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 -* <i r c proportional to 7V 4 . Therefore
i2 4 : 60.302 :: N* : qD,
and
qZ) Q = _ = .oo2 9 o8i7V 4 . . (25)
12
Making ^"=1,2, 3, 4, etc., the values of gD = Q
of the table are obtained; whence
12 THE TRANSITION-CURVE FIELD-BOOK.
The values of qD = Q at the even stations
tested by formula (24) result in Q 9 = 0.0466,
<2 4 ~ 0.7446, 6 = 3.770, <2 8 = ii. 914, and <2 10 = 2 9 .o84,
being practically the values given in the table.
x
The expression q = -, formula (i), in which x
4
represents the ordinate to the cubic parabola at Z,
applies approximately to the Transition Curve, but
not nearly enough for accurate work when Z is
beyond station 6. The actual relation between
the values of the gap q and the end ordinate x is
given in line F of the table, its values being sim-
ply those of q in decimal parts of x obtained from
the division of q by x. As both q and x vary
directly in proportion to the chord length, the ex-
pression F = is general, and in all cases
q = Fx. ...... (26)
The factor F is used in determining q when the
offset/ is fixed.
EXPLANATION OF GENERAL TABLE.
All of the quantities in each column of the table
have reference to the station of that column, ex-
cept that the quantity Q always refers to a point
midway between the station of its column and .
Line Ogives the station numbers of the Transi-
tion Curve, station o being at .
The following relations exist between N and
other quantities:
Z of table;
THE TRANSITION-CURVE FIELD-BOOK. 13
N*
= a in degrees;
2N* = A in minutes;
O.2JV = D when c = 100;
= c when JD i.
Line X gives the ordinates from the tangent SK
to the stations of the Transition Curve of loo-foot
chords.
For a chord c of any other length the station or-
dinate x = X X - .
100
The values of X in the table are the sums of the
products of the consecutive loo-foot chords mul-
tiplied each into the sine of its inclination to the
tangent SK.
X= Ftang J,
x = Q.2NX when
Line Y gives the distances on the tangent SK
from S to the station ordinates of the Transition
Curve of loo-foot chords.
For a chord c of any other length the ordinate
distance y = Y X -- .
100
The values of Y in the table are the sums of the
products of the consecutive loo-foot chords, multi-
plied each into the cosine of its inclination to the
tangent SK.
Y X cotang A\
y = Q.2NY when D = i.
14 THE TRANSITION CURVE FIELD-BOOK.
Line F gives the values of q at - in decimal parts
of x at n.
Its value is
x
Line Q gives at each station N the gap midway
from N to S, between the tangent SK and the
parallel tangent of a i curve connecting with the
Transition Curve at JV.
For any other degree of curve the gap q -^5.
The values of Q in the table are
qD 0.0029081^*;
q = Q when D = i.
As q is at the point midway between Z and 6* it
may fall at any whole or half station up to station
6, but not beyond.
Line Z gives the lengths of Transition Curves
connecting with i curves at the several stations.
2*
For any other degree of curve the length n=-=-.
The value of Z in the table is
c = 2ooa =
n Z when D = i.
Line C gives the semi-chord of the arc of a i
circular curve of which a subtends half that arc,
and is used to simplify the determination of the
position of S, the initial point of the Transition
Curve.
THE TRANSITION CURVE FIELD-BOOK. 15
Its value is
C= 573 sin
For a circular curve of other than i the value
of C is to be divided by Z>, 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 "<N "N O "V O "<N >
^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<TS8~~ 12
CASE 2. Given Z>, 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 <?, Z, C, F, ^, and a.
Find ^ from (31).
Q _ n ,,o _Z_ n _
q ~ ' i>' /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
<ier of this example is exactly as in the examples
given under Problem VII, if p a be assumed = 4.0.
THE TRANSITION-CURVE FIELD-BOOK. 47
EXAMPLE 2, Problem VIII.
Given 7=6o, #7=125, -=-=857.9.
1547
Assume/* 5, R' a == 857.9 820.5.
* OO s i
J =
13975
Take R' 819.0, Z>' = 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?///<?. Find 7?' of D' from (41), and p from (42).
Take (2 next lower than /Z>', 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<i
33
.22261
4.49215
.24100
4.14934
.25955
3.85284
.27826
3.59370
27
34
.22292
4.48600
.24131
4.14405
.25986
3.84824
.27858
3.58966
20
35
.22322
4.47986
.24162
4.13877
.26017
3.84364
.27889
3.58562
25
30
.22353
4.47374
.24193
4.13350
.26048
3.83906
.27921
3.58160
21
37
.22383
4.46764
.24223
4.12825
.26079
3.83449
.27952
3.57758
2:}
38
.22414
4.46155
.24254
4.12301
.26110
3.82992
.27983
3.57357
22
39
.22444
4.45548
.24285
4.11778
.26141
3.82537
.28015
3.56957
21
40
.22475
4.44942
.24316
4.11256
.2617'2
3^82083
.28046
3.56557
20
41
.22505
4.44338
.24347
4.10736
.26203
3.81630
.28077
3.56159
10
42
.22536
4.43735
.24377
4.10216
.26235
3.81177
.28109
3.55761
10
43
.22567
4.43134
.24408
4.09699
.26266
3.80726
1 .28140
3.55364
17
44
.22597
4.42534
.24439
4.09182
.26297
3.80276
.28172
3.54968
10
45
.22628
4.41936
.24470
4.08666
.26328
3.79827
.28203
3.54573
15
46
.22658
4.41340
.24501
4.08152
.26359
3.79378
.28234
3.54179
14
47
.22689
4.40745
.24532
4.07639
.26390
3.78931
.28266
3.53785
13
48
.22719
4.40152
.24562
4.07127
.26421
3.78485
.28297
3.53393
12
49
.22750
4.39560
.24593
4.06616
.26452
3.78040
.28329
3.53001
11
50
.22781
4.38969
.24624
4.06107
.26483
3.77595
.28360
3.52609
10
51
.22811
4.38381
.24655
4.05599
.26515
3.77152
.28391
3.52219
9
52
.22843
4.37793
.24686
4.05092
.26546
3.76709
.28423
3.51829
8
53
.22872
4.37207
.24717
4.04586
.26577
3.76268
.28454
3.51441
7
54
.22903
4.36623
.24747
4.04081
.26608
3.75828
.28486
3.51053
6
55
.22934
4.36040
.24778
4.03578
.26639
3.75388
.28517
3.50666
5
56
.22964
4.35459
.24809
4.03076
.26670
3.74950
.28549
3.50279
4
57
.22995
4.34879
.24840
4.02574
.26701
3.74512
.28580
3.49894
3
58
.23026
4.34300
.24871
4.02074
.26733
3.74075
.28612
3.49509
2
59
.23056
4.33723
.24902
4.01576
.26764
3.73640
.28643
3.49125
1
60
.23087
4.33148
.24933
4.01078
.26795
3.73205
.28675
3.48741
f
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
77
76
75
74
78
-NATURAL TANGENTS AND COTANGENTS.
