"9-NRLF 572 273 WALLS 7 LIBRARY OF THE UNIVERSITY OF CALIFORNIA. l^ceived j^) 12 1893 .189 Accessions No.l\C(C(C\L^ . Class No, H * - RETAINING- WALLS FOR EARTH. THE THEORY OF EARTH-PRESSURE AS DEVELOPED FROM THE ELLIPSE OF STRESS. AN APPENDIX PRESENTING THE THEORY OF PROF. WEYRAUCH. BY MALVERD A. HOWE, O.E., Professor of Civil Engineering, Rose Polytechnic Institute. Seconlr Btiition, 3acbtscti aitlr NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1891, *Wx^ THE 'UHIVERSITTl COPYRIGHT, 1891, BY JOHN WILEY & SONS. FERRIS BROS., ROBERT DRCMMOND, printers, Electrotype,; pearl street| 414 & 446 Pearl Street, N Mew York. CONTENTS. PART I. PAGE NOMENCLATURE, vii FORMULAS FOR THE THRUST OF EARTH, 1 FORMULAS FOR THE BREADTH OF BASE OF A WALL, .... 6 FORMULAS FOR THE DEPTH OF FOUNDATIONS 9 EXAMPLES, 11 PART II. DEMONSTRATION OF THE FORMULAS FOR THE THRUST OF EARTH, . . 27 DEMONSTRATION OF THE FORMULAS FOR THE BREADTH OF THE BASE OF A WALL, 48 DEMONSTRATION OF THE Fo MULAS FOR THE DEPTH OF FOUNDATIONS, 54 APPENDIX. WEYRAUCH'S THEORY OF EARTH-PRESSURE, ...... 59 REFERENCES, 103 DIAGRAM I, 10? TABLES, ....,,.,...,,. 109, PREFACE. THE first edition of this work was based upon the theory advanced by Prof. Weyrauch in 1878, but owing to the length of the demonstrations used by him, it was thought advisable to present different and shorter demonstrations in this edition. To show that the new demonstrations give identical results with those obtained by Prof. Weyrauch, his demonstrations have been given in an appendix as they appeared in the first edition. The new demonstrations are based upon the theory first advanced by Prof. Rankine in 1858. Those readers who are familiar with Rankine's Ellipse of Stress can omit pages 27 to 35, inclusive, in following the demonstrations. An attempt has been made to present the theory in a shape easily followed by those who have only a knowledge of algebra, geometry, and trigonometry; whenever cal- culus has been resorted to, the work has been simplified as much as possible. For convenience in practice, the formu- las have been arranged in a condensed shape in Part I, and are followed by numerous examples illustrating their application. The values of various coefficients have been computed and tabulated and will be found to very materially decrease the labor of substitution in the formulas, v vi PREFACE. It is hoped that the introduction of a brief treatment of the supporting power of earth in the case of foundations, as well as the formula for determining the breadth of the base of a retaining- wall, will prove acceptable. For valuable help in the verification of proofs of formu- las, and the critical reading of the whole text, I acknowl edge the kind assistance of Prof. Thos. Gray. M. A. H. TEBRE HAUTE, IND. , March, 1891. NOMENCLATURE. = the angle of repose, or the maximum angle which any force acting upon any plane within the mass of earth can make with the normal to the plane. e = the angle made by the surface of the earth with the horizontal; e is positive when measured above and negative when measured below the horizontal. a = the angle which the back of the wall makes with the vertical passing through the heel of the wall; a is positive when measured on the left and nega- tive when measured on the right of the vertical. 6 = the angle which the direction of the resultant earth- pressure makes with the horizontal. 0' = the angle of friction between the wall and its foun- dation. 0" = the angle of friction between the back of the wall and the earth. If = the vertical height of the wall in feet. h '= the depth of earth in feet which is equivalent to a given load placed upon the surface of the earth. B' = the width in feet of the top of the wall. B = the width in feet of the base of the wall. Q = the distance in feet from the toe of the wall to the point where R cuts the base. vii Vlll NOMENCLATURE. P the resultant earth-pressure in pounds against a ver- tical wall. E = the resultant earth-pressure in pounds against any wall. R = the resultant pressure in pounds on the base of the wall. G = the total weight in pounds of material in the wall. y = the weight in pounds of a cubic foot of earth. W = the weight in pounds of a cubic foot of wall. p = the intensity of the pressure in pounds on the base of the wall at the toe. p' the intensity of the pressure in pounds on the base of the wall at the heel. p 9 = the average intensity of the pressure in pounds on the base of the wall, x H tan a. OF THE UNIVERSITY RETAINING-WALLS FOR EARTH FORMULAS FOR EARTH-PRESSURE. IN the following formulas a and e are considered as positive, and the wall is assumed-to be one foot long. CASE I. General case of inclined earth-surface and in- clined back of wall. = % cos 2 a cos e t / \/ V . ( cos e i/cos' 2 e cos' 2 ) a in 2 a -{- cos 9 (e a) 1 ( cos e -f- I'cos* e cos 2
\ -f 2 sin esiu acos (e - a) 4 - r - _= - ( cos e + I/cos 2 e cos^ > ) ' ^ ' or V(C) tan * : sn tf cos e -f sin e cos (e oi]A , A - or tan d = - - cos (e o-) 2 RETAIN1NG-WALLS FOR EARTH. where cos e l/cos 2 e cos" , 7 . A = cos e = == . . . (a) cos e -f r cos 2 e cos 2
from to 90 and of e from to 90, vary-
ing by 5.
or for all vertical walls tlie direction of the earth-pressure
is parallel to the surface of the earth.
CASE III. The surface of the earth parallel to the surface
of repose.
E _ H^r cos (0 ^) |/sin j a -f- cos a (0 - a) ,^\
2 cos 2 a cos ^ -j- 2 sin <^ sin cos (0 fl')'
tan the
intensity q.
The following identities can be written:
P = UP + 9) + l(p - q),
and
30
RETAINING-WALLS FOR EARTH.
or the resultant intensity on the plane CB may be con-
sidered as being the resultant of two intensities, one being
the intensity of the resultant stress caused by two like prin-
cipal stresses having the same intensity |(;j -[- q), and the
other the intensity of the resultant stress caused by two
unlike principal stresses having the same intensity %(p ({)
FIG. 3.
The intensity of the resultant stress caused by the first
two principal stresses will be, by Case I, \(p + ), and the
direction of the resultant will be normal to the plane CB.
By Case II the resultant of the second pair of principal
stresses will make the angle 6 with the direction of P, and
its intensity will be ^(p q); then the resultant intensity
can be found as follows:
In Fig. 3 draw MD normal to BC, and make LD =
^(p-\-q)', with L as a centre and LD as radius, describe
an arc cutting FD at F. Then the angle LFD = LDF = 6.
Lay off LG = $(p q), and draw GD, which is the result-
THEORY OF EARTH-PRESSURE. 31
ant intensity, and the intensity of the resultant stress on
CD caused by the two principal stresses P and Q. GD
also represents the direction of the resultant stress R.
Since the intensities of the principal stresses remain con-
stant, %(p -f- q) and l(p q) will remain the same for
any inclination of the plane GB\ hence the intensity r of
the resultant depends upon the angle 6 when p and q are
given.
From Fig. 3,
GLcos2&=LM and GLsin2V=GM,
DM = DL + LM = t(p + q) + \(p - q) cos 20,
or
r = Vp* cos 2 + 2 sin* 0, . . . . (a)
which is the general expression for the intensity of the
resultant stress of a pair of principal stresses.
As the angle 6 changes, the angle ft will also change,
and it will have its maximum value when the angle
LGD 90. This is easily proven as follows:
With L as centre and GL as radius describe an arc;
then ft will have its maximum value when the line DG is
tangent to the arc; but when DG is tangent to the arc the
angle LGD is a right angle, since LG is the radius of the
arc.
(b)
from which the following can be easily obtained :
jo _ 1 sin max /?
~q ~ I + sin max"/?'
32
RETAINING- WALLS FOR EARTH.
which, expresses the limiting ratio of the intensities of the
principal stresses consistent with equilibrium, p being
greater than q.
CASE IV. Given the intensity and direction of the re-
sultant stress on any plane, and the value of max ft, to
determine the intensities and directions of the principal
stresses.
Fm. 4.
Let AD represent the given plane and GD the direction
and intensity of the resultant stress at the point D.
Draw DL normal to AD, and draw DI, making the angle
max /3 with LD. At any point J in DL describe an arc
tangent to DI, cutting GD in K and draw GL parallel
to KJ\ with L as a centre and L G as radius describe
THEORY OF EARTH-PRESSURE. 33
a circumference. This circumference will pass through G
GL
and be tangent to DI\ hence -,-ry = sin max /?.
Since sin max ft = - - - , and GL and LD are com-
P + V
ponents of r,
GL = l(p-q) and DL = (p + q);
then ND
and MD = LD- LM =
which completely determines the intensities of the principal
stresses.
According to Case III, the direction of the greater prin-
cipal stress bisects the angle between the prolongation of
LM and the line GL; hence RL represents the direction
of the greater principal stress, and that of the other is at
right angles to RL.
The above intensities and directions being determined,
the intensity of the resultant stress on any other plane
passing through D is easily determined as follows:
Let DY represent any plane passing through D, draw
DL' normal to MY and equal to %(p -f q). Draw R'D
parallel to RL, and with L' as a centre and L'D as radius
describe an arc cutting R'D at 0, and make L'G'= l-(pq)' 9
then G'D = r' = the intensity of the resultant stress on
DY.
It is clear that if the value of max /3 can be obtained
for a mass of earth that the construction of Fig. 3 can be
employed in determining the intensity of the earth-pressure
at any point in dn^ plane within the mass.
34
RETAINING- WALLS FOR EARTH.
It has been established by experiment that if a body be
placed upon a plane, that (as the plane is made to incline
to the horizontal) at some angle of inclination the body
will commence to slide down the plane, and that this angle
depends largely upon the character of the surfaces in con-
tact.
B
FIG. 5.
In Fig. 5 let AS represent a plane inclined at the angle
with the horizontal, and C any mass just on the point of
sliding down the plane. Let EC represent the weight of
the mass C, and ED and DC the components respectively
parallel and normal to the plane AB. Then DE is the
force required to just keep the mass C from sliding down
the plane, assuming the plane to be perfectly smooth, or if
the plane is rough this force represents the effect of fric-
tion.
DE
or when the mass C is about to slide, the resultant pres-
sure EC on A makes the angle > y,
~ $) s i Q LX y,
+ %) cos e DX* y.
Now DY=DG' = DG-2GX,
or
DG' - y = DG - y (p q) cos G?
= UP + 0) cos e - (p - q) cos GO,
\(P + 4) sil1 <*> :: i(/> - y) : sin e,
and
p -\- q
sin 09 = ^- ! * sin e,
P-flf
or
At . A
cos GL> = i/ 1 K i-i sin 2 e = ^
/ x
\P - fiv (^ - fl')
and since J(p + g') sin = J(^ g'),
cos oo = - - f/cos 2 e cos a 0.
gin
38 RETAINING-WALLS FOR EARTH.
Substituting this value for cos GO in the equation for
DG' . y, it becomes
1
DG' -y = %(p + q) cos e l(p q) . -. 1/cos 2 e cos 2 0,
u sin
1 p-\-q
or since - *
-r - .
sin p q
DG' . Y l(p + q) ! cos 6 ^cos 3 e cos 2 0J.
In a similar manner,
e - cos 2
and
DG' _ cos e 4/cos 2 e cos 2
DG cos e -j- 4/cos 2 e cos 2
hence
T^nr r> ^COS e 4/COS 2 COS 2
JJ(JT =
cos e -{- I/cos 2 e cos 2
Let x = the vertical distance between the two planes
and AD t then
cos e Vcos 2 e cos 2
.-.DG'-y = (x) y cose
cos e v cos e cos
which is the expression for the intensity of the resultant
earth-pressure on a vertical plane at any depth x below the
surface.
Let
cos e Vcos 2 e cos
* A = cos e - =. . . (d)
cos e -f- T cos 2 e cos 2
* See Rankine's Applied Mechanics ; Alexander's Applied Me-
chanics ; Theories of Winkler and Mohr.
THEORY OF EARTH-PRESSURE. 39
The average intensity of the resultant earth-pressure on
a vertical plane of the length x will be
and hence the total pressure will be
Since the intensities of the pressures are uniformly varying
from the surface, and increasing as x increases, the appli-
cation of the resultant thrust will be at a depth of \x be-
low the surface.
Considering the earth as an unconfined mass, the above
formula is perfectly general and can be applied under all
conditions, including the case when e is negative.
The resultant stress on any plane as AB, Fig. 6, can be
found by applying the principles of Case IV. Draw PA
parallel to RL, make AN= ZZ)and NO = LG\ then AO
represents the direction of the resultant pressure on AB.
Make AC = AO', then the area of the triangle ABC mul-
tiplied by y is the total pressure on the plane AB, and this
pressure is applied at \AE below B.
In unconfined earth this construction is perfectly gen-
eral and applies to any plane. It also applies equally well
to curved profiles. An example illustrating the applica-
tion of the method will be given in the applications. See
pages 22 and 23.
The following graphical construction, Fig. 7, is more con-
venient than that of Fig. 6.
As before, let BE represent the surface of the earth, and
40
RETAINING-WALLS FOR EARTH.
AD a, plane parallel to the surface. At any point D in
this plane, draw DE vertical and make DF DE \ draw
FG horizontal and make the angle HFD = 0.
With L as a centre, describe an arc passing through G
and tangent to MF', then with L as a centre and LF as
B|
FIG. 7.
radius, describe the circumference FON, cutting AD at N;
through N draw NO parallel to AB, then draw AC nor-
mal to AB and equal to OG. The area of the triangle
AB C multiplied by y will be the total earth-pressure on
AB. To determine the direction of the thrust prolong OG
to ft then QN is the direction of the thrust.
That this construction is equivalent to that of Fig. 6 is
THEORY OF EARTH-PRESSURE. 41
proved as follows. The triangle GLF of Fig. 7 equals the
triangle OLD of Fig. 6.
. GL'Y \(p q] and LF-y = LO-y = {(p + q).
In Fig. 6, the angle NAP = NPA = 90 $(GO e) a.
. . ONA = GO e + 2a.
In Fig. 7, the angle OLN=2e-2a. But GLN= c+e.
and GO of Fig. 7 equals A of Fig. 6.
In Fig. 7, the angle QNO = 90 - /?'.
In Fig. 6, the angle GAB = 90 - ft'.
Therefore the direction of the thrust is the same in both
constructions.
The two constructions given above are all that is re-
quired to determine the thrust of earth upon any plane
within the mass of earth, as one can be used as a check
upon the other; but as a formula is often very convenient,
a general formula will now be deduced which will enable
one to determine the values of E and d for any plane with-
in a mass of earth.
GENERAL FORMULA FOR THE THRUST OF EARTH.
In Fig. 8, let BQ represent the surface of the earth and
AB any plane upon which the earth-pressure is desired.
Draw AD parallel to BQ and let the vertical distance
42
RETAINING-WALLS FOR EARTH.
From (e) the earth-pressure upon FA is parallel to the
surface and equal to
FIG. 8.
But AF= x = //(I + tan a tan e) =
cos a cos e
2 cos
COS 8 (6 - Of)
2 ' ' '
Now the thrust P combined with the weight of the
prism ABFmn&i produce the resultant pressure upon AB.
THEORY OF EARTH-PRESSURE. 43
Then from Fig. 8,
V ~ tan a (1 -f tan a tan e)
6
H*y sin a cos (e a)
E = V( V+P sin e) a +(P cos e) a = I/ F 2 +P a + 2 FP sin e.
Substituting (/) and (g) in this it becomes
jBV cos (e - a)
-tv = - - ^ -- X
2 cos* a cos e
/ ~A A*
4/sin 2 a 4- 2 sin a sin e cos (e a) -- 1- cos 2 (e a) ,
cos e ' x cos- e
which becomes, by replacing A by its value from (d ),
2 cos 2 a cos e
-}- sin 2 a
cos e i/cos 8 e cos 2
4- 2 sin nr sin e cos (e a)
cos e-j- y cos 2 e cos 2 <> /j\
, cos e I/cos 2 e cos 2 (p ) 2
cos 2 (e a) ] ;
( cos e -f- \ cos 2 e cos- )
which is the general equation for the thrust of earth upon
any plane within the mass.
To determine the direction of the thrust of the earth,
let 8 be the angle which the direction of the thrust makes
with the horizontal; then, from Fig. 8,
y
i&n 6 = -=- - -f tan e.
P cos e '
44 RETAINING-WALLS FOR EARTH.
Substituting the values of V and P given above, this
becomes
. sin a cos e -f sm e cos (e a] A
tan d = - '- . . (la)
cos e cos (e a) A
where
cos e Vcos 2 e cos a
A = cos e =. . . (r/)
cos e -j- fcos a e cos 2
Equations (1) and (la) are readily reduced to more sim-
ple forms for special cases. These forms will be found in
Part I.
