 
 
 Sin. 91° 43', 9.999805 
 
 : ^5.376, 2.575188 
 
 :: sin.B 4r° U, 9.810167 
 
 12.385355 
 
 : JC, 243, 2.385550 
 
 Observe, that thb sine of 91° 43' is the same as the cosine of 
 1°43' 
 
21^2 PLANE TRIGONOMETRY. 
 
 To find BC. 
 
 Sin. 91° 43', 9.999805 
 
 : ul5, 376, 2.575188 
 
 : : sin. A, 48° 3', 9.871414 
 
 12.446602 
 
 : sin. 5(7, 279.8, 2.446797 
 
 PROBLEM 11. 
 
 In a plane triangle, given two sides and an angle opposite 
 one of them, to determine the other parts. 
 
 Let AD = 1751 feet, one 
 of the given sides ; the angle 
 D = 31° 17' 19" ; and the side 
 opposite, 1257.5. From these 
 data, we are required to find 
 the other side and the other 
 two angles. 
 
 In this case we do not know whether AG or AE representa 
 1257.5, because J.(7 = AE. If we take AG for the other given 
 side, then DG\^ the other required side, and DA G is the vertical 
 angle. If we take AE for the other given side, then DE is the 
 required side, and DAE is the vertical angle. In such cases we 
 determine both triangles. 
 
 To find the angle E ^ C. 
 (Prop. 4.) AG= AE ^Vlbl.b, log. 3.099508 
 
 : i), 31° 17' 19", sin. 9.715460 
 
 :: AD, 1751, log. 3.243286 
 
 12.958746 
 
 jy== C,46° 18', sin. 9.859238 
 
 From 180° take 46° 18', and the remainder is the angle DUA 
 : 133° 42'. 
 
 The angle DAG =^ AGE — D, (Th. 11, B. I); 
 that is, DAG = 46° 18' — 31° 17' 19" = 15° C iV\ 
 The angles D and E, taken from 180°, give 
 DAE == 102° 24' 41". 
 
SECTION II. 2i>3 
 
 To find DQ. 
 
 Sin. Z), 31° 17' 19", log. 9.715460 
 
 : At\ 1257.5, log. 3.099508 
 
 :: ^m.DAC 15<» 0' 41", log. 0.413317 
 
 
 12.512825 
 
 • Z>a, 626.86, 
 
 2.797165 
 
 To find BE. 
 
 
 Sin.A31°iri7", 
 
 9.715460 
 
 , : AE, 1257.5, 
 
 3.099508 
 
 :: sXn.DAEy 102° 24' 41", 
 
 9.989730 
 
 
 13.089238 
 
 ! VE, 2364.7, 3.373778 
 
 IIevark. — To make the triangle possible, ^(7 must not be less than 
 AB the sine of the angle Z>, when DA i» made radius. 
 
 PROBLEM III. 
 
 In any plane triangle^ given two sides and the included 
 angle, to find the other parts. 
 
 Let AB = 1751, (see last figure), BU = 2364.5, and 
 the included angle B — 31° 17' 19". "We are required 
 to find AEy the angle BAE, and the angle E. 
 
 Observe that the angle E must be less than the angle BAE, bo- 
 cause it is opposite a less ftide. 
 
 From 180° 
 
 Take /), 31° 17' 19", 
 
 Sum of the other two angles, = 148° 42' 41", (Th. 11, B. I), 
 J sum = 74° 21' 20". 
 
 By Proposition 7, 
 DE+BA: DE— BA -= tan. 74° 21' 20" : tarn, i(BAE^E\ 
 That is, 
 
 4115.5 ; 613.5 « tan. 74° 21' 20" : taiL.l{BAE—E) 
 25* 
 
294 
 
 PLANE TRIGONOMETRY. 
 
 Tan. 74° 21' 20'', 
 613.5. 
 
 4115.5 log. (subtracted), 
 
 10.552778 
 
 2.787815 
 
 13.340593 
 3.614423 
 
 tan.K^^^-^) tan.28° V 36", 9.726170 
 
 15 ut the half sum plus the half difference of any two quantities 
 h equal to the greater of the two; and the half sum minus the 
 half difference is equal the less. 
 
 Therefore, to 
 Add 
 
 DAE =. 
 E = 
 
 74° 21' 20", 
 28° r36", 
 
 102° 22' 56", 
 46° 19' 45", 
 
 To find AE, 
 
 Sin. E, 46° 19' 45", 
 : DA, 1751, 
 :: sin.A 31° 17' 19", 
 
 AE, 1257.2, 
 
 9.859323 
 3.243286 
 9.715460 
 
 12.958746 
 
 3.099423 
 
 PROBLEM IV. 
 Given, the three sides of a plane triangle, to find the angle$. 
 
 Let ^(7 = 1751, CB = 1257.5, AB = 2364.5, to find 
 tlie angles A, B, and 0. 
 
 II' we take the formula for cosines, we C 
 
 will compute the greatest angle, which is 
 0. To correspond with the formula, 
 
 cos. i C 
 
 = n/ 
 
 Rh (s — c) 
 
 a6 ' A B 
 
 we must take a =-- 1257.5, h = 1751, and c = 2364.6. 
 The half sum of these is, 
 
 s = 2686.5; and s — c^ 322. 
 
SECTION 
 
 II. 
 
 /?« 
 
 20.000000 
 
 8 = 2686.5 
 
 3.429187 
 
 « — c = 322 
 
 2.507856 
 
 Numerator, log. 
 
 25.937043 
 
 a 1257.5 3.099508 
 
 
 h 1751. 3.243286 
 
 
 Denominator, log. 6.342794 6.342794 
 
 2 )19.594249 
 
 K'= 51° ir 10" COS. 9.797124 
 C= 102 22 20 
 
 Tlie remaining angles may now be found by Problem 4. 
 
 PRACTICAL PROBLEMS. 
 
 Let ABC represent any oblique-angled triangle. 
 1. Given, AB 697, the angle A 81° 30' 10'', and the 
 adgle B 40° 30' 44", to find the other parts. 
 
 Ans, AC, 534; BC, 813; and L^, 57° 59' 6". 
 •J. K AC ^nO.S, l_A = 70° 5' 22", L^ = 59° 35' 
 36", required the other parts. 
 
 Ans. AB, 643.2; BC, 785.8; and [_C, 50° 19' 2". 
 
 3. Given, BC 980.1, the angle A 7° 6' 26", and the 
 angle B 108° 2' 23", to find the other parts. 
 
 Ans. AB, 7283.8; AC, 7613.1 ; and L^, 66° 51' 11". 
 
 4. Given, AB 896.2, .5(7 328.4, and the angle (7113* 
 15' 20", to find the other parts. 
 
 .(AC, 112; L-i,19°35'46"> 
 ' I and L^, 46° 38' 54".' 
 
 5. Given, ^(7= 4627, BC= 5169, and the angle A =- 
 Y0° 25' 12", to find the other parts. 
 
 . 1^5,4828; L^, 57° 29' 56"; 
 • I and L^, 52° 4' 52". 
 
296 PLANE TKIGUi\OMETE"k. 
 
 6. Given, AB 793.8, BO 481.6, and AC 500.0, to lina 
 the angles. 
 
 , ( l_A, 35° 15' 32''; L-^, 36° 49' 18"; and L^ 
 * I 107° 55' 10". 
 
 7. Given, ^^ 100.3, BO 100.3, and ^(7 100.3, lo find 
 the angles. 
 
 . f riie angle ^, 60°; the angle B, 60°; and the 
 ^ "" * I angle 6^, 60°. 
 
 8. Given, AB 92.6, BO 46.3, and ^6' 71.2, to lind thf 
 angles. 
 
 ,^ fL^, 29°17'22"; L^, 48° 47' 30"; and L^, 
 ^""'•l 101° 55' 8". 
 
 9. Given, AB 4963, BO 5124, and AO 5621, to find 
 the angles. 
 
 . /L-4, 57° 30'28"; L_-^, 67° 42' 36"; and L^, 
 ^'''- 1 54° 46' 55". 
 
 10. Given, AB 728.1, BO 614.7, and ^C 583.8, to find 
 the angles. 
 
 . f L^ = 54° 32' 52", L^= 50° 40' 58", and l_0 
 ^^' \ = 74° 46' 10". 
 
 11. Given, AB 96.74, BO 83.29, and AO 111.42, to 
 find the angles. 
 
 A j L^ = 46° 30' 45", L^ = 76° 3' 46", and L^ 
 ^''''l =57° 25' 29". 
 ^ 12. Given, AB 363.4, BO USA, and the angle B 102° 
 18' 27", to find the other parts. 
 
 . j\__A = 20° 9' 17", the side A0 = 420.8, and [_0 
 -^""''X =57° 32' 16". 
 
 1.3. Given, AB 632, BO 494, and the angle A 20° 13', 
 to find the oth^r parts, the angle (7 being acute. 
 
 r L^= 26° 18' 19", L ^ = 133° 25' 41", and 
 ^^'•\ AO=10S5X 
 14. Given, AB 53.9, AO 46.21, and the angle B 58** 
 16', to find the other parts. 
 Ans. \_A = 38° 58', l_0= 82° 46', and B0-= 34.16. 
 
isECTlUiX II. 297 
 
 15. Given, AB 2163, BC 1672, and the angle C 112° 
 18' 22", to find the other parts. 
 
 Ans. AC, 877.2; L^^, 22° 2' 16"; and L^, 45° 39' 22". 
 
 16. Given, AB 496, BO 496, and the angle B 38° 16', 
 to find the other parts. 
 
 Ans, AO, 325.1; L^, 70° 52'; and L^, 70° 52'. 
 
 17. Given, AB 428, the angle 49° 16', and {AC + 
 BC) 918, to find the other parts, the angle B being 
 obtuse. 
 
 . r The angle A = 38° 44' 48", the angle B = 91° 
 ' ^*l 59' 12", ^(7=564.5, and ^(7=353.5. 
 
 18. Given, AC 126, the angle B 29° 46', and {AB-^ 
 ^(7) 43, to find the other parts. 
 
 . JThe angle ^ = 55° 51' 32", the angle (7=94° 
 ^' \ 22' 28", AB = 253.05, and ^C = 210.05. 
 
 19. Given, AB 1269, AC 1837, and the angle A 53*» 
 16' 20", to find the other parts. 
 
 A f LJ^ = 83° 23' 47", I (7= 43° 19' 53", and BC 
 \ « 1482.16. 
 
PLANE TKIUONOMETRY. 
 
 SECTION III. 
 
 A.PI LIGATION OF TRIGONOMETRY TO MEASURING 
 HEIGHTS AND DISTANCES. 
 
 In tliis useful application of Trigonometry, a base line 
 Is always supposed to be measured, or given in length; 
 and by means of a qn ad rant, sextant, circle, theodolite, 
 or some other instrument for measuring angles, such 
 angles are measured as, connected with the base line and 
 the objects whose heights or distances it is proposed to 
 determine, enable us to compute, from the principles of 
 Trigonometry, what those heights or distances are. 
 
 Sometimes, particularly in marine surveying, horizontal 
 angles are determined by the compass ; but the varying 
 effect of surrounding bodies on the needle, even in situa- 
 tions little removed Irom each other, and the general 
 construction of the instrument itself, render it unfit to be 
 employed in the determination of angles where anything 
 like precision is required. 
 
 The following problems present sufficient variety, to 
 guide the student in determining what will be the most 
 eligible mode of proceeding, in any case that is likely to 
 occur in practice. 
 
 PROBLEM I. 
 
 Being desirous of finding the distance between two 
 listant objects, Q and Z), I measured a base, AB, of 384 
 7ards, on the same horizontal plane with the objects 
 
SECTION III. 299 
 
 and D, At A, I found the angles DAB = 48<* 12', and 
 CM^ = 89° 18'; at B, the angles ABC 46° 14', and 
 ABB 87° 4'. It is required, from these data, to com- 
 pute the distance between and D, 
 
 From the angle CAB, take the angle DAB ; the 
 remainder, 41° 6*, is the angle CAD. To the angle 
 DBAj add the angle DAB, and 44° 44', the supple- 
 ment of the sum, is the angle ADB. In the same 
 way the angle ACB, which is the supplement of 
 the sum of CAB and CBA, is found to be 44° 28'. 
 
 Hence, in the triangles ABC and ABD, we have 
 
 Sin. ACB, 44° 28', 
 : AB, 384 yards, 
 :: Bin. ABCy 46° 14', 
 
 9.845405 
 2.584331 
 
 9.858635 
 
 
 12.442966 
 
 : ACy ^95.9 yards, 
 
 2.597561 
 
 Sin. ADB, 44° 44', 
 : AB, 384 yards, 
 :: sm.ABD, 87° 4', 
 
 9.847454 
 2.584331 
 9.999431 
 
 
 12.583762 
 
 : AD, 544.9 yards. 
 
 2.736308 
 
 Then, in the triangle CAD, we have given the sides GA and AD^ 
 and the included angle CAD, to find CD) to compute which we 
 proceed thus : 
 
 The supplement of the angle CAD, is the sum of the angles 
 ACD and^Z)C; 
 
 Hence, = 69° 27'; and, by proportion w« have, 
 
 AD-^-AG (=- 940.8) 2.973497 
 
 : AD^AC (= 149) 2.173186 
 
 :; faip /^^ + ^^^ (= 69° 27') 10.426108 
 
 12.599294 
 
300 
 
 tan. 
 
 PLANE TRIGONOMETRY. 
 ACD—ADG 
 
 2 
 
 (= 22° 54') 9.625797 
 
 the angle A CD, sum, 92" 21' 
 the angle ^2) C, diff., 46° 33' 
 
 Sm.J.i)(7, 46°83', • 
 
 : ^(7, 395.9 yards, 
 : : sin. CAD, 41° 6', 
 
 CD, 358.5 yards, 
 
 9.860922 
 2.597585 
 9.817813 
 
 12.415398 
 
 2.55447B 
 
 PROBLEM II. 
 
 To determine the altitude of a lighthouse, I obsei-vtJ 
 the elevation of its top above the level sand on the sea- 
 shore, to be 15° 32' 18'' ; and measuring directly from 
 it, 638 yards along the sand, I then found its elevation 
 to be 9° 56' 26". Required the height of the lighthouse 
 
 Let CD represent the height of the light- 
 house above the level of the sand, and let B 
 be the first station, and A the second ; then 
 the angle CBD is 15° 32' 18, and the angle 
 CAB is 9° 56' 26"; therefore, the angle 
 A CB, which is the diflference of the angles 
 CBD and CAB, is 5° 35' 52". 
 
 Hence, Sin. ^ CjB, 5° 35' 52", 
 
 : AB, 638, 
 : : sin. angle A, 9° 56' 26", 
 
 8.989201 
 2.804821 
 9.237107 
 
 BCj 1129.09 yards, 
 
 Radius, 
 
 BC, 1129.09, 
 
 sin. CBD, 15° 32' 18", 
 
 /)C; 302.46 yards. 
 
 12 041928 
 3.052727 
 
 10.000000 
 3.052727 
 9.427945 
 
 12.480672 
 
 2.480672 
 
SECTION III 
 
 301 
 
 PROBLEM III. 
 
 Coming from sea, at the poiut D I observed two 
 headlands, A and B, and inland, at (7, a steeple, whicb 
 appeared between the headlands. I found, from a map, 
 that the headlands were 5.35 miles apart ; that the dis- 
 tance from A to the steeple was 2.8 miles, and from 
 B to the steeple 3.47 miles ; and I found, with a sextant, 
 that the angle J.i>(7 was 12° 15, and the angle BDQ, 15° 
 30'. Required my distance from each of the headlands, 
 and from the steeple. 
 
 CONSTRUCTION. 
 
 The angle between the two headlands is 
 the sum of 15° 30' and 12° 15', or 27° 45'. 
 Take double this sum, 55° 30'. Conceive AB 
 to be the chord of a circle, and the arc on 
 one side of it to be 55° 30' ; and, of course, 
 the other will be 304° 30'. The point D 
 will be somewhere in the circumference of 
 this circle. Consider that point as determined, and draw CD. 
 
 In the triangle ABC, we have all the sides, and, of course, we 
 can find all the angles ; and if the angle A CB is less than 180°— 
 27° 45' = 152° 15', then the circle cuts the line CD in a point 
 E, and C is without the circle. 
 
 Draw AEy BE, AD, and BD. AEBD is a quadrilateral in a 
 circle, and [_ AEB + \_ ADD = 180°. 
 
 The [^ADE= the | -4^^, because both are measured by one 
 
 half the arc AE. Also, [_ EDB = [__ EAB, for a similar reason. 
 
 Now, in the triangle AEB, its side AB, and all its angles, are 
 known; and from thence AE can be computed. Then, having the 
 two sides, AC and AE, of the triangle AEC, and the included 
 angle CAE, we can find the angle AEC, and, of course, its supple- 
 ment, AED. Then, in the triangle AED, we have the side AEy 
 and the two angles AED and ADE, from which we can find AD 
 
 The computation, at length, is as follows : 
 26 
 
302 
 
 PLANE TRIGONOMiCTRY. 
 
 To find AE. 
 
 Angle EaB = 15<^ 30' Sin. AEB, 152° 15', 9.668027 
 
 Angle fy^^ = 12° 15' : AB, 5.35, .728354 
 
 27° 45' : : sin. ABE 12° 15' 9.326700 
 
 180° 
 
 AP«5l« AEB = 152° 15' 
 
 AEy 2.438, 
 
 To find the angle BAQ. 
 
 10.055054 
 
 .387027 
 
 BG, 3.47 
 AB, 5.35 
 
 AGy 2.80 
 
 log. .728354 
 log. .447158 
 
 2 ) 11.62 
 
 1.175512 
 
 5.81 
 ^C, 2.34 
 
 log. .764176 
 log. .369216 
 20 
 
 
 21.133392 
 
 17° 41' 58" 
 
 2 ) 19.957880 
 COS. 9.978940 
 
 2 
 
 
 Angle ^^C=.- 35° 23' 56" 
 Angle EAB = 15° 30' 
 
 
 Angle EAG^ 19° 53' 56" 
 
 180° 
 
 
 2)160° 6' 4" 
 80° 3' 2" 
 
 AEG 4- AGE 
 
SECTION III 
 
 803 
 
 To find the angles AEQ and ACE, 
 
 %ng\QAEC, 
 
 AC -\- AE 
 
 : AC^AE 
 
 ^ AEC -ir AGE 
 ::tan. ^ 
 
 6.238 
 .362 
 
 80<» 3' 2" 
 21° 30' 12" 
 
 .719165 
 — 1.558709 
 
 10.755928 
 
 
 10.314637 
 
 , AEC—ACE 
 : ten. 2 
 
 9.595472 
 
 7(7, 10P33a4^^8uin 
 
 
 
 vngXeACEorACD, 58°32^50^^ diff. 
 MiiileCDA, 12° 15' 
 
 70°47'50'^ supplement 109°12a0'^ angle CAD 
 - • 35° 23^ 56'^ angle CAB 
 
 . 
 
 73°48a4'^ 
 
 To find AB. 
 
 Sin.^Z>^, 12°15', 
 : ^(7,2.8, 
 :: Bm.ACD 68° 32' 50", 
 
 9.326700 
 
 .447158 
 
 9.930985 
 
 
 10.378143 
 
 • AD 11.26 miles. 
 
 1.051443 
 
 PROBLEM IV. 
 
 The elevation of a spire at one station was 23° 50' 17 ', 
 vid the horizontal angle at this station, between the 
 spire and another station, was 93° 4' 20". The horizon- 
 tal angle at the latter station, between the spire and the 
 first station, was 54° 28' 36", and the distance between 
 the two stations was 416 feet. Required the height ot 
 the spire. 
 
304 
 
 PLANE TRIGONOMETRY. 
 
 Let CD be the spire, A the first station, and 
 B the second ] then the vertical angle CAD is y"^ 
 
 23° 50' 17"; and as the horizontal angles, CAB ^ .-^^^ / 
 and CBA, are 93° 4' 20" and 54° 28' 36", re- \ / 
 spectively, the angle ACB, the supplement of 
 their sura, is 32° 27' 4". 
 
 To find AC. 
 Sin. 5(7^, 32° 27' 3", 
 : side AB, 416, 
 :: sin.^jBa, 54°28'36", 
 
 side^C, 681, 
 
 9.729G34 
 2.619093 
 9.910560 
 
 12.529653 
 
 2.800019 
 
 To find DO. 
 
 Radius, 10.000000 
 
 side AC, 631, 2.800019 
 
 tan. DA (7, 23° 50' 17", 9.645270 
 
 DC, 278.8, 
 
 2.445289 
 
 By the application of Pro- 
 blem 4, we can compute the 
 distance between two horizon- 
 tal planes, if the same object 
 is visible from both. a^ 
 
 For example, let ilf be a 
 prominent tree or rock near 
 
 the top of a mountain, and by observations taken at A^ 
 we can determine the perpendicular Mn, By like obser* 
 vations taken at B, we can determine the perpendicular 
 Mm, The difierence between these two perpendiculars is 
 nm, or the difference in the elevation between the two 
 points A and B, If the distances between A and n, or B 
 and m, are considerable, or more than two or three miles, 
 corrections must be made for the convexity of the earth ; 
 but for less distances such corrections are not necessaiy. 
 
SECTION III. 306 
 
 PRACTICAL PROBLEMS. 
 
 1. Required the height of a wall whose augle of eleva- 
 tion, at the distance of 463 feet, is observed to be 16° 
 21'. Ans. 135.8 feet. 
 
 2. The angle of elevation of a hill is, near its bottom, 
 81° 18', and 214 yards farther off, 26° 18'. Required the 
 peq-jendieular lieight of the hill, and the distance of the 
 perpendicular from the first station. 
 
 ( The height of the hill is 565.2 yards, and the 
 ^ns. < distance of the perpendicular from the first 
 I station is 929.6 yards. 
 
 3. The wall of a tower which is 149.5 feet in height, 
 makes, with a line drawn from the top of it to a distant 
 object on the horizontal plane, an angle of 57° 21'. 
 What is the distance of the object from the bottom of 
 the tower? Ans. 233.3 feet. 
 
 4. From the top of a tower, which is 138 feet in height, 
 I took the angle of depression of two objects standing 
 in a direct line from the bottom of the tower, and upon 
 the same horizontal plane with it. The depression of the 
 nearer object was found to be 48° 10', and that of the 
 further, 18° 52'. What was the distance of each from 
 the bottom of the tower ? 
 
 4 f Distance of the nearer, 123.5 feet; and of the 
 \ further, 403.8 feet. 
 
 5. Being on the side of a river, and wishing to know 
 the distance of a house on the opposite side, I measured 
 312 yards in a right line by the side of the river, and then 
 /bund that the two angles, one at each end of this line, 
 subtended by the other end and the house, were 31° 15' 
 and 86° 27'. What vas the distance between each end 
 of the line and the house ? Ans, 351.7, and 182.8 yards. 
 
 6. Having measured a base of 260 yards in a straight 
 line, on one bank of a river, I found that the two 
 angles, one at each end of the line, subtended by the 
 
 26* U 
 
goo PLANE TKIGONOMETRY. 
 
 other end and a tree on the opposite bank, were 40® and 
 80°. What was the width of the river? 
 
 Ans. 190.1 yards. 
 
 7. From an eminence of 268 feet in perpendicular 
 height, the angle of depression of the top of a steeple 
 which stood on the same horizontal plane, was found to 
 1 e 40^ 8', and of the bottom, 06° 18'. What was the 
 height of the steeple ? Ans. 117.76 feet. 
 
 8. Wanting to know the distance between two objects 
 which were separated by a morass, I measured the dis- 
 tance from each to a point from whence both could be 
 seen ; the distances were 1840 and 1428 yards, and the 
 angle which, at that point, the objects subtended, was 36** 
 18' 24''. Required their distance. J.n«. 1090.85 yards. 
 
 9. From the top of a mountain, three miles in height, 
 the visible horizon appeared depressed 2° 13' 27". Re- 
 quired the diameter of the earth, and the distance of the 
 boundary of the visible horizon. 
 
 . . r Diameter of the earth, 7958 miles ; distance of 
 \ the horizon, 154.54 miles. 
 
 10. From a ship a headland was seen, bearing north 
 39° 23' east. After sailing 20 miles north, 47° 49' west, 
 the same headland was observed to bear north, 87° 11' 
 east. Required the distance of the headland from the 
 ship at each station. 
 
 J J At first station, 19.09 miles ; at the second, 
 ^''*- I 26.96 miles. 
 
 11. The top of a tower, 100 feet above the level of tlie 
 sea, was seen as on the surface of the sea, from the mast- 
 head of a ship, 90 feet above the water. The diameter 
 of the earth being '''960 miles, what was the distance 
 between the observei and the object? 
 
 Ans. 23.92 plus j\ for refraction — 25.76 miles. 
 
 12. From the top of a tower, by the seaside, 143 feet 
 high, it was observed that the angle of depression of a 
 
SECTION III. 307 
 
 ship's bo'tom, then at anchor, measured 35® ; what, then, 
 was the ".hip's distance from the foot of the tower ? 
 
 Ans. 204.22 feet. 
 
 13. Wanting to know tlie breadth of a river, I meas- 
 ured a base of 500 yards in a straight line on one bank; 
 and at each end of this line I found the angles subtended 
 l)y the other end and a tree on the opposite bank of the 
 river, to be 53° and 79° 12'. What, then, was the per- 
 pendicular breadth of the river? Ans. 529.48 yards. 
 
 14. What is the perpendicular height of a hill, ita 
 augle of elevation, taken at the bottom of it, being 46°, 
 and 200 yards further off, on a level with the bottom, 
 81° ? Ans. 286.28 yards. 
 
 15. Wanting to know the height of an inaccessible 
 tower, at the least accessible distance from it, on the 
 same horizontal plane, I found its angle of elevation to 
 be 58° ; then going 300 feet directly from it, I found the 
 angle there to be only 32° ; required the height of the 
 tower, and my distance from it at the first station. 
 
 307.54 feet. 
 
 f Height, 
 \ Distance, 
 
 ^ns ^32.18 " 
 
 16. Two ships of war, intending to cannonade a fort, 
 
 are, by the shallowness of the water, kept so far from it, 
 
 that they suspect their guns cannot reach it with effect. 
 
 In order, therefore, to measure the distance, they separate 
 
 fi'om each other a quarter of a mile, or 440 yards, and then 
 
 each ship observes and measures the angle wliich the 
 
 other ship and fort subtends ; these angles are 83° 45', 
 
 and 85° 15'. What, then, is the distance between each 
 
 ship and the fort? . / 2292.26 yards. 
 
 ^ns. I 2298.05 " 
 
 17. A point of land was observed by a ship, at sea, to 
 bear east-by-south ;* and after sailing north-east 12 miles, 
 
 * That is, one point south of east. A point of the compass ii 
 
308 PLANE TRIGONOMETRY. 
 
 it was found to bear south- cast-by- east. It is required to 
 determiue the place of that headland, and the ship's dis- 
 tance from it at the last observation. 
 
 Ans, Distance, 26.0728 miles. 
 
 18. Wishing to know my distance from an inaccessible 
 object, 0, on the opposite side of a river, and having 
 a chain or chord for measuring distances, but no instru- 
 ment for taking angles; from each of two stations, A 
 and B, which were taken at 500 yards asunder, I meas- 
 ured in a direct line from the object, 0, 100 yards, viz., 
 A.0 and BB, each equal to 100 yards; and I found that 
 the diagonal AB measured 550 yards, and the diagonal 
 BO 560. What, then, was the distance of the object 
 from each station A and Bt . ( AO, 536.27 yards. 
 
 ^^^' \J50, 500.14 " 
 
 19. A navigator found, by observation, that the summit 
 of a certain mountain, which he supposed to be 45 min- 
 utes of a degree distant, had an altitude above the sea 
 horizon of 31' 20". Now, on the supposition that the 
 earth's radius is 3956 miles, and the observer's dip was 
 4' 15", what was the height of the mountain ? 
 
 Ans. 3960 feet. 
 
 Remark. — This should be diminished by about one eleventh 
 part of itself, for the influence of horizontal refraction. 
 
 20. From two ships, A and B, which are anchored in 
 a bay, two objects, (7 and i), on the shore, can be seen. 
 These objects are known to be 500 yards apart. At the 
 ship A, the angle subtended by the objects was measured, 
 and found to be 41° 25' ; and that by the object B and 
 the other ship was found to be 52° 12'. At the other 
 ship, the angle subtended by the objects on shore was 
 found to be 48° 10'; and that by the object (7, and the 
 ship A, to be 47° 40'. Eequired the distance between 
 
SECTION III. 309 
 
 tJie ships, and the distance from each ship to the objecta 
 
 on shore. 
 
 {Distance between sliips, 395.7 yards. 
 From ship A to object i), 743.5 " 
 From ship A to object 0, 407.7 " 
 From ship B to object D, 590.5 " 
 
 To solve this problem, suppose the distance between the ships ta 
 be 100 yards, and determine the several distances, including the 
 distance between the objects, C and />, under this supposition; then 
 multiply the values thus found for the required distances by the 
 quotient obtained by dividing the given value of CD by the com- 
 puted value. 
 
PART II. 
 
 SPHEEICAL GEOMETRY 
 
 AND 
 
 TRIGONOMETRY. 
 
 SECTION I. 
 
 SPHERICAL GEOMETRY. 
 
 DEFINITIONS. 
 
 1. Spherical Geometry has for its object the investiga- 
 tion of the properties, and of the relations to each other, 
 of the portions of the surface of a sphere which are 
 bounded by the arcs of its great circles. 
 
 2. A Spherical Polygon is a portion of the surface of a 
 sphere bounded by three or more arcs of great circles, called 
 the sides of the polygon. 
 
 3. The Angles of a spherical polygon are the angles 
 formed by the bounding arcs, and are the same as the 
 angles formed by the planes of these arcs. 
 
 4. A Spherical Triangle is a spherical polygon having 
 hut three sides, each of which is less than a semi-circum- 
 ference. 
 
 5. A Lune is a portion of the surface of a sphere in- 
 chided between two great semi-circumferences having a 
 common diameter. 
 
 6. A Spherical Wedge, or Ungula, is a portion of the 
 solid sphere included between two great semi-circles having 
 a common diameter. 
 
feECTION I. 311 
 
 7. A Spherical Pyramid is a portion of a sphere bounded 
 by the faces of a solid angle having its vertex at the 
 center, and the spherical polygon which these faces inter- 
 cept on the surface. This spherical polygon is called the 
 base of the pyramid. 
 
 8. The Axis of a great circle of a sphere is that diameter 
 of the sphere which is perpendicular to the plane of the 
 circle. This diameter is also the axis of all small circles 
 parallel to the great circle. 
 
 9. A Pole of a circle of a sphere is a point on the sur- 
 face of the sphere equally distant from every point in the 
 circumference of the circle. 
 
 10. Supplemental, or Polar Triangles, are two triangles on 
 a sphere, so related that the vertices of the angles of 
 cither triangle are the poles of the sides of the other 
 
 PROPOSITION I. 
 
 Ani/ two sides of a spherical triangle are together greater 
 than the third side. 
 
 Let AB, AC, and BO, be the three 
 Bides of the triangle, and 2) the center 
 of the sphere. 
 
 The angles of the planes that form 
 the solid angle at D, are measured by 
 the arcs AB, AC, awdBC. But any 
 two of these angles are together greater 
 than the third angle, (Th. 18, B. VI). There lore, any two 
 Hides of the triangle are, together, greater thau the third side. 
 
 Hence the proposition. 
 
 PROPOSITION II. 
 
 TJie sum of the three sides of any spherical triingle is le9% 
 than the circumference of a great circle. 
 
 Let ABC be a spherical triangle ; the two si.les, AB 
 and AC, produced, will meet at the point which is liarae- 
 tncaUy opposite to A, and the arcs, ABB an-' ACD are 
 
312 
 
 SPHERICAL GEOMETRY. 
 
 together equal to a great circle. But, 
 by the last proposition, 5(7 is less 
 than the two arcs, BD and D C. There- 
 fore, AB -\- BQ + AO, is less than 
 ABD + AOB'y that is, less than a 
 gi-eat circle. 
 
 Hence the proposition. 
 
 PROPOSITION III. 
 
 The extremities of the axis of a great circle of a sphere 
 are the poles of the great circle, and these points are also 
 the poles of all small circles parallel to the great circle. 
 
 Let be the center of 
 the sphere, and BB the 
 axis of the great circle. 
 Cm Am''-, then will 5 and 
 2>, the extremities of the 
 axis, be the poles of the 
 circle, and also the poles 
 of any parallel small cir- 
 cle, as FnE, 
 
 For, since BB is per- 
 pendicular to the plane 
 of the circle. 
 
 Cm Am", it 
 
 is perpendicular to the lines OA, Om\ Om", etc., passing 
 through its foot in the plane, (Def. 2, B. VI); hence, all 
 the arcs, Bm, Bm\ etc., are quadrants, as are also the 
 arcs Bm, Bm\ etc. The points B and B are, therefore, 
 each equally distant from all the points in the circumfer- 
 ence, Cm Am"; hence, (Def. 9), they are its poles. 
 
 Again, since the radius, OB, is perpendicular to the 
 Diane of the circle. Cm Am", it is also perpendicular to 
 the plane of the parallel small circle, FnE, and passes 
 through its center, 0\ !N'ow, the chords of the arcs, BF, 
 Bn, BE, etc., being oblique lines, meeting the plane of 
 the small circle at equal distances from the foot of the 
 
SECTION I. 313 
 
 perpendicular, BO'^ are all equal, (Th. 4, B. VI); Leuce, 
 the arcs themselves are equal, and B is one pole of the 
 circle, F71E. In like manner we prove the arcs, DF^ Dn, 
 DF, etc., equal, and therefore I) is the other pole of the 
 same circle. 
 
 Hence the proposition, etc. 
 
 Oor. 1. A point on the surface of a sphere at the distance 
 of a quadrant from two points in the arc of a great circle, not 
 at the extremities of a diameter, is a pole of that arc. 
 
 P^or, if the arcs, Bm, Bm\ are each quadrants, the angles, 
 BOm and BOm', are each right angles; and hence, BO 
 is perpendicular to the plane of the lines, Om and Om', 
 which is the plane of the arc, mm'-, B is therefoi'e the 
 pole of this arc. 
 
 Cor. 2. The angle included between the arc of a great circle 
 and the arc of another great circle, connecting any of its points 
 with the pole, is a right angle. 
 
 For, since the radius, BO, is perpendicular to the plane 
 of the circle. Cm Am", every plane passed through this 
 radius is perpendicular to the plane of the circle ; hence, 
 the plane of the arc Bm is perpendicular to that of the 
 arc Om; and the angle of the arcs is that of their planes. 
 
 PROPOSITION IV. 
 
 The angle formed hy two arcs of great circles which inter- 
 sect each other, is equal to the angle included between the tan- 
 gents to these arcs at their point of intersection, and is meas- 
 ured by that arc of a great circle whose pole is the vertex of 
 the angle, and which is limited by the sides of the angle or 
 the sides produced. 
 
 Let AM and AN be two arcs intersecting at the 
 p(»int A, and let AE and AF be the tangentj- to these 
 arcs at this point. Take AC and AD, each quadrants, 
 and draw the arc CD, of which A is the pole, and OC 
 and OD are the radii. 
 27 
 
814 
 
 SPHERICAL GEOMETRY. 
 
 Now, since the planes of the arcs intersect in the radiui 
 OA, and AU is a tangent to one arc, and AF a tangent 
 to the other, at the common point A, ^ p 
 
 these tangents form with each other an 
 angle whicii is the measure of the angle 
 of the planes of the arcs ; bnt the angle 
 of the planes of the arcs is taken as the 
 angle included by the arcs, (Def. 3). 
 
 Again, because the arcs, A and AD, 
 are each quadrants, the angles, A 00, 
 A OB, are right angles ; hence the radii, 
 (9^ and OD, which lie, one in one face, 
 and the other in the other face, of the 
 diedral angle formed by the planes of the arcs, are 
 perpendicular to the common intersection of these faces 
 at the same point. The angle, 00 D, is therefore the 
 angle of the planes, and consequently the angle of the 
 arcs ; but the angle OOD is measured by the arc OB. 
 
 Hence the proposition. 
 
 Oor. 1. Since the angles included between the arcs oi 
 great circles on a sphere, are measured by other arcs of 
 great circles of the same sphere, we may compare such 
 angles with each other, and construct angles equal to 
 other angles, by processes which do not differ in principle 
 from those by which plane angles are compared and con- 
 structed. 
 
 Oor. 2. Two arcs of great circles will form, by their in- 
 tersection, four angles, the opposite or vertical ones of 
 " which will be equal, as in the case of the angles formed 
 by tte intersection of straight lines, (Th. 4, B. I). 
 
 PROPOSITION V. 
 
 The surface of a hemisphere may he divided into three right- 
 angled and four quadrantal triangles, and one of these right- 
 angled triayigles will be so related to the other two, that two 
 of its sides and one of its angles will he complemental to th^ 
 
SECTION I. 315 
 
 itdes of one of them, and two of its sides supplemental to two 
 of the sides of the other. 
 
 Let ABC be a right-aDgled spherical triangle, right 
 angled at B. 
 
 Produce the sides, AB and AC, and 
 they will meet at A', the opposite 
 ]»oint on the sphere. Produce BC, 
 both ways, 90° from the point B, to 
 P and P', which are, therefore, poles 
 to the arc AB, (Prop. 3). Through 
 A, P, and the center of the sphere, 
 pass a plane, cutting the sphere into 
 two equal parts, forming a great circle on the sphere, 
 which great circle will be represented by the circle 
 PAF'A^ in the figure. At right angles to this plane, 
 pass another plane, cutting the sphere into two equal 
 parts ; this great circle is represented in the figure by the 
 straight line, POP', A and A' are the poles to the great 
 circle, POP' \ and P and P' are the poles to the great 
 circle, ABA'. 
 
 Now, CPI) is a spherical triangle, right-angled at i>, 
 and its sides OP and CD are complemental respectively 
 to the sides BC and AC of the A ABC, and its side PB 
 is complemental to the arc BO, which measures the 
 [_BACof thejsame triangle. Again, the A A' BC is right- 
 angled at P, and its sides A'C, A'B, are supplemental 
 respectively to the Bides AC, AB, of the a ABC. There- 
 fore, the three right-angled A's, ABC, CPD, and A'BC, 
 Lave the required relations. In the A ACP, the side AP 
 is a quadrant, and for this reason the A is called a quad- 
 rantal triangle. So also, are the A's A'CP, ACP', and 
 P'CA', quadrantal trianficies. Hence the proposition. 
 
 Scholium. — In every triangle there are six elements, three sides and 
 tliree angles, caLed the parts of the triangle. 
 
 Now, if all the parts of the triangle ^^Care known, the parts of 
 each of th« ^*s, PCD and A^BC, are as completely known. And 
 vhen the parts ol the ^ PCD are known, the parts of the A'^ ACP 
 
316 
 
 SPHBRICAL TRIGONOMETRY. 
 
 and A^CP are also known ; for, the side PD measures each .>f the | ^^'i 
 
 P4(7and PA'C, and the angle CPD, added to the right angle A' PD, 
 
 gives the | A^PC^ and the | CPA is supplemental to this. Hence, 
 
 the solution of the A ABC'is a solution of the two right-angled and 
 four qaadrantal A's, which together with it make up the surface of 
 the hemisphere. 
 
 PROPOSITION VI. 
 
 If there he three ares of great circles whose poles are the 
 angular points of a spherical triangle, such arcs^ if produced^ 
 VJill form another triangle, whose sides will he supplemental 
 to the angles of the first triangle, and the sides of the first 
 triangle will he supplemental to the angles of the second. 
 
 Let the arcs of the three great cir- 
 cles be CrH, PQ, KL, whose poles are 
 respectively A, B, and Q. Produce the 
 three arcs until they meet in I), E, and 
 F. We are now to prove that E is the 
 pole of the arc AQ\ I) the pole of the 
 arc BQ\ F the pole to the arc AB. 
 Also, that the side EF, is supplemental 
 to the angle A ; EB to the angle (7; 
 and BF to the angle B; and also, that the side AC i» 
 supplemental to the angle E, etc. 
 
 A pole is 90° from any point in the circumference of 
 Hs great circle ; and, therefore, as A is the pole of the 
 arc Gff, the point A is 90° from the point E. As O is 
 the pole of the arc LK, is 90° from any point in 
 that arc; therefore, (7 is 90° from the point E', and 
 E being 90° from both A and (7, it is the pole of the arc 
 AC In the same manner, we may prove that B is the 
 pole of BC, and F ihQ pole of AB. 
 
 Because A is the pole of the arc GH, the arc G H 
 measures the angle A, (Prop. 4) ; for a similar reason, 
 PQ measures the angle B, and LK measures the angle (7. 
 
 Because E is the pole of the arc AQ. EH=z 90° 
 
 Or, EG+ GH=^ 90° 
 
 For a like reason, FH^ GH-^ 90° 
 
SECTION I. ai7 
 
 Ailding tliese two equations, and observing that Gfl 
 mm A, and afterward transposing one A^ we have, 
 EG 4- GH + FH^^nO'' —A. 
 
 Or, ^JP=180°— ^ ^ 
 
 In like manner, i^i) = 180°— ^ \ («) 
 
 And, BE = 180° — (7 J 
 
 But the arc (180° — A), is a supplemental arc to -A, by 
 the definition of arcs; therefore, the three sides of the 
 triangle DEF^ are supplements of the angles A, B, C, of 
 the triangle ABC. 
 
 Again, as E is the pole of the arc AC, the whole angle 
 E is measured by the whole arc LH. 
 
 But, AC -{- CIT ^ 90"* 
 
 Also, AC + AL = 90^ 
 
 By addition, AC+AC-\-CH -\- AL = 180° 
 By transposition, AC-\-Cff+AL = lSO°—AC 
 That is, LR, or ^= 180° — ^1(7 ^ 
 
 In the same manner, ^=-180°— yl5 > (^) 
 
 And, I)=1S0° — BC J 
 
 That is, the sides of the first triangle are supplemental 
 to the angles of the second triangle. 
 
 PROPOSITION VII. 
 
 The sum of the three angles of any spherical triangle, i% 
 greater than two right angles, and less than six right angles. 
 
 Add equations ( « ), of the last proposition. The first 
 member of the equation so formed will be the sum of 
 the three bides of a spherical triangle, which sum we 
 may designate by S. The second member will be 6 right 
 angles (there being 2 right angles in each 180°) less the 
 three angles A, B, and C, 
 
 That is, aS' = 6 right angles — {A -^ B + C) 
 
 By Prop. 2, the sum S! is less than 4 right angles; 
 27* 
 
318 
 
 SPHERICAL GEOMETRY. 
 
 tlierelbre, to it add s, a sufficieut quantity to make 4 
 right angles. Then, 
 
 4 right angles = 6 right angles — (A + B -^ 0) -\- s 
 
 Drop or cancel 4 right angles from both members, and 
 
 transpose {A -h B -h Cf). 
 
 Then, A + B + = 2 right angles -f 8. 
 
 That is, the three angles of a spherical triangle mako 
 a greater sum than two right angles by the indefinite 
 quantity s, which quantity is called the spherical excess, 
 and is greater or less according to the size of the triangle. 
 
 Again, the sum of the angles is less than 6 right angles 
 There are but three angles in any triangle, and each one of 
 them must be less than 180°, or 2 right angles. For, an 
 angle is the inclination of two lines or two planes ; and 
 when two pianos incline by 180°, the planes are parallel, 
 or are in one and the same plane ; therefore, as neither 
 angle can be equal to 2 right angles, the three can never 
 be equal to 6 right angles. 
 
 PROPOSITION VIII. 
 
 On the same sphere, or on equal spheres, triangles which 
 are mutually equilateral are also mutually equiangular ; and, 
 conversely, triangles which are mutually equiangular are also 
 mutually equilateral, equal sides lying opposite equal angles. 
 
 First.— Lai ABO and DEF, in 
 which AB = BE, AC=:^ DF, and 
 BO = EF, be two triangles on 
 the sphere whose center is 0; 
 then will the [_ A, opposite the 
 side BO, in the first triangle, be 
 equal the \__I), opposite the equal 
 side EF, in the second; also 
 L 5 = 1 ^, andL^==L-^. 
 
SECTION I. 819 
 
 For, drawing the radii to tlie vertices of the angles of 
 these triangles, we may conceive to he the common 
 vertex of two triedral angles, one of which is hounded 
 by the plane angles AOB, BOO, and AOO, and the other 
 by the plane angles DOE, EOF, and DOF. But the 
 plane angles bounding the one of tliese triedral angles, 
 arc equal to the plane angles hounding the other, each 
 t') ea h, since they are measured by the equal sides of the 
 two triangles. The planes of the equal arcs in the two 
 triangles are therefore equally inclined to each other, 
 (Th. 20, B. VI) ; but the angles included between the 
 planes of the arcs are equal to the angles formed by the 
 i*rcs, (Def 3). 
 
 Hence the [_ A, opposite the side BC, in the A Al 
 IS equal to the [_ ^? opposite the equal side EF, in tl ^ 
 other triangle ; and for a similar reason, the L_jB= L_^, 
 
 andtheL^(7=LJ^. 
 
 Second. — If, in the triangles ABC and DEF, being f 
 the same sphere whose center is 0, the [_A = [_D,ih 
 \_B = [_E, and the [_C = [_F\ then wall the side A B, 
 opposite the \_C,\n the first, be equal to the side DE^ 
 opposite the equal \^F, in the second; and also the si'* 
 AC equal to the side DF, and the side BQ equal to tbc 
 side EF. 
 
 For, conceive two triangles, denoted by A'B'C* and 
 D'E'F', supplemental to ABC and DEF, to be formed; 
 then will these supplemental triangles be mutually equi- 
 lateral, for their sides are measured by 180*^ less the 
 o] posite and equal angles of the triangles ABC and 
 LEF, (Prop. G); and being mutually' equilateral, they 
 are, as proved above, mutually equiangular. But the 
 triangle;! ABC and DEF are supplemental to the tri- 
 argles A*B'C' and D'E'F' ; and their sides are therefore 
 measured severally by 180° less the opposite and equal 
 aoifles of the triangles A'B'C and D'E'F\ (Prop. 6). 
 
320 SPHERICAL GEOMETRY. 
 
 Heuce the triangles ABO and DUF, which are mutUiiUy 
 equiangular, are also mutually equilateral. 
 
 Scholium. — With the three arcs of great circles, AB, AC, and BC 
 either of the two triangles, ABC, DEF, may be formed ; but 't is evi 
 dent that these two triangles cannot be made to coincide, tliough they 
 ars both mutually equilateral and mutually «quiangular. Spherical 
 triangles on the same sphere, or on equal spheres, in which the side«< 
 and angles of the one are equal to the sides and 'ingles of the other 
 eaoii to each, but are not themselves capable of superposition, dirt 
 called symmetrical triangles. 
 
 PROPOSITION IX. 
 
 On the same sphere, or on equal spheres, triangles having 
 two sides of the one equal to two sides of the other, each to 
 each, and the included angles equal, have their remaining 
 sides and angles equal. 
 
 Let ABQ and DEF be two 
 triangles, in which AB =^ BE, 
 AC = DF, and the angle A — 
 the angle D; then will the side 
 BC he equal to the side FF, 
 the L ^ = the L^, and L 
 ^[_F 
 
 For, if BF lies on the same 
 side of BF that AB does of AC, the two triangles, ABO 
 and BFF, may be applied the one to the other, and they 
 may be proved to coincide, as in the case of plane tri- 
 angles. But, if BF does not lie on the same side of BF 
 that AB does of AC, we may construct the triangle "^ hicb 
 is symmetrical with BFF; and this symmetrical triangle, 
 when applied to the triangle ABC, will exactly coincide 
 with it. But the triangle BFF, and the triangle sym- 
 metrical with it, are not only mutually equilateral, but 
 also are mutually equiangular, the equal angles lying 
 opposite the equal sides, (Prop. 8) ; and as the one or the 
 other will coincide with the triangle ABC, it follows that 
 
^^£CTION I. 321 
 
 the triangles, ABC and DEF, are either absolutely or 
 By m metrically equal. 
 
 Cor, On the same sphere, or on equal spheres, trianglei 
 having two angles of the one equal to two angles of the other^ 
 each to each, and the included sides equal, have their remain- 
 ing sides and angles equal. 
 
 For, if L^ = L A L^ = L^j and side AB = side 
 DE, the triangle BUF, or the triangle symmetrical with 
 \t, will exactly coincide with A ABC, when applied to it 
 as in the case of plane triangles; hence, the sides and 
 angles of the one will be equal to the sides and angles 
 of the other, each to each. 
 
 PROPOSITION X. 
 
 In an isosceles spherical triangle, the angles opposite the 
 equal sides are equal, 
 
 A 
 
 Let ABC be an isosceles spherical tri- 
 angle, in which AB and AC are the equal 
 sides ; then will [_B = [_ (7. 
 
 For, connect the vertex A with D, the / 
 
 middle point of the base, by the arc of a / 
 great circle, thus forming the two mutu- J L^ I — ^ 
 ally equilateral triangles, ABB and ABC. 
 They are mutually equilateral, because AB is common, 
 BB = DC hy construction, and AB=AChy supposition; 
 hence they are mutually equiangular, the equal angles 
 hidng opposite the equal sides, (Prop. 8). The angles B 
 and C, being opposite the common side AB, are there^. 
 fore equal. 
 
 Cor. The arc of a great circle which joins the vertex 
 of an isosceles spherical triangle with the middle point of 
 the base, is p Brpendicular to the base, and bisects the ver- 
 tical angle of the triangle ; and, conversely, the arc of a 
 
322 
 
 SPHERICAL GEOMETRY. 
 
 greiit circle which bisects the vertical angle of an isosceles 
 spherical triangle, is perpendicular to, and bisects the 
 base. 
 
 PROPOSITION XI. 
 
 If two angles of a spherical triangle are equal, the opposite 
 sides are also equal, and the triangle is isosceles. 
 
 In the spherical triangle, ABQ, let the i_^ = 1_(7; then 
 will the sides, AB and AC, opposite these equal angles, 
 be equal. 
 
 For, let P be the pole of the base, BC, 
 and draw the arcs of great circles, PB, 
 PQ\ these arcs will be quadrants, and at 
 right angles to BQ, (Cor. 2, Prop. 3). 
 Also, produce CA and BA to meet PB 
 and P(7, in the points E and P. Now, 
 the angles, PBP and POP, are equal, 
 because the first is equal to 90° less the 
 \_ABQ, and the second is equal to 90° 
 less the equal \_ACB\ hence, the A's, 
 PBP and PQE, are equal in all their parts, 
 since thej have the L_P common, the \__PBP = [_PCE, 
 and the side PB equal to the side PC, (Cor., Prop. 9). 
 PE is therefore equal to PF, and [_PEC= [__PFB. 
 
 Taking the equals PF and PE, from the equals PC 
 and PB, we have the remainders, PC and EB, equal; 
 and, from 180°, taking the L's PFB and PEC, we have 
 the remaining |__'s, AFC and AEB, equal. Hence, the 
 £^'^,AFCii\\diAEB, have two angles of the one equal to 
 two angles of the other, each to each, and the included 
 sides equal; the remaining sides and angles are therefore 
 equal, (Cor., Prop. 9). Therefore, J. (7 is equal to BA^ 
 and the A ABC is isosceles. 
 
 Cor. An equiangular spherical triangle is also equilat- 
 eral, and the converse. 
 
SECTION I. £23 
 
 ilEMARK. — In this demonstration, tlie pole of the base, £6', is sup- 
 posed to fall without the triangle, ABC. The same figure Daay be used 
 for the case in which the pole falls within the triangle ; the modifi- 
 cation the demonstration then requires is so slight and cbvious, that 
 it would be superfluous to suggest it. 
 
 PROPOSITION XII. 
 
 The greater of two sides of a spherical triangle is opposite 
 the greater angle ; and, conversely, the greater of two angles 
 of a spherical triangle is opposite the greater side. 
 
 Let ^-6(7 be a spherical triangle, in which the angle A 
 
 is greater than the angle B ; then is the side BQ greater 
 
 than the side AC. 
 
 Throusrh A draw the arc of a 
 
 ° \) 
 
 great circle, J.i>, making, with J.^, i^ ^^r ::- B 
 
 the angle BAB equal to the angle 
 ABB. The triangle, BAB, is isos- 
 celes, and BA = BB, (Prop. 11). 
 
 In the A ACD, CD^AD>AG, 
 (Prop. 1.) ; or, substituting for AD its equal DB, we have, 
 CD -V DB> AC. 
 
 If in the above inequality we now substitute CB for 
 CD+DB, it becomes CB > CA. 
 
 Conversely ; if the side CB be greater than the side GA, 
 then is the [_A > the |_^. For, if the [_A is not greater 
 than the \_B, it is either equal to it, or less than it. The 
 [_A is not equal to the 1_J5; for if it were, the tnanglc 
 would be isosceles, and CB would be equal to GA, whic) 
 is contrary to the hj'pothesis. The y_A is not less than 
 the [_B\ for if it were, the side (7J5 would be less than the 
 side CA, by the first part of the proposition, which is also 
 contrary to the hypothesis ; hence, the \_A must be greatei 
 than the \^B. 
 
5J24 SP:1ERI0AL GEOiJlETUY 
 
 iMiOPuSlTlON Xlll. 
 Two symmetrical spJierical triangles are equal in area. 
 
 Let ABO and DUF be two A's on the «stme sphere, 
 having the sides and angles of the one equal to the sidei 
 and angles of the other, each to 
 each, the triangles themselves 
 Dot admitting of superposition. 
 It is to be proved that these 
 ^*s have equal areas. 
 
 Let P be the pole of a small 
 circle passing through the three 
 points, ABO, and connect P 
 with each of the points, A, B, 
 
 and 0, by arcs of great circles. ISText, through B draw 
 the arc of a great circle, BP\ making the angle DBF' 
 •equal to the angle ABF. Take BF' = BF, and draw 
 the arcs of great circles, F^B, F'F. 
 
 The A's, ABF and BBF', are equal in all their parts, 
 because AB=BB, BF^EF', and the [_ABF=[_BBF', 
 (Prop. 9). Taking from^the [_ABO the \_ABF, and 
 from the [_BEF the {__BBF', we have the remaining 
 angles, FBO and F'EF, equal; and therefore the A's, 
 BOF and EFF', are also equal in all their parts. 
 
 Now, since the a's, ABF and DBF', are isosceles, they 
 will coincide when applied, as will also the A's, BOF 
 and EFF', for the same reason. The polygonal areas, 
 ABOF and BEFF', are therefore equivalent. K from 
 the first we take the isosceles triangle, FAO, and from Ine 
 second the equal isosceles triangle, F'BF, the remainders, 
 or the triangles ABO and DBF, will be equivalent. 
 
 Remark. — It is assumed in this demonstration that the pole P falls 
 without the triangle. Were it to fall within, instead of without, no 
 )ther change in the above process would be required than to add the 
 isosceles triangles, PAC, P^DF, to the polygonal areas, to g:«?l thi 
 Afoas cf the triangles, ABC, DEF. 
 
SECTION I. 
 
 325 
 
 C&r. Two spherical triangles on tlie sante sphere, or on 
 equal spheres, will be equivalent — 1st, when they are 
 mutually equilateral; — 2d, when they are mutually equi- 
 angular; — 3d, when two sides of the one are equal to 
 two sides of the other, each to each, and the incluJed 
 angles are equal; — 4th, when two angles of the one are 
 equal to two angles of the other, each to each, and the 
 included sides are equal. 
 
 PROPOSITION XIV. 
 
 If two arcs of great circles intersect each other on the sur* 
 face of a hemisphere^ the sum of either two of the opposite trir 
 angles thus formed will he equivalent to a lune whose angle ik 
 the corresponding angle formed by the arcs. 
 
 Let the great circle, AUBO, be the base of a hemi 
 sphere, on the surface of which the great semi-circumfer 
 ences, BDA and CBU^ inter- 
 sect each other at B ; then will 
 the sum of the opposite tri- 
 angles, BBO and BAU, be 
 equivalent to the lune whose 
 angle is BBC; and the sum 
 of the opposite triangles, 
 CBA and BBB, will be equiv- 
 alent to the lune whose angle 
 is CBA. 
 
 Produce the arcs, BBA and 
 CDU, until they intersect on the opposite hemisphere at IT; 
 thezi, since CBU and BEH are both semi-circumference« 
 of a great circle, they are equal. Taking from each the 
 common part BE^ we have CB =HE. In the same way 
 we prove BB «= HA, and AE — BC. The two triangles, 
 BBC and HAE^ are therefore mutually equilateral, and 
 hence they are equivalent, (Prop. 13). But the two tri- 
 anglen, HAE and ABEy together, make up the lune 
 28 
 
326 
 
 SPHERICAL GEOMETRY. 
 
 DEHAB\ hence the sum of the a's, jBi>(7and ADE, la 
 eqiiivaleni to the same luiie. 
 
 By the same course of reasoning, we prove that the 
 Bum of the opposite A's, DAQ and DBE^ is equi'^aleut 
 
 the lune J) CHAD ^ whose angle is ABG, 
 
 PROPOSITION XV. 
 
 TJu surface of a lune is to the whole surface of the sphere^ 
 as the angle of the lune is to four right angles ; or^ as the are 
 which measures that angle is to the circumference of a great 
 circle. 
 
 JjetABFOA he a lune on the 
 mrface of a sphere, and BCE 
 an arc of a great circle, whose 
 poles are A and F, the vertices 
 of the angles of the lune. The 
 arc, BO, will then measure the 
 angles of the lune. Take any 
 arc, as BB, that will be con- 
 tained an exact number of times 
 in BO, and in the whole circum- 
 ference, BOEB, and, beginning at B, divide the arc and 
 the circumference into parts equal to BD, and join the 
 points of division and the poles, by arcs of great circles. 
 We shall thus divide the whole surface of the sphere 
 into a number of equal lunes. IS'ow, if the arc BO con- 
 tains the arc BB m times, and the whole circumference 
 contains this arc n times, the surface of the lune will 
 contain m of these partial lunes, and the surface of the 
 sphere will contain n of the same ; and we shall have, 
 Surf lune : surf, sphere :: m : n. 
 
 But, m : n :: BO : circumference great circle ; 
 Lence, surf, lune : surf sphere :: BO : cir. great circle ; 
 or, surf, lune : surf, sphere :: [_B0O : 4rightangle& 
 
VECTION I. 327 
 
 This demonstration assumes that BD is a common 
 measuie of the arc, BC, and the whole circumference. It 
 may happen that no finite common measure can be 
 found ; but our reasoning would remain the same, even 
 though this common measure were to become indefinitely 
 small. 
 
 Hence the proposition. 
 
 Cor. 1. Any two lunes on the same sphere, or on equal 
 spheres, are to each other as their respective angles. 
 
 ScnoLiuM. — Spherical triangles, formed by joining the pole of an 
 arc of a great circle with the extremities of this arc by the arcs of 
 great circles, are isosceles, and contain two right angles. For this 
 reason they are called bi-reciangular. If the base is also a quadrant, 
 the vertex of cither angle becomes the pole of the opposite side, and 
 each angle is measured by its opposite side. The three angles are them 
 right angles, and the triaogle is for this reason called tri-rectangular. 
 It is evident that the surface of a sphere contains eight of its tri- 
 Tectangular triangles. 
 
 Cor, 2. Taking the right angle as the unit of angleSj 
 and denoting the angle of a lune by A, and the surface 
 of a tri-rectangular triangle by T^ we have, 
 
 surf, of lune : 82^ :: ^ : 4; 
 whence, surf, of lune = 2A x T, 
 
 Cor. 3. A spherical ungula bears the same relation to 
 the entire sphere, that the lune, which is the base of tne 
 ungula, bears to the surface of the sphere ; and hence, 
 any two spherical ungulas in the same sphere, or m 
 equal spheres, are to each other as the angles of their re- 
 Ri ective lunes. 
 
 PROPOSITION XVI. 
 
 The area of a spherical triangle is measured hy ike excess 
 of the sum of its angles over two right angles^ multiplied hy 
 the tri-rectangular triangle. 
 
 Let ^i^ (7 be a spherical triangle, and DEFLK \\\q cir- 
 cumference of the base of the hemisphere on which this 
 triangle is situa^ted. 
 
S28 SPHERICAL GEOMETRY. 
 
 Produce the sides of the tri- 
 angle until they meet this cir- 
 cumference in the points, i>, E^ 
 F^ X, K, and P, thus forming 
 the sets of opposite triangles, 
 DAE, AKL ; BEF, BPK; CFL, 
 CDF, 
 
 l^ow, the triangles of each of 
 these sets are together equal to 
 a lune, whose angle is the cor- 
 responding angle of the triangle, (Prop. 14) ; hence we 
 have, 
 
 A DAE + A AKL = 2^ X I^, (Prop. 15, Cor. 2). 
 aBEF + aBFK=: 2B X T. 
 A CFL + A QDP = 2(7 x T. 
 
 If the first members of these equations be added, it is 
 evident that their sum will exceed the surface of the 
 hemisphere by twice the triangle ABQ\ hence, adding 
 these equations member to member, and substituting for 
 the first member of the result its value, 4iT -\- 2 A ABO,, 
 we have 
 
 4T + 2aAB0 = 2A.T~h 2B.T+ 2Q.T 
 
 or, 2T -\- aABO= A.T + B,T + C.T 
 
 whence, aABO = A,T + B.T + CT—IT. 
 
 That is, aABC ={A-i- B + C—2) T. 
 
 But A -^ B + — 2 is the excess of the sum cf the 
 angles of the triangle over two right angles, and T de- 
 notes the area of a tri-rectangulur triangle. 
 
 Hence the proposition ; the area, etc. 
 
SECTIOJS 1. 329 
 
 PROPOSITION XVII. 
 
 The area of any spherical 'polygon is measured by the excess 
 of the sum of all its angles over two right angles^ taken as 
 many times, less two, as the polygon has sides, multiplied by 
 the tri-rect angular triangle. 
 
 Let ABODE be a spherical poly- 
 gon ; then will its area be meas- 
 ured by the excess of the sum of 
 the angles, A, B, C, D, and E, over 
 two right angles taken a number 
 of times wliich is two less than 
 the number of sides, multiplied by 
 T, the tri- rectangular triangle. 
 Through the vertex of any of the 
 angles, as E, and the vertices of 
 
 the opposite angles, pass arcs of great circles, thus divi- 
 ding the polygon into as many triangles, less two, as the 
 polygon has sides. The sum of the angles of the several 
 triangles will be equal to the sum of the angles of the 
 polygon. 
 
 Now, the area of each triangle is measured by the 
 excess of the sum of its angles over two right angles, 
 multiplied by the tri -rectangular triangle. Hence the 
 sum of the areas of all the triangles, or the area of the 
 polygon, is measured by the excess of the sum of all the 
 angles of the triangles over two right angles, taken as 
 many times as there are triangles, multiplied by the tri- 
 rectangular triangle. But there are as many triangles as 
 the polygon has sides, less two. 
 
 Hence the proposition ; the arex of any spherical poly- 
 gon, etc. 
 
 Ccr. K S denote the sum of the angles of any spherical 
 
 polygon, n the number of sides, and T the tri-rectan- 
 
 gular triangle, the right angle being the unit of angles j 
 
 the area of the polygon will be expressed by 
 
 [,S'— 2 (n — . 2)J X T^ {S—2n -f 4) T, 
 28* 
 
3'70 SPHERICAL TKIGONOMETRT. 
 
 SECTION II 
 
 SPHERICAL TRIGONOMETHy. 
 
 A Spherical Triangle contains six parts — three sides and 
 (Lrec angles — any three of which being given, the other 
 three may be determined. 
 
 Spherical Trigonometry has for its object to explain the 
 different methods of computing three of the six parts of 
 a spherical triangle, when the other three are given. It 
 may be divided into RigJd-angled Spherical Trigonome- 
 try, and Oblique-angled Spherical Trigonometry ; the first 
 treating of the solution of right-angled, and the second 
 of oblique-angled spherical triangles. 
 
 RIGHT-ANGLED SPHERICAL TRIG0N0METI?3«: . 
 
 PROPOSITION I. 
 
 With the sines of the sides, and the tangent of ONE side 
 of any right-angled spherical triangle, two plane triangles can 
 Reformed that will be similar, and similarly situated. 
 
 Let ABQ be a spherical triangle, 
 right-angled at B ; and let B be the 
 center of the sphere. Because the 
 angle CBA is a right angle, the plane 
 CBB is perpendicular to the plane 
 DBA. From let fall CH, perpen- 
 dicular to the plane DBA ; and us the 
 
SECTION IT. 331 
 
 plane CBD is perpendicular to the plane DBA, CH will 
 lie in \\\q plane CBB^ and be perpendicular to the line 
 DB, and perpendicular to all lines that can be di-awn in 
 the plane DBA, from the point ^(Def. 2, B. Vl). 
 
 Draw HGr perpendicular to DA, and draw GC\ GO 
 will lie wholly in the plane CDA^ and CEG is a right- 
 angled triangle, right-angled at H. 
 
 We will now demonstrate that the angle DGC is a 
 right angle. 
 
 The right-angled aCBG, gives CH'+HG' = CG' (1) 
 The right-angled aDGH, gives DG'-^HG'=DH' (2) 
 13y subtraction, CH' — DG'=CG' — DH' ( 3 ) 
 By transposition, CH' -f DH^ = CG' -f DG^ (4) 
 
 But the first member of equation (4), is equal to 
 CD^, because CDH \s a right-angled triangle; 
 
 Therefore, CD' = CG' + DG' 
 
 Hence, CD is the hypotenuse of the right-angled tri- 
 angle DGC, (Th. 39, B. I). 
 
 From the point B, draw BF at right angles to DA, 
 and BF at right angles to DB, in the plane CDB ex- 
 tended ; the point F will be in the line DC Draw FF, 
 and as F is in the plane CDA, and F is in the same 
 plane, the line FF is in the plane CD A. Now we are to 
 prove that the triangle CHG is similar to the triangle 
 BFF, and similarly situated. 
 
 As HG and BF are both at right angles to DA, they 
 are parallel ; and as HC and BF are both at right angles 
 to DB, they are parallel ; and by reason of the parallels, 
 the angles GHC and FBF are equal ; but GHC is a right 
 angle ; therefore, FBF is also a right angle. 
 
 Now, as Gff and BF are parallel, and Cff and BF 
 are also parallel, we have, 
 
 DB : DB = RG : BF 
 And. DH I DB = EC : BF 
 
S32 SPHERICAL TRIGONOMETRY. 
 
 Thei-efore, EG : BE = HC : BF (Th. 6, B. U.;, 
 Or, Ha : HO = BE I BE. 
 
 Here, then, are two triangles, having an angle in the 
 one equal to an angle in the other, and the sides about 
 the equal angles proportional; the two triangles are 
 therefore equiangular, (Cor. 2, Th. 17, B. 11) ; and they 
 are similarly situated, for their sides make eqiJil aiifjlca 
 at H and B with the same line, BB, 
 
 Hence the proposition. 
 
 Scholium. — By the definition of sines, cosines, and tanp-ents, we 
 perceive that CR is the sine of the arc J5C, BH is its cosine, and RV 
 its tangent; CG is the sine of the arc AC, and BG its cosine. Also, 
 BE is the sine of the arc AB, and BE is the cosine of the same arc. 
 With this figure we are prepared to demonstrate the following propo- 
 sitions. 
 
 PROPOSITION II. 
 
 In any right-angled spherical triangle, the sine of one Me 
 is to the tangent of the other side, as radius is to the tangent 
 of the angle adjacent to the first-mentioned side. 
 
 Or, the sine of one side is to the tangent of the other side, 
 as the cotangent of the angle adjacent to the first-mentioned 
 side is to the radius. 
 
 For the sake of brevity, we will represent the angles 
 of the triangle by A, B, Q, and the sides or arcs opposite 
 ho these angles, by a, b, c, that is, a opposite A, etc. 
 
 In the right-angled plane triangle EBE, we have, 
 EB : BE = B : tSLU.BEE 
 
 That is, sin.<? : tan.a == B : tan.J., 
 which agrees with the first part of the enunciation. By 
 reference to equation (5), Section I, Plane Trigonometry, 
 we shall find that, 
 
 tan.^ cot. J. = i2'; 
 
 B' 
 
 therefore, tan.^ = -. 
 
 cot. A 
 
SECTION II. 833 
 
 Substituting tliis value for tangent^, in tlie preceding 
 proportion, and dividing the last couplet by 72, we shall 
 have, 
 
 sin.c : tan.a = 1 : -. 
 
 cot.A 
 
 Or, sin.c : tan. a = cot.^ : B. 
 
 Or, -B sin.c = tan.a cot. J., ( I ) 
 
 which answers to the second part of the enunciation. 
 
 Cor. By changing the construction, drawing the tan- 
 gent to AB, in place of the tangent to BC, and proceed- 
 ing in a similar manner, we have, 
 
 R sin.a = tan.c cot. 0. ( 2 ) 
 
 jt»KOPOSITION III. 
 
 In any right-angled spherical triangle, the sine of the right 
 angle is to the sine of the hypotenuse, as the sine of either of 
 the other angles is to the sine of the side opposite to that angle. 
 
 The sine of 90°, or radius, is designated by B. 
 In the plane triangle, CBG, we have, 
 
 Bin.Qira : Ca =^ sm.CGR : CH 
 That is, B : sin. 5 = sin.^ : sih.a 
 
 Or, B sin.a = sin.6 sin.^ ( 3 ) 
 
 Cor. By a change in the construction of the figure, 
 drawing a tangent to AB, etc., we shall have, 
 B : sin.6 = sin.(7 : sin.c 
 Or, B sin.c = sin.h sin. (7. (4) 
 
 Scholium. — Collecting the four equations taken from this and the 
 preceding proposition, we have, 
 
 ( 1 ) i? sin.c = tan.a cot.^ 
 
 (2) ^ sin.a = tan.c cot. C 
 
 (3) i? sin.a = 8in.6 sin.^ 
 
 (4) jR sin.c = 8in.6 sin.c ^ 
 
834 
 
 SPHERICAL TRIGONOMETRr. 
 
 These equations refer to the right-angled 
 triangle, ABC; but the principles are true 
 for any right-angled spherical triangle. Let 
 us apply them to the right-angled triangle, 
 PDC, the complemental triangle to ABC. 
 
 Making this application, equation 
 
 ( 1 ) becomes i? sin. CD = tan.PD cot. C 
 ( 2 ) becomes R sin.PD = tan. CD coLP 
 ( 3 ) becomes B sin.FD = Bin.FC sin. C 
 ( 4 ) becomes B sin. CD — sin.FC sin.P 
 By obsorving that sin. CD =~ cos. AC = coa.b. 
 
 And that tan.FD = cot. ', t/ = cot.^, etc. ; and by running equa- 
 tions ( » ), (w ), ( ), and {p ), back into the triangle, ABC, we shall 
 have, 
 
 (5) Bcos.b = cot.^ cot. 6^^ 
 ( 6 ) ^ cos.vl = cot.6 tan.c 
 (7 ) R cos.A = cos.a sin. C 
 ( 8 ) i? cos.6 = cos.a cos.c 
 
 By observing equation ( 6 ), we find that the second member refers 
 to sides adjacent to the angle A. The same relation holds in respocl 
 to the angle C, and gives, 
 
 ( 9 ) i? cos.c = cot.6 tan.a. 
 Making the same observations on ( 7 ), we infer, 
 ( 10 ) i? cos.c = cos.c sin.^. 
 
 Observ-^tion 1. Several of these equations can be de- 
 duced geometrically without the least difficulty. Foi 
 example, take the figure to Proposition 1. The pamllela 
 ir the plane, DBA, give, 
 
 DB : Dir= BE : Z>(7. 
 
 That IS, R : cos.a = cos.c : coa.b, 
 
 A result identical with equation ( 8 j^ and in words it is 
 expressed thus : Radius is to cosine of one side, as the cosine 
 of the other side is to the cosine of the hypotenuse. 
 
 Observation 2. The equations numbered from (1) to 
 (10) cover every possible case that can occur in right-' 
 angled spherical trigonometry; but the r'ombinations are 
 
SECTION 11. 
 
 335 
 
 too various to be remembered, and readily applied to prac- 
 tical use. 
 
 We can remedy this inconvenience, by taking the com' 
 plement of the hypotenuse, and the complements of the two 
 oblique angles, in place of the arcs themselves. 
 
 Thus, b is the hypotenuse, and let b' be its complement. 
 
 Then, 5+ 6'=9b°; or, 6= 90° — 6'; and, sin.6 = cos.6', 
 (J0S.6 = sin. 5'; tan. 6 = cot.6'. In the same manner, if ^' 
 is the complement to A, 
 
 Then, sin.^ = cos.^'; cos. J. = sin. J.'; and, tan.^^l = 
 cot.^'; and similarly, sin.C= cos. (7'; cos. (7= sin. (7'; and 
 tan. (7= cotC^, 
 
 Substituting these values for 5, A, and C, in the fore- 
 going ten equations (a and c remaining the same), vre 
 have. 
 
 (12) 
 (13) 
 (14) 
 (15) 
 (16) 
 
 (17) 
 (18) 
 (19) 
 (2C) 
 
 NAPIE 
 
 Msin.c = 
 jRsin.a = 
 72 sin. a = 
 Rsin.c = 
 Rmn.b' = 
 Rsm.A'^ 
 Rsm.A^ = 
 Rsm.y = 
 Rsm.(7= 
 Bsm.C= 
 
 r's circular 
 
 : tan. a tan.^' 
 : tan.c tan. (7 
 : C08.6' cos.J.' 
 = cos.ft' COS. (7' 
 : tan.^' tan.(7 
 : tan.y tan.(? 
 = cos.a COS. (7' 
 = COS. a COS. (7 
 : tan. 5' tan. a 
 : cos.c cos.^' 
 
 PARTS. 
 
 Omitting the consid- 
 eration of the right an- 
 gle, there arc five parts. 
 Each part taken as a 
 middle part, is connect- 
 ed to its adjacent parts 
 by one equation, and 
 to its extreme parts by 
 another equation ; there- 
 fore, ten equations are 
 required for the combi- 
 nations of all the parts. 
 
 These equations are very remarkable, because the first 
 iiembers a^'e al. composed of radius into some sine, and 
 the riecond members are all composed of the product of 
 two tangents, or two cosines. 
 
 To condense these equations into words, for the pur- 
 pose of assisting the memory, we will refer any one of 
 them directly to the right-angled triangle, ABO, in the 
 last fi^ire. 
 
330 SPHERICAL TRIGONOMETRY. 
 
 When the right angle is left out of the question, a 
 right-angled triangle consists oi jive parts — three sides, 
 and two angles. Let any one of these parts be called a 
 middle part; then two other parts will lie adjacent to this 
 part, and two opposite to it, that is, separated from it by 
 two other parts. 
 
 For instance, take equation (H), and call c the middle 
 part; then A' and a will be adjacent parts, and (7' and 6' 
 opposite parts. Again, take a as a middle part ; then e 
 and C will be adjacent parts, and ^' and b' will be oppo- 
 site parts ; and thus we may go round the triangle. 
 
 Take any equation from (H) to (20), and consider the 
 middle part in the first member of tlie equation, and we 
 shall find that it corresponds to one of the following inva- 
 riable and comprehensive rules : 
 
 1. The radius into the sine of the middle part is equal to 
 the product of the tangents of the adjacent parts. 
 
 2. The radius into the sine of the middle part is equal to 
 the product of the cosines of the opposite parts. 
 
 These rules are known as Napier's Rules, because they 
 were first given by that distinguished mathematician, 
 who was also the inventor of logarithms. 
 
 In the application of these equations, the accent maybe 
 omitted if tan. be changed to cotan., sin. to cosin., etc. 
 Thus, if equation ( 13 ) were to be employed, it would be 
 written, in the first instance, i^ sin.a= cos.^' cos.^', to 
 insure conformity to the rule ; then, we would change it 
 into R sin.a = sin.6 sin. J.. 
 
 Remark. — We caution the pupil to be very particular to take the 
 lomplements of the hypotenuse, and the complements of the oblique 
 mgles. 
 
SECTION III. 
 
 537 
 
 SECTION III 
 
 OBLIQUE-ANGLED SPHERICAL TRIGONOMETRY. 
 
 The preceding investigations have had reference to 
 right-angled spherical trigonometry only, but the appli- 
 cation of thepe principles covers oblique-angled trigonom- 
 etry also; for, every oblique-angled spherical triangle 
 may be considered as made up of the sum or difference 
 of two right-angled spherical triangles. With this ex- 
 planatory remark, we give 
 
 PROPOSITION I. 
 
 In all spherical triangleSy the sines of the sides are to each 
 ythery as the sines of the angles opposite to them. 
 
 This was proved in relation to right-angled triangles in 
 Prop. 3, Sec. II, and we now apply the principle to ob- 
 lique-angled triangles. 
 
 Let ABQ be the triangle, and let 
 CD be perpendicular to AB, or to 
 AB produced. 
 
 Then, by Prop. 3, Sec. U, we have, 
 
 JS : sin. -4 C = sin. A : sin. CD, 
 Also, 
 
 %m,CB : R = sin.CD : sin. B. 
 29 w 
 
ilStt SPHERICAL TRIGONOMElKf. 
 
 By multiplying these two proportions together, term 
 by term, and omitting the common factor M, in the first 
 couplet, and the common factor, sin.6'i>, in the second, 
 we have 
 
 s'm.CB : sin.^C= sin. J. : &m.B. 
 
 PROPOSITION II. 
 
 In any spherical triangle^ if an are of a great circle he lei 
 fall from any angle perpendicular to the opposite side as a 
 base, or to the base produced, the cosines of the other two 
 sides will be to each other as the cosines of the segments of 
 the base. 
 
 By the application of equation 8, (Sec. 11), to the last 
 Hgure, we have, 
 
 B COS. AC = C09, AI) co3.I)0 
 
 Similarly, B cos.BQ = coa.JDO coa.BI) 
 
 Dividing one of these equations by the other, omitting 
 common factors in numerators and denominators, we 
 have, 
 
 COS. AC _ COS. AD 
 C08.B0 cos.BD 
 Or, COS. AC : cos.BO = cos. AD : cos.BJJ. 
 
 PROPOSITION III. 
 
 If from any angle of a spherical triangle, a perpendicular 
 v« let fall on the base, or on the base produced, the tangents 
 of tht segments of the base will be reciprocally propcrticnal 
 u> the cotangents of the segments of the angle. 
 
 By the application of Equation 2, (Sec. II), to the lap* 
 figure, we have, 
 
 B sin.OZ) = t2in.ADcoi.ACJ). 
 
SECTION III. 339 
 
 Similarly, R sin.Ci) = tan.BD cotBCD 
 Therefore, by equality, 
 
 tan.AD cot ACD = tan.BI) cot.BCU 
 Or, tiin,AJ) : tsm.BI) = cotBCB : i.otACI>, 
 
 PROPOSITION IV. 
 
 2^i3 same construction remaining, the cosines of the angles 
 at the extremities of the segments of the base are to each 
 Mer as the sines of the segments of the opposite angle. 
 Equation 7, (Sec. II), applied to the triangle ACD, gly£^^ 
 
 R coa.A = coa. CD sin. ACD [s] 
 Also, R cos.B = coa.CD s'm.BCD (^ 
 Dividing equation (■?) by (0, gives 
 cos.^ _ Bin. A CD 
 cos.B Qin.BCD 
 Or, cos.^ : cos. J. = em.BCD : am. A CD, 
 
 PROPOSITION V. 
 
 7^e same construction remaining, the sines of the segments 
 of the base are to each other as the cotangents of the adjacent 
 angles. 
 Equation 1, (Sec. II), applied to the triangle ACD, givea 
 
 R am. AD = tan.OZ) cot.^ (s) 
 Sv.nilarly, R am.BD == tan.OZ) Qoi.B (0 
 Dividing (a) by (0, gives 
 
 am.AD _ cot.A 
 am.BD cot.B 
 Or, am.BD : i,m.AD = cot.-B : cot.^ 
 
340 SPHERICAL TRIGONOMETRY. 
 
 PROPOSITION VI. 
 
 The same construction remaining^ the cotangents of the two 
 tides are to each other as the cosines of the segments of the 
 angle. 
 
 KquatioD 9, (Sec. II), applied to the triangle J. (7i>, givea 
 
 B cos.^ CD = cot.^ Q tan. CD [s) 
 S railarlj, B cos.BOD = cot.BO tan. 02) (0 
 Dividing («) bj (0, gives 
 
 COS. A OD _ cot. AC 
 co8,BOD ~ cotBO 
 Or, cot AC: cot.BO = coa.ACB : cos.BOD. 
 
 PROPOSITION VII. 
 
 The cosine of any side of a spherical triangle, is equal to 
 the product of the cosines of the other two sides, plus the 
 product of the sines of those sides multiplied by the cosine 
 of the included angle. 
 
 Let ABQ be a spherical triangle, 
 and CB a perpendicular from the 
 angle C to the side AB, or to 
 the side AB produced. Then, by 
 Prop. 2, 
 ios.AC:co8.CB=cos.AD'.cos.BD[\) 
 
 When CD falls within the tri- 
 angle, 
 
 BD = {AB'-AD)', 
 
 and when CD falls without the triangle, 
 BD = {AD — AB). 
 
 Hence, cos.BD = co9.{AD — AB) 
 
 Now, cos.(^i? -. ^ Z>) = co8.{AD — AB), 
 
 because each of them is equal to 
 208. AB COS. AD + sin.A^ siif.Jli), (Eq. 10, Prop. :i. 
 Sec. I, Plane Trig.). 
 
SECTION III. 341 
 
 This value of cus. BD, put in proportion (1), gives 
 cos ^ 6' : COS. CB = cos. J D : cos.^lJ5 cos.-4Z>+sin.^i5 sin.^Z> ( 2 ) 
 
 Dividing the last couplet of proportion (2) by cos.^ Z>, 
 observing that 
 
 - — Tn = tan.^i), 
 cos.AI) 
 
 and we have 
 
 COS. J 6^ : COS. as = 1 : cos.^^ + sin.^^ tan.^D (3) 
 
 By applying equation 6, (Sec. II), to the triangle A CD^ 
 taking the radius as unity, we have 
 
 COS. J. = GOt.AC tUTl.AB [k) 
 
 But, tan.^(7cot.^C=l,(Eq.5, Sec. I, Plane Trig.) {I) 
 
 Multiply equation [k] by inn. AC, observing equation 
 (0, and we have 
 
 / tan.^Ccos.^ = idiW.AB 
 
 Substituting this value of tan. J.i), in proportion (3), 
 we have 
 COS. J. C : COS. CB = 1: cos. J.^ + sin. J.^ tan. J. C co&.A ( 4 ) 
 
 Multipl^'ing extremes and means, gives 
 COS. CB=co9,.A C Qos.AB-\- sin.^^ (cos.^ C tan.-4 Q) cos. JL. 
 
 But, tan.^(7= — ^-tt^, or, cos.^C tan. J. (7= sin.^l(7. 
 
 COS.^C/ 
 
 Therefore, cos. (75 = cos. A (7 co^.AB -\- qih.AB sin. AC 
 cos.J.. 
 
 If the sides opposite the angles. A, B, and {7, be re^ 
 dpectively represented by a, b, and c, this equation 
 becomes, 
 
 coa.a= co8.^ cos.c + sin.6 sin.c cos. A. 
 
 This formula conforms to the enunciation in respect to 
 the side a. Now, by interchanging h and a, and B and A^ 
 in the last equation, we get the formula for cos.J, which is, 
 
 cos.5=cos.a cos.c+sin.a sin.c cos. J?. 
 
 29* 
 
842 
 
 SPHERICAL TRIGONOMETRY 
 
 Interchanging c and a, and C and A^ we get the formula 
 for cos.c, Avhich is, 
 
 cos.(? = cos.a COS. 6 -f sin. a sin.J cos.Ol 
 Hence, we have the three symmetrical formulae : 
 cos.a = C0S.5 cos.c + sin. ft sin.c cos.^^ 
 cos.6 = cos.a cos.c + sin.a sin.(? cos.^ > (iS') 
 cos.c = cos.a C08.6 + sin.a sin.6 cos.cj 
 
 From these, by simple transposition and division, we 
 deduce the following formulae for the cosines of the 
 angles of any spherical triangle, viz : 
 
 cos.a — cos.ft cos.c^ 
 
 cos. J. = 
 cos.^ = 
 
 COS. (7 = 
 
 sin.5 sin.c 
 
 COS. 6 — cos.a cos.c 
 
 sin. a sin.c? 
 
 COS.C — COS.« COS. 5 
 
 {S') 
 
 sin. a sin. 6 
 
 By means of these equations we can find the cosine of 
 any of the three angles of a spherical triangle in terms 
 of the functions of the sides ; but in their present form 
 they are not suited for the employment of logarithms, 
 and we should be compelled to use a table of natural 
 sines and cosines, and to perform tedious numerical ope- 
 rations, to obtain the value of the angle. 
 
 They are, however, by the following process, tians- 
 formed into others well adapted to the use of logarithms. 
 
 In Eq. 34, Sec. I, Plane Trig., we have 
 
 1 + COS. J. = 2cos.''JJ.. 
 
 cos.a — C0S.5 C0S.6* 
 
 Therefore, 2cos.'iJ. = 1 + 
 
 sin.ft sin.tf 
 (sin.J sin.{7 — cos. 5 cos.c) + cos.a 
 sin.J sin.c 
 
 (m). 
 
 But, cos.(6 -\- c) — coB.h cos.c — sin.c sin.6, (Equation 
 9. Section I, Plane Trig.). By comparing .this equation 
 
SECTION III 
 
 843 
 
 with the secoad member of equation (wi), we perceive 
 that equation ( »» ) is readily reduced to 
 
 __ cos.a — co8.( b-h c) 
 Bin.6 sin.o 
 
 2cos.«JA 
 
 Considering (b-hc) as one are, and then making appli- 
 <iation of equation ( 18 ), Plane Trigonometry, we have, 
 
 2cos.'J^ = 
 
 2sm. (—2—) 81I-. (—2—) 
 
 But, 
 
 b + c 
 
 b + c + a 
 
 2 
 b + c-^a 
 
 sin.6 sin.c 
 
 — a; and if we put S to 
 
 represent — , we shall have, 
 
 A sin.AS' sin.fAS' — a) 
 
 C08.« - = r-^-A ^. 
 
