THE
FIELD PRACTICE
OF
LAYING OUT CIRCULAR CURVES
FOR
RAI LROADS.
BY JOHN C. TRAUTWINE,
Civil Engineer.
HTL A D)ELPHIA:
B. W. BaaB1L& S Ao, Pus=iBds
1851.




Entered, according to the Act of Congress, in the year 1851, by JOHNI C. TRAUTWINE,
in the Clerk's Office of the District Court of the United States, for the Eastern District
of Pennsylvania,




PRE EFACE.
I have been induced to prepare this little volume almost entirely with
reference to the wants of the many young men who desire to qualify
themselves for field-service in an Engineer Corps. On that account, I
have endeavored, by the use of the plainest language, to render the subject intelligible to them,-dispensing with that mathematical brevity
which would have better accorded with the requirements of those who
have already attained to some degree of proficiency in elementary fieldoperations. Still, I trust that it will not prove unacceptable even to the
latter.
The Table of Natural Sines and Tangents to single minutes, in<
form sufficiently portable for field use, will supply a want which I":ave
myself frequently experience], not only in the operation of laying out
curves, but on many other occasions.
One object in preparing it, was to furnish the profession with a
Table that should be not only portable, but absolutely reliable. Those
whose occupations compel them to resort to the Tables in common use,
must have frequently experienced, like myself, the extreme embarrassment which attends the inaccuracies to which they are all subject. So
long as a Table is known to contain a single error, the position of which
is not ascertained, its employment is attended with doubt in every instance in which we are obliged to refer to it. On this account, I have
not only prepared these Tables with the most scrupulous care, while in
common type, but, in order to render their accuracy a matter of ceftainty,




II
I had them stereotyped, and afterwards revised three times with the
utmost caution. I therefore feel no hesitation in saying that they may
be depended upon absolutely. The same remark applies to the other
Tables contained in the volume.
As Hassler's and Hutton's Tables of Natural Sines and Tangents are
those most in use among the profession, it will be desirable to those persons who possess them, to be able to correct the following errors which I
detected in comparing them.
In Hutton's Tables, Fifth Edition, 1811.
Sine of 60 8', for *1063425, lead'1068425.
Page 328, at top, for 25 Deg., read 40 Deg.
Tangent of 440 60', for'1000000, read 1'000000.
In Dr. Gregory's Corrected Edition (the 8th) of Hutton's Tables, 1838.
Cosine of 400 46', for'7576751, read -7573751.
In Hassler's lables, 1830.
Cosine of 110 36', read'9795752.
Sine of 200 60',    "'3583679.
Cosine of 230 41',  "'9157795.
Cosine of 330 21',  "'8353279.
Cosine of 340 40',  ".8224751.
Cosine of 360 56',  "'7993352.
Cosine of 410 48',  ".7454760.
Cosine of 440 57',  "  -7077236.
The foregoing I believe to be all the errors in the Natural Sines and
Tangents to whole minutes, in the respective Tables. The discrepancies
of 1 in the 7th decimal, I have not considered as errors, as they are occasioned by a neglect of the value of the 8th decimal. For calculating
curves, it is not necessary to use more than 4 decimals.
It is scarcely necessary to remark that, beyond 440, the Sines, Tangeaits, &c., are read upwards, from the bottom of the page, using the
corresponding columns of minutes. To find the sine of an angle exceeding 90~, subtract the angle from 1800, and take out the sine of the
remainder-because the sine of an angle, and that of what it wants of
1800, are the same.
JOiNT C. TltRAUTWINE.




FIELD PRACTICE
OF
LAYING OUT CIRCULAR CURVES
FOR
R A I-LR   A DS.
ARTICLE' I.
PRINCIPLES OF LAYING OUT CURVES.
METHOD 1.
To Lay Out a Curve by JMeans of Tangential Angles.
If from any point B, fig. 1, in a straight line AD, we lay
off any number of equal angles, asDBs, sBt, t     B     A  
u B v, &c., and at the same
time make the chords B s,
s t, tU, u v, &c., equal to
each other, then the paints
B, s, t, u,   c., & wil be
situated in the ctldirmfer-.               h      S
ence of a circle, whtI-is l"'1
tangential to the line'A"D
at the point B. 
The first of these angles
D B s, is called the tangen- 
tial angle, as being that by
which the curve is connected with the tangent A D; but inasmuch as the others are all equal to it, they also are called
-tangential angles.
If any obstacle, as h, should prevent our seeing from B,
farther than to v, the curve may be continued by removing




2
the instrument to u, the point preceding v; thence sighting'first on v, continue to lay off additional tangential angles
v u w2, WU u x, &c., as before. Or else, moving the instrument
to v itself instead of to u, sight back to u, and lay off first
the exterior angle p v w, double the tangential angle, and
afterwards continue the tangential angles w v x, x v y, &c.,
as before, to the end of the curve.
Finally, in order to pass' from the end of the curve at y,
on to a tangent y z, place the instrument at y, and sighting
back to x, lay off the tangential angle x y o; then o y continued'owards z will be the required tangent. (See Art. iv.)
ARTICLE II,
METHOD 2,
To Lay Out a Curve by.Means of Deflexion iAngles.
Fig 2. First, having, as in method 1st, laid off a tangential angle D Bs, and measured.~...........................-..... he   chord  B   s,  remove  the  instrument to the end s of the
/    chord, and make the exterior'  angle m s t equal to twice the.        tang'ential angle, and measure
the chord s t; and so on at the
6dter points t, u, v, &e., makikig each of the exterior anAy}' 2               gles n t a, o u v, &c., equal
to twice the tangential angle,
and all the chords equal; then
will the points B, s, t, u, v, &c.,
be in the circumference of a
circle which is tangential to
the line A D at the point B, as by the first method.
But, if at any of these points, as v, we wish to pass off to
a tangent v L, employ at that point, the tangential angle
z v L, equal to half the deflexion angle z v w. (See Art. iv.)
These exterior angles, included between any chord, and
the extension of the preceding chord, are caed deflxion
angles, or angles of dejexion, or angles of curvature.  In
any given circle, the angle of deflexlon is always precisely




3
double the tangential angle, supposing the chords to be
equal. We give tables of the angles corresponding to circles
of different radii, embracing the limits of railroad practice;
and calculated for chords 100 feet in length, that being the
usual length for a measuring chain on public works.
N. B. The deflexion angle of any curve is equal to the
angle t c u, or t c s, &c., at the centre of the circle, subtended
by one of the equal chords t u, or t s.  This angle at the
centre, so subtended, is called the central angle. The tangential angle being always half the deflexion angle, is, of
course, always half the central angle.
ARTICLE III.
METHOD 3.
To Lay Out a Curve by Eye.
The deflexion angles, fig. 3, e s t, f t u, g u v, h v w, &c.,
being double, the tangential
angle DBs, the arcs e dt,.... n.
f  iu, g m v, h  &c, &c., are
double the arc D c s, since.....
the arcs:f' circles are proportionate to the angles
which they subtend; but the    Fig 3
chords e t,fu, g v, h w, &c.,
are not double the chord
D s, since the chords of arcs                a
are not proportionate to the
arcs, or to the angles which              /  I
they subtend.
The chords e t, fu, g v,
h w, &c., which subtend   L
the deflexion angles, are
called deflexion distances; and the chord D s, which subtends
the tangential angle, is called the tangential distance.
But, although, in any given circle, the deflexion distance
is not truly twice the tangential.distance; yet, the difference
is so trifling in large railroad curves, with chords of but 100
feet that it may generally be neglected in curves of more than
300 feet radius.




4
In our tables the precise length of both will be found for
different radii.
Having these respective' distances, we may frequently
trace a curve on the ground by the eye only, with very
tolerable accuracy; sufficient for, guiding theexcavations
and embankments, especially on nearly level ground. Suppose, for instance, it be required to lay out in this manner a
curve of 5730 feet radius.
First, find by the table, the deflexion distance e t or fu,
&c., corresponding to a radius of 5730 feet for a chord of
100 feet, viz.: 1'745 feet; and also the tangential distance
ds 87 3 of a foot.
Then from the starting point B, and in line with A B,
measure B D equal 100 feet; and put a pin at D. Also from
B, measure the chord B s, equal 100 feet; at the same time
measuring with a graduated rod, from the pin D, the tangential distance D s, equal to'873 of a foot; and place a
stake at 0. The pin at D may then be removed.
Next, make s e equal 100 feet, placing a pin at e, precisely
in line with s B; also from s, measure s t equal 100 feet;
at the same time measuring with the rod, from the pin e,
the deflexion distance e t, equal to 1'745 feet. Place a stake
at t, and remove the pin at e. In this manner proceed to
find other points as far as the end of the curve at v.
In order to pass from the curve, as at v, to a tangent v L,
proceed as before, only using the tangential distance h n,
instead of the deflexion distance h w. (See Art. iv.)
This method is abundantly accurate for laying out curves
on a canal, or common road; and will occasionally answer
very well, when carefully performed, for railroad curves, in
the absence of an instrument. Thin straight rods, iron
pointed, and a plumb line should be used for ranging the
points, in the latter case.
The transit instrument is the best. for tracing curves, and
running lines generally. I prefer. the graduations to run from
the same zero, right and left, to 1800 each way. There
should be two verniers, graduated to minutes; by their
means half, or even quarter minutes may generally be estimated with considerable certainty. The telescope revolving
in a vertical plane, greatly expedites the laying off of exterior




angles, after having first sighted backwards to the point
be hind.
The verniers are sometimes graduated to hundredths of a
degree; and this division is, in certain cases, the best; but
for general purposes, the division into minutes, is to be preferred, as all the printed tables of sines, tangents, &c., are
calculated for that division.
ARTICLE IV.
On Sub-Chords.
*We have hitherto spoken of curves as if they were composed of equal chords, each of 100 feet in length. It frequently happens, however, that at the end of a curve, as
at e, fig. 4, we are obliged to use
a shorter, or sub-chord d e, in order    A       B
to unite properly with the tangent
ef,%
In that case, and when using
3Method 1,.irt. I, of  laying of. 
curves by means of tangential angles, we must, in order to fix the    g4
point e, lay off a sub-tangential angle d A e, as much smaller than the
entire tangential angle B A c, or.. —d    9
c A d, &c., as the sub-chord d e is
smaller than an entire 100 feet
chord, a c, c d, &c.  Thus, if the    f
sub-chord be one-half, or onefourth, &c., of the entire chord, the sub-tangential angle
must be one half, or one-fourth, &c., of the entire tangential angle.
This method is not mathematically exact, for the reason
stated in Art. iiI; yet, for curves of 300, or more feet radius,
and with chords not exceeding 100 feet in length, the error
is not observable in practice.
In like manner, when we pass off from a sub-chord, as at
e, to a second tangent ef, we must place the instrument at e,
and lay off the same sub-tangential angle deg; or which is
better, take sight from e to c, and lay off the angle c e g equal
to the sun of a tangential and the sub-tangential angle.
1.




But, when using Method 2,.Art. ii, or JMethod 3,.flrt. III,
of drflexion angles, or distances, to calculate the sub-deflexion
angle a b e, fig. 5, and sub-deflexion distance a e, formed between a sub-chord b e, and the extension b a, of an entire chord
g b, with sufticient accuracy for curves of 300 or more feet
radius, and chords of not more than 100feet,
Rule.-Say, as an entire chord of 100 feet, is to the subchord b e, so is the deflexion angle of the curve, to a certain
angle. Add these two angles together and divide their sum
by 2, for the deflexion angle of the sub-chord.
Example.-The curve,..~f           fig. 5, has a radius of
319.6 feet, and an angle
of deflexion, fg b, of
180 for chords of 100
feet. The sub-chord b e,
is 25 feet in length; what
is the sub-deflexion an"-. gle a be; and also the
sub-deflexion distance a
e, for the sub-chord b e?
Chord Sub-chord
Here, as 100 is to 25
Def. An. of Certain
100 ft. chord, Angle
So is   180 to 4~ 30Q.
The sum of these two angles 180 and 40 30' -22~ 30',
the half of which is 110 15', the required sub-deflexion angle ab e.
Again, to find the sub-deflexion distance a e, of the subchord b e; take from the table of sines, the natural sine of
one-half the sub-deflexion angle, just found. Multiply this
natural sine by 2, and multiply that product by the length
of the sub-chord.
Example.-The sub-deflexion angle is 110 15'; one-half
of it is 50 37k', the tabular natural sine of which is,0979,
which multiplied by 2, gives'1958; and this multiplied by
the sub-chord, 25 feet, gives 4'895 feet, the required subdeflexion distance a e.
Finally, to find the sub-tangential distance, b n, by means
of which to pass from e to the tangent e m, say, as 10,000




is to-the square of the sub-chord in feet, so is the tangential
distance for a 100 feet chord, to b n.
ARTICLE V.
Ordinates for Entire Chords.
It would be both tedious, and liable to inaccuracy, to attempt to fix all the necessary points in railroad curves by
the foregoing means, which are employed only for entire
chords, or for such sub-chords as may be required at the
ends of curves.
The best method is to stretch a piece of twine a b, fig. 6,
100 feet long, between two
adjacent chord-stakes, and
measure off as nearly as           LL1          b
may be at right angles to  /]gig6~.
it, with a graduated rod,
the previously calculated ordinates c d, ef, g A, &c, placing
pegs at d,f, h, &c. Our table of ordinates is calculated
for distances apart b c, c e, e g, &c., of 5 feet; and for all
curves likely to occur in practice. The 5 feet distances
on the twine should be marked by knots or otherwise, and
those at the centre, and half way between it and the ends,
be further distinguished by tying on pieces of tape.
The 5 feet distances are only used (after the excavations
and embankments are finished) for placing pegs to guide
the laying of the rails, and then only for very sudden
curves; for large ones, distances of 10 feet are quite sufficient, or even 25 feet for very easy curves. For guiding
the curves of the cuttings and fillings, it is not necessary
to place the stakes nearer than 50 feet apart; unless for
those of less than about 1000 feet radius, when they may
be placed 25 feet apart.  Ordinates for radii intermediate
of those in the table, may either be calculated by the rules
given further on; or they may be taken proportionally intermediate of the tabular ones, with sufficient accuracy for
practice.
Ordinates for Sub-Chords.
These may readily be calculated approximately enough
for railroad practice, for curves of over 300 feet radius, and




8
for chords not exceeding 100 feet, thus: In a circle of
givenradius, not less than about 300 feet, the ordinates of
an'entire 100 feet chord may be assumed to be to those of
a sub-chord, as the square of the chord is to the square of the
sub-chord.
In all our tables the chord is supposed to be 100 feet,
the square of which is 10,000; the rule therefore becomes,
as 10,000 feet: to square of sub-chord in feet:: Ord. of
Chord: Ord. of Sub-chord approximately.
Example.-In a curve of 5730 feet radius, the middle
ordinatbe of a 100 feet chord is'218 of a foot; what will
be the length of the middle ordinate of a sub-chord of 50
feet? here,
Sq. of 100 ft. Sq. of 50 ft. Mid. Ord. of Chord. Mid. Ord. Sub-Chord.  approximately.
10,000     2500  *  218 ft.   *          0545 ft.
and so of any other ordinate, always supposing the chord
and sub-chord to be divided into the same number of parts.
ARTICLE VI.
Having given the angle a b d, fig. 7, it is required to find
the point a or d, at which to commence a curve of given
radius.
Rule.-Subtract half the angle a b d, from 900; the remainder will be the angle b c a,
6            or b c d. From the table of tangents take the natural tangent of
b c a, and multiply it by the given
it,        b radius, the product will be b a or
Pig7:7N/.
Example. —Let the angle a b d
e            be 1200, how far from b must
we begin, at a or d, to lay out a
curve a n d, of 2865 feet radius?
Here, half of the angle a b d - 600, which taken from
900 leaves the angle b ca   300~.  The natural tangent of
30~0.'5773, which multiplied by the radius.of 2865 feet,
gives 1653-96 feet for b a or b d. (See Art. XII.)




ARTICLE  VII.
Having given the angle a b d,fig. 7, and the distance from
b to a or d, at one of which we wish to commence a curve,
it is required to find what radius a c or c d, the curve
must have, in order to unite with b a and b d tangentially
at a and d.
Rule.-Subtract the angle a b c, which is half the angle a
b d, from 90~; the remainder will be the angle b c a, or b
c d. Then as nat. sine of b c a, or b c d, is to nat. sine of a
b c, so is a b or b d, to a c or c d the radius required.
Example.- Let the angle a b d be 1200, and the distance b a or b d, 1654 feet; what will be the radius a c or c
d of a circle that shall touch a and d tangentially?
Here, the angle a b c = half the angle a b d, is 600,
which taken from 900, leaves the angle b c a or b c d -
300~. Then as the nat. sine of b c a or b c d (300) -= 5000,
is to nat. sine of a b c or d b c, (60~) ='8660, so is b a or
b d (1654 feet) to a c or c d, 2865 feet, the radius required.
ARTICLE VIII.
Having given the radius a c,fig. 7, of a curve, and the angle
a b d, it is required to find the number of chords of 100
feet that will constitute the curve.
Rule.-Subtract the angle a b d from 1800, and divide
the remainder by the angle of curvature, or deflexion of
the curve., The quotient will be the required number of
chords.
Example.-Let the angle a b d be 1200, and the radius a
c, 2'865 feet.
Here the angle a b d, 120~, subtracted from 180~, leaves
a remainder of 6.0~.; whichjdivided  by 20, the angle of
deffexion for a curve of 2865 feet, gives a quotient of 30;
which is the required number of chords of 100 feet.
N. B,-Had the quotient. contained a fraction of a chord,




10
it would have indicated that we should have had to employ a sub-chord at the end of the curve; for instance, had
the number of chords been 30k, a sub-chord of 50 feet
would.have been necessary.
ARTICLE IX.
How to proceed when the end of a curve does not correctly'join
the tangent.
We sometimes find on running out a curve for the number of chords determined by the Rule in the preceding Article, that instead of
d                   uniting as it should
with the previously
af   6ds'    b             determined tangent o
iy(>                          m, fig. 8, at o, it ends
/ Fig   C       \      tangentially to a line
parallel to said tangent, either within it
as at c; or beyond it
as at b. Being first certain that no error has occurred in
tracing out the curve, ascertain with the compass the bearing of the line a d, and removing the compass to the end of
the curve at c or b, (as the case may be,) run the line b o
or c o, in the same course as a d, until it strikes the tangent
d o m; which may be ascertained by ranging two stakes placed
on the tangent,
Then measure b o or c o, (as the case may be,) and if
the curve fall twithin the tangent o m, as at c, measure forwards from t towards d, the distance t a, equal to c o; or if
the curve fall beyond the tangent, as at b, measure backwards from s, the distance s a equal to b o. -Then the
curve retraced from a, will terminate tangentially in d m
at o.
N. B.-The direction of c o or b o may be ascertained
without a compass, and better, thus: Multiply the tangential
angle of the curve by twice the number of chords run, less
one; %ubtract the product fAbm 1800, and sighting back
one chord to n, lay off the angle n c b, equal to the remainder. For example, if the tangential angle be 100, and
from t to c be4 chords, then 7 times 10~ taken from 180~




11
leaves the angle n c b = 1100. When the product exceeds
180~, it must be subtracted from 3600, for the angle n c b.
This case occurs whenever an error has been made in
measuriug the distance from d to a. If d a be made too
short, the curve s b is the result; and if too long, the curve
t c.
If the error is small, it may be divided equally among
the chords by measure, without retracing the curve with
an instrument. This method may be employed with perfect security so long as the error does not exceed 1 foot
to every chord of 100 feet; and it will never be so great
if moderate care be taken.
Thus, if-the curve be 20 chords long, and the error 20
feet; the last stake may be moved 20 feet, the next 19, the
next 18, &c., as nearly at right angles to the curve as can
be judged by eye.
The same ordinates that would have been used had the
curve been correct, will answer for the one so adjusted,
without perceptible difference. For other cases, see Article X.
ARTICLE X.
Fig. 9.              Fig. 10.''! --..........
Again, it may happen that the error is not caused by a
mismeasurement of the distance a e, figs. 9 and 10, as in
the last case; but by a mistake in obtaining-the angle a ef.
If a ef, fig. 9, be measured in excess, as a e g, then the




12
curve a b c, calculated for the incorrect angle a eg, will be
found to fall beyond the true tangent ef, as at c; and the tangents e g and e f not being parallel, the curve cannot be
adjusted by either of the methods given in the preceding
Article, unless the error be within about 1 foot to each 100
feet length of the curve; in which case, (supposing no other
error to exist,) either of those methods may be employed,
with sufficient accuracy for practice.
Also, if a ef, fig. 10, be measured too small, as a eg, then
the curve a b c, calculated for the incorrect angle a e g, will
be found to fall within the true tangent ef, as at c; when so,
the remarks contained in the preceding sentence are equally
applicable here. If the error be within 1 foot to 100 feet
length of curve, it may be equally divided among the chords.
Butif greater, we must either re-measure the angle a ef correctly, and go over the whole work again, or resort to some
other mode of obviating the difficulty.  The angle a efmay
be difficult of access; or the curve may be so long that to
retrace it would be a work of much labor. We may then
adopt the method of compound curves, by which much
trouble will be avoided, and a considerable portion of the
first part of the curve be allowed to remain as it is.
Thus, whether the curve a b c fall beyond the true tangent ef, as in fig. 9, or inside of it, as in fig. 10, place the
instrument at b, figs. 9 and 10, (the point at which the change
of radius is to take place,) and sighting back one chord to
n, lay off the tangential angle n b m of the curve a b c, and
observe where the new tangent m b continued, strikes ef, as
at o. Measure both b o and the angle b of. Half the angle
b of taken from 90~, gives the angle b h o; then say,
a t. Sine of angle b h o,    $ Nat. Sine of angle b o h
As the  Na. in of anl   X g'-is  to  opposite the required
posite the given side b o,        side b h,
So is The given side b o, to The required side, or new radius b h.
Ascertain from the table, or by calculation, the angle of
deflexion, and the tangential angle corresponding to this
new radius b h; and the new curve comme ned at b will
terminate tangentially to ef at i, as fir from o as o is from b.
For the mode of uniting two curves of different radii, so
as to form a compound curve, see Article xiI.




13
It will be observed, that when the first curve, a b c, fig. 10,
falls' inside the tangent ef, the new curve must be of greater
radius; and when beyond, fig. 9, of a less one.
ARTICLE XI.
Having given the angles a b c and b c d,.fig. 11, and the distance b c, it is required to find the greatest radius, g i or h i,
that can be employed in a reverse curve, f o i n m,for uniting
ab to c d.
Rule.-Half the angle ab c taken from 90~,
leaves the angle bg i;.FII 
and half the angle bd c d
taken from 90~, leaves."
the angle i h c.  
From the table of tangents take the natural
tangent (b i) of the angle,
b g i; and that (ic) of the
angle i h c; and add them together.
Then as the sum of these two nat. tangents is to the nat.
tang. of b g i, so is b c to b i; and b i taken from b c, gives i c.
Again, in the triangle bg i, as the nat. sine of the angle
bg i, opposite the given side bi, just found, is to the nat.
sine of the angle g b i, opposite the required side g i, so is
b i, the given side, to g i, the required side or radius.
Example.-Let the angle a b c be 710 40', the angle b c d
129~ 15', and the distance b c 950 feet. What is the length
of radius h i or g i, of the easiest reverse curve that can be
traced for uniting a b to c d?
Here, half the angle a b c (350 50') taken from 90~, leaves
the angle b g i 54~ 10'; and half the angle b c d (640 37J')
taken from 90~, leaves the angle i h c = 25~ 221'.
2




14
From the table of tangents, we have nat. tang. of bg i
(540 10') -= 13848; and nat. tang. of i h c (250 22it)'4743; their sum being 1'8591.'Then as
Sum of Tang's.    t s Tang. ofbbc                 5
1 8591.    0isto54 10,   so is  950 ft,  to  707'63 ft.
1'~8591.~1'3848,
and bi, 707'63 feet, taken from  b c, 950 feet, leaves i c
242'37 feet.
Again, as the
Nat. Sine 
of angle  i toNat. Sine of         bi       required ri-the
isto  Angle g b i  so is 707.63  to dius,ired ra51097
6 g i5854    I                   feet      dius5, 51097
-8107  J84.                                feet.
ARTICLE XII.
To obtain the angle d b e, formed by two tangents, d b, and
b e, when the point b is inaccessible.  Figs. 12, 13, 14,
and 15.
This is of frequent occurrence.
CASE 1. When the included figure, fig. 12, has but three
sides.
Rukle.-Subtract the angle a d e from 180~ for the angle
b d e; and subtract the angle d e c from 180~, for the angle
d e b.  Add together b d e, and d e b, and subtract their sum
from 1800, for the angle d b e.
Fig. 12,    Fig. 13.   Fig 14.      Fig. 15.
b          b          6           b
CASE 2. When the included figure, d b ef, figs. 13 and
14, hasfour sides.
Rule.-Subtract the sum of the three internal angles of the
igure marked by dotted segments of circles, from 3600, for:he angle d b e.




15
CASE 3. When the included figure, fig. 15, has more than
four sides.
Rule.-Add together all the internal angles, marked by
dotted segments of circles; and subtract their sum from twice
as many right angles as the figure has sides, less four, for
the angle d b e.
Example.-Let the angles denoted by the dotted segments at the different letters be as follows: That at d,
700~; at o, 2200; at i, 150~; at s, 110~; at c, 160~; at e,
100~. The sum of these is 810~. The figure has 7 sides,
and twice 7, less 4 = 10; and 10 right angles = 9000;
from which the sum of the designated internal angles (810~)
being subtracted, leaves 900, for the angle d b e.
N. B.-When the angle d b e has to be deduced from a
figure of many sides, as fig. 15, the errors spoken of in
Articles IX. and X. are apt to occur, unless the several
sides and the angles o, i, s, &c., be measured with much
care. For tracing curves with any accuracy and satisfaction,
the instrument should be divided at least into minutes; as
before remarked, the transit instrument is the best for the
purpose.  With moderate care in the preparatory measurement of the sides and angles, errors will seldom occur, that
may not be adjusted with all the accuracy required in practice, by the very simple method of' dividing them equally
among the chords, as explained in Articles IX. and X.
ARTICLE XIII.
To pass from  one curve a m b, fig. 16, to another b n e
of different radius, but running in the same direction., constitutingo a COMPOUND curve.
Rule. —Placing the instrument at b, sight back to the
other end of the 100 feet chord
at a; and lay off the tangential   d         b        e
angle a b d, of the curve a m b;                  n
then from the common tangent d
b e, lay off the tangential angle          Fie       e
e b c, of the curve b n c, making
at the same time the chord b c
equal to 100 feet.