16
17
18
19
/
Tang
Cotang
Tang
Cotang
Tang j Cotang
Tang
Cotang
.28675
3.48741
.30573
3.27085
.82492 3.07768
.34433
2.90421
60
1
.28706
3.48359
.30605
3.26745
.32524
3.07464
.34465
2.90147
59
2
.28738
3.47977
.30637
3.26406
.32556
3.07160
.34498
2.89873
58
3
.28769
3.47596
.30669
3.26067
.32588
3.06857
.34530
2.89600
57
4
.28800
3.47216
.30700
3.25729
.32621
3.06554
.34563
2.89327
56
5
.28832
3.46837
.30732
3.25392
.32653
3.06252
.34596
2.89055
55
G
.28864
3.46458
.30764
3.25055
.32685
3.05950
.34628
2.88783
54
7
.28895
3.46080
.30796
3.24719
.32717
3.05649
.34661
2.88511
53
8
.28927
3.45703
.30828
3.24383
.32749
3.05349
.34693
2.88240
52
9
.28958
3.45327
.30860
3.24049
.32782
3.05049
.34726
2.87970
51
10
.28990
3.44951
.30891
3.23714
.32814
3.04749
.34758
2.87700
50
11
.29021
3.44576
.30923
3.23381
.32846
3.04450
.34791
2.87430
49
12
.29053
3.44202
.30955
3.23048
.32878
3.04152
.34824
2.87161
48
13
.29084
3.43829
.30987
3.22715
.32911
3.03854
.34856
2.86892
47
14
.29116
3.43456
.31019
3.22384
.32943
3.03556
.34889
2.86624
46
15
.29147
3.43084
.31051
3.22053
.32975
3.03260
.34922
2.86356
45
10
.29179
3.42713
.31083
3.21722
.33007
3.02963
.34954
2.86089
44
17
.29210
3.42343
.31115
3.21392
.33040
3.02667
.34987
2.85822
43
18
.29242
3.41973
.31147
3.21063
.33072
3.02372
.85020
2.85555
42
19
.29274
3.41604
.31178
3.20734
.33104
3.02077
.35052
2.85289
41
20
.29305
3.41236
.31210
3.20406
.3313G
3.01783
.35085
2.85023
40
21
.29337
3.40869
.31242
3.20079
.331C9
3.01489
.35118
2.84758
39
22
.29368
3.40502
.31274
3.19752
.S3201
3.01196
.35150
2.84494
38
23
.29400
3.40136
.31306
3.19426
.332C3
3.00903
.35183
2.84229
37
24
.29432
3.39771
.31338
3.19100
.332C6
3.COG11
.35216
2.83965
36
25
.29463
3.39406
.31370
3.18775
.33298
3.00319
.35248
2.83702
35
26
.29495
3.39042
.31402
3.18451
.sasso
3.00028
.35281
2.83439
34
27
.29526
3.38679
.31434
3.18127
.33363
2.99738
.35314
2.83176
33
28
.29558
3.38317
.314G6
3.17804
.33395
2.99447
.35346
2.82914
32
29
.29590
3.37955
.31498
3.17481
.33427
2.C9158
.35379
2.82653
31
30
.29621
3.37594
.31530
3.17159
.33460
2.98868
.35412
2.82391
30
31
.29653
3.37234
.31562
3.16838
.33492
2.98580
.35445
2.82130
29
32
.29685
3.36875
.31594
3.16517
.33524
2.88292
.35477
2.81870
28
33
.29716
3.36516
.31626
3.16197
.33557
2.98004
.35510
2.81610
27
34
.29748
3.36158
.31658
3.15877
.33589
2.97717
.35543
2.81350
26
35
.29780
3.35800
.31690
3.15558
.33621
2.97430
.35576
2.81091
25
3G
.29811
3.35443
.31722
3.15240
.33654
2.97144
.356C8
2.80833
24
37
.29843
3.35087
.31754
3.14922
.33686
2.96858
.35641
2.80574
23
38
.29875
3.34732
.31786
3.14605
.33718
2.06573
.35674
2.80316
22
39
.29906
3.34377
.31818
3.14288
.33751
2.962C8
.85707
2.80059
21
40
.29938
3.34023
.31850
3.13972
.33783
2.96004
.35740
2.79802
20
41
.29970
3.33670
.31882
3.13656
.33816
2.95721
.35772
2.79545
19
42
.30001
3.33317
.31011
3.13341
.33848
2.95437
.358C5
2.79289
18
43
.30033
3.32965
.31946
3.13027
.33881
2.95155
.358G8
2.79033
17
44
.300G5
3.32614
.31978
3.12713
.33913
2.9487'2
.35871
2.78778
16
45
.30097
3.32264
.32010
3.12400
.33945
2.94591
.35904
2.78523
15
46
.30123
3.31914
.32042
3.12087
.33978
2.94309
.35937
2.78269
14
47
.301 GO
3.315G5
.32074
3.11775
.34010
2.94028
.35909
2.78014
13
48
.30192
3.31216
.32106
3.114G4
.34043
2.93748
.36002
2.77761
12
49
.30224
3.308G8
.32f39
3.11153
.34075
2.934G8
.36035
2.77507
11
50
.30255
3.30521
.32171
3.10843
.34108
2.93189
.36068
2.77254
10
51
.30287
3.30174
.32203
3.10532
.34140
2.92910
.36101
2.77002
9
52
.30319
3.29829
.32235
3.10223
.34173
2.92632
.36134
2.76750
8
53
.30351
3.29483
.32267
3.09914
.34205
2.92354
.36167
2.76498
7
54
.30382
3.29139
.32299
3.09606
.34238
2.92076
.36199
2.76247
6
55
.30414
3.28795
.32331
3.09298
.34270
2.91799
.36232
2.75996
5
56
.30446
3.28452
.32363
3.08991
.34303
2.91523
.36265
2.75746
4
57
.30478
3.28109
.32396
3.08685
.34335
2.91246
.36298
2.75496
3
58
.30509
3.27767
.32428
3.08379
.34308
2.90971
.36331
2.75246
2
59
.30541
3.27426
.32460
3.08073
.34400
2.90696
.36364
2.74997
1
66
.30573
3.27085
.32492
3.077G8
.34433
2.90421
.36397
2.74748
/
Cotang
Tang
Cotang
Tang
Cotanj
Tang
Cotang
Tang
/
73
72
71 II 70
79
NATURAL TANGENTS AND COTANGENTS.
20
21
22
23
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
/
.36397
2.74748
.38386
2.60509
.40403
2.47509
.42447
2.35585
60
;
.36430
2.74499
.38420
2.60283
.40436
2.47302
.42482
2.35395
59
i
.36463
2.74251
.38453
2.60057
.40470
2.47095
.42516
2.35205
58
]
.36496
2.74004
.38487
2.59831
.40504
2.46888
.42551
2.35015
57
i
.36529
2.73756
.38520
2.59606
.40538
2.46682
.42585
2.34825
56
5
.36562
2.73509
.38553
2.59381
.40572
2.46476
.42619
2.34636
55
6
.36595
2.732G3
.38587
2.59156
.40606
2.46270
.42654
2.34447
54
\
.36628
2.73017
.38620
2.58932
.40640
2.46065
.42688
2.34258
53
8
.36661
2.72771
.38654
2.58708
.40674
2.45860
.42722
2.34069
52
9
.36694
2.72526
.38687
2.58484
.40707
2.45655
.42757
2.33881
51
10
.36727
2.72281
.38721
2.58261
.40741
2.45451
.42791
2.33693
50
11
.36760
2.72036
.38754
2.58038
.40775
2.45246
.42826
2.33505
49
12
.36793
2.71792
.38787
2.57815
.40809
2.45043
.42860
2.33317
48
13
.36826
2.71548
.38821
2.57593
.40843
2.44839
.42894
2.33130
47
14
.36859
2.71305
.38854
2.57371
.40877
2.44636
.42929
2.32943
46
15
.36892
2.71062
.38888
2.57150
.40911
2.44433
.42963
2.32756
45
16
.36925
2.70819
.38921
2.56928
.40945
2.44230
.42998
2.32570
44
17
.36958
2.70577
.38955
2.56707
.40979
2.44027
.43032
2.32383
43
18
.36991
2.70335
.38988
2.56487
.41013
2.43825
.43067
2.32197
42
19
.37024
2.70094
.3902.2
2.56266
.41047
2.43623
.43101
2.32012
41
20
.37057
2.69853
.39055
2.56046
.41081
2.43422
.43136
2.31826
40
21
.37090
2.69612
.39089
2.55827
.41115
2.43220
.43170
2.31641
39
22
.37123
2.69371
.39122
2.55G08
1 .41149
2.43019
.43205
2.31456
38 i
23
.37157
2.69131
.39156
2.55389
.41183
2.42819
.43239
2.31271
37
24
.37190
2.68892
.39190
2.55170
.41217
2.42618
.43274
2.31086
36
25
.37223
2.68653
.39223
2.54952
.41251
2.42418
.43308