 2 sin.6 sm.c 
 
 Or, COS. - = 4 / 8in.^8in.(Ay— g)^ 
 
 2 ^ sin. 6 sin.c 
 
 The second member of this equation gives the value 
 of the cosine when the radius is unity.' To a greater 
 radius, the cosine would be greater; and in just the same 
 proportion as the radius increases, all the trigonometrical 
 lines increase ; therefore, to adapt the above equation to 
 our tables where the radius is R, we must write R in the 
 second member, as a factor; and if we put it under the 
 radical sign, we must write R^. 
 
 For the other angles we shall have precisely similai 
 equations : 
 
 That is, COS. 4 = 4/^5l^^l8^ZE^' 
 
 Sin. 6 sin.c 
 
 COS. ^ = . A'8iu.^8in.(>S-6) 
 
 Sin. a sin.c 
 
 COB. ^= . / -K'sin.j^ 6m.(S-e) 
 
 sin. a sin.6 
 
 (T) 
 
344 SPHERICAL TRIGONOMETRY. 
 
 To deduce from formulae {jS), formulae for the Bines of 
 the half of each of the angles of a spi^erical triangle, wc 
 proceed as follows: 
 
 From Eq. 35, Sec. I, Plane Trig., mq have 
 
 2sin.' JJ. = 1 — cos.^. 
 
 Substituting the value of cos. J., ''aken from formui« 
 (aS^), and we have, 
 
 cos.a — cos.h cos.<? 
 
 2sin.' 1 J[ = 1 
 
 sin. 6 sin.(? 
 
 (sin.5 sin.c-f C0S.6 cos.c) — cop a , . 
 
 = ^ T ^ . ^- — . [o) 
 
 sm.6 sm.c 
 
 But, cos.(5 CO c) = sin.6 sin.c + cos.J coa.c, (Ey^. 10, 
 Sec. I, Plane Trig.). 
 
 This equation reduces equation ( » ) to 
 
 sin.6 sin.<? 
 
 Considering (h co <?) as a single arc, and applying equa- 
 tion 18, Sec. I, Plane Trig., we have 
 
 2sin. ( ) sm. ( ) 
 
 8m.6 sm.c 
 
 But, = ^ c = S — c, if we put S «« 
 
 g-f ^ -f-g 
 2 * 
 
 Also, = h==S—b, 
 
 Dividing equation [o^) by 2, and making these substi 
 tutions, we have 
 
 sin.o sm.c 
 when radius is unity. 
 
SECTION III. 340 
 
 When radius is E, we have 
 
 sin. 6 sin.c 
 
 o M 1 .,7.4 /i2^sin.(AS'— a)sin.(AS'--c) 
 
 Similarly, sm.J^ = V -^. A ^ ^ 
 
 *^' -^ ^ sin. a sin.c 
 
 . ^/ jg-^8in.(^-a)sin.( S-^6 
 And, sm.JC = V ZT^TT^^TT 
 
 sin. a sin.6 
 
 {IT) 
 
 The above equations are now adapted to our tables. We 
 bIkiII show tlie application of these formulae, and those in 
 group (T), hereafter. 
 
 PROPOSITION VIII. 
 
 The cosine of any of the angles of a spherical triangle, i% 
 iqual to the product of the sines of the other two angles mul- 
 tiplied hy the cosine of the included side, minus the product 
 of the cosines of these other two angles. 
 
 Let ABC be a spherical triangle, and 
 A'B'Q' its supplemental or polar tri- 
 angle, the angles of the first being de- 
 noted by J., B, and (7, and the sides 
 opposite these angles by a, 5, (?, respect- 
 ively; A\ B', C", a\ 6', c', denoting the 
 angles and corresponding sides of the 
 second. 
 
 By Prop. 6, Sec. I, we have the following relations be 
 tween the sides and angles of these two triangles. 
 
 A' = 180° — a, B'-= 180° — h, C = 180° — e; 
 
 a' = 180° — A,V « 180° — B,c'=- 180° — C. 
 
 The first of formulift (^), Prop. 7, when applied to th« 
 polar triangle, gives 
 
 rOH a* rrr POO h* POQ /?' 4- R1T1 ^' flin.c' COS. -4' (1) 
 
346 SPHERICAL TK IGO N OME T R Y. 
 
 which, by substituting the values of a', b\ c', and A\ 
 becomes 
 
 cos.(180°-.^) = cos.(180° — J?) cos.(180°— C) + sin.(180« 
 — B) sin.(180° — Q) cos.(180° — a), ( 2 ) 
 But, 
 
 cos.(180°— J.) = — cos.Jl, etc., 8in.(180°— ^)=8in.J5,etc.; 
 
 ancl placing these values for their equals in eq. (2), and 
 
 changing the signs of both members of the resulting 
 
 equation, we get 
 
 cos. J. = sin.^ sin.(7 cos.a — cos.J5 cos. (7, 
 
 which agrees with the enunciation. 
 
 By treating the other two of formulae [S)^ Prop. 7, in 
 the same manner, we should obtain similar values for the 
 cosines of the other two angles of the triangle ABQ\ 
 or we may get them more easily by a simple permuta- 
 tion of the letters J., B, (7, a, etc. 
 
 Hence, we have the three equations 
 cos. J. = sin.^ sin. C 
 
 .A = sin.^ sin.(7cos.a — cos.^ cos.C") 
 
 .B = sin. J. sin.C^cos.i — cos. J. cos. (7 V (^) 
 
 .0 = sin. J. sin.i? cos.f — cos. J. co^.B) 
 
 cos 
 cos 
 
 By transposition and division, these equations become 
 
 ^^^ cos.-A -f cos.^ COS. (7 ,o\ 
 
 sm.^ sin. (7 
 T QOQ.B + cos.^ COS. (7 
 
 COS.6 = . T—-— 77 
 
 sin.^ sm.C^ 
 COS. (7 + COS. A C0S.5 
 
 COS.C = r-— i r — ^ 
 
 Sin. J. sm.^ 
 
 From these we can find formulae to express the sine or 
 tL^ cosii.t) of one half of the side of a spherical triangle, 
 m terms of the functions of its angles ; thus : 
 
 Add 1 to each member of eq. (3), and we have 
 
 COS. J. 4- C0S.J5 cos. (7 -f sin.^ sin. (7 
 
 1 4 cos.a = 
 
 sin.^ sin. (7 
 
SECTION .II. 847 
 
 COS.^ + C0S.(j5 — O) 
 
 %\\i.B sin. 6^ 
 But, 1 + cos.a = 2cos.' Ja ; heuce, 
 
 2cos.' \a = ^. — ^— .^^— 
 
 Qin.B sin. (7 
 
 nnd Binje cos.A + cos.(jB — C) = 2cos.J(J. + J5— (7)co8.} 
 {A\-Q—B) (Eq.l7, Sec. I, Plane Trig.), we have 
 
 2co8.' U = 2C08.KA + ^ - (7)co8.K^ + (7- ^) 
 
 sin.^ sin. (7 
 
 Make ^ + ^ + (7= 2/S'; then A + ^— (7= 2S—20, 
 A + Q—B^ 2S—2B, i{A-\-B—a) = S— 0, and i(A 
 -I g—B) = S—B; whence 
 
 sm.jB sin. (7 
 
 , /^^i:(^=0)co8.(AS'— J5) 
 ' ^ ^ sin.jBsin.(7 
 
 o- -1 1 11 \ / G0S.(S—A)C08,(S—C) V /TT/v 
 
 Similarly, cos.JS = \/ ^^-^^-> > (F') 
 
 and, COS.J. = \/2£iI?^oi;^ 
 
 ^ Sin. A sin.J5 
 
 To find the sin.Ja in terms of the ftinctions of tlie 
 angles, we must subtract each member of eq. ( 3 ) from 1, 
 by which we get 
 
 ^ cos.^ + COS.jS cos.(7 
 sin.^ sin.C? 
 
 But, 1 — cos.a= 2sin.'Ja; hence we have, 
 
 p . ,- (sin.j5 sin. (7 — cos.^ cos. (7) — cos. J. 
 
 ^Sin* *fl — — — z =r ; — — • 
 
 sm.jB sin.O' 
 
 Operating upon this in a manner analogous to that l»j 
 which cos.Ja was found, we get, 
 
348 SPHERICAL TRIGONOMETRY. 
 
 ^ \ sm.B sin. (7 / 
 
 sm. J5 = { ^. ._ V — ; \2 ) ( IT) 
 
 ( sin.^ sin. (7 J / ^ ' 
 
 Bin.i.- f-cos.^co3.(^-{7)U 
 
 L.Jc.= { 
 
 sin.^ sin.^ J 
 
 L the first equation in (W) be divided by the first m 
 { ^'^ ), we shall have, 
 
 And corresponding expressions may be obtained fof 
 tan. J 6 and tan.J^?. 
 
 NAPIER'S ANALOGIES. 
 
 If the value of cos.^?, expressed in the third equation 
 of group (aS'), Prop. 7, be substituted for cos.<?, in the 
 second member of the first equation of the same group, 
 we have, 
 
 COS. a = cos.a cos.'& -|- sin. a sin. 6 cos. J cos. C+ sin.J sin.c cos.^; 
 
 which, by writing for cos.^'J its equal, 1 — sin.'ft, becomes, 
 
 coa.a=cos.a — cos.a sin.^i-fsin.a sin. 6 cos.& cos. C-fsin.ft sin.c cos.^. 
 
 Or, = — cos.a sin.^5-f-sin.a sin.6 cos.6 cos.C+sin.i sin.c cos.J.. 
 
 Dividing through by sin. 5, and transposing, we find 
 
 cos.^ sin. (j= cos.a sin.6 — sin.a cos.5 cos. (7; 
 
 - cos.^ sin.5 — sin.a cos. 5 COS. (7 ,,, 
 
 .once, cos.^l = ; (1) 
 
 sm.(? 
 
 By substituting the value of cos . in the second of the 
 
 equations of group (S), Prop. 7 ; or, merely writing £ Ibi 
 
 A, and interchanging h and a, in the above value, foi 
 
 sK)S. A, we obtain, 
 
 ^ C0S.5 sin.a— sin.5 cos.<* cos.<7. ,, • 
 cos.^ = ; (2) 
 
SECTION III. 349 
 
 Adding equations (1) and (2), member to member, 
 we have, 
 
 ^ 8in.(a+5)— sin.fa-l-J) cos.C. 
 
 COS. J. + cos.-B = i^ '— , 
 
 sm.c 
 
 by reraembtring that sin.a cos.ft + cos.a sin.6«Bin.(a+6). 
 (See Eq. (7 ), Sec. I, Plane Trig.). 
 
 Wh 3nce, cos. J. + cos.jB = (1 — cos. C) ?^^^ — \ ( 3 ) 
 
 In any splierical triangle we have, (Prop. I), 
 
 sin.-4. : sin. J5 : : sin.a : sin. 6 ; 
 
 And therefore, sin.^ + sin.^ : sin.-B : : sin.a + sin.5 : 
 sin. 6. 
 
 „ . ^ . . T> (sin.a + sin. 5) sin.J? 
 
 Hence, sm.^ + sm.jS = ^^ . — ^-^ . 
 
 sm.6 
 
 But, ?i5:^ = «4?4 which value of ?^4. 5n the abore 
 
 sin. 6 sin.tf sm.o 
 
 equation, gives 
 
 . . . ^ (sin.a + sin.5) sin.(7 , .. 
 
 sm. J. + sm.-B = ^ -, '- . (4) 
 
 sm.c 
 
 Dividing equation (4) by equation (3), member by 
 
 member, we obtain, 
 
 sin. J. + sin.5 __ sin. (7 sin.a + 8in.6 .^. 
 
 cos.JL + cos.^ 1 — cos.(7 sin.(a4-J) 
 
 Comparing this equation with Equations (20) and (2G), 
 dec. I, Plane Trigonometry, we see that it can bo re- 
 duced to 
 
 -,.._,. . ^ /> sin.a + sin. 6 ,«v 
 
 tan.J(^ + ^)='Cot.J(7x . , . ,, (6) 
 
 ^ sm.(a + 6) 
 
 Again, from the proportion, 
 
 sin. J. : sin.J5 : : sin.a : sin.5, 
 
 we likewise have, 
 
 ein.^ — sin.^ : sin.^ :: «in.a — sin.J : ain.i; 
 
 80 
 
350 SPHERICAL TRIGONOMETRY. 
 
 neuce, sm.^ — sm.^ = (sm.a — sin.o) = (sm.a — 
 
 sin.6 
 
 8111. o) . 
 
 sm.c 
 
 Dividing this equation by equation (3), member bj 
 
 niember, we obtain, 
 
 sin.^ — sin.^ _ sin. (7 sin. a — sin. 5 
 
 cos.^ + cos.^ 1 — 008.(7 sin.(a4-6)* 
 
 Comparing this with Equations (22) and (26)^ Sec. I, 
 
 Plfcue Trigonometry, we see that it will reduce to 
 
 1/^ T^i . , /-/ sin.a — sin. 6 _. 
 
 tan.|(^ — ^) = cot.i(7 x — .— t — -j^. (7 
 ^^ ' ^ sin. (a -f h) 
 
 Now,sin.a + sin.6 = 2sin.(-2~)cos.(— ^^— ); Eq. (15), 
 Sec. I, Plane Trig.). 
 
 and, sin. (a + 5) = 2sin.("2— ) c<>s.(^^); Eq. (30), 
 
 Sec. I, Plane Trig.). 
 
 Dividing the first of these by the second, we have 
 
 a — h^ 
 sin. a + sin. 6 
 
 '^°^-(-2-) 
 
 sin.(a 4-6) /« + h^ 
 
 ^ ' cos.' 
 
 Writing the second member of this equation for ita 
 &rst member in Eq (6), that equation becomes 
 
 tan. \{A + B\ = cot. K ^^^' ^ TS ' ^^^ 
 ^ COS. J(a-f 6) 
 
 Bj a similar operation, Eq. (7) may^be reduced to 
 
 tan. i{A - .g) = cot. |(7 '!"- ^.'^"f l (9) 
 ^^ y ^ sin.i(a+^') 
 
 Equations ( 8 ) and ( 9 ) may be resolved into the pro- 
 portions 
 cos. J(a + h) : cos. J(a — 5) : : cot. J (7 : tan. J(JL + jB) ; 
 sin. J(a + 5) : sin. J(a — 5) : : cot. J (7 : tan. J(J. — B). 
 These proportions are known as Napier's 1st and 2d 
 
SECTION III. 351 
 
 Analogies, and may be advantageously used in the solu- 
 tion of spherical triangles, when two sides and the in- 
 cluded angle are given. 
 
 When expressed in language, these proportions fur- 
 nish the following rules: 
 
 1. The cosine of the half sum of any two sides of a spheri- 
 cal triangle is to the cosine of the half difference of the same, 
 tides, as the cotangent of half the included angle is to *he 
 tangent of the half sum of the other two angles. 
 
 2. The sine of the half sum of any two sides of a spheri- 
 cal triangle is to the sine of the half difference of the same 
 tides, as the cotangent of half the included angle is to the 
 tangent of the half difference of the other two angles. 
 
 The half sum, and the half difference of two angles 
 of a spherical triangle, may be found by these rules, w^hen 
 two sides and the included angle ;irc given; and by add- 
 ing the hrlf sum to the half difference, we get the 
 greater of these two angles, and by subtracting the half 
 difference from the half sum, we get the smaller. The 
 third side may then be found by proportion. 
 
 We have analogous proportions applicable to the case 
 in which two angles and the included side of a spherical 
 triangle are given. 
 
 To deduce these, let us represent the angles of the tri- 
 angle ]»y A, B, and Q, and the opposite sides by a, h, and 
 c ; A', B', (7', a', h', c', denoting the corresponding angles 
 and sides of the polar triangle. 
 
 Kow, Eq. (9) is applicable to any spherical triangle, 
 and when applied to the polar triangle, it becomes 
 
 But by Prop. 6, Sec. I, Spherical Geometry, we have 
 
 A' = ISO'^ — a,B'=- 180° — h,C' = 180° — c, 
 
 a' = 180° —A,y = 180° —B,c' = 180° — C, 
 
 Whence, l{A'-B')=^\{h-a\ l{a' + h') = 180°--4_| J^^ 
 
 j(a' - 5') - \(B - ^), JC^ = 90- - \c. 
 
852 SPHERICAL TRIGONOMETRY. 
 
 By the substitution of these -values in Eq (w), th>it 
 
 equation l)ecomes 
 
 1/7 X sin. i(5 — A) ^ 
 tan. Vb — a) = -. — f; . . ^! tan. ic, 
 ^' ^ sin. J(J. + jB) ^' 
 
 or, tan. i(a — 5) = ^^^' ?L T J ^^^' J^> ^^^ 
 ' ^^ ^ sin. i(J. + ^) ^ ' 
 
 Bince tan. J(6 — a) = — tan. J(a — 5), and sin. i(J5 —--4) = 
 — sin. i{A — ^). 
 
 By applying Eq. (8) to the polar triangle, and treating 
 the resulting equation in a manner similar to the above, 
 we find 
 
 tan. i{a + 5) = ^Q«' K^ ~ g tan. },, (q) 
 
 cos. ^{A -\- B) ^ ' 
 
 Equations {p) and (?) may be resolved into the fol- 
 lowing proportions. 
 
 sin. J(J. + B) : sin. J(J. — 5) : : tan. ^o : tan. J(a — h): 
 COS. J(J. + B) : COS. J(J. — B) :: tan. Jc : tan. J(a H 6). 
 
 These proportions are called Napier's 3d and 4th 
 Analogies, and when expressed in words become the fol- 
 lowing rules: 
 
 1. The cosine of the half sum of any two angles of a 
 spherical triangle is to the cosine of the half difference of the 
 same angles^ as the tangent of half the included side is to the 
 tangent of the half sum of the other two sides. 
 
 2. The sine of the half sum of any two angles of a spheri- 
 cal triangle is to the sine of the half difference of the sami 
 angles, as the tangent of half the included side is to the tan- 
 gent of the half difference of the other two sides. 
 
 The half sum, and the half difference of two sides of 
 tt spherical triangle, may be found by these rules, when 
 two angles and the included side are given ; and by add- 
 ing the half sura to the half difference, we get the greater 
 >f these sides, and by subtracting the half difference 
 from the half sum, we get the smaller. 
 
SECTION IV. 868 
 
 SECTION IV. 
 
 ISPHEllICAL TRIGONOMETRY APPLIED. 
 SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES. 
 
 A GOOD general conception of the sphere is essential 
 to a practical knowledge of spherical trigonometry, and 
 this conception is hest obtained by the examination of 
 an artificial globe. By tracing out upon its surface the 
 various forms of right-angled and oblique-angled tri- 
 angles, and viewing them from different points, we may 
 soon acquire the power of making a natural representa- 
 tion of them on paper, which will be found of much as- 
 sistance in the solution and interpretation of problems. 
 
 For instance, suppose one side of a right-angled 
 spherical triangle to be 56°, and the angle between this 
 side and the hypotenuse to be 24°. What is the hypote- 
 nuse, and what the other side and angle ? 
 
 A person might solve this problem by the application 
 of the proper ef|uations or proportions, without really 
 comprehending it ; that is, without being able to form a 
 distinct notion of the shape of the triangle, and of its 
 relation to the surface of the sphere on which it is 
 situated. 
 
 If we refer this triangle to the common geographical 
 globe, the side 66° may be laid off on the equator, or on 
 a meridian. In the first case, the hypotenuse will be the 
 arc of a great circle drawn through one extremity of the 
 Bide 56°, above or below the equator, and making with 
 80* X 
 
PM 
 
 SPHERICAL TKIGOyOMETRY. 
 
 it an angle of 24° ; the other side will be an arc of a 
 meridian. In the second case, the side 56° falling on a 
 meridian, the hypotenuse will be the arc of a great circle 
 drawn through one extremity of this side, on the right 
 or left of the meridian, and making with it an angle of 
 24° ; the other side will be the arc of a great circle, at 
 Tight angles to the meridian in which the given side lies. 
 
 Generally speaking, th3 apparent form of a spherical 
 triangle, and consequently the manner of representing 
 it on paper, will difler with the position assumed for tl e 
 eye in viewing it. From whatever point we look at a 
 sphere, its outline is a perfect circle in the axis of which 
 the eye is situated; and when the eye is, as will be here- 
 after supposed, at an infinite distance, this circle will be 
 a great circle of the sphere. All great circles of the 
 sphere whose planes pass through the eye, will seem to 
 be diameters of the circle which represents the outline 
 of the sphere. 
 
 We will now suppose the eye to be in the plane of the 
 equator, and proceed to construct our triangle on paper. 
 
 Let the great circle, 
 PASA^y represent the out- 
 line of the sphere, the di- 
 ameter AA^ the equator, 
 and the diameter P^S' the 
 central meridian, or the 
 m'^-ridian in whose plane 
 the eye is situated. Let 
 AB— 56°, represent the 
 given side, and^ 6^,making 
 with AB the angle BA (7= 
 24°, the hypotenuse, then will BO, the arc of a meridian, 
 be the other side at right angles to AB, and the triangle, 
 ABC, corresponds in all respects to the given triangle. 
 
 Again measure off 56° from P to Q, draw the arc DQ, 
 make the arc A'G equal to 24°, and draw the quadrant 
 PEG. The triangle FQP will also reoresent the given 
 triangle in every particular. 
 
SECTION IV. 355 
 
 "We know from the construction, that DV, = 24°, is 
 greater than BO, and that AOis greater than AB, that 
 is, greater than 56°. 
 
 In like manner, we know that >4', = 24°, is greater 
 than QRj and that FR is greater than FQ, because FB 
 18 more nearly equal toFG, =90^, thanP Q is to FA, =90° 
 
 For illustration and explanation, we also give the fol- 
 lowing example: 
 
 In a light-angled spherical triangle, there are given, 
 the hypotenuse equal to 150° 33' 20", the angle at the 
 base, 23° 27' 29", to find the base and the perpendicular. 
 Let A'BQ in the last figure, represent the triangle in 
 which A'Q= 150° 33' 20", the [__ BA'Q^ 23° 27' 29", 
 and the sides A'B and BG are required. 
 
 This problem presents a right-angled spherical tri- 
 angle, whose base and hypotenuse are each greater than 
 90° ; and in cases of this kind, let the pupil observe, 
 that the base is greater than the hypotenuse, and the oblique 
 angle opposite the base, is greater than a right angle. In 
 all cases, a spherical triangle audits supplemental triangle 
 make a lune. It is 180° from one pole to its opposite, 
 whatever great circle be traversed. It is 180° along the 
 equator ABA', and also 180° along the ecliptic ACA^, 
 The lune always gives two triangles; and when the 
 sides of one of them are greater than 90°, we take the 
 triangle having supplemental sides ; hence in this case 
 we operate on the triangle ABQ, 
 
 AC m greater than AB, therefore A*B is greater than 
 the hypotenuse AO, 
 
 The [_AQB is less than 90°; therefore, the adjacent 
 angle A'CB is greater than 90°, the two together being 
 equal to two right angles. 
 
 These facts are technically expressed, by saying, that 
 the sides and opposite angles are of the same affection,* 
 
 * Same affection : that is, both greater or both less than 90°. Dif 
 /'rent affection : the one greater, the other less than 90°. 
 
8i)6 SPHERICAL TRIGONOMETRY. 
 
 Now, if the two sides of a right-angled spherical triaj^.^le 
 are of the same affection, the hypotenuse will be less than 
 90° ; and if of different affection^ the hypotenuse will be 
 greater than 90"^. 
 
 If, in every instance, we make a natural construction 
 of the figure, and use common judgment, it will b<i im- 
 possible to doubt whether an arc must be taken gr^atet 
 or less than 90°. 
 
 We will now solve the triangle ACB, AO^ 180®—- 
 150° 33' W = 29° 26' 40". 
 
 To find BO, we use Eq. (3) or (13), Prop. 3, Sec. IT., 
 
 thus: 
 
 5, sin. 29° 26' 40" . 9.691594 
 
 A, sin. 23° 21' 29" . 9.599984 
 a,sin. 11*^ 17' 7" . 9.291578 
 
 To find AB, we use equation ( 1 ) or ( H ), thus : 
 
 a, tan. 11° 17' 7" . 9.300016 
 , A, cot. 23° 27' 29" . 10.362674 
 
 c,sin. 27° 22' 32" . 9.662690 
 
 180 
 
 ^'^=152° 37' 28" 
 
 PRACTICAL PROBLEMS IN RIGHT-ANGLED SPHERICAL 
 TRIGONOMETRY. 
 
 1. In the right-angled spherical ^^ 
 triangle ABQ, given AB = 118° 21' a^.- 
 
 4", and the angle A = 23° 40' 12", to ^^""--.^,^7 
 find the other parts. B 
 
 . { AQ, 116° 17' 56"', the angle (7, 100° 59' 26"; 
 ' I and BC, 21° 5' 42". 
 
 2. In the right-angled spherical triangle ABC, given 
 AB 53° 14' 20", and the angle A 91° 25' 53", to find 
 the other parts. 
 
 . r J.(7, 91° 4' 9"; the angle (7, 53° 15' 8"; 
 ^"*'l and J5 (7, 91° 47' 10' . 
 
SECTION IV. 357 
 
 8. In the right-angled spherical triangle ABC^ given 
 XB 102° 50' 25'', and the angle A 113° 14' 37", to 
 find the other parts. 
 
 . [AC, 84° 51' 36"; the angle (7, 101° 46' 56"; 
 ^^'l and^C, 113° 46' 27". 
 
 4. In the right-angled spherical triangle ABQ, given 
 A B 48° 24' 16", and BQ 59° 38' 27", to find the 
 other parts. 
 
 . f AC, 70° 23' 42"; the angle A, m"" 20' 40^'; 
 \ and the angle (7, 52° 32' 56". 
 
 5. In the right-angled spherical triangle ABC, given 
 AB 151° 23' 9", and BC 16° 35' U", to find the 
 other parts. 
 
 . (AC, 147° 16' 51"; the angle (7, 117° 37' 25"; 
 ^** t and the angle A, 31° 52' 49". 
 
 6. In the right-angled spherical triangle ABC, given 
 AB 73° 4' 31", and ^C86° 12' 15," to find the other 
 parts. 
 
 . f ^(7, 76° 51' 20"; the angle A, 77° 24' 23"; 
 I and the angle C, 73° 29' 40". 
 
 7. In the right-angled spherical triangle ABC, given 
 AC 118° 32' 12", and AB 47° 26' 35", to find the 
 other parts. 
 
 . r 5(7, 134° 56' 20"; the angle A, 126° 19' 2"; 
 I and the angle C, 56° 58' 44". 
 
 8. In the right-angled spherical triangle ABC, given 
 AB 40° 18' 23", and AC 100° 3' 7", to find the 
 olher parts. 
 
 . f The angle A, 98° 88' 58" ; the angle 
 I (7, 41° 4' 6" ; and 5(7, 103° 13' 52". 
 y. In the right-angled spherical triangle ABC, given 
 AC 61° 3' 22", and the angle A 49° 28' 12", to find 
 the other parts. 
 
 ? ^5, 49° 36' 6" ; the angle (7, 60° 29' 20'' ; 
 ^**\ and 5(7, 41° 41' 32". 
 10. In the right-angled spherical triangle ABC^ given 
 
SPHERICAL TRl JONOMETRr. 
 
 AB 29° 12' 50", and tlie angle 37° 26' 21", to lind 
 the other parts. 
 
 /-Ambiguous; the angle A, Qb"" 27' 57", or its 
 Ans.'l supplement; AC, 53° 24' 13", or its sup- 
 
 l plement; BO, 46° 55^ 2", or its supplement 
 
 11. In the right-angled spherical triangle ABO, given 
 AB 100° 10' 3", and the angle 90° 14' 20", to find 
 the other parts. 
 
 rAO, 100° 9' 52", or its supplement; BO, 
 Ans.< 1° 19' 55'', or its supplement; and the 
 L angle ^,1° 21' 12", or its supplement. 
 
 12. In the right-angled spherical triangle ABO, given 
 AB 54° 21' 35", and the angle 61° 2' 15", to find 
 the other parts. 
 
 rBO, 129° 28' 28", or its supplement; AC, 
 
 An8.< 111° 44' 34", or its supplement; and the 
 
 I angle A, 123° 47' 44", or its supplement. 
 
 13. In the right-angled spherical triangle ABO, given 
 AB 121° 26' 25", and the angle 111° 14' 37", to 
 find the other parts. 
 
 rThe angle A, 136° 0' 5", or its supplement; 
 AnsA AO, m"" 15' 38", or its supplement; ard 
 L BO, 140° 30' 57", or its supplement. 
 
 QUADRANTAL TRIANGLES. 
 
 The solution of right-angled spheri- 
 ral triangles includes, also, the solu- 
 noi! of quadrantal triangles, as may be 
 seeij by inspecting the adjoining fig- 
 ure When we have one quadrantal 
 triangle, we have four, which with one 
 right-angled triangle, fill up.the whole hemisphere^ 
 
 To eflfect the solution of either of the four quadrantal 
 mangles, APO, AP'O, A'PO, or J.'P'(7, it is sufficient 
 to solve the small right-angled spherical triangle ABO, 
 
SECTION IV. 359 
 
 To the half lune AF'B, we add the triangle ABC, 
 and we have the quadrantal triangle AF'C; and by sub- 
 tracting the same from the equal half lune AFB, we 
 have the quadrantal triangle PA 0, 
 
 When we have the side, AC, of the same triangle, we 
 have its supplement, A'C, which is a side of the triangles 
 A'FC, and A'F'O. When we have the side, CB, of 
 the small triangle, by adding it to 90°, we have P'C', a 
 side of the triangle A'P'C\ and subtracting it from 90*^, 
 we have P(7, a side of the triangles APC, and AlPC, 
 
 PROBLEM I. 
 
 In a quadrantal triangle, there are given the quadrantal 
 gidej 90°, a side adjacent, 42° 21', and the angle opposite 
 this last sidey equal to 36° 31'. Required the other parts. 
 
 By this enunciation we cannot decide whether the triangle APQ 
 or AP'Gf is the one required, for AC = 42° 21' belongs equally 
 to both triangles. The angle AFC = AP'C^ 36° 31' = AB 
 
 We operate wholly on the triangle ABC. 
 
 To find the angle A, call it the middle part. 
 Then, R cos. CAB = R &mPA C = cot. J. C tan.^^. 
 
 cot.^C = 42° 21' . 10.040231 
 tan.J^^ == 36° 31' . 9.869473 
 
 cos. CAB = 35° 40' 51" 9.909704 
 
 " 90° 
 
 PAC = 54° 19' 9" 
 PAC= 125° 40' 51" 
 
 To find the angle (7, call it the middle part 
 R coi. A CB = sin. CAB cos. AB, 
 
 Bin.CAB = 35° 40' 51" 
 cos.^^ = 36° 31' 
 
 9.765869 
 9.905085 
 
 COB. A CB = 62° 2' 45" 
 
 180° 
 
 9.670964 
 
 ACP « A'CP = 117° 57' 15" 
 
 
860 
 
 SPHERtCAL TRIGONOMETRY. 
 
 To find the side BO, call it the middle part. 
 Rsin.BO = tan.^j5 cot.^C^. 
 
 tan.^^ == 36° 31' 0" 
 cot ACB = 62° 2' 45" 
 
 9.869473 
 9.724835 
 
 Bin.BC = 23° 8' 11" 
 90° 
 
 9.594308 
 
 PC = 66° 51' 49" 
 PC = 113° 8' 11" 
 
 
 We now have all the sides, and all the angles of the 
 four triangles in question. 
 
 PROBLEM IT. 
 
 In a quadrantal spherical triangle , having given the qiuid- 
 rantal side, 90°, an adjacent side, 115° 09', and the included 
 angle, 115° 55', to find the other parts. 
 
 This enunciation clearly points out 
 the particular triangle A'P'O, A'P' 
 = 90° ; and conceive A'0= 115° 09'. 
 Then the angle F'A'C= 115° 55' = 
 P'D, 
 
 From the angle P'^'^take 90°, or 
 P'A'B, and the remainder is the angle 
 OA'B^ BAO^ 25° 55'. 
 
 We here again operate on the triangle ABQ, A'C, 
 taken from 180°, gives 
 
 64° 51' ^ AC. 
 
 To find EC, we call it the middle part. 
 
 R sin. 5(7= sin.^Csin.^^C. 
 
 Bm.AC = 64° 51' . 9.956744 
 
 Bin.BAG = 25° 55' . 9.640544 
 
 8in.^C = 23° 18' 19" 
 
 90° 
 
 PC « 113° 18' 19" 
 
 9.597288 
 
SECTION IV. 861 
 
 To find AB, we call it the middle part. 
 
 R^m.AB = tan.BCcoi.BAC. 
 
 tau.BC = 23° 18' 19" . y.634251 
 
 ?ot.BAC = 25° 55' . 9.313423 
 
 s'mAB = G2° 2G' 8" . 
 
 8.947674 
 
 180° 
 
 
 AB = 117° 33' 52" = 
 
 the angle AP'a 
 
 To find the angle (7, we call it the middle part. 
 
 R COS. C = QotA C tan.BC. 
 
 cot.AC = 04° 51' 
 
 9.671634 
 
 taii.^C = 23° 18' 19" . 
 
 9.634251 
 
 eos.C = 78° 
 
 9.305885 
 
 18U° 19' 53" . 
 
 
 P'CA = 101° 40' 7" 
 
 
 Thus we have found the side rC = 113° 18' 19" 
 
 The angle AI^C = 117° 33' 52" 
 
 " P'CA' = 101° 40' 7" 
 
 1 
 
 PRACTICAL PROBLEMS. 
 
 1. In a qnadrantal triangle, given the qnadrantal side, 
 )0®, a side adjacent, 67° 3', and the included angle, 49** 
 18', to find the other parts. 
 
 rThe remaining side is 53° 5' 44" ; the angle 
 Ans. < opposite the qnadrantal side, 108° 32' 29"; 
 I and the remaining angle, 60° 48' 54". 
 
 2. In a qnadrantal triangle, given the qnadrantal side, 
 90°, one angle adjacent, 118° 40' 36", and the side op- 
 posite this last-mentioned angle, 113° 2' 28", to find the 
 other p.arts. 
 
 f The remaining side is 54° 38' 57"; the angle 
 Ajis. \ opposite, 51° 2' 35"; and the angle opposite 
 
 I the qnadrantal side 72° 26' 21". 
 8. In a quadrnTital triangle, given the qnadrantal side, 
 31 
 
;62 SPHERICAL TRIGONOMETRY. 
 
 90°, and the two adjacent angles, one 69° 13' 16", the 
 other 72° 12' 4'', to find the other parts. 
 
 C One of the remaining sides is 70° 8' 39". the 
 Ans. < other is 73° 17' 29", and the angle opposite 
 
 I the qiiadrantal side is 96° 13' 23". 
 
 4. In a quadrantal triangle, given the quadrantal side, 
 90°, one adjacent side, 86° 14' 40", and the angle oppo- 
 site to that side, 37° 12' 20", to find the other parts. 
 
 rThe remaining side is 4° 43' 2"; the angle op- 
 Ans. < posite, 2° 51' 23" ; and the angle opposite 
 I the quadrantal side, 142° 42' 3". 
 
 5. In a quadrantal triangle, given the quadrantal side, 
 90°, and the other two sides, one 118° 32' 16", the other 
 67° 48' 40", to find the other parts — the three angles. 
 
 j-The angles are 64° 32' 21", 121° 3' 40", and 
 Ans, < 77° 11' 6" ; the greater angle opposite the 
 ( greater side, of course. 
 
 6. In a quadrantal triangle, given the quadranta^ side, 
 90°, the angle opposite, 104° 41' 17", and one adjacent 
 Bide, 73° 21' 6", to find the other parts. 
 
 J ( Remaining side, 49° 42' 16" ; remaining 
 ^^* I angles, 47° 32' 38", and 67° 56' 13". 
 
 SOLUTION OF OBLIQUE-ANGLED SPHERICAL TRIANGLES. 
 
 All cases of oblique-angled spherical trigonometry 
 may be solved by right-angled Trigonometry, except 
 two; because every oblique-angled spherical triangle is 
 composed of the sum, or the difference, of two right- 
 angled spherical triangles. 
 
 WJien a side and tivo of the angles, ov an angle and twit 
 of the sides are given, to find the other parts, conform to 
 the following directions : 
 
 Let a perpendicular be drawn from an extremity ol a 
 given si le, and opposite a given angle or its supplement; 
 this will form two right-angled spherical triangles : and 
 
SECTION IV. 9C8 
 
 one of them will have its hypotenuse and one of its ad- 
 jacent angles given, from which all its other parts can 
 be computed ; and some of these parts will become as 
 known parts to the other triangle, from which all its 
 parts can be computed. 
 
 To facilitate these computations, we here give a sum- 
 mary of the practical truths demonstrated in the fore- 
 going propositions. 
 
 1. The sines of the sides of spherical triangles are propor- 
 tional to the sines of their opposite angles. 
 
 2. The sines of the segments of the base, made by a per- 
 pendicular from the opposite angle, are proportional to the 
 cotangents of their adjacent angles. 
 
 3. The cosines of the segments of the base are proportional 
 to the cosines of the adjacent sides of the triangle. 
 
 4. The tayigents of the segments of the base are reciprocally 
 proportional to the cotangents of the segments of the vertical 
 angle. 
 
 5. The cosines of the angles at the base are proportional 
 to the sines of the corresponding segments of the vertical 
 angle. 
 
 6. The cosines of the segments of the vertical angle ant 
 proportional to the cotangents of the adjoining sides of the 
 triangle. 
 
 The two cases in which right-angled spherical triangles 
 are not used, are, 
 
 1st. When the three sides are given to find the angles ; 
 and, 
 
 2d. When the three angles are given to find the sides. 
 
 The first of these cases is the most important of all, 
 and for that reason great attention has been given to it, 
 and two series of equations, (T'and U, Prop. 7, Sec. Ill), 
 have been deduced to facilitate its solution. 
 
 As heretofore, let ABC represent any triangle whose 
 angles are d3noted by A, B, and C, and sides by a, b. 
 
364 SPHERICAL TRIGONOMETRY. 
 
 and c ; the side a being opposite L ^> ^^^ side b oppo- 
 site L ^> 6tc. 
 
 EXAMPLES. 
 
 1. In the triangle ABa,a = 70° 4' 18'' ; 6 = 63° 21' 27" ; 
 and c, 59° 16' 23*^ ; required the angle A, 
 
 The formula for this is the first equation in group T, 
 Prop. 7, Sec. Ill, which is 
 
 A _ /E' sin.iS'sin.f/S' — aV J 
 2 \ sin. 6 sin.c / 
 
 "We write the second member of this equation thus 
 
 v/" 
 
 (^\ (.^) (sin.S) sin.f.S'— a) 
 
 showing four distinct factors under the radical. 
 
 The logarithm corresponding to - — -. is that of sm.d 
 
 B 
 
 subtracted from 10; and of - — is that of sin.c sub- 
 
 tracted from 10, which we call sin.eomplement. 
 
 BC==a= 70° 4' 18" 
 
 AB=^c= 59° 16' 28" sin. com. .065697 
 
 J[C= 6 = 63° 21'^ 27" sin. com. .048749 
 
 2)192° 42' 8" 
 
 
 S = 96° 21' 4" sin. 
 
 9.997326 
 
 ,Sf - a = 26° 16' 46" sin. 
 
 9.646158 
 
 
 2)19.757930 
 
 U= 40° 49' 10" COS. 
 
 9.878965 
 
 2 
 
 
 A = 81° 38' 20" 
 
 WTien we apply the equation to find the angle A, we 
 /rrite a first, at the top of the column ; when we apply 
 the equation to find the angle B, we write b at the top 
 of the column. Thus, 
 
SECTION IV 865 
 
 To find the angle B. 
 
 
 sin. a sin.c 
 
 = \/Y-^) (S-) (Bm.S) sin.iS^b) 
 V sin.«/ Vsin.c/ ^ ^ ^ 
 
 b == 03° 21' 27" 
 
 c = 59° 16' 23" 
 
 sin.com. 
 
 . 
 
 .065697 
 
 a= 70° 4' 18" 
 
 sin.com. 
 
 . 
 
 .026875 
 
 2)192° 42' 8" 
 
 
 
 
 S=: 96° 21' 4" 
 
 sin. . 
 
 , 
 
 9.997326 
 
 ^Sf— i>= 32° 59' 37" 
 
 sin. . 
 
 . 
 
 9.736034 
 
 
 
 2) 
 
 19.825872 
 
 iB= 35° 4' 49" 
 
 cos. . 
 
 
 9.912936 
 
 2 
 
 
 
 
 B = 70° 9' 38" 
 
 By the other equation in formulae {T, Prop. 7, Sec. 
 Ill), we can lind tlie angle 0; but, for the sake of variety, 
 we will find the angle (7 by the application of the third 
 equation in formulae (C/, Prop. 7, Sec. III). 
 
 ^_ / R's\Ti.(S—b) sm.(S—a) 
 
 \ Kin 7i fiin ^ 
 
 =v(i£z)( 
 
 c= 59° 16' 23" 
 0.= 70° 4' 18" 
 b = 63° 21' 27" 
 
 -^)sin. (^_t)8in.(^— a; 
 sin.ft/ V > V 
 
 sin.com. .026817 
 sin.com. .048479 
 
 2)192° 42' 8" 
 
 ^=96° 21' 4" 
 
 ^—a = 26° 16' 46" 
 
 ^—6 = 32° 59' 37" 
 
 sin. 
 sin. 
 
 . 9.646158 
 . 9.736034 
 
 2 ) 19.457488 
 
 }C=32° 23' 17" 
 2 
 
 sin 
 
 . 9.778744 
 
 C7=64° 46' 34" 
 85* 
 
366 SPHERICAL TRIGONOMETRY. 
 
 To sLow the harmony and practical utility of these two 
 Bets of equations, we will find the angle A^ from the 
 equation 
 
 ^'"-^^ =\/ (si) (s-fe) "''•('^-*) ^-•('^-'')- 
 
 a = 70° 4' 18" 
 
 h = 63° 21' 27" sin.com. .048749 
 
 c= 59° 16' 23" 
 
 sin.com. 
 
 .065697 
 
 2)192° 42' 8" 
 
 
 
 .S'= 96° 21' 4" 
 
 
 
 •>S'— 6= 32° 59' 37" 
 
 sin 
 
 9.736034 
 
 ;Sf~-c= 37° 4' 41" 
 
 sin. 
 
 9.780247 
 
 
 2 " 
 
 ) 19.630727 
 
 U = 40° 49' 10" 
 2 
 
 sin. 
 
 9.815363 
 
 A = 81° 38' 20" 
 
 
 2. In a spherical triangle ABC^ given the angle J., 38' 
 19' 18"; the angle B, 48° 0' 10''; and the angle (7, 
 121° 8' 6"; to find the sides a, 6, c. 
 
 By passing to the triangle polar to this, we have, 
 (Prop. 6, Sec. I, Spherical Geometry), 
 
 A= 38° 19' 18" supplement 141° 40' 42" 
 j5=: 48° 0' 10" supplement 131° 59' 50" 
 C= 121° 8' 6" supplement 58° 51' 54" 
 
 We now find the angles to the spherical triangle, 
 the sides of which are these supplements. 
 Thus, . 141° 40' 42" 
 
 131° 59' 
 58° 51' 
 
 50" 
 54" 
 
 sin.com. 
 sin.com. 
 
 .128909 
 .067551 
 
 ) 332° 32' 
 
 26" 
 
 
 
 166° 16' 
 24° 35' 
 
 13" 
 31" 
 
 sin. • 
 sin. 
 
 9.375375 
 
 9.619253 
 
 2)19.191088 
 
 66^ 47' 
 
 37 r 
 
 cos. 
 
 9 595544 
 
SECTION IV. 867 
 
 60° 47' 37 J" 
 
 2r_ 
 
 angle = 121° 35' 15" 
 
 supp. = 58° 24' 45" = a of the original triangle. 
 IiJ the same manner we find h = 60° 14' 25"; c = 89° 1 14". 
 
 It is perhaps better to avoid this indirect process of 
 computing the sides of a spherical triangle when the 
 angles are given, by the application of the equations in 
 group V or IF, Prop. 8, Sec. III. We will illustrate 
 their use by applying the second equation in group (TF), 
 for computing the side h. This equation is 
 
 fiin \h— ( 
 
 -cos.^S' cos.{S — B)\\ 
 
 
 siu.A sin.C / 
 
 A = 38° 19' 18" 
 
 
 B = 48° 0' 10" 
 
 
 6^=121° 8' 6" 
 
 
 2 ) 207° 27' 34" 
 ^=103°43'47"--cos.>S'= -f Bin.l3° 43' 47"= 9.375376 
 B = 48° 0' 10" cos.(S-^B) = 55° 43' 37" = 9.750612 
 ( S—B) = 55° 43' 37" 2 ) 19.125988 
 
 square rooi; = 9.562994 
 sin.^ = 38° 19' 18" = 9.792445 
 sin. C == 121° 8' 6" = 9.932443 
 2)1 9.7248 88 
 square root = 9.862444 = 9.862444 
 diff. — 1.700550 
 Add 10, for radius of the table, 10 
 Tabular sin. ii = 30° 7' 14" ^TgiTOOsHo 
 
 2 
 
 h = 60° 14' 28", nearly. 
 
 PRACTICAL PROBLEMS. 
 
 1. In any triangle, ABC, whose sides are a, 5, c, given 
 6 = 118° 2' 14'', (? = 120° 18' 33", and the included angle 
 A = 27° 22' 34", to find the other parts. 
 
368 SPHERICAL TRIGONOMETRY. 
 
 Ans. 
 
 '■{' 
 
 Ans. 
 
 = 23° 57' 13'', angle B - 91° 26' 44, and C - 
 
 102° 5' 52". 
 
 2. Given, ^ = 81° 38' 17", i? = 70° 9' 38", and Q^ 
 64° 46' 32", to find tlie sides a, h, c. 
 ^^ r a == 70° 4' 13", h = 63° 21' 24", and c = 59° 16' 
 
 8. Given, the three sides, a = 93° 27' 34", h --« 100° t 
 
 26", and c = 96° 14' 50", to find the angles A, B, and (7. 
 
 ^^ r J. = 94° 39' 4", ^ = 100° 32' 19", and (7= 96° 
 
 * I 58' 35". 
 
 4. Given, two sides, b = 84° 16', e = 81° 12', and the 
 angle (7= 80° 28', to find the other parts. 
 
 ^The result is amhiguous, for we may consider 
 the angle B as acute or obtuse. If the angle 
 B is acute, then JL = 97° 13' 45'',- J5 = 83° 11' 
 24", and a = 96° 13' 33". If ^ is obtuse, then 
 A = 21° 16' 43", B = 96° 48' 36", and a = 
 21° 19' 29". 
 
 5. Given, one side, (?=64° 26', and the angles adjacent, 
 A = 49°, and B = 52°, to find the other parts. 
 
 A (h = 45° 56' 46", a = 43° 29' 49", and (7= 98<» 
 'I 28' 4". 
 
 6. Given, the three sides, a = 90°, b= 90°, (j = 90°, to 
 find the angles A, B, and 0. 
 
 Ans. A = 90°, B = 90°, and (7= dC\ 
 
 7. Given, the two sides, a = 77° 25' 11", c = 128° 13' 
 47", and the angle (7 = 131° 11' 12", to find the other 
 parts. 
 
 ^^ f 5 = 84«> 29' 20", A = 69° 13' 59' and B = 72° 28' 
 
 * I 42". 
 
 8. Given, the three sides, a = 68° 34' 13", h = 59° 
 21' 18", and c = 112° 16' 32", to find the angles A, B, 
 and a 
 
 . (A = 45° 26' 38", B = 41° 11' 30' , (7 = 134° 53' 
 
SECTION IV. 869 
 
 9. Given, a = 89^ 2V 37'', 6= 97° 18' 39", o-= 86° 53' 
 46", to find A, B, and C. 
 
 ^^^ r ^ = 88° 57' 20", B = 97° 21' 26", C = 86° 47' 
 
 10. Giv^en, a = 31° 26' 41", c = 43° 22' 13", and the 
 an^le ^ = 12° 16', to find the other parts. 
 
 j-Amhiguous; b = 73° 7' 34", or 12° 17' 40"; 
 AicsJ angle 5=157° 3' 44", or 4° 58' 30"; C= 16° 
 I 14' 27", or 163° 45' 33". 
 
 11. In a triangle, ABC, we have the angle J. =56° 18' 
 40", 5= 39° 10' 38"; AI), one of the segments of the 
 base, is 32° 54' 16". The point 2> fiills upon the base 
 AB, and the angle C is obtuse.. Required the sides of 
 the tiiangle and the angle C. 
 
 /Ambiguous; (7=135^25', or 
 
 135^ 57' ; c=122^ 29', or 
 
 ' ) 123° 19' ; a= 89° 40', or 
 
 ( 90° 20' ; h= 49° 23' 41". 
 
 12. Given, A = 80° 10' 10", B = 58° 48' 36", (7 = 91<» 
 62' 42", to find a, 5, and c. 
 
 Atu. a«n79° 38' 22", 5 = 58° 39' 16", c = 86° 12' 50". 
 
B70 
 
 SPHERICAL TKIGO^^OMETRY, 
 
 SECTION V, 
 
 APPLICATIONS OF SPHERICAL TRIGONOMETRY TO 
 ASTRONOMY AND GEOGRAPHY. 
 
 SPHERICAL TRIGONOMETRY APPLIED TO ASTRONOMY. 
 
 Sphekical Trigonometry becomes a science of incalcu- 
 lable importance in its connection with geography, navi- 
 gation, and astronomy; for neither of these subjects can 
 be understood without it ; and to stimulate the student 
 to a study of the science, we here attempt to give him a 
 glimpse at some of its points of application. 
 
 Let the lines in the 
 annexed figure represent 
 circles in the heavens 
 above and around us. 
 
 Let Z be the zenith, or 
 the point just overhead, 
 Hch the horizon, FZR 
 ihe meridian in the hea- 
 vens, and P the pole of 
 the celestial equator; Ph 
 is the latitude of the 
 observer, and PZ is the 
 
 co.latitude. Qcq is a portion of the equator, and tne 
 dotted, curved line, mS' S^ parallel to the equator, is the 
 parallel of the sun's declination at some particular time ; 
 and in this figure the sun's declination is supposed to be 
 north. By the revolution of the earth on its axis, the 
 
SECTIOK V. 371 
 
 Bun is apparently brought from ihe horizon, at S, to the 
 meridian, at m ; and from thence it is carried down on 
 the same curv^e, on the other side of the meridian ; and 
 this apparent motion of the sun (or of any other celestial 
 body,) makes angles at the pole P, which are in direct 
 proportion to their times of description. 
 
 The apparent straight line, Zc, is what is denominated, 
 in astronomy, the prime vertical; that is, the east and west 
 line through the zenith, passing through the east and west 
 points in the horizon. 
 
 "When the latitude of the place is north, and the decli- 
 nation is also north, as is represented in this figure, the 
 sun rises and sets on the horizon to the north of the east 
 and west points, and the distance is measured by the arc, 
 cSy on the horizon. 
 
 This arc can be found by means of the right-angled 
 spherical triangle cqS, right-angled at q. Sq is the sun's 
 declination, and the angle Scq is equal to the co.latitude 
 of the place ; for the angle Pch is the latitude, and the 
 angle Scq is its complement. 
 
 The side cq, a portion of the equator, measures tho 
 angle cPq, the time of the sun's rising or setting beforo 
 or after six o'clock, apparent time. Thus we perceive that 
 this little triangle, cSq, is a very important one. 
 
 When the . sun is exactly east or west, it can be deter 
 
 mined by the triangle ZPS' ', the side PZ is known, 
 
 being the co.latitude ; the angle PZS' is a right angle, 
 
 • and the side PS' is the sun's polar distance. Here, then, 
 
 'are the hj^otenuse and side of a right-angled spherica. 
 
 triangle given, from which the other parts can be com- 
 
 tputed. The angle ZPS' is the time from noon, and thp 
 side ZS' is the sun's zenith distance at that time. 
 
 The following problems are given, to illustrate ttie 
 important applications that can be made of the righfr 
 angled triangle cqS. 
 
372 SPHERICAL TRIGONOMETR^i. 
 
 PRACTICAL PROBLEMS. 
 
 1. At what time will the sun rise and set in Lat. 48* 
 NT., w^hen its declination is 21° E".? 
 
 In this problem, we must make 5'jS'=21°, P/i=48°=the angle 
 Vch. Then the angle Scq = 42°. It is required to find the arc 
 cq, and convert it into time at the rate of four minutes to a degree. 
 This will give the apparent time after six o'clock that the gun sets, 
 and the apparent time before six o'clock that the sun rises, (no 
 allowance being made for refraction). 
 Making cq the middle part, we have 
 
 R m\.cq = tan. 21° tan.48° 
 tan.21° = 9.584177 
 tan.48° = 10.045563 
 
 cj= 25° 14' 5" = 25.2846° 9.629740, rej. siting 10, 
 4 
 
 1* 40"* 56* 
 Adding to 6* 
 
 Sun sets p. M., 7* 40"* 56', apparent time, 
 
 From 6^^ 
 
 Taking 1'^ 40"' 56' 
 
 Sun rises A. M., 4* 19"* 4*, apparent time. 
 From this we derive the following rule for finding the apparent 
 time of sunrise and sunset, assuming that the declination under- 
 goes no change in the interval between these instants, which we 
 may do without much error. 
 
 RULE. 
 
 To the logarithmic tangent of the suns declination, add tht 
 logarithmic tangent of the latitude of the observer ; and, after 
 rejecting ten from the result, find from the tables the arc of 
 which this is the logarithmic sine, and convert it into time ai 
 the rate of 4 minutes to a degree. 
 
 This time, added to 6 o'clock, will give the time of sunset, 
 ^nd^ subtracted from 6 o'clock, will give the time of sunrise, 
 
SECTION V. 373 
 
 when the latitude and declination are both north or both 
 touth , but when one is north, and the other south, the addi- 
 tion gives the time of sunrise, and the subtraction the time of 
 sunset, 
 
 2. At what time will the sun set when its declination 
 is 23° 12' K, and the latitude of the place is 42° 40' K? 
 
 Ans. 7^ SS"* 4*, apparent time. 
 
 3. ^Yliat will he the time of sunset for places whose 
 latitude is 42° 40' N., when the sun's declination is 15° 
 21' south ? Ans. 5'' 1"* 23*, apparent time. 
 
 4. What will be the time of sunrise and sunset foi 
 places whose latitude is 52° 30' JS"., when the sun's decli- 
 nation is 18° 42' south ? 
 
 . f Rises 7'' 44"* 42', 1 , . . 
 
 ^^*- I Sets 4. 15. 18^; I apparent time. 
 
 5. What will be the time of sunset and of sunrise at 
 St. Petersburgh, in lat. 59° 56', north, when the sun's 
 declination is 23° 24', north? What will be its ampli- 
 tude at these instants ? Also, at what hours will it be 
 due east and west, and what will be its altitude at such 
 times ? 
 
 Sun sets at 9* 13'" 30' p.m. \ apparent 
 
 Ans. 
 
 ' P.M. 1 
 • A.M. / 
 
 Sun rises at 2" 46"* 30' a.m. J time 
 Sun rises N. of east 1 59° 9f>' 18" 
 Sun sets N. of west / 
 Sun is east at 6* 58"* 2' a.m. 
 Sun is west at 5* 1"* 58* p.m. 
 1^ Alt. when east and west is 27° 18' 57". 
 
 UN THE AlPLICATrON OF OBLIQUE-ANGLED SPHERICAL 
 TRIANGLES. 
 
 One of the most important problems in navigation 
 abd astronomy, is the determination of the formula foi 
 32 
 
874 
 
 SPIlEmCAL TRIGONOMETRY. 
 
 time. This problem will 
 be understood by the tri- 
 angle PZS. When the 
 sun is on the meridian, it 
 is then apparent noon. 
 When not on the meri- 
 dian, we can determine 
 tl e interval from noon, 
 by means of the triangle 
 FZS\ for we can know 
 all its sides; and the 
 angle at P, changed into 
 
 time at the rate of 15° to one hour, will give the time 
 from apparent noon, when any particular altitude, as 
 TS^ may have been observed. P/S'is known, by the sun's 
 declination at about the time; and FZ is known, if the 
 observer knows his latitude. 
 
 Having these three sides, we can always find the sought 
 angle at the pole, by the equations already given in 
 formulae (2^, or £/, Prop. 7, Sec. Ill); but these formulae 
 require the use of the co. latitude and the eo.altitude, and 
 the practical navigator is very averse to taking the trou- 
 ble of finding the complements of arcs, when he is quite 
 certain that formulae can be made, comprising but the 
 arcs themselves. 
 
 The practical man, also, very properly demands the 
 most concise practical results. No matter how much 
 labor is spent in theorizing, provided we arrive at prac- 
 tical brevity ; and for the especial accommodation of 
 .seamen, the following formula for finding time has been 
 deduced. 
 
 From the symmetrical formulae {S') Prop. T, Sec. HI, 
 rro have, 
 
 p cos.^>S'— cos.P^ C OS. PS 
 cos / - - -^;^pz^,^jys 
 
 Kow, in place of cos. ZS. we take sin.AS'jT, which is, in 
 
SECTION V. o76 
 
 fact, the same thing ; and in pla«^G of cob.PZ, we take 
 sin. 1 at., which is also the same. 
 
 In short, let A = the altitude of the sun, L = the la- 
 Litude of the observer, and D = the sun's polar distance. 
 
 rri^ n SUl.A sin.X .COS.i> 
 
 Then, cos.P = -.,r—, — =- — 
 
 cos.L Bin.D 
 
 But, 28in.4P = 1 — cos.P. (See Eq. 32, Prop. 2, 
 Sec. I, Plane Trig.) 
 
 Therefore, 
 
 rt . , , 75 - sin.J. — sin.X cos.i) 
 28m.» JP = 1 .^—. — y- 
 
 cos.i/ sin.i> 
 
 (cos.X sm.D + sin.i cos.i)) — sin.^ 
 
 cos.X sin.i) 
 __ sin.(i^ + B) — sin.^ 
 coa.Lain.D 
 (Considering (L + i>) as a single arc, and (appljmg 
 Equation 16, Sec. I, Plane Trig.), we have, after dividing 
 by 2, 
 
 C08.( -2 ) sin.( ). 
 
 sin.'JP= 5^—: — f{ 
 
 ^ cos,L sin.i> 
 
 ^ , L + D-A^ L + B +A . 
 
 But, 2 2 ' 
 
 and if we assume S= ^ , 
 
 , ni •HID cos.aS' 8in.(AS' — A) 
 
 we shall have, sin.' JP = =-A — jr — '- 
 
 ' cos.L sin.D 
 
 or, Bin.iP = s/'lE^^W^. 
 ' ^ ^ COS.L sm.i> 
 
 This is the final result, when the radius is unity ; when 
 ^.he radius is i? times greater, then the sin.iP will be B 
 limes greater ; and, therefore, the value of this sine, cor- 
 responding to our tables, is, 
 
 8in.iP = v/G^) (^ji) co^.SBm.(S-A). 
 * ^ Ncos.i^ \sin.i>^ 
 
576 SPHERICAL TRIGONOMETRY. 
 
 PRACTICAL PROBLEMS. 
 
 1. In lat. 39° 6' 20" I^ortli, when the sun's declination 
 was 12° 3' 10" :tTorth, the true altitude* of the sun's cen- 
 ter was observed to be 30° 10' 40", rising. What waa 
 tlie api)arent time ? 
 
 Alt. 30° 10' 30" 
 Lat. 39° 6' 20" 
 RD. 77° 56' 50" 
 
 2 ) 147° 13' 40" 
 
 S = 73° 36' 50" 
 
 cos.com. .110146 
 siu.com. .009680 
 
 COS. 9.450416 
 
 A.) = 43° 26' 20" 
 
 sin. 9.837299 
 
 30° 22' 5" 
 
 9 
 
 2 ) 19.407541 
 sin. 9.703770 
 
 R = 60° 44' 10" 
 
 This angle, converted into time at the rate of 15^ to 
 one hour, or 4 minutes to 1°, gives 4* 2"* 56' from appa- 
 rent noon; and as the sun was rising, it was before 
 noon or 
 
 T 57'"4'A.M. 
 
 If to this the equation of time were applied, we should 
 have the mean time ; and if such time were compared 
 with that of a clock or watch, we could determine its 
 error. A good observer, with a good instrument, can, 
 in this manner, determine the local time within 4 or 6 
 seconds. 
 
 2. In lat. 40° 21' North, the true altitude of the sun, m 
 tho forenoon, was found to be 36° 12^, when the decliua- 
 
 * The instrument used, the manner of taking the altitude, its cor- 
 rection for refraction, semi-diameter, and other practical or circum- 
 stantial details, do not belong to a work i>f this kind, but tf a work ob 
 Practical Astronomy or Navigation. 
 
SECTION V. 377 
 
 tion of the sun was 3° 20' South. What was the appa- 
 rent time ? Ans. 9" 42'" 40* a. m. 
 
 3. In latitude 21° 2' South, when the sun's declination 
 was 18° 32' North, the true altitude, in the afternoon, 
 was found to be 40° 8'. What was the apparent time 
 of day ? Ans. 2'^ 3"* 57" p. m. 
 
 SPHERICAL TRIGONOMETRY APPLIED TO GEOGRAPHY. 
 
 If we wish to find the shortest distance between two 
 places over the surface of the earth, when the dis- 
 tance is considerable, we must employ Spherical Trigo- 
 nometry. 
 
 Suppose the least distance between Rome and New 
 Orleans is required; we would first find the distance in 
 degrees and parts of a degree, and then multiply that 
 distance by the number of miles in one degree. 
 
 In the solution of this problem, it is supposed that we 
 have the latitude and longitude of both places. Then 
 the distances, in degrees, from the north pole of the 
 earth to Rome and to New Orleans are the two sides of 
 a spherical triangle, the difterence of longitude of the 
 two places is the angle at the pole included between 
 these sides, and the problem is, to determine the third 
 side of a spherical triangle, when we have two sides and 
 the included angle given. 
 
 Let P be the north pole, B the position of Rome, and 
 N that of New Orleans. 
 
 Lat. Long. 
 
 New Orleans, 29° 57' 30" N. 90° W. 
 
 Rome, 41° 53' 54" N. 12° 28' 40" E. 
 
 Whence, FE = 48° 6' 6", 
 
 PiV^ = 60° 2' 30". 
 
 Angle NFR = 102° 28' 40". 
 
 32* 
 
378 
 
 SPHERICAL TRIGONOxAIETItY. 
 
 We now employ Na- 
 pier's 1st and 2cl Analo- 
 gies, and find the dis- 
 tance, in degrees, to be 
 78^ 48' 15". This re- 
 duced to miles, at the 
 rate of 69.16 miles to I • ) 
 
 the d ^gree, will make '; / 
 
 I he distance 5450.1 \ / 
 
 miles. \ / 
 
 The angle at iV is ^, / 
 
 47° 48' 13" and at 11, 59° '^-^. ^^ 
 
 34' 47''. ^^- '" 
 
 The third side of a spherical triangle can be found by 
 a single formula, as we shall see by inspecting formulae 
 {S') Prop. 7, Sec. III. 
 
 Let C be the included angle, and c the unknown side 
 opposite; then, 
 
 ^ COS. 6? — COS. a cos. 5 
 
 COS. (7 = , T—, 
 
 sin. a sm.o 
 
 Adding 1 to each member, and reducing, observing at 
 
 the same time that 1 -f cos. (7= 2cos.'^|(7, we have, 
 
 sin.(X sin. 5 — cos.a cos.5 -f cos.c 
 
 2cos.'^i(7 = 
 
 sin. a sin.6 
 
 Whence, 2cos.*J(7 sin.a sin.6 = cos.c — cos.(a-f-5); 
 or, cos.(? = COS. (a + 6) + 2cos.'^ J(7 sin. a sin. 5. 
 
 The second member of this equation is the algebraic 
 sum of two decimal fractions, and expresses the value of 
 the natural cosine of the side sought. 
 
 This case of Spherical Trigonometry, namely, that in 
 which two sides and the included angle are given, to 
 find the third side, is very extensively used in practical 
 astronomy, in finding the angular distance of the moon 
 from the sun, stars, and planets. For this pui'pose, the 
 right ascension and declination of each body must be 
 
SECTION V. 
 
 879 
 
 found for the same moment of absolute time. Their 
 difference in right ascen- 
 sion gives the included 
 angle, P, at the celestial 
 pole. The declination 
 subtracted from 9.0°, if it 
 be north, and added to 
 90°, if it be south, will 
 give the sides, PZ and 
 
 Pas'. 
 
 In the following exam- 
 ples, we give the right 
 ascension and declination 
 of the bodies, and from 
 
 these the student is required to compute the distance 
 between them. 
 
 The right ascensions are given in time. Their differ- 
 ence must be changed to degrees for the included angle. 
 
 MEAN TIME GEEENWICH. 
 
 June 24, 1860. 
 
 moon's 
 
 JUPITER'S 
 
 
 R.A. 
 
 Dec. 
 
 E. A. 
 
 Dec. 
 
 Distanea. 
 
 h. m. 8. 
 
 o / tr 
 
 h. m. 8. 
 
 o / // 
 
 O If 
 
 A.t neon, 10 51 36.5 
 
 3 35 24 N. 
 
 8 4 27.6 
 
 20 51 36.8 N. 
 
 44 8 IS 
 
 « 3 h., 10 58 1 
 
 2 47 43 
 
 8 4 34.2 
 
 20 51 17.8 
 
 45 53 47 
 
 « 6 h., 11 4 24.6 
 
 1 59 56.2 
 
 8 4 40.8 
 
 20 50 58.7 
 
 47 39 18 
 
 ' 8h., 11 10 47.6 
 
 1 12 6.1 
 
 8 4 47.4 
 
 20 50 39.6 
 
 49 24 4S 
 
 
 Octohei 
 
 - 6, 1860. 
 
 ^R.A. 
 
 Dee. 
 
 R.A. 
 
 Dec. 
 
 Distance. 
 
 h. m. s. 
 
 o / /» 
 
 h. m. 8. 
 
 o 1 n 
 
 O / M 
 
 ^t noon, 5 41 20.8 
 
 26 8 ON. 
 
 12 49 29.3 
 
 5 18 42.6 S. 
 
 107 37 3 
 
 « 3 h., 6 48 30.1 
 
 26 3 20 
 
 12 49 56.7 
 
 5 21 85.4 
 
 106 8 19 
 
 " 6 h., 5 55 4« 
 
 25 57 19.4 
 
 12 50 24.1 
 
 5 24 28.2 
 
 104 39 19 
 
 « 9 h., 6 2 50.5 
 
 25 49 58.1 
 
 12 50 51.4 
 
 5 27 20.9 
 
 103 10 
 
 '•12h., 6 10 1.3 
 
 25 41 15.8 
 
 12 51 19.0 
 
 
 5 30 13.6 
 
 101 40 Sf 
 
380 SPHERICAL TllIGONOMETRY. 
 
 SECTION VI 
 
 REGULAR POLYEDRONS 
 
 A Reg^ar Polyedron is a polyedron having all its faces equa! 
 and regular polygons, and all its polyedral angles equal. 
 
 The sum of all the plane angles bounding any polyedral angle w 
 less than four right angles ; and as the angle of the equilateral tri- 
 angle is I of a right angle, we have § X 8<4, | x 4<^4, and | x 5<^4 ; 
 but I x 6=4, I X 7>4, and so on. Hence, it follows that three, 
 and only three, polyedral angles may be formed, having the equi- 
 lateral triangle for faces; namely, a triedral angle and polyedral 
 angles of four and of five faces. 
 
 There are, therefore, three distinct regular polyedrons bounded 
 by the equilateral triangle. 
 
 1. The Tetraedron, having four faces and four solid angles. 
 
 2. The Octaedron, having eight faces and six solid angles. 
 
 3. The Icosaedron, having twenty faces and twenty solid angles. 
 With right plane angles we can form only a triedral angle ; hence, 
 
 with equal squares we may bound a solid having six faces and eight 
 equal triedral angles. This solid is called the Hexaedron. 
 
 The angle of the regular pentagon being f of a right angle, we 
 have fx3<;^4; but 4x4^4; hence, with plane angles eqaal t: 
 those of the regular pentagon, we can form only a triedral angle. 
 The solid bounded by twelve regular pentagons, and having twenty 
 solid angles, is called the DodecaedroiL 
 
 There are, then, but five regular polyedrons, viz. : The tetraedron^ 
 the octaedron, and the icosaedron^ each of whi(jh has the equilateral 
 triangle for faces ; the hexaedron, whose faces are equal squares, 
 and the dodecaedron, whose faces are equal regular pentagons. 
 
 It is ob\riou8 that a sphere may be circumscribed about, or in- 
 scribed within, any of these regular solids, and converselv : and 
 
SECTION VI 
 
 381 
 
 that these spheres will have a common center, which may also be 
 taken as the center of the polycdron. 
 
 Anyreirular polyedron maybe regarded as made up of a number 
 of regular pyramids, whose bases are severally the faces of the 
 polyedron, and whose common vertex is its center. Each of these 
 pyramids will have, for its altitude, the radius of the inscribed 
 sphere; and since the volume of the pyramid is measured by one 
 third of the product of its base and altitude, it follows (hat the 
 Tolume of any regular polyedron is measured by its surface multi- 
 plied by one third of the radius of the inscribed sphere. 
 
 PROBLEM. 
 
 Given, the name of a recjular 'polyedron, and the side of the hound- 
 xng polygon, to find the inclination of its faces; the radii of the in- 
 scribed and circumscribed splicres ; the area of its surface ; and its 
 volum . 
 
 Let ABha the intersection of two adjacent fiices of the polye- 
 dron, and C and D the centers of these faces, being the center 
 of the polyedron. Draw the radii, 
 00 and 0Z>, of the inscribed, and 
 the radii OA and 0^,of the circum- 
 scribed sphere ; also from C and i) 
 let fall the perpendiculars OE and 
 DE, on the edge AB, and draw OE; 
 then will the angle DEO measure 
 the inclination of the faces of the 
 polyedron, and the angle DEO is 
 one half of this inclination. 
 
 Let 1 denote the inclination of the 
 faces, m the number of faces which 
 meet to form a polyedral angle, n the 
 nuiibbr of sides in each face, and 
 suppose the edge of the polyedron to 
 be unity. 
 
 The surface of the sphere of which is the center, and radius 
 unity, will form, by its int( rsections with the planes, AOE, AOD^ 
 DOE, the right-angled spherical triangle dae, right-angled at e 
 in the right-angled triangle DEO, the angle DOE is equal to 
 
382 SPHERICAL TRIGONOMETRY. 
 
 90° — Z)^0 = 90° — i/, 
 and is measured by the arc de. The angle daCj of the spherical 
 
 kriangle, is equal to , and the angle ade = ^ ^ 
 
 2m' ° 2n 
 
 Now, by Napier's Rules we have 
 
 QO^.dae = sin.acZe cos.c?e. 
 
 or, cos.rfe = _ ; ( 1 ) 
 
 sin.ade 
 
 and, cos.CT^ = cot.f/ae cot.a^c (2) 
 
 Substituting in eq. ( 1 ), for the angles dae and ade, their values, 
 
 we find 
 
 cos.360^ 
 
 ^m fox 
 
 - sin.d60° 
 
 Equation (3) gives the value of the sine of one half of the incli- 
 nation of the planes ; .and by means of this equation we may readily 
 find the radii of the inscribed and circumscribed spheres. 
 
 In the triangle BED, we have 
 
 nE= BE cot.BDE = icot. 5^, 
 
 2n 
 amce AB = 1, and BE = iAB. 
 In the triangle DOE, we have 
 
 on = BE tan. U = icot. ^ tan.Ji (4) 
 
 2n 
 
 From the triangle A OD, we find 
 
 cos.BOA : 1 :: OD : OA 
 
 whence OA = ?.:?__ 
 
 cos.Z^O^ 
 
 But the angle DOA is measured by the arc ad-, hence, substi- 
 tuting in this last equation the values of cos.DOA and OD, takes 
 from eqs. (2) and (4), we have 
 
 (?^ = »tan.i7cot. ?^ X ^ ^ 
 
 2n cot.360° cot.360' 
 
 2m 2n 
 
 ^tan.J/tan.??^°, (5) 
 
 2m 
 
 by writing tan. for — , and reducing, 
 cot. 
 
SECTION IV. 383 
 
 Equation (4) gives the value of OD, the radius of the inscribed 
 sphere, and equation (5) gives that of OA^ the radius of the cir- 
 cumscribed sphere. The area of one of the faces of the polyedron 
 is equal to one half of the apothegm multiplied by the perimeter. 
 
 360° 
 
 The apothegm, as found above, is equal to J cot. ; hence, we 
 
 2n 
 
 onrvo 
 
 ba'e In X \ cot , for the area of one of the faces; and multi- 
 
 2/1 
 
 plying this by the number of faces of -the polyedron, we shall have 
 
 the expression for its entire area. The expression for the surface 
 
 multiplied by one third of the radius of the inscribed sphere, gives 
 
 the measure of the volume of the polyedron. 
 
 In what precedes, we have supposed the edge of the polyedron 
 to be unity. Having found the radii of the inscribed and circum- 
 scribed spheres, the surfaces, and the volumes of such polyedrons, 
 to determine the radii, surfaces, and volumes of regular polyedrons 
 having any edge whatever, we have merely to remember that the 
 homologous dimensions of similar bodies are proportional ; their 
 surfaces are as the squares of these dimensions ; and their volumes 
 as the cubes of the same. 
 
 Formula (3) gives, for the inclination of the adjacent faces of 
 
 The Tetraedron, 70° 
 
 31^ 44^^ 
 
 " Ilexaedron, 90° 
 
 00^ 00^^ 
 
 " Octaedron, 109° 
 
 28^ 18^^ 
 
 " Dodecaedron, 116° 
 
 33^ 54^^ 
 
 " Icosaedron, 138° 
 
 IV 2y' 
 
 The subjoined table gives 
 
 the surfaces and volumes of the legalav 
 
 oolyedrons, when the edge is unity. 
 
 
 
 Surfaces. 
 
 Volnmes. 
 
 Tetraedron, 
 
 1.7320508 
 
 0.1178513 
 
 Ilexaedron, 
 
 6.0000000 
 
 1.0000000 
 
 Octaedron, . 
 
 3.4641016 
 
 0.4714045 
 
 Dodecaedron. 
 
 20.6457288 
 
 7.6631189 
 
 Icosaedron, 
 
 8.6602540 
 
 2.1816%0 
 
CONTENTS. 
 
 PART I. 
 PLANE TRIGONOMETEY. 
 
 SECTION I. Tagk 
 
 Elementnry Principles 244 
 
 Dcfiniliuns 24'} 
 
 Propositions 248 
 
 Equations for tlie 8inos of the Angles 260 
 
 Natural ^ines, Cosines, etc 265 
 
 Trigonometrical Lines for Arcs exceeding 9(P 270 
 
 S ECTION II. 
 
 Plane Trigonometry, Practically Applied 272 
 
 Logarithms 278 
 
 GENERAL APPLICATIONS 'tt-ITII TIIK rSK OF LOGARITHMS. 
 
 L Eiglit-Anded Trig«mometry 2S8 
 
 II. Oblique- Angled Triironometry 291 
 
 Practical Problems 295 
 
 SECTION I II. 
 
 Application of Trigonometry to Measuring Heights and Distimces 298 
 
 Practical Problems 305 
 
 PART II. 
 
 SPHERICAL GEOMETRY AND TRIGONOMETRY. 
 
 SECTION I. 
 
 Spherical Geometry 310 
 
 SECTION II. 
 
 Eight-Angled Spherical Trigonometry 830 
 
 Napier's Circular Parts 335 
 
 SECTION III 
 
 Oblique-Anglod Spherical Trigonometry 837 
 
 Napier's Analogies o48 
 
 SECTION I y. 
 
 Spherical Trigonometry Applied— Solution of Kight-Angled Spherical Triangles 353 
 
 Practical Problems ^<5 
 
 SHlution of Quadrantal Triangle.^ 358 
 
 Practical Problems 361 
 
 Solution of Oblique-Angled Spherical Triangles 362 
 
 Practical Problems 367 
 
 SECTION V. 
 
 Spherical Trigonom'^try applied to Astronomy 870 
 
 Application of Obliqne-Anirled Spherical Triangles 373 
 
 Spherical Trigonometry applied to Geography 37T 
 
 Table of Mean Time at Greenwich 879 
 
 SECTION YI. 
 
 Regular Polyedrons ■< 3S0 
 
LOGARITHMIC TABLES; 
 
 ALSO ▲ TABLE OV 
 
 NATURAL AND LOGARITHMIC 
 
 SINES, COSINES, AND TANGENTS 
 
 TO EVERY MINUTE OF THE QUADRANT. 
 

 1 
 
 LOGARITHMS OF NUMBERS 
 
 
 noM 
 
 
 1 TO 10000. 
 
 N. 
 1 
 
 Log. 
 
 N. 
 
 Log. 
 
 i N- 
 
 Log. 
 
 N. 
 
 Ix,g. 
 
 000000 
 
 26 
 
 1 414973 ! 61 
 
 1 707570 
 
 76 
 
 1 880814 i 
 
 2 
 
 301030 
 
 27 
 
 1 431364 1 52 
 
 1 716003 
 
 77 
 
 1 886491 ! 
 
 3 
 
 477121 
 
 28 
 
 1 447158 
 
 63 
 
 1 724276 
 
 78 
 
 1 892095 
 
 4 
 
 6020d0 
 
 29 
 
 1 462398 
 
 54 
 
 I 732394 
 
 79 
 
 1 897627 
 
 6 
 
 G98970 
 
 30 
 
 1 477121 
 
 55 
 
 1 740363 
 
 80 
 
 1 903090 
 
 6 
 
 778151 
 
 31 
 
 1 491362 
 
 56 
 
 1 748188 
 
 81 
 
 1 908485 
 
 7 
 
 845098 
 
 32 
 
 1 505150 
 
 57 
 
 1 755875 
 
 82 
 
 1 913«14 
 
 8 
 
 903090 
 
 33 
 
 1 518514 
 
 58 
 
 1 ^63428 
 
 83 
 
 1 919078 
 
 9 
 
 954243 
 
 34 
 
 1 531479 
 
 59 
 
 1 /70852 
 
 84 
 
 1 924279 
 
 10 
 
 1 000000 
 
 35 
 
 I 544068 
 
 60 
 
 1 778151 
 
 85 
 
 1 929419 
 
 11 
 
 1 041393 
 
 36 
 
 1 556303 
 
 61 
 
 1 785330 
 
 86 
 
 1 934498 
 
 12 
 
 1 079181 
 
 37 
 
 I 568202 
 
 62 
 
 1 792392 
 
 87 
 
 1 939") 19 
 
 13 
 
 1 113943 
 
 38 
 
 1 579/84 
 
 (>3 
 
 1 799341 
 
 88 
 
 1 944483 
 
 14 
 
 1 146128 
 
 39 
 
 1 591065 
 
 64 
 
 1 806180 
 
 89 
 
 1 949390 
 
 15 
 
 1 176091 
 
 40 
 
 1 602060 
 
 (>5 
 
 1 812913 
 
 90 
 
 1 954243 
 
 16 
 
 1 204120 
 
 41 
 
 1 C12784 
 
 66 
 
 I 819544 
 
 91 
 
 1 959041 
 
 17 
 
 1 230449 
 
 42 
 
 1 623249 
 
 67 
 
 1 826075 
 
 92 
 
 I 9«>3Th8 1 
 
 18 
 
 1 255273 
 
 43 
 
 1 633468 
 
 (>8 
 
 1 832509 
 
 93 
 
 1 968483 
 
 19 
 
 1 278754 
 
 44 
 
 1 643453 
 
 69 
 
 1 838849 
 
 94 
 
 1 973128 
 
 20 
 
 1 301030 
 
 45 
 
 1 653213 
 
 70 
 
 1 845098 
 
 95 
 
 1 977724 
 
 21 
 
 1 322219 
 
 46 
 
 1 662758 
 
 71 
 
 1 851258 
 
 96 
 
 1 9S2271 
 
 22 
 
 1 342423 
 
 47 
 
 1 672098 
 
 72 
 
 1 857333 
 
 97 
 
 1 9S(;7.2 1 
 
 23 
 
 1 361728 
 
 48 
 
 1 681241 
 
 73 
 
 1 863323 
 
 i;8 
 
 1 lt9122t) j 
 
 24 
 
 1 360211 
 
 49 
 
 1 690196 
 
 1 74 
 
 1 8v-9-^32 
 
 y;j 
 
 1 Ui>5i,35 1 
 
 25 
 
 1 397940 
 
 50 
 
 i 
 
 1 698970 
 
 175 
 
 1 8-50ol 
 
 100 
 
 2 000000 
 
 N 
 
 B. In the following table, in the last nine columns of each page, where 
 
 the i 
 
 irst or leading figures change from 9's to O's, points or dots are now 
 
 intrc 
 
 )duced instead of the O's through the rest of the line, to catch the eye, j| 
 
 and 
 
 to indicate that from thence the corresponding natural number in j| 
 
 the ( 
 
 irst column stands in the next lower line, and :is annexed first two 1 
 
 figUJ 
 
 es of the Logarithms iu tlie secoud columo. 
 
 I 
 
 i 
 
LOGARITHMS OF NUMBERS. 3 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 i 
 
 1 
 
 ' 1-" 
 
 9 
 
 3891 
 
 100 
 
 oooono 
 
 0134 
 
 0868 
 
 1301 
 
 1734 
 
 2166 
 
 2598 
 
 3029 3461 
 
 101 
 
 4321 
 
 4750 
 
 5181 
 
 5609 
 
 6038 
 
 6466 
 
 <i894 
 
 7321 
 
 7748 
 
 8174 
 
 102 
 
 8ti00 
 
 9026 
 
 9451 
 
 9876 
 
 .300 
 
 .724 
 
 1147 
 
 1570 
 
 1993 
 
 2415 
 
 103 
 
 012837 
 
 3259 
 
 3680 
 
 4100 
 
 4621 
 
 4940 
 
 63()0 
 
 5779 
 
 6197 
 
 6616 
 
 lOi 
 
 7033 
 
 7451 
 
 7868^ 
 
 8284 
 
 8^00 
 
 9116 
 
 9632 
 
 9947 
 
 .361 
 
 .776 
 
 105 
 
 021189 
 
 1603 
 
 2016 
 
 2428 
 
 2841 
 
 3252 
 
 3664 
 
 4075 
 
 4486 
 
 4896 
 
 10:) 
 
 630') 
 
 5715 
 
 6125 
 
 6533 
 
 6942 
 
 7350 
 
 7767 
 
 8164 
 
 8571 • 8978 
 
 107 
 
 9384 
 
 9789 
 
 .195 
 
 .600 
 
 1004 
 
 1408 
 
 1812 
 
 2216 
 
 2619 3021 
 
 108 
 
 033 124 
 
 3826 
 
 4227 
 
 4628 
 
 5029 
 
 5430 
 
 6830 
 
 6i30^ 
 
 6629 7028 
 
 109 
 
 7426 
 
 7825 
 
 8223 
 
 8620 
 
 9017 
 
 9414 
 
 9811 
 
 .207 
 
 .602 
 
 .998 
 
 110 
 
 041393 
 
 1787 
 
 2182 
 
 C576 
 
 2969 
 
 3362 
 
 3756 
 
 4148 
 
 4540 
 
 4932 
 
 111 
 
 5323 
 
 5714 
 
 6105 
 
 6495 
 
 6885 
 
 7275 
 
 7664 
 
 8053 
 
 8442 
 
 8830 
 
 11-2 
 
 9218 
 
 9606 
 
 9993 
 
 .380 
 
 . 76() 
 
 1153- 
 
 1638. 
 
 1924 
 
 2309 
 
 um 
 
 113 
 
 0530-8 
 
 34()3 
 
 3846 
 
 4230 
 
 4<J13 
 
 4996 
 
 6378 
 
 5760 
 
 6142 
 
 6524 
 
 114 
 
 6905 
 
 7286 
 
 7666 
 
 8046 
 
 8426 
 
 8805 
 
 9186 
 
 9563 
 
 9942 
 
 .320 
 
 115 
 
 030398 
 
 1075 
 
 1452 
 
 1829 
 
 2206 
 
 2582 
 
 2958 
 
 3333 
 
 3709 
 
 4083 
 
 lit) 
 
 4458 
 
 4832 
 
 6206 
 
 5580 
 
 5953 
 
 6326 
 
 6699 
 
 7071 
 
 7443 
 
 7815 
 
 117 
 
 8186 
 
 8557 
 
 8928 
 
 9298 
 
 9o(i8 
 
 ..38 
 
 .407 
 
 .776 
 
 1145 
 
 1514 
 
 118 
 
 071882 
 
 2250 
 
 2617 
 
 2985 
 
 3352 
 
 3718 
 
 4085 
 
 4451 
 
 4816 
 
 6182 
 
 119 
 
 6647 
 
 5912 
 
 6276 
 
 i5640 
 
 7004 
 
 7368 
 
 7731 
 
 8094 
 
 8467 
 
 8819 
 
 120 
 
 9181 
 
 9543 
 
 9904 
 
 .266 
 
 .626 
 
 .987 
 
 1347 
 
 1707 
 
 2067 
 
 2/,26 
 
 1-21 
 
 082785 
 
 3144 
 
 3503 
 
 3861 
 
 4219 
 
 4576 
 
 4934 
 
 5291 
 
 ' 6647 
 
 6004 
 
 1-22 
 
 63()0 
 
 6^16 
 
 7071 
 
 74-26 
 
 7781 
 
 8136 
 
 8490 
 
 8845 
 
 9198 
 
 9562 
 
 123 
 
 9905 
 
 .258 
 
 .611 
 
 .963 
 
 1315 
 
 1()67 • 
 
 2018 
 
 2370 
 
 2721 
 
 3071 
 
 124 
 
 093422 
 
 3772 
 
 4122 
 
 4471 
 
 4820 
 
 5169. 
 