16
N. B.-If running the curve by eye, use the tangential
distances instead of the angles.
ARTICLE XIV.
To pass from one curve, m n t, fig. 17, to another, t i o, of
either the same, or of a different radius, but running in an
opposite direction; constituting a REVERSE curve.
Rule.-Placing the instrument at t, sight back to the
other end of the 100 feet chord'iug17                  at m, and lay off the tangential
m,7   angle m t r, of the curve m n
r         A: L  C    t; then from the common tan-.
gent r t s, lay off the tangential
angle s t o, of the curve t i o;
making at the same time the
chord t o, equal to 100 feet.
N. B.-If running the curve by eye, use the tangential
distances, instead of the angles.
ARTI'CLE XV.
RADII.
To find the radias corresponding to any given angle of deflexion, and to equal chords of any given length.
Rule 1. Subtract the angle of deflexion from 180~, then
say as nat. sine of angle of deflexion, is to nat. sine of
half the remainder, so is the given chord to the radius
required.
Example.-Let the angle of deflexion be 20, and the
chord 100 feet, required the radius.
Here 20 subtracted from 180~, leaves 1780, the half of
which is 89~, and as
Nat. Sine of 20~  Nat. Sine of 890~   Chord. Radius
*034899    *~ 999848       ~  100 feet ~ 2865feet.
Rule 2. The radius for 100 feet chords may be found
approximately, by dividing 5730 by the deflexion angle.




17
This rule is very close for radii of not less than 500 feet.
For 500 feet it gives,% ths of a foot too little; but is more
approximate for larger radii.
Example. —What is the radius to a deflexion angle of
2~, the chords being 100 feet long?
Here, 5730 divided by 2, gives 2865 feet, the radius
required.
ARTICLE XVI.
ANGLES OF DEFLEXION.
To find the angle of deflexion corresponding to any given
radius, and to equal chords of any given length.
Rule 1. Divide half the chord by the radius; the quotient
will be the natural sine of the tangential angle. Therefore,
twice the angle correspofidingto this sine, in the table of
natural sines, will be the angle of deflexion required. By
this rule our table has been prepared.
Example.-Let the radius be 2865 feet, and the chord
100 feet; what will be the deflexion angle?
Here, half the chord. 50 feet, divided by the radius,
2865 feet, gives'01745; and the tangential angle in the
table corresponding -to the nat. sine'01745 is 10, twice
which is 2~, the deflexion angle required.
Rule 2. The deflexion angle for 100 feet chords may be
found approximately by dividing 5730 by the radius. This
is very close for curves of over 500 feet radius. For 500.
feet, it gives about 1 minute too little.
Example. —What is the deflexion angle for a radius of
2865 feet, the chords being 100 each?
Here, 5730 divided by the radius 2865, gives 20, the
deflexion angle required.




18
ARTICLE  XVII.
DEFLEXION DISTANCES.
To find the deflexion distance (exactly)Jor any- given radius,
when the chords are 100feet long.
Rule. —Divide the constant number 10000 by the radius
in feet; the quotient will be the deflexion distance required.*
Exampl. —What is the deflexion distance to a radius
of 5730 feet; the chords being 100 feet long?
Here, 10000 divided by 5730 radius, gives 1'745 feet,
the deflexion distance required.
To find the deflexion distance for any given radius, and
for equal chords of any given length.
Rule.-Divide half the given chord by radius; the quotient will be the natural sine of one-half the deflexion angle;
and double this natural sine, multiplied by the chord, will
give the deflexion distance required.  By this rule our
table was prepared.
Example. —As before, what is the deflexion distanee to
a radius of 5730- feet; the chords being 10( feet long?
Here, half the chord, 50 feet, divided; by radius 5730,
gives.008727, which is the natural sine of half the deflexion angle. Now,'008727, multiplied by 2, gives
~017454, which multiplied by the chord, 100 feet, gives
1'745 feet, the required deflexion distance, the same as in
the preceding example.
ARTICLE XVIII.
TANGENTIAL ANGLES.
For the method of finding these, see Article XVI.
Because the deflexion distance to a radius of 10000 feet, with chords
of 100 feet, is 1 foot; and the deflexion distances for other radii, increase
inrereely as the radii.




19
ARTICLE XIX.
TANGENTIAL DISTANCES.
To find the tangential distance corresponding to any given
radius, and to equal chords of any given length.
Rule.-First find the tangential angle, by Article XVI;
and take from the table of nat. sines, that corresponding to
one-half of the tangential angle. Then multiply double this
sine by the given chord, for the tangential distance. By
this rule our table was prepared.
Examnple.- Let the radius be 2865 feet, and the chords
100 feet each; what will be the tangential distance?
Here, we find by Article XVI, the tangential angle 1~ for
a radius of 2865 feet.
The natural sine corresponding to 30e minutes, or bnehalf of this tangential angle, is by the table of sines,
~008727; the double of which is'017454, which multiplied
by the chord, or 100 feet, gives 1-745 feet for the tangential
distance required.
ARTICLE XX,
ORDINATES.
To find the middle ordinate to any given radius, and to any
given chord.
Rule 1.-From  the square of the radius, subtract the
square of half the chord; and take the square root of the
remainder from the radius, for the middle ordinate.
Example.-What is the length of the middle ordinate d e,
fig. 18, the radius c a being 819 feet, and the chord a b
100 feet?
Here, the square of c a (819) is 670761; and the square
of a e (50) is 2500; which, being subtracted from the
former, leaves 668261; the square root of which is e c,
817'472; which taken from'the radius 819, leaves 1'528
feet, the required middle ordinate, d e.




20
Rule 2.-Subtract the tabular cosine of the tangential
angle from 1; and multiply the remainder by the radius.
Example. —Same as foregoing, namely, radius 819 feet;
angle of deflexion 70, to chords of 100 feet. What will be
the length of the middle ordinate?
Here, tabular cosine of 31~' (the tangential angle) is
~998135; which subtracted from 1, leaves'001865; which
multiplied by 819, the radius, gives 1'527, the middle
ordinate required.
ARTICLE XXI.
Having given the middle ordinate d e, fig. 18, it is required
to find any other one, as i n.
/d               Rule 1. —Subtract the mid~/f-~O    dle ordinate d e, from  the
/a/1    b   radius, d c, the remainder will
ae / X'       be e c; then from the square'"'-..~ ]    b /of the radius c i, subtract the; Fig ~/   square of the distance o i,
c             vrwhich the required ordinate i
n, is from the middle ordinate
d e, and extract the square root'of the remainder.  This
square root will be o c. From this square root o c, subtract e c; the remainder will be o e, which is equal to i n,
the required ordinate.
Example. —The middle ordinate d e, of a 100 feet chord
b a, to a radius of 819 being 1'528 feet, it is required to find
the length of the ordinate i n, 20 feet from the middle one.
Here, the middle ordinate d e, 1'528, subtracted from
the radius 819, leaves e c, 817'472. The square of the
radius is 670761; and the square of 20 (the distance of the
required ordinate from the middle one) is 400; which taken
from 670761, leaves 670361;- the square root of which is
818'756 or o c; from which take e c, or 817'472, and the
remainder 1'284 will be o e, which is equal to i n, the
required ordinate.




21
Rule 2. —Multiply  the ordinates of a  10 curve by  the
deflexion angle of' the curve whose ordinates are required,
(chords being 100 feet.)  This is a sufficiendly close approximation for curves of not less than 500 feet radius; and for
placing ordinates for gttidiog  tl/e excavations and enbankments, it is close enough for the smallest curves in our table.
TABLE OF RADII, &c. CHORD 100 FEET.
The Tangential Angle is always one-half of the Angle of Deflexion.
o!b                      o!
a             a~
5   68760    -145.073    2 40   2149.0  4-653   2-327
10   34380    *291.146          45   2084-0  4-799   2-399
15   22920   -4:36.218        50   2023-0  4-944   2-472
20   17190    -581    -291         55   1965-0  5-090   2-545
25   13752'727    -364    3          1910-0  5-235   2-618
30  1.1460.872    *436           5   1859-0  5-380   2-690
35    9823   1-017    -509        10   1810-0  -.526   2-763
40    8595   1-163    *582        15   1763-0   5-671   2-836
45    7640   1-308    -654        20   1719-0  5.817   2.909
50    6876   1-453    *727 1   25   1677-0  5.962   2-981
0 55    6251   1-600      S00       30   1637-0  6-108   3.054
s 1     5730   1-745    *873        35   1599-0  6.253   3-127
5    5289   1-890    -945        40   1563.0   6-398   3-199
10    4912   2-036   1-018        45   1528-0  6-544   3-272
15    4584   2-181   1-091        50   1495.0   6-689   3.345
20    4298   2-327   1-164    055   1463-0  6.835   3-418
25    4045   2-472   1-236    4        1433-0  6-980   3-490
30    3820   2-618   1-309        15   1348-0  7-416   3-708
35    3619   2-763   1-382        30   1274-0  7-853   3-927
40    3438   2-908   1454         45   1207-0  8-289   4-145
45    3274   3.054   1-527    5        11460   8.722   4-361
50    3125   3-199  1 1599        15   1092-0   9-159   4579
55  12990   3-345   1-673          30   1042-0  9-595   4-798
2       2865   3-490   1-745         45    996'8  10)030   5-015
5    27.50   3-635   1-818    6        955-4  10-470   5-235
10  2644   3-781   1-891          15    917-0  10-900   5-450
15    2547   3-926   1-963        30    882-0  11-340   5-670
20    2456   4-072   2-036    0 45    849-3  11-780   5'890
25    2371   4-217   2-109  1 7         819-0  12-210   6-105
30 1 2292   4-363   2-12          15        7918. 12-640   6-320
35    2218   4-508   2-9254       30    764-5 13-080   6-540




22
TABLE OF RADII, &c. CHORD 100 FEET.
CONTINUED.
The Tangential Angle is always one-half of the Angle of Deflexion.
_d.
0                        B
o'                             o
7 45   739'9 13510   6755  15 30   370'8   26'94   13'52
8       716'8   13950   6'975   0 45   365'0   27'37   13'73
15   695'1  14380   7190  16           359-3   27'83   13'95
30   674'6  14'810  7 405   0 30   348-4   28'70   14'38
0 45   655'5  15*250   7'625  17         338'3   29'56   14'829       637'3   15'680   7 840   0 30   328 7   30'43   15'25
15   620'2   16.120   8'060  18        319'6   31'29   15'69
30   603'8   16 550   8'275   0 30   311'0   32'50   16'12
0 45   588'4   16 990   8'495  19        302'9   33'01   16'56
10       573'7   17-430  8'715   0 30   295'3   33'87   16'99
15   559 7   17 870   8 935  20        287 9   34*73   17 43
30   546'4   18 300   9'150  21        274'4   36 44   18'30
0 45   533'8   18*730   9'365  22        262'0   38.15   19'17
1 1     521'7   19 170   9'585  23        2,508   39'87   20'02
15   510'1   19'610  9'805  24        240'5   41'58   20*91
30   499 1   20 050  10-030  25       231*0   43 28   21-77
0 45   488.5  20 500  10'250  26         222'3   44'98   22'64
12      478'3  20'940  10'470  27         214'2   46'68   23'51
15   468'7  21 360 10'690  28         206'7   48'38   24'37
30   459.3  21 790 10 900  29        I 199-7  50-07   25-24
45   450'3   22*210  11'120  30        193-2   51'76   26'11
13    1 441'7   22 640  11.340  31        187'1   53'45   26'97
15   433.4   23-070  11,560  32      181.4   55'13   27'83
30   425.5   23510  11'770  33        176'0   56'80   28'70
0 45   417-7  23940  11990  34          1710   58-47   29-56
14      410'3   24i370  12'210  35        166'3   60'14   30'42
15   403'1  24'810  12'430  36        161'8   61'80   31'29
30   396'2   25 240  12'650  37       157'6   63'46   32'15
0 45   389-6  25-670  12-860  38         153'6   65'11, 33'01
15    1 383'1   26-110  13'080  39       149'8   66'76   33'87
15   376'9   26-520  13'300  40       146'2   68'40   34'73




TABLE OF ORDINATES.
Ordinates 5 feet apart. Chord 100 feet.
a "           Distances of the Ordinates from the End of the
us ~                            o100 feet Chord.
~1~.~..........3...'a    a                                           a    a
___                    Lengths of Ordinates in Feet.
o      -  _._                                        -.- _.
6'018'018'017'016'015'014  *012  *009 *006 003
10* *036'036'035'033'031'027'023'019'013'007
15'054.054'052.049  -046'041'035'028'019'010
20'073'072'070  *066'061'055'047'037'026'014
25'091  *090'087'082'076'068'058'046'032'017
30'109'108'105'099'092'082'070'055'039'020
35'127'126'123'116'108'096'082 *065 *045'024
40'145'144'140'133 4123'110 *093.074 *052'027
45'163'161  *157  *149'137'123'105'083'058'031
50'182.180'1756  166'153'138'117'092'065.034
0 55'200'198'192'182'168'151'128'102'071'038
1     *~218'216'209.198'1831'164'140.111'078'041
5  *236'234.226'215'198'178'152'120 o 085'044
10'254'252'244'231  *214'191'163'130'09!048
15'273'270'261'248'229'205'175'139'098'051
20'291'288'279'264'244'218'187'148'104'055
25'309'306'296'281'259'232'198'157.111'058
30'327'324'314'297'275'246'210'167'117'062
35'345'342'331  -314' 290'259'221'176'1241 065
40'364'360'349'330'305'273 1233'185'130 1069
45'382'378'366'347'321'287'245'195'1371 072
50'400.396'384'364   336'300i'256'204'1441 076
o 55'418'414'401'380'351'314'268.213'1501 079
2      435'432'419'397'366.327 1280.222'1571 083
5'454.450'436'4131 382'341'291'232'1631 086
10'473.468'454'430'397'355'303'241'1701 089
15'491.486'471'446'412'368'315'250'1761 093
20'509.504'489'463.428'1382'326'260'183. 096
25'527'522'506'480'443'396'338'269'1901 100
30'545'540'524'496'458'409'350'278'1961 103
35'564.558.541'513.474'423'361'288.203  107
40  6582'576'5591 529'489 14361 373'297.2091 110
45'600'594  576 j'546'504 1450'384'306'216'.114
50'618'612'5941'562   519'464.*396   315  222.117
o 55'636'630 1'611   579'535'477'408   325'2291 121
3'654'648'629'595'550'491   419'334'235'124
5' 673'666' 646'612'565'504'431'343'2421 128
10'691'684'664'629'581'518'443'353'249'131
15 j'709'702'681'645   596 1532'454 1 362'255'134




24
TABLE OF ORDINATES.
CONTINUZD.
Ordinates 5 feet apart.  Chord 100 feet.
w) 0           Distances of the Ordinates from the end of the
E:m~~ Y                    ~~~100 feet Chord.
a    i       u I3    O   Ih      O        I,n IO'S  Ir'  I ~  C  IC CO  I3 I...~. fi )Lengths of Ordinates in feet.
o  J.
3 20   *727  *720  *699'662'611  *545  *466'371 *262 *138
25    -745.738'716'678'627'559'478'380'268 1141
30   *764  *756'734'695  *642'573  *489'390' 275 *145
35'782.774  -751'711'657  -586'501'399 *281'148
40   *800.792'769  *728'673  *600'512'408'288'152
45'818.810  *786'744'688'613'524'418 *294'155
60   *836'828'804'761   703'627  *536  *427'301 *159
65   *854'846'821'778'718'641  -'547'436.308'162
4'873  *864'839'794'734'654'559'445'314'166
15' 927.918'891'844'780'695.594'473'334'176
30'981.972'944'893.8251 736'629'501'354'186
o45  11036 1.026'996  943'871.777'664'529'373'196
5    11.091 1.080 1.048.993'917'818i 699'557'393'207
15  11146 1t134 1.100 11042'963'859'734'585 14137'217
30  1.200 I1188 11153 11.092 1t0091 900'769  -613 1432 -228
0 45  11255 11-242 11205 11141 1'0551 941.804'640'452'238
6     1.309 1.296 11258 1.191 11100'982'839'668'472'249
15  1'~364 1i350 1'310 1'240 11146 1'023'874'696.492  1259
30  1.419 11404 11362 1'290 1'192 1!064I'909'.724  511'269
45  11473 11458 1'415 1'339 11238 1'105'944'762'531'280
7     1'528 1'512 1.467 1l389 11284 1.1461 979'779.551'290
15  1'582 11566 I1-520 11'438 1'330 1'187 1'014.807'570'301
30  11637 11620 1.572 11488 1.375 1.228 11048'835.590'311
45  1'692 11674 11624 11537 1'421 11269 11083'863'6]10'321
8     1'746 11728 1'677 11587 1'467 1'310 11118'891.629'332
15  1'801 1'782 l.729 11'637 1X'513 1'351 1'153'918  649  342
30  1'855!1836 1'782 11687 11559 1'39211'188'946.669'353'  45  1'910 11890 11834 11737 1'605 11433 1'223'974'689'363
9     1.965 11.944 11886 11787 11651 1.474 11258 11002'708'373
15 i2'019 11998 11939 11'837 11696 1'515 112'3 1'030'728 1384
30  2'074 2'052 1'991 11887 ]1'742 1'556 11.328 11057  748.394
o 45  2'128 12'106 j2'044 i1'937 11'788 1'597 11'363 1'085'767'405
10    12'183 12'161 12096 11'987 11834 1'637 11'398 1114  787'415
15  2'238 12.215 12148 J2'037 11880 1.678 11.433 11142'807'425
30  2'292 2269 12201 12'087 11[926 1719 11.468 117Q's827' 436
45 12347 i2-323 12.254 2136 11972 1'761 1.503 10198  846 1446
11    12401 12'377 |2.306 12186 12018 1.802 1x538 11226'866 1457
11 5  2'456 12'432 12359 12236 12'064 11843 11574 l1254'886 1467
30 12511 12486 12411 2'286 12110 1.884 1.609 1'282'906'478
45  2-566 2-540 2'464 2'336 12.156 1'926 1.644 1310 1.926 1 488




:25
TABLE OF ORDINATES.
Ordinates 5 feet apart. Chord 100 feet.
-o
bn   Distances of the Ordinates from the end of the 100 feet Chord.
~..      Z. ~ 1   - - o
~ 
t -*             %1  42            I'P. I  0p It. 0
N.PH  O:'    I                         o e         ~ -
Lengths of Ordinates in feet.
12   2'620 2'594 2'516 2'386 2'203 1'967 1'680 1'339  *946.499
15 2'675 2'649 2'569 2'436 2'249 2'008 1'715 1'367  *966'609
30 2'730 2'703 2'621 2-485 2'295 2-049 1'750 1'395'985 *520
0 45 2'785 2'757 2.674 2'535 2'341 2'091 1'785 1.423 1'0051'530
13   2'839 2-811 2.726 2'585 2'387 2'132 1'820 1.451 1.025 *541
15 2'894. 2'865 2.779 2.635 2'433 2.173 1'855 1'479 1.045 *551
30 2'949 2'920 2.832 2.685 2.479 2.214 1.891 1.507 1.065 *562
045 3.000 2.974 2.884 2'735 2'525 2.256 1'926 1'535 1'085'572
14   3'058 3'028 2.937 2'785 25'71 2.297 1.961 1.564 1.105.583
15 3'1131 3082[ 2'989[ 2'834 2'61i8 2'338i 1.996 1'592 1'124.593
30 3'168 31361 3-0421 2884 2'664{ 2'379g 2.031 1'620 1'144 *604, 45 3.222i 3.191 3-0941 2934 2.7101 2421t 2'067 1-648 1.164 *614
l     3-277 3'245 3.147 2'9841 2'756 2'462 2'102' 1'676 1'184  625
15 3'332 3'299 3.2001 3'034i 2'8021 2503 2'137 1'704 1'204'635
30 13387 3'3541 32521 3'084 2'848 2'544 2'172, 1'732 1'224'646, ~45 3'442 3'408 3.3051'31341 28951 2586' 2'208' 1s760 1'244'656
16   3'4961 3462 3'358 3'184f 2'941 2'6271 2243 1789 1'264 *667, 30 3.606 3.571| 3'463 32841 3.0331 2710 2'314 1.845 1'304 G688
17   3.716 36801 3'569 3'384 3.125 2'792{ 2'384i 1'902 1'344'709
o 30 1-826 3'788 3'674 3.484 3.218 2'875] 2.455 1'9581 1'384.730
18   3-935 3'897 3.779 3'584 3.310j 2'958 25251 2'014 1'424 *751
o 30 4.045 4'006 3'885{ 3i684 3.4031 3040 2.596 2'071 1'464'772
19   4'155 4'115 3'990 3-784 3-4951 3123! 2'666 2.127 1.504 *793
o 30 4'2651 4.2231 4'096 3 884 3'588 3-20512q737 2'184 1'544 *814
20  1 4375 4'332 4'201 3'984/ 3.680 3'288 2'8081 2.2401 1'583'836
21  4'595 4.549 4412 -44184 3.864{ 3.454 2'9!01 2'353 1'663'879
22  f 4'815 4'768 4'6241 4386 4'050.3.6201 3'093 2'4671 1'744. 922
23  1 5'035 4'986 4.836 4'5871 423.7 3786 3'236 2'581 1'8241  965
24   5'2551 5.204 5'0481 47891 44231 3'952 3-379 2'695 1'905 1.008
25   5'476 54.22 5'260 4'989 4'6091 4119 3'522 2'809 1'986 11051
26   5'697 5'642 5'473 5'192 4'798{ 4'286 3'665 2'924 2t068 1t094
27   5'918 568601 5685 5'3931 4984{ 4'454 3'808 3'039 2'150 1.137
28   6-139 6'079 5'8981 5595 5.171 4'622 3'952 3'154 2'232 1.181
29   6'361 6'f298 6'110 5'796 5'357 4'790 4'095 3'269 2'314 1.224
30   6'582 6.517 6'323 5'999 5'544 4.958 4'239 3.385 2.39611.268
31   6.804 6'737 6.5371 6.2021 5.733 5'127 4'384 3'502 2'481 11312
32   7.027 6'957 6.7511 6406 5'922 5.297 4.530 3-619 2.565 1.356
33   7'249 7'178 6-9651 6'609 6.111 5.467 4,'676 3.737 2'649 1.401
34  7.472 7'398 7.179 6'8131 6'300 5'637 4'822 3.854 2'733 11445
35   7'694 7.619 7.393 7.0171 6.4891 5807 4.9681 3972 2'817 1.490
36   7'918 7'841 7'609 7'222 6'679 5;978 5.115 4'090 2'901 1.535
37   8'143 8,063 7'825 7'427 6-870 6'149 5-262 4'209  21985 1.581
38   8'367 8'286 8.041 7-633 7.060 6'320 5.4101 4327 3'06911'626
39   8'59.2 8-508 8'257 7'838 7'2511 6'491 5.5571 4446 3'153 1'672
40  I 88161 8'7311 8-474 8.044 7'442 6-663 5.7051 45651 3'2381 l718
3




26'
ARTICLE  XXII,.
ON'LONG CHORDS.
It is sometimes convenient in preliminary locations, to
lay off curves by chords longer than 100 feet. For instance, in fig. 19, insteadof running from a by chords a b,
FigI9
b c, c d, &c., of but 100 feet, points d f g, &c., may be
obtained with less trouble by using three times the tangentiaI or deflexion angles of the table, (as the case may
be,) and employing chords a d, d f, f g, &c., nearly three
times as long as the chords a b, b c, &c.; or if a d, d f, fg,
be either 2 or 4 stations apart, then 2 or 4 times the tangential and deflexion angles would be used; and chords
nearly- 2 or 4 times 100 feet in length.
The following table contains the precise length of chord
required'to subtend respectively 1, 2, 3, or 4 stations. It
is seldom desirable to exceed the latter limit.




TABLE OF LONG CHORDS.............+S    |         Length of Chord in feet required to subtend
~ ~|.!  C     1 Station.  2 Stations. 3 Stations. 4 Stations.
5730.0   10         100       200-0      300-0      400-0
4584'0              100       200-0      300-0       399'9
3820'0       i'100      200-0      300.0       399.9
3274-0        i     100       200-0      300'0       399-8
2865-0  120         1000      200-0      299-9       399.7
2547-0.          100       200-0      299'9       399-6
2292-0              roo00     200-0      299-8      399.5
2084-0       i      100   ~  200-0       299-8       399-4
1910'0  *30         100       23000      299-7      399-3
1763-0       4      100       200-0      299-7      399 2
1637-0       4      100       200-0      299-6t    399-1
1528-0       a      100       200 0      299-6      399-0
1433-0    40        100       199-9      299-6      398-9
1348-0              100       199-9      299-5      398'7
1274-0.          100       199-9      299-4      398-5
1207-0              100       199-9      299-3      398-3
1146-0  I5.       100       199-9      299-2      398-0
1092-0       4      100       199-8      299-1       397-8
1042-0    i         100       199-8      299-0      397-6
996-8:      100       199-7      298-9      397-5
9,95'4    6        100       199-7      298-8      397'3
19170               100     199-7      298-7'397-0
882-0              1040      199-7      296,6       396-7
849-3  I           100       199-6      29.85       396-5
819-0   7O         100       199-6      298-4       396-2
790-'8             1.00      199-6      298-3      396-0
764-5       4      100       199,6      298-2      395-7
739-9              100       199-6      298-1      395-4
716-8     o        100       199-6      29;8-0     395-1
695-1  J           100       199-5    -297-9        394-8
674-6  1           100       199,5      297-8       394-5.655-5  I          100       199-4       297-7      394-3
637-3    9 ~       100       1'99'4     297-5      394.1
620-2              100       199-4      297-4      393-7
603-8              100       199-3      297-3      393-2
588-4       a      100       199-2      297-2      392-8
573-7 l 10         100       199-2      297-0      392-4
For radii less than 573-7 feet, it is never required to use longer chords
{lhan 100 feet.