The Plane of Rupture. Although it is not necessary to
know the position of the plane of rupture in order to deter-
mine the thrust of the earth, yet it may be of interest to
know its position, which can be easily determined as fol-
lows :
The plane of rupture will be back of the wall and pass
through the heel of the wall. The resultant earth-pressure
will make the angle with the normal to this plane. Now
the tangent of the angle which the direction of the result-
ant earth -pressure on any plane makes with the horizontal
is determined from the formula
sin a
tan o = -. r r ~\- tan e.
cos (e oi)A
If GO represents the angle which the plane of rupture makes
with the vertical passing through the heel of the wall,
a = GO and d = -J- GO.
tan (0 + GO) = -. r-7 + tan e,
cos (e - GO) A
from which the value of GO can be determined for any case.
THEORY OF EARTH-PRESSURE. 46
For the case where e = 0, e being positive with respect
to the wall and negative with respect to the plane of rupture,
the above equation becomes
sin a>
tan (0 4- G?) = --- T -- 7 tan 0,
cos (0 + a?) cos
which is satisfied when GO = 90 0.
For the case where e 0,
sn
tan (0 + GO) = -
cos GO tan 2
which is satisfied when G? = 45 .
Reliability of the Preceding Theory. The preceding
theory is based upon the assumptions that the earth is a
homogeneous mass and without cohesion, and the formulas
are deduced under the assumption that the surface of the
earth is a plane.
All writers on the subject have considered the earth as a
homogeneous mass and, with a few exceptions, without
cohesion.
Old and recent experiments indicate that cohesion has
very little effect upon the pressure of the earth, which ex-
plains why it has not been considered by most writers.
The assumption of a plane earth-surface is necessary
whenever practical formulas and direct graphical construc-
tions for obtaining the thrust of the earth are obtained.
General formulas can be deduced for any character of sur-
face, but they are too complex for practical use. Tliose
graphical constructions which do not require a plane earth-
46 RETAINING- WALLS FO& EARTH.
surface are not direct in tlieir solution of the problem, but
require a series of trials to obtain the maximum thrust.
If the earth-surface is not a plane, one can be assumed
which .will give the thrust of the earth sufficiently exact
for all practical purposes.
For uncon fined earth no exceptions can be taken to the
preceding theory, the assumptions upon which it is based
being accepted, and for confined earth the theory must be
true when the direction of the principal stress passing
through the heel of the wall lies entirely within the earth.
For all cases in which a and e are positive the theories
of Rankine, Winkler, Weyraucli, and Mohr agree and give
identical results with the preceding theory, as they should,
being founded upon the same assumptions.
When a is negative Weyraiich does not consider his
theory reliable, and his equations lead to indeterminate re-
sults.
WinTcler and Mohr consider their theories reliable when-
ever the direction of the principal stress passing through
the heel of the wall lies entirely within the earth.
Rankings method of considering the case where a is
negative is equivalent to assuming that the introduction of
a wall does not affect the stresses within the mass.
It may be concluded that the preceding theory is per-
fectly exact when a and e are positive; and when a or e is
negative that the stresses obtained will be the maximum
which under any circumstances can exist.
For the case where e is negative the stress obtained
(which represents the maximum thrust the wall can have
against the earth and have equilibrium) will be considerably
larger than the actual stress (when a wall is introduced),
depending upon the magnitude of e. For small values of e
the results will be practically correct. For large values of e
THEORY OF EARTH-PRESSURE.
47
the following method can be employed in determining the
thrust of the earth. The method depends upon the assump-
tion that the pressure of the earth is normal to the back of
the wall. This may or may not be the case, but it appears
to be the most consistent assumption to make for this rare
and not important case.
Fio. 8a.
* In Fig. 80, let AB be the back of the wall and Bfi\\Q
surface of the earth. Make Ba = ab = be = cd = etc.
Some prism BAa or BAb or BAc, etc., will produce the
maximum thrust on the wall; and when this maximum
thrust is produced, the resultant pressure on the plane Aa
* See Van Nostrand's Magazine, xvn, 1877, p. 5. "New Con-
structions in Graphical Statics," by H. T. Eddy, C.E., Ph.D.
48 HETAININO-WALLS FOR EAHT8.
or A b or A c, etc., will make the angle with the normal
to the plane.
On the vertical line Ad' la,yoB.Aa'=a'b' = b'c', etc., and
draw Aa" making the angle with the normal to Aa, Ab"
making the angle with the normal to Ab, etc.; then draw
a' a", b'b", etc., perpendicular to AB, and draw a curve
through Aa", b", c" , etc. Then there will be a maximum
distance parallel to a'a" between Ad' and this curve which
will be proportional to the thrust of the earth against AB.
This maximum distance multiplied by the altitude Ac -f- 2
and the product by y, the weight of a cubic foot of earth,
will be the pressure of the earth.
This method is perfectly general and can be applied in
any case.
If the earth-pressure is assumed to have the direction
given by the formulas of the preceding theory, the con-
struction will give the same value of E, the pressure of the
earth.
Some writers assume that the direction of E makes the
angle 0" = with the normal to the back of the wall in
all cases. This assumption cannot be correct until the wall
commences to tip forward, and then it is doubtful that such
is the case unless the earth and wall are perfectly dry.
To be on the side of safety in every case, it is better to
take the direction of E as given by the above theory.
The construction of Fig. 8a will give the maximum thrust
for any assumed direction for any case.
TRAPEZOIDAL WALLS.
It will be assumed that the direction and magnitude of
the earth-pressure is known, that the position and extent
of the back of the wall and the width of the top are given,
T8EORY OF EARTH-PRESSURE.
49
to determine the width of the base for stability against over-
turning, sliding, and crushing of the material.
FIG. 9.
Stability against Overturning. Let AB CD, Fig. 9, rep-
resent a section of a trapezoidal wall, TR the direction of
the earth-thrust, JG the vertical passing through the cen-
tre of gravity of the wall, and JO the direction of the re-
sultant pressure on the base AD caused by ^and G.
As long as R cuts the base AD, the wall will be stable
against overturning. When R takes the direction JQ, the
wall may be said to be on the point of overturning; then
ON
the factor of safety against overturning is ~~., where ON
is the actual value of E, and QNthe value of E required to
make the resultant R pass through D.
Stability against Sliding. Since the wall will not slide
50
RET AININO-W ALLS FOR EARTH.
along the surface DA until the resultant R makes an angle
with the normal to DA greater than the angle of friction
0', the factor of safety against sliding can be obtained as
follows: Draw JP making the angle JMU '= 0'; then
PN
the factor of safety against sliding is -^, where PN is the
force required in the direction of E to make R make the
angle 0' with the normal to AD, and ON the actual value
of K
Stability against the Crushing of the Material. In ordi-
nary practice walls for retaining earth are not of sufficient
height to cause very large pressures at their bases, but it
is necessary to consider the subject on account of the ten-
dency of the bed-joints to open under certain conditions.
Let AB, Fig. 10, represent any bed- joint in the wall, P
the vertical resultant pressure upon the joint, and x the
distance of the point of application from the centre of the
joint.
The intensity of P can be considered as composed of a
p
uniform intensity p = j, and a uniformly varying inten-
sity p 9 ', so ih&t p x = p -\- p. Let a equal the tangent of
the angle CDE, then ;;/ ax and p x = p -j- ax.
THEORY OF EARTH-PRESSURE. 51
The pressure upon a surface (dx) the joint heing con-
sidered unity in the dimension normal to the page is
p x dx = p dx -f- axdx,
and the moment of this about DB is
(p dx -f- axdx)x.
The algebraic sum of these moments for values of x be-
tween the limits must equal Px , or
Px = (p xdx + ax*dx).
Integrating,
_
B 3
and
or -
I2xx
and if x be replaced by ^B Q, where Q is the distance
from A to the point where P cuts the base, (Fig. 11,)
and
if e=i5,
p' = and
52 RETA1NING-WALLS FOR EARTH.
from which it is seen that when R cuts the base outside
the middle third, the joint will have a tendency to open at
points which are at a maximum distance from R where it
cuts the base.
Therefore in no case should the resultant pressure be
permitted to cut the base outside the middle third. This
makes it unnecessary to consider the stability against over-
turning.
C B B
Then in designing a wall the following conditions must
exist for stability :
I. The resultant R must cut the base for stability against
overturning.
II. The resultant R must not make an angle with the
normal to the base of the wall greater than the angle of fric-
tion 0'.
THEORY OF EARTH-PRESSURE. 53
III. The resultant R must not cut the base outside of
the middle third, in order that there may be no tendency for
the bed-joints to open.
The above three conditions apply to any bed-joint of the
wall; but if they are satisfied at the base and the wall has
the section shown in Fig. 11, it will not be necessary to
consider any joints above the base unless the character of
the stone or the bonding is different.
Determination of the width of the base of a retaining-
w all under the condition that R cuts the base at a point
rom the toe of the wall.
Let H, B', x, d, and E be given to determine B.
From Fig. 11,
KF - sin -{- cos 8 -- sin tf,
DO O
_ -Bx- 2B'x - B"
nv-nn B _B* + BB>- Bx-
"~
B')
For equilibrium
E(KF) = G(HF) = E \ B ' HW(HF).
Substituting the values of A^and HFin the above and
reducing, it becomes
, . (8)
54
RETAINING -WALLS FOR EARTH.
which is the general equation for the width of the base of
a trapezoidal wall.
For a rectangular wall B' = B.
For a triangular wall B' 0.
For a wall with a vertical front B' -\-x-B or
B' = B - x.
For a wall with a vertical back x = 0.
Equation (8) is easily transformed to satisfy the require-
ments of special cases.
The width of the base can be found graphically by as-
suming a value for B and finding the value of Q-, if it is
less than %B another value of B must be assumed, and so
on until Q is equal to or greater than ^B.
Depth of Foundations. Given the angle of repose of
any earth, to determine the depth to which it is necessary
to sink a foundation to support a given load. The surface
of the earth is assumed to be horizontal.
CASE I. When the intensity of the pressure on the base
of the foundation is uniform.
In Fig. 12, let p represent the intensity of the pressure
on the base of the foundation.
THEORY OF EARTH-PRESSURE. 55
Now when the masonry is about to sink (see Eq. (c)),
p. 14- sin 1 sin
* - . ' _ ' Q| Q - ftj _ '_
q ' 1 sin 1 + sin 0'
If x' represents the depth to which the foundation extends
below the surface of the earth and y the weight of a cubic
foot of earth, then yx' equals the vertical intensity of the
earth-pressure on a plane at the depth of the lowest point
of the foundation.
When the wall is on the point of sinking, the earth must
be on the point of rising, or
q _ 1 -|- sin
yx' 1 sin 0'
or
\L + r
1 sin )
In any case p 9 must not have- a greater value than that ob-
tained from (15)
= p.
y -- sin y
_ 4>\
2J
The value of x' as obtained from (16) is the least allow-
able value consistent with equilibrium. Since x' is a func-
tion of tan 4 U5 |-j, care must be taken that is assumed
at its least value. As becomes smaller the value of x'
increases rapidly.
CASE II. When the intensity of the pressure on the base
is uniformly varying.
Let p represent the maximum intensity of the pressure
on the earth and p' the minimum intensity; then for
56 RETAINING -WALLS FOR EARTH.
equilibrium p must not exceed the value obtained from the
following equation :
Also, p r must never be less than x'y; then
7; -I-?/ X r "V (
TJ = ' ' ' \ l-|-( ~ ' ^ I \. %'y *- I "*" V" /I Q\
2 2 ( U sin0/ [ r (1 sin0) a>
which expresses the maximum value which p can have for
the equilibrium of the earth. Solving (18) for x',
I I o^2~ ^/> > .... (^/
which is the minimum value x' can have for the equilibrium
of the earth.
In order that p 9 may never be less than x'y the result-
ant pressure on the base of the foundation must cut the
base within a certain distance of the centre of the base. If
x equal this distance, then (see page 51)
Substituting the value oip a from (18) and solving for x 9 ,
1 sin
< 2 >
which is the maximum value x a can have, consistent with
the stability of the earth.
Abutting Power of Earth. Let the surface of the earth
be horizontal and the body pushing the earth have a verti-
THEORY OF EARTH-PRESSURE. 57
cal face; then at the depth x' the maximum horizontal
pressure per unit of area is (see Case I above)
, 1 -f- sin
and since q varies directly as x', the total thrust P which
the earth is capable of resisting is
_ (*')> 1 + Bin 4> m .
5} 1 - sin 0' ' W
APPENDIX.
WEYRAUCH'S
THEORY OF THE RETAINING-WALL*
Itf the following the earth is supposed without cohesion,
and its pressure is determined independently of any arbi-
trary assumptions as to direction of the earth-pressure, and
with sole reference to the three necessary conditions of
equilibrium. The single and only supposition, then, is as
follows: That the forces upon any imaginary plane-section
through the mass of earth have the same direction.
This assumption lies at the foundation of all theories of
earth-pressure against retaining-walls. For those cases,
therefore, to which the following discussion does not apply
no complete or satisfactory theory is yet possible. In
what follows, the ordinary assumption as to the direction
of the earth-pressure will be proved to be incorrect, except
for special cases.
* Zeitsclirift fur Baukunde, Band I. Ilcft 2, 1878.
60
THEORY OF THE RETAILING -WALL.
GENERAL RELATIONS.
Let the surface of the earth have any form, and the
wall AB, Fig. 1, have any inclination. The earth-pres-
sure makes any angle, d, with the normal to the wall.
Suppose through the point A the plane AC. Then the
weight G of the prism ABC is held in equilibrium by the
reaction of the wall, E, arid by the resultant, R, of all the
forces acting upon A C.
Now decompose E, G, and R into components parallel
and normal to AC; then for every unit in length of the
wall, denoting by e, g, and r the lever-arms of E, G, and
R respectively with reference to A, the sum of the forces
parallel to A C = 0, or
P-P,_P =0;
(1)
GENERAL RELATIONS. 61
the sum of the forces perpendicular to A = 0, or
Q + C. - , = o ; (3)
the sum of moments about ^4 = 0, or
Og + ^e - Rr = (3)
Equation (3) was first introduced by Prof. Weyrauch.
Further, according to the theory of friction, if cp is the
coefficient of friction for earth on earth,
P / P P Z
-^ tan ;
62 THEORY OF THE RETAINING -WALL.
and when GO refers to the surface of rupture, the earth -
pressure upon AB becomes
-p cos GO sin GO tan cp
~ sin (GO -\-a-\- d) -f- cos (GO-\- a -\- 6) tan cp
Substituting the value of tan cp or -. this becomes
cos cp
cos cp cos GO sin GO sin cp
sin (GO -j- a -f- 6) cos cp -f- cos (GO -j- a -}- d) sin <>
which becomes
cos (p + GO)
In order to refer to the surface of rupture, the following
relation must exist :
, ,'G cos GO E sin (GO -\- a -}-
fG c
\6r si
sina J + J &oo B (< B +4 1 tf) i =
^ca
Performing the differentiation indicated in the equation
to), considering 6r and <# as the variables, it becomes
+ [dG COSO) - Sin o>rfo> - J?7COS (w + a+ 8)da>l [G Sin w 4 ^COS (+ a+ )]
[d(r sin a> + cos a> +<) [1-sin 2 (a+3)],
?) cos 3 (-f
or cos* (+o v ) [sin 2
GENERAL RELATIONS.
65
which equals cos 2 ( + GL>)
cos GO sin a cos acos d sin (9? -f- GO)
-]- cos G? sin 2 a sin # sin (cp -f- 6?)
cos GO sin or cos <* sin 6 cos ( cos (or + 6)
-}- cos 2 sin 6? cos #sin (cp -{ GO)
cos a sin GO sin # sin d sin (<^ -j- GO)
+ cos 2 sin a? sin d cos (^? -f- GO)
-|-cos ^ sin GO sin # cos d cos (cp -j- G?)
HORIZONTAL EARTH-SURFACE. 75
fr
which reduces to
COS ft? COS (<>+ GO) COS S ~]
sin a cos a [sin ( ) | sin ^>,
this becomes
_ 2sin ]
cos 2 a [sin (cp-\-co) cos (^> + <*>) sin G!?COS c^
[cos 2 a cos 2 (^ -f GO) -f- sin 2 ^ cos 2 &?] tan d
-)- sin a cos [sin (cp -j- GJ) cos (<^ -f- GJ)
r- sin cy cos GO] tan ^ ,
^ = 0.
HORIZONTAL EARTH-SURFACE.
79
Now
COS 2 (q) -\- GO) COS 2 GO =
cos2( -f- G^) -j- cos 2 sin 2
Y sin 2( -\- GO) cos (cp -J- o?) tan
sin %
and
tan d cos 2a cos 2 (^? + GL>) -f- 2 tan # cos 2 a cos 8
/. . . cos 2a \
= (by making cos a = - --- 1- ^ 1
- tan 6 cos 2a [cos 2 ((p + GL>) cos 2 &?] + tan ^ cos 2
pr tan 6 1 cos 2a sin (2&7 4- q>) sin ^ -j- tan # cos 2 G? ;
HORIZONTAL EARTH-SURFACE.
81
tan d sin
[
Also, - cos 2 (cp -f- GO) tan d -f- tan $ cos 2 GO
= tan d [cos'" (cp -}- GO) cos 2 GO]
= sin ) tan #.