 5518 
 
 6866 
 
 6216 
 
 6562 
 
 125 
 
 6910 
 
 7257 
 
 7604 
 
 7951 
 
 8298 
 
 8644 
 
 8990 
 
 9335 
 
 9681 
 
 1026 
 
 126 
 
 1003/1 
 
 0715 
 
 1059 
 
 1403 
 
 1747 
 
 2091 
 
 2434 
 
 2777 
 
 3119 
 
 3462 
 
 127 
 
 3804 
 
 4146 
 
 4487 
 
 4828 
 
 5169 
 
 5510 
 
 5K51 
 
 6191 
 
 6531 
 
 6871 
 
 128 
 
 7210 
 
 7649 
 
 7888 
 
 8227 
 
 8565 
 
 8903 
 
 9241 
 
 9579 
 
 9916 
 
 .253 
 
 129 
 
 110590 
 
 0926 
 
 1263 
 
 1699 
 
 1934 
 
 2270 
 
 2606 
 
 2940 
 
 3276 
 
 3609 
 
 130 
 
 3943 
 
 4277 
 
 4611 
 
 4944 
 
 6278 
 
 5611 
 
 6943 
 
 6276 
 
 6608 
 
 6940 
 
 131 
 
 7271 
 
 7()03 
 
 7934 
 
 82(.5 
 
 8595 
 
 8926 
 
 9256 
 
 9586 
 
 9915 
 
 0245 
 
 132 
 
 120574 
 
 0903 
 
 1231 
 
 15(iO 
 
 1888 
 
 2216 
 
 2644 
 
 2871 
 
 3198 
 
 3525 
 
 133 
 
 3862 
 
 41,8 
 
 4504 
 
 4830 
 
 5156 
 
 5481 
 
 6806 
 
 6131 
 
 6456 
 
 6781 
 
 134 
 
 7105 
 
 7429 
 
 7753 
 
 8076 
 
 8399 
 
 8722 
 
 9045 
 
 9368 
 
 9690 
 
 ..12 
 
 135 
 
 130334 
 
 0fi55 
 
 0977 
 
 1298 
 
 1619 
 
 1939 
 
 2260 
 
 25R0 
 
 2900 
 
 3219 
 
 136 
 
 3539 
 
 3858 
 
 -^177 
 
 4496 
 
 4814 
 
 6133 
 
 5451 
 
 5769 
 
 6086 
 
 6403 
 
 137 
 
 6721 
 
 7037 
 
 7354 
 
 7671 
 
 7987 
 
 8303 
 
 ,8618 
 
 S934 
 
 9249 
 
 ' 9564 
 
 138 
 
 -9S79 
 
 . 194 
 
 .51)8 
 
 .822 
 
 1136 
 
 1450 
 
 1763 
 
 •2076 
 
 2389 
 
 t 27 (»2 
 
 139 
 
 143016 
 
 3327 
 
 3«,30 
 
 3961 
 
 4263 
 
 4574 
 
 4885 
 
 5196 
 
 5507 
 
 5818 
 
 140 
 
 6128 
 
 6438 
 
 6748 
 
 7058 
 
 7367 
 
 7676 
 
 7985 
 
 8294 
 
 8603 
 
 8911 
 
 141 
 
 9219 
 
 9527 
 
 9H35 
 
 .142 
 
 .449 
 
 .756 
 
 10(i3 
 
 1370 
 
 1676 
 
 1982 
 
 142 
 
 152288 
 
 2594 
 
 '-900 
 
 o205 
 
 :510 
 
 3815 
 
 4120 
 
 4424 
 
 4728 
 
 5032 
 
 143 
 
 5336 
 
 5()40 
 
 6943 
 
 (-24(- 
 
 6649 
 
 6852 
 
 7154 
 
 7457 
 
 7759 
 
 8061 
 
 144 
 
 8362 
 
 S664 
 
 89(i6 
 
 9266 
 
 b6(i7 
 
 9868 
 
 .168 
 
 .469 
 
 .769 
 
 1068 
 
 145 
 
 1 1G1368 
 
 1667 
 
 1967 
 
 2266 
 
 2564 
 
 2863 
 
 3161 
 
 3460- 
 
 3758 
 
 4055 
 
 146 
 
 ; 4353 
 
 4(.50 
 
 4947 
 
 5244 
 
 5541 
 
 5838 
 
 6134 
 
 6430 
 
 6726 
 
 7022 
 
 147 
 
 7317 
 
 7613 
 
 7908 
 
 8203 
 
 8497 
 
 8792 
 
 90b() 
 
 9380 
 
 9()74 
 
 99<)8 
 
 148 
 
 not; 62 1 0565 
 
 0S48 
 
 1141 
 
 1434 
 
 1726 
 
 2019 
 
 251 1 
 
 2(;03 
 
 2895 
 
 149 
 
 1 
 
 318ti 1 3478 
 
 37(.9 
 
 40(i0 
 
 4351 
 
 4641 
 
 4932 
 
 5222 
 
 5512 
 
 6S(I2 ■ 
 
 18 
 
4 
 
 LOGAPIITHMS 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 150 
 
 176091 
 
 6381 
 
 6670 
 
 6959 
 
 7-248 
 
 7536 
 
 78-25 
 
 8113 
 
 8401 
 
 8689 
 
 
 151 
 
 89/7 
 
 92t)4 
 
 9552 
 
 9839 
 
 .126 
 
 .413 
 
 .699 
 
 .986 
 
 1272 
 
 1568 
 
 
 152 
 
 181844 
 
 2129 
 
 2415 
 
 2700 
 
 2985 
 
 3270 
 
 3655 
 
 3c39 
 
 4123 
 
 4407 
 
 
 153 
 
 4691 
 
 4975 
 
 5-259 
 
 5542 
 
 5825 
 
 6108 
 
 6391 
 
 tK)74 
 
 6956 
 
 7239 
 
 
 154 
 
 7521 
 
 7803 
 
 8084 
 
 8366 
 
 8647- 
 281 
 
 89-28 
 
 9209 
 
 949C 
 
 9771 
 
 ..61 
 
 
 155 
 
 190332 
 
 0612 
 
 0892 
 
 1171 
 
 1451 
 
 1730 
 
 2010 
 
 2289 
 
 2667 
 
 2846 
 
 
 156 
 
 3125 
 
 3403 
 
 3681 
 
 3959 
 
 4237 
 
 4514 
 
 •4792 
 
 5069 
 
 5346 
 
 56-23 
 
 
 157 
 
 5899 
 
 6176 
 
 6453 
 
 6729 
 
 7005 
 
 7281 
 
 ',656 
 
 7832 
 
 8107 
 
 8382 
 
 
 158 
 
 8657 
 
 8932 
 
 9206 
 
 9481 
 
 9755 
 
 ..29 
 
 .303 
 
 .677 
 
 .860 
 
 1124 
 
 
 lo9 
 
 201397 
 
 1670 
 
 1943 
 
 2216 
 
 2488 
 273 
 
 2761 
 
 3033 
 
 3305 
 
 3677 
 
 3848 
 
 
 160 
 
 4120 
 
 4391 
 
 4663 
 
 4934 
 
 5204 
 
 5476 
 
 5746 
 
 6016 
 
 6286 
 
 6566 
 
 
 161 
 
 6826 
 
 7096 
 
 7365 
 
 .634 
 
 7904 
 
 8173 
 
 8441 
 
 8710 
 
 8979 
 
 9247 
 
 
 162 
 
 9515 
 
 9783 
 
 ..51 
 
 .319 
 
 .586 
 
 .853 
 
 1121 
 
 1388 
 
 1654 
 
 1921 
 
 
 163 
 
 212188 
 
 2454 
 
 2,20 
 
 2986 
 
 3252 
 
 3518 
 
 3783 
 
 4049 
 
 4314 
 
 4579 
 
 
 164 
 
 4844 
 
 6109 
 
 5373 
 
 5638 
 
 5902 
 264 
 
 6166 
 
 6430 
 
 6694 
 
 6957 
 
 7221 
 
 
 165 
 
 7484 
 
 7747 
 
 8010 
 
 8273 
 
 8636 
 
 8798 
 
 9060 
 
 9323 
 
 9585 
 
 9846 
 
 
 166 
 
 220108 
 
 0370 
 
 0631 
 
 0892 
 
 1153 
 
 1414 
 
 1675 
 
 1936 
 
 2196 
 
 2456 
 
 
 167 
 
 2716 
 
 2976 
 
 3236' 
 
 3496 
 
 3755 
 
 4015 
 
 4274 
 
 4533 
 
 4792 
 
 6051 , 
 
 
 168 
 
 5309 
 
 5568 
 
 5 26 
 
 6084 
 
 6342 
 
 6600 
 
 6858 
 
 7115 
 
 7372 
 
 7630 
 
 
 169 
 
 7887 
 
 8144 
 
 8400 
 
 8667 
 
 8913 
 
 257 
 
 9170 
 
 9426 
 
 9682 
 
 9938 
 
 .193 
 
 
 170 
 
 230449 
 
 0704 
 
 0960 
 
 121p 
 
 1470 
 
 1724 
 
 1979 
 
 2234 
 
 Q488 
 
 2742 , 
 
 
 171 
 
 2996 
 
 32 -.0 
 
 3504 
 
 37^7 
 
 4011 
 
 4264 
 
 4617 
 
 4770 
 
 5023 
 
 6276 
 
 
 172 
 
 5528 
 
 5781 
 
 ()033 
 
 6285 
 
 6537 
 
 6789 
 
 7041 
 
 7292 
 
 7644 
 
 7796 
 
 
 173 
 
 8046 
 
 8297 
 
 8548 
 
 8799 
 
 9049 
 
 9299 
 
 9550 
 
 9800 
 
 ..50 
 
 .300 
 
 
 174 
 
 240549 
 
 0799 
 
 1048 
 
 1297 
 
 1546 
 249 
 
 1795 
 
 2044 
 
 2293 
 
 2641 
 
 2790 
 
 
 175 
 
 3038 
 
 3283 
 
 3534 
 
 3782 
 
 4030 
 
 4277 
 
 4525 
 
 4772 
 
 5019 
 
 5266 
 
 
 176 
 
 6513 
 
 575) 
 
 6006 
 
 6252 
 
 6499 
 
 6745 
 
 6991 
 
 7237 
 
 7482 
 
 7728 
 
 
 177^ 
 
 7973 
 
 8219 
 
 8464 
 
 8709 
 
 8954 
 
 9198. 
 
 9443 
 
 9687 
 
 9932 
 
 .176 
 
 
 178 
 
 250420 
 
 0664 
 
 0908 
 
 1151 
 
 1895 
 
 1638 
 
 1881 
 
 2125 
 
 2368 
 
 2610 
 
 
 179 
 
 2853 
 
 3096 
 
 3338 
 
 3580 
 
 3822 
 242 
 
 4064 
 
 4306 
 
 4648 
 
 4790 
 
 6031 
 
 
 180 
 
 5273 
 
 5514 
 
 5755 
 
 6996 
 
 6237 
 
 6477 
 
 6718 
 
 6958 
 
 7198 
 
 7439 
 
 
 181 
 
 7679 
 
 7918 
 
 8158 
 
 8398 
 
 8637 
 
 8877 
 
 9116 
 
 9366 
 
 9594 
 
 9833 
 
 
 182 
 
 260071 
 
 0310 
 
 0548 
 
 0787 
 
 10-25 
 
 1263 
 
 1501 
 
 1739 
 
 1976 
 
 2214 
 
 
 183 
 
 2451 
 
 2688 
 
 2925 
 
 3162 
 
 3399 
 
 3636 
 
 3K73 
 
 4109 
 
 4346 
 
 4582 
 
 
 1% 
 
 4818 
 
 5054 
 
 5290 
 
 6526 
 
 5761 
 
 235 
 
 8110 
 
 6996 
 
 6232 
 
 6467 
 
 6702 
 
 6937 
 
 
 185 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 8344 
 
 8578 
 
 8812 
 
 9046 
 
 9279 
 
 
 186 
 
 9513 
 
 9746 
 
 9980 
 
 .213 
 
 .446 
 
 .679 
 
 .912 
 
 1144 
 
 1377 
 
 1609 
 
 
 U, 
 
 271842 
 
 2074 
 
 230.) 
 
 2538 
 
 2770 
 
 3001 
 
 3233 
 
 3464 
 
 3696 
 
 3927 
 
 
 188 
 
 4158 
 
 4389 
 
 4620 
 
 4850 
 
 5081 
 
 6311 
 
 5542 
 
 5772 
 
 6002 
 
 6232 
 
 
 189 
 
 6462 
 
 6692 
 
 6921 
 
 7151 
 
 7380 
 229 
 
 7609 
 
 7838 
 
 8067 
 
 8296 
 
 8526 
 
 
 190 
 
 8754 
 
 8982 
 
 9211 
 
 9439 
 
 9667 
 
 9896 
 
 .123 
 
 .351 
 
 .578 
 
 .806 
 
 
 191 
 
 281033 
 
 1261 
 
 1488 
 
 1715 
 
 1942 
 
 2169 
 
 2396 
 
 2622 
 
 2849 
 
 3075 
 
 
 192 
 
 3301 
 
 t627 
 
 3753 
 
 3979 
 
 4-205 
 
 4431 
 
 4656 
 
 4882 
 
 5107 
 
 5332 
 
 
 193 
 
 5557 
 
 5782 
 
 6007 
 
 6232 
 
 6456 
 
 6681 
 
 6905 
 
 7130 
 
 7354 
 
 7578 
 
 
 194 
 
 7802 
 
 8026 
 
 8249 
 
 8473 
 
 8696 
 224 
 
 8920 
 
 9143 
 
 9366 
 
 9589 
 
 9812 
 
 
 195 
 
 20C035 
 
 0257 
 
 0480 
 
 0702 
 
 0925 
 
 1147 
 
 1369 
 
 1591 
 
 1813 
 
 2034 
 
 
 196 
 
 2-256 
 
 2478 
 
 2699 2920 | 
 
 3141 
 
 3363 
 
 3584 
 
 8804 
 
 4026 
 
 4246 
 
 
 197 
 
 44()6 
 
 4687 
 
 4907 
 
 5127 
 
 5347 
 
 6567 
 
 6787 
 
 6007 
 
 6226 
 
 6446 
 
 
 198 
 
 66(55 
 
 6884 
 
 7104 
 
 73-23 
 
 7642 
 
 7761 
 
 7979 
 
 8198 
 
 8416 
 
 8635 
 
 
 199 
 
 8853 
 
 9071 
 
 9-289 
 
 9507 
 
 9725 
 
 9943 
 
 .161 
 
 .378 
 
 .595 
 
 .813 
 
 
OFNUMBERS 5 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 1 
 
 5 
 
 6 
 
 7 8 1 9 
 
 200 
 
 3010.10 
 
 1247 
 
 1464 
 
 1681 
 
 1898 
 
 2114 
 
 2331 
 
 2547 2764 29bi, 
 
 201 
 
 31SJO 
 
 3412 
 
 3628 
 
 3844 
 
 4059 1 
 
 4275 
 
 4491 
 
 47(HJ 1 4921 6136 
 
 202 
 
 5351 
 
 5566 
 
 5781 
 
 699() 
 
 6211 1 
 
 6425 
 
 6(J39 
 
 6854 70 .8 7282 
 
 203 
 
 74iJ{) 
 
 7710 
 
 7924 
 
 8137 
 
 8351 
 
 8564 
 
 8778 ' 8991 
 
 9204 1 9447 II 
 
 204 
 
 9630 
 
 9843 
 
 ..56 
 
 .268 
 
 .481 
 
 212 
 
 2600 
 
 .693 
 
 .906 
 
 1118 
 
 1330 
 
 1542 
 
 205 
 
 311754 
 
 1966 
 
 2177 
 
 2389 
 
 2812 
 
 3023 
 
 3234 
 
 3445 
 
 3656 
 
 20<) 
 
 3867 
 
 4078 
 
 4289 
 
 4499 
 
 4710 
 
 4920 
 
 5130 
 
 5340 
 
 6551 
 
 5760 
 
 1 207 
 
 5970 
 
 6180 
 
 6390 
 
 6599 
 
 6809 
 
 7018 
 
 7227 
 
 7436 7(>4() 
 
 7854 
 
 208 
 
 8063 
 
 8272 
 
 8481 
 
 8689 
 
 8898 
 
 9106 
 
 9314 
 
 9522 9730 > 
 
 9938 
 
 1 209 
 
 320146 
 
 0354 
 
 0562 
 
 0769 
 
 0977 
 
 207 
 
 1184 
 
 1391 
 
 1698 
 
 1805 
 
 2012 
 
 210 
 
 2219" 
 
 2426 
 
 2633 
 
 2839 
 
 3046 
 
 3252 
 
 3458 
 
 36)5 
 
 3871 
 
 4077 
 
 211 
 
 4282 
 
 4488 
 
 4694 
 
 4899 
 
 5105 
 
 5310 
 
 6516 
 
 6721 
 
 5926 
 
 6131 
 
 212 
 
 6336 
 
 6541 
 
 ()745 
 
 6950 
 
 7155 
 
 7369 
 
 7563 
 
 7767 
 
 7972 
 
 8176 
 
 213 
 
 83S0 
 
 85H3 
 
 8787 
 
 8991 
 
 9194 
 
 9398 
 
 9:;0l 
 
 9805 
 
 ...8 
 
 .211 
 
 214 
 
 330414 
 
 0617 
 
 0819 
 
 1022 
 
 1225 
 202 
 
 1427 
 
 1630 
 
 1832 
 
 2034 
 
 2236- 
 
 215 
 
 2438 
 
 2640 
 
 2842 
 
 3044 
 
 3246 
 
 3447 
 
 3649 
 
 3850 
 
 4051 
 
 4253 
 
 216 
 
 4454 
 
 4655 
 
 4856 
 
 5057- 
 
 6257 
 
 6458 
 
 6658 
 
 6859 
 
 6059 
 
 6260 
 
 217 
 
 64t>0 
 
 6660 
 
 6860 
 
 70i)0 
 
 7*2(i0 
 
 7459 
 
 7659 
 
 7858 
 
 8058 
 
 8257 
 
 218 
 
 8456 
 
 8(i56 
 
 8855 
 
 9054 
 
 9253 
 
 9451 
 
 9660 
 
 9849 
 
 . 47 
 
 .246 
 
 219 
 
 340444 
 
 0642 
 
 0841 
 
 1039 
 
 1237 
 
 1436 
 
 1632 
 
 1830 
 
 2028 
 
 2225 
 
 
 
 
 
 
 198 
 
 
 
 
 
 220 
 
 2423 
 
 2620 
 
 2817 
 
 3014 
 
 3212 
 
 3409 
 
 3606 
 
 3802 
 
 3999 
 
 4196 
 
 221 
 
 4392 
 
 45H9 
 
 4785 
 
 4981 ' 5178 
 
 5374 
 
 5570 
 
 5766 
 
 69.>2 
 
 (,157 
 
 222 
 
 6353 
 
 6549 
 
 6744 
 
 6939 
 
 7135 
 
 7330 
 
 7526 
 
 7720 
 
 7916 
 
 8110 
 
 223 
 
 8305 
 
 8500 
 
 8694 
 
 8889. 
 
 9083 
 
 9278 
 
 9472 
 
 9666 
 
 9860 
 
 . 54 
 
 224 
 
 350248 
 
 0142 
 
 0636 
 
 0829 
 
 1023 
 193 
 
 1216 
 
 1410 
 
 1603 
 
 1796 
 
 1989 
 
 225 
 
 2183 
 
 2375 
 
 2568 
 
 2761 
 
 2954 
 
 3147 
 
 3339 
 
 3532 
 
 3724 
 
 3916 
 
 22(> 
 
 4108 
 
 4301 
 
 4493 
 
 4685 
 
 4876 
 
 5068 
 
 6260 
 
 6452 
 
 6643 
 
 6834 
 
 227 
 
 6026 
 
 6217 
 
 6408 
 
 6599 
 
 6790 
 
 f98l 
 
 7172 
 
 7363 
 
 7554 
 
 7744 
 
 228 
 
 7935 
 
 8125 
 
 H316 
 
 8506 
 
 8696 
 
 8886 
 
 9076 
 
 9266 
 
 9456 
 
 9646 
 
 229 
 
 9835 
 
 ..25 
 
 .215 
 
 .404 
 
 .693 
 190 
 
 .783 
 
 .972 
 
 1161 
 
 1350- 
 
 1539 
 
 230 
 
 361728 
 
 1917 
 
 2105 
 
 2294 
 
 2482 
 
 2671 
 
 2859 
 
 3048 
 
 3236 
 
 r424 
 
 231 
 
 3612 
 
 3800 
 
 3988 
 
 4176 
 
 43()3 
 
 4551 
 
 4739 
 
 4926 
 
 6113 
 
 6301 
 
 232 
 
 6488 
 
 5675 
 
 5862 
 
 6049 
 
 6236 
 
 (423 
 
 6610 
 
 6796 
 
 <983 
 
 7169 
 
 233 
 
 7356 
 
 7542 
 
 7729 
 
 7915 
 
 8101 
 
 8287 
 
 8473 
 
 8659 
 
 8846 
 
 9030 
 
 234 
 
 9216 
 
 9401 
 
 9587 
 
 9772 
 
 9958 
 185 
 
 .143 
 
 .328 
 
 .513 
 
 .698 
 
 883 
 
 ■ 235 
 
 371068 
 
 1253 
 
 1437 
 
 1622 
 
 1806 
 
 1991 
 
 2175 
 
 23^0 
 
 2544 
 
 2:28 1 
 
 236 
 
 2912 
 
 3096 
 
 3280 
 
 3464 
 
 3647 
 
 3831 
 
 4015 
 
 4198 
 
 4382 4565 11 
 
 237 
 
 4748 
 
 4932 
 
 5115 
 
 5298 
 
 5481 
 
 5()64 
 
 5846 
 
 6029 6212 6."^94 1 1 
 
 j 2.J8 
 
 6577 
 
 6759 
 
 6942 
 
 7124 
 
 7306 
 
 7488 
 
 7670 
 
 7852 i 
 
 8034 8216 II 
 
 1 239 
 
 839i: 
 
 8580 
 
 8761 
 
 8943 
 
 9124 
 
 182 
 
 0934 
 
 930() 
 
 9487 
 
 9668 
 
 9b49 
 
 ..30 1 
 
 240 
 
 n-jim 1 
 
 0302 
 
 0573 
 
 0754 
 
 1115 
 
 1296 
 
 1476 
 
 1656 1837 
 
 241 
 
 2017 
 
 2197 
 
 2377 
 
 2567 
 
 2737 
 
 2917 
 
 3097 
 
 3277 
 
 3456 3636 
 
 242 
 
 3815 
 
 3995 
 
 4174 
 
 4'J53 
 
 4533 
 
 4712 
 
 4891 
 
 5070 
 
 5249 5428 
 
 243 
 
 5606 
 
 5785 
 
 5964 
 
 6142 
 
 6321 
 
 6499 
 
 6677 
 
 6856 
 
 7034 7212 
 
 244 
 
 7390 
 
 7568 
 
 7746 
 
 7923 
 
 8101 
 
 178 
 
 9875 
 
 8279 
 
 8456 
 
 8634 
 
 88U 8989 
 
 245 
 
 9166 
 
 9M3 
 
 9520 
 
 9(i98 
 
 ..51 
 
 .228 
 
 .405 
 
 .582 .759 
 
 246 
 
 390935 
 
 1112 
 
 1288 
 
 1464 
 
 1641 
 
 1817 
 
 1993 
 
 2169 
 
 2345 2521 
 
 247 
 
 2697 
 
 2873 
 
 3048 
 
 3224 
 
 3400 
 
 3575 V3751 
 
 3926 
 
 4101 4277 
 
 248 
 
 4452 
 
 4ii21 
 
 4802 
 
 4977 
 
 5152 
 
 5326 15501 
 
 6676 
 
 5850 6025 
 
 249 
 
 6199 
 
 6374 
 
 6548 
 
 672:2 
 
 6896 
 
 7071 1 7245 
 
 7419 
 
 7592 7766 
 
j7- 
 
 LOGARITHMS 
 
 
 1 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 f 
 
 7 
 
 8 
 
 9 
 
 
 '250 
 
 397940 
 
 8114 
 
 8287 
 
 84()1 
 
 8634 
 
 88i»H 
 
 hOMI 
 
 9lbl 
 
 9328 
 
 9501 
 
 
 251 
 
 96/4 
 
 9847 
 
 ..20 
 
 .192 
 
 .365 
 
 .538 
 
 .711 
 
 .883 
 
 1056 
 
 1228 
 
 
 252 
 
 401401 
 
 15<3 
 
 1745 
 
 1917 
 
 2089 
 
 22<)1 
 
 2433 
 
 •^606 
 
 2777 
 
 2949 
 
 
 253 
 
 3121 
 
 3292 
 
 3464 
 
 S635 
 
 3807 
 
 3978 
 
 4149 
 
 4320 
 
 4492 
 
 4663 
 
 
 254 
 
 4834 
 
 5006 
 
 5176 
 
 5346 
 
 r5i7 
 
 5688 
 
 5858 
 
 (.029 
 
 6199 
 
 6370 
 
 
 . 
 
 
 
 
 171 
 
 
 
 
 
 
 25E 
 
 6540 
 
 6710 
 
 6881 7051 7221 
 
 73:) 1 
 
 7561 7731 
 
 7901 -^070 
 
 
 256 , 
 
 8-240 
 
 8410 
 
 8579 
 
 S749 8918 
 
 9087 
 
 i>257 9426 , 9595 3764 
 
 
 257 
 
 9933 
 
 102 
 
 .271 
 
 .440 , .609 
 
 .777 
 
 946 1114 
 
 1283 1451 
 
 
 258' 
 
 411620 
 
 1788 
 
 1956 
 
 •..124 ^ 
 
 -293 
 
 2461 
 
 1629 
 
 2796 
 
 2964 
 
 3132 
 
 
 259 
 
 3300 
 
 ti467 
 
 3635 
 
 3803 
 
 S970 
 
 4137 
 
 4305 
 
 4 472 
 
 4639 
 
 4806 
 
 
 260 
 
 4973 
 
 140 
 
 5307 
 
 r474 
 
 5641 
 
 6808 
 
 i)974 
 
 6141 
 
 6308 
 
 6474 
 
 
 2(>1 
 
 6641 
 
 6807 
 
 6973 
 
 7139 
 
 .30a 
 
 <472 
 
 7638 
 
 7804 
 
 7970 
 
 8135 
 
 
 262 
 
 8301 
 
 467 
 
 8633 
 
 8798 
 
 8964 
 
 L129 
 
 9295 
 
 9460 
 
 9()25 
 
 9791 
 
 
 263 
 
 9956 
 
 121 
 
 .286 
 
 .451 
 
 .616 
 
 .781 
 
 .945 
 
 1110 
 
 1275 
 
 1439 
 
 
 264 
 
 421604 
 
 1788 
 
 1933 
 
 -.097 
 
 2261 
 
 2426 
 
 2590 
 
 2754 
 
 2918 
 
 3082 
 
 
 265 
 
 3246 
 
 3410 
 
 3574 
 
 3737 
 
 3901 
 
 4066 
 
 4228 
 
 4392 
 
 4556 
 
 4718 
 
 
 266 
 
 4882 
 
 5045 
 
 5208 
 
 371 
 
 .63i 
 
 L697 
 
 5860 
 
 6023 
 
 6186 
 
 6349 
 
 
 267 
 
 6511 
 
 6674 
 
 6836 
 
 6999 
 
 ,161 
 
 7324 
 
 7486 
 
 7648 
 
 7811 
 
 7973 
 
 
 268 
 
 8135 
 
 8297 
 
 8459 
 
 8621 
 
 8783 
 
 8944 
 
 9106 
 
 9268 
 
 9429 
 
 9591 
 
 
 269 
 
 9752 
 
 9914 
 
 ..75 
 
 .236 
 
 .398 
 
 .559 
 
 .720 
 
 .881 
 
 1042 
 
 1203 
 
 
 270 
 
 431364 
 
 1525 
 
 1685 
 
 1846 
 
 2007 
 
 2167 
 
 2328 
 
 2488 
 
 2649 
 
 2809 
 
 
 271 
 
 2969 
 
 3130 
 
 3290 
 
 34.J0 
 
 3610 
 
 3 770 
 
 3!>30 
 
 4090 
 
 4249 
 
 4409 
 
 
 272 
 
 4569 
 
 4729 
 
 4S88 
 
 5048 
 
 5207 
 
 53o7 
 
 5526 
 
 5«)85 
 
 6844 
 
 6004 
 
 
 273 
 
 6163 
 
 6322 
 
 6481 
 
 6640 
 
 6800 
 
 6957 
 
 7116 
 
 7i75 
 
 7433 
 
 7592 
 
 
 274 
 
 7751 
 
 7909 
 
 8067 
 
 8226 
 
 8384 
 158 
 
 8542 
 
 8701 
 
 8859 
 
 9017 
 
 9175 
 
 
 275 
 
 9333 
 
 9491 
 
 9648 
 
 9806 
 
 9964 
 
 .122 
 
 279 
 
 .437 
 
 .694 
 
 .752 
 
 
 276 
 
 440909 
 
 1066 
 
 1224 
 
 1381 
 
 1538 
 
 1695 
 
 1852 
 
 20! )9 
 
 2166 
 
 -323 
 
 
 277 
 
 2480 
 
 2637 
 
 2793 
 
 2950 
 
 3106 
 
 3263 
 
 c419 
 
 3576 
 
 3732 
 
 3889 
 
 
 278 
 
 4045 
 
 4201 
 
 .367 
 
 4513 
 
 4(i69 
 
 4825 
 
 4981 
 
 6137 
 
 6293 
 
 6449 
 
 
 279 
 
 5604 
 
 5760 
 
 5915 
 
 6071 
 
 6226 
 
 6382 
 
 6637 
 
 6692 
 
 6848 
 
 7003 
 
 
 280 
 
 7168 
 
 7313 
 
 7468 
 
 7623 
 
 7778 
 
 7933 
 
 8088 
 
 8242 
 
 8397 
 
 8552 
 
 
 281 
 
 8706 
 
 8861 
 
 9015 
 
 9170 
 
 9324 
 
 9478 
 
 9633 
 
 9787 
 
 9941 
 
 . .95 
 
 
 282 
 
 450249 
 
 0403 
 
 0557 
 
 0711 
 
 0865 
 
 1018 
 
 1172 
 
 1326 
 
 1479 
 
 1633 
 
 
 283 
 
 1786 
 
 1940 
 
 2093 
 
 2247 
 
 2400 
 
 2553 
 
 2706 
 
 2859 
 
 3012 
 
 3166 
 
 
 284 
 
 3318 
 
 3471 
 
 3624 
 
 3777 
 
 3930 
 
 4082 
 
 4236 
 
 4387 
 
 4640 
 
 4692 
 
 
 2 6 
 
 4845 
 
 4997 
 
 ri50 
 
 5302 
 
 5454 
 
 5606 
 
 6768* 
 
 6910 
 
 6062 
 
 6214 • 
 
 
 286 
 
 6366 
 
 6518 
 
 6670 
 
 6821 
 
 6973 
 
 7126 
 
 276 
 
 7428 
 
 7679 
 
 7731 
 
 
 287 
 
 7882 
 
 8033 
 
 81S4 
 
 8336 
 
 8487 
 
 8638 
 
 8789 
 
 8940 
 
 9091 
 
 9242 
 
 
 288 
 
 9392 
 
 9543 
 
 9-6 H 
 
 9845 
 
 9995 
 
 .146 
 
 .296 
 
 .417 .597 
 
 .748 
 
 
 289 
 
 460898 
 
 1048 
 
 1198 
 
 1348 
 
 1499 
 
 1649 
 
 1799 
 
 1948 .098 
 
 2248 
 
 
 29C 
 
 2398 
 
 2548 
 
 2697 
 
 2847 
 
 2997 
 
 3146 
 
 3296 
 
 3445 
 
 3594 
 
 3744 
 
 
 291 
 
 3893 
 
 4042 4191 
 
 4340 
 
 4490 
 
 4639 
 
 4788 
 
 4936 
 
 5085 
 
 5234 
 
 
 292 
 
 5383 
 
 5532 
 
 6680. 
 
 5829 
 
 5977 
 
 6126 
 
 6274 
 
 6423 
 
 6571 
 
 6719 
 
 
 293 
 
 6868 
 
 7016 
 
 7164 
 
 7312 
 
 7460 
 
 7608 
 
 7766 
 
 7904 
 
 8052 
 
 8200 
 
 
 294 
 
 8347 
 
 8495 
 
 S643 
 
 8790 
 
 8938 
 147 
 
 9086 
 
 9233 
 
 9380 
 
 9527 
 
 9675 
 
 
 295 
 
 9822 
 
 9969 
 
 .116 
 
 .263 
 
 .410 
 
 .667 
 
 .704 
 
 .851 
 
 .998 
 
 1145 
 
 
 296 
 
 471292 
 
 1438 
 
 1585 
 
 1732 
 
 1878 
 
 2025 
 
 2171 
 
 2318 
 
 i464 
 
 2610 
 
 
 297 
 
 2756 
 
 •i903 
 
 3049 
 
 3195 
 
 3341 
 
 3487 
 
 3633 
 
 3779 
 
 1^925 
 
 4071 
 
 
 298 
 
 4216 
 
 4362 
 
 4508 
 
 4()53 
 
 4799 
 
 4944 
 
 5090 
 
 6235 
 
 5381 
 
 5526 
 
 
 1 299 
 
 5671 
 
 50 16 
 
 5962 
 
 6107 
 
 6252 
 
 6397 
 
 6542 
 
 6687 
 
 6832 
 
 6976 
 
 

 OF NUMBERS. 7 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 & 
 
 
 SOU 
 
 477121 
 
 7200 
 
 7411 
 
 7555 
 
 7700 
 
 7844 
 
 7989 
 
 8133 
 
 8278 
 
 8422 
 
 
 •Ml 
 
 85t»ti 
 
 8711 
 
 8855 
 
 8999 
 
 9143 
 
 9287 
 
 9481 
 
 9575 
 
 9719 
 
 9803 
 
 
 302 
 
 480v)07 
 
 0151 
 
 02:)4 
 
 0138 
 
 0582 
 
 07-25 
 
 0809 
 
 1012 
 
 1150 
 
 1299 
 
 
 303 
 
 14-i3 
 
 15S0 
 
 1729 
 
 1.S72 
 
 2010 
 
 2159 
 
 2302 
 
 2445 
 
 2688 
 
 2731 
 
 
 304 
 
 2874 
 
 3010 
 
 3169 
 
 3oi/2 
 
 • 
 
 3445 
 142 
 
 35o7 
 
 ?730 
 
 3872 
 
 4015 
 
 4167 
 
 
 iOo 
 
 4300 
 
 4442 
 
 4585 
 
 4727 
 
 4809 
 
 5011 
 
 5153 
 
 5295 
 
 5437 
 
 6579 
 
 
 3JJ 
 
 5721 
 
 680a- 
 
 0005 
 
 0147 
 
 0289 
 
 6430 
 
 0672 
 
 6714 
 
 0855 
 
 0997 . 
 
 
 30/ 
 
 7138 
 
 7280 
 
 7421 
 
 7503 
 
 7701 
 
 7845' 
 
 7980- 
 
 8127 
 
 8209 
 
 8410 • 
 
 
 30.-i 
 
 8551 
 
 8(«92 
 
 8833 
 
 8974 
 
 9114 
 
 0255 
 
 9390 
 
 9537 
 
 9077 
 
 9818 
 
 
 j 3iy 
 
 9959 
 
 . .99 
 
 .239 
 
 .380 
 
 .520 
 
 .661 
 
 .801 
 
 .941 
 
 1081 
 
 1222 
 
 
 1 
 
 310 
 
 49 130 2 
 
 1502 
 
 1042 
 
 1782 
 
 1922 
 
 2002 
 
 2201- 
 
 2341 
 
 2481 
 
 2621 
 
 
 311 
 
 2700 
 
 29. )0 
 
 3040 
 
 3179 
 
 3319 
 
 3458 
 
 3597 
 
 3737 
 
 3870 
 
 4016 
 
 
 31-2 
 
 4155 
 
 4294 
 
 4433 
 
 4572 
 
 4711 
 
 4850 
 
 4989 
 
 5128 
 
 5-207 
 
 5406 
 
 
 313 
 
 5544 
 
 6(>83 
 
 5822 
 
 5900 
 
 e099 
 
 6238 
 
 63/6 
 
 1.516 
 
 0(53 
 
 6791 
 
 
 3U 
 
 0930 
 
 7008 
 
 7200 
 
 7344 
 
 7483 
 
 7021 
 
 7759 
 
 7897 
 
 8035 
 
 8173 
 
 
 315 
 
 8311 
 
 8448 
 
 ^•586 
 
 8724 
 
 8862 
 
 8999 
 
 9137 
 
 9276 
 
 9412 
 
 S550 
 
 
 31d 
 
 9087 
 
 9824 
 
 9Uo2 
 
 ..99 
 
 .230 
 
 .374 
 
 .511 
 
 .048 
 
 .785 
 
 .922 
 
 
 317 
 
 501059 
 
 1190 
 
 1333 
 
 1470 
 
 1007 
 
 1744 
 
 1880 
 
 'Am 
 
 2154 
 
 2291 
 
 
 31H 
 
 2427 
 
 2504 
 
 2.00 
 
 2837 
 
 29 73 
 
 3109 
 
 3240 
 
 3382 
 
 3518 
 
 3()56 
 
 
 3iy 
 
 3791 
 
 3927 
 
 40o3 
 
 4199 
 
 4336 
 
 4471 
 
 4007 
 
 4743 
 
 4878 
 
 5014 
 
 
 3-20 
 
 5150 
 
 628-) 
 
 5421 
 
 5557 
 
 5093 
 
 5828 
 
 6904 
 
 609J 
 
 (;234 
 
 6370 
 
 
 1 3-21 
 
 0505 
 
 0040 
 
 0776 
 
 0911 
 
 7040 
 
 7181 
 
 7310 
 
 7451 
 
 '<68) 
 
 7721 
 
 
 1 32-2 
 
 7850 
 
 7991 
 
 812J 
 
 8200 
 
 8395 
 
 8630 
 
 8004 
 
 8799 
 
 8934 
 
 9008 
 
 
 3-23 
 
 9203 
 
 9337 
 
 9471 
 
 90 JO 
 
 9740 
 
 9874 
 
 ...9 
 
 .143 
 
 .21, 
 
 .411 
 
 
 3-24 
 
 510545 
 
 0079 
 
 0813 
 
 0947 
 
 1081 
 134 
 
 1215 
 
 13h9 
 
 1482 
 
 1610 
 
 1/60 
 
 
 325 
 
 1883 
 
 2017 
 
 2151 
 
 2284 
 
 2418 
 
 2551 
 
 2684 
 
 2818 
 
 2951 
 
 3034 
 
 
 32ti 
 
 3218 
 
 3351 
 
 3484 
 
 3017 
 
 3750 
 
 3883 
 
 401) 
 
 4149 
 
 4282-^ 
 
 4414 
 
 
 3-2/ 
 
 4548 
 
 4081 
 
 4813 
 
 4940 
 
 5079 
 
 5211 
 
 6341 
 
 6476 
 
 5009 
 
 5741 
 
 
 328 
 
 5874 
 
 OOOrJ 
 
 0139 
 
 02/1 
 
 0403 « 
 
 C635 
 
 6008 
 
 6800 
 
 0932 
 
 7004 
 
 
 329 
 
 7190 
 
 7328 
 
 7460 
 
 7592 
 
 7724 
 
 (865 
 
 7987 
 
 8119 
 
 8251 
 
 8382 
 
 
 330 
 
 8514 
 
 8046 
 
 8777 
 
 8909 
 
 9040 
 
 9171 
 
 9303 
 
 9434 
 
 9560 
 
 9697 
 
 
 331 
 
 9828 
 
 9959 
 
 ..90 
 
 .221 
 
 .353 
 
 .484 
 
 .015 
 
 .745 
 
 .876 
 
 1007 
 
 
 33-2 
 
 521138 
 
 12o9 
 
 1400 
 
 153 J 
 
 1001 
 
 1792 
 
 1922 
 
 2053 
 
 2183 
 
 2314 
 
 
 333 
 
 2444 
 
 25/5 
 
 2705 
 
 2835 
 
 2900 
 
 309a 
 
 3220 
 
 3350 
 
 3480 
 
 3016 
 
 
 334 
 
 3740 
 
 3870 
 
 400d 
 
 4l3d 
 
 4200 
 
 4390 
 
 4520 
 
 4050 
 
 4785 
 
 4915 
 
 
 335 
 
 6045 
 
 5174 
 
 5304 
 
 6434 
 
 5603 
 
 5093 
 
 5822 
 
 5951 
 
 6081 
 
 6210 
 
 
 33ti 
 
 0339 
 
 04G9 
 
 0598 
 
 0727 
 
 0850 
 
 0985 
 
 7114 
 
 7243 
 
 7372 
 
 7501 
 
 
 337 
 
 7030 
 
 7759 
 
 7888 
 
 8010 
 
 8145 
 
 8274 
 
 8402 
 
 8531 
 
 8000 
 
 8788 
 
 
 j 338 
 
 8<il7 
 
 9045 
 
 9174 
 
 9302 
 
 9430 
 
 9559 
 
 9087 
 
 9816 
 
 9943 
 
 ..72 
 
 
 j 339 
 
 530200 
 
 0328 
 
 0450 
 
 0584 
 
 0712 
 
 0840 
 
 0968 
 
 1096 
 
 1223 
 
 lo61 
 
 
 340 
 
 1479 
 
 1607 
 
 1734 
 
 1862 
 
 1960 
 
 2117 
 
 2245 
 
 2372 
 
 2500 
 
 2027 
 
 
 341 
 
 2754 
 
 2882 
 
 3009 
 
 3130 
 
 3204 
 
 3391 
 
 3518 
 
 3045 
 
 3772 
 
 3899 
 
 
 342 
 
 4020 
 
 4153 
 
 4v80 
 
 4407 
 
 4534 
 
 4001 
 
 4787 
 
 4914 
 
 5041 
 
 6107 
 
 
 343 
 
 5294 X 
 
 5421 
 
 5547 
 
 5074 
 
 5800 
 
 5927 
 
 6063 
 
 0180 
 
 6306 
 
 6432 
 
 
 344 
 
 0658 
 
 6085 
 
 6811 
 
 0937 
 
 7003 
 12t5 
 
 7189 
 
 7316 
 
 7441 
 
 7567 
 
 7693 
 
 
 345 
 
 7819 
 
 7945 
 
 8071 
 
 8197 
 
 8322 
 
 8448 
 
 8574 
 
 8699 
 
 8825 
 
 8951 
 
 
 346 
 
 9076 
 
 9202 
 
 9327 
 
 9452 
 
 9578 
 
 9/03 
 
 9829 
 
 9954 
 
 ..79 
 
 .204 
 
 
 317 
 
 540329 
 
 0455 
 
 0580 
 
 0705 
 
 0830 
 
 0955 
 
 1080 
 
 1205 
 
 1330 
 
 1464 
 
 
 348 
 
 1579 
 
 1704 
 
 1829 
 
 1953 
 
 2078 
 
 2203 
 
 2327 
 
 2452 
 
 2676 
 
 2701 
 
 
 1 349 
 
 2826 
 
 2950 
 
 3«74 
 
 3199 
 
 3323 
 
 3447 
 
 3671 
 
 3090 
 
 3820 
 
 3944 
 
 
8 
 
 LOGARITHMS 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 350 
 
 544068 
 
 4192 
 
 4316 
 
 4440 
 
 4564 
 
 4688 
 
 4812 
 
 4936 
 
 50(iO 
 
 5183 
 
 351 
 
 5307 
 
 5431 
 
 5555 
 
 6')78 
 
 5805 
 
 6925 
 
 6049 
 
 6172 
 
 6296 
 
 6419 
 
 352 
 
 6543 
 
 6666 
 
 6789 
 
 6913 
 
 7036 
 
 7159 
 
 7282 
 
 7405 
 
 7529 
 
 7652 
 
 353 
 
 7775 
 
 7898 
 
 8021 
 
 8144 
 
 8267 
 
 8389 
 
 8512 
 
 8635 
 
 8758 
 
 8881 
 
 364 
 
 9003 
 
 9126 
 
 9249 
 
 9371 
 
 94f)4 
 
 122 
 
 0717 
 
 9616 
 
 9739 
 
 9861 
 
 9984 
 
 .196 
 
 355 
 
 560228 
 
 0351 
 
 0473 
 
 0595 
 
 0840 
 
 0962 
 
 1084 
 
 1206 
 
 1328 
 
 356 
 
 1450 
 
 1572 
 
 1694 
 
 1816 
 
 1938 
 
 2060 
 
 2181 
 
 2303 
 
 2425 
 
 2547 
 
 357 
 
 2668 
 
 2790 
 
 2911 
 
 3033 
 
 3155 
 
 3276 
 
 3393 
 
 3519 
 
 3640 
 
 3762 
 
 1 358 
 
 3883 
 
 4004 
 
 4126 
 
 4247 
 
 4368 
 
 4489 
 
 4610 
 
 4731 
 
 4852 
 
 4973 
 
 359 
 
 5094 
 
 6215 
 
 6346 
 
 6457 
 
 6678 
 
 6699 
 
 582U 1 6940 
 
 6061 
 
 6182 
 
 360 
 
 6303 
 
 6423 
 
 6544 
 
 6664 
 
 6785 
 
 6905 
 
 7026 
 
 7146 
 
 7267 
 
 7387 
 
 361 
 
 7507 
 
 7(.27 
 
 7748 
 
 7868 
 
 7988 
 
 8108 
 
 8228 
 
 8349 
 
 8469 
 
 8589 
 
 362 
 
 8709 
 
 8829 
 
 8948 
 
 9068 
 
 9188 
 
 9308 
 
 9428 
 
 9548 
 
 9667 
 
 9787 
 
 363 
 
 9907 
 
 26 
 
 .146 
 
 .265 
 
 .386 
 
 .504 
 
 .624 
 
 .743 
 
 .863 
 
 .982. 
 
 364 
 
 561101 
 
 1L21 
 
 1340 
 
 .1459 
 
 1678 
 
 1698 
 
 1817 
 
 1936 
 
 2065 
 
 2173 
 
 365 
 
 2293 
 
 2412 
 
 2531 
 
 2660 
 
 2769 
 
 2887 
 
 3006 
 
 3126 
 
 3244 
 
 3362 
 
 3(56 
 
 3481 
 
 3600 
 
 3718 
 
 3837 
 
 3955 
 
 4074 
 
 4192 
 
 4311 
 
 4429 
 
 4548 
 
 367 
 
 4666 
 
 4784 
 
 4903 
 
 5021 
 
 5139 
 
 5257 
 
 5376 
 
 6494 
 
 6612 
 
 5730 
 
 368 
 
 6848 
 
 5966 
 
 6084 
 
 6202 
 
 6320 
 
 6437 
 
 6555 
 
 6673 
 
 6791 
 
 6909 
 
 369 
 
 7026 
 
 7144 
 
 7262 
 
 7379 
 
 7497 
 
 7614 
 
 7732 
 
 7849 
 
 7967 
 
 8084 
 
 370 
 
 8202 
 
 8319 
 
 8436 
 
 8554 
 
 8671 
 
 8788 
 
 8905 
 
 9023 
 
 9140 
 
 9257 
 
 371 
 
 9374 
 
 9491 
 
 9608 
 
 9725 
 
 9882 
 
 9959 
 
 ..76 
 
 .193 
 
 .309 
 
 .426 
 
 372 
 
 670o43 
 
 0660 
 
 0776 
 
 0893 
 
 1010 
 
 1126 
 
 1243 
 
 1359 
 
 1476 
 
 1592 
 
 373 
 
 1709 
 
 1825 
 
 1942 
 
 2058 
 
 2174 
 
 2291 
 
 2407 
 
 262C 
 
 2639 
 
 2755 
 
 374 
 
 2872 
 
 2988 
 
 3104 
 
 322C 
 
 3336 
 116 
 
 3452 
 
 3568 
 
 3f-:oi4 
 
 3800 
 
 c916 
 
 375 
 
 4031 
 
 4147 
 
 4263 
 
 4379 
 
 4494 
 
 4610 
 
 4726 
 
 4841 
 
 4957 
 
 5072 
 
 376 
 
 6188 
 
 6303 
 
 6419 
 
 6534 
 
 5650 
 
 5765 
 
 6880 
 
 6996 
 
 6111 
 
 6226 
 
 377 
 
 63 il 
 
 6457 
 
 6572 
 
 6687 
 
 6802 
 
 6917 
 
 7032 
 
 7147 
 
 7262 
 
 7377 
 
 378 
 
 74:)2 
 
 7607 
 
 7722 
 
 7836 
 
 7951 
 
 8066 
 
 8181 
 
 8295 
 
 8410 
 
 8525 
 
 379 
 
 8639 
 
 8754 
 
 8868 
 
 8983 
 
 9097 
 
 9212 
 
 9326 
 
 9441 
 
 9655 
 
 9669 
 
 380 
 
 9784 
 
 9898 
 
 ..12 
 
 .126 
 
 .241 
 
 .355 
 
 .469 
 
 .683 
 
 .697 
 
 .811 1 
 
 381 
 
 580925 
 
 1U39 
 
 1153 
 
 1267 
 
 1381 
 
 1495 
 
 1608 
 
 1722 
 
 1836 
 
 1950 ' 
 
 382 
 
 20o3 
 
 2177 
 
 2291 
 
 2404 
 
 2518 
 
 2631 
 
 2745 
 
 1868 
 
 2972 
 
 3086 
 
 383 
 
 3199 
 
 3:J]2 
 
 3426 
 
 3539 
 
 3652 
 
 3765 
 
 3879 
 
 J992 
 
 4105 
 
 4218 
 
 384 
 
 4331 
 
 4444 
 
 4557 
 
 4670 
 
 4783 
 
 4896 
 
 6009 
 
 <')122 
 
 6235 
 
 6348 
 
 385 
 
 6431 
 
 5574 
 
 5686 
 
 6799 
 
 6912 
 
 6024 
 
 6137 
 
 0260 
 
 6362 
 
 6475 
 
 386 
 
 6587 
 
 6700 
 
 6812 
 
 6925 
 
 7037 
 
 7149 
 
 7262 
 
 7374 
 
 7486 
 
 7599 
 
 387 
 
 7711 
 
 7823 
 
 7935 
 
 8047 
 
 8160 
 
 8272 
 
 8384 
 
 8496 
 
 8608 
 
 8720 
 
 388 
 
 8832 
 
 8944 
 
 9056 
 
 9167 
 
 9279 
 
 9391 
 
 L503 
 
 9615 
 
 9726 
 
 9834 
 
 389 
 
 9950 
 
 . .61 
 
 .173 
 
 .284 
 
 .396 
 
 .607 
 
 .619 
 
 .730 
 
 .842 
 
 .953 
 
 390 
 
 591065 
 
 1176 
 
 1287 
 
 1399 
 
 1510 
 
 1621 
 
 1732 
 
 1843 
 
 1955 
 
 2;!66 
 
 391 
 
 2177 
 
 2288 
 
 2399 
 
 2510 
 
 2621 
 
 2732 
 
 2843 
 
 2954 
 
 3064 
 
 317i; 
 
 392 
 
 3286 
 
 3397 
 
 3508 
 
 3618 
 
 3729 
 
 3840 
 
 3950 
 
 4061 
 
 4171 
 
 4282 
 
 393 
 
 4393 
 
 4503 
 
 4614 
 
 4724 
 
 4834 
 
 4945 
 
 6055 
 
 5165 
 
 6276 
 
 5386 
 
 394 
 
 6496 
 
 6608 
 
 5717 
 
 6827 
 
 5937 
 110 
 
 6047 
 
 6157 
 
 6267 
 
 6377 
 
 6487 
 
 395 
 
 6597 
 
 6707 
 
 6817 
 
 6927 
 
 7037 
 
 7146 
 
 7256- 
 
 7366 
 
 7476 
 
 7586 
 
 396 
 
 7695' 
 
 7805 
 
 7914 
 
 ■ 8024 
 
 8134 
 
 8243 
 
 8353 
 
 8462 
 
 8572 
 
 8681 
 
 397 
 
 8791 
 
 8900 
 
 90U9 
 
 9119 
 
 9228 
 
 L337 
 
 944() 
 
 t5o6 
 
 9666 
 
 y774" 
 
 398 
 
 9883 
 
 9992 
 
 .101 
 
 .210 
 
 .319 
 
 .428 
 
 .537 
 
 .646 
 
 ,755 
 
 .864 
 
 399 
 
 600973 
 
 1082 
 
 1191 
 
 1299 
 
 1408 
 
 1517 
 
 1625 
 
 1734 
 
 1843 
 
 1951 
 
' N. 
 400 
 
 OF NUMBERS. 
 
 9 
 
 
 6020)0 
 
 1 
 
 o 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2169 
 
 2277 
 
 2386 
 
 2494 
 
 2603 
 
 2711 
 
 2819 
 
 2928 
 
 3036 
 
 401 
 
 3144 
 
 3253 
 
 3361 
 
 34()9 
 
 3573 
 
 3686 
 
 3794 
 
 3902 
 
 4010 
 
 4118 
 
 4U-2 
 
 422t) 
 
 4334 
 
 4442 
 
 4550 
 
 44;68 
 
 4766 
 
 4874 
 
 4982 
 
 6089 
 
 5197 
 
 403 
 
 5305 
 
 5413 
 
 5521 
 
 5628 
 
 5736 
 
 5844 
 
 5951 
 
 6059 
 
 6166- 
 
 6274 
 
 4C4 
 
 6381 
 
 6489 
 
 6596 
 
 6704 
 
 6811 
 108 
 
 6919 
 
 7026 
 
 7133 
 
 7241 
 
 T3H8 
 
 406 
 
 7455 
 
 7562 
 
 7669 
 
 7777 
 
 7884 
 
 7991 
 
 8098 
 
 S205 
 
 8312 
 
 8419 
 
 ' 40'-, 
 
 8526 
 
 8()33 
 
 8740 
 
 8847 
 
 8954 
 
 90<)1 
 
 !Jl67 
 
 9274 
 
 9381 
 
 9488 1 
 
 1 4U7 
 
 9594 
 
 9701 
 
 9808 
 
 9914 
 
 ..21 
 
 .128 
 
 .234 
 
 .341 
 
 .447 
 
 .554 j 
 
 4o;s 
 
 610)60 
 
 0767 
 
 0873 
 
 0979 
 
 1086 
 
 1192 
 
 1298 
 
 1405 
 
 1611 
 
 1617 ! 
 
 40y 
 
 1723 
 
 1829 
 
 1936 
 
 2042 
 
 2148 
 
 '2264 
 
 2360 
 
 2466 
 
 2672 
 
 2()78 \ 
 
 410 
 
 2784 
 
 2890 
 
 2996 
 
 3102 
 
 3207 
 
 3313 
 
 3419 
 
 3525 
 
 3630 
 
 3736 1 
 
 411 
 
 3842 
 
 3947 
 
 4053 
 
 4159 
 
 4264 
 
 4370 
 
 4475 
 
 4581 
 
 4(i86 
 
 4792 
 
 412 
 
 4897 
 
 5003 
 
 6108 
 
 5213 
 
 5319 
 
 5424 
 
 5529 
 
 6()34 
 
 5740 
 
 5845 
 
 413 
 
 5950 
 
 6055 
 
 6160 
 
 6265 
 
 6370 
 
 6476 
 
 6581 
 
 6()86 
 
 6790 
 
 6895 
 
 414 
 
 7000 
 
 7105 
 
 7210 
 
 7315 
 
 7420' 
 
 7525 
 
 7629 
 
 7734 
 
 7839 
 
 7943 
 
 415 
 
 8048 
 
 8153 
 
 8257 
 
 8362 
 
 8466 
 
 8571 
 
 8676 
 
 8780 
 
 8884 
 
 8989 
 
 41() 
 
 9293 
 
 9198 
 
 9302 
 
 9406 
 
 9511 
 
 9615 
 
 9719 
 
 9824 
 
 9928 
 
 ..32 
 
 417 
 
 620136 
 
 0240 
 
 0344 
 
 0448 
 
 0552 
 
 0656 
 
 0760 
 
 0864 
 
 0968 
 
 1072 
 
 418 
 
 1176 
 
 1280 
 
 1384 
 
 1488 
 
 1692 
 
 1695 
 
 1799 
 
 1903 
 
 -■(J07 
 
 2110 
 
 419 
 
 2214 
 
 2318 
 
 2421 
 
 2525 
 
 2628 
 
 2732 
 
 2835 
 
 2939 
 
 3042 
 
 314 J 
 
 420 
 
 3249 
 
 3353 
 
 3456 
 
 3559 
 
 3663 
 
 3766 
 
 3869 
 
 3973 
 
 4076 
 
 4179 
 
 421 
 
 4282 
 
 4385 
 
 4488 
 
 4691 
 
 4695 
 
 4798 
 
 4901 
 
 5004 
 
 6107 
 
 5210 
 
 422 
 
 5312 
 
 5415 
 
 5518 
 
 5621 
 
 6724 
 
 5827 
 
 5929 
 
 6032 
 
 6136 
 
 ()238 
 
 423 
 
 6340 
 
 6443 
 
 6546 
 
 6648 
 
 6751 
 
 6853 
 
 (i956 
 
 '1 058 
 
 7161 
 
 72()3 
 
 424 
 
 7366 
 
 7468 
 
 7571 
 
 7673 
 
 7775 
 103 
 
 7878 
 
 7980 
 
 8082 
 
 8185 
 
 8287 
 
 425 
 
 8389 
 
 8491 
 
 8593 
 
 8696 
 
 8797 
 
 8900 
 
 9002 
 
 9104 
 
 9206^ 
 
 9308 
 
 42(i 
 
 9410 
 
 9512 
 
 9613 
 
 9715 
 
 9817 
 
 9919 
 
 ..21 
 
 .123 
 
 .224 
 
 .326 
 
 427 
 
 630428 
 
 0530 
 
 0o31 
 
 0733 
 
 0835 
 
 0936 
 
 1038 
 
 1139 
 
 1241 
 
 1342 
 
 428 
 
 . 1444 
 
 1545 
 
 1647 
 
 1748 
 
 1849 
 
 1951 
 
 2052 
 
 2163 
 
 2255 
 
 2356 
 
 429 
 
 2457 
 
 2559 
 
 2660 
 
 2761 
 
 2862 
 
 2963 
 
 3064 
 
 3166 
 
 3266 
 
 3367 
 
 430 
 
 3468 
 
 3569 
 
 3070 
 
 3771 
 
 3872 
 
 3973 
 
 4074 
 
 4175 
 
 4276 
 
 4376 
 
 431 
 
 4477 
 
 4578 
 
 4()79 
 
 4779 
 
 4880 
 
 4981 
 
 50HI 
 
 5182 
 
 5283 
 
 5383 
 
 432 
 
 5484 
 
 6584 
 
 5685 
 
 5785 
 
 5886 
 
 698() 
 
 6087 
 
 6187 
 
 6287 
 
 6388 
 
 433 
 
 6488 
 
 6588 
 
 6688 
 
 6789 
 
 68H9 
 
 6989 
 
 7089 
 
 7189 
 
 7290 
 
 7390 
 
 434 
 
 7490 
 
 7590 
 
 7690 
 
 7790 
 
 7890 
 
 7990 
 
 8090 
 
 8190 
 
 8290 
 
 8389 
 
 435 
 
 8489 
 
 8589 
 
 8689 
 
 8789 
 
 BBSS 
 
 8988 
 
 9088 
 
 9188 
 
 9287 
 
 9387 
 
 1 436 
 
 9486 
 
 9586 
 
 9()86 
 
 9785 
 
 9666 
 
 9984 
 
 ..84 
 
 .183 
 
 .283 
 
 .382 1 
 
 437 
 
 G40181 
 
 0581 
 
 0680 
 
 0779 
 
 3879 
 
 0978 
 
 1077 
 
 1177 
 
 1276 
 
 1375 
 
 438 
 
 1474 
 
 1573 
 
 1672 
 
 1771 
 
 1871 
 
 1970 
 
 2069 
 
 2168 
 
 2267 
 
 2366 
 
 439 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 2860 
 
 2959 
 
 3058 
 
 3156 
 
 3255 
 
 3354 
 
 440 
 
 3453 
 
 3551 
 
 3650 
 
 3749 
 
 3847 
 
 3946 
 
 4044 
 
 4143 
 
 4242 
 
 4340 
 
 441 
 
 4439 
 
 4537 
 
 4(J36 
 
 4734 
 
 4832 
 
 4931 
 
 5029 
 
 5127 
 
 5226 
 
 5324 
 
 442 
 
 6422 
 
 5521 
 
 5619 
 
 6717 
 
 5816 
 
 5913 
 
 6011 
 
 6110 
 
 6208 
 
 6306 
 
 443 
 
 6404 
 
 6502 
 
 6600 
 
 6698 
 
 G.96 
 
 6894 
 
 6992 
 
 7C89 
 
 7187 
 
 7285 
 
 444 
 
 7383 
 
 7481 
 
 7679 
 
 7676 
 
 7774 
 
 98 
 
 8750 
 
 7872 
 
 7969 
 
 8067 
 
 8166 
 
 8262 
 
 445 
 
 8360 
 
 8458 
 
 8555 
 
 8653 
 
 8848 
 
 8945 
 
 9043 
 
 9140 
 
 9237 
 
 446 
 
 9335 
 
 9432 
 
 9530 
 
 9627 
 
 9724 
 
 9821 
 
 9919 
 
 . .16 
 
 .113 
 
 .210 
 
 447 
 
 650308 
 
 0405 
 
 0502 
 
 0599 
 
 0696 
 
 0.93 
 
 0890 
 
 0987 
 
 1084 
 
 1181 
 
 448 
 
 1278 
 
 1375 
 
 1472 
 
 1569 
 
 1666 
 
 1762 
 
 1859 
 
 195o 
 
 2053 
 
 2150 
 
 4« 
 
 2246 
 
 2343 
 
 2410 
 
 2530 
 
 2633 
 
 2730 
 
 2826 
 
 2923 
 
 3019 
 
 3116 
 
^() 
 
 
 LOGARITHxMS 
 
 
 N. 
 
 
 
 1 
 
 • 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 450 
 
 653213 
 
 3309 
 
 3405 
 
 3502 
 
 3598 
 
 3695 
 
 3791 
 
 3888 
 
 3984 
 
 4080 
 
 
 451 
 
 4177 
 
 4273 
 
 4369 
 
 44;)5 
 
 4562 
 
 4; .58 
 
 4754 
 
 4850 
 
 4946 
 
 5042 
 
 
 452 
 
 5138 
 
 5235 
 
 5331 
 
 5427 
 
 5526 
 
 5619 
 
 5715 
 
 6810 
 
 5906 
 
 6002 
 
 
 453 
 
 6098 
 
 6194 
 
 6290 
 
 6386 
 
 6482 
 
 6577 i 6673 
 
 6769 
 
 6864 
 
 6960 
 
 
 454 
 
 705d 
 
 7152 
 
 7247 
 
 7343 
 
 7438 
 
 96 
 
 8393 
 
 7634 
 
 7629 
 
 7725 
 
 7820 
 
 7916 
 
 
 455 
 
 8011 
 
 8107 
 
 8202 
 
 8298 
 
 8488 
 
 8584 
 
 8679 
 
 8774 
 
 8S70 
 
 
 456 
 
 8965 
 
 9060 
 
 5^155 
 
 9250 
 
 9346 
 
 9441 
 
 L536 j 9631 
 
 9726 
 
 bS2r 
 
 
 157 
 
 ;wit) 
 
 . .11 
 
 .106 
 
 .201 
 
 .296 
 
 .391 
 
 .486 1 .681 
 
 .676 
 
 .771 
 
 
 458 
 
 660865 
 
 0960 
 
 1055 
 
 1150 
 
 1245 
 
 1339 
 
 1434 
 
 1529 
 
 1623 
 
 1718 
 
 
 459 
 
 1813 
 
 1907 
 
 2002 
 
 20c»6 
 
 2191 
 
 2286" 
 
 2380 
 
 2476 
 
 2f()9 
 
 2663 
 
 
 460 
 
 2758 
 
 2852 
 
 2947 
 
 3041 
 
 3135 
 
 3230 
 
 3324 
 
 3418 
 
 3512 
 
 3607 
 
 
 461 
 
 3701 
 
 3 7;J5 
 
 3889 
 
 3983 
 
 4078 
 
 4172 
 
 4266 
 
 4360 
 
 4454 
 
 4648 
 
 
 462 
 
 4';i42 
 
 4736 
 
 48,0 
 
 4924 
 
 5018 
 
 5112 
 
 5206 
 
 5299- 
 
 ■5393 
 
 5487 
 
 
 463 
 
 5581 
 
 56/5 
 
 57o9 
 
 5862 
 
 5956 
 
 6050 
 
 6143 
 
 6237 
 
 6331 
 
 6424 
 
 
 464 
 
 6518 
 
 6612 
 
 67J5 
 
 6799 
 
 6892 
 
 6986 
 
 7079 
 
 7173 
 
 7266 
 
 7360 
 
 
 465 
 
 7453 
 
 7546 
 
 7640 
 
 7733 
 
 7826 
 
 7920 
 
 8013 
 
 8106 
 
 8199 
 
 8293 
 
 
 466 
 
 8386 
 
 8479 
 
 8572 
 
 8665 
 
 8759 
 
 8852 
 
 8945 
 
 9038 
 
 9131 
 
 9324 
 
 
 467 
 
 9b 17 
 
 9410 
 
 9503 
 
 9596 
 
 9689 
 
 9782 
 
 9875 
 
 9967 
 
 .60 
 
 .153 
 
 
 468 
 
 6/0241 
 
 0339 
 
 0431 
 
 0524 
 
 0*il7 
 
 0710 
 
 0802 
 
 0895 
 
 0988 
 
 1080 
 
 
 469 
 
 1173 
 
 1265 
 
 1358 
 
 1451! 
 
 1643 
 
 1636 
 
 1728 
 
 1821 
 
 1913 
 
 2006 
 
 
 470 
 
 2098 
 
 2190 
 
 2283 
 
 2376 
 
 2467 
 
 2560 
 
 2652 
 
 2744 
 
 2836 
 
 2929 
 
 
 471 
 
 3021 
 
 3113 
 
 3205 
 
 3297 
 
 3390 
 
 3482 
 
 3574 
 
 3666 
 
 3758 
 
 3850 
 
 
 472 
 
 3942 
 
 4034 
 
 4126 
 
 4218 
 
 4310 
 
 4402 
 
 4494 
 
 4586 
 
 4677 
 
 4769 
 
 
 473 
 
 4861 
 
 4953 
 
 5045 
 
 5137 
 
 5228 
 
 5320 
 
 5412 
 
 6503 
 
 5595 
 
 6687 
 
 
 474 
 
 5778 
 
 5870 
 
 5962 
 
 6033 
 
 6145 
 
 91 
 
 7059 
 
 6236 
 
 6328 
 
 6419 
 
 6511 
 
 6U02 
 
 
 475 
 
 6694 
 
 6785 
 
 6876 
 
 6908 
 
 7151 
 
 7242 
 
 7333 
 
 7424 
 
 7616 
 
 
 476 
 
 7607 
 
 7698 
 
 7789 
 
 7881 
 
 7972 
 
 8063 
 
 8154 
 
 8246 
 
 8336 
 
 8427 
 
 
 .477 
 
 8518 
 
 8609 
 
 8700 
 
 8791 
 
 8882 
 
 8972 
 
 9064 
 
 9155 
 
 9246 
 
 9337 
 
 
 478 
 
 9428 
 
 9519 
 
 9610 
 
 9700 
 
 9791 
 
 9882 
 
 9973 
 
 ..63 
 
 .154 
 
 .246 
 
 
 479 
 
 680336 
 
 0426 
 
 0517 
 
 0607 
 
 0698 
 
 0789 
 
 0879 
 
 0970 
 
 1060 
 
 1161 
 
 
 480 
 
 1241 
 
 1332 
 
 1422 
 
 1513 
 
 1603 
 
 1693 
 
 1784 
 
 1874 
 
 1964 
 
 2065 
 
 
 481 
 
 2145 
 
 2235 
 
 2326 
 
 2416 
 
 2606 
 
 2696 
 
 2686- 
 
 2777 
 
 2867 
 
 2957 
 
 
 482 
 
 3047 
 
 3137 
 
 3227 
 
 3317 
 
 3407 
 
 3497 
 
 3587 
 
 3677 
 
 3767 
 
 3857 
 
 
 483 
 
 3947 
 
 4037 
 
 4127 
 
 4217 
 5114 
 
 4307 
 
 4396 
 
 4486 
 
 4676 
 
 '4666 ' 
 
 4766 
 
 
 484 
 
 4854 
 
 4935 
 
 5025 
 
 5204 
 
 6294 
 
 6383 
 
 6473 
 
 6563 
 
 6652 
 
 
 485 
 
 5742 
 
 6831 
 
 5921 
 
 6010 
 
 6100 
 
 6189 
 
 6279 
 
 6368 
 
 6458 
 
 6547 
 
 
 486 
 
 6636 
 
 6726 
 
 6815 
 
 6904 
 
 6994 
 
 7083 
 
 7172 
 
 7261 
 
 7351 
 
 7440 
 
 
 487 
 
 7529 
 
 7618 
 
 7707 
 
 7796 
 
 7886 
 
 7975 
 
 8064 
 
 8153 
 
 8242 
 
 8331 
 
 
 488 
 
 8420 
 
 8509 
 
 8598 
 
 8687 
 
 8776 
 
 8865 
 
 8953 
 
 9042 
 
 9131 
 
 9220 
 
 
 489 
 
 9309 
 
 9398 
 
 9486 
 
 9575 
 
 9664 
 
 9753 
 
 9841 
 
 9930 
 
 ..19 
 
 .107 
 
 
 490 
 
 690196 
 
 0285 
 
 0373 
 
 0362 
 
 0550 
 
 0639 
 
 0728 
 
 0816 
 
 0905 
 
 0993 
 
 
 491 
 
 1U81 
 
 1170 
 
 1258 
 
 1347 
 
 1435 
 
 1524 
 
 1612 
 
 1700 
 
 1789 
 
 1877 
 
 
 492 
 
 165 
 
 2053 
 
 2142 
 
 2230 
 
 2318 
 
 2406 
 
 2494 
 
 2583 
 
 2671 
 
 2759 
 
 
 493 
 
 2847 
 
 2935 
 
 3023 
 
 3111 
 
 3199 
 
 3287 
 
 3375 
 
 3463 
 
 3551 
 
 3639 
 
 
 494 
 
 3727 
 
 3815 
 
 3903 
 
 3991 
 
 4078 
 
 88 
 
 4956 
 
 4166 
 
 4264 
 
 4342 
 
 4430 
 
 4517 
 
 
 . 495 
 
 4605 
 
 4693 
 
 4781 
 
 4868 
 
 5044 
 
 6131 
 
 5210 
 
 6307 
 
 6394 
 
 
 j 496 
 
 5482 
 
 5569 
 
 5657 
 
 5744' 
 
 5832- 
 
 5919 
 
 6007 
 
 6094 
 
 6182 
 
 6269 
 
 
 1 497 
 
 6356 
 
 5444 
 
 6531 
 
 6618 
 
 6706 
 
 6793 
 
 ,6880 
 
 6968 
 
 7055 
 
 714!. 
 
 
 '. 490 
 
 7229 
 
 7317 
 
 7404 
 
 7491 
 
 7578 
 
 7665 
 
 7762 
 
 7839 
 
 7926 
 
 8014 
 
 
 1 499 
 
 8iUi 
 
 8188 
 
 8275 
 
 8362 
 
 8449 
 
 8535 
 
 8622 
 
 8709 
 
 8796 
 
 8883 
 
 
OF NUMBERS. 11 
 
 1 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 600 
 
 698970 
 
 9057 
 
 9144 
 
 9231 
 
 9317 
 
 9404 
 
 9491 
 
 9578 
 
 9664 
 
 9751 
 
 501 
 
 9.038 
 
 9924 
 
 . 11 
 
 ..98 
 
 .184 
 
 .271 
 
 .358 
 
 .444 
 
 .631 
 
 .617 
 
 60-2 
 
 700/04 
 
 0790 
 
 0877 
 
 0963 
 
 1050 
 
 1136 
 
 1222 
 
 1309 
 
 1395 
 
 1482 
 
 603 
 
 15()8 
 
 1654 
 
 1741 
 
 h-.27 
 
 1913 
 
 1999 
 
 2086 
 
 2172 
 
 2258 
 
 •2344 
 
 604 
 
 2431 
 
 2517 
 
 2603 
 
 2v.89 
 
 2775 
 
 86 
 
 3635 
 
 2861 
 
 2947 
 
 3033 
 
 3119 
 
 3206 
 
 505 
 
 3291 
 
 3377 
 
 3463 
 
 3549 
 
 3721 
 
 3807 
 
 3895 
 
 3979 
 
 4066 
 
 60) 
 
 4151 
 
 4236 
 
 4322 
 
 4408 
 
 4494 
 
 4579 
 
 4665 
 
 4751 
 
 4837 
 
 4922 
 
 607 
 
 5008 
 
 5094 
 
 5179 
 
 6265 
 
 5350 
 
 6436 
 
 5522 
 
 5607 
 
 6693 
 
 5778 
 
 60o 
 
 5.S04 
 
 6949 
 
 6 '35 
 
 6120 
 
 620<i 
 
 6291 
 
 6376 
 
 6462 
 
 6547 
 
 6632 
 
 509 
 
 6718 
 
 6803 
 
 6888 
 
 6974 
 
 7059 
 
 7U4 
 
 7229 
 
 73 In 7400 
 
 7486 
 
 510 
 
 7570 
 
 7655 
 
 7740 
 
 7826 
 
 7910 
 
 7996 
 
 8081 
 
 8166 
 
 o:^61 
 
 8336 
 
 , 611 
 
 8421 
 
 8506 
 
 8591 
 
 8676 
 
 8761 
 
 8846 
 
 8931 
 
 9015 
 
 9100- 
 
 9185 
 
 612 
 
 9270 
 
 9355 
 
 9440 
 
 9524 
 
 9609- 
 
 !694' 
 
 9<79 
 
 9863 
 
 9948 
 
 .33 
 
 613 
 
 710117 
 
 0202 
 
 0287 
 
 0371 
 
 0456 
 
 0540 
 
 0625 
 
 0710 
 
 0794 
 
 0879 
 
 614 
 
 0963 
 
 1048 
 
 1132 
 
 1217 
 
 1301 
 
 1385 
 
 1470 
 
 1654 
 
 1639 
 
 1723 
 
 615 
 
 1807 
 
 1892 
 
 1976 
 
 20;>0 
 
 2144 
 
 2229 
 
 2313 
 
 2397 
 
 2481 
 
 2666 
 
 510 
 
 2()o0 
 
 2734 
 
 2818 
 
 2902 
 
 2986 
 
 3070 
 
 3154 
 
 3238 
 
 3326 
 
 3407- 
 
 617 
 
 3491 
 
 35 ;5 
 
 3659 
 
 3742 
 
 3826 
 
 3910 
 
 3994 
 
 4078 
 
 4162 
 
 4246 
 
 61b 
 
 4330 
 
 4414 
 
 4-197 
 
 4581. 
 
 4665 
 
 4749 
 
 4833 
 
 4916 
 
 5000 
 
 5084 
 
 619 
 
 5167 
 
 6251 
 
 6335 
 
 5418 
 
 5602 
 
 6586 
 
 5.69 
 
 5753 
 
 5836 
 
 5920 
 
 520 
 
 6003 
 
 6087 
 
 6170 
 
 6254 
 
 6337 
 
 6421 
 
 6504 
 
 6588 
 
 6671 
 
 6754 
 
 621 
 
 6838 
 
 6921 
 
 7004 
 
 7088 
 
 7171 
 
 7254 
 
 7338 
 
 7421 
 
 7504 
 
 7587 
 
 522 
 
 7671 
 
 7754 
 
 7837 
 
 792.J 
 
 8003 
 
 80St) 
 
 8169 
 
 8253 
 
 8336 
 
 8419 
 
 523 
 
 8502 
 
 8585 
 
 8668 
 
 8751 
 
 8834 
 
 8917 
 
 9000 
 
 9083 
 
 9165 
 
 9248 
 
 524 
 
 9331 
 
 9414 
 
 9497 
 
 9580 
 
 9663 
 
 82 
 
 0490 
 
 9745 
 
 9828 
 
 9911 
 
 9994 
 
 ..77 
 
 526 
 
 720159 
 
 0242 
 
 0325 
 
 0407 
 
 05!3 
 
 0655 
 
 0738 
 
 0821 
 
 0903 
 
 526 
 
 0986 
 
 101)8 
 
 1151 
 
 1233 
 
 1316 
 
 1398 
 
 1481 
 
 1563 
 
 1646 
 
 1728 
 
 527 
 
 1811 
 
 1893 
 
 .975 
 
 2058 
 
 2140 
 
 2222 
 
 2305 
 
 2387 
 
 2469 
 
 2552 
 
 628 
 
 2t>34 
 
 2716 
 
 2798 
 
 2881 
 
 2963 
 
 3045 
 
 3127 
 
 3209 
 
 3291 
 
 3374 
 
 529 
 
 3456 
 
 3638 
 
 3620 
 
 3702 
 
 3784 
 
 3866 
 
 3948 
 
 4030 
 
 4112 
 
 4194 
 
 530 
 
 4276 
 
 4358 
 
 4440 
 
 4622 
 
 4604 
 
 4685 
 
 4767 
 
 4849 
 
 4931 
 
 5013 
 
 531 
 
 5095 
 
 6176 
 
 5258 
 
 5340 
 
 6422 
 
 5503 
 
 6585 
 
 5667 
 
 5748 
 
 5830 
 
 532 
 
 6912 
 
 5993 
 
 6075 
 
 6156 
 
 6238 
 
 6320 
 
 6-101 
 
 6483 
 
 6564 
 
 6646 
 
 633 
 
 6727 
 
 6809 ; 6890 
 
 6972 
 
 7053 
 
 7134 
 
 7216 
 
 7297 
 
 7379 
 
 7460 
 
 534 
 
 7541 
 
 7623 
 
 7704 
 
 7785 
 
 7866 
 
 7948 
 
 8029 
 
 8110 
 
 8191 
 
 8273 
 
 635 
 
 8354 
 
 8435 
 
 8516 
 
 8597 
 
 8678 
 
 8759 
 
 8841 
 
 8922 
 
 9003 
 
 9084 
 
 53U 
 
 9165 
 
 9246 
 
 9327 
 
 9403 
 
 9489 
 
 9570 
 
 9651 
 
 9732 
 
 9813 
 
 9893 
 
 537 
 
 9974 
 
 ..^5 
 
 .136 
 
 .217 
 
 .298 
 
 .378 
 
 .459 
 
 .540 
 
 .621 
 
 .702 1 
 
 538 
 
 730^82 
 
 0863 
 
 0944 
 
 1024 
 
 1105 
 
 1186 
 
 126() 
 
 1347 
 
 1428 
 
 1608 ! 
 
 539 
 
 1689 
 
 1669 
 
 1750 
 
 1830 
 
 1911 
 
 1991 
 
 2072 
 
 2162 
 
 2233 
 
 2313 
 
 MO 
 
 2394 
 
 2474 
 
 2555 
 
 2635 
 
 2715 
 
 2796 
 
 2876 
 
 2966 
 
 3037 
 
 3117 
 
 541 
 
 3197 
 
 3278 3358 
 
 3438 
 
 3518 
 
 3598 
 
 3679 
 
 3769 
 
 3S39 
 
 3919 
 
 642 
 
 3999 
 
 4079 
 
 4160 
 
 4240 
 
 4320 
 
 4400 
 
 4480 
 
 4560 
 
 4640 
 
 4720 
 
 643 
 
 4600 
 
 4i80 
 
 4960 
 
 5040 
 
 5120 
 
 5200 
 
 5279 
 
 6359 
 
 5439 
 
 6519 
 
 544 
 
 5599 
 
 6679 
 
 6759 
 
 5838 
 
 5918 
 
 80 
 
 6715 
 
 6998 
 
 6078 
 
 6157 
 
 6237 
 
 6IM7 
 
 545 
 
 6397 
 
 6476 
 
 6556 
 
 6636 
 
 6795 
 
 6874 
 
 6954 
 
 7034 
 
 7113 
 
 54t) 
 
 7193 
 
 7272 
 
 7352 
 
 7431 
 
 7511 
 
 7690 
 
 7670 
 
 7749 
 
 7829 
 
 7908 1 
 
 647 
 
 7987 
 
 80o7 
 
 8146 
 
 8225 
 
 8305 
 
 8384 
 
 8463 
 
 8543 
 
 8622 
 
 8701 t 
 
 548 
 
 8781 
 
 88')0 
 
 8939 
 
 9018 
 
 9097 
 
 9177 
 
 9256 
 
 9335 
 
 9414 
 
 949? 
 
