JC-NRLF ma REESE LIBRARY UNIVERSITY OF CALIFORNIA. Deceived 'lass No. THE TRANSITION-CURVE FIELD-BOOK. CONTAINING FULL INSTRUCTIONS FOR ADJUSTING AND LOCATING A CURVE NEARLY IDENTICAL WITH THE CUBIC PARABOLA IN TRANSITION BETWEEN ANY CIRCULAR RAILROAD CURVE AND TANGENT. SIMPLIFIED IN APPLICATION BY THE AID OF A GENERAL TABLE, AND ILLUSTRATED BY RULES AND EXAMPLES FOR VARIOUS PROBLEMS OF LOCATION: ALSO TABLES OF RADII, SINES, TANGENTS, VERSINES AND EXTERNAL SECANTS. BY C.E. NEW YOR K.- JOHN WILEY & SONS, 53 EAST TENTH STREET. 1891. COPYRIGHT, 1891, BY JOHN WILEY & SONS. FEERIS BROS., ROBERT DRUMMOND, printers, Electrotypes 82G p earl street, 444 & 446 Pearl Street, Yorkt New York. PREFATORY. THE aim of this Field-book is to furnish plain, practical rules and examples for guidance in ad- justing and locating a curve, nearly identical with the Cubic Parabola, as a Transition Curve in con- necting circular curves with tangents. In the investigation of the principles upon which the rules are based, it will be seen that with data consisting in great part of familiar approximations used in circular-curve location, and with no mathe- matics beyond a little algebra and trigonometry, practically exact results are reached in regard to laws of the Transition Curve and its relations to circular curves. By means of Transition Curves of less than i$ of central angle, tangents can be connected with all circular curves used on railroads, their combined location presenting little, if any, more difficulty than ordinary circular-curve location. The chords of each Transition Curve are all of the same length, but this length varies with each different rate of curvature of the connecting circular curve. By the use of the General Table, however, which applies to any chord length, the Transition Curve, whether laid off by angles or ordinates, is readily adjusted so as to connect with any circular curve, the two curves having the same rate of cur- vature, and a common tangent, at their point of connection. All of the Transition Curves considered, when laid off by angles, begin with a 2-minute deflec- IV PREFATORY. tion angle for the first chord, irrespective of the length of the chord or the degree of the connect- ing circular curve ; and, consequently, the field- work of location of all Transition Curves laid off by angles and with the same number of chords is the same, except for the change of the chord length in connecting with circular curves of different degrees of curvature, every connecting circular curve re- quiring a special chord length in the Transition Curve. The ordinates for laying off any Transition Curve all depend upon its special chord length, but each ordinate of every Transition Curve can be obtained from a single line of the table by simply multiply- ing a quantity taken from it by the special chord length in decimals of 100 feet. It was at first intended to give three tables for Transition Curves, beginning with deflection angles for the first chord of i minute, 2 minutes, and 3 minutes, respectively ; but the former and latter were laid aside for the reason that the advantages of increased convenience and diminished liability to error, in having a single table to refer to, en- tirely outweigh that of being able to make rather nicer adjustments in a few cases, which is about all that would be gained by using the three tables in- stead of one. For such Transition Curves as are needed in practice the table given, based upon the initial deflection of 2 minutes, seerns fo be all that is required. The word " connect," as applied to the junction of Transition and Circular curves, means that at their point of junction the two curves have always a common tangent, and, unless otherwise specified, the same rate of curvature. PREFATORY. V The expression "central angle," when applied to the Transition Curve, means the total curvature between the tangent at the initial point and that at the other point considered. Although the law governing the relations be- tween deflection angles from the initial station of Transition Curves has been suggested heretofore, it may be as well to say that when first brought to the attention of the writer it was through results then already reached and embodied in the present General Table, and also that he alone is responsi- ble for any new departures either in the matter or the manner of this Field-book. THE TRANSITION-CURVE FIELD-BOOK. GENERAL PROPOSITION. If a tangent SK and a circular arc DA are to be connected by a cubic parabola SEL of central angle a, the two curves to have a common tangent Z,T and the same rate of curvature at their junction Z, the adjusted gap CD between tangent and circu- lar curve will be one third of the middle ordinate FD of the circular arC of 201, and one fourth of the terminal ordinate LK of the parabola, i 2 THE TRANSITION-CURVE FIELD-BOOK. Draw the circular arcAV\ the chord VL sub- tending the central angle 2a\ and the tangent LT. The cubic parabola of central angle a and terminal tangent LT will have its initial tangent SK par- allel to VL. Through F, the middle of the chord LV, draw the radius OD and prolong it to C, mak- FD ing CD = - . At L and V lay off LK and FS perpendicular to ZFand equaj to ^C. Draw .S^T, which will be parallel to L V, and will pass through C. In a cubic parabola SEL the ordinate CYs mid- way between and K is to the ordinate LK at K and CE=~. o ~ is to (SJT) 3 , or But by construction Hence CZ> = zCE, and, cowsequently, the cubic parabola SEL will pass midway between C and J9. Denoting the middle ordinate FD of 20: by m, the gap (7Z> by ^, and the end ordinate Z^T by Jt:; by substitution in last equation, m x which was to be proved. Also if another circular arc connects with the same parabola under the same conditions at any THE TRANSITION-CURVE FIELD-BOOK. 3 other point Z', the same relations between the two curves exist; and using similar notation, m' x' Denoting the radius of the circular arc LD by , the parabola length SL by n, and taking = - y from similar triangles, O N n n t? Jt : :: : m, and m = r-=. 4 2 O./E Substituting for 45840' whence _m_ Z>V = 3 ~~ ^37520' ' jpo = i375 20 ^ / v f i (5) -M^f and 34380- By similar deduction for the circular curve con- necting with the parabola at Z', - whence 4 THE TRANSITION-CURVE FIELD-BOOK. But considering x and x f as ordinates to the cubic parabola, approximately, x : x' :: n* : n n ...... (7) Therefore Dn* :D'n n ::n*:n'\ and D Q :D' ::n:n' ...... (8) The central angles a and a' turned in the pa- rabola from S to L and Z', respectively, are meas- n n* ured by the circular arcs of the length - and , and of degrees of curve JD and D fQ per 100 feet. Therefore Dn a -- , ...... (9) 200 200 and a: a' :: Dn:D'n'. Multiplying alternate terms of (8) by n and ', Dn:>'n':: n*'.n'\ Hence (10) If N signifies the number of chord lengths c in ^, and N' the number of the same chord lengths c in ;/, then n cN and n' cN'\ and substituting these values of n and n' in (7), THE TRANSITION-CURVE FIELD-BOOK. 5 (10), and (8), respectively, the following propor- tions result: x: x' :: N* : N'\ .... (n) a\ a' ::N*: N n , .... (12) Z> : D' :: N : N' (13) Or, in words: I. The ordinates at the chord stations are as the cubes of the station numbers. II. The central angles from S to the chord sta- tions are as the squares of the station numbers. III. The degrees of curves connecting at the chord stations are directly as the station numbers. (T In Fig. i, is the sine of the deflection angle^ O (JT at 6* between tangent SK and chord SL. Call this deflection angle A. The sine of one half the central angle a turned in the distance of circular arc, or n of parabola, FD ls zir But CG = 23 and FD = $q\ and taking A ** i W. sin A = 20 - - --, 2 n , n o<7 sin \a = -iff -. . 2 n 6 THE TRANSITION-CURVE FIELD-BOOK. Hence si*n A : 2 sin \a :: 4 : 12, or approximately A\ a:: i : 3, and In words : The deflection angle at S between tangent Sl and any chord station is one third of the corresponding central angle. This relation between the central and deflection angles here roughly deduced by approximations is, when applied to the Transition Curve differing slightly from the cubic parabola, almost exact up to 14 24' of central angle, the limit of the General Table. The greatest angular error is -^^ of one minute, or 9 seconds; this occurring in the deflection angle from S to station 12, which calculated by the yr formula ^ = tang A gives 4 47'i B o J V> instead of 4 48' as recorded in the table. This discrep- ancy affects the ordinate at station 12 of a Transi- tion Curve 500 feet long about 0.02; and for one of 10 stations 500 feet long the angular error re- duces to 2 seconds, affecting the end ordinate less than o.oi. The quantities in lines X and Y were originally obtained approximately from the formulas X = i \R versin a and Y= - -{- R sin a, after assuming c and calculating D by transposing (9) and a from THE TRANSITION-CURVE FIELD-BOOK. f (12) on the basis of a 6 minutes. Then equations - j^-/ x = tang i and = tang A determined the angles $ and A as now recorded in the General Table. Afterward the quantities of lines X and Y were recalculated by summing the products of each chord into the sine and cosine, respectively, of its angle of inclination to the tangent SK. The ordinates and ordinate distances X and Y thus obtained are for the chord stations of a Transition Curve of 100 foot chords; but for any other chord length the corresponding ordinates and ordinate distances .rand y are directly propor- tional. As - is the tangent of the deflection angle from S to the station corresponding to x, it follows that, when the same chord length is used, each station must necessarily fall in the same place whether laid off by ordinates or by deflection angles. The length of any long chord (/) is Y- Y' c /= - -rr- X -- , . . . . (15) COS I 100 in which Y and Y' are at the end stations of /, and / is the angle of its inclination to SK. EXAMPLE. Long chord from 5 to n when c 40. Taking F n , F 5 , and /for (5), (11) from table: /- I095.I3-499-9I x 40 __ 595-22 cos 6 42'" < ioo~ > If / begins at S (see Fig. i), formula (15) becomes Y c /= - ; X . cos A 100 THE TRANSITION-CURVE FIELD-BOOK. EXAMPLE. Long chord S to 10 when c = 25. Taking F 10 and J 10 from table, _ = 66 . ioo The length of tangent / from S to its intersection with tangent /' from L is t= (Y- ^cotang a) , . (16) 7 ioo in which XY and a refer to the station at L. EXAMPLE. L at station 10; c = 25. Taking F 10 , AT 10 , and a lo from table, / = 99^97 ~ 5^.053 X 5-671281 = 4 The length of tangent /' from L to its intersec- tion with tangent SK is ^ c i ^ = - x , . . . . (17) sin a ioo in which X and a refer to the station at Z. EXAMPLE. L at station 10; c 25. Taking X IQ and a 10 from table, t , _. 58-053 y 25 _ - x - In the triangle of sides /, /, and f (STL, Fig. i), the angles opposite these sides are respectively 180 a, zA, and A, and the correctness of the re- c\ lation A = may be tested by substituting for o THE TRANSITION-CURVE FIELD-BOOK. 9 zA arid A their values and , and recalculating o o t and /', with /and the angles as data, by the fol- lowing trigonometrical proportions: 2 sin (180 a) : sin a ::/:/; o sin (180 - a) : sin - :: /: /'. 3 Whence, taking a lQ from table, /sin 6 40' _ 249.66 X .1160929 t irr ; Q - zi^ I O O . O I , sin 10 .1736482 and _ /sin 3 20' _ 249.66 X .0581448 _ "ihiTo "" ~~~~ giving practically the same values as before. As A = , from proportion (10), 5 A N - A N '::N*:N'\ making N' = i; and with the initial deflection angle ^ = 2 minutes, Making JV = i, 2, 3, etc., the angles of line zf of the table result; and multiplying these by 3 gives the corresponding angles of line a. 10 THE TRANSITION-CURVE FIELD-BOOK. The remaining angles of inclination / of the chords may be obtained by the formula J^ J7' y _ y , = tang /, (19) in which X 9 and Y 9 are from the column giving the station at the end of the chord nearest S\ and X and Y from the column giving the other end station. For the chords from S to the other stations X 9 and Y 9 become o, and (19) reduces to 2\. A f x = tang A, (20) giving the angles of line A. As A = 2N' 2 in minutes, - A t \ = in degrees. . . (21) N* Substituting for a in (9), Dn 10 200 and / (22) Dn = 2GW* = Z of table; whence n = and * THE TRANSITION-CURVE FIELD-BOOK. II Substituting for n its value Nc, . Nc and c ~J^~- ( 2 3) As the Transition Curve o*f N stations passes through the middle of the adjusted gap q at station N g - its ordinate at that point equals; or x N = - and q = . 2 2C>N From (23), D - . Therefore qX D = 2XxX = *, . (24) 2 2 making c = 100, x = X , and ^Z> = 0.4 NX N . 2 When N '= 12, ^Z> = 12.563 X 4.8 60.302. As the values of q are proportional to those of x, or to TV^ 3 , and those of D to TV^, the values of q X -* , the degree of curve, in degrees and decimals. Line a gives the central angles turned in the Transition Curve between tangent SK and the sub-tangents at the several chord stations. Its value in degrees and decimals is N* cD G N Z a = - - ; 10 200 200 versin a 0.00002616^^. Line A gives the deflection angles between tan- gent SK and the chords from S to the several chord stations. Its value in minutes is A = 2N* = --; 10 X =-. The remaining- lines of the table give the angles of inclination /of the chords to the tangent SK. each line giving the inclination of the chords from the station beginning that line to the stations forward of it. X- X' tang / =-- -ylT'y- Each angle in line A and those following it shows the inclination of a chord to the tangent SK, l6 THE TRANSITION-CURVE FIELD-BOOK. O ooooooo oo oo n in vO O r^ co 00 M en n "^ " w ^> V O " ^00aSNO^-CNMOOO M 000000000000 O cj rtTtTflOOO t^OO OO t-l in "o ^ oT N O ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ M M < r> S JT 8' o oo d co co co rt en O co O Tf co -f- -t co O co ^OjO-noooooo vQ O c? " M ^ Tt- CO O CO 1 ""!;^ o o o o 050000000^ THE TRANSITION -CURVE FIELD-BOOK. IJ EQUIVALENTS. Z _ 2oN* _ 2OOOf n = cJM == Q = -^- ; -^-; n q c n cY o.2NY cX y == - - F r b = cotang 100 D 100 cX 0.2^^ rF (9 When r = loo, Z> = 0.2^, = loo^Y, q FX, = X,y=Y. When Z> = i, ^== 20^, n = Z, $ = Q, = v.2NX' 3 y 0.2NY. 20 THE TRANSITION-CURVE FIELD-BOOK. In this case TV= 5, and * = S = 2 -s = 2 30'. Observe that, in using the table, if TV and c are given, the other elements, x, y, n y D, and ^, are readily obtained. Thus, take from column TV the quantities X, Y, Q, and Z. Then from equivalents, x = Xx ,y = YX ,#=*, -=D,-Q=4. ioo ioo n D The angles a and A are also taken from column TV. The elements include D because it is the rate of curvature of the Transition Curve at the junc- tion Z, as well as the degree of the circular curve. If in addition to TV some other element than c as x, y, /i, D ', or q is given, from equivalents, ioo ioo n Z _ 2&Nq - x X = y ~Y := W = I?W = ~Q~"' and with N and c known, the other elements are obtained as above. If a value of the central angle a in one of the columns of the table is selected and any other element is known, the column containing the selected value of a fixes TV, and the remaining ele- ments are determined as in the last case. Methods of determining the elements with other data than those above mentioned are shown in the Problems. DEFINITIONS AND NOTES REFERRING TO PROBLEMS FOLLOWING. = Initial station of Transition Curve. . L = Point of junction of Transition and Circular Curves. THE TRANSITION-CURVE FIELD-BOOK. 21 / Angle of intersection of tangents to be con- nected by Transition and Circular Curves. 75* = Distance on tangent from S to intersection point. P Point on IS opposite the PC (or P.T.) of original Circular Curve. SP = Distance on AS* from S to P. SK Distance on tangent from S to point K, op- posite L. p = Unadjusted offset between given tangent and parallel tangent of circular curve to be connected with. q = Adjusted gap between tangent SK and par- allel tangent of connecting circular curve. 6 = Angle turned in arc from P.C.C. to P when circular curve is compounded. The radius corresponding to D is denoted by R; to Z>' by ./?'; to 7>" by .tf". When n, q, or other quantity is assumed approx- imately, this is expressed by n a , q a , etc. When such expressions as x L9 X N , K 10 , etc., are used L , N , I0 , etc., indicate stations to which the larger letters refer. The central angle a cannot exceed \I, and un- less 7 is very small it is better that it should not much exceed -J-7. When the degree of the connecting curve is less than 5, by limiting the value of q to about 2 feet per degree of curve the length of the Transition Curve may be limited to about 500 feet. In any , 60.302 case q must not exceed =- . All lengths are expressed in feet and decimals of a foot. 20 THE TRANSITION-CURVE FIELD-BOOK. In this case IV = $, and a = ^ = 2. 5 = 2 30'. 10 Observe that, in using the table, if N and c are given, the other elements, x, y, n, Z>, and q, are readily obtained. Thus, take from column N the quantities X, Y, Q, and Z. Then from equivalents, The angles a and A are also taken from column N. The elements include D because it is the rate of curvature of the Transition Curve at the junc- tion Z, as well as the degree of the circular curve. If in addition to N some other element than c as x y y, //, jO\ or q is given, from equivalents, 100 100 _ n Z _ 2oJVq = * = y~y := jv = WN^ ~Q~' and with A 7 and c known, the other elements are obtained as above. If a value of the central angle a in one of the columns of the table is selected and any other element is known, the column containing the selected value of a fixes N, and the remaining ele- ments are determined as in the last case. Methods of determining the elements with other data than those above mentioned are shown in the Problems. DEFINITIONS AND NOTES REFERRING TO PROBLEMS FOLLOWING. S = Initial station of Transition Curve. . L = Point of junction of Transition and Circular Curves. THE TRANSITION-CURVE FIELD-BOOK. 21 I ' Angle of intersection of tangents to be con- nected by Transition and Circular Curves. fS= Distance on tangent from S to intersection point. P Point on IS opposite the PC (or P.T.) of original Circular Curve. SP - Distance on / from S to P. S K = Distance on tangent from S to point K, op- posite L. p = Unadjusted offset between given tangent and parallel tangent of circular curve to be connected with. q = Adjusted gap between tangent SK and par- allel tangent of connecting circular curve. = Angle turned in arc from P.C.C. to P when circular curve is compounded. The radius corresponding to U is denoted by J?; to Z>' by R'\ to Z>" by R" . When /z, q, or other quantity is assumed approx- imately, this is expressed by n a , q a , etc. When such expressions as x Ly X N , K 10 , etc., are used >AT> I0 , etc., indicate stations to which the larger letters refer. The central angle a cannot exceed |/, and un- less /is very small it is better that it should not much exceed ^/. When the degree of the connecting curve is less than 5, by limiting the value of q to about 2 feet per degree of curve the length of the Transition Curve may be limited to about 500 feet. In any , 60.302 case q must not exceed =- . All lengths are expressed in feet and decimals of a foot. 22 THE TRANSITION-CURVE FIELD-BOOK. PROBLEM I. Transition Curve to be laid off by deflection angles. Required field-work and record of field-notes. Given initial point S, chord length c, and num- ber of chords N. EXAMPLE, c = 25, N 10. CASE i. Curve to be run from S to L. Set the instrument over S, and if all the stations can be seen from that point, take from line A of table the deflection angles 2', 8', 18', 32', 50', i 12'^ i 38', 2 08', 2 42', and 3 20', and with vernier at o on tangent SK deflect these angles successively for stations i up to 10; irsing the given chord length, 25 feet, between eacli two consecutive sta- tions. Move to station 10, and with vernier at 3 20' sight on S and turn to 10, line a, giving the com- mon tangent to Transition and Circular Curve, whence the latter is run as usual. In this case _ Z 2000 X/ v, - 25 x 10 The field-notes areas follows, reading from the bottom upwards. S is supposed to fall at R.R. station 210 -(- 24. The distance column shows the total distances from S to the several stations' measured by consecutive chords. L 212 + 74 (10) 250 3 20' io.oo ( 9 ) 22 5 2%7' (8) 200 2 08' (7) 175 i38' (6) 150 i 12' THE TRANSITION-CURVE FIELD-BOOK. (5) 125 So' (4) IOO 32' (3) 75 18' (') 5 8' (0 25 2' (o) o' 3. 20 5*210 + With instrument at (10) and vernier at 3 20' sighted on S, by turning successively to 3 42', 4 08', 4 38', etc., (column 10 of table,) the points set at i, 2, 3, 4, etc., will be tested. CASE 2. Instrument required to be set at inter- mediate stations between S and Z, as (3) and (7). Set over S and deflect from SK the angles 2', 8', and 18', line A, for stations (i), (2), and (3). Move to (3), and with vernier at 18' sight on S, and turn to 1 14', i 38', 2 06', and 2 38' (line (3), table) for stations (4), (5)^ (6), and (7). Move to (7), and with vernier at 2 38' sight on (3), and turn to 5 38', 6 26', and 7 18' (line (7), table) for stations (8), (9), and (10). Move to (10), and with vernier at 7 18' sight on (7), and turn to 10, line /r, giving common tangent at (10) or L as before. The field-notes are: L 212 -f- 74 (io)O 250 7 18" io.oo S 210 -j- 24 (9) 225 (8) 200 (7)0 175 (6) 150 (5) 125 (4) loo (3)0 75 (2) 50 (0 25 (o)O o 6 36' 5 38' 2 38' 7 18' 2 06' i 3 8' ii 4 ' 18' 2 3 8' 8' 2' o 18' 24 THE TRANSITION-CURVE FIELD-I3OOK. The vernier readings repeated as backsights are underscored. CASE 3. Curve run backward from Z to S. Same data as before, except that Z is given. Set over Z = (10), and turn into tangent to the 8 curve. With vernier at 10 and instrument on tangent at Z, turn to 9 02', 8 08', and 7 iS' (column (10), table), for stations (9), (8), and (7). Move to (7), and with vernier at 7 18' sight on (ro) or Z, and turn to 4 14', 3 38', 3 06', and 2 38' (column (7), table) for stations (6), (5), (4), and (3). Move to (3), and with vernier at 2 38' sight on (7), and turn to 38', 26', and 18' (column (3), table) for stations (2), (i), and S. Move to S, and with vernier at 18' sight on (3), and turn to o, giving tangent SK. The field-notes are: S 212 + 74 (o)O 250 ij^ o o' (0 225 26' w 200 38' (3)0 J 75 2 3 8' 1 8' (4) 150 3 06' (5) I2 5 3 38' (6) IOO 4 14' (7)0 75 7 18' 2 3 8' (8) 5 8 08' (9) 25 9 02' 24 (io)Q o o ^ 10 oo 7 18' Z 210 Whether the Transition Curve is run from S to Z or from Z to , the notes given are to be read from the bottom upward. Observe, generally, that when the Transition THE TRANSITION-CURVE FIELD-BOOK. 2$ Curve is to be run from ^S* to Z, the deflection an- gles are taken from the lines of the instrument stations in the table; and when from L to they are taken from the columns of the instrument sta- tions. Or, in other words, when the station to be sighted on is between the instrument station and L or at Z, the corresponding vernier angle is in the line of the instrument station; and when the sta- tion to be sighted on is between the instrument station and S or at S, the corresponding vernier angle is in the column of the instrument station. In all cases the vernier angle for turning into tangent is in line a and in column of instrument station. If the chord length is given and any two station points of the Transition Curve are in position on the ground, the other stations can be readily re- set. For instance, in a i2-chord Transition Curve with chord length known, and stations (4) and (9) in place, when the instrument is set over (4), with vernier at 4 26' (line (4) column (9) ), sight on (9), and for the stations from (5) to (12) turn to the corresponding angles from 2 02' to 6 56.' of line (4), and for stations (3) to (S) turn to the corre- sponding angles from i 14' to o 32' of column (4). If the instrument is set at (9), with vernier at 4 26' as before, sight on (4), and for the stations from (9) to (12) turn to the corresponding angles from 9 02' to 11 06' of line (9), and for stations (8) to tlirn to the corresponding angles from 7 14' to 2 42' of column (9). To turn into tangent at any point, with vernier reading the angle of inclination of any chord end- ing at the instrument station, sight on the station point at the other end of that chord, and turn the 26 THE TRANSITION-CURVE FIELD-BOOK. vernier to the angle of line a in the column of the instrument station. Thus, with instrument at (9), sight on any station, as (4), with vernier at 4 26', (or on (2) with vernier 3 26', or on (12) with ver- neir 11 06'), and turn vernier to 8 06' for tangent. When the instrument is at S y sight on any station with vernier at the corresponding angle in line //, and turn to o for tangent; when the instrument is at Z, sight on any station with vernier at the cor- responding angle in column L y and turn to angle a in column L for tangent. Thus with instrument at S, sight on (10) with vernier 3 20' (or (i i) with vernier 4 02'), and turn to o for tangent. With instrument at L = (10), sight on Jy with vernier 3 20' (or on (3) with ver- nier 4 38'), and turn to 10 for tangent. In running the Transition Curve, the vernier reading used in setting the station moved to is always the vernier reading to be used for back- sight on the station moved from. As all the angles in the table represent the in- clination of chords (or sub-tangents in line a) to the tangent SK, the field test of the correctness of any angle turned in accordance with the foregoing instructions is as follows: When the instrument with its proper vernier reading is sighted on a station and the vernier is then turned to o, the telescope will be parallel to tangent Sl, and the needle will give its bearing. This will in all cases give a practical answer to the question whether to set the vernier to the right or to the left of o, because when the instrument is sighted on a station with the vernier properly set, and the vernier then turned to o, the needle will give the bearing of tangent SK, whilst if set on THE TRANSITION-CURVE FIELD-BOOK. 27 the wrong side of o, the needle will show the error doubled. This test is of course applicable at any station whilst running in the Transition Curve, and by its use angular errors, if there are any, are readily found and corrected. PROBLEM IT. To lay off the Transition Curve by ordinates. Given initial point S on tangent SK, chord length c, and number of stations N. EXAMPLE, c = 25,7V^=:io. The ordinate lengths, JTX i for stations (i), (2), (3), etc., are ^-^ , 100 4 .4564 1.571 , - , etc., .015, .12, .39, .93, 1.82, 3.14,4.99, 4 4 7.44, 10.59, and 14.51. Similarly the ordinate dis- tances frofn *S measured on tangent SK are Y X ~ - 2 5> 5> 75> 99-99. 