28
When this method of laying out curves by long chords is
used, the instrument should be moved to each successive
point after it is determined, in order to fix the next one, instead of attempting to obtain more than one 1oint from one
position of the instrument; because when the chords are
longer than one chain, they cannot be measured in the right
direction by eye, but must be guided by the instrument.
It must be especially borne in mind, that in any given
ucive, only the tangential and deflexion angles increase in
the same proportion as the number of 100 feet stations subtended by the long chord.  Therefore, these long chords
cannot be used for laying out curves by eye, as their tangential and deflexion distances are not known.
When it is required to use long chords for turning a
curve by eye, they must be composed of a number of whole
chains, being made, say 200, 300, or 400, &c. feet in length.
The tangential and deflexion distances of curves of more than
500 feet radius, may then be assumed in practice, to increase as the squares of the number of chains in the length
of the long chord. For instance, to lay off a 50 curve
by chords of 200, 300, or 400 feet in length, the tangential and deflexion distances of the table must be multiplied
by 4, 9, or 16, as the case may be.'In this case the tangential and deflexion angles are unknown.
This is not mathematically correct, but will answer in
practice for the curves on a canal or common road, where
great nicety is not needed.
The only proper instrument for running lines of survey,
is the transit furnished with a compass, and with a revolving telescope.  The deflexions being measured in angles,
serve as a check to the numerous sources of error to which
the compass is liable, arising from local attraction, electrical action in the glass cover, diurnal variation, &c., &c.
Besides, when the compass alone is used, it is necessary
to test every course or bearing from each end of each station;
and this involves much loss of time.
The following is a good form of field book for the transit
and compass combined:
Station. I Distance. I Total   Coure.   Deflexion   Remarks.
Distance.   I    in Degrees.    e
In every locating party, there should be one person




29
whose duty is to obtain and record the transverse slopes
of the ground at each station. His observations will
usually extend to from 50 feet, to 100 yards on each side of
the centre stakes; depending on a variety of circumstances
of locality which cannot be alluded to here. In preliminary
locations these slopes need not be taken with very great
-nicety, as they will be used chiefly for ascertaining, approx-imately, the amount of excavation and embankment, by the
rapid process, described in my little volume on that subject,
and which dispenses with nearly all the labor of the usual
calculations.
After the final location is made, the slopes should be taken
again, with great care, to the nearest quarter of a degree;
but need not extend beyond the width actually occupied by the road. Their use in this second operation, will
be for determining the cubic contents with more precision
than before, for final estimates, and also for obtaining
the positions of the side-stakes.
Should the.duty of recording these slopes devolve upon
the compassman, (which it should not,)      Fig. 20.
it will be necessary to  add ano!her
column to his field book, after that con-          70.
taining the deflexions. In this column    g
he will insert the slopes, thus, (Fig. 20,)
the dot representing the centre stake. The degrees of slope
are written above the lines; and the distance in feet to
which they extend, below.
The slopes are taken by laying a long rod orthe ground,
at light angles to the line of survey, as nearly as may be,
by eye; and measuring the angles by means of a small
slope instrument placed upon the rod.  These are made
by most of our instrument makers.
ARTICLE XXIII.
To ADJUST A TRANSIT INSTRUMENT.
Having placed the transit firmly at a, and levelled it,
clamp all fast, and direct the cross-hairs, by means of the
tangent screw,
b                 a                c..m   ~   o, w.   *.. ea
*d




30
to some convenient object, b. Then, revolving the telescope
vertically, but without moving it in the least horizontally,
let the cross hairs fix upon a second object in the opposite
direction, as c, or if there is no such object, place one, as
for instance a chain-pin, at any convenient distance.
Then unclamp the lower clamp, and revolve horizontally the entire upper part of the instrument above the parallel plates. Clamp it again, and fix the cross-hairs upon b;
then again revolve the telescope vertically. If the sight
now strikes c, as before, it is in adjustment; but if not,
place another object, d, where it does strike; and with the
adjusting pin alter the vertical cross-hairs so as to strike
half way between d and c. The instrument will then be in
adjustment.
Two or three trials will generally be needed before the
adjustment is perfect.
With care, and on a firm floor, the operation may be
performed in a long room, or by placing the instrument in
a door way communicating with two rooms of moderate
size. Fine pins, or needles should then be used as the
objects to be sighted at. It is better, however, to adjust
out of doors, with more distant objects.  It is also a good
precaution to hang up a long plumb line, or select some
vertical object, and see whether the vertical hair coincides
with it, as the telescope is raised or lowered. If from any
accident, or carelessness in the maker it does not, the defect
must be remedied by an instrument maker.




'NATURAL  AIJN DS AND TANGENT'   TO A RADIUS 1.
O Deg.                                   0 Deg.                                   0 Deg.
/t  Sine.'ang. Cotang. Cosine.           Sine.   Tang.  Cotang. Cosine.           Sine.                    Cosin.  Sine    Tan tang sin
0 0000000. 000000 Infinite. 1-o000000 60,21 -0061086 006108 163-7001.9999813 3941 0119261 *011927 83-84350. 9999289 19
1 *0002909 000291 3437-746 1-000000 5922'0063995 *006399 156-2590.9999795 3842'0122170 *012217 81-84704 9999254 18
2 10005818 000582 1718-873'9999998 58 23'0066904 ~006690 i149'4650'9999776 37 43'0125079'012508 79'94343139999218 17
3'0008727'000872 1145-915'9999996 57 124 0069813'006981 143-2371'9999756 36 44 *01279871'012799 78'12634 *9999181 16
4 *0011636'001163 8594363'9999993 5625 0072721 007272 1375075 9999736 35045  130896013090 7639000 9999143115
5 *0014544 *001454 687-5488'9999989 55 26 0075630 *007563 132-2185 *9999714 34 6'0133805'013381 74'72916'9999105 114
6 10017453 *001745 572-9572'9999985 54 27'0078539'007854 127-3213'9999692 33 47 0136713 *013672 73.13899.9999065 13
7 *0020362'002036 49111060.9999979 53 28 0081448'008145 122-7739 *9999668 32 48 0139622.013963 71-61507 19999025 12
|8 0023271 002327 429'7175.9999973 52 29'0084357 008436 118.5401'9999644 31 49.0142530'014254 70'15334 [9998984 11
9 *0026180 |002618 |38,19709 99999966 51 30'0087265'008726 114-5886 -9999619 30 50 -0145439 -014545 68-75068' 998942  10 
10 10029089 002908 343'7737'9999958 531 0090174 0090177 710'8920.9999593 29 51 -0148348'014836 67-40185'9998900  9 29
11 *0031998 -003199.312-5213 -9999949 49032'0093083 -009308 107-4264'9999567 28 52 0151256 1015127 67610547'9998856  8
12 *0034907'003490 286.47771-9999939 4833 -0095992 009599 104-1709 19999539 2753 10154165 015418 64-85800'9998812 7
13 10037815 -003781 264'440819999928147734 0098900'009890 101.1069  9999511 26 54 0157073 -015709 63-65674 9998766  6
14'0040724 1004072 245.5519 -9999917 46 35 0101809 -010181 98-21794 9999482 25 55 0159982 -016000 62-49915 -9998'72(  5
15 *0043633 1004363 229*1816 -9999905 45 36 0104718 -010472 95.48947 19999452 24 56'0162890 1016291 61'38290 1999867T  4
16.0046542'004654 214.8576'9999892 44 37. 0107627'010763 92.90848 19999421 23 571 01657991 016582 60.30582.9998625  3
17'0049451 -004945 202-21871.9999878 43 381 0110535 -011054190.463331 9999389 122 58 0168707 -016873 59.26587'9998577 2
18'0052360'005236 190-9841.9999863 42391 0113444 -011345188.14357'999935712159 -0171616 1017164 58-26117'9998527  1
19 0055268- 005526 180 9322'9999847 41 40'0116353'011636 85-93979'9999323 20 60 -0174524 -017455 57-28996 -9998477  0
20 10058177'005817 171'8854 -9999831 40
Cosine.  Cot n.1 Tang. [ Sine. |   / Cosine.  Cotan. I Tan.  Sine.                Cosine.  Cotan. | Tang.    Sine.  
Deg. 89,                                Deg. 89.                                 Deg. 89.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
1 Deg.                                  1 Deg.                                   1 Deg.
I  Sine.  Tang. Cotang. Cosine. / i/  Sine.   Tang.  Cotang. Cosine.  II'  Sine. ITang. Cotang. Cosine. 
0 -0174524 -017455 57-28996 -9998477 60,21 -0235598 *023566 42-43346 -9997224 39 41 -0293755 -029388 34-02730 *9995684 19
1'0177432 *017746 56-35059 -9998426 5922 -0238506 *023857 41.91579 -9997156 38 42 *0296662 *029679 33-69350.9995599 18
2 10180341 -018037 55-44151 -9998374 58'23 *0241414 -024148 41-41058i-9997086 37 43 0299570.029970 33-36619.9995512 17
31 0183249.018328154-56130 -9998321 57 24 10244322 -024439 40-91741 -9997015 36 44 0302478 -030261 33-04517 19995424 16
4 -0186158 -018619s53-70858 -9998267 56 25 -0247230 *024730 40-43583 *9996943 35 45 *0305385 -030552 32-73026 -9995336 15
5'0189066 -018910 52'88211'9998213 55 26 -02501,38 -025021 39-96546 -9996871 34 46 -0308293 -030843 32-42129 19995247 14
6 -0191974 -'019201 52-08067 -9998157 54 27 -0253046 -025312 39-50589 -9996798 33 47' 0311200 -031135 32-11809 -9995157 13
7 -0194883 -019492 51-30315 -9998101 53 28 -0255954'025603 39-05677 -9996724 32 48 -0314108 -031426 31-82051 -9995066 1.2
81-0197791 -019783 50-548501-99980441 52129 -0258862 -025894 38-61773 -9996649 31 491 -0317015 -031717 31-52839 9994974 11
9 1-0200699 -020074 49-81572 -9997986 51 30 -02617609 -026185 38-18845 -9996573 30 50 -0319922 -032008 31-24157 -9994881 l10  cQ
101 -0203608 -020365 49-10388 -9997927 50 31 -0264677 -026477 37-76861 -9996497 29 51 -0322830 -032299 30-95992 -9994788} 9 
11 -02065161 020656/48-41208 -9997867149 32.0267585 -026768 37-35789 -9996419 28 52'0325737 -032591130-68330 -9994693 8
12 -0209424 -020947 47-73950.-9997807 48l33 -02704931-027059 36-95600 -9996341 27 53 -0328644 -032882 30-41158 -9994598  7
13 -0212332 -021238147-08534-9997745 14-734 |0273401'-027350 36-56265 -9996262 26 54 -0331552 -03317330-14461 -9994502  6
14 -0215241 -021529 46-44886 -9997683 46 35 l02763091-027641 36-17759 -9996182 25 55 -0334459 -033464 29-88229 -9994405 5
151 02181491 021820 145-82935-9997620 45 36 *0279216 027932 35-80055 1-9996101 124 56 0337366 1-033755 29-62449 -9994308 4
16 -0221057 -022111 45-22614 -9997556 44 37 -02821241 -028223 35-43128 -9996020 23157 -0340274 -034047 29-37110 -9994209  3
17 -0223965j1022402 44-63859 -9997492143 381-0285032 -02S514135-06954 -9995937 22 58.-0343181-'034338129-122001-9994110[ 2
18 -0226873 -022693 44-06611 -9997426 42'391-0287940 -028805 34-71511 -9995854 21 59 10346088 1034629 12887708 -9994009 1
19 -0229781 -022941 43-50812 -9997360 41140 -0290847 -029097l34-36777 -9995770 20 60 -03489951 034920128-636251 -9993908  0
20 -0232690 -023275 42-96407 -9997292 40   1       1       1        1        1    I' Cosine. Cotan.  Tang. | Sine. I''  I Cosine. Cotan.  Tang.   Sine.'    Cosine. C otan.  Tang.   Sine.
Deg 88.                                 De,g. 88.                                 Deg. 88.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
2 Deg.                                  2 Deg.                                   2 Deg.
Sine.   Tang. Cotang. C osine.'    Sine.  Tang. Cotang. Cosine.               Sine.   Tang. Cotang. Cosine.
0 *0348995'034920 28-63625'9993908 60 21'0410037'041038 24'36750 *9991590 39 41'0468159'046867 21.336'9989035 19
1 -0351902'035212 28'39939'9993806 59 22'0412944'041329 24'19571'9991470 38 42'0471065'047158 21-20494'9988899 18
2'0354809'035503 28-16642'9993704 58 23'0415850'041621 24'02632'9991350 37 43 0473970'047450121'07466'9988761 17
3 io357716'035794 27'93723 -9993600 57 24 0418757'041912 23-85927'9991228 36 44'0476876 -047741 20'94596'9988623 16
4 10360623'036085 27'71174'9993495 56 25'0421663'042203 23'69453 999 1106 35 45 10479781 1-0480331 20818821 9988484 15
5'0363530'036377 27'48985'9993390 55 26'0424569'042495 23'53205'9990983 34 46 0482687'048325 20'69322'9988344 14
6'0366437'036668 27'27148'9993284 54 27'0427475'042786 23-37177'9990859 33 47'0485592'048616 20'56911'9988203 13
71 0369344'036959 27'05655'9993177 53 28'0430382'043078 23'21366'9990734 32 48 0488498'048908 20'44648'9988061 12
8 10372251'037250 26-84498'9993069 52 29'04332881'043369 23'05767'9990609 31 491 04914031 049199 20'32530'9987919 11
9 |0375158'037542 26-63669'9992960 51 30'0436194'043660 22-90376'9990482 30 56'0494308'049491| 20'20555'99871751`0 
10' 0378065'037833 26'43160'9992851 50 31'0439100'043952122'75189'9990355 29 51'0497214'049782 20'(C8719'9987631| 9 9"
11 -0380971'038124 26'22963'9992740 49 32'0442006'044243 22'60201'9990227 28 52'0500119'050074 19-97021'9987486] 8
121-0383878'038416I 2603073'9992629 14 33'0444912'044535 22'45409'9990098 27 59'0503024i'050366 19'85459'9987340  7
13'0386785'038707 25-83482'9992517147 34 0447818'044826 22'30809'9989968 26 54'10505929'050657 19'74029'9987194  6
14 -389692'038998 25'64183'9992404 46 35'0450724'045118 12216398'9989837 25 55 [0508835'050949 19'62729'9987046 5
15 0392598'039290 25'45170'9992290 45 36'0453630'045409 22'02171'9989706 24 56 10511740'051241 19'51558'9986898 4
16 -03955051 039581 25'26436'9992176 44 37'0456536'045701 21'881251-9989573 23 57'0514645'051532119'40513'9986748  3
17 -0398-411.03987 125'07975'9992060143 381 0459442'045992121'74256'9989440122 581-0517550'051824119'295921 99865981 2
181 0401318'040164124.897821.9991944 42 391-0462347'046284 21-60563'9989306 21 591'0520455 1052116 19'18793'9986447  1
19'0404224'040455 24'71851'9991827141 01 0465253'046575 21'47040'9989171 120 60'0523360'052407 19'081131 9986295  0
20 -0407131 1-040746 24'54175'9991709 40
L'  Cosine. Cotan.  Tan.  Sine.' Cosine. Cotan.  Tan.  Sine.  /     Cosine. Cotan. Tang.   Sine.
Deg. 87.                                Deg. 87.                                Deg. 87.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
3 D  g.                               3 Deg.                                 3 Deg.
Sie.          ne. Tang. Cotang.  Cosine. Cosine.                             Sine.  Tang. Cotang. Cosine.
0.052336i'052407 19s08113.9986295 60 21 0584352'058535 1708372'9982912 39 41 -0642420'064375 15.53398 -9979343 19. 052626' -052699 18-97552.9986143 59 2 0587256'058827 16-99895 -9982742 3842.0645323'064667 15.46381'9979156 18
2'052916,'052991 18'87106 9985989 58 230590160'059119 16'91502'9982570 37 43'0648226'064959 15'39427'9978968 17
31-0532074'053282 18'76775'9985835 57 24 0593064'059410 16'83191'9982398 36 44'0651129'065251 15'32535'9978779 16
41 0534979.053574 18'66556'9985680 5625 *0595967'059702 16'74961 *9982225 35 45'0654031'065543 15'25705'9978589 15
51 0537883'053866 18'56447'9985524 5526 *0598871'059994 16'66811'9982052 34 46 *0656934 *065835 15'18934 *9978399 14
6'0540788'054158 1l846447'9985367 54 27 0601775'060286 16'58739'9981877 33 47'0659836'066127 15'12224 -9978207 13
7'0543693 054449 18-36553'998520953 28'0604678'060578 16-50745 -9981701 32'8'0662739 -066419 15'05572'99780151 12
8'05465971'054741 18.267651 9985050 52 291'0607582'060870 1642827 9981525 31 49 0665641'066712 14'98978'9977821 11
9'0549502' 055033 18' 708GC |9984891 51 30.0610485. 061162 16.349851 9981348 3C 50'06685441'067004 14.92441.'9977627 10  C.
10 -0552406'055325 18-07497.9984731 50 31 0613389. 061454 16.27217. 9981170 29 51'0671446'067296 14-85961 19977433 9
11.0555311j 055616 17.980151 9984570 49 32. 0616292. 061746 16.19522'9980991 28 52.0674349'067588 14.79537. 9977237 8
12'0558215'055908 17.88631'9984408 48 33'06191961 062038 16-118991 9980811 275?'0677251'067880 14.731671-9977040 7
131 0561119.056200 17-79344'99842454734 0622099'062330 16.04348'9980631 2654'0680153'068173 14.66852'9976843  6
14'0564024'056492 17'70152'998408146 35. 0625002'062622 15.96866 19980450 25 55'06830551 068465 14.60591.9976645  5
151 05669281 056784 17'61055'9983917 4536'0627905'062914 15'89454'9980267 24 56'0685957'068757 14'54383'9976445 4
16'0569832'057075 17-52051'9983751 4437'0630808'0632t6 15'82110'9980084 23 57'0688859'069049 14'48227'9976245 3
1 -70572736 -057367 i7A.i38'9983585I43 81'06337111'063498 15'74833'9979900122 581-06917611'069342 14'42123'9976045  2
181 0575640: 057659 17/i3415'9983418 42 39 0636614'063790 15'67623'997971621 59'0694663'069634 14'36069'9975843 1
191 0578544r057951 17.25580619983250 41 40 0639517.0640'82 15.60478 -9979530120 60'06975651 069926 14.30066'9975641 
201 05814481 058243 17'16933'99830824'' C osine! CotanI) T ang.   SiCne.S' e  Cosine. Cotan. Tang.   Sine'
Deg. 86,:.                            Deg. 86.                               Deg. 86.




NATURAL SINISf AND TANGENTS TO A RADIUS 1.
4 Deg.                                  4 Deg.                                 4 Deg.
j   Sine.  Tang.Cog   sine   T                       Cotang   Cosine.osine.    Sine.e. Tangg. Cotang. Cosine.'0697565'066926$ 14'30066' 675641 6021'0758489'076068 13'14612 19971193139-41 0816486 081922 12,20671 19966612 19
1,0700467'070219 1424113. 9975437 59 22 0761390'076360 13'09575'9970972 3842'0819385'082215 12.16323'9966374 18
2'0703368'05i] II4,18209.9975t33 58 23'0764290'076653 13.04576 19970750 37 43'0822284'082507 12.12006 9966135 17
3 0706270 o 70803 14.12353'9975028 57 24 0767190'076945 12'99616'9970528 36 44 0825183'082800 12'07719'99658951 16
4'0709171'071096 14'06545 19974822 5625'0770091'077238 12-94692'9970304 35145 10828082 1083093 12.03462.9965655 15
5'0712073'071388 14'00785'9974615 5526 1 0772991'077531 12s89805 19970080 34 46 1s830981 083386 1199234'9965414 14
6'0714974'071680 13'95071'9974408 54127'0775891'077823 12-84955'9969854 33147'0833880'083679 11'95037'9965172 13
7'0717876'071973 13.89404'9974199 5328'0776791'078116 12.80141'9969628 32 48'0836778'083972 11 90868'9964929 12
8'0720777'072265 13.83782'9973990 52 29'0781691. 078409 12.75363.9969401 31149l 08396771.084265 11.867281.9964685 11
9'0723678'072558 13.78206[ 9973780 51 3 0'0784591.078701 12.70620'9969173 30150'0842576'084558 11.826161 9964440 10  c.
101 0726580'072850 13'72673'9973569 5031'0787491 -078994 12.659121 996894 5129 151.05474'084851 11.78533'9964195  9
11'0729481'073143 13'67185 19973357 4932'0790391'079287 12.61239'9968715 28152'08483731 085144 11-744771 9963948 8
12 0732382 071*5 13'61740'9973145 48 33107932901'079579 12'56599'9968485 27 53 0851271'085437 11'70450'9963701 7
13'0735283'073727 13'56339'9972931 47 34'0796190'079872 12.51994~ 9968254 26 54 0854169'085730 1 166449'9963453 6
14'0738184'074020 13'50979'9972717146 35 0799090.080165 12'47422'9968022 25 55'0857067'086023 11'62176'9963204  5
15'0741085'074312 13'45662'997250245 36 0801989'080458 12.428831 9967789 24 56'0859966'086316 11158529 19962954 4
16'0743986'074605 13'40386'997228644 37 0804889 08SQ750 12'38376 9967555 23 57'0862864'086609 11'54609'9962704 3
17'0746887'074897 13 351511.9972069143 381.080778s81081043 12 33902 19967321 22 581'0865762,1086902 11'50715'99624521 2
18'0749787'075190 13'29957'9971851 42 39 0810687'081336 12'294601 9967085 21 59 10868660'087195 11-46847 19962200  1
19'07526881 075482 13.24803'9971633 41 40'08135871 081629 12'25050'9966849 20 601 08715571 087488 11.430051 9961947 0
201 07555891 075775 13'196P8'9971413 40
Cosine. ICotan   Tang. | Sine. |It | Cosine. Cotan. Tang.   Sine. It [  Cosine. Cotan. Tang.   Sine.
— o.i-n-, I oDe...                                D. 8.                                     D. 8..
Deg. 85.                              D)eg. 85.                                Deg. 85.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
5 Deg.                                  5 Deg.                                  5 Deg.
Sine.   Tang. Cotang. Cosine.' /    Sine.   Tang. Cotang. ICosine. - T- /    Sine.   Tang. Cotang. Cosine.
_         -I. _ _____-   _     -_ -   -.  _____ -'   -I ---      --          -       -       --       --  -
0 *0871557 1087488 114 3005 *9961947 60 21 0932395 *093647 10.67834 *9956437 39 41 0990303.099519 1004828.9950844 19
1.0874455'087781 11 139188'9961693 59 22 *0935291 093940 10-64499 *9956165 38 42 0993197 *099813 10-01871'9950556 18
2.0877353 *088074 11'35397 *9961438 58 231O0938187 094234 10.61184 *9955892137143 *0996092.100107 9-989305 *9950266 17
3 10880251'088368 11*31630 *9961183 5724 *0941083 094527 10.578891 9955620136 44 *0998986'100400 9'9600728 9949976 16
4 0883148'088661 11'27888'9960926 56 25 10943979'094821 10.54615'9955345 135 45 1001881 100694 9        E931008'9949685 15
5 *0886046'088954 11-24171 9960669 55 26 0946875'095114 10'51360'9955070134 46 *1004775'100988 9'902112.9949393 14
6.0888943 *089247 11120478 *9960411 54127 *0949771 -095408 10 48126 9954794 33 47 1007669'101282 9 873382 9949101 13
7.0891840.089540 11.16808 -9960152 53128'0952666'095701 10-44911 *9954517 32 S *1010563'101576 9-844816 *9948807 12
8'0894738 -089834 11-13163 *9959892 52 29 10955562'095995 10-41715 -9954240 31 19 *1013457'101870 9-816414 -9948513 11
9108976351 090127 11109541 -9959631 151130 -0958458 096289 1038539 9953962 30 50 -1016351 -102164 19788173 19948217 10 
10'0900532 1090420 1 105943 19959370 503 1 -0961353 196582 10-35382 19953683 129 51 1019245 -102458 9-760092 19947921 9
11.0903429.090713 11-02367 -9959107 49132 -0964248 096876 10.32244| 9953403 28152 -1022138'102752 9-732171.9947625  8
12.0906326 1091007 10-98815.9958844 14833.0967144.097169 10-29125 -9953122 27 53i 1025032 [103046 9.7044071 9947327'7
13| 0909223 1091300 10-95285 19958580 47 341 0970039'097463 10-26024 -9952840 126541 1027925 -103339 9-676800 19947028  6
14 0912119 1-0915931 10-91717 -9958315 4635 *0972934 -097757 10 22942 9952557 25 55 1030819 1-103634 9-6493471 -9946729  5
151 0915016 -091887 10-88292'9958049145136'0975829 1098050 10'19878'9952274 24 56 110337121 103928 9'622048 19946428 4
16. 0917913 1092180 10.84828 -9957783144 37.09787241 098344 10.168331 9951990123 57' 1036605'10422219.5949021 9946127 3
17 10920809'092473110'813871'9957515 431381 0981619'098638 10'13805 /9951705 22 581'1039499 1104516 9'567906. 99458251 2
18'0923706'092767 10'779671-995'7247 421391'0984514- 1098932 10'107951 9951419 21 591 1042392 1104810 9'541061'9945523  1
19.0926602.093060 10-74568 1-995697 141 40 -0987408 -099225 10Q07803 1-9951132 20 60 11045285 1-1051049514364 -9945219 0
201-09294991-093354110-7119].1-995670840.
I Cosine.! Cotan.  Tan.   Sine.            Co. Cotan.  Tang          Sine.     Coine. ICotan.  Tang.    Sine. 
Deg. 84.                                 Deg. 84.                               leg. 84.




NATURAL SINES AND TANGENTS TO A RADIUS L!
6 Deg.                                   6 Deg.                                     6 Deg.
Sine.   Tang.  Cotanlg. Cosine.   Sille.   Tang.  Cotang. Cosine.                   Sine.   Tang.  Cotang.  Cosine. /
0'1045285'105104 9-5143641 9945219 60 21 *1106017'111284 8-9859854 9938648 39 11'1163818 * 117178 8534017'9932045 19
1 -1048178 1105398 9-487814'9944914 59 22 1108908 111578 8-962266 993826 3 12'1166707 117473 8-512594'9931706 18
2'1051070 105692 9-461411 19944609 58 23 11111799'111873 8-938672 993 8003 37 13 1169596 117767 8-491277 9931367 17
3'1053963'105986 9'435153'9944303 57 24 1114689 ]112168 8-915200'9937679 36 141-1 172485'1 18062 8-47006.59931026 16
4'1056S56 1106280 9-409038 19943996 56i25 11 17580 *112462 8-891850 *993735;5 35 151-1175374 *118357 8-448957 -9930685 15
5'1059748 106575 9-383066' 9943688 55 26 *1120471 *112757 8-868620'9937029 34 ]16 1178263'118652 8-4279531 9930342 14
6'1062641 106869 93572-35 9943379 54 27 *1123361'113051 8-845510 I9936703 33 17-.1181151 1 118947 8407051 9929999 13
7 1065533 107163 9-331545 19943070 5328'1126252'113346 8.822518'9936375 32 8 -1184040 119'242 8-38625 1 9929655 12
s8 1068425'107457 19305993 -9942760152129 -I1 2914112'113641 /8799644'99.6047 31 19 118 928'119537 S-365553'9929310 11
91 107131S8 107751l 9280580'994244S 51 30 1132032'1 13935 8-776887'993571936 50'1189816'119832 8-344955'9928965 10
10~ 1074210 108046 192553031 9942136 50!31 1 1349!22'114230 18754246. 9935389 29351 1192704 112012718-3244571 9928618[ 9
11'1077102'108340 9 230162'9941823'49932391 123c'8121 114525 8 7379 993505928 52'1195593'120423 8-304058 19928271  8
12 1079994'108634 9-205156'99411510 48 33'1140702  1148198709930i719934727 27'53 -1198481'120718 8j283757'9927922  7
13'1082885 -108929 9-180283'9941195 47 34 1143592'115114 8.687008'991343951 26541 1201368' 1210138'263554' 9927573  6
14~ 1085777 110922319'1 5543 19940880 46351'11464821'15409 8'664822 9934062 125%5511204256 -12130s88243418 19972241   5
15 -10886691 1095171 9130934'9940563}45136'1149372 11157034 8612747'9933728 24.56 1207144'121603S 8223438  9926873  4
616 1091560'109812|9'106456'9940246 14437 1' 152261'1159981I-6207831 9933393 23i57 1210031 -12 1898 18203523.9926521  3
171 10944521110106 19082107.1393992843[381 1  5515  1 1.1 629318.598929'9933057 22 58i 121291291t1221948-.183704-.99261691 2
18 10973431 110401 19057886 -9939610i42 39 1158040'116588 8'577183'9932721 21 59.1215806}'12248918. 163978'9925816  1
19| 110023|4 110695 19033793'9939290 4140| 1160929"11 6883| 8555546'993238  20 60| 12t869'31122784  144346'9925162  0
20 -1103126'110989 9-009826 -9938969 401 40                                                                      _ _I
Cosine. Cotan.  Tang.  Sine.                                                 /  Cosine.   Cotan.  Tan.    Sine.  / Cosine.  Cotan.   Tang.   Sine. 
Deg. 83.                                  Deg. 83.                                 Deg. 83.