Also,
tan d sin 2 sin (cp + G?) cos (cp + GL>)
sin 2<* sin GI> cos GJ tan #
= tan d sin 2a [sin (9? -f" **) cos (^ ~i~ ^ ~ g i n ^ cos
sin 2(^> + GO) sin 2&9~|
- a ~J
= tan # sin 2(Y sin + <^>);
and hence equation (240) becomes
+ sin ) sin cp cos 2or]
sin a
J
Since
2 cos 2 f cos 2 -\- GO -f- f) sin tf sin a? ;
AKGLE e ANGLE cp. 91
dividing by cos d and transposing,
cos (ex cp) cos (a -\- 6) cos cp
cos d
-j- cos (a cp) sin (cp -}- GO -j- a) sin GO { > =
+ cos (cp -\- GO) cos ( ) COS (a -(- 8) COS ) cos or cos ^ sin (a -f- co) sin # sin ^
cos 6" cos d
\ / \ cos( -f- GO -f- tf)
-)- cos a cos 9? cos (a cp) cos ( )
cos a cos (<> -j- 62?) cos 2 (cp -{- GJ -J- a) cos ( -f
for the second member.
ANGLE e = ANGLE (p. 93
The first terms, second terms, and sixth terms cancel.
Divide the equation by cos (a cp). Terms number 3
combined give
sinw sin(w + a) [sin a sin ( ) sin GO cos (<> -f- &? -j- a) j
Substituting for GO, 90 cp, this becomes
f -j- cos nr cos ( cos a] 6?
or
[sin 2 a -\- cos ( cp) (sin 9? sin a -j- cos sin (cp 2a)]doj-
therefore
tan =
1 sin cos (Var (p)'
or
sin c> cos ( = . tan $.
1 -j- sin cp sin ( x^a)
Since HD is perpendicular isAB, the earth-pressure has
the direction GJ. Further,
FD sin a sin # cos <
sin (or -|~ , and
2 /
tan (45 -f
sin [45
K+f)
If the surface is loaded, substitute IT + 7/ 2 for A 2 , or con-
sider li to be the height of the earth increased by the
height of an amount of earth weighing as much as the
applied load.
RECAPITULA T10N OF FORM UL^fl. 101
NOMENCLATURE.
Height of wall H
Thickness at base b
Thickness at top b r
Batter in inches per foot of Hon front face. . . d
Weight per cubic foot W
Total weight of wall G
Angle of repose of earth cp
Angle made by surface of rupture with vertical GO
Weight of cubic foot of earth y
Total thrust of earth against wall E
Angle made with the horizontal by the surface
of the earth s
Angle made by rear face of wall with the ver-
tical a
Angle made with normal by E. d
Dist. of point where the resultant pressure cuts
the base from the front edge of the wall . . q
The resultant pressure due to E and G R
NOTE.
FOR the translation of Prof. Wey ranch's paper the
writer is indebted to the labor of Prof. A. J. Du Bois, of
the Sheffield Scientific School, Yale College, who had
copies printed by the electric-pen process. However,
only the leading equations of Prof. Weyrauch were given ;
hence a great deal of labor has been devoted to expanding,
verifying, and filling in the intermediate steps of the
work, and this nucleus of the mathematical part alone
has grown to about double the original quantity.
M. A. H.
REFERENCES.
A brief outline of the theories advanced by the follow-
ing writers can be found in " Neue Theorie des Erd-
druckes," Dr. E. Winkler, Wien, 1872:
D' Antony, Hoffmann, Poncelet,
Ande, Holzhey, Prony,
Andoy, de Lafont, Rankine,
Belidor, Levi, Rebhann,
Blaveau, deKoszegh Martony, Rondelet,
Bullet, Maschek, Saint-Guilhem,
Considere, Mayniel, Saint- Venant,
Coulomb, Mohr, Sallonnier,
Couplet, Montlong, Scheffler,
Culmann, Moseley, Trincaux,
Frangais, Navier, Vauban,
Gadroy, Ortmann, Winkler,
Gauthey, v. Ott, Woltmann.
Hagen, Persy,
AUDE. Poussee des Terres. Nouvelles experiences sur la
poussee des terres. Paris, 1849.
BAKER-CURIE. Note sur la brochure de M. B. Baker theorie.
Annales des Pouts et Chaussees, pp. 558-592, 1882.
The actual lateral pressure of earthwork. Van Nos-
trand's Magazine, xxv, 1881; also Van Nostrand's
Science Series, No, 5G.
103
104 REFERENCES.
BOUSSINESQ. Complement a de precedentes notes sur la
poussee des terres. *Annales P. et C., 1884.
BOUSIN. Equilibrium of pulverulent bodies. The equilib-
rium of earth when confined by a wall. fVan N., 1881.
CAIN. Modification of Weyrauch's Theory. Van N., 1880.
- Earth-pressure. Modification of Weyrauch/s Theory.
Criticism of Baker's articles. Van N., 1882.
- Uniform cross-section, and T abutments: their proper
proportions and sizes, deduced from Rankine's general
formulas. Van N., 1872.
- Practical designing of retaining-walls. Van N.
Science Series, No. 3, 1888.
CHAPERON. Observations sur le memoire de M. de Sazilly
(1851). Stabilite et consolidation des talus. Annales
P. et C., 1853.
CONSIDERS. Note sur la poussee des terres. Annales P. et
C., 1870.
COUSINERY. Determination graphique de 1'epaisseur des
inurs de soutenement. Annales P. et C., 1841.
DE LAFONT. Sur la poussee des terres et sur les dimensions
a donner, suivant leurs profils, aux murs de soutene-
ment et de reservoirs d'eau. Annales P. et C., 1866.
DE SAZILLY. Sur les conditions d'equilibre des massifs de
terre, et sur les revetements des talus. Annales P. et
C., 1851.
EDDY. Retaining-walls treated graphically. Van N., 1877.
FLAMANT. Note sur la poussee des terres. Annales P. et
C., 1882.
Resume d'articles publies par la Societe des Inge-
nieures Civils de Londres sur la poussee des terres. An-
nales P. et C., 1883.
* Annales des Fonts et Chaussees.
f Van Nostrand's Magazine.
REFERENCES. 105
FLAMANT. Note sur la poussee des terres. Annales P. et
0., 1872.
- Memoire sur la stabilite de la terre sans cohesion par
W. J. Macquorm Eankine (Extrait 1856-57). An-
nales P. et C., 1874.
GOBIN. Determination precis de la stabilite des murs de
soutenement et de la poussee des terres. Annales P.
et C., 1883.
GOULD. Theory of J. Dubosque. Van N., 1883.
- Designing. Van N., 1877.
JACOB. Practical designing of retaining-walls. Van N.,
1873; also Van N. Science Series, No. 3.
JACQUIEB. Note sur la determination graphique de la
poussee des terres. Annales P. et 0., 1882.
KLEITZ. Determination de la poussee des terres et eta-
blissement des murs de soutenement. Annales P. et
C., 1844.
LAGREUE. Note sur la poussee des terres avec ou sans sur-
charges. Annales P. et C.. 1881.
L'EvEiLLE. De 1'emploi des contre-forts. Annales P. et C.
1844.
LEYGUE. Sur les grands murs de soutenement de la ligne
de Mezamet a Bedarieux. Annales P. et C., 1887.
- Nouvelle recherche sur la poussee des terres et le
profil de revetement le plus economique. Annales P.
et C., 1885.
MERRIMAN. On the theories of the lateral pressure of sand
against retaining walls. (School of Mines Quarterly.)
Engineering News, 1888.
The theory and calculation of earthwork. Engineer-
ing News, 1885.
Theorie des Erddruckes und der Futtermauern.
Wien, 1870 and 1871,
106 REFERENCES.
SAINT-GUILHEM. Sur la poussee des terres avec ou sans
surcharge. Annales P. et C., 1858.
ScHEFFLER-FouRNiE. Traite de la stabilitc des construc-
tions. Paris, 1864.
TATE. Surcharged and different forms of retaining- walls.
Van N., 1873; also Van N. Science Series, No. V
Also published by E. & F. N. Spon.
THORNTON. Theory. Van N., 1879.
DIAGRAM I.
107
TABLES.
Table I contains the crushing-strengths and the average
weights of stone likely to be used in the construction of
retaining-walls and foundations; also the average weights
of different earths.
Table II contains the coefficients of friction, limiting
angles of friction, and the reciprocals of the coefficients of
friction for various substances.
Tables III, IV, and V contain the values of the coeffi-
cients [see equation (I')] (B), (C), (D) and (E), where
x cos (e a) /ri . . 2 ( cos (e a]
(B) ~ --, (C) = sin 2 a, (D) \
cos a cos e ( cos e
, _,. . . . cos (e a)
and (E) = 2 sm a sin e - -.
cos e
The tables were computed with a Thacher calculating in-
strument and checked by means of diagrams. It is believed
that they are correct to the second place of decimals; an
error in the third place of decimals does not affect the re-
sults for practical purposes.
Table VI contains the natural sines, cosines and tan-
gents.
109
110
TABLES.
TABLE I.
VALUES OF W.
Name of Substance.
Crushing
Lds. in tons
per sq. ft.
Average
weight in Ibs.
per cu. ft.
Alab.ister
144
Urick best pressed.
40 to 300
150
" common hard
125
" soft inferior .
100
Chalk
20 to 30
150
( 'ement loose
49 6 to 102
Flint
162
Feldspar .... .... .
160
Granite
300 to 1200
170
Gneiss
168
Greenstone, trap
187
Hornblende black
203
Limestones and Marbles ordinary
250 to 1000
j 164.4
Mortftr hardened
I 108
103
Quartz common
165
Sandstone
1 50 to 550
151
Shales . .... ....
162
Slate
400 to 800
175
Soapstone
170
VALUES OF
Name of Substance.
Average
weight in Ibs.
per cu. ft.
Earth, common loam,
Gravel
loose
72 to 80
82 92
90 100
90 106
90 106
104 120
118 129
shaken
rammed moderately
Sand
Sand nerfectlv wet .
ill
TABLE II.
* ANGLES AND COEFFICIENTS OF FRICTION.
tan <.
*
tan>
Dry masonry and brickwork
Masonry and brickwork
with damp mortar ....
O.Gto 0.7
74
31 to 35
364
1.67 to 1.43
1 35
Timber on stone
about 0.4
0.7 toO.3
22
35 tolfif
2.5
1.43 to 3 33
Timber on timber
05 "02
26 10 " lli
2 " 5
Timber on metals
06 "02
31 " 11J
1 67 " 5
Metals on metals
25 " 0.15
14 " 8-T
4 " 6 67
Masonry on dry clay
" " moist clay
0.51
33
. 27.-
18J
1.96
3
Earth on earth
Earth on earth, dry sand,
clay, and mixed earth. . . .
Earth on earth, damp clay .
Earth on earth, wet clay. .
Earth on earth, shingle and
0.25 to 1.0
0.38 "0.75
1.0
0.31
81
14 to 45
21 " 37
45
17
39 to 48
4 to 1
2.63 " 1.33
1
3.23
1 23 to 9
From Rankine's Applied Mechanics.
112
TABLES.
TABLE III.
e
a = 5
a = 6
a = 7
a = 6
a = 9
(B)
(B)
(B)
(#)
(B)
1.004
1.0U5
1.007
1.010
1.012
5
1.012
1.015
1.018
1.022
1.026
10
1.019
1.024
1.029
1.035
1.040
15
1.027
1.034
1.041
.048
1.055
20
1.036
1.044
1.052
.062
1.071
25
1.045
1.055
1.065
.076
1.088
30
1.055
1.006
1.079
.092
1.105
35
1.065
1.079
1.094
1.109
1.124
40
1.078
1.094
1.111
1 . 129
1.147
45
1.093
1 111
1.131
1.152
1.173
(C)
(O)
(C)
(C)
()
0.008
0.011
0.015
0.019
0.0-^4
TABLE IV.
a =5
a = 6
a = ?
a = 8
a = 9
(D)
(D)
(D)
(D)
(D)
0.992
0.989
0.985
0.981
0.976
5
1.008
1.008
1.006
1.005
1.003
10
1.023
1.026
1.028
1.030
1.031
15
1.040
1.046
1.051
1.056
1.060
20
1.057
1.066
1.075
1.084
1.092
25
1.075
1.089
1.102
1.114
1.125
30
1.096
1.113
1.130
1.147
1.163
35
1.118
1.140
1.164
1.183
1.204
40
1.144
1.172
1.199
1.226
1.253
45
1.174
1.208
1.242
1.276
1.309
TABLE V.
6
a = 5
a = 6
a = 7
a = 8-
a = 9
(E)
(K)
(E)
(E)
(E)
5
0.015
0.018
0.021
0.024
0.027
10
0.031
0.037
0.043
0.049
0.055
15
0.046
0.055
0.065
0.074
0.083
20
0.061
0.074
0.086
0.099
0.112
25
0.076
0.092
0.108
0.124
0.140
30
0.091
0.110
0.130
0.149
0.169
35
0.106
0.128
0.151
0.174
0.197
40
0.120
0.145
0.172
0.198
0.225
45
0.134
0.162
0.192
0.222
0.253
TABLES.
113
TABLE HI Continued.
e
a= 10
a =llo
a= 12
a= 13
a= 14
()
(#)
CB)
(fi)
(#)
1.015
1.019
1.022
1.026
1.031
5
1.031
1.037
1.041
1.047
1.053
10
1.046
1.055
1.061
1.068
1.076
15
1.063
1.073
1.081
: .090
1.100
20
1.081
1.092
1.103
.112
1.120
25
1.099
1.112
1.124
.136
1.150
30
1.119
1.135
1.151
.163
1.179
35
1.141
1.159
1.175
.195
1.211
40
1.166
1.186
1.205
.225
1.245
45
1.195
1.218
1.240
1.263
1.288
(O)
CO
(C)
(C)
(C)
0.030
0.036
0.043
0.051
0.029
TABLE IV Continued.
a= 10
a= 11
a= l:>
a = 13
a= 14
()
(C>)
CD)
(*>)
(>)
0.970
0.964
0.957
0.950
0.943
5
.000
0.997
0.993
0.988
0.983
10
.031
1.031
.030
1.028
1.026
15
.064
1.067
.069
1.061
1.072
20
.099
1.105
.110
1.116
1.121
25
.136
1.147
.156
1.165
1.173
30
.178
1.194
.204
1.220
1.232
35
.224
1.244
.262
1.281
1.300
40
.291
1.304
1.328
1.353
.1.377
45
.342
1.375
1.407
1.438
1.469
TABLE V Continued.
e
a= 10
a= 11
a= !>
a = 13
a= 14
(#)
(&')
(E)
CS)
(#)
5
0.030
0.032
0.036
0.039
0.042
10
0061
0.067
0.073
0.079
0.085
15
0.093
0.102
0.111
0.119
0.130
20
0.124
0.137
0.150
0.163
0.175
25
0.156
0.173
0.189
0.205
0.221
30
0.188
0.208
0.216
0.248
0.269
35
0.220
0.244
268
0.292
0.316
40
0.252
0.280
0.308
0.336
0.365
45
0.284
0.316
0.349
0.382
0.415
UHI7BRSIT7
114
TABLES.
TABLE III Continued.
a = 15
a= 16
a = 17
a= 18
a =20
(#)
(fi)
f/0
(JB)
OS)
1.035
.040
1.048
1.051
1.062
5
1.059
.060
1.076
1.081
1.098
10
1.084
.093
l.lt)4
1.112
1.132
15
1.110
.120
1.134
1.138
1.1(58
20
1.135
.149
1.165
1.177
1.218
25
1.165
.179
1.197
1.211
1.245
30
1.195
.212
1.233
1.248
1.288
35
1.229
.249
1.272
1.291
1.339
40
1.268
.21)1
1.317
1.340
1.389
45
1.313
.338
1.369
1.393
1.451
(O)
CO)
(C')
CO)
(CO
0.067
0.076
086
0.095
11?
TABL E IV Continued.
e
a=: 15
a= 16
a= 17
a = 18
a = 20
CM)
CD)
(D)
ID)
(>)
933
0.924
0.915
0.905
0.883
5
0.977
0.971
0.964
957
0.940
10
1.023
1.018
1.016
1.011
1 . 000
15
1.072
1.073
1.071
1 069
1.068
20
1.124
1.127
1.129
1.181
1.132
25
1.181
1.188
1.194
1.200
1.208
30
1.244
1.256
1.266
1.276
1.293
35
1.316
1.332
1.348
1.363
1.390
40
1.400
1.422
1.444
1.465
1.505
45
1.500
1.530
1.559
1.588
1.643
TABLE V- Continued.
e
a = 15
a= 16
a= 17
a = 18
a= 20
CE)
(0)
()
(K)
(#)
5
0.045
0.047
0.050
0.053
0.058
10
0.091
097
0.102
0.108
0.119
15
0.139
0.148
0.157
0.165
0.183
20
0.188
0.200
0.213
0.225
0.249
25
0.238
0.254
0.270
0.177
0.318
30
0.289
. 309
0.3>9
0.349
0.389
35
341
0.365
0.390
0.414
0.463
40
0.394
0.42,
0.452
0.481
0.539
45
0.448
0.482
0.516
0.551
0.620
TABLE VI.
NATURAL SINES, COSINES, TANGENTS
AND COTANGENTS.
NATURAL SINES AND COSINES.
Sine
Tooooo
.00029
.00058
.00087
.00116
.00145
.00175
One.