2.30902
35
26
.37256
2.68414
.39257
2.54734
.412G5
2.42218
.43343
2.30718
34
27
.37289
2.68175
.39290
2.54516
.41319
2.42019
.43378
2.30534
33
28
.37322
2.67937
.39324
2.54299
.41353
2.41819
.43412
2.30351
32
29
.37355
2.67700
.39357
2.54082
.41387
2.41620
.43447
2.30167
31
30
.37388
2.67462
.39391
2.53865
.41421
2.41421
.43481
2.29984
30
31
.37422
2.67225
.39425
2.53648
.41455
2.41223
.43516
2.29801
29
32
.37455
2.66939
.39458
2.53432
.41490
2.41025
.43550
2.29619
28
33
.37488
2.66752
.39492
2.53217
.41524
2.40827
.43585
2.29437
27
34
.37521
2.66516
.39526
2.53001
.41558
2.40629
.43620
2.29254
26
35
.37554
2.66281
.39559
2.52786
.41592
2.40432
.43654
2.29073
25
36
.37588
2.66046
.39593
2.52571
.41626
2.40235
.43689
2.28891
24
37
.37621
2.65811
.39626
2.52357
.41660
2.40038
.43724
2.28710
23
38
.37654
2.65576
.39660
2.52142
.41694
2.39841
.43758
2.28528
22
39
.37687
2.65342
.39694
2.51929
.41728
2.39645
.43793
2.28348
21
40
.37720
2.65109
.39727
2.51715
.41763
2.39449
.43828
2.28167
20
41
.37754
2.64875
.39761
2.51502
.41797
2.39253
.43862
2.27987
19
42
.37787
2.64642
.39795
2.51289
.41831
2.39058
.43897
2.27806
18
43
.37820
2.64410
.39829
2.51076
.41805
2.38863
.43932
2.27626
17
44
.37853
2.64177
.39862
2.50864
41899
2.38668
.43966
2.27447
16
45
.37887
2.63945
.39896
2.50652
.41933
2.38473
.44001
2.27267
15
46
.37920
2.63714
.39930
2.50440
.41968
2.38279
.44036
2.27088
14
47
.37953
2.63483
.39963
2.50229
.42002
2.38084
.44071
2.26909
13
48
.37986
2.63252
.39997
2.50018
.42036
2.37891
.44105
2.26730
12
49
.38020
2.63021
.40031
2.49807
.42070
2.37697
.44140
2.26552
11
50
.38053
2.62791
.40065
2.49597
.42105
2.37504
.44175
2.26374
10,
51
.38086
2.62561
.40098
2.49386
.42139
2.37311
.44210
2.26196
9
52
.38120
2.62332
.40132
2.49177
.42173
2.37118
.44244
2.26018
8
53
.38153
2.62103
.40166
2.48967
.42207
2.36925
.44279
2.25840
7
54
.38186
2.61874
.40200
2.48758
.42242
2.36733
.44314
2.25663
6
55
.38220
2.61646
.40234
2.48549
.42276
2.36541
.44349
2.25486
5
56
.38253
2.61418
.40267
2 48340
.42310
2.36349
.44384
2.25309
4
57
.38286
2.61190
.40301
2.48132
.42345
2.36158
.44418
2.25132
3
58
.38320
2.60963
.40335
2.47924
.42379
2.35967
.44453
2.24956
2
59
38353
2.60736
.40369
2.47716
.42413
2.35776
.44488
2.24780
1
60
38386
2.60509
.40403
2.47509
.42447
2.35585
.44523
2.24604
/
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
f
69
68
67
66 |
80
NATURAL TANGENTS AND COTANGENTS.
24
25 |! 26
27
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
.44523
2.24604
.46631
2.14451
.48773
2.05030
.50953
1.96261
60
1
.44558
2.24428
.46666
2.14288
.48809
2.04879
.50989
1.96120
59
2
.44593
2.24252
.46702
2.14125
.48845
2.04728
.51026
1.95979
58
3
.44627
2.24077
.46737
2.13963
.48881
2.04577
.51063
1.95838
57
4
.44662
2.23902
.4677'2
2.13801
.48917
2.04426
.51099
1.95698
56
5
.44697
2.23727
.46808
2.13639
.48953
2.04276
.51136
1.95557
55
6
.44732
2.23553
.46843
2.13477
.48989
2.04125
.51173
1.95417
54
7
.44767
2.23378
.46879
2.13316
.49026
2.03975
.51209
1.95277
53
8
.44802
2.23204
.46914
2.13154
.49062
2.03825
.51246
1.95137
52
9
.44837
2.23030
.46950
2.12993
.49098
2.03675
.51283
1.94997
51
10
.44872
2.22857
.46985
2.12832
.49134
2.03526
.51319
1.94858
50
11
.44907
2.22683
.47021
2.12671
.49170
2.03376
.51356
1.94718
49
S3
.44942
2.22510
.47056
2.12511
.49206
2.03227
.51393
1.94579
48
13
.44977
2.22337
.47092
2.12350
.49242
2.03078
.51430
1.94440
47
14
.45012
2.22164
.47128
2.12190
.49278
2.02929
.51467
1.94301
46
15
.45047
2.21992
.47163
2.12030
.49315
2.02780
.51503
1.94162
45
1C
.45082
2.21819
.47199
2.11871
.49351
2.02631
.51540
1.94023
44
17
.45117
2.21647
.47234
2.11711
.49387
2.02483
.51577
1.93885
43
18
.45152
2.21475
.47270
2.11552
.49423
2.02335
.51614
1.93746
42
19
.45187
2.21304
.47305
2.11392
.49459
2.02187
.51651
1.93608
41
20
.45222
2.21132
.47341
2.11233
.49495
2.02039
.51688
1.93470
40
21
.45257
2.20961
.47377
2.11075
.49532
2.01891
.51724
1.93332
39
22
.45292
2.20790
.47412
2.10916
.49568
2.01743
.51761
1.93195
38
23
.45327
2.20619
.47448
2.10758
.49604
2.01596
.51798
1.93057
37
24
.45362
2.20449
.47483
2.10600
.49640
2.01449
.51835
1.92920
36
25
.45397
2.20278
.47519
2.10442
.49677
2.01302
.51872
1.92782
35
2(5
.45432
2.20108
.47555
2.10284
.49713
2.01155
.51909
1.92645
34
27
.45467
2.19938
.47590
2.10126
.49749
2.01008
.51946
1.92508
33
28
.45502
2.19769
.47626
2.09969
.49786
2.00862
.51983
1.92371
32
29
.45538
2.19599
.47662
2.C9811
.49822
2.00715
.52020
1.92235
31
30
.45573
2.19430
.47698
2.09654
.49858
2.00569
.52057
1.92098
30
31
.45608
2.19261
.47733
2.09498
.49894
2.00423
.52094
1.91962
29
:>>
.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
<!!
.24168
.31870
.25317
.33899
.26489
.36034
.27683
.38280
41
42
.24187
.31903
.25336
.33934
.26509
.36070
.27703
.38319
42
43
.24206
.31936
.25356
.33968
.26528
.36107
.27723
.38357
43
44
.24225
.31969
.25375
.34003
.26548
.36143
.27743
.38396
44
45
.24244
.32002
.25394
.34038
.26568
.36180
.27764
.38434
45
46
.24262
.32035
.25414
.3407'3
.26588
.36217
.27784
.38473
46
47
.24281
.32068
.25433
.34108
.26607
.36253
.27804
.38512
47
48
.24300
.32101
.25452
.34142
.26627
.36290
.27824
.38550
48
49
.24320
.32134
.25472
.34177
.26647
.36327
.27844
.38589
49
50
.24339
.32168
.25491
.34212
.26667
.36363
.27864
.38628
50
51
.24358
.32201
.25511
.34247
.26686
.36400
.27884
.38666
51
52
.24377
.32234
.25530
.34282
.26706
.36437
.27905
.38705
52
53
.24396
.32267
.25549
.34317
.26726
.36474
.27925
.38744
53
54
.24415
.32301
.25569
.34352
.26746
.36511
.27945
.38783
54
55
.24434
.32334
.25588
.34387
.26766
.36548
.27965
.38822
55
56'
.24453
.32368
.25608
.34423
.26785
.36585
.27985
.38860
56
57
.2^472
.32401
.25627
.34458
.26805
.36622
.28005
.38899
57
58
.24491
.32434
.25647
.34493
.26825
.36659
.28026
.38938
58
59
.24510
.32468
.25666
.34528
.26845
.36696
.28046
.38977
59
60
.24529
.32501
.25686
.34563
.26865
.36733
.28066
.39016
60
NATURAL, VERSED SINES AND EXTERNAL, SECANTS.