NATURAL SINES AND TANGENTS TO A RADIUS 1,
7 Deg.                                  7 Deg.                                  7 Deg.
Sine.  Tang.  Cotang.  Cosine.          Sine.  Tang.  Cotang. Cosine.           Sine.  Tang.  Cotang.   oise.
0 11218693'122784 8-144346'9925462 60 21'1279302'128990 7'752536'9917832 39 1'1336979,134909 7,412397 9910221 19
1 11221581'123079 8'124807'9925107 59 22 11282186'129285 7"734802'9917459 38 2'1339862'135205 7'396159'9909832 18
2'1224468'123375 8'105359'9924751 58 23 1125071'129581 7'717148'9917086 37 43 1342744'135501 7'379990'9909442 17
3'1227355'123670 8-086004'9924394 57 24 1287956'129877 7'699573'9916712 36 4'1345627'135797 7'363891'9909051 16
4'1230241'123965 8.066739 9924037 56 25 11290841,130173 7'682076'9916337 35 45 1348509'136094 7'347861'9908659 15
5'1233128'124261 8'047564'9923679 55 26'1293725'130469 7664658'9915961 34 46 1351392 -136390 7'331898'9908266 14
6'1236015'124556 8.028479'9923319 54 27'1296609'130764 7.647317'9915584 33 17'1354274 *136686 7.316004'9907873 13
7'1238901'124852 8.009483'9922959 53 28'1299494'131060 7'630053'9915206 32 48 1357156'136983 7.300178'9907478 12
8.1241788" 125147 7-990575'9922599 52 29[ 1302378'131356 7.612865L 9914828 31 49 1360038'137279 7.284418 19907083 11
91 12446741'125442 7'971755,9922237151 301'1305262{'131652 7-595754  9914449 30 0 1362919 *137575 7~268725-990fi687 10 
10'1247560 1125738 7'953022'9921874 5031 11308146 [131948 7'578717 -9914069 29 51 1365801'137872 7'253098 -9906290 9 
11'1250446 1126033 7'934375'9921511 49 32 11311030 1132244 7'561756'9913688 28 5211368683'138168 7'23753  9905893 8
12'12533321 126329 7'915815'9921147 48133'1313913'13254017'544869'991330627 53'1371564'138465 7'222042'9905494 7
13'12562181 126624 17897339 9920782 47 34' 13167971 13283617528057'9912923 26 54 1374445' 138761 7'206611'9905095  6
14'1259104'126920 7.878948jl9920416 46 35'1319681'1331321 7511317'9912540 25 55 13773271 139058 7'191245'9904694 5
151 1261990'127216 7.8606421 9920049 45 36 11322564'133428 7-494651'9912155 24 6'1380208'139354 7-175943'9904293 4
16 11264875'12751 11 7842419,;9919082 44 37 1325447'133724 17478057'9911770 23 5 7'13830891 139651 7'1607051 9903891  3
17'1267761'127807 7'8242791'9919314143 38'13283301'13402017'461535'991138422 58'13859701 139947 7.145530'9903489 2
18 112706468'128103 7'806221'9918944 42 391'1331213i 1343161 74450S51 9910997 I21 9 1388850 14024417'130419'9903085  1
191 1273531'128398 7'788245 -9918574 41 40'13340961 134612 7.428706'9910610120 601391731'140540 7'1153691 9902681  0
20'1276416('12869417'7703501 991820440.                                                                           I
I -I             -      -  - I     — I' s- 1-I F- I — - ~-.-.II-1- - -I  -  -I --- -  1 --
Cosine. lCotan.  Tang'   Sine.t t   Cosine. Cotan.  Tang.   Sine. I t   Cosine. Cotan.  Tang.   Sine.
Deg, 82,.                                    82,-                                Deg. 82,




NATURAL SINES AND TANGENTS TO A RADIUS 1.
a Deg,                                  8 Deg.                                  8 Deg.
/  Sine.   Tang. Cotang.  Cosine. /       Sine.   Tan     Cotang. Cosine Sine. l:         Tang. Cotang. Cosi:ne. 
0 113917311 14054017-1153691 9902681 16021'1452197 1146775 6'8131221 98939941391411'15097331 152723 6'547767 19885378 19
1'1394612'140837 7.100382'9902275 59 22 1465075'147072 6-799356 9893572 38 42'1512608 I153021 6'535029'9884939 18
2 *1397492 *141134 7.085457, 9901869 58 23'1457953'147369 6'785644'9893148 3743'1515484'153319 6'522339'9884498 17
31 1400372'141430 7.070593'9901462 5 7 24'1460830'14766716'791986 9892723 3644'1518359'*153617t6'509698'9884057116
4'1403252'141727 7'055790'9901055 56 25'1463708 1147964 6'758382 -9892298 35 45'1521234' 153914 6'497104'9883615 15
5'1406132 i142024 7'041048'9900646 55 26'1466585'148261 6-744831'9891872 34 46 1524109 1154212 16484558'9883172 14
6'1409012'142321 7'026366'9900237 5427'1469463'148559 16731334'9891445 33 47 15269841 154510 16472059'9882728 13
7'1411892'142617 7'011744'9899826 53 28'1472340'148856 6'717889'9891017132 48'1629858'154808 16469607'9882284 12
8 1414772'142914 6'997180 19899415 52 29'14752171 149153 6'704496'989058831 49'1532733'155106 16447201'9881838 11 
9.1417651 -1432111 6982678'9899003151 30 1478094 1149451 16691156 -9890159 30 50'1635607'155404 6'434842'9881392 10 
10'1420531'143,508 16968233 19898590 5031'1480971 1149748 1667786778679889728 2951 15384821 155701 16422530'9880945 9
11 114234101 14380516'953847'9898177 49.32'1483848 -150045 16664630'9889297 28152'1541356' 155999 J6'410263'9880497 8
121 1426289. 144102 16939519'9897762 48133 11486724 I150343 6.6514441 9888865 27 53'1544230 1.56297 6.398042, 98800481 7
131 1429168 14439916.9252481 9897347147!34'1489601.150640 6.638310.9888432 26154 1547104'156595 6.3858661 9879599 6
141 1432047 1446961 69110351. 9896931 46i35'1492477. 150938 6.625225.9887998 25 55'1549978'156893 6.373735 879148 5
151 1434926'144993 6.896879'9896514 45 36'14953531 151235 16612191 19887564 24 561 15528511157191 63616361650 9878697 4
161 1437805'145290 6-8827801-9896096 44137 114982301 151533 16599208'988712823 571555725 157490 6'349609 9878245 3
17l 14406841 146587 168687371.9895677 43!381.1501106 1151830 6.586273, 9886692 22158'1558598 11577881 63376121 98777921 2
18'14435621 145884 {6854750 19895258 42139'1503981 1152128 16573389 |988625521 59 1'1561472 1158086 16325660 19877338  1
19 11446440'146181 6-840819 l9894838 4140'1506857'152426 16560553'9885817 20 60'1564345'158384 6-313751 9876883 0
20'1449319 -146478 6-826943 5 9894416 40
i   Cosine. Cotan. Tang. I Sine. |I Cosine. Cotan.  Tang. i Sine.                 C I'lCosine. Cotan.  Tang.   Sine.
Deg. 81.                               Deg. 81.                                Deg. 81.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
9 Deg.                                   9 Deg.                                  9 Deg.
|     Sine.  Tang. (Cotang-. Cosine.'t LlSine.| Tang., Cotang. Cosine.'t  / |Sine.'rang. Cotang.  Cosine.
0.1564345 *158384 6-313751'9876883 60 1'1624650 -164652 6'073397'9867143 3941'1682026'170633 5.860505'9857524 19
1'1567218'158682 6'301886'9876428 5 921627520 -164951 6'062396'9866670 38 2'1684894 170933 5-850241'9857035 18
2 *1570091'158980 6.290065'9875972 5823 1630390s 165250 6051434 *986619637 43 *1687761 1171232 5'840011 19856544 17
3'1572963'159279 6'278286'9875514 57 24 1633260'165548 6.0405 14.9865722 36 44'1690628'171532 5'829817'9856053 16
4'1575836'159577 6'266551'9875057 56 25'1636129'165847 6'029624'9865246 35 45 1693495'171831 5'8196549855561 15
5'1578708'159875 3'254858'9874598 55 26'1638999'166146 6'018777'9864770 34 46'1696362'1721330 5'809531'9855069 14
6'1581581'160174 S'243208'987413854 27'1641868'166445 6'007967'9864293 3347 71699228'172430 5'799440'9854574 13
7'1584453'160472 6'231600'9873678 53 28'1644738'166744 5'997195'9863815 32 1t8 1702095'172730 5'789382'9854079 12
81 1587325'160770 16220034'9873216 52 29'1647607'16704315'986461'9863336 31 49 11704961'173029 5'779358'9853583 11
91 1590197'161069 6-208510'9872754151 30'1650476'167342 5'975764'9862856 30 50'1707828'173329 5'769368'9853087 10   C'
10'1593069'161367 6'197027'9872291 50 31'1653345' 167641 5-965104'9862375 29 51'1710694'17362815'759412'9852590  9
11'1595940'161666 6'185586'9871827149 32'1656214'167940 5'954481'9861894128 52'1713560'173928 5'749488'9852092 8
121 1598812'161964,6'174186'9871363 48 33 1659082'168239 5'943895'9861412 27 53 1716425'174228 5'739598'9851593  7
13'1601683'162263 6'162827'9870897 47 34'1661951'168539 5'933345'9860929 26 54'1719291'174527 5'729741'9851093  6
14'1604555 162561 6'151508'9870431 46 35'1664819'168838 5'922832'9860445 25 55'1722156'174827 5'719917'9850593  5
15'1607426'162860 6'140230'98699641 45 36'1667687'1691375'912355'9859960 24 56'1725022'175127 5'710125'9850091  4
16'1610297'163159 6'128992'9869496 44 37'1670556' 169436 5'901913'9859475 2357'1727887'175427 5'700366'9849589  3
17'1613167' 163457 6'117794'9869027 43 38'1673423' 16973515'891508'9858988 22 581'1730752'175727 5'690639'98490861 2
18'1616038'163756 5'106636'986855742 39 11676291'17003515'881138'9858501 21 59'1733617'176027 5'680944'9848582  1
19'1618909'164055 6'095517'9868087 41 10'1679159'170334:5'870804'9858013 20 60'1736482' 176327 5-671281'9848078t- 0
20'1621779, 164353 6'084438'9867615 40
t Cosine. Cotan. Tang.   Sine.             Cosine. Cotan.  Tang.   Sine.    t   Cosine. C otan.  Tang.   Sine.
Deg, 80.                                 Deg. 80,                               Deg. 80.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
10 Deg.                                 10 Deg.                                 10 Deg.
/   Sine.   Tang. Cotang. Cosine.           Sine.  Tang. Cotang. Cosine. I t    Sine.   Tang. Cotang. Cosine.'
0'1736482'176327 5'671281.9848078 60 21 -1796607 *182632 5.475478 -9837286 39 1 *1853808'188650 5.300801'9826668 19
1'1739346'176626 5'661650'9847572 59 221 1799469'182933 5'466481'9836763 38 2'1856666'188952 5'292350'9826128 18
2'1742211'176926 5'652051'9847066 58 23'1802330'183233 5'457512'9836239 3  3 -1859524'189253 5'283925'9825587 17
3'1745075'177226 5.642483'9846558 57 241'1805191 -183534 /5448571,9835715136 44-1862382'189554 5-275525 19825046 16
4'1747939'177527 5'632947 9846050 56 25'1808052'183835 5'439659'9835189 3  5'1865240'189855 5'267151'9824504 15
5'1750803'177827 5.623442'9845542 5526'1810913 -184135 5-430775'9834663 34 6'1868098'190157 5'258803'9823961 14
6'1753667'178127 5-613968'9845032 54 27 1813774'184436 5'421918'9834136 33 7'1870956'190458 5'250480'9823417 13
7'1756531'178427 5.604524'9844521 53 28'1816635'184737 5'413090'9833608 32 8 -1873813'190760 5'242183'9822873 12
8'1759395'178727 5.595112'9844010 52 29.18194951.185038 5'404290 -9833079 31 49'1876670'191061 5'233911'9822327 11
91 1762258'17902715.585730'9843498 51 30 1822355 (185339 5'395517'9832549 3  0'1879528'191363 5'225664'9821781 10    -
10'11765121'179327 5.576378'9842985 50 31 18252151.185639 5'386771'983201929 51 1882385'191664 5'217442'9821234 9   I'
114.1767984 -1796281 5567057'9842471 49 321 18280751 185940 15378053'9831487 2 52'1885241'191966 5'209245'9820686 8
121 1770847 1179928 5'557766'9841956 48 33'11830935. 186241 5'369363'9830955 27 53'1888098'192268 5'201073'9820137 7
13 1773710 1180228 5.548505.9841441 47 34 -1833795. 186542 5.360699'98304222 54 1890954'19256915-192926. 9819587 6
14 1776573'180529 5'539274'9840924 46 35'1836654 -186843 5'352062'9829888 25 5'1893811'192871 5'184803:9819037 5
15'17794351 180829 5.5300  409840407 45 36'1839514'187144 5'343452'9829353    6'1896667'19317315'176705. 9818485 4
161 1782298'181129 5'520900'9839889 44 37'1842373'187446 5-334869'9828818 2 57'1899523'193474 5'168631. 9817933 3
17'1785160' 18143015-511757'9839370143 38'1845232. 187747 5'326313 -9828282 2  8'1902379'193776 5'160581'9817380 2
18'1788022'181730 5'502644'9838850 42 39'1848091'188048 5'317783 -982774421 9'1905234'194078 5'152555'9816826 1
19'17908841 182031 15493560 -9838330 41 40'1850949'188349 5-309279'982720620'1908090'1943805'144554'9816272 0
20 1793746. 182331 5'484505'9837808 40                    1
t Cosine. Cotang.  Tang.  Sine.  t'   Cosine. Cotang.  Tang.   Sine.  t | Cosine. Cotang. Tang.   Sine.  t
Deg. 79.                                 Deg. 79.                                Deg. 79.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
11 Deg.                                11 Deg.                                 11 Deg.
Sine.   Tang.  Cotang. Cosine.'    Sine.  Tang. Cotang. Cosine.'           Sine:  Tang. Cotang. Cosie.
0.'1908090'194380 5'144554 9816272 60 1 1968018'200727 4-981881 8 9804433 39 1 2025024'206786 4'835901'9792818 1
1'1910945'194682 5'136576'9815716 59 2'1970870'201030 4'974381'9803860 382'2027873'207090 4'828817'9792228 18
2 *1913801 *194984 5'128622 *9815160 58 23 1973722 201332 4966903 9803286 37 43 2030721 *207393 41821753 *9791638 171
3'1916656'196286 56120692'9814603 5724'1976573'201635 4'959447'9802712 36 4412033569'207696 4'814709'9791047 16
4'1919510'195588 5'112785'9814045 56 25'1979425'201938 4'952012'9802136 35 45'2036418'208000 4'807685'9790455 15
5'1922365'195890 5'104902'9813486 55 26'1982276'200240 4'944599'9801560 34 46'2039265'208303 4'800680:9789862 14
6'1925220'196192 5'097042'9812927 54 27 1985127 202543 4937206'9800983 33 47'2042113'208607 4'793695'9789268 13
7 192807-4'196494 5'089206'9812366 53 28'1987978 20284614*929835'9800405 32 48 2044961'208910 4'786730'9788674 12
8'1930928'19679615'081392 9811805152 29'1990829'20314914'922485'9799827 31 49'2047808 1209214 4'779783'9788079 l11
9'1933782'197098 5073602 1-9811243 51 30 1993679'20345214'915157'9799247 3 50 2050655 1209518 4'772856'9787483110   P
10'1936636 197400 516066885- 9810680 50 31 1996530'203755 4'9078491 9798667 29 511 2053502'209821 14765949'9786886 9 
11'1939490'197703 65088090'9810116 49 32 19993801 204058 4900562'9798086 28 52 2056349'210125 4'759060 9'86288  8
12'1942344'198005 5'050369'980955248 33'2002230 204361 4'893295'9797504 27 53 120591951'210429 4'752190'9785689 7
13'1945197'198307 15042670'9808986 47 34'2005080 204664 4'886049'9796921 26 54 20620421 210733 4'7453401 9785090  6
14'1948050 1 98610 5'034993'9808420 46 35'2007930'204967 4'878824'9796337 25155'2064888'211036 4'738508'9784490 5
151 1950903'198912 5027339'980785i3 45 36'20107791 205270 4'871620'9795752 24 561 20677341 21134014'731695'9783889 4
16'19537561 199214 5'019707 9807285 44 37 2013629'205573 4'864435'97951671 23157 207,05801 21164414'724901'9783287 3
17'1956609 *19951715'012098 19806716 43 38 2016478'205876 4-857271 j9794581 22 58 207-3426'211948 4'718125 J9782684 2
18'1959461'199819 5'00411'9806147 42 9'2019327'20618014'850128'9793994 21 59'20762721 212252 4'711368'9782080  1
19'1962314'200122 4.996945'9805576 41 0'2022176'2064831 4843004'97934106 20 60 20791171 21255614'704630'9781476 0
201 19651661 200424 4'98902'98050054                        I.                 I I4
Cosine. Cotang. Tang.    Sine.            Cosine. Cotang.! Tang.   Sine.' Cosine. Cotang. Tang.   Sine.  
Deg. 78.                               Deg, 78.                                Dee. 78.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
12 Deg.                                 12 Deg.                                  12-Deg.
Sine.   iTang. Cotang. Cosine.                                          | Sine.   Tang. Cotang. Cosine.' Sine.   Tang.   otang.l Coeine.
0'2079117 -212556 4'704630'9781476 6  1'2138829 -218949 4-567261'9768593 3 41 *2195624'225054 4443376'9755985 19
1.2081962 [212860 4.697910'9780871 59 22 2141671 [219254 4.560911 -9767970 38 42 -2198462 -225359 4-437350'9755345 18
2.2084807.21316414-691208.9780265 5823,2144512.219559 4-554577 -9767347 37 43 2201300.225665 4-431339'9754706 17
3.2087652 213468 4-684524 -9779658 57 24 2147353 -219864 4-548260'9766723 36 44 2204137 -225971 4425343'9754065 16
4.2090497.213773 4.677859. 9779050 56 25 2150194.220169 4-541960 9766098 35 45 2206974 *226276 4.419364'9753423 15
5'2093341 214077 4 67 t1212. 9778441 55 26 2153035.220474 4.535677'9765472 34 46 2209811'226582 4.413399'9752781 14
6'2096186 214381 4.66458319777832 547'2155876.220779 4-529410'9764845 33 47 1221648 226   888 4-407450'9752138 13
7'2099030 214685 4.657972. 9777222 53 8'2158716'221084 4.523160'9764217 32 48 2215485 -227194 4401 516'9751494 12
8S 2101874'214990 4.651378. 9776611 52 29 2161556 22138914 516926 -9763589 31 49 2218321 1227500 4-395597'9750849 11
91 2104718 -2152944.644803 19775999 51 30 2164396.22169414.510708 -9762960 30 50 12221158'227806 4389694'9750203 10  CI
10 12107561.215598 4.638245- 9775386 5031'2167236 -221999 4-504507 -9762330 29 51 12223994'228112 4.383805 -9749556 9
11 -2110405,215903 4.631705'9771773 49 32 -2170076'222305 4.498322 -9761699 28 52 2226830 -228418 4.377931 19748909 8
12 -2113248 216207 4.625183- 9774159 48 33 -2172915. 222610l 4492153. 9761067 27 53. 2229666. 228724 4:372073 -9748261  7
13 -2116091'216512 4618678 1977354447 34 -2175754.222915 4.486000'9760435 26 54 2232501. 229030 4'366229'9747612  6
14'2118934. 216816 4.612190'977292846 35'2178593. 223221 4.479863. 9759802 25 55. 2235337'229336 4.360400.9746962 5
151 2121777. 217121 46057201 9772311 45 36 2181432. 2235261 4473742. 9759168 24 56 2238172'229642 4'354586 19746311  4
i6s 2124619'217425 4-599268. 977169344 37 2184271'223831 14467637 -9758533 23 57 -2241007.2299491 4348786'9745660 3
17 -21274621 217730 4-592832'9771075143 381.2187110'224137 4.461548'9757897 22 58 2243842'230255 4.343001.974,50081 2
18 S2130304. 218035 4-586414 19770456 42 39 2189948 -224442 4-455475'9757260 21 59 -2246676. 230561 4 337231. 9744355  1
191 2133146'218340 4'580012'9769836 41 401 2192786'224748 4.449418'9756623 2060'2249511'23086814.3314751 9743701  0
201 2135988 -218644 4'573628'9769215 40                                                            ___
Cosine. Cotang    ang.   Sine.                   Cotsine. ot ang.   Sine.     Cosine. Cotan   Tang.   Sine.
Deg. 77,                               Deg. 77.                                 Deg. 77.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
13 Deg.                                 13 Deg.                                 13 Deg.
Sine.   Tang. Cotang. Cosine.''/  Sine.  Tang. Cotang. Cosine                Sine.  iTang.  Cotang. Cosine.
0'2249511'230868 4'331475'9743701 6021'2308989 *237311 4'213869'9729777 39 41'2365555 *243465 4'107356.9716180 19
1'2252346'231174 4'325734 *9743046 59 2'2311819'237618 4'208419'9729105 38 42'2368381 *243773 4'102164.9715491 18
2'2255179'231481 4'320007'9742390 58 23'2314649'237926 4'202983'9728432 37 43 2371207 *244081 4'096985.9714802 17
3'2258013'231787 4'314295'9741734 57 24 2317479'238233 4'197560 *9727759 36 44 *2374033 *244390 4'091817.9714112 16
4'2260846'232094 4'308597 *9741077 5 625 2320309 238541 4'192151'9727084 35 45 *2376859'244698 4086662 *9713421 15
5'2263680'232400 4'302913'9740119 5 26'2323138'238848 4'186754'9726409 34 46 2379684 *245006 4'081519'9712729 14
6'2266513'232707 4'297244'9739760 54 27 2325967'239156 4'181371'9725733 33 47 2382510'245315 4-076389 *9712036 13
7 *2269346 *233014 4'291588 *9739100 5 28 2328796'239463 4'176001 97250561 32 48 2385335 *245623 4'071270'9711343 12
8'2272179'233320 4'285947'9738439 5  9'2331625'239771 4'170644'9724378 31 49'2388159'245932 4066164'9710649 11
9'2275012'233627 4'280319'9737778 51 30 2334454'240078 4'165299'972369 9 30 50 2390984'246240 4'061070'9709953 10
10'2277844'233934 4274706'9737116 5 31'2337282'240386 4'159968'9723020 29 51 2393808'246549 4'055987'9709258 9
11'2280677'234241 14269107'9736453149 32'2340110'240694 4154650'9722339128 52'2396633'246857 4'050917'9708561 8
12'2283509'234547 4'263521 9735789 48 33 2342938.2410014.149344'9721658 27 53 2399457'247166 4'045859'9707863  7
13'2286341'234854 4'257950'9735124 47 34'2345766'241309 4'144051'9720976 26 54 -2402280'247475 4'040812'9707165' 6
14'2289172'235161 4'252392'9734458 46 35 2348594'241617 4'138771'9720294 25 55 2405104'247783 4'035777'9706466 5
15'2292004'235468 4'246848'9733792 4536'2351421'241925 4'133504 9719610 24561 2407927'248092 4'030755'9705766 4
16 12294835'235775 14'241317'9733125 44 37 2354248 1242233 4'128249'9718926 123 57 2410751 -248401 4'025744'9705065 3
17'2297666 j236082 4'236800J 9732457 43 38 2357075.24254114.123b07'9718240122 581 2413574 124871014'020744j 9704363  2
18'2300497'236390 4-230297'9731789 42 39 2359902'242849 41 17778'9717554121 59'2416396'2490194'015757'9703660  1
191 2303328'236697 4224808'9731119 41 40'2362729'2431574'1125611 9716867120 60 2419219'249328 4'010780'9702957 0
201 23061591 23700414219331'9730449 40,
Cosine. Cotang. Tang.   Sine.             Cosine. Cotang. Tang.   Sine.         Cosine. Cotang. Tang.   Sine.
Deg. 76.                                 Deg. 76.                              Peg. 76.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
14 Deg.                                 14 Deg.                                 14 Deg.
Sine.  Tang.  C otang. Cosine.'    Sine.  Tang.  Cotang. Cosine. / /  Sine.  T''ang. Cotang. Q Cosine.
0 *2419219'249328 4.010780'9702957 6021'2478445'255826 3'908901'9687998 3  1'2534766 *262034 3'816295 *9673415 19
1 12422041'249637 4.005816'9702253 59 22'2481263 1256136 3.904171'9687277 38 42'2537579 1262345 3'811773'9672678 18
2 -2424863 -249946 4.000863'9701548 58 23'2484081'256446 3'899451 *9686555 3 43 2540393 -262656 3.807260'9671939 17
3 *2427685'250255 3'995922'9700842 57 24'2486899'256756 3-894742'9685832136    2543206'262967 36 802758'9671200116
4'2430507'250564 3.9909092 9700135 56 25'2489716'257066 3'890044'96851081355'2546019'263278 3-798266 *9670459115
5'2433329'250873 3'986073'9699428 55 26'2492533'257376 3'885357'9684383 3446 2548832'263589 3-793783 *9669718 14
6'2436150'251182 3'981166'9698720 54127'2495350 1257686 313 880680  9683658133 47 2551645'263900 3'789310'9668977 13
7'2438971'251491 3.976271'9698011 53 28'2498167'25799713'876014 9682931 32 8'2554458'264211 3'784848.9668234 12
8'2441792 1251801 13971386'9697301 52129'2500984'25830713'871358'9682204 31 49 12557270 1264522 3'780395'9667490 11
9'2444613'25211013'9665131 9696591 51 301 2503800'258617 3'866713'9681476 30 50 12560082 264833 3'775951 9666746 10  I
10'2447433'252420013961651 19695879150 311 2506616'258928 13862078 19680748 29 51'2562894'265145 3.771518'9666001  9 
11 124502541'252729 13956801 -9695167 49 32'25094321 259238 3'S57453'9680018 28!521 25657051 265456 3.7670941 9665255 8
12'2453074'2530380 3951961 969445348 33'2512248l 2595481 3852839'9679288 27 53'25685171 265768 3 7626803 9664508i   7
13 2455894'253348 3'947133 969374047 34'2515063'25985913'848235'9678557 26 54'2571328 1266079 13'758276 9663761  6
141 24587131 253658 3'942315'9693025 46 35'2517879'260169 3'843642'9677825 2555'2574139'266390 3'753881 1-9663912  5
15'2461533i 253967 3'937509'9692309 45 36'2520694'260480r 38390591 9677092 24156'2576950'26670213-749496'9662263  4
161 2464352'254277 13932714'9691593 44 37'25235081 260791 3'834486 19676358 12357'2579760'267014 3.745120 19661513 3
17'2467171 -254587 3-927929'9690875 43 381'2526323'261101 3'829923'9675624 22 58 25825701'26732513'7407541'9660762  2
18'24699901'254896 3'9231.56'9690157 42 39'2529137 -261412 13825370'9674888121 59'2585381'267637 3'736398 19660011  1
19 12472809 j255206 3.918393'9689438 41 40 125319521 26172313.820828'9674152 20160 125881901 26794913.732050 19659258  o
20 l2475627'255516 3'913642'9688719 40
Cosine. |Cotang.' Tang.   Sine.  I' -   Cosine. ICotang. Tang. | Sine.   t   Cosine. Cotang.j Tang. | Sine.
Deg. 75.                                Deg. 75.                                Deg. 75.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
15 Deg.                                  15 Deg.                                  15 Deg.
I  Sine. { Tang. Cotang. Cosine. {    [ Sine.   Tang. Cotang. Cosine. [  /  Sine. [Tang. Cotang. Cosine. /
0 *2588190 1267949  3.732050 *9659258 60 21'2647147'274507 3'642891 *9643268 39 41'2703204 *280773 3.561590'9627704 19
1'2591000'268261 3.7277131 9658505 5922 -2649952'274820 3 638744'9642497 38 42 *2706004'281087 3-557613 *9626917 18
2'2593810'268572 3'723384'9657751 58 23 {26527571.275133 3-634606'9641726 37 43'27088g05'281401 3'553644'9626130 17
3'2596619.268884 3-719065.9656996 57 241.2655561.275445 3.630477'9640954 36 44. 271160.5 281715 3.549684 -9625342 16
4'2599428'269196 3.714756.9656240 56 25'2658366'275758 3.626356'9640181 35 45'2714404 282029 3.545732 -9624552 15
5 2602237'269508 3.710455.9655484 55 26'2661170'276071 3'622244'9639407 34 461 2717204'282343 3-541788'9623762 14
6 32605045.269820 3.706164.9654726 54 27 12663973'276385 3'618141.9638633 33 471 2720003 -282657 3.537852;9622972 13
7 2607853.27013213.701883.9653968 53 28 *2666777 276698 3'6140461 9637858 32 48 2722802 28297113-533925.9622180 12
81'2610662'270444 3'697610'9653209 52 29'2669581 *27701113'609960 9637081J31 49 -2725601'283285 13530005 9621387 11
91 2613469'270757 3-693346 19652449 51 301-2672384 1277324 3'605883'9636305 30 5012728400'288599 3'526093'9620594110
10'2616277'271069 3'689092. 965168950 31'2675187'277637 3'601814'9635527 29 51'2731198'283914 3'522190'9619800  9   C~
11'2619085'271381 3'6848471 9650927 49 32'26779897 277951 3'597754'9634748 28 52 27339971 284228 3' 18294'9619005 8
121 2621892'271694 3'680611'9650165 48 33- 2680792'278264 3'593702'9633969 27 53'2736794'284543 3'514407 96182101 7
13'2624699'272006 3'676384. 9649402 47 34'2683594'278578 3'589659 19633189 26 54'2739592'284857 3'5105271 9617413  6
14'2627506'27231-8 3'672166'9648638 46 35'2686396'278891 3'585624'963240825 555 27423901 285172 3'506655' 9616616 5
151 2630312'272631 136679571 9647873 45 36'2689198'279205 3'581597'9631626 2456'2745187'28548613'502791 9615818 4
161 2633118'272943 13663757. 9647108144 371 2692000'279518 3'577579'9630843 23 57'2747984. 285801 3'498935 -9615019  3
171 2635925I.2.3256 13659566.9646341 43 38.2694801.279832 3.5735691 9630060 22 58'2750781'286115 3.495087. 9614219 2
181 26387301 273569 136553841 9645574 42 39 2697602'280145 3'569568 19629275 21 59'2753577'286430 3'491247'9613418  1
19 12641536'273881 3.651211'9644806 41 40 2700403 1280459 13565574'9628490 20 601 2756374 1286745 3.487414. 9612617 0
20'2644342'274194 3'647046'9644037 40
Cosine. C otang.Tang.   Sine.            Cosine. Cotang.  Tang.   Sine.'' Cosine. Cotang. Tang.   Sine.
Deg. 74.                                Deg. 74.                                  Deg. 74.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
16 Deg.                                 16 Deg.'16 Deg.
I Sine.   Tang. Cotang.  Cosine.'         Sine.    Tang. Cotang.  C osine. t'  Sine.   Tang.  Cotang. Cosine.0'2756374'286745 3.4874 14 -9612617 60 21 2815042 -293368 3.408688'9595600 39 1 12870819 1299697 3'336699'9579060119
1'2759170'287060 3.483589 *9611815 59 2 12817833'293683 3'405021'9594781 38 42 -2873605,'300014 3'333173 *9578225 18
21'2761965 J287375 3'479772 -9611012 58 23 2820624'293999 3'401361'9593961 37 43 *2876391'300331 3'3296541'9577389117
3 12764761 *28769013'475963 19610208 57 24 12823415 129431613'397708 *9593140136 44'28791771'300648j3'3261411'9576552 16
4'2767556'288005 3'472161'9609403 56 25'2826205'294632 3'394063'9592318 35 45 128819631 300965 3'322636'9575714 15
5'2770352 1288320 3.468367'9608598 55 26'2828995'294948 3'390424'9591496 34 46 2884748'301283 13319137'9574875 14
6'2773147'288635 3'464581'9607792 54 27'2831785'295264 3'386793'9590672 33 47 2887533'301600 3'315645'9574035 13
7'277~941'288950 3.460802'9606984 53 28'2834575 *295580 3'383169'9589848 32 48'2890318" 301917 3'312159'9573195 12
81 27787361 289265F 3457031.9606177 52 29 128373641 295897 3.379553.9589023131 49 28931031.302235 3308681 1.9572354111
91 2781530 -289580 13453267'9605368 51 30'2840153'296213 3 3759431 9588197 30 50 o2895887'302552 3305209'957151.210,!0
101 2784324'289896 134495121 9604558150 311 2842942'29652913.372340'95873711291511 2898671 1302870 133017431 957066,9] 9 
11 12787118 1290211 3.4457631 9603748149 32 12845731 i.296846 13368745 9586543128 52 129014651 30318713'2982851 95698251 8
12 -2789911'290526 3.442022'9602937 48 33 -2848-520'297163 3-365156'9585715 |2 53'2904239 #303505 3'294833'95689811 7
131 2792704'290842 3.438289'9602125 47 34 2851308 1.297479 3.361575'9584886126 541 2907022 L303823 3'291387'9568136' 6
14 -27954971 29115713'434563'9601312146 351 2854096 297796 3358000 9584056 25 55 2909805[304141 3'287948'9567290  5
15' 27982901 291473 3.430844'9600499 45 361 28568841 29811213.354433 -9583226124 561 29125881 304458 3.2845161 9566443 4
161 2801083'29178913.427133'9599684 44 37 12859671'298429 3.350872'9582394 23 57 12915371'13047761 32810901 9565595 3
17'2803875'292104 3'423429'9598869 43 381 286245815298746,3'347319'9fi581562 22 581'2918153'305094 3'277671'95647471 2
18'28066671'292420 3'419733'9598053 42 39'2865246'299063 3'343772 9580729 21 59'2920935'305412 3'274258'9563898  1
19'2809459 -29273f6 3416044'9597236 41 40 12S680321 299380 3'3402321 9579895 20 60 29237171 305730 132708521 9563048  0
20'28122511 293052 13412362'9596418 40
Cosine. Cotang.l Tang. { Sine.  /t   Cosine. Cotang. Tang.   Sine.  /t | Cosine. ICotang.I Tang. I Sine.
Deg. 73.                                 Deg. 73.                              Deg. 73.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
17 Deg.                                  17 Deg.                                  17 Deg.
Sine.   Tang. Cotang. Cosine.'Sine.   Tang. Cotang. Cosine.                  ine.   Tang. Cotang. Cosine.
0 *2923717.305730 3.270852'9563048 60 21 12982079'312422 3200789'9545009 39 41 3037559 1318820 3o136563 19527499 19
1'2926499'306048 3'267452'9562197 59 22'2984856'312742 3'197521'9544141 38 42'3040331'319140 3'133414'9526615 18
2'2929280'306367 3.264059'9561345  5  3 12987632 -313061 3.194259'9543213 37 43.30431021.319461 3-130270'9525730 17
3'2932061'30668513.260672'9560492 15 24 29904081'313381 3'1910031-9542403 36 44'3045872'319781 3'127131 19524844 16
4'2934842'307003 3'257292 19559639 56 25'2993184'3137Q0 3'1877541 9541533 35 45'3048643'320102 3'123999'9523958 15
5 12937623'307321 3'253918'9558785 55 26 2995959'314020 3'184510 19540662 34 46 3051413'320423 3'120872 19523071 14
6'2940403'307640 13250550'9557930 54 7.'2998734.314339 3.181272'9539790 33147'3054183 3 320744 3a117750'9522183 13
7'2943183.307958 3.2471891 9557074 53.8'3001509 1314659 3 178040'9538917 32 48'3056953 -321064 3.114635'9521294 12
8.2945963'308277,3.243834 19556218 522 9'3004284.31497aj3.174814'9538044131149'30597231 321385 131 115251'9520404 111
91 2948743'308595 3'240486'9555361151 30'3007058'31529813'171594'9537170 30 501 3062492'321706 3'108421'9519514 10   4
10'2951.5221 308914 3'237143 19554502 5031  o3009832'31561813168380'953 6294   29151 13065261 1322027 3'105322'9518623  9
11'2954302'309233 30233807'9553643 4932'3012606'3159381 3165172 9535418 28 521 3068030 1322348 3'102229'9517731  8
12'2957081'309551 3'230478 1-9552784 48Lq3 -3015380'316258 3'161970'9534542127 53'3070798 1322670 3'099141'9516838  7
13 l2959859'309870 3'227154'9551923'47 43018153 316578 3.158774'9533664126 54'3073566 1322991 13096059 "9515944  6
14'2962638'310189 3-223837. 9551062 4635 13020926'316898 3'155584'9532786 25 55'3076334 1323312 13092983'951:050  5
15'2965416'310508 3-2205261.9550199 451361 3023699'31721813.152399'19531907 24 56'30791021.323633 13089912 -9514154 4
16'2968194'310827 3'217221 |'9549336144137'3026471'317538 13149220'9531027 23 571 3081869 -3239551 3086846 9513258 3
17'2970971'311146 3'213922'9548473 431381'3029244'31785913'146047 19530146122 581 308463  32427613.083786'95123611 2
181 29737491 311465 13210630'9547608 4239 13032016'318179 3.142880 19529264 21 59 308740. 324591 3.080732 -9511464  1
19 12976526'311784 13207344 19546743 4140'3034788'3 18499 13139719 19528382 20 60'309017)'324919 13077683 9510565 0o
201 29793031 S12103 3-s48       520406.3- 6 401  De 7Dg 7eg. 9576 4.
Cosine. Cotang. Tang.   Sine.       / Cosine. Cotang. Tang.   Sine.         Cosine. lCotang. Tang.    Sine.
Deg. 72.                                Deg. 72.                                  Deg. 72.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
18 Deg.                                  18 Deg.                                18 Deg.
/  Sine.   Tang. Cotang. o(Jsine.    /I Sine.   Tang. Cotang. Cosine. /'    Sine.   Tang. Cotang. Cosine.  l
0'3090170'324919 3.077683.9510565 6021'3148209'331686 3'014892'9491511 39 11 3203374'33157 2'957205'9473035 19
1'3092936 1325241 3'074640'9509666 5 22'3150969'332009 3'011960'9490595 38 42'3206130 o3384381 2-954372'9472103 18
2'3095702 1325563 3'071602'9508766 58123'3153730'332332 3'009033 19489678 37 43 3208885'338805 2'951545'9471170 17
3 13098468 5325884 3-068569.9507865 57 24 13156490 133265513.006110 -9488760 361 443211640 1339129 2'948722 19470236 16
4'3101234'326206 3.065542'9506963 56 25'3159250'332978 3'003193,9487842 3  5'3214395'3394541 2945905'9469301 15
5 -3103999 1326528 3.062520'9506061 55126'3162010'333302 3'000282'9486922 34 46'3217149'339778 2.943092'9468366 14
6 13106764'326850 3'059503'9505157 54127'3164770.333625 2'997375'9486002 33 47'3219903'340103 2-940284'9467430 13
7 13109529'327172 83056492'9504253 5 28' 3167529'333948 12994473'9485081 32 8'3222657 *340427 2-937480'9466493 12
8'3112294'327494 3-053487'9503348 52 29.3i70288'334271 2.991576'9484159 31 9 13225411 1340752 2'934682 19465555 1 1
9'3115058'3278163'050486'9502443 51 30'3173047 334595 2'988685'9483237 350'03228164'341077 2-931888'9464616 10'&
10 13117822 1328138 13047491 19501536 50131 131758051.334918 2-985798 19482313 29151 3230917 1341401 2-9290991 9463677} 9
11 83120586 1328461 3.044501'9500629 49 32 *3178563'335242 2.982916'9481389 2852'32336701'341726 12926315 19462736 8
121'3123349'328783 3-041517'94997211 48 33 3181321'335566 2'980040'9480464 27 53 3236422'342051 2'923535'9461795 7
1:3 -31261 12 329105 3038538'9498812.47 34 13184079 1335889 2977168 -947953812 154 3239174 1342376 2-920761 9460854 6
14'3128875'329428 3'035564'9497902 46 35'3186836 1336213 2-974301'9478612 2555'3241926'342701 2'917990'9459911 5
15 -3131638'32975013'032595'9496991 453 6'3189593'336537 2'9714391-9477684 24 56'3244678 1343026 12915225 94589681  4
16 -3134400 1330073 13029632 19496080 44 37 31923501 336861 12.968583 -9476756 23 57 13247429 343351 12912464 9458023  3
171 3137163'330395 3'0266731'9495168 43 38'3195106 1337185 12965731 1-9475827 22 58 3250180'343677 2'909708'9457078  2
18 31399251 330718 3a023720 19494255 42 391 31978631 337509 12962884 19474897 21 591 3252931 1344002 12906957'9456132  1
19 {'342686'331041 3'020772 19493341 41 40 132006191 3378331 2960042'9473966 20 60 13255682'344327 2'904210'9455186 0
201'3145448 t331363 3'017830 19492426 40  I       I       I _                 {       I                     _ I     I
j Cosine. ICotang.{ Tang.   Sine.'     Cosine  Cota n g.   Sine.                osine. Cotang  Tang.   Sine
Deg.71/ /1.                                         Cg                         Sineg..  -.'
Deg. 71.                               D)eg. 71,                                Deg. 71.