 649 
 
 9572 
 
 9u51 
 
 9731 
 
 9810 
 
 9889 
 
 9968 
 
 ..47 
 
 .12'6 
 
 .205 
 
 .284 
 
12 
 
 
 LOGARITHMS 
 
 
 N. 
 650 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 740363 
 
 0442 
 
 0521 
 
 0560 
 
 0678 
 
 0757 
 
 0836 
 
 0915 
 
 0994 
 
 1073 
 
 
 651 
 
 1152 
 
 1230 
 
 1309 
 
 1388 
 
 1467 
 
 1646 
 
 1624 
 
 1703 
 
 1782 
 
 1860 
 
 
 552 
 
 1939 
 
 2018 
 
 2096 
 
 2176 
 
 2254 
 
 2332 
 
 2411 
 
 2489 
 
 2668 
 
 2646 
 
 
 653 
 
 2725 
 
 2804 
 
 2882 
 
 2961 
 
 3039 
 
 3118 
 
 3196 
 
 3276 
 
 3363 
 
 3431 
 
 
 654 
 
 3510 
 
 3658 
 
 3667 
 
 3746 
 
 3823 
 
 79 
 
 4606 
 
 3902 
 
 3980 
 
 4058 
 
 4136 
 
 4215 
 
 
 555 
 
 4293 
 
 4371 
 
 4449 
 
 4628 
 
 4684 
 
 4762 
 
 4840 
 
 4919 
 
 4997 
 
 
 656 
 
 5075 
 
 5153 
 
 6231 
 
 6309 
 
 5387 
 
 5465 
 
 5543 
 
 6621 
 
 6699 
 
 6777 
 
 
 657 
 
 5855 
 
 5933 
 
 6011 
 
 6089 
 
 6167 
 
 6245 
 
 6323 
 
 6401 
 
 6479 
 
 6566 
 
 
 558 
 
 6634 
 
 6712 
 
 6790 
 
 6868 
 
 6945 
 
 7023 
 
 7101 
 
 7179 
 
 7256 
 
 7334 
 
 
 5->9 
 
 7412 
 
 7489 
 
 7667 
 
 7645 
 
 7722 
 
 7800 
 
 7878 
 
 7955 
 
 8033 
 
 8110 
 
 
 6fi(l 
 
 8188 
 
 8266 
 
 8343 
 
 8421 
 
 8498 
 
 8576 
 
 8653 
 
 8731 
 
 8808 
 
 8885 
 
 • 
 
 6(>1 
 
 8003 
 
 9040 
 
 9118 
 
 9195 
 
 9272 
 
 9360 
 
 9427 
 
 9504 
 
 9682 
 
 9659 - 
 
 
 5G2 
 
 9736 
 
 9814^ 
 
 9891 
 
 9968 
 
 ..45 
 
 .123 
 
 .200 
 
 .277 
 
 .364 
 
 .431 
 
 
 563 
 
 750608 
 
 0686 
 
 0663 
 
 .0740 
 
 0817 
 
 0894 
 
 0971 
 
 1048 
 
 1125 
 
 1202 
 
 
 564 
 
 1279 
 
 1356 
 
 1433 
 
 1510 
 
 1687 
 
 1664 
 
 1741 
 
 1818 
 
 1895 
 
 1972 
 
 
 66> 
 
 2048 
 
 2125" 
 
 2202 
 
 2279 
 
 2356 
 
 2433 
 
 2609 
 
 2686 
 
 2663 
 
 2740 
 
 
 566 
 
 2816 
 
 2893 
 
 2970 
 
 3047 
 
 3123 
 
 3200 
 
 3277 
 
 3353 
 
 3430 
 
 3506 
 
 
 567 
 
 3582 
 
 3660 
 
 3736 
 
 3813 
 
 3889 
 
 3966 
 
 4042 
 
 4119 
 
 4195 
 
 4272 
 
 
 568- 
 
 4348 
 
 4425 
 
 4501 
 
 4578 
 
 4654 
 
 4730 
 
 4807 
 
 4883 
 
 4960 
 
 6036 
 
 
 569 
 
 5112 
 
 5189 
 
 6265 
 
 6341 
 
 cm 
 
 6494 
 
 5570 
 
 6646 
 
 6722 
 
 6799 
 
 
 570 
 
 5876 
 
 5961 
 
 6027 
 
 6103 
 
 61F0 
 
 6256 
 
 6332 
 
 6408 
 
 6484 
 
 6660 
 
 
 671 
 
 6636 
 
 6712 
 
 6788 
 
 6864 
 
 6940 
 
 7016 
 
 7092 
 
 7168 
 
 7244 
 
 7320 
 
 
 672 
 
 7396 
 
 7472 
 
 7548 
 
 7624 
 
 7700 
 
 7776 
 
 7851 
 
 7927 
 
 8003 
 
 8079^ 
 
 
 673 
 
 8155 
 
 8230 
 
 8306 8382 
 
 8458 
 
 8633 
 
 8609 
 
 8685 
 
 8761 
 
 8836 
 
 
 574 
 
 8912 
 
 8988 
 
 9068 
 
 9139 
 
 9214 
 
 74 
 
 9970 
 
 9290 
 
 9366 
 
 9441 
 
 9517 
 
 9592 
 
 
 575 
 
 9658 
 
 9743 
 
 9819 
 
 9894 
 
 ..45 
 
 .121 
 
 .196 
 
 .272 
 
 .347 
 
 
 676 
 
 760422 
 
 0498 
 
 0573 
 
 0649 
 
 0724 
 
 0799 
 
 0875 
 
 0950 
 
 1025 
 
 1101 
 
 
 577 
 
 1176 
 
 1251 
 
 1326 
 
 1402 
 
 1477 
 
 1552 
 
 1627 
 
 1702 
 
 1778 
 
 1853 
 
 
 578 
 
 1938 
 
 2003 
 
 2078 
 
 2lS3 
 
 2228 
 
 2303 
 
 2378 
 
 2453 
 
 2529 
 
 2604 
 
 
 679 
 
 i679 
 
 2754 
 
 2829 
 
 2904 
 
 ■2978 
 
 3053 
 
 3128 
 
 2203 
 
 3278 
 
 3353 
 
 
 580 
 
 3428 
 
 3503 
 
 3578 
 
 3653 
 
 3727 
 
 3802 
 
 3877 
 
 3962 
 
 4027 
 
 4101 
 
 
 681 
 
 4176 
 
 4251 
 
 4326 
 
 4400 
 
 4475 
 
 4550 
 
 4624 
 
 4699 
 
 4774 
 
 4848 
 
 
 682 
 
 4923 
 
 4998 
 
 6072 
 
 5147 
 
 6221 
 
 5296 
 
 6370 
 
 6446 
 
 5520 
 
 6594 
 
 
 683 
 
 6669 
 
 5743 
 
 6818 
 
 6892 
 
 5966 
 
 6041 
 
 6116 
 
 6190 
 
 6264 J 
 
 6338 
 
 
 684 
 
 6413 
 
 6487 
 
 6562 
 
 6636 
 
 6710 
 
 6785 
 
 6859 
 
 6933 
 
 7007 
 
 7082 
 
 
 585 
 
 7156 
 
 7230 
 
 7304 
 
 7379 
 
 7463 
 
 7527 
 
 7601 
 
 7675 
 
 7749 
 
 7823 
 
 
 586, 
 
 7898 
 
 7972 
 
 8046 
 
 8120 
 
 8194 
 
 8268 
 
 8342 
 
 8416 
 
 8490 
 
 8564 
 
 
 587 
 
 8638 
 
 8712 
 
 8786 
 
 8860 
 
 8934 
 
 9008 
 
 9082 
 
 9156 
 
 9230 
 
 9303 
 
 
 588 
 
 9377 
 
 9451 
 
 9525 
 
 9599 
 
 9673 
 
 9746 
 
 9820 
 
 9894 
 
 9968 
 
 ..42 
 
 
 589 
 
 170115 
 
 0189 
 
 0263 
 
 0336 
 
 0410 
 
 0484 
 
 0567 
 
 0631 
 
 0705 
 
 0778 
 
 
 59C 
 
 0852 
 
 0926 
 
 0999 
 
 1073 
 
 1146 
 
 1220 
 
 1293 
 
 1367 
 
 1440 
 
 1514 
 
 
 591 
 
 1687 
 
 1661 
 
 1734 
 
 1808 
 
 1881 
 
 1956 
 
 2028 
 
 2102 
 
 2175 
 
 2248 
 
 
 592 
 
 2322 
 
 2395 
 
 2468 
 
 3542 
 
 2615 
 
 2688 
 
 2762 
 
 2835 
 
 2908 
 
 2981 
 
 
 593 
 
 3055 
 
 3128 
 
 3201 
 
 3274 
 
 3348 
 
 3421 
 
 3494 
 
 3567 
 
 3640 
 
 3713 
 
 
 694 
 
 3786 
 
 3860 
 
 3933 
 
 4006 
 
 4079 
 
 73 
 
 4809 
 
 4152 
 
 4225 
 
 4298 
 
 4371 
 
 4444 
 
 
 595 
 
 4517 
 
 4590 
 
 4663 
 
 4736 
 
 4882 
 
 4955 
 
 5028 
 
 5100 
 
 ol73 
 
 
 696 
 
 5246 
 
 6319 
 
 5392 
 
 5465 
 
 5538 
 
 6610 
 
 6683 
 
 5756 
 
 6829 
 
 6902 
 
 
 597 
 
 5974 
 
 6047 
 
 6120 
 
 6193 
 
 6265 
 
 6338 
 
 6411 
 
 6483 
 
 6556 
 
 6629 
 
 
 598 
 
 6701 
 
 6774 
 
 6846 
 
 6919 
 
 6992 
 
 7064 
 
 7137 
 
 7209 
 
 7282 
 
 7354 
 
 
 699 
 
 7427 
 
 7499 
 
 7572 
 
 7644 
 
 7717 
 
 7789 
 
 7862 
 
 7934 
 
 8006 
 
 8079 : 
 
 
OFNUMBERS 13 
 
 i 
 
 N 
 
 
 778151 
 
 1 
 
 2 i 3 
 
 j 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 600 
 
 8224 
 
 8296 i 5368 
 
 8441 
 
 8513 
 
 8585 
 
 8658 
 
 8730 
 
 8802 
 
 (iOl 
 
 8874 
 
 8947 
 
 9019 i 9091 
 
 9163 
 
 9236 
 
 9308 
 
 9380 
 
 9452 
 
 9524 
 
 (m 
 
 95'J6 
 
 W>()9 
 
 9741 I 9813 
 
 9885 
 
 9957 
 
 . .29 
 
 .101 
 
 .173 
 
 .245 
 
 603 
 
 780317 
 
 0389 
 
 04()1 0533 
 
 0.i05 
 
 0677 
 
 0749 
 
 0821 
 
 0893 
 
 (^65 
 
 601 
 
 1037 
 
 1109 
 
 1181 
 
 1253 
 
 1324 
 
 72 
 
 2042 
 
 1396 
 
 1468 
 
 1540 
 
 1612 
 
 1684 
 
 1 605 
 
 1755 
 
 1827 
 
 1899 
 
 1971 
 
 2114 
 
 2186 
 
 2258 
 
 2329 
 
 2401 
 
 1 61K3 
 
 2473 
 
 2544 
 
 2616 
 
 268.S 
 
 2759 
 
 28 Jl 
 
 2902 
 
 2974 
 
 3046 
 
 3117 
 
 ' 607 
 
 3189 
 
 3260 
 
 3332 
 
 3403 
 
 3475 
 
 3546 
 
 3618 
 
 3689 
 
 3761 
 
 3832 
 
 608 
 
 3904 
 
 3975 
 
 4046 
 
 4118 
 
 4189 
 
 4261 
 
 4332 
 
 4403 
 
 4476 
 
 4546 
 
 609 
 
 4617 
 
 4689 
 
 4760 
 
 4831 
 
 4902 
 
 4974 
 
 6045 
 
 5116 
 
 6187 
 
 6269 
 
 610 
 
 6330 
 
 6401 
 
 6472 
 
 5543 
 
 5616 
 
 5686 
 
 5767 
 
 6828 
 
 6899 
 
 6970 
 
 611 
 
 6041 
 
 6112 
 
 6183 
 
 6254 
 
 6325 
 
 639tJ 
 
 6467 
 
 6538 
 
 6()09 
 
 6680 
 
 612 
 
 6751 
 
 6822 
 
 6893 
 
 6964 
 
 7035 
 
 7106 
 
 7177 
 
 7248 
 
 7319 
 
 7390 
 
 613 
 
 74<>0 
 
 7531 
 
 7<J0-2 
 
 7673 
 
 7744 
 
 7815 
 
 7886 
 
 79u6 
 
 8027 
 
 8098 
 
 614 
 
 8168 
 
 8239 
 
 8310 
 
 8381 
 
 8451 
 
 8622 
 
 8593 
 
 8663- 
 
 8734 
 
 8804 
 
 615 
 
 8875 
 
 8946 
 
 9016 
 
 9087 
 
 9157 
 
 9228 
 
 9299 
 
 9369 
 
 9440 
 
 9510 
 
 616 
 
 9581 
 
 9651 
 
 9722 
 
 9792 
 
 9863 
 
 9933 
 
 ...4 
 
 ..74 
 
 .144 
 
 .215 
 
 617 
 
 790285 
 
 0356 
 
 0426 1 0496 
 
 0567 
 
 0(>37 
 
 0707 
 
 0778 
 
 0848 
 
 0918 
 
 618 
 
 0988 
 
 1059 
 
 1129 
 
 1199 
 
 1269 
 
 1340 
 
 1410 
 
 1480 
 
 1650 
 
 1620 
 
 619 
 
 1691 
 
 1761 
 
 1831 
 
 1901 
 
 1971 
 
 2041 
 
 2111 
 
 2181 
 
 2252 
 
 2322 
 
 620 
 
 2392 
 
 2462 
 
 2532 
 
 2602 
 
 2672 
 
 2742 
 
 2812 
 
 2882 
 
 2952 
 
 3022 
 
 621 
 
 3092 
 
 3162 
 
 3231 
 
 3301 
 
 3371 
 
 3441 
 
 3511 
 
 3581 
 
 3651 
 
 3721 
 
 622 
 
 3790 
 
 3860 
 
 3930 
 
 4000 
 
 4070 
 
 4139 
 
 4209 
 
 4279 
 
 4349 
 
 4418 
 
 623 
 
 4488 
 
 4558 
 
 4627 
 
 4697 
 
 4767 
 
 4836 
 
 49(J«) 
 
 4976 
 
 5045 
 
 6115 
 
 624 
 
 6185 
 
 5254 
 
 6324 
 
 6393 
 
 5463 
 
 69 
 
 6158 
 
 6532 
 
 6U02 
 
 5672 
 
 5741 
 
 6811 
 
 625 
 
 5880 
 
 6949 
 
 6019 
 
 6088 
 
 6227 
 
 6297 
 
 6366 
 
 6436 
 
 6605 
 
 626 
 
 6574 
 
 6644 
 
 6713 
 
 6782 
 
 6852 
 
 6921 
 
 6990 
 
 7060 
 
 7129 
 
 7198 
 
 627 
 
 7268 
 
 7337 
 
 7406 
 
 7475 
 
 7545 
 
 7614 
 
 7683 
 
 7752 
 
 7821 
 
 7890 
 
 628 
 
 7960 
 
 8029 
 
 8098 
 
 8167 
 
 8236 
 
 8305 
 
 8374 
 
 8443 
 
 8513 
 
 8582 
 
 629 
 
 8651 
 
 8720 
 
 8789 
 
 8868 
 
 8927 
 
 8996 
 
 9066 
 
 6134 
 
 9203 
 
 9272 
 
 630 
 
 9341 
 
 9409 
 
 9478 
 
 9547 
 
 9610 
 
 9686 
 
 9764 
 
 9823 
 
 9892 
 
 9961 
 
 631 
 
 800026 
 
 0098 
 
 0167 
 
 0236 
 
 0305 
 
 0373 
 
 0442 
 
 0511 
 
 0580 
 
 0()48 
 
 632 
 
 0717 
 
 0786 
 
 0854 
 
 0923 
 
 0992 
 
 1061 
 
 1129 
 
 1198 
 
 1266 
 
 1335 
 
 633 
 
 1404 
 
 1472 
 
 1541 
 
 1609 
 
 1678 
 
 1747 
 
 1815 
 
 1884 
 
 1952 
 
 2021 
 
 634 
 
 2089 
 
 2158 
 
 2226 
 
 2295 
 
 2363 
 
 243^ 
 
 2500 
 
 2568 
 
 2637 
 
 2705 
 
 636 
 
 2774 
 
 2842 
 
 2910 
 
 2979 
 
 3047 
 
 3116 
 
 3184 
 
 3262 
 
 3321 
 
 3389 
 
 1 636 
 
 3457 
 
 3525 
 
 3594 
 
 3662 
 
 3730 
 
 3798 
 
 3867 
 
 3936 
 
 4003 
 
 4071 
 
 1 637 
 
 4139 
 
 4208 
 
 4276 
 
 4354 
 
 4412 
 
 4480 
 
 4548 
 
 4616 
 
 4685 
 
 4753 
 
 ! 638 
 
 4821 
 
 4889 
 
 4957 
 
 5025 
 
 5093 
 
 5161 
 
 5229 
 
 6297 
 
 5365 
 
 6433 
 
 639 
 
 5501 
 
 5669 
 
 6637 
 
 6705 
 
 6773 
 
 6841 
 
 5908 
 
 6976 
 
 6044 
 
 6112 
 
 640 
 
 6180 
 
 6248 
 
 6316 
 
 6384 
 
 6451 
 
 6519 
 
 6587 
 
 6655 
 
 6723 
 
 6790 
 
 641 
 
 6858 
 
 6926 
 
 6994 
 
 7061 
 
 7129 
 
 7157 
 
 7264 
 
 7332 
 
 7400 
 
 7467 
 
 642 
 
 7535 
 
 7603 
 
 7670 
 
 7738 
 
 7806 
 
 7873 
 
 7941 
 
 8008 
 
 8076 
 
 8143 
 
 643 
 
 8211 
 
 8279 
 
 8346 
 
 8414 
 
 8481 
 
 8549 
 
 8616 
 
 8684 
 
 8751 
 
 8818 
 
 644 
 
 8886 
 
 8953 
 
 9021 
 
 9088 
 
 9156 
 
 9223 
 
 9290 
 
 9358 
 
 9425 
 
 9492 i 
 
 645 
 
 95G0 
 
 9627 
 
 9694 
 
 9762 
 
 9829 
 
 9896 
 
 9964 
 
 ..31 
 
 ..98 
 
 .165 
 
 646 
 
 810233 
 
 0300 
 
 0367 
 
 0434 
 
 0501 
 
 0596 0.i36 
 
 0703 
 
 0770 
 
 0837 
 
 647 
 
 0904 
 
 0971 
 
 1039 
 
 1106 
 
 1173 
 
 1240 1307 
 
 1374 
 
 1441 
 
 1508 
 
 648 
 
 1575 
 
 1642 
 
 1709 
 
 1776 
 
 1843 
 
 1910 1977 
 
 2044 
 
 2111 
 
 2178 , 
 
 640 
 
 2245 
 
 2312 
 
 2379 
 
 2445 
 
 2612 
 
 2679 2646 
 
 3713 
 
 2780 
 
 2847 
 
14 
 
 ^ 
 
 LOGARITHMS 
 
 N. 
 650 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 812913 
 
 2980 
 
 3047 
 
 3114 
 
 3181 
 
 3247 
 
 3314 
 
 3381 
 
 3448 
 
 3514 
 
 651 
 
 3581 
 
 3648 
 
 3714 
 
 3781 
 
 3848 
 
 3914 
 
 3981 
 
 4048 
 
 4114 
 
 4181 
 
 652 
 
 4248 
 
 4314 
 
 4381 
 
 4447 
 
 4514 
 
 4581 
 
 4647 
 
 4714 
 
 4780 
 
 4847 
 
 653 
 
 4913 
 
 4980 
 
 6046 
 
 6113 
 
 6179 
 
 6246 
 
 6312 
 
 6378 
 
 6445 
 
 6511 
 
 654 
 
 6678 
 
 6644 
 
 6711 
 
 6777 
 
 5843 
 
 67 
 
 6506 
 
 6910 
 
 5976 
 
 6042 
 
 6109 
 
 6176 
 
 655 
 
 6241 
 
 6308 
 
 6374 
 
 6440 
 
 6573 
 
 6639 
 
 6706 
 
 6771 
 
 6838 
 
 656 
 
 6904 
 
 6970 
 
 7036 
 
 7102 
 
 7169 
 
 7233 
 
 7301 
 
 7367 
 
 7433 
 
 7499 
 
 657 
 
 7565 
 
 7631 
 
 7698 
 
 7764 
 
 7830 
 
 7896 
 
 7962 
 
 8028 
 
 8094 
 
 8160 
 
 058 
 
 8226 
 
 8292 
 
 8358 
 
 8424 
 
 8490 
 
 8566 
 
 8622 
 
 8688 
 
 8754 
 
 8820 
 
 659 
 
 8885 
 
 8951 
 
 9017 
 
 9083 
 
 9149 
 
 9216 
 
 9281 
 
 9346 
 
 9412 
 
 9478 
 
 660 
 
 9544 
 
 9610 
 
 9676 
 
 9741 
 
 9807 
 
 9873 
 
 9939 
 
 ...4 
 
 ..70 
 
 .136 
 
 661 
 
 820201 
 
 0267 
 
 0333 
 
 0399 
 
 0464 
 
 0530 
 
 0595 
 
 0661 
 
 0727 
 
 0792 
 
 662 
 
 0858 
 
 0924 
 
 0989 
 
 1055 
 
 1120 
 
 1186 
 
 1251 
 
 1317 
 
 1382 
 
 1448 
 
 663 
 
 1514 
 
 1579 
 
 1645 
 
 1710 
 
 1775 
 
 1841 
 
 1906 
 
 1972 
 
 2037 
 
 2103 
 
 664 
 
 2168 
 
 2233 
 
 2299 
 
 2364 
 
 2430 
 
 2495 
 
 2560 
 
 2626 
 
 2691 
 
 2766 
 
 665 
 
 2822 
 
 2887 
 
 2952 
 
 3018 
 
 3083 
 
 3148 
 
 3213 
 
 3279 
 
 3344 
 
 3409 
 
 666 
 
 3474 
 
 3539 
 
 3605 
 
 3670 
 
 3735 
 
 3800 
 
 3865 
 
 3930 
 
 3996 
 
 4061 
 
 667 
 
 4126 
 
 4191 
 
 4256 
 
 4321 
 
 4386 
 
 4451 
 
 4516 
 
 4581 
 
 4646 
 
 4711 
 
 668 
 
 4776 
 
 4841 
 
 4906 
 
 4971 
 
 6036 
 
 6101 
 
 5166 
 
 6231 
 
 6296 
 
 6361 
 
 669 
 
 6426 
 
 6491 
 
 6566 
 
 5621 
 
 6686 
 
 6751 
 
 5815 
 
 6880 
 
 6946 
 
 6010 
 
 670 
 
 6075 
 
 6140 
 
 6204 
 
 6269 
 
 6334 
 
 6399 
 
 6464 
 
 6628 
 
 6593 
 
 6668 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6917 
 
 6981 
 
 7046 
 
 7111 
 
 7175 
 
 7240 
 
 7305 
 
 672 
 
 7369 
 
 7434 
 
 7499 
 
 7663 
 
 7628 
 
 7692 
 
 7757 
 
 7821 
 
 7886 
 
 7951 
 
 673 
 
 -8015 
 
 8080 
 
 8144 
 
 8209 
 
 8273 
 
 8338 
 
 8402 
 
 8467 
 
 8631 
 
 8595 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918 
 
 65 
 
 9661 
 
 8982 
 
 9046 
 
 9111 
 
 9175 
 
 9239 
 
 675 
 
 9304 
 
 9368 
 
 9432 
 
 9497 
 
 9625 
 
 9690 
 
 9754 
 
 9818 
 
 9882 
 
 676 
 
 9947 
 
 ..11 
 
 ..76 
 
 .139 
 
 .204 
 
 .268 
 
 .332 
 
 .396 
 
 .460 
 
 .6'25 
 
 677 
 
 830589 
 
 0653 
 
 0717 
 
 0781 
 
 0845 
 
 0909 
 
 09/3 
 
 1037 
 
 1102 
 
 1166 
 
 678 
 
 1230 
 
 1294 
 
 1358 
 
 1422 
 
 1486 
 
 1550 
 
 1614 
 
 1678 
 
 1742 
 
 1806 
 
 679 
 
 1870 
 
 1934 
 
 1998 
 
 2062 
 
 2126 
 
 2189 
 
 2263 
 
 2317 
 
 2381 
 
 2446 
 
 680 
 
 2509 
 
 2573 
 
 2637 
 
 2700 
 
 2764 
 
 2828 
 
 2892 
 
 2966 
 
 3020 
 
 3083 
 
 681 
 
 3147 
 
 3211 
 
 3275 
 
 3338 
 
 3402 
 
 3466 
 
 35o0 
 
 3593 
 
 3657 
 
 3721 
 
 682 
 
 3784 
 
 3848 
 
 3912 
 
 3975 
 
 4039 
 
 4103 
 
 4166 
 
 4230 
 
 4294 
 
 4357 
 
 683 
 
 4421 
 
 4484 
 
 4548 
 
 4611 
 
 4675 
 
 4739 
 
 4802 
 
 4866 
 
 4929 
 
 4993 
 
 684 
 
 5056 
 
 5120 
 
 5183 
 
 5247 
 
 6310 
 
 6373 
 
 6437 
 
 6500 
 
 6664 
 
 5627 
 
 685 
 
 5691 
 
 5754 
 
 6817 
 
 6881 
 
 5944 
 
 6007 
 
 6071 
 
 6134 
 
 6197 
 
 6261 
 
 686 
 
 6324 
 
 6387 
 
 6451 
 
 6514 
 
 6677 
 
 6641 
 
 6704 
 
 6767 
 
 6830 
 
 6894 
 
 687 
 
 6957 
 
 7020 
 
 7083 
 
 7146 
 
 7210 
 
 7273 
 
 7336 
 
 7399 
 
 7462 
 
 7525 
 
 688 
 
 7588 
 
 7652 
 
 7715 
 
 7778 
 
 7B41 
 
 7904 
 
 7967 
 
 8030 
 
 8093 
 
 bl56 
 
 689 
 
 8219 
 
 8282 
 
 8345 
 
 8408 
 
 8471 
 
 8534 
 
 8697 
 
 8660 
 
 8723 
 
 8786 
 
 690 
 
 8849 
 
 8912 
 
 8976 
 
 9038 
 
 9109 
 
 9164 
 
 9227 
 
 9289 
 
 9362 
 
 9415 
 
 691 
 
 9478 
 
 9541 
 
 9604 
 
 9667 
 
 9729 
 
 9792 
 
 9856 
 
 9918 
 
 9981 
 
 ..43 
 
 692 
 
 840106 
 
 0169 
 
 0232 
 
 0294 
 
 0357 
 
 0420 
 
 0482 
 
 0545 
 
 0608 
 
 0671 
 
 693 
 
 0733 
 
 0796 
 
 0859 
 
 0921 
 
 0984 
 
 1046 
 
 1109 
 
 1172 
 
 1234 
 
 1297 
 
 691 
 
 1359 
 
 1422 
 
 1485 
 
 1547 
 
 1610 
 
 62 
 
 2235 
 
 1672 
 
 1735 
 
 1/97 
 
 1860 
 
 1922 
 
 695 
 
 1985 
 
 2047 
 
 2110 
 
 2172 
 
 2297 
 
 2360 
 
 2422 
 
 2484 
 
 2547 
 
 696 
 
 2609 
 
 2672 
 
 2734 
 
 2796 
 
 2859 
 
 2921 
 
 2983 
 
 3046 
 
 3108 
 
 3170 
 
 697 
 
 3233 
 
 3295 
 
 3357 
 
 3420 
 
 3482 
 
 3644 
 
 3b06 
 
 3669 
 
 3731 
 
 3793 
 
 698 
 
 3855 
 
 3918 
 
 3980 
 
 4042 
 
 4104 
 
 4166 
 
 4229 
 
 4291 
 
 4353 
 
 4415 
 
 699 
 
 4477 
 
 4539 
 
 4601 
 
 4664 
 
 4726 
 
 4788 
 
 4850 
 
 4912 
 
 4974 
 
 5036 
 
 1 
 
1 
 
 j OFNUMBERS. 15 
 
 
 N. 
 
 
 
 ' 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 70U 
 
 845098 
 
 61()0 
 
 5222 
 
 5284 
 
 6346 
 
 5408 
 
 5470 
 
 6632 
 
 6694 
 
 5666 
 
 
 701 
 
 5718 
 
 5780 
 
 5842 
 
 5!)04 
 
 5966 
 
 6028 
 
 6090 
 
 6151 
 
 6213 
 
 6276 
 
 
 702 
 
 6337 
 
 6399 
 
 64ul 
 
 6523 
 
 6585 
 
 6646 
 
 6708 
 
 6770 
 
 6832 
 
 6894 
 
 
 7U3 
 
 6955 
 
 7017 
 
 7079 
 
 7141 
 
 7202 
 
 7264 
 
 7326 
 
 7388 
 
 7449 
 
 7511 
 
 
 704 
 
 7573 
 
 7634 
 
 7676 
 
 7758 
 
 7819 
 
 62 
 
 8435 
 
 7831 
 
 7943 
 
 8004 
 
 8066 
 
 8128 
 
 
 ' 705 
 
 8189 
 
 8251 
 
 8312 
 
 8374 
 
 8497 
 
 8559 
 
 8620 
 
 8682 
 
 8743 
 
 
 706 
 
 8805 
 
 8866 
 
 8928 
 
 8989 
 
 9051 
 
 9112 
 
 9174 
 
 9235 
 
 9297 
 
 9358 
 
 
 T>7 
 
 9419 
 
 9481 
 
 9542 
 
 9604 
 
 9665 
 
 9726 
 
 9788 
 
 9849 
 
 9911 
 
 9972 
 
 
 70iS 
 
 850033 
 
 oo:)5 
 
 0156 
 
 0217 
 
 0279 
 
 0340 
 
 0401 
 
 0462 
 
 0524 
 
 0585 
 
 
 i 709 
 
 0646 
 
 0707 
 
 0769 
 
 0830 
 
 0891 
 
 0952 
 
 1014 
 
 1076 
 
 1136 
 
 1197 
 
 
 i 710 
 
 1258 
 
 1320 
 
 1381 
 
 1442 
 
 1503 
 
 1564 
 
 1625 
 
 1686 
 
 1747 
 
 I8u9 
 
 
 i 71 i 
 
 1870 
 
 1931 
 
 1992 
 
 2053 
 
 2114 
 
 2175 
 
 2236 
 
 2297 
 
 2368 
 
 2419 
 
 
 71-2 
 
 - 2480 
 
 2541 
 
 2602 
 
 2G63 
 
 2724 
 
 2785' 
 
 2846 
 
 2907 
 
 2968 
 
 3029 
 
 
 713 
 
 3090 
 
 3150 
 
 3211 
 
 3272 
 
 3333 
 
 3394 
 
 3455 
 
 3516 
 
 3577 
 
 3637 
 
 
 714 
 
 3u98 
 
 3769 
 
 3820 
 
 3881 
 
 3941 
 
 4002 
 
 4063 
 
 4124 
 
 4185 
 
 4245 
 
 
 715 
 
 430() 
 
 4367 
 
 4428 
 
 4488 
 
 4549 
 
 4610 
 
 4670 
 
 4731 
 
 4792 
 
 4852 
 
 
 71l> 
 
 4913 
 
 4974 
 
 5034 
 
 5095 
 
 6156 
 
 5216 
 
 5277 
 
 6337 
 
 5398 
 
 6459 
 
 
 717 
 
 5519 
 
 5580 
 
 6640 
 
 5701 
 
 5761 
 
 6822 
 
 5882 
 
 6943 
 
 0003 
 
 6064 
 
 
 718 
 
 6124 
 
 6185 
 
 6245 
 
 6306 
 
 6366 
 
 6427 
 
 u487 
 
 6548 
 
 6608 
 
 6668 
 
 
 719 
 
 6729 
 
 6789 
 
 6850 
 
 6910 
 
 6970 
 
 7031 
 
 7091 
 
 7162 
 
 7212 
 
 7272 
 
 
 720 
 
 7332 
 
 7393 
 
 7453 
 
 7513 
 
 7674 
 
 7634 
 
 7694 
 
 7755 
 
 7815 
 
 7876 
 
 
 721 
 
 7935 
 
 7995 
 
 8056 
 
 8116 
 
 8176 
 
 8236 
 
 8297 
 
 8357 
 
 8417 
 
 8477 
 
 
 722 
 
 8537 
 
 8597 
 
 8657 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 8958 
 
 9018 
 
 907F 
 
 
 723 
 
 9138 
 
 9198 
 
 9258 
 
 9318 
 
 9379 
 
 9439 
 
 9499 
 
 96)9 
 
 9619 
 
 9679 
 
 
 724 
 
 9739 
 
 9799 
 
 9859 
 
 9918 
 
 9978 
 
 60 
 
 0578 
 
 ..38 
 
 ..98 
 
 .158 
 
 .218 
 
 .2V8 
 
 
 725 
 
 860338 
 
 0398 
 
 0458 
 
 0518 
 
 0637 
 
 0697 
 
 0757 
 
 0817 
 
 0877 
 
 
 720 
 
 0937 
 
 0996 
 
 1056 
 
 1116 
 
 1176 
 
 1236 
 
 1295 
 
 1355 
 
 1415 
 
 1475 
 
 
 727 
 
 1534 
 
 1594 
 
 1654 
 
 1714 
 
 1773 
 
 1833 
 
 1893 
 
 1952 
 
 2012 
 
 2072 
 
 
 728 
 
 2131 
 
 2191' 
 
 2251 
 
 2310 
 
 2370 
 
 2430 
 
 2489 
 
 2549 
 
 2608 
 
 2668 
 
 
 729 
 
 2728 
 
 2787 
 
 2847 
 
 2906 
 
 2966 
 
 3025 
 
 3086 
 
 3144 
 
 3204 
 
 3263 
 
 
 730 
 
 3323 
 
 3382 
 
 3442 
 
 3501 
 
 3561 
 
 3620 
 
 3680 
 
 3739 
 
 3799 
 
 3858 
 
 
 731 
 
 3917 
 
 3977 
 
 4036 
 
 40!Xi 
 
 4155 
 
 4214 
 
 4274 
 
 4333 
 
 4uy2 
 
 4452 
 
 
 732 
 
 4511 
 
 4570 
 
 4630 
 
 4689 
 
 4148 
 
 4808 
 
 4867 
 
 4926 
 
 4985 
 
 5046 
 
 
 733 
 
 5104 
 
 5163 
 
 5222 
 
 6282 
 
 5341 
 
 6400 
 
 6469 
 
 6619 
 
 6578 
 
 6637 
 
 
 734 
 
 5696 
 
 5756 
 
 5814 
 
 6874 
 
 5933 
 
 5992 
 
 6051 
 
 6110 
 
 6169 
 
 6228 
 
 
 735 
 
 6287 
 
 6346 
 
 6405 
 
 6465 
 
 6624 
 
 6683 
 
 6642 
 
 6701 
 
 6760 
 
 6819 
 
 
 73« 
 
 6878 
 
 6937 
 
 6996 
 
 7056 
 
 7114 
 
 7173 
 
 7232 
 
 7291 
 
 7350 
 
 7409 
 
 
 737 
 
 7467 
 
 7526 
 
 7585 
 
 7644 
 
 7703 
 
 7762 
 
 7821 
 
 7880 
 
 7939 
 
 7998 
 
 
 1 738 
 
 8056 
 
 8115 
 
 8174 
 
 8233 
 
 8292 
 
 8350 
 
 8409 
 
 84t>8 
 
 8527 
 
 8586 
 
 
 1 739 
 
 8644 
 
 8703 
 
 8762 
 
 8821 
 
 8879 
 
 8938 
 
 8997 
 
 9056 
 
 9114 
 
 9173 
 
 
 740 
 
 9232 
 
 9290 
 
 9349 
 
 9408 
 
 9466 
 
 9625 
 
 9684 
 
 9642 
 
 9701 
 
 9760 
 
 
 741 
 
 9818 
 
 9877 
 
 9935 
 
 9994 
 
 ..53 
 
 .111 
 
 .170 
 
 .228 
 
 0287 
 
 .345 
 
 
 742 
 
 870404 
 
 0462 
 
 0521 
 
 0679 
 
 0638 
 
 0696 
 
 0755 
 
 0813 
 
 087i 
 
 0930 
 
 
 743 
 
 0989 
 
 1047 
 
 1106 
 
 1164 
 
 1223 
 
 1281 
 
 1339 
 
 1398 
 
 1456 
 
 1615 
 
 
 744 
 
 1573 
 
 1631 
 
 1690 
 
 1748 
 
 1806 
 
 59 
 
 2389 
 
 1865 
 
 1923 
 
 1981 
 
 2040 
 
 2098 
 
 
 745 
 
 2156 
 
 2215 
 
 2273 
 
 2331 
 
 2448 
 
 2506 
 
 2664 
 
 2622 
 
 2681 
 
 
 746 
 
 2739 
 
 2797 
 
 2855 
 
 2913 
 
 2972 
 
 3030 
 
 3088- 
 
 3146 
 
 3204 
 
 3262 
 
 
 747 
 
 3321 
 
 3379 
 
 3437 
 
 3495 
 
 3553 
 
 3611 
 
 3669 
 
 3727 
 
 37«6 
 
 3844 
 
 
 748 
 
 3902 
 
 396C 
 
 4018 
 
 4076 
 
 4134 
 
 4192 
 
 4250 
 
 4308 
 
 4360 
 
 4434 
 
 
 749 
 
 4482 
 
 4540 
 
 4698 
 
 4656 
 
 4714 
 
 4772 
 
 4830 
 
 4888 
 
 ^^945 
 
 uooa 
 
 
/6 
 
 LOGARITHMS 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 750 
 
 875031 
 
 6119 
 
 6177 
 
 6235 
 
 6293 
 
 5351 
 
 5403 
 
 5466 
 
 5624 
 
 6582 
 
 
 751 
 
 5640 
 
 j 6S98 
 
 5756 
 
 5813 
 
 5871 
 
 5929 
 
 5987 
 
 ()045 
 
 6102 
 
 6160 
 
 
 752 
 
 6218 
 
 6276 
 
 6333 
 
 6391 1 6449 
 
 6507 
 
 6564 
 
 6622 
 
 6680 
 
 6737 
 
 
 753 
 
 6795 
 
 6853 
 
 6910 : 6968 
 
 7026 
 
 7083 
 
 7141 
 
 7199 
 
 7256 
 
 7314 
 
 
 764 
 
 7371 
 
 7429 
 
 7487 
 
 7544 
 
 7602 
 
 57 
 
 8177 
 
 7659 
 
 7717 
 
 7774 
 
 7832 
 
 7889 
 
 
 755 
 
 7947 
 
 8004 
 
 8062 
 
 8119 
 
 8234 
 
 8292 
 
 8349 
 
 8407 
 
 8464 
 
 
 756 
 
 8522 
 
 8579 
 
 8637 
 
 8694 
 
 8752 
 
 8809 
 
 8866 
 
 8924 
 
 8981 
 
 9039 
 
 
 757 
 
 9096 
 
 9153 
 
 9211 
 
 9268 j 9325 
 
 9383 
 
 9440 
 
 9497 
 
 9655 
 
 9612 
 
 
 758 
 
 9669 
 
 9726 
 
 9784 
 
 9841 1 9898 
 
 9956 
 
 . .13 
 
 ..70 
 
 .127 
 
 .185 
 
 
 , 759 
 
 S80242 
 
 0299 
 
 0356 
 
 0413 
 
 0471 
 
 «528 
 
 0580 
 
 0o42 
 
 0699 
 
 0756 
 
 
 760 
 
 0ai4 
 
 0871 
 
 0928 
 
 0985 
 
 1042 
 
 1099 
 
 1156 
 
 1213 
 
 1271 
 
 1328 
 
 
 761 
 
 1385 
 
 1442 
 
 1499 
 
 1556 
 
 1613 
 
 1670 
 
 1727 
 
 1784 
 
 1841 
 
 1898 
 
 
 762 
 
 1955 
 
 2U12 
 
 2069 
 
 2126 
 
 2183 
 
 2240 
 
 2297 
 
 2354 
 
 2411 
 
 2468 
 
 
 763 
 
 2525 
 
 2581 
 
 2638 
 
 2695 
 
 2762 
 
 2809 
 
 2866 
 
 2923 
 
 2980 
 
 3037 
 
 
 764 
 
 3093 
 
 3150 
 
 3207 
 
 3264 
 
 3321 
 
 3377 
 
 3434 
 
 3491 
 
 3548 
 
 3605 
 
 
 765 
 
 3661 
 
 3718 
 
 3775 
 
 3832 
 
 3888 
 
 3945 
 
 4002 
 
 4069 
 
 4115 
 
 4172 
 
 
 766 
 
 4229 
 
 4285 
 
 4342 
 
 4399 
 
 4466 
 
 4612 
 
 4669 
 
 4625 
 
 4682 
 
 4739 
 
 
 767 
 
 4795 
 
 4852 
 
 4909 
 
 4965 
 
 6022 
 
 5078 
 
 5135 
 
 5192 
 
 6248 
 
 5305 
 
 
 768 
 
 5361 
 
 5418 
 
 6474 
 
 6631 
 
 6587 
 
 6644 
 
 6700 
 
 6767 
 
 5813 
 
 6870 
 
 
 769 
 
 6926 
 
 5983 
 
 6039 
 
 6096 
 
 6162 
 
 6209 
 
 6266 
 
 6321 
 
 6378 
 
 6434 
 
 
 770 
 
 6491 
 
 6547 
 
 6604 
 
 6660 
 
 6716 
 
 6773 
 
 6829 
 
 6885 
 
 6942 
 
 6998 
 
 
 771 
 
 7054 
 
 7111 
 
 7167 
 
 7233 
 
 7280 
 
 7336 
 
 7392 
 
 7449 
 
 7505 
 
 7561 
 
 
 772 
 
 7617 
 
 7674 
 
 7730 
 
 7786 
 
 7842 
 
 7898 
 
 7955 
 
 8011 
 
 8067 
 
 8123 
 
 
 773 
 
 8179 
 
 8236 
 
 8292 
 
 8348 
 
 8404 
 
 8460 
 
 8516 
 
 8573 
 
 8629 
 
 8666 
 
 
 774 
 
 8741 
 
 8797 
 
 8853 
 
 8909 
 
 8965 
 
 56 
 
 9526 
 
 9021 
 
 9077 
 
 9134 
 
 9190 
 
 9246 
 
 
 775 
 
 9302 
 
 9358 
 
 9414 
 
 9470 
 
 9582 
 
 9638 
 
 9694 
 
 9760 
 
 9806 
 
 
 776 
 
 9862 
 
 9918 
 
 0974 
 
 ..30 
 
 ..86 
 
 .141 
 
 .197 
 
 .253 
 
 .309 
 
 .365 
 
 
 777 
 
 890421 
 
 0477 
 
 0533 
 
 0589 
 
 0645 
 
 0700 
 
 0756 
 
 0812 
 
 0868 
 
 0924 
 
 
 778 
 
 0980 
 
 1035 
 
 1091 
 
 1147 
 
 1203 
 
 1269 
 
 1314 
 
 1370 
 
 1426 
 
 1482 
 
 
 779 
 
 1537 
 
 1593 
 
 1649 
 
 1706 
 
 1760 
 
 1816 
 
 1872 
 
 1928 
 
 1983 
 
 2039 
 
 
 780 
 
 2095 
 
 2150 
 
 2206 
 
 2262 
 
 2317 
 
 2373 
 
 2429 
 
 2484 
 
 2540 
 
 2696 
 
 
 781 
 
 2651 
 
 2707 
 
 2762 
 
 2818 
 
 2873 
 
 2929 
 
 2985 
 
 3040 
 
 3096 
 
 3151 
 
 
 782 
 
 3207 
 
 3262 
 
 3318 
 
 3373 
 
 3429 
 
 3484 
 
 3540 
 
 3595 
 
 3651 
 
 3706 
 
 
 783 
 
 3702 
 
 3817 
 
 3873 
 
 3928 
 
 3984 
 
 4039 
 
 4094 
 
 4150 
 
 4205 
 
 4261 
 
 
 784 
 
 4816 
 
 4371 
 
 4427 
 
 4482 
 
 4638 
 
 4693 
 
 4648 
 
 4704 
 
 4769 
 
 4814 
 
 
 785 
 
 4870 
 
 4926 
 
 4980 
 
 5036 
 
 5091. 
 
 5146 
 
 5201 
 
 6267 
 
 6312 
 
 5367 
 
 
 786 
 
 6423 
 
 6478 
 
 5533 
 
 6588 
 
 6644 
 
 6699 
 
 6754 
 
 6809 
 
 5864 
 
 592C 
 
 
 787 
 
 5975 
 
 6030 
 
 6085 
 
 6140 
 
 6195 
 
 6251 
 
 6306 
 
 6361 
 
 6416 
 
 6471 
 
 
 788 
 
 6526 
 
 6581 
 
 6636 
 
 6692 
 
 6747 
 
 6802 
 
 6857 
 
 6912 
 
 6967 
 
 7022 
 
 
 789 
 
 7077 
 
 7132 
 
 7187 
 
 7242 
 
 7297 
 
 7362 
 
 7407 
 
 7462 
 
 7617 
 
 7672 
 
 
 790 
 
 7627 
 
 7683 
 
 7737 
 
 7792 
 
 7847 7902 
 
 7957 
 
 8012 
 
 8067 
 
 8122 
 
 
 791 
 
 8176 
 
 8231 
 
 8286 
 
 8341 
 
 8396 8461 
 
 8506 
 
 8561 
 
 8615 
 
 8670 
 
 
 792 
 
 8725 
 
 8780 
 
 8835 
 
 8890 
 
 8944 8999 
 
 9054 
 
 9109 
 
 9164 
 
 9218 
 
 
 793 
 
 9273 
 
 9328 
 
 9383 
 
 9437 
 
 9492 
 
 9547 
 
 9602 
 
 9666 
 
 9711 
 
 9766 
 
 
 794 
 
 98'^1 
 
 9875 
 
 9930 
 
 9986 
 
 ..39 
 
 55 
 
 0586 
 
 ..94 
 
 .149 
 
 .203 
 
 .268 
 
 .312 
 
 
 795 
 
 900367 
 
 0422 
 
 0476 
 
 ^531 
 
 0640 
 
 0695 
 
 0749 
 
 0804 
 
 0859 
 
 
 796 
 
 0913 
 
 0968 
 
 1022 
 
 1077 
 
 1131 
 
 1186 
 
 1240 
 
 1295 
 
 1349 
 
 1404 
 
 
 797, 
 
 1458 
 
 1513 
 
 1667 
 
 1622 
 
 1676 
 
 1736 
 
 1785 
 
 1840 
 
 18:^4 
 
 1948 
 
 
 798 
 
 2003 
 
 2057 
 
 2112 
 
 2166 
 
 2221 
 
 2275 2329 
 
 2384 
 
 2438 
 
 2492 
 
 
 799 
 
 2547 
 
 2601 
 
 2665 
 
 2710 
 
 2764 
 
 2818 2873 
 
 2927 
 
 2981 
 
 3036 
 
 
OF NUMBERS. 17 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 ! 9 
 
 SOU 
 
 9031)90 
 
 3144 
 
 3199 
 
 3253 
 
 3307 
 
 3361 
 
 3416 
 
 3470 
 
 3524 , 3578 
 
 801 
 
 3633 
 
 3.i87 
 
 3741 
 
 3795 
 
 3849 
 
 3904 
 
 395« 
 
 4012 
 
 4iKi6 ; 4120 
 
 802 
 
 4174 
 
 4229 
 
 4283 
 
 4337 
 
 4391 
 
 4445 
 
 4499 
 
 4563 
 
 4607 46t;i 
 
 803 
 
 4716 
 
 4770 
 
 4824 
 
 4878 
 
 4932 
 
 4986 
 
 5040 
 
 50J4 
 
 5148 , 62112 
 
 804 
 
 5256 
 
 5310 
 
 5364 
 
 5418 
 
 5472 
 
 54 
 
 6012 
 
 5526 
 
 5580 
 
 6u34 
 
 668.-> j 6742 
 
 ^05 
 
 5796 
 
 6850 
 
 6904 
 
 6968 
 
 60f>6 
 
 6119 
 
 6173 
 
 1 
 6227 ' 62SI 
 
 806 
 
 6335 
 
 6389 
 
 6443 
 
 6497 
 
 6551 
 
 6604 
 
 6(i58 
 
 6712 
 
 67(56 . 6h20 . 
 
 807 
 
 6874 
 
 6927 
 
 6981 
 
 7035 
 
 7089 
 
 7143 
 
 7196 
 
 7250 
 
 7304 
 
 7358 
 
 808 
 
 7411 
 
 7465/ 7519 
 
 7573 
 
 7626 
 
 7680 
 
 7734 
 
 7787 
 
 7841 
 
 7895 
 
 Koy 
 
 7949 
 
 &00i 8056 
 
 1 
 
 8110 
 
 8163 
 
 8217 
 
 8270 
 
 8324 
 
 8378 
 
 8431 
 
 810 
 
 8485 
 
 8539 
 
 8692 
 
 8646 
 
 8699 
 
 8753 
 
 8807 
 
 8860 
 
 8914 
 
 8967 
 
 811 
 
 9021 
 
 9074 
 
 9128 
 
 9181 
 
 9235 
 
 9289 
 
 9342 
 
 9396 
 
 9449 
 
 9503 
 
 81-2 
 
 9556 
 
 9.;io 
 
 9«i(>3 
 
 9716 
 
 9770 
 
 9823 
 
 9877 
 
 9930 
 
 9984 
 
 ..37 
 
 813 
 
 910091 
 
 0144 
 
 0197 
 
 0251 
 
 0304 
 
 0358 
 
 0411 
 
 0464 
 
 0518 
 
 0571 
 
 814 
 
 0624 
 
 0678 
 
 0731 
 
 0784 
 
 0838 
 
 0891 
 
 0944 
 
 0998 
 
 1051 
 
 1104 
 
 815 
 
 1158 
 
 1311 
 
 1264 
 
 1317 
 
 1371 
 
 1424 
 
 1477 
 
 1630 
 
 1684 
 
 1637 
 
 81G 
 
 1690 
 
 1743 
 
 1797 
 
 1850 
 
 1903 
 
 1956 
 
 2009 
 
 2063 
 
 2115 
 
 2169 
 
 817 
 
 2222 
 
 2275 
 
 2323 
 
 2381 
 
 2435 
 
 2488 
 
 2541 
 
 2594 
 
 2645 
 
 2700 
 
 818 
 
 2<53 
 
 280v) 
 
 2859 
 
 2913 
 
 2966 
 
 3019 
 
 30-2 
 
 3r^5 
 
 3178 
 
 3231 
 
 819 
 
 3284 
 
 3337 
 
 3390 
 
 3443 
 
 3496 
 
 3649 
 
 3602 
 
 3655 
 
 3708 
 
 3761 
 
 8'20 
 
 3814 
 
 3867 
 
 3920 
 
 3973 
 
 4026 
 
 4079 
 
 4132 
 
 4184 
 
 4237 
 
 4290 
 
 821 
 
 4343 
 
 439(i 
 
 4449 
 
 4502 
 
 4555 
 
 4ti08 
 
 4660 
 
 4713 
 
 4766 
 
 4819 
 
 S22 
 
 4872 
 
 4925 
 
 4977 
 
 5030 
 
 50.S3 
 
 5136 
 
 6189 
 
 5241 
 
 6294 
 
 5347 
 
 823 
 
 5400 
 
 5453 
 
 6505 
 
 5658 
 
 5611 
 
 5664 
 
 5716 
 
 5769 
 
 5822 
 
 6876 
 
 824 
 
 5927 
 
 59d0 
 
 6033 
 
 6085 
 
 6138 
 
 6191 
 
 6243 
 
 6296 
 
 6349 
 
 6401 
 
 826 
 
 6454 
 
 6607 
 
 6559 
 
 6612 
 
 66(>4 
 
 6717 
 
 6770 
 
 6822 
 
 6876 
 
 6927 
 
 826 
 
 6980 
 
 7033 
 
 7085 
 
 7138 
 
 71;iO 
 
 7243 
 
 7295 
 
 7348 
 
 7400 
 
 7453 
 
 827 
 
 7506 
 
 7558 
 
 7611 
 
 7()»)3 
 
 7716 
 
 7768 
 
 7820 
 
 7873 
 
 7925 
 
 7978 
 
 828 
 
 8030 
 
 8083 
 
 8185 
 
 8188 
 
 8240 
 
 8293 
 
 8345 
 
 8397 
 
 8450 
 
 8502 
 
 829 
 
 8655 
 
 8607 
 
 8669 
 
 8712 
 
 8764 
 
 8816 
 
 8869 
 
 8921 
 
 8973 
 
 9026 
 
 830 
 
 9078 
 
 9130 
 
 9183 
 
 9235 
 
 9287 
 
 9340 
 
 9392 
 
 9444 
 
 9496 
 
 9649 
 
 831 
 
 9601 
 
 9.)53 
 
 9706 
 
 9758 
 
 9810 
 
 9862 
 
 9914 
 
 9967 
 
 .19 
 
 . .71 
 
 832 
 
 920123 
 
 0176 
 
 022?S 
 
 0280 
 
 0332 
 
 0384 
 
 0136 
 
 0489 
 
 0541 
 
 0593 - 
 
 833 
 
 0645 
 
 Od9 7 
 
 0749 
 
 0801 
 
 0853 
 
 0906 
 
 0958 
 
 1010 
 
 1062 
 
 1114 
 
 834 
 
 1166 
 
 1218 
 
 1270 
 
 1322 
 
 1374 
 
 1426 
 
 1478 
 
 1630 
 
 \oH2 
 
 1634 
 
 835 
 
 1686 
 
 1738 
 
 1790 
 
 1842 
 
 1894 
 
 1946 
 
 1998 
 
 2050 
 
 2102 
 
 2154 
 
 836 
 
 2206 
 
 2258 
 
 2310 
 
 2362 
 
 2414 
 
 2466 
 
 2518 
 
 2570 
 
 2622 
 
 2674 
 
 837 
 
 2726 
 
 2777 
 
 2829 
 
 2881 
 
 2933 
 
 2985 
 
 3037 
 
 3089 
 
 3140 
 
 3192 
 
 838 
 
 3244 
 
 3296 
 
 3348 
 
 3399 
 
 3451 
 
 3503 
 
 3555 
 
 3607 
 
 3t>58 
 
 3711 
 
 839 
 
 3762 
 
 3814 
 
 3865 
 
 3917 
 
 3969 
 
 4021 
 
 40/2 
 
 4124 
 
 4147 
 
 4228 
 
 840 
 
 4279 
 
 4331 
 
 4383 
 
 4434 
 
 448«J 
 
 4538 
 
 4589 
 
 4641 
 
 4693 
 
 4744 
 
 841 
 
 4796 
 
 4848 
 
 4899 
 
 4951 
 
 6003 
 
 5054 
 
 5106 
 
 5157 
 
 5209 
 
 6261 
 
 842 
 
 6312 
 
 6364 
 
 5415 
 
 5467 
 
 5518 
 
 5570 
 
 5621 
 
 5673 
 
 5725 
 
 6776 
 
 843 
 
 682H 
 
 5874 
 
 5931 
 
 5982 
 
 6034 
 
 6085 
 
 6137 
 
 6188 
 
 6240 
 
 6291 
 
 844 
 
 6342 
 
 6394 
 
 6445 
 
 6497 
 
 6648 
 
 52 
 
 70(i2 
 
 6600 
 
 6651 
 
 6702 
 
 6754 
 
 6805 
 
 845 
 
 6857 
 
 6908 
 
 6959 
 
 7011 
 
 -U4 
 
 7165 
 
 7216 
 
 7268 
 
 7319 
 
 846 
 
 7370 
 
 7422 
 
 7473 
 
 7524 
 
 7576 
 
 7627 
 
 7678 
 
 7730 
 
 7783 
 
 7832 
 
 84 7 
 
 7883 
 
 7935 
 
 7986 
 
 8037 
 
 80.S8 
 
 8140 
 
 8191 
 
 8242 
 
 8293 
 
 8346 
 
 848 
 
 839: i 
 
 8447 
 
 8498 
 
 8549 
 
 8601 
 
 8652 
 
 8703 
 
 8754 
 
 8805 
 
 8857 
 
 849 
 
 1 
 
 8908 
 
 8959 
 
 9010 
 
 9061 
 
 9112 
 
 9163 
 
 9216 
 
 92()6 
 
 9317 
 
 9368 
 
 1 
 
1 '' 
 
 1 
 
 
 LOGARITHMS '! 
 
 1 
 
 N. 
 850 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 1 
 
 929419 
 
 9473 
 
 9521 
 
 9572 
 
 9623 
 
 9674 
 
 9725 
 
 9776 
 
 9827 
 
 9879 
 
 H51 
 
 9930 
 
 9981 
 
 ..32 
 
 ..83 
 
 .134 
 
 .185 
 
 .236 
 
 .287 
 
 .338 
 
 .389 
 
 8c2 
 
 930440 
 
 0491 
 
 0542 
 
 0592 
 
 0643 
 
 0694 
 
 0745 
 
 0/96 
 
 0847 
 
 0898 
 
 853 
 
 0949 
 
 1000 
 
 1051 
 
 1102 
 
 1163 
 
 1204 
 
 1254 
 
 1305 
 
 1356 
 
 1407 
 
 854 
 
 1458 
 
 1509 
 
 1560 
 
 1610 
 
 1661 
 
 51 
 
 2169 
 
 1712 
 
 1763 
 
 1814 
 
 -1865 
 
 1915 
 
 855 
 
 1966 
 
 2017 
 
 i968 
 
 2118 
 
 2220 
 
 2271 
 
 2322 
 
 2372 
 
 1 
 242J 
 
 1 856 
 
 2474 
 
 2524 
 
 2575 
 
 2626 
 
 2677 
 
 2727 
 
 . 2778 
 
 2829 
 
 2879 
 
 2930 
 
 ' 857 
 
 2981 
 
 3031 
 
 3082 
 
 3133 
 
 3183 
 
 3234 
 
 3286 
 
 3335 
 
 3386 
 
 $437 
 
 858 
 
 3487 
 
 3538 
 
 3589 
 
 3639 
 
 3690 
 
 3740 
 
 3791 
 
 3841 
 
 3892 
 
 3943 
 
 ( 859 
 
 3993 
 
 4044 
 
 4094 
 
 4145 
 
 4195 
 
 4246 
 
 4269 
 
 4347 
 
 4397 
 
 4448 
 
 860 
 
 4498 
 
 4549 
 
 4599 
 
 4650 
 
 4700 
 
 4751 
 
 4801 
 
 4852 
 
 4902 
 
 4963 
 
 861 
 
 50UJ 
 
 5054 
 
 5104 
 
 5164 
 
 6205 
 
 5265 
 
 6306 
 
 5356 
 
 6406 
 
 6467 
 
 862 
 
 65U< 
 
 5558 
 
 5608 
 
 5658 
 
 5/09 
 
 5759 
 
 6809 
 
 5b60 
 
 6910 
 
 5960 
 
 863 
 
 6U11 
 
 606 1 
 
 6111 
 
 6162 
 
 6212 
 
 6262 
 
 6313 
 
 6363 
 
 6413 
 
 6463 '■ 
 
 864 
 
 6514 
 
 6564 
 
 6614 
 
 6665 
 
 6/15 
 
 6765 
 
 6815 
 
 6865 
 
 6916 
 
 6966 
 
 865 
 
 7016 
 
 7066 
 
 7117 
 
 7167 
 
 7217 
 
 7267 
 
 7317 
 
 7367 
 
 7418 
 
 7468 
 
 866 
 
 7518 
 
 7568 
 
 7618 
 
 7668 
 
 7718 
 
 7/69 
 
 7819 
 
 7869 
 
 7919 
 
 7969 
 
 867 
 
 8019 
 
 8069 
 
 8119 
 
 8169 
 
 8219 
 
 8269 
 
 8320 
 
 8370 
 
 8420 
 
 8470 
 
 868 
 
 8520 
 
 8570 
 
 8620 
 
 8670 
 
 8720 
 
 8770 
 
 8820 
 
 8870 
 
 8919 
 
 8970 
 
 869 
 
 9020 
 
 9070 
 
 9120 
 
 9170 
 
 9220 
 
 92.0 
 
 9320 
 
 9369 
 
 9419 
 
 9469 
 
 870 
 
 9519 
 
 9569 
 
 9616 
 
 9669 
 
 9719 
 
 9769 
 
 9819 
 
 9869 
 
 9918 
 
 9968 
 
 Hi\ 
 
 940018 
 
 0068 
 
 0118 
 
 0168 
 
 0218 
 
 0267 
 
 0317 
 
 036 7 
 
 0417 
 
 0467 
 
 8.2 
 
 0516 
 
 0566 
 
 0616 
 
 0666 
 
 0/16 
 
 0/65 
 
 0815 
 
 0865 
 
 0915 
 
 0964 
 
 873 
 
 1014 
 
 1064 
 
 1114 
 
 1163 
 
 1213 
 
 1263 
 
 1313 
 
 1362 
 
 1412 
 
 1462 
 
 8/4 
 
 1511 
 
 1561 
 
 1611 
 
 1660 
 
 1710 
 
 1760 
 
 1809 
 
 ia59 
 
 1909 
 
 1958 
 
 875 
 
 2008 
 
 2058 
 
 2107 
 
 2157 
 
 2207 
 
 2256 
 
 2306 
 
 2356 
 
 2405 
 
 2455 
 
 8/6 
 
 2504 
 
 2554 
 
 2603 
 
 26,3 
 
 2702 
 
 2762 
 
 2801 
 
 2851 
 
 2901 
 
 2960 
 
 877 
 
 3000 
 
 3049 
 
 3099 
 
 3148 
 
 3198 
 
 3247 
 
 3297 
 
 3346 
 
 3396 
 
 3445 
 
 878 
 
 3495 
 
 3644 
 
 3693 
 
 3643 
 
 3692 
 
 3742 
 
 3791 
 
 3841 
 
 3890 
 
 3939 
 
 879 
 
 3989 
 
 4038 
 
 4088 
 
 4137 
 
 4186 
 
 4236 
 
 4286 
 
 4336 
 
 4384 
 
 4433 
 
 1 
 
 880 
 
 4483 
 
 4532 
 
 4581 
 
 4631 
 
 4680 
 
 4729 
 
 4779 
 
 4828 
 
 4877 
 
 4927 
 
 881 
 
 4976 
 
 5025 
 
 6074 
 
 6124 
 
 5173 
 
 5-^22 
 
 5272 
 
 5321 
 
 537(» 
 
 5419 
 
 882 
 
 6469 
 
 5518 
 
 6567 
 
 6616 
 
 6665 
 
 5716 
 
 6764 
 
 5813 
 
 6862 
 
 5912 
 
 883 
 
 5961 
 
 6010 
 
 6059 
 
 6108 
 
 6157 
 
 6207 
 
 6266 
 
 6305 
 
 6354 
 
 6403 
 
 884 
 
 6452 
 
 6501 
 
 6561 
 
 6600 
 
 6649 
 
 6698 
 
 6747 
 
 6/96 
 
 6845 
 
 6894 
 
 885 
 
 6943 
 
 6992 
 
 7041 
 
 7090 
 
 7140 
 
 7189 
 
 7238 
 
 7287 
 
 7336 
 
 7385 
 
 886 
 
 7434 
 
 7483 
 
 7532 
 
 7581 
 
 7630 
 
 7679 
 
 7728 
 
 7777 
 
 7826 
 
 7875 
 
 887 
 
 7924 
 
 79 73 
 
 8022 
 
 80/0 
 
 H119 
 
 8168 
 
 8217 
 
 8266 
 
 8ol5 
 
 8365 
 
 888 
 
 8413 
 
 8462 
 
 8511 
 
 8560 
 
 8609 
 
 8657 
 
 8706 
 
 «755 
 
 8«04 
 
 8853 
 
 889 
 
 8902 
 
 8951 
 
 8999 
 
 9048 
 
 9097 
 
 9146 
 
 9195 
 
 9244 
 
 9292 
 
 9341 
 
 890 
 
 9390 
 
 9439 
 
 9488 
 
 9536 
 
 9586 
 
 9634 
 
 9683 
 
 9731 
 
 9780 
 
 9829 ' 
 
 891 
 
 98/8 
 
 9926 
 
 9975 
 
 ..24 
 
 ..73 
 
 .121 
 
 .170 
 
 .219 
 
 .26 7 
 
 .31u 
 
 892 
 
 950365 
 
 0414 
 
 0462 
 
 0511 
 
 0560 
 
 0o08 
 
 0607 
 
 0/06 
 
 0754 
 
 0803 
 
 893 
 
 0851 
 
 0900 
 
 0949 
 
 0997 
 
 1016 
 
 10i^6 
 
 1143 
 
 1192 
 
 1240 
 
 12o9 
 
 894 
 
 1338 
 
 1386 
 
 1435 
 
 1483 
 
 1532 
 48 
 
 2or 
 
 1580 
 
 162J 
 
 1677 
 
 - 
 
 1 726 
 
 17.6 
 
 895 
 
 1823 
 
 1872 
 
 1920 
 
 1969 
 
 2066 
 
 2114 
 
 2163 
 
 2211 
 
 2260 
 
 896 
 
 2308 
 
 2356 
 
 2105 
 
 2453 
 
 250i 
 
 2550 
 
 2599 
 
 'Jb47 
 
 5d9o 
 
 2/44 
 
 897 
 
 2792 
 
 2841 
 
 2889 
 
 2938 
 
 2986 
 
 3034 
 
 3063 
 
 3131 
 
 3180 
 
 322o 
 
 898 
 
 3276 
 
 3325 
 
 3373 
 
 34-1 
 
 3470 
 
 3518 
 
 ■6oo6 
 
 bol5 
 
 366a 
 
 3711 
 
 899 
 
 1 
 
 3760 
 
 3808 
 
 3856 
 
 3905 
 
 3953 
 
 4001 
 
 40.9 
 
 40j» 
 
 4146 
 
 4194 
 

 OF NUMBERS. 
 
 
 19 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 QUO 
 
 954243 
 
 4291 
 
 4339 
 
 4387 
 
 4435 
 
 4484 
 
 4532 
 
 4580 
 
 4628 
 
 4677 
 
 
 9Ul 
 
 4/25 
 
 4773 
 
 4821 
 
 4869 
 
 4918 
 
 4966 
 
 5014 
 
 5062 
 
 5110 
 
 6168 
 
 
 90-2 
 
 520? 
 
 5255 
 
 5303 
 
 5351 
 
 5399 
 
 5447 
 
 5496 
 
 5543 
 
 5592 
 
 5640 
 
 
 903 
 
 5688 
 
 5736 
 
 5784 
 
 5S32 
 
 6S80 
 
 5928 
 
 5976 
 
 6024 
 
 6072 
 
 612U 
 
 
 904 
 
 6168 
 
 U216 
 
 6265 
 
 6313 
 
 6361 
 48 
 
 6409 
 
 6457 
 
 6505 
 
 6653 
 
 6601 
 
 
 905 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 6888 
 
 6936 
 
 6984 
 
 7032 
 
 7080 
 
 
 90ti 
 
 7128 
 
 7176 
 
 7224 
 
 7272 
 
 7320 
 
 7368 
 
 7416 
 
 7464 
 
 7512 
 
 7659 
 
 
 907 
 
 7607 
 
 7655 
 
 7703 
 
 7751 
 
 7799 
 
 7847 
 
 7894 
 
 7942 
 
 7990 
 
 8o;>8 
 
 
 908 
 
 8086 
 
 8134 
 
 8181 
 
 8229 
 
 8277 
 
 8325 
 
 8373 
 
 8421 
 
 8468 
 
 8516 
 
 
 909 
 
 8664 
 
 8612 
 
 8659 
 
 8707 
 
 8756 
 
 8803 
 
 8860 
 
 8898 
 
 8946 
 
 8994 
 
 
 910 
 
 9041 
 
 9089 
 
 9137 
 
 9185 
 
 9232 
 
 9280 
 
 9328 
 
 9376 
 
 9423 
 
 9471 
 
 
 911 
 
 9518 
 
 9566 
 
 9614 
 
 9661 
 
 9709 
 
 9757 
 
 9804 
 
 9852 
 
 9900 
 
 9947 
 
 
 912 
 
 99^5 
 
 ..42 
 
 ..90 
 
 .138 
 
 .185 
 
 .233 
 
 .280 
 
 .328 
 
 .376 j .423 
 
 
 913 
 
 960471 
 
 0518 
 
 0566 
 
 0613 
 
 0()61 
 
 0709 
 
 0756 
 
 0804 
 
 0851 
 
 0899 
 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 
 1136 
 
 1184 
 
 1231 
 
 1279 
 
 1326 
 
 1374 
 
 
 915 
 
 1421 
 
 1469 
 
 1516 
 
 1563 
 
 1611 
 
 1658 
 
 1706 
 
 1763 
 
 1801 
 
 1848 
 
 
 91G 
 
 1895 
 
 1943 
 
 1990 
 
 2038 
 
 2085 
 
 2132 
 
 2180 
 
 2227 
 
 2275 
 
 2322 
 
 
 917 
 
 2369 
 
 2417 
 
 2464 
 
 2511 
 
 2559 
 
 260d 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 
 918 
 
 . 2843 
 
 2890 
 
 2937 
 
 2985 
 
 3032 
 
 3079 
 
 3126 
 
 3174 
 
 3221 
 
 3268 
 
 
 919 
 
 3316 
 
 3363 
 
 3410 
 
 3457 
 
 3504 
 
 3552 
 
 3599 
 
 3646 
 
 3693 
 
 3741 
 
 
 920 
 
 3788 
 
 3835 
 
 3882 
 
 3929 
 
 3977 
 
 4024 
 
 4071 
 
 4118 
 
 4165 
 
 4212 
 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4(»84 
 
 
 922 
 
 4731 
 
 4778 
 
 4825 
 
 4S72 
 
 4919 
 
 4966 
 
 5013 
 
 5061 
 
 5108 
 
 5155 
 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 
 
 5390 
 
 543 7 
 
 5484 
 
 5531 
 
 55/8 
 
 6625 
 
 
 924 
 
 5672 
 
 5719 
 
 5766 
 
 5813 
 
 5860 
 
 5907 
 
 5954 
 
 6001 
 
 6048 
 
 6095 
 
 
 925 
 
 6142 
 
 6189 
 
 6236 
 
 6283 
 
 6329 
 
 6376 
 
 6423 
 
 6470 
 
 6517 
 
 6564 
 
 
 926 
 
 6611 
 
 6653 
 
 6705 
 
 6752 
 
 6799 
 
 6845 
 
 6892 
 
 6939 
 
 6986 
 
 7033 
 
 
 927 
 
 7080 
 
 7127 
 
 7173 
 
 7220 
 
 7267 
 
 7314 
 
 7361 
 
 7408 
 
 7464 
 
 7501 
 
 
 9-28 
 
 7548 
 
 7595 
 
 7642 
 
 7688 
 
 7736 
 
 7782 
 
 7829 
 
 7875 
 
 7922 
 
 7969 
 
 
 929 
 
 8016 
 
 8062 
 
 8109 
 
 8156 
 
 8203 
 
 8249 
 
 8296 
 
 8343 
 
 8390 
 
 8436 
 
 
 930 
 
 8483 
 
 8530 
 
 8576 
 
 8623 
 
 8670 
 
 8716 
 
 8763 
 
 8810 
 
 8856 
 
 8903 
 
 
 931 
 
 8950 
 
 8996 
 
 9043 
 
 9U90 
 
 9136 
 
 91o3 
 
 9229 
 
 9276 
 
 9323 
 
 9369 
 
 
 932 
 
 9416 
 
 9463 
 
 9509 
 
 9556 
 
 9602 
 
 9649 
 
 9695 
 
 9742 
 
 9789 
 
 9835 
 
 
 933 
 
 9882 
 
 9928 
 
 9975 
 
 ..21 
 
 ..68 
 
 .114 
 
 .161 
 
 .207 
 
 .254 
 
 .300 
 
 
 934 
 
 1)70347 
 
 0393 
 
 0440 
 
 (I486 
 
 0533 
 
 0579 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 
 935 
 
 0812 
 
 0858 
 
 0904 
 
 0951 
 
 0997 
 
 1044 
 
 1090 
 
 1137 
 
 1183 
 
 1229 
 
 
 936 
 
 12y6 
 
 1322 
 
 1369 
 
 1415 
 
 1461 
 
 1508 
 
 1554 
 
 1601 
 
 1647 
 
 1693 
 
 
 937 
 
 1740 
 
 1786 
 
 1832 
 
 1879 
 
 1925 
 
 1971 
 
 2018 
 
 2064 
 
 2110 
 
 2157 
 
 
 938 
 
 2203 
 
 2249 
 
 2295 
 
 2342 
 
 2388 
 
 2434 
 
 2481 
 
 2527 
 
 2673 
 
 2619 
 
 
 939 
 
 2666 
 
 2712 
 
 2768 
 
 2804 
 
 2851 
 
 2897 
 
 2943 
 
 2989 
 
 3036 
 
 3082 
 
 
 94C 
 
 3128 
 
 3174 
 
 3220 
 
 3266 
 
 3313 
 
 3359 
 
 3405 
 
 3461 
 
 3497 
 
 3643 
 
 
 941 
 
 3590 
 
 3636 
 
 3682 
 
 3728 
 
 3774 
 
 3820 
 
 3866 
 
 3913 
 
 3959 
 
 4005 
 
 
 942 
 
 4051 
 
 4097 
 
 4143 
 
 4189 
 
 4235 
 
 4281 
 
 4327 
 
 .4374 
 
 4420 
 
 4466 
 
 
 943 
 
 4512 
 
 4558 
 
 4604 
 
 4650 
 
 4(i96 
 
 4742 
 
 4788 
 
 4834 
 
 4880 
 
 4926 
 
 
 944 
 
 4972 
 
 6018 
 
 6064 
 
 5110 
 
 6156 
 
 46 
 
 6616 
 
 5202 
 
 5248 
 
 5294 
 
 5340 
 
 6386 
 
 
 946 
 
 6432 
 
 5478 
 
 5524 
 
 5570 
 
 5662 
 
 5707 
 
 5753 
 
 5799 
 
 5845 
 
 
 946 
 
 6891 
 
 5937 
 
 5983 
 
 6029 
 
 6075 
 
 6121 
 
 6167 
 
 6212 
 
 6268 
 
 6304 
 
 
 947 
 
 6350 
 
 6396 
 
 6442 
 
 6488 
 
 6533 
 
 6579 
 
 6626 
 
 6671 
 
 6717 
 
 6763 
 
 
 948 
 
 6808 
 
 6854 6900 
 
 6946 
 
 6992 
 
 7037 
 
 7083 
 
 7129 
 
 7175 
 
 7220 
 
 
 949 
 
 7266 
 
 7312 7358 
 
 7403 
 
 7449 
 
 7495 
 
 7541 
 
 7586 
 
 7632 
 
 7678 
 
 
 19 
 
20 
 
 LOGARITHMS 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 7906 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 .950 
 
 977724 
 
 7769 
 
 7815 
 
 7861 
 
 7952 
 
 7998 
 
 8043 
 
 8089 
 
 8135 
 
 951 
 
 8181 
 
 8226 
 
 8272 
 
 8317 
 
 ^363 
 
 840!) 
 
 8454 
 
 8500 
 
 8546 
 
 8591 
 
 952 
 
 8637 
 
 8683 
 
 8728 
 
 8774 
 
 8819 
 
 88G5 
 
 8911 
 
 8956 
 
 9002 
 
 9047 
 
 953 
 
 9093 
 
 9138 
 
 9184 
 
 9230 
 
 9275 
 
 9321 
 
 9366 
 
 9412 
 
 9457 
 
 9503 
 
 954 
 
 9548 
 
 9594 
 
 9639 
 
 9685 
 
 9730 
 
 9776 
 
 9821 
 
 9667 
 
 9912 
 
 9968 
 
 955 
 
 980003 
 
 0049 
 
 0094 
 
 014C 0185 
 
 0231 
 
 0276 
 
 0322 
 
 0367 
 
 0412 
 
 956 
 
 0458 
 
 0503 
 
 0549 
 
 0594 0o40 
 
 0685 
 
 0730 
 
 0776 
 
 0821 
 
 0867 
 
 957 
 
 0J12 
 
 0957 
 
 1003 
 
 1048 
 
 1093 
 
 1139 
 
 1184 
 
 1229 
 
 1275 
 
 1320 
 
 958 
 
 1366 
 
 1411 
 
 1456 
 
 1501 
 
 1.547 
 
 1592 
 
 1637 
 
 1683 
 
 1728 
 
 1773 
 
 959 
 
 1819 
 
 1864 
 
 1909 
 
 1954 
 
 2000 
 
 2045 
 
 2090 
 
 2135 
 
 2181 
 
 2226 
 
 960 
 
 2271 
 
 2316 
 
 2362 
 
 2407 
 
 2452 
 
 2497 
 
 2543 
 
 2588 
 
 2633 
 
 2678 
 
 961 
 
 2723 
 
 2769 
 
 2814 
 
 2859 
 
 2904 
 
 2949 
 
 2994 
 
 3040 
 
 3085 
 
 3130 
 
 932 
 
 3175 
 
 3220 
 
 3265 
 
 3310 
 
 3356 
 
 3401 
 
 3446 
 
 3491 
 
 3536 
 
 3581 
 
 963 
 
 3626 
 
 3671 
 
 3716 
 
 3762 
 
 3807 
 
 3852 
 
 3897 
 
 3942 
 
 3987 
 
 4032 
 
 984 
 
 4077 
 
 4122 
 
 4167 
 
 4212 
 
 4257 
 
 43v;2 
 
 4347 
 
 4392 
 
 4437 
 
 4482 
 
 965 
 
 4527 
 
 4572 
 
 4617 
 
 4662 
 
 4707 
 
 4752 
 
 4797 
 
 4842 
 
 4887 
 
 4932 
 
 966 
 
 4977 
 
 5022 
 
 5067 
 
 5112 
 
 5157 
 
 5202 
 
 5247 
 
 5292 
 
 5337 
 
 5382 
 
 967 
 
 5426 
 
 5471 
 
 5516 
 
 5561 
 
 6608 
 
 6651 
 
 6699 
 
 5741 
 
 5786 
 
 6830 
 
 968 
 
 5875 
 
 6920 
 
 5965 
 
 6010 
 
 6055 
 
 6100 
 
 6144 
 
 6189 
 
 6234 
 
 6279 
 
 969 
 
 6324 
 
 6369 
 
 6413 
 
 6458 
 
 6503 
 
 6548 
 
 6593 
 
 6637 
 
 6682 
 
 6727 
 
 970 
 
 6772 
 
 6817 
 
 6861 
 
 6906 
 
 6951 
 
 6996 
 
 7040 
 
 7085 
 
 7130 
 
 7176 
 
 971 
 
 7219 
 
 7264 
 
 7309 
 
 7353 
 
 7398 
 
 7443 
 
 7488 
 
 7532 
 
 7577 
 
 7622 
 
 972 
 
 7666 
 
 7711 
 
 7756 
 
 7800 
 
 7845 
 
 7890 
 
 7934 
 
 7979 
 
 8024 
 
 8068 
 
 973 
 
 8113 
 
 8157 
 
 8202 
 
 8247 
 
 8291 
 
 8336 
 
 8381 
 
 8425 
 
 8470 
 
 8514 
 
 974 
 
 8559 
 
 8604 
 
 8648 
 
 8693 
 
 8737 
 
 8782 
 
 8826 
 
 8871 
 
 8916 
 
 8960 
 
 975 
 
 9005 
 
 9049 
 
 9093 
 
 9138 
 
 9183 
 
 9227 
 
 9272 
 
 9316 
 
 9361 
 
 9405 
 
 976 
 
 9450 
 
 9494 
 
 9539 
 
 9583 
 
 9628 
 
 9672 
 
 9717 
 
 9761 
 
 9806 
 
 9850 
 
 977 
 
 9895 
 
 9939 
 
 9983 
 
 ..28 
 
 ..72 
 
 .117 
 
 .161 
 
 .203 
 
 .250 
 
 .294 
 
 978 
 
 990339 
 
 0383 
 
 0428 
 
 0472 
 
 0516 
 
 0561 
 
 0605 
 
 0650 
 
 0694 
 
 0738 
 
 979 
 
 0783 
 
 0827 
 
 0871 
 
 0916 
 
 0960 
 
 1004 
 
 1049 
 
 1093 
 
 1137 
 
 1182 
 
 980 
 
 1226 
 
 1270 
 
 1315 
 
 1359 
 
 1403 
 
 1448 
 
 1492 
 
 1536 
 
 1580 
 
 1625 
 
 981 
 
 1669 
 
 1713 
 
 1758 
 
 1802 
 
 1846 
 
 1890 
 
 1935 
 
 1979 
 
 2023 
 
 2067 
 
 982 
 
 2111 
 
 2156 
 
 2200 
 
 2244 
 
 2288 
 
 2333 
 
 2377 
 
 2421 
 
 2465 
 
 2509 
 
 983 
 
 2554 
 
 2598 
 
 2()42 
 
 2686 
 
 2730 
 
 2774 
 
 2819 
 
 2863 
 
 2907 
 
 2951 
 
 984 
 
 2995 
 
 3039 
 
 3083 
 
 3127 
 
 3172 
 
 3216 
 
 3260 
 
 3304 
 
 3348 
 
 3392 
 
 985 
 
 3436 
 
 3480 
 
 3524 
 
 3568 
 
 3613 
 
 3657 
 
 3701 
 
 3746 
 
 3789 
 
 3833 
 
 986 
 
 3877 
 
 3921 
 
 3965 
 
 4009 
 
 4053 
 
 4097 
 
 4141 
 
 4185 
 
 4229 
 
 4273 
 
 987 
 
 4317 
 
 4361 
 
 4405 
 
 4449 
 
 4493 
 
 4537 
 
 458 i 
 
 4625 
 
 4669 
 
 4713 
 
 988 
 
 4757 
 
 4801 
 
 4845 
 
 4886 
 
 4933 
 
 4977 
 
 5021 
 
 5065 
 
 5108 
 
 6152 
 
 989 
 
 5196 
 
 5240 
 
 5284 
 
 6328 
 
 5372 
 
 5416 
 
 5460 
 
 6504 
 
 6547 
 
 5591 
 
 ' 990 
 
 5635 
 
 5679 
 
 5723 
 
 5767 
 
 5811 
 
 5854 
 
 6898 
 
 5942 
 
 5986 
 
 6030 
 
 991 
 
 6074 
 
 6117 
 
 6161 
 
 6205 
 
 6249 
 
 6293 
 
 6337 
 
 6380 
 
 6424 
 
 6468 
 
 992 
 
 6512 
 
 6555 
 
 6599 
 
 6643 
 
 6687 
 
 6731 
 
 6774 
 
 6818 
 
 6862 
 
 6906 
 
 993 
 
 6949 
 
 6993 
 
 7037 
 
 7080 
 
 7124 
 
 7168 
 
 7212 
 
 7255 
 
 7299 
 
 7343 
 
 994 
 
 7386 
 
 7430 
 
 7474 
 
 7517 
 
 7561 
 
 44 
 
 7998 
 
 7605 
 
 7648 
 
 7692 
 
 7736 
 
 7779 
 
 995 
 
 7823 
 
 7867 
 
 7910 
 
 7954 
 
 8041 
 
 8085 
 
 8129 
 
 8172 
 
 8216 
 
 996 
 
 8259 
 
 8303 
 
 8347 
 
 8390 
 
 8434 
 
 8477 
 
 8521 
 
 8564 
 
 8608 
 
 8652 
 
 997 
 
 8695 
 
 8739 
 
 8792 
 
 8826 
 
 8869 
 
 8913 
 
 8956 
 
 9000 
 
 9043 
 
 9087 
 
 998 
 
 9131 
 
 9174 
 
 9218 
 
 9261 
 
 9305 
 
 9348 
 
 9392 
 
 9435 
 
 9479 
 
 9522 
 
 999 
 
 9565 
 
 9609 
 
 9652 
 
 9696 
 
 9739 
 
 9783 
 
 9826 
 
 9870 
 
 9913 
 
 9957 i 
 
TAIJl.K II. I.og. Sines and Taiigenis. ((P) Natural Snes. 21 
 
 
 
 
 S.iie. ID 10" 
 
 Co>.iiH. |I). IO"l Taiig. 
 
 D 10" 
 
 Coiang. 
 
 N.pine 
 
 N. COS. 
 
 
 
 Neg.iuiiiiite' 
 
 lO.oaQooo! 
 
 0.000000 
 
 
 Infinite. 
 
 00000 
 
 100000 
 
 60 
 
 
 1 
 
 u.-io.irZti 
 
 
 000000 
 
 
 6.4<;3726 
 
 
 13.536274 
 
 00029 
 
 KJOOiU 
 
 59 
 
 
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 7()475.) 
 
 
 o.))ooo 
 
 
 764756 
 
 
 235244 
 
 \ 00058 
 
 100. 10 J 
 
 58 
 
 
 3 
 
 94(H4? 
 
 
 OOJOi)0 
 
 
 940847 
 
 
 059153 
 
 ,000.S7 
 
 lOOO.K) 
 
 57 
 
 
 4 
 
 7.0)5 786 
 
 
 000)00 
 
 
 7.0eJ5786 
 
 
 12.934214 
 
 OJIH) 
 
 100l»>)0 
 
 5() 
 
 
 6 
 
 162696 
 
 
 000000 
 
 
 162()96 
 
 
 837304 
 
 100145 
 
 lOOOOiJ 
 
 65 
 
 
 6 
 
 24 IS 77 
 
 
 9; 999999 
 
 
 241878 
 
 
 758122 
 
 100175 
 
 io:);>oo 
 
 54 
 
 
 7 
 
 30S824 
 
 
 999999 ! 
 
 308825 
 
 
 39 1175 
 
 1 00204 
 
 l(KJ0.)t)153 
 
 
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 366816 
 
 
 999999 
 
 
 366817 
 
 
 633183 
 
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 100. too; 52 
 
 
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 417968 
 
 
 99i>999 
 
 
 417970 
 
 
 582030 
 
 1 002(i2 
 
 lOOO.K) 51 
 
 
 10 
 
 463725 
 
 
 99i)998 
 
 
 4()3727 
 
 
 536273 
 
 00291 
 
 lOOIKJO 50 
 
 
 11 
 
 7.5UoIl8 
 
 
 9.999998 
 
 
 7.505120 
 
 
 12.494880 
 
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 99999 49 
 
 
 i 12 
 
 542906 
 
 
 999997 
 
 
 542909 
 
 
 4570<)1 
 
 00349 
 
 99i)99 48 
 
 
 13 
 
 577668 
 
 
 999997 
 
 
 577672 
 
 
 422328 
 
 ■ 00378 
 
 J9999 47 
 
 
 14 
 
 60.')8o3 
 
 
 9999<)6 
 
 
 609857 
 
 
 390143 
 
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 9999!) I 4<) 
 
 
 15 
 
 639816 
 
 
 999996 ; 
 
 639820 
 
 
 360180 
 
 00436 
 
 99!)99 
 
 45 
 
 
 \H 
 
 667845 
 
 
 999995 
 
 
 667849 
 
 
 332151 
 
 004(55 
 
 9!>999 
 
 44 
 
 
 1 I' 
 
 6941 73 
 
 
 999995 
 
 
 694179 
 
 
 305821 ; 00495 
 
 99999 
 
 43 
 
 
 IS 
 
 718997 
 
 
 999994 
 
 
 719003 
 
 
 280997 ; 00524 
 
 99999 
 
 42 
 
 
 19 
 
 742477 
 
 
 999993 
 
 
 742484 
 
 
 257516! 00553 
 
 99998 
 
 41 
 
 
 20 
 
 764754 
 
 
 999993 
 
 
 764761 
 
 
 235239; 00582 
 
 99998 
 
 40 
 
 
 21 
 
 7.7S5943 
 
 
 9.999992 
 
 
 7.785951 
 
 
 12.214049 !:oo()n 
 
 99998 
 
 39 
 
 
 22 
 
 806146 
 
 
 999991 
 
 
 806155 
 
 
 193845 
 
 ' 00640 
 
 99998 
 
 38 
 
 
 23 
 
 825451 
 
 
 999990 
 
 
 8254()0 
 
 
 174540 
 
 00(i69 
 
 99998 
 
 37 
 
 
 24 
 
 843934 
 
 
 999989 
 
 
 843944 
 
 
 15(i056 
 
 : 00(398 
 
 99998 
 
 36 
 
 
 26 
 
 861663 
 
 
 999988 
 
 
 861674 
 
 
 138326 
 
 00727 
 
 99997 
 
 36 
 
 
 26 
 
 878()95 
 
 
 999988 
 
 
 878708 
 
 
 121292 
 
 0075() 
 
 99997 
 
 34 
 
 
 27 
 
 895085 
 
 
 999987 
 
 
 895099 
 
 
 104901 
 
 00785 
 
 9!)997 
 
 33 
 
 
 28 
 
 91 OS 79 
 
 
 9999S6 
 
 
 910894 
 
 
 089106 
 
 00814 
 
 99997 
 
 32 
 
 
 29 
 
 926119 
 
 
 999985 
 
 
 926134 
 
 
 073866 
 
 00844 
 
 99996 
 
 31 
 
 
 30 
 
 940842 
 
 
 999983 
 
 
 940858 
 
 
 059142 
 
 00873 
 
 99996 
 
 30 
 
 
 31 
 
 7.95.J082 
 
 2298 
 2227 
 2161 
 
 9.999982 
 
 0.2 
 0.2 
 
 7.955100 
 
 2298 
 2227 
 2161 
 2098 
 2039 
 1983 
 1930 
 1880 
 1833 
 1787 
 1744 
 1703 
 1664 
 1627 
 1591 
 
 12.044900 
 
 00902 
 
 99996 
 
 29 
 
 
 32 
 
 9(>8870 
 
 999981 
 
 968889 
 
 031111 
 
 j 00931 
 
 99996 
 
 28 
 
 
 1 .-Kl 
 
 982233 
 
 999980 
 
 982253 
 
 017747 
 
 00960 
 
 99995 
 
 27 
 
 
 134 
 
 995198 
 
 999979 
 
 0.2 
 
 995219 
 
 004781 
 
 00989 
 
 99995 
 
 26 
 
 
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 8.0J7787 
 
 2098 
 
 999977 
 
 0-2 
 0-2 
 0-2 
 
 8.007809 
 
 11.992191 
 
 01018 
 
 99995 
 
 25 
 
 
 1 3(i 
 
 0-20021 
 
 2039 
 
 999976 
 
 020045 
 
 979955 
 
 01047 
 
 99995 
 
 24 
 
 
 1 37 
 
 031919 
 
 1983 
 
 999975 
 
 031945 
 
 968055 
 
 01076 
 
 99994 
 
 23 
 
 
 38 
 
 043501 
 
 1930 
 1880 
 1832 
 1787 
 1744 
 1703 
 1664 
 1626 
 
 999973 
 
 0-2 
 0-2 
 0-2 
 0-2 
 
 043527 
 
 956473' 01105 
 
 99994 
 
 22 
 
 
 39 
 
 054781 
 
 999972 
 
 054S()9 
 
 945191 |i 01 134 
 
 99994 
 
 21 
 
 
 40 
 
 0.i5776 
 
 999971 
 
 0(i5806 
 
 9341941 01164 
 
 99993 
 
 20 
 
 
 41 
 
 8.076500 
 
 9.999969 
 
 8.076531 
 
 11.923469 01193 
 
 99993 
 
 19 
 
 
 42 
 
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 9999()8 
 
 0'2 
 
 086997 
 
 913003 
 
 012-22 
 
 99993 
 
 IB 
 
 
 43 
 
 097183 
 
 999966 
 
 0'2 
 02 
 
 o;3 
 
 097217 
 
 902783 i 
 
 01251 
 
 99992 
 
 17 
 
 
 44 
 
 107167 
 
 999964 
 
 107202 
 
 892797 i 
 
 01280 
 
 99992 
 
 16 
 
 
 46 
 
 116926 
 
 999963 
 
 1169(S3 
 
 883037 
 
 01 30-^ 
 
 99991 
 
 15 
 
 
 46 
 
 126471 
 
 1591 
 1557 
 1524 
 1492 
 1462 
 
 999961 
 
 0.3 
 
 126510 
 
 873490; 0133*^ 
 
 99991 14 
 
 
 . 47 
 
 135810 
 
 999959 
 
 03 
 0.3 
 0.3 
 0.3 
 0.3 
 0.3 
 0.3 
 0.3 
 0.3 
 
 135851 
 
 1557 
 1524 
 1493 
 1463 
 1434 
 1406 
 1379 
 1353 
 1328 
 1304 
 1281 
 1269 
 1238 
 1217 
 
 864149 
 
 01367 
 
 9999 li 3 i 
 
 
 48 
 
 144953 
 
 999958 
 
 144996 
 
 855004 
 
 01396 
 
 99990 
 
 12 
 
 
 ; 49 
 
 153907 
 
 99995(> 
 
 153952 
 
 846048! 
 