'
44
45
46'
47
'
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.28066
.39016
.29289
.41421
.30534
.43956
.31800
.46628
1
.28086
.39055
.29310
.41463
.30555
.43999
.31821
.46674
1
2
.28106
.39095
.29330
.41504
.30576
.44042
.31843
.46719
2
3
.28127
.39134
.29351
.41545
.30597
.44086
.31864
.46765
3
4
.28147
.39173
.29372
.41586
.30618
.44129
.31885
.46811
4
5
.28167
.39212
.29392
.41627
.30639
.44173
.31907
.46857
5
G
.28187
.39251
.29413
.41669
.30660
.44217
.31928
.46903
6
7
.28208
.39291
.29433
.41710
.30681
.44260
.31949
.46949
7
8
.28228
.39330
.29454
.41752
.30702
.44304
.31971
.46995
8
9
.28248
.39369
.29475
.41793
.30723
.44347
.31992
.47041
9
10
.28268
.39409
.29495
.41835
.30744
.44391
.32013
.47087
10
11
.28289
.39448
.29516
.41876
.30765
.44435
.32035
.47134
11
12
.28309
.39487
.29537
.41918
.30786
.44479
.32056
.47180
12
13
.28329
.39527
.29557
.41959
.30807
.44523
.32077
.47226
13
14
.28350
.39566
.29578
.42001
.30828
.44567
.32099
.47272
14
15
.28370
.39606
.29599
.42042
.30849
.44610
.32120
.47319
15
16
.28390
.39646
.29619
.42084
.30870
.44654
.32141
.47365
16
17
.28410
.39685
.29640
.42126
.30891
.44698
.32163
.47411
17
18
.28431
.39725
.29661
.42168
.30912
.44742
.32184
.47458
18
19
.28451
.39764
.29681
.42210
.30933
.44787
.32205
.47504
19
20
.28471
.39804
.29702
.42251
.30954
.44831
.32227
.47551
20
21
.28492
.39844
.29723
.42293
.30975
.44875
.32248
.47598
21
22
.28512
.39884
.29743
.42335
.30906
.44019
.32270
.47644
22
23
.28532
.39924
.29764
.42377
.31017
.44963
.32291
.47691
23
24
.28553
.39963
.29785
.42419
.31038
.45007
.32312
.47738
24
25
.28573
.40003
.29805
.42461
.31059
.45052
.32334
.4778-1
25
26
.28593
.40043
.29826
.42503
.31080
.45096
.32355
.47831
26
27
.28614
.40083
.29847
.42545
.31101
.45141
.32377
.47878
27
28
.28634
.40123
.29868
.42587
.31122
.45185
.32398
.47925
28
29
.28655
.40163
.29888
.42630
.31143
.45229
.32420
,47972
29
80
.28675
.40203
.29909
.42672
.31105
.45274
.32441
.48019
30
31
.28695
.40243
.29930
.42714
.31186
.45319
.32462
.48066
31
32
.28716
.40283
.29951
.42756
.31207
.45363
.32484
.48113
32
33
.28736
.40324
.29971
.42799
.31223
.45408
.32505
.48160
33
34
.28757
.40304
.29992
.42841
.31249
.45452
.32527
.48207
34
35
.28777
.40404
.30013
.42883
.31270
.45497
.32548
.48254
35
36
.28797
.40444
.30034
.42926
.31291
.45542
.32570
.48301
36
37
.28818
.40485
.30054
.42968
.31312
.45587
.32591
.48349
37
38
.28838
.40525
.30075
.43011
.31334
.45631
.32613
.48396
38
39
.28859
.40565
.30096
.43053
.31355
.45676
.32634
.48443
39
40
.28879
.40606
.30117
.43096
.31376
.45721
.32656
.48491
40
41
.28900
.40646
.30138
.43139
.31397
.45766
.32677
.48538
41
42
.28920
.40087
.30158
.43181
.31418
.45811
.32699
.48586
42
43
.28941
.40727
.30179
.43224
.31439
.45856
.32720
.48633
43
44
.28961
.40768
.30200
.43267
.81461
.45901
.32742
.48681
44
45
.28981
.40808
.30221
.43840
.31482
.45946
.32763
.48728
45
46
.29002
.40849
.30242
.43352
.31503
.45992
.32785
.48776
46
47
.29022
.40890
.30263
.43395
.31524
.46037
.32806
.48824
47
48
.29043
.40930
.30283
.43438
.31545
.46082
.32828
.48871
48
49
.29063
.40971
.30304
.43481
.31567
.46127
.32849
.48919
49
50
.29084
.41012
.30325
.43524
.31588
.46173
.32871
.48967
50
51
.29104
.41053
.30346
.43567
.31609
.46218
.32893
.49015
51
52
.29125
.41093
.30367
.43610
.31630
.46263
.32914
.49063
52
53
.29145
.41134
.30388
.43653
.31651
.46309
.32936
.49111
53
54
.29166
.41175
.30409
.43696
.31673
.46354
.32957
.49159
54
55
.29187
.41216
.30430
.43739
.31694
.46400
.32979
.49207
55
56
.29207
.41257
.30451
.43783
.31715
.46445
.83001
.49255'
56
57
.29228
.41298
.30471
.43826
.31736
.46491
.33022
.49303
57
58
.29248
.41339
.30492
.43869
.31758
.46537
.33044
.49351
58
59
.29269
.41380
.30513
.43912
.31779
.46582
.33065
.49399
59
60
.29289
.41421
.30534
.43956
.31800
.46628
.33087
.49448
60
98
NATURAL VERSED SINES AND EXTERNAL SECANTS.