NATURAL SINES AND TANGENTS TO A RADIUS 1
19 Deg.                                 19 Deg.                                 19 Deg.
I'T Sine.   Tang. Cotang. Cosine.            Sine. I Tang. Cotang. Cosine.' /  Sine.  Tang. Cotang. | Cosine.i
0 -3255682 -344327 2-904210 9455186 60 21 13313379'351175 2'847583'9435122 39 41 *3368214'357723 2.795453'9415686 19
1'3258432 *344653 2.901468 *9454238 59 22 13316123 1351501 2.844935'9434157 38142.3370953'358051 2'792891'9414705 18
2'3261182 1344978 2'898731'9453290 58 23 13318867'351828 2.842292'9433192 37143'3373691'358380 2'790333'9413724 17
3'3263932'345304 2'895998'9452341 57 24'3321611'352155 2'839653'9432227 36 44 3376429'358708 2'787780'9412743 16
4'3266681'345629 2'893270'9451391 56 25 3324355'352482 2'837019'943126035 45'3379167'359036 2'785230'9411760 15
5 *3269430 1345955 2'890546 19450441 55 26'3327098'352809 2-834389 19430293 34 46'3381905'359365 2'782685'9410777 14
6'3272179'346281 2'887827'9449489 54 27'33298411-353136 2-831763'9429324 33 47.3384642'359693 2'780144'9409793 13
7'3274928'346606 2'885113'9448537 53 28 33325841 353464 2.829142'9428355 32 48 3387379'360022 2.7776061 9408808 12
81 32776761 346932 12882403'9447584 52 29.3335326,.353791 12826525'9427386131 9'3390116 [360350 2.7750731 9407822 11
9'3280424 1347258 2'879697'9446630 51] 30 3338069 1354118 2.823912 -9426415 30 50 3392852'360679 2'772544 9406835| 10  c
101 32831721 347584 2.8769971 9445675 50 31'33408101 354446 12821304'9425444 29 51 3395589'361008 2.770019'9405848 9
11'32859191 3479101 2874300'9444720 49 32.33435521 354773 2.818700'9424471128 52.3398325.361337 12767499I 9404860 8
12'3288666'348236 12871608'9443764148 33 i3346293 S3551011 2816100 i9423498 27 53'3401060I 361666 2.764982.9403871  7
131 3291413 134856312'868921.944280747 34 }3349034'355428 2'813504 19422525 26 54.34037961 361994 2,762469 19402881 6
14'3294160 1348889 2'866238.9441849146 351 335177513 i5756 2-81091391 9421550o25 553406531 -362324 2.7599601,9401891 5
15 l3296906'349215 2l8635601 9440890 45 3613354516'356084 2o808326a 9420575 24 56 3409265'3626531 27574561 9400s99  4
16'3299653'349542 2'860886'9439931144 371 33572351.356411 12805743 194195981231571 34120001'362982 2'754955'9399907 3
171l3302398 -349868 128582161'9438971 43 38'3359996'356739 2'803164'9418621 22158'3414734 -363311 2'7524-58'9398914t 2
18, 3305144'350195 2'855551 1943801014  9'3362735'357067 2'800590'9417644 21 59 13417468'363640 2'749966 9397921 1
191'3307889'350521 2-852891 ['9437048 41 401 3365475 1357395 2'798019'9416665i20 60 134202011 363970 2'747477'93969261 0
201'3310634'35084812'850234'9436085 4             II                       I,                    1        1
ang  Tang. Cosine.e. Congang. Tng.   Sine..                               /  Cosine. Cotang.. Tang.   Sine. 
Deg. 70.                                Deg. 70,                                Deg, 70,




NATURAL SINES AND TANGENTS- TO A RADVIIS 1.
g0 f)eg,                                 20 Deg.                                  20 Deg.
Sine.  Tang.  Clotang. Cosine.'  Sine.  Tang.;Cotang.               Sine..ang. Cotang. 
T-ag__tan,     I Ci- *' l
0.3420201'363970 2.747477",9396926 6C 21'3477540 1370903 2-696118,9375858 39 41 3532027 1377536 2-648753'9355468 19
1 13422935'364299 2'744992 19395931 59 22'3480267'37123412,693714'9374846 3  2.3534748.377868 2,64642'3.9354440 18
2 -3425668 *364629 2.742512'9394935 58 23.3482994 *371565 2,691314.9373833 37 43'3537469,378201 2.644096 19353412 171
31'3428400 1364958 2-740035 i9393938 57 24'3485720'371896 2.688919.9372820136 44'35401 i 01378533 1641774 19352382 10
4'3431133'365288 2'737562'9392940 56 25 13488447'372227 2'686526'9371806 3545'3542910'378866 2'639454'9351352 15
5'3433865 1365618 2'735093'93919421 55 26 3491173 1372559 2'684138'9370790 34 46 3545630 379198 12637139.9350321 114
6'34365927 365948 2.732628.*9390942 54127'3493898 1372890 2.681753'9369774 33 47 13548350 1379531 2,634827 19349289 13
7'3439329 -366277 2'730167 19389948 5328 -3496624'373221 12679372.9368758 3248 13551070'379864 2-632518.9345257 12
8'34420601 366607 2'727710 19388942 5219 1'3499349 1373553 12676995 19367740 31 4-9 13553789'380197 2630213 1'9347223 11
91 3444791 1366937 2-725256 19387940 51130 13502074.3738842.674621'9366722 30 50 13556508'380530 2.6279121 9346189 10 Co
10 -3447521;367268 2'722807 19386938 50[31 13504798 137421612-672251 1'9365703 29 51'3559226 138086312'625614 93461541  9 
11'3450252'367598! 2720362 19385934 |493  1'3507523 137454712.66988r.9364683 28!52 13561944 1381196 2'623319 193441199 8
12'3452982'367928 2'717920.93849304J833' 3510246.374879 2-667522.936366227 53'3564662'381529 2'621028'93430821 7
13'3455712'368258 2'715482.9383925 4734'.3512970 -375211 12666163'9362641 26 54'3567801'381862 2'618741'9342045 6
14'3458441'368589 2'713048.9382920 4-6135'3515693'375543 2.662808'9361618 25551 3570097'382196 2'616457 -9341007 5
15.3461171]-368919 2,710618'93819]13 45136'3518416'37587512'660456'936059512456 -3572814'382529 2-614176'9339968  4
16'34639001 36925012'108192 -9380906144 371'35211391'376207 2-6581 08'9359571 23571 3575531 1'382863 2'6113991 9338928  3
17 13466628 1369580 2.705769 19379898 4301,38'3523862 1376539 26655764'9358547122158 -3578248s383196 2.609625;9s337888  2
18'3469357'369911 2'703351'9378889 4239'3526584'376871 2,6534231'9357521 21159,3580964 -383630 2,607355 -9336846  1
191'3472085'37024212.700936 -9377880 41!40'35293061 37720312,6510861 9356495120160,3583679 -383864 2-6050891 9335804  0
201'3474812'370572,2'698525'9376869 401                                    - - - - _      I. I  I
CIosine. Clotang. Tang.  Sine.' I   Cosine. Cotang. Tang-    Sine.              Cosine. Cotang. Tang.   Sine.
Deg. C9.                                Deg. 69.,                                 Deg. 69.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
21 Deg.                                  21 Deg.                                 21 Deg.'  Sine.   Tang. Cotang. Cosine.',:  Sine.  Tang. Cotang. - Cosine  / -    Sine.' Tang.  Cotang. - Cosine.'
0'3583679'383864 2'605089'9335804 60 1'3640641'390889 2'558268 *9313739 39 41 -3694765 397611 2'515018'9292401 19
1'3586395'384197 2'602825'9334761 592 3643351'391224 24556075'9312679 3842'36974681-397948 2'512889'9291326 18
2'3589110'384531 2 600565 9333718 58 233646059'391560 28553885'9311619 37 43 3700170 398285 2'510762'9290250 17
3'35918251'384865 2'598309'9332673 5  4.36487681'39189512-551699.9310558 3644'3702872'39862212'508639'9289173 16
4'3594540'385199 2'596056'933162S 56 25.3651476 1392231 2549516'9309496 35 45 3705574'398959 2'506519'9288096 15
5'3597254'385533 2'593806'9330582 55 26 3654184'392567 2'547335'9308434 3446'3708276 -399296 2'504402'9287017 14
61 3599968'385867 2:591560'9329535 54 27.3656891:392902 2-545159.9307370 3347'3710977'399634 2-502289'9285938 13
7 3602682.386202 2.589317'9328488 53 28. 3659599'393238 2-542985.9306306 32 18'3713678'399971 2500178'9284858 12
81 3605395 3,8653612.587078'9327439 52 29.3662306 393574 2.540815 -9305241 31 49'3716379'400308 2'498070'9283778 11
91 3608108' 386870 2'584842'9326390 51 30'3665012'393910 2-538647.9304176 30 50'3719079'400646 2'495966'9282696 10  Cm'
10'3610821'387205 12582609'9325340 50 31 3667719'394246 12536483'9303109 2951'3721780'400994 2-493864'9281614 9
11'3613534'387539 2'580380'9324290 49 321 3670425.394582 2534323'9302042128 52'3724479 -401321 12491766'9280531  8
121'3616246'387874 2 578153'9323238 48 33 3673130 [394918)2.5321651 9300974 27 53'37271791 401659 2.4896701 92794471 7
13'3618958'388209 2.575931 19322186 4734 3675836.395255 [2.530011 -9299905126 54'3729878.401997 2.487578'9278363  6
14'3621669. 388543 2.5737111-9321133 46 351 3678541'395591 22527859.9298835 2555'3732577. 402335 2.485488'9277277 5
15'3624380. 388878 2.571495'9320079 45136 13681246. 395928 2.525711'9297765124 561 3735275 402673 2'483402. 9276191  4
16'3627091'389213 2'569283'9319024 44137'3683950 1396264 2.5235661 9296694 23 57'3737973 403011 2'481319 19275104 3
17'3629802. 389548 2.5670731J9317969 43 381 3686654. 396601 2.521424'9295622 22 581 3740671'40334912'479238.92740161 2
18'3632512'38988312'564867'9316912 42 39'3689358'396937 2.519286'9294549 21 59'3743369'403687 2'477161 1'9272928  1
191'36352221 390218 2'562664 -9315855 41 40'3692061'397274 2-517150'9293475 20 60'3746066 1404026 2'475086'9271839  0
201 36379321 390554 2'560464'9314797 40
1  Cosine.. Cotang., Tang. I Sine.'   Cosine. Cotang. Tang.   Sine             Cosine. Cotang. Tang    Sine.
Deg. 68.                                 Deg. 68.                               Deg. 68.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
22 Deg.                                  22 Deg.                                 22 Deg.
1    Sine.   Tang. Cotang. Cosine. / It  Sine.   Tang. Cotang. Cosine.''  Sine.   Tang. Cotang. Cosine. 
0 13746066'404026 2'475086'9271839 60 21 3802634'411149 2432204'9248782 39 41'3856377 417967 2392531 1.9226503 19
1'3748763 -404364 2.473015'9270748 59 22'3805324 1411489 2'430193.9247676 38 42 13859060.418309 2.390576. 9225381 18
2.37514591 404703 2.470947 9269658 58123'3808014'411830 2'428186 *9246568 37 43 3861744'418650 2'388625'9224258 17
3'3754156'405041 2.468881 *9268566 571241'3810704.412170 2'426181'9245460136 441 3864427 418992 2-386675 19223134 16
4 13756852'405380 2.466819'9267474 56 25.3813393 1412510 2.424180'9244351 35 45 13867110.419334 2'384729'9222010 151
5 53759547'405719 12464759'9266380 55 26 3816082'412851 2'422181'9243242 34 46'3869792'419676 2'382785'9220884 14
6'3762243'406057 2'462703 *9265286 5427 3818770'413191 2'420185 9242131 33 47 3872474'420019 2'380844'9219768 13
71 3764938'406396 2460649'9264192 53 28'3821459'413632 2'418191 19241020 32 48'3875156'420361 2'3789061'9218632 12
81 3767632'406735 12458598'9263096152 29'3824147 413872 124162011-9239908 31 49 -3877837 1420703 2-376970'9217504 111
9'3770327'407074 12456551'9262000151 30'38268341 41421312'4142131 9238795 30 5013880518'421046 2375037'9216376 10  CA
10 13773021'4074131 2454506'9260902 50 311 3829522.414554 2'412228'9237682 29 51 13883199'421388 123731061 921524b  9   C
11 13775714'407753 2'452464.9259805 49 32 13832209'41489512'410246'9236567 28 52 -38858800 4217311 2371179 19214116| 8
12'3778408.408092i2.450425'9258706 48 33 3834895.41523612.408267.9235452 27 53 3888560'422073 2'3692541'9212G86 7[
131 3781101'408431 12448389.9257606 47134'3837582'415577 2.406290.9234336 26 541 3891240. 422416 2.367331.9211851 6
15 137864861 40911012.444325 19255405 1451361 3842953'416259 12402345 [9232102 124561 38965981 423102 123634941 9209589 4
16'378917S4'40945012'442298 19254303 44137 13845639 1416601 2'400377'9230984 23157 13899277. 423445 2.361580'9208455 3
17'3791870 1409790 2'440273'9253201 43 38'3848324'416942 2'398411 -9229865 22 581 3901955'423788 2'359668'92073201 2
18'3794562'410129 2'438251'9252097 42 391 -38510081.417284 12'3964491 9228745 21 59'3904633'424131 2'357759 19206185  1
191 3797253 1410469 12436233'9250993 41 40'3853693'417625 2'394488'9227624 20 60'3907311'42447412-355852'9205049  0
20 1'3799944'410809 12434217 -9249888 40
Cosine. C otang. Tang.    Sine.''  Cosine. Cotang.( Tang.   Sine.           Cosie. Cotang., Tang.   Sine.
Deg. 67.                                Deg. 67.                               Deg. 67.