 014-25 
 
 99990 
 
 11 1 
 
 
 ; 50 
 
 162681 
 
 999954 
 
 162727 
 
 837273 ' 
 
 01454 
 
 99989 
 
 10 
 
 
 , 51 
 
 8.171280 ;■;;',? 
 
 !™iJ 3 9 
 18/980 ,„ 
 
 9.99!)952 
 
 8.171328 
 
 11.828672 
 
 01483 
 
 99989 
 
 9 
 
 
 152 
 
 999950 
 
 179763 
 
 820237 : 
 
 01513 
 
 99989 
 
 8 
 
 
 1 63 
 
 999948 
 
 188036 
 
 811964' 
 
 01542 
 
 99988 
 
 7 
 
 
 54 
 
 196102 ^e;';J ; 999946 
 204U70 l^;^. 99!>944 
 211895 ^^^' 999942 
 219581 ;*?/ • 999940 
 
 227134 .;:?•; 999938 
 23455 7, .;:;: 999936 
 
 196156 
 
 803844 
 
 01571 
 
 99988 
 
 6 
 
 
 55 
 
 204126 
 
 795874 01600 
 
 99f>87 
 
 6 
 
 
 56 
 
 0.3 
 0.4 
 0.4 
 n A 
 
 211953 
 
 788047 : 
 
 01629 
 
 99987 
 
 4 
 
 
 57 
 
 219641 
 
 780359 , 
 
 01658 
 
 9998(i 
 
 3 
 
 
 58 
 
 227195 
 
 772805 
 
 01687 
 
 9998<) 
 
 2 
 
 
 59 
 
 ^l 234621 
 ^■^ 241921 
 
 765379 
 
 01716 
 
 99985 
 
 1 
 
 
 60 
 
 241855 i'^'*^ 
 
 fOH If 
 
 9!)9934 
 
 768079" 
 
 01745 
 
 99985 
 
 
 
 
 S :;•' 
 
 1 Cn-nng. 
 
 
 Tan?. N. ros. 
 
 \. .«^ine- 
 
 
 S9 D-grees. 
 
 
 
 
 
22 
 
 Log. Sines and Tangents. (1°) Natural Sines. TaBLK 11. 
 
 08 
 
 1 
 
 4 
 5 
 6 
 7 
 8 
 9 
 ID 
 11 
 12 
 13 
 ,4 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 ! 50 
 ,51 
 I 52 
 i53 
 64 
 55 
 56 
 57 
 68 
 59 
 60 
 
 .241855 
 
 249033 l]^^ 
 256094 }}^' 
 2iS3042 ^8 
 269881 ^ 
 276)14 
 . 283243 
 289773 
 296-207 
 30254 
 30879 
 
 D 10 
 
 1122 
 1105 
 1088 
 1072 
 1056 
 , 1041 
 7 1027 
 
 .dl49o4 ,f),r, 
 
 321027 ^^^J 
 327016 III 
 332924 07? 
 338753 ^'^ 
 344504 
 350181 
 355783 
 
 361315 Qifj 
 
 366777 
 .372171 
 377499 
 382762 
 387962 
 393101 
 
 959 
 946 
 934 
 922 
 
 899 
 888 
 877 
 867 
 856 
 
 398179 007 
 4031991 007 
 4081611 ofA 
 
 .422/17: 70, 
 
 427462! ioi 
 
 432156} ^°4 
 
 436800 '^^ 
 
 441394 !^ll 
 
 445941 1 '^2 
 
 450440 7?.2 
 
 15^1^35 
 
 4o9301 i ,727 
 
 463665 70,, 
 
 .467985 '^To 
 
 472263 7A;; 
 
 476498 '"° 
 
 480693 l^. 
 
 484848 ^of^ 
 
 488963 TZa 
 
 493040 ^1^ 
 
 497078 ^^; 
 
 501080 ^^J 
 
 505045 ^"^ 
 
 ,508974 III 
 
 ']?4l 643 
 016/26 j:.„_ 
 
 520551 ^^' 
 
 524343 ^^^ 
 
 528102 l^\ 
 
 S"S 616 
 53oo23 ^. . 
 
 539186 l;,t^ 
 
 642819 606 
 
 Cosine. 
 
 .999934 
 999932 
 999929 
 999927 
 999925 
 999922 
 999920 
 
 -999918 
 999915 
 999913 
 999910 
 
 9.9999D7 
 999905 
 
 '999902 
 999899 
 999897 
 999894 
 999891 
 999888 
 999885 
 999882 
 
 1.999879 
 999876 
 999873 
 999870 
 999867 
 999864 
 999861 
 999858 
 999854 
 999851 
 
 K 999848 
 999844 
 999841 
 999838 
 999834 
 999831 
 999827 
 999823 
 999820 
 999816 
 
 ). 999812 
 999809 
 k'999805 
 999801 
 999797 
 999793 
 999790 
 999786 
 999782 
 999778 
 99977. 
 999769 
 999765 
 999761 
 999757 
 999753 
 999748 
 999744 
 999740 
 999735 
 
 D.IO^' 
 
 0.4 
 0.4 
 0.4 
 0.4 
 0.4 
 0.4 
 
 0-4 
 0.4 
 0.5 
 0.5 
 0.5 
 0.5 
 5 
 05 
 0.6 
 0.6 
 0.6 
 0.6 
 0.6 
 0.6 
 0.5 
 0.5 
 0.6 
 0.5 
 0.6 
 06 
 0.6 
 0.6 
 0-6 
 0.6 
 0.6 
 0.6 
 0.6 
 0.6 
 0.6 
 0.6 
 0-6 
 0.6 
 0.6 
 
 0.7 
 0.7 
 0.7 
 0.7 
 
 Tan^r. 
 
 .241921 
 249102 
 256165 
 263115 
 269956 
 276691 
 283323 
 289856 
 296292 
 302634 
 308884 
 .315046 
 321122 
 327114 
 333025 
 333856 
 344610 
 360289 
 355895 
 361430 
 366895 
 .372292 
 377622 
 382889 
 388L192 
 393234 
 398316 
 403338 
 408304 
 413213 
 418068 
 .422869 
 427618 
 432315 
 436962 
 441560 
 446110 
 450613 
 455070 
 459481 
 463849 
 .468172 
 472454 
 476693 
 480892 
 485050 
 489170 
 493250 
 497293 
 501298 
 505267 
 .509200 
 513098 
 516961 
 520790 
 624586 
 528349 
 532080 
 535779 
 539447 
 543084 
 
 D 10' 
 
 1197 
 1177 
 1158 
 1140 
 1122 
 1105 
 1089 
 1073 
 1057 
 1042 
 1027 
 1013 
 999 
 985 
 972 
 959 
 946 
 934 
 922 
 911 
 899 
 888 
 879 
 867 
 857 
 847 
 837 
 828 
 818 
 809 
 800 
 791 
 783 
 774 
 7()6 
 758 
 760 
 743 
 736 
 728 
 720 
 713 
 7U7 
 700 
 693 
 686 
 680 
 674 
 668 
 661 
 655 
 650 
 644 
 638 
 633 
 627 
 622 
 616 
 611 
 606 
 
 Coiiii)^'. 
 
 Co 
 
 !aiig. 
 
 N. sine;. N. cos 
 
 11 
 
 11 .7.58079 
 750898 
 743835 
 736885 
 730044 
 723309 
 716677 
 710144 
 703708 
 697366 
 691116 
 684954 
 678878 
 672886 
 666975 
 661144 
 655390 
 649711 
 644105 
 638570 
 633105 
 
 11 .627708 
 622378 
 617111 
 611908 
 606766 
 601685 
 596662 
 591696 
 586787 
 581932 
 
 11.577131 
 672382 
 567685 
 563038 
 558440 
 553890 
 549387 
 544930 
 540519 
 636151 
 
 11.531828 
 527546 
 523307 
 519108 
 514950 
 510830 
 506750 
 602707 
 498702 
 494733 
 490800 
 486902 
 483039 
 479210 
 476414 
 471651 
 467920 
 464221 
 460553 
 466916 
 
 P 01742 
 01774 
 :; 01803 
 ! 01832 
 I 01862 
 01891 
 
 99985 60 
 )i)984 69 
 999S4 5S 
 99983 57 
 99983 66 
 99982 66 
 
 ! I 01949 
 ji 01978 
 i02U07 
 I 02036 
 
 I 02065 
 ; 02094 
 i, 02123 
 j; 02152 
 ii 02181 
 I' 02211 
 
 I I 02240 
 'I 02269 
 ! 02298 
 II 02327 
 j' 02356 
 1102385 
 :, 02414 
 ;j 02443 
 I 02472 
 I 02501 
 
 02530 
 
 02560 
 
 02589 
 
 ' 02618 
 
 H 02647 
 
 i 02676 
 
 ! 02705 
 
 02734 
 
 02763 
 
 02792 
 
 :: 02850 
 1 102879 
 1! 02908 
 :' 02938 
 
 01920 99982 54 
 99^81 63 
 99980, 68 
 99980 51 
 99979 50 
 9y979 49 
 y9978 48 
 99977 47 
 9^)977, 46 
 99976' 45 
 99976 44 
 99975 43 
 99974! 42 
 99974' 41 
 99973^ 40 
 9y972! 39 
 99972 38 
 99971 37 
 99970 36 
 y9y69! 35 
 99969 34 
 99968J 33 
 999671 32 
 99966; 31 
 99966 30 
 99965 29 
 99964 28 
 99963! 27 
 99963! 26 
 999o2! 25 
 99961; 24 
 
 02821 99960! 23 
 
 99959I 22 
 99959 21 
 99958; 20 
 99957' 19 
 
 11 
 
 ! I 02967 99956 I8 
 '!02b9b 99955 17 
 03025 99954: 16 
 03054 99953 15 
 ( 03083 99952: 14 
 i 031 12 99^52 13 
 103141 99951; 12 
 ; 03170 99950! 11 
 103199 99949110 
 103228 999481 9 
 03257 999471 
 99946! 
 9J9i5 
 99944 
 99943 
 99942 
 99941 
 99940 
 99939 
 
 0328(i 
 03316 
 jl03<i45 
 1 1 03374 
 I 03403 
 ,03432 
 ! I 03401 
 ! 03490 
 
 Tail"-. i! N. COS. N 
 
 88 DegreeB. 
 
TABI.K II. Log. Siiit.» and Tangents. (i°) Natural Sines. 
 
 23 
 
 .542819 
 54G4^>2 
 54999.5 
 553589 
 557054 
 6(i0540 
 6G3999 
 5«)74:n 
 67083(> 
 574214 
 5775(i() 
 
 .5H()89-2 
 584193 
 
 • 5874ti9 
 590721 
 593948 
 597152 
 600332 
 603489 
 606623 
 609734 
 
 .612823 
 615891 
 618937 
 6219(>2 
 624965 
 627948 
 630911 
 633854 
 d.3()776 
 63^680 
 
 .642563 
 645428 
 648274 
 651102 
 653911 
 656702 
 659475 
 662230 
 664968 
 667(i89 
 
 1.670393 
 673080 
 675751 
 678405 
 681043 
 683665 
 686272 
 688863 
 691438 
 693998 
 
 !.69t)543 
 699073 
 701589 
 704090 
 706577 
 709049 
 711507 
 713952 
 716383 
 718800 
 
 D. 10' 
 
 r/nn 9. 
 
 603 
 595 
 591 
 586 
 681 
 576 
 572 
 5(i7 
 5()3 
 559 
 554 
 550 
 546 
 542 
 538 
 534 
 530 
 526 
 522 
 519 
 515 
 511 
 508 
 504 
 501 
 497 
 494 
 490 
 487 
 484 
 481 
 477 
 474 
 471 
 468 
 465 
 4(i2 
 459 
 45(> 
 453 
 451 
 448 
 445 
 442 
 440 
 437 
 434 
 432 
 429 
 427 
 424 
 422 
 419 
 417 
 414 
 412 
 410 
 407 
 405 
 403 
 
 Cosine. 
 
 .999735 
 999731 
 999726 
 995)722 
 999717 
 999713 
 999708 
 9997t)4 
 999699 
 9!)9694 
 999689 
 
 .999()85 
 !W9<)80 
 9<)9()75 
 
 "999670 
 999665 
 9996t)0 
 999(555 
 999650 
 999645 
 999640 
 
 .999(135 
 999629 
 999324 
 999{J19 
 9f)96l4 
 999608 
 999()03 
 999597 
 999592 
 999586 
 
 .999581 
 999575 
 999570 
 9995()4 
 999558 
 999553 
 999547 
 9})9541 
 9J)9535 
 999529 
 
 .999524 
 999518 
 999512 
 999506 
 999500 
 999493 
 999487 
 99iJ481 
 999175" 
 939469 
 
 •.9!;9+()3 
 9.'i9456 
 999 150 
 999143 
 999437 
 999431 
 9994-:4 
 999418 
 9i>9411 
 999404 
 
 D. 10' 
 
 0.7 
 0.7 
 0.7 
 0-8 
 0-8 
 0-8 
 0.8 
 0.8 
 0.8 
 0-8 
 0.8 
 0-8 
 0.8 
 0.8 
 0.8 
 0.8 
 0.8 
 0.8 
 0.8 
 0.8 
 0.9 
 0.9 
 0.9 
 0.9 
 0.9 
 0-9 
 0-9 
 0.9 
 0.9 
 0.9 
 0.9 
 0.9 
 0.9 
 0.9 
 
 Sii 
 
 1.0 
 1.0 
 1-0 
 1-0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 
 1. 
 
 1 
 
 1, 
 
 1, 
 
 1. 
 
 1, 
 
 1 
 
 1, 
 
 I 
 
 Tang. 
 
 .543084 
 54(>()91 
 5502ti8 
 553817 
 557336 
 560828 
 564291 
 667727 
 571137 
 574520 
 577877 
 
 ,581208 
 584514 
 587795 
 591051 
 594283 
 597492 
 600677 
 603839 
 606978 
 610094 
 
 .613189 
 616262 
 619313 
 622343 
 625352 
 628340 
 631308 
 634256 
 637184 
 640093 
 
 .642982 
 645863 
 648704 
 651537 
 654352 
 657149 
 659928 
 662689 
 6(i5433 
 668160 
 
 .670870 
 673563 
 67()239 
 678900 
 681544 
 684172 
 6 6784 
 689381 
 691963 
 694529 
 
 .697081 
 699617 
 702139 
 704246 
 707140 
 709«)18 
 702083 
 714534 
 71(>y?2 
 719396 
 
 Coiiuit 
 
 D. 10' 
 
 602 
 59.) 
 591 
 587 
 582 
 577 
 573 
 668 
 664 
 559 
 555 
 551 
 547 
 543 
 639 
 635 
 53 r 
 527 
 523 
 619 
 616 
 612 
 508 
 505 
 601 
 498 
 495 
 491 
 488 
 485 
 482 
 478 
 475 
 472 
 469 
 466 
 463 
 460 
 457 
 464 
 453 
 449 
 446 
 443 
 442 
 438 
 436 
 433 
 430 
 428 
 425 
 423 
 420 
 418 
 415 
 413 
 411 
 408 
 406 
 404 
 
 Colling. i|N. Bine.lN cos, 
 
 11.456916 
 
 453309 
 449732 
 446183 
 442(i64 
 439172 
 435709 
 432273 
 4288()3 
 425480 
 422123 
 
 11.418792 
 415486 
 412205 
 408949 
 405717 
 402508 
 399323 
 396161 
 393022 
 389906 
 
 11.386811 
 383738 
 380687 
 377657 
 374648 
 3716(J0 
 368692 
 365744 
 362816 
 359907 
 
 11.357018 
 354147 
 351296 
 348463 
 345648 
 342851 
 340072 
 337311 
 334567 
 331840 
 
 11.329130 
 326437 
 323761 
 321100 
 318456 
 315828 
 313216 
 310()19 
 308037 
 30.5471 
 
 11.302919 
 3fj0383 
 297861 
 295354 
 292860 
 290382 
 287917 
 2854()5 
 283028 
 280ti04 
 
 ;■ 03490199939 60 
 
 ; 0351;) 99938 59 
 ,03548 199937 58 
 03577 
 : 0360() 
 i 03635 
 
 9993( 
 99935 
 99934 55 
 03664 99933 54 
 (t3(J93 199932 53 
 03723 99931 
 0375299930 
 03781 {99929 
 03810 99927 
 03839,99926 
 (J38ti8 ,99925 
 
 I 03926 '99923 
 103955199922 
 
 1O39.S4I9992I 
 
 04013 
 04042 
 04071 
 04100 
 03129 
 04159 
 
 99919 
 99918 
 99917 
 99916 
 99915 
 99913 
 
 04188 99912 
 •04217199911 
 1 04246 J999 10 
 04275199909 
 0430499907 
 04333 !9990e) 
 04362)99905 
 0439 1|9 9904 
 ; 0442099902 
 04449199901 
 0447H|'999(J0 
 04507 [99898 
 04536 199897 
 0456.>9989li 
 04594:99894 
 04623 y9S9o 
 04()53]99892 
 04(i82 '99890 
 047 1 1 i99889 
 04740 [;J9d8h 
 047(i9rJ9SH() 
 0479hl9.J885 
 ;04S2/7J9883 
 0485()!9JJ8h2 
 04H85I99881 
 049141998 
 
 24 
 
 ,23 
 
 )-7 
 
 ' 0494.') 
 049 i 2 
 
 ! 05001 
 05030 
 0505!) 
 05088 
 05117 
 
 9.9878 
 99876 
 99875 
 99873 
 99872 
 99870 
 998(i9 
 
 ()5146 9-j8()7 
 
 0517; 
 
 9986(i 
 
 05205 998()4 
 
 05234 
 
 Tant 
 
 N. cofi. X.sint' 
 
 998()3 
 
 87 Degrees. 
 
24 
 
 Log. Sines and Tangeiils. {^) Natural Sines. TABLE II. 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 J7 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 60 
 61 
 52 
 53 
 64 
 65 
 56 
 57 
 68 
 69 
 60 
 
 Sine, 
 
 .718800 
 721204 
 723595 
 725972 
 728337 
 730088 
 733027 
 735354 
 737667 
 739989 
 742259 
 
 .744536 
 746802 
 749055 
 751297 
 753528 
 755747 
 757955 
 760151 
 762337 
 764511 
 
 .766675 
 768828 
 770970 
 773101 
 775223 
 777333 
 779434 
 781524 
 783605 
 785675 
 
 .787736 
 789787 
 791828 
 793859 
 795881 
 797894 
 799897 
 801892 
 803876 
 805852 
 
 ,807819 
 809777 
 811726 
 813667 
 815599 
 817522 
 819436 
 821343 
 823240 
 825130 
 
 .827011 
 828884 
 830749 
 832607 
 834456 
 836297 
 838130 
 839956 
 841774 
 843585 
 
 Cosine. 
 
 D. 10' 
 
 401 
 398 
 396 
 394 
 392 
 390 
 388 
 386 
 384 
 382 
 380 
 378 
 376 
 374 
 372 
 370 
 368 
 366 
 364 
 362 
 361 
 359 
 357 
 355 
 353 
 352 
 350 
 348 
 347 
 345 
 343 
 342 
 340 
 339 
 337 
 335 
 334 
 332 
 331 
 329 
 328 
 326 
 325 
 323 
 322 
 320 
 319 
 318 
 316 
 315 
 313 
 312 
 311 
 309 
 308 
 307 
 306 
 304 
 303 
 302 
 
 Cosiuu. 
 
 9.999404 
 999.'W8 
 999391 
 999384 
 999378 
 999371 
 999364 
 999357 
 999350 
 999343 
 999336 
 
 9.999329 
 999322 
 999315 
 999308 
 999301 
 999294 
 999286 
 999279 
 999272 
 999266 
 
 9.999257 
 999250 
 999242 
 999235 
 999227 
 999220 
 999212 
 999205 
 999197 
 999189 
 
 9.999181 
 999174 
 999166 
 999158 
 999150 
 999142 
 -999134 
 999126 
 999118 
 999110 
 
 9.999102 
 999094 
 999086 
 999077 
 999069 
 999061 
 999053 
 999044 
 999036 
 999027 
 999019 
 999010 
 999002 
 998993 
 998984 
 998976 
 998967 
 998958 
 998950 
 998941 
 
 Sine. 
 
 D. 1 
 
 Tanji. 
 
 S. 7 19396 
 721806 
 724204 
 726588 
 728959 
 731317 
 7336()3 
 735996 
 738317 
 740626 
 742922 
 .745207 
 747479 
 749740 
 751989 
 764227 
 766453 
 758i)68 
 760872 
 763065 
 765246 
 ,767417 
 769678 
 771727 
 773866 
 775995 
 778114 
 780222 
 782320 
 784408 
 786486 
 
 8.788554 
 790613 
 792662 
 794701 
 796731 
 798752 
 800763 
 802765 
 804858 
 806742 
 ,808717 
 810683 
 812641 
 814589 
 816529 
 818461 
 820384 
 822298 
 824205 
 826103 
 .827992 
 829874 
 831748 
 833613 
 835471 
 837321 
 839163 
 840998 
 842825 
 844644 
 
 Cot.in2. 
 
 D. 10" Cotanj?. |(N. sine. N. cos. 
 
 402 
 399 
 397 
 395 
 393 
 391 
 389 
 387 
 385 
 383 
 381 
 379 
 377 
 375 
 373 
 371 
 369 
 367 
 366 
 364 
 362 
 360 
 358 
 356 
 355 
 353 
 351 
 350 
 348 
 346 
 345 
 343 
 341 
 340 
 338 
 337 
 335 
 334 
 332 
 331 
 329 
 328 
 326 
 325 
 323 
 322 
 320 
 319 
 318 
 316 
 315 
 314 
 312 
 311 
 310 
 308 
 307 
 306 
 304 
 303 
 
 11.280604 
 278194 
 275796 
 273412 
 271041 
 2(i8683 
 266337 
 264004 
 261683 
 259374 
 257078 
 
 11.254793 
 252521 
 250260 
 248011 
 245773 
 243547 
 241332 
 239128 
 236935 
 234754 
 
 11.232583 
 230422 
 228273 
 226134 
 224006 
 221886 
 219778 
 217680 
 215592 
 213514 
 
 11.211446 
 209387 
 207338 
 205299 
 203269 
 201248 
 199237 
 197235 
 195242 
 193258 
 
 11.191283 
 189317 
 187359 
 185411 
 183471 
 181539 
 179616 
 177702 
 175796 
 173897 
 
 11.172008 
 170126 
 168252 
 166387 
 164529 
 162679 
 160837 
 159002 
 157175 
 1563.56 
 
 05234 99863 
 
 05263 
 05292 
 05321 
 05350 
 05379 
 
 05408 99854 
 
 05437 
 1 105466 
 j 05495 
 i 05524 
 I 05653 
 j 105582 
 l! 05611 
 ! 05640 
 I' 05669 
 1105698 
 {! 05727 
 I 06766 
 
 I i 05785 
 
 I I 05814 
 
 I j 05844 
 1106873 
 1 105902 
 
 I I 05931 
 ij 05960 
 
 {05989 
 ! 106018 
 1106047 
 i I 06076 
 li 06106 
 1106134 
 i! 06163 
 ;| 06192 
 1 1 08221 
 i 106260 
 i^ 06279 
 jj 06308 
 N 06337 
 i, 06366 
 
 99861 
 99860 
 99868 
 99857 
 99855 
 
 99852 
 99851 
 99849 
 99847 
 99846 
 99844 
 99842 
 99841 
 99839 
 99838 
 99836 
 99834 
 99833 
 99831 
 99829 
 99827 
 99826 
 99824 
 99822 
 99821 
 99819 
 99817 
 99815 
 99813 
 99812 
 99810 
 99808 
 99806 
 99804 
 99803 
 998OI 
 99799 
 99797 
 
 06395'99795 
 06424 997931 
 O6453I99792 
 {)<i482i99790 
 0651199788 
 ■ 06640!99786 
 I 0856919978 1 
 1 0659819978; 
 
 08627 
 06656 
 j 06685 
 106714 
 
 997SC' 
 99778 
 99776 
 99774 
 
 106743199772 
 I 06773 99770 
 p 06802 99768 
 i' 06831 199766 
 ;' 06860 99764 
 1,06889 99762 
 :: 069 18 99760 
 ,i 06947 99758 
 '! 06976 99756 
 
 Tang. 
 
 N. COS. N'.sine 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 ' 
 
 50 ; 
 
 49 i 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 16 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
TABLK II. . Log. Sines and Taiigeiils. (4°) Naiuiul Sines. 
 
 25 
 
 18.843585 
 84.53^7 
 84i-!H3 
 848971 
 8507.^)1 
 85'i5"J5 
 854-291 
 8.)t.U49 
 857801 
 .io954() 
 8()1-J83 
 
 8.8ii3014 
 8(^4738 
 8G()455 
 %81(i5 
 8t;9S(;8 
 871.'j<i5 
 873255 
 874938 
 87t)tJl5 
 878t2«5 
 879949 
 881»)07 
 883'i58 
 884^)03 
 88G54-2 
 888174 
 889801 
 8914'->1 
 893035 
 8941)43 
 
 8.89{>-.'4(i 
 89784-2 
 899432 
 901U17 
 902596 
 9041(i9 
 90573«i 
 907297 
 908853 
 910404 
 
 8.911949 
 913488 
 915022 
 9l»i5;j0 
 918073 
 919591 
 9': 1103 
 922(il0 
 924112 
 925(i0}) 
 ; 927100 
 928587 
 9300-i8 
 931544 
 933015 
 934481 
 935942 
 937398 
 938860 
 94t/29U 
 CoMJnu. 
 
 D. lU ' 
 
 300 
 29i) 
 298 
 2*7 
 2)5 
 294 
 293 
 292 
 291 
 2 
 288 
 287 
 286 
 285 
 284 
 283 
 282 
 281 
 279 
 279 
 277 
 276 
 275 
 274 
 273 
 272 
 271 
 270 
 269 
 268 
 2()7 
 26() 
 265 
 264 
 263 
 262 
 2S1 
 2(i0 
 259 
 258 
 257 
 257 
 256 
 255 
 254 
 253 
 252 
 251 
 250 
 249 
 249 
 248 
 247 
 246 
 245 
 244 
 243 
 243 
 242 
 241 
 
 Co.'sine. 
 
 1.95)8941 
 998932 
 99S923 
 998914 
 998.905 
 99889(> 
 998887 
 998878 
 998869 
 99S860 
 99S851 
 
 1.998841 
 998832 
 998823 
 998813 
 998S(M 
 998795 
 998785 
 998776 
 998766 
 99H757 
 
 I 998747 
 998738 
 998728 
 998718 
 998708 
 998()99 
 998(i89 
 998679 
 })98(;()f> 
 9!)8()59 
 
 1.998649 
 998(i3*> 
 998629 
 998619 
 998(i09 
 998599 
 99H589 
 998578 
 998568 
 998558 
 
 1.998548 
 998537 
 998527 
 .998516 
 
 -99850() 
 998495 
 998485 
 998474 
 9984(i4 
 998453 
 
 •.998442 
 998431 
 9;<8421 
 99S410 
 i)98399 
 
 • f 98388 
 9i)8377 
 998366 
 998355 
 99>>344 
 
 D. 10" 
 
 1 
 
 1 
 
 1 
 
 I 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 I 
 
 1 
 
 1 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1-6 
 
 1.6 
 
 1-6 
 
 1.6 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1 
 
 I 
 
 1 
 
 7 
 
 7 
 
 7 
 
 •1.8 
 
 1.8 
 1.8 
 
 1.8 
 1.8 
 1.8 
 1.8 
 
 Taug. 
 
 .844644 
 846455 
 84.vJ(il) 
 h50>)57 
 851846 
 853628 
 855403 
 857171 
 858932 
 8(J0i;86 
 8<>2433 
 
 ; 864173 
 8t)590t) 
 867632 
 869351 
 871064 
 872770 
 8744()9 
 87(il()2 
 877849 
 879529 
 
 l.h81202 
 882869 
 884530 
 886185 
 887833 
 889476 
 891112 
 892742 
 8943(i() 
 8!)5984 
 
 ;. 897596 
 899-203 
 900803 
 902398 
 903987 
 905670 
 907147 
 908719 
 910285 
 911846 
 
 ;. 913401 
 914961 
 916496 
 918034 
 919668 
 921096 
 922619 
 924136 
 925()49 
 927 156 
 
 1.928«i58 
 930155 
 931647 
 933134 
 934()16 
 93<)()93 
 937665 
 939032 
 940494 
 941!j52 
 
 ColJing. 
 56 D egreen. 
 
 30-2 
 301 
 299 
 298 
 29, 
 29 i 
 295 
 293 
 292 
 29! 
 -290 
 289 
 288 
 287 
 285 
 284 
 283- 
 282 
 281 
 280 
 279 
 278 
 277 
 276 
 276 
 274 
 273 
 272 
 271 
 -270 
 -269 
 268 
 267 
 266 
 '265 
 264 
 263 
 262 
 261 
 260 
 259 
 258 
 257 
 256 
 256 
 255 
 254 
 253 
 252 
 251 
 250 
 249 
 249 
 248 
 247 
 24() 
 245 
 244 
 244 
 243 
 
 Cotang. N. sine.lN. rort.l 
 
 11.155366 
 153545 
 151740 
 149943 
 148154 
 14(i372 
 144597 
 142829 
 141068 
 139314 
 137567 
 
 11.135827 
 134094 
 132368 
 130649 
 l-:-936 
 127-230 
 125531 
 123S38 
 122151 
 120471 
 
 11.118798 
 117131 
 115470 
 113815 
 112167 
 110524 
 108888 
 107258 
 105634 
 104016 
 
 11.102404 
 100797 
 099197 
 097602 
 006013 
 094430 
 092853 
 091281 
 089715 
 088154 
 
 11,086599 
 085049 
 083505 
 081966 
 080432 
 0.8904 
 077381 
 075864 
 074361 
 072844 
 
 11.071342 
 069845 
 0<)8353 
 0{)6866 
 0(i5384 
 0»j3907 
 062435 
 0(i09(i8 
 05956t> 
 058048 
 I'ang. 
 
 (J697(j 
 07005 
 07034 
 07063 
 
 1 99756 60 
 99754 59 
 99752 58 
 ; 99750 57 
 070<)2;99748 5(i 
 07121,99746:55 
 07150 99744 54 
 07 179 [99 742, 53 
 07208 iK)740: 52 
 07237, !W738i 51 
 07-266'99<3<i 50 
 07295 99734 49 
 ; 07324 99731 i 48 
 t 99729 47 
 
 07353 
 0*382 
 
 997271 46 
 6741199726145 
 07440!y"723, 44 
 07469:99721 43 
 07498199719, 42 
 07527 99716 41 
 07556199714140 
 07585J99712!39 
 0761499710 38 
 07643 99708 37 
 07672 9;)705 36 
 07701 99703 35 
 077.30 99701 34 
 07759 99(i99 33 
 07788:99696 32 
 07817i99(i94 31 
 0^846 99692. 30 
 07875 99()89| 29 
 07904i^J9o87| 28 
 07933 199685; 27 
 '■ 07962 99(i83: -26 
 07991 |99{i80 -25 
 080201)9678, 24 
 08049 J99676 23 
 08078 99673 '22 
 08107^9671 21 
 08136|9f>()«;8 -iO 
 081(i5jiW666: 19 
 08194 99()()4 18 
 08-223 99661' 17 
 08252199659 16 
 08-281 1996571 15 
 08310 99654' 14 
 08339 9.9652 13 
 0H3f, 8 991,49 12 
 08397 ^)9(f47l 1 1 
 08426 9i)()44 10 
 0845.^i99(i42i 9 
 
 08484 
 0H513 
 08542 
 08571 
 08()00 
 U8()29 
 08(i58 
 08(i87 
 08716 
 
 99()39: 
 
 :>9637 
 99(i35 
 99()32 
 99(>30 
 
 9:,()V7 
 99(i25 
 9M>22 
 99619 
 
 N. ros. N'.fine.l 
 
26 
 
 Log. Sines and Tangents. (5°) Natural Sines. *■ TABLE II. 
 
 2 
 3 
 
 4 
 6 
 6 
 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 26 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 36 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 4() 
 47 
 48 
 49 
 50 
 61 
 52 
 53 
 64 
 55 
 56 
 i~i 
 68 
 59 
 60 
 
 Sine. 
 
 ,940296 
 941738 
 943174 
 944606 
 946034 
 947466 
 948874 
 950287 
 951ii9o 
 95310a 
 964499 
 ,955894 
 957284 
 958;)70 
 960052 
 9ol429 
 962801 
 934170 
 965534 
 96()893 
 968249 
 969600 
 970947 
 972289 
 973628 
 974962 
 976293 
 977619 
 978941 
 980259 
 981573 
 982883 
 984189 
 985491 
 986789 
 988083 
 989374 
 990)60 
 991943 
 993222 
 994497 
 995768 
 997036 
 998299 
 999560 
 
 '.00J816 
 0J20;i9 
 003318 
 0045()3 
 005805 
 007044 
 
 I 003278 
 009510 
 01073 7 
 0lia'>/. 
 01318--^ 
 0144:)0 
 015613 
 016824 
 018031 
 019235 j 
 
 Cosine. | 
 
 D. 10' 
 
 8. 
 
 240 
 239 
 239 
 238 
 237 
 I 236 
 235 
 235 
 234 
 233 
 232 
 232 
 231 
 230 
 229 
 229 
 228 
 227 
 227 
 226 
 225 
 224 
 224 
 223 
 222 
 222 
 221 
 220 
 220 
 219 
 218 
 218 
 217 
 216 
 216 
 215 
 214 
 214 
 213 
 212 
 212 
 211 
 211 
 210 
 209 
 209 
 208 
 208 
 207 
 2ij6 
 206 
 205 
 206 
 204 
 203 
 203 
 2J2 
 202 
 201 
 201 
 
 Cosine. 
 
 9,998344 
 998333 
 998322 
 998311 
 998300 
 998289 
 998277 
 998266 
 998265 
 998243 
 998232 
 
 9.998220 
 998209 
 998197 
 998186 
 998174 
 998163 
 998151 
 998139 
 998128 
 998116 
 
 9.998104 
 998092 
 998080 
 998068 
 998056 
 998044 
 998032 
 998020 
 998008 
 997996 
 997984 
 997972 
 997959 
 997947 
 997936 
 997922 
 997910 
 997897 
 997886 
 997872 
 
 9 997860 
 997847 
 997835 
 997822 
 997809 
 997797 
 997784 
 997771 
 997758 
 997745 
 997732 
 997719 
 997706 
 997693 
 997680 
 997667 
 997664 
 997641 
 997628 
 997614 
 
 D. 10' 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 i.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 Sine. 
 
 TanS_ 
 
 8.941952 
 943404 
 944852 
 946295 
 947734 
 949168 
 950597 
 952021 
 953441 
 954856 
 956267 
 
 8.957674 
 959075 
 960473 
 961866 
 9{j3266 
 964639 
 966019 
 967394 
 968766 
 970133 
 
 8.971496 
 972865 
 974209 
 975560 
 976906 
 978248 
 979586 
 980921 
 982251 
 983577 
 
 8.984899 
 986217 
 987532 
 988842 
 990149 
 991451 
 992750 
 994045 
 995337 
 996()24 
 997908 
 999188 
 
 9.0i)0465 
 001 738 
 003007 
 004272 
 005534 
 006792 
 008047 
 009298 
 
 9.010546 
 011790 
 013031 
 014268 
 016502 
 016732 
 017959 
 019183 
 020403 
 021620 
 
 D. 10' 
 
 Co tang. 
 
 242 
 
 241 
 
 240 
 
 240 
 
 239 
 
 238 
 
 237 
 
 237 
 
 236 
 
 235 
 
 234 
 
 234 
 
 233 
 
 232 
 
 231 
 
 231 
 
 230 
 
 229 
 
 229 
 
 228 
 
 227 
 
 226 
 
 226 
 
 226 
 
 224 
 
 2iJ4 
 
 223 
 
 222 
 
 222 
 
 221 
 
 220 
 
 220 
 
 219 
 
 218 
 
 218 
 
 217 
 
 216 
 
 216 
 
 215 
 
 215 
 
 214 
 
 213 
 
 213 
 
 212 
 
 211 
 
 211 
 
 210 
 
 210 
 
 209 
 
 208 
 
 208 
 
 207 
 
 207 
 
 206 
 
 206 
 
 205 
 
 204 
 
 204 
 
 203 
 
 203 
 
 Cotanj^. N. sine 
 
 11.058048 
 066696 
 056148 
 053705 
 0522()6 
 050832 
 049403 
 047979 
 046669 
 045144 
 043733 
 
 11.042326 
 040925 
 039527 
 038134 
 036745 
 035361 
 033981 
 032606 
 031234 
 029867 
 
 11.028504 
 027145 
 025791 
 024440 
 02S094 
 021752 
 020414 
 019079 
 017749 
 016423 
 
 11.015101 
 013783 
 012468 
 011168 
 009861 
 008649 
 007260 
 005956 
 004663 
 003376 
 002092 
 000812 
 
 10.999535 
 998262 
 996993 
 995728 
 99W<)6 
 993208 
 991053 
 990/02 
 
 10.989454 
 988210 
 686969 
 985732 
 984498 
 983268 
 983041 
 980817 : 
 979697 i 
 978380 
 
 |i 08716 
 
 ii 08745 
 
 108774 
 
 ii 08803 
 
 i' 08831 
 
 : 08860 
 
 08889 
 
 ,08918 
 
 08947 
 
 ! 08976 
 
 09005 
 
 09034 
 
 09063 
 
 09092 
 
 cos 
 
 99619 
 99617 
 99614 
 99612 
 99()09 
 99607 
 99()04 
 99602 
 99599 
 9959(1 
 99594 
 99691 
 99588 
 99586 
 
 0912199583 
 I 09160199580 
 '09179 99578 
 109208199576 
 :09237!99572 
 ; 09266199570 
 j 0929o!99567 
 ' 09324i99564 
 \ 09353199562 
 i 09382199559 
 ; 0941 1199556 
 1 09440J99563 
 1094(1999.551 
 j 09498|99548 
 i 09527199646 
 
 i 0955(JJ99642 31 
 09685199540 
 09614199537 
 09()42;99534 
 0?>67 199531 
 09700199528 
 09729 99526 
 06< 68 99523 
 09787 9!)520 
 0981(,9951 
 09816 99514 
 09874 99511 
 09903 9f)508 
 09932 9950 
 0996199503 
 0.>99t)995U0 
 10019 99497 
 10i>48 99494 
 10077 99491 
 1010:)994.S8 
 10135 99485 
 10164 99482 
 10192 994,9 
 10221994/6 
 10-60 99473 
 10279 99470 
 10308 99467 
 10337199464 
 10366 9£Kil 
 10395 99458 
 10424 99,55 
 10453 991.52 
 
 Tan-?. : N. cop. N. sine. 
 
 St Df^srreeR. 
 
 -j 
 
TABLE II. 
 
 Log. Sinee and Taugente. (6^) Natural Sines. 
 
 Tang. iD. 10"| Cotang. iN. sine. 
 
 
 
 1 
 Q 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 \Q 
 
 i i: 
 1-: 
 
 13 
 
 14 
 15 
 l(i| 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2d 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 36 
 3(i 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 40 
 47 
 48 
 19 
 50 
 51 
 52 
 63 
 64 
 6.- 
 6- 
 67 
 58 
 58 
 (>0 
 
 ID. 10' 
 
 .019235 
 020435 
 021()32 
 022825 
 0241)1 () 
 025203 
 02()38t» 
 C>275()7 
 028744 
 029918 
 031089 
 032257 
 ^31421 
 /J4582 
 035741 
 03()89G 
 038048 
 039197 
 040342 
 041485 
 042(>25 
 
 .043702 
 044895 
 04(i02() 
 047154 
 048279 
 049400 
 050519 
 051035 
 052749 
 053859 
 
 . 054900 
 050071 
 057172 
 058271 
 059307 
 (Ki04(i0 
 001651 
 0;)2(i39 
 0(i3724 
 0()480(i 
 
 .005885 
 0t>«)902 
 008030 
 0)9107 
 070170 
 071242 
 0-2300 
 0733(;() 
 074424 
 0754.^0 
 
 I 070533 
 0i7583 
 078031 
 079070 
 O.S0719 
 08^59 
 0.vii97 
 083832 
 0818()4 
 085«94 
 
 CosinH. 
 
 200 
 199 
 199 
 198 
 198 
 197 
 197 
 196 
 190 
 195 
 195 
 194 
 194 
 193 
 192 
 192 
 191 
 191 
 190 
 190 
 189 
 189 
 180 
 188 
 187 
 187 
 186 
 186 
 185 
 185 
 184 
 184 
 184 
 183 
 183 
 182 
 182 
 181 
 181 
 180 
 l.'^O 
 179 
 171- 
 179 
 178 
 178 
 177 
 177 
 176 
 176 
 175 
 175 
 175 
 174 
 174 
 173 
 
 i;3 
 
 li2 
 172 
 172 
 
 Cosine. D. 10' 
 
 9.997614 
 997601 
 997588 
 997574 
 997561 
 997547 
 997534 
 997620 
 997507 
 997493 
 W97480 
 
 9.997466 
 997452 
 997439 
 997425 
 -^»7411 
 997397 
 997383 
 997309 
 997355 
 997341 
 
 9.997327 
 997313 
 997299 
 997286 
 997271 
 997267 
 997242 
 997228 
 997214 
 997199 
 997185 
 997170 
 997166 
 997141 
 997127 
 997112 
 997098 
 997083 
 997008 
 997(>53 
 .997039 
 997024 
 997009 
 99109941 
 996979 j 
 996964 ] 
 990949 ! 
 990934 
 990919 
 996J»04 
 
 9.990889 
 990874 
 996858 
 990843 
 996828 
 996812 
 <n)()797 
 99()782 
 99()766 
 996751 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.3 
 
 2.3 
 
 2.3 
 
 2-3 
 
 2-3 
 
 2.3 
 
 2-3 
 
 2.3 
 
 2.3 
 
 2.3 
 
 2-3 
 
 2-3 
 
 2-3 
 
 23 
 
 2.3 
 
 2.4 
 
 2-4 
 
 2.4 
 
 2.4 
 
 2-4 
 
 2.4 
 
 2.4 
 
 2.4 
 
 2.4 
 
 2-4 
 
 2-4 
 
 2.4 
 
 2.4 
 
 2-4 
 
 2.4 
 
 2.4 
 
 2.4 
 
 2.6 
 
 2.5 
 
 2 
 
 2 
 
 I 2 
 
 l2 
 
 ; 2 
 2 
 
 2-6 
 2.6 
 2.6 
 3.6 
 2.5 
 2.6 
 2.6 
 2.5 
 2.5 
 2.6 
 2 6 
 2 
 2 6 
 2 6 
 
 Sine. 
 
 9.021620 
 022834 
 024044 
 025251 
 02(i455 
 027055 
 028852 
 030046 
 031237 
 032425 
 033009 
 034791 
 035969 
 037144 
 038316 
 039485 
 040051 
 041813 
 042973 
 044130 
 046284 
 
 9.04(^434 
 047582 
 048727 
 049869 
 051008 
 052144 
 053277 
 054407 
 055535 
 05<)(>59 
 067781 
 058900 
 000016 
 001130 
 002240 
 01)3348 
 004453 
 005556 
 006666 
 067762 
 
 i9.(>68846 
 (169038 
 071027 
 072113 
 073197 
 074278 
 075356 
 070432 
 077505 
 078576 
 
 9.079044 
 080710 
 081773 
 082833 
 083891 
 084i)47 
 t)8(i000 
 087050 
 088098 
 089144 
 Cotang. 
 
 202 
 202 
 201 
 201 
 200 
 199 
 199 
 198 
 198 
 197 
 197 
 196 
 190 
 196 
 196 
 194 
 194 
 193 
 193 
 192 
 192 
 191 
 191 
 190 
 190 
 189 
 189 
 188 
 188 
 187 
 187 
 186 
 186 
 186 
 186 
 185 
 184 
 184 
 183 
 183 
 182 
 182 
 181 
 181 
 181 
 180 
 IbO 
 179 
 179 
 178 
 178 
 178 
 177 
 177 
 176 
 176 
 175 
 175 
 176 
 174 
 
 10.978380 
 977100 
 975950 
 974749 
 973545 
 973345 
 971148 
 909954 
 908763 
 967675 
 966391 
 
 10.966209 
 904031 
 9()2850 
 901()84 
 900616 
 959349 
 958187 
 957027 
 955870 
 954716 
 
 10.953560 
 952418 
 951273 
 960131 
 948992 
 947850 
 94^i723 
 945593 
 944405 
 943341 
 
 10.942219 
 941100 
 939984 
 938870 
 937760 
 936652 
 935547 
 934444 
 933345 
 932248 
 
 10.935154 
 930062 
 928973 
 927887 
 926803 
 925722 
 9-2'i644 
 923668 
 922496 
 921424 
 
 10.920356 
 919290 
 918227 
 917107 
 9l(;i09 
 915053 
 914000 
 912950 
 911J02 
 91086() 
 Tang."" 
 
 0453 
 Olh'J 
 0511 
 054(1 
 05()9 
 0597 
 062() 
 0()65 
 0')84 
 vAI3 
 0742 
 0771 
 08<H) 
 0829 
 
 N. cog. : 
 
 19452 . 60 
 
 ;!)449:59 
 <):)44() 58 
 99443 67 
 )9440 I 5() 
 99437 I 65 
 9i)434 64 
 99431 I 63 
 9^-428 u-2 
 99424 I 61 
 99421160 
 9y418 ' 49 
 1^9416:48 
 99412147 
 0858{9y4W) ; 46 
 <J887I9940()I45 
 
 09 1() 
 0945 
 0!)73 
 
 99402 1 44 
 99399 ! 43 
 99396 42 
 1002199393 i 41 
 1031199390 40 
 1000;99a86'39 
 1089199383 38 
 1118199380137 
 1147 99377! 36 
 1170 99374135 
 1205|993T0 34 
 1234;{'9367i33 
 1263!J)9364 32 
 129ll99300:31 
 1320l99i57 130 
 
 1349 
 1378 
 1407 
 143t) 
 1465 
 1494 
 1523 
 1552 
 1.580 
 1609 
 1038 
 1()67 
 169() 
 1726 
 1754 
 1783 
 1812 
 1840 
 18()9 
 189K 
 1927 
 195(. 
 19vS5 
 2014 
 204;i 
 2071 
 '21()(( 
 212!) 
 2158 
 218; 
 
 99354 ! 29 
 99351 : 28 
 )9347 i 27 
 >9o44 ; 20 
 99341 25 
 99337 i 24 
 99334 23 
 99331 I ii2 
 95)327 i 
 99324 I 
 99320 I 
 99317 I 
 99314 
 99310' 
 99307 
 99303 
 f)9300 
 
 99297 
 )9293 
 99290 
 99286 
 9{)283 
 (9279 
 99276 
 )9272 
 )92()9 
 992 .5 
 ii92{i2 
 99268 
 >f)255 
 
 N COS. N.sinf. 
 
 83 D«irree-g. 
 
28 
 
 Log, Sines and Tangents. (7°) Natural Sines. 
 
 TABLE II. 
 
 Sine. 
 9.085894 
 
 08794 7 
 088970 
 089990 
 09IO;)8 
 ^•2024 
 093037 
 094047 
 09505t) 
 096062 
 
 9.097065 
 098056 
 099066 
 100062 
 101056 
 102048 
 103037 
 104025 
 105010 
 105992 
 
 9.106973 
 107951 
 108927 
 109901 
 110873 
 111842 
 112809 
 113774 
 114737 
 115698 
 1 16656 
 117613 
 118567 
 119519 
 120469 
 121417 
 122362 
 123306 
 124248 
 125187 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 b 
 
 7 
 
 8 
 
 9 
 10 
 11 
 12 
 13 
 14 
 16 
 16 
 17 
 18 
 19 
 ^20 
 21 
 22 
 23 
 24 
 26 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 19.126126 
 
 42 I 
 
 t3l 
 
 44 
 
 45 I 
 
 46 
 
 47 
 
 48 
 
 4P 
 
 51 i 
 
 51 9 
 
 52 
 
 53 
 
 64 
 
 65 
 
 66 
 
 67 
 
 6S 
 
 59 
 
 60 
 
 127060 
 127993 
 128^26 
 129tt64 
 130781 
 131706 
 132630 
 133551 
 1344/0 
 
 . 135387 
 136303 
 137216 
 138128 
 139037 
 139944 
 140850 
 141754 
 142655 
 143556 
 
 Cosit!^. 
 
 w-w 
 
 171 
 171 
 170 
 170 
 170 
 169 
 169 
 168 
 168 
 168 
 167 
 167 
 166 
 166 
 166 
 166 
 166 
 164 
 164 
 164 
 163 
 163 
 163 
 162 
 162 
 162 
 161 
 161 
 160 
 160 
 160 
 159 
 169 
 169 
 168 
 168 
 168 
 157 
 167 
 157 
 166 
 166 
 166 
 156 
 166 
 164 
 164 
 154 
 153 
 153 
 153 
 152 
 162 
 152 
 152 
 151 
 151 
 151 
 160 
 150 
 
 Cosiuf. 
 
 9.996761 
 996735 
 996720 
 996704 
 996688 
 996673 
 996657 
 996641 
 996626 
 996610 
 996594 
 
 9.996678 
 996562 
 996546 
 996530 
 996514 
 996498 
 996482 
 996466 
 996449 
 996433 
 996417 
 996400 
 996384 
 996368 
 996361 
 996335 
 996318 
 996302 
 996286 
 996269 
 
 9.996252 
 996236 
 996219 
 996202 
 996186 
 996168 
 996161 
 996134 
 996117 
 996100 
 99()083 
 996066 
 996049 
 996032 
 996015 
 996998 
 995980 
 995963 
 99594() 
 995928 
 995911 
 995894 
 995876. 
 995859 
 995841 
 995823 
 995806 
 995788 
 995771 
 995753 
 
 Sine. 
 
 D. lu' 
 
 2.0 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.6 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.7 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.8 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 2.9 
 
 Tang. 
 
 .089144 
 090187 
 091228 
 092266 
 093302 
 094336 
 095367 
 096395 
 097422 
 098446 
 099468 
 
 .IOL'487 
 101504 
 102519 
 103532 
 104542 
 105650 
 106656 
 107559 
 108660 
 109559 
 
 .110556 
 111661 
 112643 
 11363a 
 114621 
 115507 
 116491 
 117472 
 118462 
 119429 
 
 . 120404 
 121377 
 122348 
 123317 
 124284 
 125249 
 126211 
 127172 
 128130 
 129087 
 
 .130041 
 130994 
 131944 
 132893 
 133839 
 134(84 
 136(26 
 136 567 
 137d06 
 138542' 
 1894/6 
 140409 
 141340 
 142269 
 143196 
 144121 
 145044 
 145966 
 146885 
 14/803 
 
 Lotiing. 
 
 174 
 173 
 173 
 173 
 172 
 172 
 171 
 171 
 171 
 170 
 170 
 169 
 169 
 169 
 168 
 168 
 168 
 167 
 167 
 166 
 166 
 166 
 166 
 166 
 165 
 164 
 164 
 164 
 163 
 163. 
 162 
 162 
 162 
 161 
 161 
 161 
 160 
 160 
 160 
 159 
 159 
 159 
 158 
 158 
 158 
 157 
 167 
 157 
 166 
 166 
 166 
 156 
 155 
 155 
 154 
 154 
 154 
 153 
 153 
 163 
 
 C(4aiig. 
 
 10.910856 
 909813 
 908772 
 907734 
 906698 
 905664 
 904633 
 903605 
 902578 
 901564 
 900532 
 
 10.899613 
 898496 
 897481 
 896468 
 896458 
 894450 
 893444 
 892441 
 891440 
 890441 
 
 10.889444 
 888449 
 887467 
 886467 
 885479 
 884493 
 883509 
 882628 
 881548 
 880571 
 
 10.879596 
 878623 
 877652 
 876683 
 876716 
 874751 
 873789 
 872828 
 871870 
 870913 
 
 10.869959 
 86900P 
 868056 
 867107 
 866161 
 866216 
 864274 
 8633;i3 
 862396 
 861458 
 
 10.860624 
 859591 
 868660 
 857731 
 856804 
 855879 
 854956 
 854034 
 853116 
 852197 
 
 N. siiii! 
 
 1218/- 
 
 12216 
 
 ; 12245 
 
 12274 
 
 12302 
 
 12331 
 
 12360 
 
 I 12389 
 
 : 12418 
 
 i 12447 
 
 ! 12476 
 
 I 12504 
 
 12633 
 
 99255 
 99251 
 99248 
 99244 
 99240 
 99237 
 99233 
 99230 
 9226 
 99222 
 99219 
 99215 
 99211 
 125(i2 199208 
 12.591 199204 
 12620^99200 
 
 12649 
 12678 
 12706 
 12735 
 12764 
 12793 
 12822 
 12851 
 12880 
 
 99197 
 99193 
 99189 
 99186 
 99182 
 99178 
 99176 
 99171 
 99167 
 
 12908 99163 
 
 12937 
 129(]6 
 12995 
 13024 
 13053 
 13081 
 13110 
 13139 
 13168 
 13197 
 13226 
 13254 
 
 99160 
 99156 
 99152 
 99148 
 99144 
 99141 
 99137 
 99133 
 99129 
 99125 
 99122 
 99118 
 
 13283199114 
 13312 99110 
 
 13341 
 
 99106 
 
 13370 99102 
 13(;99 199098 
 
 1:13427 
 
 I i 1345b 
 
 i 13i85 
 
 ; 13514 
 
 l! 13543 
 
 |il35'/2 
 
 13(00 
 
 13629 
 
 13658 
 
 13687 99059 
 13711.99055 
 
 Tang. 
 
 13744 
 137-3 
 
 13802 
 13831 
 13860 
 13889 
 1391. 
 N. ros. 
 
 99094 
 99091 
 99087 
 99083 
 990/9 
 990*5 
 J90/1 
 99067 
 99063 
 
 99051 
 i9047 
 J9043 
 99039 
 99035 
 1903 1 
 j9027 
 
 82 Degrees. 
 
Log. Sines and Tangents. (8°) Natural Sii 
 
 29 
 
 
 
 1 
 
 3 
 4 
 6 
 6 
 
 7 
 8 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 in 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 3-2 
 
 33 
 
 34 
 
 35 
 
 3<i 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 46 
 
 4(3 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 62 I 
 
 53 
 
 54 
 
 55 
 
 56 
 
 51 
 
 58 
 
 69 
 
 60 
 
 bine. 
 
 43555 
 44453 
 45349 
 l4(i24J 
 47136 
 48026 
 48915 
 49802 
 5U()86 
 51,>69 
 52451 
 53.i30 
 54208 
 55i)83 
 55957 
 56830 
 57;O.J 
 58569 
 59435 
 60301 
 61164 
 62026 
 62885 
 63743 
 [64600 
 (J5454 
 66307 
 67159 
 l()8008 
 68856 
 69702 
 70547 
 71389 
 72230 
 73070 
 73908 
 74744 
 755 i^8 
 7«)411 
 77242 
 78072 
 78900 
 79726 
 80551 
 181374 
 82196 
 83016 
 83834 
 84651 
 854l)(i 
 8()280 
 8/092 
 87903 
 88712 
 895 19 
 90326 
 91130 
 191933 
 92734 
 93534 
 94332 
 Cos in I'. 
 
 D. 10' 
 
 150 
 149 
 149 
 149 
 148 
 148 
 148 
 147 
 147 
 147 
 147 
 146 
 146 
 146 
 145 
 145 
 145 
 144 
 144 
 144 
 144 
 143 
 143 
 143 
 142 
 142 
 142 
 142 
 141 
 141 
 141 
 140 
 140 
 140 
 140 
 139 
 139 
 139 
 139 
 138 
 138 
 138 
 137 
 137 
 137 
 137 
 136 
 136 
 136 
 136 
 136 
 135 
 135 
 135 
 134 
 134 
 134 
 134 
 133 
 133 
 
 Cosine. |D. 10" 
 
 .995753 
 
 9fJ5735 
 
 995717 
 
 995()99 
 
 995681 
 
 995(ili4 
 
 995646 
 
 995(i28 
 
 995610 
 
 995591 
 
 995573 
 1.995556 
 
 995o37 
 
 995619 
 
 995601 
 
 995482 
 
 995464 
 
 995446 
 
 995427 
 
 995409 
 
 995390 
 .995372! 
 
 995353 ; 
 
 J95334 , 
 
 995316, 
 
 995297 
 
 995278 
 
 995260 
 
 995241 ! 
 
 995222 
 
 995203 
 
 995184! 
 
 995165 
 
 995146, 
 
 995127} 
 
 995108 
 
 995089 
 
 99.5070 
 
 995051 
 
 995032 
 
 995013 
 .994993 
 
 994974 
 
 994955 
 
 994935 
 
 994916 
 
 994896 
 
 994877 
 
 994857 
 
 994838 
 
 994818 
 .994i98 
 
 994779 
 
 994769 
 
 994739 
 
 994719 
 
 994700 
 
 994(i80 
 
 994660 
 
 99 4640 
 
 994620 
 ~Sine! 
 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.1 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 3.3 
 
 Tang. 
 
 .147803 
 148718 
 149(i32 
 150544 
 151454 
 152363 
 1.532(i9 
 154174 
 156077 
 155978 
 156877 
 
 .157776 
 16867 1 
 159565 
 160457 
 161347 
 162236 
 163123 
 164008 
 164892 
 165774 
 
 . 166654 
 167532 
 1684<J9 
 169284 
 17015/ 
 1710-29 
 171899 
 17276/ 
 173634 
 174499 
 
 .175362 
 176224 
 177084 
 177942 
 178799 
 179655 
 180508 
 181360 
 182211 
 183059 
 
 . 183907 
 184752 
 185597 
 18li439 
 187280 
 188120 
 188958 
 189794 
 190629 
 191462 
 
 . 192294 
 193124 
 193953 
 194780 
 196()06 
 196430 
 19;253 
 198074 
 198894 
 199713 
 
 Colang. 
 
 D. 10' 
 
 153 
 152 
 152 
 152 
 151 
 151 
 151 
 150 
 150 
 150 
 IlO 
 149 
 149 
 149 
 148 
 148 
 148 
 148 
 147 
 147 
 147 
 146 
 146 
 146 
 145 
 145 
 145 
 145 
 144 
 144 
 144 
 144 
 143 
 143 
 143 
 142 
 142 
 142 
 142 
 141 
 141 
 141 
 141 
 140 
 140 
 140 
 14(.» 
 139 
 139 
 139 
 139 
 138 
 138 
 138 
 138 
 137 
 137 
 1^7 
 137 
 136 
 
 Cotang. I N. sine.lN. c««.| 
 
 10.852197 
 851282 
 850368 
 849456 
 848546 
 847637 
 846731 
 845826 
 844923 
 844022 
 843123 
 
 10 842225; 
 841329 
 840435 
 839543 
 838653 : 
 837764 ; 
 836877! 
 835992 
 835108 
 834226 
 
 10.833346 
 
 832468 
 
 831591 
 
 830716 
 
 829843 
 
 828971 
 
 828101 ' 
 
 827233 ' 
 
 8263661 
 
 825501 ' 
 
 10.824638 I 
 823776! 
 822916 
 822058! 
 821^01 
 820345 ' 
 819492; 
 8186401 
 817789 
 816941 [ 
 
 10-816093 
 8152481 
 814403! 
 813561 ' 
 812720 • 
 811880 
 811(542' 
 81020() 
 809371 
 80o538 , 
 
 10.807706', 
 806876 i I 
 806047; 
 805220 ' 
 804394 
 803570 i 
 802747 ' 
 801926 
 801106 
 800287 
 
 3917|99027 60 
 394(i; 99023 69 
 39/599019 58 
 40m!99015 67 
 4033;9i>()ll 66 
 4061 !»90iH) 55 
 4090 99002 54 
 4119198998 53 
 4148198994 62 
 41 77 198990 51 
 42059H986 50 
 42; ;4. 98! 182 \ 49 
 4263 9S978 | 48 
 -I2;i2'98973 | 47 
 43209h9(i9 46 
 4349i989()5 
 4378'98961 
 440/198957 
 
 443()98953 
 4464 98948 
 
 4493 
 4522 
 4561 
 4580 
 4608 
 4637' 
 
 98944 
 9h940 
 98936 
 98931 37 
 9h927 36 
 98923 ' 35 
 4666198919; 34 
 4695198914 133 
 4723i98910!32 
 
 4752 
 
 4781 
 
 4810 
 
 4838 
 
 48(>7 
 
 489() 
 
 4925 
 
 4954 
 
 4982 
 
 5011 
 
 5040 
 
 5069 
 
 5097 
 
 6126 
 
 5155 
 
 5184 
 
 5212 
 
 5241 
 
 5270 
 
 5299 
 
 532 
 
 6356 
 
 5385 
 
 5414 
 
 5442 
 
 5471 
 
 5500 
 
 5529 
 
 5657 
 
 6586 
 
 5615 
 
 5643 
 
 98906131 
 )b902 ! 30 
 9o897 ' 29 
 98«93 ' 28 
 9b8b9 \ 27 
 98884 ! 26 
 98880 • 25 
 L>8876 24 
 
 Tang. 
 
 J8871 
 98867 
 988()3 
 9h858 20 
 9hh64 19 
 98849! 18 
 98845' 17 
 98841 I 16 
 J8836 i 16 
 98832 i 14 
 98827 I 13 
 98823 ' 12 
 
 98818 
 98814 
 98809 
 98805 
 98800 
 i879«i 
 98791 
 J8787 
 98782 
 98778 
 98773 
 y8769 
 
 N. COS. N.fline. 
 
 81 Degrees. 
 
'30 
 
 Log, Sines and Tangents. (9°) Natural Sines. 
 
 TABm II. 
 
 Sine. D. 10' 
 
 7 
 8 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 2.3 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 19 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 ' 54 
 ■ 55 
 i 56 
 l57 
 I 58 
 
 59 
 
 60 
 
 .194332 
 195129 
 19.5925 
 196719 
 197511 
 19830-2 
 199091 
 1998 79 
 20i).)66 
 201451 
 2U-223-i 
 
 .2J3017 
 203797 
 204577 
 205354 
 2C»613l 
 206906 
 207679 
 208452 
 209222 
 209992 
 
 .210760 
 211526 
 212291 
 2130.55 
 213818 
 214579 
 21.5338 
 216097 
 216854 
 217609 
 
 .218363 
 219116 
 219868 
 220618 
 221367 
 222115 
 2228<)1 
 223606 
 224349 
 225092 
 
 .225833 
 226573 
 227311 
 228048 
 228784 
 229518 
 230252 
 230984 
 231714 
 232444 
 
 .233172 
 233899 
 234625 
 235349 
 236073 
 236795 
 237515 
 238235 
 238953 
 239670 
 
 Co.sine. 
 
 133 
 133 
 132 
 132 
 132 
 132 
 131 
 131 
 131 
 131 
 130 
 130 
 130 
 130 
 129 
 129 
 129 
 129 
 128 
 128 
 128 
 128 
 127 
 127 
 127 
 127 
 127 
 126 
 126 
 126 
 126 
 125 
 125 
 125 
 125 
 125 
 124 
 124 
 124 
 124 
 123 
 123 
 123 
 123 
 123 
 122 
 122 
 122 
 122 
 122 
 121 
 121 
 121 
 121 
 120 
 120 
 120 
 120 
 120 
 119 
 
 Co.sine. D. lU 
 
 .994620 
 
 994600 
 
 994580 
 
 994560 
 
 994540 
 
 994519 
 
 994499 
 
 994479 
 
 994459 
 
 994438 
 
 99W18 
 9.994397 
 
 994377 
 
 994357 
 
 994336 
 
 994316 
 
 994295 
 
 994274 
 
 994254 
 
 994233 
 
 991212 
 ,994191 
 
 994171 
 
 994150 
 
 994129 
 
 994108 
 
 994087 
 
 994066 
 
 994045 
 
 994024 
 
 994003 
 ,993981 
 
 993960 
 
 993939 
 
 993918 
 
 993896 
 
 993875 
 
 993854 
 
 993832 
 
 993811 
 
 993789 
 9.993768 
 
 993746 
 
 993725 
 
 993703 
 
 993681 
 
 993660 
 
 993638 
 
 993616 
 
 993594 
 
 9935-2 
 
 993550 
 
 99^528 
 
 99^506 
 
 993484 
 
 993462 
 
 9.J3440 
 
 993418 
 
 993396 
 
 993374 
 
 993351 
 
 Sine. 
 
 3.3 
 3.3 
 3.3 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.4 
 3.5 
 3.5 
 3.6 
 3.5 
 3.6 
 3.5 
 3.6 
 3.6 
 3.6 
 3.5 
 3.6 
 3.6 
 3.5 
 3.5 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.6 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 
 Tauu 
 
 9.199713 
 
 20)529 
 201345 
 202159 
 20-2971 
 203782 
 204592 
 205400 
 206207 
 207013 
 207817 
 
 9.208619 
 209420 
 210220 
 211018 
 211815 
 212611 
 213405 
 214198 
 214989 
 215780 
 
 9.216568 
 217356 
 218142 
 218926 
 219710 
 220492 
 221272 
 222052 
 222830 
 223ti06 
 
 9.224382 
 225156 
 225929 
 226700 
 227471 
 228239 
 229007 
 229773 
 230539 
 231302 
 
 9.232066 
 232826 
 233586 
 234345 
 235103 
 235859 
 236614 
 237368 
 238120 
 238872 
 
 9.239622 
 240371 
 241118 
 241866 
 242610 
 243354 
 244097 
 244839 
 245579 
 246319 
 Cotiing. 
 
 D. 10' 
 
 136 
 136 
 136 
 135 
 135 
 136 
 135 
 134 
 134 
 134 
 134 
 133 
 133 
 133 
 133 
 133 
 132 
 132 
 132 
 132 
 131 
 131 
 131 
 131 
 130 
 130 
 130 
 130 
 130 
 129 
 129 
 129 
 129 
 129 
 128 
 128 
 128 
 128 
 127 
 127 
 127 
 127 
 127 
 126 
 126 
 126 
 126 
 126 
 125 
 126 
 126 
 126 
 126 
 124 
 124 
 124 
 124 
 124 
 123 
 123 
 
 Cotang. !;N. sine. N. CO.X 
 
 10.800287 
 799471 
 798655 
 797841 
 797029 
 796218 
 795408 
 794(i00 
 793793 
 792987 
 792183 
 
 10.791381 
 790580 
 789780 
 788982 
 788185 
 787389 
 786595 
 785802 
 735011 
 784220 
 
 10.783432 
 782«)44 
 7818.58 
 781074 
 780290 
 779508 
 778728 
 777948 
 777170 
 776394 
 
 10.775618 
 774844 
 774071 
 773300 
 772529 
 771761 
 770993 
 770227 
 769461 
 7(>8698 
 
 10.767936 
 767174 
 766414 
 766656 
 764897 
 764141 
 763386 
 762632 
 761880 
 761128 
 
 10.760378 
 759629 
 758882 
 758135 
 767390 
 756646 
 755903 
 755161 
 754421 
 753681 
 Tang. 
 
 15643 
 
 15672 
 
 15701 
 
 15730 
 
 15758 
 
 I 15787 
 
 ! 15810 
 
 15845 
 
 i 15873 
 
 i 15902 
 
 15931 
 
 I 15959 
 
 15988 
 
 16017 
 
 ; 16046 
 
 1 16074 
 
 I 16103 
 
 116132 
 
 i 16160 
 
 98769 
 98764 
 98760 
 98755 
 98751 
 9874() 
 98741 
 )8737 
 98732 
 98728 
 98723 
 98718 
 98714 
 98709 
 
 98704 46 
 
 98700 
 
 98695 
 
 9H690 
 
 98686 
 
 116189 98681 
 
 ! 16218;98676 
 
 i 16246198671 
 
 16275 '98667 
 
 16304 98()62 
 
 16333 98667 
 
 ; 16361 9H652 
 
 ; 16390 98648 
 
 116419 98643 
 
 ; 16447i98638 
 
 16476198633 
 
 116505 98629 
 
 16533!98624 
 
 16562 98619 
 
 16591198614 
 
 1662098609 
 
 l()648i98604 
 
 16677198600 
 
 16706i98595 
 
 16734 ""590122 
 
 16763:.48o85 21 
 
 16792 98580 20 
 
 116820 98576 19 
 
 16849 98570 
 
 16878 98565 
 
 16906 98561 
 
 1 16935 98556 
 
 i 16964 98551 
 
 ,16992 98546 
 
 11702198541 
 
 I 17050 98536 
 
 1117078 98531 
 
 {17107 98526 
 
 ii 17136198521 
 
 i 1716498516 
 
 'ini93|9S5ll 
 
 17222 98606 
 
 !l7250'98501 
 
 ,! 17279198496 
 
 17308:98491 
 
 i 17336198486 
 
 1 17365198481 
 
 Ij N. cos.lN.sine. 
 
 80 Degrees. 
 
TABLE 11. 
 
 Log. Sinefl and Tangents. (10°) Natarnl Sines. 
 
 31 
 
 
 
 1 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 10 
 II 
 1-2 
 13 
 14 
 16 
 Iti 
 17 
 18 
 
 ly 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 2t) 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 3o 
 
 3u 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 4!» 
 
 60 
 
 51 1.9 
 
 62 
 
 53 
 
 54 
 
 65 
 
 o(i 
 
 57 
 
 58 
 
 59 
 
 60 
 
 Sine. 
 
 .239670 
 240386 
 241101 
 241814 
 242526 
 243237 
 243947 
 2-W(i5l) 
 245363 
 246069 
 246775 
 247478 
 248181 
 218H83 
 249.JH3 
 250JH2 
 250980 
 251677 
 252373 
 2530o7 
 253761 
 
 .254453 
 255144 
 255ri34 
 251)523 
 257211 
 25 7898 
 25h583 
 2592()8 
 259951 
 260t.33 
 
 .261314 
 261994 
 262(i73 
 263351 
 264027 
 264703 
 265377 
 26t>051 
 266723 
 267395 
 
 . 268065 
 268734 
 269402 
 2i00t)9 
 270.35 
 271400 
 272064 
 272726 
 273388 
 274049 
 
 .374i08 
 276367 
 276024 
 276681 
 27733 7 
 277991 
 278644 
 279297 
 279948 
 280599 
 
 OMioe. 
 
 ir| Cosine. 
 
 119 
 119 
 119 
 119 
 118 
 118 
 118 
 118 
 118 
 117 
 117 
 117 
 117 
 117 
 116 
 116 
 116 
 116 
 116 
 116 
 115 
 115 
 115 
 115 
 115 
 114 
 114 
 114 
 114 
 114 
 113 
 113 
 113 
 113 
 113 
 113 
 112 
 112 
 112 
 112 
 112 
 HI 
 111 
 111 
 111 
 HI 
 111 
 110 
 HO 
 110 
 110 
 110 
 110 
 109 
 109 
 109 
 109 
 109 
 109 
 108 
 
 9.993351 
 993329 
 993307 
 993285 
 993262 
 993240 
 993217 
 993195 
 993172 
 993149 
 993127 
 
 9.993104 
 993081 
 993U59 
 99303t) 
 993013 
 992990 
 992967 
 992944 
 992921 
 992898 
 .992875 
 992852 
 992829 
 992806 
 992783 
 992759 
 992736 
 992713 
 992690 
 992666 
 .992')43 
 992619 
 992596 
 992572 
 992549 
 992525 
 992501 
 992478 
 992454 
 992430 
 .992406 
 992382 
 992359 
 992335 
 992311 
 992287 
 992263 
 -992239 
 992214 
 992190 
 
 9.992166 
 992142 
 992117 
 992093 
 992069 
 992044 
 9920.20 
 991996 
 991971 
 991947 
 
 Sine. 
 
 D. 10" 
 
 3.7 
 
 3.7 
 3.7 
 3.7 
 3.7 
 3.7 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 3.8 
 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 3.9 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.0 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 
 9. 
 
 Tang. 
 
 ,216319 
 247057 
 217794 
 218630 
 249264 
 249998 
 250730 
 261461 
 252191 
 252920 
 263648 
 
 .254374 
 256100 
 265824 
 256547 
 257269 
 257990 
 258710 
 259429 
 260146 
 260863 
 
 .261578 
 262292 
 263005 
 263717 
 264428 
 265138 
 265847 
 26«)555 
 267261 
 267967 
 2<i8671 
 269375 
 270077 
 270779 
 271479 
 272178 
 272876 
 273573 
 274269 
 274964 
 275658 
 276351 
 277043 
 277734 
 278424 
 279113 
 279801 
 2«0488 
 281174 
 281858 
 282542 
 283225 
 283907 
 284588 
 285268 
 285947 
 286<J24 
 287301 
 287977 
 288652 
 
 Co tang. 
 79 Degree! 
 
 D. 10"' Cotang. iiN.(«ine.|N. oot:. 
 
 123 
 123 
 123 
 122 
 122 
 122 
 122 
 122 
 121 
 121 
 121 
 121 
 121 
 120 
 120 
 120 
 120 
 120 
 120 
 119 
 119 
 119 
 119 
 119 
 118 
 118 
 118 
 118 
 118 
 118 
 117 
 117 
 117 
 117 
 117 
 116 
 116 
 116 
 116 
 116 
 116 
 115 
 115 
 116 
 116 
 115 
 116 
 114 
 114 
 114 
 114 
 114 
 114 
 113 
 113 
 113 
 113 
 113 
 113 
 112 
 
 10.753681 
 752943 
 752206 
 761470 
 760736 
 760002 
 749270 
 748539 
 747809 
 747080 
 746352 
 
 10.745(i26 
 744900 
 744176 
 743453 
 742731 
 742010 
 741290 
 740671 
 739854 
 739137 
 
 10.738422 
 737708 
 736995 
 736283 
 735572 
 7348(>2 
 734153 
 733445 
 732739 
 732033 
 
 10.731329 
 730625 
 729923 
 729221 
 728621 
 727822 
 727124 
 726427 
 726731 
 725036 
 
 10.724342 
 723649 
 722957 
 722266 
 721576 
 720887 
 720199 
 719612 
 718826 
 718142 
 
 10.717458 
 716775 
 716093 
 716412 
 714732 
 714053 
 713376 
 712699 
 712(^23 
 711348 
 
 I 17366'9K481 
 P393 98476 
 17422 98471 
 174 
 
 II 174 
 
 I; ITP 
 
 18138 98341 
 
 18166 98336 
 
 18195 98331 
 
 1822498325 
 
 18252i98320 
 
 18281 198315 
 
 1830998310 
 
 18338 98304 
 
 18367 98299 
 
 18396198294 
 
 18424i982fe8 
 
 18452198283 
 
 1848198277 
 
 18509|98272 
 
 1 1 18638:98267 
 
 1856798261 
 
 18595 98256 
 
 18(124 98250 
 
 18652 98245 
 
 1868 1198240 
 
 1871098234 
 
 18738 98229 
 
 ! 18767 98223 
 
 : 18795 98218 
 
 ! 1882498212 
 
 1 18852 98207 
 
 [18881198201 
 
 118910 98196 
 
 118938,98190 
 
 ! 18967198185 
 
 1 18995:98179 
 
 j 19024 98174 
 
 Tang. 
 
 19052 
 19081 
 
 98168 
 98163 
 
 N. COP. N.fine. 
 
32 
 
 Log, Sines and Tangents. (11°) Natural Sines. 
 
 TABLK II. 
 
 
 
 I 
 
 2 
 3 
 4 
 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 2; 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 
 3e 
 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 61 
 52 
 63 
 54 
 55 
 56 
 57 
 68 
 59 
 60 
 
 Sine. 
 
 .280599 
 281248 
 281897 
 282644 
 283190 
 283836 
 284480 
 285124 
 285/66 
 286408 
 287048 
 
 .287687\ 
 288326 
 288964 
 289600 
 290236 
 290870 
 291504 
 292137 
 292768 
 293399 
 
 .294029 
 294658 
 295286 
 295913 
 290539 
 297164 
 297788 
 298412 
 299034 
 299«55 
 
 .300276 
 300895 
 301514 
 302132 
 302748 
 303364 
 303979 
 304593 
 305207 
 305819 
 
 .306430 
 307041 
 307650 
 308259 
 308867 
 309474 
 310080 
 310685 
 311289 
 311893 
 
 .312495 
 313097 
 313698 
 314-.;97 
 31489/ 
 315495 
 316092 
 316689 
 317284 
 317879 
 
 Cosine. ' 
 
 D. W 
 
 lOS 
 108 
 108 
 108 
 108 
 107 
 107 
 107 
 107 
 107 
 107 
 106 
 106 
 106 
 106 
 106 
 106 
 105 
 105 
 105 
 106 
 105 
 105 
 104 
 104 
 104 
 104 
 104 
 104 
 104 
 103 
 103 
 103 
 103 
 103 
 103 
 102 
 102 
 102 
 102 
 102 
 102 
 102 
 101 
 101 
 101 
 101 
 101 
 101 
 100 
 100 
 100 
 100 
 100 
 100 
 100 
 100 
 99 
 99 
 99 
 
 U.S. UP. 
 
 .991947 
 991922 
 9;' 1897 
 991873 
 991848 
 991823 
 991799 
 991774 
 991749 
 991/24 
 991699 
 
 1.991674 
 991649 
 991624 
 991599 
 991574 
 991549 
 991524 
 991498 
 991473 
 991448 
 
 1.991422 
 991397 
 991372 
 991346 
 991321 
 991295 
 9912/0 
 991244 
 991218 
 991193 
 
 1.991167 
 991141 
 991115 
 991090 
 991064 
 991038 
 991012 
 990986 
 990960 
 990934 
 
 ' 990908 
 990882 
 990855 
 990829 
 990803 
 99077 7 
 990750 
 990724 
 990697 
 990t)71 
 
 1.990644 
 990ol8 
 990591 
 990565 
 990538 
 
 -990511 
 990485 
 990458 
 990431 
 990404 
 
 ~Sine. 
 
 P. I./' 
 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 4.1 
 4.2 
 4.2 
 
 .2 
 .2 
 4.2 
 4.2 
 4.2 
 4.2 
 4.2 
 4.2 
 4.2 
 4.3 
 4.3 
 4.3 
 4.3 
 
 4.3 
 4.3 
 4.3 
 4.3 
 4.3 
 4.3 
 4.3 
 4.3 
 4.3 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4.4 
 4,4 
 4.4 
 4.5 
 4.5 
 4.6 
 4.6 
 
 Tan-. |D. lu 
 
 .288662 
 2o9326 
 2^9999 
 290(i71 
 291342 
 292013 
 292{)82 
 293350 
 294017 
 294684 
 295349 
 
 i. 296013 
 296677 
 297339 
 298001 
 298662 
 299322 
 299980 
 
 112 
 112 
 112 
 112 
 112 
 111 
 111 
 111 
 111 
 111 
 111 
 111 
 110 
 110 
 110 
 i 110 
 110 
 110 
 
 3006381 ,,„> 
 301295 ^"-^ 
 301951 
 
 302607 
 303261 
 303914 
 304567 
 305218 
 305869 
 306519 
 307168 
 30/815 
 308463 
 309109 
 309754 
 310398 
 3110-42 
 311685 
 31232/ 
 312967 
 313608 
 314247 
 314886 
 9.315523 
 316159 
 316796 
 317430 
 318064 
 318o97 
 319329 
 319961 
 320592 
 321222 
 ,321861 
 322479 
 323106 
 323 733 
 324368 
 324983 
 325607 
 326231 
 326»53 
 327476 
 
 109 
 109 
 109 
 109 
 109 
 109 
 108 
 i 108 
 I 108 
 108 
 108 
 108 
 i 107 
 107 
 107 
 107 
 10/ 
 107 
 107 
 106 
 106 
 106 
 106 
 106 
 106 
 106 
 106 
 105 
 106 
 106 
 105 
 106 
 106 
 104 
 104 
 104 
 104 
 104 
 104 
 104 
 104 
 
 Cotang. 
 I)egr.-es. 
 
 10.711348 
 710674 
 710UJ1 
 709329 
 708668 
 707987 
 70/318 
 706650 
 705983 
 705316 
 704051 
 
 10.70^9'<7 
 703323 
 702661 
 701999 
 701338 
 700678 
 700020 
 699362 
 698705 
 698049 
 
 10-697393 
 696739 
 696086 
 695433 
 694/82 
 694131 
 693481 
 692832 
 692185 
 691537 
 
 10-690891 
 690246 
 689602 
 688958 
 688315 
 687673 
 687033 
 686392 
 686753 
 685115 
 
 10-684477 
 683841 
 683206 
 682570 
 681936 
 681303 
 680u71 
 680039 
 679408 
 678/78 
 0.678149 
 677621 
 676894 
 676267 
 675642 
 675017 
 674393 
 673769 
 673147 
 672526 
 
 119081 
 
 |il910}:« 
 
 j;iJl38 
 
 !: 19167 
 
 !il9195 
 
 119224 
 
 1 19252 
 
 ,192U 
 
 'l93tt9 
 
 i 1933b 
 
 ! 1936t) 
 
 , 19395 
 
 i 19423 
 
 ■ 19462 
 
 '19481 
 
 19509 
 
 19538 
 
 1 9566 
 
 19595 
 
 19623 
 
 19652 
 
 19680 
 
 19/09 
 
 19737 
 
 19766 
 
 197^4 
 
 ; 19823 
 
 19851 
 
 198b0 
 
 19908 
 
 1 1993/ 
 
 ' 19965 
 
 19994 
 
 ' 20022 
 
 20051 
 
 200/9 
 
 I 20108 
 
 I 20136 
 
 2U165 
 
 20193 
 
 '■20222 
 
 ,20250 
 
 1202/9 
 
 2030/ 
 
 \ 20336 
 
 20364 
 
 20393 
 
 ; 1^0421 
 
 I v0460 
 
 20478 
 
 2050/ 
 
 1 20535 
 
 205()3 
 
 J 20592 
 
 i|20u20 
 
 i! 20649 
 
 :2(r;77 
 
 20.06 
 20734 
 :20,63 
 20791 
 
 98163 
 
 I981B7! 
 
 ^J8l52 
 
 9«146' 
 
 98140 
 
 98135 1 
 
 9812i'j 
 
 981241 
 
 98118! 
 
 ..(8112 I 
 
 .^>8107j 
 
 98101 
 
 98096 
 
 98090 
 
 98084 
 
 98079 
 
 98073 ' 
 
 1)8067 I 
 
 98061 I 
 
 98056 ' 
 
 98050 
 
 98044 i 
 
 98039 \ 
 
 98033 I 
 
 98027 I 
 
 98021 I 
 
 98016 ' 
 
 98010: 
 
 98004 j 
 
 97998 1 
 
 97992; 
 
 9 7987! 
 
 9/981 i 
 
 97975 1 
 
 97969 I 
 
 97963 
 
 97958 i 
 
 97962 I 
 
 97946 j 
 
 9.940' 
 
 9/ 934 I 
 
 97928 ! 
 
 97922 : 
 
 97916, 
 
 97910; 
 
 9/905 
 
 978991 
 
 978i;3; 
 
 |9,887: 
 
 9/8bll 
 
 978761 
 
 97809' 
 
 97863, 
 
 97857! 
 
 9/851! 
 
 97845 
 
 9.839 
 
 97b3ci 
 
 9/827 
 
 97821 
 
 97815 
 
 Tang. 
 
 N. C03. X.t^i 
 
 60 
 
 59 
 
 5b 
 
 57 
 
 6(i 
 
 55 
 
 64 
 
 53 
 
 62 
 
 51 
 
 :( 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 '34 
 
 33 
 
 I 32 
 
 I 31 
 
 30 
 
 129 
 
 i28 
 
 I 27 
 
 26 
 
 126 
 
 124 
 
 i23 
 
 ,22 
 
 '21 
 
 I 20 
 
 !l9 
 
 18 
 
 ,17 
 
 i 16 
 
 15 
 
 I 14 
 
 ;13 
 : -2 
 
 \ 
 10 
 
 1 
 
TABLE IJ 
 
 Log. Sines and Tangpote. (12°) Natural Sines. 
 
 3J 
 
 Sine. 
 