/
48*
49
50-
51*
'
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.33087
.49448
.34394
.52425
.35721
.55572
.37068
.58902
1
.33109
.49496
.34416
.52476
.35744
.55626
.37091
.58959
1
2
.33130
.49544
.34438
.52527
.35766
.55680
.37113
.59016
2
3
.33152
.49593
.34460
.52579
.35788
.55734
.37136
.59073
3
4
.33173
.49641
.34482
.52630
.35810
.55789
.37158
.59130
4
5
.33195
.49690
.34504
.52681
.35833
.55843
.37181
.59188
5
6
.33217
.49738
.34526
.52732
.35855
.55897
.37204
.59245
6
7
.33238
.49787
.34548
.52784
.35877
.55951
.37226
.59302
7
8
.33260
.49835
.34570
.52835
.35900
.56005
.37249
.59360
8
9
.33282
.49884
.34592
.52886
.35922
.56060
.37272
.59418
9
10
.33303
.49933
.34614
.52938
.35944
.56114
.37294
.59475
10
11
.33325
.49981
.34636
.52989
.35967
.56169
.37317
.59533
11
12
.33347
.50030
.34658
.53041
.35989
.56223
.37340
.59590
12
13
.33368
.50079
.34680
.53092
.38011
.56278
.37362
.59648
13
14
.33390
.50128
.34702
.53144
.36034
.56332
.37385
.59706
14
15
.33412
.50177
.34724
.53196
.36056
.56387
.37408
.59764
15
16
.33434
.50226
.34746
.53247
.36078
.56442
.37430
.59822
16
17
.33455
.50275
.34768
.53299
.36101
.56497
.37453
.59880
17
18
.33477
.50324
.34790
.53351
.36123
.56551
.37476
.59938
18
19
.33499
.50373
.34812
.53403
.36146
.56606
.37498
.59996
19
20
.33520
.50422
.34834
.53455
.36168
.56661
.37521
.60054
20
21
.33542
.50471
.34856
.53507
.36190
.56716
.37544
.60112
21
22
.33564
.50521
.34878
.53559
.36213
.56771
.37567
.60171
22
23
.33586
.50570
.34900
.53611
.36235
.56826
'.37589
.60229
23
24
.33607
.50619
.34923
.53663
.36258
.56881
.37612
.60287
24
25
.33629
.50669
.34945
.53715
.36280
.56937
.37635
.60346
25
26
.33651
.50718
.34967
.53768
.36302
.56992
.37658
.60404
26
27
.33673
150767
.34989
.53820
.36325
.57047
.37680
.60463
27
28
.33694
.50817
.35011
.53872
.36347
.57103
.37703
.60521
28
29
.33716
.50866
.35033
.53924
.36370
.57158
.37726
.60580
29
30
.33738
.50916
.35055
.53977
.36392
.57213
.37749
.60639
30
31
.33760
.50966
.35077
.54029
.36415
.57269
.37771
.60698
31
32
.33782
.51015
.35099
.54082
.36437
.57324
.37794
.60756
32
33
.33803
.51065
.35122
.54134
.36460
.57380
.37817
.60815
33
34
.33825
.51115
.35144
.54187
.36482
.57436
.37840'
.60874
34
35
.33847
.51165
.35166
.54240
.36504
.57491
.37'862
.60933
35
36
.33869
.51215
.35188
.54292
.36527
.57547
.37885
.60992
36
37
.33891
.51265
.35210
.54345
.36549
.57603
.37908
.61051
37
38
.33912
.51314
.35232
.54398
.36572
.57659
.37931
.61111
38
39
.33934
.51364
.35254
.54451
.36594
.57715
.37954
.61170
39
40
.33956
.51415
.35277
.54504
.36617
.57771
.37976
.61229
40
41
.33978
.51465
.35299
.54557
.36639
.57827
.37999
.61288
41
42
.34000
.51515
.35321
.54010
.36662
.57883
.38022
.61348
42
43
.34022
.51565
.35343
.54663
.36684
.57939
.38045
.61407
43
44
.34044
.51615
.33365
.54716
.36707
.57995
.38068
.61467
44
45
.34065
.51665
.35388
.54769
.36729
.58051
.38091
.61526
45
46
.34087
.51716
.35410
.54822
.36752
.58108
.38113
.61586
46
47
.34109
.51766
.35432
.54876
.36775
.58164
.38136
.61646
47
48
.34131
.51817
.35454
.54929
.36797
.58221
.38159
.61705
48
49
.34153
.51867
.35476
.54982
.36820
.58277
.38182
.61765
49
50
.34175
.51918
.35499
.55036
.36842
.58333
.38205
.61825
50
51
.34197
.51968
.35521
.55089
.36865
.58390
.38228
.61885
51
52
.34219
.52019
.35543
.55143
.36887
.58447
.38251
.61945
52
53
.34241
.52069
.35565
.55196
.36910
.5850",
.38274
.62005
53
54
.34262
.52120
.35588
.55250
.36932
.585GO
.38296
.62065
54
55
.34284
.52171
.35610
.55303
.36955
.58617
.38319
.62125
55
56
.34306
.52222
.35632
.55357
.36978
.58674
.38342
.62185
56
57
.34328
.52273
.35654
.55411
.37000
.58731
.38365
.62246
57
58
.34350
.52323
.35677
.55465
.37023
.58788
.38388
.62306
58
59
.34372
.52374
.35699
.55518
.37045
.58845
.38411
.62366
59
60
.34394
.52425
.35721
.55572
.37068
.58902
.38434
.62427
60
99
NATURAL VERSED SINES AND EXTERNAL SECANTS,
52
53
54
55
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.38434
.62427
.39819
.66164
.41221
.70130
.42642
.74345
1
.38457
.62487
.39842
.66228
.41245
.70198
.42666
.74417
1
o
.38480
.62548
.39865
.66292
.41269
.70267
.42690
.74490
2
3
.38503
.62609
.39888
.66357
.41292
.70335
.42714
.74562
3
4
.38526
.62669
.39911
.66421
.41316
.70403
.42738
.74635
4
5
.38549
.62730
.39935
.66486
.41339
.70472
.42762
.74708
5
6
.38571
.62791
.39958
.66550
.41363
.70540
.42785
.74781
6
7
.38594
.62852
.39981
.66615
.41386
.70609
.42809
.74854
7
8
.38617
.62913
.40005
.66679
.41410
.70677
.42833
.74927
8
9
.38640
.62974
.40028
.66744
.41433
.70746
.42857
.75000
9
10
.38663
.63035
.40051
.66809
.41457
.70815
.42881
.75073
10
11
.38686
.63096
.40074
.66873
.41481
.70884
.42905
.75146
11
12
.38709
.63157
.40098
.66938
.41504
.70953
.42929
.75219
12
13
.38732
.63218
.40121
.67003
.41528
.71022
.42953
.75293
13
14
.38755
.63279
.40144
.67068
.41551
.71091
.42976
.75366
14
15
.38778
.63341
.40168
.67133
.41575
.71160
.43000
.75440
15
16
.38801
.63402
.40191
.67199
.41599
.71229
.43024
.75513
16
17
.38824
.63464
.40214
.67264
.41622
.71298
.43048
.75587
17
18
.38847
.63525
.40237
.67329
.41646
.71368
.43072
.75661
18
19
.38870
.6358?
.40261
.67394
.41670
.71437
.43096
.75734
19
20
.38893
.63648
,40284
.67460
.41693
.71506
.43120
.75808
20
21
.38916
.63710
.40307
.67525
.41717
.71576
.43144
.75882
21
22
.38939
.63772.
.40331
.67591
.41740
.71646
.43168
.75956
22
23
.38962
.63834
.40354
.67656
.41764
.71715
.43192
.76031
23
24
.38985
.63895
.40378
.67722
.41788
.71785
.43216
.76105
24
25
.39009
.63957
.40401
.67788
.41811
.71855
.43240
.76179
25
26
.39032
.64019
.40424
.67853
.41835
.71925
.43264
.76253
26
27
.39055
.64081
.40448
.67919
.41859
.71995
.43287
.76328
27
28
.39078
.64144
.40471
.67985
.41882
.72065
.43311
.76402
28
29
.39101
.64206
.40494
.68051
.41906
.72135
.43335
.76477
29
30
.39124
.64268
.40518
.68117
.41930
.72205
.43359
.76552
30
31
.39147
.64330
.40541
.68183
.41953
.72275
.43383
.76626
31
32
.39170
.64393
.40565
.68250
.41977
.72346
.43407
.76701
32
33
.39193
.64455
.40588
.68316
.42001
.72416
.43431
.76776
33
34
.39216
.64518
.40611
.'68382
.42024
.72487
.43455
.76851
34
35
.39239
.64580
.40635
.68449
.42048
.72557
.43479
.76926
35
36
.39262
.64643
.40658
.68515
.42072
.72628
.43503
.77001
36
37
.39286
.64705
.40682
.68582
.42096
.72698
.43527
.77077
37
38
.39309
.64768
.40705
.68648
.42119
.72769
.43551
.77152
38
39
.39332
.64831
.40728
.68715
.42143
.72840
.43575
.77227
39
40
.39355
.64894
M0752
.68782
.42167
.72911
.43599
.77303
40
41
.39378
.64957
.40775
.68848
.42191
.72982
.43623
.77378
41
42
.39401
.65020
.40799
.68915
.42214
.73053
.43647
.77454
42
43.
.39424
.65083
.40822
.68982
.42238
.73124
.43671
.77530
43
44
.39447
.65146
.40846
.69049
.42262
.73195
.43695
.77606
44
45
.39471
.65209
.40869
.69116
.42285
.73267
.43720
.77681
45
46
.39494
.65272
.40893
.69183
.42309
.73338
.43744
.77757
46
47
.39517
.65336
.40916
.69250
.42333
.73409
.43768
.77833
47
48
.39540
.65399
.40939
.69318
.42357
.73481
.43792
.77910
48
49
.39563
.65462
.40963
.69385
.42381
.73552
.43816
.77986
49
50
.39586
.65526
.40986
.69452
.42404
.73624
.43840
.78062
50
51
.39610
.65589
.41010
.69520
.42428
.73696
.43864
.78138
51
52
.39633
.65653
.41033
.69587
.42452
.73768
.43888
.78215
52
53
.39656
.65717
.41057
.69655
.42476
.73840
.43912
.78291
53
54
.39679
.65780
.41080
.69723
.42499
.73911
.43936
.78368
54
55
.39702
.65844
.41104
.69790
.42523
.73983
.43960
.78445
55
56
.39726
.65908
.41127
.69858
.42547
74056 !