NATURAL SINES AND TANGENTS TO A RADIUS 1L
23 Deg.                                  23 Deg.                                  23 Deg.
6slne.'l'ang. Cotang. Cosine.         Sine.   Tang. Cotang. Cosine.'    Sine.  Tang. I Cotang. Cosine.
0'3907311 *424474 2-355852 9205049 6 21'3963468'431703 2'316407'9181009 39 41 4016814 *438622 2'279865'9157795 19
1 13909989 1424818 2'353948'9203912 5 22'3966139.432048 2.314557'9179855 38 42 4019478.438969 2.278063'9156626 18
2'3912666'4251611 2352046 9202774 58 23.3968809'432393 2.312709 9178701 3743'4022141.439316 2.276264 *9155456 17
3 13915343 1425505 21350148 9201635 5 24 13971479. 432738 2-310863.9177546 36144 140248041 439663 2.274467 9154286 16
4'3918019'425848 2'348251'9200496 561 5'3974148'43308412.309020'9176391 35145 40274671 440010 2.272672'9153115 15
5 3920695 1426192 2'346358'9199356 55 26 3976818'43342912.307180'9175234 34146'4030129'44035712.270880'9151943 14
6'3923371'426536 2'3444671 9198215 54 27'3979486.43377512-305342'9174077 33 47'4032791'140705 2'269090'9150770 13
7:3926047 1426880 2'3425781 9197073 5 28'3982155'434120 2-303506'9172919 32 48 4035453'441052 2'267303'91495971 12
8'3928722'427223 2.340692'9195931 52 29'3984823 1434466 12301673.9171760131 49 4038114'44140012.265518 9148422 11
9'3931397 -427568 2.338809'9194788 51 30'3987491.43*12 2-299842'9170601 30 50'4040775'441747 12263735 -9147247 10 1
101 3934071'427912 12336928'9193644 50131 139901581 435158 12298014'91694401291511 4043436' 442095 12261955 19146072  9  "
111 3936745'42825612'3350501 919249949 32 3992825'435504 2.296188.9168279 28152'4046096| 44244312.260177.9144895  8
12.3939419.428600'2333174 19191353 481331.39954921'435850 2.294365.9167118 2753 4a3048756'44279112'2584011'9143718  7
13 3942093'428944 2'331301'919020714734.3998158'436196 2.292544'9165955126154 140514161 413139 2.256F628 9142540  6
141 39447661 42928912.329431'918906014635 14000825'436542 122907251 9164791 251551 4054075'41348712'254857 l9141361  5
15 13947439 1429633 2!327563.9187912 4536'4003490 443688912288909'9163627 24614056734'44383512.2530881 9140181  4
16'3950111'429978 2'325697 9186763 4437'4006156'437235 2'287095 9162462 2357'4059393'444183 2'251322 -9139001  3
17'3952783'43032312-323834 -9185614143 381.4008821 1'43758212'285284'916129712215814062051 1'444531 2'249558 -91378191 2
181-39554551 430668 2321974.9184464,42139i.4011486'437928 2'283475'9160130121 59'4064709i'444880 2'247796.91366375  1
19  sl'9,8127.431012 2. ol,3l.0:6s13313 41 0                         940141501.4325281669 9158963 20 60.4067366!445228 2246036 l.935455l 0
I-i-           -      -_-II!   -Cosine.I-It-    -    I                    T        S    -In
C Cosine. oCotang. Tang.Tan  Sine.                                          t "  t  Cosine. CCotang.l Tang.  Sine.  it
Deg. 66.                                De. 66.                                   Deg. 66.




NATURAL SINES AND TANG'ENTS TO A RADIUS 1.
24 Deg.                                 24 Deg.                                  24 Deg.
Sine.   Tang. Cotang. Cosine. -'        Sine.   Tang. C otang. Cosine. /!       Sine.  Tang. C otang. Cosine.[
0.4067366.445228 2.246036 19135455 60 21'4123096 *452568 2-209611 19110438 39 41.4176028'459596 2'175822'9086297 19
1 *4070024'445577 2'244279'9134271 59122 14125745 1452918 2.207901 *9109238 38142.4178671 *459948 2.174155'9085082 18
2'4072681.445926 2'242524'9133087 58 23'4128395 *453269 2'206193 9108038 37 43.4181313'460301 2'172491'9083866 17
3 *4075337.446274 2-240772'9131902 57 24'4131044 *453620 2 204487'9106837 36144.4183956 460653 2'170828 19082649 16
4 *4077993 *446623 2239021'9130716 56 25'4133693 6453970 2-202784 19105635135 45.4186597 1461006 2 169167'9081432 15
5'4080649 *446972 2 237273'9129529 55 26 *4136342'454321 2201083 19104432 34 46.4189239'461359 2'167509'9080214 14
6 4083305.447321 |2-235528'9128342154127'4138990'454672 2 199384'9103228 33 47.4191880 -461711 2'165852'9078995 13
7 14085960 1447670 12233784'91271541 53 28 4141638'455023 2'197687'9102024 32148.4194521.'462064 2'1641981 9077775 12
8'4088615.448020 12232043'9125965 52129 1'442851 455375 12195992'9100819131149.4197161 1462417 2'1625461 9076554 111
91 40912691 448369 2-2303041 9124775 51 30 14146932 1455726 12194299'9099613 30 50 14199801 1'46277J 2,1608951 90753331 10 
10'4093923.448718 2.228567'9123584fi 5031 4149579 456077 2192609 9098406 29 51'4202441 1463124 2'159247'9074111 9
11'4096577) 4490681 2226833'9122393 49 321 4152226'456429 2-1909211 9097199 28152 14205080'4634771 2157601 i9072888  8
12 4099230 449417 122252100'9121201 148 33  4154872'456780 12189234 19095990 27 53 4207719 1463831 21559571'90716651 7
13 14101883!449767 2-223370  9120008 47 34 -'41575 17 457132 2 187551'9094781 26 541 4210358'464184| 2154315 19070 t40  6
14104536'4501117 2221643'911881546 35 4160163'457483 2185869 19093572 j25 55 4212966'464538 2152675'9069215 5
15 14107189 145046712.2199171 9117620 45 36 -41628081'457835 2'184189'9092361 24 56'4215634 1464891 2'151037'9067989 4
161 4109841 [45081712'218194 19116425144137 14165453'458187 2'182511 -9091150(23 57 142182721'465245 121494021 9066762  3
171.41124921 451167!2.216473 19115229143138 141680971 458539 2 180836'9089938 22 58'4220909 1465599 2'147768 19065535  2
18'4115144'451517 2'2147511'9114033142139 14170741 1458891 2'179163 19088725 21159 142235461 465953 12146136 190643071 1
19'4117795 451867 2 213037'9112835|41 40'4173385'459243 2'177492'9087511 20 60'4226183'466307 2 144506.90630781 0
20.4120445.452217 2.2113231 9111637140
Cosine. Cotang.l Tang.   Sine.         Cosine. Cotang.  Tang.   Sine.           Cosine. Cotangg.  Tang.   Sine.
Deg. 65.                                 Deg. 65.                                 Deg. 65.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
25 Deg.                                 25 Deg.                                25 Deg.
/  Sine.   Tang. Cotang. Cosine. / /I Sine.   Tang. Cotang. I Cosine.             Sine. /  Tang. Cotang. JCosine. 7
0'4226183'466307 2'144506 -9063078 6 21 14281467'473765 211 10747'9037093 3941'4333970'480909 2.079394'9012031 19
1'4228819'466661 2'142879.9061848 59 22 4284095'474122 2.109161.9035847 38 42 4336591'481267 2'077846'9010770 18
21'4231455 467016 2'141253'9060618 58 23 4286723.474478 2.107577.9034600 37 3'4339212'481625 2'076300'9009508 17
3,4231090 467370 2'1396301.9059386 57 24'4289351'474834 2'105995.9033353 36 4'4341832'481984 2'074756'9008246116
4'4236725 -467725 2-138008.9058154 56 25 -4291979 4751 91 2 104415 *9032105 35 5'4344453'482342 2'073214'9006982 15
5'4239360 -468079 2'136389 9056922 55 26'4294606'47554812.102836.9030856 34 6'4347072;482701 2'071674'9005718 14
6'4241994'468434 2 134771.9055688154 27'4297233'475904 2'101260'9029606 33 47'4349692'483060 2'070135'9004453 13
7 14244628'468789 2'133155'9054454 53 28'4299859'47626112'099686 *9028356 32 8'4352311'483418 2.068599'9003188 12
81'4247262 -469143 2'131542'9053219 52 29'4302485'476618 2.098114.9027105fi 31 9'4354930'483777,2.067064'9001921 11
9 4249895'469498 2.129930. 9051983 51 30'4305111'47697512.096543'9025853 30 50 4357548'484136 2'065531'9000654 10  Cn
10'4252528'46985312'128321. 9050746 50 31'43077361.477332 2.094975. 9024600 29 51 4360166'484495 2'064000'8999386  9
11'4255161'470209 2.126713'9049509 49 32'4310361'47768912.093408.9023347 2852'4362784'484855 2.062471'8998117 8
121'42577931 470564i2.125108.'9048271 48 331'4312986'47804712'091843 9022092 2753'4365401 1485214 2.060944'8996848 7
13'4260425'47091912'123504.9047032147 341 43156101 478404 2'090280.9020838 26 54'4368018'485573 2'059418'8995578  6
14'4263056'471275 2'121903 -90457924635 4318234 478762 2'088720 90195822.5 5'4370634'485933 2'057895'8994307 5
15,42656871 471630 12120303'9044551 45 36 4320857'479119 2087161'9018325 245 61'4373251'486293 2'056373I'8993035 A4
161 4268318'471986 2118705'904331044371'43723481'479477 2 085603'9017068 2357'4375866'486652 2'054853'8991763 3
171 4270949 -472342 2 117110'9042068 43 38'43261031'479835 2'084048'9015810 22 58 4378482'487012 2'053334j'8990489 2
181 4273579'472697 2'115516'904082542 39'4328726'480193 2'082495'9014551 21 59'4381097 1487372 2'051818'8989215  1
19'4276208 -473053 2'113924 -9039582 41 40'4331348'480551 2'080943 -9013292 200'4383711'487732 2'050303 -8987940 0
20!'42788381-473409 2' 112334. 9038338 40'  Cosine. Cotang.l Tang.  Sine. | - I- Cosine lCotang.l  Tang.   Sine. |' Cosine. ICotang.l Tang. | Sine.
Deg. 64.                              Deg. 64.                                 Deg. 64.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
26 Deg.                                  26 Deg.                                 26 Deg.
Sine.  T'ang.  Cotang. Cosine.          Sine.  Tang.  Cotang. Cosine.   Sine.  Tang. Cotang. Cosine. i
0 14383711'487732 2.050303 -8987940 60 21.4438534'495317 2;018908 18960994 39 11 14490591 1502583 1 989720 18935021 19
1'4386326'488092 2'048791'8986665 59 22 *4441140 *495679 2 017433'8959703 38 12'4493190'502947 1'988278 8933714 18
2'4388940'488453 120472801 8985389 58 23'4443746 1496041 2.015959.8958411 37143 14495789'503312 1.986838.8932406 17
3-*4391553'488813 2'045770 -8984112 57 24 4446352.496404 2'014486'8957118 36 44'4498387'503676 1.985400'8931098 16
4'4394166'489 173 2044263'8982834 56 25 4448957 *496766 2'013016 *8955824 35 45 4500984'504041 1-983963'8929789115
5'4396779'489534 2'042757'8981555 55 26'4451562'497129 2'011547 895452913446 64503582 504406 1'982528'8928480 14
6'4399392 1489894 2'041254 8980276 5427'4454167'497492 2'010080'895323413 17 4506179'504771 1'981095'8927169 13
7'4402004'490255 2'039751 18978996 53 28'4456771'497855 2'008615'8951938132 48'4508775 1505136 1'979663'8925858 12
8'4404615 1490616 2'038251 18977715 52 29 4459375 -498218 12007151!-8950641 31 191.4511372- 505501 1-9782331.8924546 11
9'4407227'4909771 2036753'8976433 51 301 44619781 498581 120056891 894934430 5045 13967'505866 1-976805 -89232341 10- C7
10 -4409838'491338 2.0352.56'8975151 50 31'4464581 1498944 2.004229 -8948045129 51 14516563'506232 1.9753781.8921920 9
1.1'44124481 49169912.033761 18973868 49 32 14467184'499308 2.002771 18946746 1252 145191581}506597 1.9739531.8920606 8
12 14415059'492061 2'032268 18972584148 33'4469786'599671 2'001314'8945446 27 53'45217531'506963 1'972529 -89192911 7
131 4417668'492422 2.030776'8971299,47 34 14472388'500035 1.999859i'8944146 26 54/'4524347'507329 11971107'8917975  6
14'4420278 4 9278312 029287'8970014 46 35'4474990'500398 1'998405'8942844 25-55 4526941 507694 19696871 8916659 55
15'4422887'4931451 2027799'8968727 45 36'44775911 500762 11996953'8941542 24561 4529535'508060 11968268'8915342 -
16'4425496'49350712.026313'8967440 44 37 14480192.501126 11995503'8940240 23571 45321281 508426 1 9668511 8914024 3a
17'4428104'493868 2-0248281 8966153 43 381 4482792'501490 1.994055'8938936 22 58145347211 508792l1 1965436'8912705 2
18'4430712'494230 2'023346'8964864 42 39'4485392 1501854 1'992608 18937632 21 59'45373131'509159 1'9640221-8911385  1
191 4433319'494592 2'021865'896357541 401 -4487992'502218 199 1 163 18936326120 60 4539905 509525 1'9626101-8910065  0
20 4435927'494954 2020386'8962285 40 I
Cosine. ICotang. Tang. | Sine.'' Cosine. Cotang.- Tang.   Sine.             C  osine. Cotang. Tang.   Sine.
Deg. 63.                                 Deg. 63.                              Deg. 63.




NATURAL SINES AND TANGENTS TO A RADIUJS 1.
27 Deg.                                    27 Deg.                                   27 Deg.
Sine.   Tang. Cotaig. Cosine.             Sine.   Tang.  Cotang.   C osine      /  Sine.  Tang.  Cotang. Cosine. 
~eJ~nq.Cuts           llg./  Codne-/~!LIS~h~._(Tn. Coting.   -Cosin. -I -1              --- -
0.4539905 599525 1-962610 8910 65 60121 -4591248 517244 1-933323 -8882166 39 11.4645845 *524640 1 906066 -8855288 19
1.4542497 509891 1'9612C0 8908744 59 2,4598 2 5176 L2 1931945 *8880830 i8 -12 4648420 *525011 1.904719'8853936 18
2.4545088 510258 11959791  8907423 58 23*4599415 517981 1'930569 8879492 37 13 *4650996 525382 1.9033731.8852584 17
3-4547679 *510625 19583831*89061.00 5724 *4601998 *518350 1'929195'8878154 36 14 *4653571 *525754 1'902029'8851230 16
4.4550269'510991 1'956978:8904777 56 25 4604580 *518719 1'927822'8876815 35 45 *4656145 *526125 1'900687'8849876 15
5[.4552859'511358 1'955573:8903453155126 -4607162.5190891 926451 8875475134 46 465 8719.526496 1.899346.8848522 14
61.4555449  511725 1954171:8902128154 27: 4609744:519458 1.925081 18874134{33 47 4661293 *526868 1.898006.8847166 13
7r.4558038 512093 1'952770'8900803153 28'4612325 *519827 1-9237131 8872793132 18 *4663866 *527240 1'896668'8845810 12
81.4560627.512460  1'951371'8899476152291'46149061'520197 1-.9223471 8871451 131 49 *4666439'52761211'895332'8844453 11
9.4563216-51282'7 1949973'8898149 51 30 14617486 1520567 1.920982 18870108 30 50 14669012'527983 1'893997'8843095 10  0C
10{.45658041513195 1'948577 8896822 50 31'4620066'520936 1-919618'8868765 29 51 14671584 1 528356 1'892663'8841736  9   0
1 1.4568392-513562 1'947182'1895493 49 32 14622646{'521306 1-918256'8867420{28 521.4674156'528728 1'891331 -8840377  8
12.4570979.513930 1.945789.8894164 48 33.4625225 -521676 1-916896.8866075 27 53.4676727 -529100 1.890000.18839017  7
1 34573566 -514298 1.944398'8892834 47 34 -4627804 1522046 1-915537 -8864730 26 54.4679298 528 9472 1888671 18837656  6
1414576153'514665 1'9430081-8891503 46 3514630382'522417 11914179'8863383 25 55 146818691 529845 11887343'8836295  5
151'4578739.-515033 1-941620'8890171 45 36 14632960 -522787 1.912823'8862036 24 56'4684439 -530217 1-886017 -8834933  4
161.4581325. 515401 1'9402331 8888839 44 37 4635538'523157 11911469 18860688 23 571 4687009'530590 11884692.   8833569  3
17'4583910'515770( 1938848'8887506 43 38j 46381151-523528f 1'910116'8859339 22 58 14689578'530963 1'883369'8832206I  2
18]'4586496'5161381 1937464 18886172 42 39'4640692,523899 1908764 -8857989 21 591 4692147-531336 1.8820471.8830841  1
191'4589080'5165061 1936082 -88S9838 41 401 -4643269 1-524269 1907414 -8856639 20 60.46947161-531709 1'880726'8829476 0
201'4591665 1516875 1'9347021 -8883503 40                                                                __.
Cosine.- Cotang. Tang.   Sine.'   t Cosine. Cotang. Tang.   Sine.  /    Cosine. Cotang.- Tang. I Sine.
ne..                         DegI  2.e..
Deg. 62.                                 Deg. 62.                                   Deg. 62.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
29 Deg.                                  29 Deg.                                 29 Deg.
Sine. I Tang.  Cotang. Cosine. / l Sine.   Tang. Cotang. Cosine.' T    Sine.  Tang. Cotang. Cosine.'|
0'4848096'554309 1'804047'8746197 60 21 4901433 562321 1'778340'8716419 39 41 4952060'570004 1'754372 8687756 19
1'4850640'554689 1-802810'18744786 59 22 4903968 5625704 1777130 8714993 3842 4954587 570389 1753186 8686315 18
2 *4853184'555069 1'801575'8743375 58 23'49065031-563087 1-775921 *8713566137 431'4957113 *57077511'752002'8684874 17
3'4855727'555450 1'800340'8741963 57 24 14909038'563471 1'774714'8712138136 441'4959639'571161 l'1750819'8683431 16
4'4858270'555831 1'799107'874050 56 25'4911572'563854 1'773507'8710710 35 45'4962165'571547 1'749637'8681988 15
5'4860812'556211 1'797875'8739137 55 26'4914105'564237 1'772302'8709281 34 46'4964690'571933 1'748456'8680544 14
6'4863354'556592 1'796645'8737722 54 27'4916638'564621 1'771098'8707851 33 47'4967215 *572319 1'747276'8679100 13
7.4865895 *556973 1'795416'8736307 53 28.4919171'565005 1'769895'870642C 32 18'4969740'57270511'746098'8677655 12
8'4868436 *557355 1'794188'8734891 52 29'4921704 -565388 11768694'8704989 31 19'49722641'573091 1'744921'8676209 11
9'4870977'557736| 1'792961'8733475 51 301*4924236'565772 |1*767494'8703557 30 501.4974787'573478 117437451 8674762 10  c
10'4873517'558117 1'791736'8732058 50 311 4926767'566156 1'766295'8702124 29 51 4977310 -573864 1     1742570-8673314  9 
11 1-4876057]'558499 1.790512 -8730640 49 321.4929298'566541 1.765097'8700691 28 521 49798331.574251 1.741396['8671866 8
12['4878597['558881 1'789289'8729221 48 33'4931829 -566925 1.763900.86992256 27 53 -4982355 -574638 1.740224'8670417 7
13'4881136-559262 1'7880671'8727801 47 34'4934359'567309 1'762705'8697821 26 54'4984877'575025 1.7390531 8668967 6
14.4883674,'559644 1.786847'8726381 46 35 -4936889'567694 11761511 [8696386 25 55 -4987399[.575412 1.737883'8667517 5
15-4886212['560026 1'785628['8724960 45 361-4939419[.568079 11760318 -8694949 24 56['4989920'57579911'736714[-'666066 4
16'4888750'560409 1.784410'8723538 44 37'4941948'568463 1.759126 -8693512 23 57 -4992441'576187 11735546 -8664614 3
17-4891288['560791 1'783194['8722116143 38 -4944476'1-568848 1'757936'8692074 22 58'1-4994961'576574 1'7343801'86631611 2
18-4893825 -561173 1'781979 -'8720693 42 39'49470051.569233 1l7567471 -8690636 21 59 499748 1 -576962 t-7332141 -8661708  1
19 -4896361 1561556 1.780765 -8719269141 40 -49495321- 569619 1.7555591 -8689196 20 66 -5000000 -577356 1.7320501.8660254 o0
20148988971561939 1.779552 -8717844 40         I:' Cosine. Cotang.1 Tang.   Sine. l- -Cosine. Cotang./ Tang.: Sine.  H'  Cosine. Cotang.: Tang.   Sine.'Deg. 60.                                 Deg. 60.                                Deg. 60.