124.98, 149.94, 174.87, i99-75> 224.55, an d 249.24. Measure the ordinate distances from S, and set off the corresponding ordinate lengths at right an- gles to SK. The station points should be tested by consecutive chord measurements. The rule for laying off the Transition Curve by ordinates applies to all cases when S, c, and N are determined. The ordinate lengths and dis- tances vary with every change in the chord length, but are a Iways obtained as in the example just given.* * In the problems following, the solutions are given for locating by deflection angles, but after determining c, S, and N, the Transition Curves in all cases may be laid off by ordinates as in Problem II, 28 THE TRANSITION-CURVE FIELD-BOOK. PROBLEM III. The angle of intersection I of two tangents being given, to connect them with a circular curve of any given degree of curvature D by Transition Curves having a common tangent and the same rate of curvature D at their points of junction L and L' with the cir- cular curve. The example in each of the five cases following gives the elements of only one of the two Transi- tion Curves, but applies to both. In figure 2, IS-IP-PK+KS. But IP = OP tang i/ = (OB + BP) tang 7 tang |7; KS = y - Yx \ 100 and by substitution in the first equation, tang i/+ --5. . (27) In formula (27), which applies to all the cases of this problem (but not in the formulas of other problems following), -- may be substituted for THE TRANSITION-CURVE FIELD-BOOK. 29 30 THE TRANSITION-CURVE FIELD-BOOK. CASE i. Given the degree of circular curve Z>, the number of stations N, and the intersection an- gle/. Rule. Take from column N of table the quanti- ties Y, <2, Z, and C. Z Q Then from equivalents j^-^=.c y Nc = n, -~ =^. Find fS by (27), and lay off the Transition Curve as in Problem I or II. EXAMPLE. Z> = 8, N = 12, / 50. From column .A 7 ^ of table, F 12 = 1192.47, Q^ ~ 60.302, Z 12 = 2880, C 12 = 1424.9- Z 2880 - = c 30; 30 X 12 = n 360; 8 X 12 IS = (X + f) tang K+ - = ( 7I 6.8 + 7.5)0.4663! = 337-75 + 357-75 - J 78.i = 51 7-4- Measure 517.4 from / to S and S f on the two tangents. From 6* lay off the Transition Curve by angles (Problem I) or by ordinates (Problem II) with chord length 30. From L (sta. N^ the 8 curve is to be run in for the angle 50 2a 21 14' to L f of the second Transition Curve, which is to be run backward to S' on the second tangent : Or both Transition Curves may be run forward from S and 5', and the common tangents at L and THE TRANSITION-CURVE FIELD-BOOK. 31 Z' run to intersection before putting in the 21 14' of 8 curve. If the Transition Curves are located by ordi- nates, in order to get the common tangent at Z set the instrument over Z, and with the vernier at Z/ 12 sight on S and turn to # 12 for the tangent at Z: or if S cannot be seen from Z, set the vernier at / for any long chord ending at Z, sight on the sta- tion at the other end of the long chord and turn to a 12 . In the present case, for example, if S can be seen from Z, set the vernier at 4 48', sight on S and turn to 14 24' for the common tangent: or if (4) is the nearest station to S that can be seen, set the vernier at / for chord (4) (12), or at 6 56', sight on (4), and turn to 14 24' for tangent. To get common tangent at L' , substitute Z' and S' for Z and S in above description. Similar field-work is required in all of the cases of Problem III. A very convenient test of the arithmetical values of c and q as calculated, in all of the problems, is cNQ ~ A in minutes. In the present case 30 X 12 X 60.302 _ _ ~Jo, 7, and approximate length of Transition Curve n a . Rule. Take from the table the value of Z nearest to Z> X n a , and from the same column JV, Q, C, and Y. Then from equivalents -^ = ,- = r, -^ = q. For the rest proceed as in Case i. 32 THE TRANSITION-CURVE FIELD-BOOK. EXAMPLE. > = 8, n a 300, 7= 42, D X n a = 8 X 300 = 2400, and nearest Z= 2420 from column 7V n . Takeout (? 11 =4 2 -597> K n 1095.13, C=i2oi.o. 2420 302.5 -- = =302.5; - =f = (9 42. = 7~) JLS is = (1146.3 + 5-3) -38386 + ?H^~~^ : = 442.05 + 301.16 240.2 = 503.0. CASE 3. Given Z>, /, and a assumed as equal to one of its values in the table. If / is less than about 40, assume a between ^/and |7. Rule. Find the desired value of a in the table, and from the same column take out the quantities Z, Q, C, and F, and proceed as in Case 2. EXAMPLE. D 5, / 20 20', and a from col- umn JV 6 6 24'. Z s = 1280, Q B = 11.912, C 6 = 638.7, F 8 = 799.01; 1280 256 11.912 -_ = = 256; -|- = c = 32; -y- ==,? = 2.382, IS = (1146.3 + 2.4)0.17933 + 799 X 0.32 ~ = 206.0 + 255.7 - 127.7 = 334.0. CASE 4. Given Z>, 7, and approximate gap < Note that q a x Z^ must not exceed about 60.302. THE TRANSITION-CURVE FIELD-BOOK. 33 Rule. Take from table the value of Q nearest to q a X -Z? , and from same column Z, C, and Y. Pro- ceed as before. EXAMPLE. D~ 20, 7 =50, q a = 1.5, 1.5X20=30. Nearest Qirom column ^^ = 29.081; and Z 10 = C*,o = 994-9> F !o = 99 6 -97- 2Q.o8l 2000 100 q = 1.454; = = 100; - = <; = 10; 20 y ^ 20 10 IS= (287.9 + i-S) i-WIS + 997 X o.i - = 344.8 + 99.7 49.7 = 394.8. CASE 5. Given /, N, and chord length c. In this case D cannot be assumed. Rule. From column. TV" take Q, Z, C, and F, and from equivalents, Proceed as before. EXAMPLE. Given 7=40, C 30, N 9. From column 9 of table (^=.19.080, Z a = 1620, C 9 = 807.3, F a = 808.22. 30 X 9 ^ * = no; = ^ = 6*; =fe=i8; 7^ = (955.4 + 3.2) 0.36397 + 898.2 X 0.3 - 5^ 348.9 + 269.5 34 THE TRANSITION-CURVE FIELD-BOOK. PROBLEM IV. To locate the Transition Curve when both circular curve and tangent are fixed in position, and the offset p 60 is less than about -^. Given /, ), and/. In the figure, R the radius of the given curve, and BP = the given offset/. Also LK = the end ordinate x, and JB'P' = gap q. eo THE TRANSITION-CURVE FIELU-BOOK. 35 LK - x MB + BP R versin a and q Fx F(R versin +/). . . . (28) SP = SK-KP; SK=\ 100 Therefore Rule. Take from table the value of Q next lower than p X Z>, and from same column Z, jF, F, C, and a:. Find ^ by (28). Find 5^ by (29). EXAMPLE. / = 36, D = 6, / = 9. Then/Z> = 54, and next lower value of Q (from table) is (? n = 42.577; and Z n 2420, F^ = .25072, F^ 1095.13, C = I2OI.O, ^ = 12 06'. ? = 1 572(955-4 X .022217 + 9) = 7.578; 2420 430.72 = = 430.72; -- = ,= 39.16; sp _ 39- 16 X 1095. 13 _ I20I.O 100 6 428.85 200.17 = 228.68, 36 THE TRANSITION-CURVE FIELD-BOOK. At L the Transition Curve rate is 5 37', and the circular curve rate 6. NOTE. In cases where/ X D is nearly equal to the nearest value of Q, that and the corresponding table quantities should be taken even when Q is greater than / X >. The resulting rate of curvature of the Transition Curve at L will be some- what greater than that of the connecting circular curve, and will call for a greater elevation of the outer rail; but this differ- ence is too slight to be of any practical importance. On the other hand, the actual difference between the rates of curvature of the two curves at L will be less, in such cases, than when the next lower value of Q and the corresponding table quanti- ties are taken. SECOND EXAMPLE under Problem IV (see pre- ceding note). 7= 36, D = 6, / = 9, fJD = 54 as before. The nearest value of Q is Q 12 = 60.302; and F^ = .25105, Z 12 = 2880, <7 ia = 1424-9) YH = ii9 2 -47, a l2 = 14 24'; and from (28), q F(R versin a +/) =- 2 5 I0 5(955-4 X .03142 + 9) = 9-79S 6 ; = ZX = 6 .156-6 en*'; - 9.7956 2880 476.83 6^6-==* = 467-83;^ =; = 38.99; on _ 38.99 X 1192.47 I424-9 o./ ioo 6 = 464-94 237.48 227.46. At Z the Transition Curve rate is 6 09^' and the circular curve rate 6. THE TRANSITION-CURVE FIELD-BOOK. 37 PROBLEM V. To locate the Transition Curve when both circular curve and tangent are fixed in position and the offset p is 60 greater than about -$. Given /, Z> and/. In Fig. 4, B'P 9 =/; BP = /; 7? = radius of given curve Z>; and 0= central angle from P.C.C. to point of tangent parallel to SK. M'B' - MB = BP - B'P' = p - /. But M'B' = R' versin 8 and MB R versin ft Therefore (^' 7?) versin =/ / and . _ ,f versin = J>T^T^> ..... (3) A: = LK = R versin a and ^ Fx = (J? versin +p')F. . . (31) SP= SX- KP' + P'K' - K'P e ' (32) . Assume Z>' about o.gZ> and/' not greater 60 than about -jyt>- Find # from (30). Take from column giving Q next lower than p' X D' Q the quantities ' /5 ~ ' ~N~ THE TRANSITION-CURVE FIELD-BOOK. Find SP from (32)0 THE TRANSITION-CURVE FIELD-BOOK. 39 EXAMPLE. /= 60, Z> = 8,/ = 16. Assume Z>' = 7; /' = 9. From (30) - &- =r = 0.06846 = versin 819 02 716.24 102.24 versin 21 19'. 7 X 9 = 6 3- From TV^ , 19 = 60.302, Z ia P? 2880, <7 I9 = 1424-9. ^, a 14 24'. (819.02 X 0.03142 -4- 9) 0.25105 = q = 8.72. 8.72 2*880 416.5 _ = = 4i6. S ; = * = 34-71. SP = * ^^2 ^ +(8i 9 .o2- 7 i6.2 4 ).3635 = 413-92 203.56 + 37.16 = 247.5. Measure 247.5 ^ eet from P and put in S, Run in Transition Curve from 6" with 12 chords each 34.71 long. From L (station 12) run 7 curve for 21 19' 14 24' = 6 55' to P.C.C.y there connecting with the 8 curve. The point S may be determined on the ground without calculating SP, by running in the 6 55' of 7 curve from P. C. C. to Z; turning off 90 a outwardly from the tangent at Z and intersecting 40 THE TRANSITION-CURVE FIELD-BOOK. cY SK at K\ and thence measuring - - = 413.92 to S: TOO or the Transition Curve may be run in backward from L without first determining the position of S. When convenient, however, it is better to fix both vS*and L before running in SL. In this example the rate of curvature of the Transition Curve is 6 55' at Z, its junction with the 7 curve. The difference in the rates of the Transition and Circular Curves at L need not absolutely be less than at other P C.C.'s of the line; but in order properly to adjust the superelevation of the outer rail, the differences in the rates of curvature at all points of compound curve should be as small as circumstances will permit. Problem V is limited by the conditions that must be less than |7 and greater than a. If 6 as calculated is greater than -J-/, increase either the assumed value of/' or of the radius of D'\ and if the calculated 6 is Jess than a, diminish one or the other of these assumed quantities. It is always practicable, and in some cases pref- erable, to find the external distance HI from the middle of the curve D to the intersection of tan- gents (HI (R -{-p) exsec / + /), and substitute Problem VII for Problem V. PROBLEM VI. Line to be held over the middle portion of a circular curve of large central angle connecting two tangents, to shift curve inwards at the ends and put in Transi- tion Curves. Given JD and point of curve (or tangent) P. In the figure, BP' = q. THE TRANSITION-CURVE FIELD-BOOK. 01 q = MP - M'B = R versin - R' versin 0, and versin (9 = ^4 .... (33) SP = SK - KP' + P'K' - K'P, SK = , ^^' F= 7^6, /"A7 = ^' sin 6>, and I CO JJ K'P = R sin ft 42 THE TRANSITION-CURVE FIELD-BOOK. Therefore SP = C -^ Q -~-(R-R')sm6. . (34) Rule. Assume the curve D' at the ends about i.i Z>, and q a = 4 feet or less. Take from table Q nearest to q a X Z)', and from same column Z, C, and Y. Find ff by (33), and SP by (34). EXAMPLE. Z> = 8, ^ 3. Assume D' 8.8; 3 X 8.8 26.4. From column 1V 10 , Q = 29,081; and Z 10 = 2000, ,o = 994-9. ^,o = 996.97. 20.081 -- From (33) ^ ^ : - = versin 6 = 0.05077 = versin 18 20'; / * D * / 2000 227.3 = = 227.3; - = < = 22.73. From (34) a> = =Ji^H-^- > .X 0.3-454 = 226.6 H3.I 20.5 = 93-0. From StoP.C.C. via P = 93 + - 8 ^ ^ I0 - =322.2. it " L= 227.3 + 8.8 322.0. THE TRANSITION-CURVE FIELD-BOOK. 43 Run Transition Curve of 10 chords of 22.73 eacJl from S to Z; then the 8 48' curve for 6 a r=i8 20' 10 = 8 20' to P.C.C. PROBLEM VII. Given the intersection angle I of two tangents connected by the circular curve D, to change the degree of curve without moving the position of its middle point, and insert Transition Curves between tangents and circular curve. 44 TME TRANSITION-CURVE FIELD-I3OOK. From the figure, M' B MP BP'. But M'B = R' versin |7, J/7> R versin 7, and .#/>' =/; hence R' versin \I = R versin \I p. . (35) > Assume/", and R' = R- P. . (36) versin \I Find the value of Z>' from radius R' a to the nearest even minute, and take the exact radius R' corresponding to Z>'. Then from (35), / = (R - 7?')versin |7. . . . (37) x = LK = AB+ BP' = R' versin a +/, and q = Fx F(R' versin a+p). . (38) IS = IP 9 - P'K + KS = (*'+/) tang i/-^o + ^- . . ' . (39) Rule. Find R' of D' from (36), and/ from (37). Take Q next lower than /Z>', and from the same column Z, F, Y, C, and of. Find ^ from (38). Q Z 7=^ #^y"* Find 7-5 from (39). THE TRANSITION-CURVE FIELD-BOOK. 45 EXAMPLE. /= 40, D = 6. Assume p a 4, then R' a 955.4 -- - .060308 955-4 "~ 66.3 889.1. To the nearest even min- ute Z>' = 6 26' and R' = 891.1. P = (955-4 - 8 9 r.i).o6o3o8 = 3.878; 3.878 X 6.43 = 24.9. Next lower value of Q is Q 9 19.08; and F 9 = .25028, a 9 = 8 06', Z 9 = 1620, C. = 807.3, Y 9 = 898.22. 0.25028 (891.1 X 0.009976 + 3.878) = q 3.1955; 1620 271.36 = n = 271.36; - * = ^ 30.11;. 5-97 9 ro to \ . 8 7-3 , 30-I5 is = (8 9 i.i + 3-9)0.36397 - -^ + ' = 325.7 - 125.5 + 270.8 = 471.0. At L the Transition Curve with rate of curva- ture D" = 5 59' connects with circular curve D' = 6 26'. EXAMPLE. Same as last, except that the value of Q taken is nearest top X D'. ^'891.1 and / = 3.878 are found as before. The nearest Q to 24.9 is Q 10 29.081; and F^ = 0.25047, a 10 = 10, Z 19 = 2000, C 10 = 994-9, ^,o = 99 6 -97- 46 THE TRANSITION-CURVE FIELD-BOOK. .25047 (891.1 X 0.015192 + 3-878) q ^ = D"" = 6 .66 7-6 40'; 4.362 OO 3OO = " = ;= ' = = ,8,,, + 3.9) 0.363,7 - = 3 2 5-7 !54-6 + 299.1 = 470.2, At L the Transition Curve with rate of curva- ture Z>" = 6 40' connects with circular curve D' = 6 26'. PROBLEM VIII. To pass a circular curve through a point H (Fig. 6) given by its radial distance HI from I, //fo circular curve to be connected by Transition Curves with tan- gents intersecting at I. Rule. Find R, the radius of the circular curve touching both tangents, by the formula R TT T , and proceed as in Problem VII. ex sec !/ EXAMPLE. / = 40, HI = 61.32, R = ~^r^g 61.32 X. 93960 , ^ , ~ u J yj ? y 955.4, and D 6 . The remain- .06031 ' = 7; then (Form a 3 P= (857-9-819.0) .133975 = 5-212. 5212 X 7 36.7. Next lower Q is (? JO = 29.081; and F l(} = 0.25047, ar lo = 10, Z 10 =2000, 6^ = 994.9, ^,0 = 99 6 -97- .25047 (819.0 X 0.01592 + 5.212) = q 4-57 1 ; ^^ 1 = jy> = 6 o 62 = 4.571 2OOO ^14.4 ' /^ = (8 I9 .o + 5 . 2 ) .57735 - ? + 52 = 475.8 - I42.I + 313.5 = 647.2. At L the Transition Curve rate is 6 22', the circular curve 7. PROBLEM IX. With track laid over a circular curve of small central angle connecting two tangents, to shift the middle of curve outward and ends of curve inward and connect with same tangents by Transition Curves. In order to hold the road-bed as nearly as may be, if the shift at the middle is h outward, that at the ends should be about 2/1 inward. 48 THE TRANSITION-CURVE FIELD-BOOK. In the figure (7), OP - R of Z>, and O' B' of D'\ From the figure, M* B' MB AB. But M'B'^R' versin -|7, MB=(R+K) versin $/,. and ^4-Z? =/ -(- /^; hence 7?' versin \I = (R + ti) versin |7 (/ + X). . (40) Assume/ a =: 2/2; then versin \I THE TRANSITION-CURVE FIELD-BOOK. 49 Find the value of D' from radius R' a to the nearest even minute, and take the exact radius R' corresponding to Z>'. Then, from (40), /=(.# + // R') versin \I h. . . (42) x = LK GB' + AP = R' versin a +/; q Fx = F(R' versin a +/). . . . (43) SP = SK - KP' + P'K' - K'P\ SK= ', KP' = R' sin a = ^5 100 // /"A"' = ^' sin |7; K'P = (^ + //) sin |7. Therefore ^ - f^ o - ^o - (^ + h - R'} sin |7. (44) J?///', and from the same column X, F, Y y C, and a. Find q from (43). <2 _ ry,* Z _ n __ -- > ^775- , Tvr- Find 5P from (44). EXAMPLE. D = 5, 7 30, h -- 2. Radius of D = 1146.3, and 1146.3 4- 2 = ^? /a = 072. 2. .034074 Take R'= 971.5, corresponding to Z>' = 5 54'= 5. 9. Then (1146.3 4-2 971.5) .034074 2 =/ = 50 THE TRANSITION-CURVE FIELD-BOOK. 4.024; pD' 4.024 X 5.9 = 23.24. From column ^9> (? 9 = 19.08; and Z 9 = 1620, C 9 = 807.3, F 9 = 898.22, JF 9 = .25028. .25028 (971.5 X .00998 + 4.024) = q 3.434; -^ - = ^" = 5-55 6 = 5 33?'; _ . _ . J 'JJ J JOJ 7 1620 201.6 __ = , = 29 , 6; ___ = , = ==. 291.0 136.8 45-8 = 108.4. From ^ to middle of curve > = SP + ^4,-= 108.4 H ~ :=: 108.4 H~ 3 == 408.4. From 6 1 to middle of curved' = n + ^ -=^=291.6+ - = 291.6 + 116.9 408.5. The mid- o * v die of D' being outside of D makes the new line o.i the longer. In this example the rate of curvature of the Transition Curve at L (sta. 9) is 5 33^', connecting there with a 5 54' circular curve. THE TRANSITION-CURVE FIELD-BOOK. 51 PROBLEM X. With track laid over two circular curves turning in opposite directions and a tangent connecting them, to change the direction of the tangent and insert Transi- tion Curves between the circular curves and the new tangent. Given the degrees of the two curves D and D' the positions of the P. T. of Z> and P.C. of Z>', and the length of tangent from P. T. to P.C. Rule. Assume approximately the gaps q a and q' a between the circular curves and the new position of tangent. Take from table Q and Q' nearest to q a D^ and q' a D fc ', and from same columns Z, C, and F, andZ', C', and Y' . Z Z' If tangent length T is less than about -7^54^75, 2.D 2JJ adopt a lesser value of q a or q' a or of both, and proceed as before. Then for Transition Curve connecting with Z>, and for Transition Curve connecting with Z>', To find the positions of Z, S, S', Z', and the dis- tance from ^ to *$*', use the formulas following Fig. 8. In Fig. 8, R and R r are the radii of the given circular curves D and JD' connected by tangent T from P. T of D to f,C, of D', 52 THE TRANSITION-CURVE FIELD-BOOK. From centre O with radius R -f- ^ and from centre 0' with radius R' -\- q' ^ draw arcs as in figure and connect them by tangent PP'. Prolong radius R' at P.C. and intersect it at E by OE par- allel to tangent 7 T ; and prolong radius R 9 + ^' at /^ and intersect it at G by OG parallel to new tangent PP* '. THE TRANSITION-CURVE FIELD-BOOK. 53 __ sin 00' E sin OO'E OO' X sin OO'G OG = PP'. Designate the angle OO'E OO'G by ft. Then a ft = angle from P.T. back to Z, and a' /3 = angle from P.C. forward to Z'. To find K and K' on tangent PP f prolonged, set the instrument over Z, turn 90 a outward from tangent at Z, and measure on this course LK . 100 Similarly at Z' turn 90 a' and measure L' K' cX' IOO* cY From K measure KS = on tangent KK ', and 100 put in point S\ similarly from K f measure K' S' = c'Y' and put in point S'. IOO SS' = PP 9 + XP + P'K' - SK - S'K' r ~no T" ~7v"o ~~ ~- (45) From Z to L' via old line = ^" + IPO (a 9 - /?) jyo From Z to Z' via new line = n + 6*5' -f- ;/. 54 THE TRANSITION-CURVE FIELD-BOOK. EXAMPLE. Given a 10 curve to the right and a. 5 curve to the left with a tangent 500 feet long between them, to change the tangent so that its direction will pass about 6 feet outside of the 10 curve and about 4 feet outside of the 5 curve, both circular curves to be connected by Transition Curves with the new tangent. For the 10 curve J? = 573.7, q a = 6, q a D = 6 x 10 = 60. From column -A^ 12 , Q IZ = 60.302; and C 12 = 1424.9, F 19 = 1192.5. 60.302 2880 - - q 6.0302: - == n = 288.0: 10 10 288.0 - = c = 24. For the 5 curve R' = 1146.3, q' a 4, 'q' Z>' = 4 x 5 = 20. From column N Q , Q 9 19.08; and = 807.3, y. = 898.2. . 08 1620 = n' = 324.0; D 9 -g, = tang OO'E = = tang 16 12^'; T sin Uu L .279131 = COS 15 THE TRANSITION-CURVE FIELD-BOOK. 55 OO'X sin OO'G=- 1791.2 X 0.25952 = 464.8 = PP'; OO'E - OO'G = 16 12 J' - 15 o2-i' = 1 10' = ft. a. ft = 14 24' i 10' = 13 14' = angle from P. T. back to L. a' - ft = 8 06' - i 10' = 6 56' = angle for- ward from P.C. to L 9 . Fix the point L by going back from P. T. for 13 14' of arc, or 132.3 feet. Set instrument over L, turn 90 a 90 14 24' = 75 36' outward from cX tangent at L. and on this course measure - - = 100 24 X 100.083 = 24.02 to point K. 100 Similarly for L 9 turn 81 54' outward from tan- r'Yf gent and measure - - = 15.25 to K 9 . cY From K measure = 286.2 on KK* to S\ and 100 c'Y' from K' measure - - = 323.4 on K 9 K to S'. TOO C r" rV r'V ocr __ ppr _i _T_ i _T_ " 7~) "^ 7~)' ~" = 464.8 + 142.5 -f 161.5 3 2 3-4 286.2 = 159.2. Except as a test of the correctness of the rest of the work it is not necessary to calculate or meas- ure SS'. From L to L 9 via old line = IO Xl 3- 2 33 + 5QQ + 10 .00x6.933 100 56 THE TRANSITION-CURVE FIELD-BOOK. From L to L' via new line = 288.0 -f- 159.2 -f 324.0 = 771.2. The line from L to L' may be run in contin- uously by angles, or the two Transition Curves may be laid off from S to L and from S f to Z' either by angles or ordinates. PROBLEM XL When a Transition Curve is run in and does not prop- erly connect at L or at S, as the case may be, to make the proper adjustment and location. (See Fig. i.) Rule. Set the instrument over a trial L on the circular curve, turn into tangent, and intersect the given tangent at T. Move to T and measure the intersection angle LTK. If L TK = a, the trial point set over is correct for L, and the distance LT multiplied by sin a and divided by the correspond- ing value of X from the table gives the chord length c with which the Transition Curve from L to S is to be rerun. If the intersection angle LTK as measured is not equal to a, make it equal by moving the trial L backward or forward on the circular curve through a central angle equal to the difference between the angles LTK and a. When L is cor- rectly located, set over it and turn into tangent. From this tangent turn off outwardly the angle 90 a, and on this course (perpendicular to SK} intersect the given tangent in K and measure LK. Then : TOO X~ cY = f, ana THE TRANSITION-CURVE FIELD-BOOK. 57 Be careful to take table quantities from the col- umn giving the proper value of a. Super-elevation of outer rail on curves. The difference of elevation e between the inner and outer rails on Transition Curves should in- crease regularly from at -S 1 to its maximum at L. This difference of elevation for a 4' 8y gauge (and proportionally for other gauge widths) is closely approximated for all velocities and rates of -curvature by taking ^ = .000056 V*D, in which V is the velocity in miles per hour and D Q the degree of curve. For example: on a Transition Curve of 9 stations connecting with a 6 curve, the superelevation e of the outer rail at L and around the 6 curve, adjusted for 40 miles an hour, will be e = .000056 X 4 9 X 6 = 0.5376; and for stations i, 2, 3, 4, etc., of the Transition Curve the differences in elevation are 0.5376 X -J-, X f X |, X |, etc. In surfacing track the inner rail should be lowered , and the outer rail raised ; and this is 2 2 conveniently done by setting stakes at a distance equal to the width of the gauge plus the width of the rail-head on each side of the centre stake with the top of the latter at its proper elevation, and the tops of the inner and outer stakes respectively e below and e- above it TABLES. RADII OF DEGREES OF CURVE. / 1 2 40 50 / Hadius. Radius. Radius. Radius. Radius. Radius. (I Infinite" 5729 65 2864 . 