.43984
.78521
56
57
.39749
.65972
.41151
.69926
.42571
.74128 i
.44008
.78598
57
58
.39772
.66036
.41174
.69994
.42595
.74200 i
.44032
.78675
68
59
.39795
.66100
.41198
.70062
.42619
.74272
.44057
.78752
59
60
.39819
.66164
.41221
.70130
.42642
.74345
.44081
.78829
60
100
NATURAL VERSED SINES AND EXTERNAL SECANTS.
t
56
57
58
59"
/
Vers.
Ex. sec
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.44081
.78829
.45536
.83608
.47008
.88708
.48496
.94160
1
.44105
.78906
.45560
.83690
.47033
.88796
.48521
.94254
1
2
.44129
.78984
.45585
.83773
.47057
.88884
.48546
.94349
2
3
.44153
.79061
.45609
.83855
.47082
.88972
.48571
.94443
3
4
.44177
.79138
.45634
.83938
.47107
.89060
.48596
.94537
4
5
.44201
.79216
.45658
.84020
.47131
.89148
.48621
.94632
5
6
.44225
.79293
.45683
.84103
.47156
.89237
.48646
.94726
6
7
.44250
.79371
.45707
.84186
.47181
.89825
.48671
.94821
7
8
.44274
.79449
.45731
.84269
.47206
.89414
.48696
.94916
8
9
.44298
.79527
.45756
.84352
.47230
.89503
.48721
.95011
9
10
.44322
.79604
.45780
.84435
.47255
.89591
.48746
.95106
10
11
.44346
.79682
.45805
.84518
.47280
.89680
.48771
.95201
11
12
.44370
.79761
.45829
.84601
.47304
.89769
.48796
.95296
12
13
.44395
.79839
.45854
.84685
.47329
.89858
.48821
.95302
13
14
.44419
.79917
.45878
.84768
.47354
.89948
.48846
.95487
14
15
.44443
.79995
.45903
.84852
.47379
.90037
.48871
.95583
15
16
.44467
.80074
.45927
.84935
.47403
.90126
.48896
.95678
16
17
.44491
.80152
.45951
.85019
.47428
.90216
.48921
.95774
17
18
.44516
.80231
.45976
.85103
.47453
.90305
.48946
.95870
18
19
.44540
.80309
.46000
.85187
.47478
.90395
.48971
.95966
19
20
.44504
.80388
.46025
.85271
.47502
.90485
.48996
.96062
20
21
.44588
.80467
.46049
.85355
.47527
.90575
.49021
.96158
21
22
.44612
.80546
.46074
.85439
.47552
.90605
.49046
.96255
22
23
.44637
.80625
.46098
.85523
.47577
.90755
.49071
.96351
23
24
.44661
.80704
.46123
.85608
.47601
.90845
.49096
.96448
24
25
.44685
.80783
.46147
.85692
.47626
.90935
.49121
.96544
25
26
.44709
. 80862
.46172
.85777
.47651
.91026
.49146
.96641
26
27
.44734
. $0942
.46196
.85861
.47676
.91116
.49171
.967'38
27
28
.44758
.81021
.46221
.85946
.47701
.91207
.49196
.96835
28
29
.44782
.81101
.46246
.86031
.47725
.91297
.49221
.96932
29
30
.44806
.81180
.46270
.86116
.47750
.91388
.49246
.97029
30
31
.44831
.81260
.46295
.86201
.47775
.91479
.49271
.97127
31
32
.44855
.81340
.46319
.86286
.47800
.91570
.49296
.97224
32
33
.44879
.81419
.46344
.86371
.47825
.91661
.49321
.97322
33
34
.44903
.81499
.46368
.86457
.47&49
.91752
.49346
.97420
34
35
.44928
.81579
.46393
.86542
.47874
.91844
.49372
.97517
35
36
.44952
.81659
.46417
.86627
.47899
.91935
.49397
.97615
36
37
.44976
.81740
.46442
.86713
.47924
.92027
.49422
.97713
37
38
.45001
.81820
.46466
.86799
.47949
.92118
.49447
.97811
38
39
.45025
.81900
.46491
.86885
.47974
.92210
.49472
.97910
39
40
.45049
.81981
.46516
.86990
.47998
.92302
.49497
.98008
40
41
.45073
.82061
.46540
.87056
.48023
.92394
.49522
.98107
41
42
.45098
.82142
.46565
.87142
.48048
.92486
.49547
.98205
42
43
.45122
.82222
.46589
.87229
.48073
.9.2578
.49572
.98304
43
44
.45146
.82303
.46614
.87315
.48098
.92670
.49597
.98403
44
45
.45171
.82384
.46639
.87401
.48123
.92762
.49623
.98502
45
46
.45195
.82465
.46663
.87488
.48148
.92855
.49648
.98601
46
47
.45219
.82546
.46688
.87574
.48172
.92947
.49673
.98700
47
48
.45244
.82627
.46712
.87661
.48197
.93040
.49698
.98799
48
49
.45268
.82709
.46737
.87748
.48222
.93133
.49723
.98899'
49
50
.45292
.82790
.46762
.87834
.48247
.93226
.49748
.98998
50
51
.45317
.82871
.46786
.87921
.48272
.93319
.49773
.99098
51
52
.45341
.82953
.46811
.88008
.48297
.93412
.49799
.99198
52
53
.45365
.83034
.46836
.88095
.48322
.93505
.49824
.99298
53
54
.45390
.83116
.46860
.88183
.48347
.93598
.49849
.99398
54
55
.45414
.831^8
.46885
.88270
.48372
.93692
.49874
.99498
55
56
.45439
.83*80
.46909
.88357
.48396
.93785
.49899
.99598
56
57
.45463
.83362
.46934
.88445
.48421
.93879
.49924
.99698
57
58
.45487
.83444
.46959
.88532
.48446
.93973
.49950
.99799
58
59
.45512
.83526
.46983
.88620
.48471
.94066
.49975
.99899
59
60
.45536
.83608
.47008
.88708
.48-496
.94160 1
.50000
.00000
60
101
NATURAL VERSED SINES AND EXTERNAL SECANTS,
6
6
1
6
2"
6
3
Vers.