 .3178.!) 
 318473 
 al9»), 
 31}>658 
 3-20249 
 3-2.JS 40 
 3-214:}0 
 3-2-2Uiy 
 3-2-2607 
 3-23194 
 3-2;1780 
 3-24361) 
 32495U 
 3-25534 
 3-2iill7 
 3-2()700 
 3-27-281 
 32 7H()'2 
 3-2844-2 
 3-29.J21 
 3-29599 
 
 9.3301 7t) 
 330/53 
 3313-29 
 331903 
 3324/8 
 333051 
 333624 
 334195 
 3347(i(i 
 335337 
 
 9. 33590 J 
 33G475 
 33/043 
 3376l0 
 33817b 
 33874-2 
 339306 
 3398 ;i 
 340434 
 340;).>»(i 
 
 9.341658 
 342119 
 342679 
 343239 
 34379; 
 S44355 
 344912 
 345469 
 346024 
 34()579 
 
 i^. 347134 
 347687 
 348240 
 348792 
 349343 
 349893 
 3 0443 
 3 ).rJ92 
 3=^1540 
 352088 
 Oosinu. 
 
 D. 10' 
 
 J9.0 
 9S.8 
 98.7 
 98.6 
 98.4 
 98.3 
 98.2 
 98.0 
 97.9 
 97.7 
 97.6 
 97.5 
 97.3 
 97.2 
 97.0 
 96.9 
 96.8 
 96.6 
 96.5 
 96.4 
 9ii.2 
 96.1 
 96.0 
 95.8 
 95.7 
 95.6 
 95.4 
 95.3 
 95.2 
 95.0 
 94.9 
 94.8 
 94.6 
 94.5 
 94.4 
 94.3 
 94.1 
 94.0 
 93.9 
 93.7 
 93.6 
 93.5 
 93.4 
 93.2 
 93.1 
 93.0 
 92.9 
 92.7 
 92.6 
 92.6 
 92.4 
 92.2 
 92.1 
 92.0 
 91.9 
 91.7 
 91 .t) 
 91.6 
 91.4 
 91.3 
 
 Cosine. 
 
 1.990404 
 990378 
 99035 1 
 99J324 
 99)297 
 9;i02;0 
 9i)J243 
 990215 
 990188 
 99)161 
 990134 
 
 1.990107 
 990079 
 990052 
 990025 
 989997 
 989970 
 989942 
 98)J15 
 989887 
 9898a0 
 
 1.989832 
 989804 
 989777 
 989749 
 989/21 
 989J93 
 989665 
 989(i37 
 989609 
 989582 
 
 .989553 
 989526 
 989497 
 989469 
 989441 
 989413 
 989384 
 989356 
 98932» 
 989300 
 
 .989271 
 989243 
 989214 
 989186 
 989167 
 989128 
 989100 
 989071 
 989042 
 989014 
 
 .988986 
 988956 
 988927 
 988898 
 988869 
 988840 
 988811 
 988782 
 988763 
 988724 
 Sine. 
 
 D. 10' 
 
 4.5 
 4.5 
 4.5 
 4.5 
 4.5 
 4.5 
 4.5 
 4.5 
 4.5 
 4.6 
 4.5 
 4.6 
 4.6 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.9 
 
 4.9 
 
 4.9 
 
 Tan g. 
 
 1.327474 
 3-28aJo 
 8M8715 
 329334 
 3-29953 
 330570 
 331187 
 331803 
 332418 
 333033 
 333646 
 
 >, 334-259 
 334871 
 335482 
 3360)3 
 336702 
 337311 
 337919 
 33852/ 
 339133 
 339739 
 
 >. 340344 
 340948 
 341662 
 342155 
 342757 
 343358 
 343958 
 344558 
 345157 
 345755 
 
 1.346353 
 346949 
 347545 
 348141 
 348735 
 349329 
 349922 
 360514 
 351106 
 361697 
 
 1.352287 
 362876 
 353465 
 354053 
 354640 
 355327 
 355813 
 356398 
 356982 
 367666 
 
 1.368149 
 358731 
 359313 
 359893 
 3uOl74 
 361053 
 361632 
 362210 
 362787 
 363364 
 
 "CotHng. 
 
 D.J_0" 
 
 103 
 
 103 
 
 103 
 
 103 
 
 103 
 
 103 
 
 103 
 
 102 
 
 102 
 
 102 
 
 102 
 
 102 
 
 l.'>5 
 
 102 
 
 102 
 
 101 
 
 101 
 
 101 
 
 101 
 
 101 
 
 101 
 
 101 
 
 101 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 99.4 
 99.3 
 99.2 
 99.1 
 99.0 
 98.8 
 98.7 
 98.6 
 98.6 
 98.3 
 98.2 
 98 
 
 98.0 
 97.9 
 97.7 
 97.6 
 97.5 
 97.4 
 97.3 
 97.1 
 97.0 
 96.9 
 96.8 
 96.7 
 96.6 
 96 
 96.3 
 96.2 
 96. 
 
 CoLaiiif. N. siiK'. N. co>s. 
 
 10 
 
 10.65 
 
 10 
 
 lU 
 
 10 
 
 6726-26 
 671905 
 671-285 
 670666 
 670047 
 6()9430 
 668813 
 668197 
 667582 
 66(>9()7 
 66()354 
 6(i5741 
 6651-29 
 664518 
 663907 
 663298 
 662689 
 662081 
 661473 
 660867 
 660'2(;i 
 659656 
 659052 
 658448 
 657845 
 657243 
 666642 
 66()042 
 655442 
 654843 
 664245 
 653(>47 
 653051 
 652455 
 651859 
 661-265 
 650671 
 650078 
 649486 
 648894 
 648303 
 647713 
 647124 
 64{)535 
 646947 
 646360 
 644773 
 644187 
 643b02 
 643018 
 642434 
 641851 
 641269 
 640687 
 640107 
 639526 
 638947 
 6383(>8 
 637790 
 637213 
 636636 
 
 20791 97815 
 208-20 97809 
 20848 97803 
 20877 97797 
 20905 97791 
 20933 9v784 
 20962 97778 
 20990 97772 
 21019 97766 
 21047 97760 
 21076 97754 
 21104 9774- 
 21132 97742 
 ilH^I 97736 
 21109 97729,46 
 21218 97723145 
 
 21246 97717 
 21-275 97711 
 21303 97705 
 2133197698 
 ,21360 97692 
 21388 97(i86 
 21417 97680 
 
 21445 97673 137 
 21474 97667 
 21602 97661 
 21530 97656 
 21559 97648 
 21587 97642 
 21616 97636 
 21644 97630 
 21672 97623 
 2170197617 
 21729 97611 
 21768 97604 
 21 78<) 97698 
 2181491592 
 21843 97686 
 2187197679 
 21899 97673 
 21928 97666 
 21956 97560 
 21985 97653 
 32013 97547 
 2204197641 
 22070 97534 
 22098 97528 
 221-26 9,521 
 22165 97615 
 22183 97508 
 22212 97602 
 22240 9,496 
 22268 9 ;4«9 
 22297 97483 
 22325 97476 
 22363 97470 
 22382 95463 
 22410 97467 
 22438 97450 
 22467 9? 444 
 224'J597437 
 
 1: 
 
 Tang. N. cos.:X.8iDe. 
 
 77 Degrees. 
 
34 
 
 Log. Sines and TanKents. (13°) Natural Sines. 
 
 TABLE n. 
 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 i: 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 
 Sine. 
 
 .352088 
 352635 
 353181 
 353726 
 354271 
 354815 
 355358 
 355901 
 356443 
 356984 
 35 7524 
 
 . 3580t)4 
 358603 
 35.^:41 
 36^678 
 360216 
 360752 
 3612H7 
 361K22 
 362j:>6 
 3b-.^889 
 
 .363422 
 363954 
 364486 
 365016 
 365546 
 366075 
 366604 
 367131 
 367659 
 368185 
 
 .368711 
 369236 
 369761 
 370286 
 370808 
 371330 
 371852 
 372373 
 372894 
 373414 
 
 .373933 
 374452 
 3749/0 
 375487 
 376003 
 376519 
 377035 
 377649 
 378063 
 378577 
 
 ..3790b9 
 379601 
 380113 
 380624 
 381134 
 381643 
 382152 
 382661 
 383168 
 383675 
 Cosine. 
 
 D. 10"! Cosine. 
 
 91.1 
 
 91.0 
 
 90.9 
 
 90.8 
 
 90.7 
 
 90.5 
 
 90.4 
 
 90.3 
 
 90.2 
 
 90.1 
 
 89.9!, 
 
 89.8!' 
 
 89.7; 
 
 89.6 i 
 
 89.5 
 89.3 
 
 89.2 
 89.1 
 89.0 
 88.9 
 88.8 
 88.7 
 88.5 
 88.4 
 88.3 
 88.2 
 88.1 
 88.0 
 87.9 
 87.7 
 87.6 
 87.5 
 87.4 
 87.3 
 87.2 
 87.1 
 87.0 
 86.9 
 86.7 
 86.6 
 86.5 
 86.4 
 86.3 
 86.2 
 86.1 
 86.0 
 85.9 
 85.8 
 85.7 
 85.6 
 85.4 
 85.3 
 85.2 
 85.1 
 85.0 
 84.9 
 84.8 
 84.7 
 84.6 
 84.5 
 
 .988724 
 938695 
 988666 
 988636 
 988607 
 988578 
 988548 
 988619 
 988489 
 988460 
 988430 
 
 .988401 
 988371 
 988342 
 988312 
 988282 
 988252 
 988223 
 988193 
 988163 
 988133 
 
 .988103 
 988073 
 988043 
 988013 
 987983 
 987953 
 987922 
 987892 
 987862 
 987832 
 
 .987801 
 987771 
 987740 
 987710 
 987679 
 987649 
 987618 
 987588 
 987557 
 987526 
 
 .987496 
 987466 
 987434 
 987403 
 987372 
 987341 
 987310 
 98727S 
 987248 
 987217 
 
 .987186 
 987155 
 987124 
 987092 
 987061 
 987030 
 986998 
 9S6967 
 98(i936 
 986904 
 
 Sine. 
 
 D. 10" 
 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 6.0 
 6.0 
 
 5.0 
 6.0 
 5.0 
 6.0 
 6.0 
 6.0 
 6.0 
 5.0 
 6.1 
 6.1 
 6.1 
 6.1 
 6.1 
 5.1 
 6.1 
 6.1 
 6.1 
 
 6.2 
 6.2 
 £.2 
 6.2 
 5.2 
 6.2 
 6.2 
 6.2 
 5.2 
 5.2 
 5.2 
 5.2 
 6.2 
 5.2 
 6.2 
 
 T ang. 
 
 9.363364 
 363940 
 364515 
 365090 
 365664 
 366237 
 366810 
 367382 
 367953 
 368624 
 369094 
 
 9.369663 
 370232 
 370799 
 371367 
 371933 
 372499 
 373064 
 373629 
 374193 
 374766 
 
 9.376319 
 376881 
 376442 
 377003 
 377563 
 378122 
 378681 
 379239 
 379797 
 380354 
 
 9.380910 
 381466 
 382020 
 382576 
 383129 
 383682 
 384234 
 384786 
 386337 
 385888 
 
 9.386438 
 386987 
 387536 
 388084 
 388631 
 389178 
 389724 
 390270 
 390816 
 391360 
 
 9.391903 
 392447 
 392989 
 393531 
 394073 
 394614 
 395154 
 395694 
 396233 
 396 771 
 
 96. 
 95. 
 95. 
 95. 
 95. 
 95. 
 95. 
 95. 
 95. 
 95. 
 94. 
 94. 
 94. 
 94. 
 94. 
 94. 
 94. 
 94. 
 94. 
 93. 
 93. 
 93. 
 93. 
 93. 
 93. 
 93. 
 93. 
 93. 
 92. 
 92. 
 92. 
 92. 
 92 
 92 
 92. 
 92. 
 93. 
 92. 
 !91. 
 91. 
 91. 
 91. 
 91. 
 91. 
 91. 
 91. 
 91. 
 90. 
 90. 
 90. 
 90. 
 90. 
 90. 
 90. 
 90. 
 90. 
 90. 
 89. 
 89. 
 89. 
 
 Cotiin" 
 
 Cotang. N.sinej'N. «os. 
 
 10.636636 
 636060 
 635485 
 634910 
 634336 
 633763 
 633190 
 632618 
 632047 
 631476 
 630906 
 
 10.630337 
 629768 
 629201 
 628633 
 628067 
 627501 
 626936 
 626371 
 626807 
 625244 
 
 10.624681 
 624119 
 623558 
 622997 
 622437 
 621878 
 621319 
 620761 
 620203 
 619646 
 
 10.619(>90 
 618534 
 617980 
 617425 
 616871 
 616318 
 615766 
 615214 
 0*4663 
 614112 
 
 10.613562 
 613013 
 612464 
 611916 
 611369 
 610822 
 610276 
 60y/3iJ 
 60J185 
 60--ii)40 
 
 10.608097 
 60^.553 
 607011 
 606469 
 605927 
 605386 
 604846 
 604306 
 603767 
 603229 
 
 22495197437 
 2252397430 
 22562 97424 
 2268097417 
 22608 97411 
 22637,97404 
 22666 97398 
 22693197391 
 j 22722 197384 
 ! 22750:9/378 
 22778 !iT2^» 
 i22807;9736o 
 i 22835 197368 
 22863 97351 
 ; 22892 197345 
 973.8 
 97331 
 97325 
 97318 
 9/311 
 97304 
 97298 
 97291 
 97^84 
 
 j 22920 
 , 22948 
 
 22977 
 ' 23006 
 123033 
 123062 
 1 2^090 
 
 23118 
 
 23146 
 |23175!972'/8 
 
 23203 97271 
 1 2323 1197264 
 123260 9-25 
 
 i 232HH 
 
 97261 
 
 23316 9/244 
 
 ! 1 23346 
 1 1 23373 
 i 23401 
 i 23429 
 I 23468 
 I 23486 
 j 23514 
 23642 
 ! 236/1 
 1^23599 
 I j 23627 
 123656 
 1 1 23684 
 I 23712 
 '123740 
 
 Tan.i 
 
 23797 
 
 97237 
 97230 
 9/223 
 97217 
 97210 
 97203 
 97 96 
 97189 
 7182 
 97176 
 971o9 
 97162 
 y7166 
 97148 
 97141 
 
 23769 97134 
 
 97127 
 
 23825 (97] 20 
 23853 97113 
 
 23882 
 23910 
 23938 
 
 9710.) 
 97 100 
 iJ7093 
 
 23966 970.^6 
 
 23995 
 24023 
 24051 
 24079 
 24108 
 24131) 
 
 24164 97037 
 
 2419V.' 
 
 97079 
 i)to,2 
 97U66 
 9,06) 
 9/051 
 97044 
 
 9-030 
 
 •OS. N.sine. 
 
 JH Decrees. 
 
Log. Sines and Tangents. (14°) Natural Sines. 
 
 35 
 
 
 
 I 
 
 2 
 3 
 4 
 6 
 6 
 7 
 8 
 9 
 10 
 
 14 
 15 
 U> 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 
 25 
 
 2(> 
 
 27 
 
 28 
 
 39 
 
 30 
 
 31 
 
 3-2 
 
 33 
 
 34 
 
 35 
 
 3b 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 46 
 
 4U 
 
 47 
 
 48 
 
 49 
 
 60 
 
 61 
 
 62 
 
 53 
 
 64 
 
 66 
 
 6b 
 
 57 
 
 58 
 
 59 
 
 W 
 
 iliiie. 
 
 9.383675 
 384182 
 38-k«8'" 
 
 385697 
 386201 
 386704 
 387207 
 387709 
 388210 
 388711 
 9.385)211 
 389711 
 390210 
 390708 
 391206 
 391 i 03 
 392199 
 392695 
 393191 
 393685 
 394179 
 394()73 
 395166 
 395658 
 396150 
 396641 
 397132 
 397621 
 398111 
 398t»00 
 9.399088 
 3995/5 
 400062 
 400549 
 401035 
 401620 
 402005 
 402489 
 402972 
 403455 
 9.403938 
 404420 
 404901 
 406382 
 405862 
 406341 
 40t)820 
 40/299 
 407777 
 40.S254 
 9.408 731 
 409207 
 4(J9682 
 410157 
 410632 
 411106 
 411679 
 412052 
 412521 
 412996 
 Cofline. 
 
 D - 10'' Co.sine. D. lU" Tang. D. 10"| Cotang. IX.sine.lN. cos 
 
 S4.4 
 84.3 
 
 8i.l 
 
 8l.il 
 
 83. ) 
 
 «3 i 
 
 83 7 
 
 83 6 
 
 83.6 
 
 83.4 
 
 83.3 
 
 83.2 
 
 83.1 
 
 83.0 
 
 82.8 
 
 82.7 
 
 82.6 
 
 82.5 
 
 82.4 
 
 82.3 
 
 82.2 
 
 82.1 
 
 82.0 
 
 81.9 
 
 81.8 
 
 81.7 
 
 81.7 
 
 81.6 
 
 81.6 
 
 81.4 
 
 81.3 
 
 81.2 
 
 81.1 
 
 81.0 
 
 80.9 
 
 80.8 
 
 80.7 
 
 80.6 
 
 80.6 
 
 80.4 
 
 80.3 
 
 80.2 
 
 80.1 
 
 80.0 
 
 79.9 
 
 79.8 
 
 79.7 
 
 7y.6 
 
 79.5. 
 
 79.4 
 
 79.4 
 
 79.3 
 
 79.2 
 
 79.1 
 
 79.0 
 
 78.9 
 
 78. 'J 
 
 78.7 
 
 78.6 
 
 9.986904 
 
 986873 
 
 986841 
 
 986809 
 
 9867 78 
 
 98674<J 
 
 986714 
 
 986683 
 
 986651 
 
 986619 
 
 9W6587 
 
 9 . 986556 
 
 986523 
 
 986491 
 
 986459 
 
 986427 
 
 986395 
 
 986363 
 
 986331 
 
 986299 
 
 986266 
 
 9.986234 
 
 986202 
 
 986169 
 
 986137 
 
 980104 
 
 986072 
 
 986039 
 
 986007 
 
 985974 
 
 985942 
 
 9 985909 
 
 985876 
 
 985843 
 
 985811 
 
 985778 
 
 985745 
 
 9^5712 
 
 985679 
 
 985646 
 
 986613 
 
 9.9;S5580 
 
 985547 
 
 985514 
 
 986480 
 
 985447 
 
 985414 
 
 985380 
 
 985347 
 
 985314 
 
 985280 
 
 9.085247 
 
 985213 
 
 985180 
 
 985146 
 
 985H3 
 
 985079 
 
 985045 
 
 985011 
 
 984978 
 
 984944 
 
 6.2 
 
 6.3 
 
 6.3 
 
 5.3 
 
 6 3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 5.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 5.3 
 
 5.3 
 
 6.4 
 
 5.4 
 
 5.4 
 
 6.4 
 
 6.4 
 
 5.4 
 
 6.4 
 
 5.4 
 
 6.4 
 
 6.4 
 
 5.4 
 
 6.4 
 
 5.4 
 
 5.4 
 
 5.6 
 
 6.5 
 
 6.6 
 
 6.6 
 
 5,5 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.5 
 
 6.5 
 
 5.5 
 
 5.6 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.6 
 
 5.6 
 
 5.6 
 
 6.6 
 
 5.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 5.6 
 
 5,6 
 
 6.6 
 
 5.6 
 
 9.396771 
 397309 
 397846 
 398383 
 398919 
 399455 
 399990 
 400624 
 401058 
 401591 , „y ^ 
 40212412?'^ 
 9.402656 
 403187 
 403718 
 404249 
 404778 
 406308 
 405836 
 406364 
 406892 
 407419 
 9.407945 
 408471 
 408997 
 409521 
 410045 
 410569 
 411092 
 411615 
 412137 
 412658 
 413179 
 413699 
 414219 I 
 414738 
 415267 
 415 7 75 
 416293 
 416810 
 417326 
 41 7842 
 9.418358 
 418873 
 419387 
 419901 
 420415 
 420927 
 421440 
 421952 
 422463 
 422974 
 9.423484 
 423993 
 424503 
 42601 1 
 425519 
 426(/27 
 426634 
 427041 
 427647 
 428062 
 
 Cotang. 
 
 89.6 
 89.6 
 89.5 
 89.4 
 89.3 
 89.2 
 89.1 
 89.0 
 88.9 
 
 88.7 
 188.6 
 188.6 
 '88.4 
 88.3 
 88.2 
 88.1 
 88.0 
 87.9 
 87.8 
 87.7 
 87.6 
 87.6 
 87,4 
 87.4 
 87.3 
 87.2 
 87.1 
 87.0 
 86.9 
 86.8 
 86.7 
 86.6 
 86.5 
 86.4 
 86.4 
 86.3 
 86.2 
 86.1 
 86.0 
 86.9 
 86.8 
 86.7 
 86.6 
 86.5 
 86.5 
 86.4 
 86.3 
 86.2 
 86.1 
 85.0 
 84.9 
 84.8 
 84.8 
 84.7 
 84.6 
 84.6 
 84.4 
 84.3 
 84.3 
 
 10.603229 
 602(i91 
 602154 
 601617 
 601081 
 600645 
 600010 
 599476 
 698942 
 598409 
 597876 
 10.597344 
 696813 
 696282 
 695751 
 595222 
 594692 
 594164 
 693636 
 593108 
 592581 
 10.592055 
 691529 
 691003 
 690479 
 589965 
 589431 
 688908 
 588385 
 587863 
 587342 
 10.586821 
 686301 
 685781 
 585262 
 684743 
 584225 
 583707 
 583190 
 682674 
 582158 
 10.581642. 
 681127 
 680613 
 580099 
 679586 
 679073 
 678560 
 578048 
 677637 
 677026 
 10.676516 
 576007 
 575497 
 674989 
 574481 
 673973 
 573466 
 572959 
 572453 
 571948 
 
 24192 97030,60 
 
 24220,97023 I 69 
 24249,97015168 
 24277 97008.67 
 1 2430o'^7001 ' 66 
 24333'9(>994 ! 55 
 243(i2!f)6987 [ 64 
 24390J 96980 53 
 24418 96973; 52 
 24446^96966 1 61 
 24474 96959 | 60 
 24503 196952 , 49 
 
 24531196945 
 24559 196937 
 
 i 24587 
 i24<il5 
 ' 24644 
 124672 
 24700 
 24728 
 
 96930 
 96923 
 96916 
 9<>909 
 96902 
 96894 
 
 24756 96887 
 24784196880 1 39 
 24813 96873 38 
 
 124841 
 124869 
 
 2489 
 ! 24925 
 I 24954 
 : 24982 
 126010 
 ; 25038 
 i 250i)6 
 1 25094 
 125122 
 125151 
 
 25179 
 125207 
 i 25235 
 1 25263 
 '25291 
 , 25320 
 
 9()866 i 37 
 
 9<)858 I 36 
 
 96851 I 35 
 
 96844 
 
 96837 
 
 96829 
 
 96822 
 
 96815 
 
 96807 
 
 96800 
 
 96793 
 
 96786 
 
 >6778 
 
 96771 
 
 96764 
 
 96766 
 
 96749 
 
 96742 
 
 25348196734 
 
 125376 
 25404 
 26432 
 125460 
 25488 
 125516 
 i 25545 
 25573 
 25<>01 
 25629 
 i 25667 
 25685 
 25713 
 25741 
 25766 
 25798 
 258- Ui 
 25854 
 
 96727 
 96719 
 9(J712 
 96705 
 96697 
 96690 
 
 34 
 
 33 
 
 32 I 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 96682 1 1 
 
 25S8' 
 
 96670 
 96667 
 96660 
 96653 
 96645 
 9()638 
 96t)30 
 96623 
 96615 
 96t)08 
 96600 
 J6593 
 
 Tang. 
 
 N. cofuN.mnH. 
 
 75 liegrttes 
 
36 
 
 Log. Sines and Tangents. (15*^) Natural ^mes. 
 
 TAULE II. 
 
 
 
 I 
 2 
 3 
 4 
 6 
 6 
 7 
 8 
 9 
 10 
 U 
 1-2 
 13 
 14 
 15 
 i6 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2ti 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 60 
 61 
 52 
 53 
 64 
 i 55 
 56 
 57 
 68 
 59 
 60 
 
 Sine. 
 
 9. 41 299 J 
 413467 
 41393S 
 414408 
 4148/8 
 415347 
 415815 
 416283 
 416751 
 417217 
 41 ;684 
 
 9.418150 
 418615 
 419079 
 419544 
 420007 
 420470 
 420933 
 421395 
 421857 
 422318 
 
 9.4227 78 
 423238 
 4236.^7 
 424166 
 s24615 
 425073 
 425630 
 425987 
 426443 
 426899 
 
 9.427li54 
 427809 
 428263 
 428717 
 429170 
 429623 
 430076 
 430627 
 430978 
 431429 
 
 9.4318/9 
 432329 
 432778 
 43b226 
 433676 
 434122 
 434569 
 435016 
 435462 
 436908 
 .436353 
 436798 
 437242 
 43 7686 
 438129 
 438572 
 439014 
 439456 
 439897 
 440338 
 Cosine. 
 
 D. W 
 
 78.5 
 78.4 
 78.3 
 78.3 
 
 7S <: 
 
 78 1 
 78.0 
 77.9 
 77.8 
 77.7 
 77.6 
 77.5 
 77.4 
 77.3 
 77.3 
 77.2 
 77.1 
 77.0 
 76.9 
 75.8 
 76.7 
 76.7 
 76.6 
 76.6 
 76.4 
 76.3 
 76.2 
 76.1 
 76.0 
 76.0 
 75.9 
 75.8 
 75.7 
 75.6 
 76.6 
 76.4 
 76.3 
 75.2 
 76.2 
 76.1 
 76.0 
 74.9 
 74.9 
 74.8 
 74.7 
 74 6 
 74.6 
 74.4 
 74.4 
 74.3 
 74.2 
 174.1 
 174.0 
 174.0 
 73. 9 
 73.8 
 73.7 
 73.6 
 73 6 
 73.5 
 
 Uasiiic. 
 
 9 984944 
 984910 
 984876 
 984842 
 984808 
 984774 
 984740 
 984706 
 984ri72 
 984637 
 984603 
 
 9.984569 
 984535 
 984500 
 -984466 
 984432 
 984397 
 984363 
 984328 
 984294 
 984259 
 
 9.984224 
 984190 
 984155 
 984120 
 984085 
 984050 
 984U15 
 983981 
 983946 
 983911 
 
 9.983875 
 983840 
 983805 
 983770 
 9837S6 
 983 < 00 
 983664 
 983629 
 983594 
 983558 
 
 9.983523 
 983487 
 983462 
 983416 
 983381 
 983346 
 983309 
 983273 
 983238 
 983202 
 983166 
 983130 
 983094 
 983058 
 983022 
 982986 
 982960 
 S82914 
 982878 
 982842 
 
 5.7 
 5.7 
 5.7 
 5.7 
 6.7 
 6.7 
 6.7 
 6.7 
 6.7 
 6.7 
 6.7 
 6.7 
 6.7 
 5.7 
 5.7 
 6.8 
 5.8 
 
 D. lU" Tan g. 
 
 9.428052 
 
 428557 
 429062 
 42966;> 
 4300 7 IJ 
 430573 
 431075 
 431677 
 432079 
 432580 
 433080 
 
 9.433580 
 434080 
 434579 
 435078 
 435576 
 436073 
 436670 
 437067 
 437563 
 438059 
 
 9.438554 
 439048 
 439543 
 440036 
 440529 
 441022 
 441614 
 442006 
 442497 
 442988 
 
 9.443479 
 143968 
 444458 
 444947 
 445435 
 445923 
 446411 
 446898 
 447384 
 447870 
 .448356 
 448841 
 449326 
 449810 
 460294 
 450777 
 451260 
 451743 
 462225 
 452706 
 
 9.453187 
 453668 
 454148 
 454628 
 455107 
 466586 
 456064 
 456542 
 457019 
 467496 
 
 6.8 
 6.8 
 5.8 
 5.8 
 6.8 
 6.8 
 6.8 
 6.8 
 5.8 
 5.8 
 5.8 
 6.8 
 6.8 
 5.8 
 5.8 
 6.9 
 5.9 
 5.9 
 6.9 
 6.9 
 5.9 
 
 6.9 
 5.9 
 6.9 
 5.9 
 5.9 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 6.0 
 
 D. 10^' 
 
 84 2 
 
 84 
 
 84 . 
 
 h3.9 
 
 83.8 
 
 83.8 
 
 83.7 
 
 83.6 
 
 83.6 
 
 83.4 
 
 83.3 
 
 83.2 
 
 83.2 
 
 83.1 
 
 83.0 
 
 82.9 
 
 82.8 
 
 82.8 
 
 82.7 
 
 82.6 
 
 82.6 
 
 82.4 
 
 82.3 
 
 82.3 
 
 82.2 
 
 82.1 
 
 82.0 
 
 81.9 
 
 81.9 
 
 81.8 
 
 81.7 
 
 81.6 
 
 81.6 
 
 81.5 
 
 81.4 
 
 81.3 
 
 81.2 
 
 81.2 
 
 8i.l 
 
 81.0 
 
 80.9 
 
 80.9 
 
 80.8 
 
 80.7 
 
 80.6 
 
 80.6 
 
 80.6 
 
 80.4 
 
 80.3 
 
 80.2 
 
 80.2 
 
 80.1 
 
 80.0 
 
 79.9 
 
 79.9 
 
 79.8 
 
 79.7 
 
 79.6 
 
 79.6 
 
 79.5 
 
 25882 
 2591 J 
 2593o 
 2596c> 
 25994 
 
 26079 
 2610/ 
 26136 
 26163 
 26191 
 26219 
 26247 
 26276 
 
 26331 
 26369 
 26387 
 26415 
 
 Cotang. 
 li De grees. 
 
 cotaug. N. siu. 
 
 10.571948 
 
 671443 
 
 670938 
 
 670434 
 
 669930 
 
 6t)9427 
 
 5ii8925 
 
 5(i8423 
 
 567921 
 
 567420 
 
 666920 
 10.566420 
 
 6()5920 
 
 61)6421 
 
 664922 
 
 664424 
 
 6(i3927 
 
 5b3430 
 
 56-2933 
 
 662437 
 
 561941 
 10.661446 
 
 660952 
 
 660457 
 
 559964 
 
 559471 
 
 568978 
 
 658486 
 
 657994 
 
 567503 
 
 557012 
 10.556521 
 
 556032 
 
 655542 
 
 555053 
 
 554565 
 
 654077 
 
 &53589 
 
 553102 
 
 552616 
 
 652130 
 10.561644 
 
 551159 
 
 550674 
 
 650190 
 
 549706 
 
 549223 
 
 648740 
 
 548257 
 
 6477 i 6 
 
 6472§4 
 10.646813 
 
 646332 
 
 645852 
 
 645372 
 
 544893 
 
 644414 
 
 54393G 
 
 543458 
 
 642981 
 
 542504 
 
 96593 
 yiioH.") 
 9657-. 
 96570 
 96662 
 
 26022196566 
 26050|9()547 
 
 96540 
 96632 
 96624 
 96517 
 96509 
 96502 
 96494 
 96486 
 
 2630Ji 96479 
 
 96471 
 96463 
 9()456 
 964-18 
 
 !! 26443 96440 
 
 26471 
 26500 
 26528 
 26556 
 26584 
 26612 
 26640 
 26668 
 26696 
 26724 
 26/52 
 26780 
 2680b 
 26836 
 26864 
 
 96433 
 96425 
 96417 
 96410 
 96402 
 96394 
 06386 
 96379 
 96371 
 96363 
 96355 
 96347 
 96340 
 96332 
 96324 
 
 26892 96316 
 
 26920 
 
 26948 96301 
 
 126976 
 i 27004 
 
 I 27032 
 j 2/060 
 ! 27088 
 ,27116 
 127144 
 ; 27172 
 27200 
 1 27228 
 ■27256 
 ,127284 
 127312 
 
 27340 96190 
 
 2736b 
 
 i 27396 96174 
 127424 96166 
 
 9630b 
 
 96293 
 96285 
 96277 
 96269 
 96261 
 96-253 
 96241) 
 96238 
 96230 
 96-.i22 
 96214 
 96206 
 96198 
 
 96lb2 
 
 27452 '96158 
 
 j 27480 
 127608 
 127 536 
 
 127564 96126 
 
 96150 
 ybl42 
 96134 
 
 Tang. !] N. cos.|X.sine. ' 
 
Log. Sinus and Tangents. (16°) Katural Sines. 
 
 37 
 
 D. 10^ ' 
 
 73.4 
 73.3 
 73.2 
 73-1 
 73.1 
 73.0 
 72.9 
 72.8 
 72.7 
 72.7 
 72. G 
 72.6 
 ;2.4 
 72.3 
 72.3 
 72.2 
 72.1 
 72. U 
 72. U 
 71.9 
 71. « 
 71.7 
 71.6 
 71. (i 
 71.5 
 71.4 
 71.3 
 71.3 
 71.2 
 71.1 
 71.0 
 71.0 
 ?0.9 
 70.8 
 70.7 
 70.7 
 70. G 
 70.6 
 70.4 
 70.4 
 70.3 
 70.2 
 70.1 
 70.1 
 70.0 
 G9.9 
 Gy.8 
 G9.8 
 (,9.7 
 G9.G 
 G9.5 
 t)9.5 
 G9.4 
 1,9.3 
 GO. 3 
 09.2 
 G9.1 
 G9.0 
 ()1>.0 
 G8.9 
 
 Sim 
 
 
 1 
 
 9 
 3 
 4 
 5 
 
 7 
 
 8 
 
 9 
 10 
 II 
 12 
 13 
 14 
 15 
 IG 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2G 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 3G 
 37 
 38 
 39 
 
 JO 
 41 
 42 
 43 
 44 
 45 
 4G 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 55 
 5G 
 57 
 58 
 59 
 
 GO 
 
 _j 
 
 :).4l0338 
 440/^8 
 441218 
 44lo58 
 44209(> 
 442635 
 442973 
 443410 
 443847 
 444284 
 4 44720 
 '445165 
 445590 
 446025 
 441)459 
 44()S93 
 447 32G 
 4-4/769 
 448191 
 44SG23 
 449054 
 449485 
 '449916 
 450345 
 450775 
 451204 
 451G32 
 4620., 
 452488 
 452915 
 453342 
 ,453708 
 464194 
 454ol9 
 456044 
 4.-64G9 
 456893 
 45G31G 
 45G739 
 457 1G2 
 45; 584 
 
 9 458U0J 
 458427 
 458848 
 4592G8 
 459G88 
 4<>0108 
 4G0527 
 4G094G 
 4()13G4 
 4G1782 
 
 !>. 402199 
 4G2G1G 
 4«i3032 
 4G3448 
 4()38G4 
 4G42;9 
 4G4()94 
 4G5108 
 4()o622 
 4G5936 
 Cosiii". 
 
 Cosine. 
 
 ► .982812 
 982^05 
 982;(i9 
 982733 
 982t)9ti 
 982G(iO 
 982(i24 
 982587 
 982651 
 982514 
 982477 
 
 ) 982441 
 
 ^ 982404 
 9823G7 
 982331 
 982294 
 982257 
 982220 
 982183 
 9H214G 
 982109 
 
 (1982072 
 982035 
 981998 
 981 90 1 
 981924 
 9818«G 
 981849 
 981812 
 981774 
 981737 
 
 I.981G99 
 981ou2 
 y81G25 
 981587 
 981549 
 981 ,12 
 9hl474 
 98143G 
 981399 
 9bl3Gl 
 
 ). 98 1323 
 981286 
 981247 
 9812U9 
 9811/1 
 981133 
 981096 
 981057 
 981019 
 y80i>hl 
 
 ). 9809 4 2 
 980904 
 9808GG 
 980827 
 980789 
 98IJ750 
 980712 
 980o73 
 9aoG35 
 98069G 
 Sine. 
 
 li.O 
 li.O 
 (i.l 
 G.l 
 G.l 
 G.l 
 6.1 
 G.l 
 G.l 
 6.1 
 6.1 
 6.1 
 6.1 
 6.1 
 G.l 
 6.1 
 G.l 
 6.2 
 6.2 
 G.2 
 G.2 
 6.2 
 6.2 
 6.2 
 6.2 
 6.2 
 6.2 
 
 9. 
 
 6.3 
 6.3 
 6.3 
 6.3 
 G.3 
 G.3 
 6.3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 6.4 
 
 6.4 
 
 G.4 
 
 6.4 
 
 6.4 
 
 G.4 
 
 6.4 
 
 6.4 
 
 'fang. 
 
 45749G 
 457953 
 458449 
 458925 
 459400 
 459875 
 4G0349 
 4G0823 
 4612^7 
 4Gli70 
 402242 
 462714 
 4G3186 
 4G3G58 
 4G4129 
 4(i459y 
 4G50oy 
 465539 
 406008 
 4GG476 
 46G945 
 467413 
 467880 
 4G8347 
 468814 
 4G9280 
 4G9;4tJ 
 470211 
 4/0076 
 471141 
 471G05 
 472058 
 472532 
 472996 
 473457 
 473919 
 474381 
 474842 
 476303 
 4/6/03 
 476223 
 476683 
 477142 
 477001 
 478069 
 478517 
 478975 
 479432 
 479889 
 480345 
 480801 
 481257 
 481712 
 482 1G7 
 48202 1 
 483076 
 483529 
 483982 
 484436 
 484887 
 486339 
 
 D. 10' 
 
 Cotang. 
 
 Cota n-, 
 
 10.542604 
 642027 
 541551 
 6410/5 
 640G00 
 540125 
 539ii51 
 639177 
 638703 
 538230 
 537758 
 
 10.537286 
 636814 
 53G342 
 536871 
 536401 
 634931 
 5344G1 
 633992 
 533524 
 633055 
 
 10.532587 
 632120 
 631063 
 531186 
 630720 
 530264 
 629/89 
 629324 
 628869 
 628395 
 
 10.527932 
 527468 
 627006 
 526543 
 626081 
 525619 
 526168 
 524097 
 524237 
 623777 
 
 10.623317 
 522858 
 522399 
 521941 
 621483 
 521026 
 6205G8 
 5201 1 1 
 519065 
 519199 
 
 10.518743 
 618288 
 pl7833 
 517379 
 616925 
 516471 
 516018 
 5156G5 
 615113 
 614661 
 
 N. 
 
 ,,X. CO8. 
 
 27664'9G126 60 
 27692,901 18 159 
 2/020 90110 68 
 ,27G48!9(,102|67 
 27076190094 56 
 
 2770419G086 
 27731 I9G078 
 27769 9(i0;0 
 27787!9(,0G2'52 
 27815;yo054i61 
 27843|9004G:60 
 
 i 2787 1 
 I 27899 
 2792/ 
 ! 27955 
 127983 
 128011 
 : 28039 
 |2bOG7 
 , 26095 
 
 90(»3'? ; 49 
 9G029 I 48 
 90021 47 
 90013 46 
 9u005 46 
 95997 I 44 
 95989 I 43 
 95981 42 
 959/2 41 
 
 28123196904 40 
 
 28150 9695G 
 : 281 i 8 95948 
 ! 2820(i|95940 
 i28234j95931 
 28202|95923 
 28290J96916 
 28318|95907 
 : 28340196898 
 28374195890 ■ 31 
 ; 28402|958h2 I 30 
 i 28429:95874 29 
 2845/95805 
 28486 96857 
 ! 28513195849 
 '2854195841 
 
 2860995832 24 
 
 28597 96824 
 
 ': 28025 
 28052 
 ■ 28080 
 ; 287 08 
 ' 28730 
 
 95810 
 »5807 
 95799 
 95:91 
 95782 
 
 28704195774 
 95700 
 
 28792 
 2882U 
 28847 
 28875 
 '2890^ 
 ! 28931 
 '289.59 
 
 95767 
 95;49jl4 
 95740 I 13 
 95732 j 12 
 95724 
 957 1 5 
 
 28987 ■J5/ 07 
 
 29015 
 2904-^ 
 290/0 
 29098 
 2blvJ0 
 29154 
 29182 
 292 O.v 
 29247 
 
 Tang. 
 
 N. COS. N.sinc. 
 
 95(i98 
 96090 
 95()8l 
 95073 
 95(iG4 
 95056 
 95047 
 951.39 
 96G30 
 
 73 Dirgrees. 
 
38 
 
 Log. Sines and Tangents. (17°) Natural Sines. 
 
 TABLE IL 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 9.465935 
 466348 
 466761 
 467173 
 467585 
 467996 
 4()8407 
 468817 
 469227 
 469(>37 
 470046 
 9.470455 
 470863 
 471271 
 471679 
 472086 
 472492 
 472898 
 473304 
 473710 
 474115 
 9.474519 
 474923 
 475327 
 475730 
 476133 
 476536 
 476938 
 477340 
 477741 
 478142 
 9.478542 
 478942 
 479342 
 479741 
 480140 
 480539 
 480937 
 481334 
 481731 
 482128 
 9.482525 
 482921 
 483316 
 483712 
 484107 
 484501 
 484895 
 485289 
 485682 
 486075 
 1.486467 
 486860 
 487251 
 487643 
 488034 
 488424 
 488814 
 489204 
 489593 
 4899H2 
 Cosine 
 
 D. 10"( Co.sine. 
 
 68.8 
 
 68.8 
 
 68.7 
 
 68.6 
 
 68.5 
 
 68.5 
 
 68.4 
 
 68.3 
 
 68.3 
 
 68.2 
 
 68.1 
 
 68.0 
 
 68.0 
 
 67.9 
 
 67.8 
 
 67.8 
 
 67.7 
 
 67.6 
 
 67,6 
 
 67.5 
 
 67.4 
 
 67.4 
 
 67.3 
 
 67.2 
 
 67.2 
 
 67.1 
 
 67.0 
 
 66.9 
 
 66.9 
 
 66.8 
 
 66.7 
 
 66.7 
 
 66.6 
 
 66.5 
 
 66.5 
 
 66.4 
 
 66.3 
 
 66.3 
 
 66.2 
 
 66.1 
 
 66.1 
 
 66.0 
 
 65.9 
 
 65.9 
 
 65.8 
 
 65.7 
 
 65.7 
 
 65.6 
 
 65.5 
 
 65.5 
 
 65.4 
 
 66.3 
 
 65.3 j 
 
 65.21 
 
 65. l! 
 
 65.1; 
 
 65.0 1 
 
 65.0 
 
 64.9 
 
 64.8 
 
 .980596 
 980558 
 980519 
 980480 
 980442 
 980403 
 980364 
 980325 
 980286 
 980247 
 980208 
 980169 
 980130 
 980091 
 980052 
 980012 
 979973 
 979934 
 979895 
 979855 
 979816 
 
 9.979776 
 979737 
 979697 
 979658 
 979618 
 979579 
 979539 
 979499 
 979459 
 979420 
 
 9.979380 
 979340 
 979300 
 979260 
 979220 
 979180 
 9791401 
 9791001 
 979059 
 979019 
 
 9.978979 
 978939 
 978898 
 978858 
 978817 I 
 978777 
 978736 I 
 978696 I 
 978655 I 
 9786151 
 
 9.978574 
 978533 
 978493 I 
 978452 
 978411 
 9783701 
 978329 ' 
 978288 
 978247 
 978206 
 
 D. 10" 
 
 Sine. 
 
 6.4 
 
 6.4 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 Tang. 
 
 9.485339 
 485791 
 486242 
 486693 
 487143 
 487593 
 488043 
 488492 
 488941 
 489390 
 489838 
 
 9.490-'86 
 490733 
 491180 
 491627 
 4926 73 
 492519 
 49296i 
 493410 
 493854 
 494299 
 
 9.494743 
 495186 
 495630 
 496073 
 496515 
 496957 
 497399 
 497841 
 468282 
 498722 
 
 9.499163 
 499603 
 600042 
 500481 
 600920 
 501359 
 501797 
 502235 
 502672 
 503109 
 
 9.503646 
 503982 
 504418 
 504854 
 505289 
 605724 
 506159 
 606593 
 607027 
 607460 
 
 9.507893 
 508326 
 508759 
 509191 
 609622 
 6100.34 
 510485 
 610916 
 511346 
 511776 
 Co tans'. 
 
 f' Oegrces. 
 
 I). 10" | Cotang. I N. sine N. cos.' 
 ;:29237J95630 
 
 75.3 
 75.2 
 75.1 
 75.1 
 75.0 
 74.9 
 74.9 
 74.8 
 74.7 
 74.7 
 74.6 
 74.6 
 74.5 
 74.4 
 74.4 
 74.3 
 74.3 
 74.2 
 74.1 
 74.0 
 74.0 
 74.0 
 73.9 
 73.8 
 73.7 
 73.7 
 73.6 
 73.6 
 73.6 
 73.4 
 73.4 
 73.3 
 73.3 
 73.2 
 73.1 
 73.1 
 73.0 
 73.0 
 72.9 
 72.8 
 72.8 
 72.7 
 72.7 
 72.6 
 72.5 
 72.5 
 72.4 
 72.4 
 72.3 
 72.2 
 2.2 
 72.1 
 72,1 
 72.0 
 71.9 
 71.9 
 71.8 
 71.8 
 71.7 
 71.6 
 
 10.514661 
 
 614209 
 
 613758 
 
 513307 
 
 512857 
 
 612407 
 
 611957 
 
 511508 
 
 511059 
 
 510610 
 
 510162, 
 
 10.609714 
 
 509267 
 
 508820 
 
 508373 I 
 
 507927 ! 
 
 507481 : 
 
 507035 
 
 506590 
 
 506146 ; 
 
 505701 I 
 
 10.505257! 
 
 604814 
 
 604370 
 
 503927 ! 
 
 503485 i 
 
 603043; 
 
 6026011" 
 
 602159 it 
 
 501718: 
 
 50127811 
 
 10.600837 I 
 
 500397' 
 
 499958!! 
 
 499519 L 
 
 499080!^ 
 
 498641 i 
 
 498203 ij 
 
 497765 ij 
 
 497328 I 
 
 496891 Ij 
 
 10.496454!: 
 
 496018 ji 
 
 495582!! 
 
 495146!; 
 
 494/11 
 
 494276 m 
 
 493841 i 
 
 493407 1 1 
 
 49297311 
 
 492640 
 
 10.492107 ;i 
 
 491674|! 
 
 49124111 
 
 490809;! 
 
 4903 78;; 
 
 489946 
 
 489515 
 
 489084 
 
 488654 
 
 488224 
 
 29265195622 
 29293 !956 13 
 2932195605 
 29348195596 
 293 76;95588 
 '29404 95579 
 29432 95571 
 2946095562 
 29487195554 
 295 15 [95545 
 29543 !95536 
 2957195528 
 29599 '955 19 
 ! 29626 i955 11 
 ■ 29654 195502 
 29682 '95493 
 29710 954«5 
 29737195476 
 29766195467 
 29793I95459 
 29821 195450 
 29849J95441 
 2987() 195433 
 29904 !95424 
 29932:95416 
 29960195407 
 
 2998, 
 30015 
 300-i3 
 300/1 
 30098 
 30126 
 30154 
 30182 
 30209 
 3023/ 
 30265 
 30292 
 30320 
 3034b 
 30376 
 30403 
 304^1 
 30^59 
 
 95398 
 95389 
 95380 
 95372 
 95363 
 95354 
 95345 
 95337 
 95328 
 95319 
 95310 
 95301 
 95293 
 95284 
 95276 
 95266 
 95257 
 95248 
 
 30 486 [95210 
 30o 14 95231 
 30542 195222 
 305/0195213 
 3059 / 95204 
 30625|95195 
 
 30663 
 306«0 
 
 95186 
 951/7 
 
 30 1 08 95168 
 3073b 195 169 
 
 3iJ7(Ji 
 30791 
 30819 
 30846 
 308/4 
 3090- 
 
 Tang. 
 
 N. CO--. N. 
 
 95150 
 95142 
 95133 
 95124 
 95115 
 95106 
 
 60 
 69 
 58 
 57 
 56 
 55 
 64 
 53 
 52 
 51 
 60 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
TABLE II. 
 
 Log. Sines awl Tangents. (18°) Natural Sines. 
 
 39 
 
 Tang. |D. lO 
 
 .511776 
 OV2206 
 5l'2blio 
 613064 ^ 
 
 5i3y'2i I;;-* 
 
 614349 ■'^••* 
 614777 
 516204 
 615631 
 616057 
 .616484 
 516910 
 517335 
 517761 
 6181«5 
 618610 
 619034 
 519458 
 51 9882 
 6'20306 
 
 Sine. 
 
 .489982 
 490371 
 4W0'<59 
 491147 
 491535 
 491922 
 492308 
 492t)95 
 493081 
 49346() 
 493851 
 .494236 
 494^21 
 495005 
 495388 
 4957 ?2 
 496154 
 496537 
 496919 
 497301 
 497()82 
 .4980<)4 
 498444 
 498825 
 499204 
 499584 
 499963 
 600342 
 600721 
 601099 
 6014;6 
 .501854 
 602231 
 60260? 
 602984 
 603360 
 603736 
 604110 
 604485 
 604860 
 605234 
 .505608 
 605981 
 506354 
 506727 
 60-099 
 60/471 
 60-843 
 508214 
 508585 
 508956 
 1.509326 
 - 609()9() 
 510065 
 510434 
 610803 
 611172 
 611540 
 511907 
 512275 
 512642 
 Cosiuf. 
 
 p. 10" 
 
 64.8 
 '64.8 
 64.7 
 64.6 
 64,6 
 64.5 
 64.4 
 MA 
 64.3 
 64. Q 
 64.2 
 64.1 
 64.1 
 64.0 
 63.9 
 63.9 
 63.8 
 63.7 
 63.7 
 63.6 
 63.6 
 63.6 
 63.4 
 63.4 
 63.3 
 63.2 
 63.2 
 63.1 
 63.1 
 63.0 
 62,9 
 62.9 
 62.8 
 62.8 
 62,7 
 62,6 
 62.6 
 62.5 
 62.5 
 62.4 
 62.3 
 62,3 
 62.2 
 62,2 
 62.1 
 62.0 
 62.0 
 61.9 
 61.9 
 61.8 
 61.8 
 61.7 
 61.6 
 61.6 
 61,5 
 61.5 
 61.4 
 61.3 
 61.3 
 61.2 
 
 Co.sinc 
 
 D. 1(F 
 
 .978206 
 9781»)5 
 978124 
 978083 
 978042 
 978001 
 977959 
 977918 
 977877 
 977836 
 977794 
 
 .977762 
 977711 
 977669 
 977628 
 977686 
 977644 
 977603 
 977461 
 977419 
 977377 
 
 .977335 
 977293 
 977251 
 977209 
 977167 
 977125 
 977083 
 977041 
 976999 
 97()967 
 976914 
 976872 
 976830 
 976787 
 976745 
 976702 
 976660 
 976617 
 976574 
 976532 
 
 .976489 
 97644<> 
 976404 
 976361 
 97ti318 
 976275 
 976232 
 976189 
 976146 
 970103 
 .9760()0 
 97b017H 
 9769/4 
 975930 
 976887 
 975844 
 975800 
 976.67 
 975714 
 975670 
 
 Sine. 
 
 6.8 
 6.8 
 6.8 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 6.9 
 7.0 
 7.0 
 7.0 
 7.0 
 7.0 
 7.0 
 7.0 
 7,0 
 7.0 
 7,0 
 7,0 
 7.0 
 7,0 
 7.0 
 7.0 
 7.0 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7,1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.1 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7.2 
 7,2 
 7.2 
 7.2 
 
 71.6 
 71.6 
 71.6 
 
 71.3 
 71.2 
 71.2 
 71.1 
 71.0 
 71.0 
 70.9 
 70,9 
 70.8 
 70.8 
 70.7 
 70.6 
 70.6 
 70.5 
 
 .620/28 r^,.' , 
 /0.4 
 
 70.3 
 
 621151 
 521573 
 621996 
 622417 
 522838 
 623259 
 623680 
 624100 
 624520 ^^, „ 
 
 .524939 r^.-r. 
 
 625359 
 
 625778 
 
 626197 
 
 626616 
 
 627033 
 
 627451 
 
 627868 
 
 628286 
 
 62tt702 
 .629119 
 
 629635- 
 
 629950 
 
 630366 
 
 630781 
 
 631196 
 
 631611 
 
 532025 
 
 6324u9 
 
 632853 
 .633266 
 
 633(i79 
 
 634092 
 
 634504 
 
 534916 
 
 535328 
 
 635739 
 
 5ci6150 
 
 636561 
 
 5;i6972 
 
 Ck)tan};, 
 
 il Degrees, 
 
 70.3 
 70.3 
 70.2 
 70.2 
 70.1 
 70.1 
 70.0 
 
 69.9 
 69.8 
 69.8 
 69.7 
 69,7 
 69.6 
 69.6 
 69.5 
 69.5 
 69.4 
 69,3 
 69.3 
 69.3 
 69.2 
 69.1 
 69.1 
 69.0 
 69.0 
 68.9 
 68.9 
 68.8 
 68.8 
 68.7 
 68.7 
 68.6 
 68,6 
 68.5 
 68.5 
 68.4 
 
 Cotang. N. pine.jX. cos.l 
 
 10.488224 
 487794 
 487365 
 486936 
 486507 
 486079 
 485b51 
 485223 
 484796 
 484369 
 483943 
 
 10.483516 
 483090 
 482665 
 482239 
 481816 
 481390 
 480966 
 480542 
 480118 
 479695 
 
 10.479272 
 478S49 
 478427 
 478005 
 477583 
 477162 
 476741 
 476320 
 475900 
 475480 
 
 10.475061 
 474641 
 474222 
 473803 
 473385 
 472967 
 472549 
 472132 
 471715 
 471298 
 
 10.470881 
 470465 
 470050 
 469634 
 4(i9219 
 468804 
 468389 
 4679/5 
 467561 
 467147 
 
 10.466734 
 466321 
 4b5908 
 465496 
 465084 
 464672 
 464261 
 463850 
 463439 
 463028 
 
 "r^ng- 
 
 30902196106,60 
 3092995097 i 69 
 30957 195088 i 58 
 309h6;950;9[67 
 31012195070166 
 31040 950<)1 155 
 
 31068 
 31095 
 31123 
 
 95052 I 54 
 95043 53 
 95033 62 
 31151195024; 51 
 3] 178 95015 50 
 
 : 31206 
 i 31233 
 
 31261 
 ; 31289 
 131316 
 
 31344 
 
 i 3 1372194052 
 
 31399 
 31427 
 131454 
 
 95006 49 
 94J>97 48 
 94988 { 47 
 94979 I 46 
 94970 1 46 
 94961 I 44 
 
 94943 
 94933 
 94924 
 : 31482 94915 
 i31.010 94906|38 
 '31537194897 I 37 
 ,31505 94888 36 
 ; 3159394878 I 36 
 31620 948t>9 34 
 3 1648 94860 i 33 
 31t)75 94851 '32 
 ,31703 94842131 
 
 3173094832 30 
 
 31758 94823 29 
 
 31786 94814 28 
 
 r3l813 94805|27 
 
 31841 94795! 26 
 
 ,31868 94786125 
 
 131896 94777 I 24 
 
 131923 94768123 
 
 ;i 31951 94758 122 
 
 ,31979 94749121 
 
 32006 94740 I "20 
 
 [32034 94730 19 
 
 1132061 94i21 18 
 
 ! 32089 94712 17 
 
 i| 321 16 94702 16 
 
 li 32144 94693 I 15 
 
 Ii3217194(.84!l4 
 
 ; 132 199 94674; 13 
 
 32227194666 I 12 
 
 32250 94(i56 i 1 1 
 
 32282 94()46 
 
 32309 94637 
 
 32337 :94627 
 
 32364'94()18 
 
 32392:94009 
 
 32419i94599 
 
 32447 J94590 
 
 32474-94580 
 
 32502194571 
 
 32529 94501 
 
 3255. 94552 
 
 N. W8. N'.sine. 
 
TABLE II. 
 
 Log. Sines and Tangent8. (20°) Natural Sineb. 
 
 41 
 
 SJue. ID. 10' 
 
 1.53405-2 
 634399 
 634745 
 63509-2 
 6354:J8 
 635783 
 5361-29 
 53j474 
 63(i8lH 
 .53/l(i3 
 53750? 
 
 I.537S51 
 538 194 
 638538 
 
 -638880 
 539-2-23 
 53951)5 
 539907 
 540-249 
 540590 
 540931 
 
 L54l-2r'2 
 541013 
 541953 
 642-293 
 54-2(i3-2 
 54-2971 
 543310 
 643()49 
 543987 
 5443-25 
 
 I.544<)(i3 
 545000 
 545338 
 545074 
 54001 1 
 54()347 
 54()ti83 
 647019 
 547354 
 647089 
 
 1.5480-24 
 648359 
 548093 
 5490-27 
 549300 
 549093 
 5500-.0 
 650359 
 550i)9'2 
 6510-24 
 651350 
 56108; 
 55-2018 
 55-2349 
 55-2<>80 
 653010 
 553341 
 5.53070 
 654000 
 564329 
 
 Cosine. 
 
 57.8 
 67.7 
 57.7 
 57-7 
 57.0 
 57.6 
 57.5 
 57.4 
 57.4 
 57.3 
 57.3 
 57.2 
 57.2 
 57.1 
 57.1 
 57.0 
 57.0 
 50.9 
 50.9 
 50.8 
 50.8 
 50.7 
 50.7 
 50.0 
 50.0 
 50.5 
 50.5 
 60.4 
 50.4 
 50.3 
 50.3 
 50.2 
 5(i . 2 
 50.1 
 50.1 
 50.0 
 50.0 
 55.9 
 55.9 
 55.8 
 65.8 
 55.7 
 55.7 
 55.0 
 55.0 
 55.5 
 55.6 
 55.4 
 55.4 
 55.3 
 65.3 
 55.2 
 65.2 
 65.2 
 55. 1 
 55. 1 
 65.0 
 55.0 
 54.9 
 54.9 
 
 Co.«iue. 
 
 D. 10' 
 
 9.9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9.9 
 9 
 9 
 v9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9.0 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9.9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9.9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9.9 
 9 
 9 
 
 y 
 
 9 
 9 
 9 
 9 
 9 
 9 
 
 2980 
 
 2940 
 
 •2894 
 
 -2H48 
 
 -2802 
 
 2755 
 
 2/0 < 
 
 2003 
 
 •2()17 
 
 -2570 
 
 2524 
 
 2478 
 
 -2431 
 
 2385 
 
 2338 
 
 ■2-291 
 
 2245 
 
 2198 
 
 2151 
 
 2105 
 
 2058 
 
 -2011 
 
 1904 
 
 1917 
 
 1870 
 
 1823 
 
 1770 
 
 1729 
 
 1082 
 
 1035 
 
 1588 
 
 1540 
 
 1493 
 
 1440 
 
 1398 
 
 1351 
 
 1303 
 
 1-260 
 
 1-208 
 
 1101 
 
 1113 
 
 1000 
 
 1018 
 
 09 70 
 
 0922 
 
 0874 
 
 0827 
 
 0779 
 
 0731 
 
 0083 
 
 iJ036 
 
 0586 
 
 0538 
 
 0490 j 
 
 0442! 
 
 0094 1 y 
 
 0345 I **■ 
 0297 
 0249 
 0200 
 0152 
 .Sin«. 
 
 rang. 
 
 9.501(X)0 
 
 6til459 
 
 601S51 
 
 602244 
 
 502030 
 
 5f>3028 
 
 503419 
 
 50.3811 
 
 604202 
 
 604592 
 
 604983 
 9.5«.5373 
 
 6ti5703 
 
 600153 
 
 600542 
 
 600932 
 
 607320 
 
 607709 
 
 508098 
 
 608480 
 
 608873 
 9.509-201 
 
 509048 
 
 570035 
 
 5704-22 
 
 670809 
 
 671195 
 
 571581 
 
 571907 
 
 672352 
 
 672738 
 9.6731-23 
 
 573607 
 
 573892 
 
 674270 
 
 574000 
 
 576044 
 
 575427 
 
 575810 
 
 570193 
 
 670570 
 9.576958 
 
 577341 
 
 677723 
 
 5/8104 
 
 5/8480 
 
 578807 
 
 579*248 
 
 6790-29 
 
 5800*J9 
 
 580389 
 9.580709 
 
 581149 
 
 6815-28 
 
 681907 
 
 682280 
 
 582W;5 I 
 
 583043 I 
 
 6r;31-22l":;" 
 
 68?.800 ^'^•" 
 
 D. 10" 
 
 05.6 
 05.4 
 t)5.4 
 ()5 3 
 05.3 
 ()5.3 
 ()5.2 
 05.2 
 05.1 
 05.1 
 05.0 
 05.0 
 04.9 
 04.9 
 04.9 
 04 8 
 04.8 
 04.7 
 04.7 
 04.0 
 04.0 
 04.5 
 04.5 
 04.5 
 04.4 
 04.4 
 04.3 
 04.3 
 04.2 
 04.2 
 04.2 
 04.1 
 04.1 
 04.0 
 04.0 
 03.9 
 03.9 
 03.9 
 03.8 
 03.8 
 03.7 
 03.7 
 63.6 
 03.6 
 63.6 
 63.5 
 03.6 
 03.4 
 03.4 
 03.4 
 03.3 
 63.3 
 63.2 
 63.2 
 63.2 
 63.1 
 63.1 
 
 684177 
 Co tang. 
 
 62.9 
 
 Cotang. N. sinr.iN. co8. 
 
 10.438934 
 438541 
 438149 
 437750 
 437364 
 430972 
 430581 
 430189 
 436798 
 436408 
 435017 
 
 10.434027 
 434237 
 433847 
 433468 
 433008 
 432080 
 432-291 
 431902 
 431514 
 431127 
 
 10.430739 
 430352 
 4299o5 
 429578 
 4-29191 
 428805 
 428419 
 428033 
 427048 
 427-202 
 
 10.426877 
 426493 
 4-26108 
 4257-24 
 425340 
 424950 
 4-24673 
 424190 
 423807 
 4234-24 
 
 10.4-23041 
 422669 
 -42*2277 
 421896 
 421614 
 421133 
 420752 
 420371 
 419991 
 419011 
 
 10.419*231 
 418851 
 418472 
 , 418093 
 4^17714 
 417335 
 410957 
 410578 
 410200 
 4158-23 
 
 Tang. ' 
 
 34202|939()9 1 60 
 
 34229;93959 69 
 
 34-257(93949 
 
 34-284193939 
 
 343111939*29 
 
 343^9193919 
 
 34360 
 34393 
 344-21 
 34418 
 34475 
 34503 
 34530 
 34557 
 34584 
 3401-2 
 34039 
 34000 
 34094 
 34721 
 34748 
 
 93909 
 93899 
 93889 
 93879 
 93809 
 93859 
 93849 
 93839 
 93829 
 93819 
 i>380y 
 93799 
 93789 
 93779 
 93709 
 
 34775 93759 
 34803193748 
 3483093738 
 34857 93728 
 34884 93718 
 34912|93708 
 34939l93«i98 
 3490o!93()88 
 
 34993 
 35021 
 35048 
 350-5 
 35102 
 36130 
 3515< 
 35184 
 35211 
 35239 
 35-200 
 35293 
 36320 
 3534/ 
 35375 
 3540^ 
 3545^9 
 35450 
 35484 
 
 930/ i 
 930t)7 
 93057 
 93()47 
 93037 
 93(i20 
 93010 
 93000 
 93590 
 9o5o5 
 33675 
 93506 
 93665 
 9o644 
 93534 
 935*24 
 93614 , 
 93603 i 14 
 93493 j 13 
 35511193483' 12 
 1 36538193472 ' 1 1 
 36505I93402 '< 1 D 
 13569*^193452 
 
 35019 
 
 3504 
 
 35()V4 
 
 36701 
 
 367*28 
 
 35756 
 
 3578J 
 
 35810 
 
 3583 
 
 I N. COS. N.sinf 
 
 93441 
 
 93431 
 
 ^J34*20 
 
 93410 
 
 93400 I 
 
 93.*i89 I 
 
 93379 
 
 933t)8 
 
 93368 
 
 69 Degrees. 
 
42 
 
 ^og. Sinea and Tangents. (21°) Natural Sines. 
 
 TABLE IE. 
 
 Sine. 
 
 9.554329 
 654658 
 654987 
 6553 1 5 
 655(>43 
 555971 
 656-299 
 556626 
 556953 
 557280 
 557606 
 557932 
 558258 
 558583 
 658909 
 659234 
 65y55S 
 659883 
 56020/ 
 560531 
 560855 
 
 9.561178 
 661501 
 561824 
 562146 
 562468 
 562790 
 563112 
 563433 
 663755 
 564075 
 56439() 
 564716 
 565036 
 665356 
 565676 
 665995 
 566314 
 566632 
 666951 
 667269 
 567587 
 567904 
 568222 
 568539 
 668856 
 669172 
 569488 
 5()9804 
 570120 
 670435 
 
 9.570751 
 571066 
 571380 
 571695 
 672009 
 572323 
 572636 
 572950 
 573263 
 673575 
 Cosine. 
 
 D. 10' 
 
 54.8 
 54.8 
 54.7 
 54.7 
 64.6 
 64.6 
 54.6 
 54,5 
 54.4 
 54.4 
 54.3 
 54.3 
 54.3 
 54.2 
 54.2 
 54.1 
 54.1 
 64.0 
 54.0 
 53.9 
 53.9 
 53.8 
 53.8 
 53.7 
 53.7 
 53.6 
 53.6 
 53.6 
 53.5 
 53.5 
 53.4 
 53.4 
 63.3 
 63.3 
 63.2 
 53.2 
 53.1 
 63.1 
 63.1 
 53.0 
 53.0 
 52.9 
 52.9 
 52.8 
 52.8 
 52.8 
 52.7 
 52.7 
 52.6 
 52.6 
 52.6 
 52.5 
 52.4 
 62.4 
 52.3 
 52.3 
 52.3 
 52.2 
 52.2 
 52.1 
 
 Cosine. 
 
 9.970152 
 970103 
 970055 
 970006 
 969957 
 969909 
 969860 
 9()98ll 
 969762 
 969714 
 9()9665 
 .9()9616 
 969567 
 969518 
 969469 
 969^^20 
 969370 
 969321 
 969272 
 969223 
 969173 
 
 9.969124 
 969076 
 969025 
 968976 
 968926 
 968877 
 968827 
 968777 
 968728 
 968678 
 
 9.968628 
 968578 
 968528 
 968479 
 968429 
 968379 
 968329 
 968278 
 968228 
 968178 
 
 9.968128 
 968078 
 968027 
 967977 
 967927 
 967876 
 967826 
 967775 
 967725 
 967674 
 
 9.967624 
 967573 i 
 967522 
 967471 
 967421 i 
 967370 
 967319 
 967268 
 967217 
 96.166 
 "sine; 
 
 D. 10' 
 
 8.1 
 8.1 
 8.1 
 8.1 
 8.1 
 8.1 
 8.1 
 
 1 
 1 
 1 
 2 
 2 
 2 
 2 
 2 
 2 
 8.2 
 8.2 
 8.2 
 8.2 
 8.2 
 8.2 
 8.2 
 8.2 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.3 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.4 
 8.5 
 
 Tang. 
 
 9.584177 
 ,584565 
 584932 
 585309 
 586686 
 586062 
 686439 
 686815 
 687190 
 687666 
 587941 
 
 9.588316 
 588691 
 6890o6 
 589440 
 689814 
 590188 
 690562 
 590935 
 691308 
 591681 
 692054 
 592426 
 692798 
 593170 
 593542 
 593914 
 594286 
 694666 
 595027 
 595398 
 
 9.695768 
 69ol38 
 696508 
 596878 
 597247 
 597616 
 597985 
 598364 
 598722 
 599091 
 
 9.599459 
 699827 
 600194 
 600562 
 600929 
 601296 
 601662 
 602029 
 602395 
 602761 
 
 9.603127 
 603493 
 603858 
 604223 
 604588 
 601953 
 605317 
 605682 
 t 06046 
 606410 
 Cotang. 
 
 D. 10"i Cotanir. iN.sine. N. cc8, 
 
 62.9 
 62.9 
 62.8 
 62.8 
 62.7 
 62.7 
 62.7 
 62.6 
 62.6 
 62.6 
 162.5 
 62.5 
 62.4 
 62.4 
 62.3 
 62.3 
 62.3 
 62.2 
 62.2 
 62.2 
 62.1 
 62.1 
 62.0 
 62.0 
 61.9 
 61.9 
 61.8 
 61.8 
 61.8 
 61.7 
 61.7 
 61.7 
 61.6 
 61.6 
 61.6 
 61.6 
 61.5 
 61.5 
 61.4 
 61.4 
 61.3 
 61 
 61 
 61 
 61 
 61 
 61 
 61.1 
 61.0 
 61.0 
 61.0 
 60.9 
 60.9 
 60.9 
 60.8 
 60.8 
 60.7 
 60.7 
 60.7 
 60.6 
 
 10.415823 
 415445 
 415068 
 414691 
 414314 
 413938 
 413561 
 413185 
 412810 
 412434 
 412059 
 
 10.411684 
 411309 
 410934 
 410560 
 410186 
 409812 
 409438 
 409065 
 408692 
 408319 
 
 10.407946 
 407574 
 407202 
 406829 
 406458 
 406086 
 405715 
 405344 
 404973 
 404602 
 
 10.404232 
 403862 
 403492 
 403122 
 402753 
 402384 
 402015 
 401646 
 401278 
 400909 
 
 10.400541 
 400173 
 399806 
 399438 
 399071 
 398704 i 
 398338 
 397971 
 397605 
 397239 
 
 10.396873 
 396507 
 396142 
 395777 
 395412 
 395047 
 394683 
 394318 
 393954 
 393590 
 Tang. 
 
 35837 
 i 35864 
 35891 
 
 93368 
 93348 
 93337 
 
 35918193327 
 35945 93316 
 36973 [93306 
 36000 ;93295 
 ' 36027 ;93285 
 136054193274 
 |3608Ii93264 
 
 36108 
 136135 
 136162 
 : 36190 
 
 3621' 
 
 93253 
 93243 
 93232 
 93222 
 93211 
 
 136244 93201 
 113627193190 
 1 1 36298 J93 180 
 ; 36325^3169 
 !!36352;93159 
 ii36379|93l48 
 1 1 36406 193 137 
 136434193127 
 I' 364611931 16 
 i 36488 193106 
 136515193095 
 i 36542193084 
 1136569193074 
 !|36596 930t)3 
 1136623 93052 
 3665093042 
 I ! 3667-7 93031 
 ; 36704193020 
 ij 36731 193010 
 |l36758j92999 
 36785 192988 
 ; 368 12 192978 
 1 1 36839 92967 
 36867 92956 
 36894 92945 
 3692192936 
 136948 92926 
 36975 92913 
 ! 37002 92902 
 ,37029 92892 
 37056 92881 
 37083 92870 
 37110192859 
 37137 92849 
 '37164 92838 
 3719192827 
 37218 92816 
 37245 92805 
 37272 b2794 
 37299 92784 
 37326192773 
 37353 92762 
 3738092751 
 37407 92740 
 
 37434 
 37461 
 
 92729 
 92718 
 
 N. COS. N.pinc. 
 
 60 
 59 
 58 
 57 
 66 
 65 
 64 
 53 
 52 
 51 
 I 50 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 3 7 
 36 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 26 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 16 
 14 
 13 
 12 , 
 11 
 
 10 ; 
 
 9 
 
 8 
 
 7 
 
 68 Degrees. 
 
TABLE II. 
 
 Log. Sines and Tangents. (22°) Natural Sinea. 
 
 T\ 
 
 Sine. 
 
 9.573575 
 67SyS8 
 574200 
 574512 
 5748-24 
 575 13() 
 675447 
 
 Drm 
 
 52. 
 52. 
 52. 
 51. 
 51. 
 51, 
 51. 
 
 57(iUfi9!?J- 
 57(i379^,- 
 57GG89J^, 
 9.57(^999 I °, 
 577309 ' ?, 
 577t)18i?|- 
 577927 ? ■ 
 578236 ""^ 
 
 578545 
 578853 
 579162 
 579470 
 579777 
 9.580085 
 580392 
 680099 
 581005 
 581312 
 681618 
 681924 
 582229 
 582535 
 582840 
 583145 
 
 583449 1 
 583754 ""■ 
 684058 J"" 
 584361^"- 
 584665 I J,, 
 584968 i J,,' 
 
 50. 
 50. 
 50. 
 50. 
 60. 
 50, 
 150. 
 50. 
 
 685574 
 685877 
 9.686179 
 586482 
 686783 
 687085 
 587386 
 587688,, , 
 68 7989!?^; 
 688289 I ?^ 
 588590 ! ?" 
 588890 ^" 
 
 9.689190 
 689489 
 589789 
 590088 
 .•90387 
 690686 
 5909-^4 
 691282 
 69158(> 
 5;,»1878 
 Co.sine. 
 
 CJosine. 
 
 D. lU" 
 
 9.967166 
 967115 
 967064 
 967013 
 966961 
 9W)910 
 96^i859 
 966808 
 966756 
 96<)705 
 9(i6t)53 
 
 9.9()()602 
 9()655() 
 966499 
 966447 
 9<>6395 
 96(>344 
 96()292 
 
 -966240 
 
 ^9t;6188 
 966136 
 
 9.9(>6085 
 9t)6>i33 
 965981 
 965928 
 965876 
 965824 
 965772 
 965720 
 965t)68 
 965()15 
 9655(J3 
 965511 
 965458 
 9()6406 
 966353 
 965301 
 965248 
 965195 
 965143 
 965090 
 965037 
 964984 
 9t)4931 
 964879 
 964826 
 964773 
 964719 
 964666 
 964613 
 964560 
 9G4507 
 964454 
 964400 
 964347 
 964294 
 964240 
 964187 
 964133 
 W)4080 
 964026 
 
 Sin.i 
 
 8.5 
 8.6 
 8.6 
 8.5 
 8.6 
 8.6 
 8.6 
 8.5 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.6 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.7 
 8.8 
 8.8 
 8.8 
 8.8 
 8.8 
 8.8 
 8.8 
 8.8 
 
 8.9 
 8.9 
 8.9 
 8.9 
 8.9 
 8.9 
 8.9 
 8.9 
 
 Tan g. 
 
 |.60<>410 
 60<J773 
 607137 
 607500 
 607863 
 608225 
 608688 
 608960 
 609312 
 609674 
 610036 
 
 1.610397 
 610759 
 611120 
 611480 
 611841 
 612201 
 612561 
 612921 
 613281 
 613641 
 
 1.614000 
 614359 
 614718 
 616077 
 615435 
 615793 
 616151 
 616509 
 616867 
 617224 
 
 .617582 
 617939 
 618295 
 618652 
 619008 
 619364 
 619721 
 620076 
 620432 
 620787 
 
 1.621142 
 621497 
 621852 
 622207 
 622561 
 622915 
 623269 
 623623 
 623976 
 624330 
 
 1.624683 
 625036 
 625388 
 625741 
 K26093 
 o26445 
 626797 
 627149 
 627601 
 627852 
 
 Co tang. 
 iK-greeff . 
 
 D. 10^^ 
 
 60.6 
 60.6 
 60.5 
 60.5 
 60.4 
 60.4 
 60.4 
 60.3 
 60.3 
 60.3 
 60.2 
 60.2 
 60.2 
 60.1 
 60.1 
 60.1 
 60.0 
 60.0 
 60.0 
 59.9 
 59.9 
 59.8 
 59.8 
 59.8 
 69.7 
 59.7 
 59.7 
 59.6 
 59.6 
 59.6 
 59.5 
 59.5 
 59.5 
 59.4 
 59.4 
 59.4 
 59.3 
 59.3 
 59.3 
 59.2 
 59.2 
 59.2 
 59.1 
 59.1 
 59.0 
 59.0 
 69.0 
 58.9 
 58.9 
 58.9 
 58.8 
 58.8 
 58.8 
 58.7 
 58.7 
 58.7 
 58.6 
 58.6 
 58.6 
 58,5 
 
 37461 
 37488 
 37515 
 37542 
 37569 
 37595 
 37622 
 
 Cota ng. NTsine. N, cos. 
 
 10.393590 
 
 393227 
 
 392S63 
 
 392500 
 
 392137 
 
 391775 
 
 391412 
 
 391050 
 
 390688 
 
 390326 
 
 3899()4 
 10.389603 
 
 389241 
 
 388880 
 
 388520 
 
 388159 
 
 387799 
 
 387439 
 
 387079 
 
 386719 
 
 .386369 
 10.3860(»0 
 
 385641 
 
 385282 
 
 384923 
 
 384565 
 
 384207 
 
 383849 
 
 383491 
 
 383133 
 
 382776 
 10-382418 
 
 3820ol 
 
 381i05 
 
 381348 
 
 380992 
 
 380636 
 
 380279 
 
 379924 
 
 379568 
 
 379213 
 1 J. 378858 
 
 378503 
 
 378148 
 
 377793 
 
 377439 
 
 377085 
 
 376731 
 
 376377 
 
 376024 
 
 375670 
 10-37.5317 
 
 374964 
 
 374612 
 
 374259 
 
 373907 
 
 373555 
 
 37G263 
 
 373851 
 
 372499 
 
 372148 
 
 92718 
 92707 
 92697 
 92U86 
 92ti75 
 92664 
 92653 I 54 
 37649|92()42 53 
 37676192631 62 
 37703|92620 51 
 37730|92609 50 
 37757 92598 149 
 37784'92.587 48 
 37811'92576|47 
 37838 92565 4(i 
 37865 92554 45 
 37892:92543 144 
 37919 92532 143 
 3794692521 | 42 
 37973 92510 141 
 37999 92499 
 38026^92488 
 38053 92477 
 ' 38080 92466 
 1 3810792455 
 38134 92444:35 
 J381()l 92432 134 
 38188 92421 | 33 
 38215 92410 32 
 I 38-241! 92399 1 31 
 1 38268 92388 1 30 
 38295 9-2377 129 
 : 383-22 92366 28 
 i 38349192355 27 
 ; 3837692343 j 26 
 38403 92332 ' 25 
 138430 92321 [24 
 138456 92310 123 
 138483 9-2299 22 
 
 38510 92287 
 
 |38537!92276 
 j 38564 92265 
 
 38591 92254 
 38617 92243 
 '3864492231 
 3867192220 115 
 38<i98 92209 '14 
 
 13 
 
 Tang. 
 
 38725 92198 
 , 38752 92-i 86! 12 
 38778 92175 I 11 
 : 38805 92164 
 38832 92152 
 38859 92141 
 38S86 92130 
 38912 92119 
 38939 92107 
 38966 92096 
 38993 92085 
 39020 92ij, 3 
 , 3904«) 9-2062 
 J39073|92050 
 I X. cos J N.8ine. 
 
44 
 
 Log. Sinea and Tangents. (23°) Natural Sines. 
 
 TABLE II, 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 Sine. 
 
 9.591878 
 59217<i 
 592473 
 592770 
 6930o7 
 593363 
 593659 
 593955 
 594251 
 594547 
 594S42 
 
 9.595137 
 695432 
 595727 
 59J021 
 596315 
 596609 
 696903 
 597196 
 597490 
 597783 
 
 9.598U75 
 598368 
 598660 
 598952 
 599244 
 699536 
 599827 
 600118 
 600409 
 600/00 
 
 9.600990 
 601280 
 601570 
 601860 
 602150 
 602439- 
 602728 
 60301 7 
 603305 
 603594 
 
 9.603882 
 604170 
 604457 
 604/45 
 605032 
 605319 
 605606 
 605892 
 60.il 79 
 60-O465 
 
 9.606751 
 607036 
 607322 
 607607 
 60/892 
 608177 
 608461 
 60874) 
 609029 
 609313 
 C!osine. | 
 
 D. W 
 
 49.6 
 
 49.5 
 
 49.5 
 
 49.5 
 
 49.4 
 
 49.4 
 
 49.3 
 
 49.3 
 
 49.3 
 
 49.2 
 
 49.2 
 
 49.1 
 
 49.1 
 
 49.1 
 
 49.0 
 
 49.0 
 
 48.9 
 
 48.9 
 
 48.9 
 
 48.8 
 
 48.8 
 
 48.7 
 
 48.7 
 
 48.7 
 
 48.6 
 
 ^18.6 
 
 48.6 
 
 48.5 
 
 48.5 
 
 48.4 
 
 48.4 
 
 48.4 
 
 48.3 
 
 48.3 
 
 48.2 
 
 48.2 
 
 48.2 
 
 48.1 
 
 48.1 
 
 48.1 
 
 48.0 
 
 48.0 
 
 47.9 
 
 47.9 
 
 47.9 
 
 47 8 
 
 47.8 
 
 47.8 
 
 47.7 
 
 47.7 
 
 47.6 
 
 47.6 
 
 47.6 
 
 47.5 
 
 47-5 
 
 47.4 
 
 47.4 
 
 47.4 
 
 47 3 
 
 47.3 
 
 Cosine. 
 
 1.964026 
 963972 
 903919 
 9G3865 
 963811 
 963757 
 963704 
 963650 
 963596 
 963542 
 963488 
 1.963434 
 963379 
 963325 
 963271 
 963217 
 • 963163 
 963108 
 9()3054 
 962999 
 962945 
 .962890 
 962836 
 962781 
 962727 
 9(r2672 
 962617 
 962562 
 962508 
 962453 
 962398 
 .962343 
 9()2288 
 962233 
 962178 
 962123 
 962067 
 962012 
 961957 
 961902 
 961846 
 .961791 
 961735 
 961680 
 961624 
 961569 
 961513 
 961458 
 961402 
 961346 
 961290 
 ,961235 
 961179 
 961123 
 961037 
 961011 
 9(i0955 
 9t>0899 
 960843 
 960786 
 9 JO 730 
 
 "I Sine. 
 
 8.9 
 
 8.9 
 
 8.9 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.0 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9.1 
 
 9 2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9,3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.3 
 
 9.4 
 
 9.4 
 
 'i' ang. 
 
 9.627852 
 628203 
 628554 
 628905 
 629256 
 6296U6 
 629956 
 630306 
 630656 
 631005 
 631355 
 .631704 
 632053 
 632401 
 632760 
 633098 
 633447 
 633795 
 634143 
 634490 
 634838 
 9.635186 
 635532 
 635879 
 636226 
 636572 
 636919 
 637266 
 637611 
 637956 
 638302 
 ). 638647 
 638992 
 639337 
 639682 
 640027 
 640371 
 640716 
 641060 
 641404 
 641747 
 1.642091 
 642434 
 642777 
 643120 
 643463 
 643806 
 644148 
 644490 
 644832 
 645174 
 ,64.5516 
 645857 
 64(il99 
 646540 
 646881 
 647222 
 647562 
 647903 
 648243 
 648583 
 Cotang. 
 
 D. 10"| Coking, N. 8ine.|N. cos, 
 
 58.5 
 
 68.5 
 
 58.5 
 
 58,4 
 
 58.4 
 
 68.3 
 
 58,3 
 
 58.3 
 
 58.3 
 
 68.2 
 
 68.2 
 
 68.2 
 
 68,1 
 
 58.1 
 
 68.1 
 
 58,0 
 
 68.0 
 
 58.0 
 
 57.9 
 
 67.9 
 
 67.9 
 
 57.8 
 
 67.8 
 
 67.8 
 
 67.7 
 
 57.7 
 
 57.7 
 
 57.7 
 
 67,. 6 
 
 57.6 
 
 57.6 
 
 57,6 
 
 57.5 
 
 67.5 
 
 67,4 
 
 67,4 
 
 57.4 
 
 57.3 
 
 67.3 
 
 57.3 
 
 57.2 
 
 67.2 
 
 57.2 
 
 57.2 
 
 57.1 
 
 67.1 
 
 67,1 
 
 57.0 
 
 67,0 
 
 67.0 
 
 56.9 
 
 66.9 
 
 56.9 
 
 56.9 
 
 56.8 
 
 56.8 
 
 66,8 
 
 56.7 
 
 56.7 
 
 56.7 
 
 10.372148 
 371797 
 371446 
 371095 
 370745 
 370394 
 370044 
 369694 
 369344 
 368995 
 368645 
 10.368296 
 367947 
 367599 
 367250 
 36(J902 
 366553 
 366205 
 365857 
 36.5510; 
 305162 I 
 10.364815 
 364468 - 
 364121 I 
 363774 ! 
 363428 ! 
 363081 j 
 362735 I 
 362389 1 
 362044 ! 
 3616981 
 10.361353 j 
 3610081 
 360663 I 
 360318! 
 359973 i 
 369629 I 
 359284 1 
 368940 
 358596 
 358253 
 10.367909 
 357566 
 357223 
 356880 
 366537 
 356194 
 355852 
 355510 
 355168 
 354826 
 10.354484 
 364143 
 353801 
 353460 
 353119 
 352778 
 352438 
 352097 
 351757 
 351417 
 "fi^ng^ 
 
 39G73 192050 
 
 39100 
 39127 
 391.53 
 39180 
 39207 
 
 92039 
 92028 
 92016 
 92005 
 91994 
 
 39234 [91 982 
 39260 91971 
 
 39287 
 39314 
 39341 
 39367 
 39394 
 39421 
 39448 
 39474 
 
 91959 
 91048 
 
 91914 
 91902 
 91891 
 91879 
 
 39501 91868 
 ■91866 
 
 3952H 
 39555 
 39581 
 39608 
 39636 
 39661 
 39688 
 
 91845 
 91833 
 91822 
 91810 
 91799 
 91787 
 
 3971591775 
 3974191764 
 
 ! 39768 
 I 39795 
 ; 39822 
 ; 39848 
 I 39875 
 i 39902 
 I 39928 
 i 39955 
 , 39982 
 : 40008 
 ! 40035 
 
 400(i2|9l625 
 40088|91613 
 40115191601 
 40141 i91590 
 40168J915.8 
 40195 91666 
 
 40221 
 40248 
 40275 
 
 40301 91519 
 40328:91508 
 40355191496 
 40381191484 
 40408191472 
 4043491461 
 40461191449 
 40488'91437 
 40514 91425 
 
 40641 
 40567 
 40594 
 
 406211913/8 
 40(J47 91366 
 40674 913,55 
 N. COS. N.sine, 
 
 919S6 150 
 91925 I 49 
 48 
 47 
 46 
 46 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 36 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 
 8 I 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
 91752 
 91741 
 91729 
 91718 
 91706 
 91694 
 91683 
 91671 
 91660 
 91648 
 91636 
 
 91555 
 91543 
 91531 
 
 91414 
 91402 
 91o90 
 
 66 Degrees. 
 
Log. Sines and Tangents. (24°) Natural Sines. 
 