93 1910 08 1432.69 1146 28 1 343; 75 5635.72 2841.26 1899.53 1426.74 1142.47 1 2 171887 5544 83 2817.97 1889.09 1420.85 1138.69 2 y 1 1 4592 5456.82 2795.06 1878.77 1415.01 1134.94 3 4 85913.7 5371.56 2772.53 1868.56 1409.21 1131.21 4 5 68754.9 5288 . 92 2750.35 1858.47 1403.46 1127.50 5 6 57295.8 5208.79 2728.52 1848.48 1397.76 1123.82 6 "j 1 49110.7 5131.05 2707 04 1838.59 1392.10 1120.16 7 8 42971.8 5055.59 2685.89 1828 82 1386.49 1116.52 8 9 3S197.2 49.S2 33 2665.08 1819.14 1380.92 1112.91 9 10 34377.5 4911.15 2644.58 1809.57 1375.40 1109.33 10 11 31-252.3 4S41.98 2624.39 1800.10 1369.92 1105.76 11 12 ! 28617.8 4774 . 74 2604 51 1790.73 1364.49 1102.22 12 13 26444.2 4709.33 2584 . 93 1781.45 1359.10 1098.70 13 14 24555.4 4645.69 2565 . 65 1772.27 1353.75 1095.20 14 15 22918.3 4583.75 2546.64 1763.18 1348.45 1091.73 15 1> 21485.9 4523.44 2527.92 1754.19 1343.18 1068.28 16 17 20222.1 4464.70 2509.47 1745.29 1337 96 1084.85 17 18 19098 6 4407.46 2491.29 1736.48 1332.77 1081.44 18 in 18093 4 4351 67 2473.37 1727.75 1327.63 1078 05 19 20 17188.8 4297.28 2455 . 70 1719.12 1322.53 1074.68 20 21 16370.2 4244.23 2438.29 1710.56 1317.46 1071.34 21 22 15626.1 4192.47 2421.12 1702 10 1312.43 1068.01 22 2-5 14946 7 4141 96 2104.19 1693.72 1307.45 1064.71 23 21 14323.6 4092 66 2387 50 1685.42 1302.50 1061.43 24 25 13751.0 4041.51 2371 04 1677 20 1297.58 1058.16 25 26 J3*.l 3997.4'.) 2354 80 1669.06 1292 71 1054.92 26 27 12732.4 3951.54 2338.78 1661.00 1287.87 1051.70 27 28 12277.7 3906. 64 232-2.98 1653.01 1283.07 1048.48 28 29 11854.3 3S62.71 2307.39 KJ45.11 1278.30 1045.31 29 30 11459.2' 3819.83 2292.01 1637.28 1273.57 1012.14 30 31 11089 6 3777.85 2276.84 1629.52 1268.87 1039.00 31 32 10743.0 3736 7i) 2261.86 1621.84 1264.21 1035.87 32 33 10417.5 3696.61 2247.08 1614.22 1259.58 1032.76 33 31 10111.1 3657.29 2232.49 1606.68 1254.98 1029.67 34 35 9822.18 36 16. SO 221 S. 09 1599.21 1250.42 1026.60 35 36 9549. 3 t 3581 10 2203.87 1591.81 1245.89 1023.55 36 37 9291.29 3544.19 2189.84 1584.48 1241.40 1020.51 37 8H 9046.75 3508 0-3 2175.98 1577.21 1236.94 1017.49 38 39 8814.78 3472 59 2162.30 1570.01 1232.51 1014.50 39 40 8594. 4 i 3 437. 7 2148.79 1562.88 1228.11 1011.51 40 41 8384.80 3403.83 2135.44 1555.81 1223.74 1008.55 41 42 8185.16 3370.46 2122.26 1548.80 1219.40 1005.60 42 43 7994.81 3387.74 2109 24 1541.86 1215.09 1002.67 43 41 7813.11 3305. 5 2096.39 1534.98 , 1210.82 999.76 44 45 7639.49 3274.17 2083.68 1528.16 1206.57 996.87 45 46 7473.42 3243.29 2071.13 1521.40 1202.36 993.99 46 47 7*14.41 3-212.98 2058.73 1514.70 1198.17 991.13 47 4S 7162.03 3183.23 2046.48 1508.06 1194.01 988.28 48 4'.) 7015.87 3151.03 2034.37 1501.48 1189.88 985.45 49 50 6875.55 3125. 3J 2022.41 1494.95 1185.78 982.64 50 M 6740 74 3097.20 2010.59 1488.48 1181.71 979.84 51 6611.12 3069.55 1998.90 1482.07 H77.C6 977.06 52 M 6486. 3S 3042.39 1987.35 1475.71 1173.65 974.29 53 54 6366.26 3015.71 1975.93 1469.41 1169.66 971.54 54 55 6250.51 2989.48 1964.64 1463.16 1165.70 968.81 55 50 6138.90 2963.71 1953.48 1456.96 1161.76 966.09 56 57 6031 . 20 2938.39 1942.44 1450.81 1157 85 963.39 57 58 5927.22 2913.49 1931.53 1444.72 1153.97 960.70 58 59 5826.76 28H9.01 1920.75 1438.68 1150.11 958.02 59 60 5729.65 28G1.93 1910.08 1432 (50 1146.28 955.37 60 RADII OF DEGREES OF CURVE. / 6 7 ! 8 9 1O 11 t Radius. Radius. ! Radius. Radius. Radius. Radius. ~F 955.37 819.02 716.78 637.28 573.69 521 . 67 ~TT i 952.72 817.08 715.29 636.10 572.73 520.89 i 2 950.09 815.14 713.81 634.93 571.78 520.10 2 3 947.48 813.22 712.33 633.^6 570.84 519.32 3 4 944.88 811.30 710.87 632. <>0 509.90 518.54 4 5 942.29 809.40 709 40 631.44 568.96 517.76 5 6 939.72 807.50 707.95 630.29 568.02 516.99 6 7 937.16 805.61 706.49 629.14 567.09 516.21 7 8 934.62 803.73 705.05 0*7.99 566.16 515.44 8 9 932.09 801.86 703.61 626.85 565.23 514.68 9 10 929.57 800.00 702.18 625.71 564.30 513.91 10 11 927.07 798.14 700.75 624.58 563.38 513.15 11 12 924.58 796.30 699.33 623.45 562.47 512.38 1* 13 922.10 794.46 697.91 622.33 561.55 511.63 13 14 919.64 792.63 696 50 621.20 560.64 510.87 14 15 917.19 790.81 695.09 620.09 559.73 510.12 15 16 914.75 789.00 693.70 618.97 558.82 509.36 16 17 912.33 787.96 692.30 617.87 557.92 508.61 17 18 909. 02 785 40 6',)0.91 616.76 557.02 507.87 18 19 907.52 783.62 689.53 615.66 556.12 507.12 19 20 905.13 781.84 688.16 614.56 555.23 506.38 20 21 902.76 780.07 686.78 613.47 554.34 505.63 21 2:2 900.40 778.31 685.42 612.38 553.45 504.90 22 23 898.05 776 55 684.06 611.30 552.56 504.16 23 24 895.71 774.81 682.70 610.21 551.68 503.42 24 25 893.39 773.07 681.35 609.14 550.80 502.69 25 26 891.08 771.34 680.01 608.06 549 92 501 . 96 26 27 888.78 769.61 678.67 606.99 549.05 501.23 27 28 886.49 767.90 677.34 605.93 548.17 500.51 28 29 884.21 766.19 676.01 604.86 547.31 499.78 29 30 881.95 764.49 674.69 603.81 546.44 499.06 30 31 879.69 762.80 673.37 602.75 545.58 498.34 31 32 877.45 761.11 672.06 601.70 544.71 497.62 32 33 875.22 759.43 670.75 600.65 543.86 496.91 33 34 873.00 757.76 669.45 599.61 543.00 496.19 34 35 870.79 756.10 668.15 598.57 542.15 495.48 35 36 868 60 754.45 666.86 597.53 541.30 494.77 36 37 866.41 752.80 665.57 596.50 540.45 494.07 37 38 864.24 751.16 664.29 595.47 539.61 493.36 38 39 862.07 749 52 663.01 594.44 538.76 492.66 39 40 859.92 747.89 661.74 593 42 537.92 491.96 40 41 857.78 746.27 660.47 592.40 537.09 491.26 41 42 855.65 744.66 659.21 591.38 536.25 490.56 42 43 853.53 743.06 657.95 590.37 535.42 489 86 43 44 851.42 741.4*6 656.69 589.36 534.59 489.17 44 45 849.32 739.86 655.45 588.36 533.77 488.48 45 46 847.23 738.28 654.20 587.36 532.94 487.79 46 47 845.15 736.70 652.96 586.36 532.12 487.10 47 48 843.08 735.13 651.73 585.36 531.30 486.42 48 49 841.02 733.56 650.50 584.37 530.49 485.73 49 50 838.97 732.01 649.27 583.39 529.67 485.05 50 51 836.93 730.45 648.05 582.40 528 86 484.37 51 52 834.90 728.91 646.84 581.42 528.05 483.69 52 53 832.89 727.37 645.63 580.44 527.25 483.02 53 54 830 88 725.84 644.42 579.47 526.44 482.34 54 55 828.88 724.31 643.22 578.49 525.64 481.67 55 56 826.89 722.79 642.02 577 53 524.84 481.00 56 57 824 91 721.28 640.83 576.56 524.05 480.33 57 58 822.93 719.77 639.64 575.60 523.25 479.67 58 59 820.97 718.27 638.46 574.64 522.46 479.00 59 60 819.02 - 716.78 637.28 | 573.69 521.67 478.34 60 RADII OF DEGREES OF CURVE. r 12 13 14 15 10 17 f Radius. Radius. Radius. Radius. Radius. Radius. ~0~ 478.34 441.68 410 28 383.06 359.27 338.27 ~cT 1 477.68 4*1.12 409 79 382.64 358.89 337.94 i 2 477.02 443 56 409.31 382.22 3.'8.52 337.62 2 3 476.36 410.00 408.82 381.80 358.15 337.29 3 4 475.71 430.44 408 -34 381.38 357.78 336.96 4 5 475.05 438.88 407 . 86 380.96 357.42 336.04 5 6 471.40 433.33 407.38 380.54 357.05 336 31 6 7 473.75 437.77 406.90 380 13 356.68 335 . 1)9 7 8 473.10 43?. 22 406.42 379 71 356.31 335.66 8 9 472.46 436 67 405.95 379.29 355.05 335.34 9 10 471.81 436.12 405.47 378.88 355.59 335.01 10 11 471.17 435.57 405.00 378.47 355 .23 334.69 11 14 470.53 435.02 404.53 378.05 354 >6 334.37 12 13 460 89 434.47 404.05 377.64 854.50 334.05 13 14 469.25 433.93 403. 53 377.23 354.13 333.73 14 15 468.61 433.39 403.11 3 IT,. 82 353.77 333.41 15 16 467.98 432.84 402.65 3J6.41 353.41 333.09 16 17 467.35 432.30 402.18 376.(0 353.05 332.77 17 13 466.72 431.76 401.71 375.60 352.70 332.45 18 19 466.09 431.23 401.25 275.19 352.34 332.13 19 20 465.46 430.69 400.78 37'4.79 351.98 331.82 20 21 461. as 430.15 400.32 374.38 351.62 331.50 21 22 464.21 429.62 399.86 373.98 S51.27 331.18 22 23 463.59 429.09 399.40 373.57 350.91 330.87 23 24 462.97 428.56 398.94 373.17 350.56 330.56 24 25 462 35 428 03 398.48 372.77 350.21 330.24 25 26 461.73 427.50 39802 372.37 319.85 329.93 26 27 461.11 426.97 397.56 371.97 349.50 329.62 27 28 460.50 426.45 397.11 371.57 349.15 329.30 28 29 459 89 425.92 396.65 371.18 348.80 328.99 29 30 459.28 425.40 396.20 370.78 348.45 328.68 30 31 458.67 424.87 395.75 370.38 348.10 328.37 31 32 458 06 424.35 395.30 369.99 347.75 328.06 32 33 457.45 423.83 394.85 369 60 347.40 327.75 33 34 456.85 423.32 394.40 369.20 347.06 327.44 34 35 456.25 422.80 393.95 368.81 346.71 327.14 35 36 455.65 422.28 393.50 368.42 346.37 326.83 36 37 455.05 421.77 393.05 368.03 346.02 326.52 37 38 454.45 421.26 392.61 367.64 345.68 326.22 38 39 453.85 420.74 392.16 367.25 345.33 325.91 39 40 453.26 420.23 391.72 366.86 344.99 325.60 40 41 452.66 419.72 391.28 366.47 344.65 325.30 41 42 452.07 419.22 390.84 366.09 344.31 325.00 42 43 451.48 418.71 390.40 365.69 343.97 324.70 43 44 450.89 418.20 389.96 365.31 343.63 324 39 44 45 450.31 417.70 389.52 364.93 343.29 324.09 45 46 449 72 417.20 389.08 364.55 342.95 323.79 46 47 449.14 416.69 388.65 364.16 342.61 323.49 47 48 448.56 416.19 388.21 363.78 342.27 323.18 48 49 447.97 415.69 387.78 363.40 341.93 322.89 49 50 447.39 415.19 387.34 363.02 341.60 322.59 50 51 446.82 414.70 386.91 362.64 341.26 322.29 51 52 446.24 414.20 386.48 362.26 340.93 321.99 52 53 445.67 413.71 386.05 361.89 340.59 321.69 53 54 445.09 413.21 385.62 361.51 340.26 321.39 54 55 444.52 412.72 385.19 361.13 339.93 321.10 55 56 443 95 412.23 384.77 360.76 339.59 320.80 56 57 443.38 411.74 384.34 360.38 339.26 320.51 57 58 442.81 411.25 383.91 360.01 338.93 320.21 58 59 442.25 410.76 383.49 359.64 338.60 319.92 59 60 441.68 410. 2S 383.06 359.27 338.27 319.62 60 RADII OF DEGREES OF CURVE. f 18 19 20 21 33 23 / Radius. Radius. Radius, l Radius Ktidius. ; Fad us. 319.62 302.94 287. '.14 274.37 J02 04 ! 250 79 ~T 1 319 33 302.68 287.70 274 1C) 261.85 ! 250.61 i 2 319 04 302.42 287.40 273 94 261.05 250. 4 d 2 a 318.74 302.10 287.23 273.73 201.45 25026 3 4 318.45 801.89 x86 99 273 51 201 20 250.08 4 5 318.16 301. (53 286 76 273.30 261.06 249.90 5 6 317.87 301.37 26.52 373.08 360 87 249 72 7 317.58 301.11 286.29 272. H7 260.68 249 54 7 8 317.29 300 85 286.05 272.66 260 48 249 87 8 9 317.00 300.59 285.82 272 45 200 29 249 19 9 10 316.71 300.33 285.58 272.23 260.10 2-19.01 10 n 316.43 300.07 285.35 272.02 259.90 249.84 11 12 316 14 299.82 285.12 271.81 259.71 248.06 12 13 315.85 299.56 284.88 271.60 259.52 248.48 13 14 315.57 299 30 284.65 271.39 *6.83 248.31 14 15 315.28 299.04 284.42 271.18 259.13 248.13 15 16 315.00 298.79 284.20 270.97 258 94 247.96 16 17 314 71 298.53 283.96 270.76 258.75 247.78 17 18 314 43 29S.28 283.73 270.55 258.56 247.61 18 19 3 4.14 298.02 283.50 270.34 258.37 247.43 19 20 313.60 297.77 283.27 270.13 258.18 247.20 20 21 313.58 297.51 283.04 269.92 257 99 247.08 21 22 313.29 297 . 20 282.81 269 71 257.80 246.91 22 23 313 01 297.01 282 58 269.51 257.61 246.74 23 24 312.73 290. 75 282.35 269.30 257.42 240.50 24 25 312.45 296.50 282.12 269 09 257.23 240 39 25 20 312.17 296 25 281.89 268.89 257 04 246.22 26 27 311.89 296 00 281.67 268.68 256.85 246.04 27 28 311.01 295 75 281.44 268.47 250.07 245.87 28 29 311.33 295 50 281.21 268.27 256.48 245.70 29 30 311.00 295.25 280.99 268 00 256.29 245.53 30 31 310.78 295.00 280.76 267.86 256 .10 245.36 31 32 310.50 294.75 280.54 267.6*) 255.92 245.19 32 33 310.23 294.50 280 31 207.45 255.73 245.02 33 34 309 95 294 25 280.09 207. 24 255.54 244.84 34 3:> 309.67 294.00 279.86 207.04 255.36 244.07 35 36 309.40 293.76 279.64 266.84 255.17 244.50 36 37 309.12 293.51 279.42 260. 6^ 254 99 244.33 37 38 308.85 293 26 279.19 260.43 254.80 244.16 38 39 308.58 293 01 278.97 206.23 254 62 243.99 39 40 308.30 29->.77 278.75 266.02 254.43 243.83 40 41 308.03 292.52 278.52 265.82 254.25 243 66 41 42 307.76 292.28 278.30 265.62 254.06 243.49 ! 42 43 307.49 292.03 278.08 265.42 254.88 243.32 43 44 307.22 291.79 277.86 265.22 253.70 243.15 44 45 306 95 291 . 55 277.64 265 02 23.51 242.98 45 46 306 68 291.30 277.42 264.82 253.33 242 U 46 47 306.41 291.06 277.20 264.62 253 14 242.64 47 48 306.14 290.82 276.98 264.42 252.96 242.48 48 49 305.87 290.58 276.76 264.22 252.78 242.31 49 50 305. 6C 290.33 276.54 264.02 252.60 242.14 50 51 305.33 290.09 276.32 263.82 252.42 241.98 51 52 305 06 289.85 276.10 263.62 252 24 241.81 52 53 304.80 289.61 275.89 263.42 252.05 241.64 53 51 304 53 289.37 275.67 263.22 251.87 241.48 54 55 304.27 289.13 275.45 263.03 251.69 241.31 55 56 304.00 288.89 275.23 262.83 251.51 241 15 56 57 303.73 288.65 275.02 262.63 251.33 240.98 57 58 303.47 288.41 274.80 262.44 251.15 240.82 58 59 303.21 288.18 274.58 262.24 250.97 240.65 59 60 302.94 287.94 274 37 262.04 250 79 240.49 60 RADII OP DEGREES OF CURVE. / 34 25 36 27 38 29 if Radius Radius. Radius. Radius. Radius. Radius. "(T 240.49 231.01 222.27 214.18 206.68 199.70 "IT 1 240.32 230.86 222.13 214.05 206.56 199.58 1 2 240.16 230.71 221.99 213.92 206.44 199.47 2 3 239 09 230.56 221.85 213.79 206.32 199.36 3 4 239.83 230.41 221.71 2i3.ru 206 20 199.25 4 5 239.67 230.26 221.57 213.54 206.08 199 14 5 6 239.50 230.11 221.43 213.41 205.96 199.02 6 7 239 34 229.96 221.30 213.28 205.84 198.91 7 8 239.18 229.81 221.16 213.15 205.72 198.80 8 9 239.01 229.66 221.02 213.02 205.60 198.69 9 10 238.85 229.51 220.88 212.89 205.48 198.58 10 11 238.69 229.36 220.74 212.77 205.36 198.47 11 12 238.53 229.21 220.60 212.64 205.24 198 36 12 13 238.37 229.06 220.47 212.51 205.12 198 25 13 14 238.21 228.91 220.33 212.38 205.00 198.14 14 15 238.04 228.76 220.19 212.25 204.89 198 03 15 16 237.88 228.61 220.05 212.13 204.77 197.92 16 17 237.72 228.46 219.92 212.00 204.65 197.81 17 18 237.56 228.32 219.78 211.87 204.53 197.70 18 19 237.40 228.17 219.64 211.75 204.41 197.59 19 "20 237.24 228.02 219.51 211.62 204.30 197.48 20 21 237.08 227.87 219.37 211 .49 204.18 197.37 21 22 236.92 227.73 219.23 211.37 204.06 197.26 22 23 236.76 227.58 219.10 211.24 203.94 197.15 23 24 236.60 227.43 218.96 211.11 203.83 197.04 24 25 236.44 227.29 218.82 210.99 203.71 196.93 25 26 236.28 227.14 218.69 210.86 203.59 196.82 26 27 236.13 226.99 218.56 210.74 203.48 196.71 27 28 235.97 226.85 218.42 210 61 203.36 196.60 28 29 235.81 226.70 218.29 210.49 203.24 196.49 29 30 235.65 226.55 218.15 210.36 203.13 196.39 30 31 235.49 . 226.50 218.01 210.25 203.01 196.28 31 32 235.34 226.26 217.88 210.11 202.89 196.17 32 33 235.18 226.12 217.75 209.99 202.78 196 06 33 34 235.02 225.97 217.61 209.86 202.66 195.95 34 35 234.87 225.83 217.48 209.74 202.54 195.84 35 36 234.71 225.68 217.34 209.61 202.43 195.74 36 37 234.55 225.54 217.21 209.49 202.31 195.63 37 38 234.40 225.40 217.08 209.37 202.20 195.52 38 9 234.24 225.25 216.94 209.24 202.08 195.41 39 40 234.08 225.11 216.81 209.12 201.97 195.31 40 41 233.93 224.96 216.68 208.99 201.85 195.20 41 42 233.77 224.82 216.54 208.87 201.74 195.09 42 43 233.62 224.68 216.41 208.75 201.62 194.99 43 44 233.46 224.53 216.28 208.63 201.51 194.88 44 45 233.31 224.39 216.15 208.50 201.40 194 77 45 46 233.15 224.25 216.02 208.38 201.28 194.67 46 47 233.00 224.11 215.88 208.26 201.17 194.56 47 48 232.84 223.96 215.75 208.14 201.05 194.45 48 49 232.69 223.82 215.62 208.01 200.94 194.34 49 50 232.54 223.68 215.49 207.89 200.83 194.24 50 51 232.38 223.54 215.36 207.77 200.71 194.13 51 52 232.23 223.40 215.23 207.65 200.60 194.03 52 53 962.08 223.26 215.10 207.52 200.49 193.92 53 54 231.92 223.11 214.96 207.40 200.37 193.82 54 55 231.77 222.97 214.83 207.28 200. 2 i 193 71 55 56 231.62 222.83 214.70 207.16 200.15 193.61 56 57 231.47 222.69 214.57 207.04 200.03 193.50 57 58 231.31 222.55 214.44 206.92 199.92 193.40 58 59 231.16 222.41 214.31 206.80 199.81 193.29 59 60 231.01 222.27 214.18 206.68 199.70 193.19 60 65 NATURAL SINES AND COSINES. 1 2 3 4 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ' .00000 One. .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756 60 1 .00029 One. .01774 .99984 .03519 .99938 .05263 .99861 .07005 .99754 59 2 .00058 One. .01803 .99984 .03548 .99937 .05292 .99860 .07034 .99752 58 3 .00087 One. .01832 .99983 .03577 .99936 .05321 .99858 .07063 .99750 57 4 .00116! One. .01862 .99983 .03606 .99935 .05350 .99857 .07092 .99748 56 5 .00145 One. .01891 .99982 .03635 .99934 .05379 .99855 .07121 . 99746 ! 55 6 .00175 One. .01920 .99982 .03664 .99933 .05408 .99854 .07150 .99744! 54 7 .00204 One. .01949 .99981 .03693 .99932 .05437 .99852 .07179 .99742 53 8 .00233 One. .01978 .99980 .03723 .99931 .05466 .99851 .07208 .99740 52 9 .00262 One. .02007 .99980 .03752 .99930 .05495 .99849 .07237 .99738' 51 10 .00291 One. .02036 99979 .03781 .99929 .05524 .99847 .07266 .99736 50 11 .00320 .99999 .02065 .99979 .03810 .99927 .05553 .99846 .07295 .99734 49 12 .00349 .99999 .02094 .99978 .03839 .99926 .05582 .99844 .07324 .99731 48 13 .00378 .99999 .02123 .99977 .03868 .99925 .05611 . 99842 ! .07353 .99729 47 14 .00407 .99999 .02152 .99977 .03897 .99924 .05640 .99841 i .07382 .99727 46 15 .00436 .99999 .02181 .99976 .03926 .99923 .05669 .99839! .07411 .99725 45 16 .00465 .99999 .02211 .99976 .03955 .99922 .05698 .99838: .07440 .99723 44 17 .00495 .99999 .02240 .99975 .03984 .99921 .05727 .99836 .07469 .99721 43 18 .00524 .99999 .02269 .99974 .04013 .99919 .05756 .99834 .07498 .99719 42 19 .00553 .99998 .02298 .99974 .04042 .99918 .05785 .998331 .07527 .99716 41 20 .00582 .99998 .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356 .99972 .04100 .99916 .05844 .99829 .07585 .99712 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .05873 .99827 .07614 .99710 38 23 .00669 .99998 .02414 .99971 .04159 .99013 .05902 .99826 .07643 .99708 37 24 .00698 .99998 .02443 .99970 .04188 .99912 .05931 .99824; .07672 .99705 36 25 .00727 .99997 .02472 .99969 .04217 .99911 .05960 .99822: .07701 .99703 35 26 .00756 .99997 .02501 .99969 .04246 .99910 .05989 .99821 .07730 .99701 34 27 .00785 .99997 .02530 .99968 .04275 .99909 .06018 .99819! .07759 .99699 33 28 .00814 .99997 .02560 .99967 .04304 .99907 .06047 .99817! .07788 .99696 32 29 .00844 .99996 .02589 .99966 .04333 .99906 .06076 .99815 .07817 .99694 31 30 .00873 .99996 .02618 .99966 .04362 .99905 .06105 .99813 .07846 .99692 30 31 .00902 .99996 .02647 .99965 .04391 .99904 .06134 .99812 ! .07875 .99689 29 32 .00931 .99996 .02676 .99964 .04420 .9D9C2 .06103 .99810 .07904 .99687 28 33 .00960 .99995 .02705 .99963 .04449 .99901 .06192 .99808 .07933 .99685 27 34 .00989 .99995 .02734 .99963 .04478 .99900 .06221 .99806 .07962 .99683 26 35 .01018 .99995 .02763 .99962 .04507 .99898 .06250 .99804 .07991 .99680 25 36 .01047 .99995 .02792 .99961 .04536 .99897 .06279 .99803 .08020 .99678 24 37 .01076 .99994 .02821 .99960 .04565 .99896 .06308 .99801! .08049 .99676 23 38 .01105 .99994 .02850 .99959 .04594 .99894 .06337 .99799 .08078 .99673 22 39 .01134 .99994 .02879 .99959 .04623 .99893 .06366 .99797 .08107 .99671 21 40 .01164 .99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 20 41 .01193 .99993 .02938 .99957 .04682 .99890 .06424 .99793 .08165 .99666 19 42 .012221.99993 .02967 .99956 .04711 .99889 .06453 .99792 .08194 .99664 18 43 .01251 .99992 .02996 .99955 .04740 .99888 .06482 .99790; .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 .99886 .06511 .99788; .08252 .99659 16 45 .01309 .99991 .03054 .99953 .04798 .99885 .06540 .99786! .08281 .99657 15 46 .01338 .99991 .03083 .99952 .04827 .99883 .06569 .99784' .08310 .99654 14 47 .013671.99991 .03112 .99952 .04856 .99882 .06598 .99782 .08339 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99647 11 50 .01454 .99989 .03199 .99949 .04943 .99878 .06685 .99776 .08426 .99644 10 51 .01483 .99989 .03228 .99948 .04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989 .03257 .99947 .05001 .99875 .06743 .99772 .08484 .99639 8 53 .01542 .99988 .03286 .99946 .05030 .99873 .0677-3 .99770, .08513 .99637 7 54 .01571 .99988 .03316 .99945 .05059 .99872 .06802 .99768 .08542 .99635 6 55 01600 .99987 .03345 .99944 .05088 .99870 .06831 .99766 .08571 .99632 5 56 .01629 .99987 .Oa374 .99943 .05117 .99869 .06860 .99764 .08600 .99630 4 57 .016581.99986 .03403 .99942 .05146 .99867 .06889 .99762 .08629 .99627 3 58 .01 687 i. 99986 .03432 .99941 .05175 .99866 .06918 .99760; .08658 .99625 2 59 .01716 .99985 .03461 .99940 .05205 .99864 .06947 .99758 .08687 99622 1 60 .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756 . 08716 j. 99619 / Cosin Sine Cosin Sine Cosm Sine Cosin Sine Cosin i Sine t 89" 88 *7 86 85' 66 NATURAL SINES AND COSINES. / 5 6 7 8* 9 / Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ~0 .08716 .99619 .10453 .99452 .12187 .99255 .13917 .99027 .15643 .98769 60 1 .08745 .99617 .10482 .99449! .12216 .99251 .13946 .99023 .15672 .98764! 59 2 .08774 .99614 '.10511 . 99446 ! .12245 .99248 .13975 .99019 .15701 .98^60 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004 .99015 .15730 .98755 5? 4 .08831 .99609 .10569 .99440 .12302 .99240 .14033 .99011 .15758 .98751! 56 5 .08860 .99607 .10597 .99437 .12331 .99237 .14061 .99006 .157871.98746 55 6 .08889 .99604 .10626 .99434 .12360 .99233! .14090 .99002 .15816 .98741 54 7 .08918 .99602 .10655 .99431 .12389 . 99230 j .14119 .98998 .15845 .98737 53 8 .08947 .99599 .10684 .99428 .12418 .99226 .14148 .98994 .15873 .987321 52 9 .08976 .99596 .10713 .99424 .12447 .99222 .14177 .98990 .15902 .98728 51 10 .09005 .99594 .10742 .99421 .12476 .99219 .14205 .98986 .15931 .98723 50 11 .09034 .99591 .10771 .99418 .12504 .99215 .14234 .98982 .15959 .98718 4G 12 .09063 .99588 .10800 .99415 .12533 .99211 .14263 .98978 .15988 .98714 4t 13 .09092 .99586 .10829 .99412 .12562 .99203] .14292 .98973 .16017 .98709 4? 14 .09121 .99583 .10858 .99409 .12591 .99204 .14320 .98969 .16046 .98704 41 15 .09150 .99580 .10887 .99406 .12620 .99200 .14349 .98965 .16074 .98700 4.: 16 .09179 .99578 .10916 .99402 .12849 .99197 .14373 .98961 .161031.98695 44 17 .09208 .99575 .10045 .99399 .12678 .99193 .14407 .98957 .161321.98690 49 18 .09237 .99572 .10973 .99393 .12706 .99189 .14436 .98953 .16160L98686 4i -19 .09266 .99570 .11002 .99393 .127,35 .991C6 .14464 .98948 .16189 .98681 41 20 .09295 .99507 .11031 .99390 .12764 .99182 .14493 .98944 .16218 .98676 41 21 .09324 .99564 .11060 .99386 .12793 .99178 .14522 .98940 .16246 .98671 3C 22 .09353 .99562 .11030 .99333, .12822 .90175 .14551 .98936 .16275 .98667 3f- 23 .09382 .99559 .11118 .99380 .12351 .99171 .14580 .98931 .16304 .98662 37 24 .09411 .99556 .11147 .99377 .12330 .99167 .14608 .98927 .16333 . 98657 31 25 .09440 .99553 .11176 .9D374 .12908 .99163 .14637 .98923 .16361 .98652 35 26 .09469 .99551 .11205 .99370 .12937 .99160 .14666 .98919 .16390 .98648' 34 27 .09498 .99548 .11234 .99367 .12966 .99156 .14695 .98914 .16419 .98643! 