One.
One.
One.
One.
One.
One.
One.
One.
One.
One.
.01745
.01774
.01803
.01832
.01862
.99983
.99983
.99982
.99982
.99981
.99980
.99980
.99935
.99934
.99933
.99932
.99931
.99930
.01920
.01949
.01978
.02007
99846'
99844
99842
99841
00320
00349
00378
00407
00436
00465
00495
00524
00553
00582
00611
00640
02065
02094
02123
02152
02181
02211
02240
02269
02298
02327
99979
99978
99977
99977
99976
99976
99975
99974
99974
.99888
.99830
.99834
.99833
.99831
02414
02443
02472
02501
02530
99824
99822
99821
99819
99817
99815
99813
00727
00756
00785
00814
00844
00873
0-.W.I
02618
02647
02676
02705
02734
02763
02792
02821
02850
06134
06163
06192
06221
06250
06279
06308
06337
06366
06395
01018
01047
01076
01105
01134
01164
01193
01222
01251
01280
01309
01338
01367
01396
01425
01454
99957
99956
99955
99954
99953
99952
90962
06424
06453
06482
06511
06540
06569
.02967
.02996
.03025
.03054
99991
99991
99991
99990
.03112
.03141
.03170
.03199
06627
06656
06685
06714
06743
06773
.01483
.01513
.01542!
.01571
.01600 1
.01629
.01658
.01687
,01716
01745
99876
99875
99873
99872
99870
.99774
.99772
.99770
.99768
.99766
.99764
.99762
.99760
.99758
.99756
03257
03286
03316
03345
03374
03403
03432
03461
.99947
.99946
.99945
.99944
.99943
.99942
89'
85
NATURAL SINES AND COSINES.
117
Sine
08716
08745
08774
08831
.08976
.09005
.09034
.09063
09092
.09121
.09150
.09179
Cosin
.99617
.99614
.99612
.99607
.99604
.09237
.09353
.09382
.09411
.09440
.09498
.09527
.09556
.09585
.09614
.09642
.09671
.09700
.09729
.09758
.09787
.09816
.09845
.09874
.09903
.09932
.09961
.09990
.10019
.10048
.10077
.10106
.10135
.10164
.10192
.10221
.10250
.10279
.10308
.10337
10366
.10395
.10424
.10453
Cosin
99580
99578
99575
99572
99570
99567
99564
99562
99559
99556
99553
99551
99548
99545
99542
99540
99537
99534
99531
99528
99526
99523
99520
,99517
,99514
,99511
,99506
,99503
.99500
,99497
,99494
,99491
,99488
,99485
,99482
,99479
.99476
.99473
.99470
.99467
.99464
.99461
.99458
.99455
.99452
Sine
Sine
10453
.10511
.10540
.10597
.10626
.10655
.10684
.10713
.10742
.10771
.10800
.10858
.10887
.10916
.10945
.10973
.11002
.11031
.11060
.11089
.11118
.11147
.11176
.11205
.11234
.11263
.11291
.11320
.11349
.11378
.11407
.11436
.11465
.11494
.11523
.11552
.11580
.11609
.11638
.11667
.11696
.11725
.11754
.11783
.11812
.11840
.11869
.11898
.11927
.11956
.11985
.12014
.12043
.12071
.12100
.12129
.12158
.12187
Cosin
Cosin
99443
99440
99437
99434
99431
99428
99424
99421
99418
99415
99412
99409
99406
99402
99393
99377
99374
99370
99367
99364
99357
99354
99351
99347
99344,
99341
99337
99334
99331
99327
99324
99317
99314
99310
99307
.99297
.99293
.99283
.99279
.99276
.99272
.99265
.99262
^99255
Sine
83'
Sine
.12187
.12216
.12245
.12274
.12302
.12331
.12360
.12389
.12418
.12447
.13476
.12504
.12533
.12562
.12591
.12620
.12649
.12678
.12706
.12735
.12764
.12793
.12822
.12851
.12880
,12937
,12966
,12995
,13024
,13053
,13081
,13110
,13139
,13168
,13197
,13226
,13254
,13283
,13312
13341
13370
,13399
,13427
,13450
,13485
,13514
,13543
,13572
,13600
,13629
,13658
,13687
.13716
.13744
.13773
.13802
.13831
.13917
Cosin
Cosin
.99248
.99244
.99240
.99237
.99233
.99226
.99222
.99219
99215
99211
99208
99204
99197
99193
,99182
,99178
,99175
,99171
,99167
,99163
,99156
,99152
,99144
,99141
,99137
,99133
,99129
.99125
,99122
.99118
,99114
.99110
.99106
.99102
99094
,99091
,99087
,99083
,99079
,99075
,99071
,99067
.99059
,99055
,99051
,99047
,99043
,99039
.99031
.99027
Sine
82-
^
13917
13946
13975
14004
14033
14061
14090
14119
14148
14177
14205
.14234
.14263
.14292
.14320
.14349
.14378
.14407
.14436
.14464
.14493
.14522
.14551
.14580
.14608
.1463'
.14666
.14695
.14723
.1475
.14781
.14810
.14838
.14867
.14925
.14954
.14982
.15011
.15040
.15069
.15097
.15126
.15155
.15184
.15212
.15241
.15270
.1529
.15327
.15356
.15385
.15414
.15442
.15471
.15500
.15529
.15557
.15586
.15615
.15643
Cosin
Cosin
.99019
.99015
.99011
.99006
.99002
.98998
.98990
,98978
,98973
,98969
,98965
,98961
,98957
.98944
98927
.98914
,98910
,98876
,98871
,98867
,98863
,98854
,98849
,98845
.98832
,98818
.98814
,98796
,98791
,98787
,98782
98778
98773
Sine
.15643
.15672
.15701
.15730
.15758
.15787
.15816
.15845
.15873
.15902
.15931
.15959
.15988
.16017
.16046
.16074
.16103
.16132
.16160
.16189
.16218
.16246
.16275
.16304
.16333
.16361
.16390
.16419
.16447
.16476
.16505
.16533
.16562
.16591
.16620
.16648
.16677
.16706
.16734
.16763
.16792
.16820
.16849
.16878
.16906
.16935
.16964
.16992
.17021
.17050
.17078
.17107
.17136
.17164
.17193
.17222
.17250
.17279
.17308
.17336
.17365
Cosin
!osin
.98764
.98760
.98755
.98751
.98746
.98741
.98737
.98732
.98728
.98723
.98718
.98714
.98709
.98704
.98700
.98695
.98676
.98671
.98667
.98662
.98657
.98652
98629
98624
98619
98614
.98585
.98575
.98570
.98565
.98561
.98556
.98551
.98546
.98541
.98536
.98531
.98526
.98521
.98516
.98511
.98506
.98501
.98496
.98491
.98486
.98481
Sine
80'
118
NATtRAL SINES AND COSINES.
34
37
10
^
.17365
.17393
.17422
.17451
.17479
.17508
.17587
.17565
.17594
.17623
.17651
17708
17737
17766
17794
17823
17852
17880
17909
17937
17966
17995
18023
18052
18081
18109
18138
18166
18195
18224
18367
18395
18452
18481
18509
18538
18567
18595
18624
18652
18681
18710
18738
18767
18795
18910
18967
19024
19052
19081
Cosin
Cosin
.98481
.98476
.98471
.98466
.98461
.98455
.98450
.98445
.98435
.98430
98425
98420
98414
98404
98399
98394
98378
98373
98368
98362
98352
98347
98341
98336
98320
98315
98304
98277
98272
98256
98250
98245
98240
98234
98229
98223
98218
98190
98185
98179
.98174
.98168
.98163
^790
11
Sine
.19081
.1910S
.19188
-19167
19195
19224
19252
19:
19;
19423
19452
19481
19509
19566
19595
19652
19709
19737
19766
19794
19823
19851
19880
19908
19937
19965
19994
20022
20051
20079
20108
20136
20165
20279
20307
20421
.20450
'.20478
.20507
.20563
.20592
.20620
: 20849
'.20677
".20706
f. 20734
>20763
'.20791
Cosin
.98157
.98153
.98146
.98140
.98135
.98129
.98124
.98118
.98112
.98107
98101
,98080
98084
98079
98073
98050
98044
93033
98027
98021
98016
98010
98004
97998
97992
97987
97981
97975
97969
97963
97958
97952
97940
97934
97922
97916
97910
97905
97887
97881
97875
97863
97857
97851
'.97845
: 97839
'.97833
'.97827
.97821
,97815
k>sin Sinej
* 178-r
Sine
.20791
.20848
.20877
.20905
.20933
.20962
.20990
.2101
.21047
.21076
.21104
.21132
.21161
.21189
.21218
.21246
.21275
.21303
.21331
.21388
.21417
.21445
.21474
.21502
.21530
.21559
.21587
.21616
.21644
.21672
.21701
21729
.21753
21786
.21814
.21843
.21871
.21899
121956
'.21985
'.22013
-.22041
.22070
.22098
.22126
.22155
; 22183
'.22212
'.22297
f. 22325
* 22353
*22410
'.22438
22467
f. 22495
Cosin
.97815
.97809
.97803
.97797
.97791
.97784
.97778
.97772
.97766
.97760
.97754
.97748
.97742
.97735
.97729
.97723
.97717
.97711
.97705
97673
97667
97661
97655
97648
97642
97G36
97630
97623
.97017
97611
97604
9^598
97592
.97585
97579
97573
.97566
.'97560
.97553
.97547
.97541
.97534
.97528
97521
.97515
.97508
97502
?97496
'.97489
.97483
'.97476
.97470
'.97463
197457
.97450
; 97444
197437
Cosin !
' '77
13 C
Sine
.22495
.22523
.22552
.22665
.22693
.22722
.22750
.82778
.22807
,22977
,23005
.23118
.23146
.23175
.23260
.23283
.23316
.23345
.23373
.23401
.23423
.23458
.23486
.23514
.23542
.23571
.23599
.23627
$23656
.23712
.23740
.23769
'.23797
'.23853
.24023
.24051
.24079
.24108
.24164
".24192
Cosin
Cosin
.97437
.97430
.97424
.97417
.97411
.97404
,.07398
.97391
.97384
.97378
.97371
.97365
.97358
.97351
.97345
.97338
.97331
.97325
.97318
.97311
.97304
.97298
.97291
.97284
.97278
.97271
.97264
.97257
.97251
.97244
.97237
97230
97223
97217
97210
97203
97196
,97189
97182
97176
97169
,97162
,97155
,97148
.97141
,97134
.97127
.97120
.97113
.97106
.97100
.97093
,97086
,97079
,97072
,97065
.97058
.97051
,97044
97037
Sine
176-
14-
Sine
.24192
.24220
.24249
.24277
.84305
.24333
.84362
.84390
.84418
.84446
.24474
.84503
.24531
.24559
.84587
.24615
.84644
.84672
.24700
.24728
.84756
.24784
.24813
.24841
.24869
.24897
.24925
.24954
.24982
.25010
.25038
.85066
.25094
.25122
.25151
.25179
.25207
.25235
.25263
.25291 !
.25320J
.25348!
.25376
.25404
.25432,
.25460
.25488
.25516
.25545
.25573
.25601 1
.25629
.25657
.256851
.25713
.25741
.25769
.25798
.25826
.25854
Cosin
.97030
.97023
.97015
.97008
.97001
96945
96894
.96873
.96851
.96844
.96837
.96815
.96807
.96778
96771
96764
.96756
96749
.96742
96734
.96727
.96719
.96712
.96705
.96675
.96667
.96660
.96653
.96645
.96638
96630
.96623
96615
96608
96600
Cosin | Sine
NATURAL SINES AND COSINES.
119
21
15
Sine
25883
25910
25966
Coein
.96578
.96570
.96562
.96555
,260501.96547
.96540
16-
Sine
26107
26135 .96524
.96517
.26191 1.96509
.26219
96502
96494
.26359
.262751.96486
96479i
96471
96463
.96456
.26415 .96448
.26443 .96440
.26471
.26500 .96425
.96417
,96410
.26556
.26584
.26612
.26724
.26752
.26780
.87004
.27032
.27060
.27068
.27116
.27144
.27172
.27200
.27256
.27284
.27812
.27340
.27396
.27424
.27452
.27480
.27508
.27586
.27564
%379
96371
96347
96340
.96301
.96277
.96269
.96261
,96214
Cosin I
.96198
.96190
.96182
.96174
.96166
.96158
.96150
.96142
.96184
.98188
Sine
74
.27564
.27592
,27620
,27648
,27676
,27704
,27731
,27759
,27787
,27815
,27843
,27871
,27899
,27927
,27955
,27983
,28011
.28039
,28067
Cosin
.96126
.96118
.96110
.96102:
.96094
.96078
96070
.96062
.96054
.28123
.28150
.28178
28234
,28200
,28318
28346
,28374
,28429
,28457
,28485
,28513
.28541
,28569
,28597
,28625
,28652
,28736
,28764
17
,28875
,29015
,29042
,29070
,29098
.29126
,29154
.29182
9G021
96013
96005
95997
95981
95972
95956
95948
95940
,95931
,95923
,95915
,95907
,95898
,95890
.95874
,95863
,95857
,95849
,95841
.95832
.95824
.95816
.95807
.95799
.95791
,95782
.95774
.95766
,95757
.95749
.95740
.95732
.95724
,95715
,95707
,95681
,95673
Cosin Sine
.95656
.95647
.82887
.82914
.32942
.32969
.82997
.33024
.33031
.83079
.83106
.83134
.33161
.83189
.83216
.83244
.83271
.33298
.83326
.83353
.83381
.94979
.94970
.94961
.94952
.94943
95450
95441
95433
93424
95415
95407
.94888
.94878
.94869
.94860
.94851
.94842
.94832
.94303
.94293
.94284
.94274
.94264
.94254
.94245
.83408
.83436
.3346?
.83490
.83518
.83545
.3357?
.83600
.33627
.33655
.94814
.94805
.94793
.94786
.94777
.94768
.94758
.94740
.94740
.95319
.95310
.95301
.952C3
.94730
.94721
.94712
.94702
.94693
.94684
.94674
.94665
.94656
.9404o
.95275
.93260
.95257
.95248
.95240
.95231
.95222
.95213
.95204
.95195
.95186
.95177
.95168
.95159
.95150
.95142
.951&3
.95124
.95115
.95106
Sine"
.84120
.84147
.84175
34202
120
NATURAL SINES AND COSINES.
20*
34202
34229
34257
34284
34311
34339
34366
34393
34421
34448
84475
.93337
.359181.93327
.93316
.35973
.36000
.36027
.37757
.37784
.37811
.36135
.36162
.36190
.36217
.36244
.36271
.36298
.36325
.36352
.36379
.34530
.34557
.34584
.34612
.34639
.93232
.93222
.93211
.93201
.93190
.93180
.93169
.93159
.93148
.37865
.37892
.37919
.37946
.37973
.37999
.39474
.39501
.39528
.39555
.39581
93799
93789
93779
93769
93759
93748
.34830
.34857
.34884
.34912
.34939
.34966
.34993
93728
93718
93708
93698
.39715
.39741
.39768
.39795
.39822
.36515
.36542
.36569
.36590
.36623
.86650
93677
93667
93657
93647
93637
36677
36704
36731
36758
36785
.93031
.93020
.93010
.92999
350T5
35102
35130
35157
35184
35211
.38349
.38376
.384C3
.38430
.38450
.40008
.40035
.40062
.40088
.40115
.40141
.92978
.92967
.92956
.92945
.92935
93606
93596
93585
93575
93565
.38564
.38591
.38617
.38644
.38071
.38G03
93555
93544
93534
93524
93514
93503
93493
93483
93472
93402
.40195
.40221
.40248
.40275
.40301
.40320
.40355
.40381
.40408
35347
35375
35402
35429
35456
35484
35511
35538
35565
36975
37002,
87029!
37056
37083
37110
37137
37164
37191
92881
92870
92859
92849
92838
92837
.92816
.92805
.92794
.92784
.92773
.92762
.92751
.92740
.92729
.92718
35619
35647
35674
35701
35728
35755
35782
35810
.93441
.93431
.93420
.93410
.93400
.38912
.38939
.38966
.38993
.39020
.39046
.39073
Cosin
02060
.92016
.92005
.91994
.91982
.91971
.91959
.91948
91925
91914
91902
91891
91879
91868,
918561
91845 |
91833
91822 1
91810 !
91799
917871
91775
91764
91752
91741
91729
91718
91706
.91683
.91671
.91660
.91648
.91636
.91625
.91613
.91601
.91590
.91578
.91566
.91555
.91543
.91531
.91519
.91508
.91496
.91484
.91472
.40434 '.91461
.91449
.91437
.91425
.91414
.91402
.91378
.40647 .91366
. 40674 1. 91355
Cosin | Sine
_
.40674
.40700
.40727
.40753
.40780
.40806
Cosin
.91355 60
.91343 ! 59
.91331 ! 58
.91319
.91307
.91295
.91283
.40860 .91272
.40886 .91260
.409131.91248
.40939 .91236
91224
.40992
.41019
.41045
.41072
.41098
.41125
.41151
.41178
.41204
.41231
.41257
.41284
.41310
.41337
.41363
.41390
.41416
.41443
.41469
.41496
.41522
.41549
.41575
.41602
.41628
.41655
.41681
.41707
.41734
.41760
.41787
.41813
.41840
.41866
.41892
.41919
.41945
.41972
.41998
.42024
.42051
.42077
.42104
.42130
.42156
.42183
.42209
.42235
.42262
.91212
.91200
.91188
.91176
.91164
.91152
.91140
.91128
.91116
.91104
.91092
.91080
.91068
.91056
.91044
.91032
.91020
.91008
,90972
,90960
,90911
,90875
.90851
.90814
.90802-
.90790
.90778
.90766
.90753
.90741
.90729
.90717
.90704
.90692
.90680
.90668
.90655
.90643
.90631
69
Cosin Sine
65
NATURAL SINES AND COSINES.
25
26"
27
28
29
9
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
$
"o
.42262
.90631
.43837
.89879
.45399
.89101
.46947
.88295
.48481
.87462
60
i
.42288
.90618
.43863
.89867
.45425
.89087
.46973
.88281
.48506
.87448
59
2
.42315
.90606
.43889
.89854
.45451
.89074
.46999
.88267
.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
.42446
.90545
.44020
.89790
.45580
.89008
.47127
.88199
.48659
.87363
53
8
.42473
.90532
.44046
.89777
.45606
.88995
.47153
.88185
.48684
.87349
52
9
.42499
.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
T47229
T88144
.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
.90403
.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
.45968
.88808
*.47511
.87993
.49040
.87150
38
23
.42867
.90346
.44437
.89584
.45994
.83795
.47537
.87979
.49065
.87136
37
24
.42894
.90334
.44464
.89571
.46020
.88782
'.47562
.87965
.49090
.87121
36
25
.42920
.90321
.44490
.89558
.40046
.88708
147588
.87951
.49116
.87107
35
26
.42946
.90309
.44516
.89545
.46072
.88755
.47614
.87937
.49141
.87093
34
27
.42972
.90296
.44542
.89532
.4G097
.88741
.47639
.87923
.49166
.87079
33
23
.42999
.90284
.44568
.89519
.40123
.88728
.47665
.87909
.49192
.87064
32
29
.43025
.90271
.44594
.89506
.46149
.88715
.47690
.87890
.49217
.87050
31
30
.43051
.90259
.44620
.89493
.46175
.88701
.47716
.87882
.49242
.87036
30
31
.43077
.90246
f44646
.89480
146201
.'88688
f47741
.87868
.49268
.87021
29
32
.43104
.90233
.44672
.89467
.40226
.88674
.47767
.87854
.49293
.87007
28
33
.43130
.90221
.4-1698
.89454
.40252
.88061
.47793
.87840
.49318
.86993
27
34
.43156
.90203
.44724
.89441
.46278
.88647
.47818
.87826
.49344
.86978
26
35
.43182
.90196
.44750
.89428
.40304
.88634
.47844
.87812
.49369
.86964
25
38
.43209
.90183
.44776
.89415
.40330
.88620
.47869
.87703
.49394
.86949
24
37
.43235
.90171
.44802
.89402
.46355
.88007
.47895
.87784
.49419
.86935
23
38
.43261
.90158
.44828
.89389
.40381
.88593
.47020
.87770
.49445
.86921
22
39
.43287
.90146
.44854
.89376
.40407
.88580
.47940
.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
.40484
.8G539
.48022
.87715
.49546
.86863
18
43
.43392
.90095
.44958
.89324
.40510
.88526
.48048
.87701
.49571
.86849
17
44
.43418
.90082
.44984
.89311
.40536
.88512
.48073
.87687
.49596
.86834
16
45
.43445
.90070
.45010
.89298
.40501
.88409
.48099 .87673
.49622
.86820
15
46
.43471
.90057
.45036
.89285
.46587
.88485
.48124
.87659
.49647
.86805
14
47
.43497
.90045
.45062
.89272
.46613
.88472
.48150
.87645
.49672
.80791
13
43
.43523
.90032
.45088
.89259
.46639
.88458
-.48175
.87631
.49697
.86777
12
49
.43549
.90019
.45114
.89245
.46064
.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
.40742
.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
.86090
6
55
.43706
.89943
.45269
.89167
.46819
.88363
.48354
.87532
.49874
.86675
5
56
.43733
.89930
.45295
.89153
.46844
.88349
.48379
.87518
.49899
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4
57
.43759
.89918
.45321
.89140
.46870
.88336
.48405
.87504
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.86646
3
58
.43785
.89905
.45347
.89127
.46896
.88322
.48430
.87490
.49950
.86632
2
59
.43811
.89892
.45373
.89114
.46921
.88308
.48456
.87476
.49975
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1
60
.43837
.89879
.45399
.89101
.46947
.88295
.48481
.87462
.50000
86603
_0
I
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
i
64
63
62 1
61 1
60
NATURAL SINES AND COSINES.
30*
31
32
33
34*
/
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
/
"o
.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 .8C804
.56016
.82839 56
5
.50126
.86530
.51628
85642
.53115
.84728
.54586 .83788
.56040
.828221 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
.84666
.54683 .83724
.56136
.82757
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
.53283
.84619
.54750
.83676
.56208
.82708 48
13
.50327
.86413
.51828
85521
.53312
.84604 1 .54781
.83660
.50232
.82692 47
14
.50352
.86398
.51852
.85506
.53337
.84588!
.54805
.83045
.56256
.82675 1 46
15
.50377
.86384
.51877
.85491
.53361
.84573
.54829
.83029
.56280
.82659 1 45
16
.50403
.86369
.51902
.85476
.53386
.84557
.54854
.83013
.56305
.82643 44
17
.50428
.86354
.51927
.85461
.53411
.84542
.54878
.83597
.56329
.82626 43
18
.50453
.86340
.51952
.85446
.53435
.84526
.54902
.83581
.50353
.82610 42
19
.50478
.86325
.51977
.85431
.534GO
.84511
.64927
.83505
.50377
.82593 41
20
.60503
.86310
.52002
.85416
.53484
.84495
.54951
.83549
.56401
.82577
40
21
.50528
86295
.52026
.85401
.53509
.84480
.54975
.83533
.56425
.82561
89
22
.50553
.86281
.52051
.85383
.53534
.84464
.54999
.83517
.56449
.82544! 38
23
.50578
.86266
.52076
.85370
.53558
.84448
.5502-1
.83501
.56473
.82528
37
24
.50603
.86251
.52101
.85355
.53583
.84433
.55043
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.56497
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36
25
.50628
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.52126
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.53007
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.82495
35
26
.50654
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.85325
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34
27
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.52175
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.84386
.55121
.83437
.56509
.82462
33
28
.50704
.86192
.52200
.85294
.53681
.84370
.5514-5
.83421
.56593
.82446
32
29
.50729
.86178
.52225
.85279
.53705
.84355
.55109
.83405
.50017
.82429
31
80
.50754
.86163
.52250
.85264
.53730
.84339
.55194
.83389
.56641
.82413
36
31
.50779
.86148
.52275
.85249
.53754
.84324
.55218
.83373
.56665
.82396
29
82
.50804
.86133
.52293
.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
.55291
.83324
.56730
.82347
26
35
.50879
.86089
.52374
.85188
.53853
.84261
.55315
.83308
.56760
.82330
25
36
.50904
.86074
.52399
.85173
.53877
.84245
.55339
.83292
.56784
.82314
24
37
.50929
.86059
.52423
.85157
.53902
.84230
.55303
.83276
.56808
.82297
23
38
.50954
.86045
.52448
.85142
.63926
.84214
.55388
.83200
.56832
.82281
22
39
.50979
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.52473
.85127
.03951
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.83244
.56856
.82264
21
40
.51004
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.85113
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20
41
.51029
.86000
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.54000
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.55460
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19
42
.51054
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43
.51079
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.82198 17
44
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.85956
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.83103 .66976
.82181 16
45
.51129
.85941
.52621
.85035
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.84104
.65557
.83147
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46
.51154
.85926
.52646
.85020
.54122
.84088
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47
.51179
.85911
.52671
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48
.51204
.85896
.52698
.84989
.54171
.84057
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.83098 : ! .57071
.82115
12
49
.51229
.85881
.52720
.84974
.64195
.84041
.55654
.83082J .67095
.82098
11
50
.51254
.85866
.52745
.84959
.54220
.84025
.55678
.83Q66
.57119
.82082
10
51
.51279
.85851
.52770
.84943
.54244
.84009
.55702
.83050
.57143
.82065
9
52
.51304
.85836
.52794
.84928
.54209
.83994
.55726
.83034
.67107
.82048
8
53
.51329
.85821
.52819
.84913
.54293
.83978
.65750
.83017
.57191
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7
54
.51854
.85806
.52844
.84897
.54317
.83902
.55775
.83001
.57215
.82015
6
55
.51379
.85792
.52869
.84882
.54342
.83946
.55799
.82985
.57238
.81999
5
56
.51404
.85777
.52893
.84866
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.83930
.55823
.82909
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4
57
.51429
.85762
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.57286
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3
58
.51454
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.83899
.55871
.82936
.67310
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2
59
.51479
.85732
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.83883
.65895
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.57334
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1
60
.51504
.85717
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.84805
.54464
.83867
.55919
.82904
.57358
.81915
/
Sine
Cosin
Sine
Cosin
Sine I Cosin
Sine
Cosin
Sine
I
59
58
57 56
55
NATURAL SINES AND COSINES.
35 |
SB*
37*
SB
39*
9
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
~o
.57358
.81915'
.58779
.80902
.60182
.79864
.61566
.78801
.62932
.77715
60
i
.57381
.81899
.58802
.80885
.60205
.79846
.61589
.78783
.62955
.77696
69
2
.57405
.81882
.58826
.80867
.60228
.79829
.61612
.78765
.62977
.77678
58
3
.57429
.81865!
.58849
.80850
.60251
.79811
.61635
.78747
.63000
.77660
67
4
.57453
.81848
.58873
.80833
.60274
.79793
.61658
.78729
.63022
.77641
56
5
.57477
.818321
.58896
.80816
.60298
.79776
.61681
.78711
.63045
.77623
55
6
.57501
.81815
.58920
.80799
.60321
.79758
.61704
.78694
.63068
.77605
54
7
.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
61
10
.57596
.81748
.59014
.80730
.60414
.79688
.61795
.78622
.63158
.77531
60
11
.57619
.81731
.59037
.80713
.60437
.79671
'61818
.78604
.63180
.77513
40
12
.57043
.81714
.59061
.80096
.60460
.79653
.61841
.78586
.63203
.77494
48
13
.57667
.81698
.59084
.80679
.60483
.79635
.61864
.78508
.63225
.77476
47
14
.57691
.81681
.59108
.80662
.60506
.79618
.61887
.78550
.63248
.77458
46
15
.57715
.81664
.59131
.80644
.60529
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.61909
.78532
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45
16
.57738
.81647
.59154
.80627
.60553
.79583
.61932
.78514
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44
17
.57762
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.59178
.80610
.G0576
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.78496
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43
18
.57786
.81614
.59201
.80593
.60599
.79547
.61978
.78478
.63338
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42
19
.57810
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.59225
.80576
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.79530
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41
.57833
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.59248
.80558
.60645
.79512
.62024
.78442
.63383
.77347
40
21
.57857
.81563
.59272
.80541
.60668
.79494
162046
.78424
.63406
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39
23
.57881
.81546
.59295
.80524
.60691
.79477
.62069
.78405
.63428
.77310
33
23
.57904
.81530
.59318
.80507
.60714
.79459
.62092
.78387
.63451
.77292
87
24
.57928
.81513
.59342
.80489
.60738
.79441
.62115
.78369
.63473
.77273
86
25
.57952
.81406
.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
23
.58047
.81423
.59459
.80403
C0853
.79353
.62229
.78279
.63585
.77181
31
SO
.58070
.81412
.59483
.80386
.60876
.79335
.62251
.78261
.63608
.77162
30
31
.58094
.81395
.59506
.80368
.60899
.79318
.62274
.78243
.63630
.77144
20
82
.58118
.81378
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.80351
.60922
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.62297
.78225
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23
33
.58141
.81301
.59552
.80334
.60945
.79282
.62320
.78206
.63675
.77107
27
34
.58165
.81344
.59576
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.60968
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.62342
.78188
.63698
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2G
85
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25
36
.58212
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24
37
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23
38
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23
39
58283
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21
40
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.76977
go
41
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19
42
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18
43
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.76921
17
44
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.59809
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16
45
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.59832
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15
46
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.59856
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.79051
.62615
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.76868
14
47
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.59879
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.62638
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13
48
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12
49
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.59926
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.62683
.77916
.64033
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11
50
.58543
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.59949
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.78980
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10
51
.58567
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.77879
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9
52
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.64100
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8
53
.58614
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.79986
.61406
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.62774
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7
54
.58637
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6
55
.58661
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6
56
.58684
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4
57
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X
58
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.78837
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2
59
.58755
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1
60
.58779
.80902
.60182
.79864
.61566
.78801
.62932
.77715
.64279
.76604
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin,
Sine
Cosin
Sine
i
i
54
53
52
51-
50
124
NATURAL SINES AND COSINES.
40
41
42
43"
44
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
Sine
Cosin
i
"o
.64279
.76604
.65606
.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
.64399
.76511
.65716
.75375
.67021
.74217
.68306
.73036
.69570
.T1833
55
6
.64412
.76492
.65738
.75356
.6704S
.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
.76433
.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
13
.64548
.7G330
.65339
.75241
.67172
.74080
.68455
.72897
.69717
.71091
48
13
.64533
.76331
.65891
.75222
.67194
.74061
.68476
.72877
.69737
.71671
47
14
.64590
.76342
.65913
.75203
.67215
.74041
.68497
.72857
.69758
.71650
46
15
.64612
.76323
.65935
.75184
.67237
.74022
.68518
.72837
.69779
.71630
45
16
.64635
.76304
.65953
.75165
.67258
.74002
.68539
.72817
.69800
.71610
44
17
.64857
.76238
.65078
75146
.67280
.73983
.68561
.72797
.69821
.71590
43
18
.64679
.76287
.63000
75123
.67301
.73963
.68582
.72777
.69842
.71569
42
19
.64701
.76248
.63022
75107
.67323
.73944
.68603
.72757
.69862
.71549
41
20
.64723
.70229
.66044
75088
.67344
.73924
.68624
.72737
.69883
.71529
40
21
.64746
.76210
.66066
75069
.67366
.73904
.68645
.72717
.69904
.71508
39
22
.64768
.76192
.63033
75050
. 67307
.73885
.68GG3
.72697
.69925
.71488
33
23
.64790
76173
.63103
75030
.67409
.73805
.68683
.72677
.69946
.71468
37
24
.64812
76154
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75011
.67430
.73846
.68709
.72657
.69966
.71447
30
25
.64834
76135
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74992
.67452
.73823
.68730
.72637
.69987
.71427
35
26
.64856
76116
.68175
74973
.67473
.73803
.68751
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.70008
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34
27
.64878
76097
.66197
74953
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.68772
.72597
.70029
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33
28
.64901
76078
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74934
.67516
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32
29
.64923
76059
.68240
74915
.67533
.73747
.68814
.72557
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31
30
.64945
76041
.66262
74896
.67559
.73728
.68835
.72537
.70091
.71325
30
31
.64967
76022
.66284
74876
67580
.73708
.68857
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29
32
.64989
76003
.63303
74857
67602
.73683
.68878
.72497
70132
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23
33
.65011
75984
.68327
74833
67C23
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.72477
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27
34
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75965
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74818
67645
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.72457
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26
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7594S
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74799
67006
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.72437
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35
38
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75927
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74780
67G33
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.72417
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24
37
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75903
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74760
67709
73590
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23
S3
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74741
67730
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22
39
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75870
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74722
67752
73551
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21
40
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75851
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74703
67773
73531
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20
41
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75832
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74683
67795
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19
42
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75813
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74664
678fG
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18
43
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75794
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74644
67837
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17
44
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75775
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74625
67859
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16
45
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75756
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74608
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15
46
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75738
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74588
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.73413
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14
47
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75719
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74567
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13
48
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75700
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74548
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.73373
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12
49
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75680
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74523
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.73353
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11
50
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75661
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74509
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.73333
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10
51
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74489
.68008
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.69277
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9
52
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74470
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8
53
.65452
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74451
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7
54
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6
55
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70813
5
56
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4
57
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3
58
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.71974
.70670
.70752
2
59
.65584
.75490
.66891
.74334
.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
49
48
47
40 i
45
|
NATURAL TANGENTS AND COTANGENTS.
125
1
2
8
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
/
.00000
Infinite.
.01746
57.2900
.03492
28.6363
.05241
19.0811
60
1
.00029
3437.75
.01775
56.3506
.03521
28.3994
.05270
18.9755
59
2
.00058
1718.87
.01804
55.4415
.03550
28.1664
.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
.03096
27.0566
.05445
18.3G55
53
8
.00233
429.718
.01978
50.5485
.03725
26.8450
.05474
18.2G77
52
.00262
381.971
.02007
49.8157
.03754
26.63G7
.05503
18.1708
51
10
.00291
&43.T74
.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
236.478
.02095
47.7395
.03342
26.0307
.05591
17.88G3
48
13
.00378
234.441
.02124
47.0853
.03871
25.8348
.05620
17.7934
47
14
.00407
245.552
.02153
46.4489
.03900
25.6418
.05649
17.7015
46
15
.00436
229.182
.02182
45.8294
.03929
25.4517
.05678
17.6106
45
1(5
.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
100.984
.02269
44.0661
.04016
24.8978
.05766
17.3432
42
19
.00553
180.932
.02298
43.5081
.04046
24.7185
.05795
17.2558
41
80
.00583
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
82
.00040
156.259
.02386
41.9158
.04133
24.1957
.05883
16.9990
38
23
,OOGG9
149.465
.02415
41.4106
.04162
24.0263
.05912
16.9150
37
21
. 00698
143.237
.02444
40.9174
.04191
23.8593
.05941
16.8319
36
x).>
.00727
137.507
.02473
40.4358
.04220
23.6945
.05970
16.7496
35
';;
.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
2s
.00315
122.774
.02560
39.05G8
.04308
23.2137
.06058
16.5075
32
20
.00844
118.540
.02589
38.6177
.04337
23.0577
.06087
16.4283
31
CO
.00873
114.589
.02619
38.1885
.04366
22.9038
.06116
16.3499
30
81
.00002
110.892
.02648
37.7683
'.04395
22.7519
.06145
16.2722
29
32
.00931
107.426
.02677
37.3579
.04424
22.6020
.06175
16.1952
28
88
.009GO
104.171
.02706
36.9560
.04454
22.4541
.06204
16.1190
27
84
.00989
101.107
.02735
36.5G27
.04483
22.3081
.06233
16.0435
26
85
.01018
98.2179
.02764
36.1776
.04512
22.1640
.06262
15.9687
25
36
.01047
95.4895
.02793
35.8006
.04541
22.0217
.06291
15.8945
24
87
.01076
93.9085
.02822
35.4313
.04570
21.8813
.06321
15.8211
23
::s
.01105
90.4633
.02851
35.0005
.04599
21.7426
.06350
15.7483
22
.7;l
.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
11
.01193
83.8435
.02939
34.0273
.04687
21.3369
.06437
15.5340
19
12
.01222
81.8470
.029G8
33.6935
.04716
21.2049
.06467
15.4638
18
ta
.01251
79.9434
.02997
33.3662
.04745
21.0747
.06496
15.3943
17
41
.01280
78.1263
.03026
33.0452
.04774
20.9460
.06525
15.3254
16
45
.01309
76.3900
.03055
32.7303
.04803
20.8188
.06554
15.2571
15
16
.C1338
74.7292
.03084
32.4213
.04833
20.6932
.06584
15.1893
14
17
.01367
73.1390
.03114
32.1181
.048G2
20.5691
.06613
15.1222
13
48
.01396
71.6151
.03143
31.8205
.04891
20.4465
.06642
15.0557
12
ID
.01425
70.1533
.03172
31.5284
.04920
20.32.53
.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
.06817
14.6685
6
55
.01600
62.4992
.03346
29.8823
.05095
19.6273
.06847
14.6059
5
56
.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
GO
.01746
57.2900
.03492
28.6363
.05241
19.0811
.06993
14.3007
/
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
t
89
88
87* -
86
126
NATURAL TANGENTS AND COTANGENTS.
4
5
6
70
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
/
.06993
14.3007
.08749
11.4301
.10510
9.51436
.12278
8.14435
GO
1
.07022
14.2411
.08778
11.3919
.10540
9.48781
.12308
8.12481
58
2
.07051
14.1821
.08807
11.3540
.10569
9.46141
.12338
8.10536
58
3
.07080
14.1235
.08837
11.3163
.10599
9.43515
.12307
8.08600
57
4
.07110
14.0655
.08866
11.2789
.10628
9.40904
.12397
8.06674
56
5
.07139
14.0079
.08895
11.2417
.10657
9.38307
.19426
8.04756
K
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
9
.07256
13.7821
.09013
11.0954
.10775
9.28058
.12544
7.97176
51
10
.07285
13.7267
.09042
11.0594
.10805
9.25530
.12574
7.95302
60
11
.07314
13.6719
.09071
11.0237
.10834
9.23016
.12603
7.93438
40
12
.07344
13.6174
.09101
10.9882
.1CSG3
9.20516
.12CS3
7.91582
48
18
.07373
13.5634
.09130
10.9529
.10893
9.18028
.12602
7.89734
47
14
.07402
13.5098
.09159
10.9178
.1CC22
9.15554
.12692
7.87895
46
15
.07431
13.4566
.09189
10.8829
.10952
9.13093
.12722
7.86064
45
1G
.07461
13.4039
.09218
10.8483
.10901
9.10(k6
.12751
7.84242
44
IT
.07490
13.3515
.09247
10.8139
.11011
9.08211
.12781
7.82428
48
IS
.07519
13.2996
.09277
10.7797
.11040
9.057C9
.12810
7.80022
42
19
.07548
13.2480
.09306
10.7457
.11C70
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
.00394
10.6450
.11158
8.90227
.12929
7.73480
3!<
23
.07665
13.0458
.09423
10.6118
.11187
8.93867
.12958
7.71715
37
21
.07695
12.9962
.09453
10.5789
.11217
8.91520
.12988
7.C9957
86
25
.07724
-12.9469
.09482
10.54G2
.11246
8.89185
.13017
7.C8208
85
20
.07753
12.8981
.09511
10.5136
.11276
8.86862
.13047
7.CG4G6
34
27
.07782
12.8496
.C9541
10.4813
.11305
8.84551
.13076
7.64732
88
28
.07812
12.8014
.09570
10.4491
.11335
8.82252
.13100
7.63005
82
29
.07841
12.7536
.oceoo
10.4172
.11364
8.79904
.13136
7.01287
81
oO
.07870
12.7062
.09629
10.3854
.11394
8.77689
.13165
7.59575
30
81
.07899
12.6591
.09658
10.3538
.11423
8.75425
.13195
7.57872
20
:J2
.07929
12.6124
.09688
10.3224
.11452
8.73172
.13224
7.56176
28
33
.07958
12.EGGO
.00717
10.2913
.11482
8.70931
.13254
7.54487
27
34
.07987
12.5199
.09746
10.2602
.11511
8.68701
.13284
7.52806
26
35
.08017
12.4742
.09776
10.2294
.11541
8.66482
.13313
7.51132
25
30
.08046
12.4288
.09805
10.1988
.11570
8.64275
.13343
7.49465
24
37
.08075
12.3838
.09834
10.1683
.11600
8.02078
.13372
7.47806
23
38
.08104
12.3390
.09864
10.1381
.11629
8.59893
.13402
7.46154
22
39
.08134
12.2946
.00893
10.1080
.11659
8.57718
.13432
7.44509
31
40
.08163
12.2505
.09923
10.0780
.11688 8.55555
.13461
7.42871
20
41
.08192
12.2067
.09952
10.0483
.11718
8.53402
.13491
7.41240
10
42
.08221
12.1632
.09981
10.0187
.11747
8.51259
.13521
7.39616
18
43
.08251
12.1201
.10011
9.98931
.11777
8.49128
.13550
7.37999
17
44
.08280
12.0772
.10040
9.96007
.11806
8.47007
.13580
7.36389
16
45
.08309
12.0346
.10069
9.93101
.11836
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
.13669
7.31600
13
48
.08397
11.9087
.10158
9.84482
.11924
8.38G25
.13698
7.30018
12
41)
.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
.12i01
8.26355
.13876
7.206G1
G
55
.08002
11.6248
.10363
9.64935
.12131
8.24345
.13906
7.19125
6
50
.08032
11.5853
.10393
9.62205
.12100
8.22344
.13935
7.17594
4
57
.08661
11.5461
.10422
9.59490
.12190
8.20352
.13965
7.16071
3
58
.08690
11.5072
.10452
9.56791
.12219
8.18370
.13995
7.14553
2
5!)
.08720
11.4685
.10481
9.54106
.12249
8.16398
.14024
7.13042
1
GO
.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.
127
8
<
9
1
1
1-
/
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
~0
.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
c
.14113
7.08546
.15898
6.29007
.17693
5.65205
.19498
5.12862
58
s
.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
.1420S
7.04105
.15988
6.25486
.17783
5.62344
.19589
5.10490
55
G
.14232
7.02637
.16017
6.24321
.17813
5.61397
.19619
5.09704
54
7
.14262
6.91174
.16047
6.23160
.17843
5.60452
.19640
5.08921
53
8
.14291
6.99718
.16077
6.22003
.17873
5.59511
.19680
5.08139
52
9
.14321
6.982G8
.16107
6.20851
.17903
5.58573
.19710
5.07360
51
10
.14351
6.96823
.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
.10190
6.17419
.17993
5.55777
.19801
5.05037
48
13
.14440
6.92525
.16226
6.10283
.18023
5.54851
.19831
5.042G7
47
14
.14470
6.91104
.16256
6.15151
.18053
5.53927
.19861
5.03499
46
15
.14499
6.89388
.16286
6.14023
.18033
5.53007
.19891
5.02734
45
1C
.14529
6.88278
.16316
6.12899
.18113
5.52090
.19921
5.01971
44
17
.14559
6.8G874
'.16346
6.11779
.18143
5.51176
.19952
5.01210
43
18
.14588
6.85475
; 16376
6.10604
.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
116465
6.07340
.18263
5.47548
.20073
4.98188
39
22
.14707
6.79936
.16495
6.06240
.18293
5.46648
.20103
4.97438
38
23
.14737
6.73564
.16525
6.05143
.18323
5.45751
.20133
4.96690
37
24
.14767
6.77199
.16555
6.04051
.18353
5.44857
.20164
4.95945
36
23
.14796
6.75838
.16585
6.02902
.18384
5.43966
.20194
4.95201
35
23
.14826
6.74483
.16615
6.01878
.18414
5.43077
.20224
4.94460
34
27
.14356
6.73133
.16645
6.00797
.18444
5.42192
.20254
4.93721
as
28
.14386
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
SO
.14945
6.69116
.16734
5.97576
.18534
5.39552
.20345
4.91516
30
81
.14975
6.67787
.16764
5.96510
.18564
5.38677
'.20376
.90785
29
32
.15005
6.66463
.16794
5.95448
.18594
5.37805
.20406
.90056
28
33
.15034
6.65144
.16824
5.94390
.18624
5.36936
.20436
.89330
27
34
.15064
6.63831
.16854
5.93365
.18654
5.36070
.20466
.88605
26
35
.15094
6.62523
.16884
5.92283
.18684
5.35206
.20497
.87882
25
30
.15124
6.61219
.16914
5.91236
.18714
5.34345
.20527
.87162
24
37
.15153
6.50021
.16944
5.90191
.18745
5.33487
.20557
.86444
23
30
.15183
6.53627
.16974
5.89151
.18775
5.32631
.20588
.85727
22
CO
.15213
6.57339
.17094
5.88114
.18805
5.31778
.20618
.85013
21
40
.15243
6.56055
.17033
5.87080
.18835
5.30928
.20648
.84300
20
41
.15272
6.54777
.17063
5.86051
.18865
5.30080
.20679
.83590
19
42
.15302
6.53503
.17093
5.85024
.18895
5.29235
.20709
.82882
18
43
.15332
6.52234
.17123
5.84001
.18925
5.28393
.20739
.82175
17
44
.15362
6.50970
.17153
5.82982
.18955
5.27553
.20770
.81471
16
45
.15391
6.49710
.17183
5.81966
.18986
5.26715
.20800
.80769
15
46
.15421
6.48456
.17213
5.80953
.19016
5.258SO
.20830
.80068
14
47
.15451
6.47206
.17243
5.79944
.19046
5.25048
.20861
.79370
13
48
.15481
6.45961
.17273
5.78938
.19076
5.24218
.20891
.78673
12
49
.15511
6.44720
.17303
5.77936
.19106
6.23391
.20921
.77978
11
50
.15540
6.43484
.17333
5.76937
.19136
5.22566
.20952
.77286
10
51
.15570
6.42253
.17363
5.75941
.19166
5.21744
.20982
.76595
9
52
.15600
6.41026
.17393
5.74949
.19197
5.20925
.21013
.75906
8
53
.15630
6.39804
.17423
5.73960
.19227
5.20107
.21043
.75219
7
54
.15660
6.38587
.17453
5.72974
.19257
5.19293
.21073
.74534
6
55
.15689
6.37374
.17483
5.71992
.19287
5.18480
.21104
.73851
5
56
.15719
6.30165
.17513
5.71013
.19317
5.17671
.21134
.73170
4
57
.15749
6.31961
.17543
5.70037
.19347
5.16863
.21164
.72490
3
53
.15779
6.33761
.17573
5.69064
.19378
5.16058
.21195
.71813
2
59
.15809
6.32566
.17603
5.68094
.19408
5.15256
.21225
.71137
1
60
.15838
6.31375
.17633
5.67128
.19438
5.14455
.21256
.70463
f
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
/
8
! 1
8
7
9
7
B .
128
NATURAL TANGENTS AND COTANGENTS.
12
13
14 ]
15
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
f
.21256
4.70463
.23087
4.33148
.24933
4.01078
.26795
3.73205
60
1
.21286
4.69791
.23117
4.32573
.24904
4.00582
.26826
3.72771
59
2
.21316
4.G9121
.23148
4.32001
.24995
4.00086
.26857
3.72338
58
3
.21347
4.C8452
.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
G
.21438
4.6G458
.23271
4.29724
.25118
3.98117
.2G982
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
.21529
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
13
.21621
4.62518
.23455
4.26352
.25304
3.95196
.27109
3.680G1
48
13
.21G51
4.61868
.23485
4.25795
.25335
3.94713
.27201
3.67638
47
14
.21G82
4.61219
.23516
4.25239
.253G6
3.94232
.27232
3.67217
46
15
.21712
4.60572
.23547
4.24685
.25397
3.93751
.272G3
3.66796
45
1C
.21743
4.59927
.23578
4.24132
.25428
3.93271
.27294
3.66376
44
17
.21773
4.59283
.23608
4.23580
.25459
3.92793
.27326
3.65957
43
10
.21004
4.58641
.23639
4.23030
.25490
3.92316
.27357
3.65538
42
19
.21834
4.58001
.23G70
4.22481
.25521
3.91639
.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
S.90890
.27451
3.64289
39
23
.21925
4.5G091
.23762
4.20842
.25014
3.90417
.27482
3.63874
38
23
.21956
4.55458
.23793
4.20298
.25045
3.89945
.27513
3.63461
37
21
.21986
4.54826
.23823
4.19756
.25076
3.89474
.27545
3.63048
36
23
.22017
4.54196
.23854
4.19215
.25707
3.89004
.27576
3.62636
35
23
.22047
4.53568
.23885
4.18675
.25738
3.88536
.27007
3 62224
34
27
.22078
4.52941
.23916
4.18137
.25769
3.88068
.27038
3.61814
33
23
.22108
4.52316
.23946
4.17600
.25800
3.87601
.27670
3.61405
32
29
.22139
4.51693
.23977
4.17064
.25831
3.87136
.27701
3.60996
31
30
.22169
4.51071
.24008
4.16530
.25862
3.86671
.27733
3 60588
30
31
.22200
4.50451
.24039
4.15997
.25893
3.86208
.27764
3.60181
29
S3
.22231
4 49832
.24069
4.15465
.25924
3.85745
.27795
3.59775
28
83
.22261
4.49215
.24100
4.14934
.25955
3.85284
.27826
3.59370
27
24
.22292
4.48600
.24131
4.14405
.25986
3.84824
.27858
3.58966
26
33
.22322
4.47986
.24163
4.13877
.26017
3.84364
.27889
3.58562
25
30
.22353
4.47374
.24193
4.13350
.26048
3.83906
.27921
3.58100
24
37
.22383
4.46764
.24223
4.12825
.26079
3.83449
.27952
3.57758
23
38
.22414
4.46155
.24254
4.12301
.26110
3.82992
.27983
3.57357
22
29
.22444
4.45548
.24285
4.11778
.26141
3.82537
.28015
3.56957
21
40
.22475
4.44942
.24316
4.11256
.26172
3.82083
.28046
3.56557
20
41
.22505
4.44338
.24347
4.10736
.26203
3.81630
.28077
3.56159
19
42
.22536
4.43735
.24377
4.10216
.26235
3.81177
.28109
3.55761
18
13
.22567
4.43134
.24408
4.09699
.26266
3.80726
.28140
3.55364
17
44
.22597
4.42534
.24439
4.09182
.26297
3.80276
.28172
3.54968
16
43
.22028
4.41936
.24470
4.08666
.26328
3.79827
.28203
3.54573
15
40
.22058
4.41340
.24501
4.08152
.26359
3.79378
.28234
3.54179
14
47
.22089
4.40745
.24532
4.07639
.26390
3.78931
.282G6
3.53785
13
<18
.22719
4.40152
.24562
4.07127
.26421
3.78485
.28297
3.53393
12
49
.22750
4.39560
.24593
4.06616
.26452
8.78040
.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
.22842
4.37793
.24686
4.05092
.26546
3.76709
.28423
3.51829
8
Ha
.22872
4 37207
.24717
4.04586
.26577
3.76268
.28454
3.51441
7
51
.22903
4.3G623
.24747
4.04081
.26608
3.75828
.28486
3.51053
6
">
.22934
4 30040
.24778
4.03578
.26639
3.75388
.28517
3.50666
5
DC
.22904
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
CO
.23087
4.33148
.24933
4.01078
.2G795
3.73205
.28675
3.48741
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
/
77
76
75
74"
NATURAL TANGENTS AND COTANGENT?.
129
16
17
18
19
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
.28675
3.48741
.30573
3.27085
.32492
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
6
.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
.30800
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
1-3
.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
3.032GO
.34922
2.86356
45
16
.29179
3.42713
.31083
3.21722
.33007
3.02963
.34954
2.86089
44
17
.29210
3.42343
.31115
3.21392
.33040
3.02GG7
.34987
2.85822
43
18
.29242
3.41973
.31147
3.21063
.33072
3.02372
.35020
2.85555
42
19
.29274
3.41604
.31178
3.20734
.33104
3.02077
.35C52
2.85289
41
20
.29305
3.41236
.31210
3.20406
.33136
3.01783
.35085
2.85023
40
21
.29337
3.40869
.31242
3.20079
.33169
3.01489
.35118
8.84758
29
22
.29368
3.40502
.31274
3.19752
.33201
3.011C6
.351CO
2.84494
38
23
.29400
3.40136
.31208
3.19426
.33233
3.00C03
.35183
2.84229
37
24
.29432
3.39771
.31338
3.10100
.33266
3.00611
.35216
2.83965
36
25
.29463
3.39406
.31370
3.18775
.33298
3.00319
.35248
2.83702
35
2G
.29495
3.39042
.314C2
3.10451
.33330
3.00028
.35231
2.83439
34
27
.29526
3.38679
.31434
3.10127
.33363
2.99738
.35314
2.83176
33
28
.29558
3.38317
.31466
3.17804.
.33305
2.99447
.35346
2.82914
32
29
.29590
3.37955
.31498
8.17401
.33427
2.99158
.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
33
.29685
3.36375
.31594
3.16517
.33524
2.98292
.35477
2.81070
28
33
.29716
3.3G516
.31626
3.16197
.33557
2.98004
.35510
2.81610
27
34
.29748
3.30158
.31658
3.15877
.33589
2.97717
.35543
2.81350
26
85
.29780
3.35800
.31690
3.15558
.33621
2.97430
.35576
2.81091
25
36
.29811
3.35443
.31722
3.15240
.33654
2.97144
.35603
2.80833
24
37
.29843
3.35087
.31754
3.14922
.33686
2.96850
.35641
2.80574
23
38
.29875
3.34732
.31786
3.14605
.33718
2.96573
.35674
2.80316
22
39
.29906
3.34377
.31018
3.14288
.33751
2.96288
.35707
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
.31914
3.13341
.33848
2.95437
.35805
2.79289
18
43
.30033
3.32965
.31946
3.13027
.33881
2.95155
.35838
2.79033
17
44
.30065
3.32614
.31978
3.12713
.33913
2.94872
.35871
2.78778
16
45
.30097
3.32264
.32010
3.12400
.33945
2.94591
.35904
2.78523
15
^6
.30128
3.31914
.32043
3.12087
.33978
2.94309
.35937
fc. 78269
14
4,
.30160
3.31565
.32074
3.11775
.34010
2.94028
.85969
2.78014
13
48
.30192
3.31216
.32106
3.11464
.34043
2.93748
.36002
2.77761
12
49
.30224
3.30868
.32139
3.11153
.34075
2.934C8
.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
.34368
2.90971
.36331
2.75246
2
59
.30541
3.27426
.32400
3.08073
.34400
2.90696
.36364
2.74997
1
GO
.30573
3.27085
.32492
3.07768
.34433
2.90421
.36397
2.74748
/
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
/
73
72
71
70
130 NATURAL TANGENTS AND COTANGENTS.
80-
21*
22
23
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
/
.36397
2.74748
.38386
2.60509
.40403
2.47509
.42447
2.35585
60
1
.36430
2.74499
.38420
2.60283
.40436
2.47302
.42482
2.35395
59
2
.36463
2.74251
.38453
2.60057
.40470
2.47095
.42516
2.35205
58
3
.36496
2.74004
.38487
2.59831
.40504
2.4G888
.42551
2.35015
57
4
.36529
2.73756
.38520
2.59606
.40538
2.46082
.42585
2.34825
56
5
.36562
2.73509
.38553
2.59381
.40572
2.46476
.42619
2.34636
55
6
.36595
2.73263
.38587
2.59156
.40606
2.46270
.42654
2.34447
54
7"
.36628
2.73017
.38620
2.58932
.40643
2.46065
.42688
2.34258
53
8
.36661
2.72771
.38654
2.58708
.40074
2.45860
.42722
2.34069
52
9
.30694
2.72526
.38687
2.58484
.40707
2.45655
.42757
2.83881
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
2.71793
.38787
2.57815
.40809
2.45043
.42800
2.33317
48
13
.30826
2.71548
.38821
2.57593
.40843
2.44839
.42894
2.33130
47
14
.30859
2.71305
.38854
2.57371
.40877
2.44G36
.42929
2.32943
46
15
.30892
2.71062
.38888
2.57150
.40911
2.44433
.42963
2.32756
45
16
.33925
2.70819
.38921
2.56928
.40945
2.44230
.42998
2.32570
44
17
.30958
2.70577
.38955
2.56707
.40979
2.44027
.43032
2.32383
43
10
.30991
2.70335
.88988
2.56487
.41013
2.43825
.43067
2.32197
42
19
.37024
2.70094
.39023
2.56266
.41047
2.43623
.43101
2.32012
41
20
.37057
2.69853
.89055
2.56046
.41081
2.43422
.43136
2.31826
40
21
.87090
2.69612
.39089
2.55827
.41115
2.43220
.43170
2.31641
39
22
.37123
2.CD371
.39123
2.55608
.41149
2.43019
.43205
2.31456 38
23
.37157
2.C3131
.39156
2.55389
.41183
2.42819
.43239
2.31271 |37
24
.37190
2.63392
.89190
2.55170
.41217
2.42018
.43274
2.31086 36
25
.3?223
2.63G53
.39223
2.54952
.41251
2.42418
.43308
2.30902
3T>
20
.37256
2.C3414
.39257
2.54734
.412C5
2.42218
.43343
2.30718
34
27
.37289
2.C3175
.39290
2.54516
.41319
2.42019
.43378
2.30534 33
23
.37323
2.67037
.39324
2.54299
.41353
2.41819
.43412
2.30351 32
29
.37355
2.67700
.39357
2.54082
.41307
2.41620
.43447
2.30167 31
30
.37388
2.67462
.89391
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
2.63989
.33458
2.53432
.41490
2.41025
.43550
2.29619 28
33
.37488
2.GG752
.33492
2.53217
.41524
2.40827
.43585
2.29437 27
34
.37521
2.63516
.39526
2.53001
.41553
2.40029
.43620
2 29254
26
35
.37554
2.GG281
.39559
2.53708
.41592
2.40432
.43654
2.. 29073
25
36
.37588
2.60046
.39593
2.52571
.416C6
2.40235
.43689
2.28891
24
37
.37621
2.65811
.39626
2.52357
.41600
2.40038
.43724
2.28710
23
33
.37654
2.65576
.39060
2.52142
.41694
2.89841
.43758
2.28528
22
39
.37687
2.C5342
.39094
2.51929
.41728
2.39045
.43793
2.28348
21
40
.37720
2.65109
.39727
2.51715
.41763
2.39449
.43828
2.28167
20
41
.37754
2.64875
.89761
2.51502
.41797
2.39253
.43862
2.27987
19
42
.37787
2.61G42
.39795
2.51289
.41831
2.39058
.43897
2 27806
18
43
.37820
2.64410
.39829
2.51076
.41805
2.38803
.43932
2.27626
17
44
.37853
2.64177
.39802
2.50804
41899
2.38008
.43966
2.27447
16
45
.37887
2.63045
.39898
2.50652
.41933
2.38473
.44001
2.27267
15
46
.37920
2.63714
.39930
2.50440
.41908
2.38279
.44036
2.27088
14
47
.37953
2.63483
.33963
2.50229
.42003
2.38084
.44071
2.20909
13
48
.37986
2.63252
.39997
2.50018
.42036
2.37891
.44105
2.26730
12
49
.33020
2.63021
.40031
2.49807
.42070
2.37097
.44140
2.26552
11
50
.38053
2.62791
.40065
2.49597
.42105
2.87504
.44175
2.26374
10
51
.38086
2.62561
.40098
2.49386
.42139
2.87311
.44210
2.26196
9
52
33120
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
.33186
2.61874
.40200
2.48758
.42242
2.36733
.44314
2.25603
6
55
.38220
2.61646
.40234
2.48549
.42276
2.36541
.44349
2.25486
5
56
.38253
2.61418
.40267
248340
.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
GO
.38386
2.60509
.40403
2.47509
.42447
2.35585
.44523
2.24604
j
'/
Cotang
Tang,
Cotang
Tang
Cotang
Tang
Cotang Tang
/
69
> 68
67' II 66
NATURAL TANGENTS AND COTANGENTS.
131
2
40
2
5
2
6'
2
7 o
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
.46772
2.13801
.48917
2.04426
.51099
1.95698
56
.44697
2.23727
.46808
2.13639
.48953
2.04276
.51136
1.95557
55
G
.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
2.12833
.49134
2.03526
.51319
1.94858
50
11
144907
2.22683
.47021
2.12671
.49170
2.03376
.51356
1.94718
49
12
.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
18
.45082
2.21819
.47199
2.11871
.49351
2.02631
.51540
1.94023
44
ir
.45117
2.21647
.47234
2.11711
.49337
2.02403
.51577
1.93885
43
18
.45152
2.21475
.47270
2.11552
.49423
2.02335
.51614
1.93746
42
19
.45187
2.21304
.47C05
2.11392
.49459
2.02187
.51651
1.93608
41
90
.45222
2.21133
.47341
2.11233
.49495
2.02039
.51688
1.93470
40
21
.45257
2.20961
.47377
2.11075
.49532
2.01891
.51724
1.93332
39
29
.45292
2.20790
.47412
2.10916
.495C8
2.01743
.51761
1.93195
38
28
.45327
2.20619
.47448
2.10758
.49604
2.01596
.51798
1.93057
37
J4
.45362
2.20449
.47483
*. 10600
.49640
2.01449
.51835
1.92920
36
JJ5
.45397
2.20278
.47519
2.10442
.49677
2.01302
.51872
1.92782
35
26
.45432
2.20108
.47555
2.10284
.49713
2.01155
.51909
1.92645
34
2T
.45407
2.19938
.47590
2.10126
.49749
2.01008
.51946
1.92508
33
28
.45502
2.19709
.47626
2.09909
.49786
2.00862
.51983
1.92371
32
29
.45538
2.19599
.47062
2.09811
.49822
2.00715
.52020
1.92235
31
30
.45573
2.19430
.47698
2.09854
.49858
2.00569
.52057
1.92093
30
31
.45608
2.19261
.47733
2.09498
.49894
2.00423
.52094
1.91962
29
3-2
.45643
2.19092
.47769
2.09341
.49931
2.00277
.52131
1.91826
28
83
.45678
2.18923
.47005
2.09184
.49967
2.00131
.52168
1.91690
27
84
.45713
2.18755
.47840
2.09028
.50004
.99986
.52205
1.91554
26
35
.45748
2.18587
.47876
2.08872
.50040
.99841
.52242
1.91418
25
3G
.45784
2.18419
.47912
2.08716
.50076
.99695
.52279
1.91282
24
37
.45819
2.10251
.47948
2.08560
.50113
.9550
.52316
1.91147
23
88
.45854
2.18084
.47984
2.08405
.50149
.C3406
.52353
1.91012
22
39
.45889
2.17916
.43019
2.03250
.50185
.C9261
.52390
1.90876
21
40
.45924
2.17749
.48055
2.08094
.50222
.93116
.52427
1.90741
20
41
.45960
2.17582
.48091
2.07939
.50258
.98972
.52464
1.90607
19
42
.45995
2.17416
.43127
2.07785
.50295
.93828
.52501
1.90472
18
43
.40030
2.17249
.48163
2.07630
.50331
.93684
.52538
1.90337
17
41
.4G065
2.17083
.48198
2.07476
.50368
.98540
.52575
1.90203
16
45
.40101
2.10917
.48234
2.07321
.50404
.08396
.52613
1.90069
15
46
.40136
.2.10751
.48270
2.07167
.50441
.88253
.52650
1.89935
14
47
.40171
2.10585
.48306
2.07014
.50477
.93110
.52687
1.8S801
13
48
.46206
2.10420
.48342
2.00860
.50514
.97966
.52724
1.89667
12
49
.46242
2.10255
.43378
2.06706
.50550
.97823
.52761
1.89533
11
50
.46277
2.16090
.48414
2.06553
.50587
.97681
.52798
1.89400
10
51
.46312
2.15925
.48450
2.06400
.50623
.97538
.52836
1.8926B
9
52
.40348
2.157GO
.43486
2.00247
.50660
.97395
.52873
1.89133
8
53
.46383
2.15596
.43521
2.00094
.50696
.97253
.52910
1.89000
7
54
.46418
2.15432
.43557
2.05942
.50733
.97111
.52947
1.88807
6
68
.40454
2.15268
.48593
2.05790
.50769
.96969
.52985
1.88734
5
B6
.40489
2.15104
.48629
2.05637
.50806
.96827
.53022
1.88602
4
57
.46525
2.14940
.48665
2.05485
.50843
.96685
.53059
1.88469
3
58
.40560
2.14777
.48701
2.05333
.50879
.96544
.53096
1.88337
2
59
.46595
2.14614
.48737
2.05182
.50916
.96402
.53134
1.88205
1
60
.46631
2.14451
.48773
2.05030
.50953
.96261
.53171
1.88073
J)
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
/
6
5
6
4
6
3
6
z- 1
132 NATURAL TANGENTS AND COTANGENTS.
f
28
29
30
31
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
j
"o
.53171
1.88073
.55431
1.80405
.57735
1.73205
.60086
1.66428
60
1
.53208
1.87941
.55469
1.80281
.57774
1.73089
.60126
1.66318
59
2
.53246
1.87809
.55507
1.80158
.57813
1.72973
.60165
1.66209
58
3
.53283
1.87677
.55545
1.80034
.57851
1.72857
.60205
1.66099
57
4
.53320
1.87546
.55583
1.79911
.57890
1 .72741
.60245
1.65990
56
i
.53358
1.87415
.55621
1.79788
.57929
1.72625
.60284
1.65881
55
6
.53395
1.87283
.55659
1.79665
.57968
1.72509
.60324
1.65772
54
7
.53432
1.87152
.55697
1.79542
.58007
1.72393
.60364
1.65663
53
6
.53470
1.87021
.55736
1.79419
.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
.55813
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
.56003
1.78563
.58318
1.71473
.60681
1.64795
45
16
.53769
1.85979
.56041
1.78441
.58357
1.71358
.60721
1.64687
44
17
.53807
1.85850
.56079
1.78319
.58396
1.71244
.60761
1.64579
43
18
.53844
1.85720
.56117
1.78198
.58435
1.71129
.60801
1.64471
42
19
.53882
1.85591
.56156
1.78077
.58474
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
22
.53995
1.85204
.56270
1.77713
.58591
1.70673
.60960
1.64041
38
23
.54032
1.85075
.56309
1.77592
.58631
1.70560
.61000
1.63934
37
24
.54070
1.84946
.56347
1.77471
.58670
1.70446
.61040
1.63826
36
25
.54107
1.84818
.56385
1.77351
.58709
1.70332
.61080
1.63719
35
26
.54145
1.84689
.56424
1.77230
.58748
1.70219
.61120
1.63612
34
27
.54183
1.84561
.56462
1.77110
.58787
1.70106
.61160
1.63505
33
28
.54220
1.84433
.56501
1.76990
.58826
1.69992
.61200
1.63398
32
29
.54258
1.84305
.5C539
1.76869
.588C5
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
1.76510
.58983
1.69541
.61360
1.62972
28
33
.54409
1.83794
.56693
1.76390
.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
.56769
1.76151
.59101
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
1.68979
.61561
1.62442
23
38
.54597
1.83159
.56885
1.75794
.59218
1.68866
.61601
1.62336
22
39
.54635
1.83033
.56923
1.75675
.59258
1.68754
.61641
1.62230
21
40
.54673
1.82906
.56962
1.75556
.59297
1.68643
.61681
1.62125
20
41
.54711
1.82780
.57000
1.75437
.59336
1.68531
.61721
1.62019
19
42
.51748
1.82654
.57039
1.75319
.59376
1.68419
.61761
1.61914
18
43
.54786
1.82528
.57078
1.75200
.59415
1.68308
.61801
1.61808
17
44
.5^964
1.82402
.57116
1.75082
.59454
1.68196
.61842
1.61703
16
45
.54862
1.82276
.57155
1.74964
.59494
1.68085
.61882
1.61598
15
46
.54900
1.82150
.57193
1.74846
.59533
1.67974
.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
60
.55431
1.80405
.57735
1.73205
.60086
1.66428
.62487
1.60033
/
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
i 61-
60
59
58 i
NATURAL TANGENTS AND COTANGENTS.