 45 
 
 
 
 1 
 2 
 3 
 
 t 
 
 6 
 
 7 
 
 8 
 
 9 
 
 TO 
 
 II 
 
 1-2 
 
 13 
 
 14 
 
 15 
 
 U> 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 2« 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 3(i 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 64 
 
 55 
 
 5(i 
 
 67 
 
 58 
 
 59 
 
 60 
 
 Sino. 
 
 GiD313 
 (iO'Jo'Ji 
 609 SSO 
 t)10U)4 
 (ilUU,- 
 t)10.'29 
 yilJl-2 
 611294 
 611576 
 61 1858 
 fi]-2l\W 
 6124-21 
 612702 
 61298^ 
 613264 
 613545 
 613825 
 614105 
 614385 
 6146(i6 
 614944 
 9.615223 
 615502 
 616781 
 616060 
 616338 
 616616 
 616894 
 617172 
 617450 
 617727 
 6I80OI 
 618281 
 618558 
 6IS834 
 619110 
 6193»6 
 619662 
 619938 
 620213 
 620488 
 
 19.620763 
 621038 
 621313 
 621587 
 621861 
 622135 
 622409 
 622082 
 62-29 ')6 
 623229 
 
 19.6-23512 
 6-23774 
 624047 
 624319 
 624591 
 624863 
 625135 
 62540J 
 6'2.5u7 7 
 625948 
 
 D. 10' 
 
 47.3 
 
 47.2 
 
 47.2 
 
 47.2 
 
 47.1 
 
 47.1 
 
 47.0 
 
 47.0 
 
 47.0 
 
 4o.9 
 
 46.9 
 
 46.9 
 
 46.8 
 
 46.8 
 
 46.7 
 
 46.7 
 
 46.7 
 
 46.6 
 
 46.6 
 
 4«}.6 
 
 46.5 
 
 46.5 
 
 46.5 
 
 46.4 
 
 46.4 
 
 46.4 
 
 46.3 
 
 46.3 
 
 46.2 
 
 46.2 
 
 46.2 
 
 4<).l 
 
 4(i.l 
 
 46,1 
 
 46.0 
 
 46.0 
 
 46.0 
 
 45.9 
 
 45.9 
 
 45.9 
 
 45.8 
 
 45.8 
 
 45.7 
 
 45.7 
 
 45.7 
 
 45,6 
 
 45.6 
 
 45.6 
 
 45,6 
 
 45.5 
 
 45.6 
 
 45.4 
 
 45.4 
 
 45.4 
 
 45.^ 
 
 45.3 
 
 46.3 
 
 45.2 
 
 45,2 
 
 45.2 
 
 Cosini;. 
 
 1.960730 
 960674 
 9;K)6 18 
 9o0561 
 960505 
 9;)0448 
 960392 
 96J335 
 960279 
 960222 
 960166 
 1.960109 
 960052 
 959995 
 959938 
 959882 
 9598-25 
 959768 
 959711 
 959654 
 959596 
 .959539 
 959482 
 959425 
 95'J3(>8 
 959310 
 959-253 
 959195 
 959138 
 959081 
 9690-23 
 .958965 
 958908 
 958850 
 958792 
 958 i 34 
 958677 
 958619 
 958561 
 958503 
 958445 
 .958387 
 958329 
 968271 
 958213 
 958154 
 968096 
 958038 
 957979 
 957921 
 957863 
 .957804 
 95774(> 
 95 7687 
 957628 
 95*570 
 957511 
 957452 
 957393 
 .■■5/335 
 957276 
 
 D. 10' 
 
 Cosin 
 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.4 
 9.5 
 y.6 
 9.6 
 9.5 
 9.5 
 9.6 
 9.5 
 9.5 
 9.5 
 9.5 
 9.5 
 9.6 
 9.5 
 9.5 
 9.6 
 9.6 
 9.6 
 9.6 
 
 Tiin;^. 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 9.6 
 
 9,6 
 
 9.6 
 
 9.7 
 
 9.7 
 9.7 
 9.7 
 9.7 
 9.7 
 9.7 
 9.7 
 9.7 
 9.8 
 9.8 
 9.8 
 9.8 
 9.8 
 9.8 
 9.8 
 9.8 
 
 9.648583 
 648923 
 649263 
 64^)602 
 64994-2 
 650281 
 650i)20 
 650959 
 651-297 
 651636 
 651974 
 65-2312 
 652()50 
 65-2988 
 65332(> 
 6536(>3 
 654000 
 654337 
 654174 
 655011 
 655348 
 9.655684 
 656020 
 66(i356 
 656692 
 657028 
 667364 
 667699 
 658034 
 658369 
 I 658(04 
 [9.659039 
 659373 
 659708 
 660042 
 66(X3 76 
 660710 
 661043 
 6<il377 
 661710 
 662043 
 9.662376 
 662709 
 663042 
 663375 
 663707 
 664039 
 664371 
 664703 
 GtioOlid 
 665366 
 9.665(j97 
 666029 
 666360 
 666691 
 667021 
 6(i7352 
 667682 
 668013 
 668343 
 t)686i2 
 
 D. 10 
 
 Coti.nt,'. 
 
 56.6 
 
 56.6 
 
 56,6 
 
 56 6 
 
 56.5 
 
 56.5 
 
 59.6 
 
 56.4 
 
 56.4 
 
 .56.4 
 
 56.3 
 
 56.3 
 
 56.3 
 
 56.3 
 
 56,2 
 
 56 2 
 
 56,2 
 
 56,1 
 
 56,1 
 
 56.1 
 
 56.1 
 
 56.0 
 
 56.0 
 
 56.0 
 
 56.9 
 
 55.9 
 
 55.9 
 
 56.9 
 
 55.8 
 
 55.8 
 
 65.8 
 
 55.8 
 
 55.7 
 
 55.7 
 
 55.7 
 
 55.7 
 
 55.6 
 
 65.6 
 
 66.6 
 
 55.5 
 
 .55,5 
 
 55,5 
 
 55,4 
 
 56.4 
 
 55.4 
 
 65,4 
 
 55,3 
 
 65.3 
 
 55.3 
 
 55.3 
 
 65.2 
 
 55.2 
 
 55.2 
 
 65.1 
 
 55.1 
 
 5.5. 1 
 
 55. 1 
 
 55.0 
 
 55.0 
 
 55.0 
 
 Cotaiij;. 
 
 10.351417 
 351077' 
 350737: 
 360398 
 350J58 
 349; 19 
 3493^0 
 349041 I 
 348703 
 3483«)4 ! 
 3480-26 
 
 10.347688 
 347350 
 347012; 
 346674 i 
 346337 > 
 346000 ; 
 345663 
 345326 
 344989 
 344652 j! 
 
 10.344316 r 
 
 343980 
 
 343644 
 
 343308' 
 
 342972 
 
 342636' 
 
 342301 
 
 341966 ^ 
 
 341631 I 
 
 341-296;. 
 
 10.340961 '; 
 340627 
 3402921' 
 3399581 
 3396241 
 339290 ii 
 338957 I i 
 338623 i 
 3382901 1 
 337957'; 
 
 10.337624 
 3372911: 
 336958 jl 
 3366251 
 336293,' 
 335961 
 3366291 
 335297 j 
 334966 : 
 334634 i 
 
 10.3343031 
 333971 ! 
 3S3()->0 
 33o309 ! 
 3329,9! 
 33'2648 1 
 332318 
 331987 
 331667 
 331328 
 
 4080* lb 1295 
 408o3;91283 
 408()U9l2/2 
 40b86;9l260 
 40913 91248 
 40939 91:^36 
 4n966!9r224 
 4099^2 a 1-21-.; 
 41019191200 
 41045 '91 18» 
 410/2191176 
 41098;^ 1 164 
 411i>5 9ll52 
 4115191140 
 41178 911-28 
 4120491 116 
 4123191104 
 41257 9109-,^ 
 41284 91080 
 413109106b 
 41337 91056 
 4136391044 
 41390 91032 
 41416 9lO*-iO 
 4144391008 
 41469 9099() 
 41496 90984 
 41522 9097-2 
 41649 90960 
 41576 90948 
 41602 9093b 
 41628 90924 
 141655 90911 
 i 41681 9089y 
 I41707;9u88i 
 ,41734:9()8;5 
 '417(i(l90863 
 : 4178 7 90b51 
 41813 90t5oU 
 41840 908-^6 
 41866 9O0U 
 41892 90bO-z 
 419199U-90 
 41945 9077 b 
 4 1.972 90 1 66 
 41998 907 5o 
 42U24 90i-il 
 42051 90729 
 42077 90/1/ 
 42104 90704 
 421o0 9069-^ 
 421.56 9U-bO 
 42l8o9lX>t)b 
 42209 90655 
 42-235 9vX>4o 
 422(i2;90.,31 
 N . c<is.|N.siiic. 
 
 65 iHHrreee. 
 
46 
 
 Log. Sines and Tangents. (25"^) Natural Bines. 
 
 TABLE II. 
 
 "Tang. ID. 10"i Cotang. | N.f"iiio.|N. cos.i 
 
 422(i2|90G31 Uo 
 
 Sine. D. 10" Cosine. • D. 10" 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 19 
 50 
 61 
 62 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 
 .625948 
 626219 
 626490 
 6267,0 
 627030 
 627300 
 627570 
 627840 
 628109 
 628378 
 628647 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 44 
 
 44 
 
 44 
 
 144 
 
 144 
 628916 I T^ 
 
 629185 Jl 
 
 629453 l^ 
 
 629721 
 629989 
 630257 
 630524 
 630792 
 631059 
 631326 
 
 .631593 
 631859 
 632125 
 632392 
 632658 
 632923 
 633189 
 633454 
 633719 
 633984 
 
 .634249 
 634514 
 634778 
 635042 
 635303 
 635570 
 635834 
 636097 
 636360 
 636623 
 
 .63i)8S6 
 637148 
 637411 
 637673 
 637935 
 638197 
 638458 I 
 638/20 
 638981 
 639242 
 
 |.6;J9503 
 639764 
 640024 
 640284 
 640544 
 6 10804 
 641ij64 
 641324 
 641584 
 641842 
 
 Cosine. 
 
 9.957276 
 957217 
 957158 
 957099 
 957040 
 956981 
 956921 
 956862 
 956803 
 956744 
 956684 
 
 9.956625 
 956566 
 9565 J6 
 956447 
 956387 
 956327 
 956268 
 956208 
 956148 
 956089 
 
 9.956029 
 955969 
 955909 
 955849 
 955789 
 955729 
 955669 
 955609 
 955548 
 955488 
 955428 
 955368 
 955307 
 955247 
 9r,5l86 
 955126 
 955065 
 9550J5 
 954944 
 954883 
 .954823 
 954762 
 954701 
 954640 
 954579 
 954518 
 954457 
 954396 
 954335 
 954274 
 
 9 954213 
 954152 
 954090 
 954029 
 953968 
 953906 
 953845 
 953783 
 953722 
 953660 
 
 9.8 
 9.8 
 9.8 
 9.8 
 9,8 
 9.8 
 9.9 
 9.9 
 9.9 
 9 9 
 
 9.9 
 9.9 
 9.9 
 9 9 
 
 9.y 
 
 9.9 
 9.9 
 9.9 
 10. 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10.0 
 10. 
 10 
 10 
 10 
 10 
 10 
 10 
 
 10.1 
 10.1 
 10.1 
 10.1 
 10.1 
 10.1 
 10.1 
 10.1 
 10.1 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.2 
 10.3 
 
 9.668673 
 669002 
 609332 
 669{)61 
 669991 
 670320 
 670649 
 670977 
 671306 
 671634 
 671963 
 
 9.672291 
 672619 
 672947 
 673274 
 673602 
 673929 
 674257 
 674584 
 674910 
 675237 
 
 9.675564 
 675890 
 676216 
 676543 
 676869 
 677194 
 677520 
 677846 
 678171 
 678496 
 
 9.678821 
 679146 
 679471 
 679795 
 680120 
 680444 
 680768 
 681092 
 681416 
 681740 
 
 9.682063 
 68-'387 
 682710 
 683033 
 683366 
 683679 
 684001 
 684324 
 684646 
 684968 
 1.685290 
 685612 
 685934 
 686255 
 686577 
 686898 
 687219 
 687540 
 6878(>1 
 688182 
 Cotang. 
 
 65.0 
 54.9 
 64.9 
 54.9 
 54.8 
 54.8 
 64.8 
 54.8 
 54.7 
 54.7 
 64.7 
 54.7 
 64.6 
 54.6 
 64.6 
 54.6 
 54.5 
 64.6 
 54,5 
 54.4 
 64.4 
 64.4 
 54.4 
 64.3 
 54.3 
 64.3 
 54.3 
 54.2 
 54.2 
 54.2 
 64.2 
 64.1 
 54.1 
 64.1 
 54.1 
 54.0 
 54.0 
 54.0 
 64.0 
 53.9 
 63.9 
 53.9 
 53.9 
 53.8 
 53.8 
 63.8 
 53.8 
 53.7 
 53.7 
 53.7 
 
 &). 331327 
 
 330998 ■ 
 330368 
 330339 
 330009 ; 
 329680 
 329351 ' 
 329023 
 328694 ' 
 328366 i 
 328037 I 
 
 10.3277091 
 327381 i 
 327053 ! 
 326726 ; 
 326398 ■ 
 326071 
 325743 : 
 325416 1 
 325090 
 324763' 
 
 10.324436! 
 324110 
 323784' 
 323457: 
 323131 
 322806: 
 322480! 
 322154 
 321829 i 
 321 504 : 
 
 10.321179 
 320854' 
 320529 
 320206 
 319880 
 319556 
 319232, 
 318908 
 318584 
 318260 
 
 10.317937 
 317613 
 317290 
 316967 
 316644 
 316321 
 315999 
 315676 
 315354 
 -o 7 315032 
 °^i'r 10.314710 
 314388 
 
 63.6 
 
 53.6 
 
 53.6 
 
 53 
 
 63 
 
 53 
 
 53 
 
 53 
 
 63 
 
 4228b 
 42315 
 42341 
 42367 
 42394 
 42420 
 42446 
 42473 
 42499 
 42525 
 42652 
 42578 
 42604 
 42631 
 42667 
 42683 
 42709 
 42736 
 42762 
 42788 
 42816 
 42841 
 4286/ 
 42894 
 
 yj613 !69 
 90()06 I 68 
 90594 1 57 
 90582 i 66 
 90669 { 65 
 90557 1 54 
 90545 j 63 
 90532 ' 52 
 90520; 61 
 905(i7 I 50 
 90495 1 49 
 90483 , 48 
 904/0147 
 90458 i 46 
 90446 1 45 
 90433 1 44 
 90421 !43 
 90408 1 42 
 
 90396 
 90383 
 90371 
 90368 } 38 
 90346 37 
 
 90334 
 
 42920J90321 
 42946 y030ii 
 42972 90296 
 42999 90284 
 43025 90271 
 43051 90259 
 43077 90246 
 43104i90233 
 43 130j 90221 
 43156190208 
 431S2'90196 
 43209190183 
 43235 90171 
 43261 90158 
 43287190 146 
 43313190133 
 43340190120 
 43366190108 
 43392190095 
 43418|900ei2 
 434451900/0 
 
 43471 
 43497 
 43623 
 43549 
 
 90057 
 9004c 
 90032 
 90019 
 
 43575 90007 
 
 314066; 
 
 313745 
 
 313423 
 
 313102 
 
 312781 
 
 3124o0 
 
 312139 
 
 3118181 
 
 43602 
 .43, 2b 
 43654 
 43680 
 4370O 
 43733 
 43/59 
 43785 
 
 89994 
 89981 
 89968 
 89956 
 8994c 
 89930 
 89918 
 89905 
 
 Tang. 
 
 ^3bli;8yb92 
 43837 89879 
 N. cos. .\.8ir<'. 
 
 fi4 Deirrctis. 
 
Lo}?. Sines and Tangents. (26°) Natural Sines. 
 
 47 
 
 Sine. 
 
 
 1 
 o 
 
 3 
 4 
 5 
 G 
 7 
 8 
 9 
 lU 
 U 
 1-2 
 13 
 14 
 15 
 Hi 
 17 
 IH 
 1!) 
 20 
 21 
 22 
 23 
 24 
 26 
 26 
 27 
 28 
 29 
 30 
 31 
 3-2 
 33 
 34 
 35 
 36 
 
 3; 
 
 3rt 
 
 3y 
 
 40 
 41 
 42 
 4^J 
 44 
 45 
 46 
 47 
 48 
 
 4y 
 
 60 
 61 
 62 
 63 
 64 
 55 
 66 
 6< 
 58 
 
 6y 
 
 60 
 
 D. lO'l Cosine. D. lo" 
 
 9.641842 
 642101 
 6423ti0 
 642618 
 642877 
 643135 
 643393 
 643()50 
 643908 
 644165 
 (>44423 
 
 *J 644tJ80 
 644936 
 645193 
 645450 
 645706 
 615962 
 646218 
 646474 
 646729 
 64<i984 
 
 9.647240 
 647494 ! 
 647749 
 648004 
 64.S258 
 648512 
 648766 
 649020 
 6492? 4 
 64:i527 
 
 9. 649 /hi 
 
 650J34j^:j 
 650287 If 
 650539 ; 7Z 
 650,92,*'; 
 
 651044 ;^r 
 
 651297 ;4j 
 
 6515 49:^;; 
 65ihoo,;r. 
 
 652052 ^^ 
 
 9.65-.^304 
 652555 
 652806 
 65305 7 
 653308 
 65o558 ; 
 65381.8 
 6540J9 
 654;iOJ 
 (i54558 , 
 
 £ ,6.34808 , 
 655058 j 
 65530/ j 
 655556 I 
 (»5;J8U5 I 
 65o054 
 656302 \ 
 65*i5Jl i 
 (i'iyt'i'J'J ; 
 65.017 j 
 Cos i lie. I 
 
 9.953660 
 953699 
 953537 
 953475 
 953413 
 953362 
 953290 
 953228 
 953166 
 963104 
 953042 
 
 9.952980 
 952918 
 952856 
 952793 
 952731 
 952669 
 952606 
 952544 
 952481 
 952419 
 
 9.952356 
 952294 
 952231 
 952168 
 952106 
 952043 
 951980 
 951917 
 951&54 
 951791 
 951728 
 951665 
 951602 
 951639 
 951476 
 951412 
 951349 
 961286 
 9512-.^2 
 951159 
 961090 
 951032 
 950968 
 950905 
 95J841 
 9507/8 
 950714 
 950650 
 950586 
 950522 
 .950458 
 950394 
 950350 
 950->66 
 95J202 
 950 13h 
 9500 i 4 
 950011) 
 949945 
 949861 
 
 9.96 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.5 
 
 10.5 
 
 10.6 
 
 10.5 
 
 10.6 
 
 10.5 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10.5 
 
 10.6 
 
 10,6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10,6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10.6 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 10.7 
 
 l\i.7 
 
 Tang. 
 
 D. 10' 
 
 63.4 
 53.4 
 63.4 
 53.3 
 53.3 
 63.3 
 53.3 
 53.3 
 63.2 
 63.2 
 53.2 
 63.1 
 53.1 
 53.1 
 
 .688182 
 
 (588502 
 
 688823 
 
 689143 
 
 6894^>3 
 
 689783 
 
 690103 
 
 690423 
 
 690742 
 
 691062 
 
 691381 
 .691700 
 
 692019 I 
 
 692338 I 
 
 692o66 L.> , 
 
 6929751^:^- 
 
 693293 °^-\ 
 
 693612 ^3-^ 
 
 693930 I °^H 
 
 694248 
 
 694566 
 ,694883 
 
 695201 
 
 695518 
 
 695836 
 
 696153 
 
 696470 
 
 696787 
 
 697103 
 
 697420 
 
 697736 
 .698053 
 
 63.0 
 53.0 
 52.9 
 62.9 
 52.9 
 62.9 
 62.9 
 52.8 
 [52.8 
 [52.8 
 !52.8 
 152.7 
 152.7 
 
 698369 ^'i'l 
 
 52.6 
 52.6 
 52.6 
 52.6 
 62.6 
 52.6 
 
 698(i86 
 699001 
 699316 
 699632 
 (i99947 
 700-63 
 70vi6i8 
 700893 
 
 •.701-.^08 
 701523 
 701837 
 702162 
 70.;4b6 
 702,80 
 703095 
 703409 
 703,23 
 704036 
 
 1. 701350 
 704O63 
 704977 
 705290 
 705(i03 
 705916 
 70i/i28 
 70t641 
 70v*h54 
 70i lo6 
 
 fcliiiig- 
 (41 iH'grees. 
 
 52.5 
 52.6 
 62.4 
 52.4 
 62.4 
 52.4 
 52.4 
 52.3 
 52.3 
 52.3 
 52.3 
 52.2 
 52.2 
 52.2 
 52.2 
 52.2 
 52.1 
 5-J.l 
 52.1 
 52 . 1 
 52.1 
 
 CoUmg. N. sine. N. eo«. 
 
 10.311818 
 311498 
 311177 
 310857 
 310537 
 310217 
 309897 
 309577 
 309258 
 308938 
 308619 
 
 10.308300 
 307981 
 307662 
 307344 
 307026 
 306707 
 306388 
 306070 
 305752 
 305434 
 
 10.305117 
 304799 
 304482 
 3041()4 
 303847 
 30J5oO 
 303213 
 302897 ' 
 302680 
 302264 : 
 
 10-301947 
 301631 ' 
 301315 
 300999; 
 300()84 
 300368 
 300053 
 299737 : 
 -299422 ' 
 299107 i 
 
 10-2 :8<92! 
 'z98477 j 
 298163 
 2:r,848 ■ 
 29 < 634' 
 29,220 
 29()9U6 
 2i'6691 ; 
 ic9(i277 I 
 2959(54 I 
 295650 
 295337 I 
 295023 I 
 294710; 
 294:^97 j 
 2940h4 ! 
 t:93772 i 
 293469 1 
 29M46 
 292834 
 Tang. 
 
 89879100 
 89867 69 
 89854 58 
 89841 157 
 898-28 I 56 
 89816 155 
 43994 89803 64 
 44020 89790 53 
 89777 ! 52 
 44072 897(i4j 61 
 44098 89752 60 
 89739 49 
 89726 48 
 89713 
 89.00 
 89(i87 
 89674 
 89()62 
 89649 j 42 
 144333 89636 1 41 
 i 44359 89(>23 ' 40 
 8!)610i39 
 89697 38 
 89584 I 37 
 89571 136 
 89558 1 35 
 89545 1 34 
 89532 \ 33 
 89519 132 
 89506 31 
 89493 1 30 
 89480 '' 29 
 
 44124 
 44151 
 44177 
 44203 
 44229 
 44255 
 44281 
 44307 
 
 I 44385 
 44411 
 : 44437 
 {44464 
 ' 44490 
 j 445 16 
 j 44542 
 I 446(.8 
 I 44594 
 1 44620 
 I 44646 
 144672 
 ! 44698 
 I 44724 
 44750 
 i 44776 
 ! 44802 
 
 89467 128 
 89454 27 
 
 10 
 
 89441 [26 
 89428 : 26 
 8b4l6|24 
 89402 23 
 
 44828 89389,22 
 l44854jH9376t2\ 
 i 44880189303 ; 20 
 ' 44 91 R. 189360. 19 
 |4493-j!89337!18 
 i4495hih;M24 ; 17 
 1 449841893 II ; 16 
 ■45010:89298 15 
 
 460y(ii89285 14 
 
 450,,'jjh92,2;i3 
 
 45088ih9259 12 
 i45114i89245|ll 
 i 45 1 40l89232 1 
 '4516(iih9219| " 
 |45l92ih9206 
 l46218!^9193 
 
 46343 h9 180 
 !4.o2()9,IS9l')7 
 
 45296 h9 153 
 '453vl:hyl40 
 
 45347 h9 127 
 
 453 73 
 
 S. COS. N.sine. 
 
 89114 
 89101 
 
48 
 
 liOfT. SiiKis and Tangents. (27°) Natural Sines. 
 
 TABLE II. 
 
 a 
 
 
 
 1 
 
 2 
 3 
 4 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 9 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 
 40 I 
 
 41 ;9 
 42 
 43 
 44 
 45 
 16 
 47 
 48 
 49 
 50 
 61 
 62 
 53 
 54 
 55 
 5() 
 67 
 58 
 b'J 
 60 
 
 657017 
 ()67295 
 657542 
 657790 
 658037 
 658284 
 658531 
 658778 
 659 J25 
 659271 
 659517 
 659763 
 660J09 
 6(i0255 
 660501 
 660/46 
 660991 
 66123a 
 661481 
 661726 
 6619/0 
 .662214 
 662459 
 662703 
 66294tj 
 663190 
 663433 
 663677 
 663920 
 664163 
 6t)4406 
 .6641)48 
 664891 
 665133 
 665375 
 665617 
 665859 
 666100 
 666342 
 ()66583 
 6(i6824 
 .667065 
 667305 
 667516 
 667786 
 668027 
 668267 
 668o0<) 
 668746 
 668986 
 669225 
 669464 
 6t>9703 
 669942 
 6701K1 
 670419 
 670)58 
 670896 
 671134 
 671372 
 6/1609 
 Cfxsino. 
 
 I), lu' 
 
 41.3 
 
 41.3 
 
 41.2 
 
 41.2 
 
 41.2 
 
 41.2 
 
 41.1 
 
 41.1 
 
 41.1 
 
 41.0 
 
 41.0 
 
 41.0 
 
 40.9 
 
 40.9 
 
 40.9 
 
 40.9 
 
 40.8 
 
 40.8 
 
 40.8 
 
 40.7 
 
 40.7 
 
 40.7 
 
 40.7 
 
 40.6 
 
 40.6 
 
 40.6 
 
 40.5 
 
 40.6 
 
 40.5 
 
 40.5 
 
 40.4 
 
 40.4 
 
 40.4 
 
 40.3 
 
 40.3 
 
 40.3 
 
 40.2 
 
 40.2 
 
 40.2 
 
 40.2 
 
 40.1 
 
 40.1 
 
 40.1 
 
 40.1 
 
 40.0 
 
 40 
 
 40.0 
 
 39.9 
 
 39.9 
 
 39.9 
 
 39.9 
 
 39.8 
 
 39.8 
 
 39.8 
 
 39.7 
 
 39-7 
 
 39.7 
 
 39.7 
 
 39 6 
 
 39.6 
 
 L 
 
 L/U.smo. 
 
 .9498.S1 
 94Jrilj 
 949752 
 94 ^688 
 949623 
 949558 
 949494 
 949429 
 949364 
 949300 
 949235 
 
 1.949170 
 949105 
 949040 
 948975 
 948910 
 948845 
 948780 
 948715 
 948650 
 948584 
 
 1.948519 
 948454 
 948388 
 948323 
 948257 
 948192 
 948126 
 948060 
 947995 
 947929 
 
 1.947863 
 947797 
 947731 
 947666 
 947600 
 947533 
 947467 
 947401 
 947335 
 9472(>9 
 
 1.947203 
 947 '36- 
 947070 
 947004 
 946937 
 946871 
 946804 
 946738 
 94(>671 
 946604 
 
 ^ 946538 
 946471 
 946404 
 946337 
 946270 
 946203 
 946136 
 946069 
 94tJ002 
 945935 
 Sine. 
 
 8 
 8 
 ,8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 ,8 
 8 
 .9 
 .9 
 ,9 
 ,9 
 .9 
 .9 
 .9 
 ,9 
 ,9 
 ,9 
 ,9 
 ,9 
 ,0 
 ,0 
 .0 
 .0 
 ,0 
 ,0 
 .0 
 ,0 
 .0 
 .0 
 .0 
 .0 
 .0 
 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 .1 
 
 .2 
 
 .2 
 
 11.2 
 
 11.2 
 
 1). lu ' Colau-;. N. siiK; 
 
 9.707166 
 
 707478 
 
 707790 
 
 708102 
 
 708414 
 
 708726 
 
 709037 
 
 70J349 
 
 709660 
 
 709971 
 
 710282 
 9.710593 
 
 71 0904 
 
 711215 
 
 711525 
 
 711836 
 
 712146 
 
 712456 
 
 712766 
 
 713076 
 
 713386 
 9.713696 
 
 714005 
 
 714314 
 
 714624 
 
 714933 
 
 715242 
 
 715651 
 
 715860 
 
 716168 
 
 716477 
 9.716785 
 
 717093 
 
 717401 
 
 717709 
 
 718017 
 
 7183-5 
 
 718633 
 
 718940 
 
 719248 
 
 719555 
 9.719862 51.2 
 
 720169 
 
 720476 
 
 720783 
 
 721089 
 
 721396 
 
 721/02 
 
 722009 
 
 722315 
 
 722621 
 9,722927 
 
 723232 
 
 723538 
 
 723^44 
 
 724149 
 
 724454 
 
 724759 
 
 7250-)5 
 
 725369 
 
 725674 
 "Cotanj;. 
 
 52.0 
 52.0 
 52.0 
 52.0 
 51.9 
 51.9 
 51.9 
 51.9 
 51.9 
 61.8 
 61.8 
 51.8 
 61.8 
 51.8 
 51.7 
 51.7 
 61.7 
 61.7 
 51.6 
 61.6 
 61.6 
 51.6 
 51.6 
 61.5 
 51.6 
 61.5 
 61.5 
 51.4 
 61.4 
 61.4 
 61.4 
 51.4 
 61.3 
 61.3 
 61.3 
 61.3 
 61.3 
 51.2 
 61.2 
 61.2 
 
 61.2 
 51.1 
 61.1 
 51.1 
 51.1 
 61.1 
 61.0 
 51.0 
 51.0 
 51.0 
 61.0 
 60.9 
 50.9 
 50.9 
 50.9 
 60.9 
 50.8 
 60.8 
 60.8 
 
 N. cos, 
 
 89101 I 60 I 
 89J87 ■ 69 
 
 10.292834 45399 
 292522 45425 
 292210 45451 89074 
 291898 45477 89061 
 291586 45503 8904« 
 291274 45529 !890o5 
 2909;)3 45554 !890il 
 2901)51 45580 89U08 
 290340 45t)06 88995 
 290029 45632 188961 
 289718 45658188968 
 
 10.289407 i 46684J88955 
 289096; 4571 0|88942 
 288785 45736188928 
 288475' 45 762 J889 15 
 288164 45787 88902 
 28/854 458 13 188888 
 287644 45»39i888 75 
 
 286924 45891188848 
 286614' 45917188835 
 
 10 . 286304 45942188822 
 285995 45968188808 
 286686 : 45994188795 
 285376 ; 46020188782 
 285067': 4604688768 
 284758' 46072188 765 
 284449!; 46097 188741 
 284140; 46123188728 
 283832 i!46149j887 16 
 283523-46175188701 
 
 10.283215 4620188688 
 282907 j' 4622b 88674 
 282599 j! 46252 88661 
 282291 1146278 88647 
 2819831' 46304 88634 
 281676! 4()330 88620 
 
 281367 II 46365 
 
 88607 
 
 28 1 060!! 46381 
 
 280752/46407 
 
 280445 46433 
 
 10.280138 '4t)4o8 
 
 279»31 4u484 88539 
 279524 46510i88526 
 279217|i4lj536bh51 
 
 88593 
 
 88680 
 8S566 
 88653 
 
 278911!! 46561 
 
 278604 
 278298 
 27/991 
 277686 
 277379 
 10.277073 
 276768 
 
 46587 
 
 !4(;613 
 
 I 46639 
 j 46664 
 I 46690 
 j 467 16 
 46742 
 
 88499 
 88485 
 884/2 
 88458 
 88445 
 «843l 
 88417 
 88404 
 
 2764u2i: 46767 88390. 
 
 276156 
 2 75851 
 275546 
 275241 
 
 46793 8837/ 
 46819 88363 
 4684,^88349 
 4b87W 88336 
 274^)36 1 41,^96188322 
 274031 4jk>92ll88308 
 2 74326 4t)94/|88'.^95 
 
 Tanji 
 
 S. roK. -^-'-'iJ' 
 
 <)2 Df^gret'S. 
 
TABLE II. 
 
 Log. Sines and Tangents. (28°) Natural Sines. 
 
 49 
 
 
 I 
 
 2 
 3 
 4 
 6 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 l(i 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 21) 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 3G 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 Mi 
 47 
 48 
 49 
 60 
 31 
 i 52 
 53 
 54 
 55 
 5(> 
 67 
 68 
 69 
 60 
 
 Siuu. 
 
 .6716)9 
 671847 
 6720S4 
 672321 
 672558 
 672795 
 673032 
 6732,iH 
 673505 
 673741 
 673977 
 
 .674213 
 674448 
 674684 
 674919 
 675155 
 ()75390 
 675 i24 
 675859 
 6761)9 I 
 676328 
 
 .676562 
 67679o 
 677030 
 6772()4 
 677498 
 677731 
 677964 
 678197 
 678430 
 6786i)3 
 
 .678895 
 679128 
 679360 
 679592 
 679W24 
 680.)5t) 
 680288 
 680519 
 680750 
 68iJ982 
 
 .681213 
 ()81443 
 68lt)74 
 681905 
 682135 
 6823()5 
 ()82595- 
 68-^825 
 683055 
 683284 
 
 .683514 
 683743 
 6839-2 
 684201 
 684430 
 684()58 
 684887 
 6851 15 
 685343 
 685571 
 
 Oo.sine. 
 
 D. 10' 
 
 39.6 
 39.6 
 39.5 
 39.6 
 39.5 
 39.4 
 39.4 
 39.4 
 39.4 
 39.3 
 39.3 
 39.3 
 39.2 
 39.2 
 39.2 
 39.2 
 39.1 
 39.1 
 39.1 
 39.1 
 39.0 
 39.0 
 39.0 
 39.0 
 38.9 
 38.9 
 38.9 
 38.8 
 38.8 
 38.8 
 38.8 
 38.7 
 38.7 
 38.7 
 38.7 
 38.6 
 38.6 
 38.6 
 38.5 
 38.5 
 38.5 
 38.5 
 38.4 
 
 Cosini' 
 
 38.4 
 38.4 
 38.4 
 38.3 
 38.3 
 38.3 
 38,3 
 38.2 
 38.2 
 38.2 
 38.2 
 38.1 
 38.1 
 38.1 
 38.0 
 ;i8.0 
 38.0 
 
 1.945935 
 9458(iS 
 94580J 
 945733 
 94566;; 
 945598 
 9)5531 
 945464 
 945396 
 945328 
 945261 
 
 1.945193 
 945125 
 945058 
 944990 
 944922 
 944854 
 944786 
 944718 
 944()50 
 944582 
 
 .944514 
 94444<j 
 944377 
 944309 
 944241 
 944172 
 944104 
 944036 
 943967 
 943899 
 
 ^ 943830 
 943761 
 943(i93 
 943624 
 943555 
 943486 
 943417 
 943348 
 913279 
 943210 
 
 1.943141 
 943072 
 943003 
 942934 
 942864 
 942795 
 94272*) 
 94265() 
 942587 
 942617 
 
 .942448 
 942378 
 94230.8 
 94-J239 
 942169 
 942099 
 942029 
 941959 
 941889 
 941819 
 Sine. 
 
 D. 10" Tang. D. 10" Cotarig. N. sino.jN. co. 
 
 9.725674 
 725979 
 726284 
 72()588 
 72()892 
 727197 
 727501 
 727805 
 728109 
 728412 
 728716 
 
 9.72902C 
 729323 
 729626 
 729929 
 730233 
 730535 
 73.J838 
 731141 
 731444 
 731746 
 
 9.732048 
 732351 
 732653 
 732955 
 733257 
 733558 
 733860 
 734162 
 734463 
 734764 
 
 9.735066 
 735367 
 735668 
 735969 
 736269 
 736570 
 736871 
 737171 
 737471 
 737771 
 °;9. 738071 
 ° 738371 
 738671 
 738971 
 739271 
 739570 
 739870 
 740169 
 740468 
 740767 
 1.741066 
 741365 
 741664 
 741962 
 742261 
 742559 
 742858 
 74315!) 
 743454 
 743752 
 CotiiiTiir 
 
 50.8 
 50.8 
 50 7 
 50.7 
 50.7 
 50.7 
 50.7 
 50.6 
 50.6 
 50.6 
 50.6 
 50.6 
 50.5 
 50-5 
 50 5 
 50. 6 
 50.5 
 50.4 
 50.4 
 50.4 
 50.4 
 50.4 
 503 
 50.3 
 50.3 
 50.3 
 50. 3 
 50.2 
 50.2 
 50-2 
 50.2 
 50.2 
 50.2 
 50.1 
 50.1 
 50.1 
 50.1 
 60.1 
 50.0 
 50.0 
 50.0 
 50.0 
 50.0 
 49.9 
 49.9 
 49.9 
 49.9 
 49.9 
 49.9 
 49.8 
 49.8 
 49.8 
 49.8 
 49.8 
 49.7 
 49.7 
 49.7 
 49.7 
 49.7 
 49.7 
 
 10.274.326 
 274021 
 273716 
 273412 
 273108 
 272803 
 272499 
 272195 
 271891 
 2715881 
 271284 
 
 10.270980 
 270677 
 270374 
 270071 
 2697(>7 
 2694()5 
 269162 
 268859 
 268556 
 268264 
 
 10.267962 
 267649 
 267347 
 267045 
 26(i743 
 26(5442 
 266140 
 26.5838 
 265537 
 265236 
 
 10.264934 
 264633 
 264332 : 
 264031 ' 
 263731 
 263 130 
 263129 
 262829 
 262529 
 2(J2229 : 
 
 10.261929 
 261629 
 261329 
 261029 
 260729 
 260430 
 260130 
 259831 
 259532 
 259233 
 
 10.258934 
 258635 ' 
 258336 
 258038 
 257739 
 257441 
 257142 
 25(i844 
 25664() 
 256248 
 
 T:in), 
 
 4(5947 88295 
 46973 88281 
 4(5999188267 
 ^7024:882.54 
 4.050 88240 
 4707(i!88226 
 47 1 01 [882 13 
 47127188199 
 471.53188185 
 47178|88172 
 4720468158 
 47229 188144 
 472.55188130 
 47281 [881 17 
 47306188103 
 47332:88089 
 47358^8075 
 47383(88062 
 47409 '88048 
 47434,88034 
 47460 188020 
 47486 188006 
 47511 87993 
 47537 87979 
 475(52 8 79(55 
 47.588:87951 
 4761487937 
 47639187923 
 47(i65 '87909 
 47690 87896 
 477J6'87882 
 47741 '878(58 
 477(57 87854 
 47793 87840 
 47818 87826 
 47844 87812 
 47869:87798 
 4789587784 
 47920187770 
 47946i87756 
 4797187743 
 47997187729 
 48022 87715 
 48048 87701 
 48073 8768 7 
 4809987(573 
 48124 87(559 
 481 5U 87(545 
 48175 87(531 
 48201 87617 
 48226187603 
 48262187589 
 48277187575 
 483031875(51 
 4832818754(5 
 48364 187532 
 48379J87518 
 48405187504 
 48430l87490 
 48456187476 
 48 181(874(52 
 N. COS. In .sine. 
 
 61 Degrees. 
 
50 
 
 Log. Sines and Tangents. (29°) Natural Sines. 
 
 TABLE II. 
 
 D. 10" 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11, 
 12 
 131 
 14 
 1.5 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 56 
 56 
 57 
 58 
 
 5a 
 
 60 
 
 .685571 
 685799 
 686027 
 686254 
 686482 
 686709 
 686936 
 687163 
 687389 
 687616 
 687843 
 688069 
 688295 
 688521 
 688747 
 688972 
 689198 
 689423 
 689648 
 689873 
 690098 
 
 .690323 
 690o48 
 690772 
 690996 
 691220 
 691444 
 691668 
 691892 
 692116 
 692339 
 
 .692562 
 092785 
 693008 
 693231 
 693453 
 693676 
 693898 
 694120 
 694342 
 694564 
 
 .694786 
 695007 
 695229 
 695460 
 695671 
 
 . 695892 
 696113 
 696334 
 696554 
 696775 
 
 i,69b9i^5 
 697215 
 697435 
 697654 
 697874 
 698094 
 698313 
 6«8532 
 698761 
 698970 
 
 38.0 
 37.9 
 37.9 
 37.9 
 37.9 
 37.8 
 37,8 
 37.8 
 37.8 
 37.7 
 37.7 
 37.7 
 37.7 
 37.6 
 37.6 
 37.6 
 37.6 
 37.5 
 37.5 
 37.6 
 37.5 
 37.4 
 37.4 
 37.4 
 37.4 
 37.3 
 37.3 
 37.3 
 37.3 
 37.5 
 37.2 
 37.2 
 37.1 
 37.1 
 37.1 
 37.1 
 37.0 
 37.0 
 37.0 
 37.0 
 36.9 
 36 9 
 3h:.9 
 36.9 
 36.8 
 30.8 
 36.8 
 36.8 
 36.7 
 36.7 
 36.7 
 36.7 
 36.6 
 36.6 
 36.6 
 36.6 
 36.5 
 36.5 
 36.5 
 36.5 
 
 Co sine. } D. lO^^ I Tan g. 
 
 Cosine. 
 
 941819 
 941749 
 941679 
 941609 
 941539 
 941469 
 941398 
 941328 
 941258 
 941187 
 941117 
 ,941046 
 940975 
 940905 
 940834 
 940763 
 940693 
 940622 
 940561 
 940480 
 940409 
 ,940338 
 940267 
 940196 
 940126 
 940054 
 939982 
 939911 
 939840 
 939768 
 939697 
 .939626 
 939554 
 939482 
 939410 
 939339 
 939267 
 939196 
 939123 
 939062 
 938980 
 .938908 
 938836 
 938763 
 938691 
 938619 
 938547 
 938476 
 938402 
 938330 
 938268 
 .938186 
 938113 
 938040 
 937967 
 937895 
 937822 
 937749 
 937676 
 937604 
 937531 
 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.7 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 11.8 
 
 Sine. 
 
 11.8 
 11.8 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 11.9 
 12.0 
 12,0 
 12.0 
 12,0 
 12,0 
 12.0 
 12.0 
 12.0 
 12.0 
 12.0 
 12,0 
 12.0 
 12,1 
 12.1 
 12,1 
 12,1 
 12.1 
 12,1 
 12,1 
 12,1 
 12.1 
 12.1 
 12.1 
 12.1 
 
 9.743752 
 744050 
 744348 
 744645 
 744943 
 745240 
 745538 
 745836 
 746132 
 746429 
 746726 
 747023 
 747319 
 747616 
 747913 
 748209 
 748505 
 748801 
 749097 
 749393 
 749689 
 
 9.749985 
 750281 
 750576 
 760872 
 751167 
 751462 
 751757 
 762052 
 752347 
 752642 
 
 9.752937 
 763231 
 753626 
 763820 
 754115 
 754409 
 754703 
 754997 
 755291 
 765685 
 
 9.755878 
 766172 
 766466 
 756759 
 757062 
 757345 
 757638 
 767931 
 758224 
 758517 
 
 9.758810 
 -759102 
 759395 
 759687 
 759979 
 760272 
 760564 
 760856 
 761148 
 761439 
 
 D. 10' 
 
 Cotang. 
 
 49.6 
 49.6 
 49.6 
 49.6 
 49.6 
 49.6 
 49.5 
 49.5 
 49.5 
 49.6 
 49.5 
 49.4 
 49.4 
 49.4 
 49.4 
 49.4 
 49.3 
 49.3 
 49,3 
 49.3 
 49,3 
 49.3 
 49.2 
 49.2 
 49,2 
 49.2 
 49,2 
 49,2 
 49.1 
 49.1 
 49,1 
 49,1 
 49.1 
 49,1 
 49.0 
 49,0 
 49.0 
 49.0 
 49,0 
 49.0 
 48.9 
 48.9 
 48,9 
 48,9 
 48,9 
 48,9 
 48.8 
 48.8 
 48.8 
 48.8 
 48.8 
 48.8 
 48.7 
 48.7 
 48.7 
 48.7 
 48.7 
 48.7 
 48.6 
 48.6 
 
 10.256248! 48481 187462 
 255950 i,485fX):87448 
 265652 J 48532J87434 
 255,3551,48557 87420 
 255067! 148583:87406 
 
 Cotang. N. HJne.iN. COS. 
 
 254760 
 254462 
 264165 
 '253868 
 
 1 48608187391 
 4863487377 
 4865987363 
 48684'87349 
 
 253671 '48710,87335 
 253274 j 48735:87321 
 
 10.252977 i 4876 li87 306 
 252681 ! 48786187292 
 252384 14881187278 
 252087 1 48837187264 
 251791 148862:87250 
 251496 148888 87235 
 251199 l4891.3;87221 
 260903 j 48938:87207 
 250607 ! 48964187193 
 250311 j 48989 87178 
 
 10.250015 1149014 87164 
 
 249719 
 249424 
 249128 
 248833 
 248538 
 248243 
 247948 
 247633 
 247358 
 10.247063 
 246769 
 246474 
 
 49040;87150 
 49065J87136 
 49090,87121 
 49116|87107 
 4914187093 
 49166i87079 
 49192J87064 
 49217187050 
 49242J87036 
 4926887021 
 4929o87007 
 4931886993 
 2461801149344 86978 
 245885114936986964 
 245591 1 1 49394 86949 
 245297 1 14941986935 
 245003!; 49445 86921 
 2447091,49470,86906 
 244416 1149495 86892 
 10.244122^14952186878 
 243828 '149546 86863 
 2436361 49571 186849 
 243241 ji 49596 86834 
 242948! 149622,86820 
 242655 1 1 49647 186805 
 242362 ij 49672 86791 
 242069 1 49697186777 
 241776 [1 49723186762 
 2414831:49748 86748 
 10.241190: 49;73 86733 
 240898 49798 86719 
 2406051.49824 86704 
 240313 !| 49849 86690 
 240021!! 49874186676 
 2397281:49899,86661 
 239436 ii 49924 86646 
 239144 j I 49960;86632 
 238852 149975|86617 
 238561 ! 6000U 86603 
 
 Tang. Ii N. co>i.|N.f;ii 
 
 Degrees. 
 
TABLK IL 
 
 Log. Siaes aaJ Taagonta. (30°) Natural Sines. 
 
 51 
 
 Sine. D. 10' 
 
 ) G98!i70!„p 
 
 6M&2'6 ,!! 
 (i9!)8-t4 , ::^ 
 
 7Ui)jt)-2 ::^ 
 
 70U-280 ; ^,. 
 7004981^^ 
 700716'^^ 
 700933 rz 
 701151 ;3« 
 
 1.701368 i^,. 
 701583 I :J^' 
 70180-2 j^^' 
 70-2019 q.. ■ 
 702'236:^^- 
 70-2462 • Tf. 
 70-2ti«9.^^ 
 702885,^° 
 7031011^^' 
 703317 ^ 
 
 1.703533 !:? 
 703749 :;? 
 703964 r° 
 704179;:;? 
 704395 ! .^2 
 7046101:;? 
 7048-25!^? 
 
 705040;:;? 
 
 705-254;:^? 
 705469':;? 
 
 1.705683 :;? 
 
 705898 ^? 
 706112 ^?- 
 
 706326 :;?• 
 706539 i:;? • 
 
 706 753 M? • 
 70696/!:^?- 
 
 70/180 i:;? • 
 707393 i:;^ 
 
 707606^5, 
 
 1.707819 :;? 
 
 708032 ^? 
 
 708245 :;? 
 
 708458 :;? 
 
 70^6.0 :;? 
 
 708882 :;? 
 
 70J0J4 :;? 
 
 70J30J :;? 
 
 709518 :;?• 
 7097301^? 
 
 709941 :;? 
 71015J :;? 
 
 710>64 ^? 
 710575 -^^ 
 710786 
 710J67 
 711-208 
 711419 r" 
 7116-29 ^? 
 711839 "^^ 
 
 Cosine. 
 
 Cosine. 
 
 .937531 
 93 7458 
 937385 
 937312 
 937238 
 937165 
 937092 
 937019 
 936946 
 936872 
 936799 
 .936725 
 936()52 
 936578 
 936505 
 936431 
 936357 
 936-284 
 936210 
 936136 
 936062 
 
 S. 935988 
 935914 
 935840 
 935766 
 935692 
 935618 
 935543 
 935469 
 935395 
 935320 
 ,935-246 
 935171 
 935097 
 935022 
 934948 
 934873 
 934798 
 9347-23 
 934649 
 934574 
 934499 
 934424 
 934349 
 934274 
 934199 
 9341-23 
 9340 4-S 
 9339 73 
 933898 
 933822 
 
 9.933747 
 933671 
 933596 
 933520 
 933445 
 933369 
 933293 
 933217 
 933141 
 933066 
 
 Sine. 
 
 D. 10' 
 
 12.1 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.2 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.3 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.4 
 12.5 
 12.5 
 12.5 
 12.5 
 12.5 
 12.5 
 12.5 
 12.5 
 12.5 
 12.3 
 12.5 
 12.5 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 12.6 
 
 Tang. 
 
 .761439 
 761731 
 762023 
 76-2314 
 762t»06 
 76-2897 
 763188 
 763479 
 763770 
 764061 
 764352 
 
 . 764(i43 
 764933 
 766224 
 765514 
 765806 
 766095 
 766385 
 766676 
 766965 
 767255 
 
 . 767545 
 767834 
 768124 
 768413 
 768703 
 768992 
 769-281 
 769570 
 769860 
 770148 
 
 .770437 
 7707-26 
 771015 
 771303 
 771592 
 771880 
 772168 
 77-2457 
 772745 
 773033 
 773321 
 773608 
 773896 
 774184 
 774471 
 774769 
 775046 
 775333 
 775621 
 775908 
 '.776ly5 
 776482 
 776769 
 777055 
 777342 
 77 76-28 
 777915 
 778201 
 778487 
 778774 
 
 Colatig. 
 
 9. 
 
 D. 10" 
 
 48.6 
 48.6 
 48.6 
 48. «i 
 48.5 
 48.5 
 48.5 
 48.5 
 48.5 
 48.5 
 48.4 
 48.4 
 48.4 
 48.4 
 48". 4 
 48.4 
 48.4 
 48.3 
 48.3 
 48.3 
 48.3 
 48.3 
 48.3 
 48.2 
 48.2 
 48.2 
 48.2 
 48.2 
 48.2 
 48.1 
 48.1 
 48.1 
 48.1 
 48.1 
 48.1 
 48.1 
 48.0 
 48.0 
 48.0 
 48.0 
 48.0 
 48.0 
 47.9 
 47.9 
 47.9 
 47.9 
 47.9 
 47.9 
 47.9 
 47.8 
 47.8 
 47.8 
 47.8 
 47.8 
 47.8 
 47.8 
 47.7 
 47.7 
 47.7 
 47.7 
 
 Cotiing. 
 
 10.-238561 
 238269 
 23 7977 
 237686 
 237394 
 237103 
 236812 
 236521 
 236230 
 235939 
 235648 
 
 10.235357 
 2350J7 
 234776 
 234406 
 234195 
 233905 
 2336 ' s 
 2333.46 
 233035 
 232745 
 
 10.23-2455 
 232166 
 231876 
 231587 
 231297 
 231008 
 230719 
 230430 
 230140 
 2-29H52 
 
 10-2-29563 
 229274 
 2-2-985 
 2-28697 
 228408 
 228120 
 227832 
 227543 
 227255 
 2-26967 
 
 10.2-26679 
 2-26392 
 2-26104 
 2-25816 
 225529 
 2-25-241 
 224954 
 224667 
 224379 
 224092 
 
 10-2-23805 
 2-23618 
 223231 
 222945 
 222658 
 2-2-2372 
 2-22085 
 221799 
 221612 
 2212-26 
 
 N. siiio. N. cos 
 
 8(5603 
 86588 
 ■S6573 
 8(i559 
 86544 
 86530 
 8(i515 
 .S6r,01 
 8(i486 
 86471 
 86457 
 86442 
 86427 
 86413 
 8t)398 
 86384 
 8()369 
 8{i354 
 86340 
 86325 
 86310 
 8ti295 
 86-281 
 86266 
 86-251 
 86-237 
 86222 
 
 ; 60000 
 600 
 50051) 
 60076 
 60101 
 50126 
 50151 
 50176 
 5,J201 
 
 ,50-22/ 
 50252 
 50277 
 60302 
 50327 
 
 1 60352 
 50377 
 
 1 150403 
 60428 
 60453 
 60478 
 60503 
 56528 
 60553 
 60578 
 
 |5(Xi03 
 
 ' 50628 
 
 1 6mi54 
 
 ; 60079 86-207 
 ; 60704^192 
 
 i 507-29 
 60754 
 160779 
 : 50804 
 1 50829 
 60854 
 : 60879 
 i 50904 
 
 86178 
 86163 
 86148 
 86133 
 86119 
 86104 
 86089 
 86074 
 
 ■ 50929 86059 
 i 50954 86045 
 
 509(9 86030 
 '51004 86015 
 ,610-29 86000 
 151054 85985 
 
 51079 85970 
 
 -i! 
 
 61104 
 611-29 
 61154 
 61179 
 
 151-204 
 61229 
 
 rt5906 
 85941 
 859-26 
 85911 
 85896 
 85881 
 
 61254|85866 
 61279185851 
 6130^85836 
 513-29 «5821 
 
 Tang. 
 
 161354 
 51379 
 51404 
 51429 
 '51451 
 ; 61479 
 :fl504 
 I N. CCS. 
 
 85806 
 85792 
 85777 
 85762 
 86747 
 85732 
 8^717 
 .N.gine 
 
 59 iJejprees. 
 
 21 
 
52 
 
 Log. Sines and Tangents. (31°) Natural Sines. 
 
 TABLE I] 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 lOl 
 
 'M 
 
 12 
 
 13' 
 
 14' 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 1 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 3b 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 60 
 
 51 
 
 62 
 
 63 
 
 64 
 
 65 
 
 Sine. |D. 10"| Cosine. 
 
 K711839L. 
 
 7120501^? 
 
 712260 J- 
 
 712469 ^J] 
 
 712679 tl 
 
 71-2889 t; 
 
 713098 t; 
 
 713308 3^ 
 
 713517 o^ 
 
 713726 q^ 
 
 713935 r,. 
 ) 714144 ^^ 
 
 714352 ^T 
 
 714561 t: 
 
 714769 r.t 
 
 714978 t; 
 
 715186 tl 
 
 715394 t: 
 
 715602 t: 
 
 715809 5;t 
 
 716017 t: 
 
 1.716224 ^2' 
 
 716432 t: 
 
 716639 ^7 
 
 716846 I ^7 ■ 
 
 717053 i^^' 
 
 717259^^ 
 
 717466^^ 
 
 717673 :5^ 
 
 717879 :5^' 
 
 718085 r.1' 
 >. 718291 T: 
 
 718497 ij;: 
 
 718703'^;: 
 
 718909 t: 
 
 719114 t: 
 
 719320 r.T 
 71y525 "i^ 
 
 719730 ::;: 
 
 719935 "^T 
 
 720140 ::]:■ 
 
 1.720346 :^t' 
 
 720549 '^T 
 
 720754 ^J' 
 
 720958 ^T 
 
 721162 ^J' 
 
 721366 ^J 
 
 721570 ^l' 
 
 721774 ^*' 
 
 721978 ii 
 
 722181 ^^ 
 1.722385 ^^ 
 
 722588 a 
 
 •722191 a 
 
 722994 ^5 
 
 723197 t^ 
 
 723400 ^1;' 
 
 723603 r,i 
 
 723805 -^t 
 
 724007 ^l 
 _7242_l0 ^ 
 
 Ccsine. 
 
 9.9330;)6 
 932990 
 932914 
 932838 
 932762 
 932686 
 932609 
 932533 
 932467 
 932380 
 932304 
 
 9.932228 
 932161 
 932075 
 931998 
 931921 
 931845 
 931768 
 931691 
 931614 
 931537 
 
 9.931460 
 931383 
 931306 
 931229 
 931152 
 931076 
 930998 
 930921 
 930843 
 930766 
 
 9.93(J688 
 930611 
 930533 
 930466 
 930378 
 930300 
 930223 
 930145 
 930067 
 929989 
 .929911 
 929833 
 929756 
 - 929677 
 929699 
 929521 
 929442 
 929364 
 929286 
 929207 
 .929129 
 929060 
 928972 
 928893 
 928816 
 928736 
 928657 
 928578 
 928499 
 928420 
 ~Siue.~ 
 
 D. 10' 
 
 12.6 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.7 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.8 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12.9 
 12,9 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.0 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 13.1 
 
 Tang. D. 10" 
 
 778774 
 7790a0 
 779346 
 779632 
 779918 
 780203^ 
 780489- 
 780775 
 781060 
 781346 
 781631 
 
 9.781916 
 782201 
 782486 
 782771 
 783066 
 783341 
 783626 
 783910 
 784195 
 784479 
 784764 
 785048 
 785332 
 785616 
 785900 
 786184 
 786468 
 786762 
 787036 
 787319 
 
 9.787603 
 787886 
 788170 
 788453 
 788736 
 789019 
 789302 
 789585 
 789868 
 790151 
 .790433 
 790716 
 790999 
 791281 
 791663 
 791846 
 , 792128 
 792410 
 792692 
 792974 
 1.793266 
 793638 
 793819 
 794101 
 794383 
 794664 
 794946 
 795227 
 795508 
 795-789 
 Cotaug. 
 
 47.7 
 47.7 
 47.6 
 47.6 
 47.6 
 47.6 
 47.6 
 47.6 
 47.6 
 47.6 
 47.6 
 47.5 
 47.5 
 47.5 
 47.6 
 47.6 
 47.6 
 47.4 
 47.4 
 47.4 
 47.4 
 47.4 
 47.4 
 47.3 
 47.3 
 47.3 
 47,3 
 47.3 
 47.3 
 47.3 
 47.2 
 47.2 
 47.2 
 47.2 
 47.2 
 47.2 
 47.2 
 47.1 
 47.1 
 47.1 
 47.1 
 47.1 
 47.1 
 47.1 
 47.1 
 47.0 
 47.0 
 47.0 
 47.0 
 47.0 
 47.0 
 47.0 
 46.9 
 46.9 
 46.9 
 46.9 
 46.9 
 46.9 
 46.9 
 46.8 
 
 ^ u«- I 
 
 Cotaug. iiN.sine.lN. cos. 
 
 161604186717 
 I 5 1629186 7 02 
 51554185687 
 61679185672 
 ' 61604!86667 
 ; 61628185642 
 61653185627 
 I51678I861J12 
 : 61703 185597 
 '61 728 '85582 
 61763 '85567 
 161778185551 
 i 51803185636 
 i 5182b 18552 1 
 151852185506 
 ! 61877186491 
 I 61902185476 
 , 51927 186461 
 l! 51952185446 
 ,;61977!8o43l 
 ! 62002185416 
 P 62026i864(il 
 il6205ll853S5 
 i62076!86370 
 ; 62101185356 
 62126JH6340 
 ,521511853261 
 ! 62175!863lOi 
 i;62200i8o294| 
 ]i62226|S62v9! 
 1 62250|852o4 | 
 1 62275i86249 I 
 1 62299185234 
 !62324|85218 
 j 52349 85203 I 
 152374185188 1 
 |62a99jh5l73| 
 i624--;3!85l67i 
 i 52448ib5l42' 
 i 6247 3185127 
 ; 62498i85112 
 j; 62522185096 
 'i 52647185081 
 I 6'-<;57".i 85006 
 |: 6259/ 185051 
 I 62(j21|850ii5 
 
 10.221226 
 220940 
 220654 
 2203^68 
 220082 
 219 797 
 219611 
 219225 
 218940 
 218654 
 218369 
 
 10.218084 
 217799 
 217514 
 217229 
 216944 
 216659 
 216374 
 216090 
 215805 
 215521 
 
 10.216236 
 214952 
 214668 
 214384 
 214100 
 213816 
 213632 
 213248 
 212964 
 212681 
 
 10.212397 
 212114 
 211830 
 211547 
 2 11264 
 210981 
 210by8 
 210415 
 210132 
 209849 
 
 10.209567 
 209284 
 209001 
 208719 
 208437 
 208154 
 207872 
 207 590 
 207308 
 207026 
 
 10.206744 
 206462 
 206181 
 205899 
 205617 
 205336 
 205055 
 204773 
 204492 
 204211 
 
 52646185020 
 5267lib5«»«>5 
 52()96i849b9 
 62720j84974 
 52745i849j9 
 6277<»l84943 
 627941849-8 
 52biy[b49l3 
 62b44!b4by7 
 528o9l84b82 
 52b;'oi84b06 
 5-^918J84b5l 
 62943 i84»3t) 
 62967i84b20 
 52992184805 
 
 Tang. '' ^^ co.>^.|n sine. 
 
 :J 
 
 68 Degrees. 
 
TABUS II. 
 
 Log. Sines iiud Tangents. {3'2P) Natural Sines. 
 
 53 
 
 Sine. 
 
 9.724210 
 724412 
 724 j1 4 
 724811) 
 725U17 
 725219 
 725420 
 725u22 
 725823 
 72G024 
 72()225 
 720421) 
 720ii20 
 720827 
 727027 
 727228 
 727428 
 727028 
 727828 
 728027 
 728227 
 
 9. 72842/ 
 728020 
 728825- 
 729024 
 729223 
 729422 
 729021 
 729820 
 730018 
 730210 
 
 9.730415 
 730013 
 730811 
 731009 
 731200 
 731404 
 731002 
 731799 
 731990 
 732193 
 
 9.732390 
 732587 
 732784 
 732980 
 733177 
 733373 
 733509 
 733705 
 733901 
 734157 
 
 |9. 734353 
 734549 
 734744 
 734939 
 735135 
 735330 
 735525 
 735719 
 735914 
 73()109 
 Cosine. 
 
 D. 10" ! Co sine. | D. 10" [ Titng 
 
 33,7 
 
 33.7 
 33.6 
 33.6 
 33.6 
 33.6 
 33.5 
 33.5 
 33.5 
 33.5 
 33.5 
 33.4 
 33.4 
 
 33.4 
 33.4 
 33.4 
 33.3 
 33.3 
 33.3 
 33.3 
 33.3 
 33.2 
 33.2 
 33.2 
 33.2 
 33.1 
 33.1 
 33.1 
 33.1 
 33.0 
 33.0 
 33.0 
 33.0 
 33.0 
 32.9 
 32.9 
 32.9 
 32.9 
 32.9 
 32.8 
 32.8 
 32.8 
 32.8 
 32.8 
 32.7 
 32.7 
 32.7 
 32.7 
 32.7 
 32.6 
 32.6 
 32.6 
 32.6 
 32.5 
 32.5 
 32.5 
 32.5 
 32.5 
 32.4 
 32.4 
 
 9.928420 
 9283 12 
 928203 
 928183 
 928104 
 928025 
 927946 
 927807 
 927787 
 92 7 7 OS 
 927629 
 
 9.927549 
 927470 
 927390 
 927310 
 927231 
 927151 
 927071 
 92(i991 
 92091 1 
 920831 
 
 9.920751 
 920()71 
 920591 
 92051 1 
 926431 
 920351 
 920270 
 920190 
 920110 
 920029 
 
 9.925949^ 
 925808 
 925788 
 925707 
 925020 
 925545 
 926405 
 925384 
 -925303 
 925222 
 
 9.925141 
 925000 
 924979 
 924897 
 924816 
 924735 
 924654 
 924572 
 924491 
 924409 
 
 9.924328 
 924246 
 924164 
 924083 
 924001 
 923919 
 923837 
 923755 
 923073 
 923591 
 Sine. 
 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.2 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.3 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 13.4 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13.5 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.6 
 
 13.7 
 
 13.7 
 
 9. 
 
 ,795789 
 79ii0T0 
 
 790032 
 79ti913 
 797194 
 797475 
 797755 
 798036 
 798316 
 798596 
 ,798877 
 799157 
 799437 
 799717 
 799997 
 800277 
 800557 
 800836 
 8((1116 
 801396 
 801076 
 801955 
 802234 
 802513 
 802792 
 803072 
 803351 
 803(,30 
 803908 
 804187 
 
 >.8044<)6 
 804745 
 805023 
 805302 
 805580 
 805859 
 800137 
 806415 
 800093 
 800971 
 
 >. 807249 
 80.627 
 807805 
 80;-5083 
 808361 
 80:-^638 
 808916 
 8(»9193 
 809471 
 809748 
 
 >.810(»25 
 810302 
 810580 
 810857 
 811134 
 811410 
 811687 
 811964 
 812241 
 812517 
 
 D. 10"l CoUmg. N. sine.jN. I'O.^. 
 
 10.204211 i 52992184805 
 203:^30 63017 84789 
 203(i49 53041;b4774 
 203308' 63000|84759 
 203087; 63U9 184743 
 202800! 53115184728 
 2u2525i 63140'84712 
 202245 63164 84(i97 
 201904 ,63189 84081 
 2016841 63214184660 
 2014041 63238!84()50 
 
 10.201123; 63203184035 
 200843 ,63288 84019 
 200563 ; 6331 2i84()04 
 2002831 63337184588 
 200003] 63301:84573 
 199723 6338():84557 
 199443 1 6341 ri84542 
 199104 I 6343584520 
 1988841 63400 84511 
 198004 63484 84495 
 
 10.198325 1 53509 84480 
 198045' 63534 84404 
 197766 j: 53558 8444S 
 197487 ' 53583 84433 
 197208 53(i-j: 84417 
 190928 I 53032 H4402 
 190049 I 63C;50 84280 
 1963701 63681 84370 
 1900921! 63705 84355 
 195813; 63730 84339 
 
 10.195634 5376484324 
 195256 63779 84308 
 194977 163804 84292 
 194098 6382884277 
 194420 63853-84201 
 194141 63877 84245 
 193803 '53902 84230 
 193585 63920 84214 
 193307-6395184198 
 193029. 63975 84182 
 
 10.192751' 64000 84107 
 192473 '54024 84151 
 192195 '64049 84135 
 191917 116407384120 
 191039! 6409 i 84104 
 191302 I' 64122 84088 
 191084, 6414«> 84072 
 190807 i 6417184057 
 1905291 64195 84041 
 1902621164220 84025 
 
 10. 189976;! 64244 84009 
 189098 I 64209,83994 
 189420 64293*83978 
 189143 ,'64317 83962 
 l888<)6li543428o946 
 188590;' 64300 83930 
 188313 64;-9r8o915 
 188030; 64415 8o 899 
 187769 :,6444U 83883 
 187483 I 64404,83807 
 Tang. , N. ros.|N.sinc, 
 
 57 Degrt^B. 
 
r 
 
 54 
 
 Log. Sines and Tangents. (330) Natural Bines. TABLE II. 
 
 9. 
 
 Sine. 
 
 736303 
 736498 
 736692 
 73t)886 
 737080 
 737274 
 737467 
 737661 
 737855 
 738048 
 .738241 
 738434 
 738627 
 738820 
 739013 
 739206 
 739398 
 739590 
 739783 
 739976 
 
 .740167 
 740359 
 740550 
 740742 
 740934 
 741125 
 741316 
 741508 
 741699 
 741889 
 
 .742080 
 742271 
 742462 
 742652 
 742842 
 743033- 
 743223 
 743413 
 743602 
 743792 
 743982 
 
 -744171 
 744361 
 744550 
 744739 
 744928 
 745117 
 745306 
 745494 
 745683 
 7458/1 
 746059 
 746248 
 746436 
 746624 
 746812 
 74699y' 
 747187 
 747374 
 747562 
 
 Cosine. 
 
 D. 10" 
 
 32,4 
 32.4 
 32.4 
 32.3 
 32.3 
 32.3 
 32.3 
 32.3 
 32.2 
 32.2 
 32.2 
 32.2 
 32.2 
 32.1 
 32.1 
 32.1 
 32.1 
 32.1 
 32.0 
 32.0 
 32.0 
 32.0 
 32.0 
 31.9 
 31.9 
 31.9 
 31.9 
 31.9 
 31.8 
 31.8 
 31.8 
 31.8 
 31.8 
 
 31;7 
 
 31.7 
 31.7 
 31.7 
 31.7 
 31.6 
 31.6 
 31.6 
 31.6 
 31.6 
 31.5 
 31.5 
 31.5 
 
 Cosine. 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31.4 
 
 31.3 
 
 31.3 
 
 31.3 
 
 .31.3 
 
 31.3 
 
 31.2 
 
 31.2 
 
 ,923591 
 923509 
 923427 
 923345 
 923263 
 923181 
 923098 
 923016 
 922933 
 922851 
 922768 
 922686 
 922603 
 922520 
 922438 
 922355 
 922272 
 922189 
 
 I 922106 
 922023 
 921940 
 
 9.921857 
 921774 
 921691 
 921607 
 921524 
 921441 
 921357 
 921274 
 921190 
 921107 
 
 9\ 92 1023 
 920939 
 920856 
 920772 
 920688 
 920604 
 920520 
 920436 
 920352 
 920268 
 
 9.920184 
 920099 
 920015 
 919931 
 
 " 91^846 
 919762 
 919677 
 919593 
 919508 
 919424 
 
 9.919339^ 
 919254 
 919169 
 919085 
 919000 
 918915 
 9188.i0 
 918745 
 918669 
 918574 
 ~Sine7~ 
 
 D. 10' 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.7 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.8 
 
 13.9 
 
 13.9 
 
 13.9 
 
 13.9 
 
 13.9 
 
 13.9 
 
 13.0 
 
 13.9 
 
 13.9 
 
 13.9 
 
 13.9 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.0 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.1 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.2 
 
 Tung. 
 
 9.812517 
 812794 
 813070 
 813347 
 813623 
 813899 
 814175 
 814452 
 814728 
 815004 
 816279 
 9.815555 
 815831 
 816107 
 816382 
 816(J58 
 816933 
 817209 
 817484 
 817759 
 818035 
 818310 
 818585 
 818860 
 819135 
 819410 
 819684 
 819959 
 820234 
 820508 
 820783 
 9.821057 
 821332 
 821606 
 821880 
 822154 
 822429 
 822703 
 822977 
 823250 
 823524 
 9.823798 
 824072 
 824345 
 824619 
 824893 
 825166 
 825439 
 825713 
 825986 
 826259 
 9,826532 
 826805 
 827078 
 827351 
 827624 
 82 7897 
 828170 
 828412 
 828 716 
 82898 7 
 C^^oUin'T. 
 
 D. 10 
 
 46.1 
 46.1 
 46 1 
 46.0 
 46.0 
 46.0 
 46.0 
 46.0 
 46.0 
 46.0 
 46.0 
 4 1.9 
 45.9 
 45.9 
 45.9 
 45.9 
 45.9 
 45.9 
 45.9 
 45.9 
 45.8 
 45.8 
 45.8 
 45.8 
 45.8 
 45.8 
 45.8 
 45.8 
 45.8 
 45.7 
 45.7 
 45.7 
 45.7 
 45.7 
 
 Cotang. j N. .sine.lN. cos. 
 
 10, 
 
 10 
 
 187482 
 
 187206 
 
 186930 
 
 186653 i 
 
 186377 j 
 
 186101 : 
 
 185825 
 
 185548, 
 
 185272 
 
 184996 ' 
 
 184721 
 
 184445 
 
 184169 
 
 183893: 
 
 183618: 
 
 183342 i 
 
 183067 : 
 
 182791 : 
 
 5446483867 
 64488 83851 
 64513 83835 
 64537 83819 
 6456183804 
 54586183788 
 64610183772 
 54635,83756 
 54659 83740 
 54683 83724 
 54708:83708 
 64'} 32 ,83692 
 54756 83676 
 54781 83660 
 54805 83645 
 C482y;83629 
 54854:83613 
 54878 83597 
 
 45 
 
 46 
 
 45 
 
 45 
 
 45 
 
 45 
 
 46 
 
 46 
 
 45 
 
 45.6 
 
 45.6 
 
 45.6 
 
 46.6 
 
 46.5 
 
 45.5 
 
 45.6 
 
 45.6 
 
 45.6 
 
 45.6 
 
 45.6 
 
 45.5 
 
 45.5 
 
 46.4 
 
 45.4 
 
 45.4 
 
 45.4 
 
 182516: 54902 83581 
 18224] 54927 83565 
 181965 ' 54951:83549 
 
 10.181690 64975 83533 
 181415 54999 83517 
 181140 560^4 83501 
 180865 55048 83486 
 180590 55072 83469 
 180316 55097 83453 
 180041 5512183437 
 179/66 65145 83421 
 179492 55169 83405 
 179217 !i 55194 83389 
 
 10.178943: 55218 83373 
 178668 55-42 83356 
 178394! 55266 83340 
 178120: 55^9183324 
 177846: 55315 83308 
 177671 j 55339 83292 
 177297!; 55363183276 
 177023 i 56388 83260 
 176760 i 1 55412 83244 
 176476 1 155436 '83228 
 
 10. 176202 ji 65460 83212 
 176928 i|55484!83195 
 175655 ji 65509 83179 
 176381s 65533 j83163 
 176107]: 65557183147 
 1748341! 55581 183131 
 174561 155605 83115 
 174287 1! 55630 83098 
 1740141'! 5566 1 83082 
 173741 i k.5678 830ij6 
 
 10.1734681 
 1731951 
 172922 I 
 172649'! 
 172376 
 172103 |i 
 171830: 
 171558 
 171-.^85 
 171013 
 Tang. 
 
 55702 83060 
 55726 83034 
 55760 83017 
 5577583U01 
 55799 82L85 
 65823 821.69 
 : 55847 :82*j53 
 j 55871 82036 
 j 5.5895 ;82!:>20 
 55^19 82904 
 
 l_ 
 '60 
 59 
 '58 
 57 
 56 
 56 
 54 
 53 
 52 
 51 
 50 
 49 
 148 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 16 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 <) 
 
 T 
 
 
 
 56 Degrees. 
 
Log. Sines and Tangents. (34°) Natural Sines. 
 
 55 
 
 M _Sint 
 
 019 7475G2 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 4i. 
 41 
 42 
 43 
 44 
 45 
 Mi 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 64 
 55 
 56 
 57 
 68 
 59 
 60 
 
 747749 
 74793!) 
 748123 
 748310 
 7'1849 7 
 74H683 
 748870 
 749056 
 749243 
 749426 
 
 9.74!>()15 
 749801 
 749987 
 750172 
 750358 
 760543 
 750729 
 750914 
 751099 
 751284 
 
 9.751469 
 751654 
 751839 
 752023 
 7522(W 
 752392 
 752576 
 752760 
 752944 
 753128 
 
 9.753312 
 753495 
 753679 
 7638()2 
 754046 
 754229 
 754412 
 754595 
 754778 
 754960 
 ,755143 
 755326 
 755508 
 755()y0 
 75r>872 
 756054 
 756236 
 756418 
 756(iO0 
 756782 
 
 9.756y()3 
 757144 
 75732(> 
 757507 
 7576.S8 
 7578()9 
 758050 
 768230 
 758411 
 758591 
 Cosine. 
 
 D. 10' 
 
 31.2 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31 
 
 31.0 
 
 31.0 
 
 31.0 
 
 31.0 
 
 30.9 
 
 30.9 
 
 30.9 
 
 30.9 
 
 30.9 
 
 30.8 
 
 30.8 
 
 30.8 
 
 30.8 
 
 30.8 
 
 30.8 
 
 30.7 
 
 30.7 
 
 30.7 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30.5 
 
 30 5 
 
 30.5 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30 
 
 30.3 
 
 30.3 
 
 30.3 
 
 30.2 
 
 30.2 
 
 30.2 
 
 30.2 
 
 30.2 
 
 30.1 
 
 30.1 
 
 30.1 
 
 30.1 
 
 30.1 
 
 Co.«ine. 
 
 >. 9 18574 
 918480 
 918404 
 918318 
 918233 
 918147 
 9180)2 
 917976 
 917891 
 -917805 
 917719 
 1.917634 
 917548 
 917462 
 917376 
 917290 
 917204 
 917118 
 917032 
 916046 
 916859 
 .916773 
 916687 
 916600 
 916514 
 916427 
 916341 
 916264 
 916167 
 916081 
 915994 
 .915907 
 915820 
 915733 
 915646 
 915559 
 915472 
 915386 
 916297 
 915210 
 915123 
 .915035 
 914948 
 914860 
 914773 
 914685 
 914598 
 914510 
 914422 
 914334 
 914246 
 .914168 
 9140i0 
 913982 
 913894 
 913806 
 913718 
 913630 
 913541 
 913453 
 913365 
 
 D. 10" 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.2 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14.6 
 
 14.5 
 
 14.6 
 
 14.5 
 
 14.6 
 
 14.6 
 
 14.5 
 
 14.6 
 
 14.5 
 
 14.5 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.6 
 
 14.7 
 
 14.7 
 
 14.7 
 
 14 
 
 14.7 
 
 " Tang. 
 
 9.828987 
 82!i260 
 829532 
 829805 
 830iJ77 
 830349 
 830o2l 
 830893 
 831165 
 831437 
 831709 
 
 ?. 831981 
 832253 
 832525 
 832796 
 833068 
 833339 
 833611 
 833882 
 834164 
 834425 
 
 ). 834696 
 834967 
 835238 
 835509 
 835780 
 836051 
 836322 
 836593 
 836864 
 837134 
 
 ). 837405 
 837676 
 837946 
 838216 
 838487 
 838767 
 839027 
 83.9297 
 839568 
 839838 
 
 1.840108 
 840378 
 840647 
 840917 
 841187 
 841467 
 841726 
 84199<) 
 842266 
 842536 
 
 1.842806 
 843074 
 843343 
 843612 
 843882 
 844151 
 844420 
 844689 
 8449.^)8 
 845227 
 Cotang. 
 65 Degrees. 
 
 D. 10" 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45.3 
 
 45.3 
 
 45.3 
 
 45.3 
 
 45.3 
 
 45.3 
 
 45.2 
 
 45.2 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45.2 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.1 
 
 45.0 
 
 45.0 
 
 45.0 
 
 45.0 
 
 46.0 
 
 45.0 
 
 45.0 
 
 45.0 
 
 45.0 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.9 
 
 44.8 
 
 44.8 
 
 44.8 
 
 44.8 
 
 44.8 
 
 Cotang. I A. sine 
 
 10 
 
 1710131 
 
 170740 
 
 1704()8' 
 
 170195! 
 
 169923 ! 
 
 169651 
 
 169379 
 
 65919 
 55943 
 559o8 
 
 .N. COS. 
 
 S2904 
 
 H2KH7 
 82871 
 
 I 56064 
 169107:156088 
 
 55992 82855 
 56016182839 
 56040 82822 
 82806 
 
 82790 
 82773 
 8275' 
 
 168835 !| 561 12 
 
 168563 ij 56136, 
 
 lti829li| 66160 82741 
 10.1680191156184 82724 
 
 167747 
 167476 
 167204 
 166932 
 166661 
 166389 
 166118 
 166846 
 165576 
 10.165304 
 166033 
 164762 
 
 6(i208 
 56232 
 56256 
 56280 
 56305 
 i 56329 
 ' 56353 
 I 56377 
 156401 
 t 56425 
 56449 
 56473 
 
 164491 1 1 66497 
 164220 I j 66521 
 1639491156545 
 163678!; 56569 
 
 163407 
 163136 
 162866 
 10.162595 
 162325 
 162054 
 1617841:56736 
 161613 ji 66760 
 161243 1 56784 
 
 66593 
 '56617 
 
 56641 
 j 56665 
 
 66689 
 
 56713 
 
 82708 
 82()92 
 82675 
 82(i59 
 82»i43 
 82()26 
 82610 
 82593 
 82577 
 82561 
 82544 
 82528 
 82511 
 82495 
 82478 
 82462 
 82446 
 82429 
 82413 
 82396 
 82380 
 82363 
 82347 
 82330 
 82314 
 
 160973 1156808 8229 
 
 160703 il568b2 
 160432 156856 
 160162 166880 
 10.159892 Ij 56904 
 159622 I 56928 
 159353 1 56962 
 159083! 156976 
 158813; I 670U0 
 158543 
 158274 
 158004 
 167734 
 167465 
 157195 
 156926 
 156657 
 166388 
 
 10 
 
 57024 
 57047 
 57071 
 57095 
 57119 
 67143 
 67167 
 57191 
 57215 
 
 82281 
 82264 
 82248 
 82231 
 82214 
 82198 
 82l8l! 16 
 821651 15 
 82148' 14 
 82132! 13 
 82115; 12 
 82098,11 
 820821 10 
 
 15611811 67238 
 1558491 57262 
 155580;! 67286 
 155311 157310 
 
 165042 
 154773 
 
 Tnng. 
 
 57334 
 67368 
 
 82065 
 82048 
 H2032 
 82015 
 81999 
 81982 
 81966 
 81949 
 81932 
 81916 
 
 9 
 S 
 7 
 6 
 6 
 4 
 3 
 2 
 
 6| 
 
 N. COS. N sine. 
 
 J 
 
56 
 
 Log. Sines and Tangents. (35°) Natural Smes. 
 
 TAIJLK II. 
 
 
 1 
 
 2 
 3 
 4 
 5 
 
 n 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 26 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 3S 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 55 
 66 
 67 
 68 
 59 
 60 
 
 9. 
 
 75S591 
 758772 
 758952 
 759132 
 759312 
 759492 
 759d72 
 759852 
 760031 
 760211 
 760390 
 76^569 
 760748 
 7b0927 
 761106 
 761285 
 761464 
 761642 
 761821 
 761999 ! 
 762177! 
 762356 j 
 762534 I 
 7627121 
 762889 i 
 763067 i 
 763245 i 
 763422 
 763600 
 763777 
 763954 
 764131 I 
 764308 
 764485 
 764662 
 764838 
 765015 
 765191 
 765367 
 765544 
 765720 
 765896 
 766072 
 766247 
 766423 
 766598 
 766774 
 766949 
 767124 
 767300 
 767475 
 .767649 
 767824 
 767999 
 768173 
 768348 
 768522 
 768697 
 768871 
 769045 
 769219 
 Cosine. 
 
 1 ). lU' 
 
 30.1 
 30.0 
 30. U 
 30 
 30. u 
 30.0 
 29.9 
 29.9 
 29.9 
 29.9 
 29.9 
 29.8 
 29.8 
 29.8 
 29.8 
 29.8 
 29.8 
 29.7 
 29.7 
 29.7 
 29.7 
 29.7 
 29.6 
 29.6 
 29.6 
 29.6 
 29.6 
 29.6 
 29.5 
 29.5 
 29.5 
 29.5 
 29.5 
 29.4 
 29.4 
 29.4 
 29.4 
 29.4 
 29.4 
 29.3 
 29.3 
 29.3 
 29.3 
 29.3 
 29.3 
 29.2 
 29.2 
 29.2 
 29.2 
 29.2 
 29.1 
 29.1 
 29.1 
 29.1 
 29.1 
 29.0 
 29.0 
 29.0 
 29.0 
 29.0 
 
 Cosine. 
 
 1.913365 
 913276 
 913187 
 913099 
 913010 
 912922 
 912833 
 912744 
 912655 
 912566 
 912477 
 
 1.912388 
 912299 
 912210 
 912121 
 912031 
 911942 
 911853 
 911763 
 911674 
 911584 
 
 1.911495 
 911405 
 911315 
 911226 
 911136 
 911046 
 910956 
 910866 
 910776 
 910J86 
 
 .91U596 
 910506 
 910415 
 910325 
 910236 
 910144 
 910054 
 909963 
 909873 
 909782 
 
 .909691 
 909501 
 909510 
 909419 
 909328 
 90923 7 
 909146 
 909055 
 908964 
 90>"i873 
 
 1.908781 
 908690 
 9085,9 
 908507 
 908416 
 908324 
 908233 
 908141 
 908049 
 907958 
 Sine. 
 
 L>. iU 
 
 4.7 
 4.7 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.8 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4.9 
 4,9 
 4.9 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.0 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.1 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.2 
 5.3 
 5.3 
 5.3 
 5.3 
 
 T:iii>j. D. 10 
 
 9.845227 
 84549.) 
 845 764 
 846033 
 846302 
 846570 
 846839 
 847107 
 847376 
 847644 
 847913 
 
 .848181 
 848449 
 848717 
 848986 
 849254 
 849522 
 849790 
 850058 
 850325 
 850593 
 
 .850851 
 851129 
 851396 
 851664 
 851931 
 852199 
 852466 
 852733 
 853001 
 853268 
 
 .853635 
 8538i)2 
 854069 
 854336 
 854603 
 854870 
 855137 
 855404 
 865671 
 855938 
 
 .856204 
 856471 
 856737 
 857004 
 
 ^57270 
 857537 
 867803 
 868069 
 858336 
 868602 
 9.868858 
 859134 
 859400 
 859666 
 859932 
 860198 
 860464 
 86J730 
 86J995 
 861261 
 
 Cotiin'T. 
 