33 28 .09527 .99545 .11263 .99364 .12995 .99152 .14723 .98910 .16447 .98638] 32 29 .09556 .99542 .11291 .99360 .13024 .99148 .14752 .98006 .16476 .98633! 31 30 .09585 .99540 .11320 .99357 .13053 .99144 .14781 .98902 .16505 .98629 3C 31 .09614 .99537 .11349 .99354 .13081 .99141 .14810 .98897 .16533 .98624 2C 32 .09642 .99534 1.11378 .93351 .13110 .9oicr .14338 .98893 .16562 .98619 2J. 33 .09671 .99531 .11407 .99347 .13139 .99133 .14867 .98889 .16591 .98614! 2? 34 .09700 .99528 .11436 .93344 .13168 .99123 .14896 .98884 .16620 .98609 2C 35 .09729 .99526 .11465 .99341 .13197 .99125 .14925 .98880 .16648 .98604! 2* 36 .09758 .99523 1 .11494 .99337 .13226 .99123 .14954 .98876 1 .16677 .98600! 24 37 .09787 .99520 .11523 .99334 .13254 .99118 .14982 .98871 .16706 .98595 2? 38 .09816 .99517 .11552 .99331 .13283 .99114 .15011 .98867 .16734 .98590 2S 39 .09845 .09514 .11580 .93327 .13312 .99110 .15040 .98863 .16763 .98585 21 40 .09874 .99511 .11009 .99324 .13341 .99106 .15069 .98858 .16792 .98580 21 41 .09903 .99508 .11638 .99320 .13370 .99102 .15097 .98854 .16820 .98575 li 42 .09932 .99500 .110G7 .99317 .13399 .99003 .15126 .98849 .16849 .98570 If 43 .09961 .99503 .11090 .99314 .13427 .90094 .15155 .98845 .168781.98565 r 44 .09990 .99500 .11725 .99310 .13456 .99091 .15184 .98841 .16906 |.98561 K 45 .10019 .99497 .11754 99307 .13485 .99087 .15212 .98836 .169351.98556 it 46 .10048 .99494 .11783 .99303 .13514 .99083 .152-41 .98832 .16964 L98551 14 47 ,10077 .99491 .11812 .99300 .13543 .99079 .15270 .98827 .16992 .98546 48 .10106 .99488 .11840 .99297 .13572 .99075 .15299 .98823 .17021 .98541 IS 49 .10135 .99485 .11869 .99293 .13600 .99071 .15327 .98818 .17050 .98536 11 50 .10164 .99482 .11898 .99290 .13629 .99067 .15356 .98814 .17078 .98531 K 51 .10192 .99479 .11927 .99286 .13658 .99063 .15385 .98809 .17107 .98526 1 52 .10221 .99476 .11956 .99283 .13687 .99059 .15414 .98805 .17136 .98521 | 53 .10250 .99473 .11985 .99279 .13716 .99055 .15442 .98800 .17164 .98516 54 .10279 .99470 .12014 .99276 .13744 .99051 .15471 .98796 .17193 .98511 I 55 .10308 .99467 .12043 .99272 .13773 .99047 .15500 .98791 .17222|.98506 56 .10337 .99464 .12071 .99269 .13802 .99043 .15529 .98787 .17250 .98501 4 57 10366 .99461 .12100 .99265 .13831 .99039 .15557 .98782 . 172791. 98496 i i 58 .10395 .99458 .12129 .99262 .13860 .99035 .15586 .98778 .17308 . 98491 i S 59 .10424 .99455 .12158 .99258 .13889 .99031 .15615 .98773 ! .17,336 .98486' ] 60 .10453 .99452 .12187 .99255 .13917 .99027 .15643 .98769 . 1 7365 j. 984811 ( / Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin | Sine / 84 83* 82 9 81 80 67 NATURAL SINES AND COSINES. 10 11 12 13 14 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ' .17365 .98481 .19081 .98163 .20791 .97815 .22495 .97437 .24192 .97030 60 1 .17393 .98476 .19109 .98157 .20820 .97809 .22523 .97430 .24220 .97023 59 2 .17422 .98471 .19138 .98152 .20848 .97803 .22552 .97424 .24249 .97015 58 3 .17451 .98466 .19167 .98146 .20877 .97797 .22580 .97417 .24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 .97791 .22608 .97411 .24305 .97001 56 5 .17508 .98455 .19224 .98135 .20983 .97784 .22637 .97404 .24333 .96994 55 6 .17537 .98450 .19252 .98129 .20962 .97778 .22665 -07398 .24362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .22693 .97391 .24390 . 96980 53 8 .17594 .98440 .19309 .98118 .21019 .97766 .22722 .97384 .24418 .96973 52 9 .17623 .98435 .19338 .98112 .21047 .97760 .22750 .97378 .24446 .96966 51 10 .17651 .98430 .19366 .98107 .21076 .97754 .22778 .97371 .24474 .96959 50 11 .17680 .98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 12 .17708 .98420 .19423 .98096 .21132 .97742 .22835 .97358 .24531 .96945 48 13 .17737 .98414 . 19452 .98C90 .21101 .97735 .22863 .97351 .24559 . 96937 47 14 .17766 .98409 .19481 .98084 .21189 .97729 .22892 .97345 .24587 .96930 46 15 .17794 .98404 .19509 .98079; .21218 .97723 .22920 .97338 .24615 .96923 45 16 .17823 .98399 .19538 .98073 .21246 .97717 .22948 .97331 .24644 .96916 44 17 .17852 .98394 .19566 .98067 .21275 .97711 .22977 .97325 .24672 .96909 43 18 .17880 .98339 .19595 .980611 .21303 .97705 .23005 .97318 .24700 .96902 42 19 .17909 .98383 .19623 .98056 .21331 .97698 .23033 .97311 .24728 .96894 41 20 .17937 .98378 .19652 98050 .21360 .97692 .23062 .97304 .2-4756 .96887 40 21 .17966 .98373 .19680 .98044 .21388 .97686 .23090 .97298 .24784 .96880 39 22 .17995 .98368 .19709 98039 .21417 .97680! .23118 .97291 .24813 .96873 38 23 .18023 .98362 .19737 98033 .21445 .97673 .23146 .97284 .24841 .96866 37 24 .18052 .98357 .19766 98027 .21474 .97667 .23175 .97278 .24869 .96858 36 25 .18081 .98352 .19794 98021 .21502 .97661 .23203 .97271 .24897 .96851 35 26 .18109 .98347 .19823 98016 .21530 .97655 .23231 .97264 .24925 .96844 34 27 .18138 .98341 .19851 98010 1 .21559 .97648 .23260 .97257 .24954 .96837 33 28 .18166 .98a36 .19880 980041 .21587 .97642 .23288 .97251 .24982 .96829 32 29 .18195 .98331 .19908 97998 .21016 .97035 .23316 .97244 .25010 .96822 31 30 .18224 .98325 .19937 97992 .21644 .97630 .23345 .97237 .25038 .96815 30 31 .18252 .98320 .19965 97987 .21672 .97623 .23373 .97230 .25066 .96807 29 32 .18281 .98315 .19994 97981 .21701 .97017; .23401 .97223 .25094 .96800 28 33 .18309 .98310 .20022 9T975| .21729 .97611 .23429 .97217 .25122 .96793 27 34 .18338 .98304 .20051 979G9 .21758 .97604! .23458 .97210 .25151 .96786 26 35 .18367 .98299 .20079 97983 i .21786 .97593: .23486 .97203 .25179 .96778 25 36 .18395 .98294 .20108 97958 .21814 .97592 .23514 .97196 .25207 .96771 24 37 .18424 .98288 .20136 97952 .21843 .97535 .23542 .97189; .25235 .96764 23 38 .18452 .98283! .20165 97946 .21871 .97579 .23571 .97182 .25263 .96756 22 39 .18481 .98277 .20193 97940 .21839 .97573 .23599 .97176 .25291 .96749 21 40 .18509 .98272 .20222 97934 .21928 . 97566 i .23627 .97169 : .25320 .96742 20 41 .18538 .98267 .20250 97928 .21956 .97560 .23656 .97162! .253481.96734 19 42 .18567 .98261 .20279 97922 .21985 .97553 .23684 .97155 .25376 .96727 18 43 .18595 .98256 .20307 97916 .22013 .97547 .23712 .97148 .25404 .96719 17 44 .18624 .98250 .20336 97910 .22041 .97541 .23740 .97141 .25432 .96712 16 45 .18652 .98245 .20364 97905 .22070 .97534 .23769 .97134- .25460 .96705 15 46 .18681 .98240 .20393 97899 .22098 .97528 .23797 .97127 .25488 .96697 14 47 .18710 .98234 .20421 97893 .22126 .97521 .23825 .97120 ! .25516 .96690 13 48 .18738 .98229 .20450 97887 .22155 .97515 .23853 .97113 i .25545 .96682 12 49 .18767 .98223) .20478 97881 .22183 .97508 .23882 .97106 1 .25573 .96675 11 50 .18795 .98218 .20507 97875 .22212 .97502 .23910 .97100 .25601 .96667 10 51 .18824 .98212 .20535 97869 .22240 .97496 .23938 .97093 .25629 .96660 9 52 .18852 .98207 .20563 97863 .22268 .97489 .23966 .97086 .25657 .96653 8 53 .18881 .98201 1 .20592 .97857 .22297 .97483 .23995 .97079 .25685 .96645 7 54 .18910 .98196 1 .20620 .97851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 .18838 .98190 .20649 .97845 .22353 .97470 .24051 .970651 .257411.96630 5 56 .189671.98185 .20677 .97839 .22382 .97463 .24079 .97058| .257691.96623 4 57 .189951.98179 .20706 .97833 .22410 .97457 .24108 .97051 .25798.96615 3 58 .19024 .98174 .20734 .97827 .22438 .97450 .24136 .97044 .25826 .96608 2 59 .19052 .98168 .20763 .97821 .22467 .97444 .24164 .97037 .25854 .96600j 1 60 .19081 .98163 .20791 .97815 .22495 .97437 .24192 .97030 . 25882 ,. 96593| f Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine f 79 78 77 i "76 ! 75 -NATURAL SINES AND 15 16 17 18 19 / Sine 1 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ~0 .25882 .96593 .27564 .96126 .29237 .95630 .30902 .95106 .32557 .94552 60 1 .25910 .96585 .27592 .96118 .29265 .956221 .30929 .95097 .32584 .94542 59 2 .25938 .96578 .27620 .96110 .29293 .95613 , .30957 .95088 .32612 .94533 58 3 .25966 .96570 .27648 .96102 .29321 .95605 1.30985 .95079 .32639 .94523 57 4 .259941.96562 .27676 .96094 .29348 .95596 1 .34012 .95070 .32667 .94514 56 5 .26022 .96555 .27704 .96086 .29376 .95588 ! .31040 .95061 .32694 .94504 55 6 .26050 .96547 .27731 .96078 .29404 .95579 .31068 .95052 .32722 .94495 54 7 .260791.96540 .27759 96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107J. 96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .96054 .29487 .95554 .31151 .95024 .32804 .94466 51 10 .26163 .96517 .27843 .96046 .29515 .95545 .31178 .95015 .32832 .94457 50 11 .26191 .96509 .27871 .96037 .29543 .95536 .31206 .95000 .32859 .94447 49 12 .26219 .96502 .27899 .96029 .29571 .95528 .31233 .94997 .32887 .94438 48 13 .26247 .96494 .27927 .96021 .29599 .95519 .31201 .94988 1 .32914 .94428 47 14 .26275 .96485 .27955 .96013 .29026 .95511 .31289 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .95502 .31310 .94970 .32969 .94409 45 1G .26331 .96471! .28011 .95997 .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359 .96463 .28039 .95989 .29710 .95485 .31372 .94952 .33024 .94390 43 18 .2638? .96456 .28067 .95981 .29737 .95476 .31399 .94943 .33051 .94380 42 19 .26415 .96448 .28095 .95972 .29765 .95467 .31427 .94933 .33079 .94370 41 20 .26443 . 96440 ( .28123 .95964 .29793 .95459 .31454 .94924 .33106 .94361 40 21 .26471 .96433 .28150 .95956 .29821 .95450 .31482 .94915 .33134 .94351 39 22 .26500 .96425 .28178 .95948 .29849 .95441 .31510 .94900 .33161 .94342 38 23 .26528 .96417 .28206 .95940 .20876 .95433 .31537 .94897 .33189 .94332 37 24 .26556 .96410 .28234 .95931 .29904 .95424 .31505 .94888 .a3216 .94322 36 25 .28584 .96402 .28262 .95923 .29932 .95415 .31593 .94878 .33244 .94313 35 26 .26612 .96394 .28290 .95915 .29900 .95407 .31620 .94809 .33271 .94303 34 27 .26640 .96386 .28318 .95907 .29987 .95398 .31048 .94800 .33298 .94293 33 28 .26668 .96379 .28346 .95898 .30015 .95389 .31075 .94851 .33326 .94284 32 29 .28696 .96371 .28374 .95890 .30043 .95380 .31703 .94842 .33353 .94274 31 30 .20724 .96.363 .28402 .95882 .30071 .95372 .31730 .94832 .33381 .94264 30 31 .20752 .96355 .28429 .95874 .30098 .95363 .31750 .94823 .33408 .94254 29 82 .26780 .9034? .28457 .95805 .30120 .95354 .31780 .94814 .33436 .94245 28 S3 .26808 .90340 .28485 .95857 .30154 .95345 .31813 .94805 .33463 .94235 27 34 .26836 .96332 .28513 .95849 .30182 .95337 .31841 .94795 .33490 .94225 26 5 .26864 .96324 .28541 .95841 .30209 .95328 .31808 .94786 .33518 .94215 25 36 .26892 .96316: .28569 .95832 .30237 .95319 .31890 .94777 .33545 .94206 24 37 .26920 .98303 .28597 .95824 .30205 .95310 .31923 .94768 .33573 .94196 23 38 .26948 .90301 .28625 .95816 .30292 .95301 .31951 .94758 .33000 .94186 22 39 .26976 .98293 .28652 .95807 .30320 .95293 .31979 .94749 .33627 .94176 21 40 .27004 .96285 .28680 .95799 .30348 .95284 .3200C .94740 .33655 .94167 20 41 .27032 .96277 .28708 .95791 .30376 .95275 .32034 .94730 .33682 .94157 19 42 .27060 .96269 .28736 .95782 .30403 .95203 .320G1 .94721 .33710 .94147 18 43 .27088 .96261 .28764 .95774 .30431 .95257 .32080 .94712 .33737 .94137 17 44 .27116 .96253 .28792 .95760 .30459 .95248 .32110 .94702 .33764 .94127 16 45 .27144 .96246 .95757 .30486 .95240 .32144 .94693 .33792 .94118 15 46 .27172 .96238 * 28847 .95749 .30514 .95231 .32171 .94684 .33819 .94108 14 47 .27200 .96230 .'28875 .95740 .30542 .95222 .32199 .94674 .33846 .94098 13 48 .27228 .96202 .28903 .95732 .30570 .95213 .32227 .94665 .33874 .94088 12 49 .27256 .96214' .28931 .95724 .30597 .95204 .32254 .94656 33901 .94078 11 50 .27284 .96206 .28959 .95715 .30625 .95195 .32282 .94640' .33929 .94068 10 51 .27312 .96198 .28987 .95707 .30653 .95186 .32309 .94637 .33956 .94058 9 52 .27340 .96190, .29015 .95698 .30680 .9517? .32337 .94627 .33983 .94049 8 53 .27368 .96182 .29042 .95690 .30708 . 95108 i .32364 .94618 .34011 .94039 7 54 .27396 .96174 .29070 .95681 .3073(3 .95159 .32392 .94609 .34038 .94029 6 55 .27424 .96166 .29098 .95673 .30763 .95150 .32419 .94599 .34065 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142 .32447 .94590 .34093 .94009 4 57 .27480 .96150 .29154 .95656 .30819 .95133 .32474 .93580 .34120 .93999 3 58 .27508;. 96142 .29182 .95647 .30846 .95124 .32502 .94571 .34147 .93989 2 59 .27536 .96134 .29209 .95639 .30874 .95115 .32529 .94561 .34175 .93979 1 60 .27564 .96126 .29237 .95630 .30902 .95106 .32557 .94552 .34202 .93969 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine / i 74 73 72 71 70 69 NATURAL SINES AND COSINES. 20 21 1 22 23 24 Sine Cosin Sine ! Cosin Sine Cosin Sine Cosin Sine Cosin .34202 .93969 .358371.93358] .37461 .92718 .39073 .92050 .40674 .913551 60 1 .34229 .93959 .35864 .93348 .37488 .92707 .39100 .92039 .40700 .91343 59 2 .34257 .93949 .35891 .93337 .37515 .92697 .39127 .92028 .40727 .91331 58 3 .34284 .93939 .35918 .93327 .37542 .92686 .39158 .92016 .40753 .91319 57 4 .34311 .93929 .35945 .93316 .37569 .92675 .39180 .92005 .40780 . 91307 i 56 5 .34339 .93919 .35973 .93306 .37595 .92664 .39207 .91994 .40806 .91295 55 6 .34366 .93909 .36000 .93295 .37622 .92653 .39234 .91982 .40833 .91283 54 7 .34393 .93899 .36027 .93285 .37649 .92642 .39260 .91971 .40860 .91272; 53 8 .34421 .93889 .36054 .93274 .37676 .92631 .39287 .91959 .40886 .91260 52 9 .34448 .93879 .36081 .93264 .37703 .92620 .39314 .91948 .40913 .91248 51 10 .34475 .93869 .36108 .93253 .37730 .92609 .39341 .91936 .40939 .91236 50 11 .34503 .93859 .36135 .93243 .37757 .92598 .39367 .91925 .40966 .91224 49 12 .34530 .93849 .36162 .93232 .37784 .92587 .39394 .91914 .40992 .91212 48 13 .34557 .93839 .36190 .93222 .37811 .92576 .39421 .91902 .41019 .91200 47 14 .34584 .93829 .36217 .93211 .37838 .92565 .39448 .91891 .41045 .91188 46 15 .34612 .93819 .36244 '.93201 .37865 .92554 .39474 .91879 .41072 .91176 45 16 .34639 .93809 .362711.93190 .37892 .92543 .39501 .91868 .41098 .91164 44 17 .34666 .93799 .36298:. 93180 .37919 .92532 .39528 .91856 .41125 .91152 43 18 .34694 .93789 .363251.93169 .37946 .92521 .39555 .91845 .41151 .911401 42 19 .34721 .93779 .36352 .93159 .37973 .92510 .39581 .91833 .41178 .91128 41 20 .34748 .93769 .363791.93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93759 .36406 .93137 .38026 .92488 .39635 .91810 .41231 .91104 39 22 .34803 .93748 .36434 .93127 .38053 .92477 .39661 .91799 .41257 .91092! 38 23 .34830 .93738 .36461 .93116 .38080 .92466 .39688 .91787 .41284 91080 37 24 .34857 .93728 .36488 .93106 .38107 .92455 .39715 .91775 .41310 .910681 36 25 .34884 .93718 .36515 .93095 .38134 .92444 .39741 .91764 .41337 .91056 35 26 .34912 .93708 .36542 .93084 .38161 .92432 .39768 .91752 .41363 .91044 34 27 .34939 .93698 .36569 .93074 .38188 .92421 .39795 .91741 .41390 .91032! 33 28 .34966 .93688 .36590 .93063 .38215 .92410 .39822 .91729 .41416 .91020; 32 29 .34993 .93677 .36623 .93052 .. C 8241 .92399 .39848 .91718 .41443 .91008 31 30 .35021 .93667 .36650 .93042 .38268 .92388 .39875 .91706 .41469 : 90996 30 81 .35048 .9365? .36677 .93031 .38295 .92377 .39902 .91694 .41496 .90984 29 32 .35075 .93647 .36704 .93020 .38322 .92360 .39928 .91683 .41522 .90972; 28 33 .35102 .93637 .36731 .93010 .38349 .92355 .39955 .91671 .41549 .90960 27 34 .35130 .93626 .36758 .92999 .38376 .92343 .39982 .91660 .41575 .90948 26 35 .35157 .93616 .36785 .92988 .38403 .92332 .40008 .91648 .41602 .90936| 25 36 .35184 .93606 .36812 .92978 .38430 .92321 .40035 .91636 .41628 .90924 24 37 .35211 .93596 .36839 .92967 .38456 .92310 .40062 .91625 .41655 .90911 23 38 .35239 .93585 .36867 .92956 .38483 .92299 .40088 .91613 .41681 .90899 22 39 .35266 .93575 .36894 .92945 .38510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 .93565 .36921 .92935 .38537 .92276 .40141 .91590 .41734! 90875 go 41 .35320 .93555 .36948 .92924 .38564 .92265 .40168 .91578 .41760 .90863! 19 42 .35347 .93544 .36975 .92913 .38591 .92254 .40195 .91566 .41787 .90851 18 43 .35375 .93534 .37002 .92902 .38617 .92243 .40221 .91555 .41813 .90839 17 44 .35402 .93524 .37029 .92892 .38644 .92231 .40248 .91543 .41840 .90826' 16 45 .35429 .93514 .37056 .92881 .38671 .92220 .4027'5 .91531 .41866 .90814 15 46 .35456 .93503, .37083 .92870 .38698 .92209 .40301 .91519 .41892 .90802 14 47 .35484 .93493: .37110 .92859 .38725 .92198 .40328 .91508 .41919 .90790 13 48 .35511 .93483; .37137 .92849 .38752 .92186 .40355 .91496 .41945 .90778 12 49 .35538 .93472 .37164 .92838 .38778 .92175 .40381 .91484 .41972 .90766 11 50 .35565 .93462 .37191 .92827 .38805 .92164 .40408 .91472 .41998 .90753 10 51 .35592 .93452 .37218 .92816 .38832 .92152 .40434 .91461 .42024 .90741 9 52 .35619 .93441 .37245 .92805 .38859 .92141 .40461 .91449 .42051 .90729 8 53 .35647 .93431 .37272 .92794 .38886 .92130 .40488 .91437 .42077 .90717 7 54 .35674 .93420 .37299 .92784 .38912 .92119 .40514!. 91425 .42104 .90704 6 55 .35701 .93410! .37326 .92773 .38939 .92107 .40541 j. 91414 .42130 .90692 5 56 .35728 .93400 .37353 .92762 .38966 .92096 .405671.91402 .42156 .90680 4 57 .35755 .93389; .37380 .92751 .389931.92085 .40594 .91390 .42183 .90668 3 58 .35782 .93379; .37407 .92740 .39020 .92073 .40621 .91378 .42209 .90655 2 59 .35810 .93368 .37434 .92729 .39046 .92062 .40647 .91366 .42235 .90643 1 60 .35837 .93358 .37461 .92718 .39073 .92050 .40674 .91355 .42262 .90631 f Cosin Sine Cosin Sine Cosin "Sine Cosin ! Sine Cosin Sine / 69 68 67 66. 65 70 NATURAL SINES AND COSINES. 25 26 27 28 : 29 / Sine iCosin Sine Cosin Sine Cosin Sine Cosin Sine , Cosin ~o .422621.90631 .43837 ". 89879 .45399 .89101 .46947i.88295 .48481 .87462 60 1 .42288 '.9061 8 .43863 .89867 .45425 .89087 .46973 .882811 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .882671 .48532 .87434 58 3 .42341 .90594 .43916 .89841 .45477 .89061 .47024 .88254 .48557 .87420 57 4 .42367 .90582 .43942 .89828 .45503 .89048 .47050 .88240 .48583 .87406 56 5 .42394 .90569 .43968 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213 .48634 .87377 54 7 .424461.90545 .44020 .89790 .45580 .89008 .47127 .88199 .48659 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185 .48684 .87349 52 9 .424991.90520 .44072 .89764 .45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578 .90483 .44151 .89726 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45963 .83803 .47511 .87993 .49040 .87150 38 23 .42867 .90346 .44437 .89581 .45994 .83795 .47537 .87979 .49065 .87136 37 24 '.42894 .90334 .44464 .89571 .46020 .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 .89558 .46046 .88768 .47588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .46097 .88741 .47639 .87923 .49166 .87079 33 28 .42999 .90284 .44503 .89519 .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .93271 .44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .44672 .89467 .46223 .83074 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .46252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90208' .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196- .44750 .89428 .46304 .88634 .47844 .87812 .49369 .86964 25 36 .43209 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88607 .47895 .87784 .49419 .86935 23 38 .43261 .90158 .44828 .89389 .46381 .88593 .47920 .87770 .49445 .86921 22 39 .43287 .90146 .44854 .89376 .46407 .88580 .47946 .87756 .49470 .86906 21 40 .43313 .90133 .44880 .89363 .46433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 .88553 .47997 .87729 .49521 .86878 19 42 .43366 .90108 .44932 .89337 .46484 .88539 .48022 .87715 .49546 .86863 18 43 .43392 .90095 .44958 .89324 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90082 .44984 .89311 .46536 .88512 .48073 .87687 .49596 .86834 16 45 .43445 .90070 .45010 .89298 .46561 .88499 .48099 .87673 .49622 .86820 15 46 .43471 .90057 .45036 .89285 .46587 .88485 .48124 .87G59 .49647 .86805 14 47 .43497 .90045 .45062 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 48 .43523 .90032 .45088 .89259 .46639 .88458 .48175 .87631 .49697 .86777 12 49 .43549 .90019 .45114 .89245 .46664 .88445 .48201 .87617 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89981 .45192 .89206 .46742 .88404 .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390 .48303 .87561 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86690 6 55 .43706 .89943 .45269 .89167 .46819 .88363 .48354 .87532 .49874 .86675 5 56 .43733 .89930 .45295 .89153 .46844 .88349 .48379 .87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 .46870 .88336 .48405 .87504 .49924 .86646 3 58 .43785 .89905 .45347 .89127 .46896 .88322 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308 .48456 .87476 .49975 .86617 1 60 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 .50000 .86603 / Cosin Sine Cosin "Sine" Cosin Sine Cosin Sine Cosin Sine / 64 63 1 62 61 60 71 NATURAL SINES ANB COSINES. 30 31 32 33 34 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin .50000 .86603 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 60 1 .50025 .86588 .51529 .85702 .53017 .84789; .54488 .83851 .55943 .82887 59 2 .50050 .86573 .51554 .85687 .53041 .84774 .54513 .83835 .55968 .82871 58 3 .50076 .86559 .51579 .85672 .53066 .84759 .54537 .83819 .55992 .82855; 57 4 .50101 .86544 .51604 .85657 .53091 .84743 .54561 .83804 .56016 .82839 56 5 .50126 .86530 .51628 .85642 .53115 .84728 .54586 .83788 .56040 .82822 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .51678 .85612 .53164 .84697! .54635 .83756 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54659 .83740 .56112 .82773 52 9 .50227 .86471 .51728 .85582 .53214 .846661 .54683 .83724 .56136 .827571 51 10 .50252 .86457 .51753 .85567 .53238 . 84650 \ .54708 .83708 .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635' .54732 .83692 .56184 .82724 49 12 .50302 .86427 .51803 .85536 .53288 .84619 .54756 .83676 .56208 .82708 48 13 .50327 .86413 .51828 .85521 .53312 .84604 .54781 .83660 .56232 .82692' 47 14 .50352 .86398 .51852 .85506 .53337 .84588 .54805 .83645 .56256 .82675 46 15 .50377 .86384 .51877 .85491 .53361 .84573 .54829 .83629 .56280 .82659 | 45 16 .50403 .86369 .51902 .85476 .53386 .84557 .54854 .83613 .56305 .82643 44 17 .50428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .56353 .82610 42 19 .50478 .86325 .51977 .85431 .53460 .84511 .54927 .83565 .56377 .82593 41 20 .50503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 22 .50553 .86281 .52051 .85385 .53534 .84464 .54999 .83517 .5G449 .82544 38 23 .50578 .86266 .52076 .85370 .53558 .84448 .55021 .83501 .56473 .82528! 37 24 .50603 .86251 .52101 .85355 .53583 .84433 .55040 .83485 .56497 .82511136 25 .50628 .86237 .52126 .85340 .53607 .84417 .55072 .83469 .56521 . 82495 i 35 26 .50654 .86222 .52151 .85325 .53632 .84402 .55097 .83453 .56545 .82478 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437 .56569 .82462 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 .83421 .56593 . 82446 i 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .551G9 .83405 .5GG17 .82429 31 30 .50754 .86163 .52250 .85264 .53730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .56665 .82396! 29 32 .50804 .86133 .52299 .85234 .53779 .84308 .55242 .83356 .56689 .82380! 28 33 .50829 .86119 .52324 .85218 .53804 .84292 .55206 .83340 .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277 .55201 .83324 .56730 .82347 26 35 .50879 .86089 .52374 .85188 .53853 .84261 .55315 .83308 .66760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 .55339 .83292 .56784 .82314 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .55363 .83276 .56808 .82297 23 38 .50954 .86045 .52448 .85142 .53926 .84214 .55383 .83260 .56832 .82281! 22 39 .50979 .86030 .52473 .85127 .53951 .84198 .55412 .83244 .56856 .82264 21 40 .51004 .86015 .52498 .85112 .53975 .84182 .55436 .83228 .56880 .82248 20 41 .51029 .86000 .52522 .85096 .54000 .84167 .55460 .83212 .56904 .82231 19 42 .51054 .85985 .52547 .85081 .54024 .84151 .55484 .83195 .56928 .82214 18 43 .51079 .85970 .52572 .85066 .54049 .84135 .55509 .83179 .56952 .82198 17 44 .51104 .85956 .52597 .85051 .54073 .84120 .55533 .83163 .56976 .82181 16 45 .51129 ; 85941 .52621 .85035 .54097 .84104 .55557 .83147 .57000 .82165 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .55605 .83115 .57047 .82132 13 48 .51204 .85896 .52696 .84989 .54171 .84057 .55630 .83098 .57071 .82115 12 49 .51229 .85881 .52720 .84974 .54195 .84041 .55654 .83082 .57095 .82098 11 50 .51254 .85866 .52745 .84959 .54220 .84025 .55678 .83066 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009 .55702 .83050 .57143 .82065 9 52 .51304 .85836 .52794 .84928 .54269 .83994 .55726 .83034 .57167 .82048 8 53 .51329 .85821 .52819 .84913 .54293 .83978 .55750 .83017 .57191 .82032 7 54 .51354 .85806 .52844 .84897 .54317 .83962 .55775 .83001 .57215 .82015 6 55 .51379 .85792 .52869 .84882 .54342 .83946 .55799 .82985 .57238 .81999 5 56 .51404 .85777 .52893 .84866 .54366 .83930 .55823 .82969 .57262 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915 .55847 .82953 .57286 .81965 3 58 .51454 .85747 .52943 .84836 .54415 .83899 .55871 .82936 .57310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .55895 .82920 .57334 .81932 1 60 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 .57358 .81915 / Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosiu ~SimT / 59 58 57 56 55 NATURAL SINES AND COSINES. 