Ex .se
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.50000
1.00000
.51519
1.06267
.53053
1.13005
.54601
1.20209
1
.50025
1.00101
.51544
1.06375
.53079
1.13122
.54627
1.20395
1
2
.50050
1.00202
.51570
1.00483
.53104
.13239
.54653
1.20521
2
3
.50076
1.00303
.51595
1.06592
.53130
.13356
.54679
1.20647
3
4
.50101
.00404
.51621
1.00701
.53156
.13473
.54705
1.20773
4
5
.50126
.00505
.51646
.06809
.53181
.13590
.54731
1.20900
5
6
.50151
.00607
.51672
.06918
.53207
.13707
.54757
1.21026
6
7
.50176
.00708
.51697
.07027
.53233
.13825
.54782
1.21153
7
8
.50202
.00810
.51723
.07137
.53258
.13942
.54808
1.21280
8
.50227
.00912
.51748
.07246
.53284
.14060
.54834
1.21407
10
.50252
.01014
.51774
.07356
.53310
.14178
.54860
1.21535
10
11
.50277
.01116
.51799
.07465
.53336
.14296
.54886
1.21662
11
12
.50303
.01218
.51825
.07575
.53301
.14414
.54912
1.21790
12
13
.50328
.01320
.51850
.07685
.53387
.14533
.54938
1.21918
13
14
.50353
.01422
.51876
.07795
.53413
.14651
.54904
1.22045
14
15
.50378
.01525
.51901
.07905
.53439
.14770
.54990
1.22174
15
10
.50404
.01628
.51927
.08015
.53404
.14889
.55016
1.22302
10
17
.50429
.01730
.51952
.08126
.53490
.15008
.55042
1.22430
17
18
.50454
.01833
.51978
.08236
.53516
.15127
.55008
.22559
18
19
.50479
.01936
.52003
.08347
.53542
.15246
.55094
.22688
19
20
.50505
.02039
.52029
.08458
.53567
.15366
.55120
.22817
20
21
.50530
.02143
.52054
.08569
.53593
.15485
.55146
.22946
21
22
.50555
.02246
.52080
.08680
.53619
.15605
.55172
.23075
22
23
.50581
.02349
.52105
.08791
.53645
.15725
.55198
.23205
23
24
.50000
.02453
.52131
.08903
.5367'0
1.15845
.55224
.23334
21
25
.50031
.02557
.52156
.09014
.53696
1.15965
.55250
.23464
25
20
.50056
.02661
.52182
1.09126
.53722
1.16085
.55276
.23594
20
27
.50682
.02765
.52207
1.09238
.53748
1.10206
.55302
.23724
27
28
.50707
.02869
.52233
1.09350
.53774
1.16326
.55328
.23855
28
29
.50732
.02973
. 52259
1.09462
.53799
1.16447
.55354
.23985
29
30
.50758
.03077
.52284
1.09574
.53825
1.16568
.55380
.24116
30
31
.50783
.03182
.52310
1.09686
.53851
1.16689
.55406
.24247
31
32
.50808
.03286
.52335
1.09799
.53877
1.16810
.55432
.24378
32
33
.50834
.03391
.52361
1.09911
.53903
1.10932
.55458
.24509
33
34
.50859
.03496
.52386
1.10024
.53928
1.17053
.55484
1.24640
34
35
.50884
.03601
.52412
1.10137
.53954
1.17175
.55510
1.24772
35
30
.50910
.03706
.52438
1.10250
.53980
1.17297
.55536
1.24903
36
37
.50935
.03811
.52463
1.10363
.54006
1.17419
.55563
1.25035
37
:-;s
.50960
.03916
.52489
1.10477
.54032
1.17541
.55589
1.25167
38
39
.50986
.04022
.52514
1.10590
.54058
1.17003
.55015
1.25300
39
40
.51011
.04128
.52540
1.10704
.54083
1.17786
.55641
1.25432
40
41
.51036
.04233
.52566
1.10817
.54109
1.17909
.55667
1.25565
41
42
.51062
.04339
.52591
1.10931
.54135
1.18031
.55093
1.25697
42
43
.51087
.04445
.52617
1.11045
.54161
1.18154
.55719
.25830
43
44
.51113
.04551
.52642
1.11159
.54187
1.18277
.55745
.25963
44
45
.51138
.04658
.52668
1.11274
.54213
1.18401
.55771
.26097
45
40
.51163
.04764
.52694
1.11388
.54238
1.18524
.55797
.26230
40
47
.51189
.04870
.52719
1.11503
.54264
1.18648
.55823
.26364
47
48
.51214
.04977
.52745
1.11617
.54290
.18772
.55849
.26498
48
49
.51239
.05084
.52771
1.11732
.54316
.18895
.55876
.26032
49
50
.51265
.05191
.52796
1.11847
.54342
.19019
.55902
.26766
50
51
.51290
.05298
.52822
1.11963
.54368
.19144
.55928
.26900
51
52
.51316
.05405
.52848
1.12078
.54394
.19268
.55954
.27035
52
53
.51341
.05512
.52873
1.12193
.54420
.19393
.55980
.27169
53
54
.51366
.05619
.52899
1.12309
.54446
.19517
.56006
.27304
54
55
.51392
.05727
.52924
1.12425
.54471
.19642
.56032
1.27439
55
50
.51417
.05835
.52950
1.12540
.54497
. 19767
.56058
1.27574
50
57
.51443
.05942
.52976
1 . 12657
.54523
.19892
.56084
1.27710
57
58
.51468
.06050
.53001
1.12773
.54549
.20018
.56111
1.27845
58
59
.51494
.06158
.53027
1.12889
.54575
.20143
.56137
1.27981
59
60
.51519
.06267
.53053
1.13005
.54601
.20269
.56163
1.28117
60
103
NATURAL VERSED SINES AND EXTERNAL SECANTS,
G
40
6
5
6
6
6
7
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
Vers.
Ex. sec.
.56163
.28117
.57738
1.36620
.59326
1.45859
.60927
1.559J30
1
.56189
.28253
.57765
1.36768
.59353
1.46020
.60954
1.56106
1
2
.56215
.28390
.57791
1.36916
.59379
1.46181
.60980
1.56282
2
3
.56241
.28526
.57817
1.37064
.59406
1.46342
.61007
1.56458
3
4
.56267
.28663
.57844
1.37212
.59433
1.46504
.61034
1.56634
4
5
.56294
1.28800
.57870
1.37361
.59459
1.46665
.61061
1.56811
5
6
.56320
1.28937
.57896
1.37509
.59486
1.46827
.61088
1.56988
6
7
.56346
1.29074
.57923
1.37658
.59512
1.46989
.61114
1.57165
7
8
.56372
1.29211
.57919
1.37808
.59539
1.47152
.61141
1.57342
8
9
.56398
1.29349
.57976
1.37957
.59566
1.47314
.61168
1.57520
9
10
.56425
1.29487
.58002
1.38107
.59592
1.47477
.61195
1.57698
10
11
.56451
1.29625
.58028
1.38256
.5961D
1.47640
.61222
1.57876
11
12
.56477
1-29763
.58055
1.38406
.59645
1.47804
.61248
1.58054
12
13
.56503
1.29901
.58081
1.38556
.59672
1.47967
.61275
1.58233
13
11
.56529
1.30040
.58108
1.38707
.59699
1.48131
.61302
1.58412
14
15
.56555
1.30179
.58134
1.38857
.59725
1.48295
.61329
1.58591
15
16
.56582
1.30318
.58160
1.39008
.59752
1.48459
.61356
1.58771
1G
17
.56608
1.30457
.58187
1.39159
.59779
1.48624
.61383
1.58950
17
18
.56634
1.30596
.58213
1.39311
.59805
1.48789
.61409
1.59130
18
1!)
.566(50
1.30735
.58240
1.39462
.59832
1.48954
.61436
1.59311
19
20
.56687
1.30875
.58206
1.39614
.59859
1.49119
.61403
1.59491
20
m
.56713
1.31015
.58293
1.39766
.59885
1.49284
.61490
1.59672
21
22
.56730
1.31155
.58319
1 .33918
.59912
1.49450
.61517
1.59853
22
33
.56765
1.31295
.58345
1.40070
.59938
1.49616
.61544
1.60035
23
21
.5679!
1.31436
.58372
1.40222'
.59965
1.49782
.61570
1.60217
24
25
.56818
1.31576
.58398
1.40375
.59992
1.49948
.61597
1.60399
25
20
.56844
1.31717
..58425
1.40528
.60018
1.50115
.61624
1.60581
2G
27
.56870
1.31858
.58451
1.40681
.60045
1.50282
.61651
1.60763
27
28
.56896
1.31999
.58478
1 .40835
.60072
1.50449
.61678
1.60946
28
2!)