NAT URAL SINES AND TANGENTS TO A RADIUS 1.
28 Deg.                                  28 Deg.                                28 Deg.
Sine.   Tang. Cotang. Cosine. I           Sine.   Tang. Cotang. Cosine. /          Sine.  Tang. Cotang. Cosine. 
01'4694716'531709 1.880726 8829476 60 211 4748564'539570 1-853325 18800633 39 41 -4799683 -547106 1-827799 8772858 19
1'4697284 1532082 1.879407 18828 110 59 2'4751] 124'539946 11852035 8799251 38 42'4802235'547484 1.826537 8771462 18
2'4699852'532455 1-878089.8826743 58 23'4753683 540322 11850747 8797869 37 43 4804786'547862 1'825276 *8770064 17
3'47024191.532829 1'8767731.8825376 57 24 4756242'540698 1.849461" 8796486 36 44'4807337 -548240 1'824017'8768666 1(t
4 14704986 533202 1*875458 18824007 56o25 o4758801'541074 1 848176'8795102 35 45 14809888 *54861S 1S822759 18767268 15
5 14707553  533576 1'874145 18822638 55326'4761359 *541450 1'846892'8793717 34146'4812438'54899711.821502'8765868 14
6'4710119'533950 1.872833 18821269 54127'4763917 1541826 11845609.8792332 33 47 *4814987 1549375 1.820247 8764468 11i
7'4712685 *534324 1.871523.8819898 53i28 *4766474'542202 1.844328.8790946 32 48 4817537 1549754 1-818993.8763067 12
8"4715250.534698j 1870214.8818527 52-294 *4769031 154257911 84304.9 8789559 131 49 48200861 550133. 1817740'8761665 11
91 4717815'535072 1.8689061 8817155 511301 47715881 542955 11841770 18788171 30 501 4822634 l550512 1-816489'8760263 10  C
101 47203801 535446 1'867600 18815782 50131'4774144-1543332 1 840494'8786783 29151 14825182'550891 1-815239'8758859  9
11'4722944'535820 1.866295.8814409 4932.4776700'543709 1-839218 18785394 28'52'4827730'551270 1-813990 -8757455 8
12'4725508'536195 1'864992'8513035,48133 4779255'544086 1'837944'8784004 27!53'4830277'551650 1'812743'8756051  7
13'4728071'53656'9 1'863690'8811660 4'4781810'544463 1.836671'8782613 26!54'48328241 552029 1-811496'8754645  6
14'4730634'536944 1'862389'8810284 4635 4784364'544840 1'835399'8781222 25155'4835370'552409 18 180252'8753239  5
15'4733197'537319 1'861090'8808907 45'361 47869191'545217 118341291 8779830 24,56'4837916'552789 1'809008'8751832 4
161 4735759'537694 1'859792.8807530 44137.4789472'545595 11832861'8778437 23157.48404621 553168 1.80776618750425 3
17.4738321'538069 1.858496 [8806152 43a381.4792026 [545972 1-831593 -8777043 22:581.4843007 155354811-806525 -87490161 2
18| 4740882'538444 1.857201 -8804774 421391 47945791'546350 1.830327'8775649 211591 4845552'553928 1.805286.8747607 1
19'4743443'538819 1'855908'8803394 41140'47971311 -5467281 829062'8774254120 601 4848096'554309 1'804047'8746197  C
20'47460041 539195 1'854615 -'8802014 401   C      C        T                   _                           ________
|   Cosine. Cotang.] Tang.   Sine.        Cosine. Cotang. Tang.   Sine.  t    Cosine. Cotang.  Tang.   Sine.
Deg. 61.                               Deg. 61.                                  Deg. 61.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
30 Deg.                                 30 Deg.                                  30 Deg.'  Sine.  Tang.  Cotang. Cosine.' /  Sine.   Tang. C otang. Cosine.                 Sine.  Tang. Cotang. Cosine.
0 ol5oooooo00000 5773,o  1 732050 I8660254 60 21 5052809 l585524 1*707871 s8629549 39 41 5102928.593363 1 685308 s8600007 19
1 5002519 577738 1 730887 *8658799 59 22 5055319.585914 1.706732 *8628079 38 42 5I05429'593756 1'684191'8598523 18
2.5005037 578126 12729726 *8657344 58 23 5057828 *586305 1705595 8626608 37 43 *5107930 594150 1-683076 -8597037 17
3 5007556 578514 1.728565 -8655887 57 24 5060338 *586696 1 704458  8625137 36 44 5110431i 594543 1P681962 8595551 16
4.5010073 *578902 1'727406 *8654430 56 25.5062846 *587087 1.703323 *8623664 35 45 *5112931 [594937 1.680848.8594064 15
5'50'259'1579291 1.726247'8652973 55 26 *5065355'587478 11702 189 -8622191 34 46 *5115431'595331 1'679736'8592576 14
6'5015107'579679 11725090'865I514 54 27'5067863 *587870 1'701055 ]8620717 33 47'5117930'595725 1'67S625 *8591088 13
7 5017624 *580068 1.723934 *8650055 5328 *5070370 *588261 1 699923 *8619243 3248 5120429 *596119 1.677515 18589599 12
8 *5020140.580457 1.722779 [8648595 52 29 15072877'5S8653 11698792.8617768131 49.5122927[ 596514 1'676406- 8588109[ 11
9 -5022655 |'580846 1-721626 [ 8647134 51 30'50753841 589045 1'697663.8616292 30 50'51254251 596908 1'675298| 8586619 10
101 50251701 581235 1 7204731 8645673 50 31'5077890'589436 1'6965341 8614815 29 51 -5127923 1597303 11 674192'8585127 9
11 150276855 581624 11719322 -8644211 149 32 50803961 589828 1'6954061 8613337 28 52[ 5130420'597697 1~673086'8583635  8
12.5030199.582013l 1718172.8642748 48 33' 5082901 1590221 1'694280 18611859 27 53 -5132916['598092 1'671981 18552143  7
13 15032713'582403 1.717023'8641284 47 34 5085406 1590613 1.693155 18610380 26 54 i5135413 [598487 1'670878'8580649 6
141 5035227.582793 1.715875 18639820 146 35.5087910 591005 1l692030'8608901 25 55[ 5137908'598882 1'669775'8579155  5
15 15037740'583182 1.7147281 8638355 45 36.5090414'591398 1'690907. 8607420 24156'51404041 599278 1'668674'8577660 4
16'50402521 583572 1'713582'8636889144 37'5092918 1591791 1'689785'8605939123 571 5142899.599673 1.667574.8576164 3
171'50427651-583962 1.712438 18635423 43138|'5095421'592183 1.688664'8604457 22158'5145393'600069 1'666474'8574668[    2
18'5045276'584352 1.711294 18633956 42 39, 50979241 592576 1.6875441 8602975 21 59 15147887 1600464 1'665376 l85731711 1
19'5047788.584743 1.7101521 8632488 41 40 5100426'592969 1.686426.86014911201601 5150381 1600860 11664279 85716731 0
20. 5050298'585133 1.709011. 8631019 40                            
-']Cosine ICotang.I  Tang.   - _ine. I 1_ I'-I-__ s-I-I —' Cosine. |Cotang. Tang.   Sine.          Cosine. Cotang  Tang.   Sine.             Cosine. Cotang  Tang. I Sine.
Deg. 59,                               Deg. 59.                                 Deg. 59,




NATURAL SINES AND TANGENTS TO A RADIUS 1.
31 Deg.                                  31 Deg.                                 31 Deg.'  Sine.  Tang. C Cotang. Cosine.           Sine.   Tang. Cotang. Cosine.           Sine.   Tang. Cotang. Cosine. 
0'5150381 -600860 1'664279 18571673 60 21 [5202646 1609205 1 641482'8540051 39 41 5252241'617210 11620192'8509639 19
1 5152874 601256 1'663183'8570174 59 22 5205130'609604 1640408'8538538 38 42 5254717 6176712 1619138'8508111 18
2'5155367'601452 1'662088 18568675 58 23'5207613'610003 1'639335 18537023 37143'5257191 t618014 1'618085 -8506582 17
31 5157859 1602049 1'660994 *8567175 57 24. 5210096'610402 1'638263 18535508 36 44'5259665 *618416 1-617033'8505053 16
4'5160351'602445 1'659901 8565674156 25'5212579.610801 1'637191'8533992 35 45'5262139'618818 1615982'8503522 15
5'5162842'602841 1'658809'8564173 55 26'5215061'61120111 636121'8532475 34 46'5264613'619221 1'614932'8501991 14
6'5165333'603238 1'657718'8562671 54 27'5217543'611601 1-635052'8530958 33 17 5267085'619623 1'613882'8500459 13
7 51678241'603635 1'6566291 8561168 53 281.5220024.612000 1.633984 *8529440 32 8!.5269558.620026 1.612834 8498927 12
8s 5170314 4604032 1.655540 18559664152 29 5222505'612400 1.632917 18527921 31 91 52720301'6204291 1611787 18497394 11
9'51728041'604429 11654452.8558160 51 30l 5224986'612800 1631851 8526402 3050 5274502 1620832 1 6107411 8495860 10   S
101 5175293.604826 1.653366 8556655150 31 15227466 16132011 1630786'8524S81295 1, 5276973 621235 1'6096961'8494325 9
11 -5177782 -605224 1'652280 -8555149 49  2 -52299451 613601 L[629722 -8523360 28 5215279443'621638 1'608652-8492790  8
12'51802701'605621 1'6511961 8553643148 331'5232424'614001 1'6286591 8521839 27153'5281914'622041 1607609'84912541 7
13t'51%2758'606019 1'650112'8552135147134'5234903| 61440211 627597'8520316126 54 5284383'622445 1'6065671'8489717 6
14- 5185246- 606417 1'649030'855062746 35'5237381'614803 1626536'8518793 2 555'5286853- 622848 1 605526'8488179  5
15/'51877331'606814 1'6479491 8549119 45136'5239859'615204 11625476'8517269 24156     65289322'623252 1'604485'8486641  4
16'5190219'60721311'6468681 8547609 44 37 5242336'615605 11624417.8515745 2357'52917901 623656 1'603446'8485102 3
171 51927051'607611 11645789'8546099 43 381 5244813' 616006 11a623359'8514219 22[58'5294258'6240601 i602408'8483562 2
181 51951911 608009 1'644711'8544588 42 9'5247290'l616407 11622302'8512693 211591 5296726 162446511'6013701'8482022  1
19'51976761'608408 1'6436331 8543077 41 401'5249766 1616809 1'621246'8511167i20 60'5299193'624869 11600334'8480481  0
201.5200161 1608806 1.642557.8541564140' osine. Cotang. Tang.    SinSine.                                              Cosie. Cotanl CoTang.   Sine.g.  S i
Deg. 58.                                 Deg. 58,                                 Deg..58
Deg. 58-                               Peg, 68,                                  Deg. 58.




NATURAL SINES AND TANGENTS TO A RADItUS 1.
M  Deg,                                  32 Deg.                                  32 Deg.
/   Sine.   Tang. C(otang. Cosine. /  Sine.   Tang.  Cotang. Cosine.''   Sine.   Tang.  Cotang. Cosine.  f
0 *5299193 1624869 1-600334 -8480481 60 21 5350898.633395 1-578791'8447952 39 41 5399955 641577 1-558657'8416679 19
1 15301659 *625273 1-599299 18478939 59 22.5353355.633803 1'577776 -8446395 38 42'5402403 641988 11557660'8415108 18
2 -5304125'625678 1.598264 8477397 58 23 -5355812'634211 1.576761'8444838 37 43 15404851'642399 1'556663'8413536 17
3.5306591.62608311.597231 *8475853 57 24 *5358268 *634619 1'575747 *8443279 36 44 5407298 64281 0 1-555668'8411963 16
4'5309057 -626488 1'596198 18474309 56 25'5360724 635027 1'574735'8441720 35 45 5409745]'643221 1'554674'8410390 15
51 5311521'626893 1'595167 18472765 55126'5363179'635435 1-573723'8440161 34 461 5412191 643632 1'553680 8408816 14
6'5313986'627298 11594136'8471219 54 271 5365634.635844 1-572712.8438600 33 471 5414637'64444 4 1.552688 *8407241 13
7'5316450'627704 1.593107'8469673 53 28 5368089'636252 1.571702'8437039 32 48 5417082. 644456 1.551696 84056666 12
81.5318913.628109 1.592078 -8468126 52 29j.5370543'636661 1.570693.8435477 31 49.5419527 644867 11550705 18404090 11
9'53213761-628515 1-5910500 8466579 51 301 5372996 6370707 1569685.8433914 30 50 -5421971.645279 1.549715'8402513 10:
101 53238391'628921 115900231 8465030 50131 15375449 1637479 1.5686781 8432351129 51 154244151 645691 11 548726 18400936  9
11-53263011 629327 11588997 8463481 49 321 5377902'637888 1'567672'8430787 28152'5426859'646104 1'5477381'8399357 8
12- 5328763~ 629733 1'587973'S461932148 33'5380354'638297 1'566666'8429222 27 53'54293021'646516 1'546751'8397778  7
13 -53312241-630139 11556949 18460381 147 341 53828061 63870711'565662'8427657 26 54!'54317441 646929 1'545764'8396199  6
[14'15333685'630546[ 1'585926  8458830 146 35 5385257'639116 1'564659  8426091 25 55 554341871'647341 1'544779 18394618, 5
15 15336145'630953 1'584904'8457278 145 36'5387708'639526 1'563656'8424524 24 56'5436628 1647754 1'5437941 8393037 4
161-5338605-'631359 1-583883'8455726 44 37 -5390158[ 639936 1.5626541 8422956 23 57'5439069{ 648167 11542810;83914551   3
171 5341065'631766 1'5828621'8454172 43 38 -5392608!640346 1'561654 -8421388 22 58'5441510'648580 1-541828'8389873 2
181[5343523'632173 1-581843 5 8452618142 39 -539o058'6400756 15606541 8419819121 159'54439511 648994 11540846.83882901 1
19| 5345982| 632581 1580825'8151064 41 40 -5397507 -641167l 1559655  -841824920 601-5446390 e649407 1 539865'8386706  0
20'5348440 63298811 579807'84495058 40.    _.        ____
1  _____1_1_.-I I- -- ___________                                             I -        I -I I1
Cosine. Cotang. Tang.    Sine.           Cosine. lCotang. Tang. I Sine. I'    Cosine. Cotang.  Tang. I Sine. 
Deg,. 56.7.                              Deg. 57.                               Deg. 57.




NATURAL SINES AND TANGENTS TO A RADIIJS 1.
33 Deg.                                  33 Deg.                                   33 Deg.
Sine.   Tang. Cotang. Cosine.           Sine.   Tang. Cotang. Cosine. /           Sine.   Tang. Cotang. Cosine. 
01 5446390 -649407 1539865 8386706 60221- 54975220 658127 1519463 -8353279 39 41 5546024 G66496 1'500382 -8321155 19
1'5448830'649821 1 538884'8385121 59122 5499950 1658544 1'518501 18351680 38 2 5548444 666917 1'499436 *8319541 18
21 5451269'650235 1'537905 8383536 58123 5502379 658961 1-517540'8350080 37 43'5550864'667337 1498492'8317927 17
3 6453707 650649 1'536927'8381950 57 24'5504807'659378 1 516579'8348479 36 14'5553283 667758 1'497548'8316312 16
41 5456145'651063 1-535949 8380363 56 25'5507236'659796 11515620'8346877 35 45'5555702'668178 1496605'8314696 15
5 5458583 651477 1-534972:8378775 55 26'5509663'660213 1 514661'8345275 34 16'5558121'668599 1 495663 8313080 14
61 5461020 651891 1'533996 *8377187 54 27 5512091'660631 1513703'8343672133 71 5560539'669020 1'494722 8311463 13
71.5463456 6523061 1533021'8375598 53 28'5514518'661049 11512746'8342068 32 48'5562956'669441 114937821 8309845 12
8- 5465892'652721 1532047'8374009 52 29'5516944'661467 15117901 8340463131 49'55653731669863 1'492842- 8308226 11
91 5468328'653136 1531074'8372418 51 30 15519370'661885 1l510835 18338858 30 50 15567790 670284 11491903 8306607 10
10- 5470763 653551 1.530102'8370827 50 31 -5521795 1662304 1'509880'8337252 29 51 -5570200 -670706 1'490965'8304987  9
11'5473198 653966 11529130'8369236149 32'5524220'662722 1508927.8335646128 52l 5572621 671128 1'4900281 8303366  8:
121'54756321-654381 1.528160'8367643148 33'5526645'663141 1-5079741 8334038127 53 15575036'671550 14890921'8301745  7
131'54780661 654797 1'527190( 8366050 47 341 5529069'663560 1'507022'8332430 26 541 55774511 671972 14881571 8300123  6
141'5480499'655212 1'526221 18364456 46 351 5531492 1663979. 15060711 8330822 25 55'5579865'672394 14872221 8298500  5
151 5482932'655628 1'5252531 8362862 45 36'5533915 1664398 1'505121'8329212 241561 5582279'672816 1'486288 8296877 4
16'5485365'656044 1.524286'8361266 44 371.5536338'664817 1.504171 18327602 23 57'55846921 673239 1'4853551 8295252  3
17 -5487797'656460 1       4523320j 8359670 43 381'55387601*665237|1'503222 -8325991 22 58 -5587105 673662 1'4844231|8293628  2
18'5490228 656877 1'522354'8358074142 39'5541182'665657 11502275'8324380 21 59'5589517 6'74085 1'483491'8292002  1
19'5492659'657293 1521389'8356476 41 401 5543603'666076 1'501328'8322768 2o 60 5591929'674508 1'482561'8290376  0
20'54950901 657710 1520426 -8354878 40
Cosine. Cotang. Tang.    Sine.   /        Cosine. Cotang. Tang.    Sine.           Cosine. Cotang. Tang.   Sine. 
Deg. 56.                                Deg. 56.                                  Deg. 56.




NArURAL SINES AND TANGENTS TO A RADItS 1.
14 Deg.                                  34 Deg.                                 34 Deg.
/  Sine.  Tang.ang.    otang. Cosine. /    Sine.  Tang. Cotang. Cosine. / /   Sine.  Tang.  Cotang. Cosine. /
0 *5591929 674508 1I482561'8290376 60 21 *5642467'683433 1463200 *8256062 39 1'569040' *692002 1'445081 18223096 19
11 5594340 674931 1'481631 8288749 59 22.5644869'6fi83860 1'462287'8254420 38 42'56927)95 1692432 1'444183'8221440 18
2'5596751 *675355 1'480702 *8287121 58[23'5647270 *684287 12461374 18252778 37 43 5695187'692863 1'443286'8219784 17
31'5599162 1675779 1'47977318285493 57 24 5649670 *684714 1.46046f3.8251135136 44 5697577. 693293 1'442389 *8218127 16
4 5601572'676202 1.478846'8283864 56 25 15652070 -685141'459552 18249491 35 5'56999681'693724 11441494'8216469 15
5'5603981 1676626 1'477919 18282234 55 26 -5654469'685569 1'458642 *8247847 34 6'5702357'694156 1'440599'8214811 14
6'5606390'677050 1.476993'8280603 54 27'5656868'685996 1'457732'8246202 33 h'5704747'694586 1-439704 18213152 13
7'5608798'677475 1.476068 8278972 53 28 5659267'686424 1'456824'8244556 32 8'5707136'695018 1'438811 18211492 12
8'56112061 677899 1-475144'8277340 52129 15661665'686852 114559161.8242909 31 49 57095241'695449 1 4379181 8209832 11
9'5613614'6783241 1474221'8275708 51 301 5664062'687281 1'455009'8241262 30560'57119121'695881 1'437026['8208170 10 0
10'56160211 67874911.4732981 8274074 50 31'5666459'687709 11454102'8239614 29 51'57142991'696313 114361351 8206509  9  CT
11 15618428 1679174 1.472376 18272440 49 32 15668856.688137 1.453197'8237965 285I21 57166861'696745 1'4352451'8204846 8
12'56208341 67959911'4714551 8270806 48 331 5671252'688566 1'452292'8236316 27531'57190731'697177 1'4343551'8203183 7
13'5620239'680024 1'470535 8269170 47 34.56736418 688995 1'451388'8234666 2654'5721459'697609 1'433466'8201519  6
141'56256451'680450 11469615'8267534 4635'5676043'689424 1'450485'8233015 2555'5723844.698042 1'4325781 8199854 5
15'56280491 680875 1'468696 18265897145.36'5678437'68985311'449582 8231364124156 57262291 69847411.431690J'8198189 4
1656630453 681301 1.467778 8264260 4437 15680832 -690283 1.448680. 822971212357'57286141.698907 1.4308038196523  3
17'5632857168172711.4668611 8262622 43 381.5683225'690712 1.447779 8228059122581 5730998 169934011.4299171 8194856 2
18'5635260'682153 11465945'8260983 42 39'5685619'691142 1'446879'8226405 2159- 5733381'699774 1.429032'8193189  1
19 5637663'682580 1'465029'8259343 41 40'5688011'69157211'445980'8224751120160'5735764'70020711.428148'8191520  0
20'5640066 1'683006 1.464114 8257703 40
j Cosine. lCotang., Tang.                         Sine.'                         osine.  Cotang. Tang.   Sine.
Deg. 55.                                Deg. 55.                                Deg. 55




NATURAL SINES AND TANGENTS TO A RADIUS 1.
35 Deg.                                  35 Deg.                                 35 Deg.
Sine.'rang.  Cotang. Cosine.'           Sine.   Tang.  Cotang.  Cosine'       Sine.   Tang. Cotang. Cosine. 
0'5735764'700207 1'428148'8191520 60 21 5785696'709350 1'409740'8156330 3941'.833050'718131 1'392501'8122532 19
1 5738147'700641 1 427264'8189852 59 2'5788069'709787 1'408871'8154647 38 42'5835412'718572 1'391647'8120835 18
2 *5740529'701074 1'426381 -8188182 58 3 5790440 *710225 1'408003 *8152963 37 43'5837774'719014 11390793.8119137 17
3 *5742911.70150811.425498'8186512 5 24 5792812.71066311407136'815127836 44 5840136 *719455 11389940'8117439 16
4 *5745292 701943 11424617'8184841 56 5.5795183'711100 11406270'8149593 35 45 5842497.719897 11389087'8115740 15
5'5747672'702377 1423736'8183 169 5526 65797553'711539 1'405404'8147906134 46 5844857'720338 1 388235 18114040 14
6 (5750053'702811 1'422856'8181497 5427 -5799923 -711977 1'404539'8146220 33 47.5847217'720780 1.387384'8112339 13
7 5752432'703246 11421976'8179824 53128, 58022921 712415 11403674'8144532 32 48 5849577.721222 1-386534'8110638 12
8'5754811'703681 1421097'8178151 52 29.5804661 1712854 1.402811'8142844[31 49 5851936.721665 1.385684'8108936 11
9'5757190'704116 11420220'8176476 51 301 5807030'713293 11401948'8141155 30150'5854294['722107 1'384835'8107234 10   C4
10- 5759568'70455111'4193421'8174801 50 31'58093971 713732 11401086'8139466 29 5115856652.'722550 1'3839861 8105530 9
111 5761946'704986 11 418466'8173125 49132.5811765'714171 11400224'8137775128 521 5859010.722993 1.383139.8103826  8
12'-5764323.705422 1.417590'-8171449 48 33'5814132 1714610 11399363'8136084 27 581.58613671'723436 1.382292'8102122  7
13 5766700'705858 1'416715'8169772 47 34 5816498'715050501'398503'8134393 26 545863724'723879 1.3814451 8100416 6
14'5769076 1706294 1'4158401 8168094 46 351 5818864'715489 1.397644'8132701 25 555866080 724322 1380600 8098710  5
1515771452'706730 1'414967'8166416 45 361 5821230'7159291'396785 18131008 2456t'5868435'724766 1-379755'8097004  4
16'57738271 707166 14140941 8164736 44 37 15823595 -716369 11 395927.812931423 571 5870790 725210 1.378910.8095296 3
171 5776202'707602 1'413222'8163056 43 38 -5825959'716810 1'395069'8127620 22 8. 5873145'725654 1.3780671'8093588 2
Deg. 54.                                Deg. 54.                                 Deg. 54.
18 5778576'708039  1l412350]'8161376 4239'5828323'717250 1.394 213'8125925`2198 97 9             32'1'0'8 l691 9 1 86 7672'2 9 6 5 8 75 8 7'9 0
j  Cosine. Cotang. Tang.   Sine.'' Cosine./Cotang. Tang.   | Sine.'' Cosine. Cotang. Tang.    Sine.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
36 Deg.                                36 Deg.                                36 Deg.
Sine.   Tang. Cotang. Cosine.!  Sine.   Tang. Cotang. - Cosine. -! i    Sine.  Tang. I Cotang. Cosine. 
05877853 726542 11-376381 1-8090170 0160 1 -59271631 -735971 1-358848 -8054113 391 1-5973919 -7449241 1-342417 -8019495119
11 -5880206 -726987 1-375540 8088460 59 22 -5929505 -736366 1-358020 -8052389 3842 -5976251 *745377 1-341602 -8017756 18
2 -5882558 -727431 1-374699 -8086749 5 23 -5931847 -736814 1-357193.8050664137 43.5978583'745829 1-340788 -8016018 17
3 -5884910 -727876 1-373859 -8085037 57 24 -5934189 -737263 1L356367 -8048938 36 44 -5980915 -746282 1-339975 -8014278 16
4 -5887262 -728321 1-373019 -8083325 56 25 -5936530 -737712 1-355541 -8047211 35 45 -5983246 -746735 1-3391621-8012538 15
5 5889613 -728767 1-372180 -8081612 55]26 -5938871 -738162 1-354716 -8045484 34 46 -5985577 -747188 1-3383501-8010797 14
6 -5891964 -729212 1-371342 -8079899 5427 -5941211 -738611 1-353891 -8043756 33 47 -5987906 -747642 1337538-8009056 13
7 -5894314 -729658 1-370504 -8078185 5328.5943550 -739061 1-35'3068 -8042028 32l48 5990236 -748095 1-336727 -8007314112
8 5896663 730104 1-369667 -8076470 52129.59458891'739511 1-352244 -8040299 3149 -5992565 -748549 1-335917 -8005571 11
9 -5899012'1730550 1-368831 -8074754 51130.5948228 -739961 1-3514221 8038569 30[501 59948931 74900311-3351071 -8003827!10
10 o5901361 -7309961 13679951 8073038 50311.59505661-740411 113506001 8036838 29o51 -5997221 -7494571 1334298 -8002083 9
11 -5903709 -731442 1-367161 -8071321 49j321.5952904-'740861 1-349779 -8035107 28]52 -5999549 -749911 1.333490.8000338 8
12 -5906057 -731889 1-366326 -8069603 48331 -5955241 -741312 1-348958j-8033375127 53 -6001876 -750366 1.3326821.7998593  7
13 -1908404 -732336 1-365493 -8067885 47j341 5957577 -741763 1-348139- -8031642 26 54 -6004202 -750821 113318751-7996847 6
141 5910750 -732783 1-364660 -8066166 46j35.5959913 -74221411-347319 -8029909 25155 -60065281 -751276 1-331068 -79951001 5
151 -5913096 -733230 1-363827 -8064446 45 36 -5962249 -742665 1-346501 -8028175 24156 -160088541-751731 1-330262 -7993352  4
161 -5915442 733677 1-362996 -8062726144 37 -5964584 -7431171].345683 -8026440123571 -60111791 -752186 1-329457 -7991604 3
171 59177871-734125 1-362165]'8061005143 381-5966918 -743568 1.344865!'80247051221581 6013503 -r5264211-328652 -79898551 2
18 -5920132 -734b573[ 1-361335 -8059283 42i391-5969252 -74402011.344049 -8022969 21 59 -6015827 -753098 1-327848 -7988105  1
191 -59224761-735021 1-360505 -8057560 4140l -5971586 -7444721 1.343233 -8021232 20 60 -6018150 -753554 1-3270441-7986355 0
20-.5924819 735469 1-359676 -8055837 40J                                                 I
1  -  I        -1       -I       I,       -I     — I     — I       -  II       —. -!    I        I       -IIt                                        C Cosine. C Totan g. Tang.   Sine.'  Cosine. Cotang. Tang. I Sine.
Deg. 53.                              Deg. 53.                                Deg. 53.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
37 Deg.                                 37 Deg.                                 37 Deg.
Sine.   T                                        aan.  Cotang. Cosine. I   Sine.  Tang.  Cotang. Cosine   Sine.  Tang. Cotang. Cosine.
0'6018150 *753554 1 327044'7986355 60 1'6066824'763175 131031.4 7949444 39 41'6112969'772423 1'294627'7914014 19
1'6020473'754010 1-326242 -7984604 59 2 16069136 *763636 1-309523 *7947678 38 42 6115270'772887 1'293848'7912235 18
2'6022795'754466 1 -325439'7982853 58123'6071447'764096 1'308734'7945913137 143 6117572'773352 1'293071'7910456 17
3.6025117 *754923 11324638'7981100 57 24 *6073758'764557 1'3079451'7944146,36 441'6119873 1773817 1'292294'7908676 16
4'6027439'755379 1.323837.7979347 56 25'6076069'765018 11307157 -7942379135 45 16122173 *774282 1.291517'7906896 15'6029760'755836 1'3'23036'7977594 55 26'6078379'765480 1'306369'7940611 134 46 6124473'774748 1'290742'7905115 14
i 6'60320801 7562941 1'322237'7975839 54127'6080689'765941 1'305582'7938843133 47'6126772'775213 1289966] 7903333 13
7'60344001756751 1'321437'7974084 53 28'6082998'766403 1;304796'7937074 3248 6129071. 775679 1.2891921 7901550 12
8'6036719'757209 1'320639 r7972329 52 29 d 6085306'766864 1304010'7935304 31 49.6131369'776145 1'288418'7899767 11
9'6039038'757666 11319841'7970572 51 3  6087614'7673271 1303225 7933533 30 50 6133666'77661111 287644'7897983 10
10'6041356 758124 1'319044'7968815 50 31'6089922'767789 1.302440.7931762 29 11 6135964.777078 1.286871'7896198 9
11'60436741 758582 11318247'7967058149 321.6092229'768251 1'301656'7929990{28 52'6138260'777544 1.286099'7894413 8
12'6045991 1759041 1'317451 17965299 48]331 6094535 1768714 1'300873'7928218 127 5316140556 1778011 112853271'7892627 7
13'6048308'759499 11316655'7963540 4734'6096841'769177 1'300090'7926445 2654 -6142852'778478| 1'284556'789Q841 6
14 -60506241 759958 11315861'7961780 46351 60991471 7696401.299308'7924671 2 55 16145147'778946 1'2837861 7889054 5
15 -60529401 760417 1.315066'7960020 45 6.6101452 77010 11298526 7922896 24 5616147442 7794.13 12830161788   7266 4
16'6055255'760876 1'314273 17958259 44137 6103756' 770567 1'297745'7921121 2357 61497361 779881 112822461 7885477 3
17 -60575701.76133611.313480'7956497 43 38'6106060.771030 1.296964 17919345 22 581 61520291 780349 1.2814771 78836881 2
18'60598841 761795 1.312687.795473542 39'6108363'771494 1'296185'7917569 21 59'6154322'7808171 12807091 7881898  1
191 6062198 1-762255 1.3118951.7952972{41 40'6110666 1771958 11295405'7915792 20 60'61566151'781285 1'279941'7880108 0
20'60645111 762715 1.311104i'7951208{40.   j                I         {.'_
_ Cosine.,Cotang., Tang. 1 Sine.  I'     Cosine. Cotang.{ Tang.   Sine.  I'   Cosine. Cotang.' Tang.   Sine.
Deg. 52'                               Deg. 52.                                Deg. 52.