35 36 37 38 39 / Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin o .57358 7819151 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 60 1 .57381 . 81899 i .58802 .80885 .60205 .79846 .61589 .78783 .62955 .77696 59 2 .57405 .81882 .58826 .80867 .60228 .79829! .61612 .78765 .62977 .77678 58 3 .57429 .81865 .58849 .808501 .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453 .81848: .58873 .80833 .60274 .79793 .61658 .78729 .63022 .77641 56 5 .57477 .81832, .58896 .80816 .60298 .79776 .61681 .78711! .63045 ,77623 55 6 .57501 .81815 .58920 .80799 .60321 .79758 .61704 .78694) .63068 .77605 54 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782; .58967 .80765 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .77531 50 11 .57619 .81731 .59037 .80713 .60437 .79671 .61818 .78604 .63180 .77513 40 12 1 .57643 .81714 .59061 . 80696 i .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 .59084 .80679^ .60483 .79635 .61864 .78568 .632251.77476 47 14 .57691 .81681 .59108 .80662 .60506 .79618| .61887 .78550 .63248J.77458 46 15 .57715 .81664 .59131 .80644 .60529 .79600 .61909 .78532 .63271 .77439 45 16 .57738 .81647 .59154 .80627 .60553 .79583 .61932 .78514 .63293 .77421 44 17 .57762 .81631 .59178 .80610 .60576 .79565 .61955 .78496 .63316 .77402 43 18 .57786 .81614 .59201 . 80593 i .60599 .79547 .61978 .78478 .63338 .77384 42 19 .57810 .81597 .59225 .80576 .60622 .79530! .620011.78460 .633611.77366 41 20 .57833 .81580 .59248 .80358 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .57857 .81563 .59272 .80541 .60668 .79494 .62046 .78424 .63406 .77329 39 22 .57881 .81546 .59295 .805241 .60691 .79477 .62069 .78405 .63428 .77310 38 23 24 .57904 .57928 .81530 .81513 .59318 .59342 .80507 .80489 .60714 .60738 .79459 .79441 .62092 .62115 .78387 .78369 .63451 .77292 .63473 .77273 37 36 25 .57952 .81496 .59365 .80472 .60761 .79424 .62138 .78351 .63496 .77255 35 26 .57976 .81479 .59389 .80455 .60784 .79406 .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438 .60807 .79388 .62183 .78315 .63540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 .62206 .78297 .63563 .77199 32 29 .58047 .81428 .59459 .80403 60853 .79353 .62229 .78279 .63585 .77181 31 30 .58070 .81413 .59482 .80386 .60876 .79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .81378 .59529 .80351 .60922 .79300 .62297 .78225 .63653 .77125 28 33 .58141 .81361 .59552 .80334 .60945 .79282 .62320 .78206 .63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264 .623421.78188 .63698 .77088 26 35 .58189 .81327 .59599 .80299 .60991 .79247 .62365 .78170 .637201.77070 25 36 .58212 .81310 .59622 .80282 .61015 .79229 .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59646 .80264 .61038 .79211 .62411 .78134 .63765 .77033 23 38 .58260 .81276 .59669 .802471 .61061 .79193 .62433 .78116 .63787 .77014 22 39 .58283 .81259 .59693 .80230 .61084 .79176 .62456 .78098 .63810 .76996 21 40 .58307 .81242 .59716 .80212 .61107 .79158 .62479 .78079 .63832 .76977 20 41 .58330 .81225 .59739 .80195 .61130 .79140 .62502 .78061 .63854 .76959 19 42 .58354 .81208 .59763 .80178 .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .62547 .78025 .63899 .76921 17 44 .58401 .81174 .59809 .80143 .61199 .79087 .62570 .78007 .63922 .76903 16 45 .58425 .81157 .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 15 46 .58449 .81140 .59856 .80108 .61245 .79051 .62615 .77970 .63966 .76866 14 47 .58472 .81123 .59879 .80091 .61268 .79033 .62638 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62660 .77934 .64011 .76828 12 49 .58519 .81089 .59926 .80056 .61314 .78998 .62683 .77916 .64033 .76810 11 50 .58543 .81072 .59949 .80038 .61337 .78980 .62706 .77897 .64056 .76791 10 51 .58567 .81055 .59972 .80021 .61360 .78962 .62728 .77879 .64078 .76772 9 52 .58590 .81038 .59995 .80003 .61383 .78944 .62751 .77861 .64100 .76754 8 53 .58614 .81021 .60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824, .64145 .76717 6 55 .58661 .80987 .60065 .79951 .61451 .78891 .62819 .77806' .64167J. 76698 5 56 .58084 .80970 .60089 .79934 .61474 .78873 .62842 .77788; .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61497 .78855 .62864 .777691 .64212 .766611 3 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 . 77751 1 .64234 .76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819 .62909 .777331 .64256 .76623 1 60 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 .64279 .76604 o i Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin | Sine 7 54 53 52 51* 50 73 NATURAL SINES AND COSINES. 40 41 42 43' 44 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ' .64279 .76604 .6500b .75471 .66913 .74314 .68200 .73135 .69466 .71934 60 1 .64301 .76586 .65628 .75452 .66935 .74295 .68221 .73116 .69487 .71914 59 2 .64323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 58 3 .64346 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 .64368 .76530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 56 5 .64390 .76511 .65716 .75375 .67021 .74217 .68306 .73036 .69570 .71833 55 6 .64412 .76492 .65738 .75356 .67043 .74198 .68327 .73016 .69591 .71813 54 7 .64435 .76473 .65759 .75337 .67064 .74178 .68349 .72996 .69612 .71792 53 8 .64457 .76455 .65781 .75318 .67086 .74159 .68370 .72976 .69633 .71772 52 9 .64479 .76436 .65803 .75299 .67107 .74139 .68391 .72957 .69654 .71752 51 10 .64501 .76417 .65825 .75280 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 12 .64546 .76380 .65869 .75241 .67172 .74080 .68455 .72897 .69717 .716911 48 13 .64568 .7G361 .65891 .75222 .67194 .74061 .68476 .72877 .69737 .71671 47 14 .64590 .70342 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71650 46 15 .64612 .76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 .71630 45 16 .64635 .76304 .65956 .75165 .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64657 .76286 .65978 .75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .64679 .76267 .66000 .75126 .67301 .739G3 .68582 .72777 .69842 .71569 42 19 .64701 .76248 .66022 .75107 .67323 .73944 .68G03 .72757 .69862 .71549 41 20 .64723 .76229, .66044 75088 .67344 .73924 .68624 .72737 .69883 .71529 40 21 .64746 .76210 .66066 75089 .67366 .73904 .68645 .72717 .69904 .71508 39 22 .64768 .76192 .66088 75050 .67387 .73885 .68GGO .72697 .69925 .71488 38 23 .64790 .76173 .66109 75030 .67409 .738G5 .68683 .72677 .69946 .71468 37 24 .64812 .76154 .66131 75011 .67430 .7'3846 .68709 .72657 .69966 .71447 36 25 .64834 .76135 .66153 74992 .67452 .73820 .68730 .7'2637 .699871.71427 35 26 .64856 .76116 .66175 74973 .67473 .73800 .68751 .72617 .70008 .71407 34 27 .64878 .76097 .66197 74953 .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 76078 .68218 74934 .G7516 .73767 .68793 .72577 .70049 .71366 32 29 .64923 76059 .66240 74915 .G7538 .73747 .68814 .72557 .70070 .71345 31 30 .64945 76041 .66262 74896 .67559 .73728 .68835 .72537 .70091 .71325 30 31 .64967 76022 .66284 74876 .67580 .73708 .68857 .72517 .70112 .71305 29 32 .64989 76003 .66306 74857 .67602 .73688 .68878 .72497 70132 .71284 28 33 .65011 759841 .66327 74838 .67623 .73669 .68899 .72477 .70153 .71264 27 34 .65033 75965 .66349 74818 .67645 .73649 .68920 .72457 .70174 .71243 26 35 .65055 75946 .66371 74799 .67666 .73629 .68941 .72437 .70195 .71223 35 36 .65077 75927 .66393 74780 .67688 .73610 .68962 .72417 .70215 .71203 24 37 .65100 75908 .66414 74760 .67709 .73590 .68983 .72397 .70236 .71182 23 38 .65122 75889 .66436 74741 .67730 73570 .69004 .72377 .70257 .71162 22 39 .65144 75870 .66458 74722 .67752 73551 .69025 .72357 .70277 .71141 21 40 .65166 75851 .66480 74703 .67773 .73531 .69046 .72337 .70298 .71121 20 41 .65188 75832 .66501 74683 .67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 75813 .66523 74664 .67816 .73491 .69088 .72297 .70339 .71080 18 43 .65232 75794 .66545 74644 .67837 .73472 .69109 .72277 .70360 .71059 17 44 .65254 75775 .66566 74625 .67859 .73452 .69130 .72257 .70381 .71039 16 45 .65276 75756 .66588 74606 .67880 .73432 .69151 .72236 .70401 .71019 15 46 .65298 75738 .66610 74586 .67901 .73413 .69172 .72216 .70422 .70998 14 47 .65320 75719 .66632 74567 .67923 .73393 .69193 .72196 .70443 .70978 13 48 .65342 75700 .66653 74548 .67944 .73373 .69214 .72176 .70463 .70957 12 49 .65364 75680 .66675 74528 .67965 .73353 .69235 .72156 .70484 .70937 11 50 .65386 75661 .66697 74509 | .67987 .73333 .69256 .72136 .70505 .70916 10 51 .65408 75642 .66718 74489! .68008 .73314 .69277 .72116 .70525 .70896 9 52 .65430 75623 .66740 74470 i .68029 .73294 .69298 .72095 .70546 .70875 8 53 .65452 75604 .66762 74451 .68051 .73274 .69319 72075 .70567 .70855 7 54 .65474 75585 .66783 74431 .68072 .73254 .69340 72055 .70587 .70834 6 55 .65496 .75566 .66805 74412 .68093 .73234 .69361 72035 .70608 .70813 5 56 .65518 .75547! .66827 .743921 .68115 .73215 .69382 .72015 .706281.70793 4 57 .65540 .75528 .66848 .74373! .68136 .73195 .69403 .71995 .70649 .70772 3 58 .65562 .75509 .66870 74353 .68157 .73175 .69424 .71974 .70670 .70752 2 59 .65584 .75490 .66891 . 74334 l .68179 .73155 .69445 .71954 .70690 .70731 1 60 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 .70711 .70711 ' Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine f 49 48 47 46 45 NATURAL TANGENTS AND COTANGENTS. 1 I! 2 |l 3 Tang Cotang Tang Cotang Tang Cotang Tang I Cotang .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 (-0 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1064 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 7 .00204 491.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2677 52 9 .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 51 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47.7395 .03842 26.0307 .05591 17.8863 48 13 .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 40 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 10 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.0661 .04016 24.8978 .05706 17.3432 42 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 156.259 .02386 41.9158 .04133 24.1C57 .05883 16.9990 38 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 2(; .00756 132.219 ,02502 39.9G55 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 33 28 .00815 122.774 .02560 39.0568 .04308 23.2137 .00058 16.5075 32 29 .00844 118.540 .02589 38.G177 .04337 23.0577 .00087 16.4283 31 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 28 33 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 26 35 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 86 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 88 .01105 90.4633 .02851 35.0695 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 42 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 10 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 15 40 .01338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 7 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .00817 14.6685 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 50 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 4 57 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 60 .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 f Cotang Tang Cotang Tang Cotang Tang Cotang | Tang / 89 88 \\ 87 86 75 NATURAL TANGENTS AND COTANGENTS. 4 5 6 7 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 00 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12481 59 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .12338 8.10536 58 3 .07080 14.1235 .08837 11.3163 .10599 9.43515 .12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 .12397 8.06674 56 6 .07139 14.0079 .08895 11.2417 .10657 9.38307 .12426 8.04756 55 6 .07168 13.9507 .08925 11.2048 .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 .12515 7.99058 52 8 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 51 10 .07'285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 7.93438 49 12 .07344 13.6174 .09101 10.9882 .108G3 9.20516 .12033 7.91582 48 13 .07373 13.5634 .09130 10.9529 .10803 9.18C28 .12602 7.89734 47 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12092 7.87895 40 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12722 7.86064 45 16 .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 7.84242 44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 43 18 .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 7.80622 42 10 .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 30 22 .07636 13.0958 .09394 10.6450 .11153 8.90227 .12929 7.73480 38 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .C9453 10.5789 .11217 8.91520 .12988 7.69957 30 25 .07724 12.9469 .C9482 10.5462 .11246 8.89185 .13017 7.68208 35 20 .07753 12.8981 .C9511 10.5136 .11270 8.86862 .13047 7.66466 : ; 1 27 .07783 12.8496 .C9541 10.4813 .11305 8.84551 .13076 7.64732 88 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.63005 32 29 .07841 12.7536 .C96CO 10.4172 .11864 8.79964 .13136 7.61287 31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.59575 30 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 20 32 .07929 12.6124 .C9GC8 10.3224 .11452 8.73172 .12224 7.50176 33 .07958 12.5660 .C9717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 20 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.49465 24 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 38 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 39 .08134 12.2946 .C9893 10.1080 .11659 8.57718 .13432 7.44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 .11718 T. 53402 .13491 7.41240 10 42 .08221 12.1632 .C0981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .C8251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11800 8.47007 .13580 7.36389 10 45 .08309 12.0346 .10069 9.93101 .11830 8.44896 .13609 7.34786 15 40 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13609 7.31600 13 48 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 7.30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 50 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 55 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 7.19125 5 56 .08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .121CO 8.20352 .13965 7.16071 3 58 .08690 11.5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 60 .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 85 84 83 82 NATURAL TANGENTS AND COTANGENTS. 8 9 10 11 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 60 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 56 5 .14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 54 7 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 52 9 .14321 6.98268 .16107 6.20851 I .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 1 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.04267 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10G64 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 J2 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 \ r '3 .14737 6.78564 .16525 6.05143 .18323 5.45751 I .20133 4.96690 37 * .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 36 .14796 6.75838 .16585 6.02962 .18384 5.43966 .20194 4.95201 35 .26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 ;28 .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18504 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.878S2 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5.83151 .18775 5.32631 .20588 4.85727 22 39 .15213 6.57339 .17004 5.83114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 6.54777 .17063 5. 86051 .18865 5.30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15511 6.44720 .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 10 01 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 7 54 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6.36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72490 3 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.32566 .17603 5.68094 .19408 5.15256 .21225 4.71137 1 C>0 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 t Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 81 80 79 78 77 NATURAL TANGENTS AND COTANGENTS. 12 13 14 1 15 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .21256 4.70463 .23087 4.33148 .24933 4.01078 .26795 3.73205 60 1 .21286 4.69791 .23117 4.32573 .24964 4.00582 .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.00086 .26857 3.72338 58 3 .21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 6 .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .2159 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.68061 48 13 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 14 .21682 4.61219 .23516 4.25239 .25366 3.94232 .27232 3.67217 40 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.66796 45 16 .21743 4.59927 .23578 4.24132 .25428 3.93271 .27294 3.66376 41 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27326 3.65957 43 18 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21834 4.58001 .23670 4.22481 .25521 3.01839 .27388 3.65121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 2:3 .21925 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 88 23 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27513 3.63461 37 24 .21986 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 36 25 .22017 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 20 .22047 4.53568 .23885 4.18675 .25738 3.88536 .27607 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.88068 .27638 3.61814 33 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 20 .22139 4.51693 .23977 4.17064 .25831 3.87136 .27701 3.60996 31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 2> .45643 2.19092 .47769 2.09341 .43931 2.00277 .52131 1.91826 28 33 ,45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47912 2.08716 .50076 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.03405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07'630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 .50368 1.98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 1.98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 1.98110 .52687 1.8C801 13 48 .46206 2.16420 .48342 2.06860 .50514 1.97966 .52724 1.89667 12 49 .46242 2.16255 .48378 2.06706 .50550 1.97823 .52761 1.89533 11 50 .46277 2.16090 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46383 2.15596 .48521 2.06094 .50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 5 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 .53022 1.88602 4 57 .46525 2.14940 .48665 2.05485 | .50843 1.96685 .53059 1.88469 3 58 .46560 2.14777 .48701 2.05333 1 .50879 1.96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 1.96402 .53134 1.88205 1 GO .46631 2.14451 .48773 2.05030 1 .50953 1.96261 .53171 1.88073 / Cotang Tang Cotang Tang i Cotang Tang Cotang Tang f 65 64 63 62 81 NATURAL TANGENTS AND COTANGENTS. 2 go 2 9 3 3 i Tang Cotang Tang Cotang Tang Cotang Tang Cotang .53171 .88073 .55431 1.80405 .57735 1.73205 .60086 1.66428 60 1 .53208 .87941 .55469 .80281 .57774 1.73089 .60126 1.66318 59 2 .53246 .87'809 .55507 .80158 .57813 1.72973 .60165 1.66209 58 3 .53283 .87677 .55545 .80034 .57851 1.72857 .60205 1.66099 57 4 .53320 .87546 .55583 .79911 .57890 1.72741 .60245 1.65990 56 5 .53358 .87415 .55621 .79788 .57929 1.72625 .60284 1.65881 55 6 .53395 .87283 .55659 .79665 .57968 1.72509 .60324 1.65772 54 7 .53432 .87152 .5569? .79542 .58007 1.72393 .60364 1.65663 53 8 .53470 .87021 .55736 .79419 .58046 1.72278 .60403 1.65554 52 9 .53507 1.86891 .55774 1.79296 .58085 1.72163 .60443 1.65445 51 10 .53545 1.86760 .55812 1.79174 .58124 1.72047 .60483 1.65337 50 11 .53582 1.86630 .55850 1.79051 .58162 1.71932 .60522 1.65228 49 12 .53620 1.86499 .55888 1.78929 .58201 1.71817 .60562 1.65120 48 13 .53657 1.86369 .55926 1.78807 .58240 1.71702 .60602 1.65011 47 14 .53694 1.86239 .55964 1.78685 .58279 1.71588 .60642 1.64903 46 15 .53732 1.86109 .5G003 1.78563 I .58318 1.71473 .60681 1.64795 45 16 .53769 1.85979 .56041 1.78441 .58357 1.71358 .00721 1.64687 44 17 .53807 1.85850 .50079 1.78319 .58396 1.71244 .60761 1.64579 43 18 .53844 1.85720 .50117 1.78198 .58435 1.71129 .60801 1.64471 42 19 .53882 1.85591 .50156 1.78077 .50474 1.71015 .60841 1.64363 41 20 .53920 1.85462 .56194 1.77955 ! .58513 1.70901 .60881 1.64256 40 21 .53957 1.85333 .56232 1.77834 .58552 1.70787 .60921 1.64148 39 .53995 1.85204 .50270 1.77713 .58591 1.70673 .60960 1.64041 38 23 .54032 1.85075 .50309 1.77592 .58631 1.70500 .61000 1.63934 37 24 .54070 1.84946 .50347 1.77471 .58670 1.70446 .61040 1.63826 36 25 .54107 1.84818 .50385 1.77351 .58709 1.70332 .61080 1.63719 35 20 .54145 1.84689 .56424 1.77230 .58743 1.70219 .61120 1.63612 34 27 .54183 1.84561 .50462 1.77110 .58787 1.70106 .61160 1.63505 33 28 .54220 1.84433 .50501 1.76990 .58826 1.69992 .61200 1.63398 32 29 .54258 1.84305 .5C5S9 1.70869 .58865 1.69879 .61240 1.63292 31 30 .54296 1.84177 .56577 1.76749 .58905 1.69766 .61280 1.63185 30 31 .54333 1.84049 .56616 1.76629 .58944 1.69653 .61320 1.63079 29 32 .54371 1.83922 .50654 1.70510 .58983 1.69541 .61360 1.62972 28 33 .54409 1.83794 .50693 1.7G390 .59022 1.69428 .61400 1.62866 27 34 .54446 1.83667 .56731 1.76271 .59061 1.69316 .61440 1.62760 26 35 .54484 1.83540 .50769 1.76151 .5910JL 1.69203 .61480 1.62654 25 36 .54522 1.83413 .56808 1.76032 .59140 1.69091 .61520 1.62548 24 37 .54560 1.83286 .56846 1.75913 .59179 .68979 .61561 1.62442 23 38 .54597 1.83159 .56885 1.75794 .59218 .68806 .61601 1.62336 22 39 .54635 1.83033 .50923 1.75675 .59258 .68754 .61641 1.62230 21 40 .54673 1.82906 .56962 1.75556 .59297 .68643 .61681 1.62125 20 41 .54711 1.82780 .57000 1.75437 .59336 .68531 .61721 1.62019 19 42 .54748 1.82654 .57039 1.75319 .59376 .68419 .61761 1.61914 18 43 .54786 1.82528 .57078 1.75200 .59415 .68308 .61801 1.61808 17 44 .54824 1.82402 .57116 1.75082 .59454 .68196 .61842 1.61703 16 45 .54862 1.82276 .57155 1.74964 .59494 .68085 .61882 1.61598 15 46 .54900 1.82150 .57193 1.74846 .59533 .67974 .61922 1.61493 14 47 .54938 1.82025 .57232 1.74728 .59573 1.67863 .61962 1.61388 13 48 .54975 1.81899 .57271 1.74610 .59612 1.67752 .62003 1.61283 12 49 .55013 1.81774 .57309 1.74492 .59651 1.67641 .62043 1.61179 11 50 .55051 1.81649 .57348 1.74375 .59691 1.67530 .62083 1.61074 10 51 .55089 1.81524 .57386 1.74257 .59730 1.67419 .62124 1.60970 9 52 .55127 1.81399 .57425 1.74140 .59770 1.67309 .62164 1.60865 8 53 .55165 1.81274 .57464 1.74022 .59809 1.67198 .62204 1.60761 7 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 .62245 1.60657 6 55 .55241 1.81025 .57541 1.73788 .59888 1.66978 .62285 1.60553 5 56 .55279 1.80901 .57580 1.73671 .59928 1.66867 .62325 1.60449 4 57 .55317 1.80777 .57619 1.73555 .59967 1.66757 .62366 1.60345 3 58 .55355 1.80653 .57657 1.73438 .60007 1.66647 .62406 1.60241 2 59 .55393 1.80529 .57696 1.73321 .60046 1.66538 .62446 1.60137 1 60 .55431 1.80405 .57735 1.73205 .60086 1.66428 .62487 1.60033 / Cotang Tang j Cotang Tang Cotang | Tang Cotang Tang / 6 _ , j 6 1 5 3 5 3 NATURAL TANGENTS AND COTANGENTS. 32 33 34 35 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .62487 1.60033 .64941 1.53986 .67451 1.48256 .70021 1.42815 00 1 .62527 1.59930 .64982 1.53888 .67493 1.48163 .70064 1.42726 59 2 .62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3 .62608 1.59723 .65065 1.53693 .67578 1.47977 .