.56923
1.32140
.58504
1.40988
.60098
1.50617
.61705
1.61129
29
30
.56949
1.32282
.58531
1.41142
.60125
1.50784
.61732
1.61313
30
31
.56975
1.32424
.58557
1.41296
.60152
1.50952
.61759
.61496
31
32
.57001
1.32566
.58584
1.41450
.60178
1.51120
.61785
.61680
32
33
.57028
1.32708
.58610
1.41605
.60205
1.51289
.61812
.61864
33
34
.57054
1.32850
.58637
1.41760
.60232
1.51457
.61839
.62049
34
35
.57080
1.32993
.58663
1.41914
.60259
1.51626
.61866
.62234
35
36
.57106
1.33135
.58690
1.42070
.60285
1.51795
.61893
.62419
36
37
.57133
1.33278
.58716
1.42225
.60312
1.51965
.61920
.62604
ffl
38
.57159
.33422
.58743
1.42380
.60339
1.52134
.61947
.62790
38
39
.57185
.33565
.58769
1.42536
.60365
1.52304
.61974
.62976
39
40
.57212
.33708
.58796
1.42692
.60392
1.52474
.62001
.63162
40
41
.57238
.33852
.58822
1.42848
.60419
1.52645
.62027
.63348
41
42
.57264
.33996
.58849
1.43005
.60445
1.52815
.62054
.63535
42
43
.57291
.34140
.58875
1.43162
.60472
1.52986
.62081
.63722
43
44
.57317
.34284
.58902
1.43318
.60499
1.53157
.62108
.63909
44
45
.57343
.34429
.58928
1.43476
.60526
1.53329
.62135
.64097
45
46
.57369
.34573
.58955
1.43633
.60552
1.53500
.62162
.64285
46
47
.57396
.34718
.58981
1.43790
.60579
1.53672
.62189
.64473
47
48
.57422
.34863
.59008
1.43948
.60606
1.53845
.62216
1.64662
48
49
.57448
.35009
.59034
1.44106
.60633
1.54017
.62243
1.64851
49
50
.57475
.35154
.59061
J. 44264
.60659
1.54190
.62270
1.65040
50
51
.57501
.35300
.59087
.44423
.60686
1.54363
.62297
1.65229
51
52
.57527
.35446
.59114
.44582
.60713
1.54536
.62324
.65419
52
53
.57554
.35592 !
.59140
.44741
.00740
1.54709
.62351
.65609
53
54
.57580
.35738
.59167
.44900
.60766
1.54883
.62378
.65799
54
55
.57606
.35885
.59194
.45059
.60793
1.5505?
.62405
.65989
55
56
.57633
.36031
.59220
.45219
.60820
1.55231
.62431
.66180
56
57
.57659
.36178
.59247
.45378
.60847
1.55405
.62458
.66371
57
58
.57685
.36325
.59273
1.45539
.60873
1.55580
.62485
.66563
58
59
.57712
.36473
.59300
1.45699
.60900
1.55755
.62512
.66755
59
60
.57738
.36620
.59326
1.45859
.60927
1.55930
.62539
.66947
60
103
NATURAL VERSED SINES AND EXTERNAL SECANTS.
/
68-
(
;o
i
'0
i
n
Vers.
Ex. sec
Vers.
Ex. sec
Vers.
Ex. sec
Vers.
Ex. sec.
.62539
1.66947
.64163
1.79043
.65798
1.92380
.67443
2.07155
.62566
1.67139
.64190
1.79254
.65825
1.92614
.67471
2.07415
]
.62593
1.67332
.64218
1.79466
.65853
1.92849
.67498
2.07675
j
.62620
1.67525
.64245
1.79679
.65880
1.93083
.67526
2.07936
j
<
.62647
1.67718
.64272
1.79891
.65907
1.93318
.67553
2.08197
i
.62674
1.67911
.64299
1.80104
.65935
1. 935*4
.67581
2.08459
t
1
.62701
1.68105
.64326
1.80318
.65962
1.93790
.67608
2.08721
6
'
.62728
1.68299
.64353
1.80531
.65989
1.94026
.67636
2.08983
7
.62755
1.68494
.64381
1.80746
.66017
1.94263
.67663
2.09246
8
J
.62782
1.68689
.64408
1.80960
.66044
1.94500
.67691
2.09510
9
10
.62809
1.68884
.64435
1.81175
.66071
1.94737
.67718
2.09774
10
11
.62836
1.69079
.64462
1.81390
.66099
1.94975
.67746
2.10038
1
12
.62863
1.69275
.64489
1.81605
.66126
1.95213
.67773
2.10303
<
13
.62890
1.69471
.64517
1.81821
.66154
1.95452
.67801
2.10568
j
14
.62917
1.69667
.64544
1.82037
.66181
1.95691
.67829
2.10834
^
15
.62944
1.69864
.64571
1.82254
.66208
1.95931
.67856
2.11101
t
16
.62971
1.70061
.64598
1.82471
.06236
1.96171
.67884
2.11367
16
17
.62998
1.70258
.64625
1.82688
.66263
1.96411
.67911
2.11635
17
18
.63025
1.70455
.64653
1.8S906
.66290
1.96652
.67939
2.11903
18
19
.63052
1.70653
.64680
1.83124
.66318
1.96893
.67966
2.12171
19
20
.63079
1.70851
.64707
1.83342
.66345
1.97135
.67994
2.12440
20
21
.63106
1.71050
.64734
1.83561
.66373
1.97377
.68021
2.12709
21
22
.63133
1.71249
.64761
1.83780
.G6400
1.97619
.08049
2.12979
22
2?
.63161
1.71448
.64789
1.83999
.66427
1.97862
.68077
2.13249
23
24
.63188
1.71647
.64816
1.84219
.66455
1.98106
.68104
2.13520
24
25
.63215
1.71847
.64843
1.84439
.66482
1.98349
.68132
2.13791
25
26
.63243
1.72047
.64870
1.84659
.66510
1.98594
.68159
2.14063
26
27
.63269
1.72247
.04898
1.84880
.66537
1.98838
.68187
2.14335
27
28
.63298
1.7'2448
.64925
1.85102
.66504
1.99083
.68214.
2.14608
28
29
.63323
1.72649
.64952
1.85323
.66592
1.99329
.68242
2.14881
29
30
.63350
1.72850
.64979
1.85545
.66619
1.99574
.68270
2.15155
30
31
.63377
1.73052
.65007
1.85767
.66647
1.99821
.68297
2.15429
31
32
.63404
1.73254
.65034
1.85990
.66674
2.00067
.68325
2.15704
32
33
.63431
1.73456
.65061
1.86213
.66702
2.00315
.68352
2.15979
33
34
.63458
1.73659
.65088
1.86437
.66729
2.00562
.68380
2.16255
34
35
.63485
1.73862
.65116
1.86661
.66756
2.00810
.68408
2.16531
35
36
.63512
1.74065
.65143
1.86885
.66784
2.01059
.68435
2.16808
36
37
.63539
1.74269
.65170
1.87109
.66811
2.01308
.68463
2.17085
37
38
.63566
1.74473
.65197
1.87334
.66839
2.01557
.68490
2.17363
38
39
.63594
1.74677
.65225
1.87560
.66866
2.01807
.68518
2.17641
39
40
.63621
1.74881
.65252
1.87785
.66894
2.02057
.68546
2.17920
40
41
.63648
1.73086
.65279
1.88011
.66921
2.02308
.68573
2.18199
41
42
.63675
1.75292
.65306
1.88238
.66949
2.02559
.68601
2.18479
42
43
.63702
1.75497
.65334
1.88465
.6697'6
2.02810
.68628
2.18759
43
44
.63729
1.75703
.65361
1.88692
.67003
2.03062
.68656
2.19040
44
45
.63756
1.75909
.65388
1.88920
.67031
2.03315
.68684
2.19322
45
:6
.63783
1.76116
.65416
1.89148
.67058
2.03568
.68711
2.19604
46
:7
.63810
1.76323
.65443
1.89376
.67086
2.03821
.68739
2.19886
47
48
.63838
1.76530
.65470
1.89605
.67113
2.04075
.68767
2.20169
48
49
.63865
1.76737
.65497
1.89834
.67141
2.04329
.68794
2.20453
49
50
.63893
1.76945
.65525
1.90063
.67168
2.04584
.68822
2.20737
50
51
.63919
1.77154
.65552
1.90293
.67196
2.04839
.68849
2.21021
51
52
.63946
1.77362
.65579
1.90524
.67223
2.05094
.68877
2.21306
52
53
.63973
1.77571
.65607
1.90754
.67251
2.05350
.68905
2.21592
53
54
.64000
1.77780
.65634
1.90986
.67278
2.05607
.68932
2.21878
54
55
.64027
1.77990
.65661
1.91217
.67306
2.05864
.68960
2.22165
55
56
.64055
1.78200
.65689
1.91449
.67333
2.06121
.68988
2.22452
)6
57
.64082
1.78410
.65716
1.91681
.67361
2.06379
.69015
2.22740
>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