NATURAL SINES AND TANGENTS TO A RADITJS,1.
38 Deg,                                  38 Deg.                                  38 Deg.
Sine.   Tang. Cotang. Cosine.           Sine.  Tang  Cotang. Cosine.''    Sine.   Tang. Cotang. Cosine.
0 o6156615'781285 112799-4i'7880108 6C 21 *6204636 *791170 1-263950'7842352 39 41 6250156'800673 1248948 7806123 19
11 6158907'781754 1'279174 7878316 59 22'6206917'791643 1*263195'7840547 38 42 6252427'801151 1'248204'7804304 18
2'6161198'782222 1-278407'7876524 58 23'6209198 792116 1-262440'7838741 37 3'6254696 1801628 1-247460'7802485 17
31 6163489'782691 11277641'7874732 57124 -62114781.79259011.261686.7836935J 364 6256966 802106 1-246716 7800665 16
4'6165780'783161 1*276876 17872939 56 25 *6213757'793064 11260932*i7835127 3  5'6259235 802584 1 245974'7798845 15
5'6168069 -783630 1S276111'7871145 55 26'6216036.793537 1.260179 -7833320 34 46 *6261503 803063 1-245232'7797024 14
6'6170359'784100 1'275347'7869350 54 27'6218314'794012 1'2594261 -7831511 33 47 *6263771 *803541 11244490'7795202 13
7'6172648'784570 1-274583'7867555 153 28'6220592 1794486 1-258674 17829702 32 48'6266038 1804020 1-243749'7793380 12
8'6174936'785040 11273820'7865759 52 296222870.79 1961 11257923 -7827892 31 49 -62683051 804499 1'243008 7791557 11
91'6177224'785510 1'273057'7863963 51] 301-6225146'795435 11257172 17826082130 50 16270571 1804979 1'242268'7789733 10  C
10-'6179511'78598011'272295'7862165 50131'62274231 795911 1'2564211-7824270 29151 16272837l 805458 1'241529'7787909  91 Z
11'6181798 1786451 112715341 7860367149 32'62296981 796386 1'255672"17822459 28 52'6275102'805938 1 2407901 7786084 |
12 e6184084'786922 1'270773'7858569 48 33'6231974'796861 1'254922 -7820646 27 53'6277366 -806418i 11240051'7784258 7
13r 6186370o.787393 1.270013'7856770 47 34'6234248 -797337 1.254174 -7818833126 541'6279631 -806898 1.239313 17782431  6
141 6188655 -787864 15269253'7854970 46 351 6236522 -797813 1.2534261 7817019 2.5 55'6281894'807378 1'238576'7780604  5
151-6190939'788336 11268494 7853169 45 36'6238796.798289 1'252678.7815205 24 56 16284157'807859 11237839 7778777  4
161I'6193224 788808 1'2677351'7851368144 37'6241069.798765 1.25193l 1-7813390 23 571 6286420l -808340 1'2371031-'77769491 3
171'6195507]'789280 1-'266977!'7849566 43 38'6243342.79924211'2511841'7811574122 58'6288682'808821.! 1'2363671'7775120. 2
1 8.6197790'789752 112662191 7847764 42 39 -6245614'799719 1-250438 17809757 21 59'6290943 |809302 *1235631'7773290  1 
19 1-6200073]-790224 1'265462'7845961 41 40'6247885'800196 1'249693.7807940 20 601'6293204 -809784 1'234897,'7771460  0!
201'6202355['790697 11264706 -7844157 40                     Tag    S           I            C        T        Sn   I 
ll|Cosine. Cotang. Tang. I Sine.'  Cosine. Cotang.l Tang.   Sine. — lCosine. lCotang. Tang. | Sine.*Deg. 51.                                Deg. 51.                                  Deg. 51.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
39 Deg.                                   39 Deg.                                 39 Deg.
i    Sine.   Tang.  Cotang. Cosine.          Sine.   Tang. C otang. Cosine.   /   Sine.   Tang. C otang. - Cosine.'
0 6293204'809784 1'234897'7771460 60 21 "6340559 819948 1-219588 *7732872:39 41 6385440 *829724 1.205219'7695853 19
1'6295464.810265 11234162 17769629 59 22 *6342808 1820435 1-218865 17731027 38 42'6387678 830216 11204505 17693996 18
2 [6297724 *810747 1.233429 *7767797 58 23 *6345057'8'20922 1.218142 17729182 37 43 6389916 830707 11203793 *7692137 17
3!.6299983 1811230 1.232696.7765965 57 24 16347305 1821409 1.217419.7727336 36 44 *6392153 -831199 1.203081.7690278116
4 *6302242'811712 1-231963.7764132156125 *6349553.821896 1-216698'7725489 35 45 16394390. 831691 1-202369'7688418 15
5'6304500.812195 1i'231231.7762298 55 26 16351800 1822384 1-2 15976 17723642 34 46 *6396626.832183 1.201658 57686558 14
6'6306758'812678 1'230499.7760464 54 27'6354046'822871 1'215256'7721794133 47 16398862.832675 1'200947 7684697 13
7'6309015 1813161 1'229768'7758629 53 28'6356292'823359 1'214535'7719945 32 48'6401097 *833168 1'200237'7682835 12
8'6311272 1813644 1'229038'7756794I52 29'6358537 1823847 1-213816'7718096131 49'6403332 [8336611 1 199527'7680973 11
9 163135281 814128 1'228308 -7754957 51 0 06360782'824336 1*213097 17716246 30150 -6405566 1834154 11198818 17679110 10 
10'6315784'814611 11227578'7753121 50 31 16363026 -824825 1-212378 -7714395 29 51'6407799 1834648 111981091'7677246  9
1 16318039 815095 1226849 17751283 49 32'63652701 825314 1-2 11660 17712544 28 521 6410032 8351411 1197401 17675382 8
121 63202931 815580 1'226121'7749445 48 33'6367513.82580311 210942'7710692127 531 64122641 835635 1'1966931'76735171 7
131 6322547'816064 1'225393'7747606 47 34'6369756'826292 11210225'17708840 26 54 16414496'836129 1'195986'7671652  6
14'63248001.816549 11224665'7745767 46 35 16371998'826782 11209508 17706986 25 55 164167281 836624 1'1952791 7669785 5
15{'6327053'817034 1'2239381'7743926 45 36 -6374240'827271 1'208792'7705132 24 56'6418958'837118 111945731'7667918 4
16 16329306'81751911.2232121 7742086 44 37.6376481 -827762 112080761 7703278 23 57'64211891 837613 1.1938671 7666051  3
171 63315571 818004 11222486'7740244 43 381 6378721'828252 1.2073611.7701423 22 58'64234181 838108 1.1931621 7664183  2
181 63338091 818490 11221761'7738402 42 39'6380961'828742[ 1-2066461,7699567 21 591 64256471 838604 111924571 7662314  1
191 6336059'818976 11221036 17736559 41 401 6383201'829233 112059321 7697710 20 60'64278761 83909911. 191753176604441 0
20 16338310'819462 1'220312'7734716 40
J Cosine. ICotang,l  Tang.   Sine.  I  / Cosine. Cotang.! Tang.   Sine.   t    Cosine. Cotang.  Tang. { Sine.  t
Deg. 50.                                 Deg. 50.                               Deg. 50.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
40 Deg.                                  40 Deg.                                  40 Deg.
t  Sine.   Tang. Cotang. Cosine.'!    Sine.   Tang. I Cotang. Cosine. i _       Sine.   Tang. Cotang. Cosine.'
0 6427876 839099 1191753'7660444 60 21'6474551'849563 1'177075'7621036 39 41 *6518778'859629 1.163291 7583240 19
1'6430104 839595 1.191049'7658574 59122'6476767 *850064 1'176382'7619152 3842 6520984 *860135 11 62607'7581343 18
2'64323321 840091 1.1 90346'7656704 58 23'6478984 *850565 1117568 7617268 37 3 6523189'860641 1'161923'7579446 1 7
3.64345591'840587 1.189643 -7654832 57 24 16481199.85106611.1749961.7615383 36 44 6525394.861148 1.161240'7577548 16
4'6436785'841084 1.188941'7652960 56 25'6483414.851568 1'174303'7613497 35 45 6527598.861655 1'160557'7575650 15
5 6439011'841581 1'188239'7651087 55 26'6485628 852070 11173612 -7611611 13446 *6529801'862162 1-159874 7573751 14
6 6141236'842078 1'187538'7649214 54 27'6487842'852572 11172920'7609724 33 471 6532004 862669 1'1591921 7571851 13
7'6443461 1842575 11186837'7647340 53 28 S6490056.853075 1'172229 17607837 32 48 6534206.863176 1.158511 17569951 12
81'64456855 843073 11861361 764546515229'6492268i'85357711171539'7605949 31 491 6536408 863684 1'157830 -7568050 11
9 1.6447909'843570 1185437'7643590 51 30'6494480 1854080 11170849 176040601 30 501 6538609 -864192 1'157149 7566148 10
tl 0'64501321'844068 1'184737'7641714 50 311 64966921 S545s3 111701601 7602170 29 51 -6540810.864700 11156469. 7564246  9
111.64523551844567 1184038'7639838 49 321 64989031855087 1.169471.760028028 5216543010'865209 11557891 7562343  8
121.64545771 845065[ 1.183340'7637960148 33'65011141-855591 1'168782'7598389 27 53 6545209 -865718 1'155110'7560439  7
1431 64567981 8459646 1182642'7636082 47 34'6503324-'856095 1'1680941 7596498 26 541 6547408'866227 1'154431'7558535  6'14'6459019l-84606311'181944'7634204146 35'6505533'856599 11167407'7594606 25 55'6549607'866736 1'153753'7556630  5
115'6461240'846-562 1181247'7632325 45 36'6507742'-5710311'1667201 7592713 24 56 }6551804'86724611'153075'7554724 4
1161'64634601'847062 11180551'7630445144 371-6509951,85760811,166033.7590820 23 57'6554002, 867755 1 152397. 7552818  3
117| 64656791847561 11179855'7628564t43 38'*6512158|-858113  1'165347.17588926 22 58 16556198'868265 1'151721'75509111 2
18 -646'7898-848061 1-179159 -7626683142 39'6514366 [858618 1'164661 -7587031 21 59 -6558395 -868776 1151044 7549004  1
19'6470116'848561 |1'178464 -7624802141 40'6516572'859124 1'163976'7585136120|60'6560590'869286 1150368'-7547096 0
20'6472334'849062 1-177769'7622919 40
t Cosine.Cotang. Tang.  Sine tCosine.  Cotang. Tang.   S  Sine.    a   T        /   cosine. Cotang. Tang.   Sine. 
Deg. 49.                                   eg. 49.  Deg. 49.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
41 Deg.                                  41 Deg.                                 41 Deg.
/  Sine.   Tang. Cotang. Cosine.'   Sine.   Tang. Cotang. Cosine.'    Sine.'rang.  Cotang. Cosine. 
0 6560590 1869286 1-150368'7547096 60 21 *6606570.880068 1.136274'7506879 39 41 *6650131'890445 1123032.7468317 19
1 6562785'869797 1'149692'7545187 59 22 *6608754'880585 1-135608'7504957 38 42'6652304 890967 1'122375'7466382 18
2'6564980'870308 11149017'7543278 58 23'6610936'881101 1'134942 17503034137143'6654475'891489 1'121718'7464446 17
3'6567174'870820 11148342 17541368 57 24'6613119 1881618 1'134277'7501111 36144'66566461 892011 1'121061!7462510 16
4 6569367 871331 1'147668 -7539457 56 25 6615300'882135 1-133612'7499187 3545'6658817'892534 1'1204051'7460574 15
51 6571560'871843 11146994 *7537546 551261 6617482 1882653 1'132947'7497262 34 46'6660987 1893056 1-119749'7458636114
6 6573752'872355 1'146321'7535634 54 7.6619662 1883170 1.1322831.7495337 33 47 6663156'893579 1'119094'7456699 13
7'6575944'872868 1'145648'7533721 53 28, 6621842'883688 1'131620'7493411 32 48'6665325 [894103 1'118439 7454760112
8['65781351'873380 111449761 7531808 521291'6624022'884206 1'130957'749148421 491 6667493.894626 1'117784'7452821 111
91'6580326'873893 11144304'7529894 51 30 66262001'884725 11302941 7489557130150. 6669661'895150 1.117130'7450811 10 
10 6582516'874406 1'1436321 7527980 50 331 6628379'885244 1'129632'7487629 29 51 16671828 1895674 1'116476'7448941  9
11'6584706'874920 1142961'7526065 49 32 66305571 885763l 1128970'7485701 28 521 66739941 896199 1115823'7446999] 8
12'6586895'875433 11142290'7524149 48133'6632734'886282 1-1283081 7483772 27153'6676160'896723 1'115170'7445058  7
131 6589083'875947 11141620'7522233147 34'6634910'88680111.127647 -7481842126 54'6678326'897248 1'114518.7443115 6
14 6591271'876462 1'140950'7520316146 35'6637087'887321 1'16987'7479912 25 55'6680490'897773 1113866'7441173  5
15 165934581 876976 1'140281'7518398145 36'66392621 8878411' 1126327'7477981 24 561 66826551 898299 1'113214'7439229  4
16 16595645'877491 11139612'7516480144 371 6641437 85883611 1125667'7476049 23 57'66848181 898825 1-1 12563'7437285  3
171'6597831'878006 11138944 -7514561143 381'6643612'888882 11125008'7474117122 58 6686981'8993511 1111912'7435340  2
1816600017'878521 1'1382761 7512641 42 39'6645785.889403 11 124349'7472184121 591 6689144[ 899877 1'111262'7433394  1
19'6602202-'879037 1'137608. 7510721 41 40'66479591 889924 11123690'7470251 20 60'66913061 900404 L1110612'7431448 0
201 -66043861 -879552 1'136941'7508800140                                           i.   on 
1 Cosine. Cotang. Tang.   Sine.' |         Cosine. Cotang.1 Tang.   Sin.    ne.                       Tang.   Sine.
Deg. 48.                                Deg. 48.                                 Deg. 48.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
42 Deg.                                  42 Deg,                                  42 Deg.
S Sine.  Tang. Cotang.  Cosine.'/   Sine.   T Tlang. Cotang. Cosine.    /   Sine.   Tang. Cotang. Cosine.
0 *6691306'900404 1 110612'7431448 60 21'6 36577'911526 1'097060.7390435f39 1 6779459'922235 1.084322'7351118 19
1'6693468'900930 1-109963 7429502 59 22 6738727'912059 1.096420 *7388475 38 42 *6781597'922773 1.083689'7349146 18
2'6695628'901458 1-109314'7427554 58 23 6740876.91259211.095779 -7386515 37 3 6783734'923312 11083057 *7347173 17
3 *6697789'901985 1  108665.7425606 57 24'6743024  913125 1.095139 7384553 36 44.6785871.923851 1082425 *7345199 16
41 6699948'902513 1-108017'7423658 56 25'6745172 9 13659 11094500. 7382592 35 45 67880071 924390 1.081793 17343225115
5 6702108 903041 1-107369'7421708 155 26 67473319'914192 1'093861 7380629 3446'6790143 924930 1'081162'73412501 14
6 67042661 903569 1 106721'7419758 54 27 6749466'914727 1.093222.7378666 33 47 6792278 1925470 11080532 -7339275 13
7.6706424 904097 1.106075'7417808 53328. 6751612'915261 1.092584 I7376703 32 48'6794413'926010 1'079901'7337299 12
816708582'904626 11105428'7415857 52929 16753757'915796 1'091946'7374738131 491 6796547'92655011 0792711 7335322 11
9.6710739'905155 11104782 7413905151 30 167559021 916331 1.091308'7372773130 50'6798681 1927091 11078642 17333345 10   CZ
10 6712895'905685 11104136 1 7411953150 31'6758046'916866 1.090671.7370808 29 51 168008131 927632 110780131'7331367  9
11 16715051 -906214 11103491 17410000149 32 167601901 917402 11090034'7368842 28 52'68029461 928173 1'0773841'7329388  8
12 -6717206 90674411'102846'7408046 48 33'6762333.917937 1.089398'7366875 27 5316805078'928715 1'076756 17327409  7
13'6719361'907274 11102201'740609247 34 6764476'918474 1088762 -7364908 26 54 6807209'92925711'076128'7325429  6
14 -67215151 9078051 1101557'7404137146 35 6766618'919010 11088126'7362940 25 5516809339'929799 110755001 7323449  5
151 67236681 908336 1.1009141 7402181 45 361 6768760.919547 11087491 17360971 24 56 6811469 1930342 110748731 7321467  4
161'6725821'908867 1'100270'7400225 4437'6770901 92008411 086857'7359002123 57 6813599'9308841 1074246'731948s6  3
17 -6727973'909398 1'099628'7398268 43 38'67730411'920621 1.0862221'7357032 22 58f'6815728 1931428 11073620'7317503J 2
18 167301251 909930 1098985'7396311 42 39 -6775181 1921159 11085588s,7355061 21 59'6817856 1931971' 1072994 17315521  1
191-67322761-910461 11098343 17394353141 40 6777320'921696 1.084955j'7353090 20 60 16819984 1932515 1'072368'7313537  0
20 167344271 910994 11097702 17392394 401' Cosine. Cotang.l  Tang. I Sine.          Cosine. C otang. Tang.   Sine.           Cosine. Cotang.1 Tang.   Sine.
Deg. 47.                                Deg. 47.                                  Deg. 47.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
43 Deg.                                  43 Deg.                                  43 Deg.
Sine.  Tang.  Cotang. Cosine. t'        Sine.   Tang. Cotang. Cosine.' _        Sine.   Tang. [Cotang. Cosine.'
0'6819984! 932515 1'072368'7313537 60 21j 68'64532'944001 11059320" 7271740 39 1 6906721 955064 1'047049( 7231681 19
1 *6822111, 933059 11071743 }7311553 59 22 *6866647 *944551 1.058703'7269743 38 2'6908824 *955620 11046440'7229671 18
2'6824237'933603 11071118 *7309568 58 23'6fi868761.945102 1.058086'7267745 373'6910927 *956177 1.045831 *7227661 17
31.6826363.934147 11070494'7307583 57 24 6870875.945653 1.057470 *7265747136 44 6913029 *956734 1,045222.7225651116
4 16828489'934692 1.069870.7305597 56i25.6872988. 946204 1.056854'7263748 35 5'6915131 957291 1.044613 7223640115
5'6830613.935238 11069246 17303610 55 6.6875101.946755 1.056238'7261748 34 6 6917232'957849 1.044005.7221628 14
6 6832738'*935783 1'068623'7301623 54 27.6877213'94'7307 1'055623 17259748 33 47'6919332 *958407 1'043397'7219615 13
7'6834861 1936329 1'068000'7299635 53 28 *6879325'947859 1055008 *7257747 32 48 6921432'958965 1'042790 7217602 12
8'6836984.1936875 1.067377'7297646 52 29.6881435'94841111'054394 17255746 31 49 6923531'959524 1.0421831 7215589 11
9 6839107 937421 1.066755. 7295657 51 30.68835461 948964 1'0537801 7253744 30 50 69256301 96008211'041576'7213574 10
10'6841229'937968 1.0661341 7293668 50 31 16885655.949517 1.053166 17251741 2951'6927728| 960642 110409701 7211559 9
111'6843350'938515 1'065512 17291677149 32.6887765'950070 1052553'7249738 28 52'6929825 961201 110403641 7209544 8
121'6845471'939062 1'064891'7289686148 33 -6889873'950624 11.5139401'7247734 27 53169319221 961761 11039758'7207528  7
131'6847591'939610 1'064271'7287695 47 34.6891981'951178 1-051227'724572926 54 69340181'962321 11039153'7205511  6
141-6849711. 940157 1.0636511 7285703 46 351 6894089. 951732 1.0507151 72437242 55'69361141.962881 110385481.7203494  5
151'6851830'940706 1'063031'72837101451361 6896195'952287 1'050103'7241719 24 61'69382091'963442 l'037944'7201476  4
161'68539481'941254 1'062411 -7281716144 37 16898302 1952842 110494921 7239712123 571'e940304J.96400311'037340'7199457 3
17'6856066'941803,1 0617921 7279722 43 38'6900407.953397 1.048880'723770522 581'6942398.964565 11036736 7197438 2
18,16858184. 942352 1'061174'7277728 42 39 169025121 953952 1'048270'7235698 21 59'6944491 196512611 0361331 7195418  1
19 1 68603001'942901 1'060556  7275732 41 40 6G904617 -954508 120476591 7233690 260'6946584'965688 11035530'7193398  0
201 68624161 943451 11059938'72737361401                   1
Cosine. Cotang. Tang. I Sine.         -' Cosine. Cotangng. Tang    Sine.                           Tang    Sine.
Deg. 16.                                Deg. 46.                                  Deg. 46.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
44 Deg.                                  44 Deg.                                 44 Deg.
Sine.   Tang. Cotang. Cosine.'  Sine.   Tang. C.otang. Cosine.            Sine.   Tang. Cotang. Cosine.
01.,97'96 _.__ 0 _0_1
0 -6946584 *965688 11035530'7193398 601 6990396 977564 1-022950 7150830 39 41 -7031879 -989006 1-011115 -7110041 19
1 -6948676 -966251 1-034927 *7191377 59 2 6992476 978133 1-022355 -7148796 38 42 -7033947 -989582 1-010527 -7107995 18
2 -6950767 -966813 1034325 *7189355 5S 23 6994555 -978702 1-021760 714676237 43 7036014 990158 1-009939 -7105948 17
3 6952858 -967376 11033723 -7187333 57 24 6996633 -979272 1-021166 -7144727 36 44 -038081.990734 1'0093521-7103901 16
4'6954949 -967939 1-.033122 7185310156 25 -6998711 -979842 1020572 -7142691 35 45 -7040147 -991311 1-0087641 7101854 15
5 *6957039 968503 1-032520 718328755 26 -7000789 -980412 1-019978 17140655 34 46 -7042213 -991888 1-0081781-7099806 14
6 -6959128 -969067 1-031919 -7181263 5427 -7002866 1980983 1-019385 -7138618 33 47 -7044278'992465 1-007591 17097757 13
7 169612171.996631 1-031319 -717923853 28 7004942 981554 1-018792 -7136581 32481 7046342 -99304211-007005 7095707 12
816963305 -970196 1-030719 -7177213152 29 7007018 -982125 1-018199 -7134543 31 49 -7048406 -993620 1-006420.7093657 11
916965392 -970761 1-030119 -7175187 51 30 7009093 -982697 1-017607 -7132504 30 50 -7050469 -994199 1005834 -7091607 10 
10-6967479 -971326 1-02952017173161 50 311 -70111671 -983269 1-017015 -7130465 29511 -7052532.994777 1-005249 17089556 9
11 -6969565 -9718911 1028921 -717113449 32 -7013241 -983841 1-016423 -7128426128521-7054594 -99,i356 1-004665 -7087504 8
12'.6971651 -972457 1-028322-1716910648 33 -7015314 -984414 110158321 7126385 27 53 7056655'-995935 1-004080 -7085451  7
13 -6973736 -973023 1-027724 -7167078 47 34 -7017387 -984987 1-015241 -7124344 26 54 7058716 -996515 1-003496-'7083398 6
14-6975821 -973590! 1-027126 -7165049 46 35 -70194591 985560 11014651 1-7122303 25 551 -7060776 -997095 1-002913 -7081345  5
15-6977905-974156fi 10265281 -7163019 14536 -7021531 -986133 1-014061 -7120260 24 56 -7062835 -997675] 1-002329 17079291  4
161-6979988 -974724 1-025931 -7160989 14437 -7023601.98670711-013471 -7118218 23157 -7064894 -998256 1-0017461-7077236  3
17 -6982071 -975291 1-025334 -7158959 43 381-70256721.987282 1-0128811-7116174 22158 -7066953 -998837 1-001-164 -70751801 2
18 1-6984153 -975859 1-024738 -7156927 42 391-7027741.987856' 1-012292 -7114130121 59 -70690 11 -999418 1-000581 -7073124  1
191-6986234 -9764-27 1-024141 -7154895141 0 -7029811 1.988431 1-011703 -7112086 20 60 -7071068 1-00000 1-000000 -7071068 0
201 -6988315-976995 1-023546 -7152863 40   l 
Cosine. -Cotang.-  Tang.;  Sine. |;I;   Cosine. lCotang.l Tang. I Sine.  I| -Cosine. lCotang. Tang.  Sine. 
Deg. 45.                                 Deg. 45.                                Deg. 45