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 50 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 6 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7 .62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 52 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 .62933 1.58900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 .62973 1.58797 .65438 1.52816 ! .67900 1.47146 .70542 1.41759 48 3 .63014 1.58695 .65480 1.52719 .68002 1.47053 .70586 1.41672 47 14 .63055 1.58593 .65521 1.52622 .68045 1.40902 .70629 1.41584 40 15 .63095 1.58490 .65563 1.52525 1 .68088 1.46870 .70673 1.41497 45 16 .63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1.41409 44 17 .63177 1.58286 .65646 1.52332 .68173 1.46686 .70760 1.41322 48 18 .63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.52139 .68258 1.46503 .70848 1.41148 41 SO .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.51946 .68343 1.46320 .70935 1.40974 39 22 .63380 1 . 57778 .65854 1.51850 .08386 1.46229 .70979 1.40887 :;S 23 .63421 1.57676 .65896 1.51754 .68429 1.40137 .71023 1.40800 37 24 .63462 1.57575 .65938 1.51058 .68471 1.4G046 .71066 1.40714 3i J 25 .63503 1.57474 .65980 1.51502 .68514 1.45955 .71110 1.40627 35 26 .63544 1.57372 .66021 1.51,466 I .68557 1.45864 .71154 1.40540 ;4 27 .63584 1.57271 .06063 1.51370 .68600 1.45773 .71198 1.40454 33 28 .63625 1.57170 .06105 1.51375 .68642 1.45682 .71242 1.40367 :5-3 29 .63666 1.57069 .00147 1.51179 .68685 1.45592 .71285 1.40281 31 30 .63707 1.56969 .00189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1 .45410 .71373 1.40109 29 32 .63789 1.50707 .GG27'2 1.50893 .68814 1.45320 .71417 1.40022 :JS 33 .63830 1.50667 .06314 1.50797 .68857 1.45229 .71461 1.39936 27 34 .63871 1.56566 .00356 1.50702 .68900 1.45139 .71505 1.39850 20 35 .63912 1.56466 .66398 1.50007 .68942 1.45049 .71549 1.39764 25 36 .63953 1.56366 .66440 1.50512 .08985 1.44958 .71593 1.39679 21 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .G4035 1.56165 .66524 1.50322 .09071 1.44778 .71681 1.39507 122 39 .64076 1.56065 .66506 1.50228 .69114 1.44688 .71725 1.39421 i21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .GGG92 1.49944 .69243 1.44418 .71857 1.39165 il8 43 .64240 1.55666 .06734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.38994 116 45 .64322 1.55467 .66818 1.49661 .69372 1.44149 .71990 1.38909 115 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 '14 47 .64404 1.55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69502 1.43881 .72122 1.38653 12 49 .64487 1.55071 .06986 1.49284 .69545 1.43792 .72167 1.38568 ill 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 52 .64610 1.54774 i .07113 1.49003 .69675 1.43525 .72299 1.38314 8 53 .64652 1.54675 .07155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64693 1.54576 ! .07197 1.48816 .69761 1.43347 .72388 1.38145 6 55 .64734 1.54478 i .67239 1.48722 .69804 1.43258 .72432 1.38060 5 56 .64775 1.54379 .07282 1.48629 .69847 1.43169 .72477 1.37976 4 57 .64817 1.54281 .07324 1.48536 .69891 1.43080 .72521 1.37891 3 5! .64858 1.54183 ! .67366 1.48442 .69934 1.42992 . 72565 1.37807 2 59 .64899 1.54085 i .67409 1.48349 .69977 1.42903 .72610 1.37723 1 60 .64941 1.53986 i .67451 1.48256 .70021 1.42815 .72054 1.37638 Cotang j Tang Cotang Tang Cotang Tang i Cotang ^fang 9 57 56 55 i 54 88 NATURAL TANGENTS AND COTANGENTS. 36 37 38 39 Tang Cotang Tang | Cotang Tang Cotang Tang Cotang .72654 1.37638 .75355 1.32704 .78129 1.27994 .80978 1.23490 60 1 .72699 1.37554 .75401 1.32624 .78175 1.27917 .81027 1.23416 59 2 .72743 1.37470 .75447 1.32544 .78222 1.27841 .81075 1.23343 58 3 .72788 1.37386 .75492 1.32464 .78269 1.27764 .81123 1.23270 57 4 .72832 1.37302 .75538 1.32384 .78316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 .78363 1.27611 .81220 1.23123 55 6 .72921 1.37134 ! .75629 1.32224 .78410 1.27535 .81268 1.23050 54 7 .72966 1.37050 .75675 1.32144 .78457 1.27458 .81316 1.22977 53 8 .73010 1.36967 .75721 1.32064 .78504 1.27382 .81364 1.22904 52 9 .73055 1.30)883 i .75767 1.31984 .78551 1.27306 .81413 1.22831 51 10 .73100 1.36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 50 11 .73144 1.36716 .75858 1.31825 .78645 1.27153 .81510 1.22685 49 12 .73183 1.8G633 .75904 1.31745 .78692 1.27077 .81558 1.22612 48 13 .73234 1.36549 .75950 1.31666 .78739 1.27001 .81606 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 1.22394 45 16 .73368 1.36300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 17 .73413 1.36217 .76134 1.31348 .78928 1.26698 .81800 1.22249 43 18 .73457 1.36134 .76180 1.31269 .78975 1.26622 .81849 1.22176 42 19 .73503 1.30051 .76226 1.31190 .79022 1.26546 .81808 1.22104 41 20 .73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 1.21959 39 22 .73637 1.35802 .7'6364 1.30952 .79104 1.26319 .82044 1.21886 38 23 .73681 1.35719 .76410 1.S0873 .79212 1.20244 .82092 1.21814 37 24 .73726 1.35637 .76456 1.30795 .79259 1.26169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 35 26 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 27 .73861 1.35389 .76594 1.30558 .79401 1.25943 .82287 1.21526 33 28 .73906 1.35307 .76640 1.30480 .79449 1.25867 .82336 1.21454 32 29 .73951 1.35224 .76686 1.30401 .79496 1.25792 .82385 1.21382 31 30 .73996 1.35142 .76733 1.30323 .79544 1.25717 .82434 1.21310 30 31 .74041 1.35060 .76779 1.30244 .79591 1.25642 .82483 1.21238 29 32 .74083 1.34978 .76825 1.30166 .79639 1.25567 .82531 1.21166 28 33 .74131 1.34896 .76871 1.30087 .70686 1.25492 .82580 1.21094 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25417 .82629 1.21023 26 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1.34160 .77289 1.29385 .80115 1.24820 .83022 1.20451 18 43 .74583 1.34079 .77335 1.29307 .80163 1.24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 .74674 1.33916 .77428 1.29152 .80258 1.24597 .83169 1.20237 15 46 .74719 1.33835 .77475 1.29074 .80306 1.24523 .83218 1.20166 14 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 112 49 .74855 1.33592 .77615 1.28842 .80450 1.24301 .83366 1 . 19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 1.19740 8 53 .75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 1.19669 7 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 .83613 1.19599 6 55 .75128 1.33107 .77895 1.28379 .80738 1.23858 .83662 1.19528 5 56 .75173 1.33026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 1.19387 3 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .83811 1.19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.23563 .83860 1 . 19246 1 60 .75355 1.32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 / Cotang Tang Cotang Tang Cotang Tang ; Cotang Tang t 53 52 51 50 84 NATURAL TANGENTS AND COTANGENTS. 4 D 4 L 4 2 4 3 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .83910 1.19175 .86929 1.15037 .90040 1.11061 .93252 1.07237" 60 1 .83960 1.19105 .86980 1.14969 .90093 .10996 .93306 1.07174 59 2 .84009 1.19035 .87031 1.14902 .90146 .10931 .93360 1.07112 58 3 .84059 1.18964 .87082 1.14834 .90199 .10867 .93415 1.07049 57 4 .84108 1.18894 .87133 1.14767 .90251 .10802 .93469 1.06987 56 5 .84158 1.18824 .87184 1.14699 .90304 .10737 .93524 1.06925 55 6 .84208 1.18754 .87236 1.14632 .90357 .10672 .93578 1.06862 54 7 .84258 1.18684 .87287 1.14565 .90410 .10607 .93633 1.06800 53 8 .84307 1 . 18614 .87338 1.14498 .90463 .10543 .93688 1.06738 52 9 .84357 1.18544 .87389 1.14430 .90516 .10478 .93742 1.06676 51 10 .84407 1.18474 .87441 1.14363 .90569 .10414 .93797 1.06613 50 11 .84457 1.18404 .87492 .14296 .90621 .10349 .93852 1.06551 49 12 .84507 1.18334 .87543 .14229 .90674 .10285 .93906 1.06489 4H 13 .84556 1 . 18264 .87595 .14162 .90727 .10220 .93961 1.06427 47 14 .84606 1.18194 .87646 .14095 .90781 .10156 .94016 1.06365 40 15 .84656 1.18125 .87698 .14028 .90834 .10091 .94071 1.06303 45 16 .84706 1.18055 .87749 .13961 .90887 .10027 .94125 1.06241 41 17 .84756 1 . 17986 .87801 1.13894 .90940 .09963 .94180 1.06179 43 18 .84806 1.17916 .87852 1.13828 .90993 .09899 .94235 1.06117 42 19 .84856 1.17846 .87904 1.13761 .91046 .09834 .94290 1.06056 41 20 .84906 1.17777 .87955 1.13694 .91099 .09770 .94345 1.05994 40 2i .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 1.05932 39 22 .85006 1.17638 .88059 1.13561 .91206 1.09642 .94455 1.05870 88 23 .85057 1.17569 .88110 1.13494 .91259 1.09578 .94510 1.05809 87 24 .85107 1 . 17500 .88162 1.13428 .91313 1.09514 .94565 1.05747 36 25 .85157 1.17430 .88214 1.13361 .91366 1.09450 .94620 1.05685 36 26 .85207 1.17361 .88265 1.13295 .91419 1.09386 .94676 1.05624 34 27 .85257 1.17292 .88317 1.13228 .91473 1.09322 .94731 1.05562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 1.05501 :52 29 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 1.05439 31 30 .85408 1.17085 .88473 1.13029 .91633 1.09131 .94896 1.05378 30 31 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 1.C5317 29 32 .85509 1.16947 .88576 1.12897 .91740 1.09003 .95007 1.05255 28 33 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 . .05194 27 34 .85609 1.16809 .88680 1.12765 .91847 1.08876 .95118 .05133 20 35 .85660 1.16741 .88732 1.12699 .91901 .08813 .95173 .05072 35 36 .85710 1.16672 .88784 1.12633 .91955 .08749 .95229 .05010 24 37 .85761 1.16603 .88836 1.12567 .92008 .08686 .95284 .04949 23 38 .85811 1.16535 .88888 1.12501 .92062 1.08622 .95340 .04888 22 39 .85862 1.16466 .88940 1.12435 .92116 1.08559 .95395 .04827 Ul 40 .85912 1.16398 .88992 1.12369 .92170 1.08496 .95451 .04766 20 41 .85963 1.16329 | .89045 1.12303 .02224 1.08432 .95506 .04705 10 42 .86014 1.16261 .89097 1.12238 .92277 1.08369 .95562 .04644 18 43 .86064 1.16192 .89149 1.12172 .92331 1.08306 .95618 .04583 17 44 .86115 1.16124 .89201 1.12106 .92385 1.08243 .95673 .04522 16 45 .86166 1.16056 .89253 1.12041 .92439 1.08179 .95729 .04461 15 46 .86216 1.15987 .89306 1.11975 .92493 1.08116 .95785 .04401 It 47 .86267 1.15919 .89358 1.11909 .92547 1.08053 .95841 .04340 18 48 .86318 1.15851 .89410 1.11844 .92601 1 -07990 .95897 .04279 12 49 .86368 1.15783 .89463 1 11778 .92655 1.07927 .95952 .04218 11 50 .86419 1.15715 .89515 1.11713 .92709 1.07864 .96008 .04158 10 51 .86470 1.15647 .89567 1.11648 .92763 1.07801 .96064 .04097 9 52 .86521 1.15579 .89620 1.11582 .92817 1.07738 .96120 .04036 8 53 .86572 1.15511 .89672 1.11517 .92872 1.07676 .96176 .03976 7 54 .86623 1.15443 .89725 1.11452 .92926 1.07613 .96232 .03915 6 55 .86674 1.15375 .89777 1.11387 .92980 1.07550 .96288 .03855 5 56 .86725 1.15308 .89830 1.11321 .93034 1.07487 .96344 .03794 4 57 .86776 1.15240 .89883 1.11256 .93088 1.07425 .96400 .03734 3 58 .86827 1.15172 .89935 1.11191 .93143 1.07362 .96457 .03674 2 59 .86878 1.15104 .89988 1.11126 .93197 1.07299 .96513 1.03613 1 60 .86929 1.15037 .90040 1.11061 .93252 1.07237 .96569 1.03553 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang 4 t9 4 8 4 7 4 6 85 NATURAL TANGENTS AND COTANGENTS. t 4 4 ^ 14 1 4 4 Tang Cotang Tang Cotang Tang Cotang .96569 1.03553 60 20 .97700 1.02355 40 (40 .98843 1.01170 30 1 .96625 1.03493 59 21 .97756 1.02295 39 41 .98901 : .01112 u 2 .96681 1.03433 58 22 .97813 1.02236 88 i42 .98958 .01053 18 3 .96738 1.03372 57 23 .97870 1.02176 37 43 .99016 .00994 17 4 .96794 1.03312 56 24 .97927 1.02117 88 44 .99073 .00935 16 5 .96850 1.G3252 55 25 .97984 1.02057 ,35 45 .99131 .00876 15 6 .96907 1.03192 54 26 .98041 1.01998 84 I 46 .99189 .00818 14 7 .96963 1.03132 53 27 .98098 1.01939 33 47 .99247 : .00759 13 8 .97020 1.03072 52 28 .98155 1.01879 32 48 .99304 .00701 12 9 .97076 1.03012 51 29 .98213 1.01820 81 i4D .99362 : .00642 11 10 .97133 1.02952 50 30 .98270 1.01761 30 50 .99420 .00583 10 11 .97189 1.02892 49 31 .98327 1.01702 29 51 .99478 .00525 9 12 .97246 1.02832 48 32 .98384 1.01642 28 52 .99536 .00467 8 13 .97302 1.02772 47 33 .98441 1.01583 27 53 .99594 .00408 7 14 .97359 1.02713 46 34 .98499 1.01524 20 54 .99652 : .00350 6 15 .97416 1.02653 45 35 .98556 1.01465 25 55 .99710 .00291 5 16 .97472 1.02593 44 86 .98613 1.01406 24 56 .99768 .00233 4 1? .97529 1.02533 43 37 .98671 1.01347 3s 57 .99826 .00175 3 18 .97586 1.02474 42 38 .98728 1.01288 22 58 .99884 .00116 2 19 .97643 1.02414 41 39 .98786 1.01229 21 59 .99942 1.00058 1 20 .97700 1.02355 40 40 .98843 1.01170 20 60 1.00000 1.00000 Cotang Tang Cotang Tang / Cotang Tang / 4 5 4 5 4 5 NATURAL VERSED SINES AND EXTERNAL SECANTS. 1 2 3 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .00000 .00000 .00015 .00015 .00061 .00061 .00137 .00137 1 .00000 .00000 .00016 .00016 .00062 .00062 .00139 00139 1 2 .00000 .00000 .00016 .00016 .00063 .00063 .00140 .00140 2 3 .00000 .00000 .00017 .00017 .00064 .00064 .00142 .00142 3 4 .00000 .00000 .00017 .00017 .00065 .00065 .00143 .00143 4 5 .00000 .00000 .00018 .00018 .00066 .00066 .00145 .00145 5 6 .00000 .00000 .00018 .00018 .00067 .00067 .00146 .00147 6 7 .00000 .00000 .00019 .00019 .00068 .00068 .00148 .00148 7 8 .00000 .00000 .00020 .00020 .00069 .00069 .00150 .00150 8 9 .00000 .00000 .00020 .00020 .00070 .00070 .00151 .00151 9 10 .00000 .00000 .00021 .00021 .00071 .00072 .00153 .00153 10 11 .00001 .00001 .00021 .00021 .00073 .00073 .00154 .00155 11 12 .00001 .00001 .00022 .00022 .00074 .00074 .00156 .00156 12 13 .00001 .00001 .00023 .00023 .00075 .00075 .00158 .00158 13 14 .00001 .00001 .00023 .00023 .00076 .00076 .00159 .00159 14 15 .00001 .00001 .00024 .00024 .00077 .00077 .00161 .00161 15 16 .00001 .00001 .00024 .00024 .00078 .00078 .00162 .00163 16 17 .00001 .00001 .00025 .00025 .00079 .00079 .00164 .00164 17 18 .00001 .00001 .00026 .00026 .00081 .00081 .00166 .00166 18 19 .00002 .00002 .00026 .00026 .00082 .00082 .00168 .00168 19 20 .00002 .00002 .00027 .00027 .00083 .00083 .00169 .00169 20 21 .00002 .00003 .00028 .00028 .00084 .00084 .00171 .00171 21 22 .00002 .00002 .00028 .00028 .00085 .00085 .00173 .00173 22 23 .00002 .00002 .00029 .00029 .00087 .00087 .00174 .00175 23 24 .00002 .00002 .00030 .00030 .00088 .00088 .00176 .00176 24 25 .00003 .00003 .00031 .00031 .00089 .00089 .00178 .00178 25 23 .00003 .00003 .00031 .00031 .00090 .00090 .00179 .00180 26 27 .00003 .00003 .00032 .00032 .00091 .00091 .00181 .00182 27 28 .00003 .00003 .00033 .00033 .00093 .00093 .00183 .00183 28 23 .00004 .00004 .00034 .00034 .00094 .00094 .00185 .00185 29 30 .00004 .00004 .00034 .00034 .00095 .00095 .00187 .00187 30 31 .00004 .00004 .00035 .00035 .00096 .00097 .00188 .00189 31 32 .00004 .00004 .00036 .00036 .00098 .00098 .00190 .00190 32 S3 .00005 .00005 .00037 .00037 .00099 .00099 .00192 .00192 33 34 .00005 .00005 .00037 .00037 .00100 .00100 .00194 .00194 34 35 .00005 .00005 .00038 .00038 .00102 .00102 .00196 .00196 35 3G .00005 .00005 .00039 .00039 .00103 .00103 .00197 .00198 36 37 .00006 .00006 .00040 .00040 .00104 .00104 .00199 .00200 37 38 .00006 .00006 .00041 .00041 .00106 .00106 .00201 .00201 38 39 .00006 .00006 .00041 .00041 .00107 .00107 .00203 .00203 39 40 .00007 .00007 .00042 .00042 .00108 .00108 .00205 .00205 40 41 .00007 .00007 .00043 .00043 .00110 .00110 .00207 .00207 41 42 .00007 .00007 .00044 .00044 .00111 .00111 .00208 .00209 42 43 .00008 .00008 .00045 .00045 .00112 .00113 .00210 .00211 43 44 .00008 .00008 .00046 .00046 .00114 .00114 .00212 .00213 44 45 .00009 .00009 .00047 .00047 .00115 .00115 .00214 .00215 45 46 .00009 .00009 .00048 .00048 .00117 .00117 .00216 .00216 46 47 .00009 .00009 .00048 .00048 .00113 .00118 .00218 .00218 47 48 .00010 .00010 .00049 .00049 .00119 .00120 .00220 .00220 48 49 .00010 .00010 .00050 .00050 .00121 .00121 .00222 .00222 49 50 .00011 .00011 .00051 .00051 .00122 .00122 .00224 .00224 50 51 .00011 .00011 .00052 .00052 .00124 .00124 .00226 .00226 51 52 .00011 .00011 .00053 .00053 .00125 .00125 .00228 .00228 52 53 .00012 .00012 .00054 .00054 .00127 .00127 .00230 .00230 53 54 .00012 .00012 .00055 .00055 .00128 .00128 .00232 .00232 54 55 .00013 .00013 .00056 .00056 .00130 .00130 .00234 .00234 55 56 .00013 .00013 .00057 .00057 .00131 .00131 .00236 .00236 56 57 .00014 .00014 .00058 .00058 .00133 .00133 .00238 .00238 57 58 .00014 .00014 .00059 .00059 .00134 .00134 .00240 .00240 58 59 .00015 .00015 .00060 .00060 .00136 .00136 .00242 .00242 69 60 .00015 .00015 .00061 .00061 .00137 .00137 .00244 .00244 60 NATURAL VERSED SINES AND EXTERNAL SECANTS / 4 5 e a 7 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .00244 .00244 ~0038T .00382 .00548 .00551 .00745 .00751 1 .00246 .00246 .00383 .00385 .00551 .00554 .00749 .00755 1 2 .00248 .00248 .00386 .00387 .00554 .00557 .00752 .00758 2 3 .00250 .00250 .00388 .00390 .00557 J00560 .00756 .00762 3 4 .00252 .00252 .00391 .00392 .00560 .00563 .00760 .00765 4 5 .00254 .00254 .00393 .00395 .00563 .00566 .00763 .00769 5 6 .00256 .00257 .00396 .00397 .00566 .00569 .00767 .00778 6 7 .00258 .00259 .00398 .00400 .00569 .00573 .00770 .00776 7 8 .00260 .00261 .00401 .00403 .00572 .00576 .00774 .00780 8 9 .00262 .00263 .00404 .00405 .00576 .00579 .00778 .00784 9 10 .00264 .00265 .00406 .00408 .00579 .00582 .00781 .00787 10 11 .00266 .00267 .00409 .00411 .00582 .00585 .00785 .00791 11 12 .00269 .00269 .00412 .00413 .00585 .00588 .00789 .00795 12 13 .00271 .00271 .00414 .00416 .00588 .00592 .00792 .00799 13 14 .00273 .00274 .00417 .00419 .00591 .00595 .00796 .00802 14 15 .00275 .00276 .00420 .00421 .00594 ..00598 .00800 .00806 15 16 .00277 .00278 .00422 .00424 .00598 .00601 .00803 .00810 16 17 .00279 .00280 .00425 .00427 .00601 .00604 .00807 .00813 17 18 .00281 .00282 .00428 .00429 .00604 .00608 .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 .00814 .00821 19 20 .00286 .00287 .00433 .00435 .00610 .00614 .00818 .00825 20 21 .00288 .00289 .00436 .00438 .00614 .00617 .00822 .00828 21 22 .00290 .00291 .00438 .00440 .00617 .00621 .00825 .00832 22 23 .00293 .00293 .00441 .00443 .00620 .00624 .00829 .00836 23 24 .00295 .00296 .00444 .00446 .00623 .OOG27 .00833 .00840 24 25 .00297 .00298 .00447 .00449 .00626 .00630 .00837 .00844 25 26 .00299 .00300 .00449 .00451 .00630 .C0634 .C0840 .00848 26 27 .00301 .00302 .00452 .00454 .00633 .00637 .00844 .00851 27 28 .00304 .00305 .00455 .00457 .00636 .00640 .00848 .00855 28 29 .00306 .00307 .00458 .00460 .00640 .00644 .00852 .00859 29 80 .00308 .00309 .00460 .00463 .00643 .00647 .00856 .00863 30 31 .00311 .00312 .00463 .00465 .00646 .00650 .00859 .00867 31 82 .00313 .00314 .00466 .00468 .00649 .00654 .00863 .00871 32 33 .00315 .00316 .00469 .00471 .00653 .00657 .00867 .00875 33 84 .00317 .00313 .00472 .00474 .00656 .00660 .00871 .00878 34 35 .00320 .00321 .00474 .00477 .00659 .00664 .00875 .00882 35 36 .00322 .00323 .00477 .00480 .00663 .00667 .00878 .00886 36 37 .00324 .00326 .00480 .00482 .00666 .00671 .00882 .00890 37 38 .00327 .00328 .00483 .00485 .00669 .00674 .00886 .00894 38 39 ' .00329 .00330 .00486 .00488 .00673 .00677 .00890 .00898 39 40 .00332 .00338 .00489 .00491 .00676 .00681 .00894 .00902 40 41 .00334 .00335 .00492 .00494 .00680 .00684 .00898 .00906 41 42 .00336 .00337 .00494 .00497 .00683 .00688 .00902 .00910 42 43 .00339 .00340 .00497 .00500 .00686 .00691 .00906 .00914 43 44 .00341 .00342 .00500 .00503 .00690 .00695 .00909 .00918 44 45 .00343 .00345 .00503 .00506 .00693 .00698 .00913 .00922 45 46 .00346 .00347 .00506 .00509 .00697 .00701 .00917 .00926 46 47 .00348 .00350 .00509 .00512 .00700 .00705 .00921 .00930 47 48 .00351 .00352 .00512 .00515 .00703 .00708 .00925 .00934 48 49 .00353 .00354 .00515 .00518 .00707 .00712 .00929 .00938 49 50 .00356 .00357 .00518 .00521 .00710 .00715 .00933 .00942 50 51 .00358 .00359 .00521 .00524 .00714 .00719 .00937 .00946 51 52 .00361 .00362 .00524 .00527 .00717 .00722 .00941 .00950 52 53 .00363 .00364 .00527 .00530 .00721 .00726 .00945 .00954 53 54 .00365 .00367 .00530 .00533 .00724 .00730 .00949 .00958 54 55 .00368 .00369 .00533 .00536 .00728 .00733 .00953 .00962 55 56 .00370 .00372 .00536 .00539 .00731 .00737 .00957 .00966 56 57 .00373 .00374 .00539 .00542 .00735 .00740 .00961 .00970 57 58 .00375 .00377 .00542 .00545 .00738 .00744 .00965 .00975 58 59 .00378 .00379 .00545 .00548 .00742 .00747 .00969 .00979 59 60 .00381 .00382 .00548 .00551 .00745 .00751 1 .00973 .00983 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. 8 9 10 11 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. ] Ex. sec. .00973 .00983 .01231 .01247 .01519 .01543 .01837 .01872 1 .00977 .00987 .01236 .01251 .01524 .01548 .01843 .01877 1 2 .00981 .00991 .01240 .01256 .01529 .01553 .01&48 .01883 2 3 .00985 .00995 ' .01245 .01261 .01534 .01558 .01854 .01889 3 4 .00989 .00999 .01249 .01265 .01540 .01564 .01860 .01895 4 5 .00994 .01004 .01254 .01270 .01545 .01569 .01865 .01901 5 6 .00998 .01008 .01259 .01275 .01550 .01574 .01871 .01906 6 7 .01002 .01012 .01263 .01279 .01555 .01579 .01876 .01912 7 8 .01006 .01016 .01268 .01284 .01560 .01585 .01882 .01918 8 9 .01010 .01020 .01272 .01289 .01565 .01590 .01888 .01924 9 10 .01014 .01024 .01277 .01294 .01570 .01595 .01893 .01930 10 11 .01018 .01029 .01282 .01298 .01575 .01601 .01899 .01936 11 12 .01022 .01033 .01286 .01303 .01580 .01606 .01904 .01941 12 13 .01027 .01037 .01291 .01308 .01586 .01611 .01910 .01947 13 14 .01031 .01041 .01296 .01313 .01591 .01616 .01916 .01953 14 15 .01035 .01046 .01300 .01318 .01596 .01622 .01921 .01959 15 16 .01039 .01050 .01305 .01322 .01601 .01627 .01927 .01965 16 17 .01043 .01054 .01310 .01327 .01606 .01633 .01933 .01971 17 18 .01047 .01059 .01314 .01332 .01612 .01638 .01939 .01977 18 19 .01052 .01063 .01319 .01337 .01617 .01643 .01944 .01983 19 20 .01056 .01067 .01324 .01342 .01622 .01649 .01950 .01989 20 21 .01060 .01071 .01329 .01346 .01627 .01654 .01956 .01995 21 22 .01064 .01076 .01333 .01351 .01632 .01659 .01961 .02001 22 23 .01069 .01080 .01338 .01356 .01638 .01665 .01967 .02007 23 24 .01073 .01084 .01343 .01361 .01643 .01670 .01973 .02013 24 25 .01077 .01089 .01348 .01366 .01648 .01676 .01979 .02019 25 26 .01081 .01093 .01352 .01371 .01653 .01681 .01984 .02025 26 27 .01086 .01097 .01357 .01376 .01659 .01687 .01990 .02031 27 28 .01090 .01102 .01362 .01381 .01664 .01692 .01996 .02037 28 29 .01094 .01106 .01307 .01386 .01669 .01698 .02002 .02043 29 30 .01098 .01111 .01371 .01391 .01675 .01703 .02008 .02049 30 31 .01103 .01115 .01376 .01395 .01680 .01709 .02013 .02055 31 32 .01107 .01119 .01381 .01400 .01685 .01714 .02019 .02061 32 33 .01111 .01124 .01386 .01405 .01690 .01720 .02025 .02067 33 34 .01116 .01128 .01391 .01410 .01696 .01725 .02031 .02073 34 35 .01120 .01133 .01396 .01415 .01701 .01731 .02037 .02079 35 36 .01124 .01137 .01400 .01420 .01706 .01736 .02042 .02085 36 37 .01129 .01142 .01405 .01425 .01712 .01742 .02048 .02091 37 38 .01133 .01146 .01410 .01430 .01717 .01747 .02054 .02097 38 39 .01137 .01151 .01415 .01435 .01723 .01753 .020GO .02103 39 40 .01142 .01155 .01420 .01440 .01728 .01758 .02066 .02110 40 41 .01146 .01160 .01425 .01445 .01733 .01764 .02072 .02116 41 42 .01151 .01164 .01430 .01450 .01739 .01769 .02078 .02122 42 43 .01155 ,01169 .01435 .01455 .01744 .01775 .02084 .02128 43 44 .01159 .01173 .01439 .01461 .01750 .01781 .02090 .02134 44 45 .01164 .01178 .01444 .01466 .01755 .01786 .02095 .02140 45 46 .01168 .01182 .01449 .01471 .01760 .01792 .02101 .02146 46 47 .01173 .01187 .01454 01476 .01766 .01798 .02107 .02153 47 48 .01177 .01191 .01459 .01481 .01771 .01803 .02113 .02159 48 49 .01182 .01196 .01464 .01486 .01777 .01809 .02119 ,02165 49 50 .01186 .01200 .01469 .01491 .01782 .01815 .02125 .02171 50 51 .01191 .01205 .01474 .01496 .01788 .01820 .02131 .02178 51 52 .01195 .01209 .01479 .01501 .01793 .01826 .02137 .02184 52 53 .01200 .01214 .01484 .01506 .01799 .01832 .02143 .02190 53 54 .01204 .01219 .01489 .01512 .01804 .01837 .02149 .02196 54 55 .01209 .01223 .01494 .01517 .01810 .01843 .02155 .02203 55 56 .01213 .01228 .01499 .01522 .01815 .01849 .02161 .02209 56 57 .01218 .01233 .01504 .01527 .01821 .01854 .02167 .02215 57 58 .01222 .01237 .01509 .01532 .01826 .01860 .02173 .02221 58 59 .01227 .01242 .01514 .01537 .01832 .01866 .02179 .02228 59 60 .01231 .01247 .01519 .01543 .01837 .01872 .02185 .02234 60 NATURAL VERSED SINES AND EXTERNAL SECANTS, f 12 13 14 15 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .02185 .02234 .02563 .02630 .02970 .03061 .03407 .03528 1 .02191 .02240 .02570 .02637 .02977 .03069 .03415 .03536 1 2 .02197 .02247 .02576 .02644 .02985 .03076 i .03422 .03544 2 3 .02203 .02253 .025&3 .02651 .02992 .03084 .03430 .03552 3 4 .02210 .02259 .02589 .02658 .02999 .03091 ! .03438 .03560 4 5 .02216 .02266 .02596 .02665 .03006 .03099 j .03445 .03568 5 6 .02222 .02272 .02602 .02672 .03013 .03106 .03453 .03576 6 7 .02228 .02279 .02609 .02679 .03020 .03114 .03460 .03584 7 8 .02234 .02285 .02616 .C2686 .03027 .03121 .03468 .03592 8 9 .02240 .02291 .02622 .02693 .03034 .03129 .03476 .03601 9 10 .02246 .02298 .02629 .02700 .03041 .03137 .03483 .03609 10 11 .02252 .02304 .02635 .02707 .03048 .03144 .03491 .03617 11 12 .02258 .02311 .02642 .02714 .03055 .03152 .03498 .03625 12 13 .02265 .02317 .02649 .02721 .03063 .03159 .03506 .03633 13 14 .02271 .02323 .02655 .02728 .03070 .03167 .03514 .03642 14 15 .02277 .02330 .02662 .02735 .03077 .03175 .03521 .03650 15 16 .02283 .02336 .02669 .02742 .03084 .03182 .03529 .03658 16 17 .02289 .02343 .02675 .02749 .03091 .03190 .03537 .03666 17 18 .02295 .02349 .02682 .02756 .03098 .03198 .03544 .03674 18 19 .02302 .02356 .02689 .02763 .03106 .03205 .03552 .03683 19 20 .02308 .02362 .02696 .02770 .03113 .03213 .03560 .03691 20 21 .02314 .02369 .02702 .02777 .03120 .03221 .03567 .03699 21 22 .02320 .02375 .02709 .02784 .03127 .03228 .03575 .03708 22 23 .02327 .02382 .02716 .02791 .03134 .03236 .03583 .03716 23 24 .02333 .02388 .02722 .02799 .03142 .03244 .03590 .03724 24 25 .02339 .02395 .02729 .02806 .03149 .03251 .03598 .03732 25 26 .02345 .02402 .02736 .02813 .03156 .03259 .03006 .03741 26 27 .02352 .02408 .02743 .02820 .03163 .03267 .03614 .03749 27 28 .02358 .02415 .02749 .02827 .03171 .03275 .03621 .03758 28 29 .02364 .02421 .02756 .02834 .03178 .03282 .03629 .03766 29 30 .02370 .02428 .02763 .02842 .03185 .03290 .03637 .03774 30 31 .02377 .02435 .02770 .02849 .03193 .03298 .03645 .03783 31 32 .02383 .02441 .02777 .02856 .03200 .03306 .03653 .03791 32 33 .02389 .02448 .02783 .02863 .03207 .03313 .03660 .03799 33 34 .02396 .02454 .02790 .02870 .03214 .03321 .03668 .03808 34 35 .02402 .02461 .02797 .02878 .03222 .03329 .03676 .03816 35 36 .02408 .02468 .02804 .02885 .03229 .03337 .03684 .03825 36 37 .02415 .02474 .02811 .02892 .03236 .03345 .03692 .03833 37 38 .02421 .02481 .02818 .02899 .03244 .03353 .03699 .03842 38 39 .02427 .02488 .02824 .02907 .03251 .03360 .03707 .03850 89 40 .02434 .02494 .02831 .02914 .03258 .03368 .03715 .03858 40 41 .02440 .02501 .02838 .02921 .03266 .03376 .03723 .03867 41 42 .02447 .02508 .02845 .02928 .03273 .03384 .03731 .03875 42 43 .02453 .02515 .02852 .02936 .03281 .03392 .03739 .03884 43 44 .02459 .02521 .02859 .02943 .03288 .03400 .03747 .03892 44 45 .02466 .02528 .02866 .02950 .03295 .03408 .03754 .03901 45 46 .02472 .02535 .02873 .02958 .03303 .03416 .03762 .03909 46 47 .02479 .02542 .02880 .02965 .03310 .03424 .03770 .03918 47 48 .02485 .02548 .02887 .02972 .03318 .03432 .03778 .03927 48 49 .02492 .02555 .02894 .02980 .03325 .03439 .03786 .03935 49 50 .02498 .02562 .02900 .02987 .03333 .03447 .03794 .03944 50 51 .02504 .02569 .02907 .02994 .03340 .03455 .03802 .03952 51 52 .02511 .02576 .02914 .03002 .03347 .03463 .03810 .03961 52 53 .02517 .02582 .02921 .03009 .03355 .03471 .03818 .03969 53 54 .02584 .02589 .02928 .03017 ! .03362 .03479 .03826 .03978 54 55 .02530 .02596 .02935 .03024 .03370 .03487 .03834 .03987 55 56 .02537 .02603 .02942 .03032 .03377 .03495 .03842 .03995 56 57 .02543 .02610 .02949 .03039 .03385 .03503 .03850 .04004 57 58 .02550 .02617 .02956 .03046 .03392 .03512 .03858 .04013 58 59 .02556 .02624 .02963 .03054 .03400 .03520 .03866 .04021 59 60 .02563 .02630 .02970 .03061 .03407 .03528 .03874 .04030 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. / 16* lt a 18* 19 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex, sec. Vers. Ex. sec. .03874 .04030 .04370 .04569 .04894 .05146 .05448 .057'62 1 .03882 .04039 .04378 .04578 .04903 .05156 .05458 .05773 1 2 .03890 .04047 .04387 .04588 .04912 .05166 .05467 .05783 2 3 .03898 .04056 .04395 .04597 .04921 .05176 .05477 .05794 3 4 .03906 .04065 .04404 .04606 .04930 .05186 .05486 .05805 4 5 .03914 .04073 .04412 .04616 .04939 .05196 .05496 .05815 5 6 .03922 .04082 .04421 .04625 .04948 .05206 .05505 .05826 6 7 .03930 .04091 .04429 .04635 .04957 .05216 .05515 .05836 7 8 .03938 .04100 .04438 .04644 .04967 .05226 .05524 .05847 8 9 .03946 .04108 .04446 .04653 .04976 .05236 .05534 .05858 9 10 .03954 .04117 .04455 .04663 .04985 .05246 .05543 .05869 10 11 .03963 .04126 .04464 .04672 .04994 .05256 .05553 .05879 11 12 .03971 .04135 i .04472 .04682 .05003 .05266 .05562 .05890 12 13 .03979 .04144 ! .04481 .04691 .05012 .05276 .05572 .05901 13 14 .03987 .04152 I .04489 .04700 .05021 .05286 .05582 .05911 14 15 .03995 .04161 .04498 .04710 .05030 .05297 i .05591 .05922 15 16 .04003 .04170 .04507 .04719 .05039 .05307 I .05601 .05933 16 17 .04011 .04179 .04515 .04729 .05048 .05317 i .05610 .05944 17 18 .04019 .04188 .04524 .04738 .05057 .05327 .05620 .05955 18 19 .04028 .04197 .04533 .04748 .05067 .C5337 .05630 .05965 19 20 .04036 .04206 .04541 .04757 .05076 .05347 .05639 .05976 20 21 .04044 .04214 .04550 .04767 .05085 .05357 .05649 .05987 21 22 .04052 .04223 .04559 .04776 .05094 .C5367 .05658 .05998 22 23 .04060 .04232 .04567 .04786 .05103 .05378 .05668 .06009 23 24 .04069 .04241 .04576 .04795 .05112 .05388 ! .05678 .06020 24 25 .04077 .04250 .04585 .04805 .05122 .C5398 .05687 .06030 25 26 .04085 .04259 .04593 .04815 .05131 .C5408 .05697 .06041 26 27 .04093 .04268 .04602 .04824 .05140 .05418 .05707 .06052 27 28 .04102 .04277 .04611 .04834 .C5149 .05429 .05716 .06063 28 29 .04110 .04286 .04620 .04843 .05158 .C5439 .05726 .06074 29 30 .04118 .04295 .04628 .04853 .05168 .05449 .05736 .06085 30 31 .04126 .04304 .04637 .04863 .05177 .05460 .05746 .06096 31 32 .04135 .04313 .04646 .04872 .C5186 .C5470 .05755 .06107 32 33 .04143 .04322 .04655 .04882 .05195 .05480 .05765 .06118 33 34 .04151 .04331 .04663 .04891 .05205 .05490 .05775 .06129 34 35 .04159 .04340 .04672 .04901 .05214 .C5501 .05785 .06140 35 36 .04168 .04349 .04681 .04911 .05223 .05511 .05794 .06151 36 37 .04176 .04358 .04690 .04920 .05232 .05521 .05804 .06162 37 38 .04184 .04367 .04699 .04930 .05242 .05532 .05814 .06173 38 39 .04193 .04376 .04707 .04940 .05251 .C5542 .05824 .06184 39 40 .04201 .04385 .04716 .04950 .05260 .05552 .05833 .06195 40 41 .04209 .04394 .04725 .04959 .05270 .05563 .05843 .06206 41 42 .04218 .04403 .04734 .04969 .C5279 .05573 .05853 .06217 42 43 .04226 .04413 .04743 .04979 .05288 .05584 .05863 .06228 43 44 .04234 .04422 .04752 .04989 .05298 .05594 .05873 .06239 44 45 .04243 .04431 .04760 .04998 .05307 .05604 .05882 .06250 45 46 .04251 .04440 .04769 .05008 .05316 .05615 ! .05892 .06261 46 47 .04260 .04449 .04778 .05018 .05326 .05625 .05902 .06272 47 48 .04268 .04458 .04787 .05028 .05335 .05636 .05912 .06283 48 49 .04276 .04468 .04796 .05038 .05344 .05646 .05922 .06295 49 50 .04285 .04477 .04805 .05047 .05354 .05657 | .05932 .06306 50 51 .04293 .04486 .04814 .05057 .05363 .05667 .05942 .06317 51 52 .04302 .04495 .04823 .05067 .05373 .05678 .05951 .06328 52 53 .04310 .04504 .04832 .05077 .05382 .05688 .05961 .06339 53 54 .04319 .04514 .04841 .05087 .05391 .05699 .05971 .06350 54 55 .04327 .04523 .04850 .05097 .05401 .05709 .05981 .06362 55 56 .04336 .04532 .04858 .05107 .05410 .05720 .05991 .06373 56 57 .04344 .04541 .04867 .05116 .05420 .05730 .06001 .06384 57 58 .04353 .04551 .04876 .05126 .05429 .05741 .06011 .06895 58 59 .04361 .04560 .04885 .05136 .05439 .05751 .06021 .0&407 59 60 .04370 .04569 .04894 .05146 .05448 .05762 .06031 .06418 60 91 NATURAL VERSED SINES AND EXTERNAL SECANTS, / 20 21' 22 23 i Vers. Ex. sec. Vers. Ex. sec. i Vers. Ex. sec. Vers. Ex. sec. ~T .06031 .06418 .06642 .07115 .07282 .07853 .07950 .08636 i .06041 .06429 .06652 .07126 .07293 .07866 .07961 .08649 1 2 .06051 .06440 .06663 .07138 .07303 .07879 .07972 .08663 2 3 .06061 .06452 .06673 .07150 .07314 .07892 .07984 .08676 3 4 .06071 .06463 .06684 .07162 .07325 .07904 .07995 .08690 4 5 .06081 .06474 .06694 .07174 .07336 .07917 .08006 .08703 5 6 .06091 .06486 .06705 .07186 .07347 .07930 .08018 .08717 6 7 .06101 .06497 .06715 .07199 .07358 .07943 .08029 .08730 7 8 .06111 .06508 .06726 .07211 .07369 .07955 .08041 .08744 8 9 .06121 .06520 .06736 .07223 .07380 .07968 .08052 .08757 9 10 .06131 .06531 .06747 .07235 .07391 .07981 .08064 .08771 10 11 .06141 .06542 .06757 .07247 .07402 .07994 .08075 .08784 11 12 .06151 .06554 .06768 .07259 .07413 .08006 .08086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 .06171 .06577 .06789 .07283 .07435 .08032 .08109 .08825 14 15 .06181 .06588 .00799 .07295 .07446 .08045 .08121 .08839 15 16 .06191 .06600 .06810 .07307 .07457 .08058 .08132 .08852 16 IT .06201 .06611 .06820 .07320 .07468 .08071 .08144 .08866 17 18 .06211 .06622 .06831 .07332 .07479 .08084 .08155 .08880 18 19 .06221 .06634 .06841 .07344 .07490 .08097 .08167 .08893 19 20 .06231 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .06252 .06668 .06873 .07380 .07523 .08135 .08201 .08934 22 23 .06262 .06680 .06884 .07393 .07534 .08148 .08213 .08948 23 24 .06272 .06691 .06894 .07405 .07545 .08161 .08225 .08962 24 25 .06282 .06703 .06905 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 .07568 .08187 .08248 .08989 26 27 .06302 .06726 .06926 .07442 .07579 .08200 .08259 .09003 27 28 .06312 .06738 .06937 .07454 .07590 .08213 .08271 .09017 28 29 .06323 .06749 .06948 .07466 .07601 .08226 .08282 .09030 20 30 .06333 .06761 .06958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .06773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06353 .06784 .06980 .07503 .07634 .08265 .08317 .09072 32 33 .06363 .'06796 .06990 .07516 .07645 .08278 .08329 .09086 33 34 .06374 .06807 .07001 .07528 .07657 .08291 .08340 .09099 34 35 .06384 .06819 .07012 .07540 .07668 .08305 .08352 .09113 35 36 .06394 .06831 .07022 .07553 .07679 .08318 .08364 .09127 36 37 .06404 .06843 .07033 .07565 .07690 .08331 .08375 .09141 37 38 .06415 .06854 .07044 .07578 .07701 .08344 .08387 .09155 33 39 .06425 .06866 .07055 .07590 .07713 .08357 .08399 .09169 39 40 .06435 .06878 .07065 .07602 .07724 .08370 .08410 .09183 40 41 .06445 .06889 .07076 .07615 .07735 .08383 .08422 .09197 41 42 .06456 .06901 .07087 .07627 .07746 .08397 .08434 .09211 42 43 .06466 .06913 .07098 .07640 .07757 .08410 .08445 .09224 43 44 .06476 .06925 .07108 .07652 .07769 .08423 .08457 .09238 44 45 .06486 .06936 .07119 .07665 .07780 .08436 .08469 .09252 45 46 .06497 .06948 .07130 .07677 .07791 .08449 .08481 .09266 46 47 .06507 .06960 .07141 .07690 .07802 .08463 .08492 .09280 47 48 .06517 .06972 .07151 .07702 .07814 .08476 .08504 .09294 43 49 .06528 .06984 .07162 .07715 .07825 .08489 .08516 .09308 40 50 .06538 .06995 .07173 .07727 .07836 .08503 .08528 .09323 50 51 .06548 .07007 .07184 .07740 .07848 .08516 .08539 .09a37 51 52 .06559 .07019 .07195 .07752 .07859 .08529 .08551 .09351 52 53 .06569 .07031 .07206 .07765 .07870 .08542 .08563 .09365 53 54 .06580 .07043 .07216 .07778 .07881 .08556 .08575 .09379 54 55 .06590 .07055 .07227 .07790 .07893 .08569 .08586 .09393 55 56 .06600 .07067 .07238 .07803 .07904 .08582 .08598 .09407 56 57 .06611 .07079 .07249 .07816 .07915 .08596 .08610 .09421 57 58 .06621 .07091 .07260 .07828 .07927 .08609 .08622 .09435 58 59 .06632 .07103 .07271 .07841 .07938 .08623 .08634 .09449 59 60 .06648 .07115 .07282 .07853 .07950 .08636 .08645 .09464 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. / 24 25 26 27 / Vers. Ex. sec. Vers. Ex. cec. Vers. Ex. sec. Vers. Ex. sec. .08645 .09464 .09369 .10338 .10121 .11260 .10899 .12233 1 .08657 .09478 .09382 .10353 .10133 .11276 .10913 .12249 1 2 .08669 .09492 .09394 .10368 .10146 .11292 .10926 .12266 2 3 .08681 .09506 .09406 .10383 .10159 .11308 .10939 .12283 3 4 .08693 .09520 .09418 .10398 .10172 .11323 .10952 .12299 4 5 .08705 .09535 .09431 .10413 .10184 .11339 .10965 .12316 5 G .08717 .09549 .09443 .10428 .10197 .11355 .10979 .12333 6 7 .08728 .09563 .09455 .10443 .10210 .11371 .10992 .12349 7 8 .08740 .09577 .09468 .10458 .10223 .11387 .110C5 .12366 8 9 .08752 .09592 .09480 .10473 .10236 .11403 .11019 .12383 9 10 .08764 .09606 .09493 .10488 .10248 .11419 .11032 .12400 10 11 .08776 .09620 .09505 .10503 .10261 .11435 .11045 .12416 11 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .12433 12 13 .08800 .09649 .C9530 .10533 .10287 .11467 .11072 .12450 13 14 .08812 .09663 .09542 .10549 .10300 .11483 .11085 .12467 11 15 .08824 .09678 .09554 .10564 .10313 .11499 .11098 .12484 15 16 .08836 .09692 .09567 .10579 .10326 .11515 .11112 .12501 1G 17 .08848 .09707 .09579 .10594 .10328 .11531 .11125 .12518 17 18 .08860 .09721 .09592 .10609 .10351 .11547 .11138 .12534 13 19 .08872 .09735 .09604 .10625 .103G4 .11563 .11152 .12551 19 20 .08884 .09750 .09617 .10640 .10377 .11579 .11165 .12568 20 21 .08896 .09764 .09629 .10655 .10390 .11595 .11178 .12585 21 22 .08903 .09779 .C9G42 .10670 .10403 .11611 .11192 .12602 22 23 .08920 .09793 .09654 .10686 .10416 .11627 .11205 .12619 23 24 .08932 .09808 .09666 .10701 .10429 .11643 .11218 .12636 24 25 .08944 .09822 .09679 .10716 .10442 .11659 .11232 .12653 25 26 .08956 .09837 .09691 .10731 .10455 .11675 .11245 .12670 26 27 .08968 .00851 .09704 .10747 .10468 .11691 .11259 .12687 27 28 .08980 .09866 .09716 .10762 .10481 .11708 .11272 .12704 28 29 .08992 .09880 .C9729 .10777 .10494 .11724 .11285 .12721 29 30 .09004 .09895 .09741 .10793 .10507 .11740 .11299 .12738 30 31 .09016 .09909 .09754 .10808 .10520 .11756 .11312 .12755 31 32 .09028 .09924 .09707 .10824 .10533 .11772 .11326 .12772 32 33 .09040 .09930 .08779 .10839 .10546 .11789 .11339 .12789 33 34 .09052 .09953 .09792 .10854 .10559 .11805 .11353 .12807 34 85 .09064 .09908 .09804 .10870 : ; .10572 .11821 .11366 .12824 35 6 .09076 .09982 .09817 .10885 .10585 .11838 .11380 .12841 3G 37 .09089 .09997 .09829 .10901 .10598 .11854 .11393 .12858 37 88 .09101 .10012 .09842 .10916 .10611 .11870 .11407 .12875 38 39 .09113 .10026 .09854 .10932 .10624 .11886 .11420 .12892 30 40 .09125 .10041 .09867 .10947 .10637 .11903 .11434 .12910 40 (1 .09137 .10053 .09880 .10963 .10650 .11919 .11447 .12927 41 42 .09149 .10071 .C9892 .10978 .106G3 .11936 .11461 .12944 42 43 .09161 .10085 .09905 .10994 .10676 .11952 .11474 .12961 43 44 .09174 .10100 .09918 .11009 .10689 .11968 .11488 .12979 44 45 .09186 .10115 .09930 .11025 .10702 .11985 .11501 .12996 45 46 .09198 .10130 .09943 .11041 .10715 .12001 .11515 .13013 46 47 .09210 .10144 .09955 .11056 .10728 .12018 .11528 .13031 47 48 .09222 .10159 .09968 .11072 .10741 .12034 .11542 .13048 48 49 .09234 .10174 .09981 .11087 .10755 .1.2051 .11555 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11569 .13083 50 51 .09259 .10204 .10006 .11119 .10781 .12084 .11583 .13100 51 52 .09271 .10218 .10019 .11134 .10794 .12100 .11596 .13117 52 53 .09283 .10233 .10032 .11150 .10807 .12117 .11610 .13135 53 54 .09296 .10248 .10044 .11166 .10820 .12133 .11623 .13152 54 55 .09308 .10263 .10057 .11181 .10833 .12150 .11637 .13170 55 56 .09320 .10278 .10070 .11197 .10847 .12166 .11651 .13187 56 57 .09332 . 10293 .10082 .11213 .10860 .12183 .11664 .13205 57 58 .09345 .10308 .10095 .11229 .10873 .12199 .11678 .13222 58 59 .09357 .10323 .10108 .11244 .10886 .12216 .11692 .13240 59 60 .09369 . 10338 . 10121 .11260 .10899 .12233 .11705 .13257 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 28 29 30 31 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .11705 .13257 .12538 .14335 .13307 .15470 .14283 .16663 i .11719 .13275 .12552 .14354 .13412 .15489 .14298 .16684 1 2 .11733 .13292 .12566 .14372 .13427 .15509 .14313 .16704 2 ft .11746 .13310 .12580 .14391 .13441 .15528 .14328 .16725 3 4 .11700 .13327 .12595 .14409 .13456 .15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .1347'0 .15567 .14358 .16766 5 C .11787 .13362 .12623 .14446 .13485 .15587 .14373 .16786 6 7 .11801 .13380 .12637 .14465 .13499 .15006 .14388 .16806 7 8 .11815 .13398 .12651 .14483 .13514 .15626 .14403 .16827 8 9 .11823 .13415 .12665 .14502 .13529 .15045 .14418 .16848 9 10 .11842 . 13433 .12679 .14521 .13543 .15665 .14433 .16868 10 11 .11850 .13451 .12694 .14539 .13558 .15684 .14449 .16889 11 12 .11670 .13408 .12708 .14558 .13573 .15704 .11404 .16909 12 13 .11883 .13486 .12722 .14576 .13587 .15724 .14479 .16930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .1-1404 .16950 14 15 .11911 .13521 .12750 .14614 .13616 .15763 .11509 .16971 15 16 .11925 .13539 .12765 .14632 .13631 .15782 .14524 .16992 16 17 .11933 .13557 .12779 .14651 .13646 .15802 .11539 .17012 17 1 18 .11952 .13575 .127103 .14670 .13660 .15822 .14554 .17033 18 19 .11366 .13593 .12807 .14689 .13075 .15841 .14569 .17C54 19 20 .11980 .13610 .12822 .14707 .13690 .15861 .14584 .17075 20 21 .11994 .13628 .12836 .14726 .13705 .15881 .14599 .17095 21 22 .12007 .13646 .12850 .11745 .13719 .15901 .14615 .17116 22 23 .12021 .13664 .12864 .14704 .13734 .15920 .14630 .17137 23 21 .12035 .13682 .12879 .14782 .13749 .15940 .14645 .17158 24 25 .12049 .13700 .12893 .14801 .13703 .15960 .14660 .17178 25 28 .12063 .13718 .12907 .14820 .13778 .15980 .14675 .17199 26 27 .12077 .13735 .12921 .14839 .13793 .16000 .14690 .17220 27 28 .12091 .13753 .12936 .14858 .13803 .10019 .14703 .17241 28 29 .12104 .13771 .12950 .1-1077 .13822 .16039 .14721 .17262 29 .12118 .13789 .12964 .14896 .13837 .10059 .14736 .17283 30 El .12132 .13807 .12979 .14914 .13852 .16079 .14751 .17304 31 32 .12146 .13825 .12993 .14933 .13807 .10003 .14703 .17325 32 83 .12160 .13843 .13007 .14952 .13881 .10119 .14782 .17346 33 34 .12174 .13861 .13022 .14971 .13893 .10139 .14797 .17367 34 So .12188 .13879 .13036 .14990 .13911 .101.9 .14812 .17388 35 36 .12202 .13897 .13051 .15009 .13923 .10179 .14827 .17409 36 37 .12216 .13916 .13065 .15028 .13941 .16199 .14843 .17430 37 38 .12230 .13934 .13079 .15047 .13955 .16219 .14853 .17451 38 9 .12244 .13952 .13094 .15066 .13970 .162:- 9 .14873 . 17472 39 40 .12257 .13970 .13108 .15085 .13985 .10259 .14888 .17493 40 41 .12271 .13983 .13122 .15105 .14000 .16279 .14904 .17514 41 42 .12285 .14006 .13137 .15124 .14015 .16299 .14919 .17535 42 43 .12299 .14024 .13151 .15143 .14030 .16319 .14904 .17556 43 44 .12313 .14042 .13166 .15162 .14044 .16339 .14949 .17577 44 45 .12327 .14061 .13180 .15181 .14059 .16359 .14905 .17598 45 46 .12341 .14079 .13195 .15200 .14074 .16380 .14980 .17620 4(5 47 .12355 .14097 .13209 .15219 .14089 .16400 .14905 .17641 47 48 .12369 .14115 .13223 .15239 .14104 .16420 .15011 .17662 43 49 .12383 .14134 .13238 .15258 .14119 .10440 .15023 .17683 49 50 .12397 .14152 .13252 .15277 .14134 .16460 .15041 .17704 50 51 .12411 .14170 .13267 .15*96 .14149 .16481 .15057 .17726 51 52 .12425 .14188 .13281 .15315 .14164 .16501 .15072 .17747 52 53 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 53 54 .12454 .14225 .13310 .15354 . 14194 .16541 .15103 .17790 54 55 .12468 .14243 .13325 .15373 .14208 .16562 .15118 .17811 55 56 .12482 .14262 .13339 .15393 .14223 .16582 .15134 .17832 56 57 .12496 .14280 .13354 .15412 .14238 .16602 .15149 .17854 57 58 .12510 .14299 .13368 .15431 .14253 .16623 .15164 .17875 58 59 .12524 .14317 .13383 .15451 .14268 .16643 .15180 .17896 59 60 .12538 .14335 .13397 .15470 .14283 . 16663 .15195 .17918 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. / 32" 33 34 35 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .15195 .17918 .16133 .19236 .17096 .20622 .18085 .22077 1 .15211 .17939 .16149 .19259 .17113 .20645 .18101 .22102 1 2 . 15226 .17961 .16165 .19281 .17129 .20669 .18118 .22127 2 3 .15241 .17982 .16181 .19304 .17145 .20693 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19349 .17178 .20740 .18168 .22202 5 6 .15288 . 18047 .16228 .19372 .17194 .20764 .18185 .22227 6 7 .15303 .18068 .16244 .19394 .17210 .20788 .18202 .22252 r 8 .15319 .18090 .16260 .19417 .17227 .20812 .18218 .22277 8 9 .15334 .18111 .16276 .19440 .17243 .20836 .18235 .22302 9 10 .15350 .18133 .16292 .19463 .17259 .20859 .18252 .22327 10 11 .15365 .18155 .16308 .194&5 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17292 .20907 .18286 .22377 13 13 .15396 .18198 .16340 .19531 .17308 .20931 .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 .20955 .18319 .22428 14 15 .15427 .18241 .16371 .19576 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18353 .22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .18369 .22503 17 18 .15474 .18307 .16419 .19645 .17390 .21051 .18386 .22528 18 19 .15489 .18328 .16435 .19668 .17407 .21075 .18403 .22554 19 20 .15505 .18350 .16451 .19691 .17423 .21099 .18420 .22579 20 21 .15520 .18372 .16467 .19713 .17439 .21123 .18437 .22604 21 22 .15536 .18394 .16483 .19736 .17453 .21147 .18454 .22629 22 23 .15552 .18416 .16499 .19759 .17472 .21171 .18470 .22655 23 24 .15567 .18437 .16515 .19782 .17483 .21195 .18487 .22680 24 25 .15583 .18459 .16531 .19805 .17505 .21220 .18504 .22706 25 26 .15598 .18481 .16547 .19828 .17522 .21244 .18521 .22731 26 27 .15614 .18503 .16563 .19851 .17533 .21268 .18538 .22756 27 28 .15630 .18525 .16579 .19874 .17554 .21292 .18555 .22782 28 29 . 15645 .18547 .16595 .19897 .17571 .21316 .18572 .22807 29 30 .15661 .18569 .16611 .19920 .17587 .21341 .18588 .22833 30 31 .15676 .18591 .16627 .19944 .17604 .21365 .18605 .22858 31 32 .15692 .18613 .16644 .19967 .17620 .21389 .18622 .22884 32 33 .15708 .18635 .16660 .19990 .17637 .21414 .18639 .22909 33 34 .15723 .18657 .16676 .20013 .17653 .21438 .18656 .22935 34 35 .15780 .18679 .16692 .20036 .17670 .21462 .18673 .22960 35 36 .15755 .18701 .16708 .20059 .17686 .21487 .18690 .22986 36 37 .15770 .18723 .16724 .20083 .17703 .21511 .18707 .23012 37 38 .15786 .18745 .16740 .20106 .17719 .21535 .18724 .23037 38 39 .15803 .18767 .16756 .20129 .17736 .21560 .18741 .23063 39 40 .15818 .18790 .16772 .20152 .17752 .21584 .18758 .23089 40 41 .15833 .18812 .16788 .20176 .17769 .21609 .18775 .23114 41 42 .15849 .18834 .16805 .20199 .17786 .21633 .18792 .23140 42 43 .15865 .18856 .16821 .20222 .17802 .21658 .18809 .23166 43 44 .15880 .18878 .16837 .20246 .17819 .21682 .18826 .23192 44 45 .15896 .18901 .16853 .20269 .17835 .21707 .18843 .23217 45 46 .15912 .18923 .16869 .20292 .17852 .21731 .18860 .23243 46 47 ,15923 .18945 .16885 .20316 .17868 .21756 .18877 .23269 47 48 .15943 .18967 .16902 .20339 .17885 .21781 .18894 .23295 48 49 .15959 .18990 .16918 .20363 .17902 .21805 .18911 .23321 49 50 .15975 .19012 .16934 .20386 .17918 .21830 .18928 .23347 50 51 .15991 .19034 .16950 .20410 .17935 .21855 .18945 .23373 51 52 .16006 .19057 .16966 .20433 .17952 .21879 .18962 .23399 52 53 .16022 .19079 .16983 .20457 .17968 .21904 .18979 .23424 53 54 .16038 .19102 .16999 .20480 .17985 .21929 .18996 .23450 54 55 .16054 .19124 .17015 .20504 .18001 .21953 .19013 .23476 55 56 .16070 .19146 .17031 .20527 .18018 .21978 .19030 .23502 56 57 .16085 .19169 . 17047 .20551 .18035 .22003 .19047 .23529 57 58 .16101 .19191 .17064 .20575 .18051 .22028 .19064 .23555 58 59 .16117 .19214 .17080 .20598 .18068 .22053 .19081 .23581 59 60 .16133 .19236 i .17096 .20622 .18085 .22077 .19098 .23607 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. / 36 37 38 39 t Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .19098 .23607 .20136 .25214 .21199 .26902 .22285 .28676 1 .19115 .23633 .20154 .25241 .21217 .26931 .22304 .287'06 1 2 .19133 .23659 .20171 .25269 .21235 .26960 .22322 .28737 2 3 .19150 .23685 .20189 .25296 .21253 .26988 .22340 .28767 3 4 .19167 .23711 .20207 .25324 .21271 .27017 .22359 .28797 4 5 .19184 .23738 .20224 .25351 .21289 .27046 .22377 .28828 5 6 .19201 .23764 .20242 .25379 .21307 .27075 .22395 .28858 6 7 .19218 .23790 .20259 .25406 .21324 .27104 .22414 .28889 7 8 .19235 .23816 .20277 .25434 .21342 .27133 .22432 .28919 8 9 .19252 .23843 .20294 .25462 .21360 .27162 .22450 .28950 9 10 .19270 .23869 .20312 .25489 .21378 .27191 .22469 .28980 10 11 .19287 .23895 .20329 .25517 .21396 .27221 .22487 .29011 11 12 .19304 .23922 .20347 .25545 .21414 .27250 .2250G .29042 12 13 .19321 .23948 .20305 .25572 .21432 .27279 .22524 .29072 13 14 .19338 .23975 .20382 .25600 .21450 .27308 .22542 .29103 14 15 .19356 .24001 .20400 .25628 .21468 .27337 .22561 .29133 15 16 .19373 .24028 .20417 .25656 .21486 .27366 .22579 .29164 16 17 .19390 .24054 .20435 .25683 .21504 .27396 .22598 .29195 17 38 .19407 .24081 .20453 .25711 .21522 .27425 .22616 .29226 18 19 .19424 .24107 .20470 .25739 .21540 .27454 .22634 .29256 19 .19442 .24134 .20488 .25767 .21558 .27483 .22653 .29287 20 21 .19459 .241 GO .20506 .25795 .21576 .27513 .22671 .29318 21 22 .19476 .24187 .20523 .25823 .21505 .27542 .22690 .29349 22 23 .19493 .24213 .20541 .25851 .21613 .27572 .22708 .29380 23 24 .19511 .24240 .20559 .25879 .21631 .27601 .22727 .29411 24 25 .19528 .24267 .20576 .25907 .21649 .27630 .22745 .29442 25 26 .19545 .24293 .20594 .25935 .21667 .27660 .22764 .29473 26 27 . 