 44.8 
 
 44.8 
 
 44.8 
 
 44.8 
 
 44.8 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.7 
 
 44.6 
 
 44.6 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44.5 
 
 44.5 
 
 44.5 
 
 44.6 
 
 44.6 
 
 44.5 
 
 44.5 
 
 44.5 
 
 44.5 
 
 44.5 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.4 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 44.3 
 
 10, 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 154; 73 
 154504 
 154236 
 153967 
 153698 
 153430 
 153161 
 152893 
 152(i24 
 152356 
 15C087 
 151819 
 151551 
 151283 
 151014 
 150746 
 150478 
 150210 
 149942 
 149675 
 149407 
 149139 
 148871 
 148604 
 148336 
 148059 
 147801 
 147534 
 147267 
 146999 
 146732 
 146465 
 146198 
 145931 
 145664 
 145397 
 146130 
 144863 
 144596 
 14432y 
 144062 
 143796 
 143529 
 143263 
 142996 
 142730 
 142463 
 142197 
 141931 
 141664 
 141398 
 141132 
 140866 
 140500 
 140334 
 140068 
 139802 
 139536 
 139270 
 1390J5 
 138739 
 
 I N. sine. IN. cos. I 
 
 ;. 57358 81915 60 
 ' 57381 181899 59 
 I 57 405 181 b&2 158 
 I 57429.8 186{> 
 ||57453J81a48 
 1 157477181832 
 1!67'^01J81815 
 I' 57624181 / 98 53 
 ji57648|81782|52 
 I i 67572,81 765 '51 
 |!57596i8174e 
 'l57619|81731 
 i 67643!81714 
 !' 67667 I81C98 
 !' 57691181681 
 i 57715181664 
 i:57738!81647 
 ii 57762181631 
 !' 67786 81614 
 1:5781081597 
 
 57833 81580 
 i 57857 81563 
 i: 67881 81546 
 ^57904 81530 
 ! 57928 81513 
 !' 57952 81496 
 
 57976 81479 
 167999 81462 
 j 58023 81445 
 158047:81428 
 i580/0;81412 
 I 58094 8 1395 
 i58118'813V8 
 158141181361 
 :68165|81344 
 
 58189|81327 
 ,58'212|81310 
 I 68236 [8 1293 
 158260 81276 
 j 58283 '8 1259 
 158307181242 
 I 5833081226 
 '58354|81208 
 '58378:81191 
 
 68401 181 174 
 ;58425'811c7 
 
 58449 
 58472 
 58496 
 
 81140 
 81123 
 81105 i 12 
 11 
 
 i 5851981089 
 I 58543 810^2 
 58667 181055 
 I 58590 8 1058 
 
 1 158614 
 
 I 58637 
 i 158661 
 i: 68684 
 j' 58708 
 
 81021 
 81004 
 80987 
 80Ji0 
 80953 
 
 !58731|80j36 
 i58755|80919 
 i 58779,80902 
 
 N. COS. N. sine. 
 
 54 Degrees. 
 
TAIJLi: II. 
 
 Log. Siiii'B and Tangents, (36°) NatKral Sines. 
 
 57 
 
 D. 10" 
 
 
 
 9.7t)9219 
 
 1 
 
 769393 
 
 2 
 
 7(i9ot)ti 
 
 3 
 
 709740 
 
 4 
 
 769JJ13 
 
 6 
 
 77008/ 
 
 6 
 
 77t)2(>0 
 
 7 
 
 770433 
 
 8 
 
 770* UK) 
 
 y 
 
 770779 
 
 10 
 
 7701);Vi 
 
 11 
 
 9.771125 
 
 12 
 
 771298 
 
 13 
 
 771470 
 
 14 
 
 771043 
 
 15 
 
 771815 
 
 16 
 
 771987 
 
 17 
 
 772159 
 
 18 
 
 - 772331 
 
 19 
 
 772503 
 
 20 
 
 772675 
 
 21 
 
 9.77284 7 
 
 22 
 
 773018 
 
 23 
 
 773190 
 
 24 
 
 773361 
 
 25 
 
 773533 
 
 2t) 
 
 773704 
 
 27 
 
 773875 
 
 28 
 
 774046 
 
 29 
 
 774217 
 
 30 
 
 774388 
 
 31 
 
 9.774558 
 
 32 
 
 774729 
 
 33 
 
 774899 
 
 34 
 
 775070 
 
 35 
 
 775240 
 
 3(> 
 
 775410 
 
 3? 
 
 775580 
 
 38 
 
 775750 
 
 39 
 
 775920 
 
 40 
 
 776090 
 
 41 
 
 9.776259 
 
 42 
 
 776429 
 
 43 
 
 776598 
 
 44 
 
 7767()8 
 
 45 
 
 776937 
 
 -Hi 
 
 77710;> 
 
 47 
 
 777275 
 
 48 
 
 777444 
 
 49 
 
 777613 
 
 50 
 
 777781 
 
 51 
 
 1). 7 7 7950 
 
 52 
 
 7:8119 
 
 53 
 
 778287 
 
 54 
 
 778455 
 
 55 
 
 778624 
 
 56 
 
 778792 
 
 57 
 
 7789vi0 
 
 5S 
 
 779128 
 
 59 
 
 779295 
 
 GO 
 
 779463 
 
 29.0 
 
 28.9 
 
 28.9 
 
 28.9 
 
 28.9 
 
 28.9 
 
 28.8 
 
 28.8 
 
 28.8 
 
 28.8 
 
 28.8 
 
 28.8 
 
 28.7 
 
 28.7 
 
 28.7 
 
 28.7 
 
 28.7 
 
 28.7 
 
 28.6 
 
 28.6 
 
 28.6 
 
 28.6 
 
 28.6 
 
 28.6 
 
 28.6 
 
 28.5 
 
 28.5 
 
 28.5 
 
 28.5 
 
 28.5 
 
 28.4 
 
 28.4 
 
 28.4 
 
 28.4 
 
 28.4 
 
 -.8.4 
 
 28.3 
 
 28.3 
 
 28.3 
 
 28.3 
 
 28.3 
 
 28.3 
 
 28.2 
 
 28.2 
 
 28.2 
 
 28.2 
 
 28.2 
 
 28.1 
 
 28.1 
 
 28.1 
 
 28.1 
 
 28.1 
 
 28.1 
 
 28.0 
 
 28.0 
 
 28.0 
 
 28.0 I 
 
 28.0 
 
 28 
 
 27 9 
 
 Co.sine. 
 
 .907958 
 90;8(;6 
 907774 
 90?(i82 
 907590 
 907498 
 907401) 
 907314 
 907222 
 907129 
 907037 
 
 1.906945 
 90()852 
 906760 
 906667 
 906575 
 90()482 
 906389 
 90()296 
 906204 
 906111 
 
 1.906018 
 905925 
 905832 
 905739 
 905646 
 905552 
 905459 
 905366 
 905272 
 905179 
 
 1.905085 
 904992 
 904898 
 904804 
 904711 
 904G17 
 904523 
 904429 
 904335 
 904241 
 
 ). 904 147 
 904053 
 903959 
 9.)3S64 
 903770 
 903676 
 903581 
 903487 
 903392 
 903298 
 
 i. 903202 
 903108 
 9J3014 
 902919 
 902824 
 902729 
 902634 
 902539 
 902444 
 902349 
 
 D. 10' 
 
 Sine. 
 
 15.3 
 
 15.3 
 
 15.3 
 
 15.3 
 
 15.3 
 
 15.3 
 
 15.3 
 
 15.4 
 
 15.4 
 
 15.4 
 
 15.4 
 
 16.4 
 
 16.4 
 
 15.4 
 
 15.4 
 
 16.4 
 
 15.4 
 
 15.6 
 
 16.6 
 
 15.5 
 
 15.6 
 
 15.6 
 
 15.6 
 
 15 ' 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 1 ) 
 
 15 
 
 15 
 
 15.6 
 
 15.6 
 
 15.6 
 
 15.6 
 
 16.6 
 
 15.6 
 
 15.7 
 
 15.7 
 
 15.7 
 
 15.7 
 
 15.7 
 
 16.7 
 
 16.7 
 
 15.7 
 
 15.7 
 
 15.7 
 
 16.7 
 
 16.8 
 
 16.8 
 
 16.8 
 
 16.8 
 
 16.8 
 
 15.8 
 
 15.8 
 
 16.8 
 
 16.8 
 
 16.9 
 
 15.9 
 
 9. 
 
 Tang. 
 
 861261 
 861527 
 861792 
 8(i205S 
 8(i2323 
 862589 
 862854 
 863119 
 863385 
 8()3650 
 863915 
 
 .864180 
 864445 
 864710 
 864975 
 8t)5240 
 865505 
 865770 
 866035 
 866300 
 86()5u4 
 
 .866829 
 867094 
 867358 
 867623 
 867887 
 868152 
 8684 1^ 
 868680 
 868945 
 869209 
 
 .869473 
 86973 7 
 870001 
 870265 
 870529 
 870793 
 871057 
 871321 
 871685 
 871849 
 
 .872112 
 872376 
 8/2640 
 872903 
 873167 
 873430 
 873694 
 873957 
 874220 
 874484 
 
 1.874747 
 875010 
 875273 
 875536 
 h.5800 
 876063 
 876326 
 876689 
 876861 
 877114 
 Cotajig. 
 
 D. 10"| Cotaiig. N. sine.jN. CU8 
 
 44.3 
 44.3 
 44 2 
 44.2 
 44.2 
 44.2 
 44.2 
 44.2 
 44.2 
 44.2 
 44.2 
 44.2 
 44 2 
 44.2 
 44 1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.1 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 44.0 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.9 
 43.8 
 43.8 
 43.8 
 43.8 
 43.8 
 43.8 
 43.8 
 
 10. 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 138739 
 138473 
 138208 
 137942 
 137677 
 137411 
 137146 
 136881 ! 
 136615 
 136350 
 13(i086 
 135820 
 1 35555 
 136290 
 135025 
 134760; 
 134495 
 134230 
 1339()5 
 133700 
 133436' 
 133171 1 
 1326061 
 132642 j 
 132377 
 132113! 
 131848; 
 131584 
 131320' 
 131055 
 130/91 
 130527- 
 13026311 
 129999: 
 129735 
 129471' 
 129207 ' 
 128943 1 ■ 
 128679 i 
 128415! 
 128151 I 
 , 1278881, 
 127624 I' 
 12:360 : 
 127097 I 
 126833: 
 126570' 
 126306 I 
 1260431 
 125780' 
 125516; 
 .125253 
 124990. 
 124727 
 124464 : 
 124200,: 
 123937 
 123674 I 
 123411' 
 123149! 
 122886' 
 
 'ring. !; 
 
 68779 80902 
 58802.80885 
 68826 808()7 
 68849 80850 
 68873 80h33 
 58896 80F 16 
 68920 80799 
 58943 80782 
 58967 80765 
 58990 80748 
 59014 80730 
 59037 80713 
 69061 80()96 
 6y(k*^4H(Kj7y 
 69 lOO 80662 
 69131 80644 
 69154 80627 
 69178 80()10 
 59201 80593 
 59225 80576 
 59248 80568 
 69272 80541 
 69295 80524 
 5931880507 
 69342 80489 
 59365 80472 
 69389 80455 
 59412 80438 
 59436 80422 
 59459 80403 
 59-482 80386 
 59506 80368 
 59529 80361 
 59552 80334 
 6j5/6b03l6 
 59599 80299 
 69622 80282 
 69640 80264 
 696()9:80247 
 69693 |802o0 
 59;1()80-j1--^ 
 59739 80 196 
 59(63:801 1 8 
 59786!80160 
 59809 80143 
 69832 8U 1 25 
 59856 80108 
 69879 80091 
 69902 !800 73 
 69926 '80051) 
 59949 80038 
 5!:»9;2 80021 
 59995 800"3 
 60019 79986 
 60042 1/9968 
 60065 i79961 
 60089 17 9934 
 60I12J/9916 
 601351:9899 
 60158i/9f)81 
 6018279864 
 
 N. 
 
 to?., 
 
 A' .sine 
 
 53 Degn^fl. 
 
58 
 
 Log. Sines and Tangents. (37°) Natural Sinos. 
 
 TABLE II. 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ~7 
 
 8 
 
 9 
 
 \0 
 
 11 
 
 12 
 
 13 
 
 H 
 15 
 l(j 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2t> 
 27 
 28 
 29 
 80 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 61 
 52 
 53 
 54 
 55 
 56 
 67 
 58 
 59 
 60 
 
 Sine. D. 10' 
 
 9.779463 
 
 779o31 
 779798 
 7799o6 
 780133 
 780300 
 780457 
 7801)34 
 780801 
 780£>o8 
 781134 
 
 9.781301 
 781468 
 781634 
 781800 
 78 19b 6 
 782132 
 782298 
 782464 
 782630 
 782796 
 1.782961 
 783127 
 783292 
 783458 
 783623 
 783/88 
 783t»53 
 784118 
 784282 
 -78444/ 
 
 9.784612 
 784776 
 784941 
 785105 
 785269 
 785433 
 785597 
 785761 
 785925 
 786089 
 
 9.786252 
 786416 
 7865/9 
 786 742 
 786906 
 7870o9 
 78.232 
 787395 
 78755/ 
 787720 
 787883 
 788045 
 
 27.9 
 27.9 
 27.9 
 27.9 
 27.9 
 27.8 
 27.8 
 27.8 
 27.8 
 27.8 
 27.8 
 27.7 
 27.7 
 27.7 
 27.7 
 27.7 
 27.7 
 27.6 
 27.6 
 27.6 
 27.6 
 27.6 
 27.6 
 27. c 
 27.5 
 27.5 
 2 7.6 
 27.5 
 27.5 
 27.4 
 27.4 
 27.4 
 27.4 
 27.4 
 27.4 
 27.3 
 27.3 
 27.3 
 27.3 
 27.3 
 27.3 
 27.2 
 27.2 
 27.2 
 27.2 
 27.2 
 27.2 
 
 Cosine. |D. 10' 
 
 27.1 
 27.1 
 27.1 
 27.1 
 27.1 
 27.1 
 
 788208, 
 788370 '"^'-^ 
 
 788532 
 788694 
 788856 
 789018 
 789180 
 789342 
 I Ccsine. 
 
 27.0 
 27.0 
 27.0 
 27.0 
 27.0 
 27.0 
 
 902349 
 
 902253 
 
 902158 
 
 902063 
 
 90196 7 
 
 901872 
 
 901776 
 
 901681 
 
 901585 
 
 9014^0 
 
 901394 
 
 901298 
 
 901202 
 
 901106 
 
 901010 
 
 900914 
 
 900818 
 
 900722 
 
 900626 
 
 900529 
 
 900433 
 
 900337 
 
 900242 
 
 900144, 
 
 900047 
 
 899951 
 
 899854 
 
 899757 
 
 899660 
 
 899564 
 
 899467 
 
 8993/0 
 
 8992 /3H 
 
 899176 
 
 899078 
 
 898981 
 
 898884 
 
 898787 
 
 898689 
 
 898592 
 
 898494 
 
 9.898397 
 898299 
 898202 
 898104 
 898006 
 897908 
 89/810 
 "897712 
 897614 
 897516 
 
 9.897418 
 89/320 
 897222 
 897123 
 89/025 
 
 -896926 
 896828 
 896/29 
 89v><.)31 
 896532 
 
 15.9 
 15.9 
 15.9 
 15 9 
 15 9 
 15.9 
 15 9 
 15 9 
 15.9 
 
 15 9 
 
 16 
 16 
 16 
 16 
 16 
 16 
 16 
 16'0 
 16 
 16'0 
 16 1 
 16 1 
 16 1 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.3 
 
 16.3 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16.4 
 
 16.4 
 
 16.4 
 
 16.4 
 
 16.4 
 
 16.4 
 
 16,4 
 
 16.4 
 
 Tan<2 
 
 D. 10"! Cotang. jjN.sine.jN. t-os 
 
 9.877114 
 877377 ^^' 
 877640^^- 
 877903 I ^^' 
 878165^^' 
 
 . 878428 2^ • 
 
 878691 ;:: • 
 
 878953 K:; • 
 879216 I ^• 
 879478 'TX- 
 879741 ^ 
 
 9.88fX)03 
 880265 
 880528 
 880790 
 881052 
 881314 
 881576 
 881839 
 882101 
 882363 
 
 9.882625 
 882887 
 883148 
 883410 
 883672 
 883934 
 884196 
 884457 
 884/19 
 884980 
 
 9.885242 
 885503 
 885/65 
 886026 
 886288 
 886549 
 886810 
 887072 
 88/333 
 887594 
 
 9.887^f5 
 888116 
 8883 77 
 888639 ! 
 888900 
 889160 
 889421 
 889682 
 889943 
 890204 
 890465 
 890725 
 890986 
 891247 
 891507 
 891 768 
 892028 
 892289 
 892549 
 892810 
 
 43, 
 43, 
 43, 
 43 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43 
 43 
 43, 
 43, 
 43, 
 43, 
 43. 
 43, 
 43, 
 43, 
 43, 
 43, 
 43 
 43 
 43 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43, 
 43. 
 43. 
 43, 
 43. 
 143. 
 43. 
 43. 
 
 10 
 
 10 
 
 10. 
 
 10 
 
 10 
 
 10 
 
 122886 
 122623 
 122360 
 122097 
 121835 
 121572 
 121309 
 121047 
 120784 
 120522 
 120259 
 119997 
 119735 
 119472 
 119210 
 118948 
 118686 
 118424 
 118161 
 117899 
 117637 
 117375 
 117113 
 116852 
 116590 
 116328 
 116066 
 115804 
 115543 
 115281 
 115020 
 114758 
 114497 
 114235 
 113974 
 113712 
 113451 
 113190 
 112928 
 112667 
 112406 
 112145 
 111884 
 111623 
 111361 
 111100 
 110840 
 110579 
 110318 
 110057 
 109796 
 10J535 
 109275 
 109014 
 108753 
 1 (>8493 
 108232 
 10-972 
 107'Jn 
 10/451 
 10719U 
 
 |Q«(^ !«fn«ol 
 
 60182 79864 
 
 60205 
 60228 
 60251 
 60274 
 60298 
 60321 
 60344 
 60367 
 : 60390 
 160414 
 
 79846 
 79829 
 79811 
 79793 
 79776 
 79758 
 79741 
 79723 
 79706. 
 79688 
 
 60437 179671 
 604ti0 79658 
 
 60483 
 1 60506 
 1 60529 
 { 60553 
 160576 
 ! 60599 
 1 60622 
 1 60645 
 ' 60668 
 160691 
 ,60714 
 160738 
 160761 
 '60/84 
 J6080/ 
 
 60830 
 
 ! 60853 
 608/6 
 j 60699 
 I 60J22 
 ! 60945 
 i 60968 
 160991 
 161015 
 ,61038 
 '61061 
 
 79635 
 79618 
 79600 
 79583 
 79565 
 79547 
 79530 
 79512 
 79494 
 79477 
 79459 
 79441 
 79424 
 79406 
 79388 
 79371 
 79353 
 79335 
 79318 
 79300 
 79282 
 /9264 
 79247 
 79229 
 79211 
 79193 
 
 ,61084, /9r76 
 
 :6110/|79158 
 61130179140 
 6115379122 
 61J 76 79105 
 6119917908/ 
 61222/9069 
 61245 79051 
 61268/9033 
 612911/9016 
 61314|<8998 
 6133/78^80 
 61 360; 78962 
 61383178944 
 61406 78926 
 6142976908 
 61451 
 61474 
 6149/ 
 
 161520 
 
 I f 1543 
 
 i eViio 
 
 I N cn.s. X.vine. 
 
 76891 
 78873 
 8655 
 8837 
 6819 
 8801 
 
 60 
 59 
 58 
 57 
 56 
 55 
 54 
 53 
 52 
 51 
 50 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 
 52 Dc-grees. 
 
TABLK II. 
 
 Log. Sines and Tangents. (38°) Natural Sinw. 
 
 5S 
 
 DTTo^ 
 
 D. 10") Cotang. N.sine.lN. cos. 
 
 ob. 789342 or 
 
 1 789504 ;^ 
 
 2 78J(>tio ;"• 
 789827 ;"• 
 789988 ^■ 
 
 790149 ;:!• 
 
 791)bl0 *^ 
 
 790471 f^ 
 
 79J632 ^^• 
 
 9 79J793 ,,'• 
 
 JO 790954 -^• 
 
 11 9.791115 ^^.• 
 
 12 791275 ;;,^- 
 
 13 791430 ^" 
 
 14 ""'--'"^ 
 
 791430 ;, • 
 791590 ^.^• 
 
 15 J791757 ;^- 
 
 16 •"""•" ^^ 
 
 791917 il 
 17 792077 20 
 
 792077 -" 
 
 18 792237 .j^" 
 
 19 792397 ^.' 
 
 20 792557 ^^• 
 
 21 9.792710 ;^- 
 
 22 792870 -Tf/ 
 
 23 - "- ^^ 
 
 24 
 
 792870 - 
 
 23 793035 „p 
 
 24 793195 fy 
 
 25 793354 ;;"• 
 20 793514 ^^• 
 
 27 793073 I Zi 
 
 28 793832 ! *^ 
 
 29 793991 I ;.^ 
 
 30 794150 ^r 
 
 31 9.794308,^^- 
 
 32 794407 ;^- 
 334. 794020 ;:p' 
 
 34 794784^^- 
 
 35 794942';)^" 
 30 795101 ijp 
 
 37 795259 ^^: 
 
 38 795417 ^r 
 
 39 795575 ^J^' 
 
 40 795733 ^^" 
 
 41 9.795891 g^" 
 
 42 790049 ^J" 
 
 43 - 790200 f^ 
 
 44 79«J304 ;}i 
 
 45 790621 ^^ 
 790079 OK 
 7»ii836 ^ 
 790993 ^ 
 797150 .':^ 
 
 40 
 
 47 
 48 
 49 
 60 
 
 52 
 63 
 64 
 55 
 50 
 57 
 58 
 69 
 00 
 
 79/150 -^^. 
 
 797307 ,,2 
 
 61 9.797404 T^ 
 
 Cosine. 
 
 9,896532 
 890433 
 890335 
 890230 
 890137 
 890038 
 895939 
 895840 
 895741 
 895041 
 895542 
 ,89.5443 
 895343 
 895244 
 895145 
 895045 
 894945 
 894840 
 894740 
 894040 
 894540 
 .894440 
 894340 
 894240 
 894140 
 894040 
 893940 
 893840 
 893745 
 893045 
 893544 
 1.893444 
 893343 
 893243 
 -893142 
 893041 
 892940 
 892839 
 892739 
 892038 
 892530 
 
 9.892435 
 892334 
 892233 
 892132 
 892030 
 891929 
 891827 
 891720 
 891024 
 891523 
 
 .797404 ;" 
 
 797021 ^ 
 
 797777 1^. 
 
 797934 20 
 
 798091 OK 
 
 798247 ;^ 
 
 798403 il 
 
 798500 - 
 
 798710 ;^ 
 
 798872 '^^ 
 
 8 
 .8 
 8 
 8 
 8 
 8 
 7 
 7 
 7 
 7 
 7 
 7 
 6 
 6 
 
 .0 
 6 
 G 
 6 
 5 
 6 
 5 
 5 
 6 
 .4 
 4 
 4 
 4 
 4 
 4 
 A 
 3 
 3 
 ■ 3 
 3 
 3 
 3 
 3 
 2 
 2 
 2 
 2 
 2 
 1 
 
 ;}j9. 891421 
 
 Cosine. 
 
 891319 
 891217 
 891115 
 891013 
 89091 1 
 890809 
 890707 
 890005 
 890503 
 
 Sine. 
 
 D. 10" 
 
 16.4 
 10.5 
 10.5 
 10.5 
 10.5 
 10.5 
 10.6 
 10.5 
 10.6 
 10.6 
 10.5 
 16.6 
 16.0 
 10.6 
 10.0 
 10.0 
 10.0 
 10.0 
 16.6 
 10.6 
 
 10.6 
 
 16.7 
 
 16.7 
 
 16.7 
 
 10.7 
 
 16.7 
 
 16.7 
 
 16.7 
 
 16.7 
 
 16.7 
 
 16.7 
 
 16.8 
 
 16.8 
 
 10.8 
 
 10.8 
 
 16.8 
 
 10.8 
 
 10.8 
 
 10.8 
 
 10.8 
 
 10.8 
 
 16.9 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 17.0 
 
 Tang. 
 
 .892810 
 893070 
 893331 
 893591 
 893851 
 894111 
 894371 
 894632 
 894892 
 895152 
 895412 
 
 .895672 
 895932 
 890192 
 890452 
 890712 
 890971 
 897231 
 897491 
 897751 
 898010 
 
 .898270 
 898530 
 898789 
 899049 
 899308 
 899508 
 899827 
 900080 
 900340 
 900005 
 
 .900804 
 901124 
 901383 
 901042 
 901901 
 902100 
 902419 
 902079 
 902938 
 903197 
 
 .903455 
 903714 
 903973 
 904232 
 904491 
 904750 
 905008 
 905207 
 905520 
 
 ■ 906784 
 
 i. 900043 
 900302 
 900500 
 900819 
 907077 
 907336 
 907594 
 907852 
 908111 
 908369 
 
 Cotiuig. 
 
 61772;78040 
 
 61804 
 
 01887 
 
 01909 
 
 61932 
 
 61956J78490 
 
 61978"'"-° 
 
 62001 
 
 62024 
 
 62040 
 
 02009 78405 
 78387 
 78309 
 78351 
 78333 
 78315 
 78297 
 8279 
 T8201 
 78243 
 78225 
 ;8200 
 78188 
 78170 
 
 62388 78152 
 
 02411 
 02433 
 62450 
 
 78134 
 78110 
 78098 
 
 62479 78079 
 02502 78001 
 02524 78043 
 62547 78025 
 02570 78007 
 62592 77988 
 02015 77970 
 02038177952 
 
 Tung. 
 
 N. COS. N.«ine 
 
 51 Dugre«a. 
 
60 
 
 Log. Sines and Tangents. (39*^) Natural Sines. 
 
 TABLE IL 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 lU 
 
 H 
 
 12 
 
 13 
 
 14 
 
 15 
 
 IG 
 
 17 
 
 18 
 
 I 19 
 2U 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 3-2 
 33 
 34 
 35 
 3ti 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 40 
 47 
 48 
 49 
 60 
 51 
 52 
 53 
 54 
 55 
 50 
 57 
 58 
 59 
 00 
 
 Sine. 
 
 9.798772 
 799028 
 799184 
 799339 
 799495 
 799051 
 799806 
 799902 
 800117 
 8002/2 
 800427 
 9. 800582 
 800737 
 800892 
 801047 
 801201 
 801356 
 801511 
 801665 
 801819 
 801973 
 9.802128 
 802282 
 802436 
 802589 
 802743 
 8028y7 
 803050 
 803204 
 803357 
 80351 1 
 9.803664 
 803817 
 803970 
 804123 
 804276 
 804428 
 804581 
 804734 
 804880 
 805039 
 9.805191 
 805343 
 805495 
 805647 
 805799 
 805951 
 806103 
 806254 
 80o406 
 80J557 
 9.806/09 
 806860 
 807011 
 807163 
 807314 
 807465 
 80/615 
 807766 
 807917 
 808067 
 Cosine. 
 
 D. 10"| Cosine. 
 
 26.0 
 26.0 
 26.0 
 25.9 
 25.9 
 25.9 
 25.9 
 25.9 
 25.9 
 25.8 
 25.8 
 25.8 
 25.8 
 25.8 
 25.8 
 25.8 
 25.7 
 25.7 
 25.7 
 25.7 
 25.7 
 25.7 
 25.6 
 25.6 
 26.6 
 25.6 
 25.6 
 25.6 
 25.6 
 25.5 
 25.5 
 25.6 
 
 25 
 
 25.5 
 25.5 
 25.4 
 25.4 
 25.4 
 25.4 
 25.4 
 25.4 
 25.4 
 25.3 
 25.3 
 25.3 
 25.3 
 25.3 
 25.3 
 25.3 
 25.2 
 25.2 
 .2 
 25.2 
 25.2 
 25.2 
 25.2 
 25.1 
 25.1 
 25.1 
 25.1 
 
 9.890503 
 890400 
 890298 
 890195 
 89J0J3 
 889990 
 889888 
 88978) 
 -889682 
 889579 
 889477 
 
 9.889374 
 889271 
 889168 
 8S9064 
 8&S961 
 888858 
 888755 
 888651 
 888548 
 888444 
 .888341 
 888237 
 888134 
 888030 
 887926 
 887822 
 887718 
 887614 
 887510 
 887406 
 
 9.887302 
 887198 
 887093 
 886989 
 88(>885 
 886780 
 886676 
 886571 
 886466 
 886362 
 
 9.886257 
 886152 
 886047 
 885942 
 885837 
 885732 
 885627 
 885522 
 885416 
 885311 
 
 9.8^5205 
 885100 
 884994 
 88-1889 
 884783 
 884677 
 884572 
 884466 
 884360 
 884254 
 
 D. 10" 
 
 17.0 
 17.1 
 17.1 
 17.1 
 17.1 
 17.1 
 17.1 
 17.1 
 17. J 
 17.1 
 17.1 
 17.2 
 17.2 
 17.2 
 17.2 
 17.2 
 17.2 
 17.2 
 17.2 
 17.2 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.3 
 17.4 
 17.4 
 
 Tang. 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17.4 
 
 17.4 
 
 17.5 
 
 17.5 
 
 17.6 
 
 17.5 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 17.6 
 
 9.903369 
 90S;)28 
 908886 
 90J144 
 90J4J2 
 909660 
 909918 
 910177 
 910435 
 910 J93 
 910951 
 9.911209 
 911467 
 911724 
 911982 
 912240 
 912498 
 912756 
 913014 
 913271 
 913529 
 9.913787 
 914044 
 914302 
 914560 
 914817 
 915075 
 915332 
 915590 
 915847 
 916104 
 9.916362 
 916619 
 916877 
 917134 
 917391 
 917648 
 917905 
 918163 
 918420 
 918677 
 9.918934 
 919191 
 919448 
 919705 
 919962 
 920219 
 920476 
 920733 
 920990 
 921247 
 9.921503 
 921760 
 922017 
 922274 
 922530 
 922787 
 923044 
 923300 
 92355 7 
 923813 
 Co tan-'. 
 
 D. 10' 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 43.0 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42.9 
 
 42,9 
 
 42.9 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42,8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.8 
 
 42.7 
 
 IN. .siiic.)N. cofl.l 
 
 62932 
 
 77715 
 
 62955 77696 
 
 10.091631 
 
 091372 
 
 091114 
 
 090856 
 
 090598 
 
 090340 
 
 090J82 
 
 089823 
 
 089565 
 
 089307 
 
 089049 
 10.088791 
 
 088533 
 
 088276 
 
 088018 
 
 087760 
 
 087502 
 
 087244 
 
 086986 
 
 086729 
 
 086471 
 10.086213 
 
 085956 
 
 085698 
 
 085440 
 
 085183 
 
 084925 
 
 084668 
 
 084410 
 
 084153 
 
 083896 
 10-083638 
 
 083381 
 
 083123 
 
 082866 
 
 082609 
 
 082352 
 
 082095 
 
 081837 
 
 081580 
 
 081323 !| 63832 769 
 10 081066 1,63854 
 
 0808091! 6387, 
 
 080552 i 163899 
 
 080295 j I 63922 
 
 0800381163944 
 
 079781 11 63966 (7o866 
 
 0795241 6398 j|76847 
 
 079267 i! 6401 176828 
 
 62977 
 6300J 
 63022 
 63045 
 63068 
 
 1 63090 
 63113 
 63135 
 63158 
 93180 
 63203 
 63225 
 63248 
 63271 
 63293 
 63316 
 63338 
 63361 
 63383 
 63406 
 63428 
 63451 
 63473 
 63496 
 
 163518 
 
 163.540 
 63563 
 
 1 63585 
 63608 
 
 63630 
 63653 
 63675 
 63698 
 163720 
 1 63 742 
 ! 6371)5 
 ! 63787 
 ,63810 
 
 77678 
 77660 
 77641 
 77623 
 77605 
 77586 
 7751)8 
 77550 
 77631 
 77513 
 77494 
 77476 
 77458 
 77439 
 77421 
 77402 
 77384 
 77366 
 77347 
 77329 
 77310 
 77292 
 77273 
 77255 
 77236 
 77218 
 77199 
 77181 
 
 162 
 77144 
 77125 
 77107 
 
 0o8 
 77070 
 77051 
 
 0J3 
 77014 
 76996 
 
 76959 
 7t>940 
 76921 
 76903 
 
 76884 
 
 079010 ''64033 
 078753 1164056 
 10.0784971164078 
 
 078240' 
 
 07/9831 
 
 077726 
 
 077470 
 
 077213 
 
 076956 
 
 0/6700 
 
 076443 
 
 076187 
 
 64100 
 6412o 
 64145 
 6416 7 
 64190 
 04212 
 64234 
 64256 
 64279 
 
 Tan* 
 
 N. COS. N.t^iiu 
 
 ,6810 
 .6791 
 76772 
 76754 
 76735 
 76717 
 7u698 
 76679 
 76661 
 ,6642 
 76623 
 i6604 
 
 60 
 69 
 58 
 67 
 66 
 55 
 64 
 53 i 
 62. 
 51 
 60 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 6 
 4 
 3 
 2 
 1 
 
 
 50 Degrees. 
 
TAIiLli 11. 
 
 l.o;;. .>iiies and Tangents. (40°) Natural Sines. 
 
 61 
 
 Of J. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 ti 
 
 7l 
 
 loi 
 
 lll9 
 V2 
 13 
 14 
 15 
 lo 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 21J 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 I 4ti 
 47-1 
 48 
 49 
 50 
 51 
 52 
 63 
 64 
 65 
 5(i 
 57 
 58 
 59 
 60 
 
 Sine. 
 
 80-iOJ7 
 808il8 
 80-}3.)8 
 8a8jl9 
 80.-iJ6'^ 
 8j>-i«19 
 8J89d9 
 80J119 
 8i)J2U9 
 8JJ419 
 80J569 
 809718 
 8l)98o8 
 810017 
 
 -8l01l)7 
 810316 
 810465 
 810614 
 811)763 
 810912 
 811061 
 
 .S11210 
 811358 
 811507 
 811655 
 811804 
 811952 
 81210J 
 812248 
 812396 
 812544 
 
 .812692 
 812840 
 812988 
 813135 
 813-^83 
 813430 
 813578 
 813725 
 8138/2 
 814019 
 
 .814166 
 814313 
 814460 
 814607 
 8U763 
 814900 
 81504<) 
 815193 
 815339 
 815485 
 
 .815631 
 815-78 
 815924 
 8160u9 
 816215 
 816361 
 816507 
 816652 
 816798 
 816943 
 Cosine. | 
 
 D. W l Co atne. |D. 10"| Tang, 
 
 25.1 
 
 25.1 
 
 25.1 
 
 25.0 
 
 25.0 
 
 25.0 
 
 25.0 
 
 25.0 
 
 25.0 
 
 24.9 
 
 24.9 
 
 24.9 
 
 24.9 
 
 24.9 
 
 24 9 
 
 24.8 
 
 24.8 
 
 24.8 
 
 24.8 
 
 24.8 
 
 24.8 
 
 24.8 
 
 24.7 
 
 24.7 
 
 24.7 
 
 24.7 
 
 24.7 
 
 24.7 
 
 24.7 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.5 
 
 24.5 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.6 
 
 24.5 
 
 24.4 
 
 24.4 
 
 24.4 
 
 24.4 
 
 24.4 
 
 24.4 
 
 24.4 
 
 24.3 1, 
 
 21.3 
 
 24.3 
 
 24.3 
 
 24.3 
 
 24.3 
 
 24.3 
 
 24.2 
 
 24.2 
 
 24.2 
 
 .884254 
 
 884148 
 
 884042 
 
 88393( 
 
 883829 
 
 883723 
 
 883617 
 
 883510 
 
 883404 
 
 88329; 
 
 883191 
 .883084 
 
 882977 
 
 882871 
 
 882764 
 
 88265/ 
 
 8S2550 
 
 882443 
 
 882336 
 
 882229 
 
 882121 
 .882014 
 
 88190/ 
 
 881799 
 
 881692 
 
 881584 
 
 881477 
 
 88l3«i9 
 
 881261 
 
 881153 
 
 881046 
 .880938 
 
 880830 
 
 880 ; 22 
 
 880613 
 
 880505 
 
 8t;0397 
 
 880-289 
 
 880180 
 
 8800/2 
 
 879963 
 .879855 
 
 879746 
 
 879637 
 
 879629 
 
 879420 
 
 879311 
 
 8792(J2 
 
 879093 
 
 878984 
 
 8788/6 
 .878766 
 
 878G56 
 
 878647 
 
 878438 
 
 878328 
 
 878219 
 
 878109 
 
 877999 
 
 877890 ,„ o 
 _877780 ^^'^ 
 "sine. 
 
 17.7 
 17.7 
 .'J 17.7 
 W.7 
 17.7 
 17.7 
 17.7 
 17.7 
 17.7 
 17.8 
 17.8 
 17.8 
 17.8 
 17.8 
 17.8 
 17.8 
 17.8 
 17.8 
 17.9 
 17.9 
 17.9 
 17.9 
 [7.9 
 17.9 
 7.9 
 17.9 
 17.9 
 17.9 
 18.0 
 18.0 
 18.0 
 18.0 
 18.0 
 18.0 
 18.0 
 18.0 
 18.0 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.1 
 18.2 
 18.2 
 18.2 
 18.2 
 18.2 
 18.2 
 18.2 
 18.2 
 18.2 
 18.3 
 18.3 
 18.3 
 
 .923813 
 924070 
 924327 
 924583 
 924840 
 92509t) 
 925352 
 925«)09 
 9258()5 
 926122 
 926378 
 
 .926634 
 926890 
 927147 
 927403 
 927659 
 927915 
 928171 
 928427 
 928b83 
 928940 
 
 .929196 
 929452 
 929708 
 929964 
 930220 
 930475 
 930731 
 930987 
 931243 
 931499 
 
 ••931755 
 932010 
 932266 
 932522 
 932778 
 933033 
 933289 
 933545 
 933800 
 934056 
 
 1.934311 
 934567 
 934823 
 935078 
 935333 
 935689 
 936844 
 936100 
 936355 
 936610 
 
 1.936866 
 937121 
 937376 
 937632 
 937887 
 938142 
 938398 
 938()53 
 938908 
 939163 
 
 Cotarig. 
 
 D. lO'^ l Cota ng. I jN.Binc. 
 
 '64279 
 64301 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42.7 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42.7 
 
 42.7 
 
 42,7 
 
 42.7 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.6 
 
 42.5 
 
 42,5 
 
 42.6 
 
 42.5 
 
 42.6 
 
 10.076187 
 076930 
 0756,3 
 075417 
 075160 
 074904 
 074648 
 074391 
 074136 
 073878 
 073622 
 
 10.073366 
 073110 
 072853 
 072597 
 072341 
 072085 
 071829 
 071573 
 071317 
 071060 
 
 10.070804 
 070548 
 070292 
 070036 
 069780 
 069525 
 0ii92()9 
 069013 
 068767 
 068501 
 
 10.068245 
 067990 
 067734 
 067478 
 06.222 
 066967 
 066711 
 066456 
 066200 
 0ti5944 
 
 10.065689 
 065433 
 065177 
 064922 
 (I64()ti7 
 064411 
 064156 
 063900 
 063645 
 063390 
 
 10.0ti3134 
 062879 
 062624 
 062368 
 0O2113 
 061858 
 061602 
 Ot)1347 
 06109'2 
 06083/ 
 
 N. COS. 
 
 76604 
 765>}6 
 64323' 76567 
 64346176548 
 64308 1 76530 
 64390 76511 
 
 64412, 
 64435 
 64457 
 64479 
 64501 
 64524 
 64546 
 
 76492 
 76473 
 76456 
 7<)436 
 76417 
 76398 
 76380 
 
 645681 763ul 
 
 1 64690 
 164612 
 I (J4635 
 164657 
 1 64679 
 i 64701 
 1 64723 
 j 04746 
 
 64768 
 164790 
 164812 
 I 64834 
 
 64856 
 1 64878 
 164901 
 i 64923 
 i 64945 
 j 64967 
 i (>4989 
 I65U11 
 
 65033 
 
 7()342 
 7()323 
 7(io04 
 7()286 
 76267 
 76248 
 76229 
 76210 
 76192 
 76173 
 76154 
 76135 
 76116 
 76097 
 76078 
 76059 
 76041 
 76022 
 76UU3 
 75984 
 759u5 
 
 65055] /594b 
 |65077|769-^7 
 165100175908 
 I 65 1-^2 176889 
 66144 75 ~^'0 
 ;5ft51 
 75832 
 75813 
 76794 
 76775 
 /5«56 
 76738 
 75/19 
 ,6iOO 
 /56b0 
 /5061 
 , 604VJ 
 75623 
 /5o04 
 .5565 
 /55o6 
 7 554 / 
 / 5528 
 .5509 
 /54b0 
 /54/1 
 
 '6oltj6 
 
 65188 
 165210 
 i 66232 
 : 65:^54 
 
 65276 
 ' 65-iy8 
 
 653 V 
 1 65342 
 
 65364 
 
 ()5b8o 
 
 6540b 
 
 65430 
 
 65452 
 
 65474 
 
 65496 
 
 6551b 
 
 05540 
 
 655<jVf 
 
 655b ^ 
 
 65i>lk> 
 
 N. fos. N.tsiue 
 
 49 Degree*. 
 
62 
 
 —] 
 
 Log:. Sines and Tangents. (41°) Natural Sines. 
 
 TABLE n. 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 t8 
 49 
 50 
 51 
 62 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 
 Sine. 
 
 3.816943 
 817088 
 817233 
 817379 
 817524 
 817668 
 817813 
 817958 
 818103 
 818247 
 818392 
 
 3.818536 
 818681 
 818825 
 818969 
 819113 
 819257 
 819401 
 819545 
 819689 
 819832 
 
 >. 819976 
 826120 
 820263 
 826406 
 82055U 
 820693 
 820836 
 820979 
 821122 
 821265 
 
 >. 821407 
 821560 
 821693 
 821835 
 821977 
 822120 
 822262 
 822404 
 822646 
 822688 
 
 1,822830 
 822972 
 823114 
 823255 
 823397 
 823639 
 8i3680 
 823821 
 823963 
 824104 
 
 1.824245 
 824386 
 824527 
 824668 
 8248 iJ8 
 824949 
 825090 
 825230 
 826371 
 825511 
 
 Cosine. 
 
 D. 10" Cosine. 
 
 24.2 
 24.2 
 24.2 
 24.2 
 24.1 
 24.1 
 24.1 
 24.1 
 24.1 
 24.1 
 24.1 
 24.0 
 24.0 
 24.0 
 24.0 
 24.0 
 24.0 
 24.0 
 23.9 
 23.9 
 23.9 
 23.9 
 23.9 
 23.9 
 23.9 
 23.8 
 23.8 
 23.8 
 23.8 
 23.8 
 23.8 
 23.8 
 23.8 
 23.7 
 23.7 
 23.7 
 23.7 
 23.7 
 23.7 
 23.7 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.6 
 23.5 
 23.5 
 23.4 
 23.4 
 23.4 
 23.4 
 23.4 
 23.4 
 
 1.877780 
 877670 
 877560 
 877450 
 877340 
 877230 
 877120 
 877010 
 876899 
 876789 
 876678 
 .876568 
 876457 
 876347- 
 876236 
 876125 
 876014 
 875904 
 875793 
 876682 
 875571 
 .875469 
 875348 
 875237 
 875126 
 875014 
 874903 
 874791 
 874680 
 874568 
 874456 
 .874344 
 874232 
 874121 
 874009 
 873896 
 873784 
 873672 
 873560 
 873448 
 873335 
 .873223 
 873110 
 872998 
 872885 
 872772 
 872659 
 872647 
 872434 
 872321 
 872208 
 .872096 
 871981 
 871868 
 871765 
 871641 
 871528 
 871414 
 871o01 
 871187 
 871073 
 Sine. 
 
 D. 10' 
 
 18.3 
 18.3 
 18.3 
 18.3 
 18.3 
 18.4 
 18.4 
 18.4 
 18.4 
 18.4 
 18.4 
 18.4 
 18.4 
 18.4 
 18.5 
 18.5 
 18.6 
 18.5 
 18.6 
 18.5 
 18.5 
 18.5 
 18.5 
 18.6 
 18.6 
 18.6 
 18.6 
 18.6 
 18.6 
 18.6 
 18.6 
 18.6 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.7 
 18.8 
 18.8 
 18.8 
 18.8 
 18.8 
 18.8 
 18.8 
 18.8 
 18.8 
 18.9 
 18.9 
 18.9 
 18.9 
 18.9 
 18.9 
 18.9 
 18.9 
 18.9 
 
 Tana;. 
 
 3.939163 
 939418 
 939673 
 939928 
 940183 
 940438 
 940o94 
 940949 
 941204 
 941458 
 941714 
 
 3,941968 
 942223 
 942478 
 942733 
 942988 
 943243 
 943498 
 943752 
 944007 
 944262 
 
 3.944617 
 944771 
 945026 
 945281 
 945535 
 945790 
 946045 
 946299 
 946554 
 946808 
 
 3.9470j3 
 947318 
 947672 
 947826 
 948081 
 948336 
 948590 
 948844 
 949099 
 949363 
 
 3.949607 
 949862 
 9501 16 
 950370 
 950625 
 950879 
 951133 
 951388 
 951642 
 951896 
 
 ),95216U 
 952405 
 952659 
 952913 
 963167 
 953421 
 953676 
 953929 
 954183 
 954437 
 Cotanj'. 
 
 D. 10' 
 
 42.5 
 
 42.5 
 
 42.5 
 
 42.6 
 
 42.5 
 
 42.6 
 
 42.5 
 
 42.6 
 
 42.5 
 
 42.6 
 
 42.5 
 
 42.5 
 
 42.6 
 
 42.5 
 
 42 6 
 
 42.5 
 
 42.6 
 
 4-2.6 
 
 42.5 
 
 42.5 
 
 42.6 
 
 42.6 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.4 
 
 42.3 
 
 42.3 
 
 42.3 
 
 42.3 
 
 42.3 
 
 Cctang. I, N. sine. 
 
 10.030837 1 166606 
 0605821! 65628 
 060327 I j 66650 
 
 75471 
 75452 
 75433 
 
 060072 165672 76414 
 059817'! 66691 
 059562;! 65716 
 059306 ,65738 
 
 75395 
 75375 
 75366 
 
 059051 l65759|75337 
 
 058796 65781 
 058542 j 165803 
 058286 '165825 
 
 10.058032 1 1 65847 
 05777711 65869 
 057522! I 65891 
 057267 65913 
 057012;! 65935 
 056767 '65956 
 056502 165978 
 0562481 66000 
 055993 I' 66022 
 055 738! 166044 
 
 10.055483; 66066 
 055229 H 66088 
 064974! 66109 
 054719 I! 66131 
 054465 1 166153 
 054210 66175 
 053955' 66197 
 0537011 66218 
 063446 '66240 
 053192' 66262 
 
 10.052937' 66284 
 05J682 ' 66306 
 0524281166327 
 0521741:66349 
 051919 i| 66371 
 051664' 66393 
 051410 66414 
 051156 66436 
 050901 1 66458 
 050647 '66480 
 
 10.050393 I 66501 
 050138 I 66523 
 049884 ! 66545 
 0496301 665b6 
 049376' 66588 
 049121 1 66610 
 0488671 66632 
 0486121 66653 
 0483581 66675 
 048104' 66697 
 
 10.047860 66718 
 047696; 66740 
 047341 '66762 
 
 uo.. 
 
 75318 
 
 75299 
 
 75280 
 
 75261 
 
 75241 
 
 75222 
 
 75203 
 
 75184 
 
 75165 
 
 75146 
 
 75126 
 
 75107 
 
 75088 
 
 76069 
 
 75050 
 
 75030 
 
 76011 
 
 74992 
 
 74973 
 
 74953 
 
 74934 
 
 74915 
 
 74896 
 
 74876 
 
 74857 
 
 74838 
 
 74818 
 
 4799 
 
 4780 
 
 4760 
 
 74741 
 
 4722 
 
 74703 
 
 4683 
 
 74663 
 
 4644 
 
 4625 
 
 74606 
 
 74586 
 
 74567 
 
 74548 
 
 4522 
 
 4509 
 
 74489 
 
 74470 
 
 4451 
 
 04708 7 1 66783174431 
 046833 1166805 
 0465 79 '66827 
 046325' 66848 
 046071 66870 
 045817 -'66891 
 045563' 66913 
 
 Tanj^. 
 
 74412 
 74392 
 74373 
 74353 
 74334 
 74314 
 
 60 
 
 59 
 68 
 57 
 56 
 55 
 54 
 ,63 
 I 52 
 61 
 60 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 36 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 26 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 ; 
 
 8 1 
 
 7 ' 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
 48 Dej'rees. 
 
TA1JL1-: II. 
 
 Ix)g. Sines and Tangents. (42") Natural ciinc'S. 
 
 63 
 
 019 
 
 I 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 11 
 12 
 13 
 14 
 13 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2(i 
 27 
 28 
 29 
 30' 
 31 9 
 32 
 33 
 
 .825511 
 
 825661 
 
 825791 
 
 825 ).l 
 
 82(3 (71 
 
 82t).ll 
 
 82G];)! 
 
 820491 
 
 820031 
 
 820 7.0 
 
 820910 
 
 8270.9 
 
 837189 
 
 8273-8 i 
 
 8274i)7 I 
 
 827600 
 
 82 7; 45 
 
 8278S4 
 
 828023 
 
 828102 
 
 828301 
 .828439 
 
 828578 
 
 828710 
 
 828855 
 
 828993 
 
 829131 
 
 82^209 
 
 829407 
 
 829545 
 
 829683 
 .829821 
 
 829959 
 
 830097 
 
 830234 . ;. 
 
 830372 ^ 
 
 830509 
 830.)40 
 830784 
 830921 
 831058 
 .831195 
 831332 
 831409 
 
 831o06 -; 
 831742 ^^ 
 
 22 
 22 
 22 
 22 
 22 
 22 
 22 
 22 
 22 
 
 833105 22 
 
 833241 t, 
 
 833377 t; 
 
 83 3512 ** 
 
 833048 ,■;.'; 
 
 833783 ^^ 
 Oxsine. 
 
 831879 
 832U15 
 832152 
 83-2288 
 832425 
 832661 
 832097 
 832833 
 83-2969 
 
 9.871073 
 870.>0iJ 
 870840 
 870732 
 870618 
 870501 
 870^90 
 870270 
 870161 
 8;0J47 
 8 .99 }3 
 
 9.869818 
 -869704 
 869589 
 869474 
 869360 
 869245 
 869130 
 809015 
 8)8900 
 -868785 
 
 9.868070 
 868555 
 808440 
 8J«324 
 808209 
 808093 
 867978 
 867862 
 867747 
 
 ^ 807031 
 
 19.867515 
 807399 
 807-283 
 807167 
 807051 
 800935 
 800819 
 860703 
 80()586 
 866470 
 860353 
 800-237 
 8061-20 
 860004 
 86.5887 
 866770 
 805()53 
 865530 
 865419 
 865302 
 
 19.865185 
 805068 
 804950 
 864833 
 804716 
 864598 
 8(>4481 
 864303 
 864245 
 804127 
 .Sine. 
 
 19. 
 19. 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19 
 19, 
 19, 
 19, 
 19, 
 19 
 19, 
 19, 
 19 
 19 
 19 
 19 
 19 
 19 
 19, 
 19, 
 19, 
 19, 
 19. 
 19, 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19. 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19, 
 19 
 19, 
 19, 
 19, 
 19, 
 19 
 19 
 19 
 19, 
 
 ' Sine. D. 10" Co.sin«. D. 10"( Tang. D. 10" Cotung. |,N. sine. [ N. cos. 
 
 .954437 
 954691 
 954945 
 955-200 
 955454 
 955707 
 955901 
 950215 
 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 
 950409 to'" 
 
 950723 
 950977 
 
 .957231 
 957485 
 957739 
 957993 
 958246 
 958600 
 958754 
 959008 
 959202 
 959516 
 
 .959769 
 900023 
 960277 
 900531 
 900784 
 961038 
 961291 
 961545 
 901799 
 902052 
 
 .962300 
 962560 
 962813 
 963067 
 903320 
 903574 
 963827 
 904081 
 964335 
 904588 
 
 .904842 
 965095 
 905349 
 905602 
 965855 
 900109 
 9603(i2 
 906616 
 966869 
 967123 
 
 .967376 
 967629 
 967883 
 968136 
 968389 
 968643 
 968896 
 969149 
 969403 
 969656 
 
 Cotaug. 
 47 Degreey. 
 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42. 3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.3 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 42.2 
 
 10.045563 
 045309 
 045056 
 044800 
 044540 
 044293 
 0+1039 
 043785 
 043531 
 043277 
 043023 
 
 10.042769 
 (H2615li 
 042-261 : 
 042007 
 041764 
 041500 
 041240 
 040992; 
 04U738 
 040484 
 
 10.040231: 
 039977: 
 0b9723l 
 0394691 
 03921611 
 038962 
 038709 
 038455 
 038201 1 1 
 03794811 
 
 10.037694 i 
 037440 
 037187 
 03(i933 
 036680 
 030426 
 036173 |l 
 035919'. 
 035666 i! 
 035412 'i 
 
 10.035158: 
 034905 
 
 6691374314 
 6693574295 
 66956 174276 
 06978174256 
 6699974237 
 67021 '74217 
 67043174198 
 07064174178 
 67086.74169 
 67107174139 
 67129 74120 
 67151174100 
 67172174080 
 67194174061 
 67215:74041 
 67-237 1740-22 
 6725874002 
 6728073983 
 67301173963 
 673-23173944 
 673441739-24 
 67366 73904 
 67387173885 
 6V409'73865 
 67430:73846 
 67452173826 
 67473'73806 
 67495:73787 
 67516:73767 
 67538173747 
 67659173728 
 6768073708 
 
 67602 
 67623 
 67645 
 67666 
 
 73688 
 73669 
 73649 
 73()-29 
 
 034e;51 " 
 034o98 
 034145 
 033891 
 033638 
 033384 
 033131 
 03-2877:' 
 10.03-26-24; 
 032371 i 
 0321 17 j 
 0318641 
 031611 I 
 031357 
 031104 
 03U851 
 030597 
 030344 
 Tang. 
 
 67688173610 
 
 67709 73590 
 
 67730 7357 
 
 67762J73551 
 
 67773173531 
 
 67796173511 
 
 67816173491 
 
 67837173472 
 
 67859'73452 
 
 6788073432 
 
 67901 
 
 67923 
 
 67944 
 
 67965 
 
 i3413 
 73393 j 
 73373; 
 73 53j 
 67987173333; 
 
 68U08 
 68029 
 68051 
 680/2 
 68093 
 68115 
 68136 
 68157 
 68179 
 68200 
 
 73314 
 3294 
 3274 
 73264 
 73234 
 73215 
 73195 
 73176 
 73155 
 73136 
 
 N. io.«. N.. sine. I 
 
64 
 
 Log. Sines and Tangents. (43°) Natural Sines. 
 
 TABLE II. 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 t 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 It) 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 2(j 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 3d 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 55 
 56 
 67 
 58 
 59 
 60 
 
 Slue. 
 
 1.833783 
 
 833919 
 
 834054 
 
 834189 
 
 83 4325 
 
 834460 
 
 834595 
 
 834730 
 
 834865 
 
 834999 
 
 835134 
 ,835269 
 
 «^35403 
 
 835538 
 
 835672 
 
 8358 J7 
 
 835941 
 
 8360/5 
 
 836209 
 
 836343 
 
 836477 
 .836611 
 
 836745 
 
 8368/8 
 
 837012 
 
 837146 
 
 837279 
 
 837412 
 
 837546 
 
 837679 
 
 837812 
 .83794 J 
 
 8380/8 
 
 S3»211 
 
 838344 
 
 838477 
 
 838610 
 
 838 742 
 
 8388-6 
 
 839007 
 
 839140 
 .8392 72 
 
 839404 
 
 839536 
 
 839668 
 
 83980J 
 
 839932 
 
 840064 
 
 840196 
 
 840328 
 
 840159 
 9.84U591 
 
 840722 
 
 840854 
 
 840985: 
 
 84iii6l;rJ 
 
 841247 1^ 
 841378 I „j 
 841509 
 841640 
 8417/1 
 sin". 
 
 D. 10" 
 
 Co.siiie. 
 
 9.864127 
 864010 
 863892 
 863774 
 863656 
 863538 
 863419 
 863301 
 863183 
 863064 
 862946 
 
 9.862827 
 862709 
 862590 
 862471 
 862353 
 862234 
 862115 
 861996 
 861877 
 861758 
 
 3.861638 
 
 -861519 
 861400 
 861280 
 861161 
 861041 
 860922 
 860802 
 860682 
 860562 
 
 9.860442 
 860322 
 8602U2 
 860082 
 859962 
 859842 
 859721 
 8596U1 
 859480 
 859360 
 
 9.859239 
 859119 
 858998 
 8588 7 7 
 858756 
 858635 
 858514 
 858393 
 858272 
 858151 
 
 9.858029 
 857908 
 857786 
 85 7665 
 857543 
 85/422 
 857300 
 857178 
 857056 
 85J934 
 Sine. 
 
 P. 10' 
 
 19.6 
 19.6 
 19.7 
 19.7 
 19.7 
 19.7 
 19.7 
 19.7 
 19.7 
 19.7 
 19,8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.8 
 19.9 
 19.9 
 19.9 
 19.9 
 19.9 
 19.9 
 19.9 
 19.9 
 19.9 
 20.0 
 20.0 
 20.0 
 20.0 
 20.0 
 20.0 
 20.0 
 20.0 
 20.1 
 20.1 
 20.1 
 20.1 
 2U.1 
 20.1 
 20.1 
 20.1 
 20.2 
 20.2 
 20.2 
 20.2 
 20.2 
 20.2 
 20.2 
 2U,2 
 20. 2 
 20.3 
 20.3 
 20,3 
 20.3 
 20.3 
 20.3 
 
 Tang. 
 
 9.969656 
 969909 
 970162 
 970416 
 970669 
 970922 
 971175 
 971429 
 971682 
 971935 
 972188 
 
 9.972441 
 972694 
 972948 
 973201 
 973454 
 973707 
 973960 
 974213 
 974466 
 974719 
 974973 
 975226 
 976479 
 975732 
 975985 
 976238 
 976491 
 976744 
 976997 
 977250 
 
 9.977503 
 977756 
 978009 
 978262 
 978515 
 978768 
 979021 
 979274 
 979527 
 979780 
 .980033 
 980286 
 980538 
 980791 
 981044 
 981297 
 981550 
 981803 
 982056 
 982309 
 
 9.982562 
 982814 
 983067 
 9J3320 
 983573 
 983826 
 984079 
 984331 
 984584 
 98483; 
 Cotaii'j;. 
 
 D. 10" 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42,2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42,2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.2 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 42.1 
 
 Cotaug. iJ.V .siuc.|N. 
 
 10.030344 
 030091 
 029838 
 029584 
 029331 
 029078 
 028825 
 028571 
 028318 
 028065 
 027812 
 
 10.027559 
 027306 
 027052 
 026799 
 026546 
 026293 
 026040 
 025787 
 025534 
 025281 
 
 10.025027 
 024774 
 024521 
 024268 
 024015 
 023762 
 023509 
 023256 
 023003 
 022750 
 
 10.022497 
 022244 
 021991 
 021738 
 021485 
 021232 
 020979 
 020726 
 020473 
 020220 
 
 10.019967 
 019714 
 019462 
 019209 
 018956 
 018703 
 018450 
 018197 
 017944 
 017691 
 
 10.017438 
 017186 
 016933 
 016680 
 016427 
 016174 
 015921 
 015669 
 015416 
 115163 
 Tanij. 
 
 168200 73135 
 
 68221 
 i 68242 
 1 68264 
 168285 
 1 68306 
 
 68327 
 
 73116 
 73096 
 73076 
 73056 
 73036 
 73016 
 
 ! 1 68349172996 
 
 ! 1 683 70172976 
 
 '68391 72957 
 
 I 68412172937 
 
 68434172917 
 
 i 1 68455172897 
 
 i!684/6'72877 
 
 1 1 6849 7 172857 
 
 j 68518172837 
 
 j] 68539 72817 
 
 jj 68561 72797 
 
 168582 72777 
 
 '168603 72757 
 
 68624 72737 
 
 68645 72717 
 
 68666172697 
 
 68688 172677 
 
 68709172657 
 
 68730r72637 
 
 68751|72617 
 
 68772'72597 
 
 68793:725 77 
 
 68814|72557 
 
 68835 172537 
 
 68857j72517 
 
 68878172497 
 
 68899 72477 
 
 68920 72457 
 
 68941 172437 
 
 68962|7241 7 
 
 68983|72397 
 
 : 1 69004172377 
 
 I 69025^72357 
 1169016 72337 
 |i69067!72317 
 
 |69088!72297 
 i I 6910972277 
 1169130 72257 
 '6915172236 
 ;! 69172 72216 
 6^)193 72196 
 ■69214 72176 
 {69235 72156 
 169256 72136 
 16927/172116 
 '169298 72096 
 69319 72075 
 I 69340 172055 
 i! 69361 -72035 
 : 1 69382 7201 5 
 
 II 69403 71995 
 69424 71974 
 
 i 69445 7 1951 
 ! 69466^71931 
 
 N. ous.iN. 
 
 46 Degrees. 
 
TABLE II. 
 
 Log. Sines and Tangents. (U°) Natural Sine*. 
 
 65 
 
 u 
 
 9.841771 
 
 1 
 
 841902 
 
 •2 
 
 S42033 
 
 ;j 
 
 8421(i3 
 
 4 
 
 842294 
 
 6 
 
 842424 
 
 G 
 
 842555 
 
 7 
 
 842.iH5 
 
 8 
 
 842815 
 
 9 
 
 84294b 
 
 lU 
 
 84307b 
 
 11 
 
 9.84320(i 
 
 1-2 
 
 843336 
 
 13 
 
 8434bb 
 
 14 
 
 84;J696 
 
 16 
 
 843/25 
 
 lb 
 
 843855 
 
 17 
 
 843984 
 
 18 
 
 844114 
 
 19 
 
 844243 
 
 20 
 
 844372 
 
 21 
 
 9.844502 
 
 22 
 
 844631 
 
 23 
 
 844760 
 
 24 
 
 844889 
 
 26 
 
 845018 
 
 2G 
 
 845147 
 
 27 
 
 845276 
 
 28 
 
 845405 
 
 29 
 
 845533 
 
 30 
 
 845662 
 
 31 
 
 9.845790 
 
 32 
 
 845919 
 
 33 
 
 846047 
 
 34 
 
 846175 
 
 35 
 
 846304 
 
 3G 
 
 846432 
 
 37 
 
 846560 
 
 38 
 
 846688 
 
 39 
 
 846816 
 
 40 
 
 84*i944 
 
 41 
 
 9.847071 
 
 42 
 
 847199 
 
 43 
 
 847327 
 
 44 
 
 847454 
 
 45 
 
 847582 
 
 Hi 
 
 847709 
 
 47 
 
 847836 
 
 48 
 
 84/964 
 
 49 
 
 8480JI 
 
 60 
 
 848218 
 
 5! 
 
 9.848345 
 
 62 
 
 848472 
 
 63 
 
 848599 
 
 64 
 
 848 r26 
 
 65 
 
 H I8tt52 
 
 66 
 
 8489.9 
 
 67 
 
 849106 
 
 58 
 
 819232 
 
 69 
 
 849359 
 
 60^ 
 
 849485 
 
 Cosine. 
 
 Cotang. I N. sino. N. cos. 
 
 10.0151G3 
 0'4910 
 014657 
 014104 
 014152 
 013899 
 013646 
 013393 
 013140 
 012888 
 012635 
 
 10.012382 
 012129 
 011877 
 011b24 
 011371 
 011118 
 010866 
 010613 
 010360 
 010107 
 
 10.0U9855 
 009602 
 009349 
 009097 
 008844 
 0118591 
 ■ ^338 
 0u<->08b 
 007833 
 007580 
 
 10 007328 
 00.0/5' 
 006822 I 
 006570 i 
 006317 i 
 00o064! 
 005811) 
 005559 ' 
 005306 j 
 005053 
 
 10- 004801 I 
 004548 ! 
 004295 
 004043 ' 
 0J3.90 
 003537 i 
 003285 
 003032 
 002779 
 002527 ; 
 
 10 002274 
 002021 ■ 
 001769 
 00151b I 
 001263 
 001011 i 
 000758 ! 
 000505, 
 00i>253!' 
 000000 I 
 
 6;>4()t) 
 6948 
 '>95US 
 ; 69529 
 69549 
 69570 
 ()9591 
 69(il2 
 691533 
 69654 
 69676 
 69()9( 
 69il1 
 69737 
 : 69 758 
 69779 
 6980.) 
 69821 
 69842 
 6986' 
 69o83 
 69901 
 69925 
 69946 
 ()99()6 
 69987 
 70008 
 700-29 
 70049 
 700/0 
 70091 
 70112 
 7013 
 70103 
 70174 
 70195 
 70215 
 70236 
 70257 
 702/7 
 70298 
 70319 
 70339 
 703()0 
 70..81 
 70401 
 70422 
 70443 
 70463 
 70484 
 70505 
 70525 
 70546 
 7056 < 
 70587 
 70(i08 
 70628 
 70649 
 70670 
 70690 
 70711 
 
 71934 
 71914 
 71894 
 71873 
 718i,3 
 71833 
 71813 
 71792 
 71772 
 71752 
 71732 
 71711 
 71691 
 71671 
 71650 
 71630 
 71610 
 71590 
 71569 
 71549 
 71529 
 71508 
 71488 
 
 71468 137 
 
 71447 
 71427 
 71407 
 71386 
 71366 
 71345 
 /1 325 
 71305 
 71284 
 71264 
 71243 
 71223 
 71203 
 71182 
 71162 
 71141 
 /1 121 
 71100 
 1080 
 
 Tang. 11 N. coc. N.ciin 
 
 1059 17 
 /1038j 16 
 71019' 16 
 70998 14 
 709/8 
 70967 
 
 0937 
 70916 
 7LI896 
 
 0876 
 70855 
 70834 
 70813 
 70793 
 /0772 
 i0752 
 70731 
 /O.ll 
 
 45 Degrees. 
 
36 
 
 
 LOGARITHMS 
 
 
 
 
 
 TABLE III. 
 
 
 
 
 
 LOGARITHMS OF NUMBERS. 
 
 
 
 
 From I to 200, 
 
 
 
 
 INCLUDING 
 
 TWELVE DECIMAL 
 
 PLACES. ( 
 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 
 1 
 
 000000 000000 
 
 41 
 
 612783 856720 
 
 81 
 
 908485 018879 
 
 
 2 
 
 301029 995664 
 
 42 
 
 623249 290398 
 
 82 
 
 913813 852384 
 
 
 3 
 
 477121 254720 
 
 43 
 
 633468 455580 
 
 83 
 
 919078 092376 
 
 
 4 
 
 60-2039 991328 
 
 44 
 
 643452 676486 
 
 84 
 
 924279 286062 
 
 
 6 
 
 698970 004336 
 
 45 
 
 653212 513775 
 
 85 
 
 929418 925714 
 
 
 6 
 
 778151 250384 
 
 46 
 
 662757 831682 
 
 86 
 
 934498 451244 
 
 
 7 
 
 845098 040014 
 
 47 
 
 672097 857926 
 
 87 
 
 939519 252619 
 
 
 8 
 
 903089 98()992 
 
 48 
 
 681241 237376 
 
 88 
 
 944482 672150 
 
 
 9 
 
 954242 609439 
 
 49 
 
 690196 080028 
 
 89 
 
 949390 006645 
 
 
 10 
 
 yame as to 1. 
 
 50 
 
 Samo as to 5. 
 
 90 
 
 Same as to 9. 
 
 
 11 
 
 041392 685158 
 
 51 
 
 707570 176098 
 
 91 
 
 959041 392321 
 
 
 12 
 
 079181 246048 
 
 62 
 
 716003 343635 
 
 92 
 
 963787 827346 
 
 
 13 
 
 113943 352307 
 
 63 
 
 724275 8()9n01 
 
 93 
 
 968482 948554 
 
 
 14 
 
 146128 035678 
 
 54 
 
 732393 759823 
 
 94 
 
 973127 853600 
 
 
 16 
 
 176091 269056 
 
 55 
 
 740362 689494 
 
 96 
 
 977723 605889 
 
 
 16 
 
 204119 982656 
 
 56 
 
 748188 027006 
 
 96 
 
 982271 233040 
 
 
 17 
 
 230448 921378 
 
 57 
 
 755874 855672 
 
 97 
 
 986771 734266 
 
 
 18 
 
 255272 505103 
 
 68 
 
 763427 993563 
 
 98 
 
 991226 075692 
 
 
 19 
 
 278:53 600953 
 
 69 
 
 770852 011642 
 
 99 
 
 995635 194598 
 
 
 20 
 
 Same as to 2. 
 
 60 
 
 Same as to 6 
 
 100 
 
 Same as to 10, 
 
 
 21 
 
 322219 2947 
 
 61 
 
 785329 835011 
 
 101 
 
 004321 373783 
 
 
 22 
 
 342422 680822 
 
 62 
 
 792391 699498 
 
 102 
 
 008600 171762 
 
 
 23 
 
 , 361727 836018 
 
 63 
 
 799340 549453 
 
 103 
 
 012837 224705 
 
 
 24 
 
 380211 241712 
 
 64 
 
 806179 973984 
 
 104 
 
 017033 339299 
 
 
 25 
 
 397940 008672 
 
 65 
 
 812913 356643 
 
 105 
 
 021189 299070 
 
 
 26 
 
 414973 347971 
 
 66 
 
 819543 935542 
 
 103 
 
 025305 865265 
 
 
 27 
 
 431363 764159 
 
 67 
 
 826074 802701 
 
 107 
 
 029383 777685 
 
 
 28 
 
 447158 031342 
 
 68 
 
 832508 912706 
 
 108 
 
 033 423 755487 
 
 
 29 
 
 462397 997899 
 
 69 
 
 838849 090737 
 
 109 
 
 037426 497941 1 
 
 
 30 
 
 8ime as to 3. 
 
 70 
 
 Same as to 7. 
 
 110 
 
 Same a* to 11. | 
 
 
 31 
 
 491361 693834 
 
 71 
 
 851258 348719 
 
 111 
 
 045322 978787 \ 
 
 
 32 
 
 505149 978320 
 
 72 
 
 857332 496431 
 
 112 
 
 049218 022670 
 
 
 33 
 
 518513 939878 
 
 73 
 
 863322 8G0120 
 
 113 
 
 053078 4434H3 
 
 
 34 
 
 631478 917042 
 
 74 
 
 869231 719731 ' 
 
 114 
 
 056904 851336 
 
 
 36 
 
 544068 044350 
 
 75 
 
 875 J61 263392 
 
 115 
 
 060397 840354 
 
 
 36 
 
 556302 500767 
 
 76 
 
 880813 692281 
 
 116 
 
 064457 989227 
 
 
 37 
 
 568201 724!)67 
 
 77 
 
 886490 725172 
 
 117 
 
 068185 861746 
 
 
 38 
 
 579783 596617 
 
 78 
 
 892094 602690 
 
 118 
 
 071882 007306 
 
 
 39 
 
 691064 607026 
 
 79 
 
 897627 091290 
 
 119 
 
 075546 961393 
 
 
 ^ 40 
 
 Same a« to 4. 
 
 80 
 
 Same as to 8. 
 
 120 
 
 Same a^ to 12. 
 
 
 
 • 
 
 
 
 - ^ 
 
 

 OF NUMBERS. 
 
 
 67 
 
 N. 
 
 121 
 
 082786 370316 
 
 , N. 
 
 Log. 
 
 N. 
 
 Log 
 
 148 
 
 170261 715395 
 
 176 
 
 243038 048686 
 
 122 
 
 086359 830675 
 
 149 
 
 173186 268412 
 
 176 
 
 245612 1 1678 14 
 
 123 
 
 0b990t 111439 
 
 ; 150 
 
 176091 259056 
 
 177 
 
 241973 2(i6362 
 
 124 
 
 093421 685162 
 
 151 
 
 178976 947293 
 
 178 
 
 250420 002309 
 
 125 
 
 096910 01 3008 
 
 1 152 
 
 181843 687945 
 
 179 
 
 252863 030980 
 
 126 
 
 100370 645118 
 
 ' 163 
 
 184691 430818 
 
 180 
 
 256272 505103 
 
 127 
 
 103803 7-20956 
 
 1 154 
 
 187520 710836 
 
 181 
 
 257678 574^69 
 
 128 
 
 10/209 969(i48 
 
 155 
 
 190331 698170 
 
 182 
 
 2(i0071 387<.85 
 
 129 
 
 110589 710-299 
 
 166 
 
 193124 588354 
 
 183 
 
 262451 089730 
 
 130 
 
 Same as to 13. 
 
 [ 157 
 
 195899 662409 
 
 184 
 
 264817 823010 
 
 131 
 
 117271 295656 
 
 158 
 
 198657 086954 
 
 186 
 
 267171 728403 
 
 132 
 
 120573 931-206 
 
 159 
 
 201397 124320 
 
 186 
 
 2(i9512 944218 
 
 133 
 
 123851 64091)7 
 
 160 
 
 204119 982656 
 
 187 
 
 271841 60fi536 
 
 134 
 
 127104 798365 
 
 161 
 
 206826 876032 
 
 188 
 
 274157 849264 
 
 136 
 
 130333 768495 
 
 162 
 
 209516 014543 
 
 189 
 
 276461 804173 
 
 136 
 
 133538 908370 
 
 163 
 
 212187 604404 
 
 190 
 
 278753 600953 
 
 137 
 
 136720 567156 
 
 164 
 
 214843 848048 
 
 191 
 
 281033 367248 
 
 138 
 
 139879 086401 
 
 166 
 
 217483 944214 
 
 192 
 
 283301 228704 
 
 139 
 
 143014 800-254 
 
 166 
 
 220108 088040 
 
 193 
 
 285557 309008 
 
 140 
 
 146128 035678 
 
 167 
 
 222716 471148 
 
 194 
 
 287801 729930 
 
 141 
 
 149219 112655 
 
 168 
 
 225309 281726 
 
 195 
 
 290034 611362 
 
 142 
 
 152288 344383 
 
 169 
 
 227886 704614 
 
 196 
 
 292256 071356 
 
 143 
 
 155336 037465 
 
 170 
 
 230448 921378 
 
 197 
 
 2944(;6 226162 
 
 144 
 
 158362 49-2095 
 
 171 
 
 232996 110392 
 
 198 
 
 296666 1902<j2 
 
 145 
 
 161368 002235 
 
 172 
 
 236528 446908 
 
 199 
 
 298863 07641 u 
 
 146 
 
 164352 855784 
 
 173 
 
 238046 103129 
 
 
 
 147 
 
 167317 334748 
 
 174 
 
 240549 248283 
 
 
 
 LOG 
 
 ARITHMS 
 
 ] 
 
 OF 
 
 •"ROM 
 
 THE PRI^ 
 200 TO 1543, 
 
 IE ]^ 
 
 lUMBERS 
 
 IN 
 
 CLUDING 
 
 rwE 
 
 LVE DECIM 
 
 AL I 
 
 'LACES. 
 
 N. 
 201 
 
 Log. 
 
 N. 
 277 
 
 Log. 
 
 N. 
 
 Log. 
 
 303196 057420 
 
 "442479 769064 i 
 
 379 
 
 578()39 209968 
 
 203 
 
 307496 037913 
 
 281 
 
 448706 319905 
 
 383 
 
 5831JJ8 773968 
 
 207 
 
 315970 345457 
 
 283 
 
 451786 435524 
 
 389 
 
 589949 (i01326 
 
 209 
 
 320146 286111 
 
 293 
 
 466867 620354 
 
 397 
 
 598790 50;i763 
 
 211 
 
 324282 455298 
 
 307 
 
 487138 375477 j 
 
 401 
 
 603144 372620 1 
 
 223 
 
 348304 863048 
 
 311 
 
 492760 389027 
 
 409 
 
 611723 308007 ' 
 
 227 
 
 356025 857193 
 
 313 
 
 495544 337546 , 
 
 419 
 
 622214 U22966 j 
 
 229 
 
 359835 482340 
 
 317 
 
 601059 262218 , 
 
 421 
 
 624282 095836 1 
 
 233 
 
 367356 92UK26 
 
 331 
 
 519827 993776 
 
 431 
 
 634477 270161 • 
 
 239 
 
 378397 900948 
 
 337 
 
 6276-29 900871 
 
 433 
 
 636487 89(>363 
 
 241 
 
 3820' 7 042575 
 
 347 
 
 640329 474791 
 
 439 
 
 642424 520242 
 
 261 
 
 399f»73 721481 
 
 349 
 
 642826 4-26959 ' 
 
 443 
 
 64()403 72(i223 
 
 267 
 
 409P33 123331 
 
 353 
 
 47774 705388 
 
 449 
 
 65-2246 341003 
 
 263 
 
 419955 748490 
 
 359 
 
 565094 448578 
 
 457 
 
 669916 200070 1 
 
 269 
 
 429 ?52 2800 J-2 
 
 367 
 
 664666 064252 , 
 
 461 
 
 663:00 9-25390 
 
 271 
 
 4?;2969 290874 
 
 373 
 
 571708 831809 
 
 463 
 
 665580 991018 
 
68 
 
 LOGARITHMS 
 
 N. 
 
 Log. 
 
 N. 
 
 Loi?. 
 
 N. 
 1171 
 1181 
 1187 
 11J3 
 1 1201 
 
 Log. 
 
 467 
 479 
 487 
 491 
 499 
 
 6i9.Jilu 580506 
 68033') 513414 
 68/(:28 961215 
 691081 492123 
 69ril00 545623 
 
 821 
 823 
 827 
 829 
 839 
 
 914343 157119 
 915399 835212 
 917505 50c)553 
 918554 530550 
 923761 960829 
 
 06bbi6 896073 
 0/2249 807613 
 0744-0 718955 
 076640 443670 
 0.9543 007385 
 
 503 
 509 
 521 
 523 
 641 
 
 701 507 985056 
 706717 782337 
 716837 '/ 23300 
 718501 688867 
 733197 2o5107 
 
 853 
 857 
 859 
 863 
 877 
 
 930949 031168 
 932980 821923 
 938993 163831 
 936010 795715 
 942999 593356 
 
 1213 
 1217 
 1223 
 1229 
 1231 
 
 0838 30 800845 
 085290 578-210 
 087426 458017 
 089551 882866 
 090258 052912 
 
 547 
 557 
 563 
 569 
 671 
 
 737987 326333 
 745855 195174 
 75U5U8 394851 
 755112 26(;393 
 766636 1082io 
 
 881 
 883 
 887 
 907 
 911 
 
 944975 908412 
 945960 703578 
 947923 619832 
 957607 287060 
 959518 376973 
 
 1237 
 1249 
 1259 
 
 1277 
 1279 
 
 092369 699609 
 096662 438356 
 100026 729204 
 103190 896808 
 1068/0 642460 
 
 577 
 587 
 593 
 599 
 601 
 
 761175 813156 
 768638 101248 
 773U54 693364 
 777426 8-^2389 
 778874 472002 
 
 919 
 929 
 937 
 941 
 947 
 
 963315 511386 
 968015 713994 
 971739 590888 
 973589 623427 
 976349 979003 
 
 1283 
 1-.89 
 1291 
 1297 
 1301 
 
 108226 656362 
 110252 917337 
 110926 242517 
 112939 986(K)6 
 114277 296540 
 
 607 
 613 
 617 
 619 
 631 
 
 783138 6910^5 
 787460 474518 
 790285 164033 
 791690 649020 
 80J0-29 359244 
 
 963 
 
 967 
 
 9;i 
 
 977 
 983 
 
 979092 900638 
 985426 474083 
 987219 229908 
 989894 563719 
 992553 517832 
 
 1303 
 1307 
 1319 
 1321 
 
 1327 
 
 114944 415712 
 116276 687564 
 120244 795568 
 120902 817604 
 122870 922849 
 
 641 
 643 
 647 
 653 
 659 
 
 806858 f)29519 
 808210 972924 
 8109J4 280569 
 814913 181275 
 818885 414594 
 
 991 
 
 997 
 
 1009 
 
 1013 
 
 1019 
 
 996073 654485 
 998695 158312 
 003891 166237 
 005609 445360 
 008174 184006 
 
 1361 
 1367 
 1373 
 1381 
 1399 
 
 133858 125188 
 135768 614554 
 137670 537223 
 140193 678544 
 145817 714122 
 
 661 
 673 
 677 
 683 
 691 
 
 810201 459486 
 828015 064224 
 830588 668685 
 834420 703682 
 839478 047374 
 
 1021 
 1031 
 1033 
 1039 
 1049 
 
 009025 742087 
 U 13258 665284 
 014100 321520 
 016615 547557 
 020775 488194 
 
 1409 
 1423 
 1427 
 1429 
 1433 
 
 148910 994096 
 153204 896557 
 154424 012366 
 155032 228774 
 156246 402184 
 
 701 
 
 709 
 719 
 727 
 733 
 
 845718 017967 
 850646 235183 
 856728 89U383 
 861534 410859 
 865103 974742 
 
 1051 
 1061 
 1063 
 1069 
 1087 
 
 021602 716028 
 025715 383901 
 026533 264523 
 028977 705209 
 036229 544086 
 
 1439 
 1447 
 1451 
 1453 
 1459 
 
 158060 793919 
 160468 531109 
 161667 412427 
 162265 614286 
 164055 291883 
 
 739 
 743 
 751 
 767 
 761 
 
 868644 488395 
 870988 813761 
 855639 937004 
 879095 879500 
 881384 656771 
 
 1091 
 1093 
 1097 
 1103 
 1109 
 
 037824 750588 
 038620 161950 
 040206 627575 
 042595 512440 
 044931 546119 
 
 1471 
 
 1481 
 1483 
 1487 
 1489 
 
 167612 672629 
 170555 058512 
 171141 151014 
 172310 968489 
 172894 731332 
 
 769 
 773 
 
 787 
 797 
 809 
 
 885926 339S01 
 888179 493918 
 895974 732359 
 901458 321396 
 907948 521612 
 
 1117 
 1123 
 1129 
 1151 
 1153 
 
 018053 173116 
 050379 756261 
 052693 941925 
 061075 323630 
 061829 307295 
 
 1493 
 1499 
 1611 
 1523 
 1531 
 
 174059 807708 
 175801 632866 
 179264 464329 
 182699 903324 
 184976 190807 
 
 811 
 
 909020 854211 
 
 1163 
 
 066679 714728 
 
 1643 
 
 188366 926063 
 
OF NUMBERS 
 
 69 
 
 AUXILIARY LOGARITHMS 
 
 N. 
 
 ,Oity 
 
 .008 
 ,007 
 008 
 005 
 
 1.003 
 1.002 
 1.001 
 
 Log. ;l N. 1 
 
 003891 1()()237 - 
 
 
 1.0)J9 
 
 003400532110 
 
 
 1.0003 
 
 003029470554 
 
 
 1.0)07 
 
 00259S08li(Jb5 
 
 
 i.oooy 
 
 0U21U(j0fil76() 
 
 M 
 
 1.0005 
 
 001-33712775 
 
 
 1.0004 
 
 001300933020 
 
 
 1.0003 
 
 0i)0.S«)7721529 
 
 
 1.0u02 
 
 000434077479 J 
 
 
 l.OUOl 
 
 Log. 
 
 00039l)»i89248 
 0i)034:29()(84 
 000';03899,84 
 00j2(J0^9.^547 
 OO.J-jU 0929,0 
 000I73(i83067 
 0001302()8804 
 00008085, »2 11 
 0U0;J434'J7277 
 
 N. 
 
 Log. 
 
 1.00009 
 1.00008 
 1.00007 
 1.00006 
 1.00005 
 1.00004 
 1.00003 
 1.00002 
 1.00001 
 
 000039083266 
 00003474tH)91 
 0(X)030398072 
 00002()0.55410 
 000021712704 
 00;)017371430 
 000013028638 
 000008()85802 
 000004342923 
 
 N. 
 
 1.000009 
 
 1 . oouo.js 
 
 1.000007 
 1.000006 
 1 . 000005 
 1.000004 
 1.000003 
 1.000002 
 1.000001 
 
 Log. 
 
 000003908(528 
 000003474338 
 000003040047 
 00000-'(i05756 
 0000021 7 14(J4 
 000001737173 
 000001302880 
 OU00008G8587 
 000000434294 
 
 1.0000001 
 1.00000001 
 [. 00001)0001 
 , 000000 JOOl 
 
 Log- 
 
 000000043429 
 000000004343 
 000000000434 
 000000000043 
 
 (n) 
 (o) 
 (P) 
 
 (q) 
 
 m=0.43429448l9 log. —1.637784298. 
 
 By the preceding tables — and the auxiliaries A, B, and 
 C, we can tind the logarithm of any number, true to at leant 
 ten decimal places. 
 
 But some may prefer to use the following direct formula, 
 which may be found in any of the standard works on algebra: 
 
 Log. (z-|-l)=log.2r+0.8685889638/^_L \ 
 
 if z be 
 
 The result will be true to twelve decimal places, 
 over 2000. 
 
 The log. of composite numbers can be determined by the 
 combination of logarithms, already in the table, and the prime 
 numbers from the formula. 
 
 Thus, the number 5083 is a prime number, find its log'a- 
 rithm. 
 
 We first find the log. of the number 3082. By factoring, 
 we discover thai this is the product of 46 into 67. 
 
70 NUMBERS 
 
 Log. 46, 1.6627578316 
 
 Log. 67, 1.8260748027 
 
 Log. 3082 3.4888326343 
 
 Log.3083=3.4888326343+«-^^^^^^96'^« 
 
 6165 
 
 NUMBERS AND THEIR LOGARITHMS, 
 
 OFTEN USED IN COMPUTATIONS. 
 
 Circamference of a circle to dia. 1 ) Log. 
 
 Surface of a sphere to dmmeter IV =3.14159265 0.4971499 
 Area of a circle to radiun 1 ) 
 
 Area of a circle to diameter 1 -— .7853982 —1.8950899 
 
 Capacity of a sphere to diameter 1 = .6236988—1.7189986 
 Capacity of a sphere to radius 1 =4.1887902 0.6220886 
 
 Arc of any circle equal to the radius =57°29578 1.7581226 
 Arc equal to radius expressed in sec. =:206264"8 5.3144251 
 Length of a degree, (radius unity) = .01745329 —2.2418773 
 
 12 hours expressed in seconds, = 43200 4.6354837 
 
 Complement of the same, =0.00002315 — 5.3645163 
 
 360 degrees expressed in seconds, = 1 296000 6. 1 1 26050 
 
 A gallon of distilled water, when the temperature is 62° 
 Fahrenheit, and Barometer 30 inches, is ^ILf^-J^j cubic 
 inches. 
 
 ^277.274=16.651542 nearly. 
 
 277.274 
 
 .775398 
 
 = 1 8.78925284 ^ 231 = 1 5. 1 98684. 
 
 V 282 =16.792855. 
 
 ^".785398 
 
 The French Metre=3.2808992, English feet linear mea- 
 sure, =39.3707904 inches, the length of a pendulum vi- 
 brating seconds. 
 
 ^^^- _= 18.948708. 
 
VB 35967 
 
 924227 
 
 ^A 53/ 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY