THE
FIELD PRACTICE
OF
LAYING OUT CIRCULAR CURVES
FOR
RAILROADS.
BY JOHN C. TRAUTWINE,
CIVIL ENGINEER.
FOURTH THOUSAND.
PHILADELPHIA:
PUBLISHED BY WILLIAM HAMILTON,
HALL OF THE FRANKLIN INSTITUTE.
1854.




Entered according to Act of Congress, in the year 1851, by
JOHN C. TRAUTWINE,
in the Clerk's Office of the District Court of the United States for the Easern
District of Pennsylvania.
STEREOTYPED BY L. JOHNSON AND CO.
PHILADELPHIA.




PREFACE.
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 field operations. Still, I trust that it will not
prove unacceptable even to the latter.
The Table of Natural Sines and Tangents to single minutes, in a
form sufficiently portable for field use, will supply a want which I
have myself frequently experienced, 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 oocupations 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 certainty, 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
3




4
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, read'1068425.
Page 328, at top, for 25 Deg., read 40 Deg.
Tangent of 440 60', for'1000000, read 1'000000.
Tangent of 41~ 60', for'8994040, read'9004040.
In Dr. Gregory's Corrected Edition (the 8th) of Hutton's Tables, 1838
Sine of 490 14', for'7576751, read'7573751.
In Hassler's Tables, 1830.
Sine of 78~ 24', read'9795752.
Sine of 200 60',  "'3583679.
Sine of 660 19',  "'9157795.
Sine of 560-39, "'8353279.
Sine of 550 20', "'8224751.
Sine of 530 4', "'7993352.
Sine of 480 12', "'7454760.
Sine of 450  3/, "   7077236.
The foregoing I believe to be all the errors in the Natural Sinee
_Wnd Tangents to whole minutes, in the respective tables, The disr
erepancies 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 screely necessary to remark that, beyond.44~, the Sinqe,,
Tangents, &c. are read upwards, from the bottom of the page, using
the corresponding column of minutes.  To find the sine of an angle
exceeding 900~, 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 180~, are the same.
In this edition pages 39 to 42 have been added.
JOHN C. TRAUTWINE.
ERRATA. — None.




FIELD PRACTICE
OF
LAYING OUT CIRCULAR CURVES
FOR
RAILROADS.
ARTICLE I.
PRINCIPLES OF LAYING OUT CURVES.
METHOD 1.
To lay out a Curve by means of Tangential Angles.
IF from any point B, fig. 1, in a straight line A D, we
lay off any number of equal
angles, as  B, s B t, tB u,           B.-.- D
u B v, &c., and at the same time
make the chords B s, s t, t u,
X v, &c. equal to each other,
then the points B, s, t, u, v, &c.
-will be situated in the circum-   Z,. J   i
ference of a circle, which is tangential to the line A ID at the
point B.
The first of these angles, z  
DB s, is called the tangential                 p
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 1, should prevent our seeing from B
farther than to v, the curve may be continued by removing
1*                5




6
the instrument to u, the point preceding v; thence sighting
first on v, continue to lay off additional tangential angles
v u w, w 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, equal to double the tangential
angle, and afterward continue the tangential angles w v x,
x v g, &c., as before, to the end of the curve.
Finally, in order to pass from the end of the curve at g,
on to a tangent g z, place the instrument at g, and sighting
back to x, lay off the tangential angle x g o; then o g continued toward z will be the required tangent. (See Art. IV.)
For the tangential angles corresponding to different radii,
and chords of 100 feet, see page 25.
ARTICLE II.
METHOD 2.
To lay out a Curve by means of Deflection Angles.
Fig. 2. First, having, as in method 1, laid off a tangential angle D B s, and measured
A._.. -'-P   the chord B s, remove the instrument to the end s of the
/  7Th chord, and make the exterior
angle m s t equal to twice the
f............. tangential angle, and measure
the chord s t; and so on at the
"n other points t, u, v, &c., making
Fi. 2  each of the exterior angles n tu,
//-i   o u v, &c. equal to twice the tan/0;u       gential angle, and all the chords
to     equal; then will the points B,
/  8, t,, v, &c. be in the circumL   /7             ference of a circle which is tamgential to the line A D at the point B, as by the firist
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 deflection angle z v w. (See Art. IV.)
These exterior angles, included between any chord and
the extension of the preceding chord, are called deflectim




7
angles, or cangles f deflection, or angles of curvature. In
any given circle, the angle of deflection is always precisely
double the- tangential angle, supposing the chords to be'equal. At page 25, 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 deflection angle of any curve is equal to the
angle t a 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 deflection angle,
is, of course, always half the central angle.
ARTICLE III.
METHOD 3.
To lay out a Curve by Eye.
The deflection angles, fig. 3, e 8 t, f t u, g u v, h v w, &c.,
being double, the tangential
angle D B s, the arcs e d t,,, D
f i u,gmv, h nw, &c., are            Tic
double the arc D c s, since              I
the arcs of circles are proportionate to the angles which
they subtend; but the chords
t, f u,, g v, h w, &c. are not    Zig. 3
double the chord D s, since                  
the chords of arcs are not
proportionate to the arcs, or
to the angles which they sub-._!
tend.                                   m9 
The chords e t, f u, g v,
h w, &c., which subtend the L 
deflection angles, are called
deflection distances; and the chord D s, which subtends the
tangential angle, is called the tangential distance.
But although, in any given circle, the deflection distance
is not truly twice the tangential distance, yet the difference




8
is so trifing in large railroad curves, with chords of but 100
feet, that it may generally be neglected in curves of more
than 800 feet radius.
In our tables the precise length of both will be found for
different radii, and for chords of 100 feet.
Having these respective distances, we may frequently
trace a curve on the ground by the eye only, with very tolerable accuracy, sufficient for guiding the excavations 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, page 25, or by Art. XVII, the
deflection distance e t or f u, &c., corresponding to a radius
of 5730 feet for a chord of 100 feet, viz. 1'745 feet; and
also the tangential distance d 8'873 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 a, equal to'873 of a foot; and place
a stake at s. 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 8 t equal 100
feet; at the same time measuring with the rod from the pin e,
the deflection distance e t, equal to 1I745 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 deflection 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, ironpointed, 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 exte



9
riot angles, after having first sighted backward to the point
behind.
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 pre,
ferred, as all the printed tables of sines, tangents, &:., 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.
to unite properly with the tangent
ef. 
In that case, and when using
Method 1., Art. I., of laying off
curves by means of tangential angles, we must, in order to fix the P.UA  
point e, lay off a sub-tangential angle d A e, as much smaller than the           /
entire tangential angle B A c, or            / 
c A d, &c., as the sub-chord d e is
smaller than an entire 100 feet             e
chord, a c, c d, &c. Thus if the
sub-chord be one-half, or one-fourth, &c. of the entire
chord, the sub-tangential angle must be one-half, or onefourth, &c. of the entire tangential angle.
This method is not mathematically exact, for the reason
stated in Art. III. (viz. that the chords subtending different
angles are not proportional to those angles;) 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, e f, we must place the instrument
at e, and lay off the same sub-tangential angle d e g; or
which is better, take sight from e to c, and lay off the angle
c e g, equal to the sum of a tangential and thesub-tangential -angle.




1-0
But when using Method 2, Art. II. of deflection angles, or
Method 3, Art. III. of deflection distances, we may calculate
the sub-deflection angle, a s e, fig. 5, and sub-deflection distance a e, formed between a sub-chord s e, and the extension
s a, of an entire chord g s, with sufficient accuracy for curves
of 300 or more feet radius, and chords of not more than
100 feet, thus:
Rule.-Say, as an entire chord of 100 feet is to the subchord s e, so is the deflection angle of the curve, to a certain
angle. Add these two angles together and divide their sum
by 2, for the sub-deflection angle a s e, of the sub-chord.
Example.-The curye,
XZ'f:fig. 5, has a radius of
319'6 feet, and an angle
of deflection, fg s, of 180
for chords of 100 feet.
The sub-chord s e is 25
/  Fi~p    feet in length; what is the,,a sub-deflection angle as 8 e!.... and also the sub-deflection
Zatv5        ei     distance a e, for the subchord s e?
Chord.  Sub-Chord.
Here, as 100 is to 25,
-mz,           Def. An. of  Certain
100 ft. chord.  Angle.
So is     180 to 46 30'.
The sum of these two angles, 180 and 40 30' = 220 30',
the half of which is 110 15', the required sub-deflection
angle a s e.
Again, to find the sub-deflection distance a e, of the subchord s e; take from the table of sines, the natural sine of
one-half the su*-deflection angle a s e, just found. Multiply
this natural sine by 2, and multiply that product by the
length of the sub-chord..Example.-The sub-deflection angle is 110 15'; one-half
of it, is 50~ 37J, 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 subdeflection distance a e.
Finally, to find the sub-tangential distance s n, by means
of which to pass from e to the tangent e m, say, as 10000




ii
is to the square of the sub-chord in feet; so is the tangential
distance for a 100 feet chord, to s n. In this instance, we
have as 10000 is to 625, so is 15'69 feet to'980 feet, or s 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 ad.
jacent chord-stakes, and mea-      I
sure off as nearly as may be,       l
at right angles to it, with a /   F i/g.o             e 
graduated rod, the previously
calculated ordinates, c d, e f, g h, &c., placing pegs at
d, f, h, &c.* Our table of ordinates, page 28, is calculated
for distances apart b c, c e, eg, &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 center, 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 those of large radii, 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
* On the tops of these stakes, small tacks are driven to define the precise
point in the oeuve.




12
for railroad practice, for curves of over 300 feet radius, and
for chords not exceeding 100 feet, thus: In a circle of
given radius, 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 10000; the rule therefore becomes,
as 10000 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
ordinate 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,
Mid. Ord.    Mid. Ord. Sub-Chord
Sq. of 100 ft.: Sq. of 51) ft.::    of Chord.    Mid. rd. b-Choate
of Chord.     approximatelv.
10000: 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.
ifaving 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,
or b c d. From the table of tangents take the natural tangent of
b c a, and multiply it by the given
radius; the product will be 6 a, or
b d.
FTg7 \/.Example.-Let the angle a b d
e t        be 120~, 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 = 60", which taken from
900 leaves the angle b c a = 300.  The natural tangent:of
300 _=  5773, which multiplied by the radius of 2865 feet,
gives 1653'96 feet for b a or 6 d. (See Art. XII.)




13
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 tangen
tially at a and d.
Rule.-Subtract the angle a b c, which is half the angle
a b d, from 900; the remainder will be the angle b c a, or b
c d.  Then as nat. sine of b c a,* is to nat. sine of a b c,t
so is a b to a c, the radius required.
Ezample.-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 (30~) - -5000 is to
nat. sine of a b c, (600) -= 8660, so is b a (1654 feet) to
a c, (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 deflection 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, 2865 feet.
Here the angle a b d, 1200, subtracted from 1800, leaves
a remainder of 60~; which, divided by 20, the angle of
"deflection 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 afraction of a chord,
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 30,7 a sub-chord of 50 feet
(very approximately) would have been necessary.
* The angle opposite the given side, a b.
t The angle opposite the required side, a c.
2




ARTICLE  IX.
How to proceed when the end of a curve does not correctly
join the tangent.
We sometimes find, in running out- a curve for the number of chords determined by the Rule in the preceding Article, that instead of uniting as it should with the previously determined tangent d m, fig. 8, at o, it ends tangentially to a lineparallel 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 tangent 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.
The  measueg. 8. or c(s         y
Then measure b x, or c o, (as the case may be,) and if
the curve fall within 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 back-wards 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; subtract the product from 1800, and sighting back
one chord to n or r, lay off the angle n c b, or r b v, equal to
the remainder. For example, if the tangential angle be
100, and from t to c be 4 chords, then 7 times 100 taken




from 1800 leaves the angle n c b, or r b v=1100. When the
product exceeds 180~, it must be subtracted from 360~,
or the angle n c b, or r bv.*
This case occurs whenever an error has been made in
measuring 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 the 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 Art. X.
ARTICLE X.
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 mistake in obtaining the angle a e f.
Big. 9.                  Fig. 10.,
/ li zf.-,,
Ji.........!.,,,*"                   /
Ii               2,9'
If a e f, fig. 9, be measured in excess, as a e g, then the
* In both oeases the angle is measured outwardly from the curve; but when
the curve falls beyond the tangent, as at b, then b v must be continued inwardly
asbo.




16
curve a b c, calculated for the incorrect angle a e g, will be
found to fall beyond the true tangent e f, 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 e f, fig. 10, be measured too small, as a e g, then
the curve a b c, calculated for the incorrect angle a e g, will
be found to fall within the true tangent e f, 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.
But if greater, we must either remeasure 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 ef may 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, (see
Art. XIII.,) 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 e f, 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 tangent m b continued, strikes ef, as
at o. Measure both b o, and the angle b o f. Half the angle
b of taken from 900, gives the angle b h o; then say,
As the Nat. Sine of angle b h o op-  Nat. Sine of angle b o 
the  posite the given side, b o, is to  opposite the required
side b A,
So is The given side b o, to The required side, or new radius b A.
Ascertain from the table, or by calculation, the angle of
deflection, and the tangential angle corresponding to this
new radius b h; and the new curve commencing at b will
terminate tangentially to e f at i, as far 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 XIII.




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.
IHaving 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, (see
Article XIV) f o i n m, for uniting a b to c d.
Rule.-Half the. angle
a b c taken from  900,
leaves the angle b g i; and  Aj.            A
half the angle b c d taken        6      //
from 90', leaves the angle  
ihc.'                        /
From the table of tan-      \   \          7
gents take the natural tan-.. /
gent (b i) of the angle b g i;
and that (i c) of the angle
i It c; and add them together.
Then as the sum of these two nat. tangents is to the nat.
tang. of bg i, so is b c to b i; and b i taken from b c, gives ic.
Again, in the triangle bg i, as the nat. sine of the angle
b g i, opposite the given side b i, 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
1290 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 900, leaves
the angle bg i 540 10'; andhalf the angle b c d (640 37I')
taken from 90~, leaves the angle i h c = 25~ 22k'.
From the table of tangents, we have nat. tang. of b g i
(540 10')= 1;3848; and nat. tang. of i h e (25~ 22'0 =
~4743; their sum being 1'8591.
2*




18
Then as
Sum of Tang's.is t agong. of              
1S8591    i to540 10' so is 950 ft.,  to  707'63 ft.
1'3848, )    (     )
and o i, 707T63 feet, taken from  b c, 950 feet, leaves i c
242'37 feet.
Again, as the
Nat. Sine       Nat. sine of                    g i or A i, the
of angle  is to  Angle g b i  so is            required rabg i                              707'63      dius, 510'97
s8107             5854,            feet       feet.
ARTICLE XII.
To obtain the angle d b e, formed by two tangents, d b, and
b e, when thepoint 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.
Rule.-Subtract the angle a d e from 1800 for the angle
6 d e; and subtract the angle d e c from 1800, for the angle
d e b. Add together b d e and d e b6, and subtract their
sum from 1800~ for the angle d b e.
Fig. 12.    Fig. 13.    Fig. 14.    Fig. 15.
b          6          6            6
Id         d I"":~j~,  d7.      e 0
CASE 2. When the included figure, d b ef, figs. 13 and
14, has four sides.
Rule.-Subtract the sum of the three internal angles of
the figure marked by dotted segments of circle, from 360~,
for the angle d b e.
CASE 3. When the included figure, 1.5, has more than
four sides.




19
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, 1500; at 8, 1100; at c, 1600~; at e,
1000. 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
(8100) 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 c,
of different radius, but running in the same direction,
constituting a COMPOUND curve.
R1ule. —Placing the instrument at b, sight back to the
other end of the 100 feet chord at
a; and lay off the tangential angle   e
a bd- dof the curve a m b;:then   m
from the common tangent d b e, lay -
off the tangential angle e b c, of the     i 
curve b n.c, making at the same
time the chord b c equal to 100 feet.
N. B.-If running the curve by eye, use the tangential
distances instead of the angles.




20
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 it
an opposite direction; constituting a REVERSRE   curve.
Rule.-Placing the instrument at t, sight back to the
other end of the 100 feet chord
I7  at m, and lay off the tangential
)'i,g17           angle m t r, of the curve m n t;
9"n'         -t  then from the common tangent
r t s, lay off the tangential angle
8 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.
ARTICLE XV.
RADII.
To find the radius corresponding to any given angle of deflection, and to equal chords of any given length.
Rule 1. —Subtract the angle of deflection from 1804,
then say, as nat. sine of angle of deflection, is to nat. sine
of half the remainder, so is the given chord to the radius
required.
Example.-Let the angle of deflection be 20, and the
chord 100 feet, required the radius.
Here 2~ subtracted from 180~, leaves 178~, the half of
which is 89', and as
Nat. Sine of 20  Nat. Sine of 890  Chord   Radius
~034899    ~    999848    *. 100 feet * 2865 feet.
Rule 2.-The radius for 100 feet chords may be found
approximately, by dividing 5730 by the deflection angle.




21
This rule is very close for radii of not less than 500 feet.
Eor 500 feet it gives eight-tenths of a foot too little, but is
more approximate for larger radii.
Example.-What is the radius to a deflection angle of
20, the chords being 100 feet long?
Here, 5730 divided by 2, gives 2865 feet, the radius
required.
ARTICLE XVI.
TANGENTIAL AND DEFLECTION ANGLES.
To find either the Tangential or Deflection Angle corresponding to any given radius, and to equal chords of any
given length.
iule 1. —Divide half the chord by the radius; the quotient will be the natural sine of the tangential angle.
Therefore, the angle corresponding to this sine, in the table
of natural sines, will be the tangential angle required; and
the tangential angle multiplied by 2 will give the deflection
angle.
Example. —Let the radius be 2865 feet, and the chord
100 feet; what will be the tangential and deflection angles?
Here, half the chord, (50 feet,) divided by the radius,
(2865 feet,) gives'01745; and the tangential angle in the
table corresponding to the natural sine'01745 is 10, twice
which is 20, the deflection angle required.
Rule 2.-The deflection 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 one minute too little.
Example.-What is the deflection angle for a radius of
2865 feet, the chords being 100 each?
Here, 5730 divided by the radius 2865, gives 20, the
deflection angle required.




22
ARTICLE XVII.
DEFLECTION DISTANCES.
To find the Deflection Distance (exactly) for any given
radius, when the chords are 100 feet long.
Rule.-Divide the constant number 10000 by the radius
in feet; the quotient will be the deflection angle required.*
Example.-What is the deflection distance to a radius
of 5730 feet, the chords being 100 feet long?
Here, 10000 divided by 5730 radius, gives 1'745 feet,
the deflection distance required.
To find the Deflection 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 deflection
angle; and double this natural sine, multiplied by the chord,
will give the deflection distance required. By this rule our
table was prepared.
E.xanple. — As before, what is the deflection distance to
a radius of 5730 feet, the chords being 100 feet long?
Here, half the chord, (50 feet,) divided by radius, (5730
feet,) gives'008727, which is the natural sine of half the
deflection angle. Now'008727, multiplied by 2, gives'017454, which, multiplied by the chord, (100 feet,) gives
1'745 feet, the required deflection distance, the same as in
the preceding example.
ARTICLE XVIII.
TANGENTIAL DISTANCES.
To find the Tangential Distance corresponding to any givegiv
radius, and to equal chords of any given- length.
Rule.-First find the tangential angle by Article XVL,
and take from the table of natural sines, that correspond~ Because the deflection distance to a radius of 10000 feet, with chords of
100 feet, is 1 foot; and the deflection distances for other radii increase inversely
as the radii.




ing 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..Example. —Let the radius be 2865 feet, and the chords
100 feet each; what will be the tangential distance?
Here we find, by Article XVL, the tangential angle 1~
for a radius of 2865 feet.
The natural sine corresponding to 30 minutes, or onehalf 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 XIX.
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
ab 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.
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 deflection 70, to chords of 100 feet. What will be
the length of the middle ordinate?
Here, tabular cosine of 3~0 (the tangential angle) is
~998135; which, subtracted from 1, leaves'001865; which,
multiplied by 819, the radius, gives 1'527, the middle
ordinate required.




24
ARTICLE XX.
Having given the Middle Ordinate d e, fig. 18, it is required
to find any other one, as i n.
d  *      MRule 1.-Subtract the middle
0  Be         ordinate d e, from the radius d c,
N4-   e      n —-/' \    the remainder will be ec: then
from  the square of the radius
i, subtract the square of the
\.;j/    Fig18  distance oi, which 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; 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
6 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 in, the
required ordinate.
Rule 2.-M-Iultiply the ordinates of a 10 curve by the
deflection angle of the curve whose ordinates are required,
(chords being 100 feet.) This is a sufficiently close approximation for curves of not less than 500 feet radius; and for
placing ordinates forguiding the excavations and embankments, it is close enough for the smallest curves in our table.




25
TABLE  OF RADII, &C.-CHORD 100 FEET.
The Tangential Angle is always one-half of the Angle of Deflection.
Angle of  Radius  Deflection Tangential  Angle of  Radius Deietaion  Tangentatl
Defleion,  e.  distance  dista                    ist aee   dists
Deflection. inffeet.             Deflection. in feet.  in feet.  in feet.
in feet.  in feet. o             - 
1  343775.029.014            44     7814   1'279'639
2  171887.058'029          45     7640   13808'654
3  114692   *087'043        46     7474  1-337'668
4   85944'116.058          47     7315  1'366'683
5   68756   -145.072        48     7162  1l395    8697
6   67296   *174.087          49     7016   1-424'712
7   49111   -203'101        50     6876   1'453    -726
8   42972   *232.116        51     6741   1'482'741
9   38197   -262    -131        52     6611   1-511.755
10   34378   -291.145          63    6487   1-540    -770
11   31254'320'160        54     6367  1.569    -784
12 - 28648.349'174        55     6251   1 598.799,
13   26444.378    *189          56     6139  1.627.813
14   24556   -407'203        57     6032  1.656'828
15   22918'436'218        58     5928  1.685    *842
16   21485'465'232    o 59       5827   1-715.857
17   20222.494'247    1            5730  1.745'872
18   19098'523'261         2     6545   1'802'901
19   18093'552'276         4     5372   1'860'930
20   17189   -681    -290         6     5209   1'918'959
21   16371'610'305         8     6056   1'976'988
22   15627'639.319        10     4912 I 2-036   1.018
23   14947'668'334        12     4775  2-094   1'047
24   14324'697'348        14     4646  2'152   1'076
25   18751'727:363        16     4524  2-210   1'105
26   13222'756'378        18     4408  2'268   1'134
27   12732    785'392        20     4298  2.326   1 163
28   12278'814'407        22     4193  2-384   1.192
29   11855'843.421          24     4093  2-443   1.221
30   11459   -872'436        26     3998  2-501   1 250
81   11089'900'450        28     3907  2'559   1'279
82   10743'930'465        30     3820  2.617   1'308
33   10418'959'479        32     3737  2'676   1'388
34   10111'988'494        34     3657  2'734   1'367
85    9822  1-017'508        36     3581   2'793   1'396
86    9549  1'046'523        38     3508  2'851   1'425
87    9291  1-075'537          40     3438  2'908   1'454
38    9046  1'104'552        42     3370  2'967   1'483
89    8814  1'133'566        44     3306  3'8025   1'512
40    8594  1'162'581        46     3243  3 083   1'541
41    8384  1 191'595        48    8183   3'141   1'570
42    8185  1'221'610        50     3126   8'199   1'599
43     7994  1'250'625       52     3069   3'258   1'629
3




26
TABLE OF RADII, &c. —CHOR) 100 FEET.
CONTINUED.
nTo Tangential Angle is always one-half of the Angle of Deflection.
Deflection Tangential  of  adi   Deflection Tangential
| menb   Radis dlist    Taien.e   Angleof  Radiun f a.  s
Agi in feaet.,         Deflection.  in feet.  disntanfee
ni n f eet.  in feet.  in feet.  in feet.  in feet
o                               o 0
I 64    8016  3'316   1G658    3 26    1719.    5'817  2'908
66    2964  38374   1 687       22    1702    6 875  2'937
68    2914  3.432   1*716       24    1685   56983  2-966
2865  3 490   1-745      26   1669.   5992  2.996
2    2818  3'548   1'774       28   1653    6 050   3 025
4    2772  3'606   1'803       80   1687    6'108  38064
6    2729  3'665   1-832       32   1621    6'166  38083
8    2686  3'723   1'861       34   1606    6'224  83112
10    2644  3-781   1*890       36   1691    6*282  3'141
12    2604  3-839   1 919       38   1577    6-340  38170
14    2566  3'897   1 948       40    1563    6 398  8'199
16    2528  3-956   1 978       42~  1549    6-456  3 228
18    2491  4'014   2 007       44   1534    6'515  83257
20    2456  4'072   2'036       46    1521    6'574  3-287
22    2421  4-130   2'065       48   1508    6*632  8-316
24    2387  4-188   2'094       50   1495    6'690  8.345
26    2355  4'246   2'123       52   1482    6'748  3'874
28    2323  4'305   2'152       54   1469    6'806  38403
80    2292  4'863   2-182       56   1457    6-864  83 432
82    2262  4'421   2-210       58   1445    6'922  83461
34    2232  4'479   2-239   4         1433    6'980  8'490
86    2204  4'538   2-269        6   1403    7-125  35662
88    2176  4-696   2'298       10   1375    7 270  38635
40    2149  4'658   2-326       16   1348    7'416  38708
42    2122  4'712   2'356       20.   1322    7'563  83781
44    2096  4'770   2*385       25   1298    7 708  3'854
46    2071  4.828   2-414       80   1274    7 853  3.927
48    2046  4.886   2.443       85.  1251    7.998  3 999
60    2023  4*944   2.472       40   1228    8.143  4.071
62    1999  6 002   26501       45   1207    8.289  4.146
54    1976  6 060   2 530       50   1185    8.432  4.216
56    1953 5.118   2-559        655   1166    8.577  4.288
58    1932  5'176   2-588   5         1146    8-722  4.361
1910  65235   2-618       6   1127    8-869  4-434
2    1889  5'293   2.646       10   1109    9'014  45607
4    1868  65351   2'676       15   1092    9'159  4-579
6    1848 5.409   2.704        20   1074    9.304  4.652
8    1828  65468   2-734       25    1058    9 449  4*724
10    1810  5.526   2.763       80   1042    9.595  4.798
12    1790  5 584   2.792       85   1026    9*740  4-870
14    1772  65642   2.821       40   1011    9 885  4.942
16    1754  56700   2.850       45     996.8 1003  65.015
18    1736  6.758   2.879       60    982.7 1018  65-090...,     _ -.~~~~~~~~




27
TABLE  OF RADII, &c.-CHoRD 100 Frm.,
CONTINUED.
The Tangential Angle i alwaya one-half of the Angle of Defection.
Rad ius  n  Deflection Tsoi
Angle of  Radius Deflection Tangential  Angle of  Radis  dtn
Deflection. in feet.  distanc e distance  Deflection. in feet.  distance
in feet.  in feet.    in feet.  ii feeL.
0 t                             0  O
5 55  969-0   10-32   5-160   12  0  4593    21-79  10-90
6     955.4   10-47   5.285    o 45  450-3   22-21  11-12
"5  947-5   10-62   5-310   13        441-7   22-64  11-384
10  989-7   10-76   5-380       15  433.4    23-07  11-56
15  917-0   10-90   5-450       30  425.5    23-51  11-77
20  -905.0   11;04   5-520;   45  417,7   23-94  11-99
25  893-5   11-20   5-600  14        410-3   24-37  12-21
30  882.0   11-84   5-670       15  403-1   24-81  12-48
85  870-7   11-.48   5-740      30  396-2   25-24  12-66
40  859-5   11'-63   5-815    o 45  389-6   25-67   12-86
45  849-3   11-78   5-890   16       383-1   26-11   138-08
50  838-9   11-92   5-960       15  876-9   26-52  18-80
55  828-9   12-06   6-030      80  370.8   26-94  13-52
819-0   12-21   6-105   045  365-0   27-37  13-73
5  813-3   12-36   6-180   16       359-3   27-83  18-95
10  807-4   12-50   6-250   o 30  848-4   28-70  14-38
15  790-8   12-64   6-320   17       838.3   29-56  14-82
20  781-9   12-79   6-895  o  30  328-7   30-43  15-25
25  778-2   12-94   6-470   18       319-6   81-29  15-69
80  764-5   13-08   6-540  o  30  311.0   32-15  16-12
35  756-1   13-22   6-610   19    8302-9   33-01  16-56
40  748-0   13-37   6-685       30  295-3   83-87  16-99
45  739-9   13-51   6-755   20       287-9   34-73  17-43
50  732-0   13-66   6-830   21       274-4   36-44  18-80
55  724-8   13-80   6-900   22       262-0  838-15  19-17
716-8   18-95   6-975   23      250-8   39-87  20-02
15  695-1    1 4-88   7-190   24     240-5   41-58  20-91
80  674-6   14-81   7-405   25       231-0   43-28  21-77
o 45  655-5   16-26   7-625   26       222-3   44-98  22-64
9     637-3   15-68   7-840   27       214-2   46-68  23-51
15  620-2   16-12   8-060   28       206-7   48-38  24-37
30  6038-8   16-55   8-275   29      199-7   50-07  25-24
46  588-4   16,99   8-495   80       1938-2   51-76  26-11
1A)    578-7   17-43   8-715   31       187-1   58-45  26-97
15  559-7   17-87   8-935   32       181-4   55-18  27-88
30  546-4   183-80   9-150   83      176-'0    56.80  28-70
45  5833-8   18-73   98-365   34      171-0   58-47  29-566
11     521-7   19-17  - 9-585  386      166-8   6014  380-42
15  510'1   19-61   9-805   86       161,8   61-80  31-29
80  499-1   20-05  10-03   837       167-6   68-46  82-15
45  488      20-50  10-25    382    168-6   65-11  33-01
2  478-8   20-94  10-47    89       149-8   66-76  83-87
16  468-7   21-36  10-69    40       146-2   68-40  34-73




TABLE OF ORDINATES.
Ordinates five feet apart.-Chord one hundred feet.
Distances of the Ordinates from the end of the 100 feet Chord.
Angle of Middle, 4 feet. 40 feet. 35 feet. 30 feet. 25 feet. 20 feet. 15 feet. 10 feet. 5 feet.
Dedi'n. 50 feet.
0o
2   007'007.007  *006'006'005  -003'003  *002  -001
4  * 014  014'014'018'012'010  -008'008  *005'003
6  *021'021'021'020'019'016  -013'011'008'004
8  *029'029'028  *026  *024'022'018'015'010'005
10'036.036  *035.033'031  -027.023  *019  -013'007
12'043'043.041.038  *037'033'028'022'015'008
14'050  -050'048'044'043'038.032'026'017'010
16  *058'058'056'052'049.044  -037'030  *020'011
18'065'065.063'059'055'050  *042  *033.023'013
20'073.072'070  *066'061.055.047  *037.026'014
22'080'079'076  *071  *067.060  -051  *041'029  [016
24'087'086'083  *077'074'066.056'045'031'017
26'094'093'090.084'080'071'060'048'034'018
28  *102  -101 -'098.092'086'077'065'052'036'019
30'109'108  -105'099'092.082.070'055'039'020
82.116'115'112  *106'098'088.075'058'042 -022
34'123  -122  *118.111'104.094'079'062'044'023
86'131'130'126.119'110'099'084'066'047'024
88'1838  -137'133'126'116'105'089'070'049'025
40'145'144'140'133'123'110'093'074'052  -027
42'152'150'146.138'128.115'098'077'055'028
44'160'158'153.145  -135.121.103'081'057'030
46'167'165'160'152'141.126'107'085'060'032
48'174'172'167'158'147'132'112'088'062'033
50'182'180'175'166'153.138'117'092'065,034
52'189'187'181'171'159'143'122'095'068'035
54'196'194'188'178'165'148'126'099'070'036
66'204'202'195'185'171.154'131'108'073.038
5 68'211'209'202'192'177'159'136'107'075.039
1'218'216.209'198'183.164'140'111'078'041
2'225'223'216'204'189.169'145'114  -081.042
4'283'231'223'211'196'175'150'118'083'043
6'240'238'230'217 ".202'180'155'121'086.046
8'247j'245, -237  -224'208'186'159'125'088'046
10'2.64 -:':252g::244'231'214  -191'163'130'091'048
12:262  "r260;252'237'220'196  4168'183'094.        049
14!.269 -? 266T::255'244'226  *'202'173'136'096.050
16,27  I174 —26 -...251   232 *207.i77 o140 o099:052
18'284    82'278'257.238'213.182'144'101  *068
20.291  - 288'279'264'244.218.187.148'104.056




TABLE OF ORDINATES —CCoNTINUD.
Ordinate8 five feet apart. —Chord one Aundred feet.
Distanoes of the Ordinates from the end of the 100 feet Chord.
Angle of Middle, 46 feet. 40 feet. 36 feet. 30 feet. 25 feet. 20 fet. 15 eet. 10 fet.  feet..e1o'n.  50 feet.
1 22'298'295'286'270'250'224'192  *161'107  *056
24'306'303'293'277'256'229'197'155'109  0567
26'313'310  3'800'284'263'235'201'169'112'059
28'320'317'307'291'269'240'206'163'114'060
80'327'324'314'297'275'246'210'167'117'062
32'334'331'321'304'281'251'215'171'120'063
34  3'841   338  3'828  3'810'287'257'219'174'122'065
86'349'345'335  3'817'293'262'224'178'125  -066
88   8656  363  3'842  3'823  299'268'228'182'127'068
40'8364  3'860  3849'330.305'273'233'185'130'069
42  -371  -867'356'337  -312'278'238'189'133'070
44'378'8374  3'863   343'318  -284'242'192'1835'07
46   385  3'882  -370'360  -324'289'247'196'138  -073
48  3'893  3'889  3'877'356  8330'295'251'200'141'075
50'400  3'896  3'884  364  8336  -300'266'204'144'076
62'407  -403'391  -370'842  -305  -261'208'147'077
54'414'410'398  376'348  3'811'2656  211'149'079
66'422'418'405  -3883  354  3'816  270'215'162'080
o 58'429'425'412   3889'360  3'822'275'219  -154'082
2'436  -432'419'397'366'827'280  -222'167  083
2  *443'439'426  -402'373  -332'284  -226'160'084
4'451'446  -433.409'379'338  -289  -230  -162'086
6'458'454  *440'416  -385  -343'293  -234 -166'087
8'465  -461'447'426.391  -349  -298'287'167'088
10'473'468'454  -430.397'355'303  -241'170'089
12  -480'475'461  -437.403  8360'308'245 -173'090
14'487'482'468  -443.409'366'312'248  -175'092
16'495'490'475'450.415'871'317'252'178.093
18'602'497'482'456'421  -377'321  2566'180'096
20'509'504'489'463'428'382'326'260'183'096
22'516'511  -496.470  -434'387'830'264'186'097
24'623'618  -603'476  -440  -393  8334'267'188'099
26  6531  -626  -510'483'446.898'338'271'191.100
28  5688  6533'617'489'452  -404  8346 -276'194'102
80  -546'640  -524  -496'458  -409.3560 1278'196'103
82  6552'647  5631'5038  465'416  -356 -282'199.104
84'560   6554'638'509'.471'420  6359'286;201'106
86'567'562  -545  -616  -477'426'364'289'204'107
88'674'669  5662'522'488  -431'368'293'206'109
40  6582  576  55669  6-529'489'486'878'297'209'110




TABLE OF ORDINATES-CoNrmUnn.
Ordinate fivi feet apart.-C-hord one hundred feet.
Distances of the Ordinates from the end of the 100 feet Chord.
Angl'e o~ Si5ddle,  45 feet. 40 feet. 35 feet. 30 feet. 25 feet. 20 feet. 15 feet. 10 feet. 5 feet.
Ded'n. 50 feet.
2 42'589.583'566.536.495  -441.378.301'212.111
44  5696'590.573  *542.501.447.382  -3804  -214.113
46  -603'598   *580.549'507.452  *387.308.217'114
48'611.605'587.555'513.458'391'312.219'116
50'618  -612'594.562'519'464'396'315.222'117
52'.625   619'601'669'526'469'401'319'225'118
654   632'626'608.575.532.474'405'322.227'119
56'640.634.616.582  -538.480.410  *326'230.121, 58'647.641.622'588.544.485.414  *330.232'123
3      ~654.648.629.595.550.491  -419.334.235'124
2'661.655'636  -602.556.496  -424'3388  238  -126
4'669'662'643'608'562.502'428'341'240'127
6'676'670'650'615'568'507.433.345'243'128
8'683.677'657'621'574'512'438'349'246'130
10.691'684'664.629'581'518.443.353.249'131
12.698'691'671'635'587'523.448.357'251'132
14.706'698'678'642.593.529'452'360.254'134
16'713'706'685'649.599.534.457'364  *257'135
18'720.713'692.655'605.540'462'368'259.137
20'727.720'699'662.611.545  -466'371  -262'138
22.734.727  -706'668'617  -550.471'375  -264'139
24'742.734'713'675  -623'556'475'378'267'141
26.749'742'720.682  -629.561.480'382  -270'142
28  -756'749'727  -688'635'567'485'386'272'144
30   *7 64.756'734'695.642  -573.489'390'275  -145
32'771.763'741'702  -648.578.494'394  -278'146
34.779.770'748.708'654;584'498'397'280.148
36'786.777'755  -716'660  -589.503'401.283.149
38.793.785'762'721  -666.594'508'405.285  -151
40'800.792'769.728'673.600'512'408.288.152
42.807.799'776.734.679  -605  -517'412'291.153
44.814'806'783.741.685.611'521'415.293'155
46'822'814'790.748'691'616'526'419  -296  -156
48'829.821'797.754'697.621.          531'423'298.158
50  -836  -828'804'761'703'627'536'427.301'159
5a2   843.835'811'768'709.632'541'431.304'160
54'850:842'818.774'716.638.545  -484'306'162
56'858.860.825'781'721'643'550'438.309'163
58.865,857'832'787.728.648  -555  442.311.165
878'864'839'794  -734 -655.559.445.314'166




TABLE -OF ORDINATES —CONTINuED.
Ordiates five feet apart.-Chord one hnmdred feet.
Distanees of the Ordinates from the end of the 100 feet Chord.
Angl of Middle, 45 feet 40 feet. 35 feet. 30 feet. 25 feet. 20 feet. 15 feet. 10 feet. 5 feet.
4  6  *891  *882.856.810'749.668'571.454.320'169
10.909'900  *874  *827.764  *682.582  *464  *327.173
15'927  *918  *891  *844  *780.695  6594'473'334  -176
20'945'936'909'860'795'709'606.482'340.179
25'963'954'926'877'810'723'617.491'347'183
80'981  -972'944'893.825'736'629'501'354'186
35.999'990'961'909'840'750'640.510'860.189
40 1'017 1-008'979'926'855  -764'652'519'367'193
45 1.036 1'026'996'943'871'777'664'529'373'196
50 1'054 1-044 1'014'959'886  -791  -676'538'380'199
65 1072 1-062 1'031'     976' 901  -804'  687'547'386'203
6     1'091 1080 1'048'993'917  -818  -699 1 557'393'207
5 1'109 1'098 1'065 1.009'932'831'711'566'400'210
10 1.127 1.116 11083 11026'947'8456  722  6576  -406'214
15 11146 1.134 1'100 1 042'963'859'734.585  -413.217
20 1.164 1-152 1'118 1'058'978'872'746.594'419.220
25 1'182 1'170 1'135 1'075'993'886'757.603'426.224
30 1'200 1'188 1-153 1-092 1-009'900  -769'613'432'228
385 1218 1'206 1-170 1'108 1' 024'913'781.622'438'231
40 1.236 1'224 1-188 1'124 1-039  -927'792'631'445'235
45 1.255 1-242 1'205 1'141 1'055'941'804'640'452.238
50 1.273 1'260 1'223 1'157 1'070'954'816'649'458  -241
66 1*291 1*278 1*240 1'174 1'085  -967'827'658'465'245
1809 1'296 1'258 1'191 1'100'982'839'668'472'248
5 1-327 1'314 1'275 1'207 1'115.995  -851'677'478.251
10 1'345 18332 1'293 1.224 1'130 1'009'862'686'485'255
15 1'364 1'350 1'310 1'240 1'146 1'023'874'696'492'259
20 1'382 1'368 1'328 1'256 1'161 1.036'886'705'498'262
25 1*400 1'386 1'345 1'273 1'176 1-050'897'714'505.266
30 1-419 1'404 1'362 1'290 1'192 1'064'909'724'511'269
35 1'437 1'422 1.379 1'306 1'207 1'077'921'733'517'272
40 1'455 1'440 1"397 1'323 1'222 1.091'932'742'524'276
45 1'473 1-458 1'415 1'339 1'238 1'105.944'752'531'280
50 11491 11476 1.432 1.355 1.253 1.118'956.761'537'283
55 1 509 1.494 1.450 1.372 1.268 1.132.967'770'544'287
1.528 1-512 1.467 1.389 1.284 1.146'979'779'551'290
5 1'646 1'530 1'484 1'405 1'299 1'159.991'788'557'293
1o 1'564 1548 1.502 1.422 1.314 1.1173 1,002'798'564'297
15 16582 1.566 1'520 1.438 1.330 1.187 1014'807'570'301
20 1-600 1.584 1.537 1.454 1.345 1200 i-026.816'576'304




TABLE OF ORDINATES-CoNTnvm).
Ordinate. five feet apart. —Chord one hundred feet.
Distanoes of the Ordinates from the end of the 100 feet Chord.
Angle of 5M9iddle, 45 feet. 40 feet. 35 feet. 30 feet. 25 feet. 20 feet. 15 feet. 10 feet. 5 fet.
DefA'. 50 feet.
o t
7 26 1'618 1'602 1'55 1'471 13860 1'214 1'037   825'5883'808
80 1'637 1'620 1'572 1'488 18375 1'228 1'048'8386  690  3811
83  1'655 1'638 1'589 1'504 1'390 1'241 1.060'844'596'814
40 1'673 1'656 1'607 1'521 1'405 1'265  1'071  *864  *603'818
45 1'692 1'674 1-624 1'537 1'421 1'269 1'083'863'610'821
50 1'710 1'692 1'641 1'553 1'436 1'282 1'095'872  *616'824
o 56 1-728 1'710 1'659 1'570 1'451 1'296 1'106'881'623 *328
8    1'746 1'728 1'677 1'587 1'467 1'310 1'118'891  *629'882
15 1-801 1-782 1'729 1-637 1'513 18351 1'153'918'649 *342
30 1'855 1'836 1'782 1'687 1'559 1'392 1'188  -946'669'85363
45 1910 1890 1890 1834 1'737 1'605 1'433 1'223 8974  689 3'868
9    1'965 1'944 1'886 1'787 1'651 1'474 1'258 1'002  *708'378
16 2'019 1'998 1'939 1'837 1'696 1'515 1'293 1'030  *728'884
30 2'074 2'052 1'991 1'887 1'742 1'556 1'328 1'067  *748 *894
45 2'128 2'106 2'044 1'937 1'788 1'597 1'363 1'085'767'405
10    2-183 2'161 2'096 1'987 1'834 1'637 1'398 1'114  *787'415
15 2'238 2'215 2'148 2'037 1'880 1'678 1'433 1'142.807 -425
80 2'292 2'269 2'201 2' 087 1' 926 1'719 1'468 1'170'827'436
[  45 2'847 2'323 2'254 2'136 1'972 1'761 1'503 1'198'846'446
11    2'401 2'377 2'306 2'186 2'018 1-802 1'538 1'226 *866'467
15 2'456 2'432 2'359 2'236 2'064 1'843 1'574 1'254'886'467
30 2'511 2'486 2'411 2'286 2'110 1'884 1'609 1'282'906'478
45 2'566 2'540 2'464 2'336 2'156 1.926 1'644 1'310  *926'488
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 18367'966 *509
80 2'730 2'703 2'621 2'485 2'295 2'049 1'750 11395'985'520
45 2'785 2' -757 2'674 2'535 2'341 2'091 1'786 11'423 1 005  5830
1~8   2'839 2'811 2'726 2'586 2-387 2'182 1'820 1'461 1'025'541
15 2'894 2'865 2'779 2'636 2'433 2'173 1'856 1'479 1'045 6561
30 2'949 2'920 2'832 2'686 2-479 2'214 1'891 1'507 1'065 6'562
46 3'000 2'974 2'884 2'735 2'5626 2256 1'926 15365 1'085'572
14    3'058 3'028 2'937 2'785 2'571 2'297 1'961 1'564 1'105'588
15 3'113 3-082 2'989 2'834 2'618 2-338 1'996 1'592 1'124'593
80 3'168 13136 3'042 2'884 2'664 2'379 2'031 1'620 1'144'604
0 45 3'222 3-191 83094 2'934 2'710 2'421 2'067 1'648 1'164'614
16    3'277 3'245 83147 2'984 2'766 2'462 2'102 1'676 1'184'625
15 3'332 3'299 3'200 83034 2'802 2'503 2'137 1'704 1'204'685
30 8'387 3'854 83'252 3'084 2'848 2'544 2'172 1'782 1'224'646
45 83442 38408'83305 3'134 28956 2'586 2'208 1'760 1'244'666
16    8'496 3'462 18'3658 8184 2'941 2'627 2'243 1'789 1'264'667




33
TABLE OF ORDINATES —CONTINUED.
Ordinates  five feet apart.-Chord one hundred feet.
Dietanoes of the Ordinates from the end of the 100 feet Chord.
Dnglon  Mide, 45 feet. 40feet. 35feet. 30feet. 25feet. 20feet. 5 feet. 10 feet. 5 fet.
D0eetn. 50 feet.                      25 feet. 0 feet. 15 feet. 10 feet.  e
0,
16 30 3-606 3.'511 3.463 38284 3.033 2.716 2-314 1-845 1.304'688
17   38716 3.680 3.569 38384 3.125 2.792 2.384 1.902 1.344'709
30 3.826 38788. 3674 38484 3.218 2.875 2.455 1.958 1.384.730
18    3-935 3'897 38779 3-584 3-310 2.958 2-525 2.014 1.424'751
30 4045 4.006 3'886 3-684 38403 3.040 2.596 2.071 1.464'772
19   4.165 4.116 38990 38784 38-495 3.123 2.666 2.127 1.504.793
30 4.265 4.223 4.096 3.884 38588 38206 2.737 2-184 1.544'814
iPO    48.75 4:332 4-201 38984 38680 3-288 2.808 2.240 1.583'836
21    4.595 4.549 4.412 4.184 38864 3.454 2.950 2'353 1-663'879
22    4-815 4.768 4-624 4.386 4-050 3-620 3a-98  2-467 1'744'922
23    5.035 4-986 4-836 46587 4,237 3.786 3-236 2.581 1.824  -965
24  5.255 5.204 5.048 4-789 4'423 38952 38379 2.695 1.905 1-008
25  5.476 5.422 5.260 4-989 4.609 4-119 38522 2.809 1'986 11051
26    5.697 5-642 5.473 5-192 4.798 4.286 3.665 2-924 2.068 1 094
27  5.918 5.860 5.685 5.393 4.984 4.454 3.808 3.039 2.150 1.137
28    6.139 6.079 5.898 5-595 5.171 4-622 38952 3.154 2-232 1-181
29    68361 6'298 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 2-396 1.268
31    6.804 6'737 6.537 6.202 5.733 5-127 4.384 35602 2.481 1.312
32    7-027 6-957 6-751 6'406 5'922 65297 4-530 3.619 2.565 1.356
33    7'249 7.178 6'965 6.609 6-111 5.467 4.676 3.737 2-649 1 401
34    7'472 7'398 7.179 6.813 6.300 5-637 4.822 3-854 2-733 1.445
36    7-694 7.619 7.393 7.017 6'489 5.807 4'968 3-972 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 2'985 1.581
38    8-367 8'286 8-041 7'633 7.060 6.320 5.410 4.327 3-069 1.626
39    8.592 8.508 8.257 7-838 7.251 6.491 566557 4.446 3.158 1.672
40    8.816 8.731 8'474 8.044 7'442 6.663 5.705 4.565 3'238 1.718




34
ARTICLE XXI.
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, instead of running from a by chords a b,
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 tangential
or deflection angles of the table, (as the case may be,) and
employing chords a d, df, fg, &c. nearly three times as
long as the chords a b, b c, &c.; or if a d, df, fy be either
2 or 4 stations apart, then 2 or 4 times the tangential and
deflection 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, 8, or 4 stations. It
is seldom desirable to exceed the latter limit.




35
TABLE OF LONG CHORDS.
RardiusIn fe et  Angle of  Length of Chord in feet required to subtend
Deflection. 1 Station.   2 Stations.   3 Stations.   4 Stations.
6780-0     1~      100       200'0      80040       400 0
4684'0        i    100       200'0      300.0       899'9
8820'0   I         100       200'0      800'0       399.9
8274'0        j    100       200'0      800'0       399'8;286560    20       100      200'0      299'9        899'7
2647 0        ~    100       200-0      299'9       899'6
2292'0        r    100       200'0      299'8       899'5
2084'0      -      100       200'0      299'8       899'4
1910'0     30      100       200'0      299'7       899'3
1763'0        ~    100       200'0      299'7       399'2
1637'0             100       200'0      299'6       399'1
1528'0        Q    100       200'0      299'6       899.0
143880     40~     100       199'9      299'6       398'9
1348'0        i   100        199'9      299'6       898'7
1274'0             100       199*9      299'4       398'6
1207'0        i    100       199'9      299'3       898'3
114640     50      100       199'9      299'2       398-0
1092'0        1    100       199'8      299'1       897'8
1042-0        {    100       199'8      299-0       897'6
996'8     ~       100       199-7      298-9       89786
966-4     60      100       199'7      298-8       397'83
917'0        ~    100       199'7      298'7       397'0
882'0        {    100       199'7      298'6       896'7
849'8     100               199'6      298'8       39676
819'0     70      100       199'6      298'4       39672
79088     2       100       199'6      298'3       396'0
7649'6            100       199-6      298'2       89657
739'9     i       100       199'6      298'1       89564
716-8     80      100       199'6      298-0       89651
-69'51,   100       199.6      29789        89458
674'6      9      100       199.6      297.8       894.4
665-6     ~       100       199.4      297'7       894'3
6873      go90    100       199.4      297.6       894.1
620'2    ~        100       199'4      297'4       893'7
603.8     i       100       199'3      297'3       393'2:588'4   1         100       199'2      297'2       892'8
573.7    10~      100       199.2      297-0  -   892. 4
For radii less than 673.7 feet, it is never required to use longer
chords than 100 feet.




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 point 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
curve, only the tangential and deflection 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 deflection 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 deflection 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 deflection distances of the table must be multiplied by
4, 9, or 16, as the case may be. In this case the tangential and deflection 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 deflections 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 loss of time.
The following is a good form of field-book for the transit
and compass combined.
~Tot<~ IDeflection   The right hand page iS
Station. Distance. Distance. Course. in Degrees. lef blank for Remarks,
ItLeft. I ight.n Sketches of TopograLeft. I Right.   phy.




37
In every locating party there should be one person whose
duty is to obtain, and record the transverse slopes of the
ground at each station. His observations will usually extend to from fifty feet, to one hundred 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, approximately, 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 another
column to his field-book, after that con-     A -
tainirg the deflections.  In this column, L   70
he will insert the slopes, thus, (Fig. 20.)  10~
the dot representing the center 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 on the ground,
at right angles to the line of survey, as nearly as may be
judged 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 XXII.
TO ADJUST A TRANSIT INSTRUMENT.
Having placed the transit firmly at a, fig. 21, and levelled it, clamp all fast, and direct the cross-hairs, by means
of the tangent screw, 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
4




38
object in the opposite direction, as c; or, if there be no
such object, place one, as for instance a chain-pin, at any
convenient distance.
l b.d'
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 teleecope 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-hair so as to strike
halfway between d and c. The instrument will then be
in adjustment.
Two or more 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 doorway 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 its construction, it does not, the defect must
be remedied by an instrument-maker.
ARTICLE XXIII.
Frequently it may be necessary to commence a curve at
a point A, which is less than 100 feet from the preceding station M; and in this case it conduces to convenience in
drawing the profile of the work, to make the first part of
the curve a subchord A C, of such a length as will just
make up the 100 feet from M. The stations will then coincide with the vertical lines on the engraved profile-paper.
Although the method of proceeding in this case is extremely
simple, and readily deducible from Article IV., page




39
still those who have not yet acquired a facility in applying
the various modifications, will not object to the following
illustrationON COMMENCING A CURVE WITH A SUBCHORD.
Place the instrument at A, Fig. 22, the commencement of
the curve, and first sighting back along the tangent A M, lay
off the subtangential angle O A C, bearing the same proportion to the entire tangential angle of the curve that the
subchord A C, bears to the entire chord C D of 100 feet.
Then with the instrument still remaining at A, continue
the curve by laying off entire tangential angles, C A D,
D A E, &c., and entire chords C D, D E as usual.
Fig. 22.
A
Or if, in consequence of obstructions to the view, the instrumnent has to be removed to the end C of the subchord
A C, first sight back to the beginning of the curve at A,
and lay off for a defiexion angle, S C D the sum of the subtangential angle, and an entire tangential angle, making
C D an entire chord; and continue the curve as before,
with entire tangential angles and chords.
ARTICLE XXIV.
ON THE CORRECTION OF COMPOUND CURVES.*
Having laid out a curve from A to B, Fig. 28, and there
changed it to a compound curve by adding B C —and finding that at C it terminates tangentially to a( N, parallel to
F M, instead of terminating in F M,-it is required to find
a new changing-point K, so that the compound curve,
* This has been introduced at the suggestion of Mr. JOSEPH D. POTTS,
my Principal Assistant on the Coal Run Railroad, to whom I am indebted for it.




40
traced with the same radii may terminate at F tangentially to F M.
There are four cases, viz:
1st. When the shorter radius follows the longer, and the
tangent sought is exterior to the tanfgent traced.
2d. When the shorter radius follows the longer, and the
tangent sought is interior to the tangent traced.
Figure 23 illustrates cases 1 and 2.
3d. When the longer radius follows the shorter, and the
tangent sought is exterior to the tangent traced.
4th. Where the longer radius follows the shorter, and the
tangent sought is interior to the tangent traced.
RULE.-First find the distance OF on the ground, measured at right angles to the parallel tangents C N and F M.
Fig. 23.
\\ 
Then multiply the number of chords run in the last se
Then multiply the number of chords run in the last section of the compound curve, by the angle of deflexion of
said curve, and call the resulting angle m.
Then say, as rad. 1: nat. cosine of angle mi:: the difference between the 2 radii to a certain quantity, which
call q.
In cases 1 and 4, add O F to q; or in cases 2 and 3,
subtract O F fiom q, calling the sum, or difference, (as the
case-may be,) x.




41
Then, as the difference between the radii is to x, so is
radius 1, to a certain tabular cosine. The angle opposite
this cosine in the table will be the number of degrees to
be run in the last part of the new compound curve; and
when subtracted from the sum of the entire angles subtended
by the two parts of the compound curve, will give the number of degrees that the first part of the curve must be previously run, before beginning the second part.
-Example.-Let Fig. 23 represent case 1.
Let A B be a 40 curve; its radius therefore 1433 feet.
Let B C be a 6~ curve; its radius therefore 955 feet.
Difference of radii, or E D, 478 feet.
Also suppose B C to have been traced for 710 feet, or
7-1 chords;  giving for the angle B  D  C or D E H,
6~ x A1 = 42~'6 = 42~ 36'; and let O F be found to be
20 feet.
Now to find E GA( Rd 1.   Nat. Cos. of.  Diff. between the    E G or q of the
As Ra..  * iangle D E    *  2 Rad. (E D)     rule.
That isas Rad. 1.  *  7361     *     478 feet   *     351-9
Now to find E P or x of the rule, since the example
is case 1, add 0 F or 20 feet to 351'9 or q, making 371'9
feet = E P or x of the rule.
Now to find the angle I E H, or K I P, of which E P
is the cosine, say
As E I or diff. of Radii. E P or x..  Rad. 1. a Tabular Cosine, or
478 feet  ~    3719. *    Rad. 1   ~  7780
Now the angle found in the table opposite the natural
cosine'7780 is 380 55', which is the number of degrees
that the last section K F of the compound curve must be
traced from K.
Finally, suppose that A B, the first part of the 40 curve,
had been traced for 6 chords; then the angle A E B
would be 40 x 6   240; and as the angle B D C is 420 36'
their sum, or the total angle of the compound curve, is
66~ 36'; therefore if from 660 36' we subtract the new angle
K I F=380 55', we have 270 41' as the number of degrees
that the curve A  B K must first be traced in order to
reach K, the point of change of curve.
4*




42
DEMONSTRATION OF CASE 1.
Fig. 23 represents cases 1 and 2, and by drawing a
figure illustrative of cases 3 and 4, the demonstration on
the same principle will be evident.
With a radius D E equal to the difference between the
two radii E B and D B of the compound curve, describe
the curve D H, and make the angle D E H  equal to the
angle B D C.  From  D  draw D  G at right angles to
E H: also make G P equal to 0 F the perpendicular
distance between the two tangents C N  and F M, and
from P draw P I at right angles to E H; also from I
draw I F at right angles to the two tangents. F will be
the point of termination of the new curve. Finally, continue E I indefinitely towards K.
Now the angle D E H having been made equal to the
angle B D C, E H is parallel to D C.  Also F I and ID  C
both being at right angles to the parallel tangents, are
parallel to each other, and consequently F I is parallel to
E II; therefore the angle K I F is equal to the angle I E H.
Now the point required is to determine the number of
degrees in I E H or K I F, in order to obtain the distance
of the new point K, of change of curvature, from F.
To do this, we have in the triangle D E G, as radius 1
is to the natural cosine of the angle D E G (m of the rule)
so is E D, the difference between the 2 radii, to G E (q
of the rule.)
Now to G E, thus obtained, add G P = 0 F, previously
measured on the ground; this gives E P (x of the rule.)
Then in the triangle I E  P we have, as E I, the difference between the 2 radii, is to E P (x of the rule,) so
is the tabular radius 1 to the cosine of the angle I E H,
or its equal angle K I P.
NOTE.-In that excellent little work, " IIENCK'S FIELD BooK roR
RAILROAD ENGINEERS," recently published, Mr. Henck applies the term
" tangent distance" to the sine of the tangential angle, whereas I have in
this book applied it-to the chord of the tangential angle. Had I anticipated the double application, I should have avoided it by the use of
some other term, and thus have prevented the confusion likely to arise
from it.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
0eg.                   0                0Deg.                                    0 Deg.
Sine.  T'ang.  Cotang. Cosine.            Sine.  Tang.  Cotang. Cosine.             ine.  Tang. Cotang. Cosine.
0'0000000'000000 Infinite. 1'000000 50 1'0061086'006108 163'7001'9999813 39 1'0119261'011927 83'84350'9999289 19
1 000o2909'000291 3437'746 1'000000 59 2 0063995 *006399 156'2590 -9999795 38 2'0122170'012217 81-84704'9999254 18
2'0005818 *000582 1718'873'9999998 58 3 *0066904'006690 149'4650 *9999776 3743'0125079'012508 79 943431'9999218 17
3'0008727 1000872 1145.915 19999996 57 4'0069813'006981 143.2371. 9999756 36 44 0127987 1012799 78s12634 9999181 16
4.0b11636'001163 859'4363 19999993 5 25'0072721 1007272 137.5075'9999736 35 5.0130896'013090 76.39000'9999143 15
5 10014544 1001454 687.5488'9999989 55 26 10075630 1007563 132'2185'9999714 34 6 0133805'013381 74'72916'9999105 14
6 10017453 10017451572-9572'9999985 54 27'0078539'007854 127'3213'9999692 33 7'0136713'013672 73-13899'9999065 13
7'0020362'0020361491'1060 *9999979153 8 10081448'008145 122'7739'9999668 32 8'0139622'013963 71'61507'9999025 12
8'0023271'0023271429'7175'9999973152 29 0084357'008436 118'5401'9999644 31 9'01425301 014254 70'15334'9998984 11
9'0026180'002618 381.9709'9999966 51 30 10087265 1008726 114 5886 -9999619 30 50 0145439 10145451 6875008 19998942 1l0
10'0029089'0029081343'77371 9999958150 31 10090174 1009017 110'89201 9999593 29151 101483481 014836167'40185'99989001 9  C1
11'0031998'0031991 312'5213'999994949 321 40093083'009308 107'4264'9999567 28 52 0151256 0-15127 66'10547'9998856  8
12'0034907'0034901286.47771'9999939 14833'00959921'009599 104'17091 9999539 27531 0154165j.015418 64.85800'9998812 7
13 100378151 003781 1264'4408'9999928 47 341 00989001 0098901101-10691 9999511 26 541 0157073'015709 63'656741 9998766, 6
141 0040724'004072 245.55191 9999917146 351 0101809 0101811980217941'9999482125 51 0159982'016000 62'499151 9998720 5
15 10043633'0043631229.1816 19999905 45 36'0104718'010472 95.48947 19999452 24 561 01628901 016291161,38290 19998673  4
16'00465421 0046541214'8576'9999892 44 371 0107627'010763 92'90848 19999421 23571 01657991 016582 60'305821 9998625 3
17'0049451 1004945 202-21871"9999878 43 38 101105351 011054190.463331 9999389 22 58 0168707'0168-3 159.26587'9998577 2
18'0052360 1005236 190.98411 9999863 42 91 0113444 1011345 88 143571 9999357 21 59 -01716161 017164 58 26117 9998527 1
19 10055268 1005526 1809322 19999847 41 0'0116353 3011636 18593979'9999323 120 00174524'017455 57-28996'9998477 0
201 0058177 1005817 171.88541 9999831 40
Cosine.  Cot n.| Tanz. I Sine.'' Cosine. Cotan. - Tang. I Sine. I t         Cosine. Cotan. I Tang.  ISine.
Deg. 89                                DPet. 89.                                 Deg. 89




NATURAL SINES AND TANGENTS TO A RADIUS 1
1 Deg.                                  1 Deg.                                  1 Deg.
r   Sine.   Tang. Cotang. Cosine. t o   Sine.   Tang. Cotang. Cosine.'  /  Sine.   Tang. Cotang. Cosine.
0 O0174524 *017455 57 28996'9998477 6 21'0235598'023566 42-43346'9997224 39 41'0293755'029388 34'02730'9995684 19
1'0177432'017746 56'35059'9998426 59 22 0238506'023857 41'91579'9997156 38 42'0296662'029679 33'69350 *9995599 18
2'0180341'018037 55'44151 19998374 58 23'0241414'024148 41 41058i*9997086 37 43 0299570'029970 33'36619'9995512 17
3 "0183249'018328154-56130'9998321 57 24'0244322'024439 40'91741'9997015 36 44 0302478'030261 33'04517'9995424 16
4'0186158 *01861915370858'9998267 56 25'0247230'024730 40'43583'9996943 35 45 0305385'030552 32'73026'9995336 15
5 s0189066'018910 52-88211 9998213 5 26 0250138 025021 3996546 9996871 3446 0308293 030843 3242129 9995247 14
6'0191974'019201 52008067'9998157 54 27'0253046'025312 39'50589 09996798 33 47'0311206'031135 3211809 *9995157 13
7'0194883'019492 51'30315'9998101 53 28'0255954'025603 39'05677'9996724 3248'0314108'031426 31'82051'9995066 12
8'0197791'019783 50'54850'9998044 52 29'0258862'025894 38'61773'9996649 31 49'0317015'031717 31'52839'9994974 11
9'0200699'020074 49'91572'9997986 51 30'0261769'026185 38'18845'9996573 30 50'0319922'032008 31'24157'9994881 10.
10'0203608'020365 49'10388'9997927 50 31'0264677'026477137-76861'19996497 29 51 10322830'032299 30'95992'9994788 9   "
11'0206516'020656 48'41208'9997867 49 32'0267585'026768137'35789'9996419 28 521 0325737'032591 30'68330'9994693  8
12'0209424'020947 47'73950'9997807 48 33 -0270493'027059 136'95600'9996341 27 53'0328644'032882 30'41 158'9994598 7
13'0212332'021238147'08534' 9997745 47 34'0273401'027350 36'56265'9996262 26 54 0331552'033173 30-14461'9994502  6
14 10215241'021529 46'44886'999768346 35 1 02763091.027641 36'17759'9996182 25 55 0334459'033464 29'88229'9994405  5
15'0218149'021820 45'82936'9997620145 36'0279216'027932 3580055'9996101 124 56 0337366 1033755129'62449'9994308 4
16'0221057 -022111 45'22614'9997556144 37'0282124 02822 23 3543128 -9996020 23 57 0340274'034047 29'37110'9994209 3
17'0223965'022402144'63859'9997492 43 381'0285032'028514 35'06954'9995937 22 581.0343181'034338 29'12200'9994110 2
181 0226873'022693 44'06611'9997426142 39 0287940'028805 34'71511 -9995854121 59 0346088'034629 28'87708'9994009 1
19'0229781'022984 43'50812 9997360 41 40 -0290847'029097 34'36777'9995770 20 60 0348995'034920 28'63625'9993908 0
20'0232690' 023275 42'96407'9997292 40' Cosine. |Cotan.  Tang.   Sine.'   Cosine. Cotan.  Tang.   Sine. |              Cosine. Cotan.  Tang.   Sine.
Deg 88.                                 Deg. 88.                                 Deg. 88.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
2 Deg.                                  2 Deg.                                   2 g.  2 Deg.
S  ine.   T ang.  Cotang. Cosine.     /  Sine.   Tang. Cotang. Cosine. / [/  Sine.   Tang. Cotang. Cosine. /7t
lot.                         O                  O _                         _ _.
0 0348995 *034920 28-63625 *9993908 60 21 *0410037 *041038 24-36750 *9991590 39 41 0468159 *046867 21.33685 *9989035 19
1 0351902 *035212 28-39939'9993806 59122 *0412944 *041329 24-19571 *9991470 38 2'0471065 *047158 21-20494.9988899 18
2 0354809.035503 28-16642 19993704 58 23.0415850 *041621 24-02632 *9991350 3743 10473970 -0474-50 21-07466'9988761 17
3035771 6 1035794 27.93723 19993600 57124 104-18757 1041 912 23 85927 19991228 36!4'0476876 1047741 I20-94596 19988623 116
41:0360623'036085 27-71174'9993495 56f25.0421663 -042203 23-69453 *9991106 35]45'0479781'048033 20.81882'9988484 15
5  0363530 *036377 27.48985 9993390 55 26 0424569 *042495 23-53205.9990983 34146'0482687 0483225 20.69322.9988344 146 0366437'036668 27-27148.9993284 54 27 *0427475 *042786 23-37177'9990859 33 47 104855921 048616 20'56911 t9988203 13
7-0369344.036959 27.05655 *9993177 53 28 0430382 *043078 23-21366'9990734 3248.0488498 1048908 20-44648 1 9988061 12
8I s037225 11037250 12684498s 19993069 52 29.0433288 1043369 23-05767- 9990609 3149 04914031 049199 20.32530 19987919 1 1
91 037515S8 037542 26 63669 -9992960151 301 04361941 043660122 90376'9990482 13050 0494308 1-049491 120205551 9187'775 1O 
10103780651 037833 26.43160.9992851 50 31 104391001 043952122.75189 19990355129151 10497214 -049782120-087191 9987631  9t
11'03809711 038124 26'229631 9992740 149 321 04420061 0442431 2260201 19990227 28152 -0500119 1050074 19.97021'9987486 8
12.0383878.038416 26.03073.9992629 48 331 0444912. 044535 22.45409.9990098 2753 i-0503624 050366 19.85459'9987340 7
13'0386785 -038707125'83482'9992517147 341 04478181 044826122'30809'9989968 26154 1 0505929'050657 19-740291 9987194  (3,
14 03896921 038998 12564183'9992404 46 351 04507241 045118 22 163981 9989837125 551 0508835 10509491 19627291 9987046  5
1510392598'039290 25.451701 9992290145 361 0453630 1045409 22.021711 998970612456 1-0511740'051241 19-515581 9986898  4
161 03955051 -039581 125264361 -9992176 44371 0456536'045701 21'88125 19989573 23157'0514645'0515'32 19-40513'9986748  3
117.0398111 -039872 25-07975'9992060143138 -0459442'045992121-74256 -9989440122 581 0517550 -051824119-29592'99865981 2
18 0401318'040164 24'89782'9991944 42 391 -0462347'046284 21-60563 -9989306 21 59 -0520455 -052116119-18793 -99864471 1
19'04042241 040455 24.71851 199918271 41 40 04652531 046575 21.470401 9989171 120i60 10523360'052407 19-08113  9986295  O
201'0407131'040746 12454175'9991709 40
/ I Cosine. |Cotan.  Tan.  Sine. I  It Cosine. Cotan.  TanI. / Sine. I  I/  Cosine. Cotan./  Tanr.  Sine.   /
Deg. 87.                                Deg. 87.                                Deg. 87.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
3 Deg.                                  3 Deg.                                   3 Deg.
-   Sia;e.  Tang. Cotang.  Cosine. I  /  Sine.  Tang. Cotang. Cosine.'    Sine.  Tang. Cotang. Cosine.0 *052336(t 052407 1 908113'9986295 60 1'0584352'058535 17.08372 -9982912 39 41 0642420'064375 15.53398'9979343 19
1'052626'*052699 18-97552'9986143 59 2  0587256'058827 16-99895' 9982742 38 2'0645323'064667 15'46381'9979156 18
2'0529169'052991 18-87106'9985989 5823 *0590160 *059119 16'91502'9982570 37 43 0648226'064959 15'39427'9978968 17
3 10532074'053282 18-76775'9985835 57 4'0593064'059410 16'83191'9982398 36 44'0651129'065251 15'32535'9978779 16
4'0534979.053574 18'66556'9985680 5 25 0595967'059702 16'74961'9982225 3545'0654031'065543 15-25705'9978589 15
5 -0537883'053866 18'56447'9985524 55 26 0598871'059994 16'66811'9982052 34 46'0656934'065835 15'18934'9978399 14
6'0540788 0564158 18046447'9985367 5 27'0601775'060286 1658739 *9981877 33 7'0659836'066127 15'12224'9978207 13
7'0543693'054449 18036553 9985209 53 8'0604678'060578 16'50745 *9981701 32 18'0662739'066419 15005572'9978015 12
81 0546597'054741 18-267651 9985050152 29'0607582'060870 16'42827.9981525131 9'0665641'066712 14-98978'9977821 11
91 0549502'055033 18'17080'9984891 51 30'0610485'061162 16 349851 9981348 30 0'06685441 067004 14092441'9977627 10 
101 05524061 055325 18'07497'9984731 50 31 0613389'061454 1627217'9981170 29 1'0671446'06729611485961'9977433 9
11 -0555311'055616 17'98015'9984570 49 32'0616292'061746 16'19522 -9980991 28 2'0674349'067588 14-79537'9977237 8
121 05582151 055908 17.886311 9984408 48 33 0619196 *062038 1601 1899'9980811 27 3 0677251'067880 14'7316719977040  7
13'0561119'056200 17.79344. 9984245 47 34'0622099 062330w 16'04348 -9980631 26  - 0680153'068173 14'66852'9976843  6
141 0564024~ 056492 17.70152'9984081 46 35. 0625002 062622 15-96866'9980450 25 55'0683055'068465 14'60591'9976645  5
15j 05669281 056784 17'61055 998391745 36 0627905 062914 15'894541 9980267 2456'0685957 -068757 14'54383'9976445 4
16'0569832'057075 17'52051'9983751 44 37 0630808.063206 15.82110. 9980084 2357 -0688859'069049 14-48227'9976245  3
17'0572736 -057367 17'43138. 9983585 43 381.06337111,063498 15-74833'9979900 22 58'06917611-069342 14.42123,'9976045  2
181 05756401 057659 17'34315'9983418 4 39'0636614 063790 15'676231 9979716 21 9'0694663'069634 14'36069 -9975843 1
191 05785441-057951 17.25580'9983250 4140 0639517 4064082 15,60478. 9979530 20 60'0697565'069926114'30066'9975641 0
20'0581448 1058243 17'16933'9983082 4
tI Cosine. ne. Cotan.   Tang.T  Sine. C osine.                                  t Cotan.  Tang. S            Sine'
Deg. 86.                               Deg. 86.                                 Deg. 86.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
4 Deg.                                  4 Deg.                                   4 Deg.
/  Sine.   Tang. Cotang. Cosine'         Sine.  Tang.  Cotang. Csine.            Sine.   Tang. Cotang. Cosine.
0 *0697565 *069926 14'30066 -9975641 60 211 0758489 -076068 13'14612'9971193 39 41 10816486 1081922 12'20671'9966612 19
1'0700467. 070219 14'24113 19975437 59 22 10761390'076360 13'09575'9970972 38 42'0819385'082215 12'16323 19966374 18
2'0703368'070511 14'18209'9975233 58 23 10764290'076653 13.04576'9970750 37143'0822284 }082507 12'12006'9966135 17
3'0706270'070803 14'12353'9975028 57 24 10767190 1076945 12'99616'9970528 36 44 0825183'0828 800 1207719'9965895 16
4'0709171'071096 14'06545'9974822 56 25'0770091'077238 12'94692'9970304 35145'0828082'083093 12'03462'9965655 15
5'0712073'071388 14'00785'9974615 55 26'0772991'077531 12-89805 19970080 34146 -0830981'0833.86 11'99234'9965414 14
6'0714974'071680 13'95071 19974408 54127'0775891'077823 12-84955'9969854 33147 10833880 1083679 11'95037'9965172 13
7' 0717876'071973 13'89404 19974199 53 28 0778791 -078116 12'80141'9969628 32 48'0836778'083972 11'90868 *9964929 12
8'0720777'072265 13'83782'9973990 52 29 0781691 1078409 12-75363'9969401 31 49 108396771 084265 11'86728 19964685 11
91 0723678 072558 13-78206 9973780 51 30'0784591'078701 1270620'9969173 30 50 0842576'084558 11'82616'9964440 10
10'0726580 1072850 13'72673'9973569 50 31'0787491 1078994 12.65912 19968945 29151'0845474'084851 11'78533'9964195  9
111 0729481'073143 13'67185'9973357 49 32'0790391'079287 1261239'9968715 28 52'0848373'085144 1174477'99639481 8
12'0732382'073435 13'61740'9973145 48 33'0793290 [079579 12'56599'996848512 253'0851271'08543711'70450'9963701 7
13'0735283.073727i 13-56339 19972931 47 34'0796190'0798721 1251994'9968254 261541 0854169'085730 11'66449'9963453 6
14'07381841'074020 13'50979'9972717 46 35'0799090.080165 12'47422'9968022 25155'0857067 1086023 11162176'9963204 5
15'07410851'0743121 L3'45662'9972502 45 36'0801989'0804581 12'42883'9967789 24156 10859966 ['086316 1 158529 19962954  4
16'07439861-074605 113403861 9972286144 37'0804889'080750 12-38376 19967555 123 57'0862864 086609 11'54609 19962704 3
17'0746887'074897 13'35151'9972069143 38 108077881'081043 1233902 19967321 22 58'0865762 -086902 11150715 199624521 2
18'0749787,-075190 13'29957'9971851 42 39'08106871-081336 12'29460'9967085 21 591 0868660,087195 114-68471 9962200  1
19. 0752688'075482113-248031 997163341 40 108135871081629 12-25050 5019966849120 601 0871557 1087488 11.430051 9961947 0
201'0755589 -075775 1319688 19971413 40I
S Cosine. I Cotan.  Tang.   Sine.  t                                            Cosine. Cotan.  Tang.   Sine.
Deg. 85.                               Deg. 85.                                 Deg. 85.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
5 Deg.                                  5 Deg.                                  5 Deg.
I Sine.   Tang. { Cotang.  Cosine. -    - Sine. - Tang.  Cotang. I Cosine. [' L    Sine. { Tang.  Cotang.  Cosine. 
0 *0871557 -087488 11'43005 -9961947 6021 *0932395'093647 10-67834'9956437 39 1'0990303'099519 10'04828'9950844 19
1'0874455'087781 11.39188'9961693 59 22 -0935291'093940 10.64499 *9956165 38 42'0993197.099813 10.01871'9950556 18
2.08773531 088074 11135397 *9961438 58 23 10938187'094234 10-61184.9955892137 43'0996092'100107 9 989305'9950266 17
3'0880251 088368 11 31630 *9961183 57 24'0941083 *094527 10'57889L-9955620 36 44'0998986 100400 9960072 994949976 16
4'0883148'088661 11'27888'9960926 56 25 0943979'094821 10'54615.9955345 35 45 *1001881 Q100694 9-931008'9949685 15
5'0886046'088954 11-24171 -9960669 55 26'0946875'0995114 10'51360'9955070 34146'1004775'100988 9'902112'9949393 14
6'0888943'089247 11.20478 *9960411 54127'0949771'095408 10.48126 -9954794 33 17'1007669'101282 9'873382'9949101 13
7 0891840 089540 11-16808'9960152153 28'0952666'095701 10'44911'9954517 32 18 1010563'101576 9844816'9948807 12
8.0894738]-089834 11113163.9959892152 29.0955562 09599551 1041715'9954240 31 19'1013457'101870 9.816414 1994851311
9'0897635 1090127 11.09541'9959631 151301 09584581096289 110'38539'9953962130 501 1016351'102164 9'788173'9948217{10   FP
10| 0900532| 090420 11105943| 9959370 50 31'0961353 1096582 10'353821 9953683 29 511 1019245'102458 9.760092 19947921| 9' 11 0903429'090713 11'02367'9959107 49 321 0964248'096876 10'3224,4'9953403 28 52'1022138'102752 9'732171'9947625 8
12'-0906326 l-0910071098815 -9958844148 33'09671441'097169 10-29125j 9953122 27 53 1025032'103046 9'704407'9947327 7
13 0909223'091300 10'95285 19958580 47 341 0970039'097463 10'26024'9952840126 54' 10279251 103339 9'676800'9947028  6
1t4-0912119 -091593 10.91777'9958315146 351 09729341 097757 10.229421 9952557 25 55 10308191 103634 9.649347'9946729  5
15'0915016'091887 10.882921 9958049 45 36'09758291 098050110'198781 9952274 24 561 1033712'103928 9'622048 -9946428 4
16'-0917913'-092180 10'84828'9957783 44 37'0978724'098344 10 168331-9951990 23 57'1036605 -10422219.594902 -9946127 3
17.0920809.-092473110813871.9957515 43 381.0981619'098638110.13805'9951705 22 581'1039499 1104516 9'567906{'9945825 2
18.09237061 092767 10.779671-9957247 42 39 -09845141 098932 10.107951 9951419 21 59'1042392'104810 9'541061'9945523  1
19.0926602 1093060 10'74568 -9956978 41 40 10987408'099225 10'07803 -9951132 20 60 10452851 105104 9'514364 -9945219 0
201 0929499'093354 10'711911 9956708140
Cosine. Cotan.  Tan-.   Sine. |        Cosine. Cotan.  Tang.   Sine.' Cosine. I Cotan.  Tang. I Sine. |
Deg. 84.                                Deg. 84.                               Deg. 84.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
6 Deg.                                 6 Deg.                                  6 Deg.
Sine.  Tang. CI otang. Cosine. -' 7      Sine.  Tang. Cotang. Cosine.            Sine.  Tang. Cotang. Cosine..0'1045285'105104 9'514364'9945219 60 1 -1106017  1 1284 8-985984 19938648 39 41 1163818'117178 8.534017'9932045 19
1'1048178'105398 9'487814'9944914 59 22'1108908'111578 8'962266'9938326 38 42'1166707'117473 8'512594'9931706 18
2'1051070'105692 9'461411'9944609 58 23'1111799'111873 8'938672'9938003 3743'1169596 1 17767 8'491277'9931367 17
3'1053963 1105986 9.435153'9944303 57 4 -1114689'112168 8.915200 1993767936 44 1 172485 1 18062 8-470065 19931026 16
4'1056856'106280 9.409038'9943996 56i25 1117580'1 12462 8'891850 {9937355 35 45 1175374'118357 8'448957'9930685 15
5 *10.59748 1 06575 9-383066'9943688 55 6 1 120471  112757 8'868620 9937029 346  1178263 11 18652 8'427953 *9930342 14
6 61062641'106869 9'357235 9943379 54 7'1123361 1 13051 8'8455101 9936703 33471 11811511 118947 8'407051'9929999 13
7'1065533 107163 9-331545'9943070 53 28'1126252'113346 8'822518'9936375 32 48 1 184040'1 19242 8'386251 -9929655 12
8.10684251 107457 9.305993 19942760 52 29'1129142 1113641 8.799~44'9936047 31 9 -1186928'119537 8.365553'9929310 11
9110713181 107751 19280580 1994244s851 301 1132032'113935 18776887 19935719 30 50 11898161 1198321 8344955'9928965 10 
10' 10742101 108046 19255303'9942136150 31 1134922 111423018'754246'9935389 29 51 1192704'120127 8 324457'9928618 9
11'1077102'108340 9'230162'9941823149 32'1 137812'1 14525 8'731719'9935058 2852'1195593'120423 8'304058'9928271 8
12 1079994'1l08634 9205156'9941510 48 33 1140702'11481918'709307'9934727 27 53 1198481'120718 8'2837571 9927922 7
13'1082885'108929 9.180283'9941195 471341 1143592'115114 18687008 -9934395 261541 12013681 121013 8'263554'9927573 6
14'1085777'109223 9.1555431 9940880 46 35'1146482'115409 8'664822'9934062 25 55'1204256'121308 8'243448'9927224 5
15'1088669'109517 9.130934'9940563 45 36 -1149372'115703 8'642747'9933728 24 56'1207144'121603 8'223488'9926873 4
161 10915601 109812 9.1064561 9940246 44 371 l 152261 1115998 18620783 -9933393123157' 1210031'1121898 18203523 19926521 3
17'10944521.1 10 0619.0821071.9939928143 38.1155151'11629318s598929 19933057 22 58 1212919'122194 8'1837041 9926169  2
18'1097343'110401 9057886'9939610142 39'1158040 116588 8'577183'9932721 21 59'1215806'122489 8'163978'9925816  1
19' 1100234'110695 9'033793'999939290 41 0 1160929'116883 8'555546 -9932384 20 60' 1218693'122784 8'144346'9925462 0
20'-1103126 1110989 190098261'9938969 40
Cosine. ICotan.  Tang. I Sine.  t   | Cosine.  Cotan.  Tang. | Sine.  t {t" Cosine. Cotan.| Tang.  Sine. |t
Deg. 83.                               Deg. 83.                               Deg. 83.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
7 Deg.                                   7 Deg.                                  7 Deg.
_ r   Sine. - Tang.  Cotang. - Cosine. i' L/'  Sine.   Tang. ICotang. - Cosine.'/ - Sine.  Tang.  Cotang. Cosine. -
0.1218693 *122784 8-144346.9925462 60 21 *1279302 *128990 7.752536i 9917832 39 41 *1336979 -134909 7'412397 *9910221 19
1 1221581 *123079 8~124807 ~9925107 59 22 1282186'129285 7'734802 -9917459 38 42 ~1339862 *135205 7-396159 9909832 18
2 */1224468 *123375 8-105359'9924751 58123'1285071 129581 7 717148 *9917086 37 43 *1342744 *135501 7'379990 *9909442 17
31*1227355 *123670 8-086004 *9924394 57 24 *12879,56 *129877 7 699573 *9916712 3644 1345627 *135797 7-363891 9909051 16
4'1230241 *123965 8-066739 9924037 56 256 1290841 *130173 7.682076 19916337 35145.1348509 *136094 7-347861.9908659115
5 *1233128  124261 8'047564 *9923679 55 26 1293725 1130469 7.664658 *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 19
7 *1238901 124852 8-009483 ~99229f9 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 7.612865.9914828 31 19 1360038 *137279 7.284418.9907083 11
9 -1244674 125442 7-971755 9922237 51 30 1305262 131652 7'5957541 9914449 30 50 1362919 137575 7'268725 9906687 10  C
101'1247560 -125738 7-953022 -9921874 50131 11308146.131948 7.578717. 9914069 29 51 -1365801 137872 7.253098 9906290  9
11.1-250446 l126033 7'934375 9921511 49 32 1311030 5132244 7'561756'9913688 28152' 1368683 138168 7-237537'9905893  8
12'1253332 -126329 7     915815 992114.7 48 33 13139131 132540 175448639 9913306 27153 -1371564138465 7222042 9905494 7
13'1256218  12662417'897339'9920782147 34 I13167 97-132836 17528057 i991292' 26 54.13744451 138761 7-206611 -9905095  6
141-1259104~ 1269201 78789488 9920416 4635 *1319681 1.331320 751131 7 9912540 25155 -1377327 -139058 7-1912458 9904694  5
151-1261990- 127216 7 8606421.9920049 14536 -1322564 -133128 7'494651 |9912155 1214 56 1380208  139354 7-1759431 990429.3  4
161-1264875 ~127511 7-842419 -99196824437 11325447l133724 7478057 1 9911770 23157 -1383089 11396517'1607051 9903891  3
17.1267761.12780717.824279.19919314 4338 13283301-13402017461535.9911384 22 58 1385970 Q139947 7 14553019903489  2
18'1270646 -128103 178062 2]1 -9918944 42139'1-331213 i134316 7'445085 1991099 7 21 59.1388850 *140244 7-130419 19903085  1
1 9 12735311-1128398 7-788245  991857414140 1-1334096 -134612 7'428706 19910610 20 60 -1391731 140540) 7115369 9902681
20 -1276416 128694 7770350 -9918204 40 1                                                                       ___
Cosine. Cot.an. |Tan -.   Sine.  /|' I   Cosine. C (otan.  Tang.   Sine.      I /  Cosine.' Cotan.  Ta;ng.   Sine.
Deg. 82.                                 Deg. 82.                                  Deg. 82.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
8 Deg.                                  8 Deg.                                   8 Deg.
I Sine.   Tang. Cotang. Cosile.            Sine.   Tan,.  Cotang.        neCosine.   /    Sine.''ang. Cotang. Cosine. i
0'1391731 -140540 7'115369'9902681 60 1 1452197'146775 6'813122'9893994 3941 *1509733'152723 16547767'9885378 19
1 -1394612'140837 7'100382'9902275 59 22'1455075 147072 86799356'9893572 38 42'1512608 -153021 6'535029 -9884939 18
2 11397492'141134 7'085457 19901869 58 23'1457953'147369 6'785644'9893148 37 43'1515484'153319 6'522339'9884498 17
3 11400372 1441430 7'070593'9901462 57 24'1460830'147667 6'791986'9892723 36 44 *1518359 *153617 6'509698'9884057 16
4 -1403252 ~141727 7-055790'9901055 56 56 1463708'14796416'758382'9892298 35 45'1521234 1153914 6'497104 19883615 15
5'1406132'142024 7.041048'9900646 55 26'1466585'148261 16744831'9891872 34 46'1524109'154212 6-484558 -9883172 14
6 1409012 -142321 7'026366'9900237 5427'14604631 14855916'731334'9891445 33 47'1526984'154510 6'472059'9882728113
7'1411892'142617 7'011744'9899826 53 8'1472340'148856 6'717889'9891017 32 48 1529858'154808 6-459607'9882284 12
8 1414772'1429141 6997180'9899415 52 291 14752171 149153 6'704496'9890588 31149 l1532733'155106 6'44'7201'988183g 11,
9'1417651 -143211 6'982678'9899003 51 30'1478094. 149451 16691156'9890159 3050'15356071 155404 6-434842'9881392 10 
10'1420531'143508 6'968233'9898590 50 3111480971 1149748 6'677867 19889728 2951 11538482'155701 6.4225301 9880945 9
111 1423410'143805 16953847'9598177 49 2 11483848'150045 6.664630'9889297 2852'1541356'155999 6'410263 19880497 8
12 11426289 1144102 16939519'9897762 48 33'1486724 1150343 6.651444J.9888865 2753 1544230'156297 6-398042'9880048  7
13 11429168'14439916'925248'9897347147 34 14896011 150640 6.638310 19888432 2654 -1547104'156595 6-385866'9879599  6
141'14320471 144696 6.911035 19896931 461351 14924771 150938 6.625225'9887998 2555'1549978'156893 6373735'9879148 5
15 1'434926'14499316'896879'9896514 45 361 1495353'1512351 6612191 19887564 241561 1552851 115719116'361650'9878697 4
16'1437805'145290 6-882780'9896096 44 7' 1498230'15153316.599208'9887128 2357'15557251 157490 16349609'9878245 3
17 11440684 1145587 6'8687371'9895677143 381'1501106 1151830 6.586273'9886692 22 581]1558598 1.157788 6.337612198777921 2
181'1443562'145884 6-854750'9895258 42 391'1503981 1'1521981 65733891 9886255 21 59 1561472 -158086 6'325660'98773381 1
191'1446440'146181 6'840819 19894838 41 140.1506857 1152426 6.560553 19885817 20 60'1564345-'158384 16313751 19876883 0
201'1449319' 146478 6'826943'9894416 40-                          
Cosine. ICotan. Tang.   Sine.           Cosine. e. Cotan. I Tang.   Sine.       Cosine. Cotan.  Tang. I 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.'  /  Sine.  Tang. Cotang. Cosine. i'    Sine.   Tang. Cotang. Cosine.
0'1564345'158384 6'313751'9876883 60 21 1624650'164652 6'073397'9867143 39 41'1682026'170633 5'860505'9857524 19
1'1567218'158682 6'301886'9876428 59 22 1627520'164951 6'062396'9866670 38 42'1684894'170933 5'850241'9857035 18
2'1570091'158980 6'290065'9875972 58 231630390 1165250 6'051434'9866196 37 43'16877611'171232 5'840011'9856544 17
3'1572963'159279 6'278286'9875514 57 24 1633260'165548 6'0405101'9865722 36 44 16906281'171532 5'829817'9856053 16
4'1575836'159577 6'266551'9875057 56 25 1636129'165847 6'029624'9865246 35 45 1693495'171831 5-819657'9855561 15
5'1578708'159875 6'254858'9874598 55 26'1638999'166146 6'018777'9864770 34 46'1696362'172130 5'809531'9855068 14
6'1581581'160174 6'243208'9874138 54 27 1641868'166445 6'007967'9864293 33 7'1699228'172430 5'799440'9854574 13
7'1584453'160472 6'231600'9873678 53 28'1644738'166744 5'997195'9863815 32 18'1702095'172730 5'789382'9854079 12
8!'1587325'160770 6.220034'9873216152 29'1647607.16704315'986461'9863336 31 9 11704961'173029 5'779358'9853583 11
91 1590197'161069 6'208510'9872754 51 30'1650476'167342 5'975764'9862856 30 50'1707828'173329 5'769368'9853087 10
10'1593069'161367 6'197027'9872291 50 31 11653345'167641 5'965104'9862375 29 51 11710694'173628 5'759412'98525940 9
11'1595940'161666 6-185586'9871827 49 32'1656214'167940 5'954481'986189412852'1713560'173928 5'749488'9852092  8
121 1598812i'161964 6'174186'9871363 48 33'1659082'168239 5'943895 986141227 53 1716425'174228 5'739598'9851593 7
13'1601683'162263 6'162827'9870897 47 34'1661951'168539 5'933345'9860929 26 14'1719291'174527 5-729741'9851093  6
14 11604555'162561 6'151508'9870431 4I 35 11664819'168838 15922832'9860445 25 55'1722156'174827 5'719917'9850593  5
15'1607426'162860 6'40230'9869964 45 36'1667687'169137 5'9 12355'9859960 24 56'1725022'175127 5'710125'9850091 4
16'1610297'163159 6'128992'9869496 44 371'1670556'169436 5'901913'9859475 23 57'1727887'175427 5'700366'9849589 3
171IS 13167 163457 6'117794'9869027 43 381 1673423 l16973515'891508'9858988 22 581'1730752'175727 5'690639'98490861 2
18';1616038 1163756 1'106636'9868557 12 39'1676291'170035 5'881138'9858501 1  91733617'176027 5'680944'9848582  1
19'1618909'16405516'095517'9868087 11 10'1679159'170334 5'870804'9858013 20 30'1736482'176327 5'671281'9848078  0
2-1621779'164353 6'084438'9867615 40'Cosine. Cotan.  Tan-g.   Sine.          Cosine.'Cotan.  Tang.   Sine.                                       Sine.
Deg. 80.                                 Deg. 80.                                Deg. 80.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
10 Deg.                                 10 Deg.                                10 Deg.
t  Sine.'l'ang. Cotang. Cosine.      Sine. I Tang. Cotang. Cosine.    /   Sine.   Tang. Cotang. Cosine. 
0 1.736482'176327 5.67-1281'9848078 6012 1 1796607 1 82632 5'475478 -9837286 3  1.1853808 1 88650 5.300801'9826668 19
1'1739346'176626 5'661650'9847572 59 22 1799469 1182933 5'466481'9836763 38 42 1856666'188952 5-292350'9826128 18
2 1.742211'176926 5'652051'9847066 5823'1802330 *183233 5457512 19836239 37 43 1859524'189253 5'283925'9825587 17
3'1745075'177226 5'642483'9846558 57 24 11805191'1835341 5448571 *983571513  4'1862382'189554 5'275525'9825046 16
41 1747939 ~177527 5'6329471 9846050 5625'1808052 -183835 5-439659'98351893  5 51865240'189855 5'267151'9824504 1
5'1750803.17782715-623442'9845542 55 26'1810913'18413515'430775'9834663 34 46 1868098'190157 5'258803'98239fi61 14
6'1753667'178127 5'613968 19845032 54 27'181 3774 1 184436 542l918'9834136 33 7'1870956'190458 53250480'9823417 13
7'1756531 178427 15604524 9844524 21 53 28'1816635'184737| 5413090 19833608 32 48'1873813'190760 5242183' 9822873 12
8'1759395'17872715'595112'9844010 5229 118194951 1850381 5404290 1 9833079 31 19 11876670'191061 5233911'9822327 111
9'1762258'179027 5'585730 19843498 51 30'1822355'185339 5'395517'9832549 30 501 1879528'191363 5-225664 19821781 1 10
10'1765121'17932715'576378'984298615031'18252151 185639 5'386771'9832019 291'1882385 *191664 5'2174421 9821234 9
* 11'1767984 -1796281 5567057 19842471 49 32.1828075' 18594015'3780531 9831487 28 521 1885241'191966 5'209245'9820686  8
12'17708471 179928 5'5577664 9841956 i48 433 183093.5 1186241 5'369363 -9830955 27 53 -1888098j 192268 5'2010731 9820137 7
13'1773710i 18022815'548505'9841441147 341'1833795. 18654215'360699 -9830422 26 54.1890954, 192569 5'192926'9819587
14 14776573 1l80529 15539274 9840924146 35 1836654'186843 5'352062'9829888 255 51893811. 192871 5'184803.9819037 5
151 17794351 180829 5'530072[ 9840407 45 361 1839514'1871441 5343452'9829353 24 561 1896667'193173 5176705'9818485 44
16'1782298 181129 5'520900'9839889 44 37 11842373'187446 5'334869'9828818 2 57 1l899523 193474 5'168631 -9817933  3
17- 17851, Go18143015.511757 -9839370 43 38.18452321 187747 5'326313'9828282 2 58. 19023791,193776 5'160581'9817380 2
18'17880221 181730 155026441 9838850 12 39'1848091 188048 5'317783'9827744121 59'19052341 1940788 5 152555'98168266 1
191 1790884'182031 5-4935fi60 -9838330141 40'1850949 -188349 153092791 9827206 20 0'19080901 19438015'144554'9816272 0
20'17937461 182331 154845051 -9837808 40                             I                            ___
i Cosine. Cotang.  Tang.   Sine.  t t  Cosine. Cotimn.   Tang. I Sine.     / |Cosine. Cotang. Tang. | Sine.
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.'l'ang. - Uotang. Cosine.
0. 1908090'194380 5.144554'9816272 60 1'1968018'200727 4'981881'9804433 39 1 -2025024'206786 4-835901 -9792818 1.9
1'1910945'194682 5.136576'9815716 59 22 1970870'201030 49743.81'9803860 38 2'2027873'207090 4s828817'9792228 18
2'1913801'194984 5.128622 98 15160 58 23 *19737221 201332 4.966903 9803286 37 43 2030721'207393 41821753'9791638 17
3'1916656'195286 5'120692'9814603 5724'1976573'201635 4'959447 *9802712 36 441*2033569'207696 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 1'95890 5'104902'9813486 55 26 1982276'202240 4'944599'9801560 34 46'2039265'208303 4'800680'9789862 14
6'1925220'196192 5'097042'9812927 54 27'1985127'202543 4-937206'9800983 33 47'2042113'208607 4'793695'9789268 13
7 *1928074'196494 5-089206'9812366 53 28  1987978 *202846 4'929835'9800405 3248'2044961'208910 4'786730 *9788674 12
8'1930928 -196796 5'081392 -9811805 52 29'1990829'20314-914'922485'9799827 31 49 -20478081 209214 4-779783'9788079 11
9'1933782'197098 5'073602 -9811243 51 301 1993679'203452 4.9151571 9799247 30 501'2050655' 209518 4'772856'9787483 10 
10'1936636'197400 5'065835 -9810680 50 31'19965301 203755 4'907849'9798667 29 51'2053502 1209821 14765949'9786886 9
11'1939490'197703 5'058090. 9810116 49 32'1999380. 204058 4'900562'9798086 28 52'20563149 210125 4'759060 -9786288 8
12'1942344 -198005 5'050369. 9809552 48 33'2002230'204361 4'893295'9797504 27 53'2059195'210429 4'752190'9785689[ 7
13'1945197'198307 5'042670'9808986 47 34'2005080'204664 4'886049'9796921 26 54'2062042' 210733 4'745340' 9785090' 6
14'1948050 1 98610 5.034993. 9808420 46 35'2007930'204967 4'878824 9796337 25 55'2064888'211036'-738508. 9784490 5
151 1950903' 198912 5.027339'9807853 45 36 -2010779'205270 4'871620'9795752 2456 -2067734 121134014'7316951 9783889  4
16. 1953756'199214 5-019707 -98072851 44  7 2013629'205573 4'864435'9795167 23 57 -2070580'211644 4'724901'9783287 3
17 l1956609'19951715'012098.98067   3     01201 6478'205876 4857271.9794581 22 58 2073426'21194814'71812519782684 2
18'1959461 -199819 5'0045111 9806147 42 39 2019327 -206180 4'8r0128'9793994 21 59'2076272'212252 4'711368'9782080  1
19 11962314'200122 4'996945'9805576 41 40 2022176'206483 4-843004'9793406 260 2079117'212556 4'704630'9781476 0
20~ 1965166'200424 4-989402'9805005 40
Cosine. Cotang. Tang.   Sine.'      Cosine. Cotang. Tang.   Sine.             Cosine. Cotgn. Tang.   Sine. |
Deg. 78.                                Deg. 78.                                 Deg. 78.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
12 Deg.                                 12 Deg.                                  12 )eg.
me.   Tang. Cotang. Cosine.'    Sine.   Tang. Cotang. Cosine.'  Sine.   Tang. Cotang. Cosine. 
0'2079117 1212556 4'704630'9781476 6  1 12138829'218949 4'567261'9768593 39 41'2195624'225054 4'443376'9755985 19
1I 2081962  212860 1-697910'9780871 59 2'2141671'219254 4'560911'9767970 38 42 2198462'225359 4'437350'9755345 18
2'2084807'213164 4'691208'9780265 58 3'2144512'219559 4'554577'9767347 37 43'2201300'225665 4'431339'9754706 17
31 2087652'213468 4'684524'9779658 57 4'2147353'219864 4-548260'9766723 36 44'2204137'225971 4'425343'9754065 16
4'2090497'213773 4'677859'9779050 56 25 2150194'220169 4'541960'97660 98 35 45'2206974'226276 4'419364'9753423 15
5'2093341'214077 4'67 1212'9778441 55 6'2153035'220474 4'535677'9765472 34 46'2209811 1226582 4'413399'9752781 14
6'2096186'214381 4'664583'9777832 5 27'2155876'220779 4'529410'9764845 33 47'2212648'226888 4'407450 19752138 13
7'2099030'214685 4'657972'9777222 5 28'2158716'221084 4-523160'9764217 32 48 2215485'227194 4'4015161 9751494 12
8!'21018741'21499014'651378'9776611 52 29'2161556'22138914'516926'9763589 31 49'2218321'227500 4'395597'9750849 11
91 2104718'215294 4'644803'9775999 51 30'21643961 221694 4'510708'9762960 30 50 2221158'227806 4'389694' 9750203 10
101 2107561'215598 4'6382451 9775386 5 31'2167236'221999 4'504507'9762330 29 51'2223994'228112 4'383805'9749556 9
111 2110405'215903 4'631705'9771773 49 2'2170076'222305 4'498322'9761699 28 52'2226830'228418 4'377931'9748909 8
12'2113248'216207 4'625183'9774159 4 33'2172915'222610 4'492153'9761067 27 53'2229666'228724 4'372073'9748261 7
13'2116091'216512 4'618678'9773544 434 2175754'222915 4'486000'9760435 26 54'2232501'229030 4'366229'9747612 6
14'2118934'216816 4'6121 90 9772928 46 35'2178593'2232214'479863'9759802 25 55'2235337'229336 4'360400'9746962. 5
15' 2121777'217121 4'6057201 9772311 4536'2181432'223526 14473742'9759168 24 56'2238172'229642 4'354586. 9746311  4
16. 2124619 1217425 4.599268.9771693J44  7. 2184271 1223831 4'467637 -9758533 23 57. 2241007.22994914.348786'9745660  3
171 2127462'217730 4.592832'9771075 43 38.2187110 -224137 4'461548 -9757897 22 58'2243842'23025514.343001,9745008j 2
18' 2103004. 218035 4.586414'9770456 42391.2189948 1224442 4'455475'9757260 21 59'2246676'230561 14337231 19744355  1
t9.2133146 l 218340 4.580012'9769836 414  0.2192786. 224748 4.449418. 9756623 20 60 2249511 1230868 4.331475 19743701  0
20 2135988 -218644 4-573628 -9769215                                         1 
I Cosine. ICotang. Tan.   Sine.         Cosine. Cotang. Tang.   Sine.         - Cosine. Cotang.l Tang      Sine.
Deg. 77.                                Deg. 77.                                 Deg. 77,




NATURAL SINES AND TANGENTS TO A RADIUS 1l
13 Deg.                                 13 Deg.                                 13 Deg.
L   Sine. [ Tang.[ Cotang. Cosine.           Sine.    Tang. Cotang. Cosine /        Sine.  ITang.  Cotang.  Cosine.  
0'2249511 *230868 4'331475'9743701 60 21 12308989 *237311 14213869 19729777 39 41 23655555 243465 4107356.9716180 19
1'2252345'231174 4-325734 9743046 59 22'2311819 *237618 4 208419'9729105 38 42 2368381'243773 4'102164.9715491 18
2'2255179 231481 4-320007'9742390 5823 12314649.237926 4'202983'9728432 37 43.2371207.244081 4.096985.9714802 17
3 12258013'231787 4'314295 19741734 7 21"4 23174791238233 4.197560.9727759 36 44 2374033 1244390 4.091817.9714112 16
4'2260846'232094 4'308597 9741077 56 25 1.23203091 238541 14'192151- 972708413545 52376859.244698 4'086662.9713421 15
5'2263680.232400 4.302913 9740,119 55 26'2323138. 238S48 4'186754 19726409 34 46 2379684 245006 4'081519'9712729 14
6 12266513'232707 4.297244'9739760 54 27'2325967.239156 4.181371 9725733 33 47 2382510.245315 4-076389 9712036 13
71 22693461*233014 4-291588'9739100 53 28 2328796.239463 4'176001'9725056 132 48 2385335.245623 4'071270 -9711343 12
81 22721791 233320 4.2859471 9738439 52 29 2331625 -239771 4.1706441 9724371 31 9 {23881591 245932 140661641 9710649 11
9 12275012 1233627 14280319 19737778 51 301 23344541-240078 4-165299 9723699 130 501 2390984'246240 140610701 9709953 10 Cl
101 2277844'23393414'274706 -97371161 50 31'2337282 12403864'-159968'97230201 29 511 2393808 1246549 14055987 197092581 9
111 2280677 123424114.269107 197364531491321 2340110.240694;4.154650.9722339128 52.2396633 124685714.050917.9708561  8
12'2283509'234547 4-26352 1 9735789 48 33'2342938 |241001 41 49344 1 9721658 27 53 2399457 1247166 4'045859 9707863 7
131-2286341 12348541 4257950 19735124 47 34 23457661-241309 4.144051 -9720976 26 541 2402280.247475 4.040812. 9707165t 6
14 |2289172.235161 4-252392 19734458 46 35 -2348594 -241617 14138771 -9720294 25 55 12405104 1247783 14035777'9706466| 5
15"'22920041 235468 14246848 9733792 145 36 12351421 1241925 4'133504'9719610 24 56 12407927'24809214'030755 19705766 4
16 122948351'235775 14241317 19733125 144 37 2354248'242233 4'128249 -9718926 23 57.2410751'248401 4'025744'9705065 3
17, 22976661{236082 4.235800.-9732457 43 38'23570751 2425414 1123007 -9718240 22 581.24135741'24871014 020744'9704363  2
18 -2300497'236390 4-230297'9731789 42 39'2359902 1242849 4'117778'9717554 21 59'2416396'249019 4'015757'9703660  1
191 2303328 1236697 1'2248081 9731119 41 40'23627291'243157 4'1125611 9716867120 601 2419219'249328 4'010780'9702957[ 0
201 2306159 12370041 4219331 19730449140
Cosine.Cota         Tang    Sine. Cosine. Cotang./         Tang.  Sine.    /        Cosine. Cotang. Tang.    Sine. 
Deg. 76.                                 Deg. 76.                              Deg. 76.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
14 Deg.                                 14 Deg.                                14 Deg.
Sine.  Tang.  Cotang. Cosine.'jSine.  Tang. Cotang. Cosine.                Sine.  Tang. Coang.    osine. /
_I.ine...      ITang.              _otang _                  _        _____
0'2419219 *249328 4.010780 *9702957 60 1 2478445 255826 3'9~8901 9687998 39 1'2534766'262034 3'816295'9673415 19
1'2422041'249637 4'005816'9702253 59 22 12481263'256136 3'904171'9687277 38 42 2537579 262345 3.811773 19672678 18
2'2424863'2499461 4000863'9701548 58 23'2484081'256446 3'899451 *9686555 37 3 -2540393 *262656 3'807260'9671939 17
3'2427685'250255 3.995922 *9700842 57 24'2486899'256756 3.894742 9685832136 44'2543206,262967 3'802758'9671200 it
4'2430507 1 250564 3'990992 19700135 56125'2489716'257066 3'890044'9685108 3545'2546019'263278 3'798266'9670459115
5 12433329'250873 3.986073'9699428 55 26 12492533 1257376 3.885357 19684383 1346 12548832'263589 13793783 9669718 14
6'2436150'251182 3'981166'9698720 54 27'2495350'257686 3'88068(; 968365833147'2551645.263900 3.789310'9668977 13
7'2438971 -2514911 3976271'9698011 53 28'2498167'257997 3-876014'9682931 13 48 2554458'264211 3'784848.9668234 12
8'24417921 251801 3'9713861 9697301 52129'2500984'2583073'871358!9682204 31 49'2557270'264522 3'780395'9667490 11
9'2444613 -25211013-9665131 9696591 51 301 25038001 258617 31866713 19681476 30 50 25600821 264833 3.7759511 9666746 10  C
101 24474/33'25242013'961651 19695879 50 31 125066161 25892813'862078'9680748129 1 51'2562894'265145 3'7715181 9666001 9
11'2450'254'252729 3'956801l 9695167 49132 12509432'259238 3'857453'9680018 28 521 2565705'265456 3'7670941 9665255 8
12'2453074'253038 3'9519611 9694453 48 33 12512248'25954813'852839'9679288 27 53 2568517 -265768 3'762680'9664508  7
13 12455894] 253348 3.9471331 9693740 47 34 -2515063'259859 13848235'9678557 26 54 2571328. 2660791 3758276'9663761  6
141 2458713'253658 3.942315'9693025 4.6 35'2517879'260169 3'843642 -9677825 25 55 -2574139.266390 3.753881'9663012 5
15 12461533j.253967 l3937509i-9692309 45 36'2520694'260480 3.839059'9677092 24 561 25769501 266702 3'749496. 9662263 4
161 2464352.25427713.932714.9691593 44 37'2523508'260791 3'834486'9676358 2'- 571 25797601 2670141 3745120'9661513 3
17'2467171'254587 3'927929'9690875 43 381 2526323'261101 3'8299231 9675624 22 58 -25825701 26732513.7407541 9660762  2
18'2469990'254896 3-9231.56'9690157 42 39 25291371'261412 3-825370'9674888 21 59'2585381'267637 3~736398'9660011 1
19 12472809 1255206 3.918393'9689438 41 40 125319521 261723 3'820828'9674152 2 60'25881901 267949 3.732050'9659258 0
20'2475627'255516 3'913642'9688719 40
Cosine. Cotang. Tang. I Sine.'   Cosine. Cotang. Tang.   Sine.  -          Cosine. Cotang. Tang.   Sine.
Deg. 75.                                Deg. 75.                                Deg. 75.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
15 Deg.                                 15 Deg.                                 15 Deg.'Sine.   Tan. Cotang. Cosine.              Sine.   Tang. Cotang. Cosine.'   Sine.   Tang. Cotang. Cosine. /
0'2588190.267949 3'732050.9659258 6021 *2647147'274507 3.642891. 9643268 39 1 *2703204 *280773 3'561590'9027704 19
1'2591000'268261 3'727713'9658505 59 22'2649952'274820 3'638744  9642497 38 2'2706004'281087 3'557613'9626917 18
2'2593810'268572 3'723384'9657751 58 23'26527571'275133 3'634606'9641726 37 3'2708805 128140113'553644'9626130 17
3'2596619'268884 3-719065'9656996 57 241'2655561'275445 3'630477'9640954 3 6 442711605 281715 3549684 9625342 16
4 -2599428'269196 3'714756'9656240 56 25'2658366'275758 3'626356'9640181 35 45'2714404'282029 3545732'9624552 15
5 2602237'269508 3'710455'9655484 55126'2661170'276071 3'622244'9639407 34 46'2717204'282343 34541788'9623762 14
61 2605045'269820 3'706164'9654726 54 27'2663973 -276385 3'618141'9638633 33 47 2720003'282657 3'537852'9622972 13
7'2607853'270132 3-701883 -9653968 53 28'2666777 -276698 3'614046 19637858 3248'2722802 *282971 3.533925'9622180 12
8'2610662'270444 13697610'9653209 52 29'2669581'27701113'609960'963708131 49'2725601'28328513-530005 -9621387 11
9'2613469'270757 3'693346'9652449 51 30'2672384'277324 3'605883'9636305 30 501'2728400'283599 3526093'9620594110  C
10'2616277'271069 3'689092'9651689 50 31 2675187'277637 3'601814'9635527 29 51'2731198'283914 3'522190'9G 19800  9
11'2619085 -271381 3'684847'9650927 49 3212677989'277951 13597754'9634748 28 521 2733997'284228 3518294'9619005  8
121'2621892'271694 3'680611'9650165 48 331 2680792'27826413'593702.9633969 27 53'2736794'2845431 3514407 I 9618210, 7
13'2624699 1272006 3'676384'9649402 47 3412683594'278578 3-589659 1-9633189 26 54 2739592'284857 3 510527'9617413.6
14'2627506'272318 3-672166'9648638 46 35 12686396'278891 3'585624'9632408 25 55 2742390 1285172 3'506655'9616616 5
15'2630312'272631 3'667957 19647873 45 36 2689198'279205 3'581597'9631626 24561 2745187 1285486 3-502791'9615818 4
16'26331181'272943 3'663757 -9647108 44 37-2692000'279518 3'577579. 9630843 23 57 2747984 1285801 3.498935'9615019 3
171 2635925I.273256 3'659566'9646341 43 381 2694801'279832 13573569'9630060 22 581 2750781'286115 3.495087. 9614219 2
18 12638730i 273569 3'655384. 9645574 42 39 2697602'280145 3.569568'9629275 21 59'27535771 286430 3.491247'9613418  1
19 12641536,.273881 3'651211'9644806 41 402700403'280459 3-565574'9628490 2060 12756374'286745 3.487414'9612617 0
20'26443421'274194!3'647046'9644037 40'Cosine. Cotang.l 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.
1  Sine.    Tang.  Cotang. Cosine.'  /  Sine.  Tang. Cotanlg.  Cosine.''  Sine.  Tang.  Cotang. Cosine. 
0'2756374'286745 3.487414 19612617 6021'2815042 -293368 3.408688'9595600 39 41'2870819 1299697 3.336699'9579060 19
1'2759170'287060 3.483589 19611815 59 2'2817833'293683 3.405021'9594781 38 42'2873605'300014 3.333173'9578225 18
2 12761965'287375 3.479772 19611012 58 3'2820624'293999 3.401361.9593961 37 43 2876391'300331 3.329654!.9577389117
3'2764761'287690 3'475963 969610208 57 24 2823415 29431613'397708'9593140 36 44.28791771.3006481j33261411 9676552 16
4'2767556'288005 3'472161'9609403 56 25 2826205'294632 13394063'9592318 35 45 2881963'300965 13322636'9575714 15
5'2770352'288320 3'468367'9608598 55 26 2828995'29494813'390424 19591496 34 46 28847481 301283 3'319137'9574875 14
6 i2773147 1288635 3.464581'9607792 54 27 12831785'295264 133867931 9590672 33 47 12887533 1301600 133156451 9574035 13
7.2775941 1.288950 3.460802 9606984 53 28 -28345751.295580 13383169'9589848 32 48.28903181 301917 3.312159'9573195 12
8.2778736'289265! 3.457031'9606177 52 29'2837364 1295897 3.379553 19589023 31 49.2893103'302235 13308681'.9572354 11
9 2781530[ 289580 3.453267'9605368 51 30 12840153'2962131 3375943'9588197 30 501 2895887'302552 3-305209'9571512 10  CT
)101 27843241 2389896 13449512 9604558 50 311 2842942] 296529 3.372340'9587371 29151 -2898671'3028701 3301743 19570669 9  e
I11'2787118'290211 13445763'9603748 49 32 [28457311 296846 3'368745'9586543128 52'2901455'303187 3'298285'9569825 8
121 2789911'290526 3.442022'9602937 48 33. 2848520 297163 3-365156. 9585715 27 53 2904239'303505 3'294833'9568981  7
13[ 2792704 129084213'438289. 9602125 47 34'28513081 297479 3'361575 -9584886 265 41 2907022. 303823 3.291387'9568136 6
14 127954971 291157 3-434563'9601312 46 35 12854096 1297796 13358000 -9584056 25 55.2909805 1304141 13287948 19567290 5
15'27982901 2914731 3430844a 9600499 45 361 28568841 298112 13354433'9583226 24 56 129125881 3044581 3284516'9566443 4
16 12801083 1291789 3.4271313 9599684 44 37 2859671 -29842913.350872'9582394 23 57129153711-304776 132810901 9565595 3
171'2803875I2921041 3423429'9598869 43 38 128624581 298746 3.347319'9.581562 22 581.2918153'30509413l277671.9564747 2
18'2806667'292420 3.419733'9598053 42 39 2865246'299063 3'343772'9580729 21 59'2920935'305412 3'274258'9563898.1
19'2809459'292736 3.416044 19597236 41 40'2568032'299380 3'340232'9579895 20 60 2923717'305730 3-270852'9563048  0
20'28122.51 129305213'412362'9596418140                                                                          
t Cosine. Cotang.l Tang.   Sine.'' Cosine. Cotang.! Tang.   Sine.  t II Cosine. Cotang. Tang.   Sine. It
Deg. 73.                                Deg. 73.                              Deg. 73.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
17 Deg.                                 17 Deg.                                 17 Deg.
X   Sine. -Tang. Cotang. Cosine. I/_I Sine.   Tang.  Cotang. Cosine.''   Sine.   Tang. Cotang.  Cosine. 
0'2923717'305730 3'270852 9563048 6 21 2982079'312422 3'200789'9545009 3941'3037559 *318820 3'136563'9527499 19
1.2926499'306048.-267452'9562197 59 22 2984856 *312742 3'197521'9544141 3 42'3040331'319140 3'133414'9526615 18
2.2929280'306367 3.264059'9561345 58 3 12987632'313061 3.194259'9543273 37 43'3043102.319461 3.130270'9525730 17
3'2932061 130668513.260672'9560492 5 24 29904081 313381 3.1910031'9542403 36 44'3045872'319781 t3127131'9524844 16
4'2934842'307003 3.257292'9559639 5  5 252993184 1313700 3.187754'9541533 35 45'3048643'320102 3.123999'95233958 15
5'2937623'307321 3.253918'9558785 55126 29595959 314020 3.184510 19540662 34 46 -3051413 3320423 3.120872'9523071 14
6'2940403'307640 3-250550'9557930 5  7'2998734 1314339 3'181272 19539790 33 47'3054183'320744 3'117750'9522183 13
7 *2943183 307958 3.247189'9557074 53 8'3001509'314659 3-178040'9538917 32 48 3056953 f321064 3 114635'9521294 12
8.2945963 1308277 3'243834'9556218 5 29'3004284.31497913'174814'9538044 31 49'3059723 1321385131 11525'9520404 11
91'2948743'308595 3'240486 9555361 51 30 3007058'315298 3'171594'9537170 30 50 3062492'321706 3'108421'9519514 10    C
101 2951522'3089141 3237143 9554502 501   31 3009832 -31561813 168380 -9536294 29 511 3065261'322027 3'105322'9518623  9
111 2954302'30923313-233807'9553643 49 32 3012606 1315938 3-165172 -9535418 28 52'306803,'322348 3'102229'9517731  8
12 12957081'309551 3'230478 -9552784 48 33 i3015380'316258 3'161970'9534542 27 53. 3070798'322670 3'099141'9516838  7
13'2959859'309870 3'227154 l9551923 4734 1-3018153. 316578 3.158774.9533664 26 54 3073566'322991 3.096059. 9515944 6
141-2962638'310189 l32238371 9551062 46 3513020926'316898 13155584 19532786 2555 1-3076334 1323312 3.092983 -951.50501 5
151 2966416'310508 3'220526'95501994 36 -3023699'31721813'152399 -9531907 2456'3079102'323633 3'089912'9514154  4
16 -2968194'310827 3'2 17221 1.9549336f    7'3026471'317538 3'149220 19531027 23 57 13081869'323955 3'086846'9513258  3
171 2970971'311146 3-213922'9548473143 81-3029244. 317859{3.146047'9530146 22 58 -3084636'324276 13083786'95123611 2
181 2973749'311465 3.210630'9547608 42 39'3032016 -31817913.142880'9529264 21 9'.3087403'324598 3.080732'9511464  1
19 12976526'311784 13207344.9546743 41 10'3034788 1318499 13139719 19528382 20 60 30901701 324919 13077683. 9510565 0
20 12979303 1312103 3204063'9545876 4I
Cosine. Cotang. Tang.   Sine.' Cosine. Cotang.! Tang.   Sine.       I Cosine. Cotang. 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. C osine.'__ Sine.   Tang. Cotang. Cosine.   /    Sine.   Tang. Cotang. Cosine. 
0'3090170'324919 3'077683 l9510565 60 21'3148209 1331686 3.014892 19491511 39 1'32033741 338157 2.957205'9473035 19
1 *3092936 1325241 3074640 -9509666 59922'3150969 -332009 3'011960'9490595 38 42 3206130'338481 2'954372'9472103 18
2'3095702'325563 3'071602'9508766 58 23 3153730 -332332 3'009033'9489678 3  3'3208885'338805 2'951545'9471170 17
3'3098468 325884 3.068569'9507865 57 24'3156490'332655 3'006110 19488760 36 44 3211640'339129 2'948722'9470236 16
4 3101234 1326206 3'065542'9506963 56 25 13159250'332978 3.003193 19487842 35 5 *3214395'339454 2.945905'9469301 15
5 33103999'326528 3.062520'9506061155 26'3162010.33330213.000282'9486922 3146 3217149'339778 2.943092'9468366 14
6'3106764'326850 3.059503'9505157 54 27'3164770 *333625 2'997375'9486002 3  7'3219903'340103 2-940284'9467430 13
7 13109529'327172 3.056492'9504253 53 28.3 167529'3339482994473 9485081 3 13214 8 3222657'340427 2.937480'9466493 12
8'3112294'327494 3.053487 -9503348 5229 1-3170288 334271 2.991576 19484159 31 49'3225411'340752 2.934682 9465555 11
o  9'3115058'327816 3'050486'9502443 51 30 3173047'334595 2'988685'94832373 350 3228164'341077 2'931888'9464616 10 
10'3117822'328138 33047491 -9501536 50 1 13175805'334918 2'9857981 948231329 1 13230917'341401 2'929099'9463677 9 
11 1'3120586 1328461 3'044501'9500629 149 32 3178563'3352422'982916'9481389 28 2'32336701'341726 2'926315 9462736 8
121'31233491 328783 13041517 -9499721 48 3313181321'335566 2.9800401 9480464 27153 13236422 -342051 2923535'9461795  7
13'.31261121 329105 3.038538 19498812 47134 31840791 335889 12977168 -9479538 26154132391741 342376 2.920761 19460854  6
14'3128875'329428 13035564'9497902 46 35 3186836'336213 2'974301'9478612 j2. 551 3241926'342701 2'917990'9459911  5
151'31316381 32975013'0325951 9496991 45 36 31895931 336537 2'971439'9477684 24 56'3244678'343026 2'915225'9458968 4
161 3134400'33007313'029632'9496080 44 37 31923501 336861 2'968583'9476756 123 57 3247429'343351 2'912464'945802-31 3
171 3137163'33039513'026673j.9495168 43 38'3195106'337185 2'965731'9475827 22 58'3250180'343677 2'909708'9457078  2
18 3139925 1330718 3'023720 1-9494255142 39 3197863 1'3375091 2962884'9474897 21 59'3252931'344002 2'906957 -9456132  1
191'31!42686'331041 130207721 9493341 14140 32006191'3378331 2960042'9473966 2060'3255682'344327 2'904210 9455186  0
201 3145448'331363 13017830 1949242640                                                     I                 I        I 
CosinelCotang. Tang. [ Sine.  l-    Cosine  Cotang.l Tang. I Si               ne.         Cotang. Tang.   Sine.
Deg. 71.                                D"g. 71.                                  Deg. 71.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
19 Deg.                                19 Deg.                                 19 Deg.
/  Sine. ITang. Cotang. Cosine. / /  Sine.   Tang. Cotang. Cosine. I /    Sine.  Tang.  Cotang. Cosine.  I
0'3255682'344327 2'904210'9455186 60 21'3313379'351175 2'847583'9435122 39 41'3368214'357723 2'795453'9415686 19
1 13258432 -344653 2.901468 19454238 59 22'3316123'351501 2.844935'9434157 38 42 3370953 *358051 2.792891'9414705 18
2 *3261 i82 *344978 2s898731 *9453290158 23'3318867 *351828 2.842292'9433192 37 43 *3373691'358380 2.790333 *9413724117
3 13263932 1345304 2'895998'9452341 57 24!3321611'352155 2'839653'9432227 36 44'3376429 358708 2.787780'9412743 1 6
4 13266681 *345629 2'893270 19451391 56 25 13324355 1352482 2'837019 19431260 35 45 *3379167'359036 2'785230'9411760 15
5'3269430 0 345955 2.890546 19450441 55 26 3327098'35280912 834389'9430293 3446 *3381905'359365 2'782685'9410777 14
6'3272179'346281 2 887827'9449489 54 27'3329S41.35313612.831763'9429324 33 47.3384642'359693 2 780144'9409793 13
7'3274928'346606 2'885113'9448537 5328 *3332584 f353464 2829142'9428355 32 48 3387379'360022 2'777606'9408808 12
8 132776761 346932 2'882403'9447584152 29 133353261'353791 2'826525 19427386 131 9.3390116'360350 12775073'9407822 11
91 32804241 347258 2'879697'944663051 30'3338069 J354118 2'823912'9426415 3050 13392852 1360679 2772544 9406835 10l
101 3283172 1347584 2'876997'9445675 501311 33408101 354446 2'8213041 9425444 2951.33955891 361008 127700191 9405846 9
11'3285919'347910 2'874300'9444720 4932'3343552j 354773 2'818700'9424471 28 52 3398325'361337 2'767499'9404860 8
12'3288666'348236 2'871608'9443764 48 33'33462931 355101 2'816100'9423498 2753 3401060'361666 2'764982'9403871 7
13 13291413'348563 12868921'9442807147134'3349034'1355428 2.813504 19422525 26 5413403796'361994 12762469 19402881 6
15'3296906'349215 2.8635601 9440890 45 36~ 33545161 356084 2'808326'9420575124 56 3409265'362653 2'757456'9400899 4
161 3299653'349542 12860886 19439931 44137 13357256 1356411 2.805743 19419598 23157'3412000'362982 12754955 -9399907 3
171 3302398'1349868 12858216'9438971 1431381 3359996'356739 2'803164 19418621 122 58 3414734'363311 12752458'9398914  2
18 l3305144 1350195 12855551 19438010 42139 3362735'357067 128005901 9417644 21 59 134174681 363640 2'749966 19397921  1
191 3307889'350521 12'852891'19437048 41 40 133654751 357395 127980191 9416665120 60 13420201 1363970 12747477'9396926 0
201 3310634'350848 2'850234 -9436085 40
]jtI Cosine. [Cotang.l Tang.   Sine.     t Cosine. Cotang., Tang.   Sine.   Cosine. Cotang. Tang.   Sine.
Deg. 70.                                Deg. 70.                               Deg. 70.




NATURAL SINES AND TANGENTS TO A RADIVJS 1.
20 Deg.                                 20 Deg.                                 20 Deg.
Stine.  Tang.  Cotang. Cosine.           Sine.  Tang. Cotang. Cosine.ll/   Sine.  Tang. Cotang. Cosine.'
0'3420201'363970 2'747477'9396926 60 1'3477540'370903 2'696118'9375858 39 1'3532027'377536 2-648753'9355468 19
1'3422935'364299 2'744992'9395931 59 22'3480267'371234 2'693714'9374846 38 42'3534748 -377868 2'646423 J9354440 18
2'3425668'364629 2'742512'9394935 58 23'3482994'371565 2.691314'9373833 37 3'3537469'378201 2'644096'9353412 17
3'3428400'364958 2'740035 9393938 57 24'3485720'371896 2'688919' 9372820 36 4'3540190'37853312'641774 *9352382 16
4'3431133'365288 2'737562'9392940 56 25'3488447'372227 2-686526'9371806 35 45 3542910'378866 2'639454'9351352 15
5'3433865'365618 2'735093'9391942 55 26 3491173'372559 2'684138'9370790 3  6'35456301 379198 2'637139'9350321 14
6'3436597'365948 2'732628'9390943 54 27'3493898 5372890 2.681753'9369774 33 7'35483501 379531 2'634827'9349289 13
7'3439329'366277 2'730167'9389943 53 28'3496624'373221 2'679372'9368758 32 8'3551070'379864 2'632518.9348257 12
8'34420601 366607 2'727710'938S942 52 9'3499349'373553 12676995'9367740 31 9'3553789 380197 2'630213'93472233 11
91 3444791'366937 2.725256'9387940 51 30.3502074 j373884 2.674621. 9366722 3) 0  35565081 380530 2.627912.9346189 10 
101 3447521'367268 2'722907'9386938 50 31'3504798'374216 2-6722511 9365703 2 51 -3559226'380863 2'625614'9345154  9'
111 34502521 367598 2'720362J.9385934 49132'3507523'374547 2'669885'9364683 28 521 3561944 1 381196 12623319 19344119J 8
121 3452982 1.367928 2.717920'9384930 48 33.3510246.374879 12667522'9363662 2 75313564662'381529 2'6210281'9343082 7
13'3455712 -368258 2'715482'9383925 47 34'3512970.375211 2.665163'9362641 26 4'3567380'381862 2.618741 19342045 6
14, 34584411 368589 2.713048'9382920 4635 35135693'375543 2-662808'9361618 2555'3570097'382196 12616457'93410071 5
15'3461171'368919 2-710618 19381913 45 36 13518416 1375875 2'660456 19360595 2  61 3572814'382529 12614176'9339968 4
16'3463900.369250'2'708192'9380906 44 37'3521139'376207 2.658108 -9359571 23 57 3575531 1382863 2.611899. 9338928  3
17'3466628 5369580 2'705769'937989843 38'3,523862'376539 2.655764'9358547 22 58 3578248 5383196 2.609625'9337888 2
18'3469357'369911 2'703351'9378889142 39'3526584'376871 126534231 9357521 21 591'35809641 383530 2'607355'9336846 1
191 3472085'370242 2'700936'9377880 41 40 3529306'377203 2-651086'9356495 20 60'3583679'38386412'605089'9335804  0
20'3474812 1370572 2'6985251 9376869 40
I I -  — II — I-.  D      -            —.- I                                  - -. 1T  Cosine. CCotang. Tang. | Sine.'   I Cosine. Cotang. Tang.  Sine.             Cosine. Cotang. Tang.  Sine.
Deg. 69.                               Deg. 69.                                Deg. 69.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
21 Deg.                                  21 Deg.                                 21 Deg.
w   Sine.   T'ang. Cotang. Cosine. / [ /  Sine. I ang.  (Cotang. l  Cosine  i /    Sine.  T'an.  Cotang. Cosine. / t
0 3583679'383864 2605089 9335804 6021'3640641'390889 2 558268 931372l9 39 41 36914765 397611 2 51501P  9292401 19
1 3586395'384197 2-602825 9334761 59 22 -3643351 -391224 2.556075 -9312679 38 42 1'36974681 397948 2-512889.9291326 18
2'3589110'384531 2'600565 9333718 58 23 3646059 -391560 2553885.9311619j37 43 3700170  398285 2-510762 -9290250 17
3'359 tS251-38486512-598309'9332673 57j24 -3648768 -391895 2-551699'9310558136 44 3702872 *398622 2-508639 19289173 16
4'3594540 -385199 2-596056'933162S 56 25.3651476 139223 1 2549516.9309496 35 45 -3705574.398959 2-506519.9288096 15
5'3597254. 385533 2-593806'9330582 55 26 13654184 -392567 2.547335 -9308434 34 46 1.3708276 1399296 2.50440'2 9287017 14
6 -3599968- 385867 2-591560'9329535154127'3656891 -392902 2-545159.9307370 33 47'3710977'399634 2-502289 /9285938 13
71 3602682'386202 2-589317'9328458153128'3659599 -393238 2-542985 -93063061 32 18.3713678'399971 2-500178'9284858 12
8'3605395 -38653612-587078 19327439 52129 -3662306'39357412.540815 -9305241 31 49'3716379'400308 2-498070 -9283778 11 
91 36081081 386870 2-584842 19326390 51130 -3665012'393910 12538647'9304176 130 50'3719079 1400646 12495966 -92826961101
101 36108211 387205 12582609 19325340 5031 -3667719 -394246 12536483- 9303109129 51'37217801 400984 2-493864 19281614 9
11'3613534 1387539 2-580380'9324290 49132'3670425 -394582 2-5343231 9302042 28 521 3724479 -401321 2-491766 -9280531  8
12 3616246'387874 2-578153'9323238 48 33 -3673130 -394918)2-532165 -9300974 27 53'3727179'-401659 124896701-9279447 7
13 361895 I388209 2-575931'9322186 47 34 3675836 3952551 2-530011 -9299905 26 54 13729878 -401997 2-487578i 9278363  6
141 3621669 -388543 2-573711 19321133 146135 3678541.395591 2-527359.9298835 25 55 -3732577 -402335 2-4854881 -r277277  5
151 36243801 388878 2-571495 -9320079 45136 13681246 -395928 2.525711 -9297765 24 56 -37352751 402673 2-483402 -9276191 I 4
16 -3627091 -389213 2-569283 -9319024 44 37-.3683950 1396264 12523566 -9296694 23 571.3737973 -403011 2-491319 19275104  3
171'3629802 1389548 2-5670731-9317969 4313s8'36866541-396601 2-521424 -9295622 22 58)'3740671 -403349 2'479238 192740161 2
18'3632512 -38988312-564867 -9316912 42139'3689358 139693712-519286 t9294549 21 591 -3743369 -403687 2-477161 192729281 1
19 13635222 -390218 2-562664 -19315855 41 01 3692061'397274 12517150 -9293475 20 60 -3746066 -404026 2-475086 -19271839  0
20 -3637932 -390554 2-560464'9314797 40
/ Cosine. Cotang.  Tang.   S Sine.         Costine. 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.
I    Sine.   Tang. Cotang. Cosine.         Sine.  Tang. Cotang. Cosine.'   Sine.  Tang. Cotang. Cosine.  /
0'3746066'404026 2'475086'9271839 6021'3802634'411149 2.432204 19248782 39 41 3856377'417967 2'392531 19226503 19
1'3748763'404364 2'473015'9270748 59 22 1805324'411489 2'430193'9247676 38 2'3859060'418309 2-390576'9225381 18
2.37514591.404703 2'470947'9269655818 23 *3808014'411830 2'428186 *9246568 37 43 3861744 *418650 2'388625'9224258 17
3'3754156'405041 2'468881'9268566 57 2413810704'412170 2'426181'9245460136 44 3864427 *418992 2'386675 9223134 16
4'3756852'405380 2'466819 19267474 56 251 3813393'412510 2-424180'9244351 35 15 3867110.41933412'384729'9222010 15
5 *3759547'40571912'464759'9266380 55 261'3816082'412851 2'422181 -9243242 34 6'3869792.41967612'382785'9220884 14
6'3762243'4060571 2'462703 9265286 54127 i3818770'413191 2'4201]851 9242131 33 7.38724741420019 2.380844'9219758 13
7'3764938'406396 2'460649 19264192 53 28'3821459'413532 2'418191 19241020 32 48'38751561 420361 123789061 9218632 12
8'3767632 *406735 2*458598'9263096 52 9 3824147'413872 2'4162()1 19239908 31 49 3877837'420703 2 376970 *9217504 1 1
9 13770327 14070741 2456551'9262000151 30 38268341 4142134 2414213'9238795 30 50'3880518 421046 2'3750371 9216375 10  C
10'37730211 40741312'454506'926090215031 13829522'414554 2'412228 9237682 29 511 3883199'42138812'373106 1921524 - 9
11'37757141 407753 2'452464'925980514932'3832209'41489512410246'9236567 28 521 38858801 421731 2'371179 19214116  8
12'3778408'408 092 2.450425'9258706 48 331.3834895'415236 2'408267'9235452 27 53 13888560'422073 2'369254 1922861  7
13'3781101 -408431 12448389'925760614734'3837582'415577 2'406290'9234336 26 54 3891240'422416 2'367331'9211851 6
14'3783794 1408771 2'446355'9256506 46 35'3840268 415918 2'404316 9233220 25 55 3893919'422759 2'365411 9210722  5
15'3786486 1409110 2'444325'9255405 45636'3842953'416259 2'402345'9232102 241561 3896598'423102 2'363494'9209589 4
16'3789178'40945012'442298 19254303 441371 38456391 416601 2'400377'9230984 23 57 3899277 l4234451 2361580 -9208455 3!17'37918701'409790 2'440273 -9253201 43138'3848324'416942 2'398411'.9229865 22 581.3901955 1423788 2.359668'92073273   2
18'3794562 1-410129 12438251 1925209714239 1-3851008 1417284 12396449'9228745'21 591-3904633 1424131 12357759'9206185i 1
19'3797V53'410469 2'436233'9250993 41 40 3853693 -417625 2'394488 -9227624 20 60'3907311 1424474 2'3558521 9205049 0
2043799944 410809 12434217 79249888140,                           
| Cosine. Cotang. Tang.   Sine.         Cosine. Cotan      an Sine.             C C:osine. Cotang. Tang.   Sine.
Deg. 67.                                Deg. 67.                              Deg. 67.




NATURAL SINES AND TANGENTS TO A RADIU'S 1.
23 Deg.                                  23 Deg.                                 23 Deg.
Sine.   T'ang.  Cotang.  Cosine. I-I Sine.   Tang. Cotang.  Cosine.              S   ine.   Tang. [ Cotang.  Cosine.   |
O0 3907311 *424474.2355852 9205049 60 1'3963468'431703 2'316407 *9181009 39 41'4016814'438622 2'279865'9157795 19
1 3909989'424818 12353948'9203912 59 2 -3966139'432048 2.314557'9179855 38 42 4019478'438969 2-278063'9156626 18
2'3912666'425161 2'352046'9202774 58 3'3968809'432393 2'312709'9178701 37 43'4022141  439316 2'276264'9155456 17
3 13915343 1425505 2'350148'9201635 57 4 13971479 1432738 2-310863 9 177546136144 14024804'439663 2'274467'9154286 16
4 13918019 1425848 2.348251'9200496 56 5'3974148'433084 2'309020'9176391 35145 14027467'440010 2.272672 9153115 15
5 3920695'426192 2'346358'9199356 55 6'3976818'433429 2'307180'9175234 34 46'4030129'440357 2270880'9151943 14
6 13923371'426536 2'3444'67 19198215 54 7'3979486'43377512'305342'9174077 33 47'4032791'440705 2'269090'9150770 13
7'3926047'426880 2-342578'9197073 53 28 3982155'43412012.303506'9172919 32 48'4035453'441052 2.267303'9149597 12
8'3928722!'42722312.340692'9195931 52 9 13984823'43446612.3016731'9171760 31 49'4038 14'441400 2'265518'9148422 11
91 39313971 427568 12338809'9194788 51 30 13987491 434812 122998421 9170601 130150 14040775'441747 2.263735 -9147247 10 {
10'39340.71 142791212.336928 19193644 50 31 -39901581'435158 12298014 19169440 29 51 14043436"'442095 2.261955 [9146072 9!
11'3936745'428256 2'335050'919249949 32 3992825'435504 2'296188 -9168279 28 52 4046096'442443 2-260177'9144895  8
121 3939419'428600'2.333174'9191353 483.31 3995492'435850 2.294365'916711827 53'4048756'442791, 2-258401.9143718  7
13'3942093'428944 2.331301 19190207 47 34'3998158'436196 2'292544 -9165955 26 54'4051416 413139 2'256628'9142540! 6
14'3944766'429289 2.329431 -9189060 46 35'4000825' 436542 2-290725i 9164791 125 55 140540751 413487 122548571 9141361  5
15'3947439'429633 2'327563[ 918791  45 36 14003490.436889 2.288909'9163627124 56 4056734.4438351 2253088'9140181  4
161'3950111'42997812'3256971 9186763 4 371-4006156'43723512.2870951 9162462 23157 14059393 -444183 2.251322 19139001  3
171 3952783.43032312-323834 -9185614 43 381.4008821'437582 2.285284.9161297 22 581-4062051.444531 2.249558'91378191 2I
181i3955455'430668 21321974191 84464 -239.401 1486'437928 2'2834751 9160130 21 91.4064709 1444880 2.247796 191366371 1
19 -3958127 1431012 2'320116 |9183313 41 040114150 438275 2.281669 9 158963 20 60 4067366 |445228 2s246036'9135455 0
20'39607981 431357 12318260 19182161 40                                       I  I                            I     1
tCosine. Cotang.l Tang.    Sine.        I Cosine. Cotang.l Tang. I Sine.   I    Cosine.  Cotang. Tang.   Sine. 
Deg. 66.                                Deg. 66.                                 Deg. 66.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
24 Deg.                                24 Deg.                                 24 Deg.
Sine.  Tang. Cotang. Cosine.   /   Sine.   Tang. C otang. Cosine. e'    Sine.  Tang. C otang. Cosine.
0.4067366.445228 2246036'9135455 60 21'4123096'452568 2'209611 *9110438 39 41.4176028'459596 2'175822'9086297 19
1 14070024.445577 2'244279 19134271 59 22'4125745 1452918 2'207901 *9109238 38 42.4178671 1459948 2:174155 9085082 18
21 4072681.445926 2'242524'9133087 58 23'4128395'453269 2'206193 *9108038 37 43.4181313 460301 25172491'9083866 17
3'4075337.446274 2.240772 19131 902 57 24 14131044'453620 2'204487'9106837 36 44.4183956 460653 2'170828 19082649 16
4'4077993 *446623 2'239021'9130716 56 25 14133693'453970 2'202784'9105635 35 45.4186597'461006 2'1691671 9081432 15
51 4080649.446972 2.237273'9129529 55126'4136342 1454321 2'201083 19104432 34146.4189239 1461359 2'167509'9080214 14
6 4-0833051 44732112.235528'9128342 15427 14138990'454672 2'199384 19103228 33147.4191880'461711 12'165852! 9078995 13
7'4085960.447670 2.233784 9 127154 53 28 4141 638'455023 2-197687 9102024 32 48.4194521.462064 12164198'9077775 12
8'4088615.448020 2.2320431 9125965 52 291 4144285'455375 2-195992'9100819 31 49'41971611 462417 2 162546'9076554 11
91 4091269'448369 2'230304'9124775 51 30'4146932 45572612'194299'9099613 130 50.4199801'462771 2'160895 -90753331 10  C1
10 14093923 1448718 2.2285671 9123584 150 31'4149579'45607712'192609'9098406 29 51.4202441'46312412.1592471 90741111 9 /
111 40965771 449068 2'226833'9122393149 32'4152226'456429 2'190921'9097199 28 521.4205080'463477 2'157601 i9072888 8
12'4099230 )449417 2.225100'9121201 48 33 14154872 i456780 12189234'9095990 27153'4207719'463831 2'1559571 9071665  7
13'4101883.449767 2'223370'9120008 147 34 41575171 457132 2'187551'9094781 26 54 4210358 1464184 2'154315'9070 140 6
14'4104536'450117 2'221643 9 118815146 351 4160163'457483 2'185869'9093572 j25 551 42129661 464538 2'1526751 9069215 5
151 4107189 1450467 2'219917 19117620 45 3614162808'457835 2'184189'9092361 12456'42156341 464891 2'151037 19067989  4
16 -4109841 1450817 2-218194 19116425 44371 4165453'458187 2 1825111 9091150 23 57. 4218272'465245 2 149402'9066762 3
171-4112492 -451167 2.2164731 9115229 43138'4168097'458539 2.1808361 9089938 22 581 4220909. 465599 2'147768'9065535 2
8i41 15144'451517 2'214754'9114033 42 39 4170741 1458891 2'179163'9088725 21 59 -4223546'465953 2'146136'9064307 1
191.41177951.4518671 22130371 9112835 41 401 4173385.459243 2.177492.9087511120 601 4226183'466307 12'144506 19063078 0
120141204451 452217 12211323 19111637140                                    I             I                         V 
|  Cosine. |Cotang.  Tang. I Sine. I' Cosine. lCotang. Tang.   Sine.         Cosine. Cotang. Tang.   Sine.
Deg. 65.                                Deg. 69.                                Deg. 65.




NATUJRAL SINES AND TANGENTS T  A RADIUS L,
25 Ileg.                                 25 Deg.                                  25 Dego
/I Sine.  Tang. Cotang. Cosine. I       Sine. -Tang.             Cosine. / /  Sine.   Tang. Cotang. Cosine..1'_3'44.                                           -] _  _ie T. Cotang. __                                 Cos _, _  S..t __ jfC. 
0 -4226183 *466307 2-144506 -9063078 60 21.4281467'473765 2 110747 19037093 39 41 4333970 *480909 2-079394 -9012031 19
1 -4228819 466 661 21 42879 9061848 59 22 -4284095 -474122 2 09 161 69035847 38 42 4336591 481267 2 077846 9010770 18
2 -4231455 -467016 21L41253'9060618 58 23 -42867233 474478 2'107577 -9034600 37 143 4339212 -481625 2-076300 -9009508 17
3 -4234090'467370 2-1396301 9059386 57124 -4289351'474834 2'105995 -9033353 36 44 -4341832'481984 2-074756 *9008246 16
4 -4236725 -467725 2-138008 9058154 56 25 *4291979 475191 2-104415 9032105 35 45 4344453 482342 2-073214 -9006982 15
5 -4239360 -468079 2-136389 -9056922 55 26 -4294606 ~475548 2-102836.-9030856 34 46 4347072'482701 2-071674 -9005718 14
6 o4241994 -468434 2 134771.9055688 54 27 -4297233.4759041 21012601 9029606 33 47 o4349692 -483060 2-0701351-9004453 13
7 -4244628.468789 2-133155.9054454153128 4299859.4762611 2099686 19028356 32 48 4352311 1483418 2-068599 9003188 12
8 -4247262- 469143 2-131542 -9053219 52 29'4302485 -476618 20981 14 902710 31 49 4-354930 4S3777 120670641 9001921 11
9 -4249895 -469498 2.129930 -9051983 51 30 -4305111 -4769751 2096543 1.9025853 30150 -4357548 484136 2-065531 9000654 10  c
10 -4252528 -469853 2-128321 19050746 50 31 143977361 477332 120949751 9024600 29 51 143601661 484495 i2064000'8999386  9
11 -4255161'470209 2'126713 19049509!49 321'4310361'477689 2'093408.9023347 28 52 -43627841484855 2-062471 -8998117  8
121-4257793 -470564 2'125108 -9048271 48 33 -4312986'478047,2'091843'9022092127 53 -4365401'485214 12060944 -8996848i 7
131 4260425 -470919 2-123504 -9047032 47 34 -4315610 ]4784041 2090280 19020838 126 54 -4368018 -485'573 12059418 8995578  6
14 -4263056 -471275 2-121903.9045792 46135'43182341 478762 208S-720,9019582 12.555 543706341 485933 2-057895 -8994307 5
15; 4265687'471630 2-120303 -9044551 14536 -4320857 -479119 2-087161 -9018325 24 56 -4373251'486293 2'056373 -8993035 4
16 -42683181 471986 2'118705 -904331014437 -'4323481 -479477 2-0856031 9017068 23 57'43758661 -486652 12054853 -8991763  3
17 -4270949 1472342 2'1171101-9042068 43138 -4326103 479835 2-084048 9015810 22 58 -4378482 -487012 2-053334 8990489  2
181 4273579 1-472697 2'1155161 90408251421391-43287261 480193 2'0824951-9014551 21 59 -43810971 48737212'051818 -8989215 1
19 -42762081 4730531 21139241 90395821411401-4331348 1480551 2-0809431 9013292 20 60 -43837111 -487732 12050303 -8987940 0
120 4278838 1-47340912- 12334 -90383381401                                          ______
o sIiC  sn —                                      Ct        Tg       Sin. -     / CI Ine  -.I —
I Cosine.l Cotang.l  Tang. / sine. | I-  Cosine   ICotang   Tang.   Sine.                    C  ~Cosine. g  Cotang.l  Tang.   Sine. |
Deg. 64.                                Deg. 64.                                  Deg. 64.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
26 Deg.                                  26 Deg.                                  26 Deg.
|-   Sine. 1Tang. iCotang. Cosin.           Sine.. Cotan      osine.          Sine.  Tang. Cotang.  Cosine.
O 4383711 |487732 2-050303 -8987940 60 21 4438534 -495317 2 018908- 8960994 39 11 4490591 -502583 1-989720 8935021 19
1 -4386326 -488092 2-048791 -8986665 59 22 -4441140 -495679 2017433 -8959703 38 2 *4493190 -502947 1988278 -8933714 18
2 *4388940 *488453 2-047280 *8985389 58 23 -4443746 -496041 2-015959'8958411 37 13.4495789 -503312 1-986838 8932406 17
3'*4391553 1488813 2-045770( 8984112 57 24 -444;6352 -496404 2-014486' 8957118 36 4 44498387'503676 1-985400 18931098 16
4 -4394166 -48917312-044263'8982834 56 25'444-8957 -49676612.013016 8955824 35 15 -4500984'504041 1-983963 8929789 15
5 [4396779 -489534 2'042757' 898155 5 55 26 -4451562 -497129 2-011547'8954529 34 46 4503582 -504406 1-982528 8928480 14
6 -4399392.489894 2-041254[ 8980276 54127 4454167 -497492 2-010080'8953234 33 17.4506179 *504771 1-981095 -8927169 13
7'4402004 -490255 2-039751 -8978996 53[28 -44A56771 -4978551 2008615 -8951938 32 18.4508775 -505136 1-979663 8925858 12
8 -44046151-490616 2-038251 -8977715 52 29'4459375'4982182.-007151 *S950641 31 19 -4511372 -50550! 1-978233(-8924546 1 1
9 -4407227 -490977 2-036753 -8976433 51 30'4461978 -498581 2-005689'8949344 30 50 45139671-505866 1-9768051-8923234 10
10 -4409838 -491338 2-035256 -8975151 50:31'4464581 -498944 2-0042291'8948045 29 51 -4516563[-506232 1-975378'8921920  9
11 144124481 491699 2-033761 -8973868 49132 -4467184 -499308 2-002771 -8946746 2 52[-4519158 -506597 1-973953 1 8920606  8
12 -4415059 -492061 2-032268 -8972584 48 331 4469786 -59967112 001314'8945446 127 31-4521753 506963 1-972529 18919291 7
13 -4417668 -492422 2-030776 -8971299 47 34 *4472388 -500035 1-9998591,8944146 26 54-4524347 -507329 1-971107 -8917975 6
14 -4420278 -4 92783 2-029287 -8970014 46 35 -4474990 -500398 1-998405 -8942844 25155 -4526941 -507694 1-969687]-8916659  5i
15 -4422887 -493 145 12027799 -8968727 45 36 14477591 -500762 1-996953 -8941542 24 56 -4529535 -508060 1-968268 -8915342 4 a
161 4425496- 493507 2-026313 -8967440 44 37 -4480192'50112611-995503'8940240 23 571 4532128 -508426 1-96685118914024  3
17 -4428104 -493868 2-024828 -8966153 43 381 4482792 -501490 1-9940551-8938936 22 58 -45347211-50879211-965436 -8912705  2
18 14430712- 494230 2-023346 -8964864142 391 44853921 501854 1-9926081 -8937632 21 591'45373131 509159 11964022 -8911385  1
19 -4433319 -494592 2-021865 -8963575 41 40 -4487992 1502218 11-991163-8936326 20 60 -4539905 509525 1-962610 -8910065 0
20 -4435927 -494954 12-020386'8962285 40
Cosine. Cotang. Tang.    Sine.'   Cosine. Cotang. Tang.   Sine.   /' Cosine. Cotang.  Tang.   Sine. 
Deg. 63.                                  Deg. 63.                               Deg. 63.




NATURAL SINES AND TANGENTS TO A RADITJS 1.
27 Deg.                                 27 Deg.                                 27 Deg.'  Sine.   Tang. Cotang. Cosine.'' 1    Sine.  Tang. Cotang. Cosine''  Sine.   Tang. I Cotang. Cosiue. I
0.4539905 509525 1-962610'8910065 6021'4594248 517244 1933323'8882166 39 41 *4646845'524640 1-906066'88i55288 19
1.4542497 *509891 11961200 *8908744 5922'4596832 517612 1.931945'8880830 38 42.4648420'525011 1.904719 -8853936 18
2.4545088 5610258 1.959791.8907423 5823'4599415 517981 1'930569'8879492 37 3'4650996'525382 1.903373.8852584 17
31 4547679 510625 1. 9583831-8906100 57 24 4601998 1518350 1-929195 *8878154 36 44.4653571'525754 1.902029 8851230 1 6
4 -4550269 -510991 1'956978'8904777 56 25 4604580'518719 11927822'8876815 35 5.4656145'526125 1'9006871 8849876 15
5.4552859.511358 1.955573.8903453 5526'4607162'519089 1.926451'8875475 34 46 4 658719 *526496 1.899346'8848522 14
61.4555449.511725 1 9541711 8902128 54 27 4609744'519458 1.925081.8874134 33 17 -4661293 *526868 1 898006'8847166 13
7.4558038 512093 1'952770 18900803153 28 4612325 519827 1.9237131 8872793 32 18 4663866.527240 1.896668'8845810 12
8.45606271 512460 1'951371'8899476 5229 4614906.520197 1.922347 -8871451131 9'4666439'527612 1.895332 1884445311
9{.45632161-512827{ 1'9499731 8898149 51 30 14617486 1520567{ 1.920982 -8870108 30150 4669012'527983{ 1'893997'8843095 101'
10.4565804'513195 1'948577'8896822 50 31 146200661 520936 1-919618'8868765 29151'46715841 528356 1.8926631 8841736 9
11.4568392 -513562 1'947182'8895493 49 32'4622646'521306 1-918256'8867420 28 52'4674156'528728 1'891331'8840377 8
12 145709791 513930 1-945789'8894164 48 33'46252251 52167611.916896'8866075 27 531'4676727'529100 1.8900001 8839017 7
13.4573566'514298 1.944398'8892834 47 34.4627804'522046 11915537'8864730 26 54.4679298.529472 11888671'88376561 6
14.4576153 -514665 1.9430081 8891503 46 351 4630382'522417 1.914179'8863383 25 551-4681869'529845 1.887343'8836295 5
15 14578739'515033 1'941620 18890171 45 36'4632960l 522787 1'912823 -8862036 24 56l4684439 l530217 1'886017'8834933 4
161'46813251'515401 1-940233'8888839 44137'46355381 523157 1'911469 18860688 23 571'4687009'530590 1'884692'8833569 3
171'4583910 -515770l 1938848'8887506 43 381.46381 i51{52352811.9101 16'8859339122 581 4689578'530963 11883369'8832206! 2
18 -4586496 -516138 1'937464'8886172 42 39 46406921 523899 1-9087641 885 7989 21 59'4692147-'53133611'882047 18830841  1
19'4589080 1516506 1'936082 18884838 41 40'4643269 1524269 1'907414'8856639 20 60'4694716['53170911.880726 18829476  o
201-45916651.516875 1.934702[-8883503 40
I Cosine. Cotang. Tang. I Sine. |ti  Cosine. Cotang.{ Tang. I Sine.              C osine. Cotang.  Tang.   Sine.
Deg. 62.                               Deg. 62.                                Deg. 62




NATURAL SINES AND TANGENTS TO A RADIUS 1.
28 Deg.                                  28 Deg.                                 28 Deg.'  Sine.   Tang. Cotang. Cosine.'          Sine.   Tang. Cotang. Cosine.''   Sine.  Tang. Cotang. Cosine. ('
o0 4694716 *531709 1'880726'8829476 60 21 4748564'539570 1.853825 18800633 39 1'4799683'647106 1'827799'8772858 19
1 *4697284'532082 1.879407 8828  110 59 2.475 1124 539946 1.852035 18799251 32 4802235'547484 1*826537'8771462 18
2 -4699852'532455 11878089'8826743 5 23'4753683'540322 1l850747 18797869 37 43 4804786'547862 1'825276 18770064 17
3'4702419'532829 1'876773'8825376 5 24'4756242'540698 1'849461'8796486 36 44'4807337 *548240 1'824017'8768666 16
4'4704986'533202 1'875458'8824007 5 25'4758801'541074 1'848176'8795102 35 45 4809888'548618 1'822759'8767268 15
5'4707553 533576 1'874145'8822638 5526 4761359'541450 1 846892'8793717 34 6'4812438'548997 1'821502'8765868 14
6'4710119'533950 1'872833'8821269 5  7'4763917'541826 1-845609'8792332 33 17'4814987'549375 1820247'8764468 13
7 4712685'534324 1871523 8819898 53 8 4766474'542202 1,844328'8790946   48 4817537'549754 1'818993 *8763067 12
8-~4715250. 534698 1 ~870214 8818527 52 294769031'542579 1'843049'8789559 31 49 4820086.550133 1'817740 *8761665 11
9'4717815'535072 1.868906'8817155 51 30 -4771588 542955 1'841770 8788171 30 501 4822634'550512 1'816489'8760263 10 
10'4720380'535446 1'867600'8815782 50 31 4774144 -543332 1.840494 8786783 29 51'4825182 -550891 1'815239'8758859 9
11'4722944'535820 1'866295'8814409 49 32 4776700 543709 1'839218'8785394 28 521'4827730'551270 1'813990 -8757455 8
12'4725508'536195 1.864992'8813035 48 33 4779255'544086 1.837944'878400{ 2753'4830277'551650 1'812743'8756051 7
13'4728071'536569 1'863690'8811660 4  4'4781810'544463 1'836671'8782613 26 54'4832824'552029 1'811496'8754645 6
14'4730634'536944 1'862389 8810284 4 35. 4784364.544840 1'835399  8781222 25'5j4835370'552409 1'810252 8753239  5
15'4733197'537319 1'861090'8808907 4 36'47869191'545217 1'834129'8779830 24 5154837916'552789 1-809008'8751832 4
16'4735759'537694 1'859792.8807530J44 37'4789472'545595 1.832861 8778437 23571 4840462'553168 1'807766'8750425 3
17'4738321'538069 1.858496 388061524 38'4792026'545972 1'831593'8777043 2258'4843007'553548 1'806525'8749016) 2
18 -4740882'538444 1'857201'8804774 42 39  4794579'546350 1'830327'8775649 21 9'4845552 -553928 1'805286'8747607  1
19'4743443'538819 1.855908'8803394 41 401'4797131'546728 1-829062'8774254 2060'4848096'554309 1'8040471 8746197 0
20'47460041'539195 1'854615'8802014 4   
Cosine. Cotang. Tang.  Sine.   t'  Cosine. ICotang. Tang.   Sine.'t t Cosine. Cotang. Tang.   Sine.
Deg. 61.                                Deg. 61.                                 Deg. 61.




NATURAL SINES AND TANGENTS TO A RADIUS i,
29 Dego                                  29 Deg.                                  29 Deg.'I Sine.  Tanl.  Cotan. Cosine.L           Sine.   Tang. Co Tang. - Cosine.   1-  Cotan. Cosine. 
0 8S48096'554309 1-804047 874G6197 60 21 *4901433'562321 1 778340 8716419 39 41'4952060'570004 1754372 i8687756 19
1.4850640. 554689 1i802810'8744786 59 22.4903968 -562704 1.777130.8714993 38142 4954587'570389 1.753186'8686315 18
2'4853184 555069 1'801575'8743375 58 23'4906503 t563087 1 775921 18713566 374 13 4957113 570775 1*752002 *8684874 17'3 48557 27 555450 18003-4,0,8741963 57 24t'4909038'563471 1 7174'714. 8712138 36 44 4959639 1571161 1750819 18683431'16
4'4858270'555831 1799107 -874 0550 56 25 -4911572 -563854  i773507,-8710710 35 45 4962165'571547 1'749637 -8681988 15
5.4860812 556211 1-797875'8739137 55 26 -4914105'564237 1.772302' i709281 134-146 4964690'571933 1.748456 -8680544 14
/6 4863354 556592 1'796645'8737722 54 27'4916638'564621 1'771098'8707851 334-71'4967215'572319 1'747276'8679100|13
72'4865895' 556973 1'7954-16 8736307 53 28.4919171'565005 1769895'870642C 132 18'4969740'572705 11746098'8677655 12
8 14868436 557355  11794188'8734891 52 29'4921704'565388 1'768694'8704989  311'19'4972264'573091 1744921-'8676209 11'9|4870977'557736 1'792961'87334-7551 301'4924236'565772 1'767494'8703557 30 50- 4974787'573478 1'743745'8674762 10
10 14873517'558117 1'791736'8732058 50 31i14926767'566156 1'766295'8702124 29 l51'4977310'573864 1'742570'8673314  9
11 4876057| 558499 1.790512'8730640 49 32'4929298'5665411 1-765097 -8700691 28 52. 4979833.574251 1'741396'8671866  8
12 4878597'558881 1-789289 8729221 48338 4931829'566925 1-763300 -8699256 2753  4982355 57463811740224'8670417  7
13 48811361 559262 11788067'8727801 47 34 4934359'567309 l'762705 18`97821 265 4'4984877 575025 1'739053'8668967 6
14'4883674'559644 11786847'8726381 46 35'4936839'567694 1 7615 1 18696380 25 5 5'4987399'575412  1 737883 86675 1 7 5
15'4886212 -560026 1'785628 8'724960 415 36'4939419'568079 1.760318.869 949 24  1  4989920'575799 11736714 8666066 4
1 6'488787501 560409 1'784410 8723538 44:37'4941948'568463 1 759126'86933512123 57  -4992441'576187 11 735546'8664614 3
17' 4891 288 560791 1 7831941 8722116 4 3 38'4944476'5658848 1'757936'8692074 22; 58'4994961 -576574 1'734380'8663161| 2
181 4893825 1561173 1;781979'8720693 42 39 14947005'569233 1 1'756747 186906362 1 59 4997481'576962 1'733214'8661708 1
19'48963611-561556 1780765'871V269 41 40'49495F32 569619 11755559 8689192060  500000157735C73205018660254 
20'48988971'5619391 1779552 "8717844 40                                              11731050 I
Cosine. Cotang.   Tang.l Sine. |- Cosine, Cotagi-  Tan.   Sine.'/'  Cosine. Cotang.  Tang.   Sine.
Deg. 60.                                 Deg. 60.                                 Deg. 60.




NATURAL SINES AND TANGENTS TO A RADIUS lo
30 Deg.                                 30 Deg.                                  30 Deg.
-- Sine.  Tang.  Cotang. Cosine. i          Sine.   Tang. C otang. Cosine.'t  Sine.  Tang. C otang. Cosine.   1
0 5000000 577350 ~1732050 18660254 60 21 15052809 1585524 117078711 8629549 39 1,5102928 {593363 1-685308,8600007 19
1 50025 19 577738 1'730887'8658799 59 22 5055319'585914 1'706732 8628079 38 42 5105429' 593756 1'684191'8598523 18
2 5005037'578126 1'729726'8657344 58 23'5057828'586305 1705595'8626608 37 43 5107930'594150 P1683076'8597037117
315007556 15785141 1'728565'8655887157 24'5060338'586696 1'704458'8625137 36 44151 1043 1 594543 1'681962'8595551 16
4['5010073'578902 1'727406' 8654430 56 25'5062846'587087 1'703323 18623664 35 45'5 112931 1594937 1'680848 18594064 1 5
6 5012591 579291 1'726247'8652973 55 26'5065355'587478 1'7021891'8622191 34 46'51 15431'595331 1'679736 18592576 14
6'5015107'579679 1'725090'8651514 54 27'5067863'587870 l-701055 18620717 33 47'51179301'595725 1'678625'8591088 13
7'5017624'580068 1-723934'8650055 53 28 5070370'588261 1.699923'8619243132 38 5120429'596119 1.6775151'8589599 12
8 8'5020140'580457 1'722779'8648595 52 29'5072877'588653 1.698792'8617768 31 9'5122927'596514[ 1'6764061 8588109 11
9'5022655- 580846 1'721626'8647134 151 30'5075384'589045 1697663.8616292 30 50'51254251 596908 1'675298- 8586619 10
10'50251701 581235 1'720473'8645673 50 31'5077890 1589436 11696534 18614815 29 51 5127923'597303 116741921 85851271 9
11'5027685 1581624 1'7193221 864421114932'5080396'589828 1' 695406 8613337J28 52'5130420'597697 1'6730861 8583635  8
12'5030199'582013 1'718172'8642748148 33'5082901'590221 1o694280'8611859 27 53'5132916'59809211'6719811'8582l431 7
131'50327131-5824031 171702381'864128447 3  5085406 -590613 1-693155 86103801265451354134598487 1670878'18580649    6
141'5035227'582793 1'715875'8639820 46 351 5087910'5910051 1'692030'8608901 25 55 51379081 5988821 1'669775'8579155 5
15' 50377401'583182 1'714728'8638355 45361 5090414'591398 1'690907'8607420124 56'5140404'599278 11668674'8577660  4
161'50402521 583572 1'713582'8636889144 371 5092918'591791 1L6897851 8605939 123 57 5142899 1599673 11667574'8576164[ 3'17'50427651 583962 1-7124381 863542314338 215095421'592183 1'6886641~86044571221581 5145393 1600069 1,666474 -85746681 2
181o5045276'584352 1.711294'8633956142 39 5097924'592576 1'687544'8602975 21 591'5147887 1600464 1.665376'8573171 1
19'5047788' 584743 11710152 18632488 41 40 -5100426'592969 1'6864261 8601491 120 601 510381 60086011'664279'8571673  0
20'5050298'585133 1'709011'8631019140   I        _      I                                         ____-I
Cosine. Cotang. Tang.   Sine.   /'[ Cosine. Cotang.  Tang. I Sine, I/'Cosine. Cotang.  Tang.   Sine.
Deg. 59.                               Deg. 59.                                  Deg. 59.




NATURAL SINES AND TANGENTS TO A RADIUS 1.:31 Deg.                                 31 Deg.                                 31 Deg.
Sine.   Tang. Cotang.Sine.   Tang. Cotang.  Cosine.                             Sine.   Tang. Cotang. Cosine.
0.5150381'600860 1-664279'8571673 6 21'5202646'609205 1-641482'8540051 39 1'5252241'617210 1'620192 8509639 19'
1.5152874 *601256 1!6631831 8570174 59 2'5205130'609604 1.640408.8538538 38 2'5254717'617612 1'619138'8508111 18
2 *5155367 *601652-11'662088'8568675 5 23 15207613'610003 1:639335 *8537023 37 3'5257191'618014 1.618085 -8506582 17
3'5157859 602049 1i660994'8567175 57 24.5210096'610402 1.638263'8535508 36 4 15259665 1618416 11617033 *8505053'16
4'5160351'602445 1'659901'8565674 56 25.5212579.610801 1 637191'8533992 35 5'5262139'618818 1.615982'8503522 15
5'5162842'602841 11 658809'8564173 55 26'5215061 1611201 1'636121'8532475 34 6'5264613'619221 1'614932'8501991 14
6'5165333'603238 11657718'8562671 54 27 15217543'611601 1.635052'8530958 3347'.5267085'619623 1'613882'8500459 13
7'51678241'603635 1'656629'8561168 53 28'5220024'612000 1'633984'8529440 328'5269558'620026 11612834'8498927 12
8'51703141'604032 1'655540 18559664 52 29 5222505'612400 11632917 185-27921 31 91'5272030 1620429 1'611787 18497394 11
9'.51728041'604429 11654452 1-8558160 51 30 5224986'61280011'631851'8526402 30 50 5274502 1620832 1'610741 18495860 10  ~
10'51752931 604826 1'6533661 8556655 5031'5227466 -61320 111630786 l8524881!29  1 -52769731 621235 1.6096961-84943251 9
11 -51777821'605224 1.652280'8555149 49 2.5229945['613601 1. 629722 18523360 28 5215279443 1621638J 1.6086521 8492790 8
12'5180270'605621 1.651196'8553643 48 335232424'614001 1.628659'8521839 27 31.5281914.622041 1-607609 18491254| 7
131'5182758'606019 11650112 1'8552135147 345234903'614402 1'627597 18520316126 i4.5284383'622445 1'6065671'8489717 6
141 5185246'606417 1'649030 18550627146 5'5237381 -614803 1'626536'8518793 25 551'52868.53 6228481 1-605526 18488179 5
151'5187733'606814 1'647949'8549119'45 6'5239859'615204 1'6254761'8517269 24 6'5289322'6232521 1'604485 18486641 4
16'5190219 1607213 1'6468681-8547609 4437'5242336 1615605 11624417 18515745 23 57 5291790 623656 1-603446 8485102{ 3
17:5192705.607611 116457891 8546099 4338:52448131.616006 1 6233591:8514219 2  8 15294258 1624060 1.6024081.8483562  2
181'5195191'608009 11644711 18544588142139 -52472901'6164071 1622302'8512693121591'5296726-624465 116013701-8482022  1
191 5197676'608408 1 643633'8543077 41  0'5249766'616809 1i621246'8511167 20 01.5299193'624869 1'600334'8480481 0
20 152001611 608806 1.6425571 8541564140                            1        1     I  a'i| Cosine. |Cotang.| T'ang. I Sine.       Cosine Cotang. Tang.  Sine.'         Cosine. Cotang. Tang. Sine.
Deg. 58'                               Deg. 58,                                 De,. 58.




NATURAL SINES AND TANGENTI$ TO A RADIUS 1.
32 Deg.                                   32 Deg.                                  32 Deg.
/   Sine.   Tang. Cotanrg. Cosine. 7    - Sine.   Tang. Cotang. Cosine.    /    Sine.   T'an,. (Cotang. C*osine. 
0'5299193 *624869 1'600334 *8480481 60 21 5350898 *633395 1578791'8447962 3941 5399955'641577 1558657 8416679 19
1.5301659.625273 1 5992991 8478939 59 22'5353355 *633803 1,577776 18446395 38 42 54024003 641988 1-557660. 8415108 18
2'5304125'625678 1.598264 18477397 58 23'5355812 1634211 1 576761 18444838 37 43'5404851'642399 1-5566631 8413536 17
3 *5306591 *626083 11597231'8475853 5724'5358268'634619 1575747 *8443279 36 44 5407298 *642810 1.555668.8411963 16
4'5309057 626488 1.596198'8474309 56 25'5360724 /635027 11574735'8441720 35 45.54097451643221 1]5546741 8410390 15
5 5311521'626893 1 595167 8472765 55 26 5363179'635435 1573723 8440161 34 46'5412191 -643632 1'5536801 840SS16114
6 5313986 -627298 1'594136 8471219 54 27'5365634.635844 1'572712 -8438600 33 47'5414637'6440441 1552688.8407241 13
7'5316450'627704j 1'593107'8469673 53 28'5368089'636252 1 571702'8437039 32 481 5417082j 644456 1'551696'8405666112
8 5318913'628109 1'5920781 8468126 52 291 53705431 636661 1'570693'8435477 31 491.5419527'644867 1'5507051'840409011
91 5321376'628515 1'59 1050'8466579 51 30'5372996 1637070 1'569685 -8433914 30 50'5421971'645279 1'5497151 8402513 10
101 5323839'628921 1.590023 -8465030 50 31 5375449| 637479 1-568678'8432351 29 51'5424415'645691 1548726 8400936  9
111 5326301'629327 1 588997 8463481 49132'5377902'637888 1'5676721 8430787 28 52'5426859'646104 1'547738 8399357 8
12.532876316629733 1'587973'8461932 48 33'5380354'638297 1 56666'842922227 53 -54293021 646516 1.5467511.8397778 7
131'5331224'630139 1'586949 184603811 47 34 53828061 63870711'565662 8S427657 26 54'5431744 -646929 1'545764'8396199 6
14'53336851 630546 1'5859261 8458830 14635'5385257'63911611 5646591'8426091 25155 15434187'647341 l'5447791'8394618 5
151'53361451 630953 11584904'8457278 45 36'53877081 63952611-563656'8424524 124156 15436628'647754 1'543794'8393037 4
16 5338605'631359 1'583883'8455726 44 371i5390158'639936 11562654'8422956 23157'5439069'64816'7 1'542810]'8391455  3
1 71-5341 065'631766 1.582862.84541 721431 3815392608'.640346 1-15616541-84213881221581 5441510'648580 1.541828'8389873  2
18 -5343523' 63217311 5818431 8452618 142139'53950581 640756 1.560654 18419819121 591'5443951'648994 1'540846'8388290  1
191-5345982'632581 1'5808251 8451064141 40'5397507'641167 1.5596551 8418249 20 60'5446390'649407 1'539865 8386706 0
10 1-53484401 632988 11579807 -8449508 40
|   Cosine. Cotang. Tan. } Sine.             Cosine. Cotang. Tang. I Sine.   / / Cosine. Cotang.  Tang.   Sine.
Deg. 57.                                  Deg. 57.                               Deg. 57.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
33 Deg.                                33 Deg.                                33 Deg.
S' ine.   T Tang. Cotang. Cosine.        Sine.   Tang. Cotang. Cosine. / /  Sine.   Tang. Cotang. Cosine. I
0 154463901 649407 1-539865 -8386706 6  1 -5497520 -658127 1-519463 -8353279 39 1 -5546024 -666496 1-500382 -8321155 19
1 -5448830 -649821 1-538884 -8385121 5 22 -5499950 -658544 1-518501 18351680 38 42 5548444 666917 1-499436 -8319541 18i.2 -5451269 -650235 1-537905 -8383536 5823 -5502379 -658961 1 517540 18350080 37 3 5550864 -667337 1-498492 -8317927 171
3 -5453707 -650649 1536927 -8381-950 5724 -5504807 -659378 1-5165791 83484i9 36 141 5553283 -667758 1-497548 -8316312 16
4 -5456145 -651063 1-535949 8380363 56125 -55072366 -659796 1-515620- 8346877 35 45 -5555702 -668178 1-496605 -8314696 15
5 65458583 *651477 1-5349721 8378775 5 26 *555096 633 660213   1'514661 *8345275 34 l6 *5558121 x668599 1'495663'8313080 14
6 -5461020 -651891 1-533996 -8377187 54127 -5512091 -660631 1-513703 -8343672 33 47 5560539 -669020 1-494722 -8311463 13
7 *5463456 -652306 1-533021 -8375598 53 8 -5514518 -661049 1-512746 -8342068 32 8 -5562956 1669441 1493782 -8309845 12
8 -5465892 -652721 1-532047 -8374009 5 291 -5516944 -661467 1-511790 -8340463 31 9 -15565373 -669863 1-492842 -8308226 11
91'5468328 -653t36. 1 5310741 83724181  5 30]'5519370 *6618851 1'510835'8338858 30 50 5567790'670284 1'491903'83066071 10 
10 -5470763 -653551 11530102 -8370827 50 1 -5521795 -662304 1-509880 -8337252 2 51 5570206 -670706 1-490965 -8304987 9  C
11 -5473198 1653966 1-529130 -8369236 49 32-5524220 -662722 1-508927 -8335646 2 52 5572621 -671128 1-490028 -8303366 8
12 -5475632 -1654381 1-528160-8367643 4 33-5526645-663141 15079741-8334038127535575036-671550 1-489092-8301745 7
13 -54780661.654797 1-527190-.8366050147 4-'5529069!-663560 1-507022 1833243012  415577451 -671972 11488157 -8300123 6
14. -5480499 -655212 1-5262211-8364456 4  5 -5531492 -663979 1-506071 -8330822 2  5-5579865 -672394 1-4872221-8298500 5
15 -5482932 -655628 1-5252531 8362862 45 361-5533915 -664398 1-505121 -8329212 2 56 5582279 -672816 1-486288 -8296877 4:
16-.5485365.65604411 5242861836126644 37.5536338 1664817 11504171 -8327602 23 57-5584692 -673239 1-4853551 8295252 3
17-.5487797 -6564601 1-.23320|-8359670 4.381-5538760 -665237 1-503222 -83'5991 22 81 5587105 -673662 1-4844231-82936281 2
18 -5490228 -656877 1 5223541 8358074142 915541182 -665657 1-502275 -8324380 21 5915589517 -674085 1-483491 -82920021 
19,-5492659 -6572931 1i521389.8356476j41 40j'5543603 -666076  1501328'8322768 2 6015591929 -6745C8| 1-482561 -8290376i 020 -5495090 -657710   1-520426 -8354878 4
-5osine. Cotang.0  Tang. I Sine.                         Tang. I   Sine.       Cosine. Cotang. Tang. |   Sine.
eDog. 5                               Deg.-56.                               Deg. 56.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
34 Deg.                                 34 Deg.                                 34 Deg.
n                                                 ang.         i
K.  Sine.  Tang. Cotaung.  Cosine.                                           3 1 SSine. / Tang.  Cotang.  Cosine.  Sine.  Tang. Cotang. Cosine
0'5591929 67450  1-482561'8290376 60 1 *5642467.683433 1.463200'8256062 3  1'56904031.692002 1.445081.88223096 19
1 5594340 674931 I 481631'8288749 5  22 5644869 *683860 11462287 *8254420 3  2'5692795'692432 11444183'8221440 18
2'5596751 67535F, 1'480702'8287121 58 23'5647270'684287 1'461374 *8252778 3  3'5695187'69286311'443286'8219784 17
3 -5599162'675779 1'479773'8285493 57 24'5649670'684714 1-460463'8251135136 44 5697577'693293 1'442389 -8218127 16'4'5601572 -676202 1'478846'8283864 56 25'5652070'685141 1'459552'8249491 135 5 5699968{'693724 1'441494'8216469 15
6 -5603981'676626 1'477919'8282,34 55 26 -5654469 -685569 1'458642'8247847 134 6 6'57023571'694155 1"440599{'8214811 I14
6 15606390'677050 1,476993'8280603 54 27'5656868 *685996, 1457732'8246202 13347 5704747. 694586 1'439704'8213,152 13
7'6608798'677475 1.476068'8278972 653 28 5659267'686424 1'456824'8244556 32 8'5707136'695018 1'438811'8211492 12
8 -5611206'677899 1.475144'8277340 52 29'5661665'686852| 1'4559161.8242909 31 49'5709524'695449 1'437918'8209832 11
9'5613614 1678324| 1.474221'18275708 51 30'5664062'687281 1'455009. 8241262 13 50 -5711912 1'95881 1'437026'8208170 10 
* 10'6616021 1-678749 1'473298'8274074 50 31'156664591 6877091!-454102'8239614 29[5 1 5714299 1696313 1'4361351'82065091  9
111 5618428'1679174 1.4723761 8272440 49 321 5668856'688137 1453197'8237965 2E52 -57166861'696745 1'4352451 8204846 8
12.56208341 679599 1.47 1455 |'8270806 48 33| 5671252'688566 1.452292'8236316 2 753 5719073'697177 1'4343551'8203183 7
13 -56623239'680024| 1-470535 8269170 47 4   56736181'688995 1'451388'8234666 2 54'5721459'697609 11433466'8201519 {
141 5625646'680460 1-469615'8267534 46 35'56760431 689424 1.450485'823301512  5'5723844.698042 114325781'8199854 5
15 15628049 -6fi80875 1-468696 -8265897 45 36 -56784371 689853 1.449582 18231364 24 561 57262291 698474 1.431690'81981891 4
16 15630453.6813f01- 1467778'8264260 44 37 -5680832'690283{ 1.44868Q'8229712 2357'5728614.698907{ 1.430803 18196523  3
17'5632857 -681727' 1466i861 8262622 43 8} 5683225'690712 1-447779 -8228059 22 58 -5730998'699340 1.429917.'8194856 2
18 563526-0 -682153 1465945'8260983 4239'5685619'691142 1.446879. 82264052159'5733381'699774 1.429032 8193189  1
19'5637663.682580 1-465029'8259343 41 401 5688011 1691572 1     1445980'8224751 20 60'5735764'700207 1.428148'89!520 0
20'5640066'683006 11464114'8257703 40
[ {Cosine. CotagTang.   Sine.'     Coine. Cota           Tang. I Sine.'' Cosine. Cotang. Tang.   Sine.
Deg. 55.                                Deg. 55.                                Deg. 55




NATURAL SINES AND TANGENTS TO A RADIUS 1.
35 Deg.                                 35 Deg.                                 35 Deg.
Sine.  Tang.  Cotang. Cosine.            ine.  Tang.  Cotalg. Cosine'    Sine. I Tang. Cotang. Cosine.'
0'573f64 700207 1-428148'8191520 60 21'5785696 709350 1'409740'8156330 39 1'5833050'718131 1'392501'8122532 19
-1 57381-47'700641 [1427264 [8189852 59 2'5788069'709787 1'408871'8154647 38 42 5835412'718572 1-391647'8120835 1s
2'5740529'701074 1'426.381 8188182 58'5790440'710225 1'408003'8152963 37 43 5837774'719014 1'390793'8119137 17
3'5742911'70150811-425498'8186512 57 4 5792812'710663 11407136'8151278 36 4'5840136'719455 11389940'8117439 16
4{'5745292'701943 1'424617'8184841    5.5795183'711100 1'406270'8149593 35 45'5842497'719897 1'389087'8115740 15
5 65747672'702377 1-'423736'81831 69 5 26'5797553'711539 1'405404'8147906 34 46 5844857'720338 1'388235'8114040 14
61 5750053'702811 1'422856'8181497 57 5799923'711977 1'404539'8146220 33 47'5847217'720780 1'387384 18112339 13
7 5752432'703246 1'421976'8179824 5328'5802292'712415 1'403674'8144532 32 48'5849577'721222 1'396534'8110638 12
8'5754811'703681 1'421097'8178151 52 9'5804661'712854 1'402811 18142844 31 9'5851936'721665 11385684'8108936 11
9'5757190'7041 161.420220'8176476 51 30'58070301 713293 1'401948'8141155 30 0'5854294'722107 1'384835'8107234 101   s
01' 5759568'704551 1'41934-2'8174801 50 31 15809397'713732 1'401086'8139466 29 1.5856652'722550 1'383986'8105530 9
11'5761946'704986 1'418466'8173125 49 32'58117651'714171 1-400224'8137775128 2'5859010'722993 1'383139'8103826  8
12'5764323'705422 1'417590 18171449'48 33'5814132 1714610 11399363'8136084 27 53 5861367'723436 1'382292 *8102122 7
18 -5766700'705858 1-416715" 8169772 47'58164981 7150.50 1'398503'8134393 26 -545863724'723879 1'3814459'8100416 -6
14 -5769076'706294 1'4159401 8168094 4 35'5818864'715489 1'397644 18132701 25 5'5866080'724322 1'380600'8098710 5
15'5771452'706730 1'414967'8166416 45 36'5821230'715929 1'396785'8131008 24 56.5868435'724766 1'379755 18097004 4
16'5773827'707166 1'414094 -8164736   37'582359.5 716369 1'395927'8129314 23 5715870790'725210 1'378910'8095296 3
17'5776202'707602 1-413222'8163056 43 38'5825959'716810 1'395069'8127620 22 5815873145'7256541 3780671 8093588 2
18'5778576'708039' 1-412350 -8161376 42 39'5828323 -717250 1'394213'8125925 21 9'5875499'726098 1'3772241:8091879  1
19'5780950'708476 1'411479 8159695 41 40'5830687'717691 1-39335:'8124229 20 01-5877853'726542 1'376381'8090170 0
20'5783323'708913 1-410609 8159013 40
Cosine. ICotang. Tang.   Sine.         Cosie. Cotang  Tan.   Sine.    t   Cosine. Cotang. Tang.   Sine.  I
Deg. 54.                               Deg. 54.                                Deg. 54.




NATURAL SINES AND TANGENTS TO A RADIUS 1.
36 Deg.                               36 Deg.                               36 Deg.
Si|ne.   Tang. Cotang. Cosine.        Sine.   Tang. Cotang. Cosine.'   Sine.  Tang. Cotang. Cosine. r
I.._.                - _   __  --'   - I —'____ -        - -    -'*"  t
0'58778531 726542 1'376381'8090170 6  1'5927163'735917 1'358848'8054113 39 1'5973919'744924 1342417'8019496 19
1'5880206 *726987 1-375540 18088460 59 22'5929505'736366 1'358020'8052389 38 2'5976251'745377  1341602 -8017756 18
2'58825581 727431 1.374699 18086749 58 3'5931847'736814 1'357193.8050664 3 43'5979583'745829 1-340788!8016018 17
3'58849101 727876 1.373859 18085037 5  4'5934189 737263 11I356367 18048938 36 44 5980915'746282 1'339975 18014278 16
4'5887262'728321 1.373019'8683325 56 25 5936530'737712 1.355541'8047211 3545'5983246'746735 1'339162'8012538 15
6 6889613'728767 1.372180'8081612 55 26.5938871'738162 1.354716 18045484 34 46 5985577 1747188 1.338350. 8010797w 14
6'5891964 1729212 1.371342'8079899 54 7.5941211'738611 1.353891 18043756 33 47 5987906'747642 1.337538.8009056113
7.5894314i.729658 1.370504'8078185 53 285943550'739061 1.353068 *8042028 32 48 -5990236'748095 1.336727'8007314 12
85896663 730104 1369667'8076470 52 29.5945889'739511 1.352244.8040299 3149 -59925665 748549 1.33591718005571 1 1
91'58990121 730550 1.368831'8074754 51 01 5948228'739961 1351422 18038569 3 50 15994893l 749003 1-3351071 8003827 10 
10 -5901361 1730996 1136799518073038 5 31.5950566'740411- 1350600 8036838 29 51'5997221 749467 l1334298'8002083 9
11 15903709'7314421'367161 18071321 4 32 15952904'740861 11349779'8035107 28152 15999549 1749911 1-3334901 80003938 8
12 -5906057'731889I 13663261 8069603 4 33 15955241'741312 1.3489581.8033375 2753 -6001876'750366 1-332682 17998593 7
13 15908404 s732l36 1.3654933 806788514 34 5957577'741763 1.348139 -8031642 26 541-6004202'750821  1-31875 7996847 6
14'5910750'732788 1'34660'8066166   351 5959913'742214 1-J47319.8029909 25 551 6006528 -751276 1.331068i-7995100 5
6 -55913096] 733230 -13638271 8064446 45 [61.5962249.742665 134650'8028175 2 56'608854'761731 1330262 7993352 4
16'5915442'733677 1'362996'8062726 14  7'.5964584 -7431171.345683-1'8026440 2 57'16011179t'76216 1-3294157-79916041 3
171 59177871.734125 1.362165'8061005J4 381:5966918'743568 1.3448651 8024705 22 8 601-350314572642I1328652.79898551
18 t5920132' 734b73 1.361335 1-8059283 4 391.5969252 1-74402011.344049 802296921 59 -6015827 1753098 I13274S7988105  1
1 5922476 -735021 1'360505'8057560 41 01'5971586'744472 1.343233 802123212 60'6018150'753554 I'327044'798635 
20'5924819 735469 1'359676'8055837  _I4I                                             
T  Cosine. Cotang.i Tang.   Sine.    " 1 Cosine. Cotang. Tang.   Sine.'  Cosine. lCotang.I Tang.   Sine.'
Deg. 53.                             Deg. 53.                               Deg. 53.




,NATIURIAL SINES AND TAN'GENTS TO A RADIUS ]o
37 Deg.                                 37 Deg.                                  37 Deg.
-          Sine.'Jan,. t     Cotang.I- Cosine. r    Sine.   Tan g. Cotang.  Cosine. /    Sine.   Tang.  Cotang. Cosine. /
0 -6018150 753554 1 327044'7986355 601 21'6066824'763175 1 310314'7949444 39 41'6112969 *772423 1'294627'7914014 19
1 6020473'754010 1-326242 7984604 59 22 *6069136 1763636 1-309523 7947678 38 42 *6115270'772887 1*293848*7912235 18
2 6022795 -754466 11325439 17982853 58 23 -6071447'764096 1-308734 794591337 43 6117572'773352 1'293071'7910456 17
3.6025117'75492311-324638'7981100 57 241.6073758'764557 1-3079451.7944146,36 44*6119873'.773817 1'292294'7908676 16
4 6027439'755379 1'323837  7979347 5f6 25 60760691 765018 1.307157 7942379 35 451 61221731774282 1.291517'7906896 15
5 60297601-755836 1'323036'7977594.55 26'6078379'765480 1306369 17940611 34 461 6124473'774748 1'290742'7905115 14
6'60320S01'756294 1-322237'7975839 54 27.6080689.765941 11305582'7938843 33 47'6126772'775213 1 289966 j7903,33 13
7?60344001756751 1,321437 7974084 53 28 96082998 6766403 1,304796, 7937074 3248 96129071'1775679 1 2891921 7901550 12
8 ]6036719 757209 11320639 7972329 52 29 6085306 1766864 1'304010'7935304131 49 6131369'776145 1.288418 17899767 11
9 6039038 757666 11319841  7970572 51 301 6087614 1767327 1'303225'7933533 30 50 6133666 776611 1.287644 17897983 10 
10 6041356 758124 1 319044 7968815 50 311 6089922 -767789 1-302440 17931762 29 51 -61359641 777078 1-286871'7896198  9
11/,6043674 758582 1,318247, 7967058|49 321 6092229'76825111 301656 /7929990 28 52 6138260'777544 11286099 7894413  8
12| 6045991 759041 1,317451'7965299148 331 6094535,76871411,3008'73 7928218 2753 -6140556'778011 1-285327'78926271 7
13'6048308'759499 1,3166551 7963540 47 34 1(6096841 |769177 1,300090 17926445 26 54 6142852 778478 11284556| 7890841  6
14'6050624'759958 1-3158611 7961780 46 35,6099147 1769640 112993081 7924671 125 55'6145147,778946 112837861 7889054[ 5
15'6052940'760417 1.315066 17960020 145136 6101452 -770103 1-298526 -7922896 124 56 6147442 -779413 11283016'7887266 4
16 160552551'76087611'314273 17958259 44 37'6103756 770567 1-297745 17921121 23 57 61497361 779881 1.2822461.7885477  3
17 16057.570!.761336 11.313480.7956497143 38'6106060) 771030 1-296964 -7919345 22 58 16152029'780349 1.281477 7883688  2
18'60598841'761795 11.3126871 7954735 42 39'6108363'771494 12961851-7917569 21 59'6154322'780817 1'280709 17881898  1
19 -6062198'762255 1.311895.7952972 41140'6110666'771958 1'2954051 7915792 20 60'6156615'781285 1'279941 -78801081 0
20'6064511 -762715 1.31110457951208 409                          — 1                                
Cosine. Cotang. I Tang.  Sine.   t  | Cosine. Cotang.l Tang.   Sine. |'   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.  Tan. Cotang.       Cosine./'  Sine.   Tang. Cotang.  Cosie.l
0'6156615'781285 1'279941'7880108 6 21'6204636'791170 1'263950'7842352 39 1'6250156 800673 1'248948'7806123 19
1 16158907'781754 1'279174'7878316 5 22 16206917'791643 1;263195'7840547 3  2'6252427'801151 1'248204'7804304 18
2 6161198'782222 l'278407'7876524 58 23 6209198'792116 1'2624140'7838741 37 3'6254696'801628 1'247460 *7802485 171
3.6163489'782691 1.277641'7874732 57 24 62  1478 *792590 1.261686 *7836935 36 44'6256966'802106 1'246716 7800665116
4 -6165780 1783161 1'276876'7872939 56 25 16213757'793064 1.260932'7835127135 5 6259235'802584 1 245974 *7798845 15
5'6168069 -783630 1.276111 7871145 55 26 6216036.793537 112601791 7833320 34 46 6261503].803063 1'246232 7797024 14
6'6170359.784100 1.275347'7869350 54 27'6218314'794012 1.259426 17831511 33  7 6263771'803541 1.244490.*7795202 13
7'6172648 784570 1274583 7867555 53 28 6220592'794486 112586741'7829702 3  81 6266038'804020 1'243749'7793380 12;
81 6174936'785040 11273820'7865759 52 2916222870'791961 1-257923 -7827892 3  91462683051 804499 1.243008 17791557 11
9'6177224| 785510o1 12730571 7863963|5] 301'6225146 1795435 11257172 17826082 3  0o 6270571'80497911 242268'7789733 10.z
101 6179511 1785980 1.272295 17862165 50 31j'6227423 17959111 11256421 17824270 29 1'62728371 805458 11241529'7787909 9
11'6181798'786451  1'2715341'7860367149 321'6229698]'796386 1'255672'7822459  2 2-6275102 j805938 1'240790!'7786084J 8[
12 16184s084'786922 1'270773'7858569.48 33 16231974'796861 1'2549221 7820646 27 31'6277366'806418 1"2400511'7784258 7
13'6186370 787393 1.270013 -7856770 47 34 16234248. 79733711 254174 1-7818833 26 54'627963 1 806898 1'239313'7782431  6
14 -6188655 1787864 1'2692531 7854970 46 351 6236522'797813 1.253426'7817019 2. 55 62818941-807378 1'238576'7780604] 5
15 -6190939'788336 1268494 17853169 45 36'6238796.798289 1'252678 -7815205 24 56.6284157'807859 1'237839- 7778777 4
16'6193224 7888081 1267735'7851.368 44 371 6241069 798765  1'251931'7813390 23 571 6286420'808340 1.2371031 7776949 3
17 -6195507'789280 1'266977,-7849566 43 38 1 6243342 -799242J11251184'7811574 22 58 -6288682'808821 1'236367['7775120 2a
18.6197790 i789752 1-266219'784776442 391 62456141 799719 1'250438 -780975721 1591-6290943 1809302 112356311*7773290 1
191 6200073 -790224 1'265462'7845961 41 0 -62478851 -800196 1'2496931 -7807940 2 60'6293204-809784 1234897-'7771460 0
20t 62023551-790697 1'264706 -7844157 40
t Csine. Cotang.l Tan.   Sine.  / / Cosine. Cotang.  Tang.   Sine.             Cosine. Cotang. Tang.   Sine. I /
Deg. 51.                               Deog. 51.                               Deg. 51.




NATURAL SINES AND TANGENTS T'1O A RADIUS 1.
39 Deg.                                   39 Deg.                                  3'9 De-.
Sine.   Tang.  Cotang. Cosine. I /  Sine.   Tang. C otan            osi.          Sie.    ang. C/ otang. Cosinle. 
01'6293204 -809784 1-234897 -7771460 616021 6340559 -819948 1.219588 -7732872 39 411 -6385440 [829724 1-205219 *7695853 19
11 6295464 -810265 1-234162 17769629 59 22 *6342808 -820435 1.218865 7731027 38 42 -6387678 830216 1-204505 17693996 18
2 6297724 1810747 1-233429 7767797 58123 *6345057 -820922 11218142 -7729182 37 43 6389916 830707 11203793 -7692137 17
31'6299983 1811230 1-232696 -7765965157 24 -6347305 -821409 1-217419'7727336 36 44- 6392153 -831199 1-203081'7690278116
4 *6302242 -811712 1-231963 -7764132 56 25 -6349553 -821896 1-216698 1772548913545 6394390 *831691 11-202369 -768841 8 15
5/.6304500 -812195 1-231231 -7762298 55 26 -6351800 -822384 12 15976 -7723642 34 46 -6396626 -83218.3 1201658 -7686558 14
6 6306758 -812678 1-230499 7760464 54 27 -6354046 822871 1-215256 7721794133 47 -6398862 -832675 1-200947 -7684697 13
7 6309015 813161 1-229768 7758629 53128 6356292 -823359 1-214535 -7719945132 48.6401097 -833168 1.200237 -7682835 12
8 -6311272 -813644 1-2290381.7756794 52[29 -6358537 -823847 1213816 7718096 31 49.6403332.-83366111-1995271-7680973 11
I 96313528 -814128 1-228308 77549571 51 30 6360782 -S24336 1213097 -7716246 30 50 -6405566 4834154 1-198818 -76791 010 10 
10 163157841-814611 11227578.7753121 50 31 -6363026 1-824825 112123781 -7714395{29 51 -6407799 -834648 1-1981091-7677246 9
11 -6318039|-815095 11226849 -7751283 49|32 -6365270 -825314 1-211660.7712544 28 52 -6410032 -835141 1-197401 -7675382 8
12 -63202931-815580 1-226121 -7749445 48133 -6367513 -825803i1.210942 -7710692 2753 6412264 -8S35635 1-196693 -76735171 7
13 -6322547 -816064 1-225393 -7747606 47 34 -6369756 -826292 1-210225 -7708840 126 54 -6414496 -836129 1- 195986 7671652  6
141-6324800 -816549 1-224665 -7745767 14635 -6371998 -826782 1-209508 -7706986 i25 551 6416728 -836624 1-1952791 7669785  5
151 6327053 -81703411-223938 -7743926145136 -6374240 -827271 11208792 -7705132124 56 -6418958 -837118  1-1945731 7667918  4
16 -6329306 -81751911-223212 -7742086 144137 -6376481 i827762 1-2080761'77032781 23157 -6421189 -83761311-1938671-7666051{ 3
17 -6331557 -818004 1-222486 -7740244 431381 6378721 -828252 1-207361 7701423 122 58 6423418/-838108 111931621-76641831 2
I18 6333809 -818490 1-2217611 -7738402 142 39 -6380961 -828742 1-2066461-769956721 259 6425647 -838604 1-192457 -76623141 1
19l-63360591.818976 1122 1036 -7736559 41 01 -6383201 - 829233 1-2059321 7697710120 60 -6427s76 -839099 1-1917531-7660444! 0
201 63383101-819462 1-220312 -7734716140
- Cosine. Cotang.. Tang. | Sine.  /'   Cosine. Cotang.  Tang.   Sine.                osine. Cotang.  Tang.   Sine. 
Deg. 50.                                  Deg. 50.                               Deg. 50.




NATURAL SINES AND TANGENTS TO A RADIUS 1,
40 Deg,                                    40 Deg.                                   40 Deg.
f  Sine.   Tang. Cotang. - Cosine. /'     Sine.   Tan,. (Cotang.  Cosine.  /  /   Sine.   Tang. Cotang. Cosine. 
0 16427876 *839099 1o191753  7660444 60 21 6474551 -849563 1:177075 *7621036 39 41 *65-18778 -859629 1 163291  7583240 19
116430104 "839595 1'191049'7658574 59 22 "6476767,850064 1.176382 17619152F42 3842 6520984 860135 1162607 7581343 18
2.64323321840091 1.190346 "7656704 58 231 6478984.850565 11175688 *7617268 37 43 6523189 *860641 1-161923 7579446 1i
3 64345591 840587 1'189643'7654832 57 24 "6481199'85106611'1749961 7615383 36 44'6525394 "861148 1-161240 "757754-8 16
4 6436785 "S41084 1-188941'7652960 56 25 "6483414'851568 111743031 7613497135 45 "6527598 -861655 11160557 7575650 15
5 6439011 *841581 1188239 17651087 55126 "6485628  S52070 1.173612 17611611 34 46.6529801 1862162 1*159874 17573751 14
6'6441236 "842078 1-187538 "7649214 54127'6487842 E852572 1'1729201 7609724 33 47'65320041 86266911'159192 -7571851 13l
7 6443461 *842575 11'86837'7647340 53 28 16490056 1853075 1 172229 "7607837 32 48 *6534206 *863176 1-158511 {7569951112
8.6445685 "843073 1-186136 17645465 52{  2   64922681-853577 1-171539  7605949 31 49 6536408.863684 1157830-7568050 11
91-6447909 1-8435701 1-185437 -7643590 51 30 1-6494480 854080   1*1708499 7604060 30j50 6538609 "864192 1"157 149"7566148 10   0
10 6450132 -844068 1,184737 -7641714 50 31 1-64966921-854583 1-170160 17602170 29 51 -65408101-864700 11156469 -75642461 9  
1 1-6452355 "844567 1-184038 17639838 49 32 -6498903  855087 1 1 694 71 -7600280128 52 16543010 1865209 11 55789 17562343 8
12 16454577 845065 1'183340 -7637960 48 33 16501114 1855591 11168782 17598389127 53 16545209 "865718 11155110' 7560439  7
13  6456798 -845564 1 182642  7636082 47 34 "6503324 -856095!1-168094 17596498126 54 "6547408 -866227 1-154431 "7558535  6
14 |6459019 1846063 1-181944 "7634204 46 35 "6505533 |856599 1167407 "7594606 25 55 "6549607 "866736 1"153753 "7556630  5
15 1 6461240 "846562 1 11812471 -7632325 45 36 -6507742 1857103 1 1 66720 "7592713 24 56  655 1 804 1 867246 1 -1530751 7554724 4
117]-6465679 8 47561 1-179855 17628564143 381-6512158 -858113 1-1653471-75S8926 22 58 -6556198 -868265 1151721 1-7550911  2
181-6467898 1848061 11179159.7626683142 39.6514366 8S58618 1.164661 i7587031 21 59 16558395 -868776 1151044 17549004 1
19 16470 116 848561 11'78464 17624802141 40 -6516572 1859124 1-163976 -7585136 20 60 16560590 *869286 1'150368 -75470961 0
20o 64723341.84906211.1777697622919 40   I                                                                        I         I I
-i Cosine. cotang.  Tang-    Sine.        / Cosine  Cotang. Tang.   Sine. "'  Cosine. Cotang. Tang.   Sin-e.
Deg 4 9.                                   De. 49                                   DPeg. 439




NATURAL SINES AND TANGENTS TO A RADIUS 1.
41 Deg.                                 41 Deg.                                 41 Deg.'  Sine.   Tang. Cotang. Cosine.''  Sine.   Tang. Cotang. Cosine.               ine.    Tang. Cotang. Coine. 
Sine.             Cotang.  Cosine. -
0.6560590'869286 1-150368 -7547096 60 21 6606570'880068 1.136274'7506879 39 41 6650131'890445 1-123032'7468317 19
1'6562785 869797 1.149692!7545187 59 226608754 880585 1.135608'7504957 38 42 6652304 890967 1122375'7466382 18
2'6564980'870308 11149017 -7543278 58 23 6610936'881101 1.134942'7503034 3743'6654475'891489 1'121718 17464446 17
3 6567174 870820 1'148342'7541368 5 24 6613119 881618 1 134277 -7501111 36 44 66566461,892011 1'121061 7462510 16
4'6569367 -871331 1'147668'7539457 56 25 16615300 -882135 1'133612 -7499187 35 45'6658817'892534 1'120405'7460574 15
5'6571560 -871843 11146994'7537546 55 261 6617482'882653 1'132947'7497262 34 46 6660987'893056 1'119749'7458636 14
6'6573752'872355 1'146321'7535634 54 7'6619662'883170 1132283 1-7495337 33 47'6663156'893579 1119094 7456699 13
7'6575944'872868 1'145648'7533721 53 281 6621842'883688 1'131620'7493411 32 48 16665325'894103 1'118439'7454760 1-2
8 -65781351'87338011'144976'7531808 52 91'6624022'884206 1'130957!7491484 31 49 -6667493.894626 1'117784I'7452821 11
9'6580326'873893 1'144304'7529894 51 30 6626200'884725 1'130294'7489557 30 501 6669661'895150 1'117130'7450881 10 
10'6582516'874406 1'143632'7527980 50 311 6628379'885244 1'129632'7487629 29 51 16671828'895674 1'116476 17448941 9
11 -6584706'874920 1'142961'7526065 49 32 6630557'885763 1'128970'7485701 28 52'6673994'896199 1.115823'74469991 8
12'6586895'875433 1'142290'7524149 4 331 6632734'886282 1'128308'7483772 2753'6676160'896723 1I 15170'7445058`"
18'6589083'875947 1114162017522233 47 34 6634910'886801 1'127647'7481842 26 54 6678326'897248 1'1 14518'7443115 6
14'6591271 1876462 11140950'7520316 46 5'6637087'887321 1'126987'7479912 25 55'6680490'897773 1'113866'7441173 5
15'6593458'87676611 140281'7518398 |4 36'6639262. 887841 1 126327 17477981 24 56 -6682655'898299 1 113214'7439229 4
16-6595645'877491 139612751648    37'6641437888361 11256671'7476049 2357 668418 898825 1112563'7437285 3
17 -6597831'878006 i1'38944'7514561 4  81'6643612'888882 1'125008'7474117 22 58 6686981'8993511 1'111912'7435340 2
18t'6600017'878521i 1'1382761'7512641[ 4  91 6645785'889403 1'1241349'7472184 21 59166891441'899877 1' 11262'7433394  1
19'i6602202'18790371 i137608J'75107214 41 0I6647959l 889924 1123690  7470251 20' 6691306'900404 L1106121'7431448 0
201 6604386 879552' 1136941'75088004                                        1     1                        
Cosine. Cotang.l Tang.   Sine.'   I Cosine. Cotang.' Tang.   Sine.'        Cosine. Coang  Tang.   Sine.
Deg. 48.                               Deg. 48.                                 Deg. 48.




NATURAL SINES ANDt TANGENTS TO A RADIUS 1.
42 Deg.                                 42 Deg.                                  42 Deg.
/  Sine.  Tang. Cotang. Cosine.''        Sine.   Tang. Cotang. Cosine.            Sine.  Tang. Cotang. Cosine.;
0 -6691306 900404 11 10612.7431448 60 1 *6736577 *911526 1-097060 *7390436 39 1 6779459'922235 1.084322'7351-18 19
1:6693468 900930 1109963, 7429502 59 22 6738727.912059 1.096420.7388475 38 2.6781597 -922773 1.083689.7349146 18
2.6695628'901458 1.1093141 7427554 58 3 -6740876.91259211.095779 *7386515 37 43 6783734'923312 1.083057'7347173 17
3 *6697789 901985 1.108665 7425606 57 24 6743024 913125 1.095139 7384553 36 46785871'923851 1.082425j 7345199 16
4'6699948'9025f13 1.108017 *7423658 5  5'6745172. 913659 1-094500. 7382592 35 45 6788007'924390 1.081793. 7343225 15
5 6702108 903041 1.107369 7421708 5526'6747319 914192 1.093861 7380629 34 6'6790143'924930 1.081162 *7341250 14
6'6704266 903569 1.106721'7419758 54 27 6749466'914727 11093222'7378666 33 7'6792278. 925470 1.080532. 7339275 13
7 6706424'904097 1-1060756 7417808 5328'6751612. 915261 1.092584 7376703 32 8 -6794413. 926010 1.079901 7337299 12
8'6708582'904626 1.105428'7415857 5229'6753757'915796 1.091946'7374738 31 49 6796547. 926550 1.079271 7335322 11.9 6710739 905155 1.104782 7413905 51 30'6755902 916331 1.091308 7372773 30 50-6798681 927091 1.078642 7333345 10
10'6712895.905685 1.1041361 7411953 50 31'6758046 916866 1.090671 7370808 29 51 6800813 927632 10780180137331367 9
11'6715051.906214 1.103491. 7410000 49132'6760190'917402 1 090034. 7368842 28 52 6802946- 928173 1.0773841 7329388 8
12.67172061.906744 1'102846. 7408046 4833'6762333'917937 1.0893981 7366875 27 53.6805078'928715 1.0767561 7327409  7
13'6719361. 907274 1.102201'7406092 4 34'6764476. 918474 1.088762 17364908 26 56807209 929257 11076128. 7325429 6
14 6721515 -907805 1'101557. 7404137 46 35 -6766618 919010 1.088126 7362940 25 55 6809339 929799 11075500 7323449 5
15'6723668. 908336 1.100914. 7402181 45 36'6768760 919547 1.087491. 7360971 24 561 6811469. 930342 1.074873'7321467 4
16 6725821 -908867 1100270l 7400225 4 4376770901'920084 1.086857.73590021 23 57 6813599. 930884 1.074246 7319486 3
17 6727973 909398 1.099628 7398268 4338'6773041- 920621 10862227357032 22 5816815728 931428 1-073620 7317503 2
1-8 6730125. 909930 1.098985. 7396311 42 39 6775181'921159 1.085588 7355061 21 59.6817856 -931971 1.072994'7315521  1
19'6732276. 910461 1-098343 17394353 41 40 6777320'921696 10849551735309020 606819984.1932515 11072368'7313537
20'6734427. 9109941 1097702' 7392394 40
Cosine. Cotang.l Tang.   Sine.            Cosine. Cotang  Tang. I Sine.           osine. Cotang  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.'         Sine.  Tang. Cotang. Cosine.         Sine.  Tang. Cotang. Cosine. 
0'6819984'932515 1'072368'7313537 60 1'6864532'944001 1.059320'7271740 39 1'6906721.955064 1.047049'7231681119
1'6822111'933059 1071743'7311553 5  2'6866647.944551 1'058703'7269743 38 2'6908824.955620 1-046440'7229671 18
2'6824237'933603 1071118'7309568 58 3'6868761.945102 1 058086'7267745 3'6910927'956177 1-045831'7227661 17
3'6826363'934147 1'070494'7307583 57 46870875.945653 1-057470.7265747136 46913029'956734 1'045222.7225651 16
4'6828489'934692 1069870'7305597 56 25 *6872988.946204 1.056854'7263748 35 45 6915131'957291 1'044613'7223640 15
5'6830613,935238 1'069246'7303610 55 26 6875101.946755 1.056238'7261748 34 46 6917232'957849 1'044005'7221628 14
6'6832738!935783 1'068623'7301623 54 27 6877213 947307 1'055623'7259748 33  7 6919332'958407 1'043397'7219615 13
7'6834861'936329 1.068000 7299635 53 8.6879325 *947859 1.055008'7257747 32 8'6921432'958965 1.042790'7217602 12
8'6836984'936875 1-067377 7297646 52 296881435 *948411 1.054394'7255746 31 9'6923531'959524 1'0421831 72155891 i
9'6839107'937421 1'066755'7295657 51 30 6883546.948964 1053780 7253744 30 50 6925630'960082 10415761 72135741 10
10'6841229 1937968 1'066134'7293668 56 311 6885655.949517 1.053166'7251741 2951'6927728'960642 1'0409701'7211559 9
11'6843350'938515 1'065512'7291677 49 326887765 950070 1.052553.17249738 2852 -6929825'961201 1'040364 172095441 8
12'6845471'939062 1'064891.7289686 48 3 368898731.950624 1'051940'72477342  31'6931922'961761 110397581j7207528 7
13'6847591'939610 1'064271'7287695 47 34.6891981'951178 1.051327'7245729 2 54'6934018.'962321 1'039153'7205511  6
14'6849711'940157 1:063651'7285703 46 35 6894089'951732 1.050715 1724372425 5569361141962881 1'038548.7203494  5
15*6851830 *940706 1-063031 *7283710 4536.6896195.-952287 10501031 7241719 24 56'6938209.963442 1-037944  7201476 4
16 6853948'941254 1,062411'7281716 44 37 6898302 1952842 1'049492'7239712 2 571 69403041 964003 1'037340'7199457
17'686066'941803 1'061792'7279722 43    6900407'953397 1-048880 -7237705 22 58169423981'964565 1'036736 -7197438 2
18'686184'942352 1,061174'7277728 42 B9 6902512'953952 1.048270'7235698 21 59'69444911'965126 1'036133r'7195418  1
191'6860300'942901 1'060656'7275732 41 0 690461'1954508 1'047659-'7233690 2060 6946584'965588 1'035530 7193398 0
20l6862416J'943451 1'059938'7273736 40
tCosine. | Cotang. Tang.   Sine.         Cosine. ICotang.' Tang.   Sine. 1 t    Cosine. Cotang. i 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. Cotang. Cosine. I'    Sine.  Tang. Cotang.  Cosine.
-...            _   __   -..-   --, I.1    -              -    -    -' -
0 -6946584. 965688 1-035530 -7193398 6  1 -6990396 -977564 1-022950 -7150830 39 1 -7031879 -989006 1-011115 -7110041 19
1 -6948676 -966251 1-034927 -7191377 5  2 -6992476 -978133 1-022355 -7148796 38 2 -7033947'989582 1-010527 -7107995 18.
2 -6950767 -966813 1-034325 -7189355 5 23 -6994555 -978702 1-021760 -7146762 37 4a -7036014 -990158 1-009939 -7105948 17
3 6952858 -967376 1-033723 -7187333 5  4 -6996633 -979272 1-021166 -7144727 36 44 -038081 990734 1-009352 -7103901 16
4-6954949-967939 1-033122 -7185310 56 25 -6998711 -979842 1-020572 -7142691 35 45 -7040147 -991311 1-008764 -7101854 15
5 -6957039 -968503 1-032520 7183287 5. 26 *7000789 *980412 1-019978 7140655 34 46 7042213 1991888 1-008178 7099806 14
6 -6959128 -969067 1-031919 -7181263 5 27 -7002866 -980983 1-019385 -7138618 33 47 -7044278 -992465 1-007591 -7097757 13
7 69612171l969631 11-031319-7179238 ~58 -7004942 -981554 11018792 *713658 132 4  70463421 993042 11007005 ~7095707112
8 16963305 -970196 1-030719 -7177213 529 -7007018 -982125 1-018199 -7134543 31 49 -7048406 -993620 1-006420 -7093657 11
91 -6965392 -970761 1-030119 -7175187 51 0 -7009093 -982697 1-017607 -7132504 30 50 -7050469 -994199 1-005834 -7091607 10  -
101 -6967479- 9713261 1-029520 -7173161150  1 -7011167 -983269 1-017015 -7130465 29 51 -7052532 -994777 1-005249 -7089556 9
11 -6969565 -971891 1-028921 -7171134149 2 -7013241 -1983841 1-016423 -7128426 28 52 -7054594 -995356 1-004665 -7087504 8
12 -6971651 -972457 1-028322 -7169106148 33 -7015314 -984414 1-015832 -7126385 27 53 -7056655 -995935 1-004080 -7085451 7
13 -6973736 -973023 1-027724 -71670784 34 -70173871 -984987 1-015241 -7124344 26 54 -7058716 996515 1-003496 -7083398 6
141 -6975821 -973590 1-027126 -7165049 146 35 -7019459 -985560 1-014651 -7122303 25 55 -7060776 -997095 1-002913 -7081345 5
15 -6977905 -974156 1-026528 -7163019 4  6 -7021531 -986133 1-014061 -7120260 2456 -7062835 -997675 1-002329 -7079291 4
161-69799881-974724 1-025931 -7160989 4  7 -7023601.986707 1-013471 -7118218 23 57 -7064894 -998256 1-001746 -7077236 3
17 -6982071 -97529111-025334 -7158959 43 8 -7025672.987282 1-012881 -7116174 22 58 -7066953 -998837 1-001164 -70751801 2
181 -69841531 -975859 1-024738 -715692714 39 -7027741.987856 1-012292 -7114130 21 59 -7069011 -999418 1-000581 7073124 1
19 -6986234 -976427 1-024141 -7154895 41 0 -70298111.98843111-011703 -7112086 20 60 -7071068 1-00000 1-000000 7071068 0
20 -69883151 -976995 1-023546 -7152863 4                                      _I
— 1  -   I   I.         I____  — I.      -    1-     1  _______ -   -._ _ 1 ______  --   -   -I  — 1
Cosine. Cotang. Tang.   Sine.       I Cosine. Cotang. Tang.   Sine.          Cosine. Cotang. Tang.    Sine. 
Deg. 45.                              Deg. 45.                              Deg. 45