19562 .24320 .20612 .25963 ,21685 .27689 .22782 .29504 27 28 .19580 .24347 .20629 .25991 .21703 .27719 .22801 .29535 28 29 .19597 .24373 .20647 .26019 .21721 .27748 .22819 .29566 29 30 .19614 .24400 .20665 .26047 .21739 .27778 .22838 .29597 30 31 .19632 .24427 .20682 .26075 .21757 .27807 .22856 .29628 31 32 .19649 .24454 .20700 .26104 .21775 .27837 .2287'5 .29659 32 33 .19666 .24481 .20718 .26132 .21794 .27867 .22893 .29690 33 34 .1^684 .24508 .20736 .26160 .21812 .27896 .22912 .29721 34 35 .19?'01 .24534 .20753 .26188 .21830 .27926 .22930 .29752 35 36 .19718 .24561 .20771 .26216 .21848 .27956 .22949 .29784 36 37 .19736 .24588 .20789 .26245 .21866 .27985 .22907 .29815 37 38 .19753 .24615 .20807 .26273 .21884 .28015 .22986 .29846 38 39 .19770 .24642 .20824 .26301 .21902 .28045 .23004 .29877 39 40 .19788 .24669 .20842 .26330 .21921 .28075 .23023 .29909 40 41 .19805 .24696 .20860 .26358 .21939 .28105 .23041 .29940 41 42 .19822 .24723 .20878 .26387 .21957 .28134 .23060 .29971 42 43 .19840 .24750 .20895 .26415 .21975 .28164 .23079 .30003 43 44 .19857 .24777 .20913 .26443 .21993 .28194 .23097 .30034 44 45 .1987'5 .24804 .20931 .26472 .22012 .28224 .23116 .30066 45 46 .19892 .24832 .20949 .26500 .22030 .28254 .23134 .30097 46 47 .19909 .24859 .20967 .26529 .22048 .28284 .23153 .30129 47 48 .19927 .24886 .20985 .26557 .22066 .28314 .23172 .30160 48 49 .19944 .24913 .21002 .26586 .22084 .28344 .23190 .30192 49 50 .19962 .24940 .21020 .26615 .22103 .28374 .23209 .30223 50 51 .19979 .24967 .21038 .26643 .22121 .28404 .23228 .30255 51 52 .19997 .24995 .21056 .26672 .22139 .28434 .23246 .30287 52 53 .20014 .25022 .21074 .26701 .22157 .28464 .23265 .30318 53 54 .20032 .25049 .21092 .26729 .22176 .28495 .23283 .30350 54 55 .20049 .25077 .21109 .26758 ! .22194 .28525 .23302 .30382 55 56 .20066 .25104 .21127 .26787 | .22212 .28555 .23321 .30413 56 57 .20084 .25131 .21145 .26815 .22231 .28585 .23339 .30445 57 58 .20101 .25159 .21163 .26844 .22249 .28615 .23358 .30477 58 59 .20119 .25186 .21181 .26873 .22267 .28646 .23377 .30509 59 60 .20136 .25214 .21199 .26902 .22285 .28676 .23396 .30541 60 NATURAL VERSED SINES AND EXTERNAL SECANTS. 40 41 42 43 / Vers. Ex. sec Vers. Ex. sec. Vers. Ex, sec. Vers. Ex. sec. .23396 .30541 .24529 .32501 .25686 .34563 .26865 .36733 1 .23414 .30573 .24548 .32535 .25705 .34599 .26884 .36770 1 2 .23-133 .30605 .24567 .32568 .25724 .34634 .26904 .36807 2 3 .23452 .30636 .24586 .32602 .25744 .34669 .26924 .36844 3 4 .23470 .306G8 .24605 .32636 .25763 .34704 .26944 .36881 4 5 .23489 .30700 .24625 .32669 .25783 .34740 .26964 .36919 5 6 .23508 .30732 .24644 .32703 .25802 .34775 .26984 .36956 6 7 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 .32770 .25841 .34846 .27024 .37030 8 9 .235G4 '.30829 .24701 .32804 .25861 .34882 .27043 .37068 9 10 .23583 .30861 .24720 .32838 .25880 .34917 .27063 .37105 10 11 .23602 .30893 .24739 .32872 .25900 .34953 .27083 .37143 11 12 .23620 .30925 24759 .32905 .25920 .34988 .271C3 .37180 12 13 .23639 .30957 .*24778 .32939 .259C9 .35024 .27123 .37218 13 14 .23658 .30989 .24797 .32973 .25959 .35060 .27143 .37255 14 15 .23677 .31022 .24816 .33007 .25978 .35095 .27163 .37293 15 16 .23696 .31054 .24835 .33041 .25993 .35131 .27183 .37330 16 17 .23714 .31086 .24854 .33075 .26017 .35167 .27203 .37368 17 18 .23733 .31119 .24874 .33109 .26037 .35203 .27223 .37406 18 19 .23752 .31151 .24803 .33143 .26056 .35238 .27243 .37443 19 20 .23771 .31183 .24912 .33177 .26076 .35274 .27263 .37481 20 21 .23790 .31216 .24931 .33211 .26096 .35310 .27283 .37519 21 22 .23808 .31248 .24950 .33245 .26115 .35346 .27303 .37556 22 23 .23827 .31281 .24970 .33279 .26135 .35382 .27323 .37594 23 24 .23846 .31313 .24989 .33314 .26154 .35418 .27343 .37632 24 25 .23865 .31346 .25008 .33348 .26174 .35454 .273C3 .37670 25 26 .23884 .31378 .25027 .33382 .26194 .35490 .27383 .37708 26 27 .23903 ' .31411 .25047 .33416 .26213 .35526 .27403 .37746 27 28 .23922 .31443 .25066 .33451 .26233 .35562 .27423 .37784 28 29 .23941 .31476 .25085 .33485 .26253 .35598 .27443 .37822 29 30 .23959 .31509 .25104 .33519 .26272 .35634 .27463 .37860 30 31 .23973 .31541 .25124 .33554 .26292 .35670 .27483 .37898 31 32 .23997 .31574 .25143 .33588 .26312 .35707 .27503 .37936 32 33 .24016 .31607 .251G2 .33622 .26331 .35743 .27523 .37974 33 34 .24035 .31610 .25182 .33657 .26351 .35779 .27543 .38012 34 35 .24054 .31672 .25201 .33691 .26371 .35815 .27563 .38051 35 36 .24073 .31705 .25220 .33726 .26390 .35852 .27583 .38089 36 37 .24092 .31738 .25240 .33760 .26410 .35888 1 .27603 .38127 37 38 .24111 .31771 .25259 .33795 .26430 .35924 .27623 .38165 33 39 .24130 .31804 .25278 .33830 .26449 .35961 .27643 .38204 39 40 .24149 .31837 .25297 .33864 .26469 .35997 .27663 .38242 40 7 58 .64109 1.78621 .65743 1.91914 .67388 2.06637 .69043 2.23028 58 59 .64136 1.78832 .65771 1.92147 .67416 2.06896 .69071 2.23317 59 60 .64163 1.79043 .65798 1.92380 .67443 2.07155 .69098 2.23607 50 104 NATURAL VERSED SINES AND EXTERNAL SECANTS. > 72 73 74 75 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .69098 2.23607 .70763 2.42030 .72436 2.62796 .74118 2.86370 1 .69126 2.23897 .70791 2.42356 .72464 2.63164 .74146 2.86790 1 2 .69154 2.24187 .70818 2.42683 .72492 2.63533 .74174 2.87211 2 3 .69181 2.24478 .70846 2.43010 .72520 2.63903 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 .74231 2.88056 4 5 .69237 2.25062 .70902 2.43666 .72576 2.64645 .74259 2.88479 5 6 .69264 2.25355 .70930 2.43995 .72604 2.65018 .74287 2.88904 6 7 .69292 2.25648 .70958 2.44324 .72632 2.65391 .74315 2.89330 7 8 .69320 2.25942 .70985 2.44655 .72660 2.65765 .74343 2.89756 8 9 .69347 2.26237 .71013 2.44986 .72688 2.66140 .74371 2.90184 9 10 .69375 2.26531 .71041 2.45317 .72716 2.66515 .74399 2.90613 10 11 .69403 2.26827 .71069 2.45650 .72744 2.66892 .74427 2.91042 11 13 .69430 2.27123 .71097 2.45983 .72773 2.67269 .74455 2.91473 12 13 .69458 2.27420 .71125 2.46316 .72800 2.67647 .74484 2.91904 13 14 .69486 2.27717 .71153 2.46651 .72828 2.68025 .74512 2.92337 14 15 .69514 2.28015 .71180 2.46986 .72856 2.68405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 .72884 2.68785 .74568 2.93204 16 17 .69569 2.28612 .71236 2.47658 .72912 2.69167 .74596 2.93640 17 18 .69597 2.28912 .71264 2.47995 .72940 2.69549 .74624 2.94076 18 19 .69624 2.29212 .71292 2.48333 .72968 2.69931 .74652 2.94514 19 20 .69652 2.29512 .71320 2.48671 .72996 2.70315 .74680 2.94952 20 31 .69680 2.29814 .71348 2.49010 .73024 2.70700 .74709 2 95392 21 22 .69708 2.30115 .71375 2.49350 .73052 2.71085 .74737 2^95832 22 28 .69735 2.30418 .71403 2.49691 .73080 2.71471 .74765 2.96274 23 24 .69763 2.30721 .71431 2.50032 .73108 2.71858 .74793 2.96716 24 25 .69791 2.31024 .71459 2.50374 .73136 2.72246 .74821 2.97160 25 20 .69818 2.31328 .71487 2.50716 .73164 2.72635 .74849 2.97604 20 27 .69846 2.31633 .71515 2.51060 .73192 2.73024 .74878 2.98050 27 28 .69874 2.31939 .71543 2.51404 .73220 2.73414 .74906 2.98497 28 29 .69902 2,32244 .71571 2.51748 .73248 2.73806 .74934 2.98944 29 30 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 30 31 .69957 2.32858 .71626 2.52440 .73304 2.74591 .74990 2.99843 31 32 .69985 2.33166 .71654 2.52787 .73332 2.74984 .75018 3.00293 32 33 .70013 2.33474 .71682 2.53134 .73360 2.75379 .75047 3.00745 33 34 .70040 2.33783 .71710 2.53482 .73388 2.75775 .75075 3.01198 34 35 .70068 2.34092 .71738 2.53831 .73416 2.76171 .75103 3.01652 35 36 .70096 2.34403 . 71766 2.54181 .73444 2.76568 .75131 3.02107 36 37 .70124 2.34713 .71794 2.54531 .73472 2.76966 .75159 3.02563 37 38 .70151 2.35025 .71822 2.54883 .73500 2.77365 .75187 3.03020 38 39 .70179 2.35336 .71850 2.55235 .73529 2.77765 .75216 3.03479 39 40 .70207 2.35649 .71877 2.55587 .73557 2.78166 .75244 3.03938 40 41 .70235 2.35962 .71905 2.55940 .73585 2.78568 .75272 3.04398 41 42 .70263 2.36276 .71933 2.56294 .73613 2.78970 .75300 3.04860 42 43 .70290 2.3G590 .71961 2.56649 .7'3641 2.79374 .75328 3.05322 43 44 .70318 2.36905 .71989 2.57005 .73669 2.7977S .75356 3.05786 44 45 .70346 2.37221 .72017 2.57361 .73697 2.80183 .75385 3.06251 45 46 .70374 2.37537 .72045 2.57718 .73725 2.80589 .75413 3.06717 40 47 .70401 2.37854 .72073 2.58076 .73753 2.80996 .75441 3.07184 47 48 .70429 2.38171 .72101 2.58434 .73781 2.81404 .75469 3.07652 48 49 .70457 2.38489 .72129 2.58794 .73809 2.81813 .75497 3.08121 49 50 .70485 2.38808 .72157 2.59154 .73837 2.82223 .75526 3.08591 50 51 .70513 2.39128 .72185 2.59514 .73865 2.82633 .75554 3.09063 51 52 .70540 2.39448 .72213 2.59876 .73893 2.83045 .75582 3.09535 52 53 .70568 2.39768 .72241 2.60238 .73921 2.83457 .75610 3.10009 53 54 .70596 2.40089 .72269 2.60601 .73950 2.83871 .75639 3.10484 54 55 .70624 2.40411 .72296 2.60965 .73978 2.84285 .75667 3.10960 55 56 .70652 2.40734 .72324 2.61330 .74006 2.84700 .75695 3.11437 50 57 .70679 2.41057 .72352 2.61695 .74034 2.85116 .75723 3.11915 57 58 .70707 2.41381 .72380 2.62061 .74062 2.85533 .75751 3.12394 58 59 .70735 2.41705 .72408 2.62428 .74090 2.85951 .75780 3.12875 59 GO .70763 2.42030 .72436 2.62796 .74118 2.86370 .75808 3.13357 60 105 NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 76 77 78 79 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .75808 3.13357 .77505 3.44541 .79209 3.80973 .80919 4.24084 1 .75836 3.13839 .77533 3.45102 .79237 3.81633 .80948 4.24870 1 2 .75864 3.14323 .77562 3.45664 .79266 3.82294 .80976 4.25658 2 3 .75892 3.14809 .77590 3.46228 .79294 3.82956 .81005 4.26448 3 4 .75921 3.15295 .77618 3.46793 .79323 3.83621 .81033 4.27241 4 5 .75949 3.15782 .77647 3.47360 .79351 3.84288 .81062 4.28036 5 6 .75977 3.16271 .77675 3.47928 .79380 3.84956 .81090 4.28833 6 7 .76005 3.16761 .77703 3.48498 .79408 3.85627 .81119 4.29634 7 8 .76034 3.17252 .77732 3.49069 .79437 3.86299 .81148 4.30436 8 9 .76062 3.17744 .77760 3.49642 .79465 3.86973 .81176 4.31241 9 10 .76090 3.18238 .77788 3.50216 .79493 3.87649 .81205 4.32049 10 11 .76118 3.18733 .77817 3.50791 .79522 3.88327 .81233 4.32859 11 12 .76147 3.19228 .77845 3.51368 .79550 3.89007 .81262 4 . 33671 12 13 .76175 3.19725 .77874 3.51947 .79579 3.89689 .81290 4.34486 13 14 .76203 3.20224 .77902 3.52527 .79607 3.90373 .81319 4.35304 14 15 .76231 3.20723 .77930 3.53109 .79636 3.91058 .81348 4.36124 15 16 .76260 3.21224 .77959 3.53692 .79664 3.91746 .81376 4.36947 10 17 .76288 3.21726 .77987 3.54277 .79693 3.92436 .81405 4.37772 IT 18 .76316 3.22229 .78015 3.54863 .79721 3.93128 .81433 4.38600 18 19 .76344 3.22734 .78044 3.55451 .79750 3.93821 .81402 4.39430 11) 20 .76373 3.23239 .78072 3.56041 .79778 3.94517 .81491 4.40263 20 21 .76401 3.23746 .78101 3.56632 .79807 3.95215 .81519 4.41099 21 22 .76429 3.24255 .78129 3.57224 .79835 3.95914 .81548 4.41937 22 23 .76458 3.24764 .78157 3.57819 .79864 3.96616 .81576 4.42778 23 24 .76486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 25 .76514 3.25787 .78214 3.59012 .79921 3.98025 .81633 4.44468 25 20 .76542 3.26300 .78242 3.59611 .79949 3.98733 .81662 4.45317 26 27 .76571 3.26814 .78271 3.60211 .79978 3.99443 .81691 4.46169 27 28 .76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 28 29 .76627 3.27847 .78328 3.61417 .80035 4.00869 .81748 4.47881 29 30 .76655 3.28366 . 7835S 3.62023 .80063 4.01585 .81776 4.48740 30 a .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 32 .76712 3.29406 .78413 3.63238 .80120 4.03024 .81834 4.50468 32 33 .76740 3.29929 .78441 3.63849 .80149 4.03746 .81862 4.51337 33 34 .76769 3.30452 .78470 3.64461 .80177 4.04471 .81891 4.52208 34 35 .76797 3.30977 .78498 3.65074 .80205 4.05197 .81919 4.53081 35 30 .76825 3.31503 .78526 3.65690 .80234 4.05926 .81948 4.53958 36 37 .76854 3.32031 .7'8555 3.66307 .80263 4.06657 .81977 4.54837 37 38 .76882 3.32500 .78583 3.66925 .80291 4.07390 .82005 4.55720 38 39 .76910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 40 .76938 3.33622 .78640 3.68167 .80348 4.08863 .82063 4.57493 40 41 .76967 3.34154 .78669 3.68791 .80377 4.09602 .82091 4.58383 41 42 .76995 3.34689 .78697 3.69417 .80405 4.10344 .82120 4.59277 42 43 .77023 3.35224 .78725- 3.70044 .80434 4.11088 .82148 4.60174 43 44 .77052 3.35761 .78754 3.70673 .80462 4.11835 .82177 4.61073 44 45 .77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61976 45 40 .77108 3.36839 .78811 3.71935 .80520 4.13334 .82234 4.62881 46 47 .77137 3.37380 .78839 3.72569 .80548 4.14087 .82263 4.63790 47 48 .77165 3.37923 .78868 3.73205 .80577 4.14842 .82292 4.64701 48 49 .77193 3.38466 .78896 3.73843 .80605 4.15599 .82320 4.65616 49 50 .77222 3.39012 .78924 3.74482 .80634 4.16359 .82349 4.66533 50 51 .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 51 52 .77278 3.40106 .78981 3.75760 .80691 4.17886 .82406 4.68377 52 53 .77307 3.40656 .79010 3.76411 .80719 4.18652 .82435 4.69304 53 54 .77335 3.41206 .79038 3.77057 .80748 4.19421 .82403 4.70234 54 55 .77363 3.41759 .79067 3.77705 .80776 4.20193 824?)' > 4.71166 55 56 .77392 3.42312 .79095 3.78355 .80805 4.20966 '.82521 4.72102 56 57 .77420 3.42867 .79123 3.79007 .80833 4.21742 .82549 4.73041 57 58 .77448 3.43424 .79152 3.79661 .80862 4.22521 .82578 4.73983 58 59 .77477 3.43982 .79180 3.80316 .80891 4.23301 .82607 4.74929 59 60 .77505 3.44541 .79209 3.80973 .80919 4.24084 .82635 4.75877 60 106 NATURAL VERSED SINES AND EXTERNAL SECANTS. 80 81 82 83 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .82635 4.75877 .84357 5.39245 .86083 6.18530 .87813 7.20551 1 .82664 4.76829 .84385 5.40422 .86112 6.20020 .87842 7.22500 1 2 .82692 4.77784 .84414 5.41602 .86140 6.21517 .87871 7.24457 2 3 .82721 4.78742 .84443 5.42787 .86109 6.23019 .87900 7.26425 3 4 .82750 4.79703 .84471 5.43977 .86198 6.24529 .87929 7.28402 4 5 .82778 4.80667 .84500 5.45171 .86227 6.20044 .87957 7.30388 5 6 .82807 4.81G35 .84529 5.46369 .86256 6.27566 .87986 7.32384 6 7 .82836 4.82606 .84558 5.47572 .86284 6.29095 .88015 7.34390 7 8 .82864 4.83581 .84586 5.48779 .86313 6.30630 .88044 7.36405 8 9 .82893 4.84558 .84615 5.49991 .86342 6.32171 .88073 7.38431 9 10 .82922 4.85539 .84644 5.51208 .86371 6.33719 .88102 7.40466 10 11 .82950 4.86524 .84673 5.52429 .86400 6.35274 .88131 7.42511 11 12 .82979 4.87511 .84701 5.53G55 .80428 6.36835 .88160 7.44566 12 18 .83008 4.88502 .84730 5.54886 .86457 6.38403 .88188 7.46632 13 14 .8:3036 4.89497 .84759 5.56121 .86486 6.39978 .88217 7.48707 14 15 .83065 4.90495 .84788 5.57361 .86515 6.41560 .88246 7.50793 15 1G .8:5094 4.91496 .84816 5.58606 .86544 6.43148 .88275 7.52889 16 17 .83122 4.92501 .84845 5.59855 .86573 6.44743 .88304 7.54996 17 18 .83151 4.93509 .84874 5.61110 .86601 6.46346 .88333 7.57113 18 1!) .83180 4.94521 .84903 5.62369 .86630 6.47955 .88362 7.59241 1!) 20 .83208 4.95536 .84931 5.63G33 .86659 C.49571 .88391 7.61379 20 2! .83237 4.96555 .84960 5.64902 .86688 6.51194 .88420 7.63528 21 oo .83366 4.97577 .84989 5.6G1?'6 .86717 6.52825 .88448 7.65688 22 3 .83294 4.98603 .85018 5.67454 .86746 6.54462 .88477 7.67859 23 24 .83323 4.99633 .85046 5.68738 .86774 6.56107 .88506 7.70041 21 25 .8:3352 5.00666 .85075 5.70027 .86803 6.57759 .88535 7.72234 25 20 .83380 5.01703 .85104 5.71321 .86832 6.59418 .88564 7.74438 20 27 .83409 5.02743 .85133 5.72620 .86861 6.61085 .88593 7.76653 27 28 .8:3438 5.03787 .85162 5.73924 .86890 6.62759 .88622 7.78880 28 29 .83467 5.04834 .85190 5.75233 .86919 6.64441 .88651 7.81118 2!) 30 .83405 5.05886 .85219 5.76547 .8694T G.66130 .88680 7.83367 30 31 .83524 5.06941 .85248 5.77866 .86976 6.67826 .88709 7.85628 31 32 '| .83553 5.08000 .85277 5.79191 .87005 6.09530 .88737 7.87901 32 33 .83581 5.09062 .85305 5.80521 .87034 6.71242 .88766 7.90186 33 34 .83610 5.10129 .85334 5.81856 .87063 6.72962 .88795 7.92482 34 35 .83639 5.11199 .85363 5.83196 .87092 6.74689 .88824 7.94791 35 36 .83667 5.12273 .85392 5.84542 .87120 6.76424 .88853 7.87111 30 37 .83696 5.13350 .85420 5.85893 .87149 6.78167 .88882 7.99444 37 38 .83725 5.14432 .85449 5.87250 .87178 6.79918 .88911 8.01788 88 39 .83754 5.15517 .85418 5.88612 .87207 6.81677 .88940 8.04146 30 40 .83782 5.16607 .85507 5.89979 .87236 6.83443 .88969 8.06515 40 41 .83811 5.17700 .85536 5.91352 .87265 6.85218 .88998 8.08897 41 42 .83840 5.18797 .85564 5.92731 .87294 6.87001 .89027 8.11292 42 43 .83868 5.19898 .85593 5.94115 .87322 6.88792 .89055 8.13699 43 44 .83897 5.21004 .85622 5.95505 .87351 6.90592 .89084 8.16120 44 45 .83926 5.22113 .85651 5.96900 .87380 6.92400 .89113 8.18553 45 46 .83954 5.23226 .85680 5.9&301 .87409 6.94216 .89142 8.20999 40 47 .83983 5.24343 .85708 5.99708 .87438 6.96040 .89171 8.23459 47 48 .84012 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 49 .84041 5.26590 .85766 6.02538 .87496 6.99714 .89229 8.28417 49 50 .84069 5.27719 .85795 6.03962 .87524 7.01565 .89258 8.30917 50 51 .84098 5.28853 .&5823 6.05392 .87553 7.03423 .89287 8.33430 51 52 .84127 5.29991 .85852 6.06828 .87582 7.05291 .89316 8.35957 52 53 .84155 5.31133 .85881 6.08269 .87611 7.07167 .89345 8.38497 53 54 .84184 5.32279 .85910 6.09717 .87640 7.09052 .89374 8.41052 54 55 .84213 5.33429 .85939 6.11171 .87669 7.10946 .89403 8.43620 55 56 .84242 5.34584 .85967 6.12630 .87698 7.12849 .89431 8.46203 50 57 .84270 5.35743 .85996 6.14096 .87726 7.14760 .89460 8.48800 57 58 .84299 5.36906 .86025 6.15568 .87755 7.16681 .89489 8.51411 58 50 .84328 5.38073 .86054 6.17046 .87784 7.18612 .89518 8.54037 59 60 .84357 5.39245 .86083 6.18530 .87813 7.20551 .89547 8.56677 60 107 NATURAL VERSED SINES AND EXTERNAL SECANTS. / 84 85 86 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .89547 8.56677 .91284 10.47371 .93024 13.33559 1 .89576 8.59332 .91313 10.51199 .93053 13.39547 1 2 .89605 8.62002 .91342 10.55052 .93082 13.45586 2 3 .89634 8.64687 .91371 10.58932 .93111 13.51676 3 4 .89663 8.67387 .91400 10.62837 .93140 13.57817 4 5 .89692 8.70103 .91429 10.66769 .93169 13.64011 5 6 .89721 8.72833 .91458 10.70728 .93198 13.70258 6 7 .89750 8.75579 .91487 10.74714 .93227 13.76558 7 8 .89779 8.78341 .91516 10.78727 .93257 13.82913 8 9 .89808 8.81119 .91545 10.82768 .93286 13.89323 9 10 .89836 8.83912 .91574 10.86837 .93315 13.95788 10 11 .89865 8.86722 .91603 10.90934 .93344 14.02310 11 12 .89894 8.89547 .91632 10.95060 .93373 14.08890 12 13 .89923 8.92389 .91661 10.99214 .93402 14.15527 13 14 .89952 8.95248 .91690 11.03397 .93431 14.22223 14 15 .89981 8.98123 .91719 11.07610 .93460 14.28979 15 16 .90010 9.01015 .91748 11.11852 .93489 14.35795 16 17 .90039 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90068 9.06849 .91806 11.20427 .93547 14.49611 18 19 .90097 9.09792 .91885 11.24761 .93576 14.56614 19 20 .90126 9.12752 .91804 11.29125 .93605 14.63679 20 21 .90155 9.15730 .91893 11.33521 .93634 14.70810 21 22 .90184 9.18725 .91922 11.37948 .93663 14.78005 22 23 .90213 9.21739 .91951 11.42408 .93692 14.85268 23 24 .90242 9.24770 .91980 11.46900 .93721 14.92597 24 25 .90271 9.27819 .92009 11.51424 .93750 14.99995 25 26 .90300 9.30887 .92038 11.55982 .93779 15.07462 26 27 .90329 9.33973 .92067 11.60572 .93808 15.14999 27 28 .90358 9.37077 .92096 11.65197 .93837 15.22607 28 29 .90386 9.40201 .92125 11.69856 .93866 15.30287 29 30 .90415 9.43343 .92154 11.74550 .93895 15.38041 30 31 .90444 9.46505 .92183 11.79278 .93924 15.45869 31 32 .90473 9.49685 ,92212 11.84042 .93953 15.53772 32 33 .90502 9.52886 .92241 11.88841 .93982 15.61751 33 34 .90531 9.56106 .92270 11.93677 .94011 15.69808 34 35 .90560 9.59346 .92299 11.98549 .94040 15.77944 35 36 .90589 9.62605 .92328 12.03458 .94069 15.86159 36 37 .90618 9.65885 .92357 12.08040 .94098 15.94456 37 38 .90647 9.69186 .92386 12.13388 .94127 16.02835 38 39 .90676 9.72507 .92-115 12.18411 .94156 16.11297 39 40 .90705 9.75849 .92444 12.23472 .94186 16.19843 40 41 .90734 9.79212 .92473 12.28572 .94215 16.28476 41 42 .90763 9.82596 .92502 12.33712 .94244 16.37196 42 43 .90792 9.86001 .92531 12.38891 .94273 16.46005 43 44 .90821 9.89428 .92500 12.44112 .94302 16.54903 44 45 .90850 9.92877 .92589 12.49373 .94331 16.63893 45 46 .90879 9.96348 .92618 12.54676 .94360 16.72975 46 47 .90908 9.99841 .92647 12.60021 .94389 16.82152 47 48 .90937 10.03356 .92676 12.65408 .94418 16.91424 48 49 .90966 10.06894 .92705 12.7'0838 .94447 17.00794 49 50 .90995 10.10455 .92734 12.76312 .94476 17.10262 50 51 .91024 10.14039 .92763 12.81829 .94505 17.19830 51 52 .91053 10.17646 .92792 12.87391 .94534 17.29501 52 53 .91082 10.21277 .92821 12.92999 .94563 17.39274 53 54 .91111 10.24932 .92850 12.98651 .94592 17.49153 54 55 .91140 10.28610 .92879 13.04350 .94621 17.59139 55 56 .91169 10.32313 .92908 13.10096 .94650 17.69233 56 57 .91197 10.36040 .92937 13.15889 .94679 17.79438 57 58 .91226 10.39792 .92966 13.21730 .94708 17.89755 58 59 .91255 10.43569 .92995 13.27620 .94737 18.00185 59 60 .91284 10.47371 .93024 13.33559 .94766 18.10732 60 108 NATURAL VERSED SINES AND EXTERNAL SEOANT8. / 87 88 89 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .94766 18-J0732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 2 .94825 18-32182 .96568 28.13917 .98313 58.27431 2 3 .94854 18.43088 .96597 28.38812 .98342 59.31411 3 4 .94883 18.54119 .96626 28.64137 .98371 60.39105 4 5 .94912 18.65275 .96655 28.89903 .98400 61.50715 5 6 .94941 18.76560 .96684 29.16120 .98429 62.66460 6 7 .94970 18.87976 .96714 29.42802 .98458 63.86572 7 8 .94999 18.99524 .96743 29.69960 .98487 65.11304 8 9 .95028 19.11208 .96772 29.97607 .98517 66.40927 9 10 .95057 19.23028 .96801 30.25758 .98546 67.75736 10 11 .95086 19.34989 .96830 30.54425 .98575 69.16047 11 12 .95115 19.47093 .96859 30.83623 .98604 70.62285 12 13 .95144 19.59341 .96888 31.13366 .98633 72.14583 13 14 .95173 19.71737 .96917 31.43671 .98662 73.73586 14 15 .95202 19.84283 .96946 31.74554 .98691 75.39655 15 16 .95231 19.96982 .96975 32.06030 .98720 77.13274 16 17 .95260 20.09838 .97004 32.38118 .98749 78.94968 17 18 .95289 20.22852 .97033 32.70835 .98778 80.85315 18 19 .95318 20.36027 .97062 33.04199 .98807 82.84947 19 20 .95347 20.49368 .97092 33.38232 .98836 84.94561 20 21 .95377 20.62876 .97121 33.72952 .98866 87.14924 21 22 .95406 20.76555 .97150 84.08380 .98895 89.46886 22 23 .95435 20.90403 .97179 34.44530 .98924 91.91387 23 24 .95464 21.04440 .97208 34.81452 .98953 94.49471 24 25 .95493 21.18653 .97237 35.19141 .98982 97.22303 25 26 .95522 .21.33050 .97266 35.57633 .99011 100.1119 26 27 .95551 21.47635 .97295 35.96953 .99040 103.1757 27 28 .95580 21.62413 .97324 36.37127 .99069 106.4311 28 29 .95609 21.77386 .97353 36.78185 .99098 109.8966 29 30 .95638 21.92559 .97382 37.20155 .99127 113.5930 30 31 .95667 22.07935 .97411 37.63068 .C9156 117.5444 31 32 .95696 22.23520 .97440 38.06957 .99186 121.7780 32 33 .95725 22.39316 .97470 38.51855 .99215 126.3253 33 34 .95754 22.55329 .97499 38.97797 .99244 131.2223 34 35 .95783 22.71563 .97528 39.44820 .99273 136.5111 35 36 .95812 22.88022 .97557 39.92963 .99302 142.2406 36 37 .95842 23.04712 .97586 40.42266 .99331 148.4684 37 38 .95871 23.21637 .97615 40.92772 .99360 155.2623 38 39 .95900 23.38802 .97644 41.44525 .99389 162.7033 39 40 .95929 23.56212 .97673 41.97571 .99418 170.8883 40 41 .95958 23.73873 .97702 42.51961* .99447 179.9350 41 42 .95987 23.91790 .97731 43.07746 .99476 189.9868 42 43 .96016 24.09969 .97760 43.64980 .99505 201.2212 43 44 .96045 24.28414 .97789 44.23720 .99535 213.8600 44 45 .96074 24.47134 .97819 44.84026 .99564 228.1839 45 46 .96103 24.66132 .97848 45.45963 .99593 244.5540 46 47 .96132 24.85417 .97877 46.09596 .99622 263.4427 47 48 .96161 25.04994 .97906 46.74997 .99651 285.4795 48 49 .96190 25.24869 .97935 47.42241 .99680 311.5230 49 50 .96219 25.45051 .97964 48.11406 .99709 342.7752 50 51 .96248 25.65546 .97993 48.82576 .99738 380.9723 51 52 .96277 25.86360 .98022 49.55840 .99767 428.7187 52 53 .96307 26.07503 .98051 50.31290 .99796 490.1070 53 54 .96336 26.28981 .98080 51.09027 .89825 671.9581 54 55 .96365 26.50804 .98109 51.89156 .99855 686.5496 55 56 .96394 26.72978 .98138 52.71790 .99884 858.4369 56 57 .96423 26.95513 .98168 53.57046 .99913 1144.916 57 58 .96452 27.18417 .98197 54.45053 .99942 1717.874 58 59 .96481 27.41700 .98226 55.35946 .99971 3436.747 59 60 .96510 27.65371 .98255 56.29869 1.00000 Infinite 60 109