133
32"
83
34 |
35'
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
~o
62487
1. 600*3
.64941
1.53986
.67451
1.48256 i
.70021
1.42815
60
i
62527
1.59930
.64983
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
56
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
g
62852
1.59105
.65314
1.53107
.67832
1.47422
.70412
1.42022
51
10
62893
1.59003
.65355
1.53010
.67875
1.47330
.70455
1.41934
50
11
62933
1.58900
.65397
1.52913
.67917
1.47238
.70499
1.41847
49
12
62973
1.58797
.65438
1.52816
.67900
1.47146
.70542
1.41759
48
13
63014
1.58695
.65480
1.52719
.68002
1.47053
.70586
1.41672
47
14
63055
1.58593
65521
1.52623
.68045
1.46962
.70629
1.41584
46
15
63095
1.58490
.65563
1.52525
.68088
1.4G870
.70673
1.41497
45
16
.63136
1.58388
! 65604
1.52429
.68130
1.46778
.70717
1.41409
44
17
.63177
1.58286
.65646
1.52333
.68173
1.46G86
.707GO
1.41322
43
18
.63217
1.581&4
.65G88
1.52235
.68215
1.46595
.70804
1.41235
42
19
.63258
1.58083
.65729
1.E2139
.68258
1.4G503
.70S48
1.41148
41
20
.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
1.577T3
.65854
1.51850
.68386
1.4G229
.70979
1.40887
38
23
.63421
1.57076
.65893
1.51754
.68429
1.46137
.71023
1.40800
37
24
.63462
1.57575
.65938
1.51058
.68471
1.4G046
.71066
1.40714
3G
25
.63503
1.57474
.65980
1.61563
.68514
1.45955
.71110
1.40G27
35
26
.63544
1.57372
.66021
1.51406
.68557
1.458G4
.71154
1.40540
S4
27
.63584
1.57271
.66003
1.51370
.68600
1.45773
.71108
1.40454
33
28
.63625
1.57170
.66105
1.51275
.68642
1.45683
.71242
1.40367
32
29
.63666
1.57009
.66147
1.51179
.68685
1.45593
.71285
1.40281
31
30
.63707
1.56969
.66189
1.51084
.68728
1.45501
.71329
1.40195
30
31
.63748
1.56868
.66230
1.50988
.68771
1.45410
.71373
1.40109
29
12
.63789
1.56767
.66272
1.50893
.68814
1.45320
.71417
1.40022
28
33
.63830
1.56667
.66314
1.50797
.68857
1.45229
.71461
1.39936
27
34
.63871
1.56566
.66356
1.50702
.G8900
1.45139
.71505
1.39850
2G
35
.63912
1.56466
.66398
1.60607
.68943
1.45049
.71549
1.39764
25
36
.63953
1.56366
.66440
1.50513
.68985
1.44958
.71593
1.39679
24
37
.63994
1.56265
.66482
1.50417
.69028
1.44868
.71637
1.39593
23
38
.64035
1.56165
.66524
1.50323
.C9071
1.44778
.71681
1.39507
22
39
.64076
1.56065
.66566
1.50228
.C9114
1.44688
.71725
1.39421
21
40
.64117
1.55966
.66608
1.50133
.69157
1.44598
.71769
1.39336
20
41
.W158
1.55866
.66650
1.50038
.69200
1.44508
.71813
1.39250
19
42
.64199
1.55766
.66692
1.49944
.C9243
1.44418
.71857
1.89165
18
43
.64240
1.55666
.66734
1.49849
.69286
1.44329
.71901
1.39079
17
44
.64281
1.55567
.66776
1.49755
.69329
1.44239
.71946
1.38994
1G
45
.64322
1.55467
.66818
1.49661
.69372
1.14149
.71990
1.38909
15
46
.64363
1.55368
.66860
1.49566
.69416
1.44060
.72034
1.38824
14
47
.64404
1.55263
.66902
1.49473
.69459
1.43970
.72078
1.38738
13
48
.64446
1.55170
.66944
1.49378
.69503
1.43881
.72122
1.38G53
12
4(
.64487
1.55071
.66CCS
1.49284
.69545
1.43793
.72167
1.385C8
11
&
.64528
1.54972
.67023
1.49190
.69588
1.43703
.72211
1.38484
10
5
.64560
1.54873
.67071
1.49097
.69631
1.43614
.72255
1.38399
9
5
.64610
1.54774
.67113
1.49003
.69675
1.43525
.72299
1.38314
8
53
.64652
1 .54675
.67115
1.48909
.69718
1.43436
.72344
1.38229
54
.64693
1.54576
.67197
1.48816
.69761
1.43347
.72388
1.38145
6
5o
.64734
1.54478
.67239
1.48722
.69804
1.43358
.72432
1.88060
5
5
.64775
1.54379
.67282
1.48629
.69847
1.43169
.72477
1.37976
4
ti
.64817
1.54281
.67324
1.48536
.69891
1.43080
.72521
1.37801
3
5*
.64858
1.54183
.67CGe
1.48442
.69934
1.42992
.72565
1.37807
2
5
.64899
1.54085
.67409
1.48349
.69977
1.42903
.72610
1.37722
1
(X
.64941
1.53986
.67451
1.48256
.70021
1.42815
.72654
1.37638
(
Cotang
Tang
i Cotang
Tang
Cotang
Tang
, Cotang
Tang
57'
56
55 II 54
134
NATURAL TANGENTS AND COTANGENTS.
31
5
3
7
3
B"
3
90 1
Tangf
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
.72654
1.37638
.75355
1.32704
.78129
1.27994
.80978
.23490
00
1
.72699
1.37554
.75401
1.32624
.78175
1.27917
.81027
.23416
59
2
.72743
1.37470
.75447
1.32544
.78222
1.27841
.81075
.23343
58
3
.72788
1.37386
.75492
1.32464
.78269
1.27764
.81123
.23270
57
4
.72832
1.37302
.75538
1.32384
.78316
1.27688
.81171
.23196
56
5
.72877
1.37218
.75584
1.32304
.78363
1.27611
.81220
.23123
55
6
.72921
1.37134
.75629
1.32224
.78410
1.27535
.81268
.23050
54
7
.72966
1.37050
.75675
1.32144
.78457
1.27458
.81316
.22977
53
8
.73010
1.36967
.75721
1.32064
.78504
1.27382
.81364
.22904
52
9
.73055
1.36883
.75767
1.31984
.78551
1.27306
.81413
.22*31
51
10
.73100
1.36800
.75812
1.31904
.78598
1.27230
.81461
.22758
50
11
.73144
1.36716
.75858
1.31825
.78645
1.27153
.81510
.22685
49
12
.73189
1.36633
.75904
1.31745
.78692
1.27077
.81558
.22612
48
13
.73234
1.36549
.75950
1.31666
.78739
1.27001
.81606
.22539
47
14
.73278
1.36466
.75996
1.31586
.78786
1.26925
.81655
.22467
46
15
.73323
1.36383
.76042
1.31507
.78834
1.26849
.81703
.22394
45
16
.73368
1.36300
.76088
1.31427
.78881
1.26774
.81752
.22321
44
17
.73413
1.36217
.76134
1.31348
.78928
1.26698
.81800
.22249
43
18
.73457
1.36134
.76180
1.31269
.78975
1.26622
.81849
.22176
42
19
.73502
1.36051
.76226
1.31190
.79022
1.26546
.81898
.22104
41
20
.73547
1.35968
.76272
1.31110
.79070
1.26471
.81946
.22031
40
21
.73592
1.35885
.76318
1.31031
.79117
1.26395
.81995
.21959
39
22
.73637
1.35802
.76364
1.30952
.79164
1.26319
.82044
.21886
38
23
.73681
1.35719
.76410
1.30873
.79212
1.26244
.82092
.21814
37
24
.73726
1.35637
.76456
1.30795
.79259
1.26169
.82141
.21742
36
25
.73771
1.35554
.76502
1.30716
.79306
1.26093
.82190
.21670
35
26
.73816
1.35472
.76548
1.30637
.79354
1.26018
.82238
.21598
34
27
.73861
1.35389
.76594
1.30558
.79401
1.25943
.82287
.21526
33
28
.73906
1.35307
.76640
1.30480
.79449
1.25867
.82336
.21454
32
29
.73951
1.35224
.70686
1.30401
.79496
1.25792
.82385
.21382
31
30
.73996
1.35142
.76733
1.30323
.79544
1.25717
.82434
.21310
30
31
.74041
1.35060
.76779
1.30244
.79591
1.25642
.82483
.21238
29
32
.74086.
1.34978
.76825
1.30166
.79639
1.25567
.82531
.21166
28
33
.74131
1.34896
.76871
1.30037
.79686
1.25492
.82580
.21094
27
34
.74176
1.34814
.76918
1.30009
.79734
1.25417
.82629
.21023
26
35
.74221
1.34732
.76964
1.29931
.79781
1.25343
.82678
.20951
25
36
.74267
1.34650
.77010
1.29853
.79829
1.25268
.82727
.20879
24
37
.74312
1.34568
.77057
1.29775
.79877
1.25193
.82776
.20808
23
38
.74357
1.34487
.77103
1.29696
.79924
1.25118
.82825
.20736
22
39
.74402
1.34405
.77149
1.29618
.79972
1.25044
.82874
.20665
21
40
.74447
1.34323
.77196
1.29541
.80020
1.24969
.82923
.20593
20
41
.74492
1.34242
.77242
1.29463
.80067
1.24895
.82972
.20522
19
42
.74538
1.34160
.77289
1.29385
.80115
1.24820
.83022
.20451
18
43
.74583
1.34079
.77335
1.29307
.80163
1.24746
.83071
.20379
17
44
.74628
1.33998
.77382
1.29229
.80211
1.24672
.83120
.20308
16
45
.74674
1.33916
.77428
1.29152
.80258
1.24597
.83169
.20237
15
46
.74719
1.33835
.77475
1.29074
.80306
1.24523
.83218
.20166
14
47
.74764
1.33754
.77521
1.28997
.80354
1.24449
.83268
.20095
13
48
.74810
1.33C73
.77568
1.28919
.80402
1.24375
.83317
.20024
12
49
.74855
1.33592
.77615
1.28842
.80450
1.24301
.83366
.19953
11
50
.74900
1.33511
.77661
1.28764
.80498
1.24227
.83415
.19882
10
51
.74946
1.33430
.77708
1.28687
.80546
1.24153
.83465
.19811
9
52
.74991
1.33349
.77754
1.28610
.80594
1.24079
.83514
.19WO
8
53
.75037
1.33268
.77801
1.28533
.80642
1.24005
.82564
.19669
7
54
.75082
1.33187
.77848
1.28456
.80690
1.23931
.83613
.19599
6
55
.75128
1.33107
.77895
1.28379
.80738
1.23858
.83662
.19528
5
56
.75173
1.33026
.77941
1.28302
.80786
1.23784
.83712
.19457
4
57
.75219
1.32946
.77988
1.28225
.80834
1.23710
.83761
.19387
8
58
.75264
1.32865
.78035
1.28148
.80882
1.23637
.83811
.19316
2
59
.75310
1.32785
.78082
1.28071
,80930
1.23563
.83860
.19246
1
60
.75355
1.32704
.78129
1.27994
.80978
1.23490
.83910
.19175
t
Cotang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
t
I
3
C
2"
1
!
j
0*
NATURAL TANGENTS AND COTANGENTS. 135
40
Tang
.83910
.84009
.84108
.84158
.84258
.84307
.84357
.84407
.84457
.84507
.84556
.84606
.84656
.84706
.84756
.84806
.84056
.85057
.85107
.85157
.85207
.85257
.85358
.85408
.85458
.85660
.85761
.85811
.85862
.85963
.86014
.86115
.86216
.86419
.86470
.86572
.86674
.86725
.86776
.86878
Cotang
Cotang
.19175
.19105
.19035
.18964
.18824
.18754
.18614
.18544
.18474
.18404
.18264
.18194
.18125
.18055
.17986
.17916
.17846
.17777
.17708
.17638
.17500
.17430
.17361
.17292
.17154
.17085
.17016
.16947
.16878
.16809
.16741
.16672
.16535
.16466
.16329
.16192
.16124
.16056
.15987
.15919
.15851
.15783
.15715
.15647
.15579
.15511
.15443
.15375
.15308
.15240
.15172
.15104
.15037
Tang
49
41
Tang
.87031
.87082
.87133
.87184
.87287
.87338
.87389
.87441
.87492
.87543
.87595
.87646
.87749
.87801
.87904
.87955
.88007
.88214
.88265
.88421
.88524
.88576
.88784
.89410
.89463
.89515
.89777
.89935
.89988
.90040
Cotang
Cotang
1.15037
1.14969
1485*
14767
.14565
.14498
.14430
.14363
.14296
.14229
.14162
.14095
.13761
.13627
.13561
.13494
.13428
.13361
.13096
.13029
.12963
.12897
.12831
.12765
.12633
.12567
.12501
.12435
.12303
.12172
.12106
.12041
.11975
.11909
.11844
11778
.11713
.11648
.11582
.11517
.11452
.11387
.11321
.11256
.11191
.11126
.11061
Tang
Tang
.90040
.90146
.90199
.90251
.90304
.90357
.90410
.90516
.90674
.90727
.90781
.90940
.91046
.91099
.91153
.91206
.91259
.91313
.91419
.91473
.91580
.91687
.91740
.91794
.91847
.91901
.91955
.92062
.92116
.92170
.92224
.92277
.92331
.92493
.92547
.92601
.92655
.92709
.92817
.92872
.93143
.93197
.93252
Cotang
Cotang
1.11061
1
.10802
.10737
.10672
.10607
.10543
.10478
.10414
.10349
.10285
.10156
.10091
.10027
.09770
.09706
.09578
.09514
.09450
.09195
.09131
.09067
.08876
.C3813
.C8749
.03559
.084%
.CS369
.08306
.08243
.C3179
.08116
.08053
.07090
.07927
.07864
.07801
.07738
.07676
.07613
.07550
.07487
.07425
.07209
.07237
Tang
48
47
43
Tang
.93415
.93742
.93797
.94016
.94071
.94125
.94180
.94345
.94400
.94455
.94510
.94565
.94620
.94676
.94731
.94786
.94841
.94952
.95007
.95118
.95173
.95229
.95451
.95506
.95562
.95618
.95673
.95729
.95785
.95841
.95897
.95952
.96008
.96064
.96120
.96176
.96400
.96457
.96513
.96569
Cotang
Cotang
.07237
.07174
.07112
.07049
.06738
.06676
.06613
.06551
.06489
.06427
.06179
.06117
.06056
.05994
.05870
.05809
.05747
.05685
.05624
.05562
.05501
.05439
.05378
.05317
.05255
.05194
.05133
.05072
.05010
.04949
.04766
.04705
.04644
.04583
.04522
.04461
.04401
.04310
.04279
.04218
.04158
.04097
.04036
.03976
.03915
.03855
.03794
.03734
03553
Tang
46
136
NATURAL TANGENTS AND COTANGENTS.
440
440
44.
Tang
Cotang
Tang
Cotang
Tang
Cotang
.96569
.03553
60
20
.97700
.02355
40
40
.98843
.01170
20
1
.96625
.03493
59
21
,97756
.02295
'39
41
.98901
.01112
19
2
.96681
.03433
58
22
.97813
.02236
38
42
.98958
.01053
18
8
.96738
.03372
57
23
.97870
.02176
87
43
.99016
.00994
17
4
.9679-1
.03312
56
24
.97927
.02117
36
44
.99073
.00935
16
5
.96850
.03252
55
25
.97984
.02057
35
45
.99131
.00876
15
6
.96907
.03192
54
26
.98041
.01998
34
46
.99189
.00818
14
.96963
.0313*
53
27
.98098
.01939
33
47
.99247
.00759
13
8
.97020
.03072
52
28
.98155
.01879
32
48
.99304
.00701
12
9
.97076
.03012
51
29
.98213
.01820
81
49
.99362
.00642
11
10
.97133
.02952
50
30
.98270
.01761
30
50
.99420
.00583
10
11
.97189
.02892
49
31
.98327
.01702
20
51
.99478
.00525
9
12
.97246
.02832
48
32
.98384
.01642
28
52
.99536
.00467
8
13
.97302
.02772
47
.98441
.01583
27
53
.99594
.00408
7
14
.97359
.02713
46
34
.98499
.01524
26
54
.99652
.00350
6
15
.97416
.02653
45
.35
.98556
.01465
25
55
.99710
.00291
5
16
.97472
1.02593
44
36
.98613
.01406
24
56
.99768
.00233
4
17
.97529
1.02533
43
37
.98671
.01347
23
57
.99826
.00175
3
18
.97586
1.02474
42
38
.98728
.01288
22
58
.99884
.00116
2
19
90
.97643
.97700
1.02414
1.02355
41
40
39
40
.98786
.98843
.01229
.01170
21
20
59
60
.99942
1.00000
.00058
.00000
1
Cotang
Tang
'
Cotang
Tang
t
Cotang
Tang
/
45
45-
45-
/&' OF THE^^^.
{TJ'HIVEKSITTJ
X^JFogl^:
UNIVERSITY OF CALIFORNIA LIBRARY
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
7
11
APR 16 1946
30m 6/14
YB 11067